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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nImage classification and segmentation are fundamental tasks in many visual recognition applications involving natural and medical images. Given a large image dataset annotated with global image-level labels for classification or with pixel-level labels for segmentation, deep learning (DL) models achieve state-of-the-art performances for these tasks  \\citep{dolz20183d,Goodfellow-et-al-2016,krizhevsky12,LITJENS201760,LongSDcvpr15,Ronneberger-unet-2015}. However, the impressive accuracy of such fully-supervised learning models comes at the expense of a considerable cost for collecting and annotating large image data sets. While the acquisition of global image-level annotation can be relatively inexpensive, pixel-wise annotation involves a laborious process, a difficulty further accrued by the requirement of domain expertise, as in medical imaging, which increases the annotation costs.\n\\begin{figure}[t!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.49]{proposal}\n  \\caption{Proposed AL framework with weak annotator.}\n  \\label{fig:fig-proposal}\n\\end{figure}\n\nWeakly-supervised learning (WSL) has recently emerged as a paradigm that relaxes the need for dense pixel-wise annotations \\citep{rony2019weak-loc-histo-survey,zhou2017brief}. WSL techniques depend on the type of application scenario and annotation, such as global image-level labels  \\citep{belharbi2019weakly,kim2017two,pathak2015constrained,teh2016attention,wei2017object}, scribbles \\citep{Lin2016,ncloss:cvpr18}, points \\citep{bearman2016}, bounding boxes \\citep{dai2015boxsup,Khoreva2017} or global image statistics such as the target-region size \\citep{bateson2019constrained,jia2017constrained,kervadec2019curriculum,kervadec2019constrained}. This paper focuses on learning using only image-level labels, which enables to classify an image while yielding pixel-wise scores (i.e., segmentations), thereby localizing the regions of interest linked to the image-class predictions.\n\n\nSeveral CNN visualization and interpretation methods have recently been proposed, based on either perturbation, propagation or activation approaches, and allowing to localize the salient image regions responsible for a CNN's predictions \\citep{fu2020axiom}. In particular, WSL techniques \\citep{rony2019weak-loc-histo-survey} rely on activation-based methods like CAM and, more recently, Gradient-weighted Class Activation Mapping (Grad-CAM), Grad-CAM++, Ablation-CAM and Axiom-based Grad-CAM \\citep{fu2020axiom, lin2013network, PinheiroC15cvpr}. Trained with only global image-level annotations, these methods locate the regions of interest (ROIs) of the  corresponding class in a relatively inexpensive and accurate way. However, while these WSL techniques can provide satisfying results in natural images, they typically yield poor segmentations in relatively more challenging scenarios, for instance, histology data in medical image analysis \\citep{rony2019weak-loc-histo-survey}.\nWe note two limitations associated with CAMs: (1) they are obtained in an unsupervised way (\\emph{i.e.}\\@\\xspace without pixel-level supervision under an ill-posed learning problem \\citep{choe2020evaluating}); and (2) they have low resolution. For instance, CAMs obtained from ResNet models \\citep{heZRS16} have a resolution of ${1\/32}$ of the input image. Interpolation is required to restore full image resolution. Both of these limitation with CAM-based methods lead to high false-positive rates, which may render them impractical \\citep{rony2019weak-loc-histo-survey}.\n\nEnhancing deep WSL models with pixel-wise annotation, as supported by a recent study in weakly-supervised object localization \\citep{choe2020evaluating}, can improve localization and segmentation accuracy, which is the central goal of this paper. To do so, we introduce a deep WSL model that allows supervised learning for classification, and active learning for segmentation, with the latter providing more accurate and high-resolution masks. We assume that the images in the training set are globally annotated with image-class labels.  Relevant images are \\emph{gradually} labeled at the pixel level through an oracle that respects a low  annotation-budget constraint. Consequently, this leads us to an active learning (AL) paradigm \\citep{settles2009active}, where an oracle is requested to annotate pixels in a subset of samples.\n\nDifferent sample-acquisition techniques have been successfully applied to deep AL for classification based on, e.g., certainty \\citep{ducoffe2015qbdc,gal2017deep,kirsch2019batchbald} or representativeness \\citep{kim2020task,sinha2019variational}. However, very few deep AL techniques were investigated in the context of segmentation \\citep{gaur2016membrane,gorriz2017active,lubrano2019deep}. Most AL techniques focus mainly on the sampling criterion (Fig.\\ref{fig:fig-proposal}, left) to populate the labeled pool using an oracle. During training, only the labeled pool is used, while the unlabeled pool is left dormant. Such an AL protocol may limit the accuracy of DL models under constrained oracle-supervision budget in real-world applications for multiple reasons:\n\n\\textbf{(1)} Standard AL protocols may be relevant to small\/shallow models that can learn and provide reliable queries using a few training samples. Since training accurate DL models typically depends on large training sets, large numbers of queries may be needed to build reliable DL models, which may incur a high annotation cost.\n\n\\textbf{(2)} In most AL work, the experimental protocol starts with a large labeled pool, and acquires a large number of queries for sufficient supervision, neglecting the workload placed on the oracle.\nThis typically reaches a plateau-performance of a DL quickly, hampering a reliable study of the impact of different AL selection techniques. Moreover, model-based sampling techniques are inconsistent \\citep{gaur2016membrane} in the sense that the model is used to query samples while it is still in an early learning stage.\n\n\\textbf{(3)} Segmentation and classification problems are associated with different properties and challenges, such as decision boundaries and uncertainty, which provide additional challenges to AL.  For instance, the class boundaries defined by different classification methods \\citep{ducoffe2018adversarial,settles2009active,tong2001support} are not defined in the context of segmentation, making such a branch of methods inadequate for segmentation.\n\n\\textbf{(4)}\nIn critical fields such as medical imaging, acquiring a sample itself can be very expensive\\footnote{\nFor instance, prior to a diagnosis of breast cancer from a histological sample, a patient undergoes a bilateral diagnostic mammogram and breast ultrasound that are interpreted by a radiologist, one to several needle biopsies (with low risks under ${1\\%}$ of hematoma and wound infection) to further assess areas of concern, surgical consultation and pre-operative blood work, and surgical excision of the positive tissue for breast cancer cells. The biopsy and surgical tissues are processed (fixation, embedding in parraffin, H\\&E staining) and interpreted by a pathologist. Depending on the cancer stage, the patient may undergo additional procedures. Therefore, accounting for all the steps required for breast cancer diagnosis from histological samples, a rough estimation of the final cost associated with obtaining a Whole Slide Image (WSI) is about \\$400 (Canadian dollars, by 1999) \\citep{will1999diagnostic}. Moreover, some cancer types are rare \\citep{will1999diagnostic}, adding to the values of these samples. All these procedures are conducted by highly trained experts, with each procedure taking from a few minutes to an hour and requiring expensive specialized medical equipment.}.\nThe time and cost associated with each sample makes them valuable items. Such considerations may be overlooked for large-scale data sets with almost-free samples, as in the case of natural images. Given this high cost, keeping the unlabeled pool dormant during learning may be ineffective.\n\n\nBased on the aforementioned arguments, we advocate that focusing solely on the sample acquisition and supervision pool may not be an efficient way to build\nhigh-performing DL models in an AL framework for segmentation. Therefore, we consider augmenting the labeled pool using the model as a second source of annotation, in a self-learning fashion \\citep{mao2020survey} (Fig.\\ref{fig:fig-proposal}, right). This additional annotation might be less accurate (\\emph{i.e.}\\@\\xspace, weak\\footnote{Not to be confused with the weak annotation of data in weakly supervised learning frameworks.}) compared to the oracle that provides strong but expensive annotations. However, it is expected to fast-improve the model's performance \\citep{mao2020survey}, while using a few oracle-annotated samples, reducing the annotation cost.\n\nOur main contributions are the following.\n\\textbf{(1) Architecture design}: As an alternative to CAMs, we propose using a segmentation mask trained with pixel-level annotations, which yields more accurate and high-resolution ROIs. This is achieved through a convolutional architecture capable of simultaneously classifying and segmenting images, with the segmentation task trained using annotations acquired using an AL framework.\nAs illustrated in Fig.\\ref{fig:fig-archi}, the architecture combines well-known DL models for classification (ResNet \\citep{heZRS16}) and segmentation (U-Net \\citep{Ronneberger-unet-2015}), although other architectures could also be used.\n\\textbf{(2) Active learning}: We augment the size of the labeled pool by weak-annotating a large number of unlabeled samples based on predictions of the DL model itself, providing a second source of annotation (Fig.\\ref{fig:fig-proposal}). This enables rapid improvements of the segmentation accuracy, with less oracle-based annotation. Moreover, our method can be integrated on top of any sample-acquisition method.\n\\textbf{(3) Experimental study}: We conducted comprehensive experiments over two challenging benchmarks -- high-resolution medical images (histology GlaS data for colon cancer) and natural images (CUB-200-2011 for bird species). Our results indicate that, by simply using random sample selection, the proposed approach can significantly outperform state-of the-art CAMs and AL methods, with an identical oracle-supervision budget.\n\n\n\\section{Related work}\n\\label{sec:related-work}\n\n\\noindent \\textbf{Deep active learning:}\nAL has been studied for a long time in machine learning, mainly for classification and regression, using linear models in particular  \\citep{settles2009active}. Recently, there has been an effort to transfer such techniques to DL models for classification tasks by mimicking their intuition or by adapting them, taking into consideration model specificity and complexity. Such methods include, for instance, different mechanisms for uncertainty  \\citep{beluch2018power,ducoffe2015qbdc,ducoffe2018adversarial,gal2017deep,kim2020task,kirsch2019batchbald,lakshminarayanan2017simple,wang2016cost,yoo2019learning} and representativeness estimation \\citep{kim2020task,sener2018coreset,sinha2019variational}. However, most deep AL techniques are validated on synthetic, simple or tiny data, which does not explore their full potential in real applications.\n\nWhile deep AL for classification is rapidly growing, deep AL models for segmentation are uncommon in the literature. In fact, the very few methods in the literature mostly focused on the direct application of deep AL classification methods. The limited research in this area may be explained by the fact that segmentation tasks bring challenges to AL, such as the additional spatial information and the fact that a segmentation mask lies in a much larger dimension than a classification prediction. In classification, AL often deals with one output that is used to drive queries \\citep{huang2010active}. The spatial information in segmentation does not naturally provide a direct scoring function that can indicate the overall quality or certainty of the output. Most of deep AL methods for segmentation consider pixels as classification instances, and apply standard AL techniques to each pixel.\n\nFor instance, the authors of \\citep{gaur2016membrane} exploit a variant of entropy-based acquisition at the pixel level, combined with a distribution-based term that encodes diversity using a complex hierarchical clustering algorithm over sliding windows, with application to microscopic membrane segmentation. Similarly, \\citep{gorriz2017active,lubrano2019deep} apply Monte-Carlo dropout uncertainty \\citep{gal2017deep} at the pixel level, with application to myelin segmentation using spinal cord and brain microscopic histology images. In \\citep{roels2019cost}, the authors experiment with five acquisition functions of classification for a segmentation task, including entropy-based, core-set \\citep{sener2018coreset}, k-mean and Bayesian \\citep{gal2017deep} sampling,  with application to electron microscopy segmentation. Entropy-based methods seem to be dominant over multiple datasets. In \\citep{yang2017suggestive}, the authors combine two sampling terms for histology image segmentation. The first employs bootstrapping over fully convolutional networks (FCN) to estimate uncertainty, where a set of FCNs are trained on different subsets of samples. The second term is a representation-based term that selects the most representative samples. This is achieved by solving an optimization of a generalization version of the maximum cover set problem \\citep{feige1998threshold} using sample description extracted from an FCN. Despite the obtained promising results, this approach remains complex and impractical due to the use of bootstrapping over DL models and an optimization step. Moreover, the authors of \\citep{yang2017suggestive} do not provide a comparison to other acquisition functions. The work in \\citep{casanova2020} considers a specific case of AL using reinforcement learning for \\emph{region-based} AL for segmentation, where only a selected region of the image is labeled. This method is adequate for data sets with large and unbalanced classes, such as street-view images. While the method in \\citep{casanova2020} outperforms random and Bayesian \\citep{gal2017deep} selection, surprisingly, it performs close to entropy-based selection.\n\n\n\\noindent \\textbf{Weak annotators:}\nThe AL paradigm does not prohibit the use of unlabelled data for training \\citep{settles2009active}, but it mainly constrains the oracle-labeling budget. The standard AL experimental protocol (Fig.\\ref{fig:fig-proposal}, left) was inherited from AL of simple\/linear ML models, and adopted in subsequent works. Budget-constrained oracle annotation may not be sufficient to build effective DL models, due to the lack of labeled samples. Furthermore, several studies in AL for classification have successfully leveraged the unlabelled data to provide additional supervision  \\citep{lin2017active,long2008graph,vijayanarasimhan2012active,wang2016cost,zhou2010active,zhualsslharmo2003}.\n\nFor instance, the authors of \\citep{lin2017active,wang2016cost} propose to pseudo-label a part of the unlabeled pool. The latter is selected using dynamic thresholding based on confidence, through the model itself, so as to learn a better embedding. Furthermore, a theoretical framework for AL using strong and weak annotators for classification task is introduced in \\citep{zhang2015active}. Their results suggest that using multiple annotators can reduce the cost of oracle annotation, in comparison to one annotator. Multiple sources of annotations that include both strong and weak annotators were used in AL, crowd-sourcing, self-paced learning and other interactive learning scenarios for classification to help reducing the number of requests for the strong annotator \\citep{kremer2018robust,malago2014online,mattsson2017active,murugesan2017active,urner2012learning,yan2016active,zhang2015active}.\nUsing the model itself for pseudo-annotation is motivated mainly by the success of deep self-supervised learning \\citep{mao2020survey}.\n\n\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n\\centering\n\\includegraphics[scale=0.65]{knn}\n\\caption{The $k$-nn method for selecting $\\mathbb{U}^{\\prime\\prime}$ subset to be speudo-labeled. Assumption to select $\\mathbb{U}^{\\prime\\prime}$: since $\\mathbb{U}^{\\prime\\prime}$ lives nearby supervised samples, it is more likely to be assigned good segmentation by the model.\nWe consider measuring $k$-nn for each \\textbf{unlabeled} sample. In this example, using $k=4$ allows ${\\abs{\\mathbb{U}^{\\prime\\prime}} = 14}$. If $k$-nn is considered for each \\textbf{labeled} sample: $\\abs{\\mathbb{U}^{\\prime\\prime}} = 8$. ${\\abs{\\cdot}}$ is the set cardinal. Note that ${k}$-nn is only considered between samples of the \\emph{same class}.}\n  \\label{fig:fig-knn}\n\\vspace*{-0.1in}\n\\end{wrapfigure}\n\n\n\\noindent \\textbf{Label Propagation (LP):}\nOur approach is also related to LP methods \\citep{bengiolabelprop2010,zhou2004learning,zhulp2002} for classification, which aim to label unlabeled samples using knowledge from the labeled ones (Fig.\\ref{fig:fig-knn}). However, while LP propagates labels to unlabeled samples through an iterative process, our approach bypasses this using the model itself. In our case, the propagation is limited to the neighbors of labeled samples defined through $k$-nearest neighbors ($k$-nn) (Fig.\\ref{fig:fig-knn}). Using $k$-nn has been also studied to combine AL and domain adaptation \\citep{berlind2015active}, where the goal is to query samples from the target domain. Such an approach is connected to the recently developed core-set method for deep AL \\citep{sener2018coreset}. Our method intersects with \\citep{berlind2015active} only in the sense of predicting the labels to query samples using their labeled neighbors.\n\nIn contrast to state-of-the-art DL models for AL segmentation, we consider increasing the unlabeled pool through pseudo-annotated samples (Fig.\\ref{fig:fig-proposal}, right). To this end, the model is used for pseudo-labeling samples within the neighborhood of samples with strong supervision (Fig.\\ref{fig:fig-knn}). From a self-learning perspective, the works in \\citep{lin2017active,wang2016cost} on face recognition are the closest to ours. While both rely on pseudo-labeling, they mainly differ in the sample selection for pseudo-annotation. In \\citep{lin2017active,wang2016cost}, the authors considered model confidence, where samples with high confidence are pseudo-labeled, while low-confidence samples are queried. While this yields good results, it makes the overall method strongly dependent on model confidence. As we consider segmentation tasks, model-confidence is not well-defined. Moreover, using the expected pixel-wise values can be less representative for model confidence.\n\nOur approach relies on the spatial assumption in Fig.\\ref{fig:fig-knn}, where the samples to pseudo-label are selected to be near the labeled samples, and expected to have good pseudo-segmentations. This makes the oracle-querying technique independent from the pseudo-labeling method, giving more flexibility to the user. Our pseudo-labeled samples are added to the labeled pool, along with the samples annotated by the oracle. The underlying assumption is that, given a sample labeled by an oracle, the model is more likely to produce good segmentations of images located nearby that sample. Our assumption is verified empirically in the experimental section of this paper. This simple procedure enables to rapidly increase the number of pseudo-labeled samples, and helps improving segmentation performance under a limited amount of oracle-based supervision.\n\n\n\n\n\n\\section{Proposed approach}\n\\label{sec:proposal}\n\n\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n\\centering\n \\includegraphics[scale=0.75]{archi}\n \\caption{Our proposed DL architecture for classification and segmentation  composed of: (1) a shared \\textbf{backbone} for feature extraction; (2) a \\textbf{classification head} for the classification task; (3) and a \\textbf{segmentation head} for the segmentation task with a U-Net style \\citep{Ronneberger-unet-2015}. The latter merges representations from the backbone, while gradually upscaling the feature maps to reach the full image resolution for the predicted mask, similarly to the U-Net model.}\n \\label{fig:fig-archi}\n\\vspace*{-0.1in}\n\\end{wrapfigure}\n\n\nWe consider an AL framework for training deep WSL models, where all the training images have class-level annotations, but no pixel-level annotations. Due to their high cost, pixel annotations are gradually acquired for training through oracle queries. It propagate pixel-wise knowledge encoded in the model though the labeled images.\n\nActive learning training consists of sequential training rounds. At each round $r$, the total training set ${\\mathbb{D}}$ that contains $n$ samples with unlabeled and labeled subsets (Fig.\\ref{fig:fig-proposal}).\n\\textbf{(1) Unlabeled subset:} contains samples without pixel-wise annotation (unlabeled samples) ${\\mathbb{U} = \\{\\bm{x}_i, y_i, -\\-\\}_{i=1}^u}$ where ${\\bm{x} \\in \\mathcal{X}}$ is the input image, ${y}$ is its global label; and the pixel label is missing.\n\\textbf{(2) Labeled subset:} contains samples with full supervision ${\\mathbb{L}=\\{\\bm{x}_i, y_i, \\bm{m}_i\\}_{i=1}^l}$ where ${\\bm{m}}$ is the pixel-wise annotation of the sample. ${\\mathbb{L}}$ is initially empty. It is gradually populated from ${\\mathbb{U}}$ by querying the oracle using an acquisition function.\nLet ${f(\\cdot: \\bm{\\theta})}$ denotes a DL model that is able to classify and segment an image ${\\bm{x}}$ (Fig.\\ref{fig:fig-archi}). For clarity, and since we focus on the segmentation task, we omit the notation for the classification task (to simplify the presentation). Therefore, ${f(\\bm{x})}$ refers to the predicted segmentation mask.\nLet ${\\mathbb{U}^{\\prime} \\subseteq \\mathbb{U}}$ and ${\\mathbb{U}^{\\prime\\prime} \\subseteq \\mathbb{U}}$ denote two subsets (Fig.\\ref{fig:fig-proposal}), with ${ \\mathbb{U}^{\\prime} \\cap \\mathbb{U^{\\prime\\prime}} = \\varnothing}$.\nIn our method, we introduce ${\\mathbb{P}}$ as a subset holder for pseudo-labeled samples, which is initially empty and gradually replenished (Fig.\\ref{fig:fig-proposal}, right).\nTo express the dependency of each subset on round ${r}$, we introduce the following notations: ${\\mathbb{U}_r, \\mathbb{L}_r, \\mathbb{P}_r, \\mathbb{U}^{\\prime}_r, \\mathbb{U}^{\\prime\\prime}_r}$.  The samples in ${\\mathbb{P}_r}$ are denoted as ${\\{\\bm{x}_i, y_i, \\hat{\\bm{m}}_i\\}}$. The following holds: ${\\forall r: \\mathbb{D} = \\mathbb{L}_r \\cup \\mathbb{U}_r \\cup \\mathbb{P}_r}$.\n\n\n\nAlg.\\ref{alg:alg-0} describes the overall AL process with our pseudo-annotation method.\nFirst, ${\\mathbb{U}^{\\prime}_r}$ is queried, then labeled by the oracle, and added to ${\\mathbb{L}_r}$.\nUsing $k$-nn, ${\\mathbb{U}^{\\prime\\prime}_r}$ is selected based on their proximity to ${\\mathbb{L}_r}$ (Fig.\\ref{fig:fig-knn}); and pseudo-labeled by the model, then added to ${\\mathbb{P}_r}$. To fast-increase the size of ${\\mathbb{L}_r}$, ${\\mathbb{P}_r}$ is protected from being queried for the oracle until it is inevitable. In the \\emph{latter case}, queried samples from ${\\mathbb{P}_r}$ are used to fill ${\\mathbb{U}^{\\prime}}$; and they are no longer considered pseudo-labeled since they will be assigned the oracle annotation.\n\nTo measure image similarity for the $k$-nn method, we used the color distribution to describe image content. This can be a flexible descriptor for highly unstructured images such as histology images. Note that the ${k}$-nn method is considered \\emph{only} for pairs of samples of the \\emph{same class}. The underlying assumption is that samples of the same class, with similar color distributions, are likely to contain relatively similar objects. Consequently, labeling representative samples could be a proxy for supervised learning based on the underlying data distribution. This can increase the likelihood of the model to provide relatively good segmentations of the other samples. The proximity between two images ${(\\bm{x}_i, \\bm{x}_j)}$ is measured using the Jensen-Shannon divergence between their respective color distributions (measured as normalized histograms). For an image with multiple color planes, the similarity is formulated as the sum of similarities, one for each plane.\n\nAt round $r$, the queried and pseudo-labeled samples are both used in training by optimizing the following loss function:\n\\begin{equation}\n\\label{eq:eq-1}\n    \\min_{\\bm{\\theta}} \\sum_{\\bm{x}_i \\in \\mathbb{L}_{r-1}} \\mathcal{L}(f(\\bm{x}_i), \\bm{m}_i) + \\lambda \\sum_{\\bm{x}_i \\in \\mathbb{P}_{r-1}} \\mathcal{L}(f(\\bm{x}_i), \\hat{\\bm{m}}_i),\n\\end{equation}\nwhere ${\\mathcal{L}(\\cdot, \\cdot)}$ is a segmentation loss, and ${\\lambda}$ a positive scalar. Eq.(\\ref{eq:eq-1}) corresponds to training the model (Fig.\\ref{fig:fig-archi}) solely for the segmentation task. Simultaneous training for classification and segmentation in this AL setup is avoided due to the unbalance between the number of samples that are labeled globally and at the pixel level. Therefore, we consider training the model for classification first. Then, we freeze the classifier parameters. Training for the segmentation tasks is resumed later. This yields the best classification performance, and allows a better study of the impact of the queried samples on the segmentation task.\n\nConsidering the relation of our method and label propagation algorithm \\citep{bengiolabelprop2010,zhou2004learning,zhulp2002}, we refer to our proposal as Label\\_prop.\n\n\n\\begin{center}\n\\begin{minipage}{0.6\\linewidth}\n\\IncMargin{0.04in}\n\\begin{algorithm}[H]\n    \\SetKwInOut{Input}{Input}\n   \n    \\Input{\n    ${\\mathbb{P}_0 = \\mathbb{L}_0 = \\varnothing}$\n    \\\\\n    ${\\bm{\\theta}^0}$: Initial parameters of ${f}$ trained on the classification task.\n    \\\\\n    ${\\texttt{maxr}: \\textrm{Maximum number of AL rounds}}$.\n    }\n    \\vspace{0.1in}\n    Select ${\\mathbb{U}^{\\prime}_0}$ randomly and label them by an oracle. \\\\\n    \\vspace{0.025in}\n    $  \\mathbb{L}_0 \\ \\leftarrow\\  \\mathbb{U}^{\\prime}_0$. \\\\\n    \\For{${r \\in 1 \\cdots \\texttt{maxr}}$} \n    {\n       \n        ${\\bm{\\theta} \\ \\leftarrow\\  \\bm{\\theta}^0}$. \\\\\n        \\vspace{0.03in}\n        Train $f$ using ${\\mathbb{L}_{r-1}}$ \\colorbox{mybluelight}{${\\cup\\; \\mathbb{P}_{r-1}}$} and the loss in Eq. (\\ref{eq:eq-1}). \\\\\n        \\vspace{0.03in}\n        Select ${\\mathbb{U}^{\\prime}_r}$ and label them by an oracle. \\\\\n        \\vspace{0.03in}\n        $  \\mathbb{L}_r \\ \\leftarrow\\  \\mathbb{L}_{r-1} \\cup \\mathbb{U}^{\\prime}_r$. \\\\\n        \\vspace{0.03in}\n        \\colorbox{mybluelight}{Select ${\\mathbb{U}^{\\prime\\prime}_r}$.} \\\\\n        \\vspace{0.03in}\n        \\colorbox{mybluelight}{$ \\mathbb{P}_r  \\ \\leftarrow\\   \\mathbb{P}_{r-1} \\cup \\mathbb{U}^{\\prime\\prime}_r$.} \\\\ \n        \\vspace{0.03in}\n        \\colorbox{mybluelight}{Pseudo-label ${\\mathbb{P}_r}$.}\n    }\n    \\vspace{0.1in}\n    \\caption{Standard AL procedure and our proposal. The extra instructions associated with our method are indicated with a \\colorbox{mybluelight}{blue background}.\n    }\n    \\label{alg:alg-0}\n\\end{algorithm}\n\\DecMargin{0.04in}\n\\end{minipage}\n\\end{center}\n\n\n\n\\section{Results and discussion}\n\\label{sec:experiments}\n\n\n\n \\subsection{Experimental methodology:}\n\n \\begin{figure}[!b]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{samples-datasets}\n    \\caption{\\textbf{Top row}: GlaS dataset \\citep{sirinukunwattana2017gland}. \\textbf{Bottom row}: CUB dataset \\citep{WahCUB2002011}. (Best visualized in color.)}\n  \\label{fig:fig-datasets}\n\\end{figure}\n\n\\noindent \\textbf{a) Datasets.}\nFor evaluation, datasets should have global and pixel-wise annotation. We consider two public datasets including both medical (histology) and natural images (Fig.\\ref{fig:fig-datasets}).\n\\begin{inparaenum}[(1)]\n\n  \\item \\textbf{GlaS dataset}: This dataset contains histology images for colon cancer diagnosis\\footnote{GlaS: \\href{https:\/\/warwick.ac.uk\/fac\/sci\/dcs\/research\/tia\/glascontest}{warwick.ac.uk\/fac\/sci\/dcs\/research\/tia\/glascontest}.} \\citep{sirinukunwattana2017gland}. It includes 165 images derived from 16 Hematoxylin and Eosin (H\\&E) histology sections of two grades (classes): benign and malignant. It is divided into 84 samples for training and 80 samples for testing.\n  The ROIs to be segmented are the glandes.\n \n \n  \\item \\textbf{CUB-200-2011 dataset (CUB)}\\footnote{CUB: \\href{http:\/\/www.vision.caltech.edu\/visipedia\/CUB-200-2011.html}{www.vision.caltech.edu\/visipedia\/CUB-200-2011.html}} \\citep{WahCUB2002011} is a dataset for bird species with ${11,788}$ samples ($5,994$ for training and $5,794$ for testing) and ${200}$ species. The ROIs to be segmented are the birds.\n \n  \\end{inparaenum}\n  %\n  In GlaS and CUB datasets, we randomly select $80\\%$ of the training samples for effective training, and $20\\%$ for validation (with full supervision) to perform early stopping. The splits are identical to the ones used in \\citep{belharbi2019unimoconstraints,rony2019weak-loc-histo-survey} (split 0, fold 0), and are publicly available.\n\n\\noindent \\textbf{b) Active learning setup.}\nTo assess the performance of different AL acquisition methods, we consider a realistic scenario with respect to the number of samples to be labeled at each AL round, accounting for the load imposed on the oracle.\nTherefore, only a few samples are selected at each round for oracle annotation, and  ${\\mathbb{L}}$ is slowly replenished. This allows better comparison between AL selection techniques since we spend more time\nin a phase where ${\\mathbb{L}}$ holds a few samples. Such a phase allows to better measure the impact of the selected samples. Filling  ${\\mathbb{L}}$ quickly brings the model's performance to a plateau that hides the impact of newly selected samples.\nThe initial replenishment ($r=1$) is achieved by randomly selecting a few samples. The same samples are used for all AL techniques at round $r=1$ for a fair comparison. To avoid any bias from selecting unbalanced classes that can directly affect the segmentation performance, and hinder AL evaluation, the same number of samples is selected from each class (since the global annotation is known beforehand for all the samples). Note that the oracle is used only to provide pixel-wise annotation. Tab.\\ref{tab:tab0} provides the selection details.\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Number of samples selected for the oracle per round.}\n\\label{tab:tab0}\n\\centering\n   \\resizebox{0.7\\linewidth}{!}{\n\\begin{tabular}{l|ccc}\n    Dataset  &  \\makecell[c]{\\#selected samples \\\\ \\emph{per-class} ($r=1$)} & \\makecell[c]{\\#selected samples \\\\ \\emph{per-class} ($r > 1$)} & \\makecell[c]{Max AL rounds \\\\ (\\texttt{maxr} in Alg.\\ref{alg:alg-0})}\\\\\n    \\toprule\n    GlaS   &   $4$   & $1$  & $25$ \\\\\n    CUB   &   $1$  & $1$    & $20$ \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\\noindent \\textbf{c) Evaluation.}\nWe report the classification accuracy obtained by the classification head (Fig.\\ref{fig:fig-archi}). Average Dice index is used to measure the segmentation quality at each AL round forming a Dice index curve over all the rounds. To better assess the \\emph{dominance} of each method \\citep{settles2009active}, the Area Under the Dice index Curve is used (AUC). This provides a fair indicator of the dominant curve, but contrasts with standard AL works, where one or multiple specific operating points in the curve are selected (leading to a biased and less accurate protocol). The average and standard deviation of Dice index curve and AUC metric are reported based on 5 replications of a complete AL session, using a different seed for each session. An AL session across different methods uses the same seed.\n\nWhile our approach, referred to as (Label\\_prop), can operate on top of any AL selection criterion, we demonstrate its efficiency using simple random selection, which is often a baseline for AL experiments. Note that our pseudo-annotations are obtained from the segmentation head shown in Fig.\\ref{fig:fig-archi}. Our method is compared to three different AL selection approaches for segmentation:\n\\textbf{(1) random selection (Random)}: the samples are randomly selected;\n\\textbf{(2) entropy-based selection (Entropy)}: the scoring function per sample is the average entropy at the pixel level \\citep{gaur2016membrane}. Samples with high entropy are selected; and\n\\textbf{(3) Monte-Carlo dropout uncertainty (MC\\_Dropout)}: we use Monte-Carlo dropout \\citep{gorriz2017active,lubrano2019deep} at the pixel level to compute the uncertainty score per sample. Samples are forwarded ${50}$ times in the model, where dropout is set to ${0.2}$ \\citep{gorriz2017active,lubrano2019deep}. Then, the pixel-wise variance is estimated. Samples with high mean variance are selected.\n\n\n\\noindent  \\textbf{Lower bound performance (WSL)}: We consider the segmentation performance obtained by WSL method as a lower bound. It is trained using only global annotation. CAMs are used to extract the segmentation mask. WILDCAT method is considered \\citep{durand2017wildcat} (Fig.\\ref{fig:fig-archi}) at the classification head to obtain the CAMs. For WSL method, a pre-trained model over ImageNet \\citep{imagenetcvpr09} is used to initialize the weights of the backbone, which is then fined-tuned.\nThe model is trained over the entire dataset, where samples are labeled globally only.\nThe obtained classifier using seed=0 is frozen and used as a backbone for \\emph{all} the other methods.\n\n\\noindent \\textbf{Upper bound performance (Full\\_sup)}: Fully supervised segmentation is considered as an upper bound on performance. The model in Fig.\\ref{fig:fig-archi} is trained for segmentation only using the entire dataset, where samples are labeled at the pixel level.\n\nFor a fair comparison, all the methods are trained using the same hyper-parameters over the same dataset. WSL and Full\\_sup methods have minor differences. Due to space limitation, all the hyper-parameters are presented in the supplementary material.\nIn Alg.\\ref{alg:alg-0}, notice that for our method, ${\\mathbb{P}_r}$ is not used at the current round $r$ but until the next round ${r+1}$. To take advantage of ${\\mathbb{P}_r}$ at round $r$, instructions from line-4 to line-10 are repeated twice in the provided results.\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\\label{sub:results}\n\n\n\n\\begin{figure}[htbp]\n  \\centering\n \\begin{subfigure}[b]{.5\\linewidth}\n  \\centering\n  \\includegraphics[scale=0.39]{glas-dice-idx-test}\n  \\caption{}\n  \\label{fig:fig-al-0}\n \\end{subfigure}%\n \\begin{subfigure}[b]{.5\\linewidth}\n  \\centering\n  \\includegraphics[scale=0.39]{Caltech-UCSD-Birds-200-2011-dice-idx-test}\n  \\caption{}\n  \\label{fig:fig-al-1}\n \\end{subfigure}%\n\\caption{Average Dice index of the proposed and baseline methods over test sets.\n(\\protect\\subref{fig:fig-al-0}) GlaS.\n(\\protect\\subref{fig:fig-al-1}) CUB.\n}\n\\label{fig:fig-al-results}\n\\end{figure}\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Classification accuracy over of the proposed deep WSL model on GlaS and CUB test datasets.}\n\\label{tab:tab-cl-acc}\n\\centering\n   \\resizebox{.5\\linewidth}{!}{\n\\begin{tabular}{l|ccc}\n    Dataset  &  \\multicolumn{1}{c}{GlaS} & \\multicolumn{1}{c}{CUB}\\\\\n    \\toprule\n    \\makecell{Classification \\\\ accuracy (\\%)}   &   $99.50 \\pm 0.61$\n                                                 &   $73.22 \\pm 0.19$\n                                                 \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Average AUC and standard deviation (Fig.\\ref{fig:fig-al-results}) for Dice index performance over GlaS and CUB test sets.}\n\\label{tab:tab-seg-perf}\n\\centering\n   \\resizebox{.6\\linewidth}{!}{\n\\begin{tabular}{l|ccc}\n    Dataset  &  \\multicolumn{1}{c}{GlaS} & \\multicolumn{1}{c}{CUB} \\\\\n    \\toprule\n    \\toprule\n    WSL   &    $66.44 \\pm 0.20$  &   $39.22 \\pm 0.19$  \\\\\n    \\midrule\n    Random   &    $78.57 \\pm 0.93$  &   $68.15 \\pm 0.61$  \\\\\n    Entropy   &   $79.13 \\pm 0.26$  &   $68.25 \\pm 0.29$  \\\\\n    MC\\_Dropout   &   $77.92 \\pm 0.49$  &   $67.69 \\pm 0.27$  \\\\\n    \\rowcolor{mybluelight}\n    \\textbf{Label\\_prop (ours)}   &   $\\bm{81.48 \\pm 1.03}$  &   $\\bm{71.73 \\pm 0.67}$ \\\\\n    \\midrule\n    Full\\_sup   &   $86.53 \\pm 0.31$  &   $75.29 \\pm 1.50$  \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\n\\begin{table*}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Readings of Dice index (mean $\\pm$ standard deviation) from Fig.\\ref{fig:fig-al-results} over test set for the \\textbf{first 5 queries} formed by each method. We start from the second query since the first query is random but identical for all methods.}\n\\label{tab:tab-dice-q}\n\\centering\n   \\resizebox{0.9\\linewidth}{!}{\n\\begin{tabular}{l|cccccc}\n   \\textbf{Queries}  &  \\textbf{q2} & \\textbf{q3} & \\textbf{q4} & \\textbf{q5} & \\textbf{q6} \\\\\n   \\toprule\n   \\toprule\n   \\multicolumn{6}{c}{GlaS} \\\\\n    \\toprule\n    WSL   &    \\multicolumn{4}{c}{$66.44 \\pm 0.20$}  \\\\\n    \\midrule\n    Random   &   $70.26 \\pm 3.02$ &   $71.58 \\pm 3.14$ & $71.43 \\pm 1.83$ & $74.05 \\pm 3.14$ &    $75.36 \\pm 3.45$\\\\\n    Entropy   &   $\\bm{72.75 \\pm 2.96}$ &   $70.93 \\pm 3.58$ & $72.60 \\pm 1.44$ & $73.44 \\pm 1.38$ &   $75.15 \\pm 1.63$\\\\\n    MC\\_Dropout   &   $68.44 \\pm 2.89$  &   $69.70 \\pm 1.96$ &   $69.97 \\pm 1.95$ & $72.71 \\pm 2.21$ & $73.00 \\pm 1.04$\\\\\n    \\rowcolor{mybluelight}\n    \\textbf{Label\\_prop (ours)}   &   $71.02 \\pm 4.19$  &   $\\bm{74.07 \\pm 3.93}$ &   $ \\bm{76.52 \\pm 3.49}$ & $\\bm{77.63 \\pm 2.73}$ & $\\bm{78.41 \\pm 1.23}$\\\\\n  \n    Full\\_sup   &   \\multicolumn{4}{c}{$86.53 \\pm 0.31$}\\\\\n    \\toprule  \\toprule\n   \\multicolumn{6}{c}{CUB} \\\\\n    \\toprule\n    WSL   &    \\multicolumn{4}{c}{$39.22 \\pm 0.18$}  \\\\\n    \\midrule\n    Random   &   $56.86 \\pm 2.07$ &   $61.39 \\pm 1.85$ & $62.97 \\pm 1.13$ & $63.56 \\pm 4.02$ &    $66.56 \\pm 2.50$\\\\\n    Entropy   &   $53.37 \\pm 2.06$ &   $59.11 \\pm 2.50$ & $60.48 \\pm 3.56$ & $63.81 \\pm 2.75$ &   $63.59 \\pm 2.34$\\\\\n    MC\\_Dropout   &   $57.13 \\pm 0.83$  &   $59.98 \\pm 2.06$ &   $63.52 \\pm 2.26$ & $63.02 \\pm 2.68$ & $64.68 \\pm 1.41$\\\\\n    \\rowcolor{mybluelight}\n    \\textbf{Label\\_prop (ours)}  &   $\\bm{62.58 \\pm 2.15}$  &   $\\bm{66.32 \\pm 2.34}$ &   $ \\bm{67.01 \\pm 2.85}$ & $\\bm{69.40 \\pm 3.40}$ & $\\bm{68.28 \\pm 1.60}$\\\\\n    \\midrule\n    Full\\_sup   &   \\multicolumn{4}{c}{$75.29 \\pm 1.50$}\\\\\n    \\bottomrule\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\nWe report the classification and segmentation performances following the training the proposed deep WSL model in Fig.\\ref{fig:fig-archi}. Tab.\\ref{tab:tab-cl-acc} reports the Classification accuracy of the classification head using WSL, which is\nclose to the results reported in \\citep{belharbi2019weakly,rony2019weak-loc-histo-survey}. The results of GlaS suggest that it is an easy dataset for classification.\n\nThe segmentation results are reported in Tabs. \\ref{tab:tab-dice-q} and \\ref{tab:tab-seg-perf}, and in Fig \\ref{fig:fig-al-results}.\n\nFig. \\ref{fig:fig-al-0} compares Dice accuracy on the \\textbf{GlaS dataset}. On the latter, we observe that adding more labels increases Dice index for all AL methods, yielding, as expected, better performance than the WSL method.  Reading from Tab.\\ref{tab:tab-dice-q}, randomly labeling only 4 samples per class enables to easily outperform WSL. This means that using our approach in Fig.\\ref{fig:fig-archi}, with limited supervision, can lead to more accurate masks compared to using CAMs in the WSL method. From Fig.\\ref{fig:fig-al-0}, one can also observe that Random, Entropy, and MC\\_Dropout methods grow relatively in the same way, leading to the same overall performance, with the Entropy method slightly ahead. Considering the overall behavior of the curves, one may conclude that using advanced selection techniques such as MC\\_Dropout and Entropy provides an accuracy similar to simple random selection. On the one hand, since both methods have shown substantial improvements in AL for classification, and based on the results in Fig.\\ref{fig:fig-al-0}, one may conclude that all samples are equivalently informative for the model. Therefore, there is no better order to acquire them. On the other hand, using simply random selection and pseudo-labeled samples allowed our method to substantially improve the overall performance,  demonstrating the benefits of self-learning.\n\nFig.\\ref{fig:fig-al-1} and Tab.\\ref{tab:tab-dice-q} compare Dice accuracy on the \\textbf{CUB dataset}, where labeling only one sample per class yielded a large improvement in Dice index, in comparison to WSL. Adding more samples increases the performance of all the methods. One can observe similar pattern as for GlaS: Random, Entropy and MC\\_Dropout methods yield similar curves, while the AUC performances of Random and Entropy methods are similar, and slightly ahead of MC\\_Dropout.\nSimilar to GlaS analysis, and based on the results of these three methods, one can conclude that none of the methods for ordering the samples is better than simple random selection. Using self-labeled samples in our method shows again its benefits. Simple random selection combined with self-annotation yields an overall best performance. Using two datasets, our empirical results suggest that self-learning, under limited oracle-annotation, has the potential to provide a reliable second source of annotation, which can efficiently enhance model performance, while using simple sample acquisition techniques.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=.7\\linewidth]{dice-pseudo-labeled}\n  \\caption{Average Dice index over the pseudo-labeled samples of our method in \\textbf{each} AL round.}\n  \\label{fig:fig-dice-pseudo-labeled}\n\\end{figure}\n\n\\noindent \\textbf{Pseudo-annotation performance}. Furthermore, the proposed approach is assessed on the pseudo-labeled samples at each AL round. Fig.\\ref{fig:fig-dice-pseudo-labeled} shows that the model provides good segmentations at the initial rounds. Then, the more supervision, the more accurate the pseudo-segmentation, as expected. This figure shows the interest and potential of self-learning in segmentation, and confirms our assumption that samples near the labeled ones are likely to achieve accurate pseudo-segmentation by the model.\n\n\\noindent \\textbf{Hyper-parameters}. Our approach requires two main hyper-parameters: ${ k \\ \\text{and } \\lambda}$. We conducted an ablation study over ${k}$ on GlaS dataset, and over ${\\lambda}$ on both datasets. Results, which are presented in the supplementary material, suggest that our method is less sensitive to ${k}$. ${\\lambda}$ plays an important role, and\nbased on our study, we recommend using small values of this weighting parameter. In our experiments, we used $\\lambda=0.1$ for Glas and $\\lambda = 0.001$ for CUB. We set ${k=40}$. We note that hyper-parameter tuning in AL is challenging due to the change of the size of the data set, which in turn changes the training dynamics. In all the experiments, we used fixed hyper-parameters across the AL rounds.\nFig.\\ref{fig:fig-dice-pseudo-labeled} suggests that a dynamic ${\\lambda(r)}$ that is increased through AL rounds could be more beneficial. However, this requires a principled update protocol for ${\\lambda}$, which was not explored in this work. Nonetheless, using a fixed value seems to yield promising results overall.\n\n\\noindent \\textbf{Supplementary material}. Due to space limitation, we deferred the hyper-parameters used in the experiments, results of the ablation study, visual results for the similarity measure and examples of predicted masks to the supplementary materials.\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nDeep WSL models trained with global image-level annotations can play an important role in CNN visualization and interpretability. However, they are prone to high false-positive rates, especially for challenging images, leading to poor segmentations.\nTo alleviate this issue, we considered  using pixel-wise supervision provided gradually through an AL framework. This annotation is integrated into training using an adequate deep convolutional model that allows supervised learning for both tasks: classification and segmentation. Through a few pixel-supervised samples, such a design is intended to provide full-resolution and more accurate masks compared to standard CAMs, which are trained without pixel supervision and often provide coarse resolution. Therefore, it enables a better CNN visualization and interpretation of CNN predictions.\nFurthermore, and unlike standard deep AL methods that focus solely on the acquisition function, we considered using self-learning as a second source of supervision to fast-improve the model segmentation.\nEvaluating our method using a realistic AL protocol over two challenging benchmarks, our results indicate that:\n(1) using a \\emph{few} supervised samples, the proposed architecture yielded more accurate segmentations compared to CAMs, with a large margin using different AL methods. Thus, it provides a solution to enhance pixel-wise predictions\nin real-world visual recognition applications.\n(2) using self-learning with random selection yielded substantial improvements. Self-learning under a limited oracle-budget can, therefore, provide a cost-effective alternative to standard AL protocols, where most of the effort is spent on the acquisition function.\n\n\\section*{Acknowledgment}\nThis research was supported in part by the Canadian Institutes of Health Research, the Natural Sciences and Engineering Research Council of Canada, Compute Canada, MITACS, and the Ericsson Global AI Accelerator Montreal.\n\n\\clearpage\n\\newpage\n\n\\appendices\n\n\n\n\\section{Supplementary material for the experiments}\nDue to space limitation, we provide in this supplementary material detailed hyper-parameters used in the experiments, results of the ablation study, visual results to the similarity measure, and examples of predicted masks.\n\n\\subsection{Training hyper-parameters}\n\\label{subsec:hyper-params}\nTab.\\ref{tab:tabx-tr-hyper-params} shows the used hyper-parameters in all our experiments.\n\n\n\\begin{table}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Training hyper-parameters.}\n\\label{tab:tabx-tr-hyper-params}\n\\centering\n\\resizebox{0.7\\linewidth}{!}{\n\\begin{tabular}{lccc}\n    Hyper-parameter  &  GlaS  & CUB\\\\\n    \\toprule\n    Model backbone  & \\multicolumn{2}{c}{ResNet-18 \\citep{heZRS16}}\\\\\n    WILDCAT \\citep{durand2017wildcat}: && \\\\\n    ${\\alpha}$ &  \\multicolumn{2}{c}{${0.6}$} \\\\\n    ${kmin}$ &  \\multicolumn{2}{c}{${0.1}$} \\\\\n    ${kmax}$ &  \\multicolumn{2}{c}{${0.1}$} \\\\\n    modalities &  \\multicolumn{2}{c}{${5}$} \\\\\n    \\midrule\n    Optimizer &  \\multicolumn{2}{c}{SGD}\\\\\n    Nesterov acceleration & \\multicolumn{2}{c}{True}\\\\\n    Momentum & \\multicolumn{2}{c}{$0.9$} \\\\\n    Weight decay & \\multicolumn{2}{c}{$0.0001$}\\\\\n    Learning rate (LR) & ${0.1}$ (WSL: $10^{-4}$) & ${0.1}$ (WSL: $10^{-2}$)\\\\\n    LR decay & ${0.9}$  & ${0.95}$ (WSL: ${0.9}$) \\\\\n    LR frequency decay & $100$ epochs & $10$ epochs  \\\\\n    Mini-batch size & ${20}$ & ${8}$ \\\\\n    Learning epochs & ${1000}$ & ${30}$ (WSL: ${90}$) \\\\\n    \\midrule\n    Horizontal random flip & \\multicolumn{2}{c}{True} \\\\\n    Vertical random flip & True & False  \\\\\n   \n    Crop size & \\multicolumn{2}{c}{${416 \\times 416}$}\\\\\n    \\midrule\n    ${k}$ & \\multicolumn{2}{c}{${40}$}\\\\\n    ${\\lambda}$ & ${0.1}$ & ${0.001}$ \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\\begin{figure}[h!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.5]{knn-ablation-glas-Label_prop-best-k-40}\n  \\caption{Ablation study over GlaS dataset (test set) over the hyper-parameter $k$ (x-axis). y-axis: AUC of Dice index (\\%) of \\textbf{25 queries for one trial}.\n  AUC average $\\pm$ standard deviation: ${81.49 \\pm 0.59}$.\n  Best performance in red dot: $k=40, AUC=82.41\\%$.\n  }\n  \\label{fig:abl-glas-knn}\n\\end{figure}\n\n\n\n\\begin{figure}[h!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.5]{lambda-ablation-glas-Label_prop-best-lambda-0-dot-1}\n  \\caption{Ablation study over GlaS dataset (test set) over the hyper-parameter ${\\lambda}$ (x-axis). y-axis: AUC of Dice index (\\%) of \\textbf{15 queries for one trial}.\n  Best performance in red dot: $\\lambda=0.1, AUC=79.15\\%$.\n  }\n  \\label{fig:abl-glas-lambda}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.5]{lambda-ablation-Caltech-UCSD-Birds-200-2011-Label_prop-best-lambda-0-dot-001}\n  \\caption{Ablation study over CUB dataset (test set) over the hyper-parameter ${\\lambda}$ (x-axis). y-axis: AUC of Dice index (\\%) of \\textbf{5 queries for one trial}.\n  Best performance in red dot: $\\lambda=0.001, AUC=66.94\\%$.\n  }\n  \\label{fig:abl-cub-lambda}\n\\end{figure}\n\n\n\\subsection{Ablation study}\n\\label{sub:ablation}\n\nWe study the impact of $k$ and ${\\lambda}$ on our method. Results are presented in Fig.\\ref{fig:abl-glas-knn}, \\ref{fig:abl-glas-lambda} for GlaS over ${k, \\lambda}$; and in Fig.\\ref{fig:abl-cub-lambda} for CUB over ${\\lambda}$. Due to the expensive computation time required to perform AL experiments, we limited the experiments (${k, \\lambda}$, number of trials, and \\texttt{maxr}).\nThe obtained results of this study show that our method is less sensitive to ${k}$ (standard deviation of ${0.59}$ in Fig.\\ref{fig:abl-glas-knn}).\nIn other hand, the method shows sensitivity to ${\\lambda}$ as expected from penalty-based methods \\citep{bertsekas1999nonlinear}.\nHowever, the method seems more sensitive to ${\\lambda}$ in the case of CUB than GlaS. CUb dataset is more challenging leading to more potential erroneous pseudo-annotation. Using Large ${\\lambda}$ will systematically push the model to learn on the wrong annotation (Fig.\\ref{fig:abl-cub-lambda}) which leads to poor results. In the other hand, GlaS seems to allow obtaining good segmentation where using large values of ${\\lambda}$ did not hinder the performance quickly (\\ref{fig:abl-glas-lambda}).\nThe obtained results recommend using small values that lead to better and stable performance. Using high values, combined with the pseudo-annotation errors, push the network to learn erroneous annotation leading to overall poor performance.\n\n\n\n\n\n\\subsection{Similarity measure}\n\\label{subsec:similarity}\nIn this section, we present some samples with their nearest neighbors. Although, it is difficult to quantitatively evaluate the quality of such measure.\nFig.\\ref{fig:glas-sim} shows the case of GlaS. Overall, the similarity shows good behavior of capturing the general stain of the image which is what was intended for since the structure of such histology images is subject to high variation. Since the stain variation is one of the challenging aspects in histology images \\citep{rony2019weak-loc-histo-survey}, labeling a sample with a common stain can help the model in segmenting other samples with similar stain.\nThe case of CUB, presented in Fig.\\ref{fig:cub-sim}, is more difficult to judge the quality since the images contain always the same species within their natural habitat. Often, the similarity succeeds to capture the overall color, background which can help segmenting the object in the neighbors and also the background. In some cases, the similarity captures samples with large zoom-in where the bird color dominate the image.\n\n\n\\begin{figure*}[hbt!]\n\\centering\n  \\centering\n  \\includegraphics[width=1.\\linewidth]{glas}\n  \\caption{Examples of ${k}$-nn over GlaS dataset. The images represents the 10 nearest images to the first image in the extreme left ordered from the nearest.}\n  \\label{fig:glas-sim}\n\\end{figure*}\n\n\n\\begin{figure*}[hbt!]\n\\centering\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{Caltech-UCSD-Birds-200-2011}\n  \\caption{Examples of ${k}$-nn over CUB dataset. The images represents the 10 nearest images to the first image in the extreme left ordered from the nearest.}\n  \\label{fig:cub-sim}\n\\end{figure*}\n\n\n\\subsection{Predicted mask visualization}\n\\label{subsec:mask-vis}\nFig.\\ref{fig:cub-results-visu} shows several test examples of predicted masks of different methods over CUB test set at the first AL round (${r=1}$) where only one sample per class has been labeled by the oracle. This interesting functioning point shows that by labeling only one sample per class, the performance of the average Dice index can go from ${39.08 \\pm 08}$ for WSL method up to ${62.58 \\pm 2.15}$ for Label\\_prop and other AL methods. The figure shows that WSL tend to spot small part of the object in addition to the background leading high false positive. Using few supervision in combination with the proposed architecture, better segmentation is achieved by spotting large part of the object with less confusion with the background.\n\n\\begin{figure*}[h!]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{sample-test-Caltech-UCSD-Birds-200-2011} \\\\\n  \\includegraphics[width=0.95\\linewidth]{code}\n  \\caption{Qualitative results (on several CUB test images) of the predicted binary mask for each method after being trained in the first round ${r=1}$ (\\emph{i.e.}\\@\\xspace after labeling 1 sample per class) using seed=0.  The average Dice index over the test set of each method is: ${40.16\\%}$ (WSL), ${55.32\\%}$ (Random), ${55.41\\%}$ (Entropy), ${55.52\\%}$ (MC\\_Dropout), ${59.00\\%}$ (Label\\_prop), and ${75.29\\%}$ (Full\\_sup). (Best visualized in color.)}\n  \\label{fig:cub-results-visu}\n\\end{figure*}\n\n\\FloatBarrier\n\n\n\n\n\n\n\n\\bibliographystyle{apalike}\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nImage classification and segmentation are fundamental tasks in many visual recognition applications involving natural and medical images. Given a large image dataset annotated with global image-level labels for classification or with pixel-level labels for segmentation, deep learning (DL) models achieve state-of-the-art performances for these tasks  \\citep{dolz20183d,Goodfellow-et-al-2016,krizhevsky12,LITJENS201760,LongSDcvpr15,Ronneberger-unet-2015}. However, the impressive accuracy of such fully-supervised learning models comes at the expense of a considerable cost for collecting and annotating large image data sets. While the acquisition of global image-level annotation can be relatively inexpensive, pixel-wise annotation involves a laborious process, a difficulty further accrued by the requirement of domain expertise, as in medical imaging, which increases the annotation costs.\n\\begin{figure}[t!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.49]{proposal}\n  \\caption{Proposed AL framework with weak annotator.}\n  \\label{fig:fig-proposal}\n\\end{figure}\n\nWeakly-supervised learning (WSL) has recently emerged as a paradigm that relaxes the need for dense pixel-wise annotations \\citep{rony2019weak-loc-histo-survey,zhou2017brief}. WSL techniques depend on the type of application scenario and annotation, such as global image-level labels  \\citep{belharbi2019weakly,kim2017two,pathak2015constrained,teh2016attention,wei2017object}, scribbles \\citep{Lin2016,ncloss:cvpr18}, points \\citep{bearman2016}, bounding boxes \\citep{dai2015boxsup,Khoreva2017} or global image statistics such as the target-region size \\citep{bateson2019constrained,jia2017constrained,kervadec2019curriculum,kervadec2019constrained}. This paper focuses on learning using only image-level labels, which enables to classify an image while yielding pixel-wise scores (i.e., segmentations), thereby localizing the regions of interest linked to the image-class predictions.\n\n\nSeveral CNN visualization and interpretation methods have recently been proposed, based on either perturbation, propagation or activation approaches, and allowing to localize the salient image regions responsible for a CNN's predictions \\citep{fu2020axiom}. In particular, WSL techniques \\citep{rony2019weak-loc-histo-survey} rely on activation-based methods like CAM and, more recently, Gradient-weighted Class Activation Mapping (Grad-CAM), Grad-CAM++, Ablation-CAM and Axiom-based Grad-CAM \\citep{fu2020axiom, lin2013network, PinheiroC15cvpr}. Trained with only global image-level annotations, these methods locate the regions of interest (ROIs) of the  corresponding class in a relatively inexpensive and accurate way. However, while these WSL techniques can provide satisfying results in natural images, they typically yield poor segmentations in relatively more challenging scenarios, for instance, histology data in medical image analysis \\citep{rony2019weak-loc-histo-survey}.\nWe note two limitations associated with CAMs: (1) they are obtained in an unsupervised way (\\emph{i.e.}\\@\\xspace without pixel-level supervision under an ill-posed learning problem \\citep{choe2020evaluating}); and (2) they have low resolution. For instance, CAMs obtained from ResNet models \\citep{heZRS16} have a resolution of ${1\/32}$ of the input image. Interpolation is required to restore full image resolution. Both of these limitation with CAM-based methods lead to high false-positive rates, which may render them impractical \\citep{rony2019weak-loc-histo-survey}.\n\nEnhancing deep WSL models with pixel-wise annotation, as supported by a recent study in weakly-supervised object localization \\citep{choe2020evaluating}, can improve localization and segmentation accuracy, which is the central goal of this paper. To do so, we introduce a deep WSL model that allows supervised learning for classification, and active learning for segmentation, with the latter providing more accurate and high-resolution masks. We assume that the images in the training set are globally annotated with image-class labels.  Relevant images are \\emph{gradually} labeled at the pixel level through an oracle that respects a low  annotation-budget constraint. Consequently, this leads us to an active learning (AL) paradigm \\citep{settles2009active}, where an oracle is requested to annotate pixels in a subset of samples.\n\nDifferent sample-acquisition techniques have been successfully applied to deep AL for classification based on, e.g., certainty \\citep{ducoffe2015qbdc,gal2017deep,kirsch2019batchbald} or representativeness \\citep{kim2020task,sinha2019variational}. However, very few deep AL techniques were investigated in the context of segmentation \\citep{gaur2016membrane,gorriz2017active,lubrano2019deep}. Most AL techniques focus mainly on the sampling criterion (Fig.\\ref{fig:fig-proposal}, left) to populate the labeled pool using an oracle. During training, only the labeled pool is used, while the unlabeled pool is left dormant. Such an AL protocol may limit the accuracy of DL models under constrained oracle-supervision budget in real-world applications for multiple reasons:\n\n\\textbf{(1)} Standard AL protocols may be relevant to small\/shallow models that can learn and provide reliable queries using a few training samples. Since training accurate DL models typically depends on large training sets, large numbers of queries may be needed to build reliable DL models, which may incur a high annotation cost.\n\n\\textbf{(2)} In most AL work, the experimental protocol starts with a large labeled pool, and acquires a large number of queries for sufficient supervision, neglecting the workload placed on the oracle.\nThis typically reaches a plateau-performance of a DL quickly, hampering a reliable study of the impact of different AL selection techniques. Moreover, model-based sampling techniques are inconsistent \\citep{gaur2016membrane} in the sense that the model is used to query samples while it is still in an early learning stage.\n\n\\textbf{(3)} Segmentation and classification problems are associated with different properties and challenges, such as decision boundaries and uncertainty, which provide additional challenges to AL.  For instance, the class boundaries defined by different classification methods \\citep{ducoffe2018adversarial,settles2009active,tong2001support} are not defined in the context of segmentation, making such a branch of methods inadequate for segmentation.\n\n\\textbf{(4)}\nIn critical fields such as medical imaging, acquiring a sample itself can be very expensive\\footnote{\nFor instance, prior to a diagnosis of breast cancer from a histological sample, a patient undergoes a bilateral diagnostic mammogram and breast ultrasound that are interpreted by a radiologist, one to several needle biopsies (with low risks under ${1\\%}$ of hematoma and wound infection) to further assess areas of concern, surgical consultation and pre-operative blood work, and surgical excision of the positive tissue for breast cancer cells. The biopsy and surgical tissues are processed (fixation, embedding in parraffin, H\\&E staining) and interpreted by a pathologist. Depending on the cancer stage, the patient may undergo additional procedures. Therefore, accounting for all the steps required for breast cancer diagnosis from histological samples, a rough estimation of the final cost associated with obtaining a Whole Slide Image (WSI) is about \\$400 (Canadian dollars, by 1999) \\citep{will1999diagnostic}. Moreover, some cancer types are rare \\citep{will1999diagnostic}, adding to the values of these samples. All these procedures are conducted by highly trained experts, with each procedure taking from a few minutes to an hour and requiring expensive specialized medical equipment.}.\nThe time and cost associated with each sample makes them valuable items. Such considerations may be overlooked for large-scale data sets with almost-free samples, as in the case of natural images. Given this high cost, keeping the unlabeled pool dormant during learning may be ineffective.\n\n\nBased on the aforementioned arguments, we advocate that focusing solely on the sample acquisition and supervision pool may not be an efficient way to build\nhigh-performing DL models in an AL framework for segmentation. Therefore, we consider augmenting the labeled pool using the model as a second source of annotation, in a self-learning fashion \\citep{mao2020survey} (Fig.\\ref{fig:fig-proposal}, right). This additional annotation might be less accurate (\\emph{i.e.}\\@\\xspace, weak\\footnote{Not to be confused with the weak annotation of data in weakly supervised learning frameworks.}) compared to the oracle that provides strong but expensive annotations. However, it is expected to fast-improve the model's performance \\citep{mao2020survey}, while using a few oracle-annotated samples, reducing the annotation cost.\n\nOur main contributions are the following.\n\\textbf{(1) Architecture design}: As an alternative to CAMs, we propose using a segmentation mask trained with pixel-level annotations, which yields more accurate and high-resolution ROIs. This is achieved through a convolutional architecture capable of simultaneously classifying and segmenting images, with the segmentation task trained using annotations acquired using an AL framework.\nAs illustrated in Fig.\\ref{fig:fig-archi}, the architecture combines well-known DL models for classification (ResNet \\citep{heZRS16}) and segmentation (U-Net \\citep{Ronneberger-unet-2015}), although other architectures could also be used.\n\\textbf{(2) Active learning}: We augment the size of the labeled pool by weak-annotating a large number of unlabeled samples based on predictions of the DL model itself, providing a second source of annotation (Fig.\\ref{fig:fig-proposal}). This enables rapid improvements of the segmentation accuracy, with less oracle-based annotation. Moreover, our method can be integrated on top of any sample-acquisition method.\n\\textbf{(3) Experimental study}: We conducted comprehensive experiments over two challenging benchmarks -- high-resolution medical images (histology GlaS data for colon cancer) and natural images (CUB-200-2011 for bird species). Our results indicate that, by simply using random sample selection, the proposed approach can significantly outperform state-of the-art CAMs and AL methods, with an identical oracle-supervision budget.\n\n\n\\section{Related work}\n\\label{sec:related-work}\n\n\\noindent \\textbf{Deep active learning:}\nAL has been studied for a long time in machine learning, mainly for classification and regression, using linear models in particular  \\citep{settles2009active}. Recently, there has been an effort to transfer such techniques to DL models for classification tasks by mimicking their intuition or by adapting them, taking into consideration model specificity and complexity. Such methods include, for instance, different mechanisms for uncertainty  \\citep{beluch2018power,ducoffe2015qbdc,ducoffe2018adversarial,gal2017deep,kim2020task,kirsch2019batchbald,lakshminarayanan2017simple,wang2016cost,yoo2019learning} and representativeness estimation \\citep{kim2020task,sener2018coreset,sinha2019variational}. However, most deep AL techniques are validated on synthetic, simple or tiny data, which does not explore their full potential in real applications.\n\nWhile deep AL for classification is rapidly growing, deep AL models for segmentation are uncommon in the literature. In fact, the very few methods in the literature mostly focused on the direct application of deep AL classification methods. The limited research in this area may be explained by the fact that segmentation tasks bring challenges to AL, such as the additional spatial information and the fact that a segmentation mask lies in a much larger dimension than a classification prediction. In classification, AL often deals with one output that is used to drive queries \\citep{huang2010active}. The spatial information in segmentation does not naturally provide a direct scoring function that can indicate the overall quality or certainty of the output. Most of deep AL methods for segmentation consider pixels as classification instances, and apply standard AL techniques to each pixel.\n\nFor instance, the authors of \\citep{gaur2016membrane} exploit a variant of entropy-based acquisition at the pixel level, combined with a distribution-based term that encodes diversity using a complex hierarchical clustering algorithm over sliding windows, with application to microscopic membrane segmentation. Similarly, \\citep{gorriz2017active,lubrano2019deep} apply Monte-Carlo dropout uncertainty \\citep{gal2017deep} at the pixel level, with application to myelin segmentation using spinal cord and brain microscopic histology images. In \\citep{roels2019cost}, the authors experiment with five acquisition functions of classification for a segmentation task, including entropy-based, core-set \\citep{sener2018coreset}, k-mean and Bayesian \\citep{gal2017deep} sampling,  with application to electron microscopy segmentation. Entropy-based methods seem to be dominant over multiple datasets. In \\citep{yang2017suggestive}, the authors combine two sampling terms for histology image segmentation. The first employs bootstrapping over fully convolutional networks (FCN) to estimate uncertainty, where a set of FCNs are trained on different subsets of samples. The second term is a representation-based term that selects the most representative samples. This is achieved by solving an optimization of a generalization version of the maximum cover set problem \\citep{feige1998threshold} using sample description extracted from an FCN. Despite the obtained promising results, this approach remains complex and impractical due to the use of bootstrapping over DL models and an optimization step. Moreover, the authors of \\citep{yang2017suggestive} do not provide a comparison to other acquisition functions. The work in \\citep{casanova2020} considers a specific case of AL using reinforcement learning for \\emph{region-based} AL for segmentation, where only a selected region of the image is labeled. This method is adequate for data sets with large and unbalanced classes, such as street-view images. While the method in \\citep{casanova2020} outperforms random and Bayesian \\citep{gal2017deep} selection, surprisingly, it performs close to entropy-based selection.\n\n\n\\noindent \\textbf{Weak annotators:}\nThe AL paradigm does not prohibit the use of unlabelled data for training \\citep{settles2009active}, but it mainly constrains the oracle-labeling budget. The standard AL experimental protocol (Fig.\\ref{fig:fig-proposal}, left) was inherited from AL of simple\/linear ML models, and adopted in subsequent works. Budget-constrained oracle annotation may not be sufficient to build effective DL models, due to the lack of labeled samples. Furthermore, several studies in AL for classification have successfully leveraged the unlabelled data to provide additional supervision  \\citep{lin2017active,long2008graph,vijayanarasimhan2012active,wang2016cost,zhou2010active,zhualsslharmo2003}.\n\nFor instance, the authors of \\citep{lin2017active,wang2016cost} propose to pseudo-label a part of the unlabeled pool. The latter is selected using dynamic thresholding based on confidence, through the model itself, so as to learn a better embedding. Furthermore, a theoretical framework for AL using strong and weak annotators for classification task is introduced in \\citep{zhang2015active}. Their results suggest that using multiple annotators can reduce the cost of oracle annotation, in comparison to one annotator. Multiple sources of annotations that include both strong and weak annotators were used in AL, crowd-sourcing, self-paced learning and other interactive learning scenarios for classification to help reducing the number of requests for the strong annotator \\citep{kremer2018robust,malago2014online,mattsson2017active,murugesan2017active,urner2012learning,yan2016active,zhang2015active}.\nUsing the model itself for pseudo-annotation is motivated mainly by the success of deep self-supervised learning \\citep{mao2020survey}.\n\n\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n\\centering\n\\includegraphics[scale=0.65]{knn}\n\\caption{The $k$-nn method for selecting $\\mathbb{U}^{\\prime\\prime}$ subset to be speudo-labeled. Assumption to select $\\mathbb{U}^{\\prime\\prime}$: since $\\mathbb{U}^{\\prime\\prime}$ lives nearby supervised samples, it is more likely to be assigned good segmentation by the model.\nWe consider measuring $k$-nn for each \\textbf{unlabeled} sample. In this example, using $k=4$ allows ${\\abs{\\mathbb{U}^{\\prime\\prime}} = 14}$. If $k$-nn is considered for each \\textbf{labeled} sample: $\\abs{\\mathbb{U}^{\\prime\\prime}} = 8$. ${\\abs{\\cdot}}$ is the set cardinal. Note that ${k}$-nn is only considered between samples of the \\emph{same class}.}\n  \\label{fig:fig-knn}\n\\vspace*{-0.1in}\n\\end{wrapfigure}\n\n\n\\noindent \\textbf{Label Propagation (LP):}\nOur approach is also related to LP methods \\citep{bengiolabelprop2010,zhou2004learning,zhulp2002} for classification, which aim to label unlabeled samples using knowledge from the labeled ones (Fig.\\ref{fig:fig-knn}). However, while LP propagates labels to unlabeled samples through an iterative process, our approach bypasses this using the model itself. In our case, the propagation is limited to the neighbors of labeled samples defined through $k$-nearest neighbors ($k$-nn) (Fig.\\ref{fig:fig-knn}). Using $k$-nn has been also studied to combine AL and domain adaptation \\citep{berlind2015active}, where the goal is to query samples from the target domain. Such an approach is connected to the recently developed core-set method for deep AL \\citep{sener2018coreset}. Our method intersects with \\citep{berlind2015active} only in the sense of predicting the labels to query samples using their labeled neighbors.\n\nIn contrast to state-of-the-art DL models for AL segmentation, we consider increasing the unlabeled pool through pseudo-annotated samples (Fig.\\ref{fig:fig-proposal}, right). To this end, the model is used for pseudo-labeling samples within the neighborhood of samples with strong supervision (Fig.\\ref{fig:fig-knn}). From a self-learning perspective, the works in \\citep{lin2017active,wang2016cost} on face recognition are the closest to ours. While both rely on pseudo-labeling, they mainly differ in the sample selection for pseudo-annotation. In \\citep{lin2017active,wang2016cost}, the authors considered model confidence, where samples with high confidence are pseudo-labeled, while low-confidence samples are queried. While this yields good results, it makes the overall method strongly dependent on model confidence. As we consider segmentation tasks, model-confidence is not well-defined. Moreover, using the expected pixel-wise values can be less representative for model confidence.\n\nOur approach relies on the spatial assumption in Fig.\\ref{fig:fig-knn}, where the samples to pseudo-label are selected to be near the labeled samples, and expected to have good pseudo-segmentations. This makes the oracle-querying technique independent from the pseudo-labeling method, giving more flexibility to the user. Our pseudo-labeled samples are added to the labeled pool, along with the samples annotated by the oracle. The underlying assumption is that, given a sample labeled by an oracle, the model is more likely to produce good segmentations of images located nearby that sample. Our assumption is verified empirically in the experimental section of this paper. This simple procedure enables to rapidly increase the number of pseudo-labeled samples, and helps improving segmentation performance under a limited amount of oracle-based supervision.\n\n\n\n\n\n\\section{Proposed approach}\n\\label{sec:proposal}\n\n\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n\\centering\n \\includegraphics[scale=0.75]{archi}\n \\caption{Our proposed DL architecture for classification and segmentation  composed of: (1) a shared \\textbf{backbone} for feature extraction; (2) a \\textbf{classification head} for the classification task; (3) and a \\textbf{segmentation head} for the segmentation task with a U-Net style \\citep{Ronneberger-unet-2015}. The latter merges representations from the backbone, while gradually upscaling the feature maps to reach the full image resolution for the predicted mask, similarly to the U-Net model.}\n \\label{fig:fig-archi}\n\\vspace*{-0.1in}\n\\end{wrapfigure}\n\n\nWe consider an AL framework for training deep WSL models, where all the training images have class-level annotations, but no pixel-level annotations. Due to their high cost, pixel annotations are gradually acquired for training through oracle queries. It propagate pixel-wise knowledge encoded in the model though the labeled images.\n\nActive learning training consists of sequential training rounds. At each round $r$, the total training set ${\\mathbb{D}}$ that contains $n$ samples with unlabeled and labeled subsets (Fig.\\ref{fig:fig-proposal}).\n\\textbf{(1) Unlabeled subset:} contains samples without pixel-wise annotation (unlabeled samples) ${\\mathbb{U} = \\{\\bm{x}_i, y_i, -\\-\\}_{i=1}^u}$ where ${\\bm{x} \\in \\mathcal{X}}$ is the input image, ${y}$ is its global label; and the pixel label is missing.\n\\textbf{(2) Labeled subset:} contains samples with full supervision ${\\mathbb{L}=\\{\\bm{x}_i, y_i, \\bm{m}_i\\}_{i=1}^l}$ where ${\\bm{m}}$ is the pixel-wise annotation of the sample. ${\\mathbb{L}}$ is initially empty. It is gradually populated from ${\\mathbb{U}}$ by querying the oracle using an acquisition function.\nLet ${f(\\cdot: \\bm{\\theta})}$ denotes a DL model that is able to classify and segment an image ${\\bm{x}}$ (Fig.\\ref{fig:fig-archi}). For clarity, and since we focus on the segmentation task, we omit the notation for the classification task (to simplify the presentation). Therefore, ${f(\\bm{x})}$ refers to the predicted segmentation mask.\nLet ${\\mathbb{U}^{\\prime} \\subseteq \\mathbb{U}}$ and ${\\mathbb{U}^{\\prime\\prime} \\subseteq \\mathbb{U}}$ denote two subsets (Fig.\\ref{fig:fig-proposal}), with ${ \\mathbb{U}^{\\prime} \\cap \\mathbb{U^{\\prime\\prime}} = \\varnothing}$.\nIn our method, we introduce ${\\mathbb{P}}$ as a subset holder for pseudo-labeled samples, which is initially empty and gradually replenished (Fig.\\ref{fig:fig-proposal}, right).\nTo express the dependency of each subset on round ${r}$, we introduce the following notations: ${\\mathbb{U}_r, \\mathbb{L}_r, \\mathbb{P}_r, \\mathbb{U}^{\\prime}_r, \\mathbb{U}^{\\prime\\prime}_r}$.  The samples in ${\\mathbb{P}_r}$ are denoted as ${\\{\\bm{x}_i, y_i, \\hat{\\bm{m}}_i\\}}$. The following holds: ${\\forall r: \\mathbb{D} = \\mathbb{L}_r \\cup \\mathbb{U}_r \\cup \\mathbb{P}_r}$.\n\n\n\nAlg.\\ref{alg:alg-0} describes the overall AL process with our pseudo-annotation method.\nFirst, ${\\mathbb{U}^{\\prime}_r}$ is queried, then labeled by the oracle, and added to ${\\mathbb{L}_r}$.\nUsing $k$-nn, ${\\mathbb{U}^{\\prime\\prime}_r}$ is selected based on their proximity to ${\\mathbb{L}_r}$ (Fig.\\ref{fig:fig-knn}); and pseudo-labeled by the model, then added to ${\\mathbb{P}_r}$. To fast-increase the size of ${\\mathbb{L}_r}$, ${\\mathbb{P}_r}$ is protected from being queried for the oracle until it is inevitable. In the \\emph{latter case}, queried samples from ${\\mathbb{P}_r}$ are used to fill ${\\mathbb{U}^{\\prime}}$; and they are no longer considered pseudo-labeled since they will be assigned the oracle annotation.\n\nTo measure image similarity for the $k$-nn method, we used the color distribution to describe image content. This can be a flexible descriptor for highly unstructured images such as histology images. Note that the ${k}$-nn method is considered \\emph{only} for pairs of samples of the \\emph{same class}. The underlying assumption is that samples of the same class, with similar color distributions, are likely to contain relatively similar objects. Consequently, labeling representative samples could be a proxy for supervised learning based on the underlying data distribution. This can increase the likelihood of the model to provide relatively good segmentations of the other samples. The proximity between two images ${(\\bm{x}_i, \\bm{x}_j)}$ is measured using the Jensen-Shannon divergence between their respective color distributions (measured as normalized histograms). For an image with multiple color planes, the similarity is formulated as the sum of similarities, one for each plane.\n\nAt round $r$, the queried and pseudo-labeled samples are both used in training by optimizing the following loss function:\n\\begin{equation}\n\\label{eq:eq-1}\n    \\min_{\\bm{\\theta}} \\sum_{\\bm{x}_i \\in \\mathbb{L}_{r-1}} \\mathcal{L}(f(\\bm{x}_i), \\bm{m}_i) + \\lambda \\sum_{\\bm{x}_i \\in \\mathbb{P}_{r-1}} \\mathcal{L}(f(\\bm{x}_i), \\hat{\\bm{m}}_i),\n\\end{equation}\nwhere ${\\mathcal{L}(\\cdot, \\cdot)}$ is a segmentation loss, and ${\\lambda}$ a positive scalar. Eq.(\\ref{eq:eq-1}) corresponds to training the model (Fig.\\ref{fig:fig-archi}) solely for the segmentation task. Simultaneous training for classification and segmentation in this AL setup is avoided due to the unbalance between the number of samples that are labeled globally and at the pixel level. Therefore, we consider training the model for classification first. Then, we freeze the classifier parameters. Training for the segmentation tasks is resumed later. This yields the best classification performance, and allows a better study of the impact of the queried samples on the segmentation task.\n\nConsidering the relation of our method and label propagation algorithm \\citep{bengiolabelprop2010,zhou2004learning,zhulp2002}, we refer to our proposal as Label\\_prop.\n\n\n\\begin{center}\n\\begin{minipage}{0.6\\linewidth}\n\\IncMargin{0.04in}\n\\begin{algorithm}[H]\n    \\SetKwInOut{Input}{Input}\n   \n    \\Input{\n    ${\\mathbb{P}_0 = \\mathbb{L}_0 = \\varnothing}$\n    \\\\\n    ${\\bm{\\theta}^0}$: Initial parameters of ${f}$ trained on the classification task.\n    \\\\\n    ${\\texttt{maxr}: \\textrm{Maximum number of AL rounds}}$.\n    }\n    \\vspace{0.1in}\n    Select ${\\mathbb{U}^{\\prime}_0}$ randomly and label them by an oracle. \\\\\n    \\vspace{0.025in}\n    $  \\mathbb{L}_0 \\ \\leftarrow\\  \\mathbb{U}^{\\prime}_0$. \\\\\n    \\For{${r \\in 1 \\cdots \\texttt{maxr}}$} \n    {\n       \n        ${\\bm{\\theta} \\ \\leftarrow\\  \\bm{\\theta}^0}$. \\\\\n        \\vspace{0.03in}\n        Train $f$ using ${\\mathbb{L}_{r-1}}$ \\colorbox{mybluelight}{${\\cup\\; \\mathbb{P}_{r-1}}$} and the loss in Eq. (\\ref{eq:eq-1}). \\\\\n        \\vspace{0.03in}\n        Select ${\\mathbb{U}^{\\prime}_r}$ and label them by an oracle. \\\\\n        \\vspace{0.03in}\n        $  \\mathbb{L}_r \\ \\leftarrow\\  \\mathbb{L}_{r-1} \\cup \\mathbb{U}^{\\prime}_r$. \\\\\n        \\vspace{0.03in}\n        \\colorbox{mybluelight}{Select ${\\mathbb{U}^{\\prime\\prime}_r}$.} \\\\\n        \\vspace{0.03in}\n        \\colorbox{mybluelight}{$ \\mathbb{P}_r  \\ \\leftarrow\\   \\mathbb{P}_{r-1} \\cup \\mathbb{U}^{\\prime\\prime}_r$.} \\\\ \n        \\vspace{0.03in}\n        \\colorbox{mybluelight}{Pseudo-label ${\\mathbb{P}_r}$.}\n    }\n    \\vspace{0.1in}\n    \\caption{Standard AL procedure and our proposal. The extra instructions associated with our method are indicated with a \\colorbox{mybluelight}{blue background}.\n    }\n    \\label{alg:alg-0}\n\\end{algorithm}\n\\DecMargin{0.04in}\n\\end{minipage}\n\\end{center}\n\n\n\n\\section{Results and discussion}\n\\label{sec:experiments}\n\n\n\n \\subsection{Experimental methodology:}\n\n \\begin{figure}[!b]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{samples-datasets}\n    \\caption{\\textbf{Top row}: GlaS dataset \\citep{sirinukunwattana2017gland}. \\textbf{Bottom row}: CUB dataset \\citep{WahCUB2002011}. (Best visualized in color.)}\n  \\label{fig:fig-datasets}\n\\end{figure}\n\n\\noindent \\textbf{a) Datasets.}\nFor evaluation, datasets should have global and pixel-wise annotation. We consider two public datasets including both medical (histology) and natural images (Fig.\\ref{fig:fig-datasets}).\n\\begin{inparaenum}[(1)]\n\n  \\item \\textbf{GlaS dataset}: This dataset contains histology images for colon cancer diagnosis\\footnote{GlaS: \\href{https:\/\/warwick.ac.uk\/fac\/sci\/dcs\/research\/tia\/glascontest}{warwick.ac.uk\/fac\/sci\/dcs\/research\/tia\/glascontest}.} \\citep{sirinukunwattana2017gland}. It includes 165 images derived from 16 Hematoxylin and Eosin (H\\&E) histology sections of two grades (classes): benign and malignant. It is divided into 84 samples for training and 80 samples for testing.\n  The ROIs to be segmented are the glandes.\n \n \n  \\item \\textbf{CUB-200-2011 dataset (CUB)}\\footnote{CUB: \\href{http:\/\/www.vision.caltech.edu\/visipedia\/CUB-200-2011.html}{www.vision.caltech.edu\/visipedia\/CUB-200-2011.html}} \\citep{WahCUB2002011} is a dataset for bird species with ${11,788}$ samples ($5,994$ for training and $5,794$ for testing) and ${200}$ species. The ROIs to be segmented are the birds.\n \n  \\end{inparaenum}\n  %\n  In GlaS and CUB datasets, we randomly select $80\\%$ of the training samples for effective training, and $20\\%$ for validation (with full supervision) to perform early stopping. The splits are identical to the ones used in \\citep{belharbi2019unimoconstraints,rony2019weak-loc-histo-survey} (split 0, fold 0), and are publicly available.\n\n\\noindent \\textbf{b) Active learning setup.}\nTo assess the performance of different AL acquisition methods, we consider a realistic scenario with respect to the number of samples to be labeled at each AL round, accounting for the load imposed on the oracle.\nTherefore, only a few samples are selected at each round for oracle annotation, and  ${\\mathbb{L}}$ is slowly replenished. This allows better comparison between AL selection techniques since we spend more time\nin a phase where ${\\mathbb{L}}$ holds a few samples. Such a phase allows to better measure the impact of the selected samples. Filling  ${\\mathbb{L}}$ quickly brings the model's performance to a plateau that hides the impact of newly selected samples.\nThe initial replenishment ($r=1$) is achieved by randomly selecting a few samples. The same samples are used for all AL techniques at round $r=1$ for a fair comparison. To avoid any bias from selecting unbalanced classes that can directly affect the segmentation performance, and hinder AL evaluation, the same number of samples is selected from each class (since the global annotation is known beforehand for all the samples). Note that the oracle is used only to provide pixel-wise annotation. Tab.\\ref{tab:tab0} provides the selection details.\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Number of samples selected for the oracle per round.}\n\\label{tab:tab0}\n\\centering\n   \\resizebox{0.7\\linewidth}{!}{\n\\begin{tabular}{l|ccc}\n    Dataset  &  \\makecell[c]{\\#selected samples \\\\ \\emph{per-class} ($r=1$)} & \\makecell[c]{\\#selected samples \\\\ \\emph{per-class} ($r > 1$)} & \\makecell[c]{Max AL rounds \\\\ (\\texttt{maxr} in Alg.\\ref{alg:alg-0})}\\\\\n    \\toprule\n    GlaS   &   $4$   & $1$  & $25$ \\\\\n    CUB   &   $1$  & $1$    & $20$ \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\\noindent \\textbf{c) Evaluation.}\nWe report the classification accuracy obtained by the classification head (Fig.\\ref{fig:fig-archi}). Average Dice index is used to measure the segmentation quality at each AL round forming a Dice index curve over all the rounds. To better assess the \\emph{dominance} of each method \\citep{settles2009active}, the Area Under the Dice index Curve is used (AUC). This provides a fair indicator of the dominant curve, but contrasts with standard AL works, where one or multiple specific operating points in the curve are selected (leading to a biased and less accurate protocol). The average and standard deviation of Dice index curve and AUC metric are reported based on 5 replications of a complete AL session, using a different seed for each session. An AL session across different methods uses the same seed.\n\nWhile our approach, referred to as (Label\\_prop), can operate on top of any AL selection criterion, we demonstrate its efficiency using simple random selection, which is often a baseline for AL experiments. Note that our pseudo-annotations are obtained from the segmentation head shown in Fig.\\ref{fig:fig-archi}. Our method is compared to three different AL selection approaches for segmentation:\n\\textbf{(1) random selection (Random)}: the samples are randomly selected;\n\\textbf{(2) entropy-based selection (Entropy)}: the scoring function per sample is the average entropy at the pixel level \\citep{gaur2016membrane}. Samples with high entropy are selected; and\n\\textbf{(3) Monte-Carlo dropout uncertainty (MC\\_Dropout)}: we use Monte-Carlo dropout \\citep{gorriz2017active,lubrano2019deep} at the pixel level to compute the uncertainty score per sample. Samples are forwarded ${50}$ times in the model, where dropout is set to ${0.2}$ \\citep{gorriz2017active,lubrano2019deep}. Then, the pixel-wise variance is estimated. Samples with high mean variance are selected.\n\n\n\\noindent  \\textbf{Lower bound performance (WSL)}: We consider the segmentation performance obtained by WSL method as a lower bound. It is trained using only global annotation. CAMs are used to extract the segmentation mask. WILDCAT method is considered \\citep{durand2017wildcat} (Fig.\\ref{fig:fig-archi}) at the classification head to obtain the CAMs. For WSL method, a pre-trained model over ImageNet \\citep{imagenetcvpr09} is used to initialize the weights of the backbone, which is then fined-tuned.\nThe model is trained over the entire dataset, where samples are labeled globally only.\nThe obtained classifier using seed=0 is frozen and used as a backbone for \\emph{all} the other methods.\n\n\\noindent \\textbf{Upper bound performance (Full\\_sup)}: Fully supervised segmentation is considered as an upper bound on performance. The model in Fig.\\ref{fig:fig-archi} is trained for segmentation only using the entire dataset, where samples are labeled at the pixel level.\n\nFor a fair comparison, all the methods are trained using the same hyper-parameters over the same dataset. WSL and Full\\_sup methods have minor differences. Due to space limitation, all the hyper-parameters are presented in the supplementary material.\nIn Alg.\\ref{alg:alg-0}, notice that for our method, ${\\mathbb{P}_r}$ is not used at the current round $r$ but until the next round ${r+1}$. To take advantage of ${\\mathbb{P}_r}$ at round $r$, instructions from line-4 to line-10 are repeated twice in the provided results.\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\\label{sub:results}\n\n\n\n\\begin{figure}[htbp]\n  \\centering\n \\begin{subfigure}[b]{.5\\linewidth}\n  \\centering\n  \\includegraphics[scale=0.39]{glas-dice-idx-test}\n  \\caption{}\n  \\label{fig:fig-al-0}\n \\end{subfigure}%\n \\begin{subfigure}[b]{.5\\linewidth}\n  \\centering\n  \\includegraphics[scale=0.39]{Caltech-UCSD-Birds-200-2011-dice-idx-test}\n  \\caption{}\n  \\label{fig:fig-al-1}\n \\end{subfigure}%\n\\caption{Average Dice index of the proposed and baseline methods over test sets.\n(\\protect\\subref{fig:fig-al-0}) GlaS.\n(\\protect\\subref{fig:fig-al-1}) CUB.\n}\n\\label{fig:fig-al-results}\n\\end{figure}\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Classification accuracy over of the proposed deep WSL model on GlaS and CUB test datasets.}\n\\label{tab:tab-cl-acc}\n\\centering\n   \\resizebox{.5\\linewidth}{!}{\n\\begin{tabular}{l|ccc}\n    Dataset  &  \\multicolumn{1}{c}{GlaS} & \\multicolumn{1}{c}{CUB}\\\\\n    \\toprule\n    \\makecell{Classification \\\\ accuracy (\\%)}   &   $99.50 \\pm 0.61$\n                                                 &   $73.22 \\pm 0.19$\n                                                 \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Average AUC and standard deviation (Fig.\\ref{fig:fig-al-results}) for Dice index performance over GlaS and CUB test sets.}\n\\label{tab:tab-seg-perf}\n\\centering\n   \\resizebox{.6\\linewidth}{!}{\n\\begin{tabular}{l|ccc}\n    Dataset  &  \\multicolumn{1}{c}{GlaS} & \\multicolumn{1}{c}{CUB} \\\\\n    \\toprule\n    \\toprule\n    WSL   &    $66.44 \\pm 0.20$  &   $39.22 \\pm 0.19$  \\\\\n    \\midrule\n    Random   &    $78.57 \\pm 0.93$  &   $68.15 \\pm 0.61$  \\\\\n    Entropy   &   $79.13 \\pm 0.26$  &   $68.25 \\pm 0.29$  \\\\\n    MC\\_Dropout   &   $77.92 \\pm 0.49$  &   $67.69 \\pm 0.27$  \\\\\n    \\rowcolor{mybluelight}\n    \\textbf{Label\\_prop (ours)}   &   $\\bm{81.48 \\pm 1.03}$  &   $\\bm{71.73 \\pm 0.67}$ \\\\\n    \\midrule\n    Full\\_sup   &   $86.53 \\pm 0.31$  &   $75.29 \\pm 1.50$  \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\n\\begin{table*}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Readings of Dice index (mean $\\pm$ standard deviation) from Fig.\\ref{fig:fig-al-results} over test set for the \\textbf{first 5 queries} formed by each method. We start from the second query since the first query is random but identical for all methods.}\n\\label{tab:tab-dice-q}\n\\centering\n   \\resizebox{0.9\\linewidth}{!}{\n\\begin{tabular}{l|cccccc}\n   \\textbf{Queries}  &  \\textbf{q2} & \\textbf{q3} & \\textbf{q4} & \\textbf{q5} & \\textbf{q6} \\\\\n   \\toprule\n   \\toprule\n   \\multicolumn{6}{c}{GlaS} \\\\\n    \\toprule\n    WSL   &    \\multicolumn{4}{c}{$66.44 \\pm 0.20$}  \\\\\n    \\midrule\n    Random   &   $70.26 \\pm 3.02$ &   $71.58 \\pm 3.14$ & $71.43 \\pm 1.83$ & $74.05 \\pm 3.14$ &    $75.36 \\pm 3.45$\\\\\n    Entropy   &   $\\bm{72.75 \\pm 2.96}$ &   $70.93 \\pm 3.58$ & $72.60 \\pm 1.44$ & $73.44 \\pm 1.38$ &   $75.15 \\pm 1.63$\\\\\n    MC\\_Dropout   &   $68.44 \\pm 2.89$  &   $69.70 \\pm 1.96$ &   $69.97 \\pm 1.95$ & $72.71 \\pm 2.21$ & $73.00 \\pm 1.04$\\\\\n    \\rowcolor{mybluelight}\n    \\textbf{Label\\_prop (ours)}   &   $71.02 \\pm 4.19$  &   $\\bm{74.07 \\pm 3.93}$ &   $ \\bm{76.52 \\pm 3.49}$ & $\\bm{77.63 \\pm 2.73}$ & $\\bm{78.41 \\pm 1.23}$\\\\\n  \n    Full\\_sup   &   \\multicolumn{4}{c}{$86.53 \\pm 0.31$}\\\\\n    \\toprule  \\toprule\n   \\multicolumn{6}{c}{CUB} \\\\\n    \\toprule\n    WSL   &    \\multicolumn{4}{c}{$39.22 \\pm 0.18$}  \\\\\n    \\midrule\n    Random   &   $56.86 \\pm 2.07$ &   $61.39 \\pm 1.85$ & $62.97 \\pm 1.13$ & $63.56 \\pm 4.02$ &    $66.56 \\pm 2.50$\\\\\n    Entropy   &   $53.37 \\pm 2.06$ &   $59.11 \\pm 2.50$ & $60.48 \\pm 3.56$ & $63.81 \\pm 2.75$ &   $63.59 \\pm 2.34$\\\\\n    MC\\_Dropout   &   $57.13 \\pm 0.83$  &   $59.98 \\pm 2.06$ &   $63.52 \\pm 2.26$ & $63.02 \\pm 2.68$ & $64.68 \\pm 1.41$\\\\\n    \\rowcolor{mybluelight}\n    \\textbf{Label\\_prop (ours)}  &   $\\bm{62.58 \\pm 2.15}$  &   $\\bm{66.32 \\pm 2.34}$ &   $ \\bm{67.01 \\pm 2.85}$ & $\\bm{69.40 \\pm 3.40}$ & $\\bm{68.28 \\pm 1.60}$\\\\\n    \\midrule\n    Full\\_sup   &   \\multicolumn{4}{c}{$75.29 \\pm 1.50$}\\\\\n    \\bottomrule\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\nWe report the classification and segmentation performances following the training the proposed deep WSL model in Fig.\\ref{fig:fig-archi}. Tab.\\ref{tab:tab-cl-acc} reports the Classification accuracy of the classification head using WSL, which is\nclose to the results reported in \\citep{belharbi2019weakly,rony2019weak-loc-histo-survey}. The results of GlaS suggest that it is an easy dataset for classification.\n\nThe segmentation results are reported in Tabs. \\ref{tab:tab-dice-q} and \\ref{tab:tab-seg-perf}, and in Fig \\ref{fig:fig-al-results}.\n\nFig. \\ref{fig:fig-al-0} compares Dice accuracy on the \\textbf{GlaS dataset}. On the latter, we observe that adding more labels increases Dice index for all AL methods, yielding, as expected, better performance than the WSL method.  Reading from Tab.\\ref{tab:tab-dice-q}, randomly labeling only 4 samples per class enables to easily outperform WSL. This means that using our approach in Fig.\\ref{fig:fig-archi}, with limited supervision, can lead to more accurate masks compared to using CAMs in the WSL method. From Fig.\\ref{fig:fig-al-0}, one can also observe that Random, Entropy, and MC\\_Dropout methods grow relatively in the same way, leading to the same overall performance, with the Entropy method slightly ahead. Considering the overall behavior of the curves, one may conclude that using advanced selection techniques such as MC\\_Dropout and Entropy provides an accuracy similar to simple random selection. On the one hand, since both methods have shown substantial improvements in AL for classification, and based on the results in Fig.\\ref{fig:fig-al-0}, one may conclude that all samples are equivalently informative for the model. Therefore, there is no better order to acquire them. On the other hand, using simply random selection and pseudo-labeled samples allowed our method to substantially improve the overall performance,  demonstrating the benefits of self-learning.\n\nFig.\\ref{fig:fig-al-1} and Tab.\\ref{tab:tab-dice-q} compare Dice accuracy on the \\textbf{CUB dataset}, where labeling only one sample per class yielded a large improvement in Dice index, in comparison to WSL. Adding more samples increases the performance of all the methods. One can observe similar pattern as for GlaS: Random, Entropy and MC\\_Dropout methods yield similar curves, while the AUC performances of Random and Entropy methods are similar, and slightly ahead of MC\\_Dropout.\nSimilar to GlaS analysis, and based on the results of these three methods, one can conclude that none of the methods for ordering the samples is better than simple random selection. Using self-labeled samples in our method shows again its benefits. Simple random selection combined with self-annotation yields an overall best performance. Using two datasets, our empirical results suggest that self-learning, under limited oracle-annotation, has the potential to provide a reliable second source of annotation, which can efficiently enhance model performance, while using simple sample acquisition techniques.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=.7\\linewidth]{dice-pseudo-labeled}\n  \\caption{Average Dice index over the pseudo-labeled samples of our method in \\textbf{each} AL round.}\n  \\label{fig:fig-dice-pseudo-labeled}\n\\end{figure}\n\n\\noindent \\textbf{Pseudo-annotation performance}. Furthermore, the proposed approach is assessed on the pseudo-labeled samples at each AL round. Fig.\\ref{fig:fig-dice-pseudo-labeled} shows that the model provides good segmentations at the initial rounds. Then, the more supervision, the more accurate the pseudo-segmentation, as expected. This figure shows the interest and potential of self-learning in segmentation, and confirms our assumption that samples near the labeled ones are likely to achieve accurate pseudo-segmentation by the model.\n\n\\noindent \\textbf{Hyper-parameters}. Our approach requires two main hyper-parameters: ${ k \\ \\text{and } \\lambda}$. We conducted an ablation study over ${k}$ on GlaS dataset, and over ${\\lambda}$ on both datasets. Results, which are presented in the supplementary material, suggest that our method is less sensitive to ${k}$. ${\\lambda}$ plays an important role, and\nbased on our study, we recommend using small values of this weighting parameter. In our experiments, we used $\\lambda=0.1$ for Glas and $\\lambda = 0.001$ for CUB. We set ${k=40}$. We note that hyper-parameter tuning in AL is challenging due to the change of the size of the data set, which in turn changes the training dynamics. In all the experiments, we used fixed hyper-parameters across the AL rounds.\nFig.\\ref{fig:fig-dice-pseudo-labeled} suggests that a dynamic ${\\lambda(r)}$ that is increased through AL rounds could be more beneficial. However, this requires a principled update protocol for ${\\lambda}$, which was not explored in this work. Nonetheless, using a fixed value seems to yield promising results overall.\n\n\\noindent \\textbf{Supplementary material}. Due to space limitation, we deferred the hyper-parameters used in the experiments, results of the ablation study, visual results for the similarity measure and examples of predicted masks to the supplementary materials.\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nDeep WSL models trained with global image-level annotations can play an important role in CNN visualization and interpretability. However, they are prone to high false-positive rates, especially for challenging images, leading to poor segmentations.\nTo alleviate this issue, we considered  using pixel-wise supervision provided gradually through an AL framework. This annotation is integrated into training using an adequate deep convolutional model that allows supervised learning for both tasks: classification and segmentation. Through a few pixel-supervised samples, such a design is intended to provide full-resolution and more accurate masks compared to standard CAMs, which are trained without pixel supervision and often provide coarse resolution. Therefore, it enables a better CNN visualization and interpretation of CNN predictions.\nFurthermore, and unlike standard deep AL methods that focus solely on the acquisition function, we considered using self-learning as a second source of supervision to fast-improve the model segmentation.\nEvaluating our method using a realistic AL protocol over two challenging benchmarks, our results indicate that:\n(1) using a \\emph{few} supervised samples, the proposed architecture yielded more accurate segmentations compared to CAMs, with a large margin using different AL methods. Thus, it provides a solution to enhance pixel-wise predictions\nin real-world visual recognition applications.\n(2) using self-learning with random selection yielded substantial improvements. Self-learning under a limited oracle-budget can, therefore, provide a cost-effective alternative to standard AL protocols, where most of the effort is spent on the acquisition function.\n\n\\section*{Acknowledgment}\nThis research was supported in part by the Canadian Institutes of Health Research, the Natural Sciences and Engineering Research Council of Canada, Compute Canada, MITACS, and the Ericsson Global AI Accelerator Montreal.\n\n\\clearpage\n\\newpage\n\n\\appendices\n\n\n\n\\section{Supplementary material for the experiments}\nDue to space limitation, we provide in this supplementary material detailed hyper-parameters used in the experiments, results of the ablation study, visual results to the similarity measure, and examples of predicted masks.\n\n\\subsection{Training hyper-parameters}\n\\label{subsec:hyper-params}\nTab.\\ref{tab:tabx-tr-hyper-params} shows the used hyper-parameters in all our experiments.\n\n\n\\begin{table}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Training hyper-parameters.}\n\\label{tab:tabx-tr-hyper-params}\n\\centering\n\\resizebox{0.7\\linewidth}{!}{\n\\begin{tabular}{lccc}\n    Hyper-parameter  &  GlaS  & CUB\\\\\n    \\toprule\n    Model backbone  & \\multicolumn{2}{c}{ResNet-18 \\citep{heZRS16}}\\\\\n    WILDCAT \\citep{durand2017wildcat}: && \\\\\n    ${\\alpha}$ &  \\multicolumn{2}{c}{${0.6}$} \\\\\n    ${kmin}$ &  \\multicolumn{2}{c}{${0.1}$} \\\\\n    ${kmax}$ &  \\multicolumn{2}{c}{${0.1}$} \\\\\n    modalities &  \\multicolumn{2}{c}{${5}$} \\\\\n    \\midrule\n    Optimizer &  \\multicolumn{2}{c}{SGD}\\\\\n    Nesterov acceleration & \\multicolumn{2}{c}{True}\\\\\n    Momentum & \\multicolumn{2}{c}{$0.9$} \\\\\n    Weight decay & \\multicolumn{2}{c}{$0.0001$}\\\\\n    Learning rate (LR) & ${0.1}$ (WSL: $10^{-4}$) & ${0.1}$ (WSL: $10^{-2}$)\\\\\n    LR decay & ${0.9}$  & ${0.95}$ (WSL: ${0.9}$) \\\\\n    LR frequency decay & $100$ epochs & $10$ epochs  \\\\\n    Mini-batch size & ${20}$ & ${8}$ \\\\\n    Learning epochs & ${1000}$ & ${30}$ (WSL: ${90}$) \\\\\n    \\midrule\n    Horizontal random flip & \\multicolumn{2}{c}{True} \\\\\n    Vertical random flip & True & False  \\\\\n   \n    Crop size & \\multicolumn{2}{c}{${416 \\times 416}$}\\\\\n    \\midrule\n    ${k}$ & \\multicolumn{2}{c}{${40}$}\\\\\n    ${\\lambda}$ & ${0.1}$ & ${0.001}$ \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\\begin{figure}[h!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.5]{knn-ablation-glas-Label_prop-best-k-40}\n  \\caption{Ablation study over GlaS dataset (test set) over the hyper-parameter $k$ (x-axis). y-axis: AUC of Dice index (\\%) of \\textbf{25 queries for one trial}.\n  AUC average $\\pm$ standard deviation: ${81.49 \\pm 0.59}$.\n  Best performance in red dot: $k=40, AUC=82.41\\%$.\n  }\n  \\label{fig:abl-glas-knn}\n\\end{figure}\n\n\n\n\\begin{figure}[h!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.5]{lambda-ablation-glas-Label_prop-best-lambda-0-dot-1}\n  \\caption{Ablation study over GlaS dataset (test set) over the hyper-parameter ${\\lambda}$ (x-axis). y-axis: AUC of Dice index (\\%) of \\textbf{15 queries for one trial}.\n  Best performance in red dot: $\\lambda=0.1, AUC=79.15\\%$.\n  }\n  \\label{fig:abl-glas-lambda}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\centering\n  \\centering\n  \\includegraphics[scale=0.5]{lambda-ablation-Caltech-UCSD-Birds-200-2011-Label_prop-best-lambda-0-dot-001}\n  \\caption{Ablation study over CUB dataset (test set) over the hyper-parameter ${\\lambda}$ (x-axis). y-axis: AUC of Dice index (\\%) of \\textbf{5 queries for one trial}.\n  Best performance in red dot: $\\lambda=0.001, AUC=66.94\\%$.\n  }\n  \\label{fig:abl-cub-lambda}\n\\end{figure}\n\n\n\\subsection{Ablation study}\n\\label{sub:ablation}\n\nWe study the impact of $k$ and ${\\lambda}$ on our method. Results are presented in Fig.\\ref{fig:abl-glas-knn}, \\ref{fig:abl-glas-lambda} for GlaS over ${k, \\lambda}$; and in Fig.\\ref{fig:abl-cub-lambda} for CUB over ${\\lambda}$. Due to the expensive computation time required to perform AL experiments, we limited the experiments (${k, \\lambda}$, number of trials, and \\texttt{maxr}).\nThe obtained results of this study show that our method is less sensitive to ${k}$ (standard deviation of ${0.59}$ in Fig.\\ref{fig:abl-glas-knn}).\nIn other hand, the method shows sensitivity to ${\\lambda}$ as expected from penalty-based methods \\citep{bertsekas1999nonlinear}.\nHowever, the method seems more sensitive to ${\\lambda}$ in the case of CUB than GlaS. CUb dataset is more challenging leading to more potential erroneous pseudo-annotation. Using Large ${\\lambda}$ will systematically push the model to learn on the wrong annotation (Fig.\\ref{fig:abl-cub-lambda}) which leads to poor results. In the other hand, GlaS seems to allow obtaining good segmentation where using large values of ${\\lambda}$ did not hinder the performance quickly (\\ref{fig:abl-glas-lambda}).\nThe obtained results recommend using small values that lead to better and stable performance. Using high values, combined with the pseudo-annotation errors, push the network to learn erroneous annotation leading to overall poor performance.\n\n\n\n\n\n\\subsection{Similarity measure}\n\\label{subsec:similarity}\nIn this section, we present some samples with their nearest neighbors. Although, it is difficult to quantitatively evaluate the quality of such measure.\nFig.\\ref{fig:glas-sim} shows the case of GlaS. Overall, the similarity shows good behavior of capturing the general stain of the image which is what was intended for since the structure of such histology images is subject to high variation. Since the stain variation is one of the challenging aspects in histology images \\citep{rony2019weak-loc-histo-survey}, labeling a sample with a common stain can help the model in segmenting other samples with similar stain.\nThe case of CUB, presented in Fig.\\ref{fig:cub-sim}, is more difficult to judge the quality since the images contain always the same species within their natural habitat. Often, the similarity succeeds to capture the overall color, background which can help segmenting the object in the neighbors and also the background. In some cases, the similarity captures samples with large zoom-in where the bird color dominate the image.\n\n\n\\begin{figure*}[hbt!]\n\\centering\n  \\centering\n  \\includegraphics[width=1.\\linewidth]{glas}\n  \\caption{Examples of ${k}$-nn over GlaS dataset. The images represents the 10 nearest images to the first image in the extreme left ordered from the nearest.}\n  \\label{fig:glas-sim}\n\\end{figure*}\n\n\n\\begin{figure*}[hbt!]\n\\centering\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{Caltech-UCSD-Birds-200-2011}\n  \\caption{Examples of ${k}$-nn over CUB dataset. The images represents the 10 nearest images to the first image in the extreme left ordered from the nearest.}\n  \\label{fig:cub-sim}\n\\end{figure*}\n\n\n\\subsection{Predicted mask visualization}\n\\label{subsec:mask-vis}\nFig.\\ref{fig:cub-results-visu} shows several test examples of predicted masks of different methods over CUB test set at the first AL round (${r=1}$) where only one sample per class has been labeled by the oracle. This interesting functioning point shows that by labeling only one sample per class, the performance of the average Dice index can go from ${39.08 \\pm 08}$ for WSL method up to ${62.58 \\pm 2.15}$ for Label\\_prop and other AL methods. The figure shows that WSL tend to spot small part of the object in addition to the background leading high false positive. Using few supervision in combination with the proposed architecture, better segmentation is achieved by spotting large part of the object with less confusion with the background.\n\n\\begin{figure*}[h!]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{sample-test-Caltech-UCSD-Birds-200-2011} \\\\\n  \\includegraphics[width=0.95\\linewidth]{code}\n  \\caption{Qualitative results (on several CUB test images) of the predicted binary mask for each method after being trained in the first round ${r=1}$ (\\emph{i.e.}\\@\\xspace after labeling 1 sample per class) using seed=0.  The average Dice index over the test set of each method is: ${40.16\\%}$ (WSL), ${55.32\\%}$ (Random), ${55.41\\%}$ (Entropy), ${55.52\\%}$ (MC\\_Dropout), ${59.00\\%}$ (Label\\_prop), and ${75.29\\%}$ (Full\\_sup). (Best visualized in color.)}\n  \\label{fig:cub-results-visu}\n\\end{figure*}\n\n\\FloatBarrier\n\n\n\n\n\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\\label{sec1}\nStar formation in galaxies is maintained through the continued accretion of gas from the intergalactic medium. In turn, the chemical properties of the interstellar medium within galaxies are modulated by the processes of stellar evolution, stellar feedback, and by the abundances of elements within the accreted material. Studies of large samples of galaxies have demonstrated correlations between the gas-phase oxygen abundance in the interstellar medium of galaxies and the total stellar mass \\citep{Tremonti2004}, star formation rate \\citep{Mannucci2010} and gravitational potential \\citep{D'Eugenio2018}. \n\nThese empirical correlations between global galaxy properties and metallicity hint at three processes that can influence the metal content of galaxies. With the buildup of stellar mass through star formation, heavy elements are synthesised in the cores of stars and deposited in the interstellar medium through stellar winds and the terminal phases of the stellar life cycle \\citep[see e.g.][]{Maiolino2019}. Assuming that this enriched material mixes efficiently in to the ISM, the metallicity will then depend on the the metal yield from the stars and the amount of gas into which these metals are dispersed. In other words, the metallicity depends on the degree to which the nucleosynthetic products of stellar evolution are diluted in the ISM \\citep[e.g.][]{Larson1972,Moran2012,Bothwell2013}. Meanwhile, winds driven by the energy or momentum injected into the ISM through stellar feedback have been shown to remove enriched material from galaxies \\citep{Heckman1990}. The magnitude of the effect of metal loss from galaxies due to outflows on setting their metallicity is not known precisely. This is determined by both the mass loading factor, $\\lambda$, which quantifies the total mass lost for a given rate of star formation, and also the metal loading factor, which describes the metal content of the outflow. Recent work by \\cite{Chisholm2018} found that the the metallicity of outflowing material is independent of the stellar mass of a galaxy, and therefore of the ISM metallicity. However, it is often assumed that the expelled gas has the same metallicity as the ISM \\citep{Erb2008,Finlator2008}.\n\nThe tradeoff between these different galactic processes has been captured by a number of different attempts to model the chemical evolution of galaxies \\citep[e.g.][]{Finlator2008,Peeples2011,Lilly2013}. The \\cite{Lilly2013} model makes the simplifying assumption that the evolution of a galaxy`s star formation and metallicity are determined almost entirely by the total gas content. Making simple assumptions about the flow of gas into and out of galaxies, these models are able to reproduce the global mass-metallicity relation, along with the star formation rate - stellar mass relation.\n\nGiven the success of analytic models in describing the global properties of galaxies, there has been recent effort to extend the global scaling relations to kpc scales within galaxies \\citep[e.g.][]{RosalesOrtega2012,BarreraBallesteros2016,CanoDiaz2016,Medling2018}. These studies have been enabled by the introduction of large-scale spatially-resolved spectroscopic surveys such as the Calar Alto Legacy Integral Field Area Survey \\citep[CALIFA][]{Sanchez2012}, the Sydney-AAO Multi-object Integral Field Spectrograph \\citep[SAMI][]{Croom2012,Bryant2015} and Mapping Nearby Galaxies at Apache Point Observatory \\citep[MaNGA][]{Bundy2015}. \\cite{CanoDiaz2016} found that the gas phase metallicity in kpc-sized regions of galaxies is tightly correlated with the stellar mass surface density, $\\Sigma_{*}$. However, \\cite{BarreraBallesteros2016} showed that this correlation is also dependent on the total stellar mass of the galaxy. That is, at fixed $\\Sigma_{*}$ the gas-phase metallicity is correlated with the integrated stellar mass. In direct analogy the argument connecting the global MZR to the increasing depth of the gravitational potential well in more massive galaxies, \\cite{BarreraBallesteros2018} showed that the local metallicity also correlates with the local escape velocity. This may explain the connection of the local $\\Sigma_{*}$-metallicity relation to the integrated stellar mass.\n\n\nThe existence of local scaling relations are an indicator that some kind of regulatory process is at play, and that processes occurring on local scales may be able to explain the results seen in single-fibre spectroscopic surveys \\citep{BarreraBallesteros2016}. Indeed work by \\cite{Carton2015} and \\cite{BarreraBallesteros2018} found that the \\cite{Lilly2013} gas regulator model is able to fit the metallicity given a reasonable estimate of the local gas fraction and mass loading factors.\n\nThe ability of gas regulated models to achieve an equilibrium is in part determined by the rate at which gas is accreted and the metallicity of that gas. Historically it was often assumed that the gas fueling star formation is accreted in a chemically pristine state \\citep[e.g.][]{Larson1972,Quirk1973,Finlator2008,Mannucci2010}. However, there is mounting evidence that this is not the case \\citep{Oppenheimer2010,Rubin2012,Brook2014,Kacprzak2016,Gupta2018}. While work such as that done by \\cite{Peng2014} relies on the inference of the properties of gas in the intergalactic medium from modeling, there is a growing body of observations that are able to directly probe the metallicity of gas outside of galaxies by measuring the absorption of background quasar light by extragalactic clouds \\citep{Lehner2013,Prochaska2017}. These clouds are observed to be cool and metal rich, and are expected to be accreted onto their host galaxies in the future. Indeed, hydrodynamic simulations \\citep{Oppenheimer2010} have shown that below $z \\sim 1$ the dominant source of accreted gas onto galaxies is material the was previously ejected.\n\nIn addition to internal processes such as star formation and the expulsion of gas by feedback, there are a number of ways in which the gas content of a galaxy could be diminished by external environmental processes. Starvation \\citep{Larson1980}, which can occur when a galaxy enters a dense environment and the acquisition of new gas is prevented. In this instance, the existing gas reservoir is consumed by star formation over several Gyr and the metallicity of the ISM increases. Starvation can also be initiated through the heating of gasses in the intergalactic medium by galactic feedback \\citep[e.g.][]{Fielding2017}, or by the shock heating of accreted material \\citep[e.g.][]{Birnboim2003}. Alternatively, a kinetic interaction between the interstellar medium of a galaxy and the intergalactic medium can result in the ram pressure stripping of gas from a galaxy \\citep{Gunn1972}. Occurring on relatively short ($<100 \\, \\mathrm{Myr}$) timescales, ram pressure stripping is not expected to alter the chemical abundances in a galaxy before it is fully quenched. \n\nStudies of the environmental effect on galaxy metallicities generally find a small, but significant dependence. For example, \\cite{Cooper2008} find that approximately $15 \\%$ of the scatter in the mass metallicity relation is attributable to an increase in the metallicity of galaxies in high-density environments. Observationally, there is consensus that the environment has the largest effect on low-mass galaxies \\citep{Pasquali2012,Peng2014,Wu2017}. However, the interpretation of this fact is contentious. \\cite{Wu2017} attribute the elevated metallicity at fixed stellar mass at greater local galaxy overdensity to a reduction in the gas accretion rate. However, \\cite{Peng2014} showed that even when the star formation rates of satellite galaxies in different environments is kept constant, implying no change in the total accretion rate, the metallicity offset is still evident. Their observations and modeling led them to the conclusion that satellite galaxies in dense environments must acquire more enriched gas from their surroundings.\n\nIn this paper we make use of the broad range of galaxy environments and stellar masses covered by the MaNGA survey to explore how the local metallicity scaling relations are affected by the environment for satellite and central galaxies. With MaNGA's wide wavelength coverage and spatial resolution we are able to estimate local gas-phase metallicities, gas mass fractions and escape velocities to compare the observations to a model for the gas regulation of the metallicity, and from this modeling infer environmental trends for the metallicity of gas inflows.\n\nIn Section \\ref{Methods}, we present our data and analysis techniques. In Section \\ref{Results} we investigate how galaxy environments change the local metallicity scaling relations, and in Section \\ref{Discussion}, we discuss our observations in the context of the \\cite{Lilly2013} gas regulator model. Throughout this paper we assume a flat $\\Lambda$CDM cosmology, with $H_{0}=70 \\mathrm{km \\, s^{-1} \\, Mpc^{-1}}$, $\\Omega_{\\Lambda} = 0.7$ and $\\Omega_{m} = 0.3$. Unless otherwise stated we assume a \\cite{Chabrier2003} stellar initial mass function. We will make use of two oxygen abundance indicators, which each have different absolute abundance scales. The O3N2 \\citep{Pettini2004} assumes $12+\\log(\\mathrm{O\/H})_{\\odot} = 8.69$ and the N2S2H$\\alpha$ indicator \\citep{Dopita2016} assumes $12+\\log(\\mathrm{O\/H})_{\\odot} = 8.77$.\n\n\n\\section{Methods}\\label{Methods}\nTo investigate the trends of the spatially-resolved metal distribution in galaxies with environment, we make use of the $8$th MaNGA product launch (MPL-8) internal data release, which is similar to the SDSS DR15 \\citep{Aguado2018}, but includes data from of $6507\\,$ unique galaxies. In this Section we describe the data, sample selection and methods used to perform our analysis on these data.\n\n\\subsection{MaNGA data}\nThe MaNGA survey is the largest optical integral field spectroscopic survey of galaxies to date. Run on the $2.5 \\, \\mathrm{m}$ SDSS telescope \\citep{Gunn2006} at Apache point observatory, the MaNGA survey aims to observe approximately $10,000$ galaxies. This sample comprises a primary and a secondary subsample that were selected to have approximately flat distributions in the $i$-band absolute magnitude, $M_{i}$. These provide coverage of galaxies out to $1.5$ and $2.5 \\, R_{e}$ respectively \\citep{Wake2017} and median physical resolutions of $1.37$ and $2.5 \\, \\mathrm{kpc}$.\n\nObservations of each galaxy are made with one of $17$ hexagonal optical fibre hexabundles, each comprising between $19$ and $127$ $2\\arcsec$ optical fibres, subtending between $12\\arcsec$ and $32\\arcsec$ on the sky. The fibre faces fill the bundle with $56\\%$ efficiency, so each target is observed with a three-point dither pattern with $15$-minute exposures per pointing. This pattern of observations is repeated until a median S\/N of $20 \\, fibre^{-1} \\, pixel^{-1}$ is achieved in the $g$-band, which is typically $2-3$ hours in total \\citep{Law2015,Yan2016}. Light from each hexabundle is taken from the fibres to the BOSS spectrograph \\citep{Smee2013} where it is split by a dichroic at $\\sim 6000 \\, \\mathrm{\\AA}$ into red and blue channels, then dispersed at $R \\approx 2000$. The resulting spectra are then mapped onto a regular grid with $0.5\\arcsec$ square spaxels, with continuous wavelength coverage between $3600 \\, \\mathrm{\\AA}$ and $10300 \\, \\mathrm{\\AA}$. For an in-depth discussion of the MaNGA data reduction pipeline, see \\cite{Law2016}.\n\nThe reduced data are analysed by the MaNGA Data Analysis Pipeline \\citep[\\DAP;][]{Belfiore2019,Westfall2019}, which extracts stellar kinematics, measures the strengths of continuum features, and extracts emission line fluxes, equivalent widths and kinematics for each galaxy. For this work, we make use of the emission line fluxes from the \\DAP's hybrid binning scheme. In this scheme, the data cubes are Voronoi binned \\citep{Cappellari2003} to a S\/N of at least $10$ in the continuum. Within each of these bins, {\\tt pPXF} ~\\citep{Cappellari2004} is used to fit an optimal continuum template which is made up of a linear combination of hierarchically-clustered templates from the MILES stellar library \\citep{SanchezBlazquez2006,FalconBarroso2011} as well as an $8$th degree multiplicative Legendre polynomial. This optimal continuum template constrains the stellar populations within the Voronoi bin, and is fitted by {\\tt pPXF} ~in conjunction with a set of Gaussian emission line templates to each individual spaxel in the bin. For a full description of the \\DAP ~fitting process, see \\cite{Westfall2019}, and for a discussion of the robustness of the emission line measurements, see \\cite{Belfiore2019}.\n\n\\subsection{Sample Selection}\\label{sample_selection}\nGalaxies in the MaNGA survey are selected such that the full sample has a roughly flat distribution of stellar masses. However, this parent sample contains galaxies with a wide range of morphologies and star formation rates. Our goal with this work is to make spatially-resolved measurements of the properties of the gas in galaxies as a function of the local environment. The metallicity indicators mentioned in Section \\ref{Calibrators} are only calibrated for HII regions, and therefore cannot be applied to a fraction of the spaxels in MaNGA. To make this determination, we compare the $\\mathrm{[NII]\\lambda 6584 \/ H\\alpha}$ and $\\mathrm{[OIII]\\lambda 5007 \/ H\\beta}$ emission line ratios on a \\cite{Baldwin1981} (BPT) diagram. Only spaxels with emission line ratios that satisfy both the \\cite{Kauffmann2003} and \\cite{Kewley2001} criteria for excitation of the gas by a young stellar population are included in our analysis. We further exclude spaxels for which the S\/N ratio in the emission lines used for the metallicity and determination of star formation are less than $3$. \\cite{Belfiore2019} showed that the fluxes of lines above this threshold in MaNGA are relatively robust to systematic effects. From these constraints we calculate the fraction of spaxels for a data cube for which we are able to reliably determine a metallicity. In computing this fraction, we include only spaxels where the $g$-band flux from the data cube is detected at a S\/N of $2$ or greater. This condition is imposed so that galaxies that do not fully fill the IFU field of view are not unduly excluded. Galaxies for which the fraction of spaxels with a measurable metallicity is larger than $60\\%$ are retained for our analysis.\n\nWe make a further cut on the galaxies in our sample based on our ability to robustly measure their metallicity gradients. Galaxies with an elliptical minor to major axis ratio ($b\/a$) less than $0.6$, as determined by the NASA-Sloan Atlas \\citep[NSA;][]{Blanton2011} Elliptical Petrosian photometry, were excluded to give a sample of face-on galaxies. We made a further restriction on the measured $r$-band effective radius, $R_{e}$. Galaxies with $R_{e}<4\\arcsec$ are also rejected. These criteria are motivated by the analysis performed by \\cite{Belfiore2017}, who showed that beam-smearing effects are non-negligible for highly inclined systems, or for galaxies that are small relative to the MaNGA point spread function. Similarly, \\cite{Pellegrini2019} used a set of realistic simulations to show that light-weighted quantities, such as dust attenuation, are systematically overestimated when observed in highly inclined systems.\n\n\nOur final sample consists of $1008 \\,$ galaxies, with stellar masses in the range $7.8< \\log(M_{*}\/M_{\\odot})<11.4$. For the majority of our analysis we restrict the stellar masses considered to $9< \\log(M_{*}\/M_{\\odot})<11$, and in this range our sample comprises $967$ galaxies. The distribution of galaxies in our sample the colour-mass plane is shown in Figure \\ref{M_star_cmd}.\n\n\\begin{figure}\n\\includegraphics{M_star_cmd.pdf}\n\\caption{In panel $a)$ we show the distribution of stellar mass, $\\mathrm{M_{*}}$ for the input sample (\\textit{grey}) and for the final sample (\\textit{red}). The attrition of sources occurs preferentially at high stellar mass, which is consistent with the rising fraction of passive galaxies. In $b)$ the positions of galaxies in the input sample (\\textit{grey}) and final sample (\\textit{red}) on the $u-r$ colour-mass diagram is shown. Galaxies that satisfy our selection criteria are predominantly in the blue cloud and forming stars.}\\label{M_star_cmd}\n\\end{figure}\n\n\\subsection{Gas-phase metallicities}\\label{Calibrators}\nTo probe the chemical evolution of the galaxies in our sample, we use the \\DAP ~emission line measurements to estimate the gas-phase oxygen abundances in these systems. For convenience we will use the terms `gas-phase oxygen abundance' and `metallicity' interchangeably throughout this work.\nThe estimation of gas-phase oxygen abundances with optical spectroscopy is often achieved by measuring a set of emission line ratios that vary with the conditions of the gas. The metallicity can be calculated by comparing the measured line ratios to theoretical models for HII regions \\citep[e.g.][]{Blanc2015}. Alternatively it can be estimated by using relationships that are empirically calibrated by comparing these line ratios to the spectra of HII regions for which the metallicity has been measured directly using temperature-sensitive emission line ratios, such as $\\mathrm{[OIII] \\lambda 4363 \/ [OIII] \\lambda 5007}$. While it is generally accepted that the direct method of the oxygen abundance determination is more robust than theoretical modeling or using empirical calibrations, it is generally not possible with datasets such as MaNGA as the $\\mathrm{[OIII] \\lambda 4363}$ line is typically $\\sim 100$ times fainter than the $\\mathrm{[OIII] \\lambda 5007}$ line.\n\nMany of the empirical calibrations suffer from systematics, being biased either high or low due to variations in the ionisation parameter in the gas, or contamination from diffuse ionised gas or light from an AGN. To account for this fact, we will make use of two different metallicity calibrations. \n\\subsubsection{O3N2}\nFor consistency with \\cite{BarreraBallesteros2018}, we use the \\cite{Pettini2004} $O3N2$ oxygen abundance diagnostic. This method makes use of the $\\mathrm{[OIII] \\lambda5007\/H\\beta}$ and $\\mathrm{[NII]\\lambda6584 \/ H\\alpha}$ emission line ratios, and was calibrated against a set of $137$ extragalactic HII regions for which a metallicity had been determined either by the direct $T_{e}$ method or by photoionisation modeling of their spectra. Taking $O3N2 = \\mathrm{\\log([OIII]\\lambda 5007 \/ H\\beta) - \\log([NII]\\lambda6584 \/ H\\alpha)}$, the metallicity of an HII region can be calculated as\n\\begin{equation}\n12+\\log(\\mathrm{O\/H}) = 8.73 - 0.32 \\times O3N2,\n\\end{equation} \nover the range $8.1 < 12+\\log(\\mathrm{O\/H}) < 9.05$. It should be noted that this calibration suffers from some degeneracy with variation in the ionisation parameter within the gas. For this reason it cannot be used in spectra that include significant contamination from diffuse ionised gas, or active galactic nuclei, and may also be biased by variations in $q$ between star-forming regions within a galaxy \\citep[see e.g.][]{Poetrodjojo2018}.\n\n\\subsubsection{N2S2H$\\alpha$}\nAn alternative method for calculating the metallicity of HII regions based on the relative intensities of the $\\mathrm{[NII]\\lambda 6584}$, $\\mathrm{[SII]\\lambda 6717,6731}$ and H$\\alpha$ lines was presented by \\cite{Dopita2016}. Assuming a simple relationship between N\/O and O\/H, and modeling theoretical HII regions with a variety of gas pressures and ionisation parameters using the {\\tt MAPPINGS 5.0} software, they found\n\\begin{equation}\n\\begin{aligned}\n12+\\mathrm{\\log(O\/H)} = & 8.77 + \\log(\\mathrm{[NII]\\lambda6584 \/[SII]\\lambda6717,6731}) \\\\\n\t& + 0.264 \\log(\\mathrm{[NII]\\lambda6584 \/H\\alpha}),\n\\end{aligned}\n\\end{equation}\nwhich they showed has very little dependence on the ionisation parameter and is valid over the range $8.0 \\lesssim12 + \\log(\\mathrm{O\/H}) < 9.05$. There is some evidence that the relationship between N\/O and O\/H varies with the total stellar mass of a galaxy \\citep{Belfiore2017}, but this should be negligible if analysis is carried out within narrow bins of stellar mass.\n\n\nEach of the two metallicity calibrations outlined above have different absolute abundance scalings. While it is not possible to directly compare the metallicities of galaxies derived with different calibrations, relative differences between two measurements made with the same indicator are likely to reflect real differences in the chemical composition of the galaxies in question.\n\n\n\\subsection{Gas density}\nFollowing the methodology of \\cite{BarreraBallesteros2018} we estimate the local neutral gas surface density from the dust attenuation derived from the observed Balmer line ratios. Under the assumption of a fixed gas-to-dust ratio, \\cite{BarreraBallesteros2018} utilised the observation that the gas surface density is related to the V-band attenuation via $\\Sigma_{gas} = 30 \\times \\mathrm{A_{V}} \\, \\mathrm{pc^{-2}}$. We apply a small correction to this to account for the variation in the dust-to-gas ratio with the gas phase metallicity using the relation given by \\cite{Wuyts2011},\n\\begin{equation}\n\\Sigma_{gas} = 30 \\times A_{V} \\times \\left( \\frac{Z}{Z_{\\odot}} \\right)^{-1} \\, \\mathrm{pc^{-2}},\n\\end{equation}\nfor $Z<Z_{\\odot}$ and is independent of metallicity above $Z_{\\odot}$. In this expression we calculate the total extinction in the $V$-band as\n\\begin{equation}\nA_{V} = R_{V} \\times \\frac{2.5}{k(\\lambda_{\\mathrm{H\\beta}}) - k(\\lambda_{\\mathrm{H\\alpha}})} \\log \\left(\\frac{f(\\mathrm{H\\alpha}) \/ f(\\mathrm{H}\\beta)}{2.86}\\right).\n\\end{equation}\n\nFor the \\cite{Cardelli1989} extinction law, $k(\\lambda_{\\mathrm{[OII]}}) = 4.77$, $k(\\lambda_{\\mathrm{[NII]}}) = 2.53$, $k(\\lambda_{\\mathrm{H\\alpha}}) = 2.54$ and $k(\\lambda_{\\mathrm{H\\beta}}) = 3.61$, and the ratio of total to selective extinction, $R_{V}$, is $3.1$. For our sample (described below), $95\\%$ of the spaxels have metallicity above $0.5 Z_{\\odot}$, meaning that the majority of gas surface density measurements will differ systematically from those derived by \\cite{BarreraBallesteros2018} by a factor of two at most.\n\n\n\n\n\\subsection{Stellar mass surface density}\nA large fraction of the baryons in the inner parts of galaxies have been locked away in long-lived stars. The stellar content of a galaxy is therefore a valuable diagnostic for its integrated star formation and chemical enrichment history. For this work we take the estimates of stellar mass per spaxel provided by Pace et al. (2019a, 2019b). With this method, individual spectra are fitted using a basis set of six vectors, found using principal component analysis (PCA) of a permissive set of $40,000$ star formation histories. These basis spectra were generated from the C3K library (Conroy, \\textit{in prep.}), using a \\cite{Kroupa2001} stellar initial mass function. Dust attenuation is taken into account using a two-component model following \\cite{Charlot2000}, whereby light from the younger component of the stellar population experiences a different degree of extinction than the light from the older stellar populations. Pace et al. (2019a) showed that this technique provides statistically robust estimates of the stellar mass-to-light ratio across a wide range of signal-to-noise ratios, star formation histories and metallicities, with random uncertainties typically $0.1 \\, \\mathrm{dex}$ or less for spectra with $\\mathrm{S\/N}>2$.\n\nWe have rescaled the estimated stellar masses from a \\cite{Kroupa2001} to a \\cite{Chabrier2003} initial mass function by dividing by $1.06$ \\citep{Zahid2012}. To calculate stellar mass surface density ($\\Sigma_{*}$) in a spaxel, we take these values and divide them by the projected area of a $0\\farcs 5$ square spaxel at the systemic redshift the host galaxy. A small correction for inclination of the galaxy is applied by multiplying these surface densities by the elliptical Petrosian minor-to-major axis ratio.\n\n\n\n\\subsection{Estimating the local escape velocity}\nWe estimate the local escape velocity from the halo assuming a spherically-symmetric dark matter halo that is described by a \\cite[NFW]{Navarro1997} profile using the same procedure as \\cite{BarreraBallesteros2018}. This method assumes that the star-forming gas is confined to a thin disk coplanar with the optical disk, and is in a circular orbit around the centre of the galaxy. We extract a rotation curve by taking the maximum and minimum measured line-of-sight velocity within a $30^{\\circ}$ wedge along the photometric major axis of the galaxy, similar to \\cite{BarreraBallesteros2014}. This rotation curve is corrected for the galaxies inclination to our line of site using the $r$-band elliptical Petrosian major to minor axis ratio. We fit the resulting rotation curve using the \\cite{Bohm2004} parametrisation,\n\n\\begin{equation}\nV(r_{depro}) = V_{max} \\frac{r_{depro}}{\\left(R_{turn}^{\\alpha}  + r_{depro}^{\\alpha}    \\right)^{1\/\\alpha}},\n\\end{equation}\nwhere $V_{max}$ is the maximum velocity of rotation, $r_{depro}$ is the deprojected radius, $R_{turn}$ is the radius at which the rotation curve flattens, and $\\alpha$ is a parameter that determines the shape of the rotation curve. This formulation is a special case of the phenomenological model presented by \\cite{Courteau1997}. The fitted parameters $V_{max}$ and $R_{turn}$ are then used to derive the local escape velocity using the following formula:\n\\begin{equation}\n V_{esc}^{2} = \n\t\\begin{cases}\n\t\tV_{esc,in}^{2} + V_{esc,out}^{2} & r_{depro} < R_{turn} \\\\\n\t\tV_{esc,out}^{2} & r_{depro} > R_{turn}\n\t\\end{cases} \n\\end{equation}\nwhere\n\\begin{equation*}\nV_{esc,in} = \\left(V_{max}\/R_{turn}\\right)^{2} \\left(R_{turn} - r_{depro} \\right)^{2} + V_{esc,out}^{2}\n\\end{equation*}\nand\n\\begin{equation*}\nV_{esc,out}^{2} = 2V_{max}^{2} \\ln \\left( R_{vir}\/r_{depro}     \\right) + 2V_{max}^2.\n\\end{equation*}\nIn this relation, $R_{vir}$ is the virial radius of the galaxy's halo, which we obtain by estimating the galaxy's total halo mass from its stellar mass using the relation derived by \\cite{Behroozi2010}. This computation of the local escape velocity assumes the galaxy potential to be spherically symmetric. \\cite{BarreraBallesteros2018} tested how a more complicated two-component halo, that includes a contribution to the gravitational field from the baryons in galaxy disk and found that this causes a deviation in the estimated escape velocity of only $\\sim 5\\%$  from the simpler spherical case.\n\n\n\n\\subsection{Environment}\nThe aim of our current work is to explore how the local environments that galaxies are in today impact their chemical evolution. While there are many different ways of characterising environment, each capable of tracing a variety of different physical processes that can occur during a galaxy's lifetime, we will use the satellite\/central classification of our sample as the primary metric for environment. This has been shown to be a good predictor of the star-forming properties of galaxies at fixed stellar mass \\citep[e.g.][]{Peng2012}. We make use of the \\cite{Tempel2017} catalogue, which uses a friends-of-friends algorithm to provide estimates of group membership, group richness, and dark matter halo mass for galaxies in SDSS DR12 \\citep{Eisenstein2011, Alam2015}. Of the $6507\\,$ galaxies in MPL-8, $5333 \\,$ were associated with groups in the \\cite{Tempel2017} catalogue, of which $3447$ are identified as the centrals of their halo and $1886$ are satellites. This sample comprises a wide range of group masses, $M_{200}$, which is the mass contained within $R_{200}$ of the group centre, the radius at which the density of an NFW profile drops to $200$ times the average density of the Universe. Groups within this catalogue contain as few as two galaxies and up to $254$ members for the most massive halo. We show the distribution of halo masses for our final sample in Figure \\ref{M_200_dist}.\n\\begin{figure}\n\\includegraphics{M_200_dist.pdf}\n\\caption{The distribution of group halo mass, $M_{200}$ for the input sample (\\textit{grey}) and for the final sample (\\textit{red}). The loss of sources from the input sample at higher halo mass is more severe than in low mass halos due to the higher fraction of passive galaxies in the most dense environments. }\\label{M_200_dist}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics{RP_vs_mass.pdf}\n\\end{center}\n\\caption{The median metallicity radial profile for galaxies in the stellar mass ranges indicated by the legend at the top of panel $a)$. The solid curves represent the median profiles, while the shaded regions of the same colour represent the $1\\sigma$ error range on the median. In panel $a)$ we show the metallicity median profiles made using the \\cite{Pettini2004} O3N2 indicator and in panel $b)$ we show the results from the \\cite{Dopita2016} $\\mathrm{N2S2H\\alpha}$ indicator. Note that each metallicity indicator has a different abundance scaling on the y-axis, and in each panel we mark the assumed solar abundance with a grey dashed line.}\\label{RP_vs_mass}\n\\end{figure*}\n\n\\subsection{Calculating metallicity radial profiles}\nTo characterise the radial dependence of metallicity in the galaxies in our sample, we construct the radial profile in the following way. In each galaxy, the deprojected distance from the centre of the galaxy has been calculated by the \\DAP based on the $r$-band surface brightness distribution in the SDSS imaging, and assuming each galaxy to be a tilted thin disk. We measure the metallicity as a function of radius and take the average value for spaxels in $0.5\\arcsec$-wide bins. The metallicity measurements are included only if they are classified as star-forming on the BPT diagram, if they have $\\mathrm{S\/N} >3$ in all emission lines utilised, and if the $\\mathrm{H\\alpha}$ equivalent width is greater than $6 \\, \\mathrm{\\AA}$ in emission. This $\\mathrm{H}\\alpha$ equivalent width criterion is consistent with that chosen by \\cite{BarreraBallesteros2018} to minimize contamination of the emission line fluxes from diffuse ionised gas.\n\nTo understand the behaviour of an ensemble of galaxies we measure what we will call the `median profile' for the metallicity. For this, we take the median of the individual radial profiles within radial bins that are $0.2 \\, R_{e}$ wide. Once the median has been calculated we perform a bootstrap resampling of the galaxies, recalculating the median profile for $1000$ realisations of the sample. At each radius, the uncertainty on the sample median is estimated to be the standard deviation of the bootstrapped median profiles. \n\n\\section{Results}\\label{Results}\n\n\n\n\n\\subsection{Metallicity profiles}\nFollowing \\cite{Belfiore2017} we calculate the median metallicity profiles in narrow bins of stellar mass using two different oxygen abundance indicators. The median profiles for all galaxies in our sample are shown in Figure \\ref{RP_vs_mass}. In panel $a)$ of this figure, we show the \\cite{Pettini2004} O3N2 oxygen abundance median profiles in $0.5 \\, \\mathrm{dex}$-wide bins of stellar mass. The profiles shown here are consistent with those shown in Figure 3 of \\cite{Belfiore2017}, in particular with the steepening of the metallicity gradient at higher stellar mass. Panel $b)$ shows the profiles for the same galaxies derived using the \\cite{Dopita2016} $\\mathrm{N2S2H\\alpha}$ indicator. This indicator shows qualitatively different results to O3N2 in the centres of galaxies above $\\log(M\/M_{\\odot}) \\sim 10.25$. While the $\\mathrm{N2S2H\\alpha}$ metallicities both continue to rise in the centres of massive galaxies, the O3N2 indicator shows a flattening. The origin of the mismatch in the behaviour of the metallicity profiles in the centres of massive galaxies between different abundance calibrations may be due to the fact that the \\cite{Dopita2016} metallicity calibration is strongly tied to the N\/O ratio. \\cite{Belfiore2017} showed N\/O to increase towards the centres of massive galaxies, while O\/H does not.\n\nAgain, we caution that the interpretation of metallicity radial profiles from datasets with kpc-scale physical resolution such as MaNGA is subject to the flattening of gradients by the observational point spread function \\citep{Yuan2013,Mast2014}. \\cite{Carton2017} presented a method to account for this effect, however they reported that it was not always robust in the presence of clumpy star formation distributions. Our galaxy size and inclination selection criteria that were outlined in Section \\ref{sample_selection} should mitigate the most severe resolution effects \\citep{Belfiore2017}, but we note that the most accurate determinations of metallicity gradients require observations with finer spatial resolution. The core conclusions of this work, particularly those based on local scaling relations, will be only minimally affected by this issue. \n\n\n\\subsubsection{Satellites vs. centrals}\nTo investigate the environmental dependence of the radial distribution of the oxygen abundance, we split our sample into satellites and centrals, then re-calculate the median profiles with each metallicity indicator. We show these radial profiles in Figure \\ref{sat_cen_rp_all}. At fixed stellar mass there are minor qualitative and quantitative differences between the radial distributions of the oxygen abundance. At all radii, the absolute differences in the median metallicity at fixed stellar mass are less than $\\sim 0.05 \\, \\mathrm{dex}$ for O3N2, and less than $\\sim 0.1 \\, \\mathrm{dex}$ for $\\mathrm{N2S2H\\alpha}$. We note that in the highest stellar mass bins, there are very small numbers of star-forming satellites, and their distribution is biased towards the lower stellar masses. For this reason the differences for the most massive galaxies are not robust in this sample. At lower stellar masses, however, the stellar mass distributions of satellites and centrals are similar, the sample sizes are larger and a fairer comparison can be made.\n\nSince each metallicity indicator has different systematics and biases, we only deem a difference to be real if it is reflected in both the O3N2 and $\\mathrm{N2S2H\\alpha}$ data. For galaxies in the mass range  $9.4 < \\log(M_{*}\/M_{\\odot})<10.2$, there is a systematic offset in the metallicity, with satellite galaxies being more metal rich at all radii sampled, but with no significant difference in the gradient. In the lowest mass range, the O3N2 indicator shows a change in the metallicity gradient, however this is not evident in the $\\mathrm{N2S2H\\alpha}$ metallicity. \n\n\n\\begin{figure*}\n\\includegraphics{sat_cen_rp_all.pdf}\n\\caption{The median metallicity radial profiles in bins of stellar mass split into satellites and centrals. In the upper row, we show the profiles for the O3N2 indicator, while in the lower row the results for $\\mathrm{N2S2H\\alpha}$ are shown. The profiles for central galaxies are shown in the left column, and for the satellite galaxies in the middle column. On the right we show the difference between the satellites and centrals. Satellite galaxies in the range $9.4 < \\log(M_{*}\/M_{\\odot})<10.2$ are systematically more metal rich than centrals of the same mass in both metallicity indicators.}\\label{sat_cen_rp_all}\n\\end{figure*}\n\n\\subsection{Local scaling relations}\\label{local_scaling_relations}\nIn addition to the global scaling relations relating a global metallicity measurement to the integrated stellar mass of a galaxy, there are also local correlations between the stellar mass surface density \\citep{Moran2012,RosalesOrtega2012} and the local gas-phase metallicity. These local scaling relations capture the response of the chemical abundance of gas to processes occuring on local $\\sim \\mathrm{kpc}$ scales. \n\n\\begin{figure*}\n\\includegraphics{sig_mass_met_PP04.pdf}\n\\includegraphics{sig_mass_met_D16.pdf}\n\\caption{The relationship between local stellar mass surface density and metallicity in bins of total stellar mass for satellite and central galaxies. The greyscale background represents the density of data points for central galaxies, while the red contours represent the distribution of data points from satellite galaxies. For clarity, the distribution for satellite galaxies was smoothed to make the contours less subject to noise. Blue points represent the median values of metallicity in the central galaxies at a fixed $\\Sigma_{*}$, and the red points are the medians for satellite galaxies. These are only calculated where there are sufficient data. We include bootstrapped standard errors of the median, but these uncertainties are often smaller than the data points. In the top row we show the results for the \\cite{Pettini2004} O3N2 indicator, while in the bottom row we show the result for the \\cite{Dopita2016} N2S2H$\\alpha$ indicator. The metallicity of satellite galaxies at fixed stellar mass surface density is slightly higher $(\\sim 0.01 \\, \\mathrm{dex})$ than for central galaxies.}\\label{sig_mass_met}\n\\end{figure*}\n\n\\subsubsection{Metallicity and Stellar Density}\nAs more stars form and the stellar surface density ($\\Sigma_{*}$) increases, the amount of enrichment of the ISM also increases. We explore this relation in Figure \\ref{sig_mass_met}, where we show the relationship between $\\Sigma_{*}$ and metallicity for satellite and central galaxies. \\cite{Hwang2019} showed that in addition to the relationship between the local $\\Sigma_{*}$ and $12+\\log(\\mathrm{O\/H})$, there is a secondary dependence on the total stellar mass. For this reason we have split our analysis into $0.5 \\, \\mathrm{dex}$-wide bins of integrated $M_{*}$. Using both the O3N2 and N2S2H$\\alpha$, there is a small ($\\sim 0.01 \\, \\mathrm{dex}$) difference between the metallicity at fixed $\\Sigma_{*}$ between satellites and centrals, particularly in the $9.5 < \\log(\\mathrm{M_{*}\/M_{\\odot}})<10$ interval. While the formal uncertainties on the medians indicate that these differences are statistically significant, they are a factor of ten smaller than the standard deviations of the metallicity distributions.  We note that for systems of low stellar mass, a satellite galaxy may occupy a group with a wide range of possible halo masses, corresponding to very different environments.\n\nGiven that the largest differences in the metallicity between satellites and centrals occurs at the lowest stellar masses, we show the impact of varying the stellar mass of the central in Figure \\ref{sig_mass_met_cen_mass}. Choosing satellite galaxies between $9<\\log(\\mathrm{M_{*}\/M_{\\odot}})<10$, we find a large systematic offset in metallicity for satellites of more massive central galaxies, corresponding to more massive group halos. Satellite galaxies associated with centrals more massive than $\\log(M_{*}\/\\mathrm{M_{\\odot}}) = 10.5$ have metallicities that are on average $0.08 \\pm 0.009 \\, \\mathrm{dex}$ higher than those galaxies which are satellites of centrals with $\\log(M_{*}\/\\mathrm{M_{\\odot}}) < 10$. To eliminate the possibility that a different distribution of total stellar masses  for the targeted galaxies within the central stellar mass bin is responsible for the discrepancy, we perform a two-sample Kolmogorov-Smirnov test \\citep{Smirnov1939}. This test returns a statistic of $D=0.17$ with $p=0.71$, indicating no statistically significant difference in the total stellar masses.\n\n\n\\begin{figure}\n\\includegraphics{sig_mass_met_cen_mass_PP04.pdf}\n\\includegraphics{sig_mass_met_cen_mass_D16.pdf}\n\\caption{The local $\\Sigma_{*}-\\mathrm{O\/H}$ relation for satellite galaxies with $9<\\log(M_{*}\/\\mathrm{M_{\\odot}})<10$ split by the mass of the galaxy which is central to their halo. In the upper panel, we show the results for the PP04 indicator, and in the lower panel we show the results for the D16 indicator. Blue points show the median metallicity at a given $\\Sigma_{*}$ for satellites of low-mass centrals ($\\log(M_{*}\/\\mathrm{M_{\\odot}})<10$). These points trace the median of the grey-shaded distribution. Red points are the median metallicity as a function of $\\Sigma_{*}$ for satellites of high-mass centrals, shown by the red contours. The oxygen abundance is systematically higher for satellites of more massive centrals. }\\label{sig_mass_met_cen_mass}\n\\end{figure}\n\n\\subsubsection{Metallicity and gas fraction}\nModels predict \\citep[e.g.][]{Lilly2013}, and observations show \\citep{Mannucci2010, Moran2012}, that if low-metallicity gas is accreted onto the galaxy and the local gas fraction rises, then the metal content is diluted and the total metallicity of the gas will decrease. This relationship is investigated in Figure \\ref{mu_met}, where we show the gas-phase metallicity as a function of the local gas fraction, $\\mu$ in intervals of total stellar mass. In narrow bins of stellar mass, we find a tight correlation between the local gas fraction and the metallicity of the ISM. Once again, there is a small difference in the metallicities between satellites and centrals, with the difference being largest in galaxies between $10^{9.5}$ and $10^{10} \\, \\mathrm{M_{\\odot}}$. \n\n\n\n\\begin{figure*}\n\\includegraphics{mu_met_PP04.pdf}\\\\\n\\includegraphics{mu_met_D16.pdf}\n\\caption{The dependence of $12+\\log(\\mathrm{O\/H})$ on the local gas fraction, $\\mu$ in different bins of stellar mass. The contours, greyscale and coloured points are the same as in Figure \\ref{sig_mass_met}. The difference in metallicity at fixed $\\mu$ between satellites and centrals is largest in the range $9.5 < \\log(M_{*}\/\\mathrm{M_{\\odot}}) < 10$, where it reaches $\\sim 0.015 \\, \\mathrm{dex}$.  }\\label{mu_met}\n\\end{figure*}\n\n\nFocussing again on the lower-mass satellite galaxies in our sample, we see in Figure \\ref{mu_met_cen_mass} that the satellites of massive galaxies are more enriched at fixed $\\mu$ than the satellites of less massive centrals. Comparing the contours of spaxels in the $\\mu$-metallicity plane, we see that on average, the satellites of more massive centrals have a lower inferred gas fraction. Nevertheless, at fixed $\\mu$ the offset in O\/H remains.\n\n\n\\begin{figure}\n\\includegraphics{mu_met_cen_mass_PP04.pdf}\\\\\n\\includegraphics{mu_met_cen_mass_D16.pdf}\n\\caption{The relationship between $\\mu$ and O\/H for satellites of low-mass centrals (grey background with blue points indicating the median) and high-mass (red contours with red points indicating the medians) for PP04 O3N2 (top) and D16 N2S2H$\\alpha$ (bottom). At fixed gas fraction, the median metallicity is $\\sim 0.1 \\, \\mathrm{dex}$ higher for the satellites of massive centrals.}\\label{mu_met_cen_mass}\n\\end{figure}\n\n\n\n\\section{Discussion}\\label{Discussion}\n\\subsection{The impact of environment on local scaling relations}\nWe have shown that the metallicity versus stellar mass, local escape velocity and gas fraction local scaling relations vary with the environment that galaxies inhabit. In this study we utilised the mass of the largest galaxy in the group as our estimate of environment. This quantity is correlated with the total mass of the halo \\citep{Behroozi2010}, though does not suffer from the large uncertainties involved in estimating halo dynamical masses from spectroscopy \\citep[see][for an excellent discussion of this point]{Robotham2011}. For satellite galaxies, the magnitude of the difference in metallicity appears to be a function of the stellar mass of the galaxy that is central to the group. In Figures \\ref{sig_mass_met_cen_mass} and \\ref{mu_met_cen_mass} we showed that the satellites of central galaxies more massive than $\\log{\\mathrm{M_{*}}\/M_{\\odot}}>10.5$ have local metallicities that are enhanced by $\\sim 0.1 \\, \\mathrm{dex}$ over similar galaxies which are satellites of less massive ($\\log{\\mathrm{M_{*}}\/M_{\\odot}}<10$) centrals. This enhancement appears to be independent of the local gas fraction and escape velocity. \n\n\n\\subsection{Accounting for outflows with the gas-regulator model}\nWhile differences in the metallicity of satellite galaxies at fixed $\\Sigma_{*}$ and $\\mu$ may be suggestive of some intrinsic difference between the chemical evolution of satellites in different mass halos, these simple scaling relations taken individually are unable to account for all factors that may influence the oxygen abundance. Neither of these scaling relations explicitly accounts for the loss of metals and corresponding reduction in oxygen abundance through outflows. To control for all of these factors at once, we fit the gas regulator model of \\cite{Lilly2013} to the data. \n\nThe gas regulator model for galaxy evolution makes the simple assumption that a galaxy's current star formation and rate and metallicity are largely determined by the present day gas fraction. While it was originally devised to apply to entire galaxies as a whole, some authors have recently showed that it can be applied to galaxies locally on $\\sim \\mathrm{kpc}$ scales \\citep{Carton2015, BarreraBallesteros2018}. In their derivation of this model, \\cite{Lilly2013} showed that the metal content of galaxies will reach an equilibrium on timescales shorter than the time it takes for their total gas content to be depleted. At equilibrium, the metallicity is\n\n\\begin{equation}\\label{gas_regulator}\nZ_{eq} = Z_{0} + \\frac{y}{1 + r_{gas} + (1-R)^{-1} \\left(\\lambda + \\epsilon^{-1}\\frac{d \\ln(r_{gas})}{dt} \\right)},\n\\end{equation}\nwhere $r_{gas}$ is the ratio of gas to stellar mass, $R$ is the fraction of gas returned from stars to the ISM by stellar evolution, and $\\epsilon$ is the star formation efficiency. While this equation contains several unknown quantities, we can fix these to sensible values based on previous estimates from the literature. We adopt a value of $R = 0.4$, which is consistent with the predictions of stellar population synthesis models \\citep{Bruzual2003}, and is in line with the assumptions underlying previous work on this topic \\citep{Lilly2013,Carton2015,BarreraBallesteros2018}. Further, based on fitting the mass-metallicity relation for SDSS galaxies, \\cite{Lilly2013} were able to constrain the product $ \\epsilon^{-1}\\frac{d\\ln( r_{gas})}{dt} = -0.25$. The nucleosynthetic yield, $y$, is also not well known. The yield per stellar generation (and gas return fraction, $R$) is dependent on the stellar initial mass function, which some suggest may not be universal \\citep[e.g.][]{Gunawardhana2011,Parikh2018}. \\cite{Finlator2008} estimate the yield to be in the range $0.008 \\leq y \\leq 0.023$, but we assume the value near the middle of this range of $0.014$. This is the value calculated by \\cite{BarreraBallesteros2018} based on both theoretical modeling using { \\tt STARBURST99} \\citep{Leitherer2014} and by closed-box modeling of galaxy cluster data \\citep{Renzini2014}. We assume that this value is constant and valid throughout our entire sample. For a rigorous discussion of the impact of variations of the assumed yield on the calibration and interpretation of metallicities, see \\cite{Vincenzo2016}.\n\n\n\n\nIn the gas regulator model, the outflows are described by the mass-loading factor, $\\lambda$, which is the ratio of the star formation rate to the rate of mass loss due to stellar feedback and winds. We parametrized the mass-loading factor similar to the formulation of \\cite{Peeples2011}, and assuming the metallicity of outflows is the same as the metallicity in the ISM of the galaxy\n\n\\begin{equation}\\label{lambda}\n\\lambda = \\left( \\frac{v_{0}}{v_{esc}(r)}\\right)^{\\alpha}.\n\\end{equation}\n\n\\cite{Peeples2011} suggest either $\\alpha=1$ or $2$, however we note that the they parametrise $\\lambda$ in terms of the virial velocity of galaxy halos. The relationship between the virial velocity and the local escape velocity is complicated, and so to account for this we allow $\\alpha$ to vary freely. \\cite{BarreraBallesteros2018} also include an additive constant in their parametrisation of $\\lambda$, which imposes a minimum level of outflows from even the deepest potential well. For a reasonable choice of yield, we find that this has the effect of limiting the maximum metallicity that a gas-regulated system can achieve.\n\n\n\\subsubsection{Fitting the \\cite{Pettini2004} metallicity}\nIn order to constrain the values of $v_{0}$ and $\\alpha$ for our model, we fit equation \\ref{gas_regulator} to the metallicity, gas fraction and local escape velocities inferred from the MaNGA spaxel data for all galaxies in our sample. We find $v_{0}=368 \\, \\mathrm{km \\, s^{-1}}$, and $\\alpha = 0.52$. In their work on deriving the $v_{esc}$-dependence of $\\lambda$, \\cite{BarreraBallesteros2018} noted that the gas regulator model does not necessarily provide a good fit to the data, though it was preferred to the leaky-box model of \\cite{Zhu2017} on the grounds that it provided more realistic estimates of $\\lambda$. In our fits of the gas-regulator model to the MaNGA data on kpc scales, we find smaller residuals at low gas fraction. This is a direct result of our choice not to include an additive constant in our parametrisation of $\\lambda$. In the analysis that follows we fix the dependence of $\\lambda$ on the escape velocity, reducing this problem to fitting only one variable, the metallicity of accreted gas, $Z_{0}$.\n\nWe perform a least-squares fit of the gas regulator model to the O3N2-based gas-phase metallicities, fitting for $Z_{0}$ in the sub-populations where the largest difference in metallicity is seen. This is for satellite galaxies with stellar masses in the range $9<\\log (M_{*}\/\\mathrm{M_{\\odot}})<10$, split based on the stellar mass of the corresponding central galaxy. For satellites of low-mass centrals ($\\log(M_{*}\/\\mathrm{M_{\\odot}})<10$), the metallicity of the gas precipitating onto their discs inferred from the modeling is $Z_{0} = (4.68 \\pm 0.11) \\times 10^{-4}$, corresponding to $12+\\log(\\mathrm{O\/H})=7 .46 \\pm 0.01$. For the gas being accreted onto the $9<\\log(M_{*}\/\\mathrm{M_{\\odot}})<10$ satellites of high-mass central galaxies, we derive a metallicity of $Z_{0}= (1.17 \\pm 0.001) \\times 10^{-3}$, or $12+\\log(\\mathrm{O\/H})=7.87 \\pm 0.003$. \n\n\n\\subsubsection{Fitting the \\cite{Dopita2016} metallicity}\nThe metallicities derived from the \\cite{Dopita2016} N2S2H$\\alpha$ calibration have a different absolute abundance scaling and cover larger range for the same set of spectra. This can be seen by comparing the $y$-axes in Figure \\ref{sig_mass_met}. Using the same values of the yield and gas return fraction as were used to fit the O3N2 metallicities, the data favours a negative value of $Z_{0}$, which is unphysical. With this indicator, the values of $\\lambda$ allowed by this parametrisation that also gives an appropriate shape to the distribution of modeled data in the $\\mu -Z $ plane, are too large. Using the $\\lambda$ parametrisation of \\cite{BarreraBallesteros2018}, $\\lambda=\\left(v_{0}\/v_{esc} \\right)^{\\alpha} + \\lambda_{0}$, we find $Z_{0} = (3.0 \\pm 0.12 )\\times 10^{-4}$ or $12+\\log(\\mathrm{O\/H})=7.27 \\pm 0.01$ for the satellites of low-mass centrals. For the satellites of high-mass centrals we find  $Z_{0} = (8.6 \\pm 0.1 )\\times 10^{-4}$ or $12+\\log(\\mathrm{O\/H})=7.86 \\pm 0.003$. \n\n\n\nWe note that the absolute value for these inferred quantities is correlated with a number of unconstrained parameters including the nucleosynthetic yield of oxygen, $y$, the gas return fraction from stars, $R$, and the precise form of the mass loading factor, $\\lambda$. Nevertheless, we argue that the assumption that these parameters do not vary between star-forming galaxies in a relatively narrow mass range is reasonable, and that the choice to fix them for this comparison is justified. With this limitation it is not possible to derive an absolute abundance for the accreted gas, but the existence of a difference is robust. As was shown in Figures \\ref{sig_mass_met_cen_mass} and \\ref{mu_met_cen_mass}, the difference in local metallicity scaling relations is significant between these two galaxy subpopulations. The gas regulator model provides an interpretive framework to describe these differences in terms of the variation of the metallicity of the intergalactic medium in different environments, while controlling for small differences in the estimated local gas fraction and escape velocity.\n\n\n\n\\subsection{Can starvation explain our results?}\\label{Starvation}\nOur results are analogous to those seen by previous studies of the environmental dependence of the global mass-metallicity relation \\citep[e.g.][]{Cooper2008,Pasquali2012,Peng2014,Wu2017}. While different studies have found qualitatively similar results, there is considerable disagreement in the interpretation. \\cite{Peng2014}, argue that the primary driver of this trend must be the elevation of the metallicity of gas being accreted onto galaxies in dense environments. This argument hinges on their observation that the distribution of star-formation rates in their sample is independent of environment.\nThis conclusion contrasts starkly with the interpretation of \\cite{Wu2017}, who suggested that the environmental variation of the mass-metallicity relation can be explained by the reduction in the gas-fractions of galaxies with the local galaxy overdensity. \n\n\\begin{figure}\n\\includegraphics{sig_M_sig_SFR_cen_mass.pdf}\n\\caption{The star formation rate surface density as a function of stellar mass surface density for galaxies with $9<\\log(M_{*}\/\\mathrm{M_{\\odot}})<10$. The greyscale shows the distribution of measurements from satellites of low-mass centrals ($\\log(M_{*,cen}\/\\mathrm{M_{\\odot}})<10$), with the median of this distribution shown by blue points, and the $16$th and $84$th percentiles shown by blue lines. The red contours indicate the $\\Sigma_{*} - \\Sigma_{SFR}$ distribution for satellites of massive galaxies ($\\log(M_{*,cen}\/\\mathrm{M_{\\odot}})>10.5$), with the red points showing the median and the red lines marking the $16$th and $84$th percentile of the distribution. There is very little difference in the two distributions for the vast majority of spaxels in the two samples.}\\label{sig_M_sig_SFR_cen_mass}\n\\end{figure}\n\n\nStarvation, whereby the accretion of gas onto galaxies is curtailed and the gas reservoir is not replenished following star formation \\citep{Larson1980}, will have the effect of increasing the gas-phase metallicity of a galaxy or a region of the galaxy. This is a natural consequence of maintaining a constant metal yield from stellar evolution, while reducing the replenishment of the reservoir with relatively low-metallicity gas. Within the framework of gas-regulated galaxy evolution, this implies an anti-correllaton between the gas surface density or star formation rate surface density and the metallicity in the gas. Starvation has been suggested as a key component for determining the star-forming properties of galaxies today \\citep{Peng2015,Trussler2018}, with environment appearing to play a role in instigating this process \\citep{vonderlinden2010, Davies2016}. \n\nWhile we do infer gas fractions that are, on average, lower for galaxies that are in more extreme environments (for example low-mass satellites of high-mass galaxies), we find that at fixed gas fraction the metallicity is higher for satellites relative to centrals, even in spaxels with high $\\mu$. The differing distributions of $\\log(\\mu)$ evident in Figure \\ref{mu_met_cen_mass} are largely driven by the differences in the distributions of $\\Sigma_{*}$. Although the distributions of total $M_{*}$ between the two subsamples used are not significantly different, the distributions of local stellar mass surface densities are. In Figure \\ref{sig_M_sig_SFR_cen_mass} we show the joint distributions for $\\Sigma_{*}$ and $\\Sigma_{SFR}$ for low-mass satellites, split by the stellar mass of the central galaxy. At fixed $\\Sigma_{*}$, the difference between the means of the $\\Sigma_{SFR}$ is smaller than $0.03$ dex, except above $\\log(\\Sigma_{*})=8.1$, but this range accounts for only $\\sim 10 \\% $ of the data and therefore has a minimal impact on our model fitting. \n\nIt is possible that the distribution of star formation has also changed. \\cite{Schaefer2019} showed that in dense environments, the outer parts of a galaxy can be quenched, leaving star formation in the inner regions unaffected. This transformation was nevertheless accompanied by a reduction in the total specific star formation rate (sSFR). To test for this, we perform a KS-test on the integrated specific star formation rates of the two subsamples. This yields  $KS = 0.17$ with $p=0.58$. Furthermore, the median of the sSFR for the satellites of high mass galaxies is $-10.12 \\pm 0.07 \\, \\mathrm{yr^{-1}}$ and the median for the satellites of more massive galaxies is $-10.21 \\pm 0.04 \\, \\mathrm{Gyr^{-1}}$, where the error on the median has been estimated using a bootstrap resampling. The difference of medians is within the error margin. \n\nThe similar distributions of $\\Sigma_{SFR}$ and  sSFR disfavour the interpretation that the changing gas fraction due to starvation is responsible to the environmental differences in metallicity on kpc scales within our sample. This is not to say that starvation does not occur in dense environments; our sample selection simply favours the most star-forming galaxies, which are unlikely to have had their star formation rates reduced by environmental effects yet. Environmental differences in metallicity may occur in satellite galaxies before the onset of environment quenching.\n\n\n\\subsection{Comparison to simulations}\nNumerical simulations of galaxy evolution are beginning to show that the gas being accreted onto galaxies cannot be assumed to be pristine in all environments \\citep{Oppenheimer2010,Gupta2018}. In the simulations, the origin of accreted gas is observed to be highly dependent on redshift, with cosmological accretion of low-metallicity gas dominating at high redshift. However, as time progresses, feedback from star formation and AGN activity expel gas from the interstellar media of galaxies, which enriches their local environment with material that subsequently falls onto their neighbours. In the FIRE simulations, \\cite{AnglesAlcazar2017} found that the exchange of gas between galaxies that is facilitated by galactic winds dominates the accretion budget by $z=0$. \n\n\\cite{Gupta2018} explored this effect using data from the IllustrisTNG simulations. They showed that the enrichment of the intergalactic medium and the associated accretion onto galaxies is dependent on both the halo mass and whether a galaxy is infalling into its host halo or whether it has been a satellite for some time. At $z<0.5$, they find that the metallicity of gas being accreted onto galaxies with $9<\\log(M_{*}\/M_{\\odot})<10$ that are infalling into clusters is approximately $0.35 \\, Z_{\\odot}$, which is $1.5$ - $2$ times more metal rich than for similar galaxies in the field. This is consistent with the metallicity difference that we have inferred between the satellites of low-mass and high-mass centrals, though the absolute abundances differ. We again note that the value of $Z_{0}$ returned when the model represented by Equation \\ref{gas_regulator} is fitted to the data is sensitive to the precise values of the yield, $y$, and the gas return fraction, $R$, which are not well constrained by observations. The choice of these values will change the estimate of $Z_{0}$, but the relative difference between subsamples will not be greatly affected.\n\n\n\\subsection{Other studies of the environmental dependence of metallicity}\nThe impact of environment on galaxy evolution, in particular star formation and metallicity, is subtle. For this reason there have been very few observational works that measure the impact of environment on the spatial patterns of chemical abundances in galaxies. It has only been recently that large enough samples of integral field spectroscopic data have become available to adequately measure these effects. \n\nIn a recent study, Lian et al. (\\emph{in prep.}) used MaNGA data to study the metallicity gradients of galaxies as a function of the local environmental overdensity. They find that the metallicity gradients in low-mass satellite galaxies are shallower in dense environments, with a higher metallicity in their outer parts than similar galaxies in the field. They also find that the star formation rate gradients in galaxies in dense environments are steeper and conclude that the most likely explanation for these observations is a variation in the gas accretion timescale in different environments. Superficially this would seem to contradict our results, but we argue that this apparent disagreement can be resolved by noting the differences between the samples of galaxies considered. Lian et al. place less stringent constraints on the number of star-forming spaxels than we do, meaning that their galaxies have lower specific star formation rates on average. They therefore study galaxies that are likely to have inhabited their host halos for a longer period of time, and are more affected by environment quenching processes. This point is made in Section \\ref{Starvation}, where we rule out starvation as the primary driver of the environmental effects discussed in this work.  Additionally, we note that Lian et al. placed no constraints on the inclination of galaxies in their sample to the line of sight. This may explain the differences in the metallicity gradients to those reported in our work.\n\n\n\n\n\\section{Conclusions}\nWe have estimated local metallicities, gas fractions, escape velocities and star formation rate surface densities for a sample of nearly face-on star-forming galaxies observed by MaNGA. In this sample we have explored the impact of the environment on local scaling relations between these estimated quantities, with a particular focus on satellite galaxies. \nWe find\n\\begin{itemize}\n\\item{At fixed stellar mass, we find a small but global offset of $0.025 \\, \\mathrm{dex}$ (for O3N2) or $0.05 \\, \\mathrm{dex}$ (for N2S2H$\\alpha$) in the metallicities of galaxies between satellites and centrals. For our sample we find little evidence for changes in the metallicity gradient between satellites and centrals.}\n\\item{The disparity between the metallicity of satellites and centrals is also evident in the O\/H -- $\\Sigma_{*}$ and O\/H -- $\\mu$ local scaling relations. We find the greatest offset when we split our satellite sample by the stellar mass of the galaxy that is central to the respective halo. For satellite galaxies in the range $9<\\log(M_{*}\/M_{\\odot})<10$, the local scaling relations are $\\sim 0.1 \\, \\mathrm{dex}$ more oxygen rich for satellites of hosts more massive than $10^{10.5} \\, M_{\\odot}$ than for hosts less massive than $10^{10} \\, M_{\\odot}$.}\n\\item{The offset in metallicity for satellite galaxies is found to exist between different environments at constant stellar mass surface density, gas mass fraction and star formation rate surface density. From these we conclude that the observed differences cannot be explained by gas starvation occurring in satellites around more massive centrals. Interestingly, the impact of environment on the chemical enrichment of galaxies appears to precede the onset of the quenching of star formation in their disks. }\n\\item{Measured on kpc-scales, local metallicities and gas fractions are found to be quantitatively consistent with the gas regulator model of \\cite{Lilly2013}. We assume that the mass loading factor describing outflows in galaxies is a function of the local escape velocity. Within the framework of the gas regulator model the only explanation for the elevated metallicity is an increase in $Z_{0}$ for satellites of high-mass galaxies. We estimate that the oxygen abundance in the inflowing gas changes from $12+\\log(\\mathrm{O\/H}) = 7.54 \\pm 0.01$ to $7.86 \\pm 0.003$ using the \\cite{Pettini2004} O3N2 indicator, or $12+\\log(\\mathrm{O\/H}) = 7.27 \\pm 0.01 $ to $7.86 \\pm 0.003 $ for N2S2H$\\alpha$.}\n\n\n\\end{itemize}\n\nGiven these conclusions, we interpret the enhanced metallicity of the satellites of more massive centrals to be evidence for the exchange of enriched gas between galaxies. In this picture, which has been motivated by both observations \\citep{Peng2014} and simulations \\citep{Oppenheimer2010,AnglesAlcazar2017,Gupta2018}, feedback driven winds expel metal-rich gas from a massive star-forming central which is subsequently accreted onto nearby satellites. While our estimates for the metallicity of gas accreted onto satellite galaxies are a factor of $\\sim 3$ lower than the predictions of \\cite{Gupta2018}, we note that the values returned by our modeling are subject to inherent uncertainties in the nucleosynthetic yield, the gas return fraction from stellar evolution, and the absolute abundance scaling of the strong-line metallicity diagnostics. Notwithstanding these systematic effects, the inferred differential in the metallicity of gas accreted onto satellites in different environments is qualitatively in good agreement with the simulations.\n\nThe impact of environment on the gas phase metallicity distribution of galaxies is likely to be complicated and multifaceted. In addition to the accretion of enriched gas in dense environments that we have studied here, tidal interactions, mergers and ram pressure can influence the distribution of metals in a galaxy. A complete understanding of the effect of environment on gas phase metallicities must take these other processes into account. A more comprehensive analysis of the detailed environmental dependence of the chemical properties of galaxies will be made possible when the full MaNGA sample becomes available, or with future integral field spectroscopic surveys such as HECTOR \\citep{Bryant2016}.\n\n\n\n\\acknowledgements \nWe would like to thank the anonymous referee, whose constructive comments were extremely helpful in clarifying some aspects of this paper. \nALS, ZJP and CT acknowledge NSF CAREER Award AST-1554877. AJ acknowledges NSF Award 1616547. RM acknowledges ERC Advanced Grant 695671 \"QUENCH\" and support from the the Science and Technology Facilities Council (STFC).\n\nThis research made use of \\texttt{Astropy}, a community-developed core \\texttt{python} package for astronomy \\citep{Astropy2013,Astropy2018}; \\texttt{matplotlib} \\citep{Matplotlib}, an open-source \\texttt{python} plotting library; and \\texttt{LMFIT} \\citep{LMFIT}, an interface for non-linear optimization in \\texttt{python}.\nFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrof\\'{i}sica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut f\\\"{u}r Astrophysik Potsdam (AIP), Max-Planck-Institut f\\\"{u}r Astronomie (MPIA Heidelberg), Max-Planck-Institut f\\\"{u}r Astrophysik (MPA Garching), Max-Planck-Institut f\\\"{u}r Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\\'{o}rio Nacional \/ MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\\'{o}noma de M\\'{e}xico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\n\n\n\n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCharalambos presented the $q-$ deformed Vandermonde and Cauchy formulae. Moreover, the $q-$ deformed univariate discrete probability distributions were investigated.Their properties and limiting distributions were derived \\cite{CA1}.\n\nFurthermore, the $q-$ deformed multinomial coefficients was defined and their recurrence relations were deduced. Also, the $q-$ deformed multinomial  and negative $q-$ deformed multinomial probability distributions of the first and second kind  were presented \\cite{CA2}. \n\nThe same author extended the multivariate $q-$ deformed vandermonde and Cauchy formulae. Also, the multivariate $q-$ Pol\\'ya and inverse  $q-$ Pol\\'ya were constructed \\cite{CA3}.\n\nLet $p$ and $q$ be two positive real numbers such that $ 0<q<p<1.$ We consider a meromorphic function ${\\mathcal R}$ defined on $\\mathbb{C}\\times\\mathbb{C}$ by\\cite{HB}:\\begin{equation}\\label{r10}\n\\mathcal{R}(u,v)= \\sum_{s,t=-l}^{\\infty}r_{st}u^sv^t,\n\\end{equation}\nwith an eventual isolated singularity at the zero, \nwhere $r_{st}$ are complex numbers, $l\\in\\mathbb{N}\\cup\\left\\lbrace 0\\right\\rbrace,$ $\\mathcal{R}(p^n,q^n)>0,  \\forall n\\in\\mathbb{N},$ and $\\mathcal{R}(1,1)=0$ by definition. We denote by $\\mathbb{D}_{R}$ the bidisk \\begin{eqnarray*}\n\t\\mathbb{D}_{R}\n\n\n\t&=&\\left\\lbrace w=(w_1,w_2)\\in\\mathbb{C}^2: |w_j|<R_{j} \\right\\rbrace,\n\\end{eqnarray*}\nwhere $R$ is the convergence radius of the series (\\ref{r10}) defined by Hadamard formula as follows:\n\\begin{eqnarray*}\n\t\\lim\\sup_{s+t \\longrightarrow \\infty} \\sqrt[s+t]{|r_{st}|R^s_1\\,R^t_2}=1.\n\\end{eqnarray*}\nFor the proof and more details see \\cite{TN}. \nWe denote by \n${\\mathcal O}(\\mathbb{D}_R)$ the set of holomorphic functions defined\non $\\mathbb{D}_R.$\n\nThe  $\\mathcal{R}(p,q)-$ deformed numbers is defined by \\cite{HB}: \n\\begin{eqnarray}\\label{Rpqn}\n[x]_{\\mathcal{R}(p,q)}:= \\mathcal{R}(p^x,q^x),\\qquad x\\in\\mathbb{N},\n\\end{eqnarray}\nthe $\\mathcal{R}(p,q)-$ deformed factorials and binomial coefficients are given  as:\n\\begin{eqnarray}\\label{Rpqf}\n[x]!_{\\mathcal{R}(p,q)}:= \\left\\{\\begin{array}{lr} 1 \\quad \\mbox{for} \\quad n=0 \\quad \\\\\n\\mathcal{R}(p,q)\\cdots \\mathcal{R}(p^x,q^x) \\quad \\mbox{for} \\quad x\\geq 1, \\quad \\end{array} \\right.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{Rpqbc}\n\\left[\\begin{array}{c} x  \\\\ y\\end{array} \\right]_{\\mathcal{R}(p,q)}:=\n\\frac{[x]!_{\\mathcal{R}(p,q)}}{[y]!_{\\mathcal{R}(p,q)}[x-y]!_{\\mathcal{R}(p,q)}},\\quad x, y= 0, 1, 2, \\cdots;\\quad x\\geq y.\n\\end{eqnarray}\n\nWe consider the following linear operators   on ${\\mathcal O}(\\mathbb{D}_R)$  given by:\n\\begin{eqnarray}\\label{operat}\n&&\\quad Q: \\varphi \\longmapsto Q\\varphi(z) := \\varphi(qz)\n\\nonumber \\\\\n&&\\quad P: \\varphi \\longmapsto P\\varphi(z): = \\varphi(pz)\n,\n\\end{eqnarray}\nleading to define  the $\\mathcal{R}(p,q)-$ deformed derivative:\n\\begin{equation}{\\label{deriva1}}\n\\partial_{{\\mathcal R},p,q} := \\partial_{p,q}\\frac{p - q}{P-Q}{\\mathcal R}(P, Q)\n= \\frac{p - q}{pP-qQ}{\\mathcal R}(pP, qQ)\\partial_{p,q},\n\\end{equation}\nwhere $\\partial_{p,q}$ is the $(p,q)-$ derivative:\n\\begin{eqnarray}\n\\partial_{p,q}:\\varphi \\longmapsto\n\\partial_{p,q}\\varphi(z) := \\frac{\\varphi(pz) - \\varphi(qz)}{z(p-q)}.\n\\end{eqnarray}\nThe quantum algebra associated with the $\\mathcal{R}(p,q)-$ deformation, denoted by \n${\\mathcal A}_{\\mathcal{R}(p,q)}$ is generated by the\nset of operators $\\{1, A, A^{\\dagger}, N\\}$ satisfying the following\ncommutation relations \\cite{HB1}:\n\\begin{eqnarray}\n&& \\label{algN1}\n\\quad A A^\\dag= [N+1]_{\\mathcal{R}(p,q)},\\quad\\quad\\quad A^\\dag  A = [N]_{\\mathcal{R}(p,q)}.\n\\cr&&\\left[N,\\; A\\right] = - A, \\qquad\\qquad\\quad \\left[N,\\;A^\\dag\\right] = A^\\dag\n\\end{eqnarray}\nwith its realization on ${\\mathcal O}(\\mathbb{D}_R)$ given by:\n\\begin{eqnarray*}\\label{algNa}\n\tA^{\\dagger} := z,\\qquad A:=\\partial_{\\mathcal{R}(p,q)}, \\qquad N:= z\\partial_z,\n\\end{eqnarray*}\nwhere $\\partial_z:=\\frac{\\partial}{\\partial z}$ is the usual derivative on $\\mathbb{C}.$\n\nThe $\\mathcal{R}(p,q)-$ deformed numbers \\eqref{Rpqn} can be rewritten as follows \\cite{HMD}:\n\\begin{eqnarray}\n[x]=\\frac{\\tau^{x}_1-\\tau^{x}_2}{ \\tau_1-\\tau_2}, \\quad \\tau_1\\neq \\tau_2,\n\\end{eqnarray}\nwhere $\\big(\\tau_i\\big)_{i\\in\\{1,2\\}}$ are functions depending of the parameters deformations $p$ and $q.$\n\nThe following relations hold \\cite{HMRC}:\n\\begin{eqnarray}\\label{011}\n[x]_{\\mathcal{R}(p^{-1},q^{-1})} &=& (\\tau_1\\,\\tau_2)^{1-x}\\,[x]_{\\mathcal{R}(p,q)},\\\\\n\\,[r]_\n{\\mathcal{R}(p^{-1},q^{-1})}!&=& (\\tau_1\\,\\tau_2)^{- {r  \\choose 2}}\\,\\,[r]_{\\mathcal{R}(p,q)}!,\\label{014}\\\\\n\\,{[x]_{r,\\mathcal{R}(p^{-1},q^{-1})}}\n&=& (\\tau_1\\,\\tau_2)^{-xr + {r +1 \\choose 2}}\\,{[x]_{r,\\mathcal{R}(p,q)}}\\label{015}.\n\\end{eqnarray}\nFurthermore, the generalized Vandermonde, Cauchy formulae and univariate probability distributions induced from the $\\mathcal{R}(p,q)-$ deformed quantum algebras are investigated in \\cite{HMD}. \n\nOur aims is to construct the multinomial coeficient, multivariate Vandermonde, and Cauchy formulae, multivariate probability distributions, and properties associated to the  $\\mathcal{R}(p,q)-$ deformed quantum algebras \\cite{HB1}.  \n\nThis paper is organized as follows: In section $2,$ we investigate the $\\mathcal{R}(p,q)-$ multinomial coefficient, the  multivariate Vandermonde formula associated to the $\\mathcal{R}(p,q)-$ deformed quantum algebras. Inverse multivariate $\\mathcal{R}(p,q)-$ Vandermonde and negative multivariate $\\mathcal{R}(p,q)-$ Vandermonde formula are computed. Moreover,  the $\\mathcal{R}(p,q)-$ deformed Cauchy formula is deduced. Section $3,$ is reserved to the construction of the multivariate probability distributions. We derive the case of bivariate distributions and compute their $\\mathcal{R}(p,q)$-factorial moments and covariance. Relevant particular cases corresponding to the quantum algbras known in the literature are derived from the general formalism.\n\\section{$\\mathcal{R}(p,q)$-  multivariate Vandermonde and  Cauchy formulae }\nIn this section, we investigate the generalized  multinomial coefficients,  multivariate Vandermonde formula, and  multivariate Cauchy formula  associated to the $\\mathcal{R}(p,q)-$ deformed quantum algebras. Their recurrence relations are also derived. \n\\begin{proposition}\n\tThe generalized  multinomial coefficient\n\t\\begin{eqnarray}\\label{eq2.1}\n\t\\left[\\begin{array}{c}x \\\\r_1,r_2,\\ldots,r_k\\end{array} \\right]_{\\mathcal{R}(p,q)}=\\frac{[x]_{r_1+r_2+\\cdots+r_k,\\mathcal{R}(p,q)}}{ [r_1]_{\\mathcal{R}(p,q)}![r_2]_{\\mathcal{R}(p,q)}!\\cdots[r_k]_{\\mathcal{R}(p,q)}!}\n\t\\end{eqnarray} \n\tsatisfies the recursion relation:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t&&\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\ldots,r_k}_{{\\mathcal R}(p,q)}=\\tau_1^{s_k}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\ldots,r_k}_{{\\mathcal R}(p,q)}\n\t\t+\\tau_2^{x-m_1}\\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\\ldots,r_k}_{\\mathcal {R}(p,q)}\\cr&& \\quad\\quad\\quad\n\t\t+\\tau_2^{x-m_2}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\\ldots,r_k}_{{\\mathcal R}(p,q)}+\\cdots+\\tau_2^{x-m_k}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\ldots,r_k-1}_{{\\mathcal R}(p,q)}.\n\t\t\\end{eqnarray*}\n\t\tEquivalently,\n\t\t\\begin{eqnarray*}\n\t\t&&\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\ldots,r_k}_{\\mathcal {R}(p,q)}=\\tau_2^{s_k}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\ldots,r_k}_{{\\mathcal R}(p,q)}\n\t\t+\\tau^{x-m_1}_1\\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\\ldots,r_k}_{{\\mathcal R}(p,q)}\\cr&&\\quad\\quad\n\t\t+\\tau^{x-m_2}_1\\tau_2^{s_1}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\\ldots,r_k}_{\\mathcal {R}(p,q)}\n\t\t+\\cdots+\\tau^{x-m_k}_1\\tau_2^{s_{k-1}}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\ldots,r_k-1}_{{\\mathcal R}(p,q)},\n\t\t\\end{eqnarray*}\n\t\twhere $r_j\\in\\mathbb{N}$ and $j\\in\\{1,2,\\ldots,k\\}$, with $m_j=\\sum_{i=j}^kr_i$ and $s_j=\\sum_{i=1}^jr_i.$\n\t\\end{small}\n\\end{proposition}\n\\begin{proof} \n\tIt is straightforwad by computation. $\\hfill{\\square}$\n\\end{proof}\n\nFrom the relations (\\ref{011}), (\\ref{014}) and (\\ref{015}), we obtain another expression of the genralized multinomial coefficient in the simpler form:\n\\begin{small}\n\t\\begin{equation}\\label{eq2.2}\n\t\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\cdots,r_k}_{\\mathcal{R}(p^{-1},q^{-1})}\n\t=(\\tau_1\\tau_2)^{-\\displaystyle\\sum_{j=1}^k r_j(x-m_j)}\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\cdots,r_k}_{\\mathcal{R}(p,q)}\\end{equation}and\n\t\\begin{equation}\\label{eq2.2a}\n\t\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\cdots,r_k}_{\\mathcal{R}(p^{-1},q^{-1})}=(\\tau_1\\tau_2)^{-\\displaystyle\\sum_{j=1}^k r_j(x-s_j)}\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\cdots,r_k}_{\\mathcal{R}(p,q)},\n\t\\end{equation}\n\twhere $s_j=\\displaystyle\\sum_{i=1}^jr_i,$  $m_j=\\displaystyle\\sum_{i=j}^kr_i,$ $r_j\\in\\mathbb{N},$ $j\\in\\{1,2,\\cdots,k\\}$ and $k\\in\\mathbb{N}.$ \n\\end{small}\n\nAnother  recurrence relations can be obtained by  using the expression  (\\ref{eq2.2}), respectively. Thus, we get:\n\\begin{small}\n\t\\begin{eqnarray*} \\label{eq2.5}\n\t\t\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\cdots,r_k}_{\\mathcal {R}(p,q)}&=&\\tau_2^{m_1}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\cdots,r_k}_{\\mathcal {R}(p,q)}\n\t\t+\\tau_2^{m_2}\\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\\cdots,r_k}_{{\\mathcal R}(p,q)}\\nonumber\\\\\n\t\t&+&\\tau_2^{m_3}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\\cdots,r_k}_{\\mathcal{R}(p,q)}\n\t\t+\\cdots+\\tau^x_1\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\cdots,r_k-1}_{\\mathcal{R}(p,q)}\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray}\\label{eq2.5a}\n\t\t\\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\\cdots,r_k}_{{\\mathcal R}(p,q)}&=&\\tau^x_1\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\cdots,r_k}_{{\\mathcal R}(p,q)}\n\t\t+\\tau_2^{x-s_1}\\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\\cdots,r_k}_{{\\mathcal R}(p,q)}\\nonumber\\\\\n\t\t&+&\\tau_2^{x-s_2}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\\ldots,r_k}_{{\\mathcal R}(p,q)}\n\t\t+\\cdots+\\tau_2^{x-s_k}\\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\\ldots,r_k-1}_{{\\mathcal R}(p,q)}.\n\t\\end{eqnarray}\n\\end{small}\n\\begin{theorem}\\label{thm2.1}\n\tThe generalized multivariate  Vandermonde formula is given by the following relations:\n\t\\begin{small}\n\t\t\\begin{equation}\\label{eq2.7}\n\t\t\\Big[\\sum_{i=1}^{k+1}x_i\\Big]_{n,\\mathcal {R}(p,q)}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{\\mathcal{R}(p,q)}\n\t\t\\mathcal{V}(\\tau_1,\\tau_2,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,{\\mathcal{R}(p,q)}}\n\t\t\\end{equation}\n\t\tand\n\t\t\\begin{equation}\\label{eq2.8}\n\t\t\\Big[\\sum_{i=1}^{k+1}x_i\\Big]_{n,\\mathcal {R}(p,q)}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{\\mathcal{R}(p,q)}\n\t\t\\mathcal{V}(\\tau_2,\\tau_1,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,\\mathcal{R}(p,q)},\n\t\t\\end{equation}\n\t\\end{small}\n\twhere $\\mathcal{V}(\\tau_1,\\tau_2)=\\tau_1^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\tau_2^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)},$ $s_j=\\sum_{i=1}^jr_i,$ $z_j=\\sum_{i=j+1}^{k+1}x_i,$ $j\\in\\{1,2,\\cdots,k\\},$ $r_{k+1}=n-s_k$, and  $\\sum_{i=1}^kr_i\\leq n.$\n\\end{theorem}\n\\begin{proof}\nWe use the same procedure as Theorem in \\cite{HMD}.\n$\\hfill{\\square}$\n\\end{proof}\n\\begin{proposition}\n\tThe negative generalized multivariate  Vandermonde formula is described  by the following relations:\n\t\\begin{small}\n\t\t\\begin{equation}\n\t\t\\Big[\\sum_{i=1}^{k+1}x_i\\Big]_{-n,\\mathcal {R}(p,q)}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{-n}{r_1,\\cdots,r_k}_{\\mathcal{R}(p,q)}\n\t\t\\mathcal{V}(\\tau_1,\\tau_2,-n)\\prod_{j=1}^{k+1}[x_j]_{r_j,{\\mathcal{R}(p,q)}}\n\t\t\\end{equation}\n\t\tand\n\t\t\\begin{equation}\n\t\t\\Big[\\sum_{i=1}^{k+1}x_i\\Big]_{-n,\\mathcal {R}(p,q)}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{-n}{r_1,\\cdots,r_k}_{\\mathcal{R}(p,q)}\n\t\t\\mathcal{V}(\\tau_2,\\tau_1,-n)\\prod_{j=1}^{k+1}[x_j]_{r_j,\\mathcal{R}(p,q)},\n\t\t\\end{equation}\n\t\\end{small}\n\twhere $s_j=\\sum_{i=1}^jr_i$, $z_j=\\sum_{i=j+1}^{k+1}x_i$, $j\\in\\{1,2,\\cdots,k\\},$ $r_{k+1}=n-s_k,$ and $\\sum_{i=1}^kr_i\\leq n.$\n\\end{proposition}\n\\begin{proof}\n\tWe use the same procedure as the proof of Theorem \\eqref{thm2.1}. \n\\end{proof}\n\\begin{remark}\nBy replacing $x_j$ by $-x_j$, for $j\\in\\{1,2,\\ldots,k+1\\},$ in the relations (\\ref{eq2.7}) and (\\ref{eq2.8}), and using the following expressions:\n\\begin{eqnarray}\n[-x]_{r,\\mathcal{R}(p,q)}=(-1)^x(\\tau_1\\tau_2)^{-xr-\\binom{r}{2}}[x+r-1]_{r,\\mathcal{R}(p,q)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{eq2.id}\n\\binom{n}{2}=\\sum_{j=1}^{k+1}\\binom{r_j}{2}+\\sum_{j=1}^kr_j(n-s_j),\n\\end{eqnarray}\nthe generalized multivariate   Vandermonde formula can be expressed as:\n\\begin{small}\n\t\\begin{eqnarray}\\label{eq2.7a}\n\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,\\mathcal {R}(p,q)}&=&\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{\\mathcal {R}(p,q)}\\tau_1^{\\sum_{j=1}^{k}x_j(n-s_j)}\n\t\\tau_2^{\\sum_{j=1}^{k}r_jz_j}\\nonumber\\\\ &\\times& \\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,\\mathcal {R}(p,q)},\n\t\\end{eqnarray}\n\tand\n\t\\begin{eqnarray}\\label{eq2.8a}\n\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,\\mathcal {R}(p,q)}&=&\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{\\mathcal {R}(p,q)}\\tau_1^{\\sum_{j=1}^{k}r_jz_j}\n\t\\tau_2^{\\sum_{j=1}^{k}x_j(n-s_j)}\\nonumber\\\\&\\times&\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,\\mathcal {R}(p,q)},\n\t\\end{eqnarray}\n\\end{small}\nwhere $s_j=\\sum_{i=1}^jr_i$, $z_j=\\sum_{i=j+1}^{k+1}x_i$, $j\\in\\{1,2,\\ldots,k\\}$, and $r_{k+1}=n-s_k$, and  $\\sum_{i=1}^kr_i\\leq n$.\n\\end{remark}\n\\begin{theorem}\\label{thm2.2}\n\n\tThe generalized multivariate inverse  Vandermonde formula is presented by the following relations as follows:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\\label{eq2.11}\n\t\t\\frac{1}{[x_{k+1}]_{n,{\\mathcal R}(p,q)}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{{\\mathcal R}(p,q)}\\frac{\\tau_1^{\\sum_{j=1}^{k}r_j(z_j-s_k+s_j-n+1)}}{\n\t\t\\tau_2^{\\sum_{j=1}^{k}(-n-s_k+s_j)(x_j-r_j)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,{\\mathcal R}(p,q)}}{[\\sum_{i=1}^{k+1}x_i]_{n+s_k,{\\mathcal R}(p,q)}},\n\t\t\\end{eqnarray*}\n\t\t%\n\t\tprovided $|\\big({\\tau_2\\over \\tau_1}\\big)^{-x_{k+1}}|<1$, and\n\t\t%\n\t\t\\begin{eqnarray}\\label{eq2.12}\n\t\t\\frac{1}{[x_{k+1}]_{n,\\mathcal {R}(p,q)}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\cdots,r_k}_{\\mathcal {R}(p,q)}\\frac{\\tau_1^{\\sum_{j=1}^{k}(n+s_k-s_j)(x_j-r_j)}}{\n\t\t\\tau_2^{\\sum_{j=1}^{k}r_j(-z_j+s_k-s_j+n-1)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,\\mathcal{R}(p,q)}}{[\\sum_{i=1}^{k+1}x_i]_{n+s_k,\\mathcal {R}(p,q)}},\\quad\n\t\t\\end{eqnarray}\n\t\\end{small}\n\tprovided $|\\big({\\tau_2\\over \\tau_1}\\big)^{x_{k+1}}|<1,$ where $s_j=\\sum_{i=1}^jr_i$ and $z_j=\\sum_{i=j+1}^{k+1}x_i,$ for $j\\in\\{1,2,\\cdots,k\\}.$\n\\end{theorem}\n\\begin{proof}. From the  inverse ${\\mathcal R}(p,q)$-Vandermonde formula \\cite{HMRC}, we get:\n\\begin{small}\n\\begin{eqnarray}\n\\frac{1}{[x_{k+1}]_{n,{\\mathcal R}(p,q)}}=\\sum_{r_k=0}^\\infty\\genfrac{[}{]}{0pt}{}{n+r_k-1}{r_k}_{{\\mathcal R}(p,q)}\\tau^{r_k(x_{k+1}-n+1)}_1\n\\tau_2^{n(x_k-r_k)}\\frac{[x_k]_{r_k,{\\mathcal R}(p,q)}}{[x_k+x_{k+1}]_{n+r_k,{\\mathcal R}(p,q)}}.\n\\end{eqnarray}\nSimilarly,\n\t\\begin{eqnarray}\n\t\\frac{1}{[x_k+x_{k+1}]_{n+r_k,{\\mathcal R}(p,q)}}&=&\\sum_{r_{k-1}=0}^\\infty\\genfrac{[}{]}{0pt}{}{n+r_k+r_{k-1}-1}{r_{k-1}}_{{\\mathcal R}(p,q)}\\tau^{r_{k-1}(x_k+x_{k+1}-n-r_k+1)}_1\\nonumber\\\\&\\times&\n\t\\frac{\\tau_2^{(n+r_k)(x_{k-1}-r_{k-1})}[x_{k-1}]_{r_{k-1},\\mathcal {R}(p,q)}}{[x_{k-1}+x_k+x_{k+1}]_{n+r_k+r_{k-1},\\mathcal {R}(p,q)}}\n\t\\end{eqnarray}\nand finally,\n\t\\begin{eqnarray}\n\t\\frac{1}{[x_2+x_3+\\cdots+x_{k+1}]_{n+s_k-s_1,{\\mathcal R}(p,q)}}&=&\\sum_{r_1=0}^\\infty\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1}_{{\\mathcal R}(p,q)}\\tau^{r_1(x_2+x_3+\\ldots+x_{k+1}-n-s_k+s_1+1)}_1\\nonumber\\\\&\\times&\n\t\\frac{\\tau_2^{(n+s_k-s_1)(x_1-r_1)}[x_1]_{r_1,\\mathcal {R}(p,q)}}{[x_1+x_2+\\cdots+x_{k+1}]_{n+s_k,{\\mathcal R}(p,q)}}.\n\t\\end{eqnarray}\nApplying these $k$ expansions, one after the other in the inner sum of each step, and using the relation:\n\\begin{small}\n\t\\begin{eqnarray}\n\t\\genfrac{[}{]}{0pt}{}{n+r_k-1}{r_k}_{\\mathcal {R}(p,q)}\\genfrac{[}{]}{0pt}{}{n+r_k+r_{k-1}-1}{r_{k-1}}_{\\mathcal {R}(p,q)}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1}_{\\mathcal {R}(p,q)}\n\t=\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,r_2,\\ldots,r_k}_{\\mathcal {R}(p,q)},\n\t\\end{eqnarray}\n\t\\end{small}\nexpansion (\\ref{eq2.11}) is obtained. The alternative expansion (\\ref{eq2.12}), is similarly deduced by using the following inverse ${\\mathcal R}(p,q)$-Vandermonde expansions \\cite{HMRC}:\n\t\\begin{eqnarray}\n\t\\frac{1}{[x_{j+1}+\\cdots+x_{k+1}]_{n+s_k-s_j,{\\mathcal R}(p,q)}}&=&\\sum_{r_j=0}^\\infty\\genfrac{[}{]}{0pt}{}{n+s_k-s_{j-1}-1}{r_j}_{{\\mathcal R}(p,q)}\\nonumber\\\\&\\times&\n\t\\frac{\\tau_2^{r_j(z_j-s_k+s_j-n+1)}[x_j]_{r_j,{\\mathcal R}(p,q)}}{[x_j+\\cdots+x_{k+1}]_{n+s_k-s_{j-1},{\\mathcal R}(p,q)}},\n\t\\end{eqnarray}\nfor $j\\in\\{1,2,\\cdots,k\\},$ with $s_0=0$.\n\\end{small}\n$\\hfill{\\square}$\n\\end{proof}\n\nNow, we investigated the generalized multivariate   Cauchy formula and its related formulae. \n\\begin{corollary}\\label{cor2.1}\n\tThe generalized multivariate Cauchy formula is furnished by:\n\t\\begin{small}\n\t\t\\begin{eqnarray}\\label{eq2.9}\n\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i}{n}_{{\\mathcal R}(p,q)}=\\sum_{r_j=0}^{n} \\tau_1^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\tau_2^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{{\\mathcal R}(p,q)}\n\t\t\\end{eqnarray}\n\t\tand\n\t\t\\begin{equation}\\label{eq2.10}\n\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i}{n}_{{\\mathcal R}(p,q)}=\\sum_{r_j=0}^{n} \\tau_1^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)} \\tau_2^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\,\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{{\\mathcal R}(p,q)},\n\t\t\\end{equation}\n\t\twhere $s_j=\\sum_{i=1}^jr_i,$ $z_j=\\sum_{i=j+1}^{k+1}x_i,$ $j\\in\\{1,2,\\cdots,k\\},$ and $r_{k+1}=n-s_k$  with $\\sum_{i=1}^kr_i\\leq n.$\n\t\\end{small}\n\\end{corollary}\n\\begin{remark}\nSeveral formulae can be deduced as follows:\n\\begin{enumerate}\n\\item[(a)] The generalized multivariate  Cauchy formula can be re-written as:\n\\begin{small}\n\t\\begin{eqnarray*}\\label{eq2.9a}\n\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{\\mathcal {R}(p,q)}=\\sum_{r_j=0}^{n}\\tau_1^{\\sum_{j=1}^{k}x_j(n-s_j)} \\tau_2^{\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{{\\mathcal R}(p,q)}\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{{\\mathcal R}(p,q)}\n\t\t=\\sum_{r_j=0}^{n}\\tau_1^{\\sum_{j=1}^{k}r_jz_j} \\tau_2^{\\sum_{j=1}^{k}x_j(n\\!-\\!s_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{{\\mathcal R}(p,q)}.\n\t\\end{eqnarray*}\n\\end{small}\n\\item[(b)] Another expressions of the generalized multivariate  Cauchy formula are:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{{\\mathcal R}(p,q)}=\\sum_{r_j=x_j}^{r}\\tau_1^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)} \\tau_2^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{{\\mathcal R}(p,q)}\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\t\\,\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{{\\mathcal R}(p,q)}\n\t\t=\\sum_{r_j=x_j}^{r}\\tau_1^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)} \\tau_2^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{{\\mathcal R}(p,q)},\n\t\\end{eqnarray*}\n\twhere $s_j=\\sum_{i=1}^jr_i$, $y_j=\\sum_{i=1}^{j}x_i$, $j\\in\\{1,2,\\ldots,k\\}$, $r_{k+1}=r-s_k$, $x_{k+1}=r-y_k$,  with $\\sum_{i=1}^kr_i\\leq r.$\n\\end{small}\n\\end{enumerate}\n\\end{remark}\n\\begin{remark}\n\tHere, we determine the particular case of the generalized multivariate Vandermonde and Cauchy formulae corresponding to the quantum deformed algebras known in the literature.\n\t\\begin{enumerate}\n\t\\item[(a)]\\begin{small}\n\t\tSetting $\\mathcal{R}(x)=\\frac{x-x^{-1}}{q-q^{-1}},$ we deduce the  multivariate Vandermonde and Cauchy formulae from  the quantum algebra\\cite{BC,M}:\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]_{n,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{q}\n\t\t\t\\mathcal{V}(q,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,q}\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]_{n,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{q}\n\t\t\t\\mathcal{V}(q,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,q},\n\t\t\\end{eqnarray*}\n\t\twhere $\\mathcal{V}(q,n)=q^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{-\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}.$ Moreover, \n\t\tthe negative multivariate  $q$ -Vandermonde formula is presented  by:\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]_{-n,q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}_{q}\\,\\mathcal{V}(q,-n)\n\t\t\t\\prod_{j=1}^{k+1}[x_j]_{r_j,q}\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]_{-n,q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}_{q}\\,\\mathcal{V}(q,-n)\n\t\t\t\\prod_{j=1}^{k+1}[x_j]_{r_j,q},\n\t\t\\end{eqnarray*}\n\t\twhere $\\mathcal{V}(q,-n)=q^{\\sum_{j=1}^{k}r_j(z_j-(-n-s_j))}\\,q^{-\\sum_{j=1}^{k}(-n-s_j)(x_j-r_j)}.$ Also, \n\t\tthe multivariate $q-$  Vandermonde formula can be rewritten as:\n\t\t\\begin{equation*}\n\t\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{q}q^{\\sum_{j=1}^{k}x_j(n-s_j)}\n\t\tq^{-\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,q},\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{q}q^{\\sum_{j=1}^{k}r_jz_j}\n\t\tq^{-\\sum_{j=1}^{k}x_j(n-s_j)}\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,q},\n\t\t\\end{equation*}\n\t\tand \n\t\tthe multivariate inverse $q-$ Vandermonde formula as:\n\t\t\\begin{equation*}\n\t\t\\frac{1}{[x_{k+1}]_{n,q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{q}\\frac{q^{\\sum_{j=1}^{k}r_j(z_j-s_k+s_j-n+1)}}{\n\t\t\tq^{-\\sum_{j=1}^{k}(-n-s_k+s_j)(x_j-r_j)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,q}}{[x_1+\\cdots+x_{k+1}]_{n+s_k,q}},\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\\frac{1}{[x_{k+1}]_{n,q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{q}\\frac{q^{\\sum_{j=1}^{k}(n+s_k-s_j)(x_j-r_j)}}{\n\t\t\tq^{-\\sum_{j=1}^{k}r_j(-z_j+s_k-s_j+n-1)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,q}}{[x_1+\\cdots+x_{k+1}]_{n+s_k,q}}.\n\t\t\\end{equation*}\n\t\tFurthermore, the multivariate $q$ - Cauchy formula is furnished by:\n\t\t\\begin{equation*}\n\t\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}_{q}=\\sum_{r_j=0}^{n} q^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{-\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{q}\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}_{q}=\\sum_{r_j=0}^{n} q^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)} q^{-\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\,\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{q}.\n\t\t\\end{equation*}\n\t\tMoreover, the multivariate $q$ - Cauchy formula can be re-written as:\n\t\t\\begin{equation*}\n\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{q}=\\sum_{r_j=0}^{n}q^{\\sum_{j=1}^{k}x_j(n-s_j)} q^{-\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{q}\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{q}\n\t\t=\\sum_{r_j=0}^{n}q^{\\sum_{j=1}^{k}r_jz_j} q^{-\\sum_{j=1}^{k}x_j(n\\!-\\!s_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{q}.\n\t\t\\end{equation*}\n\t\tAnother expressions of the $q-$ deformed multivariate Cauchy formulae are furnished by :\n\t\t\\begin{equation*}\n\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{q}=\\sum_{r_j=0}^{n}q^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}q^{-\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{q}\n\t\t=\\sum_{r_j=0}^{n}q^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}q^{-\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}.\n\t\t\\end{equation*}\n\t\\end{small}\n\t\\item[(b)] \n\t\\begin{small}\n\tSetting $\\mathcal{R}(x,y)=\\frac{x-y}{p-q},$ we deduce the  multivariate Vandermonde and Cauchy formulae associated to the quantum algebra\\cite{JS}:\n\t\\begin{eqnarray*}\n\t[x_1+\\cdots+x_{k+1}]_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{p,q}\n\t\\mathcal{V}(p\\,q,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,p,q}\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t[x_1+\\cdots+x_{k+1}]_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{p,q}\n\t\\mathcal{V}(q\\,p,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,p,q},\n\t\\end{eqnarray*}\n\twhere $\\mathcal{V}(p\\,q,n)=p^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}.$ Moreover, \n\tthe negative multivariate  $(p,q)$ -Vandermonde formula is presented  by:\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]_{-n,p,q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}_{p,q}\\,\\mathcal{V}(p\\,q,-n)\n\t\t\t\\prod_{j=1}^{k+1}[x_j]_{r_j,p,q}\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]_{-n,p,q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}_{p,q}\\,\\mathcal{V}(q\\,p,-n)\n\t\t\t\\prod_{j=1}^{k+1}[x_j]_{r_j,p,q},\n\t\t\\end{eqnarray*}\n\twhere $\\mathcal{V}(p\\,q,-n)=p^{\\sum_{j=1}^{k}r_j(z_j-(-n-s_j))}\\,q^{\\sum_{j=1}^{k}(-n-s_j)(x_j-r_j)}.$ Also, \n\t\tthe multivariate $(p,q)-$  Vandermonde formula can be rewritten as:\n\t\t\t\\begin{equation*}\n\t\t\t\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{p,q}p^{\\sum_{j=1}^{k}x_j(n-s_j)}\n\t\t\t\tq^{\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,p,q},\n\t\t\t\\end{equation*}\n\t\t\tand\n\t\t\t\\begin{equation*}\n\t\t\t\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{p,q}p^{\\sum_{j=1}^{k}r_jz_j}\n\t\t\t\tq^{\\sum_{j=1}^{k}x_j(n-s_j)}\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,p,q},\n\t\t\t\\end{equation*}\n\tand \n\tthe multivariate inverse $(p,q)-$ Vandermonde formula as:\n\t\t\\begin{equation*}\n\t\t\t\\frac{1}{[x_{k+1}]_{n,p,q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{p,q}\\frac{p^{\\sum_{j=1}^{k}r_j(z_j-s_k+s_j-n+1)}}{\n\t\t\tq^{\\sum_{j=1}^{k}(-n-s_k+s_j)(x_j-r_j)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,p,q}}{[x_1+\\cdots+x_{k+1}]_{n+s_k,p,q}},\n\t\t\\end{equation*}\n\t\t and\n\t\t\\begin{equation*}\n\t\t\t\\frac{1}{[x_{k+1}]_{n,p,q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{p,q}\\frac{p^{\\sum_{j=1}^{k}(n+s_k-s_j)(x_j-r_j)}}{\n\t\t\tq^{\\sum_{j=1}^{k}r_j(-z_j+s_k-s_j+n-1)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,p,q}}{[x_1+\\cdots+x_{k+1}]_{n+s_k,p,q}}.\n\t\t\\end{equation*}\n\t\nFurthermore, the multivariate $(p,q)$ - Cauchy formula is furnished by:\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}_{p,q}=\\sum_{r_j=0}^{n} p^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{p,q}\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}_{p,q}=\\sum_{r_j=0}^{n} p^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)} q^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\,\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{p,q}.\n\t\t\\end{equation*}\nMoreover, the multivariate $(p,q)$ - Cauchy formula can be re-written as:\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{p,q}=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}x_j(n-s_j)} q^{\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{p,q}\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{p,q}\n\t\t\t=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}r_jz_j} q^{\\sum_{j=1}^{k}x_j(n\\!-\\!s_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{p,q}.\n\t\t\\end{equation*}\n\tAnother expressions of the $(p,q)-$ deformed multivariate Cauchy formulae are furnished by :\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{p,q}=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}q^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{p,q}\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{p,q}\n\t\t\t=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}q^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{p,q}.\n\t\t\\end{equation*}\n\t\\end{small}\n\t\\item[(c)]\n\t\\begin{small}\n\tSetting $\\mathcal{R}(x,y)=\\frac{1-xy}{(p^{-1}-q)x},$ we deduce the multivariate Vandermonde and Cauchy formulae related to the quantum algebra \\cite{CJ}:\n\t\\begin{eqnarray*}\n\t\t[x_1+\\cdots+x_{k+1}]_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{p^{-1},q}\n\t\t\\mathcal{V}(p\\,q,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,p^{-1},q}\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\t[x_1+\\cdots+x_{k+1}]_{n,p^{-1},q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}_{p^{-1},q}\n\t\t\\mathcal{V}(q\\,p,n)\\prod_{j=1}^{k+1}[x_j]_{r_j,p^{-1},q},\n\t\\end{eqnarray*}\n\twhere $\\mathcal{V}(p\\,q,n)=p^{-\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}.$ Moreover, \n\tthe negative multivariate  $(p^{-1},q)$ -Vandermonde formula is presented  by:\n\t\\begin{eqnarray*}\n\t\t[x_1+\\cdots+x_{k+1}]_{-n,p^{-1},q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}_{p^{-1},q}\\,\\mathcal{V}(p\\,q,-n)\n\t\t\\prod_{j=1}^{k+1}[x_j]_{r_j,p^{-1},q}\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\t[x_1+\\cdots+x_{k+1}]_{-n,p^{-1},q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}_{p^{-1},q}\\,\\mathcal{V}(q\\,p,-n)\n\t\t\\prod_{j=1}^{k+1}[x_j]_{r_j,p^{-1},q},\n\t\\end{eqnarray*}\n\twhere $\\mathcal{V}(p\\,q,-n)=p^{-\\sum_{j=1}^{k}r_j(z_j-(-n-s_j))}\\,q^{\\sum_{j=1}^{k}(-n-s_j)(x_j-r_j)}.$ Also, \n\tthe multivariate $(p^{-1},q)-$  Vandermonde formula can be rewritten as:\n\t\\begin{equation*}\n\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,p^{-1},q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{p^{-1},q}p^{-\\sum_{j=1}^{k}x_j(n-s_j)}\n\tq^{\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,p^{-1},q},\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]_{n,p^{-1},q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}_{p^{-1},q}p^{-\\sum_{j=1}^{k}r_jz_j}\n\tq^{\\sum_{j=1}^{k}x_j(n-s_j)}\\prod_{j=1}^{k+1}[x_j+r_j-1]_{r_j,p^{-1},q},\n\t\\end{equation*}\n\tand \n\tthe multivariate inverse $(p^{-1},q)-$ Vandermonde formula as:\n\t\\begin{equation*}\n\t\\frac{1}{[x_{k+1}]_{n,p^{-1},q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{p^{-1},q}\\frac{p^{-\\sum_{j=1}^{k}r_j(z_j-s_k+s_j-n+1)}}{\n\t\tq^{\\sum_{j=1}^{k}(-n-s_k+s_j)(x_j-r_j)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,p^{-1},q}}{[x_1+\\cdots+x_{k+1}]_{n+s_k,p^{-1},q}},\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\frac{1}{[x_{k+1}]_{n,p^{-1},q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}_{p^{-1},q}\\frac{p^{-\\sum_{j=1}^{k}(n+s_k-s_j)(x_j-r_j)}}{\n\t\tq^{\\sum_{j=1}^{k}r_j(-z_j+s_k-s_j+n-1)}}\\frac{\\prod_{j=1}^k[x_j]_{r_j,p^{-1},q}}{[x_1+\\cdots+x_{k+1}]_{n+s_k,p^{-1},q}}.\n\t\\end{equation*}\n\t\n\tFurthermore, the multivariate $(p^{-1},q)$ - Cauchy formula is furnished by:\n\t\\begin{equation*}\n\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}_{p^{-1},q}=\\sum_{r_j=0}^{n} p^{-\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{p^{-1},q}\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}_{p^{-1},q}=\\sum_{r_j=0}^{n} p^{-\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)} q^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\,\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}_{p^{-1},q}.\n\t\\end{equation*}\n\tMoreover, the multivariate $(p^{-1},q)$ - Cauchy formula can be re-written as:\n\t\\begin{equation*}\n\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{p^{-1},q}=\\sum_{r_j=0}^{n}p^{-\\sum_{j=1}^{k}x_j(n-s_j)} q^{\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{p^{-1},q}\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}_{p^{-1},q}\n\t=\\sum_{r_j=0}^{n}p^{-\\sum_{j=1}^{k}r_jz_j} q^{\\sum_{j=1}^{k}x_j(n\\!-\\!s_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_{p^{-1},q}.\n\t\\end{equation*}\n\tAnother expressions of the $(p^{-1},q)-$ deformed multivariate Cauchy formulae are furnished by :\n\t\\begin{equation*}\n\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{p^{-1},q}=\\sum_{r_j=0}^{n}p^{-\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}q^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{p^{-1},q}\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{p^{-1},q}\n\t=\\sum_{r_j=0}^{n}p^{-\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}q^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{p^{-1},q}.\n\t\\end{equation*}\n\\end{small}\n\t\\item[(d)]\n\tPutting $\\mathcal{R}(x,y)=\\frac{xy-1}{(q^{-1}-p)y},$ we derive the multivariate Vandermonde and Cauchy formulae related to the quantum algebra \\cite{Hounkonnou&Ngompe07a}:\n\t\\begin{eqnarray*}\n\t[x_1+\\cdots+x_{k+1}]^Q_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}^Q_{p,q}\n\t\\mathcal{V}(p\\,q,n)\\prod_{j=1}^{k+1}[x_j]^Q_{r_j,p,q}\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t[x_1+\\cdots+x_{k+1}]^Q_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\cdots,r_k}^Q_{p,q}\n\t\\mathcal{V}(q\\,p,n)\\prod_{j=1}^{k+1}[x_j]^Q_{r_j,p,q},\n\t\\end{eqnarray*}\n\twhere $\\mathcal{V}(p\\,q,n)=p^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{-\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}.$ It's  can be rewritten as:\n\t\t\\begin{small}\n\t\t\t\\begin{equation*}\n\t\t\t\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]^Q_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}^Q_{p,q}p^{\\sum_{j=1}^{k}x_j(n-s_j)}\n\t\t\t\tq^{-\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}[x_j+r_j-1]^Q_{r_j,p,q},\n\t\t\t\\end{equation*}\n\t\t\tand\n\t\t\t\\begin{equation*}\n\t\t\t\t\\Big[\\sum_{i=1}^{k+1}x_i+n-1\\Big]^Q_{n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\\ldots,r_k}^Q_{p,q}p^{\\sum_{j=1}^{k}r_jz_j}\n\t\t\t\tq^{-\\sum_{j=1}^{k}x_j(n-s_j)}\\prod_{j=1}^{k+1}[x_j+r_j-1]^Q_{r_j,p,q},\n\t\t\t\\end{equation*}\n\t\t\\end{small}\n\tFurthermore, \n\tthe corresponding  negative multivariate  Vandermonde formula is presented  by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]^Q_{-n,p,q}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}^Q_{p,q}\\,\\mathcal{V}(p\\,q,-n)\n\t\t\t\\prod_{j=1}^{k+1}[x_j]^Q_{r_j,p,q},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\t[x_1+\\cdots+x_{k+1}]^Q_{-n,p,q}=\\sum\\genfrac{[}{]}{0pt}{}{-n}{r_1,r_2,\\cdots,r_k}^Q_{p,q}\\,\\mathcal{V}(q\\,p,-n)\n\t\t\t\\prod_{j=1}^{k+1}[x_j]^Q_{r_j,p,q},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $\\mathcal{V}(p\\,q,-n)=p^{\\sum_{j=1}^{k}r_j(z_j-(-n-s_j))}\\,q^{\\sum_{j=1}^{k}(-n-s_j)(x_j-r_j)},$ and \n\tthe multivariate inverse  Vandermonde formula as:\n\t\\begin{small}\n\t\t\\begin{equation*}\n\t\t\t\\frac{1}{[x_{k+1}]^Q_{n,p,q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}^Q_{p,q}\\frac{p^{\\sum_{j=1}^{k}r_j(z_j-s_k+s_j-n+1)}}{\n\t\t\tq^{-\\sum_{j=1}^{k}(-n-s_k+s_j)(x_j-r_j)}}\\frac{\\prod_{j=1}^k[x_j]^Q_{r_j,p,q}}{[x_1+\\cdots+x_{k+1}]^Q_{n+s_k,p,q}},\n\t\t\\end{equation*}\n\t\t\\begin{equation*}\n\t\t\t\\frac{1}{[x_{k+1}]^Q_{n,p,q}}=\\sum_{r_j=0}^{n}\\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,\\ldots,r_k}^Q_{p,q}\\frac{p^{\\sum_{j=1}^{k}(n+s_k-s_j)(x_j-r_j)}}{\n\t\t\tq^{-\\sum_{j=1}^{k}r_j(-z_j+s_k-s_j+n-1)}}\\frac{\\prod_{j=1}^k[x_j]^Q_{r_j,p,q}}{[x_1+\\cdots+x_{k+1}]^Q_{n+s_k,p,q}}.\n\t\t\\end{equation*}\n\t\\end{small}\nMoreover, \n\t\tthe Hounkonnou-Ngompe generalized $q-$ Quesne multivariate Cauchy formula is furnished by:\n\t\\begin{small}\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}^Q_{p,q}=\\sum_{r_j=0}^{n} p^{\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}q^{-\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}^Q_{p,q},\n\t\t\\end{equation*}\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{x_1+x_2+\\cdots+x_{k+1}}{n}^Q_{p,q}=\\sum_{r_j=0}^{n} p^{\\sum_{j=1}^{k}(n-s_j)(x_j-r_j)} q^{-\\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\\,\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j}{r_j}^Q_{p,q}. \n\t\t\\end{equation*}\n\t\\end{small}\nIt's  can be re-written as:\n\t\\begin{small}\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}^Q_{p,q}=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}x_j(n-s_j)} q^{-\\sum_{j=1}^{k}r_jz_j}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}^Q_{p,q}\n\t\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{\\sum_{i=1}^{k+1}x_i+n-1}{n}^Q_{p,q}\n\t\t\t=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}r_jz_j} q^{-\\sum_{j=1}^{k}x_j(n\\!-\\!s_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}^Q_{p,q}.\n\t\t\\end{equation*}\n\t\\end{small}\n\tAnother expressions of the Hounkonnou-Ngompe generalized $q-$ Quesne  multivariate Cauchy formulae are furnished by :\n\t\\begin{small}\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}^Q_{p,q}=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}q^{-\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}^Q_{p,q}\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\genfrac{[}{]}{0pt}{}{r+k}{n+k}^Q_{p,q}\n\t\t\t=\\sum_{r_j=0}^{n}p^{\\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}q^{-\\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}^Q_{p,q}.\n\t\t\\end{equation*}\n\t\\end{small}\n\t\\end{enumerate}\n\\end{remark}\n\\section{Mutivariate probability distributions from $\\mathcal {R}(p,q)-$ deformed quantum algebras }\nIn this section, we construct some multivariate probability distributions ( P\\'olya, inverse P\\'olya, hypergeometric and negative hypergeometric) from the generalized quantum algebras \\cite{HB1}. The corresponding bivariate probability distributions and properties are also investigated.\n\\subsection{Multivariate $\\mathcal {R}(p,q)-$ deformed P\\'{o}lya distribution}\\label{sec3}\nWe denotes by $Y_\\mu$  the number of balls of color $c_\\mu$ drawn in $n$ $\\mathcal{R}(p,q)$-drawings in a multiple $\\mathcal{R}(p,q)$-P\\'olya urn model, with conditional probability of drawing a ball of color $c_\\mu$ at the $i$th $\\mathcal{R}(p,q)$-drawing, given that $j_\\mu-1$ balls of color $c_\\mu$ and a total of $i_{\\mu-1}$ balls of colors $c_1,c_2,\\cdots,c_{\\mu-1}$ are drawn in the previous $i-1$ $\\mathcal{R}(p,q)$-drawings, is given by \n\\begin{eqnarray}\\label{eq3.1}\np_{i,j_\\mu}(i_{\\mu-1})=\\frac{\\tau_2^{s_{\\mu-1}+mi_{\\mu-1}}[r_\\mu+m(j_\\mu-1)]_{\\mathcal\n\t\t{R}(p,q)}}{[r+m(i-1)]_{\\mathcal\n\t\t{R}(p,q)}}\n,\\quad \\mu\\in\\{1,2,\\cdots,k\\}\n\\end{eqnarray}\n\n The probability distribution of the $\\mathcal{R}(p,q)$- random vector $\\underline{Y}=\\big(Y_1,Y_2,\\cdots,Y_k\\big)$ may be called multivariate $\\mathcal{R}(p,q)$- P\\'olya distribution, with parameters $n,$ $(\\beta_1,\\beta_2,\\cdots,\\beta_k)$, $\\beta$, $p,$  and $q.$\n\\begin{theorem}\\label{thm3.1}\n\tThe mass function of the  multivariate  $\\mathcal{R}(p,q)$-P\\'olya  distribution, with parameters $n$, $(\\beta_1,\\beta_2,\\ldots,\\beta_k)$, $\\beta$, $p,$  and $q$, is presented as follows:\n\t\\begin{small}\n\t\t\\begin{eqnarray}\\label{eq3.2}\n\t\tP(\\underline{Y}=\\underline{y})=\\Psi_k(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1,\\ldots,y_k}_{\\mathcal{R}(p^{-m},q^{-m})}\\frac{\\prod_{j=1}^{k+1}[\\beta_j]_{y_j,\\mathcal{R}(p^{-m},q^{-m})}}{ [\\beta]_{n,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\t\\end{eqnarray}\n\t\\end{small}\n\twhere $\\Psi_k(p,q)=\\tau^{-m\\sum_{j=1}^{k}y_{j}(\\beta_{j+1}-y_{j+1})}_1\\tau_2^{-m\\sum_{j=1}^{k}(n-x_j)(\\beta_j-y_j)},$\n\t $y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$  $\\sum_{j=1}^k y_j\\leq n$, $y_{k+1}=n-\\sum_{j=1}^k y_j$, $\\beta_{k+1}=\\beta-\\sum_{j=1}^k \\beta_j$, and $y_j=\\sum_{i=1}^{j}y_i.$\n\\end{theorem}\n\\begin{proof}\n For the proof, we use the total probability theorem. Moreover, \n\t  the probabilities (\\ref{eq3.2}) sum to unity according to the  multivariate $\\mathcal\n\t{R}(p,q)$-Vandermonde formula (\\ref{eq2.7}) , the multivariate $\\mathcal\n\t{R}(p,q)$-Cauchy formula (\\ref{eq2.9}). \n\t $\\hfill{\\square}$\n\\end{proof} \n\\begin{remark}\nAnother relation of the multivariate P\\'olya distribution form generalized quantum algeras is interested.\n\t\\begin{enumerate}\n\t\t\\item[(i)]\n\t\tThe multivariate $\\mathcal{R}(p,q)$-P\\'{o}lya probability  distribution (\\ref{eq3.2}) can be rewritten in the form:\n\t\t\\begin{eqnarray}\\label{eq3.2bis}\n\t\t\tP(Y_1=y_1,\\ldots,Y_k=y_k)=\\frac{\\tau^{-m\\sum_{j=1}^{k}y_{j}(\\beta_{j+1}-x_{j+1})}_1}{\\tau_2^{m\\sum_{j=1}^{k}(n-x_j)(\\beta_j-y_j)}}\n\t\t\t\\frac{\\prod_{j=1}^{k+1} \\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal{R}(p^{-m},q^{-m})}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray}\n\t\t Indeed, it's derived  from \\eqref{eq3.2} according to  the relations\n\t\\begin{eqnarray*}\n\t\\genfrac{[}{]}{0pt}{}{n}{y_1,y_2,\\cdots,y_k}_{\\mathcal\n\t\t{R}(p,q)}=\\frac{[n]_{\\mathcal\n\t\t\t{R}(p,q)}!}{[y_1]_{\\mathcal\n\t\t\t{R}(p,q)}![y_2]_{\\mathcal\n\t\t\t{R}(p,q)}!\\cdots[y_k]_{\\mathcal\n\t\t\t{R}(p,q)}![y_{k+1}]_{\\mathcal\n\t\t\t{R}(p,q)}!}\n\t\\end{eqnarray*}\n\tand \n\t\\begin{eqnarray*}\n\t\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal\n\t\t{R}(p,q)}=\\frac{[\\beta_j]_{y_j,{\\mathcal\n\t\t\t\t{R}(p,q)}}}{[y_j]_{\\mathcal\n\t\t\t{R}(p,q)}!}.\n\t\\end{eqnarray*}\n\t\t\\item[(ii)] Taking $k=1,$ we obtain the $\\mathcal{R}(p,q)-$ P\\'olya distribution given in\\cite{HMD}.\n\t\\end{enumerate}\n\\end{remark}\n\nWe assume that the $\\mathcal{R}(p,q)$- random vector $\\big(Y_1,Y_2,\\ldots,Y_k\\big)$ satisfy the multivariate  $\\mathcal{R}(p,q)$- P\\'olya probability distribution with parameters $n,$ $(\\beta_1,\\beta_2,\\ldots,\\beta_k)$, $\\beta$, $p,$  and $q.$ Then:\n\\begin{theorem}\\label{thm3.2}\n\tFor $\\mu\\in\\{1,2,\\cdots,k\\},$ then, the  marginal distribution of the $\\mathcal{R}(p,q)$- random vector $(Y_1,Y_2,\\cdots,Y_\\mu)$ is a  $\\mathcal{R}(p,q)$-P\\'olya distribution of order $\\mu,$ with parameters $n,$ $(\\beta_1,\\beta_2,\\cdots,\\beta_\\mu)$, $\\beta$, $p,$ and $q.$ \n\t\n\tMoreover, for $\\mu\\in\\{1,2,\\cdots,k-t\\}$ and $t\\in\\{1,2,\\cdots,k-1\\},$  the conditional distribution of the $\\mathcal{R}(p,q)$-random vector$(Y_{\\mu+1},Y_{\\mu+2},\\ldots,Y_{\\mu+t})$, given that $(Y_1,Y_2,\\ldots,Y_{\\mu})=(y_1,y_2,\\cdots,y_{\\mu})$ is a  $\\mathcal{R}(p,q)$-P\\'olya probability distribution of order $t,$ with parameters $n$, $(\\beta_{\\mu+1},\\beta_{\\mu+2},\\cdots,\\beta_{\\mu+t})$, $\\beta-\\beta_1-\\beta_2-\\cdots-\\beta_{\\mu}$, $p,$ and $q.$ \n\\end{theorem}\n\\begin{proof} For\tthe proof, we use the probability \\eqref{eq3.2bis}. Summing it for $y_j\\in\\{0,1,\\cdots,\\linebreak n-x_{\\mu}\\},$ $j\\in\\{\\mu+1,\\mu+2,\\cdots,k\\},$ with\n$y_{\\mu+1}+y_{\\mu+2}+\\cdots+y_k\\leq n-x_\\mu$,  and according to the relation (\\ref{eq2.9}), we have:\n\\begin{eqnarray*}\nP(Y_1=y_1,Y_2=y_2,\\cdots,Y_{\\mu}=y_{\\mu})&=&\\Psi_k(p,q)\\prod_{j=1}^{\\mu}\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal{R}(p^{-m},q^{-m})}\n\\sum \\Phi_k(p,q)\\nonumber\\\\&\\times&\\prod_{j=\\mu+1}^{k}\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{_{\\mathcal{R}(p^{-m},q^{-m})}}\n\\frac{\\genfrac{[}{]}{0pt}{}{\\beta-\\beta_1-\\cdots-\\beta_k}{n-y_1-\\cdots-y_k}_{_{\\mathcal{R}(p^{-m},q^{-m})}}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{_{\\mathcal{R}(p^{-m},q^{-m})}}}\\nonumber\\\\\n&=&\\Psi_k(p,q)\\prod_{j=1}^{\\mu}\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{_{\\mathcal{R}(p^{-m},q^{-m})}}\n\\frac{\\genfrac{[}{]}{0pt}{}{\\beta-\\beta_1-\\cdots-\\beta_\\mu}{n-y_1-\\cdots-y_\\mu}_{\\mathcal{R}(p^{-m},q^{-m})}\n}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p^{-m},q^{-m})}},\n\\end{eqnarray*}\nwhich is the probability function of a  $\\mathcal{R}(p,q)$- P\\'{o}lya distribution of order $\\mu,$  with parameters $n,$ $(\\beta_1,\\beta_2,\\cdots,\\beta_\\mu)$, $\\beta$, $p,$ and $q.$\n\nFurthermore, according to the relation, \n\\begin{align*}\nP(Y_\\mu=y_\\mu,\\cdots,Y_{\\mu+k-1}&=y_{\\mu+k-1}|Y_1=y_1,Y_2=y_2,\\cdots,Y_{\\mu-1}=y_{\\mu-1})\\\\\n&=\\frac{P\\big(Y_1=y_1,Y_2=y_2,\\ldots,Y_{\\mu+k-1}=y_{\\mu+k-1})}{P(Y_1=y_1,Y_2=y_2,\\ldots,Y_{\\mu-1}=y_{\\mu-1}\\big)},\n\\end{align*}\nwe obtain,  the density function of the conditional distribution of the $\\mathcal{R}(p,q)$- random vector $(Y_{\\mu},Y_{\\mu+1},\\cdots,Y_{\\mu+k-1})$, given that $(Y_1,Y_2,\\cdots,Y_{\\mu-1})=(y_1,y_2,\\cdots,y_{\\mu-1}):$\n\\begin{align*}\nP(Y_\\mu&=y_\\mu,\\ldots,Y_{\\mu+k-1}=y_{\\mu+k-1}|Y_1=y_1,Y_2=y_2,\\ldots,Y_{\\mu-1}=y_{\\mu-1})\\\\\n&=\\frac{\\Psi_k(p,q)\\frac{\\prod_{j=1}^{\\mu+k-1}\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal{R}(p^{-m},q^{-m})}\n\t\\genfrac{[}{]}{0pt}{}{\\beta-\\beta_1-\\cdots-\\beta_{\\mu+k-1}}{n-y_1-\\cdots-y_{\\mu+k-1}}_{\\mathcal{R}(p^{-m},q^{-m})}}{\n\t{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p^{-m},q^{-m})}}}}\n\t{\\Psi_k(p,q)\\frac{\\prod_{j=1}^{\\mu-1}\n\t\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{\\beta-\\beta_1-\\cdots-\\beta_{\\mu-1}}{n-y_1-\\cdots-y_{\\mu-1}}_{\\mathcal{R}(p^{-m},q^{-m})}\n\t}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p^{-m},q^{-m})}}}\\\\\n&=\\Psi_k(p,q)\\frac{\\prod_{j=\\mu}^{\\mu+k-1}\\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal{R}(p^{-m},q^{-m})}\n\\genfrac{[}{]}{0pt}{}{\\beta-\\beta_\\mu-\\cdots-\\beta_{\\mu+k-1}}{n-y_\\mu-\\cdots-y_{\\mu+k-1}}_{\\mathcal{R}(p^{-m},q^{-m})}\n}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p^{-m},q^{-m})}},\n\\end{align*}\nwhich is the mass function of a  $\\mathcal{R}(p,q)$-P\\'olya distribution of order $k,$ with parameters $n,$ $(\\beta_\\mu,\\beta_{\\mu+1},\\ldots,\\beta_{\\mu+k-1})$, $\\beta-\\beta_1-\\beta_2-\\cdots-\\beta_{\\mu-1}$, $p,$ and $q$. $\\hfill{\\square}$\n\\end{proof}\n\n\\subsubsection{Bivariate $\\mathcal{R}(p,q)$- P\\'olya distribution}\nFor $k=2$ in the multiple $\\mathcal{R}(p,q)$- urn model, we obtain the following results: Then, we denote by $\\underline{Y}=\\big(Y_1,Y_2\\big)$\n the $\\mathcal{R}(p,q)$- random vector. Also, $\\underline{Y}$ follows the  bivariate $\\mathcal{R}(p,q)$- P\\'olya distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$ Its probability function is derived  by the following relation:\n\\begin{small}\n\t\\begin{equation*}\\label{bp1}\n\tP(Y_1=y_1,Y_2=y_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1,y_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,\\mathcal{R}(p^{-m},q^{-m})}}{ [\\beta]_{n,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{equation*}\n\tequivalently,\n\t\\begin{eqnarray*}\n\tP(Y_1=y_1,Y_2=y_2)=\\Psi_2(p,q)\n\t\\frac{\\prod_{j=1}^{3} \\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{\\mathcal{R}(p^{-m},q^{-m})}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\Psi_2(p,q)=\\tau^{-m\\sum_{j=1}^{2}y_{j}(\\beta_{j+1}-y_{j+1})}_1\\tau_2^{-m\\sum_{j=1}^{2}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2\\},$  $y_1+y_2\\leq n$, $y_{3}=n-y_1+y_2$, $\\beta_{3}=\\beta-\\beta_1-\\beta_2$, $x_1=y_1,$\nand $x_2=y_1+y_2.$\n\nThe properties of the bivariate $\\mathcal{R}(p,q)$- P\\'olya distribution  are presented as: \n\\begin{theorem}\n\tThe $\\mathcal{R}(p,q)$-factorial moments of the bivariate $\\mathcal{R}(p,q)$-P\\'olya probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ are given by:\n\t\\begin{small}\n\t\\begin{eqnarray}\\label{bp2}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}\\big)=\\frac{[n]_{i_1,\\mathcal{R}(p^m,q^m)}[\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray}\n\t\\begin{eqnarray}\\label{bp3}\n\tE\\big([Y_2]_{i_1,\\mathcal{R}(p^m,q^m)}|Y_1=y_1\\big)=\\frac{[n-y_1]_{i_2,\\mathcal{R}(p^m,q^m)}[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}},\\,i_2\\in\\{0,1,\\cdots,n-y_1\\},\n\t\\end{eqnarray}\n\t\\begin{eqnarray}\\label{bp4}\n\tE\\big([Y_2]_{i_2,\\mathcal{R}(p^m,q^m)}\\big)=\\frac{[n]_{i_2,\\mathcal{R}(p^m,q^m)}[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\tau^{-mi_2\\beta_1}_2}{[\\beta]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray}\n\tand\n\t\\begin{small}\n\t\\begin{eqnarray}\\label{bp5}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[Y_2]_{i_2,\\mathcal{R}(p^m,q^m)}\\big)=\\frac{[n]_{i_1+i_2,\\mathcal{R}(p^m,q^m)}[\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{mi_2\\beta_1}_2[\\beta]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray}\n\\end{small}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\n\\end{theorem}\n\\begin{proof}\nThe  $\\mathcal{R}(p,q)$- random variable  $Y_1$ follows  a $\\mathcal{R}(p,q)$-P\\'olya distribution, with mass function \\cite{HMD}:\n\\begin{eqnarray*}\n\tP(Y_1=y_1)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})}\\frac{[\\beta_1]_{y_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1]_{n-y_1,\\mathcal{R}(p^{-m},q^{-m})}}{ [\\beta]_{n,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray*}\n\twhere $\\Psi_2(p,q)=\\tau^{-my_{1}(\\beta-\\beta_1+y_1-n)}_1\\tau_2^{-m(n-y_1)(\\beta-y_1)}$ and $y_1\\in\\{0,1,\\cdots,n\\}.$\n\t\n\tThus, from \\cite{HMD}, the $\\mathcal{R}(p,q)$-factorial moments of the $\\mathcal{R}(p,q)$-random variable $Y_1$ are determined by \\eqref{bp2}. Moreover, the conditional distribution of the $\\mathcal{R}(p,q)$-random variable $Y_2,$ given that $Y_1=y_1,$ is a $\\mathcal{R}(p,q)$- P\\'olya distribution, with mass function\n\t\\begin{eqnarray*}\n\tP(Y_2=y_2|Y_1=y_1)&=&\\tau^{-my_{1}(\\beta-\\beta_1+y_1-n)}_1\\tau_2^{-m(n-y_1)(\\beta-y_1)}\\genfrac{[}{]}{0pt}{}{n-y_1}{y_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\frac{[\\beta_2]_{y_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2]_{n-y_1-y_2,\\mathcal{R}(p^{-m},q^{-m})}}{ [\\beta-\\beta_1]_{n-y_1,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray*}\n\twhere  $y_2\\in\\{0,1,\\cdots,n-y_1\\}.$ Once, using\\cite{HMD}, the conditional $\\mathcal{R}(p,q)$- factorial moments of $Y_2,$ given that $Y_1=y_1,$ are furnished by \\eqref{bp3}.\n\t\n\tNow, we compute the $\\mathcal{R}(p,q)$- factorial moments  $E\\big([Y_2]_{i_2, \\mathcal{R}(p^m,q^m)}\\big),\\, i_2\\in\\{0,1,\\cdots,n\\}.$\n\tSince, \n\t\\begin{eqnarray*}\n\tE\\big([Y_2]_{i_2, \\mathcal{R}(p^m,q^m)}\\big)&=&E\\bigg(E\\big([Y_2]_{i_2, \\mathcal{R}(p^{-m},q^{-m})}|Y_1\\big)\\bigg)\\nonumber\\\\\n\t&=&\\frac{[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}E\\big([n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)\n\t\\end{eqnarray*}\n\tand \n\t\\begin{eqnarray*}\n\tE\\big([n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&\\sum_{y_1=0}^{n-i_2}[n-y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\genfrac{[}{]}{0pt}{}{n}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})}\\Psi_2(p,q)\\nonumber\\\\&\\times&\\frac{[\\beta_1]_{y_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1]_{n-y_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\\end{eqnarray*}\n\tFrom the relations \n\t\\begin{eqnarray*}\n\t[n-y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2(n-\\beta_1)-m{i_2+1\\choose 2}}\\big(\\tau_1\\tau_2\\big)^{mi_2(\\beta_1-y_1)}[n-y_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t[n]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2\\,n-m{i_2+1\\choose 2}}[n]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t[n-y_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})}=[n]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n-i_2}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\twe obtain the folllowing expression\n\t\\begin{eqnarray*}\n\tE\\big([n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&\\frac{[n]_{i_2,\\mathcal{R}(p^{m},q^{m})}[n-\\beta_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{mi_2\\alpha_1}_2[\\beta]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&\\times&\\sum_{y_1=0}^{n-i_2}[n-y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n-i_2}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\frac{[\\beta_1]_{y_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-i_2]_{n-i_2-y_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-i_2]_{n-i_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\\end{eqnarray*}\n\tUsing the $\\mathcal{R}(p,q)$- Vandermonde formula \\cite{HMD},\\cite{HMRC}, we have:\n\t\\begin{eqnarray*}\n\tE\\big([n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&\\frac{[n]_{i_2,\\mathcal{R}(p^{m},q^{m})}[n-\\beta_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{mi_2\\alpha_1}_2[\\beta]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\\end{eqnarray*}\n\tThus, the relation \\eqref{bp4} follows.\n\t\n\tAnalougsly, we calculate the joint $\\mathcal{R}(p,q)$-factorial moments $E\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[Y_2]_{i_2,\\mathcal{R}(p^m,q^m)}\\big),$\n\twhere $i_1\\in\\{0,1,\\cdots,n\\}$ and  $i_2\\in\\{0,1,\\cdots,n-i_1\\}.$ They  can  computed according to the relation\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[Y_2]_{i_2,\\mathcal{R}(p^m,q^m)}\\big)&=&E\\bigg(E\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[Y_2]_{i_2,\\mathcal{R}(p^m,q^m)}|Y_1\\big)\\bigg)\\nonumber\\\\\n\t&=&\\frac{[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}E\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big).\n\t\\end{eqnarray*}\n\tSince\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&\\sum_{y_1=0}^{n-i_2}[y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[n-y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\genfrac{[}{]}{0pt}{}{n}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\Psi_2(p,q)\\frac{[\\beta_1]_{y_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1]_{n-y_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\\end{eqnarray*}\n\t\\end{small}\n\tand using the relations \n\t\\begin{small}\n\t\\begin{eqnarray*}\n\t[y_1]_{i_1,\\mathcal{R}(p^m,q^m)}=\\big(\\tau_1\\tau_2\\big)^{mi_1\\,y_1-m{i_1+1\\choose 2}}[y_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t[n-y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2(n-\\beta_1)-m{i_2+1\\choose 2}}\\big(\\tau_1\\tau_2\\big)^{mi_2(\\beta_1-y_1)}[n-y_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t[n]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2\\,n-m{i_2+1\\choose 2}}[n]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t[y_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[n-y_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n}{y_1}_{\\mathcal{R}(p^{-m},q^{-m})}=[n]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n-i_1-i_2}{y_1-i_1}_{\\mathcal{R}(p^{-m},q^{-m})},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t[n]_{i_1+i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{m(i_1+i_2)n-{i_1+i_2+1\\choose 2}}[n]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}\n\t\\end{eqnarray*}\n\t\\end{small}\n\twe have\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&\\frac{[n]_{i_1+i_2,\\mathcal{R}(p^{m},q^{m})}[\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[n-\\beta_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{mi_2\\alpha_1}_2[\\beta]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&\\times&\\sum_{y_1=0}^{n-i_2}[n-y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n-i_2-i_1}{y_1-i_1}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\frac{[\\beta_1-i_1]_{y_1-i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-i_2]_{n-i_2-y_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-i_2-i_1]_{n-i_2-i_1,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\\end{eqnarray*}\n\tFrom the $\\mathcal{R}(p,q)$- Vandermonde formula \\cite{HMD},\\cite{HMRC}, we get:\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)=\\frac{[n]_{i_1+i_2,\\mathcal{R}(p^{m},q^{m})}[\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[n-\\beta_1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{mi_2\\alpha_1}_2[\\beta]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\\end{eqnarray*}\n\tThus, \n\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[Y_2]_{i_2,\\mathcal{R}(p^m,q^m)}\\big)\n\t&=&\\frac{[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&\\times&E\\big([Y_1]_{i_1,\\mathcal{R}(p^m,q^m)}[n-Y_1]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)\\nonumber\\\\\n\t&=&\\frac{[n]_{i_1+i_2,\\mathcal{R}(p^{m},q^{m})}[\\beta_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{mi_2\\alpha_1}_2[\\beta]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\\end{eqnarray*}\t\n\tand the relation \\eqref{bp5} holds.\n\t\\end{small}\n\t$\\hfill{\\square}$\n\\end{proof}\n\\begin{corollary}\n The $\\mathcal{R}(p,q)$- covariance of $[Y_1]_{\\mathcal{R}(p^m,q^m)}$ and $[Y_2]_{\\mathcal{R}(p^m,q^m)}$ is given by the relation:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([Y_1]_{\\mathcal{R}(p^m,q^m)},[Y_2]_{\\mathcal{R}(p^m,q^m)}\\big)=\\frac{[n]_{\\mathcal{R}(p^m,q^m)}[\\beta_1]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{m\\beta_1}_2[\\beta]_{\\mathcal{R}(p^{-m},q^{-m})}}\\Delta(n,m,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,m,\\beta)=\\frac{[n-1]_{\\mathcal{R}(p^m,q^m)}}{[\\beta-1]_{\\mathcal{R}(p^{-m},q^{-m})}}-\\frac{[n]_{\\mathcal{R}(p^m,q^m)}}{[\\beta_1]_{\\mathcal{R}(p^{-m},q^{-m})}}\n\\end{eqnarray*}\nand  $\\underline{Y}=\\big(\nY_1,Y_2\\big)$  a $\\mathcal{R}(p,q)$-random vector verifying   the bivariate $\\mathcal{R}(p,q)$-P\\'olya  distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\\end{corollary}\n\\begin{proof}\nBy definition, we get:\n\\begin{eqnarray*}\nCov\\big([Y_1]_{\\mathcal{R}(p^m,q^m)},[Y_2]_{\\mathcal{R}(p^m,q^m)}\\big)&=&E\\big([Y_1]_{\\mathcal{R}(p^m,q^m)}[Y_2]_{\\mathcal{R}(p^m,q^m)}\\big)\\nonumber\\\\&-&E\\big([Y_1]_{\\mathcal{R}(p^m,q^m)}\\big)E\\big([Y_2]_{\\mathcal{R}(p^m,q^m)}\\big).\n\\end{eqnarray*}\nTaking $i_1=i_2=1,$ in the relations \\eqref{bp2}, \\eqref{bp4}, and \\eqref{bp5}, we obtain:\n\\begin{eqnarray*}\nE\\big([Y_1]_{\\mathcal{R}(p^m,q^m)}[Y_2]_{\\mathcal{R}(p^m,q^m)}\\big)&=&\\frac{[n]_{2,\\mathcal{R}(p^m,q^m)}[\\beta_1]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{m\\beta_1}_2\\,[\\beta]_{2,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\\n&=& \\frac{[n]_{\\mathcal{R}(p^m,q^m)}[n-1]_{\\mathcal{R}(p^m,q^m)}[\\beta_1]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{\\mathcal{R}(p^{-m},q^{-m})}}{\\tau^{m\\beta_1}_2\\,[\\beta]_{\\mathcal{R}(p^{-m},q^{-m})}\\,[\\beta-1]_{\\mathcal{R}(p^{-m},q^{-m})}}\n\\end{eqnarray*}\nand \n\\begin{eqnarray*}\nE\\big([Y_1]_{\\mathcal{R}(p^m,q^m)}\\big)E\\big([Y_2]_{\\mathcal{R}(p^m,q^m)}\\big)=\\frac{[n]^2_{\\mathcal{R}(p^m,q^m)}[\\beta_1]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{\\mathcal{R}(p^{-m},q^{-m})}\\tau^{-m\\beta_1}_2}{[\\beta]^2_{\\mathcal{R}(p^{-m},q^{-m})}}.\n\\end{eqnarray*}\nAfter computation, the proof is achieved.\n$\\hfill{\\square}$\n\\end{proof}\n\\begin{remark}\nThe multivariate P\\'olya probability distribution and properties generated by quantum algebras are presented in the sequel:\n\\begin{enumerate}\n\t\\item[(a)]The mass function of the  multivariate  $q$-P\\'olya  distribution, with parameters $n$, $(\\beta_1,\\beta_2,\\ldots,\\beta_k)$, $\\beta$,  and $q$, is presented as follows:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(\\underline{Y}=\\underline{y})=\\Psi_k(q)\\genfrac{[}{]}{0pt}{}{n}{y_1,\\ldots,y_k}_{q^{-m}}\\frac{\\prod_{j=1}^{k+1}[\\beta_j]_{y_j,q^{-m}}}{ [\\beta]_{n,q^{-m}}},\n\t\t\\end{eqnarray*}\n\tor\n\t\\begin{eqnarray*}\n\tP(Y_1=y_1,\\ldots,Y_k=y_k)=\\Psi_k(q)\n\t\\frac{\\prod_{j=1}^{k+1} \\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{q^{-m}}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{q^{-m}}},\n\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $\\Psi_k(q)=q^{-m\\sum_{j=1}^{k}y_{j}(\\beta_{j+1}-y_{j+1})}q^{m\\sum_{j=1}^{k}(n-x_j)(\\beta_j-y_j)},$\n\t$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$  $\\sum_{j=1}^k y_j\\leq n$, $y_{k+1}=n-\\sum_{j=1}^k y_j$, $\\beta_{k+1}=\\beta-\\sum_{j=1}^k \\beta_j$, and $x_j=\\sum_{i=1}^{j}y_i.$ Taking $k=2,$\n\tthe probability distribution of the $q$- random vector $\\underline{Y}=\\big(Y_1,Y_2\\big)$ is called the bivariate $q$- P\\'olya distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ and $q.$ Also, it's probability function is given by the following relation:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tP(Y_1=y_1,Y_2=y_2)=\\Psi_2(q)\\genfrac{[}{]}{0pt}{}{n}{y_1,y_2}_{q^{-m}}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,q^{-m}}}{ [\\beta]_{n,q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $\\Psi_2(q)=q^{-m\\sum_{j=1}^{2}y_{j}(\\beta_{j+1}-y_{j+1})}q^{m\\sum_{j=1}^{2}(n-x_j)(\\beta_j-y_j)},$\n\t$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $y_1+y_2\\leq n$, $y_{3}=n-y_1+y_2$, $\\beta_{3}=\\beta-\\beta_1-\\beta_2$, $x_1=y_1,$\n\tand $x_2=y_1+y_2.$\n\tBesides, its \n\t$q$-factorial moments  are given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([Y_1]_{i_1,q^m}\\big)=\\frac{[n]_{i_1,q^m}[\\beta_1]_{i_1,q^{-m}}}{[\\beta]_{i_1,q^{-m}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([Y_2]_{i_1,q^m}|Y_1=y_1\\big)=\\frac{[n-y_1]_{i_2,q^m}[\\beta_2]_{i_2,q^{-m}}}{[\\beta-\\beta_1]_{i_2,q^{-m}}},\\,i_2\\in\\{0,1,\\cdots,n-y_1\\},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([Y_2]_{i_2,q^m}\\big)=\\frac{[n]_{i_2,q^m}[\\beta_2]_{i_2,q^{-m}}q^{-mi_2\\beta_1}}{[\\beta]_{i_2,q^{-m}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([Y_1]_{i_1,q^m}[Y_2]_{i_2,q^m}\\big)=\\frac{[n]_{i_1+i_2,q^m}[\\beta_1]_{i_1,q^{-m}}[\\beta_2]_{i_2,q^{-m}}}{q^{-mi_2\\beta_1}\\,[\\beta]_{i_1+i_2,q^{-m})}},\n\t\t\\end{eqnarray*}\n\t\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\t\\end{small}\n\tMoreover,  the $q$- covariance of $[Y_1]_{q^m}$ and $[Y_2]_{q^m}$ is given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tCov\\big([Y_1]_{q^m},[Y_2]_{q^m}\\big)=\\frac{[n]_{q^m}[\\beta_1]_{q^{-m}}[\\beta_2]_{q^{-m}}}{q^{-m\\beta_1}[\\beta]_{q^{-m}}}\\Delta(n,m,\\beta),\n\t\t\\end{eqnarray*}\n\t\twhere \n\t\t\\begin{eqnarray*}\n\t\t\t\\Delta(n,m,\\beta)=\\frac{[n-1]_{q^m}}{[\\beta-1]_{q^{-m}}}-\\frac{[n]_{q^m}}{[\\beta_1]_{q^{-m}}}.\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\\item[(b)]The mass function of the  multivariate  $(p,q)$-P\\'olya  distribution, with parameters $n$, $(\\beta_1,\\beta_2,\\ldots,\\beta_k)$, $\\beta$, $p,$  and $q$, is presented as follows:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(\\underline{Y}=\\underline{y})=\\Psi_k(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1,\\ldots,y_k}_{p^{-m},q^{-m}}\\frac{\\prod_{j=1}^{k+1}[\\beta_j]_{y_j,p^{-m},q^{-m}}}{ [\\beta]_{n,p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\tor\n\t\\begin{eqnarray*}\n\t\tP(Y_1=y_1,\\ldots,Y_k=y_k)=\\Psi_k(p,q)\n\t\t\\frac{\\prod_{j=1}^{k+1} \\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{p^{-m},q^{-m}}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\Psi_k(p,q)=p^{-m\\sum_{j=1}^{k}y_{j}(\\beta_{j+1}-y_{j+1})}\\,q^{-m\\sum_{j=1}^{k}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$  $\\sum_{j=1}^k y_j\\leq n$, $y_{k+1}=n-\\sum_{j=1}^k y_j$, $\\beta_{k+1}=\\beta-\\sum_{j=1}^k \\beta_j$, and $x_j=\\sum_{i=1}^{j}y_i.$\n\nThe probability distribution of the $(p,q)$- random vector $\\underline{Y}=\\big(Y_1,Y_2\\big)$ is called the bivariate $(p,q)$- P\\'olya distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$ Also, it's probability function is given by the following relation:\n\\begin{small}\n\t\\begin{eqnarray*}\n\tP(Y_1=y_1,Y_2=y_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1,y_2}_{p^{-m},q^{-m}}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,p^{-m},q^{-m}}}{ [\\beta]_{n,p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\Psi_2(p,q)=p^{-m\\sum_{j=1}^{2}y_{j}(\\beta_{j+1}-y_{j+1})}q^{-m\\sum_{j=1}^{2}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $y_1+y_2\\leq n$, $y_{3}=n-y_1+y_2$, $\\beta_{3}=\\beta-\\beta_1-\\beta_2$, $x_1=y_1,$\nand $x_2=y_1+y_2.$\nBesides, its \n\t$(p,q)$-factorial moments  are given by:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,p^m,q^m}\\big)=\\frac{[n]_{i_1,p^m,q^m}[\\beta_1]_{i_1,p^{-m},q^{-m}}}{[\\beta]_{i_1,p^{-m},q^{-m}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([Y_2]_{i_1,p^m,q^m}|Y_1=y_1\\big)=\\frac{[n-y_1]_{i_2,p^m,q^m}[\\beta_2]_{i_2,p^{-m},q^{-m}}}{[\\beta-\\beta_1]_{i_2,p^{-m},q^{-m}}},\\,i_2\\in\\{0,1,\\cdots,n-y_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([Y_2]_{i_2,p^m,q^m}\\big)=\\frac{[n]_{i_2,p^m,q^m}[\\beta_2]_{i_2,p^{-m},q^{-m}}q^{-mi_2\\beta_1}}{[\\beta]_{i_2,p^{-m},q^{-m}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]_{i_1,p^m,q^m}[Y_2]_{i_2,p^m,q^m}\\big)=\\frac{[n]_{i_1+i_2,p^m,q^m}[\\beta_1]_{i_1,p^{-m},q^{-m}}[\\beta_2]_{i_2,p^{-m},q^{-m}}}{q^{mi_2\\beta_1}\\,[\\beta]_{i_1+i_2,p^{-m},q^{-m})}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nMoreover,  the $(p,q)$- covariance of $[Y_1]_{p^m,q^m}$ and $[Y_2]_{p^m,q^m}$ is given by:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([Y_1]_{p^m,q^m},[Y_2]_{p^m,q^m}\\big)=\\frac{[n]_{p^m,q^m}[\\beta_1]_{p^{-m},q^{-m}}[\\beta_2]_{p^{-m},q^{-m}}}{q^{m\\beta_1}[\\beta]_{p^{-m},q^{-m}}}\\Delta(n,m,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,m,\\beta)=\\frac{[n-1]_{p^m,q^m}}{[\\beta-1]_{p^{-m},q^{-m}}}-\\frac{[n]_{p^m,q^m}}{[\\beta_1]_{p^{-m},q^{-m}}}.\n\\end{eqnarray*}\n\\end{small}\n\\item[(c)]The mass function of the  multivariate  $(p^{-1},q)$-P\\'olya  distribution, with parameters $n$, $(\\beta_1,\\beta_2,\\ldots,\\beta_k)$, $\\beta$, $p,$  and $q$, is presented as follows:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(\\underline{Y}=\\underline{y})=\\Psi_k(p^{-1},q)\\genfrac{[}{]}{0pt}{}{n}{y_1,\\ldots,y_k}_{p^{m},q^{-m}}\\frac{\\prod_{j=1}^{k+1}[\\beta_j]_{y_j,p^{m},q^{-m}}}{ [\\beta]_{n,p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\tor\n\t\\begin{eqnarray*}\n\t\tP(Y_1=y_1,\\ldots,Y_k=y_k)=\\Psi_k(p^{-1},q)\n\t\t\\frac{\\prod_{j=1}^{k+1} \\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}_{p^{m},q^{-m}}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}_{p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\Psi_k(p^{-1},q)=p^{m\\sum_{j=1}^{k}y_{j}(\\beta_{j+1}-y_{j+1})}\\,q^{-m\\sum_{j=1}^{k}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$  $\\sum_{j=1}^k y_j\\leq n$, $y_{k+1}=n-\\sum_{j=1}^k y_j$, $\\beta_{k+1}=\\beta-\\sum_{j=1}^k \\beta_j$, and $x_j=\\sum_{i=1}^{j}y_i.$\n\nThe probability distribution of the $(p^{-1},q)$- random vector $\\underline{Y}=\\big(Y_1,Y_2\\big)$ is called the bivariate $(p^{-1},q)$- P\\'olya distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$ Also, it's probability function is given by the following relation:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(Y_1=y_1,Y_2=y_2)=\\Psi_2(p^{-1},q)\\genfrac{[}{]}{0pt}{}{n}{y_1,y_2}_{p^{-m},q^{-m}}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,p^{m},q^{-m}}}{ [\\beta]_{n,p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\Psi_2(p^{-1},q)=p^{m\\sum_{j=1}^{2}y_{j}(\\beta_{j+1}-y_{j+1})}q^{-m\\sum_{j=1}^{2}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $y_1+y_2\\leq n$, $y_{3}=n-y_1+y_2$, $\\beta_{3}=\\beta-\\beta_1-\\beta_2$, $x_1=y_1,$\nand $x_2=y_1+y_2.$\nBesides, its \n$(p^{-1},q)$-factorial moments  are given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tE\\big([Y_1]_{i_1,p^{-m},q^m}\\big)=\\frac{[n]_{i_1,p^{-m},q^m}[\\beta_1]_{i_1,p^{m},q^{-m}}}{[\\beta]_{i_1,p^{m},q^{-m}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([Y_2]_{i_1,p^{-m},q^m}|Y_1=y_1\\big)=\\frac{[n-y_1]_{i_2,p^{-m},q^m}[\\beta_2]_{i_2,p^{m},q^{-m}}}{[\\beta-\\beta_1]_{i_2,p^{m},q^{-m}}},\\,i_2\\in\\{0,1,\\cdots,n-y_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([Y_2]_{i_2,p^{-m},q^m}\\big)=\\frac{[n]_{i_2,p^{-m},q^m}[\\beta_2]_{i_2,p^{m},q^{-m}}q^{-mi_2\\beta_1}}{[\\beta]_{i_2,p^{m},q^{-m}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\tE\\big([Y_1]_{i_1,p^{-m},q^m}[Y_2]_{i_2,p^{-m},q^m}\\big)=\\frac{[n]_{i_1+i_2,p^{-m},q^m}[\\beta_1]_{i_1,p^{m},q^{-m}}[\\beta_2]_{i_2,p^{m},q^{-m}}}{q^{mi_2\\beta_1}\\,[\\beta]_{i_1+i_2,p^{m},q^{-m})}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nMoreover,  the $(p^{-1},q)$- covariance of $[Y_1]_{p^{-m},q^m}$ and $[Y_2]_{p^{-m},q^m}$ is given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tCov\\big([Y_1]_{p^{-m},q^m},[Y_2]_{p^{-m},q^m}\\big)=\\frac{[n]_{p^{-m},q^m}[\\beta_1]_{p^{m},q^{-m}}[\\beta_2]_{p^{m},q^{-m}}}{q^{m\\beta_1}[\\beta]_{p^{m},q^{-m}}}\\Delta(n,m,\\beta),\n\t\\end{eqnarray*}\n\twhere \n\t\\begin{eqnarray*}\n\t\t\\Delta(n,m,\\beta)=\\frac{[n-1]_{p^{-m},q^m}}{[\\beta-1]_{p^{m},q^{-m}}}-\\frac{[n]_{p^{-m},q^m}}{[\\beta_1]_{p^{m},q^{-m}}}.\n\t\\end{eqnarray*}\n\\end{small}\n\\item[(d)]The mass function of the  multivariate  Hounkonnou-Ngompe generalized $q$- Quesne P\\'olya  distribution, with parameters $n$, $(\\beta_1,\\beta_2,\\ldots,\\beta_k)$, $\\beta$, $p,$  and $q$, is presented as follows:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(\\underline{Y}=\\underline{y})=\\Psi_k(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1,\\ldots,y_k}^Q_{p^{-m},q^{-m}}\\frac{\\prod_{j=1}^{k+1}[\\beta_j]^Q_{y_j, p^{-m},q^{-m}}}{ [\\beta]^Q_{n, p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\tor\n\t\\begin{eqnarray*}\n\t\tP(Y_1=y_1,\\ldots,Y_k=y_k)=\\Psi_k(p,q)\n\t\t\\frac{\\prod_{j=1}^{k+1} \\genfrac{[}{]}{0pt}{}{\\beta_j}{y_j}^Q_{p^{-m},q^{-m}}}{\\genfrac{[}{]}{0pt}{}{\\beta}{n}^Q_{p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\Psi_k(p,q)=p^{-m\\sum_{j=1}^{k}y_{j}(\\beta_{j+1}-y_{j+1})}\\,q^{m\\sum_{j=1}^{k}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$  $\\sum_{j=1}^k y_j\\leq n$, $y_{k+1}=n-\\sum_{j=1}^k y_j$, $\\beta_{k+1}=\\beta-\\sum_{j=1}^k \\beta_j$, and $x_j=\\sum_{i=1}^{j}y_i.$\n\nThe probability function of the bivariate   Hounkonnou-Ngompe generalized $q$- Quesne P\\'olya distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$  is given by:\n\\begin{small}\n\t\\begin{equation*}\n\tP(Y_1=y_1,Y_2=y_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{y_1,y_2}^Q_{p^{-m},q^{-m}}\\frac{\\prod_{j=1}^{3}[\\beta_j]^Q_{y_j,p^{-m},q^{-m}}}{ [\\beta]^Q_{n,p^{-m},q^{-m}}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p,q)=p^{-m\\sum_{j=1}^{2}y_{j}(\\beta_{j+1}-y_{j+1})}q^{m\\sum_{j=1}^{2}(n-x_j)(\\beta_j-y_j)},$\n$y_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $y_1+y_2\\leq n$, $y_{3}=n-y_1+y_2$, $\\beta_{3}=\\beta-\\beta_1-\\beta_2$, $x_1=y_1,$\nand $x_2=y_1+y_2.$\nBesides, its \n\trelated factorial moments  are:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]^Q_{i_1,p^m,q^m}\\big)=\\frac{[n]^Q_{i_1,p^m,q^m}[\\beta_1]^Q_{i_1,p^{-m},q^{-m}}}{[\\beta]^Q_{i_1,p^{-m},q^{-m}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([Y_2]^Q_{i_1,p^m,q^m}|Y_1=y_1\\big)=\\frac{[n-y_1]^Q_{i_2,p^m,q^m}[\\beta_2]^Q_{i_2,p^{-m},q^{-m}}}{[\\beta-\\beta_1]^Q_{i_2,p^{-m},q^{-m}}},\\,i_2\\in\\{0,1,\\cdots,n-y_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([Y_2]^Q_{i_2,p^m,q^m}\\big)=\\frac{[n]^Q_{i_2,p^m,q^m}[\\beta_2]^Q_{i_2,p^{-m},q^{-m}}q^{mi_2\\beta_1}}{[\\beta]^Q_{i_2,p^{-m},q^{-m}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([Y_1]^Q_{i_1,p^m,q^m}[Y_2]^Q_{i_2,p^m,q^m}\\big)=\\frac{[n]^Q_{i_1+i_2,p^m,q^m}[\\beta_1]^Q_{i_1,p^{-m},q^{-m}}[\\beta_2]^Q_{i_2,p^{-m},q^{-m}}}{q^{-mi_2\\beta_1}\\,[\\beta]^Q_{i_1+i_2,p^{-m},q^{-m})}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nMoreover,  the  covariance of the random variables  $[Y_1]^Q_{p^m,q^m}$ and $[Y_2]^Q_{p^m,q^m}$ is deduced as :\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([Y_1]^Q_{p^m,q^m},[Y_2]^Q_{p^m,q^m}\\big)=\\frac{[n]^Q_{p^m,q^m}[\\beta_1]^Q_{p^{-m},q^{-m}}[\\beta_2]^Q_{p^{-m},q^{-m}}}{q^{-m\\beta_1}[\\beta]^Q_{p^{-m},q^{-m}}}\\Delta^Q(n,m,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta^Q(n,m,\\beta)=\\frac{[n-1]^Q_{p^m,q^m}}{[\\beta-1]^Q_{p^{-m},q^{-m}}}-\\frac{[n]^Q_{p^m,q^m}}{[\\beta_1]^Q_{p^{-m},q^{-m}}}.\n\\end{eqnarray*}\n\\end{small}\n\\end{enumerate}\n\\end{remark}\n\\subsection{Multivariate inverse $\\mathcal{R}(p,q)$- P\\'olya distribution}\\label{sec4}\nLet $W_\\nu$ be the number of balls of color $c_\\nu$ drawn until the $n$th ball of color $c_{k+1}$ is drawn in a multiple $\\mathcal{R}(p,q)$-P\\'{o}lya urn model, with the conditional probability of drawing a ball of color $c_\\nu$ at the $i$th $\\mathcal{R}(p,q)$-drawing, given that $j_\\nu-1$ balls of color $c_\\nu$ and a total of $i_{\\nu-1}$ balls of colors $c_1,c_2,\\ldots,c_{\\nu-1}$ are drawn in the previous $i-1$ $\\mathcal{R}(p,q)$-drawings, given by (\\ref{eq3.1}), for $\\nu\\in\\{1,2,\\cdots,k\\}.$ \n\\begin{theorem}\\label{thm4.1}\n\tThe probability function of the multivariate inverse $\\mathcal{R}(p,q)-$ deformed P\\'{o}lya distribution, with parameters $n$, $\\varTheta$ $(\\vartheta_1,\\vartheta_2,\\cdots,\\vartheta_k)$,   $p,$ and $q$, is given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray}\\label{eq4.1}\n\t\tP(W_1=w_1,\\ldots,W_k=w_k)\n\t\t&=&F_k(p,q)\\genfrac{[}{]}{0pt}{}{n+w_k-1}{w_1,w_2,\\cdots, w_k}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^k[\\beta_j]_{w_j,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_{k+1}]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta]_{n+w_k,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\t\\end{eqnarray}\n\t\\end{small}\n\twhere $F_k(p,q)=\\tau_2^{-m\\sum_{j=1}^{k}(n+w_k-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2,\\cdots,k\\},$ \n\n\t$\\beta_{k+1}=\\beta-\\sum_{j=1}^k\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2,\\cdots,k\\}.$\n\\end{theorem}\n\\begin{proof} The probability function of the $k$-variate inverse $\\mathcal{R}(p,q)$-P\\'{o}lya distribution is  connected to the probability function $k$-variate $\\mathcal{R}(p,q)$-P\\'{o}lya distribution. Specifically,\n\\[\nP(W_1=w_1,W_2=w_2,\\cdots,W_k=w_k)=p_{n+u_k-1}(w_1,w_2,\\cdots, w_k)p_{n+u_k,n},\n\\]\nwhere $p_{n+u_k-1}(w_1,w_2,\\cdots, w_k)$ is the probability of drawing $w_\\nu$ balls of color $c_\\nu$, for all $\\nu\\in\\{1,2,\\cdots,k\\},$ and $n-1$ balls of color $c_{k+1}$ in $n+w_k-1$ $\\mathcal{R}(p,q)$-drawings and $p_{n+w_k,n}=q^{-m(\\beta_k-w_k)}[a_{k+1}-n+1]_{\\mathcal{R}(p^{-m},q^{-m})}\/[a-n-w_k+1]_{\\mathcal{R}(p^{-m},q^{-m})}$ is the conditional probability of drawing a ball of color $c_{k+1}$ at the $(n+w_k)$th $q$-drawing, given that $n-1$ balls of color $c_{k+1}$ and a total of $w_k$ balls of colors $c_1,c_2,\\ldots,c_k$ are drawn in the previous $n+w_k-1$ $q$-drawings. Thus using (\\ref{eq3.2}), expression (\\ref{eq4.1}) is deduced. Note that the multivariate inverse $\\mathcal{R}(p,q)$-Vandermonde formula (\\ref{eq2.11}) guarantees that the probabilities (\\ref{eq4.1}) sum to unity.\n$\\hfill{\\square}$\n\\end{proof}\n\n\\subsubsection{Bivariate inverse $\\mathcal{R}(p,q)-$ P\\'{o}lya distribution}\nThe mass function of the bivariate  inverse $\\mathcal{R}(p,q)-$ deformed P\\'olya distribution, with parameters $n$,  $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is derived as :\n\t\\begin{small}\n\t\t\\begin{eqnarray}\\label{bip1}\n\t\tP(W_1=w_1,W_2=w_2)\n\t\t&=&F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_1,w_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^2[\\beta_j]_{w_j,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_{3}]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\t\\end{eqnarray}\n\t\\end{small}\n\twhere $F_2(p,q)=\\tau_2^{-m\\sum_{j=1}^{2}(n+w_2-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2\\},$ \n\n\t$\\beta_{3}=\\beta-\\sum_{j=1}^2\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2\\}.$\n\t\\begin{theorem}\nFor $i_1\\in\\mathbb{N}\\cup\\{0\\}$ and $i_2\\in\\mathbb{N}\\cup\\{0\\},$ \tthe $\\mathcal{R}(p,q)$-factorial moments of the bivariate inverse $\\mathcal{R}(p,q)$-P\\'olya probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ are given by:\n\t\\begin{small}\n\t\\begin{eqnarray}\\label{bip2}\n\tE\\big([W_2]_{i_2,\\mathcal{R}(p^m,q^m)}\\big)=\\frac{[n+i_2-1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray}\n\t\\begin{eqnarray}\\label{bip3}\n\tE\\big([W_1]_{i_1,\\mathcal{R}(p^m,q^m)}|W_2=w_2\\big)=\\frac{[n+w_2+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1+i_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray}\n\t\\begin{eqnarray}\\label{bip4}\n\tE\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^m,q^m)}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)=\\frac{[n+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{\\Theta(n,m,\\beta)},\n\t\\end{eqnarray}\n\t and \n\t\\begin{eqnarray}\\label{bip5}\n\tE\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^m,q^m)}[W_2]_{i_2,\\mathcal{R}(p^m,q^m)}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)&=&\\frac{[n+i_1+i_2-1]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}}{\\Theta(n,m,\\beta)}\\nonumber\\\\&\\times& \\frac{[\\beta_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}},\n\t\\end{eqnarray}\n\twhere \\begin{eqnarray*}\n\t\\Theta(n,m,\\beta)=[\\beta-\\beta_1+i_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{m},q^{m})}.\n\\end{eqnarray*}\t \n\t\\end{small}\n\t\\end{theorem}\n\t\\begin{proof}\nThe marginal distribution of the $\\mathcal{R}(p,q)$- random variable is an inverse $\\mathcal{R}(p,q)$-P\\'olya distribution, with probability function:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(W_1=w_1)\n\t\t&=&F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{[\\beta_2]_{w_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_{2}]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\tThe $\\mathcal{R}(p,q)$- factorial moments $E\\big([W_2]_{i_2,\\mathcal{R}(p,q)}\\big),\\,i_2\\in\\mathbb{N}\\cup\\{0\\}$ are given by:\n\t\t\\begin{eqnarray}\\label{bip6}\n\t\tE\\big([W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&\\sum_{w_2=i_2}^{\\infty}[w_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{[\\beta_2]_{w_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_{2}]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray}\n\t\tUsing the relations\n\t\t\\begin{eqnarray*}\n\t\t[w_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}=[n+i_2-1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2-i_2}_{\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[w_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2\\,w_2-m{i_2+1\\choose 2}}\\,[w_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2\\,\\beta_2-m{i_2+1\\choose 2}}\\,[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t-mn(\\beta_2-w_2)+mi_2\\,w_2-mi_2\\,\\beta_2=-m(n+i_2)(\\beta_2-w_2),\n\t\t\\end{eqnarray*}\n\t\tthe relation \\eqref{bip6} can be rewritten as:\n\t\t\\begin{eqnarray*}\n\t\tE\\big([W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)&=&[n+i_2-1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\sum_{w_2=i_2}^{\\infty}F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2-i_2}_{\\mathcal{R}(p^{-m},q^{-m})}\n\t\t\\frac{[\\beta_2-i_2]_{w_2-i_2,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\tFrom the negative $\\mathcal{R}(p,q)$- Vandermonde formula \\cite{HMD},\\cite{HMRC}, we have \n\t\t\\begin{eqnarray*}\n\t\tE\\big([W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}\\big)=\\frac{[n+i_2-1]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-\\beta_2+i_2]_{n+i_2,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t\\end{eqnarray*}\n\t\tand the relation \\eqref{bip2} follows by using \n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-\\beta_2+i_2]_{n+i_2,\\mathcal{R}(p^{-m},q^{-m})}=[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}.\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\tFurthermore, the conditional distribution $W_1,$ given that $W_2=w_2,$ is an inverse $\\mathcal{R}(p,q)$-P\\'olya probability distribution, with mass function:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(W_1=w_1|W_1=w_1)\n\t\t&=&F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2+w_1-1}{w_1}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{[\\beta_1]_{w_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta]_{n+w_2+w_1,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\tFrom the same procedure using to obtain \\eqref{bip2}, the relation \\eqref{bip3} holds.  Moreover, the expected value of the $\\mathcal{R}(p,q)$-function of $\\underline{W}=\\big(W_1,W_2\\big)$ \n\t\t\\begin{eqnarray*}\n\t\t\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\n\t\t\\end{eqnarray*}\n\t\tcan be computed by using\n\t\t\\begin{eqnarray*}\n\t\t E\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)=E\\bigg(E\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}|W_2\\bigg)\\bigg).\n\t\t\\end{eqnarray*}\n\t\tSince \n\t\t\\begin{eqnarray*}\n\t\tE\\bigg(\\frac{[n+i_1+W_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)&=& \\sum_{w_2=0}^{\\infty}\\frac{[n+i_1+w_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}F_2(p,q)\\nonumber\\\\&\\times&\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{[\\beta_2]_{w_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_{2}]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t\\end{eqnarray*}\n\t\tand from the relations\n\t\t\t\\begin{eqnarray*}\n\t\t[n+i_1+w_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}&=&[n+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\genfrac{[}{]}{0pt}{}{n+i_1+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}=\\frac{[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{mi_1(\\beta-\\beta_1-n-w_2)-m{i_2+1\\choose 2}}},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1]_{n+i_1+w_2,\\mathcal{R}(p^{-m},q^{-m})}=[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\twith the  negative $\\mathcal{R}(p,q)$- Vandermonde formula \\cite{HMD},\\cite{HMRC}, we obtain:\n\t\t\\begin{eqnarray*}\n\t\tE\\bigg(\\frac{[n+i_1+W_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)&=& \\frac{[n+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{-mi_1(\\beta-\\beta_1-n-\\beta_2)+m{i_2+1\\choose 2}}}\\nonumber\\\\&\\times&\\sum_{w_2=0}^{\\infty}F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+i_2+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{[\\beta_2]_{w_2,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta-\\beta_1]_{n+i_1+w_2,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&=&\\frac{[n+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{-mi_1(\\beta-\\beta_1-n-\\beta_2)+m{i_2+1\\choose 2}}}\\nonumber\\\\&\\times&\\frac{[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-\\beta_2]_{n+i_1,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\tAlso, according to \n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-\\beta_2]_{n+i_1,\\mathcal{R}(p^{-m},q^{-m})}=[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}\n\t\t\\end{eqnarray*}\n\t\tand \n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{m},q^{m})}=\\frac{[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{mi_1(\\beta-\\beta_1-\\beta_2-n)-m{i_2+1\\choose 2}}},\n\t\t\\end{eqnarray*}\n\t\twe get\n\t\t\\begin{eqnarray*}\n\t\tE\\bigg(\\frac{[n+i_1+W_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)= \\frac{[n+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{m},q^{m})}}.\n\t\t\\end{eqnarray*}\n\t\tThen, \n\t\t\\begin{eqnarray*}\n\t\tE\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)&=&\\frac{[\\beta_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1+i_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&\\times&E\\bigg(\\frac{[n+i_1+W_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)\\nonumber\\\\\n\t\t&=&  \\frac{[n+i_1-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{m},q^{m})}[\\beta-\\beta_1+i_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\t\\end{small}\n\t\tIn addition, we compute the mean of the $\\mathcal{R}(p,q)$- function\n\t\t\\begin{eqnarray*}\n\t\t\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}[W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}.\n\\end{eqnarray*}\t\nFor instance, we use the relation\n\\begin{small}\n\\begin{eqnarray*}\n\t  E\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}[W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)&=&E\\bigg(E\\bigg(\\frac{[W_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}[W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}|W_2\\bigg)\\bigg)\\nonumber\\\\\n\t  &=&\\frac{[\\beta_1]_{i_1,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1+i_1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&\\times&E\\bigg(\\frac{[W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}[n+i_1+W_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)\n\t  \\end{eqnarray*}\n\t  Since\n\t  \\begin{eqnarray*}\n\t  E\\big(\\overline{W}\\big)&:=& E\\bigg(\\frac{[W_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}[n+i_1+W_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-W_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}\\bigg)\\nonumber\\\\\n\t  &=& \\sum_{w_2=0}^{\\infty}\\frac{[w_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}[n+i_1+w_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}}F_2(p,q)\\nonumber\\\\&\\times&\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}\n\t\t\\frac{[\\beta_2]_{w_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_{2}]_{n,\\mathcal{R}(p^{-m},q^{-m})}}\n\t\t{[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t  \\end{eqnarray*}\n\t  Using the relations\n\t  \\begin{eqnarray*}\n\t\t[w_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}[n+i_1+w_2-1]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_2}_{\\mathcal{R}(p^{-m},q^{-m})}&=&[n+i_1+i_2-1]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\\genfrac{[}{]}{0pt}{}{n+i_1+w_2-1}{w_2-i_2}_{\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[w_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2\\,w_2-m{i_2+1\\choose 2}}\\,[w_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}=\\big(\\tau_1\\tau_2\\big)^{mi_2\\,\\beta_2-m{i_2+1\\choose 2}}\\,[\\beta_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{m},q^{m})}=\\frac{[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{mi_1(\\beta-\\beta_1-n-\\beta_2)-m{i_2+1\\choose 2}}},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1]_{n+i_1+w_2,\\mathcal{R}(p^{-m},q^{-m})}=[\\beta-\\beta_1]_{n+w_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-n-w_2]_{i_1,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\twith the  negative $\\mathcal{R}(p,q)$- Vandermonde formula \\cite{HMD},\\cite{HMRC}, we obtain:\n\t\t\\begin{eqnarray*}\n\tE\\big(\\overline{W}\\big)&=& \\frac{[n+i_1+i_2-1]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{-mi_1(\\beta-\\beta_1-n-\\beta_2)+m{i_2+1\\choose 2}}}\\nonumber\\\\&\\times&\\sum_{w_2=0}^{\\infty}F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+i_1+w_2-1}{w_2-i_2}_{\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&\n\t\t\\frac{[\\beta_2-i_2]_{w_2-i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}\n\t\t{[\\beta-\\beta_1]_{n+i_1+w_2,\\mathcal{R}(p^{-m},q^{-m})}}\\nonumber\\\\&=&\\frac{[n+i_1+i_2-1]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{\\big(\\tau_1\\tau_2\\big)^{-mi_1(\\beta-\\beta_1-n-\\beta_2)+m{i_2+1\\choose 2}}[\\beta-\\beta_1-\\beta_2+i_2]_{n+i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\tFurthermore, using \n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-\\beta_2+i_2]_{n+i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}&=&[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}\\nonumber\\\\&\\times&[\\beta-\\beta_1-\\beta_2]_{n+i_1,\\mathcal{R}(p^{-m},q^{-m})},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-\\beta_2]_{n+i_1,\\mathcal{R}(p^{-m},q^{-m})}=[\\beta-\\beta_1-\\beta_2]_{n,\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}.\n\t\t\\end{eqnarray*}\n\t\tand \n\t\t\\begin{eqnarray*}\n\t\t[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{m},q^{m})}=\\frac{[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{-m},q^{-m})}}{\\big(\\tau_1\\tau_2\\big)^{mi_1(\\beta-\\beta_1-\\beta_2-n)-m{i_2+1\\choose 2}}},\n\t\t\\end{eqnarray*}\n\t\twe obtain\n\t\t\\begin{eqnarray*}\n\tE\\big(\\overline{W}\\big)= \\frac{[n+i_1+i_2-1]_{i_1+i_2,\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{i_2,\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-\\beta_2-n]_{i_1,\\mathcal{R}(p^{m},q^{m})}[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,\\mathcal{R}(p^{-m},q^{-m})}}.\n\t\t\\end{eqnarray*}\n\t\tThen, the relation \\eqref{bip5} follows and the proof is achieved.\n\t\t\\end{small}\n\t$\\hfill{\\square}$\n\t\\end{proof}\n\t\\begin{corollary}\n\tThe $\\mathcal{R}(p,q)$- covariance of the $\\mathcal{R}(p,q)$- random variable $\\ddot{W}=[W_2]_{\\mathcal{R}(p^{m},q^{m})}$  and $\\widehat{W}=\\frac{[W_1]_{\\mathcal{R}(p^{m},q^{m})}}{[\\beta-\\beta_1-n-W_2]_{\\mathcal{R}(p^{m},q^{m})}}$  is given by:\n\t\\begin{small}\n\t\\begin{eqnarray}\\label{bip7}\n\tCov\\big(\\widehat{W}, \\ddot{W}\\big)&=&\\frac{[n]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{\\mathcal{R}(p^{m},q^{m})}[\\beta_1]_{\\mathcal{R}(p^{m},q^{m})}}{\\nabla(n,\\beta)}\\nonumber\\\\&\\times&\\bigg([n+1]_{\\mathcal{R}(p^{-m},q^{-m})}-[n]_{\\mathcal{R}(p^{-m},q^{-m})}\\bigg),\n\t\\end{eqnarray}\n\twhere $$\\nabla(n,\\beta)=[\\beta-\\beta_1+1]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta-\\beta_1-\\beta_2-n]_{\\mathcal{R}(p^{m},q^{m})}[\\beta-\\beta_1-\\beta_2+1]_{\\mathcal{R}(p^{-m},q^{-m})}$$ and the $\\mathcal{R}(p,q)$-random vector $\\big(W_1,W_2\\big)$ satisfy the bivariate $\\mathcal{R}(p,q)$- P\\'olya distribution, with parameters $n,$ $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\t\\end{small}\n\t\\end{corollary}\n\t\\begin{proof}\n\tThe $\\mathcal{R}(p,q)$-covariance of $\\widehat{W}$ and $\\ddot{W},$ is defined by\n\t\\begin{eqnarray*}\n\tCov\\big(\\widehat{W}, \\ddot{W}\\big)= E\\big(\\widehat{W}\\ddot{W}\\big)-E\\big(\\widehat{W}\\big)E\\big(\\ddot{W}\\big).\n\t\\end{eqnarray*}\n\tFrom the relations \\eqref{bip2}, \\eqref{bip4}, and \\eqref{bip5}, with $i_1=i_2=1,$ we have:\n\t\\begin{eqnarray*}\n\tE\\big(\\widehat{W}\\ddot{W}\\big)=\\frac{[n+1]_{\\mathcal{R}(p^{-m},q^{-m})}[n]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_1]_{\\mathcal{R}(p^{m},q^{m})}[\\beta_2]_{\\mathcal{R}(p^{m},q^{m})}}{\\nabla(n,\\beta)} \n\t\\end{eqnarray*}\n\tand \n\t\\begin{eqnarray*}\n\tE\\big(\\widehat{W}\\big)E\\big(\\ddot{W}\\big)=\\frac{[n]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_2]_{\\mathcal{R}(p^{m},q^{m})}[n]_{\\mathcal{R}(p^{-m},q^{-m})}[\\beta_1]_{\\mathcal{R}(p^{m},q^{m})}}{\\nabla(n,\\beta)}.\n\t\\end{eqnarray*}\n\tAfter computation, the relation \\eqref{bip7} holds and the proff is achieved.\n\t$\\hfill{\\square}$\n\t\\end{proof}\n\t\\begin{remark}\n\tParticular cases of multivariate inverse P\\'olya distribution are deduced as:\n\t\\begin{enumerate}\n\t\\item[(a)]\n\tThe probability function of the multivariate inverse $q-$ deformed P\\'olya distribution, with parameters $n$, $\\varTheta$ $(\\vartheta_1,\\vartheta_2,\\cdots,\\vartheta_k)$, and $q$, is given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(W_1=w_1,\\ldots,W_k=w_k)\n\t\t&=&F_k(q)\\genfrac{[}{]}{0pt}{}{n+w_k-1}{w_1,w_2,\\cdots, w_k}_{q^{-m}}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^k[\\beta_j]_{w_j,q^{-m}}[\\beta_{k+1}]_{n,q^{-m}}}\n\t\t{[\\beta]_{n+w_k,q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $F_k(q)=q^{m\\sum_{j=1}^{k}(n+w_k-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2,\\cdots,k\\},$ \n\t$\\beta_{k+1}=\\beta-\\sum_{j=1}^k\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2,\\cdots,k\\}.$\n\t\n\tThe probability function of the bivariate  inverse $q-$  P\\'olya distribution, with parameters $n$,  $\\underline{\\beta}=(\\beta_1,\\beta_2),$  and $q,$ is  :\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tP(W_1=w_1,W_2=w_2)\n\t\t\t&=&F_2(q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_1,w_2}_{q^{-m}}\\nonumber\\\\&\\times&\n\t\t\t\\frac{\\prod_{j=1}^2[\\beta_j]_{w_j,q^{-m}}[\\beta_{3}]_{n,q^{-m}}}\n\t\t\t{[\\beta]_{n+w_2,q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $F_2(q)=q^{-m\\sum_{j=1}^{2}(n+w_2-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2\\},$ \n\t$\\beta_{3}=\\beta-\\sum_{j=1}^2\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2\\}.$\n\tMoreover, \n\tfor $i_1\\in\\mathbb{N}\\cup\\{0\\}$ and $i_2\\in\\mathbb{N}\\cup\\{0\\},$ its\tfactorial moments  are given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([W_2]_{i_2,q^m}\\big)=\\frac{[n+i_2-1]_{i_2,q^{-m}}[\\beta_2]_{i_2,q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([W_1]_{i_1,q^m}|W_2=w_2\\big)=\\frac{[n+w_2+i_1-1]_{i_1,q^{-m}}[\\beta_1]_{i_1,q^{m}}}{[\\beta-\\beta_1+i_1]_{i_1,q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\bigg(\\frac{[W_1]_{i_1,q^m}}{[\\beta-\\beta_1-n-W_2]_{i_1,q^{m}}}\\bigg)=\\frac{[n+i_1-1]_{i_1,q^{-m}}[\\beta_1]_{i_1,q^{m}}}{[\\beta-\\beta_1+i_1]_{i_1,q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{i_1,q^{m}}},\n\t\t\\end{eqnarray*}\n\t\tand \n\t\t\\begin{eqnarray*}\n\t\t\tE\\bigg(\\frac{[W_1]_{i_1,q^m}[W_2]_{i_2,q^m}}{[\\beta-\\beta_1-n-W_2]_{i_1,q^{m}}}\\bigg)&=&\\frac{[n+i_1+i_2-1]_{i_1+i_2,q^{-m}}}{[\\beta-\\beta_1+i_1]_{i_1,q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{i_1,q^{m}}}\\nonumber\\\\&\\times& \\frac{[\\beta_1]_{i_1,q^{m}}[\\beta_2]_{i_2,q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,q^{-m}}}.\n\t\t\\end{eqnarray*} \n\t\\end{small}\n\tFurthermore, the $q$- covariance of the $q$- random variable $\\ddot{W}=[W_2]_{q^{m}}$  and $\\widehat{W}=\\frac{[W_1]_{q^{m}}}{[\\beta-\\beta_1-n-W_2]_{q^{m}}}$  is determined by :\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tCov\\big(\\widehat{W}, \\ddot{W}\\big)=\\frac{[n]_{q^{-m}}[\\beta_2]_{q^{m}}[\\beta_1]_{q^{m}}}{\\nabla(n,\\beta)}\\bigg([n+1]_{q^{-m}}-[n]_{q^{-m}}\\bigg),\n\t\t\\end{eqnarray*}\n\t\t\n\t\twhere $$\\nabla(n,\\beta)=[\\beta-\\beta_1+1]_{q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{q^{m}}[\\beta-\\beta_1-\\beta_2+1]_{q^{-m}}$$the $q$-random vector $\\big(W_1,W_2\\big)$ satisfy the bivariate inverse $q$- P\\'olya distribution, with parameters $n,$ $\\underline{\\beta}=(\\beta_1,\\beta_2),$  and $q.$\n\t\\end{small}\n\t\\item[(b)]The probability function of the multivariate inverse $(p,q)-$ deformed P\\'{o}lya distribution, with parameters $n$, $\\varTheta$ $(\\vartheta_1,\\vartheta_2,\\cdots,\\vartheta_k)$,   $p,$ and $q$, is given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(W_1=w_1,\\ldots,W_k=w_k)\n\t\t&=&F_k(p,q)\\genfrac{[}{]}{0pt}{}{n+w_k-1}{w_1,w_2,\\cdots, w_k}_{p^{-m},q^{-m}}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^k[\\beta_j]_{w_j,p^{-m},q^{-m}}[\\beta_{k+1}]_{n,p^{-m},q^{-m}}}\n\t\t{[\\beta]_{n+w_k,p^{-m},q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $F_k(p,q)=q^{-m\\sum_{j=1}^{k}(n+w_k-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2,\\cdots,k\\},$ \n\t$\\beta_{k+1}=\\beta-\\sum_{j=1}^k\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2,\\cdots,k\\}.$\n\t\n\t The probability function of the bivariate  inverse $(p,q)-$  P\\'{o}lya distribution, with parameters $n$,  $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is  :\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(W_1=w_1,W_2=w_2)\n\t\t&=&F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_1,w_2}_{p^{-m},q^{-m}}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^2[\\beta_j]_{w_j,p^{-m},q^{-m}}[\\beta_{3}]_{n,p^{-m},q^{-m}}}\n\t\t{[\\beta]_{n+w_2,p^{-m},q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $F_2(p,q)=q^{-m\\sum_{j=1}^{2}(n+w_2-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2\\},$ \n\n\t$\\beta_{3}=\\beta-\\sum_{j=1}^2\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2\\}.$\n\tMoreover, \nfor $i_1\\in\\mathbb{N}\\cup\\{0\\}$ and $i_2\\in\\mathbb{N}\\cup\\{0\\},$ it's\tfactorial moments  are given by:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([W_2]_{i_2,p^m,q^m}\\big)=\\frac{[n+i_2-1]_{i_2,p^{-m},q^{-m}}[\\beta_2]_{i_2,p^{m},q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([W_1]_{i_1,p^m,q^m}|W_2=w_2\\big)=\\frac{[n+w_2+i_1-1]_{i_1,p^{-m},q^{-m}}[\\beta_1]_{i_1,p^{m},q^{m}}}{[\\beta-\\beta_1+i_1]_{i_1,p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\bigg(\\frac{[W_1]_{i_1,p^m,q^m}}{[\\beta-\\beta_1-n-W_2]_{i_1,p^{m},q^{m}}}\\bigg)=\\frac{[n+i_1-1]_{i_1,p^{-m},q^{-m}}[\\beta_1]_{i_1,p^{m},q^{m}}}{[\\beta-\\beta_1+i_1]_{i_1,p^{-m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{i_1,p^{m},q^{m}}},\n\t\\end{eqnarray*}\n\t and \n\t\\begin{eqnarray*}\n\tE\\bigg(\\frac{[W_1]_{i_1,p^m,q^m}[W_2]_{i_2,p^m,q^m}}{[\\beta-\\beta_1-n-W_2]_{i_1,p^{m},q^{m}}}\\bigg)&=&\\frac{[n+i_1+i_2-1]_{i_1+i_2,p^{-m},q^{-m}}}{[\\beta-\\beta_1+i_1]_{i_1,p^{-m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{i_1,p^{m},q^{m}}}\\nonumber\\\\&\\times& \\frac{[\\beta_1]_{i_1,p^{m},q^{m}}[\\beta_2]_{i_2,p^{m},q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,p^{-m},q^{-m}}}.\n\t\\end{eqnarray*} \n\t\\end{small}\n\tFurthermore, the $(p,q)$- covariance of the $(p,q)$- random variable $\\ddot{W}=[W_2]_{p^{m},q^{m}}$  and $\\widehat{W}=\\frac{[W_1]_{p^{m},q^{m}}}{[\\beta-\\beta_1-n-W_2]_{p^{m},q^{m}}}$  is determined by :\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tCov\\big(\\widehat{W}, \\ddot{W}\\big)=\\frac{[n]_{p^{-m},q^{-m}}[\\beta_2]_{p^{m},q^{m}}[\\beta_1]_{p^{m},q^{m}}}{\\nabla(n,\\beta)}\\bigg([n+1]_{p^{-m},q^{-m}}-[n]_{p^{-m},q^{-m}}\\bigg),\n\t\\end{eqnarray*}\n\t\n\twhere $$\\nabla(n,\\beta)=[\\beta-\\beta_1+1]_{p^{-m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{p^{m},q^{m}}[\\beta-\\beta_1-\\beta_2+1]_{p^{-m},q^{-m}}$$the $(p,q)$-random vector $\\big(W_1,W_2\\big)$ satisfy the bivariate inverse $(p,q)$- P\\'olya distribution, with parameters $n,$ $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\t\\end{small}\n\\item[(c)] The probability function of the multivariate inverse $(p^{-1},q)-$ deformed P\\'{o}lya distribution, with parameters $n$, $\\varTheta$ $(\\vartheta_1,\\vartheta_2,\\cdots,\\vartheta_k)$,   $p,$ and $q$, is given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(W_1=w_1,\\ldots,W_k=w_k)\n\t\t&=&F_k(p^{-1},q)\\genfrac{[}{]}{0pt}{}{n+w_k-1}{w_1,w_2,\\cdots, w_k}_{p^{m},q^{-m}}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^k[\\beta_j]_{w_j,p^{m},q^{-m}}[\\beta_{k+1}]_{n,p^{m},q^{-m}}}\n\t\t{[\\beta]_{n+w_k,p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $F_k(p^{-1},q)=q^{-m\\sum_{j=1}^{k}(n+w_k-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2,\\cdots,k\\},$ \n$\\beta_{k+1}=\\beta-\\sum_{j=1}^k\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2,\\cdots,k\\}.$\n\nThe probability function of the bivariate  inverse $(p^{-1},q)-$  P\\'olya distribution, with parameters $n$,  $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is  :\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(W_1=w_1,W_2=w_2)\n\t\t&=&F_2(p^{-1},q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_1,w_2}_{p^{m},q^{-m}}\\nonumber\\\\&\\times&\n\t\t\\frac{\\prod_{j=1}^2[\\beta_j]_{w_j,p^{m},q^{-m}}[\\beta_{3}]_{n,p^{m},q^{-m}}}\n\t\t{[\\beta]_{n+w_2,p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $F_2(p^{-1},q)=q^{-m\\sum_{j=1}^{2}(n+w_2-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2\\},$ \n$\\beta_{3}=\\beta-\\sum_{j=1}^2\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2\\}.$\nMoreover, \nfor $i_1\\in\\mathbb{N}\\cup\\{0\\}$ and $i_2\\in\\mathbb{N}\\cup\\{0\\},$ its\tfactorial moments  are given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tE\\big([W_2]_{i_2,p^{-m},q^m}\\big)=\\frac{[n+i_2-1]_{i_2,p^{m},q^{-m}}[\\beta_2]_{i_2,p^{-m},q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([W_1]_{i_1,p^{-m},q^m}|W_2=w_2\\big)=\\frac{[n+w_2+i_1-1]_{i_1,p^{m},q^{-m}}[\\beta_1]_{i_1,p^{-m},q^{m}}}{[\\beta-\\beta_1+i_1]_{i_1,p^{m},q^{-m}}},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\bigg(\\frac{[W_1]_{i_1,p^{-m},q^m}}{[\\beta-\\beta_1-n-W_2]_{i_1,p^{-m},q^{m}}}\\bigg)=\\frac{[n+i_1-1]_{i_1,p^{m},q^{-m}}[\\beta_1]_{i_1,p^{-m},q^{m}}}{[\\beta-\\beta_1+i_1]_{i_1,p^{m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{i_1,p^{-m},q^{m}}},\n\t\\end{eqnarray*}\n\tand \n\t\\begin{eqnarray*}\n\t\tE\\bigg(\\frac{[W_1]_{i_1,p^{-m},q^m}[W_2]_{i_2,p^{-m},q^m}}{[\\beta-\\beta_1-n-W_2]_{i_1,p^{-m},q^{m}}}\\bigg)&=&\\frac{[n+i_1+i_2-1]_{i_1+i_2,p^{m},q^{-m}}}{[\\beta-\\beta_1+i_1]_{i_1,p^{m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{i_1,p^{-m},q^{m}}}\\nonumber\\\\&\\times& \\frac{[\\beta_1]_{i_1,p^{-m},q^{m}}[\\beta_2]_{i_2,p^{-m},q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]_{i_2,p^{m},q^{-m}}}.\n\t\\end{eqnarray*} \n\\end{small}\nFurthermore, the $(p^{-1},q)$- covariance of the $(p^{-1},q)$- random variable $\\ddot{W}=[W_2]_{p^{-m},q^{m}}$  and $\\widehat{W}=\\frac{[W_1]_{p^{-m},q^{m}}}{[\\beta-\\beta_1-n-W_2]_{p^{-m},q^{m}}}$  is determined by :\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tCov\\big(\\widehat{W}, \\ddot{W}\\big)=\\frac{[n]_{p^{m},q^{-m}}[\\beta_2]_{p^{-m},q^{m}}[\\beta_1]_{p^{-m},q^{m}}}{\\nabla(n,\\beta)}\\bigg([n+1]_{p^{m},q^{-m}}-[n]_{p^{m},q^{-m}}\\bigg),\n\t\\end{eqnarray*}\n\t\n\twhere $$\\nabla(n,\\beta)=[\\beta-\\beta_1+1]_{p^{m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]_{p^{-m},q^{m}}[\\beta-\\beta_1-\\beta_2+1]_{p^{m},q^{-m}}$$the $(p^{-1},q)$-random vector $\\big(W_1,W_2\\big)$ satisfy the bivariate inverse $(p^{-1},q)$- P\\'olya distribution, with parameters $n,$ $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\t\\item[(d)]\n\tThe probability function of the multivariate inverse Hounkonnou-Ngompe generalized $q$- Quesne P\\'{o}lya distribution, with parameters $n$, $\\varTheta$ $(\\vartheta_1,\\vartheta_2,\\cdots,\\vartheta_k)$,   $p,$ and $q$, is given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tP(W_1=w_1,\\ldots,W_k=w_k)\n\t\t\t&=&F_k(p,q)\\genfrac{[}{]}{0pt}{}{n+w_k-1}{w_1,w_2,\\cdots, w_k}^Q_{p^{-m},q^{-m}}\\nonumber\\\\&\\times&\n\t\t\t\\frac{\\prod_{j=1}^k[\\beta_j]^Q_{w_j,p^{-m},q^{-m}}[\\beta_{k+1}]^Q_{n,p^{-m},q^{-m}}}\n\t\t\t{[\\beta]^Q_{n+w_k,p^{-m},q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $F_k(p,q)=q^{m\\sum_{j=1}^{k}(n+w_k-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2,\\cdots,k\\},$ \n\t$\\beta_{k+1}=\\beta-\\sum_{j=1}^k\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2,\\cdots,k\\}.$\n\t\n\tThe mass function of the bivariate  inverse  Hounkonnou-Ngompe generalized $q$- Quesne   P\\'{o}lya distribution, with parameters $n$,  $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is  :\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(\\underline{W}=\\underline{w})\n\t\t=F_2(p,q)\\genfrac{[}{]}{0pt}{}{n+w_2-1}{w_1,w_2}^Q_{p^{-m},q^{-m}}\n\t\t\\frac{\\prod_{j=1}^2[\\beta_j]^Q_{w_j,p^{-m},q^{-m}}[\\beta_{3}]^Q_{n,p^{-m},q^{-m}}}\n\t\t{[\\beta]^Q_{n+w_2,p^{-m},q^{-m}}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $F_2(p,q)=q^{-m\\sum_{j=1}^{2}(n+w_2-w_j)(\\beta_j-w_j)}$for $w_j\\in\\mathbb{N}\\cup\\{0\\},$ $j\\in\\{1,2\\},$ \n\n\t$\\beta_{3}=\\beta-\\sum_{j=1}^2\\beta_j,$ and $w_j=\\sum_{i=1}^jw_i,$ for $j\\in\\{1,2\\}.$\n\tMoreover, \nfor $i_1\\in\\mathbb{N}\\cup\\{0\\}$ and $i_2\\in\\mathbb{N}\\cup\\{0\\},$ \tthe corresponding factorial moments  are given by:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([W_2]^Q_{i_2,p^m,q^m}\\big)=\\frac{[n+i_2-1]^Q_{i_2,p^{-m},q^{-m}}[\\beta_2]^Q_{i_2,p^{m},q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]^Q_{i_2,p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([W_1]^Q_{i_1,p^m,q^m}|W_2=w_2\\big)=\\frac{[n+w_2+i_1-1]^Q_{i_1,p^{-m},q^{-m}}[\\beta_1]^Q_{i_1,p^{m},q^{m}}}{[\\beta-\\beta_1+i_1]^Q_{i_1,p^{-m},q^{-m}}},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\bigg(\\frac{[W_1]^Q_{i_1,p^m,q^m}}{[\\beta-\\beta_1-n-W_2]^Q_{i_1,p^{m},q^{m}}}\\bigg)=\\frac{[n+i_1-1]^Q_{i_1,p^{-m},q^{-m}}[\\beta_1]^Q_{i_1,p^{m},q^{m}}}{[\\beta-\\beta_1+i_1]^Q_{i_1,p^{-m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]^Q_{i_1,p^{m},q^{m}}},\n\t\\end{eqnarray*}\n\t and \n\t\\begin{eqnarray*}\n\tE\\bigg(\\frac{[W_1]^Q_{i_1,p^m,q^m}[W_2]^Q_{i_2,p^m,q^m}}{[\\beta-\\beta_1-n-W_2]^Q_{i_1,p^{m},q^{m}}}\\bigg)&=&\\frac{[n+i_1+i_2-1]^Q_{i_1+i_2,p^{-m},q^{-m}}}{[\\beta-\\beta_1+i_1]^Q_{i_1,p^{-m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]^Q_{i_1,p^{m},q^{m}}}\\nonumber\\\\&\\times& \\frac{[\\beta_1]^Q_{i_1,p^{m},q^{m}}[\\beta_2]^Q_{i_2,p^{m},q^{m}}}{[\\beta-\\beta_1-\\beta_2+i_2]^Q_{i_2,p^{-m},q^{-m}}}.\n\t\\end{eqnarray*} \n\t\\end{small}\n\tFurthermore, the  covariance of the random variable $\\ddot{W}=[W_2]^Q_{p^{m},q^{m}}$  and $\\widehat{W}=\\frac{[W_1]^Q_{p^{m},q^{m}}}{[\\beta-\\beta_1-n-W_2]_{p^{m},q^{m}}}$  is given by:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tCov\\big(\\widehat{W}, \\ddot{W}\\big)=\\frac{[n]^Q_{p^{-m},q^{-m}}[\\beta_2]^Q_{p^{m},q^{m}}[\\beta_1]^Q_{p^{m},q^{m}}}{\\nabla^Q(n,\\beta)}\\bigg([n+1]^Q_{p^{-m},q^{-m}}-[n]^Q_{p^{-m},q^{-m}}\\bigg),\n\t\\end{eqnarray*}\n\t\n\twhere $$\\nabla^Q(n,\\beta)=[\\beta-\\beta_1+1]^Q_{p^{-m},q^{-m}}[\\beta-\\beta_1-\\beta_2-n]^Q_{p^{m},q^{m}}[\\beta-\\beta_1-\\beta_2+1]^Q_{p^{-m},q^{-m}}.$$\n\t\\end{small}\n\t\\end{enumerate}\n\t\\end{remark}\n\t\n\\subsection{Multivariate $\\mathcal{R}(p,q)-$ hypergeometric distribution}\nIn this section, the multivariate $\\mathcal{R}(p,q)$- hypergeometric and the negative multivariate $\\mathcal{R}(p,q)$- hypergeometric distribution are determined.\n \nWe \nconsider a sequence of independent Bernoulli trials and suppose that the probability of success at the $i$th trial is given as follows:\n\\begin{equation*}\np_i=\\frac{\\theta \\tau_2^{i-1}}{\\tau^{i-1}_1+\\theta \\tau_2^{i-1}}, \\ \\ i\\in\\mathbb{\nN},\\,\\, \\ 0<\\theta<1.\n\\end{equation*}\n\nWe denote by  $H_j$  the number of successes after the $(s_{j-1})$th trial and until the $(s_j)$th trial, with $j\\in\\{1,2,\\ldots,k+1\\},$ $s_0=0$, $s_j=\\sum_{i=1}^jr_i$, $j\\in\\{1,2,\\ldots,k+1\\}$, and $s_{k+1}=r$. Thus, the $\\mathcal{R}(p,q)$ - random  variables $H_j$\n are independent and the probability distribution of the $\\mathcal{R}(p,q)$- binomial probability distribution of the first kind is given by:\n\\begin{equation}\\label{rpqbf}\nP(H_j=h_j)=\\genfrac{[}{]}{0pt}{}{r_j}{h_j}_{\\mathcal{R}(p,q)}\\frac{(\\theta \\tau_2^{s_{j-1}})^{h_j}\\tau^{\\binom{r_j-h_j}{2}}_1\\tau_2^{\\binom{h_j}{2}}}{\\big(\\tau_1^{s_{j-1}}\\oplus\\theta \\tau_2^{s_{j-1}}\\big)^{r_j}_{\\mathcal{R}(p,q)}},\n\\ \\ h_j\\in\\{0,1,\\ldots,r_j\\}.\n\\end{equation}\n\\begin{theorem}\\label{thm3.4}\n\t The conditional probability function of the $\\mathcal{R}(p,q)$- random vector $(H_1,H_2,\\ldots,H_k)$, given that $H_1+H_2+\\ldots+H_{k+1}=n$, is the multivariate $\\mathcal{R}(p,q)$-hypergeometric distribution. Its  probability distribution is given by:\n\t \\begin{small}\n\t \t\\begin{equation}\\label{eq3.2a}\n\t \tP(H_1=h_1,\\ldots,H_k=h_k)=\\mathcal{H}_k(p,q)\\genfrac{[}{]}{0pt}{}{n}{h_1,\\cdots,h_k}_{\\mathcal{R}(p,q)}{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{\\beta_j}{h_j}_{\\mathcal{R}(p,q)}\n\t \t\t\\over \\genfrac{[}{]}{0pt}{}{\\beta}{n}_{\\mathcal{R}(p,q)}},\n\t \t\\end{equation}\n\t \\end{small}\n\t where $\\mathcal{H}_k(p,q)=\\tau^{\\sum_{j=1}^{k}h_{j}(\\beta_{j+1}-h_{j+1})}_1\\tau_2^{\\sum_{j=1}^{k}(n-y_j)(\\beta_j-h_j)}$\n\t for $h_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2,\\cdots,k\\},$ with $\\sum_{j=1}^k h_j\\leq n$, $h_{k+1}=n-\\sum_{j=1}^k h_j$, $\\beta_{k+1}=\\beta-\\sum_{j=1}^k \\beta_j$, and $y_j=\\sum_{i=1}^{j}h_i$, for $j\\in\\{1,2,\\cdots,k\\}.$\n\\end{theorem}\n\\begin{proof} \nFrom the relation (\\ref{rpqbf}), the probability distribution of the sum $Y=H_1+H_2+\\cdots+H_{k+1}$, which is the number of successes in $r$ trials, is\n\\begin{equation*}\nP(Y=n)=\\genfrac{[}{]}{0pt}{}{r}{n}_{\\mathcal{R}(p,q)}\\frac{\\theta^n\\tau_1^{\\binom{r-n}{2}} \\tau_2^{\\binom{n}{2}}}{\\big(1\\oplus \\theta \\big)^{r}_{\\mathcal{R}(p,q)}},\\ \\ n\\in\\{0,1,\\ldots,r\\}.\n\\end{equation*}\nThen, the joint conditional probability distribution of the $\\mathcal{R}(p,q)$- random vector $(H_1,H_2,\\ldots,H_k)$, given that $Y=n,$\n\\begin{equation*}\nP(H_1=h_1,\\ldots,H_k=h_k|Y=n)=\\frac{P(H_1=h_1)\\ldots P(H_k=h_k)P(H_{k+1}=n-y_k)}{P(Y=n)},\n\\end{equation*}\non using these expressions, is obtained as:\n\\begin{equation*}\nP(H_1=h_1,H_2=h_2,\\ldots,H_k=h_k|Y=n)=\\big(\\tau_1\\tau_2\\big)^{c_k}\\frac{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{\\mathcal{R}(p,q)}}{\\genfrac{[}{]}{0pt}{}{r}{n}_{\\mathcal{R}(p,q)}},\n\\end{equation*}\nwhere\n\\[\nc_k=\\sum_{i=1}^kx_is_{i-1}+(n-y_k)s_k+\\sum_{j=1}^k\\binom{x_j}{2}+\\binom{n-y_k}{2}-\\binom{n}{2}.\n\\]\nThus, after some algebraic manipulations, it reduces to\n\\begin{align*}\nc_k&=n\\sum_{j=1}^kr_j-\\sum_{i=1}^kx_i(s_k-s_{i-1})+\\sum_{j=1}^k\\binom{x_j}{2}+\\binom{y_k+1}{2}-ny_k\\\\\n&=\\sum_{j=1}^kr_j(n-y_j)-\\sum_{j=1}^kx_j(n-y_j)=\\sum_{j=1}^k(n-y_j)(y_j-x_j),\n\\end{align*}\nand the proof is completed. $\\hfill{\\square}$\n\\end{proof}\n\\begin{remark}\n\tThe multivariate $\\mathcal{R}(p,q)$- hypergeometric can also be obtained by taking $m=-1$ in the relation (\\ref{eq3.2}).\n\n\\end{remark}\n\\subsubsection{Bivariate $\\mathcal{R}(p,q)$-hypergeometric distribution}\nThe probability distribution of the $\\mathcal{R}(p,q)$- random vector $\\underline{H}=\\big(H_1,H_2\\big)$ is called the $\\mathcal{R}(p,q)$- P\\'olya distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$ Its probability function is given by the following relation:\n\\begin{small}\n\t\\begin{equation}\\label{bh1}\n\tP(H_1=h_1,H_2=h_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{h_1,h_2}_{\\mathcal{R}(p,q)}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{h_j,\\mathcal{R}(p,q)}}{ [\\beta]_{n,\\mathcal{R}(p,q)}},\n\t\\end{equation}\n\\end{small}\nwhere $\\Psi_2(p,q)=\\tau^{\\sum_{j=1}^{2}h_{j}(\\beta_{j+1}-h_{j+1})}_1\\tau_2^{\\sum_{j=1}^{2}(n-x_j)(\\beta_j-h_j)},$\n$h_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $h_1+h_2\\leq n$, $h_{3}=n-h_1+h_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=h_1,$\nand $x_2=h_1+h_2.$\n\\begin{proposition}\n\tThe $\\mathcal{R}(p,q)$-factorial moments of the bivariate $\\mathcal{R}(p,q)$-hypergeometric  distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ are presented as follows:\n\t\\begin{small}\n\t\\begin{eqnarray}\\label{bh2}\n\tE\\big([H_1]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}\\big)=\\frac{[n]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}[\\beta_1]_{i_1,\\mathcal{R}(p,q)}}{[\\beta]_{i_1,\\mathcal{R}(p,q)}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray}\n\t\\begin{eqnarray}\\label{bh3}\n\tE\\big([H_2]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}|H_1=h_1\\big)=\\frac{[n-h_1]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}[\\beta_2]_{i_2,\\mathcal{R}(p,q)}}{[\\beta-\\beta_1]_{i_2,\\mathcal{R}(p,q)}},\\,i_2\\in\\{0,1,\\cdots,n-h_1\\},\n\t\\end{eqnarray}\n\t\\begin{eqnarray}\\label{bh4}\n\tE\\big([H_2]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}\\big)=\\frac{[n]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}[\\beta_2]_{i_2,\\mathcal{R}(p,q)}\\tau^{i_2\\beta_1}_2}{[\\beta]_{i_2,\\mathcal{R}(p,q)}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray}\n\tand\n\t\\begin{eqnarray}\\label{bh5}\n\tE\\big([H_1]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}[H_2]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}\\big)=\\frac{[n]_{i_1+i_2,\\mathcal{R}(p^{-1},q^{-1})}[\\beta_1]_{i_1,\\mathcal{R}(p,q)}[\\beta_2]_{i_2,\\mathcal{R}(p,q)}}{\\tau^{-i_2\\beta_1}_2\\,[\\beta]_{i_1+i_2,\\mathcal{R}(p,q)}},\n\t\\end{eqnarray}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\n\\end{proposition}\n\\begin{corollary}\n The $\\mathcal{R}(p,q)$- covariance of $[H_1]_{\\mathcal{R}(p^{-1},q^{-1})}$ and $[H_2]_{\\mathcal{R}(p^{-1},q^{-1})}$ is derived by:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([H_1]_{\\mathcal{R}(p^{-1},q^{-1})},[H_2]_{\\mathcal{R}(p^{-1},q^{-1})}\\big)=\\frac{[n]_{\\mathcal{R}(p^{-1},q^{-1})}[\\beta_1]_{\\mathcal{R}(p,q)}[\\beta_2]_{\\mathcal{R}(p,q)}}{\\tau^{-\\beta_1}_2[\\beta]_{\\mathcal{R}(p,q)}}\\Delta(n,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,\\beta)=\\frac{[n-1]_{\\mathcal{R}(p^{-1},q^{-1})}}{[\\beta-1]_{\\mathcal{R}(p,q)}}-\\frac{[n]_{\\mathcal{R}(p^{-1},q^{-1})}}{[\\beta_1]_{\\mathcal{R}(p,q)}}\n\\end{eqnarray*}\nand  $\\underline{H}=\\big(\nH_1,H_2\\big)$  a $\\mathcal{R}(p,q)$-random vector verifying   the bivariate $\\mathcal{R}(p,q)$-hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\\end{corollary}\n\\begin{remark}\nDifferents  multivariate hypergeometric distributions are deduced as:\n\\begin{enumerate}\n\t\\item[(a)]\n\tThe multivariate $q$-hypergeometric distribution. Its  probability distribution is given by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\tP(H_1=h_1,\\ldots,H_k=h_k)=\\mathcal{H}_k(q)\\genfrac{[}{]}{0pt}{}{n}{h_1,\\cdots,h_k}_{q}{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{\\alpha_j}{h_j}_{q}\n\t\t\t\\over \\genfrac{[}{]}{0pt}{}{\\alpha}{n}_{q}},\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\twhere $\\mathcal{H}_k(q)=q^{\\sum_{j=1}^{k}h_{j}(\\alpha_{j+1}-h_{j+1})}\\,q^{-\\sum_{j=1}^{k}(n-y_j)(\\alpha_j-h_j)}$\n\tfor $h_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$ with $\\sum_{j=1}^k h_j\\leq n$, $h_{k+1}=n-\\sum_{j=1}^k h_j$, $\\alpha_{k+1}=\\alpha-\\sum_{j=1}^k \\alpha_j$, and $y_j=\\sum_{i=1}^{j}h_i$, for $j\\in\\{1,2,\\cdots,k\\}.$\n\t\n\tThe probability function of the  bivariate $(p,q)$- hypergeometric distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$  and $q.$ is given by the following relation:\n\t\\begin{small}\n\t\t\\begin{equation*}\n\t\tP(H_1=h_1,H_2=h_2)=\\Psi_2(q)\\genfrac{[}{]}{0pt}{}{n}{h_1,h_2}_{q}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{h_j,q}}{ [\\beta]_{n,q}},\n\t\t\\end{equation*}\n\t\\end{small}\n\twhere $\\Psi_2(q)=q^{\\sum_{j=1}^{2}h_{j}(\\beta_{j+1}-h_{j+1})}\\,q^{-\\sum_{j=1}^{2}(n-x_j)(\\beta_j-h_j)},$\n\t$h_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $h_1+h_2\\leq n$, $h_{3}=n-h_1+h_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=h_1,$\n\tand $x_2=h_1+h_2.$\n\t\n\tBesides, its  $q$-factorial moments are presented as follows:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([H_1]_{i_1,q^{-1}}\\big)=\\frac{[n]_{i_1,q^{-1}}[\\beta_1]_{i_1,q}}{[\\beta]_{i_1,q}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([H_2]_{i_1,q^{-1}}|H_1=h_1\\big)=\\frac{[n-h_1]_{i_2,q^{-1}}[\\beta_2]_{i_2,q}}{[\\beta-\\beta_1]_{i_2,q}},\\,i_2\\in\\{0,1,\\cdots,n-h_1\\},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([H_2]_{i_2,q^{-1}}\\big)=\\frac{[n]_{i_2,q^{-1}}[\\beta_2]_{i_2,q}q^{-i_2\\beta_1}}{[\\beta]_{i_2,q}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([H_1]_{i_1,q^{-1}}[H_2]_{i_2,q^{-1}}\\big)=\\frac{[n]_{i_1+i_2, q^{-1}}[\\beta_1]_{i_1,q}[\\beta_2]_{i_2,q}}{q^{i_2\\beta_1}\\,[\\beta]_{i_1+i_2,q}},\n\t\t\\end{eqnarray*}\n\t\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\t\\end{small}\n\tFurthermore, \n\tthe $q$- covariance of $[H_1]_{q^{-1}}$ and $[H_2]_{q^{-1}}$ is derived by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tCov\\big([H_1]_{q^{-1}},[H_2]_{q^{-1}}\\big)=\\frac{[n]_{q^{-1}}[\\beta_1]_{q}[\\beta_2]_{q}}{q^{\\beta_1}[\\beta]_{q}}\\Delta(n,\\beta),\n\t\t\\end{eqnarray*}\n\t\twhere \n\t\t\\begin{eqnarray*}\n\t\t\t\\Delta(n,\\beta)=\\frac{[n-1]_{q^{-1}}}{[\\beta-1]_{q}}-\\frac{[n]_{q^{-1}}}{[\\beta_1]_{q}}.\n\t\t\\end{eqnarray*}\n\t\\end{small}\n\\item[(b)] The probability distribution of the  multivariate $(p,q)$-hypergeometric distribution  is given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\tP(H_1=h_1,\\ldots,H_k=h_k)=\\mathcal{H}_k(p,q)\\genfrac{[}{]}{0pt}{}{n}{h_1,\\cdots,h_k}_{p,q}{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{\\alpha_j}{h_j}_{p,q}\n\t\t\\over \\genfrac{[}{]}{0pt}{}{\\alpha}{n}_{p,q}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\mathcal{H}_k(p,q)=p^{\\sum_{j=1}^{k}h_{j}(\\alpha_{j+1}-h_{j+1})}\\,q^{\\sum_{j=1}^{k}(n-y_j)(\\alpha_j-h_j)}$\nfor $h_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$ with $\\sum_{j=1}^k h_j\\leq n$, $h_{k+1}=n-\\sum_{j=1}^k h_j$, $\\alpha_{k+1}=\\alpha-\\sum_{j=1}^k \\alpha_j$, and $y_j=\\sum_{i=1}^{j}h_i$, for $j\\in\\{1,2,\\cdots,k\\}.$\n\n The probability function of the  bivariate $(p,q)$- hypergeometric distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$ is given by the following relation:\n\\begin{small}\n\t\\begin{equation*}\n\tP(H_1=h_1,H_2=h_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{h_1,h_2}_{p,q}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{h_j,p,q}}{ [\\beta]_{n,p,q}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p,q)=p^{\\sum_{j=1}^{2}h_{j}(\\beta_{j+1}-h_{j+1})}\\,q^{\\sum_{j=1}^{2}(n-x_j)(\\beta_j-h_j)},$\n$h_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $h_1+h_2\\leq n$, $h_{3}=n-h_1+h_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=h_1,$\nand $x_2=h_1+h_2.$\n\nBesides, its  $(p,q)$-factorial moments are presented as follows:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([H_1]_{i_1,p^{-1},q^{-1}}\\big)=\\frac{[n]_{i_1,p^{-1},q^{-1}}[\\beta_1]_{i_1,p,q}}{[\\beta]_{i_1,p,q}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([H_2]_{i_1,p^{-1},q^{-1}}|H_1=h_1\\big)=\\frac{[n-h_1]_{i_2,p^{-1},q^{-1}}[\\beta_2]_{i_2,p,q}}{[\\beta-\\beta_1]_{i_2,p,q}},\\,i_2\\in\\{0,1,\\cdots,n-h_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([H_2]_{i_2,p^{-1},q^{-1}}\\big)=\\frac{[n]_{i_2,p^{-1},q^{-1}}[\\beta_2]_{i_2,p,q}q^{i_2\\beta_1}}{[\\beta]_{i_2,p,q}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([H_1]_{i_1,p^{-1},q^{-1}}[H_2]_{i_2,p^{-1},q^{-1}}\\big)=\\frac{[n]_{i_1+i_2, p^{-1},q^{-1}}[\\beta_1]_{i_1,p,q}[\\beta_2]_{i_2,p,q}}{q^{-i_2\\beta_1}\\,[\\beta]_{i_1+i_2,p,q}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nFurthermore, \nthe $(p,q)$- covariance of $[H_1]_{p^{-1},q^{-1}}$ and $[H_2]_{p^{-1},q^{-1}}$ is derived by:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([H_1]_{p^{-1},q^{-1}},[H_2]_{p^{-1},q^{-1}}\\big)=\\frac{[n]_{p^{-1},q^{-1}}[\\beta_1]_{p,q}[\\beta_2]_{p,q}}{q^{-\\beta_1}[\\beta]_{p,q}}\\Delta(n,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,\\beta)=\\frac{[n-1]_{p^{-1},q^{-1}}}{[\\beta-1]_{p,q}}-\\frac{[n]_{p^{-1},q^{-1}}}{[\\beta_1]_{p,q}}.\n\\end{eqnarray*}\n\\end{small}\n\\item[(c)]The probability distribution of the  multivariate $(p^{-1},q)$-hypergeometric distribution  is given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(H_1=h_1,\\ldots,H_k=h_k)=\\mathcal{H}_k(p^{-1},q)\\genfrac{[}{]}{0pt}{}{n}{h_1,\\cdots,h_k}_{p^{-1},q}{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{\\alpha_j}{h_j}_{p^{-1},q}\n\t\t\t\\over \\genfrac{[}{]}{0pt}{}{\\alpha}{n}_{p^{-1},q}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\mathcal{H}_k(p^{-1},q)=p^{-\\sum_{j=1}^{k}h_{j}(\\alpha_{j+1}-h_{j+1})}\\,q^{\\sum_{j=1}^{k}(n-y_j)(\\alpha_j-h_j)}$\nfor $h_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$ with $\\sum_{j=1}^k h_j\\leq n$, $h_{k+1}=n-\\sum_{j=1}^k h_j$, $\\alpha_{k+1}=\\alpha-\\sum_{j=1}^k \\alpha_j$, and $y_j=\\sum_{i=1}^{j}h_i$, for $j\\in\\{1,2,\\cdots,k\\}.$\n\nThe probability function of the  bivariate $(p^{-1},q)$- hypergeometric distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q$ is given by the following relation:\n\\begin{small}\n\t\\begin{equation*}\n\tP(H_1=h_1,H_2=h_2)=\\Psi_2(p^{-1},q)\\genfrac{[}{]}{0pt}{}{n}{h_1,h_2}_{p^{-1},q}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{h_j,p^{-1},q}}{ [\\beta]_{n,p^{-1},q}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p^{-1},q)=p^{-\\sum_{j=1}^{2}h_{j}(\\beta_{j+1}-h_{j+1})}\\,q^{\\sum_{j=1}^{2}(n-x_j)(\\beta_j-h_j)},$\n$h_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $h_1+h_2\\leq n$, $h_{3}=n-h_1+h_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=h_1,$\nand $x_2=h_1+h_2.$\n\nBesides, its  $(p^{-1},q)$-factorial moments are presented as follows:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tE\\big([H_1]_{i_1,p^{1},q^{-1}}\\big)=\\frac{[n]_{i_1,p^{1},q^{-1}}[\\beta_1]_{i_1,p^{-1},q}}{[\\beta]_{i_1,p^{-1},q}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([H_2]_{i_1,p^{1},q^{-1}}|H_1=h_1\\big)=\\frac{[n-h_1]_{i_2,p^{1},q^{-1}}[\\beta_2]_{i_2,p^{-1},q}}{[\\beta-\\beta_1]_{i_2,p^{-1},q}},\\,i_2\\in\\{0,1,\\cdots,n-h_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([H_2]_{i_2,p^{1},q^{-1}}\\big)=\\frac{[n]_{i_2,p^{1},q^{-1}}[\\beta_2]_{i_2,p^{-1},q}q^{i_2\\beta_1}}{[\\beta]_{i_2,p^{-1},q}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\tE\\big([H_1]_{i_1,p^{1},q^{-1}}[H_2]_{i_2,p^{1},q^{-1}}\\big)=\\frac{[n]_{i_1+i_2, p^{1},q^{-1}}[\\beta_1]_{i_1,p^{-1},q}[\\beta_2]_{i_2,p^{-1},q}}{q^{-i_2\\beta_1}\\,[\\beta]_{i_1+i_2,p^{-1},q}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nFurthermore, \nthe $(p^{-1},q)$- covariance of $[H_1]_{p^{1},q^{-1}}$ and $[H_2]_{p^{1},q^{-1}}$ is derived by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tCov\\big([H_1]_{p^{1},q^{-1}},[H_2]_{p^{1},q^{-1}}\\big)=\\frac{[n]_{p^{1},q^{-1}}[\\beta_1]_{p^{-1},q}[\\beta_2]_{p,q}}{q^{-\\beta_1}[\\beta]_{p^{-1},q}}\\Delta(n,\\beta),\n\t\\end{eqnarray*}\n\twhere \n\t\\begin{eqnarray*}\n\t\t\\Delta(n,\\beta)=\\frac{[n-1]_{p^{1},q^{-1}}}{[\\beta-1]_{p^{-1},q}}-\\frac{[n]_{p^{1},q^{-1}}}{[\\beta_1]_{p^{-1},q}}.\n\t\\end{eqnarray*}\n\\end{small}\n\\item[(d)]The probability distribution of the  multivariate Hounkonnou-Ngompe generalized $q$-Quesne hypergeometric distribution  is given by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tP(H_1=h_1,\\ldots,H_k=h_k)=\\mathcal{H}^Q_k(p,q)\\genfrac{[}{]}{0pt}{}{n}{h_1,\\cdots,h_k}^Q_{p,q}{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{\\alpha_j}{h_j}^Q_{p,q}\n\t\t\t\\over \\genfrac{[}{]}{0pt}{}{\\alpha}{n}^Q_{p,q}},\n\t\\end{eqnarray*}\n\\end{small}\nwhere $\\mathcal{H}^Q_k(p,q)=p^{\\sum_{j=1}^{k}h_{j}(\\alpha_{j+1}-h_{j+1})}\\,q^{-\\sum_{j=1}^{k}(n-y_j)(\\alpha_j-h_j)}$\nfor $h_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2,\\ldots,k\\},$ with $\\sum_{j=1}^k h_j\\leq n$, $h_{k+1}=n-\\sum_{j=1}^k h_j$, $\\alpha_{k+1}=\\alpha-\\sum_{j=1}^k \\alpha_j$, and $y_j=\\sum_{i=1}^{j}h_i$, for $j\\in\\{1,2,\\cdots,k\\}.$\n\n The probability function of the  bivariate Hounkonnou-Ngompe generalized $q$-Quesne  hypergeometric distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q$ is given by the following relation:\n\\begin{small}\n\t\\begin{equation*}\n\tP(H_1=h_1,H_2=h_2)=\\Psi^Q_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{h_1,h_2}^Q_{p,q}\\frac{\\prod_{j=1}^{3}[\\beta_j]^Q_{h_j,p,q}}{ [\\beta]^Q_{n,p,q}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi^Q_2(p,q)=p^{\\sum_{j=1}^{2}h_{j}(\\beta_{j+1}-h_{j+1})}\\,q^{-\\sum_{j=1}^{2}(n-x_j)(\\beta_j-h_j)},$\n$h_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $h_1+h_2\\leq n$, $h_{3}=n-h_1+h_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=h_1,$\nand $x_2=h_1+h_2.$\n\nBesides, its  factorial moments are presented as follows:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([H_1]^Q_{i_1,p^{-1},q^{-1}}\\big)=\\frac{[n]^Q_{i_1,p^{-1},q^{-1}}[\\beta_1]^Q_{i_1,p,q}}{[\\beta]^Q_{i_1,p,q}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([H_2]^Q_{i_1,p^{-1},q^{-1}}|H_1=h_1\\big)=\\frac{[n-h_1]^Q_{i_2,p^{-1},q^{-1}}[\\beta_2]^Q_{i_2,p,q}}{[\\beta-\\beta_1]^Q_{i_2,p,q}},\\,i_2\\in\\{0,1,\\cdots,n-h_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([H_2]^Q_{i_2,p^{-1},q^{-1}}\\big)=\\frac{[n]^Q_{i_2,p^{-1},q^{-1}}[\\beta_2]^Q_{i_2,p,q}q^{-i_2\\beta_1}}{[\\beta]^Q_{i_2,p,q}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([H_1]^Q_{i_1,p^{-1},q^{-1}}[H_2]^Q_{i_2,p^{-1},q^{-1}}\\big)=\\frac{[n]^Q_{i_1+i_2, p^{-1},q^{-1}}[\\beta_1]^Q_{i_1,p,q}[\\beta_2]^Q_{i_2,p,q}}{q^{i_2\\beta_1}\\,[\\beta]^Q_{i_1+i_2,p,q}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nFurthermore, \nthe  covariance of $[H_1]^Q_{p^{-1},q^{-1}}$ and $[H_2]^Q_{p^{-1},q^{-1}}$ is derived by:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([H_1]^Q_{p^{-1},q^{-1}},[H_2]^Q_{p^{-1},q^{-1}}\\big)=\\frac{[n]^Q_{p^{-1},q^{-1}}[\\beta_1]^Q_{p,q}[\\beta_2]^Q_{p,q}}{q^{\\beta_1}[\\beta]^Q_{p,q}}\\Delta(n,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,\\beta)=\\frac{[n-1]^Q_{p^{-1},q^{-1}}}{[\\beta-1]^Q_{p,q}}-\\frac{[n]^Q_{p^{-1},q^{-1}}}{[\\beta_1]^Q_{p,q}}.\n\\end{eqnarray*}\n\\end{small}\n\\end{enumerate}\n\\end{remark}\n\\subsection{Multivariate negative $\\mathcal{R}(p,q)$-hypergeometric distribution}\nWe consider a sequence of independent Bernoulli trials and suppose that the conditional probability of success at a trial, given that $j-1$ successes occur in the previous trials, is determined by:\n\\begin{equation*}\np_j=1-\\theta \\tau_1^{1-j}\\tau_2^{j-1}, \\ \\ j\\in\\mathbb{N},\\,\\, \\ 0<\\theta<1.\n\\end{equation*}\n\nWe denotes by $V_j$  the number of failures after the $(s_{j-1})$th success and until the occurrence of the $(s_j)$th success, for $j\\in\\{1,2,\\cdots,k+1\\}$, with $s_0=0$, $s_j=\\sum_{i=1}^jr_i$, $j\\in\\{1,2,\\cdots,k+1\\}.$\n Thus, the $\\mathcal{R}(p,q)$- random  variables $V_j$ are independent and the $\\mathcal{R}(p,q)$- binomial probability distribution of the second kind  is presented by: \n\\begin{small}\n\\begin{equation}\\label{rpqbs}\nP(V_j=v_j)=\\genfrac{[}{]}{0pt}{}{r_j+v_j-1}{v_j}_{\\mathcal{R}(p,q)}(\\theta \\tau_2^{s_{j-1}})^{v_j}\\big(\\tau_1^{s_{j-1}} \\ominus\\theta \\tau_2^{s_{j-1}}\\big)^{r_j}_{\\mathcal{R}(p,q)},\n\\,  v_j\\in\\mathbb{N}\\cup\\{0\\}.\n\\end{equation}\n\\end{small}\n\\begin{theorem}\\label{thm3.5}\n\t The conditional probability function of the $\\mathcal{R}(p,q)$- random vector $(V_1,V_2,\\cdots,V_k)$, given that $V_1+V_2+\\cdots+V_{k+1}=n$, is the multivariate negative $\\mathcal{R}(p,q)$-hypergeometric distribution with probability function:\n\t\\begin{equation}\\label{eq3.9}\n\tP(V_1=v_1,\\cdots,V_k=v_k)=\t\\big(\\tau_1\\tau_2\\big)^{\\sum_{j=1}^kr_j(n-y_j)}\\frac{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j+v_j-1}{v_j}_{\\mathcal{R}(p,q)}\n\t}{\\genfrac{[}{]}{0pt}{}{r+n-1}{n}_{\\mathcal{R}(p,q)}}.\n\\end{equation}\nEquivalently, \n\\begin{eqnarray}\n\tP(V_1=v_1,\\cdots,V_k=v_k)&=&\\genfrac{[}{]}{0pt}{}{n}{v_1,v_2,\\cdots, v_k}_{\\mathcal{R}(p,q)}\n\t\\big(\\tau_1\\tau_2\\big)^{\\sum_{j=1}^{k}r_j(n-y_j)}\\nonumber\\\\&\\times&\\frac{\\prod_{j=1}^{k+1}[r_j+v_j-1]_{v_j,\\mathcal{R}(p,q)}}{[r+n-1]_{n,\\mathcal{R}(p,q)}},\n\t\\end{eqnarray}\n\t%\n\tfor $x_j\\in\\{0,1,\\cdots,n\\},$ $j\\in\\{1,2,\\cdots,k\\},$ with $\\sum_{j=1}^k x_j\\leq n$, and, where $x_{k+1}=n-\\sum_{j=1}^k x_j$, $r_{k+1}=r-\\sum_{j=1}^k r_j$, and $y_j=\\sum_{i=1}^{j}x_i$, $j\\in\\{1,2,\\cdots,k\\}$.\n\\end{theorem}\n\\begin{proof}  According to the relation (\\ref{rpqbs}),  the probability function of the sum $U=V_1+V_2+\\cdots+V_{k+1}$, which is the number of failures until the occurrence of the $r$th success, is\n\\begin{equation*}\nP(U=n)=\\genfrac{[}{]}{0pt}{}{r+n-1}{n}_{\\mathcal{R}(p,q)}\\theta^n \\big(1\\ominus \\theta \\big)^{r}_{\\mathcal{R}(p,q)},\\ \\ n\\in\\mathbb{N}\\cup\\{0\\}.\n\\end{equation*}\nThen, the joint conditional probability function of the random vector $(W_1,W_2,\\ldots,W_k)$, given that $V=n$,\n\\[\nP(W_1=w_1,\\ldots,W_k=w_k|V=n)=\\frac{P(W_1=w_1)\\cdots P(W_k=w_k)P(W_{k+1}=n-v_k)}{P(V=n)},\n\\]\non using these expressions, is obtained as\n\\begin{equation*}\nP(W_1=w_1,\\ldots,W_k=w_k|V=n)=\\big(\\tau_1\\tau_2\\big)^{c_k}\\frac{\\prod_{j=1}^{k+1}\\genfrac{[}{]}{0pt}{}{r_j+w_j-1}{w_j}_{\\mathcal{R}(p,q)}}{\\genfrac{[}{]}{0pt}{}{r+n-1}{n}_{\\mathcal{R}(p,q)}},\n\\end{equation*}\nwhere\n\\[\nc_k=\\sum_{i=1}^kw_is_{i-1}+(n-v_k)s_k, \\ \\ v_j=\\sum_{i=1}^jw_i,\\ \\ j\\in\\{1,2,\\ldots,k\\}.\n\\]\nThus, after some algebra, it reduces to\n\\[\nc_k=n\\sum_{j=1}^kr_j-\\sum_{i=1}^kw_i(s_k-s_{i-1})=n\\sum_{j=1}^kr_j-\\sum_{j=1}^kr_jv_j=\\sum_{j=1}^kr_j(n-v_j)\n\\]\nand the proof of (\\ref{eq3.9}) is completed.\n\\end{proof}\n\\subsubsection{Bivariate negative $\\mathcal{R}(p,q)$-hypergeometric distribution}\nLet $\\underline{V}=\\big(V_1,V_2\\big)$ be the $\\mathcal{R}(p,q)$- random vector. Then, the probability function  of  the bivariate  negative $\\mathcal{R}(p,q)$- hypergeometric probability distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is given by the following relation:\n\\begin{small}\n\t\\begin{equation*}\n\tP(V_1=v_1,V_2=v_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{v_1,v_2}_{\\mathcal{R}(p^{-1},q^{-1})}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{x_j,\\mathcal{R}(p^{-1},q^{-1})}}{ [\\beta]_{n,\\mathcal{R}(p^{-1},q^{-1})}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p,q)=\\tau^{-\\sum_{j=1}^{2}v_{j}(\\beta_{j+1}-v_{j+1})}_1\\tau_2^{-\\sum_{j=1}^{2}(n-x_j)(\\beta_j-v_j)},$\n$v_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $v_1+v_2\\leq n$, $v_{3}=n-v_1+v_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=v_1,$\nand $x_2=v_1+v_2.$\n\\begin{proposition}\n\tThe $\\mathcal{R}(p,q)$-factorial moments of the bivariate negative  $\\mathcal{R}(p,q)$-hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ are derived as follows:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([V_1]_{i_1,\\mathcal{R}(p,q)}\\big)=\\frac{[n]_{i_1,\\mathcal{R}(p,q)}[\\beta_1]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}}{[\\beta]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([V_2]_{i_1,\\mathcal{R}(p,q)}|V_1=v_1\\big)=\\frac{[n-v_1]_{i_2,\\mathcal{R}(p,q)}[\\beta_2]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}}{[\\beta-\\beta_1]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}},\\,i_2\\in\\{0,1,\\cdots,n-v_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([V_2]_{i_2,\\mathcal{R}(p,q)}\\big)=\\frac{[n]_{i_2,\\mathcal{R}(p,q)}[\\beta_2]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}\\tau^{-i_2\\beta_1}_2}{[\\beta]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([V_1]_{i_1,\\mathcal{R}(p,q)}[V_2]_{i_2,\\mathcal{R}(p,q)}\\big)=\\frac{[n]_{i_1+i_2,\\mathcal{R}(p,q)}[\\beta_1]_{i_1,\\mathcal{R}(p^{-1},q^{-1})}[\\beta_2]_{i_2,\\mathcal{R}(p^{-1},q^{-1})}}{\\tau^{i_2\\beta_1}_2\\,[\\beta]_{i_1+i_2,\\mathcal{R}(p^{-1},q^{-1})}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\n\\end{proposition}\n\\begin{corollary}\n The $\\mathcal{R}(p,q)$- covariance of $[V_1]_{\\mathcal{R}(p,q)}$ and $[V_2]_{\\mathcal{R}(p,q)}$ is determined by:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([V_1]_{\\mathcal{R}(p,q)},[V_2]_{\\mathcal{R}(p,q)}\\big)=\\frac{[n]_{\\mathcal{R}(p,q)}[\\beta_1]_{\\mathcal{R}(p^{-1},q^{-1})}[\\beta_2]_{\\mathcal{R}(p^{-1},q^{-1})}}{\\tau^{\\beta_1}_2[\\beta]_{\\mathcal{R}(p^{-1},q^{-1})}}\\Delta(n,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,\\beta)=\\frac{[n-1]_{\\mathcal{R}(p,q)}}{[\\beta-1]_{\\mathcal{R}(p^{-1},q^{-1})}}-\\frac{[n]_{\\mathcal{R}(p,q)}}{[\\beta_1]_{\\mathcal{R}(p^{-1},q^{-1})}},\n\\end{eqnarray*}\nand  $\\underline{V}=\\big(\nV_1,V_2\\big)$ is a $\\mathcal{R}(p,q)$-random vector obeying   the bivariate negative $\\mathcal{R}(p,q)$-hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\\end{corollary}\n\\begin{remark}\nThe bivariate negative hypergeometric distribution related to the quantum algebras is also interesting for the lecture. Then, \n\\begin{enumerate}\n\t\\item[(a)]The mass function  of  the bivariate  negative $q$- hypergeometric probability distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$  and $q,$ is given by the following relation:\n\t\\begin{small}\n\t\t\\begin{equation*}\n\t\tP(V_1=v_1,V_2=v_2)=\\Psi_2(q)\\genfrac{[}{]}{0pt}{}{n}{v_1,v_2}_{q^{-1}}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,q^{-1}}}{ [\\beta]_{n,q^{-1}}},\n\t\t\\end{equation*}\n\t\\end{small}\n\twhere $\\Psi_2(q)=q^{-\\sum_{j=1}^{2}v_{j}(\\beta_{j+1}-v_{j+1})}q^{\\sum_{j=1}^{2}(n-x_j)(\\beta_j-v_j)},$\n\t$v_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $v_1+v_2\\leq n$, $v_{3}=n-v_1+v_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=v_1,$\n\tand $x_2=v_1+v_2.$\n\tFurthermore, its \n\tfactorial moments  are derived as follows:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([V_1]_{i_1,q}\\big)=\\frac{[n]_{i_1,q}[\\beta_1]_{i_1,q^{-1}}}{[\\beta]_{i_1,q^{-1}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([V_2]_{i_1,q}|V_1=v_1\\big)=\\frac{[n-v_1]_{i_2,q}[\\beta_2]_{i_2,q^{-1}}}{[\\beta-\\beta_1]_{i_2,q^{-1}}},\\,i_2\\in\\{0,1,\\cdots,n-v_1\\},\n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([V_2]_{i_2,q}\\big)=\\frac{[n]_{i_2,q}[\\beta_2]_{i_2,q^{-1}}q^{i_2\\beta_1}}{[\\beta]_{i_2,q^{-1}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\t\\end{eqnarray*}\n\t\tand\n\t\t\\begin{eqnarray*}\n\t\t\tE\\big([V_1]_{i_1,q}[V_2]_{i_2,q}\\big)=\\frac{[n]_{i_1+i_2,q}[\\beta_1]_{i_1,q^{-1}}[\\beta_2]_{i_2,q^{-1}}}{q^{-i_2\\beta_1}\\,[\\beta]_{i_1+i_2,q^{-1}}},\n\t\t\\end{eqnarray*}\n\t\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\t\\end{small}\n\tMoreover, \n\tthe $q$- covariance of $[V_1]_{q}$ and $[V_2]_{q}$ is determined by:\n\t\\begin{small}\n\t\t\\begin{eqnarray*}\n\t\t\tCov\\big([V_1]_{q},[V_2]_{q}\\big)=\\frac{[n]_{q}[\\beta_1]_{q^{-1}}[\\beta_2]_{q^{-1}}}{q^{-\\beta_1}[\\beta]_{q^{-1}}}\\Delta(n,\\beta),\n\t\t\\end{eqnarray*}\n\t\twhere \n\t\t\\begin{eqnarray*}\n\t\t\t\\Delta(n,\\beta)=\\frac{[n-1]_{q}}{[\\beta-1]_{q^{-1}}}-\\frac{[n]_{q}}{[\\beta_1]_{q^{-1}}},\n\t\t\\end{eqnarray*}\n\t\tand  $\\underline{V}=\\big(\n\t\tV_1,V_2\\big)$ is a $q$-random vector obeying   the bivariate negative $q$-hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$  and $q.$\n\t\\end{small}\n\\item[(b)]\nThe mass function  of  the bivariate  negative $(p,q)$- hypergeometric probability distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is given by the following relation:\n\\begin{small}\n\t\\begin{equation*}\n\tP(V_1=v_1,V_2=v_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{v_1,v_2}_{p^{-1},q^{-1}}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,p^{-1},q^{-1}}}{ [\\beta]_{n,p^{-1},q^{-1}}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p,q)=p^{-\\sum_{j=1}^{2}v_{j}(\\beta_{j+1}-v_{j+1})}q^{-\\sum_{j=1}^{2}(n-x_j)(\\beta_j-v_j)},$\n$v_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $v_1+v_2\\leq n$, $v_{3}=n-v_1+v_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=v_1,$\nand $x_2=v_1+v_2.$\nFurthermore, its \n\t $(p,q)$-factorial moments  are derived as follows:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([V_1]_{i_1,p,q}\\big)=\\frac{[n]_{i_1,p,q}[\\beta_1]_{i_1,p^{-1},q^{-1}}}{[\\beta]_{i_1,p^{-1},q^{-1}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([V_2]_{i_1,p,q}|V_1=v_1\\big)=\\frac{[n-v_1]_{i_2,p,q}[\\beta_2]_{i_2,p^{-1},q^{-1}}}{[\\beta-\\beta_1]_{i_2,p^{-1},q^{-1}}},\\,i_2\\in\\{0,1,\\cdots,n-v_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([V_2]_{i_2,p,q}\\big)=\\frac{[n]_{i_2,p,q}[\\beta_2]_{i_2,p^{-1},q^{-1}}q^{-i_2\\beta_1}}{[\\beta]_{i_2,p^{-1},q^{-1}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([V_1]_{i_1,p,q}[V_2]_{i_2,p,q}\\big)=\\frac{[n]_{i_1+i_2,p,q}[\\beta_1]_{i_1,p^{-1},q^{-1}}[\\beta_2]_{i_2,p^{-1},q^{-1}}}{q^{i_2\\beta_1}\\,[\\beta]_{i_1+i_2,p^{-1},q^{-1}}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nMoreover, \n the $(p,q)$- covariance of $[V_1]_{p,q}$ and $[V_2]_{p,q}$ is determined by:\n\\begin{small}\n\\begin{eqnarray*}\nCov\\big([V_1]_{p,q},[V_2]_{p,q}\\big)=\\frac{[n]_{p,q}[\\beta_1]_{p^{-1},q^{-1}}[\\beta_2]_{p^{-1},q^{-1}}}{q^{\\beta_1}[\\beta]_{p^{-1},q^{-1}}}\\Delta(n,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta(n,\\beta)=\\frac{[n-1]_{p,q}}{[\\beta-1]_{p^{-1},q^{-1}}}-\\frac{[n]_{p,q}}{[\\beta_1]_{p^{-1},q^{-1}}},\n\\end{eqnarray*}\nand  $\\underline{V}=\\big(\nV_1,V_2\\big)$ is a $(p,q)$-random vector obeying   the bivariate negative $(p,q)$-hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\\item[(c)]\nThe mass function  of  the bivariate  negative $(p,q)$- hypergeometric probability distribution with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is given by the following relation:\n\\begin{small}\n\t\\begin{equation*}\n\tP(V_1=v_1,V_2=v_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{v_1,v_2}_{p^{-1},q^{-1}}\\frac{\\prod_{j=1}^{3}[\\beta_j]_{y_j,p^{-1},q^{-1}}}{ [\\beta]_{n,p^{-1},q^{-1}}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p,q)=p^{-\\sum_{j=1}^{2}v_{j}(\\beta_{j+1}-v_{j+1})}q^{-\\sum_{j=1}^{2}(n-x_j)(\\beta_j-v_j)},$\n$v_j\\in\\{0,1,\\ldots,n\\}$, $j\\in\\{1,2\\},$  $v_1+v_2\\leq n$, $v_{3}=n-v_1+v_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=v_1,$\nand $x_2=v_1+v_2.$\nFurthermore, its \n$(p,q)$-factorial moments  are derived as follows:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tE\\big([V_1]_{i_1,p,q}\\big)=\\frac{[n]_{i_1,p,q}[\\beta_1]_{i_1,p^{-1},q^{-1}}}{[\\beta]_{i_1,p^{-1},q^{-1}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([V_2]_{i_1,p,q}|V_1=v_1\\big)=\\frac{[n-v_1]_{i_2,p,q}[\\beta_2]_{i_2,p^{-1},q^{-1}}}{[\\beta-\\beta_1]_{i_2,p^{-1},q^{-1}}},\\,i_2\\in\\{0,1,\\cdots,n-v_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tE\\big([V_2]_{i_2,p,q}\\big)=\\frac{[n]_{i_2,p,q}[\\beta_2]_{i_2,p^{-1},q^{-1}}q^{-i_2\\beta_1}}{[\\beta]_{i_2,p^{-1},q^{-1}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\tE\\big([V_1]_{i_1,p,q}[V_2]_{i_2,p,q}\\big)=\\frac{[n]_{i_1+i_2,p,q}[\\beta_1]_{i_1,p^{-1},q^{-1}}[\\beta_2]_{i_2,p^{-1},q^{-1}}}{q^{i_2\\beta_1}\\,[\\beta]_{i_1+i_2,p^{-1},q^{-1}}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\\end{small}\nMoreover, \nthe $(p,q)$- covariance of $[V_1]_{p,q}$ and $[V_2]_{p,q}$ is determined by:\n\\begin{small}\n\t\\begin{eqnarray*}\n\t\tCov\\big([V_1]_{p,q},[V_2]_{p,q}\\big)=\\frac{[n]_{p,q}[\\beta_1]_{p^{-1},q^{-1}}[\\beta_2]_{p^{-1},q^{-1}}}{q^{\\beta_1}[\\beta]_{p^{-1},q^{-1}}}\\Delta(n,\\beta),\n\t\\end{eqnarray*}\n\twhere \n\t\\begin{eqnarray*}\n\t\t\\Delta(n,\\beta)=\\frac{[n-1]_{p,q}}{[\\beta-1]_{p^{-1},q^{-1}}}-\\frac{[n]_{p,q}}{[\\beta_1]_{p^{-1},q^{-1}}},\n\t\\end{eqnarray*}\n\tand  $\\underline{V}=\\big(\n\tV_1,V_2\\big)$ is a $(p,q)$-random vector obeying   the bivariate negative $(p,q)$-hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\\item[(d)]The probability  function  of  the bivariate  negative generalized $q$- Quesne  hypergeometric probability distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q,$ is deduced as :\n\\begin{small}\n\t\\begin{equation*}\n\tP(V_1=v_1,V_2=v_2)=\\Psi_2(p,q)\\genfrac{[}{]}{0pt}{}{n}{v_1,v_2}^Q_{p^{-1},q^{-1}}\\frac{\\prod_{j=1}^{3}[\\beta_j]^Q_{y_j,p^{-1},q^{-1}}}{ [\\beta]^Q_{n,p^{-1},q^{-1}}},\n\t\\end{equation*}\n\\end{small}\nwhere $\\Psi_2(p,q)=p^{-\\sum_{j=1}^{2}v_{j}(\\beta_{j+1}-v_{j+1})}q^{\\sum_{j=1}^{2}(n-x_j)(\\beta_j-v_j)},$\n$v_j\\in\\{0,1,\\cdots,n\\}$, $j\\in\\{1,2\\},$  $v_1+v_2\\leq n$, $v_{3}=n-v_1+v_2$, $\\beta_{3}=\\beta-\\beta_1+\\beta_2$, $x_1=v_1,$\nand $x_2=v_1+v_2.$\nFurthermore, its \n\t generalized $q-$ Quesne factorial moments  are derived as follows:\n\t\\begin{small}\n\t\\begin{eqnarray*}\n\tE\\big([V_1]^Q_{i_1,p,q}\\big)=\\frac{[n]^Q_{i_1,p,q}[\\beta_1]^Q_{i_1,p^{-1},q^{-1}}}{[\\beta]^Q_{i_1,p^{-1},q^{-1}}},\\,i_1\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([V_2]^Q_{i_1,p,q}|V_1=v_1\\big)=\\frac{[n-v_1]^Q_{i_2,p,q}[\\beta_2]^Q_{i_2,p^{-1},q^{-1}}}{[\\beta-\\beta_1]^Q_{i_2,p^{-1},q^{-1}}},\\,i_2\\in\\{0,1,\\cdots,n-v_1\\},\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\tE\\big([V_2]^Q_{i_2,p,q}\\big)=\\frac{[n]^Q_{i_2,p,q}[\\beta_2]^Q_{i_2,p^{-1},q^{-1}}\\,q^{i_2\\beta_1}}{[\\beta]^Q_{i_2,p^{-1},q^{-1}}},\\,\\,i_2\\in\\{0,1,\\cdots,n\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\tE\\big([V_1]^Q_{i_1,p,q}[V_2]^Q_{i_2,p,q}\\big)=\\frac{[n]^Q_{i_1+i_2,p,q}[\\beta_1]^Q_{i_1,p^{-1},q^{-1}}[\\beta_2]^Q_{i_2,p^{-1},q^{-1}}}{q^{-i_2\\beta_1}\\,[\\beta]^Q_{i_1+i_2,p^{-1},q^{-1}}},\n\t\\end{eqnarray*}\n\twhere $i_1\\in\\{0,1,\\cdots,n-i_2\\}$ and  $i_2\\in\\{0,1,\\cdots,n\\}.$\n\t\nBesides, \n the generalized $q$- Quesne covariance of $[V_1]^Q_{p,q}$ and $[V_2]^Q_{p,q}$ is given by:\n\\begin{eqnarray*}\nCov\\big([V_1]^Q_{p,q},[V_2]^Q_{p,q}\\big)=\\frac{[n]^Q_{p,q}[\\beta_1]^Q_{p^{-1},q^{-1}}[\\beta_2]^Q_{p^{-1},q^{-1}}}{q^{-\\beta_1}[\\beta]^Q_{p^{-1},q^{-1}}}\\Delta^Q(n,\\beta),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n\\Delta^Q(n,\\beta)=\\frac{[n-1]^Q_{p,q}}{[\\beta-1]^Q_{p^{-1},q^{-1}}}-\\frac{[n]^Q_{p,q}}{[\\beta_1]^Q_{p^{-1},q^{-1}}},\n\\end{eqnarray*}\nand  $\\underline{V}=\\big(\nV_1,V_2\\big)$ is a generalized $q$- Quesne random vector verifying   the bivariate negative generalized $q$- Quesne hypergeometric  distribution, with parameters $n$, $\\underline{\\beta}=(\\beta_1,\\beta_2),$ $p$ and $q.$\n\\end{small}\n\\end{enumerate}\n\\end{remark}\n\\section*{Acknowledgements}\nThis research was partly supported by the SNF Grant No. IZSEZ0\\_206010.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe say that a tree is \\emph{labeled} if each node is assigned a\ncharacter from an alphabet $\\Sigma$. Given two sequences of\nlabeled nodes $p$ and $t$, we say that $p$ is a \\emph{subsequence}\nof $t$, denoted $p \\sqsubseteq t$, if $p$ can be obtained by\nremoving nodes from $t$. Given two rooted, labeled trees $P$ and\n$T$ the \\emph{tree path subsequence problem} (TPS) is to determine\nwhich paths in $P$ are subsequences of which paths in $T$. Here a path\nbegins at the root and ends at a leaf. That is, for each path $p$ \nin $P$ we must report all paths $t$ in $T$ such that $p \\sqsubseteq t$.\n\nThis problem was introduced by Chen~\\cite{Chen2000} who gave an\nalgorithm using $O(\\min(l_{P}n_{T} + n_P, n_{P}l_{T}+n_T))$ time and\n$O(l_{P}d_{T} + n_P + n_T)$ space. Here, $n_{S}$, $l_{S}$, and $d_{S}$ denotes\nthe number of nodes, number of leaves, and depth, respectively, of\na tree $S$. Note that in the worst-case this is quadratic time and\nspace. In this paper we present improved algorithms giving the following result:\n\\begin{theorem}\\label{main}\nFor trees $P$ and $T$ the tree path subsequence problem can be \nsolved in $O(n_{P} + n_{T})$ space with the following running times:\n\\begin{equation*}\n\\min\n\\begin{cases}\n    O(l_{P}n_{T} + n_P) , \\\\\n     O(n_{P}l_{T}+n_T) , \\\\\n     O(\\frac{n_{P}n_{T}}{\\log n_{T}}+ n_T + n_P \\log n_P).\n\\end{cases}\n\\end{equation*}\n\\end{theorem}\nThe first two bounds in Theorem~\\ref{main} match the previous time bounds while improving the space to linear. This is achieved using a algorithm that resembles the algorithm of Chen~\\cite{Chen2000}. At a high level, the algorithms are essentially identical and therefore the bounds should be regarded as an improved analysis of Chen's algorithm. The latter bound is obtained by using an entirely new algorithm that improves the worst-case quadratic time. Specifically, whenever $\\log n_{P} = O(n_{T}\/ \\log n_{T})$ the running time is improved by a logarithmic factor. Note that -- in the worst-case -- the number of pairs consisting of a path from $P$ and a path $T$ is $\\Omega(n_{P}n_{T})$, and therefore we need at least as many bits to report the solution to TPS. Hence, on a RAM with logarithmic word size our worst-case bound is optimal. Most importantly, all our algorithms use linear space. For practical applications this will likely make it possible to solve TPS on large trees and improve running time since more of the computation can be kept in main memory.\n\n\n\\subsection{Applications} We propose TPS as a useful query\nprimitive for XML data. The key idea is that an XML document $D$\nmay be viewed as a rooted, labeled tree (different nodes can be assigned the same label).\n\\begin{figure}[t]\n\\begin{center}\n  \\begin{psmatrix}[colsep=0.5cm,rowsep=0.3cm,labelsep=1pt, nodesep=1pt]\n  &&&&&&&&[name=cat] catalog \\\\\n \n  &&[name=book] book &&&&&&[name=book2] book & [name=book3] & [name=book4]\\\\\n \n  & [name=author] author &&[name=chapter] chapter&&\n  [name=author2] author && [name=chapter2] chapter && [name=chapter3] chapter \\\\\n \n  & [name=john] John & [name=paul] Paul & [name=xml] XML &&\n  [name=name] name && [name=title] title  & [name=section] section & [name=title2] title \\\\\n \n  &&&&&  [name=john2] John && [name=DB] databases & [name=xml2] XML & [name=queries] queries \\\\\n \n  && (a) &&&&&& (b)\n \n  \\ncline{author}{book} \\ncline{chapter}{book}\n  \\ncline{john}{author} \\ncline{paul}{author} \\ncline{xml}{chapter}\n \n  \\ncline{cat}{book2}\\psset{linestyle=dashed,dash=2pt 4pt}\\ncline{cat}{book3} \\ncline{cat}{book4}\n  \\psset{linestyle=solid}\n  \\ncline{book2}{author2}\\ncline{book2}{chapter2}\\ncline{book2}{chapter3}\n  \\ncline{author2}{name}\\ncline{chapter2}{title}\\ncline{chapter2}{section}\\ncline{chapter3}{title2}\n  \\ncline{name}{john2}\\ncline{title}{DB}\\ncline{section}{xml2}\\ncline{title2}{queries}\n  \\end{psmatrix}\n   \\caption{(a) The trie of queries 1,2,3, or the tree for query 4. (b)\n   A fragment of a catalog of books.}\n  \\label{xmlexample}\n  \\end{center}\n\\end{figure}\nFor example, suppose that we want to maintain a catalog of books\nfor a bookstore. A fragment of a possible XML tree, denoted $D$,\ncorresponding to the catalog is shown in Fig.~\\ref{xmlexample}(b).\nIn addition to supporting full-text queries, such as find all\ndocuments containing the word ``John'', we can also use the tree\nstructure of the catalog to ask more specific queries, such as the\nfollowing examples:\n\\begin{enumerate}\n  \\item Find all books written by John,\n  \\item find all books written by Paul,\n  \\item find all books with a chapter that has something to do with XML, or\n  \\item find all books written by John and Paul with a chapter that has something to do with XML.\n\\end{enumerate}\nThe queries 1,2, and 3 correspond to a \\emph{path query} on $D$,\nthat is, compute which paths in $D$ that contains a specific path\nas a subsequence. For instance, computing the paths in $D$ that\ncontain the path of three nodes labeled ``book'', ``chapter'', and\n``XML'', respectively, effectively answers query 3. Most XML-query\nlanguages, such as XPath \\cite{CD1999}, support such queries.\n\nUsing a depth-first traversal of $D$ a path query can be solved in linear time.\nMore precisely, if $q$ is a path consisting of $n_{q}$ nodes, answering the path query on $D$ takes $O(n_{q} + n_{D})$ time. Hence, if we are given path queries $q_{1}, \\ldots, q_{k}$ we can\nanswer them in $O(n_{q_{1}} + \\cdots + n_{q_{k}} + kn_{D})$ time. However, we can do better by constructing the \\emph{trie}, $Q$, of $q_{1}, \\ldots, q_{k}$. The trie $Q$ has labels on the nodes and is constructed such there is one node for every common prefix of $q_{1}, \\ldots, q_{k}$. \nAnswering all path queries now correspond to solving TPS on $Q$ and $D$. As an example the queries 1,2, and 3 form the trie shown in Fig.~\\ref{xmlexample}(a). As $l_Q \\leq k$, Theorem~\\ref{main} gives us an algorithm with running time \n\\begin{equation}\n\\label{bound}\nO\\left(n_{q_{1}} + \\cdots + n_{q_{k}} + \\min\\left(kn_D + n_Q, n_Ql_D + n_D, \\frac{n_Qn_D}{\\log n_D} + n_D + n_Q\\log n_Q\\right)\\right). \n\\end{equation}\nSince $n_Q \\leq n_{q_{1}} + \\cdots + n_{q_{k}}$ this is at least as good as answering the queries individually and better in many cases. If many paths share a prefix, i.e., queries 1 and 2 share \"book\" and \"author\", \nthe size of $n_Q$ can be much smaller than $n_{q_{1}} + \\cdots + n_{q_{k}}$. Using our solution to TPS we can efficiently take advantage of this situation since the latter two terms in \\eqref{bound} depend on $n_Q$ and not on $n_{q_{1}} + \\cdots + n_{q_{k}}$.  \n\nNext consider query 4. This query cannot be answered by solving a\nTPS problem but is an instance of the \\emph{tree inclusion\nproblem} (TI). Here we want to decide if $P$ is \\emph{included} in\n$T$, that is, if $P$ can be obtained from $T$ by \\emph{deleting}\nnodes of $T$. Deleting a node $y$ in $T$ means making the children\nof $y$ children of the parent of $y$ and then removing $y$. It is\nstraightforward to check that we can answer query 4 by deciding if\nthe tree in Fig.~\\ref{xmlexample}(a) can be included in the tree\nin Fig.~\\ref{xmlexample}(b).\n\nRecently, TI has been recognized as an important XML query\nprimitive and has recieved considerable attention, see e.g.,\n\\cite{SM2002, YLH2003,YLH2004, ZADR03, SN2000, TRS2002}.\nUnfortunately, TI is NP-complete in general \n\\cite{KM1995} and therefore the existing algorithms are based on\nheuristics. Observe that a necessary condition for $P$ to included\nin $T$ is that all paths in $P$ are subsequences of paths in $T$.\nHence, we can use TPS to quickly identify trees or parts of trees that cannot be\nincluded $T$. We believe that in this way TPS can be used as an\neffective \"filter\" for many tree inclusion problems that occur in\npractice.\n\nWe note that for \\emph{ordered} trees, that is, a left-to-right\nordering among siblings is given, the tree inclusion problem can\nbe solved in polynomial time~\\cite{KM1995, BG2005}. In this case\ndeleting a node $y$ inserts the children of $y$ in the place of in\nthe left-to-right order among the siblings of $y$.\n\n\\subsection{Technical Overview} Given two strings (or labeled\npaths) $a$ and $b$, it is straightforward to determine if $a$ is a\nsubsequence of $b$ by scanning the character from left to right in\n$b$. This uses $O(|a| + |b|)$ time. We can solve TPS by applying\nthis algorithm to each of the pair of paths in $P$ and $T$,\nhowever, this may use as much as $O(n_Pn_T(n_P + n_T))$ time.\nAlternatively, Baeza-Yates \\cite{BaezaYates1991} showed how to\npreprocess $b$ in $O(|b|\\log |b|)$ time such that testing whether\n$a$ is a subsequence of $b$ can be done in $O(|a|\\log |b|)$ time.\nUsing this data structure on each path in $T$ we can solve the TPS\nproblem, however, this may take as much as $O(n^2_{T}\\log n_{T} +\nn_{P}^{2}\\log n_{T})$. Hence, none of the availiable subsequence algorithms\non strings provide an immediate efficient solution to TPS. \n\nInspired by the work of Chen~\\cite{Chen2000} we take another\napproach. We provide a framework for solving TPS. The main\nidea is to traverse $T$ while maintaining a subset of nodes in\n$P$, called the \\emph{state}. When reaching a leaf $z$ in $T$ the\nstate represents the paths in $P$ that are subsequences of the\npath from the root to $z$. At each step the state is updated using a simple\nprocedure processing a subset of nodes. The result of\nTheorem~\\ref{main} is obtained by taking the best of two\nalgorithms based on our framework: The first one uses a simple\ndata structure to maintain the state. This leads to an algorithm\nusing $O(\\min(l_{P}n_{T} + n_P, n_{P}l_{T}+n_T))$ time. At a high level this\nalgorithm resembles the algorithm of Chen~\\cite{Chen2000} and\nachieves the same running time. However, we improve the analysis\nof the algorithm and show a space bound of  $O(n_P +\nn_T)$. This should be compared to the worst-case quadratic space\nbound of $O(l_{P}d_{T} + n_P + n_T)$ given by Chen~\\cite{Chen2000}. Our second\nalgorithm takes a different approach combining several \ntechniques. Starting with a simple quadratic time and space\nalgorithm, we show how to reduce the space to $O(n_P \\log n_T)$\nusing a decomposition of $T$ into disjoint paths. We then divide\n$P$ into small subtrees of logarithmic size called \\emph{micro\ntrees}. The micro trees are then  preprocessed such that subsets of\nnodes in a micro tree can be maintained in constant time and\nspace. Intuitively, this leads to a logarithmic improvement of the time and space bounds.\n\n\n\n\\subsection{Notation and Definitions} In this section we define\nthe notation and definitions we will use throughout the paper. For\na graph $G$ we denote the set of nodes and edges by $V(G)$ and\n$E(G)$, respectively. Let $T$ be a rooted tree. The root of $T$ is\ndenoted by $\\ensuremath{\\mathrm{root}}(T)$. The \\emph{size} of $T$, denoted by $n_T$,\nis $|V(T)|$. The \\emph{depth} of a node $y\\in V(T)$, $\\ensuremath{\\mathrm{depth}}(y)$,\nis the number of edges on the path from $y$ to $\\ensuremath{\\mathrm{root}}(T)$ and the\ndepth of $T$, denoted $d_T$, is the maximum depth of any node in\n$T$. The parent of $y$ is denoted $\\ensuremath{\\mathrm{parent}}(y)$. A node with no\nchildren is a leaf and otherwise it is an internal node. The\nnumber of leaves in $T$ is denoted $l_T$. Let $T(y)$ denote the\nsubtree of $T$ rooted at a node $y \\in V(T)$. If $z\\in V(T(y))$\nthen $y$ is an ancestor of $z$ and if $z\\in V(T(y))\\backslash\n\\{y\\}$ then $y$ is a proper ancestor of $z$. If $y$ is a (proper)\nancestor of $z$ then $z$ is a (proper) descendant of $y$. We say\nthat $T$ is \\emph{labeled} if each node $y$ is assigned a\ncharacter, denoted $\\ensuremath{\\mathrm{label}}(y)$, from an alphabet $\\Sigma$. The path\nfrom $y$ to $\\ensuremath{\\mathrm{root}}(T)$, of nodes $\\ensuremath{\\mathrm{root}}(T) = y_1, \\ldots, y_k =\ny$ is denoted $\\ensuremath{\\mathrm{path}}(y)$. Hence, we can formally state TPS as\nfollows: Given two rooted tree $P$ and $T$ with leaves $x_1,\n\\ldots, x_r$ and $y_1, \\ldots, y_s$, respectively, determine all\npairs $(i,j)$ such that $\\ensuremath{\\mathrm{path}}(x_{i}) \\sqsubseteq \\ensuremath{\\mathrm{path}}(y_{j})$.\nFor simplicity we will assume that leaves in $P$ and $T$ are\nalways numbered as above and we identify each of the paths by the\nnumber of the corresponding leaf.\n\nThroughout the paper we assume a unit-cost RAM model of computation\nwith word size $\\Theta(\\log n_T)$ and a standard instruction set including bitwise boolean operations, shifts, addition and multiplication. All space complexities refer to the number of words used by the algorithm.\n\n\\section{A Framework for solving TPS}\nIn this section we present a simple general\nalgorithm for the tree path subsequence problem. The key\ningredient in our algorithm is the following procedure. For any $X\n\\subseteq V(P)$ and $y \\in V(T)$ define:\n\\begin{relate}\n\\item[$\\ensuremath{\\textsc{Down}}(X,y)$:] Return the set\n    $\\ensuremath{\\textsc{Child}}(\\{x \\in X \\mid \\ensuremath{\\mathrm{label}}(x) = \\ensuremath{\\mathrm{label}}(y)\\})\n    \\cup \\{x \\in X \\mid \\ensuremath{\\mathrm{label}}(x) \\neq \\ensuremath{\\mathrm{label}}(y)\\}$.\n\\end{relate}\nThe notation $\\ensuremath{\\textsc{Child}}(X)$ denotes the set of children of $X$. \nHence, $\\ensuremath{\\textsc{Down}}(X,y)$ is the set consisting of nodes in $X$ with a\ndifferent label than $y$ and the children of the nodes $X$ with\nthe same label as $y$. We will now show how to solve TPS using\nthis procedure.\n\nFirst assign a unique number in the range $\\{1,\\ldots,l_{P}\\}$ to\neach leaf in $P$. Then, for each $i$, $1\\leq i \\leq l_{P}$, add a\n\\emph{pseudo-leaf} $\\bot_{i}$ as the single child of the $i$th\nleaf. All pseudo-leaves are assigned a special label $\\beta\n\\not\\in \\Sigma$. The algorithm traverses $T$ in a depth first\norder and computes at each node $y$ the set $X_{y}$. We call this\nset the \\emph{state} at $y$. Initially, the state consists of\n$\\{\\ensuremath{\\mathrm{root}}(P)\\}$. For $z \\in \\ensuremath{\\mathrm{child}}(y)$, the state $X_z$ can be\ncomputed from state $X_y$ as \n\\begin{equation*}\nX_{z} =   \\ensuremath{\\textsc{Down}}(X_{y}, z).\n\\end{equation*}\nIf $z$ is a leaf we report the number of each pseudo-leaf in $X_z$\nas the paths in $P$ that are subsequences of $\\ensuremath{\\mathrm{path}}(z)$. See\nFigure~\\ref{fig:framework}  for an example. To show\nthe correctness of this approach we need the following lemma.\n\n\\begin{figure}[tb]\n\\begin{center}\n  %\n\\begin{psmatrix}[colsep=0.6cm,rowsep=0.3cm,labelsep=3pt]\n \n  && \\cnode{.2}{rootp}\\rput(0,0){$a$}\\rput(0,.4){$\\ensuremath{\\mathrm{root}}(P)$} &&&&&&\n \n  \\cnode{.2}{roott}\\rput(0,0){$a$}\\rput(0,.4){$\\ensuremath{\\mathrm{root}}(T)$}\n  \\\\\n \n  &\\cnode{.2}{v1}\\rput(0,0){$c$}\\rput(.5,0){$x_1$} & &\n  \\cnode{.2}{v2}\\rput(0,0){$b$}\\rput(.5,0){$x_2$}\n \n  &&&&& \\cnode{.2}{w1}\\rput(0,0){$c$}\\rput(.4,0){$1$}\\\\\n \n &\\cnode{.2}{v3}\\rput(0,0){$a$}\\rput(.5,0){$x_3$} & &\n  [name=b2]\\psframebox{}\\rput(.3,0){$\\bot_2$}\n \n  &&&&\\cnode{.2}{w2}\\rput(0,0){$a$}\\rput(.4,0){$2$} & &\n  \\cnode{.2}{w3}\\rput(0,0){$b$}\\rput(.4,0){$4$}\n  \\\\\n \n  &[name=b1] \\psframebox{}\\rput(.3,0){$\\bot_1$}\n \n  &&&&&&\\cnode{.2}{w4}\\rput(0,0){$b$}\\rput(.4,0){$3$} & &\n  \\cnode{.2}{w5}\\rput(0,0){$b$}\\rput(.4,0){$5$}\n  \\\\\n  && $P$ &&&&&& $T$\n\n \n  \\ncline{rootp}{v1}\\ncline{rootp}{v2}\n  \\ncline{v1}{v3}\\ncline{v2}{b2}\\ncline{v3}{b1}\n  \\ncline{roott}{w1}\n  \\ncline{w1}{w3}\\ncline{w1}{w2}\\ncline{w2}{w4}\n    \\ncline{w3}{w5}\n  \\end{psmatrix}\n  \\caption{The letters inside the nodes are the labels, and the\n  identifier of each node is written outside the node. Initially we have $X=\\{\\ensuremath{\\mathrm{root}}(P)\\}$.\n  Since $\\ensuremath{\\mathrm{label}}(\\ensuremath{\\mathrm{root}}(P))=a=\\ensuremath{\\mathrm{label}}(\\ensuremath{\\mathrm{root}}(T))$ we replace $\\ensuremath{\\mathrm{root}}(P)$ with is children and\n  get\n  $X_{\\ensuremath{\\mathrm{root}}(T)}=\\{x_1,x_2\\}$. Since\n  $\\ensuremath{\\mathrm{label}}(1)=\\ensuremath{\\mathrm{label}}(x_1)\\neq\\ensuremath{\\mathrm{label}}(x_2)$ we get\n  $X_1=\\{x_3,x_2\\}$. Continuing this way we get\n  $X_2=\\{\\bot_1,x_2\\}$, $X_3=\\{\\bot_1,\\bot_2\\}$, $X_4=\\{x_3,\\bot_2\\}$,\n  and $X_5=\\{x_3,\\bot_2\\}$. The nodes $3$ and $5$ are leaves of $T$ and we thus\n  report paths $1$ and $2$ after computing $X_3$ and path $2$ after computing\n  $X_5$.}\\label{fig:framework}\n  \\end{center}\n\\end{figure}\n\\begin{lemma}\\label{lem:invariant}\nFor any node $y \\in V(T)$ the state $X_{y}$ satisfies the\nfollowing property:\n$$x \\in X_{y} \\Rightarrow \\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(x)) \\sqsubseteq \\ensuremath{\\mathrm{path}}(y)\\;.$$\n\\end{lemma}\n\\begin{proof}\nBy induction on the number of iterations of the procedure.\nInitially, $X = \\{\\ensuremath{\\mathrm{root}}(P)\\}$ satisfies the property\n since $\\ensuremath{\\mathrm{root}}(P)$ has no parent. Suppose that\n$X_{y}$ is the current state and $z\\in \\ensuremath{\\mathrm{child}}(y)$ is the next node\nin the depth first traversal of $T$. By the induction hypothesis\n$X_{y}$ satisfies the property, that is, for\nany $x \\in X_{y}$, $\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(x)) \\sqsubseteq \\ensuremath{\\mathrm{path}}(y))$. Then, \n\\begin{equation*}\nX_{z} = \\ensuremath{\\textsc{Down}}(X_{y},z) =  \\ensuremath{\\textsc{Child}}(\\{x \\in X_{y} \\mid \\ensuremath{\\mathrm{label}}(x) =\n\\ensuremath{\\mathrm{label}}(z)\\}) \\cup \\{x \\in X_{y} \\mid \\ensuremath{\\mathrm{label}}(x) \\neq \\ensuremath{\\mathrm{label}}(z)\\}\\;.\n\\end{equation*}\nLet $x$ be a node in $X_y$. There are two cases. If $\\ensuremath{\\mathrm{label}}(x) =\n\\ensuremath{\\mathrm{label}}(z)$ then $\\ensuremath{\\mathrm{path}}(x) \\sqsubseteq \\ensuremath{\\mathrm{path}}(z)$ since\n$\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(x)) \\sqsubseteq \\ensuremath{\\mathrm{path}}(y)$. Hence, for any child\n$x'$ of $x$ we have $\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(x')) \\sqsubseteq \\ensuremath{\\mathrm{path}}(z)$. On\nthe other hand, if $\\ensuremath{\\mathrm{label}}(x) \\neq \\ensuremath{\\mathrm{label}}(z)$ then $x \\in X_z$. Since\n$y=\\ensuremath{\\mathrm{parent}}(z)$ we have $\\ensuremath{\\mathrm{path}}(y) \\sqsubseteq \\ensuremath{\\mathrm{path}}(z)$, and hence\n$\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(x)) \\sqsubseteq \\ensuremath{\\mathrm{path}}(y) \\sqsubseteq \\ensuremath{\\mathrm{path}}(z)$.\n\\end{proof}\nBy the above lemma all paths reported at a leaf $z\n\\in V(T)$ are subsequences of $\\ensuremath{\\mathrm{path}}(z)$. The following lemma shows that the paths reported at a leaf $z \\in\nV(T)$ are \\emph{exactly} the paths in $P$ that are subsequences of $\\ensuremath{\\mathrm{path}}(z)$.\n\\begin{lemma}\\label{lem:correct}\nLet $z$ be a leaf in $T$ and let $\\bot_i$ be a pseudo-leaf in $P$.\nThen,\n\\begin{equation*\n\\bot_i \\in X_{z} \\Leftrightarrow\n\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(\\bot_i)) \\sqsubseteq \\ensuremath{\\mathrm{path}}(z)\\;.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nIt follows immediately from Lemma~\\ref{lem:invariant} that $\\bot_i\n\\in X_{z} \\Rightarrow \\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(\\bot_i)) \\sqsubseteq\n\\ensuremath{\\mathrm{path}}(z)$. It remains to show that $\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(\\bot_i))\n\\sqsubseteq \\ensuremath{\\mathrm{path}}(z) \\Rightarrow \\bot_i \\in X_{z}$. Let\n$\\ensuremath{\\mathrm{path}}(z)=z_1,\\ldots,z_k$, where $z_1=\\ensuremath{\\mathrm{root}}(T)$ and $z_k=z$, and\nlet $\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(\\bot_i))=y_1,\\ldots,y_\\ell$, where\n$y_1=\\ensuremath{\\mathrm{root}}(P)$ and $y_\\ell=\\ensuremath{\\mathrm{parent}}(\\bot_i)$. Since\n$\\ensuremath{\\mathrm{path}}(\\ensuremath{\\mathrm{parent}}(\\bot_i)) \\sqsubseteq \\ensuremath{\\mathrm{path}}(z)$ there are nodes\n$z_{j_i}=y_i$ for $1\\leq i\\leq k$, such that (i) $j_i< j_{i+1}$\nand (ii) there exists no node $z_j$ with $\\ensuremath{\\mathrm{label}}(z_j)=\\ensuremath{\\mathrm{label}}(y_i)$,\nwhere $j_{i-1}<j<j_i$. Initially, $X=\\{\\ensuremath{\\mathrm{root}}(P)\\}$. We have\n$\\ensuremath{\\mathrm{root}}(P) \\in X_{z_j}$ for all $j<j_{1}$, since $z_{j_1}$ is the\nfirst node on $\\ensuremath{\\mathrm{path}}(z)$ with label $\\ensuremath{\\mathrm{label}}(\\ensuremath{\\mathrm{root}}(P))$. When we get\nto $z_{j_1}$, $\\ensuremath{\\mathrm{root}}(P)$ is removed from the state and $y_2$ is\ninserted. Similarly, $y_i$ is in all states $X_{z_j}$ for $j_{i-1}\n\\leq j <j_i$. It follows that $\\bot_i$ is in all states $X_{z_j}$\nwhere $j\\geq j_\\ell$ and thus $\\bot_i \\in X_{z_k}=X_z$.\n\\end{proof}\nThe next lemma can be used to give an upper bound on the number of\nnodes in a state.\n\\begin{lemma}\\label{lem:statesize}\nFor any $y\\in V(T)$ the state $X_y$ has the following property:\nLet $x \\in X_y$. Then no ancestor of $x$ is in $X_y$.\n\\end{lemma}\n\\begin{proof}\n By induction on the length of $\\ensuremath{\\mathrm{path}}(y)$.\nInitially, the state only contains $\\ensuremath{\\mathrm{root}}(P)$. Let $z$ be the\nparent of $y$, and thus $X_y$ is computed from $X_z$. First we\nnote that for all nodes $x\\in X_y$ either $x\\in X_z$ or\n$\\ensuremath{\\mathrm{parent}}(x)\\in X_z$. If $x\\in X_z$ it follows from the induction\nhypothesis that no ancestor of $x$ is in $X_z$, and thus no\nancestors of $x$ can be in $X_y$. If $\\ensuremath{\\mathrm{parent}}(x)\\in X_z$ then due\nto the definition of \\ensuremath{\\textsc{Down}}\\ we must have $\\ensuremath{\\mathrm{label}}(x)=\\ensuremath{\\mathrm{label}}(y)$. It\nfollows from the definition of \\ensuremath{\\textsc{Down}}\\ that $\\ensuremath{\\mathrm{parent}}(x)\\not \\in\nX_y$.\n\\end{proof}\nIt follows from Lemma~\\ref{lem:statesize} that $|X_y|\\leq l_P$ for\nany $y\\in V(T)$. If we store the\nstate in an unordered linked list each step of the depth-first\ntraversal takes time $O(l_{P})$ giving a total $O(l_{P}n_{T} +n_{P})$\ntime algorithm.\nSince each state is of size at most $l_P$ the space used is\n$O(n_{P} + l_P n_{T})$. In the following sections we show how to improve these bounds.\n\n\\section{A Simple Algorithm}\nIn this section we consider a simple\nimplementation of the above algorithm, which has running time\n$O\\left(\\min(l_Pn_T +n_P, n_Pl_T + n_T)\\right)$ and uses $O(n_P + n_T)$ space. We assume\nthat the size of the alphabet is $n_T + n_P$ and each character in\n$\\Sigma$ is represented by an integer in the range $\\{1,\\ldots,n_T\n+ n_P\\}$. If this is not the case we can sort all characters in\n$V(P) \\cup V(T)$ and replace each label by its rank in the sorted\norder. This does not change the solution to the problem, and\nassuming at least a logarithmic number of leaves in both trees it\ndoes not affect the running time. To get the space usage down to\nlinear we will avoid saving all states. For this purpose we\nintroduce the procedure \\ensuremath{\\textsc{Up}}, which reconstructs the state $X_z$\nfrom the state $X_y$, where $z=\\ensuremath{\\mathrm{parent}}(y)$. We can thus save space\nas we only need to save the current state.\n\nWe use the following data structure to represent the current state $X_y$: A\n\\emph{node dictionary} consists of two dictionaries denoted\n$X^{c}$ and $X^{p}$.  The dictionary $X^c$ represents the node set\ncorresponding to $X_y$, and the dictionary $X^p$ represents the\nnode set corresponding to the set $\\{x \\in X_z \\mid x \\not\\in X_y\n\\text{ and } z \\text{ is an ancestor of } y\\}$.  That is, $X^c$\nrepresents the nodes in the current state, and $X^p$ represents\nthe nodes that is in a state $X_z$, where $z$ is an ancestor of\n$y$ in $T$, but not in $X_y$. We will use $X^p$ to reconstruct\nprevious states. The dictionary $X^c$ is indexed by $\\Sigma$ and\n$X^p$ is indexed by $V(T)$.\nThe subsets stored at each entry are represented by doubly-linked\nlists. Furthermore, each node in $X^{c}$ maintains a pointer to\nits parent in $X^{p}$ and each node $x'$ in $X^{p}$ stores a\nlinked list of pointers to its children in $X^{p}$.\nWith this representation the total size of the node dictionary is\n$O(n_{P}+n_T)$.\n\nNext we show how to solve the tree path subsequence problem in our\nframework using the node dictionary representation. For\nsimplicity, we add a node $\\top$ to $P$ as a the parent of\n$\\ensuremath{\\mathrm{root}}(P)$. Initially, the $X^{p}$ represents $\\top$ and $X^{c}$\nrepresents $\\ensuremath{\\mathrm{root}}(P)$. The $\\ensuremath{\\textsc{Down}}$ and $\\ensuremath{\\textsc{Up}}$ procedures are\nimplemented as follows:\n\\begin{relate}\n\\item[$\\ensuremath{\\textsc{Down}}((X^{p}, X^{c}), y)$:]\\begin{enumerate}\n\\item Set\n$X:=X^c[\\ensuremath{\\mathrm{label}}(y)]$ and $X^c[\\ensuremath{\\mathrm{label}}(y)]:=\\emptyset$.\n\n\\item For each $x\\in X$ do:\n\\begin{enumerate}\n\\item Set\n$X^p[y] := X^p[y] \\cup \\{x\\}$.\n\\item For each $x' \\in \\ensuremath{\\mathrm{child}}(x)$ do:\n\\begin{enumerate}\n  \\item Set $X^c[\\ensuremath{\\mathrm{label}}(x')] := X^c[\\ensuremath{\\mathrm{label}}(x')] \\cup \\{x\\}$.\n  \\item Create pointers between $x'$ and $x$.\n\\end{enumerate}\n\\end{enumerate}\n\\item Return $(X^{p}, X^{c})$.\n\\end{enumerate}\n\\item[$\\ensuremath{\\textsc{Up}}((X^{p}, X^{c}), y)$:]\n\\begin{enumerate}\n\\item Set $X:=X^p[y]$ and $X^p[y]:=\\emptyset$.\n\n\\item For each $x \\in X$ do:\n\\begin{enumerate}\n\\item Set\n$X^c[\\ensuremath{\\mathrm{label}}(x)] := X^c[\\ensuremath{\\mathrm{label}}(x)] \\cup \\{x\\}$.\n\\item For each $x' \\in \\ensuremath{\\mathrm{child}}(x)$ do:\n\\begin{enumerate}\n\\item Remove pointers between $x'$ and $x$.\n\\item Set $X^c[\\ensuremath{\\mathrm{label}}(x')] := X^c[\\ensuremath{\\mathrm{label}}(x')] \\setminus \\{x'\\}$.\n\\end{enumerate}\n\\end{enumerate}\n\\item Return $(X^{p}, X^{c})$.\n\\end{enumerate}\n\\end{relate}\nThe next lemma shows that\n\\ensuremath{\\textsc{Up}}\\ correctly reconstructs\nthe former state.\n\\begin{lemma}\\label{lem:updown}\nLet $X_z=(X^c,X^p)$ be a state computed at a node $z\\in V(T)$, and\nlet $y$ be a child of $z$. Then,\n\\begin{equation*}\nX_z=\\ensuremath{\\textsc{Up}}(\\ensuremath{\\textsc{Down}}(X_z,y),y)\\;.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nLet $(X^c_1,X^p_1)= \\ensuremath{\\textsc{Down}}(X_z,y)$ and $(X^c_2,X^p_2)=\n\\ensuremath{\\textsc{Up}}((X^c_1,X^p_1),y)$.\nWe will first show that $x \\in X_z \\Rightarrow x \\in\n\\ensuremath{\\textsc{Up}}(\\ensuremath{\\textsc{Down}}(X_z,y),y)$.\n\nLet $x$ be a node in $X^c$. There are two cases. If $x \\in\nX^c[\\ensuremath{\\mathrm{label}}(y)]$, then it follows  from the implementation of \\ensuremath{\\textsc{Down}}\\\nthat $x \\in X^p_1[y]$. By the implementation of \\ensuremath{\\textsc{Up}}, $x \\in\nX^p_1[y]$ implies $x \\in X^c_2$. If $x \\not\\in X^c[\\ensuremath{\\mathrm{label}}(y)]$ then\n$x\\in X^c_1$. We need to show $\\ensuremath{\\mathrm{parent}}(x) \\not\\in X^p_1[y]$. This\nwill imply $x\\in X^c_2$, since the only nodes removed from $X^c_1$\nwhen computing $X^c_2$ are the nodes with a parent in $X^p_1[y]$.\nSince $y$ is unique it follows from the implementation of \\ensuremath{\\textsc{Down}}\\\nthat $\\ensuremath{\\mathrm{parent}}(x)\\in X^p_1$ implies $x\\in X^c[\\ensuremath{\\mathrm{label}}(y)]$.\n\nLet $x$ be a node in $X^p$. Since $y$ is unique we have $x \\in\nX^p[y']$ for some $y'\\neq y$.  It follows immediately from the\nimplementation of \\ensuremath{\\textsc{Up}}\\ and \\ensuremath{\\textsc{Down}}\\ that\n$X^p[y']=X^p_1[y']=X^p_2[y']$, when $y'\\neq y$, and thus\n$X^p=X^p_2$.\n\n\nWe will now show $x \\in \\ensuremath{\\textsc{Up}}(\\ensuremath{\\textsc{Down}}(X_z,y),y) \\Rightarrow x \\in\nX_z$.\nLet $x$ be a node in $X^c_2$.  There are two cases. If $x \\not \\in\nX^c_1$ then it follows from the implementation of \\ensuremath{\\textsc{Up}}\\ that $x \\in\nX^p_1[y]$. By the implementation of \\ensuremath{\\textsc{Down}}, $x \\in X^p_1[y]$\nimplies $x \\in X^c[\\ensuremath{\\mathrm{label}}(y)]$, i.e., $x\\in X^c$. If $x\\in X_1^c$\nthen by the implementation of \\ensuremath{\\textsc{Up}}, $x \\in X^c_2$ implies\n$\\ensuremath{\\mathrm{parent}}(x) \\not\\in x^p_1[y]$. It follows from the implementation\nof \\ensuremath{\\textsc{Down}}\\ that $x\\in X^c$. Finally, let $x$ be a node in $X^p_2$.\nAs argued above $X^p=X^p_2$, and thus $x \\in X^p$.\n\\end{proof}\nFrom the current state $X_{y}=(X^c,X^p)$ the next state $X_{z}$ is\ncomputed as follows:\n\\begin{equation*}\nX_{z} =\n\\begin{cases}\n   \\ensuremath{\\textsc{Down}}(X_{y}, z)   & \\text{if $y = \\ensuremath{\\mathrm{parent}}(z)$}, \\\\\n   \\ensuremath{\\textsc{Up}}(X_{y},y)  & \\text{if $z = \\ensuremath{\\mathrm{parent}}(y)$}.\n\\end{cases}\n\\end{equation*}\nThe correctness of the algorithm follows from\nLemma~\\ref{lem:correct} and Lemma~\\ref{lem:updown}. \nWe will now analyze the running time of the algorithm. The procedures\n\\ensuremath{\\textsc{Down}}\\ and \\ensuremath{\\textsc{Up}}\\ uses time linear in the size of the current state\nand the state computed.\nBy Lemma~\\ref{lem:statesize} the size of each state is $O(l_P)$.\nEach step in the depth-first traversal thus takes time $O(l_P)$,\nwhich gives a total running time of $O(l_P n_T+n_P)$. On the other\nhand consider a path $t$ in $T$. We will argue that the\ncomputation of all the states along the path takes total time\n$O(n_P+n_t)$, where $n_T$ is the number of nodes in $t$. \nTo show this we need the following lemma.\n\\begin{lemma}\\label{lem:Tpath}\nLet $t$ be a path in $T$. During the computation of the states\nalong the path $t$, any node $x\\in V(P)$ is inserted into $X^c$ at\nmost once.\n\\end{lemma}\n\\begin{proof}\nSince $t$ is a path we only need to consider the \\ensuremath{\\textsc{Down}}\\\ncomputations. The only way a node $x\\in V(P)$ can be inserted into\n$X^c$ is if $\\ensuremath{\\mathrm{parent}}(x)\\in X^c$. It thus follows from\nLemma~\\ref{lem:statesize} that $x$ can be inserted into $X^c$ at\nmost once.\n\\end{proof}\nIt follows from Lemma~\\ref{lem:Tpath} that\nthe computations of the all states when $T$ is a path takes time\n$O(n_P+n_T)$. Consider a path-decomposition of $T$. A\npath-decomposition of $T$ is a decomposition of $T$ into disjoint\npaths. We can make such a path-decomposition of the tree $T$\nconsisting of $l_T$ paths. Since the running time of \\ensuremath{\\textsc{Up}}\\ and\n\\ensuremath{\\textsc{Down}}\\ both are linear in the size of the current and computed\nstate it follows from Lemma~\\ref{lem:updown} that we only need to\nconsider the total cost of the \\ensuremath{\\textsc{Down}}\\ computations on the paths in\nthe path-decompostion. Thus, the algorithm uses time at most\n$\\sum_{t\\in T}O(n_p + n_t)=O(n_P l_T+ n_T)$. \n\n\nNext we consider the space used by the algorithm. Lemma~\\ref{lem:statesize}\nimplies that $|X^c|\\leq l_P$. Now consider the size of $X^p$. A\nnode is inserted into $X^p$ when it is removed from $X^c$. It is\nremoved again when inserted into $X^c$ again. Thus\nLemma~\\ref{lem:Tpath} implies $|X^p| \\leq n_P$ at any time. The\ntotal space usage is thus $O(n_P+n_T)$.\nTo summarize we have shown,\n\\begin{theorem}\\label{simple} For trees $P$ and $T$ the tree path subsequence\nproblem can be solved in $O\\left(\\min(l_Pn_T + n_P, n_Pl_T+n_T)\\right)$ time and $O(n_P + n_T)$ space.\n\\end{theorem}\n\n\n\\section{A Worst-Case Efficient Algorithm}\nIn this section we consider the worst-case\ncomplexity of TPS and present an algorithm using subquadratic\nrunning time and linear space. The new algorithm works within our framework but does not use the $\\ensuremath{\\textsc{Up}}$\nprocedure or the node dictionaries from the previous section.\n\nRecall that using a simple linked list to represent the states we\nimmediately get an algorithm using $O(n_{P}n_{T})$ time and space.\nWe first show how to modify the traversal of $T$ and discard\nstates along the way such that at most $O(\\log n_{T})$ states are\nstored at any step in the traversal. This improves the space to\n$O(n_{P}\\log n_{T})$. Secondly, we decompose $P$ into small\nsubtrees, called \\emph{micro trees}, of size $O(\\log n_{T})$. Each micro tree can be represented\nin a single word of memory and therefore a state uses only\n$O(\\ceil{\\frac{n_{P}}{\\log n_{T}}})$ space. In total the space used to\nrepresent the $O(\\log n_{T})$ states is $O(\\ceil{\\frac{n_{P}}{\\log\nn_{T}}} \\cdot \\log n_{T}) = O(n_{P} + \\log n_T)$. Finally, we show how to\npreprocess $P$ in linear time and space such that computing the\nnew state can be done in constant time per micro tree.\nIntuitively, this achieves the $O(\\log n_{T})$ speedup.\n\\subsection{Heavy Path Traversal} \nIn this section we present the\nmodified traversal of $T$. We first partition $T$ into disjoint\npaths as follows. For each node $y\\in V(T)$ let $\\ensuremath{\\mathrm{size}}(y) =\n|V(T(y))|$. We classify each node as either \\emph{heavy} or\n\\emph{light} as follows. The root is light. For each internal node\n$y$ we pick a child $z$ of $y$ of maximum size among the children\nof $y$ and classify $z$ as heavy. The remaining children are\nlight. An edge to a light child is a \\emph{light edge}, and an\nedge to a heavy child is a \\emph{heavy edge}. The heavy child of a\nnode $y$ is denoted $\\ensuremath{\\mathrm{heavy}}(y)$. Let $\\ensuremath{\\mathrm{lightdepth}}(y)$ denote the\nnumber of light edges on the path from $y$ to $\\ensuremath{\\mathrm{root}}(T)$.\n\\begin{lemma}[Harel and Tarjan \\cite{HT1984}]\\label{lightdepth}\nFor any tree $T$ and node $y\\in V(T)$,  $\\ensuremath{\\mathrm{lightdepth}}(y) \\leq \\log\nn_{T} + O(1)$.\n\\end{lemma}\nRemoving the light edges, $T$ is partitioned into \\emph{heavy\npaths}. We traverse $T$ according to the heavy paths using the\nfollowing procedure. For  node $y \\in V(T)$ define:\n\\bigskip\n\\begin{relate}\n  \\item[$\\ensuremath{\\textsc{Visit}}(y)$:]  \n  \\begin{enumerate}\n  \\item If $y$ is a leaf report all leaves in $X_y$ and return.\n  \\item Else let $y_{1}, \\ldots, y_{k}$ be the light children of $y$ and let $z = \\ensuremath{\\mathrm{heavy}}(y)$.\n  \\item For $i:= 1$ to $k$ do:\n  \\begin{enumerate}\n    \\item Compute $X_{y_{i}} := \\ensuremath{\\textsc{Down}}(X_{y}, y_{i})$\n    \\item Compute $\\ensuremath{\\textsc{Visit}}(y_{i})$.\n  \\end{enumerate}\n  \\item Compute $X_{z} := \\ensuremath{\\textsc{Down}}(X_{y}, z)$.\n  \\item Discard $X_{y}$ and compute $\\ensuremath{\\textsc{Visit}}(z)$.\n\\end{enumerate}\n\\end{relate}\nThe procedure is called on the root node of $T$ with the initial\nstate $\\{\\ensuremath{\\mathrm{root}}(P)\\}$. The traversal resembles a depth first\ntraversal, however, at each step the light children are visited\nbefore the heavy child. We therefore call this a \\emph{heavy path\ntraversal}. Furthermore, after the heavy child (and therefore all\nchildren) has been visited we discard $X_{y}$. At any step we have\nthat before calling $\\ensuremath{\\textsc{Visit}}(y)$ the state $X_{y}$ is availiable,\nand therefore the procedure is correct. We have the following\nproperty:\n\\begin{lemma}\\label{traversal}\nFor any tree $T$ the heavy path traversal stores at most $\\log\nn_{T} + O(1)$ states.\n\\end{lemma}\n\\begin{proof}\nAt any node $y \\in V(T)$ we store at most one state for each of\nthe light nodes on the path from $y$ to $\\ensuremath{\\mathrm{root}}(T)$. Hence, by\nLemma~\\ref{lightdepth} the result follows.\n\\end{proof}\nUsing the heavy-path traversal immediately gives an $O(n_Pn_T)$ time and $O(n_P\\log n_T)$ space algorithm. In the following section we improve the time and space by an additional $O(\\log n_T)$ factor.\n\n\\subsection{Micro Tree Decomposition} \nIn this section we present\nthe decomposition of $P$ into small subtrees. A \\emph{micro tree}\nis a connected subgraph of $P$. A set of micro trees $MS$ is a\n\\emph{micro tree decomposition} iff $V(P) = \\cup_{M \\in MS} V(M)$\nand for any $M, M' \\in MS$, $(V(M) \\backslash \\{\\ensuremath{\\mathrm{root}}(M)\\}) \\cap\n(V(M') \\backslash \\{\\ensuremath{\\mathrm{root}}(M')\\}) = \\emptyset$.\n Hence, two micro trees in a decomposition share at most one node and this node must be the root in at least one of the micro trees. If $\\ensuremath{\\mathrm{root}}(M') \\in V(M)$ then $M$ is the \\emph{parent} of $M'$ and $M'$ is the \\emph{child} of $M$. A micro tree with no children is a \\emph{leaf} and a micro tree  with no parent is a \\emph{root}. Note that we may have several root micro trees since they can overlap at the node $\\ensuremath{\\mathrm{root}}(P)$. We decompose $P$ according to the following classic result:\n\\begin{lemma}[Gabow and Tarjan \\cite{GT1985}]\\label{clustering}\nFor any tree $P$ and parameter $s > 1$, it is possible to build  a\nmicro tree decomposition $MS$ of $P$ in linear time such that\n$|MS| = O(\\ceil{n_P\/s})$ and $|V(M)| \\leq s$ for any $M \\in MS$\n\\end{lemma}\n\n\\subsection{Implementing the Algorithm} In this section we show\nhow to implement the $\\ensuremath{\\textsc{Down}}$ procedure using the micro tree\ndecomposition. First decompose $P$ according to\nLemma~\\ref{clustering} for a parameter $s$ to be chosen later.\nHence, each micro tree has at most $s$ nodes and $|MS| =\nO(\\ceil{n_P\/s})$. We represent the state $X$ compactly using a bit vector\nfor each micro tree. Specifically, for any micro tree $M$ we store\na bit vector $X_M = [b_{1}, \\ldots, b_{s}]$, such that $X_M[i] = 1$\niff the $i$th node in a preorder traversal of $M$ is in $X$. If\n$|V(M)| < s$ we leave the remaining values undefined. Later we\nchoose $s= \\Theta(\\log n_{T})$ such that each bit vector can be\nrepresented in a single word.\n\nNext we define a $\\ensuremath{\\textsc{Down}}_{M}$ procedure on each micro tree $M\\in\nMS$. Due to the overlap between micro trees the $\\ensuremath{\\textsc{Down}}_{M}$\nprocedure takes a bit $b$ which will be used to propagate\ninformation between micro trees. For each micro tree $M \\in MS$,\nbit vector $X_M$, bit $b$, and $y\\in V(T)$ define:\n\\begin{relate}\n\\item[$\\ensuremath{\\textsc{Down}}_{M}(X_M, b, y)$:] Compute the state  $X'_M := \\ensuremath{\\textsc{Child}}(\\{x \\in X_M \\mid \\ensuremath{\\mathrm{label}}(x) = \\ensuremath{\\mathrm{label}}(y)\\}) \\cup \\{x \\in X_M \\mid \\ensuremath{\\mathrm{label}}(x) \\neq \\ensuremath{\\mathrm{label}}(y)\\}$. If $b=0$, return $X_M'$, else return $X_M' \\cup \\{\\ensuremath{\\mathrm{root}}(M)\\}$.\n\\end{relate}\nLater we will show how to implemenent $\\ensuremath{\\textsc{Down}}_{M}$ in constant time\nfor $s = \\Theta(\\log n_{T})$. First we show how to use $\\ensuremath{\\textsc{Down}}_M$ to\nsimulate $\\ensuremath{\\textsc{Down}}$ on $P$. We define a recursive procedure $\\ensuremath{\\textsc{Down}}$\nwhich traverse the hiearchy of micro trees. For micro tree\n$M$, state $X$, bit $b$, and $y \\in V(T)$ define:\n\\begin{relate}\n\\item[$\\ensuremath{\\textsc{Down}}(X,M,b,y)$:] Let $M_1, \\ldots, M_k$ be the children of $M$.\n\\begin{enumerate}\n\\item Compute  $X_M := \\ensuremath{\\textsc{Down}}_{M}(X_M, b, y)$.\n\\item For $i:=1$ to $k$ do:\n\\begin{enumerate}\n\\item Compute $\\ensuremath{\\textsc{Down}}(X, M_i, b_{i}, y)$, where $b_{i} = 1$\niff\n\n$\\ensuremath{\\mathrm{root}}(M_i) \\in X_M$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{relate}\nIntuitively, the $\\ensuremath{\\textsc{Down}}$ procedure works in a top-down fashion\nusing the $b$ bit to propagate the new state of the root of micro\ntree. To solve the problem within our framework we initially\nconstruct the state representing $\\{\\ensuremath{\\mathrm{root}}(P)\\}$. Then, at each\nstep we call $\\ensuremath{\\textsc{Down}}(R_{j}, 0, y)$ on each root micro tree $R_{j}$.\nWe formally show that this is correct:\n\\begin{lemma}\nThe above algorithm correctly simulates the $\\ensuremath{\\textsc{Down}}$ procedure on\n$P$.\n\\end{lemma}\n\\begin{proof}\nLet $X$ be the state and let $X' :=\\ensuremath{\\textsc{Down}}(X, y)$. For\nsimplicity, assume that there is only one root micro tree $R$.\nSince the root micro trees can only overlap at $\\ensuremath{\\mathrm{root}}(P)$ it is\nstraightforward to generalize the result to any number of roots.\nWe show that if $X$ is represented by bit vectors at each micro\ntree then calling $\\ensuremath{\\textsc{Down}}(R, 0, y)$ correctly produces the new\nstate $X'$.\n\nIf $R$ is the only micro tree then only line 1 is executed. Since\n$b = 0$ this produces the correct state by definition of\n$\\ensuremath{\\textsc{Down}}_{M}$. Otherwise, consider a micro tree $M$ with children\n$M_{1}, \\ldots, M_{k}$ and assume that $b = 1$ iff $\\ensuremath{\\mathrm{root}}(M) \\in\nX'$. Line 1 computes and stores the new state returned by\n$\\ensuremath{\\textsc{Down}}_{M}$. If $b=0$ the correctness follows immediately. If\n$b=1$ observe that $\\ensuremath{\\textsc{Down}}_{M}$ first computes the new state and\nthen adds $\\ensuremath{\\mathrm{root}}(M)$. Hence, in both cases the state of $M$ is\ncorrectly computed. Line 2 recursively computes the new state of\nthe children of $M$.\n\\end{proof}\n\nIf each micro tree has size at most $s$ and $\\ensuremath{\\textsc{Down}}_{M}$ can be\ncomputed in constant time it follows that the above algorithm\nsolves TPS in $O(\\ceil{n_{P}\/s})$ time. In the following section we show\nhow to do this for $s = \\Theta(\\log n_{T})$, while maintaining\nlinear space.\n\n\\subsection{Representing Micro Trees} In this section we show\nhow to preprocess all micro trees $M \\in MS$ such that\n$\\ensuremath{\\textsc{Down}}_{M}$ can be computed in constant time. This preprocessing\nmay be viewed as a ``Four Russian Technique''~\\cite{ADKF1970}. To\nachieve this in linear space we need the following auxiliary\nprocedures on micro trees. For each micro tree $M$, bit vector\n$X_M$, and $\\alpha \\in \\Sigma$ define:\n\\begin{relate}\n\\item[$\\ensuremath{\\textsc{Child}}_{M}(X_M)$:] Return the bit vector of nodes in $M$ that are children of nodes in $X_M$.\n\\item[$\\ensuremath{\\textsc{Eq}}_{M}(\\alpha)$:] Return the bit vector of nodes in  $M$ labeled $\\alpha$.\n\\end{relate}\nBy definition it follows that:\n\\begin{equation*}\n\\begin{aligned}\n\\ensuremath{\\textsc{Down}}_{M}(X_M,b, y) &=\n\\begin{cases}\n\\ensuremath{\\textsc{Child}}_{M}(X_M \\cap \\ensuremath{\\textsc{Eq}}_M(\\ensuremath{\\mathrm{label}}(y)))\\; \\cup \\\\\n\\quad (X_M \\backslash (X_M \\cap  \\ensuremath{\\textsc{Eq}}_M(\\ensuremath{\\mathrm{label}}(y)))      & \\text{if $b = 0$}, \\\\\n\\ensuremath{\\textsc{Child}}_{M}(X_M \\cap \\ensuremath{\\textsc{Eq}}_M(\\ensuremath{\\mathrm{label}}(y))) \\; \\cup \\\\\n\\quad (X_M \\backslash (X_M \\cap \\ensuremath{\\textsc{Eq}}_M(\\ensuremath{\\mathrm{label}}(y))) \\cup \\{\\ensuremath{\\mathrm{root}}(M)\\}\n& \\text{if $b=1$}.\n\\end{cases} \\\\\n\\end{aligned}\n\\end{equation*}\nRecall that the bit vectors are represented in a single word. Hence,\ngiven $\\ensuremath{\\textsc{Child}}_{M}$ and $\\ensuremath{\\textsc{Eq}}_{M}$ we can compute $\\ensuremath{\\textsc{Down}}_M$ using\nstandard bit-operations in constant time.\n\nNext we show how to efficiently implement the operations. For each\nmicro tree $M \\in MS$ we store the value $\\ensuremath{\\textsc{Eq}}_{M}(\\alpha)$ in a\nhash table indexed by $\\alpha$. Since the total number of\ndifferent characters in any $M\\in MS$ is at most $s$, the hash\ntable $\\ensuremath{\\textsc{Eq}}_{M}$ contains at most $s$ entries. Hence, the total\nnumber of entries in all hash tables is $O(n_{P})$. Using perfect\nhashing we can thus represent $\\ensuremath{\\textsc{Eq}}_{M}$ for all micro trees, $M\\in\nMS$, in $O(n_{P})$ space and $O(1)$ worst-case lookup time. The\npreprocessing time is expected $O(n_{P})$ w.h.p.. To get a worst-case bound we \nuse the deterministic dictionary of Hagerup et. al.\n\\cite{HMP2001} with $O((n_{P})\\log (n_{P}))$ worst-case preprocessing\ntime. \n\nNext consider implementing $\\ensuremath{\\textsc{Child}}_{M}$. Since this\nprocedure is independent of the labeling of $M$ it suffices to\nprecompute it for all \\emph{topologically} different rooted trees\nof size at most $s$. The total number of such trees is less than\n$2^{2s}$ and the number of different states in each tree is at\nmost $2^{s}$. Therefore $\\ensuremath{\\textsc{Child}}_{M}$ has to be computed for a\ntotal of $2^{2s}\\cdot 2^{s} = 2^{3s}$ different inputs. For any\ngiven tree and any given state, the value of $\\ensuremath{\\textsc{Child}}_{M}$ can be\ncomputed and encoded in $O(s)$ time. In total we can precompute\nall values of $\\ensuremath{\\textsc{Child}}_{M}$ in $O(s2^{3s})$ time. Choosing the\nlargest $s$ such that $3s + \\log s \\leq n_{T}$ (hence $s =\n\\Theta(\\log n_{T})$) we can precompute\nall values of $\\ensuremath{\\textsc{Child}}_{M}$ in $O(s2^{3s}) = O(n_{T})$ time and space. Each of\nthe inputs to $\\ensuremath{\\textsc{Child}}_{M}$ are encoded in a single word such that\nwe can look them up in constant time.\n\nFinally, note that we also need to report the leaves of a state efficiently since this is needed in line 1 in the $\\ensuremath{\\textsc{Visit}}$-procedure. To do this compute the state $L$ corresponding to all leaves in $P$. Clearly, the leaves of a state $X$ can be computed by performing a bitwise AND of each pair of bit vectors in $L$ and $X$. Computing $L$ uses $O(n_{P})$ time and the bitwise AND operation uses $O(\\ceil{n_{P}\/s})$ time.\n\nCombining the results, we decompose $P$, for $s$ as described\nabove, and compute all values of $\\ensuremath{\\textsc{Eq}}_{M}$ and $\\ensuremath{\\textsc{Child}}_{M}$. Then, we solve TPS using the heavy-path\ntraversal. Since $s = \\Theta(\\log n_{T})$, from Lemmas~\\ref{traversal} and \\ref{clustering} we have the following\ntheorem:\n\\begin{theorem}\\label{faster} For trees $P$ and $T$ the tree path subsequence problem can be solved in $O(\\frac{n_Pn_T}{\\log n_T} +n_T+ n_P\\log n_P)$ time and $O(n_P + n_T)$ space.\n\\end{theorem}\nCombining the results of Theorems~\\ref{simple} and \\ref{faster} proves Theorem~\\ref{main}.\n\n\\section{Acknowledgments}\nThe authors would like to thank Anna {\\\"O}stlin Pagh for many helpful comments.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMost of the problems that arise in different disciplines of science can be\nformulated by the equations in the for\n\\begin{equation}\nFx=0\\text{,}  \\label{eqn1}\n\\end{equation\nwhere $F$ is some function. The equations given by (1) can easily be\nreformulated as the fixed point equations of type \n\\begin{equation}\nTx=x\\text{.}  \\label{eqn2}\n\\end{equation\nwhere $T$ is a self-map of an ambient space $X$ and $x\\in X$. These\nequations are often classified as linear or nonlinear, depending on whether\nthe mappings used in the equation is linear with respect to the variables.\nOver the years, a considerable attention has been paid to solving such\nequations by using different techniques such as direct and iterative\nmethods. In case of linear equations, both direct and iterative methods are\nused to obtain solutions of the equations. But in case of nonlinear\nequations, due to various reasons, direct methods can be impractical or fail\nin solving equations, and thus iterative methods become a viable\nalternative. Nonlinear problems are of importance interest to\nmathematicians, physicists and engineers and many other scientists, simply\nbecause most systems are intrinsically nonlinear in nature. That is why,\nresearchers in various disciplines of sciences are often faced with the\nsolving such problems. It would be hard to fudge that the role of iterative\napproximation of fixed points have played in the recent progress of\nnonlinear science. Indeed, the phrase iterative approximation has been\nintroduced to describe repetition-based researches into nonlinear problems\nthat are inaccessible to analytic methods. For this reason, the iterative\napproximation of fixed points has become one of the major and basic tools in\nthe theory of equations, and as a result, numerous iterative methods have\nbeen introduced or improved and studied for many years in detail from\nvarious points of aspects by a wide audience of researchers, see, [1-14,\n16-42, 44, 45].\n\nIn this paper, we show that a Picard-S iteration method \\cite{Gursoy} can be\nused to approximate fixed point of weak-contraction mappings. Also, we show\nthat this iteration method is equivalent and converges faster than CR\niteration method \\cite{CR} for the aforementioned class of mappings.\nFurthermore, by providing an example, it is shown that the Picard-S\niteration method converges faster than CR iteration method and hence also\nfaster than all Picard \\cite{Picard}, Mann \\cite{Mann}, Ishikawa \\cit\n{Ishikawa}, Noor \\cite{Noor}, SP \\cite{SP}, S \\cite{S} and some other\niteration methods in the existing literature when applied to\nweak-contraction mappings. Finally, a data dependence result is proven for\nfixed point of weak-contraction mappings with the help of the Picard-S\niteration method.\n\nThroughout this paper the set of all positive integers and zero is shown by \n\\mathbb{N}\n$. Let $B$ be a Banach space, $D$ be a nonempty closed convex subset of $B$\nand $T$ a self-map of $D$. An element $x_{\\ast }$ of $D$ is called a fixed\npoint of $T$ if and only if $Tx_{\\ast }=x_{\\ast }$. The set of all fixed\npoint of $T$ denoted by $F_{T}$. Let $\\left\\{ a_{n}^{i}\\right\\}\n_{n=0}^{\\infty }$, $i\\in \\left\\{ 0,1,2\\right\\} $ be real sequences in $\\left[\n0,1\\right] $ satisfying certain control condition(s).\n\nRenowned Picard iteration method \\cite{Picard} is formulated as follo\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\np_{0}\\in D\\text{, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \np_{n+1}=Tp_{n}\\text{, }n\\in \n\\mathbb{N}\n\\text{,\n\\end{array\n\\right.   \\label{eqn3}\n\\end{equation\nand generally used to approximate fixed points of contraction mappings\nsatisfying: for all $x$, $y\\in B$ there exists a $\\delta \\in \\left(\n0,1\\right) $ such tha\n\\begin{equation}\n\\left\\Vert Tx-Ty\\right\\Vert \\leq \\delta \\left\\Vert x-y\\right\\Vert \\text{.}\n\\label{eqn4}\n\\end{equation\nThe following iteration methods are known as Noor \\cite{Noor} and SP \\cit\n{SP} iterations, respectively\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\omega _{0}\\in D\\text{, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ } \\\\ \n\\omega _{n+1}=\\left( 1-a_{n}^{0}\\right) \\omega _{n}+a_{n}^{0}T\\varpi _{n\n\\text{, \\ \\ } \\\\ \n\\varpi _{n}=\\left( 1-a_{n}^{1}\\right) \\omega _{n}+a_{n}^{1}T\\rho _{n}\\text{, \n} \\\\ \n\\rho _{n}=\\left( 1-a_{n}^{2}\\right) \\omega _{n}+a_{n}^{2}T\\omega _{n}\\text{, \n}n\\in \n\\mathbb{N}\n\\text{,\n\\end{array\n\\right.   \\label{eqn5}\n\\end{equation\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nq_{0}\\in D\\text{,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ } \\\\ \nq_{n+1}=\\left( 1-a_{n}^{0}\\right) r_{n}+a_{n}^{0}Tr_{n}\\text{, \\ \\ } \\\\ \nr_{n}=\\left( 1-a_{n}^{1}\\right) s_{n}+a_{n}^{1}Ts_{n}\\text{,} \\\\ \ns_{n}=\\left( 1-a_{n}^{2}\\right) q_{n}+a_{n}^{2}Tq_{n}\\text{, }n\\in \n\\mathbb{N}\n\\text{,\n\\end{array\n\\right.   \\label{eqn6}\n\\end{equation}\n\n\\begin{remark}\n(i) If $a_{n}^{2}=0$ for each $n\\in \n\\mathbb{N}\n$ , then the Noor iteration method reduces to iterative method of Ishikawa \n\\cite{Ishikawa}.\n\n(ii) If $a_{n}^{2}=0$ for each $n\\in \n\\mathbb{N}\n$ , then the SP iteration method reduces to iterative method of Thianwan \n\\cite{iam}. \n\n(iii) When $a_{n}^{1}=$ $a_{n}^{2}=0$ for each $n\\in \n\\mathbb{N}\n$, then both Noor and SP iteration methods reduce to an iteration method due\nto Mann \\cite{Mann}.  \n\\end{remark}\n\nRecently, G\\\"{u}rsoy and Karakaya \\cite{Gursoy} introduced a Picard-S\niterative scheme as follows: \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nx_{0}\\in D\\text{, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nx_{n+1}=Ty_{n}\\text{, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ } \\\\ \ny_{n}=\\left( 1-a_{n}^{1}\\right) Tx_{n}+a_{n}^{1}Tz_{n}\\text{, \\ \\ \\ \\ \\ \\ }\n\\\\ \nz_{n}=\\left( 1-a_{n}^{2}\\right) x_{n}+a_{n}^{2}Tx_{n}\\text{, }n\\in \n\\mathbb{N}\n\\text{,\n\\end{array\n\\right.   \\label{eqn7}\n\\end{equation\nThe following definitions and lemmas will be needed in obtaining the main\nresults of this article.\n\n\\begin{definition}\n\\cite{Berinde} Let $\\left\\{ a_{n}\\right\\} _{n=0}^{\\infty }$ and $\\left\\{\nb_{n}\\right\\} _{n=0}^{\\infty }$ be two sequences of real numbers with limits \n$a$ and $b$, respectively. Assume that there exist\n\\begin{equation}\n\\underset{n\\rightarrow \\infty }{\\lim }\\frac{\\left\\vert a_{n}-a\\right\\vert }\n\\left\\vert b_{n}-b\\right\\vert }=l\\text{.}  \\label{eqn8}\n\\end{equation}\n\n(i) If $l=0$, the we say that $\\left\\{ a_{n}\\right\\} _{n=0}^{\\infty }$\nconverges faster to $a$ than $\\left\\{ b_{n}\\right\\} _{n=0}^{\\infty }$ to $b$.\n\n(ii) If $0<l<\\infty $, then we say that $\\left\\{ a_{n}\\right\\}\n_{n=0}^{\\infty }$ and $\\left\\{ b_{n}\\right\\} _{n=0}^{\\infty }$ have the same\nrate of convergence.\n\\end{definition}\n\n\\begin{definition}\n\\cite{Berinde} Assume that for two fixed point iteration processes $\\left\\{\nu_{n}\\right\\} _{n=0}^{\\infty }$ and $\\left\\{ v_{n}\\right\\} _{n=0}^{\\infty }$\nboth converging to the same fixed point $p$, the following error prediction\n\\begin{equation}\n\\left\\Vert u_{n}-p\\right\\Vert \\leq a_{n}\\text{ for all }n\\in \n\\mathbb{N}\n\\text{,}  \\label{eqn9}\n\\end{equation\n\\begin{equation}\n\\left\\Vert v_{n}-p\\right\\Vert \\leq b_{n}\\text{ for all }n\\in \n\\mathbb{N}\n\\text{,}  \\label{eqn10}\n\\end{equation\nare available where $\\left\\{ a_{n}\\right\\} _{n=0}^{\\infty }$ and $\\left\\{\nb_{n}\\right\\} _{n=0}^{\\infty }$ are two sequences of positive numbers\n(converging to zero). If $\\left\\{ a_{n}\\right\\} _{n=0}^{\\infty }$ converges\nfaster than $\\left\\{ b_{n}\\right\\} _{n=0}^{\\infty }$, then $\\left\\{\nu_{n}\\right\\} _{n=0}^{\\infty }$ converges faster than $\\left\\{ v_{n}\\right\\}\n_{n=0}^{\\infty }$ to $p$.\n\\end{definition}\n\n\\begin{definition}\n\\cite{Vasile} Let $\\left( B,\\left\\Vert \\cdot \\right\\Vert \\right) $ be a\nBanach space. A map $T:B\\rightarrow B$ is called weak-contraction if there\nexist a constant $\\delta \\in \\left( 0,1\\right) $ and some $L\\geq 0$ such tha\n\\begin{equation}\n\\left\\Vert Tx-Ty\\right\\Vert \\leq \\delta \\left\\Vert x-y\\right\\Vert\n+L\\left\\Vert y-Ty\\right\\Vert \\text{, for all }x\\text{, }y\\in B\\text{.}\n\\label{eqn11}\n\\end{equation}\n\\end{definition}\n\n\\begin{definition}\n\\cite{Vasile} Let $T$,$\\widetilde{T}:B\\rightarrow B$ be two operators. We\nsay that $\\widetilde{T}$ is an approximate operator of $T$ if for all $x\\in B\n$ and for a fixed $\\varepsilon >0$ we have \n\\begin{equation}\n\\left\\Vert Tx-\\widetilde{T}x\\right\\Vert \\leq \\varepsilon .  \\label{eqn12}\n\\end{equation}\n\\end{definition}\n\n\\begin{lemma}\n\\cite{Weng}Let $\\left\\{ \\beta _{n}\\right\\} _{n=0}^{\\infty }$ and $\\left\\{\n\\rho _{n}\\right\\} _{n=0}^{\\infty }$ be nonnegative real sequences satisfying\nthe following inequality\n\\begin{equation}\n\\beta _{n+1}\\leq \\left( 1-\\lambda _{n}\\right) \\beta _{n}+\\rho _{n}\\text{,}\n\\label{eqn13}\n\\end{equation\nwhere $\\lambda _{n}\\in \\left( 0,1\\right) $, for all $n\\geq n_{0}$, \n\\dsum\\nolimits_{n=1}^{\\infty }\\lambda _{n}=\\infty $, and $\\frac{\\rho _{n}}\n\\lambda _{n}}\\rightarrow 0$ as $n\\rightarrow \\infty $. Then \n\\lim_{n\\rightarrow \\infty }\\beta _{n}=0$.\n\\end{lemma}\n\n\\begin{lemma}\n\\cite{Data Is 2} Let $\\left\\{ \\beta _{n}\\right\\} _{n=0}^{\\infty }$ be a\nnonnegative sequence for which one assumes there exists $n_{0}\\in \n\\mathbb{N}\n$, such that for all $n\\geq n_{0}$ one has satisfied the inequality \n\\begin{equation}\n\\beta _{n+1}\\leq \\left( 1-\\mu _{n}\\right) \\beta _{n}+\\mu _{n}\\gamma _{n\n\\text{,}  \\label{eqn15}\n\\end{equation\nwhere $\\mu _{n}\\in \\left( 0,1\\right) ,$ for all $n\\in \n\\mathbb{N}\n$, $\\sum\\limits_{n=0}^{\\infty }\\mu _{n}=\\infty $ and $\\gamma _{n}\\geq 0$, \n\\forall n\\in \n\\mathbb{N}\n$. Then the following inequality holds \n\\begin{equation}\n0\\leq \\lim \\sup_{n\\rightarrow \\infty }\\beta _{n}\\leq \\lim \\sup_{n\\rightarrow\n\\infty }\\gamma _{n}.  \\label{eqn16}\n\\end{equation}\n\\end{lemma}\n\n\\section{Main Results}\n\n\\begin{theorem}\nLet $T:D\\rightarrow D$ be a weak-contraction map satisfying condition (11)\nwith $F_{T}\\neq \\emptyset $ and $\\left\\{ x_{n}\\right\\} _{n=0}^{\\infty }$ an\niterative sequence defined by (7) with real sequences $\\left\\{\na_{n}^{i}\\right\\} _{n=0}^{\\infty }$, $i\\in \\left\\{ 1,2\\right\\} $ in $\\left[\n0,1\\right] $ satisfying $\\sum_{k=0}^{\\infty }a_{k}^{1}a_{k}^{2}=\\infty $.\nThen $\\ \\left\\{ x_{n}\\right\\} _{n=0}^{\\infty }$ converges to a unique fixed\npoint $u^{\\ast }$of $T$.\n\\end{theorem}\n\n\\begin{proof}\nUniqueness of $u^{\\ast }$ comes from condition (11). Using Picard-S\niterative scheme (7) and condition (11), we obtai\n\\begin{eqnarray}\n\\left\\Vert z_{n}-u^{\\ast }\\right\\Vert  &\\leq &\\left( 1-a_{n}^{2}\\right)\n\\left\\Vert x_{n}-u^{\\ast }\\right\\Vert +a_{n}^{2}\\left\\Vert Tx_{n}-Tu^{\\ast\n}\\right\\Vert   \\notag \\\\\n&\\leq &\\left( 1-a_{n}^{2}\\right) \\left\\Vert x_{n}-u^{\\ast }\\right\\Vert\n+a_{n}^{2}\\delta \\left\\Vert x_{n}-u^{\\ast }\\right\\Vert +a_{n}^{2}L\\left\\Vert\nu^{\\ast }-Tu^{\\ast }\\right\\Vert   \\notag \\\\\n&=&\\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert\nx_{n}-u^{\\ast }\\right\\Vert \\text{,}  \\label{eqn17}\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert y_{n}-u^{\\ast }\\right\\Vert  &\\leq &\\left( 1-a_{n}^{1}\\right)\n\\left\\Vert Tx_{n}-Tu^{\\ast }\\right\\Vert +a_{n}^{1}\\left\\Vert Tz_{n}-Tu^{\\ast\n}\\right\\Vert   \\notag \\\\\n&\\leq &\\left( 1-a_{n}^{1}\\right) \\delta \\left\\Vert x_{n}-u^{\\ast\n}\\right\\Vert +a_{n}^{1}\\delta \\left\\Vert z_{n}-u^{\\ast }\\right\\Vert \\text{,}\n\\label{eqn18}\n\\end{eqnarray\n\\begin{equation}\n\\left\\Vert x_{n+1}-u^{\\ast }\\right\\Vert \\leq \\delta \\left\\Vert y_{n}-u^{\\ast\n}\\right\\Vert \\text{.}  \\label{eqn19}\n\\end{equation\nCombining (16), (17) and (18\n\\begin{equation}\n\\left\\Vert x_{n+1}-u^{\\ast }\\right\\Vert \\leq \\delta ^{2}\\left[\n1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert x_{n}-u^{\\ast\n}\\right\\Vert \\text{.}  \\label{eqn20}\n\\end{equation\nBy inductio\n\\begin{eqnarray}\n\\left\\Vert x_{n+1}-u^{\\ast }\\right\\Vert  &\\leq &\\delta ^{2\\left( n+1\\right)\n}\\dprod\\nolimits_{k=0}^{n}\\left[ 1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta \\right)\n\\right] \\left\\Vert x_{0}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\leq &\\delta ^{2\\left( n+1\\right) }\\left\\Vert x_{0}-u^{\\ast }\\right\\Vert\n^{n+1}e^{-\\left( 1-\\delta \\right) \\dsum\\nolimits_{k=0}^{n}a_{k}^{1}a_{k}^{2}\n\\text{.}  \\label{eqn21}\n\\end{eqnarray\nSince $\\sum_{k=0}^{\\infty }a_{k}^{1}a_{k}^{2}=\\infty $\n\\begin{equation}\ne^{-\\left( 1-\\delta \\right)\n\\dsum\\nolimits_{k=0}^{n}a_{k}^{1}a_{k}^{2}}\\rightarrow 0\\text{ as \nn\\rightarrow \\infty \\text{,}  \\label{eqn22}\n\\end{equation\nwhich implies $\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-u^{\\ast\n}\\right\\Vert $.\n\\end{proof}\n\n\\begin{theorem}\nLet $T:D\\rightarrow D$ with fixed point $u^{\\ast }\\in F_{T}\\neq \\emptyset $\nbe as in Theorem 1 and $\\{q_{n}\\}_{n=0}^{\\infty }$, $\\{x_{n}\\}_{n=0}^{\\infty\n}$ two iterative sequences defined by SP (6) and Picard-S (7) iteration\nmethods with real sequences $\\left\\{ a_{n}^{i}\\right\\} _{n=0}^{\\infty }$, \ni\\in \\left\\{ 0,1,2\\right\\} $ in $\\left[ 0,1\\right] $ satisfying \n\\sum_{k=0}^{n}a_{k}^{1}a_{k}^{2}=\\infty $. Then the following are equivalent:\n\n(i) $\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-u^{\\ast }\\right\\Vert =0$;\n\n(ii) $\\lim_{n\\rightarrow \\infty }\\left\\Vert q_{n}-u^{\\ast }\\right\\Vert =0$.\n\n\\begin{proof}\n(i)$\\Rightarrow $(ii): It follows from (6), (7), and condition (11) tha\n\\begin{eqnarray}\n\\left\\Vert x_{n+1}-q_{n+1}\\right\\Vert  &=&\\left\\Vert \\left(\n1-a_{n}^{0}\\right) \\left( Ty_{n}-r_{n}\\right) +a_{n}^{0}\\left(\nTy_{n}-Tr_{n}\\right) \\right\\Vert   \\label{eqn23} \\\\\n&\\leq &\\left( 1-a_{n}^{0}\\right) \\left\\Vert Ty_{n}-r_{n}\\right\\Vert\n+a_{n}^{0}\\left\\Vert Ty_{n}-Tr_{n}\\right\\Vert   \\notag \\\\\n&\\leq &\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] \\left\\Vert\ny_{n}-r_{n}\\right\\Vert +\\left[ 1-a_{n}^{0}\\left( 1-L\\right) \\right]\n\\left\\Vert y_{n}-Ty_{n}\\right\\Vert \\text{,}  \\notag\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert y_{n}-r_{n}\\right\\Vert  &=&\\left\\Vert \\left( 1-a_{n}^{1}\\right)\n\\left( Tx_{n}-s_{n}\\right) +a_{n}^{1}\\left( Tz_{n}-Ts_{n}\\right) \\right\\Vert \n\\label{eqn24} \\\\\n&\\leq &\\left( 1-a_{n}^{1}\\right) \\left\\Vert Tx_{n}-s_{n}\\right\\Vert\n+a_{n}^{1}\\left\\Vert Tz_{n}-Ts_{n}\\right\\Vert   \\notag \\\\\n&\\leq &\\left( 1-a_{n}^{1}\\right) \\left\\Vert Tx_{n}-s_{n}\\right\\Vert\n+a_{n}^{1}\\delta \\left\\Vert z_{n}-s_{n}\\right\\Vert +a_{n}^{1}L\\left\\Vert\nz_{n}-Tz_{n}\\right\\Vert \\text{,}  \\notag\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert Tx_{n}-s_{n}\\right\\Vert  &=&\\left\\Vert \\left( 1-a_{n}^{2}\\right)\n\\left( Tx_{n}-q_{n}\\right) +a_{n}^{2}\\left( Tx_{n}-Tq_{n}\\right) \\right\\Vert \n\\label{eqn25} \\\\\n&\\leq &\\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert\nx_{n}-q_{n}\\right\\Vert +\\left[ 1-a_{n}^{2}\\left( 1-L\\right) \\right]\n\\left\\Vert x_{n}-Tx_{n}\\right\\Vert \\text{,}  \\notag\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert z_{n}-s_{n}\\right\\Vert  &\\leq &\\left( 1-a_{n}^{2}\\right)\n\\left\\Vert x_{n}-q_{n}\\right\\Vert +a_{n}^{2}\\left\\Vert\nTx_{n}-Tq_{n}\\right\\Vert   \\label{eqn26} \\\\\n&\\leq &\\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert\nx_{n}-q_{n}\\right\\Vert +a_{n}^{2}L\\left\\Vert x_{n}-Tx_{n}\\right\\Vert \\text{.}\n\\notag\n\\end{eqnarray\nCombining (22), (23), (24), and (25\n\\begin{eqnarray}\n\\left\\Vert x_{n+1}-q_{n+1}\\right\\Vert  &\\leq &\\left[ 1-a_{n}^{0}\\left(\n1-\\delta \\right) \\right] \\left[ 1-a_{n}^{1}\\left( 1-\\delta \\right) \\right]\n\\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert\nx_{n}-q_{n}\\right\\Vert   \\label{eqn27} \\\\\n&&+\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] \\left\\{ \\left(\n1-a_{n}^{1}\\right) \\left[ 1-a_{n}^{2}\\left( 1-L\\right) \\right]\n+a_{n}^{1}a_{n}^{2}\\delta L\\right\\} \\left\\Vert x_{n}-Tx_{n}\\right\\Vert  \n\\notag \\\\\n&&+\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] a_{n}^{1}L\\left\\Vert\nz_{n}-Tz_{n}\\right\\Vert +\\left[ 1-a_{n}^{0}\\left( 1-L\\right) \\right]\n\\left\\Vert y_{n}-Ty_{n}\\right\\Vert \\text{.}  \\notag\n\\end{eqnarray\nIt follows from the facts $\\delta \\in \\left( 0,1\\right) $ and $a_{n}^{i}\\in\n\\left[ 0,1\\right] $, $\\forall n\\in \n\\mathbb{N}\n$, $i\\in \\left\\{ 0,1,2\\right\\} $ tha\n\\begin{equation}\n\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] \\left[ 1-a_{n}^{1}\\left(\n1-\\delta \\right) \\right] \\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right) \\right]\n<1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\text{.}  \\label{eqn28}\n\\end{equation\nHence, inequality (26) become\n\\begin{eqnarray}\n\\left\\Vert x_{n+1}-q_{n+1}\\right\\Vert  &\\leq &\\left[ 1-a_{n}^{1}a_{n}^{2\n\\left( 1-\\delta \\right) \\right] \\left\\Vert x_{n}-q_{n}\\right\\Vert \n\\label{eqn29} \\\\\n&&+\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] \\left\\{ \\left(\n1-a_{n}^{1}\\right) \\left[ 1-a_{n}^{2}\\left( 1-L\\right) \\right]\n+a_{n}^{1}a_{n}^{2}\\delta L\\right\\} \\left\\Vert x_{n}-Tx_{n}\\right\\Vert  \n\\notag \\\\\n&&+\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] a_{n}^{1}L\\left\\Vert\nz_{n}-Tz_{n}\\right\\Vert +\\left[ 1-a_{n}^{0}\\left( 1-L\\right) \\right]\n\\left\\Vert y_{n}-Ty_{n}\\right\\Vert \\text{.}  \\notag\n\\end{eqnarray\nDenote tha\n\\begin{eqnarray}\n\\beta _{n} &:&=\\left\\Vert x_{n}-q_{n}\\right\\Vert \\text{, \\ \\ }  \\label{eqn30}\n\\\\\n\\lambda _{n} &:&=a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\in \\left(\n0,1\\right) \\text{, \\ }  \\notag \\\\\n\\rho _{n} &:&=\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] \\left\\{\n\\left( 1-a_{n}^{1}\\right) \\left[ 1-a_{n}^{2}\\left( 1-L\\right) \\right]\n+a_{n}^{1}a_{n}^{2}\\delta L\\right\\} \\left\\Vert x_{n}-Tx_{n}\\right\\Vert  \n\\notag \\\\\n&&+\\left[ 1-a_{n}^{0}\\left( 1-\\delta \\right) \\right] a_{n}^{1}L\\left\\Vert\nz_{n}-Tz_{n}\\right\\Vert +\\left[ 1-a_{n}^{0}\\left( 1-L\\right) \\right]\n\\left\\Vert y_{n}-Ty_{n}\\right\\Vert \\text{.}  \\notag\n\\end{eqnarray\nSince $\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-u^{\\ast }\\right\\Vert =0$\nand $Tu^{\\ast }=u^{\\ast }\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-Tx_{n}\\right\\Vert\n=\\lim_{n\\rightarrow \\infty }\\left\\Vert y_{n}-Ty_{n}\\right\\Vert\n=\\lim_{n\\rightarrow \\infty }\\left\\Vert z_{n}-Tz_{n}\\right\\Vert =0\\text{,}\n\\label{eqn31}\n\\end{equation\nwhich implies $\\frac{\\rho _{n}}{\\lambda _{n}}\\rightarrow 0$ as $n\\rightarrow\n\\infty $. Therefore, inequality (28) perform all assumptions in Lemma 1 and\nthus we obtain $\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-q_{n}\\right\\Vert\n=0$. Sinc\n\\begin{equation}\n\\left\\Vert q_{n}-u^{\\ast }\\right\\Vert \\leq \\left\\Vert x_{n}-q_{n}\\right\\Vert\n+\\left\\Vert x_{n}-u^{\\ast }\\right\\Vert \\rightarrow 0\\text{ as }n\\rightarrow\n\\infty \\text{,}  \\label{eqn32}\n\\end{equation\n$\\lim_{n\\rightarrow \\infty }\\left\\Vert q_{n}-u^{\\ast }\\right\\Vert =0$.\n\n(ii)$\\Rightarrow $(i): It follows from (6), (7), and condition (11) tha\n\\begin{eqnarray}\n\\left\\Vert q_{n+1}-x_{n+1}\\right\\Vert  &=&\\left\\Vert\nr_{n}-Ty_{n}+a_{n}^{0}\\left( Tr_{n}-r_{n}\\right) \\right\\Vert   \\label{eqn33}\n\\\\\n&\\leq &\\delta \\left\\Vert r_{n}-y_{n}\\right\\Vert +\\left( 1+a_{n}^{0}+L\\right)\n\\left\\Vert r_{n}-Tr_{n}\\right\\Vert \\text{,}  \\notag\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert r_{n}-y_{n}\\right\\Vert  &\\leq &\\left( 1-a_{n}^{1}\\right)\n\\left\\Vert s_{n}-Tx_{n}\\right\\Vert +a_{n}^{1}\\left\\Vert\nTs_{n}-Tz_{n}\\right\\Vert   \\label{eqn34} \\\\\n&\\leq &\\left( 1-a_{n}^{1}\\right) \\left\\Vert s_{n}-Tx_{n}\\right\\Vert\n+a_{n}^{1}\\delta \\left\\Vert s_{n}-z_{n}\\right\\Vert +a_{n}^{1}L\\left\\Vert\ns_{n}-Ts_{n}\\right\\Vert \\text{,}  \\notag\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert s_{n}-Tx_{n}\\right\\Vert  &\\leq &\\left\\Vert\nTs_{n}-Tx_{n}\\right\\Vert +\\left\\Vert s_{n}-Ts_{n}\\right\\Vert   \\label{eqn35}\n\\\\\n&\\leq &\\delta \\left\\Vert s_{n}-x_{n}\\right\\Vert +\\left( 1+L\\right)\n\\left\\Vert s_{n}-Ts_{n}\\right\\Vert   \\notag \\\\\n&\\leq &\\delta \\left\\Vert q_{n}-x_{n}\\right\\Vert +\\delta a_{n}^{2}\\left\\Vert\nTq_{n}-q_{n}\\right\\Vert +\\left( 1+L\\right) \\left\\Vert\ns_{n}-Ts_{n}\\right\\Vert \\text{,}  \\notag\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert s_{n}-z_{n}\\right\\Vert  &\\leq &\\left( 1-a_{n}^{2}\\right)\n\\left\\Vert q_{n}-x_{n}\\right\\Vert +a_{n}^{2}\\left\\Vert\nTq_{n}-Tx_{n}\\right\\Vert   \\label{eqn36} \\\\\n&\\leq &\\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert\nq_{n}-x_{n}\\right\\Vert +a_{n}^{2}L\\left\\Vert q_{n}-Tq_{n}\\right\\Vert \\text{.}\n\\notag\n\\end{eqnarray\nCombining (32), (33), (34), and (35\n\\begin{eqnarray}\n\\left\\Vert q_{n+1}-x_{n+1}\\right\\Vert  &\\leq &\\delta ^{2}\\left[\n1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert\nq_{n}-x_{n}\\right\\Vert   \\label{eqn37} \\\\\n&&+\\delta ^{2}a_{n}^{2}\\left[ 1-a_{n}^{1}\\left( 1-L\\right) \\right]\n\\left\\Vert q_{n}-Tq_{n}\\right\\Vert   \\notag \\\\\n&&+\\left( 1+a_{n}^{0}+L\\right) \\left\\Vert r_{n}-Tr_{n}\\right\\Vert +\\delta\n\\left( 1-a_{n}^{1}+L\\right) \\left\\Vert s_{n}-Ts_{n}\\right\\Vert \\text{.} \n\\notag\n\\end{eqnarray\nSince $\\delta \\in \\left( 0,1\\right) \n\\begin{equation}\n\\delta ^{2}\\left[ 1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\right]\n<1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\text{.}  \\label{eqn38}\n\\end{equation\nHence, inequality (36) become\n\\begin{eqnarray}\n\\left\\Vert q_{n+1}-x_{n+1}\\right\\Vert  &\\leq &\\left[ 1-a_{n}^{1}a_{n}^{2\n\\left( 1-\\delta \\right) \\right] \\left\\Vert q_{n}-x_{n}\\right\\Vert \n\\label{eqn39} \\\\\n&&+\\delta ^{2}a_{n}^{2}\\left[ 1-a_{n}^{1}\\left( 1-L\\right) \\right]\n\\left\\Vert q_{n}-Tq_{n}\\right\\Vert   \\notag \\\\\n&&+\\left( 1+a_{n}^{0}+L\\right) \\left\\Vert r_{n}-Tr_{n}\\right\\Vert +\\delta\n\\left( 1-a_{n}^{1}+L\\right) \\left\\Vert s_{n}-Ts_{n}\\right\\Vert \\text{.} \n\\notag\n\\end{eqnarray\nDenote tha\n\\begin{eqnarray}\n\\beta _{n} &:&=\\left\\Vert q_{n}-x_{n}\\right\\Vert \\text{, \\ \\ }  \\label{eqn40}\n\\\\\n\\lambda _{n} &:&=a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\in \\left(\n0,1\\right) \\text{, \\ }  \\notag \\\\\n\\rho _{n} &:&=\\delta ^{2}a_{n}^{2}\\left[ 1-a_{n}^{1}\\left( 1-L\\right) \\right]\n\\left\\Vert q_{n}-Tq_{n}\\right\\Vert   \\notag \\\\\n&&+\\left( 1+a_{n}^{0}+L\\right) \\left\\Vert r_{n}-Tr_{n}\\right\\Vert +\\delta\n\\left( 1-a_{n}^{1}+L\\right) \\left\\Vert s_{n}-Ts_{n}\\right\\Vert \\text{.} \n\\notag\n\\end{eqnarray\nSince $\\lim_{n\\rightarrow \\infty }\\left\\Vert q_{n}-u^{\\ast }\\right\\Vert =0$\nand $Tu^{\\ast }=u^{\\ast }\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\left\\Vert q_{n}-Tq_{n}\\right\\Vert\n=\\lim_{n\\rightarrow \\infty }\\left\\Vert r_{n}-Tr_{n}\\right\\Vert\n=\\lim_{n\\rightarrow \\infty }\\left\\Vert s_{n}-Ts_{n}\\right\\Vert =0\\text{,}\n\\label{eqn41}\n\\end{equation\nwhich implies $\\frac{\\rho _{n}}{\\lambda _{n}}\\rightarrow 0$ as $n\\rightarrow\n\\infty $. Therefore, inequality (38) perform all assumptions in Lemma 1 and\nthus we obtain $\\lim_{n\\rightarrow \\infty }\\left\\Vert q_{n}-x_{n}\\right\\Vert\n=0$. Sinc\n\\begin{equation}\n\\left\\Vert x_{n}-u^{\\ast }\\right\\Vert \\leq \\left\\Vert q_{n}-x_{n}\\right\\Vert\n+\\left\\Vert q_{n}-u^{\\ast }\\right\\Vert \\rightarrow 0\\text{ as }n\\rightarrow\n\\infty \\text{,}  \\label{eqn42}\n\\end{equation\n$\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-u^{\\ast }\\right\\Vert =0$.\n\\end{proof}\n\\end{theorem}\n\nTaking R. Chugh et al.'s result (\\cite{CR}, Corollary 3.2) into account,\nTheorem 2 leads to the following corollary under weaker assumption:\n\n\\begin{corollary}\nLet $T:D\\rightarrow D$ with fixed point $u^{\\ast }\\in F_{T}\\neq \\emptyset $\nbe as in Theorem 1. Then the followings are equivalent:\n\n1)The Picard iteration method (3) converges to $u^{\\ast }$,\n\n2) The Mann iteration method \\cite{Mann} converges to $u^{\\ast }$,\n\n3) The Ishikawa iteration method \\cite{Ishikawa} converges to $u^{\\ast }$,\n\n4) The Noor iteration method (5) converges to $u^{\\ast }$,\n\n5) S-iteration method \\cite{S} converges to $u^{\\ast }$,\n\n6) The SP-iteration method (6) converges to $u^{\\ast }$,\n\n7) CR-iteration method \\cite{CR} converges to $u^{\\ast }$,\n\n8) The Picard-S iteration method (7) converges to $u^{\\ast }$.\n\\end{corollary}\n\n\\begin{theorem}\nLet $T:D\\rightarrow D$ with fixed point $u^{\\ast }\\in F_{T}\\neq \\emptyset $\nbe as in Theorem 1. Suppose that $\\left\\{ \\omega _{n}\\right\\} _{n=0}^{\\infty\n}$, $\\left\\{ q_{n}\\right\\} _{n=0}^{\\infty }$ and $\\left\\{ x_{n}\\right\\}\n_{n=0}^{\\infty }$ are iterative sequences, respectively, defined by Noor\n(5), SP (6) and Picard-S (7) iterative schemes with real sequences $\\left\\{\na_{n}^{i}\\right\\} _{n=0}^{\\infty }\\subset \\left[ 0,1\\right] $, $i\\in \\left\\{\n0\\text{,}1\\text{,}2\\right\\} $ satisfying\n\n(i) $0\\leq a_{n}^{i}<\\frac{1}{1+\\delta }$,\n\n(ii) $\\lim_{n\\rightarrow \\infty }a_{n}^{i}=0$.\n\nThen the iterative sequence defined by (7) converges faster than the\niterative sequences defined by (5) and (6) to a unique fixed point of $T$,\nprovided that the initial point is the same for all iterations.\n\\end{theorem}\n\n\\begin{proof}\nFrom inequality (20), we hav\n\\begin{equation}\n\\left\\Vert x_{n+1}-u^{\\ast }\\right\\Vert \\leq \\delta ^{2\\left( n+1\\right)\n}\\left\\Vert x_{0}-u^{\\ast }\\right\\Vert ^{n+1}\\dprod\\nolimits_{k=0}^{n}\\left[\n1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta \\right) \\right]   \\label{eqn43}\n\\end{equation\nUsing (6) we obtai\n\\begin{eqnarray}\n\\left\\Vert q_{n+1}-u^{\\ast }\\right\\Vert  &=&\\left\\Vert \\left(\n1-a_{n}^{0}\\right) r_{n}+a_{n}^{0}Tr_{n}-u^{\\ast }\\right\\Vert   \\label{eqn44}\n\\\\\n&\\geq &\\left( 1-a_{n}^{0}\\right) \\left\\Vert r_{n}-u^{\\ast }\\right\\Vert\n-a_{n}^{0}\\left\\Vert Tr_{n}-Tu^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\left[ 1-a_{n}^{0}\\left( 1+\\delta \\right) \\right] \\left\\Vert\nr_{n}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\left[ 1-a_{n}^{0}\\left( 1+\\delta \\right) \\right] \\left\\{ \\left(\n1-a_{n}^{1}\\right) \\left\\Vert s_{n}-u^{\\ast }\\right\\Vert -a_{n}^{1}\\delta\n\\left\\Vert s_{n}-u^{\\ast }\\right\\Vert \\right\\}   \\notag \\\\\n&=&\\left[ 1-a_{n}^{0}\\left( 1+\\delta \\right) \\right] \\left[\n1-a_{n}^{1}\\left( 1+\\delta \\right) \\right] \\left\\Vert s_{n}-u^{\\ast\n}\\right\\Vert   \\notag \\\\\n&\\geq &\\left[ 1-a_{n}^{0}\\left( 1+\\delta \\right) \\right] \\left[\n1-a_{n}^{1}\\left( 1+\\delta \\right) \\right] \\left\\{ \\left( 1-a_{n}^{2}\\right)\n\\left\\Vert q_{n}-u^{\\ast }\\right\\Vert -a_{n}^{2}\\delta \\left\\Vert\nq_{n}-u^{\\ast }\\right\\Vert \\right\\}   \\notag \\\\\n&=&\\left[ 1-a_{n}^{0}\\left( 1+\\delta \\right) \\right] \\left[\n1-a_{n}^{1}\\left( 1+\\delta \\right) \\right] \\left[ 1-a_{n}^{2}\\left( 1+\\delta\n\\right) \\right] \\left\\Vert q_{n}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\cdots   \\notag \\\\\n&\\geq &\\left\\Vert q_{0}-u^{\\ast }\\right\\Vert ^{n+1}\\prod\\limits_{k=0}^{n}\n\\left[ 1-a_{k}^{0}\\left( 1+\\delta \\right) \\right] \\left[ 1-a_{k}^{1}\\left(\n1+\\delta \\right) \\right] \\left[ 1-a_{k}^{2}\\left( 1+\\delta \\right) \\right] \n\\text{.}  \\notag\n\\end{eqnarray\nUsing now (42) and (43\n\\begin{equation}\n\\frac{\\left\\Vert x_{n+1}-u^{\\ast }\\right\\Vert }{\\left\\Vert q_{n+1}-u^{\\ast\n}\\right\\Vert }\\leq \\frac{\\delta ^{2\\left( n+1\\right) }\\left\\Vert\nx_{0}-u^{\\ast }\\right\\Vert ^{n+1}\\dprod\\nolimits_{k=0}^{n}\\left[\n1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta \\right) \\right] }{\\left\\Vert\nq_{0}-u^{\\ast }\\right\\Vert ^{n+1}\\prod\\limits_{k=0}^{n}\\left[\n1-a_{k}^{0}\\left( 1+\\delta \\right) \\right] \\left[ 1-a_{k}^{1}\\left( 1+\\delta\n\\right) \\right] \\left[ 1-a_{k}^{2}\\left( 1+\\delta \\right) \\right] }\\text{.}\n\\label{eqn45}\n\\end{equation\nDefine \n\\begin{equation}\n\\theta _{n}=\\frac{\\delta ^{2\\left( n+1\\right) }\\dprod\\nolimits_{k=0}^{n\n\\left[ 1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta \\right) \\right] }\n\\dprod\\nolimits_{k=0}^{n}\\left[ 1-a_{k}^{0}\\left( 1+\\delta \\right) \\right]\n\\left[ 1-a_{k}^{1}\\left( 1+\\delta \\right) \\right] \\left[ 1-a_{k}^{2}\\left(\n1+\\delta \\right) \\right] }\\text{.}  \\label{eqn46}\n\\end{equation\nBy the assumtio\n\\begin{eqnarray}\n&&\\lim_{n\\rightarrow \\infty }\\frac{\\theta _{n+1}}{\\theta _{n}}  \\label{eqn47}\n\\\\\n&=&\\lim_{n\\rightarrow \\infty }\\frac{\\frac{\\delta ^{2\\left( n+2\\right)\n}\\dprod\\nolimits_{k=0}^{n+1}\\left[ 1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta\n\\right) \\right] }{\\dprod\\nolimits_{k=0}^{n+1}\\left[ 1-a_{k}^{0}\\left(\n1+\\delta \\right) \\right] \\left[ 1-a_{k}^{1}\\left( 1+\\delta \\right) \\right]\n\\left[ 1-a_{k}^{2}\\left( 1+\\delta \\right) \\right] }}{\\frac{\\delta ^{2\\left(\nn+1\\right) }\\dprod\\nolimits_{k=0}^{n}\\left[ 1-a_{k}^{1}a_{k}^{2}\\left(\n1-\\delta \\right) \\right] }{\\dprod\\nolimits_{k=0}^{n}\\left[ 1-a_{k}^{0}\\left(\n1+\\delta \\right) \\right] \\left[ 1-a_{k}^{1}\\left( 1+\\delta \\right) \\right]\n\\left[ 1-a_{k}^{2}\\left( 1+\\delta \\right) \\right] }}  \\notag \\\\\n&=&\\lim_{n\\rightarrow \\infty }\\frac{\\delta ^{2}\\left[\n1-a_{n+1}^{1}a_{n+1}^{2}\\left( 1-\\delta \\right) \\right] }{\\left[\n1-a_{n+1}^{0}\\left( 1+\\delta \\right) \\right] \\left[ 1-a_{n+1}^{1}\\left(\n1+\\delta \\right) \\right] \\left[ 1-a_{n+1}^{2}\\left( 1+\\delta \\right) \\right] \n}  \\notag \\\\\n&=&\\delta ^{2}<1\\text{.}  \\notag\n\\end{eqnarray\nIt thus follows from ratio test that $\\sum\\limits_{n=0}^{\\infty }\\theta\n_{n}<\\infty $. Hence, we have $\\lim_{n\\rightarrow \\infty }\\theta _{n}=0$\nwhich implies that the iterative sequence defined by (7) converges faster\nthan the iterative sequence defined by SP iteration method (6).\n\nUsing Noor iteration method (5), we ge\n\\begin{eqnarray}\n\\left\\Vert \\omega _{n+1}-u^{\\ast }\\right\\Vert  &=&\\left\\Vert \\left(\n1-a_{n}^{0}\\right) \\omega _{n}+a_{n}^{0}T\\varpi _{n}-u^{\\ast }\\right\\Vert \n\\label{eqn48} \\\\\n&\\geq &\\left( 1-a_{n}^{0}\\right) \\left\\Vert \\omega _{n}-u^{\\ast }\\right\\Vert\n-a_{n}^{0}\\left\\Vert T\\varpi _{n}-Tu^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\left( 1-a_{n}^{0}\\right) \\left\\Vert \\omega _{n}-u^{\\ast }\\right\\Vert\n-a_{n}^{0}\\delta \\left\\Vert \\varpi _{n}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\left[ 1-a_{n}^{0}-a_{n}^{0}\\delta \\left( 1-a_{n}^{1}\\right) \\right]\n\\left\\Vert \\omega _{n}-u^{\\ast }\\right\\Vert -a_{n}^{0}a_{n}^{1}\\delta\n^{2}\\left\\Vert \\rho _{n}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\left\\{ 1-a_{n}^{0}-a_{n}^{0}\\delta \\left( 1-a_{n}^{1}\\right)\n-a_{n}^{0}a_{n}^{1}\\delta ^{2}\\left[ 1-a_{n}^{2}\\left( 1-\\delta \\right)\n\\right] \\right\\} \\left\\Vert \\omega _{n}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\left\\{ 1-a_{n}^{0}-a_{n}^{0}\\delta \\left[ 1-a_{n}^{1}\\left( 1-\\delta\n\\right) \\right] \\right\\} \\left\\Vert \\omega _{n}-u^{\\ast }\\right\\Vert   \\notag\n\\\\\n&\\geq &\\left[ 1-a_{n}^{0}\\left( 1+\\delta \\right) \\right] \\left\\Vert \\omega\n_{n}-u^{\\ast }\\right\\Vert   \\notag \\\\\n&\\geq &\\cdots   \\notag \\\\\n&\\geq &\\left\\Vert \\omega _{0}-u^{\\ast }\\right\\Vert\n^{n+1}\\prod\\limits_{k=0}^{n}\\left[ 1-a_{k}^{0}\\left( 1+\\delta \\right) \\right]\n\\text{.}  \\notag\n\\end{eqnarray\nIt follows by (42) and (47) tha\n\\begin{equation}\n\\frac{\\left\\Vert x_{n+1}-u^{\\ast }\\right\\Vert }{\\left\\Vert \\omega\n_{n+1}-x_{\\ast }\\right\\Vert }\\leq \\frac{\\delta ^{2\\left( n+1\\right)\n}\\left\\Vert x_{0}-u^{\\ast }\\right\\Vert ^{n+1}\\dprod\\nolimits_{k=0}^{n}\\left[\n1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta \\right) \\right] }{\\left\\Vert \\omega\n_{0}-u^{\\ast }\\right\\Vert ^{n+1}\\prod\\limits_{k=0}^{n}\\left[\n1-a_{k}^{0}\\left( 1+\\delta \\right) \\right] }\\text{.}  \\label{eqn49}\n\\end{equation\nDefine \n\\begin{equation}\n\\theta _{n}=\\frac{\\delta ^{2\\left( n+1\\right) }\\dprod\\nolimits_{k=0}^{n\n\\left[ 1-a_{k}^{1}a_{k}^{2}\\left( 1-\\delta \\right) \\right] }\n\\prod\\limits_{k=0}^{n}\\left[ 1-a_{k}^{0}\\left( 1+\\delta \\right) \\right] \n\\text{.}  \\label{eqn50}\n\\end{equation\nBy the assumptio\n\\begin{eqnarray}\n\\lim_{n\\rightarrow \\infty }\\frac{\\theta _{n+1}}{\\theta _{n}}\n&=&\\lim_{n\\rightarrow \\infty }\\frac{\\delta ^{2}\\left[\n1-a_{n+1}^{1}a_{n+1}^{2}\\left( 1-\\delta \\right) \\right] }{\\left[\n1-a_{n+1}^{0}\\left( 1+\\delta \\right) \\right] }  \\label{eqn51} \\\\\n&=&\\delta ^{2}<1\\text{.}  \\notag\n\\end{eqnarray\nIt thus follows from ratio test that $\\sum\\limits_{n=0}^{\\infty }\\theta\n_{n}<\\infty $. Hence, we have $\\lim_{n\\rightarrow \\infty }\\theta _{n}=0$\nwhich implies that the iterative sequence defined by (7) converges faster\nthan the iterative sequence defined by Noor iteration method (5).\n\\end{proof}\n\nBy use of the following example due to \\cite{QR}, it was shown in (\\cite{CR\n, Example 4.1) that CR iterative method \\cite{CR} is faster than all of \\\nPicard (3), S \\cite{S}, Noor (5) and SP (6) iterative methods for a\nparticular class of \\ operators which is included in the class of\nweak-contraction mappings satisfying (11). In the following, for the sake of\nconsistent comparison, we will use the same example as that of \\ (\\cite{CR},\nExample 4.1) in order to compare the rates of convergence between the\nPicard-S iterative scheme (7) and the CR iteration method \\cite{CR} for\nweak-contraction mappings. In the following example, for convenience, we use\nthe notations $\\left( PS_{n}\\right) $ and $\\left( CR_{n}\\right) $ for the\niterative sequences associated to Picard-S (7) and CR \\cite{CR} iteration\nmethods, respectively.\n\n\\begin{example}\n\\cite{QR} Define a mapping $T:\\left[ 0,1\\right] \\rightarrow \\left[ 0,1\\right]\n$ as $Tx=\\frac{x}{2}$. Let $a_{n}^{0}=a_{n}^{1}=a_{n}^{2}=0$, for \nn=1,2,...,24$ and $a_{n}^{0}=a_{n}^{1}=a_{n}^{2}=\\frac{4}{\\sqrt{n}}$, for\nall $n\\geq 25$.\n\nIt can be seen easily that the mapping $T$ satisfies condition (11) with the\nunique fixed point $0\\in F_{T}$. Furthermore, it is easy to see that Example\n1 satisfies all the conditions of Theorem 1. Indeed, let $x_{0}\\neq 0$ be an\ninitial point for the iterative sequences $\\left( PS_{n}\\right) $ and \n\\left( CR_{n}\\right) $. Utilizing Picard-S (7) and CR \\cite{CR} iteration\nmethods we obtain \n\\begin{eqnarray}\nPS_{n} &=&\\frac{1}{2}\\left( \\frac{1}{2}-\\frac{4}{n}\\right) x_{n}  \\notag \\\\\n&=&\\cdots   \\notag \\\\\n&=&\\dprod\\limits_{k=25}^{n}\\left( \\frac{1}{4}-\\frac{2}{k}\\right) x_{0}\\text{\n}  \\label{eqn52}\n\\end{eqnarray\n\\begin{eqnarray}\nCR_{n} &=&\\left( \\frac{1}{2}-\\frac{1}{\\sqrt{n}}-\\frac{4}{n}+\\frac{8}{n\\sqrt{\n}}\\right) x_{n}  \\notag \\\\\n&=&\\cdots   \\notag \\\\\n&=&\\dprod\\limits_{k=25}^{n}\\left( \\frac{1}{2}-\\frac{1}{\\sqrt{k}}-\\frac{4}{k}\n\\frac{8}{k\\sqrt{k}}\\right) x_{0}\\text{.}  \\label{eqn53}\n\\end{eqnarray\nIt follows from (51) and (52) tha\n\\begin{equation*}\n\\frac{\\left\\vert PS_{n}-0\\right\\vert }{\\left\\vert CR_{n}-0\\right\\vert }\n\\frac{\\dprod\\limits_{k=25}^{n}\\left( \\frac{1}{4}-\\frac{2}{k}\\right) x_{0}}\n\\dprod\\limits_{k=25}^{n}\\left( \\frac{1}{2}-\\frac{1}{\\sqrt{k}}-\\frac{4}{k}\n\\frac{8}{k\\sqrt{k}}\\right) x_{0}}\n\\end{equation*\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ }=\\dprod\\limits_{k=25}^{n}\\frac{\\frac{1}{4}-\\frac{2}{k\n}{\\frac{1}{2}-\\frac{1}{\\sqrt{k}}-\\frac{4}{k}+\\frac{8}{k\\sqrt{k}}}\n\\end{equation*\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }=\\dprod\\limits_{k=25}^{n}\\frac\n\\left( k-8\\right) \\sqrt{k}}{2\\left( k\\sqrt{k}-2k-8\\sqrt{k}+16\\right) }\n\\end{equation*\n\\begin{eqnarray}\n\\text{ \\ \\ \\ \\ \\ \\ } &=&\\dprod\\limits_{k=25}^{n}\\frac{\\left( k-8\\right) \n\\sqrt{k}}{2\\left( \\sqrt{k}-2\\right) \\left( k-8\\right) }  \\notag \\\\\n&=&\\dprod\\limits_{k=25}^{n}\\frac{\\sqrt{k}}{2\\left( \\sqrt{k}-2\\right) }\\text{\n}  \\label{eqn54}\n\\end{eqnarray\nFor all $k\\geq 25$, we hav\n\\begin{eqnarray}\n\\frac{\\left( k-2\\right) \\left( \\sqrt{k}-4\\right) }{4} &>&1  \\notag \\\\\n&\\Rightarrow &\\left( k-2\\right) \\left( \\sqrt{k}-4\\right) >4  \\notag \\\\\n&\\Rightarrow &k\\left( \\sqrt{k}-4\\right) >2\\left( \\sqrt{k}-2\\right)   \\notag\n\\\\\n&\\Rightarrow &\\frac{\\sqrt{k}-4}{2\\left( \\sqrt{k}-2\\right) }>\\frac{1}{k} \n\\notag \\\\\n&\\Rightarrow &\\frac{\\sqrt{k}}{2\\left( \\sqrt{k}-2\\right) }<1-\\frac{1}{k}\\text\n,}  \\label{eqn55}\n\\end{eqnarray\nwhich yield\n\\begin{equation}\n\\frac{\\left\\vert PS_{n}-0\\right\\vert }{\\left\\vert CR_{n}-0\\right\\vert \n=\\dprod\\limits_{k=25}^{n}\\frac{\\sqrt{k}}{2\\left( \\sqrt{k}-2\\right) \n<\\dprod\\limits_{k=25}^{n}\\left( 1-\\frac{1}{k}\\right) =\\frac{24}{n}\\text{.}\n\\label{eqn56}\n\\end{equation\nTherefore, we hav\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\frac{\\left\\vert PS_{n}-0\\right\\vert }{\\left\\vert\nCR_{n}-0\\right\\vert }=0\\text{,}  \\label{eqn57}\n\\end{equation\nwhich implies that the Picard-S iterative scheme (7) is faster than the CR\niteration method \\cite{CR}$.$\n\\end{example}\n\nHaving regard to R. Chugh et al.'s result (\\cite{CR}, Example 4.1), L.B.\nCiric et al.'s results \\cite{Ciric} and Example 1 above, we conclude that\nPicard-S iteration method is faster than all Picard (3), Mann \\cite{Mann},\nIshikawa \\cite{Ishikawa}, S \\cite{S}, Noor (5) and SP (6) iterative methods.\n\nWe are now able to establish the following data dependence result.\n\n\\begin{theorem}\nLet $T$ with fixed point $u^{\\ast }\\in F_{T}\\neq \\emptyset $ be as in\nTheorem 1 and $\\widetilde{T}$ an approximate operator of $T$. Let $\\left\\{\nx_{n}\\right\\} _{n=0}^{\\infty }$ be an iterative sequence generated by (7)\nfor $T$ and define an iterative sequence $\\left\\{ \\widetilde{x}_{n}\\right\\}\n_{n=0}^{\\infty }$ as follow\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\widetilde{x}_{0}\\in D\\text{, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\widetilde{x}_{n+1}=\\widetilde{T}\\widetilde{y}_{n}\\text{, \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\widetilde{y}_{n}=\\left( 1-a_{n}^{1}\\right) \\widetilde{T}\\widetilde{x\n_{n}+a_{n}^{1}\\widetilde{T}\\widetilde{z}_{n}\\text{, \\ \\ \\ \\ \\ \\ } \\\\ \n\\widetilde{z}_{n}=\\left( 1-a_{n}^{2}\\right) \\widetilde{x}_{n}+a_{n}^{2\n\\widetilde{T}\\widetilde{x}_{n}\\text{, }n\\in \n\\mathbb{N}\n\\text{,\n\\end{array\n\\right.   \\label{eqn58}\n\\end{equation\nwhere $\\left\\{ a_{n}^{i}\\right\\} _{n=0}^{\\infty }$, $i\\in \\left\\{\n1,2\\right\\} $ be real sequences in $\\left[ 0,1\\right] $ satisfying (i) \n\\frac{1}{2}\\leq a_{n}^{1}a_{n}^{2}$ for all $n\\in \n\\mathbb{N}\n$, and (ii) $\\sum\\limits_{n=0}^{\\infty }a_{n}^{1}a_{n}^{2}=\\infty $. If \n\\widetilde{T}\\widetilde{u}^{\\ast }=\\widetilde{u}^{\\ast }$ such that \n\\widetilde{x}_{n}\\rightarrow \\widetilde{u}^{\\ast }$ as $n\\rightarrow \\infty \n, then we hav\n\\begin{equation}\n\\left\\Vert u^{\\ast }-\\widetilde{u}^{\\ast }\\right\\Vert \\leq \\frac\n5\\varepsilon }{1-\\delta }\\text{,}  \\label{eqn59}\n\\end{equation\nwhere $\\varepsilon >0$ is a fixed number.\n\\end{theorem}\n\n\\begin{proof}\nIt follows from (7), (11), (12), and (57) tha\n\\begin{eqnarray}\n\\left\\Vert z_{n}-\\widetilde{z}_{n}\\right\\Vert &\\leq &\\left(\n1-a_{n}^{2}\\right) \\left\\Vert x_{n}-\\widetilde{x}_{n}\\right\\Vert\n+a_{n}^{2}\\left\\Vert Tx_{n}-\\widetilde{T}\\widetilde{x}_{n}\\right\\Vert  \\notag\n\\\\\n&\\leq &\\left[ 1-a_{n}^{2}+a_{n}^{2}\\delta \\right] \\left\\Vert x_{n}\n\\widetilde{x}_{n}\\right\\Vert +a_{n}^{2}L\\left\\Vert x_{n}-Tx_{n}\\right\\Vert\n+a_{n}^{2}\\varepsilon \\text{,}  \\label{eqn60}\n\\end{eqnarray\n\\begin{eqnarray}\n\\left\\Vert y_{n}-\\widetilde{y}_{n}\\right\\Vert &\\leq &\\left(\n1-a_{n}^{1}\\right) \\delta \\left\\Vert x_{n}-\\widetilde{x}_{n}\\right\\Vert\n+a_{n}^{1}\\delta \\left\\Vert z_{n}-\\widetilde{z}_{n}\\right\\Vert  \\notag \\\\\n&&+\\left( 1-a_{n}^{1}\\right) L\\left\\Vert x_{n}-Tx_{n}\\right\\Vert\n+a_{n}^{1}L\\left\\Vert z_{n}-Tz_{n}\\right\\Vert  \\notag \\\\\n&&+\\left( 1-a_{n}^{1}\\right) \\varepsilon +a_{n}^{1}\\varepsilon \\text{,}\n\\label{eqn61}\n\\end{eqnarray\n\\begin{equation}\n\\left\\Vert x_{n+1}-\\widetilde{x}_{n+1}\\right\\Vert \\leq \\delta \\left\\Vert\ny_{n}-\\widetilde{y}_{n}\\right\\Vert +L\\left\\Vert y_{n}-Ty_{n}\\right\\Vert\n+\\varepsilon \\text{.}  \\label{eqn62}\n\\end{equation\nFrom the relations (59), (60), and (61\n\\begin{eqnarray}\n\\left\\Vert x_{n+1}-\\widetilde{x}_{n+1}\\right\\Vert &\\leq &\\delta ^{2}\\left[\n1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert x_{n}\n\\widetilde{x}_{n}\\right\\Vert  \\notag \\\\\n&&+\\left\\{ a_{n}^{1}a_{n}^{2}\\delta ^{2}L+\\left( 1-a_{n}^{1}\\right) \\delta\nL\\right\\} \\left\\Vert x_{n}-Tx_{n}\\right\\Vert  \\notag \\\\\n&&+L\\left\\Vert y_{n}-Ty_{n}\\right\\Vert +a_{n}^{1}\\delta L\\left\\Vert\nz_{n}-Tz_{n}\\right\\Vert  \\notag \\\\\n&&+a_{n}^{1}a_{n}^{2}\\delta ^{2}\\varepsilon +\\left( 1-a_{n}^{1}\\right)\n\\delta \\varepsilon +a_{n}^{1}\\delta \\varepsilon +\\varepsilon \\text{.}\n\\label{eqn63}\n\\end{eqnarray\nSince $a_{n}^{1}$, $a_{n}^{2}\\in \\left[ 0,1\\right] $ $\\ $and $\\frac{1}{2\n\\leq a_{n}^{1}a_{n}^{2}$ for all $n\\in \n\\mathbb{N}\n\n\\begin{equation}\n1-a_{n}^{1}a_{n}^{2}\\leq a_{n}^{1}a_{n}^{2}\\text{,}  \\label{eqn64}\n\\end{equation\n\\begin{equation}\n1-a_{n}^{1}\\leq 1-a_{n}^{1}a_{n}^{2}\\leq a_{n}^{1}a_{n}^{2}\\text{,}\n\\label{eqn65}\n\\end{equation\n\\begin{equation}\n1\\leq 2a_{n}^{1}a_{n}^{2}\\text{.}  \\label{eqn66}\n\\end{equation\nUse of the facts $\\delta $, $\\delta ^{2}\\in \\left( 0,1\\right) $, (63), (64),\nand (65) in (62) yields \n\\begin{eqnarray}\n\\left\\Vert x_{n+1}-\\widetilde{x}_{n+1}\\right\\Vert &\\leq &\\left[\n1-a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\right] \\left\\Vert x_{n}\n\\widetilde{x}_{n}\\right\\Vert  \\notag \\\\\n&&+a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\left\\{ \\frac{L\\delta \\left(\n1+\\delta \\right) \\left\\Vert x_{n}-Tx_{n}\\right\\Vert }{1-\\delta }\\right. \n\\notag \\\\\n&&\\left. +\\frac{2L\\left\\Vert y_{n}-Ty_{n}\\right\\Vert +2\\delta L\\left\\Vert\nz_{n}-Tz_{n}\\right\\Vert +5\\varepsilon }{1-\\delta }\\right\\} \\text{.}\n\\label{eqn67}\n\\end{eqnarray\nDefin\n\\begin{eqnarray}\n\\beta _{n} &:&=\\left\\Vert x_{n}-\\widetilde{x}_{n}\\right\\Vert \\text{,}\n\\label{eqn68} \\\\\n\\mu _{n} &:&=a_{n}^{1}a_{n}^{2}\\left( 1-\\delta \\right) \\in \\left( 0,1\\right) \n\\text{,}  \\notag \\\\\n\\gamma _{n} &:&=\\frac{L\\delta \\left( 1+\\delta \\right) \\left\\Vert\nx_{n}-Tx_{n}\\right\\Vert +2L\\left\\Vert y_{n}-Ty_{n}\\right\\Vert +2\\delta\nL\\left\\Vert z_{n}-Tz_{n}\\right\\Vert +5\\varepsilon }{1-\\delta }\\geq 0\\text{.}\n\\notag\n\\end{eqnarray\nHence, the inequality (66) perform all assumptions in Lemma 2 and thus an\napplication of Lemma 2 to (66) yields \n\\begin{eqnarray}\n0 &\\leq &\\lim \\sup_{n\\rightarrow \\infty }\\left\\Vert x_{n}-\\widetilde{x\n_{n}\\right\\Vert  \\label{eqn69} \\\\\n&\\leq &\\lim \\sup_{n\\rightarrow \\infty }\\frac{L\\delta \\left( 1+\\delta \\right)\n\\left\\Vert x_{n}-Tx_{n}\\right\\Vert +2L\\left\\Vert y_{n}-Ty_{n}\\right\\Vert\n+2\\delta L\\left\\Vert z_{n}-Tz_{n}\\right\\Vert +5\\varepsilon }{1-\\delta }. \n\\notag\n\\end{eqnarray\nWe know from Theorem 1 that $\\lim_{n\\rightarrow \\infty }x_{n}=u^{\\ast }$ and\nsince $Tu^{\\ast }=u^{\\ast }\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\left\\Vert x_{n}-Tx_{n}\\right\\Vert\n=\\lim_{n\\rightarrow \\infty }\\left\\Vert y_{n}-Ty_{n}\\right\\Vert\n=\\lim_{n\\rightarrow \\infty }\\left\\Vert z_{n}-Tz_{n}\\right\\Vert =0\\text{.}\n\\label{eqn70}\n\\end{equation\nTherefore the inequality (68) becomes \n\\begin{equation}\n\\left\\Vert u^{\\ast }-\\widetilde{u}^{\\ast }\\right\\Vert \\leq \\frac\n5\\varepsilon }{1-\\delta }.  \\label{eqn71}\n\\end{equation}\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe observation of a superfluid (SF) to Mott insulator (MI) transition in an \noptical lattice \\cite{greiner_02} have opened a new paradigm to explore the\nphysics of quantum many-body systems. Optical lattices are clean and highly\ncontrollable; in contrast, the condensed matter systems of interest are never\ndevoid of impurities. Thus, some of the fundamental questions in condensed \nmatter physics are related to quantum phase transitions in the presence of \ndisorder. The presence of disorder constrains the evolution of a quantum \nsystem in the Hilbert space and gives rise to quantum glassy phases \nlike Bose glass (BG) \\cite{fisher_89,lin_12} and phenomena like Anderson \nlocalization \\cite{anderson_58, schulte_05, billy_08, roati_08}. The early\ntheoretical investigations of disordered Bose Hubbard model (DBHM) \n\\cite{fisher_89, giamarchi_88} showed that there is no MI-SF transition in \npresence of diagonal disorder as the BG phase always occurs as the intermediate\nphase. The theorem of inclusion \\cite{pollet_09, gurarie_09} agrees well with \nthis prediction while identifying BG phase as a Griffiths phase containing \nthe rare regions. In these rare-regions, the energy gap of adding \nanother boson to the system vanishes and thus can be identified as SF islands.\n\n  The DBHM have been studied with diverse techniques: \nmean field \\cite{krutitsky_06}, projected Gutzwiller method \\cite{lin_12}, \nsite independent and multisite mean-field method \n\\cite{buonsante_07, pisarski_11}, stochastic mean field \\cite{bissbort_10},\nquantum Monte Carlo \\cite{gimperlein_05, soyler_11, sengupta_07}, \ndensity matrix renormalisation group (DMRG) \\cite{rapsch_99, gerster_16} for \n1D system and numerous others \n\\cite{pai_96, nikolay_04, kruger_09, kruger_11,carrasquilla_10}.\nIn all the cases the introduction of disorder leads to the emergence of BG \nphase which is characterized by finite compressibility and zero \nsuperfluid stiffness. In the present work, we study 2D DBHM at finite \ntemperatures using single site Gutzwiller and cluster Gutzwiller mean field \ntheories. More importantly, we examine the effect of the artificial gauge\nfields in DBHM. Here, it must be emphasized that most of the theoretical \ninvestigations of DBHM are at zero temperatures, but the experimental \nrealizations are at finite temperatures. This gap is addressed in the present \nwork by examining the consequent effects of thermal fluctuations to the BG \nphase. One key finding is the presence of normal fluid (NF) phase at finite \ntemperatures and melting of Bose glass phase. The latter is consistent with \nthe findings reported in ref. \\cite{thomson_16}. \n\n  In optical lattices it is possible to create an equivalent of Lorentz force \nwith artificial gauge fields\n\\cite{lin_09,lin_11,dalibard_11,hof_76, garcia_12, aidelsburger_11} and\nis associated with what is referred to as synthetic magnetic field.\nThe introduction of the artificial gauge field breaks time reversal symmetry\nand modify the band structure. Through the introduction of tunable artificial\ngauge field it has been possible to observe the single particle mobility \nedge \\cite{gadway_18} in zig-zag chains. Apart from the transport properties, \nthe localization effect of the artificial gauge field can enhance the glassy \nfeatures of DBHM. Indeed, our study reveals that localization in DBHM\ncan be controlled through the artificial gauge field. For this we use\nEdward Anderson order parameter (EAOP) as a measure of localization \nwhile tuning the strength of artificial gauge field. The EAOP is a measure of \nnumber fluctuation over disorder realizations and it is finite for the BG \nphase, but zero and close to zero for the MI and SF\nphases, respectively. From the values of EAOP we find that\nthere is enhancement of the BG phase in the presence of artificial gauge\nfield. From the experimental point of view this is important as it could\nfacilitate detailed studies of the BG phase.\n\n  Experimentally, DBHM can be realized by the addition of speckle type of \ndisorder \\cite{clement_05, clement_08, white_09}, or by the generation of \nincommensurate multichromatic lattice \\cite{damski_03,fallani_07}. Indirect\nmeasurements on SF-BG transition have been reported in 1D \\cite{gadway_11} and \n3D \\cite{pasienski_10,meldgin_16} systems through transport and coherence \nmeasurements. In 2D, the observation of center of mass dynamics \\cite{yan_17} \nhas been theoretically proposed as a method to detect the BG phase \nwhile ref. \\cite{delande_09} suggests measuring the radius of the atomic cloud.\nReplica symmetry breaking \\cite{thomson_14, morrison_08} also has been \nproposed as a possible detection scheme. In spite of these various \nproposals and progresses towards the realization of a Bose glass, a clear \nand unambiguous experimental evidence of BG phase is yet to be achieved. \nIn future studies, quantum gas microscopes \\cite{bakr_09} could probe the\nproperties of the BG phase as it can study the SF islands in BG phase.\nAnd, recent work has proposed  it as an experimental tool to detect \nBG phases \\cite{thomson_16}. \n  \n  This paper is organized as follows. In the Section II we give an account of \nthe single site and cluster Gutzwiller mean field theories. This is then \nfollowed by a description of the artificial gauge field and observable \nmeasures to distinguish different phases in Section III and IV. Then, in \nSection V we provide detailed description of the results obtained from our\nstudies and discuss our observations. And, we then conclude in Section VI.\n\n\n\n\\section{Model and Gutzwiller mean field theory}\n\\label{model}\n\nThe DBHM for a square lattice with nearest neighbour hopping is defined by the \nHamiltonian\n\\begin{eqnarray}\n  \\hat{H} &= &-\\sum_{p,q}\\left [ J_x\\left( \\hat{b}_{p+1, q}^{\\dagger}\n              \\hat{b}_{p,q} + {\\rm H.c.} \\right ) \n              + J_y\\left( \\hat{b}_{p, q+1}^{\\dagger}\n              \\hat{b}_{p,q} + {\\rm H.c.}\\right ) \\right ]\n                          \\nonumber \\\\\n          && + \\sum_{p,q}\\hat{n}_{p,q} \\left [\\frac{U}{2}(\\hat{n}_{p,q}\n            -1) -\\tilde\\mu_{p,q} \\right],\n  \\label{dbhm}\n\\end{eqnarray}\nwhere $p$ ($q$) is the lattice index along $x$ ($y$) axis, \n$\\hat{b}_{p,q}^{\\dagger}$ ($\\hat{b}_{p,q}$) is the creation (annihilation) \noperator for a a boson at the $(p,q)$ lattice site, and $\\hat{n}_{p,q}$ is the boson \ndensity operator; $J_x$ ($J_y$) is the hopping strength between two nearest \nneighbour sites along $x$ ($y$) axis,  $U>0$ is the on-site inter-atomic \ninteraction strength, and $\\tilde\\mu_{p,q} = \\mu - \\epsilon_{p,q}$ is the local\nchemical potential. The disorder is introduced through the random energy \noffset $\\epsilon_{p,q}$ which are uniformly distributed independent random \nnumbers $r_{p,q} \\in [-D, D]$ and $D$ is bound of \nrandom numbers. Depending on the ratio of $J$ and $U$ the above Hamiltonian \ncan describe three possible phases of the system --- MI, BG and \nSF~\\cite{fisher_89}. In the strong on-site interaction limit \n$(J\/U\\rightarrow 0)$ the system is either in the MI phase (gapped phase), or \nin the BG phase. Whereas the system is in SF phase when the tunneling overcomes \nrepulsive interaction. \n \n\n\n\n\\subsection{Zero temperature Gutzwiller mean-field theory}\n\\label{gmf}\n\n  In the present work we employ the Gutzwiller mean-field theory to compute the \nproperties of the DBHM. In this section we describe two variants of the \nGutzwiller mean field theory: First is the single site Gutzwiller mean-field \n(SGMF) method, where the lattice sites are correlated through a scalar mean \nfield $\\phi$ and cannot describe entangled states such as the quantum Hall state. \nAnd, the second is the cluster Gutzwiller mean field (CGMF) method, which \nincorporates the correlation within a cluster of neighbouring lattice sites \nexactly and inter-cluster correlation through $\\phi$. A larger cluster captures the\ncorrelation effects better but at the cost of higher computation.\n\n\n\n\\subsubsection{SGMF method}\n\\label{sgmf}\n\nIn the SGMF method, $\\hat{b}_{p, q}$ ( $\\hat{b}^\\dagger_{p, q}$) at a \nparticular lattice site $(p,q)$ is decomposed into mean field \n$\\phi_{p, q}$ ($\\phi^{*}_{p, q}$) and fluctuation $\\delta \\hat{b}_{p, q}$\n($\\delta \\hat{b}^{\\dagger}_{p, q}$) parts as\n\\begin{subequations}\n\\begin{eqnarray}\n  \\hat{b}_{p, q}           &=& \\phi_{p,q} + \\delta \\hat{b}_{p, q},       \\\\\n  \\hat{b}^{\\dagger}_{p, q} &=& \\phi^{*}_{p, q} + \\delta \\hat{b}^{\\dagger}_{p, q}\n \\label{decompose} \n\\end{eqnarray}\n\\end{subequations}\nwhere, $\\phi_{p,q} = \\langle\\hat{b}_{p,q}\\rangle$, and $\\phi^{*}_{p, q} = \n\\langle\\hat{b}^{\\dagger}_{p,q}\\rangle$ are the mean field and its complex\nconjugate, respectively. The expectations are defined with respect to the \nground state of the system. Employing this decomposition, the Hamiltonian\nin Eq. (\\ref{dbhm}) is reduced to the SGMF Hamiltonian\n\\begin{eqnarray}\n  \\hat{H}^{\\rm MF} &=& \\sum_{p, q}\\Biggr\\{-J_x\n                        \\bigg [ \\Big(\\hat{b}_{p + 1, q}^{\\dagger}\\phi_{p, q} \n                        + \\phi_{p + 1, q}^{*}\\hat{b}_{p, q}  \n                        - \\phi_{p+1,q}^{*}\\phi_{p, q}\\Big) \n                   \\nonumber\\\\   \n                   && + {\\rm H.c.}\\bigg ] \n                      - J_y\\bigg [ \\Big(\\hat{b}_{p, q+1}^{\\dagger} \\phi_{p, q} \n                      + \\phi_{p, q+1}^{*}\\hat{b}_{p, q} \n                      - \\phi_{p, q+1}^{*}\\phi_{p, q}\\Big)\n                   \\nonumber\\\\   \n                   && + {\\rm H.c.}\\bigg ] \n                      + \\biggr[\\frac{U}{2}\\hat{n}_{p, q}\n                      (\\hat{n}_{p, q}-1) - \\tilde{\\mu}_{p, q}\n                      \\hat{n}_{p, q}\\biggr] \\Bigg \\},\n\\label{mf_hamil}\n\\end{eqnarray}\nwhere terms up to linear in fluctuation operators are considered and those\nquadratic in fluctuation operators are neglected. The total Hamiltonian \nin the above expression can be rewritten as \n$\\hat{H}^{\\rm MF} = \\sum_{p,q}\\hat{H}^{\\rm MF}_{p,q}$, where \n$\\hat{H}^{\\rm MF}_{p,q}$ is the single site mean field Hamiltonian. The mean \nfield $\\phi_{p, q}$ can be identified as the SF order parameter which\ndefines the MI to BG phase-transition in DBHM. Thus, $\\phi_{p, q}$ is\nzero, when the  system is in MI phase, and finite in BG as well as in the SF phase. \n\nTo compute the ground state of the system the Hamiltonian matrix of \n$\\hat{H}^{\\rm MF}_{p,q}$ can be diagonalized for each lattice site $(p, q)$ separately. \nAnd, then the ground state of the system is direct product of the single\nsite ground states $\\ket{\\psi}_{p,q}$. Using the Gutzwiller approximation, the \nground state of the system is\n\\begin{eqnarray}\n \\ket{\\Psi_{\\rm GW}} = \\prod_{p, q}\\ket{\\psi}_{p, q}\n                       = \\prod_{p, q} \\sum_{n = 0}^{N_{\\rm b}}c^{(p,q)}_n\n                         \\ket{n}_{p, q},\n \\label{gw_state}\n\\end{eqnarray}\nwhere $N_b$ is the maximum allowed occupation number basis (Fock space basis),\nand $c^{(p,q)}_n$ are the coefficients of the occupation number state $\\ket{n}$\nat the lattice site $(p,q)$. From $\\ket{\\Psi_{\\rm GW}}$  we can calculate\n$\\phi_{p, q}$, the SF order parameter, as\n\\begin{equation}\n\\phi_{p, q} = \\langle\\Psi_{\\rm GW}|\\hat{b}_{p, q}|\\Psi_{\\rm GW}\\rangle \n            = \\sum_{n = 0}^{N_{\\rm b}}\\sqrt{n} \n              {c^{(p,q)}_{n-1}}^{*}c^{(p,q)}_{n}.\n\\label{gw_phi}              \n\\end{equation}\nFrom the above expression it is evident that $\\phi_{p, q}$ is zero in the MI \nphase as only one occupation number state $\\ket{n}$ contributes to \n$\\ket{\\psi}_{p,q}$ and hence only one $c^{(p,q)}_n$ has nonzero value. \nSimilarly, the occupancy and number fluctuation at a lattice site are\n\\begin{eqnarray}\n   \\langle \\hat{n}_{p,q}\\rangle &=& \n    \\sum_{n = 0}^{N_{\\rm b}} | c_n^{(p,q})|^2 n_{p,q},\\label{number} \\\\\n      \\delta n _{p,q} &=& \\sqrt{\\langle \\hat{n}_{p,q}^2\\rangle \n                       - \\langle \\hat{n}_{p,q}\\rangle ^2 }\n  \\label{deltan} \n\\end{eqnarray}\nIn the MI phase $\\delta n _{p,q}$ is zero, which makes MI phase incoherent. In \nthe BG and SF phase $\\delta n _{p,q}$ has nonzero value, but the value of \n$\\delta n _{p,q}$ in the BG phase is very small which arises due to\nthe presence of SF islands in the BG phase. The nonzero and \nrelatively large $\\delta n _{p,q}$ in the SF phase indicates strong phase \ncoherence. Thus $\\delta n _{p,q}$ can also be considered as the\norder parameter for MI-BG phase transition.  \n\n\n\n\n\\subsubsection{CGMF method}\n\\label{cgmf}\n In the CGMF method, to incorporate the hopping term exactly and hence improve\nthe correlation effects, the total lattice considered is partitioned into \nclusters. That is, for an optical lattice of dimension $K\\times L$, we can \nseparate it into $W$ clusters ($C$) of size $M\\times N$, that is \n$W=(K\\times L)\/(M\\times N)$. Thus, the case of CGMF with $M = N = 1$ is \nequivalent to SGMF.  In CGMF, the kinetic energy or the hopping term is decomposed\ninto two types. First is the intra-cluster or hopping within the lattice sites\nin a cluster, and second is the inter-cluster which is between neighbouring \nlattice sites which lie on the boundary of different clusters. The details of\nthe present implementation of the CGMF method is reported in ref. \\cite{bai_18} \nand the Hamiltonian of a single cluster is\n\\begin{eqnarray}\n \\hat{H}_C & =& -{\\sum_{p, q \\in C}}'\\biggr[J_x \n              \\hat{b}_{p+1, q}^{\\dagger}\\hat{b}_{p, q} \n              + J_y \\hat{b}_{p, q+1}^{\\dagger}\\hat{b}_{p, q}\n              + {\\rm H.c.}\\biggr]\\nonumber\\\\\n              &&-\\sum_{p, q\\in \\delta C}\n              \\biggr[J_x (\\phi^c_{p+1,q})^{\\ast}\\hat{b}_{p, q} \n              + J_y (\\phi^c_{p,q+1})^{\\ast}\\hat{b}_{p, q}\n              + {\\rm H.c.}\\biggr]\\nonumber\\\\\n           && +\\sum_{p, q \\in C}\n              \\biggr[\\frac{U}{2}\\hat{n}_{p, q}(\\hat{n}_{p, q}-1) - \n              \\tilde{\\mu}_{p, q}\\hat{n}_{p, q}\\biggr] \n\\label{cg_hamil}         \n\\end{eqnarray}\nwhere $(\\phi^c_{p,q})^{\\ast} = \\sum_{p^{'},q^{'}\\not\\in  C} \\langle \nb_{p^{'},q^{'}}\\rangle$ is the SF order parameter at the lattice site $(p, q)$ which lies\nat the boundary of neighbouring cluster. The prime in the summation of the \nfirst term is to indicate that the $(p+1,q)$ and $(p,q+1)$ lattice points are\nalso within the cluster. And, in the second term $\\delta C$ denotes the lattice\nsites at the boundary of the cluster. The matrix element of $\\hat{H}_C$ is\ndefined in terms of the cluster basis states\n\\begin{equation}\n  \\ket{\\Phi_c}_\\ell = \\prod_{q=0}^{N-1}\\prod_{p=0}^{M-1} \\ket{n_p^q},\n\\end{equation}\nwhere $\\ket{n_p^q}$ is the occupation number basis at the $(p,q)$ lattice\nsite, and $\\ell \\equiv \\{n_0^0, n_1^0, \\ldots, n_{M-1}^0, n_0^1, n_1^1,\\ldots\nn_{M-1}^1, \\ldots, n_{M-1}^{N-1}\\}$ is \nthe index quantum number to identify the cluster state. After diagonalizing \nthe Hamiltonian, we can get the ground state of the cluster as\n\\begin{equation}\n   |\\Psi_c\\rangle = \\sum_{\\ell} C_\\ell\\ket{\\Phi_c}_\\ell.\n\\end{equation}\nwhere $C_\\ell$ is the coefficient of the cluster state.\nThe ground state of the entire $K\\times L$ lattice, like in SGMF, is the direct\nproduct of the cluster ground states\n\\begin{equation}\n \\ket{\\Psi^c_{\\rm GW}} = \\prod_k\\ket{\\Psi_c}_k\n \\label{cgw_state}\n\\end{equation}\nwhere, $k$ is the cluster index and varies from 1 to \n$W$. The SF order parameter $\\phi$ is computed similar \nto Eq.~(\\ref{gw_phi}) as\n\\begin{equation}\n   \\phi_{p,q} = \\bra{\\Psi^c_{\\rm GW}}\\hat{b}_{p,q}\\ket{\\Psi^c_{\\rm GW}}.\n\\label{cgw_phi}              \n\\end{equation}\nWith respect to cluster basis, the average occupancy and number fluctuation \nof lattice sites in the $k$th cluster are\n\\begin{eqnarray}\n   \\langle \\hat{n}\\rangle _k &=& \\frac{\\sum_{p,q\\in C} \\langle\\hat{n}_{p,q}\n                                 \\rangle _k}{MN} \\label{cnumber}  \\\\  \n   (\\delta n)_k &=& \\sqrt{\\langle\\hat{n}^2\\rangle _k - \n                 \\langle\\hat{n}\\rangle^2_k}.\n   \\label{cdeltan} \n\\end{eqnarray}\nFor the entire lattice, the average density can be defined as the mean of the\naverage occupancy of the clusters.\n\n\n\n\n\\subsection{Finite temperature Gutzwiller mean field theory}\n\\label{gutz_t}\n\n To incorporate finite temperature effects we require the entire set of \neigenvalues and eigenfunctions obtained from the diagonalization of the mean\nfield Hamiltonian. So, in the case of SGMF we use all the single site \neigenvectors $\\ket{\\psi}^l_{p,q}$ and corresponding eigenvalues $E^l_{p,q}$ to\ndefine the single site partition function\n\\begin{eqnarray}\n  Z = \\sum_{l=1}^{N_b}e^{-\\beta E_l},\n  \\label{pf}\n\\end{eqnarray}\nwhere $\\beta = 1\/k_BT$, $T$ being the temperature of the system. Since the \nenergy $E_l$ is scaled with respect to $U$, $T$ is in units of $U\/k_{\\rm B}$ or \nin other words in the rest of the paper temperature is defined in terms\nof the dimensionless unit $k_{\\rm B}T\/U$. In a similar way, for the CGMF we \ncan define the cluster partition function in terms of the eigenfunctions \n$\\ket{\\Psi_c}^l$ and the corresponding eigenvalues. \nUsing the above description, the thermal average of the SF order parameter at \nthe $(p,q)$ lattice site is\n\\begin{equation}\n   \\langle \\phi_{p,q}\\rangle = \\frac{1}{Z}\\sum_{l}{_k^l\\bra{\\Psi_c}}\n                \\hat{b}_{p,q} e^{-\\beta E_l} \\ket{\\Psi_c}_k^l,\n\\label{phi_t}\n\\end{equation}\nwhere $\\langle\\ldots\\rangle$ is used to represent thermal averaging and \n$\\ket{\\Psi_c}_k^l$ is the $l$th excited state of the $k$th cluster within \nwhich the $(p,q)$ lattice site lies. Similarly, the occupancy or the density \ncan be computed as \n\\begin{equation}\n   \\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle =  \\frac{1}{Z}\\sum_{l}\n                      {_k^l\\bra{\\Psi_c}}\\hat{n}_{p,q}e^{-\\beta E_l}\n                      \\ket{\\Psi_c}^l_k,\n   \\label{number_t}\n\\end{equation}\nwhere, following the notations in Eq. (\\ref{number}) and (\\ref{cnumber}), the \nadditional $\\langle\\ldots\\rangle$ represents thermal averaging. Once we \nobtain $\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$, the average density or \noccupancy is \n$\\langle\\rho\\rangle=\\langle n \\rangle \n= \\sum_{p,q}\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle \/(K\\times L)$. Then, \nlike defined earlier, the number fluctuation is \n\\begin{equation}\n   \\delta n_{p,q} = \\sqrt{\\langle\\langle \\hat{n}^2_{p,q}\\rangle\\rangle\n                 -\\langle\\langle\\hat{n}_{p,q}\\rangle\\rangle^2}.\n\\end{equation}\nA new feature of considering finite temperature effects is, it is possible to \nhave vanishing $\\langle \\phi_{p,q}\\rangle$ but with  non-integer \n$\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$. This heralds a new \nphase in the phase diagram and is referred to as the normal fluid (NF). \nThus, at finite temperatures SF order parameter can act as the order parameter \nfor the NF-BG transition. \nCompared to the NF phase, the MI on the other hand has vanishing \n$\\langle \\phi_{p,q}\\rangle$ and integer \n$\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$. So, with vanishing \n$\\langle \\phi_{p,q}\\rangle$  the change from \ninteger value to non-integer $\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$\ncan be identified as MI-NF transition.\n\n\n\n\n\\section{Artificial gauge field}\n\\label{art_gauge}\n  Artificial gauge fields \\cite{lin_09,lin_11,dalibard_11} engineered through \noptical fields can create synthetic magnetic fields for charge neutral \nultracold atoms in optical lattices. This generates an equivalent of Lorentz \nforce for these atoms, and optical lattice is, then, endowed with properties\nanalogous to the quantum Hall system. Such a system is an excellent model \nsystem to study the physics of strongly correlated states like quantum Hall\nstates and their properties. The same logic also applies to the DBHM.\nIn the Hamiltonian description, the presence of an artificial gauge \nfield induces a complex hopping parameter $J \\rightarrow J\\exp(i\\Phi)$ and\naccordingly the SGMF Hamiltonian in Eq.~(\\ref{mf_hamil}) is modified to\n\\begin{eqnarray}\n  \\hat{H}^{\\rm MF} &=& \\sum_{p, q}\\Biggr\\{-J_x e^{i\\Phi}\n                        \\bigg [ \\Big(\\hat{b}_{p + 1, q}^{\\dagger}\\phi_{p, q}  \n                        + \\phi_{p + 1, q}^{*}\\hat{b}_{p, q}  \n                                  \\nonumber\\\\   \n                   && - \\phi_{p+1,q}^{*}\\phi_{p, q}\\Big) \n                      + {\\rm H.c.}\\bigg ] \n                      - J_y\\bigg [ \\Big(\\hat{b}_{p, q+1}^{\\dagger} \\phi_{p, q} \n                                  \\nonumber\\\\   \n                   && + \\phi_{p, q+1}^{*}\\hat{b}_{p, q} \n                      - \\phi_{p, q+1}^{*}\\phi_{p, q}\\Big)\n                      + {\\rm H.c.}\\bigg ] \n                                  \\nonumber\\\\   \n                   && + \\biggr[\\frac{U}{2}\\hat{n}_{p, q}\n                      (\\hat{n}_{p, q}-1) - \\tilde{\\mu}_{p, q}\n                      \\hat{n}_{p, q}\\biggr] \\Bigg \\},\n\\label{mf_hamil_gauge}\n\\end{eqnarray}\nwhere, $\\Phi$ is the phase an atom acquires when it traverses around a unit\ncell or plaquette. The artificial gauge field is considered in the Landau gauge \nand the phase for hopping along $x$ direction arises via the Peierl's\nsubstitution \\cite{hof_76, garcia_12}. The artificial gauge field, then,\ncreates a staggered synthetic magnetic flux \\cite{aidelsburger_11} along \n$y$ direction.  The phase can also be defined in terms of the $\\alpha$, the \nflux quanta per plaquette, through the relation $\\Phi = 2\\pi\\alpha q$, and \nthe flux quanta is restricted in the domain $0\\le \\alpha\\le 1\/2$.  In the present work, we \nexamine the properties of bosons in presence of artificial gauge field while \nexperiencing a random local chemical potential. Although, the effect of an \nartificial gauge field on BHM is quite well studied, the same is not true of \nDBHM. \n\n    \n\n\n\\section{Characterization of states}\n\n  Each of the low temperature phases supported by DBHM has special properties \nand this leads to unique combinations of order parameters as signatures of \neach phase. The values of these order parameters also determine\nthe phase boundaries. In Table.~\\ref{table:tab}, we list the order parameters\ncorresponding to each phase.  \n\n\n\n\n\\subsection{Superfluid stiffness and compressibility}\n\nPhase coherence is a characteristic property of the SF phase, and it is \nabsent in the other phases (MI, NF and BG) supported by DBHM. Thus in the \nSF phase it requires finite amount of energy to alter the phase coherence, or \nin other words, it acquires stiffness towards phase change. This property is \nreferred to as the superfluid stiffness $\\rho_s$, and hence plays an important\nrole in determining the phase boundary between BG and SF phase. To \ncompute $\\rho_s$, a twisted boundary condition (TBC) is imposed on the state. \nIf the TBC is applied along the $x$ direction, the hopping term in the \nDBHM is transformed as \n\\begin{equation}\n\\!\\!\\!\\!J_x(\\hat{b}_{p+1,q}^{\\dagger}\\hat{b}_{p,q} + {\\rm H.c})\\rightarrow \n             J_x(\\hat{b}_{p+1,q}^{\\dagger}\\hat{b}_{p,q}e^{i2\\pi\\varphi\/L} \n                              + {\\rm H.c}),\n\\label{twist}\n\\end{equation}\nwhere, $\\varphi$ is the phase shift or twist applied to the periodic boundary\ncondition, $L$ is the size of the lattice along $x$ direction, and \n$2\\pi\\varphi\/L$ is phase shift of an atom when it hops between nearest \nneighbours. Accordingly, $\\rho_s$ is computed employing the following \nexpression \\cite{gerster_16}\n\\begin{equation}\n  \\rho_s=\\frac{L}{8\\pi^2}\\frac{\\partial^2E_0}{\\partial\\varphi^2}|_{\\varphi = 0}.\n\\label{stiff}\n\\end{equation}\nwhere $E_0$ is the ground state energy with TBC.\nThe SF phase is a compressible state as $\\delta n$ is finite. However, MI phase\nand strongly correlated phase like quantum Hall states are incompressible. \nThus, the compressibility $\\kappa$ is a property of the system which can be\nemployed a diagnostic to support the phases determined through the order\nparameters. By definition, $\\kappa$ is given by\n\\begin{equation}\n   \\kappa=\\frac{\\partial\\langle\\hat{n}\\rangle}{\\partial\\mu}.\n   \\label{sfactor}\n\\end{equation}\nThat is, $\\kappa$ is the sensitivity of $n$ to the change of $\\mu$.\n\n\n\n\\begin{figure}\n  \\includegraphics[width=8cm]{eaop_dord.jpg}\n  \\caption{$q_{\\text{\\tiny{EA}}}$ as a function of $\\mu\/U$ and $J\/U$ at zero \n          temperature. (a)-(d) show $q_{\\text{\\tiny{EA}}}$ using SGMF method\n          and (e)-(h) are obtained employing the CGMF method with 2$\\times$ 2\n          cluster. (c)-(d) show the enhancement of the BG phase region in the \n          presence of an artificial gauge field with $\\alpha = 1\/4$ compared to \n          (a)-(b) corresponding to $\\alpha = 0$ with disorder strengths\n           $D\/U= 0.2$ and $0.6$ respectively. This enhancement is also\n           captured in (g)-(h) for $\\alpha = 1\/4$ compared to (e)-(f) using the\n           CGMF method.The increase of BG regions with an increase of $D\/U$ is\n           also notable for both in the presence and in the absence of an artificial\n           gauge field. In all the above figures, $q_{\\text{\\tiny{EA}}}$ is\n           obtained by averaging over 50 different disorder distributions. }\n    \\label{eaop-t0}\n\\end{figure}\n\n\n\\subsection{Edwards-Anderson order parameter}\n\nFor a disordered system the natural and hence, more appropriate order parameter\nis the Edwards-Anderson order parameter (EAOP). It can distinguish the \nGriffiths phase by its non zero value and can describe the effect of disorder \nbetter than other properties like  $\\rho_s$, $\\kappa$, structure factor, etc.\nIn the studies with mean field theory, EAOP was first introduced to detect the \nnon trivial breaking of ergodicity. Since then various type of EAOP have been \nproposed in literature \\cite{morrison_08,graham_09,thomson_14,khellil_16}. In \nour study we consider the EAOP of the following form \\cite{thomson_14}\n\\begin{eqnarray}\n   q_{\\text{\\tiny{EA}}} = \\overline{\\langle{\\hat{n}}_{p,q}\\rangle^2}\n       -\\overline{\\langle{\\hat{n}}_{p,q}\\rangle}^2,\n\\label{eaop}\n\\end{eqnarray}  \nwhere, $n_{p,q}$ is the number of atoms at the $(p,q)$ lattice site. The above \nexpression involves two types of averages: $\\langle\\cdots\\rangle$ represents\nthermal; and $\\overline{\\cdots}$ indicates average over disorder distribution.\nFor the $\\langle\\cdots\\rangle$ we consider all the excited states.\nFrom the definition, as the MI phase is identified by integer \nvalues of $\\langle{\\hat{n}}_{p,q}\\rangle$ $q_{\\text{\\tiny{EA}}}$ is zero. In \nthe SF phase $\\langle{\\hat{n}}_{p,q}\\rangle$ is real and $\\delta n_{p,q}$ is \nfinite, however, for the clean system $q_{\\text{\\tiny{EA}}}$ is \nzero as $\\langle{\\hat{n}}_{p,q}\\rangle$ is homogeneous. With disorder, \n$\\langle{\\hat{n}}_{p,q}\\rangle$ is inhomogeneous in the SF phase and hence, \n$q_{\\text{\\tiny{EA}}}$ is finite but small $O(10^{-3})$ \\cite{thomson_16}. In \nthe BG phase $q_{\\text{\\tiny{EA}}}$ is relatively large due to correlation \nbetween number density and disorder. Thus using $q_{\\text{\\tiny{EA}}}$ the BG \nphase is distinguishable from MI and NF phases in the present in the system. In \nzero temperature limit we define $q_{\\text{\\tiny{EA}}}$  as\n\\begin{eqnarray}\n   q_{\\text{\\tiny{EA}}}|_{(T=0)} = \\overline{\\langle{\\hat{n}}_{p,q}\\rangle^2}\n       -\\overline{\\langle{\\hat{n}}_{p,q}\\rangle}^2,\n   \\label{eaop0}\n\\end{eqnarray}  \nwhere we consider expectations only for the ground state.\n\\begin{table}[h!]\n  \\begin{ruledtabular}\n    \\begin{tabular}{lr} \n      \\textbf{Quantum phase} & \\textbf{Order parameter} \\\\\n      \\colrule\n      Superfluid (SF) &  $q_{\\text{\\tiny{EA}}} > 0$, $\\rho_s > 0$, $\\kappa > 0$, $\\phi\\ne 0$ \\\\\n      Mott insulator (MI) & $q_{\\text{\\tiny{EA}}} = 0$, $\\rho_s = 0$, $\\kappa = 0$, $\\phi = 0$ \\\\\n      Bose glass (BG) & $q_{\\text{\\tiny{EA}}} > 0$, $\\rho_s = 0$, $\\kappa > 0$, $\\phi\\ne 0$ \\\\\n      Normal fluid (NF) & $q_{\\text{\\tiny{EA}}} > 0$, $\\rho_s = 0$, $\\kappa > 0$, $\\phi= 0$\\\\\n    \\end{tabular}\n    \\caption{ Classification of quantum phases and the order parameters \n              supported by DBHM at zero and finite temperatures.}\n      \\label{table:tab}\n  \\end{ruledtabular}\n\\end{table}\n\n\n\n\\section{Results and Discussions}\n\\label{results}\n\n  To compute the ground state of the system and determine the phase diagram,\nwe scale the parameters of the DBHM Hamiltonian with respect to the\ninteraction strength $U$. So, the relevant parameters of the model are\n$J\/U$, $\\mu\/U$ and $D\/U$. We, then, determine the phase diagram of the DBHM\nin the $J\/U-\\mu\/U$ plane for different values of $D\/U$, and one unique feature\nof the model is the emergence of the BG phase. The local glassy nature of\nthe BG phase leads to very different properties from the incompressible and\ngapped MI phase, and compressible and gapless SF phase. Thus as mentioned\nearlier, one of the key issues in the study of DBHM is to identify appropriate\norder parameters to distinguish different phases. And, in particular, to\ndetermine the BG phase without ambiguity based on its local properties. To\nconstruct the phase diagram, we consider a $12\\times 12$ square\nlattice superimposed with a homogeneous disorder distribution. \n\nIn DBHM, depending on the magnitude of $D\/U$, the phase diagrams can be\nclassified into three broad categories. First, at low disorder strength\n$D\/U \\leqslant 0.1$, BG phase emerge in the phase diagram. Second, at moderate\ndisorder strengths $0.2\\leqslant D\/U \\leqslant 1$, the domain of BG phase is \nenhanced.  This is the most important regime to explore the physics of BG \nphase. The distinctive features in this range consist of shrinking of MI phase\nand enhancement of the BG phase. Finally, at very high disorder strengths \n$D\/U > 1$, the MI phase disappears and DBHM supports only two phases, BG and SF.\nFor reference the selected zero temperature results are shown in the \nAppendix.\n\n\n\n\n\\subsection{Zero temperature results}\n\\label{t0}\n\n  The synthetic magnetic field arising from the introduction of the artificial \ngauge field localizes the bosons and suppresses their itinerant property. \nThis manifests as a larger MI lobe in the presence of artificial gauge field.  \nHowever, locally the combined effect of disorder and artificial gauge field \nfavours the formation of SF islands. This synergy, then, creates a larger domain \nof BG phase in the phase diagram. In terms of identifying the phase boundaries, \nunlike in the $\\alpha=0$ where  $\\rho_s$ has linear dependence\non $J\/U$ in the SF domain, $\\rho_s$ cannot be used here as it exhibits no dependence \non $J\/U$. The two possible causes of this are: the TBC required to compute \n$\\rho_s$ modifies the magnetic unit cell associated with the chosen value of \n$\\alpha$; and with $\\alpha\\neq 0$ the SF phase contains vortices which reduce \nthe SF phase coherence. So, we use $q_{\\text{\\tiny{EA}}}$ as the order \nparameter to distinguish BG phase from the MI and SF phases. For consistency \nwe compute  $q_{\\text{\\tiny{EA}}}$ both for $\\alpha = 0$ and $\\alpha = 1\/4$ \nemploying SGMF and the results are shown in Fig.~\\ref{eaop-t0}(a)-(d), \nwhere  $q_{\\text{\\tiny{EA}}}$ is shown as a function of $\\mu\/U$ and $J\/U$. \nThe general trend is that  $q_{\\text{\\tiny{EA}}}$ is \nzero in MI and  O$(10^{-3})$ in the SF phase, and O$(10^{-1})$ in BG phase.  From \nthe figure, the presence of the BG phase between different MI lobes is \ndiscernible from the finite values of $q_{\\text{\\tiny{EA}}}$ and it is \nconsistent with the phase diagram determined from $\\rho_s$ shown in \nFig.~\\ref{ph-dia-al0}(g)-(j) in Appendix. We can define sharp MI-BG and SF-BG \nboundaries in the phase diagram by defining a threshold value \nof $q_{\\text{\\tiny{EA}}}$ between the Mott lobes, however, this is \nnon-trivial for the patina of BG phase present at the tip of Mott lobes. \nThis is the domain where the MI-SF quantum phase transition is driven by phase \nfluctuations and consequently, the number fluctuation is highly suppressed. As \na result the value of $q_{\\text{\\tiny EA}}$ is negligible and it cannot be \nused to distinguish BG and SF phases \\cite{buonsante_07,bissbort_10}. Thus, to \nidentify the BG domain it is essential to complement the results from \n$q_{\\text{\\tiny EA}}$ with those of other quantities.\n\n  For $\\alpha = 1\/4$, the region with finite values of $q_{\\text{\\tiny{EA}}}$ \nincreases significantly. This is discernible from the plot of \n$q_{\\text{\\tiny{EA}}}$ in Fig.~\\ref{eaop-t0}(d). For the case of $D\/U = 0.6$, \nwhen $\\alpha = 1\/4$, the $q_{\\text{\\tiny{EA}}}$ is finite with a value \nof $\\approx 0.2$ upto $J\/U \\approx 0.03$. Whereas, with $\\alpha=0$ as shown in \nFig.~\\ref{eaop-t0}(b), $q_{\\text{\\tiny{EA}}}$ has similar value only \nupto $J\/U \\approx 0.02$. This indicates the enhancement of BG region in the\npresence of the artificial gauge field. Employing CGMF method with \n$2 \\times 2$ cluster, the values of the $q_{\\text{\\tiny{EA}}}$ obtained are \nshown in Fig.~\\ref{eaop-t0}(e)-(h). One important change is that, \n$q_{\\text{\\tiny EA}}$ is no longer zero in the MI phase, but it is of \nO$(10^{-6})$. This is due to the presence of particle-hole \nexcitations in the cluster states. And, the non-zero value of \n$q_{\\text{\\tiny EA}}$ is consistent with the results reported in a previous \nwork \\cite{morrison_08}. The figures show similar trends of artificial \ngauge field induced enhancement of the BG region in the phase diagram. The\nincrease of BG regions with the increase of $D\/U$ is also notable for both \n$\\alpha = 0$ and $\\alpha = 1\/4$. Another observation is that, \n$q_{\\text{\\tiny{EA}}}$ obtained \nfrom the CGMF method contains less fluctuations and thus describes the boundary \nof SF-BG transition better compared to the SGMF method. Increasing the cluster \nsize CGMF can describe the BG-SF boundary more accurately but at the cost of\nmuch higher computational resources. \n\\begin{figure}[H]\n\t\\centering\n \\includegraphics[width=7.5cm]{phd-finite-t}\\\\\n\\vskip 0.1cm\n \\includegraphics[width=8cm]{eaop-al0}\n    \\caption{Finite temperature phase diagram using SGMF method\n             in absence of artificial gauge\n             field for six different temperatures (a) $T = 0.005 U\/k_B$, \n             (b)$T = 0.01U\/k_B$, (c) $T = 0.02U\/k_B$, (d) $T = 0.03U\/k_B$,\n             (e)$T = 0.05U\/k_B$ and (f) $T = 0.1U\/k_B$ .\n             Disorder strength is fixed at $D = 0.6U$ and each data in the plot\n             is obtained by averaging over 500 different disorder\n             distributions. (g) shows finite temperature effects on \n             Edward-Anderson order parameter ( $q_{\\text{\\tiny{EA}}}$) \n             with $D\/U = 0.6$ and $J\/U$ being fixed at 0.01. The magnitude \n             of $q_{\\text{\\tiny{EA}}}$ gradually decreases with increase of \n             temperature.}\n    \\label{ph-dia-t}\n\\end{figure}\n\n\n\n\n\n\\subsection{Finite temperature results}\n\\label{ftemp}\n The important outcome of finite temperature is the emergence of a new phase,\nthe NF phase. This new phase, like the SF phase, has real commensurate \nnumber of particles per site. But, unlike SF $\\phi$ is zero. So, the NF phase \nhas some features common to both the MI and SF phases. Previous works reported\nthe appearance of the NF phase at finite temperatures in the case of the \ncanonical Bose-Hubbard model \\cite{gerbier_07}, and extended\nBose-Hubbard model with nearest neighbour interactions \\cite{ng_10, lin_17}.\n\n\n\n\n\\subsubsection{$\\alpha=0$}\n\n The effect of the thermal fluctuations to the $q_{\\text{\\tiny{EA}}}$, in\nabsence of artificial gauge field ( $\\alpha=0$), is shown in \nFig.~\\ref{ph-dia-t}(g).  The \nresults presented in the figure correspond to $D\/U=0.6$  and \neach plot is an average over 500 realizations of disorder distributions. With \nincreasing temperature there is a monotonic decrease in $q_{\\text{\\tiny{EA}}}$,\nwhich indicates the {\\em melting} of BG phase. Along with the BG phase the MI \nphase also melts, however, this is not apparent from the values of \n$q_{\\text{\\tiny{EA}}}$. And, the extent of melting can be inferred from the \nphase diagram. To illustrate this point the phase diagram of DBHM at different \ntemperatures are shown in Fig~\\ref{ph-dia-t}(a-f). As mentioned earlier, \nprevious studies have also reported the melting of MI phase due to thermal \nfluctuations \\cite{gerbier_07}. But a clear theoretical description and phase \ndiagram incorporating finite temperature effects are lacking. Our present work \nshows that the BG phase also melts due to thermal fluctuations. Here, the key \npoint is the SF islands, which are hallmark of the BG phase, melts into NF.\nThis arises from \nthe local nature of the SF islands in BG phase, which as a result is affected \nby the local nature of the thermal fluctuations. The bulk SF phase, on the \nother hand, has long range phase correlations and is more robust against \nlocal fluctuations stemming from finite temperatures. \n\n\\begin{figure}[H]\n\t\\centering\n \\includegraphics[width=8cm]{phd-ctemp}\n    \\caption{Finite temperature phase diagram using CGMF for $2\\times 2$\n             cluster in absence of artificial gauge\n             field for two different temperatures (a) $T = 0.01 U\/k_B$, \n             (b)$T = 0.02U\/k_B$;\n             Disorder strength is fixed at $D = 0.6U$ and each data in the plot\n             is obtained by averaging over 50 different disorder distributions.\n             }\n    \\label{cphd-ft}\n\\end{figure}\n\n \n   In the plots the region within the black line is MI phase, whereas, the \nregion bounded by the black and green lines is the NF phase, where $\\phi$ is \nclose to zero  $\\phi \\leqslant 10^{-6}$. The BG phase lies in the region \nbounded by the green and orange lines, and the area right of the orange line \nis the SF phase. As the temperature is increased, due to the increased thermal \nfluctuations, the phase diagrams undergo several changes. First, the  MI \nlobes shrink and at $k_{\\rm B}T\/U = 0.02$, MI lobes disappear from the phase \ndiagram. This is due to the melting of MI phase and conversion into NF phase. \nSo, as discernible from the comparison of Fig.~\\ref{ph-dia-t}(a) and (b), the \nMI lobe with $\\rho=1$ is bounded and lies in the domain \n$0.40\\leqslant \\mu\/U\\leqslant 0.6$ at $k_{\\rm B}T\/U = 0.005$,\nbut it shrinks to 0.47 $\\leqslant \\mu\/U\\leqslant 0.53 $ at $k_{\\rm B}T\/U = 0.01$.\nSecond, the region of the BG phase is reduced with increasing temperature. The \nchange is more prominent in the regions which lie between the MI lobes. For \nexample, at $\\mu=0$ the BG phase exists in the domain \n$0.004\\leqslant J\/U\\leqslant 0.014$ for $k_{\\rm B}T\/U=0.005$. But, it is \nreduced to  $0.008\\leqslant J\/U\\leqslant 0.015$ when the temperature is \nincreased to $k_{\\rm B}T\/U = 0.01$. As discernible from Fig. ~\\ref{ph-dia-t}(f)\nat $k_{\\rm B}T\/U = 0.1$  the domain is reduced to \n$0.04\\leqslant J\/U\\leqslant 0.043$.  And, third, at finite temperatures the \nMI lobes are bounded from top and bottom by straight lines in the SGMF \nresults. But, as visible from Fig. \\ref{cphd-ft}, the MI boundary is not a \nstraight line with CGMF results. This is on account of the better correlation \neffects in CGMF, in contrast, SGMF tends to support sharp NF-MI boundaries \nas a function of $\\mu\/U$ due to short range coupling through $\\phi$.\n\\begin{figure}[H]\n\t\\centering\n \\includegraphics[width=8cm]{eaop-mu-jp02}\n    \\caption{ $q_{\\text{\\tiny{EA}}}$ as a function of $\\mu\/U$ for four\n             different values of $\\alpha$ at (a) $T = 0$ (b) $T = 0.03U\/k_B$; \n             with fixed disorder strength $D = 0.6U$ and\n             hopping strength $J = 0.02U$.\n             In each subfigure  $q_{\\text{\\tiny{EA}}}$ are calculated for \n             $\\alpha = 0, 1\/12, 1\/6$ and $1\/2$\n             and averaged over 500 different disorder distributions. \n             }\n    \\label{eaop-al}\n\\end{figure}\n\n\n Based on the above observations of the phase diagrams at different \ntemperatures, the NF-BG and BG-SF phase boundaries shift toward higher \n$J\/U$ with increasing temperature. This is due to higher hopping energy \nrequired to prevail over thermal fluctuations. So that the SF phase is present \nas islands or homogeneous network in BG and SF phases, respectively. The \nother important point is that, the SF phase does not melt directly to NF phase.\nIn other words, the BG phase advances into the SF phase with ever decreasing \nwidth with increasing temperature. Thus, the BG phase is an intermediate \nphase between the NF and SF phases. This is the finite temperature equivalent\nof the zero temperature phase structure, where BG phase is an intermediate\nphase between the MI and SF phases.\n\n To improve the accuracy of the phase diagram by incorporating additional \ncorrelation effects, we compute the phase diagram with CGMF using \n$2\\times 2$ cluster, and the resulting phase diagram is shown in \nFig.~\\ref{cphd-ft}. The results are for the temperatures $k_{\\rm B}T\/U = 0.01$ \nand $0.02$, and for better illustration the phase diagrams of only upto \n$\\mu\/U = 1.0$ are shown in the figure. As to be expected the MI lobes are \nlarger in the CGMF results, but the one important change is that the \nenvelope of BG phase around the MI and NF phases is more pronounced. \nConsequent to the larger MI lobes, the NF and BG phases encompass\nregions with higher $J\/U$ compared with the SGMF results. In particular, \nat $\\mu =0$ the BG phase occurs in the domain \n$0.011\\leqslant J\/U \\leqslant 0.018$ and $0.018\\leqslant J\/U \\leqslant 0.022$  \nfor the $k_{\\rm B}T\/U = 0.01$ and $k_{\\rm B}T\/U = 0.02$, respectively.\n\n\n\n\n\\subsubsection{$\\alpha \\ne 0$}\n\n The thermal fluctuations delocalize the atoms through the entire lattice,\nand melt MI phase. This tends to reduce $\\phi$. Whereas, as mentioned earlier,\nartificial gauge field localizes the atoms, and thereby enhances the\nMI lobes. So, these two have opposing effects on the DBHM, and the combined\neffects of these two physical factors on the $q_{\\text{\\tiny{EA}}}$ are shown \nin Fig.~\\ref{eaop-al}. In the figure 4 the plots of $q_{\\text{\\tiny{EA}}}$ for \n$k_{\\rm B}T\/U =0$ and $0.03$ are shown for different $\\alpha$ as a function \n$\\mu\/U$ at $J\/U=0.02$. From the figures it is apparent that the effect of the\nartificial gauge field is negligible in the region between the $\\rho=0$ and\n$\\rho=1$ Mott lobes. However, in the regions between other Mott lobes there is \nan enhancement of the BG phase as indicated by the increase in \n$q_{\\text{\\tiny{EA}}}$. As discernible from Fig.~\\ref{eaop-al}(a) the \nvalue of  $q_{\\text{\\tiny{EA}}}$ increases from $0.13$ to $0.19$ for the \nregion between $\\rho=1$ and $\\rho=2$ corresponding to \n$0.65 \\leqslant \\mu\/U\\leqslant 1.36 $ for non-zero $\\alpha$ at \n$k_{\\rm B}T\/U =0$. From the figure it is also evident that \n$q_{\\text{\\tiny{EA}}}$ gradually increases with the increase of $\\alpha$.\nConsequently, the enhancement of BG phase region in DBHM depends on the\nstrength of artificial gauge field. As a quantitative measure of it, for\n$\\alpha = 0, 1\/12, 1\/6$ and $1\/2$ $q_{\\text{\\tiny{EA}}}$\ntakes the value $0.139$, $0.148$, $0.164$ and $0.187$ respectively around \n$\\mu = U$. To demonstrate the combined effect of finite \ntemperature and artificial gauge field, the phase diagram in terms of \n$q_{\\text{\\tiny{EA}}}$ is shown in Fig. \\ref{eaop-ft}. As the \nfigure is based on $50$ disorder realizations, the general trends of \n$q_{\\text{\\tiny{EA}}}$ observable in Fig.~\\ref{eaop-al} are not apparent. \nHowever, from the figure the enlargement of the BG phase region between the \nMI lobes is discernible. Thus, this implies that the enhancement of the BG \nphase in presence of artificial gauge field is stable against thermal \nfluctuations.\n\\begin{figure}\n\t\\centering\n  \\includegraphics[width=8cm]{eaop_al1b4-cl}\\\\\n    \\caption{ $q_{\\text{\\tiny{EA}}}$ as a function of $\\mu\/U$ and $J\/U$ for\n             $\\alpha =1\/4$ for two different values\n             of temperature $T = 0.01 U\/k_B$ (a) and $T = 0.03 U\/k_B$. (b) \n             Disorder strength is kept fixed at $D= 0.6 U$ and \n             $q_{\\text{\\tiny{EA}}}$ are averaged over 50 different disorder \n             distributions with CGMF method. }\n    \\label{eaop-ft}\n\\end{figure}\n\n\n\n\n\\section{Conclusions}\n\\label{conc}\nAt finite temperatures, the thermal fluctuations lead to melting of the BG \nphase and formation of NF phase. The emergence of the NF phase at finite \ntemperatures necessitates using a combination of order parameters and \nproperties to identify each phase without ambiguity. More importantly, \nthe BG phase is an intermediate phase between the NF and SF phases. \nThis is similar to the zero temperature phase where the BG phase is an \nintermediate phase between the MI and SF phases. At higher temperatures the \nmelting of MI phase is complete and\nonly three phases NF, BG and SF phases exist in the system. The addition of \nartificial gauge field brings about a significant change in the phase diagram \nby enhancing the BG phase domain, which is observed in the trends of \nthe $q_{\\text{\\tiny{EA}}}$ without any ambiguity. This implies that such \nenhancements would be observable in quantum gas microscope experiments. To get \naccurate results with mean field theories it is desirable to use the CGMF \ntheory. It incorporates correlation effects better and the phase diagrams \nobtained from CGMF are quantitatively different from those obtained from SGMF.\n\n\n\n\n\\begin{acknowledgments}\n\nThe results presented in the paper are based on the computations\nusing Vikram-100, the 100TFLOP HPC Cluster at Physical Research Laboratory, \nAhmedabad, India.\n\n\\end{acknowledgments}\n\n\n\n\n\\section*{Appendix}\n\nTo determine the MI-BG phase boundary, we consider number fluctuation \n($\\delta n$) as the property which distinguishes the two phases. In the \nMI phase $\\delta n$ is zero for $D\/U=0$, however, for $D\/U\\neq 0$, it is \nnon-zero but small due to the disorder. We set $\\delta n < 10^{-6}$ as the \ncriterion to identify the MI phase in our computations. On the other hand, to \ndefine the BG-SF boundary, we compute the superfluid stiffness ($\\rho_s$). \nIn BG phase as the SF phase exists as islands the phase coherence is limited\nto these, so the $\\rho_s$ small, and we consider $\\rho_s < 10^{-2}$ as the \nthreshold to distinguish the BG from SF phase. In the SF phase as there is \nphase coherence throughout the system $\\rho_s$ is large and it is $O(1)$.  \n\n\\begin{figure}[H]\n~ \\includegraphics[width=7.8cm]{gstate-t0}\\\\\n  \\includegraphics[width=7.5cm]{phd-ss}\n    \\caption{ Order parameter $\\phi$ of DBHM at zero temperature for \n              $J\/U = 0.01$ and $D\/U$ keeping fixed at 0.6 (a)-(c) without and \n             (d)-(f) \n              with ($\\alpha=1\/4$) artificial gauge field. (a) \\& (d) MI phase \n              with $\\mu\/U = 0.5$; (b) \\& (e) BG phase with  $\\mu\/U = 0.1$; \n              and  (c) \\& (f)SF phase with $\\mu\/U = 1.0$. \n              (g)-(j) equilibrium phase diagram of DBHM using SGMF method at \n              zero temperature in absence of artificial gauge field\n              ($\\alpha=0$) for disorder strengths $D\/U=0$, $0.2$, $0.6$ and\n              $1.2$, respectively for 500 different disorder realizations.\n              }\n    \\label{ph-dia-al0}\n\\end{figure}\n\n The phase diagrams of DBHM with $\\alpha=0$ at  different values of $D\/U$ have \ndistinctive features \\cite{lin_12}. As examples, the phase \ndiagrams for the case of $D\/U = 0$, $0.2$, $0.6$ and $1.2$ obtained from the\nSGMF method are shown in Fig.~\\ref{ph-dia-al0}(g)-(j).  With $D\/U=0$, the phase \ndiagram as shown in Fig.~\\ref{ph-dia-al0}(g) consists of only two phases MI \nand SF. With non-zero $D\/U$ BG appears in the phase diagram, and as shown in \nFig.~\\ref{ph-dia-al0}(h) for $D = 0.2$ the domain of the MI phase shrinks and an\nenvelope of BG phase emerges around the MI lobes. From \nFig.~\\ref{ph-dia-al0}(h), it is clear that the BG phase is most prominent in\nbetween the MI lobes.  These are the domains with large density \nfluctuations and small disorder is sufficient to render the bosons itinerant \nto create islands of SF phase. This, then, leads to the formation of BG phase. \nWhen the $D\/U$ is increased to $0.6$, as shown in Fig.~\\ref{ph-dia-al0}(i), \nthe MI lobes shrink further and the area of the BG phase is enlarged. At \nsufficiently high disorder strength, $D = 1.2U$, the MI phase disappears and \nphase diagram Fig.~\\ref{ph-dia-al0}(j) is composed of only SF and BG phases. \n \n\nThere is an improvement in the phase diagram, which is apparent from the \nenlarged MI lobes, when the phase diagram is computed using CGMF. In particular,\nwe consider $2\\times 2$ cluster and the phase diagrams so obtained are\nshown in  Fig.~\\ref{cph-dia-al0}. The overall structure of the phase diagram\nis qualitatively similar to the SGMF case. However, there are few quantitative\nchanges. For comparison, consider the case of $D\/U = 0.6$, based on our \nresults and as visible in Fig.~\\ref{ph-dia-al0}(i) and \nFig.~\\ref{cph-dia-al0}(b), there are three important difference due to \nbetter correlation effects encapsulated in the CGMF method. First, the tip of \nthe Mott lobe $\\rho=1$ extends upto $0.035$ while it was $0.032$ with SGMF.  \nSecond, at $\\mu\/U \\simeq 0$, the SF-BG transition occurs at $J\/U\\approx  0.022$, \nwhich in the case of SGMF is at $ J\/U\\approx 0.014$. This is due to the \nassociation of BG phase with islands of SF phase, and CGMF captures the phase \ncorrelations in these islands better. The SGMF, on the other hand, tends to \nfavour long range phase correlations through the $\\phi$ coupling between the \nlattices sites. And, finally, around the tip of the Mott lobes, the area of BG \nphase increases in CGMF method.  \n\\begin{figure}[H]\n \\includegraphics[width=8cm]{c-ph-dia}\n    \\caption{Equilibrium phase diagram of DBHM using CGMF method with cluster\n             size 2$\\times$2 at zero temperature in absence of artificial\n             gauge field for disorder strength (a) $D\/U = 0.2$, and  \n             (b) $D\/U = 0.6$ for 50 different disorder realizations.}\n    \\label{cph-dia-al0}\n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nSoftware-Defined Networking (SDN) decouples the network control plane from the data plane via a well-defined programming interface (such as OpenFlow~\\cite{mckeown2008}).\nThis decoupling allows the control logic to be logically centralized, easing the implementation of network policies, enabling advanced forms of traffic engineering (e.g., \\\\Google's B4~\\cite{jain2013}), and facilitating innovation (network virtualization~\\cite{koponen2014} being a prominent example).\n\nThe controllers are the crucial enabler of the SDN paradigm: they maintain the logically centralized network state to be used by applications and act as a common intermediary with the data plane.\nFigure \\ref{fig:intro_sdn} shows the normal execution in an SDN environment.\nUpon receiving a packet it does not know how to handle, the switch sends an \\emph{event} to the controller.\nThe controller delivers the event to the applications, which afterwards apply their logic based on this event, and eventually instruct the controller to send \\emph{commands} to the switches (e.g., to install flow rules).\n\n\\begin{figure}[h!]\n    \\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/intro_sdn.png}\n    \\caption[SDN flow execution]{SDN flow execution. Switches send events to the controller as needed and the controller replies with one or more commands that modify the switch's tables.}\n    \\label{fig:intro_sdn}\n\\end{figure}\n\nA trivial implementation of SDN using a centralized controller would lead to an undesirable outcome: a single point of failure.\nTo guarantee the required availability of network control, it is necessary the controller platform to be made fault-tolerant.\nFault tolerance demands transparency: for a controller that claims having such ability -- in other words, for it to be logically centralised -- applications that run on it should operate correctly in the presence of faults.\nThis is a fundamental requirement, as in case of controller failures the network view needs to be maintained consistent, otherwise applications will operate in a stale network view, leading to network anomalies that can have undesirable consequences (e.g., security breaches)~\\cite{levin2012, katta2015}.\n\nTo address this problem, traditional replication techniques are usually employed~\\cite{oki1988, lamport1998, ongaro2014}.\nHowever, building a consistent network view in the controllers is not enough to offer consistent logically centralized control that prevents the above-mentioned anomalies.\nIn SDN, it is necessary to include switch state into the system model to achieve this goal~\\cite{katta2015}.\nSince switches are programmed by controllers (and controllers can fail), there must be mechanisms to ensure that the entire event-processing cycle of SDN is handled consistently.\n\nA correct, fault-tolerant SDN environment needs to ensure \\emph{observational indistinguishability}~\\cite{katta2015} between an ideal central controller and a replicated controller platform.\nInformally, to ensure observational indistinguishability the fault-tolerant system should behave the same way as a fault-free SDN for its users (end-hosts and network applications).\nFot this purpose, it is necessary the following properties to be met:\n\n\\begin{itemize}\n\\item \\textbf{Total Event Ordering:} Controller replicas should process events in the same order and subsequently all controller application instances should reach the same internal state\n\\item \\textbf{Exactly-Once Event Processing:} All the events are processed, and neither are lost nor processed repeatedly.\n\\item \\textbf{Exactly-Once Execution of Commands:} Any given series of commands are executed once, and only once on the switches.\n\\end{itemize}\n\nTo the best of our knowledge, the problem of correct, fault-tolerant SDN control has only been addressed in the work by Katta~\\emph{et al.}~\\cite{katta2015}.\nInstead of just keeping the controller state consistent, the authors propose Ravana, a fault-tolerant SDN controller platform that handles the entire event-processing cycle as a transaction -- either all or none of the components of this transaction are executed.\nThis enables Ravana to correctly handle switch state and thus guarantee SDN correctness even under fault.\n\nTo achieve these properties, however, Ravana requires modifications to the OpenFlow protocol and to existing switches.\nSpecifically, switches need to maintain two buffers, one for events and one for commands, and four new messages need to be added to the protocol.\nThese modifications preclude the adoption of Ravana on existing systems and hinder the possibility of it being used in the near future (as there are no guarantees these messages be added to OpenFlow anytime soon, for instance).\n\nFaced with this challenge, we propose Rama, a fault-tolerant SDN controller platform that, similar to Ravana, offers a transparent control plane that allows unmodified network applications to run in a consistent and fault-tolerant environment.\nThe novelty of the solution lies in Rama not requiring changes to OpenFlow nor to the underlying hardware, allowing immediate deployment.\nFor this purpose, Rama exploits existing mechanisms in OpenFlow and orchestrates them to achieve its goals.\n\nThe main contributions of this work can be summarized as follows:\n\n\\begin{itemize}\n\\item A protocol for fault-tolerant SDN that provides the correctness  guarantees of a logically centralised controller \\emph{without} requiring changes to OpenFlow or modifications to switches.\n\\item The implementation and evaluation of a prototype controller -- Rama -- that demonstrates the overhead of the solution to be modest.\n\\end{itemize}\n\n\\section{Fault tolerance in SDN}\n\\label{motivation}\n\nKatta~\\emph{et al.} have experimentally shown~\\cite{katta2015} that traditional techniques for replicating controllers do not ensure correct network behaviour in case of failures.\nThe reason is that these techniques address only part of the problem: maintaining consistent state in controller replicas.\nBy not considering switch state (and the interaction controller-switches) inconsistencies may arise, resulting in potentially severe network anomalies.\nIn this section we present a summary of the problems of using techniques that do not incorporate switches in the system model, which lead to the design requirements of a \\emph{correct} fault-tolerant SDN solution. \nWe also present Ravana~\\cite{katta2015}, the first fault-tolerant controller that achieves the required correctness guarantees for SDN.\n\n\\subsection{Inconsistent event ordering}\n\nSince OpenFlow 1.3, switches can maintain TCP connections with multiple controllers.\nIn a fault-tolerant configuration switches can be set to send all their events to all known controller replicas.\nAs replicas process events as they are received, each one may end up building a different internal state.\nAlthough TCP guarantees the order of events delivered by each switch, there are no ordering guarantees between events sent to controllers by the different switches, leading to the problem.\n\nConsider a simple scenario with two controller replicas (c1 and c2) and two switches (s1 and s2) that send all events to both controllers.\nSwitch s1 sends two events -- e1 and e2, in this order -- and switch s2 sends two other events --  e3 and e4, in this order.\nOne possible outcome where both controllers receive events in a different order while respecting the TCP FIFO property is c1 receiving events in the order e1, e3, e2, e4 and c2 receiving in the order e3, e4, e1, e2.\nUnfortunately, an inconsistent ordering of events can lead to incorrect packet-processing decisions~\\cite{katta2015}.\nAs a result of this consistency problem we derive the first design goal for a fault-tolerant and correct SDN controller:\n\n\\textbf{Total event ordering:} controllers replicas should process the same (total) order of events and subsequently all controller application instances should reach the same internal state.\n\n\\subsection{Unreliable event delivery}\n\nIn order to achieve a total ordering of events between controller replicas two approaches can be used:\n\n\\begin{enumerate}\n\\item The master (primary) replica can store controller state (including state from network applications) in an external consistent data-store (as in Onix \\cite{koponen2010}, ONOS \\cite{berde2014}, and SMaRtLight \\cite{botelho2014});\n\\item The controller state can be kept consistent using replicated state machine protocols.\n\\end{enumerate}\n\nAlthough both approaches ensure a consistent ordering of events between controller replicas, they are not fault-tolerant in the standard case where only the master controller receives all events.\n\nIf we consider \u2013- for the first approach -- that the master replica can fail between receiving an event and finishing persisting the controller state in the external data-store (which happens after processing the event through controller applications), that event will be lost and the new master (i.e., one of the other controller replicas) will never receive it.\nThe same can happen in the second approach: the master replica can fail right after receiving the event and before replicating it in the shared log (which in this case happens before processing the event through the controller applications).\nIn these cases, since only the crashed master received the event, the other controller replicas will not have an updated view of the network.\nAgain, this may cause severe network problems~\\cite{katta2015}. \nSimilar problems can occur in case of repetition of events.\nThese problems lead to the second design goal:\n\n\\textbf{Exactly-once event processing:} All the events sent by switches are processed, and are neither lost nor processed repeatedly.\n\n\\subsection{Repetition of commands}\n\nIn either traditional state machine replication or consistent storage approaches, if the master controller fails while sending a series of commands, the new elected master may send repeated commands.\nThis may happen when the old master fails before informing the slave replica of its progress.\nSince some commands are not idempotent~\\cite{katta2015}, its duplication can lead to undesirable network behaviour. \nThis problem leads to the third and final design goal:\n\n\\textbf{Exactly-once command execution:} any series of commands are executed only once on the switches.\n\n\\subsection{Ravana}\n\nRavana~\\cite{katta2015} is the first controller to provide correct fault-tolerant SDN control.\nTo achieve this, Ravana processes control messages transactionally and exactly once (at both the controllers and the switches) using a replicated state machine approach, but without involving the switches in an expensive consensus protocol.\n\nThe protocol used by Ravana is show in Figure \\ref{fig:ravana-protocol}.\nSwitches buffer events (as they may need to be retransmitted) and send them to the master controller that will replicate them in a shared log with the slaves.\nThe controller will then reply back to the switch acknowledging the reception of the events.\nThen, events are delivered to applications that may after processing require one or more commands to be sent to switches.\nSwitches reply back to acknowledge the reception of these commands and buffer them to filter possible duplicates.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/ravana-protocol.jpg}\n  \\caption[Ravana Protocol]{Ravana protocol. In Ravana switches maintain two buffers (displayed on the left) to re-transmit events and filter repeated commands in case of master failure. New acknowledge messages (\\texttt{ack\\_event} and \\texttt{ack\\_cmd}) are exchanged between the switch and the master to guarantee the consistency requirements.}\n  \\label{fig:ravana-protocol}\n\\end{figure}\n\nWhile Ravana allows unmodified applications to run in a fault-tolerant environment, it requires modifications to the OpenFlow protocol and to switch hardware.\nNamely, Ravana leverages on buffers implemented on switches to retransmit events and filter possible repeated commands received from the controllers.\nAlso, explicit acknowledgement messages must be added to the OpenFlow protocol so that the switch and the controller acknowledge received messages.\nUnfortunately, these requirements preclude immediate adoption of Ravana.\nFor instance, it is not antecipated OpenFlow to be extended to include the required messages anytime soon.\nThese limitations are the main motivation for our proposal, which we present next.\n\n\\section{Rama design}\n\\label{design}\n\nThe goal of our work is to build a strongly consistent and fault-tolerant control plane for SDN to be used transparently by unmodified applications.\nThis section describes the architecture and protocol for such control plane, which is driven by the following four requirements. First, reliability: the system should maintain a correct and consistent state even in the presence of failures (in both the controllers and switches). Second, transparency: the consistency and fault-tolerance properties should be completely transparent to applications. Third, performance: the performance of the system should not degrade as the number of network elements (events and switches) grows. Fourth, immediate deployability: the solution should work with existing switches and not require new additions to the OpenFlow protocol.\n\n\\subsection{Architecture}\n\nThe high-level architecture of our system, Rama\\footnote{In the Hindu epic Ramayana, Rama is the hero whose wife (Sita) is abducted by Ravana.}, is depicted in Figure \\ref{fig:arquitectura_alto_nivel}.\nIts main components are: (i) OpenFlow enabled switches (switches that are implemented according to the OpenFlow specification), (ii) controllers that manage the switches and (iii) a coordination service.\nIn our model, we consider only one network domain with one primary controller and one or more backup controllers, depending on the number of faults to tolerate.\nEach switch connects to one primary controller and multiple (\\textit{f} to be precise) backup controllers (to tolerate up to \\emph{f} crash controller faults).\nThis primary\/backup model is supported by OpenFlow in the form of master\/slave and allows the system to tolerate controller faults.\nWhen the master controller fails, the remaining controllers will elect a new leader to act as the new master for the switches managed by the crashed master.\nThis election is supported by the coordination service.\n\n\\begin{figure}[t]\n    \\includegraphics[width=0.2\\textwidth]{figures\/architecture.png}\n    \\centering\n    \\caption[High level architecture of the system]{High level architecture of the system.}\n    \\label{fig:arquitectura_alto_nivel}\n\\end{figure}\n\nThe coordination service offers strong consistency and abstracts controllers from complex primitives like fault detection and total order, making them simpler and more robust.\nNote that the coordination system requires a number of replicas equal to \\textit{2f+1}, with \\textit{f} being the number of faults to tolerate.\nThe strong consistency model assures that updates to the coordination service made by the master will only return when they are persistently stored.\nThis means that slaves will always have the fresh modifications available as soon as the master receives confirmation of the update.\nThis results in a consistent network view among all controllers even if some fail.\nThe need for agreement between several replicas make the coordination service the system bottleneck~\\cite{hunt2010}.\nIn addition to the controllers' state, the switches also maintain state that must be handled consistently in the presence of faults.\nFulfilling this request is the main goal of the protocol we present next.\n\n\\subsection{Rama protocol}\n\\label{sec:ft_protocol}\n\nIn an SDN setting, switches generate events (e.g., when they receive packets or when the status of a port changes) that are forwarded to controllers.\nThe controllers run multiple applications that process the received events and may send commands to one or more switches in reply to each event.\nThis cycle repeats itself in multiple switches across the network as needed.\n\nIn order to maintain a correct system in the presence of faults, one must handle the state in the controllers and the state in the switches consistently.\nTo ensure this, the entire cycle presented in Figure \\ref{fig:control_loop} is processed as a \\emph{transaction}: either all or none of the components of this transaction are executed.\nThis means that\n\\begin{inlinelist}\n\\item the events are processed exactly once at the controllers,\n\\item all controllers process events in the same (total) order to reach the same state, and\n\\item the commands are processed exactly once in the switches.\n\\end{inlinelist}\nBecause the standard operation in OpenFlow switches is to simply process commands as they are received, the controllers must coordinate to guarantee the required exactly-once semantics.\nRavana~\\cite{katta2015} does not need this coordination because the (modified) switches can simply buffer the commands received and discard repeated commands (i.e., those with the same identifier) sent by the new controller.\n\n\\begin{figure}[h]\n    \\includegraphics[width=0.45\\textwidth]{figures\/control_loop.png}\n    \\centering\n    \\caption[Control loop]{Control loop of (1) event delivery, (2) event ordering, (3) event processing, and (4) command execution. Events are delivered to the master controller, which decides a total order on the received events. The events are processed by applications in the same order in all controllers. Applications issue commands to be executed in the switches.}\n    \\label{fig:control_loop}\n\\end{figure}\n\nBy default, in OpenFlow a master controller receives all asynchronous messages (e.g., \\texttt{OFPT{\\_}PACKET{\\_}IN}), whe\\-re\\-as the slaves controllers only receive a subset (e.g., port modifications).\nWith this configuration only the master controller would receive the events generated by switches.\nThere are two options to solve this problem.\nOne is for slaves to change this behaviour by sending an \\texttt{OFPT{\\_}SET{\\_}ASYNC} message to each switch that modifies the asynchronous configuration.\nAs a result, switches send all required events to the slaves.\nAlternatively, all controllers can set their role to \\texttt{EQUAL}.\nThe OpenFlow protocol specifies that switches should send all events to every controller with this role.\nThen, controllers need to coordinate between themselves who the master is (i.e., the one that processes and sends commands).\nWe have opted for the second solution and use the coordination service for leader election amongst controllers.\n\nThe fault-free execution of the protocol is represented in Figure \\ref{fig:rama-protocol}.\nIn the figure we consider a switch to be connected with one master controller and a single slave controller.\nThe main idea is that switches must send messages to \\textit{all controllers}, so that they can coordinate themselves even if some fail at any given point.\nIn Ravana, because switches simply buffer events (so that they can be retransmitted to a new master if needed), switches can send events only to the current master, instead of to every controller.\n\n\\begin{figure}[ht!]\n    \\includegraphics[width=0.45\\textwidth]{figures\/rama-protocol.png}\n    \\centering\n    \\caption[Fault-free case of the protocol]{Fault-free case of the protocol. Switches send generated events to all controllers so that no event is lost. The master controller replicates the event in the shared log and then feeds its applications with the events in log order. Commands sent are buffered by the switch until the controller sends a Commit Request. The corresponding Commit Reply message is forwarded to all controllers.}\n    \\label{fig:rama-protocol}\n\\end{figure}\n\nThe master controller then replicates the event in a shared log with the other controllers, imposing a total order on the events received (to simplify, the coordination service is omitted from the figure).\nWhen the event is replicated to the shared log controllers, it is processed by the master controller applications, which will generate zero or more commands.\nTo guarantee exactly-once semantics, the commands are sent to the switches in bundles (a feature introduced in OpenFlow 1.4, see Figure \\ref{fig:openflow-bundles}).\nWith this feature a controller can open a bundle, add multiple commands to it and then instruct the switch to commit all commands present in the bundle in an atomic and ordered fashion.\n\n\\begin{figure}[ht!]\n    \\includegraphics[width=0.45\\textwidth]{figures\/openflow-bundles.png}\n    \\centering\n    \\caption[OpenFlow Bundles]{OpenFlow Bundles.}\n    \\label{fig:openflow-bundles}\n\\end{figure}\n\nRama uses bundles in the following way.\nWhen an event is processed by all modules, the required commands are added by the master controller to a bundle.\nThe master then sends an \\texttt{OFPBCT\\_COMMIT\\_REQUEST} message to each switch affected by the event.\nThe switch processes the request and tries to apply all the commands in the bundle in order.\nAfterwards, it then sends a reply message indicating if the Commit Request was successful or not.\nThis message is used by Rama as an acknowledgement.\n\nAgain, we need to make sure that this reply message is sent to all controllers.\nThis is a challenge, because Bundle Replies are Controller-to-Switch messages and hence are only sent to the controller that made the request (using the same TCP connection).\nTo overcome this challenge we introduce a new mechanism in Rama.\nThe way we inform other controllers if the bundle was committed or not (so that they can decide later if they need to resend specific commands) is by including one \\texttt{OFPT{\\_}PACKET{\\_}OUT} message in the end of the bundle with the action \\texttt{output=controller}.\nThe outcome is that the switch will send the information included in the \\texttt{OFPT{\\_}PACKET{\\_}OUT} message to all connected controllers in a \\texttt{OFPT{\\_}PACKET{\\_}IN} message.\nThis message is set by the master controller to inform slave controllers about the events that were fully processed by the switch (in this bundle).\nThis prevents a new master from sending repeated commands, thus guaranteeing exactly-once semantics.\nRavana does not need to rely on bundles since switches buffer all received commands so that they can discard possible duplicates from a new master.\n\nThe master finishes the transaction by replicating an \\texttt{event} \\texttt{processed} \\texttt{message} in the log, informing backup controllers that they can safely feed the corresponding event in the log to their applications.\nThis is done to simply bring the slaves to the same updated state as the master controller (the resulting commands sent by the applications are naturally discarded).\n\n\\subsubsection{Fault cases}\n\\label{subsec:fault-cases}\n\nWhen the master controller fails, the backup controllers will detect the failure (by timeout) and run a leader election algorithm to elect a new master for the switches.\nUpon election, the new master must send a Role Request message to each switch, to register as the new master.\nThere are three main cases where the master controller can fail:\n\n\\begin{enumerate}\n\\item Before replicating the received event in the distributed log (Figure \\ref{fig:rama-protocol-fault-case-1});\n\\item After replicating the event but before sending the Commit Request (Figure \\ref{fig:rama-protocol-fault-case-2});\n\\item After sending the Commit Request message.\n\\end{enumerate}\n\n\\begin{figure}[h]\n    \\includegraphics[width=0.45\\textwidth]{figures\/rama-protocol-fault-case-1.png}\n    \\centering\n    \\caption[Failure case 1 of the protocol]{Case of the protocol where the master fails before replicating the event received. Because the slaves buffer all events, the event is not lost and the new master can resume the execution of the failed controller.}\n    \\label{fig:rama-protocol-fault-case-1}\n\\end{figure}\n\nIn the first case, the master failed to replicate the received events to the shared log.\nAs slave controllers receive and buffer all events, no events are lost.\nFirst, the new master must finish processing any events logged by the older master.\nNote that events marked as processed have their resulting commands filtered.\nThis makes the new master reach the same internal state as the previous one before choosing the new order of events to append to the log (this is valid for all other fault cases).\nThe new elected master then appends the buffered events in order to the shared log and continues operation (feeding the new events to applications and sending commands to switches).\n\nIn the cases where the event was replicated in the log (cases 2 and 3), the master that crashed may or may not have issued the Commit Request message.\nTherefore, the new master must carefully verify if the switch has processed everything it has received before re-sending the commands and the Commit Request message.\nTo guarantee ordering, OpenFlow provides a Barrier message, to which a switch can only reply after processing everything it has received before.\nIf a new master receives a Barrier Reply message without receiving a Commit Reply message (in form of \\texttt{OFPT{\\_}PACKET{\\_}OUT}), it can safely assume that the switch did not receive nor execute a Commit Request for that event from the old master (case 2)\\footnote{This relies on the FIFO properties of the controller-switch TCP connection.}.\nEven if the old master sent all commands but did not send the Commit Request message, the bundle will never be committed and will eventually be discarded.\nTherefore, the new master can safely resend the commands.\nIn case 3, since the old master sent the Commit Request before crashing, the new master will receive the confirmation that the switch processed the respective commands for that event and will not resend them (guaranteeing exactly-once semantics for commands).\n\n\\begin{figure}[ht]\n    \\includegraphics[width=0.45\\textwidth]{figures\/rama-protocol-fault-case-2.png}\n    \\centering\n    \\caption[Failure case 2 of the protocol]{Case of the protocol where the master fails after replicating the event. The first part of the protocol is identical to the fault-free case and is omitted from the figure. In this case, the crashed master may have already sent some commands or even the Commit Request to the switch.}\n    \\label{fig:rama-protocol-fault-case-2}\n\\end{figure}\n\n\\renewcommand{\\arraystretch}{1.7}\n\\newcolumntype{A}{ >{\\centering\\arraybackslash} m{.28\\linewidth} }\n\\newcolumntype{B}{ >{\\centering\\arraybackslash} m{.32\\linewidth} }\n\\newcolumntype{C}{ >{\\centering\\arraybackslash} m{.32\\linewidth} }\n\\begin{table*}[t!]\n\\begin{tabular}{ABC}\n\\hline\n\\vspace{-4px}\\textbf{Property} & \\vspace{-4px}\\textbf{Ravana} & \\vspace{-4px}\\textbf{Rama}\\\\\n\\hline\n\\textit{At least once events} & Buffering and retransmission of switch events & Switches send events to every controller with role EQUAL\\\\\n\\hline\n\\textit{At most once events} & \\multicolumn{2}{c}{Event IDs and filtering in the log}\\\\\n\\hline\n\\textit{Total event order} & \\multicolumn{2}{c}{Master appends events to a shared log}\\\\\n\\hline\n\\textit{At least once commands} & RPC acknowledgments from switches & \\multirow{2}{0.3\\textwidth}[-0.1cm]{\\centering{Bundle commit is known by every controller by piggybacking PacketOut in OpenFlow Bundle}}\\\\\n\\cline{1-2}\n\\textit{At most once commands} & Command IDs and filtering at switches & \\\\\n\\hline\n\\end{tabular}\n\\caption{How Rama and Ravana achieve the same consistency properties using different mechanisms}\n\\label{tab:rama-properties}\n\\end{table*}\n\n\n\\section{Correctness}\n\nThe Rama protocol we propose in this paper was designed to guarantee correctness of fault-tolerant SDN control.\nWe define correctness as in~\\cite{katta2015}, where the authors introduce the concept of observational indistinguishability in the SDN context, defined as follows:\n\n\\emph{Observational indistinguishability:} If the trace of observations made by users in the fault-tolerant system is a possible trace in the fault-free system, then the fault-tolerant system is observationally indistinguishable from a fault-free system.\n\nFor observational observability, it is necessary to guarantee transactional semantics to the entire control loop, including (i) exactly-once event delivery, (ii) event ordering and processing, and (iii) exactly-once command execution.\nIn this section we summarize how the mechanisms employed by our protocol fulfil each of these necessary requirements.\nFor a brief comparison with Ravana, see Table \\ref{tab:rama-properties}.\n\n\\textbf{Exactly once event processing:} events cannot be lost (processed \\textit{at least once}) due to controller faults nor can they be processed repeatedly (they must be processed \\textit{at most once}).\nContrary to Ravana, Rama does not need switches to buffer events neither that controllers acknowledge each received event to achieve \\textit{at-least once event processing} semantics.\nInstead, Rama relies on switches sending the generated events to \\textit{all (f+1)} controllers (considering that the system tolerates up to \\emph{f} crash faults) so that at least one will known about the event.\nUpon receiving these events, the master replicates them in the shared log while the slaves add the events to a buffer.\nAs such, in case the master fails before replicating the events, the new elected master can append the buffered events to the log.\nIf the master fails after replicating the events, the slaves will filter the events in the buffer to avoid duplicate events in the log.\nThis ensures \\textit{at-most once event processing} since the new master only processes each event in the log once.\nTogether, sending events to all controllers and filtering buffered events ensures \\textit{exactly-once event processing}.\n\n\\textbf{Total event ordering:} to guarantee that all controller replicas reach the same internal state, they must process any sequence of events in the same order.\nFor this, both Rama and Ravana rely on a shared log across the controller replicas (implemented using the external coordination service) which allows the master to dictate the order of events to be followed by all replicas.\nEven if the master fails, the new elected master always preserves the order of events in the log and can only append new events to it.\n\n\\textbf{Exactly once command execution:} for any given event received from one switch, the resulting series of commands sent by the controller are processed by the affected switches exactly \\textit{once}.\nHere, Ravana relies on switches acknowledging and buffering the commands received from controllers (to filter duplicates).\nAs this requires changes to the OpenFlow protocol and to switches, Rama relies on OpenFlow Bundles to guarantee transactional processing of commands.\nAdditionally, the Commit Reply message, which is triggered after the bundle finishes, is sent to \\textit{all} controllers and thus acts as an acknowledgement that is independent of controller faults. If the master fails, the new master needs to know if it should resend the commands for the logged events or not.\nA Packet Out message at the end of the bundle acts as a Commit Reply message to the slave controllers.\nThis way, upon becoming the new master, the controller replica has the required information to know if the switch processed the commands inside the bundle or not, without relying on the crashed master.\nFurthermore, the new master sends a Barrier Request message to the switch.\nReceiving the corresponding Barrier Reply message guarantees that neither the switch nor the link are slow (because a message was received and TCP maintains FIFO order) and thus there is no possibility of the Packet Out being delayed.\nTherefore, the use of Bundles that include a Packet Out at the end, in addition to the Barrier message ensures that commands will be processed by the switches \\textit{exactly-once}.\n\nIt is important to note that we also consider the case where switches fail.\nHowever, this is not a special case of the protocol because it is already treated by the OpenFlow protocol under normal operation.\nA switch failure will generate an event in the controller which will be delivered to applications, for them to act accordingly (e.g., re-route traffic around the failed switch).\nA particularly relevant case is when a switch fails before sending the Commit Reply to the master and the slave controllers.\nImportantly, this event does not result in transaction failure.\nSince this is a normal event in SDN, the controller replicas simply mark pending events for the failed switch as processed and continue operation.\n\nWhile we detail our reasoning as to why our protocol meets the correctness requirements of observational indistinguishability in SDN, modelling the Rama protocol and giving a formal proof is left as future work and out of the scope of this paper.\n\n\\section{Implementation}\n\\label{implementation}\n\nWe have built Rama on top of Floodlight~\\cite{floodlight}.\nFor coordination, we opted for ZooKeeper~\\cite{hunt2010}.\nThis service abstracts controllers from fault detection, leader election, and event transmission and storage (for controller recovery).\nRama introduces two main modules into Floodlight: the \\textit{Event Replication} module (Section \\ref{sec:event-replication}) and the \\textit{Bundle Manager} module (Section \\ref{sec:bundle-manager}).\nAdditionally, the Floodlight architecture was optimised for performance by introducing parallel network event collection and logging (Rama's multi-thread architecture is shown in Figure \\ref{fig:rama-threads}) and by batching events (Section \\ref{sec:event-batching}).\nThe multi-thread paralelism is introduced carefully, not to break TCP FIFO order of event processing, as will be explained next.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/rama-architecture.png}\n  \\caption{Rama thread architecture}\n  \\label{fig:rama-threads}\n\\end{figure}\n\n\nIn the original Floodlight, worker threads are used to collect network events and to process the modules pipeline (in Floodlight network applications are called ``modules'').\nThis design precludes event batching and other optimisations.\nIdeally, we want to free the threads that collect network events as soon as possible so that they can keep collecting more events.\nFor this purpose, the worker threads' only job in Rama is to push events to the Replication Queue.\nEvents for a particular switch are collected always by the same thread (although each thread can be shared by several switches) and thus TCP FIFO order is guaranteed in the Replication Queue.\nNext, the Rama runtime imposes a total order on the events by giving them a monotonically increasing ID.\nAs such, several Replication threads can then take events from this queue and execute the logic in the Event Replication module, which will send the events to ZooKeeper in batches, without breaking the required total order for correctness.\nThis technique is equivalent to Ravana's parallel event logging~\\cite{katta2015}. \nWhen ZooKeeper replies to the request, the events are added to the Pipeline Queue to be processed by the Floodlight modules.\nA single thread is used in this step, to guarantee the total order.\nThe slave replicas also follow the total order from the IDs assigned by the master.\n\nOne of our requirements was to make the control plane transparent for applications to execute unmodified.\nThe Event Replication module is transparent to other modules as it acts before the pipeline.\nThe modules will continue to receive events as usual in Floodlight and process them by changing their internal structures and sending commands to switches.\nThe process of sending messages inside OpenFlow Bundles as required by Rama is also made completely transparent to Floodlight modules, as will be explained in Section \\ref{sec:bundle-manager}.\n\n\\subsection{Event Replication and ZK Manager}\n\\label{sec:event-replication}\n\nThe Event Replication module is the bridge between receiving events from the worker threads and pushing them into the pipeline queue to be processed by Floodlight modules.\nEvents are only added to the pipeline queue after being stored in ZooKeeper.\nTo separate tasks, Event Replication leverages on the ZK Manager, an auxiliary class that acts as ZooKeeper client (establishing connection, making requests and processing replies) and keeps state regarding the events (an event log and an event buffer in case of slaves) and switch leadership.\nEvent Replication and the ZK Manager work together to attain exactly-once event delivery and total order as follows.\n\nWhen an event arrives at the Event Replication module, we check whether the controller is in master or slave mode.\nIn master mode the event is replicated in ZooKeeper and added to its in-memory log.\nThis log is a collection of \\texttt{RamaEvent} objects which, apart from the switch information and message content, contains the unique event identifier explained before.\nThe events are replicated in ZooKeeper in batches (see Section \\ref{sec:event-batching}), so each replication thread simply adds an event to the current batch and becomes free to process a new event.\nEventually the batch will be sent to ZooKeeper containing one or more events to be stored.\nUpon receiving the reply, the events are pushed to the pipeline queue, ordered according to the identifier given by the master to guarantee total order.\n\nIn slave mode, the event is simply buffered in memory (to be used in the case where the master controller fails).\nA special case is when the event received is the Packet Out that the master controller included in the bundle.\nIn this case, the slave marks that this switch already processed all commands for this event.\nSlaves also keep an event log as the master, but only events that come from the master are added to it.\nEvents from the master arrive via \\textit{watches} set in ZooKeeper nodes.\nSlaves set watches and are notified when the master creates event nodes under that node. \nNew events are added to the in memory log (so it is kept up-to-date with the log maintained by the master) and the events are added to the pipeline queue in the same way as in the master controller.\nAn important detail is that event identifiers are set by the master controller, and when slaves deserialize the data obtained from nodes stored in ZooKeeper they get the same \\texttt{RamaEvent} objects created by the master.\nTherefore, the events will be queued in the same order as they were in the master controller replica.\n\n\\subsection{Bundle Manager}\n\\label{sec:bundle-manager}\n\\begin{sloppypar}\nThe Bundle Manager module keeps state related to the open bundles for each switch (as result of an event) and is responsible for adding messages to the bundle, closing and committing it.\nTo guarantee transparency to applications, we modified the write method in \\texttt{OFSwitch.java} (the class that is used by all modules to send commands to switches) to call the Bundle Manager.\nThis module will wrap the message sent by application modules in a \\texttt{OFPT{\\_}BUNDLE{\\_}ADD\\_MESSAGE} and send it to the switch.\nThis process is transparent because applications are unaware of the Bundle Manager module.\n\\end{sloppypar}\n\\begin{sloppypar}\nIn the end of the pipeline, the Bundle Manager module is thus called to prepare and commit the bundles containing the commands instructed by the modules as a response to this event.\nNote that one event may cause modules to send commands to multiple switches, so in this step the Bundle Manager may send \\texttt{OFPBCT\\_COMMIT\\_REQUEST} to one or more switches.\nBefore committing the bundle, the Bundle Manager also adds a \\texttt{OFPT\\_PACKET\\_OUT} message to it, so that slave controllers will know if the commands for an event were committed or not in the switch (as explained in Section \\ref{sec:ft_protocol}). \nThis message will be received by the slave controllers as a \\texttt{OFPT\\_PACKET\\_IN} message with the required information set by the master controller.\n\\end{sloppypar}\n\n\\subsection{Event batching}\n\\label{sec:event-batching}\n\nFloodlight thread architecture was modified to allow event batching, for performance reasons.\nConsidering that ZooKeeper is running on a separated machine from the master controller replica, sending one event at a time to ZooKeeper would significantly degrade performance.\nTherefore, the ZKManager groups events before sending them to ZooKeeper in batches.\nBatches are sent to ZooKeeper using a special request called \\texttt{multi}, which contains a list of operations to execute (e.g., create, delete, set data).\nFor event replication, the multi request will have a list with multiple create operations as parameter.\nThis request is sent after reaching the maximum configured amount of events (e.g., 1000) or some time after receiving the first event in the batch (e.g., 50ms).\nThis means that each event has a maximum delay bound (regarding event batching).\n\n\\section{Evaluation}\n\\label{evaluation}\n\nIn this section we evaluate Rama to understand its viability and the costs associated with the mechanisms used to achieve the desired consistency properties (without modifying the OpenFlow protocol or switches).\n\n\\subsection{Setup}\n\nFor the evaluation we used 3 machines connected to the same switch via 1Gbps links as shown in Figure \\ref{fig:setup}.\nEach machine has an Intel Xeon E5-2407 2.2GHz CPU and 32 GB (4x8GB) of memory.\nMachine 1 runs one or more Rama instances, machine 2 runs ZooKeeper 3.4.8, and machine 3 runs Cbench to evaluate controller performance.\nThis setup tries to emulate a scenario similar to a real one with ZooKeeper on a different machine for fault-tolerance purposes, and Cbench on a different machine to include network latency. \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/setup.png}\n  \\caption{Experiment setup}\n  \\label{fig:setup}\n\\end{figure}\n\n\\begin{figure*}[h!]\n\t\\centering\n    \\begin{subfigure}[t]{0.25\\linewidth}\n        \\centering\n        \\resizebox{\\columnwidth}{!}{\n            \\begin{tikzpicture}\n\t\t\t\t\\begin{axis}[\n\t\t\t\t\t\tybar=10pt,\n\t\t\t\t\t\tbar width=12pt,\n\t\t\t\t\t\tx=1.8cm,\n\t\t\t\t\t\tymin=0,\n\t\t\t\t\t\taxis on top,\n\t\t\t\t\t\tymax=60,\n\t\t\t\t\t\tylabel=Throughput (K Responses\/s),\n\t\t\t\t\t\txtick=data,\n\t\t\t\t\t\tenlarge x limits=0.6,\n\t\t\t\t\t\n                        symbolic x coords={Ravana,Rama},\n\t\t\t\t\t\taxis lines*=left,\n\t\t\t\t\t\tclip=false,\n\t\t\t\t\t\ttranspose legend,\n\t\t\t\t\t\tlegend style={draw=none,at={(0.5,1.3)},anchor=north},\n                        nodes near coords,\n                        cycle list name=black white,\n                        every axis plot\/.append style={fill=gray,no markers}\n\t\t\t\t\t]\n\t\t\t\t\t\\addplot coordinates {(Ravana,46.4) (Rama,28.3)};\n\t\t\t\t\\end{axis}\n\t\t\t\\end{tikzpicture}\n        }\n        \\caption{Fault-tolerant controllers throughput}\n        \\label{fig:controllers-throughput}\n    \\end{subfigure}\n    \\hspace{1cm}\n    \\begin{subfigure}[t]{0.25\\linewidth}\n        \\centering\n        \\resizebox{\\columnwidth}{!}{\n            \\begin{tikzpicture}\n\t\t\t\t\\begin{axis}[\n\t\t\t\t\t\tybar=0pt,\n\t\t\t\t\t\tbar width=12pt,\n\t\t\t\t\t\tx=1.7cm,\n\t\t\t\t\t\tymin=0,\n\t\t\t\t\t\taxis on top,\n\t\t\t\t\t\tymax=60,\n\t\t\t\t\t\tylabel=Throughput (K Responses\/s),\n\t\t\t\t\t\txtick=data,\n\t\t\t\t\t\tenlarge x limits=0.6,\n\t\t\t\t\t\tsymbolic x coords={Ravana,Rama},\n\t\t\t\t\t\taxis lines*=left,\n\t\t\t\t\t\tclip=false,\n\t\t\t\t\t\ttranspose legend,\n\t\t\t\t\t\tlegend style={draw=none,at={(0.5,1.3)},anchor=north, column sep=1.5mm},\n                        legend cell align=left,\n                        nodes near coords=\\pgfmathfloatifflags{\\pgfplotspointmeta}{0}{}{\\pgfmathprintnumber{\\pgfplotspointmeta}},\n                        every node near coord\/.append style={rotate=90, anchor=west},\n\t\t\t\t\t]\n\t\t\t\t\t\\addplot coordinates {(Ravana,52) (Rama,35.6)};\\label{legend-blue}\n\t\t\t\t\t\\addplot coordinates {(Ravana,0) (Rama,50.1)};\\label{legend-red}\n\t\t\t\t\t\\addplot coordinates {(Ravana,46) (Rama,28.3)};\\label{legend-brown}\n\t\t\t\t\t\\legend{Exactly-once events, Exactly-once commands, Both}\n\t\t\t\t\\end{axis}\n\t\t\t\\end{tikzpicture}\n        }\n        \\caption{Throughput with different consistency guarantees}\n        \\label{fig:ravana-vs-rama}\n    \\end{subfigure}\n    \\hspace{1cm}\n    \\begin{subfigure}[t]{0.35\\linewidth}\n        \\centering\n        \\resizebox{\\columnwidth}{!}{\n            \\begin{tikzpicture}\n    \\begin{axis}[\n    xlabel={Number of switches},\n    ylabel={Throughput (Responses\/s)},\n    xmin=1, xmax=7,\n    ymin=0, ymax=60,\n    xtick={1,2,3,4,5,6,7},\n    xticklabels={1,2,4,8,16,32,64},\n    ytick={0,10,20,30,40,50,60},\n    yticklabels={0K,10K,20K,30K,40K,50K,60K},\n   \n    legend style={draw=none,at={(0.5,1.3)},anchor=north},\n    ymajorgrids=true,\n    grid style=dashed,\n    ]\n    \\addplot[\n    color=blue,\n    mark=*,\n    ]\n    coordinates {\n    (1,22.0)(2,23.5)(3,28)(4,31.8)(5,35.6)(6,36.5)(7,36.1)\n    };\n    \\addplot[\n    color=red,\n    mark=*,\n    ]\n    coordinates {\n    (1,32.0)(2,36.8)(3,44.2)(4,45.3)(5,50.7)(6,51.9)(7,51.0)\n    };\n    \\addplot[\n    color=brown,\n    mark=*,\n    ]\n    coordinates {\n    (1,15.3)(2,16.4)(3,22.1)(4,24.3)(5,28.3)(6,28.7)(7,29.1)\n    };\n    \\legend{Exactly-once events, Exactly-once commands, Both}\n    \\end{axis}\n\t\t\t\\end{tikzpicture}\n        }\n        \\caption{Rama throughput with different number of switches}\n        \\label{fig:rama-throughput-switches}\n    \\end{subfigure}\n\\caption{Throughput} \n\\label{fig:rama-throughput}\n\\end{figure*}\n\n\\subsection{Rama performance}\n\nWe have compared the performance of Rama against Ravana~\\cite{katta2015}.\nFigure \\ref{fig:controllers-throughput} shows the throughput for each controller (for Ravana we use the results reported in \\cite{katta2015}, as its authors considered a similar setup).\nFor Rama measurements we run Cbench emulating 16 swit\\-ches.\n\nRama achieves a throughput close to 30K responses per second.\nThis figure is lower than Ravana's, as our solution incurs in higher costs compared to Ravana for the consistency guarantees provided.\nThe additional overhead is caused by two requirements of our protocol.\nFirst, current switches' lack of mechanisms to allow temporary storage of OpenFlow events and commands require Rama to instruct switches to send all events to all replicas, increasing network overhead.\nSecond, the lack of acknowledgement messages in OpenFlow leads Rama to a more expensive solution -- bundles -- to achieve similar purposes.\nThe overhead introduced by these mechanisms is translated into reduced throughput when compared with Ravana.\n\nIn figure \\ref{fig:ravana-vs-rama} we show, separately, throughput results considering the different levels of consistency provided by both Rama and Ravana.\nThe exactly-once events consistency level (\\ref{legend-blue}) ensures that no events are lost and that controllers do not process repeated events.\nAdditionally, controllers must agree on a total order of events to be delivered to applications.\nFor the latter, both Rama and Ravana rely on ZooKeeper to build a shared log across controllers.\nIn our case, the master controller batches events in multiple requests to ZooKeeper, waits for replies, and orders the events before adding them to the Pipeline Queue.\nNote that neither Rama nor Ravana wait for ZooKeeper to persistently store requests on disk (they both use ZooKeeper in-memory).\nIn our case, the multi-request is sent asynchronously (i.e., threads are freed to continue operation) and a callback function is registered. \nThis function will be activated when ZooKeeper replies to our multi request and enqueues the logged events (in order) in the Pipeline Queue to be processed by the modules.\nIn Ravana the processing is equivalent.\n\nThe Exactly-once commands semantics (\\ref{legend-red}) ensures that commands sent by controllers are not lost and that switches do not receive duplicate commands.\nRavana relies on switches to explicitly acknowledge each command and filter repeated ones.\nFor Rama, this includes maintaining state of all opened bundles for switches, and sending additional messages to the switches.\nInstead of replying only with a Packet Out as in Floodlight, Rama must send messages to open the bundle, add the Packet Out to it, close the bundle and commit it.\nTo evaluate this case, we modified Cbench to make switches increase their counters only when they receive a Commit Request message from the controller.\nThis allows a fair evaluation of the performance of Rama in a real system -- indeed, in Rama a packet will only be forwarded after committing the bundle on the switch to guarantee consistent processing.\n\nAs show in Figure \\ref{fig:ravana-vs-rama}, some guarantees are costlier to ensure than others\\footnote{Note that we do not include the results from Exactly-once commands in Ravana as these are not available in~\\cite{katta2015}. It is possible, however, to extrapolate that the results will be inline with the rest of the analysis.}.\nFor instance, the cost of providing Exactly-once events semantics is higher than Exactly-once commands semantics.\nThis result brings with it an important insight: the system bottleneck is the coordination service.\nIn other words, the additional mechanisms Rama uses to guarantee the desired consistency properties add overhead but, crucially, system performance is not limited by these mechanisms.\n\nFigure \\ref{fig:rama-throughput-switches} shows how maintaining multiple switch connections affects Rama throughput.\nAs switches send events at the highest possible rate, the throughput of the system saturates with around 16 switches.\nImportantly, the throughput does not decrease with a higher number of switches.\n\n\n\\subsection{Event batching}\n\nRama batches events to reduce the communication overhead of contacting ZooKeeper.\nIn practice, events are sent to ZooKeeper after reaching a configurable number of events in the batch (batching size) or after a configurable timeout (batching time).\n\nTo evaluate batching we conducted a series of tests with different configurations to understand how the batching size and time affects Rama performance (Figure \\ref{fig:rama-batch-size}).\nIntuitively, a larger batching size will increase throughput, but as downside will also increase latency.\nAs batching size increases, throughput increases due to the reduction of RPC calls required to replicate events.\n\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{.75\\linewidth}{!}{\n  \\begin{tikzpicture}\n  \\begin{axis}[\n  xlabel={Batch size},\n  ylabel={Throughput (Responses\/s)},\n  xmin=1, xmax=7,\n  ymin=10, ymax=30,\n  xtick={1,2,3,4,5,6,7},\n  xticklabels={10,100,200,400,600,800,1000},\n  ytick={15,20,25,30},\n  yticklabels={15K,20K,25K,30K},\n  ymajorgrids=true,\n  grid style=dashed,\n  ]\n  \\addplot[\n  color=blue,\n  mark=*,\n  ]\n \n \n  coordinates {\n  (1,16.0)\n  (2,16.9)\n  (2.5,17.3)\n  (3,17.9)\n  (3.5,18.4)\n  (4,19.1)\n  (4.5,20.5)\n  (5,22.6)\n  (5.5,24)\n  (6,24.9)\n  (6.5,25.5)\n  (7,28.3)\n  };\n  \\end{axis}\n  \\end{tikzpicture}\n  }\n  \\caption{Variation of Rama throughput with batch size}\n  \\label{fig:rama-batch-size}\n  \n\\end{figure}\n\n\\subsection{Failover Time}\n\nTo measure the time for Rama to react to failures we use mininet~\\cite{mininet}, OpenvSwitch~\\cite{pfaff2015}, and iperf.\nWe setup a simple topology in Mininet with one switch and two hosts, one to act as iperf server and another as client.\nWe start the client and server in UDP mode, with the client generating 1 Mbit\/sec for 10 seconds.\nThe switch connects to two Rama instances and sends all events to both controllers.\nEach Rama instance is connected to the ZooKeeper server running on another machine (as before) with a negotiated session timeout of 500ms.\nTo make sure that no rules are installed on the switch -- so that events are sent to the controllers each time a packet arrives -- we run Rama with a module that only forwards packets (using Packet Out messages) without modifying the switch's tables.\n\nFigure \\ref{fig:rama-failover} shows the reported bandwidth from the iperf server and indicates the time taken by Rama to react to failures. \nNamely, the slave replica takes around 550ms to react to faults.\nThis includes the time for: (a) ZooKeeper to detect the failure and notify the slave replicas (500ms); (b) electing a new leader for the swit\\-ches; (c) the new leader to transition to master (finish processing logged events from the old master to reach the same internal state); (d) append buffered events to the log and start delivering unprocessed events in the log to applications so they start sending commands to the switches.\nAs is clear, the major factor affecting failover time is the time ZooKeeper needs to detect the failure of the master controller.\n\n\\begin{figure}[h!]\n  \\centering\n  \\resizebox{.75\\linewidth}{!}{\n    \\begin{tikzpicture}\n    \\begin{axis}[\n\t\txlabel={Time (s)},\n\t\tylabel={Bandwidth (Mbits\/sec)},\n\t\txmin=1, xmax=10,\n\t\tymin=0, ymax=1.5,\n\t\txtick={1,2,3,4,5,6,7,8,9,10},\n\t\n\t\n\t\tytick={0,1},\n\t\n\t\n\t\n\t\tlegend style={legend pos=south east},\n\t\tymajorgrids=true,\n\t\tgrid style=dashed,\n\t\t]\n\t\t\\addplot[\n\t\n\t\tcolor=blue,\n\t\tmark=none,\n\t\tline width=0.5mm\n\t\t]\n\t\tcoordinates {\n        (1,1)\n        (3,1)\n        (3.4,0.6)\n        (3.5,0)\n        (4,0)\n        (4,1)\n        (10,1)\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t};\n        \\draw[gray, dashed, thick] (3.4,0) -- (3.4,1.5);\n        \\draw[gray, dashed, thick] (4.0,0) -- (4.0,1.5);\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\legend{}\n    \\end{axis}\n  \\end{tikzpicture}\n  }\n\\caption{Rama failover time}\n\\label{fig:rama-failover}\n\\end{figure}\n\n\n\n\\subsection{Summary}\n\nRama comes close, but does not achieve the performance of Ravana.\nThis is due to the fact that our system incurs in higher costs. \nRama requires more messages to be sent over the network and introduces new mechanisms, such as bundles, which increase the overhead of the solution in order to achieve the same properties as Ravana.\nDespite the (relatively small) loss in performance, the value proposition of Rama of guaranteeing consistent command and event processing without requiring modifications to switches or to the OpenFlow protocol still makes it an effective enabler for immediate adoption of fault-tolerant SDN solutions.\n\n\\section{Related work}\n\\label{related}\n\n\\textbf{Consistent SDN.} Levin et al. \\cite{levin2012} have explored the trade-offs of state distribution in a distributed control plane, motivating the importance of strong consistency in applications' performance.\nOn the one hand, view staleness affects the correct operation of applications, which may lead to poor network performance.\nOn the other, applications need to be more complex in order to be aware of possible network inconsistencies.\n\nHaving a strongly consistent network view across the controllers may be critical to the operation of some applications (e.g., load balancing) in terms of correctness and performance \\cite{levin2012}.\nHowever, as noted in the CAP theorem, a system can not provide availability while also achieving strong consistency in the presence of network partitions.\nBecause of this, fault-tolerant and distributed SDN architectures must use techniques to explicitly handle partitions in order to optimize consistency and availability (and thus achieving a tradeoff between them)~\\cite{brewer2012}.\n\nPart of the strong consistency in the controllers comes from a consistent packet processing (i.e., packets received from switches). OF.CPP \\cite{perevsini2013} explores the consistency and performance problems associated with packet processing at the controller and proposes the use of transactional semantics to solve them.\nThese semantics are achieved by using multi-commit transactions, where each event is a sub transaction, which can commit or abort, of the related packet (the main transaction).\nHowever, this transactional semantics in packet processing is not enough: controllers should also coordinate to guarantee the same semantics in the switches' state. \nSpecifically, the commands sent by the controllers should be processed exactly once by the corresponding switches -- a problem our work addresses.\n\n\\textbf{Consistent network updates.} The concepts of per-packet and per-flow consistency in SDN were introduced in \\cite{reitblatt2011} to provide a useful abstraction for applications: consistent network updates.\nWith consistent updates, packets or flows in flight are processed exclusively by the old or by the new network policy (never a mix of both).\nFor example, with per-packet consistency,  every packet traversing the network is processed by exactly one consistent global network configuration.\nThe authors extend this work in \\cite{reitblatt2012} and implement Kinetic, which runs on top of NOX \\cite{gude2008} to offer these abstractions in a control plane to be used by applications.\nThe main mechanism used to guarantee consistent network updates is the use of a two-phase protocol to update the rules on the switches.\nFirst, the new configuration is installed in an unobservable way (no packets go through these rules yet).\nAfterwards, the switch's ingress ports are updated one-by-one to stamp packets with a new version number (using VLAN tags).\nOnly packets with the new version number are processed by the new rules.\n\nIn \\cite{canini2013}, Canini et al. extend Kinetic to a distributed control plane and formalize the notion of fault-tolerant policy composition.\nTheir algorithm also requires a bounded number of tags, regardless of the number of installed updates, as opposed to the unbounded number of tags in \\cite{reitblatt2012}.\n\nThis class of proposals addresses consistent network updates, which is an orthogonal problem to the one addressed here.\n\n\\textbf{Fault-tolerance in SDN.} Botelho et al.~\\cite{botelho2014} and Katta et al.~\\cite{katta2015} both address fault tolerance in the control plane while achieving strong consistency.\nIn~\\cite{botelho2014} the authors proposes SMaRtLight, a fault-tolerant controller architecture for SDN. \nTheir architecture uses a hybrid replication approach: passive replication in the controllers (one primary and multiple backups) and active replication in an external distributed data store, to achieve durability and strong consistency.\nThe controllers are coordinated through the data store and ca\\-ching mechanisms are employed to achieve acceptable performance.\nIn~\\cite{botelho2016} the authors extend their solution to a distributed deployment.\nIn contrast to our solution, SMaRtLight requires applications to be modified to use the data store directly.\nMore importantly, the solution does not consider the consistency if switch state in the system model.\nRavana~\\cite{katta2015} was the first fault-tolerant controller that integrates switches into the problem.\nThe techniques proposed by its authors guarantee correctness of event processing and command execution in SDN.\nThe main differentiating factor of our work to Ravana is that our solution does not require changes to the OpenFlow protocol nor to switches.\n\n\\textbf{Distributed SDN controllers}. The need for scalability and dependability has been a motivating factor for distribution and fault-tolerance in SDN control planes.\nOnix~\\cite{koponen2010}, the first distributed, dependable, production-level solution considered these problems from the outset.\nAs the choice of the ``right'' consistency model was perceived as fundamental by its authors, Onix offered two data stores to maintain the network state: an eventually consistent and a strong consistent option.\nONOS~\\cite{berde2014} is an open-source solution that shares with Onix the fact that controller state is stored in an external consistent data-store.\nBoth approaches ensure a consistent ordering of events between controller replicas, but they do not include switch state and hence can lead to the network anomalies of traditional replication solutions. \n\n\\textbf{Traditional fault-tolerance techniques.} Viewstamped Replication~\\cite{oki1988}, Paxos~\\cite{lamport1998}, and Raft~\\cite{ongaro2014} are well-known distributed consensus protocols used for replication of state machines in client-server models.\nNone of these widely-used protocols is directly applicable in the context of SDN, where to guarantee correctness it is necessary not only to have consistent controller state, but also switch state.\n\n\\section{Conclusions}\n\nIn a fault-tolerant SDN, maintaining consistent controller state is not enough to achieve correctness.\nUnlike traditional distributed systems, in SDN it is necessary to consistently handle switch state to avoid loss or repetition of commands and events under controller failures.\nTo address these challenges we propose Rama, a consistent and fault-tolerant SDN controller that handles the entire event processing cycle transactionally.\n\nRama differs from the existing alternative, Ravana ~\\cite{katta2015}, by not requiring modifications to the OpenFlow protocol nor to switches.\nThis comes at a cost, as the techniques introduced in Rama incur in a higher overhead when compared to Ravana.\nAs the overhead leads to a relatively modest decrease in performance, we expect, in practice, this to be compensated by the fact that our solution is immediately deployable.\nWe make our software available open source\\footnote{https:\/\/github.com\/fvramos\/rama} to further foster adoption of fault-tolerant SDN.\n\nAs for future work, besides devising a formal proof on the consistency guarantees Rama provides, we plan to address correctness in distributed SDN deployments and to consider richer fault models. \n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\\setcounter{equation}{0}\nGeneral Relativity is presently the most successful theory for describing the gravitational interaction\nat the classical level.\nIts own failure is marked by the prediction of the formation of geodesic singularities whenever a trapped surface arises from the gravitational collapse of a compact object.\\footnote{We\nalso recall that pointlike sources are mathematically incompatible with the Einstein field equations~\\cite{geroch}.}\nSuch considerations open up the possibility that significant departures from General Relativity\nmight occur where our experimental data do not yet place strong enough constraints,\nlike for example in regions of strong gravity near a very massive source.\nHowever, Einstein's field equations are not linear and this makes it difficult to modify the laws of gravity in the\nstrong-field regime without affecting also the weak-field behavior, since these regimes are likely to be related\nnontrivially in any nonlinear theories.\n\\par\nThe bootstrapped Newtonian gravity~\\cite{BootN,Casadio:2020mch} is an attempt at\ninvestigating these issues in a somewhat simplified context.\nThe approach, based on Deser's conjecture~\\cite{deser}, consists of retrieving the full Einstein's theory including gravitational self-coupling terms in the Fierz-Pauli action.\\footnote{This\nidea is indeed older, see {\\em e.g.}~Ref.~\\cite{Feynman}.} \nThese additional terms must be consistent with diffeomorphism invariance, in order to preserve the covariance of any (modified) metric theory.\nWe can obtain different modified gravitational theories depending on the choice of boundary conditions in the reconstruction procedure~\\cite{rubio}.\nA key observation is that a practically effective dynamics can be derived only starting with a ``small'' contribution of matter sources.\nFor large astrophysical sources, this implies that the matter source must also be included in\na nonperturbative way.\nIn the present approach this task is addressed starting from the Fierz-Pauli action corresponding to the potential generated by an arbitrarly large static source, and putting in extra terms representing gravitational self-coupling.\nFurthermore, the coupling constants for the additional terms are not fixed to their Einstein-Hilbert values in order to accommodate for diverse underlying dynamics.\nThis approach then results in a nonlinear equation including pressure effects and the gravitational self-interaction terms to next-to-leading order in the Newton constant, whose solution is the gravitational potential operating on test particles at rest.\nSuch equation was useful to investigate compact objects~\\cite{Casadio:2019cux,Casadio:2020kbc,Casadio:2019pli} and coherent quantum states~\\cite{Casadio:2016zpl,Casadio:2017cdv}.\\footnote{These quantum states show some of the\nproperties~\\cite{ciarfella} found in the corpuscular model of black holes~\\cite{DvaliGomez}.\nHowever, we shall not discuss quantum aspects in this work.}\n\\par\n The motion of (test) particles and photons in the surrondings of a compact object represents the most immediate tool to gather information on the gravitational potential in which they revolve.\nIn Ref.~\\cite{Casadio:2021gdf}, a full (effective) metric tensor was obtained from the bootstrapped\nNewtonian potential, which allows one to study these trajectories in general, and to compare them with\nresults from General Relativity.\nThe requirement that the resulting theory of gravity is covariant is satisfied by the use of an effective metric tensor, since the bootstrapped Newtonian dynamics is implicitly assumed to be invariant after coordinate transformations.\nNonetheless, the particular metric found in Ref.~\\cite{Casadio:2021gdf} differs from the Schwarzschild \ngeometry; hence, it is not a solution of the Einstein equations in the vacuum.\nAn effective fluid is therefore present, as was already noted in the cosmological context~\\cite{cosmo}.\n\\par\nThe bootstrapped effective metric is given as a function of parameterized\npost-Newtonian (PPN) parameters~\\cite{weinberg} in the weak-field expansion.\nThese parameters can be consistently chosen so as to minimize deviations from the Schwarzschild metric\nonly up to a point.\nIn fact, some of the PPN parameters are uniquely related, and at the PPN order determined in\nRef.~\\cite{Casadio:2021gdf}, they can be expressed in terms of one free parameter. \nIn this work, we report on a phenomenological investigation aiming at placing bounds on \nthis remaining free parameter from the measured precessions in the Solar System ~\\cite{DeMartino2018,Moyer200,Will2018}and from the study\nof S-star orbits around the black hole in the center of the Galaxy ~\\cite{Eckart1996,Eckart1997,Gillessen:2009,Gillessen2009L,Ghez1998,Ghez:2008}.\n\\par\nThe paper is organized as follows.\nIn Sec.~\\ref{s:boot}, we briefly review the equation for the bootstrapped Newtonian \npotential and its solution in the vacuum.\nWe then just recall the full effective metric reconstructed from this potential, which is \nthen used to analyze Solar System data and S-star motions in Sec.~\\ref{s:app}. We conclude with comments and an outlook in Sec.~\\ref{s:conc}.\n\\section{Bootstrapped Newtonian vacuum}\n\\label{s:boot}\n\\setcounter{equation}{0}\nWe shall only review briefly the derivation of the bootstrapped Newtonian equation, since all the\ndetails can be found in Refs.~\\cite{Casadio:2017cdv,BootN,Casadio:2019cux,Casadio:2019pli}. We shall use units with\nthe speed of light $c=1$ in this section.\nWe start from the Lagrangian for the Newtonian potential $V=V(r)$ generated by a static\nand spherically symmetric source of density $\\rho=\\rho(r)$, to wi\n\\begin{equation}\nL_{\\rm N}[V]\n=\n-4\\,\\pi\n\\int_0^\\infty\nr^2\\,\\d r\n\\left[\n\\frac{\\left(V'\\right)^2}{8\\,\\pi\\,G_N}\n+V \\rho\n\\right]\n\\ .\n\\label{LVn}\n\\end{equation}\nThe corresponding Euler-Lagrange field equation is given by Poisson's\n\\begin{equation}\n\\dfrac{1}{r^2}\\,\\dfrac{d}{d r}\n\\left(r^2\\,\\frac{dV}{dr}\\right)=\n4\\,\\pi\\,G_N\\,\\rho\n\\ ,\n\\label{poisson}\n\\end{equation}\nwhere we recall that the radial coordinate $r$\nis the one obtained from harmonic coordinates~\\cite{weinberg,Casadio:2021gdf}.\nWe next couple $V$ to a gravitational current proportional to its own energy density,\n\\begin{equation}\nJ_V\n\\simeq\n4\\,\\frac{d U_{ N}}{d \\mathcal{V}} \n=\n-\\dfrac{\\left[V'(r)\\right]^2}{2\\,\\pi\\,G_N}\n\\ ,\n\\end{equation}\nwhere $\\mathcal{V}$ is the spatial volume and $U_{\\rm N}$ is the Newtonian potential energy.\nWe also add the ``one loop term'' $J_{\\rho}\\simeq-2\\,V^2$, which couples to $\\rho$, and\nthe pressure term $p$~\\cite{Casadio:2019cux}.\nThe total Lagrangian then reads \n\\begin{align}\nL[V]\n=&\n-4\\,\\pi\n\\int_0^\\infty\nr^2\\,\\d r\n\\left[\n\\frac{\\left(V'\\right)^2}{8\\,\\pi\\,G_N}\n\\left(1-4\\,q_V\\,V\\right)\n\\right.\\nonumber\\\\&\\left.+\\left(\\rho+3\\,q_p\\,p\\right)V\n\\left(1-2\\,q_\\rho\\,V\\right)\\right]\\,\n\\label{LagrV}\n\\end{align}\nwhere the coupling constants $q_V$, $q_p$ and $q_\\rho$ can be used to track the effects of the different\ncontributions.\nFor instance, the case $q_V=q_p=q_\\rho=1$ reproduces the Einstein-Hilbert action at next-to-leading order\nin perturbations around Minkowski~\\cite{Casadio:2017cdv,Casadio:2019cux,Casadio:2019pli}.\nFinally, the bootstrapped Newtonian field equation reads\n\\begin{eqnarray}\n\\dfrac{1}{r^2}\\,\\dfrac{d}{d r}\n\\left(r^2\\,\\dfrac{d V}{d r}\\right)\n&&=\n4\\,\\pi\\,G_N\n\\dfrac{1-4\\,q_\\rho\\,V}{1-4\\,q_V\\,V}\n\\left(\\rho+3\\,q_p\\,p\\right)\n\\nonumber\\\\&&+\\dfrac{2\\,q_V\\left(V'\\right)^2}\n{1-4\\,q_V\\,V}\n\\ ,\n\\label{EOMV}\n\\end{eqnarray}\nwhich must be solved along with the conservation equation $p' = -V'\\left(\\rho+p\\right)$. \n\\subsection{Vacuum potential}\nIn vacuum, we have $\\rho=p=0$, and Eq.~\\eqref{EOMV} simplifies to\n\\begin{equation}\n\\frac{1}{r^2}\\,\\frac{d}{d r}\n\\left(r^2\\,\\frac{d V}{d r}\\right)\n=\n\\frac{2\\,q \\left(V'\\right)^2}{1-4\\,q\\,V}\n\\ ,\n\\label{EOMV0}\n\\end{equation}\nwhere we renamed $q\\equiv q_V$ for simplicity.\nThe exact solution was found in Ref.~\\cite{BootN} and reads\n\\begin{eqnarray}\n\\label{potential}\nV(r)\n=\n\\frac{1}{4\\,q}\n\\left[1-\\left(1+\\frac{6\\,q\\,G_N\\,M}{r}\\right)^{2\/3}\\right]\n\\ .\n\\end{eqnarray}\nThe asymptotic expansion away from the source yields\n\\begin{equation}\nV_{2}\n\\simeq\n-\\frac{G_N\\,M}{r}\n+q\\,\\frac{G_N^2\\,M^2}{r^2}\n-q^2\\,\\frac{8\\,G_N^3\\,M^3}{3\\,r^3}\n\\ ,\n\\label{Vasy}\n\\end{equation}\nso that the Newtonian behavior is always recovered (for $q=0$) and the post-Newtonian terms\nare seen to depend on the coupling $q$ (see Fig.~\\ref{f:V}).\n\\\n\\\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[scale=0.9]{FIG1.pdf}\n    \\caption{Bootstrapped Newtonian potential $V$ in Eq.~\\eqref{potential} compared to\n    its expansion $V_2$ from Eq.~\\eqref{Vasy} and to the Newtonian potential $V_{\\rm N}$\n    (for $q=1$).}\n    \\label{f:V}\n\\end{figure}\n\n\\subsection{Vacuum effective metric}\n\\label{ss:metric}\nA complete spacetime metric was reconstructed from the vacuum potential~\\eqref{potential}\nin Ref.~\\cite{Casadio:2021gdf}.\nThe procedure is rather cumbersome, and we shall only recall here a few main steps leading\nto the necessary expressions in the weak-field regime. We explicitly show the speed of light\n$c$ from here on.\nOne starts from the PPN form~\\cite{weinberg\n\\begin{widetext}\n\\begin{equation}\nds^2\\simeq\\left[1-\\alpha\\dfrac{2R_g}{\\bar{r}}+(\\beta-\\alpha\\gamma)\\dfrac{2R_g^2}{\\bar{r}^2}+(\\zeta-1)\\dfrac{8R_g^3}{\\bar{r}^3}\\right]c^2dt^2 +\\left[1+\\gamma\\dfrac{2R_g}{\\bar{r}}+\\xi\\dfrac{4R_g^2}{\\bar{r}^2}+\\sigma\\dfrac{8R_g^3}{\\bar{r}^3}\\right]d\\bar{r}^2+\\bar{r}^2d\\Omega^2\n\\end{equation}\n\\end{widetext}\nwhere $R_g=\\frac{G_N\\,M}{c^2}$ and $\\bar{r}$ is the areal radius, which differs from the radial coordinate $r$ in which\nthe potential~\\eqref{potential}\nis expressed.\nThe latter is obtained from harmonic coordinates and the two radial coordinates are\nrelated by~\\cite{Casadio:2021gdf}\n\\begin{eqnarray}\nr\n\\simeq&&\n\\bar{r}\n+\n(1-3\\,\\gamma)\\,\\frac{R_g}{2}\n+\\nonumber\\\\\n&&+(1-3\\,\\gamma+2\\,\\gamma^2-2\\,\\Xi)\\,\n\\frac{R_g^2}{\\bar{r}}\n\\ ,\n\\end{eqnarray}\nin which $\\Xi$ is a free parameter.\nFurthermore, we have\n\\begin{equation}\nq\n=\n\\beta+\\frac{\\gamma-1}{2}\n\\ .\n\\label{eq:q}\n\\end{equation}\n\\par\nWe can next set $\\alpha=1$ by simply absorbing this coefficient in the definition of the mass $M$~\\cite{adm},\nand  $\\beta=\\gamma=1$ in order to satisfy the experimental constraints $|\\gamma-1|\\simeq|\\beta-1|\\ll 1$.\nFrom Eq.~\\eqref{eq:q}, this is tantamount to setting $q=1$, as expected.\nThe higher order PPN parameters are then fully determined by $\\Xi$ according to \n\\begin{eqnarray}\n\\xi\n&\\!\\!=\\!\\!&\n1+\\Xi\n\\label{eq:xi}\n\\\\\n\\zeta\n&\\!\\!=\\!\\!&\n1-\\frac{5+6\\,\\Xi}{12}\n=\n\\frac{13-6\\,\\xi}\n{12}\n\\label{eq:zeta}\n\\\\\n\\sigma\n&\\!\\!=\\!\\!&\n\\frac{9+14\\, \\Xi}\n{4}\n\\ .\n\\label{eq:sigma}\n\\end{eqnarray}\nAs already noted in Ref.~\\cite{Casadio:2021gdf}, the General Relativistic PPN combination $\\xi=\\zeta=1$\ncannot be obtained for any value of $\\Xi$, and the bootstrapped metric for which we have the minimum deviation\nfrom the Schwarzschild form is thus given by \n\n\\begin{eqnarray}\nds^2\n\\simeq&&\n-\\left[\n1\n-\\frac{2\\,R_g}{r}\n-(5+6\\,\\Xi)\\,\\frac{2\\,R_g^3}{3\\,c^6\\,r^3}\n\\right]\nc^2\\,dt^2\n\\nonumber\n\\\\\n&&\n+\n\\left[\n1\n+\n\\frac{2\\,R_g}{r}\n+\n(1+\\Xi)\\,\\frac{4\\,R_g^2}{r^2}\n\\right.\\nonumber\\\\&&\\left.\n+\n(9+14\\, \\Xi)\\frac{2\\,R_g^3}{r^3}\n\\right]\ndr^2\n+\nr^2\\,d\\Omega^2\n\\ ,\n\\label{eq:g}\n\\end{eqnarray}\nin which we drop the bar from the areal coordinate for simplicity from now on.\nWe can see that there are contributions in the metric coefficients which cannot be reduced to the\nSchwarzschild expressions.\nThis deviation from the Schwarzschild solution is encoded by the free parameter $\\Xi$,\nwhose value is \\textit{a priori} unknown and must be constrained by observations.\nIn particular, we will test these corrections by analyzing the \nplanets in the Solar System and S-stars motion around Sgr~A*.\nThe geodesic equation\n\\begin{equation}\n\\Ddot{x}^{\\mu}+\\Gamma^{\\mu}_{\\alpha\\beta}\\,\\dot{x}^{\\alpha}\\,\\dot{x}^{\\beta}\n=\n0,\n\\end{equation}\nwhere a dot indicates the derivative with respect to the proper time, can be equivalently computed using the Euler-Lagrange equations\n\\begin{equation}\n\\dfrac{d}{d s}\n\\dfrac{\\partial\\mathcal{L}}{\\partial\\dot{x}^{\\mu}}\n-\\dfrac{\\partial\\mathcal{L}}{\\partial x^{\\mu}}\n=\n0\n\\ , \n\\end{equation}\nwith $\\mathcal{L}=g_{\\alpha\\beta}\\,\\dot x^\\alpha\\,\\dot x^\\beta=-1$ for a massive object.\nFrom the metric in Eq.~\\eqref{eq:g}, one then finds  \n\\par\n\\begin{widetext}\n\\begin{align}\n\\Ddot{r}\n&\n=\n\\frac{R_g\n\\left\\{\n4\\, (1+\\Xi)\\, R_g\\,r\\,\\dot{r}^2\n+R_g^2\n\\left[3\n\\left(\n9+14\\Xi\n\\right)\n\\dot{r}^2\n-c^2\n\\left(5+6\\Xi\\right)\n\\dot{t}^2\n\\right]\n+r^2\\left(\\dot{r}^2-c^2\\,\\dot{t}^2\\right)\n\\right\\}\n+r^5\\,(\\dot{\\theta}^2+\\dot{\\phi}^2\\,\\sin^2\\theta)}\n{r\n\\left[2\\, (9+14\\, \\Xi)\\, R_g^3+4\\,(1+\\Xi)\\, R_g^2\\, r+2\\,R_g\\,r^2+\\,r^3\n\\right]}\n\\label{eq:ddr}\n\\\\\n\\Ddot{\\theta}\n&=\n\\dot{\\phi}^2\\,\\sin\\theta\\,\\cos\\theta\n-\\frac{2\\,\\dot{r}\\,\\dot{\\theta}}{r}\n\\\\\n\\Ddot{\\phi}\n&=\n-\\frac{2\\,\\dot{\\phi}}{r}\\,(\\dot{r}+r\\,\\dot{\\theta}\\,\\cot\\theta)\n\\\\\n\\Ddot{t}\n&\n=\n\\frac{6\\,\\dot{r}\\,\\dot{t}\n\\left[\n(5+6\\,\\Xi)\\, R_g^3\n+R_g\\,r^2\n\\right]}\n{2\\,(5+6\\,\\Xi)\\, R_g^3\\,r\n+6\\,R_g\\,r^3\n-3\\,r^4}\n\\ .\n\\label{eq:ddt}\n\\end{align}\n\\end{widetext}\n The third and fourth equations are the usual conservation equations for the angular momentum\nand energy conjugated to $t$, respectively.\nSpherical symmetry as usual implies that the orbital motion occurs on a plane which we can\narbitrarily set at $\\theta=0=\\dot\\theta$.\n\n\nThe above parametric system of nonlinear differential equations can be integrated numerically\nin order to study the orbits.\n\n\\subsubsection{Precession}\n\\label{ss:precession}\n\\par\nIt is easy to express the perihelion precession in terms of the PPN parameters~\\cite{weinberg}.\nAt leading order, one has\n\\begin{equation}\n\\label{eq:eq1}\n\\Delta\\phi^{(1)}\n=\n2\\,\\pi\n\\left(2-\\beta+2\\,\\gamma\\right)\n\\dfrac{R_g}{\\ell}\n\\ ,\n\\end{equation}\nwhere $\\ell=a\\,(1-e^2)$ is the \\textit{semilatus rectum}, $a$ is the semimajor axis\nand $e$ is the eccentricity.\nFor $\\beta=\\gamma=1$, Eq.~(\\ref{eq:eq1}) reproduces the General Relativistic result\n\\begin{equation}\n\\label{eq:eq2}\n\\Delta\\phi_S^{(1)}\n=\n6\\,\\pi\\,\\frac{R_g}{\\ell}\n\\ .\n\\end{equation}\nThe second order correction depends on $\\xi$ and $\\zeta$, and for $\\beta=\\gamma=1$,\nit reads~\\cite{Casadio:2021gdf}\n\\begin{align}\n\\Delta\\phi^{(2)}\n&\n=\n\\pi\\left[(41+10\\xi-24\\,\\zeta)\n+\n(16\\,\\xi-13)\\,\\frac{e^2}{2}\\right]\n\\frac{R_g^2}{\\ell^2}\n\\nonumber\n\\\\\n&\n\\simeq\n\\pi\\left[(37+22\\,\\Xi)+(3+16\\,\\Xi)\\,\\frac{e^2}{2}\\right]\n\\frac{R_g^2}{\\ell^2}\n\\nonumber\n\\\\\n&\n\\simeq\n\\Delta\\phi_S^{(2)}\n+\n2\\,\\pi\\left[\n11\\,\\xi-7+4\\,(\\xi-1)\\,e^2\\right]\n\\frac{R_g^2}{\\ell^2}\n\\ ,\n\\end{align}\nwhere the General Relativistic result $\\Delta\\phi_S^{(2)}$ corresponds to $\\xi=\\zeta=1$.\nFrom Eqs.~\\eqref{eq:xi} and \\eqref{eq:zeta}, it follows that we cannot have $\\xi=\\zeta=1$ for any value of $\\Xi$,\nand a deviation from General Relativity remains. \n\\section{Astronomical tests}\n\\label{s:app}\n\\setcounter{equation}{0}\nIn order to constrain the free parameter of the bootstrapped Newtonian potential, $\\Xi$,\nwe confronted the theoretical results exposed in Sec.~\\ref{ss:metric} with astronomical \ndata.\n\nTo infer a range of validity for $\\Xi$, we compared the analytical expression of the precession with the observed values of the perihelion advance of Solar System's planets (Sec.~\\ref{ss:precession2}).\n\nThen, we turned our attention to the Galactic Center, and we studied the motion of S-stars\norbiting around Sgr A*.\nTo constrain $\\Xi$, we let it vary in a given range and fit the corresponding simulated\norbits to astrometric observations.\nIn particular, we adopted a fully relativistic approach which consists of integrating numerically\nEqs.~\\eqref{eq:ddr}-\\eqref{eq:ddt} in order to get the mock orbits, instead of solving Newton's\nlaw with the standard potential replaced by the modified one. \n\\subsection{Perihelion precession in the Solar System}\n\\label{ss:precession2}\nIn order to constrain $\\Xi$ we can start from the Solar System planets whose orbital precession\nhas been measured, namely Mercury, Venus, Earth, Mars, Jupiter and Saturn~\\cite{2015MNRAS.451.3034N}.\nThe confidence region for $\\Xi$ can be identified as the set of values such that the precession\n\\begin{equation}\n\\Delta\\phi=\\Delta\\phi^{(1)}+\\Delta\\phi^{(2)}\n\\label{eq:eq3}\n\\end{equation}\nis compatible with the observations.\nThe planetary parameters\\footnote{The reported values are taken from NASA fact sheet at\nhttps:\/\/nssdc.gsfc.nasa.gov\/planetary\/factsheet\/.},\nthe corresponding observed values of the precession~\\cite{2015MNRAS.451.3034N} and\nthe General Relativistic value obtained by Eq.~(\\ref{eq:eq2}) are reported in Table~\\ref{tab:tab1}\nfrom first to seventh columns.\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{cccccccc}\n    \\hline\\hline\n    Planet & $a(10^6km)$ & $P(years)$ & $i(^{\\circ})$ & $e$ & $\\Delta\\phi_{obs}(''\/cy)$ & $\\Delta\\phi_S(''\/cy)$ & $[\\Xi_{min};\\Xi_{max}]$ \\\\\n    \\hline\n    $\\textbf{Mercury}$&$57.909$&$0.24$&$7.005$&$0.2056$&$43.1000\\pm0.5000$&$42.9822$&$[-89708.7;144995]$\\\\\n    $\\textbf{Venus}$&$108.209$&$0.61$&$3.395$&$0.0067$&$8.6247\\pm0.0005$&$8.6247$&$[-1149.67;1167.47]$\\\\\n    $\\textbf{Earth}$&$149.596$&$1.00$&$0.000$&$0.0167$&$3.8387\\pm0.0004$&$3.83881$&$[-3660.86;2094.96]$\\\\\n    $\\textbf{Mars}$&$227.923$&$1.88$&$1.851$&$0.0935$&$1.3565\\pm0.0004$&$1.35106$&$[155248.;179879.]$\\\\\n    $\\textbf{Jupiter}$&$778.570$&$11.86$&$1.305$&$0.0489$&$0.6000\\pm0.3000$&$0.0623142$&$[5.46709\\times10^8;1.92679\\times10^9]$\\\\\n    $\\textbf{Saturn}$&$1433.529$&$29.45$&$2.485$&$0.0565$&$0.0105\\pm0.0050$&$0.0136394$&$[-1.57315\\times10^8;3.59618\\times10^7]$\\\\\n    \\hline\\hline\n    \\end{tabular}\n\\caption{Values of semimajor axis ($a$), orbital period ($P$), tilt angle ($i$), eccentricity ($e$), observed orbital precession\n($\\Delta\\phi_{obs}$), orbital precession as predicted by General Relativity ($\\Delta\\phi_S$) and constraints on $\\Xi$ for Solar System's planets.}\n    \\label{tab:tab1}\n\\end{table*}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.6]{Fig2.pdf}\n    \\caption{Bootstrapped orbital precession as a function of the parameter $\\Xi$.\n    Black lines give the theoretical prediction from Eq.~\\eqref{eq:eq3}, blue dashed lines represent\n    the measurements adapted from Ref.~\\cite{2015MNRAS.451.3034N} and red lines depict the General Relativistic\n    values as in Eq.~\\eqref{eq:eq2}.\n    Confidence regions for $\\Xi$ are shaded in gray.}\n    \\label{fig:fig1}\n\\end{figure*}\nThe allowed region of $\\Xi$ for each planet is defined as the range of values compatible with data, having as extremes the\nvalues of $\\Xi$ solving the equation\n\\begin{equation}\n\\Delta\\phi=\\Delta\\phi_{obs}\n\\ .\n\\end{equation}\nThe inferred lower and upper limits on $\\Xi$ are reported in the last column of Table~\\ref{tab:tab1}, and the included area\nis depicted in Fig.~\\ref{fig:fig1} for each planet (gray shades).\nIt is worth noticing the discrepancy between the General Relativistic value (the red line) and the observed precession\n(blue dashed lines) for Mars and Jupiter; it could be attributed to the incomplete subtraction of nonrelativistic effects\nfrom the observed value, complicated by the presence of the asteroid belt between Mars and Jupiter, and the presence\nof an anomalous residual precession \\cite{2015MNRAS.451.3034N,2013MNRAS.432.3431P}.\n\\par\nThe tightest interval on the parameter $\\Xi$ is obtained with Venus, for which it can vary between $-1149.67$ and $1167.47$.\nWe can use the values defining such an interval to predict the precession for Uranus, Neptune and Pluto,\nfor which no observation is available.\nThe results, summarized in Table~\\ref{tab:tab2}, show that the bootstrapped theory predictions are in perfect agreement\nwith General Relativity. \n\\begin{table*}\n    \\centering\n    \\begin{tabular}{ccccccc}\n    \\hline\\hline\n    Planet & $a(10^6km)$ & $P(years)$ & $i(^{\\circ})$ & $e$ &  $\\Delta\\phi_{S}(''\/cy)$ & $[\\Delta\\phi_{min};\\Delta\\phi_{max}]$ \\\\\n    \\hline\n    $\\textbf{Uranus}$&$2872.463$&$84.01$&$0.772$&$0.0457$&$0.00238404$&$[0.00238404;0.00238405]$\\\\\n    $\\textbf{Neptune}$&$4495.060$&$164.786$&$1.769$&$0.0113$&$0.000775374$&$[0.000775373;0.000775375]$\\\\\n    $\\textbf{Pluto}$&$5869.656$&$247.936$&$17.16$&$0.2444$&$0.000419669$&$[0.000419669;0.00041967]$\\\\\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{Orbital parameters from Nasa Fact Sheet, the General Relativistic prediction for the precession in the sixth column\n    and the values predicted by the bounds on the parameter $\\Xi$ of the bootstrapped theory deduced for Venus (see Table~\\ref{tab:tab1}).}\n    \\label{tab:tab2}\n\\end{table*}\n\\par\nNow it is useful to move to a different scale and analyze $S2$ (see Table \\ref{tab4}), the only one among the S-stars whose precession was observed \\cite{GRAVITY:2020gka}.\nWe can next calculate the precession for Mars, Jupiter, and $S2$ with the values of $\\Xi$ as obtained by Mercury, Venus, Earth\nand Saturn to check agreement with the corresponding Schwarzschild value and with the observations (Table \\ref{tab3}). The results confirm the compatibility of our predictions with General Relativity.\nThe mean value of the parameter $\\Xi$ such that\n\\begin{equation}\n    \\Delta\\phi\n    =\n    \\Delta\\phi_S\n\\end{equation}\nis given by\n\\begin{equation}\n\\Xi\n=\n-1.64236\\pm 0.10305\n\\ .\n\\label{eq:c2}\n\\end{equation}\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{cccccccc}\n    \\hline\\hline\n    Star & $a(AU)$ & $P(years)$ & $i(^{\\circ})$ & $e$ & $\\Delta\\phi_{obs}(''\/\\text{orbit})$ & $\\Delta\\phi_S(''\/\\text{orbit})$ & $[\\Xi_{min};\\Xi_{max}]$ \\\\\n    \\hline\n    $\\textbf{S2}$&$1031.32$&$16.0455$&$134.567$&$0.884649$&$730.382\\times(1.10\\pm0.19)$&$730.382$&$[-103.066;326.398]$\\\\\n\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{For the star $S2$, orbital parameters~\\cite{GRAVITY:2020gka}, observed orbital precession ($\\Delta\\phi_{obs}$),\n    orbital precession as predicted by General Relativity ($\\Delta\\phi_S$), and constraints on $\\Xi$.}\n    \\label{tab4}\n\\end{table*}\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{cccccc}\n    \\hline\\hline\n   Object  & $\\Delta\\phi_S$ & $\\Delta\\phi(\\Xi_{Mercury})$ &$\\Delta\\phi(\\Xi_{Venus})$&$\\Delta\\phi(\\Xi_{Earth})$&$\\Delta\\phi(\\Xi_{Saturn})$\\\\\n    \\hline\n    $\\textbf{Mars}$&$1.35106$&$[1.34814;1.35577]$&$[1.35102;1.3511]$&$[1.35094;1.35113]$&$[-3.75855;2.5191]$\\\\\n    $\\textbf{Jupiter}$&$0.0623142$&$[0.0622752;0.0623773]$&$[0.0623137;0.0623147]$&$[0.0623126;0.0623151]$&$[-0.00607962;0.0779489]$\\\\\n    $\\textbf{S2}$&$730.382$&$[-57243.9;94435.7]$&$[-11.7295;1485.75]$&$[-1634.61;2085.15]$&$[-1.01666*10^8;2.32414*10^7]$\\\\\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{Precession for Mars, Jupiter, and $S2$ as predicted by confidence regions for $\\Xi$ inferred from Mercury, Venus, Earth and Saturn.}\n    \\label{tab3}\n\\end{table*}\n\n\\subsection{S-star dynamics}\n\\label{ss:dynamic}\nWe can confirm the bounds on $\\Xi$ deduced from orbital precessions by comparing them with results deduced from the analysis\nof stellar orbits at the Galactic Center.\nThis further analysis consists in comparing simulated orbits in bootstrapped Newtonian gravity, obtained by integrating numerically\nEqs.~\\eqref{eq:ddr}-\\eqref{eq:ddt}, with observed orbits of three S-stars constructed by astrometric observations\n(see Sec.~\\ref{sec:data}).\nIn particular, we focused on stars $S2$, $S38$ and $S55$ for two main reasons:\namong the brightest stars they are those with the shortest period.\nThese properties are desired because highly bright stars are less prone to being confused with other sources,\nand a short period allows us to observe a larger part of the orbit in a given observation session.\nFor simplicity, we neglected perturbations from other members of the cluster and any extended matter structures. \n\\subsubsection{Astrometric data}\n\\label{sec:data}\nAstrometric data are taken from Ref.~\\cite{Gillessen:2017}~\\footnote{Data are publicly available on the electronic version of\nRef.~\\cite{Gillessen:2017} at the link https:\/\/iopscience.iop.org\/article\/10.3847\/1538-4357\/aa5c41\/meta.}\nand cover $25$ years of observations performed in the near-infrared (NIR), where the interstellar extinction amounts to\nabout three magnitudes.\nDifferent instruments have been used, which we briefly describe below.\n\\begin{enumerate}\n\\item\nSHARP.- First high-resolution data of the Galactic Center were obtained in $1992$ with the SHARP camera at the European Southern Observatory's\n(ESO) $3.5\\,$m New Technology Telescope (NTT) in Chile, operating in Speckle mode with exposure times of $0.3\\,$s, $0.5\\,$s and $1.0\\,$s.\nThe data, described in detail in Ref.~\\cite{Schodel:2003gy}, led to the detection of high proper motion near the central massive object.\n\\item\nNACO.- The first Adaptive Optics (AO) imaging data were produced by Naos-Conica (NACO) system,\nmounted at the telescope Yepun ($8.0\\,$m) of the VLT and starting to operate in $2002$.\nIt followed a great improvement due to larger telescope aperture, and the higher Strehl ratio (about $40\\%$).\n\\item\nGEMINI.- The dataset includes observations obtained by the $8\\,$m telescope Gemini North in Mauna Kea, Hawaii.\nThese images, obtained using the AO system in combination with the NIR camera Quirc, were processed by the Gemini team.\n\\end{enumerate}\nThe astrometric calibration of these data, treated in Ref.~\\cite{Gillessen:2009ht}, consists in the following steps:\nobtaining high-quality maps of the S-stars, extracting pixel positions, and transforming them to a common astrometric\ncoordinate system.\nIn particular, the astrometric reference frame is implemented relating the S-stars positions to a set of selected reference\nstars, which are in turn related to a set of Silicon Monoxide (SiO) maser stars whose positions relative to Sgr~A* is known.\n\\subsubsection{Fitting procedure}\nThe first step of the fitting procedure is the numerical integration of the system of parametric nonlinear\ndifferential equations~\\eqref{eq:ddr}-\\eqref{eq:ddt} to produce stellar simulated orbits in bootstrapped\nNewtonian gravity.\n\\begin{table*}\n\\begin{centering}\n    \\begin{tabular}{cccc}\n    \\hline\\hline\nParameter           & S2                     & S38               & S55               \\\\ \\hline\n$a$ (mas)           & $125.058\\pm0.041$      & $141.6\\pm0.2$     & $107.8\\pm1.0$     \\\\\n$\\Omega$ ($^\\circ$) & $228.171\\pm0.031$      & $101.06\\pm0.24$   & $325.5\\pm4.0$     \\\\\n$e$                 & $0.884649\\pm0.000066$  & $0.8201\\pm0.0007$ & $0.7209\\pm0.0077$ \\\\\n$i$ ($^\\circ$)      & $134.567\\pm0.033$     & $171.1\\pm2.1$     & $150.1\\pm2.2$     \\\\\n$\\omega$ ($^\\circ$) & $66.263\\pm0.031$       & $17.99\\pm0.25$    & $331.5\\pm3.9$     \\\\\n$t_p$ (yr)          & $2018.37900\\pm0.00016$ & $2003.19\\pm0.01$  & $2009.34\\pm0.04$  \\\\\n$T$ (yr)            & $16.0455\\pm0.0013$     & $19.2\\pm0.02$     & $12.80\\pm0.11$  \\\\ \n$m_K$     &\n13.95          &\n17.       &\n17.5     \\\\\nRef.          &\n\\cite{GRAVITY:2020gka} &\n\\cite{Gillessen:2017}              &\n\\cite{Gillessen:2017} \\\\\n\\hline\\hline\n    \\end{tabular}\n        \\caption{Orbital parameters of $S2$, $S38$, and $S55$:\n        semimajor axis  $a$, eccentricity $e$, inclination $i$, angle of the line of node $\\Omega$,\n        angle from ascending node to pericenter $\\omega$, orbital period $T$, and the time of the\n        pericenter passage $t_p$. }\n        \\label{tab1}\n        \\end{centering}\n\\end{table*}\n\\begin{table*}\n\\begin{centering}\n    \\begin{tabular}{cccc}\n    \\hline\\hline\n     Star&$M(M_{\\odot})$&$R(kpc)$&Ref.\\\\\n    \\hline\n    S2&$(4.261\\pm0.012)\\times10^6$&$8.2467\\pm0.0093$&GRAVITY Collaboration \\cite{GRAVITY:2020gka} \\\\\n    S38&$(4.35\\pm0.13)\\times10^6$&$8.33\\pm0.12$&Gillessen et al. \\cite{Gillessen:2017}\\\\\n    S55&$(4.35\\pm0.13)\\times10^6$&$8.33\\pm0.12$&Gillessen et al. \\cite{Gillessen:2017}\\\\\n    \\hline\\hline\n    \\end{tabular}\n        \\caption{Parameters of the central BH: the mass $M$ and the distance $R$.} \\label{tab2}\n        \\end{centering}\n\\end{table*}\n\\par\nPreliminarily, we fix the Keplerian elements and the parameters of the central mass to the values reported\nin Tables~\\ref{tab1} and \\ref{tab2}.\nIn particular, for the study of $S2$, we used the values obtained by the GRAVITY Collaboration~\\cite{GRAVITY:2020gka},\nand for $S38$ and $S55$, we used those obtained in Ref.~\\cite{Gillessen:2017}.\nIn order to have a well-defined Cauchy problem, we must provide initial conditions for the four-dimensional coordinates\nand their derivatives with respect to the proper time: \\{$r(0)$, $\\dot{r}(0)$, $\\theta(0)$, $\\dot{\\theta}(0)$, $\\phi(0)$,\n$\\dot{\\phi}(0)$, $t(0)$, $\\dot{t}(0)$\\}. \nWe assume that the star initially lies on the equatorial plane of the reference system, for which $\\theta(0)=\\pi\/2$,\nand that its initial velocity is parallel to the equatorial plane, that is $\\dot{\\theta}(0)=0$.\nIt then follows that $\\ddot{\\theta}(0)=0$ identically.\nIn particular, we set the initial conditions for $r$ and $\\phi$ at the time of passage of the apocenter,\nwhen the Cartesian coordinates of the star expressed in the orbital plane are given by\n\\begin{equation}\n(x_{orb},y_{orb})\n=\n\\left(-a\\,(1+e),0\\right)\n\\end{equation}\nand the Cartesian components of its velocity read\n\\begin{equation}\n(v_{x,orb},v_{y,orb})\n=\n\\left(0,\\dfrac{2\\,\\pi\\,a^2}{T\\,r}\\,\\sqrt{1-e^2}\\right)\n\\ .\n\\end{equation}\nThe initial condition for $\\dot{t}$ can be retrieved from the normalization of four-velocities requiring\nthat the geodesic is timelike.\n\\par\nStarting from the initial conditions of each star, we proceed with an explicit Runge-Kutta numerical integration\nof the relativistic equations of motion.\nThe results are the stars mock orbit in the orbital plane, described by a four-dimensional array\n$\\{t(\\tau),r(\\tau),\\theta(\\tau),\\phi(\\tau)\\}$.\nTo compare the theoretical orbits with those observed from the Earth, we must project any point\n$(x_{\\rm orb}, y_{\\rm orb})$ on the orbital plane into the point $(x, y)$ on the observer's sky plane.\nSuch a transformation is realized by applying the Thiele-Innes formulas~\\cite{1930MNRAS,aitken}:\n\\begin{align}\n    x&=l_1\\,x_{\\rm orb}+l_2\\,y_{\\rm orb}\n    \\\\\n    y&=m_1\\,x_{\\rm orb}+m_2\\,y_{\\rm orb}\n    \\ .\n\\end{align}\nThe Thiele-Innes elements $l_1$, $l_2$, $m_1$ and $m_2$ depend on the Keplerian elements by according to\n\\begin{align}\n    l_1\n    &=\n    \\cos\\Omega\\,\\cos\\omega-\\sin\\Omega\\, \\sin\\omega\\, \\cos i\n    \\\\\n    l_2\n    &=\n    -\\cos \\Omega\\, \\sin\\omega-\\sin\\Omega\\, \\cos\\omega \\cos i\n    \\\\\n    m_1\n    &=\n    \\sin\\Omega\\, \\cos\\omega+\\cos\\Omega\\, \\sin\\omega \\cos i\n    \\\\\n    m_2\n    &=\n    -\\sin\\Omega\\,\\sin\\omega+\\cos\\Omega\\, \\cos\\omega\\, \\cos i\n     .\n     \\end{align}\n\\par     \nThe second step consists in the fitting procedure itself, and has the aim to constrain the parameter $\\Xi$.\nGuided by the results obtained from the precession in Sec.~\\ref{ss:precession}, we let it vary freely in an\nappropriate range including the value~\\eqref{eq:c2}.\nFor each value of $\\Xi$ we repeated the aforementioned procedure to get the true positions $(x_i,y_i)$\nand velocities $(\\dot{x}_i,\\dot{y}_i)$ of the stars at all the observed epochs.\nAfter transforming the true positions into the apparent positions $(x_i^{th},y_i^{th})$, we computed the\nreduced-$\\chi^2$ distribution to quantify the discrepancy between theory and observations as\n\\begin{equation}\n  \\chi^2_{\\rm red}\n  =\n  \\frac{1}{2\\,N-1}\\,\n  \\sum_i^N\\left[\\left(\\frac{x_i^{\\rm obs}-x_i^{\\rm th}}{\\sigma_{x_i^{\\rm obs}}}\\right)^2\n  +\\left(\\frac{y_i^{\\rm obs}-y_i^{\\rm th}}{\\sigma_{y_i^{\\rm obs}}}\\right)^2\n  \\right]\n  \\ ,\n\\end{equation}\nwhere $(x_i^{\\rm obs},y_i^{\\rm obs})$ and $(x_i^{\\rm th},y_i^{\\rm th})$ are respectively the observed and\nthe predicted positions, $N$ is the number of observations and $(\\sigma_{x_i^{\\rm obs}},\\sigma_{y_i^{\\rm obs}})$\nare the observative uncertainties.\nFinally, we calculated the likelihood probability distribution, $2\\,\\log\\mathcal{L}=-\\chi^2_{\\rm red}(\\Xi)$.\nThe best-fit value for $\\Xi$ was derived as the point that maximizes the likelihood distribution.\n\\subsubsection{Results}\n\\begin{table}\n    \\centering\n    \\begin{tabular}{cc}\n    \\hline\\hline\n     Star&$\\Xi$\\\\\n    \\hline\n    $S2$&$-5900_{-44964.9}^{+39358.8}$\\\\\n    $S38$&$25500_{-23312.88}^{+22607.1}$\\\\\n    $S55$&$60400_{-87446.9}^{+81386}$\\\\\n    Multi-star&$17400_{-32244.3}^{+30555.6}$\\\\\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{Best-fit values for $\\Xi$.}\n    \\label{tab5}\n\\end{table}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.45]{Fig3.pdf}\n    \\caption{Comparisons between the NTT\/VLT astrometric observations with their errors (black circles)\n    and the theoretical best-fit orbits using parameters reported in the first three rows of Table~\\ref{tab5}.\n    The results for $S2$, $S55$ and $S38$ are illustrated respectively in the top left, top right, and bottom panels.}\n    \\label{fig:fig3}\n\\end{figure*}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.6]{Fig4.pdf}\n    \\caption{Top panels show the comparison between the observed and fitted coordinates, and\n    bottom panels show the corresponding (O-C) residuals for $S2$, $S38$ and $S55$.}\n    \\label{fig:fig4}\n\\end{figure*}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.7]{Fig5.pdf}\n    \\caption{Best relativistic multistar orbit fit of $S2$, $S38$ and $S55$.}\n    \\label{fig:fig5}\n\\end{figure*}\nOur results are summarized in Table~\\ref{tab5} and represented in Figs.~\\ref{fig:fig3}, \\ref{fig:fig4} and \\ref{fig:fig5}.\n\\par\nIn Fig.~\\ref{fig:fig3} we show the comparison between best fit and observed orbits of the selected stars:\nthe top left panel, the top right panel, and the bottom panel illustrate the results respectively for $S2$, $S55$ and $S38$.\nAstrometric data are reported with their own error bars to note the effectiveness of our fitting procedure.\n\\par\nFigure~\\ref{fig:fig4} depicts the comparisons between the observed and simulated coordinates with the corresponding residuals.\nThe left column contains the right ascension (RA), while the right column reports the declination (Dec).\nIt is worth noticing that in all stars and for both coordinates, residuals are larger at the beginning observing epochs,\nand decrease as astrometric accuracy improves.\n\\par\nFinally, we show in Fig.~\\ref{fig:fig5} the orbits of the studied S-stars corresponding to the best multistar fit for\n$\\Xi=17400_{-32244.3}^{+30555.6}$ (last row of Table~\\ref{tab5}).\nAs expected, the parameter $\\Xi$ is compatible with the the mean value~\\eqref{eq:c2} such that the bootstrapped\nNewtonian precession recovers General Relativity.\n\\section{Conclusions}\n\\label{s:conc}\n\\setcounter{equation}{0}\nIn this paper we tested astronomically the bootstrapped Newtonian gravity.\nThe starting point is the complete spacetime metric~\\eqref{eq:g} derived in Ref.~\\cite{Casadio:2021gdf}.\nThe leading order deviation from the Schwarzschild solution cannot be eliminated and is encoded in the free parameter\n$\\Xi$, which is not \\textit{a priori} known and must be constrained by observations.\n\\par\nFirst, we show that bounds on $\\Xi$ can be deduced from the comparison between the measurements\nof the orbital precession of Solar System bodies and the theoretical predictions arising from bootstrapped\nNewtonian metric computed in Ref.~\\cite{Casadio:2021gdf}.\nThe inferred confidence region for $\\Xi$ for each planet is reported in Table~\\ref{tab:tab1} and graphically\ndepicted in Fig.~\\ref{fig:fig1}.\nBased on the tightest interval obtained with Venus, we found that $\\Xi$ lies in the range $[-1149.67\\,;\\,+1167.47]$.\nWith these values of the parameter $\\Xi$ we predicted the orbital precession for Uranus, Neptune and Pluto,\nand we found a theoretical precession in great agreement with the General Relativistic value.\nSuch a compatibility was confirmed by turning our attention to the Galactic Center and repeating the same\nanalysis for the star $S2$~\\cite{GRAVITY:2020gka}.\nThe mean value of the parameter $\\Xi$ such that the bootstrapped Newtonian precession equals the Schwarzschild\nvalue is \n\\begin{equation}\n\\Xi=-1.64236\\pm0.10305\n\\ .\n\\end{equation}\n\\par\nWe next focused on the Galactic Center scale to constrain $\\Xi$ by investigating the orbital motion of S-stars.\nWe used a fully relativistic approach based on an agnostic method:\nfor each value of $\\Xi$, we solved the geodesic equations numerically starting from initial conditions at the\napocenter.\nAfter applying the Thiele-Innes formulas to the mock positions, we were able to compare the resulting solution\nwith the observed stellar orbits.\nFinally, we quantified the discrepancy between the simulated and observed orbits performing a $\\chi^2$-statistics.\nThe inferred confidence region for $\\Xi$ is compatible with the bounds obtained by the precession analysis,\nand thus with General Relativity.\nIndeed we found $17400_{-32244.3}^{+30555.6}$.\nSince S-stars are at a distance of about  $r>1000\\, R_g$ from the source, strong-field effects are not relevant,\nand such a result was expected. \n\\par\nThe proposed approach is completely general and represents a useful tool in the classification of extended\ntheories of gravity.\nMoreover, this approach has already been used to test a Yukawa-like gravitational potential by means of dynamical tests at the\nGalactic Center~\\cite{yuk1,yuk2,yuk3,DellaMonica2021}, where no significant deviations from General Relativity were found.\nNevertheless, the definitive confirmation\/exclusion of a given extended theory of gravity requires the improvement\nof the constraints on its free parameters based on the observation of various strong-field effects.\nThis task can be accomplished taking advantage of the increasing high accuracy observations of second\ngeneration instruments like GRAVITY~\\cite{Gillessen:2010ei}.\n\\par\nIn particular, we focus on finding stars with short semimajor axis and highly eccentric orbits within the\npericenter of $S2$.\nThe existence of such a population of stars can be inferred from the recent discovery of the sources $S62$,\n$S4711$ and $S4714$~\\cite{peissker2020,peissker20202}.\nObserving stars at smaller radii is essential to detect strong-field effects, which become no longer negligible\nfor distances of the pericenter $r\\simeq 10\\,R_g$, and therefore any deviations from General Relativity\nto find out the underlying gravitational theory.\n\n\n\n\n\\begin{acknowledgments}\nR.C.~is partially supported by the INFN grant FLAG. M.D.L. and A.D. acknowledges INFN Sez. di Napoli (Iniziativa Specifica TEONGRAV). \nA.G.~is supported by the European Union's Horizon 2020 research and innovation programme under the \nMarie Sk\\l{}odowska-Curie Actions (grant agreement No.~895648--CosmoDEC).\nThe work of R.C.~and A.G~has also been carried out in the framework\nof activities of the National Group of Mathematical Physics (GNFM, INdAM). \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA Borel set $E$ in an Abelian Polish group $X$ is said to be \\emph{Haar Null} if there is a Borel probability measure $\\mu$ on $X$ such that $\\mu(x+E)=0$ for every $x\\in X$. Haar null sets were introduced for the first time by J.P.R.\\ Christensen in \\cite{chris_1972} in order to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups, where the Haar measure is not defined. Indeed, Haar null sets and sets with zero Haar measure agree on locally compact Abelian Polish groups and, as in the locally compact case, Haar null sets form a $\\sigma$-ideal in the Borel $\\sigma$-algebra of $X$. We say that a Borel set $E\\subseteq X$ is \\emph{Haar positive} if $E$ is not Haar null. The reader is invited to have a look at the survey papers \\cite{bog_2018} and \\cite{en_2020}, as well as at \\cite{benlin}, Chapter 6, for a detailed exposition of this topic.\n\nAlthough the definition of Haar positive sets is measure-theoretical in nature, in \\cite{daverava2} a measure-free characterisation of Haar positive closed, convex sets in separable Banach spaces is provided.\n\n\\begin{theo}\n\\label{theo:intro}\nLet $C$ be a closed and convex set in a separable Banach space $X$ with unit ball $B_X$. The following assertions are equivalent.\n\\begin{enumerate}\n\\item $C$ is Haar positive.\n\\item There is $r>0$ with the property that, for every compact set $K\\subseteq rB_X$, there is $x\\in X$ such that $x+K\\subseteq C$.\n\\end{enumerate}\n\\end{theo} \n\\noindent Equivalently, a closed, convex set $C$ in a separable Banach space $X$ is Haar null if and only if $C$ is \\emph{Haar meagre}. That is, there is a compact metric space $M$ and a continuous function $\\func{f}{M}{X}$ such that $f^{-1}(x+C)$ is meagre in $M$ for every $x\\in X$. For a more general treatment of Haar meagre sets, we refer the reader to \\cite{dar_2013}, where they were first introduced.\n\nTheorem \\ref{theo:intro} motivates the introduction of several geometric radii associated to such sets. These are defined in Section \\ref{sec:radii} and multiple properties about these quantities are shown. In Section \\ref{sec:weaks} we exploit these radii to prove a new characterisation of Haar positive, closed, convex sets. Namely, a closed, convex set $C$ in a separable Banach space $X$ is Haar positive if and only if its weak$^*$ closure in the second dual $X^{**}$ has nonempty interior with respect to the norm topology. This improves the well-known fact that closed, convex subsets of a Euclidean space $\\mathbb{R}^d$ have positive Lebesgue measure if and only if their interior is nonempty and a theorem of Eva Matou\\v{s}kov\\'{a} (\\cite{eva3}) which states that, in separable, reflexive Banach spaces, closed convex sets are Haar positive if and only if they have nonempty interior. As a corollary, it is shown in Section \\ref{sec:top} that the family of Haar positive, closed, convex and bounded sets is open in the space of all nonempty, closed, convex and bounded subsets of $X$, endowed with the Hausdorff distance.\n\nThe standard notation of Banach space theory is used throughout the paper. Given a Banach space $X$, $B_X$ and $S_X$ stand for the closed unit ball and the unit sphere of $X$ respectively. $X^*$ denotes the dual of $X$, whereas $X^{**}$ is the second dual. The closure of a set $E\\subseteq X$ is denoted by $\\textup{cl}(E)$ and in a dual space we denote by $\\textup{w}^*\\textup{-cl}(E)$ the closure of $E$ in the weak$^*$ topology. We use the notation $\\mathcal{C}(X)$ for the space of all nonempty, closed, convex and bounded subsets of $X$. This turns into a complete metric space if endowed with the Hausdorff distance $d_\\textup{H}$ given by\n\\[ d_\\textup{H}(C,D)=\\inf\\{\\epsilon>0\\,:\\,C\\subseteq\\textup{cl}(D+\\epsilon B_X) \\text{ and }D\\subseteq\\textup{cl}(C+\\epsilon B_X)\\}. \\]\nAll Banach spaces are assumed to be real.\n\n\\section{Geometric radii of closed, convex and bounded sets} \n\\label{sec:radii}\nGiven $\\rho>0$ and a nonempty, closed, convex set $C$ in a Banach space $X$, we denote by $\\textup{ir}(C,\\rho)$ the $\\rho\\,$-\\emph{inner radius} of $C$. That is, the supremum of all $r\\geq 0$ such that $C$ contains a closed ball of the form $x+rB_X$, where $x\\in\\rho B_X$. Clearly, $C$ has nonempty interior if and only if $\\textup{ir}(C,\\rho)>0$ for some $\\rho>0$. The $\\rho\\,$-\\emph{compact radius} $\\textup{kr}(C,\\rho)$ is defined as follows: it is the supremum of all $r\\geq 0$ with the property that, for every compact set $K\\subseteq rB_X$, there is $x\\in\\rho B_X$ such that $x+K\\subseteq C$. The $\\rho\\,$-\\emph{finite radius} $\\textup{fr}(C,\\rho)$ is defined similarly. Namely, it is the supremum of all $r\\geq 0$ with the property that, for every finite set $F\\subseteq rB_X$, there is $x\\in\\rho B_X$ such that $x+F\\subseteq C$. Finally, we introduce the $\\rho\\,$-\\emph{loose radius} $\\textup{lr}(C,\\rho)$ as follows: it is the supremum of all $r\\geq 0$ with the property that, for every finite set $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ and every $\\epsilon>0$, there are a finite set $G=\\{y_1,\\dots,y_n\\}$ and $z\\in\\rho B_X$ such that $\\norm{x_j-y_j}{X}<\\epsilon$ for every $j\\in\\{1,\\dots,n\\}$ and $z+G\\subseteq C$.\n\n\\begin{theo}\n\\label{theo:radii}\nLet $C$ be a nonempty, closed, convex set in a Banach space $X$. The following inequalities hold for every $\\rho>0$.\n\\[ \\textup{ir}(C,\\rho)\\leq\\textup{kr}(C,\\rho)\\leq\\textup{fr}(C,\\rho)\\leq\\textup{lr}(C,\\rho)\\leq 2\\textup{kr}(C,\\rho). \\]\n\\end{theo}\n\\begin{proof}\nThe only nontrivial inequality is the last one and we will prove it using a variation of the Banach-Dieudonn\\'{e} Theorem, similar to the one which appears in \\cite{eva1}. If $\\textup{lr}(C,\\rho)=0$, the claim is obvious. Assume that $\\textup{lr}(C,\\rho)>0$. We aim to show that, for every $r\\geq 0$ such that $2r<\\textup{lr}(C,\\rho)$, every compact set $K\\subseteq rB_X$ can be translated into $C$ via some $z\\in\\rho B_X$. Let $K$ be a compact subset of $rB_X$ and find a finite set $F_1=\\{x_{1,1},\\dots,x_{1,k(1)}\\}\\subseteq 2rB_X$ such that $2^{-1}F_1$ is an $(r\/4)$-net for $K$. Find a further finite set $G_1=\\{y_{1,1},\\dots,y_{1,k(1)}\\}$ and $z_1\\in\\rho B_X$ such that $\\norm{y_{1,j}-x_{1,j}}{X}<r\/2$ for every $j\\in\\{1,\\dots,k(1)\\}$ and such that $z_1+G_1\\subseteq C$. Notice that $2^{-1}G_1$ is an $(r\/2)$-net of $K$. Define \n\\[ K_1=2\\Bigl((K-2^{-1}G_1)\\cap\\frac{r}{2}B_X\\Bigr), \\]\nand observe that $K_1$ is a compact subset of $rB_X$ such that $K\\subseteq 2^{-1}G_1+2^{-1}K_1$. Working by induction, let $n$ be a positive integer and assume that we have already found finite sets $G_1,\\dots,G_n\\subseteq X$, $z_1,\\dots,z_n\\in\\rho B_X$ and a compact set $K_n\\subset rB_X$ such that $z_j+G_j\\subseteq C$ for every $j\\in\\{1,\\dots,n\\}$ and \n\\begin{equation}\n\\label{eq:loose1}\nK\\subseteq \\sum_{j=1}^n2^{-j}G_j+2^{-n}K_n.\n\\end{equation}\nLet $F_{n+1}=\\{x_{n+1,1},\\dots,x_{n+1,k(n+1)}\\}\\subseteq 2rB_X$ be a finite set such that $2^{-1}F_{n+1}$ is an $(r\/4)$-net for $K_n$. Find a further finite set $G_{n+1}=\\{y_{n+1,1},\\dots,y_{n+1,k(n+1)}\\}$ and $z_{n+1}\\in\\rho B_X$ such that $\\norm{y_{n+1,j}-x_{n+1,j}}{X}<r\/2$ for every $j\\in\\{1,\\dots,k(n+1)\\}$ and $z_{n+1}+G_{n+1}\\subseteq C$. Notice again that $2^{-1}G_{n+1}$ is an $(r\/2)$-net of $K_n$. Define \n\\[ K_{n+1}=2\\Bigl((K_n-2^{-1}G_{n+1})\\cap\\frac{r}{2}B_X\\Bigr), \\]\nand observe that $K_{n+1}\\subseteq rB_X$ is compact and that $K_n\\subseteq 2^{-1}G_{n+1}+2^{-1}K_{n+1}$ holds. Now we have \n\\[ K\\subseteq \\sum_{j=1}^{n+1}2^{-j}G_j+2^{-(n+1)}K_{n+1}, \\]\ntherefore we can claim by induction that we can find a sequence ${(z_n)}_{n=1}^\\infty$ in $\\rho B_X$, a sequence ${(G_n)}_{n=1}^\\infty$ of finite subsets of $X$ and a sequence ${(K_n)}_{n=1}^\\infty$ of compact subsets of $rB_X$ such that $z_n+G_n\\subseteq C$ and (\\ref{eq:loose1}) hold for every $n$. In particular, this implies \n\\[ K\\subseteq\\textup{cl}\\biggl(\\,\\sum_{n=1}^\\infty 2^{-n}G_n\\biggr). \\]\nPut $z=\\sum_{n=1}^\\infty 2^{-n}z_n$ and observe that $z\\in\\rho B_X$. Then\n\\[ z+K\\subseteq\\sum_{n=1}^\\infty 2^{-n}z_n+\\textup{cl}\\biggl(\\,\\sum_{n=1}^\\infty 2^{-n}G_n\\biggr)=\\textup{cl}\\biggl(\\,\\sum_{n=1}^\\infty 2^{-n}z_n+\\sum_{n=1}^\\infty 2^{-n}G_n\\biggr)\\subseteq C, \\]\nwhere the last inclusion holds because $C$ is closed and convex. Since $K\\subseteq rB_X$ is arbitrary, we are done.\n\\end{proof}\n\nThe following argument also appears with some minor differences in \\cite{eva1}. We report it here for completeness and formulate it in the language we have just introduced.\n\\begin{lemma}\n\\label{lemma:kr}\nLet $C$ be a closed, convex set in a separable Banach space $X$. $C$ is Haar positive if and only if there is $\\rho>0$ such that $\\textup{kr}(C,\\rho)>0$. It follows in particular that $C$ is Haar positive if and only if $\\textup{lr}(C,\\rho)>0$ for some $\\rho>0$.\n\\end{lemma}\n\\begin{proof}\nIt is a direct consequence of Theorem \\ref{theo:intro} that $\\textup{kr}(C,\\rho)=0$ for every $\\rho>0$ if $C$ is Haar null. To show the converse statement, assume by way of contradiction that, for every positive integer $n$, there is a compact set $K_n\\subseteq n^{-1}B_X$ such that, whenever $x+K_n\\subseteq C$ for some $x\\in X$, we have $\\norm{x}{X}>n$. By Theorem \\ref{theo:intro}, there is $r>0$ such that every compact subset of $rB_X$ can be translated into $C$. In particular, this holds for\n\\[ K=\\{0\\}\\cup\\bigcup_{n>r^{-1}}K_n. \\]\nIf $x$ is such that $x+K\\subseteq C$, then $x+K_n\\subseteq C$ for every $n>r^{-1}$. It then follows by the choice of $K_n$ that $\\norm{x}{X}>n$ for every positive integer $n$, which does not make any sense. The last assertion is a consequence of Theorem \\ref{theo:radii}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:lr-fr}\nGiven $\\rho>0$ and a nonempty, closed, convex set $C$ in a Banach space $X$, we have\n\\[ \\textup{lr}(C,\\rho)=\\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr). \\]\n\\end{lemma}\n\\begin{proof}\nSet $r=\\textup{lr}(C,\\rho)$. Pick $\\epsilon>0$ and find $F=\\{x_1,\\dots,x_n\\}\\subset(r+\\epsilon)B_X$ and $\\delta_0>0$ such that every finite set $G=\\{y_1,\\dots,y_n\\}$ which fulfills the condition $\\norm{y_j-x_j}{X}<2\\delta_0$ for every $j\\in\\{1,\\dots,n\\}$ cannot be translated into $C$ via some $z\\in\\rho B_X$. Suppose that there is $z\\in\\rho B_X$ such that $z+F\\subset\\textup{cl}(C+\\delta_0B_X)$. This would imply that, for every $j\\in\\{1,\\dots,n\\}$, we can find $w_j\\in C$ such that $\\norm{z+x_j-w_j}{X}<2\\delta_0$. Put $y_j=w_j-z$ for every $j$. Then the set $G=\\{y_1,\\dots,y_n\\}$ satisfies $z+G\\subseteq C$ and $\\norm{x_j-y_j}{X}<2\\delta_0$ for every $j$, in contradiction with the choice of $F$ and $\\delta_0$. Thus, $F$ cannot be translated into $\\textup{cl}(C+\\delta_0B_X)$ via some $z\\in\\rho B_X$, which yields\n\\[ \\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr)\\leq\\textup{fr}\\bigl(\\textup{cl}(C+\\delta_0B_X),\\rho\\bigr)\\leq r+\\epsilon. \\]\nSince $\\epsilon$ is arbitrary, we get\n\\[ \\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr)\\leq r. \\]\nTo show the opposite inequality, observe that it is obvious if $r=0$. Assume that $r>0$, choose $\\delta\\in(0,r)$ and pick a finite set $F=\\{x_1,\\dots,x_n\\}\\subset(r-\\delta)B_X$. Find $G=\\{y_1,\\dots,y_n\\}$ and $z\\in\\rho B_X$ such that $z+G\\subseteq C$ and $\\norm{x_j-y_j}{X}<\\delta$ for each $j$. Notice that\n\\[ z+x_j=z+y_j+x_j-y_j\\in C+\\delta B_X. \\]\nfor each $j$, hence $z+F\\subseteq\\textup{cl}(C+\\delta B_X)$. Since $\\delta>0$ and $F\\subset (r-\\delta)B_X$ are arbitrary, we conclude that\n\\[ \\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr)\\geq\\lim_{\\delta\\to 0^+}(r-\\delta)=r, \\]\nas wished.\n\\end{proof}\n\n\\section{Weak$^*$ closures of Haar positive closed, convex sets}\n\\label{sec:weaks}\nGiven a Banach space $X$ and a positive integer $n$, recall that we can endow the Banach space $X^n$, the product of $n$ copies of $X$, with the $\\infty$-product norm:\n\\[ \\norm{x}{X^n}=\\max_{1\\leq j\\leq n}\\norm{x_j}{X} \\]\nfor every $x=(x_1,\\dots,x_n)\\in X^n$. In this way we have $B_{X^n}={(B_X)}^n$. We denote by $\\func{\\Delta}{X}{X^n}$ the diagonal embedding $x\\mapsto(x,\\dots,x)$. The second dual space of $X^n$ is simply ${(X^{**})}^n$, endowed with the same $\\infty$-product norm. Notice that, in ${(X^{**})}^n$, we have $\\textup{w}^*\\textup{-cl}(\\Delta(\\rho B_X))=\\Delta(\\rho B_{X^{**}})$. We are now ready to state our main theorem.\n\\begin{theo}\n\\label{theo:hpws}\nLet $\\rho>0$, let $C$ be a nonempty, closed, convex set in a Banach space $X$ and let $r\\geq 0$. The following assertions are equivalent.\n\\begin{enumerate}\n\\item $\\textup{lr}(C,\\rho)\\geq r$.\n\\item For every finite $F\\subseteq rB_X$ there is $z\\in\\rho B_{X^{**}}$ such that $z+F\\subseteq\\textup{w}^*\\textup{-cl}(C)$.\n\\item There is $z_0\\in\\rho B_{X^{**}}$ such that $z_0+rB_{X^{**}}\\subseteq\\textup{w}^*\\textup{-cl}(C)$.\n\\end{enumerate}\nIn particular, $\\textup{lr}(C,\\rho)=\\textup{ir}(\\textup{w}^*\\textup{-cl}(C),\\rho)$ and, in case $X$ is separable, $C$ is Haar positive if and only if $\\textup{w}^*\\textup{-cl}(C)$ has nonempty interior in the norm topology of $X^{**}$.\n\\end{theo}\n\\begin{proof}\n(1)$\\Rightarrow$(2). Let $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ be a finite set and, for each positive integer $k$, find $G_k=\\{y_{k,1},\\dots,y_{k,n}\\}$ and $z_k\\in\\rho B_X$ such that $\\norm{x_{k,j}-y_{k,j}}{X}<k^{-1}$ for each $j\\in\\{1,\\dots,n\\}$ and $z_k+G_k\\subseteq C$. Since ${(z_k)}_{k=1}^\\infty$ is a bounded sequence, it admits a subsequence with a weak$^*$ limit $z\\in\\rho B_{X^{**}}$. Now, it is not hard to see that $z+F\\subseteq\\textup{w}^*\\textup{-cl}(C)$.\n\n(2)$\\Rightarrow$(3). Let $\\mathcal{U}$ be the directed set of all weak$^*$ neighbourhoods of $0$ in $X^{**}$, ordered by reverse inclusion. Since $rB_X$ is weakly$^*$ dense in $rB_{X^{**}}$ and $rB_{X^{**}}$ is weakly$^*$ compact, for every $U\\in\\mathcal{U}$ we can find a finite set $F_U\\subseteq rB_X$ and $z_U\\in \\rho B_{X^{**}}$ such that $rB_{X^{**}}\\subseteq F_U+U$ and $z_U+F_U\\subseteq\\textup{w}^*\\textup{-cl}(C)$. Since ${(z_U)}_{U\\in\\mathcal{U}}$ is a bounded net, there exist another directed set $I$ and a map $\\func{\\psi}{I}{\\mathcal{U}}$ such that ${(z_{\\psi(\\alpha)})}_{\\alpha\\in I}$ is a subnet of ${(z_U)}_{U\\in\\mathcal{U}}$ which converges in the weak$^*$ topology to some $z_0\\in\\rho B_{X^{**}}$. Assume by contradiction that there is $x_0\\in rB_{X^{**}}\\setminus(\\textup{w}^*\\textup{-cl}(C)-z_0)$. The Hahn-Banach Theorem delivers $f\\in X^*$ and $t\\in\\mathbb{R}$ such that $f(x_0)>t>f(x)$ for every $x\\in\\textup{w}^*\\textup{-cl}(C)-z_0$. Define\n\\[ s=\\frac{f(x_0)-t}{2} \\]\nand $V=\\{x\\in X^{**}\\,:\\,|f(x)|<s\\}$. Find $\\alpha_0\\in I$ such that $\\psi(\\alpha)\\subseteq V$ for every $\\alpha\\geq\\alpha_0$. Further, find $\\beta_0\\in I$ such that $z_{\\psi(\\beta)}-z_0\\in V$ for every $\\beta\\geq\\beta_0$. Take $\\gamma\\in I$ such that $\\gamma\\geq\\alpha_0$ and $\\gamma\\geq\\beta_0$ and find $y_0\\in F_{\\psi(\\gamma)}$ such that $x_0\\in y_0+\\psi(\\gamma)\\subseteq y_0+V$. Then we have $|f(y_0)-f(x_0)|<s$, which implies\n\\[ f(y_0)>f(x_0)-s=\\frac{f(x_0)+t}{2}. \\]\nAt the same time though, we have $z_{\\psi(\\gamma)}+y_0-z_0\\in\\textup{w}^*\\textup{-cl}(C)-z_0$, therefore $|f(z_{\\psi(\\gamma)})-f(z_0)+f(y_0)|<t$, which implies\n\\[ f(y_0)<t+|f(z_{\\psi(\\gamma)})-f(z_0)|<t+s=\\frac{f(x_0)+t}{2}, \\]\na contradiction.\n\n(3)$\\Rightarrow$(1). Assume by contradiction that $\\textup{lr}(C,\\rho)<r$. By Lemma \\ref{lemma:lr-fr}, there is $\\delta>0$ such that $\\textup{fr}(\\textup{cl}(C+\\delta B_X),\\rho)<r$, which in turn means that there is a finite set $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ such that $(z+F)\\setminus\\textup{cl}(C+\\delta B_X)\\ne\\varnothing$ for every $z\\in\\rho B_X$. In $X^n$ define $D=(C-x_1)\\times\\cdots\\times(C-x_n)$ and observe that\n\\[ \\textup{cl}(D+\\delta B_{X^n})=\\bigl(\\textup{cl}(C+\\delta B_X)-x_1\\bigr)\\times\\cdots\\times\\bigl(\\textup{cl}(C+\\delta B_X)-x_n\\bigr). \\]\nNotice moreover that $\\Delta(\\rho B_X)\\cap\\textup{cl}(D+\\delta B_{X^n})=\\varnothing$, otherwise it would hold that $z+F\\subseteq\\textup{cl}(C+\\delta B_X)$ for every $(z,\\dots,z)\\in \\Delta(\\rho B_X)\\cap\\textup{cl}(D+\\delta B_{X^n})$, against how $F$ has been chosen. Hence, by the Hahn-Banach theorem, there are $f\\in{(X^n)}^*$ and $t_1,t_2\\in\\mathbb{R}$ such that $t_1<t_2$, $f(x)<t_1$ for every $x\\in\\Delta(\\rho B_X)$ and $f(x)>t_2$ for every $x\\in D$. By taking the weak$^*$ closures of both sets in ${(X^n)}^{**}$, we have $f(x)\\leq t_1$ for every $x\\in\\Delta(\\rho B_{X^{**}})$ and $f(x)\\geq t_2$ for every $x\\in\\textup{w}^*\\textup{-cl}(D)$, thus $\\Delta(\\rho B_{X^{**}})\\cap\\textup{w}^*\\textup{-cl}(D)=\\varnothing$. This is a contradiction, because\n\\[ z_0\\in\\bigcap_{j=1}^n\\bigl(\\textup{w}^*\\textup{-cl}(C)-x_j\\bigr), \\]\ni.e.\\ $(z_0,\\dots,z_0)\\in(\\textup{w}^*\\textup{-cl}(C)-x_1)\\times\\cdots\\times(\\textup{w}^*\\textup{-cl}(C)-x_n)=\\textup{w}^*\\textup{-cl}(D)$.\n\nThe last statement is a consequence of Lemma \\ref{lemma:kr}.\n\\end{proof}\n\nTheorem \\ref{theo:hpws} offers an interesting connection with the theory of weak$^*$ derived sets. If $X$ is a Banach space and $E$ is a subset of $X^*$, the first weak$^*$ derived set of $E$ is given by\n\\[ E^{(1)}=\\bigcup_{n=1}^\\infty\\textup{w}^*\\textup{-cl}(E\\cap nB_{X^*}) \\]\nand corresponds to the set of all possible limits of bounded, weak$^*$ convergent nets with elements in $E$. Clearly $E^{(1)}\\subseteq\\textup{w}^*\\textup{-cl}(E)$. We refer the reader to \\cite{ostr_2023} and the references therein for a detailed account on weak$^*$ derived sets. In particular, it has to be remarked that, in general, $E^{(1)}$ and $\\textup{w}^*\\textup{-cl}(E)$ can be different. However, we have the following result.\n\n\\begin{cor}\nLet $C$ be a closed, convex set in a Banach space $X$. In the second dual $X^{**}$, $C^{(1)}$ has empty interior in the norm topology if and only if $\\textup{w}^*\\textup{-cl}(C)$ does.\n\\end{cor}\n\\begin{proof}\nOne of the implications follows from $C^{(1)}\\subseteq\\textup{w}^*\\textup{-cl}(C)$. Conversely, assume that $\\textup{w}^*\\textup{-cl}(C)$ has nonempty interior in the norm topology. Then, by Theorem \\ref{theo:hpws} and Theorem \\ref{theo:radii}, there are $\\rho>0$ and $r>0$ such that $\\textup{kr}(C,\\rho)\\geq r$. Now, it is not hard to see that $\\textup{kr}(C\\cap nB_X,\\rho)\\geq r$ for every positive integer $n>\\rho+r$, hence $\\textup{w}^*\\textup{-cl}(C\\cap n B_X)$ has nonempty interior in the norm topology for every $n>\\rho+r$ and so does $C^{(1)}$.\n\\end{proof}\n\n\\section{Haar positive sets and the Hausdorff metric}\n\\label{sec:top}\n\nLet $C$ be a closed and convex set in a Banach space $X$. Under the additional assumption that $C$ is bounded we define \n\\begin{align*} \n\\textup{bir}(C)=\\sup_{\\rho>0}\\textup{ir}(C,\\rho),&\\quad\\textup{bkr}(C)=\\sup_{\\rho>0}\\textup{kr}(C,\\rho),\\\\\n\\textup{bfr}(C)=\\sup_{\\rho>0}\\textup{fr}(C,\\rho),&\\quad\\textup{blr}(C)=\\sup_{\\rho>0}\\textup{lr}(C,\\rho). \n\\end{align*}\nAll these values are finite, as $\\textup{diam}(C)$ is an upper estimate for all of them. Moreover, the chain of inequalities \n\\begin{equation} \n\\label{eq:radii}\n\\textup{bir}(C)\\leq\\textup{bkr}(C)\\leq\\textup{bfr}(C)\\leq\\textup{blr}(C)\\leq 2\\textup{bkr}(C)\n\\end{equation}\nis a direct consequence of Theorem \\ref{theo:radii}. We call $\\textup{bir}(C)$ the \\emph{bounded inner radius} of $C$. It is the supremum of all $r\\geq 0$ such that $C$ contains a closed ball of radius $r$. $\\textup{bkr}(C)$ is the \\emph{bounded compact radius} of $C$ and corresponds to the supremum of all $r\\geq 0$ with the property that, for every compact set $K\\subseteq rB_X$, there is $x\\in X$ such that $x+K\\subseteq C$. The \\emph{bounded finite radius} $\\textup{bfr}(C)$ is, similarly, the supremum of all $r\\geq 0$ with the property that, for every finite set $F\\subseteq rB_X$, there is $x\\in X$ such that $x+F\\subseteq C$. Finally, the \\emph{bounded loose radius} $\\textup{blr}(C)$ is the supremum of all $r\\geq 0$ with the property that, for every finite set $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ and every $\\epsilon>0$, there are a finite set $G=\\{y_1,\\dots,y_n\\}$ and $z\\in X$ such that $\\norm{x_j-y_j}{X}<\\epsilon$ for every $j$ and $z+G\\subseteq C$. In case $X$ is separable, $C$ is Haar positive if and only if $\\textup{blr}(C)>0$. This follows from ($\\ref{eq:radii}$) and Theorem \\ref{theo:intro}.\n\nTheorem \\ref{theo:hpws} allows to prove that the family $\\mathcal{H}^+(X)$ of Haar positive, closed, convex and bounded subsets of a separable Banach space $X$ is an open subset of $\\mathcal{C}(X)$. This result follows from a few lemmas we are going to show.\n\n\\begin{lemma}\n\\label{lemma:hpws}\nGiven a closed, convex and bounded set $C$ in a Banach space $X$, we have $\\textup{blr}(C)=\\textup{bir}(\\textup{w}^*\\textup{-cl}(C))$.\n\\end{lemma}\n\\begin{proof}\nUsing Theorem \\ref{theo:hpws}, we have\n\\[ \\textup{blr}(C)=\\sup_{\\rho>0}\\textup{lr}(C,\\rho)=\\sup_{\\rho>0}\\textup{ir}\\bigl(\\textup{w}^*\\textup{-cl}(C),\\rho\\bigl)=\\textup{bir}\\bigl(\\textup{w}^*\\textup{-cl}(C)\\bigr). \\qedhere \\]\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:wscliso}\nGiven a Banach space $X$, the function $\\func{\\textup{w}^*\\textup{-cl}}{\\mathcal{C}(X)}{\\mathcal{C}(X^{**})}$ is an isometry.\n\\end{lemma}\n\\begin{proof}\nTake $\\epsilon>0$ and let $C,D\\in\\mathcal{C}(X)$ be such that $d_\\textup{H}(C,D)\\leq\\epsilon$. This means that $C\\subseteq\\textup{cl}(D+\\epsilon B_X)$ and $D\\subseteq\\textup{cl}(C+\\epsilon B_X)$. Take $x\\in\\textup{w}^*\\textup{-cl}(C)$ and let ${(x_\\alpha)}_{\\alpha\\in I}$ be a net in $C$ whose weak$^*$ limit is $x$. Choose $\\delta>0$ and, for each $\\alpha\\in I$, find $y_\\alpha\\in D$ and $z_\\alpha\\in\\epsilon B_X$ such that $\\norm{x_\\alpha-(y_\\alpha+z_\\alpha)}{X}\\leq\\delta$. By considering a subnet if necessary, we can assume that the nets ${(y_\\alpha)}_{\\alpha\\in I}$ and ${(z_\\alpha)}_{\\alpha\\in I}$ have weak$^*$ limits $y\\in\\textup{w}^*\\textup{-cl}(D)$ and $z\\in\\epsilon B_{X^{**}}$ respectively. Moreover, \n\\[ \\norm{x-(y+z)}{X^{**}}\\leq\\liminf_{\\alpha\\in I}\\norm{x_\\alpha-(y_\\alpha+z_\\alpha)}{X}\\leq\\delta, \\]\ni.e.\\ $x\\in\\textup{w}^*\\textup{-cl}(D)+(\\epsilon+\\delta)B_{X^{**}}$. Since $\\delta$ is arbitrary, we have\n\\[ x\\in\\bigcap_{\\delta>0}\\bigl(\\textup{w}^*\\textup{-cl}(D)+(\\epsilon+\\delta)B_{X^{**}}\\bigr)=\\textup{cl}\\bigl(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}}\\bigr). \\]\nAs $x$ is also arbitrary, we conclude that $\\textup{w}^*\\textup{-cl}(C)\\subseteq\\textup{cl}(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}})$. The inclusion $\\textup{w}^*\\textup{-cl}(D)\\subseteq\\textup{cl}(\\textup{w}^*\\textup{-cl}(C)+\\epsilon B_{X^{**}})$ is shown similarly, hence we get that $d_\\textup{H}(\\textup{w}^*\\textup{-cl}(C),\\textup{w}^*\\textup{-cl}(D))\\leq\\epsilon$. \n\nConversely, suppose that $d_\\textup{H}(C,D)>\\epsilon$ and, by swapping $C$ and $D$ if necessary, assume that there is $x_0\\in C\\setminus\\textup{cl}(D+\\epsilon B_X)$. Using the Hahn-Banach Theorem, find $f\\in X^*$ and $t\\in\\mathbb{R}$ such that $f(x_0)>t>f(x)$ for every $x\\in\\textup{cl}(D+\\epsilon B_X)$. Then $f(x)\\leq t$ for every $x\\in\\textup{w}^*\\textup{-cl}(D+\\epsilon B_X)$. Now observe that \n\\begin{align*} \n\\textup{cl}(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}}) &= \\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}}= \\\\\n&=\\textup{w}^*\\textup{-cl}(D)+\\textup{w}^*\\textup{-cl}(\\epsilon B_{X})\\subseteq\\textup{w}^*\\textup{-cl}(D+\\epsilon B_X). \n\\end{align*}\nThus we get $x_0\\notin\\textup{cl}(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}})$, from which $d_\\textup{H}(\\textup{w}^*\\textup{-cl}(C),\\textup{w}^*\\textup{-cl}(D))>\\epsilon$ follows. Since $C$, $D$ and $\\epsilon$ are arbitrary, the proof is complete.\n\\end{proof}\n\nFinally, we want to show the continuity of the bounded inner radius in the metric space of all nonempty, closed, convex and bounded subsets of a Banach space, endowed with the Hausdorff metric. Although it seems that this result cannot be found in the literature, it might be well-know and belong to the folklore. We prove it here for the sake of completeness.\n\\begin{lemma}\n\\label{lemma:ir}\nIn a Banach space $X$, the function $\\func{\\textup{bir}}{\\mathcal{C}(X)}{[0,+\\infty)}$ is Lipschitz continuous with Lipschitz constant $1$.\n\\end{lemma}\n\\begin{proof}\nThe proof is based on the following claim: if $C$ is a nonempty, closed, convex and bounded set in $X$ and $\\epsilon>0$, then\n\\begin{equation}\n\\label{eq:ir+eps}\n\\textup{bir}\\bigl(\\textup{cl}(C+\\epsilon B_X)\\bigr)=\\textup{bir}(C)+\\epsilon.\n\\end{equation}\n\nLet us see first that $\\textup{bir}(\\textup{cl}(C+\\epsilon))\\leq\\textup{bir}(C)+\\epsilon$. Set $r=\\textup{bir}(C)$ and, looking for a contradiction, assume that $\\textup{bir}(\\textup{cl}(C+\\epsilon))>r+\\epsilon$. Then there is $\\delta>0$ such that $r+\\epsilon+\\delta<\\textup{bir}(\\textup{cl}(C+\\epsilon))$ and, without loss of generality, we may assume that $(r+\\epsilon+\\delta)B_X\\subseteq\\textup{cl}(C+\\epsilon B_X)$. Since $(r+\\delta)B_X\\setminus C\\ne\\varnothing$, the Hahn-Banach Theorem provides $x_0\\in(r+\\delta)B_X$, $t\\in\\mathbb{R}$ and $f\\in S_{X^*}$ such that $f(x_0)>t>f(x)$ for every $x\\in C$. Further, we have $t<\\norm{f}{X^*}\\norm{x_0}{X}\\leq r+\\delta$. Find $\\delta'$ such that $r+\\delta-\\delta'-t>0$ and $x_1\\in(r+\\epsilon+\\delta)B_X$ such that $f(x_1)>r+\\epsilon+\\delta-\\delta'$. Since $(r+\\epsilon+\\delta)B_X\\subseteq\\textup{cl}(C+\\epsilon B_X)$, there exist $x\\in C$ and $y\\in\\epsilon B_X$ such that\n\\[ \\norm{x_1-(x+y)}{X}<r+\\delta-\\delta'-t \\]\nOn the other hand, we have\n\\[ \\norm{x_1-(x+y)}{X}\\geq f(x_1)-f(x)-f(y)>r+\\epsilon+\\delta-\\delta'-t-\\epsilon=r+\\delta-\\delta'-t, \\]\na contradiction. \n\nTo see that $\\textup{bir}(C)+\\epsilon\\leq\\textup{bir}(\\textup{cl}(C+\\epsilon B_X))$, set again $r=\\textup{bir}(C)$, choose $\\delta\\in(0,r)$ and find $x_0\\in X$ such that $x_0+(r-\\delta)B_X\\subseteq C$. Then $x_0+(r+\\epsilon-\\delta)B_X\\subseteq\\textup{cl}(C+\\epsilon B_X)$. Since $\\delta$ is arbitrary, it follows that $\\textup{bir}(\\textup{cl}(C+\\epsilon B_X))\\geq r+\\epsilon$, as wished.\n\nTo prove the statement, pick $C,D\\in\\mathcal{C}(X)$ and $\\epsilon>0$. If $d_\\textup{H}(C,D)\\leq\\epsilon$, then $D\\subseteq\\textup{cl}(C+\\epsilon B_X)$, which implies by (\\ref{eq:ir+eps}) that $\\textup{bir}(D)\\leq\\textup{bir}(C)+\\epsilon$. The inequality $\\textup{bir}(C)\\leq\\textup{bir}(D)+\\epsilon$ follows similarly. Thus, $|\\textup{bir}(C)-\\textup{bir}(D)|\\leq\\epsilon$. Since $C,D$ and $\\epsilon$ are arbitrary, this lets us conclude that $\\textup{bir}$ is $1$-Lipschitz.\n\\end{proof}\n\n\\begin{theo}\nLet $X$ be a Banach space. The function $\\func{\\textup{blr}}{\\mathcal{C}(X)}{\\mathbb{R}}$ is $1$-Lipschitz. In particular, in case $X$ is separable, the family $\\mathcal{H}^+(X)$ of all Haar positive closed, convex and bounded subsets of $X$ is open. Equivalently, a convergent sequence ${(C_n)}_{n=1}^\\infty\\subset\\mathcal{C}(X)$ of Haar null sets has a Haar null limit.\n\\end{theo}\n\\begin{proof}\nWe have $\\textup{blr}=\\textup{bir}\\circ\\textup{w}^*\\textup{-cl}$ by Lemma \\ref{lemma:hpws}, Lemma \\ref{lemma:wscliso} and Lemma \\ref{lemma:ir}, hence $\\textup{blr}$ is a composition of $1$-Lipschitz maps and therefore it is $1$-Lipschitz. Since\n\\[ \\mathcal{H}^+(X)=\\textup{blr}^{-1}\\bigl((0,+\\infty)\\bigr), \\]\nit follows immediately that $\\mathcal{H}^+(X)$ is open.\n\\end{proof}\n\n\\section*{Acknowledgements}\nThe author would like to thank Professors Eva Kopeck\\'{a} and Christian Bargetz for the many helpful remarks, and Professor Mikhail Ostrovskii for pointing out a mistake in an earlier version of the preprint. The author's research is supported by the Austrian Science Fund (FWF): P 32523-N.\n\n\\printbibliography\n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe area of high-dimensional statistics deals with estimation in the\n``large \\ensuremath{p}, small \\ensuremath{n}'' setting, where $\\ensuremath{p}$ and $\\ensuremath{n}$\ncorrespond, respectively, to the dimensionality of the data and the\nsample size. Such high-dimensional problems arise in a variety of\napplications, among them remote sensing, computational biology and\nnatural language processing, where the model dimension may be\ncomparable or substantially larger than the sample size.  It is\nwell-known that such high-dimensional scaling can lead to dramatic\nbreakdowns in many classical procedures.  In the absence of additional\nmodel assumptions, it is frequently impossible to obtain consistent\nprocedures when $\\ensuremath{p} \\gg \\ensuremath{n}$. Accordingly, an active line of\nstatistical research is based on imposing various restrictions on the\nmodel----for instance, sparsity, manifold structure, or graphical\nmodel structure----and then studying the scaling behavior of different\nestimators as a function of sample size $\\ensuremath{n}$, ambient dimension\n$\\ensuremath{p}$ and additional parameters related to these structural\nassumptions.\n\nIn this paper, we study the following problem: given $\\ensuremath{n}$ i.i.d.\nobservations $\\{X^{(\\obsind)}\\}_{\\ensuremath{k}=1}^{\\ensuremath{n}}$ of a zero mean\nrandom vector $X \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p}}$, estimate both its covariance\nmatrix \\mbox{$\\ensuremath{\\ensuremath{\\Sigma}^*}$,} and its inverse covariance or\nconcentration matrix $\\ensuremath{\\Theta^*} \\ensuremath{:  =} \\inv{\\ensuremath{\\ensuremath{\\Sigma}^*}}$.  Perhaps\nthe most natural candidate for estimating $\\ensuremath{\\ensuremath{\\Sigma}^*}$ is the\nempirical sample covariance matrix, but this is known to behave poorly\nin high-dimensional settings. For instance, when $\\ensuremath{p}\/\\ensuremath{n}\n\\rightarrow c > 0$, and the samples are drawn i.i.d. from a\nmultivariate Gaussian distribution, neither the eigenvalues nor the\neigenvectors of the sample covariance matrix are consistent estimators\nof the population\nversions~\\cite{Johnstone2001,JohnstoneLu2004}. Accordingly, many\nregularized estimators have been proposed to estimate the covariance\nor concentration matrix under various model assumptions.  One natural\nmodel assumption is that reflected in shrinkage estimators, such as in\nthe work of \\citet{LedoitWolf2003}, who proposed to shrink the sample\ncovariance to the identity matrix. An alternative model assumption,\nrelevant in particular for time series data, is that the covariance or\nconcentration matrix is banded, meaning that the entries decay based\non their distance from the diagonal. \\citet{Furrer2007} proposed to\nshrink the covariance entries based on this distance from the\ndiagonal. \\citet{Wu2003} and \\citet{Huang2006} estimate these banded\nconcentration matrices by using thresholding and $\\ell_1$-penalties\nrespectively, as applied to a Cholesky factor of the inverse\ncovariance matrix. \\citet{BickelLevina2008} prove the consistency of\nthese banded estimators so long as $\\frac{(\\log\\,\\ensuremath{p})^2}{\\ensuremath{n}}\n\\rightarrow 0$ and the model covariance matrix is banded as well, but\nas they note, these estimators depend on the presented order of the\nvariables.\n\nA related class of models are based on positing some kind of sparsity,\neither in the covariance matrix, or in the inverse covariance.\n\\citet{BickelLevina2007} study thresholding estimators of covariance\nmatrices, assuming that each row satisfies an $\\ell_q$-ball sparsity\nassumption.  In independent work, \\citet{Karoui2007} also studied\nthresholding estimators of the covariance, but based on an alternative\nnotion of sparsity, one which captures the number of closed paths of\nany length in the associated graph.  Other work has studied models in\nwhich the inverse covariance or concentration matrix has a sparse\nstructure.  As will be clarified in the next section, when the random\nvector is multivariate Gaussian, the set of non-zero entries in the\nconcentration matrix correspond to the set of edges in an associated\nGaussian Markov random field (GMRF).  In this setting, imposing\nsparsity on the concentration matrix can be interpreted as requiring\nthat the graph underlying the GMRF have relatively few edges.  A line\nof recent papers~\\citep{AspreBanG2008,FriedHasTib2007,YuanLin2007}\nhave proposed an estimator that minimizes the Gaussian negative\nlog-likelihood regularized by the $\\ell_1$ norm of the entries (or the\noff-diagonal entries) of the concentration matrix. The resulting\noptimization problem is a log-determinant program, which can be solved\nin polynomial time with interior point methods~\\citep{Boyd02}, or by\nfaster co-ordinate descent\nalgorithms~\\citep{AspreBanG2008,FriedHasTib2007}.  In recent work,\n\\citet{Rothman2007} have analyzed some aspects of high-dimensional\nbehavior of this estimator; assuming that the minimum and maximum\neigenvalues of $\\ensuremath{\\ensuremath{\\Sigma}^*}$ are bounded, they show that consistent\nestimates can be achieved in Frobenius and operator norm, in\nparticular at the rate ${\\mathcal{O}}(\\sqrt{\\frac{(\\ensuremath{s} + \\ensuremath{p}) \\log\n\\ensuremath{p}}{\\ensuremath{n}}})$.\n\n\nThe focus of this paper is the problem of estimating the concentration\nmatrix $\\Theta^*$ under sparsity conditions.  We do not impose\nspecific distributional assumptions on $X$ itself, but rather analyze the estimator in terms of \nthe tail behavior of the maximum deviation $\\max_{i,j}\n|\\estim{\\Sigma}^n_{ij} - \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}|$ of the sample and\npopulation covariance matrices.  To estimate $\\Theta^*$, we\nconsider minimization of an $\\ell_1$-penalized log-determinant Bregman\ndivergence, which is equivalent to the usual $\\ell_1$-penalized\nmaximum likelihood when $X$ is multivariate Gaussian.  We analyze the\nbehavior of this estimator under high-dimensional scaling, in which\nthe number of nodes $\\ensuremath{p}$ in the graph, and the maximum node degree\n$\\ensuremath{\\ensuremath{d}}$ are all allowed to grow as a function of the sample size\n$\\ensuremath{n}$.\n\nIn addition to the triple $(\\ensuremath{n}, \\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$, we also\nexplicitly keep track of certain other measures of model complexity,\nthat could potentially scale as well.  The first of these measures is\nthe $\\ell_\\infty$-operator norm of the covariance matrix\n$\\ensuremath{\\ensuremath{\\Sigma}^*}$, which we denote by $\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{:  =}\n\\matnorm{\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty}$.  The next quantity involves the\nHessian of the log-determinant objective function, $\\ensuremath{\\Gamma^*} \\ensuremath{:  =}\n(\\ensuremath{\\Theta^*})^{-1} \\otimes (\\ensuremath{\\Theta^*})^{-1}$.  When the distribution\nof $X$ is multivariate Gaussian, this Hessian has the more explicit\nrepresentation $\\ensuremath{\\Gamma^*}_{(j,k), (\\ell, m)} = \\operatorname{cov}\\{X_j\nX_k, \\; X_\\ell X_m \\}$, showing that it measures the covariances of\nthe random variables associated with each edge of the graph.  For this\nreason, the matrix $\\ensuremath{\\Gamma^*}$ can be viewed as an edge-based\ncounterpart to the usual node-based covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.\nUsing $\\ensuremath{S}$ to index the variable pairs $(i,j)$ associated with\nnon-zero entries in the inverse covariance.  our analysis involves the\nquantity $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} = \\matnorm{(\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}})^{-1}}{\\infty}$. Finally, we also impose a mutual\nincoherence or irrepresentability condition on the Hessian $\\ensuremath{\\Gamma^*}$;\nthis condition is similar to assumptions imposed on $\\ensuremath{\\ensuremath{\\Sigma}^*}$ in\nprevious work~\\cite{Tropp2006,Zhao06,MeinsBuhl2006,Wainwright2006_new} on\nthe Lasso.  We provide some examples where the Lasso\nirrepresentability condition holds, but our corresponding condition on\n$\\ensuremath{\\Gamma^*}$ fails; however, we do not know currently whether one\ncondition strictly dominates the other. \n\n\nOur first result establishes consistency of our estimator\n$\\estim{\\Theta}$ in the elementwise maximum-norm, providing a rate\nthat depends on the tail behavior of the entries in the random matrix\n$\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n} - \\ensuremath{\\ensuremath{\\Sigma}^*}$. For the special case of sub-Gaussian\nrandom vectors with concentration matrices having at most $d$\nnon-zeros per row, a corollary of our analysis is consistency in\nspectral norm at rate \\mbox{$\\matnorm{\\estim{\\Theta} - \\Theta^*}{2} =\n{\\mathcal{O}}(\\sqrt{(\\ensuremath{\\ensuremath{d}}^2 \\,\\log \\ensuremath{p})\/\\ensuremath{n}})$,} with high\nprobability, thereby strengthening previous\nresults~\\cite{Rothman2007}.  Under the milder restriction of each\nelement of $X$ having bounded $4m$-th moment, the rate in\nspectral norm is substantially slower---namely,\n\\mbox{$\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{2} = {\\mathcal{O}}(\\ensuremath{\\ensuremath{d}}\\,\n\\ensuremath{p}^{1\/2m}\/\\sqrt{\\ensuremath{n}})$}---highlighting that the familiar\nlogarithmic dependence on the model size $\\ensuremath{p}$ is linked to\nparticular tail behavior of the distribution of $X$.  Finally, we show\nthat under the same scalings as above, with probability converging to\none, the estimate $\\estim{\\Theta}$ correctly specifies the zero\npattern of the concentration matrix $\\Theta^*$.\n\n\nThe remainder of this paper is organized as follows.  In\nSection~\\ref{SecBackground}, we set up the problem and give some\nbackground.  Section~\\ref{SecResult} is devoted to statements of our\nmain results, as well as discussion of their consequences.\nSection~\\ref{SecProof} provides an outline of the proofs, with the\nmore technical details deferred to appendices.  In\nSection~\\ref{SecExperiments}, we report the results of some simulation\nstudies that illustrate our theoretical predictions.\n\n\n\\myparagraph{Notation} For the convenience of the reader, we summarize\nhere notation to be used throughout the paper. Given a vector $\\ensuremath{u}\n\\in \\ensuremath{{\\mathbb{R}}}^\\ensuremath{d}$ and parameter $a \\in [1, \\infty]$, we use\n$\\|\\ensuremath{u}\\|_a$ to denote the usual $\\ell_a$ norm.  Given a matrix\n$\\ensuremath{U} \\in \\ensuremath{{\\mathbb{R}}}^{p \\times p}$ and parameters $a,b \\in [1, \\infty]$,\nwe use $\\matnorm{\\ensuremath{U}}{a,b}$ to denote the induced matrix-operator\nnorm $\\max_{\\|y\\|_a = 1} \\|\\ensuremath{U} y\\|_b$; see \\citet{Horn1985} for\nbackground.  Three cases of particular importance in this paper are\nthe \\emph{spectral norm} $\\matnorm{\\ensuremath{U}}{2}$, corresponding to the\nmaximal singular value of $\\ensuremath{U}$; the\n\\emph{$\\ell_\\infty\/\\ell_\\infty$-operator norm}, given by\n\\begin{eqnarray}\n\\label{EqnLinfOp}\n\\matnorm{\\ensuremath{U}}{\\infty} & \\ensuremath{:  =} & \\max \\limits_{j=1, \\ldots, p}\n\\sum_{k=1}^p |\\ensuremath{U}_{jk}|,\n\\end{eqnarray}\nand the \\emph{$\\ell_1\/\\ell_1$-operator norm}, given by\n$\\matnorm{\\ensuremath{U}}{1} = \\matnorm{\\ensuremath{U}^T}{\\infty}$.  Finally, we use\n$\\|\\ensuremath{U}\\|_\\infty$ to denote the element-wise maximum $\\max_{i,j}\n|\\ensuremath{U}_{ij}|$; note that this is not a matrix norm, but rather a norm\non the vectorized form of the matrix. For any matrix $\\ensuremath{U} \\in\n\\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times \\ensuremath{p}}$, we use $\\ensuremath{\\operatorname{vec}}(\\ensuremath{U})$ or equivalently\n$\\widebar{\\ensuremath{U}} \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p}^2}$ to denote its \\emph{vectorized\nform}, obtained by stacking up the rows of $\\ensuremath{U}$.  We use\n$\\tracer{\\ensuremath{U}}{\\ensuremath{V}} \\ensuremath{:  =} \\sum_{i,j} \\ensuremath{U}_{ij} \\ensuremath{V}_{ij}$ to\ndenote the \\emph{trace inner product} on the space of symmetric\nmatrices. Note that this inner product induces the \\emph{Frobenius\nnorm} $\\matnorm{\\ensuremath{U}}{F} \\ensuremath{:  =} \\sqrt{\\sum_{i,j}\n\\ensuremath{U}_{ij}^2}$. Finally, for asymptotics, we use the following\nstandard notation: we write $f(n) = {\\mathcal{O}}(g(n))$ if $f(n) \\leq c\ng(n)$ for some constant $c < \\infty$, and $f(n) = \\Omega(g(n))$ if\n$f(n) \\geq c' g(n)$ for some constant $c' > 0$.  The notation\n\\mbox{$f(n) \\asymp g(n)$} means that \\mbox{$f(n) = {\\mathcal{O}}(g(n))$} and\n\\mbox{$f(n) = \\Omega(g(n))$.}\n\n\\section{Background and problem set-up}\n\\label{SecBackground}\n\nLet $X = (X_1, \\ldots, X_\\ensuremath{p})$ be a zero mean $\\ensuremath{p}$-dimensional\nrandom vector. The focus of this paper is the problem of estimating\nthe covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*} \\ensuremath{:  =} \\ensuremath{{\\mathbb{E}}}[ X X^T]$ and\nconcentration matrix $\\ensuremath{\\Theta^*} \\ensuremath{:  =} \\invn{\\ensuremath{\\ensuremath{\\Sigma}^*}}$ of the\nrandom vector $X$ given $\\ensuremath{n}$ i.i.d. observations\n$\\{X^{(\\ensuremath{k})}\\}_{\\ensuremath{k}=1}^{\\ensuremath{n}}$.  In this section, we\nprovide background, and set up this problem more precisely.  We begin\nwith background on Gaussian graphical models, which provide one\nmotivation for the estimation of concentration matrices.  We then\ndescribe an estimator based based on minimizing an $\\ell_1$\nregularized log-determinant divergence; when the data are drawn from a\nGaussian graphical model, this estimator corresponds to\n$\\ell_1$-regularized maximum likelihood.  We then discuss the\ndistributional assumptions that we make in this paper.\n\n\\subsection{Gaussian graphical models}\n\nOne motivation for this paper is the problem of Gaussian graphical\nmodel selection.  A graphical model or a Markov random field is a\nfamily of probability distributions for which the conditional\nindependence and factorization properties are captured by a graph. Let\n$X = (X_1, X_2, \\ldots, X_\\ensuremath{p})$ denote a zero-mean Gaussian random\nvector; its density can be parameterized by the inverse covariance or\n\\emph{concentration matrix} $\\ensuremath{\\Theta^*} = (\\ensuremath{\\ensuremath{\\Sigma}^*})^{-1} \\in\n\\Symconepl{\\ensuremath{p}}$, and can be written as\n\\begin{eqnarray}\n\\label{EqnDefnGaussMRF}\nf(x_1, \\ldots, x_\\ensuremath{p}; \\ensuremath{\\Theta^*}) & = & \\frac{1}{\\sqrt{(2 \\pi)^\\ensuremath{p}\n \\det((\\ensuremath{\\Theta^*})^{-1})}} \\; \\exp \\big\\{ -\\frac{1}{2} x^T \\ensuremath{\\Theta^*}\n x \\big \\}.\n\\end{eqnarray}\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\psfrag{#1#}{$1$} \\psfrag{#2#}{$2$} \\psfrag{#3#}{$3$}\n\\psfrag{#4#}{$4$} \\psfrag{#5#}{$5$}\n\\raisebox{.2in}{\\widgraph{0.3\\textwidth}{simple_gauss.eps}} & &\n\\widgraph{.25\\textwidth}{fig_simple_gauss_invcov.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{(a) Simple undirected graph.  A Gauss Markov random field has\na Gaussian variable $X_i$ associated with each vertex $i \\in \\ensuremath{V}$.\nThis graph has $\\ensuremath{p} = 5$ vertices, maximum degree $d = 3$ and $s=6$\nedges.  (b) Zero pattern of the inverse covariance $\\ensuremath{\\Theta^*}$\nassociated with the GMRF in (a).  The set $\\ensuremath{E}(\\ensuremath{\\Theta^*})$\ncorresponds to the off-diagonal non-zeros (white blocks); the diagonal\nis also non-zero (grey squares), but these entries do not correspond\nto edges.  The black squares correspond to non-edges, or zeros in\n$\\ensuremath{\\Theta^*}$.}\n\\label{FigMarkov}\n\\end{figure}\n\nWe can relate this Gaussian distribution of the random vector $X$ to a\ngraphical model as follows.  Suppose we are given an undirected graph\n$\\ensuremath{G} = (\\ensuremath{V}, \\ensuremath{E})$ with vertex set $\\ensuremath{V} = \\{1, 2, \\ldots,\n\\ensuremath{p} \\}$ and edge\\footnote{As a remark on notation, we would like to\ncontrast the notation for the edge-set $\\ensuremath{E}$ from the notation for\nan expectation of a random variable, $\\mathbb{E}(\\cdot)$.} set\n$\\ensuremath{E}$, so that each variable $X_i$ is associated with a\ncorresponding vertex $i \\in \\ensuremath{V}$. The Gaussian Markov random field\n(GMRF) associated with the graph $\\ensuremath{G}$ over the random vector $X$\nis then the family of Gaussian distributions with concentration\nmatrices $\\ensuremath{\\Theta^*}$ that respect the edge structure of the graph, in\nthe sense that $\\ensuremath{\\Theta^*}_{ij} = 0$ if $(i,j) \\notin\n\\ensuremath{E}$. Figure~\\ref{FigMarkov} illustrates this correspondence between\nthe graph structure (panel (a)), and the sparsity pattern of the\nconcentration matrix $\\ensuremath{\\Theta^*}$ (panel (b)).  The problem of\nestimating the entries of the concentration matrix $\\ensuremath{\\Theta^*}$\ncorresponds to estimating the Gaussian graphical model instance, while\nthe problem of estimating the off-diagonal zero-pattern of the\nconcentration matrix----that is, the set\n\\begin{eqnarray}\n\\label{EqnDefnEdgeSet}\n\\ensuremath{E}(\\ensuremath{\\Theta^*}) & \\ensuremath{:  =} & \\{i, j \\in \\ensuremath{V} \\mid \\, i \\neq j,\n\\ensuremath{\\Theta^*}_{ij} \\neq 0 \\}\n\\end{eqnarray}\ncorresponds to the problem of Gaussian graphical \\emph{model\nselection}. \n\nWith a slight abuse of notation, we define the \\emph{sparsity index}\n$\\ensuremath{s} \\ensuremath{:  =} |\\ensuremath{E}(\\ensuremath{\\Theta^*})|$ as the total number of non-zero\nelements in off-diagonal positions of $\\ensuremath{\\Theta^*}$; equivalently, this\ncorresponds to twice the number of edges in the case of a Gaussian\ngraphical model. We also define the \\emph{maximum degree or row\ncardinality}\n\\begin{eqnarray}\n\\label{EqnDefnDegmax}\n\\ensuremath{\\ensuremath{d}} & \\ensuremath{:  =} & \\max_{i = 1, \\ldots, \\ensuremath{p} } \\biggr|\\big \\{ j \\in\n\\ensuremath{V} \\, \\mid \\, \\ensuremath{\\Theta^*}_{ij} \\neq 0 \\big\\} \\biggr|,\n\\end{eqnarray}\ncorresponding to the maximum number of non-zeros in any row of\n$\\ensuremath{\\Theta^*}$; this corresponds to the maximum degree in the graph of\nthe underlying Gaussian graphical model.  Note that we have included\nthe diagonal entry $\\ensuremath{\\Theta^*}_{ii}$ in the degree count,\ncorresponding to a self-loop at each vertex.\n\nIt is convenient throughout the paper to use graphical terminology,\nsuch as degrees and edges, even though the distributional assumptions\nthat we impose, as described in Section~\\ref{SecDistAssum}, are milder\nand hence apply even to distributions that are not Gaussian MRFs.\n\n\n\n\\subsection{$\\ell_1$-penalized log-determinant divergence}\n\nAn important set in this paper is the cone\n\\begin{eqnarray}\n\\Symconepl{\\ensuremath{p}} & \\ensuremath{:  =} & \\big \\{ A \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times \\ensuremath{p}}\n\\mid A = A^T, \\; A \\succeq 0 \\big \\},\n\\end{eqnarray}\nformed by all symmetric positive semi-definite matrices in $\\ensuremath{p}$\ndimensions. We assume that the covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ and\nconcentration matrix $\\ensuremath{\\Theta^*}$ of the random vector $X$ are\nstrictly positive definite, and so lie in the interior of this cone\n$\\Symconepl{\\ensuremath{p}}$. \n\nThe focus of this paper is a particular type of $M$-estimator for the\nconcentration matrix $\\ensuremath{\\Theta^*}$, based on minimizing a Bregman\ndivergence between symmetric matrices.  A function is of Bregman type\nif it is strictly convex, continuously differentiable and has bounded\nlevel sets~\\cite{Bregman67a,Censor}.  Any such function induces a\n\\emph{Bregman divergence} of the form $\\Breg{A}{B} = g(A) - g(B) -\n\\trs{\\nabla g(B)}{A-B}$.  From the strict convexity of $g$, it follows\nthat $\\Breg{A}{B} \\geq 0$ for all $A$ and $B$, with equality if and\nonly if $A = B$.\n\n\nAs a candidate Bregman function, consider the log-determinant barrier\nfunction, defined for any matrix $A \\in \\Symconepl{\\ensuremath{p}}$ by\n\\begin{eqnarray}\n\\label{EqnDefnLogDet}\ng(A) & \\ensuremath{:  =} &  \\begin{cases} - \\log \\det(A)  & \\mbox{if $A \\succ 0$} \\\\\n                              + \\infty & \\mbox{otherwise.}\n                \\end{cases}\n\\end{eqnarray}\nAs is standard in convex analysis, we view this function as taking\nvalues in the extended reals $\\ensuremath{{\\mathbb{R}}}_* = \\ensuremath{{\\mathbb{R}}} \\cup \\{+\\infty \\}$.\nWith this definition, the function $g$ is strictly convex, and its\ndomain is the set of strictly positive definite matrices.  Moreover,\nit is continuously differentiable over its domain, with $\\nabla g(A) =\n- A^{-1}$; see Boyd and Vandenberghe~\\cite{Boyd02} for further\ndiscussion.  The Bregman divergence corresponding to this\nlog-determinant Bregman function $g$ is given by\n\\begin{eqnarray}\n\\label{EqnDefnBreg}\n\\Breg{A}{B} & \\ensuremath{:  =} & - \\log \\det A + \\log \\det B +\n\\tracer{B^{-1}}{A-B},\n\\end{eqnarray}\nvalid for any $A, B \\in \\Symconepl{\\ensuremath{p}}$ that are strictly positive\ndefinite.  This divergence suggests a natural way to estimate\nconcentration matrices---namely, by minimizing the divergence\n$\\Breg{\\ensuremath{\\Theta^*}}{\\Theta}$---or equivalently, by minimizing the\nfunction\n\\begin{equation}\n\\label{EqnPop}\n\\min_{\\Theta \\succ 0 } \\big \\{ \\tracer{\\Theta}{\\ensuremath{\\ensuremath{\\Sigma}^*}} - \\log\n\\det \\Theta \\big \\},\n\\end{equation}\nwhere we have discarded terms independent of $\\Theta$, and used the\nfact that the inverse of the concentration matrix is the covariance matrix\n(i.e., $(\\ensuremath{\\Theta^*})^{-1} = \\ensuremath{\\ensuremath{\\Sigma}^*}= \\ensuremath{{\\mathbb{E}}}[X X^T]$).  Of course,\nthe convex program~\\eqref{EqnPop} cannot be solved without knowledge\nof the true covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$, but one can take the\nstandard approach of replacing $\\ensuremath{\\ensuremath{\\Sigma}^*}$ with an empirical\nversion, with the possible addition of a regularization term.\n\n\nIn this paper, we analyze a particular instantiation of this strategy.\nGiven $\\ensuremath{n}$ samples, we define the \\emph{sample covariance matrix}\n\\begin{eqnarray}\n\\label{EqnDefnSamCov}\n\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n} & \\ensuremath{:  =} & \\frac{1}{\\ensuremath{n}} \\sum_{\\ensuremath{k}=1}^\\ensuremath{n}\n\\sp{X}{\\ensuremath{k}} (\\sp{X}{\\ensuremath{k}})^T.\n\\end{eqnarray}\nTo lighten notation, we occasionally drop the superscript $\\ensuremath{n}$,\nand simply write $\\ensuremath{\\widehat{\\Sigma}}$ for the sample covariance.  We also define\nthe \\emph{off-diagonal $\\ell_1$ regularizer}\n\\begin{eqnarray}\n\\ellreg{\\Theta} & \\ensuremath{:  =} & \\sum_{i \\neq j} |\\Theta_{ij}|,\n\\end{eqnarray}\nwhere the sum ranges over all $i, j = 1, \\ldots, \\ensuremath{p}$ with $i \\neq\nj$.  Given some regularization constant $\\ensuremath{\\lambda_\\ensuremath{n}} > 0$, we consider\nestimating $\\ensuremath{\\Theta^*}$ by solving the following\n\\emph{$\\ell_1$-regularized log-determinant program}:\n\\begin{eqnarray}\n\\label{EqnGaussMLE}\n\\ensuremath{\\widehat{\\Theta}} & \\ensuremath{:  =} & \\arg\\min_{\\Theta \\succ 0} \\big \\{\n\\tracer{\\Theta}{\\ensuremath{\\widehat{\\Sigma}}^\\ensuremath{n}} - \\log \\det(\\Theta) + \\ensuremath{\\lambda_\\ensuremath{n}}\n\\ellreg{\\Theta} \\big \\}.\n\\end{eqnarray}\nAs shown in Appendix~\\ref{AppLemMLECharac}, for any $\\ensuremath{\\lambda_\\ensuremath{n}} > 0$ and\nsample covariance matrix $\\ensuremath{\\widehat{\\Sigma}}^\\ensuremath{n}$ with strictly positive\ndiagonal, this convex optimization problem has a unique optimum, so\nthere is no ambiguity in equation~\\eqref{EqnGaussMLE}.  When the data\nis actually drawn from a multivariate Gaussian distribution, then the\nproblem~\\eqref{EqnGaussMLE} is simply $\\ell_1$-regularized maximum\nlikelihood.\n\n\n\\defc{c}\n\\defc{c}\n\n\\subsection{Tail conditions}\n\\label{SecDistAssum}\n\nIn this section, we describe the tail conditions that underlie our\nanalysis.  Since the estimator ~\\eqref{EqnGaussMLE} is based on using\nthe sample covariance $\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n}$ as a surrogate for the\n(unknown) covariance $\\ensuremath{\\ensuremath{\\Sigma}^*}$, any type of consistency requires\nbounds on the difference $\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n} - \\ensuremath{\\ensuremath{\\Sigma}^*}$.  In\nparticular, we define the following tail condition:\n\\begin{defns}[Tail conditions]\n\\label{DefnTail}\nThe random vector $X$ satisfies tail condition $\\ensuremath{\\mathcal{T}}(f,\nv_*)$ if there exists a constant $v_* \\in (0, \\infty]$ and\na function $f: \\mathbb{N} \\times (0,\\infty) \\rightarrow (0,\n\\infty)$ such that for any $(i,j) \\in \\ensuremath{V} \\times \\ensuremath{V}$:\n\\begin{eqnarray}\n\\label{EqnSamTail}\n\\ensuremath{\\mathbb{P}}[|\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}_{ij} - \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}| \\geq \\delta] & \\leq\n& 1\/f(\\ensuremath{n},\\delta) \\qquad \\mbox{for all $\\delta \\in (0,\n1\/v_*]$.}\n\\end{eqnarray}\nWe adopt the convention $1\/0 \\ensuremath{:  =} + \\infty$, so that the value\n$v_* = 0$ indicates the inequality holds for any $\\delta \\in\n(0,\\infty)$.\n\\end{defns}\n\n\n\\newcommand{\\ensuremath{a}}{\\ensuremath{a}}\n\nTwo important examples of the tail function $f$ are the\nfollowing:\n\\begin{enumerate} \n\\item[(a)] an \\emph{exponential-type tail function}, meaning that\n$f(\\ensuremath{n},\\delta) = \\exp(c \\, \\ensuremath{n}\\,\n\\delta^{\\ensuremath{a}})$, for some scalar $c > 0$, and exponent\n$\\ensuremath{a} > 0$; and\n\\item[(b)] a \\emph{polynomial-type tail function}, meaning that\n$f(\\ensuremath{n},\\delta) = c \\, \\ensuremath{n}^{m} \\,\n\\delta^{2m}$, for some positive integer $m \\in\n\\mathbb{N}$ and scalar $c > 0$.\n\\end{enumerate}\nAs might be expected, if $X$ is multivariate Gaussian, then the\ndeviations of sample covariance matrix have an exponential-type tail\nfunction with $\\ensuremath{a} = 2$.  A bit more generally, in the following\nsubsections, we provide broader classes of distributions whose sample\ncovariance entries satisfy exponential and a polynomial tail bounds\n(see Lemmata~\\ref{LEM_SAM_COV_BOUND_SUBG}\nand~\\ref{LEM_SAM_COV_BOUND_MOMENT} respectively).\n\nGiven a larger number of samples $\\ensuremath{n}$, we expect the tail\nprobability bound $1\/f(\\ensuremath{n},\\delta)$ to be smaller, or\nequivalently, for the tail function $f(\\ensuremath{n},\\delta)$ to\nlarger.  Accordingly, we require that $f$ is monotonically\nincreasing in $\\ensuremath{n}$, so that for each fixed $\\delta >0$, we can\ndefine the inverse function\n\\begin{eqnarray}\n\\label{EqnSamTailN}\n\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(r; \\delta) & \\ensuremath{:  =} & \\arg \\max \\big \\{ n \\; \\mid \\;\nf(\\ensuremath{n}, \\delta) \\leq r \\big \\}.\n\\end{eqnarray}\nSimilarly, we expect that $f$ is monotonically increasing in\n$\\delta$, so that for each fixed $\\ensuremath{n}$, we can define the\ninverse in the second argument\n\\begin{eqnarray}\n\\label{EqnSamTailT}\n\\ensuremath{\\widebar{\\delta}_f}(r; \\ensuremath{n}) & \\ensuremath{:  =} & \\arg \\max \\big \\{ \\delta \\; \\mid\n\\; f(\\ensuremath{n}, \\delta) \\leq r \\big \\}.\n\\end{eqnarray}\nFor future reference, we note a simple consequence of the monotonicity\nof the tail function $f$---namely\n\\begin{eqnarray}\n\\label{EqnMonot}\n\\ensuremath{n} > \\ensuremath{{\\widebar{\\ensuremath{n}}_f}}( \\delta, r) \\quad \\mbox{for some $\\delta > 0$} &\n\\Longrightarrow & \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},r) \\leq \\delta.\n\\end{eqnarray}\nThe inverse functions $\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ and $\\ensuremath{\\widebar{\\delta}_f}$ play an\nimportant role in describing the behavior of our estimator.  We\nprovide concrete examples in the following two subsections.\n\n\n\\subsubsection{Sub-Gaussian distributions}\n\\label{SecSamCovBoundSubG}\nIn this subsection, we study the case of i.i.d. observations of\nsub-Gaussian random variables.\n\\begin{defns}\nA zero-mean random variable $Z$ is \\emph{sub-Gaussian} if there exists\na constant $\\ensuremath{\\sigma}\\in (0, \\infty)$ such that\n\\begin{eqnarray}\n\\label{EqnDefnSubgauss}\n\\ensuremath{\\mathbb{E}}[\\exp(t Z)] & \\leq & \\exp(\\ensuremath{\\sigma}^2 \\, t^2\/2) \\qquad \\mbox{for all\n$t \\in \\ensuremath{{\\mathbb{R}}}$.}\n\\end{eqnarray}\n\\end{defns}\nBy the Chernoff bound, this upper bound~\\eqref{EqnDefnSubgauss} on the\nmoment-generating function implies a two-sided tail bound of the form\n\\begin{eqnarray}\n\\label{EqnSubgaussChern}\n\\ensuremath{\\mathbb{P}}[|Z| > z] & \\leq & 2 \\exp \\big(- \\frac{z^2}{2 \\ensuremath{\\sigma}^2}\\big).\n\\end{eqnarray}\nNaturally, any zero-mean Gaussian variable with variance $\\sigma^2$\nsatisfies the bounds~\\eqref{EqnDefnSubgauss}\nand~\\eqref{EqnSubgaussChern}.  In addition to the Gaussian case, the\nclass of sub-Gaussian variates includes any bounded random variable\n(e.g., Bernoulli, multinomial, uniform), any random variable with\nstrictly log-concave density~\\cite{BulKoz,Ledoux01}, and any finite\nmixture of sub-Gaussian variables.\n\nThe following lemma, proved in\nAppendix~\\ref{APP_LEM_SAM_COV_BOUND_SUBG}, shows that the entries of\nthe sample covariance based on i.i.d. samples of sub-Gaussian random\nvector satisfy an exponential-type tail bound with exponent $\\ensuremath{a}\n= 2$.  The argument is along the lines of a result due to Bickel and\nLevina~\\cite{BickelLevina2007}, but with more explicit control of the\nconstants in the error exponent:\n\\begin{lems}\n\\label{LEM_SAM_COV_BOUND_SUBG}\nConsider a zero-mean random vector $(X_1, \\ldots, X_\\ensuremath{p})$ with\ncovariance $\\ensuremath{\\ensuremath{\\Sigma}^*}$ such that each $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$\nis sub-Gaussian with parameter $\\ensuremath{\\sigma}$.  Given $\\ensuremath{n}$\ni.i.d. samples, the associated sample covariance $\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}$\nsatisfies the tail bound\n\\begin{eqnarray*}\n\\ensuremath{\\mathbb{P}} \\big[ |\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}_{ij }- \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}| > \\delta\n \\big] & \\leq & 4 \\exp \\big \\{- \\frac{\\ensuremath{n} \\delta^2}{ 128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2 }\n \\big \\},\n\\end{eqnarray*}\nfor all $\\delta \\in \\big(0, \\max_{i}(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\,8(1 + 4\n\\ensuremath{\\sigma}^2)\\big)$.\n\n\\end{lems}\nThus, the sample covariance entries the tail condition $\\ensuremath{\\mathcal{T}}(f,\nv_*)$ with $v_* = \\big[\\max_{i}(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\,8(1 + 4\n\\ensuremath{\\sigma}^2)\\big]^{-1}$, and an exponential-type tail function with\n$\\ensuremath{a} = 2$---namely\n\\begin{eqnarray}\n\\label{EqnSubgaussF}\n\\qquad f(\\ensuremath{n}, \\delta) = \\frac{1}{4} \\exp( c_{*}\n\\ensuremath{n} \\delta^2), & \\mbox{with}  & c_{*} = \\big[ 128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2 \\big]^{-1}\n\\end{eqnarray}\nA little calculation shows that the associated inverse functions take\nthe form\n\\begin{equation}\n\\label{EqnExpInverse}\n\\ensuremath{\\widebar{\\delta}_f}(r; \\ensuremath{n} ) \\, = \\, \\sqrt{\\frac{\\log(4 \\,r)}{c_{*} \\,\n\\ensuremath{n}}}, \\quad \\mbox{and} \\quad \\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(r; \\delta) \\, = \\,\n\\frac{\\log(4 \\, r)}{c_{*} \\delta^2}.\n\\end{equation}\n\n\n\n\\subsubsection{Tail bounds with moment bounds}\n\\label{SecSamCovBoundMoment}\n\nIn the following lemma, proved in\nAppendix~\\ref{APP_LEM_SAM_COV_BOUND_MOMENT}, we show that given\ni.i.d. observations from random variables with bounded moments, the\nsample covariance entries satisfy a polynomial-type tail bound. See\nthe papers~\\cite{Zhao06,Karoui2007} for related results on tail bounds\nfor variables with bounded moments.\n\\begin{lems}\n\\label{LEM_SAM_COV_BOUND_MOMENT}\nSuppose there exists a positive integer $m$ and scalar $K_{\\momentpow} \\in\n\\ensuremath{{\\mathbb{R}}}$ such that for $i = 1,\\hdots,\\ensuremath{p}$,\n\\begin{eqnarray}\n\\label{EqnBoundedMoments}\n\\ensuremath{{\\mathbb{E}}} \\biggr[ \\big(\\frac{X_i}{\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}} \\big)^{4 m}\n\\biggr] & \\leq &   K_{\\momentpow}.\n\\end{eqnarray}\nFor i.i.d. samples $\\{\\Xsam{\\ensuremath{k}}_i \\}_{\\ensuremath{k}=1}^\\ensuremath{n}$, the\nsample covariance matrix $\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}$ satisfies the bound\n\\begin{eqnarray}\n\\ensuremath{\\mathbb{P}} \\big [ \\Big| \\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}_{ij} - \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} \\Big)\\Big|\n  > \\delta \\big] & \\leq & \\frac{\\big\\{m^{2m+1}\n  2^{2m} (\\max_i \\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}) ^{2m}\\, (K_{\\momentpow}\n  + 1 ) \\big\\}}{\\ensuremath{n}^{m}\\, \\delta^{2m}}.\n\\end{eqnarray}\n\\end{lems}\nThus, in this case, the sample covariance satisfies the tail condition\n$\\ensuremath{\\mathcal{T}}(f, v_*)$ with $v_* = 0$, so that the bound\nholds for all $\\delta \\in (0,\\infty)$, and with the polynomial-type\ntail function\n\\begin{equation}\n\\label{EqnPolyF}\nf(\\ensuremath{n},\\delta) = c_{*} \\ensuremath{n}^{m}\n\\delta^{2m} \\quad \\mbox{where $c_{*} =\n1\/\\big\\{m^{2m+1} 2^{2m} (\\max_i\n\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}) ^{2m}\\, (K_{\\momentpow} + 1 ) \\big\\}$.}\n\\end{equation}\nFinally, a little calculation shows that in this case, the inverse\ntail functions take the form\n\\begin{equation}\n\\label{EqnPolyInverse}\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},r) \\, = \\,\n\\frac{(r\/c_{*})^{1\/2m}}{\\sqrt{\\ensuremath{n}}}, \\quad\n\\mbox{and} \\quad \\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,r) \\, = \\,\n\\frac{(r\/c_{*})^{1\/m}}{\\delta^{2}}.\n\\end{equation}\n\n\n\n\\section{Main results and some consequences}\n\\label{SecResult}\n\nIn this section, we state our main results, and discuss some of their\nconsequences.  We begin in Section~\\ref{SecAssumptions} by stating\nsome conditions on the true concentration matrix $\\ensuremath{\\Theta^*}$ required in\nour analysis, including a particular type of incoherence or\nirrepresentability condition.  In Section~\\ref{SecEllinf}, we state\nour first main result---namely, Theorem~\\ref{ThmMain} on consistency\nof the estimator $\\ensuremath{\\widehat{\\Theta}}$, and the rate of decay of its error in\nelementwise $\\ell_\\infty$ norm.  Section~\\ref{SecModelCons} is devoted\nto Theorem~\\ref{ThmModel} on the model selection consistency of the\nestimator.  Section~\\ref{SecInco} is devoted the relation between the\nlog-determinant estimator and the ordinary Lasso (neighborhood-based\napproach) as methods for graphical model selection; in addition, we\nillustrate our irrepresentability assumption for some simple graphs.\nFinally, in Section~\\ref{SecFrob}, we state and prove some corollaries\nof Theorem~\\ref{ThmMain}, regarding rates in Frobenius and operator\nnorms.\n\n\\subsection{Conditions on covariance and Hessian}\n\\label{SecAssumptions}\n\nOur results involve some quantities involving the Hessian of the\nlog-determinant barrier~\\eqref{EqnDefnLogDet}, evaluated at the true\nconcentration matrix $\\ensuremath{\\Theta^*}$.  Using standard results on matrix\nderivatives~\\citep{Boyd02}, it can be shown that this Hessian takes\nthe form\n\\begin{eqnarray}\n\\label{EqnDefnHess}\n\\ensuremath{\\Gamma^*} & \\ensuremath{:  =} & \\nabla^2_{\\Theta} g(\\Theta) \\Big |_{\\Theta =\n  \\ensuremath{\\Theta^*}} \\; = \\; \\invn{\\ensuremath{\\Theta^*}} \\otimes \\invn{\\ensuremath{\\Theta^*}},\n\\end{eqnarray}\nwhere $\\otimes$ denotes the Kronecker matrix product.  By definition,\n$\\ensuremath{\\Gamma^*}$ is a $\\ensuremath{p}^{2} \\times \\ensuremath{p}^{2}$ matrix indexed by vertex\npairs, so that entry $\\ensuremath{\\Gamma^*}_{(j,k), (\\ell, m)}$ corresponds to the\nsecond partial derivative $ \\frac{\\partial^2 g}{\\partial \\Theta_{jk}\n\\partial \\Theta_{\\ell m}}$, evaluated at $\\Theta = \\ensuremath{\\Theta^*}$.  When\n$X$ has multivariate Gaussian distribution, then $\\ensuremath{\\Gamma^*}$ is the\nFisher information of the model, and by standard results on cumulant\nfunctions in exponential families~\\cite{Brown86}, we have the more\nspecific expression $\\ensuremath{\\Gamma^*}_{(j,k), (\\ell, m)} =\n\\operatorname{cov}\\{X_j X_k, \\; X_\\ell X_m \\}$.  For this reason,\n$\\ensuremath{\\Gamma^*}$ can be viewed as an edge-based counterpart to the usual\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.\n\nWe define the set of non-zero off-diagonal entries in the model\nconcentration matrix $\\ensuremath{\\Theta^*}$:\n\\begin{eqnarray}\n\\ensuremath{E}(\\ensuremath{\\Theta^*}) & \\ensuremath{:  =} & \\{ (i,j) \\in \\ensuremath{V} \\times \\ensuremath{V} \\,\n\t\\mid \\, i \\neq j, \\ensuremath{\\Theta^*}_{ij} \\neq 0 \\},\n\\end{eqnarray}\nand let $\\ensuremath{S}(\\ensuremath{\\Theta^*}) = \\{ \\ensuremath{E}(\\ensuremath{\\Theta^*}) \\cup \\{(1,1),\n\\ldots, (\\ensuremath{p}, \\ensuremath{p}) \\}$ be the augmented set including the\ndiagonal.  We let $\\ensuremath{\\EsetPlus^c}(\\ensuremath{\\Theta^*})$ denote the complement of\n$\\ensuremath{S}(\\ensuremath{\\Theta^*})$ in the set $\\{1, \\ldots, \\ensuremath{p} \\} \\times \\{1,\n\\ldots, \\ensuremath{p}\\}$, corresponding to all pairs $(\\ell, m)$ for which\n$\\ensuremath{\\Theta^*}_{\\ell m} = 0$.  When it is clear from context, we shorten\nour notation for these sets to $\\ensuremath{S}$ and $\\ensuremath{\\EsetPlus^c}$,\nrespectively.  Finally, for any two subsets $T$ and $T'$ of $\\ensuremath{V}\n\\times \\ensuremath{V}$, we use $\\ensuremath{\\Gamma^*}_{T T'}$ to denote the $|T| \\times\n|T'|$ matrix with rows and columns of $\\ensuremath{\\Gamma^*}$ indexed by $T$ and\n$T'$ respectively.\n\n\nOur main results involve the $\\ell_\\infty\/\\ell_\\infty$ norm applied to\nthe covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$, and to the inverse of a sub-block\nof the Hessian $\\ensuremath{\\Gamma^*}$.  In particular, we define\n\\begin{eqnarray}\n\\label{EqnCovConst}\n\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} & \\ensuremath{:  =} & \\matnorm{\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} \\; = \\; \\Big(\n \\max_{i=1, \\ldots ,\\ensuremath{p}} \\sum_{j=1}^\\ensuremath{p} |\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}| \\Big),\n\\end{eqnarray}\ncorresponding to the $\\ell_\\infty$-operator norm of the true\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$, and\n\\begin{eqnarray}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} & \\ensuremath{:  =} & \\matnorm{(\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}})^{-1}}{\\infty} \\; = \\; \\matnorm{([ {\\ensuremath{\\Theta^*}}^{-1}\n\\otimes {\\ensuremath{\\Theta^*}}^{-1}]_{\\ensuremath{S} \\ensuremath{S}})^{-1}}{\\infty}.\n\\end{eqnarray}\nOur analysis keeps explicit track of these quantities, so that they\ncan scale in a non-trivial manner with the problem dimension $\\ensuremath{p}$.\n\n\nWe assume the Hessian satisfies the following type of \\emph{mutual\nincoherence or irrepresentable condition}:\n\\begin{asss}\n\\label{AssInco}\nThere exists some $\\mutinco \\in (0,1]$ such that\n\\begin{eqnarray}\n\\label{EqnInco}\n\\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c} \\ensuremath{S}} (\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}})^{-1}}{\\infty} & \\leq & (1 - \\mutinco).\n\\end{eqnarray}\n\\end{asss}\n\nThe underlying intuition is that this assumption imposes control on\nthe influence that the non-edge terms, indexed by $\\ensuremath{\\EsetPlus^c}$, can\nhave on the edge-based terms, indexed by $\\ensuremath{S}$.  It is worth\nnoting that a similar condition for the Lasso, with the covariance\nmatrix $\\Sigma^*$ taking the place of the matrix $\\ensuremath{\\Gamma^*}$ above, is\nnecessary and sufficient for support recovery using the ordinary\nLasso~\\cite{MeinsBuhl2006,Tropp2006,Wainwright2006_new,Zhao06}.  See\nSection~\\ref{SecInco} for illustration of the form taken by\nAssumption~\\ref{AssInco} for specific graphical models.\n\nA remark on notation: although our analysis allows the quantities\n$\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ as well as the model size $\\ensuremath{p}$ and maximum\nnode-degree $\\ensuremath{\\ensuremath{d}}$ to grow with the sample size $\\ensuremath{n}$, we\nsuppress this dependence on $\\ensuremath{n}$ in their notation.\n\n\n\\subsection{Rates in elementwise $\\ell_\\infty$-norm}\n\\label{SecEllinf}\n\nWe begin with a result that provides sufficient conditions on the\nsample size $\\ensuremath{n}$ for bounds in the elementwise\n$\\ell_\\infty$-norm.  This result is stated in terms of the tail\nfunction $f$, and its inverses $\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ and $\\ensuremath{\\widebar{\\delta}_f}$ (equations~\\eqref{EqnSamTailN} and~\\eqref{EqnSamTailT}), and so covers a general range of possible\ntail behaviors.  So as to make it more concrete, we follow the general\nstatement with corollaries for the special cases of exponential-type\nand polynomial-type tail functions, corresponding to sub-Gaussian and\nmoment-bounded variables respectively.  \n\nIn the theorem statement, the choice of regularization constant\n$\\ensuremath{\\lambda_\\ensuremath{n}}$ is specified in terms of a user-defined parameter $\\tau >\n2$.  Larger choices of $\\tau$ yield faster rates of convergence in\nthe probability with which the claims hold, but also lead to more\nstringent requirements on the sample size.\n\\begin{theos}\n\\label{ThmMain}\nConsider a distribution satisfying the incoherence\nassumption~\\eqref{EqnInco} with parameter \\mbox{$\\mutinco \\in (0,1]$,}\nand the tail condition~\\eqref{EqnSamTail} with parameters\n$\\ensuremath{\\mathcal{T}}(f, v_*)$.  Let $\\ensuremath{\\widehat{\\Theta}}$ be the unique optimum of\nthe log-determinant program~\\eqref{EqnGaussMLE} with regularization\nparameter \\mbox{$\\ensuremath{\\lambda_\\ensuremath{n}} = (8\/\\mutinco) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$} for some $\\tau > 2$.  Then,\nif the sample size is lower bounded as\n\\begin{eqnarray}\n\\label{EqnSampleBound}\n\\ensuremath{n} & > & \\ensuremath{{\\widebar{\\ensuremath{n}}_f}} \\Biggr( 1 \\Big\/\\max \\Big \\{ v_*,\\; 6\n \\big(1 + 8\\mutinco^{-1} \\big) \\: \\ensuremath{\\ensuremath{d}}\\, \\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}\n \\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\Big \\} , \\; \\;\\ensuremath{p}^{\\tau}\n \\Biggr),\n\\end{eqnarray}\nthen with probability greater than $1-1\/\\ensuremath{p}^{\\tau - 2}\n\\rightarrow 1$, we have:\n\\begin{enumerate}\n\\item[(a)] The estimate $\\ensuremath{\\widehat{\\Theta}}$ satisfies the elementwise\n$\\ell_\\infty$-bound:\n\\begin{eqnarray}\n\\label{EqnEllinfBound}\n\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty & \\leq & \\big\\{2 \\big(1 + 8 \\mutinco^{-1}\\big)\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\big\\}\\;\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray}\n\\item[(b)] It specifies an edge set $\\ensuremath{E}(\\ensuremath{\\widehat{\\Theta}})$ that is a\nsubset of the true edge set $\\ensuremath{E}(\\ensuremath{\\Theta^*})$, and includes all\nedges $(i,j)$ with $|\\ensuremath{\\Theta^*}_{ij}| >  \\big\\{2 \\big(1 + 8 \\mutinco^{-1}\\big)\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\big\\}\\; \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$.\n\\end{enumerate}\n\\end{theos}\n\nIf we assume that the various quantities $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}},\n\\mutinco$ remain constant as a function of $(\\ensuremath{n}, \\ensuremath{p},\n\\ensuremath{\\ensuremath{d}})$, we have the elementwise $\\ell_\\infty$ bound \\mbox{$\\|\n\\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty =\n{\\mathcal{O}}(\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}))$}, so that the inverse\ntail function $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$ (see\nequation~\\eqref{EqnSamTailT}) specifies rate of convergence in the\nelement-wise $\\ell_\\infty$-norm.  In the following section, we derive\nthe consequences of this $\\ell_\\infty$-bound for two specific tail\nfunctions, namely those of exponential-type with $\\ensuremath{a} = 2$, and\npolynomial-type tails (see Section~\\ref{SecDistAssum}).  Turning to\nthe other factors involved in the theorem statement, the quantities\n$\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$ and $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ measure the sizes of the entries in the\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ and inverse Hessian $(\\ensuremath{\\Gamma^*})^{-1}$\nrespectively.  Finally, the factor $(1 + \\frac{8}{\\mutinco})$ depends\non the irrepresentability assumption~\\ref{AssInco}, growing in\nparticular as the incoherence parameter $\\mutinco$ approaches $0$.\n\n\\subsubsection{Exponential-type tails}\n\nWe now discuss the consequences of Theorem~\\ref{ThmMain} for\ndistributions in which the sample covariance satisfies an\nexponential-type tail bound with exponent $\\ensuremath{a} = 2$.  In\nparticular, recall from Lemma~\\ref{LEM_SAM_COV_BOUND_SUBG} that\nsuch a tail bound holds when the variables are sub-Gaussian.\n\\defc_1{c_1}\n\\defc_2{c_2}\n\n\n\\begin{cors} \n\\label{CorEllinfSubg}\nUnder the same conditions as Theorem~\\ref{ThmMain}, suppose moreover\nthat the variables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are sub-Gaussian with\nparameter $\\ensuremath{\\sigma}$, and the samples are drawn independently.  Then if\nthe sample size $\\ensuremath{n}$ satisfies the bound\n\\begin{eqnarray}\n\\ensuremath{n} & > & \\ensuremath{C_1} \\; \\ensuremath{\\ensuremath{d}}^2 \\, (1 + \\frac{8}{\\mutinco})^2\n\\;\\big (\\tau \\log \\ensuremath{p} + \\log 4 \\big)\n\\end{eqnarray}\nwhere $\\ensuremath{C_1} \\ensuremath{:  =} \\big\\{48\\sqrt{2} \\,(1 + 4 \\csubg^2) \\,\\max_{i} (\\CovMatStar_{ii})  \\, \\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\big\\}^{2}$, then with\nprobability greater than $1-1\/\\ensuremath{p}^{\\tau -2}$, the estimate\n$\\ensuremath{\\widehat{\\Theta}}$ satisfies the bound,\n\\begin{eqnarray*}\n\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty & \\leq & \n\\big\\{16\\sqrt{2} \\,(1 + 4 \\csubg^2) \\,\\max_{i} (\\CovMatStar_{ii})  \\, (1 + 8\\mutinco^{-1})\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\big\\}\\; \\sqrt{\\frac{\\tau \\log \\ensuremath{p} + \\log 4}{\\ensuremath{n}}}.\n\\end{eqnarray*}\n\\end{cors}\n\\begin{proof}\nFrom Lemma~\\ref{LEM_SAM_COV_BOUND_SUBG}, when the rescaled variables\n$X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are sub-Gaussian with parameter\n$\\ensuremath{\\sigma}$, the sample covariance entries satisfies a tail bound\n$\\ensuremath{\\mathcal{T}}(f, v_*)$ with with $v_* = \\big[\\max_{i}(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\,8(1 + 4 \\ensuremath{\\sigma}^2)\\big]^{-1}$ \nand $f(\\ensuremath{n},\\delta) = (1\/4) \\exp(c_{*} \\ensuremath{n} \\delta^2)$, where \\mbox{$c_{*} =\n\\big[128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2\\big]^{-1}$.}  As a consequence, for this particular model, the\ninverse functions $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$ and\n$\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,\\ensuremath{p}^{\\tau})$ take the form\n\\begin{subequations}\n\\label{EqnExpTailInv}\n\\begin{eqnarray}\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) & = &\n\\sqrt{\\frac{\\log(4\\,\\ensuremath{p}^{\\tau})}{c_{*} \\, \\ensuremath{n}}} \\; = \\;\n\\sqrt{ 128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2} \\; \\sqrt{\\frac{\\tau \\log \\ensuremath{p} + \\log 4}{\\ensuremath{n}}},\\\\\n\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,\\ensuremath{p}^{\\tau}) &= &\n\\frac{\\log(4\\,\\ensuremath{p}^{\\tau})} {c_{*} \\delta^2} \\; = \\; \n128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2  \\;\n\\biggr(\\frac{\\tau \\log \\ensuremath{p} + \\log 4}{\\delta^{2}}\\biggr).\n\\end{eqnarray}\n\\end{subequations}\nSubstituting these forms into the claim of Theorem~\\ref{ThmMain} and\ndoing some simple algebra yields the stated corollary.\n\\end{proof}\n\nWhen $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco$ remain constant as a function of\n$(\\ensuremath{n}, \\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$, the corollary can be summarized\nsuccinctly as a sample size of \\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\log\n\\ensuremath{p})$} samples ensures that an elementwise\n$\\ell_\\infty$ bound \\mbox{$\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty =\n{\\mathcal{O}}\\big( \\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}}\\big)$} holds with high probability. \nIn practice, one frequently considers graphs with maximum node degrees\n$\\ensuremath{\\ensuremath{d}}$ that either remain bounded, or that grow sub-linearly with\nthe graph size (i.e., $\\ensuremath{\\ensuremath{d}} = o(\\ensuremath{p})$). In such cases, the sample\nsize allowed by the corollary can be substantially smaller than the\ngraph size, so that for sub-Gaussian random variables, the method can\nsucceed in the $\\ensuremath{p} \\gg \\ensuremath{n}$ regime.\n\n\\subsubsection{Polynomial-type tails}\n\nWe now state a corollary for the case of a polynomial-type tail\nfunction, such as those ensured by the case of random variables with\nappropriately bounded moments.\n\\begin{cors}\n\\label{CorEllinfPoly}\nUnder the assumptions of Theorem~\\ref{ThmMain}, suppose the rescaled\nvariables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have $4m^{th}$\nmoments upper bounded by $K_{\\momentpow}$, and the sampling is i.i.d.\nThen if the sample size $\\ensuremath{n}$ satisfies the bound\n\\begin{eqnarray}\n\\label{EqnPolyTailSampSize}\n\\ensuremath{n} & > & \\ensuremath{C_2} \\, \\ensuremath{\\ensuremath{d}}^{2}\\, \\big(1 + \\frac{8}{\\mutinco}\n\\big)^2\\, \\ensuremath{p}^{\\tau\/m},\n\\end{eqnarray}\nwhere $\\ensuremath{C_2} \\ensuremath{:  =} \\big\\{12 m \\,[m (K_{\\momentpow} +\n  1)]^{\\frac{1}{2m}}\\, \\max_i(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\max\n\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{4} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\big\\}^{2}$,\nthen with probability greater than $1-1\/\\ensuremath{p}^{\\tau -2}$, the\nestimate $\\ensuremath{\\widehat{\\Theta}}$ satisfies the bound,\n\\begin{eqnarray*}\n\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty & \\leq & \\{4m\n [m (K_{\\momentpow} + 1)]^{\\frac{1}{2m}}\\, \\big(1 +\n \\frac{8}{\\mutinco} \\big) \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\}\\;\n \\sqrt{\\frac{\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}.\n\\end{eqnarray*}\n\\end{cors}\n\\begin{proof}\nRecall from Lemma~\\ref{LEM_SAM_COV_BOUND_MOMENT} that when the\nrescaled variables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have bounded\n$4m^{th}$ moments, then the sample covariance $\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}$\nsatisfies the tail condition $\\ensuremath{\\mathcal{T}}(f, v_*)$ with\n$v_* = 0$, and with $f(\\ensuremath{n},\\delta) = c_{*}\n\\ensuremath{n}^{m} \\delta^{2m}$ with $c_{*}$ defined as\n$c_{*} = 1\/\\big\\{m^{2m+1} 2^{2m} (\\max_i\n\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}) ^{2m}\\, (K_{\\momentpow} + 1 ) \\big\\}$. As a\nconsequence, for this particular model, the inverse functions take the\nform\n\\begin{subequations}\n\\label{EqnPolyTailInv}\n\\begin{eqnarray}\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) & = &\n\\frac{(\\ensuremath{p}^{\\tau}\/c_{*})^{1\/2m}}{\\sqrt{\\ensuremath{n}}}\n\\,=\\, \\{2m [m (K_{\\momentpow} +\n1)]^{\\frac{1}{2m}} \\max_i \\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} \\}\\;\n\\sqrt{\\frac{\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}},\\\\\n\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,\\ensuremath{p}^{\\tau}) & = &\n\\frac{(\\ensuremath{p}^{\\tau}\/c_{*})^{1\/m}}{\\delta^{2}} \\,=\\,\n\\{2m [m (K_{\\momentpow} + 1)]^{\\frac{1}{2m}}\n\\max_i \\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} \\}^{2}\\;\n\\big(\\frac{\\ensuremath{p}^{\\tau\/m}}{\\delta^{2}}\\big).\n\\end{eqnarray}\n\\end{subequations}\nThe claim then follows by substituting these expressions into Theorem~\\ref{ThmMain} and performing some algebra.\n\\end{proof}\n\nWhen the quantities $(\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco)$ remain constant\nas a function of $(\\ensuremath{n}, \\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$,\nCorollary~\\ref{CorEllinfPoly} can be summarized succinctly as\n\\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\, \\ensuremath{p}^{\\tau\/m})$}\nsamples are sufficient to achieve a convergence rate in elementwise\n$\\ell_\\infty$-norm of the order \\mbox{$\\| \\estim{\\Theta} -\n\\ensuremath{\\Theta^*}\\|_\\infty = {\\mathcal{O}}\\big(\n\\sqrt{\\frac{\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}\\big)$,} with high\nprobability.  Consequently, both the required sample size and the rate\nof convergence of the estimator are polynomial in the number of\nvariables $\\ensuremath{p}$.  It is worth contrasting these rates with the case\nof sub-Gaussian random variables, where the rates have only\nlogarithmic dependence on the problem size $\\ensuremath{p}$.\n\n\n \n\n\n\n\\subsection{Model selection consistency}\n\\label{SecModelCons}\n\nPart (b) of Theorem~\\ref{ThmMain} asserts that the edge set\n$\\ensuremath{E}(\\ensuremath{\\widehat{\\Theta}})$ returned by the estimator is contained within the\ntrue edge set $\\ensuremath{E}(\\ensuremath{\\Theta^*})$---meaning that it correctly\n\\emph{excludes} all non-edges---and that it includes all edges that\nare ``large'', relative to the $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$\ndecay of the error.  The following result, essentially a minor\nrefinement of Theorem~\\ref{ThmMain}, provides sufficient conditions\nlinking the sample size $\\ensuremath{n}$ and the minimum value\n\\begin{eqnarray}\n\\label{EqnDefnThetaMin}\n\\ensuremath{\\theta_{\\operatorname{min}}} & \\ensuremath{:  =} & \\min_{(i,j) \\in \\ensuremath{E}(\\ensuremath{\\Theta^*})} |\\ensuremath{\\Theta^*}_{ij}|\n\\end{eqnarray}\nfor model selection consistency.  More precisely, define the event\n\\begin{eqnarray}\n\\ensuremath{\\mathcal{M}}(\\ensuremath{\\widehat{\\Theta}}; \\ensuremath{\\Theta^*}) & \\ensuremath{:  =} & \\big \\{ \\textrm{sign}(\\ensuremath{\\widehat{\\Theta}}_{ij})\n= \\textrm{sign}(\\ensuremath{\\Theta^*}_{ij}) \\quad \\forall (i,j) \\in \\ensuremath{E}(\\ensuremath{\\Theta^*})\n\\big \\}\n\\end{eqnarray}\nthat the estimator $\\ensuremath{\\widehat{\\Theta}}$ has the same edge set as $\\ensuremath{\\Theta^*}$,\nand moreover recovers the correct signs on these edges.  With this\nnotation, we have:\n\\begin{theos}\n\\label{ThmModel}\n\nUnder the same conditions as Theorem~\\ref{ThmMain}, suppose that\nthe sample size satisfies the lower bound\n\\begin{eqnarray}\n\\label{EqnNumobsModel}\n\\ensuremath{n} & > & \\ensuremath{{\\widebar{\\ensuremath{n}}_f}} \\Biggr( 1 \\big\/\\max \\Big \\{ 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} (1 +\n8\\mutinco^{-1})\\, \\ensuremath{\\theta_{\\operatorname{min}}}^{-1}, \\; v_*, \\; 6 \\big (1 +\n8\\mutinco^{-1} \\big) \\: \\ensuremath{\\ensuremath{d}}\\, \\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\Big\\} , \\; \\;\\ensuremath{p}^{\\tau}\n\\Biggr).\n\\end{eqnarray}\nThen the estimator is model selection consistent with high probability\nas $\\ensuremath{p} \\rightarrow \\infty$,\n\\begin{eqnarray}\n\\ensuremath{\\mathbb{P}} \\big[ \\ensuremath{\\mathcal{M}}(\\ensuremath{\\widehat{\\Theta}}; \\ensuremath{\\Theta^*}) \\big] & \\geq & 1 -\n1\/\\ensuremath{p}^{\\tau - 2} \\; \\rightarrow \\; 1.\n\\end{eqnarray}\n\\end{theos}\n\nIn comparison to Theorem~\\ref{ThmMain}, the sample size\nrequirement~\\eqref{EqnNumobsModel} differs only in the additional term\n$\\frac{2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} (1 + \\frac{8}{\\mutinco})}{\\ensuremath{\\theta_{\\operatorname{min}}}}$ involving the\nminimum value.  This term can be viewed as constraining how quickly\nthe minimum can decay as a function of $(\\ensuremath{n}, \\ensuremath{p})$, as we\nillustrate with some concrete tail functions.\n\n\n\\subsubsection{Exponential-type tails} \n\nRecall the setting of Section~\\ref{SecSamCovBoundSubG}, where the\nrandom variables $\\{X^{(\\obsind)}_{i}\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}\\}$ are\nsub-Gaussian with parameter $\\ensuremath{\\sigma}$.  Let us suppose that the\nparameters $(\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco)$ are viewed as constants\n(not scaling with $(\\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$.  Then, using the\nexpression~\\eqref{EqnExpTailInv} for the inverse function\n$\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ in this setting, a corollary of Theorem~\\ref{ThmModel}\nis that a sample size \n\\begin{eqnarray}\n\\label{EqnModelSampSub}\n\\ensuremath{n} & = & \\Omega \\big( (\\ensuremath{\\ensuremath{d}}^2 + \\ensuremath{\\theta_{\\operatorname{min}}}^{-2}) \\, \\tau \\log\n\\ensuremath{p} \\big)\n\\end{eqnarray}\nis sufficient for model selection consistency with probability greater\nthan $1-1\/\\ensuremath{p}^{\\tau-2}$.  Alternatively, we can state that $\\ensuremath{n}\n= \\Omega(\\tau \\ensuremath{\\ensuremath{d}}^2 \\log \\ensuremath{p})$ samples are sufficient, as\nalong as the minimum value scales as \\mbox{$\\ensuremath{\\theta_{\\operatorname{min}}} =\n\\Omega(\\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}})$.}\n\n\\subsubsection{Polynomial-type tails} \n\nRecall the setting of Section~\\ref{SecSamCovBoundMoment}, where the\nrescaled random variables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have bounded\n$4m^{th}$ moments. Using the expression~\\eqref{EqnPolyTailInv}\nfor the inverse function $\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ in this setting, a corollary of\nTheorem~\\ref{ThmModel} is that a sample size\n\\begin{eqnarray}\n\\label{EqnModelSampPoly}\n\\ensuremath{n} & = & \\Omega\\big( (\\ensuremath{\\ensuremath{d}}^2 + \\ensuremath{\\theta_{\\operatorname{min}}}^{-2})\\,\n\\ensuremath{p}^{\\tau\/m} \\big)\n\\end{eqnarray}\nis sufficient for model selection consistency with probability greater\nthan $1-1\/\\ensuremath{p}^{\\tau-2}$.  Alternatively, we can state than $\\ensuremath{n}\n= \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\ensuremath{p}^{\\tau\/m})$ samples are\nsufficient, as long as the minimum value scales as \\mbox{$\\ensuremath{\\theta_{\\operatorname{min}}} =\n\\Omega(\\ensuremath{p}^{\\tau\/(2m)}\/{\\sqrt{\\ensuremath{n}}})$.}\n\n\n\n\\subsection{Comparison to neighbor-based graphical model selection}\n\\label{SecInco}\n\nSuppose that $X$ follows a multivariate Gaussian distribution, so that\nthe structure of the concentration matrix $\\ensuremath{\\Theta^*}$ specifies the\nstructure of a Gaussian graphical model.  In this case, it is\ninteresting to compare our sufficient conditions for graphical model\nconsistency of the log-determinant approach, as specified in\nTheorem~\\ref{ThmModel}, to those of the neighborhood-based method,\nfirst proposed by \\citet{MeinsBuhl2006}. The latter method estimates\nthe full graph structure by performing an $\\ell_1$-regularized linear\nregression (Lasso)---of the form $X_i = \\sum_{j \\neq i} \\theta_{ij}\nX_j + W$--- of each node on its neighbors and using the support of the\nestimated regression vector $\\theta$ to predict the neighborhood set.\nThese neighborhoods are then combined, by either an OR rule or an AND\nrule, to estimate the full graph.  Various aspects of the\nhigh-dimensional model selection consistency of the Lasso are now\nunderstood~\\cite{MeinsBuhl2006,Wainwright2006_new,Zhao06}; for\ninstance, it is known that mutual incoherence or irrepresentability\nconditions are necessary and sufficient for its\nsuccess~\\cite{Tropp2006,Zhao06}. In terms of scaling,\nWainwright~\\cite{Wainwright2006_new} shows that the Lasso succeeds\nwith high probability if and only if the sample size scales as\n\\mbox{$\\ensuremath{n} \\asymp c (\\{\\ensuremath{\\ensuremath{d}} + \\theta_{\\operatorname{min}}^{-2}\n\\} \\log \\ensuremath{p})$,} where $c$ is a constant determined by the covariance\nmatrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.  By a union bound over the $\\ensuremath{p}$ nodes in the\ngraph, it then follows that the neighbor-based graph selection method\nin turn succeeds with high probability if $\\ensuremath{n} = \\Omega(\\{\\ensuremath{\\ensuremath{d}}\n+ \\ensuremath{\\theta_{\\operatorname{min}}}^{-2} \\} \\log \\ensuremath{p})$.\n\nFor comparison, consider the application of Theorem~\\ref{ThmModel} to\nthe case where the variables are sub-Gaussian (which includes the\nGaussian case). For this setting, we have seen that the scaling\nrequired by Theorem~\\ref{ThmModel} is $\\ensuremath{n} = \\Omega( \\{ \\ensuremath{\\ensuremath{d}}^2\n+ \\ensuremath{\\theta_{\\operatorname{min}}}^{-2} \\} \\log \\ensuremath{p})$, so that the dependence of the\nlog-determinant approach in $\\ensuremath{\\theta_{\\operatorname{min}}}$ is identical, but it depends\nquadratically on the maximum degree $\\ensuremath{\\ensuremath{d}}$.  We suspect that that\nthe quadratic dependence $\\ensuremath{\\ensuremath{d}}^2$ might be an artifact of our\nanalysis, but have not yet been able to reduce it to $\\ensuremath{\\ensuremath{d}}$.\nOtherwise, the primary difference between the two methods is in the\nnature of the irrepresentability assumptions that are imposed: our\nmethod requires Assumption~\\ref{AssInco} on the Hessian $\\ensuremath{\\Gamma^*}$,\nwhereas the neighborhood-based method imposes this same type of\ncondition on a set of $\\ensuremath{p}$ covariance matrices, each of size\n$(\\ensuremath{p} -1) \\times (\\ensuremath{p}-1)$, one for each node of the graph.  Below\nwe show two cases where the Lasso irrepresentability condition holds,\nwhile the log-determinant requirement fails.  However, in general, we\ndo not know whether the log-determinant irrepresentability strictly\ndominates its analog for the Lasso.\n\n\n\\subsubsection{Illustration of irrepresentability: Diamond graph} \nConsider the following Gaussian graphical model example from\n\\citet{Meins2008}.  Figure~\\ref{FigSimpGraph}(a) shows a\ndiamond-shaped graph $G = (V,E)$, with vertex set $V = \\{1,2,3,4\\}$\nand edge-set as the fully connected graph over $V$ with the edge\n$(1,4)$ removed.\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\psfrag{#1#}{$1$} \\psfrag{#2#}{$2$} \\psfrag{#3#}{$3$}\n\\psfrag{#4#}{$4$} \\widgraph{.3\\textwidth}{meins.eps} & \\hspace*{.2in}\n& \\psfrag{#1#}{$1$} \\psfrag{#2#}{$2$} \\psfrag{#3#}{$3$}\n\\psfrag{#4#}{$4$} \\widgraph{.3\\textwidth}{small_star.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{(a) Graph of the example discussed by~\\citet{Meins2008}.  (b)\nA simple $4$-node star graph.}\n\\label{FigSimpGraph}\n\\end{figure}\nThe covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ is parameterized by the\ncorrelation parameter $\\rho \\in [0,1\/\\sqrt{2}]$: the diagonal entries\nare set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{i} = 1$, for all $i \\in \\ensuremath{V}$; the entries\ncorresponding to edges are set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = \\rho$ for $(i,j)\n\\in \\ensuremath{E} \\backslash \\{(2,3)\\}$, $\\ensuremath{\\ensuremath{\\Sigma}^*}_{23} = 0$; and finally\nthe entry corresponding to the non-edge is set as $\\ensuremath{\\ensuremath{\\Sigma}^*}_{14} =\n2 \\rho^2$. \\citet{Meins2008} showed that the $\\ell_1$-penalized\nlog-determinant estimator $\\ensuremath{\\widehat{\\Theta}}$ fails to recover the graph\nstructure, for any sample size, if $\\rho > -1 + (3\/2)^{1\/2} \\approx\n0.23$.  It is instructive to compare this necessary condition to the\nsufficient condition provided in our analysis, namely the incoherence\nAssumption~\\ref{AssInco} as applied to the Hessian $\\ensuremath{\\Gamma^*}$.  For\nthis particular example, a little calculation shows that\nAssumption~\\ref{AssInco} is equivalent to the constraint\n\\begin{eqnarray*}\n4 |\\rho| (|\\rho| + 1) & < & 1,\n\\end{eqnarray*}\nan inequality which holds for all $\\rho \\in (-0.2017, 0.2017)$.  Note\nthat the upper value $0.2017$ is just below the necessary threshold\ndiscussed by \\citet{Meins2008}. On the other hand, the\nirrepresentability condition for the Lasso requires only that $2\n|\\rho| < 1$, i.e., $\\rho \\in (-0.5,0.5)$.  Thus, in the regime $|\\rho|\n\\in [0.2017,0.5)$, the Lasso irrepresentability condition holds while\nthe log-determinant counterpart fails.\n\n\n\\subsubsection{Illustration of irrepresentability: Star graphs} \nA second interesting example is the star-shaped graphical model,\nillustrated in Figure~\\ref{FigSimpGraph}(b), which consists of a\nsingle hub node connected to the rest of the spoke nodes. We consider\na four node graph, with vertex set $V = \\{1,2,3,4\\}$ and edge-set $E =\n\\{(1,s) \\mid s \\in \\{2,3,4\\}\\}$. The covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$\nis parameterized the correlation parameter $\\rho \\in [-1,1]$: the\ndiagonal entries are set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} = 1$, for all $i \\in V$;\nthe entries corresponding to edges are set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} =\n\\rho$ for $(i,j) \\in E$; while the non-edge entries are set as\n$\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = \\rho^2$ for $(i,j) \\notin E$.  Consequently, for\nthis particular example, Assumption~\\ref{AssInco} reduces to the\nconstraint $|\\rho| (|\\rho| + 2) < 1$, which holds for all $\\rho \\in\n(-0.414, 0.414)$. The irrepresentability condition for the Lasso on\nthe other hand allows the full range $\\rho \\in (-1,1)$. Thus there is\nagain a regime, $|\\rho| \\in [0.414,1)$, where the Lasso\nirrepresentability condition holds while the log-determinant\ncounterpart fails.\n\n\n\\subsection{Rates in Frobenius and spectral norm}\n\\label{SecFrob}\nWe now derive some corollaries of Theorem~\\ref{ThmMain} concerning\nestimation of $\\ensuremath{\\Theta^*}$ in Frobenius norm, as well as the spectral\nnorm.  Recall that $\\ensuremath{s} = |\\ensuremath{E}(\\ensuremath{\\Theta^*})|$ denotes the total\nnumber of off-diagonal non-zeros in $\\ensuremath{\\Theta^*}$.\n\\begin{cors}\n\\label{CorOperatorNorm} \nUnder the same assumptions as Theorem~\\ref{ThmMain}, with probability\nat least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$, the\nestimator $\\estim{\\Theta}$ satisfies\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{EqnPrecFrob}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{F} & \\leq & \\big\\{2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big(1\n + \\frac{8}{\\mutinco}\\big)\\big\\} \\, \\sqrt{\\ensuremath{s} +\\ensuremath{p}} \\;\n \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}), \\qquad \\mbox{and} \\\\\n\\label{EqnPrecSpectral}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{2} & \\leq & \\big\\{2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big(1\n + \\frac{8}{\\mutinco}\\big) \\big\\}\\, \\min \\{\\sqrt{\\ensuremath{s} +\\ensuremath{p}}, \\,\n \\ensuremath{\\ensuremath{d}} \\} \\; \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray}\n\\end{subequations}\n\\end{cors}\n\\begin{proof}\n\\newcommand{\\ensuremath{\\nu}}{\\ensuremath{\\nu}}\n\nWith the shorthand notation $\\ensuremath{\\nu} \\ensuremath{:  =} 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} (1 + 8\/\\mutinco)\n\\; \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$, Theorem~\\ref{ThmMain}\nguarantees that, with probability at least $1 - 1\/\\ensuremath{p}^{\\tau -\n2}$, $\\|\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}\\|_\\infty \\leq \\ensuremath{\\nu}$.  Since the edge\nset of $\\ensuremath{\\widehat{\\Theta}}$ is a subset of that of $\\ensuremath{\\Theta^*}$, and\n$\\ensuremath{\\Theta^*}$ has at most $\\ensuremath{p} + \\ensuremath{s}$ non-zeros (including the\ndiagonal), we conclude that\n\\begin{eqnarray*}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{F} & = & \\big[ \\sum_{i=1}^\\ensuremath{p}\n(\\ensuremath{\\widehat{\\Theta}}_{ii} - \\ensuremath{\\Theta^*}_{ii})^2 + \\sum_{(i,j) \\in \\ensuremath{E}}\n(\\ensuremath{\\widehat{\\Theta}}_{ij} - \\ensuremath{\\Theta^*}_{ij})^2 \\big]^{1\/2} \\\\\n& \\leq & \\ensuremath{\\nu} \\; \\sqrt{\\ensuremath{s} + \\ensuremath{p}},\n\\end{eqnarray*}\nfrom which the bound~\\eqref{EqnPrecFrob} follows.  On the other hand,\nfor a symmetric matrix, we have\n\\begin{eqnarray}\\label{EqnPrecInftyOp}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}}- \\ensuremath{\\Theta^*}}{2} & \\leq & \\matnorm{\\ensuremath{\\widehat{\\Theta}} -\n\\ensuremath{\\Theta^*}}{\\infty} \\; \\leq \\; \\ensuremath{\\ensuremath{d}} \\ensuremath{\\nu},\n\\end{eqnarray}\nusing the definition of the $\\ensuremath{\\nu}_\\infty$-operator norm, and the fact\nthat $\\ensuremath{\\widehat{\\Theta}}$ and $\\ensuremath{\\Theta^*}$ have at most $\\ensuremath{\\ensuremath{d}}$ non-zeros per\nrow.  Since the Frobenius norm upper bounds the spectral norm, the\nbound~\\eqref{EqnPrecSpectral} follows.\n\n\n\\end{proof}\n\n\\subsubsection{Exponential-type tails}\nFor the exponential tail function case where the rescaled random\nvariables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are sub-Gaussian with\nparameter $\\ensuremath{\\sigma}$, we can use the expression~\\eqref{EqnExpTailInv}\nfor the inverse function $\\ensuremath{\\widebar{\\delta}_f}$ to derive rates in Frobenius\nand spectral norms.  When the quantities $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}},\n\\mutinco$ remain constant, these bounds can be summarized succinctly\nas a sample size \\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\log \\ensuremath{p})$} is\nsufficient to guarantee the bounds\n\\begin{subequations}\n\\begin{eqnarray}\n\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{F} & = &\n{\\mathcal{O}}\\biggr(\\sqrt{\\frac{(\\ensuremath{s} + \\ensuremath{p})\\,\\log \\ensuremath{p}}{\\ensuremath{n}}}\\,\\biggr),\n\\quad \\mbox{and} \\\\\n\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{2} & = &\n{\\mathcal{O}}\\biggr(\\sqrt{\\frac{\\min\\{\\ensuremath{s} + \\ensuremath{p},\\,\\ensuremath{\\ensuremath{d}}^{2}\\} \\, \\log\n\\ensuremath{p}}{\\ensuremath{n}}}\\,\\biggr),\n\\end{eqnarray}\n\\end{subequations}\nwith probability at least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$.\n\n\\subsubsection{Polynomial-type tails}\n\nSimilarly, let us again consider the polynomial tail case, in which\nthe rescaled variates $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have bounded $4\nm^{th}$ moments and the samples are drawn i.i.d.  Using the\nexpression~\\eqref{EqnPolyTailInv} for the inverse function we can\nderive rates in the Frobenius and spectral norms.  When the quantities\n$\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco$ are viewed as constant, we are\nguaranteed that a sample size \\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\,\n\\ensuremath{p}^{\\tau\/m})$} is sufficient to guarantee the bounds\n\\begin{subequations}\n\\begin{eqnarray}\n \\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{F} & = &\n {\\mathcal{O}}\\biggr(\\sqrt{\\frac{(\\ensuremath{s} +\n \\ensuremath{p})\\,\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}\\,\\biggr),\\textrm{ and } \\\\\n\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{2} & = &\n{\\mathcal{O}}\\biggr(\\sqrt{\\frac{\\min\\{\\ensuremath{s} + \\ensuremath{p},\\,\\ensuremath{\\ensuremath{d}}^{2}\\} \\,\n\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}\\,\\biggr),\n\\end{eqnarray}\n\\end{subequations}\nwith probability at least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$.\n\n\n\\subsection{Rates for the covariance matrix estimate}\nFinally, we describe some bounds on the estimation of the covariance\nmatrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.  By Lemma~\\ref{LEM_MLE_CHARAC}, the estimated\nconcentration matrix $\\ensuremath{\\widehat{\\Theta}}$ is positive definite, and hence can\nbe inverted to obtain an estimate of the covariance matrix, which we\ndenote as $\\hat{\\CovHat} \\ensuremath{:  =} (\\ensuremath{\\widehat{\\Theta}})^{-1}$.\n\n\\begin{cors}\n\\label{CorCovBound}\nUnder the same assumptions as Theorem~\\ref{ThmMain}, with probability\nat least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$, the following bounds hold.\n\\begin{enumerate}\n\\item[(a)] The element-wise $\\ell_{\\infty}$ norm of the deviation\n $(\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*})$ satisfies the bound\n\\begin{eqnarray}\n\\label{EqnCovInftyBound}\n\\vecnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} & \\leq & \\ensuremath{C_3},\n\t[\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})] + \\ensuremath{C_4}\n\t\\ensuremath{\\ensuremath{d}}\\, [\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})]^{2}\n\\end{eqnarray}\nwhere $\\ensuremath{C_3} = 2 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\Big(1 +\n\\frac{8}{\\mutinco}\\Big)$ and $\\ensuremath{C_4} = 6 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2}\\Big(1 + \\frac{8}{\\mutinco}\\Big)^{2}$.\n\\item[(b)] The $\\ell_2$ operator-norm of the deviation $(\\hat{\\CovHat} -\n\\ensuremath{\\ensuremath{\\Sigma}^*})$ satisfies the bound\n\\begin{eqnarray}\n\\label{EqnCovSpectralBound}\n\\matnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{2} & \\leq & \\ensuremath{C_3} \\, \\ensuremath{\\ensuremath{d}} \\,\n\t[\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})] + \\ensuremath{C_4} \\ensuremath{\\ensuremath{d}}^{2}\n\t\\, [\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})]^{2}.\n\\end{eqnarray}\n\\end{enumerate}\n\\end{cors}\nThe proof involves certain lemmata and derivations that are parts of\nthe proofs of Theorems~\\ref{ThmMain} and \\ref{ThmModel}, so that we\ndefer it to Section~\\ref{SecCorCovProof}.\n\n\t\n\n\\section{Proofs of main result}\n\\label{SecProof}\n\nIn this section, we work through the proofs of Theorems~\\ref{ThmMain}\nand~\\ref{ThmModel}.  We break down the proofs into a sequence of\nlemmas, with some of the more technical aspects deferred to\nappendices.\n\nOur proofs are based on a technique that we call a \\emph{primal-dual\nwitness method}, used previously in analysis of the\nLasso~\\cite{Wainwright2006_new}.  It involves following a specific\nsequence of steps to construct a pair $(\\ThetaWitness, \\ensuremath{\\widetilde{Z}})$ of\nsymmetric matrices that together satisfy the optimality conditions\nassociated with the convex program~\\eqref{EqnGaussMLE} \\emph{with high\nprobability}.  Thus, when the constructive procedure succeeds,\n$\\ThetaWitness$ is \\emph{equal} to the unique solution\n$\\estim{\\Theta}$ of the convex program~\\eqref{EqnGaussMLE}, and\n$\\ensuremath{\\widetilde{Z}}$ is an optimal solution to its dual.  In this way, the\nestimator $\\estim{\\Theta}$ inherits from $\\ThetaWitness$ various\noptimality properties in terms of its distance to the truth\n$\\ensuremath{\\Theta^*}$, and its recovery of the signed sparsity pattern.  To be\nclear, our procedure for constructing $\\ThetaWitness$ is \\emph{not} a\npractical algorithm for solving the log-determinant\nproblem~\\eqref{EqnGaussMLE}, but rather is used as a proof technique\nfor certifying the behavior of the $M$-estimator~\\eqref{EqnGaussMLE}.\n\n\n\n\\subsection{Primal-dual witness approach}\n\\label{SecPrimalDualWitness}\nAs outlined above, at the core of the primal-dual witness method are the standard convex\noptimality conditions that characterize the optimum $\\ensuremath{\\widehat{\\Theta}}$ of the\nconvex program~\\eqref{EqnGaussMLE}.  For future reference, we note\nthat the sub-differential of the norm $\\ellreg{\\cdot}$ evaluated at\nsome $\\Theta$ consists the set of all symmetric matrices $Z \\in\n\\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times \\ensuremath{p}}$ such that\n\\begin{eqnarray}\\label{EqnSubGradDefn}\nZ_{ij} & = & \\begin{cases} 0 & \\mbox{if $i =j$} \\\\ \\textrm{sign}(\\Theta_{ij})\n& \\mbox{if $i \\neq j$ and $\\Theta_{ij} \\neq 0$} \\\\ \\in [-1, +1] &\n\\mbox{if $i \\neq j$ and $\\Theta_{ij} = 0$.}\n             \\end{cases}\n\\end{eqnarray}\nThe following result is proved in Appendix~\\ref{AppLemMLECharac}:\n\\begin{lems}\n\\label{LEM_MLE_CHARAC}\nFor any $\\ensuremath{\\lambda_\\ensuremath{n}} > 0$ and sample covariance $\\ensuremath{\\widehat{\\Sigma}}$ with strictly\npositive diagonal, the $\\ell_1$-regularized log-determinant\nproblem~\\eqref{EqnGaussMLE} has a unique solution $\\ensuremath{\\widehat{\\Theta}} \\succ 0$\ncharacterized by\n\\begin{eqnarray}\n\\label{EqnZeroSubgrad}\n\\ensuremath{\\widehat{\\Sigma}} - \\ensuremath{\\widehat{\\Theta}}^{-1} + \\ensuremath{\\lambda_\\ensuremath{n}} \\ensuremath{\\hat Z} & = & 0,\n\\end{eqnarray}\nwhere $\\ensuremath{\\hat Z}$ is an element of the subdifferential $\\partial\n\\ellreg{\\ensuremath{\\widehat{\\Theta}}}$.  \n\\end{lems}\n\nBased on this lemma, we construct the primal-dual witness solution\n$(\\ThetaWitness, \\ensuremath{\\widetilde{Z}})$ as follows:\n\\begin{enumerate}\n\\item[(a)]  We determine the\nmatrix $\\ThetaWitness$ by solving the restricted log-determinant\nproblem\n\\begin{eqnarray}\n\\label{EqnRestricted}\n\\ThetaWitness & \\ensuremath{:  =} & \\arg \\min_{\\Theta \\succ 0, \\;\n\\Theta_{\\ensuremath{\\EsetPlus^c}} = 0} \\big \\{\\tracer{\\Theta}{\\ensuremath{\\widehat{\\Sigma}}} - \\log\n\\det(\\Theta) + \\ensuremath{\\lambda_\\ensuremath{n}} \\ellreg{\\Theta} \\big \\}.\n\\end{eqnarray}\nNote that by construction, we have $\\ThetaWitness \\succ 0$, and\nmoreover $\\ThetaWitness_{\\ensuremath{\\EsetPlus^c}} = 0$.\n\\item[(b)] We choose $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{S}}$ as a member of the\nsub-differential of the regularizer $\\ellreg{\\cdot}$, evaluated\nat $\\ensuremath{\\ThetaWitness}$.\n\\item[(c)] We set $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}$ as\n\\begin{eqnarray}\n \\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}} &=& \\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}}\\big\\{-\n \\ensuremath{\\widehat{\\Sigma}}_{\\ensuremath{\\EsetPlus^c}} + [\\invn{\\ensuremath{\\ThetaWitness}}]_{\\ensuremath{\\EsetPlus^c}}\\big\\},\n\\end{eqnarray}\nwhich ensures that constructed matrices $(\\ensuremath{\\ThetaWitness},\\ensuremath{\\widetilde{Z}})$ satisfy\nthe optimality condition~\\eqref{EqnZeroSubgrad}.\n\\item[(d)]  We verify the \\emph{strict dual feasibility} condition\n\\begin{eqnarray*}\n|\\ensuremath{\\widetilde{Z}}_{ij}| & < & 1 \\quad \\mbox{for all $(i,j) \\in \\ensuremath{\\EsetPlus^c}$}.\n\\end{eqnarray*}\n\n\\end{enumerate}\nTo clarify the nature of the construction, steps (a) through (c)\nsuffice to obtain a pair $(\\ensuremath{\\ThetaWitness},\\ensuremath{\\widetilde{Z}})$ that satisfy the\noptimality conditions~\\eqref{EqnZeroSubgrad}, but do \\emph{not}\nguarantee that $\\ensuremath{\\widetilde{Z}}$ is an element of sub-differential $\\partial\n\\ellreg{\\ensuremath{\\ThetaWitness}}$.  By construction, specifically step (b) of the\nconstruction ensures that the entries $\\ensuremath{\\widetilde{Z}}$ in $\\ensuremath{S}$ satisfy\nthe sub-differential conditions, since $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{S}}$ is a member\nof the sub-differential of $\\partial \\ellreg{\\ensuremath{\\ThetaWitness}_{\\ensuremath{S}}}$.\nThe purpose of step (d), then, is to verify that the remaining\nelements of $\\ensuremath{\\widetilde{Z}}$ satisfy the necessary conditions to belong\nto the sub-differential.\n\n\nIf the primal-dual witness construction succeeds, then it acts as a\n\\emph{witness} to the fact that the solution $\\ensuremath{\\ThetaWitness}$ to the\nrestricted problem~\\eqref{EqnRestricted} is equivalent to the solution\n$\\ensuremath{\\widehat{\\Theta}}$ to the original (unrestricted)\nproblem~\\eqref{EqnGaussMLE}.  We exploit this fact in our proofs of\nTheorems~\\ref{ThmMain} and \\ref{ThmModel} that build on this: we first show\nthat the primal-dual witness technique succeeds with high-probability,\nfrom which we can conclude that the support of the optimal solution\n$\\ensuremath{\\widehat{\\Theta}}$ is contained within the support of the true $\\ensuremath{\\Theta^*}$.\nIn addition, we exploit the characterization of $\\ensuremath{\\widehat{\\Theta}}$ provided\nby the primal-dual witness construction to establish the elementwise\n$\\ell_\\infty$ bounds claimed in Theorem~\\ref{ThmMain}.\nTheorem~\\ref{ThmModel} requires checking, in addition, that certain\nsign consistency conditions hold, for which we require lower bounds on\nthe value of the minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$.\n\nIn the analysis to follow, some additional notation is useful.  We let\n$\\ensuremath{W}$ denote the ``effective noise'' in the sample covariance\nmatrix $\\ensuremath{\\widehat{\\Sigma}}$, namely\n\\begin{eqnarray}\n\\label{EqnWdefn}\n\\ensuremath{W} & \\ensuremath{:  =} & \\ensuremath{\\widehat{\\Sigma}} - (\\ensuremath{\\Theta^*})^{-1}.\n\\end{eqnarray}\nSecond, we use $\\Delta = \\ThetaWitness - \\ensuremath{\\Theta^*}$ to measure the\ndiscrepancy between the primal witness matrix $\\ensuremath{\\ThetaWitness}$ and the\ntruth $\\ensuremath{\\Theta^*}$.  Finally, recall the log-determinant barrier $g$\nfrom equation~\\eqref{EqnDefnLogDet}. We let $\\ensuremath{R}(\\Delta)$ denote the\ndifference of the gradient $\\nabla g(\\ensuremath{\\ThetaWitness}) =\n\\invn{\\ThetaWitness}$ from its first-order Taylor expansion around\n$\\ensuremath{\\Theta^*}$. Using known results on the first and second derivatives\nof the log-determinant function (see p. 641 in Boyd and\nVandenberghe~\\cite{Boyd02}), this remainder takes the form\n\\begin{eqnarray}\n\\label{EqnRdefn}\n\\ensuremath{R}(\\Delta) & = & \\invn{\\ThetaWitness} - \\invn{\\ensuremath{\\Theta^*}} +\n{\\ensuremath{\\Theta^*}}^{-1} \\Delta {\\ensuremath{\\Theta^*}}^{-1}.\n\\end{eqnarray}\n\n\\subsection{Auxiliary results}\n\nWe begin with some auxiliary lemmata, required in the proofs of our\nmain theorems.  In Section~\\ref{SecStrictDual}, we provide sufficient\nconditions on the quantities $\\ensuremath{W}$ and $\\ensuremath{R}$ for the strict dual\nfeasibility condition to hold. In Section~\\ref{SecRemainder}, we\ncontrol the remainder term $\\ensuremath{R}(\\Delta)$ in terms of $\\Delta$, while\nin Section~\\ref{SecEllinfBound}, we control $\\Delta$ itself, providing\nelementwise $\\ell_\\infty$ bounds on $\\Delta$.  In\nSection~\\ref{SecSignConsis}, we show that under appropriate conditions\non the minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$, the bounds in the earlier lemmas\nguarantee that the sign consistency condition holds.  All of the\nanalysis in these sections is \\emph{deterministic} in nature.  In\nSection~\\ref{SecNoise}, we turn to the probabilistic component of the\nanalysis, providing control of the noise $\\ensuremath{W}$ in the sample\ncovariance matrix.  Finally, the proofs of Theorems~\\ref{ThmMain}\nand~\\ref{ThmModel} follows by using this probabilistic control of\n$\\ensuremath{W}$ and the stated conditions on the sample size to show that\nthe deterministic conditions hold with high probability.\n\n\n\\subsubsection{Sufficient conditions for strict dual feasibility}\n\\label{SecStrictDual}\n\nWe begin by stating and proving a lemma that provides sufficient\n(deterministic) conditions for strict dual feasibility to hold, so\nthat $\\|\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}\\|_\\infty < 1$.\n\\begin{lems}[Strict dual feasibility]\n\\label{LemStrictDual}\nSuppose that\n\\begin{eqnarray}\n\\label{EqnDetSuff}\n\\max \\big \\{ \\|\\ensuremath{W}\\|_\\infty, \\; \\|\\ensuremath{R}(\\Delta)\\|_\\infty \\big \\}  &\n\\leq & \\frac{\\mutinco \\, \\ensuremath{\\lambda_\\ensuremath{n}}}{8}.\n\\end{eqnarray}\nThen the matrix $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}$ constructed in step (c)\nsatisfies $\\|\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}\\|_\\infty < 1$, and therefore\n$\\ThetaWitness = \\ensuremath{\\widehat{\\Theta}}$.\n\\end{lems}\n\\begin{proof}\nUsing the definitions~\\eqref{EqnWdefn} and~\\eqref{EqnRdefn}, we can\nre-write the stationary condition~\\eqref{EqnZeroSubgrad} in an\nalternative but equivalent form\n\\begin{eqnarray}\n\\label{EqnLME}\n\\invn{\\ensuremath{\\Theta^*}} \\Delta \\invn{\\ensuremath{\\Theta^*}} + \\ensuremath{W} - \\ensuremath{R}(\\Delta) +\n\\ensuremath{\\lambda_\\ensuremath{n}} \\ensuremath{\\wtil{Z}} & = 0.\n\\end{eqnarray}\nThis is a linear-matrix equality, which can be re-written as an\nordinary linear equation by ``vectorizing'' the matrices.  We use the\nnotation $\\ensuremath{\\operatorname{vec}}(A)$, or equivalently $\\myvec{A}$ for the\n$\\ensuremath{p}^2$-vector version of the matrix $A \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times\n\\ensuremath{p}}$, obtained by stacking up the rows into a single column vector.\nIn vectorized form, we have\n\\begin{equation*}\n\\ensuremath{\\operatorname{vec}} \\big(\\invn{\\ensuremath{\\Theta^*}} \\Delta \\invn{\\ensuremath{\\Theta^*}} \\big) = \\big\n(\\invn{\\ensuremath{\\Theta^*}} \\otimes \\invn{\\ensuremath{\\Theta^*}} \\big) \\myvec{\\Delta} \\; =\n\\; \\ensuremath{\\Gamma^*} \\myvec{\\Delta}.\n\\end{equation*}\nIn terms of the disjoint decomposition $\\ensuremath{S}$ and\n$\\ensuremath{\\EsetPlus^c}$, equation~\\eqref{EqnLME} can be re-written as two\nblocks of linear equations as follows:\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{EqnStatBlockS} \\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}} \\myvec{\\Delta}_{\\ensuremath{S}} + \\myvec{\\ensuremath{W}}_{\\ensuremath{S}} -\n\\myvec{\\ensuremath{R}}_{\\ensuremath{S}} + \\ensuremath{\\lambda_\\ensuremath{n}} \\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{S}} & = & 0\n\\\\\n\\label{EqnStatBlockScomp}\n\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c} \\ensuremath{S}} \\myvec{\\Delta}_{\\ensuremath{S}} +\n\\myvec{\\ensuremath{W}}_{\\ensuremath{\\EsetPlus^c}} - \\myvec{\\ensuremath{R}}_{\\ensuremath{\\EsetPlus^c}} +\n\\ensuremath{\\lambda_\\ensuremath{n}} \\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}} & = & 0.\n\\end{eqnarray}\n\\end{subequations}\nHere we have used the fact that $\\Delta_{\\ensuremath{\\EsetPlus^c}} = 0$ by\nconstruction.\n\nSince $\\ensuremath{\\Gamma^*}_{\\ensuremath{S} \\ensuremath{S}}$ is invertible, we can solve for\n$\\myvec{\\Delta}_{\\ensuremath{S}}$ from equation~\\eqref{EqnStatBlockS} as\nfollows:\n\\begin{eqnarray*}\n\\myvec{\\Delta}_{\\ensuremath{S}} & = & \\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}\n\\big[-\\myvec{\\ensuremath{W}}_{\\ensuremath{S}} + \\myvec{\\ensuremath{R}}_{\\ensuremath{S}} - \\ensuremath{\\lambda_\\ensuremath{n}}\n\\myvec{\\ensuremath{\\widetilde{Z}}_\\ensuremath{S}} \\big].\n\\end{eqnarray*}\nSubstituting this expression into equation~\\eqref{EqnStatBlockScomp},\nwe can solve for $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}$ as follows:\n\\begin{eqnarray}\n\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}} &= & -\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}}\n\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\\myvec{\\Delta}_{\\ensuremath{S}} +\n\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\myvec{\\ensuremath{R}}_{\\ensuremath{\\EsetPlus^c}} - \\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}}\n\\myvec{\\ensuremath{W}}_{\\ensuremath{\\EsetPlus^c}} \\nonumber\\\\\n& = & -\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}(\\myvec{\\ensuremath{W}}_{\\ensuremath{S}} -\n\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}) + \\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}} \\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{S}} -\n\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}} (\\myvec{\\ensuremath{W}}_{\\ensuremath{\\EsetPlus^c}} -\n\\myvec{\\ensuremath{R}}_{\\ensuremath{\\EsetPlus^c}}).\n\\end{eqnarray}\nTaking the $\\ell_\\infty$ norm of both sides yields\n\\begin{multline*}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}}\\|_{\\infty} \\leq \\frac{1}{\\lambda_n}\n\\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}}{\\infty}\n(\\|\\myvec{\\ensuremath{W}}_{\\ensuremath{S}}\\|_{\\infty} +\n\\|\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}\\|_{\\infty}) \\\\\n+ \\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}}{\\infty}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{S}}\\|_{\\infty} + \\frac{1}{\\lambda_n}\n(\\|\\myvec{\\ensuremath{W}}_{\\ensuremath{S}}\\|_{\\infty} +\n\\|\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}\\|_{\\infty}).\n\\end{multline*}\nRecalling Assumption~\\ref{AssInco}---namely, that\n$\\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}}{\\infty} \\le (1 - \\mutinco)$---we\nhave\n\\begin{eqnarray*}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}}\\|_{\\infty} & \\leq &\n\\frac{2-\\mutinco}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\,\n(\\|\\myvec{\\ensuremath{W}}_{\\ensuremath{S}}\\|_{\\infty} +\n\\|\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}\\|_{\\infty}) + (1 - \\mutinco),\n\\end{eqnarray*}\nwhere we have used the fact that $\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_\\ensuremath{S}\\|_{\\infty}\n\\leq 1$, since $\\ensuremath{\\widetilde{Z}}$ belongs to the sub-differential of the norm\n$\\ellreg{\\cdot}$ by construction.  Finally, applying\nassumption~\\eqref{EqnDetSuff} from the lemma statement, we have\n\\begin{eqnarray*}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}}\\|_{\\infty} & \\leq &\n\\frac{(2-\\mutinco)}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\, \\big( \\frac{\\mutinco \\ensuremath{\\lambda_\\ensuremath{n}}}{4}) +\n(1-\\mutinco) \\\\\n& \\leq &  \\frac{\\mutinco}{2} + (1-\\mutinco)  \\: < \\; 1,\n\\end{eqnarray*}\nas claimed.\n\n\n\\end{proof}\n\n\n\\subsubsection{Control of remainder term}\n\\label{SecRemainder}\n\nOur next step is to relate the behavior of the remainder\nterm~\\eqref{EqnRdefn} to the deviation $\\Delta = \\ensuremath{\\ThetaWitness} -\n\\ensuremath{\\Theta^*}$.\n\\begin{lems}[Control of remainder]\n\\label{LEM_R_CONV}\nSuppose that the elementwise $\\ell_\\infty$ bound $\\|\\Delta\\|_\\infty\n\\leq \\frac{1}{3 \\, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}$ holds.  Then:\n\\begin{eqnarray}\n\\label{EqnRemExpand}\n\\ensuremath{R}(\\Delta) & = & \\invn{\\opt{\\Theta}} \\Delta \\invn{\\opt{\\Theta}} \\Delta\nJ \\invn{\\opt{\\Theta}},\n\\end{eqnarray}\nwhere $J \\ensuremath{:  =} \\sum_{k=0}^{\\infty} (-1)^{k} \\big(\\invn{\\opt{\\Theta}}\n\\Delta\\big)^{k}$ has norm $\\matnorm{J^T}{\\infty} \\leq 3\/2$.  Moreover,\nin terms of the elementwise $\\ell_\\infty$-norm, we have\n\\begin{eqnarray}\n\\label{EqnRemBound}\n\\| \\ensuremath{R}(\\Delta)\\|_\\infty & \\leq & \\frac{3}{2} \\ensuremath{\\ensuremath{d}}\n\\|\\Delta\\|_\\infty^2 \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3.\n\\end{eqnarray}\n\\end{lems}\nWe provide the proof of this lemma in Appendix~\\ref{APP_LEM_R_CONV}.  It\nis straightforward, based on standard matrix expansion techniques.\n\n\\subsubsection{Sufficient conditions for $\\ell_\\infty$ bounds}\n\\label{SecEllinfBound}\n\nOur next lemma provides control on the deviation \\mbox{$\\Delta =\n\\ThetaWitness - \\ensuremath{\\Theta^*}$,} measured in elementwise $\\ell_\\infty$\nnorm.\n\\begin{lems}[Control of $\\Delta$]\n\\label{LEM_D_CONV}\nSuppose that\n\\begin{eqnarray}\n\\label{EqnDconvAss}\nr \\ensuremath{:  =} 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big( \\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}} \\big) & \\leq & \\min\n\\big \\{ \\frac{1}{3 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}, \\; \\frac{1}{3 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3\n\\;\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\ensuremath{\\ensuremath{d}}} \\big \\}.\n\\end{eqnarray}\nThen we have the elementwise $\\ell_\\infty$ bound\n\\begin{eqnarray}\n\\label{EqnDconvBound}\n\\|\\Delta \\|_\\infty = \\| \\ensuremath{\\ThetaWitness} - \\ensuremath{\\Theta^*} \\|_\\infty & \\leq & r.\n\\end{eqnarray}\n\\end{lems}\n\nWe prove the lemma in Appendix~\\ref{APP_LEM_D_CONV}; at a high level,\nthe main steps involved are the following.  We begin by noting that\n$\\ThetaWitness_{\\ensuremath{\\EsetPlus^c}} = \\ensuremath{\\Theta^*}_{\\ensuremath{\\EsetPlus^c}} = 0$, so\nthat $\\vecnorm{\\Delta}{\\infty} =\n\\vecnorm{\\Delta_{\\ensuremath{S}}}{\\infty}$.  Next, we characterize\n$\\ensuremath{\\ThetaWitness}_{\\ensuremath{S}}$ in terms of the zero-gradient condition\nassociated with the restricted problem~\\eqref{EqnRestricted}.  We then\ndefine a continuous map $F: \\Delta_{\\ensuremath{S}} \\mapsto\nF(\\Delta_{\\ensuremath{S}})$ such that its fixed points are equivalent to\nzeros of this gradient expression in terms of $\\Delta_\\ensuremath{S} =\n\\ensuremath{\\ThetaWitness}_\\ensuremath{S} - \\ensuremath{\\Theta^*}_{\\ensuremath{S}}$.  We then show that the\nfunction $F$ maps the $\\ell_\\infty$-ball\n\\begin{eqnarray}\n\\label{EqnDefnRad}\n\\ensuremath{\\mathbb{B}}(\\ensuremath{r}) & \\ensuremath{:  =} & \\{ \\Theta_{\\ensuremath{S}} \\mid \\|\\Theta_\\ensuremath{S}\n\\|_\\infty \\leq \\ensuremath{r} \\}, \\qquad \\mbox{with $\\ensuremath{r} \\ensuremath{:  =} 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big\n(\\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}} \\big)$},\n\\end{eqnarray}\nonto itself.  Finally, with these results in place, we can apply\nBrouwer's fixed point theorem (e.g., p. 161; Ortega and\nRheinboldt~\\cite{OrtegaR70}) to conclude that $F$ does indeed have a\nfixed point inside $\\ensuremath{\\mathbb{B}}(\\ensuremath{r})$.  \n\n\\subsubsection{Sufficient conditions for sign consistency}\n\\label{SecSignConsis}\n\n\nWe now show how a lower bound on the minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$, when\ncombined with Lemma~\\ref{LEM_D_CONV}, allows us to guarantee\n\\emph{sign consistency} of the primal witness matrix\n$\\ensuremath{\\ThetaWitness}_{\\ensuremath{S}}$.\n\\begin{lems}[Sign Consistency]\n\\label{LemSignConsis}\nSuppose the minimum absolute value $\\ensuremath{\\theta_{\\operatorname{min}}}$ of non-zero entries\nin the true concentration matrix $\\ensuremath{\\Theta^*}$ is lower bounded as\n\\begin{eqnarray}\n\\label{EqnThetaminLow}\n\\ensuremath{\\theta_{\\operatorname{min}}} & \\geq & 4 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big (\\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}}\n\\big),\n\\end{eqnarray}\nthen \\mbox{$\\textrm{sign}(\\ensuremath{\\ThetaWitness}_\\ensuremath{S}) = \\textrm{sign}(\\ensuremath{\\Theta^*}_\\ensuremath{S})$}\nholds.  \n\\end{lems}\nThis claim follows from the bound~\\eqref{EqnThetaminLow} combined with\nthe bound~\\eqref{EqnDconvBound} ,which together imply that for all\n$(i,j) \\in \\ensuremath{S}$, the estimate $\\ensuremath{\\ThetaWitness}_{ij}$ cannot differ\nenough from $\\ensuremath{\\Theta^*}_{ij}$ to change sign.\n\n\n\n\n\n\n\n\\subsubsection{Control of noise term}\n\\label{SecNoise}\n\nThe final ingredient required for the proofs of Theorems~\\ref{ThmMain}\nand \\ref{ThmModel} is control on the sampling noise $\\ensuremath{W} = \\ensuremath{\\estim{\\ensuremath{\\Sigma}}}\n- \\ensuremath{\\ensuremath{\\Sigma}^*}$.  This control is specified in terms of the decay\nfunction $f$ from equation~\\eqref{EqnSamTail}.\n\\begin{lems}[Control of Sampling Noise]\n\\label{LemWconv}\nFor any $\\tau > 2$ and sample size $\\ensuremath{n}$ such that\n$\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^\\tau) \\le 1\/v_*$, we have\n\\begin{eqnarray}\n\\ensuremath{\\mathbb{P}}\\biggr[ \\|\\ensuremath{W}\\|_\\infty \\geq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\biggr] & \\leq &\n\\frac{1}{\\ensuremath{p}^{\\tau - 2}} \\; \\rightarrow \\; 0.\n\\end{eqnarray}\n\n\\end{lems}\n\\begin{proof}\nUsing the definition~\\eqref{EqnSamTail} of the decay function\n$f$, and applying the union bound over all $\\ensuremath{p}^2$ entries of\nthe noise matrix, we obtain that for all $\\delta \\leq 1\/v_*$,\n\\begin{eqnarray*}\n\\ensuremath{\\mathbb{P}} \\big[\\max_{i, j} |\\ensuremath{W}_{ij}| \\geq \\delta \\big] & \\leq &\n\\ensuremath{p}^2\/f(\\ensuremath{n},\\delta).\n\\end{eqnarray*}\nSetting $\\delta = \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$ yields that\n\\begin{eqnarray*}\n\\ensuremath{\\mathbb{P}} \\big[\\max_{i, j} |\\ensuremath{W}_{ij}| \\geq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\big] & \\leq & \\ensuremath{p}^2\/\n\\big[f(\\ensuremath{n},\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})) \\big] \\; =\n\\; 1\/\\ensuremath{p}^{\\tau - 2},\n\\end{eqnarray*}\nas claimed.  Here the last equality follows since\n$f(\\ensuremath{n},\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})) =\n\\ensuremath{p}^\\tau$, using the definition~\\eqref{EqnSamTailT} of the\ninverse function $\\ensuremath{\\widebar{\\delta}_f}$.\n\\end{proof}\n\n\\def\\delta'{\\delta'}\n\\def\\numobsNew{\\ensuremath{n}'} \n\n\n\n\\subsection{Proof of Theorem~\\ref{ThmMain}}\n\\label{SecThmMain}\n\nWe now have the necessary ingredients to prove\nTheorem~\\ref{ThmMain}. We first show that with high probability the\nwitness matrix $\\ensuremath{\\ThetaWitness}$ is equal to the solution $\\ensuremath{\\widehat{\\Theta}}$ to the\noriginal log-determinant problem~\\eqref{EqnGaussMLE}, in particular by\nshowing that the primal-dual witness construction (described in in\nSection~\\ref{SecPrimalDualWitness}) succeeds with high probability.\nLet $\\mathcal{A}$ denote the event that $\\|\\ensuremath{W}\\|_\\infty \\leq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$.  Using the monotonicity of the\ninverse tail function~\\eqref{EqnMonot}, the lower lower\nbound~\\eqref{EqnSampleBound} on the sample size $\\ensuremath{n}$ implies that\n$\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\le 1\/v_*$.  Consequently,\nLemma~\\ref{LemWconv} implies that $\\ensuremath{\\mathbb{P}}(\\mathcal{A}) \\ge 1 -\n\\frac{1}{\\ensuremath{p}^{\\tau - 2}}$.  Accordingly, we condition on the\nevent $\\mathcal{A}$ in the analysis to follow.\n\nWe proceed by verifying that assumption~\\eqref{EqnDetSuff} of\nLemma~\\ref{LemStrictDual} holds.  Recalling the choice of\nregularization penalty \\mbox{$\\ensuremath{\\lambda_\\ensuremath{n}} = (8\/\\mutinco)\\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$,} we have $\\|\\ensuremath{W}\\|_\\infty\n\\leq (\\mutinco\/8) \\ensuremath{\\lambda_\\ensuremath{n}}$.  In order to establish\ncondition~\\eqref{EqnDetSuff} it remains to establish the bound\n$\\|\\ensuremath{R}(\\Delta)\\|_\\infty \\leq \\frac{\\mutinco \\,\\ensuremath{\\lambda_\\ensuremath{n}}}{8}$.  We do so\nin two steps, by using Lemmas~\\ref{LEM_D_CONV} and~\\ref{LEM_R_CONV}\nconsecutively.  First, we show that the\nprecondition~\\eqref{EqnDconvAss} required for Lemma~\\ref{LEM_D_CONV}\nto hold is satisfied under the specified conditions on $\\ensuremath{n}$ and\n$\\ensuremath{\\lambda_\\ensuremath{n}}$. From Lemma~\\ref{LemWconv} and our choice of regularization\nconstant \\mbox{$\\ensuremath{\\lambda_\\ensuremath{n}} = (8\/\\mutinco)\\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$,}\n\\begin{eqnarray*}\n2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big( \\,\\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}} \\big) & \\leq & 2\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 + \\frac{8}{\\mutinco}\\Big) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}),\n\\end{eqnarray*}\nprovided $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\le 1\/v_*$.  From\nthe lower bound~\\eqref{EqnSampleBound} and the\nmonotonicity~\\eqref{EqnMonot} of the tail inverse functions, we have\n\\begin{eqnarray}\n\\label{EqnDelPreqBound}\n2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 + \\frac{8}{\\mutinco}\\Big) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) & \\leq & \\min \\big\\{ \\frac{1}{3\n\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}, \\; \\frac{1}{3 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\;\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\ensuremath{\\ensuremath{d}}}\n\\big\\},\n\\end{eqnarray}\nshowing that the assumptions of Lemma~\\ref{LEM_D_CONV} are satisfied.\nApplying this lemma, we conclude that\n\\begin{eqnarray}\n\\label{EqnDelBound}\n\\| \\Delta \\|_\\infty & \\leq & 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big( \\, \\|\\ensuremath{W}\\|_\\infty +\n\\ensuremath{\\lambda_\\ensuremath{n}} \\big) \\; \\leq \\; 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 + \\frac{8}{\\mutinco}\\Big)\n\\, \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray}\n\nTurning next to Lemma~\\ref{LEM_R_CONV}, we see that its assumption\n$\\|\\Delta\\|_{\\infty} \\leq \\frac{1}{3 \\, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}$ holds, by\napplying equations~\\eqref{EqnDelPreqBound} and \\eqref{EqnDelBound}.\nConsequently, we have\n\\begin{eqnarray*}\n\\|\\ensuremath{R}(\\Delta)\\|_\\infty & \\leq & \\frac{3}{2} \\,\\ensuremath{\\ensuremath{d}}\\;\n\\|\\Delta\\|_\\infty^2 \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\\\\n& \\leq & 6 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^2 \\, \\ensuremath{\\ensuremath{d}} \\, \\Big(1 +\n\\frac{8}{\\mutinco}\\Big)^2 [\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})]^{2}\n\\\\\n& = & \\Biggr \\{ 6 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^2 \\, \\ensuremath{\\ensuremath{d}} \\, \\Big(1 +\n\\frac{8}{\\mutinco}\\Big)^2 \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\Biggr\n\\} \\frac{\\mutinco \\ensuremath{\\lambda_\\ensuremath{n}}}{8} \\\\\n& \\leq & \\frac{\\mutinco \\ensuremath{\\lambda_\\ensuremath{n}}}{8},\n\\end{eqnarray*}\nas required, where the final inequality follows from our\ncondition~\\eqref{EqnSampleBound} on the sample size, and the\nmonotonicity property~\\eqref{EqnMonot}.  \n\nOverall, we have shown that the assumption~\\eqref{EqnDetSuff} of\nLemma~\\ref{LemStrictDual} holds, allowing us to conclude that\n$\\ensuremath{\\ThetaWitness} = \\ensuremath{\\widehat{\\Theta}}$. The estimator $\\ensuremath{\\widehat{\\Theta}}$ then satisfies the\n$\\ell_\\infty$-bound~\\eqref{EqnDelBound} of $\\ensuremath{\\ThetaWitness}$, as claimed in\nTheorem~\\ref{ThmMain}(a), and moreover, we have\n$\\ensuremath{\\widehat{\\Theta}}_{\\ensuremath{\\EsetPlus^c}} = \\ensuremath{\\ThetaWitness}_{\\ensuremath{\\EsetPlus^c}} = 0$, as\nclaimed in Theorem~\\ref{ThmMain}(b).  Since the above was conditioned\non the event $\\mathcal{A}$, these statements hold with probability\n$\\ensuremath{\\mathbb{P}}(\\mathcal{A}) \\ge 1 - \\frac{1}{\\ensuremath{p}^{\\tau - 2}}$.\n\n\\subsection{Proof of Theorem~\\ref{ThmModel}}\n\\label{SecThmModel}\n\nWe now turn to the proof of Theorem~\\ref{ThmModel}. A little\ncalculation shows that the assumed lower bound~\\eqref{EqnNumobsModel}\non the sample size $\\ensuremath{n}$ and the monotonicity\nproperty~\\eqref{EqnMonot} together guarantee that\n\\begin{eqnarray*}\n\\ensuremath{\\theta_{\\operatorname{min}}} & > & 4 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 +\\frac{8}{\\mutinco}\\Big) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})\n\\end{eqnarray*}\nProceeding as in the proof of Theorem~\\ref{ThmMain}, with probability\nat least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$, we have the equality\n\\mbox{$\\ensuremath{\\ThetaWitness} = \\ensuremath{\\widehat{\\Theta}}$,} and also that $\\|\\ensuremath{\\ThetaWitness} -\n\\ensuremath{\\Theta^*}\\|_\\infty \\leq \\ensuremath{\\theta_{\\operatorname{min}}}\/2$.  Consequently,\nLemma~\\ref{LemSignConsis} can be applied, guaranteeing that\n$\\textrm{sign}(\\ensuremath{\\Theta^*}_{ij}) = \\textrm{sign}(\\ensuremath{\\ThetaWitness}_{ij})$ for all $(i,j) \\in\n\\ensuremath{E}$.  Overall, we conclude that with probability at least $1 -\n1\/\\ensuremath{p}^{\\tau - 2}$, the sign consistency condition\n$\\textrm{sign}(\\ensuremath{\\Theta^*}_{ij}) = \\textrm{sign}(\\ensuremath{\\widehat{\\Theta}}_{ij})$ holds for all $(i,j)\n\\in \\ensuremath{E}$, as claimed.\n\n\n\n\\def\\hat{\\Delta}} \\def\\Lin{L{\\hat{\\Delta}} \\def\\Lin{L}\n\n\\subsection{Proof of Corollary~\\ref{CorCovBound}}\n\\label{SecCorCovProof}\nWith the shorthand $\\hat{\\Delta}} \\def\\Lin{L = \\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}$, we have\n\\begin{equation*}\n\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*} = (\\ensuremath{\\Theta^*} + \\hat{\\Delta}} \\def\\Lin{L)^{-1} -\n\\inv{\\ensuremath{\\Theta^*}}.\n\\end{equation*}\nFrom the definition~\\eqref{EqnRdefn} of the residual $R(\\cdot)$, this\ndifference can be written as\n\\begin{eqnarray}\n\\label{EqnCovDev}\n\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*} & = & - \\invn{\\ensuremath{\\Theta^*}} \\hat{\\Delta}} \\def\\Lin{L\n\\invn{\\ensuremath{\\Theta^*}} + R(\\hat{\\Delta}} \\def\\Lin{L).\n\\end{eqnarray}\n\nProceeding as in the proof of Theorem~\\ref{ThmMain} we condition on\nthe event $\\mathcal{A} = \\{ \\|\\ensuremath{W}\\|_\\infty \\leq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\}$, and which holds with\nprobability $\\ensuremath{\\mathbb{P}}(\\mathcal{A}) \\ge 1 - \\frac{1}{\\ensuremath{p}^{\\tau - 2}}$.\nAs in the proof of that theorem, we are guaranteed that the\nassumptions of Lemma~\\ref{LEM_R_CONV} are satisfied, allowing us to\nconclude\n\\begin{align}\\label{EqnRequiv}\n\\ensuremath{R}(\\hat{\\Delta}} \\def\\Lin{L) = \\invn{\\opt{\\Theta}} \\hat{\\Delta}} \\def\\Lin{L \\invn{\\opt{\\Theta}}\n\\hat{\\Delta}} \\def\\Lin{L J \\invn{\\opt{\\Theta}},\n\\end{align}\nwhere $J \\ensuremath{:  =} \\sum_{k=0}^{\\infty} (-1)^{k} \\big(\\invn{\\opt{\\Theta}}\n\\hat{\\Delta}} \\def\\Lin{L\\big)^{k}$ has norm $\\matnorm{J^T}{\\infty} \\leq 3\/2$.\n\nWe begin by proving the bound~\\eqref{EqnCovInftyBound}.  From\nequation~\\eqref{EqnCovDev}, we have $\\vecnorm{\\hat{\\CovHat} -\n\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} \\leq \\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} +\n\\vecnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty}$.  From Lemma~\\ref{LEM_R_CONV}, we have\nthe elementwise $\\ell_\\infty$-norm bound\n\\begin{eqnarray*}\n\\| \\ensuremath{R}(\\hat{\\Delta}} \\def\\Lin{L)\\|_\\infty & \\leq & \\frac{3}{2} \\ensuremath{\\ensuremath{d}}\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_\\infty^2 \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3.\n\\end{eqnarray*}\nThe quantity $L(\\hat{\\Delta}} \\def\\Lin{L)$ in turn can be bounded as follows,\n\\begin{eqnarray*}\n\\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} & = &\n\\max_{ij}\\big|e_{i}^{T}\\invn{\\ensuremath{\\Theta^*}} \\hat{\\Delta}} \\def\\Lin{L \\invn{\\ensuremath{\\Theta^*}}\ne_{j}\\big|\\\\ &\\le& \\max_{i} \\|e_{i}^{T}\\invn{\\ensuremath{\\Theta^*}}\\|_{1}\n\\max_{j} \\|\\hat{\\Delta}} \\def\\Lin{L \\invn{\\ensuremath{\\Theta^*}} e_j\\|_{\\infty}\\\\ &\\le& \\max_{i}\n\\|e_{i}^{T}\\invn{\\ensuremath{\\Theta^*}}\\|_{1} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} \\|\\max_{j}\n\\|\\invn{\\ensuremath{\\Theta^*}} e_j\\|_{1}\n\\end{eqnarray*}\nwhere we used the inequality that $\\|\\hat{\\Delta}} \\def\\Lin{L u \\|_{\\infty} \\le\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} \\|u\\|_{1}$.  Simplifying further, we obtain\n\\begin{eqnarray*}\n\\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} & \\leq &\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty}\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{1}\\\\ \n& \\leq & \\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty}^{2}\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} \\\\ \n& \\leq & \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty},\n\\end{eqnarray*}\nwhere we have used the fact that $\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{1} =\n\\matnorm{[\\invn{\\ensuremath{\\Theta^*}}]^{T}}{\\infty} =\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty}$, which follows from the symmetry\nof $\\invn{\\ensuremath{\\Theta^*}}$.  Combining the pieces, we obtain\n\\begin{eqnarray}\n\\label{EqnCovDevInftya}\n\\vecnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} &\\le &\n\\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} + \\vecnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty}\\\\\n& \\leq & \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} + \\frac{3}{2} \\ensuremath{\\ensuremath{d}}\n\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\|\\hat{\\Delta}} \\def\\Lin{L\\|_\\infty^2. \\nonumber\n\\end{eqnarray}\nThe claim then follows from the elementwise $\\ell_\\infty$-norm\nbound~\\eqref{EqnEllinfBound} from Theorem~\\ref{ThmMain}.\n\n\nNext, we establish the bound~\\eqref{EqnCovSpectralBound} in spectral\nnorm.  Taking the $\\ell_{\\infty}$ operator norm of both sides of\nequation~\\eqref{EqnCovDev} yields the inequality $\\matnorm{\\hat{\\CovHat} -\n\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} \\leq \\matnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} +\n\\matnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty}$. Using the\nexpansion~\\eqref{EqnRequiv}, and the sub-multiplicativity of the\n$\\ell_{\\infty}$ operator norm, we obtain\n\\begin{eqnarray*}\n\\matnorm{\\ensuremath{R}(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} & \\leq &\n\\matnorm{\\invn{\\opt{\\Theta}}}{\\infty} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}\n\\matnorm{\\invn{\\opt{\\Theta}}}{\\infty} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}\n\\matnorm{J}{\\infty} \\matnorm{\\invn{\\opt{\\Theta}}}{\\infty}\\\\ &\\le&\n\\matnorm{\\invn{\\opt{\\Theta}}}{\\infty}^{3} \\matnorm{J}{\\infty}\n\\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}^{2}\\\\ \n& \\leq & \\frac{3}{2} \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}^{2},\n\\end{eqnarray*}\nwhere the last inequality uses the bound $\\matnorm{J}{\\infty} \\leq\n3\/2$.  (Proceeding as in the proof of Lemma~\\ref{LEM_R_CONV}, this\nbound holds conditioned on $\\mathcal{A}$, and for the sample size\nspecified in the theorem statement.)  In turn, the term $L(\\hat{\\Delta}} \\def\\Lin{L)$\ncan be bounded as\n\\begin{eqnarray*}\n\\matnorm{L(\\Delta)}{\\infty} & \\leq & \\matnorm{\\invn{\\ensuremath{\\Theta^*}}\n\\hat{\\Delta}} \\def\\Lin{L \\invn{\\ensuremath{\\Theta^*}}}{\\infty}\\\\ &\\le&\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty}^{2} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}\\\\\n& \\leq & \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty},\n\\end{eqnarray*}\nwhere the second inequality uses the sub-multiplicativity of the\n$\\ell_\\infty$-operator norm.  Combining the pieces yields\n\\begin{eqnarray}\n\\label{EqnCovDevInftyOp}\n\\matnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} & \\leq &\n\\matnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} + \\matnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} \\;\n\\leq \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty} + \\frac{3}{2} \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_\\infty^2.\n\\end{eqnarray}\nConditioned on the event $\\mathcal{A}$, we have the\nbound~\\eqref{EqnPrecInftyOp} on the $\\ell_\\infty$-operator norm\n\\begin{eqnarray*}\n\\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty} & \\leq & 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 +\n\\frac{8}{\\mutinco}\\Big) \\, \\ensuremath{\\ensuremath{d}} \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray*}\nSubstituting this bound, as well as the elementwise $\\ell_\\infty$-norm\nbound~\\eqref{EqnEllinfBound} from Theorem~\\ref{ThmMain}, into the\nbound~\\eqref{EqnCovDevInftyOp} yields the stated claim.\n\n\n\n\\section{Experiments}\n\\label{SecExperiments}\n\nIn this section, we illustrate our results with various experimental\nsimulations, reporting results in terms of the probability of correct\nmodel selection (Theorem~\\ref{ThmModel}) or the $\\ell_\\infty$-error\n(Theorem~\\ref{ThmMain}).  For these illustrations, we study the case\nof Gaussian graphical models, and results for three different classes\nof graphs, namely chains, grids, and star-shaped graphs. We also\nconsider various scalings of the quantities which affect the\nperformance of the estimator: in addition the triple $(\\ensuremath{n}, \\ensuremath{p},\n\\ensuremath{\\ensuremath{d}})$, we also report some results concerning the role of the\nparameters $\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ and $\\ensuremath{\\theta_{\\operatorname{min}}}$ that we have identified in the\nmain theorems. For all results\nreported here, we solved the resulting $\\ell_1$-penalized\nlog-determinant program~\\eqref{EqnGaussMLE} using the \\texttt{glasso}\nprogram of \\citet{FriedHasTib2007}, which builds on the block\nco-ordinate descent algorithm of~\\citet{AspreBanG2008}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\raisebox{.6in}{\\widgraph{0.25\\textwidth}{chain.eps}}& &\n\\raisebox{.0in}{\\widgraph{0.28\\textwidth}{grid4.eps}}& &\n\\raisebox{.1in}{\\widgraph{0.25\\textwidth}{stargraph.eps}}\\\\\n(a) && (b) && (c)\n\\end{tabular}\n\\caption{Illustrations of different graph classes used in simulations.\n(a) Chain ($\\ensuremath{\\ensuremath{d}} = 2$).  (b) Four-nearest neighbor grid ($\\ensuremath{\\ensuremath{d}} =\n4$) and (c) Star-shaped graph ($\\ensuremath{\\ensuremath{d}} \\in \\{1,\\hdots,\\ensuremath{p} - 1\\}$).}\n\\label{FigGraphs}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{FigGraphs} illustrates the three types of graphs used in\nour simulations: chain graphs (panel (a)), four-nearest neighbor\nlattices or grids (panel (b)), and star-shaped graphs (panel (c)).\nFor the chain and grid graphs, the maximal node degree $\\ensuremath{\\ensuremath{d}}$ is\nfixed by definition, to $\\ensuremath{\\ensuremath{d}} =2$ for chains, and $\\ensuremath{\\ensuremath{d}} =4$ for\nthe grids.  Consequently, these graphs can capture the dependence of\nthe required sample size $\\ensuremath{n}$ only as a function of the graph\nsize $\\ensuremath{p}$, and the parameters $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$,\n$\\ensuremath{\\theta_{\\operatorname{min}}}$).  The star graph allows us to vary both $\\ensuremath{\\ensuremath{d}}$ and\n$\\ensuremath{p}$, since the degree of the central hub can be varied between $1$\nand $\\ensuremath{p} -1$.  For each graph type, we varied the size of the graph\n$\\ensuremath{p}$ in different ranges, from $\\ensuremath{p} =64$ upwards to $\\ensuremath{p} =\n375$.\n\n\nFor the chain and star graphs, we define a covariance matrix\n$\\ensuremath{\\ensuremath{\\Sigma}^*}$ with entries $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} = 1$ for all $i =1,\n\\ldots, \\ensuremath{p}$, and $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = \\rho$ for all $(i,j) \\in\n\\ensuremath{E}$ for specific values of $\\rho$ specified below.  Note that these\ncovariance matrices are sufficient to specify the full model.  For the\nfour-nearest neighbor grid graph, we set the entries of the\nconcentration matrix $\\ensuremath{\\Theta^*}_{ij} = \\omega$ for $(i,j) \\in \\ensuremath{E}$,\nwith the value $\\omega$ specified below. In all cases, we set the\nregularization parameter $\\ensuremath{\\lambda_\\ensuremath{n}}$ proportional to\n$\\sqrt{\\log(\\ensuremath{p})\/\\ensuremath{n}}$, as suggested by Theorems~\\ref{ThmMain}\nand~\\ref{ThmModel}, which is reasonable since the main purpose of\nthese simulations is to illustrate our theoretical results.  However,\nfor general data sets, the relevant theoretical parameters cannot be\ncomputed (since the true model is unknown), so that a data-driven\napproach such as cross-validation might be required for selecting the\nregularization parameter $\\ensuremath{\\lambda_\\ensuremath{n}}$.\n\n\n\nGiven a Gaussian graphical model instance, and the number of samples\n$\\ensuremath{n}$, we drew $N = 100$ batches of $\\ensuremath{n}$ independent samples\nfrom the associated multivariate Gaussian distribution. We estimated\nthe probability of correct model selection as the fraction of the $N=\n100$ trials in which the estimator recovers the signed-edge set\nexactly.\n\n\\defK{K}\nNote that any multivariate Gaussian random vector is sub-Gaussian; in\nparticular, the rescaled variates $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are\nsub-Gaussian with parameter $\\ensuremath{\\sigma} = 1$, so that the elementwise\n$\\ell_\\infty$-bound from Corollary~\\ref{CorEllinfSubg} applies.\nSuppose we collect relevant parameters such as $\\ensuremath{\\theta_{\\operatorname{min}}}$ and the\ncovariance and Hessian-related terms $\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ and\n$\\mutinco$ into a single ``model-complexity'' term $K$ defined\nas\n\\begin{eqnarray}\\label{EqnKdefn}\nK & \\ensuremath{:  =} & \\left[(1 + 8 \\mutinco^{-1}) (\\max_{i}\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\n\\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2},\\frac{\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}}{\n\\ensuremath{\\ensuremath{d}}\\, \\ensuremath{\\theta_{\\operatorname{min}}}}\\}\\right].\n\\end{eqnarray}\nThen, as a corollary of Theorem~\\ref{ThmModel}, a sample size of order \n\\begin{eqnarray}\n\\label{EqnCrudeBound}\n\\ensuremath{n} & = & \\Omega\\left( K^{2} \\; \\ensuremath{\\ensuremath{d}}^2 \\, \\tau \\log \\ensuremath{p}\n\\right),\n\\end{eqnarray}\nis sufficient for model selection consistency with probability greater\nthan $1-1\/\\ensuremath{p}^{\\tau-2}$. In the subsections to follow, we\ninvestigate how the empirical sample size $\\ensuremath{n}$ required for model\nselection consistency scales in terms of graph size $\\ensuremath{p}$, maximum\ndegree $\\ensuremath{\\ensuremath{d}}$, as well as the ``model-complexity'' term $K$ defined\nabove.\n \n\\newcommand{.48\\textwidth}{.48\\textwidth}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\widgraph{.48\\textwidth}{figures\/chainprobvsn.eps} & \\hspace*{.1in} &\n\\widgraph{.48\\textwidth}{figures\/chainprobvsnlogp.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{Simulations for chain graphs with varying number of nodes\n$\\ensuremath{p}$, edge covariances $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 0.10$. Plots of\nprobability of correct signed edge-set recovery plotted versus the\nordinary sample size $\\ensuremath{n}$ in panel (a), and versus the rescaled\nsample size $\\ensuremath{n}\/\\log \\ensuremath{p}$ in panel (b). Each point corresponds\nto the average over $100$ trials. }\n\\label{FigChainProbvsNP}\n\\end{figure}\n\n\n\n\\subsection{Dependence on graph size}\n\nPanel (a) of Figure~\\ref{FigChainProbvsNP} plots the probability of\ncorrect signed edge-set recovery against the sample size $\\ensuremath{n}$ for\na chain-structured graph of three different sizes.  For these chain\ngraphs, regardless of the number of nodes $\\ensuremath{p}$, the maximum node\ndegree is constant $\\ensuremath{\\ensuremath{d}} = 2$, while the edge covariances are set\nas $\\ensuremath{\\Sigma}_{ij} = 0.2$ for all $(i,j) \\in \\ensuremath{E}$, so that the\nquantities $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\mutinco)$ remain constant.  Each of\nthe curve in panel (a) corresponds to a different graph size $\\ensuremath{p}$.\nFor each curve, the probability of success starts at zero (for small\nsample sizes $\\ensuremath{n}$), but then transitions to one as the sample\nsize is increased.  As would be expected, it is more difficult to\nperform model selection for larger graph sizes, so that (for instance)\nthe curve for $\\ensuremath{p} = 375$ is shifted to the right relative to the\ncurve for $\\ensuremath{p} = 64$.  Panel (b) of Figure~\\ref{FigChainProbvsNP}\nreplots the same data, with the horizontal axis rescaled by $(1\/\\log\n\\ensuremath{p})$.  This scaling was chosen because for sub-Gaussian tails, our\ntheory predicts that the sample size should scale logarithmically with\n$\\ensuremath{p}$ (see equation~\\eqref{EqnCrudeBound}).  Consistent with this\nprediction, when plotted against the rescaled sample size\n$\\ensuremath{n}\/\\log \\ensuremath{p}$, the curves in panel (b) all stack up.\nConsequently, the ratio $(\\ensuremath{n}\/\\log\\ensuremath{p})$ acts as an effective\nsample size in controlling the success of model selection, consistent\nwith the predictions of Theorem~\\ref{ThmModel} for sub-Gaussian\nvariables.\n\nFigure~\\ref{FigStarProbvsNP} shows the same types of plots for a\nstar-shaped graph with fixed maximum node degree $\\ensuremath{\\ensuremath{d}} = 40$, and\nFigure~\\ref{FigGridProbvsNP} shows the analogous plots for a grid\ngraph with fixed degree $\\ensuremath{\\ensuremath{d}} = 4$.  As in the chain case, these\nplots show the same type of stacking effect in terms of the scaled\nsample size $\\ensuremath{n}\/\\log \\ensuremath{p}$, when the degree $\\ensuremath{\\ensuremath{d}}$ and other\nparameters ($(\\mutinco, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}})$) are held fixed.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\widgraph{.48\\textwidth}{figures\/starprobvsn.eps} & \\hspace*{.1in} &\n\\widgraph{.48\\textwidth}{figures\/starprobvsnlogp.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{Simulations for a star graph with varying number of nodes\n$\\ensuremath{p}$, fixed maximal degree $\\ensuremath{\\ensuremath{d}} = 40$, and edge covariances\n$\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 1\/16$ for all edges. Plots of probability of\ncorrect signed edge-set recovery versus the sample size $\\ensuremath{n}$ in\npanel (a), and versus the rescaled sample size $\\ensuremath{n}\/\\log \\ensuremath{p}$ in\npanel (b).  Each point corresponds to the average over $N = 100$\ntrials.}\n\\label{FigStarProbvsNP}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\widgraph{.48\\textwidth}{figures\/gridprobvsn.eps} &\n\\widgraph{.48\\textwidth}{figures\/gridprobvsnlogp.eps} \\\\\n(a) & (b)\n\\end{tabular}\n\\end{center}\n\\caption{Simulations for $2$-dimensional lattice with\n$4$-nearest-neighbor interaction, edge strength interactions\n$\\ensuremath{\\Theta^*}_{ij} = 0.1$, and a varying number of nodes $\\ensuremath{p}$. Plots\nof probability of correct signed edge-set recovery versus the sample\nsize $\\ensuremath{n}$ in panel (a), and versus the rescaled sample size\n$\\ensuremath{n}\/\\log \\ensuremath{p}$ in panel (b).  Each point corresponds to the\naverage over $N = 100$ trials.}\n\\label{FigGridProbvsNP}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\widgraph{.48\\textwidth}{figures\/dvarstarprobvsn.eps} &\n\\widgraph{.48\\textwidth}{figures\/dvarstarprobvsnd.eps} \\\\\n(a) & (b)\n\\end{tabular}\n\\end{center}\n\\caption{ Simulations for star graphs with fixed number of nodes\n$\\ensuremath{p} = 200$, varying maximal (hub) degree $\\ensuremath{\\ensuremath{d}}$, edge\ncovariances $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 2.5\/\\ensuremath{\\ensuremath{d}}$.  Plots of probability of\ncorrect signed edge-set recovery versus the sample size $\\ensuremath{n}$ in\npanel (a), and versus the rescaled sample size $\\ensuremath{n}\/\\ensuremath{\\ensuremath{d}}$ in\npanel (b). }\n\\label{FigStarProbvsND}\n\\end{figure}\n\n\\subsection{Dependence on the maximum node degree} \n\nPanel (a) of Figure~\\ref{FigStarProbvsND} plots the probability of\ncorrect signed edge-set recovery against the sample size $\\ensuremath{n}$ for\nstar-shaped graphs; each curve corresponds to a different choice of\nmaximum node degree $\\ensuremath{\\ensuremath{d}}$, allowing us to investigate the\ndependence of the sample size on this parameter.  So as to control\nthese comparisons, the models are chosen such that quantities other\nthan the maximum node-degree $\\ensuremath{\\ensuremath{d}}$ are fixed: in particular, we\nfix the number of nodes $\\ensuremath{p} = 200$, and the edge covariance entries\nare set as $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 2.5\/\\ensuremath{\\ensuremath{d}}$ for $(i,j) \\in \\ensuremath{E}$ so\nthat the quantities $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\mutinco)$ remain constant.\nThe minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$ in turn scales as $1\/\\ensuremath{\\ensuremath{d}}$.  Observe\nhow the plots in panel (a) shift to the right as the maximum node\ndegree $\\ensuremath{\\ensuremath{d}}$ is increased, showing that star-shaped graphs with\nhigher degrees are more difficult.  In panel (b) of\nFigure~\\ref{FigStarProbvsND}, we plot the same data versus the\nrescaled sample size $\\ensuremath{n}\/\\ensuremath{\\ensuremath{d}}$.  Recall that if all the curves\nwere to stack up under this rescaling, then it means the required\nsample size $\\ensuremath{n}$ scales linearly with $\\ensuremath{\\ensuremath{d}}$.  These plots are\ncloser to aligning than the unrescaled plots, but the agreement is not\nperfect.  In particular, observe that the curve $\\ensuremath{\\ensuremath{d}}$ (right-most\nin panel (a)) remains a bit to the right in panel (b), which suggests\nthat a somewhat more aggressive rescaling---perhaps\n$\\ensuremath{n}\/\\ensuremath{\\ensuremath{d}}^\\gamma$ for some $\\gamma \\in (1,2)$---is appropriate.\n\nNote that for $\\ensuremath{\\theta_{\\operatorname{min}}}$ scaling as $1\/\\ensuremath{\\ensuremath{d}}$, the sufficient\ncondition from Theorem~\\ref{ThmModel}, as summarized in\nequation~\\eqref{EqnCrudeBound}, is $\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\log\n\\ensuremath{p})$, which appears to be overly conservative based on these data.\nThus, it might be possible to tighten our theory under certain\nregimes.\n\n\n\n\\subsection{Dependence on covariance and Hessian terms}\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\nNext, we study the dependence of the sample size required for model\nselection consistency on the model complexity term $K$ defined\nin \\eqref{EqnKdefn}, which is a collection of the quantities\n$\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ and $\\mutinco$ defined by the covariance\nmatrix and Hessian, as well as the minimum value\n$\\ensuremath{\\theta_{\\operatorname{min}}}$. Figure~\\ref{FigChainNstarvsGamma} plots the probability\nof correct signed edge-set recovery versus the sample size $\\ensuremath{n}$\nfor chain graphs.  Here each curve corresponds to a different setting\nof the model complexity factor $K$, but with a fixed\nnumber of nodes $\\ensuremath{p} = 120$, and maximum node-degree $\\ensuremath{\\ensuremath{d}} = 2$.\nWe varied the actor $K$ by varying the value $\\rho$ of the\nedge covariances $\\ensuremath{\\Sigma}_{ij} = \\rho,\\, (i,j) \\in \\ensuremath{E}$.  Notice how\nthe curves, each of which corresponds to a different model complexity\nfactor, shift rightwards as $K$ is increased so that\nmodels with larger values of $K$ require greater number of\nsamples $\\ensuremath{n}$ to achieve the same probability of correct model\nselection.  These rightward-shifts are in qualitative agreement with\nthe prediction of Theorem~\\ref{ThmMain}, but we suspect that our\nanalysis is not sharp enough to make accurate quantitative predictions\nregarding this scaling.\n\n\\newcommand{.5\\textwidth}{.5\\textwidth}\n\n\\begin{figure}\n\\begin{center}\n\\widgraph{.5\\textwidth}{figures\/gammavarchainprobvsn.eps} \\\\\n\\end{center}\n\\caption{Simulations for chain graph with fixed number of nodes $\\ensuremath{p}\n= 120$, and varying model complexity $K$. Plot of\nprobability of correct signed edge-set recovery versus the sample size\n$\\ensuremath{n}$.}\n\\label{FigChainNstarvsGamma}\n\\end{figure}\n\n\\subsection{Convergence rates in elementwise $\\ell_\\infty$-norm}\n\nFinally, we report some simulation results on the convergence rate in\nelementwise $\\ell_\\infty$-norm.  According to\nCorollary~\\ref{CorEllinfSubg}, in the case of sub-Gaussian tails, if\nthe elementwise $\\ell_\\infty$-norm should decay at rate\n${\\mathcal{O}}(\\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}})$, once the sample size\n$\\ensuremath{n}$ is sufficiently large.  Figure~\\ref{FigStarNormvsBaserate}\nshows the behavior of the elementwise $\\ell_\\infty$-norm for\nstar-shaped graphs of varying sizes $\\ensuremath{p}$.  The results reported\nhere correspond to the maximum degree $\\ensuremath{\\ensuremath{d}} = \\lceil 0.1 \\ensuremath{p}\n\\rceil$; we also performed analogous experiments for $\\ensuremath{\\ensuremath{d}} =\n{\\mathcal{O}}(\\log \\ensuremath{p})$ and $\\ensuremath{\\ensuremath{d}} = {\\mathcal{O}}(1)$, and observed\nqualitatively similar behavior.  The edge correlations were set as\n$\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 2.5\/\\ensuremath{\\ensuremath{d}}$ for all $(i,j) \\in \\ensuremath{E}$ so that the\nquantities $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\mutinco)$ remain constant.  With\nthese settings, each curve in Figure~\\ref{FigStarNormvsBaserate}\ncorresponds to a different problem size, and plots the elementwise\n$\\ell_\\infty$-error versus the rescaled sample size $\\ensuremath{n}\/\\log\n\\ensuremath{p}$, so that we expect to see curves of the form $f(t) =\n1\/\\sqrt{t}$.  The curves show that when the rescaled sample size\n$(\\ensuremath{n}\/\\log \\ensuremath{p})$ is larger than some threshold (roughly $40$ in\nthe plots shown), the elementwise $\\ell_\\infty$ norm decays at the\nrate $\\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}}$, which is consistent with\nCorollary~\\ref{CorEllinfSubg}.\n\n\\begin{figure}\n\\begin{center}\n\\widgraph{.5\\textwidth}{figures\/starcvgrate.eps} \\\\\n\\end{center}\n\\caption{Simulations for a star graph with varying number of nodes\n$\\ensuremath{p}$, maximum node degree $\\ensuremath{\\ensuremath{d}} = \\lceil 0.1 \\ensuremath{p} \\rceil$, edge\ncovariances $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}= 2.5\/\\ensuremath{\\ensuremath{d}}$. Plot of the element-wise\n$\\ell_\\infty$ norm of the concentration matrix estimate error\n$\\vecnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{\\infty}$ versus the rescaled sample\nsize $\\ensuremath{n}\/\\log (\\ensuremath{p})$.}\n\\label{FigStarNormvsBaserate}\n\\end{figure}\n\n\\section{Discussion}\n\nThe focus of this paper is the analysis of the high-dimensional\nscaling of the $\\ell_1$-regularized log determinant\nproblem~\\eqref{EqnGaussMLE} as an estimator of the concentration\nmatrix of a random vector.  Our main contributions were to derive\nsufficient conditions for its model selection consistency as well as\nconvergence rates in both elementwise $\\ell_\\infty$-norm, as well as\nFrobenius and spectral norms.  Our results allow for a range of tail\nbehavior, ranging from the exponential-type decay provided by Gaussian\nrandom vectors (and sub-Gaussian more generally), to polynomial-type\ndecay guaranteed by moment conditions.  In the Gaussian case, our\nresults have natural interpretations in terms of Gaussian Markov\nrandom fields.\n\nOur main results relate the i.i.d. sample size $\\ensuremath{n}$ to various\nparameters of the problem required to achieve consistency.  In\naddition to the dependence on matrix size $\\ensuremath{p}$, number of edges\n$\\ensuremath{s}$ and graph degree $\\ensuremath{\\ensuremath{d}}$, our analysis also illustrates\nthe role of other quantities, related to the structure of the\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ and the Hessian of the objective\nfunction, that have an influence on consistency rates.  Our main\nassumption is an irrepresentability or mutual incoherence condition,\nsimilar to that required for model selection consistency of the Lasso,\nbut involving the Hessian of the log-determinant objective\nfunction~\\eqref{EqnGaussMLE}, evaluated at the true model. When the\ndistribution of $X$ is multivariate Gaussian, this Hessian is the\nFisher information matrix of the model, and thus can be viewed as an\nedge-based counterpart to the usual node-based covariance matrix We\nreport some examples where irrepresentability condition for the Lasso\nhold and the log-determinant condition fails, but we do not know in\ngeneral if one requirement dominates the other.  In addition to these\ntheoretical results, we provided a number of simulation studies\nshowing how the sample size required for consistency scales with\nproblem size, node degrees, and the other complexity parameters\nidentified in our analysis.\n\nThere are various interesting questions and possible extensions to\nthis paper.  First, in the current paper, we have only derived\nsufficient conditions for model selection consistency. As in past work\non the Lasso~\\cite{Wainwright2006_new}, it would also be interesting\nto derive a \\emph{converse result}---namely, to prove that if the\nsample size $\\ensuremath{n}$ is smaller than some function of $(\\ensuremath{p},\n\\ensuremath{\\ensuremath{d}}, \\ensuremath{s})$ and other complexity parameters, then regardless\nof the choice of regularization constant, the log-determinant method\nfails to recover the correct graph structure.  Second, while this\npaper studies the problem of estimating a fixed graph or concentration\nmatrix, a natural extension would allow the graph to vary over time, a\nproblem setting which includes the case where the observations are\ndependent. For instance, \\citet{ZhoLafWas08} study the estimation of\nthe covariance matrix of a Gaussian distribution in a time-varying\nsetting, and it would be interesting to extend results of this paper\nto this more general setting.\n\n\n\n\\subsection*{Acknowledgements}  We thank Shuheng Zhou for helpful comments\non an earlier draft of this work.  Work was partially supported by NSF\ngrant DMS-0605165.  Yu also acknowledges support from ARO\nW911NF-05-1-0104, NSFC-60628102, and a grant from MSRA.\n\n\n\n\\subsection*{#1} \\end{center}}\n\n\\newcommand{\\ensuremath{\\mathbb{P}}}{\\ensuremath{\\mathbb{P}}}\n\n\n\n\\newcommand{\\ensuremath{\\mathcal{Q}}}{\\ensuremath{\\mathcal{Q}}}\n\n\\newcommand{\\tracer}[2]{\\ensuremath{\\langle \\!\\langle {#1}, \\; {#2}\n\\rangle 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+{"text":"\\section{Introduction}\n\n\nThe authors of Ref. \\cite{zhao} investigated the effects of the variation of the mass parameter $a$ on the thick branes. They used a real scalar field, which has a potential of the $\\phi^{6}$ model, as the background field of the thick branes. It was found that the number of the bound states (in the case without gravity) or the resonant states (in the case with gravity) increases with the parameter $a$. That work considered the simplest Yukawa coupling $\\eta\\bar{\\Psi}\\phi\\Psi$, where $\\eta$ is the coupling constant. The authors stated that as the value of $a$ is increasing, the maximum of the matter energy density splits into two new maxima, and the distance of the new maxima increases and the brane gets thicker. The authors also stated that the brane with a big value of $a$ would trap fermions more efficiently.\n\nIn this paper, we reinvestigated the effect of the variation of the mass parameter $a$ on the thick branes, because the above investigation does not analyze the zero mode in details and contains some misconceptions. We only focus attention in the case with gravity. We find that the variation of $a$ on the thick brane is associated to the phenomenon of brane splitting. From the static equation of motion, we analyze the asymptotic behavior of $A(y)$ and find that the zero mode for left-handed fermions can be localized on the brane depending on the value for the coupling constant $\\eta$ and the mass parameter $a$. We also show that as the value of $a$ is increasing the simplest Yukawa coupling does not support the localization of fermions on the brane, as incompletely argued in Ref. \\cite{zhao}.\n\n\n\n\\section{Thick brane with gravity}\n\nThe action for our system is described by \\cite{cam}\n\\begin{equation}\\label{ac1}\n    S=\\int d^4xdy\\sqrt{ -g}\\left[\\frac{1}{4}\\,R-\\Lambda-\\frac{1}{2}g^{MN}\\partial_{M}\\phi\\partial^{N}\\phi-V(\\phi) \\right],\n\\end{equation}\n\\noindent where $M,N=0,1,2,3,4$, $\\Lambda$ is the 5D bulk cosmological constant and the scalar potential $V(\\phi)$ is given by \\cite{zhao}\n\\begin{equation}\\label{pot1}\n    V(\\phi)=a\\phi^{2}-b\\phi^{4}+c\\phi^{6}\\,,\n\\end{equation}\n\n\\noindent where $a,b,c>0$. There are three minima for $V(\\phi)$, one is at $\\phi^{(1)}=0$ (local minima) corresponding to a disordered bulk phase and the other two are at $\\phi^{(2)}=-\\phi^{(3)}=v$ (global minima) with\n\\begin{equation}\\label{v}\n    v=\\sqrt{\\frac{\\sqrt{b^{2}-3ac}}{3c}+\\frac{b}{3c}}\\,.\n\\end{equation}\n\n\\noindent They are degenerated and correspond to ordered bulk phases. As $a=a_{c}$ ($a_{c}=b^{2}\/4c$), $V(\\phi^{(1)})=V(\\phi^{(2)})=V(\\phi^{(3)})$, $V(\\phi)$ has three degenerated global minima. For the case with gravity, the critical value of $a$ is not $a_{c}$ but a smaller effective critical value $a_{*}$. In this case, $a_{c}=a_{*}=0.837$ \\cite{zhao}. The line element in this model is considered as\n\\begin{equation}\\label{metric}\n    ds^{2}=g_{ab}dx^{a}dx^{b}=\\mathrm{e}^{2A(y)}\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}+dy^{2},\n\\end{equation}\n\n\\noindent where $\\mu,\\nu=0,1,2,3$, $\\eta_{\\mu\\nu}=\\mathrm{diag}(-1,1,1,1)$, and $\\mathrm{e}^{2A}$ is the so-called warp factor. We suppose that $A=A(y)$ and $\\phi=\\phi(y)$.\n\n\n\\subsection{Effects of the variation of the mass parameter $a$ on the thick brane}\n\nFor this model, the equations of motion are\n\\begin{equation}\\label{em1b}\n    \\phi^{\\prime\\prime}=-4A^{\\prime}\\phi^{\\prime}+\\frac{dV(\\phi)}{d\\phi},\n\\end{equation}\n\\begin{equation}\\label{em2b}\n    A^{\\prime\\prime}+2A^{\\prime}\\,^{2}=-\\frac{1}{3}\\,\\phi^{\\prime}\\,^{2}\n    \\frac{2}{3}\\,(V+\\Lambda)\\,,\n\\end{equation}\n\\begin{equation}\\label{em3b}\n    A^{\\prime}\\,^{2}=\\frac{1}{6}\\,\\phi^{\\prime}\\,^{2}-\\frac{1}{3}\\,(V+\\Lambda)\\,.\n\\end{equation}\n\n\\noindent It is possible to rewrite (\\ref{em2b}) and (\\ref{em3b}) as\n\\begin{equation}\\label{em3c}\nA^{\\prime\\prime}=-\\frac{2}{3}\\,\\phi^{\\prime}\\,^{2}\\,.\n\\end{equation}\n\n\\noindent The boundary conditions can be read as follows\n\\begin{equation}\\label{bc1}\n    A(0)=A^{\\prime}(0)=\\phi(0)=0\\,,\n\\end{equation}\n\\begin{equation}\\label{bc2}\n    \\phi(+\\infty)=-\\phi(-\\infty)=v.\n\\end{equation}\n\n\nThe matter energy density has the form\n\\begin{equation}\\label{de}\n    T_{00}=\\rho(y)=\\mathrm{e}^{2A(y)}\\left[ \\frac{1}{2}\\,\\left( \\frac{d\\phi}{dy\n    \\right)^{2}+V\\left(\\phi\\right) \\right].\n\\end{equation}\n\n\\noindent At this point, it is also instructive to analyze the matter energy of the toy model\n\\begin{equation}\\label{ephi}\n    E_{\\phi}=\\int^{\\infty}_{-\\infty}dy\\,T_{00}\\,,\n\\end{equation}\n\n\\noindent substituting (\\ref{de}) in (\\ref{ephi}), we get\n\\begin{equation}\\label{ephi2}\n    E_{\\phi}=\\int^{\\infty}_{-\\infty}dy\\,\\mathrm{e}^{2A(y)}\\left[ \\frac{1}{2}\\,\\left( \\frac{d\\phi}{dy\n    \\right)^{2}+V\\left(\\phi\\right) \\right]\\,,\n\\end{equation}\n\n\\noindent using (\\ref{em3b}) and (\\ref{em3c}), we obtain the value of the matter energy given by\n\\begin{equation}\\label{ephi3}\n    E_{\\phi}=\\frac{3}{2}\\,\\left[\\mathrm{e}^{2A(-\\infty)}A^{\\prime}(-\\infty)\n    \\mathrm{e}^{2A(\\infty)}A^{\\prime}(\\infty) \\right]-\\Lambda\\int^{\\infty}_{-\\infty}d\n    \\mathrm{e}^{2A(y)}.\n\\end{equation}\n\n\\noindent As $\\Lambda=0$, the value of the matter energy depends on the asymptotic behavior of the warp factor. If $y\\rightarrow\\pm\\infty$ then $\\phi^{\\prime}(\\pm\\infty)=0$ and by the analysis to Eq. (\\ref{em2b}), we can see that $A(\\pm\\infty)\\propto -|\\,y|$. Therefore, $\\mathrm{e}^{2A(\\pm\\infty)}\\rightarrow0$ and the value of the matter energy is zero. This fact is the same to the case of branes with generalized dynamics \\cite{arro}.\n\n\\noindent The scalar curvature (or Ricci scalar) is given by\n\\begin{equation}\\label{ricci}\n    R=-4(5A^{\\prime}\\,^{2}+2A^{\\prime\\prime}).\n\\end{equation}\n\n\\noindent The profiles of the matter energy density is shown in Fig. (\\ref{fde}) for some values of $a$. Figure (\\ref{fde}) clearly shows that for $a=0$ the matter energy density has not a single-peak around $y=0$. The core of the brane is localized at $y=0$ for $a=0$, because this region has a positive matter energy density. On the other hand, as the value of $a$ is increasing, we can see that the single brane splits into two sub-branes and as $a\\rightarrow a_{*}$ each sub-brane is a thick brane. This phenomenon is so-called of brane splitting \\cite{angel}. From the peak of the matter energy density is evident know where the core of the branes are located. Therefore, the brane does not get thicker with the increases of the value of the mass parameter $a$, as argued in Ref. \\cite{zhao}. The profiles of the matter energy density and the Ricci scalar are shown in Fig. (\\ref{desc}) for $a=0.8$. Note that the presence of regions with positive Ricci scalar is connected to the capability to trap matter near to the core of the brane \\cite{alm} and it reinforces the conclusion of the analyzes from the matter energy density. Also note that far from the brane, $R$ tends to a negative constant, characterizing the $AdS_{5}$ limit from the bulk.\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7cm, angle=0]{de.eps}\n\\end{center}\n\\caption{The profiles of the energy density for $b=2$, $c=1$, $a=0$ (thin line), $a=0.8$ (dashed line) and $a=0.836$ (dotted line).} \\label{fde}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7cm, angle=0]{deera08.eps}\n\\end{center}\n\\caption{The profiles of the matter energy density (thin line) and Ricci scalar (thick line) for $b=2$, $c=1$ and $a=0.8$.} \\label{desc}\n\\end{figure}\n\n\\subsection{Fermion localization}\n\nThe action for a Dirac spinor field coupled with the scalar fields by a general Yukawa coupling is\n\\begin{equation}\\label{ad}\n    S=\\int d^{5}x\\sqrt{|\\,g|}\\left[ i\\bar{\\Psi}\\Gamma^{M}\\nabla_{M}\\Psi-\\eta\\bar{\\Psi}F(\\phi)\\Psi \\right]\\,,\n\\end{equation}\n\\noindent where $\\eta$ is the positive coupling constant between fermions and the scalar field. Moreover,\nwe are considering the covariant derivative $\\nabla_{M}=\\partial_{M}+\\frac{1}{4}\\,\\omega^{\\bar{A}\\bar{B}}_{M}\\Gamma_{\\bar{A}}\\Gamma_{\\bar{B}}$\\,,\nwhere $\\bar{A}$ and $\\bar{B}$, denote the local Lorentz indices and $\\omega^{\\bar{A}\\bar{B}}_{M}$ is\nthe spin connection. Here we consider the field $\\phi$ as a background field. The equation of motion is\nobtained as\n\\begin{equation}\\label{dkp}\ni\\,\\Gamma ^{M }\\nabla_{M}\\Psi-\\eta F(\\phi)\\Psi =0.\n\\end{equation\n\nAt this stage, it is useful to consider the fermionic current. The conservation law for $J^{M}$ follows from the standard procedure and it becomes\n\\begin{equation}\\label{corr}\n\\nabla_{M}J^{M}=\\bar{\\Psi}\\left(\\nabla_{M}\\Gamma^{M}\\right)\\Psi\\,,\n\\end{equation}\n\\noindent where $J^{M}=\\bar{\\Psi}\\Gamma^{M}\\Psi$. Thus, if\n\\begin{equation}\\label{cj0}\n    \\nabla_{M}\\Gamma^{M}=0\\,,\n\\end{equation}\n\\noindent then four-current will be conserved. The condition (\\ref{cj0}) is the purely geometrical assertion that the curved-space gamma matrices are covariantly constant.\n\n\\noindent Using the same line element (\\ref{metric}) and the representation for\ngamma matrices $\\Gamma^{M}=\\left( \\mathrm{e}^{-A}\\gamma^{\\mu},-i\\gamma^{5}\\right)$,\nthe condition (\\ref{cj0}) is trivially satisfied and therefore the current is conserved.\n\nThe equation of motion (\\ref{dkp}) becomes\n\\begin{equation}\\label{em}\n\\left[ i\\gamma^{\\mu}\\partial_{\\mu}+\\gamma^{5}\\mathrm{e}^{A}(\\partial_{y}+2\\partial_{y}A\n-\\eta\\,\\mathrm{e}^{A}F(\\phi) \\right]\\Psi=0.\n\\end{equation}\n\\noindent Now, we use the general chiral decomposition\n\\begin{equation}\\label{dchiral}\n    \\Psi(x,y)=\\sum_{n}\\psi_{L_{n}}(x)\\alpha_{L_{n}}(y)+\\sum_{n}\\psi_{R_{n}}(x)\\alpha_{R_{n}}(y),\n\\end{equation}\n\\noindent with $\\psi_{L_{n}}(x)=-\\gamma^{5}\\psi_{L_{n}}(x)$ and $\\psi_{R_{n}}(x)=\\gamma^{5}\\psi_{R_{n}}(x)$.\nWith this decomposition $\\psi_{L_{n}}(x)$ and $\\psi_{R_{n}}(x)$ are the left-handed and\nright-handed components of the four-dimensional spinor field, respectively. After applying\n(\\ref{dchiral}) in (\\ref{em}), and demanding that $i\\gamma^{\\mu}\\partial_{\\mu}\\psi_{L_{n}}=m_{n}\\psi_{R_{n}}$\nand $i\\gamma^{\\mu}\\partial_{\\mu}\\psi_{R_{n}}=m_{n}\\psi_{L_{n}}$, we obtain two equations\nfor $\\alpha_{L_{n}}$ and $\\alpha_{R_{n}}$\n\\begin{equation}\\label{ea1}\n    \\left[ \\partial_{y}+2\\partial_{y}A+\\eta F(\\phi) \\right]\\alpha_{L_{n}}=m_{n}\\mathrm{e}^{-A}\\alpha_{R_{n}}\\,,\n\\end{equation}\n\\begin{equation}\\label{ea2}\n    \\left[ \\partial_{y}+2\\partial_{y}A-\\eta F(\\phi) \\right]\\alpha_{R_{n}}=-m_{n}\\mathrm{e}^{-A}\\alpha_{L_{n}}\\,.\n\\end{equation}\n\\noindent Inserting the general chiral decomposition (\\ref{dchiral}) into the action (\\ref{ad}), using (\\ref{ea1}) and (\\ref{ea2}) and also requiring that the result take the form of the standard four-dimensional action for the massive chiral fermions\n\\begin{equation}\\label{ad2}\n    S=\\sum_{n}\\int d^{4}x\\, \\bar{\\psi}_{n}\\left( \\gamma^{\\mu}\\partial_{\\mu}-m_{n} \\right)\\psi_{n},\n\\end{equation}\n\\noindent where $\\psi_{n}=\\psi_{L_{n}}+\\psi_{R_{n}}$ and $m_{n}\\ge0$, the functions $\\alpha_{L_{n}}$ and $\\alpha_{R_{n}}$ must obey the following orthonormality conditions\n\\begin{equation}\\label{orto}\n    \\int_{-\\infty}^{\\infty}dy\\,\\mathrm{e}^{3A}\\alpha_{Lm}\\alpha_{Rn}=\\delta_{LR}\\delta_{mn}.\n\\end{equation}\n\n\\noindent Implementing the change of variables\n\\begin{equation}\\label{cv}\n    z=\\int^{y}_{0}\\mathrm{e}^{-A(y^{\\,\\prime})}dy^{\\,\\prime},\n\\end{equation}\n\n\\noindent $\\alpha_{L_{n}}=\\mathrm{e}^{-2A}L_{n}$ and $\\alpha_{R_{n}}=\\mathrm{e}^{-2A}R_{n}$, we get\n\\begin{equation}\\label{sleft}\n    -L_{n}^{\\prime\\prime}(z)+V_{L}(z)L_{n}=m_{n}^{2}L_{n}\\,,\n\\end{equation}\n\\begin{equation}\\label{sright}\n    -R_{n}^{\\prime\\prime}(z)+V_{L}(z)R_{n}=m_{n}^{2}R_{n}\\,,\n\\end{equation}\n\\noindent where\n\\begin{eqnarray}\n  V_{L}(z) &=& \\eta^{2}\\mathrm{e}^{2A}F^{2}(\\phi)-\\eta\\partial_{z}\\left( \\mathrm{e}^{A}F(\\phi) \\right),\\label{vefa} \\\\\n  V_{R}(z) &=& \\eta^{2}\\mathrm{e}^{2A}F^{2}(\\phi)+\\eta\\partial_{z}\\left( \\mathrm{e}^{A}F(\\phi) \\right)\\label{vefb}.\n\\end{eqnarray}\n\\noindent Using the expressions $\\partial_{z}A=\\mathrm{e}^{A(y)}\\partial_{y}A$ and $\\partial_{z}F=\\mathrm{e}^{A(y)}\\partial_{y}F$, we can recast the potentials (\\ref{vefa}) and (\\ref{vefb}) as a function of $y$ \\cite{yu}-\\cite{yo}\n\\begin{eqnarray}\n  V_{L}(z(y)) &=& \\eta\\mathrm{e}^{2A}\\left[ \\eta F^{2}-\\partial_{y}F-F\\partial_{y}A(y) \\right],\\label{vya} \\\\\n  V_{R}(z(y)) &=& V_{L}(z(y))|_{\\eta\\rightarrow-\\eta}\\,.\\label{vyb}\n\\end{eqnarray}\n\n\\noindent It is worthwhile to note that we can construct the Schr\\\"{o}dinger potentials\n$V_{L}$ and $V_{R}$ from eqs. (\\ref{vya}) and (\\ref{vyb}).\n\nAt this stage, it is instructive to state that with the change of variable (\\ref{cv})\nwe get a geometry to be conformally flat\n\\begin{equation}\\label{cm}\n    ds^{2}=\\mathrm{e}^{2A(z)}\\left( \\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu} +dz^{2}\\right).\n\\end{equation}\n\\noindent Now we focus attention on the condition (\\ref{cj0}) for the line element (\\ref{cm}). In this case we obtain\n\\begin{equation}\\label{mc}\n    \\nabla_{M}\\Gamma^{M}= i(\\partial_{z}A(z))\\mathrm{e}^{-A(z)}\\gamma^{5}.\n\\end{equation}\n\n\\noindent Therefore, the current is no longer conserved for the line element (\\ref{cm}) \\cite{arro}. It is known that, in general, the reformulation of the theory in a new conformal frame leads to a different, physically inequivalent theory. This issue has already a precedent in cosmological models \\cite{val}.\n\n\\noindent Under this arguments, we only use the change of variable (\\ref{cv}) to have a qualitative analysis of the potential profiles (\\ref{vya}) and (\\ref{vyb}), which is a fundamental ingredient for the fermion localization on the brane.\n\nNow we focus attention on the calculation of the zero mode. Substituting $m_{n}=0$ in (\\ref{ea1}) and (\\ref{ea2}) and using $\\alpha_{L_{n}}=\\mathrm{e}^{-2A}L_{n}$ and $\\alpha_{R_{n}}=\\mathrm{e}^{-2A}R_{n}$, respectively, we get\n\\begin{equation}\\label{mzL}\n    L_{0}\\propto \\exp \\left[-\\eta\\int_{0}^{y}dy^{\\prime}F(\\phi) \\right],\n\\end{equation}\n\\begin{equation}\\label{mzR}\n    R_{0}\\propto \\exp \\left[\\eta\\int_{0}^{y}dy^{\\prime}F(\\phi) \\right].\n\\end{equation}\n\\noindent This fact is the same to the case of two-dimensional Dirac equation \\cite{luis}. At this point is worthwhile to mention that the normalization of the zero mode and the existence of a minimum for the effective potential at the localization on the brane are essential conditions for the problem of fermion localization on the brane. This fact was already reported in \\cite{yo}.\n\nIn order to guarantee the normalization condition (\\ref{orto}) for the left-handed fermion zero mode (\\ref{mzL}), the integral must be convergent, \\textit{i.e}\n\\begin{equation}\\label{cono}\n    \\int^{\\infty}_{-\\infty}dy\\exp\\left[ -A(y)-2\\eta\\int^{y}_{0}dy\\,^{\\prime}F(\\phi(y\\,^{\\prime})) \\right]<\\infty.\n\\end{equation}\n\\noindent This result clearly shows that the normalization of the zero mode is decided by the asymptotic behavior of $F(\\phi(y))$. Furthermore, from (\\ref{vya}) and (\\ref{vyb}), it can be observed that the effective potential profile depends on the $F(\\phi(y))$ choice. This fact implies that the existence of a minimum for the effective potential $V_{L}(z(y))$ or $V_{R}(z(y))$ at the localization on the brane is decided by $F(\\phi(y))$. This point will be more clear when it is considered a specific Yukawa coupling. Therefore, the behavior of $F(\\phi(y))$ plays a leading role for the fermion localization on the brane \\cite{yo}. Having set up the two essential conditions for the problem of fermion localization on the brane, we are now in a position to choice some specific forms for Yukawa couplings.\n\n\\subsection{Zero mode and fermion localization}\n\nFrom now on, we mainly consider the simplest case $F(\\phi)=\\phi$. First, we consider the\nnormalizable problem of the solution. In this case, we only need to consider the asymptotic behavior of the integrand in (\\ref{cono}). It becomes\n\\begin{equation}\\label{in}\n    I\\rightarrow\\mathrm{exp}\\left[ -A(\\pm\\infty)-2\\eta\\int^{y}_{0}\\phi(y^{\\prime})dy^{\\prime} \\right]\\,.\n\\end{equation}\n\n\\noindent By the analysis from eq. (\\ref{em2b}), we obtain that $A(\\pm\\infty)\\rightarrow-\\sqrt{|V(\\pm v)|\/3}\\,|\\,y|$. For the integral $\\int dy\\phi$, we only need to consider the asymptotic behavior of $\\phi$ for $y\\rightarrow\\pm\\infty$ \\cite{liu} and as $\\phi(\\pm\\infty)=\\pm v$ the equation (\\ref{in}) becomes\n\\begin{equation}\\label{int}\n    I\\rightarrow\\mathrm{exp}\\left[ -2\\left( \\eta\\,v-\\sqrt{|V(\\pm v)|\/12} \\right)|\\,y| \\right]\\,.\n\\end{equation}\n\n\\noindent This result clearly shows that the zero mode of the left-handed fermions is normalized only for $\\eta>\\frac{1}{v}\\,\\sqrt{\\frac{|V(\\pm v)|}{12}}$. Now, under the change $\\eta\\rightarrow-\\eta$ ($L_{0}\\rightarrow R_{0}$) we obtain that the right-handed fermions can not be a normalizable zero mode. The shape of the potentials $V_{L}$ and $V_{R}$ are shown in Fig. (\\ref{pe}) for some values of $a$. Figure \\ref{pe}(a) shows that the effective potential $V_{L}$, is indeed a volcano-like potential for $a=0$. As $a$ increases the well structure of $V_{L}$ gets a double well. Figure \\ref{pe}(b) shows that the potential $V_{R}$ has also a well structure, but the minimum of $V_{R}$ is always positive, therefore the potential does not support a zero mode. The shapes of the matter energy density, $V_{L}$ potential and $|\\,L_{0}|^{2}$ are shown in Fig. \\ref{a}. The Fig. \\ref{a}(a) ($a=0$) shows that the zero mode is localized on the brane. On the other hand, Fig. \\ref{a}(b) ($a=0.836$) clearly shows that the normalizable zero mode is localized between the two sub-branes, as a consequence the zero mode is not localized on the brane. Therefore, we can conclude that the zero mode of the left-handed fermions is localized on the brane only as $0\\leq a \\approx a_{*}$.\n\n\\begin{figure}[ht]\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{vl.eps}}\\\\\n          \\end{minipage}\\hfill\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{vr.eps}}\\\\\n           \\end{minipage}\n       \\caption{Potential profile: (a) $(V_{L}(y))_{A}$ (left) and (b) $(V_{R}(y))_{A}$ (right) for $\\eta=1$, $b=2$,  $c=1$, $a=0$ (thin line), $a=0.8$ (dashed line) and $a=0.836$ (dotted line) }\\label{pe}\n   \\end{figure}\n\n\n\\begin{figure}[ht]\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{mzervla0_3.eps}}\\\\\n          \\end{minipage}\\hfill\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{mzervla0836_3.eps}}\\\\\n           \\end{minipage}\n       \\caption{The profiles of the Ricci scalar (thin line), $V_{L}(y)$ (thick line) and $|\\,L_{0}|^{2}$ (dashed line) for $\\eta=1$, $b=2$, $c=1$; (a) $a=0$ (left) and (b) $a=0.836$ (right).}\\label{a}\n   \\end{figure}\n\n\\section{Conclusions}\n\nWe have reinvestigated the effects of the variation of the mass parameter $a$ on the thick branes as well as the localization of fermions. We showed that the variation of $a$ is associated to the phenomenon of brane splitting, therefore the brane does not get thicker with the increases of the value of $a$, as argued in Ref. \\cite{zhao}. We can conclude that the appearance of two sub-branes is associated to phase transition for $a=a_{*}$ (a disordered phase between two ordered phases). Also, we showed that the value of the matter energy depends on the asymptotic behavior of the warp factor. From the static equation of motion we have analyzed the asymptotic behavior of $A(y)$ and showed that the zero mode of the left-handed fermions for the simplest Yukawa coupling $\\eta\\bar{\\Psi}\\phi\\Psi$ is normalizable under the condition $\\eta>\\frac{1}{v}\\,\\sqrt{\\frac{|V(\\pm v)|}{12}}$ and it can be trapped on the brane only for $0\\leq a \\approx a_{*}$, because the zero mode has a single-peak at the localization of the brane. We also showed that as $a\\rightarrow a_{*}$ the zero mode has a single-peak between the two sub-branes and as a consequence the normalizable zero mode is not localized on the brane. Therefore, the brane with a big value of $a$ would not trap fermions more efficiently, in opposition to what was adverted in Ref. \\cite{zhao}. This work completes and revises the analyzing of the research in Ref. \\cite{zhao}, because in that work does not analyze the zero mode in full detail and contain some misconceptions.\n\nAdditionally, we showed that the change of variable $dz=\\mathrm{e}^{-A(y)}dy$ leads to a non conserved current, because the curved-space gamma matrices are not covariantly constant. An interesting issue will be investigate the effects of non-conserved current on resonances modes and bear out the main conclusion of Ref. \\cite{zhao}.\n\n\n\\begin{acknowledgments}\nThis work was supported by means of funds provided by CAPES.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction}\n\nComplex contact manifolds are still has many open problems. The importance of this subject is not only the complex version of real contact manifolds also one can find many important informations about complex manifolds, K\\\"ahler manifolds. In addition there are some applications in theoretical physics  \\cite{kholodenko2013applications}. Complex contact manifolds has an old history same as real contact manifolds, researchers could not give their attention to the subject. When we look at the 1980s there are very important improvements in the Riemannian geometry of complex contact manifolds. Ishihara and Konishi constructed tensorial relations for a complex almost contact structure and they also presented normality \\cite{ishihara1979real,ishihara1980complex,ishihara1982complex}. The Riemann geometry of complex almost contact metric manifolds could be divided into three notions;\n\\begin{enumerate}\n\t\\item \\textbf{\\textit{IK-normal complex contact metric manifolds}} : A complex contact manifold has a normal contact structure in the sense of Ishihara-Konishi. This type of manifolds were studied in \\cite{ishihara1979real,ishihara1980complex,ishihara1982complex,imada2014construction,imada2015complex,turgut2018h}.\n\t\\item \\textbf{\\textit{Normal complex contact metric manifolds}}: A complex contact manifold has a normal contact structure in the sense of Korkmaz.  This type of manifolds were studied in \\cite{korkmaz2000,Korkmaz2003,blair2006corrected,blair2009special,blair2011bochner,blair2012homogeneity,blair2012symmetry,vanli2015curvature,turgut2017conformal,vanli2017complex}.\n\t\\item \\textbf{\\textit{Complex Sasakian manifolds}}: A complex contact manifold with a globally complex contact form and has a normal contact structure in the sense of Korkmaz. This type of manifolds were studied in \\cite{foreman2000complex,fetcu2006harmonic,fetcuadapted}.\n\\end{enumerate}\nIn this work we study on third type of these manifolds. Firstly we adopted the definition of a complex Sasakian manifold by consider the definition of a real Sasakian manifold. Later we give some fundamental equations and we obtain curvature properties. Finally we examine some flatness conditions. We use a general tensor which is defined in \\cite{shaikh2018some} and is called by $ B- $tensor. We prove that a complex Sasakian manifold could not be $ B- $flat. \n\n\n\\section{Preliminaries}\nIn this section we give fundamental facts on complex contact manifolds. For details reader could be read \\cite{foreman1996variational,korkmaz2000,blair2010riemannian}.\n\\begin{definition}\n\tLet $N$ be a complex manifold of odd complex dimension $2p+1$ covered by an\nopen covering $\\mathcal{C}=\\left\\{ \\mathcal{A}_{i}\\right\\} $ consisting of\ncoordinate neighborhoods. If there is a holomorphic $1$-form $\\eta _{i}$\non each $\\mathcal{A}_{i}\\in \\mathcal{C}$ in such a way that for any \n\\mathcal{A}_{i},\\mathcal{A}_{j}\\in \\mathcal{C}$ and for a holomorphic function $f_{ij}$ on $\\mathcal{A}_{i}\\cap \\mathcal{A}_{j}\\neq \\emptyset $\n\\begin{equation*}\n\\eta _{i}\\wedge (d\\eta _{j})^{p}\\neq 0\\text{ in }\\mathcal{A}_{i}\\text{, }\n\\end{equation*\n\\begin{equation*}\n\\eta _{i}=f_{ij}\\eta _{j},\\text{ }\\mathcal{A}_{i}\\cap \\mathcal{A\n_{j}\\neq \\emptyset ,\n\\end{equation*\nthen the set $\\left\\{ \\left( \\eta _{i},\\mathcal{A\n_{i}\\right) \\mid \\mathcal{A}_{i}\\in \\mathcal{C}\\right\\} $ of local structures is called complex contact structure and with this structure $N$ is called a complex contact manifold.\n\\end{definition}\n\n\nThe complex contact structure determines a non-integrable distribution $H_i$ by the equation $\\eta_i =0$ such as \n\\begin{equation*}\nH_{i}=\\{X_{P}:\\eta_{i}(X_{P})=0, X_{P}\\in T_{P}N\\}.\n\\end{equation*}\nand a holomorphic vector field $ \\xi_i $\nis defined by  \n\n\\begin{equation*}\n\\eta_i(\\xi_i)=1\n\\end{equation*}\nand a complex line bundle is defined by $ E_i=Span\\{\\xi_i\\} $.\\\\\n\\qquad Let $ T^c(N) $ be complexified of tangent bundle of $ (N,J,\\eta_i) $ and let define vector fields \n\\begin{eqnarray*}\n\tU_i=\\xi_i+\\bar{\\xi_i}\\ \\ \\ \\ \\ \\ \\ V_i=-i(\\xi_i+\\bar{\\xi_i})\n\\end{eqnarray*}\nand 1-forms\n\\begin{eqnarray*}\n\tu_i=\\frac{1}{2}(\\eta_i+\\bar{\\eta_i}) \\ \\ \\ \\ \\ \\ v_i=\\frac{1}{2}i(\\eta_i-\\bar{\\eta_i}).\n\\end{eqnarray*}\nTherefore we get \n\\begin{enumerate}\n\t\\item $ V_i=-JU_i $ and $ v_i=u_i\\circ J $\n\t\\item $ U_i=JV_i $ and $ u_i=-v_i\\circ J $\n\t\\item $ u_i(U_i)=v_i(V_i)=1 $ and $ u_i(V_i)=v_i(V_i)=0 $.\n\\end{enumerate}\nThe complexified $ H_i $ and $ E_i $ is defined by \n\\begin{eqnarray*}\n{\tH_i}^c&=&\\{W\\in T^c(N)\\arrowvert u(W)=v(W)=0\\}\\\\\n{\tE_i}^c&=&Span\\{U,V\\}. \n\\end{eqnarray*}\nWe use notation $ \\mathcal{H} $ and $ \\mathcal{V} $ for the union of $ {\tH_i}^c $ and $ E_i^c $ respectively. $ \\mathcal{H} $ is called horizontal distribution and $ \\mathcal{V} $ is called vertical distribution \nand we can write \n\\begin{equation} \\label{TN=H+V}\nTN=\\mathcal{H}\\oplus\\mathcal{V}.\n\\end{equation}\n Ishihara and Konishi \\cite{ishihara1980complex} proved that $N$ admits always a complex contact structure of $ C^{\\infty} $.\n\\begin{definition} \\label{complexalmostscontact}\n\tLet $N$ be a complex manifold with complex structure $ J $, Hermitian metric $ g $ and $\\mathcal{C=}\\left\\{ \n\t\\mathcal{A}_{i}\\right\\} $ be open covering of $N$ with coordinate\n\tneighbourhoods $\\{\\mathcal{A}_{i}\\mathcal{\\}}.$ If $N$ satisfies the\n\tfollowing two conditions then it is called a \\textit{complex almost contact\n\t\tmetric manifold}:\n\t\n\t1. In each $\\mathcal{A}_{i}$ there exist $1$-forms $u_{i}$ and \n\tv_{i}=u_{i}\\circ J$, with dual vector fields $U_{i}$ and $V_{i}=-JU_{i}$ and \n\t$(1,1)$ tensor fields $G_{i}$ and $H_{i}=G_{i}J$ such that \n\t\\begin{equation} \\label{G^2veH^2}\n\tH_{i}^{2}=G_{i}^{2}=-I+u_{i}\\otimes U_{i}+v_{i}\\otimes V_{i}\n\t\\end{equation\n\t\\begin{equation*} \\label{H=GJ}\n\tG_{i}J=-JG_{i},\\quad GU_{i}=0,\\quad\n\t\\end{equation*\n\t\\begin{equation*} \\label{g(GX,Y)=-g(X,GY)}\n\tg(X,G_{i}Y)=-g(G_{i}X,Y).\n\t\\end{equation*\n\t2.On $\\mathcal{A}_{i}\\cap \\mathcal{A}_{j}\\neq \\emptyset $ we have \n\t\\begin{eqnarray*}\n\t\tu_{j} &=&au_{i}-bv_{i},\\quad v_{j}=bu_{i}+av_{i},\\; \\\\\n\t\tG_{j} &=&aG_{i}-bH_{i},\\quad H_{j}=bG_{i}+aH_{i}\n\t\\end{eqnarray*\n\twhere $a$ and $b$ are functions on $\\mathcal{U}_{i}\\cap \\mathcal{U}_{j}$\n\twith $a^{2}+b^{2}=1$ \\cite{ishihara1979real,ishihara1980complex}.\n\\end{definition}\nBy direct computation we have \n\\begin{eqnarray} \nH_{i}G_{i} &=&-G_{i}H_{i}=J_{i}+u_{i}\\otimes V_{i}-v_{i} \\otimes U_{i} \\label{HG=-GH} \\label{HG,GH}\\\\\nJ_{i}H_{i} &=&-H_{i}J_{i}=G_{i}  \\notag \\\\\nG_{i}U_{i} &=&H_{i}U_{i}=H_{i}V_{i}=0  \\notag \\\\\nu_{i}G_{i} &=&v_{i}G_{i}=u_{i}H_{i}=v_{i}H_{i}=0 \\notag \\\\\nJ_{i}V_{i} &=&U_{i}, \\ g(U_{i},V_{i})=0 \\notag\\\\\ng(H_{i}X,Y) &=&-g(X,H_{i}Y) \\notag.\n\\end{eqnarray}\n\n\nBy the local contact form $\\eta $ is $u-iv$ to within a\nnon-vanishing complex-valued function multiple and the local fields $G$ and \n\\ H$ are related to $du$ and $dv$ by \n\\begin{eqnarray*}\n\tdu(X,Y) &=&g(X,GY)+(\\sigma \\wedge v)(X,Y),~~~ \\\\\n\tdv(X,Y) &=&g(X,HY)-(\\sigma \\wedge u)(X,Y)\n\\end{eqnarray*}\nwhere $\\sigma (X)=g(\\nabla _{X}U,V)$, $\\nabla $ being the Levi-Civita\nconnection of $g$ \\cite{ishihara1980complex}. $ \\sigma $ is called IK-connection \\cite{foreman1996variational}.\n\nIshihara and Konishi \\cite{ishihara1980complex} study on normality of complex   almost contact metric manifolds. They defined local tensors\n\\begin{eqnarray*}\n\tS(X,Y) &=&[G,G](X,Y)+2g(X,GY)U-2g(X,HY)V \\\\\n\t&&+2(v(Y)HX-v(X)HY)+\\sigma (GY)HX \\\\\n\t&&-\\sigma (GX)HY+\\sigma (X)GHY-\\sigma (Y)GHX,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\tT(X,Y) &=&[H,H](X,Y)-2g(X,GY)U+2g(X,HY)V \\\\\n\t&&+2(u(Y)GX-u(X)GY)+\\sigma (HX)GY \\\\\n\t&&-\\sigma (HY)GX+\\sigma (X)GHY-\\sigma (Y)GHX\n\\end{eqnarray*}\nwhere\n\\begin{equation*}\n\\lbrack G,G](X,Y)=(\\nabla _{GX}G)Y-(\\nabla _{GY}G)X-G(\\nabla\n_{X}G)Y+G(\\nabla _{Y}G)X\n\\end{equation*\nis the Nijenhuis torsion of $G$. \n\\begin{definition}\n\\cite{ishihara1980complex} A complex{\\color{white}$ \\eta $}almost{\\color{white}$ \\eta $}contact{\\color{white}$ \\eta $}metric{\\color{white}$ \\eta $}manifold is called IK-Normal{\\color{white}$ \\eta $} if $ S=T=0 $. \n\\end{definition}\nAn IK-Normal manifold has K\\\"ahler structure. In other words a non-K\\\"ahler complex almost contact metric manifold is not to be IK-Normal such Iwasawa manifold. Iwasawa manifold is not K\\\"ahler and  it is compact manifold which has symplectic structure. Also Baikousis et al. \\cite{baikoussis1998holomorphic} obtained complex almost contact structure on Iwasawa manifold. Korkmaz gave a weaker definition for normality and Iwasawa manifold is normal in the sense of this definition. \n\\begin{definition} [Korkmaz's Definition,  \\cite{korkmaz2000} ]\n\tA complex almost contact metric manifold is called normal if\n\t\n\t$\\qquad S(X,Y)=T(X,Y)=0$ \\ for all $X,Y$ in $\\mathcal{H},$ and $\\ $\n\t\n\t$\\qquad S\\left( X,U\\right) =T\\left( X,V\\right) =0$ for all $X.$\n\\end{definition} \n\nAlso for arbitrary vector fields $ X$ on $N$ we have \\cite{ishihara1980complex,foreman1996variational,korkmaz2000} \n\\begin{align}\n\\nabla _{X}U &=-GX+\\sigma (X)V,~~  \\label{nablaXU} ,\n\\ \\ \\ \\nabla _{X}V =-HX-\\sigma (X)U,~~  \\\\\n\\nabla _{U}U &=\\sigma (U)V,~~~\\nabla _{U}V=-\\sigma (U)U  \\label{nablaUU} , \\ \\ \n\\nabla _{V}U =\\sigma (V)V,~~~~\\nabla _{V}V=-\\sigma (V)U.~~  \n\\end{align}\n\n\\begin{theorem}\n\tA complex almost\n\tcontact metric manifold is normal if and only\n\tif the covariant derivative of \\ $G$ and $H$ have the following\n\tforms: \n\t\\begin{eqnarray}\n\t(\\nabla _{X}G)Y &=&\\sigma (X)HY-2v(X)JY-u\\left( Y\\right) X\n\t\\label{Yeni normallik G} \\\\\n\t&&-v(Y)JX+v(X)\\left( 2JY_{0}-\\left( \\nabla _{U}J\\right) GY_{0}\\right)  \\notag\n\t\\\\\n\t&&+g(X,Y)U+g(JX,Y)V  \\notag \\\\\n\t&&-d\\sigma (U,V)v(X)\\left( u(Y)V-v(Y)U\\right)  \\notag\n\t\\end{eqnarray}\n\tan\n\t\\begin{eqnarray}\n\t(\\nabla _{X}H)Y &=&-\\sigma (X)GY+2u(X)JY+u(Y)JX  \\label{Yeni normallik H} \\\\\n\t&&-v(Y)X+u(X)\\left( -2JY_{0}-\\left( \\nabla _{U}J\\right) GY_{0}\\right)  \\notag\n\t\\\\\n\t&&-g(JX,Y)U+g(X,Y)V  \\notag \\\\\n\t&&+d\\sigma (U,V)u(X)\\left( u(Y)V-v(Y)U\\right)  \\notag\n\t\\end{eqnarray}\n\twhere $X=X_{0}+u(X)U+v(X)V$ and $Y=Y_{0}+u(Y)U+v(Y)V,X_{0},Y_{0}\\in $ \n\t\\mathcal{H}$ \\cite{vanli2015curvature}.\n\\end{theorem}\nFrom this theorem on a normal complex contact metric manifold we have\n\n\\begin{eqnarray*}\n\t(\\nabla _{X}J)Y &=&-2u\\left( X\\right) HY+2v(X)GY+u(X)\\left( 2HY_{0}+\\left(\n\t\\nabla _{U}J\\right) Y_{0}\\right) \\\\\n\t&&+v(X)\\left( -2GY_{0}+\\left( \\nabla _{U}J\\right) JY_{0}\\right) .\n\\end{eqnarray*}\n\nAs we have seen, there are two normality notions for a \\NCM. The other kind of \\NCM s is complex Sasakian manifold. This type of manifolds are normal due to Korkmaz's definition  and they were studied in \\cite{foreman2000complex,fetcu2006harmonic,fetcuadapted}. The fundamental difference of this type from others is globally definition of complex contact form. Kobayashi \\cite{kobayashi1959remarks} proved that a complex contact form is globally defined if and only if first Chern class of $ N $ vanishes.  Also Foreman obtained following result; \n\\begin{lemma}\n\tLet $ (N,\\eta_i) $ be a complex contact manifold. If $ \\eta_i $ is globally defined i.e $ f_{ij}=1 $ then $ \\sigma=0 $ \\cite{foreman2000complex}. \n\\end{lemma}\n A complex Sasakin manifold is defined as follow; \n\\begin{definition}\n\tLet $\\left(N,G,H,J,U,V,u,v,g\\right) $ be a normal complex contact metric\n\tmanifold and $\\eta =u-iv$ is globally defined. If fundemental 2- forms \n\t\\widetilde{G}$ and $\\widetilde{H}$ is defined by \n\t\\begin{equation*}\n\t\\widetilde{G}\\left( X,Y\\right) =du(X,Y)\\text{ and \\ }\\widetilde{H}\\left(\n\tX,Y\\right) =dv\\left( X,Y\\right)\n\t\\end{equation*\n\tthen $ N $ is called a complex Sasakian manifold, where $X,Y$ are vector fields on \n\t$N$.\n\\end{definition}\nThus we get following result;\n\\begin{theorem}\n\tLet $ N $ be a \\NCM. Then $N$ is complex Sasakian if and only\n\tif\\bigskip \n\t\\begin{eqnarray*}\n\t(\\nabla _{X}G)Y&=&-2v(X)HGY-u(Y)X-v(Y)JX+g(X,Y)U+g(JX,Y)V \\\\\n\t(\\nabla _{X}H)Y &=&-2u(X)HGY+u(Y)JX-v(Y)X-g(JX,Y)U+g(X,Y)V.\n\t\\end{eqnarray*}\n\\end{theorem}\nThis result was also given by  Ishihara-Konishi\\cite{ishihara1980complex}. But should have been considered that $ N $ is not K\\\"ahler in here. So,we have \n\\begin{eqnarray*}\n\t(\\nabla _{X}J)Y &=&-2u(X)HY+2v(X)GY.\n\\end{eqnarray*}\nFrom (\\ref{nablaXU} )  on a complex Sasakin manifold we get \n\\begin{eqnarray}\n\\nabla _{X}U =-GX,~~ \\ \\ \\ \\nabla _{X}V =-HX. \\label{NablaX-U}\n\\end{eqnarray}\nOn the other hand we have \n\\begin{corollary}\n\tAn IK-normal complex contact metric manifold could not be complex Sasakian \\cite{turgut2018h}. \n\\end{corollary} \n This result support that the geometry of complex Sasakian manifolds has some different properties. \n\\section{Curvature Properties of Complex Sasakian Manifolds}\nIn the Riemannian geometry of contact manifolds curvature properties have an important position. We use these relations for future works. The curvature relations of a complex almost contact metric manifolds were given in \\cite{foreman1996variational}, an IK-Normal manifold were given in \\cite{ishihara1980complex} and a normal complex contact metric manifold were given in \\cite{korkmaz2000,vanli2015curvature}. In this section by taking advantage from these curvature properties, we present curvature relations for complex Sasakian manifold.\nLet $ N $ be a complex Sasakian manifold. Then for $ X,Y \\in \\Gamma(TN) $ we have, \n\\begin{align}\nR\\left( U,V\\right) V&=R\\left( V,U\\right) U=0 \\label{SasakianR(UVV)} \\\\\nR(X,U)U&=X+u(X)U+v(X)V\t\\label{SasakianR(XUU)} \\\\\nR(X,V)V&=X-u(X)U-v(X)V\t\\label{SasakianR(XVV)} \\\\\nR(X,U)V&=-3JX-3u(X)V+3v(X)U \\label{SasakianR(XUV)}\\\\\nR(X,V)U&=0 \\label{SasakianR(XVU)}\\\\\nR(X,Y)U&=v(X)JY-v(Y)JX+2v(X)u(Y)V-2v(Y)u(X)V  \\label{SasakianR(XYU)}\\\\\n&+u(Y)X-u(X)Y-2g(JX,Y)V \\notag \\\\\nR(X,Y)V&=3u(X)JY-3u(Y)JX-2u(X)v(Y)U+2u(Y)v(X)U \\label{SasakianR(XYV)} \\\\\n&+v(Y)X-v(X)Y+2g(JX,Y)U \\notag \\\\\nR(U,V)X&=JX+u(X)V-v(X)U\\label{SasakianR(UVX)} \\\\\tR(X,U)Y&=-2v(Y)v(X)U+2u(Y)v(X)V-g(Y,X)U \\label{SasakianR(XUY)}\\\\\n&+u(Y)X+g(JY,X)V  \\notag \\\\\nR(X,V)Y&=3u(Y)JX+2u(Y)u(X)V+3g(JY,X)U\\label{SasakianR(XVY)} \\\\&-2v(Y)u(X)U-g(Y,X)V+v(Y)X-2u(X)JY \\notag .\n\\end{align}\n For $X,Y,Z,W$ horizontal vector fields we have \\cite{vanli2015curvature} \n\\begin{equation}\\label{r(gx,gy,gz,gw)}\ng(R(GX,GY)GZ,GW)=g(R(HX,HY)HZ,HW)=g(R(X,Y)Z,W)\n\\end{equation}\nand we have \n\t\\begin{align*}\ng(R(X,GX)GX,X)&+g(R(X,HX)HX,X)+g(R(X,JX)JX,X)=-6g\\left( X,X\\right)\\\\\ng(R(X,GX)Y,GY) &=g(R(X,Y)X,Y)+g(R(X,GY)X,GY)-2g(GX,Y)^{2} \\\\\t\n&-4g(HX,Y)^{2}-2g(X,Y)^{2} +2g(X,X)g(Y,Y)-4g(JX,Y)^{2}\\\\\ng(R(X,HX)Y,HY) &=g(R(X,Y)X,Y)+g(R(X,HY)X,HY)-2g(HX,Y)^{2} \\\\\n&-4g(GX,Y)^{2}-2g(X,Y)^{2}+2g(X,X)g(Y,Y)-4g(JX,Y)^{2} \\\\\ng(R(X,HX)JX,GX)&=-g(R(X,HX)HX,X)-4g(X,X)^{2}\\\\\ng(R(X,JX)HX,GX)&=g(R(X,JX)JX,X)-2g(X,X)^{2}.\n\\end{align*}\n\\begin{align*}\ng(R(GX,HX)HX,GX)&=g(R(X,JX)JX,X) \\\\\ng(R(GX,JX)JX,GX)&=g(R(X,HX)HX,X)\\\\\ng(R(JX,JY)JY,JX)&=g(R(X,Y)Y,X)\\\\\ng(R(X,Y)JX,JY)&=g(R(X,Y)Y,X)+4g(X,GY)^{2}+4g(X,HY)^{2}\\\\\ng(R(Y,JX)JX,Y)&=g(R(X,JY)JY,X) \\\\\ng(R(X,JY)JX,Y)&=g(R(X,JY)JY,X)+4g(X,HY)^{2}+4g(X,GY)^{2} \\\\\ng(R(X,JX)JY,Y)&=-g(R(JX,JY)X,Y)-g(R(JY,X)JX,Y) \\\\\ng(R(X,JX)JY,Y)&=g(R(X,Y)Y,X)+g(R(X,JY)JY,X)\\\\\n&+8\\left(g(X,GY)^{2}+g(X,HY)^{2}\\right). \n\\end{align*}\nWe have an nice relation as follow;\n\\begin{corollary}\n\tFor a unit horizontal vector $ X $ on $N$ we have\n\t\\begin{equation}\n\tk\\left( X,GX\\right) +k\\left( X,HX\\right) +k\\left( X,JX\\right) =6. \\label{sectionalrelation}\n\t\\end{equation}\n\\end{corollary}\nThis relation also valid for an IK-normal complex contact metric manifold \\cite{imada2014construction}. \n\\begin{corollary}\n\tLet $N$ be a complex Sasakian manifold and $ X $ be a unit  horizontal vector field on $N$. Then for the sectional curvature $ k $ we have \n\t\\begin{equation*}\n\tk(U,V)=0\\, \\ \\ \\text{and} \\ \\ \\ k(X,U)=1.\n\t\\end{equation*}\n\\end{corollary}\nTurgut Vanl\\i\\  and Unal \\cite{vanli2015curvature} presented properties of Ricci curvature tensor\nof a normal{\\color{white}$ \\theta $}complex{\\color{white}$ \\theta $}contact{\\color{white}$ \\theta $} metric manifold. For complex Sasakian case \nwe have following relations; \n \n\t\\begin{eqnarray} \\label{Ricciler}\n\t\t\\rho (U,U) &=&\\rho (V,V)=4p,\\text{ \\ }\\rho (U,V)=0\\\\\n\t\t\\rho (X,U) &=&4pu(X)  \\notag,\\ \n\t\t\\rho \\left( X,V\\right) =4pv(X)  \\notag\\\\\n\t\t\\rho (X,Y) &=&\\rho (GX,GY)+4p\\left( u(X)u(Y)+v(X)v(Y)\\right) \n\t\t\\label{Ricci(X,Y)=Ricci(GX,GY)+} \\notag\\\\\n\t\t\\rho (X,Y) &=&\\rho (HX,HY)+4p\\left( u(X)u(Y)+v(X)v(Y)\\right)  \\notag\n\t\\end{eqnarray}\nwhere $ X,Y \\in \\Gamma(TN) $.\\par \nThe sectional curvature of Riemann manifold give us important information about the geometry of the manifold. In complex manifold we have holomorphic sectional curvature which is the curvature of section is spanned by $ X $  and $ JX $. Similarly in contact manifold we have $ \\phi- $sectional curvature which is the curvature of section is spanned by $ X $  and $ \\phi X $ \\cite{blair2010riemannian}. For complex contact case we have sectional curvature, holomorphic sectional curvature and $ \\mathcal{GH}- $sectional curvature which was given by Korkmaz \\cite{korkmaz2000} as below: \n\\begin{definition}\n\t\\cite{korkmaz2000} Let $ N $ be a \\NCM. $X$ be an unit horizontal vector field  on $N$ and $a^{2}+b^{2}=1$.  A $\\mathcal{GH-}$section is a plane which is spanned by $X$ and $Y=aGX+bHX$ and the sectional curvature of this plane is called $\\mathcal{GH-}$\\textit{sectional curvature } is defined by \n\t\\mathcal{GH}_{a,b}\\left( X\\right) =k\\left( X,aGX+bHX\\right) ,$ where \n\tk(X,Y)$\\ is the sectional curvature of the plane section spanned by $X$ and $Y$.\n\\end{definition}\n$\\mathcal{GH-}$\\textit{sectional curvature } is denoted by $ \\mathcal{GH}_{a,b}  $ and we assume that it does not depend the choice of $a$ and $b$. So we\nwill use $\\mathcal{GH}\\left( X\\right) $ notation. \\par\n\nLet $ N $ be a complex Sasakian manifold. Since the complex contact form is globally defined $ \\mathcal{GH}_{a,b}  $ does not depend the choice of $a$ and $b$ in naturally. Also we have \\cite{korkmaz2000}\n\\begin{equation}\nk(X,JX)=\\mathcal{GH}\\left( X\\right) +3.  \\label{k(X,JX)=GH(X)+3}\n\\end{equation}\nThus from (\\ref{sectionalrelation}) we obtain\n\t\\begin{equation}\nk\\left( X,GX\\right) +k\\left( X,HX\\right) +\\mathcal{GH}\\left( X\\right) =3. \n\\end{equation}\n\\begin{example}\n\tThe well known example of normal complex contact metric manifold complex Heisenberg group. A globally defined complex contact form and complex almost contact structure on complex Heisenberg group was given by Baikousis et al. \\cite{baikoussis1998holomorphic}. Korkmaz obtained normality of complex Heisenberg group. Thus a complex Heisenberg group is an example of complex Sasakian manifolds. For details about complex Heisenberg group see \\cite{blair2006corrected,foreman2000complex,korkmaz2000,vanli2015curvature}.   \n\\end{example}\n\\begin{example}\n\tAn other example of complex Sasakian manifolds were given by Foreman \\cite{foreman2000complex}. Foreman obtained an example from hyperk\\\"ahler manifold. For details see \\cite{foreman2000complex}. \n\\end{example}\n\\section{Flatness on complex Sasakian manifolds}\nA real Sasakian manifold can not be flat, i.e its Riemannian curavture could not be zero identically \\cite{de2009complex}. We obtain same results for complex Sasakin manifolds. \n\\begin{theorem}\n\tA complex Sasakian manifold can not be flat.\n\\end{theorem}\n\n\\begin{proof}\n\tSuppose that a complex Sasakian manifold is flat. Then $R(X,Y)Z=0$\n\tand from that $\\rho \\left( X,Y\\right) =0.$ On the other hand from (\\ref{Ricciler}) we have \n\t\\begin{equation*}\n\t4nu(X)=0.\n\t\\end{equation*}\n\tThis is not possible. So the manifold can not be flat.\n\\end{proof}\n Shaikh and Kundu \\cite{shaikh2018some} generalized curvature tensors and gave the definition of B-tensor. They proved some equivalence relations between most of certain curvature conditions. \n\\begin{definition}\n\t$N$ be a complex Sasakian manifold . $ (0, 4) $ tensor $ B $ of $N$ is given by\n\t\\begin{eqnarray} \\label{B-TENSOR}\n\tB(X,Y,Z,W) &=& a_{0}R(X,Y,Z,W) + a_{1}R(X,Z,Y,W)\\\\\n\t&+& a_{2}\\rho(Y,Z)g(X,W)+a_{3}\\rho(X,Z)g(Y,W)\\notag\\\\\n\t&+&a_{4}\\rho(X,Y)g(Z,W)+a_{5}\\rho(X,W)g(Y,Z)\\notag \\\\\n\t&+&a_{6}\\rho(Y,W)g(X,Z)+a_{7}\\rho(Z,W)g(X,Y) \\notag \\\\\n\t&+& \\tau\\{a_{8}g(X,W)g(Y,Z)+a_{9}g(X,Z)g(Y,W) \\notag\\\\\n\t&+&_{10}g(X,Y)g(Z,W)\\} \\notag\n\t\\end{eqnarray}\n\twhere $ a_{i} $'s are scalars on $  M $ and $ X,Y,Z,W \\in \\Gamma(TN) $.\n\\end{definition}\nFor different value of $ a_{i} $, $ B $ is became projective, conformal, concircular, quasi-conformal and conharmonic etc. curvature tensors (see \\cite{shaikh2018some}). Therefore flatness of B-tensor also determine flatness of these tensors. Let's define $N$ is $ B-$flat if $ B=0 $. Then we have following. \n\n\\begin{theorem}\n\tA complex Sasakian manifold can not be $ B-$flat.   \n\\end{theorem}\n\\begin{proof}\n\tAssume that $ B=0 $. Let chose $Y=Z=U$ and $ X=W=X\\in \\mathcal{H} $ then we have \n\t\\begin{eqnarray*}\n\t\t0 &=& a_{0}R(X,U,U,X) + a_{1}R(X,U,U,X) \\\\\n\t\t&+&a_{2}\\rho(U,U)g(X,X)+a_{5}\\rho(X,X)g(U,U)\\\\\n\t\t&+& \\tau\\{a_{8}g(X,X)g(U,U)\\}.\n\t\\end{eqnarray*}\n\tand from (\\ref{SasakianR(XUU)}) and (\\ref{Ricciler})  we get\n\\begin{align}\n\\rho(X,X)=-\\frac{a_0+a_1+4pa_2+\\tau a_8}{a_5} \\label{B-tens\u00f6rro(X,X)}.\n\\end{align}\n\tTherefore, since $ \\rho(X,X) $ is constant $ \\rho(X,X)=\\rho(Y,Y) $. \n\tOn the other hand for unit and orthogonal horizontal vector fields $ X,Y $ by choosing   $Y=Z=Y \\in \\mathcal{H}$ and $ X=W=X\\in \\mathcal{H} $ under $B=0 $ condition we have \n\t\\begin{equation*}\n\tR(X,Y,Y,X)=-\\frac{a_{2}\\rho(Y,Y)+a_{5}\\rho(X,X)+a_{8}\\tau}{a_{0}+a_{1}}\n\t\\end{equation*}\n\tand thus we get \n\t\t\\begin{align*}\n\tR(X,Y,Y,X)=\\frac{a_{2}a_{0}+a_{1}a_{2}+a_{0}a_{5}+a_{1}a_{5}\n\t\t+4pa_{2}a_{2}^2+4pa_{2}a_{5}+\\tau a_{8}a_{2}}{a_{5}(a_{0}+a_{1})}.\n\t\\end{align*}\n\tThis shows us the sectional curvature is independent of $ X $ and $ Y $ and then $ k(X,JX)=\\mathcal{GH}(X) $. But from (\\ref{k(X,JX)=GH(X)+3}) there is a contradiction. So $ N $ could not be $ B- $flat. \n\\end{proof}\nIn \\cite{turgut2017conformal} two of presented authors showed that a normal{\\color{white}$ \\theta $}complex{\\color{white}$ \\theta $}contact{\\color{white}$ \\theta $}manifold is not be conformal,concircular, quasi-conformal and conharmonic flat.\n The other notion of flatness is $ \\phi-T $-flatness for a $(1,1)-  $tensor $ T $. Now we examine this notion for a complex Sasakian manifold. \n\\begin{definition}\n\tLet $N$ be a complex Sasakian manifold. If \n\t\\begin{equation}\n\tG^{2}(\\mathcal{B}(GX,GY)GZ)=0\\text{ and }H^{2}(\\mathcal{B}(HX,HY)HZ)=0\n\t\\label{gh-conformal condition}\n\t\\end{equation\n\tfor $T,Y,Z$ vector fields on $N$ then $N$ is called a $\\mathcal{GH}-$ $ \\mathcal{B}-$flat.\n\\end{definition}\n\\begin{theorem}\n\tA complex Sasakian manifold can not be $\\mathcal{GH}-$ $ \\mathcal{B}-$flat.  \n\\end{theorem}\n\\begin{proof}\n\tLet $ N $ be a $ \\mathcal{B}-$flat complex Sasakian manifold. For  $ X,Y, Z,W\\in \\Gamma(\\mathcal{H}) $ we have \n\t\\begin{eqnarray*}\ng(B(GX,GY)GZ, GW)&&=B(GX,GY,GZ,GW)\\\\\n &&= a_{0}R(GX,GY,GZ,GW) + a_{1}R(GX,GY,GZ,GW)\\\\\n&&+ a_{2}\\rho(GY,GZ)g(GX,GW)+a_{3}\\rho(GX,GZ)g(GY,GW)\\notag\\\\\n&&+a_{4}\\rho(GX,GY)g(GZ,GW)+a_{5}\\rho(GX,GW)g(GY,GZ)\\notag \\\\\n&&+a_{6}\\rho(GY,GW)g(GX,GZ)+a_{7}\\rho(GZ,GW)g(GX,GY) \\notag \\\\\n&&+ \\tau\\{a_{8}g(GX,GW)g(GY,GZ)+a_{9}g(GX,GZ)g(GY,GW) \\notag\\\\\n&&+a_{10}g(GX,GY)g(GZ,GW)\\}. \\notag\n\\end{eqnarray*}\nFrom (\\ref{r(gx,gy,gz,gw)}) and (\\ref{Ricciler}) we get \n\\begin{eqnarray*}\ng(B(GX,GY)GZ, GW)=B(GX,GY,GZ,GW)=B(X,Y,Z,W).\n\\end{eqnarray*}\nAlso we know from Theorem 4, $ B\\neq 0 $, thus we have \n\\begin{equation*}\ng(B(GX,GY)GZ, GW)=-g(GB(GX,GY)GZ, W)\\neq 0.\n\\end{equation*}\nBecause of $ W \\in \\Gamma(T\\mathcal{H}) $ then $GB(GX,GY)GZ\\neq0$ and so $G^2B(GX,GY)GZ\\neq0$. \n\\end{proof}\nBy this theorems we obtain that on a complex Sasakian manifold Weyl conformal curvature tensor, projective curvature tensor, concircular curvature tensor, conharmonic curvature tensor, quasi conformal curvature\ntensor, pseudo projective curvature\ntensor, quasi-concircular curvature tensor, pseudo\nquasi conformal curvature tensor, M-projective curvature tensor, $ W_{i} $-curvature tensor, $ i = 1, 2, . . . ,9 $, $ W_{i}* $\n-curvature, $ T $ -curvature tensor (see \\cite{shaikh2018some} for details on tensors) can not vanish. In the other words a complex Sasakian manifold can not transform to flat space under any transformation such as conformal, projective, concircular etc.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nMore than fifty years have passed since the first detection of a cosmic ray with an energy $\\,\\raisebox{-0.13cm}{$\\stackrel{\\textstyle> 10^{20}${\\rm eV} \\cite{Linsley}, yet the nature of UHECRs and the identity of their sources remain a mystery.  \nThe nature of UHECRs sources depends on their composition, and we focus here on sources capable of producing UHE protons.\nDifficulties with the leading source candidates, AGNs and GRBs,  led Farrar and Gruzinov  \\cite{fg09}  \nto propose that  UHECRs, if protons, must be produced in a new class of powerful AGN transients, as could arise from the tidal disruption of a star or an extreme accretion disk instability around a supermassive black hole at  a galactic center.  \nFollowing the detection of a relativistic outflow from the  tidal disruption event (TDE) \\textit{Swift}\\ J1644+57 \\cite{Burrows+11,Bloom+11,Zauderer+11}\nand from  \n\\textit{Swift}\\ J2058+05 \\cite{Cenko+12} we confront here the viability of a scenario in which UHECRs are protons produced in the jets of tidal disruption events (TDEs).   We begin by recalling the requirements for UHECR acceleration.  Then, we use observations and modeling of \\textit{Swift}\\ J1644+57, \na likely example of a TDE jet seen in ``blazar-mode'' (i.e., looking down the axis of the jet), to test whether individual TDE jets satisfy the Hillas criterion necessary for accelerating protons to  $10^{20}${\\rm eV}.   Finally, using the recently measured TDE rate \\cite{vfRate14}, we examine whether  TDEs  can account for the observed UHECR energy injection rate and whether they provide a sufficiently large number of active sources to explain the lack of strong clustering in the arrival direction distribution \\cite{augerSrcDen13}.      \n\n\n\\section{Conditions on Sources of UHECRs}\nIn order for a CR to be confined during the acceleration process, its Larmor radius must remain smaller than the size, $R$, of the accelerating system.  This places a strict lower bound on $B R$ for UHECR acceleration known as the Hillas criterion, valid for any acceleration mechanism that involves magnetic fields \\cite{hillas84}:  \n\\begin{equation}\\label{conf}\nB R\\,\\raisebox{-0.13cm}{$\\stackrel{\\textstyle> 3\\times 10^{17} ~ \\Gamma ^{-1} \\, Z^{-1} \\, E_{20} ~ {\\rm Gauss \\,cm},\n\\end{equation}\nwhere $B$ is the magnetic field, $\\Gamma$ is the bulk Lorentz factor of the jet,\n $Z$ is the charge of the UHECR, and $E$ the CR's energy with $E_{20} \\equiv E\/ 10^{20} {\\rm eV}$.  Eq. (\\ref{conf}) implies a lower bound on the total Poynting luminosity required to accelerate protons to UHE, for which the total bolometric luminosity can be taken as a surrogate \\cite{fg09}:\n\\begin{equation}\\label{lumi}\n L_{\\rm bol}\n \\approx {1\\over 6}c\\, \\Gamma ^4 \\, (B R)^2 \\, \\,\\raisebox{-0.13cm}{$\\stackrel{\\textstyle> \\, 10^{45}\\, \\Gamma ^2\\, (E_{20}\/Z)^2 \\, {\\rm erg\/s}.\n\\end{equation}\nIt was shown in \\cite{fg09} that if the conditions Eqs. (\\ref{conf},\\ref{lumi}) are met in an AGN-like jet,  cooling and interaction with photons prior to escape from the accelerator are not the limiting factors in the maximum energy. The strong dependence on $Z$ of  Eqs. (\\ref{conf},\\ref{lumi}) indicates that the constraints on the UHECR sources are very different for protons or nuclei:  for protons the sources must be amongst the most luminous known EM sources, while for nuclei the requirements are much more modest. \n\nDenoting the energy injection rate of UHECRs in the range $10^{18}-10^{20}$ eV by ${\\dot E}_{_{UCR}} \\equiv {\\dot E}_{44} \\,\\, 10^{44} \\,{\\rm erg} \\, {\\rm Mpc}^{-3} \\, {\\rm yr}^{-1}$, for continuous sources the density of sources implied by Eq. (\\ref{lumi}) is \n\\begin{equation}  \\label{num}\nn_{\\rm src} \\approx 3 \\times 10^{-9}  \\frac{{\\dot E}_{44}}{\\epsilon_{_{UCR}} \\, \\Gamma ^2\\, (E_{20}\/Z)^2} ~ {\\rm Mpc}^{-3},\n\\end{equation}\nwhere $\\epsilon_{_{UCR}}\\equiv L_{_{UCR}}\/L_{\\rm bol}$ is the luminosity in UHECRs relative to the Poynting luminosity.   From the observed UHECR spectrum, \\cite{katz+GRB-UHECR09} estimates ${\\dot E}_{44} = 2.3$ to 4.5 for source spectra $\\sim E^{-2} \\, {\\rm to} \\, E^{-2.5}$, with O(1) uncertainty.  Thus one continuous source with  $\\epsilon_{_{UCR}} \\sim 1$ within the GZK distance ($\\approx 200$ Mpc) would be sufficient to produce the entire observed flux. \nHowever,  the lack of clustering \\cite{augerSrcDen13} in the arrival directions of UHECRs with energies above 70 EeV, implies a constraint on the density of sources whose stringency depends on the characteristic maximum deflections of the UHECRs: for $ 30^{\\circ} $ ($3^{\\circ}$), the source density must be greater than \n\\begin{equation} \\label{nlimsrc}\nn_{\\rm min} = 2 \\times 10^{-5} ~ ( 7 \\times 10^{-4}) ~ {\\rm Mpc}^{-3}. \n\\end{equation}\nThus efficient continuous protonic sources are incompatible with the source abundance requirement \n\\cite{fg09,WL09,MuraseTakami09}. Note that  if  deflections in the Galactic and intergalactic magnetic fields are so strong that UHECR arrival directions do not reflect the direction of their sources, the bound Eq. (\\ref{nlimsrc}) does not apply. However in this case, the observed correlation \\cite{TAaniso14,augerAniso14} with local structure would not be explained.\n\nEq. (\\ref{num}) can be reconciled with the observed minimum source density, Eq. (\\ref{nlimsrc}),  if UHECR production is very inefficient with $\\epsilon_{_{UCR}} \\ll 1$.   However,  inefficiency is not a solution, since even the weakest bound in Eq. (\\ref{nlimsrc}) requires $\\approx 700$ sources within the GZK distance, whereas powerful steady sources (of any kind) with luminosity larger than $10^{45}$ erg\/s are rare \\cite{fg09}.  Another way out is via an  acceleration mechanism   which does not involve  magnetic field confinement and thus evades the luminosity requirement Eq. (\\ref{lumi}). However an efficient mechanism of this kind is not known. \n\n\nTransient sources can evade the previous conundrum.  They must satisfy the Hillas confinement condition embodied in Eqs. (\\ref{conf}) and (\\ref{lumi}), and furthermore the energy injection condition sets a limit on the energy that must be released in UHECRs in a single transient event:\n\\begin{equation}\n\\label{EUCR}\n{\\cal E}_{_{UCR}} \\equiv {\\dot E}_{_{UCR}} \\, \/ \\, \\Gamma_{_{\\rm UCRtran}} =  10^{51}\\,{\\rm erg} ~ {\\dot{E}}_{44}  ~ \\Gamma_{_{\\rm UCRtran, -7}}^{-1},\n\\end{equation}  \nwhere $\\Gamma_{_{\\rm UCRtran}} \\equiv \\Gamma_{_{\\rm UCRtran, -7}} \\, 10^{-7}$ Mpc$^{-3}$ yr$^{-1}$ is the rate that the UHECR-producing transients take place.  In addition, the number of sources contributing at a given time must be large enough.  Deflections in the extragalactic magnetic field spread out the arrival times of UHECRs from an individual transient event. \nIn the approximation that the deflections are small and many, the resultant characteristic arrival time spread is \\cite{waxmanME}:\n\\begin{equation}\n\\label{tau}\n\\tau \\approx \\frac{ D^{2} Z^2 \\langle B^2 \\lambda \\rangle}{9 E^{2}} = \n3 \\times 10^{5}\\, {\\rm yr} \n\\left(\\frac{D_{100}  \\, B_{\\rm nG}}{E_{20}\/Z}\\right)^{2}   \\lambda_{\\rm Mpc}, \n\\end{equation}\nwhere $D_{100}$ is the distance of the source divided by 100 Mpc, $B_{\\rm nG}$ is the rms random extragalactic magnetic field strength in $nG$, and $\\lambda_{\\rm Mpc}$ is its coherence length in Mpc.   To be compatible with the Auger bound,\nthe number density $n_{\\rm eff}$ of contributing transient sources at a given time, must satisfy\n\\begin{equation}\n\\label{neff}\nn_{\\rm eff} \\approx \\tau_{5} ~ \\Gamma_{_{\\rm UCRtran, -7}} \\, 10^{-2}\\,{\\rm Mpc}^{-3} \\, \\geq n_{\\rm min} = 2 \\times 10^{-5} ~ {\\rm Mpc}^{-3},\n\\end{equation}\nwhere $ \\tau_{5} \\,10^{5}\\, {\\rm yr}$ is the mean time delay averaged over sources.\n \nIn order for any transient UHECR source type to be viable it must therefore satisfy three requirements, the Hillas condition (Eqs. (\\ref{conf}) or (\\ref{lumi})) and also Eqs. (\\ref{EUCR}) and (\\ref{neff}).  \nThe classical transient source candidate, GRBs,  \\cite{waxman95} satisfy easily the first and third conditions, but \nthere is debate whether their energy output is sufficient to satisfy the second condition, Eq. (\\ref{EUCR}), unless their UHECR energy output exceeds significantly their photon output \\cite{fg09,eichler+GRB-UHECR10,katz+GRB-UHECR09}.  This, and the lack of the expected high energy neutrinos \\cite{IceCubeNatureGRBs12}, makes  GRBs less favorable source candidates.\n\nA number of  TDE flare candidates have been detected and followed up in real-time: \\textit{Swift}\\  J1644+57  \\cite{Burrows+11,Bloom+11,Zauderer+11},  PS1-10jh \\cite{gezari+Nature12} and PS1-11af \\cite{chornock+14}.  Two candidates had been found earlier in archival SDSS data \\cite{vf11} and one was subsequently found in archival Swift data  \\textit{Swift}\\ J2058.4+0516 \\cite{cenkoSw2058}.  (Still earlier TDE candidates were put forward in \\cite{donley02,Gezari08}, but AGN-flare background could not be well characterized for those observations, so the origins of the flares were uncertain.) \n \n\\section{TDEs as UHECR sources}\n\nIn this section we address whether TDE jets are good source candidates for the protonic UHECR scenario.\nWe begin with the Hillas  condition, Eq. (\\ref{conf}), which must be satisfied for any UHECR source that is based on EM acceleration. Here the recent observations of  \\textit{Swift}\\ J1644+57 provide us with an example of a TDE jet with very good multi-wavelength follow-up, enabling the Hillas criterion (Eq. \\ref{conf}) to be directly checked.\n\n\\subsection{\\textit{Swift}\\  J1644+57 and the Hillas criterion}\n\n\\textit{Swift}\\ J1644+57 was detected on March 25th 2011 by the {\\it Swift} satellite.\nIts location  at the nucleus of an inactive galaxy made it immediately a strong TDE candidate.  It  is uniquely suitable for testing whether TDE jets can satisfy the Hillas condition, Eq. (\\ref{conf}), because of the thorough multi-wavelength monitoring from its inception for more than 600 days, which has enabled detailed modeling of the conditions in its jet.  \nThe observations  \\cite{Burrows+11,Bloom+11,Zauderer+11} revealed  two different\nemission sites:  an inner emission region where the X-rays are emitted, and an outer region\nwhere the radio emission is produced. \n\n Basic models for the TDE emission follow the ideas of a gamma-ray bursts, c.f.,  e.g. \n\\cite{piran04}:  the central engine of the TDE produces a pair of relativistic jets via the\nBlandford-Znajeck \\cite{bz77} process, with internal dissipation shocks within the jets accelerating\nparticles and producing the X-ray emission at relatively short distances from the central engine.\nAt larger distances, the outflow interacts with the surrounding matter;  this slows it down and produces the radio emission, in an afterglow-like manner \\cite{Zauderer+11}.   In the following we consider both the X-ray and radio-emitting regions as possible sites for UHECR acceleration. \nWe discuss first the radio emitting region where the situation is clearer, as we can use simple equipartition arguments. We then discuss the situation within the X-ray emitting jet. \n\nThe conditions within the radio emitting region have been analyzed using relativistic equipartition considerations (see the Appendix). In the relativistic regime the equipartition analysis depends on the geometry of the emitting regions and  there are two possible solutions  \\cite{Barniol+13b} shown in  Fig. \\ref{fig:J1644_radio}.\nFor the most reasonable geometry -- a narrow jet with an opening angle 0.1 -- the equipartition value of $B R$   is slightly larger than $10^{17}$ Gauss cm.   Given \nthat the equipartition estimates  \nyield a lower limit on the energy, this can be considered compatible with the Hillas condition.  A more detailed model that takes into account the inter-relation between the X-ray jet and the radio emitting electrons  \\cite{Kumar+13} yields  $B R$  larger by a factor of a few  than the simple equipartition estimate \n\\cite{Barniol+13b} (see  Fig. \\ref{fig:J1644_radio});  at early times $B R \\approx 3 \\times 10^{17}$ Gauss cm.  \nThus,  the outer radio emitting region of the \\textit{Swift}\\ J1644+57 TDE jet appears likely to have had conditions for UHECR acceleration. \n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Figure1cropped.eps}\n\\caption{ The equipartition values of  $ B R  $  as a function of time, for different relativistic  models for the TDE \\textit{Swift}\\ J1644+57:  a narrow jet with an opening angle $0.1$ (red dots), a wide jet  (green dots) and the detailed model of \\cite{Kumar+13} (solid line). }  \\label{fig:J1644_radio} \n\\end{figure}\n\nDuring the first two days, the X-ray luminosity of \\textit{Swift}\\ J1644+57  fluctuated reaching isotropic equivalent peak luminosities of  $\\approx 10^{48}$ ergs\/sec.  Models of the X-ray emission  as arising from a relativistic blob of plasma \\cite{Burrows+11} yield \nestimates of $B R$ ranging from a few $\\times 10^{15}$  to $5 \\times 10^{17}$ Gauss cm, depending on \n the dominant energy of the jet (Poynting flux or baryonic), the emission mechanism (synchrotron or external IC), the position of the emission region and the relative contributions of the disk and the jet. \nIt is possible but not certain that the X-ray emitting regions in \\textit{Swift}\\ J1644+57 also had conditions for UHECR acceleration.  If so, from this point of view TDEs resemble powerful AGNs in the way they satisfy their Hillas condition, with UHECR acceleration being possible in two different regions.   \n\n\\subsection{Energy budget and source abundance}\nWith a total energy of $\\approx 10^{54} \\,(M_{*}\/M_{\\odot})$ erg available in a tidal disruption event, only a small fraction -- $ 10^{-3} \\, (\\langle M_{*}\\rangle\/M_{\\odot})  \\, {\\dot{E}}_{44} \\, \\Gamma_{_{\\rm UCRtran, -7}}^{-1}$, where $M_{*}$ is the mass of the tidally disrupted star --  needs to go to UHECR production in order to satisfy Eq. (\\ref{EUCR}), if the rate of TDEs producing jets capable of accelerating UHECRs is adequate.   \n\nThe rate of TDEs in {\\it inactive} galaxies has recently been determined based on the discovery of two TDEs in a search of SDSS Stripe 82 \\cite{vfRate14}.  In volumetric terms\n\\begin{equation} \\label{TDErate}\n\\Gamma_{\\rm TDE} =  (0.4-0.8) \\cdot 10^{-7\\pm 0.4}\\,{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1},\n\\end{equation}\nwhere the statistical uncertainty is in the exponent and the prefactor range reflects the  light curve uncertainty.\nThis result is roughly consistent with earlier theoretical \\cite{WangMerritt04} and observational estimates (within their uncertainties) \\cite{Gezari+09}.  However, more refined estimates are needed because not all TDEs have jets and only a fraction of those may be capable of producing UHECRs; in the following we ignore jets weaker than the \\textit{Swift}\\ events which satisfy the Hillas criteria.   \n\nAn estimate of the UHECR-producing TDE rate is obtained from the observed TDEs with jets,  \\textit{Swift}\\ J1644+57 and J2058+05.  Burrows et al. \\cite{Burrows+11} estimate that the observation of one event per seven years by \\textit{Swift}\\ corresponds to an all sky rate of $0.08-3.9$  events per year up to the detection distance of $z=0.8$ which contains a co-moving volume of $\\approx 100\\, {\\rm Gpc}^{3}$.  Their subsequent archival discovery of J2058+05 increased this rate by a factor of two and reduced somewhat the uncertainty range, leading to an estimated rate of $ \\approx 3  \\times 10^{-11}$ Mpc$^{-3}$ yr$^{-1}$ of TDE events with jets pointing towards us, or\n\\begin{equation}\n\\label{SwiftJetRate}\n\\Gamma_{_{\\rm UCRtran, -7}} \\approx 3 \\, f_b^{-1} \\, 10^{-4},\n\\end{equation}\nwhere $f_b$ is the beaming factor.  \n\nIn J1644+57, the total EM isotropic equivalent emitted energy in  X-rays was $ 3 \\times 10^{53}$ergs \\cite{Burrows+11,Bloom+11}; assuming that this is about 1\/3 of the total bolometric EM energy, they estimate that the total isotropic equivalent EM energy injection rate is $ \\approx 10^{54}$ ergs. Cenko et al. \\cite{Cenko+12} estimate a similar total  EM isotropic equivalent energy for J2058+05. \nSince the isotropic equivalent energy is a factor $f_{b}^{-1}$ larger than the true emitted energy, the beaming factor cancels between rate and energy factors, yielding an estimated \nEM energy injection rate of $\\approx 3 \\times  10^{43}$ erg \\, Mpc$^{-3}$ yr$^{-1}$ without relying on knowledge of the beaming factor.  This falls  short of the required energy injection rate for UHECRs, which is $2-4 \\times 10^{44}$ erg Mpc$^{-3}$ yr$^{-1}$. Interestingly a similar situation arises when comparing the observed EM GRB flux with the needed UHECR energy injection rate \n\\cite{fg09,eichler+GRB-UHECR10} (but see however \\cite{katz+GRB-UHECR09}).   The crux question here, is the relation between the emission observed in a particular EM band and the energy production in UHECRs.  The emission mechanisms and relevant particle energies are so different, that it is far from evident how to relate them.\n\nWe can also estimate the energy of the jet from the energy needed to produce the radio signal, using equipartition arguments and thus deriving a lower limit on the actual energy.   Different assumptions about conditions within the radio-emitting region lead to different energy estimates. \nFrom  \\cite{Barniol+13b}, we have an estimate of the minimal isotropic equivalent total energy of leptons and magnetic field that is capable of producing the radio emission produced in J1644+57, $\\approx  10^{51}$ erg;  including the energy of the accompanying protons increases the estimated minimal energy by a factor $\\sim 5$.   Unlike the relativistic inner jet producing the X-rays, the outflow has slowed to being only mildly relativistic with a  low Lorentz factor by the time it produces the observed radio, so that the radio emission is isotropic even when it arises from a jet.  Therefore, the factor $f_b^{-1}$ in the true rate of jetted-TDEs, Eq. (\\ref{SwiftJetRate}), does not cancel out and the implied energy injection rate (including the protonic contribution) is of order $ 2 \\times 10^{44}  (f_b\/10^{-3})^{-1} {\\rm erg} \\, {\\rm Mpc}^{-3}\\,{\\rm yr}^{-1} $, roughly what is needed as a UHECR flux. \n\nThe above discussion suggests that TDE jets alone could satisfy the needed UHECR injection rate, but underlines the importance of investigating what relationship should be expected between the EM and UHECR spectra and total luminosity. \n\nThe above discussion produced two independent, compatible estimates of the rate of TDEs with jets.  Using the \\textit{Swift}\\ observed rate and estimating the beaming factor to be  $f_{b} \\approx 10^{-2} - 10^{-3}$ based on the Lorentz factor of $\\sim 10-20$ estimated for \\textit{Swift}\\ J1644+57 \\cite{Burrows+11}, Eq. (\\ref{SwiftJetRate}) gives $\\Gamma_{_{\\rm UCRtran, -7}}^{-1} \\gtrsim 3 \\cdot 10^{-2}$.  Or, taking the total rate from \\cite{vfRate14} and a jet fraction $\\sim 0.1-0.2$ based on the fraction of TDEs detected in the radio \\cite{vV+Radio13}, gives $ \\Gamma_{_{\\rm UCRtran, -7}} \\lesssim 0.4$.    Using the lower estimate, \nEq. (\\ref{neff}) gives $n_{\\rm eff} \\approx   3 \\, \\tau_{5} \\,10^{-4}~ {\\rm Mpc}^{-3}$, comfortably compatible with the Auger source density limit in the low-deflection scenario, $n_{\\rm min} = 2 \\times 10^{-5} ~ {\\rm Mpc}^{-3}$.   \n\n\n\\section*{Composition}\n\nThe reader can ask why it is interesting to consider the possibility that UHECRs are predominantly or exclusively protonic, in view of the observed depth-of-shower-maximum distribution of AUGER which favors a predominantly mixed composition of intermediate mass nuclei, if interpreted with current hadronic interaction models tuned at the LHC  \\cite{augerXmaxMeas14,augerXmaxComp14}.  First, the recently finalized Auger analysis \\cite{augerXmaxComp14} finds a protonic component persisting to highest energies.  Second, a mixed composition requires a very hard injection spectrum incompatible with shock acceleration \\cite{shahamPiran13,Aloisio+13} and a composition at the source which has been argued to be strange and unlikely \\cite{shahamPiran13}.  Third, a  predominantly proton composition is of particular interest because both Auger and TA find evidence of correlations between UHECRs above 55 EeV and the local matter distribution \\cite{TAaniso14,augerAniso14}, although anisotropy {\\it per se} does not exclude mixed composition, particularly for the case of a single, nearby source \\cite{Piran11,rfICRC13,kfs14}.  The final and most compelling virtue of a predominantly proton composition is that it naturally explains the shape of the spectrum from below $10^{18}$ eV to the highest energies, including the observed ``dip'' structure around $10^{18.5}$ eV and the cutoff at highest energies, without needing an ad-hoc and fine-tuned transitional component between Galactic and extragalactic cosmic rays \\cite{berezGGdip05,AlBerezGaz12} with additional parameters to tune the composition and maximum energy by hand.  \n\nThe reader might also ask whether it is legitimate to set aside inferences on composition from current hadronic interaction models; the answer is ``yes''.  The nucleon-nucleon CM energy in the collision of a $10^{18}$ eV proton with the air -- 140 TeV -- is a factor 20 above current the LHC CM energy, so that the models must be extrapolated far into uncharted territory.  Furthermore, detailed comparisons by the Auger collaboration of the model predictions with a variety of observed shower properties reveals several discrepancies, including that the models underpredict the muon content of the ground shower by 30\\% or more \\cite{ICRC13topdown,augerHorizMuons14}, and that the model which does the best with respect to the muons at ground level (EPOS-LHC) is in the most serious contradiction with the observed depth of muon production in the atmosphere \\cite{augerMPD14}.\n\n\\section*{Summary}\n  \n\nTo conclude, we have shown that a scenario in which UHECRs are predominantly or purely protons can be realized, with acceleration occurring in transient AGN-like jets created in stellar tidal disruption events.  A well-studied example of such a TDE-jet, \\textit{Swift}\\ J1644+57, displays inner and outer emission sites in which collisionless shocks satisfy the Hillas criteria.\nThus we propose that, like in AGN models and GRB models, the  basic shock acceleration mechanism is applicable for UHECR acceleration in TDE jets.  As shown in \\cite{fg09}, the conditions in such jets are such that the radiation fields within the outflow are not large enough to cool the UHECRs before they escape.  Thus both the outer and inner emission regions in TDEs may in principle be viable UHECR sources.  \n\nWe also investigated whether the total observed flux of UHECRs is compatible with the UHECR injection rate that can be expected for TDE jets; although a more thorough theoretical understanding of the UHECR acceleration mechanism is needed for a definitive conclusion, present evidence indicates the energetics are satisfactory.  Finally, we showed that the effective number of sources predicted in the protonic-UHECRs-from-TDE-jets scenario, is compatible with the even the most stringent Auger bound, i.e., the case that typical deflections are less than $3^{\\circ}$.\n\nUnlike for GRBs, the TDE-jet model for UHECR production cannot be tested directly by association of an observed transient event with a signature of UHECRs.   In the case of GRBs, prompt neutrinos are produced via photoproduction of charged pions in the source, which arrive approximately simultaneously with the gammas.  (The UHECRs themselves arrive 10's or 100's of thousands of years after the gammas or neutrinos, due to the magnetic deflections discussed previously.)  By contrast, the level of prompt neutrino production in a TDE-jet is much lower, because the radiation field in a TDE jet is less, inhibiting photopion production.   Moreover the duration of UHECR production in a TDE jet is weeks or months, so even the prompt neutrinos are broadly spread in arrival times. \n\nThe conjecture of predominantly protonic composition, the role of transients in UHECR production, and the TDE-jet model can be tested purely observationally, as follows.  \\\\\n\\noindent $\\bullet$  Whether UHECRs are protons or nuclei can in principle be determined without relying on hadronic interaction models to infer composition, by detecting or placing sufficiently strong limits on VHE photons and neutrinos produced during the propagation, as their spectra distinguish UHECR protons from nuclei.   If UHECRs are predominantly protonic, as shown above (updating earlier arguments \\cite{fg09,WL09,MuraseTakami09}) their primary sources must be transients, with TDE jets the leading candidate.  \n\\\\\n\\noindent $\\bullet$  Whether sources are continuous or transient can be determined from the spectrum of UHECRs from a single source, because UHECRs arriving at a given epoch from a transient have similar values of rigidity $\\equiv E\/Z$ rather than displaying a power-law spectrum \\cite{waxmanME}, c.f., Eq. (\\ref{tau}).  \n\\\\\n\\noindent $\\bullet$  \nIf sources are confirmed to be transients, the presence or absence of VHE neutrinos accompanying the transient EM outburst, will distinguish between GRB and TDEs being the sources.  \n\\\\\n\\noindent $\\bullet$   As far as is presently known, the galaxies hosting TDEs are generally representative of all galaxies; if so, UHECR arrival directions would correlate (only) with the large scale structure, after taking into account Galactic and extragalactic magnetic deflections.  However if the rate of TDEs with jets is enhanced in active galactic nuclei, as conjectured in \\cite{fg09}, an enhanced correlation of UHECRs with AGNs relative to random galaxies could potentially be seen.\n\n\\noindent{\\bf Acknowledgements:}\nWe thank Rodolfo Barniol-Duran  for discussions and for help preparing the figure, and Kohta Murase for helpful discussions.  The research of GRF was supported in part by NSF-PHY-1212538; she thanks the Racah Institute for its hospitality during the initial stages of this work.  GRF is a member of the Pierre Auger Collaboration and acknowledges valuable interactions with Auger colleagues.  The research of TP was supported by the ERC advanced research grant ``GRBs''  the  I-CORE (grant No 1829\/12) and a grant from the Israel Space Agency SELA; he thanks the \nLagrange institute de Paris for hospitality while this research was concluded. \n\n\\section{Appendix:  Details of \\textit{Swift}\\ J1644+57 Analyses}\n\n\\noindent {\\bf The Radio Emitting Region:}\\\\\nRadio observations of Sw1644 began 0.9 days after the onset of the trigger \\cite{Zauderer+11}  and lasted for about $ 600$ days \\cite{Berger+12,Zauderer+13}. Wide frequency coverage began at 5 days.  At that time, the peak of the spectral energy distribution (SED) was at  $ \\nu_p \\approx 345$ GHz, with a peak flux  $F_p = 35$ mJy. The peak frequency and flux decreased  to $\\sim 5$ GHz and $0.5$ mJy at 570 days.   We begin by discussing the classical equipartition method of interpreting radio observations \\cite{Pacholczyk1970,ScottReadhead77,Chevalier98,kumarNarayan09}, and its relativistic generalization \\cite{Barniol+13a}.  A direct application to the Sw1644 observations \\cite{Zauderer+11} is likely overly naive, as we discuss subsequently.\n\nThe radio emitting region is characterized by four unknowns:  the size, $R$,   the magnetic field, $B$,  the total number of emitting electrons, $N$, and their typical Lorentz factor, $\\gamma_e$. The total energy of the emitting region is the sum of the electrons' energy  $N m_e c^2 \\gamma_e$ and the magnetic field energy $B^2 R^3 \/6$, the baryons' energy being unimportant in the Newtonian case. \nIdentifying the spectral peak as the self absorption frequency \nand using the standard expressions for the synchrotron frequency,  synchrotron  flux and  the self-absorbed flux (see e.g. \\cite{Barniol+13a} for details) one can eliminate 3 of the 4 parameters and express, e.g., $B$, $N$ and $\\gamma_e$ in terms of  $R$ and the observables $\\nu_p$, $F_p$ and $z$. One then \nobtains  $ E = C_1 \/R_{17}^6 + C_2 R_{17}^{11}$, where the first term is the electrons' energy and the second the magnetic field energy.  The constants $C_1$ and $C_2$ are given in term of the observables: \n$C_1 = 4.4 \\times 10^{50} \\,{\\rm erg} \\, ( F_{p}^4 \\, d_{28}^8 \\, \\nu_{p,10}^{-7} \\, \\, (1+z)^{-11} ) $ and \n$C_2 = 2.1 \\times 10^{46} \\,{\\rm erg} \\, (  F_{p}^{-4} \\, d_{28}^{-8} \\, \\nu_{p,10}^{10} \\, (1+z)^{14}  ), $ where $d_{28}(z)$ is the   luminosity distance in units of  $10^{28}$cm and $F_p$ is the peak flux measured in mJy. \n\nThe energy is minimized when the electrons' energy is roughly equal\nto the magnetic energy, or put differently, when the system is in equipartition. \nThe size of the system is strongly constrained, as the energy is a very steep function of $R$ both above and below the minimum.   \nAs we have three equations and four unknowns we can choose a different independent variable. For our purpose  $B R$ is most suitable and in this case one obtains $ E = \\tilde C_1 \/( B R)^{6\/5}  + \\tilde C_2 (B R)^{11\/5}$.  This dependence is less steep and hence the resulting value for $BR$ is less constrained by these considerations.  If $E$ is an order of magnitude above the minimal value, $B R$ can be a factor-10 lower or a factor-3 higher than at equipartition. \n\nWhen applying the equipartition considerations to Sw1644 one has to take into account that the outflow is relativistic \nin this case. The  relativistic equipartition estimates are somewhat more complicated than the\nNewtonian ones. A detailed equipartiton formalism for relativistic outflows was  developed recently by \\cite{Barniol+13a}.\nLike in the Newtonian case, the total energy depends very steeply on $R$, as\n$ E = \\hat C_1 \/R^6 + \\hat C_2 R^{11}$, where  $\\hat C_1$ and $\\hat C_2$  depend  on the observed quantities\nbut now also on  the outflow Lorentz factor, $\\Gamma$, and on the specific geometry of the emitting region (see \\cite{Barniol+13a} for details). \nNote that here the kinetic energy of the baryons within the relativistic outflow should also be included in the total energy of the system.  The bulk Lorentz factor, $\\Gamma$,  can be determined using time of arrival arguments; the geometrical factors involved have to be guessed.   Given the very steep dependence of the total energy on $R$, $R$ is still  well constrained by the energy-minimization, equipartition considerations. Using these arguments \\cite{Barniol+13b} find that  for the most reasonable geometry -- a narrow jet with an opening angle 0.1 -- the equipartition value of $B R$ in Sw1644 is slightly larger than $10^{17}$ Gauss cm \n(see Fig. \\ref{fig:J1644_radio}).  Given \nthat the equipartition estimates give a lower limit on the energy (that assumes maximal efficiency), this can be considered compatible with the Hillas condition.  Furthermore and independently, at the time of the last observations the minimal (equipartition)  energy is  $0.8 \\times 10^{51}$ ergs \\cite{Barniol+13b}, which is consistent with the energy required to account for the observed UHECR spectrum given the rate of TDEs, estimated above.\n\nHowever the naive equipartition analysis can be doubted, since applying it leads to the conclusion that the energy of the jet increases by about a factor of 20 from the initial observations at around 5 days, to the final observations.  The apparent increase required in the jet energy appears in other approaches as well, as noticed first\nby \\cite{Berger+12}, who analyzed the data based on GRB afterglow modeling. This interpretation  requires an energy supply to the radio emitting region. Such an energy supply is inconsistent with the continous  decrease in the X-ray luminosity during this period \\cite{Berger+12}, which supposedly reflects the activity of the inner engine and the accretion rate.  \n\nKumar et al.  \\cite{Kumar+13} suggested that the puzzling behavior comes about because the X-ray jet passes through the radio emitting region, causing the radio emitting electrons to be continuously cooled via Inverse Compton (IC) scattering with these X-ray photons.   This efficient IC cooling decreases the observed synchrotron radio flux relative to the equipartition estimate, causing the equipartition analysis to yield a lower energy than the true energy content of the system, resulting in an erroneously-low inferred value for $B R$.  At later times  the X-ray flux diminishes, the IC cooling ceases, and the synchrotron flux increases, consistent with the late-time observations and obviating the need for an increase of the \nenergy of the jet.  \\cite{Kumar+13} estimate  $B R$ to be larger by a factor of a few  than the simple equipartition estimate \n\\cite{Barniol+13b}. The results from their analysis are shown as the solid line in  Fig. \\ref{fig:J1644_radio}; at early times $B R \\approx 3 \\times 10^{17}$ Gauss cm.  This is just  the value needed to accelerate $10^{20}$ eV UHECRs.\n\nTo summarize, even with naive application of equipartition, the values of the total energy and $B R$ within the radio emitting region of TDE J1644 are marginally compatible with those needed to accelerate UHECRs.  Given \nthat the equipartition estimates give a lower limit on the energy (i.e., assumes maximal efficiency), and that the more physically satisfactory modeling of \\cite{Kumar+13} yields a larger estimate comfortably compatible with accelerating protons to UHE, we conclude that the outer region of the Sw 1644 TDE jet likely has conditions for UHECR acceleration. \n\n\\noindent{\\bf The X-ray emitting region:} \\\\\nThe isotropic equivalent X-ray luminosity is $\\approx 10^{48}$ ergs\/sec.   If this were coming from the accretion disk, then by the argument of \\cite{fg09} we would conclude that Eq. (\\ref{lumi}) was easily satisfied within the jet.  However since in \\textit{Swift}\\ J1644 we are viewing the jet close to its axis, there can be additional contributions to the X-ray emission, which must be modeled before $B R$ in the inner jet region can be inferred.   Unfortunately, the X-ray  observations are much less constraining on the conditions within the X-ray emitting regions than are the radio observations on the conditions in the outer region.  The X-ray spectrum is a power law;  if combined with the NIR observation it yields  a steep slope (steeper than 1\/3).  The  Fermi upper limits on the GeV emission suggest a suppression of the high energy emission due to photon-photon opacity. Overall, only a single component has been observed, with no clear evidence of the peak frequency and only an upper limit on the high energy IC component. Therefore there is  significant  freedom in modeling this emission. \n\nRef. \\cite{Burrows+11}  models the emission (following the blazer emission model of \\cite{GhiselliniTavecchio09}) as arising from a relativistic blob of plasma. They  \nhave put forward three models for this spectrum. \nThe models differ in the dominant energy of the jet (Poynting flux or baryonic), the emission mechanims (synchrotron or external IC), the position of the emission region and the relative contributions of the disk and the jet. \nAccording to these models the X-rays are generated between $10^{14}$ and $10^{16}$ cm from the black hole. Estimates of $BR$ range from a few $\\times 10^{15}$ for model 3 to $5 \\times 10^{17}$ for model 2.\n\n\n\\defAstrophys.\\ J.{Astrophys.\\ J.}\n\\defNature{Nature}\n\\defAstrophys.\\ J. Lett.{Astrophys.\\ J. Lett.}\n\\defAstrophys.\\ J.{Astrophys.\\ J.}\n\\defAstron.\\ Astrophys.{Astron.\\ Astrophys.}\n\\defPhys. Rev. D{Phys. Rev. D}\n\\defPhys.\\ Rep.{Phys.\\ Rep.}\n\\defMonth. Not. RAS {Month. Not. RAS }\n\\defAnnual Rev. Astron. \\& Astrophys.{Annual Rev. Astron. \\& Astrophys.}\n\\defAstron. \\& Astrophys. Rev.{Astron. \\& Astrophys. Rev.}\n\\defAstronom. J.{Astronom. J.}\n\\defJCAP{JCAP}\n\n\n\\bibliographystyle{apsrev4-1.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nElemental cerium is stark example of the complicating effects of magnetism in metals.  In principle a simple system, cerium at room temperature and ambient pressure presents a monatomic Bravais lattice (the face-centered-cubic $\\gamma$ phase).  However, under the application of modest pressures, cerium undergoes a spectacular first-order transition to a low temperature $\\alpha$ phase.  Both $\\alpha$ and $\\gamma$ are isostructural f.c.c. phases, but differ by $\\sim 15 \\%$  in volume, and have markedly different magnetic susceptibilities and resistivities.\\cite{bestreview}  There are competing explanations for the transition, which variously invoke a ``Kondo Volume Collapse''\\cite{KVC} mechanism, a ``Mott Transition'',\\cite{MT} or some entropic mechanism\\cite{entropy,jpcm_celath}.  In all cases however, it is generally agreed that the large volume $\\gamma$ phase contains relatively larger localized magnetic moments (evinced by the higher susceptibility) and relatively less itinerant electrons (evinced by the higher resistivity).  Beyond that, the nature of the transisiton remains contentious.\n\nOne of the main complications in the study of cerium is that although the {$\\gamma - \\alpha$} transition occurs on cooling over a wide range of pressures, it is avoided at ambient pressure.  In that case, an intervening dhcp phase forms from the $\\gamma$ phase, and transforms into the $\\alpha$ phase only at lower temperature.\\cite{bestreview}  There are commonly two methods used to resolve this problem.  Measurements are either performed under pressure on pure cerium, or else the cerium is alloyed with other elements known to suppress the dhcp phase (typically thorium).  Results from these two avenues of investigation are not always consistent - for example, it has been claimed that the phonon entropy plays little role in the {$\\gamma - \\alpha$} transition in a {Ce$_{0.9}$Th$_{0.1}$} alloy,\\cite{neutron1,neutron2} while studies on pure cerium under pressure have returned the opposite result.\\cite{puretron,PNASxray}  Nevertheless, the alloying of cerium has generated a family of related compounds, in which the order of the transition, the magnitude of the volume collapse, and the transition temperature can be smoothly varied.\\cite{fiskthompson,lashtricrit}\n\nOne such alloy, {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}, is of particular interest in regards to spin-lattice coupling at the phase transition.  Alloying with thorium suppresses the dhcp phase, while alloying with lanthanum causes the transition to become continuous and supresses the critical temperature.\\cite{fiskthompson, lashtricrit}  It has been reported that cooling {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} in high magnetic fields can suppress the resistivity change associated with the volume collapse to lower temperature, and that the application of pulsed magnetic fields can induce a large moment below the critical temperature.\\cite{jpcm_celath}  These measurements map out a phase boundary for the {$\\gamma - \\alpha$} transition as a function of magnetic field, and suggest that the volume collapse can be fully suppressed by $\\mu_{0} H \\lesssim 56$ Tesla.  Here we report a direct study of the structure of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}, using synchrotron x-ray powder diffraction in pulsed magnetic fields as high as 28 Tesla.  Lattice parameter and structural correlation length are extracted throughout the $H - T$ phase diagram.\n\n\n\n\\section{Experiment}\n\nSamples of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} were prepared at Los Alamos by arc-melting elemental cerium, lanthanum and thorium in a zirconium gettered argon atmosphere.  The button was melted and flipped 15 times to ensure mixing.  The final melt was cooled in a round-bottomed trough, which was 5 mm wide.  The resulting $\\sim$ 25 mm long sample was unidirectionally rolled $\\sim$ 10 times to achieve a finished thickness of 50 microns.  A section of the as-rolled material was heat treated at 450$^\\circ$C for 1 hour to remove the rolling texture before a final sample, 2 mm in diameter, was cut from the sheet with a razor blade.  The sample was subsequently mounted inside a blind hole in a sapphire bracket, and sealed with a sapphire cap and stycast epoxy, to ensure good thermal contact and prevent oxidization.  The sapphire bracket was anchored to the cold finger of the double-funnel solenoid pulsed magnet at the Advanced Photon Source (APS), which facilitates diffraction at up to 22$^{\\circ}$ of scattering angle, to temperatures as low as $\\sim 5$ K and in magnetic fields as high as $\\sim$ 30 Tesla.\\cite{solenoid1,solenoid2}  This system features a resistive magnet coil (of Tohoku design) in a liquid nitrogen bath, connected to a 40 kJ capacitor bank.  The bank is capable of discharging a 3 kV charge through the coil in a $\\sim 6$ ms half-sine wave pulse, generating a peak current of $\\sim 10$ kA and a peak magnetic field of $\\sim 30$ Tesla.  The heat generated in the resistive coil during a pulse must be allowed to dissipate before the system can be pulsed again.  At peak field values, this requires a wait time $\\gtrsim$ 8 minutes between pulses.  The sample to be studied is both thermally and vibrationally isolated from the coil.  This new instrument offers a complementary measurement geometry to the APS split-pair pulsed magnet developed previously.\\cite{firstrsi,ttopmag}  \n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.1cm]{fig1.eps}\n\\caption {Top:  The inicident x-ray pulse (the narrow pulse) is timed to arrive during the magnetic field peak (broader pulse).  In this case, a 5.75 ms magnetic field pulse to ~15.4 Tesla weighted by an x-ray exposure with a 1 ms fullwidth gives an average magnetic field value of $\\sim$ 14.9 Tesla, integrated over  $\\pm$ 0.5 Tesla for the exposure.  Middle:  Raw data collected on the image plate for a single 1 ms exposure.  Bottom:  The same data, radially integrated and converted to $|\\bf{q}|$.}\n\\label{fig:1}\n\\end{figure}\n\nExperiments were performed at the 6ID-B station at the APS.  30 keV photons were selected using a Si(111) monochromator, with the mirror removed to provide an unfocused beam.  The monochromator was detuned so as to reduce higher harmonic contamination.  The incident beam size was defined by slits, with an illuminated area on sample of 0.2 mm X 1.0 mm.  A $\\sim 1$ ms pulse of x-rays was defined using a pair of fast platinum shutters, and the arrival of this pulse at the sample was synchronized with the peak of the magnetic field pulse.  Scattered x-rays were recorded on a two-dimensional image plate detector.  In this configuration, our system is similar to the pulsed magnet instrument in operation at the European Synchrotron Radiation Facility (ESRF).\\cite{detlefs_ins}  To calibrate our instrument, we repeated the measurements\\cite{solenoid2} of Detlefs \\textit{et al.} on polycrystalline TbVO$_4$ in pulsed magnetic fields, as previously measured by the ESRF instrument.\\cite{detlefs_tbvo}  We reproduced the reported high field peak splitting in the tetragonal phase of TbVO$_4$,\\cite{detlefs_tbvo} validating our measurement scheme.\\cite{solenoid1,solenoid2}  We performed our measurements of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} with the axis of the solenoid coincident with the incident beam, and the detector positioned so as to capture the majority of the solenoid exit window.  In this way, at 30 keV the first two powder rings (corresponding to the $<$111$>$ and $<$200$>$ peaks) are completely captured by the detector, allowing a full radial integration and excellent counting statistics even for a single 1 ms exposure.  The current through the solenoid and the x-ray flux incident on the sample were monitored as a function of time by tracing the output of a high-resolution current monitor and an air-filled ion chamber on a digital storage oscilloscope.  A typical trace is shown in Fig. 1, along with data from a single exposure on the image plate.  In order to extract the correlation length and lattice parameter from these diffraction patterns, each of the two lowest lying peaks was fit individually, generating two independent sets of data, which were subsequently compared.  In all cases, the properties extracted from the two peaks aggreed within the errorbars of the measurement, and the quantities quoted in the following are the average of the two.\n\n\n\\section{Zero-Field Results}\n\nPowder diffraction patterns at room temperature show a single phase material, with symmetry and lattice constant consistent with the $\\gamma$ phase of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  On cooling, we observe the first two Bragg peaks to shift to higher $|\\bf{q}|$, while also broadening by a factor of $\\sim 2$ and exhibiting a reduction in peak intensity (see Fig. 2).  This is consistent with a collapse in volume, accompanied by a striking reduction in the structural correlation length.   It is interesting to note that we observe a single phase at all temperatures, in contrast to recent neutron diffraction measurements on {Ce$_{0.9}$Th$_{0.1}$} which reported a coexistence of both the $\\alpha$ and $\\gamma$ phases over a wide range of temperatures.\\cite{neutron2}  Our sample was initially cooled directly to 8K to characterize the magnitude of the volume collapse, then warmed to 250 K while collecting data.  The volume was observed to approach the $\\gamma$ phase value quite gradually.  Next we cooled slowly and measured the lattice parameters with a fine point density.  The resulting continuous but hysteretic volume collapse curves are plotted in Fig. 3.  Surprisingly, on our second cooling we found that the absolute size of the volume collpase was diminished - thermal cycling of the sample is detrimental to the phase transition.\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig2.eps}\n\\caption {Zero-field temperature dependence of the $<$111$>$ Bragg reflection.  The peak shifts to higher $|\\bf{q}|$ and broadens on cooling.  The sample is a single phase at all temperatures.  Resolution limited peaks at high temperature are well described by Gaussians, while a Voigt shape at low temperature is consistent with an Ornstein-Zernike scattering function with a short correlation length (solid lines).}\n\\label{fig:2}\n\\end{figure}\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig3.eps}\n\\caption {Zero-field volume collapse in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Sample was warmed from 8 K to 250 K ($\\blacksquare$), then cooled back to 10 K ($\\bullet$).  This thermal cycling lead to a reduction in the magnitude of the volume collapse, shown by the dashed lines.  The inset shows the Ornstein-Zernike correlation length extracted from the cooling measurements (see text).  The correlated regions of the collapsed phase have average size of only $\\sim 20$ nm.}\n\\label{fig:3}\n\\end{figure}\n\n\nThe observed peak broadening can be modelled phenomenologically, by assuming that the high temperature patterns show long-range ordered and resolution limited scattering, while the low temperature broadening arises from a reduction in the Ornstein-Zernike correlation length.  Our high temperature peaks are well described by Gaussians, and we are assuming a Lorentzian form for the scattering function (see Ref.[\\onlinecite{strucfluc}] for details).  Therefore, the measured scattering at low temperature should be well described by the convolution of a Lorentzian and a Gaussian - a Voigt function.  Since the width of this Voigt function is related to the width of the Gaussian (the resolution) and the width of the Lorentzian (the inverse correlation length), the numerically extracted fullwidth of the Bragg peaks can be converted to a resolution-deconvoluted correlation length.  The solid lines in Fig. 2 demontrate the degree of agreement between these lineshapes and the measured data, and the extracted correlation length is plotted in the inset of Fig. 3.  Our resolution limit is found to be on the order of a few microns, however, the low temperature collapsed phase exhibits correlation lengths which are roughly two orders of magnitude shorter than this.  Below $\\sim$ 30 K, the $\\alpha$ phase correlations are seen to extend over an average range of $\\sim$ 20 nm.  Since our alloy is $\\sim$ 80$\\%$ cerium, the mean distance between dopant atoms (either lanthanum or thorium) is on the order of 2 nm.  Thus, while the inclusion of lanthanum and thorium in the alloy does not completely hinder the formation of the $\\alpha$ phase, and each correlated region on average contains multiple dopant atoms, nevertheless these atoms act to strongly disorder the sample at the nanoscale within the $\\alpha$ phase.  Conversely, within the $\\gamma$ phase, individual grains are well correlated over micron-sized regions despite the same level of chemical disorder.\n\n\\section{results in pulsed magnetic field}\n\nMotivated by the results of Ref. [\\onlinecite{jpcm_celath}], we next sought to characterize the effect of applied magnetic field on the volume collapse transition in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Currently, the available capacitance at the APS limits our measurements to applied magnetic fields $\\lesssim$ 30 Tesla, well below the 56 Tesla estimated to fully suppress the {$\\gamma - \\alpha$} transition to zero Kelvin.\\cite{jpcm_celath}  However, inspection of the resistivity measurements of Ref. [\\onlinecite{jpcm_celath}] reveals that our highest achievable magnetic fields should be sufficient to suppress the temperature of the volume collapse inflection point by $\\sim$ 8 K.  We can maximize our sensitivity to this effect by cooling to the inflection point of the volume collapse ($\\sim$ 35 K) and pulsing to high magnetic field.  The slope of $\\frac{\\Delta V}{V}$ is large at this temperature, and therefore an 8 K shift in the inflection point should cause a $\\sim$ 0.7 $\\%$ change in volume at 35 K, well within our resolution.  However, as can be seen from Fig. 4, we failed to observe this effect.  After cooling to 35 K, we collected five diffraction patterns at zero field, which served to define the zero field Bragg peak positions to an accuracy better than $\\sim 10^{-3} \\AA^{-1}$.  Next, we collected ten patterns in pulsed field, with the average applied field over the x-ray exposure being $\\sim 28 \\pm 1$ Tesla.  We observed no measurable change in the Bragg peak positions in any individual pattern.  Nor did we observe any cumulative effect from multiple high-field pulses.  Our measurements therefore constrain any volume changes at (35 K, 28 T) to be less than $\\sim$ 0.07 $\\%$, a full order of magnitude smaller than what would be expected.\\cite{jpcm_celath} \n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig4.eps}\n\\caption {Observed and expected effects of 28 Tesla pulsed magnetic fields on the $<$111$>$ Bragg peak at T=35 K.  There is no discernable change in the position or width of the peak.  The expected curve is calculated based on the position and width change that would arise from an 8 K shift in the transition temperature, consistent with the results of Ref. [\\onlinecite{jpcm_celath}].}\n\\label{fig:4}\n\\end{figure}\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig5.eps}\n\\caption {Volume collapse ($\\Delta V\/V$) and Ornstein-Zernike correlation length ($\\xi$), comparing zero field cooled measurements and measurements where the system is repeatedly pulsed to 28 T on cooling.  No field effect is seen.}\n\\label{fig:5}\n\\end{figure}\n\nIt is important to remember that the evidence for field induced suppression of the volume collapse in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} for magnetic fields below 30 Tesla comes from resistivity curves, measured by cooling in constant applied magnetic fields.  This is procedurally distinct from the zero-field-cooled pulsed measurements reported here.  Discrepancy between field-cooled and zero-field-cooled behaviour is a hallmark of spin glass materials, wherein chemical disorder and competing interactions collude to freeze in disordered ground states.\\cite{spinglassrev}  We have already shown that chemical disorder in our alloy has a strong effect on the development of $\\alpha$-phase correlations (see Fig. 3).  It is therefore tempting to sugest that there may be some glassy behavior at play in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Direct comparison between field cooled and zero field cooled measurements are therefore highly desirable, however, a truly field-cooled measurement is not achievable with pulsed magnets.  We have attempted to approximate one by slow cooling, and pulsing in regular intervals.  In order to minimize the obscuring effects of thermal cycling, we performed two identical cooling curves, one after the other, warming to 280 K once to reset the system.  For each of these measurements, we cooled at a controlled rate of 1 K$\/$min from 280 K.  During the first of these cooldowns, we pulsed to 28 T at 10 min (10 K) intervals.  During the second cooldown, we collected zero-field data in the same manner, albeit more frequently since there was no need to wait for the coil to cool down. The results are plotted in Fig. 5.  Clearly, magnetic fields applied in this way are no more effective at suppressing the volume collapse than the zero-field-cooled methods shown in Fig. 4. \n\nWe are therefore forced to conclude that the available magnetic fields were insufficient to induce any structural effects in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  If the {$\\gamma - \\alpha$} transition were continuous for this alloy (as claimed in Ref. [\\onlinecite{fiskthompson}]), and the magnetic field induced suppression occured as claimed in Ref. [\\onlinecite{jpcm_celath}], then we would have expected to see a clear shift in $|\\bf{q}|$ as denoted in Fig. 4 for the zero-field-cooled measurements, and a clear suppression in temperature should have occurred for the repetitively pulsed measurements reported in Fig. 5.  Conversely, if the {$\\gamma - \\alpha$} transition were first order for this alloy (as claimed in Ref. [\\onlinecite{lashtricrit}]), then our measurements imply that the phase boundary was never crossed in our cooling curve.  This would require a steeper increase of the critical field on lowering temperature than was reported in Ref. [\\onlinecite{jpcm_celath}.]  It is also possible that the alloy is dynamically inhibited on the timescale of the magnetic field pulse, which would be consistent with a glassy behavior.  Finally, it is possible that the effects reported in Ref. [\\onlinecite{jpcm_celath}] were not intrinsic to {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Clearly, the issue of field induced suppression of the {$\\gamma - \\alpha$} transition requires further investigation in elemental cerium and its alloys.\n\n\n\\section{effect of thermal cycling}\n\nAs has been noted in the preceeding sections, we have observed a detrimental effect due to thermal cycling.  Our initial measurements on as-grown samples revealed well-correlated grains on the order of microns in size.  On first cooling, the volume collapsed $\\alpha$ phase was seen to exhibit nanoscale disorder, and on second cooling the absolute size of the volume collapse had diminished (see Fig. 3 and discussion in Section III).  Over the course of our week-long experiment, we observed the $\\alpha$-phase volume to gradually increase with each thermal cycling, while the $\\gamma$-phase volume was seen to gradually decrease.  In addition, while the $\\alpha$-phase consistently showed nanoscale disorder with correlation lengths of $\\sim$ 20 nm, the $\\gamma$ phase correlations were observed to shrink, from micron-sized grains in the as-grown sample to correlation lengths of only $\\sim$ 150 nm after 11 thermal cycles.  Zero-field cooling curves from the second, seventh, and eleventh cycle are plotted in Fig. 6, to illustrate the effect.  It is likely that the strains associated with the large volume change and spatial variations in doping,\\cite{fiskthompson} which are manifest in the $\\alpha$-phase disorder, act to introduce new domain boundaries at low temperature, which thereafter persist as defects.  This effect is insidious, since the suppression by thermal cycling we observe here could easily be mistaken for suppression by an external perturbation, if that perturbation was varied monotonically during the course of an experiment.  Hypothetically, had we measured cooling curves through the volume collapse in DC magnetic fields which were monotomically increased during the course of our measurements, we may easily have misinterpreted the gradual buildup of thermal cycling damage as a magnetic field induced effect.  Therefore, it is important to take extra care when mapping out phase diagrams in cerium alloys.  More generally, these measurements highlight the need for careful consideration of strain driven effects in all materials studies, which can often obfuscate results.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=8.5cm]{fig6.eps}\n\\caption {Gradual contamination by nanoscale disorder, induced by thermal cycling through the {$\\gamma - \\alpha$} transition.  The effects are clear in both the magnitude of the volume collapse ($\\Delta V\/V$) and the Ornstein-Zernike correlation length ($\\xi$).  All data was collected on cooling in zero magnetic field.  The three data sets represent the second ($\\bullet$), seventh ($\\blacksquare$), and eleventh ($\\blacktriangle$) thermal cycle.}\n\\label{fig:6}\n\\end{figure}\n\n\\section{conclusions} \n\nIn this article, we have reported a synchrotron x-ray diffraction study of the volume collapse transition in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}, as a function of temperature and pulsed magnetic field.  In this alloy, the {$\\gamma - \\alpha$} transition is smooth but hysteretic, characterized by a $\\sim 4 \\%$ volume change and a dramatic reduction in structural correlation length.  The collapsed $\\alpha$ phase is shown to be unaffected by pulsed magnetic fields $\\lesssim$ 28 Tesla, both in zero-field cooled conditions and under repetitive pulsing on cooling.  It is suggested that a deviation between truly field cooled and zero field cooled behavior in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} may account for the disagreement between our measurements and previous bulk studies.\\cite{jpcm_celath}  This implies a glassy character to the low temperature phase.  We have also shown that repeated thermal cycling through the {$\\gamma - \\alpha$} transition acts to disorder the sample on nanometer length scales, while suppressing the magnitude of the volume collapse transition.  These measurements may constitute a microscopic measure of what has been generally refered to as ``sluggish dynamics'' in cerium alloys.\\cite{jpcm_celath,fiskthompson}\n\nIt is our hope that these results will advance understanding of the exotic {$\\gamma - \\alpha$} transition in elemental cerium and its alloys.  Specifically, we believe the data presented here highlights the strong effects of disorder in these systems.  We gratefully acknowledge fruitful discussions with J. Lashley and B. Toby.  Use of the Advanced Photon Source is supported by the DOE, Office of Science, under Contract No. DE-AC02-06CH11357.  Pulsed magnet collaborations between Argonne and Tohoku University are supported by the ICC-IMR.   HN acknowledges KAKENHI No. 23224009 from MEXT.  JPCR aknowledges the support of NSERC of Canada.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGamma-ray observations of the Small Magellanic Cloud with the EGRET telescope onboard \nthe {\\it Compton Gamma Ray Observatory} have proved that the bulk of cosmic rays (CRs) \npropagating in the Milky Way are produced in Galactic sources \\cite{sre93}. \nObservations of the diffuse $\\gamma$-ray emission from our Galaxy allow to estimate \nthe total CR luminosity \\cite{dog02}:\n\\begin{equation}\nL_{\\rm CR} = L_\\gamma {x_\\gamma \\over x} \\sim 5 \\times 10^{40}~{\\rm erg~s}^{-1},\n\\label{eqvt1}\n\\end{equation}\nwhere $L_\\gamma \\sim 5 \\times 10^{39}$~erg~s$^{-1}$ is the total \nluminosity of diffuse high-energy ($>100$~MeV) $\\gamma$ rays emitted in the decay of \n$\\pi^0$ produced by CR interaction with the interstellar medium (ISM), $x_\\gamma \n\\sim 120$~g~cm$^{-2}$ is the mean grammage needed for a CR ion to produce \na $\\pi^0$ in the ISM and $x \\sim 12$~g~cm$^{-2}$ is the mean path length that CRs \ntraverse before escaping the Galaxy, which is determined from measurements of the CR\nchemical composition near Earth. In comparison, the total power supplied by Galactic \nsupernovae (SNe) is\n\\begin{equation}\nL_{\\rm SN} = E_{\\rm SN} R_{\\rm SN} \\approx 10^{42}~{\\rm erg~s}^{-1},\n\\label{eqvt2}\n\\end{equation}\nwhere $E_{\\rm SN} \\approx 1.5\\times 10^{51}$~erg is the approximate total ejecta kinetic \nenergy of a SN and $R_{\\rm SN} \\approx 2$ per century is the current epoch \nGalactic SN rate \\cite{fer98}. Thus, SNe have enough power to sustain the CR \npopulation against escape from the Galaxy and energy losses, if there is a mechanism \nfor channeling $\\sim 5$\\% of the SN mechanical energy release into relativistic \nparticles.\n\nDiffusive shock acceleration (DSA) at the blast waves generated by SN explosions can \nin principle produce the required acceleration efficiency, as well as the observed \npower-law spectrum of CRs \\cite{kry77,axf78,bel78,bla78}. In this model, a fraction \nof ambient particles entering the SN shock front can be accelerated to high energies \nduring the lifetime of a supernova remnant (SNR) by diffusing back and forth on \ncompressive magnetic fluctuations of the plasma flow on both sides of the shock. A \ncritical ingredient of the theory is the strength of the turbulent magnetic field \nin the shock acceleration region, which governs the acceleration rate and in turn \nthe maximum energy of the accelerated particles. If the turbulent field upstream of \nthe SN shock is similar to the preexisting field in the surrounding ISM ($B \\sim\n5$~$\\mu$G), the maximum total energy of an ion of charge $Z$ was estimated 25 years ago \nto be (for a quasi-parallel shock geometry) $E_{\\rm max} \\sim 10^{14} Z$~eV \\cite{lag83}. \nBut in more recent developments of the DSA theory, it is predicted that large-amplitude \nmagnetic turbulence is self-generated by streaming of accelerated particles in the \nshock region, such that the ambient magnetic field can be strongly amplified as part \nof the acceleration process \\cite{bel01,ama06,vla06}. In this case, protons might be \naccelerated in SNRs up to $3\\times10^{15}$~eV, i.e. the energy of the spectral \"knee\"   \nabove which the measured all-particle CR spectrum shows a significant steepening. \nContributions of accelerated $\\alpha$-particles and heavier species might then explain \nthe existing CR measurements up to $\\sim$10$^{17}$~eV \\cite{ber07}. Above energies of \n10$^{18}$--10$^{19}$~eV, CRs are probably of extragalactic origin.  \n\nAnother uncertain parameter of the DSA model is the fraction of total shocked \nparticles injected into the acceleration process. Although theoretical progress has \nbeen made in recent years \\cite{bla05}, the particle injection and consequently the \nacceleration efficiency are still not well known. However, theory predicts that for \nefficient acceleration the energy density of the relativistic nuclear component can \nbecome comparable to that of the postshock thermal component, in which case the \nbackreaction of energetic ions can significantly modify the shock structure and \nthe acceleration process can become highly nonlinear (e.g. \\cite{ber99}). In \nparticular, the compression ratio of a CR-modified shock is expected to be higher \nthan for a test-particle shock (i.e. when the accelerated particles have no \ninfluence on the shock structure). \nThis is because of both the softer equation of state of a relativistic (CR) gas and \nthe energy loss due to escape of accelerated particles from the shock region \n\\cite{dec00}. Moreover, the temperature of the shock-heated gas can be reduced if a \nsignificant fraction of the total available energy of the shock goes into relativistic \nparticles. Observations of these nonlinear effects \\cite{hug00,dec05,war05} provide \nindirect evidence for the efficient acceleration of ions in SN shock waves. \n\nThe acceleration of electrons in SNRs leaves no doubt, since we observe the nonthermal \nsynchrotron emission that these particles produce in the local magnetic field. Radio \nsynchrotron radiation, which in SNRs is emitted by GeV electrons, was discovered \nin the 1950's. More recent is the observation of X-ray synchrotron emission from \nyoung shell-type SNRs \\cite{koy95}, which is due to electrons accelerated to very high \nenergies, $E_e>$1~TeV. Thanks to the extraordinary spectroscopic-imaging capabilities of \nthe {\\it XMM-Newton} and {\\it Chandra} X-ray observatories, this nonthermal emission can \nnow be studied in great details and recent observations of SNRs with these satellites \nhave shed new light on the DSA rate and the maximum energy of the accelerated particles. \nThis is the subject of Section~2. \n\nIn Section~3, we discuss the origin of the TeV $\\gamma$-ray emission observed from \na handful of shell-type SNRs with atmospheric Cerenkov telescopes. For some objects, \nthe detected $\\gamma$-rays have been explained as resulting from $\\pi^0$ production in \nnuclear collisions of accelerated ions with the ambient gas. If this were true, this \nhigh-energy emission would be the first observational proof that CR ions are indeed \naccelerated in SN shock waves. However, the origin of the TeV $\\gamma$-rays \nemitted in SNRs is still a matter of debate, because at least in some cases the \nhigh-energy photons can also be produced by inverse Compton scattering of \ncosmic-microwave-background photons (and possibly optical and infrared interstellar \nphotons) by ultrarelativistic electrons. \n\nIn Section 4, we show that radio observations of extragalactic SNe can \nprovide complementary information on the DSA mechanism. As an example, we use a \nsemianalytic description of nonlinear DSA to model the radio light curves \nof SN 1993J. We choose this object because the set of radio data \naccumulated over the years \\cite{wei07} constitutes one of the most detailed sets of \nmeasurements ever established for an extragalactic SN in any wavelength range. We \nderive from these data constraints on the magnetic field strength in the environment \nof the expanding SN shock wave, the maximum energy of the accelerated particles, as \nwell as on the fractions of shocked electrons and protons injected into the \nacceleration process. Conclusions are given in Section 5.\n\n\\section{X-ray synchrotron emission from SNRs}\n\nTogether with the thermal, line-dominated X-ray emission from the shock-heated gas, \na growing number of SNRs show nonthermal, featureless emission presumably produced\nby ultrarelativistic electrons in the blast wave region via a synchrotron process. \nHigh-angular resolution observations made with the {\\it Chandra} and \n{\\it XMM-Newton} X-ray observatories have revealed very thin rims of nonthermal \nemission associated with the forward shock. In several cases, \nlike SN~1006 \\cite{koy95} and G347.3-0.5 \\cite{cas04}, the synchrotron component \ncompletely dominates the thermal X-ray emission. The measured power-law spectral \nindex of the X-ray synchrotron radiation is always much steeper than that of the \nnonthermal radio emission, which is consistent with expectation that the X-ray \ndomain probes the high-energy end of the accelerated electron distribution. The \ncomparison of radio and X-ray fluxes allows to determine the exponential cutoff \n(maximum) frequency of the synchrotron emission, $\\nu_c$, which is related to the \nmaximum energy of the accelerated electrons and the ambient magnetic field as \n(e.g. \\cite{sta06})\n\\begin{equation}\n\\nu_c = 1.26\\times10^{16} \\bigg({E_{e,\\rm max} \\over 10{\\rm~TeV}}\\bigg)^2 \n\\bigg({B \\over 10~\\mu{\\rm G}}\\bigg)~{\\rm Hz}.\n\\label{eqvt3}\n\\end{equation}\nDiffusive shock acceleration can only occur for particles whose acceleration rate \nis higher than their energy loss rate in the acceleration region. The maximum \nelectron energy, $E_{e,\\rm max}$, can be estimated by equating the synchrotron \ncooling time (e.g. \\cite{par06}),\n\\begin{equation}\n\\tau_{\\rm syn}(E_{e,\\rm max}) = {E_{e,\\rm max} \\over (dE\/dt)_{\\rm syn}} \\propto\nE_{e,\\rm max}^{-1} B^{-2},\n\\label{eqvt4}\n\\end{equation}\nwhere $(dE\/dt)_{\\rm syn}$ is the synchrotron loss rate at $E_{e,\\rm max}$, to \nthe acceleration time\n\\begin{equation}\n\\tau_{\\rm acc}(E_{e,\\rm max}) = {E_{e,\\rm max} \\over (dE\/dt)_{\\rm acc}} \\sim\n{\\kappa(E_{e,\\rm max}) \\over V_s^2},\n\\label{eqvt5}\n\\end{equation}\nwhere $(dE\/dt)_{\\rm acc}$ and $\\kappa(E_{e,\\rm max})$ are the acceleration \nrate and mean spatial diffusion coefficient of the electrons of \nenergy $E_{e,\\rm max}$ in the blast wave region and $V_s$ is the shock speed. \nWe have neglected here the dependence of $\\tau_{\\rm acc}$ on the shock \ncompression ratio (see \\cite{par06}). The value of $\\kappa(E_{e,\\rm max})$ \ndepends on the strength and structure of the turbulent magnetic field. The DSA \ntheory predicts that CRs efficiently excite large amplitude magnetic fluctuations \nupstream of the forward shock and that these fluctuations scatter CRs very \nefficiently \\cite{bel78,bel01,ama06,vla06}. It is therefore generally assumed that \nthe spatial diffusion coefficient is close to the Bohm limit:\n\\begin{equation}\n\\kappa \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} \\kappa_B={r_g v \\over 3},\n\\label{eqvt6}\n\\end{equation}\nwhere $v$ is the particle speed and $r_g$=$pc\/(QeB)$ the particle gyroradius, $p$ \nbeing the particle momentum, $c$ the speed of light, $Q$ the charge number ($Q=1$ \nfor electrons and protons), and $-e$ the electronic charge. Note that for \nultrarelativistic electrons, $\\kappa_B = r_g c\/3 \\propto E_e B^{-1}$. Equating \nequations~(\\ref{eqvt4}) and \n(\\ref{eqvt5}) and using equation~(\\ref{eqvt3}) to express $E_{e,\\rm max}$ as a \nfunction of $B$ and $\\nu_c$, we can write the ratio of the electron diffusion \ncoefficient at the maximum electron energy to the Bohm coefficient as:\n\\begin{equation}\n\\eta_\\kappa={\\kappa(E_{e,\\rm max}) \\over \\kappa_B} \\propto V_s^2 \\nu_c^{-1}.\n\\label{eqvt7}\n\\end{equation}\nThus, measurements of $V_s$ and $\\nu_c$ can allow to derive $\\eta_\\kappa$ \nwithout knowing the ambient magnetic field. Using this result, several \nrecent studies \\cite{sta06,par06,rey04,yam04} have shown that there are regions \nin young ($t<10^4$~yr) SNRs where {\\it acceleration occurs nearly as fast as the \nBohm theoretical limit} (i.e. $1<\\eta_\\kappa<10)$. This provides an important \nconfirmation of a key prediction of the DSA model. \n\nThe strength of the magnetic field in the shock acceleration region may be derived \nfrom the thickness of the nonthermal X-ray rims observed in young SNRs (e.g. \n\\cite{vin03,bal06,par06}). One of the two interpretations that have been proposed to \nexplain the thin X-ray filaments is that they result from fast synchrotron cooling \nof ultrarelativistic electrons transported downstream of the forward shock. In this \nscenario, the width of the filaments is set by the distance that the electrons cover \nbefore their synchrotron emission falls out of the X-ray band. The electron transport \nin the downstream region is due to a combination of diffusion and advection, whose \ncorresponding scale heights are \\cite{bal06} $l_{\\rm diff}=\n\\sqrt{\\kappa \\tau_{\\rm syn}}\\propto B^{-3\/2}$ (see eqs.[\\ref{eqvt4}] and [\\ref{eqvt6}]) \nand $l_{\\rm adv}=\\tau_{\\rm syn}V_s\/r_{\\rm tot}\\propto B^{-3\/2} E_X^{-1\/2} V_s\/r_{\\rm tot}$, \nrespectively. Here, $r_{\\rm tot}$ is the overall compression ratio of the shock and \n$E_X \\sim 5$~keV is the typical X-ray energy at which the rims are observed. Thus, by \ncomparing $l_{\\rm diff}$ and $l_{\\rm adv}$ to the measured width of the X-ray filaments \n(e.g. $l_{\\rm obs}\\approx 3''$ in Cas A which gives 0.05~pc for a distance of 3.4 kpc \n\\cite{vin03}) one can estimate the downstream  magnetic field. Applications of this \nmethod to {\\it Chandra} and {\\it XMM-Newton} observations of young SNRs have shown that \n{\\it the magnetic field at the forward shock is amplified by about two orders of \nmagnitude} as compared with the average Galactic field strength. This conclusion has \nbeen recently strengthened by the observations of rapid time variations ($\\sim$1~yr) in \nbright X-ray filaments of the SNR RXJ1713.7-3946 (also named G347.3-0.5), which are \ninterpreted as resulting from fast synchrotron cooling of TeV electrons in a magnetic \nfield amplified to milligauss levels \\cite{uch07}. Such a high magnetic field is \nlikely the result of a nonlinear amplification process associated with the efficient \nDSA of CRs \\cite{bel01,ama06,vla06}. \n\nThe other interpretation that has been proposed to account for the thin X-ray filaments \nis that they reflect the spatial distribution of the ambient magnetic field rather than\nthe spatial distribution of the ultrarealtivistic electrons \\cite{poh05}. In this \nscenario, the magnetic field is thought to be amplified at the shock as well, but the \nwidth of the X-ray rims is not set by $l_{\\rm diff}$ and $l_{\\rm adv}$, but by the \ndamping length of the magnetic field behind the shock. In this case, the relation given \nabove between the rim thickness and the \ndownstream magnetic field would not be valid. Comparison of high-resolution X-ray and \nradio images could allow to distinguish between the two interpretations, because \nthe synchrotron energy losses are expected to be relatively small for GeV electrons \nemitting in the radio band \\cite{vin03}. Thus, if the X-ray filaments are due to \nrapid synchrotron cooling of TeV electrons, the same structures should not be seen in \nradio images. A recent detailed study of Tycho's SNR has not allowed to draw firm \nconclusions on the role of magnetic damping behind the blast wave \\cite{cas07}. \nFurther high-resolution observations of SNRs in radio wavelengths would be very useful. \n\nThe findings that (1) DSA can proceed at nearly the maximum possible rate (i.e. the Bohm \nlimit) and (2) the magnetic field in the acceleration region can be strongly amplified, \nsuggest that CR ions can reach higher energies in SNR shocks than previously estimated \nby Lagage \\& Cesarsky \\cite{lag83}. Thus, Berezhko \\& V\\\"olk \\cite{ber07} have argued \nthat protons can be accelerated in SNRs up to the energy of the knee in the CR \nspectrum, at $3\\times10^{15}$~eV. But relaxing the assumption of Bohm diffusion used \nin the calculations of Berezhko \\& V\\\"olk, Parizot et al. \\cite{par06} have obtained \nlower maximum proton energies for five young SNRs. These authors have derived an upper \nlimit of $\\sim8\\times10^{14}$~eV on the maximum proton energy, $E_{p,\\rm max}$, and \nhave suggested that an additional CR component is required to explain the CR data \nabove the knee energy. Recently, Ellison \\& Vladimirov \\cite{ell08} have pointed out \nthat the average magnetic field that determines the maximum proton energy can be \nsubstantially less than the field that determines the maximum electron energy. This is \nbecause electrons remain in the vicinity of the shock where the magnetic field can be  \nstrongly amplified, whereas protons of energies $E_p>E_{e,\\rm max}$ diffuse farther in \nthe shock precursor region where the field is expected to be weaker \n($E_{p,\\rm max}>E_{e,\\rm max}$ because radiation losses affect the electrons only). \nThis nonlinear effect of efficient DSA could reduce $E_{p,\\rm max}$ relative \nto the value expected from test-particle acceleration. Nonetheless, recent calculations \nof $E_{p,\\rm max}$ in the framework of nonlinear DSA models suggest that SNRs might \nwell produce CRs up to the knee \\cite{ell08,bla07}.\n\n\\section{TeV gamma-ray emission from SNRs}\n\nAtmospheric Cerenkov telescopes have now observed high-energy $\\gamma$ rays from six \nshell-type SNRs: Cas A with HEGRA \\cite{aha01} and MAGIC \\cite{alb07a}, \nRX~J1713.7-3946 with CANGAROO \\cite{eno02} and HESS \\cite{aha07a}, RX~J0852.0-4622 \n(Vela Junior) with CANGAROO-II \\cite{kat05} and HESS \\cite{aha07b}, RCW~86 with HESS \n\\cite{hop07}, IC~443 with MAGIC \\cite{alb07b}, and very recently SN~1006 in deep HESS \nobservations \\cite{aha05b}. In addition, four $\\gamma$-ray sources discovered in the \nGalactic plane survey performed with HESS are spatially coincident with SNRs \\cite{aha06a}. \n\nWith an angular resolution of $\\sim0.06^\\circ$ for individual $\\gamma$ rays\n\\cite{aha07a,aha07b}, HESS has provided detailed images above 100 GeV of the extended \nSNRs RX~J1713.7-3946 and RX~J0852.0-4622 (their diameters are $\\sim1^\\circ$ and \n$\\sim2^\\circ$, respectively). In both cases the images show a shell-like structure \nand there is a striking correlation between the morphology of the $\\gamma$-ray \nemission and the morphology previously\nobserved in X-rays. For both objects the X-ray emission is completely dominated by \nnonthermal synchrotron radiation. The similarity of the $\\gamma$-ray and X-ray images \nthus suggest that the high-energy emission might also be produced by ultrarelativistic\nelectrons, via inverse Compton (IC) scattering off cosmic-microwave-background (CMB), \noptical-starlight and infrared photons. The $\\gamma$-ray radiation would then be \nproduced by electrons of energy (in the Thompson limit)\n\\begin{equation}\nE_e \\sim \\bigg({3 \\over 4}{E_\\gamma \\over E_\\star}\\bigg)^{1\/2} m_e c^2,\n\\label{eqvt8}\n\\end{equation}\nwhere $E_\\star$ is the typical energy of the seed photons, $E_\\gamma$ is the average \nfinal energy of the upscattered photons and $m_e$ is the electron  mass. For the CMB, \nwhose contribution to the total IC emission of SNRs generally dominates, $E_\\star \\sim \n3kT_{\\rm CMB}=7.1\\times10^{-4}$~eV ($k$ is the Boltzmann constant and \n$T_{\\rm CMB}=2.73$~K). Significant $\\gamma$-ray emission beyond $E_\\gamma=30$~TeV has \nbeen detected from RX~J1713.7-3946 \\cite{aha07a}. Thus, an IC origin for the high-energy \nemission would imply that electrons are accelerated to more than 90~TeV in this object. \nIn more accurate calculations that take into account the contributions of the optical \nand infrared interstellar radiation fields, the maximum electron energy is found to be \n$E_{e,\\rm max} \\sim 15$--40~TeV \\cite{por06}. This result is consistent with the value \nof $E_{e,\\rm max}$ derived from the width of X-ray filaments in RX~J1713.7-3946, \n$E_{e,\\rm max}=36$~TeV \\cite{par06}. \n\nAssuming that the same population of ultrarelativistic electrons produce both the \nobserved TeV $\\gamma$-rays and nonthermal X-rays, the mean magnetic field in the \ninteraction region can be readily estimated from the ratio of synchrotron to IC \nluminosities:\n\\begin{equation}\n{L_{\\rm syn} \\over L_{\\rm IC}}={U_B \\over U_{\\rm rad}}={B^2 \\over 8\\pi U_{\\rm rad}},\n\\label{eqvt9}\n\\end{equation}\nwhere $U_B=B^2 \/ (8\\pi)$ is the magnetic field energy density and $U_{\\rm rad}$ is \nthe total energy density of the seed photon field. With \n$U_{\\rm CMB}\\approx 0.25$~eV~cm$^{-3}$ for the CMB and \n$U_{\\rm IR}\\approx 0.05$~eV~cm$^{-3}$ for the interstellar infrared background (e.g. \n\\cite{aha06b}), we have $U_{\\rm rad}\\approx 0.3$~eV~cm$^{-3}$ (we neglect here the \ncontribution to the IC emission of the optical starlight background). Then, from the\nmeasured ratio $L_{\\rm syn} \/ L_{\\rm IC}\\approx 10$ for RX~J1713.7-3946 \\cite{aha06b}, \nwe obtain $B\\approx 11~\\mu$G. This value is significantly lower than the downstream \nmagnetic field estimated from the observed X-ray rims: $B\\sim80~\\mu$G \\cite{par06} (or \n$B>65~\\mu$G in Ref.~\\cite{ber06}). In other words, if the magnetic\nfield in the electron interaction region is as high as derived from the width of the \nX-ray filaments, IC radiation cannot account for the TeV $\\gamma$-ray data. \n\nThe magnetic field amplification is the main argument to favor a hadronic origin for \nthe high-energy $\\gamma$ rays produced in RX~J1713.7-3946 and other SNRs \\cite{ber06}. \nThe shape of the measured $\\gamma$-ray spectrum below $\\sim$1~TeV has also been used\nto advocate that the high-energy emission might not be produced by IC scattering \n\\cite{aha06b}, but the IC calculations of Ref.~\\cite{por06} reproduce the broadband \nemission of RX~J1713.7-3946 reasonably well. In the hadronic scenario, the TeV \n$\\gamma$ rays are due to nuclear collisions of accelerated protons and heavier \nparticles with ambient ions, which produce neutral pions $\\pi^0$ that decay in 99\\% \nof the cases into two photons with energies of 67.5~GeV each in the $\\pi^0$ rest \nframe ($2\\times67.5$~GeV is the $\\pi^0$ mass). At TeV energies in the observer rest \nframe, the spectrum of the $\\pi^0$-decay $\\gamma$ rays essentially reproduces,\nwith a constant scaling factor, the one of the parent ultrarelativistic particles. \nThe accelerated proton energies can be estimated from the $\\gamma$-ray spectrum \nas $E_p \\sim E_\\gamma\/0.15$ \\cite{aha07a}. The detection of $\\gamma$ rays with \n$E_\\gamma>30$~TeV in RX~J1713.7-3946 thus implies that protons are accelerated to more \nthan 200~TeV, which is still about an order of magnitude below the energy of the knee. \n\nHowever, the hadronic scenario is problematic for RX~J1713.7-3946. Due to the lack of \nthermal X-ray emission, the remnant is thought to expand mostly in a very diluted \nmedium of density $n<0.02$~cm$^{-3}$ \\cite{cas04}. It is likely that the SN \nexploded in a bubble blown by the wind of the progenitor star. The flux of $\\gamma$ \nrays produced by pion decay is proportional to the product of the number of \naccelerated protons and the ambient medium density. Thus, the total energy contained\nin CR protons would have to be large to compensate the low density of the ambient \nmedium. From the $\\gamma$-ray flux measured with HESS from the center of the remnant, \nPlaga \\cite{pla08} has recently estimated that the total CR-proton energy would have to \nbe $>4\\times10^{51}$~erg! Katz \\& Waxman \\cite{kat08} also argue against a hadronic \norigin for the TeV emission from RX~J1713.7-3946. They show that it would require that \nthe CR electron-to-proton abundance ratio at a given relativistic energy \n$K_{\\rm ep}\\lsim2\\times10^{-5}$, which is inconsistent with the limit they derived from \nradio observations of SNRs in the nearby galaxy M33, $K_{\\rm ep}\\gsim10^{-3}$. Moreover, radio \nand X-ray observations of RX~J1713.7-3946 suggest that the blast wave has recently hit \na complex of molecular clouds located in the western part of the remnant \\cite{cas04}. \nThe ambient medium density in this region has been estimated to be $\\sim300$~cm$^{-3}$. \nIn the hadronic scenario, a much higher $\\gamma$-ray flux would be expected in this \ndirection, contrary to the observations \\cite{pla08}. Thus, the $\\gamma$-ray morphology \nrevealed by HESS practically rules out pion production as the main contribution to \nthe high-energy radiation of RX~J1713.7-3946.\n\nBut then, why the magnetic field given by the ratio of synchrotron to IC luminosities \n(eq.~[\\ref{eqvt9}]) is inconsistent with the field derived from the X-ray filaments? \nThis suggests that the filamentary structures observed with {\\it Chandra} and \n{\\it XMM-Newton} are localized regions where the magnetic field is enhanced in \ncomparison with the mean downstream field \\cite{kat08}. It is possible that the \nmagnetic field is amplified at the shock as part of the nonlinear DSA process, but then \nrapidly damped behind the blast wave \\cite{poh05}. The mean field downstream the shock\nwould then not be directly related to the observed thickness of the X-ray rims. \n\nAlthough the unambiguous interpretation of the TeV observations of shell-type SNRs \nremains uncertain, the high-energy $\\gamma$-ray emissions from RX~J1713.7-3946 and \nRX~J0852.0-4622 are probably produced by IC scattering \\cite{kat08}. For Cas~A, the case \nfor a hadronic origin of the TeV radiation may be more compelling, as the density of the \nambient medium is higher \\cite{vin03}. The new source MAGIC~J0616+225 \\cite{alb07b} which \nis spatially coincident with IC~443 may also be produced by pion decay. IC~443 is one of \nthe best candidates for a $\\gamma$-ray source produced by interactions between CRs \naccelerated in a SNR and a nearby molecular cloud \\cite{tor03}. Hopefully the upcoming\n{\\it GLAST} satellite will allow a clear distinction between hadronic and electronic \n$\\gamma$-ray processes in these objects. With the expected sensitivity of the LAT \ninstrument between 30 MeV and 300 GeV, {\\it GLAST} observations of SNRs should \ndifferentiate between pion-decay and IC spectra \\cite{fun08}.\n\n\\section{Radio emission and nonlinear diffusive shock acceleration in SN 1993J}\n\nAbout 30 extragalactic SNe have now been detected at radio wavelengths\\footnote{See \nhttp:\/\/rsd-www.nrl.navy.mil\/7213\/weiler\/kwdata\/rsnhead.html.}. In a number of cases, \nthe radio evolution has been monitored for years after outburst. It is generally \naccepted that the radio emission is nonthermal synchrotron radiation from relativistic \nelectrons accelerated at the SN shock wave \\cite{che82}. At early epochs the radio flux \ncan be strongly attenuated by free-free absorption in the wind lost from the progenitor \nstar prior to the explosion. Synchrotron self-absorption can also play a role in some \nobjects \\cite{che98}. The radio emission from SNe can provide unique information on \nthe physical properties of the circumstellar medium (CSM) and the final stages of \nevolution of the presupernova system \\cite{wei02}. We show here that this emission \ncan also be used to study critical aspects of the DSA mechanism. \n\nThe type IIb SN~1993J, which exploded in the nearby galaxy M81 at a distance of \n$3.63\\pm0.34$~Mpc, is one of the brightest radio SNe ever detected (see \\cite{wei07} \nand references therein). Very long baseline interferometry (VLBI) imaging has revealed \na decelerating expansion of a shell-like radio source, which is consistent with the\nstandard model that the radio emission arises from a region behind the forward shock \npropagating into the CSM. The expansion has been found to be self-similar \n\\cite{mar97}, although small departures from a self-similar evolution have been \nreported \\cite{bar00}. The velocity of the forward shock can be estimated from the \nmeasured outer radius of the radio shell, i.e. the shock radius $r_s$, as \n$V_s=dr_s\/dt=3.35\\times10^4~t_d^{-0.17}$~km~s$^{-1}$, where $t_d$ is the time after \nshock breakout expressed in days. \n\nExtensive radio monitoring of the integrated flux density of SN~1993J has been \nconducted with the Very Large Array and several other radio telescopes \\cite{wei07}. \nFigure~\\ref{figvt1} shows a set of measured light curves at 0.3~cm (85--110 GHz), \n1.2~cm (22.5~GHz), 2~cm (14.9~GHz), 3.6~cm (8.4~GHz), 6~cm (4.9~GHz), and 20~cm \n(1.4~GHz). We see that at each wavelength the flux density first rapidly increases \nand then declines more slowly as a power in time (the data at 0.3~cm do not allow to \nclearly identify this behavior). The radio emission was observed to suddenly decline \nafter day $\\sim$3100 (not shown in Fig.~\\ref{figvt1}), which is interpreted in terms \nof an abrupt decrease of the CSM density at radial distance from the progenitor \n$r\\sim3\\times10^{17}$~cm \\cite{wei07}. The maximum intensity is reached first at lower \nwavelengths and later at higher wavelengths, which is characteristic of absorption \nprocesses. For SN~1993J, both free-free absorption in the CSM and synchrotron \nself-absorption are important \\cite{che98,fra98,wei07}. To model light curves of radio \nSNe, Weiler et al. \\cite{wei02,wei07} have developed a semi-empirical formula that \ntakes into account these two absorption mechanisms. For SN~1993J, the best fit to the \ndata using this semi-empirical model ({\\it dotted blue curves} in Fig.~\\ref{figvt1}) \nrequires nine free parameters \\cite{wei07}. \n\n\\begin{figure}\n\\includegraphics[width=1\\textwidth]{f1.eps}\n\\caption{Radio light curves for SN~1993J at 0.3, 1.2, 2, 3.6, 6, and 20~cm. The dotted\nblue lines represent the best fit semi-empirical model of Ref.~\\cite{wei07}. The \ndashed red (resp. solid green) lines show results of the present model for \n$\\eta_{\\rm inj}^p=\\eta_{\\rm inj}^e=10^{-5}$ (resp. \n$\\eta_{\\rm inj}^p=2\\times10^{-4}$ and $\\eta_{\\rm inj}^e=1.4\\times10^{-5}$; see text).\nThe data are from Ref.~\\cite{wei07} and references therein. \n}\n\\label{figvt1}\n\\end{figure}\n\nI have developed a model for the radio emission of SN~1993J, which is inspired by \nprevious works on the morphology of synchrotron emission in young SNRs \n\\cite{cas05,ell05}. The model will be presented in detail in a forthcoming publication \n\\cite{tat08} and I only give here broad outlines. First, the density profile for the \nCSM is taken as $\\rho_{\\rm CSM}(r)=\\rho_0(r\/r_0)^{-2}$ as expected for a constant wind \nmass-loss rate and terminal velocity. Here $r_0=3.49\\times10^{14}$~cm is the shock \nradius at $t=1$~day after outburst and $\\rho_0$ is a free parameter. Evidence for a \nflatter CSM density profile has been advocated ($\\rho_{\\rm CSM}\\propto r^{-s}$, \nwith $s \\sim 1.6$; see \\cite{wei07} and references therein), based on the measured time \ndependence of the optical depth to free-free absorption in the CSM, $\\tau_{\\rm ff}$. \nHowever, Fransson \\& Bj\\\"ornsson \\cite{fra98} have shown that the time \nevolution of $\\tau_{\\rm ff}$ can be explained by a decrease of the CSM temperature with \n$r$ together with the standard $r^{-2}$ distribution for the density. The results of the\npresent work provide support to this latter interpretation \\cite{tat08}. Thus, in the \npresent model, free-free absorption is calculated assuming the time dependence of \n$\\tau_{\\rm ff}$ obtained in Ref.~\\cite{fra98} and using the best-fit value of $\\rho_0$ \n(or more precisely $\\rho_0^2$) as a normalization factor. \n\nBecause synchrotron self-absorption is important in SN~1993J, the strength and evolution \nof the mean magnetic field in the region of the radio emission can be estimated from \nthe measured peak flux at different wavelengths \\cite{che98}. Using equation~(12) of \nRef.~\\cite{che98}, I obtain from the data at 1.2, 2, 3.6, 6, and 20~cm: \n\\begin{equation}\n\\langle B \\rangle = (46 \\pm 19) \\alpha^{-2\/9} t_d^{-1.01\\pm0.09}~{\\rm G}, \n\\label{eqvt10}\n\\end{equation}\nwhere $\\alpha$ is the ratio of the total energy density in relativistic electrons to\nthe magnetic energy density. The errors include the uncertainty in the contribution of \nfree-free absorption. This time dependence of $\\langle B \\rangle$ is close to that \nexpected if the magnetic field at the shock is amplified by a constant factor from \nthe available kinetic energy density\n(see \\cite{bel01}). In this case, one expects $B^2\/8\\pi \\propto \\rho_{\\rm CSM} V_s^2$,\nwhich gives $B\\propto t^{-1}$ for $\\rho_{\\rm CSM}\\propto r^{-2}$. Note that the \nflatter CSM density profile supported by Weiler et al. \\cite{wei07},  \n$\\rho_{\\rm CSM}\\propto r^{-1.6}$, would imply for the assumed scaling \n$B\\propto \\rho_{\\rm CSM}^{1\/2} V_s \\propto t^{-0.83}$, which is somewhat inconsistent \nwith the data. \n\nBased on the measured time dependence of $\\langle B \\rangle$, the immediate postshock \nmagnetic field is assumed to be of the form $B_d=B_0t_d^{-1}$, where $B_0$ is a free\nparameter expected to be in the range $\\sim$100--600~G for a typical value of $\\alpha$ \nin the range $\\sim$10$^{-5}$--10$^{-2}$ (see eq.~[\\ref{eqvt10}]). The evolution of \nthe magnetic field behind the shock is then calculated from the assumption that the \nfield is carried by the flow, frozen in the plasma, so that the parallel and \nperpendicular magnetic field components evolve conserving flux (see \\cite{cas05} and \nreferences therein). Results obtained with the alternative assumption that the \nmagnetic field is rapidly damped behind the shock wave will be given in \\cite{tat08}. \n\nThe hydrodynamic evolution of the plasma downstream the forward shock is calculated \nfrom the two-fluid, self-similar model of Chevalier \\cite{che83}, which takes into \naccount the effects of CR pressure on the dynamics of \nthe thermal gas. The overall structure of SNRs can be described by self-similar \nsolutions, if the initial density profiles in the ejected material (ejecta) \nand in the ambient medium have power-law distributions, and if the ratio of \nrelativistic CR pressure to total pressure at the shock front is constant \\cite{che83}. \nThe backreaction of energetic ions can strongly modify the shock structure of young \nSNRs, such as e.g. Kepler's remnant \\cite{dec00}. But the situation is different for \nSN~1993J because of the much higher magnetic field in the shock precursor region, \nwhich implies that energy is very efficiently transfered from the CRs to the thermal \ngas via Alfv\\'en wave dissipation \\cite{ber99}. The resulting increase in the gas \npressure ahead of the viscous subshock is found to limit the overall compression ratio, \n$r_{\\rm tot}$, to values close to 4 (i.e. the standard value for a test-particle \nstrong shock) even for efficient DSA. Thus, the hydrodynamic evolution of SN~1993J \ncan be safely calculated in the test-particle, self-similar approximation. \n\nBoth the energy spectra of the accelerated particles and the thermodynamic properties of \nthe gas just behind the shock front (i.e. the boundary conditions for the self-similar \nsolutions of the hydrodynamic evolution) are calculated with the semianalytic model of \nnonlinear DSA developed by Berezhko \\& Ellison \\cite{ber99} and Ellison et al. \n\\cite{ell00}. However, a small change to the model has been made: the Alfv\\'en waves \nare assumed to propagate isotropically in the precursor region and not only in the \ndirection opposite to the plasma flow (i.e. eqs.~(52) and (53) of Ref.~\\cite{ber99} are \nnot used). This is a reasonable assumption given the strong, nonlinear magnetic field \namplification \\cite{bel01}. Although the semianalytic model strictly applies to \nplane-parallel, steady state shocks, it has been sucessfully used in Ref.~\\cite{ell00} \nfor evolving SNRs. The main parameter of this model, \n$\\eta_{\\rm inj}^p$, is the fraction of total shocked protons in protons with momentum \n$p \\geq p_{\\rm inj}$ injected from the postshock thermal pool into the DSA process. \nThe work of Ref.~\\cite{bla05} allows us to accurately relate the injection momentum \n$p_{\\rm inj}$ to $\\eta_{\\rm inj}^p$. Similarly, we define $\\eta_{\\rm inj}^e$ for the \nelectron injection. The latter parameter is not important for the shock structure, \nbecause the fraction of total particle momentum carried by electrons is negligible, but \nit determines the brightness of optically-thin synchrotron emission for a given magnetic \nfield strength. The electrons accelerated at the shock experience adiabatic and \nsynchrotron energy losses as they are advected downstream with the plasma flow. The \nspectral evolution caused by these losses is calculated as in Ref.~\\cite{rey98}. \nFinally, once both the nonthermal electron distribution and the magnetic field in a \ngiven shell of material behind the forward shock have been determined, the synchrotron \nemission from that shell can be calculated \\cite{pac70}. The total radio emission along \nthe line of sight is then obtained from full radiative transfer calculations that \ninclude synchrotron self-absorption. \n\nWith the set of assumptions given above, the model has four free parameters: $\\rho_0$, \n$B_0$, $\\eta_{\\rm inj}^p$ and $\\eta_{\\rm inj}^e$. While the product $\\rho_0 \\times \n\\eta_{\\rm inj}^e$ is important for the intensity of optically-thin synchrotron \nemission, the CSM density normalization $\\rho_0$ also determines the level of free-free\nabsorption. The main effect of changing $B_0$ is to shift the radio light curves in time,\nbecause the turn-on from optically-thick to optically-thin synchrotron emission is \ndelayed when the magnetic field is increased. The proton injection paramater \n$\\eta_{\\rm inj}^p$ determines the shock structure and hence influences the shape of the\nelectron spectrum. \n\nCalculated radio light curves are shown in Fig.~\\ref{figvt1} for \n$\\rho_0=1.8\\times10^{-15}$~g~cm$^{-3}$, $B_0=400$~G, and two sets of injection \nparameters: $\\eta_{\\rm inj}^p=\\eta_{\\rm inj}^e=10^{-5}$ (test-particle case), and \n$\\eta_{\\rm inj}^p=2\\times10^{-4}$ and $\\eta_{\\rm inj}^e=1.4\\times10^{-5}$. We see that \nin the test-particle case, the decline of the optically-thin emission with time is too\nslow as compared to the data, except at 0.3~cm. The CR-modified shock provides a better \noverall fit to the measured flux densities, although significant deviations from the \ndata can be observed. In particular, we see that the calculated light curve at 0.3~cm \nfalls short of the data at this wavelength. It is possible that the\ndeviations of the best-fit curves from the data partly arise from the \napproximations used in the DSA model of Ref.~\\cite{ber99}, in which the nonthermal \nphase-space distribution function $f(p)$ is described as a three-component power law. \nBut it is also possible that it tells us something about the magnetic field evolution \nin the downstream region. The spatial distribution of the postshock magnetic field will \nbe studied in Ref.~\\cite{tat08} by comparing calculated synchrotron profiles with the \nobserved average profile of the radio shell. \n\n\\begin{figure}\n\\includegraphics[width=1\\textwidth]{f2.eps}\n\\caption{Calculated postshock phase space distribution functions, $f(p)$, vs. kinetic \nenergy, at day 100 after shock breakout. Following Ref.~\\cite{ber99}, $f(p)$ \nhas been multiplied by $[p\/(m_pc)]^4$ to flatten the spectra, and by $[(m_pc)^3\/n_u]$ to \nmake them dimensionless ($m_p$ is the proton mass and $n_u$ the proton number density \nahead of the shock precursor). The red lines are for protons and the black lines for \nelectrons. The two sets of injection parameters $\\eta_{\\rm inj}^p$ and $\\eta_{\\rm inj}^e$ \nare those used for the synchrotron calculations shown in Fig.~1.}\n\\label{figvt2}\n\\end{figure}\n\nFor SN~1993J, the main effect of the CR pressure is to reduce the compression ratio \nof the subshock, $r_{\\rm sub}$, whereas the overall compression ratio $r_{\\rm tot}$ \nremains nearly constant (see above). For $\\eta_{\\rm inj}^p=2\\times10^{-4}$, $r_{\\rm sub}$ \nis found to decrease from 3.58 to 3.35 between day 10 and day 3100 after outburst, whereas \n$r_{\\rm tot}$ stays between 4 and 4.04. Such a shock modification affects essentially \nthe particles of energies $<m_pc^2$ that remain in the vicinity of the subshock during \nthe DSA process. Thus, we see in Fig.~2 that the increase of $\\eta_{\\rm inj}^p$ from \n10$^{-5}$ to $2\\times10^{-4}$ steepens the phase-space distribution functions $f(p)$\nof both protons and electrons between their thermal Maxwell-Boltzmann distributions and \n$\\sim1$~GeV, as the spectral index in this energy domain,  \n$q_{\\rm sub}=3r_{\\rm sub}\/(r_{\\rm sub}-1)$ with $f(p)\\propto p^{-q_{\\rm sub}}$ \n\\cite{ber99}, increases with decreasing $r_{\\rm sub}$. The radio light curves provide \nclear evidence that the spectral index of the nonthermal electrons below 1~GeV is higher \nthan the strong-shock test-particle value $q_{\\rm sub}=4$. For the preferred injection \nparameters $\\eta_{\\rm inj}^p=2\\times10^{-4}$ and $\\eta_{\\rm inj}^e=1.4\\times10^{-5}$, \nthe calculated electron-to-proton ratio at, e.g., 10~GeV is $K_{\\rm ep}=7.8\\times10^{-4}$ \nat $t_d=100$~days (Fig.~2) and $K_{\\rm ep}=6.3\\times10^{-4}$ at $t_d=3100$~days. These values \nare roughly consistent with -- but slightly lower than -- those recently estimated for the \nblast wave of Tycho's SNR, $K_{\\rm ep}\\sim10^{-3}$ \\cite{cas07}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.7\\textwidth]{f3.eps}\n\\end{center}\n\\caption{Estimated maximum proton energy in SN~1993J as a function of time after shock \nbreakout. $E_{\\rm max}^{\\rm age}$ and $E_{\\rm max}^{\\rm size}$ are the maximum energies \ncaused by the finite shock age and size, respectively. The resulting maximum proton \nenergy ({\\it solid line}) is the minimum of these two quantities.}\n\\label{figvt3}\n\\end{figure}\n\nThe immediate postshock magnetic field obtained in this work, $B_d\\approx 400\\times \nt_d^{-1}$~G is in very good agreement with the mean field in the synchrotron-emitting \nregion previously estimated by Fransson \\& Bj\\\"ornsson \\cite{fra98}, \n$\\langle B \\rangle\\approx 370\\times t_d^{-1}$~G. As already pointed out by Bell \\& Lucek \n\\cite{bel01}, the derived magnetic field strength in the forward shock precursor is \nconsistent with a simple estimate based on nonlinear magnetic field \namplification driven by the pressure gradient of accelerated particles. For \n$\\eta_{\\rm inj}^p=2\\times10^{-4}$, the ratio of the mean magnetic energy density in the \nshock precursor to the energy density in CR protons is found to slightly decrease from 0.6 \nto 0.5 between day 10 and day 3100 after outburst \\cite{tat08}. It is interesting that \nthe magnetic field strength is very close to that given by equipartition.  \n\nAs also pointed out in Ref.~\\cite{bel01}, CRs were probably accelerated to very high \nenergies in SN~1993J short after shock breakout. The maximum proton energy $E_{p,\\rm max}$ \nis expected to be limited by the spatial extend of the shock, because high-energy \nparticles diffusing upstream far enough from the shock front can escape from the \nacceleration region. This size limitation can be estimated by assuming the existence of \nan upstream free escape boundary located at some constant fraction $f_{\\rm esc}$ of the \nshock radius: $d_{\\rm FEB}=f_{\\rm esc}r_s$. The maximum energy that particles can acquire \nbefore reaching this boundary is then obtained by equalling $d_{\\rm FEB}$ to the upstream \ndiffusion length, $l \\sim \\kappa \/ V_s$ (e.g. \\cite{ell08}). Calculated time evolution of \n$E_{p,\\rm max}$ is shown in Fig.~\\ref{figvt3} for $f_{\\rm esc}=0.05$ (see, e.g., \\cite{ell05}) \nand $\\eta_\\kappa=3$ (see eq.~\\ref{eqvt7} and, e.g., \\cite{par06}). We see that the maximum \nenergy caused by the particle  escape, $E_{\\rm max}^{\\rm size}$ \n($\\propto f_{\\rm esc}\\eta_\\kappa^{-1}$), becomes rapidly lower than the maximum energy \nassociated with the shock age, $E_{\\rm max}^{\\rm age}$ ($\\propto \\eta_\\kappa^{-1}$), which \nis obtained by time integration of the acceleration rate $(dE\/dt)_{\\rm acc}$ from an initial \nacceleration time assumed to be $t_0=1$~day after outburst. The resulting maximum proton \nenergy (i.e. the minimum of $E_{\\rm max}^{\\rm size}$ and $E_{\\rm max}^{\\rm age}$) is \nsignificantly lower than the previous estimate of Ref.~\\cite{bel01}, \n$E_{p,\\rm max}\\sim3\\times10^{17}$~eV. Moreover, we did not take into account the nonlinear \neffects recently pointed out by Ellison \\& Vladimirov \\cite{ell08}, which could further \nreduce $E_{p,\\rm max}$. However, SN~1993J may well have accelerated protons above the knee \nenergy of $3\\times10^{15}$~eV (Fig.~\\ref{figvt3}), which provides support to the scenario \nfirst proposed by V\\\"olk \\& Biermann \\cite{vol88} that the highest-energy Galactic CRs are \nproduced by massive stars exploding into their own wind. In this context, it is instructive \nto estimate the total energy acquired by the CR particles during the early phase of \ninteraction between the SN ejecta and the red supergiant wind:\n\\begin{equation}\nE_{\\rm CR} \\cong \\int_{t_d=1}^{3100} \\epsilon_{\\rm CR}(t) \\times {1 \\over 2} \n\\rho_{\\rm CSM}(r_s) V_s^3 \\times 4\\pi r_s^2 dt = 6.8 \\times 10^{49}~\\rm erg,\n\\label{eqvt11}\n\\end{equation}\nwhere $\\epsilon_{\\rm CR}(t)$ is the time-dependent fraction of total incoming energy \nflux, $F_0(r_s) \\cong 0.5\\rho_{\\rm CSM}(r_s) V_s^3$, going into CR particles. With the \nbest-fit parameters obtained from the radio light curves, $\\epsilon_{\\rm CR}(t)$ is found \nto slowly increase from 12\\% to 16\\% between day 10 and day 3100 after outburst. It is \nremarkable that the value obtained for $E_{\\rm CR}$ is very close to the mean energy per \nSN required to account for the Galactic CR luminosity, $\\approx 7.5 \\times 10^{49}$~erg \n(see Sect.~1), which suggests that SN~1993J might be typical of the SNe that produce the \nCR population in our own Galaxy. \n\n\\section{Conclusions}\n\nObservations of young shell-type SNRs with the {\\it Chandra} and {\\it XMM-Newton} X-ray \nspace observatories give more credit to the current paradigm that the bulk of Galactic \nCRs are accelerated in shock waves generated by SN explosions. Evidence is accumulating\nsuggesting that acceleration in SN shocks can be nearly as fast as the Bohm limit and \nthat self-excited turbulence in the shock vicinity can strongly amplify the ambient \nmagnetic field, thus allowing the acceleration of CR ions to energies above 10$^{15}$~eV. \nBesides, observations of nonlinear effects caused by efficient DSA support the fact that \nthe acceleration efficiency can be high enough to account for the Galactic CR luminosity. \nBut the direct and unambiguous observation of ion acceleration in SN shocks \nremains uncertain, as it is difficult at the present time to distinguish between \npion decay and IC scattering as the main mechanism responsable for the very-high-energy \n$\\gamma$-ray emission detected with ground-based atmospheric Cerenkov telescopes from \nseveral shell-type SNRs. Following Refs.~\\cite{pla08,kat08} I have argued, however, that \nthe TeV $\\gamma$ rays emitted in RX~J1713.7-3946 and RX~J0852.0-4622 are probably \nproduced by ultrarelativistic electrons, via IC scattering off CMB, optical-starlight and \ninfrared photons. One of the most critical ingredients for the interpretation of these \nobservations is the evolution of the postshock magnetic field in the downstream region \n(see Sect.~3). Hopefully, the upcoming GLAST satellite (launch scheduled for May 2008) \nwill allow a clear identification of the contributions of hadronic and electronic \n$\\gamma$-ray emission processes in SNRs, which in turn will be useful to understand the \nimportance of magnetic field damping behind the forward shock.\n\nRadio observations of extragalactic SNe can also shed light on the DSA process. The \nimpressive set of radio data available for SN~1993J make this \nobject a unique laboratory to study particle acceleration in a SN shock. Using a \nsemianalytic model of nonlinear DSA to explain the radio light curves, the results I have \nobtained suggest, in particular, that the acceleration of ions has become efficient soon \nafter outburst and that the magnetic field in the forward shock vicinity has been amplified \nto equipartition with the CR energy density. The field amplification implies that CR \nprotons may have been accelerated to energies above $3\\times10^{15}$~eV during the early \nphase of interaction between the SN ejecta and the red supergiant wind lost from the \nprogenitor star. During this time, which lasted only $\\sim$8.5~years, the shock processed \nin the expansion a total energy of $4.6\\times10^{50}$~erg and the mean acceleration \nefficiency was found to be $\\epsilon_{\\rm CR}=15\\%$. Thus, a total energy of almost \n$7\\times10^{49}$~erg has been stored up by CR particles during this phase. A significant \nfraction of this energy might have escaped into the interstellar medium after day \n$\\sim$3100, when the shock started to expand into a more diluted CSM. \n\nThe results obtained for SN~1993J suggest that massive stars exploding into a wind \nenvironment could be a major source of Galactic CRs, as first proposed by V\\\"olk \n\\& Biermann \\cite{vol88}. It is well known that most massive stars are born in OB \nassociations, whose activity formes superbubbles of hot and turbulent gas inside \nwhich the majority of core-collapse SNe explode. Turbulent re-acceleration inside \nsuperbubbles may modify the spectrum of CRs and, in particular, increase their \nmaximum energy (see \\cite{par04} and references therein). \n\n\\acknowledgements{It is a pleasure to thank Anne Decourchelle and J\\\"urgen Kiener for \ncritical reading of the manuscipt, Margarita Hernanz for numerous discussions and her \nhospitality at IEEC-CSIC, and Jordi Isern for the invitation to the conference. \nFinancial support from the Generalitat de Catalunya through the AGAUR grant \n2006-PIV-10044 and the project SGR00378 is acknowledged.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\subsection{Mean magnetic field of specific scale}\nSuppose that mean magnetic field is axially symmetrical and depends on radial and vertical coordinates only, $\\overline{\\bm B} = \\overline{\\bm B}(\\eta,z)$.\nCoordinate system $\\left( \\xi, \\eta, z\\right)$ is local in the vicinity of Lagrangian trajectory with the radius $\\rho_0$. Since we study evolution of large-scale magnetic field, we should consider dynamics of magnetic field at wavenumbers $k \\sim k_b \\ll k_f$. In the case that is under our investigation this scale is supposed to be much smaller than the size of coherent vortex, so $R_{u}^{-1} \\ll k_b \\ll k_f$,\nwhere $R_{u}$ is the characteristic size of coherent vortex \\cite{kolokolov2020structure}. Therefore in this case any partial derivative of magnetic field over spatial variable can be written in local Cartesian system of coordinates $\\left( \\xi, \\eta, z\\right)$.  Particularly, the term $\\kappa\\Delta\\overline{\\bm B}$ equals to $\\kappa\\left( \\partial_\\eta^2+\\partial_z^2 \\right)\\overline{\\bm B}$, the term $\\operatorname{rot} \\bm{\\mathcal{ E}}$ consist of first-order linear partial derivatives. Note that if the scale of mean magnetic field $k_b$ is comparable to coherent vortex size, these terms will be nonlinear.\nSo, in the case $R_u^{-1}\\ll k_b \\ll k_f$ it is possible to write down evolution of mean magnetic field in Fourier space leaving only the most significant terms:\n\\begin{eqnarray}\\label{large_magnetic_Fourier_simple}\n\\partial_t\n\\begin{pmatrix}\n  \\overline{B}_{\\bm k}^\\xi\\\\\n  \\overline{B}_{\\bm k}^\\eta\\\\\n \\overline{B}_{\\bm k}^z\n\\end{pmatrix}\n \\approx\n \\begin{pmatrix}\n  - \\kappa k^2 & -\\Sigma  & 0\\\\\n  ik_z\\alpha_{\\xi\\xi} & - \\kappa k^2 & 0\\\\\n  0 & 0 & - \\kappa k^2\\\\\n \\end{pmatrix}\n \\begin{pmatrix}\n \\overline{B}_{\\bm k}^\\xi\\\\\n \\overline{B}_{\\bm k}^\\eta\\\\\n \\overline{B}_{\\bm k}^z\n\\end{pmatrix}\n\\end{eqnarray}\nMore detailed calculations and full expression for mean magnetic field evolution are made in Supplementary materials.\nThis linear system with constant coefficients has three eigenvalues one of them is negative ($\\lambda_0 = - \\kappa k^2$) and two of them have imaginary and real part that sign depend on wavevector $\\bm k$:\n\\begin{eqnarray}\\label{eigenvalues_final}\n\\lambda_{\\pm}\\approx -\\kappa k^2 \\pm \\sqrt{\\frac{\\alpha_{\\xi\\xi}|k_z \\Sigma|}{2}}\\left( 1 - i\\cdot sgn\\left(k_z \\Sigma \\right)\\right)\n\\end{eqnarray}\nEigenvalue $\\lambda_-$ has negative real part. However another eigenvalue, $\\lambda_+$, may have positive real part that is possible only if $\\sqrt{\\alpha_{\\xi\\xi}|k_z\\Sigma|\/2}\\gtrsim\\kappa k^2$. Assuming $\\epsilon\\sim \\nu\\Sigma^2$ \\cite{kolokolov2020structure} and $\\epsilon_h \\sim \\epsilon k_f$, we can rewrite this condition with logarithmic accuracy (we do not take into account logarithm in $\\alpha_{\\xi\\xi}$):\n\\begin{eqnarray}\\label{restriction}\nEk \\lesssim Ro^2 Pr_m^2 \\left(\\frac{k_f}{k_b}\\right)^3\n\\end{eqnarray}\nNote that the sign of imaginary part of $\\lambda_+$ depends on sign of $k_z\\Sigma$ according to (\\ref{eigenvalues_final}). Then let's find eigenvector that corresponds to $\\lambda_+$ with positive real part in the limit $Ek \\lesssim Ro^2 Pr^2_m \\left(\\frac{k_f}{k_b}\\right)^3$. Here we have restriction $k_\\eta B^\\eta_k + k_z B^z_k =0$ because mean magnetic field is axially symmetric. We can also express $B^\\xi_k$ through $B^\\eta_k$ using (\\ref{large_magnetic_Fourier}). Finally, we will have following eigenvector of mean magnetic field in the limit $Ek \\ll Ro^2 Pr_m^2\\left(\\frac{k_f}{k_b}\\right)^3$:\n\\begin{eqnarray}\\label{configuration_simple}\n\\overline{\\bm B}^{(+)}_{\\bm k} \\approx\n\\begin{pmatrix}\n \n  \\frac{|\\Sigma|}{\\lambda_+}\\\\\n  1 \\\\\n  -\\frac{k_\\eta}{k_z}\n\\end{pmatrix} C_{\\bm k} e^{\\lambda_+ t}\n\\end{eqnarray}\nPre-exponent $C_{\\bm k}$ can be choosen as real. Note that in the limit $Ek \\ll Ro^2 Pr_m^2 \\left(\\frac{k_f}{k_b}\\right)^3$ relation $|\\Sigma|\/\\lambda_+$ is large: $|\\Sigma\/\\lambda_+|\\gg 1$. Thus mean magnetic field $\\overline{\\bm B}^{(+)}$ is mainly directed along $\\xi-$axis. This result is in agreement with experiments of magnetic field observation in columnar vortex in the Earth \\cite{} where field is also directed tangentially. We can see also that magnetic field configuration that corresponds to $\\lambda_\\pm$ has different sign of imagine part in the regions $k_z\\Sigma>0$ and $k_z\\Sigma<0$. Particularly, in the region $k_z\\Sigma>0$ magnetic field component $\\overline{B}^{(+),\\xi}_{\\bm k}$ is ahead of component $\\overline{B}^{(+),\\eta}_{\\bm k}$ in phase by $\\pi\/4$, in the region $k_z\\Sigma<0$ component $\\overline{B}^{(+),\\eta}_{\\bm k}$ is ahead of component $\\overline{B}^{(+),\\xi}_{\\bm k}$ in phase by $\\pi\/4$.\n\nNonstationary dynamo experiments for conducting helical flow were obtained at work \\cite{frick2002non}. Such flow with large enough magnetic Reynolds number and small magnetic Prandtl number ($Pr_m \\sim 10^{-5}$) was accelerated in toroidal chanel and then was braked up abruptly. Numerical simulation of the magnetic field evolution was performed for a helical flow with a profile extracted from the experimental data. It was shown that magnetic field increased with time at some values of parameters.\n\n\n\n\n\n\n\n\\textbf{Conclusion.} To summarise we have quantitatively characterized $\\alpha$-effect in three-dimensional coherent geostrophic helical vortex of conducting liquid with external magnetic field. We calculated each element of $\\alpha$-matrix that describes $\\alpha$-effect.\n\nThe results were obtained in considered region $\\operatorname{Ro}\\lesssim 1, \\,\\  Pr_m \\lesssim 1$ due to physical reasons of coherent vortexes existence. The expression for elements of $\\alpha$-matrix was obtained in the framework of the theory developed in \\cite{kolokolov2020structure} and \\cite{ogorodnikov2021velocity}. The elements of $\\alpha$-matrix strongly depends on order relation between Rossby $\\operatorname{Ro}$ and magnetic Prandtl $Pr_m$ numbers. The interplay of them leads to the interplay between time scales $\\tau_*$ and $\\tau_0$ that characterizes kinematic viscosity and rotation correspondingly.\n\n\\textbf{Acknowledgments.} This work was supported by the {Russian Science Foundation, Grant No. 20-12-00383.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently much attention has been given to lower dimensional gauge theories.\nSuch remarkable results as the chiral symmetry breaking \\cite{1}, quantum\nHall effect \\cite{2}, spontaneously broken Lorentz invariance by the\ndynamical generation of a magnetic field \\cite{3}, and the connection\nbetween non-perturbative effects in low-energy strong interactions and QCD$%\n_{2}$ \\cite{4}, show the broad range of applicability of these theories.\n\nIn particular, 2+1 dimensional gauge theories with fractional statistics\n-anyon systems \\cite{4a}- have been extensively studied. One main reason for\nsuch an interest has been the belief that a strongly correlated electron\nsystem in two dimensions can be described by an effective field theory of\nanyons \\cite{5}, \\cite{5a}. Besides, it has been claimed that anyons could\nplay a basic role in high-T$_{C}$ superconductivity \\cite{5a}-\\cite{6b}. It\nis known \\cite{a} that a charged anyon system in two spatial dimensions can\nbe modeled by means of a 2+1 dimensional Maxwell-Chern-Simons (MCS) theory.\nAn important feature of this theory is that it violates parity and\ntime-reversal invariance. However, at present no experimental evidences of P\nand T violation in high-T$_{C}$ superconductivity have been found. It should\nbe pointed out, nevertheless, that it is possible to construct more\nsophisticated P and T invariant anyonic models\\cite{6a}. In any case,\nwhether linked to high-T$_{C}$ superconductivity or not, the anyon system is\nan interesting theoretical model in its own right.\n\nThe superconducting behavior of anyon systems at $T=0$ has been investigated\nby many authors \\cite{6}-\\cite{15a}. Crucial to the existence of anyon\nsuperconductivity at $T=0$ is the exact cancellation between the bare and\ninduced Chern-Simons terms in the effective action of the theory.\n\nAlthough a general consensus exists regarding the superconductivity of anyon\nsystems at zero temperature, a similar consensus at finite temperature is\nyet to be achieved \\cite{8}-\\cite{16}. Some authors (see ref. \\cite{9}) have\nconcluded that the superconductivity is lost at $T\\neq 0$, based upon the\nappearance of a temperature-dependent correction to the induced Chern-Simons\ncoefficient that is not cancelled out by the bare term. In ref. \\cite{11} it\nis argued, however, that this finite temperature correction is numerically\nnegligible at $T<200$ $K$, and that the main reason for the lack of a\nMeissner effect is the development of a pole $\\sim \\left( \\frac{1}{{\\bf k}%\n^{2}}\\right) $ in the polarization operator component $\\Pi _{00}$ at $T\\neq\n0 $. There, it is discussed how the existence of this pole leads to a so\ncalled partial Meissner effect with a constant magnetic field penetration\nthroughout the sample that appreciably increases with temperature. On the\nother hand, in ref. \\cite{8}, it has been independently claimed that the\nanyon model cannot superconduct at finite temperature due to the existence\nof a long-range mode, found inside the infinite bulk at $T\\neq 0$. The long\nrange mode found in ref. \\cite{8} is also a consequence of the existence of\na pole $\\sim \\left( \\frac{1}{{\\bf k}^{2}}\\right) $ in the polarization\noperator component $\\Pi _{00}$ at $T\\neq 0$.\n\nThe apparent lack of superconductivity at temperatures greater than zero has\nbeen considered as a discouraging property of anyon models. Nevertheless, it\nmay be still premature to disregard the anyons as a feasible solution for\nexplaining high -T$_{c}$ superconductivity, at least if the reason\nsustaining such a belief is the absence of the Meissner effect at finite\ntemperature. As it was shown in a previous paper \\cite{16}, the lack of a\nMeissner effect, reported in ref. \\cite{11} for the case of a half-plane\nsample as a partial Meissner effect, is a direct consequence of the omission\nof the sample boundary effects in the calculations of the minimal solution\nfor the magnetic field within the sample. To understand this remark we must\ntake into account that the results of ref. \\cite{11} were obtained by\nfinding the magnetization in the bulk due to an externally applied magnetic\nfield at the boundary of a half-plane sample. However, in doing so, a\nuniform magnetization was assumed and therefore the boundary effects were\nindeed neglected. Besides, in ref. \\cite{11} the field equations were solved\nconsidering only one short-range mode of propagation for the magnetic field,\nwhile as has been emphasized in our previous letter \\cite{16}, there is a\nsecond short-range mode whose qualitative contribution to the solutions of\nthe field equations cannot be ignored.\n\nIn the present paper we study the effects of the sample's boundaries in the\nmagnetic response of the anyon fluid at finite temperature. This is done by\nconsidering a sample shaped as an infinite strip. When a constant and\nhomogeneous external magnetic field, which is perpendicular to the sample\nplane, is applied at the boundaries of the strip, two different magnetic\nresponses, depending on the temperature values, can be identified. At\ntemperatures smaller than the fermion energy gap inherent to the\nmany-particle MCS model ($T\\ll \\omega _{c}$), the system exhibits a Meissner\neffect. In this case the magnetic field cannot penetrate the bulk farther\nthan a very short distance ($\\overline{\\lambda }\\sim 10^{-5}cm$ for electron\ndensities characteristic of the high -T$_{c}$ superconductors and $T\\sim 200$\n$K$). On the other hand, as it is natural to expect from a physical point of\nview, when the temperatures are larger than the energy gap ($T\\gg \\omega\n_{c} $) the Meissner effect is lost. In this temperature region a periodic\ninhomogeneous magnetic field is present within the bulk.\n\nThese results, together with those previously reported in ref. \\cite{16},\nindicate that, contrary to some authors' belief, the superconducting\nbehavior (more precisely, the Meissner effect), found in the charged anyon\nfluid at $T=0$, does not disappear as soon as the system is heated.\n\nAs it is shown below, the presence of boundaries can affect the dynamics of\nthe system in such a way that the mode that accounts for a homogeneous field\npenetration \\cite{8} cannot propagate in the bulk. Although these results\nhave been proved for two types of samples, the half-plane \\cite{16} and the\ninfinite strip reported in this paper, we conjecture that similar effects\nshould also exist in other geometries.\n\nOur main conclusion is that the magnetic behavior of the anyon fluid is not\njust determined by its bulk properties, but it is essentially affected by\nthe sample boundary conditions. The importance of the boundary conditions in\n2+1 dimensional models has been previously stressed in ref.\\cite{b}.\n\nThe plan for the paper is as follows. In Sec. 2, for completeness as well as\nfor the convenience of the reader, we define the many-particle 2+1\ndimensional MCS model used to describe the charged anyon fluid, and briefly\nreview its main characteristics. In Sec. 3 we study the magnetic response in\nthe self-consistent field approximation of a charged anyon fluid confined to\nan infinite-strip, finding the analytical solution of the MCS field\nequations that satisfies the boundary conditions. The fermion contribution\nin this approximation is given by the corresponding polarization operators\nat $T\\neq 0$ in the background of a many-particle induced Chern-Simons\nmagnetic field. Using these polarization operators in the low temperature\napproximation ($T\\ll \\omega _{c}$), we determine the system's two London\npenetration depths. Taking into account that the boundary conditions are not\nenough to completely determine the magnetic field solution within the\nsample, an extra physical condition, the minimization of the system\nfree-energy density, is imposed. This is done in Sec. 4. In this section we\nprove that even though the electromagnetic field has a long-range mode of\npropagation in the charged anyon fluid at $T\\neq 0$ \\cite{8}, a constant and\nuniform magnetic field applied at the sample's boundaries cannot propagate\nthrough this mode. The explicit temperature dependence at $T\\ll \\omega _{c}$\nof all the coefficients appearing in the magnetic field solution, and of the\neffective London penetration depth are also found. In Sec. 5, we discuss how\nthe superconducting behavior of the charged anyon fluid disappears at\ntemperatures larger than the energy gap ($T\\gg \\omega _{c}$). Sec. 6\ncontains the summary and discussion.\n\n\\section{MCS Many-Particle Model}\n\nThe Lagrangian density of the 2+1 dimensional non-relativistic charged MCS\nsystem is\n\n\\begin{equation}\n{\\cal L}=-\\frac{1}{4}F_{\\mu \\nu }^{2}-\\frac{N}{4\\pi }\\varepsilon ^{\\mu \\nu\n\\rho }a_{\\mu }\\partial _{\\nu }a_{\\rho }+en_{e}A_{0}+i\\psi ^{\\dagger\n}D_{0}\\psi -\\frac{1}{2m}\\left| D_{k}\\psi \\right| ^{2}+\\psi ^{\\dagger }\\mu\n\\psi  \\eqnum{2.1}\n\\end{equation}\nwhere $A_{\\mu }$ and $a_{\\mu }$ represent the electromagnetic and the\nChern-Simons fields respectively. The role of the Chern-Simons fields is\nsimply to change the quantum statistics of the matter field, thus, they do\nnot have an independent dynamics. $\\psi $ represents non-relativistic\nspinless fermions. $N\\ $ is a positive integer that determines the magnitude\nof the Chern-$%\n\\mathop{\\rm Si}%\n$mons coupling constant. The charged character of the system is implemented\nby introducing a chemical potential $\\mu $; $n_{e}$ is a background\nneutralizing `classical' charge density, and $m$ is the fermion mass. We\nwill consider throughout the paper the metric $g_{\\mu \\nu }$=$(1,-%\n\\overrightarrow{1})$. The covariant derivative $D_{\\nu }$ is given by\n\n\\begin{equation}\nD_{\\nu }=\\partial _{\\nu }+i\\left( a_{\\nu }+eA_{\\nu }\\right) ,\\qquad \\nu\n=0,1,2  \\eqnum{2.2}\n\\end{equation}\n\nIt is known that to guarantee the system neutrality in the presence of a\ndifferent from zero fermion density $\\left( n_{e}\\neq 0\\right) $,$\\ $a\nnontrivial background of Chern-Simons magnetic field $\\left( \\overline{b}=%\n\\overline{f}_{21}\\right) $ is generated. The Chern-Simons background field\nis obtained as the solution of the mean field Euler-Lagrange equations\nderived from (2.1)\n\n\\begin{mathletters}\n\\begin{equation}\n-\\frac{N}{4\\pi }\\varepsilon ^{\\mu \\nu \\rho }f_{\\nu \\rho }=\\left\\langle\nj^{\\mu }\\right\\rangle  \\eqnum{2.3}\n\\end{equation}\n\n\\begin{equation}\n\\partial _{\\nu }F^{\\mu \\nu }=e\\left\\langle j^{\\mu }\\right\\rangle\n-en_{e}\\delta ^{\\mu 0}  \\eqnum{2.4}\n\\end{equation}\nconsidering that the system formed by the electron fluid and the background\ncharge $n_{e}$ is neutral\n\n\\end{mathletters}\n\\begin{equation}\n\\left\\langle j^{0}\\right\\rangle -n_{e}\\delta ^{\\mu 0}=0  \\eqnum{2.5}\n\\end{equation}\nIn eq. (2.5) $\\left\\langle j^{0}\\right\\rangle $ is the fermion density of\nthe many-particle fermion system\n\n\\begin{equation}\n\\left\\langle j^{0}\\right\\rangle =\\frac{\\partial \\Omega }{\\partial \\mu }, \n\\eqnum{2.6}\n\\end{equation}\n$\\Omega $ is the fermion thermodynamic potential.\n\nIn this approximation it is found from (2.3)-(2.5) that the Chern-Simons\nmagnetic background is given by\n\n\\begin{equation}\n\\overline{b}=\\frac{2\\pi n_{e}}{N}  \\eqnum{2.7}\n\\end{equation}\n\nThen, the unperturbed one-particle Hamiltonian of the matter field\nrepresents a particle in the background of the Chern-Simons magnetic field $%\n\\overline{b\\text{,}}$\n\n\\begin{equation}\nH_{0}=-\\frac{1}{2m}\\left[ \\left( \\partial _{1}+i\\overline{b}x_{2}\\right)\n^{2}+\\partial _{2}^{2}\\right]  \\eqnum{2.8}\n\\end{equation}\nIn (2.8) we considered the background Chern-Simons potential, $\\overline{a}%\n_{k}$, $(k=1,2)$, in the Landau gauge\n\n\\begin{equation}\n\\overline{a}_{k}=\\overline{b}x_{2}\\delta _{k1}  \\eqnum{2.9}\n\\end{equation}\n\nThe eigenvalue problem defined by the Hamiltonian (2.8) with periodic\nboundary conditions in the $x_{1}$-direction: $\\Psi \\left( x_{1}+L,\\text{ }%\nx_{2}\\right) =$ $\\Psi \\left( x_{1},\\text{ }x_{2}\\right) $,\n\n\\begin{equation}\nH_{0}\\Psi _{nk}=\\epsilon _{n}\\Psi _{nk},\\qquad n=0,1,2,...\\text{ }and\\text{ }%\nk\\in {\\cal Z}  \\eqnum{2.10}\n\\end{equation}\nhas eigenvalues and eigenfunctions given respectively by\n\n\\begin{equation}\n\\epsilon _{n}=\\left( n+\\frac{1}{2}\\right) \\omega _{c}\\qquad  \\eqnum{2.11}\n\\end{equation}\n\n\\begin{equation}\n\\Psi _{nk}=\\frac{\\overline{b}^{1\/4}}{\\sqrt{L}}\\exp \\left( -2\\pi\nikx_{1}\/L\\right) \\varphi _{n}\\left( x_{2}\\sqrt{\\overline{b}}-\\frac{2\\pi k}{L%\n\\sqrt{\\overline{b}}}\\right)  \\eqnum{2.12}\n\\end{equation}\nwhere $\\omega _{c}=\\overline{b}\/m$ is the cyclotron frequency and $\\varphi\n_{n}\\left( \\xi \\right) $ are the orthonormalized harmonic oscillator wave\nfunctions.\n\nNote that the energy levels $\\epsilon _{n}$ are degenerates (they do not\ndepend on $k$). Then, for each Landau level $n$ there exists a band of\ndegenerate states. The cyclotron frequency $\\omega _{c}$ plays here the role\nof the energy gap between occupied Landau levels. It is easy to prove that\nthe filling factor, defined as the ratio between the density of particles $%\nn_{e}$ and the number of states per unit area of a full Landau level, is\nequal to the Chern-$%\n\\mathop{\\rm Si}%\n$mons coupling constant $N$. Thus, because we are considering that $N$ is a\npositive integer, we have in this MCS theory $N$ completely filled Landau\nlevels. Once this ground state is established, it can be argued immediately \n\\cite{6}, \\cite{6b}, \\cite{10a}, \\cite{15}, that at $T=0$ the system will be\nconfined to a filled band, which is separated by an energy gap from the free\nstates; therefore, it is natural to expect that at $T=0$ the system should\nsuperconduct. This result is already a well established fact on the basis of\nHartree-Fock analysis\\cite{6} and Random Phase Approximation \\cite{6b},\\cite\n{10a}. The case at $T\\neq 0$ is more controversial since thermal\nfluctuations, occurring in the many-particle system, can produce significant\nchanges. As we will show in this paper, the presence in this theory of a\nnatural scale, the cyclotron frequency $\\omega _{c}$, is crucial for the\nexistence of a phase at $T\\ll \\omega _{c}$, on which the system, when\nconfined to a bounded region, still behaves as a superconductor.\n\nThe fermion thermal Green's function in the presence of the background\nChern-Simons field $\\overline{b}$\n\n\\begin{equation}\nG\\left( x,x^{\\prime }\\right) =-\\left\\langle T_{\\tau }\\psi \\left( x\\right) \n\\overline{\\psi }\\left( x^{\\prime }\\right) \\right\\rangle  \\eqnum{2.13}\n\\end{equation}\nis obtained by solving the equation\n\n\\begin{equation}\n\\left( \\partial _{\\tau }-\\frac{1}{2m}\\overline{D}_{k}^{2}-\\mu \\right)\nG\\left( x,x^{\\prime }\\right) =-\\delta _{3}\\left( x-x^{\\prime }\\right) \n\\eqnum{2.14}\n\\end{equation}\nsubject to the requirement of antiperiodicity under the imaginary time\ntranslation $\\tau \\rightarrow \\tau +\\beta $ ($\\beta $ is the inverse\nabsolute temperature). In (2.14) we have introduced the notation\n\n\\begin{equation}\n\\overline{D}_{k}=\\partial _{k}+i\\overline{a}_{k}  \\eqnum{2.15}\n\\end{equation}\n\nThe Fourier transform of the fermion thermal Green's function (2.13)\n\n\\begin{equation}\nG\\left( p_{4},{\\bf p}\\right) =\\int\\limits_{0}^{\\beta }d\\tau \\int d{\\bf x}%\nG\\left( \\tau ,{\\bf x}\\right) e^{i\\left( p_{4}\\tau -{\\bf px}\\right) } \n\\eqnum{2.16}\n\\end{equation}\ncan be expressed in terms of the orthonormalized harmonic oscillator wave\nfunctions $\\varphi _{n}\\left( \\xi \\right) $ as \\cite{Efrain}\n\n\\begin{eqnarray}\nG\\left( p_{4},{\\bf p}\\right) &=&\\int\\limits_{0}^{\\infty }d\\rho\n\\int\\limits_{-\\infty }^{\\infty }dx_{2}\\sqrt{\\overline{b}}\\exp -\\left(\nip_{2}x_{2}\\right) \\exp -\\left( ip_{4}+\\mu -\\frac{\\overline{b}}{2m}\\right)\n\\rho  \\nonumber \\\\\n&&\\sum\\limits_{n=0}^{\\infty }\\varphi _{n}\\left( \\xi \\right) \\varphi\n_{n}\\left( \\xi ^{\\prime }\\right) t^{n}  \\eqnum{2.17}\n\\end{eqnarray}\nwhere $t=\\exp \\frac{\\overline{b}}{m}\\rho $, $\\xi =\\frac{p_{1}}{\\sqrt{%\n\\overline{b}}}+\\frac{x_{2}\\sqrt{\\overline{b}}}{2}$, $\\xi ^{\\prime }=\\frac{%\np_{1}}{\\sqrt{\\overline{b}}}-\\frac{x_{2}\\sqrt{\\overline{b}}}{2}$ and $%\np_{4}=(2n+1)\\pi \/\\beta $ are the discrete frequencies $(n=0,1,2,...)$\ncorresponding to fermion fields.\n\n\\section{Linear Response in the Infinite Strip}\n\n\\subsection{Effective Theory at $\\mu \\neq 0$ and $T\\neq 0$}\n\nIn ref.\\cite{8} the effective current-current interaction of the MCS model\nwas calculated to determine the independent components of the magnetic\ninteraction at finite temperature in a sample without boundaries, i.e., in\nthe free space. These authors concluded that the pure Meissner effect\nobserved at zero temperature is certainly compromised by the appearance of a\nlong-range mode at $T\\neq 0$. Our main goal in the present paper is to\ninvestigate the magnetic response of the charged anyon fluid at finite\ntemperature for a sample that confines the fluid within some specific\nboundaries. As we prove henceforth, the confinement of the system to a\nbounded region (a condition which is closer to the experimental situation\nthan the free-space case) is crucial for the realization of the Meissner\neffect inside the charged anyon fluid at finite temperature.\n\nLet us investigate the linear response of a charged anyon fluid at finite\ntemperature and density to an externally applied magnetic field in the\nspecific case of an infinite-strip sample. The linear response of the medium\ncan be found under the assumption that the quantum fluctuations of the gauge\nfields about the ground-state are small. In this case the one-loop fermion\ncontribution to the effective action, obtained after integrating out the\nfermion fields, can be evaluated up to second order in the gauge fields. The\neffective action of the theory within this linear approximation \\cite{8},%\n\\cite{11} takes the form\n\n\\begin{equation}\n\\Gamma _{eff}\\,\\left( A_{\\nu },a_{\\nu }\\right) =\\int dx\\left( -\\frac{1}{4}%\nF_{\\mu \\nu }^{2}-\\frac{N}{4\\pi }\\varepsilon ^{\\mu \\nu \\rho }a_{\\mu }\\partial\n_{\\nu }a_{\\rho }+en_{e}A_{0}\\right) +\\Gamma ^{\\left( 2\\right) }  \\eqnum{3.1}\n\\end{equation}\n\n\\[\n\\Gamma ^{\\left( 2\\right) }=\\int dx\\Pi ^{\\nu }\\left( x\\right) \\left[ a_{\\nu\n}\\left( x\\right) +eA_{\\nu }\\left( x\\right) \\right] +\\int dxdy\\left[ a_{\\nu\n}\\left( x\\right) +eA_{\\nu }\\left( x\\right) \\right] \\Pi ^{\\mu \\nu }\\left(\nx,y\\right) \\left[ a_{\\nu }\\left( y\\right) +eA_{\\nu }\\left( y\\right) \\right] \n\\]\nHere $\\Gamma ^{\\left( 2\\right) }$ is the one-loop fermion contribution to\nthe effective action in the linear approximation. The operators $\\Pi _{\\nu }$\nand $\\Pi _{\\mu \\nu }$ are calculated using the fermion thermal Green's\nfunction in the presence of the background field $\\overline{b}$ (2.17). They\nrepresent the fermion tadpole and one-loop polarization operators\nrespectively. Their leading behaviors for static $\\left( k_{0}=0\\right) $\nand slowly $\\left( {\\bf k}\\sim 0\\right) $ varying configurations in the\nframe ${\\bf k}=(k,0)$ take the form\n\n\\begin{equation}\n\\Pi _{k}\\left( x\\right) =0,\\;\\;\\;\\Pi _{0}\\left( x\\right) =-n_{e},\\;\\;\\;\\Pi\n_{\\mu \\nu }=\\left( \n\\begin{array}{ccc}\n{\\it \\Pi }_{{\\it 0}}+{\\it \\Pi }_{{\\it 0}}\\,^{\\prime }\\,k^{2} & 0 & {\\it \\Pi }%\n_{{\\it 1}}k \\\\ \n0 & 0 & 0 \\\\ \n-{\\it \\Pi }_{{\\it 1}}k & 0 & {\\it \\Pi }_{\\,{\\it 2}}k^{2}\n\\end{array}\n\\right) ,  \\eqnum{3.2}\n\\end{equation}\n\nThe independent coefficients: ${\\it \\Pi }_{{\\it 0}}$, ${\\it \\Pi }_{{\\it 0}%\n}\\,^{\\prime }$, ${\\it \\Pi }_{{\\it 1}}$ and ${\\it \\Pi }_{\\,{\\it 2}}$ are\nfunctions of $k^{2}$, $\\mu $ and $\\overline{b}$. In order to find them we\njust need to calculate the $\\Pi _{\\mu \\nu }$ Euclidean components: $\\Pi\n_{44} $, $\\Pi _{42}$ and $\\Pi _{22}$. In the Landau gauge these Euclidean\ncomponents are given by\\cite{11},\n\n\\begin{mathletters}\n\\begin{equation}\n\\Pi _{44}\\left( k,\\mu ,\\overline{b}\\right) =-\\frac{1}{\\beta }%\n\\sum\\limits_{p_{4}}\\frac{d{\\bf p}}{\\left( 2\\pi \\right) ^{2}}G\\left( p\\right)\nG\\left( p-k\\right) ,  \\eqnum{3.3}\n\\end{equation}\n\n\\begin{equation}\n\\Pi _{4j}\\left( k,\\mu ,\\overline{b}\\right) =\\frac{i}{2m\\beta }%\n\\sum\\limits_{p_{4}}\\frac{d{\\bf p}}{\\left( 2\\pi \\right) ^{2}}\\left\\{ G\\left(\np\\right) \\cdot D_{j}^{-}G\\left( p-k\\right) +D_{j}^{+}G\\left( p\\right) \\cdot\nG\\left( p-k\\right) \\right\\} ,  \\eqnum{3.4}\n\\end{equation}\n\n\\end{mathletters}\n\\begin{eqnarray}\n\\Pi _{jk}\\left( k,\\mu ,\\overline{b}\\right) &=&\\frac{1}{4m^{2}\\beta }%\n\\sum\\limits_{p_{4}}\\frac{d{\\bf p}}{\\left( 2\\pi \\right) ^{2}}\\left\\{\nD_{k}^{-}G\\left( p\\right) \\cdot D_{j}^{-}G\\left( p-k\\right)\n+D_{j}^{+}G\\left( p\\right) \\cdot D_{k}^{+}G\\left( p-k\\right) \\right. \n\\nonumber \\\\\n&&\\left. +D_{j}^{+}D_{k}^{-}G\\left( p\\right) \\cdot G\\left( p-k\\right)\n+G\\left( p\\right) \\cdot D_{j}^{-}D_{k}^{+}G\\left( p-k\\right) \\right\\} \n\\nonumber \\\\\n&&-\\frac{1}{2m}\\Pi _{4},  \\eqnum{3.5}\n\\end{eqnarray}\nwhere the notation\n\n\\begin{eqnarray}\nD_{j}^{\\pm }G\\left( p\\right) &=&\\left[ ip_{j}\\mp \\frac{\\overline{b}}{2}%\n\\varepsilon ^{jk}\\partial _{p_{k}}\\right] G\\left( p\\right) ,  \\nonumber \\\\\nD_{j}^{\\pm }G\\left( p-k\\right) &=&\\left[ i\\left( p_{j}-k_{j}\\right) \\mp \n\\frac{\\overline{b}}{2}\\varepsilon ^{jk}\\partial _{p_{k}}\\right] G\\left(\np-k\\right) ,  \\eqnum{3.6}\n\\end{eqnarray}\nwas used.\n\nUsing (3.3)-(3.5) after summing in $p_{4}$, we found that, in the $k\/\\sqrt{%\n\\overline{b}}\\ll 1$ limit, the polarization operator coefficients ${\\it \\Pi }%\n_{{\\it 0}}$, ${\\it \\Pi }_{{\\it 0}}\\,^{\\prime }$, ${\\it \\Pi }_{{\\it 1}}$ and $%\n{\\it \\Pi }_{\\,{\\it 2}}$ are\n\n\\[\n{\\it \\Pi }_{{\\it 0}}=\\frac{\\beta \\overline{b}}{8\\pi {\\bf k}^{2}}%\n\\sum_{n}\\Theta _{n},\\;\\qquad {\\it \\Pi }_{{\\it 0}}\\,^{\\prime }=\\frac{2m}{\\pi \n\\overline{b}}\\sum_{n}\\Delta _{n}-\\frac{\\beta }{8\\pi }\\sum_{n}(2n+1)\\Theta\n_{n}, \n\\]\n\n\\[\n{\\it \\Pi }_{{\\it 1}}=\\frac{1}{\\pi }\\sum_{n}\\Delta _{n}-\\frac{\\beta \\overline{%\nb}}{16\\pi m}\\sum_{n}(2n+1)\\Theta _{n},\\qquad {\\it \\Pi }_{\\,{\\it 2}}=\\frac{1}{%\n\\pi m}\\sum_{n}(2n+1)\\Delta _{n}-\\frac{\\beta \\overline{b}}{32\\pi m^{2}}%\n\\sum_{n}(2n+1)^{2}\\Theta _{n}, \n\\]\n\n\\begin{equation}\n\\Theta _{n}=%\n\\mathop{\\rm sech}%\n\\,^{2}\\frac{\\beta (\\epsilon _{n}\/2-\\mu )}{2},\\qquad \\Delta _{n}=(e^{\\beta\n(\\epsilon _{n}\/2-\\mu )}+1)^{-1}  \\eqnum{3.7}\n\\end{equation}\n\nThe leading contributions of the one-loop polarization operator coefficients\n(3.7) at low temperatures $\\left( T\\ll \\omega _{c}\\right) $ are\n\n\\begin{equation}\n{\\it \\Pi }_{{\\it 0}}=\\frac{2\\beta \\overline{b}}{\\pi }e^{-\\beta \\overline{b}%\n\/2m},\\qquad {\\it \\Pi }_{{\\it 0}}\\,^{\\prime }=\\frac{mN}{2\\pi \\overline{b}}%\n{\\it \\Lambda },\\qquad {\\it \\Pi }_{{\\it 1}}=\\frac{N}{2\\pi }{\\it \\Lambda }%\n,\\quad {\\it \\Pi }_{\\,{\\it 2}}=\\frac{N^{2}}{4\\pi m}{\\it \\Lambda },\\qquad {\\it %\n\\Lambda }=\\left[ 1-\\frac{2\\beta \\overline{b}}{m}e^{-\\beta \\overline{b}%\n\/2m}\\right]  \\eqnum{3.8}\n\\end{equation}\nand at high temperatures $\\left( T\\gg \\omega _{c}\\right) $ are\n\n\\begin{equation}\n{\\it \\Pi }_{{\\it 0}}=\\frac{m}{2\\pi }\\left[ \\tanh \\frac{\\beta \\mu }{2}%\n+1\\right] ,\\qquad {\\it \\Pi }_{{\\it 0}}\\,^{\\prime }=-\\frac{\\beta }{48\\pi }%\n\\mathop{\\rm sech}%\n\\!^{2}\\!\\,\\left( \\frac{\\beta \\mu }{2}\\right) ,\\qquad {\\it \\Pi }_{{\\it 1}}=%\n\\frac{\\overline{b}}{m}{\\it \\Pi }_{{\\it 0}}\\,^{\\prime },\\qquad {\\it \\Pi }_{\\,%\n{\\it 2}}=\\frac{1}{12m^{2}}{\\it \\Pi }_{{\\it 0}}  \\eqnum{3.9}\n\\end{equation}\nIn these expressions $\\mu $ is the chemical potential and $m=2m_{e}$ ($m_{e}$\nis the electron mass). These results are in agreement with those found in\nrefs.\\cite{8},\\cite{14}.\n\n\\subsection{MCS Linear Equations}\n\nTo find the response of the anyon fluid to an externally applied magnetic\nfield, one needs to use the extremum equations derived from the effective\naction (3.1). This formulation is known in the literature as the\nself-consistent field approximation\\cite{11}. In solving these equations we\nconfine our analysis to gauge field configurations which are static and\nuniform in the y-direction. Within this restriction we take a gauge in which \n$A_{1}=a_{1}=0$.\n\nThe Maxwell and Chern-Simons extremum equations are respectively,\n\n\\begin{equation}\n\\partial _{\\nu }F^{\\nu \\mu }=eJ_{ind}^{\\mu }  \\eqnum{3.10a}\n\\end{equation}\n\n\\begin{equation}\n-\\frac{N}{4\\pi }\\varepsilon ^{\\mu \\nu \\rho }f_{\\nu \\rho }=J_{ind}^{\\mu } \n\\eqnum{3.10b}\n\\end{equation}\nHere, $f_{\\mu \\nu }$ is the Chern-Simons gauge field strength tensor,\ndefined as $f_{\\mu \\nu }=\\partial _{\\mu }a_{\\nu }-\\partial _{\\nu }a_{\\mu }$,\nand $J_{ind}^{\\mu }$ is the current density induced by the anyon system at\nfinite temperature and density. Their different components are given by\n\n\\begin{equation}\nJ_{ind}^{0}\\left( x\\right) ={\\it \\Pi }_{{\\it 0}}\\left[ a_{0}\\left( x\\right)\n+eA_{0}\\left( x\\right) \\right] +{\\it \\Pi }_{{\\it 0}}\\,^{\\prime }\\partial\n_{x}\\left( {\\cal E}+eE\\right) +{\\it \\Pi }_{{\\it 1}}\\left( b+eB\\right) \n\\eqnum{3.11a}\n\\end{equation}\n\n\\begin{equation}\nJ_{ind}^{1}\\left( x\\right) =0,\\qquad J_{ind}^{2}\\left( x\\right) ={\\it \\Pi }_{%\n{\\it 1}}\\left( {\\cal E}+eE\\right) +{\\it \\Pi }_{\\,{\\it 2}}\\partial _{x}\\left(\nb+eB\\right)  \\eqnum{3.11b}\n\\end{equation}\nin the above expressions we used the following notation: ${\\cal E}=f_{01}$, $%\nE=F_{01}$, $b=f_{12}$ and $B=F_{12}$. Eqs. (3.11) play the role in the anyon\nfluid of the London equations in BCS superconductivity. When the induced\ncurrents (3.11) are substituted in eqs. (3.10) we find, after some\nmanipulation, the set of independent differential equations,\n\n\\begin{equation}\n\\omega \\partial _{x}^{2}B+\\alpha B=\\gamma \\left[ \\partial _{x}E-\\sigma\nA_{0}\\right] +\\tau \\,a_{0},  \\eqnum{3.12}\n\\end{equation}\n\n\\begin{equation}\n\\partial _{x}B=\\kappa \\partial _{x}^{2}E+\\eta E,  \\eqnum{3.13}\n\\end{equation}\n\n\\begin{equation}\n\\partial _{x}a_{0}=\\chi \\partial _{x}B  \\eqnum{3.14}\n\\end{equation}\nThe coefficients appearing in these differential equations depend on the\ncomponents of the polarization operators through the relations\n\n\\[\n\\omega =\\frac{2\\pi }{N}{\\it \\Pi }_{{\\it 0}}\\,^{\\prime },\\quad \\alpha =-e^{2}%\n{\\it \\Pi }_{{\\it 1}},\\quad \\tau =e{\\it \\Pi }_{{\\it 0}},\\quad \\chi =\\frac{%\n2\\pi }{eN},\\quad \\sigma =-\\frac{e^{2}}{\\gamma }{\\it \\Pi }_{{\\it 0}},\\quad\n\\eta =-\\frac{e^{2}}{\\delta }{\\it \\Pi }_{{\\it 1}}, \n\\]\n\n\\begin{equation}\n\\gamma =1+e^{2}{\\it \\Pi }_{{\\it 0}}\\,^{\\prime }-\\frac{2\\pi }{N}{\\it \\Pi }_{%\n{\\it 1}},\\quad \\delta =1+e^{2}{\\it \\Pi }_{\\,{\\it 2}}-\\frac{2\\pi }{N}{\\it \\Pi \n}_{{\\it 1}},\\quad \\kappa =\\frac{2\\pi }{N\\delta }{\\it \\Pi }_{\\,{\\it 2}}. \n\\eqnum{3.15}\n\\end{equation}\n\nDistinctive of eq. (3.12) is the presence of the nonzero coefficients $%\n\\sigma $ and $\\tau $. They are related to the Debye screening in the two\ndimensional anyon thermal ensemble. A characteristic of this 2+1 dimensional\nmodel is that the Debye screening disappears at $T=0$, even if the chemical\npotential is different from zero. Note that $\\sigma $ and $\\tau $ link the\nmagnetic field to the zero components of the gauge potentials, $A_{0}$ and $%\na_{0}$. As a consequence, these gauge potentials will play a nontrivial role\nin finding the magnetic field solution of the system.\n\n\\subsection{Field Solutions and Boundary Conditions}\n\nUsing eqs.(3.12)-(3.14) one can obtain a higher order differential equation\nthat involves only the electric field,\n\n\\begin{equation}\na\\partial _{x}^{4}E+d\\partial _{x}^{2}E+cE=0,  \\eqnum{3.16}\n\\end{equation}\nHere, $a=\\omega \\kappa $, $d=\\omega \\eta +\\alpha \\kappa -\\gamma -\\tau \\kappa\n\\chi $, and $c=\\alpha \\eta -\\sigma \\gamma -\\tau \\eta \\chi $.\n\nSolving (3.16) we find\n\n\\begin{equation}\nE\\left( x\\right) =C_{1}e^{-x\\xi _{1}}+C_{2}e^{x\\xi _{1}}+C_{3}e^{-x\\xi\n_{2}}+C_{4}e^{x\\xi _{2}},  \\eqnum{3.17}\n\\end{equation}\nwhere\n\n\\begin{equation}\n\\xi _{1,2}=\\left[ -d\\pm \\sqrt{d^{2}-4ac}\\right] ^{\\frac{1}{2}}\/\\sqrt{2a} \n\\eqnum{3.18}\n\\end{equation}\nWhen the low density approximation $n_{e}\\ll m^{2}$ is considered (this\napproximation is in agreement with the typical values $n_{e}=2\\times\n10^{14}cm^{-2}$, $m_{e}=2.6\\times 10^{10}cm^{-1}$ found in high-T$_{C}$\nsuperconductivity), we find that, for $N=2$ and at temperatures lower than\nthe energy gap $\\left( T\\ll \\omega _{c}\\right) $, the inverse length scales\n(3.18) are given by the following real functions \n\\begin{equation}\n\\xi _{1}\\simeq e\\sqrt{\\frac{m}{\\pi }}\\left[ 1+\\left( \\frac{\\pi ^{2}n_{e}^{2}%\n}{m^{3}}\\right) \\beta \\exp -\\left( \\frac{\\pi n_{e}\\beta }{2m}\\right) \\right]\n\\eqnum{3.19}\n\\end{equation}\n\\begin{equation}\n\\xi _{2}\\simeq e\\sqrt{\\frac{n_{e}}{m}}\\left[ 1-\\left( \\frac{\\pi n_{e}}{m}%\n\\right) \\beta \\exp -\\left( \\frac{\\pi n_{e}\\beta }{2m}\\right) \\right] \n\\eqnum{3.20}\n\\end{equation}\nThese two inverse length scales correspond to two short-range\nelectromagnetic modes of propagation. These results are consistent with\nthose obtained in ref. \\cite{8} using a different approach. If the masses of\nthe two massive modes, obtained in ref. \\cite{8} by using the\nelectromagnetic thermal Green's function for static and slowly varying\nconfigurations, are evaluated in the range of parameters considered above,\nit can be shown that they reduce to eqs. (319), (3.20).\n\nThe solutions for $B$, $a_{0}$ and $A_{0}$, can be obtained using eqs.\n(3.13), (3.14), (3.17) and the definition of $E$ in terms of $A_{0,}$\n\n\\begin{equation}\nB\\left( x\\right) =\\gamma _{1}\\left( C_{2}e^{x\\xi _{1}}-C_{1}e^{-x\\xi\n_{1}}\\right) +\\gamma _{2}\\left( C_{4}e^{x\\xi _{2}}-C_{3}e^{-x\\xi\n_{2}}\\right) +C_{5}  \\eqnum{3.21}\n\\end{equation}\n\n\\begin{equation}\na_{0}\\left( x\\right) =\\chi \\gamma _{1}\\left( C_{2}e^{x\\xi\n_{1}}-C_{1}e^{-x\\xi _{1}}\\right) +\\chi \\gamma _{2}\\left( C_{4}e^{x\\xi\n_{2}}-C_{3}e^{-x\\xi _{2}}\\right) +C_{6}  \\eqnum{3.22}\n\\end{equation}\n\n\\begin{equation}\nA_{0}\\left( x\\right) =\\frac{1}{\\xi _{1}}\\left( C_{1}e^{-x\\xi\n_{1}}-C_{2}e^{x\\xi _{1}}\\right) +\\frac{1}{\\xi _{2}}\\left( C_{3}e^{-x\\xi\n_{2}}-C_{4}e^{x\\xi _{2}}\\right) +C_{7}  \\eqnum{3.23}\n\\end{equation}\nIn the above formulas we introduced the notation $\\gamma _{1}=\\left( \\xi\n_{1}^{2}\\kappa +\\eta \\right) \/\\xi _{1}$, $\\gamma _{2}=\\left( \\xi\n_{2}^{2}\\kappa +\\eta \\right) \/\\xi _{2}$.\n\nIn obtaining eq. (3.16) we have taken the derivative of eq. (3.12).\nTherefore, the solution of eq. (3.16) belongs to a wider class than the one\ncorresponding to eqs. (3.12)-(3.14). To exclude redundant solutions we must\nrequire that they satisfy eq. (3.12) as a supplementary condition. In this\nway the number of independent unknown coefficients is reduced to six, which\nis the number corresponding to the original system (3.12)-(3.14). The extra\nunknown coefficient is eliminated substituting the solutions (3.17), (3.21),\n(3.22) and (3.23) into eq. (3.12) to obtain the relation\n\n\\begin{equation}\ne{\\it \\Pi }_{{\\it 1}}C_{5}=-{\\it \\Pi }_{{\\it 0}}\\left( C_{6}+eC_{7}\\right) \n\\eqnum{3.24}\n\\end{equation}\n\nEq. (3.24) has an important meaning, it establishes a connection between the\ncoefficients of the long-range modes of the zero components of the gauge\npotentials $(C_{6}+eC_{7})$ and the coefficient of the long-range mode of\nthe magnetic field $C_{5}$. Note that if the induced Chern-Simons\ncoefficient ${\\it \\Pi }_{{\\it 1}}$, or the Debye screening coefficient ${\\it %\n\\Pi }_{{\\it 0}}$ were zero, there would be no link between $C_{5}$ and $%\n(C_{6}+eC_{7})$. This relation between the long-range modes of $B$, $A_{0}$\nand $a_{0}$ can be interpreted as a sort of Aharonov-Bohm effect, which\noccurs in this system at finite temperature. At $T=0$, we have ${\\it \\Pi }_{%\n{\\it 0}}=0$, and the effect disappears.\n\nUp to this point no boundary has been taken into account. Therefore, it is\neasy to understand that the magnetic long-range mode associated with the\ncoefficient $C_{5}$, must be identified with the one found in ref. \\cite{8}\nfor the infinite bulk using a different approach. However, as it is shown\nbelow, when a constant and uniform magnetic field is perpendicularly applied\nat the boundaries of a two-dimensional sample, this mode cannot propagate\n(i.e. $C_{5}\\equiv 0$) within the sample. This result is crucial for the\nexistence of Meissner effect in this system.\n\nIn order to determine the unknown coefficients we need to use the boundary\nconditions. Henceforth we consider that the anyon fluid is confined to the\nstrip $-\\infty <y<\\infty $ with boundaries at $x=-L$ and $x=L$. The external\nmagnetic field will be applied from the vacuum at both boundaries ($-\\infty\n<x\\leq -L$, $\\;L\\leq x<\\infty $).\n\nThe boundary conditions for the magnetic field are $B\\left( x=-L\\right)\n=B\\left( x=L\\right) =\\overline{B}$ ($\\overline{B}$ constant). Because no\nexternal electric field is applied, the boundary conditions for this field\nare, $E\\left( x=-L\\right) =E\\left( x=L\\right) =0$. Using them and assuming $%\nL\\gg \\lambda _{1}$, $\\lambda _{2}$ ($\\lambda _{1}=1\/\\xi _{1}$, $\\lambda\n_{2}=1\/\\xi _{2}$), we find the following relations that give $C_{1,2,3,4}$\nin terms of $C_{5}$,\n\n\\begin{equation}\nC_{1}=Ce^{-L\\xi _{1}},\\quad C_{2}=-C_{1},\\quad C_{3}=-Ce^{-L\\xi _{2}},\\quad\nC_{4}=-C_{3},\\quad C=\\frac{C_{5}-\\overline{B}}{\\gamma _{1}-\\gamma _{2}} \n\\eqnum{3.25}\n\\end{equation}\n\n\\section{Stability Condition for the Infinite-Strip Sample}\n\nAfter using the boundary conditions, we can see from (3.25) that they were\nnot sufficient to find the coefficient $C_{5}$. In order to totally\ndetermine the system magnetic response we have to use another physical\ncondition from where $C_{5}$ can be found. Since, obviously, any meaningful\nsolution have to be stable, the natural additional condition to be\nconsidered is the stability equation derived from the system free energy.\nWith this goal in mind we start from the free energy of the infinite-strip\nsample\n\n\\[\n{\\cal F}=\\frac{1}{2}\\int\\limits_{-L^{\\prime }}^{L^{\\prime\n}}dy\\int\\limits_{-L}^{L}dx\\left\\{ \\left( E^{2}+B^{2}\\right) +\\frac{N}{\\pi }%\na_{0}b-{\\it \\Pi }_{{\\it 0}}\\left( eA_{0}+a_{0}\\right) ^{2}\\right. \n\\]\n\n\\begin{equation}\n\\left. -{\\it \\Pi }_{{\\it 0}}\\,^{\\prime }\\left( eE+{\\cal E}\\right) ^{2}-2{\\it %\n\\Pi }_{{\\it 1}}\\left( eA_{0}+a_{0}\\right) \\left( eB+b\\right) +{\\it \\Pi }_{\\,%\n{\\it 2}}\\left( eB+b\\right) ^{2}\\right\\}  \\eqnum{4.1}\n\\end{equation}\nwhere $L$ and $L^{\\prime }$ determine the two sample's lengths.\n\nUsing the field solutions (3.17), (3.21)-(3.23) with coefficients (3.25), it\nis found that the leading contribution to the free-energy density ${\\it f}=%\n\\frac{{\\cal F}}{{\\cal A}}$ ,\\ (${\\cal A}=4LL^{\\prime }$ being the sample\narea) in the infinite-strip limit $(L\\gg \\lambda _{1}$, $\\lambda _{2}$, $%\nL^{\\prime }\\rightarrow \\infty )$ is given by\n\n\\begin{equation}\nf=C_{5}^{2}-{\\it \\Pi }_{{\\it 0}}\\left( C_{6}+eC_{7}\\right) ^{2}+e^{2}{\\it %\n\\Pi }_{\\,{\\it 2}}C_{5}^{2}-2e{\\it \\Pi }_{{\\it 1}}\\left( C_{6}+eC_{7}\\right)\nC_{5}  \\eqnum{4.2}\n\\end{equation}\n\nTaking into account the constraint equation (3.24), the free-energy density\n(4.2) can be written as a quadratic function in $C_{5}$. Then, the value of $%\nC_{5}$ is found, by minimizing the corresponding free-energy density\n\n\\begin{equation}\n\\frac{\\delta {\\it f}}{\\delta C_{5}}=\\left[ {\\it \\Pi }_{{\\it 0}}+e^{2}{\\it %\n\\Pi }_{{\\it 1}}^{\\,}\\,^{2}+e^{2}{\\it \\Pi }_{{\\it 0}}{\\it \\Pi }_{\\,{\\it 2}%\n}\\right] \\frac{C_{5}}{{\\it \\Pi }_{{\\it 0}}}=0,  \\eqnum{4.3}\n\\end{equation}\nto be $C_{5}=0$.\n\nThis result implies that the long-range mode cannot propagate within the\ninfinite-strip when a uniform and constant magnetic field is perpendicularly\napplied at the sample's boundaries.\n\nWe want to point out the following fact. The same property of the finite\ntemperature polarization operator component $\\Pi _{00}$ that is producing\nthe appearance of a long-range mode in the infinite bulk, is also\nresponsible, when it is combined with the boundary conditions, for the\nnon-propagation of this mode in the bounded sample. It is known that the\nnonvanishing of ${\\it \\Pi }_{{\\it 0}}$ at $T\\neq 0$ (or equivalently, the\npresence of a pole $\\sim 1\/k^{2}$ in $\\Pi _{00}$ at $T\\neq 0$) guarantees\nthe existence of a long-range mode in the infinite bulk \\cite{8}. On the\nother hand, however, once ${\\it \\Pi }_{{\\it 0}}$ is different from zero, we\ncan use the constraint (3.24) to eliminate $C_{6}+eC_{7}$ in favor of $C_{5%\n\\text{ }}$ in the free-energy density of the infinite strip. Then, as we\nhave just proved, the only stable solution of this boundary-value problem,\nwhich is in agreement with the boundary conditions, is $C_{5}=0$.\nConsequently, no long-range mode propagates in the bounded sample.\n\nIn the zero temperature limit $\\left( \\beta \\rightarrow \\infty \\right) $,\nbecause ${\\it \\Pi }_{{\\it 0}}=0$, it is straightforwardly obtained from\n(3.24) that $C_{5}=0$ and no long-range mode propagates.\n\nAt $T\\neq 0$, taking into account that $C_{5}=0$ along with eq. (3.25) in\nthe magnetic field solution (3.21), we can write the magnetic field\npenetration as\n\n\\begin{equation}\nB\\left( x\\right) =\\overline{B}_{1}\\left( T\\right) \\left( e^{-(x+L)\\xi\n_{1}}+e^{\\left( x-L\\right) \\xi _{1}}\\right) +\\overline{B}_{2}\\left( T\\right)\n\\left( e^{-(x+L)\\xi _{2}}+e^{\\left( x-L\\right) \\xi _{2}}\\right)  \\eqnum{4.4}\n\\end{equation}\nwhere,\n\n\\begin{equation}\n\\overline{B}_{1}\\left( T\\right) =\\frac{\\gamma _{1}}{\\gamma _{1}-\\gamma _{2}}%\n\\overline{B},\\text{ \\qquad \\quad }\\overline{B}_{2}\\left( T\\right) =\\frac{%\n\\gamma _{2}}{\\gamma _{2}-\\gamma _{1}}\\overline{B}  \\eqnum{4.5}\n\\end{equation}\n\nFor densities $n_{e}\\ll m^{2}$, the coefficients $\\overline{B}_{1}$and $%\n\\overline{B}_{2}$ can be expressed, in the low temperature approximation $%\n\\left( T\\ll \\omega _{c}\\right) $, as\n\n\\begin{equation}\nB_{1}\\left( T\\right) \\simeq -\\frac{\\left( \\pi n_{e}\\right) ^{3\/2}}{m^{2}}%\n\\left[ 1\/m+\\frac{5}{2}\\beta \\exp -\\left( \\frac{\\pi n_{e}\\beta }{2m}\\right)\n\\right] \\overline{B},\\qquad  \\eqnum{4.6}\n\\end{equation}\n\\begin{equation}\nB_{2}\\left( T\\right) \\simeq \\left[ 1+\\frac{5\\pi n_{e}}{2m^{2}}\\sqrt{\\pi n_{e}%\n}\\beta \\exp -\\left( \\frac{\\pi n_{e}\\beta }{2m}\\right) \\right] \\overline{B} \n\\eqnum{4.7}\n\\end{equation}\n\nHence, in the infinite-strip sample the applied magnetic field is totally\nscreened within the anyon fluid on two different scales, $\\lambda _{1}=1\/\\xi\n_{1}$ and $\\lambda _{2}=1\/\\xi _{2}$. At $T=200K$, for the density value\nconsidered above, the penetration lengths are given by $\\lambda _{1}\\simeq\n0.6\\times 10^{-8}cm$ and $\\lambda _{2}\\simeq 0.1\\times 10^{-4}cm$ .\nMoreover, taking into account that $\\xi _{1}$ increases with the temperature\nwhile $\\xi _{2}$ decreases (see eqs. (3.19)-(3.20)), and that $B_{1}\\left(\nT\\right) <0$ while $B_{2}\\left( T\\right) >0$, it can be shown that the\neffective penetration length $\\overline{\\lambda }$ (defined as the distance $%\nx$ where the magnetic field falls down to a value $B\\left( \\overline{\\lambda \n}\\right) \/\\overline{B}=e^{-1}$) increases with the temperature as\n\n\\begin{equation}\n\\overline{\\lambda }\\simeq \\overline{\\lambda }_{0}\\left( 1+\\overline{\\kappa }%\n\\beta \\exp -\\frac{1}{2}\\overline{\\kappa }\\beta \\right)   \\eqnum{4.8}\n\\end{equation}\nwhere $\\overline{\\lambda }_{0}=\\sqrt{m\/n_{e}e^{2}}$ and $\\overline{\\kappa }%\n=\\pi n_{e}\/m$. At $T=200K$ the effective penetration length is $\\overline{%\n\\lambda }\\sim 10^{-5}cm$.\n\nIt is timely to note that the presence of explicit (proportional to $N$) and\ninduced (proportional to ${\\it \\Pi }_{{\\it 1}}$) Chern-Simons terms in the\nanyon effective action (3.1) is crucial to obtain the Meissner solution\n(4.4). If the Chern-Simons interaction is disconnected ($N\\rightarrow \\infty \n$ and ${\\it \\Pi }_{{\\it 1}}=0$), then $a=0,$ $d=1+e^{2}{\\it \\Pi }_{{\\it 0}%\n}{}^{\\prime }\\neq 0$ and $c=e^{2}{\\it \\Pi }_{{\\it 0}}\\,\\neq 0$ in eq.\n(3.16). In that case the solution of the field equations within the sample\nare $E=0$, $B=\\overline{B}$. That is, we regain the QED in (2+1)-dimensions,\nwhich does not exhibit any superconducting behavior.\n\n\\section{High Temperature Non-Superconducting Phase}\n\nWe have just found that the charged anyon fluid confined to an infinite\nstrip exhibits Meissner effect at temperatures lower than the energy gap $%\n\\omega _{c}$. It is natural to expect that at temperatures larger than the\nenergy gap this superconducting behavior should not exist. At those\ntemperatures the electron thermal fluctuations should make available the\nfree states existing beyond the energy gap. As a consequence, the charged\nanyon fluid should not be a perfect conductor at $T\\gg \\omega _{c}$. A\nsignal of such a transition can be found studying the magnetic response of\nthe system at those temperatures.\n\nAs can be seen from the magnetic field solution (4.4), the real character of\nthe inverse length scales (3.18) is crucial for the realization of the\nMeissner effect. At temperatures much lower than the energy gap this is\nindeed the case, as can be seen from eqs. (3.19) and (3.20).\n\nIn the high temperature $\\left( T\\gg \\omega _{c}\\right) $ region the\npolarization operator coefficients are given by eq. (3.9). Using this\napproximation together with the assumption $n_{e}\\ll m^{2}$, we can\ncalculate the coefficients $a$, $c$ and $d$ that define the behavior of the\ninverse length scales,\n\n\\begin{equation}\na\\simeq \\pi ^{2}{\\it \\Pi }_{{\\it 0}}{}^{\\prime }{\\it \\Pi }_{\\,{\\it 2}} \n\\eqnum{5.1}\n\\end{equation}\n\n\\begin{equation}\nc\\simeq e^{2}{\\it \\Pi }_{{\\it 0}}{}  \\eqnum{5.2}\n\\end{equation}\n\n\\begin{equation}\nd\\simeq -1  \\eqnum{5.3}\n\\end{equation}\n\nSubstituting with (5.1)-(5-3) in eq. (3.18) we obtain that the inverse\nlength scales in the high-temperature limit are given by\n\n\\begin{equation}\n\\xi _{1}\\simeq e\\sqrt{m\/2\\pi }\\left( \\tanh \\frac{\\beta \\mu }{2}+1\\right) ^{%\n\\frac{1}{2}}  \\eqnum{5.4}\n\\end{equation}\n\n\\begin{equation}\n\\xi _{2}\\simeq i\\left[ 24\\sqrt{\\frac{2m}{\\beta }}\\cosh \\frac{\\beta \\mu }{2}%\n\\left( \\tanh \\frac{\\beta \\mu }{2}+1\\right) ^{-\\frac{1}{2}}\\right]  \n\\eqnum{5.5}\n\\end{equation}\n\nThe fact that $\\xi _{2}$ becomes imaginary at temperatures larger than the\nenergy gap, $\\omega _{c}$, implies that the term $\\gamma _{2}\\left(\nC_{4}e^{x\\xi _{2}}-C_{3}e^{-x\\xi _{2}}\\right) $ in the magnetic field\nsolution (3.21) ceases to have a damping behavior, giving rise to a periodic\ninhomogeneous penetration. Therefore, the fluid does not exhibit a Meissner\neffect at those temperatures since the magnetic field will not be totally\nscreened. This corroborate our initial hypothesis that at $T\\gg \\omega _{c}$\nthe anyon fluid is in a new phase in which the magnetic field can penetrate\nthe sample.\n\nWe expect that a critical temperature of the order of the energy gap ($T\\sim\n\\omega _{c}$) separates the superconducting phase $\\left( T\\ll \\omega\n_{c}\\right) $ from the non-superconducting one $\\left( T\\gg \\omega\n_{c}\\right) $. Nevertheless, the temperature approximations (3.8) and (3.9)\nare not suitable to perform the calculation needed to find the phase\ntransition temperature. The field solutions in this new non-superconducting\nphase is currently under investigation. The results will be published\nelsewhere.\n\n\\section{Concluding Remarks}\n\nIn this paper we have investigated the magnetic response at finite\ntemperature of a charged anyon fluid confined to an infinite strip. The\ncharged anyon fluid was modeled by a (2+1)-dimensional MCS theory in a\nmany-particle ($\\mu \\neq 0$, $\\overline{b}\\neq 0$) ground state. The\nparticle energy spectrum of the theory exhibits a band structure given by\ndifferent Landau levels separated by an energy gap $\\omega _{c}$, which is\nproportional to the background Chern-Simons magnetic field $\\overline{b}$.\nWe found that the energy gap $\\omega _{c}$ defines a scale that separates\ntwo phases: a superconducting phase at $T\\ll \\omega _{c}$, and a\nnon-superconducting one at $T\\gg \\omega _{c}$.\n\nThe total magnetic screening in the superconducting phase is characterized\nby two penetration lengths corresponding to two short-range eigenmodes of\npropagation of the electromagnetic field within the anyon fluid. The\nexistence of a Meissner effect at finite temperature is the consequence of\nthe fact that a third electromagnetic mode, of a long-range nature, which is\npresent at finite temperature in the infinite bulk \\cite{8}, does not\npropagate within the infinite strip when a uniform and constant magnetic\nfield is applied at the boundaries. This is a significant property since the\nsamples used to test the Meissner effect in high-$T_{c}$ superconductors are\nbounded.\n\nIt is noteworthy that the existence at finite temperature of a Debye\nscreening (${\\it \\Pi }_{{\\it 0}}\\,\\neq 0$) gives rise to a sort of\nAharonov-Bohm effect in this system with Chern-Simons interaction ($N$\nfinite, ${\\it \\Pi }_{{\\it 1}}\\neq 0$). When ${\\it \\Pi }_{{\\it 0}}\\,\\neq 0$,\nthe field combination $a_{0}+eA_{0}$ becomes physical because it enters in\nthe field equations in the same foot as the electric and magnetic fields\n(see eq. (3.12)). A direct consequence of this fact is that the coefficient $%\nC_{5}$, associated to the long-range mode of the magnetic field, is linked\nto the coefficients $C_{6}$ and $C_{7}$ of the zero components of the\npotentials (see eq. (3.24)).\n\nWhen $T=0$, since ${\\it \\Pi }_{{\\it 0}}\\,=0$ and ${\\it \\Pi }_{{\\it 1}}\\neq 0$%\n, eq. (3.24) implies $C_{5}=0$. That is, at zero temperature the long-range\nmode is absent. This is the well known Meissner effect of the anyon fluid at \n$T=0$. When $T\\neq 0$, eq. (3.24) alone is not enough to determine the value\nof $C_{5}$, since it is given in terms of $C_{6}$ and $C_{7}$ which are\nunknown. However, when eq. (3.24) is taken together with the field\nconfigurations that satisfy the boundary conditions for the infinite-strip\nsample (eqs. (3.17), (3.21)-(3.23) and (3.25)), and with the sample\nstability condition (4.3), we obtain that $C_{5}=0$. Thus, the combined\naction of the boundary conditions and the Aharonov-Bohm effect expressed by\neq. (3.24) accounts for the total screening of the magnetic field in the\nanyon fluid at finite temperature.\n\nFinally, at temperatures large enough ($T\\gg \\omega _{c}$) to excite the\nelectrons beyond the energy gap, we found that the superconducting behavior\nof the anyon fluid is lost. This result was achieved studying the nature of\nthe characteristic lengths (3.18) in this high temperature approximation. We\nshowed that in this temperature region the characteristic length $\\xi _{2}$\nbecomes imaginary (eq. (5.5)), which means that a total damping solution for\nthe magnetic field does not exist any more, and hence the magnetic field\npenetrates the sample.\n\n\\begin{quote}\nAcknowledgments\n\\end{quote}\n\nThe authors are very grateful for stimulating discussions with Profs. G.\nBaskaran, A. Cabo, E.S. Fradkin, Y. Hosotani and J. Strathdee. We would also\nlike to thank Prof. S. Randjbar-Daemi for kindly bringing the publication of\nref. \\cite{b} to our attention. Finally, it is a pleasure for us to thank\nProf. Yu Lu for his hospitality during our stay at the ICTP, where part of\nthis work was done. This research has been supported in part by the National\nScience Foundation under Grant No. PHY-9414509.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Log-linear models with user groups} \\label{sec:augmentation}\n\n\nAs discussed above, a user-agnostic model such as \\eqref{eq:basicModel} does not do justice to the variability of language comprehension and production across different speakers and listeners. We will therefore extend it to a model which distinguishes different \\emph{user groups}. We will not try to model why\\footnote{E.g., in the sense of explicitly modeling sociolects or the difference between novice system users vs.\\ experts.} users behave differently. Instead our model sorts users into groups simply based on the way in which they respond to stimuli, in the sense of Section~\\ref{sec:basicModel}, and implements this by giving each group $g$ its own parameter vector $\\parAdapt{g}$. As a theoretical example, Group 1 might contain users who reliably comprehend REs which use colors (``the green button''), whereas Group 2 might contain users who more easily understand relational REs (``the button next to the lamp''). These groups are then discovered at training time.\n\n\nWhen our trained NLG system starts interacting with an unseen user $u$, it will infer the group to which $u$ belongs based on $u$'s observed responses to previous stimuli. Thus as the dialogue with $u$ unfolds, the system will have an increasingly precise estimate of the group to which $u$ belongs, and will thus be able to generate language which is increasingly well-tailored to this particular user.\n\n\\subsection{Generative story}\n\n\\begin{figure} \n\\begin{center}\n\t\\input{plate_diagram.tex}\n\\end{center}\n\\caption{Plate diagram for the user group model.} \\label{fig:plates_drawn}\n\\end{figure}\n\nWe assume training data $D = \\{(b_i,s_i,u_i)\\}_i$ which contains stimuli $s_i$ together with the behaviors $b_i$ which the users $u_i$ exhibited in response to $s_i$. We write $D\\up{u} = \\{(b^u_1, s^u_1), \\ldots (b^u_N, s^u_N)\\}$ for the data points for each user $u$.\n\nThe generative story we use is illustrated in Fig.~\\ref{fig:plates_drawn}; observable variables are shaded gray, unobserved variables and parameters to be set in training are shaded white and externally set hyperparameters have no circle around them. Arrows indicate which variables and parameters influence the probability distribution of other variables.\n\nWe assume that each user belongs to a group $g \\in \\{1,\\ldots,K\\}$, where the number $K$ of groups is fixed beforehand based on, e.g., held out data. A group $g$ is assigned to $u$ at random from the distribution\n\\begin{eqnarray} \\label{eq:prior}\n\\condProb{g}{\\ensuremath{\\pi}} = \\displaystyle\n\\frac{\\exp(\\ensuremath{\\pi}_g)}{\\sum_{g'=1}^K \\exp(\\ensuremath{\\pi}_{g'})}\n\\end{eqnarray}\nHere $\\ensuremath{\\pi} \\in \\mathbb{R}^K$ is a vector of weights, which defines how probable each group is a-priori.\n\n\\begin{figure*}\n\t\\begin{eqnarray}\n\t\tP(D;\\theta) = & \\displaystyle \\left(\\prod_{u \\in U} \\sum_{g =1}^K P(g|\\ensuremath{\\pi}) \\cdot \\prod_{d \\in D\\up{u}} \\condProb{b_d}{s_d;\\parGroup{g}}\\right) \\cdot \\left(\\mathcal{N}\\left(\\ensuremath{\\pi}|0,\\sigma\\up{\\ensuremath{\\pi}}\\right) \\cdot \\prod_{g=1}^K\\mathcal{N}\\left(\\parAdapt{g}|0,\\sigma\\up{\\rho}\\right) \\right) \\label{eqn::probability} \\\\\n\t\t\\mathcal{L}(\\theta) = & \\displaystyle \\sum_{u \\in U} \\log \\sum_{g = 1}^K P(g|\\ensuremath{\\pi}) \\cdot \\prod_{d \\in D\\up{u}} \\condProb{b_d}{s_d;\\parGroup{g}} \\label{eqn::loglike} \\\\\n\t\t\\mathcal{AL}(\\theta) = & \\displaystyle \\sum_{u \\in U} \\sum_{g = 1}^K \\left( \\condProb{g}{D\\up{u};\\theta\\down{i-1}} \\cdot \\left( \\log P(g|\\ensuremath{\\pi}) + \\sum_{d \\in D_u} \\log \\condProb{b_d}{s_d;\\parGroup{g}}\\right)\\right) \\label{eqn::grad}\n\t\\end{eqnarray}\n\\end{figure*}\n\nWe replace the single parameter vector $\\rho$ of \\eqref{eq:basicModel} with group-specific parameters vectors $\\parAdapt{g}$, thus obtaining a potentially different log-linear model $\\condProb{b}{s; \\parGroup{g}}$ for each group. After assigning a group, our model generates responses $b^u_1,\\ldots,b^u_N$ at random from $\\condProb{b}{s; \\parAdapt{g}}$, based on the group specific parameter vector and the stimuli $s^u_1,\\ldots,s^u_N$. This accounts for the generation of the data.\n\nWe model the parameter vectors $\\ensuremath{\\pi} \\in \\mathbb{R}^K$, and $\\parAdapt{g} \\in \\mathbb{R}^n$ for every $1 \\leq g \\leq K$ as drawn from normal distributions $\\mathcal{N}(0,\\sigma\\up{\\pi})$, and $\\mathcal{N}(0,\\sigma\\up{\\rho})$, which are centered at $0$ with externally given variances and no covariance between parameters. This has the effect of making parameter choices close to zero more probable. Consequently, our models are unlikely to contain large weights for features that only occurred a few times or which are only helpful for a few examples. This should reduce the risk of overfitting the training set.\n\nThe equation for the full probability of the data and a specific parameter setting is given in \\eqref{eqn::probability}. The left bracket contains the likelihood of the data, while the right bracket contains the prior probability of the parameters.\n\n\\subsection{Predicting user behavior}\n\\label{sec:group-posterior}\n\nOnce we have set values $\\theta = (\\ensuremath{\\pi}, \\parAdapt{1}, \\ldots, \\parAdapt{K})$ for all the parameters, we want to predict what behavior $b$ a user $u$ will exhibit in response to a stimulus $s$. If we encounter a completely new user $u$, the prior user group distribution from \\eqref{eq:prior} gives the probability that this user belongs to each group. We combine this with the group-specific log-linear behavior models to obtain the distribution:\n\\begin{eqnarray} \\label{eq:group-loglin}\n\\condProb{b}{s; \\theta} = \n \\sum_{g = 1}^K \n \\condProb{b}{s; \\parGroup{g}} \\cdot \\condProb{g}{\\ensuremath{\\pi}}\n \\label{expr::findREUser} \\end{eqnarray}\n\n\\noindent\nThus, we have a group-aware replacement for \\eqref{eq:basicModel}.\n\nFurthermore, in the interactive setting of a dialogue system, we may have multiple opportunities to interact with the same user $u$. We can then develop a more precise estimate of $u$'s group based on their responses to previous stimuli. Say that we have made the previous observations $D\\up{u} = \\{\\tuple{s_1,b_1},\\dots,\\tuple{s_N,b_N}\\}$ for user $u$. Then we can use Bayes' theorem to calculate a \\emph{posterior} estimate for $u$'s group membership:\n\\begin{eqnarray} \\label{eq:posterior}\n\\condProb{g}{D\\up{u}; \\theta}\n\\propto\n\\condProb{D\\up{u}}{\\parGroup{g}} \\cdot \\condProb{g}{\\ensuremath{\\pi}}\n\\end{eqnarray}\n\nThis posterior balances whether a group is likely in general against whether members of that group behave as $u$ does. We can use $P_u(g) = \\condProb{g}{D\\up{u}; \\theta}$ as our new estimate for the group membership probabilities for $u$ and replace \\eqref{eq:group-loglin} with:\n\\begin{eqnarray}  \\label{eq:posterior-loglin}\n\\condProb{b}{s, D\\up{u}; \\theta} = \n \\sum_{g = 1}^K \\condProb{b}{s; \\parGroup{g}} \\cdot P_u(g)\n \\end{eqnarray}\n\n\\noindent for the next interaction with $u$.\n\nAn NLG system can therefore adapt to each new user over time. Before the first interaction with $u$, it has no specific information about $u$ and models $u$'s behavior based on \\eqref{eq:group-loglin}. As the system interacts with $u$ repeatedly, it collects observations $D\\up{u}$ about $u$'s behavior. This allows it to calculate an increasingly accurate posterior $P_u(g) = \\condProb{g}{D\\up{u}; \\theta}$ of $u$'s group membership, and thus generate utterances which are more and more suitable to $u$ using \\eqref{eq:posterior-loglin}.\n\n\\section{Log-linear models for NLG in dialog} \\label{sec:basicModel}\n\nWe start with a basic model of the way in which people produce and comprehend language. In order to generalize over production and comprehension, we will simply say that a human language user exhibits a certain \\emph{behavior} $b$ among a range of possible behaviors, in response to a \\emph{stimulus} $s$. The behavior of a speaker is the utterance $b$ they produce in order to achieve a communicative goal $s$; the behavior of a listener is the meaning $b$ which they assign to the utterance $s$ they hear.\n\nGiven this terminology, we define a basic log-linear model~\\cite{BergerPP96} of language use as follows:\n\\begin{eqnarray} \\label{eq:basicModel}\n\t\\condProb{b}{s;\\rho} = \\displaystyle \\frac{\\exp(\\rho \\cdot \\phi(b,s))}{\\sum_{b'}{\\exp(\\rho \\cdot  \\phi(b', s))}}\n\\end{eqnarray}\n\n\\noindent\nwhere $\\rho$ is a real-valued parameter vector of length $n$\nand $\\phi(b,s)$ is a vector of real-valued \\textit{feature functions} $f_1,...,f_n$ over behaviors and stimuli. The parameters can be trained by maximum-likelihood estimation from a corpus of observations $(b,s)$. In addition to maximum-likelihood training it is possible to include some prior probability distribution, which expresses our belief about the probability of any parameter vector and which is generally used for regularization. The latter case is referred to as \\emph{a posteriori} training, which selects the value of $\\rho$ that maximizes the product of the parameter probability and the probability of the data. \n\n\nIn this paper, we focus on the use of such models in the context of the NLG module of a dialogue system, and more specifically on the generation of referring expressions (REs). Using \\eqref{eq:basicModel} as a \\emph{comprehension} model, \\newcite{EngonopoulosVTK13} developed an RE generation model in which the stimulus $s=(r,c)$ consists of an RE $r$ and a visual context $c$ of the GIVE Challenge \\cite{StrDenGarGarKolThe11}, as illustrated in Fig.~\\ref{fig:GIVE}. The behavior is the object $b$ in the visual scene to which the user will resolve the RE. Thus for instance, when we consider the RE $r=$``the blue button'' in the context of Fig.~\\ref{fig:GIVE}, the log-linear model may assign a higher probability to the button on the right than to the one in the background. \\newcite{EngonopoulosK14} develop an algorithm for generating the RE $r$ which maximizes $\\condProb{b^*}{s;\\rho}$, where $b^*$ is the intended\nreferent in this setting.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\columnwidth]{screen1_wt}\n\\caption{A visual scene and a system-generated instruction from the GIVE challenge.} \n\\label{fig:GIVE}\n\\end{figure}\n\nConversely, log-linear models can also be used to directly capture how a human speaker would refer to an object in a given scene. In this case, the stimulus $s = (a,c)$ consists of the target object $a$ and the visual context $c$, and the behavior $b$ is the RE. We follow \\newcite{ferreira14:_refer} in training individual models for the different attributes which can be used in the RE (e.g., that $a$ is a button; that it is blue; that the RE contains a binary relation such as ``to the right of''), such that we can simply represent $b$ as a binary choice $b \\in \\{1,-1\\}$ between whether a particular attribute should be used in the RE or not. We can then implement an analog of Ferreira's model in terms of \\eqref{eq:basicModel} by using feature functions $\\phi(b,a,c) = b \\cdot \\phi'(a,c)$, where $\\phi'(a,c)$ corresponds to their \\textit{context} features, which do not capture any speaker-specific information. \n\n\n\n\n\\section{Conclusion}\n\nWe have presented a probabilistic model for NLG which predicts\nthe behavior of individual users of a dialog system by dynamically\nassigning them to user groups, which were discovered\nduring training\\footnote{Our code and data is available in \\url{https:\/\/bit.ly\/2jIu1Vm}}. \nWe showed for two separate NLG-related tasks,\nRE production and RE comprehension, how our model, \nafter being trained with data that is not annotated with user groups,\ncan quickly adapt to unseen users as it gets more observations \nfrom them in the course of a dialog and makes increasingly \naccurate predictions about their behavior.\n\nAlthough in this work we apply our model to tasks related to NLG,\nnothing hinges on this choice; it can also be applied to any other \ndialog-related prediction task where user variation plays a role. \nIn the future, we will also try to apply the basic principles of our \nuser group approach to more sophisticated underlying models, such \nas neural networks.   \n\\section{Evaluation} \\label{sec:evaluation}\n\nOur model can be used in any component of a dialog system for which a prediction\nof the user's behavior is needed. In this work, we evaluate it in two\nNLG-related prediction tasks: RE production and RE comprehension. In both cases\nwe evaluate the ability of our model to predict the user's behavior given a\nstimulus. We expect our user-group model to gradually improve its prediction accuracy\ncompared to a generic baseline without user groups as it sees more observations\nfrom a given user.\n\nIn all experiments described below we set the prior variances \n$\\sigma_{\\gamma}=1.0$ and $\\sigma_{\\pi}=0.3$ after trying out values between\n0.1 and 10 on the training data of the comprehension experiment.\n\n\\subsection{RE production}\\label{sub:production}\n\n\\paragraph{Task} The task of RE generation can be split in two\nsteps: \\emph{attribute selection}, the selection of the visual attributes\nto be used in the RE such as color, size, relation to other objects and\n\\emph{surface realization}, the generation of a full natural language expression. \nWe focus here on attribute selection: given a visual scene and a target object, \nwe want to predict the set of attributes of the target object that a human\nspeaker would use in order to describe it. Here we treat attribute selection in\nterms of individual classification decisions on whether to use each attribute,\nas described in Section~\\ref{sec:basicModel}.\nMore specifically, we focus on predicting whether the speaker will use a \\emph{spatial\nrelation} to another object (``landmark''). Our motivation for choosing this attribute \nstems from the fact that previous authors ~\\cite{viethen2008use,ferreira14:_refer}\nhave found substantial variation between different users with respect to their preference\ntowards using spatial relations.\n\n\n\\paragraph{Data} We use the GRE3D3 dataset of human-produced\nREs~\\cite{viethen2010speaker}, which contains 630 descriptions for 10 scenes\ncollected from 63 users, each describing the same target object in each scene.\n$35\\%$ of the descriptions in this corpus use a spatial relation. An example of\nsuch a scene can be seen in Fig.~\\ref{fig:gre3d}.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.6\\columnwidth]{gre3d} \n\\caption{A sample scene with a human-produced RE from the GRE3D3 dataset.} \n\\label{fig:gre3d} \n\\end{figure}\n\n\\paragraph{Models}\n\nWe use two baselines for comparison: \n\n\\begin{itemize}[leftmargin=0pt]\n\\item[] \\emph{Basic}: The state-of-the-art model on this task with this dataset, \nunder the assumption that users are seen in training, is\npresented in \\newcite{ferreira14:_refer}. They define context features such as\ntype of relation between the target object and its landmark, number of object of\nthe same color or size, etc., then train an SVM classifier to predict the use of each\nattribute. We recast their model in terms of a log-linear model with the same features,\nto make it fit with the setup of Section~\\ref{sec:basicModel}.\n\n\\item[] \\emph{Basic++}: \\newcite{ferreira14:_refer} also take speaker features into account.\nWe do not use speaker identity and the speaker's attribute frequency vector, \nbecause we only evaluate on unseen users. \nWe do use their other speaker features (age, gender), \ntogether with \\emph{Basic}'s context features; \nthis gives us a strong baseline which is aware of manually\nannotated user group characteristics.\n\\end{itemize}\n\nWe compare these baselines to our \\emph{Group}  model \nfor values of $K$ between 1 and 10, using the exact same features as \n\\emph{Basic}. We do not use the speaker features of \\emph{Basic++}, \nbecause we do not want to rely on manually annotated groups. \nNote that our results are not directly comparable with those \nof \\newcite{ferreira14:_refer}, because of a different training-test split:\ntheir model requires having seen speakers in training, \nwhile we explicitly want to test our model's ability to generalize to unseen users.\n\n\\paragraph{Experimental setup} We evaluate using cross-validation, splitting the\nfolds so that all speakers we see in testing are previously unseen\nin training. We use 9 folds in order to have folds of the same size (each\ncontaining 70 descriptions coming from 7 speakers). At each iteration we train\non 8 folds and test on the 9th. At test time, we process each test instance\niteratively: we first predict for each instance whether the user $u$ would use a\nspatial relation or not and test our prediction; we then add the actual\nobservation from the corpus to the set $D\\up{u}$ of observations for this particular\nuser, in order to update our estimate about their group membership.\n\n\\paragraph{Results}\n\nFigure~\\ref{fig:gre3d3-f1} shows the test F1-score (micro-averaged over all\nfolds) as we increase the number of groups, compared to the baselines. \nFor our \\emph{Group} models, these are averaged over all interactions with the user.\nOur model gets F1-scores between $0.69$ and $0.76$ for all values of $K>1$, \noutperforming both \\emph{Basic} ($0.22$) and \\emph{Basic++} ($0.23$).\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=1.0\\columnwidth]{gre3d3-f1}\n\\vspace{-1cm}\n\\caption{F1 scores on test data for values of $K$ between $1$ and $10$ in the production experiment.}\n\\label{fig:gre3d3-f1} \\end{figure}\n\n\nIn order to take a closer look at our model's behavior, we also show the\naccuracy of our model as it observes more instances at test time. We compare the\nmodel with $K=3$ groups against the two baselines.\nFigure~\\ref{fig:gre3d3-time} shows that the group model's F1-score increases dramatically after\nthe first two observations and then stays high throughout the test phase, \nalways outperforming both baselines by at least 0.37 F1-score points after the \nfirst observation. The baseline models of course are not expected to improve \nwith time; fluctuations are due to differences between the visual scenes.\nIn the same figure, we plot the evolution of the entropy of the group model's posterior\ndistribution over the groups (see (\\ref{eq:posterior})). As expected, the model is \nhighly uncertain at the beginning of the test phase about which group the user \nbelongs to, then gets more and more certain as the set $D\\up{u}$ of observations  \nfrom that user grows.   \n\n\n\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=1.0\\columnwidth]{gre3d3-time} \n\\vspace{-1cm}\n\\caption{F1-score evolution with increasing number of observations from the user in the production experiment.} \n\\label{fig:gre3d3-time} \n\\end{figure}\n\n\n\\subsection{RE comprehension}\\label{sub:comprehension}\n \n\\paragraph{Task}\nOur next task is to predict the referent to which a user will resolve an RE in\nthe context of a visual scene. Our model is given a stimulus $s=(r,\nc)$ consisting of an instruction containing an RE $r$ and a visual context\n$c$ and outputs a probability distribution over all possible referents $b$. Such\na model can be used by a probabilistic RE generator to select an RE which is\nhighly likely to be correctly understood by the user or predict potential\nmisunderstandings (see Section~\\ref{sec:basicModel}). \n\n\\paragraph{Data}\nWe use the GIVE-2.5 corpus for training and the GIVE-2 corpus for \ntesting our model (the same used by \\newcite{EngonopoulosVTK13}). These contain recorded\nobservations of dialog systems giving instructions to users who play a game in a\n3D environment. Each instruction contains an RE $r$, which is recorded in the data\ntogether with the visual context $c$ at the time the instruction was given. The object\n$b$ which the user understood as the referent of the RE is inferred by the immediately subsequent\naction of the user. In total, we extracted 2927 observations by 403 users from GIVE-25 and \n5074 observations by 563 users from GIVE-2. \n\n\\paragraph{Experimental setup} \nWe follow the training method described in\nSection~\\ref{sec:basicModel}. \nAt test time, we present the observations from each user in the order they occur \nin the test data; for each stimulus, we ask our models to predict the referent $a$ which the user \nunderstood to be the referent of the RE, and compare with the recorded observation.\nWe subsequently add the recorded observation to the dataset for the user and continue.\n\n\\paragraph{Models}\n\nAs a baseline, we use the \\emph{Basic} model described in Section\n\\ref{sec:basicModel}, with the features of the ``semantic'' model \nof \\newcite{EngonopoulosVTK13}. Those features capture\ninformation about the objects in the visual scene (e.g. salience) \nand some basic semantic properties of the RE (e.g. color, position). \nWe use those features for our \\emph{Group} model as well, and evaluate for \n$K$ between 1 and 10. \n\n\n\\paragraph{Results on GIVE data}\n\n\\emph{Basic} had a test accuracy of 72.70\\%, which was almost identical\nwith the accuracy of our best \\emph{Group} model for $K=6$ (72.78\\%).\nThis indicates that our group model does not differentiate between users. \nIndeed, after training, the 6-group model assigns\n$81\\%$ prior probability to one of the groups, and effectively gets stuck with this\nassignment while testing; the mean entropy of the posterior group distribution only falls from \nan initial 1.1 to 0.7 after 10 observations.\n\nWe speculate that the reason behind this is that the\nfeatures we use are not sensitive enough to capture the differences between the users in\nthis data. Since our model relies completely on observable behavior, \nit also relies on the ability of the features to make relevant distinctions between users.\n\n\\paragraph{Results on synthetic data}\n\nIn order to test this hypothesis, we made a synthetic dataset based on the GIVE datasets \nwith 1000 instances from 100 users, in the following way:\nfor each user, we randomly selected 10 scenes from GIVE-2, and replaced the target\nthe user selected, so that half of the users always select the target \nwith the highest visual salience, and the other half always select the one with the lowest. \nOur aim was to test whether our model is capable of identifying groups when they are\nclearly present in the data and exhibit differences which our\nfeatures are able to capture.\n\nWe evaluated the same models in a 2-fold cross-validation. \nFigure~\\ref{fig:synth-acc} shows\nthe prediction accuracy for \\emph{Basic} and the \\emph{Group} models for $K$ from 1 to 10. \nAll models for $K>1$ clearly outperform the baseline model: the 2-group model gets\n$62.3\\%$ vs $28.6\\%$ averaged over\nall test examples, while adding more \nthan two groups does not further improve the accuracy. \nWe also show in Figure~\\ref{fig:synth-time} the evolution of the accuracy as \n$D\\up{u}$ grows: the \\emph{Group} model with $K=2$ reaches a 64\\% testing accuracy \nafter seeing two observations from the same user. In the same figure,\nthe entropy of the posterior distribution over groups (see production experiment)\nfalls towards zero as $D\\up{u}$ grows.\nThese results show that our model is capable of correctly assigning a user to the group \nthey belong to, once the features are adequate for distinguishing between different \nuser behaviors.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\columnwidth]{synth-acc} \n\\vspace{-1cm}\n\\caption{Prediction accuracies in the comprehension experiment with synthetic data.} \n\\label{fig:synth-acc} \n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\columnwidth]{synth-time} \n\\vspace{-1cm}\n\\caption{Accuracy evolution with increasing number of observations from the user in the comprehension experiment with synthetic data.} \n\\label{fig:synth-time} \n\\end{figure}\n\n\n\n\\subsection{Discussion}\n\n\nOur model was shown to be successful in discovering groups \nof users with respect to their behavior, within datasets which present\ndiscernible user variation. In particular, if all listeners are influenced in a similar way\n by e.g. the visual salience of an object, then the group model cannot \nlearn different weights for the visual salience feature; if this happens \nfor all available features, there are effectively no groups for our model to discover.\n\nOnce the groups have been discovered, \nour model can then very quickly distinguish between them at test time. \nThis is reflected in the steep performance improvement even after the first\nuser observation in both the real data experiment in~\\ref{sub:production}\nand the synthetic data experiment in~\\ref{sub:comprehension}.\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction} \\label{sec:introduction}\n\nPeople vary widely both in their linguistic preferences when producing\nlanguage and in their ability to understand specific\nnatural-language expressions, depending on what they know about the\ndomain, their age and cognitive capacity, and many other factors. It\nhas long been recognized that effective NLG systems should \ntherefore \\emph{adapt} to the current user, in\norder to generate language which works well for them. \nThis adaptation needs to address all levels of the NLG pipeline,\nincluding discourse planning \\cite{paris88:_tailor}, sentence planning\n\\cite{walker07:_indiv_domain_adapt_senten_plann_dialog}, and RE generation\n\\cite{janarthanam14:_adapt_gener_dialog_system_using}, and depends on\nmany features of the user, including level of expertise and language\nproficiency, age, and gender.\n\nExisting techniques for adapting the output of an NLG system have shortcomings \nwhich limit their practical usefulness. Some systems need user-specific information \nin training \\cite{ferreira14:_refer} and therefore cannot generalize to unseen \nusers. Other systems assume that each user in the training data \nis annotated with their group, which allows them to learn a model \nfrom the data of each group.\nHowever, hand-designed user groups may not reflect the true variability \nof the data, and may therefore inhibit the system's ability to flexibly adapt to new users.\n\nIn this paper, we present a user adaptation model for NLG systems\nwhich induces user groups from training data in which these groups\nwere not annotated. At training time, we probabilistically assign users to groups and learn the language preferences for each group. At evaluation time, we assume that \nour system has a chance to interact with each new user repeatedly \n-- e.g., in the context of a dialogue system. It will then calculate \nan increasingly accurate estimate of the user's group membership\n based on observable behavior, and use it to generate \nutterances that are suitable to the user's true group.\n\nWe evaluate our model on two tasks involving the generation of referring \nexpressions (RE). First, we predict the use of spatial relations in\n humanlike REs in the GRE3D domain~\\cite{viethen2010speaker}\n using a log-linear production model in the spirit of \\newcite{ferreira14:_refer}. \nSecond, we predict the comprehension of generated REs, in a synthetic dataset\nbased on data from the GIVE Challenge domain \\cite{StrDenGarGarKolThe11} \nwith the log-linear comprehension model of \\newcite{EngonopoulosVTK13}.\nIn both cases, we show that our model discovers user groups in the training \ndata and infers the group of unseen users with high confidence after only a few \ninteractions during testing. In the GRE3D domain, our system outperformed a strong baseline \nwhich used demographic information for the users. \n\n\n\n\n\n\n\n\\section{Related Work} \\label{sec:related}\n\nDifferences between individual users have a\nsubstantial impact on language comprehension. Factors that play a role\ninclude level of expertise and spatial ability \\cite{benyon1993developing}; \nage \\cite{h\u00e4user17:_age}; gender \\cite{navi-12}; or\nlanguage proficiency \\cite{KolStrGarByrCasDalMooObe10}.\n\nIndividual differences are also reflected in the way people \nproduce language. \\newcite{viethen2008use} present a corpus study \nof human-produced REs (GRE3D3) for simple visual scenes, where they \nnote two clearly distinguishable groups of speakers, one that always uses\na spatial relation and one that never does.\n\\newcite{ferreira14:_refer} show that \na model using speaker-specific information\noutperforms a generic model in predicting the attributes used by a speaker\nwhen producing an RE. However, \ntheir system needs to have seen the particular speaker in training,\nwhile our system can dynamically adapt to unseen users.\n\\newcite{ferreira2017improving} also demonstrate\nthat splitting speakers in predefined groups and training each group separately\nimproves the human likeness of REs compared to training individual user models.\n\n\nThe ability to adapt to the comprehension and production preferences of a user\nis especially important in the context of a dialog system, where \nthere are multiple chances of interacting with the same user. \nSome methods adapt to dialog system users \nby explicitly modeling the users' knowledge state.\nAn early example is \\newcite{paris88:_tailor}; she selects a discourse plan for\na user, depending on their level of domain knowledge ranging between novice\nand expert, but provides no mechanism for inferring the group to\nwhich the user belongs. \\newcite{rosenblum93:_partic_instr_dialog} try to\ninfer what knowledge a user possesses during dialogue, based on the\nquestions they ask. \\newcite{janarthanam14:_adapt_gener_dialog_system_using}\n adapt to unseen users by using reinforcement learning with simulated users\nto make a system able to adjust to the level of the user's knowledge.  \nThey use five predefined groups from which they generate the simulated users' behavior, \nbut do not assign real users to these groups. \nOur system makes no assumptions about the user's knowledge and does not need to \ntrain with simulated users, or use any kind of information-seeking moves;\nwe instead rely on the groups that\nare discovered in training and dynamically assign new, unseen users,\nbased only on their observable behavior in the dialog. \n\n\nAnother example of a user-adapting dialog component is SPaRKy\n \\cite{walker07:_indiv_domain_adapt_senten_plann_dialog},\na trainable sentence planner that can tailor sentence plans to\nindividual users' preferences. This requires training on separate data\n for each user; in contrast to this, we leverage the similarities between users \nand can take advantage of the full training data.\n\n\n\n\n\n\n\n\n\n\\section{Task} \\label{sec:task}\n\n\n\\section{Training} \\label{sec::training}\n\nSo far we have not discussed how to find settings for the parameters $\\theta = \\ensuremath{\\pi},\\parAdapt{1},\\dots,\\parAdapt{K}$, which define our probability model. The key challenge for training is the fact that we want to be able to train while treating the assignment of users to groups as unobserved.\n\nWe will use a maximum \\emph{a posteriori} estimate for $\\theta$, i.e., the setting which maximizes \\eqref{eqn::probability} when $D$ is our training set. We will first discuss how to pick parameters to maximize only the left part of \\eqref{eqn::probability}, i.e., the data likelihood, since this is the part that involves unobserved variables. We will then discuss handling the parameter prior in section \\ref{sec::prior}.\n\n\\subsection{Expectation Maximization}\n\nGradient descent based methods \\cite{NocedalW06} exist for finding the parameter settings which maximize the likelihood for log-linear models, under the conditions that all relevant variables are observed in the training data. If group assignments were given, gradient computations, and therefore gradient based maximization, would be straightforward for our model. One algorithm specifically designed to solve maximization problems with unknown variables by reducing them to the case where all variables are observed, is the expectation maximization (EM) algorithm \\cite{NealH99}. Instead of maximizing the data likelihood from \\eqref{eqn::probability} directly, EM equivalently maximizes the log-likelihood, given in \\eqref{eqn::loglike}. It helps us deal with unobserved variables by introducing ``pseudo-observations'' based on the expected frequency of the unobserved variables.\n\n\nEM is an iterative algorithm which produces a sequence of parameter settings $\\theta\\down{1},\\dots,\\theta\\down{n}$. Each will achieve a larger value for (\\ref{eqn::loglike}). Each new setting is generated in two steps: (1) an lower bound on the log-likelhood is generate and (2) the new parameter setting is found by optimizing this lower bound. To find the lower bound we compute the probability for every possible value the unobserved variables could have had, based on the observed variables and the parameter setting $\\theta\\down{i-1}$ from the last iteration step.  Then the lower bound  essentially assumes that each assignment was seen with a frequency equal to these probabilities - these are the ``pseudo-observations''.\n\n\nIn our model the unobserved variables are the assignments of users to groups. The probability of seeing each user $u$ assigned to a group, given all the data $D\\up{u}$ and the model parameters from the last iteration $\\theta\\down{i-1}$, is simply the posterior group membership probability $\\condProb{g}{D\\up{u};\\theta\\down{i-1}}$. The lower bound is then given by \\eqref{eqn::grad}. This is the sum of the log probabilities of the data points under each group model, weighted by $\\condProb{g}{D\\up{u};\\theta\\down{i-1}}$. We can now use gradient descent techniques to optimize this lower bound.\n\n\\subsubsection{Maximizing the Lower Bound}\n\nTo fully implement EM we need a way to maximize \\eqref{eqn::grad}. This can be achieved with gradient based methods such as L-BFGS \\cite{NocedalW06}. Here the gradient refers to the vector of all partial derivatives of the function with respect to each dimension of $\\theta$. We therefore need to calculate these partial derivatives.\n\nThere are existing implementations of the gradient computations our base model such as in \\newcite{EngonopoulosVTK13}. The gradients of \\eqref{eqn::grad} for each of the $\\parAdapt{g}$ is simply the gradient for the base model on each datapoint $d$ weighted by $\\condProb{g}{D\\up{u};\\theta\\down{i-1}}$ if $d \\in D_u$, i.e., the probability  that the user $u$ from which the datapoint originates belongs to group $g$. We can therefore compute the gradients needed for each $\\parAdapt{g}$ by using implementations developed for the base model.\n\nWe also need gradients for the parameters in $\\ensuremath{\\pi}$, which are only used in our extended model. We can use the rules for computing derivatives to find, for each dimension $g$:\n\n\\begin{equation*}\n\\grad{\\mathcal{UL}(\\theta)}{\\ensuremath{\\pi}_g} = \\displaystyle \\sum_{u \\in U} P_u(g) - \\frac{\\exp\\left({\\ensuremath{\\pi}}_g\\right)}{\\sum_{g' = 1}^K \\exp\\left({\\ensuremath{\\pi}}_{g'}\\right)}\n\\end{equation*}\n\nwhere $P_u(g) = \\condProb{g}{D\\up{u};\\theta\\down{i-1}}$. Using these gradients we can use L-BFGS to maximize the lower bound and implement the EM iteration.\n\n\\subsection{Handling the Parameter Prior} \\label{sec::prior}\n\nSo far we have discussed maximization only for the likelihood without accounting for the prior probabilities for every parameter. To obtain our full training objective we add the log of the right hand side of \\eqref{eqn::probability}:\n\n\\begin{equation}\n\\log \\left(\\mathcal{N}\\left(\\ensuremath{\\pi}|0,\\sigma\\up{\\ensuremath{\\pi}}\\right) \\cdot \\prod_{g=1}^K\\mathcal{N}\\left(\\parAdapt{g}|0,\\sigma\\up{\\rho}\\right) \\right) \\nonumber\n\\end{equation}\n\ni.e., the parameter prior, to \\eqref{eqn::loglike} and \\eqref{eqn::grad}. The gradient contribution from these priors can be computed with standard techniques.\n\n\\subsection{Training Iteration}\n\nWe can now implement an EM loop, which maximizes \\eqref{eqn::probability} as follows: we randomly pick an initial value $\\theta\\down{0}$ for all parameters. Then we repeatedly compute the $\\condProb{g}{D\\up{u};\\theta\\down{i-1}}$ values and maximize the lower bound using L-BFGS to find $\\theta\\down{i}$. This EM iteration is guaranteed to eventually converge towards a local optimum of our objective function. Once change in the objective falls below a pre-defined threshold, we keep the final $\\theta$ setting.\n\nFor our implementation we make a small improvement to the approach: L-BFGS is itself an iterative algorithm and instead of running it until convergence every time we need to find a new $\\theta\\down{i}$, we only let it take a few steps. Even if we just took a single L-BFGS step in each iteration, we would still obtain a correct algorithm \\cite{NealH99} and this has the advantage that we do not spend time trying to find a $\\theta\\down{i}$ which is a good fit for the likely poor group assignments $\\condProb{g}{D\\up{u};\\theta\\down{i-1}}$ we obtain from early parameter estimates.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Conflict of interest}\nThe authors declare that they have no conflict of interest.\n\n\n\\bibliographystyle{spbasic_FS}     \n\n\n\\section{Introduction}\\label{intro}\nOur Sun is an active star and as such undergoes cyclic variations, which are related to more or less frequently occurring activity phenomena observed at the solar surface. High energetic activity phenomena, produced due to changes in the Sun's magnetic field, propagate through our solar system where they interact with the planet's atmospheres. At Earth, these interactions are well documented and known to cause geomagnetic disturbances having consequences for modern society. The influence by the Sun on our solar system is termed \\textit{Space Weather}. Therefore, solar activity needs to be permanently monitored from space and ground in order to assess times of increased influence. International space agencies created programs, such as ESA \\textit{Space Situational Awareness} (SSA) or NASA \\textit{Living with a star} (LWS) (cf. Figure~\\ref{fig:esa}), to enhance Space Weather awareness and with that support and fund on a long-term basis fundamental research and development of Space Weather forecasting tools. \n\nThis review article focuses on the following Space Weather phenomena: \n\\begin{enumerate}\n    \\item Coronal mass ejections\n    \\item Flares\n    \\item Solar Energetic Particles  \n    \\item Solar wind stream interaction regions  \n\\end{enumerate}\n\nTo properly describe these phenomena from the solar perspective, a number of processes need to be understood, such as active region and magnetic field evolution, energy build-up and release, as well as the global structuring of inner heliospheric space. Space Weather is a topic of broad interest and sustains an exciting and wealthy interdisciplinary research community\\footnote{For example, the SCOSTEP effort that resulted in excellent publications via CAWSES \\url{http:\/\/www.terrapub.co.jp\/onlineproceedings\/ste\/CAWSES2007\/index.html}, the VarSITI programs \\cite[e.g., ISEST][see \\url{http:\/\/www.varsiti.org}]{zhang18} or the international Space Weather Action Teams, iSWAT, where interdisciplinary groups gather together under \\url{https:\/\/www.iswat-cospar.org}.}. With that it fosters information and knowledge exchange between international research groups on solar-, heliospheric- and geo-space (Sun-to-impact disciplines) in order to enhance scientific knowledge for improving existing and developing new models for Space Weather forecasting. \n\n\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{intro1.pdf}\n\\caption{Solar activity phenomena (depicted here as CME) affect Earth and near-Earth space and therefore, need to be permanently monitored. Space Weather forecasting is of global interest and funded by international agencies. In the near future, satellites will observe the Sun and its dynamic phenomena from different viewpoints, such as a combined L1 and L5 position. Image courtesy: ESA}\n\\label{fig:esa}      \n\\end{figure}\n\n\nCoronal mass ejections (CMEs) are a rather recent phenomenon, discovered just about 50 years ago, but in the meantime are known as the main drivers of the most severe Space Weather disturbances \\citep[see e.g.,][]{howard06,Gopalswamy16_CME_history}. They are huge structures that manifest themselves within some tens of minutes as clouds of magnetized plasma impulsively expelled from the Sun and subsequently propagating into interplanetary space \\citep[see e.g.,][]{forbes00}. CMEs arise from usually complex and closed magnetic field structures in equilibrium that is disrupted due to some instability causing its eruption \\cite[e.g., emerging magnetic flux, remote reconfiguration of large scale magnetic field, or field rotation; see e.g.,][]{torok13,schmieder15,green18}. Instabilities in the solar magnetic field and their occurrence frequency are modulated by the 11-year activity cycle of the Sun. The most strong CME events may propagate the 1AU distance within a day \\citep[e.g.,][]{cliver90,gopalswamy05_extreme,liu14_nature}. Less strong events, on average, propagate the same distance in up to 4 days \\citep[see e.g.,][]{shanmugaraju14}. CMEs may be linked to large geomagnetic disturbances, due to shock compression and reconnection with the Earth's magnetic field. They may lead to ionospheric and geomagnetically-induced currents \\citep[see e.g.,][]{pirjola05}.  Usually the most severe geomagnetic storms are caused by fast and massive CMEs, erupting from the central region of the visible solar disk and carrying a strong southward magnetic field component that reconnects with the Earth's magnetic field \\citep[see e.g.,][]{pulkkinen07}. Consequently, CMEs are a major topic of solar and Space Weather research. \n\n\nThe power for making a CME energetic (i.e., being fast and wide) undoubtedly stems from the free magnetic energy which is released as consequence of magnetic reconnection processes. Magnetic reconnection enables to impulsively drive plasma and to accelerate particles to high energies causing on the one hand flare emission, which is observed in the solar atmosphere, and on the other hand solar energetic particles (SEPs), which are measured in interplanetary space. Energetic particles from strong SEP events may reach almost speed of light and travel the 1AU distance within about 10 minutes. High energy SEP events (about 1 GeV) may lead to enhanced proton fluxes even at ground level. Hence, most intense events can endanger life and technology on Earth and in space. Further consequences of CMEs and SEPs are disruptions of satellite operations, radio communications and ground power systems \\citep[e.g.,][]{bothmer07}. Unlike CMEs, having lead times of some tens of hours between first observational signatures and impact at Earth, flares and SEP events occur and impact almost simultaneously \\citep[see e.g.,][]{lugaz17,cairns18,Malandraki18_book}. Accordingly, to predict the occurrence of flares and SEPs one needs to predict the instabilities leading to the onset of magnetic reconnection processes, one of the big challenges in solar physics.  \n\n\n\nThe continuous solar wind flow in a quiet state (usually termed background solar wind) is represented by an alternation of slow and fast solar wind streams that interact and form stream interaction regions (SIRs). If steady in their existence and persisting over more than one solar rotation, they are called co-rotating interaction regions (CIRs). During times of low solar activity, Space Weather is dominated by CIR induced storms \\citep{tsurutani06}. Different flow speeds of the background solar wind also change the propagation behavior of CMEs in interplanetary space. This has consequences on the CME transit time and impact speed at planetary atmospheres \\citep[drag force; see][]{gopalswamy00_accelCME,vrsnak01_solph,Cargill2004OnEjections,vrsnak06}. Moreover, CMEs disrupt the continuous outflow of the solar wind and reconfigure the magnetic field on large spatial and short temporal scales altering the background solar wind. For Space Weather and CME modeling\/forecasting purposes, these ever changing conditions in interplanetary space are very challenging to tackle.  \n\n \\begin{figure}\n  \\includegraphics[width=\\textwidth]{intro2.pdf}\n\\caption{Current and past space missions carrying instruments for gathering remote sensing image data and in-situ plasma and magnetic field measurements. The majority of spacecraft is located in the ecliptic plane orbiting planets or at the Lagrangian point L1. The coronagraph field of view of the SoHO\/LASCO instrument C3 covers 30 solar radii. The background white-light image is taken from STEREO\/HI1+2 data covering about 90 degrees in the ecliptic. Not to scale.\n}\n\\label{fig:insitu}      \n\\end{figure}\n\nFor comprehensive investigations a rich source of observational data is currently available from many different instruments located at multiple viewpoints and different radial distances (see Figure~\\ref{fig:insitu}). In Earth orbit current operational missions are e.g., GOES (Geostationary Operational Environmental Satellite), SDO \\citep[SDO:][]{Pesnell2012TheSDO}, Proba-2 \\citep{santandrea13}, located at L1 - 1.5 million km upstream of Earth - there is the Solar and Heliospheric Observatory \\citep[SoHO:][]{Domingo95_SOHO}, the Advanced Composition Explorer \\cite[ACE:][]{Stone98_ACE}, the WIND spacecraft \\cite{ogilvie95}, and DSCOVR \\cite[][]{burt12}. At $\\sim$1AU with variable longitudinal angles from Earth, there is the Solar TErrestrial RElations Observatory \\cite[STEREO:][]{kaiser08_STEREO} consisting of two identical spacecraft named STEREO-Ahead and STEREO-Behind (lost signal end of 2014). The combination of remote sensing image data and in-situ measurements is found to be optimal for enhancing our knowledge about the physics of Space Weather phenomena. For better understanding large eruptive activity phenomena, multi-viewpoint and multi-wavelength data are exploited (e.g., combined L1, STEREO as well as ground-based instruments). The various available data from spacecraft orbiting around planets (e.g., VEX (2006--2014), MESSENGER (2011--2015), MAVEN (2014--), BepiColombo (2018--)) also enable to analyze the evolution of Space Weather phenomena as function of distance and longitude. \n\n\n \n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{intro3.pdf}\n\\caption{Each image shown here is a snapshot of the Sun taken every spring with the SOHO Extreme ultraviolet Imaging Telescope (EIT) in the 284\\AA\\ wavelength range. It shows the variations of the solar activity in terms of increasing and decreasing number of bright active regions visible in the corona. Courtesy: NASA\/ESA}\n\\label{fig:soho22}      \n\\end{figure}\n\n    \n    \nA flagship of international collaboration and boost for Space Weather research, is SoHO which now achieved 25 years in space. Figure~\\ref{fig:soho22} shows SOHO\/EIT \\citep{delaboudiniere95} EUV image data covering the variations of the solar corona over a full magnetic solar cycle (Hale cycle). Long-term observations are of utmost importance for monitoring and learning about the interaction processes of solar activity phenomena with Earth and other planets as well as for improving our capabilities in Space Weather forecasting. Most recent and unprecedented missions are Parker Solar Probe, launched in August 2018 \\citep{fox16}, and Solar Orbiter, launched in February 2020 \\citep{mueller20} having on-board imaging and in-situ facilities with the goal to approach the Sun as close as never before ($\\sim$0.05AU and $\\sim$0.3AU) and investigating the Sun out of the ecliptic ($\\sim$30 degrees). To support space missions and for providing valuable complementary data, we must not forget the importance of ground based observatories that observe the Sun over broad wavelength and energy ranges allied in international networks such as the Global high-resolution H$\\alpha$ network\\footnote{\\url{http:\/\/bbso.njit.edu\/Research\/Halpha\/}}, the Global Oscillation Network Group\\footnote{\\url{https:\/\/gong.nso.edu}},  the database for high-resolution Neutron Monitor measurements\\footnote{E.g., \\url{http:\/\/www01.nmdb.eu\/nest\/search.php}}, muon telescope networks, or the Worldwide Interplanetary Scintillation Stations Network \\footnote{\\url{http:\/\/helios.mexart.unam.mx\/pruebas\/wipss\/index.html}}. \n\n\nFrom the derived research results based on the observational data, over the past years a plethora of models could be developed for predicting Space Weather and their geomagnetic effects. The permanent monitoring of the Sun and provision of data in almost real-time enabled to apply those results and even to install operational services that produce forecasts mostly in an automatic manner (e.g., facilitated by ESA\/SSA\\footnote{\\url{http:\/\/swe.ssa.esa.int}}; NASA\/CCMC\\footnote{\\url{https:\/\/ccmc.gsfc.nasa.gov}}; NOAA\/SWPC\\footnote{\\url{https:\/\/www.swpc.noaa.gov}}). However, the operational services also clearly demonstrated the limitations in the forecasting accuracy as on average the errors are large and get worse with increasing solar activity. This is mainly due to the large uncertainties coming from the model input, namely observational parameters at or close to the Sun. It also reveals the complexity of the interplay between the different driving agents of Space Weather, that makes it difficult to fully capture the physics behind and to improve models. Reliable Space Weather forecasting is still in its infancy.  \n\n\n\\section{Space Weather}\n\nFrom the historical perspective, the so-called ``Carrington-event'' from September 1, 1859 is the reference event for referring to extreme Space Weather and with that the beginning of Space Weather research \\citep[see also][]{Schwenn06}. At that time only optical observations of the solar surface were performed and the observed emitted radiation in white-light for that event showed impressively the vast amount of energy that was distributed to the dense lower atmospheric layers of the Sun where it heated the photosphere. At Earth, the associated geomagnetic effects were observed in terms of aurora occurring from high to low latitudes (e.g., Honolulu at 20 degrees northern latitude) and ground-induced currents in telegraph wires \\citep[see][]{Eastwood17}. The associated SEP event is thought to be about twice as large as the huge SEP events from July 1959, November 1960, or August 1972 \\citep{cliver13}. Only several years after the Carrington event, the usage of spectroscopes enabled to regularly observe prominence eruptions revealing the dynamic changes of the solar corona and material ejections with speeds exceeding hundreds of km\/s \\citep{tandberg95}. The continuous monitoring of the Sun was intensified in the 1940's, when solar observations in radio, white-light and in the H$\\alpha$ wavelength range were performed. At that time also galactic cosmic rays were studied and found that they are anti-correlated with solar activity \\citep[so-called Forbush decrease, measured as sudden drop in the cosmic ray flux due to interplanetary disturbances; see also][]{cane00}. In the early 1960's magnetic structures driving shocks were inferred from observations in the metric radio observations and geomagnetic storm sudden commencements \\citep{gold62,fokker63}. The \\textit{transient events with mass moving through the solar corona and actually leaving the Sun}, i.e., CMEs, that were associated with the prominence\/filament eruptions were discovered only in the early 1970's with the advent of the space era \\citep[see][]{Tousey71,macqueen74}. Recent reviews on the history of prominences and their role in Space Weather can be found in \\cite[][and references therein]{vial15,Gopalswamy16_CME_history}. While most of the extreme space weather events happen during the solar cycle maximum phase, occasionally strong geoeffective events may occur close to the solar cycle minima and also during weak solar activity, provided there are appropriate source regions on the Sun \\citep[see also e.g.,][]{Vennerstrom16,hayakawa20}. For more details about the solar cycle see the Living Reviews by \\cite{hathaway10}.\n\n\nNowadays, a wealth of space and ground-based instruments are available, delivering valuable observational data, as well as modeling facilities. This enables to study in rich detail the manifold processes related to Space Weather events and to better understand the physics behind. To forecast the geomagnetic effects of an impacting disturbance at Earth (e.g., by the Dst\\footnote{The disturbance storm time (Dst) index monitors variations in the Earth's equatorial ring current.} or Kp index\\footnote{The planetary K index (Kp) monitors variations in the horizontal component of the Earth's magnetic field.}), the most common parameters we need to know in advance - and various combinations of these - are the amplitude\/orientation and variation of the north-south component of the interplanetary magnetic field ($B_{\\rm z}$), speed ($v$), and density ($n$). Especially, the electric field $vB_{\\rm s}$ ($B_{\\rm s}$ = $B_z<0$) is found to show a high correlation with the Dst storm index \\citep[see e.g.,][]{baker81,wu02,gopalswamy08}. For details on the geomagnetic effects of Space Weather phenomena as described here, see the Living Reviews by \\cite{pulkkinen07}.\n\nThe Space Weather ``chain of action'' from the solar perspective is described best by the recent example of the multiple Space Weather events that occurred in September 2017 (see Section~\\ref{sec:active}). But before that, we elaborate the physical basis. \n\n\n\\section{Magnetic reconnection: common ground}\nThe commonality that unites everything and yet produces such different dynamic phenomena is \\textit{magnetic reconnection} and the release of free magnetic energy. This leads to particle acceleration, heating, waves, etc. and to a restructuring of the (local) magnetic field in the corona by newly connecting different magnetic regimes and with that changing magnetic pressure gradients. Especially the latter shows to affect the solar corona globally.\n\n\n\\begin{figure*}\n  \\includegraphics[width=1.8\\textwidth,angle=90]{intro4.pdf}\n\\caption{Left: Stochastic acceleration model for solar flares. Magnetic field lines (green) and turbulent plasma or plasma waves (red circles) generated during magnetic reconnection. Blue arrows and areas mark accelerated particles impinging on the lower denser chromosphere where they produce Bremsstrahlung and on the upside may escape to interplanetary space where they are detected as SEPs \\citep[adapted from][]{Petrosian04,Vlahos19_SEP}. Middle: CME-flux rope configuration in the classical scenario (CSHKP) covering also the post eruptive arcade usually observed in SXR and EUV wavelength range \\citep[adapted from][]{Lin00}; Right: CME flux rope acting as driver of a bow shock (black arc) may accelerate SEPs (black dots) in the corona or heliosphere via diffusive shock acceleration \\citep[adapted from][]{Mikic06}. }\n\\label{fig:flare_CME_SEP}      \n\\end{figure*}\n\nIn order to derive a complete picture about Space Weather, we first need to understand the interrelation between these many individual processes starting at the Sun. This covers a cascade of small and large scale phenomena varying over different time scales. The primary source of Space Weather producing phenomena, i.e., CMEs-flares-SEPs (note that in the following eruptive phenomena are considered and not stealth CMEs), are active regions representing the centers of strong magnetic field and energy \\citep[more details on the evolution of active regions, see the Living Reviews by][]{van_driel15,toriumi19}. However, in detail the energy build-up and release processes are not well understood. The key-driver certainly is the magnetic field configuration below the visible surface (photosphere), that cannot be directly observed and characterized for giving reliable predictions of its status and further development. The lack of magnetic field information is also given in the upper atmospheric layers. There are currently no instruments enabling measurements of the magnetic field in the corona, hence, we need to rely on models simulating the coronal and, further out, interplanetary magnetic field \\citep[see, e.g., the Living Reviews by][on coronal and solar wind MHD modeling ]{gombosi18}. While active regions are characterized by closed magnetic field, coronal holes cover mainly open magnetic field from which high speed solar wind streams emerge. They structure interplanetary space and set the coupling processes between continuous solar wind flow and transient events. To better understand the propagation behavior of transient events, we also need to study the evolution and characteristics of the solar wind flow, and hence, the interplay between open and closed magnetic field. \n\n\nFigure~\\ref{fig:flare_CME_SEP} sketches three different time steps in the evolution of an eruptive flare event, causing a CME and SEPs, as a consequence of magnetic reconnection \\citep[see][]{Petrosian04,Lin00,Mikic06}. The left panel of Figure~\\ref{fig:flare_CME_SEP} focuses on the early evolution stage of the eruptive event, introducing stochastic acceleration processes causing high energetic particles to precipitate along magnetic field lines towards and away from the Sun. Flare emission is observed on the solar surface due to the acceleration of particles towards the Sun. Particles that escape into interplanetary space along the newly opened magnetic field, produce SEPs. The middle panel of Figure~\\ref{fig:flare_CME_SEP} shows the creation of the CME body, i.e., the production of a closed magnetic field structure (flux rope), as well as the related post-eruptive arcade which is formed below. The exact acceleration mechanism(s) of SEPs is still an open issue, hence, cartoons as shown here usually present both possible driving agents, the flare and the CME shock. To complete the picture for a flare-CME-SEP event, the right panel of Figure~\\ref{fig:flare_CME_SEP} depicts the interplanetary magnetic field and its behavior which differs from the typical Parker spiral orientation due to the propagating CME shock component causing SEP acceleration in interplanetary space. The deviation of the interplanetary magnetic field from the nominal Parker spiral is an important issue when dealing with magnetic connectivity for studying SEPs and propagation behavior of CMEs. \n\nIn the following, we will discuss in more detail the characteristics of the different manifestations occurring in an eruptive flare event.\\\\\n\\section{Solar Flares} \\label{sec:flares}\n\n\\subsection{Eruptive capability of an active region}\n\nActive regions may be classified either by the morphology of an active region using the McIntosh classification \\citep{mcintosh90} or the magnetic structure using Hale's\/K\\\"unzel's classification \\citep{kunzel60}. Due to the emergence of magnetic flux the degree of complexity in the magnetic field of an active region grows, which increases the likeliness to create strong flares and CMEs \\citep[e.g.,][]{sammis00,toriumi17}. The probability that an X-class flare is related to a CME is found to be larger than 80\\% \\citep{yashiro06}, however, there are well observed exceptions reported. So-called confined flares are neither accompanied by a CME nor a filament eruption \\citep[e.g.,][]{moore01}. Their special magnetic field configuration allows particle acceleration (observed as flare), but they do not escape into interplanetary space and, hence, do not produce SEPs \\citep{gopalswamy09}. Therefore, confined flares may produce strong X-ray emission but, presumably due to a strong bipolar overlying coronal magnetic field configuration, are not related to the opening of the large-scale magnetic field \\citep[e.g.,][]{wang07,sun15,Thalmann15}. The electromagnetic radiation of confined flares can still instantaneously cause sudden changes in the ionospheric electron density profile (disturbing radio wave communication or navigation), also known as \\textit{solar flare effect} or geomagnetic crochets \\citep{campbell03} but occurring rather rarely. However, confined flares are also potential candidates for false Space Weather alerts in terms of an erroneous forecast of geomagnetic effects due to the magnetic ejecta that would have arrived tens of hours later at Earth. \n\n\nTherefore, the manifestation of the eruptive capability of an active region is one of the prime targets for prospective forecasting of SEPs and CMEs. For example, the length of the main polarity inversion line of an active region or the magnetic shear and its sigmoidal morphology, is obtained to be highly indicative of the potential to open large scale magnetic field and to produce CMEs and SEPs \\citep[e.g.,][]{Canfield99}. Studies also showed that active regions, for which the polarity inversion line quickly changes with height into a potential field configuration, are more favorable for producing non-eruptive events \\citep{Baumgartner18}. Likewise, the decay index of the horizontal magnetic field (ratio of the magnetic flux in the lower corona to that in the higher corona) is found to be lower for failed eruptions compared to that for full eruptions \\citep[cf.,][]{torok05,fan07,guo10,olmedo10}. \n\n\nFor more details on the issue of flare-productive active regions I refer to the Living reviews by \\cite{toriumi19}. See also \\cite{forbes00}, \\cite{webb12}, \\cite{parenti14}, or \\cite{chen17} for a more theoretical approach on that issue.\n\n\\subsection{Eruptive solar flares: general characteristics}\\label{sec:flare}\n\n\\begin{figure*}\n  \\includegraphics[width=0.6\\textwidth]{flare1.pdf}\n\\caption{Flare-CME-SEP relation in time. The onset of the solar flare is indicated by the vertical red line. The grey\nshaded area marks the time difference between flare start and SEP flux increase for MeV energies. Taken from \\cite{Anastasiadis19} who adapted it from \\cite{miro03}.}\n\\label{fig:timing}      \n\\end{figure*}\n\n\nFlares are observed to release a huge amount of energy from 10$^{19}$ up to 10$^{32}$ erg over a timescale of hours\\footnote{An automatically updated list of flares is available under \\url{http:\/\/www.lmsal.com\/solarsoft\/latest_events\/} or \\url{https:\/\/www.solarmonitor.org}}. With the advent of modern ground-based and space-borne instruments, our small optical window was massively enlarged and it is now well-known that this energy is radiated over the entire electromagnetic spectrum from decameter radio waves to gamma-rays beyond 1 GeV. Figure~\\ref{fig:timing} depicts the temporal relation between flare emission, observed in different wavelength ranges, CME kinematics and SEP flux profiles in the GeV and MeV energy range. The flare activity profile consists of a so-called pre-flare phase, showing thermal emission in SXR and EUV, as well as H$\\alpha$ kernel brightenings. If related to a filament eruption, this phase partly coincides with the slow rise phase of the filament\\footnote{Filament detection and eruption catalogues can be found e.g., under \\url{http:\/\/cesar.kso.ac.at\/sn_iv\/filaments.php} or \\url{http:\/\/aia.cfa.harvard.edu\/filament\/}}. This is followed by the impulsive flare phase during which most of the energy is released and non-thermal emission in terms of hard X-ray (HXR) footpoints appears due to particles accelerating out of the localized reconnection area and bombarding the denser chromosphere where they emit Bremsstrahlung \\citep[for a review on solar flare observations see e.g.,][]{fletcher11}. At this point also the CME body forms as consequence of the closing of the magnetic field lines in the upper part of the reconnection area revealing a flux rope structure (note that the most compelling argument for an already existing flux rope is actually a filament). As the flare emission increases also the SEP flux in the GeV energy range starts to rise. After the flare reaches a maximum in intensity, the decay phase is observed during which the intensity level goes back to the background level from before the flare start. The exact timing of the rise and decay phase is dependent on the energy release and the energy range in which the flare is observed which is known as the so-called Neupert effect \\citep[the HXR flux rise phase time profile corresponds to the derivative of the SXR flux time profile; see][]{neupert68}. The last phase may have a duration of several hours or longer. During that phase also post-eruptive arcades (or loops) start to form, that may still grow over 2--20 hours. The growth of the post-eruptive arcade is hinting towards an ongoing reconnecting process, which is not energetic enough to produce a significant emission in EUV or SXR \\citep[see e.g.,][]{tripathi04}. For more details on the global properties of solar flares I refer to the review by \\cite{hudson11}.\n\nFigure~\\ref{fig:moestl08} shows the temporal evolution of a flare and erupting filament observed in H$\\alpha$ and the associated CME observed in white-light coronagraph image data. The event is classified in the emitted SXR flux as GOES M3.9 flare (corresponding to the measured power of $3.6\\times10^{-5}$ W\/m$^{2}$) which occurred on November 18, 2003 in a magnetically complex $\\beta\\gamma$ active region. The associated CME caused two days later one of the strongest geomagnetic storms of solar cycle 23 having a minimum Dst value of $-$472~nT \\citep{gopalswamy05_nov}. Inspecting the time stamps on the image data of that event, about one hour after the appearance of the flare signatures, the CME became visible in the coronagraph. The filament started to rise some tens of minutes before the flare emission occurred.\n\n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{flare2.pdf}\n\\caption{Global flare evolution and relation to CME from the November 18, 2003 event. Left panels: H$\\alpha$ filtergrams from the Kanzelh{\\\"o}he Solar Observatory (Austria). The associated erupting filament is indicated by arrows. Right panels: Temporal evolution of the CME in coronagraph images from SOHO\/LASCO. Taken from \\cite{mostle08}.}\n\\label{fig:moestl08}      \n\\end{figure*}\n\nThe orientation of the magnetic structure, especially of the $B{\\rm z}$ component, of an ICME is key to forecast its geoeffectiveness and poses the Holy Grail of Space Weather research. Knowing the flux rope orientation already at the Sun could provide information on the impact of CMEs early in advance, hence, as soon as they erupt or even before. While the handedness of flux ropes can be well observed from in-situ measurements \\citep{Bothmer98,mulligan98}, on the Sun observational proxies need to be used. Figure~\\ref{fig:palmerio17} shows several surface signatures from which the magnetic helicity (sense of twist of the flux rope: right-handed or left-handed) can be inferred. Typically, in EUV observations these are sigmoidal structures (S- or reverse S-shaped) or post-eruptive arcades (skewness of EUV loops and polarity of the underlying magnetic field), in H$\\alpha$ the fine structures of filaments are used (orientations of barbs) or statistical relations like the \\textit{hemispheric helicity rule} \\cite[see][]{wang13}. However, strong coronal channeling, latitudinal and also longitudinal deflection and\/or rotation, that the magnetic component of the CME undergoes as it evolves through the low solar corona, may change those parameters as shown in various studies by e.g., \\citet{shen11,Gui11,bosman12,Panasenco13,WangY14,kay15,moestl15}, or \\citet{Heinemann19}. Recent approaches in ICME $B{\\rm z}$ forecasting can be found in, e.g., \\cite{savani15}, \\cite{palmerio17}, or \\cite{kay17}.\n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{flare3.pdf}\n\\caption{(a) SDO\/HMI data from June 14, 2012 showing the magnetic tongues of the erupting active region revealing a positive chirality. (b) Forward\u2010S sigmoidal structure from the coronal loops observed by SDO\/AIA 131\\AA, indicating a right\u2010handed flux rope. (c) SDO\/HMI magnetogram showing the approximated polarity inversion line (red line). (d) Base\u2010difference SDO\/AIA 131\\AA~image overlaid with the HMI magnetogram contours saturated at $\\pm$200~G (blue = negative polarity; red = positive polarity). The dimming regions indicating the flux rope footpoints are marked by green circles. Panels (a--d) are adapted from \\cite{palmerio18}. (e) The cartoon shows the handedness inferred from the magnetic field and sigmoidal structures or orientation of post eruptive loops.}\n\\label{fig:palmerio17}      \n\\end{figure*}\n\n\n\n\nFor more details about the energetics and dynamics of solar flares I refer to the Living Reviews by \\cite{benz17} and for the magnetohydrodynamic processes in active regions responsible for producing a flare to the Living Reviews by \\cite{Shibata11}. \n\n\n\n\\section{Coronal mass ejections (CMEs)}\\label{subsec:cme}\n\n\\subsection{General characteristics}\n \nCMEs are optically thin large-scale objects, that quickly expand, and are traditionally observed in white-light as enhanced intensity structures. The intensity increase is due to photospheric light that is Thomson scattered off the electrons forming the CME body and integrated over the line-of-sight \\citep{Hundhausen93}. Due to strong projection effects their apparent morphology greatly depends on the viewpoint and, hence, makes CMEs a rather tricky object to measure  \\citep[see e.g.,][]{burkepile04,cremades04}.\n\nBy using coronagraphs, CMEs are visible with teardrop-like shapes that are characterized by multiple structures. Figure~\\ref{fig:CME_struct} shows SOHO\/LASCO \\citep{Brueckner1995TheLASCO} coronagraph white-light images of two CMEs having different propagation directions. For the CME that leaves the Sun in a rather perpendicular angle to the observer (left panel of Figure~\\ref{fig:CME_struct}), the various CME structures are well visible. In general, we distinguish between the shock (yellow arrow) and CME body (green arrow) that are followed by some cavity created by the expanding magnetic flux rope (red arrow) and an increased brightness structure (orange arrow). Partly these structures are detected also from in-situ measurements for the interplanetary counterparts of CMEs (ICMEs; see Section~\\ref{sec:sub3-5}). The increased brightness structure consists of prominence material \\citep{Vourlidas13} or is suggested to appear due to a brightness increase of the two overlapping CME flanks \\citep{howard17}. The sheath region behind the shock has less clear signatures in coronagraph images taken close to the Sun as it is generated later when the solar wind plasma gets piled-up in interplanetary space \\citep[see e.g.,][]{kilpua17,salman20}. For CMEs propagating in the line-of-sight towards or away from the observer (right panel of Figure~\\ref{fig:CME_struct}), the different structures are less well visible. As these CMEs are launched close to the central meridian of the observed disk, they most severely suffer from projection effects. Energetic ones are frequently observed as so-called \\textit{halo} CMEs, revealing extensive white-light signatures made of compressed plasma material surrounding the occulting disk of a coronagraph. For halo CMEs, evidence that the CME is actually moving towards the observer is given from the associated activities observed on the solar disk (such as filament eruptions, flare emission, dimming regions, or coronal wave signatures). Highly relevant for Space Weather, halo CMEs are of special interest and are diversely studied mostly by using single spacecraft data from the coronagraphs aboard SoHO.\n\n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{cme1.pdf}\n\\caption{LASCO CME excess mass images showing the expanding shock wave front (yellow arrow) and the CME leading edge density enhancement (green arrow) for two different events. For the CME propagating rather in the plane of sky (left panel), typical structures such as the cavity due to the expanding magnetic ejecta} (red arrow) followed by some intensity enhancement (orange arrow) can be observed, that is less well visible for the halo CME (right panel). The projected LASCO CME speeds are given in the legend \\citep[adapted from][]{Vourlidas13}.\n\\label{fig:CME_struct}      \n\\end{figure*}\n\n\nUp to the distance of about 30 solar radii (LASCO\/C3 field of view), statistical studies showed that CMEs undergo several phases in their dynamics. Before the actual launch a slow rising phase occurs (initiation phase), continued by the acceleration phase over which a rapid increase in speed is observed in the inner corona, that is followed by a rather smooth propagation phase as the CME leaves the Sun \\citep{Zhang2006AEjections}. On average, over the coronagraphic field of view, CME fronts reveal radial speeds in the range of 300--500~km\/s with maximum values observed up to 3000~km\/s, accelerations of the order of 0.1--10~km\/s$^2$, angular widths of about 30--65 degrees and masses of $\\sim$10$^{14}$--10$^{16}$\\,g \\citep[e.g.,][]{vourlidas10,Lamy19}. The ratio in density between the CME body and surrounding solar wind decreases from $\\sim$11 at a distance of 15 solar radii to $\\sim$6 at 30 solar radii \\citep{temmer21}.  However, CMEs vary in their occurrence rate as well as in their characteristics over different solar cycles. While flare rates and their properties have not changed much over the past solar cycles, the CME properties for solar cycle 24 are significantly different as given in recent statistics \\citep{Lamy19,dagnew20Cycle}. CMEs were found to be more numerous and wide compared to solar cycle 23. Close to the Sun, the CME expansion is driven by the increased magnetic pressure inside the flux rope, while further out they most probably expand due to the decrease of the solar-wind dynamic pressure over distance \\citep{lugaz20}. Therefore, the increased width for CMEs of cycle 24 may be explained by the severe drop ($\\sim$50\\% ) in the total (magnetic and plasma) heliospheric pressure \\citep[see e.g.,][]{mccomas13,gopalswamy14,gopalswamy15,dagnew20}. Interestingly, also the maximum sunspot relative number in cycle 24 reached only 65\\% of that from cycle 23\\footnote{\\url{http:\/\/sidc.be\/silso\/cyclesminmax}}. The different expansion behaviors have consequences also for Space Weather effects in terms of their abilities in driving shocks \\citep[see e.g.,][]{lugaz17_radial}. \n  \n\n\n\n\n\\subsection{CME early evolution}\\label{sec:sub2-5}\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{cme2.pdf}\n\\caption{STEREO-B observations of the CME from August 24, 2014. The images show combined EUVI (304\\AA) and COR1 image data. Filament plasma material is ejected into space forming the bright CME core following the cavity. Plasma that is lacking sufficient kinetic energy to escape from the Sun's gravity, falls back onto the solar surface. Credit: STEREO\/NASA. The movie is online available.\n}\n\\label{fig:euv_cor1}      \n\\end{figure*}\n\n\nBesides the traditional observations in white-light images, also EUV or SXR imagery reveal CME signatures, presumably due to compression and heating that makes it visible in filtergrams sensible for high temperatures \\citep[see e.g.,][]{glesener13}. Satellite missions that carry EUV instruments having large field of views can be effectively used with combined white-light coronagraph data to track CME structures for deriving the kinematical profile over their early evolution covering the CME main acceleration phase. The SECCHI instrument suite \\citep{Howard08_secchi} aboard STEREO provides EUV and white-light data that seamlessly overlap\\footnote{SoHO EIT and C1 also provided that possibility but C1 was lost in June 1998 due to spacecraft failure. For a couple of events the usage of combined EIT-C1 data could be shown \\citep[see][]{Gopalswamy00_earlyCME,Zhang01,cliver04b}.}. as shown in Figure~\\ref{fig:euv_cor1}. For such studies one needs to keep in mind that the observational data image different physical quantities (density and temperature in EUV, and density in white-light), hence, dark and bright features in both image data do not necessarily match. \n\nFrom combined high temporal resolution EUV and white-light data a more detailed understanding about the energy budget (see also Section~\\ref{sec:budget}) and relation between flares, filaments and CMEs is revealed providing relevant information for SEP acceleration and generation of radio type II bursts. It is found that the thermal flare emission observed in SXR and the CME speed profile show similar behavior in timing \\citep{Zhang01,Zhang04,chen03,Maricic07}. For strong eruptive events an almost synchronized behavior between flare HXR emission and CME acceleration is obtained through a feedback relation \\citep{Temmer08,temmer10}. The CME acceleration is found to peak already as low as about 0.5 solar radii above the solar surface \\citep[for statistics see][]{Bein2011ImpulsiveCharacteristics}. Figure~\\ref{fig:temmer16} gives the schematic profiles and distances over time between non-thermal (HXR) and thermal (SXR) flare energy release and CME kinematics (acceleration, speed). The flare-CME feedback loop can be well explained by the CSHKP standard model \\citep{Carmichael64,Sturrock66,Hirayama74,Kopp76} through the magnetic reconnection process underlying both activity phenomena. In a simplistic scenario, we may summarize that magnetic reconnection drives particle acceleration (neglecting details on the actual acceleration process) leading to flare emission and closes magnetic field increasing the magnetic pressure inside the presumable CME flux rope (neglecting details on the actual magnetic configuration of the active region and surrounding). For strong flares that are related to CMEs of high acceleration values, the available free magnetic energy might be larger. This occurs preferably for CMEs initiated at lower heights where the magnetic field is stronger. With that, particles get accelerated to larger energies, hence, producing stronger flares, and more poloidal flux can be added per unit time, hence, generating a stronger expansion of the flux rope and a faster CME eruption. This is supported by theoretical investigations on the feedback process, covering magnetic reconnection with the ambient coronal magnetic field \\citep[reconnective instability][]{welsch18}. More details are found in, e.g.,  \\cite{chen03,Vrsnak2007AccelerationScales,vrsnak08,jang17}.  \n\n\nAssociated to the erupting CME, we frequently observe coronal dimming regions that evolve over a few tens of minutes \\citep{1996hudson,webb00}. Core dimming regions are assumed to be located at the anchoring footpoints of the associated magnetic flux rope and reveal the loss of plasma from the corona into the CME structure adding mass to the CME body \\citep[see][]{Temmer17_flar-cme}. Secondary dimming regions most probably refer to mass depletion in the wake of the large-scale magnetic field opening as the CME fully erupts \\citep[for more details on core and secondary dimming regions, see][]{mandrini07}. Recent studies discovered a strong relation between dimming intensity and flare reconnected flux as well CME speed \\citep[e.g.,][]{dissauer18,dissauer19}. Also the final width of the CME can be estimated from the amount of magnetic flux covered by the CME associated post-eruptive flare arcade as the surrounding magnetic field prevents the CME flux rope from further expansion \\citep{moore07}. On the contrary, the CME surrounding shock as well as associated coronal waves on the solar surface, that are ignited by the lateral CME expansion, are freely propagating and are not limited in their spatial extend \\citep[for more details on globally propagating coronal waves I refer to the Living Reviews by][]{Warmuth15}.\n\n\\begin{figure*}\n  \\includegraphics[width=1.6\\textwidth, angle=90]{cme3.pdf}\n\\caption{CME-flare relation. a) schematics of the thermal (SXR) and non-thermal (HXR) flare energy release in comparison to the CME kinematical evolution close to the Sun. It is found that CME acceleration and HXR emission as well as the CME speed and SXR emission, respectively, are closely related. Taken from \\cite{Temmer16}. b) observational results for the December 22, 2009 CME event revealing the early evolution from combined EUV and coronagraph data (STEREO-B spacecraft) and GOES SXR flux profile for the related flare and derivative (proxy for HXR emission). Taken from \\cite{Bein2012ImpulsiveEruptions}.}\n\\label{fig:temmer16}      \n\\end{figure*}\n\n\nTo derive in more detail the temporal linking of flare-CME-SEP events, image data covering large field of views for observing the lower and middle corona is of utmost importance. Figure~\\ref{fig:mid-corona} shows the different field of views of currently available and future EUV instruments to observe and study the middle corona (distance up to about 4 solar radii). The Extreme EUV Imager suite aboard Solar Orbiter works at the 174\\AA~and 304\\AA~EUV passbands \\citep[EUI:][]{Rochus20_SoloEUI}. The EUVI-LGR instrument aboard ESA's \\textit{Lagrange} L5 mission (launch planned for 2027) has an extended field of view to the West limb of the Sun, that is perfectly suited to track the early evolution of Earth directed CMEs from L5 view (60 degrees separation with Earth). We must not forget the capabilities of ground-based coronagraph instruments such as the COSMO K-Cor at the Mauna Loa Solar Observatory in Hawaii (replaced in 2013 the aging MLSO Mk4 K-coronameter\\footnote{Details can be found under: \\url{https:\/\/www2.hao.ucar.edu\/cosmo\/documentation}}) observing a field of view starting as low as 1.15 solar radii, however, quite restricted in observational time compared to satellite data. \n\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{cme4.pdf}\n\\caption{EUV image from Proba-2\/SWAP combined with a LASCO\/C2 coronagraph image covering in total a field of view up to $\\sim$4 solar radii. The colored boxes mark the relative nominal field of views of different EUV observing instruments. FSI (Full Sun Imager is part of the EUI suite aboard Solar Orbiter), EUVI-LGR (aboard the  planned L5 \\textit{Lagrange} mission), and SoHO\/EIT. STEREO\/EUVI, Proba-2\/SWAP, GOES\/SUVI are instruments with the largest field of view of about 1.7 solar radii. Taken from \\url{http:\/\/middlecorona.com}.\n}\n\\label{fig:mid-corona}      \n\\end{figure*}\n\n\nFor more details on CME trigger mechanisms I refer to recent review articles by \\cite{schmieder15} or \\cite{green18}. For a more specific background on CME initiation models, see, e.g., the Living Reviews by \\cite{webb12}.\n\n\n\n\\subsubsection{Shock formation, radio bursts, and relation to SEPs}\\label{shock-form}\nClosely related to studies of the CME early evolution and acceleration profiles, are shock formation processes. To generate a shock wave, a short-duration pulse of pressure is needed. Besides the CME, acting as piston, there is also the possibility that a strong flare energy release initiates a blast wave or simple-wave shock \\citep[e.g.,][]{vrsnak08b}. At which height shocks are formed by an erupting disturbance is important for understanding particle acceleration processes. The acceleration profile derived from tracking the CME frontal part suggests its formation at rather low coronal heights $<$1.5 solar radii. The  shock formation height itself is also strongly depending on the plasma environment. From model calculations a local minimum of the Alfv{\\'e}n speed is derived for a distance of about 1.2--1.8 solar radii and a local maximum around 3.8 solar radii from the Sun \\citep{mann99,Gopalswamy01_CME+longwave_typeII,vrsnak02}. Hence, the statistical maximum of CME acceleration profiles is also in accordance with the local minimum of the Alfv{\\'e}n speed. The occurrence of such local extrema has major consequences for the formation and development of shock waves in the corona and the near-Sun interplanetary space as well as their ability to accelerate particles. \n\n\nThe most compelling argument for shock formation is the observation of a radio type II burst. In the case of being driven by a CME, they are reported not only to occur at the apex of a CME shock front, but also to originate from the lateral expansion of the CME as observed with LOFAR\\footnote{Low Frequency Array \\citep{haarlem13}. Recent attempts to use LOFAR for Space Weather purposes are reported under \\url{http:\/\/lofar4sw.eu}.} \\citep[e.g.,][]{zucca18}. Due to the large density in the lower coronal heights a large compression appears with a quasi-perpendicular geometry, favoring the shock formation process. In that respect, moving type IV radio bursts might actually represent shock signatures due to CME flank expansion, that can be used as additional diagnostics for studying the lateral evolution of a CME \\citep{morosan20_flank}. The SEP intensity is found to be correlated with the width of a CME, and as such identifies the CME flank region to be an efficient accelerator of particles \\citep[see][]{holman83,mann05,richardson15}. Comparing the CME apex and flanks, the field lines are disturbed at different heights that may lead to different onset times for the acceleration of SEPs (cf., Figure~\\ref{fig:reames09}). The time needed for shock formation also leads to a temporal delay of the onset of SEP events with respect to both, the initial energy release (flare) and the onset of the solar type II radio burst (evidence of shock formation). Hence, the timing is an important factor and has to be taken into account when relating these phenomena to each other.\n\n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{cme5.pdf}\n\\caption{Two CME events and associated coronal waves (June 12 and 13, 2010) are investigated with high cadence SDO EUV 211\\AA~data. Manually tracked positions of the wavefronts are marked by dashed black lines. The connection between coronal surface wave and CME front is nicely observed. The right panels give radio data (Learmonth observatory, Australia) revealing type III and type II bursts (upper and lower frequency band) and measurements from particle detectors at L1 (bottom panel; vertical dashed lines show AIA waves onsets). Adapted from \\cite{Kozarev11}.}\n\\label{fig:kozarev11}      \n\\end{figure*}\n\n\nShocks may also be formed at larger distances from the Sun (several tens of solar radii), depending on the acceleration phase duration, the maximum expansion velocity and the width of the CME \\citep{Zic2008CylindricalShocks}. Due to the declining magnetic field with distance \\citep[well defined band-splits in type II bursts can be used to estimate the magnetic field in the corona; see e.g.,][]{vrsnak02}, shocks forming at larger heights are related to softer SEP spectra \\citep[e.g.,][]{gopalswamy17_SEP}. As can be seen, CME acceleration, shock formation height and hardness of SEP spectra is closely connected. Compared to SEPs, which strongly depend on the magnetic connectivity with the observer, type II bursts can be observed without connectivity issues and thus, give additional information about particle acceleration processes driven by CME shocks. In that respect, type II radio bursts may be used for predicting SEPs as well as shock arrival times \\citep[e.g.,][]{gopalswamy08_typeII,cremades15}. In combination, these parameters have strong implications for Space Weather impact, revealing the importance of monitoring and studying the early evolution phase of solar eruptive events. More details on SEPs are given in Section \\ref{sec:4}.\n\n\n\nFigure~\\ref{fig:kozarev11} presents a case study about the evolution of a CME front close to the Sun by using high cadence EUV images from SDO for June 12 and 13, 2010. The derived kinematics of the CME reveals a fast acceleration of its frontal part with about 1~km\/s$^2$ over the distance range 1.1--2.0 solar radii. The almost vertical traces in the radio spectra are type III bursts, identified with streams of electrons (radio emission due to particles moving along open magnetic field lines), followed by diagonal structures of a moving type II bursts, identified with shock waves. The onset of the type II burst appears together with the CME shock front, as observed in EUV, with a bit of a delay with respect to the shock formation that occurs close to the maximum CME acceleration. By the time the CME occurs in the LASCO field of view, the CME speed decreased to the sub-Alfv{\\'e}nic regime. The event produced an enhanced proton flux at 1AU. However, the complex magnetic topology related to the active region prevents from making strong conclusions about the possible sites of particle acceleration \\citep[see also][]{Kozarev11,suli11,gopalswamy12}. Definitely, more such detailed case studies combined with improved modeling of the magnetic environment is needed for advancing our understanding in the processes that accelerate particles. \n\n\n\\subsubsection{Stealth CMEs}\nIn contrast to fast and massive CMEs and their related cascade of solar surface signatures, there exist so-called \\textit{stealth CME} events that are most probably caused by some simple (low-energetic) magnetic field reconfiguration in the upper corona releasing magnetic flux ropes of low density that usually do not exceed the solar wind flow speed. Actually, they were recognized already in the mid 1980's and were identified as \\textit{spontaneous CMEs} or \\textit{unassociated CMEs} (meaning no surface signatures) by \\cite{wagner84}. Later studies showed, that they start at very large heights in the corona without noticeable signatures, such as flare emission, filament eruptions, coronal waves, or coronal dimmings \\citep{Robbrecht09}. Stealth CMEs are potential candidates to cause \\textit{problem} storms and missed Space Weather events, as they are hardly recognized in white-light data and due to the lack of observational imprints on the solar disk. In the recent years several studies have been published on this issue discussing those events \\citep[see e.g.,][]{DHuys2014ObservationalSignatures,nitta17,vourlidas18}. \n\n\n\n\n\\subsection{Advantages due to multi-viewpoint observations}\\label{sec:sub1-5}\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{cme6.pdf}\n\\caption{Earth-directed CME from December 12, 2008 as observed from multiple perspectives. STEREO-A (left) and STEREO-B (right) are separated from Earth by an angle of about 45 degrees. The running difference image from LASCO\/C2 (middle panel) observes the CME as weak partial halo event. The inlay in the middle panel gives the spacecraft location (STEREO-A red filled circle; STEREO-B blue filled circle) with respect to Earth (green filled circle) and the CME propagation direction (yellow arrow). White arrows in each panel point to roughly similar parts of the CME observed with the different instruments.} The closer the CME propagates in the plane-of-sky of the instrument, the higher the intensity in white-light. Adapted from \\cite{Byrne10}.\n\\label{fig:multi_byrne}      \n\\end{figure}\n\n\nIn contrast to a flare, which is a rather localized phenomenon, the analyzes of CMEs and related coronal waves, propagating over large areas of the solar surface, as well as SEPs profit enormously from at least two viewpoints. The twin-spacecraft STEREO unprecedentedly provides, since its launch end of 2006, image data in EUV and white-light from multiple perspectives. STEREO consists of two identical spacecraft, STEREO Ahead (A) and Behind (B; lost signal October 2014), orbiting the Sun in a distance close to Earth, with STEREO-A being closer and STEREO-B further away from the Sun. The separation angle between the two spacecraft increases by about 45 degrees per year\\footnote{Current position of STEREO and other spacecraft can be found under \\url{https:\/\/stereo-ssc.nascom.nasa.gov\/where.shtml}}. There are four instrument packages mounted on each of the two STEREO spacecraft, SECCHI comprising EUV and white-light coronagraphs and imagers \\citep{Howard08_secchi}, IMPACT sampling the 3D distribution of solar wind plasma and magnetic field \\citep{Luhmann08_IMPACT,Acuna08_IMPACT_MAG}, SWAVES tracking interplanetary radio bursts \\citep{Bougeret08_SWAVES}, and PLASTIC measuring properties of the solar wind plasma characteristics \\citep{galvin08_PLASTIC}. Conjoined with instruments from Earth perspective, such as SoHO (1995--), Hinode (2006--) and SDO (2010--), as well as ground based observatories (covering the radio and visual wavelength range, e.g., chromospheric H$\\alpha$ and Ca II lines), the evolution of active regions together with flares, CMEs, and SEPs could be for the first time stereoscopically observed. Unfortunately, a big drawback for multi-viewpoint magnetic field investigations was the lack of magnetographs onboard STEREO \\citep[this might be overcome by the ESA \\textit{Lagrange} L5 mission planned to be launched in 2027; see also  e.g.,][]{gopal11_L5,Lavraud2016AScience}. \n\n \nBesides having more than one vantage point, STEREO carries the heliospheric (HI) instruments, enabling to seamlessly observe the entire Sun-Earth line in white-light. They provide a unique long-term, synoptic data-set to be exploited for Space Weather application. Wide-angle image data allow to undoubtedly link CMEs to their interplanetary counterparts (ICMEs) as measured in-situ and to investigate in detail the in-situ signatures caused by the different CME structures and orientations. More details on ICMEs are given in Section~\\ref{sec:sub3-5}. \n\nFigure~\\ref{fig:multi_byrne} shows an Earth-directed CME observed from multiple perspectives and over a large distance range using STEREO data. From Earth perspective (shown in the middle panel), the CME is observed as weak halo event which makes it almost impossible to reliably determine a propagation direction and its radial speed. From STEREO perspective, the CME is observed close to the plane of sky of the instruments, lowering the projection effects for deriving its radial kinematics. Hence, the multiple viewpoints and homogeneous dataset of STEREO, enable to do 3D reconstructions of solar structures and to investigate projection effects with the attempt of correcting them, or at least limit and assess the uncertainties of the projected measurements. For SEPs, the identical instruments aboard the two spacecraft bring the advantage of having the same energy threshold, allowing systematic studies of SEPs coming from the same active region but related to a different magnetic connectivity and to probe the longitudinal dependencies.\n\n\n\n\n\n\\begin{figure}\n  \\includegraphics[width=1.\\textwidth]{cme7.pdf}\n\\caption{CME from March 7, 2011: (a) Excess brightness image from STEREO-A COR2. 3D shock front (green mesh in panel (b)) projected on the image plane is shown with the dotted line. The diamond marks the geometric center of the ellipsoid model projected onto the same plane. (b) Excess brightness in panel (a) with the 3D shock front (green mesh) modeled with the ellipsoid model described in \\cite{kwon17}. (c) Geometric relation among the Sun, shell-like sheath, and line-of-sight. A partial circle around the origin O is the solar disk. A shell-like sheath is represented in gray color. Arrows in black, blue and red are the line-of-sight, the projected shock normal on the image plane, and the actual shock normal in 3D, respectively. Taken from \\cite{kwon18}.}\n\\label{fig:kwon}      \n\\end{figure}\n\nMulti-spacecraft views enable to study especially the CME geometry and its substructures in more detail. With that, the different manifestations of shock and driver could be well confirmed and it is now well acknowledged that the outer envelope of the observed CME presents the expanding shock or compressed shell that encompasses the driver \\citep[e.g.,][]{Ciaravella06,Ontiveros09,Vourlidas13}. As the different parts have a different impact on Earth, for forecasting purposes, measurements of the CME's outer front should be clearly specified (e.g., shock front versus magnetic structure). In addition, the long-standing question whether halo CMEs would be different compared to limb events \\citep[see e.g., the Living Reviews by][]{chen11} could be solved. The shock shell of the CME can be presented as sphere-like structure expanding over 360 degrees (see Figure~\\ref{fig:kwon}). It was found that especially strong events (having a large compression) can be observed as halo CME independent of the viewpoint \\citep{kwon15,kwon18}. In that respect STEREO data also showed that the outermost shock component of the CME matches well the solar surface structure of coronal EUV waves \\citep{Kienreich09,Patsourakos09,Veronig2010FirstWave,kwon17}. Therefore, observations of the surface signatures of CME related coronal waves give supportive information about the CME expansion and propagation direction and should be closely monitored for early Space Weather forecasting.\n\n\n\nThe CME speed is actually a mixture of lateral and radial expansion dynamics making it tricky to derive the ``true'' propagation behavior. Multiple viewpoints enable to separately study projected versus deprojected speeds and radial versus lateral expansion behaviors of CMEs. Comparing single and multiple spacecraft data revealed that single viewpoint measurements are definitely valid. However, especially measurements of the CME width (or lateral expansion) and speed for slow CMEs (deprojected speeds below 900km\/s) reveal high uncertainties depending on the perspective \\citep{Shen13,Balmaceda18}. Models taking into account projection effects showed to significantly decrease the uncertainties in forecasting the arrival times of CMEs \\citep[e.g.,][]{colaninno13,Mishra13,shi15,makela16,rollett16}. A well known empirical relation exists between the radial and the lateral expansion speed, $V_{\\rm rad} = 0.88 V_{\\rm exp}$, as described by \\cite{dalLago03} and \\cite{schwenn05}. Follow-on studies showed that this relation can also be described by the CME half-width, $w$ (assuming a cone model), given by $f(w)=1\/2(1+\\cot{w})$ and that the kinematics of extremely fast CMEs is better estimated by $V_{\\rm rad} \\approx V_{\\rm exp}$ \\citep{michalek09}. Moreover, statistical studies revealed that the relationship between the radial and lateral expansion speed is a linear function, hinting towards the self-similar expansion behavior of CMEs already close to the Sun \\citep{Vourlidas17,balmaceda20}. However, in the low corona, for some events a strong overexpansion is observed \\citep[e.g.,][]{Patsourakos10}. The assumption of a rather self similar expansion is found to be valid for most of CME events when propagating in interplanetary space \\citep[e.g.,][]{Bothmer98,Leitner07,demoulin08,Gulisano12,vrsnak19}. \n\n\n\\begin{figure}\n  \\includegraphics[width=1.\\textwidth]{cme8.pdf}\n\\caption{March 7--11, 2012: GCS fitting (green mesh) of two CMEs (CME1: top panels. CME2: bottom panels - note that CME1 is visible as extended bright structure in these images) using white-light data from the three spacecraft, STEREO-A, SOHO and STEREO-B. The first, second, and third columns contain coronagraph images from COR2 aboard STEREO-B, C3 aboard SOHO, and COR2 aboard STEREO-A, respectively. Taken from \\cite{patsourakos16}.}\n\\label{fig:patsourakos16}      \n\\end{figure}\n\n\n\nSince we cannot gather the full complexity of the CME structure, idealized geometries assuming self-similar expansion, act as basis of many CME models and 3D reconstruction techniques that were developed over the past years. Basic models make use of a simple cone-type geometry \\citep[e.g.,][]{St.Cyr00,schwenn05,Michalek06,Xie04}. With the availability of image data from multiple views, those tools were refined and full 3D reconstructions were enabled from which estimates of the deprojected kinematics, geometry, and propagation direction are derived. Methods comprise, e.g., inter-image tie points and triangulation in various wavelength ranges \\citep[see e.g.,][]{Harrison08,Howard08,maloney09,Reiner09,Temmer09,Liewer10}, forward models related to white-light data \\citep[e.g.,][]{wood09}, or center of mass calculations \\citep{Colaninno09}. Also online tools were made available, such as e.g., the CCMC tools StereoCat\\footnote{StereoCat \\url{https:\/\/ccmc.gsfc.nasa.gov\/analysis\/stereo\/}}. A well known and widely used technique is the graduated cylindrical shell (GCS) forward model developed by \\cite{Thernisien06,Thernisien09}. Coronagraph image data showing the CME from at least two different vantage points are required, on that an idealized flux rope structure in the form of a croissant is fitted. The GCS model depends on a number of free parameters, such as the flux-rope height and angular width as well as the aspect ratio which determines the rate of self-similar expansion. Figure~\\ref{fig:patsourakos16} gives the 3D reconstruction of two CME events using GCS applied on STEREO and LASCO data in a study by \\cite{patsourakos16}. Especially for multiple events, the investigation and determination of the cause of geoeffectiveness is rather challenging as the processes happening on the Sun are complex. In that respect, geometrical fitting methods help to derive the propagation direction of a particular solar event in order to reliably link it to a geoeffective event at Earth. In a similar way, stereoscopy can be applied also on radio data. Figure~\\ref{fig:gono_radio} shows results from so-called goniopolarimetric observations using WIND and STEREO spacecraft data studying the location of radio type II bursts. That method is used to derive the direction of arrival of an incoming electromagnetic radio wave, its flux, and its polarization.\n\n\n\n\\begin{figure}\n  \\includegraphics[width=1.\\textwidth]{cme9.pdf}\n\\caption{The CME flux rope obtained from GCS 3D reconstruction (black grid croissant). The 3D reconstruction of the radio type II burst (dark and light blue spheres) using gonopolarimetric technique. The yellow sphere represents the Sun (left panel). View of the flux rope and radio sources as seen from Earth (right panel) SOHO\/LASCO C3 image showing the CME as seen from Earth, for comparison with panel. Adapted from \\cite{magdalenic14}.}\n\\label{fig:gono_radio}      \n\\end{figure}\n\n\nBesides geometry related information, STEREO coronagraph data can also be used to derive the CME deprojected mass close to the Sun \\citep{Colaninno09,Bein2013TheObservations} which, together with the early acceleration phase, are taken for better estimating the energy budget between flares and CMEs (see also Section~\\ref{sec:budget}). STEREO and its wide-angle HI instruments also enable to derive the 3D geometry of compressed density structures like CIRs \\citep[see][]{Rouillard08,wood10}. Sometimes the disentanglement between CMEs and CIRs in HI is tricky, hence, it needs careful inspection of the data when tracking specific features \\citep[see e.g.,][]{Davis10}. \n\nUsing multi-viewpoint data and applying different reconstruction techniques we vastly gained important insight about CME characteristics. Moreover, the results stemming from STEREO observations clearly challenged existing CME and SEP models. However, using idealized geometries, the real 3D structure of a CME or SEP paths can only be approximated and we need to keep in mind that there are strong deviations from these. Especially, in interplanetary space, the geometry of the CME front clearly changes, as flanks and nose interact differently with the non-uniform solar wind \\citep[pancaking effect; see e.g.,][]{riley04,Nieves-Chinchilla12}. In addition, the CME shape might vary due to intensity changes as the relative position to the Thomson sphere changes, that makes the tracking of specific white light structures complicated \\cite[e.g.,][]{vourlidas06}. Therefore, the derived deprojected values and forecasts based on them still need to be treated with caution \\citep[see also][]{Mierla10,riley18}. For more details on the complex interactions of CMEs with their surroundings I refer to the review by \\cite{manchester17}.\n\n\n\n\nFor more information on solar stereoscopy and tomography techniques, applied to various large-scale structures in the solar corona, I refer to the Living Reviews by \\cite{aschwanden11}. For comprehensive investigations, the EU funded project HELCATS\\footnote{Heliospheric Cataloguing, Analysis and Techniques Service; \\url{https:\/\/www.helcats-fp7.eu}} established databases ready to use for analyzing STEREO 3D CME characteristics and HI CME tracks on a statistical basis \\citep[see e.g.,][]{Murray18,Barnes19}. Out of that an extensive ICME catalogue\\footnote{\\url{https:\/\/helioforecast.space\/icmecat}} was compiled by \\cite{moestl17} and \\cite{palmerio18}. A conjunction catalogue covering CME in-situ measurements by two or more radially aligned spacecraft (MESSENGER, Venus Express, STEREO, Wind\/ACE) is given by \\cite{salman20}.\n\n\n\\section{Interplanetary counterparts of CMEs: ICMEs}\\label{sec:sub3-5}\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{icme1.pdf}\n\\caption{From running difference STEREO-A HI1+2 image data the central rows are extracted (right) at each time step and rotated by 90 degrees (middle). From this a time-elongation plot (so-called Jmap) is constructed (left). The CME front is marked by a yellow arrow in the direct image as well as in the Jmap. Adapted from \\cite{davies09}.}\n\\label{fig:jmap}      \n\\end{figure*}\n\n\nNewly developed imaging capabilities clearly enhanced our understanding about the relation between solar eruptions, CMEs, and their counterparts in interplanetary space (ICMEs). SMEI, the Solar Mass Ejection Imager \\citep[SMEI:][]{Eyles03} on the Earth-orbiting Coriolis spacecraft, was the first heliospheric white-light imaging instrument covering the Sun-Earth space \\citep[for more details see the review by][]{howard13}. The successor of SMEI are the heliospheric imagers \\citep[HI:][]{Eyles09} aboard STEREO \\citep{kaiser08_STEREO}. The WISPR instrument \\citep{vourlidas19} aboard the Parker Solar Probe mission and the SoloHI instrument \\citep{howard20} aboard Solar Orbiter build upon the STEREO\/HI heritage and make similar observations of the inner heliosphere. The observational principle is like a coronagraph, but as these are wide-angle instruments, they observe much larger distances from the Sun enabling the tracking of CMEs throughout interplanetary space. The unprecedented image data facilitated the tracking of CMEs through interplanetary space and with that could unambiguously relate the CME white-light structure to in-situ measurements \\citep[see e.g.,][]{moestl09_APJL,moestl14} and moreover, to get better insight on how CMEs interact with the ambient solar wind structures. Figure~\\ref{fig:jmap} shows a so-called Jmap which is constructed from running difference white-light HI data covering the Sun-Earth distance range. By extracting the central part of the HI images in the horizontal direction, the ICME front can be rather easily followed as function of the elongation angle. Before further analysis, the elongation-time measurements need to be converted into radial distance. For that, methods assume either a certain CME geometry and apply the propagation direction of the CME \\cite[see e.g.,][]{lugaz10} or use fitting functions \\cite{Rouillard08}. These procedures cover rather high uncertainties in the derived kinematical profiles, that needs to be taken into account when interpreting CME propagation profiles for interplanetary space \\citep[e.g.,][]{rollett12,liu13}.\n\nIt is well known that CMEs during their propagation phase tend to get adjusted to the ambient solar wind flow owing to the drag force exerted by the ambient solar wind \\citep{Gopalswamy01,wang04}. As consequence, CMEs which are faster than the ambient flow speed get decelerated while those which are slower get accelerated. This alters their speed, hence, travel time and with that has impact on Space Weather forecasting. The adjustment to the ambient flow speed happens most probably in interplanetary space \\citep[e.g.,][]{Sachdeva15}. At which distance exactly depends on the competing forces acting on a CME, Lorentz versus drag force \\citep[e.g.,][]{vrsnak08}. The longer the CME is driven, hence, the longer the magnetic reconnection process is ongoing (which might be inferred from flare emission and growing post eruptive arcades), the farther away from the Sun the adjustment may occur. Empirical relations found between CME kinematics and flare properties (flare ribbons, coronal dimmings, or post-eruptive arcade regions) actually may be used to estimate the reconnected flux that empowers the CME \\citep[e.g.,][]{gopalswamy17,Temmer17_flar-cme,dissauer18,Tschernitz18}\\footnote{A database of more than 3000 solar flare ribbon events observed by SDO and reconnection flux is given in \\cite{kazachenko17}.}. The amount of drag from the solar wind depends on the relative speed and density between the solar wind and the CME as well as the CME width\/size. It is found that wide CMEs of low mass tend to adjust rather quickly to the solar wind speed and, hence, their transit time (i.e., how long a CME needs to traverse a certain distance) is determined primarily by the flow speed in interplanetary space. Narrow and massive CMEs propagating in a fast solar wind have the shortest transit times \\citep[see e.g.,][]{vrsnak10}. \n\n\n\n\n\nFigure~\\ref{fig:ICME} shows typical in-situ signatures of a well-defined ICME at 1~AU, revealing a simultaneous jump of all measured components (shock) with a subsequent sheath structure (compressed plasma) of increased density, speed and turbulence, that is followed by signatures of a smooth and enhanced magnetic field together with a rotation as observed in the vector components (changing from plus to minus or vice versa). The ICME magnetic structure is usually identified by that smooth field rotation (flux rope), a plasma-beta lower than 1 (referring to a dominant magnetic component), a low temperature, and a linearly decreasing proton speed \\citep[see also the Living Reviews by][]{kilpua17}. Sometimes that flux rope can be associated with twisted structures observed already in white-light image data. \nHaving a long-lasting southward directed magnetic field (measured in the $B_{\\rm z}$ component), flux ropes are the main contribution of strong geoeffectiveness. The passage of rather isolated magnetic ejecta at 1~AU typically takes about 1 day \\citep[cf.][]{RC10}\\footnote{A near-Earth ICME catalogue is given under: \\url{http:\/\/www.srl.caltech.edu\/ACE\/ASC\/DATA\/level3\/icmetable2.htm}}. Hence, geomagnetic disturbances may last for many hours. Flank hits, interacting CMEs and complex ejecta, can have passage durations of about 3 days at 1~AU \\citep[see][]{Burlaga02,Xie06,Marubashi2007Long-durationModels,Mostl2010STEREO2010}, affecting the Earth's atmospheric layers over a much longer time range, and, hence, causing stronger geomagnetic effects (see also Section~\\ref{sec:preconditioning}).\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{icme2.pdf}\n\\caption{STEREO-A in-situ measurements and identification of a CME together with its closed magnetic structure. From top to bottom: pitch-angle distribution data of suprathermal electrons, total magnetic field intensity, magnetic field vectors (in RTN coordinates), solar wind proton bulk speed, proton number density, proton temperature \\citep[in red the expected proton temperature is given calculated from an empirical relation to the solar wind speed as given by][]{Richardson95}, plasma-beta, total pressure, distribution of the iron charge state. Vertical dashed lines mark the shock-sheath, and the boundaries of the magnetic structure. Taken from \\cite{Jian18}.}\n\\label{fig:ICME}      \n\\end{figure*}\n\n\nBy combining remote sensing and in-situ data using multi-spacecraft reconstruction methods, it is revealed that from in-situ measurements we observe localized variations of the magnetic field behavior that may not be representative of the global structure \\citep[see][]{mostle08,moestl09_APJL,rouillard10,farrugia11,DeForest2013TrackingEjection}. Studies using multi-spacecraft encounters separated in radial distance and longitude give insight on the magnetic coherence of ICMEs on various scales and with that raise questions on the inner structure of CMEs as well as their interaction processes with the interplanetary magnetic field \\citep[see e.g.,][]{good18,lugaz18}. Using flux rope reconstructions methods applied on in-situ measurements \\citep[see e.g.,][]{al-haddad13} a comparison between the physical parameters derived close at the Sun with those measured in-situ can be performed enabling to interpret changes in the mass, flux, etc. due to the interaction with the interplanetary solar wind \\citep[see e.g.,][]{bisi10,Temmer17_flar-cme,temmer21}. Especially the reconnection of the magnetic flux rope with the interplanetary magnetic field is found to lead to either a loss of magnetic flux (so-called erosion) or adding of magnetic flux \\citep[see e.g., ][]{Dasso07,manchester14_CMEerosion,Ruffenach15}. Removing or adding magnetic flux may lead to a change in the ICME propagation behavior. Filament material is found less often from in-situ measurements (identified by low charge state species) despite the fact that most CMEs are accompanied by filament eruptions. Heating mechanisms or simply missing the cold filament material due to the localized in-situ measurements might be a reason for that \\citep{filippov02}. This is supported by findings that magnetic ejecta are only partly filled with hot plasma related to heating by the flare \\citep[e.g.,][]{gopalswamy13_filament}.\n\n\n\n\n\nMore details on the relation between white light remote sensing image data and in-situ measurements, including proper nomenclature, is given by \\cite{rouillard11}. A review on multi-point ICME encounters before and during the early years of STEREO is given by \\cite{kilpua11}. The recent review by \\cite{luhmann20} comprises a thorough overview on the ICME propagation in the inner heliosphere.\n\n\n\n\\section{Solar Energetic Particles (SEPs)}\\label{sec:4}\n\\subsection{General characteristics}\nSEP events are observed in-situ as enhanced electron, proton and heavy ion flux (and as increased level of cosmic rays on ground) largely exceeding the thermal energy levels, ranging from keV to GeV. Strong fluxes of energetic protons (so-called proton events) cause strongest geoeffective phenomena. High energy SEPs in the range of GeV reach the Earth within less than 10 minutes and may produce ground level enhancements (GLE; measurable in neutron monitors at Earth surface), that are of special interest as they have major effects on crewed spaceflight and aircraft due to the increased radiation exposure \\citep[e.g.,][]{Malandraki18_book}. \n\nIn general, there are two populations of SEP events, gradual and impulsive ones \\citep[e.g.,][cf. Figure~\\ref{fig:reames99}]{Reames99}. It is the different temporal scaling which is disentangling those two populations. Driving agents acting on longer time scales are related to CME shock acceleration mechanisms. However, gradual events seem to be accompanied as well by an impulsive part which is thought to be related to short-time magnetic reconnection processes, as observed in flares. Obviously gradual events are caused by both driving agents prolonging the acceleration process but on a less energetic level \\citep[see also][]{Anastasiadis19}. Impulsive events are also obtained to be related to those SEP events where the location of the particle accelerator is magnetically well connected to the observer. SEP\/GLE events are found to have the hardest spectra and the largest initial acceleration \\citep{Gopalswamy16_backsideSEP}. There are still many open issues about the processes leading to energetic particles as well as about their (suprathermal) seed populations in the corona and interplanetary space \\citep[e.g.,][]{Mason99,Desai06,Mewaldt12}. Clearly, the primary condition for the production of SEPs is the opening of magnetic field lines into interplanetary space (as for eruptive events) and that accelerated particles have access to that open field lines. It is confirmed that for confined flares no SEPs are observed \\citep[e.g.,][]{Trottet15}. As shocks play an important role in the acceleration of particles, coronal shock waves on the solar surface and interplanetary space related to CMEs as well as interacting CMEs are investigated in relation to SEPs \\citep[see e.g.,][and related MHD modeling results, e.g., \\citeauthor{Pomoell08}, \\citeyear{Pomoell08}]{Park13,Lario14,Miteva14}. \n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{sep1.pdf}\n\\caption{Separation into gradual and impulsive SEP events and their suggested driving mechanisms. a) Gradual SEP events result from diffusive acceleration of particles by large-scale shocks produced by CMEs and populate interplanetary space over a wide range of longitudes. b) Impulsive SEP events result from acceleration during magnetic reconnection processes in solar flares and are observed well when magnetically connected to the flare site. Intensity-time profiles of electrons and protons in c) gradual and d) impulsive SEP events. Taken from \\cite{Desai16} after \\cite{Reames99}.\n}\n\\label{fig:reames99}      \n\\end{figure*}\n\nSEP events that become Space Weather effective are controlled by many factors, such as the source region location of the eruption (longitude and latitude) and width of the CME, background solar wind, seed populations, multiple CMEs and their interaction, or magnetic field configuration near the shock. For example, narrow CMEs (slow or fast) do not efficiently accelerate particles \\citep[see e.g.,][]{kahler19}, and CMEs originating from the eastern hemisphere are less likely to create a SEP event near the Earth because of the weak magnetic connection between Sun and the Earth. The highest-energy particles are most likely accelerated close to the shock nose where the shock is strongest, while the lower energy particles are accelerated at all regions \\citep[see e.g.,][]{bemporad11,gopalswamy18}. Hence, also the ecliptic distance to the shock nose, i.e., the event source region latitude, is an important parameter for SEP prediction \\citep[see e.g.,][]{gopalswamy13}. \n\nKnowledge about the magnetic connectivity is a crucial parameter in order to detect SEPs in-situ and to relate them to the proper driving agent \\citep{reames09}. Figure~\\ref{fig:reames09} depicts the propagating idealized circular-shaped CME shock front in relation to the radially oriented magnetic field lines near the Sun. The cartoon describes a scenario in that a narrow range of heights (2--4 solar radii) exists where compression is sufficient for effective particle acceleration \\citep[e.g.,][]{Cliver04} and for having a good connection to the observer, with heights increasing at the eastern and western flanks. Concluding, the connectivity changes with distance from the Sun. On the other hand, flare locations lying close to open structures like coronal holes, have different magnetic configuration and facilitate the acceleration of particles into the heliosphere \\citep{cane88,Reames96,shen06}. In that respect, the interplay between open and closed magnetic field is important to know and due to the lack of observations needs to be supported by reliable coronal modeling. \n\n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{sep2.pdf}\n\\caption{Cartoon showing a possible acceleration scenario for SEPs. The radial field lines (black lines) are hit by the CME shock front (blue) at different heights for the nose and the flank. The solar particle release (SPR) likely begins at a 2--4 solar radii (marked by the red region) for the apex or higher up for the flanks. Taken from \\cite{reames09}.}\n\\label{fig:reames09}      \n\\end{figure*}\n\n\nFor Space Weather forecasting purposes, it is desired to derive clear signatures showing that particles associated to eruptive events were able to escape to the high corona and interplanetary space. Therefore, the monitoring of possible radio emission (from decimetre and longer waves) is found to be of utmost importance \\citep[e.g.,][]{Klein10}. Observations of flares in the high energy range provide additional information on the location, energy spectra, and composition of the flare accelerated energetic particles at the Sun that can be compared to 1AU SEP events \\citep{lin06}. RHESSI imaging capabilities could show that flare $\\gamma$-ray sources are not co-spatial with flare HXR sources \\citep{fletcher11}. \\cite{Laurenza09} developed a technique for short-term forecasting of SEPs based on flare coordinates and flare flux together with the time\u2010integrated intensity of SXRs and type III radio emission ($\\sim$1 MHz). Similar, the forecast of the occurrence of SEP events could be determined using the peak ratios in flare fluxes measured over (0.05\u20130.4 nm)\/(0.1\u20130.8 nm) as described by \\cite{kahler18}. For improved SEP forecasting, it is suggested to take into account parameters from both driving agents, flares and CMEs \\citep[see][]{Klein17}. Figure~\\ref{fig:stcyr17} shows for an eruptive event the timing between flare SXR emission (GOES flux), radio type III burst (WIND), electron and proton spectra measured at 1AU (SOHO), and combined white-light image data from the ground-based K-Cor coronagraph and LASCO\/C2 instrument. Especially, observations of the early evolution of CMEs and derivation of shock formation heights (see also Sect.~\\ref{shock-form}) as well as distribution of Mach numbers along the shock surface \\citep{rouillard16} might give some lead time for SEP forecasting. Monitoring the generation of flare associated coronal surface waves \\citep[EIT\/EUV waves; see][]{Thompson98}, gives additional hints on shocks ignited by the CME lateral expansion (for confined events no coronal waves are observed).\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{sep3.pdf}\n\\caption{In the top row, the January 1, 2016, eruptive event appears in the sequence of images from SDO AIA (gold), MLSO K-Cor (blue), and SOHO\/LASCO (red). A fast CME associated with an SEP event detected near Earth, is seen appearing off the southwest limb of the Sun. The time profiles reveal that data from the ground-based K-Cor coronagraph could be used for a timely warning of particle events as described in that case study. Taken from \\cite{stcyr17}.\n}\n\\label{fig:stcyr17}      \n\\end{figure*}\n\n \n\n\n\\subsection{SEPs observed from multiple viewpoints}\n \n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{sep4.pdf}\n\\caption{February 25, 2014 SEP event and detected proton intensities (red = STEREO-A\/HET, blue = STEREO-B\/HET, green = SOHO\/ERNE). Inset: relative locations of the STEREO spacecraft and the Earth during the event. The arrow pointing out from the Sun shows the location of the SEP producing active region (at longitude E82), and the asterisks mark the nominal Parker spiral magnetic field lines connecting each observer to the Sun. Taken from \\cite{Paassilta18}.}\n\\label{fig:sep-stereo}      \n\\end{figure*}\n\n\nUsing STEREO data, the observation of wide-spread SEP events shed new light on the possible generation mechanisms together with the lateral expansion of CMEs and their interaction with other coronal structures \\citep[see][]{rouillard12}. Figure~\\ref{fig:sep-stereo} gives for the February 25, 2014 SEP event intensity profiles as measured by different spacecraft that are separated from Earth by 152 degrees (STEREO-A) and 160 degrees (STEREO-B). The SEP producing eruptive event is located at E82 (marked by the green arrow). As STEREO-B is closest to the SEP source, that flux profile reveals the highest intensity. The related CME was observed as halo event from Earth, having a projected speed of more than 2000~km\/s and before that, other CMEs were launched from that region. SEPs may be directed to wide-spread angles by field line draping around the closed magnetic field of the CME and\/or complex magnetic fields due to CME-CME conglomerates, or CME-CIR interaction \\citep[e.g.,][]{Dresing16,Dresing18,Gomez-Herrero11,Gomez-Herrero17,Xie17,Guo2018ModelingMeasurement}. For more details on wide-spread SEP events, including a comprehensive catalogue see \\cite{Paassilta18}. \n\n\n\\begin{figure*}\n \\includegraphics[width=1.\\textwidth]{sep5.pdf}\n\\caption{Coronagraph images from SOHO\/LASCO (left panel: direct image; other panels to the right: difference images) showing the evolution of two CMEs from August 9, 2011 and from May 17, 2012 (GLE event). Inlay images are from SDO\/AIA 193\\AA~showing the solar sources. Red arrows point to the CME nose. The May 17, 2012 event remains bright over the LASCO field of view and its shock structure is located close to the CME ((g), (h)). The August 9, 2011 event reveals a smaller CME main body and a wider shock structure ((c), (d)), which is a sign of a weak shock. Taken from \\cite{gopalswamy13}.}\n\\label{fig:gopal_GLE}      \n\\end{figure*}\n\n\nDuring solar cycle 24 strong SEP events produced only two GLEs, that could be related to CMEs launched from the Sun on May 17, 2012 and September 10, 2017. \\cite{gopalswamy13} did a comparative study between the GLE May 17, 2012 and a non-GLE CME event from August 9, 2011 using STEREO image data as given in Figure~\\ref{fig:gopal_GLE}. The study revealed for the GLE event a shock formation height very low in the corona (1.38 solar radii from solar center) and that the shock remained closer to the driver structure over the coronagraphic field of view. This is indicative of a stronger shock that is driven over a longer time and, hence, can produce very energetic particles. \\cite{rouillard16} modelled for the May 17, 2012 event the background topology of the magnetic field using multiple viewpoints to derive the geometry of the shock front and to find where particles get accelerated most efficiently \\citep[see also][]{Plotnikov17,Kouloumvakos19}. Particles that get magnetically trapped in between CME structures pose a particle reservoir that may play a key role in the late acceleration of gradual SEPs related to CME-CME interaction events \\citep[e.g.,][]{lugaz17}. Despite these tremendous enhancements in our knowledge gained from stereoscopic observational data, still a major drawback in unraveling the SEP nature is the unknown configuration of the interplanetary magnetic field along which SEPs propagate and internal distribution \\citep{Kahler13}. Therefore, the acceleration process of SEPs seems not to be spatially limited but happens over a wide range of longitudes including transport before being injected at distant longitudes \\citep[e.g.,][]{Vlahos19_SEP,Kozarev17,Malandraki18_book}. \n\n\n\nFor recent reviews on SEP events covering in detail the production and acceleration processes I refer to the Living Reviews by \\cite{Desai16} or the book by \\cite{Reames17_book}. The review by \\cite{lugaz17} is focusing on SEPs with respect to CME-CME interaction events and the Living Reviews by \\cite{kilpua17} on particle acceleration due to ICME shocks.\n\n\n\n\\section{Energy budget between flares, CMEs, and SEPs}\\label{sec:budget}\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{budget1.pdf}\n\\caption{Schematic diagram showing the different energy dissipation processes of the observed activity phenomena. The chart covers the energy input (light yellow shaded boxes) and energy dissipation via primary (colored boxes) and secondary processes (white boxes). Those are the major processes identified for studying their energy closure relationship. Taken from \\cite{Aschwanden17}.}\n\\label{fig:energy_budget}      \n\\end{figure*}\n\n\nFigure~\\ref{fig:energy_budget} schematically shows the relevant components of energy build-up and dissipation processes. How much of the free energy is actually released and to which parts that released energy partitions into primary and secondary processes can only be answered by statistics. In cutting-edge studies performed by \\cite{emslie04,Emslie12}, a sample of eruptive events was investigated with respect to the energy release and its distribution into different components, concluding that about a third of the total available energy might be released in an eruptive event. Similar conclusions are drawn by \\cite{gopalswamy17_extreme} from calculating for extreme eruptive events the ratio of the total reconnected flux to the available active region flux. However, discrepancies were found from simulation studies \\citep[e.g.,][]{reeves10}. Recent extensive statistics by \\cite{Aschwanden17}, yield that $\\sim$87\\% of the magnetic energy is released. About 10\\% of the released energy drives the CME and $\\sim$80\\% goes into particle acceleration. In total about 10\\% percent of the free magnetic energy goes particularly into SEP acceleration. With respect to the CME, SEPs dissipate about 3\\% of the CME kinetic energy \\citep[similar as derived by][]{Emslie12}. The CME velocity shows strongest correlations with SEP characteristics and all that is consistent with CME-driven shock acceleration  \\citep[see][]{mewaldt06,Papaioannou16}. \n\nThe discrepancies in the results show that there might be processes that cannot be disentangled from each other, cover energy conversion (e.g., non-thermal into thermal due to cooling), or are simply not well observed. Nevertheless, the conclusion is that the free magnetic energy of an active region is sufficient to generate flare-CME-SEP events and with that confirms their common magnetic origin. For the interested reader I refer to the book by \\cite{aschwanden19}.\n\n\n\n\n\n\n\n\n\\section{Structuring of interplanetary space: the solar wind}\\label{sec:SWstructure}\n\nThe solar wind is the major hub in interplanetary space dictating how fast disturbances may evolve and guiding the motion of accelerated particles. Knowledge about the prevailing structure of the solar wind in terms of plasma and magnetic field distribution is therefore of utmost importance in order to obtain reliable Space Weather forecasts. In turn, this also leads to a better understanding and interpretation of the propagation behavior of CMEs as well as the occurrence and energetics of SEPs. Interplanetary space is strongly shaped by the interplay between slow and fast solar wind flows, causing stream interaction and compression regions, transient disturbances such as shocks and closed magnetic structures (flux ropes) of evolving CMEs. These structures pose magnetic barriers that are able to change the propagation characteristics of a specific CME and affect SEP fluxes. In the following I will focus on solar wind structures relevant for CMEs and SEPs. For more details on the heliospheric magnetic field I refer to the Living Review by \\cite{owens13} and for solar wind stream interaction regions throughout the heliosphere to the Living Review by \\cite{Richardson18}.\n\n\n\n\\subsection{General characteristics}\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{wind1.pdf}\n\\caption{Solar wind variation over the solar cycle. Outward interplanetary magnetic field in blue, inward interplanetary magnetic field in red. The bottom panel shows the timeline and line graphs of the relative smoothed sunspot number. Note also the inverted interplanetary magnetic field lines between the first and third orbit due to the reversal of the magnetic field. Courtesy: ESA}\n\\label{fig:ulysses}      \n\\end{figure*}\n\n\n\nThe solar wind is a continuous flow of charged particles propagating radially outward from the hot solar corona into interplanetary space. The characteristics of the solar wind are measured in-situ at specific locations such as the Lagrangian point L1 close to Earth (ACE, WIND, DSCOVR), from satellites orbiting planets (e.g., BepiColombo, VEX, MESSENGER, MAVEN), STEREO having varying longitudinal separation close to Earth's orbit, or PSP and Solar Orbiter with special mission trajectories in the inner heliosphere (cf. Figure~\\ref{fig:insitu}). Figure~\\ref{fig:ulysses} shows the solar wind speed measurements from spacecraft Ulysses which had the goal to examine the poles of the Sun \\citep{Wenzel92_ULYSSES}. In total, Ulysses performed three polar orbits, with each one taking six years to complete, over different phases of the solar cycle 22 and 23. The first one covers the solar minimum phase revealing slow solar wind streams over the equator and a fast wind over the poles where CHs are situated. The second orbit happened over solar maximum activity and shows the intermix of fast and slow winds at all latitudes. Three quarters of the third orbit were completed during the minimum of the next solar cycle. \n\n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{wind2.pdf}\n\\caption{Radial profiles of solar-wind parameters for the Sun-Earth distance. Median values are obtained from Helios and OMNI measurements and are extrapolated to the PSP orbit region as close as 10 solar radii. The lower edges of the shaded areas correspond to solar minimum and the upper edges to solar maximum. As comparison, overplotted are model results from \\cite{Banaszkiewicz98}, from \\cite{Sheeley97} and \\cite{WangY00} for the slow solar wind speed, from \\cite{Leblanc98} for the density and the range of temperature measurements given by \\cite{Billings59} and \\cite{Liebenberg75}. Taken from \\cite{Venzmer18}.}\n\\label{fig:sw_venzmer}      \n\\end{figure*}\n\nBesides the observed changes over latitude, the solar wind characteristics differ over distance. Early missions like Helios 1 and 2 \\citep{Schwenn75,Rosenbauer77}, achieved a perihelion of 0.29~AU and gathered valuable information about the solar wind characteristics close to the Sun. Figure~\\ref{fig:sw_venzmer} shows the radial dependence of the solar wind parameters over the Sun-Earth distance derived from HELIOS (1974--1981) and OMNI (1963--2016) observations at 1~AU. The study by \\cite{Venzmer18} extrapolates that information to regions as close as 10 solar radii, based on an empirical solar-wind model for the inner heliosphere in dependence on the solar cycle. Solar wind measurements from PSP at a distance of about 35 solar radii basically confirm the results from these earlier missions and their derived radial scalings, but also obtain that the magnetic field is very strongly fluctuating \\citep{Bale19,kasper19}\\footnote{For recent PSP result see also special issues of the Astrophysical Journal Supplement series and Astronomy \\& Astrophysics under  \\url{https:\/\/iopscience.iop.org\/journal\/0067-0049\/page\/Early_Results_from_Parker_Solar_Probe} and \\url{https:\/\/www.aanda.org\/component\/toc\/?task=topic&id=1326}.}.\n \n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{wind3.pdf}\n\\caption{Left: schematic drawing of the different large scale structures in interplanetary space \\citep[taken from][]{Yermolaev09}. Digits designate (1) heliospheric current sheet, (2) slow streams from coronal streamers, (3) fast streams from coronal holes, (4) compressed plasma (CIR on the front of fast and slow streams, and sheath region before the leading edge of a \"piston\"), (5) \"pistons\" (such as a magnetic cloud or ejecta), (6) rarefaction region. Right: Large-scale properties of the inner heliosphere (out to 1 AU) for Carrington Rotation 2068 from a global MHD solution. The two meridional slices in each panel show the radial velocity and radial magnetic-field strength, scaled to 1 AU. The slice in the equatorial plane shows the scaled number density. The sphere at 30 solar radii shows the scaled radial magnetic-field strength. Taken from \\cite{Riley10}.}\n\\label{fig:yermo_riley}      \n\\end{figure*}\n\n\nThe left panel of Figure~\\ref{fig:yermo_riley} schematically depicts various large-scale structures in the interplanetary space. We differentiate between the heliospheric current sheet, separating opposite magnetic field, solar wind streams of different speeds, and closed magnetic fields of CMEs that, due to their rapid expansion, act as pistons creating shocks. The interaction between fast and slow streams (plasma volume with frozen-in magnetic field) leads to compression, forming so-called stream interaction regions (SIRs) to the West, and rarefaction regions to the East. On the large scale it is assumed that the interplanetary dynamics can be described by ideal MHD equations. The right panel of Figure~\\ref{fig:yermo_riley} gives the results from a simulation using a 3D MHD model \\citep[see more details on coronal and solar wind MHD modeling in the Living Reviews by][]{gombosi18}. The model results obtain a warping of the heliospheric current sheet and show the latitudinal dependence of the fast solar wind stream in the radial direction \\citep{wilcox80}. All that reveals the complexity and interplay between open and closed magnetic field structures, occurring with different dynamics. For more details on the multi-scale nature of the solar wind I refer to the Living Reviews by \\cite{verscharen19}.\n\n\n\nThe observed large-scale structures of the solar wind are intimately connected to the coronal magnetic field originating from the solar photosphere. The quasi-steady fast wind ($>$ 450 km\/s) emanates from coronal holes, locations of predominantly open magnetic field, while the variable slow component is believed to originate mostly from closed magnetic field configurations around the streamer belt \\citep{McComas00_firstOrbit}. Coronal holes are observed as low density and low temperature structures, and therefore appear as dark areas in the wavelength ranges of EUV and SXR, imaging coronal temperatures of a few million Kelvin. Figure~\\ref{fig:CH-CIR} depicts the interplay between slow and fast solar wind streams. After a coronal hole passed the central part of the solar disk, in-situ measurements reveal about 1--2 days later an increase in the density and magnetic field, and about 3--4 days later in the plasma speed \\citep[][]{vrsnak07_sw}. Since coronal holes, and with that the fast component of the solar wind, are long-lived structures, SIRs can often be observed for several solar rotations, and are correspondingly called co-rotating interaction regions (CIRs) when observed more than once. The leading edge of a CIR represents a forward pressure wave and the trailing edge of a CIR a reverse pressure wave \\citep[cf.\\, right panel of Figure~\\ref{fig:CH-CIR}; for more details see the review by][]{cranmer17}. These waves may develop into shocks, and as such, large periodically recurrent coronal holes may cause geomagnetic storms roughly appearing with the frequency of the solar rotation, i.e., every 27 days \\citep[e.g.,][]{Rotter12}. During times of low solar activity, induced storms by recurrent CIRs may put equally much energy into the Earth's magnetosphere-ionosphere system as CMEs \\citep[e.g.,][]{Richardson01,tsurutani06}. On average, the strongest geomagnetic storms due to CIRs occur during the early declining phase of a solar cycle \\citep[e.g.,][]{Verbanac13,Grandin19}. Compared to CMEs, CIRs may drive prolonged geomagnetic activity and cause strong high energy particle enhancements in the Earth's radiation belts \\citep[e.g.,][]{Reeves03,Miyoshi13,Kilpua15}. As the solar wind parameters vary with the level of solar activity, so does their geoeffectiveness \\citep[see e.g.,][]{Jian11,Richardson12,Watari18}. There is substantial effort in the solar and heliospheric physics community to improve the understanding and modeling of the spatial and temporal distribution of solar wind plasma and magnetic field properties \\citep[see][]{Cranmer19}. \n\n\\begin{figure*}\n    \\centering\n\\includegraphics[width=1.\\textwidth]{wind4.pdf}\n\\caption{Left: SDO\/AIA composite image of the wavelength channels 211-193-171\\AA~from June 30, 2012 showing the reduced density region of a coronal hole (shaded area). At the time t0, the coronal hole reaches a central position. From in-situ data at 1 AU about 1 day later the maximum in the density\/magnetic field is measured and about 4 days later the maximum in the speed\/temperature. Right: Fundamental processes involved in the 3D dynamics of stream evolution, adapted from \\cite{pizzo78}.}\n\\label{fig:CH-CIR}      \n\\end{figure*}\n\nSince the source regions of slow and fast streams on the Sun, namely closed and open magnetic field, are different, their intermix affects detailed analyzes of the solar wind. Therefore, it is suggested that solar wind studies should be organized by the origin of the solar wind plasma \\citep{schwenn83,Zhao09,Borovsky19}. As minimum requirement, it is accepted to distinguish between the different solar wind structures by their in-situ measured plasma (density, speed, temperature) and magnetic field characteristics. A categorization scheme developed by \\cite{xu15} can be applied to separate the solar wind plasma into four types, namely, coronal-hole-origin (fast solar wind), streamer-belt-origin (slow solar wind), sector-reversal-region (plasma from top of helmet streamers), and ejecta (solar transients, such as CMEs). \n\n\n\\begin{figure*}\n    \\centering\n\\includegraphics[width=0.7\\textwidth]{wind5.pdf}\n\\caption{STEREO-A in-situ solar wind measurements of a SIR and identification of the stream interface (SI; given by a magenta vertical line marking the peak of the total perpendicular pressure). As blue vertical line, the heliospheric current sheet (HCS) is marking the sector structure of different interplanetary magnetic field polarity. Top to bottom panels: pitch-angle distribution data of suprathermal electrons, solar wind proton bulk speed, proton number density, proton temperature, entropy, total magnetic field intensity, the ratios of the radial and transversal component of the magnetic field, total perpendicular pressure. Taken from \\cite[][]{jian09}. }\n\\label{fig:jian09}      \n\\end{figure*}\n \n\nFigure~\\ref{fig:jian09} shows typical plasma and magnetic field characteristics for a well defined SIR at 1~AU. The SIR measurements reveal an abrupt drop in density, a simultaneous rise in the proton temperature, and an East-West flow deflection \\citep[see e.g.,][]{Jian06,Jian19}. The stream interface (SI), given by a rather symmetric profile in the total perpendicular pressure peaking shortly after the density maximum, separates originally slow, dense plasma from originally fast, thin plasma back at the Sun \\citep[e.g.,][]{Wimmer-Schweingruber97}. The behaviour of the suprathermal electrons gives additional information about the topology, hence, connectivity of the interplanetary magnetic field lines between the Sun and the observer. That also allows to investigate, e.g., the Parker spiral versus non\u2010Parker spiral orientation of the magnetic field and the related open flux versus closed or disconnected flux natures of the magnetic field \\citep[see][]{owens13}. SIR signatures differ clearly from those of ICMEs (cf.\\,Figure~\\ref{fig:ICME}) and as such, the specific characteristics can be applied to identify SIRs and CMEs from visual inspection of the in-situ solar wind plasma and magnetic field measurements. Together with the information of transit times (can be roughly derived from the average in-situ speed of the specific structure) we can link the in-situ signatures back onto the solar surface to study their relation. Table~\\ref{tab:1} gives average values and standard deviations for the different structures observed, such as slow and fast speed solar wind streams, CME sheath and magnetic ejecta. The time range used for that statistics covers the years 1976--2000 \\citep[see][]{Yermolaev09}. As can be seen, the values reveal large standard deviations reflecting the variations of the interplanetary conditions over the solar cycle. Similar as the activity level changes over time with respect to the occurrence of sunspots, active regions, flares and CMEs (see also Section\\ref{subsec:cme}), also the occurrence of coronal holes varies having implications on the global structuring of interplanetary space \\cite[e.g.,][]{harvey02, Heinemann2019StatisticalCATCH}.\n\n\n\\begin{table}\n\\caption{Properties of the different solar wind types derived from OMNI data analyzed over the period 1976\u2013-2000. Average values and standard deviations are given for the flow speed ($v_{\\rm p}$), proton density ($n_{\\rm p}$), total magnetic field ($B$), and proton temperature ($T_{\\rm p}$)}. In addition, the geoeffectiveness of the different solar wind types is given as measured by the disturbance storm time (Dst). Taken from Table 3 in \\cite{Yermolaev09}.\n\\label{tab:1}      \n\\begin{tabular}{lllll}\n\\hline\\noalign{\\smallskip}\n& fast wind & slow wind & CMEs (shock-sheath) & CME (magnetic ejecta)  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$v_{\\rm p}$, km~s$^{-1}$ & $>$ 450--500 & $<$ 400--450 & $\\sim$450$\\pm$110 & $\\sim$410$\\pm$110 \\\\\n$n_{\\rm p}$, cm$^{-3}$ & 6.6$\\pm$5.1 & 10.8$\\pm$7.1 & 14.3$\\pm$10.6 & 10.1$\\pm$8.0 \\\\\n$B$, nT & 6.4$\\pm$3.5 & 5.9$\\pm$2.9 & 8.5$\\pm$4.5 & 12.0$\\pm$5.2 \\\\\n$T_{\\rm p}$ $\\times10^4$ K & 13.1$\\pm$11.8 & 4.4$\\pm$4.4 & 12.9$\\pm$17.6 & 4.5$\\pm$6.6 \\\\\nDst, nT & $-$28.7$\\pm$25.9 & $-$10.7$\\pm$18.2 & $-$21.5$\\pm$33.0 & $-$52.1$\\pm$45.8 \\\\\n\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Background solar wind}\nOpen magnetic flux from the solar surface structures interplanetary space and with that influences CME and SEP propagation behavior. We assume that the majority of open flux originates within coronal holes. This is supported by an empirical relation linking the size of coronal holes observed on the solar surface to the in-situ solar wind plasma and magnetic field measured at 1AU a few days later \\citep[][]{vrsnak07_sw}. In detail, the width of a coronal hole, i.e., its longitudinal extension, is found to be strongly related to the in-situ measured peak speed \\citep{garton18}. Deriving the solar wind characteristics stemming from a specific coronal hole, is found to be tricky as each coronal hole evolves rather individually and the surrounding solar surface structures may play a major role in shaping the interplanetary solar wind \\citep{Heinemann20}. Still, an open discussion is the discrepancy between estimates of open solar magnetic flux from remote photospheric and in-situ spacecraft observations that may differ by as much as a factor of two \\citep[e.g.,][]{Arden14,Linker17,wallace19}. This suggests a fundamental issue in our understanding about the topology of the coronal magnetic field and the energization of plasma, hence, the acceleration of fast solar wind flows. Recent studies find that the open magnetic field within coronal holes is predominantly concentrated in unipolar magnetic flux tubes with high outflow velocities but covering only a fraction of about 10\\% of the coronal hole area \\citep[see][]{akiyama13,wiegelmann14,Hofmeister2017Characteristics24}. For more details on modeling the coronal magnetic field and open solar flux see e.g., the Living Reviews by \\cite{mackay12} and \\cite{lockwood13}. \n\n\nDue to the rather slow evolution of coronal holes, a legitimate assumption is that the solar wind parameters do not vary strongly over the duration of an entire solar rotation. Based on that, to forecast the occurrence of high speed solar wind streams at Earth, L1 spacecraft data may simply be forward shifted over one Carrington rotation period (27.28 days). These so-called persistence models are found to work remarkably well \\citep[see e.g.,][]{Owens13_persistence,Reiss2016VerificationModels}. However, spacecraft used for forecasting may be located over different latitudes and in-situ measurements gather the characteristics of rather localized structures within that large scale three dimensional objects. The spatial restrictions of such localized structures are demonstrated in Figure~\\ref{fig:gomez11}, showing how solar wind streams appear differently as measured by STEREO-A and STEREO-B spacecraft that are separated in latitude by $\\sim$10 degrees. The data cover two Carrington rotations (2076 and 2077) revealing a solar wind stream well observed by STEREO-B, but clearly missed by STEREO-A. For comparison, the top panel of Figure~\\ref{fig:gomez11} gives the photospheric magnetic field configuration on the Sun using magnetic field extrapolations from GONG data marking open and closed field lines. With such inevitable differences in the latitude between the measuring spacecraft, rather large uncertainties in the 1~AU measured speed of a particular coronal hole are obtained \\citep[see][]{Hofmeister18,Owens19}. Recent studies suggest that solar wind forecasting based on persistence models may work best when using a combination of spacecraft located behind Earth (L5 position or STEREO data from time-varying spacecraft position) and empirical or numerical solar wind modeling \\citep[see e.g.,][]{Opitz10_Multispacecraft,Temmer18,Owens19,Bailey20}.\n\n \n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{wind6.pdf}\n\\caption{Top panel: synoptic map from GONG overlaid with magnetic field extrapolation results from the PFSS \\citep[potential field solar surface; see e.g.,][]{Schrijver03} model showing the location of open field lines at the ecliptic plane. Bottom panel: solar wind bulk speed measured by PLASTIC instruments aboard STEREO-A (red) and STEREO-B (blue). Colored bands on the top part of this panel represent the interplanetary magnetic field polarity. Black arrows mark the solar wind stream observed over two rotations (CR 2076 and 2077) by STEREO-B, but missed by STEREO-A due to the latitudinal separation of the spacecraft. Taken from \\cite{Gomez-Herrero11}.}\n\\label{fig:gomez11}      \n\\end{figure*}\n\n\n\n\\subsection{Solar wind structures affecting CME and SEP evolution}\\label{sec:affect}\nCMEs are known to change shape, accelerate or decelerate, depending on the background solar wind flow properties, as well as may change their propagation direction if encountering other magnetic structures (cf.\\,Section \\ref{subsec:cme}). From recent results it is found that many coronal fine structures and small scale magnetic flux ropes are embedded in the solar wind \\citep[first PSP\/WISPR results see][]{howard19}. Likewise, PSP in-situ measurements show that low-latitude coronal holes might be an additional source of slow solar wind, causing close to the Sun strong fluctuations in the solar wind flow \\citep[][]{Bale19}. Such local solar wind dynamics, comprising of numerous voids, compact small- to large-scale sized structures (``woodgrain'' appearance), clearly have impact on the evolution of a CME and with that complicates forecasting. As example,  Figure~\\ref{fig:sw-structure} shows a disrupted CME front close to the Sun as consequence of an interaction with a high speed solar wind stream emanating from the southern polar coronal hole. The increased speed of the ambient solar wind flow and stretched magnetic field, causes a weaker compression and, hence, deviations from the shape of a circular\/elliptic CME front. The structured solar wind in the outer corona is also revealed from specially noise reduced post-processed STEREO\/HI image data applying an algorithm to dim the appearance of bright stars and dust \\citep[right panel of Figure~\\ref{fig:sw-structure}; cf.,][]{DeForest18}. See also  recent results from PSP image data resolving small-scale flux ropes, density structures and fluctuations in solar wind streamers \\citep{howard19}. Besides adjusting to the ambient solar wind flow speed, CMEs also tend to rotate for adjusting to the ambient magnetic field \\citep[e.g.,][]{Yurchyshyn01,Yurchyshyn09,Vourlidas11,Isavnin14}. \n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{wind7.pdf}\n\\caption{a) Left panel shows a STEREO\/HI difference image with the distorted CME front (red arrow). Right panel gives SDO\/AIA 193\\AA~EUV image with the CME source region (red box) and the southern coronal hole. The fast solar wind stream out of the coronal hole (depicted with blue arrows) deforms the CME front. b) Highly structured solar wind flow close to the Sun from a STEREO\/HI image after computer processing (credit: NASA's Goddard Space Flight Center\/Craig DeForest, SwRI).}\n\\label{fig:sw-structure}      \n\\end{figure*}\n\n\nSEPs propagate along the magnetic field lines, hence are directly guided by the interplanetary magnetic field structure. During CME events, SEP path lengths are found to be much longer compared to quiet solar wind conditions \\citep[e.g.,][]{masson12}. For SEPs, which are injected into the legs of a CME, the magnetic path lengths from the Sun to Earth may vary largely, up to a factor of two, depending on the specific width of the CME \\citep{reames09}. CME geometry as well as kinematics (propagation of shock front) and time-dependent changes play an important role with respect to particle acceleration processes. In that respect, SEPs can be taken as probes as they map the interplanetary magnetic field structure \\citep{Reames17_book}. \n\n\n\n\\subsection{Preconditioning of interplanetary space}\\label{sec:preconditioning}\nThe correct simulation of the prevailing background solar wind structures in interplanetary space is important for a reliable CME-SEP forecasting. Using MHD modeling to simulate the solar wind distribution works well for low solar activity. However, increased solar activity changes the magnetic field in the photosphere, which serves as main observational model input, more quickly and propagating CMEs strongly disturb the interplanetary background solar wind making models tend to fail \\citep[see e.g.,][]{Lee09,Gressl14}. This effect is commonly known as \\textit{preconditioning} of interplanetary space, which alters the initial conditions for CME and SEP evolution. It is found that single CME events may disturb the slowly evolving solar wind flow for 2--5 days \\citep{Temmer17_preconditioning,janvier19} and most strong preconditioning effects are obtained due to CME-CME or CME-CIR interacting events as they form complex magnetic structures \\citep[e.g.,][]{gopalswamy00_accelCME,Burlaga02,Harrison12,dumbovic19}. Multiple CME activity is supposed to cause a decrease of density in interplanetary space and a radial stretching of the interplanetary magnetic field. That leads to a low drag force acting on subsequently propagating CMEs \\citep[see e.g.,][]{Farrugia04,Maricic14}. The STEREO-A directed July 23, 2012 event was one of the fastest CMEs ever recorded and propagated over a 1AU distance in less than 21 hours \\citep{liu14_nature}. It could be shown that the drag parameter, due to a preceding CME from July 19, 2012, was lowered by one order of magnitude \\citep{temmer15}. If the super-fast CME from July 2012 would have been Earth directed, it would have caused an extreme Space Weather event with an estimated Dst of $-$600 to $-$1100~nT \\citep[e.g.][]{ngwira14,baker13}. Due to increased fluctuations and extended periods of negative $B_{\\rm z}$ most intense geomagnetic storms occur for complex interacting CMEs \\citep[e.g.,][]{wang03,Farrugia06,Xie06,dumbovic15,scolini20}. \n\n\nFor SEPs, the preconditioning that comes from a preceding CME has consequences in the seed population. The presence of a previous CME is found to increase the probability for the subsequent fast CME to be SEP-rich \\citep{Gopalswamy02,Gopalswamy04,Kahler05}. The amount of particles that can get accelerated is increased for multiple CMEs and the increased turbulence in the interaction region is likely to accelerate the particles more efficiently to higher energies \\citep[so-called twin-CME scenario as proposed by][]{Li05}. Furthermore, the magnetic structure configuration in the sheath region for ICMEs is changed as the CME propagates in interplanetary space, by which the magnetic connectivity is altered \\citep[see also review by][on CME sheath regions]{kilpua17}. Interacting CMEs are also found to be more often related to widespread SEPs that can be observed all around the Sun using multiple viewpoints \\citep[e.g.,][]{Dresing16,Gomez-Herrero17}. For more details on CME-CME interaction and SEPs, I refer to the review by \\cite{lugaz17}.\n\n\n\n  \nOver the solar cycle, the CME occurrence rate lies on average in the range of 0.3 per day during solar minimum phase and about 4--5 per day during solar maximum phase \\citep[e.g.,][]{St.Cyr00}. With average CME transit times from Sun to 1AU of the order of 1--4 days we can assume that during times of increased solar activity CME-CME interaction happens rather frequently. Reliably modeling these dynamic conditions in interplanetary space is therefore key for improving space weather forecasting capabilities. \n \n\n\n\\section{The chain of action on the example of the September 2017 events}\\label{sec:active}\n\n\nThe September 2017 activity phenomena are so far the most well studied strong Space Weather events in our modern space research era. Therefore, the chain of action can be described in great detail, especially with respect to the solar surface signatures and deduced parameters. The multiple disturbances can be well connected from Sun to Earth (and also up to Mars) and show how complex interactions lead to preconditioning effects and strong geoffectiveness.\n\n\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{action1.pdf}\n\\caption{High-resolution SDO\/AIA multi-wavelength imagery from September 6, 2017 (left: 171\\AA, middle: composite from 211-193-171\\AA) and SDO\/HMI line-of-sight magnetic field (right panel). The prominent active region NOAA 12673 caused the strongest eruptions during solar cycle 24 including SEPs. Courtesy of AIA team, adapted from \\url{http:\/\/suntoday.lmsal.com}.} \n\\label{fig:sdo}        \n\\end{figure}\n\nA strong emergence of magnetic flux in the southern hemisphere of the Sun rapidly led to the development of active region NOAA 12673 into a complex $\\alpha\\beta\\gamma$ magnetic field configuration. As consequence, between September 4--10, 2017 that active region released a series of major flare events, which were actually the largest in more than a decade. These multiple events caused very strong geomagnetic disturbances with a minimum Dst of $-$142 nT on September 7, 2017. Additional minor storms were produced by high speed solar wind streams that arrived together with the transient events. Figure~\\ref{fig:sdo} gives EUV image data taken with SDO\/AIA on September 6, 2017 in different wavelength ranges showing the hot corona from about 0.6 to 2~MK. The wavelength ranges cover 171\\AA\\ (left panel), and with a triple-filter 211\\AA\\ (red), 193\\AA\\ (green), and 171\\AA\\ (blue) to highlight different temperatures (middle panel). The line-of-sight magnetogram for the same day taken with SDO\/HMI reveals the magnetic field in the photosphere (right panel).  \n\n\\begin{figure}\n  \\includegraphics[width=1.9\\textwidth,angle=90]{action2.pdf}\n\\caption{September 2017 flare-CME-SEP series taken and adapted from the CDAW catalogue (proton-height\/time-X-ray plots - PHTX - can be found under \\url{https:\/\/cdaw.gsfc.nasa.gov\/CME\\_list}). Top panel shows solar energetic particle events for protons in the GOES energy channels $>$10, $>$50 and $>$100~MeV. Middle panel gives the CME height-time profile as measured from LASCO (colors give the main propagation direction - see legend to the left in the middle panel). Bottom panel gives the GOES flare SXR emission disk integrated over the wavelength ranges 0.5--4.0 and 1.0--8.00~\\AA. }\n\\label{fig:cdaw}      \n\\end{figure}\n\nFigure~\\ref{fig:cdaw} gives a combined overview on the series of activity pulses covering the two major flare-CME-SEP events during September 6--10, 2017. In total, active region NOAA 12673 produced an intense solar storm period revealing five X-class flares and 39 M-class flares (including the two largest flares from solar cycle 24, the X9.3 flare on September 6, 2017 and the X8.2 flare on September 10, 2017). The first SEP event was measured in the GOES channels at 1AU over September 6, 2017 12:15UT--September 7, 2017 23:25UT and is related to the halo CME that occurred on the Sun on September 6, 2017 12:24UT (first observation in LASCO\/C2) with a projected speed of 1570~km\/s over the coronagraph field of view. The CME has the source region location coordinates S08W33 and is launched together with a flare that started on September 6, 2017 11:53UT and reached X9.3 class in the measured GOES SXR flux. The second flare-CME-SEP event occurred September 10, 2017 (SEP: September 10, 2017 16:25UT--September 11, 2017 11:40UT; halo CME: September 10, 2017 16:00 with 1490~km\/s; flare: X8.2 class on September 10, 2017 15:35UT) with the source region located behind the west limb. For both events long-duration high-energy gamma-ray emission was detected by the Fermi-Large Area Telescope, having durations exceeding 15 hours and with that being the third and fifth largest among all detected \\citep[][]{longo17,gopalswamy18,omodei18}. \n\n \n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{action3.pdf}\n\\caption{Top panels: EUV images from September 6, 2017 observed with SDO AIA (running ratio of composite image data) and Proba-2\/SWAP (difference image of 171\\AA\\ data). Bottom panels: LASCO\/C2 and C3 coronagraphs covering a field of view up to 30 solar radii. The generated SEPs are accelerated to relativistic speeds producing spikes in the image data (``snowstorm'' effect). This event was the first flare event in a sequence of X-class flares on 6, 7, and 10 September 2017 causing strong disturbances at Earth and Mars. Note that the field of view of Proba-2 is larger compared to AIA and that due to different CCD techniques no saturation effects are visible in Proba-2. Movies for each panel are available online.}\n\\label{fig:sep17}      \n\\end{figure}\n\nFigure~\\ref{fig:sep17} impressively shows the manifestation of the eruptive event from September 6, 2017 around 12~UT in solar observations using EUV image and white-light coronagraph data. To make changes visible, different techniques are used, such as a running ratio images as applied on the SDO\/AIA composite image and difference images as applied on the Proba-2\/SWAP 171\\AA\\ image or on the LASCO white-light images. SDO and Proba-2 processed images nicely reveal the flare component (X9.3 class), a coronal surface wave that was ignited, and the CME flank on-disk as well above the limb visible in EUV due to compression and heating of the plasma. The off-disk counterpart of the coronal wave is observed in white-light as the shock wave surrounding the CME body. In white-light coronagraph data from LASCO C2 and C3 the CME is detected as partial halo event. The projected CME speed over the field of view up to 20 solar radii was measured from LASCO data with $\\sim$1500~km\/s. SEPs that are produced by relativistic protons hit the CCD camera of LASCO within minutes and generate the well-known ``snowstorm'' effect. Heavy snowstorms strongly disturb the image quality which may complicate the analysis of halo events (especially during times of increased solar activity). \n\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{action4.pdf}\n\\caption{Left panel: location of flare ribbons (orange contour), dimming areas (cyan contour) and post-eruptive arcade (PEA, magenta contour) derived for the eruptive event from September 6, 2017. The greyscale background HMI line-of-sight magnetogram is scaled $\\pm$100~G with black and white representing negative and positive polarities, respectively \\citep[adapted from][]{scolini20}. Right panel: cartoon giving the relation between flare, CME erupting flux rope, coronal wave and dimming areas. }\n\\label{fig:scolini}      \n\\end{figure}\n\n\nFigure~\\ref{fig:scolini} shows for the September 6, 2017 event the associated dimming region and post-eruptive arcade together with the magnetic field. The core dimmings are accompanied by secondary dimmings covering larger areas. The cartoon to the right in Figure~\\ref{fig:scolini} depicts the relation between different features of an eruptive event and associated solar surface signatures. E.g., measurements of the post-eruptive arcade (PEA) areas and their underlying magnetic field were taken to derive the reconnected flux to feed CME propagation models using magnetized CMEs \\citep[][]{scolini20}. The orientation of the PEA with respect to the underlying magnetic field, revealed the handedness of the flux rope. Coronal waves initiated close to the eruption side, hint towards the main compression direction of the eruption, hence, a southward propagation direction of the associated CME (confirmed by the coronagraph images; cf.\\,Figure~\\ref{fig:sep17}). Therefore, monitoring surface structures gives additional (and early) information about a potential Space Weather event that might affect Earth, and moreover, provides valuable input for modeling efforts.\n\n\nThe multiple CME events from September 6--9, 2017 preconditioned interplanetary space and were interacting, which intensified their geomagnetic effects. Figure~\\ref{fig:werner} shows the complex in-situ signatures revealing multiple shocks and magnetic ejecta regions. Note that the shock of ICME2 propagated into the magnetic structure of ICME1. These so-called ``shock-in-a-cloud'' events are found to cause stronger geomagnetic responses than isolated geoeffective CMEs \\citep{lugaz15}. For the September 2017 events, the shock compression might have enhanced the geoeffectiveness by a factor of 2 \\citep{shen18}.\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{action5.pdf}\n\\caption{Interacting in-situ signatures for the multiple CME events from September 6--9, 2017. Magnetic field and plasma data are given from WIND at L1. Top to bottom: Total magnetic field and vector components, dynamical pressure, proton temperature, and plasma-beta. Taken from \\cite{Werner2019ModelingWSA-ENLIL+Cone}. }\n\\label{fig:werner}      \n\\end{figure}\n\nAs the region rotates close to the West limb of the Sun, the event from September 10 occurred as partially occulted and, because of the favorable magnetic connectivity, generated a GLE. Figure~\\ref{fig:bruno19} gives the SEP proton fluxes measured near Earth (ACE, GOES) together with the geomagnetic Dst index and neutron monitor profiles showing the Forbush decrease related to the arrival of the closed magnetic structure \\citep[for more details see][]{bruno19}. Due to the eruption site located close to the limb, measurements of the CME kinematics were less strongly affected by projection effects. From EUV observations of the wide field of view Solar Ultraviolet Imager on board the GOES-16 spacecraft exceptionally high values for the CME acceleration and speed were derived, revealing the huge expansion in both, radial and lateral direction \\citep{Gopalswamy18_sept17,seaton18,veronig18}. This caused the initiation of a coronal wave propagating over the entire solar surface \\citep{liu18}.\n\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{action6.pdf}\n\\caption{Proton intensities measured by ACE\/EPAM, GOES\/EPEAD, and GOES\/HEPAD. Dst index and count rate variations registered by SOPO and MGDN neutron monitor stations. The vertical dotted and dashed lines mark the onset of the SEP events and the time of the shocks, respectively. The green, orange, and gray areas indicate the periods of the ICME, magnetic cloud, and high speed streams, respectively. Taken from \\cite{bruno19}.}\n\\label{fig:bruno19}      \n\\end{figure*}\n\n\nThe September events did not only affect the Earth, but were also registered at Mars. Energetic particles were observed by MAVEN orbiter, and on the surface by the Curiosity Mars rover \\citep[e.g.,][]{hassler18}. Many further aspects of the September 2017 activity period are well represented in a collection of publications in the AGU Space Weather journal\\footnote{\\url{https:\/\/agupubs.onlinelibrary.wiley.com\/doi\/toc\/10.1002\/(ISSN)1542-7390.SW-SEPT2017}}.\n\n\n\\section{Space weather forecasting models}\\label{sec:6}\n``Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.'' by \\cite{box76}. \\\\\n\n\nFor comprehensive investigations of the structure of interplanetary space, propagating transient events, and magnetic connectivity, we need to rely on modeling efforts. Having only scarcely distributed single point in-situ measurements and the lack of plasma and magnetic field information derived by applying remote sensing imaging techniques, we are clearly limited in our ability to assess the performance and reliability of these models hindering improvements. Nevertheless, the close collaboration between model developers and observational community is a need to push forward our understanding of CME-solar wind coupling and interaction processes in interplanetary space. Also on kinetic scales, particle acceleration processes and trigger mechanisms of flare-CME-SEP events need to be supported by modeling, as from observational data the information are not sufficient to make conclusive interpretations about the underlying physics. In recent years substantial progress has been made in efficiently combining models with observations. Data assimilation techniques are known to hugely improve operational forecasting models, as observational data are incorporated in a self consistent way into numerical models for increasing the model accuracy. Examples about the application to solar data can be found in e.g., \\cite{Schrijver03,arge10}. In heliospheric physics this method is difficult to apply due to the relatively sparse observations. Recent efforts comprise in-situ measurements from 1~AU to update and improve inner-boundary conditions of solar wind models \\citep[see e.g.,][]{lang19}. \n\n\nForecasting flares and SEPs with a lead time of at least a few hours is related to the forecasting of the evolution of active regions producing eruptive solar flare events. However, that means to estimate the emergence of magnetic flux from below the photosphere that is not accessible from direct observations. Therefore, statistical relations between photospheric magnetic field characteristics of active regions and flare occurrence are usually used for solar flare forecasting. More precisely, it is the time evolution of the magnetic parameters that plays a major role, but is very complex to derive. The prediction of major flares seems to be more easily achievable compared to flares of medium to low energy, however, the uncertainties coming from the statistical approach are rather large preventing from more accurate single event prediction \\citep[see e.g.,][]{schrijver07,georgoulis07,bloomfield12}. Real-time solar flare forecasting is provided by e.g., the EU project Flarecast\\footnote{\\url{http:\/\/flarecast.eu}} with a fully automated system and the NASA\/CCMC flare forecasting scoreboard provides a platform to test different forecasting methods fostering further development of the available algorithms\\footnote{\\url{https:\/\/ccmc.gsfc.nasa.gov\/challenges\/flare.php}}. In general, having standardized metrics is a necessity in order to reliably validate and cross-check the performance of the different flare forecasting tools \\citep{barnes08}. See also a review on the origin, early evolution and predictability of solar eruptions by \\cite{green18}.\n\n\nFor forecasting CME arrival times and impact, a vast amount of models is currently available in various levels of complexity \\citep[see, e.g., Table~1 in][]{riley18}. Simple models, such as empirical relations between CME transit time and speed from statistical results provide tools to derive an average behavior of CME propagation in interplanetary space \\citep{Gopalswamy01}. The forecasting accuracy can be significantly improved, when using observational data that can track the CME kinematics to beyond a distance of 50 solar radii (e.g., using image data from STEREO\/HI, or radio IPS) where the CME is assumed to evolve in a linear way \\citep[see][]{colaninno13,rollett16,hess17}. Several analytical models include the physics of drag force (viscous, aerodynamic, hybrid - that means a linear or quadratic relation between solar wind and CME speed difference) that a CME experiences in interplanetary space. A widely used analytical model is the drag-based-model \\citep[DBM; see][]{vrsnak13}. It applies the aerodynamic drag as analogon for the MHD drag force exerted on the CME embedded flux rope. \nMore sophisticated are numerical MHD models such as e.g., EUHFORIA \\citep{pomoell18_euhforia}, ENLIL \\citep{odstrcil99_enlil}, CORHEL \\citep{Riley12} or SUSANOO \\citep{shiota16}. Besides simulating transient events, MHD models also cover the variation in the background solar wind \\citep[see also][]{arge00}. A compilation of data, services and tools can be found at CDPP (Plasma physics data center in France)\\footnote{\\url{http:\/\/cdpp.irap.omp.eu}}. For CMEs, the disentanglement between shock and magnetic ejecta is an important issue highly relevant for forecasting. Numerical models usually use a simple pressure pulse to ignite a CME shock front, but do not include the magnetic structure. There are recent efforts of magnetized CMEs incorporating the observed reconnected magnetic flux at the Sun as model input parameter for the CME flux rope \\citep[e.g.][]{Scolini19,singh19}. \n\nFor validation purposes and for increasing the awareness of the limitations of the accuracy of the model results, the uncertainties of observational model input data need to be quantified. \\cite{riley18} summarized hit and miss statistics from the CCMC scoreboard\\footnote{\\url{https:\/\/kauai.ccmc.gsfc.nasa.gov\/CMEscoreboard\/}}, a CME prediction board that gives the possibility to use different models in real-time forecasting (crude facts approach that challenges models and their users). Metadata and metrics are suggested to give the community a common base for an objective inter-comparison of their models \\citep{Verbeke19}. A review on the current status and open issues on CME propagation and forecasting methods is given by \\cite{vourlidas19}.\n\nCompared to the prediction of CME arrival times and impact speeds, SEP forecasting is a more tricky issue as the accelerated particles propagate with fractions of the speed of light and the lead time is of only a few minutes. Several models are available, mostly using empirical relations based on statistical relations with CME-flare locations or type II radio burst occurrence at decametric\u2013hectometric (DH) wavelengths. Current models cover e.g., PROTONS \\citep{balch08}, PPS \\citep{kahler07}; ESPERTA \\citep{Laurenza09}, FORSPEF \\citep{anastasiadis17} as well as physics based models such as e.g., SOLPENCO \\citep{Aran06} or HESPERIA \\citep[][and references therein]{Malandraki18_book}. Incorporating CME characteristics is difficult as the real-time coronagraph image data, from which usually the CME parameters are derived, are delivered with some delay. Nevertheless, the SEPForecast tool resulting from the EU project COMESEP\\footnote{\\url{http:\/\/comesep.aeronomy.be}} also uses CME parameters as model input \\citep{Dierckxsens15}. The application of solar surface proxies for some CME parameters might overcome that drawback. A detailed comparison of false alarm rates for SEP forecasting is presented in e.g., \\cite{Alberti17}. The current status of forecasting and nowcasting of SEPs and open questions is given in a review by \\cite{Anastasiadis19}. \n\n\nFurther approaches for improving forecasting purposes also cover ensemble models incorporating the uncertainties in the observational data \\citep[e.g.,][]{lee13,mays15,Dumbovic2018ThePropagation,Amerstorfer2018EnsembleImagers}, and machine learning techniques \\citep[see e.g.,][]{camporeale19}. In addition to methods covering large statistics, case studies that model Space Weather events from Sun to Earth, give a wealth of detailed information from which we can hugely improve our understanding of flare-CME-SEP events. Community centers like ESA\/VSWMC\\footnote{\\url{https:\/\/esa-vswmc.eu}} or NASA\/CCMC\\footnote{\\url{https:\/\/ccmc.gsfc.nasa.gov}} cover the increased need of computational power and appropriate IT infrastructure and provide a platform for models to be tested and actually used. Such platforms are also the driveway for R2O (research to operation) activities and where scientists and users meet. A collection of Space Weather tools is presented at the ESA\/SSA website, including services from the European Expert Service Centers and their individual Expert Groups (cf.\\,Figure~\\ref{fig:ssa}). \n\n\\begin{figure*}\n  \\includegraphics[width=1.\\textwidth]{model1.pdf}\n\\caption{Webpage of the SSA Space Weather Service Network where Expert Service Centers provide their tools and services covering Space Weather forecast and overview from the solar-heliosphere perspective to space radiation, ionospheric and geomagnetic conditions. ESA\/SSA: \\url{http:\/\/swe.ssa.esa.int}}\n\\label{fig:ssa}      \n\\end{figure*}\n\n\n\\section{Concluding remarks}\nNowadays, the term Space Weather covers basic research as well as application and is a platform for strong interdisciplinary research bringing the domains of solar-, heliospheric, and geo-physics closer together. The real-time forecasting results of flare-CME-CIR-SEP events at Earth using operational tools reveal that we still face large uncertainties. The reason is on the one hand the huge complexity of the solar phenomena and their unknown coupling processes and propagation behavior in interplanetary space. On the other hand, there is a clear lack of accurate enough measurements to properly feed the available models. The source of high errors and inaccuracies in the measurements is due to extraction of remote sensing data which are affected by projection effects, idealized assumptions of magnetic field extrapolations (lack of coronal magnetic field measurements) and 3D reconstructions, as well as localized in-situ measurements making it tricky to validate all the results. For understanding and forecasting the geoeffectiveness of Earth affecting Space Weather events, a far bigger challenge for the future is the modeling of the interplanetary magnetic field orientation and variation.  \n\n\nA clear advantage is given by the combination of several instruments having multiple views on the Sun covering different longitudes and latitudes. STEREO-A is still operational providing data material from varying angles. PSP will be the first spacecraft that comes as close to the Sun as 10 solar radii and aims to answer questions about the origin of the solar wind, how is it accelerated or about SEP acceleration and transport processes. The mission already gathered unprecedented data enabling new interpretation and finding on the source regions of the solar wind. Solar Orbiter revealed first data in July 2020 and will study the solar surface and its magnetic field in great detail. The spacecraft will have a highly elliptical orbit to progressively move to a more inclined orbit out of ecliptic. That perspective will give new insight onto the polar regions of the Sun and is expected to improve magnetic field modeling and to better understand the solar wind from coronal holes in the polar regions.\nConjunctions with other active missions (such as STEREO, L1 missions, MAVEN), complemented by ground-based instruments, will give unique possibilities to investigate the evolution of solar flares, CMEs, or SEPs, as well as coronal holes and solar wind high speed streams. Future missions, such as the ESA \\textit{Lagrange} mission to L5 (launch is planned for 2027) or polar missions, as the proposed Solaris Solar Polar Mission \\citep{hassler20}, will provide more valuable data and fill the gaps in our understanding of magnetic field connectivity and coupling processes between open and closed magnetic field structures in interplanetary space. \n\n\nWe are facing challenging and exciting times having a wealth of data at hand and promising future missions yet to come. Many of the currently available data are well prepared in ready-to-use catalogues, waiting to be explored. However, we still need to obtain data from L4\/L5 Lagrange points revealing the necessary side views on a regular basis to fully track eruptions with low projection effects and for gaining a better understanding of their characteristics that affect Earth. Moreover, we need to improve solar wind models to be used in MHD simulations for properly determining CME propagation. With that we will successfully enhance the reliability of Space Weather forecasting tools and models.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro_sec} \n\nNon-local models are used for the description of several complex phenomena for which a local approach is inappropriate or limiting. In the past, they have been successfully employed in different fields, such as material sciences (\\cite{bates2006some}), dislocation dynamics in crystals (\\cite{dipierro2015dislocation}), image processing (\\cite{gilboa2008nonlocal}) or elasticity (\\cite{silling2000reformulation}).\n\nNon-local terms appear also in many diffusion models in biology, where they describe biological systems on scales that are convenient to observation and data collection. A classical example is the chemotaxis phenomena, which can be modeled by non-local differential  equations of diffusion type. For instance, in \\cite{burger2006keller} a non-local Keller-Segel model is used to study the chemotaxis of a cell population in which the chemicals are produced by the cells themselves. Furthermore, one can find numerous examples of other non-local diffusion phenomena in the book \\cite{okubo2013diffusion}, where they are presented several systems involving, for instance, random walks and random population diffusion. \n\nMotivated also by the aforementioned applications, these types of problems have recently raised the interest of the mathematical community. In particular, we may refer to the monograph \\cite{andreu2010nonlocal} for an extended mathematical survey containing different examples of non-local diffusion models. \n\nIn this paper, we consider a diffusion spatially non-local equation, where the non-locality is given by the presence of an integral kernel. In particular, we are interested in the analysis of control properties. \n\nControl Theory deals with dynamical systems that can be controlled, i.e. whose evolution can be influenced by some external agent. In this framework, a first problem one could face is the one of \\textit{controllability}, that is, to study the possibility of steering the system from one configuration to another by employing one or more controls applied through actuators. If controllability to a given final state is granted, one can try to reach this state by minimizing\nsome cost, thus defining an \\textit{optimal control} problem. The mathematical theory of optimal control has rapidly developed in the last decades into an important and separate field of applied mathematics. Actually, applications of optimal control of parabolic equations span a large spectrum of fields including aviation and space technology, robotics, movement sequences in sports, and the control of chemical processes.\n\nFor the non-local diffusion models that we are going to consider in the present work, which will be introduced in the next section, controllability has been already treated in some recent paper (see, for instance, \\cite{biccari2017null,fernandez2016null,lissy2018internal,micu2017local}). On the other hand, to the best of our knowledge, the analysis of optimal control problems was never addressed in this setting. Then, the main purpose of our work will be to give an insight on some classical optimal control question which arises naturally from the biological applications behind the mentioned models. In this framework, we will focus our attention on two different situations:\n\\begin{enumerate}\n\t\\item In a first moment, we will deal with a classical optimal control problem, in which we are interested in minimizing a given cost functional coming from the controllability theory. This cost functional will typically characterize the optimal control of minimal energy which allows to reach some predetermined configuration.\n\t\\item Secondly, we will consider the case of missing or incomplete initial data. This is a very typical situation when studying certain natural phenomena, for instance in ecology or genetics, where the employment of models with no missing data may lead to a misunderstanding of the underneath processes. \n\\end{enumerate} \n\nThis second problem will be addressed by using the concepts of \\textit{low-regret} or \\textit{least-regret} controls, which have been introduced by J.-L. Lions in \\cite{lions1992controle}, and may be well-adapted to the non-local problem we are considering. \n\nThese techniques were designed precisely for dealing with the optimal control of models where missing data are incorporated. They consist of transferring the optimal control problem with missing data to a classical one, relaxing the problem of control by a sequence of low-regret controls. This idea was firstly applied to linear models (\\cite{gabay1994decisions,lions1999duality}), and it was later extended to the nonlinear setting in \\cite{nakoulima2002no}. Afterwards, many other authors applied these ideas to other control problems for PDEs with incomplete data. The interested reader may refer, for instance, to \\cite{dorville2004low,jacob2010optimal,louison_1,mahoui2017pointwise}. In this work, we will see how the low-regret and least-regret approaches can be applied also in the the non-local context.\n\nThe present paper is organized as follows: in Section \\ref{sec:2}, we present the problem we are going to consider. In Section \\ref{sec:3}, we address some elementary optimal control question associated with our model, while in Section \\ref{sec:4}, we address the issue of missing initial data. Finally, Section \\ref{sec:5} is devoted to some numerical experiment. \n\n\\section{Problem formulation and existing controllability results}\\label{sec:2} \n\nLet $\\Omega$ be a bounded domain of $\\RR^d$, $d\\geq 1$, with boundary $\\partial\\Omega$ of class $\\mathcal{C}^2$. Given a time horizon $T>0$, we set $Q:=\\Omega\\times (0,T)$ and $\\Sigma:=\\partial\\Omega\\times(0,T)$. Let $K(x,\\theta,t)\\in L^\\infty(\\Omega\\times\\Omega\\times(0,T))$, and  consider the following linear non-local parabolic problem\n\\begin{align}\\label{direct_pb_y}\n\t\\begin{cases}\n\t\t\\dis\\frac{\\partial y}{\\partial t}-\\Delta y+ \\int_\\Omega K(x,\\theta,t)\\,y(\\theta,t)\\,d\\theta = v{\\bf 1}_{\\mathcal{O}}, & (x,t)\\in Q,\n\t\t\\\\\n\t\ty = 0, &(x,t)\\in \\Sigma,\n\t\t\\\\\n\t\ty(0,x) = y_0(x), & x\\in\\Omega.\n\t\\end{cases}\n\\end{align}\n\nIn \\eqref{direct_pb_y}, $y:=y(x,t)$ is the state function and $v:=v(x,t)$ is the control function. The latter acts on the system through the non-empty set $\\mathcal{O}\\in\\Omega$. Here, ${\\bf 1}_{\\mathcal{O}}$ denotes the characteristic function of $\\mathcal{O}$. \n\nMoreover, we assume that $y_0\\in L^2(\\Omega)$ and $v\\in \\mathcal{U}:=L^2(\\mathcal{O}\\times(0,T))$, so that the system \\eqref{direct_pb_y} admits a unique solution\n\\begin{align*}\n\ty\\in L^2((0,T); H^1_0(\\Omega))\\cap H^1(0,T; H^{-1}(\\Omega)),\n\\end{align*}\nwhich satisfies classical energy estimates. Actually, this remains true also if $v{\\bf 1}_{\\mathcal{O}}$ is replaced by a more general right-hand side $f\\in L^2(0,T; H^{-1}(\\Omega))$.\n\nNon-local diffusion equations in the form \\eqref{direct_pb_y} were introduced as an extension of already existing advection-diffusion-reaction models of multi-species ecosystems. They take into account the fact that, in many biological situations, local movement is also coupled with long-range influences, such as the combination of clonal growth and a dispersing phase like seeds. One example of a model for this situation was developed by Furter and Grinfeld (\\cite{furter1989local}), who examined diffusion-reaction models of single-species dynamics that incorporate a reaction term dependent on characteristics of the population as a whole.\n\nConcerning the controllability of \\eqref{direct_pb_y}, this issue has been firstly studied in \\cite{fernandez2016null}, where the authors proved an interior null controllability result by imposing analyticity assumptions on the kernel in order to obtain unique continuation properties. In this framework, also coupled systems have been treated in \\cite{lissy2018internal}. Moreover, in \\cite{micu2017local}, analogous results have been obtained for a one-dimensional equation with a kernel in separated variables, by means of classical spectral analysis techniques. Later on, these mentioned results have been extended in \\cite{biccari2017null}, where the null-controllability is proved under weaker assumptions on the kernel and for both the linear and the semilinear case, by using a Carleman approach. In particular, the authors there proved the following result.\n\n\\begin{theorem}[{\\cite[Theorem 1.1]{biccari2017null}}]\\label{theo1_bhs}\nLet $T>0$ and assume that $K\\in L^\\infty(\\Omega\\times\\Omega\\times(0,T))$ satisfies \n\\begin{align}\\label{K_est_weak}\n\t\\mathcal{K}:=\\sup_{(x,t)\\in\\overline{Q}}\\exp\\left(\\frac{\\mathcal{A}}{t(T-t)}\\right)\\int_{\\Omega} |K(x,\\theta,t)|\\,d\\theta <+\\infty,\n\\end{align}\nfor a given positive constant $\\mathcal A$ only depending on $\\Omega$. Then, for any $y_0\\in L^2(\\Omega)$, there exists a control function $v\\in \\mathcal U$ such that the associated solution $y$ of \\eqref{direct_pb_y} satisfies $y(x,T) = 0$.\t\n\\end{theorem}\t\n\nWe shall stress that, according to the hypothesis \\eqref{K_est_weak}, to get a positive controllability result the kernel $K$, as a function of $t$, should behave like \n\\begin{align*}\n\tK(\\cdot,\\cdot,t) \\sim \\exp\\left(-\\frac{\\mathcal B}{t(T-t)}\\right),\n\\end{align*}\ni.e. it should decay exponentially as $t$ goes to $0$ and $T$. \n\nThis is however a quite strong restriction on the admissible kernels. Actually, only rapidly decaying or compactly supported (in time) kernels verify \\eqref{K_est_weak}.\n\nIn some specific situations, this assumption it is actually not necessary. For instance, in the particular case when $K=|\\Omega|^{-1}\\textnormal{cst.}$, assumption \\eqref{K_est_weak} has been recently removed in \\cite{hernandez2020local}. Using the so-called shadow model (see \\cite{HR00,MCHKS18} or \\cite{HSZ20} for other applications in control), the authors have proved that the system \n\\begin{equation}\\label{eq:nonlocal_nonlinear}\n\t\\frac{\\partial y}{\\partial t}-\\Delta y + a y + b\\,  \\Xint-_{\\Omega} y(\\theta,t)d\\theta = v \\mathbf{1}_{\\mathcal O}\n\\end{equation}\nwhere $a,b\\in\\mathbb R$ and \n\\begin{align*}\n\t\\Xint-_\\Omega y(\\theta,t)d\\theta:=|\\Omega|^{-1}\\int_{\\Omega}y(\\theta,t)d\\theta,\n\\end{align*}\nis null-controllable at time $T>0$. Obviously, since the kernel $K$ is constant, the controllability result in Theorem \\ref{theo1_bhs} does not apply directly. We also emphasize that this result extends the one in \\cite{fernandez2016null} since the constant kernels do not satisfy the analyticity assumptions considered in that work. \n\nWhile the controllability of \\eqref{direct_pb_y} has been already studied in the aforementioned references, to the best of our knowledge the optimal control for this same problem is yet to be addressed. This is exactly the purpose of the present paper, in which this issue will be treated from two different viewpoints. We will firstly address a classical optimal control problem, in which we are interested in minimizing a given functional characterizing the control $v$ employed in \\eqref{direct_pb_y}. In more detail, we consider the strictly convex cost functional $J:\\mathcal U\\to\\RR$ defined by\n\\begin{equation}\\label{quadratic_cost}\n\tJ(v)={\\beta}\\norm{y(v)-\\bar{z}}{L^2(Q)}^2 +\\mu\\norm{v}{\\mathcal{U}}^2, \\qquad \\beta,\\,\\mu>0,\n\\end{equation}\nwhere we denote by $y=y(v)$ the dependence of the state in terms of the control function $v$, and we will study the optimal control problem \n\\begin{equation}\\label{min_J_part1}\n\t\\inf_{v\\in\\mathcal{U}} J(v).\n\\end{equation}\n\nTo this end, we will firstly prove the existence of a minimizer and, in a second moment, we will characterize it through a suitable optimality system.\n\nIn \\eqref{quadratic_cost}, $\\bar{z}\\in L^2(\\Omega\\times(0,T))$ is the target we aim to reach and $\\mathcal U$ is a given set of admissible controls. Here we will consider the simple case in which $\\mathcal U = L^2(\\mathcal O\\times(0,T))$, although other different choices are possible. Actually, in many practical situations realistic controls depend on the physical process that the equation is describing, and the set of admissible controls may be characterized by taking into account several constraints.\n\n\n\n\\section{Optimal control}\\label{sec:3}\n\nAs we mentioned in Section \\ref{intro_sec}, the mathematical theory of optimal control is nowadays very rich in the context of local equations, covering for instance linear and non-linear models, convex and non-convex functionals, and problems with control constraints. On the other hand, way less results are available in the framework of non-local models. Actually, to the best of our knowledge, the optimal control problem for \\eqref{direct_pb_y} has never been addressed before. \n\nIn this section, we will study the optimal control problem \\eqref{min_J_part1}. In more detail, we will firstly prove the existence of a minimizer for the functional $J$ and, in a second moment, we will give a characterization through the optimality system. Moreover, by means of classical techniques in control theory, in what follows we will need to rely on the so-called \\textit{adjoint equation}, which is given by \n\\begin{align}\\label{adjoint_pb_p}\n\t\\begin{cases}\n\t\t\\displaystyle -\\frac{\\partial p}{\\partial t}-\\Delta p+\\int_\\Omega K(\\theta,x,t)\\,p(\\theta,t)\\,d\\theta = y(v)-\\bar{z}, & (x,t)\\in Q,\n\t\t\\\\\n\t\tp = 0, & (x,t)\\in \\Sigma,\n\t\t\\\\\n\t\tp(x,T) = 0, & x\\in\\Omega.\n\t\\end{cases}\n\\end{align}\n\n\\subsection{Existence of an optimal control}\n\nWe have the following.\n\\begin{proposition}\\label{exist_prop_sect1} \nThere exists an optimal control function $u\\in\\mathcal{U}$, unique solution to the minimization problem \\eqref{quadratic_cost}, \\eqref{min_J_part1}.\n\\end{proposition}\n\n\\begin{proof}\nThe result is a consequence of the so-called \\textit{Direct Method of the Calculus of Variations} (see \\cite{brezis2010functional}), which ensures the existence and uniqueness of a minimizer provided the functional $J$ satisfies the following assumptions:\n\\begin{itemize}\n\t\\item[1.] $J$ is lower semi-continuous.\n\t\\item[2.] $J$ is strictly convex, i.e. $J((1-\\lambda)v+\\lambda w) < (1-\\lambda)J(v)+\\lambda J(w)$ for all $\\lambda\\in (0,1)$ and $v,w\\in\\mathcal U$.\n\t\\item[3.] $J$ is coercive, i.e. $\\lim_{\\norm{v}{\\mathcal U}\\to+\\infty}J(v) = +\\infty$.\t\n\\end{itemize}\n\nThe first two properties are evident since the functional is quadratic. Concerning the third one, it is enough to see that $J(v)\\geq \\mu\\norm{v}{\\mathcal U}$. This ends the proof.  \n \\end{proof}\n\n\\subsection{Characterization of the optimal control}\n\nNotice that Proposition \\ref{exist_prop_sect1} guarantees the existence of a unique solution to our optimal control problem \\eqref{quadratic_cost}-\\eqref{min_J_part1}, but it does not allow to characterize it. Notwithstanding, in practical applications it is important also an explicit knowledge of the control, which is typically computed through the so-called \\textit{optimality system}. In order to introduce this system, we will firstly need the following technical lemma.\n\n\\begin{lemma} \nLet $v\\in\\mathcal U$ be the optimal control minimizing the functional $J$. For all $w \\in \\mathcal{U}$, we have the identity\n\\begin{equation}\\label{euler_lagrange_part1}\n\t{\\beta}\\langle(y(v)-\\bar{z}),y(w)\\rangle_{L^2(Q)} + \\mu\\langle v,w\\rangle_{\\mathcal{U}} = 0.\n\\end{equation}\n\\end{lemma} \n\n\\begin{proof}\nWe use the necessary condition of Euler-Lagrange satisfied by the optimal control $v\\in \\mathcal{U}$. Thanks to the linearity of the problem with respect to the control function, for every $\\lambda>0$ and $w\\in \\mathcal{U}$ we have $y(v+\\lambda w)=y(v)+\\lambda y(w)$. Then,\n\\begin{align*}\n\tJ(v+\\lambda w)-J(v) &= {\\beta}\\norm{y(v+\\lambda w)-\\bar{z}}{L^2(Q)}^2 + \\mu\\norm{v+\\lambda w}{\\mathcal{U}}^2 - {\\beta}\\norm{y(v)-\\bar{z}}{L^2(Q)}^2 + \\mu\\norm{v}{\\mathcal{U}}^2\n\t\\\\\n\t&={\\beta\\lambda^2}\\norm{y(w)}{L^2(Q)}^2 + 2\\beta\\lambda\\langle y(v)-\\bar{z},y(w)\\rangle_{L^2(Q)} + \\mu\\lambda^2\\norm{w}{\\mathcal{U}}^2 + 2\\mu\\lambda\\langle v,w\\rangle_{\\mathcal{U}}.\n\\end{align*}\nHence, the necessary condition of Euler-Lagrange gives\n\\begin{align*}\n\t\\lim_{\\lambda\\rightarrow 0} \\left(\\frac{J(v+\\lambda w)-J(v)}{\\lambda}\\right) ={\\beta} \\langle y(v)-\\bar{z}, y(w)\\rangle_{L^2(Q)} + \\mu\\langle v,w\\rangle_{\\mathcal{U}} = 0, \\; \\mbox{ for all } w\\in\\mathcal{U}.\n\\end{align*}\n\\end{proof}\n\n\\noindent We can now give the following characterization of the optimal control.\n\n\\begin{proposition}\\label{prop_characterization_part1}\nThe optimal control $v$ is characterized by the triplet \n\\begin{align*}\n\t(v,y,p)\\in \\mathcal{U}\\times L^2(Q)\\times L^2(Q),\n\\end{align*}\nunique solution to the optimality system\n\\begin{equation}\\label{sos_part1}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{\\partial y}{\\partial t}-\\Delta y+\\int_\\Omega K(x,\\theta,t)\\,y(\\theta,t)\\,d\\theta = v{\\bf 1}_{\\mathcal{O}}, & (x,t)\\in Q,\n\t\t\\\\[9pt]\n\t\t\\displaystyle -\\frac{\\partial p}{\\partial t}-\\Delta p+\\int_\\Omega K(\\theta,x,t)\\,p(\\theta,t)\\,d\\theta = \\beta(y(v)-\\bar{z}), & (x,t)\\in Q,\n\t\t\\\\[9pt]\n\t\ty=p=0, & (x,t)\\in \\Sigma,\n\t\t\\\\[9pt]\n\t\ty(x,0)=y_0(x), \\quad p(x,T) = 0, & x\\in\\Omega,\n\t\\end{cases}\n\\end{equation}\nwith\n\\begin{equation}\\label{adjoint_state_eq_part1}\n\t p+\\mu v=0, \\qquad \\mbox{in}\\quad \\mathcal{O}\\times (0,T).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nWithout losing generality we can assume here that $y_0=0$, since in the case of non-zero initial data the result may be equivalently obtained by means of a simple change of variables. \n\nWe multiply \\eqref{direct_pb_y} by $p$ solution of \\eqref{adjoint_pb_p} and we integrate over the domain $Q$. By taking into account the boundary and initial conditions, we get by Fubini's theorem\n\\begin{align*}\n\t\\int_Q {\\beta} (y(v)-\\bar{z})y(w)\\,dxdt &=\\int_Q\\left(-\\frac{\\partial p}{\\partial t}-\\Delta p+ \\int_\\Omega K(\\theta,x,t)p(\\theta,t)\\,d\\theta\\right)y(x,t;w)\\,dxdt\n\t\\\\\n\t&=\\int_Q\\left(\\frac{\\partial y}{\\partial t}-\\Delta y+\\int_\\Omega K(x,\\theta,t)y(\\theta,t;w)\\,d\\theta\\right)p(x,t)\\,dxdt \n\t\\\\\n\t&= \\int_0^T\\int_{\\mathcal{O}}pw\\,dxdt.\n\\end{align*}\nThen, from \\eqref{euler_lagrange_part1} we immediately obtain $\\langle\\, p+\\mu v,\\, w\\rangle_{\\mathcal{U}}=0$ for all $w\\in \\mathcal{U}$,\nfrom which we get \\eqref{adjoint_state_eq_part1}.\n\\end{proof}\n\n\\section{Incomplete data problems}\\label{sec:4}\n\n\nIn this section, we consider the case of a missing or incomplete initial datum $y_0$, which is a very typical situation when studying natural phenomena governed by \\eqref{direct_pb_y}. This problem is described below and will be addressed by using the concept of no-regret controls, which has been introduced by J.-L. Lions in \\cite{lions1992controle}.\n\n\n\\subsection{Preliminaries}\n\nWe are interested in studying the control of the model \\eqref{direct_pb_y}-\\eqref{quadratic_cost} when it describes the dynamics of the density population $y(x,t)$ in the case in which the initial data is missing or incomplete. \n\nThis is a very typical situation when modeling many phenomena in physics, ecology, dynamic population, and several other fields. As a matter of fact, in those frameworks we may face the problem of incomplete data because of their practical inaccessibility, or because sometimes we have a great variety of possibilities when choosing for instance the initial conditions. In addition, boundary conditions may also be unknown or only partially known on a subset of the boundary that may be inaccessible to measurements. The same goes for source terms that can be difficult to access, or the structure of the domain, which can also be imperfectly known (for example in oil well management). \n\nIn our case, we will focus on a problem with missing initial datum. This means that this time we are assuming that $y_0$ is an unknown function, belonging to some vector closed subspace $G$ of $L^2(\\Omega)$. We are still concerned with optimal controls $v\\in\\mathcal{U}$, i.e. in solving\n\\begin{align*}\n\t\\inf_{v\\in {\\mathcal U}}J(v,y_0),\\qquad \\mbox{ for all } y_0\\in G.\n\\end{align*}\nwhere $J(v,y_0)$ denotes the explicit dependence on $y_0$ of the functional $J$ (see Lemma \\ref{lem:equiv_rest} for a precise description). Nevertheless, this minimization problem has actually no sense since $G$ is either the empty space or it has an infinite number of elements. \n\nTo deal with this problem, we use the low-regret concept of J.-L. Lions (see \\cite{lions1992controle}), which is well suited for incomplete data problems, and it is based on replacing \\eqref{min_J_part1} by\n\\begin{equation}\\label{min_J_part2}\n\t \\inf_{v\\in {\\mathcal U}}\\Bigg(\\sup_{y_0\\in G} \\Big(J(v,y_0)-J(0,y_0)-\\gamma\\|y_0\\|_G^2\\Big)\\Bigg),\n\\end{equation}\nwhere $\\gamma$ is a small parameter.\n\nThe meaning of \\eqref{min_J_part2} is to look for the control not making things worse with respect to doing nothing (i.e. case $v=0$). A solution to \\eqref{min_J_part2} is called a low-regret optimal control (see \\cite{jacob2010optimal,lions1992controle,louison_1,nakoulima2000perturbations} for further information on the method).\n\nWe show two preliminary results.\n\n\\begin{lemma}\\label{lem:equiv_rest}\nFor every $v\\in\\mathcal{U}$ and for any $y_0\\in G$, we have the property\n\\begin{equation}\\label{form0_part2}\n\tJ(v,y_0)-J(0,y_0)= J(v,0)-J(0,0)+ \\beta\\langle\\,y(v,0),y(0,y_0)\\rangle_{L^2(Q)}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe denote by $y(v,y_0):=y(x,t,v,y_0)$ the solution to \\eqref{direct_pb_y} with control $v\\in \\mathcal U$ and initial datum $y_0\\in G$. From the linearity of the problem \\eqref{direct_pb_y} we have $y(v,y_0)=y(v,0)+y(0,y_0)$. Then\n\\begin{align*}\n\tJ(v,y_0)-J(0,y_0) &= {\\beta}\\norm{y(v,y_0)-\\bar{z}}{L^2(Q)}^2 + \\mu\\norm{v}{\\mathcal{U}}^2-{\\beta}\\norm{y(0,y_0)-\\bar{z}}{L^2(Q)}^2\n\t\\\\\n\t&= {\\beta}\\norm{y(v,0)}{L^2(Q)}^2 + 2\\beta\\langle y(v,0),y(0,y_0)-\\bar{z}\\rangle_{L^2(Q)} + \\mu\\norm{v}{\\mathcal{U}}^2\n\t\\\\\n\t&= {\\beta}\\norm{y(v,0)-\\bar{z}+\\bar{z}}{L^2(Q)}^2 + 2\\beta\\langle y(v,0),y(0,y_0)-\\bar{z}\\rangle_{L^2(Q)} +\\mu\\norm{v}{\\mathcal{U}}^2\n\t\\\\\n\t&= J(v,0)-J(0,0)+ 2\\beta\\langle y(v,0),y(0,y_0)\\rangle_{L^2(Q)}.\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\\label{lem:adjoint} \nLet $\\xi:=\\xi(x,t,v,0)$ be the solution of the adjoint problem\n\\begin{equation}\\label{adjoint_pb_xi}\n\t\\begin{cases}\n\t\t\\displaystyle -\\frac{\\partial \\xi}{\\partial t}-\\Delta \\xi+\\int_\\Omega K(\\theta,x,t)\\,\\xi(\\theta,t)\\,d\\theta = y(v,0), & (x,t)\\in Q,\n\t\t\\\\\n\t\t\\xi = 0, & (x,t)\\in \\Sigma,\n\t\t\\\\\n\t\t\\xi(x,T) = 0, & x\\in\\Omega.\n\t\\end{cases}\n\\end{equation}\nThen, we have\n\\begin{equation}\\label{form_part2}\n\tJ(v,y_0)-J(0,y_0)= J(v,0)-J(0,0)+2\\beta\\langle \\xi(0)\\,,\\,y_0\\rangle_{G',G}\n\\end{equation}\nwhere $\\xi(0):=\\xi(x,0,v,0)$ is the solution of the adjoint problem (\\ref{adjoint_pb_xi}) at time $t=0$ and where $G'$ is the topological dual of $G$.\n\\end{lemma}\n\\begin{proof}\nIf we multiply the first equation of \\eqref{adjoint_pb_xi} by $y(0,y_0)$ and we integrate by parts, we obtain\n\\begin{align*}\n\t\\langle y(v,0),y(0,y_0)\\rangle_{L^2(Q)} &= \\int_Q y(v,0)y(0,y_0)\\,dxdt\n\t\\\\\n\t&=\\int_Q\\left(-\\frac{\\partial \\xi}{\\partial t}-\\Delta \\xi+ \\int_\\Omega K(\\theta,x,t)\\,\\xi(\\theta,t)\\,d\\theta\\right)y(0,y_0)\\,dxdt\n\t\\\\\n\t&=\\int_Q\\left(\\frac{\\partial y}{\\partial t}-\\Delta y+\\int_\\Omega K(x,\\theta,t)\\,y(\\theta,t,w)\\,d\\theta\\right)\\xi(x,t)\\,dxdt + \\displaystyle\\int_\\Omega \\xi(0)y_0(x)\\,dx\n\t\\\\\n\t&= \\int_\\Omega \\xi(0)y_0(x)\\,dx.\n\\end{align*}\nThen, the result follows immediately by applying \\eqref{form0_part2}.\n\\end{proof}\n\nLemma \\ref{lem:adjoint} can now be employed to transform the inf\/sup problem \\eqref{min_J_part2} into a classical minimization one. Indeed, using \\eqref{form_part2}, we obtain from \\eqref{min_J_part2} that\n\\begin{align*}\n\t\\displaystyle\\inf_{v\\in {\\mathcal U} }\\left[J(v,0)-J(0,0) + 2\\beta \\displaystyle\\sup_{y_0\\in G} \\Big(\\langle \\xi(0),y_0\\rangle_{G',G} - \\frac{\\gamma}{2\\beta}\\|y_0\\|_{G}^2\\Big)\\right].\n\\end{align*}\nThis, together with the fact that\n\\begin{align*}\n\t\\sup_{y_0\\in G} \\Big(\\langle \\xi(0),y_0\\rangle_{G',G} -\\frac{\\gamma}{2\\beta}\\|y_0\\|_{G}^2\\Big) =\\frac{\\beta}{2\\gamma}\\|\\xi(0)\\|^2_{G'},\n\\end{align*}\ngives us the following minimization problem equivalent to \\eqref{min_J_part2}\n\\begin{align}\\label{infgamma}\n\t&\\inf_{v\\in {\\mathcal U} }{\\mathcal{J}}^{\\gamma}(v) \n\t\\\\\n\t&\\mathcal{J}^{\\gamma}(v)=J(v,0)-J(0,0) + \\frac{\\beta^2}{\\gamma}\\Big\\|\\xi(0)\\Big\\|_{G'}^2. \\notag\n\\end{align}\nIn what follows, we will always focus on \\eqref{infgamma} instead of the original low-regret problem \\eqref{min_J_part2}.\n\n\\subsection{Existence of the optimal low-regret control}\n\nWe discuss here the existence of a low-regret control for \\eqref{direct_pb_y}. In particular, the main result of this section will be the following. \n\n\\begin{theorem} \nThere exists a unique low-regret optimal control function denoted by $v_\\gamma \\in{\\mathcal U}$, solution to the minimization problem \\eqref{infgamma}.\n\\end{theorem}\n\n\\begin{proof} \nFirst of all, notice that for all $v\\in {\\mathcal U}$ we have $\\displaystyle {\\mathcal J}^\\gamma(v)\\geq -J(0,0)$ and, therefore, $\\dis \\displaystyle\\inf_{v\\in {{\\mathcal U} }}{\\mathcal J}^\\gamma(v)$ exists. Let then $\\{v_\\gamma^n\\}$ be a minimizing sequence such that $\\dis d_\\gamma=\\lim_{n\\to+\\infty} {\\mathcal J}^\\gamma(v_\\gamma^n)$. We have\n\\begin{align*}\n\t-J(0,0)\\leq {\\mathcal J}^\\gamma(v_\\gamma^n) =J(v_\\gamma^n,0)-J(0,0) +\\frac{\\beta^2}{\\gamma} \\Big\\|\\xi(v_\\gamma^n,0)(0)\\Big\\|_{G'}^2 \\leq d_\\gamma+1,\n\\end{align*}\nfor $n$ sufficiently large. From this, we deduce that there exists a positive constant $c_\\gamma$, independent on $n$, such that the following estimates hold\n\\begin{equation}\n\t\\Big\\|v_\\gamma^n\\Big\\|_{\\mathcal{U}} \\leq c_\\gamma,\\quad \\Big\\|y(v_\\gamma^n,0)-\\bar{z}\\Big\\|_{L^2(Q)}\\leq \nc_\\gamma,\\quad\\mbox{and}\\;\\; \\frac{\\beta}{\\sqrt{\\gamma}} \\Big\\|\\xi(v_\\gamma^n,0)(0)\\Big\\|_{G'} \\leq c_\\gamma.\n\\end{equation}\n\nThen we deduce that the sequence $\\{v_\\gamma^n\\}$ is bounded in $\\mathcal{U}$  and therefore there exists a subsequence, converging weakly in $\\mathcal{U}$ to $v_\\gamma$. Moreover, thanks to the strict convexity of ${\\mathcal J}^\\gamma$, $v_\\gamma$ is unique.\n\\end{proof}\n\n\\subsection{Characterization of the low-regret control}\n\nThis section is devoted to a characterization of the low-regret control fro \\eqref{direct_pb_y} through the corresponding optimality system. To this end, we shall first prove the following result.\n\n\\begin{lemma}\\label{lem:EL}\nLet $v_\\gamma$ be the optimal low-regret control solution of the minimization problem \\eqref{infgamma}, and denote $y_\\gamma:=y(v_\\gamma,0)$ and $\\xi_\\gamma:=\\xi(v_\\gamma,0)$. Then, for all $w\\in\\mathcal U$ we have the inequality\n\\begin{equation}\\label{euler_lagrange_part2}\n\t\\beta\\langle y_\\gamma-\\bar{z},y(w,0)\\rangle_{L^2(Q)} + \\mu\\langle v_\\gamma,w\\rangle_{\\mathcal{U}} + \\frac{\\beta^2}{\\gamma}\\left\\langle \\xi_\\gamma(0),\\xi(w,0)(0) \\right\\rangle_{G'} = 0.\n\\end{equation} \n\\end{lemma}\n\n\\begin{proof}\nWe have\n\\begin{align*}\n\t\\frac{{\\mathcal J}^\\gamma(v_\\gamma+\\lambda w) -{\\mathcal J}^\\gamma(v_\\gamma)}{\\lambda} =&\\, J(v_\\gamma+\\lambda w,0) + \\frac{\\beta^2}{\\gamma}\\norm{\\xi(v_\\gamma+\\lambda w,0)(0)}{G'}^2 - J(v_\\gamma,0) - \\frac{\\beta^2}{\\gamma}\\norm{\\xi(v_\\gamma,0)(0)}{G'}^2\n\t\\\\\n\t=&\\, {\\beta}\\norm{(y_\\gamma-\\bar{z})+\\lambda y(w,0)}{L^2(Q)}^2 + \\mu\\norm{v_\\gamma+\\lambda w}{\\mathcal{U}}^2 -{\\beta}\\norm{y_\\gamma-\\bar{z}}{L^2(Q)}^2 - \\mu\\norm{v_\\gamma}{\\mathcal{U}}^2 \n\t\\\\\n\t&+ \\frac{\\beta}{\\gamma}\\norm{\\xi_\\gamma(0) +\\lambda \\xi(w,0)(0)}{G'}^2 -\\frac{\\beta}{\\gamma}\\norm{\\xi_\\gamma(0)}{G'}^2 \n\t\\\\\n\t=&\\, {\\lambda^2\\beta}\\norm{y(w,0)}{L^2(Q)}^2 +2\\lambda\\beta\\langle y_\\gamma-\\bar{z},y(w,0)\\rangle_{L^2(Q)} + \\mu\\lambda^{2}\\norm{w}{\\mathcal{U}}^2 +2\\mu\\lambda\\langle v_\\gamma,w\\rangle_{\\mathcal{U}} \n\t\\\\\n\t&+ \\frac{\\beta^2\\lambda^2}{\\gamma}\\norm{\\xi(w,0)(0)}{G'}^2 +\\frac{2\\beta^2\\lambda}{\\gamma}\\langle \\xi_\\gamma(0),\\xi(w,0)(0)\\rangle_{G'}.\n\\end{align*}\nIdentity \\eqref{euler_lagrange_part2} is then an immediate consequence of the Euler-Lagrange condition, which gives\n\\begin{align*}\n\t0&=\\lim_{\\lambda\\rightarrow 0} \\left(\\frac{{\\mathcal J}^\\gamma(v_\\gamma+\\lambda w) -{\\mathcal J}^\\gamma(v_\\gamma)}{\\lambda}\\right) \n\t\\\\\n\t&={\\beta} \\langle y_\\gamma-\\bar{z},y(w,0)\\rangle_{L^2(Q)} + N\\langle v_\\gamma,w\\rangle_{\\mathcal{U}} +\\frac{\\beta^2}{\\gamma} \\left\\langle \\xi_\\gamma(0),\\xi(w,0)(0) \\right\\rangle_{G'}, \\quad \\mbox{ for all } w \\in\\mathcal{U}.\n\\end{align*}\n\\end{proof}\n\n\\noindent By means of Lemma \\ref{lem:EL}, we can now obtain the following characterization of the low-regret control.\n\n\\begin{proposition}\\label{prop_characterization_part2}\nThe low-regret control $v_\\gamma$ is characterized by the quadruplet \n\\begin{align*}\n\t(v_\\gamma,y_\\gamma,\\xi_\\gamma,\\sigma_\\gamma) \\in \\mathcal{U} \\times L^2(Q)\\times L^2(Q)\\times L^2(Q),\n\\end{align*}\nunique solution to the optimality system\n\\begin{align}\\label{sos_part2}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{\\partial y_\\gamma}{\\partial t} -\\Delta y_\\gamma+ \\int_\\Omega K(x,\\theta,t)\\,y_\\gamma(\\theta,t)\\,d\\theta = v_\\gamma{\\bf 1}_{\\mathcal{O}}, & (x,t)\\in Q,\n\t\t\\\\[10pt]\n \t\t\\displaystyle -\\frac{\\partial \\xi_\\gamma}{\\partial t} -\\Delta \\xi_\\gamma+ \\int_\\Omega K(\\theta,x,t)\\,\\xi_\\gamma(\\theta,t)\\,d\\theta = y_\\gamma, & (x,t)\\in Q,\n \t\t\\\\[10pt]\n\t\t\\displaystyle\\frac{\\partial \\sigma_\\gamma}{\\partial t} -\\Delta \\sigma_\\gamma+ \\int_\\Omega K(x,\\theta,t)\\,\\sigma_\\gamma(\\theta,t)\\,d\\theta = 0, & (x,t)\\in Q,\n\t\t\\\\[10pt] \n\t\t\\displaystyle -\\frac{\\partial p_\\gamma}{\\partial t} -\\Delta p_\\gamma+ \\int_\\Omega K(\\theta,x,t)\\, p_\\gamma(\\theta,t)\\,d\\theta = {\\beta}(y_\\gamma-\\bar{z})-\\sigma_\\gamma, & (x,t)\\in Q,\n\t\t\\\\[10pt]\n\t\ty_\\gamma=\\xi_\\gamma=\\sigma_\\gamma =p_\\gamma = 0,  & (x,t)\\in \\Sigma,\n\t\t\\\\[10pt]\n\t\t\\displaystyle y_\\gamma(0)=0,\\; \\sigma_\\gamma(0)=-\\frac{\\beta^2}{\\gamma}\\xi_\\gamma(0),\\; \\xi_\\gamma(T) =p_\\gamma(T) = 0, & x\\in\\Omega,\n\t\\end{cases}\n\\end{align}\nwhere $\\sigma_\\gamma:=\\sigma(v_\\gamma,0)$ and $p_\\gamma:=p(v_\\gamma,0)$, and with the adjoint state equation\n\\begin{equation}\\label{adjoint_state_eq_part2}\n\t p_\\gamma+\\mu v_\\gamma=0, \\qquad \\mbox{in} \\quad \\mathcal{O}\\times ]0,T[.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nWe introduce $\\sigma_\\gamma=\\sigma(v_\\gamma,0)$ unique solution to the problem\n\\begin{equation}\\label{direct_pb_sigma}\n\t\\begin{cases}\n\t\t\\displaystyle\\frac{\\partial \\sigma_\\gamma}{\\partial t} -\\Delta \\sigma_\\gamma+\\int_\\Omega K(x,\\theta,t)\\,\\sigma_\\gamma(\\theta,t)\\,d\\theta = 0, & (x,t)\\in Q,\n\t\t\\\\\n\t\t\\sigma_\\gamma = 0, & (x,t)\\in \\Sigma,\n\t\t\\\\\n\t\t\\sigma_\\gamma(0,x) = -\\displaystyle \\frac{\\beta}{\\gamma}\\xi_\\gamma(0), & x\\in\\Omega.\n\t\\end{cases}\n\\end{equation}\nThen we multiply (\\ref{direct_pb_sigma}) by $\\xi(w,0)$ and we integrate by parts over $Q$, thus obtaining\n\\begin{align*}\n\t0&= \\int_Q\\left(-\\frac{\\partial \\xi}{\\partial t}-\\Delta \\xi+ \\int_\\Omega K(\\theta,x,t)\\,\\xi(\\theta,t)\\,d\\theta\\right) \\sigma_\\gamma\\,dxdt - \\int_\\Omega \\xi(w,0)(0)\\,\\sigma_\\gamma(0)\\,dx\n\t\\\\\n\t&=\\int_Q y(w,0)\\,\\sigma_\\gamma\\, dxdt - \\int_\\Omega \\xi(w,0)(0)\\,\\sigma_\\gamma(0)\\,dx.\n\\end{align*}\nHence, \n\\begin{align*}\n\t\\left\\langle \\frac{\\beta^2}{\\gamma}\\xi_\\gamma(0),\\xi(w,0)(0)\n\t\\right\\rangle_{G'} = \t \\langle\\sigma_\\gamma,y(w,0) \\rangle_{L^2(Q)}.\n\\end{align*}\nThen (\\ref{euler_lagrange_part2}) reduces to\n\\begin{equation}\\label{simplification_euler_part2}\n\t\\langle {\\beta}(y_\\gamma-\\bar{z})-\\sigma_\\gamma, y(w,0)\\rangle_{L^2(Q)} + \\mu\\langle v_\\gamma,w\\rangle_{\\mathcal{U}} \\geq 0, \\quad \\mbox{ for all } w \\in\\mathcal{U}.\n\\end{equation}\nFinally, we define the adjoint state $p_\\gamma:=(x,t,v_\\gamma)$ as the unique solution of:\n\\begin{equation}\\label{adjoint_pb_p_gamma}\n\t\\begin{cases}\n\t\t\\displaystyle -\\frac{\\partial p_\\gamma}{\\partial t}-\\Delta p_\\gamma+\\int_\\Omega K(\\theta,x,t)\\,p_\\gamma(\\theta,t)\\,d\\theta = \\beta(y_\\gamma-\\bar{z})-\\sigma_\\gamma, & (x,t)\\in Q,\n\t\t\\\\\n\t\tp_\\gamma = 0,  & (x,t)\\in \\Sigma,\n\t\t\\\\\n\t\tp_\\gamma(x,T) = 0, & x\\in\\Omega.\n\t\\end{cases}\n\\end{equation}\nWe multiply (\\ref{adjoint_pb_p_gamma}) by $y(w,0)$ and we integrate by parts over $Q$ to obtain\n\\begin{align*}\n\t\\displaystyle\\int_Q ({\\beta}(y_\\gamma-\\bar{z})-\\sigma_\\gamma)y(w,0)\\,dxdt =\\displaystyle\\int_0^T\\!\\!\\int_{\\mathcal{O}} p_\\gamma\\,w\\, dxdt.\n\\end{align*}\nThen (\\ref{simplification_euler_part2}) reads $\\langle\\, p_\\gamma+\\mu v_\\gamma,\\, w\\rangle_{\\mathcal{U}} = 0$, for all $w\\in \\mathcal{U}$,\nsince $\\mathcal{U}$ is a vector space.\n\\end{proof}\n\n\\section{Numerical experiments}\\label{sec:5}\n\nIn this section, we provide some numerical simulations and comment on the practical implementation of the optimal control and the low-regret control problems for non-local parabolic systems. In what follows, for all $a<b$ we shall use the notation $\\inter{a,b}=[a,b]\\cap \\mathbb N$. \n\n\\subsection{Discretization of the non-local model and numerical implementation of the optimal control problem}\n\nLet $\\Omega=(0,1)$ and consider the 1-d model given by\n\n\\begin{align}\\label{direct_pb_y_simp}\n\t\\begin{cases}\n\t\t\\dis\\frac{\\partial y}{\\partial t}- \\frac{\\partial^2 y}{\\partial x^2}+ \\int_{0}^{1} K(x,\\theta)\\,y(\\theta,t)\\,d\\theta = v{\\bf 1}_{\\mathcal{O}}, & (x,t)\\in Q,\n\t\t\\\\\n\t\ty = 0, &(x,t)\\in \\Sigma,\n\t\t\\\\\n\t\ty(0,x) = y_0(x), & x\\in\\Omega.\n\t\\end{cases}\n\\end{align}  \nwhere $K\\in L^\\infty(\\Omega\\times\\Omega)$. For convenience, we shall consider the case when $K$ can be written in separated variables, i.e., $K(x,\\theta)=K_1(x)K_2(\\theta)$.\n\nFor the numerical tests, system \\eqref{direct_pb_y_simp} (and its adjoint) are discretized in time by using a standard implicit Euler scheme and discretized in space by a finite-difference scheme adapted to the non-local term. For given $N,M\\in\\mathbb N^*$, we set $\\delta t=T\/M$ and $h=1\/(N+1)$ and consider the following discretization for the space and time variables\n\\begin{align*}\n\t&0=x_0<x_1<\\ldots< x_{N}<x_{N+1}=1, \n\t\\\\\n\t&0=t_0<t_1<\\ldots< T_{M}=T,\n\\end{align*}\nwhere $x_i=i h$, $i\\in\\inter{0,N+1}$, and $t_n=n\\delta t$, $n\\in\\inter{0,M}$. The numerical approximation of a function $f=f(x,t)$ at a grid point $(x_i,t_n)$ will be denoted as $f_i^n:=f(t_n,x_i)$ and, for fixed $n$, we write \n\\begin{align*}\n\tf^n=\\left(\\begin{array}{ccc}f_1^n,  \\dots, f_N^n\\end{array}\\right)^\\top\n\\end{align*}\nfor the evaluation at the interior points. With this notation, we shall consider fully discrete systems of the form\n\\begin{equation}\\label{eq:discr_nonlocal}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{y^{n+1}-y^n}{\\delta t}+\\mathcal A_{h} \\, y^{n+1}=\\mathcal B_h v^{n+1}, & n \\in \\inter{0, M-1}, \n\t\t\\\\ \n\t\ty_0^{n+1}=y_{N+1}^{n+1}=0, & n\\in\\inter{0,M-1},\t\n\t\t\\\\\t\n\t\ty^0=y^0_h,\n\t\\end{cases}\n\\end{equation}\nwhere $y=(y^n)_{n\\in\\inter{0,M}}$ and $v=(v^n)_{n\\in\\inter{1,M}}$ are the (discrete) state and control variables, $y^{0}_{h}\\in \\mathbb R^{N}$ is the projection of the given initial datum $y(\\cdot,0)$ on the space-mesh, $\\mathcal A_{h}\\in \\mathbb R^{N\\times N}$ is a suitable approximation of the differential and non-local operators, and $\\mathcal B_{h}\\in\\mathbb R^{N\\times N}$ stands for the corresponding approximation of the control operator. \nWe will compute the matrix $\\mathcal A_h\\in \\RR^{N\\times N}$ as the sum of two parts, more precisely, $\\mathcal A_h=\\mathcal A_{h,D}+\\mathcal A_{h,NL}$, the first one taking into account the second order differential operator and the second one the integral kernel. \n\nUsing a standard finite-difference method, we construct the matrix $\\mathcal A_{h,D}\\in\\mathbb R^{N\\times N}$ as the usual tridiagonal matrix coming from the discretization of the Laplacian operator $-\\partial^2_{xx}$ with homogeneous Dirichlet boundary conditions, that is, \n\\begin{align*}\n\t(\\mathcal A_h y)_{i}=-\\frac{1}{h^2}(y_{i+1}-2y_{i} +y_{i-1}), \\quad i=\\inter{1,N}. \t\n\\end{align*}\n\nTo incorporate the effects of the integral kernel, we see that for a grid point $(x_j,t_n)$ with $j\\in\\inter{1,N}$ and $n\\in\\inter{1,M}$,\n\\begin{equation}\\label{eq:kernel_discrete}\n\t\\int_{0}^{1}K(x_j,\\theta)y(\\theta,t_n)d\\theta \\approx K_{1,j} \\sum_{i=1}^{N} h K_{2,i} \\, y_{i}^{n},\n\\end{equation}\nand thus, writing in vector form, the action of the non-local term can be incorporated by means of the $N\\times N$ matrix given by\n\\begin{equation*}\n\t\\mathcal A_{h,NL}= h \\left(\\begin{array}{cccc}K_{1,1} &  &  &  \\\\  & K_{1,2} &  &  \\\\  &  & \\ddots &  \\\\  &  &  & K_{1,N} \\end{array}\\right)\\left(\\begin{array}{cccc} K_{2,1} & K_{2,2} &\\dots & K_{2,N} \\end{array}\\right)\\otimes \\left(\\begin{array}{c}1 \\\\ 1 \\\\ \\vdots \\\\ 1\\end{array}\\right) \n\\end{equation*}\nwhere $\\otimes$ stands for the usual Kronecker product in $\\RR^N$. Obviously, there are many other ways to discretize the integral \\eqref{eq:kernel_discrete} but this is enough for our purposes.  Lastly, a natural choice for the control operator $\\mathcal B_h$ is the $N\\times N$ diagonal matrix with entries \n\\begin{equation*}\n\t(\\mathcal B_h)_{i,i}= \\begin{cases}\n\t\t1 &\\text{if } x_i\\in\\omega, \\\\ 0 &\\text{if } x_i\\notin\\omega. \n\\end{cases}\n\\end{equation*}\nFor the implementation of the optimal control problem, we consider the discrete functional\n\\begin{equation}\\label{eq:discr_func}\n\tJ_{h,\\delta t}(v)= \\beta \\norm{y-\\bar{z}}{L^2_{\\delta t}(0,T;\\RR^N)}^2+\\norm{\\mathcal B_h v}{L^2_{\\delta t}(0,T;\\RR^N)}^2, \\quad \\mathcal \\beta>0,\n\\end{equation}\nwhere $y=(y^n)_{n\\in\\inter{1,M}}$ is the solution to \\eqref{eq:discr_nonlocal} and $\\bar z=(\\bar z^n)_{n\\in\\inter{1,M}}$ is a discretization of the target $\\bar z\\in L^2(\\Omega\\times(0,T))$. Without loss of generality, we have assumed that $\\mu = 1$. In \\eqref{eq:discr_func}, $L^2_{\\delta t}(0,T;\\RR^N)$ denotes the discretization of the space $L^2(\\Omega\\times(0,T))$, more precisely,\n\\begin{equation*}\n\tL^2_{\\delta t}(0,T;\\mathbb R^N):=\\Big\\{f=(f^n)_{n\\in\\inter{1,M}}, \\ f^n\\in\\mathbb R^N, \\ n\\in\\inter{1,M}\\Big\\},\n\\end{equation*}\nendowed with the norm\n\\begin{align*} \n\t\\norm{f}{L^2_{\\delta t}(0,T;\\RR^N)}:= \\left(\\sum_{n=1}^{M}\\delta t |f^n|^2\\right)^{1\/2},\n\\end{align*}\nwhere $|\\cdot|$ stands for the usual Euclidean norm in $\\RR^N$. The natural associated inner product will be defined as \n\\begin{align*}\n\t(f,g)_{L^2_{\\delta t}(0,T;\\mathbb R^N)}:= \\sum_{n=1}^{M}\\delta t (f^n,g^n),\n\\end{align*}\nwhere $(\\cdot,\\cdot)$ is the usual dot product in $\\RR^N$. For shortness, sometimes we simply write $\\|\\cdot\\|_{L^2}$ instead of $\\|\\cdot\\|_{L^2_{\\delta t}(0,T;\\RR^N)}$.\n\nOnce we have written \\eqref{eq:discr_func} in such form, the obtention of an optimal control is quite standard. For completeness, we sketch it briefly. A straightforward computation yields that\n\\begin{equation}\\label{eq:grad_J_h_delta}\n\t\\nabla J_{h,\\delta t}(v)=2 \\beta S_{h,\\delta t}^\\star(S_{h,\\delta t} v - \\bar{w}) +2  \\mathcal B_{h}^* \\mathcal B_h v .\n\\end{equation}\nIn \\eqref{eq:grad_J_h_delta}, $B_h^*$ denotes the matrix transpose of $B_h$, the operator $S_{h,\\delta t}$ is defined as\n\\begin{align*}\n\tS_{h,\\delta t}: L^2_{\\delta t}(0,T;\\mathbb R^N) &\\to L^2_{\\delta t}(0,T;\\mathbb R^N) , \\quad S_{h,\\delta t}v:=y,\n\\end{align*}\nwhere $y$ is the solution to the forward system \n\\begin{equation}\\label{eq:forward_GD}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{y^{n+1}-y^n}{\\delta t}+\\mathcal A_{h} \\, y^{n+1}=\\mathcal B_h v^{n+1}, & n \\in \\inter{0, M-1}, \n\t\t\\\\\n\t\ty^0=0,\n\t\\end{cases}\n\\end{equation}\nwhile the adjoint operator $S_{h,\\delta t}^\\star$ is defined as\n\\begin{align*}\n\tS_{h,\\delta t}^\\star: L^2_{\\delta t}(0,T;\\mathbb R^N) &\\to L^2_{\\delta t}(0,T;\\mathbb R^N), \\quad S_{h,\\delta t}^\\star z := p,\n\\end{align*}\nwhere $p$ can be found from the solution to the backward system\n\n\\begin{equation}\\label{eq:backward_GD}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{p^{n}-p^{n+1}}{\\delta t}+\\mathcal A_h^* p^{n}= z^n, \\quad n\\in\\inter{1,M}, \n\t\t\\\\\n\t\t^{M+1}=0,\n\t\\end{cases}\n\\end{equation}\nand, finally, $\\bar{w}:= \\bar{z}-\\mathring{y}$ where $\\mathring{y}$ denotes the solution to \\eqref{eq:discr_nonlocal} with $v\\equiv 0$. In \\eqref{eq:forward_GD}-\\eqref{eq:backward_GD} (and similar formulas below), we shall omit the boundary conditions since we always consider homogeneous Dirichlet ones.\n\nIn this way, the optimal control $v=(v^n)_{n\\in\\inter{1,M}}$ can be readily found by solving the linear problem\n\\begin{equation}\\label{eq:opt_control}\n\t(\\beta S_{h,\\delta t}^\\star S_{h,\\delta t}+ B_h^* B_{h})v=\\beta S_{h,\\delta t}^\\star \\bar{w}.\n\\end{equation}\nwhere $S_{h,\\delta t}^\\star S_{h,\\delta t} v$ corresponds to the evaluation of the cascade forward-backward system \\eqref{eq:forward_GD}-\\eqref{eq:backward_GD}. \n\n\\subsubsection{Numerical results for the optimal control problem}\n\nLet us set $T=1$ and consider the following parameters of the system\n\\begin{align}\\label{eq:kernel_num}\n\ty_0(x)=2\\sin(\\pi x), \\quad K_1(x)=\\sin(5\\pi x), \\quad K_2(\\theta)=20\\times \\mathbf{1}_{(0,0.5)}(\\theta)\\sin(\\pi \\theta) \n\\end{align}\n\nSince nothing changes in our theoretical analysis or the numerical implementation, in the remainder of this document we shall always consider a small diffusion parameter $\\nu=0.1$ multiplying the Laplace operator. In Figure \\ref{fig:soly_libre}, we plot the free solution of system \\eqref{direct_pb_y_simp}, that is, the solution with $v\\equiv 0$. We clearly observe the effect of the non-local kernel $K(x,\\theta)$ in the resolution of the equation. Here, we have used $M=100$ and $N=60$ for the number of points in the discrete grid. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics{.\/figures_final\/Pb_Kernel_with_Incomplete_Data_New_version-figure0.pdf}\n\\caption{Evolution in time of the uncontrolled solution.}\\label{fig:soly_libre}\n\\end{figure}\n\nNow, let us consider $\\mathcal O=(0.2,0.8)$ and set the time-independent target function $ \\bar{z}(x)=\\sin(2\\pi x)$. In Figure \\ref{fig:soly_control_beta}, we plot the solution using the control obtained by solving the linear problem \\eqref{eq:opt_control} for different values of $\\mathcal \\beta$. To solve this problem, we have used a standard Gradient Descent method. As usual in this type of problems, we observe that by increasing the parameter $\\beta$, the approximation to the target $\\bar{z}$ (i.e., $\\|y(v)-\\bar{z}\\|_{L^2}$) is overall better, but the control cost increases. \n\n\\begin{figure}[ht!]\n\\centering\n\\subfloat[$\\beta=10$. Cost of control $\\|B_h v\\|_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}=1.57137$]\n{\n\\includegraphics{.\/figures_final\/Pb_Kernel_with_Incomplete_Data_New_version-figure1.pdf}\n}\\qquad \n\\subfloat[$\\beta=10^2$. Cost of control $\\|B_h v\\|_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}=4.19168 $]\n{\n\\includegraphics{.\/figures_final\/Pb_Kernel_with_Incomplete_Data_New_version-figure2.pdf}\n} \\\\\n\\subfloat[$\\beta=10^3$. Cost of control $\\|B_h v\\|_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}=7.41933 $]\n{\n\\includegraphics{.\/figures_final\/Pb_Kernel_with_Incomplete_Data_New_version-figure3.pdf}\n}\n\\caption{Evolution in time of the controlled solution for different values of the penalization parameter $\\beta$.}\n\\label{fig:soly_control_beta}\n\\end{figure}\n\nAs for the uncontrolled solution displayed in Figure \\ref{fig:soly_libre}, we can clearly observe the effects of the non-local kernel. Indeed, even though the initial condition $y_0$, the target $\\bar z$ and the control region $\\mathcal O$ are somehow symmetric with respect to the point $x=0.5$, the distance of the solution $y$ to the target $z$ is not symmetric with respect to the point $x=0.5$ as it can be seen in Figure \\ref{fig:steady_state}.\n\n\\begin{figure}[ht!]\n\\centering\n\\subfloat[Non-local]{\n\\begin{tikzpicture}[scale=0.75]\n\n\t\\begin{axis}[xlabel={$x$},xmin=0,xmax=1,legend pos=outer north east,\n\tlegend plot pos=left,\n\tlegend style={cells={anchor=west},draw=none}]\n\t    \n\t\\pgfplotstableread{figures_final\/target_60_07-Jun-2021_13h11.org}\\nonlocal    \n\t  \n\t\\addplot[very thick, solid, red] table[x=x,y=yT2] \\nonlocal;   \n\t\\addplot[very thick,dashed] table[x=x,y=yT1] \\nonlocal;\n\n\t\\end{axis}\n\t\\end{tikzpicture}\\label{fig:bad_data1}\n}\n\\subfloat[Local]{\n\\begin{tikzpicture}[scale=0.75]\n\n\t\\begin{axis}[xlabel={$x$},xmin=0,xmax=1,legend pos=outer north east,\n\tlegend plot pos=left,\n\tlegend style={cells={anchor=west},draw=none}]\n    \n\t\\pgfplotstableread{figures_final\/target_60_07-Jun-2021_13h17.org}\\local    \n\t  \n\t\\addplot[very thick, solid, red] table[x=x,y=yT2] \\local;   \t\n\t\\addplot[very thick,dotted] table[x=x,y=yT1] \\local; \\label{local-st}\n\n\n\t\\end{axis}\n\\end{tikzpicture}\\label{fig:bad_data2}\n}\n\\caption{Comparison of the non-local steady state (dashed) and the local steady state (dotted) against the target function $\\bar z$ (red solid).}\n\\label{fig:steady_state}\n\\end{figure}\n\n\\subsection{Numerical implementation of the low-regret problem}\n\nWe turn our attention to the low regret problem. As introduced in Section \\ref{sec:4} this problem aims to solve an optimal control problem in the case when the initial data is missing or incomplete. As proposed in the classical work by J.-L. Lions, we can address this problem by replacing the classical optimal control functional \\eqref{min_J_part1} by the $\\min$-$\\max$ problem \\eqref{min_J_part2}. Although it is feasible to solve \\eqref{min_J_part2} (see e.g. \\cite{DP18} and the reference within), here we will use problem \\eqref{infgamma}, which transforms the original problem into a minimization one with two state variables. To this end, let us consider the fully-discrete version of the functional $J^\\gamma(v)$ (see eq. \\eqref{infgamma}) given by\n\\begin{align}\\label{eq:func_discr_low}\n\t&J^\\gamma_{h,\\delta t}(v)= \\beta \\|y(v,0)-\\bar z\\|_{L^2_{\\delta t}(0,T;\\mathbb R^N)}^2+\\|\\mathcal B_h v\\|^2_{L^2_{\\delta t}(0,T;\\mathbb R^N)}- \\beta\\|\\bar z\\|^2_{L^2_{\\delta t}(0,T;\\mathbb R^N)}+\\frac{\\beta^2}{\\gamma}|\\xi^{1}|^2 \\notag \n\t\\\\\n\t&v\\in L^2_{\\delta t}(0,T;\\mathbb R^N)\n\\end{align}\nwhere $y(v,0)=\\left(y^n(v,0)\\right)_{n\\in\\inter{1,M}}$ denotes the solution to \\eqref{eq:discr_nonlocal} with control $v$ and initial datum $y^0\\equiv 0$, $\\bar z=(\\bar z^n)_{n\\in\\inter{1,M}}$ is the discretization of the target $\\bar z\\in L^2(\\Omega\\times(0,T))$, and $\\xi^1$ can be found from the backward equation\n\\begin{equation}\\label{eq:backward_xi}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{\\xi^{n}-\\xi^{n+1}}{\\delta t}+\\mathcal A_h^* \\xi^{n}= y^n(v,0), \\quad n\\in\\inter{1,M}, \n\t\t\\\\\n\t\t\\xi^{M+1}=0.\n\t\\end{cases}\n\\end{equation}\nA straightforward computation gives\n\\begin{align}\\label{eq:deriv_J_gamma_discr}\n\t&\\displaystyle \\left(\\nabla J^\\gamma_{h,\\delta t}(v),\\hat{v}\\right)= 2\\beta \\left(y(v,0)-\\bar{z},y(\\hat{v},0)\\right)_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}+2\\left(\\mathcal B_h v,\\mathcal B_h \\hat{v}\\right)_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}+2\\beta\\left(\\sqrt{\\tfrac{\\beta}{\\gamma}}\\xi^1,\\sqrt{\\tfrac{\\beta}{\\gamma}}\\hat{\\xi}^1\\right), \\notag \n\t\\\\ \n\t&\\mbox{ for all } \\hat{v}\\in L^2_{\\delta t}(0,T;\\RR^N),\n\\end{align}\nwhere $\\hat{\\xi}^1$ comes from the sequence $\\hat{\\xi}=(\\hat{\\xi}^n)_{n\\in\\inter{1,M}}$, i.e., the solution to \\eqref{eq:backward_xi} with right-hand side $y^n(\\hat v,0)$.\n\nOur goal is now to find a suitable expression for the gradient $\\nabla J_{h,\\delta t}^\\gamma(v)$, since it will be used later in a gradient descent algorithm. We begin by defining the space $X_{h,\\delta t}:= \\mathbb R^N\\times L^2_{\\delta t}(0,T;\\mathbb R^N)$, endowed with the canonical inner product $\\langle\\cdot,\\cdot \\rangle_{X_{h,\\delta t}}:= (\\cdot,\\cdot)+(\\cdot,\\cdot)_{L^2_{\\delta t}(0,T;\\RR^N)}$. Consider the linear operator \n\\begin{equation*}\n\tR_{h,\\delta t}: L^2_{\\delta t}(0,T;\\mathbb R^N)\\to X_{h,\\delta t}, \\quad R_{h,\\delta t}v:=\\left(\\sqrt{\\tfrac{\\beta}{\\gamma}}\\xi^1,y\\right)\n\\end{equation*}\nwhere the pair $(\\xi^1,y)$ can be found by solving the forward-backward system\n\n\\begin{equation}\\label{eq:for-back_low}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{y^{n+1}-y^n}{\\delta t}+\\mathcal A_{h} \\, y^{n+1}=\\mathcal B_h v^{n+1}, & n \\in \\inter{0, M-1}, \n\t\t\\\\\n\t\t\\displaystyle \\frac{\\xi^{n}-\\xi^{n+1}}{\\delta t}+\\mathcal A_h^* \\xi^{n}= y^n, & n\\in\\inter{1,M}, \n\t\t\\\\\n\t\ty^0=0, \\quad \\xi^{M+1}=0\n\t\\end{cases}\n\\end{equation}\n\nWe observe that system \\eqref{eq:for-back_low} is in cascade form: given $v\\in L^2_{\\delta t}(0,T;\\mathbb R^N)$, we can solve first for $y=(y^n)_{n\\in\\inter{1,M}}$ forward in time and then use this information to solve for $\\xi=(\\xi^n)_{n\\in\\inter{1,M}}$ backwardly to recover the datum $\\xi^1$. \n\nNow, let us compute the adjoint operator $R_{h,\\delta t}^\\star$. We introduce the following system: for given $\\sigma_0\\in\\mathbb R^N$ and $f \\in L^2_{\\delta t}(0,T;\\mathbb R^N)$ we set\n\\begin{align}\\label{eq:back-for_low}\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{\\sigma^{n+1}-\\sigma^n}{\\delta t}+\\mathcal A_{h} \\, \\sigma^{n+1}=0, & n \\in \\inter{0, M-1}, \n\t\t\\\\\n\t\t\\displaystyle \\frac{p^{n}-p^{n+1}}{\\delta t}+\\mathcal A_h^* p^{n}= f^n-\\sigma^n, & n\\in\\inter{1,M}, \n\t\t\\\\\n\t\t\\sigma^0=\\displaystyle -\\tfrac{\\beta}{\\gamma}\\sigma_0, \\quad p^{M+1}=0\n\t\\end{cases}\n\\end{align}\n\nAs before, we note that \\eqref{eq:back-for_low} is in cascade form, namely, given $\\sigma_0 \\in \\mathbb R^N$ we can solve for $\\sigma=(\\sigma^n)_{n\\in\\inter{1,M}}$ and with this we can solve for $p=(p^n)_{n\\in\\inter{1,M}}$ for any given $f\\in L^2_{\\delta t}(0,T;\\mathbb R^N)$.\n\nMultiplying the first equation of \\eqref{eq:for-back_low} by $p^{n+1}$ and summing over $n$, we have by a direct computation that\n\\begin{equation}\\label{eq:duality_1}\n\t\\left(y,f-\\sigma \\right)_{L^2_{\\delta t}(0,T;\\mathbb R^N)}=(v,p)_{L^2_{\\delta t}(0,T;\\RR^N)}\n\\end{equation}\nwhere we have used that $y$ and $p$ have zero initial and final datum, respectively. Analogously, multiplying the first equation of \\eqref{eq:back-for_low} by $\\xi^{n+1}$, summing over $n$ and using the initial and final conditions, we get\n\\begin{equation}\\label{eq:duality_2}\n\t(\\sigma,y)_{L^2_{\\delta t}(0,T;\\RR^N)}=-\\left(\\tfrac{\\beta}{\\gamma}\\sigma_0,\\xi^1\\right).\n\\end{equation}\nCombining \\eqref{eq:duality_1} and \\eqref{eq:duality_2} and recalling the definition of the operator $R_{h,\\delta t}$, we have\n\\begin{equation*}\n\t\\left\\langle R_{h,\\delta t} v, \\left(\\sqrt{\\tfrac{\\beta}{\\gamma}}\\sigma_0,f\\right) \\right\\rangle_{X_{h,\\delta t}}=\\left\\langle \\left(\\sqrt{\\tfrac{\\beta}{\\gamma}}\\xi^1,y\\right),\\left(\\sqrt{\\tfrac{\\beta}{\\gamma}}\\sigma_0,f\\right) \\right\\rangle_{X_{h,\\delta t}}=(v,p)_{L^2_{\\delta t}(0,T;\\RR^N)}\n\\end{equation*}\nThus, $R_{h,\\delta t}^\\star: X_{h,\\delta t}\\to L^2_{\\delta t}(0,T;\\RR^N)$ is defined as \n\\begin{align*}\n\tR^\\star_{h,\\delta t}(\\sqrt{\\tfrac{\\beta}{\\gamma}}\\sigma_0,f):= p,\n\\end{align*}\nwhere $p=(p^n)_{n\\in\\inter{1,M}}$ can be found from the solution to the forward-backward system \\eqref{eq:back-for_low}. By linearity and using the above definitions, we can then rewrite \\eqref{eq:deriv_J_gamma_discr} as\n\\begin{align*}\n\t\\left(\\nabla J^\\gamma_{h,\\delta t}(v),\\hat{v}\\right)= 2\\beta\\left\\langle R_{h,\\delta t} v, R_{h,\\delta t}\\hat v \\right\\rangle_{X_{h,\\delta t}}+ 2\\left(\\mathcal B_h v,\\mathcal B_h \\hat{v}\\right)_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}-2\\beta(\\bar{z},S_{h,\\delta t}\\hat v)_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}\n\\end{align*}\nwhence $\\nabla J^\\gamma_{h,\\delta t}(v)= 2 \\beta R^\\star_{h,\\delta t} R_{h,\\delta t} v+2\\mathcal B_h^*\\mathcal B_h v-2\\beta S_{h,\\delta t}^\\star \\bar z$, and thus the optimal control we are looking for can be computed by solving the linear problem\n\\begin{equation}\\label{eq:lin_regret}\n\t\\left(\\beta R^\\star_{h,\\delta t} R_{h,\\delta t}+\\mathcal B_h^*\\mathcal B_h\\right) v=\\beta S_{h,\\delta t}^\\star \\bar z.\n\\end{equation}\n\nObserve that the structure of the problem is the same as in \\eqref{eq:opt_control}, but this time we have to evaluate the operator $R_{h,\\delta t}^\\star R_{h,\\delta t}$ which amounts to solve four equations instead of two.\n\n\\subsubsection{Numerical results for the low-regret optimal control problem}\n\nWe fix $T=1$ and consider the kernel functions $K_i$, $i=1,2$ given in \\eqref{eq:kernel_num}. As before, we consider the control set $\\mathcal O=(0.2,0.8)$ and the time-independent target function $\\bar z(x)=\\sin(2\\pi x)$. As for the optimal control problem, we will solve the linear problem \\eqref{eq:lin_regret} by using a standard gradient descent method. \n\nIn Figure \\ref{fig:soly_control_regret}, we show the evolution in time of the state variable $y$ and the adjoint state $\\xi$ controlled with the low-regret control $v^\\gamma$ coming from the minimization of the functional \\eqref{eq:func_discr_low}. In this case, we have tuned the parameters to $\\beta=100$ and $\\gamma=1$. We observe in Figure \\ref{fig:soly_control_regret}a, the evolution of the state variable $y$ starting from 0 and transitioning into a steady state which seems to be similar to the steady state shown in Figure \\ref{fig:soly_control_beta}a. In Figure \\ref{fig:soly_control_regret}b, we observe the backward evolution of the adjoint variable $\\xi$ starting from the zero data at time $T=1$.\n\n\\begin{figure}[ht!]\n\\centering\n\\subfloat[$\\beta=10$. Cost of control $\\|B_h v\\|_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}=1.57137 $]\n{\n\\includegraphics{.\/figures_final\/Pb_Kernel_with_Incomplete_Data_New_version-figure6.pdf}\n} \\qquad \n\\subfloat[$\\beta=10^2$. Cost of control $\\|B_h v\\|_{L^2_{\\delta t}(0,T;\\mathbb R^{N})}=4.19168 $]\n{\n\\includegraphics{.\/figures_final\/Pb_Kernel_with_Incomplete_Data_New_version-figure7.pdf}\n}\n\\caption{Evolution in time of the controlled solution $y$ and the adjoint state $\\xi$ with low-regret control with parameters $\\beta=10$ and $\\gamma=0.1$.}\n\\label{fig:soly_control_regret}\n\\end{figure}\n\nIn Table \\ref{tab:low_reg_table}, we have collected data for the low-regret control problem with different values of $\\beta$ and $\\gamma$. From there, we can conclude that as in the optimal control problem, the higher the value of $\\beta$ is, the better approximation to the target function $\\bar{z}$ is.  We also see that the parameter $\\gamma$ influences greatly the behavior of the control problem in the sense that for given $\\beta>0$, lower values of $\\gamma$ translate into a smaller $L^2$-norms of the control, affecting the overall approximation of the target $\\bar z$. \n\n\\begin{table}\n\\centering\n\\renewcommand{\\arraystretch}{1.35}\n\\begin{tabular}{ |c||c|c|c|  }\n\\hline\n\\multicolumn{4}{|c|}{$\\beta=1$} \\\\\n\\hline\n$\\gamma$ & $J_{h,\\delta t}^\\gamma(v)$ & $\\|B_h v\\|_{L^2_{\\delta t}(0,T;\\RR^N)}$ & $\\|y(v,0)-\\bar z\\|_{L^2_{\\delta t}(0,T;\\RR^N)}$ \\\\\n\\hline\n10 & -0.0132321 & 0.11325 & 0.688434 \\\\ 1 & -0.0132209 & 0.113154 & 0.68845 \\\\ 0.1 &-0.0131106 & 0.112213 & 0.688605 \\\\ 0.01 & -0.0121515 & 0.10422 & 0.689959 \\\\\n \\hline \n  \\multicolumn{4}{|c|}{$\\beta=10$} \\\\\n  \\hline\n10   & -1.03067    & 0.882627  &   0.564147  \\\\\n 1 &   -0.968289  & 0.829583   &0.572797 \\\\\n 0.1 & -0.682832   & 0.631448   & 0.61298   \\\\\n  0.01 &-0.43599   & 0.558128  & 0.648367   \\\\\n  \\hline\n    \\multicolumn{4}{|c|}{$\\beta=10^2$} \\\\\n  \\hline\n   10   & -27.1986    & 2.37894  &   0.360065  \\\\\n  1 &  -17.9241 & 2.10709 & 0.500576  \\\\\n  0.1 &-14.9387  &  2.15546 & 0.546241   \\\\\n \\hline\n\\end{tabular}\n\\caption{Optimal energy $J^\\gamma(v)$, cost of control $\\|B_h v\\|_{L^2}$ and distance to the target $\\|y(v,0)-\\bar z\\|_{L^2}$ for the low-regret control problem with different values of $\\gamma$ and $\\beta$}\n\\label{tab:low_reg_table}\n\\end{table}\n\nWe conclude the discussion by showing that the low-regret control can be used for controlling equations in the case of missing data. To this end, we take the simulation parameters $\\beta=100$, $\\gamma=1$, $T=1$, and the kernels $K_1$ and $K_2$ of \\eqref{eq:kernel_num}. Using our computational tool, we can compute the low-regret control $v^\\gamma$ and use it for controlling system \\eqref{eq:discr_nonlocal} with different initial conditions. In Table \\ref{tab:comp_1}, we have collected some information about the performance of the low-regret control $v^\\gamma$ against the \\textit{real} optimal control $v^{opt}$ (computed with $\\beta=100$) and the uncontrolled case for different initial data. \n\nWe can see that the low-regret control $v^\\gamma$ does not make things worse as compared to the uncontrolled case, which is consistent with the goal of this strategy, but it is not as good as the performance achieved with the optimal control computed with the full knowledge of the initial datum. Nonetheless, the low-regret control has the advantage that is has to be computed only once and then it can be used to control the system for a wide variety of initial data. We shall remark that there is still room for improvement as shown in Table \\ref{tab:comp_2}. There, we have increased to $\\gamma=10$ and computed the corresponding low regret control to obtain a lower optimal energy and smaller distance of the target (as compared to the last two columns of Table \\ref{tab:comp_1}). Nevertheless, how to choose effectively this parameter depends largely on the application and the experiment performed.\n\n\\begin{table}\n\\centering\n\\renewcommand{\\arraystretch}{1.35}\n\\begin{tabular} {|c|c|c|c|c|c|c|}\n\\cline{2-7} \n\\multicolumn{1}{c|}{}  & \\multicolumn{2}{c|}{Uncontrolled} & \\multicolumn{2}{c|}{Optimal control} & \\multicolumn{2}{c|}{Low-regret control} \\\\\n\\hline\nInitial datum & $J(0)$ & $\\|y(0)-\\bar z\\|_{L^2}$ & $J(v^{opt})$ & $\\|y(v^{opt})-\\bar z\\|_{L^2}$ & $J(v^\\gamma)$ & $\\|y(v^{\\gamma})-\\bar z\\|_{L^2}$ \\\\\n\\hline \n$\\sin^{10}(\\pi x)$ & 27.62901 & 0.74336 & 8.82497 & 0.28957 & 17.43680 & 0.55176  \\\\\n3 & 191.09119 & 1.95495 & 45.17162 & 0.70582 & 180.79790 & 1.88988 \\\\\n$\\mathbf 1_{(0.5,0.8)}(x)-\\mathbf 1_{(0.2,0.5)}(x)$ & 37.50553 & 0.86609 & 13.85573 & 0.38048 & 26.43313 & 0.69597  \\\\\n$\\sin(\\tfrac{1}{3}\\pi x)+0.3\\cos(\\tfrac{15}{4}\\pi x)$ & 32.88063 &0.81093 &10.83359 &0.32834 &22.34038 &0.63444 \\\\\n\\hline\n\\end{tabular}\n\\caption{Comparison of the optimal energy $J(\\cdot)$ and the distance to the target for the controlled solution $\\|y(\\cdot)-\\bar{z}\\|_{L^2}$ for system \\eqref{eq:discr_nonlocal} controlled with different control functions and initial conditions.}\n\\label{tab:comp_1}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\renewcommand{\\arraystretch}{1.35}\n\\begin{tabular} {|c|c|c|}\n\\hline\nInitial datum & $J(v^\\gamma)$ & $\\|y(v^{\\gamma})-\\bar z\\|_{L^2}$ \\\\\n\\hline \n$\\sin^{10}(\\pi x)$ &11.92807 &0.42667  \\\\\n3  &175.15713 &1.85651  \\\\\n$\\mathbf 1_{(0.5,0.8)}(x)-\\mathbf 1_{(0.2,0.5)}(x)$  &17.98742 &0.55067   \\\\\n$\\sin(\\tfrac{1}{3}\\pi x)+0.3\\cos(\\tfrac{15}{4}\\pi x)$ &15.81261 &0.50965  \\\\\n\\hline\n\\end{tabular}\n\n\\caption{Effect of the parameter $\\gamma$ on the controlled solution with low-regret control.}\n\\label{tab:comp_2}\n\\end{table}\n\n\\bibliographystyle{siam}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nNetworks define versatile mathematical tools which can be conveniently invoked for modelling a plethora of physical systems~\\cite{AlbertBarabasi,newmanbook}.  Local interactions among elementary constituents take place on isolated patches, the nodes of the network, while long-ranged exchanges crawls across the links, typically driven by diffusion, bridging thereby adjacent nodes of the collection.  Networks of coupled oscillators can self-organise to operate in unison~\\cite{Pikovsky2001,arenasreview}, a condition that is necessary to sustain normal brain activity but which can yield pathological states in case of hyper-synchronisation, i.e., the inability of neurones to desynchronise~\\cite{AECPlos2018}. In a completely different context, favouring a perfect synchrony of cyclic rhythms is vital for an optimal handling of energy production and distribution in power grids~\\cite{MMAN2013}. Moreover, spatial non homogenous stationary stable motifs can spontaneously emerge following a symmetry breaking mechanism~\\cite{Turing}, the analogue of a Turing instability for reaction-diffusion systems anchored on networks~\\cite{NM2010}, which amplifies small perturbations acting on a uniform background. \n\nThese phenomena have been mainly characterised with reference to static networks. That is, the network connections do not change over time, or, alternatively, the rate of modulation is very slow as compared to the typical time scales that regulate the dynamics of the state variables. The few attempts reported in the literature to generalise beyond this setting, focused either on fast switching networks~\\cite{SBGR2006,PABFC2017,LFCP2018} among distinct static configurations, thus implying that the natural time scale of the network evolution is extremely fast as compared to that associated to the underlying dynamical system. In~\\cite{AmritkarHu2006,BHCAKP,ZS2021} commutative temporal networks are instead invoked, a working hypothesis which may prove too restrictive because it assumes the eigenvectors of the relevant involved matrices, e.g., the Laplacian or the adjacency matrix, to not evolve in time. Other attempts to tackle the problem of a comprehensive theory of pattern formation on time dependent networks do not bear the sought generality, because they are once again implicitly based on the assumption of static, or very slowly varying, Laplace eigenvectors~\\cite{vangorder2}, as we will show in the following.\n\n All endeavours must however deal with revisiting the seminal concept of master stability equation~\\cite{Pecora}, so as to incorporate an explicit account of the inherent plasticity of the underlying network~\\cite{Holme2013,MR2016}. The intent is accomplished in this work, where the master stability formalism is expanded to include the contributions stemming from an imposed time evolution of the links weight. The theory is tested against two distinct applications designed to tackle both synchronisation and Turing settings. The general framework that we shall here address assumes (nonlinear) diffusive inter-node exchanges. This is a mandatory pre-requisite for a uniform solution of the extended system to exist and, as such, it is routinely invoked  in different realms. Another viable scenario is found when the coupling term is a nonlinear function of the difference of the state variables referred to connected nodes (e.g., the paradigmatic Kuramoto model~\\cite{Kuramoto} and its extension). This falls also under the umbrella of the developed theory, as we shall hereafter argue. As we will show the theory hereby developed confirms that synchronisation can be  enhanced by commutative temporal networks as reported in the litterature~\\cite{BHCAKP,GFRMRABB2022}; we moreover take a step forward by showing that such a claim holds true beyond the limited setting of commuting networks. Similarly, we also show that the constraint on the curvature of the master stability function recently introduced in~\\cite{ZS2021} can be relaxed once we remove the assumption of commutative temporal networks; indeed there exist time varying networks whose dynamics synchronise for a larger interval of the coupling strength without meeting the above condition. Finally, the proposed general framework contributes to significantly expand our current understanding on Turing patterns on time varying networks, beyond the settings so far considered which relied on the fast switching assumption~\\cite{PABFC2017} or on peculiar choices of the evolution of the eigenvectors of the Laplace matrix~\\cite{vangorder2}.\n \n\n\\section{Materials and Methods} \n\\label{sec:themodel}\nTo set the stage for further analysis, we will begin our discussion by inspecting the spectral characteristics of a time dependent discrete Laplacian operator. Consider a symmetric time varying network made by $n$ nodes, characterised by the adjacency matrix $\\mathbf{A}(t)$:  $A_{ij}(t)=A_{ji}(t)\\neq 0$ whenever nodes $i$ and $j$ are connected at time $t$ and $A_{ij}(t)=A_{ji}(t)=0$ otherwise. Given the adjacency matrix one can construct the (combinatorial) Laplace matrix, $L_{ij}(t)= A_{ij}(t)-\\delta_{ij} k_i(t)$, where $k_i(t)=\\sum A_{ij}(t)$ denotes the degree of node $i$ at time $t$ and $\\delta_{ij}$ is the Kronecker-$\\delta$. Since $\\mathbf{L}(t)$ is a symmetric matrix, for all $t$, one can find a (time dependent) orthonormal basis of eigenvectors, $\\vec{\\phi}^{(\\alpha)}(t)$, associated with the eigenvalues $\\Lambda^{(\\alpha)}(t)\\leq 0$ such that \n\\begin{equation*}\n\\mathbf{L}(t)\\vec{\\phi}^{(\\alpha)}(t)=\\Lambda^{(\\alpha)}(t)\\vec{\\phi}^{(\\alpha)}(t) \\quad \\forall \\alpha=1,\\dots, n \\text{ and } \\forall t\\, .\n\\end{equation*}\nMoreover \n \\begin{equation}\n \\left(\\vec{\\phi}^{(\\alpha)}(t)\\right)^\\top\\cdot \\vec{\\phi}^{(\\beta)}(t)=\\delta_{\\alpha \\beta}\\, ,\n\\label{eq:eigeq2}\n\\end{equation}\nwhere the dot represents  the scalar product and $\\left( \\right) ^\\top$ stands for the transpose operation. Finally, with no loss of generality, we order the eigenvalues in such a way that $0=\\Lambda^{(1)}(t)>\\Lambda^{(j)}(t)$ for all $j\\in\\{2,\\dots,n\\}$ and we recall that $\\vec{\\phi}^{(1)}(t)=(1,\\dots,1)^\\top\/\\sqrt{n}$.\n\nAssume that the eigenvectors evolve smoothly in time. Then, one can express the eigenvectors change rate as:\n\\begin{equation}\n\\label{eq:cab}\n\\frac{d \\vec{\\phi}^{(\\alpha)}}{dt}(t)=\\sum_{\\beta}c_{\\alpha\\beta}(t)\\vec{\\phi}^{(\\beta)}(t)\\quad\\forall \\alpha=1,\\dots, n\\, .\n\\end{equation}\nwhere $\\mathbf{c}(t)$ is a $n\\times n$ time dependent matrix that quantifies the projections on the independent eigendirections. By recalling the orthonormality condition~\\eqref{eq:eigeq2} we can straightforwardly conclude that $\\mathbf{c}$ is a real skew symmetric matrix with a null first row and first column, i.e., $c_{\\alpha\\beta}+c_{\\beta\\alpha}=0$ and $c_{1\\alpha}=0$.\n\nThe time evolution of the eigenvectors is hence self-consistently ruled by the system of ODEs~\\eqref{eq:cab}, complemented by the initial conditions, i.e., the Laplace eigenbasis at $t=0$. Notice that the case of switching networks can be also brought back to the above scenario, by approximating piecewise regular functions with smooth profiles or, alternatively, using the Fourier transform (see Appendix~\\ref{sec:smallnet}).\nFinally, we require the eigenvalues to satisfy standard conditions ensuring the existence and uniqueness of the linear system~\\eqref{eq:GLHGlinalpha3}, so in particular differentiability is no longer required as in~\\cite{vangorder2}.\n\nWe are now in a position to  elaborate on the conditions that generalise the master stability formalism to systems defined on time varying networks. To this end, consider a $d$-dimensional system described locally, i.e., at each node,  by the following ODE:\n\\begin{equation}\n\\label{eq:dotxF}\n\\frac{d\\mathbf{x}}{dt}=\\mathbf{F}(\\mathbf{x})\\quad \\mathbf{x}\\in\\mathbb{R}^d\\, ,\n\\end{equation}\nwhere $\\mathbf{F}$ is an arbitrary nonlinear function. Further, assume $n$ identical copies of the above system to be coupled via a time varying network through diffusive interactions modified with the inclusion of a nonlinear function $\\mathbf{H}$:\n\\begin{equation}\n\\label{eq:maineq}\n\\frac{d\\mathbf{x}_i}{dt}=\\mathbf{F}(\\mathbf{x}_i) +\\varepsilon \\sum_{j} L_{ij}(t) \\mathbf{H}(\\mathbf{x}_j)\\, ,\n\\end{equation}  \nwhere $\\mathbf{x}_i=(x_i^{(1)},\\dots,x_i^{(d)})^\\top$ photographs the state of the system on node $i$, $\\varepsilon>0$ is the strength of the coupling and $L_{ij}(t)$ are the entries of the time varying Laplace matrix.\n\nLet us now fix a reference orbit, $\\mathbf{s}(t)$, of the aspatial system~\\eqref{eq:dotxF}. By exploiting the obvious condition $\\sum_j L_{ij}(t)=0$ for all $i=1,\\dots, n$ and all $t$, it is immediate to conclude that $\\mathbf{s}(t)$ is also solution of Eq.~\\eqref{eq:maineq}. Namely the coupled system exhibits a spatially homogeneous synchronous solution. Assuming the latter solution to be stable for the decoupled system~\\eqref{eq:dotxF}, the question to be answered is whether it can turn unstable (or conversely, preserve its stability) when the inter-nodes couplings get activated by a small heterogeneous perturbation.  Denote by $\\delta\\mathbf{x}_i=\\mathbf{x}_i-\\mathbf{s}$ the deviations from the reference orbit and by assuming these latter small, one can derive a self-consistent set of linear ODE for tracking the evolution of the perturbation in time. To this end,  we introduce $\\delta\\mathbf{x}_i$ in Eq.~\\eqref{eq:maineq} and perform a Taylor expansion arrested to linear order, to eventually get:\n\\begin{equation}\n\\label{eq:GLHGlin}\n\\frac{d\\delta\\mathbf{x}_i}{dt}=\\mathbf{J}_\\mathbf{F}(\\mathbf{s}(t))\\delta\\mathbf{x}_i +\\varepsilon \\sum_{j} {L}_{ij}(t) \\mathbf{J}_\\mathbf{H}(\\mathbf{s}(t))\\delta\\mathbf{x}_j\\, , \n\\end{equation}\nwhere $\\mathbf{J}_\\mathbf{F}(\\mathbf{s}(t))$ (resp. $\\mathbf{J}_\\mathbf{H}(\\mathbf{s}(t))$) denotes the Jacobian matrix of the function $\\mathbf{F}$ (resp. $\\mathbf{H}$) evaluated on the trajectory $\\mathbf{s}(t)$. Remark that a completely equivalent governing equation is obtained when starting from an inter-nodes coupling of the type $\\sum_j A_{ij} \\mathbf{H}(\\mathbf{x}_j-\\mathbf{x}_i)$, with $\\mathbf{H}(\\mathbf{0})=\\mathbf{0}$, as anticipated above.\n\nTo make further progress in the study of the linear non-autonomous system (\\ref{eq:GLHGlin}), we project $\\delta\\mathbf{x}_i$ onto the orthonormal basis formed by the eigenvectors of $\\mathbf{L}(t)$, to yield $\\delta\\mathbf{x}_i=\\sum_\\alpha \\delta\\hat{\\mathbf{x}}_{\\alpha}\\phi^{(\\alpha)}_i$. By inserting the latter into~\\eqref{eq:GLHGlin} and recalling the definition of matrix $\\mathbf{c}(t)$, one obtains for all $\\beta$ (see Appendix~\\ref{sec:MSEtv}):\n\\begin{equation}\n\\label{eq:GLHGlinalpha3}\n\\frac{d\\delta\\hat{\\mathbf{x}}_{\\beta}}{dt} = \\sum_\\alpha c_{\\beta\\alpha}(t)\\delta\\hat{\\mathbf{x}}_{\\alpha}+\\left[\\mathbf{J}_\\mathbf{F}(\\mathbf{s}(t))+\\varepsilon \\Lambda^{(\\beta)}(t)\\mathbf{J}_\\mathbf{H}(\\mathbf{s}(t))\\right]\\delta\\hat{\\mathbf{x}}_{\\beta}\\, . \n\\end{equation}\n\n By introducing  $\\delta\\hat{\\mathbf{x}}=(\\delta\\hat{\\mathbf{x}}^\\top_{1},\\dots,\\delta\\hat{\\mathbf{x}}_{n}^\\top)^\\top$ one can cast Eq.~\\eqref{eq:GLHGlinalpha3} in  compact form:\n  \\begin{equation}\n\\label{eq:GLHGlinalpha3compact}\n\\frac{d\\delta\\hat{\\mathbf{x}}}{dt} =  \\left[\\mathbf{c}\\otimes \\mathbf{1}_d+\\mathbf{1}_n\\otimes \\mathbf{J}_\\mathbf{F}+\\varepsilon \\mathbf{\\Lambda}\\otimes\\mathbf{J}_\\mathbf{H}\\right]\\delta\\hat{\\mathbf{x}}:=\\mathbf{M}\\delta\\hat{\\mathbf{x}}\\, ,\n\\end{equation}\nwhere $\\otimes$ denotes the Kronecker product, $\\mathbf{\\Lambda}=\\mathrm{diag}\\left(\\Lambda^{(1)},\\dots,\\Lambda^{(n)}\\right)$ and, given any positive integer $m$, $\\mathbf{1}_m$ denotes the $m$ dimensional identity matrix. Let us observe that the latter formula and the following analysis differ from the one presented in~\\cite{vangorder2} where the perturbation is assumed to align onto a single mode, hypothesis that ultimately translates in the stationarity of the Laplace eigenvectors, that is $\\mathbf{c}=\\mathbf{0}$. The same assumption is also at the root of the results by~\\cite{ZS2021}; indeed commuting time varying networks implies to deal with a constant eigenbasis. In conclusion, Eq.~\\eqref{eq:GLHGlinalpha3} or Eq.~\\eqref{eq:GLHGlinalpha3compact} are capable to describe the projection of the linearised dynamics on a generic time varying Laplace eigenbasis, and thus allow us to draw general conclusions without simplifying assumptions.\n\nThe above system~\\eqref{eq:GLHGlinalpha3} represents the generalised version of the celebrated Master Stability Equation (MSE)~\\cite{Pecora,HCLP}, which includes an explicit account of the network evolution as encoded in matrix $\\mathbf{c}(t)$, as well as in the dependence of $\\Lambda^{(\\beta)}(t)$ against time. Its largest Lyapunov exponent quantifies the exponential rate at which an infinitesimal perturbation in the transverse subspace grows: it defines an improved version of the Master Stability Function (MSF) and enables one to draw conclusions about the stability of the reference orbit. In concrete terms, consider the matrix equation $d \\mathbf{O} \/ dt = \\mathbf{M} \\mathbf{O}$ where $\\mathbf{O}(0)=\\mathbf{1}_{nd}$. Solve the preceding equation numerically and compute $\\nu_i(t)$ ($i=1,..., nd$)  the (time dependent) eigenvalues of $\\mathbf{O}(t)$. The Lyapunov \nexponents are the computed by $\\lambda_i=\\lim_{t \\rightarrow \\infty} \\ln \\nu_i \/ t$.  Notice that~\\eqref{eq:GLHGlinalpha3compact} -- or its equivalent counterpart~\\eqref{eq:GLHGlinalpha3} from which matrix $\\mathbf{M}$ originates -- displays two independent time scales (in addition to the ones characterising the reactive dynamics), one reflecting  the dynamics of the isolated units, i.e., $\\mathbf{s}(t)$, and the other stemming from the network modulation over time, as mirrored in $\\mathbf{c}(t)$ and $\\Lambda^{(\\beta)}(t)$. Hence, the examined system cannot be managed via standard Floquet methods, not even when periodic homogeneous $\\mathbf{s}(t)$ are concerned. Further, classical MSF approaches -- carried out over static networks -- can be conveniently simplified by considering the evolution of the imposed perturbation along each independent direction (the eigenvectors), associated to different eigenvalues of the Laplacian. This path cannot be pursued here as the matrix $\\mathbf{c}(t)$ is responsible for a non trivial entanglement of different modes, as also remarked in~\\cite{vangorder3}. In general, system (\\ref{eq:GLHGlinalpha3compact}) should be hence handled numerically (see e.g.,~\\cite{HCLP} for an account of the subtleties to be faced when carrying the numerical computation). \n\nFor demonstrative purposes and to highlight the consequences resulting from the form of Eq.~\\eqref{eq:GLHGlinalpha3}, we will hereby concentrate onto two illustrative examples, which are constructed so as to yield a time independent generalised MSF. In both cases, the network is made of just three nodes, the eigenvalues are assumed constant and the time derivative of the eigenvectors  projected on the eigenbasis returns a matrix $\\mathbf{c}$ that does not change in time (details are supplied in~\\ref{sec:smallnet}). In the first example, the reference orbit is stationary and we will operate in the furrow of the Turing  instability~\\cite{Turing,NM2010}. In the second case, we will assume a set of nonlinearly coupled Stuart-Landau (SL) oscillators~\\cite{vanharten,aranson,garcamorales}, to investigate the impact of the network dynamics on the onset of synchronisation.  \n\n\n\\section{Turing instability on time varying network}\n\\label{sec:turing} \nThe reference solution $\\mathbf{s} \\equiv \\mathbf{s}_0$ is assumed stationary and the coupling linear, i.e., $\\mathbf{H}(\\mathbf{x})=\\mathbf{D}\\mathbf{x}$, where $\\mathbf{D}$ is a suitable diagonal matrix with positive entries. To reduce to the usual Turing setting we posit $d=2$, namely $\\mathbf{x}_i=(u_i,v_i)$, $\\mathbf{F}(\\mathbf{x}_i)=(f(u_i,v_i),g(u_i,v_i))$ and $\\mathbf{D}=\\mathrm{diag}(D_u,D_v)$. Then, Eq.~\\eqref{eq:maineq} rewrites for all $i=1,\\dots,n$\n\\begin{equation}\n\\label{eq:Tnet}\n\\begin{dcases}\n\\frac{du_i}{dt}&=f(u_i,v_i)+D_u\\sum_{j=1}^{n}L_{ij}(t) u_j  \\\\ \n\\frac{dv_i}{dt}&=g(u_i,v_i)+D_v\\sum_{j=1}^{n}L_{ij}(t) v_j \n\\end{dcases}\\, ,\n\\end{equation}\nwhere $D_u>0$ (resp. $D_v>0$) is the diffusion coefficients of species $u$ (reps. $v$) and ${L}_{ij}(t)$ are the elements of the above defined Laplace matrix of the time varying network. Recall that $\\mathbf{s}_0=(u^*,v^*)$ is a stable equilibrium solution for the reaction part. Therefore, $f(u^*,v^*)=g(u^*,v^*)=0$, $\\mathrm{tr}(\\mathbf{J}_0)<0$ and $\\det(\\mathbf{J}_0)>0$, where $\\mathbf{J}_0$ is the Jacobian of the reaction part evaluated at the equilibrium. Notice that $\\mathbf{s}_0$ is also an equilibrium solution for the whole system of coupled equations~\\eqref{eq:Tnet}. We are thus interested in studying its stability under an imposed node-dependent, hence heterogeneous  perturbation.\n\nBy setting $\\delta x_i=u_i-u^*$, $\\delta y_i=v_i-v^*$, one can follow the main steps of the theory presented above to eventually obtain the analogous of Eq.~\\eqref{eq:GLHGlinalpha3compact},\n\\begin{equation}\n\\label{eq:Tnetlin5}\n\\frac{d \\delta\\hat{\\mathbf{x}}}{dt}= \\left[\\mathbf{c}\\otimes \\mathbf{1}_2+\\mathbf{1}_n\\otimes \\mathbf{J}_0+\\mathbf{\\Lambda}\\otimes \\mathbf{D}\\right]\\delta\\hat{\\mathbf{x}}\\, ,\n\\end{equation}\nwhere $\\delta\\hat{\\mathbf{x}}=(\\delta\\hat{\\mathbf{x}}_1^\\top,\\dots,\\delta\\hat{\\mathbf{x}}_n^\\top)^\\top$ and  $\\delta\\hat{\\mathbf{x}}_\\beta=(\\delta \\hat{x}_{\\beta},\\delta \\hat{y}_{\\beta})$ represents the projection of the $i$-th perturbation $(\\delta x_i,\\delta y_i)$ on the $\\beta$-eigenvector. \n\nFor a definite application, we focus on the three-nodes time dependent network mentioned above and further characterised in~\\ref{sec:smallnet}. For sake of definitiveness let us hereby recall that the matrix $\\mathbf{c}$ defined by Eq.~\\eqref{eq:cab} is given by\n\\begin{equation*}\n \\mathbf{c}=\\left(\n\\begin{matrix}\n 0 & 0 & 0\\\\\n 0 & 0 & \\Omega\\\\\n 0 & -\\Omega& 0\n\\end{matrix}\\right)\\, ,\n\\end{equation*}\nfor some $\\Omega >0$. That together with the following initial Laplace eigenbasis\n\\begin{equation*}\n \\vec{\\phi}^{(1)}(0)=\\frac{1}{\\sqrt{3}}\\left(\n\\begin{smallmatrix}\n 1\\\\1\\\\1\n\\end{smallmatrix}\n\\right)\\, , \\vec{\\phi}^{(2)}(0)=\\frac{1}{\\sqrt{6}}\\left(\n\\begin{smallmatrix}\n 1\\\\-2\\\\1\n\\end{smallmatrix}\n\\right)\\, ,\\vec{\\phi}^{(3)}(0)=\\frac{1}{\\sqrt{2}}\\left(\n\\begin{smallmatrix}\n -1\\\\0\\\\1\n\\end{smallmatrix}\n\\right)\n\\end{equation*}\nreturns the following adjacency matrix\n\\begin{equation}\n A_{ij}(t)=\\left(\\begin{smallmatrix} 0 & \\frac{1}{2}-\\frac{\\cos\\left(\\frac{\\pi }{3}+2\\Omega t\\right)}{3} & \\frac{\\cos\\left(2\\Omega t\\right)}{3}+\\frac{1}{2}\\\\ \\frac{1}{2}-\\frac{\\cos\\left(\\frac{\\pi }{3}+2\\Omega t\\right)}{3} & 0 & \\frac{1}{2}-\\frac{\\cos\\left(\\frac{\\pi }{3}-2\\Omega t\\right)}{3}\\\\ \\frac{\\cos\\left(2\\Omega t\\right)}{3}+\\frac{1}{2} & \\frac{1}{2}-\\frac{\\cos\\left(\\frac{\\pi }{3}-2\\Omega t\\right)}{3} & 0 \\end{smallmatrix}\\right)\\, .\n\\label{eq:adjt}\n\\end{equation}\n\nMoreover, we assume the paradigmatic Brusselator scheme as the reference reaction model, namely $f(u,v)=1-(b+1)u+cu^2v$ and $g(u,v)=bu-cu^2v$, where $b$ and $c$ are the positive parameters. The stationary equilibrium is thus $u^* = 1$ and $v^* = b\/c$, while the Jacobian of the reaction part evaluated on the equilibrium is $\\partial_u f = b-1$, $\\partial_vf = c$, $\\partial_u g = -b$ and $\\partial_v g = -c$. To exemplify our conclusion we will further set $D_u =0.01$ and $D_v = 1$,  compute the eigenvalues of the linear system~\\eqref{eq:Tnetlin5} (here straightforward as the linear system has constant coefficients) and then characterise the region in the $(b,c)$ plane  where the instability is predicted to take place for (i) the standard Turing setting, i.e., when the network is made time independent ($\\Omega=0$, which implies $\\mathbf{c}$ to be the null matrix); (ii) the extended scenario when the network is made to evolve in time ($\\Omega >0$, hence $\\mathbf{c} \\ne \\mathbf{0}$).\n\nThe results reported in Fig.~\\ref{fig:resultsTuring} show that, for this specific choice of the parameters, the region deputed to the instability (black shadow) shrinks when the system is made to evolve on a time varying network~\\footnote{Throughout this work the numerical simulations have been performed using a $4$-th order Runge-Kutta scheme implemented in Matlab~\\cite{MATLAB2021}. The initial conditions have been realised by drawing uniformly random perturbations $\\delta$-close to the homogeneous equilibrium and the simulation time has been taken of the order of $(-\\log \\delta) \/ \\rho_{\\mathrm{MSF}}$, where $\\rho_{\\mathrm{MSF}}$ is the maximum of the MSF. This is indeed the time necessary to (possibly) increase the $\\delta$-perturbation up to a macroscopic size. In the rest of the work we set $\\delta=10^{-2}$, small enough to discriminate between the onset of the instability using a reasonable simulation time for the values of $\\rho_{\\mathrm{MSF}}$ we are dealing with.}. The outcome of the simulations corroborates the predictions, thus confirming the adequacy of the theory and the crucial role played by the extra contribution to the MSF which can be traced back to matrix $\\mathbf{c}$.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.22]{Fig1.pdf}\n\\caption{\\textbf{Turing instability on time varying networks.} Middle panels report the parameters region associated to the emergence of Turing instability (black region) for the Brusselator model, on respectively a static  (top panel) and time varying network (bottom panel, $\\Omega=2$). The remaining panels display the computed trajectories for  $b=10$, $c=15$ (left, yielding to Turing patterns just for the case of a static network) and $b=c=15$ (right, patterns develop in both considered situations). Here, $D_u=0.01$ and $D_v=1.0$.}\n\\label{fig:resultsTuring}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:sizeTuring} we report the size of the Turing instability region $\\Theta(\\Omega)$ as a function of $\\Omega$ (black dots); by visual inspection one can appreciate that indeed the instability region shrinks once $\\Omega$ increases, i.e., $\\Theta(0)>\\Theta(\\Omega)$ for all $\\Omega$. We can also observe  a non-monotone behaviour with a minimum for $\\Omega=1$. The Figure also supports the correctness of the results proved in~\\cite{PABFC2017}. Indeed,  for sufficiently fast network dynamics, the emergence of Turing pattern can be proved by looking at the behaviour of the system defined on the averaged network, whose associated size (i.e., the extension of the deputed region in the parameters plane) is given by the green line.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.33]{RelBrusselatorModel01012022_001All.pdf}\n\\vspace{-3cm}\n\\caption{\\textbf{Size of the Turing region with respect to $\\Omega$.} For a fixed range of model parameters $(b,c)\\in [0,25]\\times[0,25]$ we report the size of the region associated to the emergence of Turing patterns for a given $\\Omega$, assuming the network defined above as the underlying support for the dynamics. The horizontal green line stands for the size of the instability region as obtained for the averaged network. The diffusion coefficients have been set to $D_u=0.01$ and $D_v=1.0$.}\n\\label{fig:sizeTuring}\n\\end{figure*}\n\n\n\\section{Synchronisation of Stuart-Landau oscillators nonlinearly coupled via time varying networks}\n\\label{sec:SL}  \nConsider a Stuart-Landau (SL) oscillator, the normal form for a generic system close to a supercritical Hopf-bifurcation. It is characterised by a complex amplitude $w$ which evolves in time according to $\\dot{w}=\\sigma w-\\beta |w|^2w$, where $\\sigma=\\sigma_\\Re+i\\sigma_\\Im$ and $\\beta=\\beta_\\Re+i\\beta_\\Im$ are complex control parameters. The oscillator admits a limit cycle $\\hat{z}(t)=\\sqrt{\\sigma_\\Re\/\\beta_\\Re}e^{i\\omega t}$, where $\\omega=\\sigma_\\Im-\\beta_\\Im \\sigma_\\Re\/\\beta_\\Re$. The latter is a stable solution of an isolated SL equation, provided $\\sigma_\\Re>0$ and $\\beta_\\Re>0$, conditions that we hereby assume. To  proceed in the analysis we couple together \n$n$ identical SL oscillators, each bearing a complex amplitude $w_j$, with $j=1,...,n$:\n\\begin{equation}\n\\label{eq:maineqSL}\n\\frac{dw_j}{dt}= \\sigma w_j-\\beta w_j|w_j|^2+\\mu \\sum_{\\ell} {L}_{j\\ell}(t) H(w_\\ell)\\,  ,\n\\end{equation}\nwhere $\\mu=\\mu_\\Re+i\\mu_\\Im$ is a complex parameter that sets the strength of the coupling and where  $H(w)=w|w|^{m-1}$, for some integer $m\\geq 1$. The above system falls in the class of Eq.~\\eqref{eq:maineq}. We focus now on the extended solution $w_j=\\hat{z}(t)$ $\\forall j$ and inspect its stability, as a function of the model parameters, namely to the emergence of global synchronisation. To achieve this goal we set: \n\\begin{equation}\n\\label{eq:wpert}\nw_j(t)=\\hat{z}(t)(1+\\rho_j(t))e^{i\\theta_j(t)}\\, ,\n\\end{equation}\nwhere the real functions $\\rho_j(t)$ and $\\theta_j(t)$ are assumed to be small. \nA straightforward computation (more details can be found in~\\ref{sec:synchSL}) leads to :\n\\begin{equation}\n\\label{eq:maineqSLlinMSF}\n\\frac{d \\delta\\hat{\\mathbf{x}}}{dt}= \\left[\\mathbf{c}\\otimes \\mathbf{1}_2+\\mathbf{1}_n\\otimes \\mathbf{J}_0+\\mathbf{\\Lambda}\\otimes \\mathbf{J}_{H}\\right]\\delta\\hat{\\mathbf{x}}\\, ,\n\\end{equation}\nwhere $\\delta\\hat{\\mathbf{x}}=(\\delta\\hat{\\mathbf{x}}_1^\\top,\\dots,\\delta\\hat{\\mathbf{x}}_n^\\top)^\\top$ and  where $\\delta\\hat{\\mathbf{x}}_\\beta=(\\hat{\\rho}_{\\beta},\\hat{\\theta}_{\\beta})$ denotes\nthe projection of $(\\rho_i,\\theta_i)$ on the $\\beta$-eigenvector. The Jacobian of the isolated SL system is given by $\\mathbf{J}_0=\\left(\n\\begin{smallmatrix}\n -2\\sigma_\\Re & 0\\\\\n  -2\\beta_\\Im \\sigma_\\Re\/\\beta_\\Re & 0\n\\end{smallmatrix}\\right)$, whereas the contributions stemming from the nonlinear coupling correspond to  $\\mathbf{J}_H=\\left(\\frac{\\sigma_\\Re}{\\beta_\\Re}\\right)^{\\frac{m-1}{2}}\\left(\n\\begin{smallmatrix}\nm\\mu_\\Re & -\\mu_\\Im\\\\\nm\\mu_\\Im & \\mu_\\Re\n\\end{smallmatrix}\\right)$.\nTo illustrate the outcome of the analysis, we set $n=3$ and deal with the time dependent network introduced above (see also~\\ref{sec:smallnet}). We let $\\beta_\\Im$ and $\\mu_\\Im$ to vary freely, and freeze the other parameters to nominal values, for the sake convenience. In Fig.~\\ref{fig:resultsSynchro} we depict in black the region of the parameters plane $(\\beta_\\Im, \\mu_\\Im)$ where the generalised MSF is positive, i.e., where the uniform limit cycle solution is predicted to be unstable and thus synchronisation is excluded. The complementary domain points hence to the choice of the parameters that yields stable synchronous oscillations. The inherent dynamics of the network enhances the ability of the system to synchronise, at a global scale. Indeed the region in black shrinks when the network is made to evolve in time. However, the parameters  $\\beta_\\Im$ and $\\mu_\\Im$ can be chosen in such a way that the coupled collection of SL oscillators turns unstable when evolved on a dynamical support ($\\Omega \\ne 0$), while being stable on its static counterpart ($\\Omega = 0$).\n\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.22]{Fig2.pdf}\n\\caption{\\textbf{Synchronisation on time varying networks.} Middle panels identify the regions in the parameters plane $(\\beta_\\Im, \\mu_\\Im)$ where the synchronous solution is unstable (back domains) on, respectively, a static (top panel) and time varying network (bottom panel, $\\Omega=4$). The remaining panels show the evolution in time of the real part of \n$w_j$, for different choices of the parameters (as indicated by the arrows). The two selected working points are $\\beta_\\Im=4.9$, $\\mu_\\Im=-2.3$ and $\\beta_\\Im=4.9$, $\\mu_\\Im=-3.6$. Here,  $\\sigma = 1.0+4.3i$, $\\beta_\\Re = 1.0$, $\\mu_\\Re = 0.1$ and $m = 3$.}\n\\label{fig:resultsSynchro}\n\\end{figure*}\n\nTo study the interplay of the coupling strength and the parameter $\\Omega$ on synchronisation, let us set in Eq.~\\eqref{eq:maineqSL} $\\mu = \\varepsilon \\mu_0$ for a fixed complex parameter $\\mu_0$ (hereby fixed, without loss of generality, to $\\mu_0=0.1-0.5 i$) and a positive real parameter $\\varepsilon$. In Fig.~\\ref{fig:fig1SL} we report the behaviour of the MSF as a function of the coupling strength $\\varepsilon$ for two values of $\\Omega$ (blue curve $\\Omega=2.0$, red curve $\\Omega=0.0$, i.e., static network). Let us recall that values of $\\varepsilon$ associated to a positive MSF correspond to an unstable limit cycle solution for the SL and thus to desynchronisation. The size of such interval of desynchronisation is given by the first nonzero value of $\\varepsilon$ for which the MSF is zero, say $\\varepsilon_0$. The latter clearly  depends on $\\Omega$ and moreover $\\varepsilon_0(2.0)>\\varepsilon_0(0.0)$, namely the time varying network exhibits a larger domain of synchronisation, being the interval of instability smaller than in the static network case.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.33]{GLModel20012022_001_mod.pdf}\n\\vspace{-2cm}\n\\caption{\\textbf{Master Stability Function for the Stuart-Landau nonlinearly coupled oscillators.} We report the MSF computed for the SL system coupled with the simple $3$ nodes network as a function of $\\varepsilon$. The remaining parameters have been fixed to the values, $\\Omega=2.0$, $\\sigma = 1.0+4.3i$, $\\beta= 1.0+1.1i$, $\\mu_0 = 0.1-0.5i$ and $m = 3$. We emphasised the first nontrivial zeros of the MSF,  $\\varepsilon_0(\\Omega)$, as as a function of the coupling strength and its dependence of $\\Omega$.}\n\\label{fig:fig1SL}\n\\end{figure*}\n\nThe latter claim is further supported by the results displayed in Fig.~\\ref{fig:fig2SL} where we report the dependence of $\\varepsilon_0(\\Omega)$ as a function of $\\Omega$. One can observe that $\\varepsilon_0(0)>\\varepsilon_0(\\Omega)$ for all the considered values of $\\Omega$ and thus conclude that the time varying network exhibits a larger domain of synchronisation. We observe that such result is not limited to the small network here considered as numerically shown in~\\ref{sec:smallnet}.\n\nLet us remark that the MSF has a negative second derivative evaluated at $\\varepsilon_0(\\Omega)$ for all $\\Omega$ (see Fig.~\\ref{fig:fig1SL}), we can thus conclude that the result here presented generalises the one recently found in~\\cite{ZS2021} without resorting to the restrictive assumption of commuting time varying networks: if the MSF is convex, with respect to the coupling strength, then the time varying network can synchronise for a range of $\\varepsilon$ larger than that associated to its static analogue.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.33]{RelGLModel04022022_001_mod.pdf}\n\\vspace{-2cm}\n\\caption{\\textbf{Size of the synchronisation region for the Stuart-Landau nonlinearly coupled oscillators.} We report the value of $\\varepsilon_0(\\Omega)$ as a function of $\\Omega$. The remaining parameters have been fixed to the values, $\\sigma = 1.0+4.3i$, $\\beta= 1.0+1.1i$, $\\mu_0 = 0.1-0.5i$ and $m = 3$. Let us observe the existence of an interval of values $I=[\\Omega_1,\\Omega_2]$ such that for all $\\Omega\\in I$ there exist three values of $\\varepsilon_0(\\Omega)$ (see Appendix~\\ref{sec:synchSL}).}\n\\label{fig:fig2SL}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{sec:concl}\nSumming up we have here extended the Master Stability theory to a setting where the time evolution of the underlying network support is explicitly accounted for. The dynamics of the network is reflected in a time dependent skew symmetric matrix $\\mathbf{c}(t)$, that stems from the time evolution of the Laplacian eigenvectors. The corresponding eigenvalues can also adjust in time and contribute to shape the evolution of the imposed perturbation at a linear order of approximation. The proposed theory is general and applies to all systems for which the  Master Stability formalism was originally conceived. For illustrative purposes we have here decided to test it against two simple cases study, which can be respectively ascribed to synchronisation and Turing instability. The method applies however to more complex settings, as e.g., synchronisation of  chaotic trajectories (see Appendix~\\ref{sec:synchLorenz}). In particular, we showed that the condition of the negative curvature of the MSF invoked in~\\cite{ZS2021} to ensure that commuting time varying networks do synchronise, is no longer required when general time varying networks are accounted for. Indeed we have demonstrated that coupled Lorenz systems can easily synchronise once coupled with a time varying network, even if the associated MSF has a positive curvature (see Appendix~\\ref{sec:synchLorenz}). \n\n\\vspace{0.5cm}\n\\noindent\\textbf{Acknowledgement.} We thank Robert A Van Gorder for fruitful discussions.\n\n \\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{1}\n\nThe existence of maximally nonassociative quasigroups was an open\nquestion for quite a long time \\cite{kep,gh,dv1}. In 2018 a maximally\nnonassociative quasigroup of order nine was found \\cite{dv3}, and that\nwas the first step to realise that Stein's nearfield\nconstruction \\cite{st} can be used to obtain maximally nonassociative\nquasigroups of all orders $q^2$, where $q$ is an odd prime\npower \\cite{dl}. A recent result \\cite{dw} constructs examples of all\norders with the exception of a handful of small cases and two sparse\nsubfamilies within the case $n\\equiv 2\\bmod 4$. The main construction\nof \\cite{dw} is based upon quadratic orthomorphisms and can be used\nfor all odd prime powers $q\\ge 13$. However, it was left open how many\nquadratic orthomorphisms can be used in the construction. We provide\nan asymptotic answer to that question in this paper.\n\nThroughout this paper $q$ is an odd prime power and\n$\\mathbb{F} = \\mathbb{F}_q$ is a field of order $q$. For $a,b\\in \\mathbb{F}$ \ndefine a binary operation on $\\mathbb{F}$ by\n\\begin{equation}\\label{e11}\nu*v = \n\\begin {cases}\nu+a(v-u)&\\text{if $v-u$ is a square;}\\\\\nu+b(v-u)&\\text{if $v-u$ is a nonsquare.}\n\\end{cases}\n\\end{equation}\nThis operation yields a quasigroup \nif and only if\nboth $ab$ and $(1-a)(1-b)$ are squares, and both $a$ and $b$ are\ndistinct from $0$ and $1$, cf.~\\cite{Eva18,cyclatom}. \nDenote by $\\Sigma = \\Sigma(\\mathbb{F})$ the\nset of all such $(a,b) \\in \\mathbb{F}\\times \\mathbb{F}$ for which $a\\ne b$. \n\nFor each $(a,b) \\in \\Sigma$ denote the quasigroup $(\\mathbb{F},*)$\nby $Q_{a,b} = Q_{a,b}(\\mathbb{F})$. A quasigroup $(Q,*)$ is said to be\n\\emph{maximally nonassociative} if \n\\begin{equation}\\label{e12}\n(u*v)*w = u*(v*w)\\ \\Longrightarrow \\ u=v=w\n\\end{equation}\nholds for all $u,v,w \\in Q$. By \\cite{kep}, a maximally nonassociative \nquasigroup\nhas to be idempotent (i.e., $u*u=u$ for all $u\\in Q$). Hence in\na maximally nonassociative quasigroup the converse implication  \nto \\eref{12} holds as well.\n\nIf $a=b\\in \\mathbb{F}\\setminus\\{0,1\\}$, then \\eref{11} defines a quasigroup\nin which $u*(v*u) = (u*v)*u$ for all $u,v \\in \\mathbb{F}$. This means that\nsuch a quasigroup is never maximally nonassociative. If $q\\ge 13$,\nthen there always exists $(a,b) \\in \\Sigma(\\mathbb{F}_q)$ such that\n$Q_{a,b}$ is maximally nonassociative \\cite{dw}. This paper is concerned\nwith the density of such $(a,b)$. Our main result is as follows:\n\n\\begin{thm}\\label{11}\nFor an odd prime power $q$ denote by $\\sigma(q)$ the number\nof $(a,b) \\in \\Sigma(\\mathbb{F}_q)$ for which $Q_{a,b}$ is maximally\nnonassociative. Then \n\\begin{equation}\\label{e13}\n\\lim_{q\\to \\infty}\\frac{\\sigma(q)}{q^2} =\n\\begin{cases} \n953\\cdot2^{-15}\\approx 0.02908&\\text{for $q\\equiv 1\\bmod 4$,}\\\\\n825\\cdot2^{-16}\\approx 0.01259&\\text{for $q\\equiv 3\\bmod 4$.}\n\\end{cases}\n\\end{equation}\n\\end{thm}\n\nAs we show below, the set $\\Sigma$ consists of $(q^2-8q+15)\/4$\nelements.  Hence a random choice of $(a,b) \\in \\Sigma$ yields a\nmaximally nonassociative quasigroup with probability $\\approx 1\/8.596$ if\n$q\\equiv 1\\bmod 4$, and with probability $\\approx 1\/19.86$ if $q\\equiv\n3 \\bmod 4$. This may have an important consequence for the cryptographic\napplication described in \\cite{gh}. It means that a maximally nonassociative\nquasigroup of a particular large order can be obtained in an acceptable\ntime by randomly generating pairs $(a,b)$ until one is found for which\n$Q_{a,b}$ is maximally nonassociative.\n\n\n\nAn important ingredient in the proof of \\tref{11} is the\ntransformation described in \\pref{12}, and used in \\cref{13} to\ndetermine $|\\Sigma|$.\n\nDefine $S = S(\\mathbb{F})$ as the set of all $(x,y)\\in \\mathbb{F}\\times \\mathbb{F}$ such that\nboth $x$ and $y$ are squares, $x\\ne y$ and $\\{0,1\\}\\cap\\{x,y\\}=\\emptyset$.\n\n\\begin{prop}\\label{12}\nFor each $(a,b)\\in \\Sigma$ there exists exactly one $(x,y)\\in S$\nsuch that\n\\begin{equation}\\label{e14}\na=\\frac{x(1{-}y)}{x{-}y}, \\quad b = \\frac{1{-}y}{x{-}y}, \\quad\n1{-}a = \\frac{y(1{-}x)}{y{-}x} \\text{ \\, and \\, }\n1{-}b=\\frac{1{-}x}{y{-}x}.\n\\end{equation}\nThe mapping\n\\[\n\\Psi\\colon \\Sigma\\to S, \\quad (a,b) \\mapsto \\left ( \\frac ab,\n\\frac{1{-}a}{1{-}b}\\right)\\]\nis a bijection. If $(x,y)\\in S$, then $\\Psi^{-1} ((x,y)) = (a,b)$\nif and only if \\eref{14} holds.\n\\end{prop}\n\\begin{proof} If $x,y,a,b\\in \\mathbb{F}$ satisfy $x\\ne y$, \n$a =x(1{-}y)\/(x{-}y)$ and\n$b= (1{-}y)\/(x{-}y)$, then \n\\begin{equation} \\label{e15}\n\\text{$1{-}a = y(1{-}x)\/(y{-}x)$ and\n$1{-}b = (1{-}x)\/(y{-}x)$.}\n\\end{equation} \nDefine\\begin{equation*}\n\\Phi\\colon S\\to \\mathbb{F}\\times \\mathbb{F}, \\quad (x,y)\\mapsto \\left ( \\frac{x(1{-}y)}\n{x{-}y}, \\frac{1{-}y}{x{-}y}\\right ).\n\\end{equation*}\nSuppose that $(x,y) \\in S$ and set $b = (1{-}y)\/(x{-}y)$. Then\n$b\\ne0$ as $y\\ne 1$, and $b\\ne 1$ since $x\\ne 1$. Put $a = xb$.\nThen $a\\ne 0$ since $b\\ne 0$ and $x\\ne 0$, and $a\\ne b$ since\n$x\\ne 1$. Furthermore, $a\\ne 1$ since $y\\ne 0$ and $x\\ne 1$.\nSince $a=xb$, $ab=xb^2$ is a square. By \\eref{15}, $1{-}a = y(1{-}b)$.\nHence $(1{-}a)(1{-}b) = y(1{-}b)^2$ is a square too. This verifies that\n$\\Phi$ may be considered as a mapping  $S\\to \\Sigma$.\n\nAssume $(a,b) \\in \\Sigma$. By definition, $\\Psi((a,b))=(x,y)$, where\n$x=a\/b$ and $y= (1{-}a)\/(1{-}b)$. We have $x\\notin \\{0,1\\}$ since\n$a\\ne 0$ and $a\\ne b$. Similarly, $y\\notin \\{0,1\\}$. Furthermore, \n$x\\ne y$ since $x=y$ implies\n$a = b$. Thus $(x,y)\\in S$. By straightforward verification,\n$\\Psi\\Phi = \\operatorname{id}_S$ and $\\Phi\\Psi = \\operatorname{id}_\\Sigma$.\n\\end{proof}\n\n\\begin{cor}\\label{13}\n$|\\Sigma(\\mathbb{F}_q)| = |S(\\mathbb{F}_q)| = (q^2-8q+15)\/4$.\n\\end{cor}\n\\begin{proof} By \\pref{12}, $|\\Sigma| = |S|$. By the definition,\n$S$ contains $((q-3)\/2)^2 - (q-3)\/2$ elements. \n\\end{proof}\n\nThe definition of $Q_{a,b}$ follows the established way of defining\na quasigroup by means of an orthomorphism, say $\\psi$, of an\nabelian group $(G,+)$. Here, $\\psi$ is said to be an \\emph{orthomorphism}\nof $(G,+)$ if it permutes $G$ and if the mapping $x\\mapsto \\psi(x)-x$\npermutes $G$ as well. If $\\psi$ is an orthomorphism, then\n$x*y = x+\\psi(y-x)$ is always a quasigroup. A \\emph{quadratic orthomorphism}\n$\\psi=\\psi_{a,b}$ is defined for each $(a,b)\\in \\Sigma(\\mathbb{F}_q)$ by \n\\begin{equation}\\label{e16}\n\\psi(u) = \\begin{cases} au \\quad \\text{\\,if $u$ is a square;}\\\\\nbu \\quad \\text{\\ if $u$ is a nonsquare.}\\end{cases}\n\\end{equation}\nThe definition \\eref{11} of the quasigroup $Q_{a,b}$ thus\nfits the general scheme. See~\\cite{Eva18,diagcyc} for more \ninformation on quasigroups defined by means of orthomorphisms.\n\nThe maximal nonassociativity of $Q_{a,b}$ can be expressed via\nthe \\emph{Associativity Equation:}\n\\begin{equation} \\label{e17}\n\\psi(\\psi(u)-v) = \\psi(-v) + \\psi(u-v-\\psi(-v))\n\\end{equation}\n\n\\begin{prop}\\label{14}\nFor $(a,b)\\in \\Sigma$ put $\\psi = \\psi_{a,b}$.\nAn ordered pair $(u,v)\\in \\mathbb{F}^2$ fulfils the Associativity Equation \n\\eref{17} if and only\nif $v*(0\\,*u) = (v*\\,0)*u$. Furthermore,\n\\begin{equation}\\label{e18}\nu-v-\\psi(-v) = u - (v*0) \\quad\\text{and}\\quad \\psi(u) - v = (0*u)-v.\n\\end{equation}\nIf $(u,v)\\ne (0,0)$ fulfils \\eref{17}, then none of\n$u$, $v$, $u-v-\\psi(-v)$ and $\\psi(u)-v$ vanishes,\nand $(c^2u,c^2v)$ fulfils \\eref{17} too, for any $c\\in \\mathbb{F}$.\n\nThe quasigroup $Q_{a,b}$ if maximally nonassociative if and only\nif $(u,v)=(0,0)$ is the only solution to \\eref{17}. \n\\end{prop}\n\\begin{proof} This is a restatement of Lemmas~1.3 and~3.1 from\n\\cite{dw}. A sketch of the proof follows,\nin order to make this paper self-contained. \nSince $u \\mapsto z + u$ is an automorphism of $Q=Q_{a,b}$ for each\n$z\\in \\mathbb{F}$, the\nmaximal nonassociativity is equivalent to having no $(u,v)\\ne (0,0)$\nsuch that $u*(0*v) = (u*0)*v$. This turns into \\eref{17} \nby invoking the formula $u*v = u+\\psi(v-u)$.\nSince $x\\mapsto c^2x$\nis an automorphism of $Q$ for each $c\\in \\mathbb{F}$, $c\\ne 0$, the \nAssociativity Equation holds for $(u,v)$ if and only if it holds\nfor $(c^2u,c^2v)$. For the rest it suffices to observe\nthat in an idempotent quasigroup $u*(v*w) =(u*v) *w $ implies $u=v=w$\nif $u=v$ or $u=v*w$ or $v=w$ or $u*v = w$. \n\\end{proof}\n\nFor $(a,b)\\in \\Sigma$ denote by $E(a,b)$ the set of $(u,v)\\ne (0,0)$\nthat satisfy the Associativity Equation \\eref{17}. By \\pref{14},\n$Q_{a,b}$ is maximally nonassociative if and only if $E(a,b)= \\emptyset$.\nThe number of such $(a,b)$ may be obtained indirectly by counting\nthe number of $(a,b)\\in \\Sigma$ for which $E(a,b)\\ne \\emptyset$. \nTo this end, we will partition $E(a,b) = \\bigcup E_{ij}^{rs}(a,b)$,\nwhere $i,j,r,s\\in \\{0,1\\}$. To determine\nto which part an element $(u,v)\\in E(a,b)$ belongs,\nthe following rule is used:\n\\begin{gather*}\n\\begin{aligned}\ni=0\\quad  &\\Longleftrightarrow \\quad u \\ \\text {is a square};\\\\\nj=0\\quad &\\Longleftrightarrow \\quad -v \\ \\text {is a square};\\\\\nr=0\\quad &\\Longleftrightarrow \n\\quad \\psi_{a,b}(u)-v \\ \\text {is a square, and};\\\\\ns=0\\quad &\\Longleftrightarrow \\quad u-v-\\psi_{a,b}(-v) \\ \\text {is a square}.  \n\\end{aligned}\n\\end{gather*}\nThus, if one of the elements $u$, $-v$, $\\psi_{a,b}(u){-}v$\nand $u{-}v{-}\\psi_{a,b}(-v)$ is a nonsquare, then the respective value\nof $i$, $j$, $r$ or $s$ is set to $1$. For each $(u,v)\\in E(a,b)$ there\nhence exists exactly one quadruple $(i,j,r,s)$ such that\n$(u,v) \\in E_{ij}^{rs}(a,b)$, giving us the desired partition. \nWe will also work with sets \n\\begin{equation*}\n\\Sigma_{ij}^{rs} = \\{(a,b)\\in \\Sigma: E_{ij}^{rs}(a,b)\\ne \\emptyset\\},\n\\end{equation*}\nwhere $i,j,r,s\\in \\{0,1\\}$.\nThe next observation directly follows from the definition\nof the sets\n$\\Sigma_{ij}^{rs}$. It is recorded here for the sake of later reference.\n\n\\begin{prop}\\label{15}\nSuppose that $(a,b)\\in \\Sigma=\\Sigma(\\mathbb{F}_q)$, for an odd\nprime power $q>1$. The quasigroup $Q_{a,b}$ is maximally nonassociative\nif and only if $(a,b)\\notin \\bigcup \\Sigma_{ij}^{rs}$.\n\\end{prop}\n\n\nIf it is assumed that $(u,v) \\in E_{ij}^{rs}(a,b)$, then \nthe Associativity Equation \\eref{17} can be turned into a linear \nequation in unknowns $u$ and $v$ since each occurrence of $\\psi$ can\nbe interpreted by means of \\eref{16}. The list of these linear equations\ncan be found in \\cite{dw}. Their derivation is relatively short\nand is partly repeated in Lemmas~\\ref{24}--\\ref{27}. The approach used here\ndiffers from that of \\cite{dw} in two aspects. The symmetries induced\nby opposite quasigroups and by automorphisms $Q_{a,b}\\cong Q_{b,a}$\nare used more extensively here, and characterisations\nof $\\Sigma_{ij}^{rs}$ are immediately transformed \ninto characterisations of \n\\begin{equation}\\label{e110}\nS_{ij}^{rs} = \\Psi(\\Sigma_{ij}^{rs}).\n\\end{equation}\nAs will turn out, sets $S_{ij}^{rs}$ can be described by a requirement\nthat several polynomials in $x$ and $y$ are either squares or nonsquares.\nEstimates of $|S_{ij}^{rs}|$ can be thus obtained by means of the Weil bound\n(as formulated, say, in \\cite[Theorem 6.22]{Evans}). \nWe shall not be using the Weil bound directly, but via \\tref{16} below, \na straightforward consequence from \\cite[Theorem~1.4]{dw}.\nApplications of \\tref{16} to the intersections of \nsets $S_{ij}^{rs}$, with symmetries taken into account, yield, after\na number of computations, the asymptotic results stated in \\tref{11}. \n\nSay that a list of polynomials $p_1,\\dots,p_k$ in one variable,\nwith coefficients in $\\mathbb{F}$,\nis \\emph{square-free}\nif there exists no sequence $1\\le i_1<\\dots<i_r \\le k$ such that\n$r\\ge 1$ and $p_{i_1}\\cdots p_{i_k}$ is a square (as a polynomial\nwith coefficients in the algebraic closure $\\bar \\mathbb{F}$ of $\\mathbb{F}$).\nDefine $\\chi\\colon \\mathbb{F}\\to \\{\\pm1,0\\}$ to be the\nquadratic character extended by $\\chi(0)=0$. \n\n\\begin{thm}\\label{16}\\label{t:Weil}\nLet $p_1,\\dots,p_k\\in \\mathbb{F}[x]$ be a square-free list of polynomials of\ndegree $d_i\\ge1$, and let $\\varepsilon_1,\\dots,\\varepsilon_k\\in\\{-1,1\\}$. Denote by\n$N$ the number of all $\\alpha \\in \\mathbb{F}$ such that\n$\\chi(p_i(\\alpha))= \\varepsilon_i$, for $1\\le i \\le k$.\nThen\n$$|N-2^{-k}q|<(\\sqrt q+1)D\/2-\\sqrt{q}(1-2^{-k})<(\\sqrt q+1)D\/2\n$$\nwhere $D=\\sum_i d_i$.\n\\end{thm}\n\nThe purpose of \\secref{2} is to describe each of the sets $S_{ij}^{rs}$ \nby a list of polynomials $p(x,y)$ such that the presence\nof $(x,y)\\in S$ in $S_{ij}^{rs}$ depends upon $p(x,y)$ being\na square or nonsquare.\n\\tref{210} gives such a description for $q=|\\mathbb{F}|\\equiv 1 \\bmod 4$,\nand \\tref{211} for $q\\equiv 3 \\bmod 4$. \\secref{3} contains \nauxiliary results that make applications of \\tref{16} possible.\nNote that \\tref{16} is concerned with polynomials in only one\nvariable. To use it, one of the variables, say $y$, has to be\nfixed. If $y=c$, and $p_1(x,y),\\dots,p_k(x,y)$ are the polynomials\noccurring in Theorems~\\ref{210} and~\\ref{211}, then \\tref{16}\nmay be used without further specifications \nonly for those $c$ for which $p_1(x,c),\\dots,p_k(x,c)$\nis a square-free list. The purpose of \\secref{3} is to show\nthat this is true for nearly all $c$, and that the number of possible\nexceptional values of $c$ is very small. \\secref{4} provides \nthe estimate of $S\\setminus \\bigcup S_{ij}^{rs}$ for $q\\equiv 3\\bmod4$,\nand \\secref{5} for $q\\equiv 1 \\bmod 4$, cf.~Theorems~\\ref{46} and \\ref{53}.\n\\secref{6} consists of concluding remarks.\n\n\n\\section{Quadratic residues and the Associativity Equation}\\label{2}\n\nLet $Q^{op}_{a,b}$ denote the opposite quasigroup of $Q_{a,b}$, namely the\nquasigroup satisfying $Q^{op}_{a,b}(u,v)=Q_{a,b}(v,u)$ for all $u,v$.\nThe following facts are well known \\cite{Eva18,cyclatom} and easy to verify:\n\n\\begin{lem}\\label{21}\nIf $(a,b)\\in \\Sigma$, then \n\\begin{enumerate}\n\\item[(i)]\n$u\\mapsto u\\zeta$ is an isomorphism\n$Q_{a,b}\\cong Q_{b,a}$, for every nonsquare\n$\\zeta \\in \\mathbb{F}$;\n\\item[(ii)] $Q^{op}_{a,b} = Q_{1-a,1-b}$ if $q\\equiv 1\\bmod 4$,\nand $Q^{op}_{a,b} = Q_{1-b,1-a}$ if $q\\equiv 3\\bmod 4$.\n\\end{enumerate}\n\\end{lem}\n\nAn alternative way to express that $q\\equiv 1 \\bmod 4$ is to say\nthat $-1$ is a square. If $\\bar *$ denotes the operation of the\nopposite quasigroup, then  $(v\\,\\bar *\\, 0)\\,\\bar *\\, u = \nv\\,\\bar *\\, (0 \\,\\bar*\\, u)$ holds in $Q_{a,b}^{op}$ if and only if\n$u*(0*v) = (u*0)*v$. Hence $(u,v)\\in E(a,b)$ if and only if\n$(v,u)\\in E(a',b')$, where $(a',b')=(1{-}a,1{-}b)$ if $-1$ is a square,\nand $(a',b')= (1{-}b,1{-}a)$ if $-1$ is a nonsquare, by part (ii)\nof \\lref{21}. Similarly $(u,v)\\in E(a,b)$ $\\Leftrightarrow$\n$(\\zeta u,\\zeta v)\\in E(b,a)$. \n\nWorking out these connections with respect to being square or nonsquare\nyields the following statement. It appears without a proof since it \ncoincides with Lemmas~3.2 and~3.3 of \\cite{dw} and since the proof is \nstraightforward. \n\n\\begin{lem}\\label{22}\nAssume $(a,b)\\in \\Sigma$ and $i,j,r,s\\in \\{0,1\\}$.\nThen \n\\begin{align} \\label{e21}\n(u,v)\\in E_{ij}^{rs}(a,b) \\ &\\Longleftrightarrow \\\n(\\zeta u,\\zeta v) \\in E_{1-i,1-j}^{1-r,1-s}(b,a); \\\\ \\label{e22}\n(u,v)\\in E_{ij}^{rs}(a,b) \\ &\\Longleftrightarrow \\\n(v,u) \\in E_{ji}^{sr}(1{-}a,1{-}b) \\text{ if $-1$ is a square; and} \\\\\n\\label{e23}\n(u,v)\\in E_{ij}^{rs}(a,b) \\ &\\Longleftrightarrow \\\n(v,u) \\in E_{1-j,1-i}^{1-s,1-r}(1{-}b,1{-}a) \\text{ if $-1$ is a nonsquare.}\n\\end{align}\n\\end{lem}\n\n\\begin{prop}\\label{23}\nBoth of the mappings $(x,y)\\mapsto (y,x)$ and $(x,y)\\mapsto (x^{-1},y^{-1})$\npermute the set $S = S(\\mathbb{F})$. If $i,j,r,s\\in \\{0,1\\}$, then \n\\begin{equation*}\n(x,y) \\in S_{ij}^{rs}\\ \\Longleftrightarrow\\ (y,x) \\in S_{ji}^{sr}\n\\ \\Longleftrightarrow\\  (x^{-1},y^{-1})\\in S_{1-i,1-j}^{1-r,1-s}.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nBy definition, $(x,y)\\in S$ if and only if $x$ and $y$ are both squares,\n$x\\ne y$, and $\\{x,y\\}\\cap\\{0,1\\}=\\emptyset$. These\nproperties are retained both by the switch $(x,y)\\mapsto (y,x)$\nand by the inversion $(x,y)\\mapsto (x^{-1},y^{-1})$. These mappings thus\npermute $S$. \n\nLet $(a.b)\\in \\Sigma$ be such that $\\Psi((a,b)) = (x,y)$. Then\n$x=a\/b$, $y = (1{-}a)\/(1{-}b)$. Hence $\\Psi((b,a)) = (x^{-1},y^{-1})$\nand $\\Psi((1{-}a,1{-}b)) = (y,x)$. For the proof we thus need to show\nthat \n\\begin{equation*}\n(a,b) \\in \\Sigma_{ij}^{rs}\\ \\Longleftrightarrow\\ (1{-}a,1{-}b) \\in \n\\Sigma_{ji}^{sr}\n\\ \\Longleftrightarrow\\  (b,a)\\in \\Sigma_{1-i,1-j}^{1-r,1-s}.\n\\end{equation*}\nSuppose that $(a,b) \\in \\Sigma_{ij}^{rs}$, i.e., that there exists\n$(u,v) \\in E_{ij}^{rs}(a,b)$. If $-1$ is a square, then $(v,u)\\in\nE_{ji}^{sr}(1{-}a,1{-}b)$ by \\eref{22}. If $-1$ is a nonsquare,\nthen $(\\zeta v,\\zeta u) \\in E_{ji}^{sr}(1{-}a,1{-}b)$, by \\eref{23}\nand \\eref{21}. Thus $(1{-}a,1{-}b)\\in \\Sigma_{ji}^{sr}$ in both cases.\nWe also have $(\\zeta u,\\zeta v) \\in E_{1-i,1-j}^{1-r,1-s}(b,a)$, by \\eref{21}.\nHence $(b,a)\\in \\Sigma_{1-i,1-j}^{1-r,1-s}$.\n\\end{proof}\n\nTo determine all of the sets $S_{ij}^{rs}$ it thus suffices to know the\nsets\n\\begin{equation}\\label{e24}\nS_{00}^{00},\\ S_{00}^{01},\\ S_{00}^{11},\\ S_{01}^{00},\\ S_{01}^{01}\n\\text{ and } S_{01}^{10}.\n\\end{equation}\nWe next determine these sets via a sequence of Lemmas.\n\n\\begin{lem}\\label{24}\nIf $-1$ is a square, then $S_{01}^{00}=S_{01}^{10}=\\emptyset$, while\n\\begin{align*}\n(x,y)\\in S_{00}^{00} \\ &\\Longleftrightarrow \\\n\\text{$(1{-}x)(y{-}x)$ and $(1{-}y)(y{-}x)$ are squares; and}\\\\\n(x,y)\\in S_{00}^{11} \\ &\\Longleftrightarrow \\\n\\text{$(x^2y+xy-x^2-y^2)(y{-}x)$ and $(xy^2+xy-x^2-y^2)(y{-}x)$}\\\\\n&\\phantom{\\Longleftrightarrow \\ } \\ \\text{are nonsquares.}\n\\end{align*}\n\\end{lem}\n\n\\begin{proof} We assume that $-1$ is a square.\nIf $(u,v)\\in E_{00}^{00}(a,b)$, then\nthe Associativity Equation attains the form $a(au-v)=-av+a(u-v+av)$,\nand that is the same as $(1{-}a)(u{-}v)=0$. Since $1{-}a\\ne 0$,\nand since $u$ is assumed to be square, the set $E_{00}^{00}(a,b)$\nis nonempty if and only if it contains $(1,1)$, by \\pref{14}.\nThis takes place if and only if $1{-}a$ and $a$ are squares.\nSuppose that $(x,y) = \\Psi((a,b))$.\nThen $a=x(1{-}y)\/(x{-}y)$ is a square if and only if\n$(1{-}y)(y{-}x)$ is a square, and $1{-}a =  y(1{-}x)\/(y{-}x)$ is a square\nif and only if $(1{-}x)(y{-}x)$ is a square.\n\nIf $(u,v) \\in E_{00}^{11}(a,b)$, then $b(au-v)=-av+b(u-v+av)$\nyields $ub(a{-}1)=a(b{-}1)v$, where both $u$ and $v$ are squares. \nThus $(u,v)$ is a solution if and only if $(1,b(a{-}1)\/a(b{-}1))$\nis a solution. Since $v=b(a{-}1)\/a(b{-}1)$ is always a square,\nthe conditions for the existence of the solution are that\n$a-v$ and $1-(1{-}a)v$ are nonsquares. If $(x,y) = \\Psi((a,b))$,\nthen $v = y\/x$, $a-v = (x^2-x^2y-yx+y^2)\/x(x-y)$ and\n$1-(1{-}a)v=(xy-x^2-y^2+y^2x)\/x(y{-}x)$.\n\nIf $(u,v) \\in E_{01}^{00}(a,b)$, then $a(au-v)=-bv + a(u-v+bv)$\nand $a(a-1) u = b(a-1)v$. This implies that $uv$ is a square.\nHowever, the assumption $(u,v) \\in E_{01}^{00}(a,b)$ implies that\n$u$ is a square and $-v$ is a nonsquare. Thus $uv$ should\nbe both a square and a nonsquare, which is a contradiction.\nIf $(u,v) \\in E_{01}^{10}(a,b)$, then $b(au-v)=-bv + a(u-v+bv)$,\nand that gives $u=v$, a contradiction again.  \n\\end{proof}\n\n\\begin{lem}\\label{25}\nIf $-1$ is a nonsquare, then $S_{00}^{00}=S_{00}^{11}=\\emptyset$, while\n\\begin{align*}\n(x,y) \\in S_{01}^{10} &\\ \\Longleftrightarrow \\ (x,y) \\in S_{10}^{01};\\\\\n&\\ \\Longleftrightarrow \\ \\text{$(1{-}y)(x{-}y)$ and $(1{-}x)(y{-}x)$\nare squares; and} \\\\\n(x,y) \\in S_{01}^{00} &\\ \\Longleftrightarrow \\ \\text{$(x{-}1)(y{-}x)$\nand $(x^2{-}2x{+}y)(y{-}x)$ are squares.}\n\\end{align*}\n\\end{lem}\n\n\\begin{proof} We assume that $-1$ is a nonsquare.\nIf $E_{00}^{00}(a,b)\\ne \\emptyset$, then $(1,1)\\in E_{00}^{00}\n(a,b)$, by the same argument as in the proof of \\lref{24}. \nHowever, $(1,1)$ cannot belong to $E_{00}^{00}(a,b)$ since $-1$ is\na nonsquare. Similarly, $E_{00}^{11}(a,b) = \\emptyset$ since\n$-b(a{-}1)\/a(b{-}1)$ is a nonsquare.\n\nSuppose that $(u,v) \\in E_{01}^{00}(a,b)$. Then \\eref{17} implies $au = bv$. \nHence $(a,b)\\in \\Sigma_{01}^{00}$ $\\Leftrightarrow$ $(1,a\/b)\\in E_{01}^{00}(a,b)$.\nThe latter takes place if and only if $a-a\/b$ and $1-(1{-}b)a\/b$ are squares.\nLet $(x,y) = \\Psi((a,b))$. Then $a-a\/b = x((1{-}y)\/(x{-}y)-1) =\nx(1{-}x)\/(x{-}y)$ and $1-(1{-}b)a\/b=1-x(1-x)\/(y{-}x)=(x^2{-}2x{+}y)\/(y{-}x)$.\n\nLet $(u,v)\\in E_{01}^{10}(a,b)$. Then $u=v$ by \\eref{17}. Hence\n$(a,b)\\in \\Sigma_{01}^{10}$ if and only if $(1,1)\\in E_{01}^{10}(a,b)$.\nThe latter is true if and only if $a-1$ is a nonsquare and $b$\nis a square. If $(x,y) = \\Psi((a,b))$, then this means that\n$(x{-}1)(y{-}x)$ is a nonsquare and $(1{-}y)(x{-}y)$ is a square.\nThe symmetry of these conditions shows that\n$(x,y)\\in S_{01}^{10}\\Leftrightarrow(y,x)\\in S_{01}^{10}$. Hence \n$S_{01}^{10}=S_{10}^{01}$, by \\pref{23}.\n\\end{proof}\n\n\\begin{lem}\\label{26}\nAssume that $(x,y) \\in S$. Then $(x,y) \\in S_{00}^{01}$ if and only if\n$-xy-y+x$ and $(-x^2y+x^2+y^2-xy)(x{-}y)$ are squares,\nand $(1{-}y)(x{-}y)$ is a nonsquare. \n\\end{lem}\n\\begin{proof} In this case the Associativity Equation is equal to\n$a(au-v)=-av+b(u-v+av)$, and that is the same as $(a^2-b)u=(ab-b)v$.\nTherefore $(a,b)\\in\\Sigma_{00}^{01}$ if and only if\n$(1,(a^2{-}b)\/b(a{-}1))\\in E_{00}^{01}(a,b)$. If $(x,y) = \\Psi((a,b))$,\nthen $(a^2{-}b)(x{-}y)^2 = x^2(1{-}y)^2-(1{-}y)(x{-}y)= (1-y)(x^2-x^2y-x+y)\n=(1{-}y)(1{-}x)(y+xy-x)$,\nand $b(a{-}1)(x{-}y)^2 = (1{-}y)y(1{-}x)$. Hence \n$v=(a^2-b)\/b(a-1)=(y+xy-x)\/y$, showing that $y+xy-x$ is a square.\nIt follows that\n$a-v\n=(-x^2y+x^2+y^2-xy)\/(x-y)y$ \nand $(1-(1{-}a)v)(y{-}x)\n=x^2(y{-}1)$. Thus $(1{-}y)(x{-}y)$ has to be\na nonsquare. \n\\end{proof}\n\n\\begin{lem}\\label{27}\nAssume that $(x,y)\\in S$. \n\\begin{enumerate}\n\\item[(i)] If $y{+}1{-}x = 0 = x^2{-}x{-}1$ and $q>43$, \nthen $(x,y) \\in S_{01}^{01}$.\n\\item[(ii)] If $y{+}1{-}x \\ne 0$ or $x^2{-}x{-}1\\ne 0$,\nthen $(x,y)\\in S_{01}^{01}$ if and only if both\n$(y{+}xy{-}x)(x{-}y{-}1)$ and $(y{-}2x{+}x^2)(x{-}y)(x{-}y{-}1)$\nare nonsquares,\nwhile $(2xy{-}y^2{-}x)(x{-}y)(x{-}y{-}1)$\nis a square.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nIn this case the Associativity Equation yields $a(au-v) = -bv + b(u -\n(1{-}b)v)$. That is equivalent to $(a^2-b)u = (b^2-2b+a)v$. If there\nexists a solution $(u,v)\\in E_{01}^{01}(a,b)$, and one of the elements\n$a^2-b$ and $b^2-2b+a$ is equal to zero, then the other has to vanish\nas well. Assume that $(x,y)=\\Psi((a,b))$. Then $a^2-b=0$\nif and only if $0 = x^2(1-y)^2-(1-y)(x-y) = (1-y)(-x^2y+x^2-x+y)\n= (1-y)(1-x)(y+xy-x)$, and $b^2-2b+a = (1-b)^2-(1-a)=\n0$ if and only if $0 = (1-x)^2-y(1-x)(y-x) = (1-x)(1-x-y^2+xy)=\n(1-x)(1-y)(y-x+1)$. If $y=x-1$, then $y+xy-x=x^2-x-1$.\n\nComputations above show that \n\\begin{equation*}\na^2-b = \\frac{(1{-}y)(1{-}x)(xy{-}x{+}y)}{(x{-}y)^2}\n\\text{ \\, and \\,} b^2-2b+a = \n\\frac{(1{-}y)(1{-}x)(y{-}x{+}1)}{(x{-}y)^2}.\n\\end{equation*}\n\nSuppose now that at least one of $x^2{-}x{-}1$ and $y{-}x{+}1$ does not vanish.\nIf $y{-}x{+}1=0$, then $E_{01}^{01}(a,b)=\\emptyset$ \nand $(y{+}xy{-}x)(x{-}y{-}1)=0$, which \nis a square. Hence $y{-}x{+}1\\ne 0$ may be assumed. \nThat implies $b^2-2b+a\\ne 0$. From the Associativity Equation\nit then follows that  $(a,b)\\in \\Sigma_{01}^{01}$ if and only if\n$(1,v)\\in E_{01}^{01}(a,b)$, where $v=(a^2-b)\/(b^2-2b+a)$. Now,\n\\begingroup\n\\begin{align*}\n-v &= \\frac{b-a^2}{b^2-2b+a} = \n\\frac{y{+}xy{-}x}{x{-}y{-}1}, \\\\\n1-(1{-}b)v&= \\frac{(x{-}y{-}1)(y{-}x)+(1{-}x)(y{+}xy{-}x)}{(x{-}y{-}1)(y{-}x)}\n=\\frac{y(x^2{-}2x{+}y)}{(x{-}y{-}1)(x{-}y)}, \\text{ and}\\\\\na{-}v &= \\frac{x(1{-}y)(x{-}y{-}1) + (x{-}y)(y{+}xy{-}x)}{(x{-}y)(x{-}y{-}1)\n= \\frac{2xy{-}y^2{-}x}{(x{-}y)(x{-}y{-}1)}.\n\\end{align*}\n\\endgroup\n\nIt remains to prove that $E_{01}^{01}(a,b)$ is nearly always nonempty\nif $a^2-b=b^2-2b+a=0$.\nLet the latter be true. Then $b^2-2b+a=a^4-2a^2+a = a(a{-}1)(a^2{+}a{-}1)$.\nThus $a^2{+}a{-}1=0$. A pair $(1,v)$ is a solution to the Associativity \nEquation if $-v$ is a nonsquare, $1+(1{-}b)(-v)$ is a nonsquare, and\n$a-v$ is a square. Put $p_1(t) = t$, $p_2(t) = 1+(1{-}b)t=1+(1{-}a^2)t$,\nand $p_3(t) = a+t$. A solution $(1,v)$ exists if there exists $\\gamma=-v\\in \\mathbb{F}$\nsuch that $\\chi(p_1(\\gamma))=\\chi(p_2(\\gamma))=-1$ and \n$\\chi(p_3(\\gamma)) = 1$. Polynomials $p_2$ and $p_3$ have a common\nroot if and only if $0=1-a+a^3$. \nIf this is true, then \n$0=a^2+a^3 = a^2(1+a)$. This implies $a=-1$ and\n$0 = (-1)^2+(-1)-1 =-1$, a contradiction. The list of polynomials $p_1$, \n$p_2$, $p_3$ is therefore square-free.  \\tref{16} guarantees the \nexistence of $\\gamma$\nif $0<q\/8 -(\\sqrt q+1)(3\/2)+\\sqrt q(1-1\/8)=q\/8-5\\sqrt q\/8-3\/2$. \nThis is true for each prime power $q\\ge 47$.\n\\end{proof}\n\n\\begin{rem}\\label{28}\nLemmas~$\\ref{24}$--$\\ref{27}$ cover all sets $S_{ij}^{rs}$ that are listed\nin \\eref{24}. Up to the exceptions discussed in \\rref{29},\neach of these sets is either empty, or  is described by a \nlist of polynomials, say $p_1,\\dots,p_k\\in \\mathbb{F}[x,y]$, $k\\in \\{2,3\\}$,\nand elements $\\varepsilon_h\\in \\{-1,1\\}$, such that\n$(\\xi,\\eta)\\in S$ belongs to $S_{ij}^{rs}$ if and only if \n$\\chi(p_h(\\xi,\\eta))=\\varepsilon_h$, for $1\\le h \\le k$. This is because\nthe polynomials\n$p_h(x,y)$ have been determined in all cases in such a way \nthat if $p_h(\\xi,\\eta)=0$ and $(\\xi,\\eta) = \\Psi((a,b))$,\nthen there is no $(u,v)\\in E_{ij}^{rs}(a,b)$. Indeed if $(u,v)$ were\nsuch a solution, then $u$ or $v$ or $u-v-\\psi_{a,b}(-v)$  or\n$\\psi_{a,b}(u)-v$ would be equal to zero, and that is impossible,\nby \\pref{14}.\n\\end{rem}\n\nNote that $(\\xi,\\eta)$ was used in \\rref{28} to emphasise the\ndistinction between elements of $S$ and formal variables $x$ and $y$.\nIn the remainder of the paper, elements of $S$ will again be denoted\nby $(x,y)$. The context will always be clear.\n\n\\begin{rem}\\label{29}\nSets $S_{01}^{01}$ and $S_{10}^{10}$ behave exceptionally in the sense\nthat the regular behaviour described in \\rref{28} needs an assumption\nthat $y{+}1{-}x\\ne 0$ or $x^2{-}x{-}1\\ne 0$ (for the set\n$S_{01}^{01}$), and that $x{+}1{-}y\\ne 0$ or $y^2{-}y{-}1\\ne 0$ (for\nthe set $S_{10}^{10}$). There are at most two pairs $(x,y) \\in S$ such\nthat $y{+}1{-}x=0=x^2{-}x{-}1$ and at most two pairs $(x,y) \\in S$\nsuch that $x{+}1{-}y=0=y^2{-}y{-}1$. Hence assuming that\n\\begin{equation}\\label{e25}\n\\text{$\\bigl [y+1-x\\ne 0$ or $x^2-x-1\\ne 0\\bigr ]$ and\n$\\bigl [x+1-y\\ne 0$ or $y^2-y-1\\ne 0\\bigr ]$}\n\\end{equation}\ncauses no difficulty when estimating $\\sigma(q)$.\nIf \\eref{25} does not hold, then $(x,y) \\in S_{01}^{01}\\cup\nS_{10}^{10}$ if $q\\ge 47$, by point (i) of \\lref{27}. In fact,\nif \\eref{25} does not hold, then $(x,y) \\in \n\\bigcup S_{ij}^{rs}$ for each $q\\ge3 $, by \\cite{dw} (cf.~the application\nof \\cite[Lemma~3.4]{dw} in the proof of \\cite[Theorem~3.5]{dw}).\n\\end{rem}\n\nFor $p(x,y)\\in \\mathbb{F}[x,y]$ such that $x\\nmid p(x,y)$ and $y\\nmid p(x,y)$\ndefine the \\emph{reciprocal} polynomial \n$\\hat p(x,y)$ as $x^ny^mp(x^{-1},y^{-1})$,  where $n$ and $m$\nare the degree of the polynomial $p$ in the variables $x$ and $y$, respectively.\nNote that if $(x,y)\\in S$\nthen $\\chi(\\hat p(x,y))=\\chi(x^ny^mp(x^{-1},y^{-1}))=\\chi(p(x^{-1},y^{-1}))$\nsince $x$ and $y$ are squares.\nNote also that $\\hat{\\hat p}(x,y)= p(x,y)$, $\\widehat{1-x} = x-1$,\n$\\widehat{x-y}=y-x$ and $\\widehat{x-1-y}=y-xy-x$. Set\n\\begin{gather}\\label{e26}\n\\begin{aligned}\n&f_1(x,y)=x^2{+}y^2{-}xy{-}x, \\quad \\quad &f_2(x,y) = y^2{+}x^2{-}xy{-}y,\n\\ \\ \\ \\\\\n&f_3(x,y)=y^2x{+}xy{-}x^2{-}y^2\\ \\ \\ \\text{and}\n&f_4(x,y)=x^2y{+}xy{-}x^2{-}y^2.\n\\end{aligned}\n\\end{gather}\nThen $f_2(x,y) = f_1(y,x)$, $f_3(x,y) = -\\hat{f_1}(x,y)$\nand $f_4(x,y) = -\\hat{f_1}(y,x)=-\\hat{f_2}(x,y)=f_3(y,x)$.\n\nA description of those sets $S_{ij}^{rs}$ that do not occur in \\eref{24} can be \nderived from Lemmas~\\ref{24}--\\ref{27} by means of \\pref{23}.\nAs an example consider sets $S_{00}^{10}$ and $S_{11}^{10}$.\nBy \\lref{26}, $(x,y)\\in S_{00}^{01}$ if $\\chi(x{-}xy{-}y)=\\chi(f_4(x,y)\n(y{-}x)) = 1$ and $\\chi((1{-}y)(x{-}y))=-1$. By \\pref{23},\n$(x,y) \\in S_{00}^{10}$ if and only if $(y,x)\\in S_{00}^{01}$,\ni.e., if $\\chi(y{-}xy{-}x)=\\chi(f_3(x,y)(x{-}y))=1$ and $\\chi\n((1{-}x)(y{-}x))=-1$, and $(x,y)\\in S_{11}^{10}$ if $(x^{-1},y^{-1})\\in\nS_{00}^{01}$, i.e., if $\\chi(y{-}1{-}x)=\\chi(f_2(x,y)(y{-}x))=1$\nand $\\chi((1{-}y)(x{-}y))=-1$.\n\nFollowing this pattern a characterisation of all sets $S_{ij}^{rs}$\nmay be derived from Lemmas~\\ref{24}--\\ref{27} by means of \\pref{23}.\nThis is done in Theorems~\\ref{210} and~\\ref{211}. Since the derivation\nis straightforward, both of them are stated without a proof. Set\n\\begin{gather}\\label{e27}\n\\begin{aligned}\n&g_1(x,y)= x^2 + y -2x,&  &g_2(x,y)= y^2+x-2y,\\\\\n&g_3(x,y)= x^2 + y -2xy& \\text{and}\\quad &g_4(x,y) = y^2+x-2xy.\n\\end{aligned}\n\\end{gather}\nNote that $g_3(x,y)=\\hat{g_1}(x,y)$, $g_4(x,y) = \\hat{g_2}(x,y)=\ng_3(y,x)$ and $g_2(x,y) = g_1(y,x)$.\n\n\\begin{thm}\\label{210}\nAssume that $q\\equiv 1\\bmod 4$ is a prime power, and that\n$S = S(\\mathbb{F}_q)$. Let $(x,y)\\in S$ be such that \\eref{25} holds.\nThe sets $S_{01}^{00}$, $S_{01}^{10}$,\n$S_{01}^{11}$, $S_{10}^{00}$, $S_{10}^{01}$ and $S_{10}^{11}$\nare empty, and $S_{11}^{11} = S_{00}^{00}$. \nPut $\\varepsilon = \\chi(x{-}y)$. Then \n\\begin{align*}\n&(x,y)\\in S_{00}^{00}& &\\Longleftrightarrow& &\n\\chi(1{-}x)=\\chi(1{-}y)=\\varepsilon;\\\\\n&(x,y)\\in S_{11}^{00}& &\\Longleftrightarrow& &\n\\chi(f_1(x,y))=\\chi(f_2(x,y)) = -\\varepsilon; \\\\\n&(x,y)\\in S_{00}^{11}& &\\Longleftrightarrow& &\n\\chi(f_3(x,y))=\\chi(f_4(x,y)) = -\\varepsilon; \\\\\n&(x,y)\\in S_{11}^{01}& &\\Longleftrightarrow& &\n\\chi(1{-}x) = -\\varepsilon,\\ \\chi(y{+}1{-}x)=1 \n\\text{ and } \\chi(f_1(x,y))=\\varepsilon; \\\\\n&(x,y)\\in S_{11}^{10}& &\\Longleftrightarrow& &\n\\chi(1{-}y) = -\\varepsilon,\\ \\chi(x{+}1{-}y)=1 \n\\text{ and } \\chi(f_2(x,y))=\\varepsilon; \\\\\n&(x,y)\\in S_{00}^{10}& &\\Longleftrightarrow& &\n\\chi(1{-}x) = -\\varepsilon,\\ \\chi(x{+}xy{-}y)=1\n\\text{ and } \\chi(f_3(x,y))=\\varepsilon; \\\\\n&(x,y)\\in S_{00}^{01}& &\\Longleftrightarrow& &\n\\chi(1{-}y) = -\\varepsilon,\\ \\chi(y{+}xy{-}x)=1 \n\\text{ and } \\chi(f_4(x,y))=\\varepsilon; \\\\\n&(x,y) \\in S_{01}^{01}& &\\Longleftrightarrow& &\n\\chi(y{+}xy{-}x)=-\\eta,\\ \\chi(g_1(x,y))=-\\eta\\varepsilon\n\\text{ and } \\chi(g_4(x,y)) = \\eta\\varepsilon, \\\\\n&&&&&\\text{where }\\eta = \\chi(y{+}1{-}x); \\text{ and} \\\\\n&(x,y) \\in S_{10}^{10}& &\\Longleftrightarrow& &\n\\chi(x{+}xy{-}y)=-\\eta,\\ \\chi(g_2(x,y))=-\\eta\\varepsilon\n\\text{ and } \\chi(g_3(x,y)) = \\eta\\varepsilon, \\\\\n&&&&&\\text{where }\\eta = \\chi(x{+}1{-}y).\n\\end{align*}\n\\end{thm}\n\n\\begin{thm}\\label{211}\nAssume that $q\\equiv 3\\bmod 4$ is a prime power, and that\n$S = S(\\mathbb{F}_q)$. Let $(x,y)\\in S$ be such that \\eref{25} holds.\nSets $S_{00}^{00}$, $S_{00}^{11}$, $S_{11}^{00}$ and $S_{11}^{11}$\nare empty, and $S_{10}^{01} = S_{01}^{10}$. The pair $(x,y)$\nbelongs to a set $S_{ij}^{rs}$ listed below if and only if all values\nin the row of $S_{ij}^{rs}$ are nonzero squares.\n\\begin{align*}\nS_{01}^{10}\\colon&\\quad (1{-}y)(x{-}y) \\text{ and \\,} (1{-}x)(y{-}x);\\\\\nS_{01}^{00}\\colon&\\quad (1{-}x)(x{-}y) \\text{ and \\,}  g_1(x,y)(y{-}x); \\\\\nS_{10}^{00}\\colon&\\quad (1{-}y)(y{-}x) \\text{ and \\,}  g_2(x,y)(x{-}y); \\\\\nS_{10}^{11}\\colon&\\quad (1{-}x)(x{-}y) \\text{ and \\,}  g_3(x,y)(x{-}y); \\\\\nS_{01}^{11}\\colon&\\quad (1{-}y)(y{-}x) \\text{ and \\,}  g_4(x,y)(y{-}x); \\\\\nS_{11}^{01}\\colon&\\quad (1{-}x)(x{-}y),\\ x{-}1{-}y\n\\text{ and \\,} (x{-}y)f_1(x,y);\\\\\nS_{11}^{10}\\colon&\\quad (1{-}y)(y{-}x),\\ y{-}1{-}x\n\\text{ and \\,} (y{-}x)f_2(x,y);\\\\\nS_{00}^{10}\\colon&\\quad (1{-}x)(x{-}y),\\ y{-}xy{-}x\n\\text{ and \\,} (x{-}y)f_3(x,y);\\\\\nS_{00}^{01}\\colon&\\quad (1{-}y)(y{-}x),\\ x{-}xy{-}y\n\\text{ and \\,} (y{-}x)f_4(x,y);\\\\\nS_{01}^{01}\\colon&\\quad (x{-}xy{-}y)(x{-}1{-}y),\\ \ng_1(x,y)(y{-}x)(x{-}1{-}y)\\text{ and \\,}\ng_4(x,y)(y{-}x)(x{-}1{-}y);\n\\\\\nS_{10}^{10}\\colon&\\quad (y{-}xy{-}x)(y{-}1{-}x),\\ \ng_2(x,y)(x{-}y)(y{-}1{-}x)\\text{ and \\,}\ng_3(x,y)(x{-}y)(y{-}1{-}x).\n\\end{align*}\n\\end{thm}\n\n\\bigskip\n\n\\section{Avoiding squares}\\label{3}\n\nOur goal is to estimate the size of the set $T=S\\setminus\\bigcup\nS_{ij}^{rs}$.  Since \\tref{16} requires polynomials in one variable,\nto determine the size of $T$ it is necessary to proceed by determining\nthe sizes of slices $\\{x\\in \\mathbb{F}:(x,c)\\in T\\}$, for each square\n$c\\notin \\{0,1\\}$.  As a convention, $p(x,c)$ will mean a polynomial\nin one variable, i.e., an element of $\\mathbb{F}[x]$, for every\n$p(x,y)\\in \\mathbb{F}[x,y]$.\n\n\\tref{16} may be directly applied only when the product of the\npolynomials involved is square-free.  Thus for\n$p_1(x,y),\\dots,p_k(x,y)\\in \\mathbb{F}[x,y]$ it is necessary to set aside\nthose $c\\in \\mathbb{F}$ for which $p_1(x,c),\\dots,p_k(x,c)$ is \\emph{not} a\nsquare-free list of polynomials. An asymptotic estimate does not\ndepend upon the number of $c$ set aside if there are only a bounded\nnumber of them. Hence a possible route is to express the discriminant\nof $p_1(x,c)\\cdots p_k(x,c)$ by means of computer algebra, and then set\naside those $c$ that make the discriminant equal to zero. The route\ntaken below is elementary and is not dependent upon computer. In this\nway the number of $c$ to avoid is limited to 51.  This is a\nconsequence of the following statement, the proof of which is the goal\nof this section.\n\n\\begin{thm}\\label{31}\nLet $\\mathbb{F}$ be a field of characteristic different from $2$. \nThe list of polynomials\n\\begin{gather}\\label{e31}\n\\begin{aligned}\n&\\text{$x$, \n$x{-}1$, $x{-}c$, $x{-}1{-}c$, $x{+}1{-}c$, $(1{-}c)x-c$, $(1{+}c)x-c$,}\n\\\\ &\\text{$g_1(x,c)$, $g_2(x,c)$,\n$g_3(x,c)$, $g_4(x,c)$, $f_1(x,c)$, $f_2(x,c)$,\n$f_3(x,c)$, $f_4(x,c)$}\n\\end{aligned}\n\\end{gather}\nis square-free if the following conditions hold:\n\\begin{align}\nc&\\notin \\{-1,0,1,1\/2,2\\}; \\label{e32}\\\\\nc&\\text{ is not a root of $x^2\\pm x\\pm1$;}\n\\label{e33}\\\\\nc&\\text{ is not a root of $x^2-3x+1$;}\\label{e34}\\\\\nc&\\notin \\{-1\/3,-3,2\/3,3\/2,1\/3,3,4\/3,3\/4\\} \\text{ if } \\operatorname{char}(F)\\ne 3;\\label{e35}\\\\\nc&\\text{ is a root of neither $x^2{-}3x{+}3$ nor $3x^2{-}3x{+}1$;} \\label{e36} \\\\\nc&\\text{ is a root of neither $x^3{+}x^2{-}1$ nor $x^3{-}x{-}1$;}\\label{e37}\\\\\nc&\\text{ is not a root of $x^2{+}1$;}\\label{e38}\\\\\nc&\\text{ is a root of neither $x^2{-}2x{+}2$ nor $2x^2{-}2x{+}1$; }\\label{e39}\\\\\nc&\\text{ is a root of neither $x^3{-}x^2{+}2x{-}1$ nor $x^3{-}2x^2{+}x{-}1$; and} \\label{e310}\\\\\nc&\\text{ is a root of neither $x^3{-}2x^2{+}3x{-}1$ nor $x^3{-}3x^2{+}2x{-}1$.} \\label{e311}\n\\end{align}\n\\end{thm}\n\nThe proof requires a number of steps. As an auxiliary notion,\nwe call a list of polynomials $p_1(x,y),\\dots,p_k(x,y)\\in\n\\mathbb{F}[x,y]$ \\emph{reciprocally\nclosed} if for each $i\\in \\{1,\\dots,k\\}$ both $x\\nmid p_i(x,y)$ and\n$y\\nmid p_i(x,y)$ are true, and there exist unique\n$j\\in \\{1,\\dots,k\\}$ and $\\lambda \\in \\mathbb{F}$ such that\n$\\hat{p_i}(x,y)= \\lambda p_j(x,y)$. \n\nIf $a=\\sum a_it^i\\in \\mathbb{F}[t]$ is a nonzero polynomial of degree $d\\ge 0$,\nthen the reciprocal polynomial $\\sum a_it^{d-i}$ will be denoted\nby $\\hat a$, like in the case of two variables. A list\n$a_1(t),\\dots,a_k(t)\\in \\mathbb{F}[t]$ is \\emph{reciprocally\nclosed} if for each $i\\in \\{1,\\dots,k\\}$ the polynomial $a_i(t)$\nis not divisible by $t$, and  there exist unique\n$j\\in \\{1,\\dots,k\\}$ and $\\lambda \\in \\mathbb{F}$ such that\n$\\hat{a_i}(t)= \\lambda a_j(t)$.\n\n\\begin{lem}\\label{32}\nLet $p_1(x,y),\\dots,p_k(x,y)\\in \\mathbb{F}[x,y]$ and $a_1(t),\\dots,a_r(t)\n\\in \\mathbb{F}[t]$ be two reciprocally closed lists of polynomials.\nDenote by $\\Gamma$ the set of all nonzero roots of polynomials\n$a_1,\\dots,a_r$. Assume that \n\\begin{equation}\\label{e:imp}\np_i(0,c)=0\\ \\Longrightarrow \\ c\\in \\Gamma \\text{ or } c=0\n\\end{equation}\nholds for all $i\\in \\{1,\\dots,k\\}$. \n\nLet $i,j\\in \\{1,\\dots,k\\}$ and $\\lambda \\in \\mathbb{F}$ be\nsuch that $p_j(x,y) = \\lambda \\hat {p_i}(x,y)$ and $i\\ne j$. If\n\\begin{equation*}\n\\gcd(p_i(x,c),p_\\ell(x,c)) = 1\n\\end{equation*}\nholds for all nonzero $c\\in \\mathbb{F}\\setminus\\Gamma$ and all $\\ell\\ne i$, $1\\le\\ell\\le k$,\nthen\n\\begin{equation*}\n\\gcd(p_j(x,c),p_h(x,c)) = 1\n\\end{equation*}\nholds for all nonzero $c\\in \\mathbb{F}\\setminus\\Gamma$ and $h\\ne j$, $1\\le h \\le k$.\n\\end{lem}\n\n\\begin{proof}\nSuppose that $h\\ne j$ and $c\\in \\mathbb{F}\\setminus \\Gamma$, $c\\ne 0$, are such\nthat $p_j(x,c)$ and $p_h(x,c)$ have a common root in $\\bar \\mathbb{F}$, say\n$\\gamma$. Thus $p_j(\\gamma,c) = p_h(\\gamma,c) = 0$. By (3.12),\n$\\gamma \\ne 0$. Since the list $p_1(x,y),\\dots,p_k(x,y)$ is\nreciprocally closed, there exists $\\ell \\ne i$ such that $p_\\ell(x,y)$\nis a scalar multiple of $\\hat{p_h}(x,c)$. Since $p_j(x,y)$ is a multiple\nof $\\hat{p_i}(x,y)$ we have $p_i(\\gamma^{-1},c^{-1})=0=p_\\ell(\\gamma^{-1},c^{-1})$\nand hence $\\gcd(p_i(x,c^{-1}),p_\\ell(x,c^{-1}))\\ne1$.\nBy the assumption on $p_i$ this cannot be true unless $c^{-1}\\in \\Gamma$.\nWe shall refute the latter possibility by proving that if\n$c^{-1}\\in \\Gamma$, then  $c\\in \\Gamma$.\nThat follows straightforwardly from the assumption that the list\n$a_1,\\dots,a_r$ is reciprocally closed. Indeed, since $c^{-1} \\in \\Gamma$,\nthere exists $s\\in \\{1,\\dots,r\\}$ such that $a_s(c^{-1}) = 0$. There\nalso exists $m\\in \\{1,\\dots,r\\}$ such that $a_m$ is a scalar\nmultiple of $\\hat{a_s}$. Because of that, $a_m(c) = a_m((c^{-1})^{-1}) = 0$.\nThis implies that $c\\in \\Gamma$ since $\\Gamma$ is defined as the set of\nall nonzero roots of polynomials $a_1,\\dots,a_r$.\n\\end{proof}\n\nIf $a(t) = t-\\gamma$, $\\gamma \\ne 0$, then $\\hat a(t) = \n-\\gamma(t-\\gamma^{-1})$. Hence the list of nonzero $c$ that\nfulfil one of the conditions \\eref{32}--\\eref{311} may be\nconsidered as a set $\\Gamma$ of nonzero roots of a reciprocally\nclosed list of polynomials in one variable.\n\nNow, remove $x$ and $x{-}1$ from the list of polynomials \\eref{31} that \nare the input to \\lref{32}.\nThe remaining polynomials can be interpreted as a list\n$p_1(x,c), \\dots, p_{13}(x,c)$ such that $p_1(x,y), \\dots,p_{13}(x,y)$\nis a reciprocally closed list of polynomials in two variables.\nIt is easy to verify that if $0$ or $1$ is a root of\nany of the polynomials $p_i(x,c)$, $1\\le i \\le 13$, then \n$c$ fulfils \\eref{32}. Polynomials $x$ and $x{-}1$ can be thus\nexcised from the subsequent discussion, and \\lref{32}\nmay be used.\n\n\\lref{32} will also be applied to some sublists of \n$p_1(x,c), \\dots, p_{13}(x,c)$ that are reciprocally closed. The\nfirst such sublist are the linear polynomials occurring in \\eref{31}\n(with $x$ and $x{-}1$ being removed). These are\n$x{-}c$, $x{-}1{-}c$, $x{+}1{-}c$, $(1{-}c)x-c$, $(1{+}c)x-c$,\n$x-(2c{-}c^2)$ and $(1{-}2c)x+c^2$. The latter two polynomials\nare equal to $g_2(x,c)$ and $g_4(x,c)$. The list of these linear\npolynomials is square-free if there are no duplicates in the set of \ntheir roots\n\\begin{equation*} \\label{e39o}\nR(c) = \\left\\{c,c+1,c-1,\\frac c{1-c},\\frac c{1+c},c(2-c), \n\\frac{c^2}{2c-1}\\right\\}.\n\\end{equation*}\nThe reciprocity yields the following pairs of roots:\n\\begin{equation}\\label{e310o}\n\\left\\{c+1,\\frac c{1+c}\\right \\},\\quad  \\left\\{c-1,\\frac c{1-c}\\right\\}\n\\text{ and } \\left\\{c(2-c), \\frac{c^2}{2c-1}\\right \\}.\n\\end{equation}\n\n\nWe now prove a sequence of lemmas which explore properties of the polynomials\n\\eref{31}.\n\n\\begin{lem}\\label{33}\nIf $c\\in \\mathbb{F}$ satisfies \\eref{32}--\\eref{34}, then $|R(c)| = 7$.\n\\end{lem}\n\\begin{proof} \nIf \\eref{32} holds, then $c$ is not equal to any other element of\n$R(c)$. Any equality\nwithin the pairs in \\eref{310o} would require that \n$c^2{+}c{+}1=0$ or $c^2{-}c{+}1=0$ or $2c(c{-}1)^2=0$. \nBy \\eref{32} and \\eref{33}, none of these conditions hold.\nClearly, $c{+}1\\ne c{-}1$. If $c{+}1 = c\/(1{-}c)$, then $c^2{+}c{-}1=0$.\nFurthermore, $c{+}1=c(2{-}c)$ $\\Leftrightarrow$ $c^2{-}c{+}1=0$, and\n$c{+}1=c^2\/(2c{-}1)$ $\\Leftrightarrow$ $c^2{+}c{-}1=0$. Hence $c{+}1$\nis not equal to any other element of $R(c)$. By the reciprocity\nrelationship described in \\lref{32}, $c\/(1{+}c)$ is also not equal\nto another element of $R(c)$. If $c{-}1$ is equal to $c(2{-}c)$, then\n$c^2{-}c{-}1 =0$. If it is equal to $c^2\/(2c{-}1)$, then $c^2{-}3c{+}1=0$.\n\\end{proof}\n\n\\begin{lem}\\label{34}\nSuppose that $c\\in \\mathbb{F}$ satisfies \\eref{32} and \\eref{35}. Then\nnone of the polynomials $f_i(x,c)$, $1\\le i \\le 4$, and $g_j(x,c)$,\n$j \\in \\{1,3\\}$, possesses a double root.\n\\end{lem}\n\\begin{proof} By a reciprocity argument similar to that of \\lref{32} \nonly $f_1(x,c)$, $f_2(x,c)$ and $g_1(x,c)$ need to be tested.\nDiscriminants of these polynomials are $(c+1)^2-4c^2 = (1-c)(3c+1)$,\n$c(c-4(c-1)) = -c(3c-4)$ and $4(1-c)$. None of these may be\nzero, by the assumptions on $c$.\n\\end{proof}\n\n\\begin{lem}\\label{35}\nIf $c\\in \\mathbb{F}$ satisfies \\eref{32}, \\eref{33} and \\eref{36},\nthen none of the elements of $R(c)$\nis a root of $g_1(x,c)$ or $g_3(x,c)$.\n\\end{lem}\n\\begin{proof}\nBy \\lref{32} it suffices to consider only the polynomial $h(x)=g_1(x,c)$.\nNow, $h(c)  = c(c{-}1)$, $h(c\\pm 1)=\nc^2\\pm 2c+1-2(c\\pm 1) +c$ is equal to $c^2+c-1$ or $c^2-3c+3$,\nwhile\n\\begin{equation*}\n\\frac{(1\\pm c)^2}c h\\left (\\frac c{1\\pm c}\\right ) = c\n-2(1\\pm c) + (1\\pm c)^2 = c^2+c-1\n\\end{equation*}\nand $c^{-1} h(c(2-c))=c(2-c)^2 - 2(2-c)+1=c^3-4c^2+6c-3=\n(c-1)(c^2-3c+3)$. Finally,\n\\begin{equation*}\n\\frac{(2c-1)^2}c h\\left (\\frac {c^2}{2c-1}\\right ) = c^3\n-2c(2c-1) + (2c-1)^2 = c^3-2c+1=(c-1)(c^2+c-1).\\qedhere\n\\end{equation*}\n\\end{proof}\n\n\n\\begin{lem}\\label{norootsf}\nIf $c\\in \\mathbb{F}$ satisfies \\eref{32}, \\eref{35} and \\eref{37}--\\eref{311},\nthen none of the elements of $R(c)$\nis a root of $f_i(x,c)$ for any $i=1,2,3,4$.\n\\end{lem}\n\n\\begin{proof}\nThe proof is very similar to that of \\lref{35}, so we only give a summary.\nBy \\lref{32}, it suffices to test the polynomials $f_1(x,c)$\nand $f_2(x,c)$. Substituting an element\nof $R(c)$ in place of $x$ always yields a polynomial from\nthe indicated list.\nNote that $c^3-3c^2+4c-2 = (c-1)(c^2-2c+2)$, $3c^2-5c+2 =\n(c-1)(3c-2)$ and $3c^3-7c^2 +5c -1 = (c-1)^2(3c-1)$.\n\\end{proof}\n\n\n\\begin{lem}\\label{36}\nSuppose that $c\\in \\mathbb{F}$ satisfies \\eref{32} and \\eref{35}. Then\nfor each $i\\in \\{1,3\\}$ there exist at least three $j\\in \\{1,2,3,4\\}$\nsuch that $g_i(x,c)$ and $f_j(x,c)$ share no root in $\\bar \\mathbb{F}$.\n\\end{lem}\n\n\\begin{proof} Because of the reciprocity, $i=1$ may be assumed.\nIf $g_1(x,c)$ and $f_1(x,c)$ have a common root $x$, then $(c{+}1)x-c^2 =2x-c$,\nand that yields $(c{-}1)x=c(c{-}1)$. If $g_1(x,c)$ and $f_2(x,c)$ have a common\nroot, then $cx -c^2+c = 2x-c$, which means that $(c{-}2)x = (c{-}2)c$.\nIf $g_1(x,c)$ and $f_3(x,c)$ have a common root, then $(c^2{+}c)x -c^2 = 2x-c$,\nand $(c{-}1)(c{+}2)x = (c{-}1)c$. In such a case $c\\ne -2$ and\n$x = c\/(c{+}2)$. The latter value is a root of $g_1(x,c)$ if and only\nif $0=c^2-2c(c{+}2)+c(c{+}2)^2=c^2(c+3)$. Here, as earlier, the solutions\nfor $c$ are forbidden by the conjunction of \\eref{32} and \\eref{35}.\n\\end{proof}\n\n\n\\begin{lem}\\label{37}\nIf $c\\in \\mathbb{F}$ satisfies \\eref{32}, \\eref{37} and \\eref{38},\nand if $1\\le i < j \\le 4$, \nthen $f_i(x,c)$ and $f_j(x,c)$ share no common root in $\\bar \\mathbb{F}$.\n\\end{lem}\n\n\\begin{proof}\nThis is obvious if $(i,j) = (1,3)$. If $(i,j) = (2,4)$, then $c(x^2-1)=0$, so\n$x\\in \\{-1,1\\}$. Now, $f_2(1,c) = (c-1)^2 \\ne 0$, while\n$f_2(-1,c) = c^2+1 = -f_4(-1,c)$. This is why $c^2\\ne -1$ has to be\nassumed.\n\nFor the rest it suffices to test pairs $(1,2)$ and $(2,3)$,\nby the reciprocity described in \\lref{32}. If $cx+x-c^2 = cx - c^2+c$,\nthen $x=c$ and $f_2(c) = c(c-1) \\ne 0$. If $cx -c^2 + c =\ncx + c^2x -c^2$, then $x=c^{-1}$ and $f_2(c^{-1}) = c^{-2}-1+c^2-c\n= c^{-2}(c^4-c^3-c^2+1) = c^{-2}(c-1)(c^3-c-1)$.\n\\end{proof}\n\n\n\nWe can now bring all the pieces together to prove the main result of this\nsection.\n\n\\begin{proof}[Proof of \\tref{31}] Suppose \nthat $c$ fulfils \\eref{32}--\\eref{311}. Besides\nLemmas~\\ref{33}--\\ref{37}\nwe also use that $g_1(x,c)$ and $g_3(x,c)$ share no root\nin $\\bar F$, which can be proved by a similar method to \\lref{37}.\n\n\nLet $p_1(x,c),\\dots,p_k(x,c)$ be a nonempty sublist\nof \\eref{31} such that the product $p_1(x,c)\\cdots p_k(x,c)$ is \na square in $\\bar\\mathbb{F}[x]$. Let $J$ be the set of those $j\\in\\{1,2,3,4\\}$\nfor which there exists $h\\in \\{1,\\dots,k\\}$ such that $f_j(x,c) = p_h(x,c)$.\nThe set $J$ must be nonempty, by Lemmas \\ref{33}--\\ref{35}. Since\n$J$ is nonempty and Lemmas~\\ref{34}, \\ref{norootsf} and \\ref{37} hold,\nthere must exist $i\\in \\{1,3\\}$ such that $g_i(x,c) = p_h(x,c)$ for\nsome $h\\in \\{1,\\dots,k\\}$. Since $g_i(x,c)$ is not a scalar multiple\nof $f_j(x,c)$ for $j\\in J$, we must have $|J|\\ge 2$, by Lemmas~\\ref{34}\nand~\\ref{35}.\nHowever, even that is not viable, given \\lref{36}. \n\\end{proof}\n\n\\section{When $-1$ is a nonsquare}\\label{4}\n\nThroughout this section $\\mathbb{F}=\\mathbb{F}_q$ will be a finite field of order $q\\equiv\n3\\bmod 4$. We put $\\Sigma = \\Sigma(\\mathbb{F}_q)$ and $S=S(\\mathbb{F}_q)$. By \\cref{13},\n$|S| = |\\Sigma| = (q^2{-}8q{+}15)\/4$.  Define\n$\\bar S_{ij}^{rs}=S\\setminus S_{ij}^{rs}$ and $T=\\bigcap\\bar S_{ij}^{rs}$,\nwhere sets $S_{ij}^{rs}$ are characterised by \\tref{211}, subject\nto the assumption that \\eref{25} holds. As we will see, \\eref{25} holds\nin all cases that are relevant for our calculations. The\naim of this section is to estimate the number $\\sigma(q) =\n\\big|\\{(a,b)\\in \\Sigma:Q_{a,b}$ is maximally nonassociative$\\}\\big|$.\nBy \\pref{15}, $\\sigma(q) = |T|=(q^2{-}8q{+}15)\/4 - |\\bigcup\nS_{ij}^{rs}|$.\nPut\n\\begin{align*} T_0& = \\{(x,y)\\in T: \\chi(y-x) = 1\\}; \\\\\nT_{1,1} &= \\{(x,y)\\in T_0: \\chi(1-y)=\\chi(1-x)=1\\}; \\\\\nT_{1,-1} &= \\{(x,y)\\in T_0: \\chi(1-y)=\\chi(1-x)=-1\\}; \\\\\nT_2 &= \\{(x,y)\\in T_0: \\chi(1-x)=-1 \\text{ and } \\chi(1-y) = 1\\};\n\\end{align*}\nand define $T_0'$, $T'_{1,1}$, $T'_{1,-1}$ and $T'_2$ by exchanging\n$x$ and $y$. For example, $T_0'=\\{(x,y)\\in T:\\chi(y-x) = -1\\}$. \nPut also $T_1=T_{1,1}\\cup\nT_{1,-1}$ and $T_1'=T'_{1,1}\\cup T'_{1,-1}$.\n\n\\begin{lem}\\label{42}\n$T=T_0\\cup T_0'$, $T_0= T_1\\cup T_2$, $T_0' = T_1' \\cup T_2'$,\n$T_1=T_{1,1}\\cup T_{1,-1}$ and $T_1'=T'_{1,1}\\cup T'_{1,-1}$.\nAll of these unions are unions of disjoints sets. \n\nBoth of the mappings\n$(x,y)\\mapsto (y,x)$ and $(x,y)\\mapsto (x^{-1},y^{-1})$ permute $T$.\nBoth of them exchange $T_1$ and $T_1'$, and $T_2$ and $T_2'$.\nFurthermore, $(x,y)\\mapsto (y,x)$ sends $T_{1,\\varepsilon}$\nto $T'_{1,\\varepsilon}$, while $(x,y)\\mapsto (x^{-1},y^{-1})$ sends\n$T_{1,\\varepsilon}$ to $T'_{1,-\\varepsilon}$, for both $\\varepsilon\\in \\{-1,1\\}$.\n\\end{lem}\n\n\\begin{proof}\nRecall that by our definition of $S$, we have $1\\notin\\{x,y\\}$ and $x\\ne y$\nfor all $(x,y)\\in T$.\nBy \\pref{23} both $(x,y) \\mapsto (y,x)$ and $(x,y)\\mapsto (x^{-1},y^{-1})$\npermute $T$. The effects of these\nmappings are easy to verify. Note, for example, that\nif $\\varepsilon = \\chi(x{-}y)$, then $\\chi(x^{-1}-y^{-1}) = \\chi(y{-}x) = -\\varepsilon$.\n\nTo see that $T_0 = T_1\\cup T_2$, note that there is no \n$(x,y) \\in T_0$ with $\\chi(1{-}x)=1$ and $\\chi(1{-}y)=-1$. Indeed,\neach such $(x,y)$ belongs to $S_{01}^{10}$.\n\\end{proof}\n\nFor $c\\in\\mathbb{F}_q$ define\n$t_2(c)=\\big|\\{x\\in\\mathbb{F}_q:(x,c)\\in T_2\\}\\big|$\nand $t_{1,1}(c)=\\big|\\{x\\in\\mathbb{F}_q:(x,c)\\in T_{1,1}\\}\\big|$.\nIn the next two propositions we seek estimates of these quantities.\nIn both results we will assume that $c$ fulfils condition\n\\eref{33}. Observe that under this assumption, $c^2{-}c{-}1\\ne 0$ and for\nall $x\\in\\mathbb{F}_q$ either $x\\ne c{+}1$ or $x^2{-}x{-}1\\ne 0$, and therefore\n\\eref{25} holds for $(x,y)=(x,c)$. This will enable us to use \\tref{211}.\n\n\n\\begin{prop}\\label{44}\nSuppose that $c$ and $1-c$ are both nonzero squares in $\\mathbb{F}_q$\nand that $c$ fulfils conditions \\eref{32}--\\eref{311}.\nThen $$|t_2(c)-25\\cdot2^{-15}q|\\le (\\sqrt q+1)165\/2+21.$$\n\\end{prop}\n\n\\begin{proof}\nWe estimate $t_2(c)$ by characterising the pairs $(x,c)$ in $T_2$.\nFor a fixed $c$ there are at most 21 values of $x$ that are roots\nof any of the polynomials in \\eref{31}. So at the cost of adding\na term equal to 21 to our eventual bound, we may assume\nfor the remainder of the proof that $x$ is not a root of any\npolynomial in \\eref{31}. Then\n$\\chi(x)=1=\\chi(c)$ since $(x,c) \\in S$ and \n$\\chi(1{-}x)=\\chi(c{-}1)=\\chi(x{-}c)=-1$ by the definition of $T_2$.\n\nFrom the definitions of $\\bar S_{01}^{00}$, $\\bar S_{10}^{00}$, $\\bar S_{10}^{11}$\nand $\\bar S_{01}^{11}$ we deduce that\n$\\chi(g_1(x,c))=\\chi(g_4(x,c))=-1$ and $\\chi(g_2(x,c))=\\chi(g_3(x,c))=1$.\nNow, from $(x,c)\\in\\bar S_{01}^{01}$ we deduce that either\n\\begin{equation}\\label{efirstalts}\n\\chi(x{-}1{-}c) = 1 \\text{ or } \\chi(x-xc-c)=1.\n\\end{equation}\nIn the former case, the requirement for $(x,c)$ to be in\n$\\bar S_{11}^{01}$ forces $\\chi(f_1(x,c))=1$, whilst in the latter case\nthe requirement for $(x,c)$ to be in\n$\\bar S_{00}^{01}$ forces $\\chi(f_4(x,c))=-1$. Of course, it is also\npossible that both alternatives in \\eref{firstalts} are realised.\n\nAnalogously, $(x,c)$ belongs to $\\bar S_{10}^{10}$, so\n\\begin{equation}\\label{esecondalts}\n\\chi(c{-}1{-}x) = 1 \\text{ or } \\chi(c-xc-x)=1.\n\\end{equation}\nIn the former case, the requirement for $(x,c)$ to be in\n$\\bar S_{11}^{10}$ forces $\\chi(f_2(x,c))=-1$, whilst in the latter case\nthe requirement for $(x,c)$ to be in\n$\\bar S_{00}^{10}$ forces $\\chi(f_3(x,c))=1$. \n\nSuppose that $i=1,\\dots,9$ indexes the nine possibilities for the\nquadruple\n\\begin{equation}\\label{e:quad}\n\\big(\\chi(c{-}1{-}x),\\,\\chi(c-xc-x),\\,\\chi(x{-}1{-}c),\\,\\chi(x-xc-c)\\big)\n\\end{equation}\nthat are consistent with \\eref{firstalts} and \\eref{secondalts}.  In\neach case, let $J_i$ denote the subset of $\\{1,2,3,4\\}$ consisting of\nthose indices $j$ for which $\\chi(f_j)$ is forced.  Combining the\nabove observations, we see that there will be $4,4,1$ cases\nrespectively in which $|J_i|=2,3,4$.\n\nBy \\tref{31} and our assumptions, the list\nof polynomials in \\eref{31} is square-free. We can hence apply\n\\tref{16} for each of the 9 possibilities for \\eref{:quad}, prescribing\n$\\chi(p(x))$ for each polynomial $p(x)$ in \\eref{31} except for\nany $f_j$ with $j\\notin J_i$.\nWe find that\n\\begin{align*}\n|t_2(c)-4\\cdot2^{-13}q-4\\cdot2^{-14}q-1\\cdot2^{-15}q|\n&\\le (\\sqrt q+1)(4\\cdot17+4\\cdot19+1\\cdot21)\/2+21.\n\\end{align*}\nThe result follows.\n\\end{proof}\n\n\n\\begin{prop}\\label{45}\nSuppose that $c$ and $1-c$ are both nonzero squares in $\\mathbb{F}_q$\nand that $c$ fulfils conditions \\eref{32}--\\eref{311}.\nThen $$|t_{1,1}(c)-25\\cdot2^{-11}q|\\le 96(\\sqrt q+1)+21.$$\n\\end{prop}\n\n\\begin{proof} \nThe proof is similar to that of \\pref{44}.\nLet us consider under which conditions a pair $(x,c)$ belongs to $T_{1,1}$,\nwhere $x$ is not a root of any polynomial in \\eref{31}.\nFor $(x,c)$ to belong to each of the sets \n$\\bar S_{01}^{10}$,\n$\\bar S_{01}^{00}$, $\\bar S_{10}^{00}$, $\\bar S_{10}^{11}$,\n$\\bar S_{01}^{11}$, $\\bar S_{11}^{01}$ and $\\bar S_{00}^{10}$ it\nis necessary and sufficient that\n$\\chi(x)=\\chi(c)=\\chi(1-c)=\\chi(1-x)=\\chi(c-x)=\\chi(g_2(x,c)) = 1$ and\n$\\chi(g_4(x,c)) = -1$. Also for $(x,c)$ to be in $\\bar S_{11}^{10}$ and\n$\\bar S_{10}^{10}$ requires that\n\\begin{align*}\n&\\chi(c{-}1{-}x) = -1\\text{ or } \\chi(f_2(x,c))=-1; \\text{ and}\\\\\n&\\chi(c{-}1{-}x) = 1\\text{ or } \\chi(g_3(x,c))=-1 \\text{ or } \\chi(c-xc-x)=1.\n\\end{align*}\nBoth of these conditions are automatically \nsatisfied if $\\chi(c{-}1{-}x) = -1$ and \n$\\chi(c-xc-x)=1$. Each of the other three possibilities for the pair\n$\\big(\\chi(c{-}1{-}x),\\chi(c-xc-x)\\big)$ forces exactly one of the\nconditions $\\chi(f_2(x,c))=-1$ or $\\chi(g_3(x,c))=-1$ to hold.\n\nSimilarly, for $(x,c)$ to be in $\\bar S_{00}^{01}$ and\n$\\bar S_{01}^{01}$ requires that\n\\begin{align*}\n&\\chi(x{-}xc{-}c) = -1\\text{ or } \\chi(f_4(x,c))=-1; \\text{ and}\\\\\n&\\chi(x{-}1{-}c) = 1\\text{ or } \\chi(g_1(x,c))=1 \\text{ or } \\chi(x-xc-c)=1.\n\\end{align*}\nBoth of these conditions are automatically \nsatisfied if $\\chi(x{-}1{-}c) = 1$ and \n$\\chi(x-xc-c)=-1$. Each of the other three possibilities for the pair\n$\\big(\\chi(x{-}1{-}c),\\chi(x-xc-c)\\big)$ forces exactly one of the\nconditions $\\chi(f_4(x,c))=-1$ or $\\chi(g_1(x,c))=1$ to hold.\n\nSuppose that $i=1,\\dots,16$ indexes the sixteen possibilities for the\nquadruple \\eref{:quad}.  Let $K_i$ denote the subset of $\\{f_2,f_4,g_1,g_3\\}$\nconsisting of those polynomials $p$ for which $\\chi(p)$ is forced.\nCombining the above observations, we see that there will be $1,6,9$\ncases respectively in which $|K_i|=0,1,2$. The values of $\\chi(p)$ for\n$p\\in\\{f_1,f_2,f_3,f_4,g_1,g_3\\}\\setminus K_i$ are unconstrained. Hence,\nby applying \\tref{16} for each of the 16 possibilities for \\eref{:quad}\nwe find that\n\\begin{align*}\n|t_{1,1}(c)-1\\cdot2^{-9}q-6\\cdot2^{-10}q-9\\cdot2^{-11}q|\n&\\le (\\sqrt q+1)(1\\cdot9+6\\cdot11+9\\cdot13)\/2+21.\n\\end{align*}\nThe result follows.\n\\end{proof}\n\nWe are now ready to prove the main result for this section.\n\n\\begin{thm}\\label{46}\nFor $q\\equiv3\\bmod4$,\n\\[\n\\big|\\sigma(q)-25(2^{-11}{+}2^{-16}) q^2\\big|<138 q^{3\/2} + 235q.\n\\]\n\\end{thm}\n\n\\begin{proof}\nBy \\cite[Theorem~10.5]{Eva18} there are $(q-3)\/4$ choices for $c\\in\\mathbb{F}_q$\nsuch that both $c$ and $1-c$ are nonzero squares. At most\n$1+4+1+3+2+3+0+2+3+3=22$ of these choices do not fulfil\nconditions \\eref{32}--\\eref{311} of \\tref{31}. (To see this, note that\n$\\chi(-1)=-1$ and that if $\\chi(c)=\\chi(1-c)=1$ then $\\chi(1-1\/c)=-1$,\nwhich means that in any pair of reciprocal field elements, at most one\nof the elements will be a viable choice for $c$. This is particularly\nuseful because of the many polynomials in \\tref{31} which form\nreciprocal pairs.)\nEach $c$ that fails one of the conditions\n\\eref{32}--\\eref{311} contributes between 0 and $(q-3)\/2$ elements $(x,c)$\nto $T$.\nPutting these observations together with \\pref{44} and \\pref{45} we have\nthat\n\\begin{align*}\n\\big||T_2|-25\\cdot2^{-15}q(q-3)\/4\\big|\n&\\le165(\\sqrt{q}+1)(q-3)\/8+21(q-3)\/4+22(q-3)\/2,\\\\\n\\big||T_{1,1}|-25\\cdot2^{-11}q(q-3)\/4\\big|&\\le96(\\sqrt{q}+1)(q-3)\/4+21(q-3)\/4+22(q-3)\/2.\n\\end{align*}\nNext, notice that it follows from \\lref{42} that\n$|T_{1,1}|=|T'_{1,1}|=|T_{1,-1}|=|T'_{1,-1}|$ and $|T_2|=|T'_2|$  and\nthat $T$ is the disjoint union of $T_{1,1}$, $T'_{1,1}$, $T_{1,-1}$,\n$T'_{1,-1}$, $T_2$ and $T'_2$. Hence\n\\begin{align*}\n\\big||T|-25(2^{-16}+2^{-11})q(q-3)\\big|\n&\\le(q-3)\\Big[(165\/4+96)(\\sqrt{q}+1)+195\/2\\Big]\\\\\n&\\le q\\Big[138\\sqrt{q}+939\/4\\Big].\n\\end{align*}\nThe result then follows from simple rearrangement.\n\\end{proof}\n\n\\begin{cor}\\label{48}\nLet $q$ run through all prime powers $\\equiv 3 \\bmod 4$.\nThen $\\lim \\sigma(q)\/q^2 = 25(2^{-11}{+}2^{-16})$.\n\\end{cor}\n\n\\section{When $-1$ is a square}\\label{5}\n\nThroughout this section $\\mathbb{F}=\\mathbb{F}_q$ will be a finite field of order\n$q\\equiv1\\bmod4$. Our broad strategy for obtaining an estimate of\n$\\sigma(q)$ is similar to that used in \\secref{4}.  For\n$i,j,r,s\\in \\{0,1\\}$ define $\\bar S_{ij}^{rs}=S\\setminus S_{ij}^{rs}$\nand put $T = \\bigcap \\bar S_{ij}^{rs}$.  The set $T$ will again be\nexpressed as a disjoint union of sets the size of each of which can be\nestimated by means of the Weil bound.\nLet $\\varepsilon=\\chi(x-y)$ and define\n\\begin{align*}\nT_1 &= \\{(x,y)\\in T: \\chi(1{-}x)=\\chi(1{-}y)=-\\varepsilon\\};\\\\\nT_2 &= \\{(x,y)\\in T: \\chi(1{-}x) = \\varepsilon\\text{ and }\n\\chi(1{-}y) = -\\varepsilon\\}; \\text{ and}\\\\\nT_2' &= \\{(x,y)\\in T: \\chi(1{-}x) = -\\varepsilon\\text{ and } \\chi(1{-}y) = \\varepsilon\\}.\n\\end{align*}\nIf $\\rho_j\\in \\{-1,1\\}$ for $1\\le j \\le 4$, then define\n\\begin{align*}\nR(\\rho_1,\\rho_2,\\rho_3,\\rho_4)\\ &=\\{(x,y)\\in T: \\rho_j = \n\\varepsilon\\,\\chi(f_j(x,y))\n\\text{ for } 1\\le j \\le 4\\};\\\\\nR_1(\\rho_1,\\rho_2,\\rho_3,\\rho_4)&=T_1\\cap R(\\rho_1,\\rho_2,\\rho_3,\\rho_4);\n\\text{ and } \\\\\nR_2(\\rho_1,\\rho_2,\\rho_3,\\rho_4)&= T_2\\cap R(\\rho_1,\\rho_2,\\rho_3,\\rho_4).\n\\end{align*}\nWe will write $R(\\bar \\rho)$ as a shorthand for $R(\\rho_1,\\rho_2,\\rho_3,\\rho_4)$\nwhere $\\bar \\rho = (\\rho_1,\\rho_2,\\rho_3,\\rho_4)$. We record the following\nbasic facts about the sets just defined.\n\n\n\\begin{lem}\\label{51}\nSuppose $\\rho_j\\in\\{-1,1\\}$ for $1\\le j\\le4$.\nThe map $(x,y)\\mapsto (y,x)$ induces bijections that show that\n$|R_1(\\rho_1,\\rho_2,\\rho_3,\\rho_4)|=|R_1(\\rho_2,\\rho_1,\\rho_4,\\rho_3)|$\nand $|T_2|=|T'_2|$. Hence $|T|=|T_1|+2|T_2|$.\nThe map $(x,y)\\mapsto (x^{-1},y^{-1})$ induces bijections that show that\n$|R_i(\\rho_1,\\rho_2,\\rho_3,\\rho_4)|=|R_i(\\rho_3,\\rho_4,\\rho_1,\\rho_2)|$\nfor $i\\in\\{1,2\\}$. Also,\n$R(\\rho_1,\\rho_2,-1,-1)=R(-1,-1,\\rho_3,\\rho_4)=\\emptyset$.\n\\end{lem}\n\n\\begin{proof}\nBy \\pref{23}, we know that $(x,y)\\mapsto (x^{-1},y^{-1})$ permutes \neach of the sets $T_1$, $T_2$ and $T_2'$, while $(x,y)\n\\mapsto (y,x)$ permutes $T_1$ and swaps $T_2$ and $T_2'$.\nThis gives us a bijection between $T_2$ and $T_2'$. \nNote also that $T=T_1\\cup T_2\\cup T_2'$ since $\\chi(1{-}x)=\\chi(1{-}y)=\\varepsilon$\nimplies that $(x,y)\\in S_{00}^{00}$. Hence $|T|=|T_1|+2|T_2|$.\nThe remaining claims about bijections follow directly from\nthe definitions of $f_1$, $f_2$, $f_3$ and $f_4$ in \\eref{26}.\n\nIf $(x,y)\\in R(\\rho_1,\\rho_2,-1,-1)$, then \n$\\chi(f_j(x,y))=-\\varepsilon$ for both $j\\in \\{3,4\\}$.\nThat implies $(x,y)\\in S_{00}^{11}$, by \\lref{24}.\nHence, $R(\\rho_1,\\rho_2,-1,-1)=\\emptyset$\nand our bijection gives $R(-1,-1,\\rho_3,\\rho_4)=\\emptyset$.\n\\end{proof}\n\nOur aim is to use the $|R_i(\\bar\\rho)|$ to estimate the size of $T$.  We\nshould note that $T$ may be a proper superset of\n$\\bigcup_{\\bar\\rho}R(\\bar\\rho)$. The (small) difference arises from the\ncontribution to $T$ from roots of the polynomials $f_i$ (this\ncontribution will be accounted for later, when all roots are included\nas an error term in our bounds).  \\lref{51} \nreduces the number of $|R_i(\\bar\\rho)|$ that we need to estimate to\nonly those $\\bar\\rho$ shown in \\Tref{T:rhosmu}. The final column of that table\nshows the multiplicity $\\mu$ that we need to use for each $|R_i(\\bar\\rho)|$\nin order to obtain $|\\bigcup_{\\bar\\rho}R_i(\\bar\\rho)|$. For example,\n$R_1(1,1,1,-1)$ has $\\mu=4$ because \\lref{51} tells us \nthat $$|R_1(1,1,1,-1)|=|R_1(1,1,-1,1)|=|R_1(1,-1,1,1)|=|R_1(-1,1,1,1)|.$$\n\n\n\\begin{table}[h]\n$\\begin{array}{r|rrrr|rrrr|c|c}\ni & \\rho_1 & \\rho_2 & \\rho_3 & \\rho_4 & s_1 & s_2 & s_3 & s_4 & k_i(\\bar\\rho) & \\mu\\\\\n\\hline\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 &4& 1\\\\\n1 & 1 & 1 & 1 &-1 & 1 & 1 & 1 & 0 &3& 4\\\\\n1 & 1 & -1& 1& -1 & 1 & 0 & 1 & 0 &2& 2\\\\\n1 & 1 & -1&-1&  1 & 1 & 0 & 0 & 1 &2& 2\\\\\n2 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 &2& 1 \\\\\n2 & 1 & 1 & 1 &-1 & 0 & 1 & 0 & 0 &1& 2 \\\\\n2 & 1 & -1& 1& -1 & 0 & 0 & 0 & 0 &0& 1 \\\\\n2 & 1 & -1&-1&  1 & 0 & 0 & 0 & 1 &1& 2 \\\\\n2 & 1 & 1 & -1 &1 & 0 & 1 & 0 & 1 &2& 2 \\\\\n2 & -1 & 1& -1& 1 & 0 & 1 & 0 & 1 &2& 1 \n\\end{array}$\n\\medskip\n\\caption{\\label{T:rhosmu}Values of $s(i,\\bar\\rho)$ and associated parameters.}\n\\end{table}\n\n\n\\begin{lem}\\label{52}\nSuppose that \\eref{25} holds.\n\\begin{enumerate}\n\\item[(i)] If\n$\\chi(x{-}1{-}y)=\\chi(x{-}xy{-}y)$, then $(x,y)\\notin S_{01}^{01}$.\nIf $\\chi\\big((x{-}1{-}y)(x{-}xy{-}y)\\big)=-1$, then there exist unique\n$\\lambda_1,\\lambda_4\\in \\{-1,1\\}$ such that $(x,y)\\in S_{01}^{01}$\n$\\Leftrightarrow$ $\\chi(g_j(x,y))=\\lambda_j$ for $j\\in \\{1,4\\}$.\n\\item[(ii)] If\n$\\chi(y{-}1{-}x)= \\chi(y{-}xy{-}x)$, then $(x,y)\\notin S_{10}^{10}$.\nIf $\\chi\\big((y{-}1{-}x)(y{-}xy{-}x)\\big)=-1$, then there exist unique\n$\\lambda_2,\\lambda_3\\in \\{-1,1\\}$ such that $(x,y)\\in S_{10}^{10}$\n$\\Leftrightarrow$ $\\chi(g_j(x,y))=\\lambda_j$ for $j\\in \\{2,3\\}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nOnly case (i) needs to be proved, because of the $x\\leftrightarrow y$\nsymmetry. If $\\chi(x{-}1{-}y)= \\chi(x{-}xy{-}y)$, then  \n$(x,y) \\notin S_{01}^{01}$ by \\tref{210}.\nIf $\\chi\\big((x{-}1{-}y)(x{-}xy{-}y)\\big)=-1$, then exactly one choice\nof $(\\chi(g_1(x,y)),\\chi(g_4(x,y))$ makes $(x,y)$ an element\nof $S_{01}^{01}$, again by \\tref{210}.\n\\end{proof}\n\n\nConsider $(x,y)\\in R_i(\\bar \\rho)$ for a particular\n$i\\in\\{1,2\\}$ and $\\bar\\rho$. Membership of $R_i(\\bar \\rho)$ implies\nvalues for $\\chi(1{-}x)$, $\\chi(1{-}y)$ and $\\rho_j$ for\n$1\\le j \\le 4$. Also, $(x,y)$ must belong to the sets\n$\\bar S_{11}^{01}$, $\\bar S_{11}^{10}$, $\\bar S_{00}^{10}$\nand $\\bar S_{00}^{01}$, which implies that some of the elements $x{-}1{-}y$,\n$y{-}1{-}x$, $y{-}xy{-}x$ and $x{-}xy{-}y$ have to be nonsquares,\nwhile for the others no such condition is imposed.  Record this into a\nquadruple $s(i,\\bar\\rho)=(s_1,s_2,s_3,s_4)$, where $s_j \\in \\{0,1\\}$\nfor $1\\le j \\le 4$. Here $s_1=1$, $s_2=1$, $s_3=1$ and $s_4=1$ mean\nrespectively that the presence of $(x,y)$ in $R_i(\\bar \\rho)$\nforces $x{-}1{-}y$, $y{-}1{-}x$, $y{-}xy{-}x$ and $x{-}xy{-}y$ to be\nnonsquare. For each $i$ and $\\bar\\rho$,\nthe value of the vector $s(i,\\bar\\rho)$ is given in\n\\Tref{T:rhosmu}. Furthermore, $k_i(\\bar\\rho)$ will be used to denote\nthe number of indices $j$ for which $s_j=1$ in $s(i,\\bar\\rho)$.\n\nAs an example consider $R_2(1,1,1,1)$. In this case $\\chi(f_j(x,y))=\\varepsilon$\nfor all $j\\in\\{1,2,3,4\\}$. Since $\\chi(1{-}x) = \\varepsilon$, \n$(x,y)\\notin S_{11}^{01}$ and $(x,y) \\notin S_{00}^{10}$. Therefore\n$s_1=s_3=0$. Since $\\chi(1{-}y)=-\\varepsilon$ we must have $\\chi(y{-}1{-}x)=-1$ \nif $(x,y)$ is to belong to $\\bar S_{11}^{10}$. Therefore \n$s_2=1$. Similarly, $s_4=1$. \n\nFor $c\\in\\mathbb{F}_q$ define\n$t_1(c)=\\big|\\{x\\in\\mathbb{F}_q:(x,c)\\in T_1\\}\\big|$\nand $t_2(c)=\\big|\\{x\\in\\mathbb{F}_q:(x,c)\\in T_2\\}\\big|$.\nIn the next two propositions we seek estimates of these quantities.\nAs in \\secref{4}, we will assume that \\eref{33} holds\nwhich means that \\eref{25} applies, enabling us to use \\tref{210} and\n\\lref{52}.\n\n\n\n\\begin{prop}\\label{p:t1}\nSuppose that $c$ is a square satisfying conditions\n\\eref{32}--\\eref{311}. Then\n$$|t_1(c)-169\\cdot2^{-14}q|\\leq(\\sqrt{q}+1)1161\/2+21.$$\n\\end{prop}\n\n\\begin{proof}\nFix $c$ satisfying conditions \\eref{32}--\\eref{311} and consider a\ncandidate $(x,c)$ for membership in $T_1$.  As we did in \\pref{44}, we\ninclude the term 21 in our bound and then for the remainder of the\nproof we may assume that $x$ is not a root of any polynomial\nin \\eref{31}.\n\nOur goal is to estimate the $c$-slice of $R_1(\\bar \\rho)$ for each $\\bar\\rho$.\nWe start with a list of polynomials\nthat guarantee the presence of $(x,c)$ in $\\bar S_{00}^{00}$,\n$\\bar S_{11}^{00}$, $\\bar S_{00}^{11}$, $\\bar S_{11}^{01}$,\n$\\bar S_{11}^{10}$, $\\bar S_{00}^{10}$ and $\\bar S_{00}^{01}$.\nThese polynomials are $x$, $1{-}x$, $c{-}x$, $f_j(x,c)$,\n$1\\le j \\le 4$, and those of $x{-}1{-}c$, $c{-}1{-}x$, $c{-}cx{-}x$ \nand $x{-}cx{-}c$\nfor which the corresponding value of $s_j$ in \n$s(i,\\bar \\rho)$ is equal to $1$. In this way we will obtain a \nlist of $7+k_1(\\bar\\rho)$\npolynomials of cumulative degree $11+k_1(\\bar \\rho)$, for \n$k_1(\\bar \\rho)$ as shown in \\Tref{T:rhosmu}. \n\nIt only remains to ensure that $(x,c)$ is in $\\bar S_{01}^{01}$ and\n$\\bar S_{10}^{10}$.  The $c$-slice of $R_1(\\bar \\rho)$ forks into\nseveral disjoint subsets, according to \\lref{52}. The forking induced\nby $\\bar S_{01}^{01}$ depends upon $(s_1,s_4)$, while the forking\ninduced by $\\bar S_{10}^{10}$ depends upon $(s_2,s_3)$. It is thus\npossible to describe only the former and obtain the latter by\nexploiting the $(x,y)\\leftrightarrow(x^{-1},y^{-1})$ symmetry between\n$\\bar S_{01}^{01}$ and $\\bar S_{10}^{10}$.\n\nIf $s_1=s_4= 1$, then there is no forking since this suffices\nto conclude that $(x,y)\\notin S_{01}^{01}$.\n\nIf $s_1+s_4 = 1$, then one of $\\chi(x{-}1{-}c)=-1$ and\n$\\chi(x{-}xc{-}c)=-1$ is mandated, and \nthere are four forks. One of them specifies the\ncharacter of only one extra polynomial to ensure that\n$\\chi(x{-}1{-}c)=\\chi(x{-}xc{-}c)=-1$.\nEach of the other three forks imposes\nrestrictions on \nthree polynomials, as it establishes first that\n$\\chi(x{-}1{-}c)=-\\chi(x{-}xc{-}c)$ and then imposes values on\n$\\chi(g_1(x,c))$ and $\\chi(g_4(x,c))$.  By \\lref{52} there are three\npossibilities to consider for the pair\n$\\big(\\chi(g_1(x,c)),\\chi(g_4(x,c))\\big)$, which thus give us the three\nforks.\n\nThe forking of the case $s_1+s_4 = 1$ will be recorded by $(1,1)\\mid(3,4)^3$.\nThis means that the first fork needs one additional \npolynomial of degree one, while the other three forks\nneed three polynomials of cumulative degree $4$. \n\nIf $s_1=s_4=0$, then there are seven forks. One of them\nimposes that $\\chi\\big((x{-}1{-}c)(x{-}xc{-}c)\\big)=1$\n(which ensures that $\\chi(x{-}1{-}c)=\\chi(x{-}xc{-}c)$), while\neach of the other six establishes first the (different)\nvalues of $\\chi(x{-}1{-}c)$ and $\\chi(x{-}xc{-}c)$, and \nthen the values of $\\chi(g_1(x,c))$ and $\\chi(g_4(x,c))$.\nSymbolically, this gives $(1,2)\\mid(4,5)^6$. \n\nLet us use $\\bullet$ to express composition of two independent\nforkings. Thus \n\\begin{equation*}\n(k_1,d_1)^{m_1}\\mid\\dots \\mid(k_a,d_a)^{m_a}\n\\bullet (k'_1,d'_1)^{m'_1}\\mid \\dots \\mid (k'_b,d'_b)^{m'_b}\n\\end{equation*}\nis a list of alternatives $(k_i+k_j', d_i+d_j')^{m_im'_j}$, where\n$1\\le i \\le a$ and $1\\le j \\le b$.\n\nOur observations above allow us to symbolically describe \npolynomial lists for each of the sets $R_1(\\bar \\rho)$. We have \n\\begin{align*}\nR_1(1,1,1,1)\\colon &(11,15), \\\\\nR_1(1,1,1,-1)\\colon &(10,14)\\bullet (1,1)\\mid(3,4)^3 = (11,15)\\mid(13,18)^3, \\\\\nR_1(1,-1,1,-1)\\colon & (9,13)\\bullet (1,1)\\mid(3,4)^3 \\bullet (1,1)\\mid (3,4)^3 \n= (11,15)\\mid (13,18)^6 \\mid (15,21)^9, \\\\\nR_1(1,-1,-1,1)\\colon & (9,13)\\bullet (1,2)\\mid(4,5)^6 = (10,15)\\mid (13,18)^6. \n\\end{align*} \nCombining this information with the last column of \\Tref{T:rhosmu}, we\nreach a symbolic description of the polynomials contributing to\n$t_1(c)$ that contains \n$(10,15)$ with multiplicity $2$,\n$(11,15)$ with multiplicity $1+4+2=7$,\n$(13,18)$ with multiplicity $4\\cdot 3 + 2\\cdot 6+2\\cdot 6 = 36$, and\n$(15,21)$ with multiplicity $2\\cdot 9=18$. In each case, the list of\npolynomials involved is square-free, by \\tref{31}. Hence, we may apply\n\\tref{t:Weil} to find that\n\\begin{equation*}\n|t_1(c)-\\alpha_1q|\\le(\\sqrt{q}+1)D_1\/2+21,\n\\end{equation*}\nwhere $\\alpha_1 = 2\\cdot 2^{-10}+7\\cdot 2^{-11} +\n36\\cdot 2^{-13} + 18\\cdot 2^{-15}=169\\cdot2^{-14}$, and \nthe cumulative degree of our polynomials is\n$D_1=9\\cdot15+36\\cdot18+18\\cdot21=1161$.\n\\end{proof}\n\n\\begin{prop}\\label{p:t2}\nSuppose that $c$ is a square satisfying conditions\n\\eref{32}--\\eref{311}. Then\n$$|t_2(c)-49\\cdot2^{-11}q|\\leq(\\sqrt{q}+1)4455\/2+21.$$\n\\end{prop}\n\n\\begin{proof}\nThe proof follows the same lines as that of \\pref{p:t1}. The symbolic\ndescription of the forks is\n\\begin{align*}\nR_2(1,1,1,1)\\colon &(9,13)\\bullet (1,1)\\mid(3,4)^3 \\bullet (1,1)\\mid(3,4)^3 \n = (11,15)\\mid (13,18)^6 \\mid (15,21)^9 \\\\\nR_2(1,1,1,-1)\\colon & (8,12)\\bullet (1,1)\\mid(3,4)^3 \\bullet (1,2)\\mid(4,5)^6\n= (10,15)\\mid (12,18)^3\\mid (13,18)^6 \\mid (15,21)^{18} \\\\\nR_2(1,-1,1,-1)\\colon & (7,11)\\bullet (1,2)\\mid(4,5)^6\\bullet (1,2)\\mid(4,5)^6\n= (9,15)\\mid (12,18)^{12} \\mid (15,21)^{36} \\\\\nR_2(1,-1,-1,1)\\colon & (8,12)\\bullet (1,1)\\mid(3,4)^3 \\bullet (1,2)\\mid(4,5)^6\n= (10,15)\\mid (12,18)^3\\mid (13,18)^6 \\mid (15,21)^{18} \\\\\nR_2(1,1,-1,1)\\colon & (9,13) \\bullet (1,1)\\mid(3,4)^3 \\bullet(1,1)\\mid(3,4)^3\n= (11,15)\\mid (13,18)^6 \\mid (15,21)^9 \\\\\nR_2(-1,1,-1,1)\\colon & (9,13) \\bullet (1,1)\\mid(3,4)^3 \\bullet(1,1)\\mid(3,4)^3\n= (11,15)\\mid (13,18)^6 \\mid (15,21)^9 \n\\end{align*} \nCombining this information with the last column of \\Tref{T:rhosmu}, we\nreach a symbolic description of the polynomials contributing to\n$t_2(c)$ that contains \n$(9,15)$ with multiplicity $1$,\n$(10,15)$ with multiplicity $2\\cdot1+2\\cdot1=4$,\n$(11,15)$ with multiplicity $1+2\\cdot1+1=4$,\n$(12,18)$ with multiplicity $2\\cdot 3 + 12 + 2\\cdot 3 = 24$, and\n$(13,18)$ with multiplicity $6+2\\cdot 6+2\\cdot 6+2\\cdot 6+6 = 48$, and\n$(15,21)$ with multiplicity $9+2\\cdot18+36+2\\cdot18+2\\cdot9+9=144$.\nCombining \\tref{31} and \\tref{t:Weil}, we find that\n\\begin{equation*}\n|t_2(c)-\\alpha_2q|\\le(\\sqrt{q}+1)D_2\/2+21,\n\\end{equation*}\nwhere $\\alpha_2 = 2\\cdot 2^{-9}+4\\cdot 2^{-10}+4\\cdot 2^{-11}\n+24\\cdot 2^{-12}+48\\cdot 2^{-13} + 144\\cdot 2^{-15}=49\\cdot2^{-11}$, and \nthe cumulative degree of our polynomials is\n$D_2=9\\cdot15+72\\cdot18+144\\cdot21=4455$.\n\\end{proof}\n\nWe are now ready to prove the main result for this section.\n\n\\begin{thm}\\label{53}\nIf $q\\equiv1\\bmod4$, then\n\\[\n\\big|\\sigma(q)-953\\cdot 2^{-15}\\,q^2\\big|<2518q^{3\/2}+2623q.\n\\]         \n\\end{thm}\n\n\\begin{proof}\nThere are $(q-3)\/2$ choices for a square $c\\in\\mathbb{F}_q$ satisfying\n$c\\notin\\{0,1\\}$. At most\n$49$ of these choices do not fulfil conditions \\eref{32}--\\eref{311}\nof \\tref{31}. Each $c$ that fails one of the conditions\n\\eref{32}--\\eref{311} contributes between 0 and $(q-3)\/2$ elements $(x,c)$\nto $T$. Putting these observations together with \\pref{p:t1} and\n\\pref{p:t2} we have that\n\\begin{align*}\n\\big||T_1|-169\\cdot2^{-14}q(q-3)\/2\\big|\n&\\le1161(\\sqrt{q}+1)(q-3)\/4+21(q-3)\/2+49(q-3)\/2,\\\\\n\\big||T_2|-49\\cdot2^{-11}q(q-3)\/2\\big|\n&\\le4455(\\sqrt{q}+1)(q-3)\/4+21(q-3)\/2+49(q-3)\/2.\n\\end{align*}\nNext, by \\lref{51} we know that $|T|=|T_1|+2|T_2|$, so\n\\begin{align*}\n\\big||T|-(169\\cdot2^{-15}+49\\cdot2^{-11})q(q-3)\\big|\n&\\le(q-3)\\Big[(1161\/4+ 4455\/2)(\\sqrt{q}+1)+105\\Big]\\\\\n&<q\\Big[2518\\sqrt{q}+10491\/4\\Big].\n\\end{align*}\nThe result then follows from simple rearrangement.\n\\end{proof}\n\n\\begin{cor}\\label{54}\nLet $q$ run through all prime powers that are $1 \\bmod 4$.\nThen $\\lim \\sigma(q)\/q^2 = 953\/2^{15}$.\n\\end{cor}\n\n\\section{Conclusions}\\label{6}\nTheorems~\\ref{46} and~\\ref{53} give formulas that can be used\nas estimates of $\\sigma(q)$ for large $q$. We did not work hard to\noptimise the constants in the bounds. Even if we had,\nthe number of applications of the Weil bound is too big to allow\nthe estimates to be useful for small $q$.\n\nThe proof of existence of maximally nonassociative quasigroups\nfor small orders was obtained in \\cite{dw} by\nconsidering $(a,b)\\in \\Sigma$ that satisfy some additional condition\n$t(a,b)=0$. It might be of interest to investigate, along these lines,\nall cases $t(x,y) = 0$ when $t$ is one of the polynomials $x{-}1{-}y$,\n$x{-}xy{-}y$, $y{-}1{-}x$, $y{-}xy{-}x$, $f_j(x,y)$ and $g_j(x,y)$ for\n$1\\le j \\le 4$.\n\n\\subsection*{Acknowledgement}\nThis work was supported in part by Australian Research Council grant\nDP150100506.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIf $M$ is a homogeneous space with full isometry group $G$, then any subgroup $K$ of $G$ naturally acts on $M$.  In some instances, this action is effectively free, and hence, the quotient $M\/K$ is a smooth manifold, called a biquotient.  Alternatively, given a compact Lie group $G$ and a homomorphism $f = (f_1,f_2):H\\to G\\times G$, there is an induced action of $H$ on $G$ given by $h\\ast g = f_1(h)g f_2(h)^{-1}$.  When this action is effectively free, the orbit space, denoted $G\\bq H$, naturally has the structure of a smooth manifold and is called a biquotient.\n\nIf $G$ is endowed with its bi-invariant metric, then the $H$ action on $G$ is by isometries, and hence induces a metric on the quotient.  By O'Niell's formulas \\cite{On1}, this implies that all biquotients carry a metric of non-negative sectional curvature.  Biquotients were introduced by Gromoll and Meyer \\cite{GrMe1} when they showed that for a particular embedding of $Sp(1)$ into $Sp(2)\\times Sp(2)$, the biquotient $Sp(2)\\bq Sp(1)$ is diffeomorphic to an exotic sphere, providing the first example of an exotic sphere with non-negative sectional curvature.  Further, until the recent example due to Grove, Verdiani, and Ziller \\cite{GVZ} and Dearicott \\cite{De}, all known examples of compact manifolds with positive sectional curvature were diffeomorphic to biquotients.  See \\cite{Ber}, \\cite{AW}, \\cite{Wa}, \\cite{Es1}, \\cite{Ba1}, \\cite{PW1}.  Furthermore, all known examples of manifolds with almost or quasi-positive curvature are diffeomorphic to biquotients.  See \\cite{Wi}, \\cite{PW2}, \\cite{EK}, \\cite{Ke1}, \\cite{Ke2}, \\cite{KT}, \\cite{Ta1}, and \\cite{D1}.\n\nRecently, biquotients have been used in the classification of non-negatively curved manifolds of small dimension with a large symmetry group.  See \\cite{Sim1}, \\cite{GGK}, and \\cite{GGS} for examples already using the $5$-dimensional classification appearing in this paper.\n\nBecause each description of a manifold as a biquotient gives rise to a different family of non-negatively curved metrics, it seems desirable to not only have a classification of manifolds diffeomorphic to a biquotient, but also to classify which groups give rise to a given manifold.  Totaro \\cite{To1} has shown that if $M\\cong G\\bq H$ is a compact, simply connected biquotient, then $M$ is also diffeomorphic to $G'\\bq H'$ where $G'$ is simply connected, $H'$ is connected, and no simple factor of $H'$ acts transitively on any simple factor of $G'$.  We call such biquotients reduced, and will classify only the reduced ones.\n\nIn dimension 2 and 3, it is easy to see that only $S^2$ and $S^3$ arise and that their only description as reduced biquotients is as $S^2\\cong SU(2)\/S^1$ and $S^3 \\cong SU(2)\/\\{e\\}$.\n\nThe goal of this paper is to prove the analogous results for compact simply connected 4 and 5 dimensional biquotient.  A follow up paper will contain the classification results for $6$ and $7$ dimensional simply connected biquotients.  We have\n\n\\begin{theorem}\\label{ThmA}Suppose $M^4\\cong G\\bq H$ is a reduced compact simply connected biquotient and that the $H$ action on $G$ is not homogeneous.  Then $G$, $H$, and the image of $H$ in $G\\times G$ appear in the following table.\n\n\\begin{center}\n\n\n\\begin{tabular}{|c|c|c|c|}\n\n\\hline\n\n$M$ & $G$ & $H$ & $H\\rightarrow G\\times G$\\\\\n\n\\hline\n\n$S^4$ & $Sp(2)$ & $Sp(1)^2$ & $Sp(1)\\times \\Delta Sp(1)$\\\\\n\n$S^4$ & $SU(4)$ & $SU(3)\\times SU(2)$ & $SU(3)\\times \\Delta SU(2)$ \\\\\n\n$S^4$ & $Spin(8)$ & $Spin(7)\\times SU(2)$ & $Spin(7)\\times \\Delta SU(2)$ \\\\\n\n$S^4$ & $Spin(8)$ & $Spin(7)\\times SU(2)$ & $\\overline{Spin(7)}\\times \\overline{SU(2)}$ \\\\\n\n$S^4$ & $Spin(7)$ & $G_2\\times SU(2)$ & $G_2\\times \\overline{SU(2)}$\\\\\n\n\\hline\n\n$\\mathbb{C}P^2$ & $SU(3)$ & $SU(2)\\times S^1$ & $\\diag(zA,\\overline{z}^2)\\times \\diag(z^4, z^4, \\overline{z}^8)$ \\\\\n\n\n$\\mathbb{C}P^2$ & $SU(4)$ & $Sp(2) \\times S^1$ & $Sp(2)\\times \\diag(z,z,z,\\overline{z}^3)$\\\\\n\n\\hline\n\n$S^2\\times S^2$ & $Sp(1)^2$ & $T^2$ &  $(z^2, wz^n)\\times(1,z^n)$ $n$ even \\\\\n\n$\\mathbb{C}P^2\\sharp -\\mathbb{C}P^2$ & $Sp(1)^2$ & $T^2$ & $(z^2, wz^n)\\times (1,z^n)$ $n$ odd \\\\\n\n$\\mathbb{C}P^2\\sharp \\mathbb{C}P^2$ & $Sp(1)^2$ & $T^2$ & $(zw,zw^2)\\times(w,z)$\\\\\n\n\\hline\n\n\\end{tabular}\n\n\\end{center}\n\n\\end{theorem}\n\nHere, $\\overline{Spin(7)}$ denotes the image of the spin representation.  The subgroup $\\overline{SU(2)}$, which is isomorphic to $SU(2)$, is the inverse image of the standard block embedding of $SO(3)$ into either $SO(7)$ or $SO(8)$ under the natural projection from $Spin(7)$ or $Spin(8)$.  This notation $-\\mathbb{C}P^2$ indicates $\\mathbb{C}P^2$ with the opposite orientation and $M$ is given up to diffeomorphism.\n\nRational homotopy theory implies that if $M^4$ is diffeomorphic to a biquotient, then it must be homeomorphic to one of the $5$ spaces listed above.  It follows from our results that $M$ is, in fact, diffeomorphic to one of those $5$ spaces.  We remark that Totaro \\cite{To1} has already shown all $5$ diffeomorphism types can be written as biquotients, where only the cases of $\\mathbb{C}P^2\\pm \\sharp \\mathbb{C}P^2$ were not previously known.\n\nIn dimension $5$, we prove\n\n\\begin{theorem}\\label{ThmB}\n\n\nSuppose $M^5\\cong G\\bq H$ is a reduced compact simply connected biquotient and that the $H$ action on $G$ is not homogeneous.  Then $G$, $H$, and the image of $H$ in $G\\times G$ appear in the following table.\n\n\\begin{center}\n\n\n\\begin{tabular}{|c|c|c|c|}\n\n\\hline\n\n$M$ & $G$ & $H$ & $H\\rightarrow  G\\times G$\\\\\n\n\\hline\n\n$S^3\\times S^2$ & $Sp(1)^2$ & $S^1$ & $\\diag(z^a, z^b)\\times \\diag(z^c, z^d)_1$ \\\\\n\n$S^3\\,\\hat{\\times}\\,S^2$ & $Sp(1)^2$ & $S^1$ & $\\diag(z^a,z^b)\\times \\diag(z^c,z^d)_2$ \\\\\n\n\\hline\n\n\n\n\\end{tabular}\n\n\\end{center}\n\n\n\\end{theorem}\n\n\nIn both cases, we have $\\gcd(a,b,c,d) = 1$.  The subscript $1$ indicates $\\gcd(a^2-c^2, b^2-d^2)= 1$ while the subscript $2$ indicates $\\gcd(a^2-c^2, b^2-d^2) = 4$.  The notation $S^3\\,\\hat{\\times} \\, S^2$ indicates the unique nontrivial $S^3$ bundle over $S^2$.    The manifold $M$ is given up to diffeomorphism.\n\nIn \\cite{Pa1}, Pavlov gives a classification of $5$-dimensional biquotients up to diffeomorphism, but he does not classify all ways of expressing a given manifold as a biquotient.  We also point out a slight error Pavlov's formula for showing the manifold $S^3\\,\\hat{\\times}\\, S^2$ is a biquotient.\n\n\nThe outline of the paper is as follows.  In section 2, we will first cover some basic facts about biquotients.  Then, using previous work and some simple rational homotopy theory, we reduce Theorems A and B to classifying the effectively free actions and diffeomorphism types of biquotients of the form $(SU(2)\\times SU(2))\\bq T^2$ and of the form $(SU(2)\\times SU(2))\\bq S^1$.  Section 3 carries this out in the case of $(SU(2)\\times SU(2))\\bq T^2$ and section 4 carries this out in the case $(SU(2)\\times SU(2))\\bq S^1$.\n\nThis paper is a portion of the author's Ph.D. thesis and he is greatly indebted to Wolfgang Ziller for helpful discussions and guidance.\n\n\n\n\n\\section{Preliminaries and Reductions}\n\nA homomorphism $f = (f_1, f_2):H\\rightarrow G\\times G$, which we will always assume has finite kernel, defines an action of $H$ on $G$ by $h\\ast g = f_1(h)g f_2(h)^{-1}$.\n\nAn action is called effectively free if whenever any $h\\in H$ fixes any point of $G$ then it fixes all points of $G$.  It is called free if the only element which fixes any point is the identity.  One easily sees that a biquotient action of $H$ on $G$ is effectively free iff whenever $f_1(h)$ is conjugate to $f_2(h)$ in $G$, then $f_1(h) = f_2(h)\\in Z(G)$.  Likewise, a biquotient action of $H$ on $G$ is free iff whenever $f_1(h)$ is conjugate to $f_2(h)$ in $G$, then $f_1(h) = f_2(h) = e\\in G$.\n\n\nIn order to reduce the scope of the classification, we will use the following fact:\n\n\\begin{proposition}\\label{diffeostabilize} Suppose $f:H\\rightarrow G\\times G$ induces an effectively free action.  Then, after any of the following modifications of $f$, the new induced action is effectively free and the quotients are naturally diffeomorphic.\n\n1.  For any automorphism $f'$ of $H$, change $f$ to $f\\circ f'$\n\n2.  For any element $g=(g_1,g_2)\\in G\\times G$, change $f$ to $C_g\\circ f$, where $C_g$ denotes conjugation.\n\n3.  For any automorphism $f'$ of $G$, change $f$ to $(f',f')\\circ f$.\n\n\\end{proposition}\n\n\\begin{remark}One may think that the $(f',f')$ in $3.$ can be replaced by $(f'_1,f'_2)$ for any pair of automorphisms of $G$.  However, this is not the case.  For example, if $G = Sp(1)\\times Sp(1)$ and $H = S^1$ with the embedding $z\\rightarrow \\big( (z,1), (1,z)\\big)$, then the induced action is free.  On the other hand, if $f'_1 = \\Id$ while $f'_2$ interchanges the two $S^3$ factors, then the action induced by $(f'_1,f'_2)\\circ f$ is not even effectively free - every element of $S^1$ fixes infinitely many points.\n\n\\end{remark}\n\nWe will only classify biquotients and the corresponding actions up to these three modifications.\n\n\nFurther, Totaro \\cite{To1} has proven that if $M$ is compact, simply connected, and diffeomorphic to a biquotient, then $M$ is diffeomorphic to a biquotient $G\\bq H$ where $G$ is compact, simply connected, and semisimple, $H$ is connected, and no simple factor of $H$ acts transitively on any simple factor of $G$.  We will henceforth assume all biquotients to be in this form.\n\n\\bigskip\n\nOne of the main tools involved in the classification of biquotients is rational homotopy theory.  A manifold $M$ is said to be rationally elliptic if $\\dim \\pi_\\ast(M)\\otimes\\mathbb{Q} < \\infty$.  All Lie group are known to be rationally elliptic with all even rational homotopy groups trivial.  Further, given any fiber bundle $F\\rightarrow E\\rightarrow B$, if two of the spaces are rationally elliptic, so is the third by the long exact sequence in rational homotopy groups.  Since any biquotient $G\\bq H$ with ineffective kernel $H'$ gives rise to a principal $H\/H'$-bundle $H\/H'\\rightarrow G\\rightarrow G\\bq H$, it follows that all biquotients are rationally elliptic.\n\nIt turns out, the topology of a simply connected rationally elliptic manifold is very constrained.  In \\cite{Pa2}, Pavlov proves\n\n\\begin{proposition}(Pavlov)  Suppose $M$ is a compact simply connected rationally elliptic manifold.  If $M$ is $4$-dimensional, $M$ has the same rational homotopy type as either $S^4$, $\\mathbb{C}P^2$, $S^2\\times S^2$, or $\\mathbb{C}P^2\\#\\mathbb{C}P^2$.  If $M$ is $5$-dimensional, $M$ has the same rational homotopy type as $S^5$ or $S^3\\times S^2$.\n\n\\end{proposition}\n\nSince the cohomology rings with coefficients in $\\mathbb{Q}$ of $S^4$, $\\mathbb{C}P^2$, and $S^5$ are all generated by a single element, the same follows for any $M$ which has the same rational homotopy type.  In particular, we can use the classification of Kapovitch and Ziller \\cite{KZ} and independently Totaro \\cite{To1} which shows that, in fact, $M$ must be diffeomorphic to one of $S^4$, $\\mathbb{C}P^2$, $S^5$, or $SU(3)\/SO(3)$.  Further, if such an $M$ is diffeomorphic to a reduced biquotient $G\\bq H$, then $G$ must be simple.\n\nIn Eschenburg's Habilitation \\cite{Es2}, summarized in \\cite{Zi}, he classifies all maximal biquotient actions of maximal rank on simple groups.  It follows from his classification that our lists in Theorem A and B is complete in the case that $M$ has the rational homotopy type of a compact rank one symmetric space.  Hence, in order to prove Theorem A and Theorem B, it remains to check the cases where $M$ has the rational homotopy type of $S^2\\times S^2$, $\\mathbb{C}P^2\\#\\mathbb{C}P^2$ or $S^3\\times S^2$.\n\n\\begin{proposition} Suppose $M$ is compact and simply connected and that it has the rational homotopy type of $S^2\\times S^2$, $\\mathbb{C}P^2\\#\\mathbb{C}P^2$, or $S^3\\times S^2$.  If $M\\cong G\\bq H$ is a reduced biquotient, then $G = SU(2)\\times SU(2)$ and $H = S^1$ or $S^1\\times S^1$.\n\n\\end{proposition}\n\n\\begin{proof}\n\nSuppose $M\\cong G\\bq H$ has the rational homotopy type of $S^2\\times S^2$ or $\\mathbb{C}P^2\\#\\mathbb{C}P^2$.  From the long exact sequence or rational homotopy groups associated the fibration $$H\\rightarrow G\\rightarrow M$$ we see that $\\pi_1(H)\\otimes\\mathbb{Q}\\cong \\pi_2(M)\\otimes\\mathbb{Q}\\cong\\mathbb{Q}^2$.  It follows that $H$, up to cover, must split as $H = H'\\times S^1\\times S^1$.  Considering the circle bundle $$S^1\\rightarrow G\\bq (H'\\times S^1\\times\\{e\\}) \\rightarrow G\\bq (H'\\times S^1\\times S^1)$$ and the associated long exact sequence of rational homotopy groups, we see $G\\bq (H'\\times S^1)$ is a reduced compact simply connected $5$-dimensional manifold with the same rational homotopy type as $S^3\\times S^2$.  But Pavlov \\cite{Pa1} shows that this implies $G = SU(2)\\times SU(2)$ while $H' = \\{e\\}$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{\\texorpdfstring{$(SU(2)\\times SU(2))\\bq T^2$}{SU(2) x SU(2) \/\/ T2}}\n\nFirst, we will classify all effectively free actions of $T^2$ on $SU(2)\\times SU(2)$.  Note that by Proposition \\ref{diffeostabilize} we may assume that the image $f(T^2)$ lies in the maximal torus of $(SU(2)\\times SU(2))^2$.  Working directly with homomorphisms from $T^2$ into $(SU(2)\\times SU(2))^2$ is cumbersome, so instead, we will adopt a more geometric description of the action.  For concreteness, we take $S^1 = \\{z\\in\\mathbb{C}: |z|=1\\}$ and $S^3 = \\{(p,q)\\in \\mathbb{C}^2: |p|^2 + |q|^2 = 1\\}$.\n\nIt is easy to see that the action of $T^2$ on $SU(2)$ given by $$ (z,w)\\ast U = \\diag \\left(z^{A} w^{B}, \\overline{z}^A \\overline{w}^B\\right)U \\diag\\left(z^C w^D,  \\overline{z}^C \\overline{w}^D\\right)^{-1}$$ is equivalent to the action of $T^2$ on $S^3$ given by $$(u,v)\\ast(p,q) = (z^{A-B}w^{C-D} p, z^{A+B}w^{C+D}q)$$ under the isomorphism $SU(2)\\rightarrow S^3$ mapping $\\begin{bmatrix} a & b\\\\ -\\overline{b} & \\overline{a}\\end{bmatrix}$ to $(a,b)$.\n\nConversely, the action of $T^2$ on $S^3$ given by $$(z,w)\\ast (p,q) = (z^{2a} w^{2b} p, z^{2c} w^{2d} q) $$ is equivalent to one on $SU(2)$  with $A = a+c$, $B = b+d$, $C = a-c$, and $D = b-d$.  It follows that any linear action of $T^2$ on $S^3\\times S^3$ is orbit equivalent to a biquotient action of $T^2$ on $SU(2)\\times SU(2)$.\n\nOur first step towards classifying the effectively free actions is to simplify their description.\n\n\\begin{proposition}Consider the action $$(z,w)\\ast \\big((p_1,q_1), (p_2,q_2)\\big) = \\big( (z^a w^b p_1, z^c w^d q_1), (z^e w^f p_2, z^g w^h q_2)   \\big)$$ with $\\gcd(a,c,e,g) = \\gcd(b,d,f,h) = 1$.  Assume the action is effectively free.  Then there is a change of coordinates on $T^2$ for which the action has the form $$(z,w)\\ast \\big( (p_1,q_1), (p_2,q_2)\\big) = \\big( (z p_1, z^\\alpha w^\\beta q_1), (w p_2, z^\\gamma w^\\delta q_2) \\big).$$\n\n\\end{proposition}\n\n\\begin{proof} Set $D = \\det\\begin{bmatrix} a&e\\\\b&f\\end{bmatrix}$.  If $D=0$ then there is nontrivial integral solution $(n_1,n_2)$ to the simultaneous equations $$\\left\\{\\begin{aligned} an_1 &+ en_2  &= 0 \\\\ b n_1 &+ f n_2 &= 0\\end{aligned}\\right.$$  Then points of the form $(z^{n_1}, z^{n_2})$ all fix $\\big( (1,0), (1,0)\\big)$, so either the ineffective kernel is infinite or the action is not free.  Hence, $D\\neq 0$.\n\nIn order to find new coordinates, we interpret $(a,c,e,g)$ and $(b,d,f,h)$ as vectors in $\\mathbb{R}^4$ and consider their span.  Using the fact that $D\\neq 0$, it is easy to see there is another basis of the form $(\\nu,\\alpha,0,\\gamma)$ and $(0,\\beta,\\kappa,\\delta)$ consisting solely of integers.  In the new basis $(z,w)$, the action now looks like $$(z,w)\\ast \\big( (p_1,q_1), (p_2,q_2)\\big) = \\big( (z^{\\nu} p_1, z^\\alpha w^\\beta q_1), (w^{\\kappa} p_2, z^\\gamma w^\\delta q_2) \\big)$$ and we may assume without loss of generality that $$\\gcd(\\nu,\\alpha, \\gamma) = \\gcd(\\kappa,\\beta,\\delta) = 1.$$\n\nIf $z$ is a $\\nu$th root of unity, then $(z,1)$ fixes $\\big((1,0), (1,0)\\big)$.  Since the action is effectively free, this implies $(z,1)$ fixes every point which in turn implies that $z$ is an $\\alpha$th and $\\gamma$th root of unity.  So, every $\\nu$th root of unity is a $\\gcd(\\nu,\\alpha,\\gamma) = 1$st root of unity, so $\\nu=1$.  An analogous argument shows $\\kappa = 1$ as well.\n\n\\end{proof}\n\nUsing Proposition \\ref{diffeostabilize} we can replace $z$, $w$, $p_i$ or $q_i$ by their complex conjugates, and hence we can assume without loss of generality that $\\alpha, \\beta$, and $\\delta$ are all bigger than or equal to $0$. Also, with this description, it is clear that an action is effectively free iff it is free.  Further, the argument in first paragraph of the previous proposition implies $\\alpha\\delta -\\beta\\gamma \\neq 0$.\n\n\\begin{proposition}\\label{4free}  The action $$(z,w)\\ast \\big( (p_1,q_1), (p_2,q_2)\\big) = \\big( (z p_1, z^\\alpha w^\\beta q_1), (wp_2, z^\\gamma w^\\delta q_2) \\big)$$ is free iff $\\alpha = \\delta = 1$ and $|1-\\beta \\gamma| = 1$.\n\n\\end{proposition}\n\n\\begin{proof}\n\nThe key observation is that if an element $(z,w)$ fixes any point in $S^3\\times S^3$, this it must fix a point of the form $\\big( (1,0), (1,0)\\big)$, $\\big( (1,0), (0,1)\\big)$, $\\big( (0,1),(1,0)\\big)$, or $\\big( (0,1), (0,1)\\big)$.  So, to guarantee the action is free, it is enough to show every $(z,w)$ moves each of these $4$ points.  Note that the point $\\big( (1,0), (1,0)\\big)$ is automatically moved by all nontrivial $(z,w)$.\n\nThe point $\\big( (1,0), (0,1) \\big)$ is fixed by $(z,w)$ iff $w$ is a $\\delta$th root of unity.  So the action always moves this point iff $\\delta = 1$.  An analogous argument shows the point $\\big( (0,1),(1,0)\\big)$ is always moved iff $\\alpha = 1$.\n\nFinally, the element $(z,w)$ fixes the point $\\big( (0,1), (0,1)\\big)$ iff $zw^\\beta = 1$ and $z^\\gamma w = 1$.  Raising the second equation to the $\\beta$th power, we see that any solution of this must have $z$ equal to a $(1-\\beta\\gamma)$th root of unity.  Raising the first equation to the $\\gamma$th power shows the same of $w$.  Then, it is easy to see that the solutions are precisely the pairs of the form $(\\overline{w}^\\beta, w)$ where $w$ is any $(1-\\beta\\gamma)$th root of unity.  Thus, in order to guarantee that we have a free action, we must have $|1-\\beta\\gamma|=1$.\n\n\\end{proof}\n\nFrom here, it is obvious that, up to interchanging $z$ and $w$, either $\\gamma = 0$ and $\\beta$ is arbitrary or $\\beta = 1$ and $\\gamma = 2$.  It remains to compute the diffeomorphism type of the quotients.\n\n\\begin{proposition} The action with $\\gamma = 0$ and $\\beta$ arbitrary has quotient $S^2\\times S^2 $ iff $\\beta$ is even and quotient $\\mathbb{C}P^2\\sharp -\\mathbb{C}P^2$ iff $\\beta$ is odd.  For the exceptional action, the quotient is diffeomorphic to $\\mathbb{C}P^2\\sharp \\mathbb{C}P^2$.\n\n\\end{proposition}\n\n\\begin{proof}\nIn \\cite{To1}, Totaro shows that the exceptional action has quotient diffeomorphic to $\\mathbb{C}P^2\\sharp\\mathbb{C}P^2$, so we only prove the theorem in the case where $\\gamma = 0$.\n\nSince $\\gamma = 0$, we see that the first factor $S^1$ of $T^2$ acts only on the first factor $S^3$ via the Hopf action with quotient $S^2$.  So, $(S^3\\times S^3)\\bq (S^1\\times \\{e\\}) \\cong S^2\\times S^3$.  The projection onto the second factor $(S^2\\times S^3)\/S^1 \\rightarrow S^3\/S^1\\cong S^2$ gives $(S^3\\times S^3)\\bq T^2$ the structure of an $S^2$ bundle over $S^2$.  By a standard clutching function argument, there are precisely two such bundles up to isomorphism, one of which is $S^2\\times S^2$ and the other of which is $\\mathbb{C}P^2\\sharp -\\mathbb{C}P^2$.\n\nGeometrically, the $\\{e\\}\\times S^1$ action on $S^2\\times S^3$ is by rotating the $S^2$ factor $|\\gamma|$ times while acting via the Hopf action on the $S^3$ factor.  Since rotating $S^2$ $|\\gamma|$ times is homotopically trivial (in $SO(3)$) iff $|\\gamma|$ is even, the quotient is $S^2\\times S^2$ iff $|\\gamma|$ is even.\n\n\\end{proof}\n\nThis completes the proof of Theorem \\ref{ThmA}.\n\n\n\n\n\\section{\\texorpdfstring{$(SU(2)\\times SU(2))\\bq S^1$}{SU(2) x SU(2) \/\/ S1}}\n\nWe begin by classifying the free actions.  Up to conjugation, a general biquotient action of $S^1$ on $SU(2)\\times SU(2)$ has the form $$z\\ast(A,B) = \\left(\\begin{bmatrix} z^a & \\\\ & \\overline{z}^a\\end{bmatrix} A \\begin{bmatrix} z^c & \\\\ & \\overline{z}^c\\end{bmatrix}^{-1}, \\begin{bmatrix} z^b & \\\\ & \\overline{z}^b\\end{bmatrix}B \\begin{bmatrix} z^d & \\\\ & \\overline{z}^d\\end{bmatrix}^{-1}  \\right)$$ where we may assume without loss of generality that $\\gcd(a,b,c,d)=1$.\n\n\\begin{proposition}\\label{5free} Such an action is effectively free iff for any choice of signs, we have $\\gcd(a\\pm c, b\\pm d) = 1$ or $2$.\n\n\n\\end{proposition}\n\n\\begin{remark}\nSince the parities of $a+c$ and $a-c$ are the same, an effectively free biquotient action either has all four $\\gcd$s equal to $1$ or all four equal to $2$.  It is easy to see that all $4$ $\\gcd$s are $1$ iff $\\gcd(a^2-c^2, b^2-d^2) = 1$ and all $4$ $\\gcd$s are $2$ iff $\\gcd(a^2-c^2, b^2-d^2) = 4$.\n\\end{remark}\n\n\\begin{proof}Since two matrices are conjugate in $SU(2)$ iff they have the same eigenvalues, we see that $\\begin{bmatrix} z^a & \\\\ & \\overline{z}^a\\end{bmatrix}$ and $\\begin{bmatrix} z^c & \\\\ & \\overline{z}^c\\end{bmatrix}$ are conjugate iff $z^a = z^c$ or $z^a = \\overline{z}^c$.  So, we see that a given element $z\\in S^1$ fixes a point in $SU(2)\\times SU(2)$ iff $z^{a\\pm c} = 1 = z^{b\\pm d}$ for some choice of signs.  In order to the action to be effectively free, a necessary and sufficient condition is that whenever $z^{a\\pm c} = 1 = z^{b\\pm d}$, then $$z^a = z^c = \\pm 1 \\in Z(SU(2))$$ and likewise $$z^b=z^d = \\pm 1\\in Z(SU(2)).$$  Equivalently, the action is effectively free iff $$\\gcd(a\\pm c, b\\pm d) | \\gcd(2a,2b,2c,2d) = 2.$$\n\n\n\\end{proof}\n\nWe will call a $4$-tuple of integers $(a,b,c,d)$ admissible if $$ \\gcd(a^2-c^2, b^2-d^2) = 1 \\text{ or } 4.$$  Pavlov \\cite{Pa1} has shown that, given any biquotient $(SU(2)\\times SU(2))\\bq S^1$, the quotient is diffeomorphic to either $S^3\\times S^2$ or $S^3\\,\\hat{\\times}\\,S^2$.  His main tool is the Barden-Smale classification of $5$-dimensional manifolds, proven by Smale \\cite{Sm1} in the spin case and Barden \\cite{Ba1} in the non-spin case.  Their result implies that a compact simply connected $5$-dimensional manifold with $H_2\\cong \\mathbb{Z}$ is diffeomorphic to either $S^3\\times S^2$ or $S^3\\, \\hat{\\times}\\, S^2$ and it is easy to see that any biquotient of the form $(SU(2)\\times SU(2))\\bq S^1$ has $H_2 \\cong \\mathbb{Z}$.\n\nWhile both $S^3\\times S^2$ and $S^3\\,\\hat{\\times}\\, S^2$ have the same cohomology ring, the second Stiefel-Whitney class distinguishes them.  This class is trivial for $S^3\\times S^2$ and nontrivial for $S^3\\,\\hat{\\times}\\, S^2$.\n\nWe will follow a method of Singhof \\cite{Si1} for computing the characteristic classes of these biquotients.  Because we are interested in the Stiefel-Whitney classes, all of the following homology and cohomology groups are assumed to have $\\zmod{2}$ coefficients.  In order to apply this method, we first recall some background.  To begin with, given any compact Lie group $G$, we will let $EG$ denote a contractible space on which $G$ acts freely and $BG = EG\/G$ will be the classifying space of $G$.  If the $H$ biquotient action on $G$ is free, the projection $\\pi:G\\rightarrow G\\bq H$ is an $H$-principal bundle, hence is classified by a map $\\phi_H:G\\bq H\\rightarrow BH$.\n\nEschenburg \\cite{Es3} has shown\n\n\\begin{proposition}\\label{commute} Suppose $\\phi:H\\rightarrow G\\times G$ induces a free action of $H$ on $G$ and consider the fibration $\\sigma:G\\rightarrow B\\Delta G\\rightarrow BG\\times BG$ induced by the natural inclusion $\\Delta G\\rightarrow G\\times G$.  There is a map $\\phi_G:G\\bq H\\rightarrow B\\Delta G$ so that the following is, up to homotopy, a pullback of fibrations.\n\n\\begin{diagram}\nG &  & G \\\\\n\\dTo & & \\dTo \\\\\nG \\bq H  & \\rTo^{\\phi_G} & B{\\Delta G} \\\\\n\\dTo^{\\phi_H} && \\dTo^{B\\sigma} \\\\\nBH & \\rTo^{Bf} & BG\\times BG\\\\\n\\end{diagram}\n\n\n\\end{proposition}\n\nIn order to compute the Stiefel-Whitney classes of a biquotient, we use the notions of $2$-groups and $2$-roots of a Lie group.  A $2$-group of a compact Lie group $G$ is any subgroup isomorphic to $(\\zmod{2})^n$ for some $n$.  The $2$-roots of $G$ are defined analogously to the roots:  the adjoint representation of $G$, when restricted to a maximal torus $T_G$, breaks into root spaces.  Likewise when the adjoint representation of $G$ is restricted to a maximal $2$-group $Q_G$ it breaks into $2$-root spaces.  We can view each $2$-root as a map $Q_G\\rightarrow \\mathbb{Z}\/2\\mathbb{Z}$ which induces, via the fibration $Q_G\\rightarrow EQ_G\\rightarrow BQ_G$, a map $H_1(BQ_G)\\rightarrow \\mathbb{Z}\/2\\mathbb{Z}$, that is, as an element of $H^1(BQ_G)$.  More generally, given a basis for $Q_G$ as a $\\zmod{2}$-vector space, the dual basis can be canonically identified as generators of $H^1(BQ_G)$.  Using this identification, Singhof has shown \\cite{Si1}\n\n\\begin{theorem}(Singhof)\\label{charclass}\n\nSuppose $H\\subseteq G\\times G$ defines a free biquotient action, then the total Stiefel-Whitney class of the tangent bundle of $G\\bq H$ is given as $$w(G\\bq H) = \\phi_G^\\ast\\big(\\Pi_{\\lambda \\in \\Delta^2 G}(1+\\lambda)\\big)\\phi_H^\\ast\\big(\\Pi_{\\rho\\in\\Delta^2 H}(1+\\rho)\\big)^{-1}$$ where $\\Delta^2 G$ denotes the $2$-roots of $G$ and where $\\phi_G^\\ast$ and $\\phi_H^\\ast$ are the maps induced on cohomology with $\\zmod{2}$ coefficients.\n\n\\end{theorem}\n\nThe $2$-roots of the classical groups are listed in \\cite{BH1}.  Note that $H=S^1$ has no nontrivial $2$ roots, nor does $SU(2)\\times SU(2)$.  On the other hand, the maximal $2$-group of $U(2)\\times U(2)$ is given, up to conjugacy, by all pairs of diagonal matrices with entries $\\pm 1$.  If $\\lambda_1$ is dual to $(\\diag(-1,1), I)$, $\\lambda_2$ is dual to $(\\diag(1,-1),I)$, with $\\mu_1$ and $\\mu_2$ dual to $(I, \\diag(-1,1))$ and $(I,\\diag(1,-1))$ respectively, then the nontrivial $2$-roots of $U(2)$ are $\\lambda_1-\\lambda_2$ and $\\mu_1-\\mu_2$ each with multiplicity $2$.\n\nThe case where $\\gcd(a,b,c,d)$ is admissible with $\\gcd(a^2-c^2, b^2-d^2)=1$ describes a free action, so we can directly apply Singhof's theorem.  Since neither $H$ nor $G$ have $2$-roots, the products in Theorem \\ref{charclass} are all trivial, so their pullbacks are as well.  Hence, if $\\gcd(a^2-c^2, b^2-d^2) = 1$, then the biquotient is diffeomorphic to $S^3\\times S^2$.\n\nUnfortunately, if $(a,b,c,d)$ is admissible with $\\gcd(a^2-c^2,b^2-d^2) = 4$, then the induced action is merely effectively free, so Singhof's approach will not directly work.  In order to circumvent this, we'll define a free biquotient action on $U(2)\\times U(2)$ which preserves $SU(2)\\times SU(2)$ and which has the same orbits through these points.  Singhof's approach will apply to this setting and then we can transfer this to information about $(SU(2)\\times SU(2))\\bq H$ by considering the following commutative diagram where $G = SU(2)\\times SU(2)$, $G' = U(2)\\times U(2)$, and $H=S^1$:\n\n\\begin{diagram}\nG &  & G' & & G' \\\\\n\\dTo & & \\dTo & & \\dTo \\\\\nG \\bq H  & \\rTo^{j} & G'\\bq H &\\rTo^{\\phi_G} & B\\Delta G' \\\\\n\\dTo^{\\phi_H} & & \\dTo^{\\phi_H'} & & \\dTo^{B\\sigma} \\\\\nBH & \\rTo^{=} & BH & \\rTo^{Bf} & BG'\\times BG'\\\\\n\\end{diagram}\nThe two vertical fibrations on the right come from Proposition \\ref{commute}, the first vertical fibration comes from the free $H$ action on $G$, and where $j:G\\bq H\\rightarrow G'\\bq H$ is induced from the natural inclusion $G\\rightarrow G'$.\n\nThe action of $H$ on $G'$ is given by the next proposition.  First note that if $\\gcd(a^2-c^2,b^2-d^2) = 4$, then both $a \\pm c$ and $b\\pm d$ are even.\n\n\\begin{proposition}  Consider any admissible $4$-tuple $(a,b,c,d)$ with the property that $\\gcd(a^2-c^2, b^2-d^2) = 4$.  Then the action of $S^1$ on $U(2)\\times U(2)$ given by $$ z*(A,B) = \\left(\\begin{bmatrix} z^{a} & \\\\ & 1\\end{bmatrix} A \\begin{bmatrix}z^{\\frac{a+c}{2}} & \\\\ & z^{\\frac{a-c}{2}}\\end{bmatrix}^{-1}, \\begin{bmatrix} z^{b} & \\\\ & 1\\end{bmatrix} B  \\begin{bmatrix} z^{\\frac{b+d}{2}} & \\\\ & z^{\\frac{b-d}{2}}\\end{bmatrix}^{-1} \\right) $$  is free and the orbits through points in $SU(2)\\times SU(2)$ are the same as in the action induced by the $4$-tuple.\n\n\\end{proposition}\n\n\\begin{proof}\nObserve that if an effectively free biquotient action is determined by a homomorphism $f:H\\rightarrow G\\times G$ and $f':H\\rightarrow \\Delta Z(G) \\subseteq G\\times G$ is any homomorphism, then the action determined by the map $F:H\\rightarrow G\\times G$ with $F(h) = f(h)f'(h)$ is also effectively free with the same orbits.  In this case, we extend the action determined by the admissible $4$-tuple $(a,b,c,d)$ to one on $G'$ and then use $f':H\\rightarrow \\Delta Z(G')$ given by $$z\\mapsto \\big( \\diag(z^a,z^a), \\diag(z^b,z^b), \\diag(z^a,z^a), \\diag(z^b,z^b)\\big).$$  We see that while $-1$ is not in the kernel of $f$, it is in the kernel of $ff'$.  Dividing $S^1$ by $\\langle -1\\rangle$ gives the above action.\n\nThe action is now free because the image of $S^1$ intersects $\\Delta Z(G)$ iff $z$ is a $\\gcd\\left(a, \\frac{a+c}{2}, b, \\frac{b+d}{2}\\right)$th root of unity, that is, iff $z=1$.\n\n\\end{proof}\n\n\nWe can also easily relate the topology of $G\\bq H$ to that of $G'\\bq H$.  First recall that the map from $SU(n)\\times S^1$ to $U(n)$ given by sending $(A,z)$ to $\\diag(z,1,...,1) A$ is a diffeomorphism.  Further, if an action of $H$ on $G'$ is invariant under the determinant $\\det:G'\\rightarrow T^2$, then this action is equivalent via this diffeomorphism to an action of $H$ on $G\\times T^2$ which acts trivially on the $T^2$ factor.  If follows that $G'\/H\\cong G \/H \\times T^2$.  In particular, the map $j$ in the above commutative diagram maps $G\\bq H$ to $G\\bq H \\times \\{(1,1)\\}\\in G'\\bq H$.\n\n\nWe can now apply Singhof's method to these actions.  By Theorem \\ref{charclass}, since $H$ has no nontrivial $2$-roots, our goal is to determine whether or not $$j^\\ast\\phi_G^\\ast \\left[(1+\\lambda_1 + \\lambda_2)(1+\\mu_1 + \\mu_2) \\right]^2 = 0$$ when restricted to $H^2(G\\bq H)$.  Since $B\\sigma^\\ast: H^\\ast(B(G\\times G))\\rightarrow H^\\ast(B\\Delta G)$ maps $(\\lambda_1 + \\lambda_2)\\otimes 1$ to $\\lambda_1 + \\lambda_2$, this is equivalent to computing $$j^\\ast\\phi_G^\\ast B\\sigma^\\ast\\left[\\big(1+(\\lambda_1+\\lambda_2)\\otimes 1\\big)\\big(1+(\\mu_1+\\mu_2)\\otimes 1\\big)\\right]^2$$ when restricted to $H^2(G\\bq H)$.  Since $B\\sigma\\, \\phi_G\\, j = Bf\\, \\phi_H'\\,j = Bf\\,\\phi_H$, this is equivalent to computing $$\\phi_H^\\ast Bf^\\ast\\left[ \\big(1+(\\lambda_1+\\lambda_2)\\otimes 1\\big)\\big(1+(\\mu_1+\\mu_2)\\otimes 1\\big)\\right]^2$$ in $H^2(G\\bq H)$.\n\nBy the Gysin sequence associated to the fiber bundle $$S^1\\rightarrow SU(2)\\times SU(2)\\rightarrow (SU(2)\\times SU(2))\\bq S^1,$$ we see that the Euler class of the bundle generates $H^2((SU(2)\\times SU(2))\\bq S^1)$.  In particular, the classifying map $\\phi_H:(SU(2)\\times SU(2))\\bq S^1\\rightarrow BS^1$ induces an isomorphism on $H^2$, so is injective.  Thus, we need only determine whether or not $$Bf^\\ast \\left[ \\big(1+(\\lambda_1 + \\lambda_2)\\otimes 1\\big)\\big(1+(\\mu_1 + \\mu_2)\\otimes 1\\big)\\right]^2$$ is trivial or not in $H^2(BS^1)$.\n\n\nIn order to compute $Bf^\\ast$, we use a technique developed by Borel \\cite{Bo1}.  If $Q_G\\subseteq G$ is a maximal $2$-group, then Borel proves that the induced map $H^\\ast(BG)\\rightarrow H^\\ast(BQ_G)$ is injective and he computes in the image for each of the classical groups.  For $H=S^1$, the maximal $2$-group $Q_{H}$ is generated by the element $-1\\in H$.  If $w$ is dual to this element, then we can identify $H^\\ast(BQ_{H}) \\cong \\zmod{2}[w]$.  In this notation, Borel proves the image of $H^\\ast(BH)$ is then $\\zmod{2}[w^2]$.\n\nFor $G'= U(2)\\times U(2)$, as above, we take $Q_{G'}$ to be the collection of all diagonal matrices with entries $\\pm 1$.  We also take the dual basis mentioned above, $\\{\\lambda_1, \\, \\lambda_2, \\, \\mu_1, \\, \\mu_2\\}$.  This allows us to identify $$H^\\ast(B_{G'})\\cong \\zmod{2}[\\lambda_1, \\, \\lambda_2, \\, \\mu_1, \\, \\mu_2].$$  From here, the K\\\"{u}nneth formula gives $$H^\\ast(B(Q_{G'}\\times Q_{G'}))\\cong\\zmod{2}[\\lambda_1, \\, \\lambda_2, \\, \\mu_1, \\, \\mu_2]\\otimes \\zmod{2}[\\lambda_1, \\, \\lambda_2, \\, \\mu_1, \\, \\mu_2].$$  Since the induced map $H^\\ast(B(G'\\times G'))\\rightarrow H^\\ast(B(Q_{G'}\\times Q_{G'}))$ is injective, and since $f(Q_h)\\subseteq Q_{G'}\\times Q_{G'}$, we can compute $Bf^\\ast$ by computing it as a map $$H^\\ast(B(Q_{G'}\\times Q_{G'}))\\rightarrow H^\\ast (BQ_H).$$\n\nIn this description, notice that $Bf_2^\\ast$ is trivial when restricted to elements of the form $x\\otimes 1$, so, we need only compute $$Bf_1^\\ast \\left[\\big( 1+(\\lambda_1 + \\lambda_2)\\otimes 1\\big) \\big(1+(\\mu_1 + \\mu_2)\\otimes 1\\big)\\right]^2$$ in $H^2(BH)$.\n\nConcretely, we have $f(z) = (f_1(z), f_2(z))$ with $$f_1(z) = \\big( \\diag\\left(z^a, 1\\right), \\diag\\left( z^b, 1\\right)\\big)$$ and $$f_2(z) = \\big(\\diag\\left(z^{\\frac{a+c}{2}}, z^{\\frac{a-c}{2}}\\right), \\diag\\left(z^{\\frac{b+d}{2}}, z^{\\frac{b-d}{2}}\\right)\\big).$$  Thus, we see  $$f(-1) = \\big( \\diag((-1)^a,1), \\diag((-1)^b,1)\\big).$$   Now we note that $a$ and $b$ must have opposite parity.  For, if $a$ and $b$ are both even, then since $\\gcd(a\\pm c, b\\pm d) = 2$, we must have $c$ and $d$ even as well.  This contradicts $\\gcd(a,b,c,d)=1$.  Similarly if $a$ and $b$ are both odd, then $c$ and $d$ must both be odd.  Then, by checking cases mod $4$, we see that $4|(a\\pm c, b\\pm d)$ for some choice of signs, again giving a contradiction.  We will thus assume without loss of generality that $a$ is odd and $b$ is even, so $f_1(-1) = \\big( \\diag(-1,1), \\diag(1,1)\\big)$.  It follows that $f_1^\\ast(\\lambda_1) = w$ while $f_1^\\ast(\\lambda_2) = f_1^\\ast(\\mu_1) = f_1^\\ast(\\mu_2) = 0$.\n\nFrom here, a simple calculation shows \n\n\\begin{proposition}With the setup as above, $Bf_1^\\ast(\\lambda_1\\otimes 1) = w$ while $Bf_1^\\ast$ is $0$ on the other generators of $H^\\ast(B(Q_{G'}\\times Q_{G'}))$.  Thus, we have $$Bf_1^\\ast\\left( 1+(\\lambda_1+\\lambda_2)\\otimes 1)(1+(\\mu_1+\\mu_2)\\otimes 1\\right)^2 = w^2\\in H^\\ast(B(S^1))$$ so is nontrivial.  In particular, the biquotient $G\\bq H$ has nontrivial second Stiefel-Whitney class, so is diffeomorphic to $S^3\\,\\hat{\\times}\\,S^2$.\n\n\\end{proposition}\n\nThis completes the proof of Theorem \\ref{ThmB}.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Intro}\n\n\nThe current three-neutrino ($3\\nu$) paradigm---the simplest possibility beyond massless neutrinos \n\\cite{Bilenky:2019gzn}---assumes that\nthe states  $(\\nu_e,\\,\\nu_\\mu,\\,\\nu_\\tau)$ with definite flavor are mixed with three states  $(\\nu_1,\\,\\nu_2,\\,\\nu_3)$ with definite masses $(m_1,\\,m_2,\\,m_3)$ \\cite{Zyla:2020zbs}. \nIn standard conventions \\cite{PDG1}, the mixing matrix is parametrized by \nthree angles $(\\theta_{12},\\,\\theta_{23},\\theta_{13})$ and one CP-violating phase $\\delta$.\nNeutrino flavor oscillations depend on such parameters and on two independent squared mass differences \\cite{PDG2} that, without loss of generality, can be chosen as $\\delta m^2=m^2_2-m^2_1>0$ and $\\Delta m^2=m^2_3-(m^2_1+m^2_2)\/2$ \n\\cite{Fogli:2005cq}, with\nsign$(\\Delta m^2)=\\pm$ distinguishing normal ordering ($+$, NO) and inverted ordering ($-$, IO). Nonoscillation \nprobes of absolute neutrino masses include: $\\beta$-decay, sensitive to an effective mass \n$m_\\beta$ \\cite{PDG3}; precision cosmology within the standard $\\Lambda$CDM model \\cite{PDG4}, \nsensitive to $\\Sigma=m_1+m_2+m_3$ \\cite{PDG5}; and, if\nneutrinos are Majorana, neutrinoless double beta decay ($0\\nu\\beta\\beta$), sensitive to another effective mass $m_{\\beta\\beta}$ \\cite{PDG6}. \nWe refer, e.g.,\\ to \\cite{Fogli:2005cq} for definitions of $m_\\beta$ and $m_{\\beta\\beta}$. \n\n\nConstraints on the oscillation parameters $(\\delta m^2,\\,\\Delta m^2,\\,\\theta_{ij},\\,\\delta)$ and on the nonoscillation observables $(\\Sigma,\\,m_\\beta,\\,m_{\\beta\\beta})$ have been explored in several global neutrino data analyses, \nincluding our previous work \\cite{Capozzi:2017ipn} and the more recent papers \\cite{Capozzi:2020,deSalas:2020pgw,Esteban:2020cvm}, plus the preliminary contribution in \\cite{Marrone2021}. \nIn particular, the analyses in \\cite{deSalas:2020pgw,Esteban:2020cvm,Marrone2021} are based on largely common datasets, \nbased on updated information from the Conference {\\em Neutrino 2020\\\/} \\cite{Nu2020}. As a result, \na solid fabric for the $3\\nu$ paradigm emerges from convergent measurements of \nfive oscillation parameters $(\\theta_{12},\\,\\theta_{23},\\theta_{13},\\,\\delta m^2,\\,|\\Delta m^2|)$, with an overall accuracy ranging from $\\sim\\! 1\\%$ for $|\\Delta m^2|$  to $\\sim\\! 6\\%$ for $\\sin^2\\theta_{23}$ (dominated by the so-called $\\theta_{23}$ octant degeneracy). However, the fabric is still unfinished in $\\nu$ oscillations, as far as the $\\theta_{23}$ octant, \nthe mass ordering and the phase $\\delta$ are concerned. \nIn particular, a tension between recent long-baseline accelerator neutrino data (from T2K \\cite{T2K2020} and NOvA \\cite{NOvA2020}) \naffects all these $3\\nu$ unknowns at the same time \\cite{deSalas:2020pgw,Esteban:2020cvm,Marrone2021,Kelly:2020fkv}.\nIn addition, the Dirac-Majorana nature\nand the absolute neutrino mass scale remain undetermined in current nonoscillation searches, with $\\Sigma$, $m_\\beta$ and $m_{\\beta\\beta}$ constrained at sub-eV scales but still consistent with null values \\cite{Formaggio:2021nfz}. \n\n\nIn this work we discuss the status of the $3\\nu$ framework, including new data that have recently become available\nfrom both oscillation and nonoscillation searches, with particular attention to issues of concordance or discordance of various data sets, and to their implications on the unfinished fabric of the paradigm. By performing a global analysis of oscillation data, including the latest Super-Kamiokande atmospheric results made publicly available in 2021 \\cite{SKmap}, \nwe find an indication for normal ordering at the level of $2.5\\sigma$, as well as $90\\%$ C.L.\\ hints for $\\theta_{23}<\\pi\/4$ and for  $\\sin\\delta<0$.  We discuss the structure and interplay of such hints, especially in the light of the T2K and NOvA tension and of the complementarity among accelerator, atmospheric and reactor data. We surmise that further understanding of neutrino nuclear interactions may help to clarify some issues. \n\nConcerning nonoscillation searches, we include the KATRIN 2021 data \\cite{KATRIN2021} that, for the first time, set sub-eV upper bounds on $m_\\beta$ at 90\\% C.L. We analyze systematically all the latest $0\\nu\\beta\\beta$ decay searches probing half lives $T>10^{25}$~y (in $^{76}$Ge, $^{130}$Te and $^{136}$Xe), and translate them into $m_{\\beta\\beta}$ bounds via correlated nuclear matrix elements. In the realm of cosmology and of its consensus $\\Lambda$CDM model, increasing attention is also being paid to old and new data tensions, see e.g.\\ \\cite{DiValentino:2021izs,DiValentino:Snowmass,Challenge2021}. In this context, we focus on the \nso-called $A_\\mathrm{lens}$ anomaly affecting Planck angular spectra (that show more lensing\nthan expected in the $\\Lambda$CDM model \\cite{Aghanim:2018eyx}), and we consider two possible  \noptions, leading to different implications for absolute mass observables. On the one hand, we revisit a previously considered ``default'' scenario \\cite{Capozzi:2020},\nincluding all Planck results (irrespective of the $A_\\mathrm{lens}$ anomaly), that sets stringent upper bounds on $\\Sigma$ at the level of $\\sim\\! 10^{-1}$~eV, and further favors normal ordering, raising its overall preference to $\\sim 3\\sigma$. On the other hand, we discuss an ``alternative'' option\nthat makes use of the recent ACT CMB polarization data release 4 (ACTPol-DR4)  \\cite{Aiola:2020azj} that is consistent with standard lensing, also in combination\nwith WMAP 9-year data (WMAP9) \\cite{Bennett:2012zja} and selected data from Planck \\cite{Aghanim:2018eyx,Aghanim:2019ame,Aghanim:2018oex}; such option is insensitive to the mass ordering and prefers $\\Sigma\\sim \\mathrm{few}\\times 10^{-1}$~eV, with different  implications for $m_\\beta$ and $m_{\\beta\\beta}$ searches.  \n\n\nBuilding upon our previous works \\cite{Capozzi:2017ipn,Capozzi:2020} we elaborate upon these recent topics as follows: \nIn Sections~II and III we update and discuss the analysis of oscillation and nonoscillation data, respectively.\nWe pay particular attention to relevant correlations among various observables, and to some emerging tensions \namong different data sets. We provide a brief synthesis of oscillation and nonoscillation results in Sec.~IV.\n\n\n\\vspace*{-1mm}\n\\section{Oscillation data, analysis methods and results}\n\\label{Sec:Osc}\n\nIn this section we introduce recent oscillation data that were not included in our previous work \\cite{Capozzi:2020}, \ntogether with the methodology used for their analysis, in the light of some emerging issues in precision oscillation physics. We then discuss the resulting constraints on the parameters $(\\delta m^2,\\,\\Delta m^2,\\,\\sin^2\\theta_{ij},\\,\\delta)$,\nboth separately and in selected pairs, highlighting the concordance, discordance and complementarity of various datasets.\n\n\\vspace*{-1mm}\n\\subsection{Oscillation data update}\n\n\nThree-neutrino oscillations are currently constrained by experiments using long-baseline (LBL) accelerator, solar, long-baseline reactor (KamLAND), short-baseline (SBL) reactor and atmospheric neutrinos. With respect to \\cite{Capozzi:2020}, we update some of these datasets as follows. LBL accelerator data, in the form of neutrino and antineutrino energy spectra for flavor disappearance and appearance channels, are taken from the presentations of T2K \\cite{T2K2020} and NOvA \\cite{NOvA2020} at {\\em Neutrino 2020\\\/} \\cite{Nu2020} (and subsequent conferences). Such spectra are endowed with statistical (Poisson) errors and \nsystematic (normalization and energy scale) uncertainties, as well as with oscillation-independent backgrounds, \nin a modified version of GLOBES \\cite{GLOBES}. Solar neutrinos data from the Super-Kamiokande-IV 2970-days run (SK-IV  energy spectrum and day-night asymmetry) are taken from the presentation at {\\em Neutrino 2020\\\/} \\cite{SK2020}, while the input solar model BP16-GS98 \\cite{Vinyoles:2016djt} is unchanged.  Concerning SBL reactors, we update from {\\em Neutrino 2020\\\/} the RENO \\cite{RENO2020} and \nDouble Chooz data \\cite{DC2020}, while the\nDaya Bay data \\cite{Adey:2018zwh} are unchanged. Note that Daya Bay and and RENO measure both \n$\\theta_{13}$ and $\\Delta m^2$, while the latter parameter is not significantly constrained by\nDouble Chooz. IceCube-DeepCore (IC-DC) atmospheric data \nare taken as in \\cite{Capozzi:2020}; a new IC-DC data release  is expected in the near future \\cite{ICDC2021}.\nFinally, our oscillation dataset is completed by the recent SK-IV atmospheric results \\cite{SK2020,Jiang:2019xwn}, included through the\n$\\chi^2$ map  recently made available by the collaboration \\cite{SKmap}. \n\n\n\\vspace*{-1mm}\n\\subsection{Analysis method and emerging issues}\n \nWe adopt the methodology proposed in \\cite{Capozzi:2018ubv}, see also  \\cite{Capozzi:2017ipn,Capozzi:2020}. In particular, \nwe start with the combination of solar, KamLAND and LBL accelerator neutrino data, that represent \nthe minimal dataset sensitive to all the oscillation parameters $(\\delta m^2,\\,\\Delta m^2,\\,\\theta_{ij},\\,\\delta)$.\nWe then add SBL reactor neutrino data, that\nsharpen the constraints on $(|\\Delta m^2|,\\theta_{13})$ and indirectly affect the parameters $(\\theta_{23},\\,\\delta)$\nand sign$(\\Delta m^2)$ via correlations. We add atmospheric neutrino data at the end, for two reasons: (a) they\n provide rich but\nrather entangled information on the parameters $(\\Delta m^2,\\theta_{23},\\,\\theta_{13},\\,\\delta)$; (b) \ntheir $\\chi^2(\\Delta m^2,\\theta_{23},\\,\\delta)$ maps\nassume an input on $(\\delta m^2,\\,\\theta_{12},\\,\\theta_{13})$ from the combination of solar, KamLAND and SBL reactor data.\nFinally, a frequentist approach based on $\\chi^2$ functions is used for all datasets. Best fits\nare obtained by $\\chi^2$ minimization, while allowed regions around best fits are expanded  \nin terms of ``number of standard deviations'' $N_\\sigma=\\sqrt{\\Delta \\chi^2}$. \n\\textcolor{black}{In particular, two-dimensional contours\nare shown for $N_\\sigma=1$, 2 and 3, which, for a $\\chi^2$ distribution with two degrees of freedom, correspond\nto C.L.\\ of 39.35\\%, 86.47\\% and 98.89\\%, respectively. Their one-dimensional \nprojections provide the $N_\\sigma$ ranges for each parameter, corresponding to C.L.\\ \nof  68.27\\%, 95.45\\% and 99.73\\%, respectively.}\nThe difference $\\Delta \\chi^2_{\\mathrm{IO}-\\mathrm{NO}}$ between the minima in IO and NO may---or may not---be accounted\nfor, when reporting fit results; these two options will be clearly distinguished in each context. \n\nWe briefly discuss some issues arising in global data analyses, in \nthe era of increasingly precise measurements and of growing sensitivity to subleading effects. \nData fits  usually\ninvolve the comparison of experimental event rates $R_\\mathrm{expt}$ \nwith their theoretical predictions $R_\\mathrm{theo}$\n\\begin{equation}\n\\label{Rtheo} \nR_\\mathrm{theo} = \\int \\Phi_\\alpha \\otimes P_{\\alpha\\beta} \\otimes \\sigma_\\beta \\otimes r_\\beta \\otimes  \\varepsilon_\\beta\\ ,\n\\end{equation}\nwhere, from left to right, the integrands represent the source flux of $\\nu_\\alpha$, the probability of\n$\\nu_\\alpha\\to\\nu_\\beta$ oscillations, and the interaction cross section,  detector resolution and efficiency for \n$\\nu_\\beta$ events. Some \nfactors may be  differential functions that need multiple integrations or convolutions, \nas alluded by the cross product $\\otimes$. Integrands are endowed with various uncertainties, that may be shared\nby (i.e., correlated among) various rates $R$ in the same or different experiment(s).\nIn solar neutrino searches, all these features can be accurately implemented to a large extent\n\\cite{Fogli:2002pt}. Also short-baseline reactor experiments (Daya Bay, RENO, Double Chooz) \ngenerally provide enough public information to allow reproducible analyses, although \na more precise joint analysis by the different collaborations (accounting for minor correlated uncertainties)\nwould be desiderable \\cite{ESCAPE}. \nMore relevant issues arise in the context of long-baseline accelerator searches, currently carried out\nby T2K \\cite{T2K2020} and NOvA \\cite{NOvA2020}. \nTheir event spectra are usually given in terms of a ``reconstructed'' (unobservable) neutrino energy $E_\\nu^\\mathrm{rec}$, that\nis processed from observable event energies at far detectors, through models (for some $R_\\mathrm{theo}$ integrands)  \nconstrained by near-detector data. \nIn principle, both T2K and NOvA should share a common theoretical model for $\\sigma$ (and to some extent for $\\Phi$), \nleading to possible correlations among their uncertainties for $E_\\nu^\\mathrm{rec}$ (affecting $\\Delta m^2$) and for\nthe event\nrates $R_\\mathrm{theo}(E_\\nu^\\mathrm{rec})$ (affecting $\\theta_{23}$ and $\\theta_{13}$). In practice, the adopted models are different, and possible covariances are ignored. A joint analysis planned\nby the T2K and NOvA \ncollaborations \\cite{JOINT} might shed light on these issues.\nFinally, in current atmospheric neutrino searches at SK and IC-DC, \nthe data processing and analysis are too complex to be reproducible by external users with an acceptable accuracy. Oscillation results are\ngiven in terms of public  $\\chi^2$ maps that, when summed up, cannot account for known covariances, \nsuch as those related to the (common, in principle) input models for $\\Phi$ and $\\sigma$. Once again, joint \nanalyses or in-depth comparisons of data by different atmospheric $\\nu$ experiments would be desirable \\cite{ATMOS}. We surmise that\nglobal oscillation analyses could \nbenefit from a better control of those systematics that are shared by different experiments \n(such as model uncertainties for $\\Phi_\\alpha$ and $\\sigma_\\beta$), but whose correlations are not yet\nproperly accounted for. See also the remarks in Sec.~\\ref{sec:remarks}.\n  \n\\vspace*{-1mm}\n\\subsection{Results on single oscillation parameters}\n\nIn this Section we present the constraints on the six oscillation parameters $(\\delta m^2,\\,\\Delta m^2,\\,\\sin^2\\theta_{ij},\\,\\delta)$ for increasingly rich data sets. We explicitly account for the $\\chi^2$ difference between NO and IO, in order to show its variations. \n\nFigure~\\ref{Fig_01} shows the results for the combination of solar and KamLAND data \n(sensitive to $\\delta m^2$, $\\sin^2\\theta_{12}$, and $\\sin^2\\theta_{13}$) with LBL accelerator data \n(mainly sensitive to $\\Delta m^2$, $\\sin^2\\theta_{23}$, $\\sin^2\\theta_{13}$ and $\\delta$), for\nboth NO (blue) and IO (red). The latter mass ordering is slightly favored (by accelerator data) at the level of $\\sim\\!1\\sigma$, as also discussed later.\nThe parameters $\\delta m^2$ and $\\sin^2\\theta_{12}$ are rather precisely measured, with nearly linear and symmetrical \n(i.e., almost gaussian) uncertainties, and no significant difference between constraints in NO and IO.\nThe parameters $\\sin^2\\theta_{13}$ and $\\sin^2\\theta_{23}$ are less \naccurately constrained. In particular, the two minima in $\\sin^2\\theta_{23}$ reflect the $\\theta_{23}$ \noctant ambiguity in the $\\nu_\\mu\\to\\nu_\\mu$ disappearance searches at LBL accelerators, \ninducing two correlated minima in  $\\sin^2\\theta_{13}$ via the leading amplitude\nof $\\nu_\\mu\\to\\nu_e$ appearance ($\\propto \\sin^2\\theta_{23}\\sin^2\\theta_{13}$) \\cite{Fogli:1996pv}.\nThe phase $\\delta$ is poorly constrained, although it appears to be \nslightly favored around $\\pi$ in NO and around $3\\pi\/2$ in IO, while it is disfavored \naround $\\pi\/2$ in both cases.   \n\nFigure~\\ref{Fig_02} shows the effect of adding SBL reactor data, which are sensitive to \n$|\\Delta m^2|$ and $\\sin^2\\theta_{13}$. One can notice the strong reduction of the \n$\\sin^2\\theta_{13}$ uncertainty, inducing also correlated changes on the relative likelihood \nof the lower and upper octant of $\\theta_{23}$ via $\\nu_\\mu\\to\\nu_e$ appearance in LBL accelerators\n\\cite{Fogli:1996pv}. The synergy of SBL reactor and LBL accelerator data \nalso helps to break mass-ordering degeneracies via independent measurements of $\\Delta m^2$\n(see, e.g.,   \\cite{Huber:2003pm}) and currently flips the fit preference from IO to NO (at \nthe level of $\\sim\\!1.3\\sigma$), together with an increase of the best-fit value of $\\Delta m^2$\nwith respect to Fig.~\\ref{Fig_01}. The preference for $\\delta\\sim \\pi$ ($\\sim 3\\pi\/2$) in NO (IO) remains\nunaltered. \n\n \n\n\n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.82\\textwidth]{Fig_01.pdf}\n\\caption{\\label{Fig_01}\n\\footnotesize Global $3\\nu$ oscillation analysis of long-baseline accelerator, solar and KamLAND $\\nu$  data. \nBounds on the parameters $\\delta m^2$, $|\\Delta m^2|$, $\\sin^2\\theta_{ij}$, and $\\delta$, for NO (blue) and IO (red), in terms of $N_\\sigma=\\sqrt{\\Delta \\chi^2}$ from the global best fit. The offset between separate minima in IO and  NO, \n$\\Delta \\chi^2_\\mathrm{IO-NO}=-1.1$, favors the IO case by $\\sim\\! 1.0\\sigma$.   \n} \\end{minipage}\n\\end{figure}\n\n\\vspace*{10mm}\n\n\\begin{figure}[h!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.82\\textwidth]{Fig_02.pdf}\n\\caption{\\label{Fig_02}\n\\footnotesize As in Fig.~\\protect\\ref{Fig_01}, but adding short-baseline reactor $\\nu$ data.   \nThe offset $\\Delta \\chi^2_\\mathrm{IO-NO}=+1.8$  favors the NO case by $\\sim\\!1.3\\sigma$.   \n} \\end{minipage}\n\\end{figure}\n\n\n\n\\newpage\n\nFigure~\\ref{Fig_03} shows the effect of adding atmospheric $\\nu$ data, which add further sensitivity\nto $\\Delta m^2$ (and to its sign), as well as to $\\sin^2\\theta_{23}$ and $\\delta$. In particular, the inclusion of SK-IV \ndata  \\cite{SK2020,SKmap} corroborates the preference in favor of NO (at an overall level of $\\sim\\!2.5\\sigma$),\nflips the $\\theta_{23}$ preference from the upper to the lower octant in NO (at $\\sim 1.6\\sigma$) \nand also moves the best fit of $\\delta$ slightly above the CP-conserving value $\\pi$ (disfavored at $\\sim 1.6\\sigma$). \nThe latter hints in favor of  $\\theta_{23}<\\pi\/4$  and on $\\delta> \\pi$, \ncurrently emerging in NO at the statistical ``threshold of interest''  \nof 90\\% C.L., represent interesting updates with respect to previous global analyses not including SK-IV\natmospheric data  \\cite{deSalas:2020pgw,Esteban:2020cvm,Marrone2021}. \n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.82\\textwidth]{Fig_03.pdf}\n\\caption{\\label{Fig_03}\n\\footnotesize As in Fig.~\\protect\\ref{Fig_02}, but adding atmospheric $\\nu$ data (i.e., with\nall oscillation data included).   \nThe offset $\\Delta \\chi^2_\\mathrm{IO-NO}=+6.5$ favors the NO case by $\\sim\\!2.5\\sigma$.   \n} \n\\end{minipage}\n\\end{figure}\n\nTable~\\ref{Tab:Synopsis} reports a numerical summary of the same information shown in Fig.~3, for \nthe separate cases of NO and IO (whose $\\chi^2$ difference is reminded in the last row). \nThe two squared mass splittings $\\Delta m^2$ and $\\delta m^2$ are measured with a formal $1\\sigma$ accuracy\nof $1.1\\%$ and $2.3\\%$, respectively. The mixing parameters $\\sin^2\\theta_{13}$, $\\sin^2\\theta_{12}$ and $\\sin^2\\theta_{23}$\nare measured with an accuracy of $\\sim\\! 3\\%$, $4.5\\%$, and $\\sim 6\\%$, respectively.\nThe latter uncertainty is largely affected by the $\\theta_{23}$ octant ambiguity; if one of the two quasi-degenerate $\\theta_{23}$ options could be removed, such uncertainty would be reduced by factor of $\\sim\\!2$ in both NO and IO.\n\nSummarizing, five oscillation parameters are known with (few) percent accuracy, while \nonly some hints emerge about the remaining three oscillation ``unknowns''. In particular, \nwe find a preference for NO at $\\sim\\!2.5\\sigma$ and, in such ordering, we also find a preference at 90\\% C.L.\\ for  \n$\\theta_{23}$ in the lower octant (with respect to the secondary best fit\n in the upper octant) and for $\\delta\\simeq 1.24\\pi$ (with respect to the CP-conserving value $\\delta=\\pi$). Conversely,\nmaximal $\\theta_{23}$ mixing is disfavored at $\\sim 1.8\\sigma$ and  \nthe range $\\delta \\in [0,\\, 0.77\\pi]$ is disfavored at $>3\\sigma$ in NO. \n\n\n\\begin{table}[h!]\n\\centering\n\\resizebox{.82\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\\caption{\\label{Tab:Synopsis}\nGlobal $3\\nu$ analysis of oscillation parameters: best-fit values and allowed ranges at $N_\\sigma=1$, 2 and 3, for  either NO or  IO, including all data. The latter column shows the formal  ``$1\\sigma$ fractional accuracy'' for each parameter, defined as 1\/6 of the $3\\sigma$ range, divided by the best-fit value and expressed in percent. We recall that \n$\\Delta m^2=m^2_3-{(m^2_1+m^2_2})\/2$ and that $\\delta \\in [0,\\,2\\pi]$ (cyclic). The last row reports the  difference\nbetween the $\\chi^2$ minima in IO and NO.\n}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccccc}\nParameter & Ordering & Best fit & $1\\sigma$ range & $2\\sigma$ range & $3\\sigma$ range & ``$1\\sigma$'' (\\%) \\\\\n\\hlin\n$\\delta m^2\/10^{-5}~\\mathrm{eV}^2 $ & NO, IO & 7.36 & 7.21 -- 7.52 & 7.06 -- 7.71 & 6.93 -- 7.93 & 2.3 \\\\\n\\hlin\n$\\sin^2 \\theta_{12}\/10^{-1}$ & NO, IO & 3.03 & 2.90 -- 3.16 & 2.77 -- 3.30 & 2.63 -- 3.45 & 4.5 \\\\\n\\hlin\n$|\\Delta m^2|\/10^{-3}~\\mathrm{eV}^2 $ & NO  & 2.485 & 2.454 -- 2.508 & 2.427 -- 2.537 & 2.401 -- 2.565 & 1.1 \\\\\n                                      & IO  & 2.455 & 2.430 -- 2.485 & 2.403 -- 2.513 & 2.376 -- 2.541 & 1.1 \\\\\n\\hlin\n$\\sin^2 \\theta_{13}\/10^{-2}$ & NO & 2.23 & 2.17 -- 2.30 & 2.11 -- 2.37 & 2.04 -- 2.44 & 3.0 \\\\\n                             & IO & 2.23 & 2.17 -- 2.29 & 2.10 -- 2.38 & 2.03 -- 2.45 & 3.1 \\\\\n\\hlin\n$\\sin^2 \\theta_{23}\/10^{-1}$ & NO & 4.55 & 4.40 -- 4.73 & 4.27 -- 5.81 & 4.16 -- 5.99 & 6.7 \\\\\n                             & IO & 5.69 & 5.48 -- 5.82 & 4.30 -- 5.94 & 4.17 -- 6.06 & 5.5 \\\\\n\\hlin\n$\\delta\/\\pi$ & NO & 1.24 & 1.11 -- 1.42 & 0.94 -- 1.74  &  0.77 -- 1.97  & 16 \\\\\n             & IO & 1.52 & 1.37 -- 1.66 & 1.22 -- 1.78  &  1.07 -- 1.90  & 9 \\\\\n\\hlin\n$\\Delta \\chi^2_{\\mathrm{{IO}-{NO}}}$ & IO$-$NO & +6.5  \\\\ [1pt]\n\\end{tabular}\n\\end{ruledtabular}\n\\end{minipage}}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.34\\textwidth]{Fig_04.pdf}\n\\vspace*{-2.5mm}\n\\caption{\\label{Fig_04}\n\\footnotesize \nRegions separately allowed by solar and KamLAND data in the plane $(\\sin^2\\theta_{12},\\,\\delta m^2)$ for  \n$\\sin^2\\theta_{13}=0.02$ and NO. (The case of IO, not shown, would be almost identical). \nThe solar $\\nu$ fit includes SK-IV 2970-day data \\cite{SK2020}.}\n\\end{minipage}\n\\vspace*{-3mm}\n\\end{figure}\n\n\n  \n  \n\\subsection{Results on selected pairs of oscillation variables}\n\\vspace*{-2mm}\n\nBy studying selected pairs of variables  we can gain \nfurther insights about current unknowns (the mass ordering, the octant of $\\theta_{23}$ and the CP phase $\\delta$), \nand appreciate their interplay with known features of $3\\nu$ oscillations. We discuss the pairs \n $(\\sin^2\\theta_{12},\\,\\delta m^2)$, $(\\sin^2\\theta_{23},\\,\\sin^2\\theta_{13})$, \n\\textcolor{black}{$(\\sin^2\\theta_{23},\\,|\\Delta m^2|)$}, \n$(\\sin^2\\theta_{23},\\,\\delta)$, as well as pairs of total \n$\\nu_e$ and $\\overline\\nu_e$ events (bi-event plots) as observed in the appearance channel by T2K and NOvA.\n\n\n\n\n\n\n\nFigure~\\ref{Fig_04} shows the regions separately allowed by solar and KamLAND neutrino data \nin the plane charted by $(\\sin^2\\theta_{12},\\,\\delta m^2)$, assuming  fixed $\\sin^2\\theta_{13}=0.02$ and NO.\nThe two regions were somewhat displaced in the past, leading to a $<2\\sigma$ tension \nbetween the best-fit $\\delta m^2$ values \\cite{PDG1} (see, e.g., the analogous Fig.~4 in \\cite{Capozzi:2018ubv}). The \ncurrent regions in Fig.~\\ref{Fig_04} appear to be in very good agreement, largely as a \nresult of a slightly smaller day-night asymmetry in SK-IV 2970-day solar data, shifting\nthe solar $\\delta m^2$ best fit upwards and closer to the KamLAND one \\cite{SK2020}. \nWe find that this shift does not alter the combined solar and KamLAND constraints on \n$\\theta_{13}$, namely, $\\sin^2\\theta_{13}\\simeq 0.014\\pm 0.015$ \n (see, e.g., Fig.~5 in \\cite{Capozzi:2018ubv}). Results for IO (not shown)\nwould be almost identical for all parameters $(\\delta m^2,\\,\\sin^2\\theta_{12},\\,\\sin^2\\theta_{13})$.\nIn conclusion, solar and KamLAND data are\nnot only in very good agreement about the $(\\nu_1,\\,\\nu_2)$ oscillation parameters\n$(\\delta m^2,\\,\\sin^2\\theta_{12})$, but are also consistent with the \nmeasurement $\\sin^2\\theta_{13}\\simeq 0.02$ at SBL reactors. \n\n \n\n\n\n\n\n\nFigure~\\ref{Fig_05} shows the covariance of the pair $(\\sin^2\\theta_{23},\\,\\sin^2\\theta_{13})$\nfor increasingly rich data sets, in both NO (top) and IO (bottom), \n\\textcolor{black}{with the corresponding $\\chi^2$ functions separately minimized for each mass ordering}. \nThe $\\theta_{23}$ octant ambiguity leads to two quasi-degenerate solutions at $1\\sigma$, that generally merge \nat $\\sim \\!2\\sigma$.  The leading appearance amplitude in LBL accelerators, scaling as $\\sin^2\\theta_{23}\\sin^2\\theta_{13}$, \ninduces an anticorrelation between the two angles in the left panels: the higher $\\theta_{23}$, the smaller $\\theta_{13}$.\nIn the middle panels, the results from SBL reactors only (represented by $\\pm 2\\sigma$ error bars) tend to prefer\nslightly  the upper-octant solution\n(with lower values of $\\theta_{13}$) in both NO and IO, \nas confirmed by the combination with LBL accelerators (continuous curves). In the right panels,\nhowever, adding atmospheric data (that include SK-IV \\cite{SK2020,SKmap}) flips the octant preference in NO, while\nconfirming it in IO. We conclude that current hints about\nthe $\\theta_{23}$ octant are still rather fragile. \n\\vspace*{-4mm}\n\n\\begin{figure}[b]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.6\\textwidth]{Fig_05.pdf}\n\\vspace*{-2.5mm}\n\\caption{\\label{Fig_05}\n\\footnotesize Regions allowed in the plane $(\\sin^2\\theta_{23},\\,\\sin^2\\theta_{13})$ for increasingly rich data sets:\nSolar + KamLAND + LBL accelerator data (left panels), plus SBL reactor data (middle panels), plus Atmospheric data (right panels). Top and bottom panels refer, respectively, to NO and IO as taken separately (i.e., without any relative $\\Delta\\chi^2$ offset). The error bars in the middle panels show the $\\pm2\\sigma$ range for $\\theta_{13}$ arising from SBL reactor data only. \n}\n\\end{minipage}\n\\end{figure}\n\\newpage\n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.6\\textwidth]{Fig_06.pdf}\n\\vspace*{-2mm}\n\\caption{\\label{Fig_06}\n\\footnotesize As in Fig.~\\ref{Fig_05}, but in the plane \n\\textcolor{black}{$(\\sin^2\\theta_{23},\\,|\\Delta m^2|)$.}\n The error bars in the middle panels show the $\\pm2\\sigma$ range for \n \\textcolor{black}{$|\\Delta m^2|$} \n arising from SBL reactor data only. \n}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[b!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.6\\textwidth]{Fig_07.pdf}\n\\vspace*{-2mm}\n\\caption{\\label{Fig_07}\n\\footnotesize As in Fig.~\\ref{Fig_06}, but in the plane $(\\sin^2\\theta_{23},\\,\\delta)$.\n}\n\\end{minipage}\n\\end{figure}\n\n\n\nFigure~\\ref{Fig_06}  shows the covariance of the pair \n\\textcolor{black}{$(\\sin^2\\theta_{23},\\,|\\Delta m^2|)$}. \nIn this case, \nthere is a marked preference of SBL reactor data for relatively ``high'' values of \n\\textcolor{black}{$|\\Delta m^2|$} \n($\\pm 2\\sigma$ error bars), \nas compared with LBL accelerator data. A compromise is more easily reached for NO, featuring a smaller \ndifference between the $\\Delta m^2$ values derived from SBL reactor and LBL accelerator data. \nThis explains why the preferred mass ordering flips from inverted to normal, when \npassing from Fig.~\\ref{Fig_01} to Fig.~\\ref{Fig_02}; see also \\cite{deSalas:2020pgw,Esteban:2020cvm}.\nFor the same reason, maximal values of $\\theta_{23}$ (corresponding to the lowest values of \n\\textcolor{black}{$|\\Delta m^2|$}\nallowed by LBL accelerator data) are slightly more disfavored by adding SBL reactor data.\nThe overall preferences for NO and for nonmaximal $\\theta_{23}$ are confirmed by atmospheric data that,\nhowever, move the best fit in NO from the upper to the lower octant. \nWe emphasize that SBL reactor data, despite having no direct sensitivity to sign($\\Delta m^2$) and $\\theta_{23}$,\ncontribute to constrain (via covariances) these two  variables,  in combination with other datasets.\n\n \n\n\n\n\n\n\n\n\n\nFigure~\\ref{Fig_07} shows the covariance of the pair $(\\sin^2\\theta_{23},\\,\\delta)$. The octant\nambiguity leads to two quasi-degenerate best fits, surrounded by allowed regions that merge at $2\\sigma$ or $3\\sigma$. \nIn IO there is rather stable preference for the CP-violating case $\\delta \\simeq 3\\pi\/2$ in all data combinations, \nwith no significant correlation\nwith $\\theta_{23}$. In NO the allowed $\\delta$ range is always larger, and includes the\nCP-conserving case $\\delta \\simeq \\pi$ at $2\\sigma$; moreover, a slight negative correlation between $\\delta$ and $\\sin^2\\theta_{23}$ emerges when adding SBL\nreactor data. It is difficult to trace the origin of these null or small covariances, since\nthe interplay between $\\delta$ and $\\sin^2\\theta_{23}$ (and with $\\sin^2\\theta_{13}$) is rather subtle, see e.g.\\ \n\\cite{Minakata:2013eoa,Coloma:2014kca}. \nIn any case, the negative correlation emerging in NO slightly amplifies the effect of adding \natmospheric neutrino data, that prefer\nboth $\\delta\\sim 3\\pi\/2$ and the lower octant of $\\theta_{23}$, thus disfavoring  $\\delta \\simeq \\pi$ in a synergic way.\nQuantitatively, we find that the CP-conserving value $\\delta=\\pi$ is disfavored \nat 90\\% C.L.\\ (or $\\sim\\!1.6\\sigma$, see Fig.~\\ref{Fig_03}), while recent  analyses  not\nincluding SK-IV atmospheric data allowed this value at $<1\\sigma$ \\cite{deSalas:2020pgw,Esteban:2020cvm}.\nAlthough these covariance effects are admittedly small in current data, they are expected\nto grow with increasing statistics and accuracy\nin LBL accelerator experiments, whose results we comment in more detail through the so-called  bi-event plots, \nderived from  bi-probability plots.\n\n\\newpage\n\n\n\\begin{figure}[t]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.83\\textwidth]{Fig_08.pdf}\n\\vspace*{-5.mm}\n\\caption{\\label{Fig_08}\n\\footnotesize \nBi-event plots: Total number of $\\nu$ and $\\overline\\nu$ appearance events \nfor T2K and NOvA, in four possible combinations. The slanted ellipses represent the theoretical expectations for NO (blue) and IO (red),\nand for two representative values of $\\sin^2\\theta_{23}$: 0.45 (lower octant, thin ellipses) and 0.57 (upper octant, thick ellipses). The CP-conserving value $\\delta=\\pi$ and the CP-violating value $\\delta=3\\pi\/2$\nare marked as a circle and a star, respectively. Each gray band represents one datum with its $\\pm1\\sigma$ statistical error (from \\cite{T2K2020,NOvA2020}); the combination of any two data provides a (black dashed) $1\\sigma$ error ellipse, whose center is marked by a cross.  See the text for details.}\n\\end{minipage}\n\\vspace*{-2mm}\n\\end{figure}\n\n\nBi-probability plots, charted by the $\\nu_\\mu\\to\\nu_e$ and $\\overline\\nu_\\mu\\to\\overline\\nu_e$ appearance probabilities in LBL accelerator experiments at fixed neutrino energy,\ndisplay the cyclic dependence on $\\delta$ through ellipses \n\\cite{Minakata:2001qm} and help to understand parameter degeneracies \\cite{Minakata:2002qi}.\nSee \\cite{Kelly:2020fkv} for a related discussion, in the context of recent T2K and NOvA data.  \nAfter integration over energy (weighted by $\\nu$ fluxes and cross sections), the probabilities\ncan be converted into total number of appearance events and thus into  bi-event plots, preserving elliptic shapes\n \\cite{Mena:2006uw}. Such theoretical ellipses can be directly compared with the measured number of events; \nsee, e.g., the presentations at {\\em Neutrino 2020\\\/} \\cite{Nu2020} by T2K \\cite{T2K2020} and NOvA \\cite{NOvA2020}.  \nAlthough we use the full energy spectra (and not their integrals) in our LBL accelerator data analysis, we\nthink that bi-event plots can help to highlight some issues emerging in the comparison of current T2K and NOvA data. \n\n  \n \nFigure~\\ref{Fig_08} show the $\\nu$ and $\\overline\\nu$ appearance events \nfor T2K and NOvA, in four possible combinations. The grey bands and the black ellipses represent the data with their \n$1\\sigma$ statistical errors, while the colored ellipses represent the theoretical expectations for NO (blue) and IO (red),\nfor two representative values of $\\theta_{23}$ in the lower octant (thin) or upper octant (thick). Two representative values\nof $\\delta$ ($\\pi$ and $3\\pi\/2$) are also marked on each ellipse. \n\nWe first consider the two experiments separately, as shown in \nthe upper left panel for T2K ($\\nu$ vs $\\overline\\nu$) and in the lower right panel for NOvA ($\\nu$ vs $\\overline\\nu$). In T2K, the best agreement of theory and data is reached for NO; in this ordering, there is a clear preference for $\\delta=3\\pi\/2$, and a slight preference for the upper\noctant of $\\theta_{23}$. In NOvA, all the four\ntheoretical ellipses are close to the experimental one, but the overlap is larger in NO; in this ordering,\nthere is a preference for $\\delta=\\pi\/2$ with respect to $3\\pi\/2$, with no significant distinction of the $\\theta_{23}$ octants. \n\nWe then rearrange exactly the same information (both data and predictions)\nby combining T2K and NOvA separately in the $\\nu$ and $\\overline\\nu$ channels,\nas shown in the upper right panel for $\\overline\\nu$, and in the lower left panel for $\\nu$. \nIn both plots, the best agreement of data and theory is now reached for IO. In such ordering,\nthere is  a clear preference for $\\delta=3\\pi\/2$, as well as for the upper (lower) octant in the $\\nu$ ($\\overline\\nu$) channel.   \n\nIn conclusion, Fig.~\\ref{Fig_08} shows that, as far as the three oscillation unknowns are concerned \n(mass ordering, $\\theta_{23}$ octant, CP symmetry), separate and combined T2K and NOvA data provide us with different indications, signalling a ``tension'' between such results; see also the\ndiscussion in \\cite{deSalas:2020pgw,Esteban:2020cvm,Kelly:2020fkv}.\n\\textcolor{black}{Ultimately, the tension reflects the fact that NOvA (T2K) observes relatively symmetric (asymmetric) rates of $\\nu$ and \n$\\overline\\nu$ in their current appearance data.}\n\n\n\\subsection{Remarks}\n\\label{sec:remarks}\n\nThe current hints about the \nthree oscillation unknowns are less converging and more fragile than in the recent past \n\\cite{Capozzi:2020}, due to the T2K-NOvA tension. As for its origin,\nFig.~\\ref{Fig_08} shows that possible statistical fluctuations of the data (at the level of one or two standard deviations) might play a role. However, it makes sense to  \nspeculate if there is more than just statistics behind the tension.  \nOne possibility is to invoke nonstandard neutrino interactions, that would induce\ndifferent effects along the T2K and NOvA baselines \\cite{Chatterjee:2020kkm,Denton:2020uda}. \nBarring new physics beyond the $3\\nu$ paradigm, we surmise that standard\ninteractions of neutrinos in nuclei might also play a systematic role.\n   \nIt is widely recognized that both the total and the differential neutrino cross sections in nuclei are not known \naccurately enough for the purposes of LBL accelerator experiments \\cite{Mosel:2016cwa,Alvarez-Ruso:2017oui}. Roughly speaking,\nnormalization and energy reconstruction uncertainties in $\\nu_\\mu$ cross sections \naffect the measurement of $\\theta_{23}$ and $\\Delta m^2$, respectively, while\nsystematics on the relative $\\nu_\\mu\/\\nu_e$ and $\\nu\/\\overline\\nu$ cross sections  affect the LBL experimental sensitivity\nto $\\theta_{13}$, $\\delta$ and the mass ordering. \nNote, e.g., that a 1\\% systematic error on the reconstructed neutrino energy $E^\\mathrm{rec}_\\nu$ is transferred to $\\Delta m^2$ \nvia the leading $\\Delta m^2\/E$ dependence of the oscillation phase. The formal $\\sim\\!1 \\%$ error on $\\Delta m^2$ \nemerging from the global fit (Table~\\ref{Tab:Synopsis}) implicitly posits \nthat energy reconstruction errors are known at sub-percent level and are independent in different experiments,\nwhich may be an optimistic representation of the current uncertainties.  \n\nWith increasingly higher statistics and accuracy, cross-section\nsystematics may start to affect both known and unknown oscillation parameters\nextracted from detailed energy spectra.  Possible parameter biases have been shown to arise by swapping \ndifferent cross section models in simulations of prospective LBL data   \n\\cite{Coloma:2013rqa,Coloma:2013tba,Benhar:2015wva}.  We remark that\na percent-level bias on $\\Delta m^2$ as measured at LBL\naccelerators, in comparison with the $\\Delta m^2$ measurement at reactors, might alter the current combined preference for NO\n(see Fig.~\\ref{Fig_06} and related discussion).\nAlthough all these effects can be reduced \nby tuning interactions models to cross-section data from near detectors \\cite{T2K2020,NOvA2020}, \nas a matter of fact T2K and NOvA use two different such models, while no single  model or neutrino generator\ncan be currently tuned to agree with world cross-section data \\cite{Alvarez-Ruso:2017oui,Barrow:2020gzb}. \n\n\nSummarizing, the global analysis of current data shows a subtle interplay between the known oscillation parameters \n\\textcolor{black}{$(|\\Delta m^2|,\\,\\sin^2\\theta_{23})$} \nand the three unknowns $(\\delta,\\,\\mathrm{sign}(\\theta_{23}-\\pi\/4),\\,\\mathrm{sign}(\\Delta m^2))$, \nas discussed through Figs.~\\ref{Fig_05}--\\ref{Fig_07}. Although there are overall hints in favor of NO\n(at $2.5\\sigma$), CP violation (at $1.6\\sigma$) and lower $\\theta_{23}$ octant  (at $1.6\\sigma$), the  T2K--NOvA tension  (Fig.~\\ref{Fig_08}) warrants some caution. Neutrino interaction systematics might affect all these parameters, in a way that escapes control in\nglobal fits by external users, since the complexity of the near-to-far analysis chain\ncan be handled only by the experimental collaborations. \nIt is thus encouraging that T2K and NOvA are planning a joint analysis \\cite{JOINT}. In this context, we practically\nsuggest that these two experiments try to swap interaction models or neutrino generators in their separate simulations, \nso as to gauge the relative size of cross-section systematics and tuning effects, before attempting a combined fit. \nIn general, we suggest that experiments sharing potential relevant systematics \n(e.g., neutrino fluxes $\\Phi_\\alpha$ and interactions in water $\\sigma_\\beta$ for atmospheric neutrinos) collaborate on \na detailed comparison of such uncertainties and possibly towards joint data analyses. \nOf course, experimental developments on $\\Phi_\\alpha$ and $\\sigma_\\beta$ should be accompanied by \ncorresponding advances in nuclear theory.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Nonoscillation data, analysis methods and results}\n\\label{Sec:Nonosc}\n\n\n\n\nIn this section we introduce recent nonoscillation data that were not included in our previous work \\cite{Capozzi:2020}, together with the methodology used for their analysis in terms of the three absolute mass observables $(m_\\beta,\\,m_{\\beta\\beta},\\Sigma)$. We include the latest $m_\\beta$ upper bounds  from the KATRIN experiment \\cite{KATRIN2021} and \nintroduce a method to combine the latest $0\\nu\\beta\\beta$ constraints in terms \nof upper bounds on $m_{\\beta\\beta}$, including correlated uncertainties on their nuclear matrix elements. We also enlarge the ensemble of cosmological datasets presented in \\cite{Capozzi:2020} in the light of the current lively discussion\non tensions in cosmology \\cite{DiValentino:2021izs,DiValentino:Snowmass,Challenge2021}, that might suggest possible inconsistencies among different data (or alterations of the standard $\\Lambda$CDM model).  In particular, we consider an ``alternative'' dataset, that is exempt from the Planck lensing anomaly, at the price of being restricted to recent cosmic microwave background (CMB) anisotropy observations from ACTPol-DR4  \\cite{Aiola:2020azj} plus WMAP9 \\cite{Bennett:2012zja} \nand selected Planck data \\cite{Aghanim:2018eyx,Aghanim:2019ame,Aghanim:2018oex}. The alternative option provides a nonzero best fit and\nmore conservative upper bounds on\n$\\Sigma$, as compared with the ``default''  \ndataset described in \\cite{Capozzi:2020}. \nWe shall highlight the different sensitivities to $\\Sigma$ and to the mass ordering in the default and alternative options, as examples of admissible cosmological variants with rather different impact on global neutrino data analyses.  \n\n\n\n\\subsection{Single beta decay and constraints on $m_\\beta$}\n\n\nThe KATRIN $\\beta$-decay experiment has recently released the results of the second campaign  of measurements \\cite{KATRIN2021}. In combination with the results of the  first campaign \\cite{Aker:2019uuj}, they constrain\nat $1\\sigma$ the effective  squared mass $m^2_\\beta$  as:\n\\begin{equation}\n\\label{eq:KATRIN}\nm_\\beta^2 = 0.1 \\pm 0.3\\ \\mathrm{eV}^2,\n\\end{equation}\nwith an approximately gaussian distribution  around the best fit, currently in the physical region\n$m^2_{\\beta}>0$ \\cite{KATRIN2021} (\nwhile it was negative in the first campaign \\cite{Aker:2019uuj}). The upper bound at 90\\% C.L.\\ ($\\sim\\! 1.6\\sigma$) \ncorresponds to $m^2_\\beta<0.6$~eV$^2$ or $m_\\beta<0.8$~eV, representing the first  constraint on the effective\n$\\beta$-decay mass in the sub-eV range \\cite{KATRIN2021}. Note that variants of the statistical analysis may lead to small \ndifferences in the second significant digit of $m^2_\\beta$ or $m_\\beta$ \\cite{KATRIN2021},\nnot considered herein.  We implement the datum in Eq.~(\\ref{eq:KATRIN}) via a contribution $((x-0.1)\/0.3)^2\\in \\chi^2$, where $x=(m_\\beta\/\\mathrm{eV})^2$. \n\n\\subsection{Neutrinoless double beta decay and constraints on half-lives and $m_{\\beta\\beta}$}\n\n\n\n\nNeutrinoless double beta decay \\cite{PDG6} can be considered as the process of creation of two matter particles (electrons) \\cite{Vissani:2021gdw}, occurring if neutrinos are of Majorana type \\cite{Bilenky:2019gzn}. \nWithin the $3\\nu$ paradigm, the decay half-life $T$ is given by\n\\begin{equation}\n\\label{eq:2beta}\n\\frac{1}{T_i} = G_i |M_i|^2 m^2_{\\beta\\beta} = S_i\n\\end{equation}\nwhere the index $i$ labels the $0\\nu\\beta\\beta$ nuclide, characterized by a phase space $G_i$ and a nuclear matrix element (NME) $M_i$, while $m_{\\beta\\beta}$ is the effective Majorana mass.  The inverse half life $S_i=1\/T_i$ represents, up to a constant factor, the observable decay rate or signal strength. \n\n\n\n\nCurrent experiments are consistent with null signal ($S=0$), placing lower limits on $T$ and upper limits on $m_{\\beta\\beta}$ \\cite{Formaggio:2021nfz} via Eq.~(\\ref{eq:2beta}). In deriving separate and combined limits on $m_{\\beta\\beta}$, two issues arise: (1) experimental results are often given in terms of \n90\\% C.L.\\ bounds on $T$ (say, $T>T_{90}$), with little or no information on the probability distribution of $T$ (or of $S$);\n(2) theoretical uncertainties on the NME are rather large (and correlated among nuclides, e.g., via the axial coupling)\n\\cite{Engel:2016xgb}. See e.g.\\ \n\\cite{Biller:2021bqx} for a recent discussion. \nWe describe below our approach to these issues, in order to build first a probability distribution for half-lives $T_i$ and then, including NME's, for the effective mass $m_{\\beta\\beta}$. \n\n\nWe limit our analysis to experiments placing limits $T_{90}> 10^{25}$~y, namely: \nGERDA \\cite{Agostini:2020xta}\nand MAJORANA \\cite{Alvis:2019sil}\nfor $^{76}$Ge;\nCUORE \\cite{Adams:2021rbc}\nfor $^{130}$Te;\nKamLAND-Zen 400 \\cite{KamLAND-Zen:2016pfg}\nand 800 (preliminary) \\cite{Gando:2020cxo} \nand EXO-200 \\cite{Anton:2019wmi}\nfor $^{136}$Xe. For each experiment we need not only a limit ($T_{90}$) but the  probability distribution of $T_i$ or, equivalently, a function $\\Delta\\chi^2(S_i)$. \nUnfortunately, such detailed information is not provided by current\nexperiments in an explicit or user-friendly way; see \\cite{Biller:2021bqx,Caldwell:2017mqu} for recent attempts to \nparametric reconstructions.\n\nWe adopt a $\\Delta\\chi^2(S_i)$ parametrization inspired by \\cite{Caldwell:2017mqu} and based on the following considerations.\nIn $0\\nu\\beta\\beta$ searches with zero background and nearly null results, the likelihood $\\cal L$ of a signal $S>0$ should be a poissonian with a scaling coefficient $\\mu$ ($\\cal L \\sim \\mathrm{exp}(-\\mu S)$), leading to a linear\ndependence on $S$ ($\\Delta \\chi^2 \\sim \\ln {\\cal{L}} \\propto S$) \\cite{Biller:2021bqx}. In $0\\nu\\beta\\beta$ searches with nonnegligible background subtraction, the dependence is expected to be nearly  gaussian ($\\Delta \\chi^2 \\propto (S-S_0)^2$),\nwhere the best-fit signal $S_0$ may fall either in the physical region $(S_0\\geq 0)$\nor in the unphysical one $(S_0<0)$. All these limiting cases can be covered by a quadratic form\n\\begin{equation}\n\\label{eq:abc}\n\\Delta\\chi^2(S_i) =  a_i\\, S_i^2 + b_i\\,S_i + c_i  \\ ,\n\\end{equation}\nas previously advocated  in \\cite{Caldwell:2017mqu} on\nan empirical basis. Note that the offset $c_i$ is set by the condition that $\\Delta\\chi^2\\geq 0$ \nin the physical region $S_i\\geq0$. In particular, for $a_i>0$, a $\\Delta\\chi^2$ minimum in the physical region implies\n$b_i\\leq 0$ and $c_i=b_i^2\/4a_i$, while in the unphysical one it implies $b_i> 0$ and $c_i=0$. For $a_i=0$ one recovers the linear  limit, that implies $b_i>0$ and $c_i=0$. The case $a_i<0$ is never realized.\n\n\n\n\nIn order to assess the coefficients $(a_i,\\,b_i,\\,c_i)$, we have carefully sifted \nthe information contained in the experimental publications \n\\cite{Agostini:2020xta,\nAlvis:2019sil,\nAdams:2021rbc,\nKamLAND-Zen:2016pfg,\nGando:2020cxo,\nAnton:2019wmi} and in\navailable PhD theses conducted within EXO-200\n\\cite{Jewell,Ziegler}, KamLAND-Zen 400 \\cite{Sayuri} and KamLAND-Zen 800 (preliminary) \\cite{Ozaki}.\nWe find that the linear approximation  advocated in \\cite{Biller:2021bqx} for GERDA, can be \nroughly applied also to MAJORANA (up to subleading corrections at small $S_i$, \nneglected herein). The other experimental bounds on $T_i$ can be reasonably\napproximated by parabolic $\\Delta\\chi^2$ curves, setting the various coefficients $(a_i,\\,b_i,\\,c_i)$. We also\nrequire that our 90\\% C.L.\\ limit $\\Delta \\chi^2(S_{90})=2.706$ reproduces the $T_{90}$ limit reported by each \nexperiment, up to their quoted significant digits. \nWe have further checked that, by shifting the minimum of each $\\Delta\\chi^2$ function from $S=S_0$ to $S=0$,\nthe corresponding 90\\% C.L.\\ limits are in reasonable agreement with the reported sensitivities\nfor the null hypothesis.\nWe are thus confident that current experimental results are fairly well represented by our $\\Delta\\chi^2$'s.\nFinally, we combine different experiments probing the same nuclide,\nby adding up their $\\Delta\\chi^2$ functions. \n\n\n\n\n\n\n\\begin{table}[t!]\n\\centering\n\\resizebox{.8\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\\caption{\\label{tab:abc}\nNeutrinoless double beta decay: Details of the adopted parametrization $\\Delta\\chi^2(S_i)=a_i\\, S_i^2 + b_i\\,S_i + c_i  \n$ for the signal strength $S_i=1\/T_i$, expressed\nin units of 10$^{-26}$~y$^{-1}$. The first two columns report the nuclide and the\nname of the experiment(s). The next three columns report our evaluation of the \ncoefficients $(a_i,\\,b_i,\\,c_i)$, for the various experiments, taken either separately (upper six rows) or in combination\nfor the same nuclide (lower three rows).  The sixth column reports our 90\\% C.L.\\ \n($\\Delta\\chi^2=2.706$) half-life limits $T_{90}$ in units of $10^{26}$~y, to be compared with\nthe experimentally quoted ones in the seventh column (in the same units). Pertinent references are listed in the last column. \n}\n\\begin{ruledtabular}\n\\begin{tabular}{rlrrrccc}\nNuclide \t& Experiment(s) \t\t\t& $a_i~~~$ \t& $b_i~~~$ \t& $c_i~~~$ \t& $T_{90}\/10^{26}\\,\\mathrm{y}$ & $T_{90}$ (expt.)\n\t\t\t\t\t\t\t\t\t\t& References \\\\ \n\\hlin\n$^{76}$Ge\t& GERDA\t\t\t\t\t\t& 0.000 \t& \\textcolor{black}{4.867} \t& 0.000 \t& 1.800\t& 1.8\n\t\t\t\t\t\t\t\t\t\t& \\cite{Agostini:2020xta}\\\\\n$^{76}$Ge\t& MAJORANA\t\t\t\t\t& 0.000\t\t& \\textcolor{black}{0.731} \t& 0.000\t\t& 0.270 & 0.27 \n\t\t\t\t\t\t\t\t\t\t& \\cite{Alvis:2019sil}\\\\\n$^{130}$Te\t& CUORE\t\t\t\t\t\t& 0.245\t\t& $-0.637$ \t& 0.414\t\t& 0.216 & 0.22 \n\t\t\t\t\t\t\t\t\t\t& \\cite{Adams:2021rbc}\\\\\n$^{136}$Xe\t& KamLAND-Zen 400\t\t\t& 0.540\t\t& 2.374\t\t& 0.000\t\t& 1.065 & 1.07 \n\t\t\t\t\t\t\t\t\t\t&\\cite{KamLAND-Zen:2016pfg,Sayuri} \\\\\n$^{136}$Xe\t& KamLAND-Zen 800 prelim.\t& 1.006\t\t& $-0.169$\t& 0.007 \t& 0.580 & 0.58 \n\t\t\t\t\t\t\t\t\t\t&\\cite{Gando:2020cxo,Ozaki}\\\\\n$^{136}$Xe\t& EXO-200\t\t\t\t\t& 0.440\t\t& $-0.338$ \t& 0.065\t\t& 0.350 & 0.35 \n\t\t\t\t\t\t\t\t\t\t& \\cite{Anton:2019wmi,Jewell,Ziegler}\\\\\n\\hlin\n$^{76}$Ge\t& GERDA + MAJORANA\t\t\t& 0.000\t\t& 5.598\t\t& 0.000\t\t& 2.070 & \\textemdash \n\t\t\t\t\t\t\t\t\t\t& This work \\\\\n$^{130}$Te\t& CUORE\t(same as above)\t\t& 0.245\t\t& $-0.637$ \t& 0.414\t\t& 0.216 &  0.22 \n\t\t\t\t\t\t\t\t\t\t& \\cite{Adams:2021rbc} \\\\\n$^{136}$Xe\t& KamLAND-Zen (400 + 800 prelim.) + EXO-200\t\t\t\n\t\t\t\t\t\t\t\t\t\t& 1.986\t\t& 1.867\t\t& 0.000\t\t& 1.267 & \\textemdash \n\t\t\t\t\t\t\t\t\t\t& This work\n\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{minipage}}\n\\end{table}\n\n\nTable~\\ref{tab:abc} reports our numerical results for the coefficients $(a_i,\\,b_i,\\,c_i)$ used in Eq.~(\\ref{eq:abc}),\nfor both separate and combined bounds. [We formally keep up to four significant digits, \nto avoid accumulation of round-off errors in the analysis.] By combining GERDA and MAJORANA, \nwe evaluate a 90\\% C.L.\\ limit as high as $T>2.07\\times10^{26}$~y for the $^{76}$Ge half life, about 15\\% higher than from GERDA alone. A similar improvement is obtained for $^{136}$Xe, reaching $T>1.27\\times10^{26}$~y in the\ncombination of KamLAND-Zen and EXO data. For $^{130}$Te, CUORE alone sets the bound $T>0.22\\times10^{26}$~y.   \n\n\n\n\n\n\n\n\nFigure~\\ref{Fig_09} shows our parametrized $\\Delta\\chi^2$ functions\nin terms of $1\/T=S$ (bottom abscissa) and of $T$ (top abscissa). The left and right panels refer to separate experiments and to combinations for the same nuclide, respectively. The dotted horizontal lines intersect all curves at the 90\\% C.L.\\ limit\n$T_{90}$. Note that the hierarchy of bounds at $90\\%$ C.L.\\ is not necessarily preserved at different statistical levels, since\nsome curves cross each other. This is another reason to suggest that the experimental collaborations \nexplicitly provide their probability\nprofiles for $T$, rather than focusing on a single C.L.\\ limit ($T_{90}$), that \nprovides a poor summary of the data and a limited comparison of different results.\n\n\n\n\\begin{figure}[b!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.88\\textwidth]{Fig_09.pdf}\n\\caption{\\label{Fig_09}\n\\footnotesize \nNeutrinoless double beta decay: $\\Delta\\chi^2$ functions as defined in Eq.~(\\ref{eq:abc}),\nin terms of the half life $T$ (top abscissa) and of $S=1\/T$ (bottom abscissa). \nLeft and right panels: separate experiments and their combinations for the same nuclide, respectively.\nDotted horizontal lines intersect the curves at 90\\% C.L.}\n\\end{minipage}\n\\end{figure}\n\nIn order to translate constraints from $T_i$ to $m_{\\beta\\beta}$, one needs to know the phase space factors $G_i$ and\nthe nuclear matrix elements $|M_i|$ in Eq.~(\\ref{eq:2beta}). The $G_i$ can be accurately computed, see\n\\cite{Deppisch:2020ztt,Mirea:2015nsl} for recent calculations.  The $|M_i|$ embed complex nuclear physics, currently treated\nin a variety of approaches that, unfortunately, still carry significant uncertainties despite the \ntheoretical progress, see e.g.\\ the recent reviews in  \\cite{Engel:2016xgb,Coraggio:2020iht,Cirigliano:2020yhp,Dolinski:2019nrj,Ejiri:2019ezh,Vergados:2016hso,DellOro:2016tmg,Barea:2015kwa}. It is \ncommon practice to select a set of published values $\\{M_i\\}$ in order to obtain a set of upper  \nbounds $\\{m_{\\beta\\beta}\\}$, whose \nspread may be taken as indicative of the theoretical uncertainties; see \\cite{Biller:2021bqx} for a very recent application.\nHowever, this procedure overlooks significant correlations among the NME uncertainties of different nuclides \n\\cite{Faessler:2008xj,Faessler:2013hz,Faessler:2011rv,Lisi:2015yma} \nthat, as shown below, are as important as the uncertainties themselves in obtaining conservative bounds.\n\nIn a given nuclear model, estimating NME covariances requires massive numerical experiments to generate many $M_i$ variants. To our knowledge, this task has been performed only in \\cite{Faessler:2008xj} within the quasiparticle random phase approximation (QRPA), by  varying the axial coupling $g_A$, the short-range correlations, the model basis, and \nthe renormalization procedure, while requiring consistency with available $2\\nu\\beta\\beta$ data. Apart \nfrom the subsequent papers \\cite{Faessler:2013hz,Faessler:2011rv,Lisi:2015yma}, and\ndespite the potential relevance of NME covariance issues  \\cite{Engel:2015wha},\nwe are aware of just one independent\n(but preliminary) correlation matrix estimate in a different model \\cite{Gautam:2017bgq} \nand of a single statistical analysis \\cite{Ge:2017erv} based \non the correlations in \\cite{Faessler:2008xj}.\nIn the absence of novel estimates of NME covariances, we shall use the only available results of \\cite{Faessler:2008xj} at face value. \nWe have checked\nthat, in any case, the NME uncertainties estimated therein are conservative enough to  cover within $\\pm2\\sigma$  most\n(and within $\\pm3\\sigma$ all)  of the $M_i$ values\ncompiled in  \\cite{Engel:2016xgb,Coraggio:2020iht,Cirigliano:2020yhp,Dolinski:2019nrj,Ejiri:2019ezh,Vergados:2016hso,DellOro:2016tmg,Barea:2015kwa} for the three nuclides.\n\nWe summarize and use the results in \\cite{Faessler:2008xj} as follows. In order to deal with large $|M_i|$ variations, \npossibly hitting the unphysical range $|M_i|<0$, the (adimensional) $|M_i|$ values are parametrized in terms of the logarithms $\\eta_i$,\n\\begin{equation}\n |M_i| = e^{\\eta_i} = e^{\\overline \\eta_i+\\Delta \\eta_i} = |\\overline M_i|e^{\\Delta \\eta_i}\\ ,  \n\\end{equation}\nwhere the overlined symbols represent central values, $|\\overline M_i|=e^{\\overline\\eta_i}$. The index $i=1,\\,2,\\,3$ runs over $^{76}$Ge, $^{130}$Te, $^{136}$Xe.\n  The NME variations $\\Delta \\eta_i$ are endowed with variances $\\sigma^2_i$\nand a covariance matrix $\\sigma_{ij}=\\rho_{ij}\\sigma_i\\sigma_j$, whose inverse defines the weight matrix $w_{ij}=(\\sigma_{ij})^{-1}$. For any choice of $m_{\\beta\\beta}$ and of $\\Delta \\eta_i$, the signal strength in the\n$i$-th nuclide is given by\n\\begin{equation}\nS_i = G_i  |M_i|^2 m^2_{\\beta\\beta} = q_i m^2_{\\beta\\beta} e^{2\\Delta \\eta_i}\\ ,  \n\\end{equation}\nwhere $q_i=G_i |\\overline M_i|^2$. The strengths $S_i$ carry two $\\Delta \\chi^2$ contributions: an experimental one coming from $\\Delta \\chi^2(S_i)$,   and a theoretical one coming from NME covariances. These contributions are coupled by---and must be minimized over---the three variations $\\{\\Delta\\eta_i\\}$\n\\begin{eqnarray}\n\\chi^2(m_{\\beta\\beta}) &=& \\min_{\\Delta_i}\\left(\n\\sum_{i=1}^3 \\Delta\\chi^2(S_i)+\\sum_{i,j=1}^3 w_{ij}\\, \\Delta\\eta_i\\Delta\\eta_j\\right)\\\\\n&=& \\min_{\\Delta_i}\\left( \\sum_{i=1}^3 \\left( a_i q^2_i m^4_{\\beta\\beta}e^{4\\Delta\\eta_i}\n+b_i q_i m^2_{\\beta\\beta}e^{2\\Delta\\eta_i} + c_i\\right) + \\sum_{i,j=1}^3 w_{ij}\\, \\Delta\\eta_i\\Delta\\eta_j\\right)\\ ,\n\\end{eqnarray}\nwhere the first line holds in general, while the second line refers to the specific parametrization in\nEq.~(\\ref{eq:abc}). \nMinimization yields three coupled equations in the three $\\Delta\\eta_i$ unknowns, to be solved numerically. \nNeglecting NME correlations (or errors)\namounts to setting $\\rho_{ij} = \\delta_{ij}$ (or $\\Delta\\eta_i=0$). Bounds for a single nuclide are obtained by selecting a specific index $i$. Subtraction of a $\\chi^2_{\\min}$ offset (if any) yields\nthe desired $\\Delta\\chi^2(m_{\\beta\\beta})$. Table~\\ref{tab:Old} reports the values of $\\sigma_{ij}$ and $q_i$ used herein. \n\n\n \n  \nFigure~\\ref{Fig_10} shows the resulting bounds on $m_{\\beta\\beta}$ for \nthe three nuclides taken separately (left panel) and in combination (right panel). The solid and dotted curves refer to our estimates with and and without NME uncertainties, respectively. Currently, the most constraining results\nare obtained by combining $^{76}$Ge data, followed by weaker constraints from $^{136}$Xe and  $^{130}$Te data.\nIn the\nright panel, the case with uncorrelated NME uncertainties is also shown (dashed line).\nThe effect of correlations is noticeable and leads to more conservative bounds;\nin fact, when the NME of different nuclides are positively correlated,\nthey are more likely to become all smaller at the same time (with respect to the uncorrelated case), allowing larger \nvalues of $m_{\\beta\\beta}$. Including correlated errors, we obtain the overall bound\n$m_{\\beta\\beta}<0.11$~eV  at $2\\sigma$; the same bound was previously estimated (although in a less refined way) as $m_{\\beta\\beta}<0.14$~eV in \\cite{Capozzi:2020} and as $m_{\\beta\\beta}<0.18$~eV in \\cite{Capozzi:2017ipn}, reflecting the steady experimental progress in the last few years. \n\n \n\n\n\n\\begin{table}[b!]\n\\centering\n\\begin{minipage}{.5\\textwidth}\n\\caption{\\label{tab:Old} \\footnotesize \nNeutrinoless double beta decay: NME covariance matrix $\\sigma_{ij}$ \nand auxiliary coefficients $q_i=G_i|\\overline M_i|^2$, as derived from the results in\n\\cite{Faessler:2008xj}; see the text for details. The $q_i$\nare given in units of $10^{-26}\\,\\mathrm{y}^{-1}\\, \\mathrm{eV}^{-2}$, for $m_{\\beta\\beta}$ expressed in eV. }\n\\begin{ruledtabular}\n\\begin{tabular}{cc|ccc|c}\n$i$ & Nuclide  &  & $\\sigma_{ij}$ & & $q_i$ \\\\\n\\hline\n1 & $^{76}$Ge  & 0.0790 &        &        &  56.6 \\\\\n2 & $^{130}$Te & 0.0920 & 0.0135 &        & 210.0 \\\\\n3 & $^{136}$Xe & 0.0975 & 0.1437 & 0.1858 &  73.1 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{minipage}\n\\end{table}\n\n\n \n\\newpage\n\n\nWe emphasize that the above methodology can be applied\n to $0\\nu\\beta\\beta$ data consistent with either a null signal or a positive detection.\nIt may include generic  likelihoods  for the half-lives $T_i$ (hopefully provided by the experimental collaborations) \n and alternative evaluations of the NME covariances (possibly computed in different nuclear models). \n\n\n\n\n  \n\n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.88\\textwidth]{Fig_10.pdf}\n\\caption{\\label{Fig_10}\n\\footnotesize \nNeutrinoless double beta decay: Our estimated bounds on $m_{\\beta\\beta}$, expressed in terms of $N_\\sigma=\\sqrt \\Delta\\chi^2$. \nThe left and right panels refer, respectively, to separate and combined bounds from the three nuclides,  \nwith (solid) or without (dotted) NME uncertainties. In the right panel, \nthe case with uncorrelated uncertainties is also shown (dashed).}\n\\end{minipage}\n\\end{figure}\n\n\n\n\n\n\n\n  \n\n\n\n \n\n\n\n\n\\vspace*{-2mm}\n\\subsection{Cosmology and constraints on $\\Sigma$}\n\\vspace*{-1mm}\n\nIn this section we discuss various choices for cosmological data \ncombinations, enlarging the set of cases previously considered in \\cite{Capozzi:2020}, in order to deal\nwith known tensions about lensing data. We then focus \non two specific cases, dubbed as default and alternative options, leading to different implications for $\\Sigma$ and \nthe sensitivity to mass ordering.  \n\nWe remind that in \\cite{Capozzi:2020} we\nanalyzed the following data in various combinations:  the complete Planck 2018 data (Planck) on the\nangular CMB temperature power spectrum (TT) \nplus polarization power spectra (TE, EE) \\cite{Aghanim:2018eyx,Aghanim:2019ame}\nand  lensing reconstruction power spectrum (lensing) \\cite{Aghanim:2018oex}; \na compilation of Baryon Acoustic Oscillation (BAO) measurements, given by data from the 6dFGS~\\cite{6dFGS}, SDSS MGS~\\cite{mgs}, and BOSS DR12~\\cite{bossdr12} surveys; and the Hubble constant datum [$H_0$({\\tt R19})] from HST observations of Cepheids in the Large Magellanic Cloud measurements \\cite{Riess:2019cxk}. We assume the standard 6-parameter $\\Lambda$CDM model augmented with nonzero neutrino masses ($\\Lambda$CDM+$\\Sigma$) and, in some cases, we add\nan extra empirical parameter $A_{\\mathrm{lens}}$ to marginalize over the excess amount of gravitational lensing \nemerging in the fits of the Planck data \\cite{Aghanim:2018eyx};\n\\textcolor{black}{see also \\cite{RoyChoudhury:2019hls} for a similar approach.}\n\nHere we also consider an alternative way to deal with the Planck lensing problem, in the light of the wider debate\nabout data tensions in the standard $\\Lambda$CDM model \\cite{DiValentino:2021izs,Challenge2021}. In particular,\nrather than introducing an additional parameter to account for unknown systematics, one may consider\nrestricting the analysis to alternative CMB datasets not affected by internal or mutual tensions. \nA possibility is offered by the combination of the recent CMB anisotropy data from ACTPol-DR4\n \\cite{Aiola:2020azj} with WMAP9 data  \\cite{Bennett:2012zja}, together with a \nPlanck-derived conservative gaussian prior  on the optical depth to reionization $\\tau$\n[$\\tau_\\mathrm{prior}=0.065 \\pm 0.015$]. The ACT Collaboration explored this combination in  \\cite{Aiola:2020azj},\nfinding no evidence for a lensing anomaly, and obtaining a relaxed\nupper bound on the total neutrino mass, $\\Sigma < 1.2$~eV at $2\\sigma$. \nUsing the same data combination, we get an upper bound in excellent agreement, $\\Sigma<1.21$~eV; interestingly, we also find that the best fit is at $\\Sigma\\simeq 0.70$~eV, with no significant difference between NO and IO. If we replace the gaussian prior on $\\tau$\nwith the actual Planck polarization data at large angular scale (dubbed Planck LowE) \n\\cite{Aghanim:2019ame}, that directly constrain the value of $\\tau$, we obtain a slightly stronger upper bound  $\\Sigma<1.12$~eV (with a best fit at \n$\\Sigma\\simeq 0.58$~eV). Finally, in order to achieve a more complete combination of CMB data not in tension with each other,\n we further add to ACT and WMAP the independent Planck lensing reconstruction results  \n\\cite{Aghanim:2018oex}, that do not suffer of the lensing anomaly. In this case\nthe upper bound becomes  $\\Sigma<0.96$~eV, with a best fit at $\\Sigma\\simeq 0.54$~eV, as obtained with CMB-only data.\nOf course, by including additional data, such as BAO measurements, the upper bound on $\\Sigma$ would \nbe pushed back to $\\sim 10^{-1}$~eV (or even below as shown in \\cite{DiValentino:2021hoh}), \nat the price of a significant mutual tension among different datasets;\nthese fits, that would be more comprehensive but also more discordant, are not further discussed herein. \n\n\n\\newpage \n\nTable~\\ref{Tab:Cosmo} reports, for convenience, the bounds on $\\Sigma$ for both the cosmological inputs\nconsidered in  \\cite{Capozzi:2020} (cases 0--9) and the new ones discussed above (cases 10--12). These\ninputs can be roughly divided into two categories: a first group (cases 0--6) where one includes at face value Planck CMB results (plus\nother data) despite the Planck lensing anomaly, obtaining\nrelatively strong upper bounds on $\\Sigma$ and a noticeable sensitivity to mass ordering; \nand a second group (cases 7--12), where one ``solves'' the lensing anomaly (by either adding an extra model parameter\nor by considering alternative CMB data), with weaker upper bounds on $\\Sigma$ and no significant sensitivity\nto NO vs IO. Hereafter we focus on just two representative cases for these two different categories, namely, case \\#3 (dubbed ``default'' as in \\cite{Capozzi:2020}) and case \\#12 (dubbed ``alternative''). \n\n\n\n\n\n\\begin{table}[t]\n\\resizebox{.78\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\\caption{\\label{Tab:Cosmo} \n\\footnotesize  Results of the global $3\\nu$ analysis of cosmological data \nwithin the standard $\\Lambda\\mathrm{CDM}+\\Sigma$ model (possibly augmented with the $A_\\mathrm{lens}$ parameter). \nThe inputs numbered from 0 to 9 are the same as in \\cite{Capozzi:2020},  and refer to various combinations\nof the Planck 2018 angular CMB temperature power spectrum (TT) \nplus polarization power spectra (TE, EE), lensing potential power spectrum (lensing),  Barion Acoustic Oscillations (BAO), and the Hubble constant from HST observations of Cepheids in the Large Magellanic Cloud, $H_0$({\\tt R19}).\nThe inputs numbered from 10 to 12 are new and refer to ACTPol-DR4 and WMAP9 data, in combination with  \na prior on optical depth to reionization ($\\tau_\\mathrm{prior}$),  \nPlanck polarization data at large angular scale (lowE), and lensing data. \nFor each case we report the $2\\sigma$ upper bound on the sum of $\\nu$ masses $\\Sigma$ (marginalized over NO and IO), together\nwith the $\\Delta\\chi^2$ difference between IO and NO, using cosmology only. In the last two columns, we report the \nsame information as in the previous two columns, but using  cosmological data plus $ m_\\beta$ and  $m_{\\beta\\beta}$ constraints. The specific cases numbered 3 and 12 are dubbed as default\nand alternative, see the text for details.}\n\\vspace*{0mm}\n\\centering\n\\begin{ruledtabular}\n\\begin{tabular}{cllcc|cc}\n\\multicolumn{3}{l}{Cosmological inputs for nonoscillation data analysis}  & \\multicolumn{2}{c|}{Results: Cosmo only} & \\multicolumn{2}{c}{Cosmo + $m_\\beta$ + $m_{\\beta\\beta}$}  \\\\[1mm]\n\\# & Model & Data set & $\\Sigma$ ($2\\sigma$)   & $\\Delta\\chi^2_\\mathrm{IO-NO}$ & $\\Sigma$ ($2\\sigma$)   & $\\Delta\\chi^2_\\mathrm{IO-NO}$ \\\\[1mm]\n\\hlin\n 0 & $\\Lambda\\mathrm{CDM}+\\Sigma$\t\t\t\t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} \t\t\t\t\t\t\t\t\t& $<0.34$ eV & $ 0.9$ & $<0.32$ eV & $ 1.0$   \\\\\n\\hlin\n 1 & $\\Lambda\\mathrm{CDM}+\\Sigma$\t\t\t\t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + lensing\t\t\t\t\t\t\t& $<0.30$ eV & $ 0.8$ & $<0.28$ eV & $ 0.9$  \\\\\n 2 & $\\Lambda\\mathrm{CDM}+\\Sigma$ \t\t\t\t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + BAO\t\t\t\t\t\t\t\t& $<0.17$ eV & $ 1.6$ & $<0.17$ eV & $ 1.8$  \\\\\n 3 & $\\Lambda\\mathrm{CDM}+\\Sigma$ \t\t\t\t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + BAO + lensing\t\t\t\t\t\t& $<0.15$ eV & $ 2.0$ & $<0.15$ eV & $ 2.2$  \\\\\n\\hlin\n 4 & $\\Lambda\\mathrm{CDM}+\\Sigma$\t\t\t\t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + lensing + $H_0$({\\tt R19})\t& $<0.13$ eV & $ 3.9$ & $<0.13$ eV & $ 4.0$  \\\\\n 5 & $\\Lambda\\mathrm{CDM}+\\Sigma$ \t\t\t\t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + BAO + $H_0$({\\tt R19}) \t\t& $<0.13$ eV & $ 3.1$ & $<0.13$ eV & $ 3.2$  \\\\\n 6 & $\\Lambda\\mathrm{CDM}+\\Sigma$ \t\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + BAO + lensing + $H_0$({\\tt R19}) \\ \\\t& $<0.12$ eV & $ 3.7$ & $<0.12$ eV & $ 3.8$  \\\\\n\\hlin\n 7 & $\\Lambda\\mathrm{CDM}+\\Sigma+A_\\mathrm{lens}$\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + lensing\t\t\t\t\t\t\t& $<0.77$ eV & $ 0.1$ & $<0.66$ eV & $ 0.1$  \\\\\n 8 & $\\Lambda\\mathrm{CDM}+\\Sigma+A_\\mathrm{lens}$\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + BAO\t\t\t\t\t\t\t\t& $<0.31$ eV & $ 0.2$ & $<0.30$ eV & $ 0.3$  \\\\\n 9 & $\\Lambda\\mathrm{CDM}+\\Sigma+A_\\mathrm{lens}$\t& Planck {\\scriptsize TT,\\,TE,\\,EE} + BAO + lensing\t\t\t\t\t\t& $<0.31$ eV & $ 0.1$ & $<0.30$ eV & $ 0.2$  \\\\\n\\hlin\n 10 & $\\Lambda\\mathrm{CDM}+\\Sigma$\t& ACT + WMAP + $\\tau_\\mathrm{prior}$\t\t\t\t\t\t\t\t\t\t\t\t\t& $<1.21$ eV & $-0.1$ & $<1.00$ eV & $ 0.1$  \\\\\n 11 & $\\Lambda\\mathrm{CDM}+\\Sigma$\t& ACT + WMAP +  Planck lowE\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& $<1.12$ eV & $-0.1$ & $<0.87$ eV & $ 0.1$  \\\\\n 12 & $\\Lambda\\mathrm{CDM}+\\Sigma$\t& ACT + WMAP + Planck lowE + lensing\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& $<0.96$ eV & $0.0$  & $<0.85$ eV & $ 0.1$  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{minipage}}\n\\end{table}\n\n\n\nFigure~\\ref{Fig_11} shows the $\\Delta\\chi^2$ curves for the default and alternative cases. In the default case, cosmological data\npush $\\Sigma$ towards its lowest physical values in both IO and NO, and favors the latter at the level of $\\sim\\! 1.5\\sigma$. In the alternative case, there\nis a  preference $(<2\\sigma)$ for higher $\\Sigma$ values, with a best fit at $\\Sigma\\simeq 0.54$~eV and an upper \nlimit $\\Sigma< 0.96$~eV at $2\\sigma$, while there are only minor differences between NO and IO.\nRoughly speaking, the alternative case corresponds to a putative cosmological ``signal'' for\nneutrino masses, equivalent to $\\Sigma \\simeq 0.54 \\pm 0.22$~eV (with symmetrized $1\\sigma$ errors). \n\n\n\n\n\n\\begin{figure}[b!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.48\\textwidth]{Fig_11.pdf}\n\\vspace*{-3mm}\n\\caption{\\label{Fig_11}\n\\footnotesize \nCosmology: $\\Delta\\chi^2$ curves for NO (blue) and IO (red) for the cases numbered in Table~\\ref{Tab:Cosmo} as \\#3 (left, solid) and \\#12 (right, dashed), taken as representative of default and alternative options, respectively. The cases \n\\#10 and \\#11 (not shown) would be qualitatively similar to case \\#12.}\n\\end{minipage}\n\\end{figure}\n\n\\newpage\n\\textcolor{black}{\nWe remark that, at fixed $\\Sigma$, the small differences between NO and IO fits are due to: a slight sensitivity\nof cosmological data to the different ordering of $\\nu$ masses at small $\\Sigma$; \nthe conversion from fit probability densities $P(\\Sigma)$ to $\\chi^2(\\Sigma)$ functions, as the $P$ normalization\ncovers different physical $\\Sigma$ ranges in NO and IO; and, to a lesser extent ($\\Delta\\chi^2<0.1$), to \nnumerical fit inaccuracies for $P\\to 0$ (i.e., for high $\\chi^2)$. See also the comments in Sec.~{II~C} of \\cite{Capozzi:2017ipn}.\n} \n   \nSummarizing, the two  cases in Fig.~\\ref{Fig_11} correspond to two qualitatively different outcomes, \nthat might persist in future cosmological data analyses, as opposite examples of the tradeoff between \ncompleteness and consistency of inputs.\n On the one hand, by \ncombining various datasets at face value (regardless of possible tensions), one may obtain strong upper limits,\n$\\Sigma<O(10^{-1})$~eV, \nwith some sensitivity \nto mass ordering (typically in favor of NO, where $\\Sigma$ attains its lowest possible values).  \nOn the other hand, by combining selected (and mutually consistent) data sets, one may end up with\nmore relaxed upper bounds on $\\Sigma$,  possibly shifting the best fit towards $\\Sigma\\sim (\\mathrm{few} \\times) 10^{-1}$~eV, at the price of a reduced sensitivity to mass ordering. We think that, at present, both options deserve  to be explored, especially because they imply rather different\noutcomes in combination with other (non)oscillation neutrino data. \n\n\\vspace*{-3mm}\n\\subsection{Results on pairs of nonoscillation observables}\n\\vspace*{-1mm}\n\nThe nonoscillation observables $(m_\\beta,\\,m_{\\beta\\beta},\\,\\Sigma)$ are strongly and positively correlated, \nvia their common dependence on the absolute neutrino mass scale. Contrary to the case of oscillation parameters, it is useful to show first the \nresults on pairs of observables, and then the projections on single ones.  With respect to our\nprevious works \\cite{Capozzi:2017ipn,Capozzi:2020}, we shall use linear (rather than logarithmic) coordinates \nas, e.g., advocated for the pair $(\\Sigma,\\,m_{\\beta\\beta})$ in \\cite{DellOro:2016tmg,DellOro:2014ysa}.\nSince cosmological data play a major role in constraining the nonoscillation parameter space, we shall discuss the two different options (default and conservative) defined in the previous Section. \n\n\nFigure~\\ref{Fig_12} shows the correlation bands at $2\\sigma$ for the pairs $(\\Sigma,\\,m_\\beta)$ and  $(\\Sigma,\\,m_{\\beta\\beta})$ in linear scales, \nincluding only the constraints from oscillation data, for NO and IO taken separately\n(i.e., without the offset $\\Delta \\chi^2_{\\mathrm{{IO}-{NO}}}$). In the top panel, the bands have a  tiny width, reflecting the small fractional errors on the \noscillation parameters ($\\delta m^2,\\,\\Delta m^2,\\,\\theta_{12},\\,\\theta_{13}$) relevant for the pair\n$(m_\\beta,\\,\\Sigma)$. In the bottom panel, the widening of the bands is almost entirely due to the unknown Majorana phases\nin $m_{\\beta\\beta}$. See also \\cite{Capozzi:2020} and Fig.~2 therein for analogous correlation plots in logarithmic scales. \n\n\n\\vspace*{-5mm}\n\\begin{figure}[b!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.45\\textwidth]{Fig_12.pdf}\n\\vspace*{-4mm}\n\\caption{\\label{Fig_12}\n\\footnotesize \nConstraints at $2\\sigma$ placed by current oscillation data in the planes charted by the nonoscillation observables \n$(\\Sigma,\\,m_\\beta)$ in the top panel, and $(\\Sigma,\\,m_{\\beta\\beta})$ in the bottom panel.\nThe blue and red curves refer to NO and IO, respectively.}\n\\end{minipage}\n\\end{figure}\n\\newpage\n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.92\\textwidth]{Fig_13.pdf}\n\\caption{\\label{Fig_13}\n\\footnotesize \nConstraints at $2\\sigma$ placed by current oscillation data and nonoscillation data \nin the same planes as in Fig.~\\ref{Fig_12}, in the default (left panel) and alternative\n(right panel) options for cosmological inputs. The dots mark\nthe best fits. }\n\\end{minipage}\n\\end{figure}\n\n\\vspace*{10mm}\nFigure~\\ref{Fig_13} shows how the data from $\\beta$ decay, $0\\nu\\beta\\beta$ decay and cosmology further\nconstrain the $2\\sigma$ bands in Fig.~\\ref{Fig_12}, for the two cosmological input options considered. In the default case, \ncosmological bounds on $\\Sigma$ dominate---via  correlations---the \nconstraints on $m_\\beta$ and $m_{\\beta\\beta}$, which are squeezed to the relatively small $2\\sigma$\nregions around the best fits, located close to the lowest possible values for $\\Sigma$ in both NO and IO.\nIn the alternative case, there is an interplay between cosmological and $0\\nu\\beta\\beta$ data: \nthe first would prefer $\\Sigma\\simeq 0.58$~eV, implying relatively high values \nfor the Majorana mass ($m_{\\beta\\beta}>0.06$~eV, see Fig.~\\ref{Fig_12}); however, such values \nare disfavored by $0\\nu\\beta\\beta$ data at  $>1\\sigma$ in Fig.~\\ref{Fig_10}. A best-fit  \ncompromise is reached for intermediate values, $\\Sigma\\sim 0.4$~eV and $m_{\\beta\\beta}\\simeq 0.05$~eV,\nsurrounded by large $2\\sigma$ allowed regions. In the right panel, note that both \ncosmology and  $0\\nu\\beta\\beta$ data constrain the correlations bands from above,\nleading to a joint $2\\sigma$ bound $\\Sigma<0.85$~eV, stronger than the bound from cosmology only  \n($\\Sigma<0.96$~eV, see also Table~\\ref{Tab:Cosmo}). In all cases,\ncurrent $\\beta$-decay data play a minor role in the overall fit.\n\nThe implications of Fig.~\\ref{Fig_13} can be summarized  as follows. In the default case, \nit appears that the current KATRIN experiment (probing $m_\\beta>0.2$~eV) is\nnot expected to find any signal, while planned $0\\nu\\beta\\beta$ experiments are expected to probe at\nleast the region covered by both NO and IO ($m_{\\beta\\beta}>0.02$~eV). The region covered \nonly by NO ($m_{\\beta\\beta}<0.02$~eV) is more difficult to probe, and becomes eventually   \nprohibitive as $m_{\\beta\\beta}$ vanishes, see e.g. \\cite{Dolinski:2019nrj,Pascoli:2007qh,Penedo:2018kpc}.\nIn the alternative case, a much larger phase space is amenable to $\\beta$ decay and $0\\nu\\beta\\beta$ decay\nsearches. Cosmological searches may find a signal for $\\Sigma$ in a wide sub-eV range. \nNeutrinoless double beta decay data might find a signal for $m_{\\beta\\beta}$ anywhere \nbelow the current bounds. \nThe KATRIN experiment might find a signal in its sensitivity region $(m_\\beta>0.2)$~eV,\nor at least strengthen significantly the upper bounds on $m_\\beta$. \n\n\\newpage\n\n\n\n\n\n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.95\\textwidth]{Fig_14.pdf}\n\\caption{\\label{Fig_14}\n\\footnotesize \n$N_\\sigma$ bounds on the single nonoscillation parameters \n$\\Sigma$ (left), $m_{\\beta\\beta}$ (center) and $m_\\beta$ (right), assuming default cosmological inputs. \nThe combination of nonoscillation data induces the offset\nbetween the absolute minima in IO (red) and NO (blue).\n}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.95\\textwidth]{Fig_15.pdf}\n\\caption{\\label{Fig_15}\n\\footnotesize \nAs in Fig.~\\ref{Fig_14}, but assuming alternative cosmological inputs. \n}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Results on single nonoscillation observables}\n\nFigures~\\ref{Fig_14} and \\ref{Fig_15} show the projected bounds on single nonoscillation parameters for the \ndefault and alternative cases, respectively. In both cases we account for the NO--IO offset coming\nfrom the combination of all nonoscillation data (i.e., the rightmost numbers in rows \\#3 and \\#12 of Table~\\ref{Tab:Cosmo}\n(that were omitted in the previous Figs.~\\ref{Fig_12} and \\ref{Fig_13}). A vertical rise of $N_\\sigma$ occurs\nwhen the lower physical limits are reached.\nThese two figures quantify \nprevious considerations about the default and alternative options: the first exemplifies the case \nof strong upper bounds on $\\Sigma$ from cosmology, accompanied by some sensitivity to the mass ordering, \nand by hard-to-probe phase spaces for $m_\\beta$ and $m_{\\beta\\beta}$; the second represents the case of \nweaker upper bounds (and a possible signal) for $\\Sigma$, with scarce sensitivity to the mass ordering but \nmore optimistic expectations for $m_\\beta$ and $m_{\\beta\\beta}$ signals. Together with \n\\textcolor{black}{Fig.~\\ref{Fig_03}},  \nthe above Figs.~\\ref{Fig_14} and \\ref{Fig_15} provide a neat summary \nof what we (do not) know in the standard $3\\nu$ paradigm.%\n\\footnote{\n\\textcolor{black}{\nAt present, we stick to the viewpoint expressed in \\cite{Fogli:2004as} and prefer to project away \nunobservable quantities, such as the lightest neutrino mass $m_0$ and the two Majorana phases $\\eta_{1,2}$ (as defined in \\cite{PDG1}). Of course,\nwhen significant (and convergent) signals will emerge among the three observables\n$(m_\\beta,\\,m_{\\beta\\beta},\\,\\Sigma)$, meaningful bounds on $m_0$ (and possibly weak hints on $\\eta_{1,2}$) \nmay also be derived.}\n}\n\n\n\n\\section{Synthesis}\n\nWe conclude our work by merging the information coming from the analysis of oscillation and nonoscillation data,\nthat have one important observable in common: the mass ordering.  Figure~\\ref{Fig_16} shows a histogram with separate and combined contributions to the \n$\\Delta\\chi^2_{\\mathrm{IO-NO}}$. The first bin adds up the contributions from oscillation data, starting from\nthe negative one in the combination of LBL accelerator, solar and KamLAND data, that becomes positive by adding SBL reactor data, and further increases with atmospheric data. The second bin shows the range spanned by all the cases considered in Table~\\ref{Tab:Cosmo}, for the fit to cosmological data only.\n The third bin shows the slight change induced by adding \ncurrent constraints on $m_\\beta$ and $m_{\\beta\\beta}$, that provide an extra upward shift (see Tab.~\\ref{Tab:Cosmo}).\nThe fourth bin adds up the contents of  the first and third bins, providing an overall indication\nin favor of NO in the range $\\sim 2.5$--$3.2\\sigma$.  \n Although none of the single oscillation or nonoscillation data sets provides compelling evidence for \n normal ordering yet, their current combination clearly favors this option at a global $\\sim 3\\sigma$ level. \n\n\\begin{figure}[t!]\n\\begin{minipage}[c]{0.85\\textwidth}\n\\includegraphics[width=0.8\\textwidth]{Fig_16.pdf}\n\\vspace*{-7mm}\n\\caption{\\label{Fig_16}\n\\footnotesize \nBreakdown of contributions to the IO-NO $\\chi^2$ difference from oscillation and nonoscillation data. The latter span the range \nof all the cosmological input variants reported in Table~\\ref{Tab:Cosmo}, as indicated by the horizontal lines (the thick one \ncorresponding to the  \ndefault case).}\n\\end{minipage}\n\\end{figure}\n\n\nIn conclusion, the main results of our global analysis can be essentially summarized in terms of bounds $N_\\sigma=N_\\sigma(p)$,\nas shown in the following figures: \nFig.~\\ref{Fig_03} for the neutrino oscillation parameters \n$p=\\delta m^2,\\,\n\\textcolor{black}{|\\Delta m^2|},\n\\,\\theta_{12},\\,\\theta_{23},\\,\\theta_{13},\\,\\delta$; \nFigs.~\\ref{Fig_14} and \\ref{Fig_15} for the nonoscillation observables $p=m_\\beta,\\,m_{\\beta\\beta},\\,\\Sigma$,\nin two representative cases for cosmological inputs; and Fig.~\\ref{Fig_16} for the discrete mass ordering parameter, $p=\\mathrm{sign}(\\Delta m^2)=\\mathrm{NO(+)\/IO}(-)$.\nFinishing the fabric of the $3\\nu$ paradigm amounts to having convergent, \nnarrow and linear $N_\\sigma$ bounds, for one surviving mass\nordering, in terms of any continuous $3\\nu$ oscillation parameter and nonoscillation observable $p$ (with \nthe possible exception of $m_{\\beta\\beta}$, if neutrinos have a Dirac nature).\nAt present, this goal has been reached for $p=\\delta m^2,\\,\n\\textcolor{black}{|\\Delta m^2|},\n\\,\\theta_{12},\\,\\theta_{13}$\nand, to some extent, for $p=\\theta_{23}$ (up to an octant ambiguity). The current \nresults for $p=\\delta,\\,m_\\beta,\\,m_{\\beta\\beta},\\,\\Sigma$ and NO\/IO\nmay instead be considered as initial, shaky steps of a long march towards the characterization \nof the neutrino-antineutrino differences and of the absolute neutrino mass spectrum. \nOn the way, we shall learn a lot about neutrino properties in many different contexts, \nclarify the origin of old and new data tensions, and possibly find obstacles\nthat, tearing away the fabric of the $3\\nu$ paradigm,  may reveal hidden new physics. \n\n\n\n\n\n\\acknowledgments\n\n \n\nWe are grateful to M.\\ Nakahata for informing us about the latest public release\nof the Super-Kamiokande atmospheric neutrino (preliminary) analysis \\cite{SKmap}.\nWe thank L.\\ Pandola and M. Sisti for useful discussions about\n$0\\nu\\beta\\beta$ decay results.  This work is partly supported by the Italian Ministero dell'Universit\\`a e Ricerca (MUR) through\nthe research grant number 2017W4HA7S ``NAT-NET: Neutrino and Astroparticle Theory Network'' under the program PRIN 2017,\nand by the Istituto Nazionale di Fisica \nNucleare (INFN) through the ``Theoretical Astroparticle Physics''  (TAsP) project. \nThe work of F.C. is supported by the U.S.\\ Department of\nEnergy under the award number DE-SC0020250.\nE.D.V.\\ acknowledges the support of the Addison-Wheeler Fellowship awarded by the Institute of Advanced Study at Durham University.\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\nDissipative nonlinear wave equations are well-posed in $H^k(\\Real^n)\\times H^{k-1}(\\Real^n)$\nfor all $k\\in [1,2]$ and $n\\geq 1$ under the monotonicity condition of Lions and Strauss~\\cite{LiSt}.\nThis global result makes no exception for nonlinear dissipations of supercritical power, as determined by invariant\nscaling, and gets around the stringent conditions for well-posedness of general nonlinear wave equations; see\nPonce and Sideris~\\cite{PS}, Lindblad~\\cite{Lind}, Wang and Fang~\\cite{WaFa} and the references therein.\n\nThe monotonicity method has been less effective in studying higher regularity.\nIt is still an open question whether\nsupercritical problems are globally well-posed in Sobolev spaces with index $k>2$.\nThe purpose of this paper is to give an affirmative answer when $n=3$ and initial data have radial symmetry.\nTo state the result, let $\\Box u =u_{tt}-\\Delta u$ be the d'Alembertian in $\\Real^{3+1}$ and $Du=(\\nabla u,\\partial_t)$\nbe the space-time gradient of $u$.  Standard notations are also $\\|\\cdot\\|_q$, for the norm in $L^q(\\Real^3)$ with $q\\in[1,\\infty]$,\nand $D^\\alpha$, for the partial derivative of integer order $\\alpha=(\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4)$.\n\nWe consider the dissipative nonlinear wave equation\n\\begin{eqnarray}\n\\label{main}\n \\Box u+|u_t|^{p-1}u_t=0,& & x\\in \\Real^3,\\quad t>0,\n \\end{eqnarray}\nwith the Cauchy data\n\\begin{eqnarray}\n\\label{data}\n u|_{t=0}=u_0,\\qquad u_t|_{t=0}=u_1,& & x\\in \\Real^3.\n \\end{eqnarray}\nOur main assumptions are $p\\geq 3$ and $(u_0,u_1)\\in H^{k}_{\\rm rad}(\\Real^3)\\times H^{k-1}_{\\rm rad}(\\Real^3)$, where\nthe radially symmetric Sobolev spaces are defined as\n\\begin{eqnarray*}\nH^k_{\\rm rad}(\\Real^3)=\\{u \\in H^k(\\Real^3): (x_i\\partial_{x_j}-x_j\\partial_{x_i})u(x)=0\\ \\hbox{ for } \\ 1\\leq i<j\\leq 3\\}.\n\\end{eqnarray*}\nClearly, such $u(x)$ depend only on $|x|=(x_1^2+x_2^2+x_3^2)^{1\/2}$. The radially symmetric spaces are invariant\nunder the evolution determined by (\\ref{main}), (\\ref{data}); see \\cite{St}, \\cite{ShSt2}.\n\n\\begin{thm}\n \\label{th1}\nAssume that $p\\geq 3$ and $(u_0,u_1)\\in H^3_{\\rm\nrad}(\\Real^3)\\times H^2_{\\rm rad}(\\Real^3).$ Then problem\n(\\ref{main}), (\\ref{data}) admits a unique global solution $u,$ such\nthat\n\\[\nD^\\alpha u \\in C([0,\\infty), H^{3-|\\alpha|}_{\\rm rad}(\\Real^3)),\\qquad\n |\\alpha|\\leq 3.\n\\]\n\\end{thm}\n\n\\begin{cor}\n \\label{cor011}\nThe global solution of problem (\\ref{main}), (\\ref{data}), given by Theorem~\\ref{th1}, satisfies\nthe following uniform estimates: for $t\\geq 0$,\n\\begin{eqnarray*}\n \\sum_{1\\leq |\\alpha|\\leq 2}\\int_0^t \\|D^\\alpha u(s)\\|^2_{\\infty}\\: ds\n  & \\leq & C_1(u_0,u_1), \\\\\n \\sum_{|\\alpha|\\leq 2}\\|DD^\\alpha u(t)\\|_2 & \\leq  & C_2(u_0, u_1),\n\\end{eqnarray*}\nwhere $C_j(u_0,u_1),$ $j=1, 2,$ are finite whenever $\\|u_0\\|_{H^{3}}+\\|u_1\\|_{H^{2}}$ is finite.\n\\end{cor}\n\nIt is easy to derive the propagation of higher regularity from Theorem~\\ref{th1} and Sobolev embedding inequalities,\nif the nonlinearity allows further differentiation.\n\n\\begin{thm}\n\\label{th01}\nLet $p$ be an odd integer and $(u_0,u_1)\\in H^k_{\\rm\nrad}(\\Real^3)\\times H^{k-1}_{\\rm rad}(\\Real^3)$ with an integer $k\\geq 4$.\nProblem (\\ref{main}), (\\ref{data}) admits a unique global solution $u,$ such\nthat\n\\[\nD^\\alpha u \\in C([0,\\infty), H^{k-|\\alpha|}_{\\rm rad}(\\Real^3)),\\quad\n|\\alpha|\\leq k.\n\\]\nIn addition, the following estimates hold uniformly in $t\\geq 0$:\n\\begin{eqnarray*}\n \\sum_{1\\leq |\\alpha|\\leq k-1}\\int_0^t \\|D^\\alpha u(s)\\|^2_{\\infty}\\: ds\n  & \\leq & C_1^{(k)}(u_0,u_1), \\\\\n \\sum_{|\\alpha|\\leq k-1}\\|DD^\\alpha u(t)\\|_2 & \\leq  & C_2^{(k)}(u_0, u_1),\n\\end{eqnarray*}\nwhere $C_j^{(k)}(u_0,u_1),$ $j=1, 2,$ are finite whenever $\\|u_0\\|_{H^{k}}+\\|u_1\\|_{H^{k-1}}$ is finite.\n\nIf $(u_0,u_1)\\in C^\\infty_{\\rm rad}(\\Real^3)\\times C^\\infty_{\\rm\nrad}(\\Real^3)$ and $(u_0,u_1)$ are compactly supported, then problem\n(\\ref{main}), (\\ref{data}) admits a unique global solution\n\\[\nu \\in C^\\infty([0,\\infty), C^\\infty_{\\rm rad}(\\Real^3)),\n\\]\nsuch that $u(\\cdot,t)$ is also compactly support for all $t\\geq 0$.\n\\end{thm}\n\nThe above results rely heavily on the dissipative nonlinearity and radial symmetry.\nIn less favorable circumstances, global regularity is well beyond the reach of existing methods which\nrequire the nonlinearity be slightly weaker than the linear differential operator.\nNonlinear conservative wave equations and Schrodinger equations, for example, are well-understood only in\nsubcritical and critical cases; see Struwe~\\cite{StM}, Grillakis~\\cite{Gr1}, Shatah and Struwe~\\cite{ShSt1},\nNakanishi~\\cite{Na} for the former and Bourgain~\\cite{Bo}, Grillakis~\\cite{Gr2}, Colliander, Keel, Staffilani, Takaoka\nand Tao~\\cite{CoKeStTaTa} for the latter.\nThe supercritical nonlinearities allow mostly conditional results about the equivalence of certain norms and nature of blow up, such as Kenig and Merle~\\cite{KM} and Killip and Visan \\cite{KV}.\nExceptions are the well-posedness for log-supercritical nonlinearities and radial data by Tao~\\cite{Ta3}\nand the well-posdeness and scattering for log-log-supercritical nonlinearities by Roy \\cite{Roy2}.\n\nDissipative and conservative nonlinear wave equations behave very differently\neven in Sobolev spaces with $k\\leq 2$ derivatives. This is evident from the invariant scaling of (\\ref{main}):\n$u(x,t)\\mapsto u_\\lambda(x,t)=\\lambda^{(2-p)\/(p-1)}u(\\lambda x,\\lambda t)$, with $\\lambda>0$. Then\n\\begin{equation}\n\\label{scaling}\n\\|u_\\lambda(t)\\|_{\\dot{H}^k}=\\lambda^{(2-p)\/(p-1)+k-3\/2}\\|u(\\lambda t)\\|_{\\dot{H}^k},\n\\end{equation}\nso the critical space for well-posedness is $\\dot{H}^{k_c}(\\Real^3)\\times \\dot{H}^{k_c-1}(\\Real^3)$ with\n\\[\nk_c=3\/2+(p-2)\/(p-1).\n\\]\nHowever, the result of \\cite{LiSt} shows that $k\\in [1,2]$ is sufficient for any $p>1$.\nHere the monotone dissipation plays a decisive role, since (\\ref{main}) remains dissipative after applying $D^\\alpha$ with $|\\alpha|=1$:\n\\begin{equation}\n\\label{diff.main}\n\\Box D^\\alpha u+p|u_t|^{p-1}D^\\alpha u_t=0.\n\\end{equation}\nWe can use $u_t$ and $D^\\alpha u_t$ as multipliers for (\\ref{main}) and (\\ref{diff.main}), respectively, to obtain\n\\begin{eqnarray*}\n  \\|Du(t)\\|^2_{2} + 2\\int_0^t \\|u_s(s)\\|^{p+1}_{p+1}\\: ds & = & \\|Du(0)\\|^2_{2},\\\\\n  \\|DD^\\alpha u(t)\\|^2_{2} + 2p\\int_0^t \\||u_s(s)|^{(p-1)\/2}D^\\alpha u_s(s)\\|^2_{2}\\: ds & = & \\|DD^\\alpha u(0)\\|^2_{2}.\n\\end{eqnarray*}\nAs derivatives of order $k\\leq 2$ turn out to be {\\em a priori}\nbounded, the corresponding well-posedness results readily follow from the monotonicity method\nof \\cite{LiSt}.\n\nTwo differentiations of (\\ref{main}) yield an equation that is no longer dissipative.\nIt becomes a nontrivial task to derive uniform estimates for derivatives of order three and higher.\nSome information is provided by the invariant scaling, which predicts the non-concentration\nof second-order norms if $k=2$ and $(2-p)\/(p-1)+1\/2>0$ in (\\ref{scaling}). Thus, the range of subcritical exponents is $p<3$.\nFor the complementary range $p\\geq 3$ and $k>2$, the global well-posedness of (\\ref{main})\nin $H^k(\\Real^3)\\times H^{k-1}(\\Real^3)$ is a difficult problem.\nUntil recently, the proof has been known only in the critical case $p=3$ with radially symmetric data~\\cite{TUY}.\n\nThis paper shows that no critical exponent exists for the regularity of dissipative nonlinear wave equations with radial symmetry.\nThe development of singularities is prevented by the monotonicity of seminorms involving\nsecond-order derivatives. We actually obtain, after setting $r=|x|$ and\n$X_{\\pm}(r,t)=(\\partial_t \\pm \\partial_r)(r\\p_tu)$, that\n\\[\n\\max_{\\pm}\\sup_{r>0}|X_{\\pm}(r,t)| \\leq  \\max_{\\pm}\\sup_{r>0}|X_{\\pm}(r,0)|, \\quad t\\geq 0.\n\\]\nSuch decreasing quantities exist only for linear and dissipative nonlinear wave equations\nin $\\Real^3$ with radial data. The proof is based on differentiating (\\ref{main}) in $t$ and rewriting the equation for $\\partial_tu$ as\n\\[\n(\\partial_t \\mp \\partial_r)X_{\\pm}+\\frac{p}{2}|\\partial_t u|^{p-1}(X_++X_-)=0.\n\\]\nA. Haraux \\cite{Ha} has applied the same idea to derive $W^{2,\\infty}(\\Real)\\times W^{1,\\infty}(\\Real)$ estimates in the one-dimensional case.\nIn higher dimensions with radial symmetry, similar $W^{1,\\infty}(\\Real)\\times L^{\\infty}(\\Real)$ estimates away from $r=0$ are used by\nJoly, Metivier and Rauch~\\cite{JoMeRa}, Liang~\\cite{Li} and Carles and Rauch~\\cite{CaRa}.\nSince the one-dimensional reduction does not work for general data, the global well-posedness in $H^{k}(\\Real^3)\\times H^{k-1}(\\Real^3)$, with $k>2,$\nremains completely open.\n\n\nThe rest of this paper is organized as follows. Section~2 contains\nseveral basic facts and estimates for the wave equation with radially symmetric data in\n$\\Real^3$. The regularity problem in $H^3_{\\rm rad}(\\Real^3)\\times H^{2}_{\\rm rad}(\\Real^3)$\nis studied in Section~3.\nIn the final Section~4, we establish Theorem~1.3 about well-posedness in $H^k_{\\rm\nrad}(\\Real^3)\\times H^{k-1}_{\\rm rad}(\\Real^3)$ with $k\\geq 4$ and\n$C^\\infty_{\\rm rad}(\\Real^3)\\times C^\\infty_{\\rm rad}(\\Real^3)$ with\ncompact support.\n\n\n\n\\section{Basic Estimates}\n\n\nFirst of all, we state the energy estimates and\nStrichartz estimates for the radial wave equation in $\\Real^3\\times \\Real$.\n\n\\begin{lem}\n\\label{lem02} Let $u$ be a solution of the Cauchy problem\n in $\\Real^3\\times \\Real$\n\\[\n\\Box u=F,\\qquad u|_{t=0}=u_0,\\qquad u_t|_{t=0}=u_1.\n\\]\n\n(a) For any source and initial data, $u$ satisfies the energy\nestimate\n\\begin{eqnarray*}\n\\|Du(t)\\|_{2} \\leq C(\\|\\nabla u_0\\|_{2}+\\|u_1\\|_{2})+C\\int_0^t\n\\|F(s)\\|_{2}\\: ds\n\\end{eqnarray*}\nwith an absolute constant $C$ for all $t\\geq 0.$\n\n\n(b) For radial source and initial data, $u$ is also a radial\nfunction which satisfies\n\\begin{eqnarray*}\n\\left(\\int_0^t \\|u(s)\\|_{\\infty}^2 \\: ds \\right)^{1\/2} \\leq\nC(\\|\\nabla u_0\\|_{2}+\\|u_1\\|_{2})+C\\int_0^t \\|F(s)\\|_2\\: ds\n\\end{eqnarray*}\nfor all $t\\geq 0.$\n\\end{lem}\n\nPart (a) can be found in Strauss~\\cite{St}, H\\\"{o}rmander~\\cite{Ho}, and Shatah and Struwe~\\cite{ShSt2}.\nEstimate (b) is the so-called ``radial Strichartz estimate\" in $3D$.\nKlainerman and Machedon~\\cite{KlMa} have found the\nhomogeneous version of (b) which implies the non- homogeneous estimate stated here.\n\nThe following is a collection of useful facts concerning local solvability and other\nproperties of problem (\\ref{main}), (\\ref{data}).\nThese results can be found in \\cite{St}, \\cite{Ho} and \\cite{ShSt2}.\n\n\\begin{lem}\n\\label{lem00} Let $k\\geq 3$ and\n $(u_0,u_1)\\in H^k(\\Real^3)\\times H^{k-1}(\\Real^3).$\n\n(a) There exists $T>0,$ such that problem (\\ref{main}), (\\ref{data})\nhas a unique solution $u$ satisfying\n\\[\nD^\\alpha u \\in C([0,T], H^{k-|\\alpha|}(\\Real^3)),\\qquad |\\alpha|\\leq\nk.\n\\]\nMoreover, we have\n\\[\n\\sup_{t\\in [0,T]}\\|D^\\alpha u(t)\\|_2\\leq C_k,\n\\]\nwhere $T$ and $C_k$ can be chosen to depend continuously on\n$\\|u_0\\|_{H^k}+\\|u_1\\|_{H^{k-1}}.$\n\n(b) The continuation principle holds: if $T_*=T_*(u_0,u_1)$ is the\nsupremum of all numbers $T$ for which (a) holds, then either\n$T_*=\\infty$ or\n\\[\n\\sup_{t\\in [0,T_*)}\\|D^\\alpha u(t)\\|_2=\\infty\n\\]\nfor some $\\alpha$ with $|\\alpha|\\leq k.$\n\n(c) If the data $(u_0,u_1)$ are radially symmetric, the solution\n$(u,u_t)$ is also radially symmetric.\n\n\\end{lem}\n\nNext, we state two preliminary estimates for problem (\\ref{main}),\n(\\ref{data}). These results, called the energy dissipation laws, are already discussed in the\nintroduction.\n\\begin{lem}\n\\label{lem01} Assume that\n $(u_0,u_1)\\in H^3(\\Real^3)\\times H^{2}(\\Real^3)$ and $|\\alpha|=1.$\nLet $u$ be the solution of problem (\\ref{main}), (\\ref{data})\nfor $t\\in [0,T]$, given by Lemma~\\ref{lem00}. Then\n\\begin{eqnarray*}\n\\frac{1}{2}\\|Du(t)\\|_2^2+\\int_0^t \\|u_s(s)\\|^{p+1}_{p+1}\\: ds & = & \\frac{1}{2}\\|Du(0)\\|_2^2,\\\\\n\\frac{1}{2}\\|DD^\\alpha u(t)\\|_2^2+p\\int_0^t \\||u_s(s)|^{(p-1)\/2}D^\\alpha\nu_s(s)\\|^2_{2}\\: ds & = & \\frac{1}{2}\\|DD^\\alpha u(0)\\|_2^2,\n\\end{eqnarray*}\nfor all $t\\in[0,T].$ Thus, the following norms of $u$ are uniformly bounded:\n\\begin{eqnarray*}\n& & \\|Du(t)\\|_2\\leq \\|Du(0)\\|_2,\\quad \\ \\|DD^\\alpha u(t)\\|_2\\leq\n\\|DD^\\alpha u(0)\\|_2,\\quad |\\alpha|\\leq 1,\\\\\n& & \\int_0^T(\\|u_s(s)\\|^{p+1}_{p+1}+\\||u_s(s)|^{(p-1)\/2}D^\\alpha u_s(s)\\|^{2}_{2})\\: ds\\leq \\|Du(0)\\|_2^2+\\|DD^\\alpha u(0)\\|_2^2.\n\\end{eqnarray*}\n\\end{lem}\n\n{\\bf Proof.}\nMultiplying equation (\\ref{main}) by $u_t$, we get\n\\begin{eqnarray*}\n0=(\\Box u+|u_t|^{p-1}u_t)u_t = \\left(\\frac{|Du|^2}{2}\\right)_t\n-\\hbox{div}(u_t\\nabla u) +|u_t|^{p+1}.\n\\end{eqnarray*}\nThe first-order energy identity follows from the integration on $\\Real^3$ and divergence theorem\nif $u(x,t)$ has compact support with respect to $x.$\nMore generally, we can\napproximate $(u_0(x),u_1(x))$ with compactly supported $C^\\infty$\nfunctions and use the finite propagation speed to show that the\nboundary integral of $\\hbox{div}(u_t\\nabla u)$ is zero. Property (a)\nin Lemma~\\ref{lem00} implies that the approximations will converge\nto the actual solution.\n\nSimilarly, we can differentiate equation (\\ref{main}) and multiply\nwith $D^\\alpha u_t(x,t)$ to show the second-order energy identity. Let us recall\nthat we work with solutions whose third-order derivatives belong to\n$L^2(\\Real^3).$\n \\qed\n\nFinally, we state the strong version of Strauss inequality. The original version\ncan be found as Radial Lemma 1 of Strauss \\cite{St1}.\n\n\\begin{lem}\n\\label{lem07}\n(Strong version of Strauss Inequality \\cite{St1}) Let $U\\in H^{1}_{\\rm rad}(\\Real^3)$. There exists a constant $C>0$,\nsuch that for every $R>0$\n\\[\nR |U(R)|\\le C\\|U\\|_{H^1(|x|>R)}\\rightarrow 0 \\ \\hbox{as}\\ R\\rightarrow\\infty.\n\\]\n\\end{lem}\n\n\\section{Global existence of radial solutions in $H^{3}\\times H^2$}\n\nThe following lemma is essential for the proof of Theorem~\\ref{th1}.\nThis approach to $L^\\infty$ estimates of second-order derivatives is borrowed from Haraux~\\cite{Ha}.\n\n\\begin{lem}\n\\label{lem:Haraux} Let $(u_0,u_1)\\in H^3_{\\rm rad}(\\Real^3)\\times H^{2}_{\\rm rad}(\\Real^3)$ and\nlet $u$ be the local solution of problem (\\ref{main}), (\\ref{data})\nconstructed in  Lemma \\ref{lem00}. There exists a positive constant\n$C=C(\\|u_0\\|_{H^3},\\|u_1\\|_{H^2})$,\nsuch that\n\\begin{eqnarray}\n& & \\sup_{r\\in[0,\\infty)}r|\\p_t^2u(r,t)|\\le C,\\label{H_Est1}\\\\\n& & \\sup_{r\\in[0,\\infty)} |\\p_t u(r,t)|\\le C,\\quad\n\\sup_{r\\in[0,\\infty)}r|\\p_r\\p_{t}u(r,t)|\\le C\\label{H_Est2},\n\\end{eqnarray}\nfor all $t\\in[0,T_*)$.\n\\end{lem}\n{\\bf Proof.} Since the solution of (\\ref{main}) is radially symmetric, the equation becomes\n\\begin{eqnarray}\n\\label{main:rad}\n(\\p^2_{t}-\\partial^2_{r})(ru)+r|\\p_tu|^{p-1}\\p_tu=0,\n\\end{eqnarray}\nwhere $r=|x|$. Let us define\n\\[\nX_{\\pm}(r,t)=(\\partial_t \\pm \\partial_r)(r\\p_tu).\n\\]\nWe differentiate (\\ref{main:rad}) in $t$ to obtain an equation for $\\partial_t u$.\nThen we multiply through by $X_{\\pm}^{2k-1}$ with $k\\in\\N$ to obtain\n\\[\n\\frac{(\\partial_t-\\partial_r)}{2k}\\{X_+(r,t)\\}^{2k}\n+pr|\\partial_tu|^{p-1}\\partial^2_t u\\{X_{+}(r,t)\\}^{2k-1}=0\n\\]\nand\n\\[\n\\frac{(\\partial_t+\\partial_r)}{2k}\\{X_-(r,t)\\}^{2k}\n+pr|\\partial_tu|^{p-1}\\partial^2_tu\\{X_{-}(r,t)\\}^{2k-1}=0.\n\\]\nAdding the above two equations and using $X_+(r,t)+X_-(r,t)=2r\\partial^2_tu$, we have\n\\[\n\\frac{(\\partial_t-\\partial_r)}{2k}X_+^{2k}+\\frac{(\\partial_t+\\partial_r)}{2k}X_-^{2k}\n+p|\\partial_tu|^{p-1}\\frac{X_++X_-}{2}(X_+^{2k-1}+X_-^{2k-1})=0.\n\\]\nHere we remark that the following inequality holds for $a,b\\in \\R$ and $k \\in \\N$:\n\\[\n(a+b)(a^{2k-1}+b^{2k-1})\\ge0.\n\\]\nFrom this inequality with $a=X_+$ and $b=X_-$, we derive\n\\begin{eqnarray*}\n\\partial_t(X_+^{2k}+X_-^{2k})+\\partial_r(X_-^{2k}-X_+^{2k})\\le 0.\n\\end{eqnarray*}\nIt is easy to see that integration on $[0,R]\\times[0,t]$ implies\n\\begin{eqnarray*}\n\\int_{0}^{R}\\{X_+^{2k}(r,t)+X_-^{2k}(r,t)\\}dr & \\le &\\int_{0}^{R}\\{X_+^{2k}(r,0)+X_-^{2k}(r,0)\\}dr\\\\\n& & +\\int_{0}^{t}\\{X_+^{2k}(R,s)-X_-^{2k}(R,s)\\}ds,\n\\end{eqnarray*}\nsince $X_-^{2k}(0,t)-X_+^{2k}(0,t)=0$. We also notice that $|X_{\\pm}(R,t)|\\rightarrow 0$ as\n$R\\rightarrow \\infty$, which is a consequence of Lemma \\ref{lem07}. Thus, we get\n\\[\n\\|X_{+}(\\cdot,t)\\|_{L^{2k}(\\Real_+)}^{2k}+\\|X_{-}(\\cdot,t)\\|_{L^{2k}(\\Real_+)}^{2k}\n\\le\\|X_{+}(\\cdot,0)\\|_{L^{2k}(\\Real_+)}^{2k}+\\|X_{-}(\\cdot,0)\\|_{L^{2k}(\\Real_+)}^{2k}.\n\\]\nThe two terms on the right hand side satisfy\n\\[\n\\|X_{\\pm}(\\cdot,0)\\|_{L^{2k}(\\Real_+)}^{2k}\\leq \\|X_{\\pm}(\\cdot,0)\\|_{L^{\\infty}(\\Real_+)}^{2k-2}\\|X_{\\pm}(\\cdot,0)\\|_{L^2(\\Real_+)}^2,\n\\]\nwhere $\\|X_{\\pm}(\\cdot,0)\\|_{L^2(\\Real_+)}<\\infty$ by Lemma~\\ref{lem07}\nand $(u_0,u_1)\\in H^3_{\\rm rad}(\\Real^3)\\times H^2_{\\rm rad}(\\Real^3)$.\nThis observation yields a more convenient estimate:\n\\begin{eqnarray*}\n\\|X_{+}(\\cdot,t)\\|_{L^{2k}(\\Real_+)}^{2k}+\\|X_{-}(\\cdot,t)\\|_{L^{2k}(\\Real_+)}^{2k}&\\le& \\|X_+(\\cdot,0)\\|_{L^{\\infty}(\\Real_+)}^{2k-2}\\|X_+(\\cdot,0)\\|_{L^2(\\Real_+)}^2\\\\\n&&+\\|X_-(\\cdot,0)\\|_{L^{\\infty}(\\Real_+)}^{2k-2}\\|X_-(\\cdot,0)\\|_{L^2(\\Real_+)}^2.\n\\end{eqnarray*}\nLetting $k\\rightarrow\\infty$, we have that\n\\begin{eqnarray}\n\\label{boundedness}\n& & \\max\\{\\|X_+(\\cdot,t)\\|_{L^{\\infty}(\\Real_+)},\\|X_-(\\cdot,t)\\|_{L^{\\infty}(\\Real_+)}\\}\\nonumber\\\\\n& & \\le \\max\\{\\|X_+(\\cdot,0)\\|_{L^{\\infty}(\\Real_+)},\\|X_-(\\cdot,0)\\|_{L^{\\infty}(\\Real_+)}\\}.\n\\end{eqnarray}\n\nThe remaining part of the proof is standard.\nSince $X_{+}(r,t)+X_{-}(r,t)=2r\\p^2_{t}u$, claim (\\ref{H_Est1})\nfollows from (\\ref{boundedness}).\nWe can write $2\\p_r(r\\p_{t}u)=X_{+}(r,t)-X_{-}(r,t)$ and\n\\[\n2r|\\p_tu(r,t)|\\le \\int_{0}^{r}|X_{+}(\\rho,t)-X_{-}(\\rho,t)|d\\rho\\le Cr.\n\\]\nHence $\\|\\p_tu(\\cdot,t)\\|_{L^{\\infty}(\\Real_+)}\\leq C\/2<\\infty$,\nwhich is the first claim in (\\ref{H_Est2}).\nWe finally observe that $2r\\p_r\\p_t u(r,t)=X_{+}(r,t)-X_{-}(r,t)-2\\p_tu(r,t)$, so the second claim in\n(\\ref{H_Est2}) follows from the first one and (\\ref{boundedness}). The proof is complete.\n\\qed\n\n\\vskip 0.5truecm\n\n{\\bf Proof of Theorem~\\ref{th1}.} It is sufficient to show that $\\|u(t)\\|_{H^3}+\\|u_t(t)\\|_{H^2}$\ndoes not blow up as $t\\rightarrow T_*$, where $u$ is the local solution of (\\ref{main}) given by\nLemma~\\ref{lem00}. The first and second order norms of $u$ are {\\it a priori} bounded from Lemma~\\ref{lem01},\nso the global existence of $u$ is guaranteed by the next result about third order norms.\n\n\\begin{prop}\n\\label{pr05} Assume that\n $(u_0,u_1)\\in H^3_{\\rm rad}(\\Real^3)\\times H^{2}_{\\rm rad}(\\Real^3)$ and\nlet $u$ be the local solution of problem (\\ref{main}), (\\ref{data})\nconstructed in  Lemma~\\ref{lem00}. There exists a positive constant $C=C(u_0,u_1,T_*)$\nsuch that\n\\begin{eqnarray*}\n\\sum_{|\\alpha|=3}\\|D^{\\alpha}u(t)\\|_2  \\le C(u_0,u_1,T_*)\n\\end{eqnarray*}\nfor all $t\\in [0,T_*).$\n\\end{prop}\n\n {\\bf Proof.} Differentiating (\\ref{main}) twice, we find that\n $D^\\alpha u$ is a weak solution of\n\\[\n \\Box D^\\alpha u+p|u_t|^{p-1}D^\\alpha u_t+p(p-1)|u_t|^{p-3}u_tD^{\\alpha_1}u_tD^{\\alpha_2}u_t=0,\n\\]\nwhere $\\alpha=\\alpha_1+\\alpha_2$ with $|\\alpha_1|=1$ and $|\\alpha_2|=1$.\nThus, $D^\\alpha u$ satisfies the energy\nestimate in Lemma~\\ref{lem02} (a):\n\\begin{eqnarray*}\n \\|DD^\\alpha u(t)\\|_2 & \\leq & C\\|DD^\\alpha u(0)\\|_2\\\\\n                      &      & + C\\int_0^t \\|u_s^{p-1}(s)D^\\alpha u_s(s)\\|_2 \\: ds\\\\\n                      &      &+C\\int_0^t \\|u_s^{p-2}(s) D^{\\alpha_1}u_s(s)D^{\\alpha_2}u_s(s)\\|_2\\: ds\n\\end{eqnarray*}\nfor $t\\in [0,T_*)$. We notice that\n\\[\n\\|u_s^{p-1}(s)D^\\alpha u_s(s)\\|_2\\le C \\|D^{\\alpha}u_s(s)\\|_2,\n\\]\nby (\\ref{H_Est1}) in Lemma~\\ref{lem:Haraux}, and\n\\[\n\\|u_s^{p-2}(s) D^{\\alpha_1}u_s(s)D^{\\alpha_2}u_s(s)\\|_2\\le C \\left\\|\\frac{D^{\\alpha_1}u_s(s)}{|\\cdot|}\\right\\|_2,\n\\]\nby (\\ref{H_Est1}) and (\\ref{H_Est2}) in Lemma~\\ref{lem:Haraux}. Thus,\n\\begin{eqnarray*}\n \\|DD^\\alpha u(t)\\|_2 & \\leq & C\\|DD^\\alpha u(0)\\|_2\\\\\n                      &      & + C\\int_0^t \\|D^\\alpha u_s(s)\\|_2 \\: ds\\\\\n                      &      &+C\\int_0^t \\left\\|\\frac{D^{\\alpha_1}u_s(s)}{|\\cdot|}\\right\\|_2 \\: ds.\n\\end{eqnarray*}\nThe third term of the right hand can be estimated by Hardy's inequality:\n\\begin{eqnarray*}\n \\|DD^\\alpha u(t)\\|_2 & \\leq & C\\|DD^\\alpha u(0)\\|_2\\\\\n                      &      & + C\\int_0^t \\|D^\\alpha u_s(s)\\|_2 \\: ds\\\\\n                      &      &+C\\int_0^t\\|DD^{\\alpha_1}u_s(s)\\|_2 \\: ds.\n\\end{eqnarray*}\nWe add these estimates for all $|\\alpha|=2$ to get\n\\[\n\\sum_{|\\alpha|=3}\\|D^{\\alpha}u(t)\\|_2 \\:  \\leq C\\sum_{|\\alpha|=3}\\|D^{\\alpha}u(0)\\|\n+C\\int_{0}^{t}\\sum_{|\\alpha|=3}\\|D^\\alpha u(s)\\|_2ds.\n\\]\nMaking use of Gronwall's inequality, we finally have\n\\[\n\\sum_{|\\alpha|=3}\\|D^{\\alpha}u(t)\\|_2  \\le C\\sum_{|\\alpha|=3}\\|D^{\\alpha}u(0)\\|\ne^{Ct}.\n\\]\nThis completes the proof of global existence.\n\\qed\n\n\n\n{\\bf Proof of Corollary~\\ref{cor011}.} We can now verify the uniform estimates of $L^2$ norms and square integrability of\n$L^{\\infty}$ norms on $[0,\\infty)$.\nNotice that inequality (b) in Lemma~\\ref{lem02} holds on any interval $[t_0,t]$ for all $|\\alpha|=1$.\nHence we get\n\\begin{eqnarray*}\n  \\sum_{|\\alpha|=1}\\left(\\int_{t_0}^t \\|D^\\alpha u(s)\\|_\\infty^2\\: ds\\right)^{1\/2}\n   & \\leq & C\\sum_{|\\alpha|=1}\\|DD^\\alpha u(t_0)\\|_2\\\\\n   &      &  +C\\sum_{|\\alpha|=1}\\int_{t_0}^t \\|u_s^{p-1}(s)D^\\alpha u_s(s)\\|_2\\: ds.\n\\end{eqnarray*}\nIt is convenient to abbreviate the left hand side as\n\\[\nN_1(t)=\\sum_{|\\alpha|=1}\\left(\\int_{t_0}^t \\|D^\\alpha u(s)\\|_\\infty^2\\: ds\\right)^{1\/2},\\qquad t\\geq t_0,\n\\]\nand estimate the integrand on the right hand aside as\n$$\\|u_s^{p-1}(s)D^\\alpha u_s(s)\\|_2\\le C\\|D^{\\alpha}u(s)\\|_{\\infty}\\|u_s^{(p-1)\/2}(s)D^\\alpha u_s(s)\\|_2,$$\nfrom (\\ref{H_Est2}). Applying the Cauchy inequality on $[t_0,t]$ to the initial estimate,\n\\begin{eqnarray*}\nN_1(t) & \\leq & C\\sum_{|\\alpha|=1}\\|DD^\\alpha u(t_0)\\|_2\\\\\n   &      &  +CN_1(t)\\left(\\int_{t_0}^t \\|u_s^{(p-1)\/2}(s)Du_s(s)\\|_2^2\\: ds\\right)^{1\/2}.\n\\end{eqnarray*}\nThe above integral converges on $[0,\\infty)$ by Lemma~\\ref{lem01}. We can find $t_0,$ such that\n\\begin{equation}\n\\label{conv}\n C\\left(\\int_{t_0}^t \\|u_s^{(p-1)\/2}(s)Du_s(s)\\|_2^2\\:\nds\\right)^{1\/2}\\leq \\frac{1}{2},\\qquad t\\geq t_0.\n\\end{equation}\nThus, $\\d N_1(t)\\leq 2C\\sum_{|\\alpha|= 1}\\|DD^\\alpha u(t_0)\\|_2$ for\nall $t\\geq t_0.$ If $t<t_0$, a similar estimate for $N_1(t)$ follows from\nTheorem~\\ref{th1}. Hence $\\|D^\\alpha u(\\cdot)\\|_\\infty\\in L^2([0,\\infty))$ for $|\\alpha|=1$.\n\nNext, we show that the second order norms in Corollary~\\ref{cor011} are also uniformly bounded.\nSet\n\\[\nN_2(t)=\\sum_{|\\alpha|= 2}\\left(\\int_{t_0}^t \\|D^\\alpha\nu(s)\\|_\\infty^2\\: ds\\right)^{1\/2},\\qquad t\\geq t_0,\n\\]\nand apply estimate (b) in Lemma \\ref{lem02} to obtain\n\\begin{eqnarray*}\n \\left(\\int_{t_0}^t \\|D^\\alpha u(s)\\|_{\\infty}^2 \\: ds\n\\right)^{1\/2} & \\leq & C\\|DD^\\alpha u(t_0)\\|_{2}+C\\int_{t_0}^t\\|u_s^{p-1}(s)D^\\alpha u_s(s)\\|_2\\: ds\\\\\n & & +C\\int_{t_0}^t\\|u_s^{p-2}(s)(Du_s(s))^2\\|_2\\: ds,\n\\end{eqnarray*}\nwhere $|\\alpha|=2.$ Noticing that (\\ref{H_Est2}) gives\n\\begin{eqnarray}\n\\|u_s^{p-1}(s)D^\\alpha u_s(s)\\|_2&\\le &C\\|u_s(s)\\|_{\\infty}^2\\|D^{\\alpha}u_s(s)\\|_{2},\\label{uni_est1}\\\\\n\\|u_s^{p-2}(s)(Du_s(s))^2\\|_2 &\\le &C\\|D^{\\alpha}u(s)\\|_{\\infty}\\|u_s^{(p-1)\/2}(s)Du_s(s)\\|_2\\label{uni_est2}\n\\end{eqnarray}\nand applying the Cauchy inequality on $[t_0,t]$, we get\n\\begin{eqnarray*}\nN_2(t) & \\leq & C\\sum_{|\\alpha|=2}\\|DD^\\alpha u(t_0)\\|_2\\\\\n       &      & +C\\int_{t_0}^t\\|u_s(s)\\|_{\\infty}^{2}\n                  \\left(\\sum_{|\\alpha|=2} \\|D^\\alpha u_s(s)\\|_2 \\right)\\: ds\\\\\n   &      &  +CN_2(t)\\left(\\int_{t_0}^t \\|u_s^{(p-1)\/2}(s)Du_s(s)\\|_2^2\\: ds\\right)^{1\/2}.\n\\end{eqnarray*}\nWe use again (\\ref{conv}) to derive\n\\begin{eqnarray*}\nN_2(t) & \\leq & 2C\\sum_{|\\alpha|=2}\\|DD^\\alpha u(t_0)\\|_2\\\\\n       &      & +2C\\int_{t_0}^t\\|u_s(s)\\|_{\\infty}^{2}\n                  \\left(\\sum_{|\\alpha|=2} \\|D^\\alpha u_s(s)\\|_2 \\right)\\: ds\n\\end{eqnarray*}\nfor sufficiently large $t\\geq t_0.$ Let us introduce\n\\[\nN_3(t)=\\sum_{|\\alpha|= 2}\\sup_{s\\in[t_0,t]}\\|DD^\\alpha\nu(s)\\|_2,\\qquad t\\geq t_0.\n\\]\nThen we can write\n\\begin{eqnarray}\nN_2(t) & \\leq & 2C\\sum_{|\\alpha|=2}\\|DD^\\alpha u(t_0)\\|_2+2CN_3(t)\\left(\\int_{t_0}^t\\|Du(s)\\|_{\\infty}^2 \\:\n       ds\\right). \\label{N2(t)}\n\\end{eqnarray}\n\nWe will combine the above estimate with Lemma \\ref{lem02} (a) for the interval $[t_0,t]$:\n\\begin{eqnarray*}\n\\|DD^{\\alpha}u(t)\\|_2 & \\leq &  C\\sum_{|\\alpha|=2} \\|DD^\\alpha u(t_0)\\|_2\\nonumber\\\\\n                      &      & +  C\\int_{t_0}^t \\|u_s^{p-1}(s)D^{\\alpha}u_s(s)\\|_2 \\: ds\n                                \\nonumber\\\\\n                      &      &+ C\\int_{t_0}^t \\|u_s^{p-2}(s)(Du_s(s))^2\\|_2 \\: ds,\n                               \\nonumber\n\\end{eqnarray*}\nwhere $t\\geq t_0.$ It follows from (\\ref{uni_est1}) and (\\ref{uni_est2}) that\n\\begin{eqnarray*}\n\\|DD^{\\alpha}u(t)\\|_2 & \\leq &  C\\sum_{|\\alpha|=2} \\|DD^\\alpha u(t_0)\\|_2\\nonumber\\\\\n                      &      & +  C\\int_{t_0}^t \\|Du(s)\\|_{\\infty}^2\\|D^{\\alpha}u_s(s)\\|_2  ds\n                                \\nonumber\\\\\n                      &      &+ C\\left(\\int_{t_0}^t \\|Du_s(s)\\|_{\\infty}^2 ds\\right)^{1\/2}\n                                     \\left(\\int_{t_0}^t \\|u_s^{(p-1)\/2}(s)Du_s(s)\\|_2^2 ds\\right)^{1\/2}.\n                               \\nonumber\n\\end{eqnarray*}\nSumming over $|\\alpha|=2$ and applying Lemma~\\ref{lem01},  we simplify the estimate to\n\\begin{eqnarray}\nN_3(t) & \\leq &  C\\sum_{|\\alpha|=2} \\|DD^\\alpha u(t_0)\\|_2\\nonumber\\\\\n                      &      & +  CN_3(t)\\left(\\int_{t_0}^t \\|Du(s)\\|^2_\\infty \\: ds\\right)\n                                \\nonumber\\\\\n                      &      &+CN_2(t)\\left(\\sum_{|\\alpha|=1}\\|DD^\\alpha u(0)\\|_2\\right).\n                               \\nonumber\n\\end{eqnarray}\nSince the integral\n\\[\n\\int_{0}^\\infty\\|Du(s)\\|_{\\infty}^2 \\: ds\n\\]\nis convergent, we choose a sufficiently large $t_0$ to finally get\n\\begin{eqnarray}\nN_3(t) & \\leq &  C\\sum_{|\\alpha|=2} \\|DD^\\alpha u(t_0)\\|_2\n                                 +CN_2(t)\\left(\\sum_{|\\alpha|=1}\\|DD^\\alpha u(0)\\|_2\\right)\n                                 \\label{N3(t)}\n\\end{eqnarray}\nfor $t\\geq t_0.$ Estimates (\\ref{N2(t)}), (\\ref{N3(t)}) are sufficient to bound\nuniformly $N_2(t)+N_3(t).$ \\qed\n\n\n\n\n\n\\section{Global existence of radial solutions in $H^{k}\\times H^{k-1}$ with $k>3$}\n\nHere we consider equation (\\ref{main}) with a supercritical integer $p=2m+1$, $m\\in \\N$.\nFor $k\\geq 3$ and $t\\in[0,T_*)$, let us define\n\\[\nE_{k}(t)=\\sum_{|\\alpha|=k-1}\\|DD^{\\alpha}u(t)\\|_2,\\qquad\nN_{k}(t)=\\sum_{1\\le|\\beta|\\le k-1}\\left(\\int_{0}^{t}\\|D^{\\beta}u(s)\\|_{\\infty}^2 ds\\right)^{1\/2},\n\\]\nwhere $u$ is the local solution of (\\ref{main}), (\\ref{data}) constructed in Lemma \\ref{lem00}.\n\n{\\bf Proof of Theorem \\ref{th01}.} We use induction in $k$.\nFrom Theorem \\ref{th1}, we know that the following hold when $k=3$:\n\\begin{equation}\n\\label{ind}\nT_*=\\infty,\\quad \\sup_{t\\in[0,\\infty)}E_k(t)<\\infty,\\quad \\sup_{t\\in[0,\\infty)}N_k(t)<\\infty.\n\\end{equation}\nAssuming these claims for $k$, we will verify them for $k+1$. The proof starts from applying $D^{\\alpha}$ to (\\ref{main})\nwith $p=2m+1$ and solving for the highest derivatives:\n\\begin{eqnarray}\n\\Box D^{\\alpha} u &= &c_{m,1}u_t^{2m}D^{\\alpha}u_t\\nonumber\\\\\n                  &  &+c_{m,2}u_t^{2m-1}\\sum_{\\alpha_1+\\alpha_2=\\alpha}D^{\\alpha_1}u_tD^{\\alpha_2}u_t \\nonumber\\\\\n                  &  &+c_{m,3}u_t^{2m-2}\\sum_{\\alpha_1+\\alpha_2+\\alpha_3=\\alpha}D^{\\alpha_1}u_tD^{\\alpha_2}u_tD^{\\alpha_3}u_t\\label{main:k-th}\\\\\n                   & & +\\cdots\\nonumber\\\\\n& &+c_{m,l}u_t^{2m+1-l}\\sum_{\\alpha_1+\\alpha_2+\\cdots+\\alpha_{l}=\\alpha}D^{\\alpha_1}u_t\\cdots D^{\\alpha_{l}}u_t=0, \\nonumber\n\\end{eqnarray}\nwhere $l=\\min\\{2m+1,k\\}$, $c_{m,i}$ are constants and $|\\alpha_i|\\geq 1$ for $i=1,\\cdots, l.$\nMaking use of Lemma \\ref{lem02} (a), we obtain\n\\begin{eqnarray}\n\\|DD^{\\alpha} u(t)\\|_2 &\\le& C\\|DD^{\\alpha}(0)\\|_2 +C\\int_{0}^{t} \\|u_s^{2m}D^{\\alpha}u_s\\|_2ds\\nonumber\\\\\n&&+\\sum_{\\alpha_1+\\alpha_2=\\alpha}C\\int_0^t \\|u_s^{2m-1}D^{\\alpha_1}u_sD^{\\alpha_2}u_s\\|_2ds\\nonumber\\\\\n&&+\\sum_{\\alpha_1+\\alpha_2+\\alpha_3=\\alpha}C\\int_0^t \\|u_s^{2m-2}D^{\\alpha_1}u_sD^{\\alpha_2}u_sD^{\\alpha_3}u_s\\|_2ds\\label{k-deriv1}\\\\\n&& +\\cdots\\nonumber\\\\\n&&+\\sum_{\\alpha_1+\\alpha_2+\\cdots+\\alpha_{l}=\\alpha}C\\int_0^t \\|u_s^{2m+1-l}D^{\\alpha_1}u_s\\cdots D^{\\alpha_{l}}u_s\\|_2\\}ds.\n\\nonumber\n\\end{eqnarray}\n\nOnly the first integrand involves a derivatives of order $k+1$. From (\\ref{H_Est2}), we get\n\\begin{equation}\n\\label{k-deriv2}\n\\|u_s^{2m}D^{\\alpha}u_s\\|_2\\leq C\\|u_s\\|_{\\infty}^2E_{k+1}(s).\n\\end{equation}\n\nIn the second integrand, there are two cases: $\\max\\{|\\alpha_1|,|\\alpha_2|\\}=k-1$\nand $\\max\\{|\\alpha_1|,|\\alpha_2|\\}\\leq k-2$. We rely on (\\ref{H_Est2}) to derive\n\\begin{eqnarray}\n\\|u_s^{2m-1}D^{\\alpha_1}u_sD^{\\alpha_2}u_s\\|_2\n&\\le& \\|u_s\\|_{\\infty}^{2m-2}\\|u_s\\|_{\\infty} \\|Du_s\\|_{\\infty}E_k(s)\\nonumber\\\\\n&\\le& C\\left(\\sum_{1\\le|\\beta|\\le2}\\|D^{\\beta}u(s)\\|_{\\infty}^2\\right)E_k(s)\\label{k-deriv3}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\|u_s^{2m-1}D^{\\alpha_1}u_sD^{\\alpha_2}u_s\\|_2\n&\\le& \\|u_s\\|_{\\infty}^{2m-2}\\|u_s\\|_{\\infty}\\|D^{\\gamma}u_s\\|_{\\infty}E_{k-|\\gamma|+1}(s)\\nonumber\\\\\n&\\le& C\\left(\\sum_{1\\le|\\beta|\\le k-1}\\|D^{\\beta}u(s)\\|_{\\infty}^2\\right)E_{k-|\\gamma|+1}(s),\\label{k-deriv4}\n\\end{eqnarray}\nrespectively, where $\\gamma$ is such that $|\\gamma|=\\max\\{|\\alpha_1|,|\\alpha_2|\\}$.\n\\par\\noindent\n\nThe third integrand also admits $\\max\\{|\\alpha_1|,|\\alpha_2|,|\\alpha_3|\\}=k-2$.\nLet $\\gamma$ be the multiindex satisfying $|\\gamma|=k-2.$ Applying (\\ref{H_Est2}), we have\n\\begin{eqnarray}\n\\|u_s^{2m-2}D^{\\alpha_1}u_sD^{\\alpha_2}u_sD^{\\alpha_3}u_s\\|_2\n&\\le& \\|u_s\\|_{\\infty}^{2m-2}\\|D^{\\gamma}u_s\\|_{\\infty}\\|Du_s\\|_{\\infty}E_2(s)\\nonumber\\\\\n&\\le&C\\left(\\sum_{1\\le|\\beta|\\le k-1}\\|D^{\\beta}u(s)\\|_{\\infty}^2\\right).\\label{k-deriv5}\n\\end{eqnarray}\n\nFinally, we consider all integrands with $\\max\\{|\\alpha_{j}|: j=1,2,\\ldots, l\\}\\leq k-3.$\nSince $k\\ge4$, we can use the Sobolev embedding and (\\ref{H_Est2}) to obtain\n\\begin{eqnarray}\n&&\\|u_s^{2m+1-l}D^{\\alpha_1}u_sD^{\\alpha_2}u_s\\cdots D^{\\alpha_{j}}u_s\\|_2\\nonumber\\\\\n&\\le& \\|u_s\\|_{\\infty}^{2m+1-l}\\left(\\sum_{1\\le|\\beta|\\le k-2}\\|D^{\\beta}u(s)\\|_{\\infty}^2\\right)\n\\left(\\sum_{j=1}^{k}E_{j}(s)\\right)^{l-2}\\label{k-deriv6}\n\\\\\n&\\le& C\\left(\\sum_{1\\le|\\beta|\\le k-2}\\|D^{\\beta}u(s)\\|_{\\infty}^2\\right).\\nonumber\n\\end{eqnarray}\n\nWe combine the basic estimate (\\ref{k-deriv1}) with estimates (\\ref{k-deriv2})--(\\ref{k-deriv6}). This yields\n\\begin{eqnarray*}\n \\|DD^\\alpha u(t)\\|_2 & \\leq & C\\|DD^\\alpha u(0)\\|_2\n+ C\\int_0^t \\|u_s(s)\\|_{\\infty}^2 E_{k+1}(s) + CN_{k}(t)+C_k\n\\end{eqnarray*}\nfor all $t\\in [0,T_*)$, where the constant $C_k=C_k(\\|u_0\\|_{H^{k}},\\|u_1\\|_{H^{k-1}})$.\nAdding all such estimates with $|\\alpha|=k$ and using (\\ref{ind}), we have\n\\begin{eqnarray*}\n E_{k+1}(t)\\leq C_{k+1}+C\\int_0^t \\|u_s(s)\\|_\\infty^2 E_{k+1}(s) \\: ds,\n\\end{eqnarray*}\nfor some constant $C_{k+1}=C_{k+1}(\\|u_0\\|_{H^{k+1}},\\|u_1\\|_{H^{k}}).$ From the Gronwall inequality,\n\\[\nE_{k+1}(t)\\leq C_{k+1}\\exp\\left(\\int_{0}^{t}\\|u_s(s)\\|_\\infty^2ds\\right).\n\\]\nThis shows that $E_{k+1}(t)$ is uniformly bounded on every finite subinterval of\n$[0,T_\\ast).$ Hence, $u$ can be continued to a global solution, such that\n$\\sup_{t\\in [0,\\infty)}E_{k+1}(t)<\\infty.$\n\nIt remains to check the uniform estimates in $L_t^2L_x^{\\infty}$.\nWe apply Lemma \\ref{lem02} (b) to the solution of (\\ref{main:k-th}). The calculations are very similar to the above ones, so we\ngive only the final estimate:\n\\begin{eqnarray*}\n  \\sum_{|\\alpha|=k}\\left(\\int_{0}^t \\|D^\\alpha u(s)\\|_\\infty^2\\: ds\\right)^{1\/2}\n   & \\leq & C\\int_0^t \\|u_s(s)\\|_\\infty^2 E_{k+1}(s) \\: ds+ C_{k+1}.\n\\end{eqnarray*}\nNoticing that  $\\sup_{t\\in[0,\\infty)}E_{k+1}(t)<\\infty$ and $\\sup_{t\\in[0,\\infty)}N_1(t)<\\infty$, we obtain\nthe final estimate in (\\ref{ind}) with $k+1$, i.e., $\\sup_{t\\in[0,\\infty)}N_{k+1}(t)<\\infty$.\nThe proof of higher regularity by induction is complete.\n\nSimilarly, we can show that $C^\\infty$ regularity is preserved during the evolution of compactly supported radial data.\n \\qed\n\n\n\n\n\n\n\n\n\n\\begin{center}\nACKNOWLEDGMENTS\n\\end{center}\nThe authors are very grateful to Professor Hideo Kubo for his useful comments.\n\n\\vskip 0.5truecm\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIt is rarely possible to model realistic physical systems with exact\nsolutions to the equations of some general underlying theory.\nDespite this, many interesting problems deviate only slightly from a\nmodel problem that can be understood exactly. Such solutions are\nusually tractable only because of symmetry assumptions. Once they're\nunderstood, perturbation theory may be used to understand many\nsystems that ``almost'' satisfy the appropriate symmetry principle.\nWhile this statement has a clear intuitive meaning, quantifying it\ncan be difficult. It is also not necessarily obvious how -- or if it\nis meaningful -- to uniquely propagate a symmetry from solutions\nwhere it is exact into the perturbations that break it. These issues\nare particularly important in the context of conservation laws. As\nan example, one might want to know how to construct approximately\nconserved quantities that are unique generalizations of some exact\ncounterpart in a similar system. There would hopefully be a sense in\nwhich such quantities varied slowly for some class of small\nperturbations.\n\nSome steps towards understanding problems like these are explored\nhere in the context of affine collineations (of which Killing\nvectors are special cases) associated with curved spacetimes. While\nthese kinds of symmetries rarely exist, there are various senses in\nwhich approximate replacements can usually be introduced. One method\nfor finding vector fields that are ``almost Killing'' is to write\ndown an action whose value provides some sense for how nearly a\nparticular flow preserves the metric \\cite{Matzner}. There are\nimportant caveats to this interpretation, although the final result\nis that any vector field extremizing such an action satisfies a\nfairly simple generalization of Killing's equation. Various reasons\nhave been given for suggesting other extensions as well\n\\cite{AlmostStat,YanoBochner,Komar1,Komar2,YorkSym}. While any\ngenuine Killing vectors that might exist are solutions to all of\nthese equations, it is not usually clear how the remaining fields\nshould be interpreted. This problem arises even in flat spacetime.\n\nWhat form an approximate symmetry should take is highly dependent on\nits intended use. One application is in the estimation of a black\nhole's angular momentum. This requires finding rotational Killing\nfields on certain 2-spheres foliating a horizon. Various methods\nhave therefore been developed for defining such objects using only\nthe intrinsic geometry of these surfaces\n\\cite{S2Conformal,S2KT,S2Killing}. The concept of approximate\nKilling fields has also been adapted for use on initial data sets\nused in $3+1$ splits of Einstein's equation \\cite{DainInitData}.\n\nThe approach taken here is to define a set of vector fields in a\nfour dimensional volume that can all be viewed as analogs of known\nsymmetries in Minkowski spacetime. The physical interpretation is\nthat these fields may be viewed as generators of approximate\nsymmetries by a specified observer. Any sufficiently small region\nnear a particular point can be made to look nearly flat. Some\nstructures from Minkowski spacetime may be therefore be introduced\nvery near this point. Approximate symmetries that take advantage of\nthis fact are proposed in Sect. \\ref{Sect:Symmetries}. It is then\nshown in Sect. \\ref{Sect:GAC} that analogous objects can also be\nintroduced near an observer's worldline. These sorts of vector\nfields can actually be extended in a non-perturbative way to finite\nregions around the point or worldline from which they were\nconstructed. A well-defined subset provides a precise analog of the\nPoincar\\'{e} group. Translations, rotations, and boosts very near an\nobserver extend in a useful way to cover large portions of the\nspacetime. Any exact symmetries that may exist are included as\nspecial cases. Some connections to conservation laws are discussed\nin Sect. \\ref{Sect:ConsLaws}, and a simple example involving\ngravitational plane wave is finally presented in Sect.\n\\ref{Sect:Example}.\n\n\n\\subsection*{Exact symmetries}\n\nGiven some spacetime $(\\mathcal{M},g_{ab})$, there are several types\nof exact symmetries that may be discussed. The most common of these\ntake the form of vector fields whose associated diffeomorphisms\npreserve some geometric structure. The most ubiquitous examples are\nthe Killing fields. Their flows preserve the metric. Vector fields\n$Y^a_{\\mathrm{K}}$ with this property satisfy\n\\begin{equation}\n  \\Lie_{Y_{\\mathrm{K}}} g_{ab} = 0 . \\label{KillingDef}\n\\end{equation}\nAny solutions that may exist can be used to find conserved\nquantities associated with geodesics or matter distributions\n\\cite{Wald}, identify mass centers \\cite{SchattStreub1}, simplify\nEinstein's equation \\cite{SimplifyEinstWSyms, ExactSolns}, classify\nits solutions \\cite{ExactSolns, Hall}, and so on.\n\nThis utility has (among other reasons) motivated various\ngeneralizations of \\eqref{KillingDef}. Perhaps the simplest of these\narises from considering flows that preserve the metric only up to\nsome constant factor:\n\\begin{equation}\n  \\Lie_{Y_{\\mathrm{H}} } g_{ab} = 2 c g_{ab} .\n  \\label{HomotheticDef}\n\\end{equation}\nAny $Y^a_{\\mathrm{H}}$ satisfying this equation with constant $c$ is\nknown as a homothetic vector field or homothety. Allowing the\ndilation factor $c$ to vary would define a conformal Killing vector.\nThese objects preserve the metric up to an arbitrary multiplicative\nfactor. The standard Killing vectors are special cases of either\nclass. Like them, conformal and homothetic vector fields usually do\nnot exist. Their presence can be very useful, however. The existence\nof a proper (non-Killing) homothetic vector is often used to define\na notion of geometric self-similarity, for example. Such objects\ntherefore appear in certain models of gravitational collapse and\ncosmology. They are also related to the appearance of critical\nphenomena in general relativity \\cite{SelfSimilar}.\n\nA simple generalization of the homotheties can be found by\nconsidering vector fields that satisfy\n\\begin{equation}\n  \\nabla_a \\Lie_{Y_{\\mathrm{A}}} g_{bc} = 0 .\n  \\label{AffineDef}\n\\end{equation}\nSolutions to this equation are known as affine collineations. They\nare the generators of infinitesimal affine transformations. Killing\nand homothetic vector fields are special cases. All of the affine\ncollineations may be interpreted geometrically as preserving the\nLevi-Civita connection. This means that $\\Lie_{Y_{\\mathrm{A}}}$ and\n$\\nabla_a$ commute when acting on arbitrary tensor fields. Geodesics\nand their affine parameters are also preserved under the action of\nany $Y_{\\mathrm{A}}^a$. Although this might seem to be a significant\ngeneralization of the Killing vector concept, solutions rarely\nexist. The only non-flat vacuum spacetimes that admit non-homothetic\naffine collineations are the \\textit{pp}-waves \\cite{RareAffine}.\nSimilarly, it has been shown that proper homotheties cannot exist in\nany asymptotically flat vacuum spacetime with positive Bondi mass\n\\cite{RareHomothety}. Despite these results, interesting affine\ncollineations can occasionally be identified in geometries that are\nnot Ricci-flat. Doing so provides a number of simplifications for\nvarious problems. Some of these derive from the fact that\n\\begin{equation}\n  K_{ab} = \\Lie_{Y_{\\mathrm{A}}} g_{ab} \\label{AffineKT}\n\\end{equation}\nis a second-rank Killing tensor; i.e.\n\\begin{equation}\n  \\nabla_{(a} K_{bc)} = 0. \\label{KTDef}\n\\end{equation}\nIt should be noted that not all symmetric tensors satisfying this\nequation can be derived from affine collineations. One\ncounterexample is the Killing tensor associated with Carter\nconstants in Kerr.\n\nTransformations generated by affine collineations can be viewed as\nmapping geodesics into geodesics. They preserve the affine\nparameters of each curve. Dropping this latter requirement recovers\nthe so-called projective collineations. A precise definition may be\nfound in \\cite{Hall}, although it will not be needed here. One of\ntheir interesting consequences is that they leave invariant the\nprojective curvature tensor:\n\\begin{equation}\n  \\Lie_{Y_{\\mathrm{P}}} \\big( R^{a}{}_{bcd} - \\frac{2}{3} \\delta^{a}_{[c} R_{d]b} \\big) = 0.\n\\end{equation}\nVector fields satisfying this equation are not always projective,\nhowever.\n\nThe list of definitions here could keep growing as new fields are\nadded that preserve more and more geometric structures.\nInterestingly, the quantities introduced so far all share a very\nuseful characteristic that does not easily generalize: the space of\nvector fields in each of the mentioned classes has finite dimension.\nFurthermore, any single element is uniquely determined by its value\nand the values of its first one or two derivatives at a single\npoint. These properties are well-known for Killing fields. Four\ndimensional spacetimes (which is all that will be considered here)\nadmit a maximum of 10 linearly independent Killing vectors. At most\none homothety can exist that is not itself Killing. The maximum\nnumber of (not necessarily proper) conformal Killing vectors is 15,\nand the affine collineations total no more than 20. Finally, the\nspace of projective collineations has a maximum of 24 dimensions\n\\cite{Hall}. Properties like these do not hold for vector fields\nwhose flows leave invariant the Riemann, Ricci, or Einstein\ncurvature tensors of a given spacetime. Despite this, the class of\napproximate symmetries introduced below is constructed so as to have\nfinite dimension. Any given member is fixed by its value together\nwith the values of its first derivatives at a point. Unlike the\nexact symmetries, these objects always exist at least in some finite\nregion. After fixing a reference frame, the space of approximate\nsymmetries has exactly 20 dimensions. Ten of these will be\nidentifiable as generalized Killing vectors, while the remaining ten\nwill be related to more general affine collineations.\n\nIt has already been remarked that the presence of Killing fields\nimplies the existence of various conserved quantities. The same can\nalso be said for more general collineations. Extensive discussions\nof exact symmetries and associated integrals of the geodesic\nequation may be found in \\cite{GeodesicConsts}. Many of these\nsymmetries are non-Noetherian in the sense that they preserve the\nequations of motion, but not the action. Despite this, their\npresence allows constants of motion to also be assigned to arbitrary\nstress-energy distributions satisfying Einstein's equation\n\\cite{KatzinLevine, Collinson}. As will be discussed in Sect.\n\\ref{Sect:ConsLaws}, generalizations of these quantities can be\nassociated with any approximate affine collineations that are\nidentified.\n\n\\section{Symmetries near a point}\\label{Sect:Symmetries}\n\nGeneric symmetries in general relativity are usually discussed in\nthe context of asymptotically flat spacetimes. There then exist\napproximate notions of isometry that improve as one approaches\ninfinity \\cite{Wald}. Generalizations of these ideas also exist for\ngeometries with somewhat more complicated (but still highly\nsymmetric) asymptotic behavior like that of anti-de Sitter\n\\cite{AsymptAdS}. The assumption of a simple limiting form for the\ngeometry makes it convenient to invariantly describe certain\nproperties of a spacetime in terms of ``measurements at infinity.''\nQuantities that may be identified as a spacetime's total energy or\nangular momentum appear naturally, for example.\n\nWhile useful in many contexts, these ideas do not always translate\ninto observations made by physical observers. Measurements like\nthose expected from gravitational wave detectors do come very close\nto fitting into this formalism. Others can require a more local\ndescription. In particular, it is sometimes important to understand\nwhat given observers would experience inside strongly curved regions\nof spacetime. Abstracting the concept of an observer to a timelike\nworldline $\\Gamma$, vector fields may be introduced in (say) some\nconvex neighborhood $W$ of $\\Gamma$ that act like approximations to\nKilling fields or more general collineations. This is always\npossible, and the symmetries these vectors generalize become exact\non $\\Gamma$ itself. Limiting collineations can evidently be useful\non scales that are either very large or very small. It is much less\nclear how to easily describe systems at intermediate distances.\n\n\\subsection{Motivation}\n\nThe idea of a local symmetry just outlined is best introduced by\nfirst considering vector fields $\\psi^a(x,\\gamma)$ that generalize\nthe affine collineations in some reasonable way near a fixed\nreference point $\\gamma$. Let these objects be defined inside a\nnormal neighborhood $N$ of this point. It is intuitively obvious\nthat vector fields may always be chosen such that $\\nabla_a \\LieS\ng_{bc}$ vanishes at $\\gamma$. While this condition is reasonable to\nrequire, it is not very interesting by itself. Much more can be said\nif each vector field in this class is uniquely fixed in $N$ by\nknowledge of\n\\begin{equation}\n  \\psi^\\sfa(\\gamma,\\gamma) , \\qquad \\nabla_\n  \\sfa \\psi^{\\mathsf{b}}(\\gamma,\\gamma) . \\label{InitDataFirst}\n\\end{equation}\nIt will be assumed that each $\\psi^a$ depends linearly on this\ninitial data with no degeneracy. This implies that there are always\n$4+16=20$ linearly independent vector fields defined about any given\npoint in a four dimensional spacetime. Note that indices in\n\\eqref{InitDataFirst} have been written in a sans-serif font to\nemphasize that they are associated with the preferred point\n$\\gamma$.\n\nApproximate affine collineations with the appropriate properties may\nbe constructed by projecting symmetries of the tangent space\n$T_\\gamma N$ into $N$ using the exponential map. Consider the linear\ntransformations\n\\begin{equation}\n  X^\\sfa \\rightarrow X^\\sfa + \\epsilon B_{{\\mathsf{b}}}{}^{a} X^{\\mathsf{b}}\n  \\label{XFormVect}\n\\end{equation}\nof vectors $X^\\sfa$ in this space parameterized by an arbitrary\ntensor $B_{{\\mathsf{b}}}{}^{\\sfa}$. Being a vector space, $T_\\gamma N$ has a\npreferred origin. Adding another term to \\eqref{XFormVect} to shift\nthat origin would be awkward. The translational symmetries that such\na procedure might produce are certainly important, although\ngenerating them requires a more subtle treatment described below.\nFor now, consider only the vector fields\n\\begin{equation}\n  \\Psi^\\sfa = X^{\\mathsf{b}} B_{{\\mathsf{b}}}{}^{\\sfa}\n  \\label{AffineTangent}\n\\end{equation}\nassociated with homogeneous transformations of the given form. These\nclearly satisfy\n\\begin{equation}\n  \\frac{\\partial}{\\partial X^\\sfa} \\Lie_\\Psi g_{{\\mathsf{b}} {\\mathsf{c}}} = 2 \\frac{\\partial}{\\partial\n  X^\\sfa} \\left( g_{{\\mathsf{d}}({\\mathsf{c}}}(\\gamma) \\frac{\\partial}{\\partial X^{{\\mathsf{b}})}} \\Psi^{{\\mathsf{d}}} \\right)\n  =0,\n\\end{equation}\nso they are affine within $T_\\gamma N$ in the sense of\n\\eqref{AffineDef}. Such transformations can be made to induce shifts\n$x \\rightarrow x + \\epsilon \\psi$ in spacetime points associated\nwith vectors $X^\\sfa$ via\n\\begin{equation}\n  x = \\exp_\\gamma X .\n  \\label{ExpMap}\n\\end{equation}\n\nA simple relation between $\\psi^a$ and $\\Psi^\\sfa$ is found by\nintroducing Synge's world function $\\sigma(x,y) = \\sigma(y,x)$. This\ntwo-point scalar returns one-half of the squared geodesic distance\nbetween its arguments. The assumption that $N$ be a normal\nneighborhood of $\\gamma$ ensures that $\\sigma(x,\\gamma)$ is uniquely\ndefined for all points $x$ in this region. Many of its properties\nare reviewed in \\cite{Synge, PoissonRev}. Most importantly for the\nproblem at hand, the first derivative of the world function\neffectively inverts the exponential map. Any set $\\{ \\gamma, x, X^a\n\\}$ satisfying \\eqref{ExpMap} is related via\n\\begin{equation}\n  X_{\\sfa} = - \\sigma_{\\sfa}(x,\\gamma),\n  \\label{XDef}\n\\end{equation}\nwhere the common shorthand $\\sigma_\\sfa = \\nabla_\\sfa \\sigma =\n\\partial \\sigma\/ \\partial \\gamma^\\sfa$ has been used. The\nright-hand side of (\\ref{XDef}) generalizes the concept of a\nseparation vector between two points. It is useful in that a\nstraightforward expansion shows that linear transformations of the\nform (\\ref{XFormVect}) effectively shift spacetime points by an\namount parameterized with a vector $\\psi^{a}(x,\\gamma)$ satisfying\n\\begin{equation}\n\\Psi^{\\sfa} = - \\sigma^{\\sfa}{}_{a} \\psi^{a} . \\label{PsiTopsi}\n\\end{equation}\nIf the various components of $X^\\sfa$ as defined in (\\ref{XDef}) are\nused as coordinates, the bitensor $-\\sigma^{\\sfa}{}_{a} = - g^{\\sfa\n{\\mathsf{b}}}\n\\partial^2 \\sigma\/ \\partial x^a \\partial \\gamma^{{\\mathsf{b}}}$\nreduces to the identity. Components of $\\psi^a$ and $\\Psi^{\\sfa}$\nare therefore identical in normal coordinate systems of this type.\nIn general, it is useful to introduce\n\\begin{equation}\n  H^{a}{}_{\\sfa} = [ - \\sigma^{\\sfa}{}_{a} ]^{-1} \\label{HDef}\n\\end{equation}\nas the matrix inverse of the operator appearing in \\eqref{PsiTopsi}.\nThis always exists in the regions considered here. Using\n\\eqref{AffineTangent} now shows that\n\\begin{equation}\n \\psi^{a}(x,\\gamma) = - H^{a}{}_{\\sfa} \\sigma_{{\\mathsf{b}}} B^{{\\mathsf{b}} \\sfa} .\n \\label{PsiPart}\n\\end{equation}\nHolding $\\gamma$ fixed, this equation defines a 16-parameter family\nof vector fields generated by $B_{\\sfa {\\mathsf{b}}}=\\nabla_\\sfa \\psi_{\\mathsf{b}}\n(\\gamma,\\gamma)$. Every such $\\psi^a$ vanishes at $\\gamma$. It also\nsatisfies \\eqref{AffineDef} at this point. In flat spacetime, these\nvector fields coincide everywhere with exact affine collineations.\n\nNot all such symmetries are included in (\\ref{PsiPart}), however.\nThe four translational Killing fields are missing. These can be\nobtained by considering transformations that directly shift the base\npoint $\\gamma$. Perturbations of this form cannot leave $X^\\sfa$\nfixed, as the initial and final vectors must be elements of\ndifferent spaces. Introducing some $A^\\sfa$, we therefore demand\nthat $X^\\sfa$ be parallel-transported along the curve that $\\gamma$\nfollows under the one-parameter family of transformations\n\\begin{equation}\n  \\gamma \\rightarrow \\gamma + \\epsilon A .\n\\end{equation}\nUsing this together with \\eqref{XDef} and the homogeneous\ntransformation \\eqref{XFormVect} generates the full 20-parameter\nfamily of approximate affine collineations\n\\begin{equation}\n  \\psi^{a} = H^{a}{}_{\\sfa} ( \\sigma^{\\sfa}{}_{{\\mathsf{b}}} A^{\\mathsf{b}} - \\sigma_{\\mathsf{b}} B^{{\\mathsf{b}} \\sfa}\n  ) . \\label{JacobiFirstDer}\n\\end{equation}\nGiven any $A^{\\sfa}$ and $B^{\\sfa {\\mathsf{b}}}$, these objects all satisfy\n\\begin{equation}\n  \\nabla_\\sfa \\LieS g_{{\\mathsf{b}} {\\mathsf{c}}} (\\gamma) = 0 . \\label{JacobiAffine}\n\\end{equation}\nThe initial data\n\\begin{equation}\n  A^\\sfa = \\psi^\\sfa(\\gamma,\\gamma) , \\qquad B^{\\sfa {\\mathsf{b}}} =\n  \\nabla^\\sfa\n  \\psi^{\\mathsf{b}}(\\gamma,\\gamma) \\label{InitialData}\n\\end{equation}\ndetermine $\\psi^a(x,\\gamma)$ throughout $N$. In Minkowski spacetime,\none finds that\n\\begin{equation}\n    \\psi^\\alpha = A^\\alpha + (x-\\gamma)^\\beta B_{\\beta}{}^{\\alpha}\n\\end{equation}\nin the usual coordinates. These coincide exactly with all of the\naffine collineations in this geometry.\n\nIn general, vector fields satisfying (\\ref{AffineDef}) in a curved\nspacetime also have the form (\\ref{JacobiFirstDer}) for some\n$A^\\sfa$ and $B^{\\sfa {\\mathsf{b}}}$. This is most easily seen by noting\nthat vector fields with the given form have been obtained before as\ngeneral solutions to the equation of geodesic deviation (also known\nas the Jacobi equation) \\cite{Dix70a}\n\\begin{equation}\n  \\sigma^{b} \\sigma^{c} ( \\nabla_{b} \\nabla_{c} \\psi_{a} -\n  R_{abc}{}^{d} \\psi_{d} ) = 0. \\label{Jacobi}\n\\end{equation}\nFor any fixed $x$, this is an ordinary differential equation along\nthe geodesic connecting that point to $\\gamma$. Solving it\nrepeatedly for all geodesics in $N$ passing through this origin\nreproduces the vector fields (\\ref{JacobiFirstDer}). It is clear\nthat such solutions always exist as long as the geometry is\nreasonably smooth. These are the spacetime's Jacobi fields about\n$\\gamma$. The bitensors $H^{a}{}_{\\sfa} \\sigma^{\\sfa}{}_{{\\mathsf{b}}}$ and\n$H^{a}{}_{\\sfa} \\sigma_{\\mathsf{b}}$ are known as Jacobi propagators.\n\nSolutions to the geodesic deviation equation effectively map one\ngeodesic into another while preserving the affine parameters of both\ncurves. It was noted above that this is the defining characteristic\nof affine collineations. The difference is that such vector fields\nmust map \\textit{every} geodesic into another geodesic. This\nintuitive argument makes it clear that all affine collineations --\nor Killing fields as special cases -- must be solutions of\n(\\ref{Jacobi}). The proof follows from noting that second\nderivatives of any exact affine collineation $Y_{\\mathrm{A}}^a(x)$\nmust satisfy\n\\begin{equation}\n  \\nabla_{b} \\nabla_{c} Y^a_{\\mathrm{A}} = - R_{bdc}{}^{a} Y^d_{\\mathrm{A}} .\n    \\label{Del2Affine}\n\\end{equation}\nThis result is clear from \\eqref{Del2General}, and is actually\nequivalent to \\eqref{AffineDef}. Substituting it into (\\ref{Jacobi})\nshows that all affine collineations are indeed special cases of\nJacobi fields. As expected, $Y^a_{\\mathrm{A}}$ satisfies the\ngeodesic deviation equation along all geodesics; even those that do\nnot pass through $\\gamma$. This point illustrates precisely how\n\\eqref{Jacobi} generalizes the equation defining an affine\ncollineation. It is simply \\eqref{Del2Affine} contracted into\n$\\sigma^b \\sigma^c$. Alternatively, the Jacobi equation is\nequivalent to \\eqref{LieSig7}.\n\nTo summarize, the following is now evident:\n\\begin{theorem}\\label{Thm:JacobiBasic}\n  Let $N$ be a normal neighborhood of some point $\\gamma$. Define a Jacobi\n  field $\\psi^a(x,\\gamma)$ to be a solution of \\eqref{Jacobi} throughout this region. It is explicitly given by\n  \\eqref{JacobiFirstDer} for some initial data with the form\n  \\eqref{InitialData}. The set of all Jacobi fields about a fixed $\\gamma$ forms a 20-dimensional group in\n  four spacetime dimensions. Each element satisfies $\\mathcal{L}_\\psi \\nabla =0$ at\n  $\\gamma$. Furthermore, all affine collineations are members of\n  this group.\n\\end{theorem}\nThese properties motivate our identification of the Jacobi fields as\ngeneralizations of affine collineations near $\\gamma$.\n\nFurther results that strengthen this decision are derived in the\nappendix. Even though few Jacobi fields are genuine affine\ncollineations, all can be interpreted as exact symmetries of certain\nquantities connected with the spacetime's geometric structure:\n\\begin{theorem}\\label{Thm:JacobiExactSyms}\n  Given a Jacobi field defined as in theorem \\ref{Thm:JacobiBasic},\n  it is always true that\n  \\begin{equation}\n    \\LieS \\sigma^\\sfa = \\LieS \\sigma^a = \\LieS \\sigma^{\\sfa}{}_{a} =\n    \\LieS H^{a}{}_\\sfa =0 ,\n  \\end{equation}\n  where one of the arguments in each of these equations is taken to be the\n  origin $\\gamma$.\n\\end{theorem}\nLie derivatives on two-point tensor fields are defined to act\nindependently on each argument. See \\eqref{LieSig1}, for example.\nQuantities appearing in this theorem are all important in Riemann\nnormal coordinate systems parameterizing arbitrary points $x$ by the\ncomponents of $X^\\sfa = -\\sigma^\\sfa (x,\\gamma)$. In terms of a more\ndirect interpretation of the Jacobi fields as approximately\nsatisfying \\eqref{AffineDef}, Lie derivatives of the metric with\nrespect to an arbitrary $\\psi^a$ are strongly constrained by the\nidentities \\eqref{LieSig6}-\\eqref{LieSig9}.\n\nStatements of this sort do not exhaust the connections between the\nJacobi equation and a spacetime's symmetries. There is a sense in\nwhich higher-rank Killing tensors that may exist are also solutions\nto the geodesic deviation equation \\cite{CavigliaBasic}.\nFurthermore, projective collineations can be shown solve an\ninhomogeneous form of (\\ref{Jacobi}) proportional to $\\sigma_a$\n\\cite{Caviglia}. These observations will not be discussed any\nfurther here, although it is possible that they could be used to\ngeneralize the present framework.\n\n\\subsection{Special cases} \\label{Sect:SpecialJacobi}\n\nIt is often useful to single out a subset of the Jacobi fields\ndistinguished by antisymmetric $B_{\\sfa {\\mathsf{b}}} = \\nabla_\\sfa\n\\psi_{\\mathsf{b}}$. These may be said to generalize only the Killing fields\nof a given spacetime. Distinguishing them with a subscript ``K,''\nthey clearly satisfy $\\Lie_{\\psi_{\\mathrm{K}}} g_{\\sfa {\\mathsf{b}}}(\\gamma)\n= 0$ as well as (\\ref{JacobiAffine}). Such objects form a\n10-dimensional group that may be thought of as a generalization of\nthe Poincar\\'{e} group. They have been suggested before as useful\ngenerators for the linear and angular momenta of extended matter\ndistributions \\cite{Dix70a, Dix74, Dix79}. Fixing a hypersurface\n$\\Sigma$ that passes through $\\gamma$ and the worldtube of some\nwell-behaved spatially-compact stress-energy distribution $T^{ab}$,\nlet\n\\begin{equation}\n  p_\\sfa(\\gamma,\\Sigma) A^\\sfa + \\frac{1}{2} S_{\\sfa{\\mathsf{b}}}(\\gamma,\\Sigma) B^{[\\sfa{\\mathsf{b}}]} = \\int_\\Sigma T^{a}{}_{b}\n  \\psi^b_{\\mathrm{K}} \\rmd S_a .\n  \\label{DixMomenta}\n\\end{equation}\nVarying the 10 free parameters here determines the four linear\nmomenta $p^\\sfa$ and six angular momenta $S^{\\sfa{\\mathsf{b}}} =\nS^{[\\sfa{\\mathsf{b}}]}$. Explicit formulae are easily found using\n(\\ref{JacobiFirstDer}). They coincide with standard definitions in\nflat spacetime (where all $\\psi^a_{\\mathrm{K}}$ are Killing).\n\nTheorem \\ref{Thm:JacobiExactSyms} is easily expanded for these\nvector fields:\n\\begin{corollary}\\label{Thm:JacobiExactSymsKilling}\n    Given any Jacobi field $\\psi^a_{\\mathrm{K}}$ satisfying\n     $\\mathcal{L}_{\\psi_{\\mathrm{K}}} g_{\\sfa {\\mathsf{b}}} =0$,\n    \\begin{equation}\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma_\\sfa =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma_a =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma_{\\sfa a} =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} H^{a \\sfa} = 0.\n    \\end{equation}\n    Again, one argument in each of these equations is assumed to be $\\gamma$.\n\\end{corollary}\nThis follows from the well-known identity \\cite{Synge, PoissonRev}\n\\begin{equation}\n  \\sigma^\\sfa \\sigma_\\sfa = \\sigma^{a} \\sigma_{a} = 2 \\sigma ,\n  \\label{SigIdent2}\n\\end{equation}\nand its first derivative\n\\begin{equation}\n  \\sigma^{\\sfa}{}_{a} \\sigma^{a} = \\sigma^\\sfa .\n  \\label{SigIdent1}\n\\end{equation}\nBy definition, $\\sigma = g_{\\sfa {\\mathsf{b}}}(\\gamma) X^\\sfa X^{\\mathsf{b}}\/2$ is\none-half of the squared geodesic distance between $\\gamma$ and $x$.\nKilling-type Jacobi fields based at $\\gamma$ therefore drag both\narguments of $\\sigma(x,\\gamma)$ in such a way that distances are\npreserved.\n\nIt is also possible to identify Jacobi fields that act like\nhomotheties near $\\gamma$. These are distinguished by letting\n\\begin{equation}\n  B_{(\\sfa {\\mathsf{b}})} = \\frac{1}{2} \\Lie_{\\psi_{\\mathrm{H}}} g_{\\sfa {\\mathsf{b}}}(\\gamma) = c g_{\\sfa\n  {\\mathsf{b}}} (\\gamma). \\label{HomB}\n\\end{equation}\nAs in \\eqref{HomotheticDef}, $c$ is an arbitrary constant. For\nsimplicity, the purely Killing components of some prospective\n$\\psi^a_{\\mathrm{H}}$ can be removed by setting $A_\\sfa = B_{[\\sfa\n{\\mathsf{b}}]} = 0$ and $c \\neq 0$. Substitution into (\\ref{JacobiFirstDer})\nthen shows that\n\\begin{equation}\n  \\psi^{a}_{\\mathrm{H}} = - c H^{a}{}_{\\sfa} \\sigma^\\sfa  = c\n  \\sigma^{a}. \\label{JacobiHom}\n\\end{equation}\nThis second equality follows from contracting $\\delta^{a}_{b} =\n-H^{a}{}_{\\sfa} \\sigma^{\\sfa}{}_{b}$ with $\\sigma^{b}$ and using\n\\eqref{SigIdent1}.\n\nThe simplicity of (\\ref{JacobiHom}) is interesting, although perhaps\nnot surprising. It is consistent with the interpretation of\n$-\\sigma^\\sfa$ as a ``separation vector'' between $x$ and $\\gamma$.\nAs has been noted before, $\\sigma^{\\sfa}{}_{b} \\rightarrow -\n\\delta^{\\alpha}_{\\beta}$ in a normal coordinate system. The\ncomponents of $\\sigma^{a}$ would therefore be equal to $X^\\alpha$.\nThe normal coordinate functions themselves act as components of an\napproximately homothetic vector field. This is unique up to a\nconstant factor and the addition of Killing-type Jacobi fields. It\ngeneralizes the dilations of flat spacetime.\n\nGeneralized Killing tensors of various types can also be generated\nfrom Jacobi fields. In analogy to \\eqref{AffineKT}, let\n\\begin{equation}\n  \\mathcal{K}_{ab} = \\LieS g_{ab} .\n  \\label{KTApprox1}\n\\end{equation}\nThese objects exactly satisfy \\eqref{KTDef} at $\\gamma$, and\npresumably approximate it near this point. It is straightforward to\nwrite down other objects which also have this property. For example,\ntwo (possibly identical) Killing-type Jacobi fields $\\psi^a$ and\n$\\bar{\\psi}^a$ can be used to define\n\\begin{equation}\n  \\mathcal{K}'_{ab} = \\psi_{(a}\n  \\bar{\\psi}_{b)} .\n  \\label{KTApprox2}\n\\end{equation}\nThis expression clearly generalizes to approximate Killing tensors\nof any rank. Exact second-rank Killing tensors probably exist that\ncannot be written in either of these forms, so it is unclear how\nuseful they are.\n\nVery near $\\gamma$, it is possible to approximate the Jacobi fields\nexplicitly. This will be especially useful in Sect.\n\\ref{Sect:ConsLaws} below, where a notion of gravitational current\nis introduced with respect to a given vector field. Consider a\nTaylor expansion of $\\LieS g_{ab}$ in powers of $X^\\sfa$. The first\ntwo terms in this series are trivially obtained from\n\\eqref{JacobiAffine} and \\eqref{InitialData}. Better approximations\ninvolve third and higher derivatives of $\\LieS g_{ab}$ in the limit\n$x \\rightarrow \\gamma$. The lowest order interesting terms can be\nfound from \\eqref{Del2Lieg} and \\eqref{Del3Lieg}. Making use of\n\\eqref{Sig4Coinc}, the final results are that\n\\begin{equation}\n  \\fl \\quad \\LieS g_{ab} \\simeq \\sigma^{\\sfa}{}_{a} \\sigma^{{\\mathsf{b}}}{}_{b} \\big[ \\LieS g_{\\sfa {\\mathsf{b}}} - \\frac{1}{3} X^{\\mathsf{c}} X^{\\mathsf{d}} ( \\LieS R_{\\sfa {\\mathsf{c}} {\\mathsf{b}} {\\mathsf{d}}}\n  + \\frac{1}{2} X^{\\mathsf{f}} \\LieS \\nabla_{\\mathsf{f}} R_{\\sfa {\\mathsf{c}} {\\mathsf{b}} {\\mathsf{d}}}) \\big] + \\Or(X^4) , \\label{LieGExpandJacobi}\n\\end{equation}\nand\n\\begin{eqnarray}\n  \\fl \\quad \\nabla_c \\LieS g_{ab} \\simeq - \\frac{2}{3} \\sigma^{\\sfa}{}_{a} \\sigma^{{\\mathsf{b}}}{}_{b}\n  \\sigma^{{\\mathsf{c}}}{}_{c} X^{\\mathsf{d}} \\big[ R_{{\\mathsf{d}} {\\mathsf{c}}\n  (\\sfa}{}^{{\\mathsf{f}}} \\LieS g_{{\\mathsf{b}}) {\\mathsf{f}}} + g_{{\\mathsf{f}}(\\sfa} \\LieS R_{{\\mathsf{b}}) {\\mathsf{d}} {\\mathsf{c}}}{}^{\\mathsf{f}}  + \\frac{3}{4}\n  X^{\\mathsf{f}}  \\nonumber\n  \\\\\n  \\fl \\qquad ~ \\times \\big( \\frac{1}{3} g_{\\mathsf{h} (\\sfa} \\LieS\n  \\nabla^{\\mathsf{h}} R_{{\\mathsf{b}}) {\\mathsf{d}} {\\mathsf{f}} {\\mathsf{c}}} - g_{\\mathsf{h} \\sfa } \\LieS \\nabla_{\\mathsf{d}} R_{{\\mathsf{f}} ({\\mathsf{b}}\n  {\\mathsf{c}})}{}^{\\mathsf{h}} - g_{\\mathsf{h} {\\mathsf{b}}} \\LieS \\nabla_{\\mathsf{d}}\n  R_{{\\mathsf{f}} (\\sfa {\\mathsf{c}})}{}^{\\mathsf{h}} \\big) \\big] + \\Or(X^3).\n  \\label{LieGGradJacobi}\n\\end{eqnarray}\nLie derivatives here are evaluated at $\\gamma$, so they only involve\n$A_\\sfa$, $B_{\\sfa {\\mathsf{b}}}$, $g_{\\sfa {\\mathsf{b}}}$, $R_{\\sfa {\\mathsf{b}} {\\mathsf{c}}\n{\\mathsf{d}}}$, and its first two derivatives. The factors of\n$\\sigma^{\\sfa}{}_{a}$ in front of these equations are used as a\nconvenient means for converting tensors at $\\gamma$ into tensors at\n$x$. It is perhaps more typical to use parallel propagators\n$g^{\\sfa}{}_{a}$ for this purpose \\cite{PoissonRev}, although the\naforementioned simplicity of $\\sigma^{\\sfa}{}_{a}$ in normal\ncoordinates makes it an attractive alternative. There is very little\ndifference at low orders regardless. $\\sigma^{\\sfa}{}_{a}$ can be\nfreely interchanged with $-g^{\\sfa}{}_{a}$ in\n\\eqref{LieGGradJacobi}. This is also possible in\n\\eqref{LieGExpandJacobi} when $B_{(\\sfa {\\mathsf{b}})} = 0$.\n\nApproximations like these are not useful over regions where the\ncurvature changes significantly, or on length scales approaching the\ncurvature radius. An alternative approach is to expand the various\nbitensors built from $\\sigma$ using its definition as an integral\nalong a geodesic. Simplifications can often be introduced by\nignoring all terms nonlinear in the Riemann tensor. A general method\nfor this type of weak-field procedure may be found in \\cite{Synge,\ndeFelice}. Specific details involved with expanding the Jacobi\nfields in this way will not be given here.\n\n\\section{Symmetries near a worldline}\\label{Sect:GAC}\n\nThe Jacobi fields just discussed generalize the idea of a Killing\nfield or more general affine collineation in a normal neighborhood\nof a given point. This is useful for some purposes, although it does\nnot have a very clear physical interpretation. The choice of origin\nshould presumably correspond to a preferred point, although there\nare few of these that might arise in practice. It is often more\nuseful to base the idea of an approximate symmetry off of a given\ntimelike worldline rather than a single point. This could correspond\nto the path of some observer. In some cases, the physical system\npicks out preferred reference frames. A binary star system\nexperiencing no mass transfer can admit three center-of-mass frames\n(rigorously defined in \\cite{Dix70a, EhlRud,CM}), for example. Two\nof these are associated with the individual stars, while the third\ndescribes the system as a whole. There are also preferred observers\nin most cosmological models. Expressing a system's dynamics in terms\nof quantities associated with these frames has an obvious physical\ninterpretation. The distinction between approximate symmetries\ndefined with respect to a point versus a worldline is closely\nanalogous to the one between Riemann and Fermi normal coordinate\nsystems.\n\nThe concept of an observer here will be taken to mean a timelike\nworldline $\\Gamma$ together with a set of hypersurfaces $\\Sigma(s)$\nthat foliate a surrounding worldtube $W$. It will be assumed that\neach of these hypersurfaces is a normal neighborhood of the point\n$\\gamma(s)$ where it intersects the central worldline. Each of them\nis therefore formed by a collection of radially-emanating geodesics\nof (usually) finite length. The most typical examples would be the\npast-directed null geodesics or the spacelike set orthogonal to\n$\\dot{\\gamma}^\\sfa = \\rmd \\gamma^\\sfa\/\\rmd s$ at $\\gamma(s)$. Other\nchoices are possible, however. Regardless, a worldline and foliation\ntogether will be referred to as an observer's reference frame.\n\n\\subsection{A family of Jacobi fields}\n\nSymmetries adapted to a particular frame can be constructed using a\none-parameter family of Jacobi fields $\\psi^a(x,\\gamma(s))$. Any\nsuch family is fixed by specifying $A_\\sfa(s)$ and $B_{\\sfa{\\mathsf{b}}}(s)$\nas defined in \\eqref{InitialData}. An optimal way of connecting\ninitial data between different points on $\\Gamma$ therefore must be\nfound. Before considering this problem, it is first useful to\ncollapse the family of Jacobi fields into an ordinary vector field\n$\\xi^a(x)$. Let $\\tau(x)$ be defined so as to identify which leaf of\nthe foliation includes an arbitrary point $x$ in the worldtube $W$.\nMore concisely, it always satisfies $x \\in \\Sigma(\\tau(x))$. The\nassumption that each hypersurface is a normal neighborhood of an\nappropriate point on $\\Gamma$ implies that $\\tau$ is always\nsingle-valued. Now set\n\\begin{equation}\n  \\xi^a (x) = \\psi^a(x, \\gamma(\\tau(x))) .\n  \\label{XiDef}\n\\end{equation}\nThe generalized affine collineations (GACs) to be defined below will\nbe of this form for a particular class of families\n$\\psi^a(x,\\gamma(s))$.\n\nOne potential application for a generalized symmetry constructed\nusing a particular frame is in the definition of quantities that\nmight be approximately conserved as one moves along $\\Gamma$. As an\nexample, consider integrals of conserved stress-energy tensors\nsimilar to (\\ref{DixMomenta}). One might define the component of\nmomentum generated by a $\\xi^a$ of the form \\eqref{XiDef} to be\n\\begin{equation}\n  \\ItP_\\xi(s) = \\int_{\\Sigma(s)} T^{a}{}_{b}\n  \\xi^b \\rmd S_a . \\label{PDef}\n\\end{equation}\nThe evolution of this quantity crucially depends on how the\nparameters $A_\\sfa(s)$ and $B_{\\sfa {\\mathsf{b}}}(s)$ in (\\ref{InitialData})\nare connected along $\\Gamma$. It is well-known that $\\ItP_\\xi$ is\nconserved if $\\xi^a$ is Killing and no matter flows across the\nboundary of the worldtube. In this case, initial data for the\none-parameter family of Jacobi fields must satisfy the Killing\ntransport (KT) equations on $\\Gamma$:\n\\numparts\n\\begin{eqnarray}\n    \\mathrm{D}A_\\sfa\/\\rmd s &= \\dot{\\gamma}^{\\mathsf{b}} B_{{\\mathsf{b}} \\sfa}\n    \\label{KTA}\n    \\\\\n    \\mathrm{D} B_{\\sfa {\\mathsf{b}}} \/ \\rmd s &= - R_{\\sfa {\\mathsf{b}}\n    {\\mathsf{c}}}{}^{\\mathsf{d}} \\dot{\\gamma}^{\\mathsf{c}} A_{\\mathsf{d}} .\n    \\label{KTB}\n\\end{eqnarray}\n\\endnumparts\nIf there exists an exact Killing vector $Y^a_{\\mathrm{K}}$ such that\n$A^\\sfa =Y^\\sfa_{\\mathrm{K}}$ and $B^{\\sfa {\\mathsf{b}}} = \\nabla^\\sfa\nY^{\\mathsf{b}}_{\\mathrm{K}}$ at a given $s = s_0$, relations like these will\nhold for all $s$. Furthermore, momenta $p^\\sfa$ and $S^{\\sfa {\\mathsf{b}}}$\nidentified using (\\ref{DixMomenta}) would satisfy\n\\begin{equation}\n  0 = (\\dot{p}_{\\sfa} - \\frac{1}{2} S^{{\\mathsf{b}} {\\mathsf{c}}} R_{{\\mathsf{b}} {\\mathsf{c}} \\mathsf{d} \\sfa}\n  \\dot{\\gamma}^{\\mathsf{c}}) Y^{\\sfa}_{\\mathrm{K}} + \\frac{1}{2} (\n  \\dot{S}_{\\sfa {\\mathsf{b}}} - 2 p_{[\\sfa} \\dot{\\gamma}_{{\\mathsf{b}}]} )\n  \\nabla^{\\sfa} Y^{\\mathsf{b}}_{\\mathrm{K}} . \\label{Papapetrou}\n\\end{equation}\nIf there were a full complement of ten Killing vectors, all possible\nversions of this expression would together be equivalent to the\nPapapetrou equations. More generally, Papapetrou's result is only an\napproximation. Any deviations can be understood using a general\n10-parameter family of possibly approximate isometries. One might\nexpect these corrections to be minimized if $A_\\sfa$ and $B_{\\sfa\n{\\mathsf{b}}}$ always satisfy the KT equations even when no exact Killing\nfields exist.\n\\begin{definition}\\label{Def:GAC}\n  Let a generalized affine collineation (GAC) $\\xi^a(x)$ associated with a\n  reference frame $\\{\\Gamma, \\Sigma\\}$ be derived from a family of\n  Jacobi fields via \\eqref{XiDef}. Individual elements of the family and their\n  first derivatives satisfy the Killing transport equations \\eqref{KTA} and \\eqref{KTB} on $\\Gamma$.\n\\end{definition}\n\nAlthough this definition was motivated by the properties of\nconserved momenta in very particular spacetimes, it also arises from\nmuch more general (if less physical) arguments. Consider all\npossible initial data for vectors built from Jacobi fields using\n\\eqref{XiDef}. It is reasonable to suppose that any GAC should be\nexactly affine on $\\Gamma$; i.e.\n\\begin{equation}\n  \\nabla_\\sfa \\LieX g_{{\\mathsf{b}} {\\mathsf{c}}} |_\\Gamma = 0.\n  \\label{GACAffine}\n\\end{equation}\nIt can also be expected that $A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$ fix\n$\\xi^a$ and its first derivatives on $\\Gamma$ just as they do for\n$\\psi^a$. Generalizing \\eqref{InitialData}, let\n\\begin{equation}\n  A^{\\sfa}(s) = \\xi^\\sfa(\\gamma(s)) , \\qquad B^{\\sfa {\\mathsf{b}}}(s) =\n  \\nabla^{\\sfa} \\xi^{\\mathsf{b}}(\\gamma(s)) . \\label{InitialDataXi}\n\\end{equation}\n\nWe start with the second of these constraints. Directly\ndifferentiating (\\ref{XiDef}) implies that\n\\begin{equation}\n  \\LieX g_{ab} = \\LieS g_{ab} + 2 \\dot{\\psi}_{(a} \\nabla_{b)} \\tau ,\n  \\label{LieXivsPsi}\n\\end{equation}\nwhere the Lie derivative with respect to $\\psi^a(x, \\tau)$ on the\nright-hand side is understood (as usual) to involve only the first\nargument of this vector field. It will be assumed that the foliation\nis always sufficiently smooth that derivatives of $\\tau$ remain\nwell-defined throughout $W$, and on $\\Gamma$ in particular.\nEvaluating \\eqref{LieXivsPsi} on the central worldline requires\nknowledge of $\\dot{\\psi}^a(\\gamma,\\gamma)$. Coincidence limits like\nthese are commonly denoted with brackets. For example,\n\\begin{equation}\n  [\\dot{\\psi}^a](\\gamma) = \\lim_{x \\rightarrow \\gamma}\n  \\frac{ \\partial}{\\partial s} \\dot{\\psi}^a(x,\\gamma(s)) .\n  \\label{CoincDef}\n\\end{equation}\nThe convention of using different fonts for indices referring to $x$\nand $\\gamma$ cannot be consistently applied in expressions like\nthis. No confusion should arise, however. Limits like\n\\eqref{CoincDef} are easily computed using Synge's rule \\cite{Synge,\nPoissonRev}. In this case,\n\\begin{eqnarray}\n  [\\dot{\\psi}^a] &= \\dot{\\gamma}^{\\mathsf{b}} [\\nabla_{\\mathsf{b}} \\psi^a] \\nonumber\n  \\\\\n   &= \\dot{\\gamma}^{\\mathsf{b}} ( \\nabla_{\\mathsf{b}} [\\psi^a] - \\delta^b_{\\mathsf{b}} [\\nabla_b \\psi^a]\n  ) \\nonumber\n  \\\\\n   &= \\mathrm{D} A^a\/\\rmd s - \\dot{\\gamma}^{\\mathsf{b}} B_{{\\mathsf{b}}}{}^{a} .\n  \\label{DotPsi}\n\\end{eqnarray}\nIt follows that (\\ref{InitialDataXi}) holds for all initial data iff\n\\eqref{KTA} is satisfied.\n\nThe other KT equation arises from enforcing \\eqref{GACAffine}.\nNoting (\\ref{JacobiAffine}) and (\\ref{KTA}), it must be true that\n\\begin{equation}\n  [\\nabla_a \\dot{\\psi}^b] = 0.\n  \\label{DelDotPsiReq}\n\\end{equation}\n$[\\ddot{\\psi}^a]$ also has to vanish, although this term is equal to\n$-\\dot{\\gamma}^b [ \\nabla_b \\dot{\\psi}^a]$. Requiring\n\\eqref{DelDotPsiReq} is therefore sufficient. Using the same type of\nprocedure as in \\eqref{DotPsi} shows that\n\\begin{equation}\n  [\\nabla_a \\dot{\\psi}_b] = \\mathrm{D} B_{ab} \/ \\rmd s+ R_{abc}{}^{d}\n  \\dot{\\gamma}^c A_d . \\label{DelDotPsi}\n\\end{equation}\nDeriving this is straightforward other than noting that\n\\eqref{Del2Affine} -- although mentioned for exact affine\ncollineations -- also holds for any Jacobi field at its origin.\nRegardless, the conclusion is that the second Killing transport\nequation \\eqref{KTB} ensures that $\\nabla_\\sfa \\LieX g_{{\\mathsf{b}}{\\mathsf{c}}}$\nvanishes everywhere on $\\Gamma$.\n\nNoting that the KT equations have the same significance for general\naffine collineations as they do for ordinary Killing fields\n\\cite{Hall}, it easily follows that\n\\begin{theorem}\\label{Thm:GACBasic}\n  The class of all generalized affine collineations associated with\n  a given reference frame forms a 20-dimensional group in four\n  spacetime dimensions. Every GAC satisfies $\\LieX \\nabla = 0$ on\n  $\\Gamma$, and all exact affine collineations are members of this\n  class.\n\\end{theorem}\nThis is closely related to theorem \\ref{Thm:JacobiBasic}. It\nstrongly supports definition \\ref{Def:GAC} and the intuitive\nidentification of GACs with approximate symmetries inside $W$.\n\nAt least in principle, finding GACs associated with a particular\nreference frame is straightforward. Suppose that $A_\\sfa(s_0)$ and\n$B_{\\sfa {\\mathsf{b}}}(s_0)$ are given as initial data at some $\\gamma_0 =\n\\gamma(s_0)$. The goal is then to determine the $\\xi^a(x)$\nsatisfying \\eqref{InitialDataXi} at the appropriate point. This is\ndone by first applying the KT equations to the given parameters\nalong $\\Gamma$ from $\\gamma_0$ to $\\gamma(\\tau(x))$. The geodesic\ndeviation equation (\\ref{Jacobi}) is then integrated between this\nlatter point and $x$ using the initial conditions\n(\\ref{InitialData}). Both of these operations simply require finding\nthe solutions to well-behaved ordinary differential equations.\nAlternatively, $\\xi^a$ could also be obtained using the explicit\nexpression \\eqref{JacobiFirstDer} together with the KT equations and\n\\eqref{XiDef}.\n\nOur prescription for generalizing arbitrary affine collineations may\nappear somewhat awkward. Killing transport equations are being\napplied along $\\Gamma$, while the Jacobi equation is used on\ngeodesics intersecting that worldline. These two procedures are not\nas different as they might appear. Trying to use Killing transport\neverywhere would generically lead to inconsistencies. Derivatives of\nthe field expected from the KT equations would not usually match the\nderivatives computed from $\\xi^a$ itself. Only the tangential\ncomponents of these derivatives can be consistently fixed by\nintegrating ordinary differential equations along a collection of\nradial geodesics. Weakening the Killing transport equations to take\nthis into account exactly reproduces the geodesic deviation\nequation. This may be seen by rewriting \\eqref{Jacobi} as a pair of\nfirst order differential equations on geodesics connecting $x$ to\n$\\gamma(\\tau(x))$. Denote the unit tangent vector to one such\ngeodesic by $u^a(l)$. Also set $\\hat{A}^a = \\psi^a$ and\n$\\hat{B}_{ab} = \\nabla_a \\psi_b$ everywhere. It is then\nstraightforward to show that \\numparts\n\\begin{eqnarray}\n  \\mathrm{D} \\hat{A}^a\/\\rmd l &= u^b \\hat{B}_{ba}\n  \\\\\n  u^a \\mathrm{D} \\hat{B}_{ab}\/\\rmd l &= - R_{abc}{}^{d} u^a u^d\n  \\hat{A}_d.\n\\end{eqnarray}\n\\endnumparts\nThe first of these equations has exactly the same form as\n\\eqref{KTA}, while the second is essentially \\eqref{KTB} contracted\nwith $u^a$. Killing and Jacobi transport are therefore very closely\nrelated operations. The latter does not uniquely propagate\n$\\hat{B}_{ab}$ from given initial data, so it is weaker. These\nremarks also clarify in what sense Jacobi fields or GACs approximate\n\\eqref{AffineDef} or \\eqref{Del2Affine}.\n\n\n\\subsection{Special cases and properties of GACs}\n\nFrom a physical perspective, momenta like \\eqref{PDef} should be\ndefinable even in the absence of any exact isometries. This is most\nconveniently done with a particular class of GACs that generalize\nonly the Killing fields. In analogy to the Killing-type Jacobi\nfields discussed in Sect. \\ref{Sect:SpecialJacobi}, suppose that\n$B_{(\\sfa {\\mathsf{b}})}$ vanishes on at least one point of $\\Gamma$. It\nimmediately follows from \\eqref{KTB} that it must actually vanish\neverywhere. The Killing-type GACs therefore form a 10-dimensional\ngroup of vector fields satisfying\n\\begin{equation}\n  \\Lie_{\\xi_\\mathrm{K}} g_{\\sfa {\\mathsf{b}}} |_\\Gamma = 0,\n  \\label{GACKilling}\n\\end{equation}\nas well as \\eqref{GACAffine}. They may be thought of as generalizing\nthe Poincar\\'{e} symmetries of flat spacetime near a given observer.\n\nIt also possible to single out GACs that are approximately\nhomothetic in the sense that\n\\begin{equation}\n  \\Lie_{\\xi_{\\mathrm{H}}} g_{\\sfa {\\mathsf{b}}} |_\\Gamma = 2 c g_{\\sfa\n  {\\mathsf{b}}}. \\label{GACHomothetic}\n\\end{equation}\nThis requires setting $B_{(\\sfa {\\mathsf{b}})} = c g_{\\sfa {\\mathsf{b}}}$. While\nalways possible, the remaining components of the initial data cannot\nbe explicitly solved for except in the case when $\\Gamma$ is a\ngeodesic. It then self-consistent to choose $B_{[\\sfa {\\mathsf{b}}]}=0$. The\nobvious way of doing this is to normalize $\\dot{\\gamma}^\\sfa$ to\nunity and set\n\\begin{equation}\n  A_\\sfa = c (s - \\bar{s} ) \\dot{\\gamma}_\\sfa\n\\end{equation}\nfor some constant $\\bar{s}$. It is easily verified that the given\nparameters satisfy the KT equations. One homothetic-type GAC\nassociated with an affinely parameterized geodesic therefore has the\nform\n\\begin{equation}\n  \\xi^a_{\\mathrm{H}} = c \\big[ \\sigma^a + (\\tau - \\bar{s}) H^{a}{}_{\\sfa}\n  \\sigma^{\\sfa}{}_{{\\mathsf{b}}} \\dot{\\gamma}^{\\mathsf{b}} \\big] .\n\\end{equation}\nAs usual, Killing-type Jacobi fields may be added to this without\nspoiling \\eqref{GACHomothetic}. It should also be emphasized that\nhomothetic-type GACs are not restricted to geodesic frames. This is\njust the case where closed-form solutions of the KT equations can be\nobtained by inspection.\n\nMany of the properties derived for Jacobi fields in Sect.\n\\ref{Sect:Symmetries} and the appendix can be carried over at least\npartially for the GACs. For example,\n\\begin{equation}\n  \\LieX \\sigma^\\sfa = \\LieS \\sigma^\\sfa  = 0  \\label{GACLie1}\n\\end{equation}\nif the arguments are of the form $(x, \\gamma(\\tau(x)))$ and $\\xi^a$\nand $\\psi^a$ are related via \\eqref{XiDef}. This may be interpreted\nas stating that spatial Fermi coordinates are preserved under flows\ngenerated by $\\xi^a$. Since the hypersurfaces $\\Sigma(s)$ can be\ndescribed as a set of geodesics intersecting $\\gamma(\\tau(x))$, it\nwill always be true that $\\sigma^a \\nabla_a \\tau =0$. This means\nthat\n\\begin{theorem}\\label{Thm:GACExactSyms}\n  Given a general GAC $\\xi^a$, $\\LieX \\sigma^\\sfa = \\LieX \\sigma^a\n  =0$ when the arguments of these equations are as in\n  \\eqref{GACLie1}. Killing-type GACs $\\xi^a_{\\mathrm{K}}$ also satisfy $\\Lie_{\\xi_{\\mathrm{K}}} \\sigma = \\Lie_{\\xi_{\\mathrm{K}}} \\sigma_\\sfa =\n  0$ with the same restriction.\n\\end{theorem}\n\nThe identity \\eqref{LieSig6} serves to constrain Lie derivatives of\nthe metric with respect to Jacobi fields. A direct analog of this\nequation for an arbitrary GAC would involve an additional term.\nDespite this, contracting the result with $\\sigma^b$ leads to the\nsimple conclusion\n\\begin{equation}\n  \\sigma^a \\sigma^b \\LieX g_{ab} = 2 \\sigma^\\sfa \\sigma^{\\mathsf{b}} B_{(\\sfa\n  {\\mathsf{b}})} .\n\\end{equation}\n``Purely radial'' components of $\\LieX g_{ab}$ therefore vanish for\nKilling-type GACs.\n\nMany other results can be carried over in similar ways. One that is\nof particular interest is the behavior of $\\LieX g_{ab}$ or\n$\\nabla_a \\LieX g_{bc}$ near $\\Gamma$. As with the Jacobi fields, it\nis possible to see how close a GAC comes to being affine as its\nreference worldline is approached. Analogs of\n\\eqref{LieGExpandJacobi} and \\eqref{LieGGradJacobi} may be obtained\nusing expansions like \\eqref{LieXivsPsi} and the identity\n\\eqref{Del2General}. Simplifying terms with the Killing transport\nequations, the lowest order correction to \\eqref{LieGExpandJacobi}\nis\n\\begin{equation}\n  \\Lie_{(\\xi-\\psi)} g_{ab} \\simeq \\frac{2}{3} \\sigma^{\\sfa}{}_{a}\n  \\sigma^{{\\mathsf{b}}}{}_{b} \\nabla_{(\\sfa} \\tau g_{{\\mathsf{b}}) {\\mathsf{f}}}\n  \\dot{\\gamma}^{\\mathsf{h}}\n  X^{\\mathsf{c}} X^{\\mathsf{d}} \\LieX R_{\\mathsf{h} {\\mathsf{c}} {\\mathsf{d}}}{}^{{\\mathsf{f}}} + \\Or(X^3).\n  \\label{LieGExpandGAC}\n\\end{equation}\nSimilarly, the first interesting change to \\eqref{LieGGradJacobi}\nhas the form\n\\begin{eqnarray}\n  \\fl \\qquad \\nabla_a \\Lie_{(\\xi-\\psi)} g_{bc} \\simeq - \\frac{4}{3} \\sigma^{\\sfa}{}_a\n  \\sigma^{{\\mathsf{b}}}{}_b \\sigma^{{\\mathsf{c}}}{}_c X^{\\mathsf{d}} \\dot{\\gamma}^{\\mathsf{f}}\n  \\big[ g_{\\mathsf{h} ({\\mathsf{b}}} \\nabla_{{\\mathsf{c}})} \\tau (\\delta^{\\mathsf{l}}_\\sfa - \\dot{\\gamma}^{\\mathsf{l}} \\nabla_\\sfa \\tau) \\LieX R_{{\\mathsf{f}} (\\mathsf{l} {\\mathsf{d}})}{}^{\\mathsf{h}} \\nonumber\n  \\\\\n  \\qquad ~  + \\frac{1}{2} \\nabla_\\sfa \\tau (R_{{\\mathsf{d}} {\\mathsf{f}}\n  ({\\mathsf{b}}}{}^{\\mathsf{h}} \\LieX g_{{\\mathsf{c}}) \\mathsf{h}} - g_{\\mathsf{h} ({\\mathsf{b}}} \\LieX R_{{\\mathsf{c}}) {\\mathsf{f}} {\\mathsf{d}}}{}^{\\mathsf{h}} ) \\big] + O(X^2).\n  \\label{LieGGradGAC}\n\\end{eqnarray}\nAs before, the magnitudes of these terms depend on how close $\\xi^a$\nis to being a symmetry of the Riemann tensor on the observer's\nworldline.\n\n\\section{Mechanics and conservation laws}\n\\label{Sect:ConsLaws}\n\nIt has already been remarked that one of the main applications of\nexact symmetries in physics is to the formulation of conservation\nlaws. These take several forms. Perhaps the most basic are those\nassociated with a spacetime's geodesics. It is sometimes possible\nfor such curves to be at least partially parameterized by a number\nof constants associated with geometric symmetries. More\ninterestingly, conservation laws can also be associated with\nextended matter distributions. Assuming only that stress-energy\ntensors satisfy\n\\begin{equation}\n  \\nabla_a T^{ab} = 0 \\label{StressCons}\n\\end{equation}\ntends to lead to the definition of slowly-varying parameters like\nthose discussed in connection with \\eqref{DixMomenta} and\n\\eqref{PDef}. The situation becomes much more interesting in full\ngeneral relativity. Einstein's equation implies stress-energy\nconservation, although it also connects symmetries of the geometry\nto those of the matter distribution (and vice versa). This allows\nthe introduction of exact conservation laws in arbitrary spacetimes.\n\n\\subsection{Geodesics}\n\nIt is well-known that any Killing fields that may exist provide\nfirst integrals of the geodesic equation. These can be used both to\nderive and parameterize the geodesics of a given spacetime. While\nless commonly discussed, similar quantities can also be associated\nwith other kinds of symmetries \\cite{GeodesicConsts,KatzinGeo}.\nUnlike in the Killing vector case, the presence of more general\ncollineations sometimes implies the existence of interesting\nconserved quantities that are not linear in the geodesic's\nfour-velocity. The curve's affine parameter can also appear\nexplicitly. As a direct calculation will easily verify, two\nconstants associated with an exact affine collineation\n$Y_{\\mathrm{A}}^a$ are \\numparts\\label{GeoConsts}\n\\begin{eqnarray}\n    C_1 &= \\dot{y}^a \\dot{y}^b \\Lie_{Y_{\\mathrm{A}}} g_{ab}\n    \\label{GeoConst1}\n    \\\\\n    C_2 & = \\dot{y}_a Y^a_{\\mathrm{A}} - \\frac{1}{2} l C_1 .\n    \\label{GeoConst2}\n\\end{eqnarray}\n\\endnumparts\nThese quantities remain fixed along any affinely-parameterized\ngeodesic $y(l)$. The first becomes degenerate if $Y^a_{\\mathrm{A}}$\nis Killing. $C_2$ then reduces to the standard conserved quantity\nassociated with a Killing field. Other constants can sometimes be\nwritten down by combining $Y^a_{\\mathrm{A}}$ with an exact Killing\ntensor \\cite{GeodesicConsts, GeoExamples}. Such constructions will\nnot be discussed here.\n\nConsider instead expressions like those just given with\n$Y_{\\mathrm{A}}^a(x)$ replaced by some Jacobi field\n$\\psi^a(x,\\gamma)$. We then have\n\\begin{eqnarray}\n  \\dot{C}_1 = \\frac{\\rmd C_1}{\\rmd l} = -\\frac{2}{l} \\frac{\\rmd C_2}{ \\rmd l} = \\dot{y}^a \\dot{y}^b \\dot{y}^c \\nabla_{(a} \\LieS\n  g_{bc)} .\n\\end{eqnarray}\nThe tangent vectors are proportional to $\\sigma^a(y,\\gamma)$ for the\nspecial case of geodesics passing through $\\gamma$. It then follows\nfrom \\eqref{LieSig9} that both $C_1$ and $C_2$ remain conserved\nalong all such trajectories. Each Jacobi field generates exact\ngeodesic constants in this way. In terms of the initial data\n$A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$, these have the values\n\\begin{eqnarray}\n  C_1  &= 2 \\dot{y}^\\sfa \\dot{y}^{\\mathsf{b}} B_{(\\sfa {\\mathsf{b}})}, \\qquad\n  C_2 &= \\dot{y}^\\sfa A_\\sfa \\label{GeoConstInit}\n\\end{eqnarray}\nwhen the parameter $l$ is chosen to vanish at $\\gamma$. It is clear\nfrom this that $B_{[\\sfa {\\mathsf{b}}]}$ is irrelevant. Multiple Jacobi\nfields may therefore generate the same constants on a particular\ncurve.\n\nExact affine collineations generalize these results by also applying\nto non-radial geodesics. Expansions like \\eqref{LieGGradJacobi} can\nbe used to derive how close general Jacobi fields come to this\nideal. To lowest nonvanishing order,\n\\begin{equation}\n  \\dot{C}_1 \\simeq - \\frac{4}{3} (\\dot{y}^a \\sigma^{\\sfa}{}_{a}) (\\dot{y}^b \\sigma^{{\\mathsf{b}}}{}_{b}) (\\dot{y}^c\n   \\sigma^{{\\mathsf{c}}}{}_{c}) X^{\\mathsf{d}} R_{{\\mathsf{d}} {\\mathsf{b}}\n  {\\mathsf{c}}}{}^{{\\mathsf{f}}} \\LieS g_{\\sfa {\\mathsf{f}}} + \\Or(X^2) . \\label{Cdot}\n\\end{equation}\nThis term vanishes if $B_{(\\sfa {\\mathsf{b}})}=0$, so the Killing-type\nJacobi fields usually provide more accurate ``conservation laws''\nfor arbitrary geodesics near $\\gamma$. In these cases,\n$\\dot{C}_1(l)$ scales like $(X\/\\mathcal{R})^2\/\\mathcal{R}$, where\n$\\mathcal{R}$ is a curvature radius. This is a worst-case estimate.\n$C_1$ and $C_2$ will probably vary much more slowly if there is a\nphysical sense in which the system is approximately symmetric. Rates\nof change for $C_1$ or $C_2$ can also be constrained using\nidentities like \\eqref{LieSig7}. This effectively restricts how much\nthese parameters can vary as a geodesic moves away from $\\gamma$.\nMore of their changes tend to occur as $y(l)$ moves across rather\nthan with the radial geodesics.\n\nParameters like $C_1$ and $C_2$ can also be defined with respect to\na GAC $\\xi^a$. These should remain approximately conserved for\ngeodesics near an observer's worldline rather than geodesics near a\npoint. They are exact constants for curves passing through $\\Gamma$\nalong the reference foliation. This can be seen by using\n\\eqref{LieXivsPsi} to show that\n\\begin{equation}\n  \\dot{y}^a \\dot{y}^b \\dot{y}^c \\nabla_a \\LieX g_{bc} =   \\dot{y}^a \\dot{y}^b \\dot{y}^c \\nabla_{a} \\LieS g_{bc}\n\\end{equation}\nwhen $\\dot{y}^a \\nabla_a \\tau =0$. Values of $C_1$ and $C_2$ here\nare the same as in \\eqref{GeoConstInit} if the quantities in that\nequation are evaluated at the point where $y$ intersects $\\Gamma$.\nThis might be useful in a coordinate system constructed from some\ncollection of GACs. Alternatively, it can be viewed as a\ngeneralization of standard results near a given observer.\n\nFurther methods of parameterizing geodesics may be found by\ngeneralizing the constants associated with higher-rank Killing\ntensors. Given any exact Killing tensor $K_{a_1 \\cdots a_n} =\nK_{(a_1 \\cdots a_n)}$, the scalar\n\\begin{equation}\n  C_K = K_{a_1 \\cdots a_n} \\dot{y}^{a_1} \\cdots \\dot{y}^{a_n}\n  \\label{CK}\n\\end{equation}\nis conserved along all geodesics $y(l)$. An analog of this quantity\nfor the approximate Killing tensor \\eqref{KTApprox1} is exactly\n$C_1$ defined above. Something more interesting can be generated by\nsubstituting \\eqref{KTApprox2} into \\eqref{CK}. In the second-rank\ncase, one may define\n\\begin{equation}\n  C_K = (\\dot{y}_a \\psi^a_{\\mathrm{K}}) (\\dot{y}_b \\bar{\\psi}^b_{\\mathrm{K}}) \\label{CKEx}\n\\end{equation}\nfor some Killing-type Jacobi fields $\\psi^a_{\\mathrm{K}}$ and\n$\\bar{\\psi}^a_{\\mathrm{K}}$. This is easily generalized to involve\nan arbitrary number of products, although it is only interesting to\nconsider linearly independent collections of Jacobi fields. The\nmaximum number of useful products is therefore ten. All of these can\nbe generated just from individual terms of the form $\\dot{y}_a\n\\psi^a_{\\mathrm{K}}$. These are interpreted as approximate constants\nassociated with objects that are nearly first-rank Killing tensors\n(i.e. Killing vectors). Everything of interest here can therefore be\nderived from the behavior of\n\\begin{equation}\n  C_3 = \\dot{y}_a \\psi^a_{\\mathrm{K}} = C_2 + \\frac{1}{2} l C_1.\n  \\label{C3}\n\\end{equation}\nAlthough this depends only on the two approximate constants defined\nbefore, it may be interpreted as an additional useful parameter. It\nhas the interesting property that\n\\begin{equation}\n  \\dot{C}_3 = \\frac{1}{2} C_1 . \\label{C3Dot}\n\\end{equation}\nTime derivatives do not appear on the right hand side. Consider the\nspecial case of a geodesic that passes through $\\gamma$. Since the\nJacobi field was assumed to be Killing at its origin, $C_1 = 0$.\nThis is true everywhere, so $C_3$ also remains fixed along the\nentire geodesic. It actually coincides with $C_2$ in this case.\n\nDifferences arise when considering non-radial geodesics. It was\nremarked above that there was a sense in which $C_1$ and $C_2$ only\nvaried due to non-radial components of $\\dot{y}^a$. This type of\nstatement can be made much more precise for $C_3$. Let\n\\begin{equation}\n  \\dot{y}^a(l) = u_{||} (l) \\sigma^a(y(l),\\gamma) + u^a_\\bot(l) ,\n\\end{equation}\nwhere $u^{[a}_\\bot \\sigma^{b]} =0$. Now \\eqref{LieSig6},\n\\eqref{GeoConst1}, and \\eqref{C3Dot} show that\n\\begin{equation}\n  \\dot{C}_3 = \\frac{1}{2} u_\\bot^a u_\\bot^b \\LieS g_{ab} .\n  \\label{C3DotBeta}\n\\end{equation}\nThis result is exact for all geodesics $y(l)$. There are many cases\nwhere $u_\\bot^a$ becomes vanishingly small as $l \\rightarrow \\pm\n\\infty$ (when the geodesic exists for these parameter values), so\n\\eqref{C3DotBeta} provides a strong restriction on how much $C_3$\ncan vary in any given situation. Very near $\\gamma$,\n\\eqref{LieGExpandJacobi} can be used to show that\n\\begin{equation}\n  \\dot{C}_3 \\simeq - \\frac{1}{6} (u_\\bot^a \\sigma^{\\sfa}{}_{a}) (u_\\bot^b \\sigma^{{\\mathsf{b}}}{}_{b})\n  X^{\\mathsf{c}} X^{\\mathsf{d}} \\LieS R_{\\sfa {\\mathsf{c}} {\\mathsf{b}} {\\mathsf{d}}} + \\Or(X^3) .\n\\end{equation}\nThe lowest order contributions here scale like\n$(X\/\\mathcal{R})^2\/\\mathcal{R}$. This is similar to the result\nexpected for $\\dot{C}_1$ when computed using a Killing-type Jacobi\nfield.\n\n\\subsection{Extended matter\ndistributions}\\label{Sect:ExtendedMatter}\n\nFrom a physical perspective, it is often more interesting to\nconsider possibly approximate integrals of the equations of motion\ndescribing an extended matter distribution rather than a pointlike\ntest particle. Suppose that this matter is modeled by a conserved\nstress-energy tensor $T^{ab}$. Contracting it with any exact Killing\nvectors that may exist yields conserved currents. These are\nequivalent to some subset of the typical laws of linear and angular\nmomentum conservation known in flat spacetime. More generally,\n\\eqref{StressCons} shows that\n\\begin{equation}\n  \\nabla_a ( T^{a}{}_{b} Y^b ) = \\frac{1}{2} T^{ab} \\Lie_Y g_{ab} .\n\\end{equation}\nThis holds for any vector field $Y^a$, although it is convenient to\nassume that it is a Killing-type GAC. The source term on the\nright-hand side may then be considered small near $\\Gamma$. This\ntherefore serves as an approximate conservation law. As long as\nthere is no matter flow through $\\partial \\Sigma$, quantities like\n\\eqref{PDef} might be expected to vary slowly in time.\n\n\nStress-energy conservation can be seen as a consequence of the\ndiffeomorphism invariance of a system's underlying action.\nConstructions that are based on it are therefore useful in many\ntheories of gravity besides general relativity. They can also hold\nfor test bodies in fixed background geometries. This generality is\nquite restrictive. Much more can be said if the full Einstein\nequation is assumed to hold. Symmetries in the geometry are then\nrelated to symmetries in the matter fields. The presence of an exact\nKilling field $Y^a_{\\mathrm{K}}$ that also satisfies\n$\\Lie_{Y_\\mathrm{K}} T_{ab} = 0$ allows many interesting results to\nbe proven regarding the momenta $p_\\sfa$ and $S_{\\sfa {\\mathsf{b}}}$ defined\nin \\eqref{DixMomenta}. For example, the net force and torque on a\nbody can be written explicitly in terms of the Killing field and its\nfirst derivative at a point. If $Y^a_{\\mathrm{K}}$ is timelike, a\nbody's center-of-mass can be shown to follow one of its orbits.\nFurthermore, mass centers always lie on the central axis of\ncylindrically symmetric spacetimes \\cite{SchattStreub1,\nSchattStreub2}.\n\nMomenta defined in terms of $T^{a}{}_{b} \\xi^b$ are useful for many\npurposes, although they are not conserved in the absence of exact\nKilling fields. Determining how they vary over time requires\ndetailed knowledge of a body's internal structure. Alternative\ndefinitions for the linear and angular momenta of an extended body\narise when using the full Einstein equation rather than just\n\\eqref{StressCons}. Taking the trace of Ricci's identity and\nrearranging terms shows that\n\\begin{equation}\n  R^{a}{}_{b} Y^b = \\frac{1}{2} ( g^{ac} g^{bd} - g^{ab} g^{cd} )\n  \\nabla_b \\Lie_Y g_{cd} + \\nabla_b \\nabla^{[a} Y^{b]} .\n  \\label{RicciId}\n\\end{equation}\nThis holds for any vector field $Y^a$. Note that the second term on\nthe right-hand side is always conserved. It follows that\n\\begin{eqnarray}\n  \\nabla_a [ 2 (T^{a}{}_{b} Y^b -\\frac{1}{2} Y^a T^{b}{}_{b}) + j^a_Y] = 0, \\label{AffineConsLaw}\n\\end{eqnarray}\nwhere the ``gravitational current'' $j^a_Y$ associated with $Y^a$\nhas been defined by\n\\begin{equation}\n  j^a_Y = \\frac{1}{8 \\pi} ( g^{ab} g^{cd} - g^{ac} g^{bd} )\n  \\nabla_b \\Lie_Y g_{cd} .\n  \\label{GravCurrent}\n\\end{equation}\nIt clearly vanishes if $Y^a$ is an exact affine collineation. This\nis not the only case where the current's contribution to\n\\eqref{AffineConsLaw} disappears. Using the Bianchi identity,\n\\begin{equation}\n  \\nabla_a j^a_Y = - \\frac{1}{8\\pi} g^{ab} \\Lie_Y R_{ab}.\n  \\label{divJ}\n\\end{equation}\nAny vector field satisfying $g^{ab} \\Lie_Y R_{ab} =0$ will therefore\ngenerate conserved matter currents involving only $T^{a}{}_{b} Y^b -\nY^a T^{b}{}_b\/2$. That such ``contracted Ricci collineations''\ngenerate conservation laws for matter distributions has been noted\nbefore in \\cite{KatzinLevine, Collinson}.\n\nThe viewpoint here will be to apply \\eqref{RicciId} and\n\\eqref{AffineConsLaw} with $Y^a$ replaced by some approximate\nsymmetry.\n\\begin{definition}\n    Fix some family of Jacobi fields $\\psi^a(x,\\gamma(s))$ that generates a GAC\n    via \\eqref{XiDef}. Define the generalized Komar momentum\n    $\\ItP^*_\\psi$ associated with these fields by\n    \\begin{equation}\n        \\ItP^*_\\psi(s) = \\frac{1}{8\\pi} \\oint_{\\partial \\Sigma(s)} \\nabla^{[a}\n        \\psi^{b]} \\rmd S_{ab} . \\label{PStarDef}\n    \\end{equation}\n\\end{definition}\nAs the name suggests, this has the same form and interpretation as a\nKomar integral. It is convenient to assume that $s$ is a fixed\nparameter for the purpose of evaluating the derivative in\n\\eqref{PStarDef}. Directly using a GAC in place of $\\psi^a$ would\nadd a dependence on the reference frame. Note that no such\ndistinctions had to be made for the $\\ItP_\\xi$ defined in\n\\eqref{PDef}. Applying Stokes' theorem together with \\eqref{RicciId}\nand \\eqref{GravCurrent} shows that\n\\begin{equation}\n  \\ItP^*_\\psi =  \\int_\\Sigma \\big[ 2 (T^{a}{}_{b} \\psi^b - \\frac{1}{2}\n  \\psi^a T^{b}{}_{b} ) + j^a_\\psi \\big] \\rmd S_a . \\label{PStar2}\n\\end{equation}\nThere is a well-defined sense in which changes in this quantity are\ndetermined by a combination of ``gravitational wave'' and matter\nfluxes across the boundary $\\partial \\Sigma$. In this\ninterpretation, the amount of $\\ItP^*_\\psi$ carried away from a\nsystem via gravitational waves vanishes if the GAC associated with\n$\\psi^a$ is an affine collineation.\n\nThe 20-parameter family of scalars $\\ItP^*_\\psi$ is intended to\ndefine the linear and angular momenta of an extended body. This is\nat least the interpretation for the 10-parameter subset satisfying\n$B_{(\\sfa {\\mathsf{b}})}=0$. As in \\eqref{DixMomenta}, it is possible to\nwrite these momenta in the more conventional form of tensor fields\non $\\Gamma$. Let\n\\begin{equation}\n  \\ItP^*_\\psi = p_\\sfa^* A^\\sfa + \\frac{1}{2} S^*_{\\sfa {\\mathsf{b}}}\n  B^{\\sfa {\\mathsf{b}}}\\label{KomarMoment}\n\\end{equation}\nfor all $A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$. As written, the generalized\nangular momentum $S_{\\sfa{\\mathsf{b}}}^*$ needn't have any particular index\nsymmetries. Non-Killing Jacobi fields generate the symmetric\ncomponents of this tensor, although such generality isn't necessary.\nVarying among all combinations of initial data completely recovers\n$p^*_\\sfa$ and $S^*_{\\sfa {\\mathsf{b}}}$. Direct expressions can also be\nobtained with the use of \\eqref{JacobiFirstDer}. Continuing to work\nwith the less explicit form \\eqref{KomarMoment}, rates of change of\nthe tensor momenta may easily be extracted from $\\dot{\\ItP}^*_\\psi$.\nUsing the KT equations \\eqref{KTA} and \\eqref{KTB},\n\\begin{equation}\n  \\dot{\\ItP}^*_\\psi = ( \\dot{p}^*_\\sfa - \\frac{1}{2} S^*_{{\\mathsf{b}} {\\mathsf{c}}}\n  R^{{\\mathsf{b}} {\\mathsf{c}}}{}_{{\\mathsf{d}} \\sfa} \\dot{\\gamma}^{\\mathsf{d}} ) A^\\sfa +\n  \\frac{1}{2} ( \\dot{S}^*_{\\sfa {\\mathsf{b}}} + 2 \\dot{\\gamma}_\\sfa p^*_{\\mathsf{b}}) B^{\\sfa {\\mathsf{b}}}.\n\\end{equation}\nThis is closely analogous to \\eqref{Papapetrou}. The left-hand side\nis parameterized entirely by $A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$, so\nvarying these quantities determines all of the corrections to the\nPapapetrou equations.\n\nTwo definitions have now been suggested for the momenta of an\nextended body. The first -- summarized by \\eqref{DixMomenta} and\n\\eqref{PDef} -- is closely related to the one given by Dixon\n\\cite{Dix70a, Dix74, Dix79}. It is well-adapted to the construction\nof multipole moments for $T^{ab}$ that intrinsically take into\naccount stress-energy conservation. Mass centers defined from these\nmomenta are known to have most of the properties one might expect\n\\cite{SchattStreub1,CM}. The boundary of the worldtube $W$ isn't\nimportant as long as it lies outside of the matter distribution\nunder discussion. There is no vacuum momentum under this definition.\nUnfortunately, there does not appear to be any exact analog of\nGauss' law either. The momenta of a matter distribution (and changes\nto it) must be computed by integrating over 3-volumes. The\ngeneralized Komar integrals defined by \\eqref{PStarDef} have\ncomplementary characteristics. Their main advantage is in having a\ndirect interpretation analogous to Gauss' law. Changes in the\ncomponent of an isolated body's momentum generated by $\\psi^a$ only\ndepends on the gravitational flux $j^a_\\psi$ passing through the\nsurface $\\partial W$. The mass and angular momenta expected from\nthis definition also agree with commonly-accepted notions at least\nin appropriately symmetric spacetimes. It is potentially problematic\nthat $\\ItP^*_\\psi$ includes what is effectively a vacuum energy.\nThese scalars usually depend on the spatial extent of $W$ even when\nits boundary lies far outside of any matter distribution. This can\nmake it difficult to neatly separate the properties of disjoint\nmatter distributions, although similar situations are found even in\nordinary electromagnetism. It might be conceptually simpler to\nextend $\\partial W$ to infinity, although it is unlikely that all of\nthe bitensors used here would remain well-defined. There might also\nbe convergence problems. Related concepts presented in\n\\cite{EnergyConference} could be more useful for defining momenta\nover very large regions.\n\nSome insight into the behavior of the $\\ItP^*_\\psi$ defined here can\nbe gained by computing it for very small spheres. To be specific,\nlet $C(r,s)$ be the closed 2-surface on $\\Sigma(s)$ satisfying\n$X^\\sfa X_\\sfa = r^2$ for some $r>0$. This is effectively a sphere\nof proper radius $r$ centered at the point $\\gamma(s)$. It follows\nfrom \\eqref{GACAffine} and \\eqref{GravCurrent} that $j^a_\\psi$ is\nnegligible for small radii in the presence of matter. The momentum\ninside $C$ is approximately\n\\begin{equation}\n  P^*_\\xi \\simeq \\frac{8\\pi}{3} r^3 (T^{\\sfa}{}_{{\\mathsf{b}}} A^{\\mathsf{b}} -\n  \\frac{1}{2} A^{\\sfa} T^{{\\mathsf{b}}}{}_{{\\mathsf{b}}}) \\nabla_\\sfa \\tau + \\Or(r^4) .\n\\end{equation}\nIt is perhaps more interesting to consider regions that are locally\ndevoid of matter. These can be understood from the behavior of the\ngravitational current. Its approximate behavior near $\\Gamma$ is\neasily calculated from \\eqref{LieGGradJacobi} and\n\\eqref{GravCurrent}. To lowest nontrivial order,\n\\begin{eqnarray}\n  \\fl \\qquad \\qquad j^a_\\psi \\simeq - \\frac{1}{8 \\pi} \\sigma^{a}{}_{\\sfa} X^{\\mathsf{b}} \\big[\n  \\LieS R^{\\sfa}{}_{{\\mathsf{b}}} + \\frac{1}{3} \\big( g^{\\sfa {\\mathsf{c}}} R^{{\\mathsf{d}}}{}_{{\\mathsf{b}}} + 2 R_{{\\mathsf{b}}}{}^{{\\mathsf{c}} \\sfa {\\mathsf{d}}} \\big) \\LieS\n  g_{{\\mathsf{c}} {\\mathsf{d}}} \\big] + \\Or(X^2). \\label{GravCurrentEst}\n\\end{eqnarray}\nThis vanishes in vacuum for Killing-type Jacobi fields. A little\nmore calculation finds the same conclusion at order $X^2$ as well.\nThis contrasts sharply with other quasilocal notions of vacuum\nmomentum in general relativity. An extensive review of these\nconcepts may be found in \\cite{EnergyRev}. As remarked there, the\nenergy contained in small spheres has been calculated using several\ndifferent definitions. The generic result is that it is proportional\nto the Bel-Robinson tensor, and scales like $r^5$. If this were true\nfor the definition suggested here, terms quadratic in the curvature\nwould appear at order $X^2$ in the current. These are not found.\nVacuum momenta should really only be associated with Killing-type\nsymmetries, so the relevant currents defined here decrease at least\nas fast as $r^6$ as $r \\rightarrow 0$. Other definitions in the\nliterature find more energy in very small spheres. It is not clear\nhow to interpret this, although it might have interesting\nconsequences for the use of near zones and related concepts\nconnected to the mechanics of compact bodies.\n\n\n\\section{An example: gravitational plane waves}\\label{Sect:Example}\n\nThe discussion so far has mainly focused on the behavior of\ngeneralized symmetries near the point or worldline used to construct\nthem. With the exception of general identities like \\eqref{LieSig6},\nit has not been clear what happens to these vector fields far away\nfrom their origins. It is therefore useful to consider an example.\nGiven \\eqref{JacobiFirstDer}, the Jacobi fields can all be\ncalculated simply by differentiating the world function. This makes\nit convenient to consider spacetimes where $\\sigma$ is known\nexactly. Essentially the only examples of this type are\n\\textit{pp}-waves or assorted cosmological models (see\n\\cite{WorldFunction} and references cited therein).\n\nIn the interest of understanding the generalized Komar momenta\n\\eqref{PStarDef} in a vacuum spacetime, only \\textit{pp}-waves will\nbe considered here. Coordinates may be introduced such that the\nmetric satisfies\n\\begin{equation}\n  \\rmd s^2 = - 2 \\rmd u \\rmd v + a(u) \\rmd x^2 + b(u) \\rmd y^2 .\n  \\label{ppGeneral}\n\\end{equation}\nIt can then be shown that one-half of the geodesic distance between\npoints with coordinates $(u,v,x,y)$ and $(\\mathsf{u}, \\mathsf{v},\n\\mathsf{x}, \\mathsf{y})$ is given by \\cite{Gunther,Friedlander}\n\\begin{equation}\n  \\sigma = \\frac{1}{2} [ \\alpha (u,\\mathsf{u}) (x-\\mathsf{x})^2 + \\beta (u,\\mathsf{u}) (y-\\mathsf{y})^2 ] -\n  (u-\\mathsf{u}) (v-\\mathsf{v}), \\label{SigmaExact}\n\\end{equation}\nwhere\n\\begin{equation}\n  \\alpha(u,\\mathsf{u}) = \\frac{u-\\mathsf{u}}{\\int_{\\mathsf{u}}^{u} a^{-1}(w) \\rmd w} ; \\quad \\beta(u,\\mathsf{u}) = \\frac{u-\\mathsf{u}}{\\int_{\\mathsf{u}}^{u} b^{-1}(w) \\rmd\n  w}.\n\\end{equation}\nIt is clear by inspection that $\\partial\/\\partial x$,\n$\\partial\/\\partial y$, and $\\partial\/\\partial v$ are exact Killing\nvectors. They are not the only ones. All nontrivial spacetimes in\nthis class admit between five and seven linearly-independent Killing\nfields.\n\nMost \\textit{pp}-waves are effectively ``null dust'' solutions of\nEinstein's equation, although there are vacuum examples as well. One\nof these is given by\n\\begin{equation}\n  a(u) = \\cos^2 (\\lambda u) ; \\quad b(u) = \\cosh^2 (\\lambda u)\n , \\label{GravWave}\n\\end{equation}\nwhere we assume that $|\\lambda u|< \\pi\/2$ in order to avoid the two\ncoordinate singularities. This represents a simple plane-fronted\ngravitational wave with amplitude $\\lambda$. The only non-vanishing\ncomponents of the curvature are\n\\begin{equation}\n  C_{uxu}{}^{x} = C_{uyu}{}^{y} = \\lambda^2.\n\\end{equation}\nIt is trivial to modify this spacetime to be flat for (say) $u<0$\n\\cite{PlaneWaves}, although impulsive waves of this type will not be\ndiscussed here.\n\nGiven \\eqref{SigmaExact}, it is straightforward to explicitly\ncompute the Jacobi propagators $H^{a}{}_{{\\mathsf{b}}}\n\\sigma^{{\\mathsf{b}}}{}_{\\sfa}$ and $H^{a}{}_{\\sfa} \\sigma_{\\mathsf{b}}$ using\n\\eqref{HDef}. This can be done for any choice of $a(u)$ and $b(u)$,\nalthough we will specialize to the case defined by \\eqref{GravWave}.\nThe results are not particularly enlightening to write down in\ndetail, although they have some interesting consequences. First, all\nJacobi fields are found to be exactly Killing if their first\nderivatives vanish at the base point (denoted as usual with\nsans-serif font). The Jacobi fields $H^{a}{}_{[\\mathsf{x}}\n\\sigma_{\\mathsf{v}]}$ and $H^{a}{}_{[\\mathsf{y}}\n\\sigma_{\\mathsf{v}]}$ are also Killing. This identifies six\nindependent Killing fields. It also implies that the linear\ngravitational momentum $p^*_\\sfa $ defined in \\eqref{KomarMoment}\nmust vanish. This can be taken to imply (unsurprisingly) that the\ngravitational wave has zero rest mass: $|p^*|^2=0$.\n\nThere remain four non-Killing Jacobi fields with skew-symmetric\n$B_{\\sfa {\\mathsf{b}}}$. The one associated with spatial rotations in the\n$x-y$ plane is relatively simple to write down when $\\mathsf{u}=0$:\n\\begin{eqnarray}\n  \\fl \\qquad \\psi_{\\mathsf{xy}}^a = 2 H^{a}{}_{\\mathsf{[x}} \\sigma_{\\mathsf{y}]} =\n  -(y-\\mathsf{y})\n  \\left( \\frac{ \\tan(\\lambda u) }{ \\tanh(\\lambda u) } \\right)\n  \\frac{\\partial}{\\partial x} + (x - \\mathsf{x}) \\left( \\frac{ \\tanh(\\lambda u) }{\n  \\tan(\\lambda u) } \\right) \\frac{\\partial}{\\partial y}\n  \\nonumber\n  \\\\\n  \\qquad \\qquad ~  + \\lambda ( x -\\mathsf{x})\n  (y - \\mathsf{y}) \\left( \\frac{ \\tanh(\\lambda u) - \\tan (\\lambda u) }{ \\tan(\\lambda u) \\tanh(\\lambda u) } \\right) \\frac{\\partial}{\\partial\n  v} .\n\\end{eqnarray}\nThis clearly reduces to its expected form when $u \\rightarrow\n\\mathsf{u}$. The degree to which it succeeds in being a genuine\nKilling field may be estimated by noting that\n\\begin{equation}\n    \\fl \\qquad |\\mathcal{L}_{\\psi_{\\mathsf{xy}}} g_{ab}|^2 = \\left( \\frac{\\cos(2 \\lambda u) + \\cosh(2 \\lambda u) -2 }{\\sqrt{2} \\sin(\\lambda u) \\sinh(\\lambda\n    u)} \\right)^2 \\simeq \\frac{8}{9} (\\lambda u)^4 + \\Or(u^8)\n    \\label{LieEstimate1}\n\\end{equation}\nwhen $\\mathsf{u}=0$. The quadratic growth estimate here is typically\nquite good even near the coordinate singularities. There is little\nqualitative change in the nature of this expression if $\\mathsf{u}\n\\neq 0$. More interestingly, the gravitational current\n\\eqref{GravCurrent} associated with $\\psi^a_{\\mathsf{xy}}$ always\nvanishes. This suggests that an observer would not be compelled to\nascribe any $xy$ component of angular momentum to gravitational\nwaves with the given form. $H^{a}{}_{[\\mathsf{u}}\n\\sigma_{\\mathsf{v}]}$ has similar properties. It satisfies an\nequation almost identical to \\eqref{LieEstimate1}, and the\ngravitational current generated by it always vanishes.\n\nMore interesting are the remaining two Killing-type Jacobi fields\n$\\psi_{\\mathsf{xu}}^a = 2 H^{a}{}_{[\\mathsf{x}}\n\\sigma_{\\mathsf{u}]}$ and $\\psi_{\\mathsf{yu}}^a = 2\nH^{a}{}_{[\\mathsf{y}} \\sigma_{\\mathsf{u}]}$. Specializing again to\nthe case $\\mathsf{u}=0$,\n\\begin{eqnarray}\n  \\fl \\quad |\\mathcal{L}_{\\psi_{\\mathsf{xu}}} g_{ab}|^2 \\simeq |\\mathcal{L}_{\\psi_{\\mathsf{yu}}} g_{ab}|^2 \\simeq \\frac{2}{3}\n  \\lambda^4 u^2 \\Big(\n  [ (x - \\mathsf{x})^2+(y - \\mathsf{y})^2] + \\frac{2}{3} (v-\\mathsf{v}) u \\Big) + \\Or(u^4) .\n\\end{eqnarray}\nUnlike the expansion in \\eqref{LieEstimate1}, this approximation\nfails long before $|\\lambda u| \\rightarrow \\pi\/2$. In general, the\ntwo magnitudes on the left-hand side have distinct behaviors that\nstrongly depend on $\\mathsf{u}$. Oscillations generically arise as\n$u$ is varied, for example. See Fig. 1.\n\n\\begin{figure}\n    \\begin{center}\n\n    \\includegraphics[width=0.8\\textwidth]{LieEx.eps}\n\n    \\caption{Plots of $|\\mathcal{L}_{\\psi_{\\mathsf{xu}}} g_{ab}|^2$ for\n    a gravitational plane wave described by \\eqref{ppGeneral} and\n    \\eqref{GravWave}. The origin is assumed to be at $\\mathsf{u}=0$.\n    Both solid curves assume that $v-\\mathsf{v} = 0$. The dashed ones\n    use $\\lambda(v-\\mathsf{v})=1\/2$ instead. Both thicker curves set\n    $x-\\mathsf{x}=0$ and $\\lambda(y-\\mathsf{y})=1\/4$. The thinner ones use\n    $\\lambda(x-\\mathsf{x})=1$ and $y-\\mathsf{y}=0$. Plots for\n    $|\\mathcal{L}_{\\psi_{\\mathsf{yu}}} g_{ab}|^2$ look very similar\n    unless $\\mathsf{u} \\neq 0$.}\n    \\end{center}\n\\end{figure}\n\nThere are gravitational currents associated with both\n$\\psi^{a}_{\\mathsf{xu}}$ and $\\psi^{a}_{\\mathsf{yu}}$. Applying\n\\eqref{GravCurrent} and expanding near $\\mathsf{u}=0$,\n\\begin{equation}\n  j^a_{\\psi_{\\mathsf{xu}}} = \\frac{\\lambda^6}{180 \\pi} u^4\n  \\left[ u \\frac{\\partial}{\\partial x} + (x - \\mathsf{x}) \\frac{\\partial}{\\partial v} \\right] + \\Or (u^6) .\n\\end{equation}\nSimilarly,\n\\begin{equation}\n  j^a_{\\psi_{\\mathsf{yu}}} = - \\frac{\\lambda^6}{180 \\pi} u^4\n  \\left[ u \\frac{\\partial}{\\partial y} + (y - \\mathsf{y}) \\frac{\\partial}{\\partial v} \\right] + \\Or(u^6).\n\\end{equation}\nThese expansions are qualitatively accurate throughout the region of\ninterest. It is now clear from \\eqref{PStar2} and\n\\eqref{KomarMoment} that the only non-vanishing gravitational\nmomenta (associated with Killing-type GACs) are\n\\begin{equation}\n  S^*_{\\mathsf{x u}} = \\int_\\Sigma j^a_{\\psi_{xu}} \\rmd S_a ; \\qquad  S^*_{\\mathsf{y u}}\n  = \\int_\\Sigma j^a_{\\psi_{yu}} \\rmd S_a .\n\\end{equation}\nThe rates at which these quantities change depends on the relevant\nfluxes through $\\partial \\Sigma$. Regardless, the magnitude of the\nangular momentum tensor always vanishes. Intuitively, these\nstatements might be taken to mean that the gravitational wave has a\n``mass dipole moment'' equal to its one non-vanishing component of\nordinary angular momentum.\n\nThe results here could be straightforwardly extended to much more\ngeneral \\textit{pp}-wave (and other) spacetimes. The most\ninteresting point is perhaps the calculation of explicit\ngravitational currents in a vacuum spacetime. In these cases, the\ngeneral results obtained in Sect. \\ref{Sect:ExtendedMatter} only\nstate that $j^a$ will decrease no slower than $X^3$ as $X\n\\rightarrow 0$. The example here scales like $X^5$. Although this\nconclusion would probably not be preserved in more complicated\nspacetimes, it shows that momenta not arising from stress-energy\ntensors can sometimes be ignored in remarkably large regions.\n\n\\section{Conclusions}\n\nTwo different notions of approximate affine collineations have been\nintroduced. One has the physical interpretation of capturing\nsymmetry principles in a normal neighborhood of a point, while the\nother is adapted to the measurements of a particular observer. Flows\ngenerated by both of these objects leave $\\sigma^\\sfa(x,\\gamma)$\ninvariant. This has the simple interpretation that Jacobi fields\npreserve Riemann normal coordinates. GACs do the same for the\nspatial components of Fermi normal coordinate systems. These objects\nalways exist, and each forms a 20-dimensional group. Individual\nelements may be interpreted using the values of the field and its\nfirst derivatives at the appropriate base point. The only caveat to\nthis is that a GAC which might initially appear to be purely\ntranslational could slowly acquire some rotational and boost-type\ncomponents. This mixing is essential in order to ensure that the\nfields nearly satisfy \\eqref{AffineDef} near the observer's\nworldline.\n\nThe relevance of these definitions ultimately lies in their\napplications. The approximate symmetries introduced here have been\nused to write down analogs of the typical conservation laws applying\nto geodesics in spacetimes admitting affine collineations. Some of\nthe resulting parameters are exact constants of motion, while others\nare only expected to vary slowly near the preferred point or\nobserver. Regardless, they may be used to classify and derive\ngeodesics in certain regions. Similar results have also been\ndiscussed in connection with extended matter distributions. This led\nto natural notions for the linear and angular momenta of a spacetime\nvolume as viewed in a particular frame. There seems to be some\ndisagreement with other quasilocal notions of gravitational momenta,\nso it is not clear how the definition here should be interpreted. It\nis unknown if it has any positivity or related characteristics.\n\nConcepts discussed here might also be applied to simplify\nperturbation theory off of some background geometry possessing an\nexact affine collineation. It could be useful, for example, to\nuniquely construct GACs with respect to a center-of-mass worldline\nthat coincides with exact timelike or axial Killing fields in the\nunperturbed geometry. Center-of-mass trajectories might also be\nestimated using notions of approximate stationarity. More\nconcretely, an analysis of the quantities $\\ItP_\\xi$ defined in\n\\eqref{PDef} can be shown to provide significant insights into the\neffects of self-forces and self-torques on isolated bodies. Details\nare presented elsewhere \\cite{HarteFuture}.\n\n\\ack\n\nI am grateful for many helpful discussions and comments from Robert\nWald and Samuel Gralla. This work was supported by NSF grant\nPHY04-56619 to the University of Chicago.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{#1} #2}\n\t\\vspace{-2pt}\n}\n\n\n\\title{Fair Division of Time: Multi-layered Cake Cutting}\n\n\\author{\nHadi Hosseini$^1$\\and\nAyumi Igarashi$^2$\\and\nAndrew Searns$^1$\n\\affiliations\n$^1$Rochester Institute of Technology, US\\\\\nNational Institute of Informatics, Japan\n\\emails\nhhvcs@rit.edu,\nayumi\\_igarashi@nii.ac.jp,\nabs2157@rit.edu\n}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nWe initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each time interval. Several real-life applications exhibit such restrictions on overlapping pieces; for example, assigning time intervals over multiple facilities and resources or assigning shifts to medical professionals. We investigate the existence and computation of envy-free and proportional allocations. We show that envy-free allocations that are both feasible and contiguous are guaranteed to exist for up to three agents with two types of preferences, when the number of layers is two. We also show that envy-free feasible allocations where each agent receives a polynomially bounded number of intervals exist for any number of agents and layers under mild conditions on agents' preferences. We further devise an algorithm for computing proportional allocations for any number of agents and layers. \n\\end{abstract}\n\n\\section{Introduction}\nConsider a group of students who wish to use multiple college facilities such as a conference room and an exercise room over different periods of time.\nEach student has a preference over what facility to use at different time of the day: Alice prefers to set her meetings in the morning and exercise in the afternoon, whereas Bob prefers to start the day with exercising for a couple of hours and meet with his teammates in the conference room for the rest of the day.\n\nThe fair division literature has extensively studied the problem of dividing a heterogeneous divisible resource (aka a \\textit{cake}) among several agents who may have different preference over the various pieces of the cake~\\citep{Steinhaus48,Robertson98,Brams06}.\nThese studies have resulted in a plethora of axiomatic and existence results~\\citep{barbanel2005geometry,moulin2004fair} as well as computational solutions~\\citep{procaccia2013cake,aziz2016discretefour} under a variety of assumptions, and were successfully implemented in practice (see~\\citep{procaccia_moulin_2016,Brams96} for an overview).\nIn the case of Alice and Bob, each facility represents a layer of the cake in a \\textit{multi-layered cake cutting} problem, and the question is how to allocate the time intervals (usage right) of the facilities according to their preferences in a fair manner.\n\nOne naive approach is to treat each cake independently and solve the problem through well-established cake-cutting techniques by performing a fair division on each layer separately.\nHowever, this approach has major drawbacks: First, the final outcome, although fair on each layer, may not necessarily be fair overall. Second, the allocation may not be feasible, i.e., it may assign two overlapping pieces (time intervals) to a single agent. \nIn our example, Alice cannot simultaneously utilize the exercise room and the conference room at the same time if she receives overlapping  intervals.\nSeveral other application domains exhibit similar structures over resources: assigning nurses to various wards and shifts, doctors to operation rooms, and research equipment to groups, to name a few.\n\nIn multi-layared cake cutting, each layer represents a divisible resource. Each agent has additive preferences over every disjoint (non-overlapping) intervals. A division of a multi-layered cake is \\emph{feasible} if no agent's share contains overlapping intervals, and is contiguous if each allocated piece of a layer is contiguous. \nThere has been some recent work on dividing multiple cakes among agents~\\citep{cloutier2010two,lebert2013envy}. Yet, none of the previous work considered the division of multiple resources under feasibility and contiguity constraints. \nIn this paper, we thus ask the following question: \n\\begin{quote}\n\\textit{What fairness guarantees can be achieved under feasibility and contiguity constraints for various number of agents and layers?}\n\\end{quote}\n\n\\subsection{Our Results}\nWe initiate the study of the multi-layered cake cutting problem for allocating divisible resources, under contiguity and feasibility requirements. Our focus is on two fairness notions, \\textit{envy-freeness} and \\textit{proportionality}. Envy-freeness (EF) requires that each agent believes no other agent's share is better than its share of the cake. Proportionality (Prop) among $n$ agents requires that each agent receives a share that is valued at least $\\frac{1}{n}$ of the value of the entire cake.\nFor efficiency, we consider \\textit{complete} divisions with no leftover pieces.\n\nFocusing on envy-free divisions, we show the existence of envy-free and complete allocations that are both feasible and contiguous for two-layered cakes and up to three agents with at most two types of preferences. These cases are particularly appealing since many applications often deal with dividing a small number of resources among few agents (e.g. assigning meeting rooms). Turning our attention to the case when the contiguity requirement is dropped, we then show that envy-free feasible allocations exist for any number $n$ of agents and any number $m$ of layers with $m \\leq n$, under mild conditions on agents' preferences.\nWe further show that proportional complete allocations that are both feasible and contiguous exist when the number of layers isa power of two. \nSubsequently, we show that although this result cannot be immediately extended to any number of agents and layers, a proportional complete allocation that is feasible exists when the number of layers is at most the number of agents, and can be computed efficiently.\n\n\n\\begin{table}[t]\n\\small \n\\centering\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\nAgents ($n$)            & Layers ($m$)                         & EF & Prop &  \\\\ \\midrule\n2               & 2                         &  \\cmark (Thm. \\ref{thm:EF:two} ) &   \\cmark (Thm. \\ref{thm:exponential})    &  \\\\\n3               & 2                         &  \\cmark (Thm. \\ref{thm:EF:any}$^\\diamondsuit\\dagger$) &    \\cmark (Thm. \\ref{thm:exponential})  &  \\\\\nany $n\\geq m$              & $2^{a}$,   \\scriptsize{$a\\in\\mathbb{Z}_{+}$}   &   \\cmark (Thm. \\ref{thm:EF:any}$^\\diamondsuit\\dagger$)  &   \\cmark (Thm. \\ref{thm:exponential})   &  \\\\ \nany $n\\geq m$  & any $m$ &  \\cmark (Thm. \\ref{thm:EF:any}$^\\diamondsuit\\dagger$)  &   \\cmark (Thm. \\ref{thm:prop:feasible:any}$^\\diamondsuit$)   &  \\\\ \\bottomrule\n\\end{tabular}\n\\caption{The overview of our results. $\\dagger$ assumes continuity of value density functions. $\\diamondsuit$ indicates that existence holds without contiguity requirement. Note that when $m> n$, no complete and feasible (non-overlapping) solution exists.}\n\\end{table}\n\n\\subsection{Related Work}\nIn recent years, cake cutting has received significant attention in artificial intelligence and economics as a metaphor for algorithmic approaches in achieving fairness in allocation of resources \\citep{procaccia2013cake,branzei2019communication,kurokawa2013cut,aziz2016discrete}. \nRecent studies have focused on the fair division of resources when agents have requirements over multiple resources that must be simultaneously allocated in order to carry out certain tasks (e.g. CPU and RAM) \\citep{Ghodsi:2011:DRF:1972457.1972490,gutman2012fair,parkes2015beyond}. \nThe most relevant work to ours is the envy-free multi-cake fair division that considers dividing multiple cakes among agents with linked preferences over the cakes. Here, agents can simultaneously benefit from all allocated pieces with no constraints. They show that envy-free divisions with only few cuts exist for two agents and many cakes, as well as three agents and two cakes~\\citep{cloutier2010two,lebert2013envy,nyman2020fair}. In contrast, a multi-layered cake cutting requires non-overlapping pieces. Thus, \\cite{cloutier2010two}'s generalized envy-freeness notion on multiple cakes does not immediately imply envy-freeness in our setting and no longer induces a feasible division.\n\n\\section{Our Model}\nOur setting includes a set of {\\em agents} denoted by $N=[n]$, a set of {\\em layers} denoted by $L=[m]$, where for a natural number $s \\in \\mathbb{N}$, $[s]=\\{1,2,\\ldots,s\\}$.\nGiven two real numbers $x,y \\in \\mathbb{R}$, we write $[x,y]=\\{\\, z \\in \\mathbb{R} \\mid x \\le z \\le y \\,\\}$ to denote an interval. We denote by $\\mathbb{R}_{+}$ (respectively $\\mathbb{Z}_{+}$) the set of non-negative reals (respectively, integers) including $0$. \nA {\\em piece} of cake is a finite set of disjoint subintervals of $[0,1]$. We say that a subinterval of $[0,1]$ is a {\\em contiguous piece} of cake. An {\\em $m$-layered cake} is denoted by $\\mathcal{C}=(C_j)_{j \\in L}$ where $C_j \\subseteq [0,1]$ is a contiguous piece for $j \\in L$. We refer to each $j \\in L$ as $j$-th {\\em layer} and $C_j$ as $j$-th {\\em layered cake}. \n\nEach agent $i$ is endowed with a non-negative {\\em integrable density function} $v_{ij}:C_j \\rightarrow \\mathbb{R}_{+}$. For a given piece of cake $X$ of $j$-th layer, $V_{ij}(X)$ denotes the value assigned to it by agent $i$, i.e., $V_{ij}(X)=\\sum_{I \\in X}\\int_{x \\in I} v_{ij}(x) dx$. These functions are assumed to be {\\em normalized} over layers: for each $i \\in N$, $\\sum_{j \\in L}V_{ij}(C_j)=1$. A {\\em layered piece} is a sequence $\\mathcal{X}=(X_j)_{j \\in L}$ of pieces of each layer $j \\in L$; a layered piece is said to be {\\em contiguous} if each $X_j$ is a contiguous piece of each layer. \nWe assume valuation functions are \\emph{additive} on layers and write $V_{i}(\\mathcal{X})=\\sum_{j \\in L}V_{ij}(X_{j})$.\n\nA layered contiguous piece is said to be {\\em non-overlapping} if no two pieces from different layers overlap, i.e, for any pair of distinct layers $j,j' \\in L$ and for any $I \\in X_j$ and $I' \\in X_{j'}$, $I \\cap I'=\\emptyset$. For two layered pieces $\\mathcal{X}$ and $\\mathcal{X}'$, we say that agent $i$ {\\em weakly prefers} $\\mathcal{X}$ to $\\mathcal{X}'$ if $V_i(\\mathcal{X}) \\geq V_i(\\mathcal{X}')$. \n\nA {\\em multi-allocation} $\\mathcal{A}=(\\mathcal{A}_1,\\mathcal{A}_2,\\ldots,\\mathcal{A}_n)$ is a partition of the $m$-layered cake $\\mathcal{C}$ where each $\\mathcal{A}_i=(A_{ij})_{j \\in L}$ is a layered piece of the cake allocated to agent $i$; we refer to each $\\mathcal{A}_i$ as a {\\em bundle} of $i$. For a multi-allocation $\\mathcal{A}$ and $i \\in N$, we write $V_{i}(\\mathcal{A}_i)=\\sum_{j \\in L}V_{ij}(A_{ij})$ to denote the value of agent $i$ for $\\mathcal{A}_i$. A {\\em multi-allocation} $\\mathcal{A}$ is said to be \n\\begin{itemize}\n\\item {\\em contiguous} if each $\\mathcal{A}_i$ for $i \\in N$ is contiguous; \n\\item {\\em feasible} if each $\\mathcal{A}_i$ for $i \\in N$ is non-overlapping.\n\\end{itemize}\n\nWe focus on \\emph{complete} multi-allocations where the entire cake must be allocated.\nNotice that some layers may be disjoint (see Figure \\ref{fig:server}), and the number of agents must exceed the number of layers, i.e. $n \\geq m$; otherwise the multi-allocation will contain overlapping pieces. We illustrate our model in the following example. \n\n\\begin{example}[Resource sharing]\nSuppose that there are three meeting rooms $r_1$, $r_2$, and $r_3$ with different capacities, and three researchers Alice, Bob, and Charlie. The first room is available all day, the second and the third rooms are only available in the morning and late afternoon, respectively (see Fig.~\\ref{fig:server}). Each researcher has a preference over the access time to the shared rooms. For example, Alice wants to have a group meeting in the larger room in the morning and then have an individual meeting in the smaller one in the afternoon. \n\\end{example}\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{tikzpicture}[scale=0.5, transform shape]\n\\draw[fill=red!10, thick] (0,1.3) rectangle (10,2.3);\n\\draw[fill=blue!10, thick] (0,0) rectangle (4,1); \n\\draw[thick] (5,-1.3) rectangle (10,-0.3);\n\n\\node at (-0.5,1.8) {\\huge $r_1$};\n\\node at (-0.5,0.5) {\\huge $r_2$};\n\\node at (-0.5,-0.8) {\\huge $r_3$};\n\n\\draw[thick,->] (-0.5,-2) -- (10.5,-2);\n\\node at (11.4,-2) {\\Huge time};\n\n\\end{tikzpicture}\n\\caption{Example of a multi-layered cake. There are three meeting rooms $r_1$, $r_2$, and $r_3$ with different capacities, shared among several research groups.}\n\\label{fig:server}\n\\end{figure}\n\n\\paragraph{Fairness.}\nA multi-allocation is said to be {\\em envy-free} if no agent {\\em envies} the others, i.e., $V_{i}(\\mathcal{A}_i) \\ge V_{i}(\\mathcal{A}_{i'})$ for any pair of agents $i,i' \\in N$. A multi-allocation is said to be {\\em proportional} if each agent gets his {\\em proportional fair share}, i.e., $V_{i}(\\mathcal{A}_i) \\ge \\frac{1}{n}$ for any $i \\in N$. The following implication, which is well-known for the standard setting, holds in our setting as well.\n\n\\begin{lemma}\\label{lem:propEF}\nAn envy-free complete multi-allocation satisfies proportionality. \n\\end{lemma}\n\\begin{proof}\nConsider an envy-free complete multi-allocation $\\mathcal{A}_i=(A_{ij})_{j \\in L}$ and an agent $i \\in N$. By envy-freeness, we have that $V_{i}(\\mathcal{A}_i) \\geq V_{i}(\\mathcal{A}_j)$ for any $j \\in N$. Summing over $j \\in N$, we get $V_{i}(\\mathcal{A}_i) \\geq \\frac{1}{n}\\sum_{j \\in N}V_{i}(\\mathcal{A}_j)=\\frac{1}{n}$ by additivity. \n\\end{proof}\n\n\\paragraph{The $m$-layered cuts.}\nIn order to cut the layered cake while satisfying the non-overlapping constraint, we define a particular approach for partitioning the entire cake into diagonal pieces. Consider the $m$-layered cake $\\mathcal{C}$ where $m$ is an even number. For each point $x$ of the interval $[0,1]$, we define\n\\begin{itemize}\n\\item $LR(x,\\mathcal{C})=(\\bigcup^{\\frac{m}{2}}_{j=1}C_j \\cap [0,x]) \\cup (\\bigcup^{m}_{j=\\frac{m}{2}+1}C_j \\cap [x,1])$; \n\\item $RL(x,\\mathcal{C})=(\\bigcup^{\\frac{m}{2}}_{j=1}C_j \\cap [x,1]) \\cup (\\bigcup^{m}_{j=\\frac{m}{2}+1}C_j \\cap [0,x])$. \n\\end{itemize}\n$LR(x,\\mathcal{C})$ consists of the top-half subintervals of points left of $x$ and the lower-half subintervals of points right of $x$; similarly, $RL(x,\\mathcal{C})$ consists of the top-half subintervals of points right of $x$ and the lower-half subintervals of points left of $x$ (Fig. \\ref{fig:LR:RL}). We abuse the notation and write $LR(x)=LR(x,\\mathcal{C})$ and $RL(x)=RL(x,\\mathcal{C})$ if $\\mathcal{C}$ is clear from the context. \n\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}[scale=0.5, transform shape]\n\n\\draw[fill=blue!10, thick] (0,0) rectangle (6,2); \n\\draw[thick] (0,-2) rectangle (6,0);\n\\draw[dotted] (0,1) -- (6,1);\n\\draw[dotted] (0,-1) -- (6,-1);\n\n\\node[thick] at (0,2.5) {\\Large $x=0$};\n\\node at (3,-1) {\\huge $LR(x)$};\n\\node at (3,1) {\\huge $RL(x)$};\n\n\\node at (-0.8,1.5) {\\Large $j=1$};\n\\node at (-0.8,0.5) {\\Large $j=2$};\n\\node at (-0.8,-0.5) {\\Large $j=3$};\n\\node at (-0.8,-1.5) {\\Large $j=4$};\n\n\\begin{scope}[xshift=8cm]\n\n\\draw[thick] (0,0) rectangle (2,2);\n\\draw[fill=blue!10, thick] (0,-2) rectangle (2,0); \n\\draw[fill=blue!10, thick] (2,0) rectangle (6,2); \n\\draw[thick] (2,-2) rectangle (6,0);\n\\draw[dotted] (0,1) -- (6,1);\n\\draw[dotted] (0,-1) -- (6,-1);\n\n\\node[thick] at (2.0,2.5) {\\bf \\Large $x=\\frac{2}{5}$};\n\\node at (1,1) {\\huge $LR(x)$};\n\\node at (4,-1) {\\huge $LR(x)$};\n\\node at (4,1) {\\huge $RL(x)$};\n\\node at (1,-1) {\\huge $RL(x)$};\n\n\\node at (-0.8,1.5) {\\Large $j=1$};\n\\node at (-0.8,0.5) {\\Large $j=2$};\n\\node at (-0.8,-0.5) {\\Large $j=3$};\n\\node at (-0.8,-1.5) {\\Large $j=4$};\n\n\\end{scope}\n \n\\end{tikzpicture}\n\\caption{Examples of the partitions induced by $x=0$ and $x=\\frac{2}{5}$ for a {\\bf four-layered} cake.}\n\\label{fig:LR:RL}\n\\end{figure}\n\n\n\\paragraph{Computational model.}\nFollowing the standard {\\em Robertson-Webb Model} \\citep{Robertson98}, we introduce two types of queries: those for a cake on each layer (called a {\\em short knife}) and those for the entire cake (called a {\\em long knife}). \n\n\\paragraph{Short knife.} Short eval query: given an interval $[x,y]$ of the $j$-th layered cake $C_j$, $eval_j(i,x,y)$ asks agent $i$ for its value $[x,y]$, i.e., $V_{ij}([x,y])$.\n Short cut query: given a point $x$ and $r \\in [0,1]$, $cut_j(i,x,r)$ asks agent $i$ for the minimum point $y$ such that $V_{ij}([x,y])=r$.\n\n\\paragraph{Long knife.} Long eval query: given a point $x$, $eval(i,x)$ asks agent $i$ for its value $LR(x)$, i.e., $V_{i}(LR(x))$. \nLong cut query: given $r \\in [0,1]$, $cut(i,r)$ asks agent $i$ for the minimum point $x$ such that $V_{i}(LR(x))=r$ if such point $x$ exists.\n\n\n\\section{Existence of a switching point}\nWe start by showing the existence of a point $x$ that equally divides the entire cake into two pairs of diagonal pieces, both for the individuals and for the majority; these will serve as a fundamental property in our problem. \nWe say that $x \\in [0,1]$ is a {\\em switching point} for agent $i$ over a layered cake $\\mathcal{C}$ if $V_i(LR(x))=V_i(RL(x))$. \n\n\\begin{lemma}\\label{lem:switching}\nSuppose that the number $m$ of layers is even. Take any $i \\in N$. Let $r\\in \\mathbb{R}$ be such that $($i$)$ $V_i(LR(0)) \\geq r$ and $V_i(RL(0)) \\leq r$, or $($ii$)$ $V_i(LR(0)) \\leq r$ and $V_i(RL(0)) \\geq r$. \nThen, there exists a point $x \\in [0,1]$ such that $i$ values $LR(x)$ exactly at $r$, i.e. $V_i(LR(x)) = r$. In particular, a switching point for $i$ always exists. \n\\end{lemma}\n\\begin{proof}\nSuppose without loss of generality that $V_i(LR(0)) \\geq r$ and $V_i(RL(0)) \\leq r$. Consider the function $f(x)=V_i(LR(x))$ for $x \\in [0,1]$. Recall that $f(x)$ is a continuous function written as the sum of continuous functions:\n$\nf(x)=\\sum^{\\frac{m}{2}}_{j=1} V_{ij}(C_j \\cap [0,x])+ \\sum^{m}_{j=\\frac{m}{2}+1} V_{ij}(C_j \\cap [x,1]). \n$\nSince $f(0) \\geq r$ and $f(1) \\leq r$, there is a point $x \\in [0,1]$ with $f(x)=r$ by the intermediate value theorem, which proves the claim. Further, by taking $r=\\frac{1}{2}$, the point $x$ where $V_i(LR(x))=\\frac{1}{2}$ is a switching point for agent $i$. \n\\end{proof}\n\nWe will generalize the notion of a switching point from the individual level to the majority. For layered contiguous pieces $\\mathcal{I}$ and $\\mathcal{I}'$, we say that the majority weakly prefer $\\mathcal{I}$ to $\\mathcal{I}'$ (denoted by $\\mathcal{I} \\mathop{\\stackrel{m}{\\succeq}} \\mathcal{I}'$) if there exists $S \\subseteq N$ such that $|S| \\geq \\ceil{\\frac{n}{2}}$ and each $i \\in S$ weakly prefers $\\mathcal{I}$ to $\\mathcal{I}'$. We say that $x \\in [0,1]$ is a {\\em majority switching point} over $\\mathcal{C}$ if $LR(x) \\mathop{\\stackrel{m}{\\succeq}} RL(x)$ and $RL(x) \\mathop{\\stackrel{m}{\\succeq}} LR(x)$.\nThe following lemma guarantees the existence of a majority switching point, for any even number of layers and any number of agents. \n\n\\begin{lemma}\\label{lem:majority:switching}\nSuppose that the number of layers, $m$, is even. Then, there exists a majority switching point for any number $n \\geq m$ of agents. \n\\end{lemma}\n\\begin{proof}\nSuppose without loss of generality that the majority of agents weakly prefer $LR(0)$ to $RL(0)$. Since $LR(0)=RL(1)$ and $RL(0)=LR(1)$, this means that by the time when the long knife reaches the right-most point, i.e., $x=1$, the majority preference switches. \n\nFormally, consider the following set of points $x \\in [0,1]$ where the majority weakly prefer $LR(x)$ to $RL(x)$:  \n\\[\nM:= \\{\\, x \\in [0,1] \\mid LR(x) \\mathop{\\stackrel{m}{\\succeq}} RL(x)\\,\\}. \n\\]\nWe will first show that $M$ is a compact set. Clearly, $M$ is bounded. To show that $M$ is closed, consider an infinite sequence as follows $X=\\{x_k\\}_{k=1,2, \\ldots} \\subseteq M$ that converges to $x^*$. For each $k=1,2,\\ldots$, we denote by $S_k$ the set of agents who weakly prefer $LR(x_k)$ to $RL(x_k)$; by definition, $|S_k| \\geq \\ceil{\\frac{n}{2}}$. Since there are finitely many subsets of agents, there is one subset $S_k \\subseteqq N$ that appears infinitely often; let $S^*$ be such subset and $\\{x^*_k\\}_{k=1,2, \\ldots}$ be an infinite sub-sequence of $X$ such that for each $k$, each agent in $S^*$ weakly prefers $LR(x^*_k)$ to $RL(x^*_k)$. Since the valuations $V_i$ for $i \\in S^*$ are continuous, each agent $i \\in S^*$ weakly prefers $LR(x^*)$ to $RL(x^*)$ at the limit $x^*$, which implies that $x^* \\in M$  and hence $M$ is closed. Now since $M$ is a compact set, the supremum $t^*=\\sup M$  belongs to $M$. By the maximality of $t^*$, at least $\\ceil{\\frac{n}{2}}$ agents weakly prefer $RL(t^*)$ to $LR(t^*)$. Since $t^* \\in M$, at least $\\ceil{\\frac{n}{2}}$ agents weakly prefer $LR(t^*)$ to $RL(t^*)$ as well. Thus, $t^*$ corresponds to a majority switching point. \n\\end{proof}\n\n\\section{Envy-free multi-layered cake cutting}\nFirst, we will look into the problem of obtaining complete envy-free multi-allocations, while satisfying non-overlapping constraints. When there is only one layer, it is known that an envy-free contiguous allocation exists for any number of agents under mild assumptions on agents' preferences \\citep{Stromquist1980,Su1999}.\nGiven the contiguity and feasibility constraints, the question is whether it is possible to guarantee an envy-free division in the multi-layered cake-cutting model. \n\n\\subsection{Two agents and two layers}\nWe answer the above question positively for a simple, yet important, case of two agents and two layers. The standard protocol that achieves envy-freeness for two agents is known as the {\\em cut-and-choose} protocol: Alice divides the entire cake into two pieces of equal value. Bob selects his preferred piece over the two pieces, leaving the remainder for Alice. \n\nWe extend this protocol to the multi-layered cake cutting using the notion of a switching point. Alice first divides the layered cake into two {\\em diagonal pieces}: one that includes the top left and lower right parts and another that includes the top right and lower left parts of the cake. Our version of the cut-and-choose protocol is specified as follows:\n\n\\vspace{5pt}\n\n\\noindent\\fbox{%\n\t\\parbox{0.985\\linewidth}\n\t   {%\n\t\t\\textbf{Cut-and-choose protocol for $n=2$ agents} over a two-layered cake $\\mathcal{C}$: \\\\\n\t\t\\textit{Step 1.} Alice selects her switching point $x$ over $\\mathcal{C}$.\\\\\n\t\n\t\t\\textit{Step 2.} Bob chooses a weakly preferred layered contiguous piece among $LR(x)$ and $RL(x)$. \\\\\n\t\t\\textit{Step 3.} Alice receives the remaining piece.\n}%\n}\n\\begin{figure}[htb]\n\\centering\n\\begin{tikzpicture}[scale=0.7, transform shape]\n\\draw[thick] (0,0) rectangle (3,1);\n\\draw[fill=blue!10, thick] (3,0) rectangle (10,1); \n\\draw[fill=blue!10, thick] (0,-1) rectangle (3,0); \n\\draw[thick] (3,-1) rectangle (10,0); \n\\node at (3.0,1.3) {\\large $x$};\n\\node at (1.5,0.5) {\\large $LR(x)$};\n\\node at (6.5,-0.5) {\\large $LR(x)$};\n\\node at (1.5,-0.5) {\\large $RL(x)$};\n\\node at (6.5,0.5) {\\large $RL(x)$};\n\\end{tikzpicture}\n\\caption{Cut-and-Choose for two-layered cake}\n\\label{fig:EF:two}\n\\end{figure}\n\n\n\\begin{theorem}\\label{thm:EF:two}\nThe cut-and-choose protocol yields a complete envy-free multi-allocation that is feasible and contiguous for two agents and a two-layered cake using $O(1)$ number of long eval and cut queries.\n\\end{theorem}\n\\begin{proof}\nIt is immediate to see that the protocol returns a complete multi-allocation where each agent is assigned to a non-overlapping layered contiguous piece. The resulting allocation satisfies envy-freeness: Bob does not envy Alice since he chooses a preferred piece among $LR(x)$ and $RL(x)$. Alice does not envy Bob by the definition of a switching point. \n\\end{proof}\n\nAs we noted in Section $2$, the existence result for two agents does not extend beyond two layers: if there are at least three layers, there is no feasible multi-allocation that completely allocates the cake to two agents. \n\n\\subsection{Three agents and two layers}\nWe move on to the case of three agents and two layers. We will design a variant of Stromquist's protocol that achieves envy-freeness for one-layered cake \\citep{Stromquist1980}: The referee moves two knives: a short knife and a long knife. The short knife points to the point $y$ and moves from left to right over the top layer, gradually increasing the left-most top piece (denoted by $Y$). The long knife keeps pointing to the point $x$, which can partition the remaining cake, denoted by $\\mathcal{C}^{-y}$, into two diagonal pieces $LR(x)$ and $RL(x)$ in an envy-free manner. Each agent shouts when the left-most top piece $Y$ becomes at least as highly valuable as the preferred piece among $LR(x)$ and $RL(x)$. Some agent, say $s$, shouts eventually (before the left-most top piece becomes the top layer), assuming that there is at least one agent who weakly prefers the top layer to the bottom layer. We note that $x$ may be positioned left to $y$; see Figure \\ref{fig:EF:three} for some possibilities of the long knife's locations. \n\nWe will show that the above protocol works, for a special case when there are at most two types of preferences: In such cases, the majority switching points coincide with the switching points of an agent with the majority preference.\n\n\\begin{lemma}\\label{lem:switching:identical}\nSuppose that $m=2$, $n=3$, and there are two different agents $i,j \\in N$ with the same valuation $V$. Then, $x$ is a majority switching point over $\\mathcal{C}$ if and only if $x$ is a switching point for $i$. \n\\end{lemma}\n\\begin{proof}\nSuppose that agents $i,j \\in N$ have the same valuation $V$. Suppose that $x$ is a majority switching point over $\\mathcal{C}$. Then, at least two agents weakly prefer $LR(x)$ to $RL(x)$, meaning that at least one of the two agents $i$ and $j$ weakly prefers $LR(x)$ to $RL(x)$, which means that both agents weakly prefers $LR(x)$ to $RL(x)$ since $i$ and $j$'s valuations are identical. Similarly, both $i$ and $j$ weakly prefer $RL(x)$ to $LR(x)$. Thus, $x$ is a switching point for $i$. \nThe converse direction is immediate. \n\\end{proof}\n\nAn implication of the above lemma is that when performing Stromquist's protocol, one can point out to a switching point of an individual, instead of a majority one. This allows the value of each piece to change continuously. For a given two-layered cake $\\mathcal{C}$, we write $\\mathcal{C}^{-y}=(C^{-y}_1,C_2)$ as a two-layered cake obtained from $\\mathcal{C}$ where the first segment $[0,y]$ of the top layer is removed, i.e., $C^{-y}_1 = C_1 \\setminus [0,y]$. For each majority switching point $x$ over $\\mathcal{C}^{-y}$, we select three different agents $\\ell(x)$, $m(x)$, and $r(x)$ as follows: \n\\begin{itemize}\n\\item $\\ell(x)$ is an agent who weakly prefers $LR(x,\\mathcal{C}^{-y})$ to $RL(x,\\mathcal{C}^{-y})$;\n\\item $m(x)$ is an agent who is indifferent between $LR(x,\\mathcal{C}^{-y})$ and $RL(x,\\mathcal{C}^{-y})$; and \n\\item $r(x)$ and agent who weakly prefers $RL(x,\\mathcal{C}^{-y})$ to $LR(x,\\mathcal{C}^{-y})$. \n\\end{itemize}\n\n\n\\begin{theorem}\\label{thm:twolayers3agents}\nSuppose that $m=2$ and $n=3$. If there are two different agents with the same valuation, an envy-free complete multi-allocation that is feasible and contiguous exists. \n\\end{theorem}\n\\begin{proof}\nAssume w.l.o.g. that at least one agent prefers the top layer over the bottom layer. This means that such agent weakly prefers the top layer to any of the pieces $LR(z,\\mathcal{C}^{-y})$ and $RL(z,\\mathcal{C}^{-y})$ when $y=1$. Suppose that $i \\in N$ is one of the two different agents with the same valuations. We design the following protocol for three agents over a two-layered cake: \n\n\\vspace{3pt}\n\\noindent\\fbox{%\n\t\\parbox{0.985\\linewidth}{%\n\t\t\\textbf{Moving-knife protocol for $n=3$ agents} over a two-layered cake $\\mathcal{C}$: w.l.o.g. assume that at least one agent weakly prefers the top layer $(j=1)$ over the bottom layer $(j=2)$\\\\\n\t\t\\textit{Step 1.} The referee continuously moves a short knife from the left-most point $(y=0)$ to the right-most point $(y=1)$ over the top layer, while continuously moving a long knife pointing to a switching point over $\\mathcal{C}^{-y}$ for $i$. \n        Let $y$ be the position of the short knife and $Y$ be the top layer piece to its left. Let $x$ be the position of the long knife.\\\\\t\t\n\t\t\\textit{Step 2.} The referee stops moving the short knife when some agent $s$ {\\em shouts}, i.e., $Y$ becomes at least as highly valuable as the preferred piece among $LR(x,\\mathcal{C}^{-y})$ and $RL(x,\\mathcal{C}^{-y})$. \\\\\n\t\t\\textit{Step 3.} We allocate the shouter $s$ to the left-most top piece $Y$ and partitions the rest into $LR(x,\\mathcal{C}^{-y})$ and $RL(x,\\mathcal{C}^{-y})$. \n\\begin{itemize}\n\\item If $s=\\ell(x)$, then we allocate $LR(x,\\mathcal{C}^{-y})$ to $m(x)$ and $RL(x,\\mathcal{C}^{-y})$ to $r(x)$. \n\\item If $s=m(x)$, then we allocate $LR(x,\\mathcal{C}^{-y})$ to $\\ell(x)$ and $RL(x,\\mathcal{C}^{-y})$ to $r(x)$. \n\\item If $s=r(x)$, then we allocate $LR(x,\\mathcal{C}^{-y})$ to $\\ell(x)$ and $RL(x,\\mathcal{C}^{-y})$ to $m(x)$. \n\\end{itemize}\n\t}%\n}\n\\vspace{3pt}\n\nBy our assumption, some agent eventually shouts and thus the protocol returns an allocation $\\mathcal{A}$. Clearly, $\\mathcal{A}$ is feasible, contiguous, and complete. Also, it is easy to see that the shouter $s$ who receives a bundle $Y$ does not envy the other two agents. The agents $i \\neq s$ do not envy $s$ because the referee continuously moves both a short and a long knife. Finally, the agents $i \\neq s$ do not envy each other by the definition of a majority switching point and by Lemma \\ref{lem:switching:identical}. \n\\end{proof}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[scale=0.55, transform shape]\n\\draw[fill=red!10, thick] (0,0) rectangle (4,1);\n\\draw[fill=blue!10, thick] (0,-1) rectangle (3,0); \n\\draw[fill=blue!10, thick] (4,0) rectangle (10,1); \n\\draw[thick] (3,-1) rectangle (10,0);\n\n\\node at (2.0,0.5) {$Y$};\n\\node at (1.5,-0.5) {$RL(x)$};\n\\node at (6.5,0.5) {$RL(x)$};\n\\node at (6.0,-0.5) {$LR(x)$};\n\n\\node at (3.0,1.3) {$x$};\n\\node at (4.0,1.3) {$y$};\n\n\\begin{scope}[yshift=-3cm]\n\\draw[fill=red!10, thick] (0,0) rectangle (4,1);\n\\draw[thick] (4,0) rectangle (10,1);\n\\draw[fill=blue!10, thick] (6,0) rectangle (10,1); \n\\draw[fill=blue!10, thick] (0,-1) rectangle (6,0); \n\\draw[thick] (6,-1) rectangle (10,0);\n\n\\node at (2.0,0.5) {$Y$};\n\\node at (6.0,1.3) {$x$};\n\\node at (4.0,1.3) {$y$};\n\n\\node at (3.0,-0.5) {$RL(x)$};\n\\node at (8.0,0.5) {$RL(x)$};\n\\node at (5.0,0.5) {$LR(x)$};\n\\node at (8.0,-0.5) {$LR(x)$};\n\\end{scope}\n \n\\end{tikzpicture}\n\\caption{Moving knife protocol for three agents over a two-layered cake. Note that the position of $x$ may appear before $y$. \n}\n\\label{fig:EF:three}\n\\end{figure}\n\nIn the general case, the existence question of contiguous and feasible envy-free multi-allocations deems to be challenging due to the non-monotonicity of valuations over diagonal pieces.\\footnote{See Section \\ref{sec:discussion} for an extensive discussion.} \nIn the next subsection, we thus turn our attention to the case when the contiguity requirement is relaxed.\n\n\\subsection{Non-connected pieces}\nHaving seen that an envy-free multi-allocation that is both feasible and contiguous exists for a special case, we will consider the case when the contiguity requirement is dropped, namely, agents may receive a collection of sub-intervals of each layer. \nWe will show the existence of an envy-free multi-allocation that is feasible and uses at most poly$(n)$ number of cuts within each layer, assuming that each density function $v_{ij}$ is continuous. In what follows, we will reduce the problem to finding a `perfect' allocation of a one-layered cake. An allocation of a single-layered cake is called {\\em perfect} if each agent values every allocated piece exactly at his proportional fair share $\\frac{1}{n}$. It is known that such allocation consisting of at most poly$(n)$ number of contiguous pieces exists whenever agents' value density functions are continuous \\citep{Alon1987}. It is not surprising that the existence of a perfect allocation implies the existence of an envy-free allocation over a single-layered cake. We show that this result also implies the existence of envy-free allocations over a multi-layered cake. \n\n\\begin{theorem}\\label{thm:EF:any}\nSuppose that $m \\leq n$ and each $v_{i,j}$ for $i \\in N$ and $j \\in L$ is continuous. Then, an envy-free complete feasible multi-allocation $\\mathcal{A}=(\\mathcal{A}_i)_{i \\in N}$ where each piece $A_{ij}$ for agent $i \\in N$ and layer $j \\in L$ contains at most poly$(n)$ number of contiguous pieces exists. \n\\end{theorem}\n\\begin{proof}\nWe assume without loss of generality that each layer $C_j$ is the whole interval $[0,1]$ by just putting zero valuations on the part outside $C_j$. \nNow, we construct an instance of a single-layered cake $I=[0,1]$ as follows. First, create a dummy agent $i_j$ for each agent $i \\in N$ and each layer $j \\in L$. Each dummy agent $i_j$ has a valuation $v'_{i_j}$ determined by agent $i$'s valuation for the $j$-th layered cake, i.e., for each sub-interval $X \\subseteq I$, $v'_{i_j}(X)=v_{i,j}(X)$. \nWe will show that a perfect allocation of the artificial cake among the $mn$ dummy agents induces an envy-free multi-allocation of the original layered cake. \n\nSpecifically, take a perfect allocation $(X_1,X_2,\\ldots,X_{mn})$ of this instance where each $X_t$ for $t \\in [mn]$ contains at most poly$(n)$ number of contiguous pieces, which is guaranteed to exist \\citep{Alon1987}. We then group each consecutive $m$ sequence of pieces together: namely, let $Y_{h}=\\bigcup^{im}_{t=(i-1)m+1}X_t$ for each $h \\in [n]$. By the definition of a perfect allocation, we have\n\\begin{align}\\label{eq}\n&v_{ij}(Y_h)= \\frac{v_{ij}(C_j)}{n},\n\\end{align}\nfor any $h \\in [n]$. Now, we partition each layer into $n$ pieces using the partition $(Y_1,Y_2,\\ldots,Y_n)$ of the artificial cake and allocate to the agents so that each agent  receives exactly one piece $Y_h$ for each layer. Formally, consider a permutation $\\sigma_j:[n] \\rightarrow [n]$ where\n\\[\n\\sigma_j(i)\n=i+j -1 \\pmod{m}. \n\\]\nConstruct an multi-allocation $\\mathcal{A}=(\\mathcal{A}_i)_{i \\in N}$ where each agent $i \\in N$ is assigned to $A_{ij}=Y_{\\sigma_j(i)}$ for each layer $j \\in L$. By our construction, each $A_{ij}$ contains at most poly$(n)$ number of contiguous pieces. Also, each layered piece $\\mathcal{A}_i$ is non-overlapping as $(Y_1,Y_2,\\ldots,Y_n)$ is a partition of the interval $[0,1]$. By \\eqref{eq}, it is immediate to see that $\\mathcal{A}$ is envy-free. \n\\end{proof}\n\n\n\\section{Proportional multi-layered cake cutting}\nFocusing on a less demanding fairness notion, it turns out that a complete proportional multi-allocation that is both feasible and contiguous exists, for a wider class of instances, i.e., when the number $m$ of layers is some power of two, and the number $n$ of agents is at least $m$. Notably, we show that the problem can be decomposed into smaller instances when the number of agents is at least the number of layers and the number of layers is a power of two. Building up on the {\\em base case} of two layers, our algorithm recursively calls the same algorithm to decide on how to allocate the cake of the sub-problems. We further show that if we relax the contiguity requirement, a proportional feasible multi-allocation can be computed efficiently whenever $m \\leq n$. \n\n\\subsection{Connected pieces}\nIn this subsection, we will show that a proportional complete multi-allocation exists for any $n \\geq m$ when $m$ is some power of $2$. We start by presenting two auxiliary lemmata. We define a {\\em merge} of two disjoint contiguous pieces $I_j$ and $I_{j'}$ of layers $j$ and $j'$ as replacing the $j$-th layered cake with the union $I_j \\cup I_{j'}$ and removing $j'$-th layered cake. The {\\em merge} of a finite sequence of mutually disjoint contiguous pieces $(I_1,\\ldots,I_k)$ can be defined inductively: merge $(I_1,\\ldots,I_{k-1})$ and then apply the merge operation to the resulting outcome and $I_k$. \nNow we observe that if there are two disjoint layers, one can safely merge these layers and reduce the problem size. \n\n\\begin{lemma}\\label{lem:merge}\nSuppose that $C_j$ and $C_{j'}$ are two disjoint layers of a layered cake $\\mathcal{C}$, and $\\mathcal{C}'$ is obtained from $\\mathcal{C}$ by merging $C_j$ and $C_{j'}$. Then, each non-overlapping contiguous layered piece of $\\mathcal{C}'$ is a non-overlapping contiguous layered piece of the original cake $\\mathcal{C}$. \n\\end{lemma}\n\\begin{proof}\nSuppose that $C_j$ and $C_{j'}$ are two disjoint layers of a layered cake $\\mathcal{C}=(C_{t})_{t \\in L}$, and the layered cake $\\mathcal{C}'=(C'_{t})_{t \\in L \\setminus \\{j'\\}}$ is obtained from $\\mathcal{C}$ by merging $C_j$ and $C_{j'}$. Let $\\mathcal{X}'=(X'_{t})_{t \\in L \\setminus \\{j'\\}}$ be a non-overlapping contiguous piece of $\\mathcal{C}'$. Consider the corresponding layered piece $\\mathcal{X}=(X_{t})_{t \\in L}$ of the original cake $\\mathcal{C}$ where $X_t=X'_{t}$ for $t \\in L \\setminus \\{j,j'\\}$ and $X_t = X'_t \\cap C_{t}$ for $t \\in \\{j,j'\\}$. It is immediate to see that $\\mathcal{X}'$ is non-overlapping and contiguous, since $C_j$ and $C_{j'}$ are disjoint. \n\\end{proof}\n\nThe above lemma can be generalized further: Let $\\mathcal{C}$ be a $2m$-layered cake and $x \\in [0,1]$. We define a {\\em merge} of $LR(x)=(S_j)_{j \\in L}$ by merging the pair $(S_j,S_{j+m})$ for each $j \\in [m]$. A {\\em merge} of $RL(x)$ can be defined analogously. Such operation still preserves both feasibility and contiguity. \n\n\\begin{corollary}\\label{cor:merge}\nLet $\\mathcal{C}$ be a $2m$-layered cake and $x \\in [0,1]$. Suppose that $\\mathcal{C}'$ is a $m$-layered cake obtained by merging $LR(x,\\mathcal{C})$ or $RL(x,\\mathcal{C})$. Then, each non-overlapping contiguous layered piece of $\\mathcal{C}'$ is a non-overlapping contiguous layered piece of the original cake $\\mathcal{C}$. \n\\end{corollary}\n\\begin{proof}\nSuppose that $\\mathcal{C}'$ is a $m$-layered cake obtained by merging $LR(x,\\mathcal{C})=(S_j)_{j \\in L}$ of a $2m$-layered cake $\\mathcal{C}$. By Lemma \\ref{lem:merge}, a non-overlapping contiguous layered piece of the cake obtained from each merge of the pair $(S_j,S_{j+m})$ for $j \\in [m]$ still corresponds to a non-overlapping and contiguous piece of the original cake. Thus, the claim holds. An analogous argument applies to the case when we merge $RL(x,\\mathcal{C})$. \n\\end{proof}\n\nWe are now ready to prove that a proportional complete multi-allocation exists for any $n=m$ when $m$ is some power of $2$. In essence, the existence of a majority switching point, as proved in Lemma \\ref{lem:majority:switching}, allows us to divide the problem into two instances. We will repeat this procedure until the number of layers of the subproblem becomes $2$, for which we know the existence of a proportional, feasible, contiguous multi-allocation by Theorem \\ref{thm:EF:two}.  \n\n\\begin{theorem}\\label{thm:prop:base}\nA proportional complete multi-allocation that is feasible and contiguous exists, for any number $m$ of layers and any number $n= m$ of agents where $m=2^a$ for some $a \\in \\mathbb{Z}_{+}$. \n\\end{theorem}\n\\begin{proof}\nWe design the following recursive algorithm $\\mathcal{D}$ that takes a subset $N'$ of agents with $|N'| \\geq 2$, a $|L'|$-layered cake $\\mathcal{C}'$, and a valuation profile $(V_{i})_{i \\in N'}$, and returns a proportional complete multi-allocation of the cake to the agents which is feasible. Suppose that $m=n$. If $m=n=1$, then we allocate the entire cake to the single agent. If $m=n=2$, we run the cut-and-choose algorithm as described in the proof of Theorem \\ref{thm:EF:two}. \nNow consider the case when $m=n=2^a$ for some integers $a \\geq 1$. Then the algorithm finds a majority switching point $x$ over $\\mathcal{C}'$. We let $\\mathcal{I}_1=LR(x)$ and $\\mathcal{I}_2=RL(x)$. By definition of a majority switching point and the fact that $n$ is even, we can partition the set of agents $N'$ into $N_1$ and $N_2$ where $N_1$ is the set of agents who weakly prefer $\\mathcal{I}_1$ to $\\mathcal{I}_2$, $N_2$ be the set of agents who weakly prefer $\\mathcal{I}_2$ to $\\mathcal{I}_1$, and $|N_k| = \\frac{|N'|}{2}$ for each $k=1,2$. \nWe apply $\\mathcal{D}$ to the merge of $\\mathcal{I}_k$ with the agent set $N_k$ for each $k=1,2$, respectively. \n\nWe will show that by induction on the exponential $a$, that the complete multi-allocation $\\mathcal{A}$ returned by $\\mathcal{D}$ satisfies proportionality as well as feasibility and contiguity. This is clearly true when $m=n=2$ due to Lemma \\ref{lem:propEF} and Theorem \\ref{thm:EF:two}. Suppose that the claim holds for $m=n=2^a$ with $1 \\leq a \\leq k-1$; we will prove it for $a=k$. Suppose that the algorithm divides the input cake $\\mathcal{C}'$ via a majority switching point $x$ into $\\mathcal{I}_1=LR(x)$ and $\\mathcal{I}_2=RL(x)$. Suppose that $(N_1,N_2)$ is a partition of the agents where $N_1$ is the set of agents who weakly prefer $\\mathcal{I}_1$ to $\\mathcal{I}_2$, $N_2$ is the set of agents who weakly prefer $\\mathcal{I}_2$ to $\\mathcal{I}_1$, and $|N_k| = \\frac{|N'|}{2}$ for each $k=1,2$. Observe that each agent $i \\in N_1$ weakly prefers $\\mathcal{I}_1$ to $\\mathcal{I}_2$ and thus $V_i(\\mathcal{I}_1) \\geq \\frac{1}{2}V_i(\\mathcal{C}')$. Similarly, $V_i(\\mathcal{I}_2) \\geq \\frac{1}{2}V_i(\\mathcal{C}')$ for each $i \\in N_2$. Thus, by the induction hypothesis, each agent $i$ has value at least $\\frac{1}{|N'|}V_i(\\mathcal{C}')$ for its allocated piece $\\mathcal{A}_{i}$. Further, by Corollary \\ref{cor:merge}, each non-overlapping contiguous layered piece of the merge of $\\mathcal{I}_1$ (respectively, $\\mathcal{I}_2$) is a contiguous non-overlapping layered piece of the original cake $\\mathcal{C}$. By the induction hypothesis, the algorithm outputs a multi-allocation of each merge that is contiguous. Thus, the algorithm returns a proportional complete multi-allocation that is feasible and contiguous.\n\\end{proof}\n\nWe will generalize the above theorems to the case when the number of agents is strictly greater than the number of layers. Intuitively, when $n>m$, then there is at least one layer whose sub-piece can be `safely' allocated to some agent without violating the non-overlapping constraint. \n\n\\begin{theorem}\\label{thm:exponential}\nA proportional complete multi-allocation that is feasible and contiguous exists, for any number $m$ of layers and any number $n \\geq m$ of agents where $m=2^a$ for some $a \\in \\mathbb{Z}_{+}$. \n\\end{theorem}\n\\begin{proof}\nWe design the following recursive algorithm $\\mathcal{D}$ that takes a subset $N'$ of agents with $|N'| \\geq 2$, a $|L'|$-layered cake $\\mathcal{C}'$, and a valuation profile $(V_{i})_{i \\in N'}$, and returns a proportional complete multi-allocation of the layered cake to the agents  which is feasible. For $n=m$, we apply the algorithm described in the proof of Theorem \\ref{thm:prop:base}. Suppose that $n>m$. The algorithm first identifies a layer $C_j$ whose entire valuation is at least $\\frac{1}{n}$ for some agent; assume w.l.o.g. that $j=1$. We move a knife from left to right over the top cake $C_1$ until some agent $i$ {\\em shouts}, i.e., agent $i$ finds the left contiguous piece $Y$ at least as highly valued as his proportional fair share $\\frac{1}{n}$. The algorithm $\\mathcal{D}$ then gives the piece to the shouter. To decide on the allocation of the remaining items, we apply $\\mathcal{D}$ to the reduced instance $(N' \\setminus \\{i\\},(C'_j)_{j \\in L},(V_{i'})_{i' \\in N' \\setminus \\{i\\}})$ where $C'_j=C_j \\setminus Y$ for $j=1$ and $C'_j=C_j$ for $j \\neq 1$. \n\nWe will prove by induction on $|N'|$ that the complete multi-allocation $\\mathcal{A}=(\\mathcal{A}_1,\\mathcal{A}_2,\\ldots,\\mathcal{A}_n)$ returned by $\\mathcal{D}$ satisfies proportionality as well as feasibility and contiguity. This is clearly true when $m=|N'|$, due to Theorem  \\ref{thm:prop:base}. Suppose that the claim holds for $|N'|$ with $m \\leq |N'| \\leq k-1$; we will prove it for $|N'|=k$. Suppose agent $i$ is the shouter who gets the left contiguous piece $Y$. Clearly, agent $i$ receives her proportional share under $\\mathcal{A}$. Observe that all remaining agents have the value at least $\\frac{|N'|-1}{|N'|}V_i(\\mathcal{C}')$ for the remaining cake. Thus, by the induction hypothesis, each agent $i' \\neq i$ has value at least $\\frac{1}{|N'|}V_i(\\mathcal{C}')$ for its allocated piece $\\mathcal{A}_{i'}$. The feasibility and contiguity of $\\mathcal{A}$ are immediate by the induction hypothesis. This completes the proof. \n\\end{proof}\n\n\n\\subsection{Non-connected pieces}\nSince envy-freeness implies proportionality, Theorem \\ref{thm:EF:any} in the previous section implies the existence of a proportional feasible multi-allocation when agents' value density functions are continuous. We strengthen this result, by showing that such desirable allocation exists for a more general case and by providing an efficient algorithm for finding one. \n\n\\begin{theorem}\\label{thm:prop:feasible}\nA proportional complete multi-allocation that is feasible exists when $m =n$ and can be computed using $O(nm^2)$ number of short eval queries and $O(nm)$ number of long eval and cut queries. Further, each bundle of the resulting multi-allocation includes at most two contiguous pieces within each layer. \n\\end{theorem}\n\nBelow, we show that each agent can divide the entire cake into $n$ equally valued layered pieces. A multi-allocation $\\mathcal{A}$ is {\\em equitable} if for each agent $i \\in N$, $V_{i}(\\mathcal{A}_i)=\\frac{1}{n}$. We design a recursive algorithm that iteratively finds two layers for which one has value at most $\\frac{1}{m}$ and at least $\\frac{1}{m}$ and removes a pair of diagonal pieces of value exactly $\\frac{1}{m}$ from the two layers.\n\n\\begin{lemma}\\label{lem:equitable}\nFor any number $m$ of layers and any number $n = m$ of agents with the identical valuations, an equitable complete multi-allocation that is feasible and contiguous exists and can be found using $O(nm^2)$ number of short eval queries and $O(nm)$ number of long cut queries.  \n\\end{lemma}\n\\begin{proof}\nWe denote by $V=V_i$ the valuation function for each agent $i \\in N$. \nConsider the following recursive algorithm $\\mathcal{D}$ that takes a subset $N'$ of agents with $|N'| \\geq 1$, a $|L'|$-layered cake $\\mathcal{C}'$, and a valuation profile $(V_{i})_{i \\in N'}$, and returns an equitable complete multi-allocation of the layered cake to the agents. When $|L'|=|N'|=1$, then the algorithm allocates the entire cake to the single agent. \nSuppose that $|L'|=|N'| \\geq 2$. \nThe algorithm first finds a layer $j$ whose entire value is at most $\\frac{1}{m}$ and another layer $j'$ whose entire value is at least $\\frac{1}{m}$. The algorithm $\\mathcal{D}$ then finds a point $x \\in [0,1]$ where $V(S_{j} \\cup S_{j'})=\\frac{1}{m}$ for $S_j=C_j \\cap [0,x]$ and $S_{j'}=C_j \\cap [x,1]$; such point exists due to Lemma \\ref{lem:switching}. We allocate $S_{j} \\cup S_{j'}$ to one agent and apply $\\mathcal{D}$ to the remaining cake $\\mathcal{C}''$ with $|N'|-1$ agents where $\\mathcal{C}''$ is obtained from merging the remaining $j$-th layered cake $C_j \\setminus S_j$ and the $j'$-th layered cake $C_{j'} \\setminus S_{j'}$. The correctness of the algorithm as well as the bound on the query complexity are immediate. \n\\end{proof}\n\n\\begin{figure*}[hbt]\n\\centering\n\\begin{tikzpicture}[scale=0.6, transform shape]\n\n\\draw[thick] (0,0) rectangle (3,1);\n\\draw[fill=red!10, thick] (3,0) rectangle (6,1);\n\\draw[fill=blue!10, thick] (0,-1) rectangle (2,0); \n\\draw[fill=red!10, thick] (2,-1) rectangle (3,0);\n\\draw[thick] (3,-1) rectangle (6,0);\n\\draw[fill=red!10, thick] (0,-2) rectangle (2,-1);\n\\draw[fill=blue!10, thick] (2,-2) rectangle (6,-1); \n\n\n\\node at (4.5,0.5) {$I_{11}$};\n\\node at (2.5,-0.5) {$I_{12}$};\n\\node at (1,-1.5) {$I_{13}$};\n\n\\node at (1.5,0.5) {$I_{21}$};\n\\node at (4.5,-0.5) {$I_{22}$};\n\n\\node at (1,-0.5) {$I_{32}$};\n\\node at (4,-1.5) {$I_{33}$};\n\n\\draw[ultra thick,->] (6.5,-0.5)--(7.5,-0.5);\n\n\\begin{scope}[xshift=8cm,yshift=-0.5cm]\n\\node[fill=red!10,draw, circle](I1) at (1,1) {$\\mathcal{I}_1$};\n\\node[draw, circle](I2) at (3,1) {$\\mathcal{I}_2$};\n\\node[fill=blue!10,draw, circle](I3) at (5,1) {$\\mathcal{I}_3$};\n\n\\node[fill=gray!10,draw, circle](a1) at (1,-1) {$1$};\n\\node[fill=gray!10,draw, circle](a2) at (3,-1) {$2$};\n\\node[fill=gray!10,draw, circle](a3) at (5,-1) {$3$};\n\n\\draw[-,red, >=latex,thick] (I1)--(a1);\n\\draw[-, >=latex,thick] (I2)--(a1);\n\\draw[-,>=latex,thick] (I3)--(a1);\n\\draw[-, >=latex,thick] (I2)--(a2);\n\\draw[-, >=latex,thick] (I2)--(a3);\n\n\\draw[ultra thick,->] (6.5,0)--(7.5,0);\n\\end{scope}\n\n\\begin{scope}[xshift=16cm,yshift=0.5cm]\n\\draw[thick] (0,-1) rectangle (3,0);\n\n\\draw[fill=blue!10, thick] (0,-2) rectangle (2,-1); \n\\draw[thick] (3,-1) rectangle (6,0);\n\\draw[fill=blue!10, thick] (2,-2) rectangle (6,-1); \n\n\\node at (1.5,-0.5) {$I_{21}$};\n\\node at (4.5,-0.5) {$I_{22}$};\n\n\\node at (1,-1.5) {$I_{32}$};\n\\node at (4,-1.5) {$I_{33}$};\n\n\\draw[thick,dotted] (4,0.5)--(4,-2.5);\n\\end{scope}\n \n\\end{tikzpicture}\n\\caption{Protocol for proportionality for three agents and three layers. Agent $1$ divides the entire cake into three equally valued layered pieces $\\mathcal{I}_1$, $\\mathcal{I}_2$, and $\\mathcal{I}_3$ (the left-most picture). Here, $\\mathcal{I}_i=(I_{ij})_{j=1,2,3}$ for each $i =1,2,3$ where $I_{23}=I_{31}=\\emptyset$. In the middle picture, agent $1$ is adjacent to every piece, meaning that he has value at least proportional fair share for every piece; on the other hand, the other agents are adjacent to the second piece only. The maximum envy-free matching is an edge between $\\mathcal{I}_1$ and agent $1$ (red edge), so the algorithm allocates $\\mathcal{I}_1$ to agent $1$ and merges $\\mathcal{I}_2$, and $\\mathcal{I}_3$. Then it applies the cut-and-choose among the remaining agents (the right-most picture).}\n\\label{fig:PROP:three}\n\\end{figure*}\n\nEquipped with Lemma \\ref{lem:equitable}, we will prove Theorem \\ref{thm:prop:feasible} by recursively computing an {\\em envy-free matching} between $n$ agents and $n$ layered pieces where one agent has proportional fair share for every piece. Specifically, given a bipartite graph $G$ with one side being the set of agents and the other side being the set of items, an envy-free matching $M$ of $G$ is a matching where no unmatched agent {\\em envies} some matched agent, i.e., no unmatched agent is adjacent to any matched item in $G$. The problem of finding an envy-free matching of maximum size can be solved in polynomial time \\citep{EladErel,GAN2019}. Further, using Hall's type condition, it can be easily shown that if there is one agent who is adjacent to every item and the number of agents is at most the number of items, then there is a non-empty envy-free matching (Corollary $1.4$ $($c$)$ of \\citet{EladErel}).  \n\n\\begin{proof}[Proof of Thm. \\ref{thm:prop:feasible}]\nIn order to obtain a proportional feasible multi-allocation, we will recursively compute a non-empty envy-free matching:\nFor each iteration, let one agent partition $n$-equally valued layered pieces, find a maximum envy-free matching between agents and pieces, and assign the matched agents to the matched pieces. By Corollary $1.4$ $($c$)$ of \\citet{EladErel}, the envy-free matching computed at each step is non-empty; thus, at least one agent is matched and we will apply the same procedure to the unmatched agents and pieces. The formal description is given as follows. See Figure \\ref{fig:PROP:three} for an illustration for $m=n=3$. \n\n\\vspace{5pt}\n\\noindent\\fbox{%\n\t\\parbox{0.985\\linewidth}{%\n\t\t\\textbf{A protocol for proportional feasible multi-allocations for $n = m$ agents} over a $m$-layered cake $\\mathcal{C}$: \\\\\n\t\t\\textit{Step 1.} One agent partitions the cake into $n$ non-overlapping layered contiguous pieces $\\mathcal{I}_1,\\mathcal{I}_2, \\ldots, \\mathcal{I}_n$ which she considers of equal value, using the algorithm in the proof of Lemma \\ref{lem:equitable}.\\\\\n\t\t\\textit{Step 2.} Construct a bipartite graph $G$ with the agents being one side and the pieces being on the other side, where there is an edge between agent $i$ and $\\mathcal{I}_h$ if agent $i$ has value at least his proportional far share for $\\mathcal{I}_h$. \\\\\n\t\t\\textit{Step 3.} Compute a maximum-size envy-free matching $M$ of $G$. Assign matched agent to the corresponding piece. \\\\\n\t\t\\textit{Step 4.} For each unmatched piece $\\mathcal{I}_h$, merge all the disjoint contiguous pieces in $\\mathcal{I}_h$, and create a $m-\\ell$-layered cake $\\mathcal{C}'$ consisting of each merge of $\\mathcal{I}_h$ where $\\ell$ is the number of matched pieces. Apply the same protocol to $\\mathcal{C}'$ among the remaining unmatched agents. \n\t}%\n}\n\\vspace{5pt}\n\nWe will show by the induction on the number of agents $n=m$ that the resulting multi-allocation $\\mathcal{A}$ is proportional and feasible. The claim clearly holds for $n=m=1$. Suppose that the claim holds for $n$ with $m=n \\leq k-1$; we will prove it for $m=n=k$. We will first show that the resulting multi-allocation $\\mathcal{A}$ is proportional. Clearly, the agents who get matched has proportional fair share for his bundle. Further, each $i$ of the remaining unmatched $n-\\ell$ agents have value less than $\\frac{1}{n}$ for each matched piece and thus has at least $1-\\frac{\\ell}{n}$ for the remaining unmatched pieces; thus, $V_i(\\mathcal{A}_i)$ is at least $\\frac{1}{n}$. It can be easily verified that each bundle $\\mathcal{A}_i$ is non-overlapping. Each iteration requires $O(m^2)$ number of short eval queries and $O(m)$ number of long cut queries for the cutter, and $O(m^2)$ number of short eval queries for each agent. Further, the number of iterations is at most $n$, which proves the bound on the query complexity. \nThis completes the proof.\n\\end{proof}\n\nSimilarly to the proof for Theorem \\ref{thm:exponential}, we can generalize the above theorem to the case when the number of agents is strictly greater than the number of layers. \n\n\\begin{theorem}\\label{thm:prop:feasible:any}\nA proportional complete multi-allocation that is feasible exists and can be computed using $O(nm^2)$ number of short eval queries and $O(nm)$ number of long cut queries, for any number $m$ of layers and any number $n \\geq m$ of agents. \n\\end{theorem}\n\nIt remains open whether a proportional contiguous multi-allocation exists when the number of layers is three. A part of the reason is that our algorithm for finding an equitable multi-allocation (Lemma \\ref{lem:equitable}) may not return a `balanced' partition: The number of pieces contained in each layered piece may not be the same when the number of layers is odd. For example, one layered piece may contain pieces from three different layers while the other two parts may contain pieces from two different layers, as depicted in Figure \\ref{fig:PROP:three}. \n\n\n\\section{Discussion} \\label{sec:discussion}\nWe initiated the study of multi-layered cake cutting, demonstrating the rich and intriguing mathematical feature of the problem. There are several exciting questions left open for future work. Below, we list some of them. \n\n\\begin{itemize}\n\\item {\\bf Existence of fair allocations}:\nWe have seen that an envy-free contiguous and feasible multi-allocation of a two-layered cake exists for two or three agents with at most two types of preferences. An interesting open problem is whether such allocation also exists for any number of agents over a two-layered cake. One might expect that the Simmon-Su's technique \\citep{Su1999} using Sperner's Lemma can be adopted to our setting by considering all possible diagonal pieces. However, this approach may not work because multi-layered cake-cutting necessarily exhibits non-monotonicity in that the value of a pair of diagonal pieces may decrease when the knife moves from left to right. For proportionality, one intriguing future direction is extending our existence result for $m=2^a$ to any $m$. This requires careful consideration of contiguity and feasibility, which are often at odds with completeness.\n\n\\item {\\bf Query complexity of fair allocations}:\nThe query complexity of finding an envy-free feasible multi-allocation is open in the multi-layered cake-cutting problem. In particular, it would be challenging to extend the celebrated result of \\citet{aziz2016discrete} -- who showed the existence of a bounded protocol for computing an envy-free allocation of a single-layered cake with any number $n$ of agents --  to our setting. We expect that a direct translation may not work, due to the intricate nature of the feasibility constraint. With respect to proportionality, our existence proof implies that if there is a way to compute a majority switching point efficiently, one can compute a proportional contiguous feasible multi-allocation for special cases when the number of layers is a power of two. It is open whether such cutting point can be computed using a bounded number of queries.\n\n\\item {\\bf Approximate fairness}: \nIn the presence of contiguity requirement, it is known that no finite protocol computes an envy-free allocation even for three agents and a single-layered cake \\citep{Stromquist2008}. However, several positive results are known when the aim is to approximately bound the envy between agents \\citep{Deng2012,Goldberg2020,Arunachaleswaran19}.\nPursuing a similar direction in the context of multi-layered cake cutting would be an interesting research topic. \n\n\\item {\\bf Efficiency requirement}: \nBesides fairness criteria, another basic desideratum is economic efficiency. In the context of a single-layered cake cutting, several works studied the relation between fairness and efficiency \\citep{CohlerLPP11,Bei12, AumannDomb2010, AumannDH13}. \nThe question of what welfare guarantee can be achieved together with fairness is open in our model. In particular, it would be interesting to investigate the compatibility of the fairness notions with an efficiency requirement, under feasibility constraints. \n\n\n\\end{itemize}\n\n\\section*{Acknowledgement}\nHadi Hosseini was supported by the National Science Foundation (grant IIS-1850076).\nAyumi Igarashi was supported by the KAKENHI Grant-in-Aid for JSPS Fellows no. 18J00997 and JST, ACT-X. The authors would like to thank Yuki Tamura for introducing the problem to us and for the fruitful discussions. The authors also acknowledge the helpful comments by the GAIW and IJCAI reviewers. \nWe are grateful to an anonymous GAIW reviewer for the proof of the existence of an envy-free multi-allocation in the general case. \n\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is well known that volumetric photonic crystals have the efficacy of controlling  propagating  waves, e.g.~\\cite{joannopoulos2008molding}, both in three and two dimensions, e.g. \\cite{Kraus}. \nRecently, in parallel with thorough and general investigations of time-varying electromagnetic systems (see, e.g., Refs.~\\cite{XUCHENPRAPPLIED, SAJJADarXiv, groupEnergy}), three-dimensional (volumetric) temporal photonic crystals whose material properties are uniform in space but changing in time have attracted significant attention~\\cite{zurita2009reflection}. This is due to the intriguing effects that they have on propagating plane waves. \n\nIt appears that it is fundamentally important to explore properties of temporal metasurfaces which are 2D material sheets or boundaries with time-varying properties. In this case, the waves are surface waves, bound to the sheet or boundary. The eigenmode problem for waves along spatially uniform but time-modulated boundaries is one of the important canonical problems in electromagnetics of time-varying structures.\n\nIn this work, we solve this problem and discuss dispersion properties of surface waves over time-modulated reactive boundaries. As a simple canonical case, we consider a planar infinite boundary that is modeled by a surface capacitance which is spatially uniform over the surface and temporally modulated by an arbitrary periodical function. As a particular realization, one can consider, for example, a high-impedance surface of the type introduced by D. Sievenpiper \\cite{sievenpiper1999high}, where the capacitance between patches is modulated using varactors.\n\n\nIn addition to derivation of the dispersion relation for surface waves over such time-varying boundary, we give numerical examples of dispersion plots. Also, we explain the conditions for appearing stop bands for propagation constants and reveal phenomena of exponential field growth, reflection amplification, and radiation of space waves from those time-modulated boundaries.\n\n\n\n\\section{Theory}\n\n\\begin{figure}\n\\includegraphics[width=.95\\linewidth]{geom.pdf}\n\\caption{Geometry of the problem: A TE-polarized surface wave over a time-modulated capacitive boundary. }\n\\label{fig:geom}\n\\end{figure}\n\nLet us consider a spatially uniform and time-varying reactive boundary. Here, as an example, we assume a time-varying capacitive one which is represented by $C(t)$. As is well known, such a boundary supports surface waves which have the transverse-electric (TE) polarization with respect to the propagation direction. In other words, in free space, the electric field is perpendicular to the propagation plane, as shown in Fig.~\\ref{fig:geom}. For a stationary boundary, the dispersion equation defines a relation between the frequency $\\omega$ and the propagation constant along the surface $\\beta$. Since the capacitive boundary periodically changes in time, the eigenmode $\\beta$ will contain components at frequencies $\\omega_n=\\omega+n\\omega_{\\rm{M}}$, $n=0,\\pm 1,\\pm 2,\\dots$. Here, $\\omega_{\\rm{M}}$ is the fundamental modulation frequency. The electric field is expressed as \n\\begin{equation}\n\\mathbf{E}=\\sum_{n=-\\infty}^{+\\infty}E_n\\exp(j\\omega_nt)\\mathbf{a}_y,\n\\end{equation}\nin which \n\\begin{equation}\nE_n=A_n\\exp(j\\beta z)\\exp(-\\alpha_nx).\n\\end{equation}\nHere, $A_n$ is the amplitude of the wave corresponding to each frequency harmonic,  and $\\alpha_n$ denotes the attenuation constant along the normal direction, for each harmonic.  The plane-wave dispersion equation for free space above the boundary sets the following relation for each harmonic: \n\\begin{equation}\n\\beta^2=\\alpha_n^2+\\omega_n^2\\epsilon_0\\mu_0 .\n\\label{eq:betaalpha}\n\\end{equation}\nIt is worth mentioning that since we investigate surface waves, $\\alpha_n$ must be a real value.\n\nSimilarly to the electric field, the tangential component of the magnetic field which is directed along the $z$-axis is given by \n\\begin{equation}\n\\mathbf{H}_{\\rm{t}}=\\sum_{n=-\\infty}^{+\\infty}H_n\\exp(j\\omega_nt)\\mathbf{a}_z,\n\\end{equation}\nwhere\n\\begin{equation}\nH_n=B_n\\exp(j\\beta z)\\exp(-\\alpha_nx).\n\\end{equation}\nThe Maxwell  equations relate the amplitudes of the electric field and the tangential component of the magnetic field to each other. By applying Eq.~\\eqref{eq:betaalpha} and doing some algebraic manipulations, we obtain a matrix relation  $\\mathbf{M}\\cdot \\mathbf{A}=\\mathbf{B}$. Here, $\\mathbf{M}$ is a matrix that has $2N+1$ rows and columns, and it is a function of $\\beta$ and $\\omega_n$. The matrices $\\mathbf{A}$ and $\\mathbf{B}$ have only one column and $2N+1$ rows, and they are representing the amplitudes.  \n\nIn analogy with the circuit theory, where we explicitly express the relation between the electric current flowing through the time-varying capacitance $i(t)$ and the voltage over it $v(t)$ as \n\\begin{equation}\n\\int i(t)dt=C(t)v(t),\n\\end{equation}\nwe simply write the relation between the tangential components of the electric and magnetic fields. In fact, this is the boundary condition in the dynamic scenario. By imposing the boundary condition, we obtain another matrix equation as $\\mathbf{Y}\\cdot \\mathbf{A}=\\mathbf{B}$. The matrix $\\mathbf{Y}$ is a function of $\\omega_n$ and the Fourier coefficients of the periodic function $C(t)$ which is expressed by the Fourier series in the exponential form. Similar modeling method has been used in our recent paper \\cite{wang2020nonreciprocity}.\n\nNow, we have two matrix equations $\\mathbf{M}\\cdot \\mathbf{A}=\\mathbf{B}$ and\\break $\\mathbf{Y}\\cdot \\mathbf{A}=\\mathbf{B}$. Therefore, we conclude that $\\big[\\mathbf{Y}-\\mathbf{M}\\big]\\cdot\\mathbf{A}=0$.  The determinant of the whole matrix in the square brackets must be zero in order to allow nonzero solutions for the electric field. Consequently,  relation \n\\begin{equation}\n\\det\\Big[\\mathbf{Y}-\\mathbf{M}\\Big]=0 \\label{eq: dispersion}\n\\end{equation}\ndetermines the dispersion of the surface waves above  time-varying capacitive boundaries. In the following, we give some particular examples and numerically investigate the dispersion curves.\n\n\n\\section{Numerical Examples and Discussion}\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{dispersion_one_tune.pdf}\n\t\\caption{Dispersion plot for modulation with one harmonic tune. The figure is plotted for $C_0=1$~pF and $\\omega_{\\rm M}=3$~GHz. \n\t} \n\t\\label{fig: dispersion_one_tunes}\n\\end{figure}\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{dispersion_two_tune.pdf}\n\t\\caption{Dispersion plot for modulation with two harmonic tunes:  $C(t)=C_0[1+0.3\\cos(\\omega_{\\rm M}t)+0.2\\cos(2\\omega_{\\rm M}t)]$. \n\t} \n\t\\label{fig: dispersion_two_tunes}\n\\end{figure}\n\n\n\nFirst, we consider a capacitive boundary that is modulated in time harmonically, assuming, as an example,  $C(t)=C_0[1+0.3\\cos(\\omega_{\\rm M}t)]$. \nTo obtain the dispersion diagram, we specify the value of $\\beta$, and solve the dispersion equation Eq.~(\\ref{eq: dispersion})  for the corresponding  eigenfrequencies. The corresponding dispersion curves are shown in  Fig.~\\ref{fig: dispersion_one_tunes}. One  can see  that for a fixed value of the propagation constant $\\beta$, there are many solutions for the eigenfrequencies $\\omega$, meaning that an eigenmode contains components at many frequencies. This property means that we can excite the mode by external sources at  many possible frequencies. \nThe excitation frequency can even be above the light line (purple dot) which excites higher-order frequency harmonics below the light line (green dot).  This indicates that, in time-varying structures, it is possible to launch surface waves with an incident plane wave, as is also reported in a recent work \\cite{galiffi2020wood}.\nMost importantly, temporal modulation opens up a band gap in $k$-space, which is a dual phenomenon of spatial periodic structures where the band gap is at the frequency axis. Similar properties of bulk media have been reported in \\cite{zurita2009reflection,lustig2018topological}.\nIncreasing the modulation depth, one  can widen the band gap. Interestingly, the number of band gaps corresponds to the number of modulation tones. Figure~\\ref{fig: dispersion_two_tunes} shows that by  adding a second modulation tone, a second band gap opens up. Positions and widths of the gaps can be tuned by varying the modulation spectrum. These band gaps provide great possibilities to control the propagation of surface waves on a metasurface plane. \n\n\\begin{figure}[!h]\n\\includegraphics[width=.95\\linewidth]{amplification.pdf}\n\\caption{(a) Surface wave on a time-invariant boundary. (b) Amplified surface wave on a time-varying boundary ($\\Delta t=5T_0$).  }\n\\label{fig:amplification}\n\\end{figure}\n\nNext, we examine the wave behavior when the excited surface modes are inside a band gap. For a specified wavenumber $\\beta$ in the band gap, the solved eigenfrequencies are complex numbers $\\omega=\\omega^\\prime\\pm j\\omega^{\\prime\\prime}$, meaning that  the wave can be exponentially attenuated or amplified in time. Next, we use COMSOL to numerically simulate the time-varying structure in this regime. \nFigure~\\ref{fig:amplification}(a) illustrates   a constant-amplitude surface wave propagating along an unmodulated capacitive boundary. Then, the temporal modulation of the surface reactance is suddenly  switched on.  Evidently, as it can be seen in Fig.~\\ref{fig:amplification}(b, ) the surface mode is significantly amplified after modulating with a duration of $\\Delta t=5T_0$, where $T_0$ is the time period at the excitation frequency.  In addition, there are higher-order harmonics generated in free space, forming a standing wave pattern along the surface, due to symmetry of the structure. \n\n\n\n\n\\section{Conclusions}\nHere, we have presented the dispersion equation and example dispersion plots for a temporally modulated electromagnetic boundary. The results show that  time modulation induces band gaps in the two-dimensional $k$-space, which provides opportunities to control surface wave propagation. In the presentation, we will discuss in detail interesting properties of surface waves launched  inside a band gap.  \nLet us note that one can repeat the above path to achieve the dispersion equation associated with  time-varying inductive boundaries. The difference is that the wave has a transverse-magnetic polarization (i.e., magnetic field is perpendicular to the propagation plane). \n\n\n\\section{Acknowledgment}\nThe authors thank Dr. V.~Asadchy for useful discussions and valuable comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Highlights}\n\\begin{itemize}\n\\item Privacy-preserving multi-agent reinforcement learning is used to coordinate residential energy  \n\\item Learning from optimisations improves coordination scalability in stochastic environments\n\\item Marginal reward signals further enhance cooperation relative to previous approaches\n\\item The curse of dimensionality is mitigated by the use of fixed-size Q-tables\n\\item Case studies with large real-life datasets yield 33.7\\% local and global cost reductions \n\\end{itemize}\n\n\\begin{multicols}{2}\n\n\\section{Introduction}\nThis paper addresses the scalability issue of distributed domestic energy flexibility coordination in a cost-efficient and privacy-preserving manner.  A novel class of coordination strategies using optimisation-based multi-agent reinforcement learning (MARL\\footnote{A full nomenclature is available in \\Cref{app:nomenclature}}) with fixed Q-table size is proposed for household-level decision-making, tackling the challenge of scalability for simultaneously learning independent agents under partial observability in a stochastic environment \\citep{Matignon2012}. Multiple versions of the novel strategy are assessed to maximise the statistical expectation of system-wide benefits, including local battery costs, grid costs and greenhouse gas emissions. \n\nWidespread electrification of primary energy provision and decarbonisation of the power sector are two vital prerequisites for limiting anthropogenic global warming to 1.5$^o$C above pre-industrial levels. To reduce risks of climate-related impacts on health, livelihood, security and economic growth, intermittent renewable power supplies could be required to supply 70\\% to 85\\% of electricity by 2050 \\citep{IPCC2015}. However, this poses the challenges of the intermittency and limited controllability of resources \\citep{Bose2019}.  Therefore, a robust, decarbonised power system will rely on two structural features: decentralisation and demand response (DR) \\citep{Leautier2019}. The coordination of distributed flexible energy resources can help reduce costs for transmission, storage,  peaking plants and capacity reserves,  improve grid stability, align demand with decarbonised energy provision, promote energy independence and security, and lower household energy bills \\citep{Vazquez-Canteli2019, Pumphrey2020}.\n\nResidential sites constitute a significant share of potential DR, representing for example 38.5\\% of the 2019 UK electricity demand, and 56.4\\% of energy consumption if including transport and heat, which are both undergoing electrification \\citep{BEIS2021}. Increasing ownership of EVs and PV panels has been facilitated by regulatory changes, with many countries committing to internal combustion car phase-outs in the near future, and by plummeting costs, with an 82\\% and 87\\% levelised cost drop between 2010 and 2019 for EVs and PV panels \\citep{Agency2018,BloomberNEF2019}. This potential is so far underexploited, as DR primarily focuses on larger well-known industrial and commercial actors that require less coordination and data management \\citep{CharlesRiverAssociates2017}, with most customers still limited to trade with utility companies \\citep{Chen2019}. The primary hurdles to unlocking residential flexibility are the high capital cost of communication and control infrastructure as the domestic potential is highly fragmented \\citep{Leautier2019}, concerns about privacy and hindrance of activities \\citep{Bugden2019,Pumphrey2020}, and computational challenges for real-time control at scale \\citep{Moret2019}.  \n\nTraditionally, convex optimisation would be used to maximise global coordination objectives in convex problems with variables known ahead of time. Techniques such as least-squares and linear programming have been well-studied for over a century \\citep{Boyd2009}. However, residential energy coordination presents challenges to its application. Firstly,  optimisations that are centralised are hindered by privacy, acceptance, and communication constraints, and present exponential time complexity at the scale of millions of homes \\citep{Dasgupta2016}. Secondly, standard optimisation methods cannot be used without full knowledge of the system's inputs and dynamics \\citep{Recht2018}. In residential energy, agents only have partial observability of the system due to both the stochasticity and uncertainty of environment variables such as individual residential consumption and generation profiles, and to the privacy and infrastructure cost constraints that hinder communication between agents during implementation \\citep{FrancoisLavet2017}. Not relying on shared information may also improve the robustness of the solutions to failure of other agents, communication delays, and unreliable information, and improve adaptability to changing environments \\citep{Sen1994}. Finally, the real-life complex electricity grid environment may not be amenable to a convex model representation. Due to the heterogeneity of users and behaviours needing different parameters and models,  the large-scale use of model-based controllers is cumbersome \\citep{Ruelens2017}. A model-free approach instead avoids modelling non-trivial interactions of parameters, including private information \\citep{Dasgupta2016}.  \n\nGiven these challenges to residential energy flexibility coordination, and the specific constraints of the problem at play which renders traditional approaches unsuitable, we seek to develop a novel coordination mechanism which satisfies the following criteria,  as tested in real-life scenarios:\n\\begin{itemize}\n\\item Computational scalability: minimal and constant computation burden during implementation as the system size increases; \n\\item Performance scalability: no drop in coordination performance as the system size increases, measured in savings obtained per hour and per agent;\n\\item Acceptability: local control of appliances, no communication of personal data, thermal discomfort, or hindrance\/delay of activities.\n\\end{itemize}\n\nThe rest of this paper is organised as follows. In \\Cref{sec:gapanalysis} we motivate the novel MARL approach with a literature review and a gap analysis. In \\Cref{system}, a system model is presented that includes household-level modelling of EVs, space heating, flexible loads and PV generation.  \\Cref{RLSection} lays out the MARL methodology, with various methodological options for independent agents to learn to cooperate. In \\Cref{data}, the input data used to populate the model is presented. In \\Cref{results}, the performance of different MARL strategies is compared to lower and upper bounds in case studies. Finally, we conclude in \\Cref{conclusion}.\n\n\n\\section{MARL-based energy coordination: literature review and gap analysis}\\label{sec:gapanalysis}\nReinforcement learning (RL) can overcome the constraints faced by centralised convex optimisation for residential energy coordination, by allowing for decentralised and model-free decision-making based on partial knowledge. RL is an artificial intelligence (AI) framework for goal-oriented agents\\footnote{Here agents are independent computer systems acting on behalf of prosumers \\citep{Wooldridge2002}. Prosumers are proactive consumers with distributed energy resources actively managing their consumption, production and storage of energy \\citep{Morstyn2018_Federated}. } to learn sequential decision-making by interacting with an uncertain environment \\citep{Sutton1998}.  As an increasing wealth of data is collected in local electricity systems, RL is of growing interest for the real-time coordination of distributed energy resources (DERs) \\citep{Antonopoulos2020,Vazquez-Canteli2019}.  Instead of optimising based on inherently uncertain data, RL more realistically searches for statistically optimal sequential decisions given partial observation and uncertainty, with no \\emph{a priori} knowledge \\citep{Recht2018}.  Approximate learning methods may be more computationally scalable, more efficient in exploring high-dimensional state spaces and therefore more scalable than exact global optimisation with exponential time complexity \\citep{Schellenberg2020, Dasgupta2016}. \n\nAs classified in \\citep{CharbonnierReview}, numerous RL-based coordination methods have been proposed in the literature for residential energy coordination, though with remaining limitations in terms of scalability and privacy protection. On the one hand, in RL-based direct control strategies, a central controller directly controls individual units, and households directly forfeit their data and control to a central RL-based scheduler \\citep{ONeill2010}. While most existing AI-based DR research thus assumes fully observable tasks \\citep{Antonopoulos2020}, direct controllability of resources from different owners with different objectives and resources and subject to privacy, comfort and security concerns is challenging \\citep{Darby2020}. Moreover, centralised policies do not scale due to the curse of dimensionality as the state and action spaces grow exponentially with the system size \\citep{Powell2011}.  On the other hand, RL-based indirect control strategies consider decision-making at the prosumer level, entering the realm of MARL. This can be achieved using different communication structures, with either centralised, bilateral, or no sharing of personal information, as presented below.\n\nFirstly, agents may share information with a central entity,  which in turn broadcasts signals based on a complete picture of the coordination problem. For example, the central entity may send unidirectional price signals to customers based on information such as prosumers' costs, constraints and day-ahead forecasts. RL can inform both the dynamic price signal \\citep{Lu2019, Kim2016}, and the prosumer response to price signals \\citep{Kim2016,Babar2018}. The central entity may also collect competitive bids and set trades and match prosumers centrally, where RL algorithms are used to refine individual bidding strategies \\citep{Vaya2014, Ye2020, Dauer2013,Sun2015,Kim2020} or to dictate the auction market clearing \\citep{Chen2019,Claessens2013}. Units may also use RL to cooperate towards common objectives with the mediation of a central entity that redistributes centralised personal information \\citep{Zhang2017,Dusparic2015,Dusparic2013,Hurtado2018}. However, information centralisation also raises costs, security, privacy and scalability of computation issues. Biased information may lead to inefficient or even infeasible decisions \\citep{Morstyn2020_P2P}. \n\nSecondly, RL-based coordination has been proposed where prosumers only communicate information bilaterally without a central authority. For example, in \\citep{Taylor2014} agents use transfer learning with distributed W-learning to achieve local and system objectives. Bilateral peer-to-peer communication offers autonomy and expression of individual preferences, though with remaining risks around privacy and bounded rationality \\citep{Herbert1982}. There is greater robustness to communication failures compared situations with a single point of failure. However, as the system size increases,  the number of communication iterations until algorithmic convergence increases, requiring adequate computational resources and limited communication network latency for feasibility \\citep{Guerrero2020}. The safe way of implementing distributed transactions to ensure data protection is an ongoing subject of research \\citep{CharbonnierReview}. \n\nFinally,  in RL-based implicit coordination strategies, prosumers rely solely on local information to make decisions. For example, in \\citep{Cao2019, Yang2019}, competitive agents in isolation maximise their profits in RL-based energy arbitrage, though they do not consider the impacts of individual actions on the rest of the system, with potential negative impacts for the grid. For example, a concern is that all loads receive the same incentive, the natural diversity on which the grid relies may be diminished \\citep{Crozier2018_Mitigating}, and the peak potentially merely displaced, with overloads on upstream transformers. Implicit cooperation, which keeps personal information at the local level while encouraging cooperation towards global objectives, has been thus far under-researched beyond frequency control.  In \\citep{Rozada2020}, agents learn the optimal way of acting and interacting with the environment to restore frequency using local information only. This is a promising approach for decentralised control. However, the applicability in more complex scenarios with residential electric vehicles and smart heating load scheduling problems has not been considered. Moreover, the convergence slows down for increasing number of agents, and scalability beyond 8 agents has not been investigated. Indeed, fundamental challenges to the coordination of simultaneously learning independent agents at scale under partial observability in a stochastic environment have been identified when using traditional RL algorithms [1]: independent learners may reach individual policy equilibriums that are incompatible with a global Pareto optimal, the non-stationarity of the environment due to other concurrently learning agents affects convergence, and the stochasticity of the environment prevents agents from discriminating between their own contribution to global rewards and noise from other agents or the environment. Novel methods are therefore needed to develop this approach.\n\nWe seek to bridge this gap, using implicit coordination to unlock the so-far largely untapped value from residential energy flexibility to provide both individual and system benefits. We propose a new class of MARL-based implicit cooperation strategies for residential DR, to make the best use of the flexibility offered by increasingly accessible assets such as photovoltaic (PV) panels, electric vehicle (EV) batteries, smart heating and flexible loads. Agents learn RL policies using a data-based, model-free statistical approach by exploring a shared environment and interacting with decentralised partially observable Markov decision processes (Dec-POMDPs), either through random exploration or learning from convex optimisation results. In the first rehearsal phase \\citep{Kraemer2016} with full understanding of the system, they learn to cooperate to reach system-wide benefits by assessing the global impact of their individual actions, searching for trade-offs between local, grid and social objectives. The pre-learned policies are then used to make decisions under uncertainty given limited local information only.\n\nThis approach satifies the computational scalability, coordination scalability and acceptance criteria set out in this paper.\n\nFirstly,  the real-time control method is computationally scalable thanks to fixed-size Q-tables which avoid the curse of dimensionality, and there is only minimal, constant local computation required to implement the pre-learned policies during implementation. No further communication is required for implementation. This increases robustness to communication issues and data inaccuracy relative to when relying on centralised and bilateral communication, and cuts the costs of household computation and two-way communication infrastructure. \n\nSecondly, we address the outstanding MARL coordination performance scalability issue for agents with partial observability in a stochastic environment seeking to maximise rewards which also depend on other concurrently learning agents \\citep{Busoniu2008,Matignon2012}. The case studies in this paper show that allowing agents to learn from omniscient, stable, and consistent optimisation solutions can successfully act as an equilibrium-selection mechanism, while the use of marginal rewards improves learnability\\footnote{\\say{the sensitivity of an agent's utility to its own actions as opposed to actions of others, which is often low in fully cooperative Markov games} \\citep{Matignon2012}} by isolating individual contributions to global rewards. This novel methodological combination offers significant improvements on MARL scalability and convergence issues, with high coordination performance maintained as the number of agents increases, where that of standard MARL drops at scale. \n\nFinally,  this method tackles acceptability issues, with no interference in personal comfort nor communication of personal data. \n\nThe specific novel contributions of this paper are (a) a novel class of decentralised flexibility coordination strategies, MARL-based implicit cooperation, with no communication and fixed-size Q-tables to mitigate the curse of dimensionality; (b) a novel MARL exploration strategy for agents under partial observability to learn from omniscient, convex optimisations prior to implementation for convergence to robust cooperation at scale; and (c) the design and testing with large banks of real-world data of combinations of reward definitions, exploration strategies and multi-agent learning frameworks for assessing individual impacts on global energy, grid and storage costs. Methodologies are identified which outperform a baseline with increasing numbers of agents despite uncertainty.\n\n\\section{Local system description}\\label{system}\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.7\\linewidth]{energybalance.pdf}\n\n\\end{center}\n\\caption{Local system model. Red dotted lines denote energy balances.}\n\\label{fig:EnergyBalance}\n\\end{figure*}\n\nIn this section, the variables, objective function and constraints of the problem are described. This sets the frame for the application of the RL algorithms presented in \\Cref{RLSection}.\n\n\\subsection{Variables}\\label{variables}\nWe consider a set of time steps $t \\in \\mathcal{T} = \\{t_0,...,t_\\textrm{end}\\}$ and a set of prosumers $i \\in \\mathcal{P} = \\{1,...,n\\}$. Decision variables are \\emph{italicised} and input data are written in roman. Energy units are used unless specified otherwise. Participants have an EV, a PV panel, electric space heating and generic flexible loads.\n\nThe EV at-home availability $\\upmu_i^t$ (1 if available, 0 otherwise), EV demand for required trips $\\textrm{d}_{\\textrm{EV},i}^t$, household electric demand $\\textrm{d}_i^{t}$,  PV production $\\textrm{p}_{\\textrm{PV},i}^t$, external temperature $\\textrm{T}_{\\textrm{e}}^t$ and solar heat flow rate $\\upphi^t$ are specified as inputs for $t \\in \\mathcal{T}$ and $i \\in \\mathcal{P}$.\n\nThe local decisions by prosumers are the energy flows in and out of the battery $b_{\\textrm{in},i}^t$ and $b_{\\textrm{out},i}^t$, the electric heating consumption $h_i^t$ and the prosumer consumption $c_i^t$. These have both local and system impacts (\\Cref{fig:EnergyBalance}). Local impacts include battery energy levels $E_i^t$,  losses $\\epsilon_{\\textrm{ch},i}^t$ and $\\epsilon_{\\textrm{dis},i}^t$,  prosumer import $p_i^t$, building mass temperature $T_{\\textrm{m},i}^t$ and indoor air temperature $T_{\\textrm{air},i}^t$. System impacts arise through the costs of total grid import $g^t$ and distribution network trading. Distribution network losses and reactive power flows are not included.  \n\n\\subsection{Objective function}\\label{objfunc}\n\nProsumers cooperate to minimise system costs consisting of grid ($c_\\textrm{g}^t$), distribution ($c_\\textrm{d}^t$) and storage ($c_\\textrm{s}^t$) costs. This objective function will be maximised both in convex optimisations off-line -- to provide an upper bound for the achievable objective function, and in some cases to provide information to the learners during the simulated learning phase -- and in the learning of MARL policies for decentralised online implementation. \n\n\\begin{equation}\n\\max F = \\sum_{\\forall t \\in \\mathcal{T}}{\\hat{F}_t} = \\sum_{\\forall t \\in \\mathcal{T}}{- (c_\\textrm{g}^t + c_\\textrm{d}^t + c_\\textrm{s}^t )}\n\\end{equation}\n\n\\begin{equation}\nc_\\textrm{g}^t = \\textrm{C}_\\textrm{g}^t \\left( g^t + \\epsilon_g \\right)\n\\end{equation}\nWhere losses incurred by imports and exports from and to the main grid are approximated as\n\\begin{equation}\n\\epsilon_g = \\frac{\\textrm{R}}{\\textrm{V}^2}\\left(g^t\\right)^2\n\\end{equation}\n\nThe grid cost coefficient $\\textrm{C}_\\textrm{g}^t$ is the sum of the grid electricity price and the product of the carbon intensity of the generation mix at time $t$ and the Social Cost of Carbon which reflects the long-term societal cost of emitting greenhouse gases \\citep{ParryM}. The impacts of local decisions on upstream energy prices are neglected.  Grid losses are approximated using the nominal root mean square grid voltage $\\textrm{V}$ and the average resistance between the main grid and the distribution network $\\textrm{R}$ \\citep{Multiclass}, based on the assumption of small network voltage drops and relatively low reactive power flows \\citep{Coffrin2012}. The second-order dependency disincentivises large power imports and exports, which helps ensure interactions of transmission and distribution networks do not reduce system stability.\n\n\\begin{equation}\nc_\\textrm{d}^t = \\textrm{C}_\\textrm{d}\\sum_{i \\in \\mathcal{P}}{\\max\\left(- p_i^t,0\\right)}\n\\end{equation}\n\\noindent Distribution costs $c_\\textrm{d}^t$ are proportional to the distribution charge $\\textrm{C}_\\textrm{d}$ on exports. The resulting price spread between individual imports and exports decreases risks of network constraints violation by incentivising the use of local flexibility first \\citep{Morstyn2020_IntegratingP2P}. Distribution network losses due to power flows between prosumers are neglected so there is no second-order dependency.\n\\begin{equation}\nc_\\textrm{s}^t = \\textrm{C}_\\textrm{s}\\sum_{i \\in \\mathcal{P}}{\\left(b_{\\textrm{in},i}^t + b_{\\textrm{out},i}^t\\right)}\n\\end{equation}\n\n\\noindent Storage battery depreciation costs $c_\\textrm{s}^t$ are assumed to be proportional to throughput using the depreciation coefficient $\\textrm{C}_\\textrm{s}$, assuming a uniform energy throughput degradation rate \\citep{DufoLopez2014}.\n\n\\subsection{Constraints}\n\n\nLet $\\textrm{E}_0$,  $\\underline{\\textrm{E}}$ and $\\overline{\\textrm{E}}$ be the initial, minimum and maximum battery energy levels, $\\upeta_\\textrm{ch}$ and $\\upeta_\\textrm{dis}$ the charge and discharge efficiencies, and $\\overline{\\textrm{b}_\\textrm{in}}$ the maximum charge per time step. Demand $\\textrm{d}_{i,k}^{t_\\textrm{D}}$ is met by the sum of loads consumed $\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ at time $t_\\textrm{C}$ by prosumer $i$ for load of type $k$ (fixed or flexible) demanded at $t_\\textrm{D}$. The flexibility boolean $\\textrm{f}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ indicates if time $t_\\textrm{C}$ lies within the acceptable range to meet $\\textrm{d}_{i,k}^{t_\\textrm{D}}$. A Crank-Nicholson scheme \\citep{ISO2007} is employed to model heating, with $\\upkappa$ a 2x5 matrix of temperature coefficients, and $\\underline{\\textrm{T}}_i^t$ and $\\overline{\\textrm{T}}_i^t$ lower and upper temperature bounds. System constraints for steps $\\forall \\ t \\in \\mathcal{T}$ and prosumers $\\forall \\ i \\in \\mathcal{P}$ are:\n\n\\begin{itemize}\n\\item Prosumer and substation energy balance (see \\Cref{fig:EnergyBalance})\n\\begin{equation}\n\tp_i^t = c_i^t + h_i^t + \\frac{b_{\\textrm{in},i}^t}{\\upeta_\\textrm{ch}} - {\\upeta_\\textrm{dis}} b_{\\textrm{out},i}^t - \\textrm{p}_{\\textrm{PV},i}^t \n\\end{equation}\n\\begin{equation}\n\\sum_{i \\in \\mathcal{P}}{p_i^t} = g^t\n\\end{equation}\n\\item Battery energy balance \n\\begin{equation}\n\tE_i^{t+1} = E_i^t  + b_{\\textrm{in},i}^t - b_{\\textrm{out},i}^t - \\textrm{d}_{\\textrm{EV},i}^t \n\\end{equation}\n\\item Battery charge and discharge constraints\n\\begin{equation}\n\t \\textrm{E}_0 = E_i^{t_0}  = E_i^{t_\\textrm{end}} + b_{\\textrm{in},i}^{t_{\\textrm{end}}} - b_{\\textrm{out},i}^{t_{\\textrm{end}}} - \\textrm{d}_{\\textrm{EV},i}^{t_{\\textrm{end}}} \n\\end{equation}\n\\begin{equation}\n\t\\upmu_i^t\\underline{\\textrm{E}}_i \\leq E_i^t \\leq \\overline{\\textrm{E}}_i\n\\end{equation}\n\\begin{equation}\n\tb_{\\textrm{in},i}^t \\leq \\upmu_i^t \\overline{\\textrm{b}_\\textrm{in}}\n\\end{equation}\n\\begin{equation}\n\tb_{\\textrm{out},i}^t \\leq \\upmu_i^t \\overline{\\textrm{E}}_i\n\\end{equation}\n\n\\item Consumption flexibility --- the demand of type $k$ at time $t_\\textrm{D}$ by prosumer $i$ must be met by the sum of partial consumptions $\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ at times $t_\\textrm{C}...t_\\textrm{C}+\\textrm{n}_\\textrm{flex}$ within the time frame $\\textrm{n}_\\textrm{flex}$ specified by the flexibility of each type of demand in matrix $\\textrm{f}_{i,k,t_\\textrm{C},t_\\textrm{D}}$\n\\begin{equation}\\label{eq:demandmet}\n\t\\sum_{t_\\textrm{C}\\in\\mathcal{T}}{\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}} \\textrm{f}_{i,k,t_\\textrm{C},t_\\textrm{D}}} = \\textrm{d}_{i,k}^{t_\\textrm{D}} \n\\end{equation}\n\\item Consumption --- the total consumption at time $t_\\textrm{C}$ is the sum of all partial consumptions $\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ meeting parts of demands from current and previous time steps $t_\\textrm{D}$:\n\\begin{equation}\\label{eq:totalcons}\n\t\\sum_{t_\\textrm{D}\\in\\mathcal{N}}{\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}}= c_{i,k}^{t_\\textrm{C}} \n\\end{equation}\n\n\\item Heating --- the workings to obtain this equation are included in \\Cref{app:heating}:\n\\begin{equation}\\label{eq:main_heating}\n\\begin{bmatrix}\nT_{\\textrm{m},i}^{t+1}\\\\\nT_{\\textrm{air},i}^{t+1} \n\\end{bmatrix}\n = \\upkappa \n \\begin{bmatrix}\n1,\nT_{\\textrm{m},i}^{t},\n\\textrm{T}_{\\textrm{e}}^t,\n\\upphi^t,\nh_i^t\n\\end{bmatrix}^\\intercal\n\\end{equation}\n\\begin{equation}\n\\underline{\\textrm{T}}_i^t \\leq T_{\\textrm{air},i}^t \\leq \\overline{\\textrm{T}}_i^t\n\\end{equation}\n\\item Non-negativity constraints \n\\begin{equation}\n\tc_i^t, h_i^t,E_i^t, b_{\\textrm{in},i}^t, b_{\\textrm{out},i}^t, \\hat{c}_{i,l,t_\\textrm{C},t_\\textrm{D}} \\geq 0\n\\end{equation}\n\\end{itemize}\n\nWhile the proposed framework could accommodate the use of idiosyncratic satisfaction functions to perform trade-offs between flexibility use and users' comfort, no such trade-offs are considered in this paper, with comfort requirements for temperature and EV usage always being met. Field evaluations have shown that programmes that do not maintain thermal comfort are consistently overridden, increasing overall energy use and costs \\citep{Sachs2012}, while interference in consumption patterns and temperature set-points cause dissatisfaction \\citep{Vazquez-Canteli2019}.  Meeting fixed domestic loads,  ensuring sufficient charge for EV trips, and maintaining comfortable temperatures are therefore set constraints. \n\n\\section{Reinforcement learning methodology}\\label{RLSection}\nThe MARL approach is now presented in which independent prosumers learn to make individual decisions which together maximise the statistical expectation of the objective function in \\Cref{system}.  \n\nAt time step $t \\in \\mathcal{T}$, each agent is in a state $s_i^t \\in \\mathcal{S}$ corresponding to accessible observations (here the time-varying grid cost), and selects an action $a_i^t \\in \\mathcal{A}$ as defined in \\Cref{sec:agentdecision}. This action dictates the decision variables in \\Cref{variables} $b_{\\textrm{in},i}^t$, $b_{\\textrm{out},i}^t$, $h_i^t$ and $c_i^t$. The environment then produces a reward $r^t \\in \\mathcal{R}$ which corresponds to the share $\\hat{F}_t$ of the system objective function presented in \\Cref{objfunc} and agents transition to a state $s_i^{t+1}$. Agents learn individual policies $\\pi_i$ by interacting with the environment using individual, decentralised fixed-size Q-tables.\n\nWe first introduce the Q-learning methodology.  Then, the mapping between the RL agent action and the decision variables in \\Cref{variables} is presented. Finally, we propose variations on the learning method, with different experience sources, multi-agent structures and reward definitions.\n\n\\subsection{Q-Learning}\nWhile any reinforcement learning methodology could be used with the framework proposed in this paper, here we focus on Q-learning, a model-free, off-policy RL methodology.  Its simplicity and proof of convergence make it suited to developing novel learning methodologies in newly defined environments \\citep{Vazquez-Canteli2019}. State-actions values $Q(s,a)$ represent the expected value of all future rewards $r_t$ $\\forall \\ t \\in \\mathcal{T}$ when taking action $a$ in state $s$ according to policy $\\pi$:\n\\begin{equation}\nQ(s,a) \\triangleq E^{\\pi}{[r_{t} + \\gamma r_{t+1} + \\gamma^2r_{t+2}...|s_t = s, a_t = a ]}\n\\end{equation}\n\n\\noindent where $\\gamma$ is the discount factor setting the relative importance of future rewards. Estimates are refined incrementally as\n\\begin{equation}\n\\hat{Q}(s,a)\\leftarrow \\hat{Q}(s,a) + \\alpha\\delta\n\\end{equation}\nwhere $\\delta$ is the temporal-difference error,\n\\begin{equation}\n\\delta = \\left(r_t +\\gamma \\hat{V}(s^\\textrm{next})-\\hat{Q}(s,a)\\right)\n\\end{equation}\n$\\hat{V}$ is the state-value function estimate,\n \\begin{equation}\n\\hat{V}(s) = \\max_{a^* \\in \\mathcal{A}(s)}{\\hat{Q}(s,a^*)}\n\\end{equation}\nand $\\alpha$ is the learning rate. In this work we use hysteretic learners, i.e. chiefly optimistic learners that use an increase rate superior to the decrease rate in order to reduce oscillations in the learned policy due to actions chosen by other agents \\citep{Matignon2012, Matignon2007}. For $\\beta < 1$:\n\\begin{equation}\n  \\alpha  =\n    \\begin{cases}\n      \\alpha_0 & \\text{if $\\delta > 0$}\\\\\n      \\alpha_0\\beta & \\text{otherwise}\\\\\n    \\end{cases}       \n\\end{equation}\n\nAgents follow an $\\epsilon$-greedy policy to balance exploration of different state-action pairs and knowledge exploitation. The greedy action with highest estimated rewards is selected with probability $1-\\epsilon$ and random actions otherwise. \n\\begin{equation}\n\\label{eqn:greedy}\n  a^*  =\n    \\begin{cases}\n      \\argmax_{\\ a^* \\in \\mathcal{A}}\\hat{Q}(s,a^*) & \\text{if $x \\sim U(0,1) > \\epsilon$}\\\\\n      a \\sim p(a) = \\frac{1}{|\\mathcal{A}|} \\ \\forall \\ a \\ \\in \\mathcal{A}& \\text{otherwise}\\\\\n    \\end{cases}       \n\\end{equation}\n\nHenceforth, we refer to the estimates $\\hat{Q}$ and $\\hat{V}$ as $Q$ and $V$ to reduce the amount of notation.\n\n\\subsection{Agent state}\nThe agent state is defined by the time-dependent grid cost coefficient $\\textrm{C}_\\textrm{g}^t$, i.e. the sum of the grid electricity price and the product of the carbon intensity of the generation mix at time $t$ and the social cost of carbon.\n\nTo convert the RL policy action into local decisions, the agent also requires information on their current PV generation, battery level, flexible loads and indoor air temperature, as described below in \\Cref{sec:agentdecision}.\n\n\\subsection{Agent action}\\label{sec:agentdecision}\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{inkscape_mu_4.pdf}\n\n\\end{center}\n\\caption{Decision variable $\\psi$. Sections 1-5 denote the trade-off regimes described in \\Cref{sec:agentdecision}. At each step, the fixed requirements for loads, heat and upcoming EV trips are first met. The $\\psi$ decision then applies to the remaining flexibility, from maximal energy exports (full use of flexibility) at $\\psi = 0$, to maximal energy imports (no use of flexibility) at $\\psi = 1$. $\\textrm{d}_\\textrm{tot}$ and $\\textrm{d}_\\textrm{fixed}$ are the sum of household and heating loads with and without their flexible component. If fixed loads cannot be fully met by PV energy,  the residual is met by storage and imports (2). If there is additional PV energy after meeting all loads, it can be stored or exported (4).}\n\\label{fig:mu}\n\\end{figure*}\n\nLarge action spaces compound the curse of dimensionality in Q-learning and waste exploration resources \\citep{Powell2011}. At each time step, the decision variables in \\Cref{system} controlling the flows in and out of the battery$b_{\\textrm{in},i}^t$ and $b_{\\textrm{out},i}^t$, the electric heating consumption $h_i^t$ and the prosumer consumption $c_i^t$ for household $i$ are therefore synthesised into a single variable $\\psi \\in [0,1]$ controlling the use of available local flexibility. \\Cref{fig:mu} shows how consumption (for domestic loads and heat), imports and storage change with $\\psi$.\n\nAt each step, the fixed requirements for loads, heat and upcoming EV trips are first met. The $\\psi$ decision then applies to the remaining flexibility. In conditions deemed optimal for energy exports $\\psi = 0$, all initial storage and residual PV generation is exported and flexible loads are delayed. On the other end, a \\emph{passive} agent does not utilise its flexibility and uses the \\emph{default} action $\\psi = 1$, maximising imports with EVs charged when plugged in and no flexible loads delayed.  Intermediate imports trade-offs are mapped on \\Cref{fig:mu}:\n \n\\begin{enumerate}\n\\item From exporting all to none of the initial storage $E_i^t$\n\\item From meeting fixed loads $\\textrm{d}_{i,\\textrm{fixed}}^t$ with the energy stored to importing the required amount \n\\item From no to maximum flexible consumption $\\textrm{d}_{i,\\textrm{tot}}^t$ \n\\item From exporting to storing PV energy $\\textrm{p}_{\\textrm{PV},i}^t$ remaining after meeting loads\n\\item From importing no additional energy to filling up the battery to capacity $\\overline{\\textrm{E}}_i$\n\\end{enumerate}\n\nCostlier actions incurring battery depreciation, losses and export costs are towards either $\\psi$ extreme, only used in highly beneficial situations (convex local costs function in the lower plot of \\Cref{fig:mu}). Ranking actions consistently ensures agents do not waste resources trialling sub-optimal combinations of decisions. For example, it is more cost-efficient to first absorb energy imports by consuming flexible loads, and only use the battery (incurring costs) if imports are large.  \n\nNote that although this action space is continuous, it can be discretised into intervals for implementation in Q-learning. \n\n\\subsection{Variations of the learning method}\\label{methodologies}\nDifferent experience sources, reward definitions and MARL structures are proposed within the MARL approach. The performance of these combinations of algorithmic possibilities will be assessed in \\Cref{results} to inform effective model design.\\\\\n\n\\subsubsection{Experience sources}\nIn data-driven strategies, the learning is determined by the collected experience.\n\\begin{itemize}\n\\item \\textbf{Environment exploration}. Traditionally, agents collect experience by interacting with an environment \\citep{Sutton1998}.\n\n\\item \\textbf{Optimisations}. A novel approach collects experience from optimisations. Learning from entities with more knowledge or using knowledge more effectively than randomly exploring agents has previously been proposed, as with agents \\say{mimicking} humans playing video games \\citep{Grandmaster}. Similarly, agents learn from convex \\say{omniscient} optimisations on historical data with perfect knowledge of current and future variables.  This experience is then used under partial observability and control for stable coordination between prosumers at scale. Note in this case that, although the MARL learning and implementation are model-free, a model of the system is used to run the convex optimisation and produce experience to learn from.  A standard convex optimiser uses the same data that would be used to populate the environment explorations but solves over the whole day-horizon with perfect knowledge of all variables using the problem description in \\Cref{system}. Then, at each time step, the system variables are translated into equivalent RL $\\{s_t,a_t,r_t, s_{t+1}\\}$ tuples for each agent, which are used to update the policies in the same way as for standard Q-learning as presented below.\\\\\n\\end{itemize}\n \n\\subsubsection{MARL structures} Both the centralised and decentralised structures proposed use fixed-size $|\\mathcal{S}| \\times |\\mathcal{A}|$ Q-tables corresponding to individual state-action pairs. The size of a global Q-table referencing all possible combinations of states and actions would grow exponentially with the number of agents. This would limit scalability due to memory limitations and exploration time requirements. Moreover, as strategies proposed in this paper are privacy-preserving, only local state-action pairs are used for individual action selection, wasting the level of detail of a global Q-table.\n\n\\begin{itemize}\n\\item \\textbf{Distributed learning}. Each agent $i$ learns its $Q_i$ table with its own experience. No information is shared between agents. \n\n\\item \\textbf{Centralised learning}. A single table $Q_\\textrm{c}$ uses experience from all agents during pre-learning. All agents use the centrally learned policy for decentralised implementation.\n\\end{itemize}\n\n\\subsubsection{Reward definitions}\nThe reward definition is central to learning as its maximisation forms the basis for incrementally altering the policy \\citep{Sutton1998}.  Assessing the impact of individual actions on global rewards accurately is key to the effective coordination of a large number of prosumers. In the following,  the Q-tables $Q^0$, $Q^\\textrm{diff}$,$Q^\\textrm{A}$ and $Q^\\textrm{count}$ may be either agent-specific $Q_i$ or centralised $Q_\\textrm{c}$ based on the MARL structure. We proposed four variations of the Q-table update rule for each experience step tuple collected $(s_i^t, a_i^t, r^t,s_i^{t+1})$.\n \\begin{equation}\nQ(s_i^t, a_i^t)\\leftarrow Q(s_i^t,a_i^t) + \\alpha \\delta\n\\end{equation}\n\\begin{itemize}\n\\item \\textbf{Total reward}. The instantaneous total system reward $r^t = \\hat{F}_t$ is used to update the Q-table $Q^0$.\n\\begin{equation}\n\\delta = r^t + \\gamma V^0(s_i^{t+1}) - Q^0(s_i^t, a_i^t)\n\\end{equation}\n\\item \\textbf{Marginal reward}. The difference in total instant rewards $r^t$ between that if agent $i$ selects the greedy action and that if it selects the default action is used to update $Q^\\textrm{diff}$ \\citep{Wolpert2002}. The default action $a_\\textrm{default}$ corresponds to $\\psi = 1$, where no flexibility is used. The default reward $r^t_{a_{i}=a_\\textrm{default}}$, where all agents perform their greedy action apart from agent $i$ which performs the default action, is obtained by an additional simulation.\n\\begin{equation}\n\\delta = \\left(r^t - r^t_{a_{i}=a_\\textrm{default}}\\right) + \\gamma V^\\textrm{diff}(s_i^{t+1}) - Q^\\textrm{diff}(s_i^t, a_i^t)\n\\end{equation}\n\\item \\textbf{Advantage reward}. The post difference between $Q^0$ values when $i$ performs the greedy and the default action is used. This corresponds to the estimated increase in rewards not just instantaneously but over all future states, analogously to in \\citep{Foerster2018}. No additional simulations are required as the Q-table values are refined over the normal course of explorations.\n\\begin{equation}\n\\delta = \\left(Q^0(s_i^t, a_i^t) - Q^0(s_i^t, a_{a_i=a_\\textrm{default}})\\right) - Q^\\textrm{A}(s_i^t, a_i^t)\n\\end{equation}\n\\item \\textbf{Count}. The Q-table stores the number of times each state-action pair is selected by the optimiser. \n \\begin{equation}\n\\alpha\\delta = 1\n\\end{equation}\n\\end{itemize}\n\n\\section{Input Data}\\label{data}\n\n\\begin{table*}[!t]\n   \\newcommand{\\rule[-10pt]{0pt}{20pt}}{\\rule[-10pt]{0pt}{20pt}}\n   \\newcommand{\\rule[-5pt]{0pt}{30pt}}{\\rule[-5pt]{0pt}{30pt}}\n\\begin{tabularx}{\\textwidth}{ c|m{5cm}|m{5cm}}\n\\toprule\n& Normalised profile  &  Scaling factor \\\\ \n\\hline\n   PV \\rule[-10pt]{0pt}{20pt} &  Randomly selected from current month bank $b_{t+1}=(m)$   & \n   \\multirow{2}{=}{\\setlength\\parskip{\\baselineskip}%\n   Computed as $\\lambda_{t+1} = \\lambda_{t} + x$, where $x \\sim \\Gamma\\left(\\alpha(b_{t},b_{t+1}),\\beta(b_{t},b_{t+1})\\right)$} \\rule[-10pt]{0pt}{20pt} \\\\\n    \\cline{1-2}\n    Load \\rule[-5pt]{0pt}{30pt} &\\multirow{2}{=}{\\setlength\\parskip{\\baselineskip}%\nCluster selected based on transition probability $p(k_{t+1} | k_t, w_t, w_{t+1})$ \\newline Normalised profile  randomly selected from bank $b_{t+1} = (k_{t+1}, w_{t+1})$} \\rule[-5pt]{0pt}{30pt} &\n \\\\ \n\\cline{1-1}\n\\cline{3-3} \n   EV \\rule[-5pt]{0pt}{30pt} & & Random variable from discrete distribution $p(\\lambda_{t+1}|\\lambda_t, b_t, b_{t+1})$ \\rule[-10pt]{0pt}{20pt} \\\\\n \\bottomrule\n\\end{tabularx}\n  \\caption{Markov chain mechanism for selecting behaviour clusters, profiles and scaling factors for input data in subsequent days}\n\\label{tab:loadnextday}\n\\end{table*}\n\nThis section presents the data that is fed into the model presented in \\Cref{system}. Interaction with this data will shape the policies learned through RL \\citep{Sutton1998} and should reflect resource intermittency and uncertainty to maximise the expectation of rewards in a robust way without over-fitting. EV demand $\\textrm{d}_{\\textrm{EV},i}^t$ and availability $\\upmu_i^t$, PV production $\\textrm{p}_{\\textrm{PV},i}^t$ and electricity consumption $\\textrm{d}_i^{t}$ are drawn from large representative datasets.\n\n\\subsection{Data selection and pre-processing}\nLoad and PV generation profiles are obtained from the Customer Led Network Revolution (CLNR), a UK-based smart grid demonstration project \\citep{TC1a,TC5}, and mobility data from the English National Travel Survey (NTS) \\citep{DepartmentforTransport2019}. The NTS does not focus on EVs only and offers a less biased view into the general population's travel pattern than small-scale EV trials data, both due to the smaller volume of data available compared to for generic cars and because the self-selected EV early trial participants may not be representative of patterns once EVs become widely adopted. It is implicitly assumed that electrification will not affect transport patterns \\citep{Crozier2018}.\n\nNTS data from 82,455 households from 2002 to 2017 results in 1,272,834 full days of travel profiles. Load and PV data from 11,907 customers between 2011 and 2014 yields 620,702 and 22,670 full days of data, respectively.  Profiles are converted to hourly resolution and single missing points replaced with the figure from the same time the day or week before or after which has the lowest sum of squares of differences between the previous and subsequent point. Tested with available data, this yields absolute errors with mean 0.13 and 0.08 kWh and 99th percentile 1.09 and 0.81 kWh for PV and load data.  PV sources have nominal capacities between 1.35 and 2.02 kWp.\n\nThe at home-availability of the vehicles is inferred from the recorded journeys' origin and destination.  EV energy consumption profiles are obtained using representative consumption factors from a tank-to-wheel model proposed in \\citep{Crozier2018}, dependent on travel speed and type (rural, urban, motorway). \n\n\\subsection{Markov chain}\n\\begin{figure*}[!b]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{f_load_EV_omni.pdf}\n\\end{center}\n\\caption{Scaling factors for normalised profiles (i.e. total daily loads in kWh) in subsequent days. Linear correlation can be observed for the load profiles,  while more complex patterns are exhibited for EV consumption. $\\rho$ is the Pearson correlation coefficient.}\n\\label{fig:Corr}\n\\end{figure*}\n\nDuring learning, agents continuously receive experience to learn from. However,  numerous subsequent days of data are not available for single agents. We design a Markov chain mechanism to feed consistent profiles for successive days, using both consistent scaling factors and behaviour clusters.\n\nDaily profiles for load and travel are normalised such that $\\sum_{t=0..24}{x^t}=1$, and clustered using K-means, minimising the within-cluster sum-of-squares \\citep{Lloyd1982} in four clusters for both weekday and weekend data (with one for no travel). The features used for load profiles clustering are normalised peak magnitude and time and normalised values over critical time windows, and those for travel are normalised values between 6 am and 10 pm. PV profiles were grouped per month.\n\nProbabilistic Markov chain transition rules are shown in \\Cref{tab:loadnextday}. Transition probabilities for clusters $k$ and scaling factors $\\lambda$ are obtained from available transitions between subsequent days in the datasets for each week day type $w$ (week day or weekend day). \\Cref{fig:Corr} shows that subsequent PV and load scaling factors follow strong linear correlation, with the residuals of the perfect correlation following gamma distributions with zero mean, whereas EV load scaling factors follow more complex patterns, so transitions probabilities are computed between 50 discrete intervals.\n\n\\section{Case study results and discussion}\\label{results}\nThis section compares the performance of the residential flexibility coordination strategies presented in \\Cref{RLSection} to baseline and upper bound scenarios for increasing numbers of prosumers. The performance of traditionally used MARL strategies drops at scale, while that of the novel optimisation-based methodology using marginal rewards is maintained.\n\n\\subsection{Set-up}\nThe MARL algorithm is trained in off-line simulations using historical data prior to online implementation. This means agents do not trial unsuccessful actions with real-life impacts during learning. Moreover, the computation burden is taken prior to implementation, while prosumers only apply pre-learned policies, avoiding the computational challenges of large-scale real-time control. \n\nThe learning occurs over 50 epochs consisting of an exploration, an update and an evaluation phase. First,  the environment is explored over two training episodes of duration $|\\mathcal{T}| = 24$ hours.  Learning in batches of multiple episodes helps stabilise learning in the stochastic environment.  Then,  Q-tables are updated based on the rules presented in \\Cref{methodologies}. Finally, an evaluation is performed using a deterministic greedy policy on new evaluation data.  Ten repetitions are performed such that the learning may be assessed over different trajectories.\n\nThe Social Cost of Carbon is set at 70 \u00a3\/tCO$_2$, consistent with the UK 2030 target \\citep{Hirst2018}. Weather \\citep{WeatherWunderground2020}, electricity time-of-use prices \\citep{OctopusEnergy2019} and grid carbon intensity \\citep{NationalGridESO2020} are from January 2020, where relevant specified for London, UK. The low solar heat gains in January are neglected \\citep{Brown2020}. Other relevant parameters for the case studies are listed in \\Cref{app:inputs}.\n\nAs performed on a Intel(R) Core(TM) i7-9800X CPU @ 3.80GHz, computation time for a learning trajectory is $2^\\prime45^{\\prime\\prime}$ for one agent and $97^\\prime5^{\\prime\\prime}$ for 30 agents, including evaluation points. The policy can then be directly applied at the household level during operation.\n\nCase study results using different experience sources, reward definitions and MARL structures are presented in \\Cref{fig:results}. Acronyms for each strategy are tabulated in the legend. Positive values denote savings relative to a baseline scenario where all agents are passive, i.e. not using their flexibility with EVs charged immediately and no flexible loads delayed. As the Q-learning policies are first initialised with zero values, in the first epoch of learning completely random action values are chosen, which provides rewards far below the baseline. As agents collect experience and update their policies at each epoch, improved policies are learned, some of which are able to outperform the baseline. An upper bound is provided by results from \\say{omniscient} convex optimisations, which are however not achievable in practice for three main reasons. Firstly, they use perfect knowledge of all the environment variables in the present and future, despite uncertainty in renewable generation, mix of the grid, and customer behaviour. Optimisation with inaccurate data would lead to suboptimal results. Secondly, prosumers may not be willing to yield their data and direct control to an external entity. Finally, central optimisations become computationally expensive for real-time control of large numbers of prosumers.\n\n\\subsection{Results}\nResults presented in \\Cref{fig:results} show that only the algorithms learning from optimisations maintained stable coordination performance at scale, while the performance of traditionally used MARL algorithms would drop in this context of stochasticity and partial observation.  The optimisation-based algorithm which uses marginal rewards (MO) performed best.  We further elaborate on the results in the subsections below.\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_vs_nag_20211210.pdf}\n\n\n\\end{center}\n\\caption{The left-hand side plot shows the five-epoch moving average of evaluation rewards relative to baseline rewards for a single prosumer. The right-hand side plot shows the mean of the final 10 evaluations against the number of prosumers.  Lines show median values and shaded areas the 25th and 75th percentiles over the 10 repetitions. The best-performing MARL structure is displayed for each exploration source and reward definition pair.  The performance of the baseline MARL algorithm (TE, orange) drops as the number of concurrently learning agents in the stochastic environment increases; the best-performing alternative algorithm proposed (MO, purple) maintains high performance at scale.}\n\\label{fig:results}\n\\end{figure*}\n\n\\subsubsection{Environment exploration-based learning}\nThe centralised MARL structure is favoured for environment exploration-based learning (continuous lines in \\Cref{fig:results}).  A single policy uses experience collected by all agents, rather than each agent learning from their own experience only.  \n\n\\Cref{fig:results} shows that environment exploration-based MARL using total rewards (TE, orange), the baseline MARL framework, exhibits a high performance for a single agent. However, savings drop as the number of cooperating agents increases, down to around zero from ten agents.  Coordination challenges arise for independent learners to isolate the contribution of their actions to total rewards from the stochasticity of the environment, compounded by other simultaneously learning agents' random explorations, and the non-stationarity of their on-policy behaviour \\citep{Matignon2012}.  \n\nUsing advantage rewards (AE, grey), based on estimates of the long-term value of actions relative to that of the baseline action, yields superior results beyond two agents. However, as AE uses the total reward $Q^0$-table as an intermediary step, results similarly drops for increasing numbers of agents.\n\nUsing marginal rewards (ME, dark green), the value of each agent's action relative to the baseline action is singled out immediately by an additional simulation and used as a reward at each time step.  This improves the performance relative to TE and AE for five agents and more, though still with declining performance as the number of agents increases.\n\n\\subsubsection{Optimisation-based learning}\nOptimisation-based learning generally favours the distributed MARL structure, with agents able to converge to distinct compatible policies (dashed lines in \\Cref{fig:results}).\n\nComparing trajectories in \\Cref{fig:results},  learning from the total rewards obtained by an optimiser (TO, light blue) yields lower savings than when using environment explorations (TE). The learned policies yield negative savings, i.e. would provide worse outcomes than inflexible agents. The omniscient optimiser takes precise, extreme decisions thanks to its perfect knowledge of all current and future system variables, importing at very high $\\psi$ values when it is optimal to do so.  RL algorithms on the other hand are used under partial observability, aiming for actions that statistically perform well under uncertainty.  Agents independently picking TO-based decisive actions in a stochastic environment do not yield optimal outcomes. Assessing the long-term advantage of actions from optimisations (AO, dark blue) follows a similar trend, whilst providing marginally superior savings relative to TO.\n\nOptimisation-based learning using marginal rewards (MO, purple) offers the highest savings as the additional baseline simulations are best able to isolate the contribution of individual actions from variations caused by both the environment and other agents.  When increasing the number of agents, the strategy is able to learn from optimal, stable, consistently behaving agents. Savings of 6.18p per agent per hour, or \u00a345.11 per agent per month are obtained on average for 30 agents, corresponding to a 33.7\\% reduction from baseline costs.  65.9\\% of savings stem from reduced battery depreciation, 20.32\\% from distribution grid congestion, 11.1\\% from grid energy, and 2.7\\% from greenhouse gas emissions.\n\nThe count-based strategy learning from optimisations (CO, light green) seeks to reproduce the state-action patterns of the omniscient optimiser with perfect knowledge of system variables and perfect control of agents for local decision-making under partial observability.  It provides results lower than the high performances of MO, though with a stable performance at scale. Savings of \u00a321.09 per agent per month on average for 30 agents are obtained. The battery and distribution grid costs increase by an equivalent of 6.0\\% and 7.7\\% of total savings respectively, while grid energy and greenhouse gas emissions costs reductions represent 59.7\\% and 54.0\\% of total savings.\n\nBoth the MO and CO strategies exhibit stable performance at scale, though converging to different types of policy. The MO policy saves more by smoothing out the charging and distribution grid utilisation profiles despite smaller savings in imports and emissions costs, while CO derives a larger advantage from the grid price differentials in grid imports, though with higher battery and distribution grid costs. The weight applied to each of those competing objectives in the objective function directly impacts the policies that are learned. Examples of how the individual home energy management system decision variables (heating, energy consumption, battery charging) vary based on the controller are illustrated in \\Cref{app:example_case_study}.\n\nOverall, the new class of optimisation-based learning performs significantly better across different numbers of prosumers, with higher savings and lower inter-quartile range than environment-based learning at scale. This superior performance requires computations to run optimisations on historical data, and to perform baseline simulations to compute marginal rewards, though computational time for pre-learning is not strictly a limiting factor as it is performed off-line ahead of implementation. \n\nA fundamental challenge in MARL has been the trade-off between fully centralised value functions, which are impractical for more than a handful of agents, or, in a more straightforward approach, independent learning of individual action-value functions by each agent in independent Q-learning (IQL) \\citep{Tan1993}. However, an ongoing issue with this approach has been that of convergence at scale, as agents do not have explicit representations of interactions between agents, and each agent's learning is confounded by the learning and exploration of others \\citep{Rashid2020}. As shown in \\Cref{fig:results}, the Pareto selection, non-stationarity and stochasticity issues presented in \\Cref{sec:gapanalysis} have prevented environment exploration-based learners from achieving successful MARL cooperation at scale for agents under partial observability in a stochastic environment. This case study of coordinated residential energy management shows that the novel combination of marginal rewards, which help agents isolate their marginal contribution to total rewards, and the learning from results of convex optimisations, where agents learn successful policy equilibriums from omniscient, stable, and consistent solutions, offer significant improvements on these scalability and convergence issues.  \n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, a novel class of strategies has addressed the scalability issue of residential energy flexibility coordination in a cost-efficient and privacy-preserving manner.  The combination of off-line optimisations with multi-agent reinforcement learning provides high, stable coordination performance at scale.\n\nWe identified in the literature that the concept of RL-based implicit energy coordination, where energy prosumers cooperate towards global objectives based on local information only, had been under-researched beyond frequency droop control with limited number of agents. The scalability of such methods was identified as a key gap that we have sought to bridge. The novel coordination mechanism proposed in this paper thus satisfies the criteria for successful residential energy coordination set out in the introduction, as tested with large banks of real data in the case studies:\n\n\\begin{itemize}\n\n\\item Computational scalability: The scalability of traditional learning algorithms is significantly improved thanks to fixed-size Q-tables to avoid the curse of dimensionality, so that policies can be learned for larger number of agents. The proposed method does not require expensive communication and control appliances at the prosumer level, as pre-learned policies are directly applied with no further communication and no exponential time real-time optimisations needed. This is a crucial benefit for applications with physical limitations in hardware availability and processing time.  \n\n\\item Performance scalability: The coordination performance remains high for increasing numbers of prosumers despite the challenges of partial observability,  environment stochasticity and concurrently learning of agents, thanks to learning from the results of global omniscient optimisations on historical data, and to rewards signals that isolate individual contributions to global rewards. Significant value of \u00a345.11 per agent per month was obtained in the presented case study for 30 agents,  thanks to savings in energy,  prosumer storage and societal greenhouse gas emissions-related costs. Those savings do not drop with increasing number of agents, as opposed to with standard MARL approaches.\n\n\\item Acceptability: The approach does not rely on sharing of personal data, thermal discomfort, or hindrance\/delay of activities, and the appliances are controlled locally. This cost-efficient and privacy-preserving implicit coordination approach could help integrate distributed energy resources such as residential energy, otherwise excluded from energy systems' flexibility management. \n\n\\end{itemize}\n\nImportant future work is a more detailed assessment of the impacts of the coordination strategies on power flows, as well as an evaluation of the generalisation and adaptability potential of policies when used by other households or if household characteristics change over time.  Moreover, while all agents readily reduce individual costs through participation in the framework, further game-theoretic tools could be used to design a post-operation reward scheme.\n\n\\section*{Acknowledgement}\nThis work was supported by the Saven European Scholarship and by the UK Research and Innovation and the Engineering and Physical Sciences Research Council (award references EP\/S000887\/1, EP\/S031901\/1, and EP\/T028564\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOver the last decade topological phases of matter have attracted many attentions for providing tremendous insights both in fundamental and experimental aspects of condensed matter physics \\cite{Chiu2016}. In addition to the gapped phases which initially ignited the field, recently discovery of gapless topological phases stimulated many works towards the understanding of nontrivial topology of gapless systems \\cite{review1}. Among those, Weyl and Dirac semimetals are of particular interest due to their experimental discovery and many unique physical properties \\cite{review1}. The Dirac\/Weyl smimetals (DSMs\/WSMs) are characterized by isolated point touchings of two degenerate\/nondegenerate bands in momentum space. Weyl nodes can be generated by splitting of degenerate Dirac nodes usually via breaking of either time-reversal ($\\mathcal{T}$) or inversion ($\\mathcal{I}$) symmetries or both. In the standard form, the low energy excitations near the Weyl nodes, disperse linearly along all three momentum directions with each node carries monopole charge of $\\pm 1$. As a result, on the surface, there exists a Fermi arc that connects a pair of Weyl nodes with opposite chiralities. Recently, generalization of WSMs to multi-Weyl nodes have been proposed where each nodes have higher-order dispersion in one or more directions and consequently caries monopole charge of larger than one \\cite{MultiWeyl1,MultiWeyl2}. \\\\\n\\indent On the other hand, DSMs\/WSMs can also be classified into type-I and type-II, based on the tilting of their nodes. In the standard type-I WSMs, Fermi surfaces are point-like while in the type-II WSMs, Weyl nodes are tilted resulting in formation of electron and hole pockets producing finite density of states at the Fermi level \\cite{typeII1,typeII2}. In a conventional type-I and II WSMs, two Weyl nodes with opposite chiralities have same types, however, recently, a theoretical proposal \\cite{hybridWeyloriginal}, introduced a new WSM where a pair of Weyl nodes with different chiralities can have different types, forming the so-called \\emph{hybrid Weyl semimetals}.\\\\\n\\indent Besides the DSMs\/WSMs, another class of three-dimensional nodal semimetals are Luttinger semimetals (LSMs) where possess a quadratic band touching (QBT) point between doubly degenerate valence and conduction bands of $J=3\/2$ (effective) fermions at an isolated point in the Brillouin zone.\nThe LSM provides the low-energy description for a plethora of both strongly and weakly correlated compounds, such as the 227 pyrochlore iridates (Ln2Ir2O7, with Ln being a lanthanide element) \\cite{LSMapp1,LSMapp2, LSMapp3,LSMapp4}, half-Heusler compounds (ternary alloys such as LnPtBi, LnPdBi) \\cite{LSMapp5,LSMapp6}, HgTe \\cite{semiconductor1,semiconductor2,QBToriginal,Ruan2016,mottQBT}, and gray-tin \\cite{LSMapp7,LSMapp8}.\nMoreover, LSMs proved to show many interesting behaviors\\cite{LSMprop1,LSMprop2,LSMprop3,LSMprop4,LSMprop5}, specially in the presence of interaction, for example, investigation of the magnetic and superconducting orders actively have been explored \\cite{LSMapp9, Roy-Ghorashi2019, GhorashiPRB2017, GhorashiPRL2018, Ghorashi2019,SatoSC32,LSMBoettcher2,LSMSCLiu1,LSMBoettcher1,Szabo-Bitan32-2018}.\\\\\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{FlashpickickedLSMv4.png}\n    \\caption{Schematic picture of summary of the results obtained in this work. Starting from a Luttinger semimetal via two different nonuniform ($k$-dependent) periodic kickings, where break inversion ($\\mathcal{I}$) and time-reversal ($\\mathcal{T}$) symmetries while preserving their combinations ($\\mathcal{IT}$), we have obtained a \\emph{hybrid dispersion Dirac semimetals} (e.g., $k_z\\Gamma_5$ where $\\Gamma_i$ are Dirac matrices [see text for details]) and tilted LSM (e.g., $k_z\\Gamma_0$). Then, by applying an external magnetic field, $J_z$, in parallel to the direction of nodes (or kick direction), \\emph{hybrid Weyl phases} can be generated.}\n    \\label{fig:adpic}\n\\end{figure*}\n\\indent The coexistence of various Weyl nodes with different charges and\/or types is an interesting phenomena that could help towards understanding as well as manipulation of the properties of various Weyl nodes in an equal footing setup. Besides a few works reporting the coexistence of type-I and II Dirac\/Weyl nodes \\cite{hybridWSMexp,hybridDSM1,hybridWSM1,WSMexp1&2,weylcoexPRL}, there have been also proposals\\cite{LSMFloq1,Ghorashi2018}, claiming dynamical generation of various Weyl nodes of different types and\/or charges in one system. Application of the light is shown to be a powerful method to change the material properties \\cite{FloquetRev}. In particular, the conversion of a topologically trivial phase into a nontrivial one using periodic driving has attracted enormous attention in the past decade \\cite{FloquetRev,Floquet1, Floquet2, Floquet3}. Specifically many proposals on Floquet WSMs in various systems exist, such as Dirac semimetals \\cite{FloqWeyl1,FloqWeyl2}, band insulators \\cite{FloqWeyl3}, stacked graphene \\cite{FloqWeyl4}, line-nodal semimetals \\cite{FloqWeyl2,FloqWeyl5}, and crossing-line semimetals \\cite{FloqWeyl6,FloqWeyl7}. Also, proposals have been made to create tunable WSMs in pyrochlore iridates with Zeeman fields \\cite{Weylpyro1,Weylpyro2}. Very recently, using circular\/elliptic polarized light on Luttinger Hamiltonian in the high-frequency limit we have shown a very rich phase diagram of various Weyl semimetals, including coexistence of type-I and II as well as single and double Weyl nodes \\cite{Ghorashi2018} . \\\\\n \\indent Despite the several dynamical proposals for the generation of different Weyl phases, a promising setup for the realization of hybrid Dirac and Weyl semimetals is still lacking. In this work, we tackle this issue by an alternative way of periodic driving, in particular the periodic kicking. Using the periodic $\\delta$-function kicks can typically simplify theoretical studies by allowing to perform calculations analytically to a large extent (in contrast to sinusoidal driving or elliptical\/circular light) \\cite{ Floquet3,kicking2}.\nHowever, for the sake of comparison we also briefly discuss the smooth driving case to show that some of the features of our discussion can be hold up in smooth driving setup as well as long as the perturbation breaks inversion (uniaxially) and time-reversal symmetres but preserve their combinations. In this paper, we show two examples of such perturbations which along with an external magnetic field induce various hybrid Dirac and Weyl phases, including a new \\emph{hybrid dispersion Dirac semimetal}. Figure.~\\ref{fig:adpic} summarizes the result of this work.\n\n\\section{Model and Formalism}\n\\subsection{Model}\nWe start with reviewing the main ingredients of Luttinger Hamiltonian in the non-equilibrium limit, which can be represented as,\n\n\\begin{align}\n    H_{L}(\\vec{k})=\\int \\frac{d^3\\vec{k}}{(2\\pi)^3}\\Psi^{\\dagger}_{\\vec{k}} h_L(\\vec{k}) \\Psi_{\\vec{k}},\n\\end{align}\nwhere\n\\begin{align}\n    h_{L}(\\vec{k})=&(\\frac{k^2}{2m_0}-\\mu)\\Gamma_0-\\frac{1}{2m_1}\\sum^3_{a=1} d_a(\\vec{k})\\Gamma_a\\cr\n    -&\\frac{1}{2m_2}\\sum^5_{a=4} d_a(\\vec{k})\\Gamma_a\n\\end{align}\nwhere $k^2=k^2_x+k^2_y+k^2_z$ and,\n\\begin{align}\n    \\Psi^T_{k}= (c_{\\vec{k},3\/2}, c_{\\vec{k},1\/2}, c_{\\vec{k},-1\/2}, c_{\\vec{k},-3\/2}).\n\\end{align}\n$\\mu$ is the chemical potential measured from the band touching point. $\\Gamma_a$ are the well-known gamma matrices which are given by,\n\\begin{align}\n    \\Gamma_1=\\tau_3\\sigma_2,\\,\\, \\Gamma_2=\\tau_3\\sigma_1,\\,\\, \\Gamma_3=\\tau_2,\\,\\,\n    \\Gamma_4=\\tau_1,\\,\\, \\Gamma_5=\\tau_3\\sigma_3,\n\\end{align}\n and satisfy $\\{\\Gamma_a,\\Gamma_b\\}=\\delta_{a,b}$, while $\\Gamma_0$ is four dimensional identity matrix. $\\tau$ and $\\sigma$ denote space of sign and magnitude of spin projection $m_s\\in\\{\\pm 3\/2, \\pm 1\/2\\}$, respectively. $d_a(\\vex{k})$ are given as,\n \\begin{align}\n     d_1=\\sqrt{3}k_y k_z,\\,\\, d_2=\\sqrt{3}k_x k_z,\\,\\, d_3=\\sqrt{3}k_x k_y,\\,\\,\\cr\n     d_4=\\frac{\\sqrt{3}}{2}(k_x^2-k_y^2),\\,\\, d_5=\\frac{1}{2}(2k_z^2-k_x^2-k_y^2)\n \\end{align}\n The five mutually anticommuting $\\Gamma$ matrices can be written\nin terms of spin-$3\/2$ matrices according to,\n\\begin{align}\n    \\Gamma_1=&\\frac{1}{\\sqrt{3}}\\{J_y,J_z\\},\\,\\,\\Gamma_2=\\frac{1}{\\sqrt{3}}\\{J_x,J_z\\},\\,\\,\\Gamma_3=\\frac{1}{\\sqrt{3}}\\{J_x,J_y\\},\\cr\n    \\Gamma_4=&\\frac{1}{\\sqrt{3}}(J^2_x-J^2_y),\\,\\,\\Gamma_5=J^2_z-\\frac{5}{4},\n\\end{align}\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveup.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveopposite.pdf}\n    \\caption{The band structure for LSM along the $k_z$ axis with $\\lambda_1=0.6$ and (a) $\\lambda_2=0.1$, (b) $\\lambda_2=0.6$.}\n    \\label{fig:LSMbare}\n\\end{figure}\n Here we take the isotropic limit of $m_1=m_2 \\equiv m$. Therefore, the Luttinger Hamiltonian can also be written in an alternative way,\n\\begin{align}\n    h_L(\\vec{k})= [(\\lambda_1+5\\lambda_2\/2)k^2-\\mu]\\Gamma_0-2\\lambda_2(\\vec{J}.\\vec{k})^2\n\\end{align}\n with $\\vec{J}=(J_x,J_y,J_z)$ and $\\vec{k}=(k_x,k_y,k_z)$ and we used $\\lambda_1=1\/2m_0$ and $\\lambda_2=1\/4m$. $J_{x,y,z}$ are effective spin-$3\/2$ operators,\n \\begin{eqnarray}\n&\\,J_z=\\begin{bmatrix}\n        \\frac{3}{2} & 0 & 0 & 0 \\\\\n        0 & \\frac{1}{2} & 0 & 0\\\\\n        0 & 0 & -\\frac{1}{2} & 0 \\\\\n        0 & 0 & 0 & -\\frac{3}{2} \\\\\n      \\end{bmatrix},J_x=\\begin{bmatrix}\n        0 & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & 1 & 0\\\\\n        0 & 1 & 0 & \\frac{\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}\\cr\n      &\\,J_y=\\begin{bmatrix}\n        0 & \\frac{-i\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & -i & 0\\\\\n        0 & 1 & 0 & \\frac{-i\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}.\n\\end{eqnarray}\nThe energy dispersions are $E(k)=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}-\\mu$ for the $j=3\/2$ and the $j=1\/2$ bands, respectively. Four bands come in doubly degenerate pairs as a result of time-reversal (with antiunitary operator $\\mathcal{T}=\\Gamma_1\\Gamma_3\\mathcal{K}$ and $\\vec{k}\\rightarrow -\\vec{k}$ where $\\mathcal{K}$ is complex conjugation) and inversion ($\\mathcal{I}=I_{4\\times4}$ and $\\vec{k}\\rightarrow -\\vec{k}$) symmetries. For $\\lambda_{2}<2\\lambda_{1}$\n($\\lambda_{2}>2\\lambda_{1}$), the degenerate bands curve the same (opposite) way as shown in Figure.~(\\ref{fig:LSMbare}). In the case of both bands bending the same way, Eq. (2) is widely used to model heavy- and light-hole bands in zinc-blende semiconductors \\cite{semiconductor1} and many properties of such a dispersion have been studied in the literature, including a recent study on the realization of fully gapped topological superconductivity with \\emph{p}-wave pairing which has states with exotic cubic and linear dispersions coexisting on the surface \\cite{congjunWuPRL16,GhorashiPRB2017}. On the other hand, when bands bend oppositely, the model in Eq. (2) is known as Luttinger semimetal with QBT and is used to describe behavior of certain pyrochlore iridates as well as some doped half-Heusler alloys such as LaPtBi \\cite{Chadov2010,Lin2010,halfheusler3}.\n\n\n\\subsection{Periodic driving}\nA general time-dependent problem with $H(t)=H_0+V(t)$, can be tackled using Floquet theory when $V(t+T)=V(t)$ is periodic. To proceed, we can expand the periodic potential in a Fourier series as\n\\begin{align}\n    V(t)=V_0+\\sum^{\\infty}_{n=1} \\big(V_n e^{i\\omega nt}+V_{-n}e^{-i\\omega nt}\\big).\n\\end{align}\n In the limit of fast driving regime, in which the driving frequency is larger than any natural energy scale in the problem, one can obtain the effective Hamiltonian and Floquet operators perturbatively up to  $\\mathcal{O}(1\/\\omega^2)$ \\cite{Floquet3}. The Floquet operator, $\\mathcal{F}(t)$, is the unitary time-evolution $\\hat{U}(t)$ after one period of drive can be factorized as,\n\\begin{align}\n    \\mathcal{F}(t)=\\exp[-i \\alpha\\Lambda(\\vec{k})]\\exp[-i H_L T]=\\exp[-i H_{eff} T].\n\\end{align}\nSo the dynamics can have three stages: initial kick at $t_i$, the evolution of system with $H(t)$ in the interval $t_f-t_i$ and final kick at $t_f$ which describes the \"micromotion\" \\cite{Floquet3}. Then, the time-evolution operator can be expressed as,\n\\begin{align}\n    \\hat{U}(t_i\\rightarrow t_f)= \\hat{U}(t_f)^{\\dagger}e^{-iH_{eff}(t_f-t_i)}\\hat{U}(t_i),\n\\end{align}\nwhere $\\hat{U}(t)=e^{-i\\mathcal{F}(t)}$. $\\mathcal{F}(t)$ is a time-periodic operator with zero average over one period.  Lets set $t_i=0$, then $H_{eff}$ and $\\mathcal{F}(t)$ can be expanded as,\n\\begin{align}\n    H_{eff}=\\sum_{n=0}^{\\infty}\\frac{1}{\\omega^n}H_{eff}^n,\\,\\,\\mathcal{F}(t)=\\sum_{n=1}^{\\infty}\\frac{1}{\\omega^n}\\mathcal{F}^n.\n\\end{align}\nA generic quantum $\\delta$-kick can be implemented with following perturbation,\n\\begin{align}\n    H_{kick}(t)=\\alpha \\Lambda(\\vec{k}) \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)\n\\end{align}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5p_05_kz.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5m_05_kz.pdf}\n    \\caption{The band structure of LSM along the $k_z$ axis in the presence of uniform strain ($\\alpha \\Gamma_5$) with (a) $\\alpha=0.5$ (tensile) and (b) $\\alpha=-0.5$ (compression). $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used.}\n    \\label{fig:G5}\n\\end{figure}\nwhere $T=2\\pi\/\\omega$, $\\alpha$ is the kicking strength and $\\Lambda(\\vec{k})$ is the matrix representation of a perturbation, which in general can be a function of momentum, $\\vec{k}$, and could be used to mimic a nonuniform kicking. Following a perturbative expansion the effective Hamiltonian for $\\delta$-kick, with $H_{kick,n}=\\alpha \\Lambda(\\vec{k})\/T\\,\\text{for all}\\,\\,n$, can be obtained as \\cite{Floquet3,kicking2},\n\\begin{align}\\label{effkick}\n    H^{kick}_{eff}=&\\,H_L+\\frac{\\alpha\\Lambda(\\vec{k})}{T}+\\frac{1}{24}[[\\alpha\\Lambda(\\vec{k}),H_L],\\alpha\\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n   \n   \n\\end{align}\nWe note that the third term in Eq.~(\\ref{effkick}), can be dropped in the limit of weak kicks, i.e, to the first order of kick strength. Therefore, a dynamical kicking effectively is equivalent to directly adding the kick term as a perturbation to the bare Hamiltonian, something which is not always relevant in the equilibrium. In the rest of this work we only focus on the physics of the effective Hamiltonian and we leave \"micromotion\" physics for future studies. \\\\\nSimilarly, the effective Hamiltonian for the case of smooth driving of the form $H(t)=H_0+\\alpha \\Lambda(\\vex{k})\\cos(\\omega t)$ can be obtained as \\cite{Floquet3},\n\\begin{align}\\label{effsmooth}\n    H^{cos}_{eff}=H_0+\\frac{1}{4\\omega^2}[[\\alpha \\Lambda(\\vec{k}),H_0],\\alpha \\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n\\end{align}\nUnlike Eq.~(\\ref{effkick}) of periodic kicking, the first correction to the $H_0$ in the case of smooth driving is $\\mathcal{O}(\\alpha^2)$.\\\\\n\\indent In the following we first investigate the effect of uniform kicking ($k$-independent) and show that some of the known results can be retrieved. Then we turn to nonuniform periodic kicking and show that by introducing $k$-dependent $\\mathcal{IT}$ symmetric kicks, interestingly,  different hybrid Dirac\/Weyl semimetals can be obtained. Finally, we briefly compare the case of smooth driving of Eq.~(\\ref{effsmooth}) with the results obtained by periodic kicking.\n\n\n \\section{Uniform kicking}\nThere are many possibilities of uniform kicking, including: five $\\Gamma_j$, and ten commutators $\\Gamma_{ij}=[\\Gamma_i,\\Gamma_j]\/(2i)$ with $i > j$ , which can be expressed in terms of the products of odd number of spin-$3\/2$ matrices. Here we focus on uniform kicks which are proportional to $\\Lambda=\\Gamma_j$. The effective Hamiltonian for such kicks, in a sufficiently weak kick limit, is given by,\n\\begin{align}\\label{effG5}\n    H^j_{eff}=h_L(\\vec{k})+\\frac{\\alpha}{T}\\Gamma_j,\n\\end{align}\nwith spectrum,\n\\begin{align}\n    E^j_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2\\sum_i d^2_i T^2 - 4\\lambda_2\\alpha T d_j+\\alpha^2}}{T},\n\\end{align}\n\nOf particular interest is the $\\Gamma_5$ kicking, that breaks the rotational invariance and can be thought of as the effect of an external strain in $z$ direction \\cite{Ruan2016,QBToriginal}. It is known that a uniaxial strain, can realize two different phases in LSMs depending on the sign of $\\alpha$ \\cite{Ruan2016,QBToriginal,Weylpyro1,Weylpyro2,LSMmagnetic}. Figure.~\\ref{fig:G5}a(b) shows the bandstructure corresponding to effective Hamiltonian of Eq.~(\\ref{effG5}) with $\\Gamma_j=\\Gamma_5$ and for $\\alpha > 0$ ($\\alpha<0$) which represents a Dirac semimetal (trivial\/topological insulator).\n\nIn the case of a tensile strain, there are two Dirac nodes at $k_z=\\pm\\frac{\\alpha}{2\\lambda_2 T}$ ($\\alpha > 0 $). Now if we add an external magnetic field, we expect that each Dirac nodes split to two Weyl points and form Weyl semimetal phases due to breaking of $\\mathcal{T}$ symmetry. Interestingly, even in the topological insulator limit of $\\alpha<0$, one can realize Weyl semimetals depending on the strength of magnetic field \\cite{LSMmagnetic}. We also note that magnetic field alone can drive a Luttinger semimetal to a Weyl semimetal phase \\cite{LSMmagnetic,Weylpyro2}. Moreover, in the experimentally relevant system of pyrochlore irridates, it is shown that magnetic field can give rise to rich phase diagram of topological semimetals \\cite{Weylpyro2}. It is important to mention that an external magnetic field can have orbital effects for sufficiently large strength. However, here and in the rest of this paper, we neglect the orbital effects of magnetic field for sake of simplicity following many works in the literature \\cite{Weylpyro1, Weylpyro2,LSMFloq1} and we refer readers to some of the works which explored the effects of magnetic field in Landau levels and quantum oscillations \\cite{LSMmagnetic,LSMquantumoscillation}. The detailed analysis of the effect of magnetic field (also possible pseudomagnetic fields due to other perturbations) will be left for future studies.\n\n\\section{Nonuniform kicking}\nNow lets turn to the main part of this work, where we investigate the effect of nonuniform kicks. Similar to the uniform kicking discussed in the previous section, we can consider numerous possibilities of nonuniform driving. However, here, we restrict ourselves to a \"linear in momentum\" kicks of the form $k_i\\Gamma_i$ where $\\Gamma_i$ are time-reversal even. The very first consequence of such choice of the kicks, is the breaking of both $\\mathcal{T}$ and $\\mathcal{I}$ symmetries, while preserving their combinations $\\mathcal{IT}$, which is a necessary symmetry for keeping double degeneracy of the bands intact. Specifically, we focus on two types of nonuniform kicking: (a) $k_z\\Gamma_5$ and (b) a $k_z\\Gamma_0$.\n\n\\begin{figure*}[htb]\n \\centering\n    \\includegraphics[width=0.42\\textwidth]{Fig4a.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz05_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz07_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz11_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz15_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz20_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz30_kz-R.pdf}\\includegraphics[width=0.42\\textwidth]{kzG5_PH_kxkz.pdf}\n    \\caption{The band structure of LSM in the presence of $k_z\\Gamma_5$ with kick strength of $\\alpha=0.8$  and (a) $h=0$, $k_y-k_z$ plane showing quadratic dispersion along the $k_y$ for the node at $\\Gamma$ point, (b) $h=0.5$, (c) $h=0.7$, (d) $h=1.1$, (e) $h=1.5$, (f) $h=2$, (g) $h=3$ and (h) same as (a) but with $\\lambda_1=0$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots (except (h) where $\\lambda_1=0$).}\n    \\label{fig:kzG5}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_5$}\n\nAs it is mentioned above the $\\Gamma_5$ can be interpreted as the effect of strain in $k_z$ directions \\cite{Ruan2016}, similarly, $k_z\\Gamma_5$ also breaks the rotational invariance, then in principle we might still think of $k_z\\Gamma_5$ term as a type of nonuniform strain or strain gradient, even though full microscopic derivation of such strain could shed more light. The presence of $k_z$ suggests that microscopically on a lattice, $\\Gamma_5$ now acts non-local and directly modifies hopping terms. On the hand, one can convert the $k_z\\Gamma_5$ to the lattice model and explicitly write it down as, $\\sum_{\\sigma\\in\\{1\/2,3\/2\\}}(C^{\\dagger}_{k_{\\perp},i,\\sigma,\\uparrow}C_{k_{\\perp},j,\\sigma,\\uparrow}-C^{\\dagger}_{k_{\\perp},i,\\sigma,\\downarrow}C_{k_{\\perp},j,\\sigma,\\downarrow})\/2i+h.c$. This is similar to the spin current operator along $z$-axis. Therefore, we can also think of the $k_z\\Gamma_5$ kick as an applied spin current (or related to it) along $z$ direction. However, it is essential to note that besides the rotational invariance the $k_z\\Gamma_5$ kick also breaks inversion and time-reversal symmetries while preserving their combinations. Then the bands of driven system are still degenerate. Therefore, here, without restricting ourselves to any particular microscopic interpretation, we emphasize that to achieve the results obtained in this section one need perturbations which break both $\\mathcal{I}$ and $\\mathcal{T}$ but preserve $\\mathcal{IT}$.\\\\\nThe effective Hamiltonian for $k_z\\Gamma_5$ kick can be written as, $H_{eff}=h_{L}(\\vec{k})+\\frac{\\alpha}{T}k_z\\Gamma_5$, with following spectrum,\n\\begin{align}\n    E_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2T^2\\sum_i d^2_i - 4\\lambda_2\\alpha T k_z d_5+\\alpha^2 k^2_z}}{T}.\n\\end{align}\nFigure.~\\ref{fig:kzG5}, shows the spectrum in $k_x-k_z$ plane. There are two Dirac nodes at $k_z=0$ and $k_z=\\frac{\\alpha}{2\\lambda_2 T}$. While one of the nodes is always pinned to $\\Gamma$ point the other node can be tuned by kick parameters. Therefore, for fixed set of parameters the distance between two nodes for $k_z\\Gamma_5$ is half of the case with uniform strain $\\Gamma_5$. Moreover, we observe three main differences in compare to the case with uniform strain as is shown in Figure.~\\ref{fig:G5}. (i) There is no major difference between the tensile ($\\alpha >0 $) and  compressive ($\\alpha <0 $) strain. In both cases system drives into a Dirac semimetal phase. (ii) Unlike the uniform tensile strain, due to broken inversion symmetry two Dirac nodes reside at different energies and (iii) the most important of all, remarkably, one of the nodes is quadratic while the other is linear, realizing an unique \\emph{hybrid Dirac semimetal}. Interestingly, the \"quadratic\" Dirac node (or a QBT which is linearly dispersed in $k_z$ direction) is pinned at the $\\Gamma$ point and the \"energy difference\" as well as distance between the nodes can be controlled with kick strength. Moreover, while the linear Dirac node is tilted, the node centered around the $\\Gamma$ point no matter how strong is the kick, shows no tilt. It should be emphasized, here, we define hybrid Dirac semimetal mainly based on the different dispersion of two Dirac nodes instead of their tilts, even though as we discussed above, they show different tilting (and types) as well. Moreover, We note that such hybrid phase is unique to the Dirac semimetals because it is impossible to realize a Weyl semimetal with pair of nodes with different dispersion or (magnitude of monopole charges).\\\\\n\\indent Next, lets apply an external magnetic field $h J_z$ where $h$ denotes the strength of magnetic field. The magnetic field splits Dirac nodes to Weyl nodes in $k_z$ direction. In the presence of a magnetic field an analytical expression of eigenvalues for the general $\\lambda_1,\\lambda_2$ can not be achieved, so in the following we proceed by solving the model numerically. Figure.~\\ref{fig:kzG5} depicts the evolution of the Weyl nodes by strength of external magnetic field. Starting from $h \\ll \\alpha $, there are 8 nodes on $k_z$ axis, four single and four double Weyl nodes, which are indicated in Figure.~\\ref{fig:kzG5} by the red and blue dots, respectively. Interestingly, we observe that one or more of the Weyl pairs realize a \\emph{hybrid Weyl phases} and they can survive up to a decently strong magnetic field (Figure.~\\ref{fig:kzG5}(b-f)). While the other Weyl nodes do not possess hybrid types, they show different tilts for each nodes of the same pair. Only at a very strong magnetic field (Figure.~\\ref{fig:kzG5}(g)) all of the Weyl points show a type-I structure.\nMoreover, we see that by increasing the magnetic field, from initial eight nodes, four of them merge and gap out and we left off with only four nodes (two single and two double nodes) at $h>>\\alpha$ limit. \\\\\n\\indent Finally, we comment on the particle-hole symmetric limit of $\\lambda_1=0$. In this limit we can get energies analytically even in presence of an external magnetic field as,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nwith 8 nodes located at,\n\\begin{align}\n    k^{\\pm,\\pm}_{z,I}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 4h\\lambda_2T^2}}{4\\lambda_2 T},\\cr\n    k^{\\pm,\\pm}_{z,II}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 8h\\lambda_2T^2}}{4\\lambda_2 T}.\n\\end{align}\nAs is clear from the above equations, four out of the eight nodes, $k^{\\pm,-}_{z,I,II}$, gap out by increasing $h$. However, it is noteworthy that even though a \\emph{hybrid dispersion Dirac semimetal phase} can still be generated, but now in the limit of $\\lambda_1=0$, the two nodes have the same energies Figure.~\\ref{fig:kzG5}(h), therefore, no longer a hybrid Weyl phase can be achieved in the presence of an external magnetic field.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_kz.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz02.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz04.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz06.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz08.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz15.pdf}\n\\caption{The band structure of LSM with tilted QBT with kick strength of $\\alpha=0.6$  and (a) $h=0$, (b) $h=0.2$, (c) $h=0.4$, (d) $h=0.6$, (e) $h=0.8$ and (f) $h=1.5$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots.}\n\\label{fig:kzG0}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_0$: Tilted QBT}\n\nThe second class of nonuniform kicks which we investigate in this work, is $\\Lambda=k_z\\Gamma_0$. The effective spectrum for such driving can be obtained as,\n\\begin{align}\\label{tQBT}\n    E_{\\pm}(\\vec{k})=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}+\\frac{\\alpha k_z}{T}.\n\\end{align}\nThe mere effect of such driving is to effectively tilt the QBT in $k_z$ direction as shown in Figure.~\\ref{fig:kzG0}(a). However, the tilted QBT behaves very differently in the presence of an external perturbation. To clarify further, we rewrite the Eq.~(\\ref{tQBT}) at $k_x=k_y=0$ as,\n\\begin{align}\n     E_{\\pm}(\\vec{k})=&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^{2}+\\frac{\\alpha k_z}{T}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})(k_z+\\mathcal{A}_z)^{2}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z+(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}^2_z\\cr\n     \\simeq&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z,\n\\end{align}\nwhere in the last line we assumed $\\mathcal{A}_z<1$. Now by comparing the first and the last line of the above equation, we obtain $\\mathcal{A}_z=\\frac{\\alpha}{2(\\lambda_{1}\\mp2\\lambda_{2})T}$. Therefore, we can think of a tilted QBT as a QBT with a pseudo-electromagnetic potential $\\mathcal{A}= (0, 0, \\mathcal{A}_z)$ which is proportional to kick strength. In particular, interesting nontrivial topological phases can emerge by applying an external magnetic field in the presence of tilt, which otherwise are absent in a conventional QBT. As we mentioned previously, it is known that in the presence of a magnetic field, the QBT splits to multiple Weyl points. However, when the QBT is tilted, due to the competition between the external magnetic field and pseudo-magnetic field due to the $\\mathcal{A}$ (proportional to kick strength) there are various regimes which different types of WSM phases can be generated. Starting with the weak field limit ($\\alpha >> h$), there are two pairs of nodes, two double and two single nodes. In this regime, all Weyl nodes have a type-II nature (Figure.~\\ref{fig:kzG0}(b)). By further increasing of the field, for intermediate fields ($h \\lesssim \\alpha$), hybrid Weyl pairs are realized where one of nodes in each of single and double pairs are type-II and the others are type-I. Interestingly, the realized hybrid WSM can survive up to a very strong magnetic field. This could be experimentally beneficent as it demonstrates that hybrid WSMs generated here, are accessible in a broad range of external fields. Similar to the $k_z\\Gamma_5$ kick, we can get the analytical expression of energies and nodes position in the particle-hole limit,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z+\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nand,\n\\begin{align}\n    k^{\\pm}_{z,I}=\\pm\\sqrt{\\frac{h}{2\\lambda_2}},\\,\\,k^{\\pm}_{z,II}=\\pm\\frac{\\sqrt{h}}{2\\sqrt{\\lambda_2}}=\\frac{k^{\\pm}_{z,I}}{\\sqrt{2}}.\n\\end{align}\nUnlike the $k_z\\Gamma_5$ kick, the presence of particle-hole symmetry does not prevent the generation of hybrid Weyl phases, because does not generate any other Dirac nodes and as we mentioned before only tilts the QBT. Moreover, the position of Weyl nodes are independent of kick strength and can be tuned solely using magnetic field.\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kxkz.pdf}\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kykz.pdf}\n    \\caption{The bandstructure of LSM at $k_x-k_z$ plane ($k_y=0$) in presence of $k_z\\Gamma_5$ smooth driving with $\\alpha=5$. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:kzG5smooth}\n\\end{figure}\n\n\\subsection{Smooth driving}\nHere, we briefly discuss the case of smooth driving using Eq.~(\\ref{effsmooth}) and make comparison with the results obtained by periodic kicking in previous sections. First of all, any type of perturbation which is proportional to the identity will not modify the system in the case of smooth driving, as it obviously commutes with Hamiltonian. However, the $\\Lambda(\\vec{k})=\\alpha k_z\\Gamma_5$ can modify the system when is applied via smooth driving. We obtain the effective Hamiltonian as,\n\\begin{align}\n    H^{cos}_{eff}(\\vec{k})=&H_0+\\frac{\\lambda_2\\alpha^2k^2_z}{\\omega}\\bigg(2k_xk_z\\{J_x,J_z\\}+2k_yk_z\\{J_y,J_z\\}\\cr\n    +&(k^2_x-k^2_y)(J^2_x-J^2_y)+2k_xk_y\\{J_x,J_y\\}\\bigg).\n\\end{align}\nwith energies,\n\\begin{align}\n    E^{\\pm}(\\vec{k})=\\lambda_1 k^2\\pm \\frac{\\lambda_2\\sqrt{f(\\vec{k})\\left[k^2_z\\alpha^2-2\\omega^2\\right]+4k^4\\omega^4}}{\\omega^2},\n\\end{align}\nwhere each $f(\\vec{k})=3 (k_x^2+k_y^2)k_z^2 (k_x^2+k_y^2 +4k_z^2)\\alpha^2$ and $E^{\\pm}$ are doubly degenerate. Figure.~\\ref{fig:kzG5smooth} shows the spectrum for $k_y=0$ plane, where four linear Dirac nodes coexist with a QBT at which is located at $\\Gamma$ point. A similar plot can be obtained for $k_x=0$ plane, then the system also possesses line-nodes in $k_x-k_y$ plane at $k_z\\neq 0$. Therefore, in addition to the coexistence of Dirac nodes and QBT at $k_x-k_z$ and $k_y-k_z$ planes, a smooth driving can lead to a richer phase diagram. This analysis was only for the purpose of comparison with kicked driving results, so a detailed analysis of such models will be left for future works.\n\n\\subsection{Effect of Lattice Regularization}\n\nLets now look at the effect of lattice regularization. One of the typical effect of lattice versus continuum model is the appearance of more nodes, usually at the boundaries of the Brillioun Zone \\cite{Ghorashi2018}. However, we show that the main features discussed in the previous subsections survives in lattice models. For the sake of brevity we restrict ourselves to $h=0$ limit. We consider a cubic lattice, which captures the physics of continuum model for $\\alpha,\\omega,\\vec{k}<<1$. In the $k_z$ cut (Figure.~\\ref{fig:latt}), we find: (i) two nodes at BZ boundaries, $k_z=\\pm \\pi$, as we expected, (ii) one at the $\\Gamma$ point with quadratic dispersion in $k_i \\perp k_z$, in agreement with the result of continuum model, (iii) however, due to lattice regularization there can be two nodes (instead of the node at $k_z=\\frac{\\alpha}{2\\lambda_2 T}$ in the continuum model) with linear dispersion away from the boundaries and the $\\Gamma$ point, which are located at $k_z=\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T}), k_z=-\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T})+\\pi$. Figure.~\\ref{fig:latt}, shows the lattice bandstructure along $k_z$ for the case of $k_z\\Gamma_5$ driving. Therefore, we confirm that the generation of the hybrid dispersion Dirac semimetal (and consequently Weyl semimetal) persists in the lattice model.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_03_lattice.pdf}\n    \\caption{The $k_x=k_y=0$ cut of the bandstructure of LSM on a cubic lattice in presence of $\\alpha\\sin(k_z)\\Gamma_5$ ($\\alpha=0.3$) along the $k_z$ axis. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:latt}\n\\end{figure}\n\\section{DISCUSSION AND CONCLUDING REMARKS}\n\nWe have proposed a dynamical way to realize hybrid Dirac and Weyl semimetals in a three-dimensional Luttinger semimetal via applying a nonuniform (momentum-dependent) periodic $\\delta$-kick. We explicitly demonstrated this through two examples of nonuniform kicking which break both inversion and time-reversal symmetries while preserving their combinations. We have identified the first example of an unusual hybrid Dirac semimetal phase where two nodes not only have different types but also have different dispersions ( a linear Dirac and QBT coexist). Then by applying an external magnetic field we demonstrated the emergence of hybrid Weyl semimetals. Next, we found that the combination of a tilted QBT with an external magnetic field provides a promising setup for generation of hybrid Weyl semimetals. Moreover, by interpreting the tilted QBT as a QBT with emergent psudomagnetic field proportional to kick strength, we discussed the interplay between the kick strength and the external magnetic field. \\\\\n\\indent We note that the experimental realization of models discussed in this work can be difficult, specifically for the case of uniform strain which requires very fast time scales for periodic driving. Despite all the difficulties there are some proposals for the fast dynamical generation of strain \\cite{straindyn1,straindyn2,kickgraphene}. However, the possibility of interpreting the $k_z\\Gamma_5$ kick as an applied spin current (or proportional to it) along the $z$-direction is potentially a promising route, considering the recent developments in the field of ultrafast spintronics \\cite{ultrafastspin}. Moreover, the 2D QBT has been already proposed to be realized in optical lattices \\cite{QBToptical}, therefore, in practice a 3D QBT could be realized in optical lattices as a potential experimental setup where parameters can be tuned at will.\\\\\n\\indent The LSMs can describe the low-energy physics of many experimental candidates. Therefore, this work opens up a promising way for the realization of various hybrid Dirac and Weyl semimetals. Therefore, it could motivate further studies on the lacking investigation of physical properties of the hybrid Weyl semimetals. \\\\\nSome of the possible future directions are: the investigation of bulk and surface transport properties in the hybrid dispersion Dirac semimetals introduced here, in particular, how some of the most interesting properties of a typical Dirac semimetal, such as transport anomalies \\cite{review1}, would be different. Moreover, it would be interesting to see whether other multi-band systems with higher-spin can show similar physics.\n\n\\section*{Acknowledgement}\nWe thank Matthew Foster and Bitan Roy for useful comments. We also acknowledge useful comments and suggestions from the anonymous referees which helped to improve this manuscript. This work was supported by the U.S. Army Research Office Grant No.W911NF-18-1-0290. we also acknowledge partial support from NSF CAREER Grant No. DMR-1455233 and ONR Grant No. ONR-N00014-16-1-3158.\n\n\n\n\n\n\n\n\\section{Introduction}\nOver the last decade topological phases of matter have attracted many attentions for providing tremendous insights both in fundamental and experimental aspects of condensed matter physics \\cite{Chiu2016}. In addition to the gapped phases which initially ignited the field, recently discovery of gapless topological phases stimulated many works towards the understanding of nontrivial topology of gapless systems \\cite{review1}. Among those, Weyl and Dirac semimetals are of particular interest due to their experimental discovery and many unique physical properties \\cite{review1}. The Dirac\/Weyl smimetals (DSMs\/WSMs) are characterized by isolated point touchings of two degenerate\/nondegenerate bands in momentum space. Weyl nodes can be generated by splitting of degenerate Dirac nodes usually via breaking of either time-reversal ($\\mathcal{T}$) or inversion ($\\mathcal{I}$) symmetries or both. In the standard form, the low energy excitations near the Weyl nodes, disperse linearly along all three momentum directions with each node carries monopole charge of $\\pm 1$. As a result, on the surface, there exists a Fermi arc that connects a pair of Weyl nodes with opposite chiralities. Recently, generalization of WSMs to multi-Weyl nodes have been proposed where each nodes have higher-order dispersion in one or more directions and consequently caries monopole charge of larger than one \\cite{MultiWeyl1,MultiWeyl2}. \\\\\n\\indent On the other hand, DSMs\/WSMs can also be classified into type-I and type-II, based on the tilting of their nodes. In the standard type-I WSMs, Fermi surfaces are point-like while in the type-II WSMs, Weyl nodes are tilted resulting in formation of electron and hole pockets producing finite density of states at the Fermi level \\cite{typeII1,typeII2}. In a conventional type-I and II WSMs, two Weyl nodes with opposite chiralities have same types, however, recently, a theoretical proposal \\cite{hybridWeyloriginal}, introduced a new WSM where a pair of Weyl nodes with different chiralities can have different types, forming the so-called \\emph{hybrid Weyl semimetals}.\\\\\n\\indent Besides the DSMs\/WSMs, another class of three-dimensional nodal semimetals are Luttinger semimetals (LSMs) where possess a quadratic band touching (QBT) point between doubly degenerate valence and conduction bands of $J=3\/2$ (effective) fermions at an isolated point in the Brillouin zone.\nThe LSM provides the low-energy description for a plethora of both strongly and weakly correlated compounds, such as the 227 pyrochlore iridates (Ln2Ir2O7, with Ln being a lanthanide element) \\cite{LSMapp1,LSMapp2, LSMapp3,LSMapp4}, half-Heusler compounds (ternary alloys such as LnPtBi, LnPdBi) \\cite{LSMapp5,LSMapp6}, HgTe \\cite{semiconductor1,semiconductor2,QBToriginal,Ruan2016,mottQBT}, and gray-tin \\cite{LSMapp7,LSMapp8}.\nMoreover, LSMs proved to show many interesting behaviors\\cite{LSMprop1,LSMprop2,LSMprop3,LSMprop4,LSMprop5}, specially in the presence of interaction, for example, investigation of the magnetic and superconducting orders actively have been explored \\cite{LSMapp9, Roy-Ghorashi2019, GhorashiPRB2017, GhorashiPRL2018, Ghorashi2019,SatoSC32,LSMBoettcher2,LSMSCLiu1,LSMBoettcher1,Szabo-Bitan32-2018}.\\\\\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{FlashpickickedLSMv4.png}\n    \\caption{Schematic picture of summary of the results obtained in this work. Starting from a Luttinger semimetal via two different nonuniform ($k$-dependent) periodic kickings, where break inversion ($\\mathcal{I}$) and time-reversal ($\\mathcal{T}$) symmetries while preserving their combinations ($\\mathcal{IT}$), we have obtained a \\emph{hybrid dispersion Dirac semimetals} (e.g., $k_z\\Gamma_5$ where $\\Gamma_i$ are Dirac matrices [see text for details]) and tilted LSM (e.g., $k_z\\Gamma_0$). Then, by applying an external magnetic field, $J_z$, in parallel to the direction of nodes (or kick direction), \\emph{hybrid Weyl phases} can be generated.}\n    \\label{fig:adpic}\n\\end{figure*}\n\\indent The coexistence of various Weyl nodes with different charges and\/or types is an interesting phenomena that could help towards understanding as well as manipulation of the properties of various Weyl nodes in an equal footing setup. Besides a few works reporting the coexistence of type-I and II Dirac\/Weyl nodes \\cite{hybridWSMexp,hybridDSM1,hybridWSM1,WSMexp1&2,weylcoexPRL}, there have been also proposals\\cite{LSMFloq1,Ghorashi2018}, claiming dynamical generation of various Weyl nodes of different types and\/or charges in one system. Application of the light is shown to be a powerful method to change the material properties \\cite{FloquetRev}. In particular, the conversion of a topologically trivial phase into a nontrivial one using periodic driving has attracted enormous attention in the past decade \\cite{FloquetRev,Floquet1, Floquet2, Floquet3}. Specifically many proposals on Floquet WSMs in various systems exist, such as Dirac semimetals \\cite{FloqWeyl1,FloqWeyl2}, band insulators \\cite{FloqWeyl3}, stacked graphene \\cite{FloqWeyl4}, line-nodal semimetals \\cite{FloqWeyl2,FloqWeyl5}, and crossing-line semimetals \\cite{FloqWeyl6,FloqWeyl7}. Also, proposals have been made to create tunable WSMs in pyrochlore iridates with Zeeman fields \\cite{Weylpyro1,Weylpyro2}. Very recently, using circular\/elliptic polarized light on Luttinger Hamiltonian in the high-frequency limit we have shown a very rich phase diagram of various Weyl semimetals, including coexistence of type-I and II as well as single and double Weyl nodes \\cite{Ghorashi2018} . \\\\\n \\indent Despite the several dynamical proposals for the generation of different Weyl phases, a promising setup for the realization of hybrid Dirac and Weyl semimetals is still lacking. In this work, we tackle this issue by an alternative way of periodic driving, in particular the periodic kicking. Using the periodic $\\delta$-function kicks can typically simplify theoretical studies by allowing to perform calculations analytically to a large extent (in contrast to sinusoidal driving or elliptical\/circular light) \\cite{ Floquet3,kicking2}.\nHowever, for the sake of comparison we also briefly discuss the smooth driving case to show that some of the features of our discussion can be hold up in smooth driving setup as well as long as the perturbation breaks inversion (uniaxially) and time-reversal symmetres but preserve their combinations. In this paper, we show two examples of such perturbations which along with an external magnetic field induce various hybrid Dirac and Weyl phases, including a new \\emph{hybrid dispersion Dirac semimetal}. Figure.~\\ref{fig:adpic} summarizes the result of this work.\n\n\\section{Model and Formalism}\n\\subsection{Model}\nWe start with reviewing the main ingredients of Luttinger Hamiltonian in the non-equilibrium limit, which can be represented as,\n\n\\begin{align}\n    H_{L}(\\vec{k})=\\int \\frac{d^3\\vec{k}}{(2\\pi)^3}\\Psi^{\\dagger}_{\\vec{k}} h_L(\\vec{k}) \\Psi_{\\vec{k}},\n\\end{align}\nwhere\n\\begin{align}\n    h_{L}(\\vec{k})=&(\\frac{k^2}{2m_0}-\\mu)\\Gamma_0-\\frac{1}{2m_1}\\sum^3_{a=1} d_a(\\vec{k})\\Gamma_a\\cr\n    -&\\frac{1}{2m_2}\\sum^5_{a=4} d_a(\\vec{k})\\Gamma_a\n\\end{align}\nwhere $k^2=k^2_x+k^2_y+k^2_z$ and,\n\\begin{align}\n    \\Psi^T_{k}= (c_{\\vec{k},3\/2}, c_{\\vec{k},1\/2}, c_{\\vec{k},-1\/2}, c_{\\vec{k},-3\/2}).\n\\end{align}\n$\\mu$ is the chemical potential measured from the band touching point. $\\Gamma_a$ are the well-known gamma matrices which are given by,\n\\begin{align}\n    \\Gamma_1=\\tau_3\\sigma_2,\\,\\, \\Gamma_2=\\tau_3\\sigma_1,\\,\\, \\Gamma_3=\\tau_2,\\,\\,\n    \\Gamma_4=\\tau_1,\\,\\, \\Gamma_5=\\tau_3\\sigma_3,\n\\end{align}\n and satisfy $\\{\\Gamma_a,\\Gamma_b\\}=\\delta_{a,b}$, while $\\Gamma_0$ is four dimensional identity matrix. $\\tau$ and $\\sigma$ denote space of sign and magnitude of spin projection $m_s\\in\\{\\pm 3\/2, \\pm 1\/2\\}$, respectively. $d_a(\\vex{k})$ are given as,\n \\begin{align}\n     d_1=\\sqrt{3}k_y k_z,\\,\\, d_2=\\sqrt{3}k_x k_z,\\,\\, d_3=\\sqrt{3}k_x k_y,\\,\\,\\cr\n     d_4=\\frac{\\sqrt{3}}{2}(k_x^2-k_y^2),\\,\\, d_5=\\frac{1}{2}(2k_z^2-k_x^2-k_y^2)\n \\end{align}\n The five mutually anticommuting $\\Gamma$ matrices can be written\nin terms of spin-$3\/2$ matrices according to,\n\\begin{align}\n    \\Gamma_1=&\\frac{1}{\\sqrt{3}}\\{J_y,J_z\\},\\,\\,\\Gamma_2=\\frac{1}{\\sqrt{3}}\\{J_x,J_z\\},\\,\\,\\Gamma_3=\\frac{1}{\\sqrt{3}}\\{J_x,J_y\\},\\cr\n    \\Gamma_4=&\\frac{1}{\\sqrt{3}}(J^2_x-J^2_y),\\,\\,\\Gamma_5=J^2_z-\\frac{5}{4},\n\\end{align}\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveup.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveopposite.pdf}\n    \\caption{The band structure for LSM along the $k_z$ axis with $\\lambda_1=0.6$ and (a) $\\lambda_2=0.1$, (b) $\\lambda_2=0.6$.}\n    \\label{fig:LSMbare}\n\\end{figure}\n Here we take the isotropic limit of $m_1=m_2 \\equiv m$. Therefore, the Luttinger Hamiltonian can also be written in an alternative way,\n\\begin{align}\n    h_L(\\vec{k})= [(\\lambda_1+5\\lambda_2\/2)k^2-\\mu]\\Gamma_0-2\\lambda_2(\\vec{J}.\\vec{k})^2\n\\end{align}\n with $\\vec{J}=(J_x,J_y,J_z)$ and $\\vec{k}=(k_x,k_y,k_z)$ and we used $\\lambda_1=1\/2m_0$ and $\\lambda_2=1\/4m$. $J_{x,y,z}$ are effective spin-$3\/2$ operators,\n \\begin{eqnarray}\n&\\,J_z=\\begin{bmatrix}\n        \\frac{3}{2} & 0 & 0 & 0 \\\\\n        0 & \\frac{1}{2} & 0 & 0\\\\\n        0 & 0 & -\\frac{1}{2} & 0 \\\\\n        0 & 0 & 0 & -\\frac{3}{2} \\\\\n      \\end{bmatrix},J_x=\\begin{bmatrix}\n        0 & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & 1 & 0\\\\\n        0 & 1 & 0 & \\frac{\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}\\cr\n      &\\,J_y=\\begin{bmatrix}\n        0 & \\frac{-i\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & -i & 0\\\\\n        0 & 1 & 0 & \\frac{-i\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}.\n\\end{eqnarray}\nThe energy dispersions are $E(k)=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}-\\mu$ for the $j=3\/2$ and the $j=1\/2$ bands, respectively. Four bands come in doubly degenerate pairs as a result of time-reversal (with antiunitary operator $\\mathcal{T}=\\Gamma_1\\Gamma_3\\mathcal{K}$ and $\\vec{k}\\rightarrow -\\vec{k}$ where $\\mathcal{K}$ is complex conjugation) and inversion ($\\mathcal{I}=I_{4\\times4}$ and $\\vec{k}\\rightarrow -\\vec{k}$) symmetries. For $\\lambda_{2}<2\\lambda_{1}$\n($\\lambda_{2}>2\\lambda_{1}$), the degenerate bands curve the same (opposite) way as shown in Figure.~(\\ref{fig:LSMbare}). In the case of both bands bending the same way, Eq. (2) is widely used to model heavy- and light-hole bands in zinc-blende semiconductors \\cite{semiconductor1} and many properties of such a dispersion have been studied in the literature, including a recent study on the realization of fully gapped topological superconductivity with \\emph{p}-wave pairing which has states with exotic cubic and linear dispersions coexisting on the surface \\cite{congjunWuPRL16,GhorashiPRB2017}. On the other hand, when bands bend oppositely, the model in Eq. (2) is known as Luttinger semimetal with QBT and is used to describe behavior of certain pyrochlore iridates as well as some doped half-Heusler alloys such as LaPtBi \\cite{Chadov2010,Lin2010,halfheusler3}.\n\n\n\\subsection{Periodic driving}\nA general time-dependent problem with $H(t)=H_0+V(t)$, can be tackled using Floquet theory when $V(t+T)=V(t)$ is periodic. To proceed, we can expand the periodic potential in a Fourier series as\n\\begin{align}\n    V(t)=V_0+\\sum^{\\infty}_{n=1} \\big(V_n e^{i\\omega nt}+V_{-n}e^{-i\\omega nt}\\big).\n\\end{align}\n In the limit of fast driving regime, in which the driving frequency is larger than any natural energy scale in the problem, one can obtain the effective Hamiltonian and Floquet operators perturbatively up to  $\\mathcal{O}(1\/\\omega^2)$ \\cite{Floquet3}. The Floquet operator, $\\mathcal{F}(t)$, is the unitary time-evolution $\\hat{U}(t)$ after one period of drive can be factorized as,\n\\begin{align}\n    \\mathcal{F}(t)=\\exp[-i \\alpha\\Lambda(\\vec{k})]\\exp[-i H_L T]=\\exp[-i H_{eff} T].\n\\end{align}\nSo the dynamics can have three stages: initial kick at $t_i$, the evolution of system with $H(t)$ in the interval $t_f-t_i$ and final kick at $t_f$ which describes the \"micromotion\" \\cite{Floquet3}. Then, the time-evolution operator can be expressed as,\n\\begin{align}\n    \\hat{U}(t_i\\rightarrow t_f)= \\hat{U}(t_f)^{\\dagger}e^{-iH_{eff}(t_f-t_i)}\\hat{U}(t_i),\n\\end{align}\nwhere $\\hat{U}(t)=e^{-i\\mathcal{F}(t)}$. $\\mathcal{F}(t)$ is a time-periodic operator with zero average over one period.  Lets set $t_i=0$, then $H_{eff}$ and $\\mathcal{F}(t)$ can be expanded as,\n\\begin{align}\n    H_{eff}=\\sum_{n=0}^{\\infty}\\frac{1}{\\omega^n}H_{eff}^n,\\,\\,\\mathcal{F}(t)=\\sum_{n=1}^{\\infty}\\frac{1}{\\omega^n}\\mathcal{F}^n.\n\\end{align}\nA generic quantum $\\delta$-kick can be implemented with following perturbation,\n\\begin{align}\n    H_{kick}(t)=\\alpha \\Lambda(\\vec{k}) \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)\n\\end{align}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5p_05_kz.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5m_05_kz.pdf}\n    \\caption{The band structure of LSM along the $k_z$ axis in the presence of uniform strain ($\\alpha \\Gamma_5$) with (a) $\\alpha=0.5$ (tensile) and (b) $\\alpha=-0.5$ (compression). $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used.}\n    \\label{fig:G5}\n\\end{figure}\nwhere $T=2\\pi\/\\omega$, $\\alpha$ is the kicking strength and $\\Lambda(\\vec{k})$ is the matrix representation of a perturbation, which in general can be a function of momentum, $\\vec{k}$, and could be used to mimic a nonuniform kicking. Following a perturbative expansion the effective Hamiltonian for $\\delta$-kick, with $H_{kick,n}=\\alpha \\Lambda(\\vec{k})\/T\\,\\text{for all}\\,\\,n$, can be obtained as \\cite{Floquet3,kicking2},\n\\begin{align}\\label{effkick}\n    H^{kick}_{eff}=&\\,H_L+\\frac{\\alpha\\Lambda(\\vec{k})}{T}+\\frac{1}{24}[[\\alpha\\Lambda(\\vec{k}),H_L],\\alpha\\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n   \n   \n\\end{align}\nWe note that the third term in Eq.~(\\ref{effkick}), can be dropped in the limit of weak kicks, i.e, to the first order of kick strength. Therefore, a dynamical kicking effectively is equivalent to directly adding the kick term as a perturbation to the bare Hamiltonian, something which is not always relevant in the equilibrium. In the rest of this work we only focus on the physics of the effective Hamiltonian and we leave \"micromotion\" physics for future studies. \\\\\nSimilarly, the effective Hamiltonian for the case of smooth driving of the form $H(t)=H_0+\\alpha \\Lambda(\\vex{k})\\cos(\\omega t)$ can be obtained as \\cite{Floquet3},\n\\begin{align}\\label{effsmooth}\n    H^{cos}_{eff}=H_0+\\frac{1}{4\\omega^2}[[\\alpha \\Lambda(\\vec{k}),H_0],\\alpha \\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n\\end{align}\nUnlike Eq.~(\\ref{effkick}) of periodic kicking, the first correction to the $H_0$ in the case of smooth driving is $\\mathcal{O}(\\alpha^2)$.\\\\\n\\indent In the following we first investigate the effect of uniform kicking ($k$-independent) and show that some of the known results can be retrieved. Then we turn to nonuniform periodic kicking and show that by introducing $k$-dependent $\\mathcal{IT}$ symmetric kicks, interestingly,  different hybrid Dirac\/Weyl semimetals can be obtained. Finally, we briefly compare the case of smooth driving of Eq.~(\\ref{effsmooth}) with the results obtained by periodic kicking.\n\n\n \\section{Uniform kicking}\nThere are many possibilities of uniform kicking, including: five $\\Gamma_j$, and ten commutators $\\Gamma_{ij}=[\\Gamma_i,\\Gamma_j]\/(2i)$ with $i > j$ , which can be expressed in terms of the products of odd number of spin-$3\/2$ matrices. Here we focus on uniform kicks which are proportional to $\\Lambda=\\Gamma_j$. The effective Hamiltonian for such kicks, in a sufficiently weak kick limit, is given by,\n\\begin{align}\\label{effG5}\n    H^j_{eff}=h_L(\\vec{k})+\\frac{\\alpha}{T}\\Gamma_j,\n\\end{align}\nwith spectrum,\n\\begin{align}\n    E^j_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2\\sum_i d^2_i T^2 - 4\\lambda_2\\alpha T d_j+\\alpha^2}}{T},\n\\end{align}\n\nOf particular interest is the $\\Gamma_5$ kicking, that breaks the rotational invariance and can be thought of as the effect of an external strain in $z$ direction \\cite{Ruan2016,QBToriginal}. It is known that a uniaxial strain, can realize two different phases in LSMs depending on the sign of $\\alpha$ \\cite{Ruan2016,QBToriginal,Weylpyro1,Weylpyro2,LSMmagnetic}. Figure.~\\ref{fig:G5}a(b) shows the bandstructure corresponding to effective Hamiltonian of Eq.~(\\ref{effG5}) with $\\Gamma_j=\\Gamma_5$ and for $\\alpha > 0$ ($\\alpha<0$) which represents a Dirac semimetal (trivial\/topological insulator).\n\nIn the case of a tensile strain, there are two Dirac nodes at $k_z=\\pm\\frac{\\alpha}{2\\lambda_2 T}$ ($\\alpha > 0 $). Now if we add an external magnetic field, we expect that each Dirac nodes split to two Weyl points and form Weyl semimetal phases due to breaking of $\\mathcal{T}$ symmetry. Interestingly, even in the topological insulator limit of $\\alpha<0$, one can realize Weyl semimetals depending on the strength of magnetic field \\cite{LSMmagnetic}. We also note that magnetic field alone can drive a Luttinger semimetal to a Weyl semimetal phase \\cite{LSMmagnetic,Weylpyro2}. Moreover, in the experimentally relevant system of pyrochlore irridates, it is shown that magnetic field can give rise to rich phase diagram of topological semimetals \\cite{Weylpyro2}. It is important to mention that an external magnetic field can have orbital effects for sufficiently large strength. However, here and in the rest of this paper, we neglect the orbital effects of magnetic field for sake of simplicity following many works in the literature \\cite{Weylpyro1, Weylpyro2,LSMFloq1} and we refer readers to some of the works which explored the effects of magnetic field in Landau levels and quantum oscillations \\cite{LSMmagnetic,LSMquantumoscillation}. The detailed analysis of the effect of magnetic field (also possible pseudomagnetic fields due to other perturbations) will be left for future studies.\n\n\\section{Nonuniform kicking}\nNow lets turn to the main part of this work, where we investigate the effect of nonuniform kicks. Similar to the uniform kicking discussed in the previous section, we can consider numerous possibilities of nonuniform driving. However, here, we restrict ourselves to a \"linear in momentum\" kicks of the form $k_i\\Gamma_i$ where $\\Gamma_i$ are time-reversal even. The very first consequence of such choice of the kicks, is the breaking of both $\\mathcal{T}$ and $\\mathcal{I}$ symmetries, while preserving their combinations $\\mathcal{IT}$, which is a necessary symmetry for keeping double degeneracy of the bands intact. Specifically, we focus on two types of nonuniform kicking: (a) $k_z\\Gamma_5$ and (b) a $k_z\\Gamma_0$.\n\n\\begin{figure*}[htb]\n \\centering\n    \\includegraphics[width=0.42\\textwidth]{Fig4a.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz05_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz07_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz11_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz15_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz20_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz30_kz-R.pdf}\\includegraphics[width=0.42\\textwidth]{kzG5_PH_kxkz.pdf}\n    \\caption{The band structure of LSM in the presence of $k_z\\Gamma_5$ with kick strength of $\\alpha=0.8$  and (a) $h=0$, $k_y-k_z$ plane showing quadratic dispersion along the $k_y$ for the node at $\\Gamma$ point, (b) $h=0.5$, (c) $h=0.7$, (d) $h=1.1$, (e) $h=1.5$, (f) $h=2$, (g) $h=3$ and (h) same as (a) but with $\\lambda_1=0$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots (except (h) where $\\lambda_1=0$).}\n    \\label{fig:kzG5}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_5$}\n\nAs it is mentioned above the $\\Gamma_5$ can be interpreted as the effect of strain in $k_z$ directions \\cite{Ruan2016}, similarly, $k_z\\Gamma_5$ also breaks the rotational invariance, then in principle we might still think of $k_z\\Gamma_5$ term as a type of nonuniform strain or strain gradient, even though full microscopic derivation of such strain could shed more light. The presence of $k_z$ suggests that microscopically on a lattice, $\\Gamma_5$ now acts non-local and directly modifies hopping terms. On the hand, one can convert the $k_z\\Gamma_5$ to the lattice model and explicitly write it down as, $\\sum_{\\sigma\\in\\{1\/2,3\/2\\}}(C^{\\dagger}_{k_{\\perp},i,\\sigma,\\uparrow}C_{k_{\\perp},j,\\sigma,\\uparrow}-C^{\\dagger}_{k_{\\perp},i,\\sigma,\\downarrow}C_{k_{\\perp},j,\\sigma,\\downarrow})\/2i+h.c$. This is similar to the spin current operator along $z$-axis. Therefore, we can also think of the $k_z\\Gamma_5$ kick as an applied spin current (or related to it) along $z$ direction. However, it is essential to note that besides the rotational invariance the $k_z\\Gamma_5$ kick also breaks inversion and time-reversal symmetries while preserving their combinations. Then the bands of driven system are still degenerate. Therefore, here, without restricting ourselves to any particular microscopic interpretation, we emphasize that to achieve the results obtained in this section one need perturbations which break both $\\mathcal{I}$ and $\\mathcal{T}$ but preserve $\\mathcal{IT}$.\\\\\nThe effective Hamiltonian for $k_z\\Gamma_5$ kick can be written as, $H_{eff}=h_{L}(\\vec{k})+\\frac{\\alpha}{T}k_z\\Gamma_5$, with following spectrum,\n\\begin{align}\n    E_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2T^2\\sum_i d^2_i - 4\\lambda_2\\alpha T k_z d_5+\\alpha^2 k^2_z}}{T}.\n\\end{align}\nFigure.~\\ref{fig:kzG5}, shows the spectrum in $k_x-k_z$ plane. There are two Dirac nodes at $k_z=0$ and $k_z=\\frac{\\alpha}{2\\lambda_2 T}$. While one of the nodes is always pinned to $\\Gamma$ point the other node can be tuned by kick parameters. Therefore, for fixed set of parameters the distance between two nodes for $k_z\\Gamma_5$ is half of the case with uniform strain $\\Gamma_5$. Moreover, we observe three main differences in compare to the case with uniform strain as is shown in Figure.~\\ref{fig:G5}. (i) There is no major difference between the tensile ($\\alpha >0 $) and  compressive ($\\alpha <0 $) strain. In both cases system drives into a Dirac semimetal phase. (ii) Unlike the uniform tensile strain, due to broken inversion symmetry two Dirac nodes reside at different energies and (iii) the most important of all, remarkably, one of the nodes is quadratic while the other is linear, realizing an unique \\emph{hybrid Dirac semimetal}. Interestingly, the \"quadratic\" Dirac node (or a QBT which is linearly dispersed in $k_z$ direction) is pinned at the $\\Gamma$ point and the \"energy difference\" as well as distance between the nodes can be controlled with kick strength. Moreover, while the linear Dirac node is tilted, the node centered around the $\\Gamma$ point no matter how strong is the kick, shows no tilt. It should be emphasized, here, we define hybrid Dirac semimetal mainly based on the different dispersion of two Dirac nodes instead of their tilts, even though as we discussed above, they show different tilting (and types) as well. Moreover, We note that such hybrid phase is unique to the Dirac semimetals because it is impossible to realize a Weyl semimetal with pair of nodes with different dispersion or (magnitude of monopole charges).\\\\\n\\indent Next, lets apply an external magnetic field $h J_z$ where $h$ denotes the strength of magnetic field. The magnetic field splits Dirac nodes to Weyl nodes in $k_z$ direction. In the presence of a magnetic field an analytical expression of eigenvalues for the general $\\lambda_1,\\lambda_2$ can not be achieved, so in the following we proceed by solving the model numerically. Figure.~\\ref{fig:kzG5} depicts the evolution of the Weyl nodes by strength of external magnetic field. Starting from $h \\ll \\alpha $, there are 8 nodes on $k_z$ axis, four single and four double Weyl nodes, which are indicated in Figure.~\\ref{fig:kzG5} by the red and blue dots, respectively. Interestingly, we observe that one or more of the Weyl pairs realize a \\emph{hybrid Weyl phases} and they can survive up to a decently strong magnetic field (Figure.~\\ref{fig:kzG5}(b-f)). While the other Weyl nodes do not possess hybrid types, they show different tilts for each nodes of the same pair. Only at a very strong magnetic field (Figure.~\\ref{fig:kzG5}(g)) all of the Weyl points show a type-I structure.\nMoreover, we see that by increasing the magnetic field, from initial eight nodes, four of them merge and gap out and we left off with only four nodes (two single and two double nodes) at $h>>\\alpha$ limit. \\\\\n\\indent Finally, we comment on the particle-hole symmetric limit of $\\lambda_1=0$. In this limit we can get energies analytically even in presence of an external magnetic field as,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nwith 8 nodes located at,\n\\begin{align}\n    k^{\\pm,\\pm}_{z,I}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 4h\\lambda_2T^2}}{4\\lambda_2 T},\\cr\n    k^{\\pm,\\pm}_{z,II}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 8h\\lambda_2T^2}}{4\\lambda_2 T}.\n\\end{align}\nAs is clear from the above equations, four out of the eight nodes, $k^{\\pm,-}_{z,I,II}$, gap out by increasing $h$. However, it is noteworthy that even though a \\emph{hybrid dispersion Dirac semimetal phase} can still be generated, but now in the limit of $\\lambda_1=0$, the two nodes have the same energies Figure.~\\ref{fig:kzG5}(h), therefore, no longer a hybrid Weyl phase can be achieved in the presence of an external magnetic field.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_kz.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz02.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz04.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz06.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz08.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz15.pdf}\n\\caption{The band structure of LSM with tilted QBT with kick strength of $\\alpha=0.6$  and (a) $h=0$, (b) $h=0.2$, (c) $h=0.4$, (d) $h=0.6$, (e) $h=0.8$ and (f) $h=1.5$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots.}\n\\label{fig:kzG0}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_0$: Tilted QBT}\n\nThe second class of nonuniform kicks which we investigate in this work, is $\\Lambda=k_z\\Gamma_0$. The effective spectrum for such driving can be obtained as,\n\\begin{align}\\label{tQBT}\n    E_{\\pm}(\\vec{k})=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}+\\frac{\\alpha k_z}{T}.\n\\end{align}\nThe mere effect of such driving is to effectively tilt the QBT in $k_z$ direction as shown in Figure.~\\ref{fig:kzG0}(a). However, the tilted QBT behaves very differently in the presence of an external perturbation. To clarify further, we rewrite the Eq.~(\\ref{tQBT}) at $k_x=k_y=0$ as,\n\\begin{align}\n     E_{\\pm}(\\vec{k})=&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^{2}+\\frac{\\alpha k_z}{T}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})(k_z+\\mathcal{A}_z)^{2}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z+(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}^2_z\\cr\n     \\simeq&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z,\n\\end{align}\nwhere in the last line we assumed $\\mathcal{A}_z<1$. Now by comparing the first and the last line of the above equation, we obtain $\\mathcal{A}_z=\\frac{\\alpha}{2(\\lambda_{1}\\mp2\\lambda_{2})T}$. Therefore, we can think of a tilted QBT as a QBT with a pseudo-electromagnetic potential $\\mathcal{A}= (0, 0, \\mathcal{A}_z)$ which is proportional to kick strength. In particular, interesting nontrivial topological phases can emerge by applying an external magnetic field in the presence of tilt, which otherwise are absent in a conventional QBT. As we mentioned previously, it is known that in the presence of a magnetic field, the QBT splits to multiple Weyl points. However, when the QBT is tilted, due to the competition between the external magnetic field and pseudo-magnetic field due to the $\\mathcal{A}$ (proportional to kick strength) there are various regimes which different types of WSM phases can be generated. Starting with the weak field limit ($\\alpha >> h$), there are two pairs of nodes, two double and two single nodes. In this regime, all Weyl nodes have a type-II nature (Figure.~\\ref{fig:kzG0}(b)). By further increasing of the field, for intermediate fields ($h \\lesssim \\alpha$), hybrid Weyl pairs are realized where one of nodes in each of single and double pairs are type-II and the others are type-I. Interestingly, the realized hybrid WSM can survive up to a very strong magnetic field. This could be experimentally beneficent as it demonstrates that hybrid WSMs generated here, are accessible in a broad range of external fields. Similar to the $k_z\\Gamma_5$ kick, we can get the analytical expression of energies and nodes position in the particle-hole limit,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z+\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nand,\n\\begin{align}\n    k^{\\pm}_{z,I}=\\pm\\sqrt{\\frac{h}{2\\lambda_2}},\\,\\,k^{\\pm}_{z,II}=\\pm\\frac{\\sqrt{h}}{2\\sqrt{\\lambda_2}}=\\frac{k^{\\pm}_{z,I}}{\\sqrt{2}}.\n\\end{align}\nUnlike the $k_z\\Gamma_5$ kick, the presence of particle-hole symmetry does not prevent the generation of hybrid Weyl phases, because does not generate any other Dirac nodes and as we mentioned before only tilts the QBT. Moreover, the position of Weyl nodes are independent of kick strength and can be tuned solely using magnetic field.\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kxkz.pdf}\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kykz.pdf}\n    \\caption{The bandstructure of LSM at $k_x-k_z$ plane ($k_y=0$) in presence of $k_z\\Gamma_5$ smooth driving with $\\alpha=5$. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:kzG5smooth}\n\\end{figure}\n\n\\subsection{Smooth driving}\nHere, we briefly discuss the case of smooth driving using Eq.~(\\ref{effsmooth}) and make comparison with the results obtained by periodic kicking in previous sections. First of all, any type of perturbation which is proportional to the identity will not modify the system in the case of smooth driving, as it obviously commutes with Hamiltonian. However, the $\\Lambda(\\vec{k})=\\alpha k_z\\Gamma_5$ can modify the system when is applied via smooth driving. We obtain the effective Hamiltonian as,\n\\begin{align}\n    H^{cos}_{eff}(\\vec{k})=&H_0+\\frac{\\lambda_2\\alpha^2k^2_z}{\\omega}\\bigg(2k_xk_z\\{J_x,J_z\\}+2k_yk_z\\{J_y,J_z\\}\\cr\n    +&(k^2_x-k^2_y)(J^2_x-J^2_y)+2k_xk_y\\{J_x,J_y\\}\\bigg).\n\\end{align}\nwith energies,\n\\begin{align}\n    E^{\\pm}(\\vec{k})=\\lambda_1 k^2\\pm \\frac{\\lambda_2\\sqrt{f(\\vec{k})\\left[k^2_z\\alpha^2-2\\omega^2\\right]+4k^4\\omega^4}}{\\omega^2},\n\\end{align}\nwhere each $f(\\vec{k})=3 (k_x^2+k_y^2)k_z^2 (k_x^2+k_y^2 +4k_z^2)\\alpha^2$ and $E^{\\pm}$ are doubly degenerate. Figure.~\\ref{fig:kzG5smooth} shows the spectrum for $k_y=0$ plane, where four linear Dirac nodes coexist with a QBT at which is located at $\\Gamma$ point. A similar plot can be obtained for $k_x=0$ plane, then the system also possesses line-nodes in $k_x-k_y$ plane at $k_z\\neq 0$. Therefore, in addition to the coexistence of Dirac nodes and QBT at $k_x-k_z$ and $k_y-k_z$ planes, a smooth driving can lead to a richer phase diagram. This analysis was only for the purpose of comparison with kicked driving results, so a detailed analysis of such models will be left for future works.\n\n\\subsection{Effect of Lattice Regularization}\n\nLets now look at the effect of lattice regularization. One of the typical effect of lattice versus continuum model is the appearance of more nodes, usually at the boundaries of the Brillioun Zone \\cite{Ghorashi2018}. However, we show that the main features discussed in the previous subsections survives in lattice models. For the sake of brevity we restrict ourselves to $h=0$ limit. We consider a cubic lattice, which captures the physics of continuum model for $\\alpha,\\omega,\\vec{k}<<1$. In the $k_z$ cut (Figure.~\\ref{fig:latt}), we find: (i) two nodes at BZ boundaries, $k_z=\\pm \\pi$, as we expected, (ii) one at the $\\Gamma$ point with quadratic dispersion in $k_i \\perp k_z$, in agreement with the result of continuum model, (iii) however, due to lattice regularization there can be two nodes (instead of the node at $k_z=\\frac{\\alpha}{2\\lambda_2 T}$ in the continuum model) with linear dispersion away from the boundaries and the $\\Gamma$ point, which are located at $k_z=\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T}), k_z=-\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T})+\\pi$. Figure.~\\ref{fig:latt}, shows the lattice bandstructure along $k_z$ for the case of $k_z\\Gamma_5$ driving. Therefore, we confirm that the generation of the hybrid dispersion Dirac semimetal (and consequently Weyl semimetal) persists in the lattice model.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_03_lattice.pdf}\n    \\caption{The $k_x=k_y=0$ cut of the bandstructure of LSM on a cubic lattice in presence of $\\alpha\\sin(k_z)\\Gamma_5$ ($\\alpha=0.3$) along the $k_z$ axis. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:latt}\n\\end{figure}\n\\section{DISCUSSION AND CONCLUDING REMARKS}\n\nWe have proposed a dynamical way to realize hybrid Dirac and Weyl semimetals in a three-dimensional Luttinger semimetal via applying a nonuniform (momentum-dependent) periodic $\\delta$-kick. We explicitly demonstrated this through two examples of nonuniform kicking which break both inversion and time-reversal symmetries while preserving their combinations. We have identified the first example of an unusual hybrid Dirac semimetal phase where two nodes not only have different types but also have different dispersions ( a linear Dirac and QBT coexist). Then by applying an external magnetic field we demonstrated the emergence of hybrid Weyl semimetals. Next, we found that the combination of a tilted QBT with an external magnetic field provides a promising setup for generation of hybrid Weyl semimetals. Moreover, by interpreting the tilted QBT as a QBT with emergent psudomagnetic field proportional to kick strength, we discussed the interplay between the kick strength and the external magnetic field. \\\\\n\\indent We note that the experimental realization of models discussed in this work can be difficult, specifically for the case of uniform strain which requires very fast time scales for periodic driving. Despite all the difficulties there are some proposals for the fast dynamical generation of strain \\cite{straindyn1,straindyn2,kickgraphene}. However, the possibility of interpreting the $k_z\\Gamma_5$ kick as an applied spin current (or proportional to it) along the $z$-direction is potentially a promising route, considering the recent developments in the field of ultrafast spintronics \\cite{ultrafastspin}. Moreover, the 2D QBT has been already proposed to be realized in optical lattices \\cite{QBToptical}, therefore, in practice a 3D QBT could be realized in optical lattices as a potential experimental setup where parameters can be tuned at will.\\\\\n\\indent The LSMs can describe the low-energy physics of many experimental candidates. Therefore, this work opens up a promising way for the realization of various hybrid Dirac and Weyl semimetals. Therefore, it could motivate further studies on the lacking investigation of physical properties of the hybrid Weyl semimetals. \\\\\nSome of the possible future directions are: the investigation of bulk and surface transport properties in the hybrid dispersion Dirac semimetals introduced here, in particular, how some of the most interesting properties of a typical Dirac semimetal, such as transport anomalies \\cite{review1}, would be different. Moreover, it would be interesting to see whether other multi-band systems with higher-spin can show similar physics.\n\n\\section*{Acknowledgement}\nWe thank Matthew Foster and Bitan Roy for useful comments. We also acknowledge useful comments and suggestions from the anonymous referees which helped to improve this manuscript. This work was supported by the U.S. Army Research Office Grant No.W911NF-18-1-0290. we also acknowledge partial support from NSF CAREER Grant No. DMR-1455233 and ONR Grant No. ONR-N00014-16-1-3158.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}}\n\n  The non-uniform and turbulent nature of the interstellar\nmedium (ISM) has long been evident from scattering\/scintillation studies in the\ndirection of pulsars and compact extra-galactic sources, as well as from HI\nemission\/absorption measurements. Power spectra of the observed \ncolumn density distribution of neutral hydrogen (HI) in the \nISM reveal its fractal nature.  \nSome estimates of the slope of spatial power\nspectra for the absorbing HI distribution in the Galaxy show a range from --3.0 \n(Crovisier and Dickey 1983) to\n--2.75 $\\pm$ 0.25 (Deshpande, Dwarakanath \\& Goss 2001), while Green (1993)\nfound a range from --2.2 to --3.0 for the slopes from HI emission spectra,\nindicating steepening with distance. \nSeveral recent statistical studies of galactic and extra-galactic HI on large scales\n(e.g. Stanimirovic et al. 1999, Dickey et al. 2001, Elmegreen et al. 2001, \nStanimirovic \\& Lazarian 2001) also reveal power-law slopes characterizing\nthe spatial power spectra that are consistent with \nthe above-mentioned range of values. The prediction of Lazarian \\& Pogosyan (2000), \nthat the spatial spectra should steepen with the increasing thickness of the velocity\nslice, was confirmed by some of these analyses (e.g. Stanimirovic \\& Lazarian 2001).\nThe question we would like to address here is why the structure of the HI is \nlike this in a statistical sense.  \n\nA prominent source of energy input driving the ISM evolution is\nthought to be supernovae (SN). Many researchers have attempted\ndetailed MHD simulations of a supernova-driven ISM to explore this aspect.\nFor example, Korpi et al. (1999) used a three-dimensional, non-ideal MHD\nmodel and monitored several physical parameters.  \nThere have been several other detailed simulation studies exploring \nMHD turbulence in different regimes (Cho, Lazarian \\& Vishniac 2002; \nWada et al. 2002; Vollmer \\& Beckert 2002; Cho \\& Lazarian 2003; \nMiville-Deschenes, Levrier \\& Falgarone 2003; Slyz et al. 2005; and\nMac Low et al. 2005).\n\nOur simulation study takes\na highly simplified view of the ISM and monitors only the statistical\ndescription of the density and velocity distributions.  As a first step\ntoward assessing what the essential features of the processes underlying\na supernova-driven ISM are, we retain the kinematical and structural\ncomponents alone and assume that these are entirely determined by the\nexpanding shells of SNRs.  \n\nWe explore two basic approaches in our modeling of a\nSN-driven ISM; I) a simple simulation of an equivalent snapshot \ndistribution, and \nII) a more complex simulation incorporating time evolution.  Sections 2 and 3\ndescribe the details and limitations of these two modeling approaches separately,\nwhile Section 4 is dedicated to the analysis of both of the models.  Results are\ndiscussed in Section 5, and our conclusions on the implications of the two\nmodels are presented in Section 6.\n\n\n\\noindent{\\section{Snapshot Distribution}}\n\nWith the goal of keeping our model as simple as possible, we first model\nresultant contributions from supernovae\ndirectly as an ensemble of simple, non-overlapping, spherical bubbles \n(or voids) in what is  otherwise a homogeneous ISM.\nWe use a three-dimensional matrix to simulate a region in space that we\nwish to populate with a large number of bubbles of different radii. \nIn this picture, the bubbles of different sizes\nrelate to supernova remnants (SNRs) of correspondingly different ages \n(and in different\nphases of their evolution) that have resulted from supernovae at different epochs.\nThis method effectively generates a snapshot\nin time, and we invoke the ergodicity theorem to justify the basis of\nsuch a simulation.  Thus, while the bubbles do not evolve and grow iteratively\nin this approach,\nwe attempt to achieve the same end result by simply generating simultaneously\na random distribution of locations and sizes.  For this first simulation,\neach of the volume cells carry a binary tag (i.e. 0 or 1) indicating whether the given cell \n(or pixel) is inside a bubble or outside in the undisturbed ISM.  We wrap \nthe respective sides (\\& corners) of the array, ensuring continuation of the bubbles across \nthem, to avoid any undesirable edge-effects, e.g. discontinuities and consequent\n``mass loss\" at the edges. This procedure is indeed consistent with the wrapping that\nis implicit in the Fourier analysis of this data, a step that is performed subsequently.\nWe use a random number generator (with uniform distribution) to determine the coordinates \nand radius of each\nbubble.  Several parameters, such as the minimum bubble radius, the\nmaximum bubble radius, and the total number of bubbles, are retained as\nuser selectable variables.  Checks are made, and iterations are skipped if needed\nto ensure that new bubbles do not overlap with any of the previously input bubbles.  \nWe note that this non-overlap constraint is unphysical, but was unavoidable\nin order to ensure that the simulated volume was not entirely emptied at any stage.\nNaturally, the distribution of radii for included bubbles would deviate significantly\nfrom a uniform distribution if not explicitly constrained, \nsuch that the number of bubbles dropped rapidly with increasing radius. \nSuch a decline in the number of large holes may in fact \nbe expected if the sources of energy input are strongly clustered, and\nare of a wide range of energies.\nAn image resulting from this simulation is shown\nin Figure~\\ref{fig:bubble_image}.\nSimilar simulations are repeated a) by constraining the resultant distribution of the \nradii to be uniform, and\/or b) with smooth-edged bubbles (with density falling \ntowards center as $(1-cos(\\pi r\/r_0))\/2$, where $r_0$ is the bubble radius, \nand $r$ is the distance from the bubble center). \n\n\n\\noindent{\\section{Time Evolution}\n\nThe second approach to the simulation process involves incorporating a time axis\nand allowing planted supernovae and the consequent SNRs to evolve.  \nWe make several assumptions for this simple model.  \nTo begin with, we include only Type II supernovae in our simulation.\nWe use an average galactic rate of 1\/44 yr$^{-1}$ \n(Tammann, Loeffler \\& Schroeder 1994) and\nconsider a uniform distribution across a 20\\% range about this rate,\nalthough Type II supernovae are correlated in both space and time.\nThe progenitor stars are assumed to have\nmasses in the range 8-20 solar masses ($M_{\\sun}$), and their non-uniform\ndistribution within this range is modeled in consistency with \nthe initial mass function as\ngiven by Wheeler, Miller \\& Scalo (1980) and approximated in our range of interest\nby a power law.  As for the compact remnant mass, we assume a value of 1.4 $M_{\\sun}$.\nWe then consider two types of ejecta, diffuse and clumpy (Willingale et al.\n2003), with different velocities, \nand use the relevant values estimated by Willingale et al based on the Cas-A data.  \nFinally, we assume that the rest of the mass\n(i.e., M[progenitor] -- M[compact-remnant] -- M[ejecta]) \nwas lost prior to the explosion by the star's stellar wind and other mass loss\nmechanisms. We will denote this component by M[preSN-loss].\n\nFor this simulation, we use four 3-dimensional arrays to store\nthe spatial distribution of \nthe mass and the 3-d velocity that evolve as a function of time.\nThe spatial resolution (the volume corresponding to a pixel or a cell) and \nthe dynamic range of scales  \nare chosen by specifying, in turn, suitable values for both the\nspatial extent represented by the entire cube and the number of pixels in\nthe desired storage arrays. \nWe initialize the mass-cube with a starting ISM density, assumed\nuniform and generally of the order of 0.6 atoms\/cm$^3$.  This value is\nseveral times smaller than the average value that the ISM density would \nattain after subsequent mass input from supernovae. \n\nTo initiate a supernova within the simulated\nspace, we choose a random pixel location in 3-d, and input the mass \\& velocity distribution\nassociated with the supernova in the form of an expanding spherical \nshell.\nThe mass in this shell consists of the ejecta mass, M[ejecta],\nas well as the mass lost during the pre-supernova phase, M[preSN-loss].\nThe inclusion\nof this latter component is justified if the distance traveled by \nthis material, which is expelled in the pre-SN\nphase, is within the above mentioned cell size. This is indeed so given the typical cell\nsizes in our simulations, and in fact we adjust the spatial resolution to ensure\nthis to be the case. \nThe magnitude of the true initial velocity is estimated by conserving \ntotal kinetic energy \nbetween the two different types of ejecta (with assumed masses and velocities, \nsee Willingale et al. 2003 for details), and the mass lost during the \npre-supernova phase (i.e., M[preSN-loss]).\nWe use a basic template cube (3 x 3 x 3 pixels) to\nstore the 3-d velocity vectors directed radially outward and initially\nnormalized to a unit length.  We then multiply these vectors by the magnitude of the\ntrue initial velocity as computed above.\nTo calculate the initial mass distribution in the shell, we distribute the\navailable mass into the template cube proportionally to the surface area of a\nsphere centered in the middle pixel and of radius 1 pixel.  \nAs can be imagined, such a cube of cells is too coarse to capture the desired\nspherical symmetry of the shell. As discussed later, we have checked the sensitivity\nof our results to the size of this cube (i.e. the shell radius), and find that\nthe final results are practically unaltered even when we used a 5 x 5 x 5 matrix\nas a template cube to model the seed shell.\n\nAfter the\nfirst explosion is initiated, we evolve the simulation through a large\nnumber of iterations, each with a dynamic time step.  To calculate what\nthe time step should be for a particular iteration, we search the velocity\narrays for the maximum magnitude and use this to\nset the time step such that the ``movement\" at this speed is restricted\nto less than one pixel.  This procedure\nensures that, for a given iteration, the range of influence of any \npixel will never extend beyond a 3 x 3 x 3 cube around that pixel.  Thus, for each\niteration, we scan through the pixel set, ``evolve\" the contribution of each\npixel by considering its interaction with matter within\na small 3 x 3 x 3 influence cube around it, and store\/add that resultant contribution\nto a buffer set of arrays for mass and 3-d velocity.  \n\n\n\nWithin this smaller cube of {\\it influence}, we calculate the movement of the mass \nand use a spread function to decrease the effects of the quantization\nintroduced by the finite pixel size.  Let $\\hat{d}$ be a unit vector \ndefining the direction of the destination pixel w.r.t. the central (or source) pixel, and\nlet D be the associated distance. The displacement $\\delta$D in that direction is \nthen simply $(\\vec{V}.\\hat{d}) \\delta t$, where $\\vec{V}$ is the velocity vector\nand $\\delta t$ is the time step. We restrict the spread of the central\npixel's contribution to destination pixels with positive values of $\\delta$D,\ni.e. to only those pixels that can be reached in the `forward' direction.\nThe  fractional displacement f = $\\delta$D\/D  determines how the original mass\nis to be shared between the original pixel and the destination pixel.\nThus the fraction of the source mass transported to a destination pixel $i$ \nis $W_i.f$, where\n$W_i=A_i\/(\\sum_i A_i)$,  and $A_i = (\\vec{V}.\\hat{d}\/|V|)$ if the dot-product is\npositive, else $A_i = 0$.\nThis consideration allows us to weight the mass\ntransfer in the forward direction while still spreading it adequately laterally.\nTotal time is kept track of throughout the iterations, and another\nexplosion is initiated whenever the time after the last explosion reaches\na pre-computed interval. This entire process is\ncarried out until the input ``run time\" is complete, at which point the\nprogram switches into the ``evolution-only\" phase.  In this phase, we allow the\niterations to continue for a certain defined length of time, but\ndisallow any new supernova.  This phase of the simulation ensures the avoidance of\nany extreme densities (both high \\& low) due to a more recent SN\/SNR (closer to the \nend of the first phase) that has not yet had time to evolve and \ninteract with the surrounding matter. As our simulation is intended to \nstudy the structure of\nthe ISM, not the stars within it, this provision is necessary to make sure\nrecent explosions do not skew our results. \nFigure~\\ref{fig:col_den_sne_image} shows a sample result from a full \nsimulation in the form of a 2-d distribution of column density.\n\n\n\n\\noindent{\\section{Analysis \\& results}}\n\n  Throughout our simulation runs (numbering ten or more in each of the cases\nexplored), particularly in the ``Time Evolution\" approach, \nwe closely monitor a variety of parameters and distributions. Typically,\nfor the evolutionary model, every 50 or so\niterations, we output the results in the form of mass (or density) \\& velocity distributions\nand\nobtain power-spectral descriptions for these. \nFor estimating the power spectra, in general, a given 2-d distribution is first \nstripped of\nits ``mean\", and  then Fourier transformed. The resultant 2-d power spectrum is\nazimuthally averaged to obtain a 1-d description of the power as a function of\nthe spatial frequency (extending up to the sides, and not the corners).\nThe power estimates at\nhigher spatial frequencies benefit increasingly from the better statistics\nimplicit in azimuthal averaging as a function of radius.\nThese spectra are displayed on a log-log scale and best-fit power-law slopes\n(spectral indices) are computed over 3 (partly overlapping) ranges of the\nspatial frequencies to allow for the possibility that a given spectrum\nmay not be adequately described by a single power-law.\n\nIn the ``Snapshot\" simulation, we mainly monitor the column density \ndistribution.  We also examine the density distributions (in, say, the \nX-Y plane) corresponding to thin slices at various depths (i.e. at various Z \nvalues).  Here, the power spectra from several slices (typically 32) were \ncombined to obtain a better average description.  \n\nFor the ``Time Evolution\" simulation, we extend our analysis to include \ntwo more data sets.  The Z-velocity distribution is monitored (for thin Z-\nslices), as well as the column density distribution as a function of \nvelocity.  The latter can be most directly compared to observational results. \n\n\\noindent{\\subsection{The Snapshot Distribution}}\n\n\nThe analysis of the column density distribution in Figure~\\ref{fig:bubble_image} \nwas performed following the\nabove procedure, and the resultant power spectrum is shown in \nFigure~\\ref{fig:bubble_spectrum}a.\nAs noted earlier, the underlying 3-d distribution includes a large number \n(500,000) of bubbles\/voids. \nThe spatial scale or the size of the simulated box is not explicitly defined, however\nan approximate scale of 1 kpc may be associated with the width of the simulated box.\nThe best-fit power-law slope for the linear\nportion of this log-log plot is close to -2.9, steeper than $-8\/3$. \nInterestingly, the spectral steepness decreased monotonically (starting from a slope \nof about $-4$ for a single bubble) as the number of bubbles increased. This trend\nis understood as due to a) the improved statistics, in general,\nand b) the increased weightage for finer structure, in particular.\nAlso, importantly, the rate of change of the spectral slope is \nobserved to decrease steadily.\nThe very slow decrease in the steepness during the latter half of the simulation suggests\nthat even after a many fold increase in the bubble count, the slope may \nreduce only slightly from its\npresent value and may stabilize at $-8\/3$ or thereabouts. \nWe have repeated the simulation runs\nto assess different realizations and find the above trend consistently evident.\nIt may be worth mentioning, as an example, that the slope\napproached the Kolmogorov value ($-11\/3$) after about 10000 bubbles, \nby which time about 42\\% of the\nvolume was occupied by the voids. At the half-way mark and at the end of this \nsimulation run the \noccupancy was about 53\\% and 54\\%, respectively. \nA similar trend was apparent in the spectra\nof distributions in Z-slices of the simulated cube. \nThe slope here was consistently shallower than\nthat for the column density distribution (i.e. for a {\\it thick} slab), \nby about 1 in the power-law index, as expected.\n\n\nIn the above simulations, the sharp edges of the bubbles are expected to enhance, somewhat\nartificially, the power at higher spatial frequencies, and this would make the spectra\nshallower than we would find in the absence of sharp edges.\nTo assess this expectation, we modeled the voids with a smooth decrease in the density as\none moved from the shell boundary to void center. Exactly as anticipated, the spectral \nslopes were consistently steeper (e.g. Figure~\\ref{fig:bubble_spectrum}b)\nthan for those derived for bubbles with sharp-edges \n(e.g. Figure~\\ref{fig:bubble_spectrum}a). \nOn the average, the power-law slope approached $-3.4$ for the column-density \ndistribution and was shallower by one order, i.e. about $-2.4$, for the density in the Z-slices.\nFor both cases, i.e. the sharp and smooth bubbles, the simulations were repeated\nwith an additional constraint wherein the resultant distribution of radii was maintained\nstatistically uniform (as would be implied by a constant SN rate and SNR birth rate). \nThe resulting distributions  were\nindistinguishable in their appearance from the ones generated without the constraint.\nMore importantly, the derived spectra (and their slopes) were practically unaffected by \nthe additional constraint, implying that the results of these {\\it snap-shot} simulations\nare not sensitive to the distribution of radii of the bubbles populating the volume. \nIn these snap-shot simulations, we have treated SNRs as bubbles \"without shells\". \nSince the shells surrounding bubbles contribute significant HI column densities, we have\nsimulated distributions consisting of only shells (with a thickness to radius ratio of 0.1),\nand examined the corresponding power spectra. Here, the power-law slope approaches\n$-3$ for the total column density distribution, and $-2$ for the density in the Z-slices.\n\n\\noindent{\\subsection{Time Evolution}}\n\nFollowing the procedure described in section 3, many simulation runs\nwere carried out to study the results of a time-evolution approach. \nDue to memory and computing speed\nlimitations, we restricted the simulation volume to a (128)$^3$ pixel array, and \nwe artificially\nraised the SN rate by a factor of 100 from the rate appropriate for a (200 pc)$^3$\nvolume. The primary phase of the runs was chosen long enough so as to have every pixel\nof the volume visited by the ejecta. The nominal time scale of this primary phase\nwas a few million years, and the SN count was several hundred. \nDuring the ``evolution-only\" phase, corresponding to \na relatively longer duration of time when compared with the typical intervals \nbetween explosions, the {\\it rms} velocity reduced monotonically as shell diffusion \ncontinued without any further energy input.\n\nFigure~\\ref{fig:te_cd_spec} shows the spatial power spectrum of a sample column \ndensity distribution (such as in Figure~\\ref{fig:col_den_sne_image})\nobtained from these simulations. \nAlthough a single power-law is a poor \napproximation to this spectrum, its average (and the best-fit) power-law slope \ninterestingly is close to $-11\/3$ and does not change in any significant \nmanner during the `evolution only' phase. \nSpectra of distributions of column density in\nthin slabs (not shown here) reveal a similar non-`single-power-law' nature, \nand they show little variation in the `evolution only' phase. \nHowever, their average slope is consistently shallower than $-11\/3$,\nand closer to $-8\/3$. Power spectra for several slabs (at different Zs) \nwere averaged together to obtain statistically improved estimates.\n\n\n\nIt is also instructive to examine the velocity distribution and its spectrum. \nIn Figure~\\ref{fig:te_zv_spec}, we show a sample Z-velocity distribution in the X-Y plane at an arbitrary Z \nand an average of the spatial power spectra of such distributions at various Zs \n(computed separately and then averaged). A single power-law seems adequate to\ndescribe the spatial spectra of 1-d components of the 3-d velocity in our simulations.\nThe power-law slope is remarkably almost always $-11\/3$, or very close to it, \nand may change only slightly through\nthe `evolution only' phase as the velocity spread (range) \ndecreases by orders of magnitude.\nWhile forming these conclusions, we have of course ignored the behavior at the\nlower spatial frequency end due to its poor statistical significance.\n\n\nThe parameter and its distribution that can be most closely related to \nobservational results is the column density as a function of velocity, as in the\ncase of, for example, the 21-cm HI line observations. From the simulated distribution\nof mass and its 3-d velocity, we extract the spatial distribution \n(in, say, the X-Y plane)\nof the column density (i.e. integrated along Z) associated with a given range of the\nvelocity component along Z. In general, we examine about 30 velocity ranges\ncovering the estimated spread ($\\pm 1\\sigma$) of the 1-d velocity, \nand for each of these velocity slices, the transverse column-density distribution \nis Fourier analyzed and the \npower spectra are estimated. Such spectra for the different velocity slices \nare averaged together with natural weighting. \nFigure~\\ref{fig:te_vslice_spec} shows a sample\ncolumn-density distribution for one such velocity slice \n(across a velocity width of about 1 km\/s) \nand a power spectrum obtained by averaging power spectra for different slices.\nThe spectrum has a slope of $-8\/3$ at the higher spatial frequency end. \nIt is important to mention\nthat the same spectral slope also extended to lower spatial frequencies prior to\nentering the `evolution only' phase. Such velocity-sliced column-density\ndistributions appear to show considerable variation in their spectral signatures, \nin general, and more systematically as the `evolution only' phase progresses. \nThis is in contrast\nto the generally stable signatures evident for the other spectra we discussed above.\nHere, almost invariably, the average power-law slope is close to $-8\/3$ or shallower. \nThe latter is rare during the main-run phase, but is almost always the case during the \n`evolution only' phase. We have investigated this\naspect in greater detail and find that in this phase the spectrum becomes \nprogressively shallower, often accompanied by considerable flattening toward \nlower frequencies.  We attempt to understand this in the following way. \nAs mentioned earlier, the velocity spread ({\\it rms})\ndecreases monotonically through the `evolution only' phase in the absence \nof any further energy input. As the velocity dispersion decreases and \nmixing continues, the scale over which significant velocity coherence \nsurvives also decreases progressively. This naturally\nleads to significant additional corrugation of the transverse \ndistribution on correspondingly smaller spatial scales when viewed \nthrough a velocity filter that is narrow compared to\nthe velocity spread. In the present case, the velocity filter widths \nare chosen as a fixed fraction of the {\\it rms} velocity, and thus may help \nto reduce the corrugation effects, although the consequences of progressive shortening \nof the coherence scale seem unavoidable.\n\n\n\\noindent{\\section{Discussion}}\n\nBefore we discuss our results and their implications, it is important to \nmention that although our primary motivation related to the neutral atomic\nhydrogen distribution in the ISM, the attempted modeling is equally\nrelevant for the ionized hydrogen component. In our SN-driven\nISM, it is not far fetched to consider the existence of these two components\nas mutually exclusive in their spatial distributions. \nFor example, the HI voids correspond to the very\nspace that is ionized as a consequence of the SN event. HII regions and the thin\nHI shells around them (that are in turn surrounded by H$_2$ shells, i.e. molecular \nclouds) represent another example of the\nsomewhat mutually exclusive existence of the two\ncomponents. As long as we are concerned about only the structural features\nand spatial distributions, modeling of either of the two components\nwould have implicit and direct relevance for the other. In more practical terms,\nvoids\/bubbles could be replaced by spheres, and the answers in terms of\nspatial spectrum or structure function would be unaltered.\n\nAs was seen earlier, the density spectrum from our `snap-shot' simulation approaches a\npower-law slope of about -3.4 as the number of supernovae, and the consequent \nbubble-like SNRs, is increased, so long as the bubbles are modeled without \nunnaturally sharp edges. The results of our `time evolution' modeling reveal a spectral\nsignature of the 1-d velocity distribution that matches remarkably well with that\nexpected from Kolmogorov turbulence.  \nThis result appears significant and, in our opinion, is not\na chance coincidence. We remain however aware that though an incorrect\nspectral signature might be taken to imply an incorrect scenario,\ncorrect spectral signature does not necessarily imply the (sole) correct scenario. \n\n A Kolmogorov spectrum is theorized to\narise when energy cascades from larger scales down to smaller scales.\nImplicit in this, we believe, is a certain hierarchical \nstructure on a range of spatial scales.\nOur results from the snap-shot simulation seem to suggest that\nthe spectrum actually derived from observations\nmay primarily be the manifestation \nof structural features that are almost entirely dominated\nby the expanding voids\/shells of SN\/SNRs and\/or structures with similar morphology.  \nThis interesting result may also suggest\nthat the role any explicit interaction between the ejecta from nearby\nevents plays is probably not very crucial in defining the statistical distribution.\nThough the explicit interaction between nearby events has to occur,\nit may contribute significantly mainly at relatively larger spatial-scales,\ni.e. as in the formation of hot, connected tunnels (Cox \\& Smith, 1974).\n\nThe simple approach in our `time evolution' models also seems adequate\nto capture the key features of the evolution and the resultant structural\ndescription of the ISM. Each new supernova event is incorporated by introducing \na {\\it seed shell} with its velocity field determined \nbased on the mass and the energy input contributed by the SN. \nIt is remarkable that  momentum conservation alone\nas a governing feature in the subsequent evolution of a given shell \nas it interacts with the surrounding matter seems to yield velocity\nstructure that has a spatial spectrum similar to that for Kolmogorov turbulence.\nThis is not surprising, since momentum conservation is considered to be \nthe characteristic of the later phases (of SNR evolution) which last the longest.\n\nAlthough prompted by limitations in computing speeds to use SN rates\nthat were artificially raised, we have performed elaborate tests to verify that\nour basics results, i.e. the spatial spectral signatures, remain valid for a \nwide-range of SN rates. We have also checked and found that the exact shape\nof the SN-induced seed shell is not very critical, so long as sharp edges and corners\nare not present, the SN statistics are adequately large, and time spans are long enough\nto allow enough interaction with the surrounding matter.\n\nWe now turn to our specific results. \nFirstly, we find that our simple\nsimulations provide a more realistic\nstructural description of the ISM compared to when the distributions are \nsynthesized directly from a red spectrum across a spatial frequency range of interest \n(following a suitable power-law).\nThe present simulations are able to naturally produce 1-d and 2-d features such as\nfilaments and curved sheets in a manner that is internally consistent with \nthe diffuse component and the overall spatial spectrum. \nIn fact, such lower dimensional features apparent in our simulated density \ndistributions may be contributing to the deviations of the associated\nspatial spectrum from a single power-law nature.\nAlso, any mutual correlations between the velocity and density structures \nis implicit in our `time evolution'\nsimulations, an aspect that is rather difficult to incorporate by imposing explicit\ncorrelation properties, particularly since it could introduce an undesired bias.\n\nSecondly, it is instructive to note the significant differences between the\nspectral descriptions for the column density distributions and for\nthe distribution of 1-d components of the 3-d velocity. Despite the latter having \na consistent single power-law with Kolmogorov index, the former clearly deviates\nfrom it at both the low and high spatial frequencies. This may provide important\nclues for our understanding of apparent deviations from the Kolmogorov\nspectrum that interstellar scintillation observations have indicated\n(e.g., Roberts \\& Ables 1982), \nprompting serious need to invoke either steeper spectra (i.e., index $\\le-4$),\nor those with suitably larger `inner scale' (see, for example, \nGoodman \\& Narayan 1985; Blandford \\& Narayan 1985;\nRomani, Narayan \\& Blandford 1986).\nIf our results are to be taken at their face value,\nit appears to us that the spectral behavior needed (for column density\nacross thick or thin screens) to explain the observations\nmay not be in conflict with the underlying turbulence being of a Kolmogorov\nnature. Indeed, if the SN-induced turbulence is to be the dominant process\ndriving the ISM, there may not be any compelling need for invoking deviations\nfrom a basic Kolmogorov signature. The argument can be turned around to conclude\nthat the refractive scintillation observations are consistent with, \nand lend support to, \nthe possibility that the ISM may be primarily SN-driven.\n\nThirdly, we draw attention to our results on the distribution\nof density across narrow velocity slices. These reveal spatial spectra that\nare invariably significantly shallower than $-11\/3$, \nand rather close to $-8\/3$ or shallower,\neven while the spatial structure in velocity remains closely Kolmogorov.\nThese results can be readily compared with observations, and are found to indeed\nbe consistent with results derived from the distributions of Galactic\nHI emission and opacity observed in narrow frequency (or velocity) channels.\nThe unavoidable implications of such shallow spectra for the apparent\ncolumn density (or opacity) fluctuations on small transverse spatial scales\nhave been discussed in detail elsewhere (Deshpande 2000). \nFor the present discussion we \nfocus on the fact that the expected magnitude of such fluctuations\non a given transverse scale has a direct dependence on both the overall \nHI column-density (or opacity) and the details (such as the power-law slope)\nof its spatial spectrum. In this particular context, the noticeable\nvariation in the spectral characteristics associated with the velocity slices\nis of great significance. That is, we should not be surprised to find\na considerable variety in the distributions sampled by velocity slices,\nand consequently in the\nstrength of column density or opacity fluctuations while observing\nin different directions in the Galaxy. It may be recalled that the \nmore significant and systematic modifications of these spectra are seen to occur\nas our `evolution only' phase progresses. Had we used the expected SN rate\nfor the volume simulated (i.e. instead of an elevated rate), \nwe would expect the intervals between successive supernovae to be \nlong enough to be describable as genuine episodes of `evolution only' phase. \nGiven this, it is only reasonable to expect a rich variety of spectral characteristics\nto be evident episodically even while observing a given region of the Galaxy.\nIn practice, for observations made during a narrow epoch range, this would be\nthe case across the different regions of the Galaxy. Thus, the observed\nspatial variation of opacity in the case of even 3C138, often cited  \nas evidence for the so-called `tiny-scale structure' (Faison \\& Goss 2001, and\nreferences therein), becomes easily understandable\nas a simple manifestation of a power-law distribution of opacity across narrow\nvelocity slices.\n\nLastly, we will next extend our simple recipe such as to monitor and study\nother aspects characterizing the ISM. The effect of Galactic rotation\non the apparent distribution in velocity slices needs to be assessed, particularly\nwhen simulating much bigger volumes than in the present case. More importantly, \none hopes to keep track of other variables, such as temperature and the resulting ionized\nfraction of the ISM.  These extensions will be followed up in a separate paper.\n\n\\noindent{\\section{Conclusions\/Summary}}\n\nIn conclusion, we feel that our highly simplified approaches to the modeling of the\nISM have led to several promising and instructive results, as summarized below.\n\n \n1) Our `snap-shot' simulation of an SN-driven ISM, modeled as a set of \nnon-overlapping voids of a range of sizes, appears to be \nsuccessful in capturing the key structural features of the medium, \nincluding a power-law nature of the spatial spectrum (of slope about -3.4).\n\n2) Another, more realistic, approach following `time evolution' of the \nSN-induced shells also\nproduces a remarkable match with the structural features of the ISM derived from\navailable observations. A simple interaction with the surrounding matter,\nusing only momentum conservation, appears to be adequate to govern the\nevolution. \n\n3) Our simulations show that while the velocity distribution closely follows \na Kolmogorov spectrum, the column density\ndistribution shows deviations from this both for high and low spatial frequencies.\nThis may have important relevance to the properties of the medium as implied \nby observations of refractive scintillation.\n\n4) The spatial spectral characteristics of distributions across velocity slices \nand the significant variation between them suggest that the magnitude of the \ncolumn-density or opacity variation on small\ntransverse scales can differ considerably when observing different directions \nin the Galaxy.\n\n5) The present simple-minded approaches (i.e. the `time evolution' simulations) \nprovide a set of internally consistent \ndensity and velocity distributions that depict 1-d \\& 2-d structure in addition to the \ndiffuse component and can be very useful in detailed studies of other properties of the\nHI and ionized components of the interstellar medium.\n\n\\acknowledgments\nWe thank Chris Salter, V. Radhakrishnan, Bon-chul Koo and James Ingalls for their \ncritical reading of the manuscript and for useful comments.\nOur thanks also to an anonymous referee for several critical comments\nthat have helped us in improving the manuscript. \nWe gratefully acknowledge the excellent support from the computer groups at\nthe Arecibo Observatory and Cal Poly. \nThis work was made possible by support from the REU program of NAIC, funded\nby the NSF. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nThe purpose of this paper is to study hyperplane sections of certain schemes, including log smooth schemes. We first study the case of schemes over a field $k$ and prove two variants of Bertini's theorem for smoothness. The first one is a Bertini theorem for log smoothness. To state it, let be a monoid $Q$ and a map $Q\\to k$ such that $(k,Q)$ is a \\emph{log point} (\\ref{setup2}). Assume $X\\subset\\mathbb{P}^n_k$ is a subscheme, $H\\subset\\mathbb{P}^n_k$ a hyperplane, $X\\cap H$ the scheme-theoretic intersection, and $i:X\\cap H\\subset X$ the inclusion. Let $M$ be a log structure on $X$ and $f:(X,M)\\to (k,Q)$ a morphism of log schemes.\n\n\\begin{thm}[Bertini theorem for log smoothness (\\ref{bertini6})]\nAssume $f$ is a log smooth morphism. If $f$ is a \\emph{sharp} morphism of log schemes, then for $H$ generic $(X\\cap H, i^*M)$ is log smooth over $(k,Q)$.\n\\end{thm}\n\nThe notion of sharp morphism of log schemes is defined in \\ref{sharp}. Suffices to say here that sharpness is a quite natural condition which is mostly pertinent to positive characteristic (cf. \\ref{tame}). The theorem is optimal in the sense that we show by example that the sharpness assumption cannot be dropped in general (\\ref{cx}).\n\nThis Bertini theorem applies in particular to all varieties with toric singularities as well as singular schemes which are locally isomorphic to a normal crossings divisor in a smooth variety. In fact, the latter case was previously shown by Jannsen and Saito \\cite{jannsensaito} by reducing to the Bertini theorem for classical smoothness. By contrast, our proof is a direct logarithmic adaptation of the transcendence degree argument given by Jouanolou in \\cite{bertini}.\n\nThe second variant of Bertini's theorem, hinted at in \\cite[2.1]{fa2}, does not make any assumption on the singularities of $X$, but rather assumes the existence of a morphism of finite type $m:\\mathcal{X}\\to X$ and a vector bundle quotient $\\mathcal{E}$ of $m^*\\Omega^1_{X\/k}$.\n\n\\begin{thm}[Bertini theorem for abstract smoothness (\\ref{bertini7})]\\label{bertabstract}\nLet $\\mathcal{N}$ denote the conormal sheaf of $X\\cap H$ in $X$, and consider the natural morphism\n\\[ \\phi:m^*\\mathcal{N}\\to i^*\\mathcal{E} \\]\nof coherent sheaves on $\\mathcal{X}\\times_X(X\\cap H)$. If $\\dim\\mathcal{X}\\leq\\rk\\mathcal{E}$, then for $H$ generic the cokernel of $\\phi$ is a vector bundle of rank $\\rk\\mathcal{E}-1$ on $\\mathcal{X}\\times_X(X\\cap H)$.\n\\end{thm}\n\nIn the case $X$ is smooth and $\\mathcal{X}=X$ one recovers the classical Bertini theorem for smoothness; in fact, the proof of the latter given in \\cite{bertini} applies mutatis mutandis.\n\nWe then study hyperplane sections of singular schemes with the additional constraint that the hyperplanes all pass through a given point. In this case simple examples show that, unlike the classical smooth case, for $X$ log smooth one cannot guarantee that a generic hyperplane section through $P\\in X$ is log smooth at the point $P$ (take $X$ to be the union of coordinate axes in affine space and $P$ the origin). The interesting case is when $X$ is defined over a complete discrete valuation ring $V$ with singular special fibre. Assume the residue field $k$ of $V$ is algebraically closed and let $K$ denote the fraction field of $V$.\n\n\\begin{thm}[\\ref{maingoodnhbprop}]\\label{goodetalemap}\nAssume $\\cha(K)=0$. Let $X$ be a flat projective $V$-scheme with geometrically reduced generic fibre of dimension $d$ everywhere. Fix a closed point $P\\in X_k:=X\\otimes_Vk$. If $X$ is regular at $P$, then, up to modifying the projective embedding of $X$, for generic $V$-hyperplanes $H_1,...,H_d$ passing through $P$ the $V$-scheme $X\\cap H_1\\cap\\cdots\\cap H_d$ has \\'etale generic fibre.\n\\end{thm}\n\nThe basic idea of the proof is to find a nice model of $X_K:=X\\otimes_VK$ (in fact the blow up of $X$ at $P$) which has a convenient projective embedding for taking hyperplane sections.\n\nFinally, we use the results on hyperplane sections to construct $K(\\Pi,1)$-neighbourhoods of smooth $V$-schemes following Faltings \\cite{fa2}, and deduce that the $p$-adic nearby cycles can be computed locally as the Galois cohomology of the geometric generic fibre. For another exposition of this result, see Olsson \\cite{olsson}.\n\n\\subsection*{Acknowledgement}\nIn a previous version of this paper some claims were incorrect. I am indebted to Piotr Achinger for providing counterexamples to those claims.\n\n\\subsection*{Overview of the paper}\nIn \\S1 we define the notion of a sharp morphism of log schemes and give examples. Then we recall some facts about log smooth morphisms.\n\nIn \\S2 we set up and prove affine versions of the Bertini theorems. We first show the Bertini theorem for log smoothness \\ref{logbertini}. The idea of the proof is as follows. The argument of \\cite{bertini} is that if the generic hyperplane section of a smooth scheme $X$ were singular at a point $x$, then the resulting $k(x)$-linear relation between differentials in $\\Omega^1_{k(x)}$ would force the transcendence degrees to be too small. Our proof reduces to this very same argument. The key is that for a point $x$ on a \\emph{sharp} log scheme $X$ over a log point, the usual differentials $\\Omega^1_{k(x)}$ inject into the logarithmic differentials $\\omega^1_{k(x)}$ (\\ref{satz2}), therefore any relation between regular differentials in the latter is in fact a relation in the former. Then in \\ref{cx} we construct a counterexample which shows that the theorem is false in general for log smooth schemes which are not sharp. Afterwards we prove the Bertini theorem for abstract smoothness by applying the argument of \\cite{bertini}.\n\nIn \\S3 we show that the Bertini theorems in the quasi-projective case follow from the affine case already shown (\\ref{bertini6}, \\ref{bertini7}). Here we also follow the method sketched in \\cite{bertini}.\n\nIn \\S4, we study the case of schemes over a complete discrete valuation ring $V$. After some preliminaries in \\ref{special}, we study the case of a generic hyperplane through a given rational point $P\\in\\mathbb{P}^n(k)$. For classical smoothness, up to modifying the projective embedding there is a Bertini theorem (cf. \\cite[Exp. XI]{sga4.3}); the underlying idea is that in a suitable projective space any hyperplane corresponds to a hyperplane in $\\mathbb{P}^n_V$ through $P$, and one can apply the usual Bertini theorem there. Although one does not obtain information about the intersection at the point $P$, this suffices in the case $X$ has good reduction, see \\ref{smoothcase}. For the general case of Theorem \\ref{goodetalemap} we proceed as follows: By regularity, the intersection of a generic hyperplane through $P$ with $X$ is regular at $P$. To deal with the other points we use the Bertini theorem for abstract smoothness. Ultimately, what makes this argument work is the nice projective embedding of the blow up of $X$ at $P$ provided by \\ref{closedimmersion}.\n\nIn \\S5 we apply the previous results to construct good neighbourhoods of points on smooth projective $V$-schemes.\n\nIn \\S6 we construct the $K(\\Pi,1)$-neighbourhoods of points of smooth $V$-schemes. To finish we explain the relevance to $p$-adic nearby cycles.\n\n\\section{Sharp, tame and smooth morphisms of log schemes}\n\n\\subsection{References and notation}\nAs a general reference for logarithmic geometry we will use K. Kato's foundational articles \\cite{log}, although for a result on differentials we refer to Ogus' notes \\cite{logbook} for lack of other reference. All monoids are commutative with unit element denoted by $1$ (with the exception of $\\mathbb{N}$ and $\\mathbb{Z}$ whose unit element is traditionally denoted by $0$). Log structures are taken for the \\'etale topology. Given a morphism of monoids $Q\\to P$ and a $\\mathbb{Z}[Q]$-scheme $Y$ we write $Y_Q[P]:=Y\\times_{\\Spec(\\mathbb{Z}[Q])}\\Spec(\\mathbb{Z}[P])$.\n\n\\subsection{Log points}\\label{setup2}\nLet $(k,Q)$ be a \\emph{log point}, i.e. $k$ is field with a monoid $Q$ defining the log structure via the map $Q\\to k$ defined\n\\[ q \\mapsto\n\\begin{cases}\n1 & q=1 \\\\\n0 & q\\neq 1.\n\\end{cases} \\]\nA \\emph{monogenic log point} is the case where $Q$ is a monogenic monoid, i.e. $Q$ can be generated by a single element. The \\emph{standard log point} is the case where $Q=\\mathbb{N}$.\n\nWe consider log points as log schemes, so that a morphism of log points just means a morphism of the associated log schemes. An \\emph{extension of log points} $(k,Q)\\to (L,P)$ is a morphism of the log points, together with a map of monoids $Q\\to P$ giving a chart for this morphism.\n\n\\subsection{Sharp morphisms}\\label{sharp}\nLet $f:(X,M)\\to (Y,N)$ be a morphism of log schemes. Let $x\\to X$ be a geometric point, and $y=f(x)$. We say that $f$ is \\emph{sharp at $x$} if there is a chart $Q\\to P$ of $f$ at $x$ such that $P\\to k(x)$ and $Q\\to k(y)$ are log points.\n\nWe say that $f$ is \\emph{sharp} if $f$ is sharp at all $x\\to X$.\n\n\\begin{proposition}\\label{sharplemma}\nLet $f:(X,M)\\to (Y,N)$ be a morphism of log schemes.\n\\begin{enumerate}[(i)]\n\\item If $f$ is sharp at a geometric point $x\\to X$, then, in the above notation, $P$ and $Q$ are sharp monoids and $(k(y),Q)\\to (k(x),P)$ is an extension of log points.\n\\item If $N$ is the trivial log structure and $M$ is fine and saturated, then $f$ is sharp.\n\\item If $f$ is sharp and $g:Z\\to X$ is a morphism of schemes, then $f\\circ g:(Z,g^{\\ast}M)\\to (Y,N)$ is sharp.\n\\item If $f$ is sharp and $g:Z\\to Y$ is a morphism of schemes, then $g^*(f):(X\\times_YZ,(f^*(g))^{\\ast}M)\\to (Z,g^*N)$ is sharp.\n\\item If $f$ is sharp at a geometric point $x\\to X$ and $M$ is integral (resp. fine, resp. saturated), then, in the above notation, $P$ is integral (resp. fine, resp. saturated).\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n(i), (iii) and (iv) follow easily from the definition. For (ii), pick a geometric point $x\\to X$ and let $P=M_x\/\\mathcal{O}_{X,x}^{\\ast}$. Then it is well-known that the map $M_x\\to P$ has a section $P\\to M_x$ such that the log structure associated to $P\\to \\mathcal{O}_X$ is the log structure $\\alpha:M\\to \\mathcal{O}_X$ in a neighbourhood of $x$. This is obtained by choosing a section of $M^{\\gp}_x\\to P^{\\gp}$ which exists because $P^{\\gp}$ is a free abelian group (since $M$ is fine and saturated). Let $p\\in P$. If $p\\in\\alpha^{-1}(\\mathcal{O}_{X,x}^{\\ast})\\cong \\mathcal{O}_{X,x}^{\\ast}$, then  $p=1$ by construction. So $P\\to k(x)$ is a log point. The fact that $P$ is sharp follows immediately from its definition. Since $N$ is the trivial log structure, we may take $Q=\\{1\\}$ and this proves that $f$ is sharp at $x$.\n\nFor (v), it suffices to note that the canonical map $P\\oplus\\mathcal{O}_{X,x}^{\\ast}\\to M_x$ is an isomorphism.\n\\end{proof}\n\nThe main case of interest in this paper is when $(Y,N)$ is a log point or the spectrum of discrete valuation ring with its canonical log structure. See \\ref{exsharp} below for an example of a sharp morphism over the standard log point, and \\ref{satz1} for a whole class of log schemes over a monogenic log point which are sharp.\n\n\\subsection{Example}\\label{exsharp}\nSuppose $X$ is a $k$-scheme with a geometric point $x\\to X$ such that locally for the \\'etale topology at $x$, $X$ is isomorphic to the spectrum of the ring\n\\[ \\dfrac{k(x)[T_1,...,T_{d+1}]}{(\\prod_{i=1}^rT_i^{e_i})} \\]\nfor some $e_i\\in\\mathbb{N}$, with $T_i(x)=0$ for $i=1,...,d+1$. We can define a log structure in a neighbourhood $U$ of $x$ by the map\n\\[ \\mathbb{N}^r\\to\\mathcal{O}_U:(n_1,..,n_r)\\mapsto\\prod_{i=1}^rT_i ^{n_i} \\]\nsuch that if we give $k$ the standard log point structure (\\ref{setup2}), then the map\n\\[ \\mathbb{N}\\to\\mathbb{N}^r:1\\mapsto (e_1,...,e_r) \\]\ndefines a morphism of log schemes $(U,\\mathbb{N}^r)\\to (k,\\mathbb{N})$.\n\n\\begin{lemma}\nThe morphism $(U,\\mathbb{N}^r)\\to (k,\\mathbb{N})$ is sharp at $x$.\n\\end{lemma}\n\\begin{proof}\nIt suffices to check that the induced map $\\mathbb{N}^r\\to k(x)$ defines the structure of a log point and that the diagram\n\\[ \\begin{CD}\n\\mathbb{N} @>>> \\mathbb{N}^r \\\\\n@VVV @VVV \\\\\nk @>>> \\mathcal{O}_{X,x}\n\\end{CD} \\]\ncommutes. The latter is obvious and since $T_i(x)=0$ for all $i$, $(k(x),\\mathbb{N}^r)$ is a log point.\n\\end{proof}\n\n\\subsection{Tame morphisms}\\label{tame}\nLet $f:(X,M)\\to (Y,N)$ be a morphism of log schemes. We say that $f$ is \\emph{tame} if the sheaf of abelian groups $M^{\\gp}\/(f^{\\ast}N)^{\\gp}$ has no torsion divisible by any of the residue characteristics of $Y$. (For any sheaf of monoids $M$ we write $M^{\\gp}$ for the associated sheaf of abelian groups.)\\\\\n\nThe interest of tame morphisms is that they provide a large class of sharp morphisms over a monogenic log point, as the following result shows.\n\n\\begin{proposition}\\label{satz1}\nSuppose that $f:(X,M)\\to (k,Q)$ is a morphism of log schemes with $(X,M)$ fine and saturated and $(k,Q)$ a monogenic log point. For any geometric point $x\\to X$ there is a fine saturated and sharp monoid $P$ defining $M$ in a neighbourhood of $x$, such that:\n\\begin{enumerate}[(i)]\n\\item $P\\to k(x)$ is a log point\n\\item if $f$ is tame, then there is a map $Q\\to P$ defining a chart of $f$ at $x$.\n\\end{enumerate}\nIn particular, if $f$ is tame, then $f$ is sharp.\n\\end{proposition}\n\\begin{proof}\nWe let $P=M_x\/\\mathcal{O}_{X,x}^*$ and then (i) can be proved as in \\ref{sharplemma} (ii).\n\nFor (ii), let $q\\in Q$ be a generator. Let $p_1,...,p_r$ be a set of generators of $P^{\\gp}\\cong\\mathbb{Z}^r$ and fix a section $s:P^{\\gp}\\to M_x^{\\gp}$. Since $M_x^{\\gp}=P^{\\gp}\\oplus\\mathcal{O}_{X,x}^{\\ast}$, we may write $q=u\\prod_{i=1}^rs(p_i)^{n_i}$ for some $u\\in\\mathcal{O}_{X,x}^{\\ast}$. Let $n=(n_1,...,n_r)$ be the greatest common divisor. Since $f$ is tame, $n$ is prime to $\\cha(k)$. Let $v\\in\\mathcal{O}_{X,x}^*$ satisfy $v^n=u$. There are integers $a_1,...,a_r$ such that $\\sum_{i=1}^ra_in_i=n$. Then we have $q=\\prod_{i=1}^r(v^{a_i}s(p_i))^{n_i}$. The section $s':P^{\\gp}\\to M_{x}^{\\gp}$ defined by $s'(p_i):= v^{a_i}s(p_i)$ is such that $q$ lies in $s'(P^{\\gp})\\cap M_x=s'(P)$. The induced map $Q\\to P$ is the desired chart.\n\\end{proof}\n\nSee Example \\ref{cx} for an example of a non-tame log smooth scheme over the standard log point which is not sharp.\n\n\\subsection{Log smooth morphisms}\\label{logsmoothdef}\nRecall that a morphism $f:(X,M)\\to (Y,N)$ of fine log schemes is \\emph{log smooth} if and only if, locally for the \\'etale topology, there exists a chart $Q\\to P$ of $f$ with $P$ and $Q$ fine monoids such that\n\\begin{enumerate}[(a)]\n\\item $\\ker(Q^{\\gp}\\to P^{\\gp})$ and the torsion subgroup of $\\cok(Q^{\\gp}\\to P^{\\gp})$ are finite groups of order invertible on $Y$\n\\item the induced map $X\\to Y_Q[P]$ is \\'etale.\n\\end{enumerate}\n$f$ is \\emph{log \\'etale} if, in addition, $\\cok(Q^{\\gp}\\to P^{\\gp})$ is a finite group of order invertible on $Y$.\n\n\\begin{remark} In spite of the above characterization, log smooth morphisms are not necessarily tame (see \\ref{cx} for an example).\n\\end{remark}\n\n\\subsection{Log differentials}\nLet $f:(X,M)\\to (Y,N)$ be a morphism of fine log schemes. We write $\\omega^1_{(X,M)\/(Y\/N)}$ for the module of relative logarithmic differentials.\n\n\\subsubsection{}\\label{diff}\nGiven any morphism $g:(Y,N)\\to (Z,L)$ of fine log schemes there is a right-exact sequence of sheaves of relative logarithmic differentials\n\\[ \\begin{CD}\n0 @>>> f^{\\ast}\\omega^1_{(Y,N)\/(Z,L)} @>>> \\omega^1_{(X,M)\/(Z,L)} @>>> \\omega^1_{(X,M)\/(Y,N)} @>>> 0\n\\end{CD} \\]\nand $f$ is log smooth if $g\\circ f$ is log smooth and the sequence is left-exact and locally split (\\cite[3.12]{log}).\n\n\\subsubsection{}\\label{jacobi}\nLet $i:Z\\hookrightarrow X$ be a closed immersion of ideal sheaf $\\mathcal{I}$ and give $Z$ the inverse image log structure of $(X,M)$ (making $i$ a strict closed immersion). Then there is a right-exact sequence\n\\[ \\begin{CD}\n0 @>>>\\mathcal{I}\/\\mathcal{I}^2 @>>> i^{\\ast}\\omega^1_{(X,M)\/(Y,N)} @>>> \\omega^1_{(Z,i^{\\ast}M)\/(Y,N)} @>>> 0\n\\end{CD} \\]\nand if $f$ is log smooth, then $f\\circ i$ is log smooth if and only if the sequence is left-exact and locally split. See \\cite[IV, 3.2.2]{logbook} for the proof.\n\n\\subsection{Generic log smoothness}\\label{generic}\nLet $f:(X,M)\\to (Y,N)$ be a morphism of finite presentation of fine log schemes with $Y$ irreducible. Assume moreover there is a chart $Q\\to N$ with $Q$ a fine monoid. Let $\\xi\\in Y$ be the generic point, endowed with the inverse image log structure of $(Y,N)$. We claim that if $f^{-1}(\\xi)$ is log smooth over $\\xi$, then there exists a dense open $V\\subset Y$ such that $f|_V$ is log smooth. We may assume $Y$ is quasi-compact. For each geometric point $x\\to f^{-1}(\\xi)$ there is an \\'etale neighbourhood $Z^x\\to f^{-1}(\\xi)$ of $x$, a chart $P^x\\to M|_{Z^x}$ with $P^x$ a fine monoid, and a map $Q\\to P^x$ such that $Z^x\\to \\xi_Q[P^x]$ is \\'etale and the map $Q\\to P^x$ satisfies property (a) of \\ref{logsmoothdef}. Moreover, since $M$ is fine, $P^x$ defines the log structure of $X$ in an \\'etale neighbourhood $X^x$ of $x$. Since $f$ is of finite presentation, up to replacing $X^x$ by an \\'etale neighbourhood of $x$, there is an open neighbourhood $Y^x$ of $\\xi$ in $Y$ and a morphism of finite presentation $X^x\\to (Y^x)_Q[P^x]$. Namely, we may take $Y^x$ to be a dense open subset of the image of $X^x\\to X\\overset{f}{\\to}Y$, up to replacing $X^x$ by $X^x\\times_YY^x$. Since this morphism becomes \\'etale at the point $\\xi\\in Y^x$, by \\cite[IV, 17.7.8]{ega} there is a dense open $U^x\\subset Y^x$ such that the map $X^x\\times_YU^x\\to (U^x)_Q[P^x]$ is \\'etale. Now since $f$ is of finite presentation, there is a finite number of points $x_1,...,x_n\\in f^{-1}(\\xi)$ such that the map $\\coprod_{i=1}^nX^{x_i}\\times_Y\\xi\\to f^{-1}(\\xi)$ is an \\'etale covering. If $U:=U^{x_1}\\cap U^{x_2}\\cap\\cdots\\cap U^{x_n}$, then there is a dense open $V\\subset U$ such that $\\coprod_{i=1}^nX^{x_i}\\times_YV\\to X\\times_YV$ is an \\'etale covering, and this proves the claim.\n\n\\subsection{Notation}\nIn the sequel we will usually drop the log structures from notation when it is clear which log structures are meant. Also we will always write $\\omega^1_{X\/Y}$ for logarithmic differentials, as opposed to the usual differentials $\\Omega^1_{X\/Y}$.\n\n\\section{Hyperplane sections}\n\nLet $k$ be a field.\n\n\\subsection{Affine hyperplane sections}\\label{setup}\nLet $X$ be a $k$-scheme of finite type and $f:X\\to\\mathbb{A}^n_k$ a morphism of $k$-schemes defined by global sections $f_1,...,f_n\\in\\Gamma(X,\\mathcal{O}_X)$. Let\n\\[ Z:=\\underline{\\Spec}_X\\left(\\mathcal{O}_X[U_0,...,U_n]\/(U_0+U_1f_1+U_2f_2+...+U_nf_n)\\right). \\]\nThen we have a projection morphism $q:Z\\to\\mathbb{A}^{n+1}_k$ whose fibre over a point $u=(u_0,...,u_n)$ is the fibre $f^{-1}(H_u)$, where $H_u$ is the hyperplane of equation $u_0+u_1T_1+u_2T_2+...+u_nT_n=0$ where $T_1,...,T_n$ are the global coordinates on $\\mathbb{A}^n_k$ with images $f_1,...,f_n$ respectively in $\\Gamma(X,\\mathcal{O}_X)$.\n\nThere is a natural isomorphism\n\\[ Z\\cong X\\times_k\\mathbb{A}^n_k \\]\nobtained by the map\n\\begin{eqnarray*}\n\\mathcal{O}_X[U_0,...,U_n]\/(U_0+U_1f_1+...+U_nf_n) &\\to &\\mathcal{O}_X[U_1,...,U_n]\\\\\nU_0 &\\mapsto& -\\sum_{i=1}^nU_if_i\n\\end{eqnarray*}\nso that $Z$ can viewed as trivial vector bundle on $X$.\n\nLet $\\mathbb{K}$ be the function field of $\\mathbb{A}^{n+1}_k$. We write $Z_\\mathbb{K}$ for the generic fibre of $q:Z\\to\\mathbb{A}^{n+1}_k$, $X_\\mathbb{K}:=X\\otimes_k\\mathbb{K}$, and $i_\\mathbb{K}:Z_\\mathbb{K}\\hookrightarrow X_\\mathbb{K}$ for the closed immersion deduced from $i$. Note that by \\cite[I, 6.3.18]{bertini} $q$ is a dominant morphism if and only if $\\dim\\overline{f(X)}>0$, hence $Z_{\\mathbb{K}}$ is non-empty if and only if $\\dim\\overline{f(X)}>0$.\n\nWe refer to \\cite[I, 6]{bertini} for further details.\n\n\\subsection{Bertini theorem for log smoothness}\nLet $(k,Q)$ be a log point, with $Q$ a fine monoid. We endow $\\mathbb{A}^{n+1}_k$ with the inverse image log structure of $(k,Q)$ via the canonical morphism $\\mathbb{A}^{n+1}_k\\to\\Spec(k)$. If $(X,M)\\to (k,Q)$ is a morphism of log schemes, then we have the fibre product $X\\times_{k}\\mathbb{A}^{n+1}_k$ in the category of log schemes (the log structure is just the inverse image of $M$ under the projection $X\\times_{k}\\mathbb{A}^{n+1}_k\\to X$). Finally, we give $i:Z\\hookrightarrow X\\times_{k}\\mathbb{A}^{n+1}_k$ the inverse image log structure.\n\nEndow $\\mathbb{K}$ with the inverse image log structure of $\\mathbb{A}^{n+1}_k$, $X_{\\mathbb{K}}$ the inverse image log structure of $X\\times_k\\mathbb{A}^{n+1}_k$, and $i_\\mathbb{K}:Z_\\mathbb{K}\\hookrightarrow X_\\mathbb{K}$ with the inverse image log structure of $X_{\\mathbb{K}}$.\n\n\\begin{theorem}\\label{main}\nWith the above notation and hypothesis.  Assume that $(X,M)$ is a sharp log $(k,Q)$-scheme. If $f$ is unramified, then for any $z\\in Z_\\mathbb{K}$ the sequence\n\\[ \\begin{CD}\n0 @>>> k(z) @>{1\\mapsto \\sum_{i=1}^nU_idf_i}>> i_\\mathbb{K}^{\\ast}\\omega^1_{X_\\mathbb{K}\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{Z_{\\mathbb{K}}}}k(z) @>>> \\omega^1_{Z_\\mathbb{K}\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{Z_{\\mathbb{K}}}}k(z) @>>> 0\n\\end{CD} \\]\nis exact.\n\\end{theorem}\n\nThis theorem will be proved below. From it one derives the Bertini theorem for log smoothness.\n\n\\begin{corollary}\\label{logbertinigeneric}\nWith notation and hypothesis as in \\ref{main}. If $X$ is log smooth over $(k,Q)$, then $Z_\\mathbb{K}$ is log smooth over $(\\mathbb{K},Q)$.\n\\end{corollary}\n\\begin{proof}\nChoose $z\\in Z_{\\mathbb{K}}$. By \\ref{main} there is an ideal $I\\subsetneq\\mathcal{O}_{Z_{\\mathbb{K}},z}$ such that the sequence\n\\[ \\begin{CD}\n0 @>>> \\mathcal{O}_{Z_{\\mathbb{K}},z}\/I @>{1\\mapsto \\sum_{i=1}^nU_idf_i}>> (i_\\mathbb{K}^{\\ast}\\omega^1_{X_\\mathbb{K}\/\\mathbb{K}})_z @>>> (\\omega^1_{Z_{\\mathbb{K}}\/\\mathbb{K}})_z @>>> 0\n\\end{CD} \\]\nis exact. Since $\\omega^1_{X_\\mathbb{K}\/\\mathbb{K}}\\cong\\omega^1_{X\/k}\\otimes_k\\mathbb{K}$ is locally free, \\ref{main} implies that $\\text{Tor}_1^{\\mathcal{O}_{Z_{\\mathbb{K}},z}}((\\omega^1_{Z_{\\mathbb{K}}\/\\mathbb{K}})_z,k(z))=0$, i.e. $(\\omega^1_{Z_{\\mathbb{K}}\/\\mathbb{K}})_z$ is flat, hence so is $\\mathcal{O}_{Z_{\\mathbb{K}},z}\/I$. Thus, tensoring the exact sequence\n\\[ 0\\to I\\to \\mathcal{O}_{Z_{\\mathbb{K}},z}\\to \\mathcal{O}_{Z_{\\mathbb{K}},z}\/I\\to 0 \\]\nwith $k(z)$ we get an exact sequence\n\\[ 0\\to I\\otimes_{\\mathcal{O}_{Z_{\\mathbb{K}},z}}k(z)\\to k(z)\\to k(z)\\to 0 \\]\nand therefore $I\\otimes_{\\mathcal{O}_{Z_{\\mathbb{K}},z}}k(z)=0$, whence $I=0$ by Nakayama's lemma. It now follows from \\ref{jacobi} that $Z_{\\mathbb{K}}$ is log smooth over $(\\mathbb{K},Q)$ at $z$.\n\\end{proof}\n\n\\begin{corollary}\\label{logbertini}\nWith notation and hypothesis as in \\ref{main}. If $X$ is quasi-compact and $(X,M)\\to (k,Q)$ is log smooth, then there is a dense open $V\\subset \\mathbb{A}^{n+1}_k$ such that $q^{-1}(V)\\to V$ is log smooth over $(k,Q)$.\n\\end{corollary}\n\\begin{proof}\nThis follows from \\ref{generic}.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{main}]\nThe proof is in the style of \\cite[I, 6.3]{bertini}. We may assume that $k$ is algebraically closed. We first show a lemma.\n\n\\begin{lemma}\\label{satz2}\nLet $(k,Q)\\to (L,P)$ be an extension of log points. Then the natural map\n\\[ \\Omega^1_{L\/k}\\to\\omega^1_{L\/k} \\]\nhas a section, in particular is injective.\n\\end{lemma}\n\\begin{proof}\nLet $N=Q\\oplus k^{\\ast}$ and $M=P\\oplus L^{\\ast}$ be the log structures. Note that if $(p,u)\\in P\\oplus L^{\\ast}$ then the image of $(p,u)$ in $L$ is 0 unless $p=1$. Let $d:L\\to\\Omega^1_{L\/k}$ be the exterior derivative and let $\\partial:M\\to\\Omega^1_{L\/k}$ be the homomorphism of monoids $M=P\\oplus L^{\\ast}\\to\\Omega^1_{L\/k}:(p,u)\\mapsto u^{-1}du$. Since the map $N\\to M$ maps $Q$ to $P$, one checks easily that the pair $(d,\\partial)$ forms a log derivation of $(L,M)$ to $\\Omega^1_{L\/k}$ over $(k,N)$ (\\cite[5.1]{logsmooth}). So, by \\cite[5.3]{logsmooth}, there is a unique $L$-linear map $s:\\omega^1_{L\/k}\\to\\Omega^1_{L\/k}$ such that $d=s\\circ d$ and $\\partial=s\\circ\\dlog$, where $\\dlog:M\\to\\omega^1_{L\/k}$ is the canonical map (the hypothesis that the log structures be fine is superfluous, cf. \\cite[IV, 1.1.6]{logbook}). In particular, the composition of $s$ with the canonical map $\\Omega^1_{L\/k}\\to\\omega^1_{L\/k}$ maps $dx\\in\\Omega^1_{L\/k}$ to itself, hence is the identity map by linearity.\n\\end{proof}\n\nLet $z\\in Z_{\\mathbb{K}}$ and suppose for a contradiction that the map\n\\[ \\begin{CD}\nk(z) @>{1\\mapsto \\sum_{i=1}^n=U_idf_i}>> \\omega^1_{X_\\mathbb{K}\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{X_\\mathbb{K}}}k(z)\n\\end{CD} \\]\nis zero. Let $x\\in X$ be the image of $z\\in Z_\\mathbb{K}$ and let $d=\\td(k(x)\/k)$ be the transcendence degree of $k(x)$ over $k$. Note that $d>0$: otherwise $Z_{\\mathbb{K}}\\times_X\\Spec(k(x))=\\Spec(k(U_0,...,U_n)\/(\\sum_iU_if_i))=\\emptyset$ and there is no point $z$ lying above $x$.\n\nChoose an algebraic closure $\\overline{k(x)}$ of $k(x)$ and let $k(x)^{\\text{sep}}$ be the separable algebraic closure of $k(x)$ in $\\overline{k(x)}$. Let $\\bar{x}\\to X$ be the geometric point lying above $x$ corresponding to the inclusion $k(x)\\subset k(x)^{\\text{sep}}$. Since $X$ is sharp over $k$, by definition (\\ref{sharp}) there is a chart $Q\\to P$ of $f$ at $\\bar{x}$ such that the induced map $P\\to k(\\bar{x})$ is a log point. Then the map $(k,Q)\\to (k(\\bar{x}),P)$ is an extension of log points, so by \\ref{satz2} the quotient $\\omega^1_{k(\\bar{x})\/k}\\cong\\omega^1_{k(x)\/k}\\otimes_{k(x)}k(\\bar{x})$ of $\\omega^1_{X\/k}\\otimes_{\\mathcal{O}_X}k(\\bar{x})$ contains $\\Omega^1_{k(\\bar{x})\/k}\\cong\\Omega^1_{k(x)\/k}\\otimes_{k(x)}k(\\bar{x})$. Thus, the canonical map $\\Omega^1_{k(x)\/k}\\to\\omega^1_{k(x)\/k}$ is injective. Hence $\\sum_iU_idf_i=0$ in $\\Omega^1_{k(x)\/k}\\otimes_{k(x)}k(z)$. Since $k$ is algebraically closed, it follows that $d=\\dim_{k(x)}\\Omega^1_{k(x)\/k}$. Since $f:X\\to\\mathbb{A}^n_k$ is unramified, $\\Omega^1_{k(x)\/k}$ is generated by $df_1,...,df_n$, so without loss of generality we may assume that $df_1,...,df_d$ form a $k(x)$-basis of $\\Omega^1_{k(x)\/k}$. Thus, for $i=d+1,...,n$ we may write $df_i=\\sum_{j=1}^da_{ij}df_j$ for some $a_{ij}\\in k(x)$. Then the relation $\\sum_iU_idf_i=0$ gives the following equations\n\\[ U_j+\\sum_{i=d+1}^nU_ia_{ij}=0 \\]\nfor each $j=1,...,d$. So we deduce that $U_1,...,U_d\\in k(z)$ belong to the $k(x)$-vector space generated by $U_{d+1},U_{d+2},...,U_n$. Moreover, since $U_0=-\\sum_{i=1}^nU_if_i$, it follows that\n\\[ k(z)=k(x)(U_0,U_1,...,U_n)=k(x)(U_{d+1},U_{d+2},...,U_n) \\]\nand so\n\\[ \\td(k(z)\/k(x))\\leq n-d. \\]\nOn the other hand we have\n\\begin{eqnarray*}\n\\td(k(z)\/k(x)) &=& \\td(k(z)\/k)-\\td(k(x)\/k) \\\\\n&\\geq & n+1-d\n\\end{eqnarray*}\nwhich is the desired contradiction.\n\\end{proof}\n\n\\begin{cx}\\label{cx}\nWe give an example of a non-sharp (and non-tame) log smooth scheme over the standard log point $(k,\\mathbb{N})$ whose generic hyperplane section is nowhere log smooth. Let $0\\neq p=\\cha(k)$ and consider the morphism of monoids\n\\begin{eqnarray*}\nh:\\mathbb{N} &\\to & \\mathbb{N}\\oplus\\mathbb{Z} \\\\\n1 &\\mapsto & (p,1).\n\\end{eqnarray*}\nThen $\\cok(h^{\\gp})\\cong\\mathbb{Z}$, hence\n\\[ X:=\\Spec(k)_{\\mathbb{N}}[\\mathbb{N}\\oplus\\mathbb{Z}]\\cong\\Spec\\left(k[t,u^{\\pm 1}]\/(t^p)\\right) \\]\nwith its natural log structure is log smooth over $(k,\\mathbb{N})$. Let $M$ denote the log structure on $X$, $N$ the standard log point structure on $k$, and $f:(X,M)\\to (k,N)$ the morphism given by map $h$. Note that the image of $(1,0)\\in\\mathbb{N}\\oplus\\mathbb{Z}$ in $M^{\\gp}\/(f^{\\ast}N)^{\\gp}$ is $p$-torsion: we have $p(1,0)=(p,0)=h(1)+(0,-1)$ and $(0,-1)\\in\\mathcal{O}_X^*$. So $f$ is not tame.\n\nWe claim that $f$ is not sharp. If not, then let $x\\to X$ be a geometric point and a chart $Q:=\\mathbb{N}\\to P$ of $f$ at $x$ such that $P\\to k(x)$ is a log point. Then we have an isomorphism $P\\oplus k(x)^*\\cong M_{k(x)}$. On the other hand, one sees easily that $M_{k(x)}=\\mathbb{N}\\oplus k(x)^*$, where $\\mathbb{N}\\to k(x)$ is the standard log point with $\\mathbb{N}$ generated by $t$. From this it follows that $P\\cong\\mathbb{N}$ as monoids, hence if $n\\in P$ is a generator, its image in $M_{k(x)}$ is of the form $tv$ for some $v\\in k(x)^*$. Now let $q=1\\in\\mathbb{N}=Q$ be the generator. By definition, its image in $M_{k(x)}$ is $t^pu$. On the other hand, its image in $P$ is $(tv)^m$ for some $m$. Thus, we must have $m=p$ and $u=v^p$. In particular, if the image of $x$ in $X$ is the generic point, then $k(x)$ is a separable closure of $k(u)$ so this is impossible. This proves the claim.\n\nNow set $f_1=t,f_2=u,f_3=u^{-1}$ and use the same notation as before. Pick a point $z\\in Z_{\\mathbb{K}}$ with image $x\\in X$. Then in $\\omega^1_{X\/k}$ we have\n\\[ 0=\\dlog(q)=\\dlog(p,1)=p\\dlog(1,0)+\\dlog(0,1)=\\dlog(0,1)=\\dlog(u) \\]\nso $du=0$ in $\\omega^1_{X\/k}$. Since $dt=t\\dlog(1,0)$ and $t(x)=0$, it follows that\n\\[ U_1dt+U_2du+U_3d(u^{-1})=0 \\]\nin $\\omega^1_{X_\\mathbb{K}\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{X_\\mathbb{K}}}k(z)$, so a generic hyperplane section of $X$ is nowhere log smooth over $(k,\\mathbb{N})$.\n\\end{cx}\n\n\\subsection{A generalization of the Bertini theorem for smoothness}\\label{bertinigeneral}\nLet again $f:X\\to\\mathbb{A}^n_k$ and $Z$ be as in \\ref{setup}. Here we do not assume the presence of any log structures. Assume there is a morphism of finite type $m:\\mathcal{X}\\to X$ with $\\dim \\mathcal{X}\\leq d$ and a surjective map of $\\mathcal{O}_\\mathcal{X}$-modules\n\\[ m^*\\Omega^1_{X\/k}\\twoheadrightarrow\\mathcal{E} \\]\nwith $\\mathcal{E}$ a vector bundle of rank $d$ on $\\mathcal{X}$. Define $\\mathcal{Z}=\\mathcal{X}\\times_XZ$. As before we let $\\mathbb{K}=k(U_0,...,U_n)$. Now, the map\n\\[ \\begin{CD} \\mathcal{O}_Z @>{1\\mapsto\\sum_{i=1}^nU_idf_i}>> \\Omega^1_{X\/k}\\otimes_{\\mathcal{O}_X}\\mathcal{O}_Z \\end{CD} \\]\nextends to a map\n\\[ \\mathcal{O}_\\mathcal{Z}\\overset{\\phi}{\\to}\\mathcal{E}\\otimes_{\\mathcal{O}_\\mathcal{X}}\\mathcal{O}_\\mathcal{Z}. \\]\nSet\n\\[ \\mathcal{F}:=\\cok(\\phi) \\]\nthe cokernel of the map $\\phi$. So we have a commutative diagram with exact rows\n\\[ \\xymatrix{\n\\mathcal{O}_{\\mathcal{Z}} \\ar[r] \\ar@{=}[d] & \\Omega^1_{X\/k}\\otimes_{\\mathcal{O}_{X}}\\mathcal{O}_{\\mathcal{Z}} \\ar[d] \\ar[r] & \\Omega^1_{Z\/k}\\otimes_{\\mathcal{O}_Z}\\mathcal{O}_{\\mathcal{Z}} \\ar[d] \\ar[r] & 0 \\\\\n\\mathcal{O}_{\\mathcal{Z}} \\ar[r]^{\\phi} & \\mathcal{E}\\otimes_{\\mathcal{O}_\\mathcal{X}}\\mathcal{O}_\\mathcal{Z} \\ar[r] & \\mathcal{F} \\ar[r] & 0\n}\n\\]\n\n\\begin{theorem}\\label{bertgeneral}\nIf $f$ is unramified, then $\\mathcal{F}_{\\mathbb{K}}$ is a rank $d-1$ vector bundle quotient of $\\Omega^1_{Z_{\\mathbb{K}}\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{Z_{\\mathbb{K}}}}\\mathcal{O}_{\\mathcal{Z}_{\\mathbb{K}}}$. Moreover, if $\\mathcal{Z}_{\\mathbb{K}}\\neq\\emptyset$, then $\\dim\\mathcal{Z}_{\\mathbb{K}}\\leq d-1$.\n\\end{theorem}\n\\begin{proof}\nNote that the last statement follows from \\cite[I, 6.3]{bertini}. For the first statement we will show that $\\mathcal{F}_{\\mathbb{K}}$ is a vector bundle of rank $d-1$ by the same argument as for the classical Bertini theorem for smoothness (loc. cit.). Since $\\mathcal{E}$ is flat, it suffices to show that the map\n\\[ \\phi(z):k(z)\\to\\mathcal{E}\\otimes_{\\mathcal{O}_\\mathcal{X}}k(z) \\]\nis injective at any point $z\\in \\mathcal{Z}_{\\mathbb{K}}$. Let $x\\in \\mathcal{X}$ be the image of $z$. Since $f$ is unramified, $df_1,...,df_n$ generate $\\Omega^1_{X\/k}$, and since $\\mathcal{E}$ is a quotient of the latter on $\\mathcal{X}$, we may assume that the images of $df_1,...,df_d$ form a $k(x)$-basis of $\\mathcal{E}\\otimes_{\\mathcal{O}_\\mathcal{X}}k(x)$. So for $i=d+1,...,n$ we can write $df_i=\\sum_{j=1}^da_{ij}df_j$ for some $a_{ij}\\in k(x)$. If $\\phi(z)$ is not injective, i.e. $\\sum_{i=1}^nU_idf_i=0$ in $\\mathcal{E}\\otimes_{\\mathcal{O}_\\mathcal{X}}k(z)$, then we get equations\n\\[ U_j+\\sum_{i=d+1}^nU_ja_{ij}=0 \\]\nfor $j=1,...,d$. Together with $U_0+\\sum_{i=1}^nU_if_i=0$, these equations imply that\n\\[ k(z)=k(x)(U_{d+1},...,U_n) \\]\nhence\n\\[ \\td(k(z)\/k(x))\\leq n-d. \\]\nOn the other hand, since $\\dim \\mathcal{X}\\leq d$ we have $\\td(k(x)\/k)\\leq d$, hence\n\\[ \\td(k(z)\/k(x))=\\td(k(z)\/k)-\\td(k(x)\/k)\\geq n+1-d \\]\na contradiction.\n\\end{proof}\n\nThe theorem easily implies the following generalization of Bertini's theorem for smoothness (the classical case being $\\mathcal{X}=X$ and $\\mathcal{E}=\\Omega^1_{X\/k}$).\n\n\\begin{corollary}\nIf $f$ is unramified and $X$ quasi-compact, then there is a dense open $U\\subset\\mathbb{A}^{n+1}_k$ such that $\\mathcal{F}_U$ is a rank $d-1$ vector bundle quotient of $\\Omega^1_{Z_U\/U}\\otimes_{\\mathcal{O}_{Z_U}}\\mathcal{Z}_U$.\n\\end{corollary}\n\\begin{proof}\nIt suffices to find $U$ such that $\\mathcal{F}_U$ is a vector bundle of rank $d-1$. First assume that $\\mathcal{F}_{\\mathbb{K}}$ is free of basis $e_1,...,e_{d-1}$ say. Then we can find a dense open set $U'\\subset\\mathbb{A}^{n+1}_k$ such that $e_i\\in\\mathcal{F}_{U'}$ for $i=1,...,d-1$, and this defines a map $\\mathcal{O}_{\\mathcal{Z}_{U'}}^{d-1}\\to\\mathcal{F}_{U'}$. Since the cokernel $\\mathcal{G}$ of this map vanishes over the generic point of $\\mathbb{A}^{n+1}_k$, by a limit argument we can find a dense open $U\\subset U'$ such that $\\mathcal{G}_U=0$. Since $X$, hence $\\mathcal{Z}$, is quasi-compact, $\\mathcal{Z}\\to\\mathbb{A}^{n+1}_k$ is of finite presentation, so the general case follows easily from this.\n\\end{proof}\n\n\\subsection{Multiple hyperplane sections}\\label{multiple}\nWe can generalize the previous results to the case of $r$ hyperplanes. This time we let\n\\[ Z:=\\underline{\\Spec}_{X}\\left(\\dfrac{\\mathcal{O}_{X}[U_i^{(j)}]_{0\\leq i\\leq n, 1\\leq j\\leq r}}{\\left(U_0^{(j)}+\\sum_{i=1}^nU_i^{(j)}f_i\\right)_{j=1,...,r}}\\right) \\]\nand $q:Z\\to\\mathbb{A}^{(n+1)r}_k$ the projection. The generic fibre of $q$ will be non-empty if (and only if) $\\dim\\overline{f(X)}\\geq r$; namely if the latter condition holds then there is an irreducible component $Y$ of $X$ for which $\\dim\\overline{f(Y)}\\geq r$ and then $Z\\times_XY\\cong Y\\times_k\\mathbb{A}^{nr}_k$ is an irreducible component of $Z\\cong X\\times_k\\mathbb{A}^{nr}_k$ and the generic fibre of $q|_{Z_Y}$ is non-empty by \\cite[I, 6.6]{bertini}.\n\n\\begin{corollary}\\label{bertini1}\nEndow $\\mathbb{A}^{(n+1)r}_k$ with the inverse image log structure from $\\Spec(k)$ and $Z$ with the inverse image log structure of $X$ (via the structure maps). If $f$ is unramified and $X\\to\\Spec(k)$ is sharp log smooth, then the generic fibre of $q$ is log smooth.\n\\end{corollary}\n\\begin{proof}\nThe proof is by induction on $r$, the case $r=1$ being of course \\ref{logbertinigeneric}. Let $\\mathbb{K}'=k(U_i^{(j)})_{0\\leq i\\leq n, 1\\leq j\\leq r-1}$ and consider\n\\[ Z':=\\underline{\\Spec}_{X}\\left(\\dfrac{\\mathcal{O}_{X}\\otimes_k\\mathbb{K}'}{\\left(U_0^{(j)}+\\sum_{i=1}^nU_i^{(j)}f_i\\right)_{j=1,...,r-1}}\\right). \\]\nBy induction, $Z'$ is log smooth over $(\\mathbb{K}',Q)$. Moreover, it is sharp by \\ref{sharplemma} (iii) and (iv). So by \\ref{logbertinigeneric} the scheme\n\\[ Z''=\\underline{\\Spec}_{Z'}\\left(\\dfrac{\\mathcal{O}_{Z'}\\otimes_{\\mathbb{K}'}\\mathbb{K}'(U_0^{(r)},U_1^{(r)},...,U_n^{(r)})}{\\left(U_0^{(r)}+\\sum_{i=1}^nU_i^{(r)}f_i\\right)}\\right) \\]\nis log smooth over the field $\\mathbb{K}'(U_0^{(r)},U_1^{(r)},...,U_n^{(r)})=k(U_i^{(j)})_{0\\leq i\\leq n, 1\\leq j\\leq r}$. To complete the proof it suffices to remark that this is just the generic fibre of $q$.\n\\end{proof}\n\nLet $\\mathcal{X}$ be as in \\ref{bertinigeneral}. Define $\\mathcal{Z}=\\mathcal{X}\\times_XZ$ with $Z$ as above. There is a canonical map $\\phi:\\mathcal{N}\\otimes_{\\mathcal{O}_Z}\\mathcal{O}_{\\mathcal{Z}}\\to\\mathcal{E}\\otimes_{\\mathcal{O}_{\\mathcal{X}}}\\mathcal{O}_{\\mathcal{Z}}$, where $\\mathcal{N}$ is the conormal sheaf of $Z$ in $X\\times\\mathbb{A}^{(n+1)r}$. Set $\\mathcal{F}:=\\cok(\\phi)$.\n\n\\begin{corollary}\\label{bertini2}\nWith notation and assumptions as in \\ref{bertinigeneral} except $\\mathcal{Z}$ and $\\mathbb{K}:=k(U_i^{(j)})_{0\\leq i\\leq n, 1\\leq j\\leq r}$. If $f$ is unramified, then $\\mathcal{F}_{\\mathbb{K}}$ is a rank $d-r$ vector bundle quotient of $\\Omega^1_{Z_{\\mathbb{K}}\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{Z_{\\mathbb{K}}}}\\mathcal{O}_{\\mathcal{Z}_{\\mathbb{K}}}$ and $\\dim\\mathcal{Z}_{\\mathbb{K}}\\leq d-r$ if $\\mathcal{Z}_{\\mathbb{K}}\\neq\\emptyset$.\n\\end{corollary}\n\\begin{proof}\nThe proof is an easy induction on $r$ as in that of \\ref{bertini1} and we use the same notation. The case $r=1$ being \\ref{bertgeneral}, assume $r>1$. Let $\\mathcal{Z}'=\\mathcal{X}\\times_XZ'$ and $\\mathcal{Z}''=\\mathcal{X}\\times_XZ''$. Note that $\\mathcal{Z}''=\\mathcal{Z}_{\\mathbb{K}}$. We may assume $\\mathcal{Z}_{\\mathbb{K}}\\neq\\emptyset$. By induction on $r$ we have a rank $d-(r-1)$ vector bundle quotient of $\\Omega^1_{Z'\/\\mathbb{K}'}\\otimes_{\\mathcal{O}_{Z'}}\\mathcal{O}_{\\mathcal{Z}'}$ and since $\\mathcal{Z}_{\\mathbb{K}}\\neq\\emptyset$ we must also have $\\mathcal{Z}'\\neq\\emptyset$, hence $\\dim\\mathcal{Z}'\\leq d-(r-1)$ by induction hypothesis. Applying \\ref{bertgeneral}, we get a rank $d-r$ vector bundle quotient of $\\Omega^1_{Z''\/\\mathbb{K}}\\otimes_{\\mathcal{O}_{Z''}}\\mathcal{O}_{\\mathcal{Z}''}$. Since it is the quotient of $\\mathcal{E}\\otimes_{\\mathcal{O}_{\\mathcal{X}}}\\mathcal{O}_{\\mathcal{Z}''}$ by the submodule generated by the images of the differentials $\\sum_iU_i^{(j)}df_i$ for $j=1,...,r$, it is equal to $\\mathcal{F}_{\\mathbb{K}}$. Finally, since $\\mathcal{Z}''\\neq\\emptyset$ we have $\\dim\\mathcal{Z}''\\leq d-(r-1)-1=d-r$ by \\cite[I, 6.3]{bertini}, and this completes the induction.\n\\end{proof}\n\n\\section{Quasi-projective case}\nLet $T$ be an arbitrary irreducible affine noetherian scheme and let $f:X\\to\\mathbb{P}^n_T$ be a morphism of finite type.\n\n\\subsection{Grassmannians}\\label{setupgrass}\nLet $G_{\\mathbb{P}^n_T,r}:=\\Grass_{n+1-r}(\\Gamma(\\mathbb{P}^n_T,\\mathcal{O}_{\\mathbb{P}^n_T}(1)))$ be the grassmannian of quotients of $\\Gamma(\\mathbb{P}^n_T,\\mathcal{O}_{\\mathbb{P}^n_T}(1))\\simeq\\mathcal{O}_T^{n+1}$ which are locally free of rank $n+1-r$. Let $S$ be a $T$-scheme and $l_1,...,l_r\\in\\mathcal{O}_S(S)^{n+1}$. We say that the $l_i$ are \\emph{non-degenerate} if the element $\\wedge_{i=1}^rl_i\\in\\wedge_{i=1}^r\\mathcal{O}_S^{n+1}$ generates a direct summand. In this case it is clear that the $l_i$ induce a locally free quotient $Q$ of $\\mathcal{O}_S^{n+1}$ of rank $n+1-r$, i.e. an element in $G_{\\mathbb{P}^n_T,r}(S)$. Conversely, if $S$ is a $T$-scheme over which every vector bundle is free, then to give a quotient of $\\mathcal{O}_S^{n+1}$ or rank $n+1-r$ is the same as giving a free submodule $M\\subset\\mathcal{O}_S^{n+1}$ of rank $r$ which has a non-degenerate basis $l_1,...,l_r$. In any case, if $l_1,...,l_r$ are non-degenerate then each of the $l_i$ determines a hyperplane in $\\mathbb{P}^n_S$ so that $L=\\mathbb{P}_S(Q)\\hookrightarrow\\mathbb{P}^n_S$ is the subscheme cut out by the $l_i$, $1\\leq i\\leq r$.\n\nFix a projective space $\\mathbb{P}^n_T$ and $r\\in\\mathbb{N}$ and write $G:=G_{\\mathbb{P}^n_T,r}$. Let $\\mathcal{Q}$ be the tautological quotient on $G$. There is a canonical closed immersion\n\\[ \\Grass_1(\\mathcal{Q})\\hookrightarrow\\Grass_1(\\mathcal{O}^{n+1}_{G})\\cong\\Grass_1(\\mathcal{O}^{n+1}_T)\\times_TG=\\mathbb{P}^n_T\\times_TG \\]\nwhose image is the set of points $(x,L)\\in\\mathbb{P}^n_T\\times_TG$ such that $x\\in L$.\nSo if $f:X\\to\\mathbb{P}^n_T$ is a morphism, then the functor\n\\[ \\mathcal{H}_X=\\{ (x,L)\\in X\\times_TG: f(x)\\in L \\}  \\]\nis the scheme $\\mathcal{H}_X:=\\Grass_1(\\mathcal{Q})\\times_{\\mathbb{P}^n_T}X$. It comes equipped with a projection\n\\[ p:\\mathcal{H}_X\\to G. \\]\nIn particular, if $k$ is a field and $P\\in\\mathbb{P}^n(k)$, then the linear subvariety defined by a point $l\\in \\mathcal{H}_P(k)$ can be identified with the intersection of a set of $r$ non-degenerate $k$-hyperplanes through the point $P$, and the map $p:\\mathcal{H}_P\\to G$ is a closed immersion.\n\n\\begin{lemma}\\label{smoothincidencevariety}\nFix a fine log structure on $T$. If $X$ is log smooth over $T$, then the scheme $\\mathcal{H}_X$ (endowed with the inverse image log structure of $X$) is log smooth over $T$. In particular, $\\mathcal{H}_P$ is smooth over $k$.\n\\end{lemma}\n\\begin{proof}\nWe must check that for a nilpotent exact closed immersion $S_0\\hookrightarrow S$ of log $T$-schemes any element in $\\mathcal{H}_X(S_0)$ lifts to an element of $\\mathcal{H}_X(S)$, up to replacing $S$ by an \\'etale covering. Now we have\n\\[ \\mathcal{H}_X(S_0)=\\{(x,L)\\in X(S_0)\\times G(S_0):f(x)\\in L(S_0)\\} \\]\nand let us translate the meaning of a point $(x,L)\\in \\mathcal{H}_X(S_0)$ in terms of modules. The point $L\\in G(S_0)$ corresponds to a quotient\n\\[ \\mathcal{O}_{S_0}^{n+1}\\twoheadrightarrow Q_0 \\]\nwith $Q_0$ locally free of rank $n+1-r$, the point $f(x)\\in\\mathbb{P}^n(S_0)$ corresponds to a quotient\n\\[ \\mathcal{O}_{S_0}^{n+1}\\twoheadrightarrow P_0 \\]\nwith $P_0$ locally free of rank one, and finally the condition $(x,L)\\in \\mathcal{H}_X(S_0)$ means that the quotient onto $P_0$ factors\n\\[ \\mathcal{O}_{S_0}^{n+1}\\twoheadrightarrow Q_0\\to P_0. \\]\nNow, $X$ being log smooth, up to localizing for the \\'etale topology on $S_0$ we can find a point $x'\\in X(S)$ lifting $x$. Then $f(x')$ corresponds to a quotient\n\\[ \\mathcal{O}_{S}^{n+1}\\twoheadrightarrow P \\]\nwith $P$ locally free of rank one. Let $M:=\\ker\\left(\\mathcal{O}_{S}^{n+1}\\twoheadrightarrow P\\right)$, $N_0=\\ker\\left(\\mathcal{O}_{S_0}^{n+1}\\twoheadrightarrow Q_0\\right)$. Up to localizing on $S_0$ we may assume that $N_0$ is a free module on the basis $l_1,...,l_r$. Since $N_0\\subset M\\otimes_{\\mathcal{O}_{S}}\\mathcal{O}_{S_0}$, we can choose lifts $n_1,...,n_r\\in M$ of $l_1,...,l_r$. Then $n_1,...,n_r\\in\\mathcal{O}_{S}^{n+1}$ form part of a basis (by Nakayama's lemma), so if we let $N$ be the free submodule generated by the $n_i$, then $Q:=\\mathcal{O}_{S}^{n+1}\/N$ is a free module. We take $Q$ to be the lift of $Q_0$. By construction we have that the map $\\mathcal{O}_{S}^{n+1}\\twoheadrightarrow P$ factors\n\\[ \\mathcal{O}_{S}^{n+1}\\twoheadrightarrow Q\\to P. \\]\nThis means that $f(x')$ lies on the linear space $L'$ corresponding to $Q$, so the point $(x',L')\\in \\mathcal{H}_X(S)$ is a lift of $(x,L)$.\n\\end{proof}\n\n\\subsection{Reduction to the affine case}\nLet $M_1,...,M_{n+1\\choose r}$ be the $r\\times r$ minors of the matrix $\\left(U_i^{(j)}\\right)_{0\\leq i\\leq n,1\\leq j\\leq r}$. Consider the open $Y\\subset\\mathbb{A}^{(n+1)r}_T$ defined\n\\[ Y:=\\mathbb{A}^{(n+1)r}_T-\\bigcap_{i=1}^{{n+1\\choose r}}V(M_i) \\]\nwhere $V(M_i)\\subset\\mathbb{A}^{(n+1)r}_T=\\Spec(\\mathcal{O}_T[U_i^{(j)}])$ is the zero set of $M_i$. Then for any $T$-scheme $S$ the set $Y(S)$ is a subset of the set of affine homogeneous $S$-hyperplanes $l_1,...,l_r$ which are non-degenerate: after making choice $X_0,...,X_n$ of homogeneous coordinates an $S$-point\n\\[ (u_0^{(1)},...,u_n^{(1)},...,u_0^{(r)},...,u_n^{(r)})\\in Y(S) \\]\ncorresponds to the hyperplanes $\\sum_{i=0}^nu_i^{(j)}X_i=0$ for $1\\leq j\\leq r$. This identification depends on the choice of homogeneous coordinates. Fixing such a choice, let $Q$ be the quotient of $\\oplus_{i=0}^n\\mathcal{O}_YX_i$ by the $\\mathcal{O}_Y$-submodule generated by the elements $l_j:=\\sum_{i=0}^nU_i^{(j)}X_i$ for $1\\leq j\\leq r$. Since, by definition, at every point of $Y$ some $r\\times r$ minor of the matrix $(U_i^{(j)})$ is invertible it follows that $l_1\\wedge l_2\\wedge\\cdots\\wedge l_r$ forms part of a basis of $\\wedge^r(\\oplus_{i=0}^n\\mathcal{O}_YX_i)$, which implies that $Q$ is a vector bundle of rank $n+1-r$ on $Y$. This defines a morphism\n\\[ g:Y\\to G:=G_{\\mathbb{P}^n_T,r}. \\]\n\n\\begin{lemma}\\phantomsection\\label{dominant}\n\\begin{enumerate}[(i)]\n\\item The morphism $g:Y\\to G$ is surjective.\n\\item $G_t$ is irreducible for all $t\\in T$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $S$ be the spectrum of a field and $l\\in G(S)$. Since $S$ is the spectrum of a field, there is a set $\\{l_1,...,l_r\\}$ of non-degenerate homogeneous $S$-hyperplanes such that $l$ is the quotient of $\\oplus_{i=0}^n\\mathcal{O}_SX_i$ by the submodule generated by $l_1,...,l_r$. For each $1\\leq j\\leq r$, write $l_j=\\sum_{i=0}^nu_i^{(j)}X_i$. These hyperplanes are non-degenerate if and only if some $r\\times r$-minor of the matrix $(u_i^{(j)})$ is non-zero. Hence the point $(u_i^{(j)})\\in\\mathbb{A}^{(n+1)r}(S)$ lies in $Y(S)$, and this proves (i). Since $Y_t$ is irreducible, (i) implies (ii).\n\\end{proof}\n\n\\begin{proposition}\\label{subspaceclosed}\nAssume $T$ is the spectrum of a field $k$. Let $\\mathcal{Q}$ denote the tautological quotient of $\\Gamma(\\mathbb{P}^n_k,\\mathcal{O}(1))\\otimes_k\\mathcal{O}_G$. If $S\\subset\\Gamma(\\mathbb{P}^n_k,\\mathcal{O}(1))$ is a subspace of dimension at most $n+1-r$ and $\\mathcal{Q}':=\\im(S\\otimes_k\\mathcal{O}_G\\to\\mathcal{Q})$, then there is a dense open $U\\subset G$ such that the map $S\\otimes_k\\mathcal{O}_U\\to\\mathcal{Q}'|_U$ is an isomorphism and $\\mathcal{Q}\/\\mathcal{Q}'|_U$ is a vector bundle on $U$.\n\\end{proposition}\n\\begin{proof}\nLet $X_0,....,X_m$ be a basis of $S$ and let $X_{m+1},...,X_n$ be elements of $\\Gamma(\\mathbb{P}^n_k,\\mathcal{O}(1))$ such that $X_0,...,X_n$ form a basis. Let $g:Y\\to G$ be the morphism determined by these homogeneous coordinates. It suffices to check that the map $S\\otimes_k\\mathcal{O}_G\\to\\mathcal{Q}'$ is an isomorphism over the generic point of $G$. By \\ref{dominant} (i), for this we may base change to the fraction field $k(Y)$ of $Y$.\n\nLet $R=\\Gamma(\\mathbb{P}^n_k,\\mathcal{O}(1))\/S$. Then we have a commutative diagram with exact rows\n\\[ \\xymatrix{\n0 \\ar[r] & S\\otimes_kk(Y) \\ar[r] \\ar[d] & \\oplus_{i=0}^n k(Y)X_i \\ar[r] \\ar[d] & R\\otimes_kk(Y) \\ar[r] \\ar[d] & 0 \\\\\n0 \\ar[r] & \\mathcal{Q}'\\otimes_{\\mathcal{O}_G}k(Y) \\ar[r] & \\mathcal{Q}\\otimes_{\\mathcal{O}_G}k(Y) \\ar[r] & (\\mathcal{Q}\/\\mathcal{Q}')\\otimes_{\\mathcal{O}_G}k(Y) \\ar[r] & 0 \\\\\n} \\]\nand, by definition of the morphism $g$, the kernel $\\kappa$ of the middle vertical map is generated by the elements $\\sum_{i=0}^{n}U_i^{(j)}X_i$ for $j=1,...,r$. If $0\\neq s\\in\\kappa\\cap (S\\otimes k(Y))$, then we may write\n\\[ s=\\sum_{j=1}^ra_j\\sum_{i=0}^{n}U_i^{(j)}X_i \\]\nand\n\\[ s=\\sum_{i=0}^mb_iX_i \\]\nfor some $a_j,b_i\\in k(Y)$. Equating we find\n\\[ \\sum_{j=1}^ra_jU_i^{(j)}=0 \\]\nfor $i=m+1,...,n$. This implies that the rank of the matrix $U:=(U_i^{(j)})_{m+1\\leq i\\leq n,1\\leq j\\leq r}$ is strictly less than $r$. Since $r\\leq n-m$ by assumption, all $r\\times r$-minors of $U$ vanish. But the subset of $Y$ where all $r\\times r$-minors of $U$ vanish is a proper closed subset, so this cannot happen. Therefore, $\\kappa\\cap (S\\otimes k(Y))=0$.\n\\end{proof}\n\nWe now fix a choice of homogeneous coordinates and let $\\mathbb{A}^n_T\\subset\\mathbb{P}^n_T$ be a standard open with induced coordinates $T_1,...,T_n$. Let $X'=X\\times_{\\mathbb{P}^n_T}\\mathbb{A}^n_T$ and define\n\\[ Z:=\\underline{\\Spec}_{X'}\\left(\\dfrac{\\mathcal{O}_{X'}[U_i^{(j)}]_{0\\leq i\\leq n, 1\\leq j\\leq r}}{\\left(U_0^{(j)}+\\sum_{i=1}^nU_i^{(j)}f_i\\right)_{j=1,...,r}}\\right)\n\\]\nwhere $f_i\\in\\Gamma(X',\\mathcal{O}_{X'})$ is the image of $T_i$ for $i=1,...,n$. Let $i:Z\\hookrightarrow X'\\times_T\\mathbb{A}^{(n+1)r}_T$ be the closed immersion defining $Z$ and $q:Z\\to\\mathbb{A}^{(n+1)r}_T$ be the second projection. ($Z$ is the scheme associated to $X'$ in \\ref{multiple}, except that we assumed there that $T$ was the spectrum of a field.)\n\n\\begin{proposition}\\label{reduction}\nWith our fixed choice of coordinates.\n\\begin{enumerate}[(i)]\n\\item There is a cartesian square\n\\[\\xymatrix{\nZ\\times_{\\mathbb{A}^{(n+1)r}_T}Y \\ar[d]^{q} \\ar[r] & \\mathcal{H}_{X'} \\ar[d]^{p} \\\\\nY \\ar[r]^{g} & G\n} \\]\n\\item The closed immersion $i\\times_{\\mathbb{A}^{(n+1)r}_T}Y:Z\\times_{\\mathbb{A}^{(n+1)r}_T}Y\\hookrightarrow X'\\times_TY$ deduced from $i$ is equal, via the identification $Z\\times_{\\mathbb{A}^{(n+1)r}_T}Y=\\mathcal{H}_{X'}\\times_GY$ of (i), to the closed immersion $\\mathcal{H}_{X'}\\times_GY\\hookrightarrow X'\\times_TY$ deduced from the closed immersion $\\mathcal{H}_{X'}\\hookrightarrow X'\\times_kG$ defining $\\mathcal{H}_{X'}$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nBy definition, $Z_Y:=Z\\times_{\\mathbb{A}^{(n+1)r}_T}Y$ is the closed subscheme of $X'\\times_TY$ defined by the ideal generated by $U_0^{(j)}+\\sum_{i=1}^{n}U_i^{(j)}f_i$ for $j=1,...,r$. So for any $T$-scheme $S$, every element of $Z_Y(S)$ determines points $x\\in X'(S)$ and $(u_0^{(1)},...,u_n^{(1)},...,u_0^{(r)},...,u_n^{(r)})\\in Y(S)$ such that the image of $x$ in $\\mathbb{A}^n_T$ lies on the (non-degenerate) hyperplanes given by $u_0^{(j)}+\\sum_{i=1}^{n}u_i^{(j)}T_i=0$ for $1\\leq j\\leq r$, and conversely. This is obviously the same thing as $\\left(\\mathcal{H}_{X'}\\times_{G}Y\\right)(S)$. Since this correspondence is clearly functorial in $S$, this proves (i).\n\nFor (ii), an element of $P\\in Z_Y(S)$ determines points $x\\in X'(S)$ and $(u_0^{(1)},...,u_n^{(1)},...,u_0^{(r)},...,u_n^{(r)})\\in Y(S)$, and then $i(P)=(x,(u_0^{(1)},...,u_n^{(1)},...,u_0^{(r)},...,u_n^{(r)}))$. This is clearly equal to the image of $P$ under the composition $Z_Y=\\mathcal{H}_{X'}\\times_GY\\to X'\\times_TY$.\n\\end{proof}\n\nThis allows us to show the following Bertini theorem for log smoothness by reducing to the affine case proven before. In the following we assume $T=\\Spec(k)$ with $k$ a field.\n\n\\begin{corollary}\\label{bertini6}\nAssume $T$ has the structure of a log point. Let $f:X\\to\\mathbb{P}^n_k$ be unramified with $X$ a sharp log smooth $k$-scheme. Endow $\\mathcal{H}_X$ with the inverse image log structure of $X$ via the second projection $\\pr_2:\\Grass_1(\\mathcal{Q})\\times_{\\mathbb{P}^n_T}X\\to X$, and $G$ with the inverse image log structure from the structure morphism to $T$. There is a dense open $U\\subset G$ such that $p|_U:\\mathcal{H}_X|_{U}\\to U$ is log smooth.\n\\end{corollary}\n\\begin{proof}\nFor each standard open $\\mathbb{A}^n_k\\subset\\mathbb{P}^n_k$, let $X'=X\\times_{\\mathbb{P}^n_k}\\mathbb{A}^n_k$, so that we have a commutative diagram as in \\ref{reduction}. Endow $\\mathbb{A}^{(n+1)r}_T$ with the inverse image log structure of $T$ and $Z$ with the inverse image log structure from $X$ so that we are in the situation of \\ref{bertini1}. It is clear that with these log structures the diagram of \\ref{reduction} (i) is a cartesian square of log schemes. Hence by \\ref{bertini1} and \\ref{dominant}, the generic fibre of $\\mathcal{H}_{X'}\\to G$ becomes log smooth after the field extension $k(G)\\to k(Y)$. Since the log structures on $k(Y)$ and $k(G)$ are both induced by $Q\\to k$, it now follows easily from \\ref{jacobi} that the generic fibre of of $\\mathcal{H}_{X'}\\to G$ is itself log smooth. So by \\ref{generic} there is a dense open subset $U'\\subset G$ such that the restriction of $\\mathcal{H}_{X'}\\to G$ to $U'$ is log smooth. Hence we can take $U$ to be intersection of the (finitely many) $U'$ for each standard open $\\mathbb{A}^n_k\\subset\\mathbb{P}^n_k$.\n\\end{proof}\n\nWe also have the quasi-projective version of \\ref{bertini2}.\n\n\\begin{corollary}\\label{bertini7}\nLet $f:X\\to\\mathbb{P}^n_k$ be unramified and let $m:\\mathcal{X}\\to X$ be a morphism of finite type with $\\dim\\mathcal{X}\\leq d$ such that there is a surjective map of $\\mathcal{O}_{\\mathcal{X}}$-modules\n\\[ \\pi:m^*\\Omega^1_{X\/k}\\twoheadrightarrow\\mathcal{E} \\]\nwhere $\\mathcal{E}$ is vector bundle of rank $d$ on $\\mathcal{X}$. If $I$ denotes the ideal sheaf of $\\mathcal{H}_X$ in $X\\times_kG$, and\n\\[ \\phi:I\/I^2\\otimes_{\\mathcal{O}_X}\\mathcal{O}_{\\mathcal{X}}\\to\\mathcal{E}\\otimes_{\\mathcal{O}_{X}}\\mathcal{O}_{\\mathcal{H}_X} \\]\nthe composition of the natural map $\\left(I\/I^2\\to\\Omega^1_{X\/k}\\otimes_{\\mathcal{O}_X}\\mathcal{O}_{\\mathcal{H}_X}\\right)\\otimes_{\\mathcal{O}_X}\\mathcal{O}_{\\mathcal{X}}$ with the map $\\pi\\otimes_{\\mathcal{O}_X}\\mathcal{O}_{\\mathcal{H}_X}$, then there is a dense open $U\\subset G$ such that $\\cok(\\phi)|_U$ is a vector bundle of rank $d-r$ on $\\mathcal{H}_{\\mathcal{X}}\\times_GU$.\n\\end{corollary}\n\\begin{proof}\nSince $G$ is irreducible it suffices to show that the restriction of $\\cok(\\phi)$ to $\\mathcal{H}_{\\mathcal{X}}\\times_G\\Spec(k(G))$ is a vector bundle of rank $d-r$. We may check this locally and therefore replace $X$ by $X'$. By \\ref{reduction} we may reduce to the situation of \\ref{multiple} and apply \\ref{bertini2}.\n\\end{proof}\n\n\\section{Hyperplanes through a point over a discrete valuation ring}\nLet $V$ be a complete discrete valuation ring with uniformizer $\\pi$, algebraically closed residue field $k=V\/\\pi V$, fraction field $K=V[1\/\\pi]$, and set $T:=\\Spec(V)$. We will often make the following abuse of notation: since a point $L\\in G_{\\mathbb{P}^n_T,r}(K)=G_{\\mathbb{P}^n_T,r}(T)$ can be identified with a linear subvariety of $\\mathbb{P}^n_T$, for a subscheme $X\\subset \\mathbb{P}^n_T$ we write $X\\cap L$ for the scheme-theoretic intersection of $X$ with the linear subvariety defined by $L$.\n\n\\subsection{Specialization of points}\\label{special}\nLet $X$ be a separated $T$-scheme of finite type and $x\\in X_K$ be a point. We define the \\emph{specialization} $\\spe_X(x)$ of $x$ to be $\\overline{\\left\\{x\\right\\}}\\cap X_k$, where $\\overline{\\left\\{x\\right\\}}$ is closure of $x$ in $X$. We simply write $\\spe(x)$ for $\\spe_X(x)$ when no ambiguity can occur.\n\nWe gather here some facts on specialization.\n\n\\begin{lemma}\\label{closed}\nIf $x\\in X_K$ is a closed point, then either $\\spe(x)=\\emptyset$ or $\\spe(x)$ is a closed point. Conversely, if $\\spe(x)$ is a closed point, then $x$ is closed.\n\\end{lemma}\n\\begin{proof}\nSince $X$ is a separated $T$-scheme of finite type, by a theorem of Nagata there is a proper $T$-scheme $\\bar{X}$ and an open immersion $X\\hookrightarrow\\bar{X}$. Let $x\\in X_K$ be closed. Then there is a finite extension $K\\subset L$ such that $x\\in X(L)\\subset\\bar{X}(L)$. Then $x$ gives a point in $s\\in\\bar{X}(V_L)$, where $V_L$ is the normalization of $V$ in $L$. Since both $\\Spec(V_L)$ and $\\bar{X}$ are proper over $T$, $s(\\Spec(V_L))$ is closed in $\\bar{X}$. It follows that the closure of $x$ in $\\bar{X}$ is equal to $s(\\Spec(V_L))$. So $\\spe(x)$ is isomorphic to a subscheme of $\\Spec(V_L\\otimes_Vk)$, i.e. is a closed point or is empty.\n\nFor the converse, let $U=\\Spec(A)\\subset X$ be an affine open neighbourhood of $\\spe(x)\\in X$. If $x\\notin U$, then $x\\in X-U$, so $\\spe(x)\\in\\overline{\\left\\{x\\right\\}}\\subset X-U$, which is absurd. Hence $x\\in U$, and $\\overline{\\left\\{x\\right\\}}\\cap U$ corresponds to a quotient $A\\to B$ with $B$ reduced. Then $\\Spec(B)$ is irreducible (otherwise $x$ lies on an irreducible component which is a proper closed subset of $\\overline{\\left\\{x\\right\\}}$), so $\\Spec(B)$ is integral. Moreover, the reduction of $B\\otimes_{V}k$ is $k$. Since $B\\otimes_VK\\neq 0$, by \\cite[IV, 14.3.10]{ega} we have $\\dim(B\\otimes_{V}K)=0$. By Noether normalization $B\\otimes_{V}K$ is a finite $K$-algebra, so since $B\\otimes_VK$ is a domain it is a finite field extension of $K$.\n\\end{proof}\n\n\\begin{lemma}\\label{gen}\nLet $x\\in X_k$ be a closed point. The set $\\spe^{-1}(x)$ is equal to the set of closed points of $\\Spec(\\mathcal{O}_{X,x}\\otimes_VK)$, viewed as a subset of $X_K$.\n\\end{lemma}\n\\begin{proof}\nLet $U$ be an open neighbourhood of $x\\in X_k$. If $y\\in X_K$ is a closed point such that $\\spe(y)=x$ and $y\\notin U$, then $x\\in\\overline{\\left\\{y\\right\\}}\\not\\subset U$, a contradiction. Hence $U$ contains $\\spe^{-1}(x)$. Since $\\Spec(\\mathcal{O}_{X,x}\\otimes_VK)=\\cap_{U\\ni x}U_K$, where the intersection is taken over all open neighbourhoods of $x$, it follows that $\\Spec(\\mathcal{O}_{X,x}\\otimes_VK)$ contains $\\spe^{-1}(x)$.\n\nFor the converse, pick a closed point of $y\\in\\Spec(\\mathcal{O}_{X,x}\\otimes_VK)$. It gives a quotient $\\mathcal{O}_{X,x}\\otimes_VK\\to L$ where $L=K(y)$ is the residue field at $y$. We  first claim that $L$ is a finite extension of $K$. Let $C$ be the image of $\\mathcal{O}_{X,x}$ in $L$. It is a local ring with residue field $k(x)=k$ satisfying $C[1\/\\pi]=L$. So by a theorem of Artin-Tate (\\cite[0, 16.3.3]{ega}) $C$ is a local domain of dimension at most 1. Since $V$ is universally catenary, by \\cite[IV, 5.6.4]{ega} we have $\\dim V+\\td(L\/K)=\\dim C+\\td(k(x)\/k)\\leq 1$, hence $\\td(L\/K)=0$ as claimed.\n\nNow for every small enough affine open $\\Spec(A)=U\\subset X$ containing $x$, $y$ induces a quotient $A\\otimes_VK\\to L$. We claim that the image $V'$ of $A$ in $L$ has special fibre over $V$ consisting of a single point. Note that $\\pi$ is not a unit in $V'$ (otherwise $\\pi$ would be a unit in $C$). Now, since $V'$ is an integral domain and $V$ is henselian, it suffices to show that $V'$ is a finite $V$-module, for then we know that $\\Spec(V'\\otimes_Vk)$ is connected and finite, hence a single point. To see this, first note that since $V'$ is a one-dimensional domain ($\\dim(V'_{\\mathfrak{p}})\\leq\\dim(V)+\\td(L\/K)=1$ for any prime ideal $\\mathfrak{p}\\subset V'$), $V'\\otimes_{V}k$ is a zero-dimensional $k$-algebra of finite type, so by Noether normalization it is finite. Lifting a $k$-basis $e_1,...,e_n$ of $V'\\otimes_{V}k$ we see that $V'\\subset \\sum_{i=1}^nVe_i+\\pi V'$, hence $V'\\subset\\sum_{i=1}^nVe_i+\\pi^mV'$ for all $m\\geq 1$. Since $V'$ is $\\pi$-adically separated (by Krull's Intersection Theorem) and $V$ complete, this implies that $V'\\subset\\widehat{V'}=\\sum_{i=1}^nVe_i$, whence $V'=\\sum_{i=1}^nVe_i$. This proves the claim. Note that the specialization $\\spe(y)$ of $y$ in $U$ is given by the quotient $A\\to V'\\otimes_Vk$, hence $\\spe(y)$ is a closed point in $U_k$. Since this holds for every small enough neighbourhood $U$ of $x$, it follows that $\\spe(y)=x$, as required.\n\\end{proof}\n\n\\begin{corollary}\\label{sp}\nLet $x\\in X_k$ be closed. An open subset $U_0\\subset X_K$ is of the form $U_0=U_{K}$ for some open neighbourhood $U$ of $x$ if and only if $\\spe^{-1}_X(x)\\subset U_0$.\n\\end{corollary}\n\\begin{proof}\nWe first show the if part. Let $Y_0$ be the complement of $U_0$ in $X_{K}$ and let $Y$ be its closure in $X$. If $x\\in Y$, then by the last lemma any closed point of $y\\in\\Spec(\\mathcal{O}_{Y,x}\\otimes_{V}K)$ is a closed point of $Y_0$ satisfying $\\spe_X(y)=x$. Since $\\spe^{-1}_X(x)\\subset U_0$ by assumption, this is absurd. So $U:=X-Y$ contains $x$. For the converse, it suffices to note that if $x\\in U$, then by the last lemma $\\spe^{-1}_X(x)\\subset U$.\n\\end{proof}\n\n\\subsection{Hyperplanes through a point}\\label{hyperplanespoint}\nLet $P\\in\\mathbb{P}^n(k)$ and let $I_P\\subset\\mathcal{O}_{\\mathbb{P}^n_T}$ be its ideal sheaf. For a sheaf $\\mathcal{F}$ on $\\mathbb{P}^n_T$ we write $\\Gamma(\\mathcal{F}):=\\Gamma(\\mathbb{P}^n_T,\\mathcal{F})$ to simplify. Then for any integer $d$ we have an exact sequence\n\\[ 0\\to I_P(d)\\to\\mathcal{O}_{\\mathbb{P}^n_T}(d)\\to k(P)\\to 0 \\]\nso taking global sections we get a left-exact sequence\n\\begin{equation}\\label{exseq:globalideal}\n0\\to \\Gamma(I_P(d))\\to\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(d))\\to k(P)\\to 0.\n\\end{equation}\n\n\\begin{lemma}\\label{hyperplanelocus}\nIf $d\\geq 0$, then the sequence \\ref{exseq:globalideal} is exact.\n\\end{lemma}\n\\begin{proof}\nIt suffices to show $\\Gamma(I_P(d))\\neq\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(d))$. For $d=0$ this is obvious, so assume $d\\geq 1$. If $\\Gamma(I_P(d))=\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(d))$ is bijective, then every hypersurface of degree $d$ vanishes at $P$, equivalently every hyperplane in $\\mathbb{P}(\\Gamma(\\mathcal{O}(d)))$ vanishes on the image of $P$. But in a suitable choice of coordinates we have $P=(1:0:\\cdots:0)\\in\\mathbb{P}(\\Gamma(\\mathcal{O}(d)))\\otimes_Vk$ and the hyperplane given by the first coordinate does not vanish at $P$. So $\\Gamma(I_P(d))\\neq\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(d))$.\n\\end{proof}\n\nSince $H^1(\\mathbb{P}^n_T,\\mathcal{O}_{\\mathbb{P}^n_T}(d))=0$ for $d\\geq 0$ we immediately find\n\n\\begin{corollary}\n$H^1(\\mathbb{P}^n_T,I_P(d))=0$ for $d\\geq 0$.\n\\end{corollary}\n\nSince $I_P(d)$ is $\\pi$-torsion free, taking cohomology in the exact sequence\n\\[ 0\\to I_P(d)\\overset{\\cdot\\pi}{\\to} I_P(d)\\to I_P(d)\\otimes_Vk\\to 0 \\]\nwe obtain\n\n\\begin{corollary}\\label{corhyperplanelocus}\n$\\Gamma(I_P(d))\\otimes_Vk=\\Gamma(I_P(d)\\otimes_Vk)$ for $d\\geq 0$.\n\\end{corollary}\n\nFor $d\\geq 0$ we can tensor the exact sequence \\ref{exseq:globalideal} with $k$ to obtain exact sequences\n\\begin{equation}\\label{exseq:globalideal1} 0\\to k(P)\\to\\Gamma(I_P(d))\\otimes_Vk\\to\\Gamma(I_P(d)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})\\to 0\n\\end{equation}\n\\begin{equation}\\label{exseq:globalideal2} 0\\to \\Gamma(I_P(d)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})\\to \\Gamma(\\mathcal{O}_{\\mathbb{P}^n_k}(d))\\to k(P)\\to 0.\n\\end{equation}\n\n\\begin{proposition}\\label{global}\n$I_P(d)$ and $I_P(d)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}$ are generated by global sections for $d\\geq 1$.\n\\end{proposition}\n\\begin{proof}\nWe first show that $I_P(d)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}$ is generated by global sections for $d\\geq 1$. By \\cite[Lecture 14, Prop.]{mumford} it suffices to check that $I_P\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}$ is $d$-regular, i.e. $H^i(\\mathbb{P}^n_k,I_P\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}(d-i))=0$ for $i\\geq 1$. To see this, note that since $d\\geq 1$, by \\ref{exseq:globalideal2} for $i\\geq 1$ we have $H^i(\\mathbb{P}^n_k,I_P\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}(d-i))=H^i(\\mathbb{P}^n_k,\\mathcal{O}_{\\mathbb{P}^n_k}(d-i))$, and by Serre's computation of the cohomology of projective space we know that this is zero for $0<i<n$, and that $H^n(\\mathbb{P}^n_k,\\mathcal{O}_{\\mathbb{P}^n_k}(d-n))$ is the $k$-dual of $H^0(\\mathbb{P}^n_k,\\mathcal{O}_{\\mathbb{P}^n_k}(-d-1))=0$.\n\nFor $I_P(d)$, since $\\mathbb{P}^n_T$ is proper over $T$, by \\ref{corhyperplanelocus} and Nakayama's lemma it will suffice to show that $I_P(d)\\otimes_Vk$ is generated by global sections if $d\\geq 1$. Now, we have an exact sequence of $\\mathcal{O}_{\\mathbb{P}^n_k}$-modules\n\\[ 0\\to k(P)\\to I_P(d)\\otimes_Vk\\to I_P(d)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}\\to 0 \\]\nand since $k(P)$ is a skyscraper sheaf and $I_P(d)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}$ is globally generated for $d\\geq 1$, by \\ref{exseq:globalideal1} this implies the result.\n\\end{proof}\n\n\\begin{corollary}\\label{diagram}\nThere is a commutative diagram of canonical morphisms\n\\[ \\xymatrix{\n\\mathbb{P}^n_T\\setminus\\{P\\} \\ar[r] & \\mathbb{P}(\\Gamma(I_P(2))) \\\\\n\\mathbb{P}^n_k\\setminus\\{P\\} \\ar[r] \\ar[u] & \\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})) \\ar[u]\n} \\]\nwhere the vertical maps are closed immersions.\n\\end{corollary}\n\\begin{proof}\nLet us define the morphisms. For the top map, by \\ref{global} on $\\mathbb{P}^n_T$ we have the quotient\n\\[ \\Gamma(I_P(2))\\otimes_V\\mathcal{O}_{\\mathbb{P}^n_T}\\twoheadrightarrow I_P(2) \\]\nand $I_P(2)$ is isomorphic to the line bundle $\\mathcal{O}_{\\mathbb{P}^n_T}(2)$ on $\\mathbb{P}^n_T\\setminus\\{P\\}$, so we get a corresponding morphism $\\phi:\\mathbb{P}^n_T\\setminus\\{P\\}\\to \\mathbb{P}(\\Gamma(I_P(2)))$. Similarly for the bottom map. It is clear (cf. \\ref{exseq:globalideal}, \\ref{exseq:globalideal1}, \\ref{exseq:globalideal2}) that these maps fit in a commutative diagram as above.\n\\end{proof}\n\nIn fact, the horizontal maps of the diagram of \\ref{diagram} are immersions. This is a consequence of the following key result.\n\n\\begin{proposition}\\label{key}\nLet $\\tilde{\\mathbb{P}}$ be the blow up of $\\mathbb{P}^n_T$ at the point $P$, and $E\\subset\\tilde{\\mathbb{P}}$ the exceptional divisor.\n\\begin{enumerate}[(i)]\n\\item There is a canonical closed immersion $\\phi:\\tilde{\\mathbb{P}}\\to\\mathbb{P}(\\Gamma(I_P(2)))$ extending the morphism of \\ref{diagram}.\n\\item $\\phi(\\tilde{\\mathbb{P}})\\cap\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}))$ is isomorphic to the blow up of $\\mathbb{P}_k^n$ at $P$.\n\\item $\\phi(E)$ is a linear subvariety of $\\mathbb{P}(\\Gamma(I_P(2)))\\otimes_Vk$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nWe first define the morphism $\\phi$. Since $\\Gamma(I_P(2))$ generates $I_P(2)$ and $I_P\\cdot\\mathcal{O}_{\\tilde{\\mathbb{P}}}$ is invertible, $\\mathcal{L}:=I_P(2)\\cdot\\mathcal{O}_{\\tilde{\\mathbb{P}}}$ is a rank 1 locally free quotient of $\\Gamma(I_P(2))\\otimes_V\\mathcal{O}_{\\tilde{\\mathbb{P}}}$, i.e. a morphism $\\tilde{\\mathbb{P}}\\to\\mathbb{P}(\\Gamma(I_P(2)))$. It clearly extends the map of \\ref{diagram}.\n\nLet $x_0,...,x_n\\in\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(1))$ be homogeneous coordinates such that $P$ is the point $(1:0:\\cdots:0)\\in\\mathbb{P}^n(k)$. Then the $x_ix_j$ ($0\\leq i,j\\leq n$) form a basis of $\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))$, so one checks easily that $x_ix_j$ for $i\\neq j$ together with $x_1^2,...,x_{n}^2$ and $\\pi x_0^2$ form a basis of $\\Gamma(I_P(2))$ (cf. \\ref{exseq:globalideal}).\n\nNow we investigate the map $\\phi$ in a neighbourhood of $P$. Let $W\\subset\\mathbb{P}^n_T$ be the complement of vanishing set of $x_0\\in\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(1))$. Then $t_i:=\\frac{x_i}{x_0}$ ($1\\leq i\\leq n$) are coordinates on $W$, and over $W$ the ideal $I_P$ is generated by the $t_i$ together with $\\pi$. So $\\tilde{\\mathbb{P}}\\times_{\\mathbb{P}^n_T}W$ is covered by the spectra of the rings\n\\[ A_i:=\\dfrac{V[t_1,...,t_n,u_{1i},...,u_{ni},s_i]}{(t_iu_{1i}-t_1,...,t_iu_{ni}-t_n,s_it_i-\\pi)}=\\dfrac{V[t_i,u_{ji},s_i]_{1\\leq j\\leq n, j\\neq i}}{(s_it_i-\\pi)} \\]\nfor $1\\leq i\\leq n$, together with\n\\[ A_0:=\\dfrac{V[t_1,...,t_n,s_1^{-1},...,s_n^{-1}]}{(\\pi s_1^{-1}-t_1,...,\\pi s_n^{-1}-t_n)}=V[s_j^{-1}]_{1\\leq j\\leq n} \\]\n(here $s_i^{-1}$ is an indeterminate, not invertible). On the other hand, consider the principal open $U_i$ of $\\mathbb{P}(\\Gamma(I_P(2)))$ where $x_ix_0\\neq 0$ ($i\\neq 0$). Let $X_i\\subset\\tilde{\\mathbb{P}}$ be the maximal open subset where $x_ix_0$ generates $\\mathcal{L}$, so that $\\phi$ restricts to a morphism $X_i\\to U_i$. We claim that $\\Spec(A_i)\\subset X_i$. Indeed, we have the trivialization\n\\[ \\mathcal{O}_{\\mathbb{P}^n_T}|_{W}\\cong\\mathcal{O}_{\\mathbb{P}^n_T}(2)|_{W}:f\\mapsto x_0^2f \\]\nand the ideal $I_P\\cdot\\mathcal{O}_{\\Spec(A_i)}$ is generated by $t_i$, so $\\mathcal{L}|_{\\Spec(A_i)}=I_P(2)\\cdot\\mathcal{O}_{\\Spec(A_i)}$ is generated by $t_ix_0^2$. This is precisely the image of $x_ix_0$, and this implies the claim. Now $U_i$ has coordinates $\\frac{x_jx_k}{x_ix_0}$ and $\\frac{\\pi x_0^2}{x_ix_0}$. We have natural identifications $\\phi^{\\#}(\\frac{x_ix_j}{x_ix_0})=t_j$, $\\phi^{\\#}(\\frac{x_jx_0}{x_ix_0})=u_{ji}$, $\\phi^{\\#}(\\frac{x_jx_k}{x_ix_0})=u_{ji}t_k$ (one checks easily that $u_{ji}t_k=u_{ki}t_j$ in $A_i$) and $\\phi^{\\#}(\\frac{\\pi x_0^2}{x_ix_0})=s_i$, so the map $\\phi|_{U_i}:X_i\\to U_i$ induces a surjective homomorphism\n\\[ \\phi^{\\#}:\\Gamma(U_i,\\mathcal{O}_{U_i})\\twoheadrightarrow A_i. \\]\nSimilarly, if $U_0$ is the principal open of $\\mathbb{P}(\\Gamma(I_P(2)))$ where $\\pi x_0^2\\neq 0$, and $X_0\\subset\\tilde{\\mathbb{P}}$ the maximal open where $\\pi x_0^2$ generates $\\mathcal{L}$, then (using that $I_P\\cdot\\mathcal{O}_{\\Spec(A_0)}$ is generated by $\\pi$) one checks that $\\Spec(A_0)\\subset X_0\\to U_0$. We have $\\phi^{\\#}(\\frac{x_jx_0}{\\pi x_0^2})=s_j^{-1}$ for $j\\neq 0$ and $\\phi^{\\#}(\\frac{x_jx_k}{\\pi x_0^2})=t_js_k^{-1}$ for $j,k\\neq 0$, so we get a surjective map\n\\[ \\phi^{\\#}:\\Gamma(U_0,\\mathcal{O}_{U_0})\\twoheadrightarrow A_0. \\]\nSince it is easy to see that the principal open where $x_i^2\\neq 0$ ($i\\neq 0$) contains the principal open of $\\tilde{\\mathbb{P}}$ where $x_i\\neq 0$ as a closed subscheme, this at least shows that $\\phi$ is a local immersion. But $\\tilde{\\mathbb{P}}$ is integral and $\\phi$ is proper, so by \\cite[I, 8.2.8]{ega} $\\phi$ is a closed immersion. This proves (i). For (ii) it suffices to note that $\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}))$ is the zero set of $\\pi x_0^2$ in $\\mathbb{P}(\\Gamma(I_P(2)))\\otimes_Vk$ (cf. \\ref{exseq:globalideal1}) and that the rings $A_i\\otimes_Vk\/(s_i)$ for $i\\neq 0$ give an open covering of the blow up of $\\mathbb{P}^n_k$ at $P$ over $W$. Finally, note that the ideal of the exceptional divisor in $A_0$ is generated by $\\pi$, and the kernel of the map\n\\[ \\Gamma(U_0,\\mathcal{O}_{U_0})\\otimes_Vk=k\\left[\\tfrac{x_jx_k}{\\pi x_0^2}\\right]_{j,k}\\twoheadrightarrow k[s_j^{-1}]_{j}=A_0\\otimes_Vk \\]\nis generated by the coordinates $\\frac{x_jx_k}{\\pi x_0^2}$ for $j,k\\neq 0$. This implies that the exceptional divisor shares a dense open subset with the linear variety cut out by these coordinates, so since the exceptional divisor is integral it must be equal to this linear variety, i.e. (iii).\n\\end{proof}\n\n\\begin{corollary}\\label{closedimmersion}\nLet $X\\hookrightarrow\\mathbb{P}^n_T$ be an immersion and let $\\tilde{X}$ be the blow up of $X$ at a closed point $P\\in X_k$. There is a canonical immersion $\\phi:\\tilde{X}\\to\\mathbb{P}(\\Gamma(I_P(2)))$. Moreover, if $X$ is regular at $P$, then the image of the exceptional divisor is a linear subvariety of $\\mathbb{P}(\\Gamma(I_P(2)))\\otimes_Vk$.\n\\end{corollary}\n\\begin{proof}\nEverything except the final statement is an immediate consequence of the proposition. Now if $X$ is regular at $P$ the ideal of $P$ in $X$ can be generated by a regular subsequence of a regular sequence generating the ideal of $P$ in $\\mathbb{P}^n_T$. So the exceptional divisor of $\\tilde{X}$ is a linear subvariety of the exceptional divisor of $\\tilde{\\mathbb{P}}$, and hence is linear by the proposition.\n\\end{proof}\n\nThis description of $\\tilde{X}$ gives us good control over hyperplane sections of its special fibre in $\\mathbb{P}(\\Gamma(I_P(2)))$. The next lemma allows one to compare hyperplanes in $\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2)))$ with hyperplanes in $\\mathbb{P}(\\Gamma(I_P(2)))$, especially for those hyperplanes which pass through the point $P$.\n\nRecall (\\S\\ref{setupgrass}) that for a projective space $\\mathbb{P}^n_S$ over a scheme $S$ we write $G_{\\mathbb{P}^n_S,r}=\\Grass_{n+1-r}(\\Gamma(\\mathbb{P}^n_S,\\mathcal{O}_{\\mathbb{P}^n_S}(1)))$.\n\n\\begin{lemma}\\label{zariskiopen}\nLet $G:=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))),r}$ and $G':=G_{\\mathbb{P}(\\Gamma(I_P(2))),r}$.\n\\begin{enumerate}[(i)]\n\\item There is a canonical isomorphism $\\mathcal{H}_P=G_{\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})),r}$.\n\\item There is a canonical isomorphism $\\iota:G_K\\cong G'_K$.\n\\item There is a dense open $G'_0\\subset G'_k$ and a canonical morphism $\\rho:G'_0\\to \\mathcal{H}_{P}$. This morphism admits a section, in particular it is surjective.\n\\item For any closed $L\\in G'_K$ such that $\\spe_{G'}(L)\\in G'_0$ we have $\\spe_{G}(\\iota^{-1}(L))=\\rho(\\spe_{G'}(L))$.\n\\item $\\iota\\left(\\spe^{-1}_G(\\mathcal{H}_P)\\cap G(K)\\right)=\\spe_{G'}^{-1}(G'_0)\\cap G'(K)$.\n\\end{enumerate}\nSo we have a commutative diagram\n\\[ \\xymatrix{\nG'_K \\ar[d]^{\\spe_{G'}} & \\spe^{-1}_{G'}(G'_0) \\ar[d]^{\\spe_{G'}} \\ar@{_(->}[l] \\ar[r]^{\\iota^{-1}} & \\spe^{-1}_G(\\mathcal{H}_P) \\ar@{^(->}[r] \\ar[d]^{\\spe_G} & G_K \\ar[d]^{\\spe_G} \\\\\nG'_k & G'_0\\ar@{_(->}[l]  \\ar[r]^{\\rho} & \\mathcal{H}_P\\ar@{^(->}[r] & G_k\n} \\]\n\\end{lemma}\n\\begin{proof}\n(i) :  Consider the sequence \\ref{exseq:globalideal2}. Any quotient of $\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})$ yields a quotient of $\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_k}(2))$ which has $\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_k}(2))\/\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})\\cong k$ as quotient, and conversely. This is exactly statement (i).\n\n(ii) : This follows from the isomorphism $\\Gamma(I_P(2))\\otimes_VK=\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))\\otimes_VK$.\n\n(iii) : Let $g\\in \\Gamma(I_P(2))\\otimes_Vk$ be a generator of the image of the map $k(P)\\to\\Gamma(I_P(2))\\otimes_Vk$ of sequence \\ref{exseq:globalideal1}, and let $\\mathcal{Q}'$ be the tautological quotient on $G'_k$. Consider the commutative diagram with exact rows\n\\[ \\xymatrix{\n0 \\ar[r] & \\mathcal{O}_{G'_k}g \\ar[r] \\ar[d] & \\Gamma(I_P(2))\\otimes_V\\mathcal{O}_{G'_k} \\ar[d] \\\\\n0 \\ar[r] & \\mathcal{L} \\ar[r] & \\mathcal{Q}'\n} \\]\nwhere $\\mathcal{L}\\subset \\mathcal{Q}'$ is the image of $\\mathcal{O}_{G'_k}g$. Note that $\\mathcal{L}\\neq 0$ since the image of $g$ is non-zero in a generic quotient of $\\Gamma(I_P(2))\\otimes_Vk$. Since $G'_k$ is integral and $\\mathcal{L}$ is a torsion-free quotient of $\\mathcal{O}_{G'_k}g$, it follows easily that $\\mathcal{L}=\\mathcal{O}_{G'_k}g$. Thus, we get a commutative diagram with exact rows\n\\[ \\xymatrix{\n0 \\ar[r] & \\mathcal{O}_{G'_k}g \\ar[r] \\ar@{=}[d] & \\Gamma(I_P(2))\\otimes_V\\mathcal{O}_{G'_k} \\ar[r] \\ar[d] & \\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})\\otimes_k\\mathcal{O}_{G'_k} \\ar[r] \\ar[d] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}_{G'_k}g \\ar[r] & \\mathcal{Q}' \\ar[r] & \\mathcal{Q}'' \\ar[r] & 0\n} \\]\nand there is a dense open $G'_0\\subset G'_k$ over which the quotient $\\mathcal{Q}''$ is locally free and therefore defines a map $\\rho:G'_0\\to G_{\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})),r}=\\mathcal{H}_P$. Any choice of splitting of the exact sequence \\ref{exseq:globalideal1} provides a section of this map, so this map is surjective and this proves (iii).\n\n(iv) : Let $K_L$ denote the residue field of $L$ and $V_L$ the normalization of $V$ in $K_L$. Suppose $L$ is given by a subspace $S\\subset\\Gamma(I_P(2))\\otimes_VK_L$. We claim that the condition $\\spe_{G'}(L)\\in G'_0$ is equivalent to $S\\cap(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))\\otimes_VV_L)=S\\cap(\\Gamma(I_P(2))\\otimes_VV_L)$. To see this, setting $Q:=\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))\\otimes_VV_L\/(S\\cap \\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))\\otimes_VV_L)$ and $Q':=\\Gamma(I_P(2))\\otimes_VV_L\/(S\\cap\\Gamma(I_P(2))\\otimes_VV_L)$ we have a commutative diagram with exact rows\n\\[ \\xymatrix{\n0 \\ar[r] & \\Gamma(I_P(2))\\otimes_VV_L \\ar[r] \\ar[d] & \\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))_VV_L \\ar[r] \\ar[d] & k(P)\\otimes_VV_L \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & Q' \\ar[r] & Q \\ar[r] & k(P)\\otimes_VV_L\/M \\ar[r] & 0\n} \\]\nwhere $M=S\\cap(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))\\otimes_VV_L)\/S\\cap(\\Gamma(I_P(2))\\otimes_VV_L)$. Taking tensor product $\\otimes_Vk$ we get a commutative diagram with exact rows\n\\[ \\xymatrix{\n0 \\ar[r] & k(P)\\otimes_VV_L \\ar[r] \\ar[d] & \\Gamma(I_P(2))\\otimes_Vk\\otimes_VV_L  \\ar[d] \\\\\n0 \\ar[r] & k(P)\\otimes_VV_L\/M \\ar[r] & Q'\\otimes_Vk\n} \\]\nIn particular, the map $k(P)\\otimes_VV_L\\to Q\\otimes_Vk$ is injective if and only if $M=0$. By definition of $G'_0$ (see the proof of (iii)), this map is injective if and only if the point $l\\in G'(k\\otimes_VV_L)$ determined by $Q'\\otimes_Vk$ lies $G'_0$. Since $\\spe_{G'}(L)$ is equal to the support of $l$, we see that $M=0$ if and only if $\\spe_{G'}(L)\\in G'_0$ as claimed. In particular, since we clearly have $\\spe_{G}(\\iota^{-1}(L))\\in\\mathcal{H}_P$ if and only if $M\\neq k(P)\\otimes_VV_L$, $M=0$ implies $\\spe_{G}(\\iota^{-1}(L))\\in\\mathcal{H}_P$ i.e. (iv).\n\n(v) : With notation as in the proof of (iv), note that in the case $K_L=K$ we have $M=0$ if and only if $M\\neq k(P)$, hence $\\spe_{G'}(L)\\in G_0'$ if and only if $\\spe_{G}(\\iota^{-1}(L))\\in\\mathcal{H}_P$ as required.\n\\end{proof}\n\n\\begin{lemma}\\label{exceptionaldivisorintersection}\nWith notation as in \\ref{closedimmersion} and \\ref{zariskiopen}. Assume further that $X$ is regular of dimension $d+1$ at $P$. If $d\\geq r$, then there is a dense open $U\\subset\\mathcal{H}_P$ such that for each $L\\in G(T)$ with $\\spe_G(L)\\in U$, $X\\cap L$ is regular at $P$ of dimension $d+1-r$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathfrak{m}\\subset\\mathcal{O}_{X}$ be the ideal sheaf of $P$. The closed immersion $E\\hookrightarrow\\mathbb{P}(\\Gamma(I_P(2)))$ of \\ref{closedimmersion} is by definition that arising from the line bundle quotient\n\\[ \\Gamma(I_P(2))\\otimes_V\\mathcal{O}_{E}\\to(\\mathfrak{m}(2)\\cdot\\mathcal{O}_{\\tilde{X}})\\otimes_{\\mathcal{O}_{\\tilde{X}}}\\mathcal{O}_E. \\]\nSince $E$ is a linear subvariety of $\\mathbb{P}(\\Gamma(I_P(2)))\\otimes_Vk$ under this embedding, it follows that the induced map\n\\[ \\sigma:\\Gamma(I_P(2))\\to\\Gamma(E,\\mathfrak{m}(2)\\cdot\\mathcal{O}_{\\tilde{X}}) \\]\nis surjective and the closed immersion $E\\hookrightarrow\\mathbb{P}(\\Gamma(I_P(2)))$ is equal to the morphism $\\Proj(\\Sym_V\\sigma)$.\n\nLet $S'=\\ker(\\sigma\\otimes_Vk)$ and $S:=\\im(S\\subset\\Gamma(I_P(2))\\otimes_Vk\\to\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}))$. Let $N=\\dim\\Gamma(I_P(2))\\otimes_Vk$. Note that $\\dim S'=N-(d+1)<N-r$, hence $\\dim S\\leq N-1-r=\\dim\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})-r$. So if $\\mathcal{Q}''$ denotes the tautological quotient on $\\mathcal{H}_P$ and $\\mathcal{Q}''':=\\im(S\\otimes_k\\mathcal{O}_{\\mathcal{H}_P}\\to\\mathcal{Q}'')$, then by \\ref{subspaceclosed} there is a dense open $U\\subset \\mathcal{H}_P$ such that $S\\otimes_k\\mathcal{O}_{U}=\\mathcal{Q}'''|_{U}$ and $\\mathcal{Q}''\/\\mathcal{Q}'''|_U$ is flat. Now let $l\\in G'_0$ be a closed point such that $\\rho(l)\\in U$. If $Q'$ denotes the quotient corresponding to $l$ and $Q''$ that corresponding to $\\rho(l)$, then as in the proof of \\ref{zariskiopen} we have a commutative diagram with exact rows\n\\[ \\xymatrix{\n0 \\ar[r] & k(P) \\ar[r] \\ar@{=}[d] & \\Gamma(I_P(2))\\otimes_Vk \\ar[r]^{p} \\ar[d]^{q'} & \\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}) \\ar[r] \\ar[d]^{q''} & 0 \\\\\n0 \\ar[r] & k(P) \\ar[r] & Q' \\ar[r] & Q'' \\ar[r] & 0\n} \\]\nSince $S\\to Q''$ is injective by choice of $U$, the above diagram implies that the map $S'\\to Q'$ is injective. Hence $\\ker(q')$ injects into $\\Gamma(E,\\mathfrak{m}(2)\\cdot\\mathcal{O}_{\\tilde{X}})$ under $\\sigma\\otimes k$. Since $\\Gamma(E,\\mathfrak{m}(2)\\cdot\\mathcal{O}_{\\tilde{X}})=\\mathfrak{m}\/\\mathfrak{m}^2$, this means that $X\\cap L$ is regular at $P$ if $\\spe(L)\\in U$.\n\\end{proof}\n\nNow with the picture clarified we can show the following. For a closed point $L\\in G_K$ we denote by $K_L$ its residue field and by $V_L$ the normalization of $V$ in $K_L$.\n\n\\begin{proposition}\\label{smoothcase}\nLet $X$ be a smooth quasi-projective $T$-scheme of relative dimension $d\\geq r$, $X\\hookrightarrow\\mathbb{P}^n_T$ an immersion, and let $G:=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))),r}$. Let $P\\in X_k$. There is a dense open $U\\subset \\mathcal{H}_P$ such that for all closed $L\\in G_K$ with $\\spe_G(L)\\in \\mathcal{H}_P$ the scheme $X_{V_L}\\cap L$ is smooth of relative dimension $d-r$ over $V_L$ at the points of $X_k$.\n\\end{proposition}\n\\begin{proof}\nRecall that by \\ref{zariskiopen} (i) a hyperplane in $\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}))$ corresponds to a hyperplane in $\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_k}(2)))$ through the point $P$. By the Bertini theorem for smoothness (\\cite[I, 6.11]{bertini}), the intersection of $r$ generic hyperplanes in $\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}))$ with the image of $X_k-\\{P\\}$ is smooth of dimension $d-r$. Thus, there is a dense open $U'\\subset \\mathcal{H}_P$ such that for any $l\\in U'$, the intersection $(X_k-\\{P\\})\\cap l$ is smooth of dimension $d-r$. Now let $\\mathfrak{m}\\subset\\mathcal{O}_{X_k,P}$ be the maximal ideal. Fix homogeneous coordinates $y_0,...,y_N$ on $\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_k}(2)))$ such that $P=(1:0:\\cdots :0)$. There is a natural surjective $k$-linear map\n\\begin{eqnarray*}\n\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}) &\\to &\\mathfrak{m}\/\\mathfrak{m}^2 \\\\\n\\sum_{i=1}^Na_iy_i &\\mapsto & \\sum_{i=1}^Na_i\\frac{y_i}{y_0}.\n\\end{eqnarray*}\nLet $S$ be its kernel. If $\\mathcal{Q}$ denotes the tautological quotient on $\\mathcal{H}_P$ and $\\mathcal{Q}':=\\im(S\\otimes_k\\mathcal{O}_{\\mathcal{H}_P}\\to\\mathcal{Q})$, then by \\ref{subspaceclosed} there is a dense open $U''\\subset \\mathcal{H}_P$ such that $S\\otimes_k\\mathcal{O}_{U''}=\\mathcal{Q}'|_{U''}$ and $\\mathcal{Q}\/\\mathcal{Q}'|_{U''}$ is flat. Thus, we have\n\\[ \\ker(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k})\\otimes_k\\mathcal{O}_{U''}\\to\\mathcal{Q}|_{U''})=\\ker(\\mathfrak{m}\/\\mathfrak{m}^2\\otimes_k\\mathcal{O}_{U''}\\to\\mathcal{Q}\/\\mathcal{Q}'|_{U''}) \\]\nand so for every $l\\in U''$ the corresponding non-degenerate homogeneous linear polynomials $l_1,...,l_r$ generate an $r$-dimensional subspace of $\\mathfrak{m}\/\\mathfrak{m}^2$. Since $P\\in X_k$ is a regular point, this implies that $X_k\\cap l$ is regular of dimension $d-r$ at $P$. Hence, $X_k\\cap l$ is smooth of dimension $d-r$ for all $l\\in U:=U'\\cap U''$. Now let $L\\in G_K$ be a closed point whose specialization lies in $U$. To complete the proof it suffices to show that $X_{V_L}\\cap L$ is flat over $V_L$, which follows from the next lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{CMflat}\nLet $X\\to\\mathbb{P}^n_T$ be a morphism of finite type, $G=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(1))),r}$, $L\\in G(T)$. Suppose $P\\in X$ is a point of $(X\\times_{\\mathbb{P}^n_T}L)_k$ such that $\\dim_PX=d+1$ and $\\dim_P(X\\times_{\\mathbb{P}^n_T}L)_k=d-r$. If $X$ is Cohen-Macaulay at $P$, then $X\\times_{\\mathbb{P}^n_T}L$ is flat over $T$ at $P$.\n\\end{lemma}\n\\begin{proof}\nSince the ideal of $X\\times_{\\mathbb{P}^n_T}L$ in $X$ can be generated by $r$ elements (the hyperplanes corresponding to $L$), we have $\\dim_PX\\times_{\\mathbb{P}^n_T}L\\geq \\dim_PX-r=d+1-r$. On the other hand, we have $\\dim_PX\\times_{\\mathbb{P}^n_T}L\\leq 1+\\dim_P(X\\times_{\\mathbb{P}^n_T}L)_k=1+d-r$. Hence $\\dim_PX\\times_{\\mathbb{P}^n_T}L=d+1-r$ and so $X\\times_{\\mathbb{P}^n_T}L$ is Cohen-Macaulay since $X$ is. As $\\dim_P(X\\times_{\\mathbb{P}^n_T}L)_k=\\dim_PX\\times_{\\mathbb{P}^n_T}L-1$, it follows that $\\pi$ is not a zero divisor on $\\mathcal{O}_{X\\times_{\\mathbb{P}^n_T}L,P}$, i.e. $X\\times_{\\mathbb{P}^n_T}L$ is flat over $T$ at $P$.\n\\end{proof}\n\n\\begin{proposition}\\label{awayfromP}\nLet $X$ be a flat projective $T$-scheme of dimension $d+1$ and with geometrically reduced generic fibre $X_K$. Let $X\\hookrightarrow\\mathbb{P}^n_T$ be a closed immersion, and $G:=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))),d}$. Fix a closed point $P\\in\\mathbb{P}^n_k$. Then there is a dense open $U\\subset \\mathcal{H}_P$ such that for any closed $L\\in G_K$ such that $\\spe_G(L)\\in U$, $X_k\\cap\\spe_G(L)$ is finite over $k$ and $X_{K_L}\\cap L$ is \\'etale over $K_L$ away from $\\spe^{-1}(P)\\cap X_{K_L}\\cap L$. In particular, if $P\\notin X$, then $X_{K_L}\\cap L$ is \\'etale over $K_L$.\n\\end{proposition}\n\\begin{proof}\nLet $X^{\\sm}\\subset X$ be the open subset of points smooth of relative dimension $d$ over $T$. We first show that $\\dim(X-X^{\\sm})\\leq d$. If not, then there is an irreducible component $Y$ of $X$ with $\\dim Y=d+1$ and $Y\\cap X^{\\sm}=\\emptyset$. Since $Y_K$ is geometrically reduced this is only possible if $Y_K=\\emptyset$, which cannot happen as $\\dim X_k=d$ ($X$ is flat over $T$). Thus, $\\dim(X-X^{\\sm})\\leq d$.\n\nNow there is  $\\Omega^1_{X\/T}|_{X^{\\sm}}$ is locally free of rank $d$ and so defines a morphism $X^{\\sm}\\to\\Grass_d(\\Omega^1_{X\/T})$ of $X$-schemes, which is an open immersion since $\\Grass_d(\\Omega^1_{X\/T})\\times_XX^{\\sm}=\\Grass_d(\\Omega^1_{X^{\\sm}\/T})=X^{\\sm}$. Let $\\mathcal{X}$ be the scheme-theoretic closure of $X^{\\sm}$ in $\\Grass_d(\\Omega^1_{X\/T})$, and let $m:\\mathcal{X}\\to X$ be the canonical map. Since $\\mathcal{X}$ is reduced (because $X^{\\sm}$ is) and each of its irreducible components dominates $T$, $\\mathcal{X}$ is flat over $T$ and so $\\dim\\mathcal{X}_k=\\dim\\mathcal{X}_K$ (\\cite[IV, 14.2.5]{ega}), whence $\\dim\\mathcal{X}_k=d$. Moreover, by construction there is a vector bundle $\\mathcal{E}$ of rank $d$ on $\\mathcal{X}$ and a surjective map\n\\[ m^*\\Omega^1_{X\/T}\\twoheadrightarrow\\mathcal{E} \\]\nof $\\mathcal{O}_{\\mathcal{X}}$-modules. Since $\\mathring{X}:=X_k-\\{P\\}$ is contained in $\\mathbb{P}(\\Gamma(I_P(2)\\cdot\\mathcal{O}_{\\mathbb{P}^n_k}))$ we can apply \\ref{bertini7} to $\\mathring{\\mathcal{X}}:=\\mathcal{X}_k-m^{-1}(P)$ to find a dense open $U\\subset\\mathcal{H}_P$ such that for all closed $l\\in U$, if $I$ denotes the ideal sheaf of $\\mathring{X}$ generated by non-degenerate hyperplanes $l_1,...,l_d$ corresponding to $l$, then the map\n\\[ I\/I^2\\otimes_{\\mathcal{O}_{\\mathring{X}}}\\mathcal{O}_{\\mathring{\\mathcal{X}}}\\to\\mathcal{E}|_{\\mathring{\\mathcal{X}}\\times_{X_k}(X_k\\cap l)} \\]\nis surjective. So if $L\\in G_K$ is a closed point such that $\\spe(L)\\in U$ and $I_L\\subset\\mathcal{O}_{X_{V_L}}$ denotes the ideal generated by non-degenerate hyperplanes corresponding to $L$ then by Nakayama's lemma the map\n\\[ I_L\/I_L^2\\otimes_{\\mathcal{O}_{X}}\\mathcal{O}_{\\mathcal{X}}\\to\\mathcal{E}|_{\\mathcal{X}\\times_X(X_{V_L}\\cap L)} \\]\nis surjective at the points of $X^{\\sm}_{K_L}\\cap L$ which specialize to points of $\\mathring{X}$. This implies that $X^{\\sm}_{K_L}\\cap L$ is \\'etale over $K$ at the points which do not specialize to $P$.\n\nTo finish, we claim that we can shrink $U$ so that $Z_{K_L}\\cap L\\subset\\spe^{-1}_X(P)$, where $Z:=X-X^{\\sm}$. To see this, note that $\\dim Z\\leq d$, so if $C\\subset Z$ is an irreducible component which dominates $T$, then $\\dim C_k<d$. Hence up to shrinking $U$ we can assume $(C_k-\\{P\\})\\cap l=\\emptyset$ for all $l\\in U$, whence $C_{K_L}\\cap L\\subset\\spe^{-1}_X(P)$ for all $L\\in G_K$ with $\\spe(L)\\in U$. If $C\\subset Z$ is an irreducible component which doesn't dominate $T$, then $C_K=\\emptyset$. This proves the claim.\n\\end{proof}\n\nWe now prove Theorem \\ref{goodetalemap}. We fix $G:=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))),d}$.\n\n\\begin{theorem}\\label{maingoodnhbprop}\nAssume $\\cha(K)=0$. Let $X$ be an flat projective $T$-scheme of dimension $d+1$ and with reduced generic fibre $X_K$. Let $X\\hookrightarrow\\mathbb{P}^n_T$ be a closed immersion and fix a closed point $P\\in X_k$ with $\\dim_PX=d+1$.\n\\begin{enumerate}[(i)]\n\\item If $X$ is regular at $P$, then there is a dense open $U\\subset \\mathcal{H}_P$ such that for any $L\\in G(K)$ with $\\spe_{G}(L)\\in U$, $X_k\\cap\\spe_{G}(L)$ is finite over $k$ and $X_{K}\\cap L$ is \\'etale over $K$.\n\\item If $X$ is smooth at $P$, then there is a dense open $U\\subset \\mathcal{H}_P$ such that for any closed $L\\in G_K$ with $\\spe_{G}(L)\\in U$, $X_k\\cap\\spe_{G}(L)$ is finite over $k$ and $X_{K_L}\\cap L$ is \\'etale over $K_L$.\n\\end{enumerate}\nFurthermore, we have $X_{V_L}\\cap L\\neq\\emptyset$ for all such $L$.\n\\end{theorem}\n\\begin{proof}\n(i) : Since $X$ is regular, by \\ref{exceptionaldivisorintersection} there is a dense open $U'\\subset\\mathcal{H}_P$ such that, for all $L\\in G(T)$ with $\\spe_G(L)\\in U'$, $X\\cap L$ is regular at $P$, and flat over $T$ at $P$ of dimension 1 by \\ref{CMflat}. Thus, $X_K\\cap L$ is \\'etale over $K$ at the points specializing to $P$. Moreover, since $\\dim X_k=d$, up to shrinking $U'$ we may assume that for all $l\\in U'$ we have $\\dim (X_k-\\{P\\})\\cap l=0$, hence $X_k\\cap l$ is finite. For the remaining points, by \\ref{awayfromP} there is a dense open $U''\\subset\\mathcal{H}_P$ such that for any $L\\in G(K)$ with $\\spe(L)\\in U''$ the scheme $X_K\\cap L$ is \\'etale at the points which not do specialize to $P$.\n\nHence we can take $U=U'\\cap U''$ for the first assertion. For the assertion that $X_{K}\\cap L\\neq\\emptyset$, consider the local ring $\\mathcal{O}_{X,P}$. By assumption it has dimension $d+1$. Since $L$ is the intersection of $d$ hyperplanes passing through $P\\in X$, we have $\\mathcal{O}_{X\\cap L,P}\\neq 0$ and hence $\\dim\\mathcal{O}_{X\\cap L,P}\\geq 1$. If $\\mathcal{O}_{X\\cap L,P}\\otimes_VK=0$, then the kernel of the canonical map $\\mathcal{O}_{X,P}\\to\\mathcal{O}_{X\\cap L,P}$ contains a power of the uniformizer $\\pi\\in V$. Thus, $\\dim\\mathcal{O}_{X\\cap L,P}=\\dim\\mathcal{O}_{X_k\\cap L,P}=0$, a contradiction. Hence $\\mathcal{O}_{X\\cap L,P}\\otimes_VK\\neq 0$, completing the proof.\n\n(ii) : The proof is similar to (i), the reference to \\ref{exceptionaldivisorintersection} being replaced by \\ref{smoothcase}.\n\\end{proof}\n\n\\section{Construction of good neighbourhoods}\nIn this section we assume $\\cha(K)=0$.\n\n\\subsection{Setup}\nLet $\\bar{X}$ be a connected smooth projective $T$-scheme of finite type of relative dimension $d$. Fix a closed immersion $\\bar{X}\\hookrightarrow\\mathbb{P}^n_T$, an open subset $X\\subset\\bar{X}$, and a closed point $P\\in X_k$.\n\n\\subsection{Good hyperplanes}\\label{goodnbhdsetup}\nLet $V\\subset V'$ be a finite extension of discrete valuation rings, $K'=V'\\otimes_VK$, and $T':=\\Spec(V')$. Let $H_0,H_1,...,H_{d-1}$ be a set of $T'$-hyperplanes and $L:=H_1\\cap H_2\\cap\\cdots\\cap H_{d-1}$. Consider the following conditions:\n\\begin{enumerate}\n\\item $H_1,...,H_{d-1}$ are non-degenerate and pass through $P$, and $H_0$ does not pass through $P$\n\\item $\\bar{X}_{T'}\\cap L$ is smooth of relative dimension 1 over $T'$\n\\item $Y_{K'}\\cap L$ is \\'etale over $K'$\n\\item $\\bar{X}_{T'}\\cap L\\cap H_0$ is \\'etale over $T'$\n\\item if $D$ is the scheme-theoretic closure of $Y_K$ in $Y$, then $(D_{T'}\\cap L\\cap H_0)_k=\\emptyset$.\n\\suspend{enumerate}\nSuppose $H_0,....,H_{d-1}$ are $T'$-hyperplanes satisfying (1)-(5). Note that since $\\bar{X}_{K}$ is geometrically connected, by (2) it follows from \\cite[I, 7.1]{bertini} that $\\bar{X}_{K'}\\cap L$ is a smooth geometrically connected projective $K'$-curve. Moreover, (4) and (5) imply that $X_{K'}\\cap L\\cap H_0=\\bar{X}_{K'}\\cap L\\cap H_0$ is finite \\'etale over $K$.\n\nWe say that $T$-hyperplanes $H_0,...,H_{d-1}$ are \\emph{good} if they satisfy conditions (1)-(5) and moreover\n\\resume{enumerate}\n\\item if, for any finite extension $T'\\to T$ as above, $H_1',...,H_{d-1}'$ is any set of non-degenerate $T'$-hyperplanes with same specialization (as a point of the appropriate grassmannian) as $H_1,...,H_{d-1}$, then conditions (2)-(5) hold for the hyperplanes $(H_0)_{T'},H_1',...,H_{d-1}'$.\n\\end{enumerate}\n\n\\begin{proposition}\\label{good}\nGood $T$-hyperplanes exist, up to replacing the closed immersion $\\bar{X}\\hookrightarrow\\mathbb{P}^n_T$ by its double.\n\\end{proposition}\n\\begin{proof}\nLet $G:=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))),d-1}$ (notation as in \\S\\ref{setupgrass}). By \\ref{smoothcase} (resp. \\ref{awayfromP}), there is a dense open $U_1\\subset \\mathcal{H}_P$ such that condition (2) (resp. (3)) holds for any closed $L\\in G_K$ with $\\spe(L)\\in U_1$.\n\nPick $z\\in U(k)$. Since $G$ is smooth over $T$ we can lift $z\\in U_k$ to a point in $G(T)$ and this gives $H_1,...,H_{d-1}$.\n\nBy the Bertini theorem for smoothness one can find $H_0$ such that $X_k\\cap L\\cap H_0$ is \\'etale over $k$. Then (4) holds by \\ref{CMflat}. Moreover, since $\\dim(D_k\\cap\\spe(L))=0$, we can choose $H_{0}$ such that it does not intersect $D_k\\cap\\spe(L)$, hence (5) is satisfied. So $H_0,H_1,...,H_{d-1}$ satisfy (1)-(5).\n\nFinally, for condition (6), note that it holds for (5) automatically since it only depends on the special fibre of $L$ and $H_0$, and that it also holds for (2), (3), and (4) given our choice of $U$ and $H_0$.\n\\end{proof}\n\n\\subsection{The map to affine space} Let $H_0,H_1,...,H_{d-1}$ be good $T$-hyperplanes and let $F_i$ be a homogeneous equation of $H_i$ for $i=0,1,...,d-1$. Let $Y_1,...,Y_{d-1}$ be coordinates on $\\mathbb{A}^{d-1}_T$ and let\n\\[ \\bar{X}'=V(F_1-Y_1F_0,F_2-Y_2F_0,...,F_{d-1}-Y_{d-1}F_0) \\]\nbe the closed subscheme of $\\bar{X}\\times_T\\mathbb{A}^{d-1}_T$ of homogeneous ideal generated by the polynomials $F_1-Y_1F_0,F_2-Y_2F_0,...,F_{d-1}-Y_{d-1}F_0$. Let $p:\\bar{X}'\\to\\mathbb{A}^{d-1}_T$, and $q:\\bar{X}'\\to\\bar{X}$ be the natural projections. Note that $p$ is a projective morphism by construction. Finally, let $C:=L\\cap H_0$.\n\n\\begin{lemma}\\phantomsection\\label{loc}\n\\begin{enumerate}[(i)]\n\\item For $z\\in \\bar{X}\\cap C$ we have $q^{-1}(z)=\\mathbb{A}^{d-1}_{k(z)}$, where $k(z)$ denotes the residue field of $\\bar{X}$ at $z$.\n\\item The map $\\bar{X}'\\setminus V(F_0)\\to \\bar{X}\\setminus V(F_0)$ induced by $q$ is an isomorphism.\n\\item If we define $Z:=V(F_0)\\subset\\bar{X}'$, then $Z=(\\bar{X}\\cap C)\\times_T\\mathbb{A}^{d-1}_T$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(i) : $q^{-1}(z)$ is the closed subscheme of $\\mathbb{A}^{d-1}_{k(z)}$ cut out by the polynomials $F_1(z)-Y_1F_0(z),F_2(z)-Y_2F_0(z),...,F_{d-1}(z)-Y_{d-1}F_0(z)$. Since $z\\in\\bar{X}\\cap C$, $F_i(z)=0$ for $0\\leq i\\leq d-1$, and the claim follows.\n(ii) : $U:=\\bar{X}\\setminus V(F_0)$ is affine and $\\bar{X}'\\setminus V(F_0)$ is the closed subscheme of $U\\times\\mathbb{A}^{d-1}$ of ideal generated by $F_1\/F_0-Y_1,...,F_{d-1}\/F_0-Y_{d-1}$, which is obviously isomorphic to $U$.\n\n(iii) : $Z$ is the closed subscheme of $\\bar{X}\\times\\mathbb{A}^{d-1}$ of ideal generated by $F_0,F_1,..,F_{d-1}$, i.e. $(\\bar{X}\\cap C)\\times\\mathbb{A}^{d-1}$.\n\\end{proof}\n\nDefine $X':=\\bar{X}-(Y\\cup (\\bar{X}\\cap H_0))=X-X\\cap H_0$. Note that $P\\in X'$ since $P\\notin H_0$. Moreover, by \\ref{loc} (ii) the map $q$ induces an isomorphism $q^{-1}(X')\\cong X'$ so we shall identity $X'$ with $q^{-1}(X')$. Finally set $Y':=\\bar{X}'-X'$.\n\nNow, the fibre over the origin $0\\in\\mathbb{A}^{d-1}(T)$ of $p:\\bar{X}'\\to\\mathbb{A}^{d-1}_T$ is the scheme $\\bar{X}\\cap L$.\n\n\\begin{lemma}\\label{spread}\nLet $y\\in\\mathbb{A}^{d-1}_{K}$ be a closed point specializing to $0\\in\\mathbb{A}^{d-1}_k$.\n\\begin{enumerate}[(i)]\n\\item $p^{-1}_K(y)$ is a smooth and geometrically connected curve over $y$.\n\\item $p^{-1}_K(y)\\cap Y$ is \\'etale over $y$.\n\\item There is an open neighbourhood of $y$ in $\\mathbb{A}^{d-1}_K$ such that the restriction of $p$ to $Y'$ is finite \\'etale over this neighbourhood.\n\\item There is an open neighbourhood of $y$ in $\\mathbb{A}^{d-1}_K$ such that $p$ is smooth over this neighbourhood.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(i) and (ii) : Let $V'$ be the normalization of $V$ in the residue field $K'$ at $y$, $T'=\\Spec(V')$, and let $\\pi'\\in V'$ be a uniformizer. Since $\\spe(y)=\\spe(0)$, $y$ extends to a point $(y_1,...,y_{d-1})\\in\\mathbb{A}^{d-1}(T')$ with $y_i\\equiv 0\\mod\\pi'$. So the fibre over $(y_1,...,y_{d-1})$ is the intersection of $\\bar{X}$ with the hyperplanes $H_1',...,H_{d-1}'$ defined by the homogeneous polynomials $F_1':=F_1-y_1F_0,F_2':=F_2-y_2F_0,...,F_{d-1}':=F_{d-1}-y_{d-1}F_0$ respectively. We claim that $H_1',...,H_{d-1}'$ are non-degenerate. Since $F_i'\\equiv F_i\\mod\\pi'$, it follows that $\\wedge_{i=1}^{d-1}F_i'\\equiv \\wedge_{i=1}^{d-1}F_i\\mod \\pi'$, hence $\\wedge_{i=1}^{d-1}F_i'$ forms part of a basis of $\\wedge^{d-1}\\mathcal{O}_{T'}^{n+1}$, i.e. the $F_i'$ are non-degenerate. Let $L':=\\cap_iH_i'$. Since $H_0,...,H_{d-1}$ are good and $H_1',...,H_{d-1}'$ are non-degenerate hyperplanes with same specialization as $H_1,...,H_{d-1}$, (i) and (ii) hold by definition.\n\n(iii) : By \\ref{loc} (iii) we have $Z_K=(\\bar{X}\\cap C)_K\\times_K\\mathbb{A}^{d-1}_K$, and since $H_0,...,H_{d-1}$ are good we have $(\\bar{X}\\cap C)_K=(X\\cap C)_K$ is finite \\'etale over $K$. Thus $Z_K$ is finite \\'etale over $\\mathbb{A}^{d-1}_K$. Now, by (ii) $(Y_{T'}\\cap L')_{K'}$ is finite \\'etale over $K$. Also, since $H_0,H_1,...,H_{d-1}$ are good, for every irreducible component $E\\subset Y$ dominating $T$ we have $(E\\cap L'\\cap H_0)_k=\\emptyset$, so by properness we deduce that $E_{T'}\\cap L'\\cap H_0=\\emptyset$. Hence $(Y_{T'}\\cap L'\\cap H_0)_{K'}=\\emptyset$, and $Y'_{T'}\\cap p^{-1}_K(y)$ is the disjoint union of $p^{-1}_K(y)\\cap Z_K$ and $Y_{K'}\\cap L'$, both of which are finite \\'etale over $k(y)$ by the above. So the morphism $Y'_K\\to\\mathbb{A}^{d-1}_K$ is dominant, proper, and its fibre over $y$ is \\'etale. Since $\\mathbb{A}^{d-1}_K$ is normal it follows from \\cite[IV, 18.10.3]{ega} that this morphism is finite \\'etale in a neighbourhood of $y$.\n\n(iv): We claim that $p_K$ is dominant. Since $p_K$ is proper, $S:=p_K(\\bar{X}'_K)\\subset\\mathbb{A}^d_K$ is closed, so is equal to $\\mathbb{A}^d_K$ if and only if $\\dim S=d$. Consider the $K$-analytic space $S^{\\an}$ associated to $S$ by Berkovich \\cite[3.4]{berkovich}. It is a closed analytic subspace of $(\\mathbb{A}^d_K)^{\\an}$. Now, to show that $\\dim S=d$ it suffices to show that $W\\subset S^{\\an}$ for some analytic open $W\\subset(\\mathbb{A}^d_K)^{\\an}$ (for then at any point $y\\in W$ whose residue field is finite over $K$, the Krull dimension of $\\mathcal{O}_{W,y}$ is $d$ and $\\widehat{\\mathcal{O}}_{W,y}=\\widehat{\\mathcal{O}}_{S,y}$ \\cite[3.4.1]{berkovich}). Let $B$ be the closed unit ball around the origin in $(\\mathbb{A}^d_K)^{\\an}$, i.e. $B=\\mathcal{M}(K\\{Y_1,...,Y_d\\})$ in the notation of \\cite{berkovich}. The reduction map $r:B\\to\\mathbb{A}^d_k$ is anti-continuous and surjective (\\cite[2.4]{berkovich}) and one sees easily that $\\mathring{B}:=r^{-1}(0)$ is the open unit ball around the origin in $(\\mathbb{A}^d_K)^{\\an}$. By the above we know that $S$ contains the set $\\spe^{-1}_{\\mathbb{A}^d_T}(0)$ ($0\\in\\mathbb{A}^{d}_k$). Hence, it follows from the definitions that $S^{\\an}$ contains the set $\\Sigma$ of points of $\\mathring{B}$ whose residue field is finite over $K$. It follows from \\cite[2.1.15]{berkovich} that $\\Sigma$ is everywhere dense in $\\mathring{B}$. Thus, $S^{\\an}\\cap\\mathring{B}$ is a closed everywhere dense subset of $\\mathring{B}$, whence $\\mathring{B}\\subset S^{\\an}$, as was to be shown.\n\nNow, up to making a finite extension we may assume $y\\in\\mathbb{A}^{d-1}(K)$. By \\cite[Exp. II, 2.6]{sga1} it suffices to see that in the generic point $\\eta\\in p^{-1}_K(y)$ we have $\\dim(\\mathcal{O}_{\\bar{X}',\\eta})=d-1$ and that the maximal ideal of  $\\mathcal{O}_{\\bar{X}',\\eta}$ is generated by that of $\\mathcal{O}_{\\mathbb{A}^{d-1},y}$. Now, we claim that $X'_K\\cap L'$ is dense in $\\bar{X}'_K\\cap L'$. First note that $X_K\\cap L'$ is dense in $\\bar{X}_K\\cap L'$ since the latter is irreducible and $Y_K\\cap L'$ is finite. So $\\dim(X_{K}\\cap L')=1$ and since $(X\\cap C)_{K}$ is finite it follows that $X'_{K}\\cap L'=X_{K}\\cap L'-(X\\cap C)_{K}\\neq\\emptyset$, so $\\dim(X'_{K}\\cap L')=1$. Thus, $X'_{K}\\cap L'=\\bar{X}'_K\\cap L'-Y'_{K}\\cap L'$ is dense in $\\bar{X}'_{K}\\cap L'$ and we have $\\mathcal{O}_{\\bar{X}',\\eta}=\\mathcal{O}_{X',\\eta}$. Since $X'_{K}$ is regular and its intersection with the hyperplanes $\\left\\{y_i=F_i\/F_0\\right\\}_{1\\leq i\\leq d-1}$ is a smooth curve, it follows that $F_1\/F_0-y_1,...,F_{d-1}\/F_0-y_{d-1}$ generate the maximal ideal of $\\mathcal{O}_{X',\\eta}$. Finally, we have $F_i\/F_0-y_i=Y_i-y_i$ so that $Y_1-y_1,...,Y_{d-1}-y_{d-1}$ do indeed generate the maximal ideal of $\\mathcal{O}_{X',\\eta}$.\n\\end{proof}\n\nBy the last lemma there exists an open $U_0\\subset\\mathbb{A}^{d-1}_{K}$, containing the set $\\spe^{-1}_{\\mathbb{A}^{d-1}_T}(0)$ ($0\\in\\mathbb{A}^{d-1}_k$), such that $p$ is smooth with geometrically connected fibres over $U_0$ and its restriction to $Y'$ is finite \\'etale. It follows from \\ref{sp} that $U_0=U_{K}$ for some open neighbourhood $U$ of $0\\in\\mathbb{A}^{d-1}_k$. So $p|_{p^{-1}(U_K)}$ is an elementary fibration (in the sense of \\cite[Exp. XI]{sga4.3}) and by induction on the dimension of $X$ we deduce\n\n\\begin{theorem}\\label{elementaryfib}\nIf $X$ is an open in a smooth projective $T$-scheme and $P\\in X_k$ a closed point, then there is an open neighbourhood $X''$ of $P$ in $X$ such that $X''_{K}$ is a good neighbourhood relative to $\\Spec(K)$ in the sense of \\cite[Exp. XI]{sga4.3}.\n\\end{theorem}\n\\begin{proof}\nWe have already established that $p_K^{-1}(U_K)\\to U_K$ is an elementary fibration. By induction on $\\dim X_K$, there is an open neighbourhood $U'\\subset U$ of $0\\in\\mathbb{A}^{d-1}_k$, such that $U'_K$ is a good neighbourhood relative to $\\Spec(K)$ in the sense of \\cite[Exp. XI]{sga4.3}. So we can take $X''=X'\\cap p^{-1}(U')$, where $X'$ is defined as above.\n\\end{proof}\n\n\\subsection{$K(\\Pi,1)$ neighbourhoods}\\label{kpi1section}\nLet $X$ be a scheme and let $X_{\\Fet}$ be the finite-\\'etale site of $X$, i.e. $X_{\\Fet}$ is the subsite of the \\'etale site $X_{\\et}$ whose objects are the finite \\'etale $X$-schemes. Recall that $X$ is a $K(\\Pi,1)$ (for the \\'etale topology) if, for each integer $n$ invertible on $X$ and any sheaf $\\mathbb{L}$ of $\\mathbb{Z}\/n$-modules on $X_{\\Fet}$, the adjunction map $\\mathbb{L}\\to R\\rho_*\\rho^*\\mathbb{L}$ is a quasi-isomorphism, where $\\rho:X_{\\et}\\to X_{\\Fet}$ is the canonical morphism of sites. Intuitively, for $X$ connected this means that its \\'etale cohomology with values in such a sheaf $\\mathbb{L}$ can be computed as the Galois cohomology of the \\'etale fundamental group of $X$ with values in the stalk of $\\mathbb{L}$ at the choice of base point.\n\nNow, since $\\cha(K)=0$, a good neighbourhood of a $K$-variety is a $K(\\Pi,1)$ (\\cite[Exp. XI, 4.6]{sga4.3}; for an algebraic proof using Abhyankar's lemma see \\cite[5.5]{olsson}), so \\ref{elementaryfib} implies\n\n\\begin{corollary}\\label{corelementaryfib}\nIf $X$ is an open in a smooth projective $T$-scheme, any closed point $P\\in X_k$ has an open neighbourhood $U$ such that $U_K$ is a $K(\\Pi,1)$.\n\\end{corollary}\n\n\\section{Construction of $K(\\Pi,1)$ neighbourhoods}\nWe maintain the assumption $\\cha(K)=0$.\n\n\\subsection{Setup}\\label{setupkpi1}\nLet $X$ be a connected affine smooth $T$-scheme of relative dimension $d$. Fix a closed immersion $X\\hookrightarrow\\mathbb{A}^n_T$ and let $\\bar{X}$ be the closure of $X$ in $\\mathbb{P}^n_T$, $Y:=\\bar{X}-X$. Fix a closed point $P\\in X_k$.\n\n\\subsection{Nice hyperplanes} Let $T'$ be a finite extension of $T$ as in \\ref{goodnbhdsetup}. Let $H_0,H_1,...,H_{d}$ be $T'$-hyperplanes and $L:=H_1\\cap H_2\\cap\\cdots\\cap H_{d}$. Consider the following conditions:\n\\begin{enumerate}\n\\item $H_1,H_2,..,H_{d}$ are non-degenerate and pass through $P$, and $H_0$ does not pass through $P$\n\\item if $D$ is the scheme theoretic closure of $Y_K$ in $Y$, then $(D_{T'}\\cap L)_k=\\emptyset$\n\\item $(\\bar{X}_{T'}\\cap L)_k$ is finite over $k$ and $(\\bar{X}_{T'}\\cap L)_K$ is \\'etale over $K$ and non-empty\n\\item $(\\bar{X}_{T'}\\cap L\\cap H_0)_k=\\emptyset$.\n\\suspend{enumerate}\nWe say that $T$-hyperplanes $H_0,H_1,...,H_{d-1}$ are \\emph{nice} if they satisfy (1)-(4) and moreover\n\\resume{enumerate}\n\\item if, for any finite extension $T'\\to T$ as above, $H_1',...,H'_{d}$ are $T'$-hyperplanes with same specialization (as a point of the suitable grassmannian) as $H_1,...,H_{d}$, then conditions (2)-(4) hold for the hyperplanes $(H_0)_{T'},H_1',...,H_d'$.\n\\end{enumerate}\n\n\\begin{proposition}\nNice $T$-hyperplanes exist, up to replacing the embedding $\\bar{X}\\hookrightarrow\\mathbb{P}^n_T$ by its double.\n\\end{proposition}\n\\begin{proof}\nLet $G:=G_{\\mathbb{P}(\\Gamma(\\mathcal{O}_{\\mathbb{P}^n_T}(2))),d}$ (\\S\\ref{setupgrass}).\n\nFor (2), note that $\\bar{X}$ is integral (since $X$ is), hence flat over $T$ and of dimension $d+1$. Since the space underlying $Y$ is equal to the intersection of $\\bar{X}$ with the hyperplane at infinity, it follows that $\\dim Y=d$. So $\\dim D_k\\leq d-1$. As $P\\notin D$, there is a dense open $U_1\\subset \\mathcal{H}_{P}$ such that $D_k\\cap l=\\emptyset$ for all $l\\in U_1$.\n\nBy \\ref{maingoodnhbprop} (ii), there is a dense open $U_2\\subset \\mathcal{H}_P$ such that condition (3) holds for all closed $L\\in G_K$ with $\\spe(L)\\in U_2$.\n\nSo we can pick a closed point in $U_1\\cap U_2$, and it can be lifted to an element of $G(T)$. This point gives the hyperplanes $H_1,...,H_d$ which satisfy conditions (2) and (3).\n\nIn view of (3), we can clearly pick $H_0$ such that condition (4) is satisfied. The hyperplanes $H_0,...,H_{d-1}$ satisfy (1)-(4).\n\nFinally, condition (5) holds for (3) by choice of $U_2$ and as (2) and (4) depend only on $\\spe(L)$, (5) holds automatically for them.\n\\end{proof}\n\n\\subsection{The map to affine space} Let $H_0,H_1,...,H_{d}$ be nice $T$-hyperplanes. Let $F_i$ be a homogeneous equation of $H_i$ for $i=0,1,...,d$. Let \n\\[ \\bar{X}'=V(F_1-Y_1F_0,F_2-Y_2F_0,...,F_{d}-Y_{d}F_0) \\]\nbe the closed subscheme of $\\bar{X}\\times_T\\mathbb{A}^{d}_T$ defined by the homogeneous polynomials $F_1-Y_1F_0,F_2-Y_2F_0,...,F_{d}-Y_{d}F_0$, where $Y_1,...,Y_{d}$ are the coordinates on $\\mathbb{A}^{d}$. Let $p:\\bar{X}'\\to\\mathbb{A}^{d}_T$, and $q:\\bar{X}'\\to\\bar{X}$ be the natural projections.\n\nDefine $X':=\\bar{X}-(Y\\cup (H_0\\cap\\bar{X}))$. Note that $P\\in X'$. By the same argument as \\ref{loc} (ii), $q$ induces an isomorphism $q^{-1}(X')\\cong X'$ which we use to identify $X'$ and $q^{-1}(X')$. In fact, since $H_0,...,H_d$ are nice we have $\\bar{X}\\cap L\\cap H_0=\\emptyset$ so $V(F_0)\\cap\\bar{X}'=\\emptyset$ and the map $q$ is an open immersion.\n\n\\begin{theorem}\\label{Kpi1}\nThere is an open neighbourhood $U\\subset\\mathbb{A}^d_T$ of $0\\in\\mathbb{A}^{d}_k$ such that the map $X'_{U_K}\\to U_K$ induced by $p$ is finite \\'etale with non-empty fibres.\n\\end{theorem}\n\\begin{proof}\nLet $y\\in\\mathbb{A}^d_K$ be a closed point specializing to $0\\in\\mathbb{A}^{d}_k$. As in the proof of of \\ref{spread} (i), the fibre $p^{-1}_K(y)$ is the intersection of $\\bar{X}_{K(y)}$ with non-degenerate hyperplanes $H_1',...,H_d'$ having the same specialization as $H_1,...,H_d$ (here $K(y)$ is the residue field at $y$). Since $H_1,...,H_d$ are nice, $p^{-1}_K(y)$ is finite \\'etale and non-empty by definition.\n\nNote that $p_K$ is dominant: this can be seen as in the proof of \\ref{spread} (iv). Now since the morphism $p_K$ is proper, dominant, and $\\mathbb{A}^d_K$ is normal, it follows from \\cite[IV, 18.10.3]{ega} that there is an open $U'\\subset\\mathbb{A}^d_K$ containing all points specializing to $0\\in\\mathbb{A}^{d}_k$ such that $p_K^{-1}(U')\\to U'$ is finite \\'etale.\n\nMoreover, since $H_0,...,H_d$ are nice we have $Y_{K(y)}\\cap p_K^{-1}(y)=\\emptyset$ for all $y\\in\\mathbb{A}^d_K$ with $\\spe(y)=0\\in\\mathbb{A}^d_k$. So if $Y':=\\bar{X}'\\setminus X'$, then $Y'_{K(y)}\\cap p_K^{-1}(y)=\\emptyset$. Since $Y'_K$ is proper over $\\mathbb{A}^d_K$, up to localizing over $U'$ we may assume that $\\bar{X}'_{U'}=X'_{U'}$.\n\nFinally, since $U'$ contains the set $\\spe^{-1}_{\\mathbb{A}^d_T}(0)$ ($0\\in\\mathbb{A}^{d}_k$), by \\ref{sp} it follows that $U'=U_K$ for some open $U\\subset\\mathbb{A}^d_T$ containing $0\\in\\mathbb{A}^{d}_k$, as required.\n\\end{proof}\n\n\\begin{corollary}[Faltings \\cite{fa2}]\\label{corKpi1}\nThere is an open neighbourhood $X''$ of $P$ in $X$ such that $X''_K$ is a $K(\\Pi,1)$.\n\\end{corollary}\n\\begin{proof}\nAccording to \\ref{Kpi1} there is an open neighbourhood $X''$ of $P$ and a morphism $f:X''\\to U$ with $U$ open in $\\mathbb{A}^d_T$ such that $X''_K\\to U_K$ is a finite \\'etale covering. By \\ref{corelementaryfib}, up to shrinking $U$ around $f(P)$ we may assume that $U_K$ is a $K(\\Pi,1)$. Since a finite \\'etale covering of a $K(\\Pi,1)$ is also a $K(\\Pi,1)$ this implies the result.\n\\end{proof}\n\n\\subsection{Application to $p$-adic nearby cycles}\\label{padic}\nLet $X$ be a smooth $T$-scheme. Denote by $\\bar{K}$ an algebraic closure of $K$. Recall that Faltings \\cite{fa4} has defined a site whose objects are pairs $(U,W)$, where $U\\to X$ is \\'etale and $W\\to U_{\\bar{K}}$ is finite \\'etale. It has been noticed that the topology on this category given in loc. cit. is not quite correct. This issue has been studied in detail in the work of Abbes and Gros \\cite{abbesgros}. The right topology is the following \\cite[10]{abbesgros}: it is the topology generated by the coverings $\\{(W_i,U_i)\\to (W,U)\\}_{i\\in I}$ such that either $U_i=U$ for all $i\\in I$ and $(W_i\\to W)_{i\\in I}$ is an \\'etale covering, or $(U_i\\to U)_{i\\in I}$ is an \\'etale covering and $W_i=W\\times_UU_i$. We denote by $\\mathcal{X}$ the site of such pairs $(U,W)$ with this topology.\n\n\\begin{corollary}\nLet $w:X_{\\bar{K},\\et}\\to\\mathcal{X}$ be the morphism of sites induced by the functor $(U,W)\\mapsto W$. For any locally constant constructible \\'etale sheaf of abelian groups $\\mathbb{L}$ on $X_{\\bar{K}}$, $w$ induces a canonical isomorphism\n\\[ H^{\\ast}(\\mathcal{X},w_{\\ast}\\mathbb{L})\\cong H^{\\ast}_{\\et}(X_{\\bar{K}},\\mathbb{L}). \\]\nMore precisely, if $\\sigma:\\mathcal{X}\\to X_{\\et}$ is the morphism induced by the functor $U\\mapsto (U,U_{\\bar{K}})$ and $\\bar{j}:X_{\\bar{K},\\et}\\to X_{\\et}$ the canonical map, then the natural map $R\\sigma_{\\ast}(w_*\\mathbb{L})\\to R\\bar{j}_{\\ast}\\mathbb{L}$ is a quasi-isomorphism.\n\\end{corollary}\n\\begin{proof}\nLet $i\\in\\mathbb{N}$ and let us show that the canonical map $\\psi:R^i\\sigma_{\\ast}(w_*\\mathbb{L})\\to R^i\\bar{j}_{\\ast}\\mathbb{L}$ is an isomorphism. Let $\\mathcal{H}^i$ be the sheaf on $X_{\\et}$ associated to the presheaf\n\\[ U\\mapsto H^i_{\\Fet}(U_{\\bar{K}},\\mathbb{L}) \\]\nwhere the subscript $\\Fet$ means we take cohomology for the finite-\\'etale topology. There is a canonical map (\\cite[10.40]{abbesgros})\n\\[ \\phi:\\mathcal{H}^i\\to R^i\\sigma_{\\ast}(w_*\\mathbb{L}) \\]\nsuch that the composition $\\psi\\circ\\phi$ is the map induced by the natural transformation $H^i_{\\Fet}((-)_{\\bar{K}},\\mathbb{L})\\to H^i_{\\et}((-)_{\\bar{K}},\\mathbb{L})$. Let us show that $\\psi\\circ\\phi$ is an isomorphism. Let $x\\to X$ be a geometric point. Let $U\\to X$ be an \\'etale neighbourhood of $x$. Choose a specialization $y\\to U$ of $x$ such that the image of $y$ is closed in $U$. If $y\\in U_k$ (resp. $y\\in U_K$), then by \\ref{corKpi1} (resp. \\cite[Exp. XI]{sga4.3}) there is an open $U'\\subset U$ such that $y\\to U$ factors through $U'$ with $U'_K$ (hence also $U'_{\\bar{K}}$) a $K(\\Pi,1)$. Then the map $x\\to U$ factors through $U'$. So this implies that the set $I$ of all \\'etale neighbourhoods $U$ of $x$ such that $U_{\\bar{K}}$ is $K(\\Pi,1)$ is cofinal in the set of all \\'etale neighbourhoods of $x$. Thus,\n\\[ \\mathcal{H}^i_x=\\varinjlim_{U\\in I}H^i_{\\Fet}(U_{\\bar{K}},\\mathbb{L})=\\varinjlim_{U\\in I}H^i_{\\et}(U_{\\bar{K}},\\mathbb{L})=\\left(R^i\\bar{j}_{\\ast}\\mathbb{L}\\right)_{x} \\]\nwhich proves that $\\psi\\circ\\phi$ is an isomorphism. Since $\\phi$ is an isomorphism by \\cite[10.41]{abbesgros}, so is $\\psi$.\n\\end{proof}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDue to increasing populations living in urban areas, many cities are facing the problem of traffic congestion, which leads to increasing levels of pollution and massive waste of time and money \\cite{congestion}. \nThe problem of mitigating congestion has been tackled in the literature from two main perspectives. \nOne approach is to influence the user behaviour by incentive-design mechanisms, for instance by road tolling \\cite{brown2017studies,fleischer2004tolls,zhao2006line,cole2006much,Como.ea:16,Como.Maggistro:22}, information design \\cite{das2017reducing,meigs2020optimal,wu2019information,wu2021value} or lottery rewards \\cite{yue2015reducing}, to minimize the inefficiencies due to the autonomous uncoordinated decisions of the agents. A second approach is to intervene on the transportation network, by building new roads or enlarging the existing ones. The corresponding \\emph{network design problem} (i.e., the problem of optimizing the intervention on a transportation network subject to some budget constraints, see e.g. \\cite{leblanc1975algorithm}) is very challenging because of its bi-level nature,\ni.e., it involves a network intervention optimization problem given the flow distribution for that particular network. We assume that each link of the network is endowed with a delay function and the flow distributes according to a Wardrop equilibrium, taking paths with minimum cost, defined as the sum of the delay functions of the links along the path (see \\cite{beckmann1956studies,wardrop1952road}). A characterization of the Wardrop equilibrium is used to construct the lower level of the bi-level network design problem.\n\nIn this work we define a network design problem (NDP), and analyze in details a special instance of the problem, where the delay functions are affine, and the planner can improve the delay function of a single link. Our objective is to strike a balance between a problem that is simple enough to guarantee tractable analysis, yet rich enough to allow insights for more general classes of NDPs. We then extend the validity of the proposed method by a numerical analysis, showing that good performance are achieved even if the delay functions are non-linear.\nFor single-link affine NDPs, our first theoretical result provides an analytical characterization of the cost variation (i.e., the total travel time at the equilibrium) corresponding to an intervention on a particular link under a regularity assumption, which states that the set of links carrying positive flow remain unchanged with an intervention. This assumption, which is not new in the traffic equilibrium literature (see e.g. \\cite{steinberg1983prevalence,dafermos1984some})\nleads to a characterization of Wardrop equilibria using a system of linear equations and enables representing single-link interventions as rank-1 perturbations of the system. We show that this assumption is satisfied provided that the total incoming flow to the network is large enough and the network is series-parallel, which may be of independent interest. We exploit the structure of our characterization and the linearity of the delay functions to express the cost variation using the effective resistance of a link (i.e., between the endpoints of the link), defined with respect to a related resistor network, obtained by making the directed transportation network undirected, and assigning a conductance to each link based on the delay function of the link. Computing the effective resistance of a single link requires the solution of a linear system whose dimension scales with the network size (we indistinctly refer to the network size as the cardinality of the node and the link sets, implicitly assuming that transportation networks are sparse in a such a way that the average degree of the nodes is independent of the number of nodes, inducing then a proportionality between the number of nodes and links). Hence, solving the NDP requires the solution of $\\mathrm{E}$ of these problems, with $\\mathrm{E}$ denoting the number of links. Since this can be computationally intractable for large networks, our second main result proposes a method based on Rayleigh's monotonicity laws to approximate the effective resistance of each link with a number of iterations independent of the network size, thus leading to a significant reduction of complexity.\nThe key idea is that the effective resistance between two adjacent nodes $i$ and $j$ depends mainly on the local structure of the network around the two nodes (i.e., the set of nodes $\\mathcal{N}_{\\le d}$ that are at distance no greater than a small constant $d$ from at least one of $i$ and $j$), and may therefore be approximate by performing only local computation. \nSince typically in transportation networks the local structure of the network is independent of the network size (think for instance of a bidimensional square grid), the size of $\\mathcal{N}_{\\le d}$ does not scale with the network size, thus we can guarantee that the approximation error and computational complexity of our method also do not scale.\nOur third main result establishes sufficient conditions under which the approximation error vanishes asymptotically in the limit of infinite networks, proving that if the related resistor network is recurrent the approximation error tends to vanish for large distance $d$. In the conclusive section we conduct a numerical analysis on synthetic and real transportation networks, showing that a good approximation of the effective resistance of a link can be achieved by looking at a small portion of the network. Moreover, while several assumptions are made to establish theoretical results (e.g., affine delay functions, support of equilibrium flows not varying with the intervention), we conduct a numerical analysis showing that good performance are achieved even if some assumptions are relaxed, i.e., if the delay functions are non-linear and the support of the equilibrium is allowed to vary with interventions.\n\nIn our work we consider a special case of NDPs. These problems have been formalized in the last decades via many different formulations. Both \\textit{continuous} network design problems \\cite{chiou2005bilevel,li2012global,wang2014models},\nwhere the budget can be allocated continuously among the links, and \\textit{discrete} formulations, in which the decision variables include which new roads to build \\cite{gao2005solution}, how many lanes to add to existing roads \\cite{wang2013global}, or a mix of those two problems \\cite{poorzahedy2007hybrid}, \nhave been considered in the literature, together with \\textit{dynamical} formulations \\cite{fontaine2017dynamic}, and formulations where the optimum is achieved by removing, instead of adding, links, because of Braess' paradox \\cite{roughgarden2006severity,fotakis2012efficient}. \nFor comprehensive surveys on the literature on NDP we refer to \\cite{yang1998models,farahani2013review}. We stress that most of the literature focuses on finding polynomial algorithms to solve in approximation NDPs in their most general form. Our main contribution is to provide a tractable approach to solve a single-link network design problem in quasi-linear time, as well as providing intuition and a completely new formulation. For the future we aim at extending our techniques to more general cases, like the multiple interventions case.\nIn the setting of affine delay functions, our NDP formulation is also related to the literature on marginal cost pricing. We assume that interventions modify the linear coefficient of the delay function of link $e$ from $a_e$ to $\\tilde{a}_e$, leading to $\\tilde{\\tau}_e(f_e) = \\tau_e(f_e)-(a_e-\\tilde{a}_e)f_e$, which is equivalent to adding a negative marginal cost toll on a link. In the literature the problem of optimal toll design has been widely explored, also dealing with the problem of the support of the Wardrop equilibrium varying after the intervention, i.e., without imposing restrictive assumptions. However, most of the toll literature aim at finding conditions under which a general NP-hard problem may be solved in polynomial time. The scope of our work is instead to provide a new formulation to a more tractable problem. Moreover, to relax the regularity assumption on the support of the equilibrium, in the toll literature it is often assumed that the network has parallel links, which is unrealistic for transportation networks (see, e.g., \\cite{hoefer2008taxing,jelinek2014computing}.\nOur work is also related to \\cite{steinberg1983prevalence,dafermos1984some}, where the authors investigate the sign of total travel time variation when a new path is added to a two-terminal network, under similar assumptions to ours, providing sufficient conditions under which the Braess' paradox arises. In our work we instead compute the total travel time variation with an intervention, and suggest an efficient algorithm to select the optimal intervention.\nAs mentioned, the key step of our approach is to reformulate the NDP in terms of a resistance problem, and also exploits the parallelism between resistor networks and random walks. From a methodological perspective it is worthwhile mentioning that the relation between Wardrop equilibria and resistor networks has been first investigated in \\cite{klimm2019computing}, while the parallelism between random walks and Wardrop equilibria has been investigated in \\cite{rebeschini2018locality}, although with different purposes. The relation between random walks and resistor networks is quite standard and well-known (see e.g. \\cite{doyle1984random}). To summarize, the contribution of this paper is two-fold. From a methodological perspective, we provide a method to locally approximate the effective resistance between adjacent nodes, which may be of independent interest (effective resistance of a link is related to spanning tree centrality \\cite{hayashi2016efficient}). From NDP perspective, we provide a new formulation of the NDP in terms of resistor networks, and exploit our methodological result to approximate efficiently single-link NDPs.\n\nThe rest of the paper is organized as follows. In Section~\\ref{sec:model} we define the model and formulate the NDP as a bi-level program. In Section~\\ref{sec3} we define single-link NDPs, rephrase the problem in terms of resistor networks, and discuss the regularity assumption. In Section~\\ref{approximation} we provide our method to approximate the effective resistance of a link and exploit such a method to construct an algorithm to solve the problem. We then analyze the asymptotic performance of the proposed method in the limit of infinite networks in Section~\\ref{performance}. In Section~\\ref{sec:simulations} we provide numerical simulations. Finally, in the conclusive section, we summarize the work and discuss future research lines.\n\n\n\n\\subsection{Notation}\nWe let $\\delta^{(i)}$, $\\mathbf{1}$, $\\mathbf{0}$ and $\\mathbf{I}$ denote the unitary vector with $1$ in position $i$ and $0$ in all the other positions, the column vector of all ones, the column vector of all zeros, and the identity matrix, respectively, where the size of them may be deduced from the context. $A^T$ and $v^T$ denote the transpose of matrix $A$ and vector $v$, respectively. Given a vector $v$, we let $\\mathbf{I}_v$ denote the matrix whose off-diagonal elements are zero and with diagonal elements $(\\mathbf{I}_v)_{ii} = v_i$.\n\n\n\n\\section{Model and problem formulation}\n\\label{sec:model}\nWe model the transportation network as a directed multigraph $\\mathcal{G}=(\\mathcal{N},\\mathcal{E})$, and denote by $\\mathrm{o}, \\mathrm{d} \\in \\mathcal{N}$ the origin and the destination of the network. We assume for simplicity of notation that $\\mathcal{N}=\\{1,\\cdots,\\mathrm{N}\\}$ and $\\mathcal{E}=\\{1,\\cdots,\\mathrm{E}\\}$, and assume that $\\mathrm{o}$ and $\\mathrm{d}$ are respectively the first and the last node of the network. Every link $e$ is endowed with a tail $\\xi(e)$ and a head $\\theta(e)$ in $\\mathcal{N}$.  We allow multiple links between the same pair of nodes, and assume that every link belongs to at least a path from $\\mathrm{o}$ to $\\mathrm{d}$, otherwise such a link may be removed without loss of generality.\nLet $m > 0$ denote the throughput from the origin $\\mathrm{o}$ to the destination $\\mathrm{d}$, and $\\nu =  m(\\delta^{(\\mathrm{o})}-\\delta^{(\\mathrm{d})}) \\in \\mathds{R}^\\mathrm{N}$. Let $\\mathcal{P}=\\{1,\\cdots,\\mathrm{P}\\}$ denote the set of paths from $\\mathrm{o}$ to $\\mathrm{d}$. \nAn admissible path flow is a vector $z \\in \\mathds{R}^{\\mathrm{P}}_+$ satisfying the mass constraint\n\\begin{equation}\n\t\\label{constraints}\n\\mathbf{1}^T z=m.    \n\\end{equation}\nLet $A \\in \\mathds{R}^{\\mathrm{E} \\times \\mathrm{P}}$ denote the link-path incidence matrix, with entries $A_{ep}=1$ if link $e$ belongs to the path $p$ or $0$ otherwise. The path flow induces a unique link flow $f \\in \\mathds{R}^\\mathrm{E}$ via\n\\begin{equation}\n\tf=Az.\n\t\\label{incidence_path}\n\\end{equation}\nEvery link $e$ is endowed with a non-negative and strictly increasing delay function $\\tau_e : \\mathds{R}_+ \\to \\mathds{R}_+$. We assume that the delay functions are in the form $\\tau_e(f_e) = \\tau_e(0) + a_e(f_e)$, where $\\tau_e(0)$ is the travel time of the link when there is no flow on it, and $a_e(f_e)$ describes congestion effects, with $a_e(0)=0$.\nThe cost of path $p$ under flow distribution $f$ is the sum of the delay functions of the links belonging to $p$, i.e.,\n\\begin{equation}\n\tc_p(f)=\\sum_{e \\in \\mathcal{E}}A_{ep}\\tau_e(f_e).\n\t\\label{cost_path}\n\\end{equation}\n\\begin{definition}[Routing game]\n\tA \\emph{routing game} is a triple $(\\mathcal{G}, \\tau, \\nu)$.\n\\end{definition}\n\nA Wardrop equilibrium is a flow distribution such that no one has incentive in changing path. More precisely, we have the following definition.\n\\begin{definition}[Wardrop equilibrium]\n\tA path flow $z^*$, with associated link flow $f^*=Az^*$, is a Wardrop equilibrium if for every path $p$\n\t\\begin{equation*}\n\t\tz^*_p>0 \\implies c_p(f^*)\\le c_q(f^*), \\quad \\forall q \\in \\mathcal{P}.\n\t\\end{equation*}\n\\end{definition}\\medskip\n\nLet $B \\in \\mathds{R}^{\\mathrm{N} \\times \\mathrm{E}}$ denote the node-link incidence matrix, with entries $B_{ne}=1$ if $n=\\xi(e)$, $B_{ne}=-1$ if $n=\\theta(e)$, or $B_{ne}=0$ otherwise.\nIt is proved in \\cite{beckmann1956studies} that a link flow $f^*$ is a Wardrop equilibrium of a routing game if and only if\n\\begin{equation}\n\t\\begin{aligned}\n\t\tf^* = \\ & \\underset{f \\in \\mathds{R}_+^\\mathrm{E}, Bf=\\nu}{\\arg\\min}\n\t\t& & \\sum_{e \\in \\mathcal{E}} \\int_0^{f_e} \\tau_e(s) ds,\n\t\\end{aligned}\n\t\\label{convex_prob}\n\\end{equation}\nwhere $Bf=\\nu$ is the projection of \\eqref{constraints} on the link set.\nSince the delay functions are assumed strictly increasing, the objective function in \\eqref{convex_prob} is strictly convex and the Wardrop equilibrium $f^*$ is unique. \n\\begin{definition}[Social cost]\n\tThe social cost of a routing game is the total travel time at the equilibrium, i.e.,\n\t\\begin{equation*}\n\t\tC^{(0)}=\\sum_{e \\in \\mathcal{E}} f_e^*\\tau_e(f_e^*).\n\t\\end{equation*}\n\\end{definition}\n\\medskip\nThe social cost can be interpreted as a measure of performance by a planner that aims at minimizing the overall congestion on the transportation network.\nWe now provide an equivalent characterization of the social cost of a routing game. To this end, let $\\lambda^*$ and $\\gamma^*$ denote the Lagrangian multipliers associated to $f^*\\ge \\mathbf{0}$ and $Bf=\\nu$, respectively.\nThe KKT conditions of \\eqref{convex_prob} read:\n\\begin{align}\n\t\\label{kkt2}\n\t\\begin{cases}\n\t\t\\tau_e(f_e^*)+\\gamma_{\\theta(e)}^*-\\gamma^*_{\\xi(e)} - \\lambda^*_{e}=0 & \\forall e \\in \\mathcal{E},\\\\\n\t\t\\sum_{e \\in \\mathcal{E}: \\theta(e)=i} f_e - \\sum_{e \\in \\mathcal{E}: \\xi(e)=i} f_e + \\nu_i=0 & \\forall i \\in \\mathcal{N},\\\\\n\t\t\\lambda_{e}^*f_{e}^*=0 & \\forall e \\in \\mathcal{E},\\\\\n\t\t\\lambda_{e}^*\\ge 0 & \\forall e  \\in \\mathcal{E},\\\\\n\t\tf_{e}^*\\ge 0 & \\forall e \\in \\mathcal{E}.\n\t\\end{cases}\n\\end{align}\nThe third condition, known as complementary slackness, implies that if $\\lambda_e^*>0$, then $f_e^*=0$, i.e., link $e$ is not used at the equilibrium. We let $\\mathcal{E}_+$ denote the set of the links $e$ such that $\\lambda_e^*>0$. The next lemma shows that the social cost may be characterized in terms of the Lagrangian multiplier $\\gamma^*$.\n\\begin{lemma}\n\t\\label{cost}\n\tLet $(\\mathcal{G},\\tau,\\nu)$ denote a routing game. Then, \n\t\\begin{equation*}\n\t\t\\begin{gathered}\n\t\t\tC^{(0)}=m(\\gamma^*_\\mathrm{o}-\\gamma_\\mathrm{d}^*).\n\t\t\\end{gathered}\n\t\\end{equation*}\n\\end{lemma}\\medskip\n\\begin{proof}\n\tSee Appendix~\\ref{app:proofs}.\n\\end{proof}\n\n\n\n\n\nWe consider a NDP where the planner can improve the delay functions of the network with the goal of minimizing a combination of the social cost after the intervention and the cost of the intervention itself. Specifically, let $u \\in \\mathds{R}_+^{\\mathrm{E}}$ denote the intervention vector, with corresponding delay functions\n$$\n\\tau_e^{(u_e)}(f_e) = \\tau_e(0) + \\frac{a_e(f_e)}{1+u_e}.\n$$\nThis type of interventions may correspond for instance to enlarging some roads of the network. We let $h_e : [0,+\\infty) \\to [0,+\\infty)$ denote the cost associated to the intervention on link $e$. The goal of the planner is to minimize a combination of the social cost and the intervention cost, where $\\alpha \\ge 0$ is the trade-off parameter. More precisely, by letting $f^*(u)$ denote the Wardrop equilibrium corresponding to intervention $u$, the NDP reads as follows.\n\\begin{problem}\n\t\\label{prob:general}\n\tLet $(\\mathcal{G},\\tau,\\nu)$ be a routing game, and $\\alpha\\ge0$ be the trade-off parameter. The goal is to select $u^*$ such that\n\\begin{equation}\n\t\\label{eq:ndp}\n\t\tu^* \\in \\underset{u \\in \\mathds{R}^\\mathrm{E}_+}{\\arg\\min} \\\n\t\t\\sum_{e \\in \\mathcal{E}} f^*_e(u) \\tau_e^{(u_e)}(f^*_e(u)) +  \\alpha h(u),\n\\end{equation}\nwhere $h(u)=\\sum_e h_e(u_e)$, and\n\\begin{equation}\n\t\\begin{aligned}\n\tf^*(u) = \\ & \\underset{f \\in \\mathds{R}_+^\\mathrm{E}, Bf=\\nu}{\\arg\\min}\n\t& & \\sum_{e \\in \\mathcal{E}} \\int_0^{f_e} \\tau_e^{(u_e)}(s) ds.\n\t\\end{aligned}\n\\label{eq:f(k)}\n\\end{equation}\n\\end{problem}\n\\medskip\n\\begin{remark}\nWe stress the fact that Problem~\\ref{prob:general} is bi-level, in the sense that the planner optimizes the intervention $u$ according to a cost function that depends on the Wardrop equilibrium $f^*(u)$, which in turn is the solution of the optimization problem \\eqref{eq:f(k)}, whose objective function depends on the intervention $u$ itself.\n\\end{remark}\n\n\\begin{remark}\nProblem~\\ref{prob:general} is not equivalent to the toll design problem. The key difference between the two problems is that tolls modify the Wardrop equilibrium, but the performance of tolls is evaluated with respect to the original delay functions $\\tau_e$. On the contrary, in Problem~\\ref{prob:general} the intervention is evaluated with respect to the new delay functions $\\tau_e^{(u_e)}$.\n\\end{remark}\n\nProblem~\\ref{prob:general} is in general non-convex, and hard to solve because of its bi-level nature. For these reasons, in the next section we shall study a simplified problem where the delay functions are affine and the planner may intervene on one link only. In this setting we are able to rephrase the problem as a single-level optimization problem, and provide an electrical network interpretation of the problem.\n\n\n\n\\section{Single-link interventions in affine networks}\n\\label{sec3}\nIn this section we provide an electrical network formulation of the NDP under some restrictive assumptions. In particular, we provide a closed formula for the social cost variation in terms of electrical quantities computed on a related resistor network. To this end, we restrict our analysis to the space of feasible interventions $\\mathcal{U}$, defined as\n$$\n\\mathcal{U} := \\{u :  u_e \\delta^{(e)} \\ \\text{for a link} \\ e \\in \\mathcal{E}, u_e \\ge 0  \\}.\n$$\nIn other words, $\\mathcal{U}$ represents the space of interventions on a single link of the network. We also assume that the delay functions are affine, i.e., $\\tau_e(f_e) = a_ef_e + b_e$ for every $e$, and denote by $(\\mathcal{G},a,b,\\nu)$ routing games with affine delay functions.\nFor an intervention $u$, let $(\\mathcal{G},a^{(u)},b,\\nu)$ denote the corresponding affine routing game, $C^{(u)}$ denote the corresponding social cost, and $\\Delta C^{(u)} = C^{(0)} - C^{(u)}$ denote the social cost gain. Our problem can be expressed as follows.\n\\begin{problem}\n\tLet $(\\mathcal{G}, a, b, \\nu)$ be an affine routing game and $\\alpha \\ge0$ be the trade-off parameter. Find\n\t\\begin{equation*}\n\t\t\tu^* \\in \\ \\underset{u \\in \\mathcal{U}}{\\arg\\max}\n\t\t\t \\ (\\Delta C^{(u)} - \\alpha h(u)).\n\t\n\t\\end{equation*}\n\t\\label{prob1}\n\\end{problem}\n\nThe next example shows that the problem cannot be decoupled by first selecting the optimal link $e^*$ and then the optimal strength of the intervention $u^*_e$.\n\\begin{example}\n\t\\label{ex:ex}\n\tConsider the transportation network in Figure~\\ref{fig:ex}, with linear delay functions\n\t$\\tau_e(f_e) = a_ef_e$.\n\tBy some computation, one can prove that\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\Delta C^{(u_1\\delta^{(1)})} & = m \\frac{a_1a_2^2 u_1}{(a_1+a_2)((u_1+1) a_1 + a_2)},\\\\\n\t\t\t\\Delta C^{(u_2 \\delta^{(2)})} & = m \\frac{a_1^2a_2 u_2}{(a_1+a_2)(a_1 + (u_2+1) a_2)},\\\\\n\t\t\t\\Delta C^{(u_3 \\delta^{(3)})} & = m \\frac{u_3}{u_3+1}.\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tIn Figure~\\ref{fig:ex} the social cost variation corresponding to intervention on every link $e$ is illustrated as functions of $u_e$. Observe that the link that maximizes the social cost gain depends on $u_e$. Thus,\n\tthe problem cannot be decoupled by first selecting the optimal link $e^*$ and then the optimal $u_e^*$.\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[draw, circle] (1) at (0,0.5)  {o};\n\t\t\t\\node[draw, circle] (2) at (0,-1)  {n};\n\t\t\t\\node[draw, circle] (3) at (0,-2.5) {d};\n\t\t\t\\node (4) at (0,-3) {};\n\t\t\t\n\t\t\n\t\t\t\\path [->, >=latex]  (1) edge [bend right=30]\n\t\t\tnode [left] {$e_1$} (2);\n\t\t\t\\path [->, >=latex]  (1) edge [bend left=30]\n\t\t\tnode [right] {$e_2$} (2);\n\t\t\t\\path [->, >=latex]  (2) edge [bend left=0]\n\t\t\tnode [right] {$e_3$} (3);\n\t\t\\end{tikzpicture}\n\t\t\\includegraphics[width = 6cm]{optimal_link_kappa.png}\n\t\t\\caption{\\emph{Left}: the graph of Example~\\ref{ex:ex}. \\emph{Right}: the social cost variation corresponding to single link interventions, with assignment $a_1 = 3, a_2 = 2, a_3 = 1, m = 3$.\n\t\t\t\\label{fig:ex}}\n\t\\end{figure}\n\\end{example}\n\nOur theoretical results rely on the following technical assumption, stating that the support of the Wardrop equilibrium is not modified with an intervention.\n\\begin{assumption}\n\tLet $\\mathcal{E}_+(u)$ be the set of links $e$ such that for the routing game $(\\mathcal{G}, a^{(u)}, b, \\nu)$ the Lagrangian multiplier $\\lambda_e^*(u)>0$.\n\tWe assume that $\\mathcal{E}_+(u)=\\mathcal{E}_+$ for every $u$ in $\\mathcal{U}$.\n\t\\label{assumption}\n\\end{assumption}\n\nAssumption~\\ref{assumption} is not new in the literature \\cite{steinberg1983prevalence,dafermos1984some}.\nWe will get back to the assumption in Section~\\ref{sec:assumption}.\nWith a slight abuse of notation, from now on let $\\mathcal{E}$ denote $\\mathcal{E} \\setminus \\mathcal{E}_+$.\nWe now define a mapping from the transportation network $\\mathcal{G}$ to an associated resistor network $\\mathcal{G}_R$.\n\\begin{definition}[Associated resistor network]\n\tGiven the transportation network $\\mathcal{G}=(\\mathcal{N},\\mathcal{E})$, the \\emph{associated resistor network} $\\mathcal{G}_R=(\\mathcal{N},\\mathcal{L},W)$ is constructed as follows:\n\t\t\\begin{itemize}\n\t\t\t\\item the node set $\\mathcal{N}$ is the same.\n\t\t\t\\item $W \\in \\mathds{R}^{\\mathrm{N} \\times \\mathrm{N}}$ is the conductance matrix, with elements\n\t\t\t\\begin{equation} \n\t\t\t\tW_{ij}= \\begin{cases}\n\t\t\t\t\t\\sum_{\\substack{e \\in \\mathcal{E}: \\\\\n\t\t\t\t\t\t\t\\xi(e)=i, \\theta(e)=j, \\ \\text{or} \\\\\n\t\t\t\t\t\t\t\\xi(e)=j, \\theta(e)=i}} \\frac{1}{a_e} & \\text{if}\\ i \\neq j \\\\\n\t\t\t\t\t0 & \\text{if}\\ i=j. \n\t\t\t\t\\end{cases}\n\t\t\t\t\\label{W}\n\t\t\t\\end{equation}\n\t\t\tNote that $W$ is symmetric, thus $\\mathcal{G}_R$ is undirected. The element $W_{ij}$ has to be interpreted as the conductance between nodes $i$ and $j$.\n\t\t\t\\item Multiple links connecting the same pair of nodes are not allowed, hence every link $l$ in $\\mathcal{L}$ can be identified by a unordered pair of nodes $\\{i,j\\}$, and the set $\\mathcal{L}$ is uniquely determined by $W$. Let $\\mathrm{L}$ denote the cardinality of $\\mathcal{L}$. The mapping $M: \\mathcal{E} \\to \\mathcal{L}$ associates to every link $e$ of the transportation network the corresponding link $l = M(e) = \\{\\xi(e),\\theta(e)\\}$ of the resistor network. Note by \\eqref{W} that $M(e)$ belongs to $\\mathcal{L}$ for every $e$ in $\\mathcal{E}$. \n\t\t\\end{itemize}\n\\end{definition}\n\nNote that the coefficients $a_e$ correspond to resistances in the resistor networks. We let $w = W\\mathbf{1}$ denote the degree distribution of the resistor network, and $w^* = \\max_{i \\in \\mathcal{N}}w_i$ denote the maximal degree. Before establishing our first main result, we define two relevant quantities.\n\\begin{definition}\n\t\\label{def:voltage}\n\tLet $v \\in \\mathds{R}^\\mathrm{N}$ be the \\emph{voltage} vector on $\\mathcal G_R$ when a net electrical current $m$ is injected from $\\mathrm{o}$ to $\\mathrm{d}$, i.e., $v$ is the unique solution of\n\\begin{equation}\n\t\\label{eq:voltage}\n\t\\sum_{k \\in \\mathcal{N}} W_{hk}(v_h- v_k)=m(\\delta^{(\\mathrm{o})}-\\delta^{(\\mathrm{d})})\\qquad \\forall h\\in\\mathcal N.\n\\end{equation}\nFor a link $e$ in $\\mathcal{E}$, let $y_e$ denote the \\emph{electrical current} flowing from $\\xi(e)$ to $\\theta(e)$ on link $M(e)$ of $\\mathcal G_R$, and let $\\Delta v_e = v_{\\xi(e)}-v_{\\theta(e)}$. By Ohm's law, $\\Delta v_e = a_e y_e$.\n\\end{definition}\n\\begin{definition}\n\t\\label{def:effective}\n\tLet $\\overline v \\in \\mathds{R}^\\mathrm{N}$ be the voltage vector on $\\mathcal G_R$ when a unitary current is injected from $i$ to $j$, i.e., \n\t\\begin{equation}\n\t\t\\label{eq:potential}\n\t\t\\sum_{k \\in \\mathcal{N}} W_{hk}(\\overline v_h- \\overline v_k)=\\delta^{(i)}-\\delta^{(j)}\\qquad \\forall h\\in\\mathcal N.\n\t\\end{equation}\n    The \\emph{effective resistance} $r_{l}$ of link $l = \\{i,j\\}$ in $\\mathcal L$ is the effective resistance between $i$ and $j$, i.e., $r_{l}=\\overline{v}_i-\\overline{v}_j$. Given a link $e$ in $\\mathcal E$, we denote by $r_e$ the effective resistance of link $M(e)$ of the associated resistor network.\n\\end{definition}\n\nThe next theorem establishes a relation between the social cost gain with a single-link intervention and the associated resistor network. \n\\begin{theorem}\n\tLet $(\\mathcal{G}, a, b, \\nu)$ be an affine routing game, and let Assumption~\\ref{assumption} hold. Then,\n\t\\begin{equation}\n\t\t \\Delta C^{(u_e \\delta^{(e)})}=a_e f_e^*\\frac{y_e}{\\frac{1}{u_e}+\\frac{r_e}{a_e}}.\n\t\t \t\t\\label{el_form}\n\t\\end{equation}\n\n\t\\label{thm}\n\\end{theorem}\n\\begin{proof}\nSee Appendix~\\ref{app:proofs}.\n\\end{proof}\n\nThe ratio $r_{e}\/a_e$ belongs to $(0,1]$ and is also known as \\emph{spanning tree centrality}, which measures the fraction of spanning trees including link $M(e)$ among all spanning trees of the undirected network $\\mathcal G_R$ \\cite{hayashi2016efficient}. The spanning tree centrality of a link is maximized when removing the link disconnects the network. \nTheorem \\ref{thm} states that the social cost variation due to intervention on link $e$ is: \n\t\\begin{itemize}\n\t\t\\item proportional to $a_e f_e^*$, which measures the delay at the equilibrium due to congestion on link $e$;\n\t\t\\item decreasing in the spanning tree centrality. Intuitively speaking, the benefits of intervention on link $e$ is larger when the intervention modifies the equilibrium flows so that agents can move from paths not including $e$ to paths including $e$, namely when $f_e^*$ increases after the intervention. This phenomenon does not occur if $e$ is a bridge, i.e., if $r_{e}\/a_e=1$, and occurs largely when many paths from $\\xi(e)$ to $\\theta(e)$ exist, i.e., when $r_{e}\/a_e$ is small;\n\t\t\\item proportional to the current $y_e$. The role of this term is more clear in the special case of linear delay functions. In this case $y_e = f_e^*$ for all links $e$ in $\\mathcal{E} \\setminus \\mathcal E_+ $, hence $a_e f_e^* y_e^* = a_e (f_e^*)^2$, which is the total travel time on link $e$ before the intervention. \n\t\\end{itemize}\nThe idea behind the proof is that with affine delay functions the KKT conditions of the Wardrop equilibrium are linear, and under Assumption~\\ref{assumption} single-link interventions are equivalent to rank-1 perturbations of the system. Thus, by Lemma~\\ref{cost} we can compute the cost variation by looking at Lagrangian multiplier $\\gamma_\\mathrm{o}^*$, and then express such a variation in terms of electrical quantities.\nIn order to solve Problem~\\ref{prob1} by the electrical formulation, we need to compute \\eqref{el_form} for every link $e$ in $\\mathcal E$. The Wardrop equilibrium $f^*$ is assumed to be observable and therefore given. The voltage $v$ (and thus $y$) can be derived by solving the linear system \\eqref{eq:voltage} and has to be computed only once.\nOn the contrary, the computation of $r_{e}$ must be repeated for every link, hence it requires to solve $\\mathrm{L}$ sparse linear systems. To reduce the computational effort, in Section~\\ref{approximation} we shall propose a method to \\textit{approximate} the effective resistance of a link that, under a suitable assumption on the sparseness of the network, does not scale with the network size, allowing for a more efficient solution to Problem~\\ref{prob1}. The next result shows how to compute the derivative of the social cost variation for small interventions.\n\\begin{corollary}\nLet $(\\mathcal{G}, a, b, \\nu)$ be a routing game, and assume that for every $i$ in $\\mathcal E$ it holds either $f_i^*>0$ or $\\lambda_i^*>0$. Then,\n$$\n\\frac{\\partial \\Delta C(u)}{\\partial u_e}\\Big|_{u = \\mathbf{0}} = \n\\begin{cases}\na_e f_e^* y_e \\quad & \\text{if} \\ \\lambda_e^* = 0, \\\\\n0 & \\text{if} \\ \\lambda_e^* > 0.\n\\end{cases}\n$$\n\\end{corollary}\n\\medskip\n\\begin{proof}\nThe fact that for every link $i$ it holds either $f_i^*>0$ or $\\lambda_i^*>0$ implies that for infinitesimal interventions the support of $f^*$ is not modified. If $\\lambda_e^*=0$, then $f_e^*$ and we can derive the social cost variation in \\eqref{el_form} with respect to $u_e$. The case $\\lambda_e^*>0$ follows from continuity arguments and from the complementary slackness condition, which implies that $f_e^*(u)=0$ in a neighborhood of $u = \\mathbf{0}$.\n\\end{proof}\n\\begin{remark}\nObserve that the derivative of the social cost does not depend on the effective resistance of the link.\n\\end{remark}\n\n\\subsection{On the validity of Assumption~\\ref{assumption}}\n\\label{sec:assumption}\nIn this section we discuss Assumption~\\ref{assumption}. In particular, we show that the assumption is without loss of generality on series-parallel networks, if the throughput is sufficiently large. We first recall the definition of directed series-parallel networks, and then present the result in Proposition~\\ref{sp_prp}.\n\\begin{definition\n\tA directed network $\\mathcal{G}$ is series-parallel if and only if\n\t(i) it is composed of two nodes only ($\\mathrm{o}$ and $\\mathrm{d}$), connected by single link from $\\mathrm{o}$ to $\\mathrm{d}$, or\n\t(ii) it is the result of connecting two directed series-parallel networks $\\mathcal{G}_1$ and $\\mathcal{G}_2$ in parallel, by merging $\\mathrm{o}_1$ with $\\mathrm{o}_2$ and  $\\mathrm{d}_1$ with $\\mathrm{d}_2$, or\n\t(iii) it is the result of connecting two directed series-parallel networks $\\mathcal{G}_1$ and $\\mathcal{G}_2$ in series, by merging $\\mathrm{d}_1$ with $\\mathrm{o}_2$.\n\\end{definition}\n\n\\begin{proposition}\n\t\\label{sp_prp}\n\tLet $(\\mathcal{G}, a, b, \\nu)$ be a routing game. If $\\mathcal{G}$ is series-parallel, there exists $\\overline{m}$ such that for every $m \\ge \\overline{m}$, $\\mathcal{E}_+=\\emptyset$. Furthermore, if $b=\\mathbf{0}$, $\\mathcal{E}_+=\\emptyset$ for every $m >0$.\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix~\\ref{app:proofs}.\n\\end{proof}\n\n\\begin{remark}\n\tProposition~\\ref{sp_prp} implies that Assumption~\\ref{assumption} is without loss of generality on series-parallel networks if $m \\ge \\overline{m}$.\n\\end{remark}\n\nThe next example shows that, if the throughput is not sufficiently large, Assumption~\\ref{assumption} may be violated.\n\\begin{example}\n\tConsider the series-parallel network in Figure~\\ref{wheatstone}.\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale = 0.7]\n\t\t\t\\node[draw, circle] (1) at (0,0)  {o};\n\t\t\t\\node[draw, circle] (2) at (4,0) {d};\n\t\t\n\t\t\t\\path [->, >=latex]  (1) edge [bend right=30]\n\t\t\tnode [below] {$e_1$} (2);\n\t\t\t\\path [->, >=latex]  (1) edge [bend left=30]\n\t\t\tnode [above] {$e_2$} (2);\n\t\t\\end{tikzpicture}\n\t\t\\caption{A directed series-parallel network. If the throughput is not sufficiently large, Assumption~\\ref{assumption} is not guaranteed to hold.}\n\t\t\\label{wheatstone}\n\t\\end{figure}\n\tLet $m=1$, and consider affine delay functions $\\tau_1(f_1)=f_1+1, \\tau_2(f_2)=f_2+3\/2$. One can verify that\n\t\\begin{equation*}\n\t\tf_1^*=3\/4, \\quad f_2^*=1\/4, \\quad \\lambda_1^*=\\lambda_2^*=0.\n\t\\end{equation*}\n\tModifying $a_1$ from $1$ to $1\/3$ (i.e., with $u=2\\delta^{(1)}$), we get:\n\t\\begin{equation*}\n\t\tf_1^*(u)=1, \\quad f_2^*(u)=0, \\quad \\lambda_1^*(u)=0, \\quad \\lambda_2^*(u)=1\/6,\n\t\\end{equation*}\n\tviolating Assumption~\\ref{assumption}. Proposition~\\ref{sp_prp} proves that this does not occur if $m$ is sufficiently large.\n\n\\end{example}\n\n\n\n\\section{An approximate solution to Problem 1}\n\\label{approximation}\nAs shown in the previous section, Problem~\\ref{prob1} may be rephrased in terms of electrical quantities over a related resistor network. Solving the NDP problem in this formulation requires to solve $\\mathrm{L}$ linear systems whose dimension scales linearly with $\\mathrm{N}$. Since the voltage $v$ may be computed in quasi-linear time by solving the sparse linear system \\eqref{eq:voltage} (see \\cite{cohen2014solving} for more details), the computational bottleneck is given by the computation of the effective resistance of every link of the resistor network. \nThe main idea of our method is that, although the effective resistance of a link depends on the entire network, it can be approximate by looking at a local portion of the network only.\nWe then formulate an algorithm to solve Problem~\\ref{prob1} by exploiting our approximation method.\n\\subsection{Approximating the effective resistance}\n\\label{method}\nWe introduce the following operations on resistor networks.\n\\begin{definition}[Cutting at distance $d$]\n\t\\label{def:cutting}\n\tA resistor network $\\mathcal{G}_R$ is cut at distance $d$ from link $l = \\{i,j\\}$ in $\\mathcal{L}$ if every node at distance greater than $d$ from link $l$ (i.e., from both $i$ and $j$) is removed, and every link with at least one endpoint in the set of the removed nodes is removed. Let $\\mathcal{G}^{U_{d}}_{l}$ and $r_{l}^{U_d}$ denote such a network and the effective resistance of link $l$ on it, respectively.\n\\end{definition}\n\n\\begin{definition}[Shorting at distance $d$]\n\tA resistor network $\\mathcal{G}_R$ is shorted at distance $d$ from $l$ in $\\mathcal{L}$ if all the nodes at distance greater than $d$ from link $l$ are shorted together, i.e., an infinite conductance is added between each pair of such nodes. Let $\\mathcal{G}^{L_{d}}_{l}$ and $r_{l}^{L_d}$ denote such a network and the effective resistance of link $l$ on it, respectively.\n\\end{definition}\n\nWe refer to Figure~\\ref{quad} for an example of these techniques applied to a regular grid.\nWe next prove that $r^{U_d}_{l}$ and $r^{L_d}_{l}$ are respectively an upper and a lower bound for the effective resistance $r_{l}$ for every link $l$. 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at (1,0)  {};\n\t\t\n\t\t\t\\node[draw,fill=yellow] (3) at (-1,0) {};\n\t\t\t\\node[draw,fill=yellow] (4) at (0,1) {};\n\t\t\t\\node[draw,fill=yellow] (5) at (1,1)  {};\n\t\t\t\\node[draw,fill=yellow] (6) at (2,0)  {};\n\t\t\t\\node[draw,fill=yellow] (7) at (1,-1)  {};\n\t\t\t\\node[draw,fill=yellow] (8) at (0,-1)  {};\n\t\t\n\t\t\t\\path (1) edge node [above] {$l$} (2);\n\t\t\t\\path (1) edge (3);\n\t\t\t\\path (1) edge (4);\n\t\t\t\\path (1) edge (8);\n\t\t\t\\path (2) edge (5);\n\t\t\t\\path (2) edge (6);\n\t\t\t\\path (2) edge (7);\n\t\t\t\\path (4) edge (5);\n\t\t\t\\path (7) edge (8);\n\t\t\\end{tikzpicture}\n\t\\quad\n\t\t\\begin{tikzpicture}[scale=0.65, transform shape]\n\t\t\t\\node[draw,fill=green] (1) at (0,0)  {};\n\t\t\t\\node[draw,fill=green] (2) at (1,0)  {};\n\t\t\n\t\t\t\\node[draw,fill=yellow] (3) at (-1,0) {};\n\t\t\t\\node[draw,fill=yellow] (4) at (0,1) {};\n\t\t\t\\node[draw,fill=yellow] (5) at (1,1)  {};\n\t\t\t\\node[draw,fill=yellow] (6) at (2,0)  {};\n\t\t\t\\node[draw,fill=yellow] (7) at (1,-1)  {};\n\t\t\t\\node[draw,fill=yellow] (8) at (0,-1)  {};\n\t\t\t\\node[draw,fill=orange] (9) at \n\t\t\t(0.5,2) {s};\n\t\t\n\t\t\t\\path (1) edge node [above] {$l$} (2);\n\t\t\t\\path (1) edge (3);\n\t\t\t\\path (1) edge (4);\n\t\t\t\\path (1) edge (8);\n\t\t\t\\path (2) edge (5);\n\t\t\t\\path (2) edge (6);\n\t\t\t\\path (2) edge (7);\n\t\t\t\\path (4) edge (5);\n\t\t\t\\path (7) edge (8);\n\t\t\n\t\t\t\\path (4) edge [bend right=0]\n\t\t\tnode [below] {} (9);\n\t\t\t\\path (5) edge [bend right=0]\n\t\t\tnode [below] {} (9);\n\t\t\t\\path (3) edge [bend right=-20]\n\t\t\tnode [below] {} (9);\n\t\t\t\\path (6) edge [bend right=20]\n\t\t\tnode [below] {} (9);\n\t\t\n\t\t\t\\coordinate (ghost) at (3, 0);\n\t\t\t\\coordinate (ghost2) at (-2,0);\n\t\t\n\t\t\t\\draw[-] (7) to[out=0, in=-90] (ghost) to[out=90, in=0] (9);\n\t\t\t\\draw[-] (8) to[out=180, in=-90] (ghost2)\n\t\t\tto[out=90, in=180] (9);\n\t\t\\end{tikzpicture}\n\t\\caption{Square grid. \\emph{Above}: the yellow, orange and red nodes are at distance $1$, $2$ and $3$, respectively from the green nodes.\n\t\\emph{Bottom-left}: the grid cut at distance $1$ from link $l$.\n\t\t\\emph{Bottom-right}: the grid shorted at distance $1$ from link $l$. \n\t\tNote that in the bottom right network the links connecting yellow nodes with node $s$ do not have unitary weights.}\n\t\\label{quad}\n\\end{figure}\n\\begin{lemma}[Rayleigh's monotonicity laws \\cite{levin2017markov}]\n\tIf the resistances of one or more links are increased, the effective\n\tresistance between two arbitrary nodes cannot decrease.\n\tIf the resistances of one or more links are decreased, the effective resistance cannot increase.\n\t\\label{ray}\n\\end{lemma}\n\\begin{proposition}\n\t\t\\label{prp_ray}\n\tLet $\\mathcal{G}_R$ be a resistor network. For every link $l=\\{i,j\\}$ in $\\mathcal{L}$, \n\t\\begin{equation*}\n\t\t\\label{bounds}\n\t\tr^{U_{d_1}}_{l}\\ge r^{U_{d_2}}_{l} \\ge r_{l} \\ge r^{L_{d_2}}_{l} \\ge r^{L_{d_1}}_{l}, \\quad \\forall d_2 \\ge d_1 \\ge 1.\n\t\\end{equation*}\n\tMoreover,\n\t\\begin{equation}\n\t\t1\/w^* \\le r^{L_d}_{l}\\le r^{U_d}_{l} \\le 1\/W_{ij}, \\quad \\forall d \\ge 1.\n\t\t\\label{remark}\n\t\\end{equation}\n\\end{proposition}\\medskip\n\\begin{proof}\nCutting a network at distance $d$ is equivalent to setting to infinity the resistance of all the links with at least one endpoint at distance greater than $d$. Shorting a network at distance $d$ is equivalent to setting to zero the resistance between any pair of nodes at distance greater than $d$. Then, by Rayleigh's monotonicity laws, it follows $r^{U_d}_{l}\\ge r_{l} \\ge r^{L_d}_{l}$. Similar arguments may be used to show that, if $d_1<d_2$, then $r^{U_{d_1}}_{l} \\ge r^{U_{d_2}}_{l}$ and $r^{L_{d_1}}_{l} \\le r^{L_{d_2}}_{l}$. The right inequality in \\eqref{remark} follows from Rayleigh's monotonicity laws, by noticing that the effective resistance computed in the network with only nodes $i$ and $j$ (which is equal to $1\/W_{ij}$) is an upper bound for $r_{l}^{U_{1}}$. The left inequality follows from noticing that the effective resistance on the network in which every node except $i$ is shorted with $j$, which results in a network with only two nodes and a conductance between $i$ and $j$ not greater than $w^*$ (hence, resistance no less than $1\/w^*$) is a lower bound of $r^{L_1}_{l}$.\n\\end{proof}\n\nProposition~\\ref{prp_ray} states that cutting and shorting a network provides upper and lower bound for the effective resistance of a link. Moreover, the bound gap is a monotone function of the distance $d$.\n\n\n\\subsection{Our algorithm}\nWe here propose an algorithm to solve in approximation Problem~\\ref{prob1} based on our method for approximating the effective resistance. Our approach is detailed in Algorithm 1.\n\\begin{algorithm}\n\t\\SetAlgoNoLine\n\t\\KwIn{The affine routing game $(\\mathcal{G},a,b,\\nu)$, the cost functions $\\{h_e\\}_{e \\in \\mathcal{E}}$, and the distance $d\\ge1$ for effective resistance approximation.}\n\t\\KwOut{The optimal intervention $u^{*d}$.}\n\tConstruct the associated resistor network $\\mathcal{G}_R$.\\\\\n\tCompute $v$ and $y$ by solving \\eqref{eq:voltage}.\\\\\n\t\\For{every $l$ in $\\mathcal{L}$}\n\t{Construct $\\mathcal{G}_{l}^{U_d}$ and $\\mathcal{G}_{l}^{L_d}$. \\\\\n\t\tCompute $r_{l}^{U_d}$ and $r_{l}^{L_d}$ on $\\mathcal{G}_{l}^{U_d}$ and $\\mathcal{G}_{l}^{L_d}$.}\n\t\\For{every $e$ in $\\mathcal{E}$}\n\t{Find $u_e^{*d}$ such that\t\n\t\t\\begin{equation}\n\t\t\t\\label{step1alg}\n\t\t\t\\begin{aligned}\n\t\t\t\tu_e^{*d} \\in \\\n\t\t\t\t& \\underset{u_e \\ge 0} {\\argmax}\n\t\t\t\t& & a_e f_e^* \\frac{y_e}{\\frac{1}{u_e}+\\frac{r_e^{U_d}+r_e^{L_d}}{2a_e}} - \\alpha h_e(u_e).\n\t\t\t\\end{aligned}\n\t\t\\end{equation}}\n\tFind $e^{*d}$ such that\t\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\te^{*d} \\in \\\n\t\t\t& \\underset{e \\in \\mathcal{E}} {\\argmax}\n\t\t\t& &  a_e f_e^*\\frac{y_e}{\\frac{1}{u_e^{*d}}+\\frac{r_e^{U_d}+r_e^{L_d}}{2a_e}} - \\alpha h_e(u_e^{*d}).\n\t\t\\end{aligned}\n\t\t\\label{step2alg}\n\t\\end{equation}\n\t\\caption{}\n\t\\label{alg}\nThe optimal intervention is $u^{*d} = u_e^{*d} \\delta^{(e^{*d})}$.\n\\end{algorithm}\nNotice that the performance of Algorithm 1 depends on the choice of the parameter $d$. Specifically, the higher $d$ is the better is the approximation of the social cost variation.\n\\begin{theorem}\n\t\\label{thm2}\n\tLet\n\t$\\Delta C^{(u)}$ be the social cost gain corresponding to intervention $u = u_e \\delta^{(e)}$ as given in Theorem~\\ref{thm}, and\n\t$$\n\t\\Delta C_d^{(u)}=a_ef_e^*\\frac{y_e}{\\frac{1}{u_e}+\\frac{r_e^{U_d}+r_{e}^{L_d}}{2a_e}}$$\n\tbe the social cost gain estimated by Algorithm~\\ref{alg} for a given distance $d\\ge 1$. Then,\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\bigg|\\frac{\\Delta C^{(u)}-\\Delta C_d^{(u)}}{\\Delta C^{(u)}}\\bigg| \n\t\t\t&\\le \\frac{\\epsilon_{ed}}{2\\Big(\\frac{1}{u_e}+\\frac{r_e^{U_d}+r_e^{L_d}}{2a_e}\\Big)} \\\\\n\t\t\t&\\le \\frac{\\epsilon_{ed}}{2\\Big(\\frac{1}{u_e}+\\frac{1}{w^* \\cdot a_e}\\Big)},\n\t\t\t\\label{err_rel2}\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere\n\t\\begin{equation*}\n\t\t\\label{epsilon}\n\t\t\\epsilon_{ed}:=\n\t\t\\frac{r_e^{U_d}-r_e^{L_d}}{a_e}.\n\t\\end{equation*}\n\tFurthermore,\n\t\\begin{equation}\n\t\t\\Delta C^{(u)} \\ge a_e f_e^* \\frac{y_e}{\\frac{1}{u_e}+\\frac{r_e^{U_d}}{a_e}}.\n\t\t\\label{min_gain}\n\t\\end{equation}\n\\end{theorem}\\medskip\n\\begin{proof}\nSee Appendix~\\ref{app:proofs}.\n\\end{proof}\n\nIn the next section we provide sufficient conditions for $\\epsilon_{ed}$ to vanish for large distance $d$ in the limit of infinite networks. In the rest of this section we show that the bound gap (and therefore $\\epsilon_{ed}$) and the computational complexity of the bounds (for a fixed $d$) depend only on the local structure around link $M(e)$ of the resistor network, and do not scale with the network size, under the following assumption.\n\\begin{assumption}\n\t\\label{sparse}\n\tLet $\\mathcal{G}_R$ be the resistor network corresponding to the transportation network $\\mathcal{G}$. Let $l \\in \\mathcal L$ be an arbitrary link of $\\mathcal{G}_R$, and $\\mathcal{N}_{\\le d}$ denote the set of nodes that are at distance no greater than $d$ from link $l$.\n\tWe assume that the network $\\mathcal{G}$ is sparse in such a way that the cardinality of $\\mathcal{N}_{\\le d}$ does not depend on $\\mathrm{N}$ for any $d$.\n\\end{assumption}\n\nAssumption~\\ref{sparse} is suitable for transportation networks, because of physical constraints not allowing for the degree of the nodes to grow unlimitedly (think for instance of planar grids, where the degree of the nodes is given no matter what the size of the network is, and the local structure of the network around an arbitrary node does not depend on the network size $\\mathrm{N}$). Notice also that, under Assumption~\\ref{sparse}, $\\mathrm{N}$ and $\\mathrm{L}$ are proportional.\n\\begin{proposition}\n\tLet $\\mathcal{G}_R=(\\mathcal{N},\\mathcal{L},W)$ be a resistor network, $l = \\{i,j\\}$ in $\\mathcal{L}$, and $d \\ge 1$. Then, $r_l^{U_d}$ and $r_l^{L_d}$, and their computational complexity, depend only on the structure of $\\mathcal{G}_R$ within distance $d+1$ from $i$ and $j$ only. Furthermore, under Assumption~\\ref{sparse} they do not depend on $\\mathrm{N}$.\n\t\\label{speed}\n\\end{proposition}\n\\begin{proof} \nSee Appendix~\\ref{app:proofs}.\n\\end{proof}\n\n\\begin{remark}\n\tTo the best of our knowledge, the complexity of the most efficient algorithm to compute the spanning tree centrality (or effective resistance) of a link in large networks scales with the number of links \\cite{hayashi2016efficient}. On the contrary,\n\tProposition~\\ref{speed} states that under Assumption~\\ref{sparse} the computational time for approximating a single effective resistance does not scale with $\\mathrm{N}$. Therefore, approximating all the effective resistances requires a computational time linear in $\\mathrm{N}$. Observe that $v$ (and thus $y$) is computed via a diagonally dominant, symmetric and positive definite linear systems. The design of fast algorithms to solve this class of problem is an active field of research in the last years. To the best of our knowledge, the best algorithm has been provided in \\cite{cohen2014solving} and has complexity\n\t$O(M \\log^k \\mathrm{N} \\log \\mathrm{1\/\\epsilon})$, where $\\epsilon$ is the tolerance error, $k$ is a constant, and $M$ is the number of non-zero elements in the matrix of the linear system. Since in our case $M$ scales with $\\mathrm{L}$, and since $\\mathrm{L}$ scales with $\\mathrm{N}$ under Assumption~\\ref{sparse}, Algorithm~\\ref{alg} is quasilinear in $\\mathrm{N}$. Step \\eqref{step1alg} consists in maximizing a function of one variable. Finally, step \\eqref{step2alg} consists in taking the maximum of $\\mathrm{E}$ numbers.\n\\end{remark}\n\n\n\\section{Bound analysis}\n\\label{performance}\nIn this section we characterize the gap between the bounds on the effective resistance of a link in terms of random walks over the resistor networks $\\mathcal{G}_R$, $\\mathcal{G}_{l}^{U_d}$ and $\\mathcal{G}_{l}^{L_d}$. We then leverage this characterization to provide a sufficient condition on the network under which the bound gap vanishes asymptotically for large distance $d$. To this end, we interpret the conductance matrix $W$ of the resistor network as the transition rates of continuous-time Markov chain whose state space is the node set of the network, and introduce the following notation. Let:\n\\begin{itemize}\n\t\\item $T_\\mathcal{S}$ and $T_\\mathcal{S}^+$ denote the hitting time (i.e., the first time $t\\ge0$ such that the random walk hits the set $\\mathcal{S} \\in \\mathcal{N}$), and the return time (i.e., the first time $t>0$ such that the random walk hits the set $\\mathcal{S}$), respectively.\n\t\\item $\\mathcal{N}_d$ denote the set of the nodes that are at distance $d$ from link $l = \\{i,j\\}$, i.e., at distance $d$ from $i$ (or $j$) and at distance greater or equal than $d$ from $j$ (or $i$). Index $l$ is omitted for simplicity of notation.\n\t\\item $p_k(X)$, $p_k^{U_d}(X)$ and $p_k^{L_d}(X)$, denote the probability that event $X$ occurs, conditioned on the fact that the random walk starts in $k$ at time $0$ and evolves over the resistor networks $\\mathcal{G}_R$, $\\mathcal{G}_{l}^{U_d}$ and $\\mathcal{G}_{l}^{L_d}$, respectively.\n\\end{itemize} \nThe next result provides a characterization of the bound gap in terms of random walks over $\\mathcal{G}_R$, $\\mathcal{G}_{l}^{U_d}$ and $\\mathcal{G}_{l}^{L_d}$.\n\\begin{proposition}\n\t\\label{gap}\n\tLet $\\mathcal{G}_R=(\\mathcal{N},\\mathcal{L},W)$ be a resistor network. For every link $l = \\{i,j\\}$ in $\\mathcal{L}$,\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\tr^{U_d}_{l}-r^{L_d}_{l}\n\t\t& \\le \\frac{w_{i}}{(W_{ij})^2} \\underbrace{p_i(T_{\\mathcal{N}_d} < T_j)}_{\\text{Term 1}} \\ \\cdot \\\\\n\t\t& \\cdot \\underset{g \\in \\mathcal{N}_d}{ \\max} \\underbrace{\\big(p^{U_d}_g(T_i < T_j)-p^{L_d}_g(T_i < T_j)\\big)}_{\\text{Term 2}},\n\t\t\\end{aligned}\n\t\t\\label{equat_gap}\n\t\\end{equation}\nwhere the quantities in \\eqref{equat_gap} are computed with respect to the continuous-time Markov chain with transition rates $W$. \n\\end{proposition}\n\\begin{proof}\nSee Appendix~\\ref{app:proofs}.\n\\end{proof}\n\nIn the next sections we shall use this result to analyze the asymptotic behaviour of the bound gap for an arbitrary link $l$ in $\\mathcal{L}$ as $d \\to +\\infty$, for networks whose node set is infinite and countable. In particular, we show in Section~\\ref{rec_sec} that this error vanishes asymptotically for the class of recurrent networks. The core idea to prove this result is to show that Term 1 vanishes. To generalize our analysis beyond recurrent networks, in Section~\\ref{beyond} we study both Term 1 and 2 and provide examples showing that all combinations in Table~\\ref{table_bounds} are possible. In particular, it is possible that the bound gap vanishes asymptotically for non-recurrent networks (for which Term 1 $\\nrightarrow 0$, see \\cite[Section 21.2]{levin2017markov}) if Term 2 $\\rightarrow 0$.\n\\begin{table}\n\t\\caption{All the four cases are possible, as shown in Section~\\ref{beyond}. Term 1 $\\rightarrow 0$ under the assumption that the network is recurrent, as proved in  Section~\\ref{rec_sec}. \\label{table_bounds}}\n\t\\centering\n\t\t\\begin{tabular}{lll}\n\t\t\t& Term 2 $\\rightarrow 0$ & Term 2 $\\nrightarrow 0$ \\\\\n\t\t\t\\hline\n\t\t\tTerm 1 $\\rightarrow 0$\n\t\t\t& 2d grid\n\t\t\t& Ring \\\\\n\t\t\tTerm 1 $\\nrightarrow 0$\n\t\t\t& 3d grid\n\t\t\t& Double tree\\\\\n\t\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\\subsection{Recurrent networks} \n\\label{rec_sec}\nWe start by introducing the class of recurrent networks.\n\\begin{definition}[Recurrent random walk]\n\tA random walk is recurrent if, for every starting point, it visits its starting node infinitely often with probability one \\cite[Section 21.1]{levin2017markov}.\n\\end{definition}\n\n\\begin{definition}[Recurrent network]\n\tAn infinite resistor network $\\mathcal{G}_R=(\\mathcal{N},\\mathcal{L}, W)$ is recurrent if the random walk on the network is recurrent.\n\\end{definition}\n\nThe next theorem states that the bound gap vanishes asymptotically on recurrent networks if the degree of every node is finite. Note that the boundedness of the degree of all the nodes is guaranteed under Assumption~\\ref{sparse}.\n\\begin{theorem}\n\tLet $\\mathcal{G}_R=(\\mathcal{N},\\mathcal{L},W)$ be an infinite recurrent resistor network, and let $w^*<+\\infty$. Then, for every $l$ in $\\mathcal{L}$,\n\t\\begin{equation*}\n\t\t\\lim_{d \\rightarrow +\\infty} (r^{U_d}_{l}-r^{L_d}_{l})=0,\n\t\\end{equation*}\n\t\\label{recurrent}\n\\end{theorem}\n\\begin{proof}\nIt is proved in \\cite[Proposition 21.3]{levin2017markov} that a network is recurrent if and only if \n\\begin{equation}\n\t\\lim_{d \\rightarrow +\\infty}p_i(T_{\\mathcal{N}_d}<T_j)=0 \\quad \\forall l=\\{i,j\\} \\in \\mathcal{L}.\n\t\\label{lim_rec2}\n\\end{equation}\nObserve that, to hit any node in $\\mathcal{N}_{d+1}$, the random walk starting from $i$ has to hit at least one node in $\\mathcal{N}_d$. Hence, the sequence $\\big\\{p_i(T_{\\mathcal{N}_d}<T_j)\\big\\}_{d=1}^\\infty$ is non-increasing in $d$ and the limit in \\eqref{lim_rec2} is well defined.\nThen, from \\eqref{equat_gap}, \\eqref{lim_rec2}, from the fact that $0 \\le p^{U_d}_g(T_i < T_j)-p^{L_d}_g(T_i < T_j)\\le1$ for every node $g$, and from the assumptions $w^* < +\\infty$ and $W_{ij}>0$ (recall that $i$ and $j$ are adjacent nodes), it follows\n\\begin{equation*}\n\t\\lim_{d \\rightarrow +\\infty} r^{U_d}_{l}-r^{L_d}_{l}\\le \\frac{w^*}{(W_{ij})^2} \\lim_{d \\rightarrow +\\infty} p_i(T_{\\mathcal{N}_d}<T_j) = 0,\n\\end{equation*} \nwhich completes the proof.\n\\end{proof}\n\\begin{corollary}\nLet $\\mathcal{G}$ be a transportation network with recurrent associated resistor network $\\mathcal{G}_R$. Then, for every $u$ in $\\mathcal{U}$, \n$$\n\\lim_{d \\to + \\infty}\\bigg|\\frac{\\Delta C^{(u)}-\\Delta C_d^{(u)}}{\\Delta C^{(u)}}\\bigg|=0.\n$$\n\\end{corollary}\\medskip\n\\begin{proof}\n\tThe proof follows from Theorem~\\ref{thm2} and Theorem~\\ref{recurrent}, which imply $\\lim_{d \\rightarrow +\\infty} \\epsilon_{ed}=0$ for every $e$ in $\\mathcal{E}$.\n\\end{proof}\n\nRecurrence is a sufficient condition for the approximation error of a link effective resistance to vanish asymptotically, but is not necessary, as discussed in the next section.\n\n\n\n\\subsection{Beyond recurrence}\n\\label{beyond}\nWe here provide examples of infinite resistor networks for all of the cases in Table~\\ref{table_bounds}.\nObserve that, for every link $l = \\{i,j\\}$ in $\\mathcal{L}$, the network cut at distance $d$ from $l$ and the network shorted at distance $d$ from $l$ differ for one node only (denoted by $s$), which is the result of shorting in a unique node all the nodes at distance greater than $d$ from $l$.\nIntuitively speaking, our conjecture is that Term 2 in \\eqref{equat_gap} is small when the network has many short paths. In this case, adding the node $s$ leads to a small variation of the probability, starting from any node in $\\mathcal{N}_d$, of hitting $i$ before $j$, thus making Term 2 small. This intuition can be clarified with the next examples.\n\\subsubsection{2d grid}\nConsider an infinite unweighted bidimensional grid as in Figure~\\ref{quad3}. This network is very relevant for NDPs, since many transportation networks have similar topologies. The network is known to be recurrent \\cite[Example 21.8]{levin2017markov}, hence Theorem~\\ref{recurrent} guarantees that Term 1 vanishes asymptotically for every link $l = \\{i,j\\}$. Our conjecture, confirmed by numerical simulations, is that for every node $g$ in $\\mathcal{N}_d$,\n\\begin{equation*}\n\t\\begin{gathered}\n\t\t\\lim_{d \\rightarrow +\\infty} p^{U_d}_g(T_i < T_j) =  1\/2, \\quad\n\t\t\\lim_{d \\rightarrow +\\infty} p^{L_d}_g(T_i < T_j) =  1\/2.\n\t\\end{gathered}\n\\end{equation*} \nHence, this is recurrent network for which also Term 2 vanishes asymptotically.\n\\subsubsection{3d grid}\nConsider an infinite unweighted tridimensional grid. This network is not recurrent \\cite[Example 21.9]{levin2017markov}, therefore Term 1 does not vanish asymptotically, and we cannot conclude from Theorem~\\ref{recurrent} that for every $l = \\{i,j\\}$ the bound gap vanishes asymptotically. Nonetheless, numerical simulations show that, similarly to the bidimensional grid, for every node $g$ in $\\mathcal{N}_d$,\n\\begin{equation*}\n\t\\begin{gathered}\n\t\t\\lim_{d \\rightarrow +\\infty} p^{U_d}_g(T_i < T_j) =  1\/2, \\quad\n\t\t\\lim_{d \\rightarrow +\\infty} p^{L_d}_g(T_i < T_j) =  1\/2.\n\t\\end{gathered}\n\\end{equation*} \nHence, this is a non-recurrent network for which Term 2 (and therefore the bound gap $r_l^{U_d}-r_l^{L_d}$) vanishes asymptotically in the limit of infinite distance $d$.\n\\subsubsection{Ring}\nConsider an infinite unweighted ring network, and let us focus on nodes $5$ and $6$ in Figure~\\ref{upper_lower_ring}. Then, \n\\begin{equation*}\n\t\\begin{gathered}\n\t\tp^{U_d}_{5}(T_1 < T_2)=1, \\quad\n\t\tp^{U_d}_6(T_1 < T_2)=0.\n\t\\end{gathered}\n\\end{equation*}\nfor each $d$ (even $d \\rightarrow + \\infty$), whereas, \n\\begin{equation*}\n\t\\begin{gathered}\n\t\tp^{L_d}_5(T_1 < T_2)=\\frac{d}{2d+1} \\xrightarrow[d \\rightarrow + \\infty]{} \\frac{1}{2}, \\\\\n\t\tp^{L_d}_5(T_1 < T_2)=\\frac{d+1}{2d+1} \\xrightarrow[d \\rightarrow + \\infty]{} \\frac{1}{2},\n\t\\end{gathered}\n\\end{equation*}\nsince this case is equivalent to the gambler's ruin problem (see \\cite[Proposition 2.1]{levin2017markov}).\nHence, Term 2 does not vanish for the ring. This is due to the fact that all the paths from $5$ to $2$ in $\\mathcal G_l^{L_2}$ not including node $1$ include the node $s$. Still, Term 1 (and thus the bound gap $r_l^{U_d}-r_l^{L_d}$) vanishes asymptotically by Theorem~\\ref{recurrent}, because the ring network is recurrent.\n\n\\begin{figure}\n\t\\centering\n\t\t\\begin{tikzpicture}[scale=0.65, transform shape]\n\t\t\t\\node[draw,fill=green] (1) at (0,0)  {i};\n\t\t\t\\node[draw,fill=green] (2) at (1,0)  {j};\n\t\t\n\t\t\t\\node[draw,fill=none] (3) at (-1,0) {};\n\t\t\t\\node[draw,fill=none] (4) at (0,1) {};\n\t\t\t\\node[draw,fill=none] (5) at (1,1)  {};\n\t\t\t\\node[draw,fill=none] (6) at (2,0)  {};\n\t\t\t\\node[draw,fill=none] (7) at (1,-1)  {};\n\t\t\t\\node[draw,fill=none] (8) at (0,-1)  {};\n\t\t\n\t\t\t\\node[draw,fill=none] (9) at (-2,0) {};\n\t\t\t\\node[draw,fill=none] (10) at (-1,1) {};\n\t\t\t\\node[draw,fill=none] (11) at (0,2)  {};\n\t\t\t\\node[draw,fill=none] (12) at (1,2)  {};\n\t\t\t\\node[draw,fill=none] (13) at (2,1)  {};\n\t\t\t\\node[draw,fill=none] (14) at (3,0)  {};\n\t\t\t\\node[draw,fill=none] (15) at (2,-1)  {};\n\t\t\t\\node[draw,fill=none] (16) at (1,-2)  {};\n\t\t\t\\node[draw,fill=none] (17) at (0,-2)  {};\n\t\t\t\\node[draw,fill=none] (18) at (-1,-1)  {};\n\t\t\n\t\t\t\\node[draw,fill=red] (19) at (-3,0) {};\n\t\t\t\\node[draw,fill=red] (20) at (-2,1) {};\n\t\t\t\\node[draw,fill=red] (21) at (-1,2)  {};\n\t\t\t\\node[draw,fill=red] (22) at (0,3)  {};\n\t\t\t\\node[draw,fill=red] (23) at (1,3)  {};\n\t\t\t\\node[draw,fill=red] (24) at (2,2)  {};\n\t\t\t\\node[draw,fill=red] (25) at (3,1)  {};\n\t\t\t\\node[draw,fill=red] (26) at (4,0)  {};\n\t\t\t\\node[draw,fill=red] (27) at (3,-1)  {};\n\t\t\t\\node[draw,fill=red] (28) at (2,-2)  {};\n\t\t\t\\node[draw,fill=red] (29) at (1,-3) {};\n\t\t\t\\node[draw,fill=red] (30) at (0,-3) {};\n\t\t\t\\node[draw,fill=red] (31) at (-1,-2)  {};\n\t\t\t\\node[draw,fill=red] (32) at (-2,-1)  {};\n\t\t\n\t\t\t\\path (1) edge (2);\n\t\t\t\\path (1) edge (3);\n\t\t\t\\path (1) edge (4);\n\t\t\t\\path (1) edge (8);\n\t\t\t\\path (2) edge (5);\n\t\t\t\\path (2) edge (6);\n\t\t\t\\path (2) edge (7);\n\t\t\n\t\t\t\\path (3) edge (9);\n\t\t\t\\path (3) edge (10);\n\t\t\t\\path (10) edge (4);\n\t\t\t\\path (4) edge (11);\n\t\t\t\\path (4) edge (5);\n\t\t\t\\path (5) edge (12);\n\t\t\t\\path (11) edge (12);\n\t\t\t\\path (3) edge (9);\n\t\t\t\\path (3) edge (10);\n\t\t\t\\path (10) edge (4);\n\t\t\t\\path (5) edge (13);\n\t\t\t\\path (6) edge (13);\n\t\t\t\\path (6) edge (14);\n\t\t\t\\path (6) edge (15);\n\t\t\t\\path (15) edge (7);\n\t\t\t\\path (16) edge (7);\n\t\t\t\\path (16) edge (17);\n\t\t\t\\path (8) edge (17);\n\t\t\t\\path (8) edge (18);\n\t\t\t\\path (7) edge (8);\n\t\t\t\\path (3) edge (18);\n\t\t\n\t\t\t\\path (19) edge (9);\n\t\t\t\\path (9) edge (20);\n\t\t\t\\path (10) edge (20);\n\t\t\t\\path (10) edge (21);\n\t\t\t\\path (11) edge (21);\n\t\t\t\\path (11) edge (22);\n\t\t\t\\path (22) edge (23);\n\t\t\t\\path (12) edge (23);\n\t\t\t\\path (12) edge (24);\n\t\t\t\\path (24) edge (13);\n\t\t\t\\path (25) edge (13);\n\t\t\t\\path (25) edge (14);\n\t\t\t\\path (26) edge (14);\n\t\t\t\\path (27) edge (14);\n\t\t\t\\path (15) edge (28);\n\t\t\t\\path (15) edge (27);\n\t\t\t\\path (16) edge (28);\n\t\t\t\\path (16) edge (29);\n\t\t\t\\path (29) edge (30);\n\t\t\t\\path (17) edge (30);\n\t\t\t\\path (17) edge (31);\n\t\t\t\\path (31) edge (18);\n\t\t\t\\path (32) edge (9);\n\t\t\t\\path (32) edge (18);\n\t\\end{tikzpicture}\n\t\\caption{Bidimensional square grid, cut at distance $d=3$ from $l=\\{i,j\\}$. The red nodes belong to $\\mathcal{N}_d$. As $d$ increases, $p_g(T_1 < T_2)$ approaches $1\/2$ for every node $g$ in $\\mathcal{N}_d$. \\label{quad3}}\n\\end{figure}\n\\begin{figure}\n\t\t\t\\centering{\\hspace{0.6cm} $\\mathcal{G}_{l}^{U_2}$ \\hspace{1.8cm} $\\mathcal{G}_{l}^{L_2}$}\\\\\n\t\t\t\\vspace{0.2cm}\n\t\t\t\\begin{tikzpicture}[scale=0.75, transform shape]\n\t\t\t\t\\node[draw, circle, fill=green] (1) at (-1,0)  {$1$};\n\t\t\t\t\\node[draw, circle, fill=green] (2) at (1,0) {$2$};\n\t\t\t\t\\node[draw, circle] (3) at (-1,1)  {$3$};\n\t\t\t\t\\node[draw, circle] (4) at (1,1) {$4$};\n\t\t\t\t\\node[draw, circle, fill=red] (5) at (-1,2) {$5$};\n\t\t\t\t\\node[draw, circle, fill=red] (6) at (1,2) {$6$};;\n\t\t\t\t\\path   (1) edge [bend right=0]\n\t\t\t\tnode [above] {$l$} (2);\n\t\t\t\t\\path   (1) edge [bend right=0]\n\t\t\t\tnode [above] {} (3);\n\t\t\t\t\\path   (2) edge [bend left=0]\n\t\t\t\tnode [above] {} (4);\n\t\t\t\t\\path (3) edge [bend right=0]\n\t\t\t\tnode [above] {} (5);\n\t\t\t\t\\path   (4) edge [bend right=0]\n\t\t\t\tnode [below] {} (6);\n\t\t\t\\end{tikzpicture}\n\t\t\t\\qquad\n\t\t\t\\begin{tikzpicture}[scale=0.56, transform shape]\n\t\t\t\t\\node[draw, circle, fill=green] (1) at (-1,0)  {$1$};\n\t\t\t\t\\node[draw, circle, fill=green] (2) at (1,0) {$2$};\n\t\t\t\t\\node[draw, circle] (3) at (-1,1)  {$3$};\n\t\t\t\t\\node[draw, circle] (4) at (1,1) {$4$};\n\t\t\t\t\\node[draw, circle, fill=red] (5) at (-1,2) {$5$};\n\t\t\t\t\\node[draw, circle, fill=red] (6) at (1,2) {$6$};\n\t\t\t\t\\node[draw, circle] (7) at (0,3) {$s$};\n\t\t\t\t\\path   (1) edge [bend right=0]\n\t\t\t\tnode [above] {$l$} (2);\n\t\t\t\t\\path   (1) edge [bend right=0]\n\t\t\t\tnode [above] {} (3);\n\t\t\t\t\\path   (2) edge [bend left=0]\n\t\t\t\tnode [above] {} (4);\n\t\t\t\t\\path (3) edge [bend right=0]\n\t\t\t\tnode [above] {} (5);\n\t\t\t\t\\path   (4) edge [bend right=0]\n\t\t\t\tnode [below] {} (6);\n\t\t\t\t\\path   (5) edge [bend left=0]\n\t\t\t\tnode [below] {} (7);\n\t\t\t\t\\path  (6) edge [bend left=0]\n\t\t\t\tnode [below] {} (7);\n\t\t\t\\end{tikzpicture}\n\t\\caption{\\emph{Left}: ring cut at distance $d=2$ from $l$. \\emph{Right}: ring shorted at distance $d=2$ from link $l=\\{1,2\\}$.} \\label{upper_lower_ring}\n\\end{figure}\n\\subsubsection{Double tree network}\n\\label{section_tree}\nThe last examples illustrates an infinitely large network in which the bound gap does not vanish asymptotically. This network is not relevant for traffic applications, since it admits one path only between every pair of nodes, but provides an interesting counterexample where the bound gap does not converge asymptotically.\nThe network is composed of two infinite trees starting from node $i$ and $j$, connected by a link $l = \\{i,j\\}$ (see Figure~\\ref{counterex}), and is unweighted. It can be shown that on this network the probability that a random walk, starting from $i$, returns on $i$ is equal to the same quantity computed on a biased random walk over an infinite line (for more details see Appendix~\\ref{app:tree}). Since the biased random walk on a line is not recurrent \\cite[Example 21.2]{levin2017markov}, then the double tree network is non-recurrent, and Term 1 $\\nrightarrow 0$. Moreover, we show in Appendix~\\ref{app:tree} that\n\\begin{equation*}\n\t\\lim_{d \\rightarrow +\\infty} r^{U_d}_{l}-r^{L_d}_{l}=\\frac{1}{3},\n\\end{equation*}\nthus implying that Term 2 $\\nrightarrow 0$.\n\\begin{figure}\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.65, transform shape]\n\t\t\\node[draw, circle, fill=green] (1) at (-2,0)  {$i$};\n\t\t\\node[draw, circle, fill=green] (2) at (2,0) {$j$};\n\t\t\\node[draw, circle] (3) at (-2.8,1)  {};\n\t\t\\node[draw, circle] (4) at (-1.2,1) {};\n\t\t\\node[draw, circle] (5) at (1.2,1) {};\n\t\t\\node[draw, circle] (6) at (2.8,1) {};\n\t\t\\node[draw, circle] (7) at (-3.2,2)  {};\n\t\t\\node[draw, circle] (8) at (-2.4,2) {};\n\t\t\\node[draw, circle] (9) at (-1.6,2) {};\n\t\t\\node[draw, circle] (10) at (-0.8,2) {};\n\t\t\\node[draw, circle] (11) at (0.8,2)  {};\n\t\t\\node[draw, circle] (12) at (1.6,2) {};\n\t\t\\node[draw, circle] (13) at (2.4,2) {};\n\t\t\\node[draw, circle] (14) at (3.2,2) {};\n\t\n\t\t\\node[draw=none] (15) at (-3.5,3)  {};\n\t\t\\node[draw=none] (16) at (-2.9,3) {};\n\t\t\\node[draw=none] (17) at (-2.7,3) {};\n\t\t\\node[draw=none] (18) at (-2.1,3) {};\n\t\t\\node[draw=none] (19) at (-1.9,3)  {};\n\t\t\\node[draw=none] (20) at (-1.3,3) {};\n\t\t\\node[draw=none] (21) at (-1.1,3) {};\n\t\t\\node[draw=none] (22) at (-0.5,3) {};\n\t\t\\node[draw=none] (23) at (0.5,3)  {};\n\t\t\\node[draw=none] (24) at (1.1,3) {};\n\t\t\\node[draw=none] (25) at (1.3,3) {};\n\t\t\\node[draw=none] (26) at (1.9,3) {};\n\t\t\\node[draw=none] (27) at (2.1,3)  {};\n\t\t\\node[draw=none] (28) at (2.7,3) {};\n\t\t\\node[draw=none] (29) at (2.9,3) {};\n\t\t\\node[draw=none] (30) at (3.5,3) {};\n\t\n\t\t\\path   (1) edge [bend right=0]\n\t\tnode [above] {$l$} (2);\n\t\n\t\t\\path   (1) edge [bend right=0]\n\t\tnode [above] {} (3);\n\t\t\\path   (1) edge [bend left=0]\n\t\tnode [above] {} (4);\n\t\t\\path (2) edge [bend right=0]\n\t\tnode [above] {} (5);\n\t\t\\path   (2) edge [bend right=0]\n\t\tnode [below] {} (6);\n\t\n\t\t\\path   (3) edge [bend left=0]\n\t\tnode [below] {} (7);\n\t\t\\path  (3) edge [bend left=0]\n\t\tnode [below] {} (8);\n\t\t\\path   (4) edge [bend right=0]\n\t\tnode [above] {} (9);\n\t\t\\path   (4) edge [bend right=0]\n\t\tnode [above] {} (10);\n\t\t\\path   (5) edge [bend left=0]\n\t\tnode [above] {} (11);\n\t\t\\path (5) edge [bend right=0]\n\t\tnode [above] {} (12);\n\t\t\\path   (6) edge [bend right=0]\n\t\tnode [below] {} (13);\n\t\t\\path   (6) edge [bend left=0]\n\t\tnode [below] {} (14);\n\t\n\t\t\\path   (7) edge [dashed]\n\t\tnode [above] {} (15);\n\t\t\\path   (7) edge [dashed]\n\t\tnode [above] {} (16);\n\t\t\\path   (8) edge [dashed]\n\t\tnode [above] {} (17);\n\t\t\\path (8) edge [dashed]\n\t\tnode [above] {} (18);\n\t\t\\path   (9) edge [dashed]\n\t\tnode [below] {} (19);\n\t\t\\path   (9) edge [dashed]\n\t\tnode [below] {} (20);\n\t\t\\path  (10) edge [dashed]\n\t\tnode [below] {} (21);\n\t\t\\path   (10) edge [dashed]\n\t\tnode [above] {} (22);\n\t\t\\path   (11) edge [dashed]\n\t\tnode [above] {} (23);\n\t\t\\path   (11) edge [dashed]\n\t\tnode [above] {} (24);\n\t\t\\path (12) edge [dashed]\n\t\tnode [above] {} (25);\n\t\t\\path   (12) edge [dashed]\n\t\tnode [below] {} (26);\n\t\t\\path   (13) edge [dashed]\n\t\tnode [below] {} (27);\n\t\t\\path (13) edge [dashed]\n\t\tnode [above] {} (28);\n\t\t\\path   (14) edge [dashed]\n\t\tnode [below] {} (29);\n\t\t\\path   (14) edge [dashed]\n\t\tnode [below] {} (30);\n\t\\end{tikzpicture}\n\t\\caption{The double tree is an infinite non-recurrent network.\n\t\t\\label{counterex}}\n\\end{figure}\n\n\\section{Numerical simulations}\n\\label{sec:simulations}\nThis section is devoted to numerical simulations.\nIn Section~\\ref{sec:res_approx} we analyze the bound gap for finite distance $d$, both on real and synthetic transportation networks.\nThen, we discuss in Section~\\ref{sec:relax_ass} how to adapt our method to more general\nNDPs with non-linear delay functions, and provide numerical simulations showing that our algorithm may be applied in real scenarios even if the regularity assumption on the Wardrop equilibrium (i.e., Assumption \\ref{assumption}) is violated.\n\n\\subsection{Effective resistance approximation}\n\\label{sec:res_approx}\n\\subsubsection{Infinite grids}\nInfinite regular grids are relevant networks to test the performance of the bounds on the effective resistance, since they are good proxy for transportation networks.\nIn Table~\\ref{table} the bound gap in a square grid network with unitary conductances is shown. Similar results are obtained in any regular infinite grid.\nNumerical simulations show that for every link $l$ in $\\mathcal{L}$,\n\\begin{equation*}\n\t\\frac{r^{U_d}_{l}-r_{l}}{r_{l}}=\\frac{r_{l}-r^{L_d}_{l}}{r_{l}}=O(1\/d^2).\n\\end{equation*}\nWe emphasize that, despite the network being infinitely large, even at $d=5$ the bounds are close to the true value effective resistance, which is 1\/2 \\cite{bartis1967let}.\n\\begin{table}[]\n\t\\caption{Table of upper and lower bound in infinite square grid. \\label{table}}\n\t\\centering\n\t\\begin{tabular}{llllll}\n\t\t& $d=1$ & $d=2$ & $d=3$ & $d=4$ & $d=5$\\\\\n\t\t\\hline\n\t\t$(r_{l}^{U_d}-r_{l})\/r_{l}$ & $1\/5$ & $0.0804$ & $0.0426$ & $0.0262$ & $0.0178$\\\\\n\t\t\\hline\n\t\t$(r_{l}-r_{l}^{L_d})\/r_{l}$ & $1\/5$ & $0.0804$ & $0.0426$ & $0.0262$ & $0.0178$\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{Oldenburg\n\ttransportation network}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=5cm]{oldenburg.png}\n\t\\caption{Average relative gap of the bounds on Oldenburg network as a function of distance $d$. \\label{old_fig}}\n\\end{figure}\nIn this section we illustrate the performance of our bounds on the effective resistance of links of the resistor network associated to the transportation network of Oldenburg \\cite{brinkhoff2002framework}. The transportation network is composed of $6105$ nodes and $7035$ links, and the diameter of the associated resistor network (i.e., maximum distance between pair of nodes) is $104$. We assume for simplicity $a_e=1$ for every link $e \\in \\mathcal{E}$, but numerical results prove to be robust with respect to some variability in those parameters. \nThe average relative bound gap on the associated resistor network, defined as\n\\begin{equation*}\n\t\\Delta R_d:=\\frac{1}{\\mathrm{L}}\\sum_{l \\in \\mathcal{L}}\\frac{r^{U_d}_{l}-r^{L_d}_{l}}{r_{l}}\n\\end{equation*} is shown in Table~\\ref{oldenburg_lower_bound} and Figure~\\ref{old_fig}. \n\\begin{table}[]\n\t\\caption{Table of average relative error bound gap at distance $d$ on the Oldenburg network. \\label{oldenburg_lower_bound}}\n\t\\centering\n\t\t\\begin{tabular}{llllllll}\n\t\t\t& d=1 & d=2 & d=3 & d=4 & d=5 & d=6 & d=7\\\\\n\t\t\t\\hline\n\t\t\t$\\Delta R_d$\n\t\t\t& 0.21\n\t\t\t& 0.12\n\t\t\t& 0.079\n\t\t\t& 0.056\n\t\t\t& 0.041\n\t\t\t& 0.031\n\t\t\t& 0.024\\\\\n\t\t\t\\hline\n\t\\end{tabular}\n\\end{table}\nWe observe that also in this network the bound gap decreases quickly compared to the diameter of the network.\n\n\n\\subsection{Relaxing assumptions}\n\\label{sec:relax_ass}\nThe goal of this section is two-fold. We first show how to adapt Theorem \\ref{thm} when the delay functions are non-affine, and validate by numerical analysis the proposed method. We then show that violating Assumption \\ref{assumption} is not a practical issue in real case scenarios. The numerical example is based on the highway network of Los Angeles (see Figure~\\ref{fig:mapLA} \\cite{los_angeles}). To handle non-linear delay functions, the main idea is to adapt Theorem~\\ref{thm} by constructing a resistor network and then follow same steps as in Algorithm~\\ref{alg}. To this end, let us write the KKT conditions of \\eqref{convex_prob} as follows:\n$$\n\\left[ \n\\begin{array}{ c  c } \n\t\\text{diag}\\Big(\\Big\\{\\frac{\\tau_e(f_e^*)}{f_e^*}\\Big\\}_{e \\in \\mathcal E}\\Big) & -(B_-)^T \\\\ \n\t-B_- & \\mathbb{0} \n\\end{array} \n\\right]\n\\left[\n\\begin{array}{c}\n\tf^* \\\\ \\gamma^*_-\n\\end{array}\n\\right]\n=\n-\\left[\n\\begin{array}{c}\n\t\\tau_e(0) \\\\ \\nu_-\n\\end{array}\n\\right],\n$$\nwhere $f^*$ and $\\gamma^*$ denote the Wardrop equilibrium and the optimal Lagrangian multipliers before the intervention. The KKT conditions suggest that in non-affine routing games the term $\\tau_e(f_e^*)\/f_e^*$ plays the role of $a_e$ in affine routing games (see the proof of Theorem~\\ref{thm} in Appendix~\\ref{app:proofs} for more details). Hence, by following similar steps as in affine routing games, we construct a resistor network with conductance matrix\n\\begin{equation}\n\t\\label{eq:new_W}\nW_{ij}= \\begin{cases}\n\t\\sum_{\\substack{e \\in \\mathcal{E}: \\\\\n\t\t\t\\xi(e)=i, \\theta(e)=j, \\ \\text{or} \\\\\n\t\t\t\\xi(e)=j, \\theta(e)=i}} \\frac{f_e^*}{\\tau_e(f_e^*)} & \\text{if}\\ i \\neq j \\\\\n\t0 & \\text{if}\\ i=j. \n\\end{cases}\n\\end{equation}\nThe social cost variation for single-link interventions is then computed by using Theorem \\ref{thm} with respect to the new resistor network with conductance matrix \\eqref{eq:new_W}. Observe that, in contrast with the affine case, this method is not exact for non-linear delay functions, since the Wardrop equilibrium (and thus the elements of $W$) are modified by interventions, not allowing to leverage Sherman-Morrison theorem to compute the social cost variation.\n\nTo validate our method we assume that delay functions are in the form $\\tau_e(f_e) = a_e (f_e)^4+b_e$, and consider interventions in the form $u = 3 \\delta^{(e)}$ for every $e$ in $\\mathcal{E}$. Numerical parameters are not reported in the paper due to limited space, but the obtained results are robust with respect to a change of numerical values. For every intervention, we compare the social cost variation computed by two methods: \\emph(i) by solving the convex optimization~\\eqref{eq:f(k)} and plugging the new equilibrium $f^*(u)$ into the social cost function (\\emph{exact}); \\emph(ii) via the electrical formulation, i.e., by leveraging Theorem~\\ref{thm} with conductance matrix \\eqref{eq:new_W} and ignoring the fact that Assumption~\\ref{assumption} may be violated (\\emph{approximated}). Figure~\\ref{fig:non_linear} illustrates the social cost variation computed by the two methods corresponding to interventions on the five links of the network that yield the largest cost variation. The numerical simulations show that support of the equilibrium varies with the intervention. Nonetheless, the proposed method approximates quite well the social cost variation and selects the optimal link for the intervention. The implication of combining the results of this section and Section \\ref{sec:res_approx} is that Algorithm~\\ref{assumption} should manage to select optimal (or weakly suboptimal) interventions in large transportation networks also when the delay functions are non-linear, Assumption~\\ref{assumption} is violated, and effective resistances are computed at small distance~$d$.\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[scale=0.22]{LAmap.png}\\\\[12pt]\n\t\\begin{tikzpicture}[scale=0.55,transform shape]\n\t\t\\node[draw, circle] (1) at (1,0) {1};\n\t\t\\node[draw, circle] (2) at (4,0) {2};\n\t\t\\node[draw, circle] (3) at (6,0) {3};\n\t\t\\node[draw, circle] (4) at (10,0) {4};\n\t\t\\node[draw, circle] (5) at (14,-1) {5};\n\t\t\\node[draw, circle] (6) at (2,-4) {6}; \n\t\t\\node[draw, circle] (7) at (4,-4) {7};\n\t\t\\node[draw, circle] (8) at (6,-4) {8};\n\t\t\\node[draw, circle] (9) at (8,-3) {9};\n\t\t\\node[draw, circle] (10) at (4,-8) {10};\n\t\t\\node[draw, circle] (11) at (6,-8) {11};\n\t\t\\node[draw, circle] (12) at (8, -8) {12};\n\t\t\\node[draw, circle] (13) at  (12, -9) {13};\n\t\t\\node[draw, circle] (14) at (14, -9) {14};\n\t\t\\node[draw, circle] (15) at (6, -10) {15};\n\t\t\\node[draw, circle] (16) at (9,-10.5) {16};\n\t\t\\node[draw, circle] (17) at  (14, -11)  {17};\n\t\t\\path[->,shorten >=1pt,auto, node distance = 0.5cm, semithick]\n\t\t(1) edge node {$l_1$} (2) \n\t\t(2) edge node {$l_2$}  (3)\n\t\t(3) edge node {$l_3$} (4)\n\t\t(4) edge node {$l_4$} (5)\n\t\t(1) edge node {$l_5$} (6)\n\t\t(6) edge node {$l_6$} (7)\n\t\t(7) edge node {$l_7$} (8)\n\t\t(8) edge node {$l_8$} (9)\n\t\t(9) edge node {$l_{9}$} (13)\n\t\t(2) edge node {$l_{10}$} (7)\n\t\t(3) edge node {$l_{11}$} (8)\n\t\t(3) edge node {$l_{12}$} (9)\n\t\t(4) edge node {$l_{13}$} (9)\n\t\t(5) edge node {$l_{14}$} (14)\n\t\t(6) edge node {$l_{15}$} (10)\n\t\t(10) edge node {$l_{16}$} (11)\n\t\t(10) edge[left] node {$l_{17}$} (15)\n\t\t(7) edge node {$l_{18}$} (10)\n\t\t(8) edge node {$l_{19}$} (11)\n\t\t(9) edge node {$l_{20}$} (12)\n\t\t(11) edge node {$l_{21}$} (12)\n\t\t(12) edge node {$l_{22}$} (13)\n\t\t(13) edge node {$l_{23}$} (14)\n\t\t(11) edge node {$l_{24}$} (15)\n\t\t(13) edge[left] node {$l_{25}$} (17)\n\t\t(14) edge node {$l_{26}$} (17)\n\t\t(15) edge[below] node {$l_{27}$} (16)\n\t\t(16) edge[below] node {$l_{28}$} (17);\n\t\\end{tikzpicture}\n\t\\caption{\\emph{Top}: the highway network in Los Angeles. \\emph{Bottom}: a graph representation of the network, where node $1$ (Santa Monica) and $17$ (Santa Ana) are respectively the origin and the destination. \\label{fig:mapLA}}\n\\end{figure} \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=7cm]{quarta2-5links.png}\n\t\\caption{\\emph{Top}: Social cost variation for interventions in the form $u = 3 \\delta^{(e)}$ for a routing game on the graph of Figure~\\ref{fig:mapLA} with delay functions in the form $\\tau_e(f_e) = a_e (f_e)^4 + b_e$. The cost variation is computed by solving convex optimization (\\emph{exact}) and by adapting Theorem~\\ref{thm} to the case of non-linear delay functions (\\emph{approximated}), as explained in Section~\\ref{sec:relax_ass}. The plot illustrates the social cost variation for the five links that maximize the cost variation.\\label{fig:non_linear}}\n\\end{figure}\n\n\\section{Conclusion}\nIn this work we study a network design problem where a single link can be improved. Under the assumption that the support of the Wardrop equilibrium is not modified with an intervention, we reformulate the problem in terms of electrical quantities computed on a related resistor network, in particular in terms of the effective resistance of a link. We then provide a method to approximate such an effective resistance by performing only local computation, which may be of separate interest.\nBased on the electrical formulation and our approximation method for the effective resistance we propose an efficient algorithm to solve efficiently the network design problem. We then show by numerical examples that our method can be adapted to routing games with non-linear delay functions, and achieves good performance even if the support of the equilibrium is modified by the intervention.\n\nAn interesting direction for the future is a deeper analysis on tightness of the bounds on effective resistance for finite distance $d$. Future research lines also include extending the analysis to the case of multiple interventions. Indeed, the general problem is not submodular, thus guarantees on the performance of greedy algorithm are not given. A possible direction is to exploit the closed formula for the social cost derivative to implement gradient descents algorithms. Other directions include extending the theoretical framework to the case of multiple origin-destination pairs and heterogeneous preferences \\cite{Cianfanelli.ea:19,Cianfanelli.ea:22}.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCooling mesoscopic and microscopic resonators down to their minimum-energy state is fundamental to observe the classical-quantum transition and to exploit the quantum advantage in nanoscience~\\cite{Nanoscience,OptoReview}. The ground-state preparation is also a crucial and implicit step in quantum information processes, including but not limited to the continuous-variable quantum computations~\\cite{ContinuousVariables,YouSuperconductingCircuits,BosonSampling,\nPhononArithmetic}, the ultrahigh precision measurements~\\cite{ForceMeasureByOscillator1,ForceMeasureByOscillator2}, and the quantum interface constructions~\\cite{CoolingMagnon}. Various cooling strategies are designed to attain an effective temperature as low as possible in trapped atom and ion systems~\\cite{SidebandCooling,SelfCooling,BeatingSidebandCooling}. In atomic laser cooling, the popular strategies are consisted of the laser Doppler cooling~\\cite{CoolingMagnon,LaserCooling2013,LaserCooling1995}, the resolved-sideband cooling, and the electromagnetically induced transparency cooling (EIT)~\\cite{EITCoolingTheory,EITCoolingExperiment}.\n\nBeyond the paradigms extracting the system energy through the designed dissipation channels based on the blue-shifted (anti-Stokes) sidebands, nondeterministic methods of measurement-based cooling have been proposed in theory~\\cite{Purification,OneModeCooling} and demonstrated in experiment~\\cite{MeasurementCoolingExp}. Rather than providing a unidirectional decay channel for the target system, the nonunitary evolution induced by repeated measurements of the ground state of the detector system leads to the postselection of the ground state of the target system (typically modelled as a resonator) and the reduction of its high-energy distribution in the ensemble. In another word, the resonator is gradually steered by the outcomes of the conditional measurement (CM) to its ground state via dynamically filtering out its vibrational modes. Ranging from cooling the nonlinear mechanical resonators~\\cite{CoolingNonlinearOscillator}, cooling by one shot measurement~\\cite{OneShotMeasurement}, expanding cooling range by an external driving~\\cite{ExternalLevelCooling}, to accelerating cooling rate by optimizing the measurement intervals~\\cite{TwoModeCooling}, an unexplored weakness of all these CM strategies is their small successful probability inherited from the projective operation (postselection). An inevitable amount of cost rises with more samples in ensemble. The unconditional measurement (UM) strategy, in sharp contrast to CM, performs a nonselective and impulsive measurement in all the bases of the target bare Hamiltonian at the end of each round of the joint free evolution of the target and the detector systems~\\cite{CoolingQubit,FockState}. It is more likely to realize a unit-success-probability cooling but suffers from a much slower cooling-rate than CM, indicating more number of measurements towards the ground-state cooling. To compromise the cooling rate and the successful probability, the interpolating-configuration of the conditional and unconditional measurements becomes an optimization problem.\n\nThe integration of a small-scale quantum circuit with a classical optimizer (e.g., the neural network) provides a design paradigm by specifying a sequence of parametrized quantum operations that are well suited to implement robust and high-fidelity algorithms. Many reinforcement learning (RL) algorithms constructed by the neural network, that demonstrated remarkable capabilities in the board and video games~\\cite{GameGoNN,GameGoWithoutHuman,GameChess,HumanLevelControl}, have substantiated widely and timely interest in studying quantum physics~\\cite{MLAndPhysics}, such as quantum error correction~\\cite{ContinuousErrorCorrection,ErrorCorrectionFeedback}, quantum simulation~\\cite{DigitalQuantumSimulation,SimulationHybridNetwork}, and quantum state preparation~\\cite{PhaseTransition,DifferentPhaseControl,StatePreparationRL}, to name a few. The proximal policy optimization (PPO) algorithm, as a typical RL algorithm with a significant sample complexity, scalability, and robustness for hyperparameters, has proven to be a fruitful tool in quantum optimization control~\\cite{ModelFreeControlPPO,FeedbackControlPPO,ManyBodyStatePreparationPPO}.\n\nIn this work, we propose a measurement-based cooling architecture as a hybrid sequence of UM and CM strategies. It involves a double optimization: on each local step along the sequence, either UM or CM can be considerably improved by using the timely-optimized measurement intervals; and on the global efficiency of the sequence, its arrangement can be separably optimized through reinforcement learning. Particularly, in a typical measurement-based cooling model, i.e., the Jaynes-Cummings (JC) model, where a mechanical resonator (the target system) is coupled to a qubit (the detector system), the conditional and unconditional measurements are alternatively performed to cool down the resonator to its ground state. In parallel to the optimized measurement-interval obtained for CM~\\cite{TwoModeCooling}, we here analytically derive the optimized interval of UM. Then the free-evolution intervals between any neighboring measurements, either UM or CM, can be optimized for cooling. The global sequence of measurements or the implementing order of UM and CM can be further optimized with reinforcement learning. The optimizer is fed with the cooperative cooling performance, a function of the average population of the resonator, the successful probability of the detector in the measured subspace, and the fidelity of the resonator in the ground state, as we defined to rank the comprehensive efficiencies of various sequences of measurements. Eventually we find an optimal sequence holding an overwhelming advantage over all the others.\n\nThe rest of this work is structured as follows. We briefly revisit the general framework for the cooling protocols based on the conditional and unconditional measurements in Secs.~\\ref{CM} and \\ref{UM}, respectively. In Sec.~\\ref{UM}, an analytical expression of the optimized measurement-interval is obtained for the unconditional measurement. In Sec.~\\ref{Optimization}, we introduce the interpolation diagram for the cooling architecture based on these two measurements, define the cooperative cooling performance to comprehensively quantify various strategies, and present the optimized result through reinforcement learning. The PPO algorithm and the optimal-control procedure are provided in Appendixes~\\ref{PPOSec} and \\ref{OptSequence}, respectively. The whole work is discussed and summarized in Sec.~\\ref{Conclusion}.\n\n\\section{Conditional and unconditional measurements}\\label{CMandUM}\n\n\\subsection{Conditional Measurement}\\label{CM}\n\nThe cooling-by-measurement framework is typically established on the JC model, whose Hamiltonian in the rotating frame with respect to $H_0=\\omega_a(|e\\rangle\\langle e|+a^\\dagger a)$ reads\n\\begin{equation}\\label{Ham}\nH=\\Delta|e\\rangle\\langle e|+g(a^\\dagger\\sigma_-+a\\sigma_+).\n\\end{equation}\nHere $\\Delta\\equiv\\omega_e-\\omega_a$ is the detuning between the atomic level-spacing $\\omega_e$ and the to-be-cooled resonator frequency $\\omega_a$ and $|\\Delta|\\ll\\omega_e, \\omega_a$. $g$ is the coupling strength between the detector (qubit) and the target resonator. Pauli matrices $\\sigma_-$ and $\\sigma_+$ denote the transition operators of the qubit; and $a$ ($a^\\dagger$) represents annihilation (creation) operator of the resonator. The cooling process is described by a sequence of piecewise joint evolutions of the resonator and the detector, that are interrupted by instantaneous projective measurements on a particular subspace of the detector.\n\nThe conditional measurement-based cooling is characterised by fixing the subspace as the ground state $|g\\rangle$. Initially, the resonator is in a thermal-equilibrium state $\\rho_a^{\\rm th}$ with a finite temperature $T$, while the detector qubit starts from the ground state. Then the overall initial state has the form of\n\\begin{equation}\\label{InitialState}\n\\rho_{\\rm tot}(0)=|g\\rangle\\langle g|\\otimes\\rho_a^{\\rm th}.\n\\end{equation}\nTo cool down the resonator, a conditional or selective measurement $M_g=|g\\rangle\\langle g|$ is implemented on the detector after a free-evolution with interval $\\tau$, when the overall state becomes $\\rho_{\\rm tot}(\\tau)=\\exp(-iH\\tau)\\rho_{\\rm tot}(0)\\exp(iH\\tau)$. And then the conditional measurement yields a nondeterministic result:\n\\begin{equation}\n\\rho_a(\\tau)=\\frac{\\langle g|\\rho_{\\rm tot}(\\tau)|g\\rangle}{{\\rm Tr}\\left[\\langle g|\\rho_{\\rm tot}(\\tau)|g\\rangle\\right]}.\n\\end{equation}\nIn regard of the time-dependence of the interval $\\tau$, the conditional cooling protocols can be categorized into the equal-time-spacing and unequal-time-spacing strategies~\\cite{OneModeCooling,TwoModeCooling}. The unequal-time-spacing strategy has demonstrated a dramatic advantage on the cooling performance by setting the measurement interval as the inverse of the thermal Rabi frequency $\\tau_{\\rm opt}^c(t)=1\/\\Omega_{\\rm th}(t)$, where $\\Omega_{\\rm th}(t)\\equiv g\\sqrt{\\bar{n}(t)}=g\\sqrt{\\sum_nnp_n(t)}$ with $p_n(t)$ denoting the current population of the resonator on the Fock state $|n\\rangle$. To attain an optimal cooling performance, our cooling architecture in this work employs the unequal-time-spacing strategy. After $N$ rounds of free-evolution and instantaneous-measurement described by an ordered time sequence $\\{\\tau_1(t_1), \\tau_2(t_2), \\cdots, \\tau_N(t_N)\\}$ with $t_{i>1}=\\sum_{j=1}^{j=i-1}\\tau_j$ and $\\tau_1\\equiv1\/[g\\sqrt{{\\rm Tr}(\\hat{n}\\rho_a^{\\rm th})}]$, the resonator state becomes\n\\begin{equation}\\label{rhoace}\n\\rho_a\\left(t=\\sum_{i=1}^N\\tau_i\\right)=\\frac{\\sum_{n}\\prod_{i=1}^N|\\alpha_n(\\tau_i)|^2p_n|n\\rangle\\langle n|}{P_g(N)},\n\\end{equation}\nwhere $p_n=\\exp(-n\\hbar\\omega_a\/k_BT)\/Z$ with $Z\\equiv1\/[1-\\exp(-\\hbar\\omega_a\/k_BT)]$ is the initial population,\n\\begin{equation}\nP_g(N)=\\sum_n\\prod_{i=1}^N|\\alpha_n(\\tau_i)|^2p_n\n\\end{equation}\nis the survival or successful probability of CM, and\n\\begin{equation}\\label{CMCoolingCoeff}\n\\left|\\alpha_n(\\tau_i)\\right|^2=\\frac{\\Omega_n^2-g^2n\\sin^2(\\Omega_n\\tau_i)}{\\Omega_n^2}\n\\end{equation}\nis the cooling coefficient with $\\Omega_n=\\sqrt{g^2n+\\Delta^2\/4}$ denoting the $n$-photon Rabi frequency. The cooling coefficient in Eq.~(\\ref{rhoace}) determines the average population\n\\begin{equation}\n\\bar{n}(t)={\\rm Tr}\\left[\\hat{n}\\rho_a(t)\\right], \\quad \\hat{n}\\equiv a^{\\dagger}a,\n\\end{equation}\nby reshaping the population distributions over all the Fock states. Note in Eq.~(\\ref{CMCoolingCoeff}), the $0$th cooling coefficient is unit, $|\\alpha_0(\\tau_i)|^2=1$, meaning that the ground-state population is always under protection during the cooling process. The populations on the high-occupied Fock states are gradually reduced by $|\\alpha_n(\\tau_i)|^N<1$ with increasing $N$ unless $\\sin(\\Omega_n\\tau_i)=0$ or $\\Omega_n\\tau_i=j\\pi$ with integer $j$.\n\n\\subsection{Unconditional Measurement}\\label{UM}\n\nExpanding the measurement subspace to the whole space of the detector system, we can transform a conditional-measurement strategy into its unconditional-measurement counterpart. After a period of joint unitary evolution under the Hamiltonian~(\\ref{Ham}), the overall state can be written as\n\\begin{equation}\n\\rho_{\\rm tot}(\\tau)=\\bigoplus_np_n\\begin{pmatrix}|\\alpha_n(\\tau)|^2 & \\chi_n(\\tau)\\\\\n\\chi^*_n(\\tau)& |\\beta_n(\\tau)|^2\\end{pmatrix},\n\\end{equation}\nwhere\n\\begin{equation*}\n\\begin{aligned}\n&\\chi_n(\\tau)\\equiv\\frac{-g\\sqrt{n}[\\Delta\\sin^2(\\Omega_n\\tau)-i\\Omega_n\\sin(2\\Omega_n\\tau)]}{2\\Omega_n^2},\\\\\n&|\\beta_n(\\tau)|^2\\equiv\\frac{g^2n\\sin^2(\\Omega_n\\tau)}{\\Omega_n^2}.\n\\end{aligned}\n\\end{equation*}\nThe UM can be implemented by tracing out the degrees of freedom of the detector ${\\rm Tr}_d[\\rho_{\\rm tot}(\\tau)]$. Then the state of resonator reads\n\\begin{equation}\\label{umrhoa}\n\\rho_a(\\tau)=\\sum_{n\\geq0}\\left[|\\alpha_n(\\tau)|^2p_n+|\\beta_{n+1}(\\tau)|^2p_{n+1}\\right]|n\\rangle\\langle n|.\n\\end{equation}\nSo that after a nonselective measurement, i.e., a measurement without recording the output, a population transfer occurs in the target resonator as\n\\begin{equation}\\label{pn}\np_n\\rightarrow|\\alpha_n(\\tau)|^2p_n+|\\beta_{n+1}(\\tau)^2|p_{n+1}.\n\\end{equation}\nIn contrast to the CM strategy that is characterized by a single cooling coefficient $\\alpha_n$ in Eq.~(\\ref{CMCoolingCoeff}), the UM strategy depends subtly on an extra cooling coefficients $\\beta_n$. According to Eq.~(\\ref{pn}), the initial population of the ground state $p_0$ becomes $|\\alpha_0(\\tau)|^2p_0+|\\beta_1(\\tau)^2|p_1=p_0+|\\beta_1(\\tau)^2|p_1$. It indicates that partial population on the first excited state of the resonator is transferred to the ground state, whose original population is left untouched. Under rounds of nonselective measurements, it is intuitively to expect that the populations on the higher excited states of the resonator will keep moving to the lower states and eventually to the ground state. In practice, the cooling efficiency is however limited since the population on certain high excited states can also be fixed and even enhanced when $|\\alpha_n(\\tau)|^2=1$ and $|\\beta_{n+1}(\\tau)|^2\\geq0$, i.e., $\\Omega_n\\tau=1$ and $\\Omega_{n+1}\\tau\\geq0$. This problem can be addressed by employing the unequal-time-spacing strategy. A time-varying $\\tau$ could ensure that the populations on all the excited states are gradually reduced.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{OptimalInterval.eps}\n\\caption{The average populations of the resonator after a single unconditional measurement as a function of the measurement-interval $\\tau$ under various initial temperatures. (a) $T=0.01$ K, (b) $T=0.1$ K, (c) $T=1.0$ K and (d) $T=10$ K. The vertical black-dashed lines indicate the analytical results for the optimized intervals given by Eq.~(\\ref{Optimaltau}). The parameters for the blue-solid curves are set as $g=0.04\\omega_a$ and $\\Delta=0.01\\omega_a$. }\\label{OptimalInterval}\n\\end{figure}\n\nAnalogous to the CM case~\\cite{TwoModeCooling}, the cooling efficiency of the UM strategy depends severely on the choice of $\\tau$ between neighboring measurements. That could be observed in Fig.~\\ref{OptimalInterval} by the average populations of the resonator $\\bar{n}$ under a single measurement on the detector. With the initial temperatures across four orders in magnitude, the $\\tau$-dependence of $\\bar{n}$ demonstrates similar patterns. It is found that the average population declines gradually to a minimal point (the magnitude of the relative reduction decreases with increasing temperature) at an optimized measurement-interval $\\tau_{\\rm opt}^u$, then rebounds quickly and ends up with a stochastic fluctuation around a value slightly lower than its initial population $\\bar{n}_{\\rm th}\\equiv{\\rm Tr}(\\hat{n}\\rho_a^{\\rm th})$.\n\nTo make full use of the cooling strategy, it is desired to analytically find the optimized interval $\\tau_{\\rm opt}^u$ as a functional of the current state and the model parameters. By virtue of Eq.~(\\ref{umrhoa}) and under the resonant situation, the average population after a single unconditional measurement reads\n\\begin{equation}\\label{nbar}\n\\begin{aligned}\n\\bar{n}&=\\sum_{n\\geq0}n\\left(p_n\\cos^2\\Omega_n\\tau+p_{n+1}\\sin^2\\Omega_{n+1}\\tau\\right)\\\\\n&=\\eta+\\frac{1}{2Z}\\sum_{n\\geq0}ne^{-nx}(\\cos2\\Omega_n\\tau-e^{-x}\\cos2\\Omega_{n+1}\\tau),\n\\end{aligned}\n\\end{equation}\nwhere $\\eta\\equiv(\\bar{n}_{\\rm th}+2\\bar{n}^2_{\\rm th})\/(2+2\\bar{n}_{\\rm th})$ and $x\\equiv\\hbar\\omega_a\/k_BT$. Since the weight function $ne^{-nx}$ in Eq.~(\\ref{nbar}) is dominant around $n_d\\equiv k_BT\/\\hbar\\omega_a$, the variables $\\Omega_n$ and $\\Omega_{n+1}$ could thus be expanded around $n=n_d$. To the first order of $n-n_d$, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\cos2\\Omega_n\\tau-e^{-x}\\cos2\\Omega_{n+1}\\tau \\\\\n&\\approx\\cos2\\Omega_{d}\\tau-e^{-x}\\cos2\\Omega_{d+1}\\tau+(n-n_d) \\\\\n&\\times\\left(-\\frac{\\Omega_d\\tau\\sin2\\Omega_d\\tau}{n_d}+e^{-x}\\frac{\\Omega_{d+1}\\tau\\sin2\\Omega_{d+1}\\tau}{n_d+1}\\right),\n\\end{aligned}\n\\end{equation*}\nwhere\n\\begin{equation}\\label{Omegad}\n\\Omega_d\\equiv g\\sqrt{n_d}, \\quad \\Omega_{d+1}\\equiv g\\sqrt{n_d+1}.\n\\end{equation}\ndefine the dominant Rabi frequencies. Under the approximations that $e^{-x}=\\bar{n}_{\\rm th}\/(\\bar{n}_{\\rm th}+1)\\approx1$ and $\\Omega_{d+1}\/(n_d+1)\\approx\\Omega_d\/n_d$ appropriate for a moderate temperature, the average population in Eq.~(\\ref{nbar}) can be expressed by\n\\begin{equation}\\label{approxn1}\n\\bar{n}\\approx\\eta+\\sin\\Omega_-\\tau\\left(\\bar{n}_{\\rm th}\\sin\\Omega_+\\tau+\\eta'\\Omega_d\\tau\\cos\\Omega_+\\tau\\right),\n\\end{equation}\nwhere $\\Omega_{\\pm}\\equiv\\Omega_{d+1}\\pm\\Omega_d$ and $\\eta'\\equiv\\bar{n}_{\\rm th}(1+2\\bar{n}_{\\rm th}-n_d)\/n_d$. Note we have applied the formulas about the geometric series $\\sum_{n=0}^\\infty ne^{-nx}=e^x\/(e^x-1)^2$ and $\\sum_{n=0}^\\infty n^2e^{-nx}=e^x(1+e^x)\/(e^x-1)^3$. $\\bar{n}$ in Eq.~(\\ref{approxn1}) depends predominantly on the fast-frequency terms characterized by $\\Omega_+$ in a moderate time step $\\tau$. In the regime of $T\\sim 0.1-10$ K, the term weighted by $\\eta'\\Omega_d\\tau$ overwhelms that weighted by $\\bar{n}_{\\rm th}$. And this advantage expands with a larger $\\tau_{\\rm opt}^u$ given the initial or effective temperature of the resonator becomes lower, as evidenced by Fig.~\\ref{OptimalInterval}. We can therefore focus on the last term in Eq.~(\\ref{approxn1}) to determine how to minimize $\\bar{n}$. Consequently, we have\n\\begin{equation}\\label{Optimaltau}\n\\tau_{\\rm opt}^u=\\frac{\\pi}{\\Omega_d+\\Omega_{d+1}}.\n\\end{equation}\nThis result can be extended to the off-resonant situation by modifying the definition of $\\Omega_d$ in Eq.~(\\ref{Omegad}) to be $\\sqrt{g^2n_d+\\Delta^2\/4}$. The vertical black-dashed lines in Fig.~\\ref{OptimalInterval} denote the optimal measurement-intervals given by Eq.~(\\ref{Optimaltau}). It is found that the analytical expression is well suited to estimate the minimum values of average population in a wide temperature range. As demonstrated by both analytical and numerical results, a shorter measurement-interval is demanded to cool down a higher-temperature resonator. In the JC-like models, coupling a high-temperature resonator to a qubit could induce a faster transition from the ground state to the excited state of the qubit. A quick measurement would interrupt this process that has a negative effect on cooling.\n\nSimilar to the optimized interval $\\tau_{\\rm opt}^c(t)$ for the conditional measurement strategy~\\cite{TwoModeCooling}, here $\\tau_{\\rm opt}^u$ is also updatable by substituting the time-evolved $\\Omega_d$ and $\\Omega_{d+1}$ to Eq.~(\\ref{Optimaltau}). The dominant Fock-state-number $n_d$ determining $\\Omega_d$ in Eq.~(\\ref{Omegad}) could be understood as a function of the effective temperature during the cooling procedure, which relies uniquely on the time-varying $\\bar{n}(t)$ or $p_n(t)$.\n\n\\section{Measurement optimization}\\label{Optimization}\n\nThe thermal resonator could be steadily cooled down by the unconditional measurement strategy armed with the optimized measurement-interval in Eq.~(\\ref{Optimaltau}). And the cooling is performed with a unit successful probability in the absence of the postselection over the measurement outcome. As demonstrated by the blue-solid line with circle markers and the orange-dotted line in Figs.~\\ref{CoolingPerformance}(a) and \\ref{CoolingPerformance}(c), the complete UM and CM cooling strategies dominate in the cooling rate and the successful probability, respectively. It is therefore desired to find an optimized sequence of measurements, which is a hybrid of UM and CM, to compromise the cooling efficiency and the experimental cost. In this section, we present an algorithm that is assisted by the reinforcement learning to generate the optimized control sequence indicting when and which measurement is performed.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{Model.eps}\n\\caption{(a) RL-optimization diagram on cooling by measurement. An agent constructed by the neural network interacts with an environment. The agent chooses an action (CM or UM strategy) according to the current state of the resonator. Then the environment would take this action and return both the state under the measurement and the reward $R$ from the cooperative cooling performance $\\mathcal{C}$ in Eq.~(\\ref{cooperative}). (b) Circuit model for our cooling algorithm based on the optimized UM and CM measurements. Starting from the thermal state, the resonator (the upper line) would be gradually cooled down to its ground state with implementation of the measurement on the detector (the lower line), which starts from the ground state. The measurement sequence can be obtained by the reinforcement learning. }\\label{Model}\n\\end{figure}\n\nThe RL-optimization diagram is shown in Fig.~\\ref{Model}(a), consisting of the ``agent'' part based on a series of neural network and the ``environment'' part performing the cooling-by-measurement actions on the quantum system. In the reinforcement learning, the agent is fed with a cluster of to-be-trained parameters, which would be learned and updated using the data collected through its interaction with the environment. In our architecture, the agent would choose an action, i.e., the conditional or the unconditional measurement, on the resonator, given its current state. Then the environment takes this action and returns the updated resonator-state $\\rho_a$ and a ``reward'' $R$ after the measurement. The reward is generated by a function to estimate whether the action is good or bad, that would be used to update the parameters of the agent. During one ``episode'', the agent would interact with the environment for $N$ times, i.e., the number of measurements during the global sequence, which has been fixed from the beginning. A total reward is eventually counted. And the agent is trained to maximize the total reward through artificial episodes until the total reward converges. Then the agent could provide a realistic control sequence of the measurement strategies with their own (optimized) measurement intervals. The cooling-by-measurement sequence can be demonstrated by a circuit model in Fig.~\\ref{Model}(b). Rounds of free-evolutions and measurements are successively arranged. The evolution time between two neighboring measurements depends on the measurement strategy and the resonator state at the end of the last round. We follow the PPO algorithm in the agent structure, the data collecting methods, and the parameters updating, whose details can be found in Appendix~\\ref{PPOSec}. The interpolation algorithm of UM and CM and the implementation of the measurement sequence are illustrated by a pseudocode in Appendix~\\ref{OptSequence}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{CoolingPerformance.eps}\n\\caption{(a) Average population, (b) Fidelity of the resonator in its ground state, (c) Successful probability, and (d) Cooperative cooling performance under various sequences of cooling-by-measurement. The blue-solid lines with circle markers labeled by $S_u$ and the orange-dotted lines labeled by $S_c$ indicate the sequences completely consisting of UM and CM strategies, respectively. The green-solid lines, the red-dashed lines, and the purple-dot-dashed lines describe the hybrid sequences shown in (e), (f), and (g), and labeled by $S_1$, $S_2$, and $S_4$, respectively. The brown-solid lines with triangle markers labeled by $S_{\\rm opt}$ is the RL-optimized sequence presented in (h). The parameters are set as $\\omega_a=1.4$ GHz, $T=0.1$ K, $g=0.04\\omega_a$, and $\\Delta=0.01\\omega_a$. }\\label{CoolingPerformance}\n\\end{figure}\n\nThe performance of any cooling-by-measurement strategy can be characterized or evaluated by the cooling ratio $\\bar{n}(t)\/\\bar{n}_{\\rm th}$, the successful probability $P_g$ of the detector in the measured subspace, and the fidelity of the resonator in its ground state $F=\\langle n=0|\\rho_a(t)|n=0\\rangle$~\\cite{OneModeCooling}. To compare various interpolation sequences of UM and CM in cooling performance and to evaluate the figure of merit for the reinforcement learning, we here define a cooperative cooling quantifier as\n\\begin{equation}\\label{cooperative}\n\\mathcal{C}=FP_g\\log_{10}{\\frac{\\bar{n}_{\\rm th}}{\\bar{n}(t)}}.\n\\end{equation}\nNotably, the logarithm function is used to obtain a positive value with almost the same order as $F$ and $P_g$ in magnitude. Then $\\bar{n}(t)\/\\bar{n}_{\\rm th}$, $P_g$, and $F$ could be considered in a balanced manner. In fact, the average population could be reduced by several (normally less than $10$) orders in magnitude under an efficient cooling protocol. In the EIT cooling~\\cite{EITCoolingExperiment2020}, $\\log_{10}[\\bar{n}(t)\/\\bar{n}_{\\rm th}]\\sim(2, 3)$; and in the resolved sideband cooling ~\\cite{SidebandCooling2016}, $\\log_{10}[\\bar{n}(t)\/\\bar{n}_{\\rm th}]\\sim(4, 5)$. It is instructive by Eq.~(\\ref{cooperative}) to find that a lower average population, a larger successful probability, and a higher ground-state fidelity yield a better cooling performance.\n\nWe consider to cool down a mechanical microresonator in gigahertz~\\cite{MechanicalResonator1,MechanicalResonator2} with various interpolation sequences of UM and CM. Using the resonator-frequency $\\omega_a=1.4$ GHz, the coupling strength between the resonator and the detector $g=0.04\\omega_a$ and the initial temperature of resonator $T=0.1$ K, it is found that the average population starts from $\\bar{n}_{\\rm th}=8.85$. The cooling performances under the sequences completely consisting of UM and CM are shown by the blue-solid lines with circle markers and the orange-dotted lines in Figs.~\\ref{CoolingPerformance}(a)-(d), labeled by $S_u$ and $S_c$, respectively. It is found that under the conditional measurement strategy with $N=16$, the average population $\\bar{n}$ is reduced by five orders in magnitude [see Fig.~\\ref{CoolingPerformance}(a)] and the ground-state fidelity is over $F>0.9999$ [see Fig.~\\ref{CoolingPerformance}(b)] at the cost of less than $10\\%$ of the successful probability [see Fig.~\\ref{CoolingPerformance}(c)]. In sharp contrast, under $N=16$ unconditional measurements, $\\bar{n}$ is merely reduced to $\\bar{n}\\approx 3.36$ with a moderate fidelity $F\\approx0.78$, despite with a unit successful probability. With respect to all the individual quantifiers, i.e., $\\bar{n}$, $F$, and $P_g$, the results under the hybrid sequences of UM and CM labelled by $S_k$, $k=1,2,4$, are among the former two limits $S_u$ and $S_c$. As illustrated by Figs.~\\ref{CoolingPerformance}(e), (f), and (g), the three sequences start from a CM strategies (indicated by $1$), switch to UM (indicated by $0$) after $k$ rounds of free-evolution and measurement, switch back to CM after a single round, and then continue the periodical arrangement. In comparison to the completed UM sequence, the interpolation with CM promotes the cooling efficiency in $\\bar{n}$. A larger $k$ gives rise to a smaller proportion of the unconditional measurements and the less probability $P_g$ that the detector remains in its measured subspace.\n\nIn terms of the cooperative cooling performance indicated by $\\mathcal{C}$ [see Fig.~\\ref{CoolingPerformance}(d)], it is found that $\\mathcal{C}(S_1)>\\mathcal{C}(S_2)>\\mathcal{C}(S_4)>\\mathcal{C}(S_u)$ and yet $\\mathcal{C}(S_2)\\approx\\mathcal{C}(S_c)$. The interpolation sequences could therefore have better cooperative cooling performance than the pure conditional-measurement-based protocol. The dependence of $\\mathcal{C}$ of arbitrary hybrid sequence on its proportion of CM strategies might not be monotonic. We are then motivated to find an optimized sequence with the help of the PPO algorithm. A typical optimized sequence of cooling strategies labeled by $S_{\\rm opt}$ is described in Fig.~\\ref{CoolingPerformance}(h). With four orders reduction in the average population (close to the cooling efficiency provided by $S_c$), an almost unit ground-state fidelity $F>0.9999$, and a moderate successful probability $P_g\\approx30\\%$ (much larger than that by $S_c$), the optimized sequence achieves an overwhelming cooperative cooling performance $\\mathcal{C}=2.73$ according to Eq.~(\\ref{cooperative}) over all the other measurement sequences. In other words, we attained the compromise of the cooling rate and the successful probability through the reinforcement learning. Notably, the optimized sequence is not unique, yet the current results of $\\bar{n}$, $F$, $P_g$, and $\\mathcal{C}$ are almost invariant after dozen measurements as long as there is one CM in the first $5$ rounds.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{DifferentTemperature.eps}\n\\caption{(a) Average populations and (b) Cooperative cooling performance under the RL-optimized cooling algorithm with various initial temperatures. (c), (d), (e), and (f) describe the optimized sequences of UM and CM with $T=0.05$ K, $T=0.1$ K, $T=0.2$ K, and $T=0.3$ K, respectively. The other parameters are the same as those in Fig.~\\ref{CoolingPerformance}. }\\label{DifferentTemperature}\n\\end{figure}\n\nThe RL-optimized algorithm applies to a wide range of the initial temperature of the resonator. Starting from different $\\bar{n}_{\\rm th}$ determined by the temperature, the average populations could be reduced by three to five orders in magnitude under the optimized measurement sequences, as demonstrated in Fig.~\\ref{DifferentTemperature}(a). It is found that under a higher temperature, it is harder to suppress the transitions between the ground state and the excited states of the resonator. Then both the relative magnitude in population reduction [see Fig.~\\ref{DifferentTemperature}(a)] and the cooperative cooling performance [see Fig.~\\ref{DifferentTemperature}(b)] manifest a monotonically decreasing behavior with increasing temperatures.\n\nSimilar to Fig.~\\ref{CoolingPerformance}(h), here we present in Figs.~\\ref{DifferentTemperature}(c), (d), (e), and (f) the optimized sequences fully determined by the PPO algorithm, which still outperform any regular interpolated sequence in cooling quantifier $\\mathcal{C}$. Comparing these four sub-figures corresponding to various temperatures, it is interesting to find that a larger portion of the unconditional measurements is required in the optimized sequence for a higher temperature. It is consistent with the fact that under CM the successful probability $P_g$ to find a detector in its ground state decreases exponentially with increase of the temperature of the target resonator. Then more UMs are used to save a rapidly declining $P_g$ for obtaining a moderate $\\mathcal{C}$. In addition, for $T>0.05$ K, the RL-optimized sequence always starts from a conditional measurement, which is important to have a significant cooling rate for $\\bar{n}$ during the first several rounds of the whole sequence.\n\nIn our cooling algorithm, both UM and CM are performed on the detector following an optimized interval of unitary evolution. The nonunitary evolution induced by the conditional measurements could remarkably reduce the average population of the resonator, at the cost of a low successful probability [e.g., see the first several rounds in Fig~\\ref{CoolingPerformance}(c)]. The unconditional measurements could reduce the average population with a much slower rate yet with a unit successful probability. Thus in general we anticipate to see more UMs than CMs in the first several rounds in an RL-optimized sequence, and more CMs than UMs in the remaining rounds.\n\n\\section{Discussion and conclusion}\\label{Conclusion}\n\nWe emphasised again that the preceding hybrid cooling sequences based on the conditional and unconditional measurements could be optimized in both global and local manners. Globally, we use the reinforcement learning to find the optimized order for UM and CM. The local optimization depends on the selected measurement intervals to obtain a minimum average-population $\\bar{n}$ under a single measurement. Taking the unconditional case in Eq.~(\\ref{Optimaltau}) as an example, $\\tau_{\\rm opt}^u(t)$ is not necessarily obtained by an instant feedback mechanism during a realistic practice. The measurement sequence $\\{\\tau_1(t_1), \\tau_2(t_2), \\cdots, \\tau_N(t_N)\\}$ can be actually obtained prior to the performance of the cooling measurements. $\\tau_1(t_1)$ depends on the initial temperature as well as the population distribution $p_n$ and $\\tau_k(t_k)$, $k\\geq2$, can be calculated on the effective temperature that is uniquely determined by the dynamics of $p_n(t)$ through Eq.~(\\ref{Omegad}). In other words, we can avoid the feedback error and imprecision induced by detecting the resonator states during the experiment.\n\nIn summary, we present an optimized cooling architecture on the sequential arrangement of the conditional and unconditional measurements. We analyse and compare the advantages and disadvantages of both CM and UM on the cooling rate and the success probability. We obtain analytically for the first time an expression of the optimized unconditional measurement-interval $\\tau_{\\rm opt}^u=\\pi\/(\\Omega_d+\\Omega_{d+1})$ in parallel to that of the conditional measurement~\\cite{TwoModeCooling}. Here the dominant Rabi frequency $\\Omega_d$ depends on the dominant distribution of the resonator in the Fock state of $n_d=k_BT\/(\\hbar\\omega_a)$ and the coupling strength between the target system and the detector. The compromise of the advantages of both measurement strategies gives rise to an optimized hybrid cooling algorithm assisted by the reinforcement learning. That is justified by the cooperative cooling performance as we defined to quantify the comprehensive cooling efficiency for arbitrary cooling-by-measurement strategy. Our work therefore pushes the cooling-by-measurement to an unattained degree in regard of efficiency and feasibility. Also, it offers an appealing interdisciplinary application of quantum control and artificial intelligence.\n\n\\section*{Acknowledgments}\n\nWe acknowledge financial support from the National Science Foundation of China (Grants No. 11974311 and No. U1801661).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGalaxy ecology is an attempt to understand the environmental factors \nthat affect galaxy formation and evolution (e.g. Balogh et al. 2004, Wetzel et al. 2012). \nThere is strong evidence for\na relationship of galaxy properties and the physical conditions surrounding them\n(e.g. Martinez et al. 2008).\nAt least three important pieces of information tell us that the environment affects\ngalaxy evolution in clusters. First, the morphology-density relation shows that galaxy morphologies\ndepend on the density (or cluster-centric radius) -- the\nspiral fraction is larger at cluster outskirts than at the center (e.g. Dressler 1980; Whitmore, Gilmore \\& Jones 1993).\nFurthermore, the fraction of star-forming galaxies decreases with the increasing\ndensity of the cluster environment (e.g. Balogh et al. 2004; Kauffmann et al. 2004).\nFinally, there is a correlation between colors and environment -- the fraction of\nred galaxies decreases toward the outer part of clusters \n(e.g. Balogh et al. 2000; De Propis et al. 2004).\nAlthough these relations are connected, they correspond to\nindependent galaxy observables, and we can understand them as distinct\ngalaxy evolution indicators. Behind them, several physical mechanisms\nmay be present: tidal stripping, mergers, cannibalism,\nstrangulation, {\\it ram} pressure stripping, galaxy harassment, etc. (see e.g. Mo, van den Bosh \\& White 2010 for a good\ndescription of all these transformation processes). \n\nIn particular, the color-radius relation may be crucial for exploring the\nenvironmental effects on galaxies. Because color evolution is mainly completed\nat low redshifts (e.g. Goto et al. 2004), it would be interesting to relate the red galaxy\nfraction to the dynamical state of their host clusters.\nTo probe the color-radius relation one needs to consider \nan important aspect, that is,  the very nature of the \nred galaxies, that is whether they are real\npassive objects with no (or little) star formation currently\ntaking place (e.g. Strateva et al. 2001; Baldry et al. 2004).\nUsually, it is assumed that all red galaxies are passive, but this \nneglects the possible contribution of dusty galaxies to the red sequence, which can be significant in an intermediate environment and for low-to-intermediate stellar masses (e.g. S\\'anchez-Bl\\'asquez et al. 2009).\n\nAn additional ingredient for galaxy ecology studies\nis the role of low-luminosity galaxies. In the hierarchical scenario,\nthey have formed from small peaks in the initial power spectrum (e.g. White \\& Rees 1978). \nIf this is the case, they are expected to be the building blocks of the whole formation process (e.g. Evstgneeva et al. 2004).\nWe would find them in great numbers, and they would be less clustered than bright galaxies.\nOn the other hand, some of them may also have formed from tidal debris from galaxy\nmergers or from galaxies that underwent {\\it ram} pressure stripping as they\nfell toward the center of clusters (e.g. Conselice et al. 2002) and, therefore, \nsome of them would be expected at cluster centers (e.g. Trentham 1998, Carrasco et al. 2006). \nHence, to determine the spatial\ndistribution and fraction of low-luminosity galaxies in clusters is important for \nunderstanding different aspects of galaxy formation and evolution models.\n\nIn this work, we present a study of the relation between colors and luminosities of cluster galaxies and\nthe dynamical state of the clusters.\nOur aim is to describe the interplay between galaxy properties and  galaxy cluster evolution.\nFor this purpose, \nwe used 10,721 member galaxies of 183 clusters extracted from the Sloan Digital\nSky Survey (SDSS) from a list of NoSOCS and CIRS targets. First, we \napplied statistical tests to classify the clusters into two categories, Gaussian and non-Gaussian, according to their velocity distribution, which we took as an indicator of their dynamical state. \nThis was made irrespective of the triaxiality of the cluster halos. Even knowing that the orientation of the velocity ellipsoid is correlated with the large-scale structure and the \nanisotropic nature of infall into clusters (e.g. White et al. 2010), we assumed that these\neffects do not severely influence the statistical approach used in this work. \n\nAfter classifying clusters\ninto Gaussian and non-Gaussian according to their velocity distribution, we\nused objective criteria to split up the galaxies according to their luminosities, colors, and photometric mean stellar age. This information was finally used to evaluate how\ngalaxies evolve in their host clusters.\n\nThe paper is organized as follows: in Section 2 we describe our data; in Section\n3 we present the methods employed to assess the dynamical state of the clusters and to separate\nthe galaxies according to \ncolor and luminosity; in Section 4 we analyze our results, in Section 5 we discuss\nour findings,  and in Section 6 we present our conclusion.\n\n\\section{Data}\n\nThis work is based on the supplemental version of the Northern Sky \nOptical Cluster Survey (NoSOCS, Lopes 2003; Lopes et al. 2004). This\nsupplemental version of NoSOCS goes deeper ($r=21$ and $z \\sim 0.5$), but covers a \nsmaller region ($\\sim 2,700$ square degrees) than the main NoSOCS catalog \n(Gal et al. 2003, 2009). NoSOCS was created from the digitized version of \nthe Second Palomar Observatory Sky Survey (DPOSS\/POSS-II, Djorgovski et \nal. 2003). The photometric calibration and object classification for DPOSS \nare described in Gal et al. (2004) and Odewahn et al. (2004), respectively.\nIn the first paper of this series (Lopes et al. 2009a) a subsample of\n7,414 systems from the NoSOCS supplemental was extracted from the \nSloan Digital Sky Survey (SDSS), data release 5 (DR5).\n\nThis sample was reduced to 179  low-redshift clusters ($z \\le 0.1$) \nwith enough spectra in SDSS (at least three galaxies within 0.50 h$^{-1}$ Mpc) \nfor spectroscopic redshift determination using the gap technique \n(Katgert et al. 1996; Lopes 2007). This technique separates \ngroups after the identification of gaps in the redshift distribution larger than a \ngiven value. To select members and exclude interlopers, the shifting gapper\ntechnique (Fadda et al. 1996) was applied to all galaxies with spectra available \nwithin a maximum aperture of 2.50 h$^{-1}$ Mpc. This method works through the \napplication of the gap technique in radial bins from the cluster center. The bin size \nis 0.42 h$^{-1}$ Mpc (0.60 Mpc for h = 0.7) or larger to force the selection of at least \n15 galaxies (consistent with Fadda et al. 1996). Galaxies not associated with the\nmain body of the cluster are eliminated. After removing the interlopers, the final sample\ncomprises 127 clusters (out of 179) with at least ten member galaxies within \n2.50 h$^{-1}$ Mpc. \n\nThe line-of-sight velocity dispersion ($\\sigma_P$) for these clusters was estimated \nand then a virial analysis was performed. The latter is analogous to the procedure \ndescribed in Girardi et al. (1998), Popesso et al. (2005, 2007), and Biviano et al. (2006).\nFirst, the projected, virial radius ($R_{PV}$) is derived and a first estimate of\nthe virial mass is obtained (using equation 5 of Girardi et al. 1998). The surface \npressure correction is applied to the mass estimate and a Navarro et al. (1997) \nprofile is assumed to obtain estimates of $R_{500}$, $R_{200}$, \n$M_{500}$ and $M_{200}$.\n\nThis low-redshift sample was complemented with 56\nmore massive systems from the cluster infall regions in the SDSS (CIRS) \nsample (Rines \\& Diaferio 2006). CIRS is a collection of $z \\le 0.1$ X-ray-selected \nclusters overlapping the SDSS DR4 footprint. In this redshift range, \nour NoSOCS sample therefore comprises only relatively poor systems. The same cluster \nparameters as listed above were determined for these 56 CIRS clusters. \nHence, the combined NoSOCS plus CIRS sample comprises 183 clusters.\nThe NoSOCS clusters have velocity dispersion estimates of $100 <\n\\sigma_P < 700$ km\/s, while the CIRS systems cover the range \n$200 < \\sigma_P < 900$ km\/s (only 23\\% of CIRS objects have \n$\\sigma_P < 400$ km\/s). Because from the original CIRS member selection, velocity\ndispersion and mass estimates are the product of the caustic\ntechnique (Rines \\& Diaferio 2006), CIRS was also used for comparing \nthese properties and for the cluster scaling relations (Lopes et al. 2009ab).\nFor all the objects in our final we also estimated the sample optical luminosity ($L_{opt}$), the richness \n(N$_{gals}$), and the X-ray luminosity ($L_X$) ($L_X$ is obtained with \nROSAT All Sky Survey data; see Lopes et al. 2009ab).\n\n\nThe centroid of each NoSOCS cluster is a luminosity, weighted estimate, \nwhich correlates well with the X-ray peak \n(see Lopes et al. 2006, and more details in \nLopes et al. 2004). Lopes et al. 2009b, showed that the scaling \nrelations based only on NoSOCS or CIRS objects agree, although in \ndifferent mass ranges, which also indicates that the centroid (optical or X-ray) \ndoes not create a bias in the cluster parameters (richness, $L_{opt}$, \n$\\sigma_P$, $R_{200}$, $M_{200}$), at least for our sample and analysis. Previous\nworks have also shown a good correlation between optical and X-ray centroids \n(Adami et al. 1998; Dai et al. 2006; Man \\& Ebeling 2012). Note that Man \\& Ebeling \n(2012) selected disturbed clusters in their search for objects with large offsets between \nthe X-ray emission and the brightest cluster galaxy (BCG). Recently, centroid offsets \nhave also been investigated using the Sunyaev-Zeldovich Effect (SZE, Sehgal et al. 2013) \nand gravitational lensing (Zitrin et al. 2012). Investigating the offsets between the\ndark matter (DM) projected center and the BCG, Zitrin et al. (2012) found that some of \nthe offsets are caused by misidentifications of the BCG.\n\n\nThe redshift limit of the sample ($z = 0.1$) is \ndue to incompleteness in the SDSS spectroscopic survey for higher \nredshifts, where galaxies fainter than $M^* + 1$ are lacked, which biases \nthe dynamical analysis (see the discussion in section 4.3 of \nLopes et al. 2009a). We considered the $M^* = -20.94$ value from \nPopesso et al. (2006), converted to our cosmology.\\footnote{\nAll quantities with cosmological dependence are computed in the concordance context,\ndefined by $\\Omega_m$ = 0.3, $\\Omega_\\lambda$ = 0.7, and $H_0 = 100~h~{\\rm km~s^{-1}Mpc^{-1}}$, with $h=0.7$.} Our galaxy sample consists of 10,721 members from these 183 clusters at\n$z \\le 0.1$.\n\nTo compute the absolute magnitudes of each galaxy (in $ugriz$ bands) we \nconsidered the following formula: $M_x = m_x - DM - k_{corr} - Qz$ ($x$ is one \nof the five SDSS bands we considered), where $DM$ is the distance modulus \n(considering the redshift of each galaxy), $k_{corr}$ is the $k-$correction\nand $Qz$ ($Q = -1.4$; Yee \\& Lopez-Cruz 1999) is a mild evolutionary \ncorrection applied to the magnitudes. The \n$k-$corrections are obtained directly from the SDSS database for every \nobject in each band. Rest-frame colors are also derived for all objects.\nMore details regarding the sample (value of $M^*$, centroid \ndetermination, member selection, virial analysis, $L_X$, $L_{opt}$, and \nN$_{gals}$ estimates) can be found in Lopes et al. (2009ab).\n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=84mm]{f1.pdf}\n\\caption{Gaussian mixture model for the $M_z$ distribution. Dashed lines depict the individual components. \nThe vertical red lines mark the intervals for\nthese modes: $M_z < -22.59$ for the bright population, $M_z > -19.03$ for faint population, and intermediate values of $M_z$ for the intermediate brightness population of galaxies. }\n\\end{figure}\n\n\n\\section{Methods}\n\\subsection{Gaussian and non-Gaussian composite clusters}\n\nOur aim is to find a connection between galaxy evolution and the dynamical \nstate of clusters. We started by defining two classes for the galaxy environments according to their velocity distributions: Gaussian velocity distribution and non-Gaussian velocity distribution clusters. Hereafter we refer to them as Gaussian and non-Gaussian clusters for simplicity, which represent the relaxed and less or unrelaxed samples. Although the theoretical  line-of-sight velocity distribution expected for galaxy clusters\nis not exactly Gaussian (e.g. Merrit 1987),\nphenomenological evidence has suggested for a long time that normality can\nbe assumed for clusters in dynamical equilibrium (e.g. Yahil \\& Vidal 1977). Hence, distinguishing\ngalaxy clusters according to their velocity distributions may be an objective way to assess \ntheir dynamics through the simple use of statistical tests (see e.g. Hou et al. 2009, Ribeiro et al. 2010, 2011). \n\nDifferently from our previous works that were only based on the Anderson-Darling (AD) test (usually appropriate for\nsmall samples -- see Hou et al. 2009), we now used three more normality tests:\nthe Kolmogorov-Smirnov, the Shapiro-Wilk, and the Robust Jarque-Bera test (see Thode 2002 for\na comprehensive review on normality tests). We also used the dip test to probe the modality\nof the velocity distributions: recent works have indicated that multimodal velocity distributions \nmay be very common in galaxy clusters.\nThey probably even correspond to the\n main cause of non-normality in the velocity distribution in clusters\n(e.g. Ribeiro et al. 2011, Hou et al. 2012, Einasto et al. 2012ab, Krause et al. 2013). Therefore, the dip method  \nhelp us to better estimate which the intrinsically Gaussian velocity distributed clusters are and which are not.\nThe dip method is based on the cumulative distribution of the variable of interest (Hartigan \\& Hartigan 1985). The dip statistic is the maximum distance between the cumulative input distribution and the best-fitting unimodal distribution. In some sense, this test is similar to the Kolmogorov-Smirnov test, but the dip test specifically searches for a flat step in the cumulative distribution function, which corresponds to a ``$dip$\" in the histogram representation. The dip test has the benefit of being insensitive to the assumption of Gaussianity and is therefore a true test of modality (e.g. Pinkney et al 1996; Muratov \\& Gnedin 2010).\n\nWe defined a non-Gaussian cluster when the dip test rejected unimodality\nand at least one of the remaining tests rejected normality, taking member galaxies out to 2$R_{200}$.\nFor ten clusters with fewer than\neight members within this limit, we obtained no reliable results. These clusters were removed from\nthe sample. For the remaining clusters,\nwe found 146 Gaussian (84\\%) and 27 non-Gaussian (16\\%) clusters\nwith a total of 9,113 galaxies. This proportion is\napproximately consistent with that found by Ribeiro et al. (2010).\n\nWe then built two stacked clusters, Gaussian and non-Gaussian, containing\n6,478 and 2,635 galaxies, respectively. The distances of the \ngalaxies in these composite groups are normalized to the distances to the group centers by $R_{200}$ and their velocities refer to the cluster median velocities and are scaled by the cluster velocity dispersions\n\n\\begin{equation}\nu_i={{v_i - \\langle v \\rangle_j}\\over \\sigma_j},\n\\end{equation}\n\n\\noindent where $i$ and $j$ are the galaxy and the cluster indices, respectively. The velocity \ndispersions of the composite clusters, $\\sigma_u$, refer to the dimensionless quantity $u_i$\n(see Ribeiro et al. 2010, 2011).\n\nThe virial properties of non-Gaussian clusters were corrected after an iteratively removing galaxies without which\nthe clusters are Gaussian. On average, 15\\% of the galaxies in non-Gaussian clusters need to be\nremoved in this process.The corrected properties are just those the cluster would have if it consisted only\nof galaxies consistent with the normal velocity distribution (see Perea et al. 1990; Ribeiro et al. 2011).\nThis correction allows one to approximately compare virial properties of Gaussian and non-Gaussian clusters. After\ncomputing the corrected \nvirial properties, we returned the removed galaxies to non-Gaussian clusters for the subsequent photometric\nanalysis of the work.\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=84mm]{f2.pdf}\n\\caption{Variation of the red galaxy fraction for the luminosity classes with\nnormalized clustercentric distance $R\/R_{200}$ for\nG (filled circles) and NG (open circles) clusters. Error-bars are obtained from a bootstrap technique with 1,000 resamplings.\nThe horizontal dashed lines indicate the RGF mean behaviour for the galaxy field. Vertical dotted lines indicate the stabilization radius for each component.}\n\\end{figure}\n\n\\subsection{Color and luminosity classification}\n\nAfter setting the composite systems, we classified galaxies into blue or red objects.\nSince the distribution of galaxies in the $g - r$ versus $u - g$ color-color diagram is strongly bimodal,\nwe simply classified galaxies \nabove and below the separator developed by Strateva et al. (2001) as red and blue, respectively. \nStrateva et al. (2001) found that the blue galaxies are dominated by late types (spirals), while the red galaxies are dominated by early types (ellipticals), and that the $u - r > 2.22$ color-selection criterion for early types results in a completeness of 98\\%. \n\nWe also classified galaxies according to their \nluminosities. For this we fitted the  absolute magnitude distribution in the $z-$band ($M_z$) \nas a Gaussian mixture with $n$\ncomponents. We used the $z$-band absolute magnitude as the closest approximation to the galaxy mass\namong the five SDSS color bands. It is estimated that the variation in\nthe stellar mass-to-light ratio is only a factor of $\\sim$3 for galaxies in \ndifferent ranges of the $z$-band absolute magnitude (Kauffmann\net al. 2003). In Figure 1, we see the $M_z$ histogram and \nthe Gaussian mixture, fitted\nafter using a Bayesian regularization for normal mixture estimation  to find the\nmixture parameters (Fraley \\& Raftery 2007). The best fit is found for a mixture of\nthree normal distributions. In Figure 1, we used red vertical lines to mark the intervals for\nthese populations, corresponding to the first\nquantile of the brightest component and the third quantile of the faintest component. \nThis choice was made to avoid overlaps of the low- and high-luminosity ends\nwith the central and dominant luminosity population.\n\nWe considered a loose definition for each luminosity class, calling \nobjects with $M_z < -22.59$ the bright population (corresponding to 15\\% of all\ngalaxies), those with $M_z > -19.03$ as the faint population (7\\% of all galaxies), \nand objects with intermediate \nvalues of $M_z$ correspond to the intermediate-brightness population of galaxies (78\\% of all galaxies).\nFinally, we find that 78\\% of the bright, 72\\% of the intermediate-brightness, and 56\\% of the faint galaxies \nare in the stacked Gaussian cluster.\n\n\n\\section{Analysis}\n\n\\subsection{The red galaxy fraction}\n\nTo study the connection between galaxy evolution and the dynamical state of \nclusters, we looked for differences in the galaxy subsamples according to the\ncluster dynamical classification, i.e., a Gaussian or non-Gaussian cluster. First, we used the \nred galaxy fraction (RGF) as an indicator of galaxy evolution\nin the stacked clusters. We found that red galaxies correspond to 74\\% of all galaxies.\nIn the stacked Gaussian cluster red galaxies correspond to 76\\% of the total, \nwhile we found a somewhat lower fraction of red galaxies in the stacked non-Gaussian cluster, 70\\%.\n\nThe mean RGF for a sample of field galaxies \nwas estimated for each luminosity population and used for comparison.\nThe field sample, containing  68,375 objects, was constructed as follows. From the whole SDSS DR7 data set,\nwe selected galaxies that are not associated to a group or cluster,\nconsidering the cluster catalog from Gal et al. (2009). To be conservative,\nwe defined a galaxy to belong to the field if it is not found within 4.0 Mpc\nand does not have a redshift offset smaller than 0.06 of any cluster from\nGal et al. (2009).  Although this definition may include void galaxies in the sample,\nthis should not be a major problem, given that several studies have shown that \ncolor bimodality is remarkably similar between \nvoid and field galaxies (e.g. Patiri et al. 2006; Hoyle et al. 2012).\nFinally, we divided the galaxies into the luminosity populations defined in Section 3.2, finding\n4\\%, 92\\%, and 4\\% of field galaxies in the bright, intermediate and faint populations, respectively.\n\nIn Figure 2, we observe\nthe emergence of the color-radius relation in all cases. The \nRGF decreases from the center toward the outskirts, with a slight \nrecovering and\/or stabilization to the outer radii, when\nthe behavior is approaching the field. \n In each case we found a significant relationship between the RGF and the clustercentric distance,\nwith a Pearson correlation coefficient $\\gtrsim 0.70$ at the 95\\% confidence level.\nThis result is expected for Gaussian clusters,\nwhich are supposed to be virialized. But we found the same behavior for non-Gaussian clusters.\nThis indicates that the color-radius (density) relation is already\nset for non-Gaussian clusters, suggesting that a strong galaxy evolution happens\nbefore the dynamical equilibrium is reached. \n Since non-Gaussian clusters (as defined in this work) should be multi-modal,\nour findings suggest that galaxies may have been pre-processed in the individual modes \nthat constitute the clusters (i.e., \nthe subunits we found in their velocity distribution -- see Ribeiro et al. 2011), a scenario consistent \nwith that of Zabludoff \\& Mulchaey (1998), where galaxies were pre-processed in\ngroup environments before accretion into large clusters.\n\nWe also note in Figure 2 important differences with respect to the\npoint at which the RGF crosses the line that depicts the mean fraction of red galaxies in the field.\nOur stacked analysis shows that larger portions of fainter red galaxies are found, on average, in\nsmaller radii.\n This result is probably related to the fact that the fraction of red galaxies heavily depends\non their own stellar mass (e.g. Baldry et al. 2006; Haines et al. 2006).\nThe most massive galaxies have resided within group-sized halos for a longer time\n(e.g. MacGee et al. 2008), which could lead to a less steep trend for the RGF.\n\n\n\n\n\\subsection{The stellar population properties}\n\n To investigate the stellar \npopulation properties of galaxies in Gaussian\/non-Gaussian groups, \nwe used the stellar population synthesis\ncode STARLIGHT (Cid Fernandes et al. 2005; Asari et al. 2007).\nThis code fits the observed spectrum with a linear combination of a \nnumber of template spectra with known properties. We built a\ntemplate sample with single stellar population spectra from Bruzual \\&\nCharlot (2003) models, using a Chabrier (2003) IMF and the Padova 1994\nstellar evolution tracks. Our template sample is composed of 45\nspectra with three different metallicities ($Z=0.004$, $Z=0.02$, or\n$Z=0.05$) and 15 ages ranging from 1\\,Myr to 13\\,Gyr. Prior to the\nfits, we shifted the SDSS galaxy spectra to the restframe and\nlinearly resampled them to 1\\,\\AA~ (the spectra are expressed with\nfixed resolution in logarithmic, not linear, scale). Our results are expressed\nin terms of two parameters proposed by Cid Fernandes et al. (2005). The first\nis the photometric mean stellar age,\n\n\\begin{equation}\n<\\log{t}>=\\sum_j x_j \\log t_j,\n\\end{equation}\n\n\\noindent where $x_j$ is the fractional contribution to the galaxy\ntotal flux of the template $j$, and $t_j$ is the age of that\ncomponent (irrespective of its metallicity). The second is a measure of the\nphotometric age dispersion,\n\n\\begin{equation}\n\\sigma (\\log{t})=\\sqrt{\\sum_j x_j (\\log t_j - <\\log t>)^2},\n\\end{equation}\n\n\\noindent which is higher for galaxies with extended star formation\nhistory and zero for a single burst\nof age $t$. Notice that due to the small SDSS fibre size (3\\,arcsec)\nthe spectra measure the central\nregion of each galaxy, so that the inferred stellar population\ndistribution is not representative of the whole galaxy. \nHowever, even if a synthesis result cannot be translated into a\ndirect meaning because of this light sampling bias, we can\nnevertheless identify average differences of the galaxies in the full\nsample.\n\n\n\\begin{table}\n      \\caption[]{Mean properties of galaxies in each\ncomponent identified in the distribution of the photometric mean stellar age.}\n$$\n         \\begin{array}{p{0.25\\linewidth}ccccc}\n\t\t\\hline\n            \\noalign{\\smallskip}\n            Component & N & M_z & \\log{t} & \\sigma(\\log{t}) & u-r   \\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n            Old & 6094 & -21.43  & 9.90 & 1.07 & 2.69\\\\\n            Transition & 774 & -21.08 & 9.32 & 1.52 & 2.42\\\\\n            Young      & 2561 & -20.70 & 8.27 & 1.58 & 1.86\\\\\n            \\noalign{\\smallskip}\n            \\hline\n\\end{array}\n$$\n\\end{table}\n\n\nThe distribution of the photometric mean stellar age \ncorresponds to a mixture of three components,  identified after using the Bayesian regularization for normal mixture estimation (the same\nprocedure we used in Section 3.2 to study the luminosity distribution; see Fraley \\& Raftery 2007).\nThe components present the following\nmean parameters: $\\log{t}=9.90\\pm 0.13$ (for the old component), \n$\\log{t}=8.27\\pm 0.27$ (for the young component) and\n$\\log{t}=9.32\\pm 0.75$ (for what we call here the transition component).\nSome properties of these components are presented in Table 1. The columns are: (1) the number\n$N$ of galaxies; (2) the mean absolute magnitude in the $z$ band; (3) the\nmean age in Gyr; (3) the mean age dispersion; and (4) the mean $u-r$ color.\nThese properties indicate that the old component is redder, more luminous\nand has the lowest photometric age dispersion, on average,\nshowing that this component hosts a more homogeneous stellar population.\n\nGalaxies in the color-age diagram have a clear bimodal distribution in this plane\n-- see Figure 3 --, as expected for a mixture of blue and red galaxies. The horizontal green dashed line depicts the color separator\ndeveloped by Strateva et al. (2001), the vertical green dashed line depicts the\nmean age of the transition component. We used these lines to roughly classify galaxies into\npassive ($u-r > 2.22$ and $\\log{t} > 9.32$) and star-forming objects \n($u-r \\leq 2.22$ and $\\log{t} \\leq 9.32$).\nThe blue and red 1-$\\sigma$ ellipses encompass the peaks identified in the color-age space\nwith the MCLUST code, \na contributed R package for multivariate normal mixture modeling and model-based clustering.\n\\footnote{R is a language and environment for statistical computing and graphics. It is a GNU project that is similar to the S language and environment and was developed at Bell Laboratories -- R Development Core Team (2011).}\nIt provides functions for parameter estimation via the expectation-maximization (EM) algorithm for normal mixture models with a variety of covariance structures (Fraley \\& Raftery 2007).\nThe ellipses indicate the central regions of star-forming (blue) and passive (red) galaxies.\nWe found that passive galaxies correspond to 84\\% of all red objects (5877 out of 7020).\nThe remaining red galaxies are probably composed of dusty, star-forming\ngalaxies (e.g. Popesso et al. 2007), and were classified in the transition component.\nFinally, star-forming objects correspond to 93\\% of all blue objects (2231 out of 2409).\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=84mm]{f3.pdf}\n\\caption{Bimodal distribution of galaxies in the color-age diagram showing the \n1$\\sigma$ ellipse around the star-forming (blue) and passive (red) peaks. The\nhorizontal and vertical green dashed lines indicate the red\/blue and star-forming\/passive\nseparations, respectively.}\n\\end{figure}\n\n\n\n\n\\subsection{Galaxy evolution in the stacked clusters}\n\n\nWe now study some properties of the stellar and luminosity populations in\nthe stacked clusters. \n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=84mm]{f4.pdf}\n\\caption{Normalized clustercentric distance of galaxies as a function of their $z$-band\nabsolute magnitude. Red circles and magenta diamonds represent passive galaxies in Gaussian and\nnon-Gaussian clusters, respectively. Blue squares and cyan stars represent star-forming galaxies\nin Gaussian and non-Gaussian clusters, respectively. The vertical dashed lines indicate the separation into\nluminosity populations. \n}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=84mm]{f5.pdf}\n\\caption{Velocity dispersions of galaxies as a function of their $z$-band\nabsolute magnitude. Red circles and magenta diamonds represent passive galaxies in Gaussian and\nnon-Gaussian clusters, respectively. Blue squares and cyan stars represent star-forming galaxies\nin Gaussian and non-Gaussian clusters. The vertical dashed lines indicate the separation into\nluminosity populations.}\n\\end{figure}\n\n\n\\subsubsection{$M_z~\\times~r\/r_{200}$}\n\nFigure 4 shows the normalized clustercentric distance of galaxies \nas a function of their $z$-band absolute magnitude. Passive galaxies in Gaussian clusters always\nhave smaller clustercentric distances than star-forming galaxies in both Gaussian and non-Gaussian clusters.\nFor galaxies in the range $-22.5<M_z<-20.5$, passive galaxies in Gaussian and non-Gaussian clusters\nhave the same behavior, while galaxies brighter than $M_z\\approx -22.5$ and \nfainter than $M_z \\approx -20.5$ are more centrally located in Gaussian than in non-Gaussian clusters.\nThe clustercentric distance of passive galaxies brighter than $M_z\\approx -22.5$\nis smaller toward brighter magnitudes in Gaussian clusters. \n\nGaussian clusters seem to have a clear structure: passive galaxies in the\nvirialized region ($r<r_{200}$), and star-forming galaxies inhabiting the outer region ($r\\gtrsim r_{200}$).\nThe absence of star-forming galaxies in the center indicates that the environment was able to\ntransform nearly all galaxies into red objects in the innermost regions. \nIn fact, we see in Figure 4 that the clustercentric distance of star-forming galaxies \nbrighter than $M_z\\approx -19.5$ is systematically smaller toward fainter magnitudes in Gaussian clusters, \nsuggesting that dynamical processes\nmay be providing both reddening and mass loss as these objects fall into the\npotential well of the cluster.\n\nThe more central distribution of bright and faint passive galaxies in Gaussian clusters\ncan be intepreted as an indicator of dynamical evolution.\nLuminosity segregation describes the tendency for more luminous\/massive \ngalaxies to be located near the cluster center (e.g. Rood et al. 1972; Dressler 1978). This effect is expected from\nthe dynamics of galaxies aggregating in the potential wells of clusters (e.g. Fusco-Femiano \\& Menci 1998), and\nthus clusters with luminosity segregation are considered to be more highly \nevolved than those without this feature.\nWe also see in Figure 4 that\nfaint passive galaxies show segregation with magnitude and position. If a fraction of these faint galaxies \ncorrespond to the result of galaxy-galaxy encounters in the cluster (e.g. Conselice 2002),\nthe presence of faint galaxies in the innermost regions of clusters could be\ntaken as an additional indicator of dynamical evolution.\n\nFinally, we note that the spatial distribution of galaxies in non-Gaussian clusters \n has some similarity to their counterparts in Gaussian ones, but with certain important\ndifferences between them: (i) non-Gaussian clusters do not present luminosity segregation -- their\nbrightest galaxies are peripheral; (ii) star-forming galaxies in non-Gaussian clusters show no significant trend\nwith the clustercentric distance; (iii) there are fewer faint galaxies\nclose to the cluster center. These results may be indicating that, regarding the galaxy\npopulation, non-Gaussian clusters are less evolved environments than Gaussian \nclusters.\n\n\\subsubsection{$M_z~\\times~\\sigma_u$}\n\nThe correlation between the velocity dispersion and the absolute\nmagnitude in the $z$-band of the galaxies is shown in Figure 5.\nWe call velocity dispersion the dispersion of\nthe galaxy velocities in each bin of absolute magnitude.\nWe confirmed the segregation between passive and star-forming galaxies\nboth in the Gaussian and non-Gaussian stacked clusters: the brightest objects\n($M_z < -22.5$) show a lower velocity dispersion, which indicates that \nthey had time to decrease their velocity\nthrough dynamical interactions or friction. We also see that passive galaxies\nin Gaussian clusters have a lower velocity dispersion over the entire luminosity range.\nThis result agrees well with some studies reporting that late-type galaxies\nhave a higher velocity dispersion than early types in nearby clusters\n(e.g. Moss \\& Dickens 1977; Sodr\\'e et al. 1989; Adami et al. 1998, 2000). This is not\ntrue for non-Gaussian clusters, where passive and star-forming galaxies\nhave approximately the same velocity dispersion in the range $-22.5< M_z < -18.5$.\nWe also note that the brightest star-forming galaxies in Gaussian clusters\nhave a high velocity dispersion and large clustercentric distance, which suggests\nthat these galaxies are falling into the clusters, possibly through\nradial orbits (e.g. Biviano \\& Katgert 2004). \n\nFrom Figures 4 and 5, we see that\nbright passive galaxies in Gaussian clusters combine low-velocity dispersion with\nsmall clustercentric distance. This kind of segregation with magnitude in\nvelocity and position has been observed in many clusters (e.g. Carlberg et al. 1997, Biviano \\& Katgert 2004).\nIt can be understood with the model of Menci\n\\& Fusco-Femiano (1996), which is a solution of the\ncollisional Boltzmann-Liouville equation, and hence accounts for galaxy collisions and merging processes\nin the cluster center. However, we also see in Figures 4 and 5 that\nthe brightest passive galaxies in non-Gaussian clusters\ncombine low velocity dispersion with large clustercentric distance, indicating\nthat these systems are less segregated than Gaussian clusters.\nIndeed, it has been found that clusters present a gradation of luminosity segregation, from highly segregated to virtually not segregated, and that their different degrees could be a consequence of either the different initial conditions present at the onset of the two-body relaxation phase or the different cluster dynamical stages \n(e.g. Yepes et al. 1991, Ribeiro et al. 2010). We must also consider that observational evidence for \nluminosity segregation in distance may be weakened by subclustering\n(Biviano et al. 1992).\nAccording to our definition in Section 3.1, non-Gaussian clusters must be multi-modal,\nwhich means that the centers of these systems are ill-defined and that\ngalaxy properties may refer more to the individual modes than to the\nentire cluster. In Figure 4, we\nsee that both the brightest and the faintest passive galaxies in non-Gaussian clusters are\nmore peripheral than their counterparts in Gaussian clusters. We verified that\nthe ratio of the clustercentric distance of passive galaxies with respect to the mode center and\nto the cluster center of non-Gaussian clusters are $\\sim$0.41, $\\sim$0.94, and $\\sim$0.65\nfor the bright, intermediate, and faint populations, respectively. Thus, in the individual modes,\nbright passive galaxies are the most centrally located, just like in Gaussian clusters,\nand segregation in distance is recovered.\n\n\nThese findings agree with the idea of non-Gaussian clusters as a set of\nsubclusters that individually show a degree of dynamical evolution, regarding the\ngalaxy population. At the same time,\nGaussian clusters appear to be more evolved systems\nwith strong evidence of segregation,\nand containing many faint galaxies in the central  regions of the clusters. If there is a sufficient number of\nthese faint galaxies, they may steepen  the faint end of the luminosity function of \ngalaxies in clusters, as we show bellow.\n\n\n\\subsection{Luminosity function}\n\nThe results achieved hitherto suggest some \nimportant differences between Gaussian and non-Gaussian clusters. We can investigate \nthis point in more detail through the luminosity function \nof galaxies in our cluster samples. \nWe built the stacked central luminosity functions (LFs), inside \na radius of 0.5\\,$R_{200}$ from the center, using the\nColless (1989) cumulation method. This method consists of a summation of the number counts (corrected for the background contribution) in all magnitude bins of the individual LFs, normalized by the total number of galaxies within the magnitude completeness limit of each cluster ($M_r=-19$ for the NoSOCS sample). \nAll photometric sources classified as galaxies that are located at\na maximum projected distance of 10 Mpc to the center \nof each cluster were retrieved from the SDSS DR7 database.\nThe absolute magnitude $M_r$ of each galaxy was obtained by $M_r=m_r-5\\,\\log\\,D\/10-K,$ where $K$ is the K-correction and $D$ is  the luminosity distance of the galaxy (in pc). All galaxies in the\ndirection of a given cluster were assigned the same value of $D$, determined from the cluster redshift; this is certainly not true for the background galaxies, but our results are insensitive to this because the background contribution can be accessed independently from the individual choice of $D$. The same K-correction was applied to all galaxies in a given cluster, corresponding to the K-correction of an elliptical galaxy at the cluster redshift. De Filippis et al. (2011) have also applied this methodology for the NoSOCS sample, and showed that \nthe results are not much different when the individual K-corrections are \ndetermined for all galaxies in the photometric sample.\nK-corrections were calculated as in Chilingarian, Melchior \\& Zolotukhin (2010), using the $(g-r)$ colors of\nelliptical galaxies of Fukugita, Shimasaku \\& Ichikawa (1995).\nThe background contribution in the individual LFs was obtained in an area with 10$\\times$10\\,Mpc and outside a radius of 2.0\\,$R_{200}$, where the contribution of cluster galaxies is expected to be very low. Local overdensities were excluded from the background estimation area whenever the local number counts exceeded 3$\\sigma$ from the mean background, calculated in square cells 150\\,kpc wide. We then rescaled the background number count in each magnitude bin to the area of the central circular region and subtracted it from the number counts in this region.\n\nThe resulting LFs for the Gaussian and non-Gaussian composite clusters are shown in Figure 6. \nThis figure also shows the best-fit Schechter functions (artificially displaced for clarity) obtained by minimization of square residuals. \nFor the non-Gaussian composite cluster, the fit resulted in $M_r^\\ast=-21.31\\pm 0.16$ and\n$\\alpha=-0.97\\pm 0.06$. \nFor the Gaussian composite cluster, a brighter characteristic magnitude $M_r^\\ast=-21.90\\pm 0.14$ and\na steeper faint-end with $\\alpha=-1.14\\pm 0.03$ were obtained. This result is consistent with the work of\nMart\\'{\\i}nez \\& Zandivarez (2011), and shows that at least a fraction of low-luminosity galaxies\nis correlated with the environment.\n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=74mm]{f6.pdf}\n\\caption{Cumulative LFs of Gaussian (filled circles) and non-Gaussian (open circles) systems. The best-fit Schechter parameters are also shown.}\n\\end{figure}\n\n\n\\section{Discussion}\n\nOur analysis gives some hints for understanding the interplay between \ngalaxy evolution and the dynamical state of galaxy clusters.\nWe verified that Gaussian clusters present at least two important\nproperties expected for dynamically evolved environments:\n(i) the color-radius relation, and (ii) the luminosity segregation in velocity and position.\nWe have to jointly consider these results to understand what \nthey imply about the interplay between galaxy evolution\nand the dynamical state of galaxy clusters. First, \nit is important to note that property (i) was also found in non-Gaussian clusters,\nreinforcing the idea of a shorter timescale for color evolution than that \nof the dynamical evolution of the clusters. Goto et al. (2004) showed that\ncolor evolution is almost completed by $z\\sim0.2$ for a sample of galaxies in SDSS clusters, while\nZabludoff \\& Mulchaey (1998) suggested that galaxies are pre-processed in group\nenvironments before accretion into larger clusters. Thus, the color-radius relation is\nexpected for low redshifts, regardless of the dynamical state of the clusters, and hence \nit cannot be taken as an indicator of galaxy cluster evolution. \nOn the other hand, there is an important difference between Gaussian and non-Gaussian\nclusters with respect to the property (ii). While segregation in velocity\nis well defined for passive galaxies in both stacked clusters, segregation in position is less well-defined\nin non-Gaussian clusters due to the large clustercentric distance of their brightest passive galaxies, a\npossible consequence of the multimodal nature of non-Gaussian clusters. Hence, \nconsidering properties (i) and (ii), our results indicate that non-Gaussian clusters are less segregated, \nbecause of subclustering, but with color evolution at an advanced stage in the\nindividual modes composing them.\n\nAt the same time, we found more faint galaxies in Gaussian clusters than in non-Gaussian clusters.\nFaint galaxies in nearby clusters present a variety in their properties, with\nsignificant differences in their spatial and kinematic distributions (e.g. Lisker et al. 2006, \nToloba et al. 2009, Paudel et al. 2010). In particular, early-type dwarf galaxies play\nan important role in understanding galaxy cluster evolution. Since these galaxies\nhave low mass and density, they are more susceptible to interactions with larger galaxies\nin the cluster environment (e.g. Lisker et al. 2006). Possibly, the number of these galaxies results from \nthe mixture of survival and destruction mechanisms in the cluster environment. Part of them could\nhave a primordial origin and start to be destroyed by tidal stripping as they fall into the clusters. \nThey could also correspond to remnants of galaxy-galaxy encounters  (e.g. Dabringhausen \\& Kroupa 2013). \nAlternatively, they could correspond to\nprimordial dwarf galaxies that were transported by galaxy groups from the outside to the inside of clusters\n(e.g. S\\'anchez-Janssen et al. 2008). \nThis latter possibility  favors the idea of non-Gaussian\nclusters as a crucial step in the evolutionary process of  galaxy clusters. They could be made of\ninteracting groups, each one transporting primordial faint galaxies to the center of future\nmore regular clusters. This possibility is consistent with the assumption that the tidal-stripping \nefficiency is proportional to the ratio of the cluster mass (M)\nto the galaxy mass (m) (e.g. Kimm et al. 2009). If non-Gaussian\nclusters correspond to assemblies of galaxy groups with smaller masses, then\nwe have lower M\/m values and thus we should expect less tidal stripping in their environments.\nThat is, faint galaxies are expected to survive much longer in the modes of non-Gaussian clusters.\n\nAt the same time, if part of the faint passive galaxies was already present in non-Gaussian clusters,\nthis means that at least a fraction of these galaxies in Gaussian clusters are formed in the\ncluster environment -- remember that Gaussian clusters present\nan excess of faint galaxies. Indeed, Figure 4 shows that faint passive galaxies in Gaussian clusters\nare more central than their counterparts in non-Gaussian clusters. \nThe mean clustercentric distances are $\\langle r\/r_{200}\\rangle^{\\rm G}\\simeq 0.62\\pm 0.34$ and\n$\\langle r\/r_{200}\\rangle^{\\rm NG}\\simeq 0.91\\pm 0.23$. The Kolmogorov-Smirnov test indicates\na significant difference between them\nat the 95\\% confidence level.\\footnote{No significant difference was found for faint star-forming galaxies.} \nThis difference in\nthe spatial distribution of these galaxies may be pointing to differences concerning their origin.\nWhile slow-motion conditions are more common in low-mass groups, slow\ngalaxy\u2013-galaxy interactions and mergers are likely to be rare in virialized clusters (e.g. Ostriker 1980).\nGalaxies moving at high speeds and in random directions\nin the center of massive galaxy clusters promote multiple \nrapid encounters between galaxies, known as galaxy harassment,\na possible mechanism underlying the production of faint passive galaxies in Gaussian clusters.\nAguerri \\& Gonz\\'alez-Garc\\'{\\i}a (2009) showed that fast galaxy-galaxy interactions\nare efficient mechanisms to transform late-type galaxies into dwarf galaxies. Thus,\nthe presence of faint galaxies in the central region of Gaussian clusters (and their\nlower incidence in non-Gaussian clusters) can be taken as a possible indicator of the\ndynamical state of a galaxy cluster.\n\nA more detailed study of the different types of faint galaxies in Gaussian and\nnon-Gaussian clusters is beyond \nthe scope of the present work. We will investigate this point in a future contribution\nbased on a larger sample.\n\n\\section{Conclusion}\n\nUnderstanding the process of galaxy cluster formation\nand evolution is an important step toward refining the details of the structure growth and constraining\nthe evolution of galaxy populations in different environments. \nThe analysis presented in this work is an attempt \nto link galaxy evolution with the dynamical state of clusters.\nOur findings suggest a scenario where non-Gaussian clusters represent a previous stage in the\nlife of a galaxy cluster during which two or more galaxy groups\nmerge to eventually form Gaussian clusters. Non-Gaussian clusters\nare less segregated because of subclustering, but with \ncolor evolution already at an advanced stage in the individual modes they are composed of.\nThe modes may also be carrying faint primordial galaxies into the central regions of larger clusters, while\nthe formation of faint galaxies through dynamical interactions \nprobably takes place on a longer timescale, probably achieving the Gaussian phase.\n\n\n\\section*{Acknowledgments}\nWe acknowledge the referee for very helpful suggestions.\nALBR also thanks for the support of CNPq, grants 306870\/2010-0 and 478753\/2010-1. \nPAAL thanks for the support of FAPERJ, process 110.237\/2010. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nHigh-mass X-ray binaries (HMXBs) are X-ray sources for which \nhigh-energy emission stems from accretion onto a compact object\n(black hole or neutron star) of material coming from a massive companion\nstar. Until recently, the huge majority of known HMXBs\nwere Be\/X-ray binaries, i.e a neutron star accreting from a disc\naround a Be star. Most of these sources are transient, even if a\nfew are persistent weak X-ray emitters\n($\\textrm{L}_\\textrm{x}\\,\\sim\\,10^{34}$ erg~s$^{-{1}}$). The other\nknown HMXBs were supergiant X-ray binaries (SGXBs), composed of\na compact object orbiting around an early-type supergiant and fed by\naccretion from the strong radiative wind of the companion. These\nobjects are persistent sources\n($\\textrm{L}_\\textrm{x}\\,\\sim\\,10^{36}$ erg~s$^{-{1}}$), and their\nrelative low number compared to the population of Be\/X-ray\nbinaries was explained as the consequence of the short lifetime of\nsupergiant stars.\n\nThe launch of the \\textit{INTErnational Gammay-Ray Astrophysics\nLaboratory}\n\\citep[\\textit{INTEGRAL}, ][]{2003Winkler} in October 2002 completely\nchanged the situation, as many more HMXBs whose companion stars\nare supergiants were discovered during the monitoring\nof the Galactic centre and the Galactic plane using the onboard\nIBIS\/ISGRI instruments \\citep{2003Ubertini, 2003Lebrun}. Most of\nthese sources are reported in\n\\citet{2007Bird} and \\citet{2007Bodaghee}, and their studies\nhave revealed two main features that were not present on previously\nknown SGXBs:\n\n\\begin{itemize} \n\\item first, many of them exhibit a\n\tconsiderable intrinsic absorption, with a column density up to\n\t$N_{\\rm H}\\,\\sim\\,2\\times10^{24}\\,\\textrm{cm}^{-2}$\nin the case of IGR~J16318-4848\n\t\\citep{2003Matt}, which explains why previous\n\thigh-energy missions had not detected them.\n\n\\item second, some of these new sources reveal a transitory\nnature. They are\n\tundetectable most of the time and occasionally present a fast\n\tX-Ray transient activity lasting a few hours. Moreover, they\n\texhibit a quiescent luminosity of\n\t$\\textrm{L}_\\textrm{x}\\,\\sim\\,10^{33}$ erg~s$^{-{1}}$, well below\n\tthe persistent state of other SGXBs.\n\t\\end{itemize}\nIt then appears that the supergiant HMXBs discovered by\n\\textit{INTEGRAL} can be classified in two classes: one\nclass of considerably obscured persistent sources that we will simply\ncall obscured SGXBs in this paper and another of\nsupergiant fast X-ray transients \\citep[SFXTs,][]{2006Negueruelaa}.\n\nHigh-energy observations can give some information about the\ncompact object or about the processes that lead to the emission\nbut do not allow study of the companion star. It is therefore\nvery important to perform multi-wavelength observations of these\nsources - from optical-to-MIR wavelength - as this\nrepresents the only way to characterise the\ncompanion or to detect dust around these highly\nobscured systems. However, positions\ngiven by \\textit{INTEGRAL} are\nnot accurate enough ($\\sim$ 2\\arcmin) to identify their optical\ncounterparts, because of the large number of objects in the\nerror circle. Observations with X-ray telescopes like\n\\textit{XMM-Newton} or \\textit{Chandra} are therefore crucial\nbecause they allow\na localisation with a position accuracy of 4$\\arcsec$ \nor better, which lowers the number of possible optical\ncounterparts.\n\nWe performed optical-to-MIR wavelength observations of\nseveral candidate SGXBs\nrecently discovered with \\textit{INTEGRAL}. Optical and NIR\nobservations were carried out at ESO\/NTT using EMMI\nand SofI instruments and aimed at constraining \nthe spectral type of the companions through accurate astrometry,\nas well as the spectroscopy and photometry of the candidate\ncounterparts. They are reported in the companion\npaper \\citep[][ CHA08 hereafter]{2008Chaty}, and it is shown\nthat most of these sources are actually supergiant stars.\n\nIn this paper, we report MIR photometric observations of the companions\nof twelve \\textit{INTEGRAL} candidate SFXTs and obscured SGXBs\nthat aimed at studying the circumstellar environment of these highly\nabsorbed sources and, more particularly, at detecting any MIR excess in\ntheir emission that could be due to the absorbing material. These\nsources were chosen because they had very accurate positions\nand confirmed 2MASS counterparts.\n\nAll the sources in our sample are listed in Table 1. The total\ngalactic column density of neutral hydrogen\n$N_{\\rm H}(\\ion{H}{i})$ is computed using the web version\nof the $N_{\\rm H}$ FTOOL from HEASARC. This tool\nuses the data from\n\\citet*{1990Dickey}, who performed \\ion{H}{i} observations from\nthe Lyman-$\\alpha$ and 21~cm lines. Moreover,\n$N_{\\rm H}(\\ion{H}{i})$ is\nthe total galactic column density, which means it is\nintegrated along the line of sight over the whole\nGalaxy. Therefore, it is likely to be overestimated compared to the\nreal value at the distance of the sources.\n\nThe total galactic column density of molecular hydrogen\n$N_{\\rm H}(\\textrm{H}_{\\rm 2})$ is computed using the\nvelocity-integrated map (W$_{\\rm CO}$) and the X-ratio given in\n\\citet*{2001Dame}. It is also likely to be overestimated compared to the\nreal value at the distance of the sources, because it is\nintegrated along the line of sight over the whole Galaxy. In\ncontrast, $N_{\\rm Hx}$, the intrinsic X-ray column density of the source, is computed\nfrom the fitting of the high-energy spectral energy distribution\n(SED), so it takes all the \nabsorption into account at the right distance of the source.\n\\newline\n\nUsing these observations, the results reported\nin the companion paper (CHA08), as well as archival\nphotometric data from the\nUSNO, 2MASS, and GLIMPSE\ncatalogues when needed, we performed the broad-band\nSEDs of these sources and fitted them with a two-component\nblack body model to assess the \ncontribution of the star and the enshrouding material in the\nemission. The ESO observations, as well as our model, are described in\nSection 2. In Section 3, results of the fits for each source\nare given and these results are discussed in Section 4. We\nconclude in Section 5. \n\n\n\n\\section{Observations}\n\n\n\\subsection{MIR observations and data reduction}\n\nThe MIR observations were carried out on 2005 June 20-22 and 2006 June\n29-30 using VISIR \\citep{2004Lagage}, \nthe ESO\/VLT mid-infrared imager and spectrograph, composed of an\nimager and a long-slit spectrometer covering several filters in\nN and Q bands and\nmounted on Unit 3 of the VLT (Melipal). The standard ''chopping\nand nodding\" MIR observational technique was used to suppress the\nbackground dominating at these wavelengths. Secondary mirror-chopping was\nperformed in the north-south direction with an amplitude of\n16$\\arcsec$ at a frequency of 0.25 Hz. Nodding technique,\nneeded to compensate for chopping residuals, was chosen as\nparallel to the chopping and applied\nusing telescope offsets of 16$\\arcsec$. Because of the high\nthermal MIR background for ground-based\nobservations, the detector integration\ntime was set to 16~ms.\n\nWe performed broad-band photometry in 3 filters, PAH1\n($\\lambda$=8.59$\\pm$0.42 $\\mu$m), PAH2\n($\\lambda$=11.25$\\pm$0.59 $\\mu$m), and Q2\n($\\lambda$=18.72$\\pm$0.88 $\\mu$m) using the small field in all bands \n(19\\farcs2x19\\farcs2 and 0\\farcs075 plate scale). All\nthe observations were bracketed with standard star\nobservations for flux calibration and\nPSF determination. The weather conditions\nwere good and stable during the observations.\n\nRaw data were reduced using the IDL reduction package written by Eric\nPantin. The elementary images were co-added in real-time to obtain\nchopping-corrected data, then the different nodding positions were\ncombined to form the final image. The VISIR detector is affected by\nstripes randomly triggered by some abnormal high-gain pixels.\nA dedicated destriping method was developed (Pantin 2008,\nin prep.) to suppress them. The MIR fluxes of all observed\nsources including the 1$\\sigma$ errors are listed in Table 2.\n\n\n\\subsection{Archival data}\n\nWhen we did not have optical-to-MIR data for our sources, we\nsearched for the counterparts in 3 catalogues: \n\\begin{itemize}\n\\item in the United States Naval Observatory (USNO)\n\tcatalogues in \\textit{B}, \\textit{R}, and \\textit{I} for USNO-B1.0, \\textit{B} and \\textit{R}\n\tfor USNO-A.2. Positions and fluxes\n\taccuracies are 0\\farcs25 and 0.3 magnitudes in the case of\n\tUSNO-B.1, 0\\farcs2 and 0.5 magnitudes in the case of USNO-A.2.\n\\item in the 2 Micron All Sky Survey (2MASS), in \\textit{J}\n\t(1.25$\\pm$0.16 $\\mu$m), \\textit{H} (1.65$\\pm$0.25 $\\mu$m) and \\textit{Ks}\n\t(2.17$\\pm$0.26 $\\mu$m) bands. Position accuracy is about 0\\farcs2.\n\\item in the \\textit{Spitzer}'s Galactic Legacy Infrared Mid-Plane Survey\n\tExtraordinaire \\citep[GLIMPSE,][]{2003Benjamin}, survey of the\nGalactic plane\n\t($|b|\\,\\leq\\,1^\\circ$ and between \\textit{l}=10$^\\circ$ and\n\\textit{l}=65$^\\circ$ on both sides of the Galactic centre)\n\tperformed with the \\textit{Spitzer Space Telescope}, using the IRAC\n\tcamera in four bands, 3.6$\\pm$0.745 $\\mu$m, 4.5$\\pm$1.023 $\\mu$m,\n\t5.8$\\pm$1.450 $\\mu$m, and 8$\\pm$2.857 $\\mu$m.\n\\end{itemize}\nAll sources had a confirmed 2MASS counterpart and three of them (IGR\nJ16195-4945, IGR~J16207-5129, and IGR~J16318-4848) had a GLIMPSE\ncounterpart given in the literature. We found all the other\nGLIMPSE counterparts using the 2MASS positions and they are listed in\nTable 3. We used all the\nfluxes given in the GLIMPSE catalogue\nexcept in the case of IGR~J17252-3616, IGR~J17391-3021 and\nIGR~J17544-2619 because\ntheir fluxes were not present in the catalogue tables. Nevertheless,\nwe measured their\nfluxes on the archival images directly with aperture\nphotometry. Uncertainties on the\nmeasurements were computed in the same way on the error\nmaps given with the data.\n\n\\subsection{Absorption}\n\nAbsorption at wavelength $\\lambda$, $\\textrm{A}_\\lambda$, is a\ncrucial parameter to fit the SEDs, especially in the\nMIR. Indeed, inappropriate values can lead to a false assessment\nof the MIR excess. Visible\nabsorption A$_\\textrm{v}$ was a free parameter of the fits. An\naccurate interstellar absorption law - i.e. the wavelength\ndependence of\nthe $\\frac{\\textrm{A}_\\lambda}{\\textrm{A}_\\textrm{v}}$ ratio in\nthe line of sight - was then needed to properly fit the SEDs.\n\nIn the optical bands, we built the function with the analytical\nexpression given in\n\\citet{1989Cardelli} who derived the average\ninterstellar extinction law in the\ndirection of the Galactic centre. From 1.25 $\\mu$m to 8 $\\mu$m,\nwe used the analytical expression given in \\citet[]{2005Indebetouw}. \nThey derived it from the measurements of the mean\nvalues of the colour excess ratios\n$\\frac{(\\textrm{A}_\\lambda-\\textrm{A}_{\\rm K})}\n{(\\textrm{A}_{\\rm J} -\\textrm{A}_{\\rm K})}$\nfrom the colour distributions of observed stars in the direction of the\nGalactic centre. They used archival data from 2MASS\nand GLIMPSE catalogues, which is relevant in our case as we use\nGLIMPSE fluxes. \n\\newline\n\nAbove 8 $\\mu$m, where absorption is dominated by the silicate\nfeatures at 9.7 $\\mu$m and 18 $\\mu$m, we found several\nextinction laws in the\nliterature \\citep{1989Rieke, 1996Lutz, 2001Moneti}, which exhibit some\ndifferences. Considering the high importance of a good assessment\nto correctly fit the MIR excess, we decided to assess the ratio\n$\\frac{\\textrm{A}_\\lambda}{\\textrm{A}_\\textrm{v}}$ in 2\nVISIR bands - PAH1 and PAH2 - from our data in order to\nbuild the relevant law for our observations. \n\n\\citet*{1985Rieke} gave the interstellar extinction law up to 13\n$\\mu{\\textrm{m}}$, and from their results, we derived \n$0.043\\leq\\frac{\\textrm{A}_{\\rm PAH1}}{\\textrm{A}_\\textrm{v}}\\leq0.074$\nand\n$0.047\\leq\\frac{\\textrm{A}_{\\rm PAH2}}{\\textrm{A}_\\textrm{v}}\\,\\leq0.06$.\nTo get the best values corresponding to our data in PAH1 and\nPAH2, we proceeded\nin 3 steps. \n\\begin{itemize}\n\t\\item We first selected the sources for which we had VISIR\nfluxes in PAH1 and\/or\n\tPAH2 and fitted their SEDs with extinction laws given in\n\t\\citet{1989Cardelli} and \\citet{2005Indebetouw} from 0.36 to 8 $\\mu{\\textrm{m}}$ and\n\thalf-interval values taken in PAH1 and PAH2.\n\t\\item Then, when we did not need any MIR excess to fit the\n\tIRAC fluxes, we adjusted the \n\t$\\frac{\\textrm{A}_{\\rm PAH1}}{\\textrm{A}_\\textrm{v}}$ and\n\t$\\frac{\\textrm{A}_{\\rm PAH2}}{\\textrm{A}_\\textrm{v}}$ ratios to\n\timprove the $\\chi^2$ of our fits.\n\t\\item We finally averaged all the extinction values obtained for all\n\tsources to get what we consider as the right ratios in PAH1 and PAH2\n\tin the direction of the Galactic plane.\n\\end{itemize}\nThe resulting values are in good agreement with those given by\nthe extinction law from \\citet{1996Lutz}, so we chose\ntheir extinction law to fit our SEDs above 8\n$\\mu{\\textrm{m}}$, the Q2 filter\nincluded. The $\\frac{\\textrm{A}_\\lambda}{\\textrm{A}_\\textrm{v}}$ values we\nused in each band are listed in Table 4, and the overall\nextinction law is displayed in Fig 1.\n\n\n\\subsection{SEDs}\n\nWith all the archival and observational data from optical-to-MIR \nwavelength, we built the SEDs for these sources. We fitted them\n(using a $\\chi^2$ minimisation) with a\nmodel combining two absorbed black bodies, one representing the companion\nstar emission and a spherical one representing a possible MIR\nexcess due to the absorbing material enshrouding the companion star:\n\n\\begin{displaymath}\n\t\\lambda{F(\\lambda)}\\,=\\,\\frac{2\\pi{h}{c}^2}{{D_{\\ast}}^2{\\lambda}^4}\n\\,10^{\\textrm{\\normalsize\n\t$-{0.4A_\\lambda}$}}\\left[\\frac{{R_\\ast}^2}{\n\te^{\\textrm{\\large\n\t$\\frac{hc}{{\\lambda}k{T}_\\ast}$}}-1}+\\frac{{R_{\\rm D}}^2}{e^{\\textrm{\\large\n\t$\\frac{hc}{{\\lambda}k{T}_{\\rm D}}$}}-1}\\right]\\,\\,\\,\\,\\,\\,\\,\\textrm{in\n\tW m}^{-2}\n\\end{displaymath}\n\nWe added to the flux uncertainties systematic errors as follows:\n\n\\begin{itemize}\n\t\\item a 2$\\%$ systematic error in each IRAC band as given in\nthe IRAC\nmanual\\footnote{http:\/\/ssc.spitzer.caltech.edu\/documents\/som\/\nsom8.0.irac.pdf}\n\t\\item comparing the variations of the flux calibration\nvalues obtained from standards with VISIR during our\nobservation nights, we figured out that systematic errors\nwith VISIR were about 5$\\%$ at 10 $\\mu{\\textrm{m}}$ and\n10$\\%$ at 20 $\\mu{\\textrm{m}}$.\n\\end{itemize}\nThe free parameters of the fits were the absorption in the\nV-band A$_\\textrm{v}$,\nthe companion star black body temperature T$_\\ast$ and radius\nto distance ratio $\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$, as\nwell as the additional spherical component black body temperature\nand radius T$_{\\rm D}$ and $\\textrm{R}_{\\rm D}$.\\\\\t\n\\newline\nThe best-fitting\nparameters for individual sources, as well as corresponding\n$\\chi^2$ are\nlisted in Table 5 and the fitted SEDs are displayed in Fig 3.\nMoreover, 90\\%-confidence ranges of parameters are listed in\nTable 6. \n\nIn Table 5, along with the best-fitting parameters, we also give\nthe total galactic \nextinctions in magnitudes $\\textrm{A}_{\\ion{H}{i}}$ and\n$\\textrm{A}_{\\rm H_2}$ in the line of sight, as well as the\nX-ray extinction of the source\nin magnitudes $\\textrm{A}_\\textrm{x}$. The values of\n$\\textrm{A}_{\\ion{H}{i}}$, $\\textrm{A}_{\\rm H_2}$, and\n$\\textrm{A}_\\textrm{x}$ are computed from\n$N_{\\rm H}(\\ion{H}{i})$, \n$N_{{\\rm H}}(\\textrm{H}_{\\rm 2})$, and\n$N_{\\rm Hx}$ given in Table 1 using the relation \n$\\textrm{A}_{\\rm H}\\,=\\,\\frac{3.1}{5.8\\times10^{21}\\,\\textrm{cm}^{-{2\n}}}N_{\\rm H}$\n\\citep*{1978Bohlin,1985Rieke}.\n\n\\begin{table*}\n\t\\caption{Sample of sources studied in this paper. \n\tWe give their name, their\n\tcoordinates (J2000 and galactic), the total galactic\n\tcolumn density of neutral hydrogen\n\t($N_{\\rm H}(\\ion{H}{i})$) and the total galactic\n\tcolumn density of molecular hydrogen\n($N_{\\rm H}(\\textrm{H}_{\\rm 2})$) in the line of\n\tsight, the X-ray column\n\tdensity of the source ($N_{\\rm Hx}$), their type\n(SFXT or OBS - obscured sources) and their\n\tspectral type (SpT). Their spectral classifications come from optical\/NIR\n\tspectroscopy, reported in the following references (Ref): c:\n\\citet{2008Chaty}, f: \\citet*{2004Filliatre}, i: \\citet{2006Zand},\n\tn1: \\citet{2005Negueruela}, n2: \\citet{2006Negueruelab}, n3:\n\\citet{2007Nespoli}, p:\n\t\\citet{2006Pellizza}, t: \\citet{2006Tomsick}.}\n\t$$\n\t\\begin{array}{c c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\alpha (\\textrm{J2000})&\\delta\n(\\textrm{J2000})&l&b&N_{\\rm H}(\\ion{H}{i})(10^{22})&\nN_{\\rm H}(\\textrm{H}_{\\rm 2})(10^{22})&N_{\\rm Hx}\n(10^{22})&\\textrm{Type}&\\textrm{SpT}&\\textrm{Ref}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&16\\,\\,19\\,\\,32.20&-49\\,\\,44\\,\\,30.7&3\n33.56&0.339&2.2&5.4&7.0&\\textrm{SFXT ?}&\\textrm{O\/B}&\\textrm{t}\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&16\\,\\,20\\,\\,46.26&-51\\,\\,30\\,\\,06.0&\n332.46&-1.050&1.7&2.2&3.7&\\textrm{OBS}&\\textrm{O\/B}&\\textrm{t}\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&16\\,\\,31\\,\\,48.60&-48\\,\\,49\\,\\,00.0&\n335.62&-0.448&2.1&3.6&200.0&\\textrm{OBS}&\\textrm{sgB[e]}&\\textrm{f}\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&16\\,\\,32\\,\\,01.90&-47\\,\\,52\\,\\,27.0&\n336.30&0.169&2.1&4.4&21.0&\\textrm{OBS}&\\textrm{O\/BI}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&16\\,\\,35\\,\\,53.80&-47\\,\\,25\\,\\,41.1&\n337.01&-0.007&2.2&7.3&33.0&\\textrm{OBS}&\\textrm{sgB[e]}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&16\\,\\,41\\,\\,51.00&-45\\,\\,32\\,\\,25.0&\n339.19&0.489&1.9&3.6&10.0&\\textrm{SFXT ?}&\\textrm{O\/B}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&16\\,\\,46\\,\\,35.50&-45\\,\\,07\\,\\,04.0\n&340.05&0.135&2.1&5.9&60.0&\\textrm{SFXT}&\\textrm{B0.5I}&\\textrm{n1}\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&16\\,\\,48\\,\\,06.60&-45\\,\\,12\\,\\,08.0&\n340.16&-0.124&2.1&8.2&7.7&\\textrm{SFXT ?}&\\textrm{O\/BI}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&17\\,\\,25\\,\\,11.40&-36\\,\\,16\\,\\,58.6&\n351.50&-0.354&1.6&3.9&15.0&\\textrm{OBS}&\\textrm{O\/BI}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&17\\,\\,39\\,\\,11.58&-30\\,\\,20\\,\\,37.6&\n358.07&0.445&1.4&4.5&30.0&\\textrm{SFXT}&\\textrm{08Iab(f)}&\\textrm{n2}\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&17\\,\\,54\\,\\,25.28&-26\\,\\,19\\,\\,52.6\n&3.26&-0.336&1.4&7.8&1.4&\\textrm{SFXT}&\\textrm{O9Ib}&\\textrm{p}\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&19\\,\\,14\\,\\,04.23&+09\\,\\,52\\,\\,58.3&\n44.30&-0.469&1.7&3.7&6.0&\\textrm{OBS}&\\textrm{B1I}&\\textrm{n3}\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*}\n\n\\begin{table*}\n\t\\caption{Summary of VISIR observations of newly discovered\n\\textit{INTEGRAL}\n\tsources. We give their MIR fluxes (mJy) in the PAH1 (8.59 $\\mu$m), PAH2\n(11.25 $\\mu$m) and\n\tQ2 (18.72 $\\mu$m) filters. When we did not detect a source,\n\twe give the upper limit. When no flux nor upper limit is\n\tgiven, the source was not observed in the considered filter.}\n\t$$\n\t\\begin{array}{c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{PAH1}&\\textrm{PAH2}&\\textrm{Q2}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&<6.1&<7.8&<50.3\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&21.7\\pm1.4&9.4\\pm1.3&<53.4\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&426.2\\pm3.0&317.4\\pm3.4&180.7\\pm15.3\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&12.1\\pm1.7&6.3\\pm1.8&\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&<6.9&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&<5.8&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&6.9\\pm1.1&<5.0&\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&10.9\\pm1.2&7.0\\pm1.6&\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&6.1\\pm0.6&<5.0&\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&70.2\\pm1.6&46.5\\pm2.6&\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&46.1\\pm2.8&20.2\\pm2.1&\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&35.2\\pm1.4&19.1\\pm1.4&\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*}\n\n\\begin{table*}\n\t\\caption{List of GLIMPSE counterparts we found for 9\nsources. We give their name, their separation from the 2MASS counterparts\n\tand their fluxes in mJy.}\n\t$$\n\t\\begin{array}{c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{GLIMPSE\ncounterpart}&\\textrm{Separation}&3.6\\,\\mu\\textrm{m}&4.5\\,\\mu\n\\textrm{m}&5.8\\,\\mu\\textrm{m}&8\\,\\mu\\textrm{m}\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&\\textrm{G336.3293+00.1689}&0\\farcs17&\n48.2\\pm1.9&44.3\\pm2.1&36.0\\pm2.0&17.3\\pm1.0\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&\\textrm{G337.0994-00.0062}&0\\farcs46&\n5.9\\pm0.5&5.6\\pm0.6&5.3\\pm1.7&\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&\\textrm{G339.1889+00.4889}&0\\farcs28&\n12.5\\pm0.9&9.5\\pm0.6&5.6\\pm0.6&3.6\\pm0.4\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&\\textrm{G340.0536+00.1350}&0\\farcs16&\n45.0\\pm2.0&32.6\\pm1.5&22.0\\pm0.9&13.7\\pm0.6\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&\\textrm{G340.1630-00.1239}&0\\farcs13&\n68.6\\pm3.1&49.6\\pm2.0&41.2\\pm2.3&19.4\\pm0.9\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&&&32.6\\pm3.7&24.7\\pm4.7&21.8\\pm5.0&\n9.6\\pm6.5\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&&&375.9\\pm44.0&297.0\\pm31.0&205.0\\pm33.0&\n111.0\\pm28.0\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&&&213.9\\pm25.4&137.0\\pm18.9&99.6\\pm7.6&\n66.5\\pm12.1\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&\\textrm{G044.2963-00.4688}&0\\farcs48&\n185.0\\pm9.3&152.0\\pm11.1&103.9\\pm5.4&62.0\\pm2.1\\\\\n\t\\hline\n\t\\end{array} \n\t$$\n\\end{table*} \n\n\\begin{table*}\n\t\\caption{Adopted $\\frac{\\textrm{A}_{\\lambda}}{\\textrm{A}_{v}}$ values.}\n\t$$\n\t\\begin{array}{c c c c c c c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Filters}&\\textrm{\\textit{U}}&\\textrm{\\textit{B}}&\\textrm{\\textit{V}}&\\textrm{\\textit{R}}&\n\\textrm{\\textit{I}}&\\textrm{\\textit{J}}&\\textrm{\\textit{H}}&\\textrm{\\textit{Ks}}&3.6\\,\\mu\\textrm{m}&\n4.5\\,\\mu\\textrm{m}&5.8\\,\\mu\\textrm{m}&8\\,\\mu\\textrm{m}&8.59\\,\\mu\n\\textrm{m}&11.25\\,\\mu\\textrm{m}&18.72\\,\\mu\\textrm{m}\\\\\n\t\\hline\n\t\\frac{\\textrm{A}_{\\lambda}}{\\textrm{A}_{v}}&1.575&1.332&1&0.757&\n0.486&0.289&0.174&0.115&0.0638&0.0539&0.0474&0.0444&0.0595&0.0605&\n0.040\\\\\n\t\\hline\n\t\\end{array} \n\t$$\n\\end{table*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[angle=270,width=10cm]{.\/absorb.ps}\n\t\\caption{Adopted extinction law. We used the law given in\n\t\\citet{1989Cardelli} in the optical, the one given in\n\t\\citet{2005Indebetouw} from 1.25 $\\mu{\\textrm{m}}$ to 8\n$\\mu{\\textrm{m}}$, and the law\n\tfrom \\citet{1996Lutz} above 8 $\\mu{\\textrm{m}}$.}\n\\end{figure*} \n\n\n\\begin{table*}\n\t\\caption{Summary of best-fitting parameters of the SEDs of the\n\tsources. We give the total galactic \n\textinctions in magnitudes $\\textrm{A}_{\\ion{H}{i}}$ and\n\t$\\textrm{A}_{\\rm H_2}$, the X-ray extinction of the source\n\tin magnitudes $\\textrm{A}_\\textrm{x}$ and then the\n\tparameters themselves: the extinction in the optical\n\t$\\textrm{A}_\\textrm{v}$, the temperature\n\t$\\textrm{T}_\\ast$ and the\n\t$\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}$ ratio of the\n\tcompanion and the\n\ttemperature and radius $\\textrm{T}_{\\rm D}$ and\n$\\textrm{R}_{{\\rm D}}$ (in\n\t$\\textrm{R}_{\\ast}$ unit) of the dust component\n\twhen needed. We also add the reduced $\\chi^2$ we reach for\n\teach fit.} \n\t$$\n\t\\begin{array}{c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{A}_{\\ion{H}{i}}&\\textrm{A}_{\\rm H_2}&\n\\textrm{A}_\\textrm{x}&\\textrm{A}_\\textrm{v}&\\textrm{T}_{\\ast}\n(\\textrm{K})&\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}&\\textrm{T}_\n{\\rm D}\n(\\textrm{K})&\\textrm{R}_{\\rm D}\n\\left(\\textrm{R}_{\\ast}\\right)&\\chi^2\/\\textrm\n{dof}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&11.7&28.6&37.4&15.5&23800&5.96\\times\n10^{-{11}}&1160&5.1&3.9\/2\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&9.3&11.8&19.8&10.5&33800&9.42\\times\n10^{-{11}}&&&28.5\/9\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&11.0&19.2&1069.5&17.0&22200&3.74\\times\n10^{-{10}}&1100&10.0&6.6\/6\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&11.4&23.6&112.2&35.4&33000&1.38\\times\n10^{-{10}}&&&7.7\/6\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&11.8&39.4&176.4&17.6&24500&3.16\\times\n10^{-{11}}&810&10.1&3.6\/2\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&10.1&19.3&53.5&14.5&32800&3.77\\times\n10^{-{11}}&&&1.4\/4\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&11.3&31.4&320.7&5.9&25000&6.40\\times\n10^{-{11}}&&&13.9\/7\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&11.4&43.8&41.2&18.5&32800&1.00\\times\n10^{-{10}}&&&7.4\/6\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&8.3&20.1&80.2&20.8&32600&7.57\\times\n10^{-{11}}&&&3.8\/5\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&7.3&23.9&160.4&9.2&31400&1.80\\times\n10^{-{10}}&&&11.7\/10\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&7.7&41.5&7.70&6.1&31000&1.27\\times10^{-{10}}\n&&&6.1\/8\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&9.0&14.0&32.1&16.5&22500&1.92\\times10^\n{-{10}}&&&14.4\/6\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*} \n\n\\begin{table*}\n\t\\caption{Ranges of parameters that give acceptable\n\tfits (90\\%-confidence) for each source.\n\t}\n\t$$\n\t\\begin{array}{c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\Delta\n\t\\textrm{A}_\\textrm{v}&\\Delta\\textrm{T}_{\\ast}&\\Delta\\frac{\\textrm{R\n}_{\\ast}}{\\textrm{D}_{\\ast}}&\\Delta\\textrm{T}_{\\rm D}&\\Delta\\textrm\n{R}_{{\\rm D}}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&14.8-15.8&13100-25900&5.68\\times10^{-{11}}\n-6.68\\times10^{-{11}}&950-1460&3.9-6.6\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&10.4-10.6&25200-36000&9.36\\times10^{-{11}}\n-1.05\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&16.6-17.4&19300-24500&3.65\\times10^{-{10}}\n-3.84\\times10^{-{10}}&960-1260&8.8-11.8\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&34.8-35.5&22000-35600&1.33\\times10^{-{10}}\n-1.69\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&17.1-18.1&19500-36000&2.64\\times10^{-{11}}\n-3.52\\times10^{-{11}}&630-1020&8.0-13.8\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&13.6-14.7&10600-36000&3.58\\times10^{-{11}}\n-5.44\\times10^{-{11}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&5.0-6.1&15400-33600&5.50\\times10^{-{11}}\n-7.95\\times10^{-{11}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&18.4-18.8&26200-36000&9.48\\times10^{-{11}}\n-1.13\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&20.3-21.0&20500-36000&7.17\\times10^{-{11}}\n-9.48\\times10^{-{11}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&8.8-9.4&16100-32200&1.78\\times10^{-{10}}\n-2.63\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&6.0-6.2&26700-35500&1.18\\times10^{-{10}}\n-1.38\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&15.7-16.7&13200-28100&1.71\\times10^{-{10}}\n-2.53\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*} \n\n\n\\section{Results}\n\n\\subsection{IGR~J16195-4945}\n\nIGR~J16195-4945 was detected by \\textit{INTEGRAL} during\nobservations carried\nout between 2003 February 27 and October 19 \\citep{2004Walter} and\ncorresponds to the ASCA source AX J161929-4945\n\\citep{2001Sugizaki,2005Sidoli}. \nAnalysing \\textit{INTEGRAL} public data,\n\\citet{2005Sidoli} derive an average flux level of\n$\\sim$ 17~mCrab (20-40~keV). Performing a follow-up with\n\\textit{INTEGRAL}, \\citet{2006Sguera} show it behaves like\nan SFXT and report a peak-flux of $\\sim$ 35~mCrab (20-40~keV).\n\n\\citet{2006Tomsick} observed the source with \\textit{Chandra}\nbetween 2005 April and\nJuly and give its position with 0\\farcs6 accuracy. They fitted its\nhigh-energy emission with an absorbed\npower law and derive $\\Gamma\\sim0.5$ and\n$N_{\\rm H}\\sim7\\times10^{22}$~cm$^{-2}$. Moreover,\nusing their accurate localisation, they found its NIR and MIR\ncounterparts in\nthe 2MASS (2MASS~J16193220-4944305) and in the GLIMPSE\n(G333.5571+00.3390) catalogues and performed its NIR photometry\nusing ESO\/NTT observations. They show its spectral type is\ncompatible with an O, B, or A supergiant star. They also found possible\nUSNO-A.2 and USNO-B.1 counterparts. Nevertheless, as already\nsuggested in their paper, the USNO source is\na blended foreground object \\citep{2006Tovmassian}. \n\\newline\n\nWe observed IGR~J16195-4945 on 2006 June 30 in PAH1 during 1200~s, but\ndid not detect it. Typical seeing and airmass were\n0\\farcs88\\ and 1.07. We nevertheless fitted its SED\nusing the NIR and the GLIMPSE flux values given in \\citet{2006Tomsick}\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=15.5,\n$\\textrm{T}_*$=23800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$5.96\\times10^{-{11}}$,\n$\\textrm{T}_{\\rm D}$=1160~K,\n$\\textrm{R}_{\\rm D}$=5.1$\\textrm{R}_\\ast$, and the reduced\n$\\chi^2$ is 3.9\/2. \n\\newline\nThe best-fitting parameters without the additional component are\n$\\textrm{A}_\\textrm{v}$=16.1,\n$\\textrm{T}_*$=13800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$8.94\\times10^{-{11}}$,\nand the corresponding reduced $\\chi^2$ is 15.8\/4. We then found\na MIR excess.\n\nThe additional component is then needed to correctly fit the SED, as\nthis source exhibits a MIR excess. Nevertheless, as shown in\nFig 4, this excess is small, and the lack of data above 8\n$\\mu$m does not allow to reach definitive conclusions. Moreover,\nin both cases (with and without dust), the stellar\ncomponent is consistent with an O\/B supergiant, as already\nsuggested in \\citet{2006Tomsick}. \n\n\\subsection{IGR~J16207-5129}\n\nIGR~J16207-5129 is an obscured SGXB that was discovered by\n\\textit{INTEGRAL} during observations carried\nout between 2003 February 27 and October 19 \\citep{2004Walter}.\n\n\\citet{2006Tomsick} observed it with \\textit{Chandra} during the\nsame run as\nIGR~J16195-4945 and give its position with 0\\farcs6\naccuracy. \nThey also fitted its high-energy emission with an absorbed\npower law and derive $\\Gamma\\sim0.5$ and\n$N_{\\rm H}\\sim3.7\\times10^{22}$~cm$^{-2}$.\nThanks to their accurate localisation, they found its NIR and MIR\ncounterparts in\nthe 2MASS (2MASS~J16204627-5130060) and in the GLIMPSE\n(G333.4590+01.0501) catalogues and performed its NIR photometry\nusing ESO\/NTT observations. They show its temperature to\nbe $\\geqslant$18000~K, which indicates the system is an\nHMXB. They also found its USNO-B1.0\n(USNO-B1.0~0384-0560875) counterpart.\n\\newline\n\nWe observed IGR~J16207-5129 on 2006 June 29 in PAH1 and PAH2 during\n1200~s in each filter, and in Q2 during 2400~s. Typical seeing and\nairmass were 0\\farcs72 and 1.09. We did not detect it in Q2 but in PAH1\nand PAH2. The fluxes we derived are 21.7$\\pm$1.4~mJy and\n9.4$\\pm$1.3~mJy, respectively.\n\nUsing those values, as well as the fluxes from ESO\/NTT observations and\nthe GLIMPSE archives found in \\citet{2006Tomsick}, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=10.5,\n$\\textrm{T}_*$=33800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$9.42\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is\n28.5\/9. \\citet*{2007Negueruela} find the spectral type is earlier\nthan B1I; our parameters are therefore in good agreement with\ntheir results.\n\nThe best fit with the additional component gives a larger\nreduced $\\chi^2$ of 30\/7\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K, which is not \nsignificant, as the presence of such cold material marginally enhances\nthe MIR flux. We therefore think IGR~J16207-5129 is an O\/B\nmassive star whose enshrouding material marginally contributes\nto its MIR emission.\n\n\n\\subsection{IGR~J16318-4848}\n\nMain high-energy characteristics of this source can be found in\n\\citet*{2003Matt} and \\citet{2003Walter}. IGR~J16318-4848 was\ndiscovered by\n\\textit{INTEGRAL} on 2003 January 29 \\citep{2003Courvoisier} and\nwas then\nobserved with \\textit{XMM-Newton}, which allowed a 4$\\arcsec$ localisation. Those\nobservations showed that the source was exhibiting a strong absorption\nof $N_{\\rm H}\\sim2\\times10^{24}$~cm$^{-2}$, a temperature kT = 9~keV, and a photon index $\\sim\\,2$. \n\nUsing this accurate position, \\citet*{2004Filliatre} discovered its\noptical counterpart and confirmed the NIR counterpart proposed by\n\\citet{2003Walter} (2MASS~J16314831-4849005). They also performed\nphotometry and spectroscopy in optical and NIR on 2003 February 23-25\nat ESO\/NTT and show that the source presents a significant NIR excess\nand that it is strongly absorbed\n($\\textrm{A}_\\textrm{v}\\sim17.4$). The spectroscopy revealed an\nunusual spectrum with a continuum very rich in strong emission\nlines, which, together with the presence of forbidden lines,\npoints towards an sgB[e] companion star \\citep[see e.g. ][ for\ndefinition and characteristics of these stars]{1998Lamers,1999Zickgraf}.\n\nUsing the 2MASS magnitudes, the GLIMPSE (G335.6260-00.4477), and\nthe MSX fluxes,\n\\citet{2006Kaplan} fitted its SED with a combination of a stellar\nand a dust component black bodies, and shows that the presence\nof warm dust around the system was necessary for explaining the\nNIR and MIR excess. From their fit, they derive \n$\\textrm{A}_\\textrm{v}\\sim18.5$, $\\textrm{T}_{\\rm D}$=1030~K,\nand $\\textrm{R}_{\\rm D}$=10$\\textrm{R}_{\\ast}$.\n\\newline\n\nWe observed IGR~J16318-4848 with VISIR twice: \n\\begin{itemize}\n\\item the first time on 2005\n\tJune 21 during 300~s in PAH1 and PAH2, and 600~s in Q2. Typical seeing\n\tand airmass were 0\\farcs81 and 1.14. We detected the source\n\tin all bands, and the derived fluxes are\n\t409.2$\\pm$2.4~mJy, 322.4$\\pm$3.3~mJy, and 172.1$\\pm$14.9~mJy\nin PAH1, PAH2, and Q2, respectively.\n\\item the second on 2006 June 30 during 600~s in all bands. Typical seeing\n\tand airmass were 0\\farcs68 and 1.09. We \n\tdetected the source in all bands, and the derived fluxes are\n\t426.2$\\pm$3.0~mJy, 317.4$\\pm$3.4~mJy, and 180.7$\\pm$15.3~mJy\nin PAH1, PAH2, and Q2, respectively.\n\\end{itemize}\nThose observations show that IGR~J16318-4848 is very bright in\nthe MIR (it is actually the brightest source in our sample) and\nthat its flux was constant within a year, considering VISIR\nsystematic errors.\n\nUsing data from our last run, as well as the magnitudes given in\n\\citet*{2004Filliatre} in the optical and the NIR, and the fluxes from\nthe GLIMPSE archives, we fitted its SED\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=17,\n$\\textrm{T}_*$=22200~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$3.74\\times10^{-{10}}$,\n$\\textrm{T}_{\\rm D}$=1100~K,\n$\\textrm{R}_{\\rm D}$=10$\\textrm{R}_\\ast$, and the reduced\n$\\chi^2$ is 6.6\/6.\nThe best-fitting parameters without the additional component are\n$\\textrm{A}_\\textrm{v}$=17.9,\n$\\textrm{T}_*$=18200~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$5.1\\times10^{-{10}}$,\nand the corresponding reduced $\\chi^2$ is 425\/8. We then confirm\nthat the MIR excess is likely due to the \npresence of warm dust around the system, as already suggested by\n\\citet*{2004Filliatre} and reported in \\citet{2006Kaplan}.\n\n\\subsection{IGR~J16320-4751}\n\nIGR~J16320-4751 was detected by \\textit{INTEGRAL} on 2003\nFebruary \\citep{2003Tomsick} \nand corresponds to the ASCA source AX\nJ1631.9-4752. \\citet{2003Rodrigueza} report observations with\n\\textit{XMM-Newton}. They give an accurate localisation (3$\\arcsec$) and\nfitted its high-energy spectrum with an absorbed power law. They\nderive $\\Gamma\\sim1.6$ and\n$N_{\\rm H}\\sim2.1\\times10^{23}$~cm$^{-2}$. \n\n\\citet{2005Lutovinov} report the discovery of X-Ray pulsations\n(P$\\sim1309$~ s), which proves the compact object is a neutron\nstar. Moreover, \\citet{2004Corbet} obtained the light curve of IGR\nJ16320-4751 between 2004 December 21 and 2005 September 17 with Swift\nand report the discovery of a 8.96 days orbital period. IGR\nJ16320-4751 is then an X-ray binary whose compact object is a\nneutron star.\n\n\\citet*{2007Negueruela} searched for the NIR counterpart of the\nsource \nin the 2MASS catalogue and found its position was consistent with\n2MASS~J16320215-4752289. They also concluded that, if it was an O\/B\nsupergiant, it had to be extremely absorbed. \n\nThe optical and NIR photometry and spectroscopy of this source were\ncarried out at ESO\/NTT, and results are reported in CHA08. It is\nshown that its NIR spectrum is consistent with an O\/B supergiant\nand that its intrinsic absorption is very high, because it was not\ndetected in any of the visible bands. We searched for the MIR\ncounterpart of IGR~J16320-4751 in the GLIMPSE archives and found it\nto be consistent with G336.3293+00.1689.\n\\newline\n\nWe observed IGR~J16320-4751 with VISIR on 2005 June 20 in PAH1 and\nPAH2, and the respective exposure\ntimes were 1800~s and 2400~s. Typical seeing and\nairmass were 0\\farcs63 and 1.13. We detected it in both filters, and\nthe respective fluxes are 12.1$\\pm$1.7~mJy and 6.3$\\pm$1.8~mJy.\nUsing the ESO\/NTT NIR magnitudes given in CHA08, as well as the\nGLIMPSE and the VISIR fluxes, we fitted its SED and the best-fitting\nparameters are $\\textrm{A}_\\textrm{v}$=35.4,\n$\\textrm{T}_*$=33000~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.38\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is\n7.7\/6. This result is in good agreement with an extremely\nabsorbed O\/B supergiant as reported in CHA08.\n\nThe best fit with the additional component gives a larger\nreduced $\\chi^2$ of 8\/4\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. We therefore\nthink that IGR~J16320-4751 is an O\/B\nsupergiant whose enshrouding material marginally contributes to\nits MIR emission, even if its intrinsic absorption is extremely high.\n\n\\subsection{IGR~J16358-4726}\n\nIGR~J16358-4726 was detected with \\textit{INTEGRAL} on 2003\nMarch 19 \\citep{2003Revnivtsev} and first observed with\n\\textit{Chandra} on 2003 March 24 \n\\citep{2004Patel}. They give its position with 0\\farcs6 accuracy\nand fitted its high-energy spectrum with an absorbed power law. They\nderive $\\Gamma\\sim0.5$ and\n$N_{\\rm H}\\sim3.3\\times10^{23}$~cm$^{-2}$. They also found a\n5880$\\pm$50~s modulation, which could be either a neutron star\npulsation or an orbital modulation. Nevertheless, \\citet{2006Patel}\nperformed detailed spectral and timing analysis of this source using\nmulti-satellite archival observations and identified a 94~s spin up,\nwhich points to a neutron star origin. Assuming that this spin up was due\nto accretion, they estimate the source magnetic field is between $10^{13}$\nand $10^{15}$ G, which could support a magnetar nature for\nIGR~J16358-4726. \n\n\\citet{2003Kouveliotou} propose 2MASS~J16355369-4725398 as the\npossible NIR counterpart, and NIR spectroscopy and photometry of\nthis counterpart was performed \nat ESO\/NTT and is reported in CHA08. They show that its\nspectrum is consistent with a B supergiant belonging to the\nsame family as IGR~J16318-4848, the so-called B[e]\nsupergiants. We also found its MIR counterpart in the GLIMPSE\narchives (G337.0994-00.0062).\n\\newline\n\nWe observed IGR~J16358-4726 with VISIR on 2006 June 29 but did\nnot detect it in any filter. Using the NIR magnitudes given in CHA08\nand the GLIMPSE fluxes, we fitted its SED\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=17.6,\n$\\textrm{T}_*$=24500~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$3.16\\times10^{-{11}}$,\n$\\textrm{T}_{\\rm D}$=810~K,\n$\\textrm{R}_{\\rm D}$=10.1$\\textrm{R}_\\ast$, and the reduced\n$\\chi^2$ is 3.6\/2. The best-fitting parameters\nwithout the additional component are $\\textrm{A}_\\textrm{v}$=16.7,\n$\\textrm{T}_*$=9800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$6.05\\times10^{-{11}}$\nand the corresponding reduced $\\chi^2$ is 8.8\/4. \n\nThe additional component is then necessary to correctly fit the\nSED, since\nthis source exhibits a MIR excess (see Fig 4). Even if we lack\nMIR data above 5.8 $\\mu$m, we think this excess is real and\nstems from warm dust, as it\nis consistent with the source being a sgB[e], as reported in CHA08. \n\n\\subsection{IGR~J16418-4532}\n\nIGR~J16418-4532 was discovered with \\textit{INTEGRAL} on 2003\nFebruary 1-5\n\\citep{2004Tomsick}. Using \\textit{INTEGRAL} observations,\n\\citet{2006Sguera} report an SFXT behaviour of this source and a\npeak-flux of $\\sim\\,$80~mCrab (20-30~keV). Moreover, using\n\\textit{XMM-Newton} and\n\\textit{INTEGRAL} observations, \\citet{2006Walter} report a\npulse period of 1246$\\pm$100~s\nand derive $N_{\\rm H}\\sim10^{23}$~cm$^{-2}$. They also\nproposed 2MASS~J16415078-4532253 as its likely NIR counterpart.\nThe NIR photometry of this counterpart was performed at ESO\/NTT and\nis reported in CHA08. We also found the MIR counterpart in the\nGLIMPSE archives (G339.1889+004889).\n\\newline\n\nWe observed IGR~J16418-4532 with VISIR on 2006 June 29 but did not\ndetect it in any filter. Using The NIR magnitudes given in CHA08, as\nwell as the GLIMPSE fluxes, we fitted its SED, and the\nbest-fitting parameters are $\\textrm{A}_\\textrm{v}$=14.5,\n$\\textrm{T}_*$=32800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$3.77\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is 1.4\/4. The best fit with the\nadditional component gives a larger reduced $\\chi^2$ of 3.9\/2\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. \n\nUncertainties on the data are high, which is\nthe reason why the reduced $\\chi^2$ are\nlow. Nevertheless, parameters of the fit, as well as\nthe 90\\%-confidence ranges of parameters listed in Table 6, are\nconsistent with an O\/B massive star nature. The temperature\nof the additional component being insignificant, we\nconclude this source is an O\/B massive star whose enshrouding\nmaterial marginally contributes to its MIR emission.\n\n\\subsection{IGR~J16465-4507}\n\nIGR~J16465-4507 is a transient source discovered with\n\\textit{INTEGRAL} on 2004\nSeptember 6-7 \\citep{2004Lutovinov}. Observations were carried\nout on 2004 September 14 with \\textit{XMM-Newton}, and\n\\citet*{2004Zurita} report a\nposition with 4$\\arcsec$ accuracy, allowing identification of an\nNIR counterpart in the 2MASS catalogue\n(2MASS~J16463526-4507045=USNO-B1.0~0448-00520455). With the ESO\/NTT,\n\\citet{2005Negueruela} performed intermediate-resolution spectroscopy of\nthe source, estimate the spectral type is a B0.5I, and\npropose that it is an SFXT. Using \\textit{XMM-Newton} and\n\\textit{INTEGRAL}, \\citet{2006Walter} find a pulse period of 227$\\pm$5~s\nand derive $N_{\\rm H}\\sim6\\times10^{23}$~cm$^{-2}$.\nWe found its MIR counterpart in the GLIMPSE archives\n(G340.0536+00.1350) using the 2MASS position.\n\\newline\n\nWe observed IGR~J16465-4507 with VISIR twice: \n\\begin{itemize}\n\\item The first one on 2005\n\tJune 20 during 600~s in PAH1. Typical seeing and airmass were\n\t0\\farcs81 and 1.14. We \n\tdetected the source and the derived flux is 8.7$\\pm$1.8~mJy.\n\\item the second on 2006 June 30 during 1200~s in PAH1 and\n\tPAH2. Typical seeing\n\tand airmass were 0\\farcs68 and 1.09. We \n\tdetected the source in PAH1 but not in PAH2. The derived flux is\n6.9$\\pm$1.1\n\tmJy.\n\t\n\\end{itemize}\nThese observations show that the IGR~J16465-4507 MIR flux was constant\nduring the year.\n\nUsing the USNO-B1.0, 2MASS, and GLIMPSE flux values, as well as our\nVISIR data, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=5.9,\n$\\textrm{T}_*$=25000~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$6.4\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is\n13.9\/7. The best fit with the additional components gives a larger\nreduced $\\chi^2$ of 20\/5\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. \n\nWe then conclude that no additional component is needed to explain\nthe MIR emission of this source, and the parameters derived\nfrom our fit are in good agreement with IGR~J16465-4507 being a\nB0.5I as reported in \\citet{2005Negueruela}.\n\n\\subsection{IGR~J16479-4514}\n\nIGR~J16479-4514 was discovered with\n\\textit{INTEGRAL} on 2003 August 8-9 \\citep{2003Molkov}. \\citet{\n2005Sguera}\nsuggest it is a fast transient after they detected recurrent\noutbursts, and \\citet{2006Sguera} report a peak-flux of\n$\\sim\\,$120~mCrab (20-60~keV). \\citet{2006Walter} observed it\nwith \\textit{XMM-Newton} and gave\nits position with 4$\\arcsec$ accuracy. Moreover, they derive\n$N_{\\rm H}\\sim7.7\\times10^{22}$~cm$^{-2}$ from\ntheir observations. They also\npropose 2MASS~J16480656-4512068=USNO-B1.0~0447-0531332 as its\nlikely NIR counterpart. The NIR spectroscopy and photometry of\nthis counterpart were performed \nat ESO\/NTT and are reported in CHA08. It is shown that its\nspectrum is consistent with an O\/B supergiant. We also\nfound the MIR counterpart in the GLIMPSE archive\n(G339.1889+004889).\n\\newline\n\nWe observed IGR~J16479-4514 with VISIR on 2006 June 29 in PAH1 and\nPAH2, and the exposure\ntime was 1200~s in each filter. Typical seeing and\nairmass were 0\\farcs9 and 1.14. We detected it in both filters, and\nthe respective fluxes are 10.9$\\pm$1.2~mJy and 7.0$\\pm$1.6~mJy.\nUsing the NIR magnitudes given in CHA08, as well as the GLIMPSE\nand the VISIR fluxes, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=18.5,\n$\\textrm{T}_*$=32800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.00\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is 7.4\/6. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 9\/4\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe then do not need any additional component to fit the SED, and\nour result is consistent with IGR~J16479-4514 being an\nobscured O\/B supergiant, in good agreement with CHA08. \n\n\\subsection{IGR~J17252-3616}\n\nIGR~J17252-3616 is a heavily-absorbed persistent source\ndiscovered with \\textit{INTEGRAL} on 2004 February 9 and\nreported in \\citet{2004Walter}. It was observed with\n\\textit{XMM-Newton} on 2004 March 21, and \\citet{2006Zurita} give its\nposition with 4$\\arcsec$ accuracy. Using the\n\\textit{XMM-Newton} observations, as well\nas those carried out with \\textit{INTEGRAL}, they show the source was a\nbinary X-ray pulsar with a spin period of $\\sim\\,$413.7~s and an orbital\nperiod of $\\sim\\,9.72$ days, and derive\n$N_{\\rm H}\\sim1.5\\times10^{23}$~cm$^{-2}$. Moreover, they\nfitted its high-energy spectrum with either an absorbed compton\n(kT$\\sim$ 5.5~keV and $\\tau\\sim7.8$) or a flat power law\n($\\Gamma\\sim0.02$). \n\nIn their paper, they\npropose 2MASS~J17251139-3616575 as its \nlikely NIR counterpart, as do \\citet*{2007Negueruela}. The NIR\nspectroscopy and photometry of\nthis counterpart were performed \nat ESO\/NTT and are reported in CHA08, where it is shown that its\nspectrum is consistent with an O\/B supergiant. Using the\n2MASS position, we searched for its MIR counterpart in the GLIMPSE\ncatalogue. Unfortunately, we did not find its IRAC fluxes\nin the database. Nevertheless, we found post-Basic Calibrated\nData (post-BCD) images of the source in all\nfilters. We then reduced those data and derived fluxes directly from\nthe images. They are listed in Table 3.\n\\newline\n\nWe observed IGR~J17252-3616 with VISIR on 2006 June 30 in PAH1 and\nPAH2 and the exposure\ntime was 1200~s in each filter. Typical seeing and\nairmass were 0\\farcs97 and 1.09. We detected it in PAH1, and the\nderived\nflux is 6.1$\\pm$0.6~mJy. Using the NIR magnitudes given in CHA08,\nas well as the GLIMPSE and the VISIR fluxes, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=20.8,\n$\\textrm{T}_*$=32600~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$7.57\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is 3.8\/5. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 6.9\/3\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. We then do not\nneed any additional component to fit the SED, and\nour result is consistent with IGR~J17252-3616 to be an\nobscured O\/B supergiant, in good agreement with CHA08. \n\n\\subsection{IGR~J17391-3021}\n\nIGR~J17391-3021 is a transient source discovered with\n\\textit{INTEGRAL} on 2003\nAugust 26 \\citep{2003Sunyaev} and it corresponds to the \\textit{Rossi\nX-ray Timing Explorer} (\\textit{RXTE}) source\nXTE J1739-302. \\citet{2005Sguera} analysed archival\n\\textit{INTEGRAL} data and\nclassified the source as a fast X-ray transient presenting a typical\nneutron star spectrum. \\citet{2006Smith} observed it with\n\\textit{Chandra} on\n2003 October 15 and give its precise position with\n1$\\arcsec$ accuracy.\nThey also give its optical\/NIR counterpart\n2MASS~J17391155-3020380=USNO-B1.0~0596-0585865 and classify IGR\nJ17391-3021 as an SFXT. \\citet{2006Negueruelab}\nperformed optical\/NIR photometry and spectroscopy of the companion\nusing ESO\/NTT and find it is a O8Iab(f) star whose distance is\n$\\sim$ 2.3 kpc. CHA08 also report optical and NIR spectroscopy\nand photometry of the companion carried out at ESO\/NTT and\nconfirm the nature of the companion. Using the\n2MASS position, we searched for its MIR counterpart in the GLIMPSE\ncatalogue and as for IGR~J17252-3616, we had to reduce post-BCD data\nand derive the fluxes directly from the images. The fluxes are\nlisted in Table 3.\n\\newline\n\nWe observed IGR~J17391-3021 with VISIR on 2005 June 20 in PAH1 and\nPAH2, and the exposure\ntime was 600~s in each filter. Typical seeing and\nairmass were 0\\farcs63 and 1.13. We detected it in both filters, and\nthe derived fluxes are 70.2$\\pm$1.6~mJy and 46.5$\\pm$2.6~mJy. Using\nthe optical and the NIR magnitudes given in CHA08, as well as\nthe GLIMPSE and\nthe VISIR fluxes,\nwe fitted its SED and the best-fitting parameters are\n$\\textrm{A}_\\textrm{v}$=9.2,\n$\\textrm{T}_*$=31400~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.8\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is 11.7\/10. The best fit with the\nadditional component gives a larger\nreduced $\\chi^2$ of 15.3\/8\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe then do not need any additional component to fit the SED, and the\nparameters derived from our fit are in good agreement with\nIGR~J17391-3021 to be an O8Iab(f) supergiant star, as initially\nreported in \\citet{2006Negueruelab}.\n\n\n\\subsection{IGR~J17544-2619}\n\nIGR~J17544-2619 is a transient source discovered with\n\\textit{INTEGRAL} on 2003\nSeptember 17 \\citep{2003Sunyaev}. \\citet{2004Gonzalez-Riestra}\nobserved it with \\textit{XMM-Newton} and derive\n$N_{\\rm H}\\sim2\\times10^{22}$~cm$^{-2}$. They also confirm the\nassociation of the companion with\n2MASS~J17542527-2619526=USNO-B1.0~0636-0620933, as proposed\nin \\citet{2003Rodriguezb}. \\citet{2005Zand} report on\nobservations performed with \\textit{Chandra}, give its position\nwith 0\\farcs6\naccuracy, $N_{\\rm H}\\sim$ 1.36$\\times10^{22}$~cm$^{-2}$, and show\nthat its high-energy spectrum is typical of an accreting\nneutron star. Moreover, they identify the counterpart as a blue\nsupergiant. \\citet{2006Sguera} report a peak-flux of\n$\\sim\\,$240~mCrab. \nUsing ESO\/NTT, \\citet{2006Pellizza} performed\noptical\/NIR spectroscopy and photometry of the companion, and give\nits spectral type as O9Ib at 2.1-4.2 kpc. Using the 2MASS\nposition, we searched for its MIR counterpart in the GLIMPSE\ncatalogue, and as for IGR~J17252-3616 and IGR~J17391-3021, we had\nto reduce post-BCD data\nand derived the fluxes directly from the images. The fluxes are listed\nin Table 3.\n\\newline\n\nWe observed IGR~J17544-2619 with VISIR on 2005 June 20 in PAH1 and\nPAH2, and the exposure\ntime was 600~s in PAH1 and 1200~s in PAH2. Typical seeing and\nairmass were 0\\farcs64 and 1.13. We detected it in both filters, and\nthe derived fluxes are 46.1$\\pm$2.8~mJy and 20.2$\\pm$2.1~mJy. Using\nthe magnitudes from \\citet{2006Pellizza}, the GLIMPSE and the VISIR fluxes,\nwe fitted its SED, and the best-fitting parameters are\n$\\textrm{A}_\\textrm{v}$=6.1,\n$\\textrm{T}_*$=31000~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.27\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is 6.1\/8. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 9\/6 and $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe then do not need any additional component to fit the SED, and the\nparameters derived from our fit are in good agreement with\nIGR~J17544-2619 as an O9Ib supergiant star, as initially\nreported in \\citet{2006Pellizza}.\n\n\n\\subsection{IGR~J19140+0951}\n\nIGR~J19140+0951 is a persistent source that was discovered with\n\\textit{INTEGRAL} on 2003 March 6-7\n\\citep{2003Hannikainen}. Observations carried out with\n\\textit{RXTE} allowed $\\Gamma\\sim1.6$\nand $N_{\\rm H}\\sim6\\times10^{22}$~cm$^{-2}$ to be derived \n\\citep*{2003Swank}. Timing analysis of the \\textit{RXTE} data\nshowed a period of 13.55 days\n\\citep{2004Corbet}, which shows the binary nature of the\nsource. After a comprehensive analysis of\n\\textit{INTEGRAL} and \\textit{RXTE} data, \\citet{2005Rodriguez}\nshow the source is spending most of\nits time in a faint state but report high variations in luminosity\nand absorption column density (up to $\\sim\\,10^{23}$~cm$^{-2}$). They\nalso find evidence that the compact object is a neutron star rather\nthan a black hole. Using \\textit{Chandra} observations carried\nout 2004 May 11,\n\\citet{2006Zand} give its position with 0\\farcs6 accuracy.\nThis allowed them to find its NIR counterpart in the 2MASS\ncatalogue (2MASS~J19140422+0952577). Moreover, they searched for\nits MIR counterpart in the Mid-course Space Experiment\n\\citep[MSX][]{1994Mill}\nand found an object at 8.3 $\\mu$m. The NIR photometry and\nspectroscopy of this source were performed at\nESO\/NTT and results are reported in CHA08. It is shown that its\nspectrum is consistent with an O\/B massive star, in good\nagreement with \\citet{2007Nespoli}, who show it is a B1I supergiant.\nUsing the 2MASS position, we also found its MIR\ncounterpart in the GLIMPSE archive (G044.2963-00.4688).\n\\newline\n\nWe observed IGR~J19140+0951 with VISIR on 2006 June 30 in PAH1 and\nPAH2, and the exposure\ntime was 1200~s in each filter. Typical seeing and\nairmass were 1\\farcs12 and 1.17. We detected it in both filters,\nand the derived\nfluxes are 35.2$\\pm$1.4~mJy and 19.1$\\pm$1.4~mJy. We point out that\nthe object given as the MSX counterpart of IGR~J19140+0951 in\n\\citet{2006Zand} is a blended source. Indeed, VISIR\nimages, whose resolution is far better, clearly show there are\ntwo sources in the field,\nIGR~J19140+0951 and a very bright southern source (see Fig 2). \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=5cm]{.\/19140.eps}\n\t\\caption{VISIR image of IGR~J19140+0951 in PAH1 (8.59\n\t$\\mu$m). 19\\farcs2x19\\farcs2 field of view and 0\\farcs075\n\tplate scale. We clearly see the two sources that were blended\n\twith MSX. The MIR counterpart of IGR~J19140+0951 is the\n\tnorthern source.}\n\\end{figure}\n\nUsing the magnitudes\ngiven in CHA08, as well as the fluxes from GLIMPSE and our\nobservations with VISIR, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=16.5,\n$\\textrm{T}_*$=22500~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.92\\times10^{-{10}}$, and\nthe reduced $\\chi^2$ is 14.4\/6. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 20.2\/4 and $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe do not need any additional component to fit the SED, and the\nparameters derived from our fit are in good agreement with\nIGR~J19140+0951 to be an B1I supergiant star, as initially\nreported in \\citet{2007Nespoli}.\n\n\n\\begin{figure*}\n\t\\centering\n\t\\begin{tabular}{c c c}\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16195.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16207.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16318.eps}\\\\\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16320.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16358.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16418.eps}\\\\\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16465.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16479.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr17252.eps}\\\\\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr17391.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr17544.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr19140.eps}\\\\\n\t\\end{tabular}\n\t\\caption{\\small Optical-to-MIR absorbed (line) and unabsorbed\n\t(dotted-line) SEDs of 12 \\textit{INTEGRAL} sources,\nincluding broad-band photometric data\n\tfrom ESO\/NTT, 2MASS, GLIMPSE, and VISIR.}\n\\end{figure*} \n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[angle=-90,width=8.5cm]{.\/igr16195_star_dust.eps}\n\t\\includegraphics[angle=-90,width=8.5cm]{.\/igr16318_star_dust.eps}\n\t\\includegraphics[angle=-90,width=8.5cm]{.\/igr16358_star_dust.eps}\n\t\\caption{IR SEDs of IGR~J16195-4945,\n\tIGR~J16318-4848, and IGR~J16358-4726 in the NIR and the MIR. We\n\tshow their SEDs including the contribution of the star and the\n\tdust (line), the star only (dashed-line), and the dust only\n(dotted-line).}\n\\end{figure} \n\n\\section{Discussion}\n\n\n\\subsection{B vs B[e] supergiants stellar winds}\n\nAll SEDs were best-fitted without any dust component (even\nthe very absorbed one like IGR~J16320-4751), except three of them\n(IGR~J16195-4945, IGR~J16318-4848, and\nIGR~J16358-4726, see Fig. 3) that exhibit a MIR excess likely\ndue to the presence of dust in their stellar wind. \n\nBlue supergiants are known to\nexhibit a very strong but sparse stellar wind of high\nvelocity ($\\sim$ 1000-2000~km~s$^{-1}$). This has been\nexplained through the so-called\nradiation line-driven CAK model \\citep*{1975Castor} in which the\nwind is driven by\nabsorption in spectral lines. Hot stars emit most of\ntheir radiation in the ultraviolet (UV) where their atmosphere\nhas many absorption lines. Photons coming from the\nphotosphere of the star with the same wavelength are absorbed and\nre-emitted to the expanding medium in a random direction with\nalmost the same\nmomentum, which results in acceleration of the wind. This process\nis very effective because the line\nspectrum of the scattering ions in the wind is Doppler-shifted\ncompared to the stellar rest frame, so the scattering atoms are\nshifted with respect to their neighbours at lower velocities and\ncan interact with an unaffected part of the stellar spectrum.\n\\newline\n\nIGR~J16318-4848 was proven to belong to a particular class of B1\nsupergiants, the B[e] supergiants or sgB[e]\n\\citep*{2004Filliatre}. \nA physical definition of B[e] stars can be found in\n\\citet{1998Lamers}. We just recall two of the\ncharacteristics here: the presence of forbidden emission lines\nof [\\ion{Fe}{ii}] and [\\ion{O}{i}] in the NIR spectrum and \nof a strong MIR excess due to hot circumstellar dust \nthat re-emits the absorbed stellar radiation through\nfree-free emission.\nAn sgB[e] is defined by the B[e] phenomenon, the\nindication of mass-loss in the optical spectrum (P-cygni\nprofiles), and a hybrid spectrum characterised by the\nsimultaneous presence of narrow low-excitation lines and broad\nabsorption features of high-excitation lines. This hybrid nature\nwas empirically explained by the simultaneous presence of a\nnormal supergiant hot polar wind (fast and sparse) and\nresponsible for the broad lines and a cool equatorial\noutflowing disk-like\nwind (slow and dense) responsible for the narrow lines\n\\citep{1983Shore,1985Zickgraf,1987Shore}. This\nempirical model has received some confirmation from polarimetry\n\\citep*{1999Oudjmaier}.\n\\newline\n\nThere are a few models that explain the creation of this outflowing\ndisk, and all of them consider the\nstar rotation to be an important parameter in the process. In\nthis paper, we present only the most consistent of them,\nthe Rotation\nInduced Bi-stability mechanism (RIB), but a review can be found\nin \\citet*{2006Kraus}. \n\\newline\n\nThe lines responsible for the creation of the wind are dependent\non the ionisation structure, and a change in this structure leads to\na change in the radiative flux. This is the bi-stability jump\nfound by \\citet*{1991Lamers}, which appears\nfor B stars with effective temperatures\nof about 23000~K. Above this temperature, the wind tends to be fast\nand sparse. Below, the mass-loss rate if five times higher and\nthe terminal velocity two\ntimes slower, which leads to a wind that is ten times denser. \n\n\\citet*{1997Cassinelli} propose that\nthe same effect is important from polar to equatorial regions\nfor rapidly\nrotating B stars. Indeed, the rapid rotation leads to polar\nbrightening that increases the poles temperature to the hot\nside of the jump. At the same time, the rotation leading to gravity\ndarkening, the equatorial region may be on the cool side of the\njump. Consequently,\nthe wind in the equatorial region is denser than the\nwind in the polar region. Nevertheless, \\citet{2000Pelupessy}\nshow that the rotational velocity of the star should be very\nclose to its critical value to allow the equatorial wind to reach the\ndensity needed to create the disk. However, supergiant stars\ncannot be close to critical rotational velocity because of\nprobable disruption. Additional mechanisms are therefore\nneeded to allow the supergiant star to reach its\ncritical velocity \\citep[see\ne.g. ][]{2006Owocki}. In the particular case of an sgB[e]\nstar in an X-ray binary system, the spin-up should occur during the\nsupergiant phase of the companion, which indicates a different\nevolutionary stage from other HMXBs.\n\nThis disk itself cannot explain the strong MIR excess the sgB[e] stars\nexhibit. Nevertheless, \\citet*{1993Bjorkman} have shown\nthe existence of a zone in the disk (about 50-60 stellar radii\nfrom the star) in which the temperature is below the\ntemperature of sublimation of the dust (about 1500~K) and the density\nhigh enough to allow for its creation.\n\\newline\n\n\nIGR~J16318-4848 is the source in our sample that \nexhibits the strongest MIR excess, and we believe it is due to\nthe sgB[e] nature of its companion star. Indeed, many other\nstrongly absorbed sources in our sample do not present any MIR\nexcess. \n\nMoreover, it is suggested that IGR~J16358-4726, the second\nsource in our sample that exhibits a MIR\nexcess and whose SED needs an additional component to be properly\nfitted is an sgB[e], because its spectrum has all the\ncharacteristic features of supergiant stars plus the [\\ion{Fe}{ii}]\nfeature (CHA08). Our fit is therefore in good\nagreement with their result, and the only other source of our\nsample that definitely exhibits a MIR excess is indeed an sgB[e]\nstar.\n\nIn the case of IGR~J16195-4945, we are more\ncautious concerning the presence of warm dust that could be\nresponsible for a MIR excess, as we lack data above 8\n$\\mu$m. Indeed, Fig 4. shows that these source could exhibit\na MIR excess, but one much lower than the other two. Nevertheless,\nif this excess were to be confirmed, we\nbelieve it would be also due to the sgB[e] nature of the\ncompanion.\n\\newline\n\nWe would like to point out that, because the dust is\nmost located in an equatorial disk in an\nsgB[e] star, the simple model we used to fit the\nSEDs cannot reproduce the complex distribution of\nthe dust around these stars. Nevertheless, it allows the\ndetection of a warm MIR\nexcess because of the presence of dust in the stellar\nwinds. Finally, for all the stars in our sample, we cannot\nexclude the presence of a cold\ncomponent - responsible for their intrinsic absorption - which we\ncannot detect because of the lack of data above 20 $\\mu$m.\n\n\n\\subsection{Spectral type and distance}\n\n\\begin{table}\n\t\\caption{Summary of spectral types (SpT) and distances\n\t($\\textrm{D}_\\ast$) derived from our fits for confirmed\n\tsupergiant stars in our sample. ($^\\ast$)\n\tsources with an accurate spectral type found in the literature\n\t($^\\dagger$) confirmed supergiant stars whose\ntemperature derived from\n\tour fits was used to assess their accurate spectral type. \nReferences to the determination of the spectral type and\/or\nspectral class of these sources are found in Table 1.}\n\t$$\n\t\\begin{array}{c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{SpT}&\\textrm{D}_\\ast(\\textrm{kpc})\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}^\\ast&\\textrm{sgB[e]}&\\sim1.6\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}^\\dagger&\\textrm{O8I}&\\sim3.5\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}^\\ast&\\textrm{sgB[e]}&\\sim18.5\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}^\\ast&\\textrm{B0.5I}&\\sim9.4\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}^\\dagger&\\textrm{O8.5I}&\\sim4.9\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}^\\dagger&\\textrm{O8.5I}&\\sim6.1\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}^\\ast&\\textrm{O8I}&\\sim2.7\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}^\\ast&\\textrm{O9I}&\\sim3.6\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}^\\ast&\\textrm{B1I}&\\sim3.1\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table} \n\n\\begin{table}\n\t\\caption{Summary of the distances\n\t($\\textrm{D}_\\ast$) derived from our fits for sources\n\twith unconfirmed spectral classes.}\n\t$$ \n\t\\begin{array}{c c ccc}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{SpT}&\\multicolumn{3}{c}{\\textrm{D}_\n\\ast(\\textrm{kpc})}\\\\\n\t\\hline\n\t&&\\textrm{V}&\\textrm{III}&\\textrm{I}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&\\textrm{B1}&\\sim3.1&\\sim5.7&\\sim9.8\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&\\textrm{O7.5}&\\sim1.8&\\sim2.8&\\sim4.1\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&\\textrm{O8.5}&\\sim4.9&\\sim8.3&\\sim13\\\\\n\t\\hline\n\t\\end{array}\n\t$$ \n\\end{table}\n\nIn our sample, six sources are supergiant stars with a known\nspectral type - IGR~J16318-4848 and IGR~J16358-4726 are sgB[e],\nIGR~J16465-4507 is a B0.5I, IGR~J17391-3021 is an 08Iab(f),\nIGR~J17544-2619 is an O9Ib, and IGR J19140+0951 is a B1I - and\nthree are found to be O\/B supergiants whose temperatures derived\nfrom our fits allow an assessment of the spectral types\nusing the classification given in \\citet{2005Martins} and\n\\citet{2006Crowther} for O and B galactic\nsupergiants, respectively, given the uncertainties of\nobservational results\n($\\sim$ 2000~K) and uncertainties on the fits temperatures as given in\nTable 6. We therefore found that IGR~J16320-4751 could be an\n08I and IGR~J16479-4514 and\nIGR~J17252-3616 could be 08.5I stars. \n\\newline\n\nConcerning the last three sources whose spectral class is unknown,\nresults of the fits listed in\nTables 5 and 6 show that they are probably all O\/B massive\nstars, and we also used their derived temperatures to \nassess their spectral type using the\nclassification given in both papers quoted\nabove. IGR~J16195-4945 could be a B1 star, and as already stressed\nabove, it could be an sgB[e] due to its MIR excess, IGR~J16207-5129\nand IGR~J16418-4532 could be O7.5-O8.5 stars. \nNevertheless, even if the high intrinsic X-ray absorption of\ntheir associated compact objects points towards a supergiant\nnature since the\naccretion is likely to be wind-fed, the fits\nthemselves do not allow an assessment of their spectral\nclasses. We then consider they could be either main\nsequence, giant, or supergiant stars.\n\\newline\n\n\\citet*{2006Martins} give a UBVJHK synthetic\nphotometry of galactic OI, OIII, and OV stars, with which one can\nget the expected unabsorbed absolute magnitude in J band\n$\\textrm{M}_{\\rm J}$ for stars having a given spectral\nclassification. Using the absorbed apparent magnitudes\n$\\textrm{m}_{\\rm J}$\nof our sources and the J band absorption we derived\nfrom our fits,\n$\\textrm{A}_{\\rm J}\\,=\\,0.289\\times\\textrm{A}_\\textrm{v}$, it\nis then possible to assess the distance of O stars\nin our sample using the standard relation:\n\n\\begin{displaymath}\n\t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,D_\n\\ast\\,=\\,10^{\\textrm{\\normalsize\n$0.2(m_{\\rm J}-A_{\\rm J}-M_{\\rm J}+5)$}}\\,\\,\\,\\,\\,\\,\\textrm{in\npc}\n\\end{displaymath}\nWe did not find any synthetic\nphotometry for B supergiants. Nevertheless, expected radii of\ngalactic BI, BIII, and BV stars are given in \\citet{1996Vacca},\nand we divided these\nvalues by the $\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$ ratio\nderived from our fits to get the star distance. The derived\ndistances for the sources whose spectral\nclass is known are listed in Table 7, in Table 8 for the others.\n\n\\subsection{X-ray properties}\n\nExcept in the case of IGR~J17544-2619, X-ray absorptions\nare systematically significantly larger than the visible\nabsorptions. This \nindicates the presence in the system of a two-component\nabsorbing material: one \naround the companion star, responsible for the visible\nabsorption, and a very dense one around the compact object\ncoming from the stellar winds that\naccrete onto the compact object and are responsible for the huge\nX-ray absorption those sources exhibit. \n\nThe obscuration of the compact object by the stellar wind is\ncaused by the photoelectric absorption of the X-ray emission by the\nwind, and this absorption varies along the\norbit of the compact object. This orbital dependence has for\ninstance been observed and modelled on 4U\n1700-37 \\citep*{1989Haberl}. \n\nMoreover, an effect on the X-ray absorption by the photoionisation\nof the stellar wind in the vicinity of the compact object by its\nX-ray emission was predicted by \\citet*{1977Hatchett}. \nIndeed, in SGXBs, the compact object moves through the\nstellar wind of the companion star, and the X-rays are\nresponsible for the enhancement or the depletion of the ionised\natoms responsible\nfor the acceleration of the wind\n(e.g. \\ion{C}{iv} and \\ion{N}{v}). \nThis has a direct consequence on the velocity profile of the\nwind; when the wind enters into an ionised zone, it\nfollows a standard CAK law until it reaches a location in which it\nis enough ionised for no further radiative driving to take\nplace, and the wind velocity is ``frozen'' to a constant value\nfrom this point. This results in a lower wind\nvelocity close to the compact object and consequently a higher\nwind density that leads to a higher obscuration of the compact object.\n\\newline\n\nMost of the sources studied in this\nwork are very absorbed in the high-energy\ndomain. Nevertheless, this absorption may not be always that\nhigh. In\nthe case of very wide eccentric orbits, the column density of\nthe sources could\nnormally vary along their orbit and\nsuddenly increase when very close to the companion star because\nof the wind ionisation. In contrast, if these objects were to\nbe always very absorbed, it could mean that their orbit is \nvery close to the companion star and weakly eccentric. If this\neffect were to be observed, we think\nit could explain the difference in behaviour between\nobscured SGXBs (close quasi-circular orbits) and SFXTs (wide\neccentric orbits).\n\n\\subsection{Optical properties}\n\\begin{table}\n\t\\caption{Sample of parameters we used to fit the SEDs of the\n\tisolated supergiants. We give their\n\tgalactic coordinates, their spectral types, the interstellar\n\textinction in magnitudes $\\textrm{A}_\\textrm{i}$ and then the\n\tparameters themselves: the extinction in the optical\n\t$\\textrm{A}_\\textrm{v}$, the temperature\n\t$\\textrm{T}_\\ast$ and the\n\t$\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}$ ratio of the\n\tstar.}\n\t$$\n\t\\begin{array}{c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&l&b&\\textrm{SpT}&\\textrm{A}_\\textrm{i}&\\textrm{A}\n_\\textrm{v}&\\textrm{T}_{\\ast}\n(\\textrm{K})&\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}\\\\\n\t\\hline\n\t\\textrm{HD~144969}&333.18&2.0&\\textrm{B0.5Ia}&3.34&3.9&26000&4.01\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~148422}&329.92&-5.6&\\textrm{B0.5Ib}&0.75&0.9&24700&8.76\n\\times10^{-{11}}\\\\\n\t\\hline\n\t\\textrm{HD~149038}&339.38&2.51&\\textrm{B1Ia}&0.81&1&24000&5.30\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~151804}&343.62&1.94&\\textrm{O8Iaf}&0.83&1.3&32000&4.20\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~152234}&343.46&1.22&\\textrm{B0.5Ia}&1.17&1.5&25100&4.94\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~152235}&343.31&1.1&\\textrm{B1Ia}&3.37&3.9&24500&1.13\n\\times10^{-{9}}\\\\\n\t\\hline\n\t\\textrm{HD~152249}&343.35&1.16&\\textrm{O9Ib}&1.34&1.7&30100&2.89\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~156201}&351.51&1.49&\\textrm{B0.5Ia}&2.68&2.9&26500&2.90\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table} \n\nWe were able to fit all but three sources\nwith a simple stellar black body model. For these three\nsources, we explained that the MIR excess was probably caused by\nthe warm dust created within the stellar wind due to the\nsgB[e] nature of the companions. Therefore, it seems that the\noptical-to-MIR wavelength emission of these SGXBs corresponds\nto the emission of absorbed blue supergiants or sgB[e]. \n\nMoreover, the results of the fits listed in the Table 5 show\nthat it is \\textit{a priori} impossible to\ndifferentiate an obscured SGXB and an SFXT from their\noptical-to-MIR wavelength SEDs, and it then seems that the difference in\nbehaviour between both kinds of SGXBs only depends on the\ngeometry of the system,\ni.e. its orbital distance or its orbit eccentricity \\citep*{2006Chaty}. \n\\newline\n\nNevertheless, to assess a possible effect of the\ncompact object on the companion star, we took a sample of eight\nisolated O\/B supergiants in the direction of the Galactic\ncentre and fitted their optical-to-NIR wavelength SEDs with an absorbed\nstellar black body. The best-fitting parameters are listed in Table\n9 along to their galactic coordinates, their spectral types and\nthe interstellar \\ion{H}{i} absorption (A$_\\textrm{i}$). The\ndistances of these supergiants are known, which allowed\nus to calculate A$_\\textrm{i}$ out to their\nposition using the tool available on the MAST website\n\\citep{1994Fruscione}.\n\nWe see that their visible absorption is\nof the same order of \nmagnitude as the interstellar \\ion{H}{i} absorption and well\nbelow the level of absorption of our sources. This could mean that \nsome supergiant stars in SGXBs exhibit an excess of absorption due to\na local absorbing component. Unfortunately, the total\ninterstellar absorption out to the distance\nof our sources is unknown, and we cannot compare\ntheir visible absorptions derived from our fits to the total\ninterstellar absorption out to their position.\n\nNevertheless, if this was the case, we think that this excess of \nabsorption could also be caused partly by the photoionisation of the\nwind in the vicinity of the companion star by the high-energy\nemission of the compact object, as this would make their winds\nslower than in isolated supergiant stars. Since the wind velocity\nis lower, the medium is denser and suitable for creating a more\nabsorbant material.\n\nIndeed, in the case of persistent sources with very\nclose and quasi-circular orbits, we think that this\npossible effect could be particularly strong, since the wind around\nthe companion star\nwould be permanently photoionised and would have lower\nvelocities than in isolated supergiants. This could be the\ngeneral scheme of obscured SGXBs.\n\nOn the other hand, in the case of very wide and eccentric\norbits, the compact object would be most of the time far from the\nsecondary and its X-ray emission would not photoionise\nthe wind close to the companion star, which would not exhibit any\nvisible absorption excess until the compact object got\ncloser. This could be the general scheme of SFXTs.\n\nAt last, in both cases, it would be possible to observe a\nvariation in the P-Cygni profiles of the companion star\n(i.e. a variation in the wind velocity) with the phase angle of\nthe compact object along its orbit.\n\\newline\n\nAs a possible confirmation of this general behaviour, we point\nout that the visible absorptions derived from our fits for \nthe companion stars of the only sources in our sample that\nsurely exhibit the SFXT\nbehaviour (IGR~J16465-4945, IGR~J17391-3021, and IGR~J17544-2619)\nare far smaller than the visible absorptions of the\nothers. Moreover, concerning obscured SGXBs, the wind velocity\nof IGR~J16318-4848 was found to be $\\sim$ 410~km~s$^{-{1}}$\n\\citep*{2004Filliatre}, far lower than\nthe expected wind velocity for O\/B supergiants ($\\sim$ 1000-2000~km~s$^{-{1}}$).\n\n\\section{Conclusions}\n\nIn this paper, we presented results of observations performed\nat ESO\/VLT with VISIR, which aimed at studying the MIR emission of twelve\n\\textit{INTEGRAL} obscured HMXBs, whose\ncompanions are confirmed or candidate supergiants. Moreover,\nusing the observations performed at ESO\/NTT and reported in the\ncompanion paper (CHA08), previous optical\/NIR observations found\nin the literature and archival data\nfrom USNO, 2MASS, and GLIMPSE, we fitted the broad-band SEDs of\nthese sources using a simple two-component black body model in\norder to obtain their visible absorptions and temperatures, and\nto assess the contribution of their enshrouding material in their\nemission. \n\nWe confirmed that all these sources were likely O\/B supergiant\nstars and that,\nfor most of them, the enshrouding material marginally contributed to the\nemission. Moreover, in the case of IGR~J16318-4848,\nIGR~J16358-4726, and perhaps\nIGR~J16195-4945, the MIR excess\ncould be explained by the sgB[e] nature of the companion stars. \n\nBy comparing the optical and high-energy characteristics of\nthese sources, we showed that the distinction SFXTs\/obscured\nSGXBs does not seem to exist from optical-to-MIR\nwavelength. \nNevertheless, most of the sources in our sample are\nsignificantly absorbed in the optical, and we think that the wind\ncan be denser\naround some supergiants in SGXBs, which could be due to the\nphotoionisation by the high-energy emission \nof the compact object.\n\\newline\n\nSeveral improvements in our study are needed to\nallow definitive conclusions. Indeed, the data used to perform\nthe SEDs were not taken simultaneously, which can for instance\nlead to an incorrect assessment of the MIR excess in the emission.\nMoreover, the lack of optical magnitudes for several sources\ncould have led to an incorrect fitting of their intrinsic visible\nabsorption $\\textrm{A}_\\textrm{v}$.\nFinally, the absence of an accurate measurement of the total\ninterstellar absorption out to the distance of these sources\ndoes not allow us to say whether the\npresence of the compact object can lead to a stellar wind denser\nin some supergiants belonging to SGXBs than in isolated supergiants.\n\\newline\n\nWe then recommend further\noptical investigations of these sources to study any\npossible variation in their P-cygni profile \nwith the phase of the compact object. We also think that the\nmeasurement of the distance of these\nsources is crucial to allow a good assessment of the real\ninterstellar absorption up to their distance, in order to detect\nany local absorbing component around companion stars. Finally,\nwe recommend X-ray monitoring so as to study the\ndependence of their column density on orbital phase angle,\nwhich could help for understanding the difference between obscured\nSGXBs and SFXTs.\n\n\\begin{acknowledgements}\n\t\n\tWe are pleased to thank J\\'er\\^ome Rodriguez for his very\n\tuseful website in which all the \\textit{INTEGRAL} sources are\n\treferenced (\n\thttp:\/\/isdc.unige.ch\/$\\sim$rodrigue\/html\/igrsources.html).\n\t\\newline\n\tBased on observations carried out at the European Southern\n\tObservatory, Chile (through programmes ID. 075.D-0773 and\n\t077.D-0721). This research has made use of NASA's Astrophysics\n\tData System, of the SIMBAD and VizieR databases operated at\n\tthe CDS, Strasbourg, France, of products from the US Naval\n\tObservatory catalogues, of products from the Two\n\tMicron All Sky Survey, which is a\n\tjoint project of the University of Massachusetts and the\n\tInfrared Processing and Analysis Center\/California Institute\n\tof Technology, funded by the National Aeronautics and Space\n\tAdministration and the National Science Foundation as well as\n\tproducts from the Galactic Legacy Infrared Mid-Plane Survey\n\tExtraordinaire, which is a \\textit{Spitzer Space Telescope}\n\tLegacy Science Program.\n\\end{acknowledgements}\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of  Quantum Chromodynamics (QCD) at high temperature\nand\/or baryon density is certainly one of the most active and interesting research topic of\nmodern high energy physics. It has been predicted by means of  first principles calculations \nthat a smooth crossover exists from the low temperature hadron gas phase to a quark-gluon plasma,\nat a (pseudo)-critical temperature $T_c\\approx 150$ \nMeV \\cite{Borsanyi:2010bp,Borsanyi:2010cj,Cheng:2009zi,Bazavov:2011nk,Borsanyi:2013bia}.\nThis crossover is accompanied by the approximate restoration of chiral symmetry.  \nWhile many studies focused on the QCD phase structure in the infinite volume limit,\nit is of a certain interest to investigate the effects of finite size on the critical lines of QCD\nin the temperature-baryon chemical potential plane.\nChiral symmetry restoration at finite temperature and density, for quantization \nin a cubic box of size $L$ with periodic or antiperiodic\nboundary conditions as well as with standing waves conditions, has been treated in a number of studies,\nsee \\cite{Xu:2019gia,Xu:2019kzy,Almasi:2016zqf,Palhares:2009tf,Braun:2011iz,Braun:2005fj,Kiriyama:2006uh,Braun:2005gy,Wang:2018ovx,\nAbreu:2019czp,Magdy:2019frj,Aoki:2002uc,Guagnelli:2004ww,Orth:2005kq,Deb:2020qmx,Zhao:2019ruc,Wang:2018qyq,\nEbert:2010eq,Bhattacharyya:2015zka,Bhattacharyya:2014uxa,Bhattacharyya:2012rp}, \nsee \\cite{Klein:2017shl} for a review. \n  \n\nWe perform a study of chiral symmetry breaking in two-flavor QCD using the renormalized quark-meson (QM) \nmodel \\cite{Skokov:2010sf,Frasca:2011zn,Ruggieri:2013cya},\nwith quantization in a cubic box of size $L$ with periodic boundary conditions. \nOur work differs from previous calculations based on the same model in the way we treat the vacuum quark loop.\nIn fact, we include the vacuum term in the thermodynamic potential:\nsince this is a divergent quantity, care should be put in the regularization and then renormalization of this contribution. \nWe follow the regularization procedure of \\cite{Xu:2019gia} with Pauli-Villars regulators; \nthis regularization is suitable for calculations at finite size\nbecause it cancels the ultraviolet divergences in a very transparent manner and leaves a finite quark loop\nboth in the infinite and the finite $L$ cases. After the regularization has been done,\nwe perform renormalization requiring\nthat in the infinite volume limit, the quark loop does not shift the expectation value of the condensate\nand of the $\\sigma-$meson mass obtained from the classical potential.\nApplying these conditions fixes the counterterms in a way that does not depend of $L$.\nTherefore, the predictions of the model at finite $L$ are unaffected by the ultraviolet divergence.\n\nThe main purpose of this study is the restoration of chiral symmetry at finite quark chemical potential, $\\mu$,\nand low temperature. When the size is small enough, $L\\lesssim 5$ fm, \nan intermediate phase appears between the chiral symmetry broken phase (the hadron gas) \nand chiral symmetry restored phase (normal quark matter). In this intermediate phase,\nthe condensate experiences a decrease but its lowering is not enough to restore chiral symmetry.\nTherefore, the symmetry breaking pattern in this new phase is the same of the hadron gas.\nHowever, this phase differs from the hadron gas phase because it has a nonzero baryon density;\nthe change of density, together with the unchanged symmetry breaking pattern, is reminiscent of a\ngas-to-liquid phase transition.\nMoreover, the correlation domains of the order parameters are larger than those found in the hadron gas\nand normal quark matter phases. \nFor all these reasons, we call this new phase the \n{\\em subcritical liquid phase}. \n\n\nWe also compute $T_c$ versus $L$. Our results  agree with previous studies that implemented effective models\nwith periodic boundary conditions or infrared cutoffs  \\cite{Xu:2019gia,Palhares:2009tf,Wang:2018ovx,Magdy:2019frj}, \nnamely $T_c$ increases with $1\/L$.\nThis is understood in terms of the zero mode that contributes to the  condensate when periodic\nboundary conditions are used,\nas well as of the curvature of the thermodynamic potential which is negative for any $L$\nin the QM model.\n\nThe plan of the article is as follows. In Section II we briefly review the quark-meson model\nand describe the renormalization procedure when periodic boundary conditions are implemented. \nIn Section III we discuss the chiral phase transition at small $T$ and large $\\mu$.\n In Section IV we present a few results for the chiral restoration at finite $T$. \n Finally, in Section V we draw our conclusions.\nWe use the natural units system $\\hbar=c=k_B=1$ throughout this article.\n\n\n\n\n\n\n\\section{The quark-meson model with periodic boundary conditions}\n\n\\subsection{The lagrangian density}\n\nThe QM model is an effective model of QCD with quarks, $\\sigma$ and $\\pi$ mesons\n(in the two-flavor versions that we consider here).\nThe meson lagrangian density is\n\\begin{eqnarray}\n{\\cal L}_\\mathrm{mesons} &=& \\frac{1}{2}\\left(\n\\partial^\\mu\\sigma\\partial_\\mu\\sigma + \\partial^\\mu\\bm\\pi\\cdot\\partial_\\mu\\bm\\pi\n\\right) \\nonumber\\\\\n&&- \\frac{\\lambda}{4}\\left(\\sigma^2 + \\bm\\pi^2 - v^2\\right)^2 +h\\sigma,\\label{eq:ls1_aa}\n\\end{eqnarray}\nwhere $\\bm\\pi = (\\pi_1,\\pi_2,\\pi_3)$ corresponds to the pion isotriplet field. This lagrangian density is\ninvariant under $O(4)$ rotations. On the other hand, as long as $v^2 > 0$ the potential\ndevelops an infinite set of degenerate minima. We choose one ground state, namely\n\\begin{equation}\n\\langle\\bm\\pi\\rangle=0,~~\\langle\\sigma\\rangle\\neq 0.\\label{eq:gs_aa}\n\\end{equation}\nThe ground state~\\ref{eq:gs_aa}\nbreaks the $O(4)$ symmetry down to $O(3)$ since the vacuum is invariant only under the rotations of the pion fields.\nThis is how chiral symmetry is spontaneously broken in this model.\nBesides the spontaneous breaking, chiral symmetry is broken softly but explicitly by the term\n$h\\sigma$ in the lagrangian density; in fact, the pion mass is $m_\\pi^2 = h\/F_\\pi$.\nAt zero temperature and in the chiral limit  $\\langle\\sigma\\rangle=F_\\pi\\approx 93$ MeV \nwhere $F_\\pi$ denotes the pion decay constant in the vacuum.\n\n\nThe quark sector of the QM model is described by the lagrangian density\n\\begin{equation}\n{\\cal L}_\\mathrm{quarks} = \\bar\\psi\\left(\ni\\partial_\\mu\\gamma^\\mu - g(\\sigma +i\\gamma_5 \\bm\\pi\\cdot\\bm\\tau)\n\\right)\\psi,\\label{eq:qlg_aaa}\n\\end{equation}\nwhere $\\bm\\tau$ are Pauli matrices in the flavor space. In the ground state~\\ref{eq:gs_aa} quarks get a dynamical (that is,\na constituent) mass given by\n\\begin{equation}\nM = g\\langle\\sigma\\rangle.\\label{eq:pppAAA}\n\\end{equation}\n We notice that in Eq.~\\ref{eq:qlg_aaa} there is no explicit mass term for the quarks. As a matter of fact,\nin this effective model the explicit breaking of chiral symmetry   is achieved by\n  $h\\neq 0$ in Eq.~\\ref{eq:ls1_aa}.\n Although in Eq.~\\ref{eq:qlg_aaa}  there is no explicit mass term, quarks get a constituent mass because of the spontaneous\nbreaking of the $O(4)$ symmetry in the meson sector: this implies that the quark chiral condensate can be nonzero.\nThe total lagrangian density is given by\n\\begin{equation}\n{\\cal L}_\\mathrm{QM} = {\\cal L}_\\mathrm{quarks} + {\\cal L}_\\mathrm{mesons}.\n\\end{equation}\nIn the following, we will use the notation $\\sigma$ to denote both the field and its expectation value,\nunless from the context it is not clear which of the two we write about.\n\n\n\\subsection{Renormalized thermodynamic potential in the infinite volume limit}\n\n \n\nThe mean field effective potential of the QM model in the infinite volume is given by\n\\begin{equation}\n\\Omega = U + \\Omega_{0,\\infty} + \\Omega_T,\\label{eq:ep1aa}\n\\end{equation}\nwhere\n\\begin{equation}\nU = \\frac{\\lambda}{4}\\left(\\sigma^2 + \\bm\\pi^2 - v^2\\right)^2 - h\\sigma\\label{eq:ls1_aaMMM}\n\\end{equation}\nis the classical potential of the meson fields as it can be read from Eq.~\\ref{eq:ls1_aa}, and\n\\begin{equation}\n\\Omega_{0,\\infty} = -2N_c N_f\\int\\frac{d^3p}{(2\\pi)^3} E_p  \\label{eq:ls1_aaMMMa}\n\\end{equation}\nis the one-loop quark contribution, with\n\\begin{equation}\nE_p = \\sqrt{p^2 +M^2},~~~M=g\\sigma.\n\\end{equation}\nFinally, $\\Omega_T$ corresponds to the finite temperature quarks contribution that we specify later.\nFor regularization and renormalization we can limit ourselves to consider the zero temperature limit of $\\Omega$,\n\\begin{equation}\n\\Omega_0 = U + \\Omega_{0,\\infty}.\\label{eq:ep1}\n\\end{equation}\nEquation~\\ref{eq:ep1} represents the effective potential for the $\\sigma$ field computed at one-loop\nand after renormalization it corresponds to the renormalized condensation energy, namely the difference between\nthe energy of the state with $\\langle\\sigma\\rangle\\neq 0$ and $\\langle\\sigma\\rangle= 0$ at $T=\\mu=0$.\n\nIt is instructive to present the renormalization of $\\Omega_0$ in the infinite volume system firstly.\nIn order to do this, we have to regularize the momentum integral in Eq.~\\ref{eq:ls1_aaMMMa}.\nWe have found that for problems with quantized momenta, in which summations replace integrals, \nrenormalization based on the Pauli-Villars (PV) method is the most convenient one.\nTherefore, also in the infinite volume limit we use PV regularization and renormalization.\nThis is the same method used in \\cite{Xu:2019gia}.\n\nIn the PV scheme we replace Eq.~\\ref{eq:ls1_aaMMMa} with\n\\begin{equation}\n\\Omega_{0,\\infty} = -2N_c N_f\\int\\frac{d^3p}{(2\\pi)^3} \n\\sum_{j=0}^3c_j \\left(E_p^2 + j \\xi^2\\right)^{1\/2},  \\label{eq:ls1_aaMMMaPV}\n\\end{equation}\nwhere $\\{c_j\\}$ is a set of PV coefficients and $\\xi$  is the renormalization scale. \nThe integration is understood cut at the scale $p=\\Lambda$.\nThe coefficient $c_0=1$ by convention;\nthe additional three coefficients are needed to remove the quartic, quadratic and log-type divergences of $\\Omega_{0,\\infty}$\nthat appear in the limit $\\Lambda\\rightarrow\\infty$.\nThe PV coefficients\nwill be chosen so the aforementioned divergences cancel and the final expression does not depend on $\\Lambda$.\nPerforming the integration, it is an easy exercise to see that the choice $c_1=-3$, $c_2=3$ and $c_3=-1$ is\nenough to cancel all the divergences, in agreement with \\cite{Xu:2019gia}. The resulting finite expression\nis\n\\begin{eqnarray}\n\\Omega_{0,\\infty} &=& \\frac{3N_c N_f}{16\\pi^2}\n\\xi^4\\log\\frac{(M^2 + \\xi^2)(M^2 + 3\\xi^2)^3}{(M^2 +2 \\xi^2)^4}\\nonumber\\\\\n&+&\\frac{6N_c N_f}{16\\pi^2}\n\\xi^2 M^2\\log\\frac{(M^2 + \\xi^2)(M^2 + 3\\xi^2)}{(M^2 +2 \\xi^2)^2}\\nonumber\\\\\n&+&\\frac{N_c N_f}{16\\pi^2}\nM^4\\log\\frac{(M^2 + \\xi^2)^3(M^2 + 3\\xi^2)}{M^2(M^2 +2 \\xi^2)^3}.\\nonumber\\\\\n&&\\label{eq:desperate12}\n\\end{eqnarray}\n\nAlthough Eq.~\\ref{eq:desperate12} is finite, it potentially can shift the location of the minimum of the classical potential\nas well as the mass of the $\\sigma-$meson in the vacuum, $m_\\sigma$.\nWhile this would be not a problem since it would require a mere change of the parameters $\\lambda$ and $v$,\nit is easier to work assuming that the quark loop does not shift these quantities.\nTo this end, we add two counterterms,\n\\begin{equation}\n\\Omega_\\mathrm{c.t.} = \\frac{\\delta v}{2}M^2 + \\frac{\\delta\\lambda}{4}M^4,\\label{eq:ctUUU}\n\\end{equation}\nand we impose the renormalization conditions\n\\begin{eqnarray}\n&&\\left.\\frac{\\partial (\\Omega_{0,\\infty} + \\Omega_\\mathrm{c.t.})}{\\partial M}\\right|_{M=gF_\\pi}=0,\\label{eq:sh4aF}\\\\\n&&\\left.\\frac{\\partial^2 (\\Omega_{0,\\infty} + \\Omega_\\mathrm{c.t.})}{\\partial M^2}\\right|_{M=gF_\\pi}=0.\\label{eq:sh5aF}\n\\end{eqnarray}\nThe first condition imposes  that the quark loop does not change the\nlocation of the minimum of the classical potential, $\\sigma=F_\\pi$,\nwhile the second states that the loop does not shift $m_\\sigma$.\nThe coefficients of the two counterterms can be computed easily,\n\\begin{eqnarray}\n\\delta v &=& -\\frac{3N_c N_f}{4\\pi^2}\\xi^2\\log\\frac{(g^2F_\\pi^2 + \\xi^2)(g^2F_\\pi^2 + 3\\xi^2)}{(g^2F_\\pi^2 + 2\\xi^2)^2},\n\\label{eq:ctdv1}\\\\\n\\delta\\lambda &=&-\\frac{N_c N_f}{4\\pi^2}\n\\log\\frac{(g^2F_\\pi^2 + \\xi^2)^3(g^2F_\\pi^2 + 3\\xi^2)}{g^2F_\\pi^2(F_\\pi^2 + 2\\xi^2)^3}.\\label{eq:ctdl1}\n\\end{eqnarray}\nThe renormalized quark loop is thus given by\n\\begin{equation}\n\\Omega_{0,\\infty}^\\mathrm{ren} = \\Omega_{0,\\infty} + \\Omega_\\mathrm{c.t.}.\n\\label{eq:rty}\n\\end{equation}\n\n\\subsection{Renormalized thermodynamic potential in the finite volume case}\nIn a finite volume $V=L^3$ the $i-$component of momentum is quantized according to\n\\begin{equation}\np_i = \\frac{2\\pi}{L}n_i,~~~n_i=0,\\pm1,\\pm2,\\dots; \\label{eq:mom_quant}\n\\end{equation}\nthis leads to the obvious replacements\n\\begin{eqnarray}\n\\int \\frac{d^3p}{(2\\pi)^3} &\\rightarrow &\\frac{1}{V}\\sum_{n_x,n_y,n_z,}, \\label{eq:mom_repl}\\\\\nE_p &\\rightarrow &E_n = \\left(\nM^2 + \\frac{4\\pi^2}{L^2}n\n\\right)^{1\/2},\\label{eq:Ep_repl}\n\\end{eqnarray}\nwith $n=n_x^2 + n_y^2 + n_z^2$.\nInstead of $\\Omega_{0,\\infty}$ we have  \n\\begin{eqnarray}\n\\Omega_{0,L}(\\Lambda) &=&-\\frac{2N_c N_f}{L^3}|M|\n\\nonumber\\\\\n&& -\\frac{2N_c N_f}{L^3}\\sum_{n=1}^{a}r_3(n)\\sqrt{\\frac{4\\pi^2}{L^2}n+M^2},\n\\label{eq:alpha_2}\n\\end{eqnarray}\nwhere the subscript $L$ reminds that the potential is computed assuming quantization in a box with volume $L^3$;\nthe first addendum on the right hand side of Eq.~\\ref{eq:alpha_2} is the zero mode contribution.\nWe have put $a=\\Lambda^2 L^2\/4\\pi^2$ \nwhere $\\Lambda$ is an UV cutoff that will disappear after \nPV renormalization; \n$r_3(n)$ denotes the sum-of-three-squares function, \nthat counts how many ways it is possible to form $n$ as the sum of the squares of three integers: \nit corresponds to the degeneracy of the level with a given $n$.\nSimilarly, instead of Eq.~\\ref{eq:ep1} we have\n\\begin{equation}\n\\Omega_0(\\Lambda) = U + \\Omega_{0,L}(\\Lambda).\\label{eq:ep1_L}\n\\end{equation}\nThis is the bare potential that is divergent and needs renormalization:\nto make this evident we have made the dependence of $\\Lambda$ explicit. \n\n \n \n \nThe  renormalization  of the thermodynamic potential is performed\nfollowing the PV method delineated in the infinite volume case.\nFirstly, we introduce a set of PV coefficients and \nreplace Eq.~\\ref{eq:alpha_2} with\n\\begin{eqnarray}\n\\Omega_{0,L} &=& -\\frac{2N_c N_f}{L^3}|M|\\nonumber\\\\\n&&-\\frac{2N_c N_f}{L^3}\\sum_{n=1}^{a}r_3(n)\n\\sum_{j=0}^3 c_j\\sqrt{E_n^2 + j \\xi^2}.\n\\label{eq:alpha_2a}\n\\end{eqnarray}\nUsing the coefficients determined in the infinite volume limit is enough to get a finite expression in the $\\Lambda\\rightarrow\\infty$\nlimit. This can be proved by brute force numerically, but can also be understood analytically as follows.\nFor studying the UV divergence, we put  $X_j^2 = M^2 + j\\xi^2$ \nand we extract the $O(X_j^2)$ and $O(X_j^4)$ terms from  $\\Omega_{0,L}$,\nthat will bring a quadratic and log-type divergence respectively. Thus we can write\n\\begin{eqnarray}\n\\Omega_{0,L} &=& -\\frac{2N_c N_f}{L^3}|M|\\nonumber\\\\\n&&-\\frac{2N_c N_f}{L^3}\\sum_{n=1}^{a}r_3(n)\n\\sum_{j=0}^3 c_j\\left[a_2 X_j^2 + a_4 X_j^4\\right]\\nonumber\\\\\n&&+~\\mathrm{UV~finite~terms},\n\\label{eq:alpha_2a2}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\na_{2} &=&\\frac{L}{4\\pi n^{1\/2}},\\\\\na_{4} &=& -\\frac{L^3}{64\\pi^3 n^{3\/2}}.\n\\end{eqnarray}\nBy virtue of a numerical calculation we prove that in the large $a$ limit\n\\begin{eqnarray}\n\\sum_{n=1}^a \\frac{r_3(n)}{4\\pi n^{1\/2}} &\\approx &\\frac{a}{2},\\\\\n\\sum_{n=1}^a \\frac{r_3(n)}{4\\pi n^{3\/2}} &\\approx & \\frac{\\log a}{2},\n\\end{eqnarray}\nwhich allow to write\n\\begin{eqnarray}\n\\Omega_{0,L} &=& -\\frac{2N_c N_f}{L^3}|M|\\nonumber\\\\\n&&-\\frac{2N_c N_f}{L^3} \n\\sum_{j=0}^3 c_j\\left[a_2 X_j^2\\frac{L a}{2} - a_4 X_j^4\\frac{L^3 \\log a}{32\\pi^2}\\right]\\nonumber\\\\\n&&+~\\mathrm{UV~finite~terms}.\n\\label{eq:alpha_2a3}\n\\end{eqnarray}\nThe above equation shows that $a_2$ and $a_4$ multiply the quadratic and log-type divergence respectively. \nUsing the PV coefficients it is easy to prove that \n\\begin{eqnarray}\n\\sum_{j=0}^3 c_j X_j^2 &=& 0,\\\\\n\\sum_{j=0}^3 c_j X_j^4 &=& 0,\n\\end{eqnarray}\nwhile the $O(X_j^6)$ term is nonzero and UV-finite. \nTherefore, the PV regulator cancels the UV divergence of $\\Omega_{0,L}$\nleaving a UV-finite, $\\xi-$dependent term.\n\nThe counterterms that we have fixed in the infinite volume case can be used here as well:\nthey will implement the conditions that in the large volume limit, we recover\n$\\sigma=F_\\pi$ and the $m_\\sigma$ fixed by the classical potential.\nOn the other hand, for a finite $L$ the $\\Omega_{0,L}$ can shift both the location\nof the minimum of the total potential and $m_\\sigma$. Therefore,\nfor finite $L$ we will use\n\\begin{equation}\n\\Omega_{0,L}^\\mathrm{ren} = \\Omega_{0,L} + \\Omega_\\mathrm{c.t.},\n\\label{eq:rtyAA}\n\\end{equation}\nwith $\\Omega_\\mathrm{c.t.}$ specified bt Eq.~\\ref{eq:ctUUU} with counterterms\ngiven by Eqs.~\\ref{eq:ctdv1} and~\\ref{eq:ctdl1}. \nTaking into account the classical potential, the renormalized thermodynamic potential\nin the vacuum at finite $L$ is\n\\begin{equation}\n\\Omega^\\mathrm{ren} = U + \\Omega_{0,L}+\\Omega_\\mathrm{c.t.}.\n\\label{eq:renpot_1}\n\\end{equation}\n\nWe close this subsection with a short comment on the choice of $\\xi$. \nIn principle, we could change this arbitrarily  at a given $L$\nby requiring that the total derivative of $\\Omega$ with respect to $\\xi$, $d\\Omega\/d\\xi$,  is zero. \nThis would amount to solve a \nRenormalization Group-like equation in which the $\\partial\\Omega\/\\partial\\xi$ is balanced by terms\nproportional to $\\partial\\lambda\/\\partial\\xi$ and $\\partial g\/\\partial\\xi$ so that $d\\Omega\/d\\xi=0$. \nSolving this equation is well beyond the purpose of the study we want to do here.\nTherefore, for a given $L$ we have limited ourselves to inspect the ranges of $\\xi$ that do not change $\\Omega$ \ntoo much. For large $L$ we have found that $\\xi$ can be arbitrarily large. \nOn the other hand, for small $L$ we have found that $\\Omega$ is quite insensitive to the specific value of $\\xi$  \nas long as $\\xi \\lesssim \\gamma F_\\pi $ with $\\gamma=O(1)$. Therefore, we  fix $\\xi=F_\\pi$ in this work.\n\n\n\n\\subsection{The total thermodynamic potential}\nThe finite temperature thermodynamic potential does not need any particular treatment:\nin infinite volume it is given by the standard relativistic fermion gas contribution, namely\n\\begin{equation}\n\\Omega_T = -2N_c N_f T\\sum_{s=\\pm 1} \\int\\frac{d^3p}{(2\\pi)^3}\\log\\left(1 + e^{-\\beta(E_p-s\\mu)}\\right),\n\\end{equation}\nwhere $\\mu$ corresponds to the chemical potential. \nIn finite volume we replace the above equation with\n\\begin{equation}\n\\Omega_T = -2N_c N_f \\frac{T}{L^3}\\sum_{s=\\pm 1}\\sum_{n}r_3(n)\n\\log\\left(1 + e^{-\\beta(E_n-s\\mu)}\\right).\\label{eq:iow}\n\\end{equation}\nPutting all together, we get the renormalized thermodynamic potential of the QM model in a volume $V=L^3$\nwith periodic boundary conditions, that is\n\\begin{equation}\n\\Omega = U + \\Omega_{0,L}+\\Omega_\\mathrm{c.t.}+ \\Omega_T.\n \\label{eq:tot_om_fs}\n\\end{equation}\nFor each value of $\\mu$ and $T$ we determine $\\sigma$ by looking for the global minimum of $\\Omega$. \n\n\\subsection{Parameters of the classical potential}\nThe \nrenormalization procedure outlined above has the advantage that does not require a shift of the \nparameters of the classical potential both in the infinite volume and in the finite size cases. \nTherefore, these parameters can be computed from $U$ and are not affected by $L$.\nThese can be computed easily from the conditions that $\\partial U\/\\partial\\sigma=0$ and\n$\\partial^2 U\/\\partial\\sigma^2 = m_\\sigma^2$, where the derivatives are understood \ncomputed at $\\sigma=F_\\pi$.\nLimiting ourselves to write concrete expressions in the limit $h\\rightarrow 0$ we find\n\\begin{eqnarray}\nv &=& F_\\pi  - \\frac{h}{m_\\sigma^2},\\\\\n\\lambda  &=& \\frac{m_\\sigma^2 }{2 F_\\pi^2 } - \\frac{h}{2F_\\pi^3 }.\n\\end{eqnarray} \n \n\n\n\n \n\n\n\n\n\n\\section{Chiral phase transition at small temperature\\label{Sec:llp}}\n\nIn this section we present the results for the \ncondensate and phase structure at low temperature\nand high density, which is the domain in which the most interesting finite size effects appear. \nOur parameters set is $m_\\sigma=700$ MeV, $F_\\pi=93$ MeV,\n$m_\\pi=138$ MeV, $h=F_\\pi m_\\pi^2$.\nFinally,  we take $\\xi=F_\\pi$ and we assume $M=g F_\\pi=335$ MeV at $\\mu=T=0$ which gives $g=3.6$.\n\n\n\n\n\n\\subsection{Condensate, $m_\\sigma$ and $m_\\pi$ versus size in the vacuum} \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{vac_collect.png}\n\\end{center}\n\\caption{\\label{Fig:vacvac}Quark mass \nand $m_\\sigma$ versus $L$ in the vacuum. \nBoth quantities are measured in units of the infinite volume cases.\nRenormalization scale is $\\xi=F_\\pi$.}\n\\end{figure}  \n\n\nTo begin with, we present the behavior of the condensate, of the $\\sigma-$meson mass and pions mass\nversus $L$ in the vacuum: the results are summarized in Fig.~\\ref{Fig:vacvac}.\nAll quantities are measured in units\nof the infinite volume cases.  We find that both $M$ and $m_\\sigma$ increase with lowering $L$,\nwhile $m_\\pi$ decreases.\nThe effects of a finite $L$ on the physical quantities become noticeable for $L\\lesssim 3$ fm.\nQualitatively the results of the renormalized QM model agree with   \nthe NJL model calculations \\cite{Xu:2019gia}.  \n\n\n\n\n     \n\n\n\\subsection{Transitions at small temperature}\n\n \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{cond_m_L3.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{nb_m_L3.png}\n\\end{center}\n\\caption{\\label{Fig_cc} Condensate (upper panel)\nand $n_B\/\\rho_0$ (lower panel) versus chemical potential, for several values of $T$.\nCalculations correspond to $L=3$ fm.}\n\\end{figure}\n\nIn the upper panel of Fig.~\\ref{Fig_cc} we plot the condensate versus $\\mu$ for several temperatures\nand for $L=3$ fm. The condensate has been computed by a global minimization procedure of $\\Omega$\nfor any $(\\mu,T)$.\nFor $T=0$  $\\sigma$ has a discontinuity for $\\mu\\equiv\\mu_1\\approx 340$ MeV and drops down \nfrom its value in the vacuum \nto a smaller, still substantial value $\\sigma_1\\approx 93$ MeV.\nThis discontinuity agrees with the one found in \\cite{Xu:2019gia} where it has been shown to be driven by the zero mode\n(a direct calculation within the QM model confirms this).\nDespite the discontinuity of $\\sigma$, chiral symmetry is still spontaneously broken \nfor $\\mu>\\mu_1$ since the value of the condensate\nis still large. \nIncreasing $\\mu$ up to a second critical value $\\mu\\equiv\\mu_2\\approx 526$ MeV there is another\njump of $\\sigma$ to a $\\sigma_2\\approx 5$ MeV:\nit is fair to identify this discontinuity with the restoration of chiral symmetry. \nIt is easy to verify that the phase transition happens when the chemical potential is large enough\nto populate the first excited state, $n=1$: in fact,\nusing $\\sigma =\\sigma_2$  the energy of this state is\n$E_1 = \\sqrt{g^2\\sigma^2 + 4\\pi^2\/L^2}\\approx 415$ MeV, therefore the state $n=1$ can be populated\nfor $\\mu \\gtrsim \\mu_2$.\nIncreasing the temperature, the chiral phase transition and the jump of the condensate approach each other;\nmoreover, the discontinuity of $\\sigma$ at $\\mu=\\mu_1$ is smoothed by temperature becoming a crossover.\n\n\nAt low enough $L$ and low $T$  an intermediate phase appears between the chiral symmetry breaking phase\nat low $\\mu$, namely the hadron gas, and the high density phase that is quark matter in which chiral symmetry\nis restored. Even though the passage from the chiral symmetry broken phase to the intermediate one is not \na phase transition because symmetries are broken in the same way in the two phases, for the sake of simplicity we   \nadopt the term transition to discuss also the change of the condensate for $\\mu=\\mu_1$.\nIn fact, we aim to interpret this transition as a gas-to-liquid phase transition.\n\nIt is interesting to examine the behavior of number density around the two transitions.\nTo this end we define the baryon density, $n_B$, as\n\\begin{equation}\nn_B = \\frac{n_u + n_d}{3},\\label{eq:bnm3} \n\\end{equation}\nwhere $n_{u}$ and $n_d$ denote the densities of $u$ and $d$ quarks respectively; it is straightforward to prove that \n\\begin{equation}\nn_B =  -\\frac{1}{3}\\frac{\\partial\\Omega}{\\partial\\mu}.\\label{eq:bnm_inter}\n\\end{equation}\nIn the lower panel of Fig.~\\ref{Fig_cc} we plot $n_B$ versus $\\mu$ for several values of $T$ and $L=3$ fm;\nbaryon density is measured in units of the nuclear saturation density, $\\rho_0= 0.16$ fm$^{-3}$.\nAt small temperature, $n_B$ experiences a first jump from zero to $n_B\\approx 0.95\\rho_0\\equiv n_B^{(1)}$\nfor $\\mu=\\mu_1$, stays constant then experiences another jump to $n_B\\approx 6.46\\rho_0\\equiv n_B^{(2)}$\nfor $\\mu=\\mu_2$.\n \nThe dependence of $n_B$ on $\\mu$ at small temperature can be easily understood.\nAs a matter of fact, at zero (as well as very small but finite) temperature,\nif $\\mu$ is large enough to excite the zero as well as the first mode,\n we have\n\\begin{equation}\nn_B \\approx \\frac{2N_c N_f}{3V}\n\\left[\n\\theta(\\mu-M) + 6\\theta(\\mu-\\sqrt{M^2 + 4\\pi^2\/L^2})\n\\right];\\label{eq:bnm5}\n\\end{equation}\nthe first addendum in the right hand side of the above equation is the contribution of the zero mode,  \nwhile the second addendum\ncorresponds to the first excited state \ncounted with its degeneracy $r_3(1)=6$. \nBaryon density is constant for \n$M <\\mu < \\sqrt{M^2 + 4\\pi^2\/L^2}$ where only the zero mode contributes;\nanalogously, density is constant also for larger values of $\\mu$ until the second excited state can be populated.\nThis is qualitatively different from the behavior $n_B \\propto (\\mu-M)^{3\/2}$ of a \nrelativistic ideal massive gas, because for a finite size system there are only discrete modes in the spectrum,\nand if temperature is low enough only few of them can be occupied giving rise to $\\Omega_T\\propto\\mu$.\nOnly when the degeneracy becomes large  one can approach the continuum limit and eventually recover the aforementioned\ndependence of the density on the chemical potential.\n\n\n\n\n\nWe suggest a similitude between the jump of the chiral condensate and a liquid-gas phase transition.\nAs a matter of fact, chiral symmetry is not restored at $\\mu_1$,\ntherefore the pattern of symmetry breaking is the same at low and intermediate $\\mu$.\nIn addition to this,\nthe quark number density at the first jump has a net increase, \nand the transition is sharp at low temperature then becomes smooth at higher temperatures.\nThese aspects characterize a liquid-gas phase transition,\nwhich corresponds to a change in density and not to a change in the pattern of spontaneous symmetry breaking.\n\n\n\\subsection{Correlation length of the fluctuations of the order parameter}\n\n\n\nWe can further characterize this liquid-gas-like jump of the condensate at $\\mu_1$ by means of the correlation length\nof the static fluctuations of the condensate, that are carried by the $\\sigma-$meson. \nTo do this, firstly we have to compute the in-medium masse of the $\\sigma-$meson.\nComputing this within the quark-meson model is a well established procedure, see for example \\cite{Castorina:2020vbh}\nand references therein, and is straightforward when two-loop contributions are neglected and \nthe Hatree approximation is used to compute the effective 2-particle-irreducible potential. Within these approximations we have\n$M_\\sigma^2 = \\partial^2\\Omega\/\\partial\\sigma^2$,\nwhere the second derivative is understood at the global minimum of $\\Omega$.\nA similar equation holds for the $\\pi-$mesons, $M_\\pi^2 = \\partial^2\\Omega\/\\partial\\pi^2$,\nwhere $\\Omega$ can be augmented with the pion field by the obvious replacement $\\sigma^2\\rightarrow\\sigma^2 + \\pi^2$\nin all but the $h\\sigma$ terms and in the zero mode contribution $M\\rightarrow \\sqrt{M^2 + g^2\\pi^2}$. \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{3fm.png}\n\\end{center}\n\\caption{\\label{Fig:3fm}In-medium $\\pi-$meson and $\\sigma-$meson masses\nversus $\\mu$, for several temperatures and $L=3$ fm. Thin lines correspond to $M_\\pi$ while thick lines\ndenote $M_\\sigma$.\n}\n\\end{figure}\n\nIn   Fig.~\\ref{Fig:3fm} we plot $M_\\sigma$ and $M_\\pi$ versus $\\mu$,\nfor several values of $T$. At $\\mu=\\mu_1$ $M_\\sigma$ drops down. However, \n$M_\\pi$ is almost insensitive to the jump of the condensate. This confirms that\nthe first jump at $\\mu_1$ should not be identified with a real phase transition.\nAt $\\mu=\\mu_2$ where the condensate drops down to almost zero,\n$M_\\sigma$ and $M_\\pi$ join and increase, signaling that the $O(4)$ symmetry is restored and these particles\nbecome heavy enough that decouple from the low energy spectrum dominated by the quarks. \nThis confirms that it is $\\mu_2$ that has to be identified with chiral symmetry restoration.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{correlation2fm.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{correlation3fm.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{correlation6fm.png}\n\\end{center}\n\\caption{\\label{Fig:3fmCOR}Correlation length of the fluctuations of the order parameter versus $\\mu$, for several temperatures.\n}\n\\end{figure}\n\n\nWe can compute the correlation length of the time independent fluctuations of the condensate, \n$\\lambda$, that are transported by the $\\sigma-$meson. In fact, for time independent fluctuations the effective action\nof the $\\sigma-$meson is formally equivalent to that of a Ginzburg-Landau theory\nfor a scalar order parameter with positive squared mass, and it is textbook matter that in this case the correlation length\nof fluctuations is nothing but  $\\lambda=1\/M_\\sigma$. We show $\\lambda\/L$ versus $\\mu$ in  \nFig.~\\ref{Fig:3fmCOR} for three representative values of $L$.\nFor $L=2$ fm, at $T=10$ MeV and $\\mu=\\mu_1$ the correlation length increases and stays constant\nup to $\\mu=\\mu_2$; for larger values of $\\mu$ it decreases and approaches the value it has in the hadron gas phase.\nThe correlation length is frozen in the intermediate phase, due to the fact that only the zero mode is excited. \nQualitatively, this happens also for $T=20$ MeV and $T=40$ MeV.\nAlso notice that in this phase $\\lambda\/L\\approx 0.2$ meaning that one correlation volume occupies\nabout the twenty percent of the volume of the system; around the two transitions $\\lambda$\ndevelops two peaks; in particular, for $T=40$ MeV we find that $\\lambda\/L=O(1)$ that implies\nthat the system is close to criticality. This is what we would expect at a critical endpoint.\nFinally, for $T=60$ MeV there is only one peak of $\\lambda$ in agreement with the fact that\nthe transition to the intermediate phase is smoothed by the temperature;\nnevertheless, $\\lambda$ experiences a net increase in comparison with the value at small $\\mu$,\nthen again $\\lambda\/L=O(1)$ at the chiral phase transition.\nThe qualitative picture is the same at $L=3$ fm, see the middle panel of   Fig.~\\ref{Fig:3fmCOR},\nwhile for a larger value of $L$ the double peak structure as well as the intermediate phase disappear,\nsee the lower panel of Fig.~\\ref{Fig:3fmCOR}.\n\nThe results  summarized in Fig.~\\ref{Fig:3fmCOR} allow to understand better the intermediate phase.\nAs a matter of fact, we learn that beside the characterization of this phase in terms of the baryon density\ndiscussed in the previous subsection,\nwe can distinguish it from the vacuum and the high density normal quark matter also looking at\nthe fluctuations of the order parameter.\nIn particular, the correlations of the order parameters are substantially larger than those in the vacuum \nand in the quark matter phase at high $\\mu$, and do not change by changing $\\mu$\nin this region:  correlation volumes are frozen due to the fact that only the zero mode\nis excited. Moreover, for small $L$ \nthe correlation volumes occupy a substantial portion of the total volume of the system.\nThese facts, together with the increase of density and\nthe symmetry pattern that is unchanged at $\\mu=\\mu_1$, \nsuggest the name of {\\it subcritical liquid} for this intermediate phase.\n\n\n \n\n\n\n\\subsection{Catalysis of chiral symmetry breaking at low temperature}\n\n\n  \n\n \n \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{moresize.png}\n\\end{center}\n\\caption{\\label{Fig:moresize} Condensate versus chemical potential, for several values of $L$\nand $T=10$ MeV.}\n\\end{figure} \n\nIn Fig.~\\ref{Fig:moresize} we plot the condensate versus $\\mu$ at $T=10$ MeV, for several values of $L$.\nFinite size effects are noticeable up to $L\\approx 5$ fm although in this case the transition to the subcritical liquid phase\nis minor. For $L=6$ fm no sign of the subcritical liquid is found, and comparing the results of $L=6$ fm and $L=8$ fm\nwe notice that the effect of the finite size is almost gone and a continuum limit is reached.\nThe results collected in Fig.~\\ref{Fig:moresize}  show that lowering the size catalyzes the spontaneous chiral symmetry \nbreaking by enlarging the subcritical liquid region.  \n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{nbsl.png}\n\\end{center}\n\\caption{\\label{Fig:nbsl}$n_B^\\mathrm{s.l.}\/\\rho_0$ versus $L$ at $T=0$.\n}\n\\end{figure}\n\n\nFor completeness, we report  on\nthe baryon density in the subcritical liquid phase, $n_B^\\mathrm{s.l.}$,\nat $T=0$.\nThis can be estimated quickly because only the zero mode is populated therefore\nwe read its value from Eq.~\\ref{eq:bnm5}, namely\n\\begin{equation}\nn_B^\\mathrm{s.l.} = \\frac{2N_c N_f}{3L^3};\\label{eq:nbsl}\n\\end{equation}\nthis amounts to put $2N_c N_f$ quarks in the volume $L^3$.  \nThe results are collected in Fig.~\\ref{Fig:nbsl} in which we show $n_B^\\mathrm{s.l.}\/\\rho_0$ versus $L$,\nwhere $\\rho_0$ is the nuclear saturation density. In particular, $n_B\\approx \\rho_0$ for $L\\approx 3$ fm.\n\n \n \n \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{pdl2.png}\n\\end{center}\n\\caption{\\label{Fig:pdl2} Critical lines for $L=2$ fm. \nDotted and dot-dashed lines correspond to smooth crossovers and solid lines to first order phase transitions.\nThe green dot denotes the critical endpoint of the chiral phase transition.\nThe indigo dot corresponds to the critical endpoint for the liquid-gas-like transition to the subcritical liquid phase.\n$\\chi$SR and $\\chi$SB denote the regions in which chiral symmetry is restored and broken respectively.\nWe have shown by blue lines the critical lines for $L=10$ fm for comparison.}\n\\end{figure}  \n\nThe results discussed in this section can be summarized in the form of a phase diagram in the $\\mu-T$ plane.\nIn Fig.~\\ref{Fig:pdl2} we plot the transition lines for the case $L=2$ fm, and for comparison we also show a portion of the\ncritical lines at $L=10$ fm that correspond to the continuum limit.\nThe regions denoted with $\\chi$SR and $\\chi$SB denote the portions of the phase diagram \nin which chiral symmetry is restored and broken respectively. \nThe dots denote critical endpoints. In the figure we focus on the subcritical region phase\nthat appears as an intermediate phase between $\\chi$SB and $\\chi$SR phases.\nComparing the critical lines for $L=2$ fm and $L=10$ fm the catalysis of symmetry breaking is evident. \nWe also notice that the critical endpoint for chiral symmetry restoration moves towards higher values of $\\mu$\nand lower $T$ with the lowering of $L$.\n\nWe have verified the stability of our results by changing the number of colors:\nin particular, for $N_c=2$ the picture is unchanged. Since QCD with $N_c=2$\nand finite $\\mu$ can be simulated on the lattice, the predictions of this article can be tested by means\nof first principle calculations.\n\n \n\\section{Chiral phase transition at high temperature}\nThe chiral  phase transition at finite temperature and low $\\mu$ has been more studied in the literature,\ntherefore we limit ourselves to present a few results and compare them with those of\nother effective models. In particular, our results agree with those of \\cite{Xu:2019gia,Xu:2019kzy}\nwhere the NJL model with PV regulators has been used.\n\n\\subsection{Numerical computation of $T_c$ versus $L$}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{masses_comparison.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{masses_comparison_m300.png}\n\\end{center}\n\\caption{\\label{Fig_mm}Condensate $\\sigma$ versus temperature, for several values of $L$\nand two representative values of $\\mu$.\n}\n\\end{figure}\n\nIn Fig.~\\ref{Fig_mm} we plot the condensate, $\\sigma$, versus temperature, for $\\mu=0$ (upper panel) and $\\mu=300$ MeV\n(lower panel) and several values of $L$. \nWe can define a pseudo-critical temperature, $T_c$, by looking at the location of the maximum variation of\n$d\\sigma\/d\\beta$. At $\\mu=0$ the condensate increases with $1\/L$ and this pattern remains stable in the whole\ntemperature range examined. We conclude that our picture is consistent with the catalysis of chiral symmetry breaking\ninduced by lowering $L$. The catalysis remains also for higher values of $\\mu$, see the lower panel \nof Fig.~\\ref{Fig_mm}, in agreement with the results presented in section~\\ref{Sec:llp}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{tctc.png}\n\\end{center}\n\\caption{\\label{Fig:tctc}Critical temperature for chiral symmetry restoration versus size at $\\mu=0$.\nThe line represents a crossover for any value of $L$.\n}\n\\end{figure} \n \nWe show $T_c$  versus $L$ at $\\mu=0$ in Fig.~\\ref{Fig:tctc}.\nThe behavior for other values of $\\mu$ can be easily guessed from the results that we have shown before.\nIn the infinite volume limit $T_c\\approx 180$ MeV. The catalysis of chiral symmetry breaking is\nclear in Fig.~\\ref{Fig:tctc}. For example, for $L=3$ fm the increase of critical temperature is $\\approx 23\\%$,\nwhile it becomes $\\approx 55\\%$ for $L=2$ fm.  The qualitative behavior of $T_c$ is in agreement \nwith previous studies within the QM model~\\cite{Palhares:2009tf,Magdy:2019frj}\nas well as the NJL model with periodic boundary conditions \\cite{Xu:2019gia,Xu:2019kzy,Wang:2018ovx}.\n\n\n\n\n\n \n\n\\subsection{Critical temperature from the Ginzburg-Landau potential}\nIn this subsection we clarify the role of the zero mode \non the behavior of $T_c$ versus $L$. We do this in the chiral limit, $h=0$, and at $\\mu=0$, \nto make the discussion more transparent. \nIn these conditions the chiral transition is of the second order and the critical temperature is given by the zero\nof the coefficient $\\alpha_2$ of the Ginzburg-Landau potential,\n\\begin{equation}\n\\alpha_2 = \\left.\\frac{\\partial^2\\Omega}{\\partial\\sigma^2}\\right|_{\\sigma=0},\n\\end{equation}\nwhere  $\\Omega$ is given by Eq.~\\ref{eq:tot_om_fs}. \nThe coefficient $\\alpha_2$ corresponds to the curvature of $\\Omega$ at $\\sigma=0$.\nWe feel that this discussion is necessary because in the literature some confusion arises when different models \nare compared to each other.\nWe keep the parameters of the classical potential,\n$v$ and $\\lambda$, unchanged by $L$. This is natural within the renormalization scheme \nthat we have adopted in this article, because the finite size corrections to $\\Omega$\nare finite and do not require any additional renormalization condition. \n\n\n \nStarting from Eq.~\\ref{eq:tot_om_fs}  we get\n\\begin{eqnarray}\n\\alpha_2^\\mathrm{QM} &=& C_2^\\mathrm{QM} \\nonumber\\\\\n&&+\\frac{4 N_c N_f g^2}{2\\pi L^2}\\sum_{n=1}^\\infty r_3(n)\\frac{e^{-\\beta\\varepsilon_n}}{\\sqrt{n}(1+e^{-\\beta\\varepsilon_n})}\n\\nonumber\\\\\n&&\n-\\frac{4 N_c N_f }{ L^3 }\\frac{g^2}{4T},\\label{eq:a2QM}\n\\end{eqnarray} \nwhere we have put $\\varepsilon_n = 2\\pi\\sqrt{n}\/L$,\nand $C_2^\\mathrm{QM}$ denotes the curvature of $\\Omega$ at $T=0$ and $\\sigma=0$,\n\\begin{eqnarray}\nC_2^\\mathrm{QM} &=& -v^2\\lambda + g^2\\delta v \\nonumber\\\\\n&&-\\frac{2 N_c N_f g^2}{L^3}\n\\sum_{n=1}^\\infty r_3(n)\\sum_{j=0}^3\\frac{c_j}{\\sqrt{j\\xi^2 + 4\\pi^2n\/L^2}}.\\nonumber\\\\\n&&\\label{eq:C2qm}\n\\end{eqnarray} \nWe remind that $m_\\sigma^2=2v^2\\lambda$ corresponds to the $\\sigma-$meson mass\nin the vacuum and in the infinite volume limit.\nThe last addendum on the right hand side of \nEq.~\\ref{eq:a2QM} is the zero mode contribution to $\\alpha_2$,\nwhile the summation represents the contribution of the higher modes.\nThe zero mode contribution is negative while the sum over the higher modes is positive:\nwhile thermal fluctuations increase $\\alpha_2$ making the broken phase less stable,\nthe zero mode lowers $\\alpha_2$ causing the broken phase to be more stable.\nA numerical inspection shows that $C_2^\\mathrm{QM}$ is quite insensitive of $L$\nbecause it is dominated by the classical contribution. Moreover $C_2^\\mathrm{QM}<0$.\n\n\n  \n\nLowering $L$, the zero mode contribution grows up in magnitude, therefore it is necessary to increase\n$T_c$ to get a positive contribution from the higher modes that overcomes both the zero mode and $C_2^\\mathrm{QM}$\nto satisfy $\\alpha_2^\\mathrm{QM}(T_c)=0$. Thus $T_c$ increases by lowering $L$.\n \n\nIf we removed the zero mode from Eq.~\\ref{eq:a2QM}, for example by imposing antiperiodic boundary condtions\nor an infrared cutoff,\nwe would be left with\n\\begin{eqnarray}\n\\alpha_{2,\\mathrm{nzm}}^\\mathrm{QM} &=& C_2^\\mathrm{QM}   \\nonumber\\\\\n&&+\\frac{4 N_c N_f g^2}{2\\pi L^2}\\sum_{n=1}^\\infty r_3(n)\\frac{e^{-\\beta\\varepsilon_n}}{\\sqrt{n}(1+e^{-\\beta\\varepsilon_n})}.\n\\label{eq:a2QMapbc}\n\\end{eqnarray}\nEven in this case, the requirement $\\alpha_{2,\\mathrm{nzm}}^\\mathrm{QM}(T_c)=0$\nimplies that\nlowering $L$ has to be balanced by the increase of $T_c$,\nbecause $ C_2^\\mathrm{QM}<0$ and almost insensitive to $L$.\nThus in the QM model even without the zero mode,  $T_c$ has to increase with  $1\/L$,\nin agreement with~\\cite{Palhares:2009tf,Magdy:2019frj}.\nThe arguments above would apply also if we had not implemented\nrenormalization and had regularized the divergent quark loop via an effective cutoff: in this case,\nas long as $-v^2\\lambda <0$, $C_2^\\mathrm{QM}<0$ for any $L$. \n\n \n\nSummarizing, we have shown that within the QM model in the chiral limit and at $\\mu=0$,\n$T_c$ increases with $1\/L$ regardless of the presence of the zero mode in the spectrum or not.\nThis happens because $C_2^\\mathrm{QM}<0$ for any $L$.\nThis behavior of $T_c$ and its explanation has not been stressed enough in the literature.\n\nThe increase of $T_c$ with $1\/L$ when periodic boundary conditions are implemented is in agreement with\nthe results of the NJL model, see for example \\cite{Xu:2019gia,Wang:2018ovx}. On the other hand,\nwhen antiperiodic boundary conditions are implemented within the NJL model, $T_c$ is found to decrease with $1\/L$.\nThis is in qualitative disagreement with the QM model and needs to be clarified. \nThe very reason of the different behavior cannot be traced back to the lack of the zero mode only:\nafter all, in the QM model $T_c$ increases with $1\/L$ even when the zero mode is removed from the spectrum.\nIn fact, a difference between the QM and NJL models is the curvature of the potential\nat $\\sigma=0$ and $T=0$, due to the different classical potentials in the two models.\nIn the QM model $C_2^\\mathrm{QM}<0$.\nOn the other hand, in the NJL model there are no mesons at the tree level and the classical potential\nis merely the mean field term $\\sigma^2\/4G$.  \nThe divergent quark loop is necessary to make $\\alpha_2$ negative and break chiral symmetry:\n while this is enough to guarantee a negative curvature in the infinite volume limit,\nit is not guaranteed that it remains negative for any $L$. \n\nTo keep the treatment simple, we use the common hard cutoff scheme for the NJL model;\nin fact, the results we find here agree with those obtained within\nPV regularization \\cite{Xu:2019gia}.\nUsing a  cutoff $\\Lambda$ the curvature at $T=0$ in NJL is\n\\begin{equation}\nC_2^\\mathrm{NJL} = \\frac{1}{2G}  -\\frac{N_c N_f}{\\pi L^2}\\sum_{n=1}^a \\frac{r_3(n)}{\\sqrt{n}},\n\\end{equation}\nwhere $a=\\Lambda^2 L^2\/4\\pi^2$ and $G$ is the NJL coupling. The contribution of the thermal fluctuations to $\\alpha_2$ in NJL\nis formally equivalent to that of QM model and is not repeated here.\nWe notice that  $C_2^\\mathrm{NJL}$\nbecomes positive for small enough $L$ because the quark loop shrinks. \nWhen the zero mode is removed from the spectrum, \nthe quark loop in $C_2^\\mathrm{NJL}$ is the only source for a negative $\\alpha_2$:\nsince lowering $L$ shrinks the loop, the contribution of the excited states at $T_c$ has to be lowered to get a vanishing $\\alpha_2$\nwhich implies that $T_c$ decreases when $L$ decreases. \n\n \nThe message of this section is that the zero mode alone is not enough to explain the behavior of $T_c$ versus $L$\nin chiral models: the difference between QM and NJL appears even when the zero mode is absent in the\nspectrum of both models. The curvature of the potential is another necessary ingredient for $T_c$ and it is precisely the\ndifferent curvature that leads to different predictions of $T_c$ versus $L$ in the two models.\n\nWe remark that if we had assumed \na dependence of the classical potential of $L$, the behavior of $T_c$ versus $L$ might have been more difficult to predict\nwithin the Ginzburg-Landau coefficient because $T_c(L)$ would have depended also on \nthe additional functions $v=v(L)$ and $\\lambda=\\lambda(L)$.\n \n \n   \n\n\n\n\n\n\n\\section{Summary and Conclusions}\nWe have studied the effect of periodic boundary conditions on chiral symmetry breaking and its restoration in QCD.\nAs an effective model of the effective potential for the quark condensate, we have used the quark-meson model \nwhich couples quarks to background meson fields.\nWe have implemented periodic boundary conditions on the effective potential for a cubic box of size $L^3$,\nthen we have performed the renormalization of the divergent vacuum term in the box;\nwe have computed the behavior of the condensate at finite temperature, $T$, and quark chemical potential, $\\mu$.\nFor the implementation of the renormalization conditions at finite $L$ we have \nadopted the Pauli-Villars regulators as in \\cite{Xu:2019gia}, that are enough to cancel the divergent contributions\nin the infinite volume as well as at finite $L$.\n\nThe most interesting effects happen for the chiral phase transition at small temperature and finite chemical potential.\nWe have found that for $L\\lesssim 5$ fm, increasing $\\mu$ up to a critical value, $\\mu_1$, \nresults in a jump of the condensate\nto lower but finite values. This jump is due to the population of the zero mode.\nThe contribution of the zero mode at such moderate size is not very strong, therefore its excitation is not enough\nto restore chiral symmetry. Increasing $\\mu$ to higher values, the first mode is excited eventually\nand chiral symmetry is restored at $\\mu=\\mu_2$.\n\nWe have suggested a similitude between the jump of the condensate at $\\mu=\\mu_1$ and a liquid-gas phase transition.\nIn fact, chiral symmetry is not restored at $\\mu_1$,\ntherefore symmetries are broken in the same way at low and intermediate $\\mu$.\nMoreover, the quark number density at the first jump has a net increase.\nBoth these aspects are common to the liquid-gas phase transition.\nWe have further characterized the jump of the condensate at $\\mu_1$ by means of the correlation length\nof the fluctuations of the condensate, that are carried by the $\\sigma-$meson; in particular,\nwe have identified $\\lambda=1\/M_\\sigma$ with $\\lambda$ the correlation length and $M_\\sigma$ the in-medium\nmass of the $\\sigma-$meson. We have found that at low temperature and $\\mu=\\mu_1$ the correlation length\nincreases then stays constant up to $\\mu=\\mu_2$ where chiral symmetry is restored. Increasing temperature\nbrings the system close to criticality and this is confirmed by the increase of $\\lambda$.\n\nWe name the intermediate phase as {\\em subcritical liquid phase} because even though\nthe system is not critical in the whole $(\\mu-T)$ window, the correlation domains in this phase\nare larger than those in the hadron gas and quark matter phases, respectively at small and large $\\mu$,\nas if the system was approaching a critical point; in addition to this, baryon density is finite due to the occupation\nof the zero mode, and symmetries are broken in the same way of the hadron phase, as it would happen in the gas-to-liquid\ntransition. \n\n\nOverall, we have found that lowering $L$  and imposing periodic boundary conditions\ncatalyzes the spontaneous breaking of chiral symmetry. This has been understood as  the result of the excitation \nof the zero mode at intermediate values of $\\mu$, that lowers a bit the value of the condensate without restoring chiral symmetry, \nand the need of a large $\\mu$ to excite the first mode\nthat leads to the definitive lowering of the condensate; the smaller $L$ the larger is the value of $\\mu$ needed to excite the first mode,\nthus leading to the catalysis of chiral symmetry breaking.\n\n\nWe have completed the study by computing \nthe critical temperature,\n$T_c(L)$, versus $L$. We have found that $T_c$ decreases with $L$ \nthus supporting the catalysis of chiral symmetry breaking \nfound in previous studies where periodic boundary conditions, or effective infrared\ncutoffs in the QM model, have been implemented \\cite{Xu:2019gia,Palhares:2009tf,Wang:2018ovx,Magdy:2019frj}. \n\nThere are several ways to continue the work presented here. A straightforward extension of the work is the \ninclusion of meson fluctuations on the same line of \\cite{Castorina:2020vbh}.\nIt would be interesting to include the possibility of inhomogeneous condensates\n\\cite{Abuki:2018iqp,Takeda:2018ldi,Abuki:2013pla,\nAbuki:2013vwa,Buballa:2020xaa,Carignano:2019ivp,Buballa:2018hux,Carignano:2017meb,Nickel:2009ke,Nickel:2009wj}.\nThe inclusion of the Polyakov loop  to take trace of confinement-deconfinement phase transition via a collective field\nwould also be possible \\cite{Fukushima:2003fw,Ratti:2005jh,Abuki:2008nm}. \nFinally, it is of a certain interest to analyze the thermodynamic geometry\n\\cite{Weinhold:1975get,Weinhold:1975gtii,\nRuppeiner:1979trg,Ruppeiner:1983ntp,Ruppeiner:1983tcf,Ruppeiner:1985tcv,Ruppeiner:1995rgf,Ruppeiner:1998rgc,\n Castorina:2019jzw,Castorina:2020vbh,Zhang:2019neb} \nof the effective models of chiral symmetry breaking with finite size.\nWe plan to report on these topics in the near future.\nWe have also investigated the stability of our results by changing the number of colors:\nin particular, we have verified that for $N_c=2$ the picture is unchanged. Since QCD with $N_c=2$\nand finite $\\mu$ can be simulated on the lattice, the predictions of this article can be tested by means\nof first principle calculations.\n\n \n\n\n \n\n\n\n\n\n\n\\section*{ACKNOWLEDGEMENTS}\nM. R. acknowledges John Petrucci for inspiration.\nThe work of the authors is supported by the National Science Foundation of China (Grants No.11805087 and No. 11875153).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe goal of information theory is to find the fundamental limits imposed on information processing and transmission by the laws of physics. One of the early breakthroughs in quantum information theory was the characterisation of the capacity of a classical-quantum (c-q) channel to transmit classical information by Holevo~\\cite{holevo98,holevo73b} and Schumacher--Westmoreland~\\cite{schumacher97}. The \\emph{classical capacity} of a quantum channel is defined as the maximal rate (in bits per channel use) at which we can transmit information such that the decoding error vanishes asymptotically as the length of the code increases. However, for many practical applications there are natural restrictions on the code length imposed, for example, by limitations on how much quantum information can be processed coherently. Therefore it is crucial to go beyond the asymptotic treatment and understand the intricate tradeoff between decoding error {probability}, code rate and code length. \n\nFor this purpose, we will study families of codes that have {\\em both} a rate approaching the capacity and an {error probability} that vanishes asymptotically as the code length $n$ increases. The following tradeoff relation gives a rough illustration of our main result: if the code rate approaches capacity as $\\Theta(n^{-t})$ for some $t \\in (0,1\/2)$, then the decoding error cannot be smaller than $\\exp(- \\Theta(n^{1-2t}))$. In fact, we will show that the constants implicit in the $\\Theta$ notation are determined by a second channel parameter {beyond} the capacity, called the \\emph{channel dispersion}. We will also show that this relation is tight, i.e., there exist families of codes achieving equality {asymptotically}. \n\nOur work thus complements previous work on the boundary cases corresponding to {$t  \\in \\{0, 1\/2\\}$}. The error exponent (or reliability function) of c-q channels (see, e.g., Refs.~\\cite{holevo00,hayashi07,dalai13}) corresponds to the case $t = 0$ where the rate is bounded away from capacity and the error probability vanishes exponentially in $n$. This is also called the \\emph{large deviations} regime.  Moreover, the second-order asymptotics of c-q channels were evaluated by Tomamichel and Tan~\\cite{tomamicheltan14}. They correspond to the case $t = 1\/2$ where the rate approaches capacity as $\\Theta(n^{-1\/2})$ and the  error probability  is non-vanishing. This is also called the \\emph{small deviations} regime. \n\nIn the present work, we consider the entire regime in between, which is dubbed the \\emph{moderate deviation} regime.\\footnote{In the technical analysis, we are considering moderate deviations from the mean of a sum of independent  {log-likelihood} ratios, thus justifying the name emanating from {statistics~\\cite[Theorem~3.7.1]{dembo98}}.} The different parameter regimes are illustrated in Fig.~\\ref{fig:regimes}.\n\n\\begin{figure}[t!]\n\t\\centering\n\n\t\\includegraphics{figure-compressed.pdf}\n\t\\begin{tabular}{|l|c|c|c|c|c|}\n\t\t\\hline\n\t\t& (I) & (II) & (III) & (IV) & (V) \\\\\n\t\t\\hline\n\t\t\\multirow{2}{*}{regime} & error  & \\!moderate deviation\\! & constant error & moderate deviation & strong converse \\\\\n\t\t& exponent & (below capacity) & (second-order) & (above capacity) & exponent \\\\\n\t\t\\hline\n\t\terror prob.\\ & \\footnotesize $\\!\\exp(-\\Theta(n))\\!$ & \\footnotesize $\\exp(-o(n))$ \\& $\\omega(1)$ & \\footnotesize $\\Theta(1)$ & \\footnotesize \\!$1 - \\exp(-o(n))$ \\& \\footnotesize $1 - \\omega(1)$\\! & \\footnotesize $1 - \\exp(-\\Theta(n))$ \\\\\n\t\t\\hline\n\t\tcode rate & \\footnotesize $C - \\Theta(1)$ & \\footnotesize $C - o(1)$ \\& $C - \\omega\\big(n^{-\\frac12}\\big)$ & \\footnotesize $C - \\Theta\\big(n^{-\\frac12}\\big)$ & \\footnotesize $C + o(1)$ \\& $C + \\omega\\big(n^{-\\frac12}\\big)$ & \\footnotesize $C + \\Theta(1)$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{The figure shows the optimal error probability as a function of the rate, for different block lengths. Darker lines correspond to longer block lengths, and the capacity is denoted by $C$. The table shows the asymptotics in each region, as the blocklength $n$ goes to infinity. The functions of $n$ implicit in the $\\Theta$, $o$, and $\\omega$ notation are assumed to be positive-valued.}\n\t\\label{fig:regimes}\n\\end{figure}\n\n\\paragraph*{Main results.} \nBefore we present our main results, let us introduce the notion of a \n\\emph{moderate sequence} of real numbers, $\\{ x_n\\}_n$ for $n \\in \\mathbb{N}$, \nwhose defining properties are that $x_n \\searrow 0$ and $\\sqrt{n}\\, x_n \\to \n+\\infty$ as $n \\to \\infty$.\\footnote{As mentioned above an archetypical \n\tmoderate sequence is $x_n = \\Theta(n^{-t})$ for some $t \\in (0, \\frac12)$. The \n\tboundary cases are not included\\,---\\,in fact $t = 0$ requires a large \n\tdeviation analysis whereas $t = \\frac12$ requires a small deviation analysis.} \nOur two main results concern binary asymmetric quantum hypothesis testing and \nc-q channel coding.\n\n\\begin{enumerate}\n\t\\item The first result, presented in detail in Sect.~\\ref{sec:hypo}, concerns binary quantum hypothesis testing between a pair of quantum states $\\rho$ and~$\\sigma$. We show that for any moderate sequence $x_n$, there exists a sequence of tests $\\{ Q_n \\}_n$ such that the two kinds of errors satisfy\n\t\\begin{align}\n\t\t\\Tr \\rho^{\\otimes n} (1 - Q_n) = e^{-nx_n^2}  &~\\textrm{and}~ \n\t\t\\Tr \\sigma^{\\otimes n} Q_n = \\exp\\Big(-n \\Big( D(\\rho\\|\\sigma) - \\sqrt{2 V(\\rho\\|\\sigma) }\\, x_n  + o(x_n) \\Big)\\Big) \\,,\n\t\t\\intertext{and another sequence of tests $\\{ Q_n' \\}_n$ such that the errors satisfy\n\t\t}\n\t\t\\Tr \\rho^{\\otimes n} (1-Q_n')  = 1-e^{- n x_n^2} &~\\textrm{and}~  \n\t\t\\Tr \\sigma^{\\otimes n} Q_n' = \\exp\\Big(-n \\Big( D(\\rho\\|\\sigma) + \\sqrt{2 V(\\rho\\|\\sigma) }\\, x_n  + o(x_n) \\Big)\\Big) \\,,\n\t\\end{align}\n\twhere $D(\\cdot\\|\\cdot)$ and $V(\\cdot\\|\\cdot)$ denote the relative entropy~\\cite{umegaki62} and relative entropy variance~\\cite{tomamichel12,li12}, respectively. (The reader is referred to the next section for formal definitions of all concepts discussed here.)\n\tMost importantly, we show that both of these tradeoffs are in fact optimal.\n\t\n\t\\item The main result, covered in Sect.~\\ref{sec:channels}, concerns coding over a memoryless classical-quantum channel $\\mathcal{W}$. Let us denote by $M^*(\\mathcal{W};n,\\varepsilon)$ the maximum $M \\in \\mathbb{N}$ such that there exists a code transmitting one out of $M$ messages over $n$ uses of the channel $\\mathcal{W}$ such that the average probability of error does not exceed $\\varepsilon$. For any sequence of tolerated error probabilities $\\{\\varepsilon_n\\}_n$ vanishing sub-exponentially with $\\varepsilon_n = e^{-nx_n^2}$, we find that\n\t\\begin{align}\n\t\t\\label{eq:moderatesecondorder1}\n\t\t\\frac{1}{n} \\log M^*(\\mathcal{W};n, \\varepsilon_n) &= C(\\mathcal{W}) - \\sqrt{2 V_{\\min}(\\mathcal{W})}\\, x_n + o(x_n) \\,,\\\\\n\t\t\\label{eq:moderatesecondorder2}\n\t\t\\frac{1}{n} \\log M^*(\\mathcal{W};n, 1-\\varepsilon_n) &= C(\\mathcal{W}) + \\sqrt{2 V_{\\max}(\\mathcal{W})}\\, x_n + o(x_n) \\,,\n\t\\end{align}\n\twhere $C(\\cdot)$ denotes the channel capacity and~$V_{\\min}(\\cdot)$ and $V_{\\max}(\\cdot)$ denote the minimal and maximal channel dispersion as defined in~Ref.~\\cite{tomamicheltan14}, respectively.\n\tThis result holds very generally for channels with arbitrary input alphabet and without restriction on the channel dispersion, strengthening also the best known results for classical channels. Moreover, as in~Ref.~\\cite{tomamicheltan14}, this generality allows us to lift the above result to a statement about coding classical information over image-additive quantum channels and general channels as long as the encoders are restricted to prepare separable states.\n\\end{enumerate}\n\nSince quantum hypothesis testing underlies many other quantum information processing tasks such as entanglement-assisted classical communication as well as private and quantum communication, we expect that our techniques will have further applications in quantum information theory.\n\n\\paragraph*{Related work.}\n\nFor classical channels, Alt\\u{u}g and Wagner~\\cite{altug14} first established the best decay rate of the average error probability for a class of discrete memoryless channels (DMCs) when the code rate approaches capacity at a rate slower than $\\Theta(n^{-1\/2})$. Shortly after the conference version of Ref.~\\cite{altug14}, Polyanskiy and Verd\\'u~\\cite{polyanskiy10c} relaxed some of the  conditions on the class of DMCs and also established the moderate deviations asymptotics for other important classical channels such as the additive white Gaussian noise channel. The other main contributions to the analysis of  hypothesis testing, channel coding, quantum hypothesis testing, and c-q channel coding in the different parameter regimes are summarised in Table~\\ref{tb:relatedwork}. \n\nFrom a technical perspective the moderate deviations regime can be approached via a refined large deviations analysis (as was done in Ref.~\\cite{altug14}) or via a variation of second-order analysis via the information spectrum method (as was proposed in Ref.~\\cite{polyanskiy10c}). In our work, we mostly follow the latter approach, interspersed with ideas from large deviation theory. In particular, we build on bounds from one-shot information theory by Wang and Renner~\\cite{wang10} and use techniques developed for the second-order asymptotics in Ref.~\\cite{tomamicheltan14}. In concurrent work, Cheng and Hsieh~\\cite{cheng17} provide a moderate deviation analysis for c-q channels via a refined error exponent analysis. Their result holds for c-q channels with finite input alphabets and their techniques are complementary to ours.\n\n\\begin{table}\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\t\\hline\n\t\t&  asymmetric binary  &  channel coding &  quantum hypothesis &  classical-quantum  \\\\\n\t\t& hypothesis testing &   &  testing & channel coding \\\\\n\t\t\\hline\n\t\tlarge deviation ($<$) & \\cite{hoeffding65} & \\cite{gallager68, csiszar11} & \\cite{hayashi07,nagaoka06} & unknown\\footnotemark \\\\\n\t\t\\hline\n\t\tmoderate deviation ($<$) & \\cite{Sas11} & \\cite{altug14, polyanskiy10c} &  \\em this work & \\em this work  \\\\\n\t\t\\hline\n\t\tsmall deviation & \\cite{strassen62} & \\cite{strassen62,hayashi09,polyanskiy10}  & \\cite{li12,tomamichel12} & \\cite{tomamicheltan14}  \\\\\n\t\t\\hline\n\t\tmoderate deviation ($>$) & \\em this work & \\em this work  & \\em this work & \\em this work \\\\\n\t\t\\hline\n\t\tlarge deviation ($>$) & \\cite{csiszar71,han89} & \\cite{arimoto73, dueck79} & \\cite{mosonyiogawa13,mosonyi14} & \\cite{mosonyi14-2} \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Exposition of related work on finite resource analysis of hypothesis testing and channel coding problems. The rows correspond to different parameter regimes, labelled by the deviation from the critical rate (i.e., the relative entropy for hypothesis testing and the capacity for channel coding problems).}\n\t\\label{tb:relatedwork}\n\\end{table}\n\n\\footnotetext{In contrast to classical channels a tight characterisation of the error exponent of c-q channels remains elusive to date even for high rates. See, e.g., Refs.~\\cite{holevo00,hayashi07,dalai13} for partial progress.}\n\n\n\\section{Preliminaries}\n\n\\subsection{Notation and classical coding over quantum channels}\n\nLet $\\mathcal{H}$ be a finite-dimensional Hilbert space and denote by $\\mathcal{S}:=\\lbrace \\rho\\in\\mathcal{H}\\,|\\,\\Tr\\rho=1,\\rho\\geq 0\\rbrace$ the quantum states on $\\mathcal{H}$. We take $\\exp(\\cdot)$ and $\\log(\\cdot)$ to be in an arbitrary but compatible base (such that they are inverses), and denote the natural logarithm by $\\ln(\\cdot)$. For convenience, we will consider the dimension of this Hilbert space to be a fixed constant, and omit any dependence constants may have on this dimension. For $\\rho, \\sigma \\in \\mathcal{S}$ we write $\\rho \\ll \\sigma$ if the support of $\\rho$ is contained in the support of $\\sigma$. For any closed subset $\\mathcal{S}_{\\circ}\\subseteq \\mathcal{S}$, we will denote by $\\mathcal{P}(\\mathcal{S}_{\\circ})$ the space of probability distributions supported on $\\mathcal{S}_{\\circ}$. We equip $\\mathcal{S}$ with the trace metric $\\delta_{\\Tr}(\\rho,\\rho'):=\\frac{1}{2}\\norm{\\rho-\\rho'}_1$ and $\\mathcal{P}(\\mathcal{S})$ with a weak-convergence metric\\footnote{An example of which is the induced L\\'evy--Prokhorov metric (see, e.g., Section 6 and Theorem 6.4 in Ref.~\\cite{partha67}).} $\\delta_\\text{wc}$, such that both are compact metric spaces with\n\\begin{align}\nf:\\mathcal{S}\\to \\mathbb{R}\\text{ continuous} \\qquad\\implies\\qquad \\mathbb{P}\\mapsto \\int\\mathrm{d}\\mathbb{P}(\\rho)\\,f(\\rho)\\text{ continuous}.\n\\end{align}\n\nWe will use the \\emph{cumulative standard normal distribution} function $\\Phi$ is defined as\n\\begin{align}\n\t\\Phi(a)&:=\\int_{-\\infty}^{a}\\frac{1}{\\sqrt{2\\pi}}\\, e^{-\\frac{x^2}2}\\, \\mathrm{d}x.\n\\end{align}\n\nFollowing Ref.~\\cite{tomamicheltan14}, we consider a general \\emph{classical-quantum channel} $\\mathcal{W}: \\mathcal{X} \\to \\mathcal{S}$ where $\\mathcal{X}$ is any set (without further structure). We define the \\emph{image of the channel} as the set $\\mathop{\\mathrm{im}} \\mathcal{W} \\subset \\mathcal{S}$ of all quantum states $\\rho$ such that $\\rho = \\mathcal{W}(x)$ for some $x \\in \\mathcal{X}$. For convenience we assume that our Hilbert space satisfies\n\\begin{align}\n\t\\mathcal{H} = \\mathop{\\mathrm{Span}}_{\\rho \\in \\mathop{\\mathrm{im}} \\mathcal{W}} \\mathrm{supp}(\\rho)\n\\end{align}\nsuch that $\\sigma>0$ is equivalent to $\\rho\\ll\\sigma$ for all $\\rho\\in \\mathop{\\mathrm{im}} \\mathcal{W}$.\n\nFor $M, n \\in \\mathbb{N}$, an \\emph{$(n,M)$-code} for a classical-quantum \nchannel $\\mathcal{W}$ is comprised of an encoder and a decoder. The \\emph{encoder} is a \nmap $E: \\{1, 2, \\ldots, M \\} \\to \\mathcal{X}^n$ and the \\emph{decoder} is a positive \noperator-valued measure $\\{ D_m \\}_{m=1}^M$ on $\\mathcal{H}^{\\otimes n}$. Moreover, an \n\\emph{$(n,M,\\varepsilon)$-code} is an \\emph{$(n,M)$-code} that satisfies\n\\begin{align}\n\t\\frac{1}{M} \\sum_{m=1}^M \\Tr \\bigg( \\bigotimes_{i=1}^n \\mathcal{W}\\bigl(E_i(m)\\bigr) \n\tD_m \\bigg) \\geq 1 - \\varepsilon \\,,\n\\end{align}\ni.e.\\ the average probability of error does not exceed $\\varepsilon$.\nThe finite blocklength achievable region for a channel $\\mathcal{W}$ is the set of triples $(n,M,\\varepsilon)$ for which there exists an \\emph{$(n,M,\\varepsilon)$-code} on $\\mathcal{W}$. We are particularly interested in the boundary\n\\begin{align}\n\tM^*(\\mathcal{W};n,\\varepsilon) := \\max \\big\\{ M \\in \\mathbb{N} : \\exists \\textrm{ a $(n,M,\\varepsilon)$-code for } \\mathcal{W} \\big\\} \\,.\n\\end{align}\nSpecifically we are going to be concerned with the behaviour of the \\emph{maximum rate}, which is defined as $R^*(\\mathcal{W};n,\\varepsilon) := \\frac{1}{n}\\log M^*(\\mathcal{W};n,\\varepsilon)$.\n\n\\subsection{Channel parameters}\n\nAn important parameter of a channel is the largest rate such that there exists a code of vanishing error probability in the large blocklength limit. This critical rate is known as the \\emph{capacity} of a channel $C(\\mathcal{W})$, which is defined as\n\\begin{align}\n\tC(\\mathcal{W}):=\\inf_{\\epsilon>0}\\liminf\\limits_{n\\to\\infty}R^*(\\mathcal{W};n,\\epsilon).\n\\end{align}\nFor classical-quantum channels there exists a \\emph{strong converse} bound, which states that the capacity described the asymptotic rate not just for vanishing error probability, but those for non-zero fixed error probabilities as well~\\cite{winter99,ogawa99}. Together with the original channel coding theorem~\\cite{schumacher01,holevo98}, this yields\n\\begin{align}\n\t\\lim\\limits_{n\\to\\infty}R^*(\\mathcal{W};n,\\epsilon)=C(\\mathcal{W})\\quad \\text{for all }\\epsilon\\in (0,1).\n\\end{align}\n\nIn essence the strong converse tells us that the capacity entirely dictates the asymptotic behaviour of the maximum rate at a fixed error probability. How quickly the rate approaches this asymptotic value for arbitrarily low and high error probabilities are described by the channel \\emph{min-dispersion} $V_{\\min}(\\mathcal{W})$ and \\emph{max-dispersion} $V_{\\max}(\\mathcal{W})$, which are defined respectively as\n\\begin{align}\n\tV_{\\min}(\\mathcal{W})&:=\\inf_{\\epsilon> 0}\\limsup_{n\\to\\infty}\\left(\\frac{C(\\mathcal{W})-R^*(\\mathcal{W};n,\\epsilon)}{\\Phi^{-1}(\\epsilon)\/\\sqrt{n}}\\right)^2,\\\\\n\tV_{\\max}(\\mathcal{W})&:=\\sup_{\\epsilon<1}\\limsup_{n\\to\\infty}\\left(\\frac{C(\\mathcal{W})-R^*(\\mathcal{W};n,\\epsilon)}{\\Phi^{-1}(\\epsilon)\/\\sqrt{n}}\\right)^2.\n\\end{align}\nAs with the strong converse, the min and max-dispersions also describe the dispersion at other fixed error probabilities~\\cite{tomamicheltan14}:\n\\begin{align}\n\t\\lim_{n\\to\\infty}\\left(\\frac{C(\\mathcal{W})-R^*(\\mathcal{W};n,\\epsilon)}{\\Phi^{-1}(\\epsilon)\/\\sqrt{n}}\\right)^2=\\begin{dcases}\n\t\tV_{\\min}(\\mathcal{W}) &\\epsilon\\in(0,1\/2)\\\\\n\t\tV_{\\max}(\\mathcal{W}) &\\epsilon\\in(1\/2,1)\n\t\\end{dcases}.\n\\end{align}\n\n\\subsection{Information quantities}\n\nClassically, for two distributions $P$ and $Q$, the \\emph{relative entropy} $D(P\\|Q)$ and \\emph{relative entropy variance} $V(P\\|Q)$ are both defined as the mean and variance of the log-likelihood ratio $\\log \\bigl(P\/Q\\bigr)$ with respect to the distribution $P$. In the non-commutative case, for $\\rho,\\sigma \\in \\mathcal{S}$ with $\\rho \\ll \\sigma$, these definitions are generalised as~\\cite{umegaki62,li12,tomamichel12}\n\\begin{align}\n\tD(\\rho\\|\\sigma)&:=\\Tr\\rho\\left(\\log\\rho-\\log\\sigma\\right), \\\\\n\tV(\\rho\\|\\sigma)&:=\\Tr\\rho\\bigl(\\log\\rho-\\log\\sigma-D\\left(\\rho\\|\\sigma\\right)\\cdot\\mathrm{id}\\bigr)^2\\,.\n\\end{align}\nIf $\\rho \\not\\ll \\sigma$ both quantities are set to $+\\infty$. \n\nFollowing Ref.~\\cite{tomamicheltan14}, for a closed set $\\mathcal{S}_{\\circ}\\in \\mathcal{S}$, the \\emph{divergence radius}\\footnote{Whilst \\cref{eqn:radius} characterises the divergence radius, we will mostly rely on a more useful form presented in \\cref{defn:radius}.} $\\chi(\\mathcal{S}_{\\circ})$ is given by\n\\begin{align}\n\t\\label{eqn:radius}\n\t\\chi(\\mathcal{S}_{\\circ})\n\t=\\sup_{\\mathbb{P}\\in \\mathcal{P}(\\mathcal{S}_{\\circ})}\\int\\mathrm{d}\\mathbb{P}(\\rho)\\,D\\left( \\rho\\, \\middle\\| \\, \\int\\mathrm{d}\\mathbb{P}(\\rho')\\,\\rho' \\right).\n\\end{align}\nwhere $\\mathcal{P}(\\mathcal{S}_{\\circ})$ denotes the space of distributions on $\\mathcal{S}_{\\circ}$. If we let $\\Pi(\\mathcal{S}_{\\circ})$ denote the distributions which achieve the above supremum, we also define the \\emph{minimal and maximal peripheral variance}, $v_{\\min}(\\mathcal{S}_{\\circ})$ and $v_{\\max}(\\mathcal{S}_{\\circ})$, as\n\\begin{align}\n\tv_{\\min}(\\mathcal{S}_{\\circ}):=\\inf_{\\mathbb{P}\\in \\Pi(\\mathcal{S}_{\\circ})} \\int\\mathrm{d}\\mathbb{P}(\\rho)\\,V\\left( \\rho\\, \\middle\\| \\, \\int\\mathrm{d}\\mathbb{P}(\\rho')\\,\\rho' \\right),\\\\\n\tv_{\\max}(\\mathcal{S}_{\\circ}):=\\sup_{\\mathbb{P}\\in \\Pi(\\mathcal{S}_{\\circ})} \\int\\mathrm{d}\\mathbb{P}(\\rho)\\,V\\left( \\rho\\, \\middle\\| \\, \\int\\mathrm{d}\\mathbb{P}(\\rho')\\,\\rho' \\right).\n\\end{align}\n\nFor the image of a quantum channel, the above three information quantities correspond exactly to the three previously defined channel parameters~\\cite{tomamicheltan14}. Specifically, for $\\mathcal{S}_{\\circ}=\\overline{\\mathop{\\mathrm{im}} \\mathcal{W}}$, we have\n\\begin{align}\n\tC(\\mathcal{W})=\\chi(\\mathcal{S}_{\\circ}),\\qquad\\quad\n\tV_{\\min}(\\mathcal{W})=v_{\\min}(\\mathcal{S}_{\\circ}),\\qquad\\quad\n\tV_{\\max}(\\mathcal{W})=v_{\\max}(\\mathcal{S}_{\\circ}).\n\\end{align}\n\n\\subsection{Moderate deviation tail bounds}\n\\label{subsec:moddev}\n\nWe now discuss the relevant tail bounds we will require in the moderate deviation regime. Let $\\lbrace X_{i,n}\\rbrace_{i\\leq n}$ be independent zero-mean random variables, and define the average variance as\n\\begin{align}\n\tV_n:=\\frac{1}{n}\\sum_{i=1}^{n}\\Var[X_{i,n}].\n\\end{align}\n\nRecall that a sequence $\\lbrace t_n\\rbrace_n$ is moderate if $x_n\\searrow 0$ \nand $\\sqrt nx_n\\to+\\infty$ as $n\\to\\infty$. Given certain bounds on the moments \nand cumulants \nof these variables, which we will make explicit below, we will see that the \nprobability that the average variable $\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}$ \ndeviates from the mean by a moderate sequence $\\lbrace t_n\\rbrace_n$ decays \nasymptotically as\n\\begin{align}\n\t\\label{eqn:asymp}\n\t\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}\\geq t_n\\right]=-\\bigl(1+o(1)\\bigr)\\frac{nt_n^2}{2V_n}.\n\\end{align}\n\n\\begin{lem}[Moderate deviation lower bound]\n\t\\label{lem:moddev lower}\n\tIf there exist constants $\\nu>0$ and $\\tau$ such that $\\nu\\leq V_n$ and \n\t\\begin{align}\n\t\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}\\left[\\abs{X_{i,n}}^3\\right]\\leq \\tau\n\t\\end{align} \n\tfor all $n$, then for any $\\eta>0$ there exists a constant $N(\\lbrace t_i\\rbrace,\\nu,\\tau,\\eta)\n\t$ such that, for all $n\\geq N$, the probability of a moderate deviation is lower bounded as\n\t\\begin{align}\n\t\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^nX_{i,n}\\geq t_n\\right]\\geq \n\t-(1+\\eta)\\frac{nt_n^2}{2V_n}.\n\t\\end{align}\n\\end{lem}\n\n\\begin{lem}[Moderate deviation upper bound]\n\t\\label{lem:moddev upper}\n\tIf there exists a constant $\\gamma$ such that \n\t\\begin{align}\n\t\\frac{1}{n}\\sum_{i=1}^{n}\\sup_{s\\in[0,1\/2]}\\abs{\\frac{\\mathrm{d}^3}{\\mathrm{d}s^3}\\ln\\mathbb{E}\\left[e^{sX_{i,n}}\\right]}\\leq \\gamma,\n\t\\end{align}\n\tfor all $n$, then for any $\\eta>0$ there exists a constant $N(\\lbrace t_i\\rbrace, \\gamma,\\eta)\n\t$ such that, for all $n\\geq N$, the probability of a moderate deviation is upper bounded as\n\t\\begin{align}\n\t\\ln \\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}\\geq t_n\\right]\\leq -\\frac{nt_n^2}{2V_n+\\eta}.\n\t\\end{align}\n\\end{lem}\n\nIf $V_n$ has a uniform lower bound, then as $\\eta\\searrow 0$ the above two bounds sandwich together, giving the two-sided asymptotic scaling of Eq.~\\ref{eqn:asymp}. In this case we can see that\n\\begin{align}\n\t\\sigma\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_i\\right]=\\sqrt{V_n\/n}=\\Theta(1\/\\sqrt{n})\n\t\\qquad\\text{and}\\qquad\n\t\\sqrt{\\frac{1}{n}\\sum_{i=1}^{n}\\sigma^2\\left[X_i\\right]}=\\sqrt{V_n}=\\Theta(1),\n\\end{align}\nwhere $\\sigma\\left[\\cdot\\right]$ denotes the standard deviation. If we interpret the standard deviation as setting the `length-scale' on which a distribution decays, then the above two quantities---the deviation of the average, and average\\footnote{More specifically the root-mean-square} of the deviation---set the length-scales of small and large deviation bounds respectively. Using this intuition, we can generalise moderate deviation bounds to give tight two-sided bounds for distributions with arbitrary normalisation, in which $V_n$ is no longer bounded. To do this we will tail bound for deviations which are moderate, \\emph{in units of }$\\sqrt{V_n}$.\n\n\\begin{cor}[Dimensionless moderate deviation bound]\n\t\\label{cor:moddev nondim}\n\tIf there exists a $\\gamma$ such that\n\t\\begin{align}\n\t\t\\frac{1}{nV_n^{3\/2}}\\sum_{i=1}^{n}\\sup_{s\\in[0,1\/2]}\\abs{\\frac{\\mathrm{d}^3}{\\mathrm{d}s^3}\\ln\\mathbb{E}\\left[e^{sX_{i,n}}\\right]}\\leq \\gamma,\n\t\\end{align}\n\tfor all $n$,\n\tthen there exists a constant $N(\\lbrace t_i\\rbrace,\\gamma)$ such that, for all $n\\geq N$, we have the two-sided bound \n\t\\begin{align}\n\t\t-(1+\\eta)\\frac{nt_n^2}{2}\\leq \\ln \\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}\\geq t_n\\sqrt{V_n}\\right]\\leq -(1-\\eta)\\frac{nt_n^2}{2}.\n\t\\end{align}\n\\end{cor}\n\nWe present proofs of these lemmas in \\cref{app:tailbounds}.\n\n\\subsection{Reversing lemma}\n\nIntuitively one might expect that moderate deviation bounds can be `reversed' e.g.\\ that the bound on the probability given the deviation (see Lemmas \\ref{lem:moddev lower} and \\ref{lem:moddev upper}) of the form\n\\begin{align}\n\t\\lim\\limits_{n\\to \\infty}\\frac{V_n}{nt_n^2}\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_i\\geq t_n\\right]=-\\frac{1}{2},\n\\end{align}\nis equivalent to a bound on the deviation given the probability\n\\begin{align}\n\t\\lim\\limits_{n\\to\\infty}\\frac{1}{t_n}\\inf\\left\\lbrace t\\in\\mathbb{R} \\,\\middle|\\, \\frac{V_n}{nt_n^2}\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_i\\geq t\\right]\\leq -\\frac{1}{2} \\right\\rbrace=\n\\end{align}\n\nWe will now see that such an ability to `reverse' moderate deviation bounds is generic. We do this by considering two quantities $A$ and $B$ defined on the same domain, and considering the infimum value of each quantity for a fixed value of the other. \n\n\\begin{lem}[Reversing Lemma]\n\t\\label{lem:reverse}\n\tLet $\\lbrace A_i\\rbrace_i$ and $\\lbrace B_i\\rbrace_i$ be sequences of real functions with $\\inf_{t} A_i(t)\\leq 0$ and $\\inf_{t}B_i(t)\\leq 0$ for all $i$. If we define $\\hat{A}_n(b):=\\inf_{t} \\left\\lbrace A_n(t) \\middle| B_n(t)\\leq b \\right\\rbrace$ and $\\hat{B}_n(a):=\\inf_{t} \\left\\lbrace B_n(t) \\middle| A_n(t)\\leq a \\right\\rbrace$, then\n\t\\begin{align}\n\t\t\\lim\\limits_{n\\to\\infty}\\frac{\\hat{A}_n(b_n)}{b_n}=1, \\quad\\forall \\lbrace b_n\\rbrace\\text{ moderate}\n\t\t~~\\qquad&\\Longleftrightarrow\\qquad~~ \n\t\t\\lim\\limits_{n\\to\\infty}\\frac{\\hat{B}_n(a_n)}{a_n}=1, \\quad\\forall \\lbrace a_n\\rbrace\\text{ moderate}.\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tSee \\cref{app:reverse}.\n\\end{proof}\n\n\n\\section{Hypothesis testing}\n\\label{sec:hypo}\n\nWhilst the divergence radius characterises the channel capacity, one-shot channel bounds are characterised by a quantity known as the \\emph{$\\epsilon$-hypothesis testing divergence}~\\cite{wang10}. As the name suggests, as well as being relevant to one-shot channel coding bounds, the hypothesis testing divergence also has an operational interpretation in the context of hypothesis testing of quantum states. We will start by considering a moderate deviation analysis of this quantity.\n\n\\subsection{Hypothesis testing divergence}\n\nConsider a hypothesis testing problem, in which $\\rho$ and $\\sigma$ correspond to the null and alternative hypotheses respectively. A test between these hypotheses will take the form of a POVM $\\lbrace Q,I-Q\\rbrace$, where $0\\leq Q\\leq I$. For a given $Q$, the type-I and type-II error probabilities are given by \n\\begin{align}\n\t\\alpha(Q;\\rho,\\sigma):=\\Tr (I-Q)\\rho,\\qquad\\qquad\n\t\\beta(Q;\\rho,\\sigma):=\\Tr Q\\sigma.\n\\end{align}\nIf we define the smallest possible type-II error given a type-I error at most $\\epsilon$ as\n\\begin{align}\n\t\\beta_\\epsilon(\\rho\\|\\sigma):=\\min_{0\\leq Q\\leq \\mathbb{I}}\\left\\lbrace \\beta(Q;\\rho,\\sigma) \\,\\middle|\\, \\alpha(Q;\\rho,\\sigma)\\leq\\epsilon \\right\\rbrace,\n\\end{align}\nthen the $\\epsilon$-hypothesis testing divergence is defined as\n\\begin{align}\n\tD^\\epsilon_{\\mathrm{h}}(\\rho\\|\\sigma):=-\\log\\frac{\\beta_{\\epsilon}(\\rho\\|\\sigma)}{1-\\epsilon}.\n\\end{align}\nWe note that the denominator of $1-\\epsilon$ follows the normalisation in~\\cite{dupuis12} such that $D_{\\mathrm{h}}^\\epsilon(\\rho\\|\\rho)=0$ for all $\\rho$.\n\nAn obvious extension of this hypothesis problem is to the case of $n$ copies of each state, i.e.\\ a hypothesis test between $\\rho^{\\otimes n}$ and $\\sigma^{\\otimes n}$, or more generally between two product states $\\otimes_{i=1}^n\\rho_i$ and $\\otimes_{i=1}^n\\sigma_i$. A second-order analysis of the $\\epsilon$-hypothesis testing divergence for a non-vanishing $\\varepsilon$ was given in~\\cite{li12,tomamichel12}.\n\n\\begin{thm}[Moderate deviation of the hypothesis testing divergence]\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and states $\\lbrace \\rho_n\\rbrace_n$ and $\\lbrace \\sigma_n\\rbrace_n$ such that both $\\lambda_{\\min}(\\sigma_i)$ and $V(\\rho_i\\|\\sigma_i)$ are both uniformly bounded away from zero, the $\\epsilon_n$- and $(1-\\epsilon_n)$-hypothesis testing divergences of non-uniform product states for $\\epsilon_n=e^{-na_n^2}$ scale as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&=D_n-\\sqrt{2V_n}\\,a_n+o(a_n), \\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&=D_n+\\sqrt{2V_n}\\,a_n+o(a_n),\n\t\\end{align}\n\twhere $D_n:=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma_i)$ and $V_n:=\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma_i)$. More specifically for any $\\rho$ and $\\sigma$ such that $\\rho\\ll \\sigma$, the hypothesis testing divergences of uniform product states scale as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\rho^{\\otimes n} \\middle\\|\\sigma^{\\otimes n}\\right)&=D(\\rho\\|\\sigma)-\\sqrt{2V(\\rho\\|\\sigma)}\\,a_n+o(a_n), \\label{eq:hypo-mod}\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\rho^{\\otimes n} \\middle\\|\\sigma^{\\otimes n}\\right)&=D(\\rho\\|\\sigma)+\\sqrt{2V(\\rho\\|\\sigma)}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{thm}\n\nIn Sect.~\\ref{subsec:inward} we will bound the regularised hypothesis testing divergences towards the relative entropy (the \\emph{inward bound}), and in Sect.~\\ref{subsec:outward} we will bound them away (the \\emph{outward bound}).\n\n\n\\begin{remark}\n\tFor sequences $\\varepsilon_n$ bounded away from zero and one the second-order expansion in Refs.~\\cite{li12,tomamichel12} yields \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\rho^{\\otimes n} \\middle\\|\\sigma^{\\otimes n}\\right) = D(\\rho\\|\\sigma) + \\sqrt{\\frac{V(\\rho\\|\\sigma)}{n}}\\, \\Phi^{-1}(\\varepsilon_n)  + O\\left(\\frac{\\log n}{n}\\right) , \\label{eq:hypo-so}\n\t\\end{align}\n\twhere $\\Phi$ denotes the cumulative distribution function of the standard normal.\n\tAs already pointed out in Ref.~\\cite{polyanskiy10c}, for small $\\varepsilon_n$ we have $\\Phi^{-1}(\\varepsilon_n) \\approx \\sqrt{- 2 \\ln \\varepsilon_n}$. Ignoring all higher order terms, the substitution $\\varepsilon_n = e^{-n a_n^2}$ into~\\eqref{eq:hypo-so} then recovers the expression in~\\eqref{eq:hypo-mod}. In this sense the two results thus agree at the boundary between small and moderate deviations.\n\\end{remark}\n\n\\begin{remark}\n\tA similar argument can be sketched at the boundary between moderate and large \n\tdeviations.\n\tThe quantum Hoeffding bound~\\cite{nagaoka06,hayashi07} states that if $\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}(\\rho^{\\otimes n}\\|\\sigma^{\\otimes n})\\leq D(\\rho\\|\\sigma) - r$ for some small $r > 0$ then $\\varepsilon_n$ drops exponentially in $n$ with the exponent given by\n\t\\begin{align}\n\t\t\\sup_{0\\leq\\alpha<1}\\frac{\\alpha-1}{\\alpha}\\Bigl[D(\\rho\\|\\sigma)-r-D_{\\alpha}(\\rho\\|\\sigma)\\Bigr] , \\label{eqn:expr1}\n\t\\end{align}\n\twhere $D_{\\alpha}(\\rho\\|\\sigma)$ is the Petz' quantum R\\'enyi relative entropy~\\cite{petz86}. For sufficiently small $r$, the expression in~\\eqref{eqn:expr1} attains its supremum close to $\\alpha = 1$ and we can thus approximate $D_{\\alpha}(\\rho\\|\\sigma) \\approx D(\\rho\\|\\sigma) + \\frac{\\alpha-1}{2} V(\\rho\\|\\sigma)$ by its Taylor expansion~\\cite{lintomamichel14}. Evaluating this approximate expression yields \n\t\\begin{align}\n\t\t\\varepsilon_n = e^{- n\\frac{r^2}{2V(\\rho\\|\\sigma)}} \\,. \\label{eqn:expr2}\n\t\\end{align} \n\tup to leading order in $r$. Substituting $r = \\sqrt{2V(\\rho\\|\\sigma)}a_n$ into~\\eqref{eqn:expr2} then recovers~\\eqref{eq:hypo-mod}.\n\tAn essentially equivalent argument is also applicable to the strong converse exponent derived in Ref.~\\cite{mosonyiogawa13}.\n\\end{remark}\n\n\\subsection{Nussbaum--Szko\\l a distributions}\n\nTo allow us to apply a moderate deviation analysis to the quantum hypothesis testing divergence, we leverage the results of Ref.~\\cite{tomamichel12} which allow us to reduce the hypothesis testing divergence of quantum states to a quantity known as the information spectrum divergence of certain classical distributions, known as the Nussbaum--Szko\\l a distributions. \n\n\\begin{defn}[Nussbaum--Szko\\l a distributions~\\cite{NussbaumSzkola2009}]\n\tThe \\emph{Nussbaum--Szko\\l a distributions} for a pair of states $\\rho$ and $\\sigma$ are given by\n\t\\begin{align}\n\t\tP^{\\rho,\\sigma}(a,b)=r_a\\abs{\\braket{\\phi_a}{\\psi_b}}^2\\qquad\\text{and}\\qquad Q^{\\rho,\\sigma}(a,b)=s_b\\abs{\\braket{\\phi_a}{\\psi_b}}^2\n\t\\end{align}\n\twhere the states are eigendecomposed as $\\rho=\\sum_ar_a\\ketbra{\\phi_a}{\\phi_a}$ and $\\sigma=\\sum_bs_b\\ketbra{\\psi_b}{\\psi_b}$. \n\\end{defn}\n\nThe power of the Nussbaum--Szko\\l a distributions lies in their ability to reproduce both the divergence and variance of the underlying quantum states\n\\begin{align}\n\tD(\\rho\\|\\sigma)=D(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})\n\t,\n\t\\qquad \\text{and}\\qquad \n\tV(\\rho\\|\\sigma)=V(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})\n\t.\n\\end{align}\nAs well as capturing these asymptotic quantities, the hypothesis testing relative entropy, which arises one-shot channel coding  bounds, can also be captured by the Nussbaum--Szko\\l a distributions. Specifically this is done via the \\emph{information spectrum divergence}, which is defined for two classical distributions $P$ and $Q$ by a tail bound on the log-likelihood ratio as\n\\begin{align}\n\tD_{\\mathrm{s}}^{\\epsilon}(P\\|Q):=\\sup\\left\\lbrace R ~\\middle|~ \\Pr_{X\\leftarrow P}\\left[\\log \\frac{P(X)}{Q(X)}\\leq R\\right]\\leq \\epsilon \\right\\rbrace.\n\\end{align}\nInserting the Nussbaum--Skzo\\l a distributions, we find that the (classical) information spectrum divergence approximates the (quantum) hypothesis testing divergence.\n\\begin{lem}[Thm.~14, Ref.~\\cite{tomamichel12}]\n\t\\label{lem:infospecdiv}\n\tThere exists a universal constant $K$ such that for any states $\\rho$ and $\\sigma$ with $\\lambda_{\\min}(\\sigma)\\geq \\lambda$ and $\\epsilon< 1\/2$, we find that $D_{\\mathrm{h}}^\\epsilon(\\rho\\|\\sigma)$ is bounded as\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^\\epsilon(\\rho\\|\\sigma)&\\leq D_{\\mathrm{s}}^{2\\epsilon}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})+\\log\\frac{1-\\epsilon}{\\epsilon^3(1-2\\epsilon)}+\\log K \\lceil \\ln(1\/\\lambda)\\rceil\\\\\n\t\tD_{\\mathrm{h}}^\\epsilon(\\rho\\|\\sigma)&\\geq D_{\\mathrm{s}}^{\\epsilon\/2}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})-\\log\\frac{1}{\\epsilon(1-\\epsilon)}-\\log K \\lceil \\ln(1\/\\lambda)\\rceil,\n\t\\end{align}\n\tand $D_{\\mathrm{h}}^{1-\\epsilon}(\\rho\\|\\sigma)$ is bounded as\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{1-\\epsilon}(\\rho\\|\\sigma)&\\leq D_{\\mathrm{s}}^{1-\\epsilon\/2}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})+\\log\\frac{1-\\epsilon\/2}{\\epsilon^4}+\\log K \\lceil \\ln(1\/\\lambda)\\rceil\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon}(\\rho\\|\\sigma)&\\geq D_{\\mathrm{s}}^{1-2\\epsilon}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})-\\log\\frac{1}{\\epsilon^2}-\\log K \\lceil \\ln(1\/\\lambda)\\rceil.\n\t\\end{align}\n\\end{lem}\n\nAs the information spectrum divergence is defined in terms of a tail bound, we will bound these quantities using the moderate deviation tail bounds of Sect.~\\ref{subsec:moddev}. To do this, we will start by showing that the log-likelihood ratio of Nussbaum--Skzo\\l a distributions is sufficiently well behaved, specifically that its cumulant generating function has bounded derivatives.\n\n\\begin{lem}[Bounded cumulants]\n\t\\label{lem:boundedcumulants}\n\tFor $\\lambda>0$, there exists constants $C_k(\\lambda)$ such that the cumulant generating function $h(t):=\\ln \\mathbb{E}\\left[e^{tZ}\\right]$ of the log-likelihood ratio $Z:=\\log P^{\\rho,\\sigma}\/Q^{\\rho,\\sigma}$ for $\\lambda_{\\min}(\\sigma)\\geq \\lambda$ is smooth and has uniformly bounded derivatives in a neighbourhood of the origin\n\t\\begin{align}\n\t\\sup_{\\abs{t}\\leq 1\/2}\\abs{\\frac{\\partial^k}{\\partial t^k}h(t)} \\leq C_k.\n\t\\end{align}\n\\end{lem}\nWe present a proof of this lemma in \\cref{app:cumulant}.\n\n\\subsection{Inward bound}\n\\label{subsec:inward}\n\\begin{prop}[Inward bound]\n\t\\label{prop:HTD-inward}\n\tFor any constants $\\lambda,\\eta>0$, there exists a constant $N(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N$, the hypothesis testing divergence can be bounded for any states $\\lbrace\\rho_i\\rbrace_i$ and $\\lbrace \\sigma_i\\rbrace_i$ with\n\t$\\lambda_{\\min}(\\sigma_i)\\geq\\lambda$ as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\sqrt{2V_n}a_n-\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\sqrt{2V_n}\\,a_n+\\eta a_n.\n\t\\end{align}\n\twhere $D_n:=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma_i)$ and $V_n:=\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma_i)$.\n\\end{prop}\n\\begin{proof}\n\tFirstly, let $Z_i$ be the log-likelihood ratios \n\t\\begin{align}\n\tZ_i:=\\log\\frac{P^{\\rho_i,\\sigma_i}(A_i,B_i)}{Q^{\\rho_i,\\sigma_i}(A_i,B_i)}, \\qquad (A_i,B_i)\\leftarrow P^{\\rho_i,\\sigma_i}.\n\t\\end{align}\n\tIn terms of these log-likelihood ratios, the lower and upper bound on the $\\epsilon_n$- and $(1-\\epsilon_n)$-hypothesis testing divergences respectively from \\cref{lem:infospecdiv} become\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq \\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace-\\log \\frac{1}{\\epsilon_n(1-\\epsilon_n)}-\\log Kn\\lceil\\ln(1\/\\lambda) \\rceil,\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq \\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq 1-\\epsilon_n\/2 \\right\\rbrace+\\log \\frac{1-\\epsilon_n\/2}{\\epsilon_n^4}+\\log Kn\\lceil\\ln(1\/\\lambda) \\rceil.\n\t\\end{align}\n\t\n\tRecalling that $\\epsilon_n:=e^{-na_n^2}$, we can see that in both cases the error terms scale like $\\Theta(na_n^2)$ and $\\Theta(\\log n)$ respectively, which are both $o(na_n)$. As such, there must exist an $N_1(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N_1$, these error terms are bounded by $\\eta na_n\/2$ as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq \\frac{1}{n}\\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace-\\eta a_n\/2,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq \\frac{1}{n}\\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq 1-\\epsilon_n\/2 \\right\\rbrace+\\eta a_n \/2.\n\t\\end{align}\n\t\n\tNext we want to apply the tail bounds of Sect.~\\ref{subsec:moddev}. To this end, we will start by defining zero-mean variables $X_i:=Z_i-D(\\rho_i\\|\\sigma_i)$. In terms of these variables, the above bounds take the form\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (-X_i)\\geq t\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace-\\eta a_n\/2,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n \\left(+X_i\\right)\\geq  t\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace+\\eta a_n\/2.\n\t\\end{align}\n\t\n\tBy \\cref{lem:boundedcumulants} there exists constants $\\bar V(\\lambda)$ and $\\gamma(\\lambda)$, such that $V_i\\leq \\bar V$ and\n\t\\begin{align}\n\t\\sup_{t\\in[0,1\/2]}\\abs{\\frac{\\mathrm{d}^3}{\\mathrm{d}s^3}\\ln\\mathbb{E}\\left[e^{s(\\pm X_i)}\\right]}\\leq \\gamma\n\t\\end{align}\n\tfor all $i$. If we let $t_n:=\\left(\\sqrt{2V_n}+\\eta\/2\\right)a_n$, then \\cref{lem:moddev upper} gives an $N_2(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N_2$, the tail probability is bounded as\n\t\\begin{align}\n\t\t\\ln\\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq t_n\\Biggr]\n\t\t&\\leq \\frac{-nt_n^2}{2V_n+\\eta^2 \/3}\\\\\n\t\t&\\leq -\\frac{\\left(\\sqrt{2V_n}+\\eta\/2\\right)^2}{2V_n+\\eta^2\/5}na_n^2\\\\\n\t\t&\\leq -\\frac{2V_n+\\eta^2\/4}{2V_n+\\eta^2\/5}na_n^2\\\\\n\t\t&= -\\left(1+\\frac{\\eta^2}{40V_n+4\\eta^2}\\right)na_n^2\\\\\n\t\t&\\leq -\\left(1+\\frac{\\eta^2}{40\\bar{V}+4\\eta^2}\\right)na_n^2.\n\t\\end{align}\n\tAs $\\eta^2\/(40\\bar{V}+4\\eta)$ is a constant and $na_n^2\\to \\infty$, there must exist a constant $N_3(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such $n\\geq N_3$ implies\n\t\\begin{align}\n\t\t\\ln\\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq t_n\\Biggr]\\leq -na_n^2-1=\\ln (\\epsilon_n\/2),\n\t\\end{align}\n\tand therefore that \n\t\\begin{align}\n\t\t\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq t\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace\\leq t_n.\n\t\\end{align}\n\tPutting everything together, we get that for any $n\\geq N(\\lbrace a_i\\rbrace,\\lambda,\\eta):=\\max\\lbrace N_1,N_2,N_3\\rbrace$ we have\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\sqrt{2V_n}\\,a_n-\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\sqrt{2V_n}\\,a_n+\\eta a_n.\n\t\\end{align}\n\tas required.\n\\end{proof}\n\n\\subsection{Outward bound}\n\\label{subsec:outward}\n\n\\begin{prop}[Outward bound]\n\t\\label{prop:HTD-outward}\n\tFor any constants $\\lambda,\\eta>0$, there exists a constant $N(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N$, the hypothesis testing divergence can be bounded for any states $\\lbrace\\rho_i\\rbrace_i$ and $\\lbrace\\sigma_i\\rbrace_i$ with\n\t$\\lambda_{\\min}(\\sigma_i)\\geq\\lambda$ as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\eta\\,a_n.\n\t\\end{align}\n\twhere $D_n:=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma_i)$. Moreover, if we \n\tlet $V_n:=\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma_i)$, and there also \n\texists a constant $\\nu>0$ such that $V_i\\geq \\nu$ for all $i$, then there \n\texists an $N'(\\lbrace a_i\\rbrace,\\lambda,\\nu,\\eta)$ such that, for $n\\geq \n\tN'$, the hypothesis testing divergence is more tightly bounded as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\sqrt{2V_n}a_n+\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\sqrt{2V_n}\\,a_n-\\eta a_n.\n\t\\end{align}\n\\end{prop}\n\\begin{proof}\n\tSimilar to \\cref{prop:HTD-inward}, we will start by taking the upper and lower bounds on the $\\epsilon_n$- and $(1-\\epsilon_n)$-hypothesis testing divergences respectively from \\cref{lem:infospecdiv}. \n\tThis gives that there exists an $N_1(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N_1$, we have\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (-X_i)\\geq t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace+\\eta a_n\/2,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n \\left(+X_i\\right)\\geq  t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace-\\eta a_n\/2.\n\t\\end{align}\n\twhere $X_i:=Z_i-D(\\rho_i\\|\\sigma_i)$.\n\t\n\tFirstly, applying Chebyshev's inequality two standard deviations below the mean gives us that\n\t\\begin{align}\n\t\t\\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq  -2\\sqrt{V_n\/n}\\Biggr]\\geq 3\/4\\geq2\\epsilon_n,\n\t\\end{align}\n\tand so we conclude that\n\t\\begin{align}\n\t\t\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n \\left(\\pm X_i\\right)\\geq  t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace\\geq -2\\sqrt{V_n\/n}.\n\t\\end{align}\n\tBy \\cref{lem:boundedcumulants}, $V_n$ must be bounded $V_n\\leq \\bar{V}(\\lambda)$, and thus $\\sqrt{V_n\/n}=\\mathcal{O}(1\/\\sqrt{n})=o(a_n)$. As such, there must exist an $N_2(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that $n\\geq N_2$ implies $2\\sqrt{V_n\/n}\\leq \\eta a_n\/2$. Inserting this tail bound, we get that for any $n\\geq N(\\lbrace a_i\\rbrace,\\lambda,\\eta):=\\max\\lbrace N_1,N_2\\rbrace$ that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\eta a_n,\n\t\\end{align}\n\tas required.\n\t\n\tIf there also exists an $\\nu > 0$ such that $V_i\\geq \\nu$, then we can use a more refined moderate deviation bound. Specifically, \\cref{lem:boundedcumulants} gives us a bound on the absolute third moment of $X_i$, which allows us to apply \\cref{lem:moddev lower}. If we let $t_n:=(\\sqrt{2V_n}-\\eta\/2)a_n$ and assume $\\eta<\\sqrt{8\\nu}$ such that $\\lbrace t_n\\rbrace_n$ is moderate, then this gives us that there exists an $N_3(\\lbrace a_i\\rbrace, \\lambda,\\nu,\\eta)$ such that, for any $n\\geq N_3$, the tail probabilities are bounded\n\t\\begin{align}\n\t\t\\ln \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\n\t\t\\geq t_n\\Biggr]\n\t\t&\\geq -\\Bigl(1+\\eta\/\\sqrt{2\\bar{V}}\\,\\Bigr)\\frac{nt_n^2}{2V_n} \\\\\n\t\t&\\geq -\\frac{\\left(1+\\eta\/\\sqrt{2\\bar{V}}\\right)\\left(\\sqrt{2V_n}-\\eta\/2\\right)^2}{2V_n}na_n^2 \\\\\n\t\t&\\geq -\\frac{\\left(1+\\eta\/\\sqrt{2V_n}\\right)\\left(\\sqrt{2V_n}-\\eta\/2\\right)^2}{2V_n}na_n^2 \\\\\n\t\t&\\geq -\\left(1-\\frac{5\\eta^2}{8\\bar V}\\right)na_n^2.\n\t\\end{align}\n\tOnce again, the second term in the parenthesis is a non-zero constant, and thus there must exist an $N_4(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that\n\t\\begin{align}\n\t\t\\log \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\n\t\t\\geq t_n\\Biggr]\\geq -na_n^2+1=\\ln 2\\epsilon_n,\n\t\\end{align}\n\tallowing us to conclude $\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\n\t\\geq t_n\\right]\\geq 2\\epsilon_n$, and therefore \n\t\\begin{align}\n\t\t\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n X_i\\geq t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace\\geq t_n.\n\t\\end{align}\n\tInserting this into the above bounds, we find that for any $n\\geq N'(\\lbrace a_i\\rbrace,\\lambda,\\nu,\\eta):=\\max\\lbrace N_1,N_3,N_4\\rbrace$, we have the desired final bound\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\sqrt{2V_n}\\,a_n+\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\sqrt{2V_n}\\,a_n-\\eta a_n.\n\t\\end{align}\n\\end{proof}\n\n\n\\section{Channel Coding}\n\\label{sec:channels}\n\nWe are now going to show how the above moderate deviation bounds can be applied to the capacity of a classical-quantum channel. \n\n\\begin{thm}[Moderate deviation of c-q channels]\n\t\\label{thm:moddevcode}\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and memoryless c-q channel \n\t$\\mathcal{W}$ with capacity $C(\\mathcal{W})$ and min-dispersion $V_{\\min}(\\mathcal{W})$, being \n\toperated at error probability no larger than $\\epsilon_n:=e^{-na_n^2}$, the \n\toptimal rate deviates below the capacity as\n\t\\begin{align}\n\t\tR^*\\left(\\mathcal{W} ;n,\\epsilon_n\\right)=C(\\mathcal{W})-\\sqrt{2V_{\\min}(\\mathcal{W})}\\,a_n+o(a_n).\n\t\\end{align}\n\tConversely, if the channel has max-dispersion $V_{\\max}$ and is operated at \n\terror probability no larger than $1-\\epsilon_n$, then the optimal rate \n\tdeviates above the capacity as\n\t\\begin{align}\n\t\tR^*\\left(\\mathcal{W};n,1-\\epsilon_n\\right)=C(\\mathcal{W})+\\sqrt{2V_{\\max}(\\mathcal{W})}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{thm}\nIf either the min- or max-dispersion is non-zero, an application of \\cref{lem:reverse} gives an equivalent formulation in terms of the minimal error probability at a given rate.\n\n\\begin{cor}\n\tFor any moderate sequence $\\lbrace s_n\\rbrace_n$, the error probability for a code with min-dispersion $V_{\\min}>0$ deviating below capacity by $s_n$ scales as\n\t\\begin{align}\n\t\t\\lim\\limits_{n\\to \\infty}\\frac{1}{ns_n^2}\\ln\\epsilon^*(\\mathcal{W};n,C-s_n)=-\\frac{1}{2V_{\\min}}.\n\t\\end{align}\n\tSimilarly, for a code with max-dispersion $V_{\\max}>0$ deviating above capacity by $s_n$, the error probability scales\n\t\\begin{align}\n\t\\lim\\limits_{n\\to \\infty}\\frac{1}{ns_n^2}\\ln\\bigl( 1-\\epsilon^*(\\mathcal{W};n,C + s_n)\\bigr)=-\\frac{1}{2V_{\\max}}.\n\t\\end{align}\n\\end{cor}\n\n\\begin{remark}\n\tRecall that our definition of c-q channels does not put any restriction on the input set. In particular, this set may be comprised of quantum states itself such that the c-q channel is just a representation of a quantum channel. Hence, as pointed out in Ref.~\\cite{tomamicheltan14}, our results immediately also apply to classical communication over general image-additive channels~\\cite{wolf14} as well as classical communication over quantum channels with encoders restricted to prepare separable states. We refer the reader to Corollaries 6 and 7 of Ref.~\\cite{tomamicheltan14} for details. \n\\end{remark}\n\nWe will split the proof of \\cref{thm:moddevcode} in two, in Sect.~\\ref{subsec:coding1} we will prove a lower bound on the maximum rate (`achievability'), followed in Sect.~\\ref{subsec:coding2} by a corresponding the upper bound (`optimality'). For the rest of this section, we will fix the channel $\\mathcal{W}$, and omit any dependencies on $\\mathcal{W}$ from here on for notational convenience.\n\n\\subsection{Achievability}\n\\label{subsec:coding1}\n\nFor achievability, we will use a lower bound on the $\\epsilon$-one-shot rate that is essentially due to Hayashi and Nagaoka~\\cite{hayashi03} who analysed the coding problem using the information spectrum method.\n\n\\begin{lem}[Theorem 1 of Ref.~\\cite{wang10}]\n\t\\label{lem:wangrenner}\n\tIf we have a c-q channel which maps from a finite message space $Y$ as $y\\mapsto \\rho^{(y)}$ , then the maximum rate with error probability at most $\\epsilon$ and $1-\\epsilon$, $R^*(\\epsilon)$ and $R^*(1-\\epsilon)$ respectively, are lower bounded\n\t\\begin{align}\n\t\tR^*(\\epsilon) &\\geq \\sup_{P_Y}\n\t\tD^{\\epsilon\/2}_{\\mathrm{h}} \\left( \\pi_{YZ} \\middle\\| \\pi_Y\\otimes \\pi_Z \\right) -\\log\\frac{8(2-\\epsilon)}{\\epsilon}\\\\\n\t\tR^*(1-\\epsilon) &\\geq \\sup_{P_{Y}}\n\t\tD^{1-2\\epsilon}_{\\mathrm{h}} \\left( \\pi_{YZ} \\middle\\| \\pi_Y\\otimes\\pi_Z \\right) -\\log\\frac{8(1-\\epsilon)}{\\epsilon}\n\t\\end{align}\n\twhere $\\pi_{YZ}$ is the joint state of the input and output, with inputs chosen according to the distribution $P_{Y}$\n\t\\begin{align}\n\t\t\\pi_{YZ}:=\\sum_{y\\in Y}P_{Y}(y)\\ketbra{y}{y}_{Y}\\otimes \\rho^{(y)}_{Z}.\n\t\\end{align}\n\\end{lem}\n\n\\begin{prop}[Channel coding: Achievability]\n\t\\label{prop:codingachievability}\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and error probability $\\epsilon_n:=e^{-na_n^2}$, the rate is at least\n\t\\begin{align}\n\t\tR^*\\left(n,\\epsilon_n\\right)\\geq C-\\sqrt{2V_{\\min}}a_n+o(a_n).\n\t\\end{align}\n\tSimilarly, at error probability $1-\\epsilon_n$, the rate is at least\n\t\\begin{align}\n\t\tR^*\\left(n,1-\\epsilon_n\\right)\\geq C+\\sqrt{2V_{\\max}}a_n+o(a_n).\n\t\\end{align}\n\\end{prop}\n\\begin{proof}\n\tLet $X$ be our, possibly infinite, message space. By Lemma 3 of Ref.~\\cite{tomamicheltan14}, there exists a finite subset $Y\\subseteq X$, and a distribution $Q_{Y}$ thereon, such that $D(\\rho\\|\\sigma)=C$ and $V(\\rho\\|\\sigma)=V_{\\min}$ for states\n\t\\begin{align}\n\t\t\\rho:=\\sum_{y\\in Y}Q_{Y}(y)\\ketbra{y}{y}\\otimes \\rho^{(y)} \n\t\t\\qquad \\text{and} \\qquad \\sigma:=\\sum_{y\\in Y}Q_{Y}(y)\\ketbra{y}{y}\\otimes \\sum_{y'\\in Y}Q_{Y}(y')\\rho^{(y')}.\n\t\\end{align}\n\t\n\tClearly by restricting the message space we can only ever decrease the rate. By applying \\cref{lem:wangrenner} to the restriction of the message space to $Y$, we can lower bound the maximum rate of the full code. Applying this reasoning to $n$ memoryless applications of our channel we find\n\t\\begin{align}\n\t\tnR^*(n,\\epsilon_n) \n\t\t&\\geq \\sup_{P_{{Y}^n}} D^{\\epsilon_n\/2}_{\\mathrm{h}} \\left( \\pi_{Y^nZ^n} \\middle\\| \\pi_{Y^n}\\otimes \\pi_{Z^n} \\right) -\\log\\frac{8(2-\\epsilon_n)}{\\epsilon_n}.\n\t\\end{align}\n\tSubstituting in both the error probability, which is no larger than $\\epsilon_n=e^{-na_n^2}$, and a product distribution $Q_{{Y}^n}(\\vec{y}):= \\prod_{i=1}^n Q_{Y}(y_i)$ then we get\n\t\\begin{align}\n\t\tR^*(n,\\epsilon_n) \\geq \n\t\t\\frac{1}{n}D^{\\epsilon_n\/2}_{\\mathrm{h}} \\left( \\rho^{\\otimes n} \\middle\\| \\sigma^{\\otimes n} \\right)+\\mathcal{O}(a_n^2).\\label{ineq:rate}\n\t\\end{align}\n\tApplying \\cref{prop:HTD-inward}, we get an overall bound on the rate of\n\t\\begin{align}\n\t\tR^*(n,\\epsilon_n) \\geq \n\t\tC-\\sqrt{2V_{\\min}}\\,a_n+o(a_n).\n\t\\end{align}\n\tIf instead we were to take a distribution $Q_{Y}$ such that $V(\\rho\\|\\sigma)=V_{\\max}$, then the same arguments would allow us to use \\cref{prop:HTD-outward} to analogously give\n\t\\begin{align}\n\t\tR^*(n,1-\\epsilon_n) \\geq \n\t\tC+\\sqrt{2V_{\\max}}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{proof}\n\n\\subsection{Optimality}\n\\label{subsec:coding2}\nSimilar to the second-order analysis of Ref.~\\cite{tomamicheltan14}, we are going to do this by relating the capacity and one-shot maximum rates to geometric quantities known as the divergence radius and divergence centre. \n\n\\begin{defn}[Divergence radius and centre]\n\t\\label{defn:radius}\n\tFor some set of states $\\mathcal{S}_0\\subseteq\\mathcal{S}$, the \\emph{divergence radius} $\\chi(\\mathcal{S}_0)$ and \\emph{divergence centre} $\\sigma^*(\\mathcal{S}_0)$ are defined as \n\t\\begin{align}\n\t\t\\chi(\\mathcal{S}_0):=\\mathop{\\mathrm{inf}\\vphantom{p}}_{\\sigma\\in\\mathcal{S}} \\sup_{\\rho\\in\\mathcal{S}_0}~D(\\rho\\|\\sigma), \\qquad\\qquad \n\t\t\\sigma^*(\\mathcal{S}_0):=\\mathop\\mathrm{arg~min}\\limits_{\\sigma\\in \\mathcal{S}}\\sup_{\\rho\\in\\mathcal{S}_0}D(\\rho\\|\\sigma).\n\t\\end{align}\n\tSimilarly the\n\t$\\epsilon$-\\emph{hypothesis testing divergence radius} $\\chi_{\\mathrm{h}}^\\epsilon(\\mathcal{S}_0)$ is defined as \n\t\\begin{align}\n\t\t\\chi_{\\mathrm{h}}^{\\epsilon}(\\mathcal{S}_0):=\\mathop{\\mathrm{inf}\\phantom{p}}_{\\sigma\\in\\mathcal{S}} \\sup_{\\rho\\in\\mathcal{S}_0}~D_{h}^{\\epsilon}(\\rho\\|\\sigma).\n\t\\end{align}\n\\end{defn}\n\nWhilst we have seen that the divergence radius captures the capacity of a channel, the $\\epsilon$-hypothesis testing divergence radius approximates the one-shot capacity.\n\n\\begin{lem}[Proposition 5 of \\cite{tomamicheltan14}]\n\t\\label{lem:rate}\n\tFor $\\mathcal{I}:=\\overline{\\mathop{\\mathrm{im}} \\mathcal{W}}$, the maximum rate with error probability at most $\\epsilon$, $R^*(\\epsilon)$, is upper bounded as\n\t\\begin{align}\n\t\tR^*(\\epsilon)\\leq \\chi_{\\mathrm{h}}^{2\\epsilon}(I)+\\log\\frac{2}{1-2\\epsilon}.\n\t\\end{align}\n\tSimilarly for an error probability $1-\\epsilon$, the maximum rate is upper bounded as\n\t\\begin{align}\n\t\tR^*(1-\\epsilon)\\leq \\chi_{\\mathrm{h}}^{1-\\epsilon\/2}(I)+\\log\\frac{2(2-\\epsilon)}{\\epsilon^2}.\n\t\\end{align}\n\\end{lem}\n\nIf we take $\\mathcal{I}_n:=\\overline{\\mathop{\\mathrm{im}} \\mathcal{W}^{\\otimes n}}$ to be the closure of the image of $n$ uses of this channel, then we can extend this bound on the one-shot rate to the $n$-shot rate as\n\\begin{align}\n\tnR^*(n,\\epsilon_n)&\\leq \\chi_{\\mathrm{h}}^{2\\epsilon_n}(\\mathcal{I}_n)+\\log\\frac{2}{1-\\epsilon_n},\\\\\n\tnR^*(n,1-\\epsilon_n)&\\leq \\chi_{\\mathrm{h}}^{2\\epsilon_n}(\\mathcal{I}_n)+\\log\\frac{2(2-\\epsilon_n)}{\\epsilon_n^2}.\n\\end{align}\nAs we are considering memoryless c-q channels, $\\mathcal{I}_n$ simply consists of elementwise tensor products of $\\mathcal{I}$\n\\begin{align}\n\t\\mathcal{I}_n=\\left\\lbrace \\bigotimes_{i=1}^n\\rho_i \\,\\middle|\\, \\rho_i\\in \\mathcal{I} \\right\\rbrace.\n\\end{align}\n\nOnce again we are going to take $a_n$ to be an arbitrary moderate sequence, and $\\epsilon_n:=e^{-na_n^2}$. Expanding this out, this gives bounds on the rate of\n\\begin{align}\n\tR^*(n,\\epsilon_n)&\\leq \\inf_{\\sigma^n}\\sup_{\\lbrace \\rho_i\\rbrace\\subseteq \\mathcal{I}}~\\frac{1}{n}D^{2\\epsilon_n}_{\\mathrm{h}}\\left(\\bigotimes_{i=1}^n\\rho_i \\middle\\|\\sigma^n \\right) +\\frac{1}{n}\\log \\frac{2}{1-\\epsilon_n},\\label{ineq:converse}\\\\\n\tR^*(n,1-\\epsilon_n)&\\leq \\inf_{\\sigma^n}\\sup_{\\lbrace \\rho_i\\rbrace\\subseteq \\mathcal{I}}~\\frac{1}{n}D^{1-\\epsilon_n\/2}_{\\mathrm{h}}\\left(\\bigotimes_{i=1}^n\\rho_i \\middle\\|\\sigma^n \\right) +\\frac{1}{n}\\log \\frac{2(2-\\epsilon)}{\\epsilon_n^2}.\n\\end{align}\n\nA standard approach now is to pick a state $\\sigma^n$, such that we can bound \nthe above quantities for arbitrary sequences $\\lbrace \\rho_i\\rbrace$ using the \nmoderate deviation analysis of the hypothesis testing divergence presented in \nSect.~\\ref{sec:hypo}. To do this we need to consider two cases. The \\emph{high cases} are those in which the empirical relative entropy corresponding to \n$\\lbrace \\rho_i\\rbrace _{i=1}^n$ is close to capacity, and the \\emph{low cases} \nare those in which the empirical relative entropy corresponding to $\\lbrace \n\\rho_i\\rbrace _{i=1}^n$ is far from capacity. Specifically, for some constant \n$\\gamma$ that will be chosen later, the $n$ which correspond to high and low \ncases are denoted by $H(\\lbrace\\rho_i\\rbrace,\\gamma)$ and \n$L(\\lbrace\\rho_i\\rbrace,\\gamma)$, respectively. They are defined as\n\\begin{align}\n\tH(\\lbrace\\rho_i\\rbrace,\\gamma):=\\left\\lbrace n~ \\middle| \n\t\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)\\geq C-\\gamma \\right\\rbrace\n\t\\quad\\text{and}\\quad\n\tL(\\lbrace\\rho_i\\rbrace,\\gamma):=\\left\\lbrace n~ \\middle| \n\t\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)< C-\\gamma \\right\\rbrace\n\\end{align}\nsuch that $H(\\lbrace\\rho_i\\rbrace,\\gamma)$ and $L(\\lbrace\\rho_i\\rbrace,\\gamma)$ \nbipartition $\\mathbb{N}$ for all $\\gamma$.\n\nBefore employing a moderate deviation bound, we are going to construct a separable state $\\sigma^n$ that will allow us two different moderate deviation analyses for low and high sequences, such that we can obtain the required bounds in both cases. A convenient choice of $\\sigma^n$ would be  $\\sigma^n={\\bar\\rho_n}^{\\otimes n}$ where $\\bar\\rho_n:=\\frac{1}{n}\\sum_{i=1}^n\\rho_i$, but the order of the infimum and supremum require $\\sigma^n$ to be chosen to be independent of the sequence $\\lbrace \\rho_i\\rbrace$. Instead we are going to construct $\\sigma^n$ from a mixture of states that lie in a covering of $\\mathcal{S}$, and the divergence centre $\\sigma^*(\\mathcal{I})$. \n\nThe following lemma is based on a construction in Lemma II.4 of Ref.~\\cite{hayden04b}.\n\n\\begin{lem}[Lemma 18 of Ref.~\\cite{tomamicheltan14}]\n\tFor every $\\delta\\in(0,1)$, there exists a set $\\mathcal{C}^\\delta\\subset\\mathcal{S}$ of size\n\t\\begin{align}\n\t\\abs{\\mathcal{C}^\\delta}\\leq \\left(\\frac{20(2d+1)}{\\delta}\\right)^{2d^2}\\left(\\frac{8d(2d+1)}{\\delta}+2\\right)^{d-1}\\leq\\left(\\frac{90d}{\\delta^2}\\right)^{2d^2}\n\t\\end{align}\n\tsuch that, for every $\\rho\\in \\mathcal{S}$ there exists a state $\\tau\\in\\mathcal{C}^\\delta$ such that\n\t\\begin{align}\n\tD(\\rho\\|\\tau)\\leq \\delta\\qquad\n\t\\textrm{and}\\qquad\\lambda_{\\mathrm{min}}(\\tau)\\geq \\frac{\\delta}{8d(2d+1)+\\delta}\\geq \\frac{\\delta}{25d^2}.\n\t\\end{align} \n\\end{lem}\n\nGiven this covering upon states, we now want to take $\\sigma^n$ to be the separable state given by a mixture over such a covering, and the divergence centre\n\\begin{align}\n\t\\sigma^n(\\gamma):=\n\t\\frac{1}{2}\\sigma^*(I)^{\\otimes n}+\n\t\\frac{1}{2\\abs{\\mathcal{C}^{\\gamma\/4}}}\\sum_{\\tau\\in\\mathcal{C}^{\\gamma\/4}} \\tau^{\\otimes n}.\\label{eqn:sigmadef}\n\\end{align}\nUsing the inequality \n\\begin{align}\n\tD_{\\mathrm{h}}^\\epsilon\\bigl(\\rho\\,\\big\\|\\,\\mu\\sigma+(1-\\mu)\\sigma'\\bigr)\\leq D^\\epsilon_{\\mathrm{h}}(\\rho\\|\\sigma)-\\log \\mu\n\\end{align}\nwe will be able to bound divergences with respect to $\\sigma^n$ by those divergences with respect to either elements of $\\mathcal{C}^{\\gamma\/4}$, or $\\sigma^*$.\n\nWe will start by considering the low case. We will see that this case only accounts for hypothesis testing relative entropies which are below the capacity by a constant amount.\n\n\\begin{lem}[Low case]\n\t\\label{lem:low}\n\tFor any $\\gamma>0$, there exists a constant $N(\\lbrace a_i\\rbrace,\\gamma)$ such that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq C-\\gamma\/4,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq C-\\gamma\/4,\n\t\\end{align}\n\tfor any $\\lbrace\\rho_i\\rbrace_i\\subset I$, $n\\in \n\tL(\\lbrace\\rho_i\\rbrace,\\gamma)$ and $n\\geq N$.\n\\end{lem}\n\\begin{proof}\n\tWe are going to start by considering the $\\epsilon_n$-hypothesis testing divergence. Take $\\tau_n$ to be the closest element in $\\mathcal{C}^{\\gamma\/4}$ to $\\bar{\\rho}_n$, such that $D(\\bar{\\rho}_n\\|\\tau_n)\\leq \\gamma\/4$. Splitting out the $\\tau_n$ term from $\\sigma_n(\\gamma)$, we have\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\sigma_n(\\gamma)\\right)\n\t\t&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\tau_n^{\\otimes n}\\right)+\\log 2\\abs{\\mathcal{C}^\\gamma}\\\\\n\t\t&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\tau_n^{\\otimes n}\\right)+2d^2\\log\\left(\\frac{120d}{\\gamma^2}\\right).\n\t\\end{align}\n\tAs the final term depending on $\\abs{\\mathcal{C}^{\\gamma\/4}}$ is independent of $n$, there must exist a constant $N_1(\\gamma)$ such that $2d^2\\log(120d\/\\gamma^2)\\leq n\\gamma\/4$ for any $n\\geq N_1$, and thus that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma)\\right)\n\t\t\\leq \\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\tau_n^{\\otimes n}\\right)+\\gamma\/4.\n\t\\end{align}\n\t\n\tApplying \\cref{prop:HTD-outward} to the $\\epsilon_n$-hypothesis testing relative entropy with respect to $\\tau_n$ we get that there exists an $N_2(\\lbrace a_i\\rbrace,\\gamma)$ such that \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\tau_n^{\\otimes} \\right)\\leq \\frac{1}{n}\\sum_{i=1}^nD(\\rho_i\\|\\tau_n)+\\gamma\/4,\n\t\\end{align}\n\tfor any $n\\geq N_2$. As for the divergence terms given with respect to $\\tau_n$, we can rearrange them in terms of divergences relative to the sequence mean $\\bar\\rho_n$ using the information geometric Pythagorean theorem, yielding\n\t\\begin{align}\n\t\t\\sum_{i=1}^{n}D(\\rho_i\\|\\tau_n)\n\t\t&=\\sum_{i=1}^{n}\\Tr \\rho_i(\\log\\rho_i-\\log\\bar\\rho_n)+\\sum_{i=1}^{n}\\Tr\\rho_i(\\log\\bar\\rho_n-\\log \\tau_n)\\label{eqn:line2}\\\\\n\t\t&=\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)+nD(\\bar\\rho_n\\|\\tau_n)\\\\\n\t\t&\\leq\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)+n\\gamma\/4.\n\t\\end{align}\n\t\n\tIf we let $N(\\lbrace a_i\\rbrace, \\gamma):=\\max\\lbrace N_1,N_2\\rbrace$, then pulling the above results together we see that for any $n\\geq N$ \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma) \\right)\n\t\t&\\leq\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\tau_n^{\\otimes n}\\right)+\\gamma\/4\\\\\n\t\t&\\leq\\frac{1}{n}\\sum_{i=1}^nD(\\rho_i\\|\\tau_n)+2\\gamma\/4\\\\\n\t\t&\\leq\\frac{1}{n}\\sum_{i=1}^nD(\\rho_i\\|\\bar{\\rho}_n)+3\\gamma\/4.\n\t\\end{align}\n\tFinally, since $n\\in L(\\lbrace\\rho_i\\rbrace,\\gamma)$ the average relative \n\tentropy is bounded away from capacity, and we arrive at the bound:\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma) \\right)\\leq C-\\gamma\/4.\n\t\\end{align}\n\t\n\tAs we only relied on \\cref{prop:HTD-outward} to bound the regularised \n\thypothesis testing divergence to within a constant of the average relative \n\tentropy, we could perform a similar analysis for the \n\t$(1-\\epsilon_n)$-hypothesis testing divergence using \\cref{prop:HTD-inward} \n\tinstead, which gives\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma) \\right)\\leq C-\\gamma\/4.\n\t\\end{align}\n\\end{proof}\n\nNow that we have dealt with cases far from capacity, we turn our attention to the high cases.\n\n\\begin{lem}[High case]\n\t\\label{lem:high}\n\tFor any $\\eta>0$, there exist constants $\\Gamma(\\eta)$ and $N(\\lbrace a_i\\rbrace,\\eta)$, such that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\Gamma)\\right)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n\n\t\\end{align}\n\tfor any $\\lbrace\\rho_i\\rbrace_i\\subset I$, $n\\in \n\tH(\\lbrace\\rho_i\\rbrace,\\Gamma)$ and $n\\geq N$. Similarly, the \n\t$(1-\\epsilon_n)$-hypothesis testing relative entropy is bounded\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\Gamma)\\right)\\leq C+\\sqrt{2V_{\\max}}a_n+\\eta a_n.\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tSplitting out the $\\sigma^*$ factor within $\\sigma_n(\\gamma)$ gives\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\log 2,\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\log 2.\n\t\\end{align}\n\tAs $\\frac{1}{n}\\log 2=o(a_n)$, there exists an $N_1(\\lbrace a_i\\rbrace)$ such that $n\\geq N_1$ implies \n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3,\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3.\n\t\\end{align}\n\tWe now wish to employ a moderate deviation result. We will start by addressing the $\\epsilon_n$-hypothesis testing divergence. For the weaker bound of \\cref{prop:HTD-outward} we will have no required bounds on $\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)$, but for the stronger bound we will need a uniform lower bound.\n\t\n\tIf $V_{\\min}\\leq \\eta^2\/18$, then the weakened bound of \\cref{prop:HTD-outward} is sufficient, giving an $N_2(\\lbrace a_n\\rbrace,\\eta)$ such that $n\\geq N_2$ implies\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)+2\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n\\\\\n\t\t&\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n.\n\t\\end{align}\n\t\n\tNext we need to consider the case where $V_{\\min}>\\eta^2\/18$. To do this, we will need to establish a lower bound on $\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)$, which places it near $V_{\\min}$. The min-dispersion is defined for distributions which exactly achieve capacity; we will now consider an analogous quantity for distributions which are \\emph{near} capacity. Specifically\n\t\\begin{align}\n\t\tV_{\\min}(\\gamma):=\\inf_{P\\in\\mathcal{P}(\\mathcal{I})}\\left\\lbrace \n\t\t\\int\\mathrm{d}P(\\rho)~V\\left(\\rho\\middle\\|\\sigma^*\\right) \n\t\t~\\middle|~\n\t\t\\int\\mathrm{d}P(\\rho)~D\\left(\\rho~\\middle\\|\\int\\mathrm{d}P(\\rho')~\\rho'\\right)\\geq C-\\gamma\n\t\t\\right\\rbrace.\n\t\\end{align}\n\tBy definition of the channel dispersion we have that $V_{\\min}(0)=V_{\\min}$. By Lemma 22 of Ref.~\\cite{tomamicheltan14} we can strengthen this to $\\lim_{\\gamma\\to 0^+}V_{\\min}(\\gamma)=V_{\\min}$, and so for any $\\eta>0$ there must exist a constant $\\Gamma(\\eta)$ such that \n\t\\begin{align}\n\t\t\\sqrt{2V_{\\min}(\\Gamma)}\\geq \\sqrt{2V_{\\min}}-\\eta\/3.\\label{eqn:V}\n\t\\end{align} \n\tAs $V_{\\min}\\geq \\eta^2\/18$, this implies that $V_{\\min}(\\Gamma)>0$.\n\t\n\tNext, let $P_n$ be the empirical distribution corresponding to the set \n\t$\\lbrace\\rho_i\\rbrace_{i=1}^n$, i.e.\\ \n\t$P_n(\\rho):=\\frac{1}{n}\\sum_{i=1}^{n}\\delta(\\rho-\\rho_i)$. For all $n\\in \n\tH(\\lbrace\\rho_i\\rbrace,\\Gamma)$, these distributions are near capacity\n\t\\begin{align}\n\t\t\\int\\mathrm{d}P_n(\\rho)~ D\\left(\\rho~\\middle\\|\\int\\mathrm{d}P_n(\\rho')~\\rho'\\right)=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\bar{\\rho}_n)\\geq C-\\Gamma,\n\t\\end{align}\n\tand so we can lower bound the average variance with respect to the divergence centre\n\t\\begin{align}\n\t\t\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)=\\int\\mathrm{d}P(\\rho)~ V(\\rho\\|\\sigma^*)\\geq V_{\\min}(\\Gamma)>0.\n\t\\end{align}\n\tUsing this lower bound, we can apply the stronger bound from \n\t\\cref{prop:HTD-outward} to give a constant $N_3(\\lbrace a_i\\rbrace,\\eta)$, \n\tsuch that, for every $n\\in H(\\lbrace\\rho_i\\rbrace,\\Gamma)$ and $n\\geq N_3$, \n\tthe hypothesis testing divergence is upper bounded\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)-\\sqrt{\\frac{2}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)}a_n+2\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)-\\sqrt{2V_{\\min}(\\Gamma)}\\,a_n+2\\eta a_n\/3\\\\\n\t\t&\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n.\n\t\\end{align}\n\t\n\tPerforming a similar argument for $V_{\\max}$, we construct a function\n\t\\begin{align}\n\t\tV_{\\max}(\\gamma):=\\sup_{P\\in\\mathcal{P}(\\mathcal{I})}\\left\\lbrace \n\t\t\\int\\mathrm{d}P(\\rho)~V\\left(\\rho\\middle\\|\\sigma^*\\right) \n\t\t~\\middle|~\n\t\t\\int\\mathrm{d}P(\\rho)~D\\left(\\rho~\\middle\\|\\int\\mathrm{d}P(\\rho')~\\rho'\\right)\\geq C-\\gamma\n\t\t\\right\\rbrace,\n\t\\end{align}\n\tand define a $\\Gamma$ such that\n\t\\begin{align}\n\t\\sqrt{2V_{\\max}(\\Gamma)}\\leq \\sqrt{2V_{\\max}}+\\eta\/3.\n\t\\end{align} \n\tFollowing through the rest of the argument, and employing \\cref{prop:HTD-inward}, we also get a bound on the $(1-\\epsilon_n)$-hypothesis testing divergence\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)\n\t\t&\\leq C+\\sqrt{2V_{\\max}}a_n+\\eta a_n.\n\t\\end{align} \n\\end{proof}\n\n\\begin{prop}[Channel coding: Optimality]\n\t\\label{prop:codingconverse}\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and error probability $\\epsilon_n:=e^{-na_n^2}$, the rate is upper bounded as\n\t\\begin{align}\n\t\tR^*\\left(n,\\epsilon_n\\right)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+o(a_n).\n\t\\end{align}\n\tFor error probability $(1-\\epsilon_n)$ the rate is similarly upper bound as\n\t\\begin{align}\n\t\tR^*\\left(n,1-\\epsilon_n\\right)\\leq C+\\sqrt{2V_{\\max}}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{prop}\n\\begin{proof}\n\tApplying Lemmas \\ref{lem:low} and \\ref{lem:high}, we get that there exist constants $\\Gamma(\\eta)$ and $N_1(\\lbrace a_i\\rbrace,\\eta)$ such that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\sigma_n(\\Gamma)\\right)\\leq \n\t\t\\begin{dcases}\n\t\t\tC-\\Gamma\/4 & n\\in L(\\lbrace\\rho_i\\rbrace,\\Gamma)\\\\\n\t\t\tC-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n & n\\in H(\\lbrace\\rho_i\\rbrace,\\Gamma)\n\t\t\\end{dcases}\n\t\\end{align}\n\tfor any $n\\geq N_1$. As $\\Gamma$ is a constant, there must exist some $N_2(\\lbrace a_i\\rbrace,\\eta)$ such that $\\Gamma\/4\\geq \\sqrt{2V_{\\min}}a_n$. As such, for any $n\\geq \\max\\lbrace N_1,N_2\\rbrace$, high or low, we have \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\sigma_n(\\Gamma)\\right)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n.\n\t\\end{align}\n\tPulling this bound back to Eq.~\\ref{ineq:converse}, we have\n\t\\begin{align}\n\t\tR^*(n,\\epsilon_n)\n\t\t&\\leq \\sup_{\\lbrace \\rho_i\\rbrace\\subseteq I}\\frac{1}{n}D^{2\\epsilon_n}_{\\mathrm{h}}\\left(\\bigotimes_{i=1}^n\\rho_i \n\t\t\\middle\\|\\sigma^n(\\Gamma) \\right) +\\frac{1}{n}\\log \n\t\t\\frac{2}{1-\\epsilon_n}\\\\\n\t\t&\\leq C-\\sqrt{2V_{\\min}}a_n+\\eta a_n+\\frac{1}{n}\\log \\frac{2}{1-\\epsilon_n}.\n\t\\end{align}\n\tFinally, noting that $1\/n=o(a_n)$, there must exist a constant $N_3 (\\lbrace a_i\\rbrace,\\eta)$ such that $n\\geq N_3$ implies\n\t\\begin{align}\n\t\\frac{1}{n}\\log\\frac{2}{1-\\epsilon_n}\\leq \\eta a_n.\n\t\\end{align}\n\tWe can therefore conclude that, for $n\\geq \\max\\lbrace N_1,N_2,N_3\\rbrace $, we get the overall upper bound\n\t\\begin{align}\n\tR^*(n,\\epsilon_n)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+2\\eta a_n.\n\t\\end{align}\n\tAs this is true for arbitrary $\\eta>0$, we can take $\\eta \\searrow0$ and conclude \n\t\\begin{align}\n\tR^*(n,\\epsilon_n)\\leq C-\\sqrt{2V_{\\min}}a_n+o(a_n)\n\t\\end{align}\n\tas required. A similar analysis for the $(1-\\epsilon_n)$-error regime shows\n\t\\begin{align}\n\tR^*\\left(n,1-\\epsilon_n\\right)\\leq C+\\sqrt{2V_{\\max}}a_n+o(a_n).\n\t\\end{align}\n\\end{proof}\n\n\n\\section{Conclusion}\n\nThe main result of this paper is to give a second order approximation of the non-asymptotic fundamental limit for classical information transmission over a quantum channel in the moderate deviations regime, as in Eqs.~\\ref{eq:moderatesecondorder1} and \\ref{eq:moderatesecondorder2}:\n\\begin{align}\n\t\\frac{1}{n} \\log M^*(\\mathcal{W};n, \\varepsilon_n) &= C(\\mathcal{W}) - \\sqrt{2 V_{\\min}(\\mathcal{W})}\\, x_n + o(x_n) \\,,\\\\\n\t\\frac{1}{n} \\log M^*(\\mathcal{W};n,1- \\varepsilon_n) &= C(\\mathcal{W}) + \\sqrt{2 V_{\\max}(\\mathcal{W})}\\, x_n + o(x_n) \\,.\n\\end{align}\nAlong the lines of third and fourth order approximations for classical channel coding in the fixed error regime (see, e.g., Refs.~\\cite{polyanskiythesis10,tomamicheltan12,moulin12}), a natural question to ask is whether we can expand this further and resolve the term $o(x_n)$. A preliminary investigation suggests the conjecture that $o(x_n) = O(x_n^2) + O(\\log n)$ and that at least some of the implicit constants can be determined precisely. We leave this for future work.\n\nDue to the central importance of binary asymmetric quantum hypothesis testing \nwe expect our techniques to have applications also to other quantum channel \ncoding tasks. In particular, source coding~\\cite{datta15,leditzky16}, \nentanglement-assisted classical coding~\\cite{datta14} as well as \nquantum~\\cite{tomamichel16} and private coding~\\cite{wilde16} over quantum \nchannels have recently been analysed in the small deviations regime by relating \nthe problem to quantum hypothesis testing. An extension of these results to \nmoderate deviations using our techniques thus appears feasible.\n\n\\paragraph*{Acknowledgements.} MT is funded by an Australian Research Council Discovery Early Career Researcher Award (DECRA) fellowship (Grant Nos. CE110001013, DE160100821). Both MT and CTC acknowledge support from the ARC Centre of Excellence for Engineered Quantum Systems (EQuS). We also thank Hao-Chung Cheng and Min-Hsiu Hsieh for useful discussions and insightful comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhite dwarfs (WDs) are degenerate stellar nuclei with a mass roughly that of the\nSun and radii one hundredth that of the Sun; consequently,\ntheir surface gravity is $\\sim$$10^4$ greater than the\nSun's. \\cite{lie84} identified some odd WDs with metal-rich\natmospheres. With such a powerful gravity pulling the chemical\nelements toward the stellar nucleus, it is somewhat unusual to have a metal-rich\natmosphere. In fact, the timescales for an element heavier than\nhydrogen or helium to sink are small: $\\sim$$10^2$ yr in WDs with\nhydrogen atmospheres (DAs) and $\\sim$$10^5$ yr with helium atmospheres\n(DBs) \\citep{jur08,von07,paq86}.\n\nInterstellar material accretion onto the WD surface was one of the first\nexplanations for the metal-rich atmospheres. Knowing the diffusion\ntimescales of metals in the stellar atmospheres and the metallicity of a given\nstar, it is possible to calculate the necessary accretion rate to keep this\nmetallicity constant in time \\citep{koe06}. Typical values are\n$10^{-18}$ to $10^{-15}$ M$_\\odot$yr$^{-1}$. These values are too high to be\nexplained exclusively by interstellar accretion. Furthermore, if there were\naccretion from the interstellar medium onto the DBs, there should be a large\namount of hydrogen pollution in their spectra, but this pollution has not been\ndetected \\citep{dup1993,far2008}.\n\n\\cite{zuc87} observed an IR~excess in the spectrum of G29-38. The shape of this\nIR~excess is a bump which peaks at $\\sim$$10~\\mu m$. Its width is roughly\n$20~\\mu m$ and can be fitted with a blackbody of $T_{\\mathrm{eff}} \\sim 10^3$ K.\n\\cite{gra90} argued that an asteroid closely approaching G29-38 could explain\nthis infrared excess. When the asteroid orbit reaches the Roche radius it is\ndisrupted and forms a disk around the star. This disk is heated up by the\nstellar radiation and emits in the infrared. The disk material falls down\ncontinuously on the WD giving rise to the observed metal-rich atmospheres.\n\nUsing a disk model, Jura \\& collaborators (2003,2007,2008) were able to fit\nthe IR excess of many WDs. Through these fits they determined disk physical\nparameters. With a different approach, \\cite{rea05} fit the spectrum to a thin\ndust shell model.\n\nWhile searching for interacting binary WDs, \n\\cite{gan2006,gan2007,gan2008} observed double peaks in calcium lines\nin a few DAZs and DBZs.  Although these observations strongly suggest\nthe disk hypothesis,it is still possible that in some cases the emitting\nregion could be a torus or a shell \\citep{rea09} rather than a disk.\n\nPrevious works focused on the IR emission properties of\ndebris disks around the WD. In this work, we propose a new and complementary\nobservational test looking at the absorption and scattering properties instead.\nWe develop a simple theoretical framework to predict what will be observable\nand measurable according to the properties of the system. We also suggest an\nobservational program to reach our goal.\n\nWe investigate the possibility of detection of debris disks effects\nin the near-UV and optical. We start the analysis in the limit of an optically\nthick disk in Section~\\ref{sec:od} and in Section~\\ref{sec:ot} we extend this to\nthe optically thin limit case. In Section~\\ref{ObsTest}, we discuss the\nobservational predictions of our models. Our conclusions are presented in\nSection~\\ref{sec:conc}.\n\n\n\n\\section{Opaque disk}\n\\label{sec:od}\n\n  The optically thick limit is the natural first approach to investigate the\npossible effects of a debris disk in the spectrum of a WD. This limit can be\nachieved not only in massive disks but also in certain regions of all disks\nspecially if the disk has some gas like some recently discovered gaseous disks\n\\citep{gan2006,gan2007,gan2008}. Also, the mathematical treatment developed in\nthis section will be used in the next section when dealing with the\noptically thin limit.\n\nA completely opaque disk will not have any\nspectral features because it is totally opaque and absorbs any photon whatever\nits energy. The only effect of an obscuring disk will be a decrease in the\nreceived flux from the star. The most the disk can obscure the star is half of\nthe projected stellar surface, $\\pi R_{wd}^2\/2$. The increase in the apparent\nmagnitude of the star will be $0.75$ mag.\n\nThis value is much higher than current photometric accuracy, and if present,\nwould have been detected for those WDs which have an observed parallax, a good\nspectrum and an IR~excess. The ``good spectrum' permits a determination of\n$T_{\\mathrm{eff}}$ and $\\log g$ and hence a luminosity. Clearly the luminosity\ninferred  from the parallax should agree with the luminosity inferred from the\nspectral fit. At the very least, this procedure will allow us to affirm that\nthere is not a big opaque disk in the known WDs with IR~excess or, at least,\nthis disk is not in a favorable inclination.\n\n  For the general case of a disk with arbitrary inclination and any combination\nof inner and outer radii, the flux received at the Earth from the system\n(target) is,\n\n\\begin{equation}\n  \\begin{aligned}\n  F^{target} &=\\lefteqn { \\int_{\\Omega_{wd}} I \\cos \\theta d\\Omega }\\\\\n     &= I \\left(\\frac{R_*}{D}\\right)^2\n      \\int_{0}^{\\pi\/2} \\int_{\\phi_{min}}^{\\phi_{max}}\n      \\sin \\theta \\cos \\theta d \\phi d \\theta \n  \\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere we assumed that the intensity (I) is uniform over the stellar surface.\nThe stellar radius is $R_*$ and $D$ is the distance from the Earth to the system.\nThere are three possible projections for the disk, as seen in Figure~\\ref{3disks}.\nThis gives different values of $\\phi_{min}$ and $\\phi_{max}$:\n\n\n\\begin{figure}\n\\epsscale{0.6}\n\\plotone{3disks2.ps}\n\\caption{Debris disk and WD. For a given size and inclination of the disk,\n         different amounts of it obscure the star. The $x-$ and $y-$axes are in\n         the sky-plane and the observer is over the $z-$axis which makes an\n         angle $i$ with the normal of the disk. The\n         angles $\\phi_i$ and $\\phi_e$ show where the disk starts and stops\n         obscuring the star.\n}\n\\label{3disks}\n\\end{figure}\n\n\n\n  \\begin{itemize}\n  \\item \n  $\\int_{\\phi_{min}}^{\\phi_{max}} d \\phi =\n      2 \\left[ \\int_{-\\pi\/2}^{\\phi_i} d \\phi + \\int_{\\phi_e}^{\\pi\/2} d \\phi\n      \\right ]\n      \\\\= 2\\pi - 2( \\phi_e-\\phi_i )\n  $,\n\n  \\item\n  $\\int_{\\phi_{min}}^{\\phi_{max}} d \\phi =\n      2 \\int_{-\\pi\/2}^{\\phi_i} d \\phi    =\n      \\pi + 2\\phi_i\n  $,\n\n  \\item\n  $\\int_{\\phi_{min}}^{\\phi_{max}} d \\phi =\n      2 \\int_{-\\pi\/2}^{\\pi\/2} d \\phi     =\n      2 \\pi\n  $.\n\n  \\end{itemize}\n\n  Using the dimensionless radii $r_{\\{i\/e\\}} \\equiv R_{\\{i\/e\\}}\/R_{star}$ we\ndefine the function $g \\equiv g(\\theta,r_i,r_e)$:\n\n  \\begin{equation}\n  g = \\left\\{\n   \\begin{array}{l l}\n      \\pi - (\\phi_e-\\phi_i) & \\mbox{, $r_e \\cos i < \\sin \\theta $}\\\\\n      \\pi\/2 + \\phi_i        & \\mbox{, $r_i \\cos i < \\sin \\theta \\leq r_e\n                                                             \\cos i $}\\\\\n      \\pi                   & \\mbox{, $r_i \\cos i \\geq \\sin \\theta$}\\\\\n    \\end{array}\n    \\right.\n  \\label{eqg}\n  \\end{equation}\n\n\n\\noindent\nand write the flux:\n\n  \\begin{equation}\n  F^{target} = I \\left(\\frac{R_*}{D}\\right)^2\n            \\int_{0}^{\\pi\/2} \\sin(2\\theta) g(\\theta,r_i,r_e) d \\theta\n  \\label{eqF}\n  \\end{equation}\n\n\\noindent\nwhere\n\n  \\begin{equation}\n  \\phi_{i,e} = \\arctan \\left[ \\cos i \\arccos\\left(\n               \\frac{\\sqrt{ \\sin^2\\theta\/r_{i,e}^2 - \\cos^2 i}}{\\sin i}\n               \\right) \\right ]\n  \\label{eqphi}\n  \\end{equation}\n\n\n  The total flux received from an unobscured system is $\\pi I\n  (R_*\/D)^2$.  We call the hypothetical unobscured star ``template''\n  and the obscured star ``target''. Defining $p$ as the ratio of the\n  obscured to the total projected area:\n\n\n\n\\begin{equation}\n p \\equiv \\frac{A_{obscured}}{A_{total}}  = \\frac{A_{target}}{A_{template}}\n\\label{def:p}\n\\end{equation}\n\n\\noindent\nwe write the increase in magnitude as:\n\n\\begin{equation}\n \\Delta m = -2.5 \\log(1-p)\n\\label{eq:mp}\n\\end{equation}\n\n\nIn the completely opaque hypothesis we may obviously write that,\n\n\\begin{equation}\n  \\begin{aligned}\n p &=\\lefteqn { \\frac{F^{target}}{F^{template}}\n   = \\frac{F^{target}}{\\pi I} \\left(\\frac{D}{R_*}\\right)^2 }\\\\\n   &= \\frac{1}{\\pi} \\int_{0}^{\\pi\/2} \\sin(2\\theta) g(\\theta,r_i,r_e) d \\theta\n  \\end{aligned}\n\\label{eq:p}\n\\end{equation}\n\n  The solution of Equation \\ref{eq:p} with Equations \\ref{eqg} and \\ref{eqphi}\ngives the flux received from the system, as can been seen in\nFigure~\\ref{fig:hidden}. The probability of finding a system more inclined than\na given angle is the ratio of the solid angle occupied by these systems\nto the total solid angle: $P(i>i_0) = \\cos i_0$.\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{hiddenFlux.ps}\n\\caption{Increase in magnitude vs. inclination for opaque disks for\n   different inner ($r_i$) and outer ($r_e$) disk radius combinations.\n   Systems seen face on have $i=0^\\circ$. The fast decrease of $\\Delta m$ for\n   $i \\rightarrow 90^\\circ$ is an artifact of the mathematical model that\n   assumes an infinitely thin disk. For real disks, there would be a plateau\n   lower than curve peak, but the disk should be really thin indeed, so the\n   plateau is very close to $\\Delta m = 0$ and it actually is not a plateau,\n   but rather a single point.\n   The top axis label shows the percentage of systems more inclined than $i$.\n   The right axis label shows $p$, the ratio of the obscured to the total\n   projected area (Equation~\\ref{def:p}).}\n\\label{fig:hidden}\n\\end{figure}\n\n  Figure~\\ref{fig:hidden} shows that it is possible to detect a completely\nopaque debris disk. For an inclination angle causing any blocking, the bigger\nthe disk is, the easier it is to detect the effect. The inner and outer radii\nare based on physical constrains. If the inner radius is small, the dust\nparticles sublimate because the\ntemperature exceeds $\\sim 1200$K~\\citep{jur07a}. On the other hand, if the\nouter edge of the disk is big ($\\gtrsim 100 R_{wd}$), the dust grains will be\ncold and any emission will be undetectable in practice. The exact\nvalues depend on the dust type and grain size.\n\n  The presence of an obscuring disk might be inferred by comparing the expected\nincrease in stellar magnitude with the luminosity derived from\n$T_{\\mathrm{eff}}$ and $\\log g$ and parallax. If the observed and the expected\nmagnitudes are correct and not equal, the flux deficiency can probably be\nexplained by obscuration from a debris disk.\n\n  We use the measured parallax of GD~362 to illustrate the previous analysis\nwith one real case. \\cite{kil08} obtained d~$= 50.6_{-3.1}^{+3.5}$~pc for\nGD~362. Using simple error propagation, we have a rough estimate for the highest\nacceptable difference between expected and measured magnitude: $\\sigma_m\n\\approx 0.15$ mag.  \\cite{kil08} did not find any discrepancies between parallax\nand flux, implying no obscuration of the star.\n\nFrom a geometrical perspective this is expected since from\nFigure~\\ref{fig:hidden} $\\Delta m\\approx0.15$ mag implies an inclination higher\nthan $\\sim$$80^\\circ$ and less than $\\sim$$20\\%$ of the systems will be more\ninclined than this. Indeed, \\cite{jur07b} showed that GD~362 must be seen\nnearly face on to be able to reproduce its IR~excess flux with physically\nreasonable inner and outer radii. Assuming an almost edge on system would\nrequire a big disk and an unusual mechanism to heat it to reproduce the\nmeasured IR~excess flux. Therefore, our work is in agreement with the previous\nresults and this analysis illustrates what kind of study must be done with\nother systems which may be found to be nearly edge on.\n\n\\section{Optically thin dust disk}\n\\label{sec:ot}\n\n  After having derived the basic concepts of the problem with the optically\n  thick limit we generalize the equations to the optically thin limit.\n  The ratio ($\\xi_\\nu$) of the flux from obscured (target) to the equivalent\nstar with no obscuration (template) is composed of three main components:\n\n\\begin{equation}\n\\begin{aligned}\n   \\xi_\\nu &\\equiv \\lefteqn {\\frac{F^{target}_\\nu}{F^{template}_\\nu}}\\\\ &=\n   \\left. \\xi_\\nu \\right|_{unobscured} +\n   \\left. \\xi_\\nu \\right|_{obscured} +\n   \\left. \\xi_\\nu \\right|_{scattered}\n\\end{aligned}\n\\label{eq:3comp}\n\\end{equation}\n\n  The unobscured ratio component is simply $1-p$. The obscured ratio\n  is given by $p e^{-\\tau_\\nu^{ext} \/\\cos{i}}$. The extinction optical depth\n  $\\tau_\\nu^{ext}$ of the disk regions obscuring the star accounts for the\n  absorbed and scattered light along the line of sight to the star.\n  The scattered component comes from the disk regions which do not obscure the\n  star but scatter photons to the line of sight. There is no emission component\n  because the dust temperature is lower than the dust sublimation temperature\n  ($\\sim$1200~K) and thus the dust emission only contributes in the infrared.\n\n  Using the dimensionless extinction efficiency ($Q^{ext}_\\nu$) instead of\nextinction cross section ($[C^{ext}_\\nu] = cm^2$) we write the differential\nextinction optical depth in the disk as:\n\n\\begin{equation}\n d\\tau_\\nu = n C^{ext}_\\nu dz = n Q^{ext}_\\nu \\pi a^2 dz,\n\\end{equation}\n\n\\noindent\nwhere $n$ (cm$^{-3}$) is the number of dust grains per unit volume, $a$ (cm)\nis the grain radius and $z$ (cm) is the vertical dimension of the disk\n\n The disk volume density ($\\rho$) is related to the density of a typical dust\ngrain ($\\rho_d$) through,\n\n\\begin{equation}\n \\rho = \\frac{4}{3} \\pi a^3 \\rho_d n .\n\\label{rho}\n\\end{equation}\n\n  Assuming the disk to be vertically uniform we integrate to write\n\n\\begin{equation}\n \\tau_\\nu^{ext} = \\int^{H\/2}_{-H\/2} \\frac{3 Q^{ext}_\\nu \\rho}{4 a \\rho_d} dz\n          = \\frac{3 Q^{ext}_\\nu \\rho}{4 a \\rho_d} H\n          = \\frac{3 Q^{ext}_\\nu \\Sigma}{4 a \\rho_d},\n\\label{eq:tau}\n\\end{equation}\n\n\\noindent\nwhere $\\Sigma$ (g\/cm$^2$) is the disk surface density and $H$ is the disk\nheight.\n\n  We define:\n\n\\begin{equation}\n \\tau_0 = \\frac{3 \\Sigma}{4 a \\rho_d}\n\\label{eqt0}\n\\end{equation}\n\n\\noindent\nand write Equation~\\ref{eq:3comp} as\n\n\\begin{equation}\n \\xi_\\nu = (1-p) +\n               p e^{ -\\tau_0 Q^{ext}_\\nu \/ \\cos{i} } +\n               \\left. \\xi_\\nu \\right|_{scattered}.\n\\label{eqFd}\n\\end{equation}\n\n  To simplify the scattering term, we assume isotropic and\ncoherent scattering and also that the light is not attenuated before and after\nbeing scattered by the disk. The last hypothesis is valid in the optically thin\ncase and causes an overestimation of the scattering because we ignore the\nabsorbed photons. The scattered intensity is given by\n\n\\begin{equation}\n I^{sca}_\\nu = \\epsilon_\\nu \\frac{H}{\\cos{i}}\n             = \\pi a^2 Q^{sca}_\\nu n J^{wd}_\\nu \\frac{H}{\\cos{i}},\n\\end{equation}\n\n\\noindent\nwhere $\\epsilon_\\nu$ is the emissivity, $Q^{sca}_\\nu$ is the scattering\nefficiency and $J^{wd}_\\nu$ is the mean stellar intensity\n\n\\begin{equation}\n  J^{wd}_\\nu = \\frac{I^{wd}_\\nu \\pi R_{wd}^2}{4 \\pi r^2}\n\\end{equation}\n\n  Ignoring the disk regions hidden by the star we integrate over the disk\nsurface to get the flux:\n\n\n\\begin{equation}\n  F_\\nu = \\frac{1}{2} \\frac{3 \\Sigma}{4 a \\rho_d} Q^{sca}_\\nu\n          \\ln \\left( \\frac{re}{ri} \\right ) \\cos{i}\n          \\, \\pi I^{wd}_\\nu \\left( \\frac{R_{wd}}{D} \\right)^2\n\\end{equation}\n\n\\noindent\nusing Equation~\\ref{eqt0} and dividing by the template flux,\n\n\n\\begin{equation}\n  \\left. \\xi_\\nu \\right|_{scattered} = \\frac{1}{2} \\tau_0 Q^{sca}_\\nu\n          \\ln \\left( \\frac{re}{ri} \\right ) \\cos{i},\n\\end{equation}\n\n\\noindent\nwhich allows us to write Equation~\\ref{eqFd} as\n\n\\begin{equation}\n \\xi_\\nu = (1-p) +\n               p e^{ -\\tau_0 Q^{ext}_\\nu \/ \\cos{i} } +\n               \\frac{1}{2} \\tau_0 Q^{sca}_\\nu\n               \\ln \\left( \\frac{re}{ri} \\right ) \\cos{i}\n\\label{eqFdcp}\n\\end{equation}\n\n\nBesides the parameters $p$ and $\\tau_0$, we have the absorption efficiencies\nwhich are characteristic of the dust type. We used the tables of optical\nconstants of silicate glasses from \\cite{dor95}. The\nauthors prepared two different glasses in laboratory: pyroxene,\nMg$_x$Fe$_{1-x}$SiO$_3$, with $x$=0.4, 0.5, 0.6, 0.7, 0.8, 0.95, 1.0 and olivine,\nMg$_{2x}$Fe$_{2-2x}$SiO$_4$, with $x$=0.4 and 0.5.\n\nFigures \\ref{pyrmg70} and \\ref{pyrmg70sca} display the results from\nEquation~\\ref{eqFdcp} for different optical depths and system\ngeometries. Figure \\ref{pyrmg70} represents inclinations where the disk\nobscures the star, and light is absorbed and Figure\n\\ref{pyrmg70sca} when there is no obscuration and we see only\nscattering plus the WD light.  For observational tests, the region\nfrom $\\textrm{3000\\AA\\ to 5000\\AA}$ is the most interesting, because it\nshows a sharp change in the ratio between the target and the template\nwhich cannot be easily discarded as bad flux calibrations.\n\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{pyrmg70_sca.ps}\n\\caption{Expected ratio between the target and the template star\n          (Equation~\\ref{eqFdcp}) when the inclination is such that the disk\n          obscures\n          a part of the WD as in 1 and 2 in Figure \\ref{3disks}. The disk is\n          composed of olivine Mg$_{0.8}$Fe$_{1.2}$SiO$_4$ dust grains. To\n          calculate the scattering we used\n          $\\ln(r_e\/r_i) \\lesssim \\ln(100\/5) = 3$\n          as a superior limit in Equation~\\ref{eqFdcp} and adopted $i=85^\\circ$.}\n\\label{pyrmg70}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{pyrmg70_pureSca.ps}\n\\caption{Expected ratio between the target and the template star\n          (Equation~\\ref{eqFdcp}) when the disk does not obscure the WD\n          and we see the scattering plus the stellar light. The disk is\n          composed of olivine Mg$_{0.8}$Fe$_{1.2}$SiO$_4$ dust grains. All the\n          curves were calculated with $\\tau_0 = 0.1$ because the pure scattering\n          term in Equation~\\ref{eqFdcp} is linear in $\\tau_0$.}\n\\label{pyrmg70sca}\n\\end{figure}\n\n\n\n\\section{Observational test}\n\\label{ObsTest}\n\n  Our modeling gives rise to direct observational tests. We can test the\n  effects of the disk in the near-UV and the optical, dividing the\n  spectrum of the target by the template. In the optically thin case, the\n  result will be color dependent and can provide physical parameters for\n  the disk structure.\n\n  The template star should be as similar to the target star as\n  possible. Ideally, it would be the same WD without the obscuring\n  disk. As that is not possible to have, we need a similar WD without\n  any peculiarity in the spectrum.  Any other WD will differ from the\n  target star in $T_{\\mathrm{eff}}$ and $\\log g$ and this difference\n  can make the division of the spectra resemble the expected disk\n  effects.\n\n  In Figure~\\ref{wdRatio}, we present these effects using theoretical WD\n  spectra from \\cite{koe08}.  We assume a target star of\n  $T_{\\mathrm{eff}} = 12,000$~K and $\\log g = 8.0$, similar to G29-38\n  \\citep{rea09}. In the upper panel, we keep $T_{\\mathrm{eff}}$ fixed\n  and vary $\\log g$ by $0.05$~dex and in the lower we keep $\\log g$\n  fixed and vary $T_{\\mathrm{eff}}$ by $150$~K. According to \\cite{lie05},\n  the uncertainties in temperature are of the order of $1.2 \\% \\approx 150$~K\n  and 0.038 in $\\log g$. So, larger temperature or $\\log g$ differences would be\n  readily noticed. One can see from Figure~\\ref{wdRatio} that modification of UV\n  flux densities by disks that absorb or scatter light, as in\n  Figures \\ref{pyrmg70} and \\ref{pyrmg70sca}, can be distinguished from\n  observational uncertainties in template stars.\n\n  The White Dwarf Catalog \\citep{mcc09} currently lists 12,456 stars.\n  Therefore it is not too hard to find a template star with a temperature\n  similar to the target.  As an example, G29-38 and Ross~548 have\n  exactly the same temperature and $\\log g$.  One needs to be careful\n  about this comparison as these values of $T_{\\mathrm{eff}}$ and\n  $\\log g$ were obtained from different determinations.  When\n  comparing the target with template it will be necessary to use\n  $T_{\\mathrm{eff}}$ and $\\log g$ obtained from similar data and the\n  same models.\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{wdRatio.ps}\n\\caption{Comparison of the effects from debris disk obscuration and the effects\n   of small differences in $T_{\\mathrm{eff}}$ and $\\log g$ between the target and\n   the template stars. The gray lines show the expected effects shown in Figures\n   \\ref{pyrmg70} and \\ref{pyrmg70sca}. In the upper panel the black solid\n   lines show the ratio of WD spectra with fixed $T_{\\mathrm{eff}}$ and\n   varying $\\log g$. In the lower panel, we fixed $\\log g$ and\n   varied $T_{\\mathrm{eff}}$.}\n\\label{wdRatio}\n\\end{figure}\n\n\\subsection{Parameter determination}\n\n  It is difficult to compare directly the definition of $p$ and $\\tau_0$ with\nthe expected values for real disks. We use disk parameters obtained in earlier\nworks to give observational expectations and also to help design future\nobservations.\n  \n  For disks, the fraction of the obscured to the total projected area, $p$\n(Equation~\\ref{def:p}), varies between $0$ when there is no obscuration and 0.5 for\na disk which obscures half of the stellar surface.\nHowever, if the infrared emission region is not a disk but a shell around the\nstar \\citep{rea05} $p$ will be always $1$. Hence, this work provides an\nindependent method to test the disk hypothesis.\n\n  In addition to $p$, we can also determine $\\tau_0$. Estimates for the expected\nvalues are more uncertain, but we can get a rough idea by using some mean values\nfor the dust and disk properties. \\cite{kru03} gives $2.5$~g\/cm$^3$ as a typical\nvalue for the interstellar dust and we assume it as a good order of magnitude\nvalue for the dust in the disk. \\cite{jur07b} constrain the disk mass of GD~362\nbetween $10^{18}$ and $10^{24}$~g. Using typical disk sizes of $10~R_{wd}$ and\n$100~R_{wd}$ for the inner and outer disk radii, respectively, we get a range\nof $\\tau_0$ from $10^{-4}$ to somewhat greater than $1$, depending on the disk\nmass and the type of dust \\citep{dor95}. Therefore the parameters used in Figure\n\\ref{pyrmg70} are realistic.\n\n  The disk inclination angle can be inferred from the presence of an flux excess\ndue to scattering into the line of sight. Figure \\ref{pyrmg70sca} shows this for\ninclinations of $0^\\circ$ and $60^\\circ$. Flux excess in near-UV has already\nbeen detected by \\cite{gan2006} in SDSS~1228+1040 and could be caused by light\nscattering. For larger inclinations there is a flux deficiency due to\nabsorption and scattering out the line of sight, as shown in\nFigure \\ref{pyrmg70} computed for $i=85^\\circ$. The dividing line between\nthe first or the second case is $\\sim$$80^\\circ$.\n\n\\section{Conclusions}\n\\label{sec:conc}\n\nIn this work, we introduce a new way of looking at the cause of IR excess in\nwhite dwarf stars.\nBy looking in the near-UV and optical instead of IR we add a new constraint to\ntest the disk hypothesis.\n\nOne important distinction of our method, is the fact that the presence of disks\nwould cause flux deficiencies in some systems and flux excess in others. We also\npoint out that shells would only introduce flux deficiencies effects, and these\neffects would be detectable in all shells. If we find flux deficiencies in every\nstar we observe this would strongly indicate the presence of shells rather than\ndisks. Flux deficiencies in only some objects and flux excess in others\ncorroborate the idea of a disk.\n\nIf we are convinced disk models are more adequate, detailed\ncomparisons between disk models and data will provide disk \nmass \\citep{jur07b}, composition, optical depth and inclination\nrelative to the line of sight.\n\n\n\n\\begin{acknowledgements}\n  The authors acknowledge financial support from CNPq-MCT\/Brazil. We thank\nPaola D'Alessio, Detlev Koester, and Don Winget for helpful discussions,\nWilliam Reach for providing data, and Shashi Kanbur for reading the manuscript.\nWe are also grateful to Nikolai Voshchinnikov for his Mie-Theory program.\nFinally, we thank the anonymous referee for helpful comments on paper.\n\\end{acknowledgements}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of random walks on groups has a long history and multiple connections to almost all areas of mathematics. It is therefore natural that from the early days of the theory of topological quantum groups, random walks on them were considered. The point of view was often that of discrete groups (the problem is interesting even for duals of classical compact Lie groups). In particular, the study of probabilistic boundaries has been the subject of several works and is still an active area of research. However, there is an aspect which has attracted no attention up to very recently, even though it is an important part of the subject for classical groups : the search for explicit estimates of convergence of random walks.\n\nIn the case of classical finite groups, the first important results for us are due to P. Diaconis and his coauthors in the eighties and reveal a surprising behaviour called the \\emph{cut-off phenomenon} : for a number of steps, the total variation distance (see Subsection \\ref{subsec:randomwalks} for the definition) between the random walk and the uniform distribution stays close to one and then it suddenly drops and converges exponentially to $0$. This triggered numerous works yielding more and more examples of cut-off in various settings, but also counter-examples so that the question of why and when this happens stays largely unanswered. In the quantum setting, the only results up to now are contained in the recent thesis of J.P. McCarthy \\cite{maccarthy2017random} which studies convergence of random walks on finite quantum groups. There, the author gives explicit bounds for families of random walks on the Kac-Paljutkin and Sekine quantum groups, as well as on duals of symmetric groups. Unfortunately, the estimates are not tight enough to yield a complete cut-off statement for these examples.\n\nIn the present work, we turn to the case of infinite compact quantum groups. In particular, we will show that a specific random walk on the free orthogonal quantum groups $O_{N}^{+}$, coming from random rotations on $SO(N)$, has a cut-off with the same threshold as in the classical case, namely $N\\ln(N)\/2(1-\\cos(\\theta))$. This is the first complete cut-off result for a compact quantum group and the statement is all the more surprising that the computations involve mainly representation theory, which is very different for $SO(N)$ and $O_{N}^{+}$. Moreover, the representation theory of $O_{N}^{+}$ being in a sense simpler than that of $SO(N)$ we are able to give very precise statements for the bounds (not only up to some order) and the conditions under which they hold.\n\nUsing techniques from \\cite{hough2017cut}, we can extend our result to random mixtures of rotations provided that the support of the measure governing the random choice of angle is bounded away from $0$. We also consider other examples involving the free symmetric quantum groups $S_{N}^{+}$. In that case, the previous techniques often prove useless. One way round the problem is to compare the corresponding transition operators, which are always well-defined. There is then several options for the choice of a norm and we give results for one of the simplest choices, namely the norm as operators on the $L^{2}$ space.\n\nLet us conclude this introduction with an overview of the organization of this work. In Section \\ref{sec:preliminaries} we give some preliminaries concerning compact quantum groups and random walks on them. We have tried to remain as elementary as possible so that the paper could be readable for people outside the field of quantum groups. In Section \\ref{sec:orthogonal} we study central random walks associated to pure states on free orthogonal quantum groups and prove a kind of cut-off phenomenon in Theorem \\ref{thm:estimategreaterthan2} : for a number of steps, the walk is not comparable in total variation distance with the Haar measure and as soon as it is, it converges exponentially. Using this, we show in Theorem \\ref{thm:randomrotation} that the uniform plane Kac walk on $O_{N}^{+}$ has a cut-off with the same threshold as in the classical case. Eventually, we give in Section \\ref{sec:further} other examples connected to free symmetric quantum groups and illustrate the analytical issue mentioned above.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\nIn this section we recall the basic notions concerning compact quantum groups and random walks on them. Since the abstract setting is not really needed to perform concrete computations, we will mainly set notations and give some fundamental results.\n\n\\subsection{Compact quantum groups}\n\nCompact quantum groups are objects of noncommutative topological nature and therefore belong to the world of operator algebras. However, in the present work most things can be treated at an algebraic level which is slightly simpler to describe. We will therefore first give the main definitions in the setting of Hopf algebras and then briefly introduce in the end of this subsection the related analytical objects. We refer the reader to Parts I and II of \\cite{timmermann2008invitation} for a detailed treatment of the algebraic theory of compact quantum groups and its link to the operator algebraic theory.\n\nThe basic example to keep in mind is of course that of a classical compact group $G$. In that case, the corresponding algebraic object is the complex algebra $\\O(G)$ of \\emph{regular functions}, i.e. coefficients of unitary representations. This is a Hopf algebra with an involution given by $f^{*}(g) = \\overline{f(g)}$. Moreover, the Haar measure on $G$ yields by integration a linear form $h$ on $\\O(G)$ which is positive ($h(a^{*}a) \\geqslant 0$) and invariant under translation. Abstracting these properties leads to the following notion (with $\\otimes$ denoting the algebraic tensor product over $\\mathbb{C}$) :\n\n\\begin{de}\nA compact quantum group $\\mathbb{G}$ is given by a Hopf algebra $\\O(\\mathbb{G})$ with an involution and a unital positive linear map $h : \\O(\\mathbb{G})\\to \\mathbb{C}$ which is invariant in the sense that for all $a\\in \\O(\\mathbb{G})$,\n\\begin{equation*}\n(h\\otimes\\id)\\circ\\Delta(a) = h(a).1 = (\\id\\otimes h)\\circ\\Delta(a),\n\\end{equation*}\nwhere $\\Delta : \\O(\\mathbb{G})\\to \\O(\\mathbb{G})\\otimes \\O(\\mathbb{G})$ is the coproduct.\n\\end{de}\n\nIn the present work we will always assume that $\\mathbb{G}$ is \\emph{of Kac type}, meaning that for all $a, b\\in \\O(\\mathbb{G})$ $h(ab) = h(ba)$ (the Haar state is then said to be \\emph{tracial}). Since the fundamental work of P. Diaconis and M. Shahshahani \\cite{diaconis1981generating}, it is known that convergence of random walks can be controlled using representation theory and we will see that the same is true in the quantum setting. As for classical compact groups, the results of \\cite{woronowicz1987compact} imply that any representation of a compact quantum group is equivalent to a direct sum of finite-dimensional unitary ones, so that we will only define the latter.\n\n\\begin{de}\nA \\emph{unitary representation of dimension $n$} of $\\mathbb{G}$ is a unitary element $u\\in M_{n}(\\O(\\mathbb{G}))$ such that for all $1\\leqslant i, j\\leqslant N$,\n\\begin{equation*}\n\\Delta(u_{ij}) = \\sum_{k=1}^{n}u_{ik}\\otimes u_{kj}.\n\\end{equation*}\nA \\emph{morphism} between representations $u$ and $v$ of dimension $n$ and $m$ respectively is a linear map $T : \\mathbb{C}^{n}\\rightarrow \\mathbb{C}^{m}$ such that $(T\\otimes \\id)u = v(T\\otimes \\id)$. Two representations are said to be \\emph{equivalent} if there is a bijective morphism between them. A representation $u$ is said to be \\emph{irreducible} if the only morphisms between $u$ and itself are the scalar multiples of the identity.\n\\end{de}\n\nWe will denote by $\\Irr(\\mathbb{G})$ the set of equivalence classes of irreducible representations of $\\mathbb{G}$ and for each $\\alpha\\in \\Irr(\\mathbb{G})$ we fix a representative $u^{\\alpha}$ and denote by $d_{\\alpha}$ its dimension (which does not depend on the chosen representative). It then follows that $\\O(\\mathbb{G})$ is spanned by the coefficients $u^{\\alpha}_{ij}$ of all the $u^{\\alpha}$'s. Moreover, the Haar state induces an inner product on $\\O(\\mathbb{G})$ for which the basis of coefficients is orthogonal. More precisely, it was proven in \\cite{woronowicz1987compact} that for any $\\alpha, \\beta\\in \\Irr(\\mathbb{G})$ and $1\\leqslant i, j\\leqslant d_{\\alpha}$, $1\\leqslant k, l\\leqslant d_{\\beta}$,\n\\begin{equation*}\nh(u^{\\alpha}_{ij}u^{\\beta\\ast}_{kl}) = \\delta_{\\alpha, \\beta}\\frac{\\delta_{i, k}\\delta_{j, l}}{d_{\\alpha}}.\n\\end{equation*}\nThe key object for computations with random walks is characters of irreducible representations. Let us therefore define these :\n\n\\begin{de}\nThe \\emph{character} of a representation $u^{\\alpha}$ of a compact quantum group $\\mathbb{G}$ is defined as\n\\begin{equation*}\n\\chi_{\\alpha} = \\sum_{i=1}^{d_{\\alpha}}u_{ii}^{\\alpha}\\in \\O(\\mathbb{G}).\n\\end{equation*}\nMoreover, it only depends on $\\alpha$ and not on the chosen representative.\n\\end{de}\n\nWe conclude this subsection with some analysis. As already mentioned, the bilinear map $(a, b)\\mapsto h(b^{*}a)$ defines an inner product on $\\O(\\mathbb{G})$ and the corresponding completion is a Hilbert space denoted by $L^{2}(\\mathbb{G})$. For any element of $\\O(\\mathbb{G})$, left multiplication extends to a bounded operator on $L^{2}(\\mathbb{G})$, yielding an injective $*$-homomorphism $\\O(\\mathbb{G})\\to B(L^{2}(\\mathbb{G}))$. The closure of the image of this map with respect to the weak operator topology is a von Neumann algebra denoted by $L^{\\infty}(\\mathbb{G})$. We will also need the analogue of $L^{1}$ functions. For $a\\in L^{\\infty}(\\mathbb{G})$, set $\\|a\\|_{1} = h(\\vert a\\vert)$ where $\\vert a\\vert = \\sqrt{a^{*}a}$ is defined through functional calculus. Then, $L^{1}(\\mathbb{G})$ is defined to be the completion of $L^{\\infty}(\\mathbb{G})$ with respect to this norm.\n\n\\subsection{Random walks and central states}\\label{subsec:randomwalks}\n\nWe will now introduce some material concerning random walks on compact quantum groups and the total variation distance. For finite quantum groups the subject has been treated in great detail by J.P. MacCarthy in \\cite{maccarthy2017random}. The generalization to the compact case is not difficult so that this subsection will be rather expository. If $G$ is a compact group and $\\mu$ is a measure on $G$, the associated random walk consists in picking elements of $G$ at random according to $\\mu$ and then multiplying them. The probability of being in some measurable set after $k$ steps is then given by the $k$-th convolution power $\\mu^{\\ast k}$ of $\\mu$, which can be expressed at the level of functions as\n\\begin{equation*}\n\\int_{G}f(g) \\mathrm{d}\\mu^{\\ast k}(g) = \\int_{G^{k}}f(g_{k}\\cdots g_{1})\\mathrm{d}\\mu(g_{1})\\cdots \\mathrm{d}\\mu(g_{k}).\n\\end{equation*}\nStudying the random walk associated to $\\mu$ is therefore the same as studying the sequence of measures $(\\mu^{\\ast k})_{k\\in \\mathbb{N}}$.\n\nTurning to quantum groups, first note that measures yield through integration linear forms on $\\O(G)$. If the measure is moreover positive, then so is the linear form and if its total mass is $1$ then the linear form sends the unit of $\\O(G)$ to $1$. Thus, probability measures yield \\emph{states} in the following sense :\n\n\\begin{de}\nA state on an involutive unital algebra $A$ is a linear form $\\varphi : A\\to \\mathbb{C}$ such that $\\varphi(1) = 1$ and $\\varphi(a^{*}a) \\geqslant 0$ for all $a\\in A$.\n\\end{de}\n\nA random walk on a compact quantum group $\\mathbb{G}$ is therefore given by a state $\\varphi$ on $\\O(\\mathbb{G})$. The definition of convolution translates straightforwardly to this setting and one can for instance define $\\varphi^{\\ast k}$ by induction through the formula\n\\begin{equation*}\n\\varphi^{\\ast (k+1)} = (\\varphi\\otimes \\varphi^{\\ast k})\\circ\\Delta = (\\varphi^{\\ast k}\\otimes\\varphi)\\circ\\Delta.\n\\end{equation*}\n\nThe key tool to estimate the rate of convergence of a random walk is a fundamental result of P. Diaconis and M. Sashahani \\cite{diaconis1981generating} bounding the \\emph{total variation distance} of the difference between a measure and the uniform one. Classically, if $\\mu$ and $\\nu$ are any two Borel probability measures on $G$, then\n\\begin{equation*}\n\\|\\mu - \\nu\\|_{TV} =\\sup_{E}\\vert \\mu(E) - \\nu(E)\\vert\n\\end{equation*}\nwhere the supremum is over all Borel subsets $E\\subset G$. This definition can be extended to quantum groups thanks to the fact that Borel subsets correspond to projections in the associated von Neumann algebra. This however requires that the states extend to $L^{\\infty}(\\mathbb{G})$, which may not be the case (see for instance Lemma \\ref{lem:boundednesscriterion}).\n\n\\begin{de}\\label{de:totalvariation}\nThe total variation distance between two states $\\varphi$ and $\\psi$ on $L^{\\infty}(\\mathbb{G})$ is defined by\n\\begin{equation*}\n\\|\\varphi - \\psi\\|_{TV} = \\sup_{p\\in \\mathcal{P}(L^{\\infty}(\\mathbb{G}))}\\vert \\varphi(p) - \\psi(p)\\vert,\n\\end{equation*}\nwhere $\\mathcal{P}(L^{\\infty}(\\mathbb{G})) = \\{p\\in L^{\\infty}(\\mathbb{G}) \\mid p^{2} = p = p^{*}\\}$.\n\\end{de}\n\nAssume now that we consider a state $\\varphi$ which is \\emph{absolutely continuous} with respect to the Haar state $h$, in the sense that there exists an element $a_{\\varphi}\\in L^{1}(\\mathbb{G})$ such that $\\varphi(x) = h(a_{\\varphi}x)$ for all $x\\in \\O(\\mathbb{G})$. Then, the total variation distance can be expressed in terms of $a_{\\varphi}$. Note that the proof below uses the traciality of the Haar state.\n\n\\begin{lem}\\label{lem:totalvariation}\nIf $\\varphi$ is a state on $L^{\\infty}(\\mathbb{G})$ with an $L^{1}$-density $a_{\\varphi}\\in L^{1}(\\mathbb{G})$, then\n\\begin{equation*}\n\\|\\varphi - h\\|_{TV} = \\frac{1}{2}\\|a_{\\varphi} - 1\\|_{1}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $\\mathbf{1}_{\\mathbb{R}_{+}}$ be the indicator function of the positive real numbers and define a projection $p_{+} = \\mathbf{1}_{\\mathbb{R}_{+}}(a_{\\varphi} - 1)$ through functional calculus. We claim that the supremum in Definition \\ref{de:totalvariation} is attained at $p_{+}$. Indeed, for any projection $q\\in \\mathcal{P}(L^{\\infty}(\\mathbb{G}))$, setting $b_{\\varphi} = a_{\\varphi} - 1$ we have\n\\begin{equation*}\n\\vert\\varphi-h\\vert(q) = \\vert h(b_{\\varphi}p_{+}q) + h(b_{\\varphi}(1-p_{+})q)\\vert \\leqslant \\max(h(b_{\\varphi}p_{+}q), h(b_{\\varphi}(p_{+}-1)q))\n\\end{equation*}\nand observing that $p_{+}$ commutes with $b_{\\varphi}$ we get\n\\begin{equation*}\n\\vert\\varphi-h\\vert(q) \\leqslant \\max(h(b_{\\varphi}p_{+}qp_{+}), h(b_{\\varphi}(p_{+}-1)q(p_{+}-1))) \\leqslant \\max(h(b_{\\varphi}p_{+}), h(b_{\\varphi}(p_{+}-1))).\n\\end{equation*}\nWe conclude using the fact that $h(b_{\\varphi}) = (\\varphi - h)(1) = 1$. Now since\n\\begin{equation*}\n\\vert b_{\\varphi}\\vert = \\vert b_{\\varphi}\\vert p_{+} + \\vert b_{\\varphi}\\vert(1-p_{+}) = 2b_{\\varphi}p_{+} - b_{\\varphi}\n\\end{equation*}\nwe get\n\\begin{equation*}\n\\|a_{\\varphi} - 1\\|_{1} = h(\\vert b_{\\varphi}\\vert) = 2h(b_{\\varphi}p_{+}) - h(b_{\\varphi}) = 2(\\varphi - h)(p_{+}) = 2\\|\\varphi - h\\|_{TV}.\n\\end{equation*}\n\\end{proof}\n\nThis equality is the trick leading to the Diaconis-Shahshahani upper bound lemma which, in the end, does not involve $a_{\\varphi}$ any more. To state this result, let us first introduce a notation : if $\\varphi$ is a state and $\\alpha\\in \\Irr(\\mathbb{G})$, we denote by $\\widehat{\\varphi}(\\alpha)$ the matrix with coefficients $\\varphi(u^{\\alpha}_{ij})$ (this does not depend on the choice of a representative of $\\alpha$). We can then consider $\\widehat{\\varphi}$ as an element of the $\\ell^{\\infty}$-sum of the matrix algebras $B(H_{\\alpha})$, denoted by $\\ell^{\\infty}(\\widehat{\\mathbb{G}})$.\n\n\\begin{lem}[Upper bound lemma]\\label{lem:upperbound}\nLet $\\mathbb{G}$ be a compact quantum group and let $\\varphi$ be a state on $\\mathbb{G}$ which is absolutely continuous with respect to the Haar state. Then,\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\|_{TV}^{2} \\leqslant \\frac{1}{4}\\sum_{\\alpha\\in \\Irr(\\mathbb{G})\\setminus\\{\\varepsilon\\}}d_{\\alpha}\\Tr\\left(\\widehat{\\varphi}(\\alpha)^{* k}\\widehat{\\varphi}(\\alpha)^{k}\\right),\n\\end{equation*}\nwhere $\\varepsilon = 1\\in M_{1}(\\O(\\mathbb{G}))$ denotes the trivial representation.\n\\end{lem}\n\n\\begin{proof}\nThe proof for compact groups (assuming that the measure is central) was given in \\cite[Lem 4.3]{rosenthal1994random} and the proof for finite quantum groups was given in \\cite[Lem 5.3.8]{maccarthy2017random}. The argument here is the same so that we simply sketch it. The Cauchy-Schwartz inequality yields\n\\begin{equation*}\n\\|a_{\\varphi} - 1\\|_{1}^{2} = h(\\vert a_{\\varphi} - 1\\vert)^{2} \\leqslant h(1^{*}1)h\\left((a_{\\varphi}-1)^{*}(a_{\\varphi}-1)\\right) = \\|a_{\\varphi}-1\\|_{2}^{2}\n\\end{equation*}\nMoreover, the formula\n\\begin{equation*}\n\\widehat{h}(x) = \\sum_{\\alpha\\in \\Irr(\\mathbb{G})}d_{\\alpha}\\Tr(x)\n\\end{equation*}\ndefines a positive weight on $\\ell^{\\infty}(\\widehat{\\mathbb{G}})$. This is the analogue of the counting measure on a discrete group and one can define a Fourier transform $\\mathcal{F} : L^{2}(\\mathbb{G})\\rightarrow \\ell^{2}(\\mathbb{G})$ (see for instance \\cite[Sec 2]{podles1990quantum}) which is isometric. The conclusion now follows from the fact that the Fourier transform of $a_{\\varphi}$ is $\\widehat{\\varphi}$ and the relationship between convolution and Fourier transform.\n\\end{proof}\n\n\\begin{rem}\nBecause of the Cauchy-Schwartz inequality, $L^{2}(\\mathbb{G})\\subset L^{1}(\\mathbb{G})$ so that if $\\varphi$ is not absolutely continuous with respect to $h$, then the right-hand side of the inequality is infinite and the inequality trivially holds.\n\\end{rem}\n\nOur goal is therefore to bound $\\sum d_{\\alpha}\\Tr(\\varphi(\\alpha)^{*k}\\varphi(\\alpha)^{k})$ by an explicit function of $k$. This requires the computation of the trace of arbitrary powers of matrices which can be very complicated. As already observed in \\cite{rosenthal1994random}, things get more tractable when the measure is assumed to be central, i.e. invariant under the adjoint action, since then the Fourier transform of its density consists in scalar multiples of identity matrices. The same is true in the quantum setting, thanks to \\cite[Prop 6.9]{cipriani2012symmetries} which we recall here for convenience.\n\n\\begin{prop}\nLet $\\mathbb{G}$ be a compact quantum group and let $\\varphi : \\O(\\mathbb{G})\\to \\mathbb{C}$ be a state. Then, $\\varphi$ is invariant under the adjoint action if and only if for any irreducible representation $\\alpha\\in \\Irr(\\mathbb{G})$, there exists $\\varphi(\\alpha) \\in \\mathbb{C}$ such that $\\varphi(u_{ij}^{\\alpha}) = \\varphi(\\alpha)\\delta_{ij}$.\n\\end{prop}\n\nSuch \\emph{central states} are completely determined by their restriction to the so-called \\emph{central algebra} of $\\mathbb{G}$, which is simply the algebra $\\O(\\mathbb{G})_{0}$ generated by the characters, thanks to the equality\n\\begin{equation*}\n\\varphi(\\chi_{\\alpha}) = \\sum_{i=1}^{d_{\\alpha}}\\varphi(u_{ii}^{\\alpha}) = d_{\\alpha}\\varphi(\\alpha).\n\\end{equation*}\nIn several key examples, the central algebra is commutative, hence states exactly correspond to measures on its spectrum. A particular case is that of Dirac measures, i.e. evaluation at one point. This setting covers natural analogues of the random walk associated to the uniform measure on a conjugacy class. To see this, assume that $\\O(\\mathbb{G})$ is generated by the coefficients of a representation $u$ of dimension $N$ and let $G$ be the abelianization of $\\mathbb{G}$, that is to say the compact group such that $\\O(G)$ is the maximal abelian quotient of $\\O(\\mathbb{G})$. By construction, $G$ is realized as a group of $N\\times N$ matrices. Let $g\\in G$ and let $\\ev_{g} : \\O(\\mathbb{G})\\to \\mathbb{C}$ be the algebra map sending $u_{ij}$ to $g_{ij}$. Then,\n\\begin{equation*}\n\\varphi_{g} = h\\circ m^{(2)}\\circ(\\id\\otimes \\ev_{g}\\otimes S)\\circ\\Delta^{(2)}\n\\end{equation*}\nis a state on $\\O(\\mathbb{G})$, where $\\Delta^{(2)} = (\\id\\otimes\\Delta)\\circ\\Delta$, $m^{(2)} = m\\circ(\\id\\otimes m)$ and $S$ is the antipode. If $\\mathbb{G}$ is classical, then for any function $f$ one has\n\\begin{equation*}\n\\varphi_{g}(f) = \\int_{G}f(kgk^{-1})\\mathrm{d} k\n\\end{equation*}\nso that $\\varphi_{g}$ is the uniform measure on the conjugacy class of $g$. In the general case, the centrality of $\\varphi_{g}$ is easily checked :\n\\begin{align*}\n\\varphi_{g}(u^{\\alpha}_{ij}) & = \\sum_{k,l=1}^{d_{\\alpha}}h(u^{\\alpha}_{ik}\\ev_{g}(u^{\\alpha}_{kl})u^{\\alpha\\ast}_{jl}) = \\sum_{k,l=1}^{d_{\\alpha}}\\ev_{g}(u^{\\alpha}_{kl})h(u^{\\alpha}_{ik}u^{\\alpha\\ast}_{jl}) \\\\\n& = \\sum_{k,l=1}^{d_{\\alpha}}\\ev_{g}(u^{\\alpha}_{kl})\\frac{\\delta_{ij}\\delta_{kl}}{d_{\\alpha}} = \\delta_{ij}\\frac{\\ev_{g}(\\chi_{\\alpha})}{d_{\\alpha}}.\n\\end{align*}\nSeveral interesting random walks on compact Lie groups are of this type and we will study them in the quantum setting in the next sections.\n\n\\section{Free orthogonal quantum groups}\\label{sec:orthogonal}\n\nThe main example which we will study in this work is free orthogonal quantum groups. These objects, denoted by $O_{N}^{+}$, were first introduced by S. Wang in \\cite{wang1995free}. Here is how the associated involutive Hopf algebra is defined :\n\n\\begin{de}\nLet $\\O(O_{N}^{+})$ be the universal $*$-algebra generated by $N^{2}$ \\emph{self-adjoint} elements $u_{ij}$ such that for all $1\\leqslant i, j \\leqslant N$,\n\\begin{equation*}\n\\sum_{k=1}^{N}u_{ik}u_{jk} = \\delta_{ij} = \\sum_{k=1}^{N}u_{ki}u_{kj}.\n\\end{equation*}\nThe formula\n\\begin{equation*}\n\\Delta(u_{ij}) = \\sum_{k=1}^{N}u_{ik}\\otimes u_{kj}\n\\end{equation*}\nextends to a $*$-algebra homomorphism $\\Delta : \\O(O_{N}^{+})\\to \\O(O_{N}^{+})\\otimes \\O(O_{N}^{+})$ and this can be completed into a compact quantum group structure.\n\\end{de}\n\nThe relations defining $\\O(O_{N}^{+})$ are equivalent to requiring that the matrix $[u_{ij}]_{1\\leqslant i, j\\leqslant N}$ is orthogonal. Using this, it is easy to see that the abelianization of $O_{N}^{+}$ is the orthogonal group $O_{N}$. To compute upper bounds for random walks, we need a description of the representation theory of these objects. In fact, since we will only consider central states, all we need is a description of the central algebra which comes from the work of T. Banica \\cite{banica1996theorie}.\n\n\\begin{thm}[Banica]\nThe irreducible representations of $O_{N}^{+}$ can be labelled by positive integers, with $u^{0}$ being the trivial representation and $u^{1} = [u_{ij}]_{1\\leqslant i, j\\leqslant N}$. Moreover, the characters satisfy the following recursion relation :\n\\begin{equation}\\label{eq:charactersorthogonal}\n\\chi_{1}\\chi_{n} = \\chi_{n+1} + \\chi_{n-1}.\n\\end{equation}\n\\end{thm}\n\nIn particular, the central algebra $\\O(\\mathbb{G})_{0}$ is abelian and in fact isomorphic to $\\mathbb{C}[X]$. Moreover, the recursion relation \\eqref{eq:charactersorthogonal} is reminiscent of that of Chebyshev polynomials of the second kind. Indeed, let $U_{n}$ be these polynomials, i.e. $U_{n}(\\sin(\\theta)) = \\sin(n\\theta)$ for all $\\theta\\in \\mathbb{R}$. Then, $u_{n}(x) = U_{n}(x\/2)$ satisfies Equation \\eqref{eq:charactersorthogonal}. With this in hand, it can be proven that the map sending $\\chi_{n}$ to $u_{n}$ is an isomorphism. Moreover, we have the equality $d_{n} = u_{n}(N)$.\n\n\\subsection{Pure state random walks}\\label{subsec:purestates}\n\nA state is said to be \\emph{pure} if it cannot be written as a non-trivial convex combination of other states. Moreover, pure states on $\\O(O_{N}^{+})$ are still pure when restricted to the central algebra and it is well-known that pure states on an abelian algebra are given by evaluation at points of the spectrum. For $O_{N}^{+}$, the spectrum of $\\chi_{1}$ in the enveloping C*-algebra of $\\O(O_{N}^{+})$ is $[-N, N]$ by \\cite[Lem 4.2]{brannan2011approximation} so that for any $t\\in [-N, N]$ there is a central state $\\varphi_{t}$ on $\\O(O_{N}^{+})$ defined by $\\varphi_{t}(n) = u_{n}(t)\/d_{n}\\in \\mathbb{R}$. It follows from the definition that for a central state $\\varphi$, we have $\\varphi^{\\ast k}(n) = \\varphi(n)^{k}$, so that by Lemma \\ref{lem:upperbound}\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}^{2} \\leqslant \\frac{1}{4}\\sum_{n=1}^{+\\infty}d_{n}\\frac{u_{n}(t)^{2k}}{d_{n}^{2k-1}} = \\frac{1}{4}\\sum_{n=1}^{+\\infty}\\frac{u_{n}(t)^{2k}}{d_{n}^{2k-2}}\n\\end{equation*}\nand we will have to bound specific values of the polynomials $u_{n}$. It turns out that the behaviour of Chebyshev polynomials is very different if the argument is less than or greater than one and this will be reflected in the existence or absence of a kind of cut-off phenomenon for the associated random walks. Before turning to this, let us give general tools for the computations. Assume that $t > 2$ and let $0 < q(t) < 1$ be such that $t = q(t) + q(t)^{-1}$, i.e.\n\\begin{equation*}\nq(t) = \\frac{t - \\sqrt{t^{2} - 4}}{2}.\n\\end{equation*}\nThen, it can be shown by induction that\n\\begin{equation*}\nu_{n}(t) = \\frac{q(t)^{-n-1} - q(t)^{n+1}}{q(t)^{-1} - q(t)}.\n\\end{equation*}\nThis writing enables to efficiently bound $u_{n}(t)$ :\n\n\\begin{lem}\\label{lem:encadrement}\nFor all $n\\geqslant 1$ and $t\\geqslant 2$,\n\\begin{equation*}\ntq(t)^{-(n-1)} \\leqslant u_{n}(t)\\leqslant\\frac{q(t)^{-n}}{1-q(t)^{2}}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nConsider the sequence $a_{n} = u_{n}(t)q(t)^{n}$. Then,\n\\begin{equation*}\n\\frac{a_{n+1}}{a_{n}} = q(t)\\frac{u_{n+1}(t)}{u_{n}(t)} = q(t)\\frac{q(t)^{-n-1} - q(t)^{n+1}}{q(t)^{-n} - q(t)^{n}} = \\frac{q(t)^{-n} - q(t)^{n+2}}{q(t)^{-n} - q(t)^{n}} > 1\n\\end{equation*}\nso that $(a_{n})_{n\\in \\mathbb{N}}$ is increasing. It is therefore always greater than its first term, which is $q(t)t = 1+q(t)^{2}$ and always less than its limit, which is\n\\begin{equation*}\n\\frac{q(t)^{-1}}{q(t)^{-1} - q(t)} = \\frac{1}{1-q(t)^{2}}.\n\\end{equation*}\n\\end{proof}\n\nTo lighten notations, let us set\n\\begin{equation*}\nA_{k}(t) = \\sum_{n=1}^{+\\infty}\\frac{u_{n}(t)^{2k}}{d_{n}^{2k-2}} = \\sum_{n=1}^{+\\infty}\\frac{u_{n}(t)^{2k}}{u_{n}(N)^{2k-2}}\n\\end{equation*}\n\n\\subsubsection{Random walks associated to small pure states}\n\nWe start with the case where $t$ is less than $2$. As we will see, things are then rather simple.\n\n\\begin{prop}\\label{prop:upperboundlessthantwo}\nLet $\\vert t\\vert < 2$ be fixed. Then, for any $k\\geqslant 2$,\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{N}{2\\sqrt{1-q(N)^{2}}}\\left(\\frac{1}{N\\sqrt{1 - t^{2}\/4}}\\right)^{k}\n\\end{equation*}\nIn particular, if $t<2\\sqrt{1-N^{-2}}$ then the random walk associated to $\\varphi_{t}$ converges exponentially.\n\\end{prop}\n\n\\begin{proof}\nBecause $\\vert t\\vert\\leqslant 2$, there exists $\\theta$ such that $t = 2\\cos(\\theta)$. Thus,\n\\begin{equation*}\n\\vert u_{n}(t)\\vert = \\vert U_{n}(\\cos(\\theta))\\vert = \\left\\vert \\frac{\\sin\\left((n+1)\\theta\\right)}{\\sin(\\theta)}\\right\\vert \\leqslant \\frac{1}{\\vert\\sin(\\theta)\\vert}\n\\end{equation*}\nand\n\\begin{align*}\nA_{k}(t) & \\leqslant \\sum_{n=1}^{+\\infty}\\frac{1}{\\vert\\sin(\\theta)^{2k}\\vert u_{n}(N)^{2k-2}} \\\\\n& \\leqslant \\frac{1}{\\vert\\sin(\\theta)\\vert^{2k}}\\sum_{n=1}^{+\\infty}\\left(\\frac{q(N)^{n-1}}{N}\\right)^{2k-2} \\\\\n& = \\frac{1}{\\vert\\sin(\\theta)\\vert^{2k}N^{2k-2}}\\frac{1}{1-q(N)^{2k-2}} \\\\\n& \\leqslant \\frac{N^{2}}{1-q(N)^{2}}\\left(\\frac{1}{N\\vert\\sin(\\theta)\\vert}\\right)^{2k} \\\\\n\\end{align*}\nThe result now follows from Lemma \\ref{lem:upperbound} and the fact that $\\vert\\sin(\\theta)\\vert = \\sqrt{1-t^{2}\/4}$. Note that for $k = 1$, we get the sum of $\\vert \\sin((n+1)\\theta)\\vert\/\\vert\\sin(\\theta)\\vert$ which need not converge even though $\\varphi_{t}$ is bounded on $L^{\\infty}(O_{N}^{+})$.\n\\end{proof}\n\nProposition \\ref{prop:upperboundlessthantwo} shows that for a fixed $t$, the distance to the Haar state decreases exponentially provided $N$ is large enough and it is natural to wonder how optimal the rate $1\/N\\sqrt{1-t^{2}\/4}$ is. We give a partial answer through a lower obtained by the duality between the noncommutative $L^{1}$ and $L^{\\infty}$ spaces of a tracial von Neumann algebra. Concretely, this means that for any $a\\in L^{1}(\\mathbb{G})$,\n\\begin{equation*}\n\\|a\\|_{1} = \\sup\\{h(ax)\\mid \\|x\\|_{\\infty}\\leqslant 1\\} = \\|\\varphi\\|.\n\\end{equation*}\n\n\\begin{prop}\\label{prop:lowerbound}\nFor any $t\\in [-N, N]$ and any $k\\geqslant 1$,\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\|_{TV} \\geqslant \\frac{N}{4}\\left(\\frac{t}{N}\\right)^{k}.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nRecall that $\\varphi^{\\ast k}(n) = \\varphi(n)^{k}$ and that $h(\\chi_{n}) = 0$ for all $n\\geqslant 1$. Thus,\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\| \\geqslant \\frac{1}{2}\\sup_{n \\geqslant 1}\\frac{\\varphi^{\\ast k}(\\chi_{n})}{\\|\\chi_{n}\\|_{\\infty}} = \\frac{1}{2}\\sup_{n \\geqslant 1}\\frac{d_{n}}{\\|\\chi_{n}\\|_{\\infty}}\\left(\\frac{u_{n}(t)}{d_{n}}\\right)^{k}.\n\\end{equation*}\nTaking $n=1$ and using $\\|\\chi_{1}\\|_{\\infty} = 2$ then yields the result.\n\\end{proof}\n\nEven though this bound is very general since it works for all $t$, it yields the same exponential rate as Proposition \\ref{prop:upperboundlessthantwo} for $t = \\pm \\sqrt{2}$, meaning that the bound of Proposition \\ref{prop:upperboundlessthantwo} is rather tight.\n\n\\subsubsection{The cut-off phenomenon}\n\nWe now turn to the case when $\\vert t\\vert$ is larger than two. The corresponding states will exhibit a kind of cut-off phenomenon : for a number of steps (depending on $t$ and $N$), the state is not absolutely continuous and as soon as it is, it converges exponentially. Let us first consider the boundedness problem.\n\n\\begin{lem}\\label{lem:boundednesscriterion}\nLet $\\vert t\\vert > 2$ be fixed. Then, $\\varphi_{t}^{\\ast k}$ extends to $L^{\\infty}(O_{N}^{+})$ if and only if $q(t) > q(N)^{1-1\/k}$. Moreover, it then has an $L^{1}$-density with respect to $h$.\n\\end{lem}\n\n\\begin{proof}\nBecause $\\|\\chi_{n}\\|_{\\infty} = n+1$,\n\\begin{equation*}\n\\frac{\\varphi_{t}^{\\ast k}(\\chi_{n})}{\\|\\chi_{n}\\|_{\\infty}} = \\frac{1}{n+1}\\frac{u_{n}(t)^{k}}{u_{n}(N)^{k-1}}\n\\end{equation*}\nso that Lemma \\ref{lem:encadrement} yields\n\\begin{align*}\n\\frac{\\varphi_{t}^{\\ast k}(\\chi_{n})}{\\|\\chi_{n}\\|_{\\infty}} & \\geqslant \\frac{1}{n+1}\\left(q(N)^{n}(1-q(N)^{2})\\right)^{k-1}\\left(q(t)^{-n+1}t\\right)^{k} \\\\\n& = \\left(\\frac{q(N)^{k-1}}{q(t)^{k}}\\right)^{n}\\frac{(tq(t))^{k}(1-q(N)^{2})^{k-1}}{n+1}\n\\end{align*}\nand this is not bounded in $n$ if $q(t) < q(N)^{1-1\/k}$. If now $k$ satisfies the inequality in the statement, then a similar estimate shows that the sequence\n\\begin{equation*}\na_{t, p} = \\sum_{n=0}^{p}\\frac{u_{n}(t)^{k}}{u_{n}(N)^{k-1}}\\chi_{n}\n\\end{equation*}\nconverges in $L^{\\infty}(\\mathbb{G})\\subset L^{1}(\\mathbb{G})$ and its limit is the density of $\\varphi_{t}^{\\ast k}$.\n\\end{proof}\n\nThe previous statement may be disappointing in that for a fixed $t$, the number $k$ goes to $1$ as $N$ goes to infinity. However, in the cases coming from classical random walks, $t$ depends on $N$ and we then get a cut-off parameter which also depends on $N$, see Subsection \\ref{subsec:randomrotations}. Let us now prove that as soon as $\\varphi_{t}^{\\ast k}$ is absolutely continuous, it converges exponentially to the Haar state. This is the main result of this section.\n\n\\begin{thm}\\label{thm:estimategreaterthan2}\nLet $\\vert t\\vert > 2$ be fixed and let $k_{0}$ be the smallest integer such that $q(t) > q(N)^{1-1\/k_{0}}$. Then, for any $k\\geqslant k_{0}$,\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{1}{2}\\frac{Nq(t)^{k_{0}}}{\\sqrt{q(t)^{2k_{0}} - q(N)^{2k_{0}-2}}}\\left(\\frac{1}{Nq(t)(1-q(t)^{2})}\\right)^{k}\n\\end{equation*}\nMoreover, there exists $t_{0}$ and $t_{1}$ depending on $N$ satisfying $2 < t_{0} < 4\/\\sqrt{3} < t_{1} < N$ and such that if $t_{0} < \\vert t\\vert < t_{1}$, then the random walk associated to $\\varphi_{t}$ converges exponentially after $k_{0}$ steps.\n\\end{thm}\n\n\\begin{proof}\nWe start the computation by using Lemma \\ref{lem:encadrement} :\n\\begin{align*}\n\\frac{u_{n}(t)^{2k}}{u_{n}(N)^{2k-2}} & \\leqslant \\left(\\frac{1}{q(t)^{n}(1-q(t)^{2})}\\right)^{2k}\\left(\\frac{q(N)^{n-1}}{N}\\right)^{2k-2} \\\\\n& = \\left(\\frac{q(N)^{2k-2}}{q(t)^{2k}}\\right)^{n-1}\\frac{1}{N^{2k-2}(q(t)(1-q(t)^{2}))^{2k}}.\n\\end{align*}\nBy assumption, $q(t) > q(N)^{1-1\/k}$ so that\n\\begin{align*}\nA_{k}(t) & \\leqslant\\frac{1}{N^{2k-2}(q(t)(1-q(t)^{2}))^{2k}}\\frac{1}{1-\\frac{q(N)^{2k-2}}{q(t)^{2k}}} \\\\\n& \\leqslant\\frac{1}{N^{2k-2}(q(t)(1-q(t)^{2}))^{2k}}\\frac{1}{1-\\frac{q(N)^{2k_{0}-2}}{q(t)^{2k_{0}}}} \\\\\n& = \\frac{N^{2}q(t)^{2k_{0}}}{q(t)^{2k_{0}} - q(N)^{2k_{0}-2}}\\left(\\frac{1}{Nq(t)(1-q(t)^{2})}\\right)^{2k} \\\\\n\\end{align*}\nand the result follows by Lemma \\ref{lem:upperbound}.\n\nThe condition for exponential convergence is $q(t)(1-q(t)^{2}) > N^{-1}$. Consider the function $f : x\\mapsto x(1-x^{2})$. Elementary calculus shows that its maximum is\n\\begin{equation*}\nf\\left(\\frac{1}{\\sqrt{3}}\\right) = \\frac{2}{3\\sqrt{3}} > \\frac{1}{3} \\geqslant \\frac{1}{N}.\n\\end{equation*}\nThus, there exists an open interval $I$ containing $1\/\\sqrt{3}$ such that $f(q(t)) > 1\/N$ as soon as $q(t)$ is in $I$. Since $q(t) = 1\/\\sqrt{3}$ corresponds to $t = q(t) + q(t)^{-1} = 4\/\\sqrt{3}$, the proof is complete.\n\\end{proof}\n\nSo far our use of the term cut-off has been a little improper since we did not provide an upper bound for the total variation distance depending only on $(k-k_{0})\/N$. We will see however that when considering particular values of $t$ related to classical random walks, one can sharpen the previous result into a genuine cut-off statement. To conclude this section, let us give an explicit formula for the threshold $k_{0}$. Taking the logarithm of both sides of the equality $q(t)^{k} > q(N)^{k-1}$ and noting that $q(t) > q(N)$ yields\n\\begin{equation*}\nk_{0} = \\left\\lceil-\\frac{\\ln(q(N))}{\\ln(q(t)\/q(N))}\\right\\rceil.\n\\end{equation*}\n\n\\subsection{The quantum uniform plan Kac walk}\\label{subsec:randomrotations}\n\nIn this section we will give an explicit example of cut-off phenomenon by considering the quantum analogue of the \\emph{uniform plane Kac walk} on $SO(N)$. In the classical case, this was studied by J. Rosenthal in \\cite{rosenthal1994random} and by B. Hough and Y. Jiang in \\cite{hough2017cut} (who coined the name). In this model, a random rotation is obtained by randomly choosing a plane in $\\mathbb{R}^{N}$ and then performing a rotation of some fixed angle $\\theta$ in that plane. The corresponding measure is the uniform measure on the conjugacy class of a matrix $R_{\\theta}$ corresponding to a rotation in a plane (they are all conjugate once the angle $\\theta$ is fixed, so that the choice of the plane does not matter). As explained in Subsection \\ref{subsec:randomwalks}, this uniform measure has a natural analogue on $O_{N}^{+}$. In a sense, we are now \"quantum rotating\" the plane of $R_{\\theta}$ and the corresponding state is $\\varphi_{R_{\\theta}}$. Since\n\\begin{equation*}\n\\ev_{R_{\\theta}}(\\chi_{1}) = \\Tr(R_{\\theta}) = N-2+2\\cos(\\theta) = u_{1}(N-2+2\\cos(\\theta)),\n\\end{equation*}\nit follows by induction that $\\ev_{R_{\\theta}}(\\chi_{n}) = u_{n}(N-2+2\\cos(\\theta))$ so that $\\varphi_{R_{\\theta}} = \\varphi_{N-2+2\\cos(\\theta)}$. Note that this in a sense means that two classical orthogonal matrices are quantum conjugate if and only if they have the same trace since the uniform measure on their conjugacy classes then coincide. This illustrates the fact that there is no \"quantum $SO(N)$\" subgroup in $O_{N}^{+}$.\n\nAssuming that $\\theta$ is fixed once and for all, we will show that the corresponding random walk has a cut-off. This means that we have to prove that there exists $k_{1}$ such that for $k_{1} + cN$ steps the total variation distance decreases exponentially in $c$ while for $k_{1} - cN$ steps it is bounded below by a function which decreases slowly in $c$. We will therefore split the arguments into two parts. To simplify notations let us set $\\tau = 2(1-\\cos(\\theta))$.\n\n\\subsubsection{Upper bound}\n\nWe start with the upper bound. Since the parameter $t$ now depends on $N$, it is not even clear that the convolution powers of the state will ever extend to $L^{\\infty}(\\mathbb{G})$. To get some insight into this problem, let us first consider the threshold for $t = N-\\tau$ obtained in the previous section. For large $N$,\n\\begin{equation*}\n-\\frac{\\ln(q(N))}{\\ln(q(N-\\tau)\/q(N))} \\sim \\frac{N\\ln(N)}{\\tau}\n\\end{equation*}\nwhich is exactly the cut-off parameter conjecture by J. Rosenthal the classical case (and proven there to be valid for $\\theta = \\pi$) and later confirmed by B. Hough and Y. Jiang in \\cite{hough2017cut}. This suggests that the same phenomenon should occur for $O_{N}^{+}$. However, proving it requires some suitable estimates on the function $q(t)$. We start by giving some elementary inequalities.\n\n\\begin{lem}\\label{lem:variousbounds}\nThe following inequalities hold for all $t, N\\geqslant 4$ :\n\\begin{enumerate}\n\\item $\\displaystyle\\frac{q(N)}{q(N-\\tau)} \\leqslant \\frac{N-\\tau}{N}$,\n\\item $q(N) > 1\/N$,\n\\item $N\\ln\\displaystyle\\left(1-\\frac{\\tau}{N}\\right) \\leqslant -\\tau$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nConsider the function $f : t\\mapsto tq(t)$. Noticing that\n\\begin{equation*}\nq'(t) = \\frac{1}{2} - \\frac{1}{2}\\frac{t}{\\sqrt{t^{2} - 4}} = \\frac{\\sqrt{t^{2} - 4} - t}{2\\sqrt{t^{2}-4}} = \\frac{-q(t)}{\\sqrt{t^{2}-4}},\n\\end{equation*}\nwe see that\n\\begin{equation*}\nf'(t) = q(t) + tq'(t) = \\left(1-\\frac{t}{\\sqrt{t^{2}-4}}\\right)q(t) < 0\n\\end{equation*}\nso that $f$ is decreasing. Applying this to $N > N-\\tau$ yields the first inequality while $f(t) > \\lim_{+\\infty} f(t) = 1$ yields the second one. The third equality follows from the well-known bound $\\ln(1-x) \\leqslant -x$ valid for any $0 \\leqslant x < 1$.\n\\end{proof}\n\nIn the proof of Theorem \\ref{thm:estimategreaterthan2} we saw that the total variation distance can be bounded by\n\\begin{equation*}\n\\frac{Nq(N-\\tau)^{k}}{2\\sqrt{q(N-\\tau)^{2k} - q(N)^{2k-1}}}\\left(\\frac{1}{Nq(N-\\tau)(1-q(N-\\tau)^{2})}\\right)^{k}\n\\end{equation*}\nand the inequalities above will enable us to bound the first part of this expression. For the second part, we need to study $Nq(N-\\tau)(1-q(N-\\tau)^{2})$. Note that it is not even clear that this is greater than one and in fact, for $\\tau = 0$ it equals $1-q(N)^{4} < 1$. However, as soon as $\\tau > 0$, assuming that $N$ is large enough everything will work. To show this we will first prove two computational lemmata.\n\n\\begin{lem}\\label{lem:threefunctions}\nConsider the following functions defined for $t > 2$ :\n\\begin{equation*}\nf(t) = \\frac{\\tau^{2}}{2t(t+\\tau)^{2}} \\text{ and } g(t) = \\frac{16}{5}\\frac{1}{t^{3}(t^{2}-4)}\n\\end{equation*}\nand set\n\\begin{equation*}\nC(\\tau) = \\frac{2}{\\tau\\sqrt{5}}(2+\\sqrt{2+9\\tau^{2}})\n\\end{equation*}\nThen, $f(t) \\geqslant g(t)$ as soon as $t\\geqslant C(\\tau)$.\n\\end{lem}\n\n\\begin{proof}\nThe inequality $f(t)\\geqslant g(t)$ can be written as\n\\begin{equation}\\label{eq:functionalinequality}\n5\\tau^{2}t^{2}(t^{2}-4)\\geqslant 32(t+\\tau)^{2}.\n\\end{equation}\nBecause $t^{2}\\geqslant t^{2}-4$, the left-hand side is greater than $[\\sqrt{5}\\tau(t^{2}-4)]^{2}$ and \\eqref{eq:functionalinequality} will be satisfied as soon as $\\sqrt{5}\\tau(t^{2}-4)\\geqslant 4\\sqrt{2}(t+\\tau)$, which amounts to the quadratic inequality\n\\begin{equation*}\n\\sqrt{5}\\tau t^{2} - 4\\sqrt{2}t - 4\\tau(\\sqrt{5} + \\sqrt{2})\\geqslant 0.\n\\end{equation*}\nThe discriminant is $32 + 16\\tau^{2}(5+\\sqrt{10}) \\leqslant 32 + 16\\times 9\\times \\tau^{2}$ so that \\eqref{eq:functionalinequality} is satisfied as soon as\n\\begin{equation*}\nt\\geqslant \\frac{2\\sqrt{2}}{\\sqrt{5}\\tau} + \\frac{2}{\\sqrt{5}\\tau}\\sqrt{2+9\\tau^{2}}.\n\\end{equation*}\nThe result now follows from the observation that $C(\\tau)$ is greater than the right-hand side because $2\/3\\geqslant \\sqrt{2\/5}$.\n\\end{proof}\n\nWith this in hand we can prove the main inequality that we need.\n\n\\begin{lem}\\label{lem:hardlowerbound}\nLet $0 < \\tau \\leqslant 4$. Then, for any $N\\geqslant \\tau + C(\\tau)$,\n\\begin{equation*}\nq(N-\\tau)(1-q(N-\\tau)^{2}) \\geqslant \\frac{e^{\\tau\/N}}{N}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet us set $a_{0} = 1$, $a_{1} = 1\/2$ and for $n\\geqslant 2$,\n\\begin{equation*}\na_{n} = \\frac{1\\times 3\\times\\cdots\\times(2n-3)}{2\\times 4\\times\\cdots\\times 2n} = \\frac{(2n-3)!}{2^{n-2}(n-2)!\\times 2^{n}(n!)} = \\frac{1}{n4^{n-1}}{\\binom{2n-3}{n-1}}\n\\end{equation*}\nso that $\\sqrt{1+x} = \\sum_{n}(-1)^{n+1}a_{n}x^{n}$. It follows that\n\\begin{equation*}\nq(t) = \\frac{t}{2}\\left(1 - \\sqrt{1-\\frac{4}{t^{2}}}\\right) = \\frac{1}{t} + \\sum_{n=2}^{+\\infty}a_{n}\\left(\\frac{2}{t}\\right)^{2n-1}.\n\\end{equation*}\nMoreover, using twice the identity $1+q(t)^{2} = tq(t)$, we see that\n\\begin{equation*}\nq(t)(1-q(t)^{2}) = q(t)(2-tq(t)) = 2q(t) - t(tq(t) - 1) = 2q(t) - t^{2}q(t) + t\n\\end{equation*}\nand we deduce from this a series expansion, namely (setting $t = N-\\tau$)\n\\begin{align*}\nq(N-\\tau)(1-q(N-\\tau)^{2}) & = \\frac{2}{t} + \\sum_{n=2}^{+\\infty}2a_{n}\\left(\\frac{2}{t}\\right)^{2n-1} - t - \\sum_{n=2}^{+\\infty}a_{n}\\frac{2^{2n-1}}{t^{2n-3}} + t \\\\\n& = \\frac{1}{t} + \\sum_{n=2}^{+\\infty}(2a_{n} - 4a_{n+1})\\left(\\frac{2}{t}\\right)^{2n-1} \\\\\n& = \\frac{1}{t} - \\sum_{n=2}^{+\\infty}b_{n}\\left(\\frac{2}{t}\\right)^{2n-1}\n\\end{align*}\nwith $b_{n} = -2(a_{n} - 2a_{n+1}) = 2a_{n}(n-2)\/(n+1) > 0$. We have to find an upper bound for the sum in this expression. Using the fact that $\\binom{a}{b}\\leqslant 2^{a}$, we see that for $n\\geqslant 4$\n\\begin{equation*}\nb_{n}\\leqslant 2\\frac{n-2}{n+1}\\frac{1}{n4^{n-1}}2^{2n-3} = \\frac{n-2}{n(n+1)} \\leqslant \\frac{1}{10}\n\\end{equation*}\nwhere the last inequality comes from the fact that the sequence $(n-2)\/n(n+1)$ is decreasing for $n\\geqslant 4$. Since $b_{2}=0$ and $b_{3} = 1\/32 < 1\/10$, we have for all $t > 2$\n\\begin{align*}\n\\sum_{n=2}^{+\\infty}b_{n}\\left(\\frac{2}{t}\\right)^{2n-1} \\leqslant \\frac{1}{10}\\sum_{n=3}^{+\\infty}\\left(\\frac{2}{t}\\right)^{2n-1} = \\frac{16}{5}\\frac{1}{t^{3}(t^{2}-4)} = g(t).\n\\end{align*}\nMoreover,\n\\begin{align*}\n\\frac{1}{t} - \\frac{e^{\\tau\/(t+\\tau)}}{t+\\tau} & = \\frac{1}{t} - \\frac{1}{t+\\tau} - \\frac{\\tau}{(t+\\tau)^{2}} - \\frac{\\tau^{2}}{2(t+\\tau)^{3}} - \\sum_{k=3}^{+\\infty}\\frac{\\tau^{k}}{k!(t+\\tau)^{k+1}} \\\\\n& \\geqslant \\frac{\\tau^{2}}{(t+\\tau)^{2}}\\left(\\frac{1}{t} - \\frac{1}{2(t+\\tau)}\\right) - \\frac{\\tau^{3}}{(t+\\tau)^{4}}\\sum_{k=3}^{+\\infty}\\frac{1}{k!} \\\\\n& = \\frac{\\tau^{2}}{(t+\\tau)^{2}}\\left(\\frac{1}{2t} + \\frac{\\tau}{2t(t+\\tau)}\\right) - \\frac{\\tau^{3}}{(t+\\tau)^{4}}\\sum_{k=3}^{+\\infty}\\frac{1}{k!} \\\\\n&  \\geqslant \\frac{\\tau^{2}}{2t(t+\\tau)^{2}} + \\frac{\\tau^{3}}{2t(t+\\tau)^{3}} - \\left(e-\\frac{5}{2}\\right)\\frac{\\tau^{3}}{(t+\\tau)^{4}} \\\\\n& \\geqslant \\frac{\\tau^{2}}{2t(t+\\tau)^{2}} = f(t)\n\\end{align*}\nSumming up, by Lemma \\ref{lem:threefunctions},\n\\begin{equation}\\label{eq:hardlowerbound}\nq(t)(1-q(t)^{2}) - \\frac{e^{\\tau\/(t+\\tau)}}{t+\\tau} \\geqslant f(t) - g(t)\\geqslant 0\n\\end{equation}\nas soon as $t\\geqslant C(\\tau)$, i.e. $N\\geqslant \\tau + C(\\tau)$\n\\end{proof}\n\n\\begin{rem}\nThe condition $N\\geqslant \\tau + C(\\tau)$ could probably be sharpened by considering better bounds for the binomial coefficients and improving Lemma \\ref{lem:threefunctions}. However, it is already quite good since for instance for $\\tau = 4$ it yields $N\\geqslant 8$ and for $\\tau=2$ it yields $N\\geqslant 6$.\n\\end{rem}\n\nWe are now ready to establish the upper bound for the cut-off phenomenon announced in the beginning of this section, which is the main result of this work.\n\n\\begin{thm}\\label{thm:randomrotation}\nThe random walk associated to $0 < \\theta\\leqslant \\pi$ has an upper cut-off at\n\\begin{equation*}\n\\frac{N\\ln(N)}{2(1-\\cos(\\theta))}\n\\end{equation*}\nsteps in the following sense : if $N\\geqslant \\tau + C(\\tau)$, then for any $c_{0} > 0$ and all $c\\geqslant c_{0}$, after\n\\begin{equation*}\nk = \\frac{N\\ln(N)}{2(1-\\cos(\\theta))} + cN\n\\end{equation*}\nsteps we have\n\\begin{equation*}\n\\|\\varphi_{R_{\\theta}}^{\\ast k} - h\\|_{TV} \\leqslant \\frac{1}{2\\sqrt{1-e^{-4c_{0}(1-\\cos(\\theta))}}}e^{-2c(1-\\cos(\\theta))}.\n\\end{equation*}\n\\end{thm}\n\n\\begin{proof}\nLet us set $k_{1} = N\\ln(N)\/\\tau$,\n\\begin{equation*}\nB_{k}(N) = \\frac{Nq(N-\\tau)^{k}}{2\\sqrt{q(N-\\tau)^{2k} - q(N)^{2k-2}}} \\text{ and } B_{k}'(N) = \\left(\\frac{1}{Nq(N-\\tau)(1-q(N-\\tau)^{2})}\\right)^{k}.\n\\end{equation*}\nWe will bound each part separately and then combine them to get the desired estimate. First, using Lemma \\ref{lem:variousbounds} we see that\n\\begin{equation*}\nN\\ln\\left(\\frac{q(N)}{q(N-\\tau)}\\right) \\leqslant N\\ln\\left(\\frac{N-\\tau}{N}\\right) \\leqslant -\\tau,\n\\end{equation*}\nso that\n\\begin{equation*}\n(2k_{1} + 2cN)\\ln\\left(\\frac{q(N)}{q(N-\\tau)}\\right) - 2\\ln(q(N)) \\leqslant - 2\\tau c\n\\end{equation*}\nand it then follows that\n\\begin{align*}\nB_{k}(N)\\leqslant \\frac{N}{2\\sqrt{1-e^{-2\\tau c}}} \\leqslant \\frac{N}{2\\sqrt{1-e^{-2\\tau c_{0}}}}.\n\\end{align*}\nTurning now to $B_{k}'(N)$, we have by Lemma \\ref{lem:hardlowerbound}\n\\begin{align*}\n(k_{1} + cN)\\left(-\\ln(Nq(N-\\tau)(1-q(N-\\tau)^{2}))\\right) & \\leqslant \\ln(N) - \\tau c\n\\end{align*}\nso that\n\\begin{equation*}\nB_{k}'(N) \\leqslant \\frac{1}{N}e^{-\\tau c}\n\\end{equation*}\nGathering both inequalities eventually yields the announced estimate.\n\\end{proof}\n\n\\begin{rem}\nThe fact that the statement is not uniform in $c$ may be disappointing, but we cannot do better with the upper bound lemma in the sense that for $k = k_{1} + cN$, all the inequalities we used become equivalences as $N$ goes to infinity, so that $e^{c\\tau}A_{k}(N-\\tau)$ is not bounded above uniformly in $c$. Note that the result could also be stated in the following way : for any $\\epsilon > 0$, there is a uniform (in $c$) upper cut-off at $k_{1}(1+\\epsilon)$ steps.\n\\end{rem}\n\n\\subsubsection{Lower bound}\n\nTo have a genuine cut-off phenomenon, we must now show that if $k = N\\ln(N)\/\\tau - cN$, then the total variation distance is bounded below by something which is almost constant. Usually, such bounds are proven using the Chebyshev inequality. Note that if $x$ is a self-adjoint element in a von Neumann algebra, then it generates an abelian subalgebra which is therefore isomorphic to $L^{\\infty}(X)$ for some space $X$. Then, any state on the original algebra restricts to a measure on $X$ so that it makes sense to apply the Chebyshev inequality to $x$. In our case, we will apply it to $x = \\chi_{1}$, so that we have to estimate the expectation and variance of this element under the state $\\varphi_{R_{\\theta}}^{\\ast k}$. To keep things clear, we first give these estimates in a lemma.\n\n\\begin{lem}\\label{lem:lowerboundquantumrotation}\nFor $k = N\\ln(N)\/\\tau - cN$ and $N\\geqslant 5$, we have\n\\begin{equation*}\n\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) \\geqslant \\frac{e^{c\\tau}}{5} \\text{ and } \\var_{\\varphi_{R_{\\theta}}^{\\ast k}}(\\chi_{1}) \\leqslant 1.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nAs explained in the proof of \\cite[Thm 2.1]{rosenthal1994random}, for any $N\\geqslant 5$ and any $-4 \\leqslant \\tau \\leqslant 4$,\n\\begin{equation*}\nN\\left(1 - \\frac{\\tau}{N}\\right)^{N\\ln(N)\/\\tau}\\geqslant \\frac{1}{5}\n\\end{equation*}\nso that\n\\begin{equation*}\n\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) = \\frac{(N-\\tau)^{k}}{N^{k-1}} = N\\left(1-\\frac{\\tau}{N}\\right)^{N\\ln(N)\/\\tau}\\left(1-\\frac{\\tau}{N}\\right)^{-cN}\\geqslant \\frac{e^{c\\tau}}{5}.\n\\end{equation*}\nAs for the second inequality, first note that $\\chi_{1}^{2} = \\chi_{2} + 1$ so that\n\\begin{align*}\n\\var_{\\varphi_{R_{\\theta}}^{\\ast k}}(\\chi_{1}) & = 1 + \\frac{((N-\\tau)^{2} - 1)^{k}}{(N^{2}-1)^{k-1}} - \\left(\\frac{(N-\\tau)^{k}}{N^{k-1}}\\right)^{2} \\\\\n& \\leqslant 1 + \\frac{(N-\\tau)^{2k}}{N^{2k-2}}\\left(\\frac{\\left(1-(N-\\tau)^{-2}\\right)^{k}}{(1-N^{-2})^{k-1}} - 1\\right) \\\\\n& \\leqslant 1.\n\\end{align*}\n\\end{proof}\n\nWe are now ready for the proof of the lower bound.\n\n\\begin{prop}\\label{prop:lowercutoff}\nThe random walk associated to $0 < \\theta\\leqslant \\pi$ has a lower cut-off at\n\\begin{equation*}\n\\frac{N\\ln(N)}{2(1-\\cos(\\theta))}\n\\end{equation*}\nsteps in the following sense : for any $c > 0$, at\n\\begin{equation*}\nk = \\frac{N\\ln(N)}{2(1-\\cos(\\theta))} - cN\n\\end{equation*}\nsteps we have\n\\begin{equation*}\n\\|\\varphi_{R_{\\theta}}^{\\ast k} - h\\|_{TV} \\geqslant 1-200e^{-2c\\tau}\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nWe will evaluate the states at projections obtained by functional calculus and use the original definition of the total variation distance. Let us denote by $\\mathbf{1}_{S}$ the indicator function of a subset $S$ of $\\mathbb{R}$. The proof relies on the same trick as in the classical case (see for instance \\cite{rosenthal1994random}) using Chebyshev's inequality : noticing that because of the first inequality of Lemma \\ref{lem:lowerboundquantumrotation}, $\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert \\chi_{1}\\vert)\\leqslant \\mathbf{1}_{[e^{c\\tau}\/10, +\\infty]}(\\vert\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) - \\chi_{1}\\vert)$, we have\n\\begin{align*}\n\\varphi_{R_{\\theta}}^{\\ast k}\\left(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert)\\right) & \\leqslant \\varphi_{R_{\\theta}}^{\\ast k}(\\mathbf{1}_{[e^{c\\tau}\/10, +\\infty]}\\left(\\vert\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) - \\chi_{1}\\vert)\\right) \\\\\n& \\leqslant 100e^{-2c\\tau}\\var_{\\varphi_{R_{\\theta}}^{\\ast k}}(\\chi_{1}) \\\\\n& \\leqslant 100e^{-2c\\tau}\n\\end{align*}\nOn the other hand, since $h(\\chi_{1}) = 0$ and $h(\\chi_{1}^{2}) = 1$,\n\\begin{equation*}\nh\\left(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert)\\right) = 1 - h\\left(\\mathbf{1}_{]e^{c\\tau}\/10, +\\infty[}(\\vert\\chi_{1}\\vert)\\right) \\geqslant 1 - 100e^{-2c\\tau}.\n\\end{equation*}\nGathering these facts, we get\n\\begin{equation*}\n\\|\\varphi_{R_{\\theta}}^{\\ast k} - h\\|_{TV} \\geqslant \\frac{1}{2}\\left\\vert h(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert)) - \\varphi_{R_{\\theta}}^{\\ast k}(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert))\\right\\vert \\geqslant 1-200e^{-2c\\tau}\n\\end{equation*}\n\\end{proof}\n\nThe combination of Theorem \\ref{thm:randomrotation} and Proposition \\ref{prop:lowerbound} establishes the announced cut-off phenomenon. For $\\theta = \\pi$, J. Rosenthal proved \\cite{rosenthal1994random} that $N\\ln(N)\/4$ steps suffice to get exponential convergence, in accordance with our result (for $N\\geqslant 8$). For $\\theta\\neq \\pi$, he could only show that at least $N\\ln(N)\/2(1-\\cos(\\theta))$ steps are required and the sufficiency was proved by B. Hough and Y. Jiang in \\cite{hough2017cut}. One can also consider the random walk given by a random reflection since they form a conjugacy class. Noting that any reflection has trace $N-2$ the previous argument shows that for $N\\geqslant 6$ there is a cut-off with parameter $N\\ln(N)\/2$, in accordance with the results of U. Porod in the classical case \\cite{porod1996cut}. Note that we could also use the same computations to obtain a cut-off for $\\varphi_{t}$ as soon as $t > 2$, but we decided to stick to random walks which are connected to important classical examples.\n\nThe reader may be surprised that our random walk is defined on an analogue of $O_{N}$ instead of $SO(N)$ since we are considering the conjugacy class of a matrix with determinant one. This comes from the fact that $O_{N}^{+}$ is in a sense connected as a compact quantum group so that it has no \"$SO^{+}_{N}$\" quantum subgroup (recall that matrices with opposite determinant are quantum conjugate if they have the same trace) and this allows the random walk to spread on the whole of $O_{N}^{+}$. There is another quantum group linked to $SO(N)$, which is the quantum group of trace-preserving automorphisms of the algebra $M_{N}(\\mathbb{C})$ of $N$ by $N$ matrices. We will see in Subsection \\ref{subsec:quantumautomorphisms} that the uniform plane Kac walk on it has a cut-off with the same parameter as for $O_{N}^{+}$.\n\n\\subsection{Mixed rotations}\\label{subsec:mixedrotations}\n\nOne may also consider random walks associated to states which are \"mixed\" instead of being pure. For instance, let $\\nu$ be a probability measure on the circle $\\mathbb{T}$, and set\n\\begin{equation*}\n\\varphi_{\\nu}(x) = \\int_{\\mathbb{T}}\\varphi_{R_{\\theta}}(x)\\mathrm{d}\\nu(\\theta).\n\\end{equation*}\nThis defines a central state $\\varphi_{\\nu}$ on $O_{N}^{+}$ corresponding to a random walk where $\\theta$ is chosen randomly according to $\\nu$ and then $R_{\\theta}$ is randomly conjugated. B. Hough and Y. Jiang obtained in \\cite{hough2017cut} cut-off results for these random walks with the sole restriction that $\\nu\\neq\\delta_{0}$. In our context, a stronger assumption will be needed, due to an analytic issue. Assume for instance $\\nu(\\{0\\}) = p > 0$, then $\\varphi_{\\nu}$ can be written as\n\\begin{equation*}\n\\varphi_{\\nu} = p\\varphi_{N} + (1-p)\\varphi_{\\nu'}\n\\end{equation*}\nwhere $\\nu' = (1-p)^{-1}(\\nu - p.\\delta_{0})$. But $\\varphi_{N}$ is a very particular map called the \\emph{co-unit} and for a compact quantum group, the co-unit is bounded on $L^{\\infty}(\\mathbb{G})$ if and only if $\\mathbb{G}$ is \\emph{co-amenable}. Since co-amenability is known to fail for $O_{N}^{+}$ by \\cite[Cor 1]{banica1997groupe} and $\\varphi_{N}^{\\ast k} = \\varphi_{N}$ for any $k$, we see that $\\varphi_{\\nu}^{\\ast k}\\geqslant p^{k}\\varphi_{N}$ never extends to $L^{\\infty}(O_{N}^{+})$ so that the total variation distance between $\\varphi_{\\nu}$ and $h$ is not defined.\n\nThis suggests to assume that $\\nu(\\{0\\}) = 0$, but we were not able to prove a cut-off in this generality. However, if we assume that the support of $\\nu$ is bounded away for $0$, then everything works. The proof closely follows that of B. Hough and Y. Jiang in \\cite{hough2017cut} except for some computations, which we first treat separately.\n\n\\begin{lem}\\label{lem:boundsformixedrotations}\nFor any $N\\geqslant \\tau + C(\\tau)$ and any $0 < \\tau\\leqslant 4$,\n\\begin{equation*}\n\\varphi_{N-\\tau}(n)^{N\\ln(N)\/\\tau}\\leqslant d_{n}^{-1}.\n\\end{equation*}\nMoreover, for any $N\\geqslant 3$ and any $\\lambda > 0$,\n\\begin{equation*}\n\\sum_{n=1}^{+\\infty}d_{n}^{-\\lambda\/\\ln(N)} \\leqslant \\frac{e^{-\\lambda\/2}}{1-e^{-\\lambda\/2}}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nSetting $k = N\\ln(N)\/\\tau$, we have by Lemma \\ref{lem:hardlowerbound} and the bounds of Lemma \\ref{lem:encadrement} and Lemma \\ref{lem:variousbounds},\n\\begin{align*}\n\\varphi_{N-\\tau}(n)^{k} & = \\left(\\frac{u_{n}(N-\\tau)}{u_{n}(N)}\\right)^{k} \\leqslant \\left(\\frac{q(N)^{n-1}}{q(N-\\tau)^{n}}\\frac{1}{N(1-q(N-\\tau)^{2})}\\right)^{k} \\\\\n& \\leqslant \\left(\\frac{q(N)}{q(N-\\tau)}\\right)^{k(n-1)}\\left(\\frac{1}{Nq(N-\\tau)(1-q(N-\\tau)^{2})}\\right)^{k} \\\\\n& \\leqslant \\frac{1}{N}\\left(1-\\frac{\\tau}{N}\\right)^{k(n-1)} \\leqslant \\frac{1}{N^{n}}\\leqslant \\frac{1}{d_{n}}.\n\\end{align*}\nFor the second inequality, we use again Lemma \\ref{lem:encadrement} to get\n\\begin{equation*}\nd_{n}^{-\\lambda\/\\ln(N)} \\leqslant \\left(\\frac{q(N)^{n-1}}{N}\\right)^{\\lambda\/\\ln(N)} = e^{-\\lambda}q(N)^{(n-1)\\lambda\/\\ln(N)}.\n\\end{equation*}\nUsing $\\sqrt{x}\\leqslant 1\/2  + x\/2$, we see that $q(N) \\leqslant 2\/N$ for all $N\\geqslant 4$, so that the right-hand side of the above inequality is bounded by\n\\begin{equation*}\ne^{-\\lambda}\\exp\\left((n-1)\\lambda\\left(\\frac{\\ln(2)}{\\ln(N)} - 1\\right)\\right) \\leqslant e^{-\\lambda(n+1)\/2},\n\\end{equation*}\nfrom which the result follows.\n\\end{proof}\n\nWe are now ready for the proof of the cut-off phenomenon. For convenience, we will rather consider a measure $\\mu$ on the interval $[0, 4]$ and set\n\\begin{equation*}\n\\varphi_{\\mu} = \\int_{0}^{4}\\varphi_{N-\\tau}\\mathrm{d}\\mu(\\tau).\n\\end{equation*}\n\n\\begin{thm}\\label{thm:mixedrotations}\nLet $\\mu$ be a measure on $[0, 4]$ such that there exists $\\delta > 0$ satisfying $\\mu([\\delta, 4]) = 1$ and set $\\eta = \\int\\tau\\mathrm{d}\\mu$. Then, for any $N\\geqslant \\max_{\\tau\\in [\\delta, 4]}(\\delta + C(\\delta))$, the random walk associated to $\\varphi_{\\mu}$ has a cut-off at $N\\ln(N)\/\\eta$ steps.\n\\end{thm}\n\n\\begin{proof}\nThe proof closely follows the argument of \\cite[Sec 4]{hough2017cut} and we first treat the upper bound. Let us set $k = N\\ln(N)\/\\eta + cN$. We start by the straightforward inequality\n\\begin{equation*}\n\\|\\varphi_{\\mu}^{\\ast k} - h\\|_{TV}\\leqslant \\int_{[\\delta, 4]^{k}}\\|\\varphi_{N-\\tau_{k}}\\ast\\cdots\\ast\\varphi_{N-\\tau_{1}} - h\\|_{TV}\\;\\mathrm{d}\\mu(\\tau_{1})\\cdots\\mathrm{d}\\mu(\\tau_{n})\n\\end{equation*}\nand set $E = \\{(\\tau_{1}, \\cdots, \\tau_{k})\\in [\\delta, 4]^{k}\\mid \\sum_{i=1}^{k}\\tau_{i}\\leqslant N\\ln(N) + c\\eta N\/2\\}$. Consider the random variable $X = \\sum_{i=1}^{k}\\tau_{i}$, which has expectation $k\\eta$ under $\\mu$. The measurable set $E$ corresponds to the event\n\\begin{equation*}\nX\\leqslant \\mathbb{E}(X)\\left(1-\\frac{c\\eta}{2(\\ln(N) + c\\eta)}\\right)\n\\end{equation*}\nso that by Hoeffding's inequality (using the fact that $0\\leqslant \\tau\\leqslant 4$),\n\\begin{equation*}\n\\mu^{\\otimes k}(E)\\leqslant \\exp\\left(-\\frac{2k}{16}\\left(\\frac{c\\eta}{2(\\ln(N) + c\\eta)}\\right)^{2}\\right) = \\exp\\left(-\\frac{c^{2}\\eta N}{32(\\ln(N)+c)}\\right).\n\\end{equation*}\nThe function $x\\mapsto x\/(\\ln(x)+c)$ is increasing as soon as $x\\geqslant e\\geqslant e^{1-c}$. In particular, for $N\\geqslant 3$ it can be bounded below by $3\/(\\ln(3)+c)$. Moreover, $c^{2}\/(\\ln(3)+c) > c-\\ln(3)$ so that\n\\begin{equation*}\n\\mu^{\\otimes k}(E)\\leqslant 3^{\\eta\/32}e^{-c\\eta\/32}\\leqslant 3^{1\/8}e^{-\\eta c\/32}.\n\\end{equation*}\nWe still have to bound the integral on the complement of $E$. To do this, we apply Lemma \\ref{lem:upperbound} and Lemma \\ref{lem:boundsformixedrotations} to the integrand (recall that $\\tau_{i} \\geqslant \\delta$ for all $i$), which is therefore less than\n\\begin{equation*}\n\\frac{1}{2}\\sqrt{\\sum_{n=1}^{+\\infty}d_{n}^{2}\\prod_{i=1}^{k}\\varphi_{N-\\tau_{i}}(n)^{2}}\\leqslant \\frac{1}{2}\\sqrt{\\sum_{n=1}^{+\\infty}\\exp\\left(2\\ln(d_{n})\\left(1-\\frac{1}{N\\ln(N)}\\sum_{i=1}^{k}\\tau_{i}\\right)\\right)}.\n\\end{equation*}\nBy definition of the complement of $E$, each term is bounded by $\\exp(-\\ln(d_{n})c\\eta\/\\ln(N))$ and by Lemma \\ref{lem:boundsformixedrotations} we conclude that\n\\begin{equation*}\n\\|\\varphi_{\\mu}^{\\ast k} - h\\|_{TV}\\leqslant 3^{1\/8}e^{-\\eta c\/32} + \\frac{e^{-c\\eta\/4}}{2\\sqrt{1-e^{-c\\eta\/2}}}.\n\\end{equation*}\n\nFor the lower bound, we proceed as in Proposition \\ref{prop:lowercutoff} and all that is needed is estimates of the mean and variance of $\\chi_{1}$. Noticing that\n\\begin{equation*}\n\\varphi_{\\mu}(1) = \\frac{1}{N}\\int_{0}^{4}\\chi_{1}(N-\\tau)\\mathrm{d}\\mu = 1 - \\frac{\\eta}{N},\n\\end{equation*}\nwe get $\\varphi_{\\mu}^{\\ast k}(\\chi_{1}) = d_{1}\\varphi_{\\mu}(1)^{k} = N(1-\\eta\/N)^{k}$ and as before we conclude that this is greater than or equal to $e^{\\eta c}\/5$ for any $N\\geqslant 5$. As for the variance, it follows from Popoviciu's inequality (see \\cite[Thm 2]{bhatia2000better} for an operator algebraic statement and proof), that since $-2 < \\chi_{1} < 2$,\n\\begin{equation*}\n\\var_{\\psi}(\\chi_{1})\\leqslant (2 - (-2))^{2}\/4 = 4\n\\end{equation*}\nfor any state $\\psi$. Applying this to $\\varphi_{\\mu}^{\\ast k}$ and using the same argument as in Proposition \\ref{prop:lowercutoff} then yields\n\\begin{equation*}\n\\|\\varphi_{\\mu}^{\\ast (N\\ln(N)\/4 - cN)} - h\\|_{TV} \\geqslant 1 - 500e^{-2\\eta c}.\n\\end{equation*}\n\\end{proof}\n\nExtending the previous result seems impossible with the techniques of the present work since it is clear that our estimates for fixed $\\tau$ can only be valid for $N$ larger than a function of $\\tau$ going to infinity as $\\tau$ goes to $0$.\n\n\\section{Further examples}\\label{sec:further}\n\nIn this section we will consider random walks on other compact quantum groups which were also introduced by S. Wang in \\cite{wang1998quantum} and called \\emph{free symmetric quantum groups}. As before, we define them through a universal algebra :\n\n\\begin{de}\nLet $\\O(S_{N}^{+})$ be the universal $*$-algebra generated by $N^{2}$ \\emph{self-adjoint} elements $u_{ij}$ such that for all $1\\leqslant i, j \\leqslant N$,\n\\begin{equation*}\nu_{ij}^{2} = u_{ij} \\text{ and } \\displaystyle\\sum_{k=1}^{N}u_{ik} = 1 = \\displaystyle\\sum_{k=1}^{N}u_{kj}.\n\\end{equation*}\nThe formula\n\\begin{equation*}\n\\Delta(u_{ij}) = \\sum_{k=1}^{N}u_{ik}\\otimes u_{kj}\n\\end{equation*}\nextends to a $*$-algebra homomorphism $\\Delta : \\O(S_{N}^{+})\\to \\O(S_{N}^{+})\\otimes \\O(S_{N}^{+})$ and this can be completed into a compact quantum group structure.\n\\end{de}\n\nAs the name and notation suggest, the abelianization of $\\O(S_{N}^{+})$ is exactly $\\O(S_{N})$ and the two even coincide for $N\\leqslant 3$. However, as soon as $N\\geqslant 4$ the compact quantum group $S_{N}^{+}$ is infinite (in the sense that the algebra $\\O(S_{N}^{+})$ is infinite-dimensional) and therefore behaves very differently from the classical symmetric group. This will raise an analytic issue in the sequel. Let us now describe the representation theory of $S_{N}^{+}$, which is quite close to that of $O_{N}^{+}$. The irreducible representations are still labelled by nonnegative integers but this time the recursion relation for characters is\n\\begin{equation}\\label{eq:recursionsymetric}\n\\chi_{1}\\chi_{n} = \\chi_{n+1} + \\chi_{n} + \\chi_{n-1}.\n\\end{equation}\nTo translate this into an explicit isomorphism with $\\mathbb{C}[X]$, first note that keeping the notations of Section \\ref{sec:orthogonal}, $u_{2n}(X)$ has only even powers of $X$ for any $n$. Thus, $v_{n}(X) = u_{2n}(\\sqrt{X})$ is a polynomial in $X$ and it is easily checked that this new sequence satisfies the above recursion relation. Once again, one has $d_{n} = v_{n}(N) = u_{2n}(\\sqrt{N})$.\n\n\\subsection{Pure state random walks on free symmetric quantum groups}\n\nAs for free orthogonal quantum groups, we can study pure state random walks. In view of the link between the polynomials $u_{n}$ and $v_{n}$, estimates of the total variation distance for the random walk associated to a pure state on $S_{N}^{+}$ can be easily deduced from Proposition \\ref{prop:upperboundlessthantwo} and Theorem \\ref{thm:estimategreaterthan2}. We will therefore simply give the statements, starting with the case of small $t$.\n\n\\begin{prop}\nLet $\\vert t\\vert < 4$ be fixed. Then, for any $k\\geqslant 2$,\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{1}{2}\\sqrt{\\frac{N}{q(\\sqrt{N})^{2}(1-q(\\sqrt{N})^{4})}}\\left(\\frac{q(\\sqrt{N})}{N\\sqrt{1 - t^{2}\/4}}\\right)^{k}\n\\end{equation*}\nIn particular, if $t<2\\sqrt{1-\\left(\\frac{q(\\sqrt{N})}{N}\\right)^{2}}$ then the random walks converges exponentially.\n\\end{prop}\n\nOne can also get a lower bound like in Proposition \\ref{prop:lowerbound} : noticing that $u_{2}(X) = X^{2} - 1$ yields\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\|_{TV} \\geqslant \\frac{N-1}{6}\\left(\\frac{t-1}{N-1}\\right)^{k}.\n\\end{equation*}\n\nFor larger $t$, the proof is also the same as in Theorem \\ref{thm:estimategreaterthan2}.\n\n\\begin{prop}\nLet $\\vert t\\vert > 4$ and let $k_{0}$ be the smallest integer such that $q(t) > q(N)^{1-1\/k_{0}}$. If $k\\leqslant k_{0}$ then the state $\\varphi_{t}^{\\ast k}$ is not bounded on $L^{\\infty}(S_{N}^{+})$ and otherwise\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{1}{2}\\sqrt{\\frac{N^{2}q(\\sqrt{t})^{4k_{0}}}{q(\\sqrt{t})^{4k_{0}} - q(\\sqrt{N})^{4k_{0}-4}}}\\left(\\frac{q(\\sqrt{N})}{\\sqrt{N}q(\\sqrt{t})^{2}(1-q(\\sqrt{t})^{2})}\\right)^{k}\n\\end{equation*}\n\\end{prop}\n\nThe main point in the above statement is that the threshold $k_{0}$ is the same as for $O_{N}^{+}$, so that the cut-off parameter of a uniform random walk on a conjugacy class should be given by the same formula as before. One of the simplest examples of such a random walk is the one associated to the uniform measure on the set of transpositions, or equivalently on the conjugacy class of a transposition. Since the trace of a transposition matrix is $N-2$, this is given by the state $\\varphi_{N-2}$ and the expected cut-off parameter is $N\\ln(N)\/2$. This can be proven by the same strategy as for Theorem \\ref{thm:randomrotation} but the computations are more involved.\n\n\\begin{thm}\\label{thm:randomtranspositions}\nFor any $N\\geqslant 16$, the random walk associated to $\\varphi_{N-2}$ on $S_{N}^{+}$ has a cut-off at $N\\ln(N)\/2$ steps.\n\\end{thm}\n\n\\begin{proof}\nFor the upper bound, the part concerning $B_{k}$ is the same as in the proof of Theorem \\ref{thm:randomrotation}, so let us focus on $B_{k}'$. It is enough to prove that\n\\begin{equation*}\n\\frac{\\sqrt{N}}{q(\\sqrt{N})}q(\\sqrt{N-2})^{2}\\left(1-q(\\sqrt{N-2})^{2}\\right) \\geqslant e^{2\/N}.\n\\end{equation*}\nWriting $q(t)^{2}(1-q(t)^{2}) = (3t-t^{3})q(t) + t^{2} - 2$ and expanding we get\n\\begin{align*}\nq(t)^{2}(1-q(t)^{2}) = t\\sum_{n=2}^{+\\infty}(3a_{n} - 4a_{n+1})\\left(\\frac{2}{t}\\right)^{2n-1} \\\\\n\\end{align*}\nsince $c_{n} = 4a_{n+1} - 3a_{n} = (n-5)a_{n}\/(n+1)$, the sum splits as\n\\begin{equation*}\n\\frac{1}{t^{2}} + \\frac{1}{t^{4}} + \\frac{1}{t^{6}} - t\\sum_{n=6}^{+\\infty}c_{n}\\left(\\frac{2}{t}\\right)^{2n-1}.\n\\end{equation*}\nMoreover, the same estimate as for $b_{n}$ yields $c_{n}\\leqslant (n-5)\/2n(n+1)$ and the sequence on the right-hand side is increasing up to $n=10$ and then decreasing. Its maximum is therefore $1\/44$. Using $\\sqrt{t}$ instead of $t$ and the fact that $c_{n} \\leqslant 1\/44$, we eventually get\n\\begin{equation*}\nq(\\sqrt{t})^{2}(1-q(\\sqrt{t})^{2}) = \\frac{1}{t} + \\frac{1}{t^{2}} + \\frac{1}{t^{3}} - \\sum_{n=6}^{+\\infty}2c_{n}\\left(\\frac{4}{t}\\right)^{n-1}\\geqslant \\frac{t^{2} + t + 1}{t^{3}} - \\frac{4^{5}}{22\\times t^{4}(t-4)} .\n\\end{equation*}\nOn the other hand,\n\\begin{equation*}\ne^{2\/(t+2)} \\leqslant \\sum_{k=0}^{+\\infty}\\left(\\frac{2}{t+2}\\right)^{k} = \\frac{t+2}{t} = 1+\\frac{2}{t}\n\\end{equation*}\nso that it is enough to have (noticing that $q(x)^{-1} = (x+\\sqrt{x^{2}-4})\/2$)\n\\begin{align*}\n& \\sqrt{t+2}\\frac{\\sqrt{t+2} + \\sqrt{t-2}}{2}\\left(\\frac{t^{2} + t + 1}{t^{3}} - \\frac{4^{5}}{22\\times t^{4}(t-4)}\\right) - 1 - \\frac{2}{t} \\geqslant 0.\n\\end{align*}\nTo see when this inequality holds, let us first prove that for $t\\geqslant 12$, $1\/2t^{3} \\geqslant 4^{5}\/22(t^{4}(t-4))$. Proceeding as in the proof of Lemma \\ref{lem:threefunctions}, we reduce the problem to $11t(t-4)\\geqslant 4^{5}$, i.e.\n\\begin{equation*}\nt^{2} - 4t - \\frac{4^{5}}{11} \\geqslant 0.\n\\end{equation*}\nwhich is satisfied as soon as $t$ is greater than $2 + 2\\sqrt{1+16^{2}\/11}\\leqslant 12$. Using this, it is now enough to check that\n\\begin{align*}\n1 + \\frac{2}{t} & \\leqslant \\left(1 + \\frac{t}{2} + \\frac{\\sqrt{t^{2}-4}}{2}\\right)\\left(\\frac{1}{t} + \\frac{1}{t^{2}} + \\frac{1}{2t^{3}}\\right) \\\\\n& = \\frac{1}{t} + \\frac{1}{t^{2}} + \\frac{1}{2t^{3}} + \\frac{1}{2} + \\frac{1}{2t} + \\frac{1}{4t^{2}} + \\frac{\\sqrt{t^{2}-4}}{2t} + \\frac{\\sqrt{t^{2}-4}}{2t^{2}} + \\frac{\\sqrt{t^{2}-4}}{4t^{3}}.\n\\end{align*}\nWe will prove the stronger inequality obtained by removing the terms with $t^{3}$ at the denominator in the right-hand side. After simplifying and multiplying by $2t^{2}$ we get the inequality\n\\begin{equation*}\nt^{2} + t \\leqslant \\frac{5}{2} + t\\sqrt{t^{2}-4} + \\sqrt{t^{2}-4}.\n\\end{equation*}\nNow, the function $f: t\\mapsto (t+1)(t-\\sqrt{t^{2}-4})$ satisfies\n\\begin{equation*}\nf'(t) = t-\\sqrt{t^{2}-4} + (t+1)\\left(1-\\frac{t}{\\sqrt{t^{2}-4}}\\right) = \\frac{t-\\sqrt{t^{2}-4}}{\\sqrt{t^{2} - 4}}\\left(\\sqrt{t^{2}-4} - (t+1)\\right).\n\\end{equation*}\nThis is negative, thus $f$ is decreasing and for $t\\geqslant 14$ it is smaller than $f(14) \\approx 2.15 < 5\/2$.\n\nConcerning the lower bound, first note that the expectation and variance of $\\chi_{1}$ with respect to $h$ are respectively equal to $0$ and $1$. Moreover, by the same argument as for $O_{N}^{+}$,\n\\begin{equation*}\n\\varphi_{N-2}(\\chi_{1}) = \\frac{(N-3)^{k}}{(N-1)^{k}} \\geqslant \\frac{e^{2c}}{5}\n\\end{equation*}\nfor $k = N\\ln(N)\/2 - c$ and the variance can be bounded independently from $N$ by Popoviciu's inequality.\n\\end{proof}\n\n\\begin{rem}\nPlotting the function appearing in the study of the upper bound suggests that it is positive as soon as $t\\geqslant 12$, which would give a cut-off for all $N\\geqslant 14$. This indicates that even though they look loose, our estimates are close to optimal.\n\\end{rem}\n\nThe cut-off parameter is the same as in the classical case (see \\cite{diaconis1981generating}). We can even consider the conjugacy class of $m$-cycles for any integer $m$ and, for $N$ large enough, the cut-off will appear at $N\\ln(N)\/m$ steps, again as in the classical case \\cite{hough2016random}.\n\n\\subsection{Mixed states and transition operators}\n\nThere are many examples of mixed states on $S_{N}^{+}$ coming from classical random walks. However, their study in the quantum case is prevented by the fact that $S_{N}^{+}$ is not amenable for $N\\geqslant 5$, a phenomenon which was alluded to for $O_{N}^{+}$ in Subsection \\ref{subsec:mixedrotations}. We will now illustrate this in more details on a simple example with random transpositions as follows : assume you have a deck of $N$ cards and spread them on a table. Randomly select one card uniformly (i.e. with probability $1\/N$ for each card) and then select another one in the same way. If the same card has been selected twice, nothing is done. Otherwise, the two cards are swapped. This corresponds to the measure on $S_{N}$ giving probability $1\/N^{2}$ to all transpositions and $1\/N$ to the identity. Since transpositions form a conjugacy class, the measure can be restated as being\n\\begin{equation*}\n\\mu_{\\text{rt}} = \\frac{N-1}{N}\\mu_{\\text{tran}} + \\frac{1}{N}\\delta_{\\id}.\n\\end{equation*}\nwhere $\\mu_{\\text{tran}}$ is the uniform measure on the set of transpositions. The equation above directly gives the state on $\\O(S_{N}^{+})$ corresponding to \"random quantum transposition\" :\n\\begin{equation*}\n\\varphi_{\\text{rt}} = \\frac{N-1}{N}\\varphi_{N-2} + \\frac{1}{N}\\varphi_{N}.\n\\end{equation*}\nThe state $\\varphi_{N-2}^{\\ast k}$ is bounded on $L^{\\infty}(S_{N}^{+})$ for $k$ large enough but not $\\varphi_{N}^{\\ast k}$ since it is the co-unit and $S_{N}^{+}$ is not co-amenable for $N\\geqslant 5$. This implies that no convolution power of $\\varphi_{\\text{rt}}$ is bounded on $L^{\\infty}(S_{N}^{+})$ so that the total variation distance is never defined (it is clear that the sum in the upper bound lemma diverges since each term is greater than $N^{-k}$). This is in sharp contrast with the classical case (a finite quantum group is always amenable).\n\nHowever, it is known (see for instance \\cite[Lem 3.4]{brannan2011approximation}) that the associated transition operator $P_{\\varphi_{\\text{rt}}} = (\\id\\otimes \\varphi_{\\text{rt}})\\circ\\Delta$ always extends to a bounded linear map on $L^{\\infty}(\\mathbb{G})$. We can therefore compare it with $P_{h}$ using operator norms. In particular, we can see them as operators on $L^{2}(S_{N}^{+})$ and the corresponding norm is then easy to compute :\n\n\\begin{lem}\nLet $\\psi$ be any central linear form on a compact quantum group $\\mathbb{G}$. Then,\n\\begin{equation*}\n\\|P_{\\psi}\\|_{B(L^{2}(\\mathbb{G}))} = \\sup_{\\alpha\\in \\Irr(\\mathbb{G})}\\vert \\psi(\\alpha)\\vert.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nBy Woronowicz' Peter-Weyl theorem, the elements $u^{\\alpha}_{ij}$ form an orthogonal basis of $L^{2}(\\mathbb{G})$. Moreover, a straightforward calculation yields\n\\begin{equation*}\nP_{\\psi}(u^{\\alpha}_{ij}) = \\psi(\\alpha)u^{\\alpha}_{ij}\n\\end{equation*}\nso that $P_{\\psi}$ is diagonal in this basis and the result follows.\n\\end{proof}\n\nSince $P_{h}$ is the projection onto the linear span of $1$, the above Lemma means that the distance in operator norm is exactly given by the spectral gap of the operator $P_{\\varphi}$. In this setting it is not very difficult to prove that there is a cut-off phenomenon.\n\n\\begin{prop}\nThe random walk associated to $\\varphi_{\\text{rt}}$ has a cut-off in the $L^{2}$-operator norm at $k = N\/2$ steps.\n\\end{prop}\n\n\\begin{proof}\nWe first show that the supremum of\n\\begin{equation*}\n\\left(\\frac{N-1}{N}\\frac{v_{n}(N-2)}{v_{n}(N)} + \\frac{1}{N}\\right)^{k}\n\\end{equation*}\nis attained at $n = 1$. Let us set, for $n\\geqslant 1$, $a_{n}(t) = u_{n+1}(t)\/u_{n}(t)$. The recursion relation \\eqref{eq:recursionsymetric} implies\n\\begin{equation*}\na_{n+1}(t) = t - \\frac{1}{a_{n}(t)}\n\\end{equation*}\nfrom which it follows by induction that for all $n$, $t-1\/t \\leqslant a_{n}(t)\\leqslant t$. Using this, we see that\n\\begin{equation*}\n\\frac{u_{n+1}(\\sqrt{N-2})}{u_{n+1}(\\sqrt{N})}\\frac{u_{n}(\\sqrt{N})}{u_{n}(\\sqrt{N-2})} = \\frac{a_{n}(\\sqrt{N-2})}{a_{n}(\\sqrt{N})} \\leqslant \\frac{\\sqrt{N(N-2)}}{N-1} < 1.\n\\end{equation*}\nThus, the sequence $u_{n}(\\sqrt{N-2})\/u_{n}(\\sqrt{N})$ is decreasing and the claim is proved. As a consequence,\n\\begin{equation*}\n\\|P_{\\varphi_{\\text{rt}}^{\\ast k}} - P_{h}\\|_{B(L^{2}(S_{N}^{+}))} = \\left(\\frac{N-1}{N}\\frac{N-3}{N-1} + \\frac{1}{N}1\\right)^{k} = \\left(1 - \\frac{2}{N}\\right)^{k}.\n\\end{equation*}\nBecause $1-x\\leqslant e^{-x}$, for any $c > 0$\n\\begin{equation*}\n\\|P_{\\varphi_{\\text{rt}}}^{\\ast (N\/2+cN)} - P_{h}\\|_{B(L^{2}(S_{N}^{+}))}\\leqslant e^{-1}e^{-2c},\n\\end{equation*}\nyielding the upper bound.\n\nAs for the lower bound, using an estimate already mentioned in Lemma \\ref{lem:lowerboundquantumrotation} we have for $N\\geqslant 5$\n\\begin{align*}\n\\left(1-\\frac{2}{N}\\right)^{N\/2}\\geqslant \\left(\\frac{1}{5N}\\right)^{1\/\\ln(N)} = e^{-1-\\ln(5)\/\\ln(N)} \\geqslant e^{-2}\n\\end{align*}\nso that for $c<1$,\n\\begin{equation*}\n\\|P_{\\varphi_{\\text{rt}}}^{\\ast (N\/2-cN)} - P_{h}\\|_{B(L^{2}(S_{N}^{+}))}\\geqslant e^{2c-2}\\geqslant e^{-2}(1-e^{-2c}).\n\\end{equation*}\n\\end{proof}\n\nThe cut-off in total variation distance for the classical random walk associated to $\\mu_{\\text{rt}}$ occurs at $N\\ln(N)\/2$ steps (see \\cite{diaconis1981generating}) and was one of the first important results of the theory. Since we considered a weaker norm, we get a better cut-off parameter. However, there are other norms available for operators on a von Neumann algebra which may be closer to the total variation distance and therefore yield a different cut-off parameter. In particular, since transition operators are completely positive, it would be interesting to have estimates for the \\emph{completely bounded norm} of $P_{\\varphi_{\\text{rt}}^{\\ast k}} - P_{h}$.\n\n\\subsection{Quantum automorphisms of matrices}\\label{subsec:quantumautomorphisms}\n\nAs mentioned in the end of Subsection \\ref{subsec:randomrotations}, apart from $O_{N}^{+}$ there is another quantum generalization of $SO(N)$, called the \\emph{quantum automorphism group of $(M_{N}(\\mathbb{C}), \\mathrm{tr})$}. This means that it is a universal object in the category of compact quantum groups acting on $M_{N}(\\mathbb{C})$ in a trace-preserving way. For $N=2$, this is known to be isomorphic to $SO(3)$.\n\nIt was shown in \\cite{banica1999symmetries} that the representation theory of this quantum group is the same as $S_{N}^{+}$. The only difference is that the dimensions are given by $u_{2n}(N) = v_{n}(N^{2})$. We can therefore consider the pure states $\\varphi_{t}$ as before for $0 \\leqslant t < N^{2}$ and the same arguments as in Theorem \\ref{thm:randomtranspositions} would show that the random walk associated to random rotations with a fixed angle $\\theta$ has a cut-off at $N\\ln(N)\/2(1-\\cos(\\theta))$ steps. There is however a quicker way to this. Consider the subalgebra of $\\O(O_{N}^{+})$ generated by all products $u_{ij}u_{kl}$ of two generators. Then, this is isomorphic to the Hopf algebra of the quantum automorphism group of $(M_{N}(\\mathbb{C}), \\mathrm{tr})$. The random walk can therefore be obtained by simply restricting the state to this subalgebra and as far as Lemma \\ref{lem:upperbound} is concerned this is just restricting to the sum of even terms. The upper bound for the cut-off then trivially follows from Theorem \\ref{thm:randomrotation}. As for the lower bound, it is a computation similar to that of Proposition \\ref{prop:lowerbound} using $\\chi_{2}$ instead of $\\chi_{1}$.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nResearch in interpretable machine learning (IML) has explored means to understand how machine learning (ML) models work since the nineties ({\\it e.g}., \\citeauthor{towell1992interpretation} \\citeyear{towell1992interpretation}). Popular methods to help understand ML models are referred to as attribution methods \\citep{olah, yeh2018, koh2017understanding}, they identify features or instances responsible for a classification.\n\nWith the exception of human-centered studies \\citep{hoffman2018metrics}, the evaluation methods being used in XAI and  IML include comparison to existing methods, metrics and axioms, sensitivity analyses, gold features, and demonstration of image classification (details and references in Section Background and Related Works). The problems with these methods include that they do not indicate where current XAI approaches fail thereby preventing consistent progress of the field. They do not measure accuracy as a way to validate correctness or to produce accountable agents ({\\it e.g}., \\citeauthor{Diakopoulos2014AlgorithmicAR} \\citeyear{Diakopoulos2014AlgorithmicAR}, \\citeauthor{kroll2016accountable} \\citeyear{kroll2016accountable}, \\citeauthor{doshivelez2017rigorous} \\citeyear{doshivelez2017rigorous}), and it is practically impossible to determine whether one XAI method is better than other or what the weaknesses of existing methods are,  leaving  researchers  without  guidance  on  which  re-search questions will advance the field.\n\nThe intended purpose of this paper is to address these limitations with current XAI evaluation methods by proposing the use of data representing ground-truth explanations (GTE). In a variety of computer science tasks, it is a standard practice to treat some representation of data as ground truth. Ground-truth data, in its essence, represents data that is verifiable and considered as the most accurate against which a new system is tested against \\citep{wikilink}. Various authors agree that the lack of ground-truth for evaluating explanations is a limitation \\citep{tomsett2019sanity, hooker2019benchmark, yang2019evaluating, Montavon2019}. Consequently, we investigate the challenges in creating data representing GTEs. Our goal is to promote consistent and methodical progress of the XAI field. The scope of this paper is limited to neural networks (NN) for classification tasks.\n\nThe next section presents related methods, metrics and axioms to evaluate XAI methods. Then, we introduce how to generate three data sets representing GTEs. Next, we use the generated data to train NN models to submit to LIME \\citep{ribeiro2016should} and produce explanations while converting the GTEs into LIME's explanations format. We evaluate LIME, and analyze the evaluation, seeking support to our conclusions as a means to validate the evaluation. We conclude with a discussion of issues and benefits, and future work.\n\\begin{table*}[t]\n\\centering\n\\small\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{p{0.31\\textwidth}|p{0.69\\textwidth}}\n    \\hline\n    \\textbf{Evaluation method\/Axiom\/Metric} & \\textbf{Method proposer and\/or example authors who employed them}\\\\\n    \\hline\n    Sensitivity analysis &  \\citet{adebayo2018sanity}\\\\\n    \\hline\n    Example images &  \\citet{ribeiro2016should}\\\\\n    \\hline\n    SAT\/Model counting & \\citet{ignatiev2018, narodytska2019assessing, ignatiev2019validating} \\\\\n    \\hline\n    Correlation, completeness, and complexity &  \\citet{cui2019integrative}\\\\\n    \\hline\n    Conservation, continuity &  \\citet{Montavon2019}\\\\\n    \\hline\n    Fidelity & \\citet{alavarez2018}\\\\\n    \\hline\n    Gold features &  \\citet{ribeiro2016should}\\\\\n    \\hline\n    Post-hoc accuracy  & \\citet{chen2018learning, Bhatt2019BuildingHT, xie2019reparameterizable, bai2020attention}\\\\\n    \\hline\n    Perturbation analysis for vision   &  \\citet{zeiler2014visualizing}\\\\\n    \\hline\n    ROAR    &  \\citet{hooker2019benchmark}\\\\\n    \\hline\n    Perturbation on Time Series     &  \\citet{schlegel2019rigorous}\\\\\n    \\hline\n    Implementation invariance, sensitivity      &  \\citet{sundararajan2017axiomatic}\\\\\n    \\hline\n    Input invariance      &  \\citet{kindermans2019reliability}\\\\\n    \\hline\n    Simulated users            &  \\citet{ribeiro2016should}\\\\\n    \\hline\n    Amazon Mechanical Turk users                  &  \\citet{ribeiro2016should, chen2018learning}\\\\\n    \\hline\n    In-depth Interviews & \\citet{hoffman2018metrics}\\\\\n    \\hline\n\\end{tabular}\n}\n\\caption{Methods, metrics or axioms used to evaluate XAI and IML methods}\n\\label{alternativetable1}\n\\end{table*}\n\\section{Background and Related Works} \\label{BGDRW}\nTable 1 lists various methods currently used to evaluate both XAI and IML methods. The authors referenced in the table are those that first proposed the methods or that have used them within XAI and IML works. None of these methods use data representing ground-truth explanations. The closest to ground truth is the use of gold features \\citep{ribeiro2016should}, which are a set of features used in a model that are well-known to be the most important.\n\n\\citet{doshivelez2017rigorous} categorize IML evaluations as application-, human-, and functionally-grounded. The authors propose that any method should be evaluated along those three categories, one informing the other. \\citet{yang2019evaluating} are the only authors who actually present a reason against using ground truth to benchmark explanation methods, which is that of explanation quality is user-dependent. These authors propose three metrics for IML, namely, generalizability, fidelity, and persuasibility. Their fidelity metric aims to measure explanation relevance in the applied context. \\citet{gunning2019darpa} propose XAI approaches are evaluated along five categories, namely, {\\it explanation goodness}, {\\it explanation satisfaction}, {\\it mental model understanding}, {\\it user task performance}, and {\\it appropriate trust and reliance}. Considering that human-centered studies entail a lot of subjectivity, of those, only {\\it explanation goodness} seems as an objective category of explanation quality. All other categories are evaluated by humans or an external task. \\citet{tomsett2019sanity} conducted a meta evaluation of saliency methods by analyzing metrics previously proposed in the literature to evaluate saliency methods. To do this, they adopted psychometric tests to verify local saliency metric reliability and consistency of the saliency metrics that rely on perturbations. They conclude that saliency metrics can produce unreliable and inconsistent results, even if not always.\n\n\\begin{figure}[ht]\n\\centerline{\\includegraphics[width=\\columnwidth, height=4cm]{GTE-diagram.png}} \n\\caption{{In green, this diagram shows the steps to generate data representing GTEs and to use the data to train models. In orange are the processes for aligning GTE to LIME, and to send predictions for LIME to explain. In yellow, the evaluation compares the two orange processes}}\n\\label{gted}\n\\end{figure}\n\n\\section{Generate Data Representing Ground-Truth Explanations (GTE)}\nFigure \\ref{gted} gives an overview of the entire approach from generating the data up to evaluation. We start describing how to generate the data. We propose to generate data sets from existing processes, either natural or artificial, and identify classes from said processes. We propose to represent classes in a data set via mathematical equality or inequality constraints as a minimal canonical representation from which explanations can be aligned with the format of explanations produced by various XAI methods. We define the classes and the intervals to populate feature values to create instances. The intervals where instance feature values can be populated will determine noise and commonsense. The nature of values allowed for each feature in the generated equations will determine whether classes will remain disjoint. Overlapping classes will impact evaluations producing noise. Another consideration when defining intervals to populate feature values is commonsense. If an explanation indicates that the value of a feature is 0.3x$10^{-6}$, then the feature should not represent someone's age. Next, we describe the generation of three data sets. \n\n\\subsection{Generate Data Set Loan}\n\nThe process we chose is loan underwriting. This is a small data set, consisting of 54 instances, two classes accept and reject, and three input features characterizing three well-known factors considered in loan underwriting, namely, job condition, credit score, and debt-to-income ratio. We created this data set manually to characterize realistic instances. \nThe instance space is given by the arrangement of the three features and their four possible values, given by 4 x 4 x 4 = 64. We eliminated 10 instances from the data because they were not realistic. The data is generated with a system of two equations as follows:\n\\begin{equation}\n    f(x)= \n\\begin{cases}\n    8(x_1 - 2)^2 + 3x_2^3 - x_3^4 + 4, & \\text{\\it if } x\\ne 2\\\\\n     3x_2^3 + x_3^4 + 12,              & \\text{\\it if } x= 2\n\\end{cases}\n\\end{equation}\n\\begin{equation}\n\\begin{cases}\n    Accepted, & \\text{\\it if } f(x)\\ge 32\\\\\n    Rejected,              & \\text{\\it if } f(x)< 32\n\\end{cases}\n\\end{equation}\n\nAs stated above, we considered class overlap and commonsense when defining the allowable values for the three features. The first feature, $x_1$ corresponds to the job condition of the applicant. This feature can be populated with integer values along the interval [2, 5], where 2 represents lack of a job, and values 3, 4, and 5, respectively that applicant has a job for less than one year, less than 3 years, or more than 3 years. The second feature, $x_2$, refers to credit score, which assumes integer values in the interval [0, 3], distributed in ranges from less than 580, 650, 750 and more than 750. The third and last feature, $x_3$, refers to the ratio of debt payments to monthly income, which assumes integer values in the interval [0, 3], distributed in ranges from less than 25\\%, 50\\%, 75\\% and more than 75\\%.\n\n\\subsection{Generate Data Set Distance}\n\nWe adopt the equation used to calculate travel consumption based on travel distance. The Data Set Distance has a total of 2,600,000 instances, described through five features, and 10 classes.\nThe 5 variables are Trip Frequency (TF), Trip Distance (TD), Transportation Occupancy (TO) ,Energy Intensity (EI) and Travel Mode (m). The Data Set Distance is generated using Equation 4 for travel energy consumption based on travel distance. \n\nUsing the base equation, we created 10 unique variations with the following goals: the variations should be kept realistic, the variations are a set of operations (such as raising to an exponent or multiplying by a scalar) performed on one or more variables. Afterward, we  generate the data for the base equation by creating every permutation of 4 equation variables within a specified range using a truncated normal distribution. The 4 variables are used as features along with a 5th variable, travel mode. For each of the 10 variations, we use the set of operations on the base equation data to generate the equivalent rows for that variation.\n\\begin{equation}\n    E_m = TF \\times \\frac{TD_m}{TO_m} \\times EI_m\n\\end{equation}\n\n\\subsection{Generate Data Set Time}\n\nFor Data Set Time and Distance, we used processes from the field of energy consumption analysis that describe various realistic processes with different focuses (e.g., distance or time) and include equations with a variety of features that can receive multiple values. These characteristics facilitate the generation of large data sets, so we can create conditions similar to those faced by XAI methods in the real world. In this paper, we generate data from transportation energy consumption, which can be used to calculate travel time and travel distances related to household energy consumption.\n\nEquation 3 is the basic equation to calculate energy (E) as a function of time. The four variables are Travel Time (TT), Speed, Fuel Economy (FE) and Travel Mode (m). The Data Set Time has a total of 504,000 instances and seven classes. Each class is defined by a small tweak to the equation. Using the base equation, we create seven unique variations with the same goals and process as we did for the Distance data set.\n\n\\begin{equation}\n    E_m  = \\sum_{m=1}^{5} = TT \\times Speed_m \\times FE_m\n\\end{equation}\n\n\\subsection{Train NN Models}\nThe number of models, the type of models, and how they vary between them depends on the metrics selected in the previous step. Consider, for example, the selected metric is {\\it implementation invariance} \\cite{sundararajan2017axiomatic}. This metric requires multiple types of models. In this paper, we trained two models for the Loan and Time data sets, and one model for the Distance data set, which we summarize next (detailed architectures are given in the Github link given at the end of this paper).\n\n\\subsubsection{Models NN1 and NN2}\nThe changes from NN1 to NN2 for Loan and Time included number and type of layers. Both models built for the Loan data reached 100\\% accuracy. Given the small number of 54 instances, we did not separate testing and training. For Data Set Time, NN1 reached 97\\% accuracy and NN2 96\\%. The generated instances were set to 403,200 for training, and 100,800 for testing.\nTrain NN 2 to 96\\% accuracy 403,200 training instances, 100,800 testing instances. The accuracy obtained for Data Set Distance was much lower, 82\\%. This is certainly due to noise class overlaps that occurred during the data set generation. \n\n\\subsection{LIME Explains Predictions}\nAs depicted in the diagram in Figure \\ref{gted}, after training the models, the next steps can be concurrent. This section describes the step where LIME explains the predictions from the models. First, let us briefly review how LIME works. The Local interpretable model-agnostic explanations (LIME) is a feature attribution method formally described in \\citet{ribeiro2016should}. LIME assumes that the behavior of a sample point ({\\it e.g}., instance) can be explained by fitting a linear regression with the point (to be explained) and the point's close neighbors. LIME perturbs the values of the point to be explained and submits them to the model to obtain their prediction, thus creating its own data set. Next, LIME measures the cosine similarity between the point to be explained and the points generated from its perturbations to select a region of points around it. LIME then utilizes a hyperparameter, {\\it number of samples (num\\_sample)}, to select the number of points it will use in the final step, which is the fitting of a linear regression. The hyperparameter {\\it num\\_sample} determines how many of the perturbed points will be used with the point to be explained to fit a linear regression. This last step produces coefficients of the line that expresses LIME's explanation. \n\nFor Data Set Loan, we submit to LIME all the 54 instances to be explained, and models NN1 and NN2. Note that all these instances with both NN models will have correct predictions because both models reached 100\\% accuracy. The number of samples selected was 25 to be used in the first evaluations, but we also created GTE for 5 and 50 number of samples. The output we receive back from LIME are two sets of 54 by 3 coefficients, one coefficient for each of the three features, one set for NN1, and one set for NN2. \n\nThe Data Set Time has 100,800 instances but the models did not reach 100\\% accuracy. Consequently, we randomly selected 10,000 that both models NN1 and NN2 predicted correctly and only submitted those 10,000 to LIME together with the two models. We made sure to select the instances from those correctly predicted because sending instances incorrectly predicted by the models would mislead LIME to produce bad decisions, and we had to make sure we could provide LIME with the best data for a fair assessment. The output produced by LIME are two sets of 10,000 by 4 coefficients, accounting for the four features in this data set, one for each model NN1 and NN2. We set hyperparameter number of samples as 1,000.\n\nThe accuracy reached by the Distance model NN1 was 82\\%. The number of testing instances was 520,000, so we randomly selected 50,000 instances from this data set from those correctly predicted. The data produced by LIME for NN1 is a 50,000 by 5 matrix, given the five features in this set. For number of samples, we used 5,000.\n\n\\subsection{GTE Data Aligns with LIME Explanation Format}\n\nThis step is represented in Figure \\ref{gted}, in orange, and is concurrent to the step \"Lime Explains Predictions\". As already noted, the data we produce representing GTEs are a specific way to represent explanations. We can only use them to evaluate any target XAI method after the data representing GTEs are at the same format or have been processed under the same conditions. As a general rule of thumb, this conversion may imply to take the ground-truth data and execute the last steps of the target method.\n\nFor LIME, an explanation consists of a fitted line from a target point whose prediction we want to explain and the number of points (based on the hyperparameter number of samples, {\\it num\\_sample}) that are the closest to the target based on the cosine similarity. Ultimately, this means that evaluating LIME means to determine how realistic are the perturbed points that are the closest based on cosine similarity in the number of the number of samples. Consequently, we take the points from the data representing GTEs and execute this same process, namely, measure the cosine similarity of each target point to be explained and then fit linear functions using the same Ridge regression method and the same regression parameters used in LIME using the same number of samples as established by the {\\it num\\_sample} hyperparameter. The result is that, for each data set, we have matrices with the same number of coefficients (one per feature per instance) as produced by each model. \n\n\\section{Evaluation Measures Compare Lime Explanations Against GTE}\n\\subsection{Propose Evaluation Measures}\n\\subsubsection{Euclidean distance (ED)} We adopt the {\\it Euclidean distance} (ED), which is an obvious choice of method to measure how far two points are in $n$-dimensional spaces. For this reason, we compute, for each instance of each data set and NN model, the ED between the point described in the GTE data and the point described through LIME's explanation coefficients. The range of the ED is (-$\\infty$, +$\\infty$), however, we do normalize the ED using the maximum and minimum points obtained for each data set and parameters. The goal is to keep the ED's results within the interval [0, 1] for better visualization. The purpose of computing the ED between the GTE data and LIME's explanation coefficients is to measure {\\it accuracy} as a measure of {\\it explanation goodness}. The ED, for being a distance, produces results in the opposite direction of quality. For this reason, later we will compute the {\\it Complement of ED}, which we denote as {\\it C-of-ED}, as its mathematical complement.\n\n\\subsubsection{Implementation Invariance}  \\citet{sundararajan2017axiomatic} proposed that explanation methods should produce identical attributions for networks that produce the same outputs while receiving the same inputs, which are referred to as functionally equivalent. This is why we created models with different architectures for two of our data sets.\n\n\\subsubsection{Measures of Order} We propose to use the order of the explanation coefficients as measures of {\\it accuracy} or {\\it explanation goodness}. In LIME \\citep{ribeiro2016should}, the explanation coefficients assign importance to each feature in the sense that the feature that is assigned the highest coefficient is the most important feature in the explanation. This is related to the use of gold features to evaluate explanations, as proposed by \\citet{ribeiro2016should}. When gold features are used, the evaluation often targets the inclusion or not of a feature in an explanation. In the studies in this paper, we do not discuss the inclusion or not of features because our data sets have three, four, and five features each. At these small numbers of features, LIME includes all of them; this way we do not have to evaluate whether a feature is present, but how important it is considered. Note that the order of features is particularly proposed to evaluate LIME given the format LIME presents its explanations, although this would be an important aspect to consider when evaluating any explanation method. \n\nWe define two evaluation measures of order: {\\it Second Correct} and {\\it All Correct}. Respectively, {\\it Second Correct} indicates whether the second feature in the descending order of importance of an explanation's coefficients is correct in the sense that it is the same feature ordered as the second most important in the GTE data. Then {\\it All Correct} indicates that all the features are in the same order as the features in the GTE data. The values for these measures are counted as 1 or 0 for each instance. The comparisons include results for 100 runs, hence the values represent percentages.\n\\subsection{Comparing GTEs to LIME Explanations}\n\n\\subsubsection{NN1 vs. NN2}\nWe start by comparing ED across the two different NN architectures for data sets Loan and Time, NN1 and NN2 to assess {\\it Implementation Invariance}. Both were executed for 100 runs and thus we compute average and standard deviation across the 100 runs for each instance.\n\nWe use the parametric statistical testing t-test to measure whether the values differ significantly across the two samples NN1 and NN2. To conduct the t-test, we pose the hypothesis that the true difference between NN1 and NN2 is zero. The t-test determines that for $p$-values greater than 0.1, we cannot reject the hypothesis that the difference is zero between the samples. The $p$-values computed for Loan and Time data set are respectively, 0.979 and 0.661. These resulting $p$-values show that for both Loan and Time, the differences between NN1 and NN2 are not statistically significant. \\citet{sundararajan2017axiomatic} suggests that explanation methods should satisfy {\\it Implementation Invariance} for functionally equivalent NNs. This means that their explanations ought to be the same. As far as the t-test shows, the explanation coefficients are not significantly different, so at this level of specificity, they satisfy {\\it Implementation Invariance}. Given these results, we will use only NN1 for the remainder of the studies.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=\\columnwidth, height=6cm]{CofEDvsMeasuresOfOrder.png}} \n\\caption{C-of-ED (red), Second Correct (green), and All Correct (black) for data sets Loan, Time, and Distance}\n\\label{CofEDvs}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{p{0.1\\textwidth} |c|c|c c}\n    \\hline\n   Data Set & Ave. C-of-ED & Ave. Second & Ave. All  \\\\\n    \\hline\n  Loan &  0.47 & 0.32 &  \\textbf{0.179}\\\\\n    \\hline\n  Time &  0.76 & 0.03 &  0.0008\\\\\n    \\hline\n  Distance & \\textbf{0.82}  & \\textbf{2.88} &  0.08\\\\      \n\\end{tabular}\n\\caption{Averages (Ave.) obtained for C-of-ED, {\\it Second correct}, and {\\it All correct} for all 100 runs and all instances for NN1 for the three data sets}\n\\label{5-25-50}\n\\end{table}\n\n\\subsubsection{Comparing C-of-ED against Second Correct and All Correct}\nNow we compare the measures of order {\\it All Correct} and {\\it Second Correct} against the complement of the ED, {\\it C-of-ED}. We use the complement given that measures of order are in the opposite direction of ED. Figure \\ref{CofEDvs} shows the three measures for Loan, Time, and Distance. \n\nThe visual inspection suggests multiple ideas. First, if we look at the red line for {\\it C-of-ED}, it shows that the quality of LIME's explanations seem to increase with larger values of the number of samples hyperparameter. Note that the charts for Time and Distance show only 100 samples because showing all 10,000 and 50,000 would make them indecipherable. For this reason, we include Table \\ref{5-25-50} with the averages for all instances to help the interpretation of the charts. The averages for Loan, Time, and Distance are, respectively, 0.47, 0.76, and 0.82. Recall the numbers set for number of samples submitted to LIME for these respective data sets were respectively, 25, 1,000, and 5,000. This seems reasonable as it allows LIME more chances to populate the region of the instance to be explained, thus increasing its chances of success.\n\nSecond, the measure {\\it All Correct} (black line) in Figure \\ref{CofEDvs} represents the number of times an instance has all its feature coefficients in the same order as GTE's coefficients. It is not surprising that this is the line further down with respect to the $y$-axis as it is the more demanding than {\\it Second Correct} (in green). \n\nThird, even with limited samples in the charts for Time and Distance, we see that the quality of LIME's explanations vary. This deserves a more detailed analysis. With the small data set Loan, we can see, for example, all three measures agree that Instance 50 is lower in quality than Instance 49. But before we examine the numbers and explore potential reasons for LIME having more difficulty to explain some instances over others, let's scrutinize these measures.\n\n\\subsection{Validating Evaluation Measures}\nIn this section, we investigate whether we have any evidence to support these results by further analyzing what these measures above can tell us about LIME explanations. To do this, we now focus on the Data Set Loan because its small scale allows us to conduct a detailed and comprehensive analysis. Above when we described the experimental design for the Loan data, we mentioned we selected the parameter number of samples to be 25. We now expand the results for two more number of sample values, 5 and 50.\n\nFigure \\ref{ed-5-25-50} shows the measures {\\it C-of-ED}, {\\it Second correct}, and {\\it All correct} for the different hyperparameter number of samples used with the Loan data. We kept the colors we used in earlier charts, making lighter hues for number of samples 5, darker for 50, and kept an intermediary tone for 25. For C-of-ED, the average at 5 number of samples is the highest, 0.60 against 0.47 and 0.41 for 25, and 50. For {\\it Second correct}, the highest average is 0.35, obtained with 5 and 50 number of samples, against 0.32 with 25. For {\\it All correct}, the highest is again at 5 number of samples with 0.22 against 0.18 and 0.16 for 25 and 50.  \n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\columnwidth, height=6cm]{ED2ndAllLoan5-25-50.PNG}} \n\\caption{Comparison of C-of-ED (top), Second Correct (middle), and All Correct (bottom) for Data Set Loan and number of samples 5, 25, and 50} \n\\label{ed-5-25-50}\n\\end{figure}\n\nThe first observation is that these results do not match the conclusion above that higher number of samples lead to more accurate explanations although this observation makes sense technically. Thorough examination of the results for every instance reveals that, at 5 number of samples, data representing GTE has a very high proportion of coefficients that are zero. The exact number is 18 zeros for coefficient $x_1$, 23 for $x_2$, and 27 for $x_3$, corresponding to 18, 23, and 50\\%. These high numbers of zeros can be explained by the low number of samples that would make it hard to fit the linear regression and thus return zeros. We then examined the number of zeros in the coefficients produced by LIME and we noted that in the 100 runs of 54 instances, the total number of zeros is 26, 27, and 29, respectively for $x_1$, $x_2$, and $x_3$, representing averages of 2.6, 2.7, and 2.9 for all 54 instances (these are around 5\\%). Consequently, given that LIME coefficients do not have such an abundance of zeros, ED will artificially show better results at number of samples 5 because these distances will be shorter. A distance between a number, which can be positive or negative, and zero will be shorter than a number and another that may also be positive or negative. The zeros also cause problems in computing the measures of order.\n\nTwo observations can be made from the identification of these high volumes of zero. First, the evaluations for 5 number of samples for the Loan data are artificial. They are revealed by the measures as good but the numbers are artificial, they do not originate from better explanations. Consequently, we do not have any reason to question that higher number of samples lead to better quality explanations.\n\nSecond, these artificially produced number do indicate better quality and all the proposed measures have shown them. This supports the quality of the proposed measures.\n\nFinally, these studies suggest that the best quality of explanations from LIME for the Loan Data should be when using 50 number of samples, but the measures do not show this with consistency. Consider that 50 number of samples is almost as much as the total number of instances in the Data Set Loan. With both the data representing GTEs and LIME using 50 number of samples, what would be the cause of the difference in the coefficients? If we could tell LIME the range and precision of the allowable values for the data to use in the perturbations, with only three features and a NN with 100\\% accuracy, LIME would only generate perturbations that matched the actual data set, and given that we used the same cosine similarity and the same Ridge regression with the same parameters, LIME's perturbations would be all actual instances. When using 50, it would be 50 out of 54, exactly like the data representing GTEs. Consequently, the only point of information separating LIME from better ({\\it i.e}., more accurate) explanations is not knowing the range and precision of allowable values. In practice, in a real-world model that needs explanation, there is nothing preventing us from asking the actual values allowed in data to create more accurate perturbations. This demonstrates how the use of data representing ground-truth explanations can lead to analyses that will improve existing XAI methods. \n\n\\section{Discussion and Conclusions}\nThe methodology we describe to generate data representing {\\it ground-truth explanations} (GTEs) poses many challenges. It requires the identification of a data-generation process and needs equations to define classes. The possibility of class overlap, their benefits and limitations, and methods to avoid noise are questions for future work.\n\nThe need to align data representing GTEs with the method targeted to evaluate may pose challenges such as the one we faced when setting a low value to a hyperparameter that produced artificially good results. This suggests this approach may be far from being fully automated.\n\nThe proposing authors of {\\it implementation invariance} \\cite{sundararajan2017axiomatic} suggest that explanation methods should satisfy it, which means producing the same explanation as long as NNs are functionally equivalent. If we envisage an explanation in support of accountability reports, then we want to have methods that can distinguish when a different architecture leads to a different explanation. Furthermore, when computing {\\it implementation invariance}, we face the question of which level of specificity to compare the explanations from these models. This poses questions with respect to whether what being the same for explanations mean. Consider that this question will differ depending on how the XAI method formats explanations.\n\nWe analyzed the results of our evaluation of LIME and showed how that analysis led us to conclusions about how LIME could be improved. Although not explicitly shown, our proposed method is measurable and verifiable, allowing the comparison between two explanation approaches. Further work examining why a method performs better in a certain type of instance, such as in outliers vs. non-outlier instances can help direct how to improve said methods. Finally, this proposed approach sheds light into how to demonstrate accountability, to create benchmarks, and contribute to the progress of the field.\n\nAll data and code necessary for reproducibility is available at https:\/\/github.com\/Rosinaweber forwardslash DataRepresentingGroundTruthExplanations\/tree\/master.\n\n\\subsubsection{Acknowledgements}\nRosina Weber and Prateek Goel are supported by the National Center for Advancing Translational Sciences (NCATS), National Institutes of Health, through the Biomedical Data Translator program award {\\#}OT2TR003448. 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+{"text":"\\section{\\label{sec:Intro}Introduction}\n\nOne way of realizing theories describing gravity as a truly\nhigher-dimensional interaction becoming effectively 4D at low-enough \nenergies  is represented by the braneworld models, where the\nobservable universe is a 3-brane (domain wall) to which the\nstandard model (non-gravitational) matter fields are confined,\nwhile gravity field enters the extra spatial dimensions the size\nof which may be much larger than the Planck length scale\n$l_\\mathrm{P}\\sim 10^{-33}\\, \\mathrm{cm}$ \\cite{Ark-Dim-Dva:1998:}.\n\nAs shown by Randall and Sundrum \\cite{Ran-Sun:1999:}, gravity can be localized near the brane at low energies even with\na non-compact, infinite size extra dimension \nwith the warped spacetime satisfying the 5D Einstein equations\nwith negative cosmological constant. Then an arbitrary energy-momentum tensor\ncould be allowed on the brane \\cite{Shi-Mae-Sas:2000:}.\n\n\nThe Randall-Sundrum model gives 4D Einstein gravity in low energy\nlimit, and the conventional potential of Newtonian gravity appears\non the 3-brane with high accuracy \\cite{Ran-Sun:1999:}. Significant deviations from the\nEinstein gravity occur at very high energies, e.g., in the very\nearly universe, and in vicinity of compact objects\n\\cite{Maa:2004:,Dad-etal:2000:,Ger-Maa:2001:,Ali-Gum:2005:}.\nGravitational collapse of matter trapped on the brane results in\nblack holes mainly localized on the brane, but their horizon could\nbe extended into the extra dimension.  The high-energy effects\nproduced by the gravitational collapse are disconnected from the\noutside space by the horizon, but they could have a signature on\nthe brane, influencing properties of black holes\n\\cite{Maa:2004:}. There are high-energy effects of local\ncharacter influencing pressure in collapsing matter, and also\nnon-local corrections of ``backreaction'' character arising from\nthe influence of the Weyl curvature of the bulk space on the brane\n-- the matter on the brane induces Weyl curvature in the bulk\nwhich makes influence on the structures on the brane due to the bulk\ngraviton stresses \\cite{Maa:2004:}. The combination of\nhigh-energy (local) and bulk stress (non-local) effects alters\nsignificantly the matching problem on the brane, compared to the\n4D Einstein gravity; for spherical objects, matching no longer\nleads to a Schwarzschild exterior in general\n\\cite{Maa:2004:,Ger-Maa:2001:}. The Weyl stresses\ninduced by bulk gravitons imply that the matching conditions do\nnot have unique solution on the brane; in fact, knowledge of the\n5D Weyl tensor is needed as a minimum condition for uniqueness\n\\cite{Ger-Maa:2001:}.\\footnote{At present, no exact 5D\nsolution in the braneworld model is known.} Some solutions for\nspherically symmetric black holes \\cite{Dad-etal:2000:} and\nuniform density stars \\cite{Ger-Maa:2001:} have been discussed. It has been shown\nthat in the black hole case the matching conditions could be satisfied and the bulk effects \non the black hole spacetimes could be represented by a single ``brany`` parameter. \n\nAssuming spherically symmetric metric induced on\nthe 3-brane, the constrained effective gravitational field equations on the\nbrane could be solved, giving  Reissner-Nordstr\\\"{o}m static\nblack hole solutions endowed with a braneworld parameter $b$ having character of a ``tidal'' charge,\n instead of the standard electric charge\nparameter $Q^2$ \\cite{Dad-etal:2000:}. The tidal charge  can be both positive and negative, however, there are some\nindications that negative tidal charge should properly represent the\n``backreaction'' effects of the bulk space Weyl tensor on the\nbrane \\cite{Dad-etal:2000:}.\n\nThe stationary and axisymmetric solutions describing\nrotating black holes localized in the Randall-Sundrum braneworld\nwere derived in \\cite{Ali-Gum:2005:}, having the metric tensor of the\nKerr-Newman form with a tidal charge describing the 5D correction\nterm generated by the 5D Weyl tensor stresses. The tidal charge\nhas an ``electric'' character again and arises due to the 5D\ngravitational coupling between the brane and the bulk, reflected\non the brane through the ``electric'' part of the bulk Weyl tensor\n\\cite{Ali-Gum:2005:}, in analogy with the spherically\nsymmetric case \\cite{Dad-etal:2000:}.\n\nWhen both the tidal and electric charge are present the\nblack hole spacetime structure is much more complex and additional off-diagonal metric components $g_{r\\phi}$,\n $g_{rt}$ are relevant along with the standard $g_{\\phi t}$ component, \ndue to the combination of the local bulk effects and the\nrotational dragging. This distorts the event horizon which\nbecomes a stack of non-uniformly rotating null circles having\ndifferent radii at fixed $\\theta$ while going from the equatorial\nplane to the poles \\cite{Ali-Gum:2005:}. The uniformly\nrotating horizon is recovered for the rotation\nparameter $a$ small enough  where Kerr-Newman form of the metric tensor is allowed describing charged and slowly rotating black holes \\cite{Ali-Gum:2005:}. In the absence of rotation, the metric tensor reduces\nto the Reissner-Nordstr\\\"{o}m form with correction term of non-local origin\n \\cite{Cha-etal:2001:}.\n\nHere we restrict our attention to the Kerr-Newman type of\nsolutions describing the braneworld rotating black holes with no\nelectric charge, since in astrophysically relevant situations the\nelectric charge of the black hole must be exactly zero, or very\nsmall \\cite{MTW}. Then the results obtained in analysing the\nbehaviour of test particles and photons or test fields around the\nKerr-Newman black holes could be used assuming both positive and\nnegative values of the braneworld tidal parameter $b$ (used instead of\ncharge parameter $Q^2$).\n\nThe information on the properties of strong gravitational fields in vicinity of compact objects, namely\nof black holes, is encoded into optical phenomena of different kind that enable us to make estimates of the black hole parameters, including its tidal charge, when predictions of the theoretical models are confronted with the observed data. From this point of view, the spectral profiles of accretion discs around the black holes in galactic binaries, e.g., in microquasars, are most promising \\cite{Nar-Mcl-Sha:2007:,McCli-Nar-Sha:2007:}, along with profiled spectral lines in the X-ray flux \\cite{Laor:1991:,Bao-Stu:1992:,Stu-Bao:1992:,Kar-Vok-Pol:1992:,Mat-Fab-Ros:1993:,Zak:2003:}. Important information could also be obtained from the quasiperiodic oscillations observed in the X-ray flux of some low-mass black hole binaries of Galactic origin \\cite{Rem-McCli:2006:ARASTRA:}, some expected intermediate black hole sources \\cite{Stroh:2007a:}, or those observed in Galactic nuclei \\cite{Asch:2004:ASTRA:,Asch:2007:}. In the case of our Galaxy centre black hole Sgr A$^*$, we could be able to measure the optical phenomena in more detailed form as compared with the other sources, since it is the nearest supermassive black hole with mass estimated to be $\\sim 4\\times 10^6 M_\\odot$ \\cite{Ghez:2005:}, enabling to measure the \"silhuette`` of the black hole and other subtle GR phenomena \\cite{Bardeen:1973:,Cun-Bar:1973:}.\n\\par\nIn the present paper, we give an introductory study of the tidal charge influence on the optical phenomena near a rotating black hole. We focus our attention to some characteristic phenomena in close vicinity of the black-hole horizon, where the effects of the tidal charge could be in principle of the same order as those of the black hole mass and spin, contrary to the case of weak lensing effects. The light escape cones are given for families of astrophysically interesting sources, namely in locally non-rotating frames, and frames related to circular geodetical motion and radially free-falling sources in section 4 \\cite{SSJ:RAGTime:2005:Proceedings}. The silhuette of the black hole is determined in section 5. Images of the accretion discs are determined in section 6 using the transfer-function method. In Section 7, time delay of hot spot radiation is determined for direct and indirect images assuming circular geodetical motion in close vicinity of the black hole horizon. In Section 8 relevance of some effects is estimated for the Galaxy centre Sgr $A^*$ supermassive black hole. Concluding remarks are presented in Section 9.\n\n\n\\section{\\label{sec:GravFielEqOnBrane}Gravitational field equations on the brane}\n\nIn the 5D warped space models of Randall and Sundrum, involving a non-compact extra dimension, the gravitational field equations in the bulk can be expressed in the form \\cite{Shi-Mae-Sas:2000:,Dad-etal:2000:}\n\n\\begin{equation}\n \\tilde{G}_{AB}=\\tilde{k}^2[-\\tilde\\Lambda g_{AB}+\\delta(\\chi)(-\\lambda g_{AB}+T_{AB})],\\label{beq1}\n\\end{equation}\nwhere the fundamental 5D Planck mass $\\tilde M_P$ enters via $\\tilde{k}^2=8\\pi\/\\tilde{M}_p^3 $, $\\lambda$ is the brane tension, and $\\tilde\\Lambda$ is the negative bulk cosmological constant. Denoting $\\chi=x^4$ as the fifth dimension coordinate , $\\chi=0$ determines location of the brane in the bulk space, at the point of $Z_2$ symmetry; $g_{AB}=\\tilde{g}_{AB}-n_A n_B$ is the induced metric on the brane, with $n_A$ being the unit vector normal to the brane.\n\\par\nThe effective gravitational field equations induced on the brane are determined by the bulk field equations (\\ref{beq1}), the Gauss - Codazzi equations and the generalised matching Israel conditions with $Z_2$-symmetry. They can be expressed as modified standard Einstein's equations containing additional terms reflecting bulk effects onto the brane \\cite{Shi-Mae-Sas:2000:}\n\n\\begin{equation}\n G_{\\mu\\nu}=-\\Lambda g_{\\mu\\nu}+k^2 T_{\\mu\\nu} + \\tilde{k}^2 S_{\\mu\\nu} -\\mathcal{E}_{\\mu\\nu},\\label{beq2}\n\\end{equation}\nwhere $k^2=8\\pi\/M_P^2$, with $M_P$ being the braneworld Planck mass. The relations of the energy scales and cosmological constants are given in the form\n\n\\begin{equation}\n M_P=\\sqrt{\\frac{3}{4\\pi}}\\left(\\frac{\\tilde{M}_P^2}{\\sqrt{\\lambda}}\\right)\\tilde{M}_P;\\quad \\Lambda=\\frac{4\\pi}{\\tilde{M}_P^3}\\left[\\tilde\\Lambda+\\left(\\frac{4\\pi}{3\\tilde{M}_P^3}\\right)\\lambda^2\\right].\\label{beq3}\n\\end{equation}\nLocal bulk effects on the matter are determined by the ``squared energy-momentum'' tensor $S_{\\mu\\nu}$, \nthat reads\n\\begin{equation}\n S_{\\mu\\nu}=\\frac{1}{12}T T_{\\mu\\nu}-\\frac{1}{4}T_\\mu^{\\phantom{\\mu}\\alpha}T_{\\nu\\alpha}+\\frac{1}{24}g_{\\mu\\nu}\\left(3T^{\\alpha\\beta}T_{\\alpha\\beta}-T^2\\right),\n\\end{equation}\nwhile the non-local bulk effects are given by the tensor $\\mathcal{E}_{\\mu\\nu}$ representing the bulk Weyl tensor $\\tilde{C}_{ABCD}$ projected onto the brane, whereas\n\n\\begin{equation}\n \\mathcal{E}_{AB}=\\tilde{C}_{ABCD}n^C n^D.\\label{beq4}\n\\end{equation}\n\nSymmetries of the Weyl tensor imply that $\\mathcal{E}_{[AB]}=\\mathcal{E}_A^{\\phantom{A}A}=0$ and $\\mathcal{E}_{AB}n^B=0$. Therefore, on the brane, $\\chi\\rightarrow 0$, there is $\\mathcal{E}_{AB}\\rightarrow \\mathcal{E}_{\\mu\\nu}\\delta_A^{\\phantom{A}\\mu}\\delta_B^{\\phantom{B}\\nu}$. The $\\mathcal{E}_{\\mu\\nu}$ tensor reflects influence of the non-local gravitational effects in the bulk, including the tidal (``Coulomb``) and transverse traceless (gravitational wave) imprints of the free gravitational field of the bulk.\n\\par\nWe restrict our attention to the vacuum (at both bulk and brane) solutions of the gravitational field equations on the brane. Assuming zero cosmological constant on the brane ($\\Lambda=0$) we arrive to the condition\n\n\\begin{equation}\n \\tilde\\Lambda=-\\frac{4\\pi\\lambda^2}{3\\tilde{M}_P^2}.\\label{beq5}\n\\end{equation}\nIn the absence of matter fields, there is $T_{\\mu\\nu}=0=S_{\\mu\\nu}$, i.e., we are not interested in the properties of the squared energy-momentum $S_{\\mu\\nu}$ representing local effects of the bulk. In the vacuum case, the effective gravitational field equations on the brane reduce to the form \\cite{Shi-Mae-Sas:2000:}\n\\begin{equation}\n R_{\\mu\\nu}=-\\mathcal{E}_{\\mu\\nu},\\quad R_\\mu^{\\phantom{\\mu}\\mu}=0=\\mathcal{E}_\\mu^{\\phantom{\\mu}\\mu}\\label{beq6}\n\\end{equation}\nimplying divergence constraint \\cite{Shi-Mae-Sas:2000:}\n\n\\begin{equation}\n \\nabla^\\mu\\mathcal{E}_{\\mu\\nu}=0\\label{beq7}\n\\end{equation}\nwhere $\\nabla_{\\mu}$ denotes the covariant derivative on the brane.\n\\par\nThe equation (\\ref{beq7}) represents Bianchi identities on the brane, i.e., an integrability condition for the field equations $R_{\\mu\\nu}=-\\mathcal{E}_{\\mu\\nu}$\\cite{Ali-Gum:2005:}. For stationary and axisymmetric (or static, spherically symmetric) solutions Eqs. (\\ref{beq6}) and (\\ref{beq7}) form a closed system of equations on the brane. \n\\par\nThe 4D general relativity energy-momentum tensor $T_{\\mu\\nu}$ (with $T_\\mu^{\\phantom{\\mu}\\mu}=0$) can be formally identified to the bulk Weyl term on the brane due to the correspondence \n\n\\begin{equation}\n k^2 T_{\\mu\\nu}\\quad\\leftrightarrow\\quad -\\mathcal{E}_{\\mu\\nu}.\\label{beq8}\n\\end{equation}\nThe general relativity conservation law $\\nabla^\\mu T_{\\mu\\nu}=0$ then corresponds to the constraints equation on the brane (\\ref{beq7}). This behaviour indicates that Einstein-Maxwell solutions in general relativity should correspond to braneworld vacuum solutions. This was indeed shown in the case of Schwarzchild  (R-N) \\cite{Maa:2004:,Dad-etal:2000:} and Kerr (K-N) spacetimes \\cite{Ali-Gum:2005:}. In both of these solutions the influence of the non-local gravitational effects of the bulk on the brane are represented by a single \"braneworld\" parameter $b$. The Coulomb-like behaviour in the Newtonian potential\n\n\\begin{equation}\n \\Phi=-\\frac{M}{M^2_{P}r}+\\frac{b}{2r^2}\\label{beq9}\n\\end{equation}\ninspired the name tidal charge \\cite{Dad-etal:2000:}.\n \\par\n\n\n\n\\section{\\label{sec:NullGeo}Null geodesics in Kerr spacetime with a tidal charge}\n\n\\subsection{Geometry}\n\nFollowing the work of \\cite{Ali-Gum:2005:}, and using the standard Boyer-Linquist coordinates ($t$, $r$, $\\theta$, $\\varphi$), we can write the line element of Kerr black-hole (or naked singularity) spacetime on the three-brane  in the form \n\\begin{eqnarray}\n\t\\mathrm{d} s^2 &=& -(1-\\frac{2Mr - b}{\\Sigma})\\mathrm{d} t^2 + \\frac{\\Sigma}{\\Delta}\\mathrm{d} r^2 + \\Sigma \\mathrm{d} \\theta^2  + \\frac{A}{\\Sigma}\\mathrm{d}\\varphi^2 \\nonumber\\\\\n\t&&- 2\\frac{2Mr-b}{\\Sigma}\\sin^2\\theta\\mathrm{d} t\\mathrm{d}\\phi,\\label{eq1}\n\\end{eqnarray}\nwhere \n\n\\begin{eqnarray}\n\t\\Sigma &=& r^2 + a^2\\cos^2\\theta\\label{eq2}\\\\\n\t\\Delta &=& r^2 - 2Mr + a^2 +b\\label{eq3}\\\\\n\tA &=& (r^2 + a^2)^2 - a^2\\Delta\\sin^2\\theta\\label{eq4}.\n\\end{eqnarray}\nM is the mass parameter, $a=J\/M$ is the specific angular momentum and the braneworld prarameter $b$ is the \\emph{tidal charge} representing imprint of non-local gravitational effects from the bulk space. The metric (\\ref{eq1}) has the \nsame form as the Kerr-Newman metric, where the tidal charge is replaced by the squared electric charge, $Q^2$. \nThe stress tensor on the brane $E_{\\mu\\nu}$ takes the form \\cite{Ali-Gum:2005:}\n\n\\begin{eqnarray}\n E_t^{\\phantom{t}t}&=&-E_\\varphi^{\\phantom{\\varphi}\\varphi}=-\\frac{b}{\\Sigma^3}[\\Sigma-2(r^2+a^2)],\\\\\n E_r^{\\phantom{r}r}&=&-E_\\theta^{\\phantom{\\theta}\\theta}=-\\frac{b}{\\Sigma^2},\\\\\n E_\\varphi^{\\phantom{\\varphi}t}&=&-(r^2+a^2)\\sin^2\\theta,\\\\ \nE_t^{\\phantom{t}\\varphi}&=&-\\frac{2ba}{\\Sigma^3}(r^2+a^2)\\sin^2\\theta\n\\end{eqnarray}\nthat is fully analogical ($b\\rightarrow Q^2$) to the components of energy-momentum tensor for Kerr-Newman spacetimes in Einstein's general relativity \\cite{Ali-Gum:2005:}.\n\nThe roots of $\\Delta = 0$ identify the type of braneworld Kerr spacetime. There are two possibilities, a black hole or a naked singularity. By introducing $a^2\/M^2\\rightarrow a^2$, $b\/M^2\\rightarrow b$, $r_+\/M\\rightarrow r_+$, or putting $M=1$, we write the roots of $\\Delta = 0$ in the form\n\n\\begin{equation}\n r_+ = 1+\\sqrt{1-a^2-b},\\quad\\textrm{(outer horizon)}\\label{horeq2}\n\\end{equation}\nand\n\\begin{equation}\n r_- = 1-\\sqrt{1-a^2-b},\\quad\\textrm{(inner horizon)}.\\label{horeq3}\n\\end{equation}\nThe metric given by the line element (\\ref{eq1}) determines the geometry of rotating black hole in braneworld universe if\n\n\\begin{equation}\n \t1\\ge a^2+b.\\label{horeq1}\n\\end{equation}\nThe strong inequality refers to the case of two horizonts $r_+$ and $r_-$. For extreme black holes ($1=a^2+b$) the horizons coincide $r_+ = r_- = 1$.\n\n\\begin{figure}[!th]\n\n\\includegraphics[width=10cm]{fig1b}\n\n\\caption{\\label{fig_1}The plot of the inner horizont radius $r_-$ as a function of tidal charge parameter $b$ for three representative values of rotational parameter $a^2=0.5$, $a^2=1.0$ and $a^2=1.5$.}\n\\end{figure}\nIt is clear that for $b\\ge 0$ the loci of the inner horizon  $r_-$ are always\npositive. But for $b<0$, the loci of the inner horizon can also be at\nnegative $r$, as illustrated in  Figure \\ref{fig_1}. \n\\par\nNotice that $a^2>1$ is not allowed for standard black holes and for $b>0$ \\cite{MTW}, but such a possibility appears for $b<0$. The rotational parameter of extreme black holes is given by $a^2=1-b$.\nThe case of $1<a^2+b$ refers to the braneworld Kerr naked singularities. \n\\par\nIn this paper we focus on astrophysicaly interesting case of black holes, with emitting sources and observers located above the outer horizont.\n\n\n\n\n\\subsection{Carter's equations}\n\nIn order to study the optical effects in braneworld Kerr spacetimes, we have to solve  equations of motion of  photons given by the null geodesics of the spacetime under consideration. \nThe geodesic equation reads\n\n\\begin{equation}\n  \\frac{\\mathrm{D} k^\\mu}{\\mathrm{d} w}=0,\\label{ce1}\n\\end{equation}\nwhere $k^\\mu=\\frac{\\mathrm{d} x^\\mu}{\\mathrm{d} w}$ is the wave vector tangent to the null geodesic and $w$ is the affine parameter. The normalization condition reads $g_{\\mu\\nu}k^\\mu k^\\nu = 0$. \nSince the components of the metric tensor do not depend on $\\varphi$ and $t$ coordinates, the conjugate momenta\n\n\\begin{eqnarray}\n \tk_\\varphi=g_{\\varphi\\nu}k^\\nu\\equiv \\Phi,\\label{ce4}\\\\\n \tk_t =g_{t\\nu}k^\\nu\\equiv -E,\\label{ce5}\n\\end{eqnarray}\nare the integrals of motion. Carter found another integral of motion $K$ as a separation constant when solving Hamilton-Jacobi equation \n\n\\begin{equation}\n \tg^{\\mu\\nu}\\frac{\\partial S}{\\partial x^\\mu}\\frac{\\partial S}{\\partial x^\\nu} = 0,\\label{ce6}\n\\end{equation}\nwhere he assumed the action $S$ in separated form\n\n\\begin{equation}\n\tS=-Et+\\Phi\\varphi + S_r(r)+S_{\\theta}(\\theta).\\label{ce6a\n\\end{equation}\n\nThe equations of motion can be integrated and written separatelly in the form\n\n\\begin{eqnarray}\n \t\\Sigma\\frac{\\mathrm{d} r}{\\mathrm{d} w}&=&\\pm\\sqrt{R(r)},\\label{ce7}\\\\\n \t\\Sigma\\frac{\\mathrm{d} \\theta}{\\mathrm{d} w}&=&\\pm\\sqrt{W(\\theta)},\\label{ce8}\\\\\n \t\\Sigma\\frac{\\mathrm{d} \\varphi}{\\mathrm{d} w}&=&-\\frac{P_W}{\\sin^2\\theta}+\\frac{a P_R}{\\Delta},\\label{ce9}\\\\\n \t\\Sigma\\frac{\\mathrm{d} t}{\\mathrm{d} w}&=&-a P_W + \\frac{(r^2+a^2)P_R}{\\Delta},\\label{ce10}\n\\end{eqnarray}\nwhere \n\n\\begin{eqnarray}\n  R(r)&=&P^2_R-\\Delta K,\\label{ce11}\\\\\n  W(\\theta)&=&K-\\left(\\frac{P_w}{\\sin\\theta}\\right)^2,\\label{ce12}\\\\\n  P_R(r)&=&E(r^2+a^2)-a\\Phi,\\label{ce13}\\\\\n  P_W(\\theta)&=&aE\\sin^2\\theta - \\Phi.\\label{ce14}\n\\end{eqnarray}\nIt is usefull to introduce integral of motion $Q$ by the formula\n\n\\begin{equation}\n Q=K-(E- a\\phi)^2.\\label{ce15}\n\\end{equation}\nIts relevance comes from the fact that in the case of astrophysically most important motion in the equatorial plane ($\\Theta = \\pi\/2$) there is $Q=0$.\n\n\n\n\\subsection{Radial and latitudinal motion}\nThe photon motion (with fixed constants of motion $E$, $\\Phi$, $Q$) is allowed in regions where $R(r;E,\\Phi,Q)\\ge 0$ and $W(\\theta; E,\\Phi,Q)\\ge 0$. The conditions $R(r;E,\\Phi,Q)=0$ and $W(\\theta;E,\\Phi,Q)=0$ determine turning points of the radial and latitudinal motion, respectively, giving boundaries of the region allowed for the motion.\nDetailed analysis of the $\\theta$-motion can be found in \\cite{Bic-Stu:1976:,Fel-Cal:1972:}, while the radial motion was analysed (with restrictions implied by the $\\theta$-motion) in \\cite{Stu:1981a:} and \\cite{Stu:1981b:}. Here we extend this analysis to the case of $b < 0$.\n\\par\n\nThe radial and latitudinal Carter equations read\n\n\\begin{eqnarray}\n\t\\Sigma^2\\left(\\frac{\\mathrm{d} r}{\\mathrm{d} w'} \\right)^2 &=& [r^2+a^2-a\\lambda]^2-\\Delta[\\mathcal{L}-2a\\lambda + a^2],\\label{eq7}\\\\\n\t\\Sigma^2\\left(\\frac{\\mathrm{d} \\theta}{\\mathrm{d} w'} \\right)^2 &=& \\mathcal{L} + a^2  \\cos^2\\theta - \\frac{\\lambda^2}{\\sin^2\\theta}\\label{eq8}\n\\end{eqnarray}\nwhere we have introduced impact parameters\n\n\\begin{eqnarray}\n\t\\lambda &=& \\frac{\\Phi}{E},\\label{eq9}\\\\\n\t\\mathcal{L} &=& \\frac{L}{E^2} = \\frac{Q+\\Phi^2}{E^2} = q + \\lambda^2,\\label{eq10}\n\\end{eqnarray}\nand rescaled the affine parameter by $w^\\prime = E w$. We assume $a>0$.\n\nThe reality conditions $(\\mathrm{d} r\/\\mathrm{d} w')^2 \\ge 0$ and $(\\mathrm{d}\\theta\/\\mathrm{d} w')^2 \\ge 0$ lead to the restrictions on the impact parameter $\\mathcal{L}$\n\n\\begin{equation}\n\t\\mathcal{L}_{min} \\leq \\mathcal{L} \\leq \\mathcal{L}_{max},\\label{eq12}\n\\end{equation}\nwhere\n\n\\begin{equation}\n\t\\mathcal{L}_{max} \\equiv \\frac{(a\\lambda -2r +b)^2}{\\Delta}+ r^2+2r-b,\\label{eq13}\n\\end{equation}\nand\n\\begin{equation}\n\t\\mathcal{L}_{min}\\equiv\\left\\{ \\begin{array}{lcr} \n\t\t\t\t\\lambda^2 & \\textrm{for} & |\\lambda|\\geq a,\\\\\n\t\t\t\t2a|\\lambda|-a^2 & \\textrm{for} & |\\lambda|\\leq a. \n\t\t\t   \\end{array}\\right.\\label{eq14}\n\\end{equation}\nThe upper(lower) constraint, $\\mathcal{L}_{max}$($\\mathcal{L}_{min}$), comes from the radial-motion (latitudinal-motion) reality condition. The properties of the photon motion are determined by the behaviour of the surface $\\mathcal{L}_{max}(r;\\lambda,a,b)$, as given by (\\ref{eq13}). The extrema of the surface $\\mathcal{L}_{max}$ (giving spherical photon orbits) are determined by\n\n\\begin{eqnarray}\n\t\\lambda=\\lambda_+ &\\equiv& \\frac{r^2+a^2}{a},\\label{eq15}\\\\\n\t\\lambda=\\lambda_- &\\equiv& \\frac{r^2-b r - a^2 - r\\Delta}{a(r-1)}.\\label{eq16}\n\\end{eqnarray}\nThe values of $\\mathcal{L}_{max}$ at these extreme points are given by\n\n\\begin{eqnarray}\n\t\\mathcal{L}_{max}(\\lambda_{+})\\equiv\\mathcal{L}_+ &=& 2r^2+a^2,\\label{eq17}\\\\\n\t\\mathcal{L}_{max}(\\lambda_{-})\\equiv\\mathcal{L}_- &=&\\frac{2r(r^3-3r+4b)+a^2(r+1)^2}{(r-1)^2}\\label{eq18}.\n\\end{eqnarray}\nThe character of the extrema follows from the sign of  $\\partial^2\\mathcal{L}_{max}\/\\partial r^2$. One finds that\n\n\n\n\n\\begin{eqnarray}\n\\frac{\\partial^2 \\mathcal{L}_{max}}{\\partial r^2} &=& \\frac{8r^2}{\\Delta},\\quad\\textrm{for}\\quad \\lambda = \\lambda_+,\\label{eq19}\\\\\n\\frac{\\partial^2 \\mathcal{L}_{max}}{\\partial r^2} &=&\\frac{8r^2}{\\Delta} - \\frac{8r}{(r-1)^2},\\quad\\textrm{for}\\quad \\lambda=\\lambda_-.\\label{eq20}\n\\end{eqnarray}\nClearly, there are only minima of $\\mathcal{L}_{max}$ along for $\\lambda=\\lambda_{+}$, corresponding to unstable\nspherical orbits.\n\\begin{figure}[ht]\n\n\\begin{tabular}{cc}\n  \\includegraphics[width=6cm]{fig2a}&\\includegraphics[width=6cm]{fig2b} \n\\end{tabular}\n\n\\caption{\\label{fig2_a_b}Left: classification of Kerr spacetime in braneworld universe according to the values of $a^2+b$, $b$ and $n_{ext}$ (the number of local extrema of the curves $\\tilde\\lambda_\\pm$, which is also the number of circular photon orbits in the equatorial plane). \nThe classification regions are: I) for $a^2+b\\leq 1$ and $n_{ext}=2$, II) for $a^2+b\\leq 1$ and $n_{ext}=4$, III) $a^2+b>1$ and $b<1$ and $n_{ext}=2$, IV) for $a^2+b>1$ and $b>1$ and $n_{ext}=2$, V) for $a^2+b>1$ and $n_{ext}=0$, VI) for $a^2+b>1$ and $b<1$ and $n_{ext}=4$, VII) for $a^2+b>1$ and $b>1$  and $n_{ext}=4$. \nRight: zoom of the area in the dashed rectangle of the left plot, to cover regions VI and VII.} \n\\end{figure}\n\n\nFurther, we have to determine where the restrictions given by the latitudinal motion $\\mathcal{L}_{min}$ meet the restrictions on the radial motion $\\mathcal{L}_{max}$. We find that $\\mathcal{L}_{max}=\\lambda^2$ (for $|\\lambda|\\ge a$) is fullfilled where\n\n\\begin{equation}\n \\lambda=\\tilde\\lambda_\\pm\\equiv\\frac{a(b-2r\\pm r^2\\sqrt{\\Delta})}{r^2-2r+b},\\label{eq23}\n\\end{equation}\nwhile $ \\quad\\mathcal{L}_{max}= 2a|\\lambda| - a^2$ (for $|\\lambda|<a$) is fullfilled where\n\n\\begin{equation}\n \\lambda=\\bar\\lambda \\equiv \\frac{1}{\\Delta}[4r-r^2-2b-a^2+2\\sqrt{\\Delta(b - 2r)}].\\label{eq24}\n\\end{equation}\n\nThe extreme points of curves $\\tilde\\lambda_\\pm$, which are also the intersection points of these curves  with $\\lambda_-$, are determined by the equation\n\n\\begin{equation}\n\tf(r;a,b)\\equiv r^4-6r^3+(9+4b)r^2-4(3b+a^2)r+4b(b+a^2)=0.\\label{eq25}\n\\end{equation}\nThe equation $f(r;a,b)=0$ determines loci of the photon equatorial circular orbits; in an implicit form the radii are given by the condition\n\n\\begin{equation}\n a^2=a^2_{ph\\pm}(r;b)=\\frac{r^2(r-3)^2+4b(r^2-3r+b)}{4(r-b)}.\n\\end{equation}\nThe maxima of the curve $\\bar\\lambda$, which also determine the intersections of curves $\\bar\\lambda$ and $\\lambda_-$ are located on $r$ satisfying the equation\n\n\\begin{equation}\n 2r^3-(3+b)r^2+2br+a^2 = 0.\\label{eq25a}\n\\end{equation}\n\nThe braneworld Kerr spacetimes can be classified due to the properties of the photon motion as determined by the behaviour of the functions $\\lambda_\\pm$, $\\tilde\\lambda_\\pm$, $\\bar\\lambda$. The classification is governed  by their divergences (i.e., by existence of the horizons) and the number of local extrema determining equatorial photon circular orbits $n_{ex}$. There exist seven classes of the braneworld Kerr spacetimes, with the criteria of separation being $a^2+b^{\\phantom{i}<}_{\\phantom{i}>} 1$, $b^{\\phantom{i}<}_{\\phantom{i}>} 1$ and $n_{ex}$ . The classification is represented in Figure \\ref{fig2_a_b}. There are two different classes of the black-hole spacetimes, differing by the presence of the photon circular orbits under the inner horizon. However, in the astrophysically relevant region outside the outer horizon, both the classes are of the same character, having two unstable  equatorial photon circular orbits, one corotating (at $r_{ph1}$) and the other counter-rotating (at $r_{ph2}>r_{ph1}$). The tidal charge $b$ introduces no qualitatively new feature into the behaviour of photon motion in the Kerr spacetimes, but the quantitative impact of $b<0$ with\nhigh magnitude are quite relevant, as shown in next sections. All the braneworld Kerr black holes with tidal charge $b<0$ belong to the class II discussed in the case of standard Kerr-Newman spacetimes \\cite{Stu:1981b:}. We illustrate in Figures \\ref{fig3}-\\ref{fig5} functions $\\lambda_\\pm$, $\\tilde\\lambda_\\pm$ and $\\bar\\lambda$ for such a black hole spacetime with parameters $a=0.9$ and $b=-1.0$. In this case typical for braneworld Kerr black hole with $b<0$ there exist ten significiant values of $\\lambda$ as given in Figures \\ref{fig3} - \\ref{fig5}.\n\n\\begin{figure}[!ht]\n\n  \\includegraphics[width=12.0cm]{fig3}\n\n\\caption{\\label{fig3}The graphs of the $\\lambda_\\pm$, $\\tilde\\lambda_\\pm$ and $\\bar\\lambda$ functions are plotted for representative values of the parameters  $a=0.9$ and $b=-1.0$. The two dashed rectangle areas labeled with numbers $1$ and $2$ are zoomed in the following figures. The horizontal gray dashed lines represent special values of the impact parameter $\\lambda$, denoted according to the text as  $\\lambda_A$...$\\lambda_J$.}\n\\end{figure}\nFor each interval of $\\lambda$ as determined by the sequence of $\\lambda_A$ - $\\lambda_J$ introduced in Figure \\ref{fig3}, there exists a characteristic type of behaviour of the restricting \"radial\" function $\\mathcal{L}_{max}$ and its relation to the \"latitudinal\" restricting function $\\mathcal{L}_{min}$. They can be found in \\cite{Stu:1981b:} and will not be repeated here.\n\n\\begin{figure}[ht]\n\n\\begin{tabular}{cc}\n  \\includegraphics[width=6cm]{fig4a}&\\includegraphics[width=6cm]{fig4b}\n\\end{tabular}\n\n\\caption{\\label{fig4}Left figure is the zoom of dashed area labelled $1$ in previous figure. Right figure is the zoom of dashed area labelled $2$ in previous figure. The dashed rectangle area here is zoomed in the next figure.}\n\\end{figure}\n\n\\begin{figure}[ht]\n\n  \\includegraphics[width=6.2cm]{fig5}\n\n\\caption{\\label{fig5}The zoom of the dashed rectangle area in previous figure.}\n\\end{figure}\n\nThe allowed values of the impact parameter $\\mathcal{L}$ lie between the limiting functions  $\\mathcal{L}_{min}$ and $\\mathcal{L}_{max}$. If the minimum $\\mathcal{L}_{max}^{min}\\equiv\\mathcal{L}_{max}(r_{min},\\lambda_0)$ of the limiting function $\\mathcal{L}_{max}$ is less than the value of the limiting function $\\mathcal{L}_{min}$, an incoming photon ($k^r < 0$) travelling from infinity will return back for all values of $\\mathcal{L}_0\\in[\\mathcal{L}_{min};\\mathcal{L}_{max}]$. If $\\mathcal{L}_{max}^{min}>\\mathcal{L}_{min}$, \nthe incoming photon ($k^r < 0$) travelling from infinity returns back if its impact parameter $\\mathcal{L}_0$  \nsatisfies the condition  $\\mathcal{L}_{0}\\ge\\mathcal{L}_{max}^{min}$ and is captured by the black hole \nif $\\mathcal{L}_0<\\mathcal{L}^{min}_{max}$.  \nThe minimum $\\mathcal{L}_{max}^{min}$ determines (with the particular value of $\\lambda$) a photon spherical orbit, \ni.e., a sphere where photons move with $r=const$ but with varying latitude $\\theta$ (and, of course, varying $\\varphi$). \nWhen the condition  $\\mathcal{L}_0 = \\mathcal{L}_{min}$ is satisfied simultaneously, the spherical photon orbit is transformed \nto an equatorial photon circular orbit. Photons with $\\mathcal{L}_0=\\mathcal{L}_{max}^{min}$ coming from distant regions or \nregions close to the black hole horizon will wind up around the photon sphere. \n \n\\clearpage\n\n\n\n\\section{\\label{sec:LEC}Light escape cones}\nThe optical phenomena related to accretion processes in  the field of rotating black holes could be efficiently studied by using the notion of light escape cones of local observers (sources) that determine which portion of radiation emitted by a source could escape to infinity and, complementary, which portion is trapped by the black hole \\cite{SSJ:RAGTime:2005:Proceedings}. Here we focus our attention to four families of observers (sources) that are of direct physical relevance.\n\n\n\\subsection{Local frames of stationary and free-falling observers}\nWe consider three families of stationary frames, namely  $LNRF$ (Locally Nonrotatig Frame), $SF$ (Static Frame) and $GF_\\pm$(Circular Geodesic Frame) and one non-stationary frame, namely $RFF$ (Radially Falling Frame). \nThe $LNRF$ are of highest physical importance since the physical phenomena take the simplest form when expressed in such frames, because the rotational spacetime effects are maximally suppressed there \\cite{Bardeen:1973:,MTW}. The $GF_\\pm$ are directly related to Keplerian accretion discs in the equatorial plane of the spacetime, both corotating and counterrotating, while $RFF$ are related to free-falling spherical accretion. The $SF$ are fixed relative to distant observers. The $GF_\\pm$ and $RFF$ are geodetical frames, while $SF$ and $LNRF$ are generally accelerated frames.\n\nThe radial and latitudinal 1-forms of the three stationary frame tetrads are common for all three stationary cases and read\n\n\\begin{eqnarray}\n\t\\omega^{(r)}&=&\\left\\{0,\\sqrt{\\Sigma\/\\Delta},0,0 \\right\\},\\label{LC9}\\\\\n\t\\omega^{(\\theta)}&=&\\left\\{0,0,\\sqrt{\\Sigma},0 \\right\\}.\\label{LC10}\n\\end{eqnarray}\n$LNRF$ correspond to observers with $\\Phi=0$ (zero angular momentum observers). Their time and azimuthal 1-forms read\n\n\\begin{eqnarray}\n\t\\omega^{(t)}&=&\\left\\{\\sqrt{\\frac{\\Delta\\Sigma}{A}},0,0,0 \\right\\},\\label{LC11}\\\\\n\t\\omega^{(\\varphi)}&=&\\left\\{-\\Omega_{LNRF}\\sqrt{\\frac{A}{\\Sigma}}\\sin\\theta,0,0,\\sqrt{\\frac{A}{\\Sigma}}\\sin\\theta\\right\\}.\\label{LC12}\n\\end{eqnarray}\nwhere \n\n\\begin{equation}\n\t\\Omega_{LNRF}=\\frac{a(2Mr-b)}{A}\\label{LC13}\n\\end{equation}\nis the angular velocity of $LNRF$ as seen by observers at infinity. \n\\par\nThe tetrad of $SF$ corresponding to observers with $\\Omega=0$ ,i.e. static relative to observers at infinity, is given by the formulae\n\n\\begin{eqnarray}\n\t\\omega^{(t)}&=&\\left\\{ \\sqrt{1-\\frac{2r-b}{\\Sigma}},0,0,\\frac{a(2r-b)\\sin^2\\theta}{\\sqrt{\\Sigma^2-(2r-b)\\Sigma)}}  \\right\\},\\\\\n\t\\omega^{(\\varphi)}&=&\\left\\{ 0,0,0,\\sqrt{\\frac{\\Delta\\Sigma}{\\Sigma-(2r-b)}}\\sin\\theta  \\right\\}.\n\\end{eqnarray}\n\nThe $GF_\\pm$ observers move along $\\varphi$-direction in the equatorial plane with velocity $V_{GF\\pm}$(+...corotating, -...counterrotating) relative to the $LNRF$ and with angular velocity $\\Omega$ relative to the static observers at infinity given by \\cite{}[SK]\n\\begin{equation}\n\\Omega_\\pm=\\pm\\frac{\\sqrt{r-b}}{r^2 \\pm a\\sqrt{r-b}}. \\label{ang_vel_gf}\n\\end{equation}\n\nThe velocity $V_{GF\\pm}$ is given by\n\n\\begin{equation}\n\tV_{GF\\pm}=\\pm\\frac{(r^2+a^2)Y\\mp a(2r-b)}{\\sqrt{\\Delta}(r^2\\pm aY)}.\\label{VGF}\n\\end{equation}\nwhere $Y=\\sqrt{r-b}$. The standard Lorentz transformation of the $LNRF$ tetrad gives the tetrad of $GF_\\pm$ in the form\n\\begin{eqnarray}\n\t\\omega^{(t)}_\\pm&=&\\left\\{ \\frac{r^2-2r+b\\pm a Y}{Z_\\pm},0,0,\\mp\\frac{(r^2+a^2)Y\\mp a(2r-b)}{Z_\\pm} \\right\\},\\\\\n\\omega^{(\\varphi)}_\\pm&=&\\left\\{\\mp \\frac{\\sqrt{\\Delta}Y}{Z_\\pm},0,0,\\frac{\\sqrt{\\Delta(r^2\\pm a Y)}}{Z_\\pm}, \\right\\}\n\\end{eqnarray}\nwhere \n\n\\begin{equation}\n\tZ_\\pm = r\\sqrt{r^2-3r+2b\\pm2aY}.\n\\end{equation}\nNote that the $GF_\\pm$ family is restricted to the equatorial plane, while $LNRF$ are defined at any $\\theta$.\n\nThe $RFF$ observers have velocity\n\n\\begin{equation}\n\tV_{RFF}=\\{V^{(r)},\\,V^{(\\theta)},\\,V^{(\\varphi)}\\}\n\\end{equation} \nas measured in $LNRF$. The radially free-falling (or free-escaping) observers starting (finishing) at infinity move with $\\theta = const$. Using the results of \\cite{Stu-Bic-Bal:1999:}, we find the velocity components of the free-falling frames in  the $LNRF$ frames \n\n\n\\begin{eqnarray}\n\tV^{(r)}&=&\\pm\\sqrt{1-\\frac{\\Sigma\\Delta}{A}},\\\\\n\tV^{(\\theta)}&=&0,\\\\\n\tV^{(\\varphi)}&=& 0.\n\\end{eqnarray}\nClearly, the free-falling (free-escaping) observers move only radially in the $LNRF$, in analogy to particles radially moving in static frames of the Schwarzchild spacetimes.\n For the radially free-falling sources, the tetrad components $\\omega^{(\\theta)}$ and $\\omega^{(\\varphi)}$ coincide with those of the LNRF tetrad, while $\\omega^{(t)}$ and $\\omega^{(r)}$ are transformed. The local Lorentz transformation of the $LNRF$ to the $RFF_\\pm$ tetrad yields\n\n\\begin{eqnarray}\n\\omega_\\pm^{(t)}&=&\\left\\{ \\gamma\\frac{\\Delta\\Sigma}{A}, \\mp\\sqrt{\\frac{\\Sigma}{\\Delta}}V,0,0 \\right\\},\\\\\n\\omega_\\pm^{(r)}&=&\\left\\{\\mp\\gamma\\sqrt{\\frac{\\Delta\\Sigma}{A}}V,\\sqrt{\\frac{\\Sigma}{\\Delta}}\\gamma,0,0\\right\\},\\\\\n\\omega_\\pm^{(\\theta)}&=&\\{0,0,\\sqrt{\\Sigma},0\\},\\\\\n\\omega_\\pm^{(\\varphi)}&=&\\left\\{-\\Omega_{LNRF}\\sqrt{\\frac{A}{\\Sigma}}\\sin\\theta,0,0,\\sqrt{\\frac{A}{\\Sigma}}\\sin\\theta \\right\\}.\n\\end{eqnarray}\n\n\n\\subsection{Construction of escape cones}\n\n\\begin{figure}[ht]\n\t\\includegraphics[width=10cm]{fig8}\n\\caption{\\label{fig8}Definition of directional angles $\\alpha_0$, $\\beta_0$ and $\\gamma_0$ in a local frame. Vectors $\\vec{e}_r$, $\\vec{e}_\\theta$, $\\vec{e}_\\varphi$ are the basic tetrad vectors. Position of the observer (source) is given by the coordinates $(r_0,\\theta_0)$. Vector $\\vec{k}$ represents a photon as observed by the observer in the given tetrad and vector $\\vec{k}^\\prime$ is its projection into the plane ( $\\vec{e}_\\theta$, $\\vec{e}_\\varphi$). }\n\\end{figure}\n\n For each direction of emission in the local frame of a source, there is a corresponding pair of values of the impact parameters $\\lambda$ and $\\mathcal{L}$ which can be related to the directional cosines of the photon trajectory in the local frame at the position of the source. Of course, the analysis of the turning points of the radial motion of photons, presented in the previous section, is crucial in determining the local escape cones as the boundary of the escape cone is given by directional angles related to spherical photon orbits.\n\nProjection of a photon 4-momentum $\\vec{k}$ onto the local tetrad of an observer is given by the formulae\n\n\\begin{eqnarray}\nk^{(t)}&=&-k_{(t)}=1,\\label{LC1}\\\\\nk^{(r)}&=&k_{(r)}=\\cos\\alpha_0,\\label{LC2}\\\\\nk^{(\\theta)}&=&k_{(\\theta)}=\\sin\\alpha_0\\cos\\beta_0,\\label{LC3}\\\\\nk^{(\\varphi)}&=&k_{(\\varphi)}=\\sin\\alpha_0\\sin\\beta_0,\\label{LC4}\n\\end{eqnarray} \nwhere $\\alpha_0$, $\\beta_0$ are directional angles of the photon in the local\nframe (see Figure \\ref{fig8}) and $\\cos\\gamma_0=\\sin\\alpha_0\\sin\\beta_0$. \nIn terms of the local tetrad components of the photon 4-momentum and the related directional angles, the conserved quantities, namely, the azimutal momentum $\\Phi$, energy $E$ and $K$ read\n\n\\begin{eqnarray}\n\t\\Phi&=&k_\\varphi=-\\omega^{(t)}_{\\phantom{(t)}\\varphi}k^{(t)} + \\omega^{(r)}_{\\phantom{(r)}\\varphi}k^{(r)}+\\omega^{(\\theta)}_{\\phantom{(\\theta)}\\varphi}k^{(\\theta)}+\\omega^{(\\varphi)}_{\\phantom{(\\varphi)}\\varphi}k^{(\\varphi)},\\label{LC6}\\\\\n\tE&=&-k_t=\\omega^{(t)}_{\\phantom{(t)}t}k^{(t)} - \\omega^{(r)}_{\\phantom{(r)}t}k^{(r)}-\\omega^{(\\theta)}_{\\phantom{(\\theta)}t}k^{(\\theta)}-\\omega^{(t)}_{\\phantom{(\\varphi)}\\varphi}k^{(\\varphi)},\\label{LC7}\\\\\n   K&=&\\frac{1}{\\Delta}\\left\\{ [E(r^2+a^2)-a\\Phi]^2-(\\Sigma k^r)^2\\right\\}.\\label{LC8}\n\\end{eqnarray}\nThe impact parameters $\\lambda$ and $\\mathcal{L}$ defined by relations (\\ref{eq9}) and (\\ref{eq10}) are thus fully determined by any double, $D$, of angles from the set $M=[\\alpha_0,\\beta_0,\\gamma_0]$.\n\n Having defined the source frame, we can construct light escape cones assuming fixed coordinates of the source $r_0$, $\\theta_0$. Their construction proceedes in the following steps:\n\n\\begin{itemize}\n\\item for given $D$, say $D=[\\alpha_0,\\beta_0]$, we calculate $\\lambda=\\lambda(\\alpha_0,\\beta_0)$,\n\\item $\\lambda$ determines the behaviour of $\\mathcal{L}_{max}=\\mathcal{L}_{max}(r;\\lambda)$,\n\\item from the analysis presented in the previous section we calculate minimum of $\\mathcal{L}_{max}$, which reads $\\mathcal{L}_{max}^{min}=\\mathcal{L}_{max}(r_{min};\\lambda)$,\n\\item we search for such a double $D$ which satisfies equation $\\mathcal{L}_0(\\alpha_0,\\beta_0)=\\mathcal{L}_{max}(r_{min};\\lambda)$.\n\\end{itemize}\nHere, we present in detail  the construction of light escape cones in particular case of the $LNRF$. The procedure is analogous for the other stationary frames and simply modified for the free-falling frames, being radius dependent.\n\n\\begin{figure}[ht]\n\\begin{tabular}{ll}\n\\includegraphics[width=6cm]{fig9a} & \\includegraphics[width=6cm]{fig9b}\n\\end{tabular}\n\\caption{\\label{fig9_a_b}Left. The functions $\\mathcal{L}_{max}$ and $\\mathcal{L}_{min}=\\lambda_0^2$ are plotted together with representative constant functions $\\mathcal{L}_1$ and $\\mathcal{L}_2$ to demonstrate the construction of the photon escape cone. Right. The intersections of $\\mathcal{L}_{max}(\\gamma_0)$ with $\\lambda^2(\\gamma_0)$ give the interval of relevant values of $\\gamma_0\\in[\\gamma_{min};\\gamma_{max}]$.}\n\\end{figure}\n\n\n\n\\par\nThe impact parameter $\\lambda$  expressed in terms of the angle $\\gamma_0$, related to the $LNRF$, reads\n\n\\begin{equation}\n\t\\lambda_0=\\frac{1}{\\Omega_{LNRF0}+\\frac{\\Sigma_0\\sqrt{\\Delta_0}}{A_0\\sin\\theta_0\\cos\\gamma_0}},\n\\end{equation}\nwhere index '$0$' refers to the frame with coordinates $[r_0,\\theta_0]$. The minimum of $\\mathcal{L}_{max}$ is located at\n\n\\begin{equation}\n\tr_{min}=\\left\\{ \\begin{array}{lcr}\n\t\t\t\\sqrt{a\\lambda - a^2} & \\textrm{for} & \\lambda\\geq\\lambda_G = a\\\\\n\t\t\t1-\\frac{k_1}{k_2}+\\frac{k_2}{3} & \\textrm{for} & \\lambda<\\lambda_G = a\n\t\t\t\\end{array}\\right.\\label{eq_rmin}\n\\end{equation} \nwhere \n\n\\begin{eqnarray}\n\tk_1&=&a^2+2b+a\\lambda-3,\\\\\n\tk_2&=&\\left\\{ 27(1-a^2-b)+2\\sqrt{3}\\sqrt{27(1-a^2-b)^2+k_1^3}\\right\\}^{1\/3}.\n\\end{eqnarray}\nThe relevant values of $\\mathcal{L}$ lie between $\\mathcal{L}_{max}$ and\n$\\mathcal{L}_{min}$ determined by Eqs (\\ref{eq13}) and (\\ref{eq14}). The\nintersections of functions $\\mathcal{L}_{max}=\\mathcal{L}_{max}(\\gamma_0)$ and\n$\\mathcal{L}_{min}(\\gamma_0)$ give the relevant interval of angles\n$\\gamma\\in[\\gamma_{min},\\gamma_{max}]$ (see Figure \\ref{fig9_a_b}). For each $\\gamma$ from $[\\gamma_{min},\\gamma_{max}]$ we calculate minimal value of the photon impact parameter $\\mathcal{L}$ for which the photon reaches the turning point $r_{min}$ and escapes to infinity. This minimal value is the minimum of $\\mathcal{L}_{max}$ which is located at $r_{min}$, eg. $\\mathcal{L}_{max}=\\mathcal{L}_{max}(r_{min};\\lambda_0(\\gamma_0),a,b)$, where $r_{min}$ is given by (\\ref{eq_rmin}).\nNow we can calculate the value of $\\alpha_0$ using equation\n\n\\begin{equation}\n\t\\cos\\alpha_0=\\frac{k^{(r)}}{k^{(t)}}=\\frac{\\omega^{(r)}_{LNRF\\mu}k^\\mu}{\\omega^{(t)}_{LNRF\\mu}k^\\mu}.\n\\end{equation} \nWe arrive to the formula\n\n\\begin{equation}\n\t\\cos\\alpha_0=\\pm\\sqrt{A_0}\\frac{\\sqrt{(r_0^2+a^2-a\\lambda_0)^2-\\Delta_0(\\mathcal{L}_{max}^{min}-2a\\lambda_0+a^2)}}{-a(a\\sin^2\\theta_0-\\lambda_0)\\Delta_0+(r_0^2+a^2)(r_0^2+a^2-a\\lambda_0)},\n\\end{equation}\nwhere $A_0=A(r_0,\\theta_0)$, $\\Delta_0=\\Delta(r_0)$  and $\\mathcal{L}_{max}^{min}=\\mathcal{L}_{max}(r_{min};\\lambda_0,a,b)$. The angle $\\beta_0$ can be  calculated from the formula (\\ref{LC4}).\nIn this way we obtain angles from the arc $\\beta_0\\in\\langle -\\pi\/2; \\pi\/2\\rangle$. The remaining arc $\\beta_0\\in\\langle \\pi\/2; 3\\pi\/2\\rangle$ can be obtained by turning the arc $\\beta_0\\in\\langle -\\pi\/2; \\pi\/2\\rangle$ around the symmetry axis determined by angles $\\beta_0=-\\pi\/2$ and $\\beta_0=\\pi\/2$. This procedure can be done because photons released under angles $\\beta_0$ and $\\pi-\\beta_0$ have the same constants of motion. \nClearly, for sources under the radius corresponding to the corotating\nequatorial photon circular orbit, only outward directed photons with no\nturning point of the $r$-motion can escape. With radius of the source\napproaching the event horizon ($r_0\\rightarrow r_+$), the escape cone shrinks\nto infinitesimal extension, except the case of extreme black hole \\cite{Bardeen:1973:}. For the other frames considered here, the procedure of\nthe related light escape cone construction can be directly repeated, but with\nthe relevant tetrad 1-form components being used in the procedure.\n\nIn order to reflect properly the effect of the tidal charge $b$ on the escape cone structure, we shall give the cones for black hole sequences of two kind: first we keep the spin $a$ fixed and change $b$, second we keep fixed \"distance\" to the extreme black hole states, i.e., $a^2+b$ is fixed, and both $a$ and $b$ are changed. The positive tidal charges have tendency to slightly increase the asymmetry of the cones as compared with $b=0$ case, keeping its character similar to the case of Kerr black holes (see next section). Therefore, we focus our attention to the influence of negative tidal charges.  \n\n\\begin{figure}[ht]\n  \\begin{tabular}{ccc}\n    \\includegraphics[width=4.0cm]{escape_cones_lnrf_a0n998_b0_re6_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a0n998_bm1_re6_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a9981_bm3_re6M_th0_ok}\\\\\n  \\includegraphics[width=4.0cm]{escape_cones_lnrf_b0_r020_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_bm1_r020_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a9981_bm3_re20M_th0_ok}\n  \\end{tabular}\n\\caption{Light escape cones as seen by $LNRF$ in the vicinity of the  braneworld kerr black hole. \nTop set of images is plotted for radial coordinate of emitter $r_e=6M$ and bottom set for $r_e=20M$.\nThe rotational parameter $a=0.9981$ is fixed and the representative values of the braneworld parameter $b$ are $0$ (left), $-1$ (middle) and $-3$ (right). The shaded area represents photons captured by black hole. }\\label{LNRF_fixed_a_on_b}\n\\end{figure}\n\n\n\n\\begin{figure}[ht]\n  \\begin{tabular}{ccc}\n    \\includegraphics[width=4.0cm]{escape_cones_lnrf_a21_b0_re6M_th0_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a22_bm1_re6M_th0_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a24_bm3_re6M_th0_ok}\\\\\n   \\includegraphics[width=4.0cm]{escape_cones_lnrf_a21_b0_re20M_th0_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a22_bm1_re20M_th0_ok}& \\includegraphics[width=4.0cm]{escape_cones_lnrf_a24_bm3_re20M_th0_ok}\n  \\end{tabular}\n\\caption{Light escape cones as seen by $LNRF$ in the vicinity of the extreme braneworld kerr black hole. Top set of images is plotted for radial coordinate of emitter $r_e=6M$ and bottom set for $r_e=20M$. The representative rotational  and braneworld parameters [$a^2$,$b$] are [$1.0$,$0.0$](left), [$2.0$,$-1.0$](middle) and [$4.0$,$-3.0$](right). The shaded area represents photons captured by black hole. }\\label{LNRF_extreme_on_b}\n\\end{figure}\n\n\nBehaviour of the $LNRF$ escape cones in dependence on the braneworld parameter\n$b$ (and the spin $a$) is represented in Figures \\ref{LNRF_fixed_a_on_b} and \\ref{LNRF_extreme_on_b}.\nThe complementary  trapped cones, corresponding to photons captured by the black hole, are shaded. \n\nAt a fixed radius expressed in units of $M$ the extension of the trapped cone grows with descending of $b$ to higher negative values and fixed spin $a$ and mass $M$, demonstrating  thus the growing gravitational pull of the black hole due to growing magnitude of the negative braneworld parameter. The same statement holds also in the case of extreme Kerr black holes, when $a$ grows and $b$ descends, while $M$ is fixed. Clearly, the positive braneworld parameters have tendency to increase the asymmetry of the cones, while the negative ones symmetrize the escape cones with growing of $|b|$.  On the other hand, the asymmetry of the escape cone grows with descending of $b$ for extreme black holes (Figure \\ref{LNRF_extreme_on_b}).\n\n\n\t\\begin{figure}[ht]\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a0n9981_b0_re1n24M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a0n9981_bm1_re3n91M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a0n9981_bm3_re6n27M_ok}\\\\\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a0n9981_b0_re10M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a0n9981_bm1_re10M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a0n9981_bm3_re10M_ok}\\\\\n\t\t\t\\includegraphics[width=4cm]{esape_cones_gf_a0n9981_b0_re10rh_ok}&\n\t\t\t\\includegraphics[width=4cm]{esape_cones_gf_a0n9981_bm1_re10rh_ok}&\n\t\t\t\\includegraphics[width=4cm]{esape_cones_gf_a0n9981_bm3_re10rh_ok}\n\t\t\\end{tabular}\n\t\t\\caption{Escape cones of GF+ observers. Top images are plotted for observer (emitter) at $r=r_{ms}$, middle images $r=10M$ and bottom images for \n\t\t$r=10\\cdot r_h$. The value of $a=0.9981$ is kept fixed. The representative values of $b$ are (from left to right) $0.0$, $-1.0$ and $-3$. }\\label{GF_escape_cones}\n\t\\end{figure}\t\n\n\t\\begin{table}[ht]\n\t\t\\tbl{Table of relevant values of $r_{ms}$ and $r_{h}$ used in plots on Figs \\ref{GF_escape_cones} and \\ref{GF_escape_cones_extreme}.}\n\t\t{\\begin{tabular}{@{}cccc@{}} \n\t\t\\toprule\n\t\t$(a^2, b)$ & (0.9981,0.0) & (0.9981,-1.0) & (0.9981,-3.0)\\\\ \n\t\t\\colrule\n\t\t$r_{ms}$ & 1.24M & 3.91M & 6.27M\\\\\n\t\t\\colrule\n\t\t$r_h$ & 1.062M & 2.002M  & 2.73M\\\\\n\t\t\\botrule\n\t\t\\end{tabular}\\label{tabulka1}}\n\t\\end{table}\n\n\\begin{figure}[ht]\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a20n9999_b0_re1n062M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a21n9999_bm1_re1n06M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a23n9999_bm3_re1n06M_ok}\\\\\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a20n9999_b0_re10M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a21n9999_bm1_re10M_ok}&\n\t\t\t\\includegraphics[width=4cm]{escape_cones_gf_a23n9999_bm3_re10M_ok}\n\t\t\n\t\t\n\t\t\n\t\t\\end{tabular}\n\t\t\\caption{Escape cones of GF+ observers. Top images are plotted for observer (emitter) at $r=r_{ms}$ and bottom images for $r=10M$. The value of $a^2+b=0.9999$ is kept fixed. The representative values of $(a^2; b)$ are (from left to right) $(0.9999;0.0)$, $(1.9999;-1.0)$ and $(3.9999;-3)$.  }\\label{GF_escape_cones_extreme}\n\t\\end{figure}\t\n \n Further, we represent the influence of the braneworld parameter on the escape cones for the circular (corotating) geodesic frames in Figure \\ref{GF_escape_cones}. Assuming astrophysically relevant sources in Keplerian accretion discs, their orbits must be located above the marginally stable orbit $r_{ms}$, determined implicitly by the condition \\cite{Ali-Gum:2005:,Stu-Kot:2008}\n\n\n\\begin{equation}\n \ta=a_{ms}(r;b)\\equiv\\frac{4(r-b)^{3\/2}\\mp r \\sqrt{3r^2-2r(1+2b)+3b}}{3r-4b}.\n\\end{equation}\nTherefore, we construct the escape cones for observers at $r=r_{ms}(a,b)$ and at fixed radii. In the sequence of black holes with fixed spin  $a=0.9981$ (Figure \\ref{GF_escape_cones}) we include also  a subsequence of escape cones constructed at the same relative distance from the black hole horizon in order to better illustrate the role of the tidal charge $b$. In the sequence of near-extreme black holes with $a^2+b=0.9999$ (Figure \\ref{GF_escape_cones_extreme}) the third sequence is not necessary as the black hole horizon is fixed at $r_h=1.01M$. Figures \\ref{GF_escape_cones} and \\ref{GF_escape_cones_extreme} demonstrate that the trapped cone expands as the tidal charge descends to lower negative values, both for black holes with fixed spin $a$ and for near-extreme holes. On the other hand, considering the cones at $r_{ms}$ we can conclude that the descending tidal charge ($b<0$) symmetrizes their shape for fixed $a$, but makes them strongly asymmetric for near-extreme black holes shrinking them strongly in the direction of the black hole rotation.\n\n\nFinally we demonstrate the relevance of the tidal charge $b$ in the character of escape cones of the $RFF_-$ (comparing them with those related to $LNRF$) in Figure \\ref{fig13a_f}. We construct the escape cones for two typical values of the tidal charge ($b=0$, $b=-3$) in a sequence of radii where the free-falling source is radiating, demonstrating thus the combined growing influence of the black hole gravitational pull on the photon motion and the velocity of the free-falling source. In order to illustrate the phenomena in a clear way, we compare the $RFF_-$ escape cones to the corresponding $LNRF$ escape cones. Clearly, the tidal charge descending to higher negative values makes stronger squeezing of the free-falling  cones relative to the $LNRF$ escape cones at any fixed radius. Notice that both the $RFF_-$ and $LNRF$ cones are shifted due to the black hole rotational dragging. \nWe again observe the tendency of negative brany parameters to symmetrize and squeeze the escape cones. At a fixed $r$, the escape cones  become smaller for growing $|b|$ due to stronger gravity. For completeness we present sequence of both the $RFF_-$ and $LNRF$ escape cones at the three fixed radii for an extreme black hole with $b=-3$ and $a^2=4$. We observe that both the $RFF_-$ and $LNRF$ cones are strongly shifted in the sense of the black hole rotation in vicinity of black hole horizon due to growing influence of the spin. The symmetrizing effect of descending values of negative tidal charge is canceled by strong influence of the rotational effects due to growing black hole spin.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n \\begin{tabular}{ccc}\n  \\includegraphics[width=4cm]{escape_cones_rf_lnrf_a0n9981_b0_re5}&\\includegraphics[width=4cm]{escape_cones_rf_lnrf_a0n9981_bm3_re5}&\\includegraphics[width=4cm]{escape_cones_rf_lnrf_a2_bm3_re5}\\\\\n  \\includegraphics[width=4cm]{escape_cones_rf_lnrf_a0n9981_b0_re10}&\\includegraphics[width=4cm]{escape_cones_rf_lnrf_a0n9981_bm3_re10}&\\includegraphics[width=4cm]{escape_cones_rf_lnrf_a2_bm3_re10}\\\\\n  \\includegraphics[width=4cm]{escape_cones_rf_lnrf_a0n9981_b0_re15}&\\includegraphics[width=4cm]{escape_cones_rf_lnrf_a0n9981_bm3_re15}&\\includegraphics[width=4cm]{escape_cones_rf_lnrf_a2_bm3_re15}\n \\end{tabular}\n\\caption{\\label{fig13a_f}Comparison of the effect of the tidal charge $b$ on the shape of\n  light escape cones of locally nonrotating (dashed curves) frames  and free falling (solid curves)\n  frames. In the left column light escape cones are plotted for the tidal\n  charge parameter $b=0$ and in the middle one the light escape cones are\n  plotted for $b=-3$. The spin $a=0.9981$ is kept fixed in both columns. The right column gives the sequence of the escape cones for an extreme black hole with [$a^2=4;b=-3$]. Emitting sources in all plots are moving in the equatorial plane. The radial distances of emitter are $r_e=5M$ (top row), $r_e=10M$ (middle row) and $r_e=15M$ (bottom row).}\n\\end{figure}\n\n\n\n\\clearpage\n\n\n\\section{\\label{sec:Silhuette}Silhuette of braneworld Kerr black hole}\n\nIn principle, it is of astrophysical importance to consider a black hole in front of a source of illumination whose angular size is large compared with the angular size of the black hole \\cite{Bardeen:1973:}. A distant observer will see a silhuette of the black hole, i.e., a black hole in the larger bright source. The rim of the black hole silhuette corresponds to photon trajectories spiralling around the black hole many times before they reach the observer. Of course, the shape of the silhuette enables, in principle, determination of the black hole parameters. But we have to be aware of the strong dependency of the silhuette shape on the observer viewing angle; clearly, the shape will be circular for observers on the black hole rotation axis, and its deformation grows with observer approaching the equatorial plane.\n\nAssuming that distant observers measure photon directions relative to the symmetry center of the gravitational field, the component of the angular displacement perpendicular to the symmetry axis is given by $-p^{(\\varphi)}\/p^{(t)}$ (for black hole rotating anticlockwise relative to distant observers), while for angular displacement parallel to the axis it is given by $p^{(\\theta)}\/p^{(t)}$. These angles are proportional to $1\/r_0$, therefore, it is convenient to use the impact parameters in the form independent of $r_0$ \\cite{Bardeen:1973:}\n\n\\begin{equation}\n \\tilde{\\alpha}=-r_0\\frac{p^{(\\varphi)}}{p^{(t)}}=-\\frac{\\lambda}{\\sin\\theta_0},\\label{silalpha}\n\\end{equation}\n\nand\n\n\\begin{eqnarray}\n \\tilde{\\beta}&=&r_0\\frac{p^{(\\theta)}}{p^{(t)}}=\\left[q+a^2\\cos^2\\theta_0-\\lambda^2\\cot^2\\theta_0\\right]^{1\/2}\\nonumber\\\\\n&&=\\left[\\mathcal{L}+a^2\\cos^2\\theta-\\frac{\\lambda^2}{\\sin^2\\theta_0}\\right]^{1\/2}.\\label{silbeta}\n\\end{eqnarray}\nPhoton trajectories reaching the observer are represented by points in the $(\\tilde{\\alpha}-\\tilde{\\beta})$ plane representing a small portion of the celestial sphere of the observer.\n\nThe shape of the black hole silhuette is the boundary of the no-turning-point region, i.e., it is the curve $\\mathcal{L}=\\mathcal{L}^{min}_{max}(\\lambda)$ expressed in the $(\\tilde{\\alpha}-\\tilde{\\beta})$ plane of the impact parameters. For observers in the equatorial plane $(\\theta_0 = \\pi\/2)$, $\\tilde{\\alpha}=-\\lambda$, $\\tilde{\\beta}=(\\mathcal{L}-\\lambda^2)^{1\/2}=q^{1\/2}$.\n\n\n\\begin{figure}[ht]\n\n\\begin{tabular}{cc}\n  \\includegraphics[width=5.5cm]{silhuettes_a0n6_b}&\\includegraphics[width=5.5cm]{silhuettes_extreme_b}\n\\end{tabular}\n\n\\caption{\\label{fig15}Left figure. The $(\\bar\\alpha_0,\\bar\\beta_0)$ plots of the silhuettes of braneworld Kerr black hole on a bright background for rotational parameter $a^2=0.6$ and four representative values of tidal charge parameter $b=-3.0$, $b=-0.4$, $b=0.0$ and $b=0.4$. The observer is located at $r_0=10^4 M$ and $\\theta_0=90^\\circ$.\nRight figure. The silhuettes of extreme black holes for three representative values of braneworld parameter $b=0$ (solid), $b=-1$ (dashed) and $b=-3$ (dotted). Static observer is in equatorial plane at radial distance from the centre $r_0=10^4 M$.}\n\\end{figure}\n\n\n\n\n\n\n\nWe consider that the black hole is observed by static distant observers. Therefore, we shall use the static frames introduced above. The silhuette of the black hole is quite naturally related to their trapped (escape) light cones.\n\nThe marginal values of impact parameters $\\lambda_0$ and $\\mathcal{L}_0$(resp $q_0$) are obtained from the  light escape cone. Using the stationarity of the braneworld Kerr spacetime we ``shoot out`` virtual photons from observer (static frame at very large distance $r_0$) and we are looking for the light escape cone of this virtual source (using the results of the previous section). The trapped light cone of this virtual source is constructed from the light escape cone of the virtual source by transformations of directional angle $\\alpha_0$ to $\\bar{\\alpha}_0=\\pi - \\alpha_0$ and directional angle $\\beta_0$ to $\\bar{\\beta}_0=\\beta_0$. In this way we get marginal directions for received photons from bright background behind the black hole.  Then we can use the formulas (\\ref{LC6}), (\\ref{LC7}) and (\\ref{LC8}) to calculate the marginal values of $\\lambda_0$ and $q_0$($\\mathcal{L}_0$) in order to obtain the silhuette of the braneworld Kerr black hole in the plane $(\\tilde{\\alpha}-\\tilde{\\beta})$, i.e., the set of doubles $(\\tilde{\\alpha}_0,\\tilde{\\beta}_0)$ from equations (\\ref{silalpha}) and (\\ref{silbeta}). Here we plotted the silhuette directly from the trapped light cone $(\\bar{\\alpha}_0,\\bar{\\beta}_0)$ on the observer's  sky $(\\bar{\\alpha}_0\\sin\\bar{\\beta}_0,\\bar{\\alpha}_0\\cos\\bar{\\beta}_0)$. Note that the angle $\\bar\\alpha_0$ is the radial coordinate and the angle $\\bar\\beta_0$ is the polar coordinate in the polar graph of the silhuette. \n\n\n\\begin{figure}[ht]\n\\begin{tabular}{cc}\n\t\\includegraphics[width=6cm]{silhuettes_a0n8_b02_th}&\\includegraphics[width=6cm]{silhuettes_a0n8_b0_th}\\\\\n\\includegraphics[width=6cm]{silhuettes_a0n8_bm1_th}&\\includegraphics[width=6cm]{silhuettes_a0n8_bm3_th}\n\\end{tabular}\n\\caption{\\label{fig18_a_d}The silhuettes of rotating braneworld black hole on a bright background.  Each image contains three black hole shapes for three representative values of observer's inclination angle $\\theta_0=\\{0^\\circ(solid),45^\\circ(dashed),90^\\circ(dotted)\\}$, observer's radial coordinate $r_0=10^4 M$ and the rotational parameter $a^2=0.8$. Top left image: $b=0.2$. Top right image: $b=0.0$. Bottom left image: $b=-1.0$. Bottom right image: $b=-3.0$.}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{tabular}{cc}\n\t\\includegraphics[width=6cm]{silhuettes_a1_b0_th_extreme}&\\includegraphics[width=6cm]{silhuettes_a2_bm3_th_extreme}\n\\end{tabular}\n\\caption{The silhuettes of extreme rotating braneworld black holes on a bright background.  Each image contains three black hole shapes for three representative values of observer's inclination angle $\\theta_0=\\{0^\\circ(solid),45^\\circ(dashed),90^\\circ(dotted)\\}$, observer's radial coordinate $r_0=10^4 M$. Silhuettes on the left figure are plotted for extreme black holes with $a^2=1$ and $b=0$ and on the right side for $a^2=4$ and $b=-3$.}\\label{fig19_a_b}\n\\end{figure}\n\n\n\\par \n\tWe shall give the silhuette of the black hole for observers located at fixed radius $r_0=10^4$M that corresponds to the angular size of $\\alpha\\sim 1.4$arcsec; for higher distances the angular size falls accordingly to the $1\/r_0$ dependence. \n\nFirst, we give an illustrative picture of the tidal charge influence on the silhuette properties for maximal inclination angle $\\theta_0=90^\\circ$ when the black hole rotational effects are strongest (Figure \\ref{fig15}). We present a sequence of silhuettes for fixed black hole spin and varying $b$ (left) and for extreme black holes with $a^2+b=1$ and both $a$, $b$ varying (right). We clearly see that the positive tidal charge squeezes  magnitude of the silhuette making its shape more asymmetric, while negative tidal charge enlarges silhuette's diameter symmetrizing its shape when $a$ is fixed. For extreme black holes the silhuette asymmetry is kept but its extension grows with $b$ descending to higher negative values.\n\n\tSecond, there is a crucial effect of the viewing angle $\\theta_0$ onto the shape of the black hole silhuette, demonstrated in Figure \\ref{fig18_a_d} for representative values of $b$ and fixed spin $a$, and in Figure \\ref{fig19_a_b} for extreme black holes with parameters [$a^2=1$;$b=0$] and [$a^2=4$;$b=-3$].\n\n\n\nThe rotational effect on the shape of the silhuette grows with inclination angle growing and becomes strongest when $\\theta_0=\\pi\/2$; then the suppressing effect of the braneworld parameter is given in the most explicit form as demonstrated in Figure \\ref{fig15}. \n\n\nThe negative values of the braneworld parameter have the tendency to make the silhuette of a Kerr black hole (with $a^2$ fixed and for $r_0$, $\\theta_0$ fixed) spherical, suppressing thus the rotational effects. However the symmetrizing effect of the tidal charge could be masked by symmetrizing effect of the viewing angle $\\theta_0$. Therefore, it is very important for black hole parameter estimates to have observational limits on the value of $\\theta_0$.\n\n\\begin{figure}[ht]\n\t\\includegraphics[width=8cm]{fig14} \n\\caption{\\label{fig14}We define shift $s$ and ellipticity $\\epsilon=x\/y$ as parameters enabling us to characterize the magnitude of distorsion of Kerr black hole silhuette in braneworld universe.}\n\\end{figure}\n\n\n\nIn order to characterize the influence of the tidal charge on the silhuette of a Kerr black hole we define two quantities in principle measurable by distant observers. The \\emph{shift} $s$ of the silhuette   \n\\begin{equation}\n \ts=\\tilde\\alpha(\\beta_m)\\sin(\\beta_m - \\pi),\\label{eqA}\n\\end{equation}\nand its \\emph{ellipticity} $\\epsilon$\n\\begin{equation}\n \t\\epsilon=\\frac{\\tilde\\alpha(\\beta=90^\\circ)+\\tilde\\alpha(\\beta=270^\\circ)}{2\\tilde\\alpha(\\beta_m)\\cos(\\beta_m - \\pi)},\\label{eqB}\n\\end{equation}\nwhere $\\beta_m$ is defined by $\\tilde\\alpha(\\beta_m)\\sin(\\beta_m - \\pi)\\ge \\tilde\\alpha(\\beta)\\sin(\\beta - \\pi),\\quad \\forall \\beta\\in[\\pi\/2,3\/2\\pi]$ i.e., it defines maximal extension of the silhuette in the $x$-direction. The definition of \\emph{shift} $s$ and \\emph{elipticity} $\\epsilon$ is illustrated in Figure \\ref{fig14}.\n\nWe calculated shift $s$ and ellipticity $\\epsilon$ as functions of tidal\nparameter $b$ for the Kerr black hole with rotational parameter $a^2=0.9995$\n(see Figure \\ref{fig16_a_b}).\nClearly, these are quantities that could be measured and used for a black hole parameters estimates, if observational techniques could be developed to the level enabling the silhuette detailed measuring. We shall discuss such a possibility for the case of the supermassive black hole predicted in the Galaxy Centre (Sgr $A^*$).\n \n\n\n\n\\begin{figure}[ht]\n\\begin{tabular}{cc}\n\t\\includegraphics[width=6cm]{shift_th89n9_th45_a0n9995_r10p4} &\\includegraphics[width=6cm]{ellipticity_th89n9_th45_a0n9995_r10p4} \\\\\n\t\\includegraphics[width=6cm]{ellipticity_extreme_th090_r0104}&\\includegraphics[width=6cm]{shift_extreme_th090_r0104}\t\n\\end{tabular}\n\\caption{\\label{fig16_a_b\n Top row. Left figure: the shift $s=s(b)$ as a function of braneworld parameter $b$. Right figure: the ellipticity $\\epsilon=\\epsilon(b)$ as a function of $b$. There are two curves on each image, one for observer inclination angle $\\theta_0=45^\\circ$ and second for $\\theta_0=89.9^\\circ$. The rotational parameter of black hole is fixed to value $a=0.9995$ and the radial coordinate of observer if $r_0=10^4 M$.\nBottom row. The ellipticity $\\epsilon$ (left) and shift $s$ (right) of the extreme black hole silhuette as functions of braneworld parameter $b$. Observer's coordinates are $\\theta_0=\\pi\/2$ and $r_0=10^4 M$. }\n\\end{figure}\n\n\n\n\n\\clearpage\n\\section{\\label{sec:DirAndIndirImages}Direct and indirect images of radiating disc}\n\nModelling of spectral line profiles of a thin radiating ring rotating in the equatorial plane of a braneworld Kerr black hole or light curve of an isotropically emitting point source orbiting such a black hole will give us information about the influence of the braneworld parameter $b$ on the optical phenomena in the strong field regime \\cite{SS:b:RAGTime:2007:Proceedings}. Here we restrict our attention to images of radiating discs. We can then, at least in principle, obtain estimates on the astrophysically acceptable values of the  braneworld parameter $b$. \n\n\\subsection{Images of isoradial geodesics}\n\t\nCalculating images of an accretion disc (ring) in the equatorial plane of a braneworld Kerr black hole is the first step to calculate the optical phenomena. Generally one could obtain a direct and an indirect image (see Figures \\ref{fig19} and \\ref{fig20}), but in special cases the situation can be much more complicated due to complex character of the latitudinal and azimuthal photon motion. Here we focus our attention to the direct and indirect images of isoradial geodesics.\n\nIn order to find all relevant positions of points forming the rotating ring on observer's sky, we have to find photon trajectories between the ring particles and the observer, i.e., we seek for such  doubles of local observational angles $[\\alpha_0,\\beta_0]$ that satisfy the condition\n\n\n\n\\begin{figure}[ht]\n\n  \\includegraphics[width=10cm]{fig19}\n\n\\caption{\\label{fig19}\\emph{Direct} image of the rotating ring in the equatorial plane at $r_e=6M$ around braneworld Kerr black hole with rotational parameter $a^2=0.5$ for four representative values of tidal charge parameter $b=-3.0$, $b=-0.4$, $b=0.0$ and $b=0.4$. The observer is located at $r_0=10^4 M$ and $\\theta_0=85^\\circ$. }\n\\end{figure}\n\n\\begin{figure}[ht]\n\n  \\includegraphics[width=8cm]{fig20}\n\n\\caption{\\label{fig20}\\emph{Indirect} image of the rotating ring in the equatorial plane at $r_e=6M$ around braneworld Kerr black hole with rotational parameter $a^2=0.5$ for four representative values of tidal charge parameter $b=-3.0$, $b=-0.4$, $b=0.0$ and $b=0.4$. The observer is located at $r_0=10^4 M$ and $\\theta_0=85^\\circ$. }\\label{indirect_a_05_th85_on_b}\n\\end{figure}\n\n\n\\begin{equation}\n \tI_U(\\alpha_0,\\beta_0;n_u,u_{sgn}) - I_M(\\alpha_0,\\beta_0;n,p,s)=0.\\label{bvp}\n\\end{equation}\nHere we introduced the modified radial coordinate $u=1\/r$ and cosine of latitudinal coordinate $\\mu=\\cos\\theta$ \\cite{Rau-Bla:1994:}. In the condition (\\ref{bvp}) $n_u$ is the number of turning points in $u$ coordinate, $n$ is the number of turning points passed in $\\mu$ coordinate, $p=mod(n,2)$, $s=(1-\\mu_{sgn})\/2$. In terms of $u$ and $\\mu$ we define the functions $I_U$ and $I_M$ by\n\n\\begin{equation}\n\tI_U(\\alpha_0,\\beta_0;n_u,u_{sgn})\\equiv\\left\\{\\begin{array}{lcr}\n\t\t\t\t\t-u_{sgn}\\left(\\int^{u_0}_{u_t} +\\int^{u_e}_{u_t}\\right) & \\textrm{for} & n_u=1\\\\\n\t\t\t\t\tu_{sgn}\\int^{u_e}_{u_0} & \\textrm{for} & n_u=0\n\t\t\t\t\t\\end{array}\\right.\n\\end{equation}\nand\n\\begin{eqnarray}\n\tI_M(\\alpha_0,\\beta_0;n,p,s)&\\equiv&\\mu_{sgn}\\left[\\int^{\\mu_+}_{\\mu_0} + (-1)^{n+1}\\int^{\\mu_+}_{\\mu_e}+\\right.\\\\ \\nonumber\n&+&\\left.(-1)^s[(1-p)n+p[(1-s)(n-1)+s(n+1)]]\\int^{\\mu_+}_{\\mu_-} \\right]\n\\end{eqnarray}\n with\n\\begin{eqnarray}\n\t\\int^{u_2}_{u_1}&\\equiv&\\int^{u_2}_{u_1}\\frac{\\mathrm{d} u}{\\sqrt{U(u)}},\\label{u_int}\\\\\n\tU(u)&=&1+(a^2-\\lambda^2-q)u^2+2[(\\lambda^2-a^2)^2+q]u^3 - \\nonumber\\\\\n\t&-&[q(a^2+b)+b(a-\\lambda)^2]u^4\n\\end{eqnarray}\nand\n\n\\begin{eqnarray}\n\t\\int^{\\mu_2}_{\\mu_1}&\\equiv&\\int^{\\mu_2}_{\\mu_1}\\frac{\\mathrm{d} \\mu}{\\sqrt{M(\\mu)}},\\label{mu_int}\\\\\n\tM(\\mu)&=&q+(a^2-\\lambda^2-q)\\mu^2-a^2\\mu^4.\n\\end{eqnarray}\n\n\\subsection{Integration of photon trajectories}\n\nWe express the integrals (\\ref{u_int}) and (\\ref{mu_int}) in the form of the standard elliptic integrals of the first kind. Rauch and Blandford presented the tables of reductions of $u$-integrals and $\\mu$-integrals for the case of photons in Kerr geometry \\cite{Rau-Bla:1994:}. Here we extended those reductions for the case of nonzero braneworld parameter $b$. Because the integration of the $\\mu$-integral does not depend on braneworld parameter $b$, the transformations are the same as in the case of Kerr metric \\cite{Rau-Bla:1994:}, but we include them for completeness. \n\n\nThere are two cases we distinguish in latitudinal integral (see table \\ref{tableEIM}). In the first case there is one positive, $M_+>0$, and one negative, $M_-<0$ root of $M(m^2)$ it implies that there are two turning points located symmetrically about the equatorial plane given by $\\pm\\sqrt{M_+}$ (so called orbital motion \\cite{Bic-Stu:1976:,Fel-Cal:1972:}. In the second case there are two positive roots, $0<M_-<M_+$ of  $M(m^2)$, which implies that the latitudinal motion is constrained to the region above or below of the equatorial plane (so called vortical motion). The relevant reductions of the integral \n$\\int^m_{m_1}\\mathrm{d} m'\/\\sqrt{M(m')}=I_M$ are stored in the table \\ref{tableEIM}.\n\n For distant observers we distinguish five relevant cases of the radial integral. These cases depend on the character of roots of the quartic equation $U(u)=0$, i.e., on the number of turning points ($n_u=0$ or $n_u=1$) in the radial motion  and the value of parameter $\\tilde{q}=q(a^2+b)+b(a-\\lambda)^2$. We have arranged those transformations into table \\ref{tableEI}.\n\nDenoting roots of the quartic equation $U(u)=0$ by $\\beta_1$, $\\beta_2$, $\\beta_3$ and $\\beta_4$, the meaning of each of the five cases is the following:\n\\begin{itemize}\n\\item\nThe \\textbf{case I}: four distinct real roots of $U(u)=0$ forming the sequence $\\beta_1>\\beta_2>\\beta_2>0$ and $\\beta_4<0$. The value of modified constant of motion $\\tilde{q}>0$.\n\\item\nThe \\textbf{case II}: four real roots as in the case I but their values form the following order: $\\beta_1>\\beta_2>0$ and $\\beta_4<\\beta_3<0$. The value of modified constant of motion  $\\tilde{q}<0$. \n\\item\nThe \\textbf{case III}: two real and two complex roots of $U(u)=0$: $\\beta_1$ being a complex root, $\\beta_2=\\bar{\\beta_1}$ and $\\beta_4<\\beta_3<0$. The value of modified constant of motion $\\tilde{q}<0$.\n\\item\nThe \\textbf{case IV}: only complex roots: $\\beta_2=\\bar{\\beta_1}$ and $\\beta_4=\\bar{\\beta_3}$. The value of modified constant of motion $\\tilde{q}<0$. \n\\item\nThe \\textbf{case V}: two real and two complex roots of $U(u)=0$: $\\beta_1>0$, $\\beta_4<0$, $\\beta_2$ being a complex root and $\\beta_3=\\bar{\\beta_2}$. \n\\end{itemize}\n\n\\begin{table}[!ht]\n\\tbl{The reductions of $\\int^m_{m_1}\\mathrm{d} m'\/\\sqrt{M(m')}=I_M$} \n{\\begin{tabular}{@{}lllll@{}}\\toprule\n  \tCase & $\\tan\\Psi$ & $m$ & $c_1$ & $m_1$\\\\ \\colrule\n\t\\\\\n\t$M_-<0$ & $\\sqrt{\\frac{M_+}{m^2}-1}$ & $\\frac{M_+}{M_+-M_-}$ & $\\frac{1}{\\sqrt{a^2(M_+-M_-)}}$ & $\\sqrt{M_+}$\\\\\n\t\\\\\n\t$M_->0$ & $\\sqrt{\\frac{M_+-m^2}{m^2-M_-}}$ & $\\frac{M_+-M_-}{M_+}$ &\n        $\\frac{1}{a^2}$ & $\\sqrt{M_+}$\\\\ \\botrule\n \\end{tabular}\\label{tableEIM}}\n\\end{table} \n\n\\begin{table}[!ht]\n\\tbl{The reductions of $\\int^u_{u_1}\\mathrm{d} u'\/\\sqrt{U(u')}=I_U$}\n{\\begin{tabular}{@{}lllll@{}}\\toprule\n\n  \tCase & $\\tan\\Psi$ & $m$ & $c_1$ & $u_1$\\\\ \\colrule\n\t\n\tI & $\\sqrt{\\frac{(\\beta_1-\\beta_3)(u-\\beta_4)}{(\\beta_1-\\beta_4)(\\beta_3-u)}}$ & $\\frac{(\\beta_1-\\beta_2)(\\beta_3-\\beta_4)}{(\\beta_1-\\beta_3)(\\beta_2-\\beta_4)}$ & $\\frac{2}{\\sqrt{\\tilde{q}(b1-b3)(b2-b4)}}$ & $\\beta_4$\\\\\n\t\\\\\n\tII & $\\sqrt{\\frac{(\\beta_1-\\beta_2)(u-\\beta_3)}{(\\beta_1-\\beta_3)(\\beta_2-u)}}$ & $\\frac{(\\beta_2-\\beta_3)(\\beta_1-\\beta_4)}{(\\beta_1-\\beta_2)(\\beta_4-\\beta_3)}$ & $\\frac{2}{\\sqrt{-\\tilde{q}(b1-b2)(b3-b4)]}}$ & $\\beta_3$\\\\\n\t\\\\\n\tIII & $\\frac{2c_2(u)}{|1-c^2_2(u)|}$ & $\\frac{4c_4 c_5 - (\\beta_3 - \\beta_4)^2 - c_4 c_5}{4c_4 c_5}$ & $\\frac{1}{\\sqrt{-\\tilde{q}c_4 c_5}}$ & $\\beta_3$\\\\\n\t\\\\\n\tIV & $\\frac{u-c_3}{\\Im(\\beta_1)(1+c_2^2)+c_2(u-c_3)}$ & $1-\\left(\\frac{c_4-c_5}{c_4+c_5}\\right)^2$ & $\\frac{2}{(c_4+c_5)\\sqrt{-\\tilde{q}}}$ & $c_3$\\\\\n\t\\\\\n\tV & $\\frac{2c_2(u)}{|1-c^2_2(u)|}$ & $1-\\frac{(c_4+c_5)^2-(\\beta_1 -\n          \\beta_4)^2}{4c_4 c_5}$ & $\\frac{1}{\\sqrt{\\tilde{q}c_4 c_5}}$ &\n        $\\beta_4$\\\\ \\botrule\n \\end{tabular}\\label{tableEI}}\n\\end{table}\n\n\\begin{table}[!th]\n\\tbl{Definitions for Table \\ref{tableEI}.}\n{\\begin{tabular}{@{}lll@{}}\\toprule\n  \n  \tCase & $^1 c_2$ & $^1 c_3$\\\\ \\colrule\n\t\n\tIII &  $\\left[\\frac{c5(u-\\beta_3)}{c_4(u-\\beta_4)}\\right]^{1\/2} $ & -\\\\\n\t\\\\\n\tIV & $\\left\\{\\frac{4[\\Im(\\beta_1)]^2-(c_4-c_5)^2}{(c_4+c_5)^2-4[\\Im(\\beta_1)]^2}\\right\\}^{1\/2}$ & $\\mbox{\\fontsize{8}{10}\\selectfont $\\Re(\\beta_1)+c_2\\Im(\\beta_1)$}$\\\\\n\t\\\\\n\tV & $\\left[\\frac{c4(u-\\beta_4)}{c_5(\\beta_1-u)}\\right]^{1\/2} $& -\\\\ \\botrule\n\\end{tabular}\\label{tableEI2}}\n\\end{table}\n\n\n\\begin{table}[!th]\n\\tbl{Definitions for Table \\ref{tableEI} and Table \\ref{tableEI2}.}\n {\\begin{tabular}{@{}lll@{}}\\toprule\n  \n  \tCase & $^1 c_4$ & $^1 c_5$\\\\ \\colrule\n\t\n\tIII &   $\\mbox{\\fontsize{8}{10}\\selectfont$\\left\\{\\left[\\Re(\\beta_1)-\\beta_3\\right]^2+[\\Im(\\beta_1)]^2\\right\\}^{1\/2}$}$ & $\\mbox{\\fontsize{8}{10}\\selectfont $\\left\\{\\left[\\Re(\\beta_1)-\\beta_4\\right]^2+[\\Im(\\beta_1)]^2\\right\\}^{1\/2}$}$\\\\\n\t\\\\\n\tIV & $\\mbox{\\fontsize{8}{10}\\selectfont $\\left\\{\\left[\\Re(\\beta_1)-\\Re(\\beta_3)\\right]^2+[\\Im(\\beta_1)+\\Im(\\beta_3)]^2\\right\\}^{1\/2}$}$ & $\\mbox{\\fontsize{8}{10}\\selectfont $\\left\\{\\left[\\Re(\\beta_1)-\\Re(\\beta_3)\\right]^2+[\\Im(\\beta_1)-\\Im(\\beta_3)]^2\\right\\}^{1\/2}$}$\\\\\n\t\\\\\n\tV & $\\mbox{\\fontsize{8}{10}\\selectfont $\\left\\{\\left[\\Re(\\beta_2)-\\beta_1\\right]^2+[\\Im(\\beta_2)]^2\\right\\}^{1\/2}$}$ & $\\mbox{\\fontsize{8}{10}\\selectfont $\\left\\{\\left[\\Re(\\beta_2)-\\beta_4\\right]^2+[\\Im(\\beta_2)]^2\\right\\}^{1\/2}$}$\\\\ \\botrule\n\\end{tabular}}\n\\begin{tabular}{c}\n\t$^1$\\textit{The symbols $\\Re(x)$ and $\\Im(x)$  refer to real and imaginary part of $x$ here.}\n\\end{tabular}\n\\end{table}\n\n\n\\par\n\nUsing presented transformations we can write the integrals (\\ref{u_int}) and (\\ref{mu_int})  in the form\n\n\\begin{equation}\n \\int^{u}_{u_1}\\frac{1}{\\sqrt{U(\\tilde{u})}}\\mathrm{d} \\tilde{u} = c_1\\mathcal{F}(\\Psi;m)\\label{ellint}\n\\end{equation}\nand \n\\begin{equation}\n \\int^{\\mu}_{\\mu_1}\\frac{1}{\\sqrt{M(\\tilde{\\mu})}}\\mathrm{d} \\tilde{\\mu} = c_1\\mathcal{F}(\\Psi;m)\\label{ellintM}\n\\end{equation}\nwhere $\\mathcal{F}$ is the elliptic integral of the first kind and $u_1$(resp $\\mu_1$) depends on the case of root distribution of quartic equation $U(u)=0$ (resp. $M(\\mu)=0$) as given in Table \\ref{tableEI} (resp \\ref{tableEIM}). If, in the cases III and V, the value of $1-c_2^2(u)<0$, we have to take instead of  (\\ref{ellint}) the form\n\n\\begin{equation}\n \\int^{u}_{u_1}\\frac{1}{\\sqrt{U(\\tilde{u})}}\\mathrm{d} \\tilde{u} =  c_1(2\\mathcal{K}(m)-\\mathcal{F}(\\Psi;m)),\\label{ellint1}\n\\end{equation}\nwhere $\\mathcal{K}$ is the complete elliptic integral of the first kind. In the case that sign$(\\mu1\\cdot\\mu)<0$ we have to take instead of (\\ref{ellintM}) the form\n\\begin{equation}\n \\int^{\\mu}_{\\mu_1}\\frac{1}{\\sqrt{M(\\tilde{\\mu})}}\\mathrm{d} \\tilde{\\mu} =  c_1(2\\mathcal{K}(m)-\\mathcal{F}(\\Psi;m)),\\label{ellintM1}\n\\end{equation}\nwhere $\\Psi$, $m$ and $c_1$ are taken from table \\ref{tableEIM}.\n We consider two basic possibilities of trajectories, namely those corresponding to direct and indirect images (Figures \\ref{fig19} and \\ref{fig20}).\n\n\\subsection{Disc images}\n\n\tIt is very important to demostrate the influence of the braneworld parameter on the shape of images of rings in the equatorial plane representing parts of Keplerian accretion discs. Of course, as well known from the Kerr (and even Schwarzchild) black holes, the images strongly depend on the latitude of the observer. We calculate the direct and indirect images of flat discs and combined, full image of the disc for two representative values of viewing angle $\\theta_0$ and appropriatelly chosen extension of radiating disc area.\n\n\n\tWe include the effect of frequency shift into the calculated images of part of the Keplerian discs assumed to be radiating at a given fixed frequency. The frequency shift $g$ is determined by the ratio of observed ($E_0$) to emitted ($E_e$) photon energy\n\\begin{equation}\n g=\\frac{E_0}{E_e}=\\frac{k_{0\\mu} u_0^\\mu}{k_{e\\mu} u_e^\\mu},\n\\end{equation}\nwhere $u_0^\\mu$($u_e^\\mu$) are components of the observer (emitter) 4-velocity and $k_{0\\mu}(k_{e\\mu})$ are components of the photon 4-momentum taken at the moment of emission (observation). For distant observers $u^\\mu_0=(1,0,0,0)$. The emitter follows an equatorial circular geodesics at $r=r_e$, $\\theta_e=\\pi\/2$. Therefore, $u_e^\\mu=(u^t_e,0,0,u_e^\\varphi)$, with components given by\n\n\\begin{eqnarray}\n u_e^t&=&\\left[1-\\frac{2}{r_e}(1-a\\Omega)^2-(r_e^2+a^2)\\Omega^2+\\frac{b}{r_e^2}(1-2a\\Omega)\\right]^{-1\/2},\\\\ u_e^\\varphi&=&\\Omega u_e^t,\n\\end{eqnarray}\nwhere $\\Omega=\\mathrm{d}\\varphi\/\\mathrm{d} t$ is the Keplerian angular velocity of the emitter related to distant observers, given by equation (\\ref{ang_vel_gf}).\n\nThe frequency shift including all relativistic effects is then given by\n\n\\begin{equation}\n g=\\frac{\\left[1-\\frac{2}{r_e}(1-a\\Omega)^2-(r_e^2+a^2)\\Omega^2+\\frac{b}{r_e^2}(1-2a\\Omega)\\right]^{1\/2}}{1-\\lambda\\Omega}\n\\end{equation}\nwhere $\\lambda\\equiv-k_\\varphi\/k_t$ is the impact parameter of the photon\nbeing a motion constant for an individual photon radiated at a specific\nposition of the radiating disc; notice that $g$ is independent of the second\nphoton motion constant  (impact parameter) $q$. Of course, depending on the\nposition of the emitter along the circular orbit, the impact parameters\n$\\lambda$, $q$ of photons reaching a fixed distant observer will vary\nperiodically (see eg., \\cite{Bao-Stu:1992:}). For each position of the emitter\nthe impact parameters are determined by the procedure of integration of photon\ntrajectories. \n\nThe influence of the frequency shift in the disc images is demonstrated in Figures \\ref{fig24_a_i} and \\ref{fig26_a_i}. The role of the braneworld parameter is illustrated both for small ($\\theta_0=30^\\circ$) and high ($\\theta_0=80^\\circ$) inclination angles. We consider two cases of the radiating disc extension: first one  with fixed inner and outer radii, independent of the black hole parameters, and the second one when the inner radius is identified with the marginally stable orbits, depending on the black hole parameters. \n\n\n\n\n\\begin{figure}[ht]\n\n  \\begin{tabular}{ccc}\n    \\includegraphics[width=3.6cm]{fig23d}\n& \\includegraphics[width=3.6cm]{fig23e}\n&\\includegraphics[width=3.6cm]{fig23f}\\\\\n \\includegraphics[width=3.6cm]{fig23g}\n& \\includegraphics[width=3.6cm]{fig23h}\n&\\includegraphics[width=3.6cm]{fig23i}\\\\\n \\includegraphics[width=3.6cm]{fig24d}\n& \\includegraphics[width=3.6cm]{fig24e}\n&\\includegraphics[width=3.6cm]{fig24f}\\\\\n     \\includegraphics[width=3.6cm]{fig24g}\n& \\includegraphics[width=3.6cm]{fig24h}\n&\\includegraphics[width=3.6cm]{fig24i}\n  \\end{tabular}\n\n\\caption{\\label{fig24_a_i}Radiating Keplerian disc images with fixed inner and outer radii. The modified frequency shift $\\bar{g}=(g-g_{min})\/(g_{max}-g_{min})$, with $g_{min}=0.4$ and $g_{max}=1.5$, of the radiation emitted from the thin disk with inner radius $r_{in}=7 M$ and outer radius $r_{out}=15 M$, encoded into colors is plotted for representative values of tidal charge parameter $b=-3.0$, $0.0$ and inclination of observer $\\theta_0=30^\\circ$, $80^\\circ$. In the left column direct images are ploted, the indirect images are ploted in the central column and the composition of direct and indirect images is plotted in the right column. The first two rows of images are plotted for the observer inclination $\\theta_0=30^\\circ$ and the second two rows of images are plotted for the observer inclination $\\theta_0=80^\\circ$. Top row images are plotted for $b=0.0$, the second row images are plotted for $b=-3.0$, the third row images are plotted for $b=0.0$ and bottom row images are plotted for $b=-3.0$.}\n\\end{figure}\n\nIn order to map the frequency shift $g$ into color palete we define modified frequency shift $\\bar{g}=(g-g_{min})\/(g_{max}-g_{min})$ where $g_{min}$ ($g_{max}$) is the minimal (maximal) value of frequency shift, which is fixed in a particular set of images.\n\nWe can see from Figs. \\ref{fig24_a_i} and \\ref{fig26_a_i} that the negative tidal charge has the tendency to enlarge and symmetrize the disc images.\n\n\n\\begin{figure}[ht]\n\\begin{tabular}{ccc}\n     \\includegraphics[width=3.6cm]{fig25a}\n& \\includegraphics[width=3.6cm]{fig25b}\n&\\includegraphics[width=3.6cm]{fig25c}\\\\\n     \\includegraphics[width=3.6cm]{fig25d}\n& \\includegraphics[width=3.6cm]{fig25e}\n&\\includegraphics[width=3.6cm]{fig25f}\\\\\n     \\includegraphics[width=3.6cm]{fig26a}\n& \\includegraphics[width=3.6cm]{fig26b}\n& \\includegraphics[width=3.6cm]{fig26c}\\\\\n     \\includegraphics[width=3.6cm]{fig26d}\n& \\includegraphics[width=3.6cm]{fig26e}\n& \\includegraphics[width=3.6cm]{fig26f}\n  \\end{tabular}\n\n\n\\caption{\\label{fig26_a_i} Radiating Keplerian disc images with $r_{in}=r_{ms}$. The modified frequency shift $\\bar{g}=(g-g_{min})\/(g_{max}-g_{min})$, with $g_{min}=0.2$ and $g_{max}=1.8$, of the radiation emitted from the thin disk with inner radius $r_{in}=r_{ms}$ (with $r_{ms}(b=0;a=0.9981)=1.3$ and $r_{ms}(b=-3;a=0.9981)=6.3$) and outer radius $r_{out}=10$, encoded into colors is plotted for representative values of tidal charge parameter $b=-3.0$, $0.0$ and inclination of observer $\\theta_0=30^\\circ$, $80^\\circ$. In the left column direct images are ploted, the indirect images are ploted in the central column and the composition of direct and indirect images is plotted in the right column. The first two rows of images are plotted for the observer inclination $\\theta_0=30^\\circ$ and the second two rows of images are plotted for the observer inclination $\\theta_0=80^\\circ$. Top row images are plotted for $b=0.0$, the second row images are plotted for $b=-3.0$, the third row images are plotted for $b=0.0$ and bottom row images are plotted for $b=-3.0$. \n}\n\\end{figure}\n\n\n\n\\clearpage\n\\newpage\n \n\\section{Time delay}\n\nFor optical effects in vicinity of a black hole, the time delay in case of systems varying with time and observed along two different directions due to the light deflection in strong gravity can be important. The cordinate time that elapses from the instant of photon emission, $t_e$, to the instant of its reception, $t_o$, is integrated from the Carter equations and reads\n\n\\begin{eqnarray}\n  t_o&=&t_e+\\mu_{sgn}\\int_{\\mu_e}^{\\mu_o}{a^2\\mu^2\\frac{\\mathrm{d}\\mu}{\\sqrt{M}}}\\nonumber\\\\\n\t&&+u_{sgn}\\int^{u_o}_{u_e}{\\frac{2a(a-\\lambda)u^3+a^2u^2+1+ab(\\lambda-a)u^4}{(u\/u_{+}-1)(u\/u_{-}-1)\\sqrt{U}}}\\mathrm{d} u\n\\end{eqnarray}\nIn order to succesfully integrate this formula, one must map all the turning points in $\\mu$ and $u$ motion to correctly set up the signs $u_{sgn}$ and $\\mu_{sgn}$.\n\nSuppose that the two light beams, direct and indirect, are emitted at the same coordinate time $t_e$. They generally reach the observer at different coordinate times $t_o^{\\mathrm{dir}}$($t_o^{\\mathrm{indir}}$ resp.). By time delay we define here the difference $\\Delta t\\equiv t_o^{\\mathrm{indir}}-t_o^{\\mathrm{dir}}$.\n\n\\begin{figure}[!ht]\n \\includegraphics[width=10cm]{fig27}\n\\caption{\\label{fig27}The illustration of the impact of tidal charge parameter on the time delay $\\Delta t$ in case of direct and indirect photons emitted from emitter $E$ at coordinate time $t_e$ and azimuthal position $\\varphi_e=\\pi$. They are received at observer $O$ at coordinate times $t_o^{\\mathrm{dir}}$($t_o^{\\mathrm{indir}}$ resp.). The emittor is on circular geodesic in equatorial plane of braneworld Kerr black hole at radial coordinate $r=r_e$. The observer is far from the center of the black hole at $r=r_o$. Its inclination is $\\theta=\\theta_o$.}\n\\end{figure}\n\n\nTo demonstrate the impact of the tidal charge $b$ on the time delay we\nconsider the following situation (see Figure \\ref{fig27}). Let the isotropicaly radiating monochromatic source orbits in the equatorial plane of the braneworld Kerr black hole at radial distance $r_e$. It can be switched on and off. When it reaches the azimuthal coordinate $\\varphi=\\pi$ it is switched on and we compare the coordinate times $t^{\\mathrm{dir}}_o$ and $t^{\\mathrm{indir}}_o$ of reception of the photons from the direct and indirect images of the source. \n\n\\begin{figure}[!ht]\n \\begin{tabular}{cc}\n  \\includegraphics[width=6.2cm]{fig28a}&\\includegraphics[width=6.2cm]{fig28b}\n \\end{tabular}\n \n\\caption{\\label{fig28_a_b}The difference (``Time Delay``), $\\Delta t = t^{\\mathrm{indir}}_o -t^{\\mathrm{dir}}_o$, between coordinate times of reception of direct and indirect geodesics of photons emmited at the same coordinate time $t_e$ from the azimuthal coordinate $\\varphi=\\pi$  is plotted as a function of tidal charge $b$. Left figure: the inclination of the observer is $\\theta_0=20^\\circ$. Right figure: the inclination of the observer is $\\theta_0=80^\\circ$. }\n\n\\end{figure}\n\nThe results are demonstrated in the Figure \\ref{fig28_a_b}. We can directly see that time delay $\\Delta t$ between times of reception of the direct and indirect photons emitted at the same instant from the azimuthal position $\\varphi=\\pi$ increases as the value of the tidal charge parameter $b$ goes to higher negative values. When $b$ is fixed, the time delay $\\Delta t$ increases as the value of the inclination decreases. The same effects appear for other positions of the radiating spot ($\\varphi\\not= \\pi$). We can see that the time delay $\\Delta t$ depends strongly on the viewing angle  $\\theta_0$. Therefore, it is extremely important to have a system with precisely determined viewing angle.\n\n\\section{Optical phenomena related to Sgr $A^*$}\nThere is an enormously growing evidence that the center of our Galaxy harbors a supermassive black hole whose position could be almost surely identified with the extremely compact radio source  Sgr $A^*$. The chain of arguments seems to be very convincing; stars orbiting an unseen mass concentration on elliptical orbits with a common focal position, the  unseen mass centered  on Sgr $A^*$ that seems to be motionless at the dynamical center of the Galaxy, extremely compact emission of the center \\cite{Reid:2008:}. Recent measurements of Ghez and collaborators \\cite{Ghez-etal:2008:} from the W.M. Keck 10 - meter telescopes of a fully unconstrained Keplerian orbit of the short period  star SO-2 provide the distance $R_0=8.0\\pm 0.6$ kpc and black hole mass $M=(4.1\\pm0.6)\\times 10^6 M_\\odot$. If the black hole is assumed to be at rest with respect to the Milky Way Galaxy (i.e., has no massive companion to induce its motion) as argued by Reid \\cite{Reid:2008:}, the fit can be further constrained to $R_0=8.4\\pm 0.4$kpc and $M=(4.5\\pm 0.4)\\times 10^6M_\\odot$ \\cite{Ghez-etal:2008:}.\n\nSuch a close and huge supermassive black hole could be clearly a very convenient object, probably the best one, for testing a wide variety of optical phenomena in strong gravity in its vicinity. The time delay of accidents happening behind the black hole and observed along two directions could be in principle easily measured. We could even expect possibility of black hole silhuette measurements. In this way the influence of the tidal charge could be properly tested and its value estimated, because for the Galaxy supermassive black hole we can determine relatively precisely the inclination angle of the observer (Solar system), although it is of course very close to $\\theta_0\\simeq 90^\\circ$.\n\nFor non-rotating , Schwarzchild black holes, the silhuette diameter is given by the impact parameter of the photon circular orbit\n\n\\begin{equation}\n\tD=2\\lambda_{ph}= 6\\sqrt{3} M.\n\\end{equation}\nUsing the Sgr $A^*$ mass estimate $M\\sim 4.5\\times 10^6M_\\odot$, we find $D\\simeq 55\\mu$arcsec while interferometer finges were reported at wavelength of $1.3$ mm and fringe spacing of $0.00005$, comparable with the expected value of $D$. Shorter wavelengths should enable detailed measurements of the black hole silhuette and relatively precise estimates of the black hole parameters due to very precise knowledge of the inclination angle. The angle can be given by the measurement of the Solar system position relative to Galaxy plane $z_\\odot\\sim 14pc$\\cite{Yoshi:2007:}. Then $\\theta_0\\sim 89.9^\\circ$ or more precisely, $\\theta_0$ lies between the values of $89.8772^\\circ$ ($z_\\odot=18pc$) and $89.9318^\\circ$($z_\\odot=10pc$). Of course, considering the silhuette shape, it is quite enough to take $\\theta_0=90^\\circ$. \n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\\includegraphics[width=6.1cm]{diameter_of_silhuette_schw}\n\t\\end{center}\n\t\\caption{Diameter $D$ as a function of braneworld parameter $b$ is plotted for Schwarzchild black hole of mass $M=4.5\\times 10^6M_\\odot$. Observer is at $r_0=8.4$kpc lying in the equatorial plane.}\\label{diameter_of_silhuette_schw}\n\\end{figure}\n\nIn the case of spherically symmetric black holes, the influence of the tidal charge parameter $b$ on the silhuette diameter can be given by the simple formula for impact parameter of photon circular orbits that reads \\cite{Stu-Hle:2002:}\n\n\\begin{equation}\n\t\\lambda_{ph}(b)=\\frac{r_{ph}^2}{\\sqrt{r_{ph}-b}}M,\n\\end{equation}\nwhere\n\n\\begin{equation}\n\tr_{ph}(b)=\\frac{3}{2}\\left(1+\\sqrt{1-\\frac{8b}{9}}\\right).\n\\end{equation}\nThe resulting dependence  of the diameter $D(b)$ is illustrated in Figure \\ref{diameter_of_silhuette_schw} . The diameter grows slowly with the descending of $b$; notice that its magnitude is twice the pure Schwarzchild value for $b=-12.8428$. Of course, for rotating black holes the silhuette is maximally deformed due to the influence of rotation since the viewing angle $\\theta_0\\sim 90^\\circ$ and is given by calculations and results presented above. Testing of the combined spin and tidal charge influence would be possible with measurement precision enlarged for 1-order relative to the recently expected state mentioned above. Clearly, we can expect that the observational accuracy in near future will be high enough to measure the Sgr $A^*$ black hole silhuette implying relevant estimates of the black hole parameters.\n \n\n\n\n\n\n\n\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=6.1cm]{td_tab4}\n\t\\end{center}\n\t\\caption{\\label{td_table4}Comparizon of time delay effect as a function of braneworld parameter $b$ between two rotating black holes with rotational parametes $a=0.5$ and $a=0.998$. For each $b$ the emitter is radiating from marginally stable orbit. The relevant values of radii $r_{ms}$ of marginally stable orbits are arranged in the Table \\ref{tabulka2}. }\n\\end{figure}\n\n\\begin{table}[ht]\n\t\t\\tbl{Table of relevant values of $r_{ms}$ used in plots oin Fig \\ref{td_table4}}\n\t\t{\\begin{tabular}{@{}ccccccc@{}} \n\t\t\\toprule\n\t\t$b$ & 0.0 & -0.5 & -1.0 & -2.0 & -3.0 & -10.0\\\\ \n\t\t\\colrule\n\t\t$r_{ms}(a=0.5)$ & 4.24M & 5.05M & 5.73M & 6.88M & 7.85M & 12.88M\\\\\n\t\t\\colrule\n\t\t$r_{ms}(a=0.998)$ & 1.24M & 3.03M  & 3.91M & 5.22M & 6.28M & 11.44M\\\\\n\t\t\\botrule\n\t\t\\end{tabular}\\label{tabulka2}}\n\\end{table}\n\nConsidering the time delay effects, the exact value of $\\theta_0$ is crucial since it plays a fundamental role in determining the time delay effect whose scale is given by the value of $t\\sim 1sec$. We illustrate the influence of the tidal charge on the time delay effects at the astrophysically important radii corresponding to marginally stable circular geodesics, i.e. in the strong gravity regime, for two representative fixed values of black hole spin (see Figure \\ref{td_table4} and Table \\ref{tabulka2}). We can expect importance of the regions close to $r_{ms}$ for relevant optical effects due to the idea of the low angular momentum accretion in Sgr $A^*$ advocated by B. Czerny \\cite{Cze-etal:2007:}. Clearly, we can see in Figure \\ref{td_table4} that the time delay effects could be well measurable and the tidal charge influence could be well tested, if the black hole spin is properly estimated.\n\n\\section{\\label{sec:Conclusions}Conclusions}\nOne of the most promising ways of estimating influence of hypothetical hidden external dimensions, considered in the framework of the braneworld model with infinite external dimension as developed by \\cite{Ran-Sun:1999:}, seems to be investigation of the optical phenomena caused by the black hole backgrounds. It is so because black holes represent the only case when the non-local influence of the bulk space on the braneworld spacetime structure can be fully described by a single, braneworld parameter called tidal charge, the sign of which can be both positive and negative, with the second possibility beeing more realistic one \\cite{Ali-Gum:2005:,Dad-etal:2000:}.\n\nHere, we focused our attention to developing a theoretical background for treating the optical phenomena in vicinity of braneworld rotating black holes and bringing general tendencies of the tidal charge effect in some basical optical phenomena. \n\nWe have shown qualitatively how the braneworld tidal charge affects the basical optical phenomena, especially the black-hole silhuette, the accretion disc image with the frequency shift of area of the disc radiating at a specific frequency, and the time delay between the direct and indirect images of the hot spot orbiting the black hole. We have shown that these phenomena could be measured and used to put limits on the tidal charge in case of Galaxy Center Sgr $A^*$ supermassive black hole.\n\nWe generalized the approaches based on the transfer-function method as introduced and developed in Schwarzchild and Kerr backgrounds \\cite{fab-rees-ste-whi:1989:,Mat-Fab-Ros:1993:,Bao-Stu:1992:,Stu-Bao:1992:,Laor:1991:,Dov-Karas-Mas-Mar:2005:,Fan-Cal-Fel-Cad:1997:,Rau-Bla:1994:} where equations of photon motion are solved in terms of the elliptic integrals (see \\cite{Rau-Bla:1994:,Kra:2005:,Kra:2007:}). For purposes of the present work, the transfer-function method seems to be most efficient. Nevertheless, we prepared the ray-tracing method too, since that could be usefull in treating other optical phenomena.\n\n Generally, rising negative value of the tidal charge strenghtens the black hole field and suppresses the rotational phenomena, when the black-hole rotation parameter is fixed.  The magnitude of the optical phenomena grows with decreasing of the negatively-valued tidal charge, but the rotation induced asymmetry of the phenomena like the black-hole silhuette, or the accretion disc image, decreases. The black-hole silhuette is characterized by two parameters, namely the shift of the center  and ellipticity, that could be in principle measurable in the Galactic Center black-hole system Sgr $A^*$, after expected development of observational techniques that at present enable measurement of the black hole diameter, not details of the shape. \nThe Galaxy center (Sgr $A^*$) seems to be also a promising candidate for testing the time delay effects both for phenomena related to the accretion disc and flares observed there , and for some expected lensing phenomena connected to the observed stars orbiting the Sgr $A^*$ central black hole. \n\nWe have found that observable phenomena could be expected for the time-delay effects. Of special interest is comparison of time delays generated for sources in vicinity of the Sgr $A^*$ black hole (both stars and disc hot spots) and those related to weak lensing of some distant sources \\cite{Zak:2003:,Sereno:2006}. \n\nSimilarly, keeping rotational parameter fixed, the negative tidal charge has tendency to make the isoradial curve images (both direct and indirect) larger and less deformed while the positive tidal charge influence is of opposite character. On the other hand, for fixed rotational parameter of the black hole and disc radiating at the innermost part above the innermost stable orbit at $r=r_{ms}$, the negative tidal charge restricts the radiating ring image simply because the radius $r_{ms}$ grows with decreasing value of braneworld parameter $b$. Suppresion of the relativistic effects can be measurable also in the spectral line profiles generated by the inner hot part of the disc radiating at special X-ray line \\cite{SS:b:RAGTime:2007:Proceedings}. \n\nThe optical tests have to be confronted with the data obtained from quasiperiodic oscillations observed in some black-hole systems (microquasars \\cite{Rem-McCli:2006:ARASTRA:}). The orbital resonance model gives good estimates of the black-hole parameters \\cite{Tor-Abr-Klu-Stu:2005:,Tor:2005a:,Tor:2005b:}; this model has been recently generalized to the case of braneworld Kerr black holes \\cite{Stu-Kot:2008}. It is shown that in the case of microquasar  GRS 1915+105 and Galactic Center Sgr $A^*$ black holes with the negative braneworld parameter $b$ are allowed by the observational data \\cite{Stu-Kot:2008}. Detailed modelling of optical phenomena connected to the oscillating discs or orbiting (oscillating) hot spots and related resonant phenomena between the oscillation modes could be very promising in putting limits on allowed values of the tidal charge of the black hole. We plan to elaborate such modelling in future. \n\n\n\\section*{Acknowledgements}\nResearch supported by the Czech grant MSM 4781305903 and LC 06014. One of the authors (Zden\\v{e}k Stuchl\\'{i}k) would like to express his gratitude to the Czech Committee for Collaboration with CERN for support.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn Nielsen theory, the purpose is to find a homotopy-invariant lower bound for the number of fixed points of a map. Similarly, Nielsen periodic point theory will try to find a homotopy-invariant lower bound for the number of periodic points. Both of these theories work for all continuous maps on connected, compact absolute neighborhood retracts, but in this paper we will mainly focus on manifolds.\n\n\\medskip\n\nSo, let $f:X\\to X$ be a continuous map on a manifold. We call $x\\in X$ a periodic point if $f^n(x)=x$ for a certain integer $n>0$. The smallest $n$ for which this holds, will be called the period of $n$. In Nielsen theory, we partition the fixed point set into fixed point classes and subsequently count the number of fixed point classes that can not disappear by using a homotopy. The resulting number is the Nielsen number $N(f)$. We can do a similar thing for $f^n$ and try to approximate the number of $n$-periodic points of $f$ by the number $N(f^n)$. However, this lower bound is not sharp in general. By studying the relations between fixed point classes for different iterates of $f$ more closely, it is possible to define a better lower bound, namely the full Nielsen-Jiang periodic number $NF_n(f)$. This number has already been studied extensively for maps on tori and nilmanifolds and in this paper we will extend some of these results to infra-nilmanifolds.\n\n\\medskip\n\nIn the first two sections, we will give an introduction to the theory of infra-nilmanifolds and to Nielsen periodic point theory. Subsequently, we will prove that infra-nilmanifolds are essentially reducible to the GCD (Theorem \\ref{thmessredgcd}) and essentially toral (Theorem \\ref{thmesstor}). These structural properties also hold for maps on nilmanifolds and tori and are a necessary first step in order to be able to compute $NF_n(f)$ on infra-nilmanifolds. We will also prove that for all $n$, $NF_n(f)=N(f^n)$ for a large class of maps on infra-nilmanifolds, namely the class of semi-hyperbolic maps (Theorem \\ref{thmNF=Nf)}).\n\n\\medskip\n\nBecause maps on tori and nilmanifolds are weakly Jiang, the computation of $NF_n(f)$ turns out to be very doable on these manifolds. On infra-nilmanifolds, however, this is not the case. In the penultimate section, we will therefore develop a method that makes the computation of $NF_n(f)$ easier. Sometimes, though, the computation still can be quite hard, but this might be inherent to the problem, because, by applying our method to several examples, it becomes apparent that the expression for $NF_n(f)$ can be very complex.\n\n\\medskip\n\nIn the last section, we will look specifically at affine maps on infra-nilmanifolds. In general, these maps behave better than arbitrary continuous maps. In Theorem \\ref{theorem uiteindelijk boost inessentieel}, we prove that, under very mild conditions, these affine maps can only be Wecken (which means that $\\#\\Fix(f)=N(f)$) at every level if and only if they are semi-hyperbolic. This allows us to determine exactly for which maps $NF_n(f)=N(f^n)$, for all $n$ (Corollary \\ref{cor NF=Nf}). \n\\section{Infra-nilmanifolds}\nLet $G$ be a connected, simply connected, nilpotent Lie group. The group of affine transformations on $G$, $\\Aff(G)= G\\semi \\Aut(G)$, admits a natural left action on $G$:\n\\[ \\forall (g,\\alpha)\\in \\Aff(G),\\, \\forall h \\in G: \\;\\;^{(g,\\alpha)}h= g \\alpha(h).\\]Define $p:\\Aff(G)=G\\semi \\Aut(G) \\to \\Aut(G)$ as the natural projection onto the second factor of the semi-direct product. \n\n\\begin{definition} A subgroup $\\Gamma \\subseteq \\Aff(G)$ is called \\textbf{almost-crystallographic} if and only if $p(\\Gamma)$ is finite and $\\Gamma\\cap G$ is a uniform and discrete subgroup of $G$. The finite group $F=p(\\Gamma)$ is called the holonomy group of $\\Gamma$.\n\\end{definition}\n\nWith these properties, the natural action of such a group $\\Gamma$ on $G$ becomes properly discontinuous and cocompact. Moreover, when $\\Gamma$ is torsion-free, this action is free, which makes the resulting quotient space $\\Gamma\\backslash G$ a compact manifold. This idea leads to the following definition. \n\n\\begin{definition}\nA torsion-free almost-crystallographic group $\\Gamma\\subseteq \\Aff(G) $ is called an \\textbf{almost-Bieberbach group}, and the corresponding manifold $\\Gamma\\backslash G$ is called an \\textbf{infra-nilmanifold} (modeled on $G$).  \n\\end{definition} \n\nWhen the holonomy group is trivial, $\\Gamma$ can be considered to be a lattice in $G$ and the corresponding manifold $\\Gamma\\backslash G$ is a nilmanifold. When $G$ is abelian, i.e. $G$ is isomorphic to $\\R^n$, $\\Gamma$ will be called a Bieberbach group and $\\Gamma\\backslash G$ a compact flat manifold. When $G$ is abelian and the holonomy group of $\\Gamma$ is trivial, then $\\Gamma\\backslash G$ is a torus. Hence, infra-nilmanifolds are a natural generalization of nilmanifolds and tori.\n\n\\medskip\n\nNow, define the semigroup $\\aff(G)=G\\semi \\Endo(G)$. Note that $\\aff(G)$ acts on $G$ in a similar way as $\\Aff(G)$:\\[ (d,D): \\; G \\rightarrow G:\\; h \\mapsto d D(h).\\]The elements of this semigroup will be called affine maps, since $\\aff(G)$ is merely a generalization of the semigroup of affine maps $\\aff(\\R^n)$ to the nilpotent case. One of the main advantages of working with infra-nilmanifolds, is the fact that every continuous map lies in the same homotopy class as a map induced by such an affine map with similar properties. These maps are often easier to handle and are therefore ideal to use in proving several theorems. This strategy will be often used throughout this paper, for example in the last section of this paper.\n\n\\begin{theorem}[K.B.\\ Lee \\cite{lee95-2}]\n\\label{leemaps} Let $G$ be a connected and simply connected nilpotent Lie group and suppose that $\\Gamma, \\Gamma'\\subseteq \\Aff(G)$ are two almost-crystallographic groups modeled on $G$. \nThen for any homomorphism $\\varphi: \\Gamma\\rightarrow \\Gamma'$ there \nexists an element $  (d, D)\\in \\aff(G)$ such that \n\\[ \\forall \\gamma \\in \\Gamma: \\; \\varphi(\\gamma) (d,D) =  (d,D) \\gamma.\\] \n\\end{theorem}\n\nWe can consider the equality $ \\varphi(\\gamma) (d,D) =  (d,D) \\gamma$ in $\\aff(G)$, since $\\Aff(G)$ is a subgroup of $\\aff(G)$. With this equality in mind, when $\\Gamma$ and $\\Gamma'$ are torsion-free, it is easy to see that the affine map $(d,D)$ induces a well-defined map between infra-nilmanifolds:\\[\\overline{(d,D)}: \\Gamma \\backslash G \\rightarrow \\Gamma' \\backslash G: \\; \\Gamma h \\rightarrow \\Gamma' d D(h),\\]\nwhich exactly induces the morphism $\\varphi$ on the level of the fundamental groups.\n\n\\medskip\n\nOn the other hand, if we choose an arbitrary map $f:\\Gamma\\backslash G\\ra \\Gamma'\\backslash G$ between two infra-nilmanifolds and choose a lifting $\\tilde{f}:G \\to G$ of $f$, then there exists a morphism $\\tilde{f}_\\ast:\\Gamma\\to \\Gamma'$ such that $\\tilde{f}_\\ast(\\gamma) \\circ \\tilde{f} = \\tilde{f}\\circ \\gamma$, for all $\\gamma\\in \\Gamma$. By Theorem~\\ref{leemaps}, an affine map $(d,D)\\in \\aff(G)$ exists which also satisfies $\\tilde{f}_\\ast(\\gamma) \\circ (d,D)= (d,D)\\circ \\gamma$ for all $\\gamma\\in \\Gamma$. Therefore, the induced map $\\overline{(d,D)} $ and $f$ are homotopic. Hence, whenever we are studying homotopy-invariant properties for maps on infra-nilmanifolds, we are free to replace an arbitrary map $f$ by its affine counterpart.\n\n\\medskip\n\nThe map $(d,D)$ will be called an \\textbf{affine homotopy lift} of $f$, while we will denote the map $\\overline{(d,D)}$ as an \\textbf{affine map on an infra-nilmanifold}.\n\n\\medskip\n\nIt might be noteworthy to mention that $(d,D)$ is not unique in the sense that it depends on the choice of lifting $\\tilde{f}:G\\to G$. For example, from \\cite{lee95-2} we know that $D$ is only determined up to an inner automorphism of $G$.  \n\n\\medskip \nIn \\cite{ll09-1}, J.B. Lee and K.B. Lee gave a formula to compute Lefschetz and Nielsen numbers on infra-nilmanifolds. Pick an infra-nilmanifold $\\Gamma\\backslash G$, determined by the almost-Bieberbach group $\\Gamma\\subseteq \\Aff(G)$ and let $F\\subseteq \\Aut(G)$ denote the holonomy group of $\\Gamma$. We will write $\\lie$ for the Lie algebra of $G$. Because $G$ is a nilpotent, connected and simply connected Lie group, the map $\\exp:\\lie\\to G$ will be a diffeomorphism. Therefore, $\\Endo(G)$ and $\\Endo(\\lie)$ are isomorphic and for every endomorphism $A\\in \\Endo(G)$, we have a unique $A_\\ast\\in \\Endo(\\lie)$, which is determined by the relation $A \\circ \\exp= \\exp \\circ A_\\ast$. This $A_\\ast$ will be called the differential of $A$. Of course, $A$ is invertible if and only if $A_\\ast$ is invertible.\n\\begin{theorem}[J.B.\\ Lee and K.B.\\ Lee \\cite{ll09-1}] \\label{LeeForm}Let $\\Gamma\\subseteq \\Aff(G)$ be an almost-Bieberbach group with holonomy group $F\\subseteq \\Aut(G)$. Let $M=\\Gamma\\backslash G$ be the associated infra-nilmanifold.  If \n $f:M\\ra M$ is a map with affine homotopy lift $(d,D)$, then  \n\\[L(f)=\\frac{1}{\\# F}\\sum_{A \\in F}\\det(I-A_\\ast D_\\ast)\\]\nand\n\\[N(f)=\\frac{1}{\\# F}\\sum_{A \\in F}|\\det(I-A_\\ast D_\\ast)|.\\]\n\\end{theorem}\n\nWe will now list a couple of properties that we will need in this paper.\n\n\\medskip\n\nThe following lemma can be found in \\cite{ddm05-1}. We have adapted the formulation very slightly, but in essence it is the same lemma and it can be proved in a similar way.\n\n\\begin{lemma}\\label{Lemma Bram}\nSuppose that $F\\subset \\GL_n(\\C)$ is a finite group, $D\\in \\C^{n\\times n}$ and for all $A\\in F$, there exists a $B\\in F$, such that $DA=BD$. Take an arbitrary element $A_1\\in F$ and build the sequence $(A_j)_{j\\in \\N_0}$, such that $DA_i=A_{i+1}D$, for all $i$. Then,\n\\begin{enumerate}\n\\item $\\forall j\\in \\N_0: \\det(I-A_1D)=\\det(I-A_jD).$\n\\item $\\exists l, j\\in \\N_0: (A_jD)^l=D^l.$\n\\end{enumerate}\n\\end{lemma}\n\nIn \\cite{dp11-1}, we can find the following theorems.\n\n\\begin{theorem}\\label{thmRInf}\nLet $\\Gamma\\subseteq \\Aff(G)$ be an almost-Bieberbach group with holonomy group $F\\subseteq \\Aut(G)$. Let $M=\\Gamma\\backslash G$ be the associated infra-nilmanifold.  If \n $f:M\\to M$ is a map with affine homotopy lift $(d,D)$, then\n\\[\n  R(f)=\\infty \\iff \\exists A \\in F \\text{ such that } \\det(I - A_\\ast D_\\ast)=0.\n\\]\n\\end{theorem}\n\n\\begin{theorem}\\label{thmN=R}\nLet $f$ be a map on an infra-nilmanifold, such that $R(f)<\\infty$, then $N(f)=R(f)$.\n\\end{theorem}\n\nWe will also mention the following definition.\n\n\\begin{definition}\nLet $M$ be an infra-nilmanifold and $f:M\\to M$ be a continuous map, with $(d,D)$ as an affine homotopy lift. We say that $f$ is a \\textbf{hyperbolic} map if $D_\\ast$ has no eigenvalues of modulus $1$. We say that $f$ is \\textbf{semi-hyperbolic} if $D_\\ast$ has no eigenvalues which are roots of unity.\n\\end{definition}\n\nThis class of (semi-)hyperbolic maps contains for example the class of expanding maps and the class of Anosov diffeomorphisms.\n\n\\section{Nielsen periodic point theory}\n\nIn this section, we will mostly follow the outline of \\cite{jm06-1}. Many of the difficult aspects of Nielsen periodic point theory will disappear when working on infra-nilmanifolds. Therefore, we will try to present all the necessary results in a swift way and skip most of the proofs and unnecessary details. More information about Nielsen periodic point theory in general can be found in \\cite{hk97-1},\\cite{heat99-1},\\cite{jm06-1} or \\cite{jian83-1}.\n\n\\medskip\n\nWhen $f^n(x)=x$, we call $x$ a periodic point. If $n$ is the smallest integer for which this holds, $x$ is a periodic point of pure period $n$. We can apply similar techniques as in Nielsen fixed point theory to achieve Nielsen periodic point theory. Just like Nielsen fixed point theory divides $\\Fix(f)$ into different fixed point classes, Nielsen periodic point theory divides $\\Fix(f^n)$ into different fixed point classes, for all $n>0$ and looks for relations between fixed point classes on different levels. This idea is covered in the following definition.\n\n\\begin{definition}\\label{defboost}\nLet $f:X \\to X$ be a self-map. If $\\F_k$ is a fixed point class of $f^k$, then $\\F_k$ will be contained in a fixed point class $\\F_{kn}$ of $(f^k)^n$, for all $n$. We say that $\\F_k$ \\textbf{boosts} to $\\F_{kn}$. On the other hand, we say that $\\F_{kn}$ \\textbf{reduces} to $\\F_k$.\n\\end{definition}\n\nThis idea of boosting a fixed point class also has a more algebraic interpretation. Fix a lifting $\\tilde{f}$ of $f$ to the universal covering $(\\tilde{X},p)$ of $X$. Then $\\tilde{f}$ induces a homomorphism $f_\\ast$ on the group of covering transformations by using the following relation:$$f_\\ast (\\alpha)\\circ \\tilde{f}=\\tilde{f}\\circ \\alpha.$$Let us denote the set of Reidemeister classes of $f$ by $\\mathcal{R}(f)$. Any element of this set will be denoted by the Reidemeister class $[\\alpha]$, where $\\alpha$ is the coordinate of a lifting $\\alpha\\circ \\tilde{f}$. Let $k,n$ be integers, such that $k|n$. We then define the following boosting function:\n$$\\gamma_{nk}: \\mathcal{R}(f^k)\\to \\mathcal{R}(f^n):[\\alpha]\\mapsto [\\alpha f_\\ast^k(\\alpha)f_\\ast^{2k}(\\alpha)\\dots f_\\ast^{n-k}(\\alpha)].$$The idea behind this boosting function, is the fact that $$(\\alpha\\circ \\tilde{f}^k)^\\frac{n}{k}=\\alpha f_\\ast^k(\\alpha)f_\\ast^{2k}(\\alpha)\\dots f_\\ast^{n-k}(\\alpha)\\circ \\tilde{f}^n.$$This equality immediately shows that if $\\gamma_{nk}([\\beta])=[\\alpha]$, the fixed point class $p\\Fix(\\beta\\circ \\tilde{f}^k)$ will be contained in the fixed point class $p\\Fix(\\alpha \\circ \\tilde{f}^n)$. Hence, our algebraic definition makes sense in retrospect to Definition \\ref{defboost}.\n\n\\medskip\n\nIn this paper, we will often make a slight abuse of notation. Whenever we use the expression $[\\alpha]_k$, we will simultaneously consider the Reidemeister class $[\\alpha]\\in \\mathcal{R}(f^k)$ and the fixed point class $p(\\Fix(\\alpha\\circ \\tilde{f}^k))$. We will also often switch between both of these interpretations, whenever necessary. Note that both interpretations are essentially the same due to the one-to-one correspondence between Reidemeister classes and fixed point classes. This also means that we make a distinction between empty fixed point classes that come from a different Reidemeister classes. In a certain sense, this approach coincides with the idea of \\textit{labeled fixed point classes}, in \\cite{jian83-1}.\n\n\\medskip             \n\nNote that this description only depends on the homomorphism $f_*$. So, when $f$ and $g$ induce the same morphism on the group of covering transformations, then the structure of their periodic point classes will be the same. More specifically, this means that the whole description above (and everything that will follow) is homotopy-invariant.  \n\n\\medskip \n\nWhen $k|m|n$, an easy computation shows that $\\gamma_{nm}\\gamma_{mk}=\\gamma_{nk}$. Also, $\\gamma_{nn}=\\Id_{\\mathcal{R}(f^n)}$.\n\n\\medskip\n\nActually, to give the precise definition of Nielsen periodic point theory, we need a little more than this definition in terms of classes, namely a definition in terms of orbits.  Define the following map$$\\mathcal{R}_f:\\mathcal{R}(f^n)\\to \\mathcal{R}(f^n):[\\alpha]\\mapsto[f_\\ast(\\alpha)].$$One can easily see that this map is well-defined and that $\\mathcal{R}_f^n=\\Id_{\\mathcal{R}(f^n)}$. Furthermore, by using the commutativity property of the fixed point index on the maps $f$ and $f^{n-1}$, it is clear that this map preserves the index of the associated fixed point classes. By identifying $[\\alpha]$ with $[f_\\ast(\\alpha)]$ in $\\mathcal{R}(f^n)$, for all $\\alpha$, we find the quotient set $\\mathcal{OR}(f^n)$ of orbits of Reidemeister classes. Since the index is preserved in every orbit, it makes sense to talk about essential and inessential orbits. One can also notice that boosting functions make sense in terms of orbits, so we can talk about reducible and irreducible orbits, depending on whether they have a pre-image under a boosting function or not.\n\n\n\\begin{lemma} [\\cite{jm06-1}, 5.1.13]\\label{lemessirred}\nIf $\\mathcal{A}\\in \\mathcal{OR}(f^n)$ is essential and irreducible, then this orbit contains at least $n$ periodic points of period $n$.\n\\end{lemma}\n\nThis lemma gives us the idea for the following definition.\n\n\\begin{definition}\nWe define the \\textbf{prime Nielsen-Jiang periodic number} $NP_n(f)$ as $$n \\times \\textrm{(number of irreducible essential orbits in }\\mathcal{OR}(f^n)).$$\n\\end{definition}\n\nIf $P_n(f)$ is the set of periodic points of $f$ of pure period $n$, then Lemma \\ref{lemessirred} ensures us that $NP_n(f)$ is a homotopy-invariant lower bound of $\\# P_n(f)$.\n\n\\medskip\n\nWe would also like to find a similar lower bound for $\\# \\Fix(f^n)$. Pick an arbitrary $\\mathcal{A}\\in \\mathcal{OR}(f^n)$. We define the depth $d(\\mathcal{A})$ to be the least divisor $k$ of $n$, such that $\\mathcal{A}\\in \\textrm{Im}(\\gamma_{nk})$.\n\n\\begin{definition}\nLet $n$ be a fixed positive integer. A subset $\\mathcal{PS}\\subset \\bigcup_{k|n}\\mathcal{OR}(f^k)$ is called a \\textbf{preceding system} if every essential orbit $\\mathcal{A}$ in $\\bigcup_{k|n}\\mathcal{OR}(f^k)$ is preceded by an element of $\\mathcal{PS}$. Such a preceding system is called \\textbf{minimal} if the number $\\sum_{\\mathcal{A}\\in \\mathcal{PS}}d(\\mathcal{A})$ is minimal.\n\\end{definition}\n\n\\begin{definition}\nThe \\textbf{full Nielsen-Jiang periodic number} $NF_n(f)$ is defined as $\\sum_{\\mathcal{A}\\in \\mathcal{PS}}d(\\mathcal{A})$, where $\\mathcal{PS}$ is a minimal preceding system.\n\\end{definition}\n\n\\begin{theorem}[\\cite{jm06-1}, 5.1.18]\n$NF_n(f)$ is a homotopy-invariant lower bound for the number $\\# \\Fix(f^n)$.\n\\end{theorem}\nNote that every preceding system must contain every essential irreducible orbit of $\\mathcal{OR}(f^n)$. Since every of these orbits has a depth of $n$, we know that the following inequality holds:$$\\sum_{k|n}NP_k(f)\\leq NF_n(f).$$\n\nAn important definition that gives some structure to the boosting and reducing relations is the following.\n\n\\begin{definition}\nA self-map $f:X\\to X$ will be called \\textbf{essentially reducible} if, for all $n,k$, essential fixed point classes of $f^{kn}$ can only reduce to essential fixed point classes of $f^k$. A space $X$ is called essentially reducible if every self-map $f:X\\to X$ is essentially reducible.\n\\end{definition}\n\nIt can be shown that the fixed point classes for maps on infra-nilmanifolds always have this nice structure for their boosting and reducing relations.\n\n\\begin{theorem}[\\cite{lz07-1}]\\label{thmleezhao}\nInfra-nilmanifolds are essentially reducible.\n\\end{theorem}\n\nA nice consequence of being essentially reducible, is the following lemma.\n\n\\begin{lemma}[\\cite{jm06-1}, 5.1.22]\\label{lemNPNF}\nIf $f:X\\to X$ is essentially reducible, then it has a unique minimal preceding system, namely the set of all the essential irreducible orbits in $\\bigcup_{k|n}\\mathcal{OR}(f^k)$. As a consequence, the following equality holds:$$\\sum_{k|n}NP_k(f)=NF_n(f).$$\n\\end{lemma}\n\nOf course, by using the M\\\"{o}bius inversion formula, we can also write$$NP_n(f)=\\sum_{k|n}\\mu\\left(\\frac{n}{k}\\right)NF_k(f),$$where $\\mu$ denotes the M\\\"{o}bius function.\n\n\\medskip\n\nAs a generalization of being essentially reducible, we can define two other structures on the boosting and reducing relations.\n\n\\begin{definition}\nA map $f:X\\to X$ is called \\textbf{essentially reducible to the greatest common divisor} (GCD) if it is essentially reducible and if for every essential fixed point class $[\\alpha]_n$ that reduces to both $[\\beta]_k$ and $[\\gamma]_l$, there exists a fixed point class $[\\delta]_d$, with $d=\\gcd(k,l)$, such that $[\\alpha]_n$ reduces to $[\\delta]_d$. If this holds for every self-map on $X$, we will say that $X$ is essentially reducible to the GCD.\n\\end{definition}\n\nAn easy consequence of this definition is the following lemma.\n\n\\begin{lemma}[\\cite{jm06-1}, 5.1.26]\\label{lemessredgcd}\nIf $f:X\\to X$ is essentially reducible to the GCD, then every essential fixed point class $[\\alpha]_n$ in $\\mathcal{R}(f^n)$ is preceded by a unique irreducible essential fixed point class $[\\beta]_k$. Moreover, $d([\\alpha]_n)=k$.\n\\end{lemma}\n\nBy the length $l([\\alpha]_n)$, we mean the minimal number $l|n$, such that $\\mathcal{R}_f^l([\\alpha]_n)=[\\alpha]_n$. Alternatively, this is the number of fixed point classes in an orbit $\\mathcal{A}\\in \\mathcal{OR}(f^n)$.\n\n\\medskip\n\nIt is immediately clear that $d([\\alpha])\\geq l([\\alpha])$, because every class in an orbit that reduces to depth $d$ will be a fixed point of the map $\\mathcal{R}_f^d$.\n\n\\begin{definition}\nA map $f:X\\to X$ is called \\textbf{essentially toral} if it is essentially reducible and if the following two conditions are fulfilled:\\begin{enumerate}\n\\item For every essential fixed point class in $\\mathcal{R}(f^n)$, the length and depth coincide.\n\\item If $[\\alpha]_n$ is essential and $\\gamma_{nk}([\\beta]_k)=\\gamma_{nk}([\\gamma]_k)=[\\alpha]_n$, then $[\\beta]_k=[\\gamma]_k$.\n\\end{enumerate}\nIf this holds for every self-map on $X$, we will say that $X$ is essentially toral.\n\\end{definition}\n\nBecause lengths and depths coincide, the following lemma follows easily.\n\\begin{lemma}[\\cite{jm06-1}, 5.1.30]\\label{lemclasses}\nIf $f:X\\to X$ is essentially toral, then $NP_n(f)$ equals the number of irreducible essential fixed point classes in $\\mathcal{R}(f^n)$.\n\\end{lemma}\nThis lemma actually tells us that if we are working on an essentially toral space, we are free to replace the orbit theory by a theory in terms of classes. By combining Lemma \\ref{lemNPNF} and Lemma \\ref{lemclasses}, the following can  also be easily deduced.\n\n\\begin{corollary}\\label{coresstor}\nIf $f:X\\to X$ is essentially toral, then $NF_n(f)$ equals the number of irreducible essential fixed point classes in $\\bigcup_{k|n}\\mathcal{R}(f^k)$.\n\\end{corollary}\n\nIn \\cite{hk97-1}, the following theorem is proved.\n\n\\begin{theorem}\nNilmanifolds are essentially reducible to the GCD and essentially toral.\n\\end{theorem}\n\nNote that they actually proved a more general version of this theorem, as they showed that the theorem above also holds for solvmanifolds.\n\n\\begin{definition}\nA map $f:X\\to X$ is called \\textbf{weakly Jiang} if $N(f)=0$ or $N(f)=R(f)$. This means that all fixed point classes are simultaneously essential or inessential.\n\\end{definition}\n\n\\begin{theorem}[\\cite{hk97-1}, Theorem 5.1]\\label{thmhk}\nSuppose that $X$ is essentially toral and essentially irreducible to the GCD. If $f:X\\to X$ is a map such that $f^n$ is weakly Jiang and $N(f^n)\\neq 0$, then $$NF_n(f)=N(f^n)$$and the same formula holds for every divisor of $n$.\n\\end{theorem}\nThe idea behind this proof is very simple. Since every fixed point class at level $n$ is essential, by Lemma \\ref{lemessredgcd}, we know that every such class is preceded by a unique irreducible essential class. On the other hand, every irreducible essential fixed point class in $\\bigcup_{k|n}\\mathcal{R}(f^k)$ has to boost essentially, since there are simply no inessential fixed point classes to boost to at level $n$. Because of these observations, there exist a bijection between the essential fixed point classes at level $n$ and the irreducible essential fixed point classes in $\\bigcup_{k|n}\\mathcal{R}(f^k)$. Corollary~\\ref{coresstor} then proves the theorem.\n\n\\medskip\n\nIt is known that every map on a nilmanifold is weakly Jiang, due to the result of Anosov (\\cite{anos85-1}) or Fadell and Husseini (\\cite{fh86-1}). Consequently, Theorem \\ref{thmhk} holds for nilmanifolds. Unfortunately, not every map on an infra-nilmanifold is weakly Jiang. Even on the Klein bottle, the smallest example of an infra-nilmanifold which is not a nilmanifold, it is possible to find counterexamples.\n\n\\begin{example}\\label{Example not weakly Jiang}\nSuppose we have the following presentation of the Klein bottle group: $$<\\alpha,\\beta | \\alpha \\beta =\\beta^{-1} \\alpha>.$$Let $k\\neq 1$ be odd. Now, let $f_*:\\alpha\\mapsto \\alpha^k, \\textrm{ } \\beta\\mapsto \\beta^{-1}$ be the induced morphism for a map $f$ on the Klein bottle. One can check that this morphism indeed induces a map on the Klein bottle, for which it holds that $R(f)=\\infty$, while $N(f)\\neq 0$.\n\\end{example}\nAn algebraic argument for the fact that maps on nilmanifolds are weakly Jiang, while maps on infra-nilmanifolds are generally not, can be found by combining Theorem \\ref{LeeForm} with Theorem~\\ref{thmRInf} and Theorem \\ref{thmN=R}. When working on nilmanifolds, the formula in Theorem \\ref{LeeForm} reduces to a single determinant. By Theorem \\ref{thmRInf}, we know that this determinant will be equal to $0$ (and hence $N(f)=0$) if and only if $R(f)=\\infty$. By combining this fact with Theorem \\ref{thmN=R}, it follows that nilmanifolds are weakly Jiang. When working on infra-nilmanifolds, the sum generally consists of multiple determinants. Therefore, it is possible that some of these determinants are $0$ and some are not. If this is the case, a similar argument as before will show that the map is not weakly Jiang, as $R(f)=\\infty$, while $N(f)\\neq 0$.\n\\section{Structure on the periodic point classes of infra-nilmanifolds}\nIn this section, we will show that infra-nilmanifolds are both essentially reducible to the GCD and essentially toral. As a result of these structural properties, we will be able to show a theorem similar to Theorem \\ref{thmhk} for semi-hyperbolic maps on infra-nilmanifolds.\n\n\\medskip\n\nWe will prove both of these structural properties for affine maps on infra-nilmanifolds and because the theory described in the previous section is homotopy-invariant, this will be sufficient. As already mentioned before, affine maps are often much easier to deal with. This fact is exemplified in the following proposition, which can be found in \\cite{fl13-1}.\n\n\\begin{proposition}\\label{propaff}\nIf $\\overline{(d,D)}:M\\to M$ is an affine map on an infra-nilmanifold, then every non-empty fixed point class is path-connected and\n\\begin{enumerate}\n\\item every essential fixed point class of $\\overline{(d,D)}$ consists of exactly one point.\n\\item every non-essential fixed point class of $\\overline{(d,D)}$ is empty or consists of infinitely many points.\n\\end{enumerate} \n\\end{proposition}\n\nNow we can prove the two main theorems of this section.\n\n\\begin{theorem}\\label{thmessredgcd}\nInfra-nilmanifolds are essentially reducible to the GCD.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{thmleezhao}, we already know that infra-nilmanifolds are essentially reducible.\n\\medskip\nIt is known from \\cite{ll06-1} that every almost-Bieberbach group $\\Gamma$ has a fully characteristic subgroup $\\Lambda$ of finite index, such that $\\Lambda\\subset G$. Therefore, every infra-nilmanifold of the form $\\Gamma\\backslash G$ is finitely covered by a nilmanifold $\\Lambda\\backslash G$, such that every continuous map $f:\\Gamma\\backslash G\\to \\Gamma\\backslash G$ can be lifted to a map $\\overline{f}:\\Lambda\\backslash G\\to \\Lambda\\backslash G$. All in all, we have the following commuting diagram, where $\\beta_n$ is a covering transformation and $\\overline{\\beta}_n$ is the natural projection of $\\beta_n$ into $\\Gamma\/ \\Lambda$.\n\\begin{displaymath}\n    \\xymatrix{ G \\ar[dd]_p \\ar[dr]_{p'} \\ar[rrrrr]^{\\beta_n \\tilde{f}^n}& & & & & G \\ar[dd]^{p} \\ar[dl]^{p'} \\\\\n    & \\Lambda\\backslash G \\ar[rrr]^{\\overline{\\beta}_n\\overline{f}^n} \\ar[dl]_{\\overline{p}} & & & \\Lambda\\backslash G \\ar[dr]^{\\overline{p}} & \\\\\n               \\Gamma\\backslash G \\ar[rrrrr]^{f^n}& & & &  & \\Gamma\\backslash G  }.\n\\end{displaymath} \n\nSuppose that $g$ is the affine map on $\\Gamma\\backslash G$ that is induced by an affine homotopy lift $\\tilde{g}$ of $f$. Let $p(\\Fix(\\beta_n\\tilde{g}^n))$ be an essential fixed point class on level $n$, such that, for $r,s|n$, this fixed point class reduces to $p(\\Fix(\\beta_r\\tilde{g}^r))$ and $p(\\Fix(\\beta_s\\tilde{g}^s))$.\n\n\\medskip\n\nBecause of Proposition \\ref{propaff}, we know that there exists $x\\in \\Fix(g^n)$, such that all these fixed point classes are equal to the set $\\{x\\}$. The fixed point index is a local property and a covering map is a local homeomorphism, hence, the fixed point class $p'(\\Fix(\\beta_n\\tilde{g}^n))$ is also essential. By using Proposition \\ref{propaff} again, we know this fixed point class will consist of one point, namely a $\\overline{x}\\in \\overline{p}^{-1}(x)$. By a similar reasoning, there will exist $\\gamma_r,\\gamma_s \\in \\Gamma$ and accordingly, $\\overline{\\gamma_r},\\overline{\\gamma_s} \\in \\Gamma \/\\Lambda$ such that $$p'(\\Fix(\\beta_r\\tilde{g}^r))=\\{\\overline{\\gamma_r}\\cdot \\overline{ x}\\}\\textrm{ and }p'(\\Fix(\\beta_s\\tilde{g}^s))=\\{\\overline{\\gamma_s}\\cdot \\overline{ x}\\}.$$An easy calculation then shows that $$p'(\\Fix(\\gamma_r^{-1}\\beta_r g_\\ast^r(\\gamma_r)\\tilde{g}^r))=\\{\\overline{x}\\}\\textrm{ and }p'(\\Fix(\\gamma_s^{-1}\\beta_sg_\\ast^s(\\gamma_s)\\tilde{g}^s))=\\{\\overline{x}\\}.$$This actually means that if we choose good representatives in the Reidemeister classes of $[\\beta_r]_r$ and $[\\beta_s]_s$, $p'(\\Fix(\\beta_n\\tilde{g}^n))$ will reduce to both $p'(\\Fix(\\beta_r\\tilde{g}^r))$ and $p'(\\Fix(\\beta_s\\tilde{g}^s))$ on our nilmanifold $\\Lambda\\backslash G$. Since nilmanifolds are known to be essentially reducible to the GCD, there exists a $\\beta_d$, with $d=\\gcd(r,s)$, such that $p'(\\Fix(\\beta_r\\tilde{g}^r))$ and $p'(\\Fix(\\beta_s\\tilde{g}^s))$ both reduce to  $p'(\\Fix(\\beta_d\\tilde{g}^d))$. By applying $\\overline{p}$ to this fixed point class, the statement is proved.\n\\end{proof}\n\n\\begin{theorem}\\label{thmesstor}\nInfra-nilmanifolds are essentially toral.\n\\end{theorem}\n\\begin{proof}\nAgain, we already know that infra-nilmanifolds are essentially reducible and again, by homotopy-invariance, it suffices to prove this theorem for affine maps $g$.\n\n\\medskip\n\nLet $[\\alpha]_n$ be an essential fixed point class of $g^n$. Since we already know that $d([\\alpha]_n)\\geq l([\\alpha]_n)$, we only need to prove that the strict inequality is impossible. So, suppose that $d=d([\\alpha]_n)> l([\\alpha]_n)=l$. Because of Proposition \\ref{propaff}, we know that there exists $x\\in \\Fix(g^n)$ such that $\\{x\\}$ is the fixed point class associated to $[\\alpha]_n$. Furthermore, $\\{g(x)\\}$ will be the fixed point class associated to $\\mathcal{R}_g([\\alpha]_n)$. By definition and because there is only one fixed point in each essential fixed point class, $g^l(x)=x$. Therefore, $[\\alpha]_n$ reduces to a fixed point class on level $l$, which is a contradiction to the fact that $d>l$. This proves the first condition.\n\n\\medskip\n\nIf $[\\beta]_k$ and $[\\gamma]_k$ are both boosted to $[\\alpha]_n$, then we know that they are both essential fixed point classes. Hence, they both have the set $\\{x\\}$ as associated fixed point class, which means that $[\\beta]_k=[\\gamma]_k$. This proves the second condition of essential torality.\n\\end{proof}\n\nNow we will use these newly obtained structural properties for infra-nilmanifolds to establish a few results concerning Nielsen periodic points. We start with the following definition.\n\n\\begin{definition}\nWe say that an essential fixed point $[\\alpha]_k$ is \\textbf{(in)essentially boosted to level $n$}, if $[\\alpha]_k$ is boosted to an (in)essential fixed point class $[\\beta]_{n}$. \n\\end{definition}\n\nLet us denote the set of all irreducible fixed point classes which are inessentially boosted to level $n$ for a continuous self-map $f$ by $IIB_n(f)$. Note that this is a subset of $\\bigcup_{k|n}\\mathcal{R}(f^k)$, since this set contains all fixed point classes on all levels that will boost to level $n$.\n\n\\begin{proposition}\\label{propIIB}\nWhenever a map $f$ is essentially reducible to the GCD and essentially toral, we have that\n$$NF_n(f)=N(f^n)+\\#IIB_n(f).$$\n\\end{proposition}\n\\begin{proof}\nWhen a map $f$ is essentially toral, we know by Corollary \\ref{coresstor} that $NF_n(f)$ equals the number of irreducible essential classes in $\\bigcup_{k|n}\\mathcal{R}(f^k)$. Now, pick an arbitrary irreducible essential class. We can distinguish two disjoint cases. \n\n\\medskip\n\nOn the one hand, suppose this class boosts essentially to level $n$. As $f$ is essentially reducible to the GCD, we can apply Lemma \\ref{lemessredgcd} and we know that every essential fixed point class reduces to a unique irreducible essential fixed point class. This means that there is a bijection between the irreducible essential classes that are essentially boosted to level $n$ and the essential fixed point classes of $\\mathcal{R}(f^n)$.\n\n\\medskip\n\nIf, on the other hand, our class boosts inessentially to level $n$, it belongs to $IIB_n(f)$. Since both cases are disjoint, the equality follows.\n\\end{proof}\n\nIt is quite easy to see that this proposition is a generalization of Theorem \\ref{thmhk}. In fact the proof is a slightly adapted version where we take inessential boosting into account.\n\n\\begin{theorem}\\label{thmNF=Nf)}\nWhen $f$ is a semi-hyperbolic map on an infra-nilmanifold, then for all $n>0$ $$NF_n(f)=N(f^n).$$\n\\end{theorem}\n\\begin{proof}\nSuppose that $(d,D)$ is an affine homotopy lift of $f$. By combining Theorem \\ref{thmRInf} and Theorem \\ref{thmN=R} we know that every fixed point class on level $n$ is essential if and only if for all $A\\in F$ (where $F$ is the holonomy group of our infra-nilmanifold),$$\\det(I-A_\\ast D_\\ast^n)\\neq 0.$$By Lemma \\ref{Lemma Bram}, we know that there exists $B \\in F$ and an integer $l$, such that $$(B_\\ast D_\\ast^n)^l=D_\\ast^{ln} \\textrm{ and } \\det(I-A_\\ast D_\\ast^n)=\\det(I-B_\\ast D_\\ast^n).$$Note that $\\det(I-B_\\ast D_\\ast^n)=0$ implies that $B_\\ast D_\\ast^n$ has an eigenvalue $1$, but this would mean that $D_\\ast^{ln}$ had an eigenvalue $1$, which is in contradiction with the fact that $f$ is semi-hyperbolic. Therefore, we know that every fixed point class on level $n$ is essential, which implies that $IIB_n(f)$ is the empty set. The theorem then follows from Proposition \\ref{propIIB}.\n\\end{proof}\n\nNote that the proof of this theorem actually also proves the following proposition, since we proved that every fixed point class on every level is essential.\n\n\\begin{proposition}\nWhen $f$ is a semi-hyperbolic map on an infra-nilmanifold, then for all $n>0$, $f^n$ is a weakly Jiang map.\n\\end{proposition}\n\nWith this proposition in mind, one can easily see that Theorem \\ref{thmNF=Nf)} is a special case of Theorem \\ref{thmhk}.\n\n\\medskip\n\nLater on, in the last section, we will show, under mild conditions, that semi-hyperbolic maps are the only maps for which a non-trivial equality $NF_n(f)=N(f^n)$ holds.\n\n\\medskip\n\nTheorem \\ref{thmNF=Nf)} actually has a nice corollary in the area of dynamical zeta functions. By $N_f(z)$, we mean the Nielsen zeta function, as defined in \\cite{fels00-2} ,\\cite{fels88-1} or \\cite{fp85-1}. In \\cite{fels00-2}, the following definition of the  \\textbf{minimal dynamical zeta function} can be found: $$NF_f(z)=\\exp\\left(\\sum_{k=1}^\\infty \\frac{NF_k(f)z^k}{k}\\right).$$\n\n\nWe now have the following corollary.\n\n\\begin{corollary}\nLet $f$ be a semi-hyperbolic map on an infra-nilmanifold, then $N_f(z)=~NF_f(z)$.\n\\end{corollary}\n\nBy using the main result of \\cite{dd13-2}, which states that Nielsen zeta functions are rational for self-maps on infra-nilmanifolds, we can also conclude the following.\n\n\\begin{corollary}\nLet $f$ be a semi-hyperbolic map on an infra-nilmanifold, then $NF_f(z)$ is a rational function.\n\\end{corollary}\n\n\\section{A method for computing $NF_n(f)$}\nIn theory, we are now capable to compute $NF_n(f)$, due to Proposition \\ref{propIIB}. By using the standard formula for Nielsen numbers for maps on infra-nilmanifolds (Theorem \\ref{LeeForm}), the computation of $N(f^n)$ becomes very simple and therefore, the only thing left to check is how many fixed point classes lie in $IIB_n(f)$. \\medskip\n\nIn some cases, for example for semi-hyperbolic maps, the computation of $\\# IIB_n(f)$ becomes trivial. However, in a more general setting, this number can be a very tedious thing to compute. In this section, we will try to develop a method to make this computation a bit easier.\n\n\\subsection{$\\sim_f$-equivalence classes}\nWe start this section with the following definition.\n\\begin{definition}\nLet $f: \\Gamma\\backslash G \\to \\Gamma\\backslash G$ be a continuous map, such that $F$ is the holonomy group of $\\Gamma$. We will say that $A, B \\in F$ are \\textbf{$f$-conjugated}, if there exist $a,b\\in G$ and $\\gamma\\in \\Gamma$ such that $(a,A)$ and $(b, B)$ are elements of $\\Gamma$ and $$\\gamma\\circ (a,A) \\circ f_*(\\gamma^{-1})=(b,B).$$We will write $A\\sim_f B.$\n\\end{definition}\n\nAn alternative for this definition is given in the following lemma. In general, the definition will be more useful when one quickly wants to find elements that are $f$-conjugated. The lemma below is often more useful when it comes to finding properties of the $\\sim_f$-relation. \n\\begin{lemma}\\label{lemma alternatief definitie}\nLet $f: \\Gamma\\backslash G \\to \\Gamma\\backslash G$ be a continuous map, such that $F$ is the holonomy group of $\\Gamma$. Then $A\\sim_f B$ if and only if for all $(a,A)\\in \\Gamma$, there exist $(b,B), \\gamma \\in \\Gamma$, such that$$\\gamma\\circ (a,A) \\circ f_*(\\gamma^{-1})=(b,B).$$\n\\end{lemma} \n\\begin{proof}\nOne direction is obvious. For the other direction, pick an arbitrary $(a,A)\\in \\Gamma$ and suppose that $A\\sim_f B$. This means that there exist $a_0, b_0 \\in G$ and $\\gamma_0 \\in \\Gamma$, such that $(a_0,A),(b_0,B) \\in \\Gamma$ and $$\\gamma_0\\circ (a_0,A) \\circ f_*(\\gamma_0^{-1})=(b_0,B).$$Then $$\\gamma_0\\circ (a,A) \\circ f_*(\\gamma_0^{-1})=\\left(\\gamma_0\\circ (a_0,A) \\circ f_*(\\gamma_0^{-1})\\right) \\circ \\left(f_*(\\gamma_0) \\circ (A^{-1}(a_0^{-1}a), \\Id) \\circ f_*(\\gamma_0^{-1})\\right).$$As $\\Gamma \\cap G$ is a normal divisor of $\\Gamma$, there exists a $(c,\\Id)\\in \\Gamma\\cap G$, such that $$\\gamma_0\\circ (a,A) \\circ f_*(\\gamma_0^{-1})=(b_0,B)\\circ (c,\\Id).$$\n\\end{proof}\n\nA simple consequence of the previous lemma is the following.\n\\begin{corollary}\n$\\sim_f$ is an equivalence relation.\n\\end{corollary}\n\\begin{proof}\nThe fact that this relation is reflexive and symmetric is easy to see, so the only thing left to prove is the transitivity. Suppose that $A\\sim_f B$ and $B\\sim _f C$. By definition, there exist $a, b \\in G$ and $\\gamma_1 \\in \\Gamma$, such that $(a,A), (b,B)\\in \\Gamma$ and $$\\gamma_1\\circ (a,A) \\circ f_*(\\gamma_1^{-1})=(b,B).$$By Lemma \\ref{lemma alternatief definitie}, we know there exist $\\gamma_2, (c,C) \\in \\Gamma$, such that $$\\gamma_2\\circ (b,B) \\circ f_*(\\gamma_2^{-1})=(c,C).$$By combining both equations, we see$$(\\gamma_2 \\circ \\gamma_1)\\circ (a,A)\\circ f_*((\\gamma_2\\circ \\gamma_1)^{-1})=(c,C),$$which means that $A\\sim_f C$.\n\\end{proof}\n\nThe fact that $\\sim_f$ is an equivalence relation implies that we can partition $F$ into \\textbf{$\\sim_f$-equivalence classes}. There is an even more convenient way to look at the $\\sim_f$-equivalence classes for which we only need to work in the holonomy group $F$ of our infra-nilmanifold. This will be the ideal tool to compute these classes in a more effective way.\n\n\\begin{definition}\nBy $f_\\#(\\Id)$, we mean the set of all $A\\in F$, such that there exist $(g,\\Id), (a,A)\\in \\Gamma$, for which $f_*(g,\\Id)=(a,A)$. Analogously, $f_\\#(C)$ is the set of all $A\\in F$, such that there exist $(c,C), (a,A)\\in \\Gamma$, for which $f_*(c,C)=(a,A)$.\n\\end{definition}\n\nNote that it is known that $\\Gamma\\cap G$ is finitely generated. With this in mind, we can deduce the following lemma. \n\n\\begin{lemma}\\label{lemma f-equivalentieklassen}\nPick an arbitrary $(c,C)\\in \\Gamma$. Again, $p:\\Aff(G)=G\\semi \\Aut(G) \\to \\Aut(G)$ denotes the natural projection onto the second factor of the semi-direct product. Suppose that $(g_i,\\Id)_{i=1}^n$ is a set of generators for $\\Gamma\\cap G$. Then we can describe $f_\\#(\\Id)$ and $f_\\#(C)$ as follows:\n\\begin{itemize}\n\\item $f_\\#(\\Id)=\\grp\\{p(f_*(g_i,\\Id))\\}.$\n\\item $f_\\#(C)=p(f_*(c,C))f_\\#(\\Id)=f_\\#(\\Id)p(f_*(c,C)).$\n\\end{itemize}\n\n\\end{lemma}\n\\begin{proof}\nIt is clear that $f_\\#(\\Id)$ contains all elements $p(f_*(g_i,\\Id))$ and it is also clear that $f_*(\\Id)$ is precisely the set $p(f_*(\\Gamma\\cap G))=p(f_*(\\grp \\{ (g_i, \\Id)\\ \\| \\ i=1\\dots n\\}))$. As $p\\circ f_*$ is a morphism, this will be equal to $\\grp\\{p(f_*( g_i, \\Id)) \\ \\| \\ i=1\\dots n\\}$, which proves the first statement.\n\n\\medskip\n\nTake an arbitrary element of the form $(c_1, C)\\in \\Gamma$. It is clear that $p(f_*(c_1,C))\\in f_\\#(C)$. Now, an easy computation shows that $$p(f_*(c_1,C))=p(f_*(c,C))p(f_*(c,C)^{-1})p(f_*(c_1,C))=p(f_*(c,C))p(f_*(C^{-1}(c^{-1}c_1),\\Id)).$$As $p(f_*(C^{-1}(c^{-1}c_1),\\Id))\\in f_\\#(\\Id)$, the first equality of second statement is proved. The second equality can be proved in a similar way, by multiplying with  $p(f_*(c,C)^{-1})p(f_*(c,C))$ on the right.\n\\end{proof}\n\nAs a side remark, note that an easy consequence of this lemma is the fact that if $p\\circ f_*:\\Gamma\\to F$ is a surjective morphism, then $f_\\#(\\Id)$ will be a normal divisor of $F$.\n\n\\medskip \n\nBy using this lemma, we can derive an easier way of determining $\\sim_f$-equivalence classes.\n\n\\begin{proposition}\nSuppose $A,B$ are elements in $F$. Then, $A\\sim_f B$ if and only if there exists a $C\\in F$, such that $B\\in CAf_\\#(C)^{-1}$. Here $f_\\#(C)^{-1}$ denotes the set of all inverses of elements in $f_\\#(C)$, or equivalently $f_\\#(C^{-1})$.\n\\end{proposition}\n\\begin{proof}\nOne direction is obvious. For the other direction, suppose that there exists a $C\\in F$, such that $B\\in CAf_\\#(C)^{-1}$. Then, there exist $a,c\\in G$, such that $(c,C), (a,A)\\in \\Gamma$. By Lemma~\\ref{lemma f-equivalentieklassen}, any element in $f_\\#(C)^{-1}$ will come from an element of the form $f_*(c,C)^{-1}f_*(g,\\Id)$, with $(g,\\Id)\\in \\Gamma$. As a result, we find that there exists a $b\\in G$, such that $$(c,C)(a,A)f_*(c,C)^{-1}f_*(g,\\Id)=(b,B).$$Note that $(b,B)$ will also be an element of $\\Gamma$. By multiplying both sides on the left with $(g^{-1},\\Id)$ (which is also in $\\Gamma$), we get$$ (g^{-1},\\Id)(c,C)(a,A)f_*((g^{-1},\\Id)(c,C))^{-1}=(g^{-1},\\Id)(b,B)=(g^{-1}b,B).$$This proves that $A\\sim_f B$.\n\n\\medskip\n\nIn order to see that $f_\\#(C^{-1})=f_\\#(C)^{-1}$, note that Lemma \\ref{lemma f-equivalentieklassen} tells us that $f_\\#(\\Id)$ is a group and that $f_\\#(C)=p(f_*(c,C))f_\\#(\\Id)=f_\\#(\\Id)p(f_*(c,C))$, from which this fact follows immediately, as $$f_\\#(C)^{-1}=f_\\#(\\Id)^{-1}p(f_*(c,C))^{-1}=f_\\#(\\Id)p(f_*(c,C)^{-1})=f_\\#(C^{-1}).$$\n\\end{proof}\n\n\\begin{corollary}\nThe $\\sim_f$-equivalence class of $A$ equals the set $$\\bigcup_{C\\in F} CAf_\\#(C)^{-1}.$$\n\\end{corollary}\n\n\\begin{corollary}\\label{corollary D invertible}\nLet $(d,D)$ be an affine homotopy lift of a continuous map $f$ on an infra-nilmanifold $\\Gamma\\backslash G$. When $D$ is invertible in $\\Endo(G)$, for any $C\\in F$, $f_\\#(C)$ will be a singleton.\n\\end{corollary}\n\\begin{proof}\nTake an arbitrary element $(g,\\Id)$ of $\\Gamma\\cap G$. Then $f_*(g,\\Id)\\circ (d,D)=(d,D)\\circ (g,\\Id)$ and hence also, $p(f_*(g,\\Id))\\circ D=D$. As $D$ is invertible, $p(f_*(g,\\Id))=\\Id$. As $(g,\\Id)$ was chosen arbitrarily, this means $f_\\#(\\Id)=\\{\\Id\\}$. By Lemma \\ref{lemma f-equivalentieklassen}, $f_\\#(C)$ is also a singleton.  \n\\end{proof}\n\n\\subsection{Properties of $\\sim_f$-equivalence classes}\nA first sign that shows that elements in the same $\\sim_f$-equivalence class are strongly connected, can be found in the following lemma.\n\n\\begin{lemma}\\label{lemdet}\nWhen $A\\sim_f B$ and $(d,D)$ is an affine homotopy lift of $f$, then $$\\det(I-A_\\ast D_\\ast)=\\det(I-B_\\ast D_\\ast).$$\n\\end{lemma}\n\\begin{proof}\nAs $A\\sim_f B$, there exists a $(c,C) \\in \\Gamma$, such that $$(c,C)\\circ(a,A)\\circ f_*(c, C)^{-1}=(b,B).$$Of course, we can compose both sides with $(d,D)$. As a result, we get the following equality$$(c,C)\\circ(a,A)\\circ (d,D)\\circ (c,C)^{-1}=(b,B)\\circ(d,D).$$Therefore, we have $CADC^{-1}=BD$ in $\\Aut(G)$. From this, the statement follows easily.\n\\end{proof}\n\nThe following theorem is in a certain sense the heart of our computational method. It splits the formula from Theorem \\ref{LeeForm} into several parts, one for each $\\sim_f$-equivalence class. The proof is heavily influenced by the proofs in \\cite{kll05-1} and \\cite{ll06-1}.\n\n\\begin{theorem}\\label{thmNAf}\nLet $f:\\Gamma\\backslash G \\to \\Gamma\\backslash G$ be a continuous map on an infra-nilmanifold, with affine homotopy lift $(d,D)$ and holonomy group $F$. Let $(G,p)$ be a universal covering of $\\Gamma\\backslash G$, such that $\\tilde{f}$ is a reference lifting of $f$. If we fix $A \\in F$, then, the number of essential fixed point classes that can be written as $p(\\Fix((a,A)\\circ \\tilde{f}))$, which we will denote by $N_A(f)$, equals $$\\frac{1}{\\#F}\\sum_{B\\sim_f A}|\\det(I-B_\\ast D_\\ast)|.$$\n\\end{theorem}\n\\begin{proof}\nFrom now on, $\\Lambda$ will be the fully characteristic subgroup of $\\Gamma$ described in \\cite{ll06-1}. Similarly to Theorem \\ref{thmessredgcd}, we have the following commuting diagram:\n\n\\begin{displaymath}\n    \\xymatrix{ G \\ar[dd]_p \\ar[dr]_{p'} \\ar[rrrrr]^{(a,A) \\circ \\tilde{f}}& & & & & G \\ar[dd]^{p} \\ar[dl]^{p'} \\\\\n    & \\Lambda\\backslash G \\ar[rrr]^{\\overline{(a,A)} \\circ \\overline{f}} \\ar[dl]_{\\overline{p}} & & & \\Lambda\\backslash G \\ar[dr]^{\\overline{p}} & \\\\\n               \\Gamma\\backslash G \\ar[rrrrr]^{f}& & & &  & \\Gamma\\backslash G  }.\n\\end{displaymath} \n\nFor $\\alpha \\in \\Gamma$, we will denote the Reidemeister class of $\\alpha$ in $\\Gamma$ by $[\\alpha]_\\Gamma$. Now, define an equivalence relation $\\sim{_\\Lambda}$ on $\\Gamma$ as follows:$$\\alpha\\sim{_\\Lambda}\\beta \\text{ iff } \\exists \\lambda\\in \\Lambda: \\beta=\\lambda\\circ \\alpha\\circ f_*(\\lambda)^{-1}.$$In a similar way as before, $[\\alpha]_\\Lambda$ will denote the equivalence class with respect to $\\sim_\\Lambda$ that contains $\\alpha$. It is straightforward to prove that $\\beta\\in [\\alpha]_\\Lambda$ implies that $p'(\\Fix(\\beta\\circ \\tilde{f}))=p'(\\Fix(\\alpha\\circ \\tilde{f}))$. In a similar way, one can prove that $\\beta\\not \\in [\\alpha]_\\Lambda$ implies that $p'(\\Fix(\\beta\\circ \\tilde{f}))\\cap p'(\\Fix(\\alpha\\circ \\tilde{f}))=\\emptyset$. Note that this can also mean that $p'(\\Fix(\\beta\\circ \\tilde{f}))$ and $ p'(\\Fix(\\alpha\\circ \\tilde{f}))$ are fixed point classes for different maps on the nilmanifold $\\Lambda\\backslash G$. Now, by labeling the possibly empty fixed point sets, we can say that there is a one-to-one relation between the sets $[\\alpha]_\\Lambda$ and the fixed point classes $p'(\\Fix(\\alpha\\circ \\tilde{f}))$ of liftings of $f$ to $\\Lambda\\backslash G$. Let us denote the set of $\\sim_\\Lambda$-equivalence classes by $\\mathcal{R}_\\Lambda(f)$ and the set of Reidemeister classes of $f$ by $\\mathcal{R}(f)$. As $\\Lambda\\leq \\Gamma$, we know that the map $$\\Psi:\\mathcal{R}_\\Lambda(f)\\to \\mathcal{R}(f): [\\alpha]_\\Lambda\\mapsto [\\alpha]_\\Gamma$$is a well-defined function, which is clearly surjective. We also know that $p(\\Fix(\\alpha\\circ \\tilde{f}))$ is essential if and only if $p'(\\Fix(\\alpha\\circ \\tilde{f}))$ is essential, because the fixed point index is a local property and $\\overline{p}$ is a local homeomorphism. Hence, when $[\\alpha]_\\Gamma$ is (in)essential, then every element in $\\Psi^{-1}([\\alpha]_\\Gamma)$ corresponds to an (in)essential fixed point class of a lift of $f$ to $\\Lambda\\backslash G$. Because of this property and the fact that $\\Psi$ is surjective and well-defined, we know that \\begin{equation}\\label{ineq}\nN_A(f)\\leq \\sum_{\\begin{subarray}{c}\n\\overline{(b,B)}\\in \\Gamma \/ \\Lambda \\\\ B\\sim_f A\n\\end{subarray}}N(\\overline{(b,B)}\\circ \\overline{f}).\n\\end{equation}\n\n\\medskip\n\n\nSuppose that $\\mathbb{F}$ is a fixed point class of the desired form. Then $$\\mathbb{F}=p(\\Fix((b,B)\\circ \\tilde{f})=[(b,B)]_\\Gamma,$$with $B\\sim_f A$.\nWhen $\\F$ is an inessential fixed point class, $[(b,B)]_\\Lambda$ corresponds to an inessential fixed point class of the map $\\overline{(b,B)}\\circ \\overline{f}$ on the nilmanifold $\\Lambda\\backslash G$. Due to the main result from \\cite{anos85-1} or \\cite{fh86-1}, we now know that every fixed point class of $\\overline{(b,B)}\\circ \\overline{f}$ is inessential, so that $N(\\overline{(b,B)}\\circ \\overline{f})=0$. This also means that $\\det(I-B_\\ast D_\\ast) =0$. By Lemma \\ref{lemdet} we know that $\\det(I-C_\\ast D_\\ast)=0$ for all $C\\sim_f B$, or equivalently, for all $C\\sim_f A$. This also means that $N(\\overline{(c,C)}\\circ \\overline{f})=0$. By definition, $N_A(f)$ is a non-negative integer and hence, it follows by inequality (\\ref{ineq}) that $$N_A(f)=0=\\frac{1}{\\#F}\\sum_{B\\sim_f A}|\\det(I-B_\\ast D_\\ast)|.$$\n\nNow, suppose that $\\F=[(b,B)]_\\Gamma$ is an essential fixed point class. The fact that$$N_A(f)\\leq \\sum_{\\begin{subarray}{c}\n\\overline{(b,B)}\\in \\Gamma \/ \\Lambda \\\\ B\\sim_f A\n\\end{subarray}}N(\\overline{(b,B)}\\circ \\overline{f})$$is not necessarily an equality comes from the fact that $\\Psi$ is not injective. This is due to situations where$$[(b,B)]_\\Gamma=[(c,C)]_\\Gamma \\text{, while } [(b,B)]_\\Lambda\\neq [(c,C)]_\\Lambda.$$So, in order to find $N_A(f)$, we need to find the number of elements in $\\mathcal{R}_\\Lambda(f)$ which are mapped to the same element of $\\mathcal{R}(f)$ by $\\Psi$. First, we will show that this number has $\\left|\\Gamma\/ \\Lambda\\right|$ as an upper bound. Suppose that $\\overline{\\gamma}_1=\\overline{\\gamma}_2 \\in \\Gamma\/ \\Lambda$, then $\\gamma_2=\\lambda\\circ \\gamma_1$, for $\\lambda\\in \\Lambda$. If $$(c_1,C_1)=\\gamma_1 \\circ (b,B) \\circ f_*(\\gamma_1^{-1}) \\textrm{ and } (c_2,C_2)=\\gamma_2 \\circ (b,B) \\circ f_*(\\gamma_2^{-1}),$$then an easy computation shows that $$(c_2,C_2)=\\lambda\\circ(c_1,C_1)\\circ f_*(\\lambda^{-1}),$$which means that  $[(c_1,C_1)]_\\Lambda= [(c_2,C_2)]_\\Lambda$.\n\n\\medskip\n\nNow we will show that this upper bound is always attained by showing that $[(c_1,C_1)]_\\Lambda= [(c_2,C_2)]_\\Lambda$ implies that $\\overline{\\gamma}_1=\\overline{\\gamma}_2$ in $\\Gamma \/ \\Lambda$. Let $(d,D)$ be an affine homotopy lift of $f$. Suppose there exist a $\\lambda \\in \\Lambda$, such that $$(c_1,C_1)=\\gamma_1\\circ (b,B)\\circ  f_*(\\gamma_1^{-1})=\\lambda\\circ (\\gamma_2\\circ (b,B)\\circ  f_*(\\gamma_2^{-1}))\\circ f_*(\\lambda^{-1})=\\lambda\\circ (c_2,C_2)\\circ f_*(\\lambda^{-1}).$$As an easy consequence, $$(b,B)=(\\gamma_1^{-1}\\circ \\lambda\\circ \\gamma_2)\\circ (b,B)\\circ f_*(\\gamma_1^{-1}\\circ \\lambda\\circ \\gamma_2)^{-1}.$$Note that $p(\\Fix((b,B)\\circ \\tilde{f}))$ is an essential fixed point class and therefore $p(\\Fix((b,B)\\circ (d,D)))$ will also be an essential fixed point class. Hence, there exists an $x\\in G$ such that $(b,B)\\circ (d,D)(x)=x$. This also means that $$(\\gamma_1^{-1}\\circ \\lambda\\circ \\gamma_2)\\circ (b,B)\\circ f_*(\\gamma_1^{-1}\\circ \\lambda\\circ \\gamma_2)^{-1}\\circ (d,D) (x)=x,$$which implies that $(\\gamma_1^{-1}\\circ \\lambda\\circ \\gamma_2)^{-1}\\cdot x$ is also in $p(\\Fix((b,B)\\circ (d,D)))$. By Proposition \\ref{propaff}, we know that such a fixed point class is a singleton and hence, $(\\gamma_1^{-1}\\circ \\lambda\\circ \\gamma_2)^{-1}\\cdot x=x$. By the free action of $\\Gamma$ on $G$, this implies that $\\lambda\\circ \\gamma_2=\\gamma_1$.\n\n\\medskip\n\nNow, we know that $\\Psi$ maps $\\left|\\Gamma\/ \\Lambda\\right|$ different elements of $\\mathcal{R}_\\Lambda(f)$ to the element $[(b,B)]_\\Gamma$. As $\\F$ was chosen arbitrarily, we know that this holds for every essential fixed point class $[(c,C)]_\\Gamma$ in $\\mathcal{R}(f)$, for which $C\\sim_f A$. So, this means that $$N_A(f)=\\frac{1}{[\\Gamma:\\Lambda]}\\sum_{\\begin{subarray}{c}\n\\overline{(b,B)}\\in \\Gamma \/ \\Lambda \\\\ B\\sim_f A\n\\end{subarray}}N(\\overline{(b,B)}\\circ \\overline{f}).$$In a similar way as in the proof of Theorem 3.4 in \\cite{ll06-1}, we can now derive that$$N_A(f)=\\frac{1}{\\#F}\\sum_{B\\sim_f A}|\\det(I-B_\\ast D_\\ast)|.$$\n\\end{proof}\n\n\n\\begin{remark}\\label{remark essential of inessential}\nDuring the proof of this theorem, we actually also proved that the fixed point class $p(\\Fix((a,A)\\circ \\tilde{f})) $ is essential if and only if $\\det(I-A_\\ast D_\\ast)\\neq 0$. This is due to the fact that $p(\\Fix((a,A)\\circ \\tilde{f}))$ can be lifted to a fixed point class $p'(\\Fix((a,A)\\circ \\tilde{f}))$ with the same index. As this is a fixed point class for a map on a nilmanifold, we can use the result from \\cite{anos85-1} or \\cite{fh86-1}, which tells us that $p'(\\Fix((a,A)\\circ \\tilde{f})) $ is essential if and only if $\\det(I-A_\\ast D_\\ast)\\neq 0$.\n\\end{remark}\n\nIt might be noteworthy to mention that a fixed point class $p(\\Fix((b,B)\\circ \\tilde{f}))$ can be written as $p(\\Fix((a,A)\\circ \\tilde{f}))$ if and only if $A\\sim_f B$. So, in a certain sense, it is justified to say that $N_A(f)$ is the number of essential fixed point classes \\textbf{above the $\\sim_f$-equivalence class of $A$}. We will denote this equivalence class by $[A]$. Also note that every fixed point class above $A$ is simultaneously essential or inessential. Hence, it makes sense to talk about the (in)essential $\\sim_f$-equivalence class $[A]$. We can also generalize these notions when considering $\\sim_{f^{k}}$-equivalence classes of $A$. For these classes, we will use the notation $[A]_k$.\n\n\\medskip\n\nSome easy corollaries of Theorem \\ref{thmNAf} are the following.\n\n\\begin{corollary}\n$$N_A(f)=\\frac{\\#\\{B\\in F\\|B\\sim_{f} A\\}}{\\# F}\\cdot |\\det(I-A_\\ast D_\\ast)|.$$\n\\end{corollary}\n\\begin{proof}\nThis follows easily by combining Lemma~\\ref{lemdet} and Theorem~\\ref{thmNAf}.\n\\end{proof}\n\n\\begin{corollary}\nIf all elements of $F$ are in the same $\\sim_f$-equivalence class, then $$N(f)=N_A(f)=|\\det(I-A_\\ast D_\\ast)|=|\\det(I- D_\\ast)|=|L(f)|.$$\n\\end{corollary}\n\nThis condition is for example satisfied when $f_*:\\Gamma\\to \\Gamma$ maps every element into $\\Gamma\\cap G$. If this is the case, then $f_*$ induces the trivial morphism on the holonomy group, from which it follows easily that all elements in $F$ are in the same $\\sim_f$-equivalence class.\n\n\\begin{corollary}\nIf every $\\sim_{f}$-equivalence class in $F$ consists of a single element, then for all $A\\in F$, $\\det(I-A_\\ast D_\\ast)$ will be divisible by $\\#F$.\n\\end{corollary}\n\nThis is for example the case when $f_*$ induces the identity morphism on $F$, while $F$ itself is an abelian group. For instance, in Example \\ref{example Z6}, in section \\ref{examples}.\n\n\\subsection{$\\sim_f$-equivalence classes on different levels}\n\n\\begin{definition}\\label{definition boosting equivalence classes}\nLet $A\\in F$ be an element of the holonomy group $F$ of $\\Gamma\\backslash G$. Let $k|n$. Then we define $\\gamma_{nk}(A)$ to be the following subset of $F$:$$\\{C\\in F \\| \\textrm{ there exists } (a,A)\\in \\Gamma, \\textrm{such that } (c,C)=\\gamma_{nk}(a,A)=(a,A)f_*^k(a,A)\\dots f_*^{n-k}(a,A)\\}.$$\n\\end{definition}\n\nWith this definition in mind, we can prove the following lemma.\n\n\\begin{lemma}\\label{lemma boosting funct on f-conjugacy classes}\nIf $A\\sim_{f^k} B$, then for any $ C \\in \\gamma_{nk}(A)$, there exists a $D\\in \\gamma_{nk}(B)$ such that $C \\sim_{f^n} D$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{lemma alternatief definitie}, for any $(a,A) \\in \\Gamma$, there exist $(b,B), \\gamma \\in \\Gamma$, such that $$\\gamma\\circ (a,A)\\circ f_*^k(\\gamma^{-1})=(b,B).$$Because we picked $(a,A)$ arbitrarily, $C$, with $(c,C)=\\gamma_{nk}(a,A)$, is also chosen arbitrarily in the set $\\gamma_{nk}(A)$. Now, the following $(d,D)$ fulfills the necessary conditions:$$(d,D)=(b,B)f_*^k(b,B)\\dots f_*^{n-k}(b,B).$$Indeed, by using the relation $\\gamma\\circ (a,A)\\circ f_*^k(\\gamma^{-1})=(b,B)$, we see$$(d,D)=(\\gamma\\circ (a,A)\\circ f_*^k(\\gamma^{-1}))f_*^k(\\gamma\\circ (a,A)\\circ f_*^k(\\gamma^{-1}))\\dots f_*^{n-k}(\\gamma\\circ (a,A)\\circ f_*^k(\\gamma^{-1})).$$Since $f_*$ is a morphism, a simple computation shows that $$(d,D)=\\gamma\\circ (c,C) \\circ f_*^n(\\gamma^{-1}).$$\n\\end{proof}\n\nWith Definition \\ref{definition boosting equivalence classes}, we actually try to define boosting functions in terms of $\\sim_f$-equivalence classes. This might not necessarily be well-defined, in the sense that it might happen that for $A \\sim_{f^k} B$, not every element in $\\gamma_{nk}(A)$ is automatically $\\sim_{f^n}$-conjugated with every element in $\\gamma_{nk}(B)$. Note that Lemma \\ref{lemma boosting funct on f-conjugacy classes} tells us that this would be the case if every element in $\\gamma_{nk}(A)$ is in the same $\\sim_{f^n}$-equivalence class. Because of Corollary \\ref{corollary D invertible}, we know that $\\gamma_{nk}(A)$ will be a singleton whenever $D$ is invertible, so in that case, these boosting functions are well-defined on the equivalence classes. Note that it might also not necessarily be true that a  $\\gamma_{nk}([A]_k)$, by which we mean the set $$\\bigcup_{B\\sim_{f^{k}}A}\\gamma_{nk}(B),$$is a full $\\sim_{f^n}$-equivalence class.\n\\medskip\n\nAlthough boosting functions might generally not behave well on $\\sim_{f^k}$-equivalence classes, we can still use them as a tool for the computation of $NF_n(f)$. The reason for this is the fact that two fixed point classes above the same equivalence class $[A]$ will boost in the exact same way. So, if we know how one fixed point class behaves, every fixed point class above the same equivalence class will behave in the same way.\n\n\\medskip\n\nThe following lemma is actually all we need to prove the statement above.\n\\begin{lemma}\\label{lemma determinants for boosted}\nLet $(d,D)$ be an affine homotopy lift of $f$. If $B, C\\in \\gamma_{nk}(A)$, then $$\\det(I-B_\\ast D_\\ast^n)=\\det(I-C_\\ast D_\\ast^n).$$\n\\end{lemma}\n\\begin{proof}\nSince there exist $(a_1,A)$ and $(a_2,A)$, such that $\\gamma_{nk}(a_1,A)=(b,B)$ and $\\gamma_{nk}(a_2,A)=(c,C)$, for certain $b,c\\in G$, we know that $$((a_1,A)(d,D)^k)^{\\frac{n}{k}}=(b,B)(d,D)^n \\textrm{ and }((a_2,A)(d,D)^k)^{\\frac{n}{k}}=(c,C)(d,D)^n.$$So, by just looking at the rotational part, we see $$BD^n=(AD^k)^{\\frac{n}{k}}=CD^n.$$By taking the differential, we obtain the desired result.\n\\end{proof}\n\n\n\\begin{proposition}\\label{prop similar boosting}\nSuppose $p(\\Fix((a,A)\\circ \\tilde{f}^k))$ is a fixed point class at level $k$ of a continuous map $f$ on an infra-nilmanifold. If this fixed point class boosts (in)essentially to level $n$, then every fixed point class of the form $p(\\Fix((b,B)\\circ \\tilde{f}^k))$, with $B\\in [A]_k$ also boosts (in)essentially to level $n$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $(d,D)$ is an affine homotopy lift of $f$. By Remark \\ref{remark essential of inessential}, we know that $p(\\Fix((a,A)\\circ \\tilde{f}^k))$ is an essential fixed point class if and only if $\\det(I-A_\\ast D_\\ast^k)\\neq 0$. Take an arbitrary fixed point class $p(\\Fix((b,B)\\circ \\tilde{f}^k))$, with $B\\in [A]_k$. Because of Lemma \\ref{lemdet}, we already know that $\\det(I-B_\\ast D_\\ast^k)\\neq 0$ and that $p(\\Fix((b,B)\\circ \\tilde{f}^k))$ is an essential fixed point class.\n\n\\medskip\n\nNow suppose that $\\gamma_{nk}(a,A)=(c,C)$ and $\\gamma_{nk}(b,B)=(e,E)$. This means that $C\\in \\gamma_{nk}(A)$ and $E\\in \\gamma_{nk}(B)$. By Lemma \\ref{lemma boosting funct on f-conjugacy classes}, we know that there exists $E_0\\in \\gamma_{nk}(B)$, such that $C\\in [E_0]_n$. Lemma \\ref{lemdet} now tells us that $$\\det(I-C_\\ast D_\\ast^n)=\\det(I-E_{0 \\ast} D_\\ast^n).$$By Lemma \\ref{lemma determinants for boosted} and by the fact that $E,E_0\\in \\gamma_{nk}(B)$, we also know that $$\\det(I-E_\\ast D_\\ast^n)=\\det(I-E_{0 \\ast} D_\\ast^n).$$Because $\\det(I-C_\\ast D_\\ast^n)=\\det(I-E_\\ast D_\\ast^n)$, we know that $[(a,A)]_k$ boosts essentially to level $n$ if and only if $[(b,B)]_k$ boosts essentially to level $n$. A similar thing applies for inessential boosting.\n\\end{proof}\n\n\\subsection{Examples}\\label{examples}\n\nSo, we know that every fixed point class above $[A]$ boosts in exactly the same way and we also know that the number of essential fixed point classes above $[A]$ equals $N_A(f)$. This is a tool we can use to compute $\\# IIB_n(f)$ in a more efficient way. We show how this can be done by looking at a few examples.\n\n\\begin{example}\nLet us first try to compute the Nielsen periodic numbers of the maps in Example~\\ref{Example not weakly Jiang}. We will use the matrix description from \\cite{duga14-1}.\nLet the Klein bottle group be generated by the following two affine transformations:\n$$\\alpha=(a,A)=\\left(\\begin{pmatrix}\n\\frac{1}{2}\\\\\n\\frac{1}{2}\\\\\n\\end{pmatrix}, \\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\\right) \\textrm{ and } \\beta=(e_2, \\Id),$$where $e_2$ denotes the second element of the standard basis of $\\R^2$. Suppose that $k\\neq 1$ is odd and that $p\\in \\R$. Then, the map induced by $$(d,D)=\\left(\\begin{pmatrix}\np\\\\\n\\frac{1}{2}\\\\\n\\end{pmatrix}, \\begin{pmatrix}\nk&0\\\\\n0&-1\n\\end{pmatrix}\\right)$$will induce the same morphism described in Example \\ref{Example not weakly Jiang}.\n\n\\medskip\n\nIt is clear that $D$ commutes with both $A$ and $\\Id$, so $f_*$ induces the identity morphism on $F$ and therefore, every $\\sim_{f^n}$-equivalence class consists of precisely one element. Also, it easy to compute that $[A]_l$ is essential if and only if $l$ is even, while $[\\Id]_l$ is essential if and only if $l$ is odd. Now, suppose that $m=ql$, then a simple computation shows that $[\\Id]_l$ always boosts to $[\\Id]_m$. Also, $[A]_l$ will boost to $[A]_m$ if $q$ is odd and to $[\\Id]_m$ if $q$ is even. The reason for this, lies in the following computation:$$(AD^l)^q=A^qD^{ql}=A^qD^m.$$All together, we see that every even boost ($q$ is even) of an essential fixed point class is inessential while every odd boost is essential.\n\n\\medskip\n\nAs a consequence, we see that if $n$ is odd, $IIB_n(f)$ is the empty set and by Proposition \\ref{propIIB}, it follows that $$NF_n(f)=N(f^n).$$On the other hand, if $n$ is even, the only essential fixed point classes that boost inessentially to level $n$ pass through level $\\frac{n}{2}$. Every essential fixed point class at this level will boost inessentially to level $n$ and every element in $IIB_{\\frac{n}{2}}(f)$ will also boost inessentially to level $n$. Therefore: $$NF_n(f)=N(f^n)+NF_{\\frac{n}{2}}(f).$$\n\nFor the sake of completeness, the case where $k=1$ also gives us a map on the Klein bottle. In this case, an easy computation shows that $N(f^n)=0$ for every $n$. As a consequence, every fixed point class at every level is inessential and hence $NF_n(f)=0$, for every integer $n$.\n\\end{example}\n\nThe following example will illustrate the use of $\\sim_f$-equivalence classes a little more. \n\n\\begin{example}\\label{example Z3}\nLet $\\Gamma$ be the Bieberbach group with generators:$$(a,A)=\\left(\\begin{pmatrix}\n0\\\\\n0\\\\\n\\frac{1}{3}\\\\\n\\end{pmatrix}, \\begin{pmatrix}\n-1&1&0\\\\\n -1 & 0 & 0\\\\\n 0&0&1 \\\\\n\\end{pmatrix}\\right)\n\\textrm{, }(e_1, \\Id)\\textrm{ and }(e_2, \\Id).$$In \\cite{duga14-1} one can find that the affine map $$(d,D)=\\left(\\begin{pmatrix}\n0\\\\\n0\\\\\n0\\\\\n\\end{pmatrix}, \\begin{pmatrix}\n0&1&0\\\\\n 1 & 0 & 0\\\\\n 0&0&2 \\\\\n\\end{pmatrix}\\right)$$induces a continuous map on the flat manifold $\\Gamma\\backslash\\R^3 $.\n\n\\medskip\n\nAn easy computation shows that $DA=A^2D$ and that $f_*$ induces a morphism $\\overline{f}_*$ on the holonomy group $\\Z_3$, such that $\\overline{f}_*^2=\\Id$. So, whenever $k$ is even, every $\\sim_{f^k}$-equivalence class is a singleton.\n\\medskip\n\nTo determine $[\\Id]_k$, with $k$ odd, note that $\\overline{f}_*^k=\\overline{f}_*$. Also, $\\overline{f}_*(A)=A^2$ and $\\overline{f}_*(A^2)=A$. So, the following are certainly subsets of $[\\Id]_k$:\n$$A\\Id f_\\#(A)^{-1}=\\{A^2\\} \\textrm{ and } A^2\\Id f_\\#(A^2)^{-1}=\\{A\\}.$$Hence, $[\\Id]_k=F$ for all odd $k$. \n\n\\medskip\n\nAn easy computation shows that $[\\Id]_k$, with $k$ odd, is always inessential. As a consequence, for every odd $n$, $$NF_n(f)=N(f^n)=0.$$When $k$ is even, $[\\Id]_k$ is inessential, while $[A]_k$ and $[A^2]_k$ are essential. In this case, every element of $F$ commutes with $D^k$, as $\\overline{f}_*^2$ is the identity morphism. Hence, the class $[A^i]_k$ boosts to the class $[A^{ip}]_{pk}$. So, an essential class can only boost inessentially if $p\\equiv 0 \\mod 3$. \n\n\\medskip\n\nIn Figure \\ref{fig1}, a scheme can be found where all these boosting relations are shown up to level $6$. In this scheme, inessential and essential fixed point classes are denoted by a circle and a square respectively. Only the boosting from an essential to an inessential class are drawn, since these are the only ones that need to be considered for the computation of $NF_n(f)$.\n\n\\begin{figure}\n\n\\centering\n\\begin{tikzpicture}[line cap=round,line width = .5pt,line join=round, >=triangle 45, x=1.5cm,y=1.5cm]\n\n\\draw[color=black](0,1.2) circle (0.3);\n\\draw[color=black](1.2,1.2) circle (0.3);\n\\draw[color=black](.9,-.9) rectangle (1.5,-1.5);\n\\draw[color=black](.9,-.3) rectangle (1.5,.3);\n\\draw[color=black](2.4,1.2) circle (0.3);\n\\draw[color=black](3.6,1.2) circle (0.3);\n\\draw[color=black](3.3,-.9) rectangle (3.9,-1.5);\n\\draw[color=black](3.3,-.3) rectangle (3.9,.3);\n\\draw[color=black](4.8,1.2) circle (0.3);\n\\draw[color=black](6,1.2) circle (0.3);\n\\draw[color=black](5.7,-.9) rectangle (6.3,-1.5);\n\\draw[color=black](5.7,-.3) rectangle (6.3,.3);\n\n\\draw[color=black] (0,1.2) node {$[\\Id]_1$};\n\\draw[color=black] (1.2,1.2) node {$[\\Id]_2$};\n\\draw[color=black] (1.2,0) node {$[A]_2$};\n\\draw[color=black] (1.2,-1.2) node {$[A^2]_2$};\n\\draw[color=black] (2.4,1.2) node {$[\\Id]_3$};\n\\draw[color=black] (3.6,1.2) node {$[\\Id]_4$};\n\\draw[color=black] (3.6,0) node {$[A]_4$};\n\\draw[color=black] (3.6,-1.2) node {$[A^2]_4$};\n\\draw[color=black] (4.8,1.2) node {$[\\Id]_5$};\n\\draw[color=black] (6,1.2) node {$[\\Id]_6$};\n\\draw[color=black] (6,0) node {$[A]_6$};\n\\draw[color=black] (6,-1.2) node {$[A^2]_6$};\n\n\\draw[->,color=black] (1.55,0) -- (3.25,0);\n\\draw[->,color=black] (1.55,-1.2) -- (3.25,-1.2);\n\\draw[->,color=black] (1.55,.1) .. controls (3,.6) and (5,.6) .. (5.7,1.1);\n\\draw[->,color=black] (1.55,-1.1) .. controls (3.6,-.7) .. (5.75,1);\n\\end{tikzpicture}\n\n\\caption{A scheme of $\\sim_{f^k}$-equivalence classes at different levels for Example \\ref{example Z3}} \\label{fig1}\n\\end{figure}\n\n\\medskip\n\nSo, suppose that $n=3^pq$ is even, such that $\\gcd(3,q)=1$, then $$NF_n(f)=\\sum_{i=0}^{p} N(f^{3^iq}).$$\n\\end{example}\n\nBy the following example, we clarify the use of Theorem \\ref{thmNAf}.\n\n\\begin{example}\\label{example Z6}\nLet $\\Gamma$ be the Bieberbach group with generators:\n$$(a,A)=\\left(\\begin{pmatrix}\n0\\\\\n0\\\\\n\\frac{1}{6}\n\\end{pmatrix}, \\begin{pmatrix}\n1 & -1 &0\\\\\n1& 0 &0 \\\\\n0&0 & 1\n\\end{pmatrix}\\right), (e_1,\\Id)\\textrm{ and }  (e_2,\\Id).$$\n\nIn \\cite{duga14-1}, one can find that the following affine map induces a map on the infra-nilmanifold $\\Gamma\\backslash \\R^3$:\n\n$$(d,D)=\\left(\\begin{pmatrix}\n0\\\\\n0\\\\\n0\n\\end{pmatrix}, \\begin{pmatrix}\n0 & 1 &0\\\\\n-1& 1 &0 \\\\\n0&0 &7\n\\end{pmatrix}\\right).$$\n\nNote that every element of $F$ commutes with $D$, so every $\\sim_f$-equivalence class consists of precisely one element. This also means that the class $[A^i]_k$ boosts to the class $[A^{ip}]_{pk}$. It is also quite easy to compute that $[A^p]_k$ is inessential if and only if $p\\equiv k \\mod 6$. This boosting scheme can be found in Figure \\ref{fig2}.\n\n\\begin{figure}\n\n\\centering\n\\begin{tikzpicture}[line cap=round,line width = .5pt,line join=round, >=triangle 45, x=1.8cm,y=1.5cm]\n\n\\draw[color=black](0,2) circle (0.3);\n\\draw[color=black](-.3,.9) rectangle (.3,1.5);\n\\draw[color=black](-.3,.1) rectangle (.3,.7);\n\\draw[color=black](-.3,-.7) rectangle (.3,-.1);\n\\draw[color=black](-.3,-1.5) rectangle (.3,-.9);\n\\draw[color=black](-.3,-2.3) rectangle (.3,-1.7);\n\n\\draw[color=black](2.1,1.7) rectangle (2.7,2.3);\n\\draw[color=black](2.4,1.2) circle (0.3);\n\\draw[color=black](2.1,.1) rectangle (2.7,.7);\n\\draw[color=black](2.1,-.7) rectangle (2.7,-.1);\n\\draw[color=black](2.1,-1.5) rectangle (2.7,-.9);\n\\draw[color=black](2.1,-2.3) rectangle (2.7,-1.7);\n\n\\draw[color=black](4.5,1.7) rectangle (5.1,2.3);\n\\draw[color=black](4.5,.9) rectangle (5.1,1.5);\n\\draw[color=black](4.8,.4) circle (0.3);\n\\draw[color=black](4.5,-.7) rectangle (5.1,-.1);\n\\draw[color=black](4.5,-1.5) rectangle (5.1,-.9);\n\\draw[color=black](4.5,-2.3) rectangle (5.1,-1.7);\n\\draw[color=black] (0,2) node {$[A]_1$};\n\\draw[color=black] (0,1.2) node {$[A^2]_1$};\n\\draw[color=black] (0,.4) node {$[A^3]_1$};\n\\draw[color=black] (0,-.4) node {$[A^4]_1$};\n\\draw[color=black] (0,-1.2) node {$[A^5]_1$};\n\\draw[color=black] (0,-2) node {$[\\Id]_1$};\n\n\\draw[color=black] (2.4,2) node {$[A]_2$};\n\\draw[color=black] (2.4,1.2) node {$[A^2]_2$};\n\\draw[color=black] (2.4,.4) node {$[A^3]_2$};\n\\draw[color=black] (2.4,-.4) node {$[A^4]_2$};\n\\draw[color=black] (2.4,-1.2) node {$[A^5]_2$};\n\\draw[color=black] (2.4,-2) node {$[\\Id]_2$};\n\n\\draw[color=black] (4.8,2) node {$[A]_3$};\n\\draw[color=black] (4.8,1.2) node {$[A^2]_3$};\n\\draw[color=black] (4.8,.4) node {$[A^3]_3$};\n\\draw[color=black] (4.8,-.4) node {$[A^4]_3$};\n\\draw[color=black] (4.8,-1.2) node {$[A^5]_3$};\n\\draw[color=black] (4.8,-2) node {$[\\Id]_3$};\n\n\\draw[->,color=black] (.35,1.15) -- (2.05,-.3);\n\\draw[->,color=black] (.35,.35) -- (2.05,-1.9);\n\\draw[->,color=black] (.35,-.35) -- (2.05,1.2);\n\\draw[->,color=black] (.35,-1.1) -- (2.05,-.4);\n\\draw[->,color=black] (.35,-1.95) -- (2.05,-2);\n\n\n\\draw[->,color=black] (.35,1.2) .. controls (3,2) .. (4.45,-1.9);\n\\draw[->,color=black] (.35,.4) .. controls (3,1) .. (4.45,.5);\n\\draw[->,color=black] (.35,-.4) .. controls (2,-2.5) and (3,-1) .. (4.45,-2);\n\\draw[->,color=black] (.35,-1.2) .. controls (2,-.1) and (3.7,-1.8) .. (4.5,.2);\n\\draw[->,color=black] (.35,-2.05) .. controls (2.4,-2.5) .. (4.45,-2.05);\n\n\\end{tikzpicture}\n\n\\caption{A scheme of $\\sim_{f^k}$-equivalence classes at the lowest levels for Example \\ref{example Z6}} \\label{fig2}\n\\end{figure}\n\n\\medskip\n\nThe only classes at level $1$ that boost inessentially to level $2$ (to $[A^2]_2$), are the inessential class $[A]_1$ and the essential class $[A^4]_1$. It is therefore clear that $\\# IIB_2(f)=N_{A^4}(f)$. By Theorem \\ref{thmNAf} and Proposition \\ref{propIIB}:$$NF_2(f)=N(f^2)+\\frac{|\\det(I-A^4D)|}{6}.$$\nIn a similar way, one can see that the only classes that boost to $[A^3]_3$ are the inessential class $[A]_1$ and the essential classes $[A^3]_1$ and $[A^5]_1$. Hence,$$NF_3(f)=N(f^3)+ \\frac{|\\det(I-A^3D)|}{6}+ \\frac{|\\det(I-A^5D)|}{6}.$$\nComputing $NF_4(f)$ becomes a little more tricky, since fixed point classes at both level $1$ and $2$ can boost to inessential fixed point classes of level $4$. With an easy computation, we see that the classes that boost to $[A^4]_4$ are $[A]_1,[A^4]_1,[A^2]_2$ and $[A^5]_2$. Note that we already knew that $[A]_1$ is inessential and that the essential class $[A^4]_1$ boosts inessentially to $[A^2]_2$. Therefore, there are no essential classes at level $1$ that boosts essentially to level $2$ and inessentially to level $4$. This means that $$NF_4(f)=N(f^4)+ \\frac{|\\det(I-A^5D^2)|}{6}+\\frac{|\\det(I-A^4D)|}{6}.$$\nAs $[A^i]_k$ boosts to $[A^{ip}]_{pk}$, we know that this boosting relation is a bijection between the $\\sim_{f^k}$-equivalence classes and the $\\sim_{f^{pk}}$-equivalence classes if and only if $p$ is invertible modulo $6$. Now, suppose $n>0$ is an integer, such that $\\gcd(n,6)=1$. Note that every divisor of $n$ will also be relatively prime to $6$. Because there is only one inessential class at each level and because there is a bijection between the classes at different levels and because maps on infra-nilmanifolds are essentially reducible, every essential class that boosts to level $n$ will do so in an essential way. Therefore, if $\\gcd(n,6)=1$, $$NF_n(f)=N(f^n).$$Whenever $n$ has many prime factors $2$ and $3$, it will be much harder to compute $NF_n(f)$, because many inessential boosts occur and we have to keep track of all these boostings in order to not count some of them multiple times. As an example, let us compute $NF_6(f)$. Note that $[\\Id]_6$ is the only inessential class at level $6$. The classes that boost to $[\\Id]_6$ are $[\\Id]_3, [A^3]_3, [\\Id]_2, [A^2]_2,[A^4]_2$ and all classes at level $1$. The only essential classes at level $1$ that boost to essential classes at both level $2$ and level $3$, are $[\\Id]_1$ and $[A^2]_1$. Also, there are no essential classes at level $1$ that boost to inessential classes at both level $2$ and level $3$. Hence,\n\\begin{equation}\\nonumber\n\\begin{split}\nNF_6(f)=N(f^6)+\\frac{|\\det(I-D^3)|}{6}+\\frac{|\\det(I-D^2)|}{6}+\\frac{|\\det(I-A^4D^2)|}{6}\\\\\n-\\frac{|\\det(I-D)|}{6}-\\frac{|\\det(I-A^2D)|}{6}.\n\\end{split}\n\\end{equation}\nHere, these last two terms are precisely the number of essential fixed point classes at level $1$ that boost essentially to level $2$ and level $3$. As they are counted double, we have to subtract them once.\n\\end{example}\n\nAs one can see from this last example, it can be very hard to compute $NF_n(f)$. The tools in this section are useful, but they still require a lot of manual labor. Looking at these examples, it is not unthinkable that there might not exist a general formula for $NF_n(f)$.\n\n\\section{Some properties of affine maps on infra-nilmanifolds}\nIn the last section of this paper, we will look specifically at affine maps on infra-nilmanifolds in order to derive a nice property of these maps (Theorem \\ref{theorem uiteindelijk boost inessentieel}).\n\n\\begin{lemma}\\label{lemma alternatief lemma bram 3.1}\nIf $A,B \\in \\GL_n(\\C)$ and $D\\in \\C^{n\\times n}$, such that $DA=BD$, then, for all $n>0$ it holds that $$\\det(I-(AD)^n)=\\det(I-(BD)^n).$$\n\\end{lemma}\n\\begin{proof}\nUsing the multiplicative properties of the determinant, we find the following equalities:$$\\det(I-(AD)^n)=\\det(A^{-1})\\det(I-(AD)^n)\\det(A)=\\det(I-(DA)^n)=\\det(I-(BD)^n).$$\\end{proof}\n\nRemember that a continuous map $f$ will be called \\textbf{Wecken} if and only if $\\#\\Fix(f)=N(f)$.  \n\n\\begin{theorem}\\label{theorem uiteindelijk boost inessentieel}\nLet $\\overline{(d,D)}$ be an affine map on an infra-nilmanifold. Suppose that there exists at least one $k$, for which $N(\\overline{(d,D)}^k)\\neq 0$. Then, $D_\\ast$ is semi-hyperbolic if and only if $\\overline{(d,D)^k}$ is a Wecken map for every $k$.   \n\\end{theorem}\n\\begin{proof}\nFirst, suppose that $D_\\ast$ is semi-hyperbolic. Just like in the proof of Theorem \\ref{thmNF=Nf)}, we know that every fixed point class at every level is essential. By Proposition \\ref{propaff}, it follows that $\\overline{(d,D)^k}$ is a Wecken map.\n\n\\medskip\n\nOn the other hand, suppose that $D_\\ast$ is not semi-hyperbolic. This means that there exists an eigenvalue $\\lambda$ of $D_\\ast$, such that $\\lambda^d=1$. Now we will show that every essential fixed point class will eventually be boosted to an inessential fixed point class. Pick an essential fixed point class $[(a,A)]_k$. This is possible, because not all Nielsen numbers are $0$. Let $l$ be an arbitrary positive integer and set $m=kl$. Consider the fixed point class $\\gamma_{km}([(a,A)])$. This coincides with the set $p(\\Fix(((a,A)\\circ(d,D)^k)^l))$. Now suppose that $$(b,B)\\circ (d,D)^{m}=((a,A)\\circ(d,D)^k)^l.$$This means that $BD^{m}=(AD^k)^l$ and by Remark~\\ref{remark essential of inessential}, we now know that $\\gamma_{km}([(a,A)])$ is inessential if and only if $$\\det(I-(A_\\ast D_\\ast^k)^l)=\\det(I-B_\\ast D_\\ast^{m})=0.$$By combining Lemma~\\ref{lemma alternatief lemma bram 3.1} and Lemma \\ref{Lemma Bram}, we see there exists a $C\\in F$ and a positive integer $p$, such that $$\\det(I-(A_\\ast D_\\ast^k)^l)=\\det(I-(C_\\ast D_\\ast^k)^l)\\textrm{ and } (C_\\ast D_\\ast^k)^p=D_\\ast^{kp}.$$By taking $l=\\lcm(p,d)$, we know that $(C_\\ast D_\\ast^k)^l=D_\\ast^{kl}$. Also, $1$ is an eigenvalue of $D_\\ast^{kl}$. By combining all of the above, we see that $$\\det(I-B_\\ast D_\\ast^{m})=\\det(I-(A_\\ast D_\\ast^k)^l)=\\det(I-(C_\\ast D_\\ast^k)^l)=\\det(I-( D_\\ast^k)^l)=0.$$As there is certainly one essential fixed point class $[(a,A)]_k$, we know that it will boost to an inessential fixed point class $[(b,B)]_m$. This actually means that $[(a,A)]_k\\subset [(b,B)]_m$, which implies that the inessential fixed point class $[(b,B)]_m$ is non-empty, which implies that $\\overline{(d,D)}^m$ is not a Wecken map.\n\\end{proof}\n\n\\begin{corollary}\\label{corWecken}\nSuppose that there exists at least one $k$, such that $N(\\overline{(d,D)}^k)\\neq 0$. Whenever $\\Fix(\\overline{(d,D)}^k)$ is finite for every $k$, $\\overline{(d,D)}$ will be Wecken at every level and $D_\\ast$ is semi-hyperbolic.\n\\end{corollary}\n\\begin{proof}\nDue to Proposition~\\ref{propaff}, we know that every non-empty inessential fixed point class contains infinitely many fixed points.\n\\end{proof}\n\n\\begin{corollary}\\label{cor NF_n(f)>Nfn}\nSuppose that $f$ is a continuous map on an infra-nilmanifold that is not semi-hyperbolic. Suppose that $N(f^k)\\neq 0$ for at least one $k$. Then, at certain levels, there exist non-empty inessential fixed point classes. Also, there exist $n>0$, such that $$NF_n(f)>N(f^n).$$\n\\end{corollary}\n\\begin{proof}\nBy examining the proof of Theorem~\\ref{theorem uiteindelijk boost inessentieel}, we see that there will exist an essential fixed point class which boosts to an inessential fixed point class. Therefore, this inessential fixed point class will be non-empty. On the other hand, due to Proposition~\\ref{propIIB}, the second statement follows.\n\\end{proof}\n\n\\begin{corollary}\\label{cor NF=Nf}\nSuppose that $f$ is a continuous map on an infra-nilmanifold. Then, $NF_n(f)=N(f^n)$, for all $n$ if and only if $f$ is a semi-hyperbolic map or $N(f^n)=0$, for all $n$.\n\\end{corollary}\n\\begin{proof}\nWhen dealing with semi-hyperbolic maps, the statement follows from Theorem \\ref{thmNF=Nf)} and Corollary \\ref{cor NF_n(f)>Nfn}. Hence, the only thing left to prove is that $NF_n(f)=0$ if $N(f^n)=0$ for all $n$. This actually follows from Proposition \\ref{propIIB}. As all fixed point classes at all levels are inessential, we know that $\\#IIB_n(f)=0$. As we already knew that $N(f^n)=0$, it follows by Proposition \\ref{propIIB} that $NF_n(f)=0$.\n\\end{proof}\n\nAgain, we can translate some of these results into comparable results concerning dynamical zeta functions.\n\n\\begin{corollary}\nSuppose that $f$ is continuous map on an infra-nilmanifold. Then, $NF_f(z)=N_f(z)$ if and only if $f$ is a semi-hyperbolic map or $N(f^n)=0$, for all $n$.\n\\end{corollary}\n\\begin{proof}\nThis follows immediately from Corollary \\ref{cor NF=Nf}.\n\\end{proof}\n\nWe can actually say something more about another zeta function. In \\cite{fels88-1}, the following dynamical zeta function was defined: $$M_g(z)=\\exp\\left(\\sum_{k=1}^\\infty \\frac{\\#\\Fix(g^k)z^k}{k}\\right).$$When working with affine semi-hyperbolic maps on infra-nilmanifolds, we actually know how this zeta function looks, due to the following result.\n\n\\begin{corollary}\nSuppose that $g$ is an affine semi-hyperbolic map on an infra-nilmanifold, then $$M_g(z)=NF_g(z).$$ \n\\end{corollary}\n\\begin{proof}\nEvery such a map is a Wecken map on every level, due to Theorem \\ref{theorem uiteindelijk boost inessentieel}. From this it follows, for all $k>0$, that $$\\#\\Fix(g^k)=N(g^k)=NF_k(g).$$ \n\\end{proof}\n\nThis result partially answers a question asked in \\cite{jezi03-01} in the case of infra-nilmanifolds. Given a map $f$ on a manifold, the author of this paper asked if it would be possible to find a map $g$ homotopic to $f$, such that $$\\#\\Fix(g^k)=NF_k(g)=NF_k(f),$$for all $k$. This question is equivalent to asking whether there exists a map $g$, homotopic to $f$, such that $M_g(z)$ and $NF_g(z)$ coincide.\n\n\\section*{Acknowledgements}\n\n\nI would like to thank my advisor Karel Dekimpe for all useful comments on the first versions of this paper.\n\n\\medskip\n\nThis work was supported by the research fund KU Leuven.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acknowledgement}\n\\section{Acknowledgement}\nWe thank the reviewers for their valuable comments. The work was supported by the National Key Research and Development Program of China (No. 2019YFB1704003), the National Nature Science Foundation of China (No. 62021002 and No. 71690231), NSF under grants III-1763325, III-1909323, III-2106758, SaTC-1930941, Tsinghua BNRist and Beijing Key Laboratory of Industrial Bigdata System and Application.\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.4]{figures\/example.pdf}\n   \n    \\caption{Example images and their corresponding scene graphs. Given the query, the original scene graph (left) is modified to be the target scene graph (right). }\n    \\label{fig:example}\n   \n\\end{figure}\nA scene graph is a structural representation that captures the semantics of visual scenes by encoding object instances, attributes of objects, and relationships between objects.\n~\\citep{Johnson2015ImageRU}. As shown in Figure~\\ref{fig:example}, the scene graph encodes objects (e.g.\\ ``\\textit{Boy}'', ``\\textit{Racket}''), attributes (e.g.\\ ``\\textit{Girl is standing}''), and relations (``\\textit{Boy holding racket}''). Scene graphs are able to capture the interactions between text and images by associating objects in the graph with regions of an image and modeling the relations between objects. Therefore, it has been used in the cross modality task such as image retrieval, image captioning, and visual question answering~\\citep{Schuster2015GeneratingSP, shi2019explainable, YangTZC19, Wang2020CrossmodalSG}. \n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.63]{figures\/framework.pdf}\n   \n    \\caption{Examples of basic operations INSERT and DELETE for scene graph modification. $Q$ denotes the textual query, $G_{S}$ denotes the source scene graph, $G_{T}$ denotes the target scene graph and $G_{I}$ is the extended graph. }\n    \\label{fig:ise}\n   \n\\end{figure*}\n\nRecently, modifying the scene graph based on the input becomes an emerging research direction as cross-modal systems may need to resort to an interactive process through multiple iterations~\\citep{Ramnath2019SceneGB,He2020SceneGM}. Take text-based image retrieval as an example, users start with a query describing the main objects or topics they are looking for, then modify the query to add more constraints or provide additional information based on previous search results. Instead of directly manipulating images, scene graphs can be used to convert the image-editing problem into a graph-editing problem, conditioned on the textual query. As shown in Figure~\\ref{fig:example}, given a retrieved image from the last turn, if the user wants to see a girl rather than a boy holding a racket, he will enter the query ``\\textit{I would like to see a girl holding racket}'' to the system. According to the query, the object ``\\textit{Boy}'' in the original scene graph will be substituted with the object ``\\textit{Girl}''. The target image can be retrieved given the updated scene graph. The key challenge in this process is how to modify the corresponding partial structure in the original scene graph based on understanding the natural language query. \n\n\n\n\n\n\n\n\nPrior effort framed this scene graph modification (SGM) task as conditional graph generation~\\citep{He2020SceneGM}, where the scene graph is generated from the scratch condition on the original graph and query~\\citep{You2018GraphRNNGR,Guo2019DenselyCG,Cai2020GraphTF}. However, rebuilding the entire scene graph may not be an optimal solution, as the model has to generate the partial structure of the original graph that should be unmodified. Moreover, nodes and edges of the scene graph are constructed separately in their proposed framework, which generates all the nodes first then attaches edges between generated nodes in the second pass. Such an approach may lead to the lack of the modeling capability of interactions between node prediction and edge prediction.\n\nInstead of rebuilding the whole scene graph, we introduce a novel formulation for SGM -- incremental structure expanding (ISE), which is able to build the target graph by gradually expanding the original structure.  At each step, ISE generates the connecting edges between the existing nodes and the newly generated node, upon which the type of the new node is jointly decided. Based on the formalism, our proposed model is able to iterate between finding the relevant part in the query and reading the partially constructed scene graph, inferring more accurate and harmonious expansion decisions progressively. \nExperiments on three SGM benchmarks demonstrate the effectiveness of the proposed approach, which is able to outperform previous state-of-the-art models by large margins. To test the ability of a model under a complex scenario, we further construct a more challenging dataset from the remote sensing domain~\\citep{lu2017exploring}, which has much more modification operations based on the more complicated queries compared with the existing scene graph modification datasets. \nOur key contributions are summarized as follows: \n\n\\begin{itemize}\n    \\item We propose a novel formulation for scene graph modification, allowing incremental expansion of the source scene graph rather than the regeneration of the target graph.\n    \\item We further construct a challenging dataset that contains more complicated queries and larger scene graphs. Extensive experiments on four SGM datasets show the effectiveness of our proposed approach.\n    \\item  Experiments on four benchmarks demonstrate the effectiveness of our approach, which surpasses the previous state-of-the-art model by large margins. \n\\end{itemize}\n\n\n\n\n\n\n\n\n\\section{Incremental Structure Expanding}\n\\label{sec:ise}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.67]{figures\/model.pdf}\n    \\caption{Overview of the model architecture. }\n    \\label{fig:model}\n   \n\\end{figure*}\n\nIn scene graph modification, a node or multiple nodes can be inserted to, deleted from or replaced with other nodes in the scene graph. \\citet{He2020SceneGM} defined the scene graph modification task as a conditional graph generation problem. Formally, given the source scene graph $G_{S}$ and the natural language query $Q$, the target scene graph $G_{T}$ is generated by maximizing the conditional probability $p(G_{T}|G_{S}, Q)$. \n\nInstead of generating the entire target graph $G_{T}$, we frame the task as an incremental structure expanding, which extends the source scene graph $G_{S}$ one node at a time, as well as the edges associated with the node. Such a formulation does not require the model to rebuild the unmodified structure of the source scene graph.\n\nUnder this formulation, we first define two basic operations: INSERT and DELETE. Scene graph modification can be viewed as combining and applying these two operations multiple times. Formally, given the query $Q$, a sequence of $n$ operations ${a_{1}, a_{2}, ..., a_{n}}$ are selected from a set of graph modification operations $\\mathcal{A}$ = \\{INSERT, DELETE\\}. After applying the operations to the source scene graph $G_{s}$, the target scene graph $G_{t}$ is derived. Each operation is defined as: \n\n\\begin{itemize}\n    \\item \\textbf{INSERT}: A new node $o$ is added to $G_{s}$, and edges are attached between $o$ and existing nodes in $G_{s}$. As shown in Figure~\\ref{fig:ise} (a), the node ``\\textit{Ball}'' is added to $G_{s}$ and an edge between ``\\textit{Ball}'' and ``\\textit{Holding}'' is attached, according to the query ``\\textit{holding a racket and a ball}''.\n    \\item \\textbf{DELETE}: As shown in Figure~\\ref{fig:ise} (b). A node $o$ is removed from $G_{s}$, as well as its associated edges. As shown in Figure~\\ref{fig:ise} (b), the node ``\\textit{Blue}'' is removed from $G_{s}$ and the edge between ``\\textit{Racket}'' and ``\\textit{Blue}'' is removed either, according to the query ``\\textit{a racket not a blue}''.\n\\end{itemize}\n\n\n\n\nInspired by incremental parsing~\\citep{Nivre2004IncrementalityID,DyerBLMS15,Cai2020AMRPV,zhang2021crowdsourcing,zhang2022identifying}, we design a data structure called extended graph $G_{I}$, which can be used to model INSERT and DELETE under the graph expansion setting. As shown in Figure~\\ref{fig:ise} (a), the extended graph $G_{T}$ is identical with the target graph $G_{T}$ after applying INSERT. As for DELETE, we introduce a dummy node ``Delete'', which is attached to the node in the source graph $G_{S}$ that should be removed. For example, the dummy node ``Delete'' is attached to the node ``Blue'' in $G_{T}$. In the postprocessing stage, nodes attached with the dummy node ``Delete'' will be removed. Using this formulation, we are able to model scene graph modification by incrementally expanding the source graph $G_{S}$ to the extended graph $G_{I}$, which can be converted to the target graph $G_{T}$ without any losses. \n\nIf the modification requires multiple operations, there will exist multiple node orderings. Take node substitution as an example, replacing a node $o_{i}$ with $o_{j}$ in $G_{s}$ can be viewed as DELETE the node $o_{i}$ first, then INSERT the node $o_{j}$, or vice versa. In practice, we impose that the DELETE operation always comes before INSERT, then the breadth-first search is used to define a deterministic node ordering. \n\n\n\\section{Model Architecture}\n\\label{sec:model}\n\n\n\nIn this section, we will present the model based on the incremental structure expanding formulation. Figure~\\ref{fig:model} gives an overview of the proposed model, which consists of five components including query encoder, graph encoder, feature fusion, edge decoder and node decoder. \n\n\n\\paragraph{Query Encoder} This module is used to encode the query $Q$ by generating the representation of each token of it.\n\n\\paragraph{Graph Encoder} This module is used to encode the graph by generating the representation of each node of it. Note that the representations of the graph are constructed incrementally during the expanding progresses based on the updated graph of the last time step. The graph is the source graph $G_{S}$ at the first timestep.\n\n\\paragraph{Feature Fusion} this module aims to combine the representations from query and graph encoder, then served as a writable memory, which is updated based on the information from edge and node decoder during the incremental expansion. \n\n\\paragraph{Edge Decoder} this module is used to predict the edges between the newly generated node and existing nodes of the graph, then update the memory of the feature fusion module with edge information.\n\n\\paragraph{Node Decoder} this module is used to generate a new node of the graph, then update the memory of the feature fusion module with node information.\n\n\n\n\n\n\\subsection{Query Encoder \\& Graph Encoder}\nFor fair comparisons with the previous work~\\citep{He2020SceneGM}, our query encoder and graph encoder are based on the vanilla transformer~\\citep{VaswaniSPUJGKP17}, which consists of multi-head self attention (MSA) and position-wise feed-forward network (FFN) blocks. The FFN contains two layers with a ReLU non-linearity. Layer normalization (\\citealt{BaKH16}) is applied before every block, and residual connections~\\citep{HeZRS16} after every block. \n\nFormally, given an input query $Q$ with $n$ tokens, each token embedding is randomly initialized and positional encoding is added to the token embedding to retain positional information. The resulted embeddings are denotes as $\\mathbf{x} = \\{x_0, x_1,..., x_n\\}$. Similar to BERT~\\citep{DevlinCLT19}, a special token is appended to the query as $x_{0}$ for sentence encoding. Transformations in the query encoder can be denoted as:\n\n\\begin{align}\n    \\mathbf{x}^{l^{\\prime}} = LN(MSA(\\mathbf{x}^{l-1}) + \\mathbf{x}^{l-1}), \\\\\n    \\mathbf{x}^{l} = LN(FFN(\\mathbf{x}^{l^{\\prime}}) + \\mathbf{x}^{l^{\\prime}}).\n\\end{align}\n\nAfter stacking $L$ blocks, we obtained the contextualized token representations from the query encoder, denoted as $\\{x_0^L, x_1^L,..., x_n^L\\}$. The first vector $x_{0}$ is treated as the sentence-level representation of the query and will be used as the initial state during expansion. For clarity, we denote the vectors as $\\mathbf{x}$$\\in$$\\mathbb{R}^{(n+1) \\times d}$, where $d$ is the dimension. \n\n\nAs for the graph encoder, we treat the input graph as a sequence of nodes in the chronological order of when they are inserted into the graph as discussed in Section~\\ref{sec:ise}. Formally, given the graph $G_{t}$ at the time step $t$, we take its node sequence $\\{o_1, o_2, ..., o_{t-1}\\}$ as the input. A transformer architecture is also applied to obtain the contextualized node embeddings. Notice that the contextualized representation of the graph is constructed incrementally as the expanding progress. Therefore, we apply the vanilla transformer with masked self-attention as the graph encoder, which only allows each position in the node sequence to attend to all positions up to and including that position. For brevity, we denoted the resulted contextualized node representations as $\\mathbf{y}$$\\in$$\\mathbb{R}^{m \\times d}$.\n\n\n\n\n\\subsection{Feature Fusion}\nUnlike the conventional sequence-to-sequence model that only has one encoder, our model contains two encoders. Previous work~\\citep{He2020SceneGM} proposed to use gating mechanism and cross attention to combine the representations of resulted representations from query and graph encoders. We choose to use vanilla multi-head attention mechanism~\\citep{VaswaniSPUJGKP17} to fuse the features from these encoders. Formally, at each time step $t$, the feature fusion component combines the query and graph representations for gradually locating and collecting the most relevant information for the next expansion:\n\n\\begin{align}\n    z_t^{l} = LN(MSA(h_t^{l-1}, \\mathbf{x}) + h_t^{l-1}), \\\\\n    z_t^{l^{\\prime}} = LN(MSA(z_t^{l}, \\mathbf{y}) + z_t^{l}), \\\\\n    h_t^{l} = LN(FFN(z_t^{l^{\\prime}}) + z_t^{l^{\\prime}}).\n\\end{align}\n\nThe initial expansion state of $h_t^{0}$ is initialized with $x_{0}$. For clarity, we denote the last hidden state $h_{t}^{L}$ as $h_{t}$, which is the expansion state at the time step $t$. We now proceed to present the details of each decision stage of one expansion step.\n\n\\subsection{Edge Decoder}\nAt the $t$-th time step, the edge decoder takes the expansion state $h_{t}$ from the feature fusion module and the contextualized representation $\\mathbf{y}$ from the graph encoder as the inputs, and predicts which nodes in the current graph should be attached to the new node. Inspired by~\\citet{CaiL19a} and~\\citet{Cai2020AMRPV}, we leverage multi-head attention and take the maximum over different heads as the final edge probabilities.\nFormally, given $h_t$ and $\\mathbf{y}$, a set of attention weights can be obtained by using multi-head attention mechanism:\n$\\{ \\alpha_{t}^{g_{i}}\\}_{i=1}^{k}$, where $k$ is the number of attention heads and $\\alpha_{t}^{g_{i}}$ is the $i$-th probability vector. The probability of the edge between the new node and the node $o_{j}$ is then computed by $\\alpha_{t}^{g} = max_{i}(\\alpha_{t}^{g_{i}})$. Intuitively, each head is in charge of a set of possible relations (though not explicitly specified).  The maximum pooling reflects that the edge should be built once one relation is activated. \n\nFinally, the edge decoder passes the edge information to the feature fusion module by updating the expansion state $h_{t+1}$ as follows:\n\n\\begin{align}\n    h_{t+1} = LN(MSA(h_{t},\\mathbf{y}) + h_{t} ).\n\\end{align}\n\n\n\\subsection{Node Decoder}\nThe node decoder needs to look at the input query and determine which tokens are the most important ones. This choice is a weighted matrix that gives an attention probability between each token in the query and generated nodes in the target graph. \nConcretely, a single-head attention $\\alpha_{t}^{s}$\nis computed based on the state $h_t$ and the sentence representation $s_{1:n}$, where $\\alpha_{t}^{s}$ denotes the attention weight of the word $w_i$ in the current time step.\nThis component then updates the parser state with the alignment information via the following equation:\n\\begin{align}\n    h_{t+1} = LN(MSA(h_{t},\\mathbf{x}) + h_{t} ).\n\\end{align}\n\nWe then compute the probability distribution of the new node through a hybrid of two channels. The new node can either be a DELETE node or a token copied from the input query. First, $h_{t}$ is fed through a $softmax$ to obtain a probability distribution over a pre-defined vocabulary, which contains the DELETE node and other dummy nodes such as $\\mathsf{EOS}$. The probability of the new node is calculated as $P^{vocab} = softmax(W^{vocab}h_{t} + b^{vocab})$.\n\nSecond, we used the attention scores $\\alpha_{t}^{s}$ as the probability to copy a token from the input query as a node label similar to the copy mechanism~\\citep{GuLLL16,SeeLM17}. Therefore, the final prediction probability of a node $o$ is defined as:\n\n\\begin{align}\n    P(o) = p_{gen} \\cdot P_{vocab}(o) + p_{copy} \\cdot \\sum_{i \\in T(c)} \\alpha_{t}^{s}[i],\n\\end{align}\n\nwhere $[i]$ indexes the $i$-th element, and\n$T(c)$ are index sets of tokens respectively that have the surface form as $o$. $P(gen)$ and $P(copy)$ are the probabilities of generating  and copying a node, respectively. They are computed by using a single layer neural network with $softmax$ activation as:\n\\begin{align}\n    [p_{gen}, p_{copy}] = softmax(W^{gate}h_{t}).\n\\end{align}\n\n\nThe whole expanding procedure is terminated if the newly generated node is the special node $\\mathsf{EOS}$.\n\n\n\n\n\\section{Dataset Construction}\n\\label{sec:dataset}\n\n\\input{tables\/dataset}\n\n\nExisting SGM datasets are synthetically constructed based on scene graphs from MSCOCO~\\citep{Lin2014MicrosoftCC} and GCC~\\citep{Sharma2018ConceptualCA}, and via crowd sourcing. To construct scene graphs, \\citet{He2020SceneGM} used an in-house scene graph parser to parse a random subset of MSCOCO description data and GCC captions, thus the constructed scene graph is relatively simple. In Table \\ref{tab:stats}, the average numbers of nodes and edges for each graph are limited to 2.9 and 1.9 respectively. GCC is more complicated than MSCOCO with a larger graph, but the percentage of nodes and edges from the development\/test set that does not appear in the training set ($\\mathrm{OOV}$ Nodes, $\\mathrm{OOV}$ Edges) are still low, which will cause the model easily overfit to the dataset. To verify the generalization ability and the scalability of the model to handle more complex scene graphs, we constructed our own Scene Graph Modification dataset based on the Remote Sensing Image Captioning Dataset (RSICD)~\\citep{lu2017exploring} in the remote sensing field for remote sensing image captioning task.\n\n\n\\input{tables\/main}\n\nInspired by the modification methods proposed by~\\citet{He2020SceneGM}. First, we adopt the parser~\\citep{Schuster2015GeneratingSP} to parse the caption for each graph and generate the original scene graph $\\textbf{x}$. Then we define three types of graph modification operations $\\mathcal{A}$ = \\{INSERT, DELETE, SUBSTITUTE\\}, and randomly apply them to the original scene graph to generate query ($\\textbf{q}$) and modified scene graph ($\\textbf{y}$). The data in RSICD consists of the triples ($\\textbf{x, y, q}$).\\footnote{We give three detailed operations and examples in the Appendix \\ref{operations}.}\n\nCompared with the existing SGM dataset, each graph of RSICD has more nodes and edges, with an average of 5.9 and 3.7 on the training\/development\/test set, which is almost twice that of User Generated and MSCOCO. In addition, the dataset comes from the field of remote sensing. Due to the large number of geographical terms, the $\\mathrm{OOV}$ Nodes of the development\/test sets compared with the training set reach 12\\%\/11\\%, and the $\\mathrm{OOV}$ Edges reach 8\\%\/8\\%, which are much higher than the MSCOCO and GCC datasets. Considering the complexity of RSICD, we construct it apart from User Generated, MSCOCO and GCC to further analysis the generalization and scalability of ISE.\n\n\n\\section{Experiments and Analyses}\n\\label{sec:experiments}\n\n\n\\subsection{Data}\nWe evaluated our model on four benchmarks, including User Generated, MSCOCO and GCC proposed by~\\citet{He2020SceneGM}, and RSICD dataset proposed in this work. MSCOCO, GCC and RSICD are constructed synthetically from publicly available datasets~\\citep{Lin2014MicrosoftCC, SoricutDSG18, lu2017exploring}, while the User Generated dataset is created via crowd sourcing. Detailed statistics of datasets are shown in Table~\\ref{tab:stats}.\n\n\n\n\\subsection{Setup}\n\nFor fair comparisons, we used the same data splits for User Generated, MSCOCO and GCC datasets as in ~\\citet{weber2021extend}. For RSICD, we randomly split the data into 8K\/1K\/1K for training\/development\/test. Following~\\citet{weber2021extend}, we use three automatic metrics for the evaluation, including node-level and edge-level F1 score, and graph-level accuracy. Graph-level accuracy is computed based on exact string match, which requires the generated scene graph to be identical to the target scene graph for a correct prediction. We reported the mean score and standard deviation by using 5 models from independent runs. We refer to the Appendix \\ref{Hyper-parameters} for the detailed implementation.\n\n\n\n\n\\subsection{Baselines}\nFor comprehensive comparisons, we include six baselines as follows. Except for the CopyGraph, all of them aim to rebuild the target scene graph. \n\n\\paragraph{CopyGraph} This baseline directly copies the source scene graph as the target scene graph, which can be viewed as the lower bound.\n\n\\paragraph{Text2Text} This baseline is introduced by~\\citet{He2020SceneGM}. They used the standard sequence-to-sequence architecture by linearizing the scene graph based on depth-first search. \n\n\\paragraph{GRNN} Graph RNN~\\citep{You2018GraphRNNGR} is used as the graph encoder and edge decoder. Specifically, the edges are represented by an adjacency matrix, which is then generated in an auto-regressive manner. Both the query encoder and node decoder are based on Gated Recurrent Units~\\citep{ChoMGBBSB14}.\n\n\\paragraph{DCGCN} Densely-Connected Graph Convolutional Networks ~\\citep{Guo2019DenselyCG} are used as the graph encoder. Other components are kept the same as the GRNN.\n\n\\paragraph{GTran} Graph Transformer~\\citep{Cai2020GraphTF} is used as the graph encoder, while other modules are the same as GRNN and DCGCN.\n\n\\paragraph{STran} The sparsely-connected transformer~\\citep{He2020SceneGM} is used to encode the source graph. In addition, a cross-attention mechanism is applied to fuse the features from graph encoder and query encoder. Node decoder and edge decoder are the same as GRNN.\n\n\\paragraph{EGraph} This is the state-of-the-art model on graph modification task. Concretely, \\citet{weber2021extend} considerably increases performance on the graph modification by phrasing it as a sequence labelling task.\n\n\n\n\\subsection{Main Results}\nAccording to Table~\\ref{tab:main}, our proposed approach (ISE) significantly outperforms the state-of-the-art model~\\citep{weber2021extend} on three datasets. Specifically, ISE outperforms EGraph 1.81, 1.11 and 1.33 percentage points in terms of graph accuracy on User Generated, MSCOCO and GCC datasets, respectively. We observe that the improvement is especially prominent on the User Generated dataset, which is \nmore challenging than the other two synthetic datasets in terms of the diversity in graph semantics and natural language expressions. All baseline models suffer from performance degradation as it is much harder to rebuild the entire target scene graph on this dataset. On the other hand, ISE constructs the target scene graph by incrementally expanding the source scene graph without changing the unmodified structure. We believe this formulation is able to effectively cope with this difficulty. \n\n\nWe also observe that both EGraph and ISE achieve lower graph accuracy on the GCC dataset. The main reason is the difficulty of predicting the correct edges between generated nodes. For example, EGraph achieves 98.62 Node F1 score on GCC, higher than 97.62 Node F1 score on the User Generated dataset. However, EGraph only achieves 75.01 Edge F1 score on GCC, while it can attain 88.26 Edge F1 score on User Generated. Our proposed model has larger improvements upon EGraph in terms of Edge F1 score on the same dataset (93.06 vs. 91.64). We attribute this stronger improvement to iterations between nodes prediction and edge prediction, which allows more accurate and harmonious expansion decisions progressively. On the other hand, EGraph predicts nodes and edges at two independent stages. Such an approach may lead to the lack of the modeling capability of interactions between node prediction and edge prediction.\n\n\\input{tables\/rsicd}\n\nWe further compare our model with EGraph on the newly constructed dataset RSICD as shown in Table~\\ref{tab:rs}. ISE is able to achieve a graph accuracy of 44.20\\% and improves upon the EGraph model by 21 percentage points. However, the graph accuracy of all the models is much lower than the one attained on the previous three SGM datasets. One reason is that RSICD has more complex queries paired with larger scene graph, which brings a challenge to existing models. The RSICD dataset also suffers from the data sparsity issue where many words (39\\%) and nodes (42\\%) only appear once in the training data. Incorrect node prediction will further propagate the errors to edge prediction. Our iterative node and edge prediction paradigm help to alleviate this issue. Specifically, ISE only outperforms EGraph 9.69 percentage points on Node F1 score, while the improvement on Edge F1 score is 13.05\\%. Therefore, ISE is able to achieve a higher accuracy. In order to further address this data sparsity issue, one potential solution is transfer learning, where the model is pretrained on User Generated dataset first then fine-tuned on RSICD. However, this approach may suffer from a domain-shift problem, as RSICD is constructed based on the remote sensing domain. We leave this direction as future works.\n\n\n\n\n\n\n\\subsection{Analysis and Discussion}\nIn this section, we provided a fine-grained analysis of our proposed model. We reported all the results on the development set by using the ISE model without contextualized embeddings from BERT.\n\n\\input{tables\/ablation}\n\n\\paragraph{Ablation Study} \nAs shown in Table~\\ref{tab:ablation}, we examine the contributions of two main components used in our model. The first one is the incremental structure expanding. We use the same model architecture but try to rebuild the target scene graph similar to previous efforts. We can observe significant drops on three SGM datasets, which further confirms the effectiveness of the extending strategy. The second one is the copy mechanism, which directly copies the token from the query as nodes in the target scene graph. It plays a significant role in predicting nodes especially when the training data is limited (User Generated).\n\n\\input{tables\/Robustness}\n\\paragraph{Performance against Training Data Size} Table~\\ref{tab:size} shows the performance of STran and ISE against different training settings on MSCOCO dataset. We considered four training settings (20\\%, 40\\%, 60\\%, 80\\%, 100\\% training data). ISE consistently outperforms STran under the same amount of training data. When the size of training data decreases, we can observe that the performance gap becomes more obvious. Particularly, using 40\\% of the training data, ISE is able to achieve a graph accuracy of 88.64\\%, higher than STran trained on the whole dataset. These results demonstrate that our model is more effective in terms of using training resources and more robust when the training data is limited.\n\n\\input{tables\/SenLength}\n\\paragraph{Performance against Query Length} Table~\\ref{tab:query} shows the results of STran and ISE under different query lengths on GCC dataset. We partitioned the sentence length into three classes (\\textless5, [5, 10), $\\geq$10). In general, ISE outperforms STran against various sentence lengths. When the length of the query increases, we can observe that the performance gap becomes more obvious in terms of graph accuracy. Intuitively, with the increase of the query length, it is more challenging for the model to comprehend the sentence. This suggests that ISE is able to handle more complex instructions.\n\n\n\\paragraph{Performance against Graph Size}\nTable~\\ref{tab:graph} shows the results of STran and ISE against different target scene graph sizes on GCC dataset. We partitioned the scene into three classes (\\textless5, [5, 10), $\\geq$10).  Based on the formulation of extending the source scene graph, our model is required to deal with larger graphs. For example, deleting a node in the scene graph becomes adding a special ``Delete'' node in the extended graph. However, ISE is able to consistently outperform STran against various target graph sizes, even when the target scene graph is large. This result suggests the superiority of the proposed formulation.\\footnote{We give an error analysis in the Appendix \\ref{error}.} \n\\input{tables\/NodeLength}\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.26]{figures\/case.pdf}\n   \n    \\caption{Two cases of STran and ISE for scene graph modification on User Generated. $Q$ denotes the textual query, $G_{S}$ denotes the source scene graph, $G_{T}$ denotes the target scene graph generated by STran and ISE.}\n    \\label{fig:case}\n   \n\\end{figure*}\n\n\\paragraph{Case Study}\n\\label{case}\n\n\n\nWe give two cases in Figure \\ref{fig:case}. STran generates scene graph from the scratch conditioned on the original graph and query may lead to the lack of the modeling capability of interactions between node prediction and edge prediction. For example, in Figure \\ref{fig:case} (a), STran omitted the attribute: ``Velvet'' during the node prediction. In addition, during the edge prediction, STran redundantly generated the relation: ``Of'' in Figure \\ref{fig:case} (b). However, these structures do not need to be modified in the source scene graph. ISE can infer more accurate target graph by incrementally expanding the source graph without changing the unmodified structure. \n\\section{Related Work}\n\\label{sec:related}\n\nWe refer to the Appendix \\ref{scene graph} for the detailed related work of scene graph. Scene graph builds a bridge between image domain and text domain. Vision and natural language are all tremendously promoted by studying into scene graphs. Recently, scene graph modification becomes an emerging research\ndirection. \\citet{Chen2020GraphED} proposed a framework based on scene graph editing for text-based image retrieval.\nOn the other hand, \\citet{He2020SceneGM} took the scene graph and the textual query as inputs and modified the source graph according to the query. They viewed the task as conditional graph generation, which is further decomposed into node prediction and edge prediction. For node prediction, all the nodes in the target scene graph is generated based on a graph-to-sequence model with dual encoder~\\citep{Song2018AGM,Beck2018GraphtoSequenceLU,ZhangGTLCLB20}, then a graph RNN is adopted to predict the edges between generated nodes~\\citep{You2018GraphRNNGR}. More recently, \\citet{weber2021extend} developed an alternative formulation of this problem in which they model the modification as an auto-regressive sequence labelling task.\n\nInstead of rebuilding the entire target graph, we framed the scene graph modification task as incremental graph expansion. This formulation is related to incremental parsing, where a sentence is scanned from left-to-right and the structured is built incrementally by inserting a node or attaching an edge. Incremental parsers are widely used in semantic parsing~\\citep{ZhouXUQLG16,ChengRSL17,GuoL18, Naseem2019RewardingST,liu2022semantic} and syntactic parsing~\\citep{HuangS10,DyerBLMS15,LiuZ17a}, as they are computationally efficient, and can use machine learning to predict actions based on partially generated structures. Our feature fusion module can be viewed as the parser state as it carries the structural information and serves as a writable memory during the expansion step. Unlike \\citet{weber2021extend} linearize the scene graph and label it in an auto-regressive manner, our model iterates between finding the relevant part in the query and reading the partially constructed scene graph, inferring more accurate and harmonious expansion decisions progressively. \n\n\n\\section{Conclusion}\n\nIn this paper, we designed a novel formulation for scene graph modification, which allows us to incrementally expand the source scene graph instead of rebuilding the entire graph. Based on the formalism, we further propose a model that is able to leverage the mutual causalities between node prediction and edge prediction. Experiments on three SGM benchmarks demonstrate the effectiveness.\nTo test our model under a complex scenario, we constructed a more challenging dataset from the remote sensing domain, which has more modification operations based on the more complicated queries compared with existing SGM datasets. For future work, we would like to explore how to integrate the model into the text-based image retrieval task.\n\n\n\\section{Appendix}\n\n\n\\subsection{Operations in RSICD}\n\n\n\\input{tables\/RSICD_Case}\n\\label{operations}\nWe introduce three operations in RSICD in details:\n\\begin{itemize}\n    \\item \\textbf{\\texttt{DELETE}}: The original scece graph is $\\textbf{x}$. We randomly select a node $\\textbf{o}$ in $\\textbf{x}$, and delete it both with related edges. The deleted graph is defined as $\\textbf{y}$. We choose a random sentence from the \\textit{DELETE Template}~\\citep{manuvinakurike-etal-2018-edit}, for example, `` I do not want \\textbf{**}.'' We replace \\textbf{**} with $\\textbf{o}$ to get modification operation $\\textbf{q}$.\n    \\item \\textbf{\\texttt{INSERT}}: It is the reverse process of \\textbf{\\texttt{DELETE}}. The graph before deleting the node is regarded as $\\textbf{y}$, and the corresponding graph after deletion is treated as $\\textbf{x}$. The modification operation is randomly selected from the \\textit{INSERT Template}~\\citep{manuvinakurike-etal-2018-edit}, for example, `` Show me \\textbf{**}.'' We replace \\textbf{**} with $\\textbf{o}$ to obtain query $\\textbf{q}$.\n    \\item \\textbf{\\texttt{SUBSTITUTE}}: We randomly select a node $\\textbf{o}$, use the AllenNLP toolkit~\\citep{gardner-etal-2018-allennlp} to find the three most similar semantics nodes compared with $\\textbf{o}$. We randomly choose a node $\\textbf{m}$, and select a sentence from the \\textit{SUBSTITUTE Template}~\\citep{manuvinakurike-etal-2018-edit}, for example, `` I prefer \\textbf{@@} to \\textbf{**}, modify \\textbf{**} to \\textbf{@@}.'' We replace \\textbf{**} and \\textbf{@@} with $\\textbf{o}$ and $\\textbf{m}$, and get modification operation $\\textbf{q}$. Note that SUBSTITUTE operation could be viewed as DELETE the node $\\textbf{o}$ first and then INSERT the node $\\textbf{m}$, or vice versa.\n\\end{itemize}\n\nIn Table \\ref{tab:RSICD_case_study}, we give the simple examples in RSICD to better understand three types of graph modification operations.\n\n\n\\subsection{Implementation Details}\n\\label{Hyper-parameters}\nHyper-parameters of the model are tuned on the development set. All transformer~\\citep{VaswaniSPUJGKP17} layers share the same hyper-parameter settings. Following~\\citet{He2020SceneGM}, we randomly initialized the word and node embeddings. We also report results with contextualized embeddings from BERT~\\citep{DevlinCLT19}. Specifically, we used the BERT-base-uncased implemented by~\\citep{wolf-etal-2020-transformers}. The parameters in BERT are fixed during training. To mitigate over-fitting, we apply dropout~\\citep{SrivastavaHKSS14} with the drop rate 0.2 between different layers.  Following~\\citet{Cai2020AMRPV}, we use a special UNK token to replace the out-of-vocabulary lemmas of the input query and remove the UNK token in the generated graph. Parameter optimization is performed with the ADAM optimizer~\\citep{KingmaB14} with $\\beta_{1}$ = 0.9 and $\\beta_{2}$ = 0.999. The learning rate schedule is similar to that in~\\citet{VaswaniSPUJGKP17}, where warm-up steps being set to 2K. We used early stopping on the development set for choosing the best model. \nPlease refer to Table \\ref{tab:hyper-parameters} for the detailed hyper-parameters settings for ISE.\n\n\\begin{table}\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{lr}\n\\toprule\n\\multicolumn{2}{l}{\\textbf{Embeddings}}\\\\\n\\midrule\nconcept  & 300 \\\\\nword & 300   \\\\\nrelation  & 100\\\\\n\\midrule\n\\multicolumn{2}{l}{\\textbf{Query Encoder}}\\\\\ntransformer layers & 4   \\\\\n\\midrule\n\\multicolumn{2}{l}{\\textbf{Graph Encoder}}\\\\\ntransformer layers & 2   \\\\\n\\midrule\n\\multicolumn{2}{l}{\\textbf{Feature  Fusion}}\\\\\nheads & 8   \\\\\nhidden size & 512   \\\\\nfeed-forward hidden size & 1024   \\\\\n\\midrule\n\\multicolumn{2}{l}{\\textbf{Node Decoder\/ Edge Decoder}}\\\\\nheads & 8   \\\\\nfeed-forward hidden size & 1024   \\\\\n\n\n\\bottomrule\n\\end{tabular}}\n\n\\caption{Hyper-parameters settings for ISE.}\n\\label{tab:hyper-parameters}\n\n\\end{table}\n\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[scale=0.26]{figures\/error.pdf}\n   \n    \\caption{Two errors of ISE for scene graph modification on User Generated. $Q$ denotes the textual query, $G_{S}$ denotes the source scene graph, $G_{T}$ denotes the target scene graph generated by ISE. $G_{G}$ denotes the gold target scene graph.}\n    \\label{fig:error}\n   \n\\end{figure*}\n\n\\subsection{Scene Graph and Application}\n\\label{scene graph}\nDeep learning has significantly promoted the advancement of computer vision~\\citep{liang2017deep, ren2021comprehensive}. Simple visual understanding tasks such as object detection and recognition are no longer sufficient. To depict the relationship between objects in the scene as a driving force, higher-level visual understanding and reasoning skills are frequently necessary. Scene graphs were created specifically to address this issue. Scene graph was first proposed by~\\citet{Johnson2015ImageRU} for image retrieval, which describes objects, their attributes, and relationships in images with a graph. A complete scene graph could represent the semantics of a dataset's scenes, not just a single image or video; additionally, it contains powerful representations that encode 2D\/3D images~\\citep{Johnson2015ImageRU,armeni20193d}, and videos~\\citep{qi2018scene,wang2020storytelling} into their abstract semantic elements. Scene graph is beneficial for various downstream tasks, such as information extraction \\cite{hu2020selfore,hu2021semi,hu2021gradient,liu2022hierarchical}, natural language summarization \\cite{liu2022psp}, and natural language inference \\cite{li2022pair}.\n\nFollowing the graph representation paradigm, different methods have been proposed to generate scene graphs from images~\\citep{XuZCF17,WangLZY18, ZellersYTC18}. Many cross-modal tasks that require understanding and reasoning on image and text are able to benefit from incorporating scene graphs, such as visual question answering~\\citep{TeneyLH17,shi2019explainable}, grounding referring expressions~\\citep{wang19}, image captioning~\\citep{YangTZC19,yao2018exploring}, and image retrieval~\\citep{Wang2020CrossmodalSG,Schroeder2020StructuredQI}. \n\n\n\n\n\n\\subsection{Error Analysis} \n\\label{error}\nWe give two wrong scene graphs generated by ISE in Figure \\ref{fig:error}. We can observe in Figure \\ref{fig:error} (a) that although ISE successfully predicts the need to insert a relation between object ``Plants'' and attribute ``Surface'', since the User Generated dataset contains a total of 2078 relations and the relations have serious long-tail effects. It is difficult for ISE to learn sparseness relations with few occurrences, leading to incorrectly predicting relation ``in growing over'' as ``on''. We attempt to address the long-tail effects of relations in future work.\nSince a node can be attached to multiple nodes, when Edge Decoder determines which nodes in the current graph should be attached to the new node, a common error is predicting the wrong node that needs to be attached. As shown in Figure \\ref{fig:error} (b), ISE incorrectly connects relation ``behind'' between ``Giraffe'' and ``Tree'' instead of ``Head'' and ``Tree''.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA phased mission system (PMS) is one that performs several different tasks or functions in sequence. The periods in which each of these successive tasks or functions takes place are known as phases \\citep{xing2008reliability,la2004phased}.  Examples of PMSs can be found in many practical applications, such as electric power systems, aerospace systems, weapon systems and computer systems. A typical example of a PMS is the monitoring system in a satellite-launching mission with three phases: launch, separation, and orbiting.\n\nA PMS is considered to be functioning if all of its phases are completed without failure, and failed if failure occurs in any phase. Therefore, the reliability of a PMS with $N$ phases is the probability that it operates successfully in all of its phases:\n\\begin{equation}\n  R_S = \\mathbb{P}(\\mbox{Phase 1 works} \\cap \\mbox{Phase 2 works} \\cap \\dots \\cap \\mbox{Phase $N$ works})\n  \\label{eq:allphases}  \n\\end{equation}\n\nThe calculation of the reliability of a PMS is more complex than that of a single phase system, because the structure of the system varies between phases and the component failures in different phases are mutually dependent \\citep{xing2008reliability}.\n\nOver the past few decades, there have been extensive research efforts to analyze PMS reliability.  Generally, there are two classes of models to address such scenarios: state space oriented models \\citep{kim1994phased,chew2008phased,lu2014reliability,wang2017competing} and combinatorial methods \\citep{xing2015binary,ma1999algorithm,la2004phased,zang1999bdd,tang2006bdd,mo2009variable,reed2011improved,xing2007reliability,xing2013bdd}. The main idea of state space oriented models is to construct Markov chains and\/or Petri nets to represent the system behaviour, since these provide flexible and powerful options for modelling complex dependencies among system components. However, the cardinality of the state space can become exponentially large as the number of components increases.  The remaining approaches exploit combinatorial methods, Boolean algebra and various forms of decision diagrams for reliability analysis of PMSs.\n\nIn particular, in recent years the Binary Decision Diagram (BDD) --- a combinatorial method --- has become more widely used in reliability analysis of PMSs due to its computationally efficient and compact representation of the structure function compared with other methods. Zang \\emph{et al.}\\  \\citep{zang1999bdd} first used the BDD method to analyze the reliability of PMSs. Tang \\emph{et al.}\\  \\citep{tang2006bdd} developed a new BDD-based algorithm for reliability analysis of PMSs with multimode failures. Mo \\citep{mo2009variable} and Reed \\emph{et al.}\\  \\citep{reed2011improved} improved the efficiency of Tang's method by proposing a heuristic selection strategy and reducing the BDD size, respectively. Xing \\emph{et al.}\\  \\citep{xing2007reliability,xing2013bdd} and Levitin \\emph{et al.}\\  \\citep{levitin2013reliability} proposed BDD based methods for the reliability evaluation of PMSs with common-cause failures and propagated failures.  Wang \\emph{et al.}\\  \\citep{wang2007reliability} and Lu \\emph{et al.}\\  \\citep{lu2015reliability} studied modular methods for reliability analysis of PMSs with repairable components, by combining BDDs with state-enumeration methods.\n\nWhile the BDD method has been shown to be a very efficient combinatorial method, it is still difficult to analyze large systems without considerable computational expense \\citep{xing2008reliability,reed2011improved}.  In this paper, we propose a combinatorial analytical approach providing a new survival signature methodology for reliability analysis of PMSs.  This paper is organized as follows: \\cref{sec:PMS} gives a brief background on PMSs; \\cref{sec:survsig} first shows how the standard survival signature can be used to evaluate PMSs with similar component types in each phase, before providing a novel methodology which facilitates heterogeneity of components across the phases.  \\Cref{sec:examples} presents illustrative examples showing numerical agreement with existing literature, but where the full benefits of the interpretability of survival signatures is now available due to this work.  Finally, \\cref{sec:conclusion} presents some conclusions ideas for future work.\n\n\\section{Phased mission systems}\n\\label{sec:PMS}\n\n\\Cref{fig:pms1} shows a simple system that performs a series of functions or tasks which are carried out over consecutive periods of time to achieve a certain overall goal (or `mission').  Such a system --- where the structure (and possibly operating environment) of the system changes over time --- is known as a Phased Mission System (PMS), with each period of operation being referred to as a `phase'. Each phase therefore corresponds to one structural configuration and components in different phases are taken to be mutually dependent.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{PMS1}\n  \\caption{A PMS with similar components in each phase.}\n  \\label{fig:pms1}\n\\end{figure}\n\nLet us consider a system consisting of $N \\ge 2$ phases, with $n_i$ components in phase $i \\in \\{1, \\dots, N\\}$.  The binary state indicator variable $X_{ij}, j \\in \\{1, \\dots, n_i\\}$ denotes the operational status of the $j$th component in phase $i$:\n\\[ X_{ij} = \\begin{cases}\n1 & \\mbox{if component $j$ works for all of phase $i$} \\\\\n0 & \\mbox{if component $j$ fails before the end of phase $i$}\n\\end{cases} \\]\n\nThe vectors $\\mathbf{X}_i = (X_{i1}, \\dots, X_{in_i}), i \\in \\{1, \\dots, N\\}$, represent the states of all components in the $i$th phase and the full vector $\\mathbf{X} = (\\mathbf{X}_1, \\dots, \\mathbf{X}_N) = (X_{11}, \\dots, X_{1n_1}, \\dots, X_{N1}, \\dots, X_{Nn_N})$ represents the states of all components during the full mission.\n\nThe state of the system in each phase is also a binary random variable, which is completely determined by the states of the components in that phase.  Let $\\phi_i$ represent the system state in the $i$th phase, that is:\n\\[ \\phi_i = \\varphi_i(\\mathbf{X}_i) = \\varphi_i(X_{i1}, \\dots, X_{in_i}) \\]\nwhere $\\varphi_i(\\cdot)$ is the structure function of the system design in phase $i$.  The structure function evaluates to $\\phi_i = 1$ if the system functions for state vector $\\mathbf{X}_i$, and $\\phi_i = 0$ if not.\n\nSimilarly, the structure function of the full PMS (that is, the operational state of the system across \\emph{all} phases) is also a binary random variable, which is completely determined by the states of all the components in the PMS\n\\begin{equation}\n  \\phi_S = \\varphi_S(\\mathbf{X}) \\triangleq \\prod_{i=1}^N \\varphi_i(X_{i1}, \\dots, X_{in_i})\n  \\label{eq:strfnpms}\n\\end{equation}\n\nThe structure function as shown in \\cref{eq:strfnpms} is again a Boolean function which is derived from the truth table of the structure functions for each phase of operation.  The truth tables depend uniquely on the system configurations and simply provide a means of tabulating all the possible combinational states of each component to realise the operational state of the system in each case. The state vectors for which $\\varphi_S(\\mathbf{X})=1$ provide a logical expression for the functioning of the system, while the states when $\\varphi_S(\\mathbf{X})=0$ provide a logical expression for the failure of the system. It should be noted that, unlike non-PMSs, there exist impossible combinations of states which should be deleted from the truth table when performing a reliability analysis.  For example, if both the system and its components are non-repairable during the mission, then if a component is failed in a certain phase it cannot be working in subsequent phases.\n\nFinally, if all phases are completed successfully, the mission is a success, that is:\n\\[ \\phi_S = \\prod_{i=1}^N \\phi_i = 1 \\iff \\phi_i = 1 \\ \\forall\\,i \\]\n\n\\section{Survival signature}\n\\label{sec:survsig}\n\nFor larger systems, working with the full structure function can be complicated and as the system size grows it becomes hard to intuit anything meaningful from the particular algebraic form it takes.  In particular, one may be able to summarize the structure function when it consists of exchangeable components of one or more types \\citep{samaniego2007system,coolen2013generalizing,coolen2014nonparametric}.\n\nRecently, the concept of the survival signature has attracted substantial attention, because it provides such a summary which enables insight into the system design even for large numbers of components of differing types.  Coolen and Coolen-Maturi \\citep{coolen2013generalizing} first introduced the survival signature, using it to analyze complex systems consisting of multiple types of component. Subsequently, \\citep{coolen2014nonparametric,coolen2015predictive,Aslett2015} presented the use of the survival signature in an inferential setting, with nonparametric predictive inference and Bayesian posterior predictive inference respectively, and \\citep{feng2016imprecise} presented methods for analyzing imprecise system reliability using the survival signature. Patelli \\emph{et al.}\\  \\citep{patelli2017simulation} developed a survival signature-based simulation method  to calculate the reliability of large and complex systems and \\citep{Aslett2017} presents a simulation method which can be used if the dependency structure is too complex for a survival signature approach.  Walter \\emph{et al.}\\  \\citep{walter2017condition} proposed a new condition-based maintenance policy for complex systems using the survival signature.  Moreover, Eryilmaz \\emph{et al.}\\  \\citep{eryilmaz2016generalizing} generalized the survival signature to multi-state systems.\n\nEfficient computation of the survival signature was addressed by Reed \\citep{reed2017efficient}, using reduced order binary decision diagrams (ROBDDs).  The survival signature of a system can be easily computed by specifying the reliability block diagram as a simple graph by using the \\texttt{ReliabilityTheory} R package \\citep{Aslett2012}.\n\nIn this section, the survival signature is first shown to apply directly to full mission-length PMSs where there is a single component type in each phase.  Thereafter, an extension is presented which enables heterogeneity of component types across phases, providing novel methodology for reliability analysis of PMSs.\n\n\\subsection{PMSs with similar components in each phase}\n\\label{sec:pms.same}\n\nWe consider a system with $N \\ge 2$ phases, with $n$ components in each phase (e.g.\\ the PMS as shown in \\cref{fig:pms1}), and let phase $i \\in \\{1, \\dots, N\\}$ run from time $\\tau_i$ to time $\\tau_{i+1}$ with $\\tau_1 \\triangleq 0$ and $\\tau_i < \\tau_{i+1} \\ \\forall\\,i$.  Thus the full mission time is denoted $\\tau_{N+1}$.\n\nWe assume that the random failure times of components in the same phase are fully independent, and in addition that the components are exchangeable. Let $\\Phi(l_1, \\dots, l_N)$ denote the probability that the PMS functions by the end of the mission given that precisely $l_i, i \\in \\{1, \\dots, N\\}$, of its components functioned in phase $i$. Both the system and its components are non-repairable during the mission, so $n \\ge l_1 \\ge l_2 \\ge \\dots \\ge l_N \\ge 0$ and the number of components that function at the beginning of phase $i$ is $m_i = l_{i-1},$ with $m_1=n$ --- so all components appear in all phases.  Subject to these constraints which do not apply in a non-PMS, the survival signature can then be applied without further modification for the mission completion time.\n\nThere are $\\binom{m_i}{l_i}$ state vectors where precisely $l_i$ components function.  Because the random failure times of components in the same phase are independent and exchangeable, the survival signature is equal to:\n\\begin{equation}\n  \\Phi(l_1, \\dots, l_N) = \\left[ \\prod_{i=1}^N \\binom{m_i}{l_i}^{-1} \\right] \\sum_{\\mathbf{X} \\in \\mathcal{S}} \\varphi_S(\\mathbf{X})\n  \\label{eq:survsig0}\n\\end{equation}\nwhere $\\mathcal{S}$ denotes the set of all possible state vectors for the whole system where $l_i$ components in phase $i$ are functioning.  This step is of the same form as the standard survival signature for a static system \\citep{coolen2013generalizing}, but note one immediate subtle difference: as noted above, $m_i$ is not fixed across evaluations of $\\Phi(\\cdot)$, but rather is determined by $l_{i-1}$, since the maximum number of functioning components in the $i$th phase is determined by how many components completed phase $i-1$ still functioning.\n\nA further subtlety arises as soon as we consider any time leading up to the mission completion time, because the structure of the system changes.  Although the standard survival signature can be used in computing the reliability of a static system at any point in its life \\citep{coolen2013generalizing}, this is no longer true in this extension to PMSs.  Consequently, \\eqref{eq:survsig0} is the survival signature which represents the probability that the whole mission completes successfully given that $l_i$ components are working in phase $i$.  For the survival function of a PMS, we must extend the survival signature to create a family of survival signatures which account for the temporally changing structure.  Let $\\Phi_p(l_1, \\dots, l_p)$ denote the survival signature of a PMS up to and including phase $p \\le N$, which is the probability that the mission has not yet failed by phase $p$ given that $l_i$ components are working in phase $i \\in \\{1, \\dots, p\\}$.  Then,\n\\begin{equation}\n  \\Phi_p(l_1, \\dots, l_p) = \\left[ \\prod_{i=1}^p \\binom{m_i}{l_i}^{-1} \\right] \\sum_{\\mathbf{X} \\in \\mathcal{S}} \\prod_{i=1}^p \\varphi_i(\\mathbf{X})\n  \\label{eq:survsig0t}\n\\end{equation}\n\nWe define a function mapping mission time $t$ to the current phase\n\\begin{equation}\n  \\rho(t) : [0,\\tau_{N+1}] \\to \\{1, \\dots, N\\}, \\mbox{ as }  \\rho(t) \\triangleq \\max\\{ i \\,:\\, \\tau_i < t \\} \\label{eq:currentphase}\n\\end{equation}\n\nFrom \\cref{eq:allphases} and \\eqref{eq:survsig0t}, the reliability of the PMS at time $t$ can then be expressed pointwise as:\n\\begin{equation}\n  R(t) = \\sum_{l_1=0}^{m_1} \\cdots \\sum_{l_{\\rho(t)}=0}^{m_{\\rho(t)}} \\left[ \\Phi_{\\rho(t)}(l_1, \\dots, l_{\\rho(t)}) \\mathbb{P}\\left( \\bigcap_{i=1}^{\\rho(t)} \\left\\{ C_i(t) = l_i \\right\\} \\right) \\right]\n  \\label{eq:survsig1}\n\\end{equation}\nwhere $C_i(t)$ is the random variable denoting the number of components in phase $i$ which function at time $t \\in [\\tau_i, \\tau_{i+1})$.  If $R(t)$ is being evaluated at $t \\ge \\tau_{i+1}$ then $C_i(t) \\triangleq C_i(\\tau_{i+1})$.  By the definition of $\\rho(t)$, $R(t)$ will never be evaluated for $t < \\tau_{i}$.\n\nBecause components are of the same type they share a common lifetime distribution as long as they all appear in all phases (and hence age together).  As a result, the sequential nature of a PMS means that components in the same phase have common conditional CDF, $F_i(t)$, for phase $i$, where conditioning is on the component having worked at the beginning of phase $i$.  That is, if the components have common CDF $F(t)$ and all components appear in every phase (in possibly different configurations), then the conditional CDF in phase $i$ is:\n\\begin{align}\n  F_i(t) &= \\mathbb{P}(T < t \\,|\\, \\tau_i, \\tau_{i+1}, T > \\tau_i) \\nonumber \\\\\n  &= \\frac{1}{1-F(\\tau_i)} \\int_{\\tau_i}^{\\min \\{t, \\tau_{i+1}\\}} dF(z) \\nonumber \\\\\n  &= \\frac{F(\\min \\{t, \\tau_{i+1}\\}) - F(\\tau_i)}{1-F(\\tau_i)} \\label{eq:condcdf}\n\\end{align}\nwhere $\\tau_i$ is the start time of phase $i$ ($\\tau_1 \\triangleq 0$) and $T$ is the random variable representing component lifetime.\n\nProceeding with this conditional CDF, the last term in \\cref{eq:survsig1} can be simplified as\n\\begin{align*}\n  \\mathbb{P}\\left( \\bigcap_{i=1}^{\\rho(t)} \\left\\{ C_i(t) = l_i \\right\\} \\right) &= \\prod_{i=1}^{\\rho(t)} \\mathbb{P}\\left( C_i(t) = l_i \\right) \\\\\n  &= \\prod_{i=1}^{\\rho(t)} \\left[ \\binom{m_i}{l_i} (R_i(t))^{l_i} (1-R_i(t))^{m_i-l_i} \\right]\n\\end{align*}\nwhere\n\\begin{equation}\n  R_i(t) = 1-F_i(t) = \\frac{1-F(\\min \\{t, \\tau_{i+1}\\})}{1-F(\\tau_i)} \\label{eq:comprel}\n\\end{equation}\nis the reliability of the components at time $t$ in phase $i$.\n\nThus, \\cref{eq:survsig1} can be rewritten pointwise in $t$ as\n\\begin{align}\n  R(t) &= \\sum_{l_1=0}^{m_1} \\cdots \\sum_{l_{\\rho(t)}=0}^{m_{\\rho(t)}} \\left\\{ \\Phi_{\\rho(t)}(l_1, \\dots, l_{\\rho(t)}) \\phantom{\\prod_{i=1}^{\\rho(t)} \\left[ \\binom{m_i}{l_i} (R_i(t))^{l_i} (1-R_i(t))^{m_i-l_i} \\right]} \\right. \\nonumber \\\\\n  &\\qquad\\qquad\\qquad\\qquad \\times \\left. \\prod_{i=1}^{\\rho(t)} \\left[ \\binom{m_i}{l_i} (R_i(t))^{l_i} (1-R_i(t))^{m_i-l_i} \\right] \\right\\} \\label{eq:survsigPMS1a}\n\\end{align}\n\nSince in the general case (see special case exception in the sequel) every component appears in every phase, this can be written\n\\begin{align}\n  R(t) &= \\sum_{l_1=0}^{l_0} \\cdots \\sum_{l_{\\rho(t)}=0}^{l_{\\rho(t)-1}} \\left\\{ \\Phi_{\\rho(t)}(l_1, \\dots, l_{\\rho(t)}) \\phantom{\\prod_{i=1}^{\\rho(t)} \\left[ \\binom{m_i}{l_i} (R_i(t))^{l_i} (1-R_i(t))^{m_i-l_i} \\right]} \\right. \\nonumber \\\\\n  &\\qquad\\qquad\\qquad\\qquad \\times \\left. \\prod_{i=1}^{\\rho(t)} \\left[ \\binom{l_{i-1}}{l_i} (R_i(t))^{l_i} (1-R_i(t))^{l_{i-1}-l_i} \\right] \\right\\} \\label{eq:survsigPMS1b}\n\\end{align}\nwhere we define $l_0 \\triangleq n$.  Writing in this final form stresses the sequential dependence in the computation, in stark contrast to the standard survival signature for a static system.\n\n\\subsubsection{Special case: Exponentially distributed component lifetime}\n\nThere are two simplifications that arise when components are Exponentially distributed.  Firstly, $F_i(t) \\equiv F(t) \\ \\forall\\, i$, so that $R_i(t) = R(t) = 1-F(t-\\tau_i) \\ \\forall i$.\n\nThe second simplification is that not all components need to appear in all phases.  It may be that some components appear only in later phases (but continue to appear after the first phase they are in).  In this case, one should be careful not to use \\eqref{eq:survsigPMS1b}, but instead \\eqref{eq:survsigPMS1a} where now $m_i=l_{i-1}+m_i^\\star$ where $m_i^\\star$ is the number of components appearing in the system for the first time at phase $i$.\n\n\\subsubsection{Modelling constraints}\n\nNote that considerable care is required in the specification of --- and implicit assumptions made for --- $F_i(t)$.  In particular, when a component is not present in a phase, then whether ageing continues (i.e.\\ time passes) or not is crucial in determining whether the assumption of identical component lifetime distribution still holds in all phases.  For example, in \\cref{fig:pms1} each component appears in all phases and therefore experiences the same wear, but in \\cref{fig:pms2} each component is in precisely 2 of the 3 phases.  Consequently, even though one might assume all components are of the same type initially, if component $C$ is considered not to `age' during phase 1 (where it is not present) then it will in fact not have identical conditional lifetime distribution to $A$ and $E$ during phase 2, since the latter will have already experienced wear from phase 1.\n\nThis imposes rather unattractive modelling strictures: all components of similar type must appear in the same phases; or all components must have constant failure rate (Exponentially distributed lifetime).  These modelling strictures severely limit applicability to real world systems, thus motivating the novel methodological extension of survival signatures hereinafter.\n\n\\subsection{PMSs with different components in different phases}\n\\label{sec:pms.diff}\n\nMost practical PMSs for which the reliability is modelled consist of heterogeneous component types both within and between phases. Therefore, a more interesting challenge is to extend the methodology of survival signatures to this more general setting.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{PMS2}\n  \\caption{A PMS with multiple types of components.}\n  \\label{fig:pms2}\n\\end{figure}\n\nWe now consider this setting in generality and show that the problem again simplifies in the special case of Exponentially distributed lifetimes, which is the only case that most of the literature has addressed to date.  The only constraint we impose is that components of the same type appear in the same phases (since then the conditional CDFs within phases remain in agreement).  However, note that this does not limit the scenarios that can be modelled, since components of the same physical type can still be split into multiple `meta-types'.\n\n\\begin{definition}{(Meta-type)}\n  Components are defined to be of the same \\emph{meta-type} when they are of the same physical type and appear in the same phases.\n\\end{definition}\n\nLet there be a total of $K$ different meta-types of component.  We take the multi-type, multi-phase survival signature to be denoted by the function $\\Phi(l_{11}, \\dots, l_{1K}, \\dots, l_{N1}, \\dots, l_{NK})$, the probability that the system functions given that precisely $l_{ik}$, components of type $k$ function in phase $i$.  That is,\n\\[ \\Phi(l_{11}, \\dots, l_{1K}, \\dots, l_{N1}, \\dots, l_{NK}) = \\left[ \\prod_{i=1}^N \\prod_{k=1}^{K} \\binom{m_{ik}}{l_{ik}}^{-1} \\right] \\sum_{\\mathbf{X} \\in \\mathcal{S}} \\varphi_S(\\mathbf{X}) \\]\nwhere $\\mathcal{S}$ denotes the set of all possible state vectors for the whole system.  Not all component types need necessarily appear in all phases, so we admit the possibility that $m_{ik}=0$ when a component type is absent from a phase and observe the standard definition that $\\binom{0}{0} \\triangleq 1$ --- this simplifies notation versus having varying numbers of $l_{i\\cdot}$ for each phase.\n\nAs before, the above survival signature is only applicable to the full mission time and we define a family of survival signatures corresponding the successive phases of the mission.  Let $\\Phi_p(l_{11}, \\dots, l_{1K}, \\dots, l_{p1}, \\dots, l_{pK})$ denote the survival signature of a PMS up to and including phase $p \\le N$, which is the probability that the mission has not yet failed by phase $p$ given that $l_{ik}$ components of type $k$ are working in phase $i \\in \\{1, \\dots, p\\}$.  Then,\n\\begin{equation}\n  \\Phi_p(l_{11}, \\dots, l_{1K}, \\dots, l_{p1}, \\dots, l_{pK}) = \\left[ \\prod_{i=1}^p \\prod_{k=1}^{K} \\binom{m_{ik}}{l_{ik}}^{-1} \\right] \\sum_{\\mathbf{X} \\in \\mathcal{S}} \\prod_{i=1}^p \\varphi_i(\\mathbf{X})\n  \\label{eq:survsig0t2}\n\\end{equation}\n\nWe retain the definition of $\\rho(t)$ given in \\eqref{eq:currentphase}.  It then follows from \\cref{eq:allphases} and \\cref{eq:survsig0t2} that the reliability of the PMS can be characterised as:\n\\begin{align}\n  R(t) &= \\sum_{l_{11}=0}^{m_{11}} \\cdots \\sum_{l_{\\rho(t),K}=0}^{m_{\\rho(t),K}} \\left[ \\Phi_{\\rho(t)}(l_{11}, \\dots, l_{1K}, \\dots, l_{\\rho(t),1}, \\dots, l_{\\rho(t),K}) \\vphantom{\\mathbb{P}\\left( \\bigcap_{i=1}^N \\bigcap_{k=1}^{K_i} \\left\\{ C_{ik}(t) = l_{ik} \\right\\} \\right)} \\right. \\nonumber \\\\\n  & \\qquad\\qquad\\qquad\\qquad\\qquad \\left. \\times \\mathbb{P}\\left( \\bigcap_{i=1}^{\\rho(t)} \\bigcap_{k=1}^{K} \\left\\{ C_{ik}(t) = l_{ik} \\right\\} \\right) \\right] \\label{eq:survsig2}\n\\end{align}\nwhere $C_{ik}(t)$ is the random variable denoting the number of components of type $k$ in phase $i$ which function at time $t \\in [\\tau_i, \\tau_{i+1})$.  In the same vein as \\cref{sec:pms.same}, if $R(t)$ is being evaluated at $t \\ge \\tau_{i+1}$ then $C_{ik}(t) \\triangleq C_{ik}(\\tau_{i+1})$.  By the definition of $\\rho(t)$, $R(t)$ will never be evaluated for $t < \\tau_{i}$.\n\nWe can simplify, by defining that $\\mathbb{P}\\left( C_{ik}(t) = 0 \\right) = 1$ when $m_{ik}=0$.\n\\begin{align*}\n  \\mathbb{P}\\left( \\bigcap_{i=1}^{\\rho(t)} \\bigcap_{k=1}^{K} \\left\\{ C_{ik}(t) = l_{ik} \\right\\} \\right) &= \\prod_{i=1}^{\\rho(t)} \\prod_{k=1}^{K} \\mathbb{P}\\left( C_{ik}(t) = l_{ik} \\right) \\\\\n  &= \\prod_{i=1}^{\\rho(t)} \\prod_{k=1}^{K} \\left[ \\binom{m_{ik}}{l_{ik}} (R_{ik}(t))^{l_{ik}} (1-R_{ik}(t))^{m_{ik}-l_{ik}} \\right]\n\\end{align*}\nwith\n\\begin{equation}\n  R_{ik}(t) = \\frac{1-F_k(\\min \\{t, \\tau_{i+1}\\})}{1-F_k(\\tau_i)} \\label{eq:comprel2}\n\\end{equation}\nwhere $F_k(\\cdot)$ is the CDF of the component lifetime distribution for the meta-type $k$.\n\nConsequently, for any time $t$ during the mission, we have the reliability of the system characterised by:\n\n\\begin{align}\n  R(t) &= \\sum_{l_{11}=0}^{m_{11}} \\cdots \\sum_{l_{\\rho(t),K}=0}^{m_{\\rho(t),K}} \\left\\{ \\Phi_{\\rho(t)}(l_{11}, \\dots, l_{1K}, \\dots, l_{\\rho(t),1}, \\dots, l_{\\rho(t),K}) \\vphantom{\\prod_{i=1}^{\\rho(t)} \\prod_{k=1}^{K} \\left[ \\binom{m_{ik}}{l_{ik}} (R_{ik}(t))^{l_{ik}} (1-R_{ik}(t))^{m_{ik}-l_{ik}} \\right]} \\right. \\nonumber \\\\\n  & \\qquad\\qquad\\ \\left. \\times \\prod_{i=1}^{\\rho(t)} \\prod_{k=1}^{K} \\left[ \\binom{m_{ik}}{l_{ik}} (R_{ik}(t))^{l_{ik}} (1-R_{ik}(t))^{m_{ik}-l_{ik}} \\right] \\right\\} \\label{eq:survsigPMS2}\n\\end{align}\nwhere $m_{ik} = l_{jk}$ for $j = \\max \\{ j : j < i, m_{jk} > 0 \\}$.  That is, $m_{ik}$ is the number components which were working in the most recent preceding phase where this component meta-type appears.\n\n\\subsubsection{Special case: Exponential component lifetimes}\n\nExponentially distributed component lifetimes again provide simplifications.  Now, the $R_{ik}(t) \\equiv R_k(t)$ due to the memoryless property of the Exponential distribution.\n\nFurthermore, we can relax the definition of a meta-type of component.  The definition of component meta-types serves two purposes: (i) to ensure that $m_{ik}$ can be determined without tracking the individual functioning status of all components; and (ii) to ensure that the conditional CDFs of all components of the same meta-type in a phase are the same.  The second purpose is made entirely redundant by the memoryless nature of the Exponential distribution.  The first purpose remains, but can be achieved with a weaker definition of meta-type.\n\n\\begin{definition}{(Exponential meta-type)}\n  Components are defined to be of the same \\emph{exponential meta-type} when they are of the same Exponentially distributed physical type, and if once any pair of components of the same \\emph{exponential meta-type} appear in a phase together, they both appear in all subsequent phases where either component appears.\n\\end{definition}\n\nIn other words, components of the same exponential meta-type may first appear in the system at different phases, but thereafter should appear whenever at least one such exponential meta-type component appears.  This definition enables the determination of $m_{ik}$ as $m_{ik} = l_{jk} + m_{ik}^\\star$ for $j = \\max \\{ j : j < i, m_{jk} > 0 \\}$, where $m_{ik}^\\star$ is the number of components of exponential meta-type $k$ appearing for the first time in phase $i$.\n\nThe benefits of Exponential component lifetimes can be mixed in a system containing both meta-type and exponential meta-types since a crucial feature of survival signatures is the factorisation of such types so that they do not interact.\n\n\\section{Numerical examples}\n\\label{sec:examples}\n\n\\subsection{Example 1}\n\nWe first consider the PMS shown in \\cref{fig:pms1}. The duration of each phase is taken to be 10 hours, and the failure rate of each component in each phase is $10^{-4}$\/hour.\n\nThe survival signatures of this PMS can be obtained using \\cref{eq:survsig0}.  The elements of the survival signature which are non-zero are shown in \\cref{tab:pms1.survsig} --- that is, rows where $\\Phi(l_1)=0, \\Phi(l_1, l_2)=0$ and $\\Phi(l_1, l_2, l_3)=0$ are omitted. The table is grouped into a nested sequence of phases, with just the first phase shown, followed by the first two phases together and finally all phases --- this helps emphasise and clarify the sequential dependence of phases, where $m_k$ depends on $l_{k-1}$.\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular}{ccccccccc}\n    \\hline\n    \\multicolumn{2}{c}{First phase} & \\multicolumn{3}{c}{Phase 1+2} & \\multicolumn{4}{c}{All Phases} \\tabularnewline\n    \\multicolumn{2}{c}{$0 \\le t \\le 10$} & \\multicolumn{3}{c}{$10 < t \\le 20$} & \\multicolumn{4}{c}{$20 < t \\le 30$} \\tabularnewline\n    \\hline \n    $l_1$ & $\\Phi(l_1)$ & $l_1$ & $l_2$ & $\\Phi(l_1, l_2)$ & $l_1$ & $l_2$ & $l_3$ & $\\Phi(l_{1},l_{2},l_{3})$ \\tabularnewline\n    \\hline \n    \\hline\n    3 & 1 & 3 & 1 & 1 & 3 & 2 & 2 & $\\frac{2}{3}$ \\tabularnewline\n     & & 3 & 2 & 1 & 3 & 3 & 2 & $\\frac{2}{3}$ \\tabularnewline\n     & & 3 & 3 & 1 & 3 & 3 & 3 & 1 \\tabularnewline\n    \\hline \n  \\end{tabular}\n  \\caption{Survival signature of the PMS shown in \\cref{fig:pms1}}\n  \\label{tab:pms1.survsig}\n\\end{table}\n\nWe can obtain the conditional reliability of components using the conditional failure rate of the component in each phase.  \\Cref{eq:survsigPMS1a} then renders the reliability of the PMS as a whole. The results are shown in \\cref{tab:pms1.R} and \\cref{fig:pms1.R}.  These results concord with those found using an independent method in \\citep{zang1999bdd}.\n\nOf note is the jump discontinuity in the reliability function at $t=20$, as shown in \\cref{fig:pms1.R}. This occurs because a failure of component $A$ during phase 2 does not necessarily cause failure of the system at that point, so long as at least one of components $B$ or $C$ work.  However, in this situation the PMS will fail instantaneously upon commencing phase 3 at $t=20^+$.  Consequently, the size of the jump discontinuity in fact corresponds to the probability of the event $\\{ A$ fails in phase 2, but the system still functions$\\}$.\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular*}{0.85\\textwidth}{@{\\extracolsep{\\fill}}ccccccc@{\\extracolsep{\\fill}}}\n    \\hline\n    $t$ & $0$ & $10^{-}$ & $10^{+}$ & $20^{-}$ & $20^{+}$ & $30$ \\tabularnewline\n    $R$ & 1 & 0.99700 & 0.997700 & 0.997700 & 0.99601 & 0.99501 \\tabularnewline\n    \\hline \n  \\end{tabular*}\n  \\caption{Reliability of the PMS in example 1}\n  \\label{tab:pms1.R}\n\\end{table}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{PMS1R}\n  \\caption{Reliability of the PMS in example 1.}\n  \\label{fig:pms1.R}\n\\end{figure}\n\n\n\\subsection{Example 2}\n\nFor the PMS shown in \\cref{fig:pms2}, phases 1, 2 and 3 last for 10, 90 and 100 hours respectively.  All components in each phase are of the same type and the lifetime distribution of these components follows a two-parameter Weibull distribution.  \\Cref{tab:pms2.pars} summarises the distribution information of the components in each phase.\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular*}{0.75\\textwidth}{@{\\extracolsep{\\fill}}lccc@{\\extracolsep{\\fill}}}\n    \\hline\n    Parameter & Phase 1 & Phase 2 & Phase 3 \\tabularnewline\\hline \n    Scale & 250 & 1000 & 300 \\tabularnewline\n    Shape & 2.6 & 3.2 & 2.6 \\tabularnewline\n    \\hline \n  \\end{tabular*}\n  \\caption{Conditional distribution information of the components in each phase}\n  \\label{tab:pms2.pars}\n\\end{table}\n\nAs described in \\cref{sec:pms.diff}, if some components of the same type appear in a phase and also appear in some subsequent phases --- but not simultaneously --- then these components should be considered as different types of component. For this example, this means that despite the fact they all share a common failure rate within phases, components $A$ and $E$ need to be labelled as type 1 and the remainder as type 2, because ageing will have been different.\n\nThe survival signatures of this PMS are shown in \\cref{tab:pms2.survsig}, with rows where $\\Phi(l_{11},l_{12})=0, \\Phi(l_{11},l_{12},l_{21},l_{22})=0$ and $\\Phi(l_{11},l_{12},l_{21},l_{22},l_{32})=0$ suppressed.  The reliability of the PMS is shown in \\cref{tab:pms2.R} and \\cref{fig:pms2.R}.\n\nWe again see a jump discontinuity in the reliability curve depicted in \\cref{fig:pms2.R}, at $t=10$.  In this instance, if component $E$ fails during phase 1 the system will still function, but instantaneous failure will occur once phase 2 commences. This is evident in \\cref{tab:pms2.R}, which shows the jump discontinuity is of size $2.3 \\times 10^{-4}$.  Indeed, this should correspond to the probability that the system survives phase 1 but with component $E$ failing during that phase.  That is:\n\\begin{align*}\n   & \\mathbb{P}(A, B \\mbox{ function} \\cap E \\mbox{ fails in phase 1}) \\\\\n   & \\quad= \\mathbb{P}(E \\mbox{ fails in phase 1}) \\mathbb{P}(A, B \\mbox{ function} \\,|\\, E \\mbox{ fails in phase 1}) \\\\\n   & \\quad = \\int_0^{10} b a^{-b} t^{b-1} e^{-(t\/a)^b}\\,dt \\left(1 - \\int_0^{10} b a^{-b} t^{b-1} e^{-(t\/a)^b}\\,dt \\right)^2 \\\\\n  & \\quad \\approx 2.3 \\times 10^{-4}  \\ \\ \\mbox{for } a=250, b=2.6\n\\end{align*}\nas required.  Hence, PMS can exhibit jump discontinuities where probability mass from non-critical failures in one phase accumulate onto phase change boundaries when the system layout switches.\n\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular}{llllllllllllll}\n    \\hline\n    \\multicolumn{3}{l}{The first phase} & \\multicolumn{5}{l}{The first two phases} & \\multicolumn{6}{l}{All phases}         \\tabularnewline\n    \\hline \n    $l_{11}$        & $l_{12}$       & $\\Phi_1$       & $l_{11}$   & $l_{12}$   & $l_{21}$   & $l_{22}$  & $\\Phi_{12}$  & $l_{11}$ & $l_{12}$ & $l_{21}$ & $l_{22}$ & $l_{32}$ & $\\Phi_S$  \\tabularnewline\n    \\hline\n    \\hline\n    1           & 1          & 1        & 1      & 1      & 1      & 1     & 1\/2   & 1    & 1    & 1    & 1    & 1    & 1\/2 \\tabularnewline\n    2           & 1          & 1        & 2      & 1      & 1      & 1     & 1\/2   & 1    & 1    & 1    & 1    & 2    & 1\/2 \\tabularnewline\n                &            &          & 2      & 1      & 2      & 0     & 1     & 1    & 1    & 1    & 1    & 3    & 1\/2 \\tabularnewline\n                &            &          & 2      & 1      & 2      & 1     & 1     & 2    & 1    & 1    & 1    & 1    & 1\/2 \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 1    & 1    & 2    & 1\/2 \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 1    & 1    & 3    & 1\/2 \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 2    & 0    & 1    & 1   \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 2    & 0    & 2    & 1   \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 2    & 1    & 1    & 1   \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 2    & 1    & 2    & 1   \\tabularnewline\n                &            &          &        &        &        &       &       & 2    & 1    & 2    & 1    & 3    & 1  \\tabularnewline\n    \\hline\n  \\end{tabular}\n  \\caption{Survival signature of the PMS shown in \\cref{fig:pms2}}\n  \\label{tab:pms2.survsig}\n\\end{table}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{PMS2R}\n  \\caption{Reliability of the PMS in example 2.}\n  \\label{fig:pms2.R}\n\\end{figure}\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular*}{0.85\\textwidth}{@{\\extracolsep{\\fill}}ccccccc@{\\extracolsep{\\fill}}}\n    \\hline\n    $t$ & $0$ & $10^{-}$ & $10^{+}$ & $100^{-}$ & $100^{+}$ & $200$ \\tabularnewline\n    $R$ & 1 & 0.999768 & 0.999536 & 0.999086 & 0.999086 & 0.998910 \\tabularnewline\n    \\hline \n  \\end{tabular*}\n  \\caption{Reliability of the PMS in example 2.}\n  \\label{tab:pms2.R}\n\\end{table}\n\n\\subsection{Example 3}\n\nIn this final example, we replicate the space application mission discussed by Zang \\citep{zang1999bdd} and Mural \\citep{mural1999dependability}.  This example includes the full complexity of real-world PMSs, where there is now heterogeneity of component types within phases.  This means that multiple component types arise necessarily and not merely as a side effect of identical components appearing in differing phases.  There are five phases involved in this space mission: launch is the first phase, followed by Hibern.1, Asteroid, Hibern.2, and finally Comet.  The reliability block diagram is shown in \\cref{fig:pms3}. The five phases last for 48, 17520, 672, 26952 and 672 hours, respectively. The failure rates of the components in each phase are given in \\cref{tab:pms3.lambda}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1\\textwidth]{PMS3}\n  \\caption{Reliability block diagram of the space application.}\n  \\label{fig:pms3}\n\\end{figure}\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular}{llllll}\n    \\hline\n                   & Phase1  & Phase 2 & Phase 3 & Phase 4 & Phase 5 \\tabularnewline\n    \\hline\n    $H_a$, $H_b$, $H_c$, $H_d$ & $10^{-5}$  & $10^{-6}$  & $10^{-5}$  & $10^{-6}$  & $10^{-5}$ \\tabularnewline\n    $L_a$, $L_b$         & $5 \\times 10^{-5}$  & 0       & 0       & 0       & 0      \\tabularnewline\n    $A_a$, $A_b$         & 0       & 0       & $10^{-5}$  & 0       & 0      \\tabularnewline\n    $C_a$, $C_b$         & 0       & 0       & 0       & 0       & $10^{-4}$ \\tabularnewline\n    \\hline\n  \\end{tabular}\n  \\caption{Failure rates of the components.}\n  \\label{tab:pms3.lambda}\n\\end{table}\n\nAs shown in \\cref{tab:pms3.types}, in order to calculate the reliability of the PMS, the 4 `real' component types must be divided into 5 types when using the methodology presented in this paper.  That is, although $H_a, H_b, H_c,$ and $H_d$ have homogeneous failure rates throughout all phases, because they do not always appear together they will exhibit different ageing.  Consequently, these are split into two `pseudo' types.\n\nThe result of analysing the reliability of this PMS is shown in \\cref{tab:pms3.R} and \\cref{fig:pms3.R}. The results found using the new methodology we have presented in this paper are in agreement with the entirely independent method in \\citep{zang1999bdd}. \n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\begin{tabular*}{0.85\\textwidth}{@{\\extracolsep{\\fill}}ccccc@{\\extracolsep{\\fill}}}\n    \\hline\n    Type 1 & Type 2 & Type 3 & Type 4 & Type 5 \\tabularnewline\n    $H_a$, $H_b$ & $H_c$, $H_d$ & $L_a$, $L_b$ & $A_a$, $A_b$ & $C_a$, $C_b$ \\tabularnewline\n    \\hline \n  \\end{tabular*}\n  \\caption{Types of components in example 3.}\n  \\label{tab:pms3.types}\n\\end{table}\n\n\\begin{table}\n  \\centering\\renewcommand{\\arraystretch}{1.25}\n  \\resizebox{0.9\\textwidth}{!}{%\n  \\begin{tabular}{ccccccccccc}\n    \\hline\n    $t$ & $0$ & $48^{-}$ & $48^{+}$ & $17568^{-}$ & $17568^{+}$ & $18240^{-}$ & $18240^{+}$ & $45192^{-}$ & $45192^{+}$ & $45864$ \\tabularnewline\n    $R$ & 1 & 0.99999 & 0.99999 & 0.99968 & 0.99964 & 0.99862 & 0.99862 & 0.99670 & 0.99600 & 0.98943 \\tabularnewline\n    \\hline \n  \\end{tabular}}\n  \\caption{Reliability of the PMS in example 3.}\n  \\label{tab:pms3.R}\n\\end{table}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{PMS3R}\n  \\caption{Reliability of the PMS in example 3, with inset graph providing blown-up detail of first 200 hours of operation.}\n  \\label{fig:pms3.R}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nComputing the reliability of a PMS is considerably more complex than that of a non-PMS, due to the variation in system structure between phases and the dependencies between component failures in different phases.  Consequently, reliability analysis of PMSs has become one of the most challenging topics in the field of system reliability evaluation and maintenance engineering in recent decades.  Despite some progress towards efficient and effective methods for measuring the reliability of PMS, it is still difficult to analyze large systems without considerable computational expense and even where it is possible, many methods fail to convey intuition about the reliability of the system.\n\nIn this paper, a new and efficient method for reliability analysis of PMS is proposed using survival signature. Signatures have been proven to be an efficient method for estimating the reliability of systems. A new kind of survival signature is derived to represent the structure function of the PMS. Then the proposed survival signature is applied to calculate the reliability of the PMS. Reliability analysis of a system using signatures could separate the system structure from the component probabilistic failure distribution. Therefore, the proposed approach is easy to be implemented in practice and has high computational efficiency.\n\n\nNote that reliability analysis of PMSs with multiple failure mode components is not studied in this paper. In practice the components may perhaps have more than one failure mode. In ongoing work, the authors are considering component importance analysis, extending work such as \\cite{feng2016imprecise, eryilmaz2018marginal} to PMSs.\n\\section*{Acknowledgements}\n\nThe authors gratefully acknowledge the support of National Natural Science Foundation of China (51575094), China Postdoctoral Science Foundation (2017M611244), China Scholarship Council (201706085013) and Fundamental Research Funds for the Central Universities (N160304004).\n\nThis work was performed whilst the first author was a visitor at Durham University.\n\n\\section*{References}\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\tThe classical binomial process  has been studied by  \\citet{jakeman} and has been used to model fluctuations in a train of events in quantum optics.\n\tRecall that the classical binomial process $ \\mathcal{N}(t)$, $t\\ge 0 $, with birth rate $\\lambda>0$ and\n\t\tdeath rate $\\mu>0$, has  state probabilities\n\t\t$p_n(t) = \\Pr \\{ \\mathcal{N}(t) = n | \\mathcal{N}(0) = M \\}$ which solve the following Cauchy problem: \n\t\t\\begin{align}\n\t\t\t\\label{state}\n\t\t\t\\begin{cases}\n\t\t\t\t\\frac{\\mathrm d}{\\mathrm dt} p_n(t) = \\mu (n+1) p_{n+1}(t) - \\mu n p_n(t) - \\lambda\n\t\t\t\t(N-n) p_n(t) \\\\\n\t\t\t\t\\qquad \\qquad + \\lambda (N-n+1) p_{n-1}(t), \\qquad \\qquad \\qquad 0 \\leq n \\leq N, \\\\\n\t\t\t\tp_n(0) =\n\t\t\t\t\\begin{cases}\n\t\t\t\t\t1, & n=M, \\\\\n\t\t\t\t\t0, & n \\neq M.\n\t\t\t\t\\end{cases}\n\t\t\t\\end{cases}\n\t\t\\end{align}\n\t\tThe initial number of individuals is $M \\geq 1$, and $N \\geq M$.\n\n\t\tNotice that the binomial process has a completely different behaviour compared to the\n\t\tclassical linear birth-death process. Here the birth rate is proportional to the\n\t\tdifference between a larger fixed number  and the number of\tindividuals present while the\n\t\tdeath rate remains linear. The whole evolution of the binomial process develops\n\t\tin the region $[0, N]$.\n\t\tFurthermore it is shown that at large times, an equilibrium is reached and displays a binomial distribution.\n\t\n\t\tFrom \\eqref{state}, it is straightforward to realise that the generating function\n\t\t\\begin{align}\n\t\t\tQ(u,t) = \\sum_{n=0}^N (1-u)^n p_n(t), \\qquad |1-u| \\leq 1,\n\t\t\\end{align}\n\t\tis the solution to\n\t\t\\begin{align}\n\t\t\t\\label{fgp}\n\t\t\t\\begin{cases}\n\t\t\t\t\\frac{\\partial}{\\partial t} Q(u,t) = -\\mu u \\frac{\\partial}{\\partial u} Q(u,t) - \\lambda u\n\t\t\t\t(1-u) \\frac{\\partial}{\\partial u} Q(u,t) -\\lambda N u Q(u,t), \\\\\n\t\t\t\tQ(u,0) = (1-u)^M.\n\t\t\t\\end{cases}\n\t\t\\end{align}\nMoreover, \\citet{jakeman}  showed that at large times, the evolving population follows a  binomial distribution with parameter $\\lambda \/ (\\lambda  + \\mu)$.  \n\nIn this paper, we  propose a fractional generalisation of the classical binomial process. The fractional generalization includes non-markovian and rapidly dissipating or bursting birth-death processes at small and regular times. We also derive more statistical and related properties of the newly developed fractional stochastic process, which are deemed useful in real applications.  Note that the theory and results presented here may have applications beyond quantum optics and may be of interest in other disciplines. \tAs in the preceding works on fractional Poisson process (e.g.\\ \\citet{laskin}) and other\n\t\tfractional point processes (see e.g.\\ \\citet{cah,pol}), fractionality is obtained by replacing the\n\t\tinteger-order derivative in the governing differential equations with a fractional-order derivative. \tIn particular, we use the Caputo fractional derivative of a well-behaved function $f(t)$ and is defined as\n\t\t\\begin{align}\n\t\t\t\\label{caputo}\n\t\t\t\\frac{\\mathrm d^\\nu}{\\mathrm dt^\\nu} f(t) = \\frac{1}{\\Gamma(m-\\nu)} \\int_0^t \\frac{\\frac{\\mathrm d^m}{\n\t\t\t\\mathrm d \\tau^m}f(\\tau)}{(t-\\tau)^{\\nu-m+1}}\n\t\t\t\\mathrm d\\tau, \\qquad m=\\lceil \\nu \\rceil,\n\t\t\\end{align}\n\twhere ``$\\lceil y \\rceil$\" is the smallest integer that is not less than  $y$. Note that the Caputo fractional derivative operator is in practice a convolution of the standard derivative\nwith a power law kernel  which  adds more memory in the process. This characteristic  is  certainly an improvement  from a physical viewpoint. By simple substitution, we obtain the following initial value problems for the probability generating function and the state\n\t\tprobabilities:\n\t\t\\begin{align}\n\t\t\t\\label{fgpfrac}\n\t\t\t\\begin{cases}\n\t\t\t\t\\frac{\\partial^\\nu}{\\partial t^\\nu} Q^\\nu(u,t) = -\\mu u \\frac{\\partial}{\\partial u} Q^\\nu(u,t) - \\lambda u\n\t\t\t\t(1-u) \\frac{\\partial}{\\partial u} Q^\\nu(u,t) -\\lambda N u Q^\\nu(u,t), \\\\\n\t\t\t\tQ^\\nu(u,0) = (1-u)^M, \\qquad \\qquad \\qquad |1-u|\\leq 1,\n\t\t\t\\end{cases}\n\t\t\\end{align}\t\t\n\t\t\\begin{align}\n\t\t\t\\label{statefrac}\n\t\t\t\\begin{cases}\n\t\t\t\t\\frac{\\mathrm d^\\nu}{\\mathrm dt^\\nu} p_n^\\nu(t) = \\mu (n+1) p_{n+1}^\\nu(t) - \\mu n p_n^\\nu(t) - \\lambda\n\t\t\t\t(N-n) p_n^\\nu(t) \\\\\n\t\t\t\t\\qquad \\qquad \\qquad + \\lambda (N-n+1) p_{n-1}^\\nu(t), & 0 \\leq n \\leq N, \\\\\n\t\t\t\tp_n^\\nu(0) =\n\t\t\t\t\\begin{cases}\n\t\t\t\t\t1, & n=M, \\\\\n\t\t\t\t\t0, & n \\neq M,\n\t\t\t\t\\end{cases}\n\t\t\t\\end{cases}\n\t\t\\end{align}\n\t\twhere $\\nu \\in (0,1]$.\n\nWe organized the rest of the paper as follows. In Section 2, the statistical properties of the fractional binomial process are derived by solving the preceding initial-value problem.  Section 3 explored the sub-models that are directly extractable from the fractional binomial process. We then conclude the paper  by providing more discussions and future extensions of the study in Section 4.\n\n\n\t\\section{Main properties of the fractional binomial process}\n\t\t\\label{se}\n\t\t\n\t\tFirstly, we prove a subordination relation which is of fundamental importance to deriving many of our results.\n\n\t\t\\begin{thm}\n\t\t\t\\label{sub}\n\t\t\tThe fractional binomial process $\\mathcal{N}^\\nu(t)$ has the following one-dimensional representation:\n\t\t\t\\begin{align}\n\t\t\t\t\\mathcal{N}^\\nu(t) \\overset{\\text{d}}{=} \\mathcal{N}(V_t^\\nu), \n\t\t\t\\end{align}\n\t\t\twhere $\\mathcal{N}(t)$ is a classical binomial process, $V_t^\\nu$, $t\\ge 0$, is the inverse process\n\t\t\tof the $\\nu$-stable subordinator (see e.g.\\ \\citet{meer}),  $t \\ge 0$, and $\\nu \\in (0,1]$.\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tLet $\\text{Pr} \\{ V_t^\\nu \\in \\mathrm ds \\} = h(s,t) \\, \\mathrm ds$\n\t\t\t\tbe the law of the inverse $\\nu$-stable subordinator. We now show that\n\t\t\t\t\\begin{align}\n\t\t\t\t\tQ^\\nu(u,t) = \\sum_{n=0}^N (1-u)^n p_n^\\nu(t) = \\int_0^\\infty Q(u,s) \\, h(s,t) \\, \\mathrm ds\n\t\t\t\t\\end{align}\n\t\t\t\tsatisfy the fractional differential equation \\eqref{fgpfrac}. We can then write\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\frac{\\partial^\\nu}{\\partial t^\\nu} \\int_0^\\infty Q(u,s) h(s,t) \\mathrm ds\n\t\t\t\t\t= \\int_0^\\infty Q(u,s) \\frac{\\partial^\\nu}{\\partial t^\\nu} h(s,t) \\mathrm ds.\n\t\t\t\t\\end{align}\n\t\t\t\tSince it can be easily verified that $h(s,t)$ is a solution to the fractional equation\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\frac{\\partial^\\nu}{\\partial t^\\nu} h(s,t) = - \\frac{\\partial}{\\partial s} h(s,t),\n\t\t\t\t\\end{align}\n\t\t\t\twe readily obtain\n\t\t\t\t\\begin{align}\n\t\t\t\t\t& \\frac{\\partial^\\nu}{\\partial t^\\nu} Q^\\nu(u,t) \\\\\n\t\t\t\t\t& = - \\int_0^\\infty Q(u,s)\n\t\t\t\t\t\\frac{\\partial}{\\partial s} h(s,t) \\mathrm ds \\notag \\\\\n\t\t\t\t\t& = \\left. - h(s,t) Q(u,s) \\right|_{s=0}^\\infty + \\int_0^\\infty h(s,t) \\frac{\\partial}{\\partial s}\n\t\t\t\t\tQ(u,s) \\mathrm ds \\notag \\\\\n\t\t\t\t\t& = \\int_0^\\infty \\left[ -\\mu u \\frac{\\partial}{\\partial u} Q(u,s)\n\t\t\t\t\t-\\lambda u(1-u) \\frac{\\partial}{\\partial u} Q(u,s)\n\t\t\t\t\t-\\lambda NuQ(u,s) \\right] h(s,t) \\mathrm ds \\notag \\\\\n\t\t\t\t\t& = -\\mu u \\frac{\\partial}{\\partial u} Q^\\nu(u,t)\n\t\t\t\t\t-\\lambda u(1-u) \\frac{\\partial}{\\partial u} Q^\\nu(u,t)\n\t\t\t\t\t-\\lambda NuQ^\\nu(u,t). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\\end{proof}\n\t\t\t\\begin{flushright} \\qed \\end{flushright}\n\t\t\\end{thm}\n\t\t\n\t\tIn the following theorem, we derive the expected number of individuals $\\mathbb{E}\\,\\mathcal{N}^\\nu(t)$ or the expected population size of the fractional binomial process  at any time $t \\ge 0$.\n\t\t\\begin{thm}\n\t\t\tFor the fractional binomial process $\\mathcal{N}^\\nu(t)$, $t \\ge 0$, $\\nu \\in (0,1]$, we have\n\t\t\t\\begin{align}\n\t\t\t\t\\label{mea}\n\t\t\t\t\\mathbb{E}\\,\\mathcal{N}^\\nu(t) = \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)\n\t\t\t\tE_{\\nu,1} \\left( -(\\lambda+\\mu) t^\\nu \\right) + N\\frac{\\lambda}{\\lambda+\\mu},\n\t\t\t\\end{align}\n\twhere\n\t\\[\n\tE_{\\alpha, \\beta} \\left( \\xi \\right) = \\sum\\limits_{r=0}^\\infty \\frac{\\xi^r}{\\Gamma (\\alpha r + \\beta) }\t\n\t\\]\nis the Mittag-Leffler function.\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tBy considering that\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\label{sharp}\n\t\t\t\t\t\\left. -\\frac{\\partial}{\\partial u} Q^\\nu(u,t) \\right|_{u=0} = \\mathbb{E}\\,\\mathcal{N}^\\nu(t)\n\t\t\t\t\\end{align}\n\t\t\t\tand on the base of \\eqref{fgpfrac},\n\t\t\t\twe can write\n\t\t\t\t\\begin{align}\n\t\t\t\t\t-\\frac{\\partial^\\nu}{\\partial t^\\nu} \\frac{\\partial}{\\partial u} Q^\\nu(u,t) = {} &\n\t\t\t\t\t\\mu \\left( \\frac{\\partial}{\\partial u} Q^\\nu(u,t) + u \\frac{\\partial^2}{\\partial u^2}\n\t\t\t\t\tQ^\\nu(s,t) \\right) \\\\\n\t\t\t\t\t& + \\lambda \\left( \\frac{\\partial}{\\partial u} Q^\\nu(u,t)\n\t\t\t\t\t+ u \\frac{\\partial^2}{\\partial u^2} Q^\\nu(u,t) \\right) \\notag \\\\\n\t\t\t\t\t& - \\lambda \\left( 2 u \\frac{\\partial}{\\partial u} Q^\\nu(u,t) + u^2\n\t\t\t\t\t\\frac{\\partial^2}{\\partial u^2}Q^\\nu(u,t) \\right) \\notag \\\\\n\t\t\t\t\t& + \\lambda N \\left( Q^\\nu(u,t)\n\t\t\t\t\t+ u \\frac{\\partial}{\\partial u} Q^\\nu(u,t) \\right), \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tthus leading to the Cauchy problem\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\label{bel}\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\frac{\\mathrm d^\\nu}{\\mathrm dt^\\nu} \\mathbb{E} \\mathcal{N}^\\nu(t) =\n\t\t\t\t\t\t-(\\mu +\\lambda) \\mathbb{E} \\mathcal{N}^\\nu(t) + \\lambda N, \\\\\n\t\t\t\t\t\t\\mathbb{E} \\mathcal{N}^\\nu(0) = M.\n\t\t\t\t\t\\end{cases}\n\t\t\t\t\\end{align}\n\t\t\t\tThe solution to \\eqref{bel} can be written as (using formula (4.1.65) of \\citet{kilbas})\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\mathbb{E}\\,\\mathcal{N}^\\nu(t)\n\t\t\t\t\t= {} & M E_{\\nu,1} \\left(-(\\lambda+\\mu)t^\\nu\\right) \\\\\n\t\t\t\t\t& + \\int_0^t (t-s)^{\\nu-1}\n\t\t\t\t\tE_{\\nu,\\nu} \\left(-(\\lambda+\\mu)(t-s)^\\nu\\right) \\lambda N \\mathrm ds \\notag \\\\\n\t\t\t\t\t= {} & M E_{\\nu,1} \\left(-(\\lambda+\\mu)t^\\nu\\right) + \\lambda N \\int_0^t y^{\\nu-1} E_{\\nu,\\nu}\n\t\t\t\t\t\\left(-(\\lambda+\\mu) y^\\nu\\right) \\mathrm dy \\notag \\\\\n\t\t\t\t\t= {} & M E_{\\nu,1} \\left(-(\\lambda+\\mu)t^\\nu\\right) + \\lambda N \\biggl| -\\frac{\n\t\t\t\t\tE_{\\nu,1}\\left( -(\\lambda+\\mu)y^\\nu \\right)}{(\\lambda+\\mu)} \\biggr|_0^t \\notag \\\\\n\t\t\t\t\t= {} & M E_{\\nu,1} \\left(-(\\lambda+\\mu)t^\\nu\\right) - \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\tN \\left( E_{\\nu,1} \\left( -(\\lambda+\\mu) t^\\nu \\right) -1 \\right) \\notag \\\\\n\t\t\t\t\t= {} & \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)E_{\\nu,1}\n\t\t\t\t\t\\left( -(\\lambda+\\mu) t^\\nu \\right) + N\\frac{\\lambda}{\\lambda+\\mu}. \\notag\n\t\t\t\t\\end{align}\n\t\t\t\\end{proof}\n\t\t\t\\begin{flushright} \\qed \\end{flushright}\n\t\t\\end{thm}\n\t\tFigure \\ref{afig} shows the mean value \\eqref{mea} in both cases $\\left[ M-N\\lambda\/(\\lambda+\\mu)\n\t\t\\right] < 0$ and $\\left[ M-N\\lambda\/(\\lambda+\\mu)\n\t\t\\right] > 0$ for specific values of the remaining parameters.\n\t\tNote also that when $M=N\\lambda\/(\\lambda+\\mu)$ the mean value $\\mathbb{E}\\,\n\t\t\\mathcal{N}^\\nu(t) = N\\lambda\/(\\lambda+\\mu)$ is constant.\n\t\n\t\t\\begin{figure}[h!t!tb!p!]\n\t\t\t\\centering\n\t\t\t\\includegraphics[height=2.5in, width=2.2in]{p1.pdf}\n\t\t\t\\includegraphics[height=2.5in, width=2.2in]{p2.pdf}\n\t\t \n\t\t\n\t\t\t\\caption{\\label{afig}The mean value of the fractional binomial process $\\mathbb{E}\\,\n\t\t\t\t\\mathcal{N}^\\nu(t)$. For both graphs we have $N=100$, $M=40$, $\\nu=0.7$. The rates are respectively\n\t\t\t\t$(\\lambda,\\mu)=(1,1)$ (left) and $(\\lambda,\\mu)=(1,3)$ (right).}\n\t\t\\end{figure}\n\n\t\n\t\tWe now proceed to deriving the variance $\\mathbb{V}\\text{ar} \\, \\mathcal{N}^\\nu(t)$ of the fractional\n\t\tbinomial process, starting from the second factorial moment.\n\t\t\n\t\t\\begin{thm}\t\t\n\t\t\t\t\tFor the fractional binomial process $\\mathcal{N}^\\nu(t)$, $t \\ge 0$, $\\nu \\in (0,1]$, we have\n\t\t\t\\begin{align}\n\t\t\t\t\\label{va}\n\t\t\t\t\\mathbb{V}\\text{ar} & \\, \\mathcal{N}^\\nu(t) \\\\\t\t\t\n\t\t\t\t= {} & \\left( \\frac{\\lambda^2 N(N-1)}{(\\lambda +\\mu)^2} -\\frac{2\\lambda M(N-1)}{\\lambda+\\mu}\n\t\t\t\t+ M(M-1) \\right) E_{\\nu,1} (-2(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t& + \\left( \\frac{2\\lambda^2 N}{(\\lambda+\\mu)^2} - \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t(N+2M) +M \\right)\n\t\t\t\tE_{\\nu,1} (-(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t& - \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)^2 \\left( E_{\\nu,1}(-(\\lambda+\\mu)t^\\nu) \\right)^2\n\t\t\t\t+ \\frac{N \\lambda\\mu}{(\\lambda+\\mu)^2}. \\notag\n\t\t\t\\end{align}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tFrom \\eqref{fgpfrac}, we have\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\frac{\\partial^\\nu}{\\partial t^\\nu} \\frac{\\partial^2}{\\partial u^2} Q^\\nu(u,t) = {} &\n\t\t\t\t\t-\\mu \\frac{\\partial^2}{\\partial u^2} Q^\\nu(u,t) -\\mu \\left( \\frac{\\partial^2}{\\partial u^2}\n\t\t\t\t\tQ^\\nu(u,t) + u \\frac{\\partial^3}{\\partial u^3} Q^\\nu(u,t) \\right) \\\\\n\t\t\t\t\t& -\\lambda \\left( (1-2u) \\frac{\\partial^2}{\\partial u^2} Q^\\nu(u,t) -2 \\frac{\\partial}{\\partial u}\n\t\t\t\t\tQ^\\nu(u,t) \\right) \\notag \\\\\n\t\t\t\t\t& - \\lambda \\left( (1-2u) \\frac{\\partial^2}{\\partial u^2}Q^\\nu(u,t)\n\t\t\t\t\t+(u-u^2)\\frac{\\partial^3}{\\partial u^3} Q^\\nu(u,t) \\right) \\notag \\\\\n\t\t\t\t\t& -\\lambda N \\frac{\\partial}{\\partial u} Q^\\nu(u,t) - \\lambda N \\left( \\frac{\\partial}{\\partial u}\n\t\t\t\t\tQ^\\nu(u,t) + u \\frac{\\partial^2}{\\partial u^2} Q^\\nu(u,t) \\right). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tRecalling \\eqref{sharp} and the equality\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\left. \\frac{\\partial^2}{\\partial u^2} Q^\\nu(u,t) \\right|_{u=0} = \\mathbb{E}\n\t\t\t\t\t\\left( \\mathcal{N}^\\nu(t)(\\mathcal{N}^\\nu(t) -1) \\right) = H^\\nu(t),\n\t\t\t\t\\end{align}\n\t\t\t\twe obtain\n\t\t\t\t\t\\begin{align}\n\t\t\t\t\t\\label{sarp}\n\t\t\t\t\t\\frac{\\mathrm d^\\nu}{\\mathrm dt^\\nu} H^\\nu(t) & =\n\t\t\t\t\t-2 \\mu H^\\nu(t) -2\\lambda H^\\nu(t) -2\\lambda \\mathbb{E}\\,\\mathcal{N}^\\nu(t) + 2\\lambda N\n\t\t\t\t\t\\mathbb{E}\\,\\mathcal{N}^\\nu(t) \\\\\n\t\t\t\t\t& = -2 (\\lambda+\\mu) H^\\nu(t) + 2 \\lambda (N-1) \\mathbb{E}\\,\\mathcal{N}^\\nu(t). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tBy substituting \\eqref{mea} into \\eqref{sarp}, we arrive at the Cauchy problem\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\label{cacio}\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\frac{\\mathrm d^\\nu}{\\mathrm dt^\\nu} H^\\nu(t) = -2 (\\lambda+\\mu) H^\\nu(t)\n\t\t\t\t\t\t+ 2 \\lambda (N-1) \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)\n\t\t\t\t\t\tE_{\\nu,1} \\left( -(\\lambda+\\mu)t^\\nu \\right) \\\\\n\t\t\t\t\t\t\\qquad \\qquad \\qquad + 2 \\lambda^2 N (N-1) \\frac{1}{\\lambda+\\mu} \\\\\n\t\t\t\t\t\tH^\\nu(0) = M(M-1),\n\t\t\t\t\t\\end{cases}\n\t\t\t\t\\end{align}\n\t\t\t\tthat can be solved using the Laplace transform $\\widetilde{H}^\\nu(z) = \\int_0^\\infty\n\t\t\t\te^{-zt} H^\\nu(t)\\, \\mathrm dt$ as follows:\n\t\t\t\t\\begin{align}\n\t\t\t\t\tz^\\nu \\widetilde{H}^\\nu(z) &- z^{\\nu-1} M(M-1) \\\\\n\t\t\t\t\t= {} & -2 (\\lambda+\\mu) \\widetilde{H}^\\nu(z) + 2 \\lambda (N-1)\n\t\t\t\t\t\\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right) \\frac{z^{\\nu-1}}{z^\\nu + (\\lambda+\\mu)} \\notag \\\\\n\t\t\t\t\t& + \\frac{1}{z} 2 \\lambda^2 N(N-1)\\frac{1}{\\lambda+\\mu}. \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tThe Laplace transform then reads\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\label{elastic}\n\t\t\t\t\t\\widetilde{H}^\\nu(z) = {} & M(M-1) \\frac{z^{\\nu-1}}{z^\\nu+2(\\lambda+\\mu)} \\\\\n\t\t\t\t\t& +2\\lambda (N-1)\n\t\t\t\t\t\\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right) \\frac{z^{\\nu-1}}{(z^\\nu+(\\lambda+\\mu))\n\t\t\t\t\t(z^\\nu+2(\\lambda+\\mu))} \\notag \\\\\n\t\t\t\t\t& + 2 \\lambda^2 N(N-1) \\frac{1}{\\lambda+\\mu} \\cdot \\frac{z^{-1}}{z^\\nu\n\t\t\t\t\t+2(\\lambda+\\mu)} \\notag \\\\\n\t\t\t\t\t= {} & M(M-1) \\frac{z^{\\nu-1}}{z^\\nu+2(\\lambda+\\mu)} \\notag \\\\\n\t\t\t\t\t& + \\frac{2\\lambda(N-1)}{\\lambda+\\mu}\n\t\t\t\t\t\\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right) \\left( \\frac{z^{\\nu-1}}{z^\\nu+(\\lambda+\\mu)}\n\t\t\t\t\t- \\frac{z^{\\nu-1}}{z^\\nu + 2(\\lambda+\\mu)} \\right)  \\notag \\\\\n\t\t\t\t\t& + 2 \\lambda^2 N(N-1) \\frac{1}{\\lambda+\\mu} \\cdot \\frac{z^{-1}}{z^\\nu\n\t\t\t\t\t+2(\\lambda+\\mu)} \\notag.\n\t\t\t\t\\end{align}\n\t\t\t\tEquation \\eqref{elastic} then implies that\n\t\t\t\t\\begin{align}\n\t\t\t\t\tH^\\nu&(t) \\\\\n\t\t\t\t\t= {} & M(M-1) E_{\\nu,1} \\left( -2(\\lambda+\\mu)t^\\nu \\right) \\notag \\\\\n\t\t\t\t\t& +\\frac{2\\lambda (N-1)}{\\lambda+\\mu} \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)\n\t\t\t\t\t\\left( E_{\\nu,1}(-(\\lambda+\\mu)t^\\nu) - E_{\\nu,1}(-2(\\lambda+\\mu)t^\\nu) \\right) \\notag \\\\\n\t\t\t\t\t& + 2\\lambda^2 N(N-1)\\frac{1}{\\lambda+\\mu} t^\\nu E_{\\nu,\\nu+1} \\left( -2(\\lambda+\\mu)t^\\nu\n\t\t\t\t\t\\right). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tConsidering that $t^\\nu E_{\\nu,\\nu+1}(at^\\nu) = a^{-1} (E_{\\nu,1}(at^\\nu)-1)$, we obtain\n\t\t\t\t\\begin{align}\n\t\t\t\t\tH^\\nu(t) = {} & M(M-1) E_{\\nu,1} \\left( -2(\\lambda+\\mu)t^\\nu \\right) \\\\\n\t\t\t\t\t& +\t\\frac{2\\lambda(N-1)}{\\lambda+\\mu} \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)\n\t\t\t\t\tE_{\\nu,1} (-(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& - \\frac{2\\lambda (N-1)}{\\lambda+\\mu} \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)\n\t\t\t\t\tE_{\\nu,1} (-2(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& -\\frac{\\lambda^2}{(\\lambda+\\mu)^2} N(N-1) E_{\\nu,1} (-2(\\lambda+\\mu)t^\\nu)\n\t\t\t\t\t+ \\frac{\\lambda^2}{(\\lambda+\\mu)^2} N(N-1) \\notag \\\\\n\t\t\t\t\t= {} & M(M-1) E_{\\nu,1} \\left( -2(\\lambda+\\mu)t^\\nu \\right) \\notag \\\\\n\t\t\t\t\t& +\n\t\t\t\t\t\\frac{2\\lambda M(N-1)}{\\lambda+\\mu} E_{\\nu,1}(-(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& - \\frac{2 \\lambda^2 N(N-1)}{(\\lambda+\\mu)^2} E_{\\nu,1} (-(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& - \\frac{2\\lambda M(N-1)}{\\lambda+\\mu} E_{\\nu,1} (-2(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& + \\frac{2\\lambda^2 N(N-1)}{(\\lambda+\\mu)^2} E_{\\nu,1}(-2(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& -\\frac{\\lambda^2 N(N-1)}{(\\lambda+\\mu)^2} E_{\\nu,1}(-2(\\lambda+\\mu)t^\\nu)\n\t\t\t\t\t+\\frac{\\lambda^2 N(N-1)}{(\\lambda+\\mu)^2} \\notag \\\\\n\t\t\t\t\t= {} & \\frac{\\lambda^2 N(N-1)}{(\\lambda+\\mu)^2} \\notag \\\\\n\t\t\t\t\t& + E_{\\nu,1}(-2(\\lambda+\\mu)t^\\nu)\n\t\t\t\t\t\\left( \\frac{\\lambda^2 N(N-1)}{(\\lambda + \\mu)^2} - \\frac{2\\lambda M(N-1)}{\\lambda+\\mu}\n\t\t\t\t\t+M(M-1) \\right) \\notag \\\\\n\t\t\t\t\t& - E_{\\nu,1} (-(\\lambda+\\mu)t^\\nu) \\left( \\frac{2 \\lambda^2 N(N-1)}{(\\lambda+\\mu)^2}\n\t\t\t\t\t-\\frac{2\\lambda M (N-1)}{\\lambda+\\mu} \\right). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tThe variance can thus be written as\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\mathbb{V}\\text{ar} & \\, \\mathcal{N}^\\nu(t) \\\\\n\t\t\t\t\t= {} & H^\\nu(t) + \\mathbb{E}\\, \\mathcal{N}^\\nu(t)\n\t\t\t\t\t- (\\mathbb{E}\\, \\mathcal{N}^\\nu(t))^2 \\notag \\\\\n\t\t\t\t\t= {} & H^\\nu(t) + \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right) E_{\\nu,1} (-(\\lambda+\\mu)t^\\nu)\n\t\t\t\t\t\\notag \\\\\n\t\t\t\t\t& + N \\frac{\\lambda}{\\lambda+\\mu} - \\left( M-\\frac{\\lambda}{\\lambda+\\mu} \\right)^2\n\t\t\t\t\t\\left( E_{\\nu,1}(-(\\lambda+\\mu)t^\\nu) \\right)^2 \\notag \\\\\n\t\t\t\t\t& - N^2 \\frac{\\lambda^2}{(\\lambda+\\mu)^2} - 2 \\frac{N \\lambda}{\\lambda+\\mu}\n\t\t\t\t\t\\left( M-N \\frac{\\lambda}{\\lambda+\\mu} \\right) E_{\\nu,1}(-(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\lambda^2 N(N-1)}{(\\lambda +\\mu)^2} -\\frac{2\\lambda M(N-1)}{\\lambda+\\mu}\n\t\t\t\t\t+ M(M-1) \\right) E_{\\nu,1} (-2(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& + \\left( \\frac{2\\lambda^2 N}{(\\lambda+\\mu)^2} - \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\t(N+2M) +M \\right)\n\t\t\t\t\tE_{\\nu,1} (-(\\lambda+\\mu)t^\\nu) \\notag \\\\\n\t\t\t\t\t& - \\left( M-N\\frac{\\lambda}{\\lambda+\\mu} \\right)^2 \\left( E_{\\nu,1}(-(\\lambda+\\mu)t^\\nu) \\right)^2\n\t\t\t\t\t+ \\frac{N \\lambda\\mu}{(\\lambda+\\mu)^2}. \\notag\n\t\t\t\t\\end{align}\n\t\t\t\\end{proof}\n\t\t\t\\begin{flushright} \\qed \\end{flushright}\n\t\t\\end{thm}\n\t\t\n\t\tExploiting Theorem \\ref{sub}, we derive the explicit expression of the extinction\n\t\tprobability $p_0^\\nu(t) = \\text{Pr} \\{ \\mathcal{N}^\\nu(t) = 0 | \\mathcal{N}^\\nu(0) = M \\}$ below.\n\t\t\n\t\t\\begin{thm}\n\t\t\tThe extinction probability $p_0^\\nu(t) = \\text{Pr} \\{ \\mathcal{N}^\\nu(t) = 0 | \\mathcal{N}^\\nu(0) = M \\}$\n\t\t\tfor a fractional binomial process $\\mathcal{N}^\\nu(t)$, $t \\ge 0$ is\n\t\t\t\\begin{align}\n\t\t\t\t\\label{extinction}\n\t\t\t\tp_0^\\nu(t) = {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\sum_{r=0}^{N-M} \\binom{N-M}{r} \\left( \n\t\t\t\t\\frac{\\lambda}{\\mu} \\right)^r \\\\\n\t\t\t\t& \\times \\sum_{h=0}^M \\binom{M}{h} (-1)^h E_{\\nu,1}(-(r+h)(\\lambda+\\mu)t^\\nu). \\notag\n\t\t\t\\end{align}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tIt is known \\citep{jakeman} that the generating function $Q(u,t) = \\sum_{n=0}^N (1-u)^n p_n(t)$\n\t\t\t\tfor the classical binomial process can be written as\n\t\t\t\t\\begin{align}\n\t\t\t\t\tQ(u,t)\n\t\t\t\t\t= {} & \\left[ 1-\\left( 1-e^{-(\\mu+\\lambda)t} \\right)\\frac{\\lambda}{\\lambda+\\mu}u \\right]^{N-M} \\\\\n\t\t\t\t\t& \\times \\left[ 1-\\left( \\left(1-e^{-(\\mu+\\lambda)t}\\right)\\frac{\\lambda}{\\lambda+\\mu} +\n\t\t\t\t\te^{-(\\mu+\\lambda)t} \\right)u \\right]^M. \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tThis suggests that the extinction probability for the classical case can be written as\n\t\t\t\t\\begin{align}\n\t\t\t\t\tp_0(t) = {} & \\left[1-\\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\t+ \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\te^{-(\\mu+\\lambda)t}\\right]^{N-M} \\\\\n\t\t\t\t\t& \\times \\left[ 1-\\left(\\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\t-e^{-(\\mu+\\lambda)t} \\frac{\\lambda}{\\lambda+\\mu}+e^{-(\\mu+\\lambda)t}\\right) \\right]^M \\notag \\\\\n\t\t\t\t\t= {} & \\left[ \\frac{\\mu}{\\lambda+\\mu} + \\frac{\\lambda}{\\lambda+\\mu} e^{-(\\mu+\\lambda)t} \\right]^{N-M}\n\t\t\t\t\t\\left[ \\frac{\\mu}{\\lambda+\\mu} - \\frac{\\mu}{\\lambda+\\mu}e^{-(\\mu+\\lambda)t} \\right]^M \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{1}{\\lambda+\\mu} \\right)^N \\mu^M \\left( \\mu+\\lambda e^{-(\\lambda+\\mu)t} \\right)^{N-M}\n\t\t\t\t\t\\left( 1-e^{-(\\lambda+\\mu)t} \\right)^M \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\left( 1+\\frac{\\lambda}{\\mu}\n\t\t\t\t\te^{-(\\lambda+\\mu)t} \\right)^{N-M} \\left( 1-e^{-(\\lambda+\\mu)t} \\right)^M \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\sum_{r=0}^{N-M} \\binom{N-M}{r} \\left( \n\t\t\t\t\t\\frac{\\lambda}{\\mu} \\right)^r e^{-r(\\lambda+\\mu)t} \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{h=0}^M\n\t\t\t\t\t\\binom{M}{h} (-1)^h e^{-h(\\lambda+\\mu)t}\n\t\t\t\t\t\\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\sum_{r=0}^{N-M} \\binom{N-M}{r} \\left( \n\t\t\t\t\t\\frac{\\lambda}{\\mu} \\right)^r \\sum_{h=0}^M \\binom{M}{h} (-1)^h e^{-(r+h)(\\lambda+\\mu)t}. \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tUsing Theorem \\ref{sub}, we now obtain\n\t\t\t\t\\begin{align}\n\t\t\t\t\tp_0^\\nu(t) = {} & \\int_0^\\infty p_0(s) h(s,t) \\mathrm ds \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\t\\sum_{r=0}^{N-M} \\binom{N-M}{r}\\left( \\frac{\\lambda}{\\mu} \\right)^r \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{h=0}^M \\binom{M}{h}\n\t\t\t\t\t(-1)^h \\int_0^\\infty e^{-(r+h)(\\lambda+\\mu)s} q(s,t) \\mathrm ds \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\sum_{r=0}^{N-M} \\binom{N-M}{r} \\left( \n\t\t\t\t\t\\frac{\\lambda}{\\mu} \\right)^r \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{h=0}^M \\binom{M}{h} (-1)^h\n\t\t\t\t\tE_{\\nu,1}(-(r+h)(\\lambda+\\mu)t^\\nu). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\\end{proof}\n\t\t\t\\begin{flushright} \\qed \\end{flushright}\n\t\t\\end{thm}\n\n\t\t\\begin{thm}\n\t\t\tThe state probabilities $p_n^\\nu(t) = \\text{Pr} \\{ \\mathcal{N}^\\nu(t) = n | \\mathcal{N}^\\nu(0)\n\t\t\t= M \\}$, $\\lambda >0$, $\\mu > 0$, have the following form:\n\t\t\t\\begin{align}\n\t\t\t\t\\label{stato}\n\t\t\t\tp_n^\\nu(t) =\n\t\t\t\t\\begin{cases}\n\t\t\t\t\t\\sum_{r=0}^n g_{n,r}^\\nu(t), & 0 \\leq n < \\min(M,N-M), \\\\\n\t\t\t\t\t\\sum_{r=0}^{N-M} g_{n,r}^\\nu(t), & N-M \\leq n < M, \\: M > N-M, \\\\\n\t\t\t\t\t\\sum_{r=n-M}^n g_{n,r}^\\nu(t), & M \\leq n < N-M, \\: M < N-M, \\\\\n\t\t\t\t\t\\sum_{r=n-M}^{N-M} g_{n,r}^\\nu(t), & \\max(M,N-M) \\leq n \\leq N,\n\t\t\t\t\\end{cases}\n\t\t\t\\end{align}\n\t\t\tand\n\t\t\t\\begin{align}\n\t\t\t\tp_n^\\nu(t) =\n\t\t\t\t\\begin{cases}\n\t\t\t\t\t\\sum_{r=0}^n g_{n,r}^\\nu(t), & 0 \\leq n< M, \\\\\n\t\t\t\t\t\\sum_{r=n-M}^M g_{n,r}^\\nu(t), & M \\leq n \\leq N,\n\t\t\t\t\\end{cases}\t\t\t\n\t\t\t\\end{align}\n\t\t\twhen $N-M=M$, and where\n\t\t\t\\begin{align}\n\t\t\t\tg_{n,r}^\\nu(t)\n\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\binom{N-M}{r} \\binom{M}{n-r} \\\\\n\t\t\t\t& \\times \\sum_{m_1=0}^r \\binom{r}{m_1} (-1)^{m_1}\n\t\t\t\t\\sum_{m_2=0}^{N-M-r} \\binom{N-M-r}{m_2} \\left( \\frac{\\lambda}{\\mu} \\right)^{m_2} \\notag \\\\\n\t\t\t\t& \\times \\sum_{m_3=0}^{n-r} \\binom{n-r}{m_3} \\left( \\frac{\\lambda}{\\mu} \\right)^{n-m_3}\n\t\t\t\t\\sum_{m_4=0}^{M-n+r} \\binom{M-n+r}{m_4} (-1)^{m_4} \\notag \\\\\n\t\t\t\t& \\times E_{\\nu,1} \\left(\n\t\t\t\t-(m_1+m_2+m_3+m_4)(\\mu+\\lambda)t^\\nu \\right). \\notag\n\t\t\t\\end{align}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tWe start by rewriting  the probability generating function of the classical binomial\n\t\t\t\tprocess  as\n\t\t\t\t\\begin{align}\n\t\t\t\t\tQ(u,t)& \\\\\n\t\t\t\t\t= {} & \\left[ 1-\\left( 1-e^{-(\\mu+\\lambda)t} \\right)\\frac{\\lambda}{\\lambda+\\mu}u \\right]^{N-M}\\notag\\\\\n\t\t\t\t\t& \\times \\left[ 1-\\left( \\left(1-e^{-(\\mu+\\lambda)t}\\right)\\frac{\\lambda}{\\lambda+\\mu} +\n\t\t\t\t\te^{-(\\mu+\\lambda)t} \\right)u \\right]^M \\notag \\\\\n\t\t\t\t\t= {} & \\left[ (1-u) \\left( \\frac{\\lambda}{\\lambda+\\mu} - \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\te^{-(\\mu+\\lambda)t}\t\\right) + \\frac{\\mu}{\\lambda+\\mu} + \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\t\te^{-(\\mu+\\lambda)t} \\right]^{N-M} \\notag \\\\\n\t\t\t\t\t& \\times \\left[ \\frac{\\lambda}{\\lambda+\\mu}(1-u) + \\frac{\\mu}{\\lambda+\\mu} + \\frac{\\mu}{\\lambda+\\mu}\n\t\t\t\t\te^{-(\\mu+\\lambda)t} \\right. \\notag \\\\\n\t\t\t\t\t& \\left. -\\frac{\\mu}{\\lambda+\\mu} e^{-(\\mu+\\lambda)t}\n\t\t\t\t\t-u \\frac{\\mu}{\\lambda+\\mu} e^{-(\\mu+\\lambda)t} \\right]^M \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\lambda}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\t\\left[ (1-u) \\left( 1-e^{-(\\mu+\\lambda)t} \\right) + \\frac{\\mu}{\\lambda}\n\t\t\t\t\t+e^{-(\\mu+\\lambda)t} \\right]^{N-M} \\notag \\\\\n\t\t\t\t\t& \\times \\left[ (1-u) \\left( 1+\\frac{\\mu}{\\lambda}e^{-(\\mu+\\lambda)t} \\right)\n\t\t\t\t\t+ \\frac{\\mu}{\\lambda} - \\frac{\\mu}{\\lambda} e^{-(\\mu+\\lambda)t} \\right]^M \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\lambda}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\t\\sum_{r=0}^{N-M} \\binom{N-M}{r} (1-u)^r \\notag \\\\\n\t\t\t\t\t& \\times \\left( 1-e^{-(\\mu+\\lambda)t} \\right)^r\n\t\t\t\t\t\\left( \\frac{\\mu}{\\lambda} + e^{-(\\mu+\\lambda)t} \\right)^{N-M-r} \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{h=0}^M \\binom{M}{h} (1-u)^h \\left( 1+\\frac{\\mu}{\\lambda}\n\t\t\t\t\te^{-(\\mu+\\lambda)t} \\right)^h \\left( \\frac{\\mu}{\\lambda}-\\frac{\\mu}{\\lambda}\n\t\t\t\t\te^{-(\\mu+\\lambda)t} \\right)^{M-h} \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\sum_{r=0}^{N-M} \\sum_{j=r}^{M+r}\n\t\t\t\t\t(1-u)^j \\binom{N-M}{r} \\binom{M}{j-r} \\left( \\frac{\\lambda}{\\mu}\n\t\t\t\t\t-\\frac{\\lambda}{\\mu} e^{-(\\mu+\\lambda)t} \\right)^r \\notag \\\\\n\t\t\t\t\t& \\times \\left( 1+\\frac{\\lambda}{\\mu} e^{-(\\mu+\\lambda)t} \\right)^{N-M-r}\n\t\t\t\t\t\\left( \\frac{\\lambda}{\\mu} + e^{-(\\mu+\\lambda)t} \\right)^{j-r} \\notag \\\\\n\t\t\t\t\t& \\times \\left( 1-e^{-(\\mu+\\lambda)t} \\right)^{M-j+r}. \\notag\n\t\t\t\t\\end{align}\n\t\t\tLetting\n\t\t\t\t\\begin{align}\n\t\t\t\t\tg_{j,r}(t) = {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\t\\binom{N-M}{r} \\binom{M}{j-r} \\left( \\frac{\\lambda}{\\mu}\n\t\t\t\t\t-\\frac{\\lambda}{\\mu} e^{-(\\mu+\\lambda)t} \\right)^r \\\\\n\t\t\t\t\t& \\times \\left( 1+\\frac{\\lambda}{\\mu} e^{-(\\mu+\\lambda)t} \\right)^{N-M-r}\n\t\t\t\t\t\\left( \\frac{\\lambda}{\\mu} + e^{-(\\mu+\\lambda)t} \\right)^{j-r} \\notag \\\\\n\t\t\t\t\t& \\times \\left( 1-e^{-(\\mu+\\lambda)t} \\right)^{M-j+r}, \\notag\n\t\t\t\t\\end{align}\n\t\t\t\twe have\n\t\t\t\t\\begin{align}\n\t\t\t\t\tQ(u,t) =\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\sum_{j=0}^{M-1} (1-u)^j \\sum_{r=0}^j g_{j,r}(t) \\\\\n\t\t\t\t\t\t\\qquad + \\sum_{j=M}^{N-M-1}\n\t\t\t\t\t\t(1-u)^j \\sum_{r=j-M}^j g_{j,r}(t) \\\\\n\t\t\t\t\t\t\\qquad + \\sum_{j=N-M}^N (1-u)^j \\sum_{j-M}^{N-M} g_{j,r}(t),\n\t\t\t\t\t\t& M < N-M, \\\\\n\t\t\t\t\t\t\\sum_{j=0}^{N-M-1} (1-u)^j \\sum_{r=0}^j g_{j,r}(t) \\\\\n\t\t\t\t\t\t\\qquad + \\sum_{j=N-M}^{M-1}\n\t\t\t\t\t\t(1-u)^j \\sum_{r=0}^{N-M} g_{j,r}(t)\\\\\n\t\t\t\t\t\t\\qquad + \\sum_{j=M}^N (1-u)^j \\sum_{r=j-M}^{N-M} g_{j,r}(t),\n\t\t\t\t\t\t& M > N-M, \\\\\n\t\t\t\t\t\t\\sum_{j=0}^{M-1} (1-u)^j \\sum_{r=0}^j g_{j,r}(t) \\\\\n\t\t\t\t\t\t\\qquad + \\sum_{j=M}^N (1-u)^j \\sum_{r=j-M}^{M} g_{j,r}(t), & N-M=M.\n\t\t\t\t\t\\end{cases}\t\t\t\t\t\t\n\t\t\t\t\\end{align}\n\t\t\t\tThe classical state probabilities therefore read\n\t\t\t\t\\begin{align}\n\t\t\t\t\tp_n(t) =\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\sum_{r=0}^n g_{n,r}(t), & 0 \\leq n < \\min(M,N-M), \\\\\n\t\t\t\t\t\t\\sum_{r=0}^{N-M} g_{n,r}(t), & N-M \\leq n < M, \\: M > N-M, \\\\\n\t\t\t\t\t\t\\sum_{r=n-M}^n g_{n,r}(t), & M \\leq n < N-M, \\: M < N-M, \\\\\n\t\t\t\t\t\t\\sum_{r=n-M}^{N-M} g_{n,r}(t), & \\max(M,N-M) \\leq n \\leq N,\n\t\t\t\t\t\\end{cases}\n\t\t\t\t\\end{align}\n\t\t\t\twhich reduce to\n\t\t\t\t\\begin{align}\n\t\t\t\t\tp_n(t) =\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\sum_{r=0}^n g_{n,r}(t), & 0 \\leq n< M, \\\\\n\t\t\t\t\t\t\\sum_{r=n-M}^M g_{n,r}(t), & M \\leq n \\leq N,\n\t\t\t\t\t\\end{cases}\t\t\t\n\t\t\t\t\\end{align}\n\t\t\t\twhen $N-M=M$.\n\t\t\t\t\n\t\t\t\tExploiting Theorem \\ref{sub}, we can derive the state probabilities for the\n\t\t\t\tfractional binomial process $\\mathcal{N}^\\nu(t)$, $t \\ge 0$, as\n\t\t\t\t\\begin{align}\n\t\t\t\t\tp_n^\\nu(t) & = \\int_0^\\infty p_n(s) h(s,t) \\mathrm ds \\\\\n\t\t\t\t\t& =\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\sum_{r=0}^n g_{n,r}^\\nu(t), & 0 \\leq n < \\min(M,N-M), \\\\\n\t\t\t\t\t\t\\sum_{r=0}^{N-M} g_{n,r}^\\nu(t), & N-M \\leq n < M, \\: M > N-M, \\\\\n\t\t\t\t\t\t\\sum_{r=n-M}^n g_{n,r}^\\nu(t), & M \\leq n < N-M, \\: M < N-M, \\\\\n\t\t\t\t\t\t\\sum_{r=n-M}^{N-M} g_{n,r}^\\nu(t), & \\max(M,N-M) \\leq n \\leq N,\n\t\t\t\t\t\\end{cases} \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tor\n\t\t\t\t\\begin{align}\n\t\t\t\t\tp_n^\\nu(t) = \\int_0^\\infty p_n(s) h(s,t) \\mathrm ds =\n\t\t\t\t\t\\begin{cases}\n\t\t\t\t\t\t\\sum_{r=0}^n g_{n,r}^\\nu(t), & 0 \\leq n< M, \\\\\n\t\t\t\t\t\t\\sum_{r=n-M}^M g_{n,r}^\\nu(t), & M \\leq n \\leq N,\n\t\t\t\t\t\\end{cases}\t\t\t\t\t\t\n\t\t\t\t\\end{align}\t\t\t\t\n\t\t\t\tfor $N-M = M$. Note that\n\t\t\t\t\\begin{align}\n\t\t\t\t\tg_{n,r}^\\nu(t) = {} & \\int_0^\\infty g_{n,r}(s) h(s,t) \\mathrm ds \\\\\n\t\t\t\t\t= {} & \\int_0^\\infty \\left[ \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\t\\binom{N-M}{r} \\binom{M}{n-r} \\left( \\frac{\\lambda}{\\mu}\n\t\t\t\t\t-\\frac{\\lambda}{\\mu} e^{-(\\mu+\\lambda)s} \\right)^r \\right. \\notag \\\\\n\t\t\t\t\t& \\left. \\times \\left( 1+\\frac{\\lambda}{\\mu} e^{-(\\mu+\\lambda)s} \\right)^{N-M-r}\n\t\t\t\t\t\\left( \\frac{\\lambda}{\\mu} + e^{-(\\mu+\\lambda)s} \\right)^{n-r} \\right. \\notag \\\\\n\t\t\t\t\t& \\times \\left. \\left( 1-e^{-(\\mu+\\lambda)s} \\right)^{M-n+r} \\right] h(s,t) \\mathrm ds \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\binom{N-M}{r} \\binom{M}{n-r}\n\t\t\t\t\t\\sum_{m_1=0}^r \\binom{r}{m_1} (-1)^{m_1} \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{m_2=0}^{N-M-r} \\binom{N-M-r}{m_2} \\left( \\frac{\\lambda}{\\mu} \\right)^{m_2} \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{m_3=0}^{n-r} \\binom{n-r}{m_3} \\left( \\frac{\\lambda}{\\mu} \\right)^{n-m_3}\n\t\t\t\t\t\\sum_{m_4=0}^{M-n+r} \\binom{M-n+r}{m_4} (-1)^{m_4} \\notag \\\\\n\t\t\t\t\t& \\times \\int_0^\\infty e^{\n\t\t\t\t\t-(m_1+m_2+m_3+m_4)(\\mu+\\lambda)s} h(s,t) \\mathrm ds \\notag \\\\\n\t\t\t\t\t= {} & \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\binom{N-M}{r} \\binom{M}{n-r}\n\t\t\t\t\t\\sum_{m_1=0}^r \\binom{r}{m_1} (-1)^{m_1} \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{m_2=0}^{N-M-r} \\binom{N-M-r}{m_2} \\left( \\frac{\\lambda}{\\mu} \\right)^{m_2} \\notag \\\\\n\t\t\t\t\t& \\times \\sum_{m_3=0}^{n-r} \\binom{n-r}{m_3} \\left( \\frac{\\lambda}{\\mu} \\right)^{n-m_3}\n\t\t\t\t\t\\sum_{m_4=0}^{M-n+r} \\binom{M-n+r}{m_4} (-1)^{m_4} \\notag \\\\\n\t\t\t\t\t& \\times E_{\\nu,1} \\left(\n\t\t\t\t\t-(m_1+m_2+m_3+m_4)(\\mu+\\lambda)t^\\nu \\right). \\notag\n\t\t\t\t\\end{align}\n\t\t\t\tThis concludes the proof.\n\t\t\t\\end{proof}\n\t\t\t\\begin{flushright} \\qed \\end{flushright}\n\t\t\\end{thm}\n\t\t\n\t\t\\begin{remark}\n\t\t\tFrom \\eqref{stato},  we retrieve the extinction probability \\eqref{extinction} when $n=0$.\n\t\t\\end{remark}\n\t\t\n\t\t\\begin{remark}\n\t\t\tAs $t \\rightarrow \\infty$, the population in a fractional binomial\n\t\t\tprocess obeys a binomial distribution, i.e., \n\t\t\t\\begin{align}\n\t\t\t\t& \\lim_{t \\rightarrow \\infty} Q^\\nu(u,t) \\\\\n\t\t\t\t& = \\sum_{r=0}^{N-M} \\sum_{j=r}^{M+r} (1-u)^j \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\\binom{N-M}{r} \\binom{M}{j-r} \\left( \\frac{\\lambda}{\\mu} \\right)^j \\notag \\\\\n\t\t\t\t& = \\sum_{r=0}^{N-M} \\sum_{h=0}^M (1-u)^{h+r} \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N\n\t\t\t\t\\binom{N-M}{r} \\binom{M}{h} \\left( \\frac{\\lambda}{\\mu} \\right)^{h+r} \\notag \\\\\n\t\t\t\t& = \\left( \\frac{\\mu}{\\lambda+\\mu} \\right)^N \\left( 1+(1-u)\\frac{\\lambda}{\\mu} \\right)^{N-M}\n\t\t\t\t\\left( 1+(1-u)\\frac{\\lambda}{\\mu} \\right)^M \\notag \\\\\n\t\t\t\t& = \\left( \\frac{\\mu}{\\lambda+\\mu} + \\frac{\\lambda}{\\lambda+\\mu}(1-u) \\right)^N \\notag \\\\\n\t\t\t\t& = \\left( 1- \\frac{\\lambda}{\\lambda+\\mu}u \\right)^N, \\notag\n\t\t\t\\end{align}\n\t\t\tis the probability generating function of a binomial random variable of parameter $\\lambda\/(\\lambda+\\mu)$\n\t\t\tand must be compared with equation $\\mathrm{(9)}$ of \\citet{jakeman}.\n\\end{remark}\nNote also that as $t \\rightarrow \\infty$, indeed we observe (from \\eqref{mea} and \\eqref{va}) that\n\t\t\t\\begin{align}\n\t\t\t\t\\mathbb{E}\\, \\mathcal{N}^\\nu(t) \\longrightarrow N \\frac{\\lambda}{\\lambda+\\mu}\n\t\t\t\\end{align}\nand \n\t\t\\begin{align}\n\t\t\t\t\\mathbb{V}ar\\, \\mathcal{N}^\\nu(t) \\longrightarrow N \\frac{\\lambda}{\\lambda+\\mu}\\left(1- \\frac{\\lambda}{\\lambda+\\mu} \\right),\n\t\t\t\\end{align}\nwhich are the mean and variance of the binomial equilibrium process. This suggests that the fractional generalization still preserves the binomial limit. \n\t\t\n\t\\section{Related fractional stochastic processes}\n\t\n\t\tIn this section, we focus our attention to two pure branching processes which are in fact sub-models of the more general fractional binomial process described in Section \\ref{se}. These are\n\t\tthe fractional linear pure death process and the saturable fractional pure birth process. More specifically,\tthese processes can be directly obtained from the fractional binomial process by letting $\\mu=0$ and \t\t$\\lambda=0$, respectively. The main motivation underlining the analysis of these specific cases\n\t\tis that they are widely used in practice particularly in modeling  evolving populations \tin interacting environment possibly causing extintion or saturation.\n\t\tOur discussion on the fractional linear pure death process complements that of \\citet{sakhno}'s. Instead, we analyze the saturable fractional pure birth\n\t\tprocess in more detail.\n\t\n\t\tWhen $\\lambda=0$, we obtain the mean value \\eqref{mea} of the fractional linear pure\n\t\tdeath process $\\mathcal{N}^\\nu_d(t)$, $t \\ge 0$, $\\nu \\in (0,1]$, (see \\citet{sakhno}):\n\t\t\\begin{align}\n\t\t\t\\mathbb{E}\\,\\mathcal{N}^\\nu_d(t) = M E_{\\nu,1}(-\\mu t^\\nu), \\qquad t \\ge 0. \n\t\t\\end{align}\n\t\tFigure \\ref{bfig} shows the mean value of the fractional linear pure death process (left)\n\t\tfor specific values of the parameters $\\mu$ and $\\nu$.\n\t\n\t\t\t\\begin{figure}[h!t!b!p!]\n\t\t\t\\centering\n\t\t\t\\includegraphics[height=2.5in, width=2.2in]{p3.pdf}\n\t\t\t\\includegraphics[height=2.5in, width=2.2in]{p4.pdf}\n\t\t \n\t\t\n\t\t\t\\caption{\\label{bfig}The mean value of the fractional binomial process $\\mathbb{E}\\,\n\t\t\t\t\\mathcal{N}^\\nu(t)$ in the two different cases of pure death (left, $(\\lambda,\\mu)=(0,1)$)\n\t\t\t\tand pure birth (right, $(\\lambda,\\mu)=(1,0)$).\n\t\t\t\tFor both cases we have $N=100$, $M=40$, $\\nu=0.7$.}\n\t\t\\end{figure}\n\t\n\t\tThe variance can be  easily determined using \\eqref{va} as\n\t\t\\begin{align}\n\t\t\t\\mathbb{V}\\text{ar}\\,\\mathcal{N}^\\nu_d(t) = {} & M(M-1) E_{\\nu,1} \\left( -2\\mu t^\\nu \\right)\n\t\t\t+ M E_{\\nu,1}\\left( -\\mu t^\\nu \\right) \\\\\n\t\t\t& - M^2 \\left[ E_{\\nu,1}\\left( - \\mu\n\t\t\tt^\\nu \\right) \\right]^2, \\qquad t \\ge 0, \\: \\nu \\in (0,1], \\notag\n\t\t\\end{align}\n\t\tand this reduces for $\\nu = 1$ to the variance of the classical process (see \\citet{bailey}, page 91, formula (8.32)):\n\t\t\\begin{align}\n\t\t\t\\mathbb{V}\\text{ar}\\,\\mathcal{N}^1_d(t) = M e^{-\\mu t}\\left( 1-e^{-\\mu t} \\right).\n\t\t\\end{align}\n\t\tFurthermore, the extinction probability of the fractional linear pure death\n\t\tprocess $\\mathcal{N}^\\nu_d(t)$, $t \\ge 0$ (see \\citet{sakhno}, page 73, formula (2.1)),\n\t\t\\begin{align}\n\t\t\t\\text{Pr} \\{ \\mathcal{N}^\\nu_d(t) = 0 | \\mathcal{N}^\\nu_d(0) = M \\}\n\t\t\t= \\sum_{h=0}^M \\binom{M}{h} (-1)^h E_{\\nu,1} \\left( -h \\mu t^\\nu \\right), \\qquad t \\ge 0,\n\t\t\\end{align}\n\t\tcan also be derived directly from \\eqref{extinction}.\n\t\t\n\t\tFor the saturable fractional pure birth process $\\mathcal{N}^\\nu_b(t)$, \n\t\tthe mean value reduces to\n\t\t\\begin{align}\n\t\t\t\\label{satur}\n\t\t\t\\mathbb{E}\\,\\mathcal{N}^\\nu_b(t) = N - \\left( N-M \\right)E_{\\nu,1}\n\t\t\t\\left( -\\lambda t^\\nu \\right).\n\t\t\\end{align}\n\t\tFigure \\ref{bfig} shows the expected value of the saturable fractional pure birth process (right)\n\t\tdetermined for specific values of the parameters $\\lambda$ and $\\nu$. The variance instead remains rather complicated and can be written by specialising \\eqref{va} as\n\t\t\\begin{align}\n\t\t\t\\mathbb{V}\\text{ar}\\,\\mathcal{N}^\\nu_b(t) = {} &\n\t\t\t\\left[ M(M-1)-N(N-1) \\right] E_{\\nu,1}\\left( -2\\lambda t^\\nu \\right) \\\\\n\t\t\t& - (N-M)(4N-1) E_{\\nu,1}\\left( -\\lambda t^\\nu \\right) - (M-N)^2 \\left[ E_{\\nu,1}\\left( -\\lambda t^\\nu\n\t\t\t\\right) \\right]^2. \\notag\n\t\t\\end{align}\nAs $t\\rightarrow \\infty$, \n\t\t\\begin{equation}\n\t\t\t\t\t\\mathbb{E}\\,\\mathcal{N}^\\nu_b(t) \\longrightarrow   N \n\t\t\\end{equation}\n\t\tand\n\t\\begin{equation}\n\t\t\t\t\t\\mathbb{V}\\text{ar}\\,\\mathcal{N}^\\nu_b(t) \\longrightarrow  0\n\t\t\\end{equation}\t\t\nas expected. \n\n\t\nWe now determine the state probabilities $p_{n,b}^\\nu(t) = \\Pr \\{ \\mathcal{N}^\\nu_b(t)=n\n\t\t| \\mathcal{N}^\\nu_b(0) = M \\}$.\n\t\tWhen $\\mu = 0$, the state probabilities can be derived from those of a\n\t\tnonlinear fractional pure birth process of \\citet{pol},  and is given as\n\t\t\\begin{equation}\n\t\t\tp_{n,b}^\\nu(t) =\n\t\t\t\\begin{cases}\n\t\t\t\t\\prod_{j=M}^{n-1} \\lambda_j \\sum_{m=M}^n \\frac{1}{\\prod_{l=M,l \\neq m}^n(\\lambda_l-\\lambda_m)}\n\t\t\t\tE_{\\nu,1}\\left( -\\lambda_m t^\\nu \\right), & M < n \\leq N ,\\\\\n\t\t\tE_{\\nu,1} \\left( -\\lambda_M t^\\nu \\right), & n=M.\n\t\t\t\\end{cases}\n\t\\end{equation}\n\tSubstituting the rates $\\lambda_j= \\lambda(N-j)$, we obtain\n\t\t\\begin{align}\n\t\t\tp_{n,b}^\\nu&(t) \\\\\n\t\t\t= {} & \\prod_{j=M}^{n-1} \\lambda(N-j) \\sum_{m=M}^n\n\t\t\t\\frac{E_{\\nu,1}\\left( -\\lambda (N-m) t^\\nu \\right)}{\\prod_{l=M,l \\neq m}^n(\\lambda (N-l)-\\lambda(N-m))}\n\t\t\t\\notag \\\\\n\t\t\t= {} & \\sum_{m=M}^n \\frac{(N-M)(N-M-1)\\dots (N-n+1)}{(m-M)(m-M-1)\\dots (m-m+1)\n\t\t\t(m-m-1)\\dots (m-n)} \\notag \\\\\n\t\t\t& \\times E_{\\nu,1}\\left( -\\lambda (N-m) t^\\nu \\right) \\notag \\\\\n\t\t\t= {} & \\sum_{m=M}^n \\frac{(N-M)!}{(N-n)!(m-M)!(n-m)!} (-1)^{n-m}\n\t\t\tE_{\\nu,1}\\left( -\\lambda (N-m) t^\\nu \\right) \\notag \\\\\n\t\t\t= {} & \\binom{N-M}{N-n} \\sum_{m=M}^n \\binom{n-M}{m-M} (-1)^{n-m}\n\t\t\tE_{\\nu,1}\\left( -\\lambda (N-m) t^\\nu \\right), \\quad M \\leq n \\leq N. \\notag\n\t\t\\end{align}\n\t\tThis and formula \\eqref{satur} show\tthat the behaviour of the saturable fractional pure birth\n\t\tprocess is subtantially different from that of the fractional Yule process. Similarly, the inter-birth waiting time $T_j^\\nu$, i.e.\\ the random\n\t\ttime separating the $j$th and $(j+1)$th birth, has law\n\t\t\\begin{align}\n\t\t\t\\Pr \\{ T_j^\\nu \\in \\mathrm ds \\} = \\lambda(N-j) s^{\\nu-1} E_{\\nu,\\nu} \\left( -\\lambda (N-j)s^\\nu \\right)\n\t\t\t\\mathrm ds, \\qquad j \\ge M, \\: s \\ge 0.\n\t\t\\end{align}\n\tThe figure below shows the sample paths of the saturable fractional (bottom) and classical (top) linear pure birth processes. Apparently, the proposed model naturally includes processes or populations that saturate faster than the classical  linear pure birth process. The figure also indicates that saturation of the fractional binomial process is faster due to the explosive growth\/birth bursts  at small times and as $\\nu \\to 0$.  Note that the parameters of these related fractional point processes can be estimated using the procedures of  \\citet{cap11b}.\n\n\t\t\\begin{figure}[h!t!b!p!]\n\t\t\t\\centering\n\t\t\t\\includegraphics[height=3in, width=4.5in]{satbirth.pdf}\n\t\t\n\t\t\t\\caption{\\label{bfig3}Sample trajectories of classical (top)  and fractional (bottom)  saturable  linear pure birth processes  using values $(N,M,\\lambda, \\nu)=(100,5,1,1)$  and $(N,M,\\lambda, \\nu)=(100,5,1,0.75)$, correspondingly.}\n\t\t\\end{figure}\n\t\n\n\\section{Concluding Remarks}\n    We have proposed a generalization of the binomial process using the techniques of fractional calculus. The fractional generalization In addition, more statistical properties of the fractional binomial process were derived. One interesting property of the fractional binomial process  was that it still preserved the binomial limit at large times while enlarging the class of models at  small and regular times that naturally include non-Markovian fluctuations with long memory. This potential made the proposed fractional binomial process appealing for real applications especially  to the quantum optics community.  New sub-models such as the saturable fractional pure birth process could also be automatically extracted from the proposed model. The generated sample trajectories of the saturable fractional linear pure birth process showed interesting features of the process such as the isolated bursts of the population growth particularly at small times. Overall, the fractional binomial process could be considered as a viable generalization of the classical binomial process.\n    \n    Although theoretical investigations have been done in the present paper, a number of issues are still left undone which could be considered as possible research extensions\nof the current exploration. These may include: application of this model to rapidly saturable binomial processes, and the formalization of the parameter estimation procedures of the proposed model.  \n        \n\t\t\t\t\n\t\t\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\\label{sec:intro}\n\\input{intro.tex}\n\n\n\\section{Related Works}\\label{sec:relwork}\n\\input{relwork.tex}\n\n\\section{Problem Definition}\\label{sec:problemDefinition}\n\\input{probdef.tex} \n\n\\section{Metric Embedding of Node-Pairs}\\label{sec:Methodology}\n\\input{methodology.tex}\n\n\\section{Experiments and Results}\\label{sec:experiemnts}\n\\input{experiments.tex}\n\n\\section{Conclusion}\\label{sec:conclusion}\n\\input{conclusion.tex}\n\n\n\n\\bibliographystyle{spmpsci}\n\n\\subsection{Dataset Descriptions}\\label{sec:DataSet}\n\nHere we discuss the construction and characteristics of the datasets used for\nexperiments. \n\n\\noindent {\\em  Enron} email corpus \\cite{priebe2005scan} consists of email\nexchanges between Enron employees. The Enron dataset has $11$ time\nstamps and $16,836$ possible node-pairs; the task is to use first $10$ snapshots for predicting links in the\n$11^{th}$ snapshot. Following \\cite{KevinS:2013}, we aggregate data into time\nsteps of $1$ week. We use the data from weeks $147$ to $157$ of the data trace\nfor the  experiments. The reason for choosing that window is that the snapshot\nof the graph at week $157$ has the highest number of edges. \\\\\n\n{\\noindent} {\\em Collaboration} dataset has $10$ time stamps with author collaboration information about $49,455$ author-pairs. The Collaboration dataset is constructed from citation\ndata containing $1.4$ million papers \\cite{Tang:2008}. We process the data to\nconstruct a network of authors with edges between them if they co-author a\npaper. Considering each year as a time stamp, the data of years $2000$-$2009$\n($10$ time stamps) is used for this experiment, where the data from the first\nnine time stamps is used for training and the last for prediction.  Since this\ndata is very sparse, we pre-process the data to retain only the active authors,\nwho have last published papers on or after year $2010$; moreover, the selected\nauthors participate in at least two edges in seven or more time stamps.  \\\\\n\n{\\noindent} {\\em  Facebook1 and Facebook2} are network of Facebook \nwall posts \\cite{viswanath2009evolution}. Each vertex is a Facebook user account and\nan edge represents the event that one user posts a message on the wall of another user.\nBoth Facebook1  and Facebook2 has $9$ time stamps. Facebook1 has\n$219,453$ node-pairs. Facebook2 is an extended version of Facebook1 dataset with $883,785$\nnode-pairs. For pre-processing Facebook1 we follow the same setup as is discussed in\n\\cite{KevinS:2015}; wall posts of 90 days are aggregated in one time step.\n\nWe filter out all people who are active for less than 6 of the 9 \ntime steps, along with the people who have degree less than 30. \nFacebook2 is created using a similar\nmethod, but a larger sample of Facebook wall posts is used for this dataset.\n\n\\subsection{Evaluation Metrics}\nFor evaluating the proposed method we use two metrics, namely, area under\nPrecision-Recall (PR) curve (PRAUC) \\cite{Davis:2006} and an information\nretrieval metric, Normalized Discounted Cumulative Gain (NDCG). PRAUC is best suited for evaluating two class classification performance when class\nmembership is skewed towards one of the classes. This is exactly the case for\nlink prediction; the number of edges $(|E|)$ is very small compared to the\nnumber of possible node-pairs ${|V|\\choose 2}$  In such scenarios, area under the Precision-Recall curve (PRAUC) gives a more informative assessment of the algorithm's performance than other metrics such as, accuracy.  The reason why PRAUC is more suitable for the skewed problem is that it does not factor in  the count of true negatives in its calculation. In skewed data where the number of negative\nexamples is huge compared to the number of positive examples, true negatives\nare not that meaningful. \n\nWe also use NDCG, an information\nretrieval metric (widely used by the recommender systems community) to evaluate\nthe proposed method.  NDCG measures the performance of link prediction system\nbased on the graded relevance of the recommended links. $NDCG_k$ varies from\n0.0 to 1.0, with 1.0 representing ideal ranking of edges. Here, $k$ is a parameter\nchosen by user representing the number of links ranked by the method. We use\n$k=50$ in all our experiments. \n\n\nSome of the earlier works on link prediction have used area under the ROC curve\n(AUC) to evaluate link prediction works \\cite{Gunes2015,Wang:2007}. But recent\nworks \\cite{Yang2015} have demonstrated the limitations of AUC and argued\nin favor of PRAUC over AUC for evaluation of link prediction. So we have not used AUC in this work.\n\n\\subsection{Competing Methods for Comparison}\n\nWe compare the performance of {\\sc DyLink2Vec}\\ based link prediction method with methods from four categories: (1) topological feature based methods, (2) feature time series\nbased methods~\\cite{Gunes2015}, (3) a deep learning based method, namely\nDeepWalk~\\cite{Perozzi:2014}, and (4) a tensor factorization based method\nCANDECOMP\/PARAFAC (CP)~\\cite{Dunlavy:2011}. \n\nBesides these four works, there are two other existing works for link\nprediction in dynamic network setting; one is based on deep Learning\n\\cite{Li:2014} (Conditional Temporal Restricted Boltzmann machine) and the other is based on a signature-based nonparametric method \\cite{ICML2012Sarkar_828}. We did not compare with these models as implementations of their models are not readily available, besides, both of these methods have numerous parameters which will make reproducibility of their results highly improbable and thus, conclusion derived from such experiments may not align with true understanding of the usefulness of the methods. Moreover, none of these methods give unsupervised feature representation for node-pairs in which we claim our main contribution. \n\n\nFor \\textbf{topological feature based methods}, we consider four prominent topological\nfeatures: Common Neighbors ($CN$), Adamic-Adar ($AA$), Jaccard's\nCoefficient ($J$) and Katz measure ($Katz$). \nHowever, in existing works, these features are defined for\nstatic networks only; so we adapt these features for the dynamic network setting\nby computing the feature values over the collapsed\\footnote{Collapsed network is\nconstructed by superimposing all network snapshots(see Figure \\ref{fig:StaticLimitaion}).} dynamic network. \n\nWe also combine the above four features to construct a combined\nfeature vector of length four (\\textbf{J}accard's\nCoefficient, \\textbf{A}damic-Adar, \\textbf{C}ommon Neighbors and \\textbf{K}atz), which we call $JACK$ and use it with a classifier to build a\nsupervised link prediction method, and include this model in our comparison.\n\nSecond, we compare {\\sc DyLink2Vec}\\ with \\textbf{time-series based} neighborhood similarity scores proposed in \\cite{Gunes2015}. In this work, the authors consider several\nneighborhood-based node similarity scores combined with connectivity\ninformation (historical edge information). Authors use time-series of\nsimilarities to model the change of node similarities over time. Among $16$\nproposed methods, we consider $4$ that are relevant to the link prediction task\non unweighted networks and also have the best performance. \n$TS\\mbox{-}CN\\mbox{-}Adj$ represents time-series on normalized score of Common\nNeighbors and connectivity values at time stamps $[1,t]$.  Similarly, we get\ntime-series based scores for Adamic-Adar ($TS\\mbox{-}AA\\mbox{-}Adj$), Jaccard's\nCoefficient ($TS\\mbox{-}J\\mbox{-}Adj$) and Preferential Attachment\n($TS\\mbox{-}PA\\mbox{-}Adj$). \n\nThird, we compare {\\sc DyLink2Vec}\\ with \\textbf{DeepWalk} \\cite{Perozzi:2014}, a latent node\nrepresentation based method.  We use DeepWalk to construct latent\nrepresentation of nodes from the collapsed dynamic\nnetwork. Then we construct latent representation of node-pairs by computing\ncross product of latent representation of the participating nodes. For example,\nif the node representations in a network are vectors of size $l$, then the\nrepresentation of a node-pair $(u,v)$ will be of size $l^2$, constructed from\nthe cross product of $u$ and $v$'s representation. The DeepWalk based\nnode-pair representation is then used with a classifier to build a supervised\nlink prediction method. We choose node representation size $l=2,4,6,8,10$ and \nreport the best performance.\n\nFinally, we compare {\\sc DyLink2Vec}\\ with a tensor factorization based method, called\n\\textbf{CANDECOMP\/PARAFAC (CP)} \\cite{Dunlavy:2011}. In this method,\nthe dynamic network is represented as a three-dimensional tensor \n$\\mathcal{Z}(n\\times n\\times t)$. Using CP decomposition $\\mathcal{Z}$ is factorized\ninto three factor matrices. The link prediction score is computed by using \nthe factor matrices.\nWe adapted the CP link prediction method for unipartite networks; which has \noriginally been developed for bipartite networks.\n\n\n\n\n\\subsection{Implementation Details}\n\nWe implemented {\\sc DyLink2Vec}\\ algorithm in Matlab version $R2014b$.  The learning method runs for a maximum of $100$ iterations\nor until it converges to a local optimal solution.  We use coding size $l=100$\nfor all datasets\\footnote{We experiment with different coding sizes\nranging from 100 to 800. The change in link prediction performance is not\nsensitive to the coding size. At most 2.9\\% change in PRAUC was observed for\ndifferent coding sizes.}.  For supervised link prediction step we use several\nMatlab provided classification algorithms, namely, AdaBoostM1, RobustBoost, and\nSupport Vector Machine (SVM).  We could use neural network classifier. But, as our main goal is to evaluate the quality of unsupervised feature representation, so, we use simple classifiers. Supervised neural network architecture may result in superior performance, but, it is out of scope of the main goal of the paper. We use Matlab for computing the feature values (CN, AA, J, Katz) that we use in other competing methods. Time-series methods are implemented using Python. We use the ARIMA (autoregressive integrated\nmoving average) time series model implemented in Python module\n\\textbf{statsmodels}.  The DeepWalk implementation is provided by the authors\nof~\\cite{Perozzi:2014}. We use it to extract node features and  extend it for\nlink prediction (using Matlab). Tensor factorization based method CP was\nimplemented using Matlab Tensor Toolbox. \n\n\n\n\\subsection{Performance Comparison Results with Competing Methods}\nIn Figure \\ref{fig:CompareAll} we present the performance comparison results\nof {\\sc DyLink2Vec}\\ based link prediction method with the four kinds of competing methods that we have discussed\nearlier. The figure have eight bar charts.  The bar charts from the top to the\nbottom rows display the results for Enron, Collaboration, Facebook1 and\nFacebook2 datasets, respectively.  The bar charts in a row show comparison\nresults using PRAUC (left), and  $NDCG_{50}$ (right) metrics.  \nEach chart has twelve bars, each representing a link prediction method, where the height\nof a bar is indicative of the performance metric value of the corresponding\nmethod. In each chart, from left to right, the first five bars (blue) correspond to the\ntopological feature based methods, the next four (green) represent time series\nbased methods, the tenth bar (black) is for DeepWalk, the eleventh bar (brown)\nrepresents tensor factorization based method CP, and the final bar (purple)\nrepresents the proposed method {\\sc DyLink2Vec}.\n\n\\subsubsection{{\\sc DyLink2Vec}\\ vs. Topological} \nWe first analyze the performance comparison between {\\sc DyLink2Vec}\\ based method and topological feature based methods\n(first five bars).  The best of the topological feature based methods have a\nPRAUC value of 0.30, 0.22, 0.137 and 0.14 in Enron, Collaboration, Facebook1,\nand Facebook2 dataset (see Figures\n\\ref{fig:CompareAll}(a), \\ref{fig:CompareAll}(c), \\ref{fig:CompareAll}(e) and\n\\ref{fig:CompareAll}(g)), whereas the corresponding PRAUC values for {\\sc DyLink2Vec}\\ are\n0.531, 0.362, 0.308, and 0.27, which translates to 77\\%, 65\\%, 125\\%, and\n93\\% improvement of PRAUC by {\\sc DyLink2Vec}\\ for these datasets.  Superiority of {\\sc DyLink2Vec}\\\nover all the topological feature based baseline methods can be attributed to\nthe capability of Neighborhood based feature representation to capture temporal\ncharacteristics of local neighborhood. Similar trend is observed using \n$NDCG_{50}$ metric, see Figures \\ref{fig:CompareAll}(b),\n\\ref{fig:CompareAll}(d), \\ref{fig:CompareAll}(f) and \\ref{fig:CompareAll}(h).\n\n\\subsubsection{{\\sc DyLink2Vec}\\ vs. Time-Series} \nThe performance of time-series based\nmethod (four green bars) is generally better than the topological feature based\nmethods. The best of the time-series based method has a PRAUC value of 0.503,\n0.28, 0.19, and 0.19 on these datasets, and {\\sc DyLink2Vec}'s PRAUC values are better than\nthese values by 6\\%, 29\\%, 62\\%, and 42\\% respectively.  Time-series\nbased methods, though model the temporal behavior well, probably fail to\ncapture signals from the neighborhood topology of the node-pairs. Superiority of\n{\\sc DyLink2Vec}\\ over Time-Series methods is also similarly indicated by information retrieval\nmetric $NDCG_{50}$.\n\n\\subsubsection{{\\sc DyLink2Vec}\\ vs. DeepWalk} \nThe DeepWalk based method (black bars in Figure \\ref{fig:CompareAll}) performs much poorly\nin terms of both PRAUC and $NDCG_{50}$---even poorer than the topological based method in all four\ndatasets. Possible reason could be the following: the latent encoding of nodes\nby DeepWalk is good for node classification, but the cross-product of those\ncodes fails to encode the information needed for effective link prediction.\n\n\\subsubsection{{\\sc DyLink2Vec}\\ vs. CANDECOMP\/PARAFAC (CP)} \n\nFinally, the tensor factorization based method CP performs marginally better\n(around 5\\% in PRAUC, and 6\\% in $NDCG_{50}$) than {\\sc DyLink2Vec}\\ in small and simple\nnetworks, such as Enron (see Figure \\ref{fig:CompareAll}(a, b)). \nBut its\nperformance degrades on comparatively large and complex networks, such as Collaboration, Facebook1\nand Facebook2. On Facebook networks, the performance of CP is\neven worse than the time-series based methods (see Figures \\ref{fig:CompareAll}(e)\nand \\ref{fig:CompareAll}(g)). {\\sc DyLink2Vec}\\ comfortably outperforms CP on larger\ngraphs, see Figures \\ref{fig:CompareAll}(c, d, e, f, g, h). In terms of PRAUC,\n{\\sc DyLink2Vec}\\ outperforms CP by 28\\%, 94\\%, and 120\\% for Collaborative, Facebook1 and Facebook2 networks respectively.\nThis demonstrates the superiority of {\\sc DyLink2Vec}\\ over one of the best state-of-the-art\ndynamic link prediction. A reason for CP's bad\nperformance on large graphs can be its inability to capture network structure \nand dynamics using high-dimensional tensors representation. \n\n\\subsubsection{Performance across datasets}\nWhen we compare the performance of all the methods across different\ndatasets, we observe varying performance. For example, for both the metrics,\nthe performance of dynamic link prediction on Facebook graphs are lower than the\nperformance on Collaboration graph, which, subsequently, is lower than the\nperformance on Enron graph, indicating that link prediction in Facebook data is\na harder problem to solve. In these harder networks, {\\sc DyLink2Vec}\\ perform substantially\nbetter than all the other competing methods that we consider in this experiment.\n\n\\begin{figure}[H]\n\\centering\n\\subfloat[Collaboration Network]{%\n\\includegraphics[width=.5\\linewidth]{DBLP2TW.eps}\n}\n\\subfloat[Facebook1 Network]{%\n\\includegraphics[width=.5\\linewidth]{FaceBookTW.eps}\n}\n\\caption{Change in link prediction performance with number of time stamps. X-axis represents size of training window used\nfor link prediction.  Largest possible window size depends on number of time\nstamps available for the dataset.\n}\n\\label{fig:TimeWindow}\n\\vspace{-0.3in}\n\\end{figure}\n\n\\subsection{Performance with varying length of Time Stamps}\n\nBesides comparing with competing methods, we also demonstrate the performance\nof {\\sc DyLink2Vec}\\ with varying number of available time snapshots.  For this\npurpose, we use {\\sc DyLink2Vec}\\ with different counts of past snapshots. For example,\nCollaboration dataset has 10 time stamps. The task is to predict links at time\nstamp 10.  The largest number of past snapshots we can consider for this data\nis 8, where $\\mathbf{\\widehat{E}}$ is constructed using time stamps $[1-8]$,\nand $\\mathbf{\\overline{E}}$ is constructed using time stamps $[2-9]$. The\nsmallest number of time stamps we can consider is 1, where\n$\\mathbf{\\widehat{E}}$ is constructed using $[8-8]$, and\n$\\mathbf{\\overline{E}}$ is constructed using $[9-9]$.\nIn this way, by varying the length of historical time stamps, \nwe can evaluate the effect of time stamp's length on the performance of a link prediction method.\n\n\nThe result is illustrated in Figure \\ref{fig:TimeWindow}. The x-axis represents\nthe number of time stamps used by {\\sc DyLink2Vec} , the left y-axis represents the\n$NDGC_{50}$ and the right y-axis represents the\nPRAUC.  Figures \\ref{fig:TimeWindow}(a),\nand \\ref{fig:TimeWindow}(b) corresponds to the results obtained on Collaboration and Facebook1, respectively. \n\n\nWe observe from Figure \\ref{fig:TimeWindow} that the performance ($NDGC_{50}$\nand PRAUC) of link prediction increases with increased number of time stamps.\nBut beyond a given number of snapshots, the performance increment becomes\ninsignificant. The performance starts to deteriorate after certain number of snapshots (see Figure \\ref{fig:TimeWindow}(a)).  This may be because of the added complexity of the optimization framework with increased number of time stamps.  We also observe\nconsistent improvement of performance with the number of snapshots for the\nFacebook1 data (Figure \\ref{fig:TimeWindow}(b)),\nwhich indicates that for this dataset link information from distant\nhistory is useful for dynamic link prediction. We do not show results of Enron and Facebook2  for this experiment, \nbecause of space constraint, however, they show similar trends.\n\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.8\\linewidth]{DBLP2.eps}\n\\caption{Effect of class imbalance in link prediction performance on Collaboration network.}\n\\label{fig:classImbalance}\n\\vspace{-0.3in}\n\\end{figure}\n\n\\subsection{Effect of Class Imbalance on Performance}\nIn link prediction problem, class imbalance is a prevalent issue. The class imbalance problem appears in a classification task, when the dataset contains imbalanced number of samples for different classes. In link prediction problem, the number of positive node-pairs (with an edge) is very small compared to the number of negative node-pairs (with no edge), causing class imbalance problem. \n\nTo demonstrate the effect of class imbalance in link prediction task, we perform link prediction using {\\sc DyLink2Vec}\\ embeddings with different level of class imbalance in the training dataset.\nWe construct the training dataset by taking all positive node-pairs and sampling from the set of negative node-pairs. \nFor a balanced dataset, the number of negative samples will be equal to the number of all positive node-pairs considered. Thus, the balanced training dataset has positive node-pairs to negative node-pairs ratio $1:1$. At this point, the only way to increase the size of the data is to increase the sample size for negative node-pairs. Consequently, the ratio of classes also increases towards negative node-pair.\nFigure \\ref{fig:classImbalance} shows gradual decrease in link prediction performance in Collaboration network with the increase of imbalance (see ratios in X-axis) in the dataset (despite the fact that the dataset gets larger by adding negative node-pairs). \n\nThis result advocates towards the design choice of under-sampling \\cite{Lichtenwalter:2010} of negative node-pairs by uniformly sampling from all negative node-pairs, so that the training set has equal numbers of positive and negative node-pairs. Under-sampling, helps\nto mitigate the problem of class imbalance while also reducing the size of the training dataset.\n\n\n\n\n\\subsection{Optimization framework for DyLink to Vec}\\label{subsec:UnsupervisedFeatureExtraction}\n\nIn this section, we discuss the optimization framework which obtains the\noptimal metric embedding of a node pair by learning an optimal coding function\n$h$. For this\nlearning task, let's assume $\\mathbf{\\widehat{E}}$ is the training dataset\nmatrix containing a collection of node-pair feature vectors. Each row of this\nmatrix represents a node-pair (say, $u$ and $v$) and it contains the feature\nvector $\\mathbf{e}^{uv}$ which stores information about neighborhood and link\nhistory, as we discussed earlier. The actual link status of the node-pairs in $\\mathbf{\\widehat{E}}$ \nin $G_{t+1}$ is not used for the learning of $h$, so the metric embedding process\nis unsupervised. In subsequent discussion, we write $\\mathbf{e}$ to represent\nan arbitrary node pair vector in $\\mathbf{\\widehat{E}}$.\n\nNow, the coding function $h$ compresses $\\mathbf{e}$ to a code vector $\\mathbf{\\alpha}$\nof dimension $l$, such that $l < k$. Here $l$ is a user-defined parameter which\nrepresents the code length and $k$ is the size of feature vector. Many\ndifferent coding functions exist in the dimensionality reduction literature,\nbut for {\\sc DyLink2Vec}\\ we choose the coding function which incurs the minimum\nreconstruction error in the sense that from the code $\\mathbf{\\alpha}$ we can\nreconstruct $\\mathbf{e}$ with the minimum error over all $\\mathbf{e} \\in\n\\mathbf{\\widehat{E}}$.  We frame the learning of $h$ as an optimization\nproblem, which we discuss below through two operations: Compression  and\nReconstruction. \\\\\n\n\\noindent {\\bf Compression:} It obtains $\\mathbf{\\alpha}$ from $\\mathbf{e}$.\nThis transformation can be expressed as a nonlinear function of linear weighted\nsum of the entries in vector $\\mathbf{e}$.\n\\begin{equation}\n\\mathbf{\\alpha} = f(\\mathbf{W}^{(c)}\\mathbf{e} + \\mathbf{b}^{(c)})\n\\label{eq:Compression}\n\\end{equation}\n$\\mathbf{W}^{(c)}$ is a $(k \\times l)$ dimensional matrix. It represents the weight matrix for compression\nand $\\mathbf{b}^{(c)}$ represents biases. $f(\\cdot)$ is the Sigmoid function, $f(x)=\\frac{1}{1+e^{-x}}$. \\\\\n\n\\noindent {\\bf Reconstruction:} It performs the reverse operation of compression, i.e., it \nobtains $\\mathbf{e}$ from $\\mathbf{\\alpha}$ (which was constructed during the compression \noperation).\n\\begin{equation}\n\\mathbf{\\beta} = f(\\mathbf{W}^{(r)}\\mathbf{\\alpha} + \\mathbf{b}^{(r)})\n\\label{eq:Reconstruction}\n\\end{equation}\n$\\mathbf{W}^{(r)}$ is a matrix of dimensions $(l \\times k)$ representing the weight matrix for reconstruction, \nand $\\mathbf{b}^{(r)}$ represents biases. \n\nThe optimal coding function $h$ constituted by the compression and reconstruction operations is defined by the parameters  $(\\mathbf{W},\\mathbf{b}) = (\\mathbf{W}^{(c)},\\mathbf{b}^{(c)},\\mathbf{W}^{(r)},\\mathbf{b}^{(r)})$.\nThe objective is to minimize the reconstruction error.\nReconstruction error for a neighborhood based feature vector $\\mathbf(e)$ is defined as,\n$J(\\mathbf{W,b,e}) = \\frac{1}{2}\\parallel \\mathbf{\\beta}-\\mathbf{e} \\parallel^2$.\nOver all possible feature vectors, the average reconstruction error augmented with a regularization\nterm yields the final objective function $J(\\mathbf{W},\\mathbf{b})$:\n\\begin{equation}\n\\begin{split}\nJ(\\mathbf{W},\\mathbf{b}) = \\frac{1}{{|\\mathbf{\\widehat{E}}|}} \\sum_{\\mathbf{e} \\in \\mathbf{\\widehat{E}}}(\\frac{1}{2}\\parallel \\mathbf{\\beta}^{uv}-\\mathbf{e}^{uv} \\parallel^2)\n\\\\+\\frac{\\lambda}{2} (\\parallel\\mathbf{W}^{(c)}\\parallel_F^2 +  \\parallel\\mathbf{W}^{(r)}\\parallel_F^2 )\n\\end{split}\n\\label{eq:lossfun2}\n\\end{equation}\n\nHere, $\\lambda$ is a user assigned regularization parameter, responsible for\npreventing over-fitting. $\\parallel\\cdot\\parallel_F$ represents the \nFrobenius norm of a matrix. In this work we use $\\lambda = 0.1$. \n\nTo this end, we discuss the motivation of our proposed optimization framework\nfor learning the coding function $h$. Note that,\nthe dimensionality of $\\mathbf{\\alpha}$ is much smaller than $\\mathbf{e}$, so the\noptimal compression of the vector $\\mathbf{e}$ must extract patterns composing of\nthe entries of $\\mathbf{e}$ and use them as high-order latent feature in\n$\\mathbf{\\alpha}$. In fact, the entries in $\\mathbf{e}$ contain the neighborhood (sum\nof adjacency vector of the node pair) and link history of a node-pair for all\nthe timestamps; for a real-life network, this vector is sparse and substantial\ncompression is possible incurring small loss. Through this compression the\ncoding function $h$ learns the patterns that are similar across different node-pairs (used in $\\mathbf{\\widehat{E}}$).\nThus the function $h$ learns a metric embedding of the node-pairs that packs node-pairs\nhaving similar local structures in close proximity in the embedded feature space.\nAlthough function $h$ acts as a black-box, it captures patterns involving neighborhood\naround a node pair across various time stamps, which obviates the manual construction \nof a node-pair feature---a cumbersome task for the case of a dynamic network. \n\n\n\\subsubsection{Optimization}\n\nThe training of optimal coding defined by parameters $(\\mathbf{W,b})$ begins with\nrandom initialization of the parameters. Since the cost function\n$J(\\mathbf{W},\\mathbf{b})$ defined in Equation \\eqref{eq:lossfun2} is non-convex in\nnature, we obtain a local optimal solution using the gradient descent approach.\nSuch approach usually provides practically useful results (as shown in the Section \n\\ref{sec:experiemnts}).\nThe parameter updates of the gradient descent are similar to the parameter updates\nfor optimizing Auto-encoder in machine learning.  \nOne iteration of gradient descent updates the parameters using following equations:\n\n\\begin{equation}\n\\begin{split}\nW_{ij}^{(c)} = W_{ij}^{(c)} - \\sigma \\frac{\\partial}{\\partial W_{ij}^{(c)}}J(W,b) \\\\\nW_{ij}^{(r)} = W_{ij}^{(r)} - \\sigma \\frac{\\partial}{\\partial W_{ij}^{(r)}}J(W,b) \\\\\nb_{i}^{(c)} = b_{i}^{(c)} - \\sigma \\frac{\\partial}{\\partial b_{i}^{(c)}}J(W,b) \\\\\nb_{i}^{(r)} = b_{i}^{(r)} - \\sigma \\frac{\\partial}{\\partial b_{i}^{(r)}}J(W,b) \n\\end{split}\n\\label{eq:updatefun1}\n\\end{equation}\n\nHere, $l$ appropriately identifies the weight and bias parameters $l \\in  \\{1,2\\}$. $\\sigma$ is the learning rate. $W_{ij}^{(1)}$ is the weight of connection between node $j$ of the input layer to node $i$ of the hidden layer. \n\nNow, from Equation \\eqref{eq:lossfun2}, the partial derivative terms in equations \\eqref{eq:updatefun1} can be written as,\n\\begin{equation}\n\\begin{split}\n\\frac{\\partial}{\\partial W_{ij}^{(c)}}J(W,b) =  \\frac{1}{|\\mathbf{\\widehat{E}}| } \\sum_{\\mathbf{e} \\in \\mathbf{\\widehat{E}}}\\frac{\\partial}{\\partial W_{ij}^{(c)}}J(\\mathbf{W,b,e})+\\lambda W_{ij}^{(c)} \\\\\n\\frac{\\partial}{\\partial W_{ij}^{(r)}}J(W,b) =  \\frac{1}{|\\mathbf{\\widehat{E}}|} \\sum_{\\mathbf{e} \\in \\mathbf{\\widehat{E}}}\\frac{\\partial}{\\partial W_{ij}^{(r)}}J(\\mathbf{W,b,e})+\\lambda W_{ij}^{(r)} \\\\\n\\frac{\\partial}{\\partial b_{i}^{(c)}}J(W,b) =  \\frac{1}{|\\mathbf{\\widehat{E}}| } \\sum_{\\mathbf{e} \\in \\mathbf{\\widehat{E}}}\\frac{\\partial}{\\partial b_{i}^{(c)}}J(\\mathbf{W,b,e}) \\\\\n\\frac{\\partial}{\\partial b_{i}^{(r)}}J(W,b) =  \\frac{1}{|\\mathbf{\\widehat{E}}| } \\sum_{\\mathbf{e} \\in \\mathbf{\\widehat{E}}}\\frac{\\partial}{\\partial b_{i}^{(r)}}J(\\mathbf{W,b,e})  \\\\\n\\end{split}\n\\label{eq:derivative}\n\\end{equation}\n\n\nThe optimization problem is solved by computing partial derivative of cost function $J(\\mathbf{W,b,e})$ using the back propagation approach \\cite{Rumelhart:1986}. \n\nOnce the optimization is done,\nthe  metric embedding of\nany node-pair ($u,v$) can be obtained by taking the outputs of compression\nstage (Equation \\eqref{eq:Compression}) of the trained optimal coding\n($\\mathbf{W,b}$).\n\n\\begin{equation}\n\\begin{split}\n\\mathbf{\\alpha}^{uv} = f(\\mathbf{W}^{(c)}\\mathbf{e}^{uv} + \\mathbf{b}^{(c)})=h(\\mathbf{e}^{uv})\n\\end{split}\n\\label{eq:unsupervisedFeatures}\n\\end{equation}\n\n\\subsubsection{Complexity Analysis}\nWe use Matlab implementation of optimization algorithm L-BFGS (Limited-memory\nBroyden-Fletcher-Goldfarb-Shanno) for learning optimal coding.  We execute the\nalgorithm for a limited number of iterations to obtain unsupervised features\nwithin a reasonable period of time. Each iteration of L-BFGS executes two tasks\nfor each node-pair: back-propagation to compute partial differentiation of cost\nfunction, and change the parameters $\\mathbf{(W,b)}$. \nTherefore, the time complexity\nof one iteration is $O(|NP_t|kl)$. Here, $NP_t$ is the set on node-pairs used to construct the training dataset $\\mathbf{\\widehat{E}}$.\n$k$ is the length of $\\mathbf{e}$ (dimensionality of initial edge features), and $l$ is\nlength of $\\mathbf{\\alpha}$ (optimal coding).\n\n\n\\section{Link Prediction using proposed metric embedding}\\label{subsec:SupervisedLinkPredictionModel}\nFor link prediction task in a dynamic network, $\\mathbb{G} = \\{G_1,G_2, \\dots ,G_t\\}$; we split the snapshots into two overlapping time windows, $[1,t-1]$ and $[2,t]$.\nTraining dataset, $\\mathbf{\\widehat{E}}$ is feature representation for time\nsnapshots $[1,t-1]$, the ground truth ($\\widehat{\\mathbf{y}}$) is constructed from\n$G_t$. {\\sc DyLink2Vec}\\ learns optimal embedding $h(\\cdot)$ using training dataset $\\mathbf{\\widehat{E}}$.\nAfter training a supervised classification model using\n$\\widehat{\\mathbf{\\alpha}}$=$h(\\widehat{\\mathbf{E}})$ and $\\widehat{\\mathbf{y}}$, prediction\ndataset $\\mathbf{\\overline{E}}$ is used to predict links at $G_{t+1}$. For this\nsupervised prediction task, we experiment with several classification\nalgorithms. Among them SVM (support vector machine) and\nAdaBoost perform the best.  \n\n\n\\begin{algorithm}[h]\n\t\\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\\begin{algorithmic}[1]\n\t\t\\Procedure{LP{\\sc DyLink2Vec}}{$\\mathbb{G},t$}\n\t\t\\Input{$\\mathbb{G}$: Dynamic Network, $t$: Time steps}\n    \t\t\\Output{$\\overline{y}$: Forecasted links at time step $t+1$}\n    \t\t\n\t\t\\State $\\widehat{\\mathbf{E}}$=NeighborhoodFeature($\\mathbb{G}$,$1$,$t-1$)\n\t\t\\State $\\widehat{\\mathbf{y}}$=Connectivity($G_t$)\n\t\t\\State $\\overline{\\mathbf{E}}$=NeighborhoodFeature($\\mathbb{G}$,$2$,$t$)\n\t\t\\State $h$=LearningOptimalCoding($\\widehat{\\mathbf{E}}$)\n\t\t\\State $\\widehat{\\mathbf{\\alpha}}$=$h(\\widehat{\\mathbf{E}})$\n\t\t\\State $\\overline{\\mathbf{\\alpha}}$=$h(\\overline{\\mathbf{E}})$\n\t\t\\State $C$=TrainClassifier($\\widehat{\\mathbf{\\alpha}},\\widehat{\\mathbf{y}}$)\n\t\t\\State $\\overline{\\mathbf{y}}$=LinkForecasting($C,\\overline{\\mathbf{\\alpha}}$)\n\t\t\\State \\textbf{return} $\\overline{\\mathbf{y}}$\n\t\t\\EndProcedure\n\t\t\\end{algorithmic}\n\t\t\\caption{Link Prediction using {\\sc DyLink2Vec}}\\label{alg:LPUFE}\n\\end{algorithm}\n\n\nThe pseudo-code of {\\sc DyLink2Vec}\\ based link prediction method is given in Algorithm \\ref{alg:LPUFE}.  For training\nlink prediction model, we split the available network snapshots into two\noverlapping time windows, $[1,t-1]$ and $[2,t]$. Neighborhood based features\n$\\widehat{\\mathbf{E}}$ and $\\overline{\\mathbf{E}}$ are constructed in Lines 2 and 4,\nrespectively. Then we learn optimal coding for node-pairs using neighborhood\nfeatures $\\widehat{\\mathbf{E}}$ (in Line 5). Embeddings\nare constructed using learned optimal coding (Lines 6 and 7) using output of\ncompression stage (Equation \\ref{eq:unsupervisedFeatures}). Finally, a\nclassification model $C$ is learned (Line 8), which is used for predicting\nlinks in $G_{t+1}$ (Line 9).\n\n\n\\section{MLJ Contribution Information Sheet}\n{\\em 1. What is the main claim of the paper? Why is this an important contribution to the machine learning literature?}\n\n\\noindent \\textbf{Our Answer:}  The temporal dynamics of a complex system such as a social network or a\ncommunication network can be studied by understanding the patterns of link\nappearance and disappearance over time. A critical task along this\nunderstanding is to predict the link state of the network at a future time\ngiven a collection of link states at earlier time points. In existing literature,\nthis task is known as link prediction in dynamic networks. In this work, we propose {\\sc DyLink2Vec}, a scalable method for finding the metric embedding of arbitrary node-pairs (both edges or non-edges) of a  \ndynamic network with many temporal snapshots. This work is different from existing \nworks of representation learning on graphs in two ways:\nfirst, existing works find the embedding of vertices whereas we consider embedding\nof arbitrary node-pairs; second, our work considers a dynamic network, which has\nmany temporal snapshots, whereas existing works consider only a static network. Therefore, \nour embedding method is particularly suitable for link forecasting\nin a dynamic networks, and thus, would be an important contribution to the machine learning literature.\\\\\n\n\\noindent {\\em 2. What is the evidence you provide to support your claim? Be precise.}\n\n\\noindent  \\textbf{Our Answer:} We present following evidences: \n\\begin{itemize}\n\\item We validate the effectiveness of our representation learning method ({\\sc DyLink2Vec}) by utilizing it for link prediction on four real-life dynamic networks- {\\em Enron}, {\\em Collaboration}, and two version of {\\em Facebook} networks.\n\\item We compare the quality of embedding generated from {\\sc DyLink2Vec}\\ (our algorithm) with multiple state-of-the-art methods for dynamic link prediction task. Our comparison results show that the proposed method is superior than all the competing methods.\n\\end{itemize}\n\n\\noindent {\\em 3. What papers by other authors make the most closely related contributions, and how is your paper related to them?}\n\n\\noindent  \\textbf{Our Answer:} A key challenge of link prediction in a dynamic network is to find a suitable feature representation of the node-pair instances which are used for training the prediction model. For the static link prediction problem, various topological metrics (common neighbors, Adamic-Adar, Jaccard's coefficient) are used as features, but they can not be extended easily for the dynamic setting having multiple snapshots of the network. In this work, we propose a very general method for constructing unsupervised feature representation in \ndynamic setting. The following papers also provide methods for generating useful node-pair (edge) feature representation, and thus they are related to our work. \n\\begin{itemize}\n\\item Temporal Link prediction using matrix and tensor factorization by D.M. Dunlavy, T. G. Kolda, and E. Acar, in ACM Trans. Knowl. Discov. Data, 2011\n\n\\item Link prediction using time series of neighborhood-based node similarity scores by\nG{\\\"u}ne{\\c{s}}, {\\.I}smail and G{\\\"u}nd{\\\"u}z-{\\\"O}{\\u{g}}{\\\"u}d{\\\"u}c{\\\"u}, {\\c{S}}ule and {\\c{C}}ataltepe, Zehra, in\nData Mining and Knowledge Discovery, 2015\n\n\\item DeepWalk: Online Learning of Social Representations by\nPerozzi, Bryan and Al-Rfou, Rami and Skiena, Steven, in Proc.\nof SIGKDD, 2014. \n\n\\end{itemize}\n\n\\noindent {\\em 4. Have you published parts of your paper before, for instance in a conference? If so, give details of your previous paper(s) and a precise statement detailing how your paper provides a significant contribution beyond the previous paper(s).}\n\n\\noindent \\textbf{Our Answer:} No, we did not publish parts of our paper anywhere.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe steady decrease of estimates of metal abundances in\nthe solar atmosphere  over the\nlast years (Holweger \\cite{hol01}, Lodders \\cite{lod03}, Asplund et al. \\cite{ags04})\nis  a serious {\\rm challenge} not only {\\rm to} solar physics, but\nalso to dust modelling.\n This calls for new {\\rm dust} models  able to produce the same extinction\nwith a smaller amount of solid material.\nA solution to the problem could be provided by\nan ``admixture of vacuum'' i.e. by the porosity of interstellar grains.\n\nGrain aggregates with {\\rm large} voids can {\\rm form} during\nthe growth of interstellar grains due to their coagulation\nin dense molecular cloud cores (Dorschner \\& Henning \\cite{dh95}).\n The internal structure of such composite grains can\nbe very complicated, but\ntheir optical properties are {\\rm often} calculated using the Mie theory\nfor homogeneous spheres with an average refractive index\nderived from effective medium theory\n(EMT; see, e.g., Mathis \\&  Whiffen \\cite{mw89},\nJones \\cite{jones88}, Ossenkopf \\cite{oss91},\nMathis \\cite{m96}, Li \\& Greenberg \\cite{li:gre98}, Il'in \\& Krivova \\cite{ik00}).\n\n Another approach  to calculate the optical properties of\nsuch aggregates is the application of complex, computationally time consuming\nmethods such as the discrete dipole approximation (DDA;\nsee, e.g., Wright \\cite{wr87}, Kozasa et al.~\\cite{kbm92}, \\cite{ko93},\nWolff et al.~{\\cite{wo94}}, Stognienko et al.~\\cite{st95},\nKimura \\& Mann \\cite{ki98}).\n\n Using the DDA,  Voshchinnikov et al.~(\\cite{vih05}) examined\nthe ability of the EMT-Mie approach to treat porous\nparticles of different structure.\nThey show that the latter approach  can give relatively\naccurate results only if the very porous particles have small\n(in comparison with the wavelength of incident radiation) ``Rayleigh''\ninclusions.\nOtherwise, the approach becomes inaccurate\nwhen the porosity exceeds $\\sim$0.5.\n At the same time,  the optical properties of heterogeneous\nspherical particles having inclusions of various sizes\n(Rayleigh and non-Rayleigh) and very large porosity\nwere found to closely resemble  those of\nspheres with a large number ($\\ga 15-20$) of different layers.\nThe errors in extinction efficiency factors are smaller than 10~--~20\\% if\nthe size parameter $\\la 15$ and porosity is equal to $0.9$.\nNote that this consideration was restricted by spheres,\nnot very absorbing materials (silicate and amorphous carbon)\nand the integral scattering characteristics\n(extinction, scattering, absorption efficiency factors,  albedo\nand asymmetry parameter)\nbut not the differential cross sections or elements of the\nscattering matrix.\nNevertheless, very simple computational models instead of\ntime-consuming DDA calculations give us a useful way to treat\ncomposite grains of different structure.\n\nIn this paper,  we apply the\nparticle models of porous interstellar dust grains\nbased on the EMT-Mie and layered-sphere calculations.\nThe models described\nin Sect.~\\ref{model} are assumed to represent composite\nparticles with small (Rayleigh) inclusions and inclusions\nof different sizes (Rayleigh and non-Rayleigh).\n The wavelength dependence of extinction is discussed in Sect.~\\ref{ext1}.\n Sections~\\ref{st_ext} and \\ref{ir_ext}\n contain the application of the models to\ncalculations of the extinction curves in the directions\nof two stars,  using new solar abundances\nand the near-infrared (IR) extinction in the directions along the\nGalactic plane.\n The next sections deal with grain temperatures\n(Sect.~\\ref{temp}), profiles of IR silicate bands (Sect.~\\ref{ir_b}),\nand grain opacities at $\\lambda = 1$\\,~mm (Sect.~\\ref{opa}).\nThese quantities are especially important for the analysis of observations\nof protoplanetary discs (Henning et al. \\cite{heal05}).\nConcluding remarks are presented in Sect.~\\ref{concl}.\n\n\\section{Particle models}\\label{model}\n\nInformation about the structure of grains can be included in light\nscattering calculations directly (layered spheres) or can be\nused to find the optical constants (EMT-Mie model).\nWe consider models of both types.\n\nFollowing  previous papers (Voshchinnikov \\&  Mathis \\cite{vm99},\nVoshchinnikov et al. \\cite{vih05}),\nwe construct layered grains as particles consisting of many concentric\nspherical layers of various materials, each with a specified\nvolume fraction $V_i$ ($\\Sigma_i V_i \/V_{\\rm total} = 1)$.\nVacuum can be one of the materials, so a\ncomposite particle may have a central cavity or voids in the form\nof concentric layers.\nThe number of layers is taken to be 18 {\\rm since}\nVoshchinnikov et al.~(\\cite{vih05})  have shown that this was enough\nto preclude an influence of the order of materials on the results.\nFor a larger number of layers,\none can speak of the optical characteristics determined  by\nthe volume fractions of different constituents only.\n\nIn the case of the EMT-Mie model, an average (effective) refractive index\nis calculated using  the popular rule of Bruggeman\n(see, e.g., Ch\\'ylek et al. \\cite{cval00}, Kr\\\"ugel \\cite{kru03}).\nIn this case, the  average dielectric permittivity\n$\\varepsilon_{\\rm eff}$$^($\\footnote{The dielectric permittivity\nis related to the refractive index as $\\varepsilon=m^2$.}$^)$\nis calculated from\n\\begin{equation}\n\\sum_i f_{i} \\frac{\\varepsilon_{i} - \\varepsilon_{\\rm eff}}{\\varepsilon_{i} + 2 \\varepsilon_{\\rm eff}} = 0,\n\\label{bru}\n\\end{equation}\nwhere $f_{i}=V_i \/V_{\\rm total}$ is the volume fraction\nof the $i$th material with the permittivity $\\varepsilon_{i}$.\n\nThe amount of vacuum in a particle can be characterized by its porosity\n${\\cal P}$ ($0 \\leq {\\cal P} < 1$)\n\\begin{equation}\n{\\cal P} = V_{\\rm vac} \/V_{\\rm total}\n= 1 - V_{\\rm solid} \/V_{\\rm total}. \\label{por}\n\\end{equation}\nTo compare the optical properties of porous and compact particles,\none should consider the  porous particles of radius (or size parameter)\n\\begin{equation}\nr_{\\rm porous} = \\frac{r_{\\rm compact}}{(1-{\\cal P})^{1\/3}}\n= \\frac{r_{\\rm compact}}{(V_{\\rm solid} \/V_{\\rm total})^{1\/3}}. \\label{xpor}\n\\end{equation}\n\n As ``basic'' {\\rm constituents}, we choose\namorphous carbon (AC1; Rouleau \\& Martin \\cite{rm91})\nand astronomical silicate (astrosil; Laor \\&  Draine \\cite{laordr93}).\n The refractive indices for {\\rm these materials and some others} considered\nin Sect.~\\ref{ab_ext} were taken from the Jena--Petersburg Database of\nOptical Constants (JPDOC) described by Henning et al.~(\\cite{heal99})\nand J\\\"ager et al.~(\\cite{jetal02}).\n\nThe application of the standard EMT is known to correspond to the case\nof particles having small randomly located inclusions of\ndifferent materials (Bohren \\& Huffman \\cite{bh83}).\n The optical properties of such particles have been well studied\n(see, e.g., Voshchinnikov \\cite{v02} and references therein).\n\n However, one needs the DDA or other computationally\ntime consuming techniques to treat  particles with inclusions\nlarger  than the wavelength of incident radiation.\nThe difference in the optical characteristics of particles with\nsmall and large inclusions has been discussed in\nprevious studies (e.g., Wolff et al.~\\cite{wo94}, \\cite{wo98}).\nThe  fact that this difference drastically\ngrows with the porosity ${\\cal P}$ and becomes quite essential\nalready for $ {\\cal P} \\ga 0.5$ has been discovered\n{\\rm only recently} by Voshchinnikov et al.~(\\cite{vih05}).\n They also found that the scattering properties of\nparticles with inclusions of different sizes (including\nthose with sizes comparable to the wavelength) were\nvery close to those of layered\nparticles, having the same size and material volume fractions.\nA similar conclusion was reached for an ensemble of  particles\nwhere each particle has inclusions of one (but different) size only.\nIn both cases of ``internal'' or ``external'' mixing, the model\nof layered spheres can be applied.\n\nThese results are of particular importance for applications including the\nastronomical ones where we deal with very porous particles.\nNote that instead of time consuming calculations with DDA-like\ncodes,  one can use the exact solution to the light scattering\nproblem for layered spheres, which is as quick as Mie theory, to estimate\nthe properties of particles with Rayleigh\/non-Rayleigh inclusions.\n\nThus, the models of the homogeneous sphere (with EMT) and layered spheres,\nboth having fast implementations, allow one to probe the difference\nin optical properties caused by the different structure of\nscatterers.\n\n\\section{Interstellar extinction and interstellar abundances}\\label{ab_ext}\n\n\n\\subsection{Wavelength dependence of extinction}\\label{ext1}\n\n\n\nAs it is well known,\nthe wavelength dependence of  interstellar extinction $A(\\lambda)$\nis determined by the wavelength dependence of the\nextinction efficiencies $Q_{\\rm ext}(\\lambda)$.\n This quantity is shown in Fig.~\\ref{ext_w} for particles of the same mass,\nbut different porosity.\nThe volume fractions of AC1 and astrosil are equal, i.e.\n$V_{\\rm AC1} \/V_{\\rm total} = V_{\\rm astrosil} \/V_{\\rm total} =\n1\/2 \\, (V_{\\rm solid} \/V_{\\rm total}) = 1\/2 \\, (1 - {\\cal P})$.\n The radius of compact grains is  $r_{\\rm s,\\,compact}=0.1\\,\\mu{\\rm m}$.\n The dependence of $Q_{\\rm ext}$ on $\\lambda$ for compact particles\nis close to that of the average interstellar extinction curve in the\nvisible--near UV ($1 \\,\\mu{\\rm m}^{-1} \\leq\\lambda^{-1} \\leq 3 \\,\\mu{\\rm m}^{-1}$)\n{\\rm where it} can be approximated by the power law\n$A(\\lambda) \\propto \\lambda^{-1.33}$\n(see discussion in Voshchinnikov~\\cite{v02}).\n From Fig.~\\ref{ext_w} one can conclude that the  extinction\nproduced by particles with small inclusions and layers\n differs   little  for compact\nand slightly porous particles. The difference becomes {\\rm most} pronounced in\nthe near and far--UV (for $\\lambda^{-1} \\ga 2.5 - 3 \\,\\mu{\\rm m}^{-1}$).\n\\begin{figure}[htb]\n\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f1.eps}}\n\\caption{Wavelength dependence of the extinction efficiency {\\rm factor}\nfor spherical particles with $r_{\\rm s,\\,compact}=0.1\\,\\mu{\\rm m}$.\nThe particles are of the same mass but of different porosity.\nUpper panel: calculations based on the EMT-Mie theory.\nLower panel: calculations based on the layered-sphere theory.\n}\\label{ext_w}\\end{center}\n\\end{figure}\n\nAs follows from Fig.~\\ref{ext_w}, the wavelength\ndependence of extinction  flattens as porosity increases.\nIt is well known (see, e.g., Greenberg \\cite{g78}) that\n{\\rm different particles produce comparable extinction if the products of their\nsize $r$ and refractive index are close, i.e.}\n\\begin{equation}\n r \\, |m-1| \\approx {\\rm const.}\n\\label{mr}\\end{equation}\n The average refractive index of particles with a larger fraction of vacuum\nis closer to 1.\nDespite a larger radius (e.g., from Eq.~(\\ref{xpor}) follows that\n$r_{\\rm s}=0.22 \\,\\mu{\\rm m}$ if ${\\cal P}=0.9$ and $r_{\\rm s,\\,compact}=0.1 \\,\\mu{\\rm m}$),\nthe product given by Eq.~(\\ref{mr}) decreases because of {\\rm an}\ndrop of $|m-1|$. This implies that a steeper extinction\nwith the wavelength dependence closer to $\\propto \\lambda^{-1.33}$\nis produced by  for compact particles with larger radii and,\nconsequently, a larger amount of solid material.\nThis behaviour is {\\rm also observed for} particles of other masses (i.e.,\ncompact spheres of other radii). Therefore, an interpretation\nof the observed interstellar extinction curve\nusing only very porous grains should\nnot give any gain in dust-phase abundances and would contradict\nthe wavelength behaviour.\n\n\\begin{figure}\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f2.eps}}\n\\caption{\nThe same as in Fig.~\\ref{ext_w} but now for normalized extinction cross section.\n}\\label{cn_w}\\end{center}\n\\end{figure}\nThe role of porosity in extinction is better seen from\nFig.~\\ref{cn_w} where\n{\\rm we give} the wavelength dependence of the normalized cross section\n\\begin{eqnarray}\nC_{\\rm ext}^{\\rm (n)} = \\frac{C_{\\rm ext}({\\rm porous \\, grain})}\n{C_{\\rm ext}({\\rm  compact \\, grain \\, of \\, same \\, mass})} = \\nonumber \\\\\n\\,\\,\\,\\,\\,\\,\\, (1-{\\cal P})^{-2\/3}\\,  \\frac{Q_{\\rm ext}({\\rm porous \\, grain})}\n{Q_{\\rm ext}({\\rm  compact \\, grain \\, of \\, same \\, mass})}. \\label{cn}\n\\end{eqnarray}\nThis quantity shows how porosity influences the extinction cross section.\nAs follows from Fig.~\\ref{cn_w}, both models predict a growth of\nextinction of porous particles in the far-UV and a decrease in the\nvisual--near-UV part  with growing ${\\cal P}$. However, the wavelength interval\nwhere $C_{\\rm ext}^{\\rm (n)} <1$ is  narrower and the minimum is\nless deep in the case of layered spheres.\nIn comparison with compact grains and particles with Rayleigh inclusions,\nparticles with Rayleigh\/non-Rayleigh inclusions can also produce rather\nlarge extinction in the near-IR part of spectrum.\nThis is especially\nimportant for the explanation of the flat extinction across the $3 - 8\\,\\mu{\\rm m}$\nwavelength range measured for several lines of sight\n(see Sect.~\\ref{ir_ext}).\nAt the same time, as follows from Fig.~\\ref{pp09},\nfor particles with $r_{\\rm s,\\,compact}>0.1 \\,\\mu{\\rm m}$\nthe normalized UV cross sections grow faster for the\nBruggeman--Mie theory than for layered spheres.\nThus, an addition of vacuum into particles does not mean an automatic\ngrowth of extinction at all wavelengths and a saving in terms\nof solid-phase elements.\nEvidently, the final decision can be made after fitting the theoretical\ncalculations with observations at many wavelengths.\n\\begin{figure}\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f3.eps}}\n\\caption{Wavelength dependence of the normalized extinction cross section\nfor spherical particles with the same  porosity ${\\cal P}=0.9$.\nUpper panel: calculations based on the EMT-Mie theory.\nLower panel: calculations based on the layered-sphere theory.\n}\\label{pp09}\\end{center}\n\\end{figure}\n\n\\subsection{Extinction in the directions\nto the stars $\\zeta$ Oph and $\\sigma$ Sco}\\label{st_ext}\n\nThe basic requirement for any model of interstellar dust is\nthe explanation of the observed extinction law along with\nthe dust-phase abundances of elements in the interstellar medium.\nThese abundances are obtained as the difference between the\ncosmic reference abundances and the observed gas-phase abundances.\nHowever, the cosmic abundances are not yet\nconclusively established and usually this causes a problem.\nFor many years, the solar abundances were used as the reference ones,\nuntil the photospheres of the early-type stars were found\nnot to be as rich in heavy elements as the solar\nphotosphere was (Snow \\& Witt \\cite{sw96}).\nThese stellar abundances caused the so-called ``carbon crisis''.\nAbundances of the most important dust-forming elements (C, O, Mg, Si, Fe)\nrequired by the  dust models were larger than available.\nHowever, during the past several years the solar abundances\ndropped and now they approach the stellar ones\n(see Asplund et al. \\cite{ags04}).\nEvidently, some abundances determined by Sofia \\& Meyer (\\cite{sm01})\nfor F and G stars must be revised downward\nas it has been done recently\nfor the Sun. This should lead to the agreement between abundances found\nfor stars of different types.\nNote also that the current solar abundances of oxygen and iron\n(see Table~\\ref{ida}) are close to those found from\nhigh-resolution \\mbox{X-ray} spectroscopy.\nJuett et al.~(\\cite{jsc03}) investigated the oxygen\nK-shell interstellar absorption edge in seven X-ray binaries and evaluated\nthe total O abundances. These abundances lie between\n467~ppm$^{(}$\\footnote{parts per million hydrogen atoms}$^{)}$ and 492~ppm.\nSchulz et al.~(\\cite{schet02}) evaluated the\ntotal abundance of iron towards the object Cyg~X-1 to be\n$[{\\rm Fe}\/{\\rm H}]_{\\rm cosmic} \\approx 25$~ppm.\n\nWe applied the model of multi-layered porous particles\nto explain the absolute extinction in the direction to the two stars.\nA first estimate has been made in order to find a possibility\nto enlarge the extinction per unit  mass and to minimize the amount\nof solid phase material.\nSeveral materials as components of composite grains were considered.\nAmong the carbonaceous species,\nthe amorphous carbon in the form of Be1 (Rouleau \\& Martin \\cite{rm91})\nwas found to produce the largest extinction. Also the extinction of iron\noxides strongly increases with the growth of porosity.\nAlthough there are no very good constraints on the abundance of oxides,\nFeO is considered as a possible carrier of the 21~$\\mu{\\rm m}$ emission\nobserved in the spectra of post-AGB stars (Posch et al. \\cite{pma04}).\nVery likely, such particles particles can be produced in redox\nreactions (Duley \\cite{dul80}, Jones \\cite{jones90}).\n\n\\begin{figure}[htb]\n\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f4.eps}}\n\\caption\n{Observed and calculated extinction in the direction to\n$\\zeta$ Oph.\nThe errors of the observations are the result of a parameterization\nof the observations (see Fitzpatrick \\&  Massa \\cite{fm90}).\nThe contribution to the theoretical extinction from different components\nis also shown.\n{\\rm The dot-dashed curve is the approximation with the observed value\n$R_{\\rm V}$ as suggested by Cardelli et al. (\\cite{ccm89}).}\n}\n\\label{zeta}\\end{center}\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f5.eps}}\n\\caption\n{The same as in Fig.~\\ref{zeta} but now for  $\\sigma$ Sco.\nThe observational data were taken from  Wegner~(\\cite{ww02}).\n}\\label{sigma}\\end{center}\\end{figure}\n\nWe fitted the observed extinction  toward the stars\n$\\zeta$~Oph  and $\\sigma$ Sco.\nFor these stars there exist well determined extinction curves and\ngas-phase abundances. It is also important that the major\npart of extinction in these directions is produced in one\ndiffuse interstellar cloud (Savage \\&  Sembach \\cite{ss96},\nZubko et al. \\cite{zkw96}). This allows us to exclude possible large\nvariations in dust composition along the line of sight.\n\nObserved and calculated extinction curves are plotted in\nFigs.~\\ref{zeta} ($\\zeta$~Oph)  and \\ref{sigma} ($\\sigma$ Sco).\nAs  follows from the previous Section,\nthe use of only porous or only compact grains apparently\ndoes not result significant benefit in the   solid-phase abundances.\nTherefore, our models are the combination of compact and\nporous particles. They consist of three or four grain populations:\n\n(I). Porous composite (multi-layered) particles\n(Be1 --- 5\\%,\npyroxene, Fe$_{0.5}$Mg$_{0.5}$SiO$_3$ --- 5\\% for $\\zeta$ Oph or\nforsterite, Mg$_2$SiO$_4$ --- 5\\% for $\\sigma$ Sco and\nvacuum --- 90\\%)\nwith the power-law size distribution having an exponential decay\n$n_{\\rm d}(r_{\\rm s})\\propto r_{\\rm s}^{-2.5}\\exp (-10\/r_{\\rm s})$.\nThe lower\/upper cut-off in the size distribution is\n$0.015\\,\\mu{\\rm m}$\/$0.25\\,\\mu{\\rm m}$ and $0.05\\,\\mu{\\rm m}$\/$0.50\\,\\mu{\\rm m}$\nfor $\\zeta$~Oph and $\\sigma$ Sco, respectively.\n\n(II). Small compact graphite\\footnote{\nThe calculations for graphite were made in the ``2\/3--1\/3'' approximation:\n$Q_{\\rm ext} = {2}\/{3}\\, Q_{\\rm ext}(\\varepsilon_{\\bot}) +\n              {1}\/{3}\\, Q_{\\rm ext}(\\varepsilon_{||})$,\nwhere $\\varepsilon_{\\bot}$ and $\\varepsilon_{||}$ are the dielectric\nfunctions for two cases of orientation of the electric field relative\nto the basal plane of graphite.} grains\nwith a narrow power-law size distribution\n($n_{\\rm d}(r_{\\rm s})\\propto r_{\\rm s}^{-2.5}$,\n$r_{\\rm s}=0.01 - 0.02\\,\\mu{\\rm m}$).\n\n(III). Porous composite grains of magnetite\n(Fe$_3$O$_4$ --- 2\\%, vacuum --- 98\\% for $\\zeta$ Oph and\nFe$_3$O$_4$ --- 8\\%, vacuum --- 92\\%  for $\\sigma$ Sco)\nwith a power-law size distribution\n$n_{\\rm d}(r_{\\rm s})\\propto r_{\\rm s}^{-2.5}$.\nThe lower\/upper cut-off in the size distribution is\n$0.005\\,\\mu{\\rm m}$\/$0.25\\,\\mu{\\rm m}$ and $0.05\\,\\mu{\\rm m}$\/$0.35\\,\\mu{\\rm m}$\nfor $\\zeta$~Oph and $\\sigma$ Sco, respectively.\n\n(IV). Compact grains of forsterite (Mg$_2$SiO$_4$)\nwith the power-law size distribution (only for $\\zeta$ Oph,\n$n_{\\rm d}(r_{\\rm s})\\propto r_{\\rm s}^{-3.5}$,\n$r_{\\rm s,\\,min}=0.10\\,\\mu{\\rm m}$, $r_{\\rm s,\\,max}=0.25\\,\\mu{\\rm m}$). \\\\\n\n\\begin{table*}[htb]\n\\begin{center}\n\\caption[]{Contribution of different grain populations\nto $A_{\\rm V}$ and dust-phase abundances\nfor the model of $\\zeta$ Oph (in ppm)}\\label{da-oph}\n\\begin{tabular}{llccccc}\n\\hline\n\\noalign{\\smallskip}\nComponent & $A_{\\rm V}$  & C & O & Mg& Si & Fe \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n(I)   Be1\/pyroxene\/vacuum & 0\\fm58 & 123  &  68  & 11.3 & 22.6 & 11.3  \\\\\n(II)  Graphite & 0\\fm075&  ~96  &      &      &      &      \\\\\n(III) Magnetite\/vacuum & 0\\fm22&      &  33  &      &      & 24.8 \\\\\n(IV) Forsterite & 0\\fm065&      &  23  & 11.4 &  5.7 &      \\\\\n\\noalign{\\smallskip}\\hline\n\\noalign{\\smallskip}\nTotal & 0\\fm94 & 219 & 124 & 22.7 & 28.2 & 36.1\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\\end{center}\n\\end{table*}\n\nFigures~\\ref{zeta} and \\ref{sigma} also contain\nthe extinction curves calculated using the approximation\nsuggested by Cardelli et al. (\\cite{ccm89}) with\nthe coefficients revised by O'Donnell (\\cite{odonn94}).\nCardelli et al. (\\cite{ccm89}) found that\nthe extinction curves from the UV through the IR\ncould be characterized as a one-parameter family dependent\non the ratio of the total extinction to the selective one\n$R_{\\rm V}=A_{\\rm V}\/E({\\rm B-V})$.\nWe used the observed values of $R_{\\rm V}$ in order\nto plot the CCM approximation. It is seen that\nthis relation describes quite well the extinction for\n$\\zeta$~Oph  but not for $\\sigma$ Sco.\n\n\\begin{table*}[htb]\n\\begin{center}\n\\caption[]{Contribution of different grain populations\nto $A_{\\rm V}$ and dust-phase abundances\nfor the model of $\\sigma$ Sco (in ppm)}\\label{da-sco}\n\\begin{tabular}{llccccc}\n\\hline\n\\noalign{\\smallskip}\nComponent & $A_{\\rm V}$ & C & O & Mg& Si & Fe \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n(I)   Be1\/forsterite\/vacuum & 0\\fm50 & 58  &  35.4 & 17.7 & 8.8 &   \\\\\n(II)  Graphite              & 0\\fm11&  79  &      &      &      &      \\\\\n(III) Magnetite\/vacuum      & 0\\fm52&      &  35.4 &      &      & 26.6 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nTotal & 1\\fm13 & 137 & 71 & 17.7 & 8.8 & 26.6\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\\end{center}\n\\end{table*}\n\nThe contributions from different components to the calculated extinction\nare given in Tables~\\ref{da-oph} and \\ref{da-sco}\nand shown in Figs.~\\ref{zeta} and \\ref{sigma}.\nThe Tables contain also the dust-phase abundances\nof five dust-forming elements for several grain populations.\nThey were calculated for ratios of the extinction cross-section\nto particle volume averaged over grain size distribution\n(see Eq.~(3.36) in Voshchinnikov~\\cite{v02}).\n\nTable~\\ref{ida} gives the current solar abundances\nof five dust-forming elements according to Asplund et al.~(\\cite{ags04})\nas well as the ``observed''\n(solar minus gaseous)  and model  abundances.\n\nThe dust-phase abundances in the line of sight to the star\n$\\zeta$~Oph (HD~149757) were taken from\nTable~2 of Snow \\& Witt~(\\cite{sw96}).\nIn our calculations, we adopted\nthe following quantities\nfor $\\zeta$~Oph: a total extinction\n$A_{\\rm V}=0\\fm94^($\\footnote{This value was obtained from the\nrelation $A_{\\rm V}= 1.12 \\, E({\\rm V-K})$ (Voshchinnikov  \\& Il'in \\cite{vi87})\nand a colour excess  $E({\\rm V-K})=0\\fm84$ (Serkowski et al. \\cite{smf75}).}$^)$,\n colour excess $E({\\rm B}-{\\rm V})=0\\fm32$ and\ntotal hydrogen column density $N({\\rm H})=1.35\\,10^{21}\\,{\\rm cm}^{-2}$\n(Savage \\&  Sembach, \\cite{ss96}).\nThe extinction curve was reproduced according to the parameterization\nof Fitzpatrick \\&  Massa~(\\cite{fm90}).\n\nFor $\\sigma$ Sco (HD~147165), we used the extinction curve,\nthe colour excess $E({\\rm B}-{\\rm V})=0\\fm35$ and\nthe total extinction $A_{\\rm V}=1\\fm13$ according to Wegner~(\\cite{ww02}).\nThe hydrogen column density\n$N({\\rm H})=2.46 \\, 10^{21}\\, {\\rm cm}^{-2}$ was adopted from\nZubko et al.~(\\cite{zkw96}). The gas-phase abundances were taken from\nAllen et al.~(\\cite{asj90}).\n\nThe dust-phase abundances required by the model  are larger\nthan the observed ones in the direction to $\\zeta$~Oph\n(for C and Fe) and\nsmaller than the observed abundances in the direction to $\\sigma$ Sco.\nNote that for $\\sigma$ Sco the required amount of C and Si in dust\ngrains is the lowest in  comparison with  previous\nmodelling. This is due to the use of highly porous particles\nwhich give  considerable extinction in the UV and near-IR (see\nFigs.~\\ref{cn_w} and \\ref{pp09}) and  allow one to ``save''  material.\nFor example, the extinction model of $\\sigma$~Sco with compact grains\npresented by Zubko et al.~(\\cite{zkw96})\nrequires 240~--~260 ppm of C and 20~--~30 ppm of Si\nand the model of Clayton et al.~(\\cite{cletal03})\nneeds 155 ppm of C and 36 ppm of Si\n(cf. 137 ppm and 8.8 ppm from Table~\\ref{ida}).\n\nThe models presented above are based on the light scattering\ncalculations for particles with  Rayleigh\/non-Rayleigh inclusions\n(layered spheres). It is evident that the observed extinction\ncan be also reproduced if we use the particles with  Rayleigh inclusions\n(i.e., if we apply the EMT-Mie theory). Our estimates show that\ndespite a larger extinction in the UV\nthis model requires more material in the solid phase\nin comparison with layered spheres because of\na smaller extinction in the visual-near IR part of the spectrum.\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption[]{Observed and model dust-phase  abundances (in ppm)}\\label{ida}\n\\begin{tabular}{cccccc}\n\\hline\n\\noalign{\\smallskip}\nElement & Solar$^\\ast$  & \\multicolumn{2}{c} {$\\zeta$ Oph} &\\multicolumn{2}{c} {$\\sigma$ Sco}\\\\\n & abundance& obs & model & obs & model \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n~C&              245  & 110  &  219  &     176    & 137    \\\\\n~O&              457  & 126  &  124  &     ~85    & ~71    \\\\\nMg$^{\\ast\\ast}$& ~33.9 & ~31.9&  ~22.7&   ~30.9    & ~~~17.7 \\\\\nSi&             ~34.2 & ~32.6&  ~28.2&  ~~32.4    & ~~~8.8  \\\\\nFe&             ~28.2 & ~28.2&  ~36.1& ~~~27.9    & ~~~26.6  \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\\end{center}\n\\noindent $^\\ast$ According to Asplund et al.~(\\cite{ags04}). \\\\\n\\noindent $^{\\ast\\ast}$ The abundance of Mg was recalculated with the  oscillator\nstrengths from Fitzpatrick~(\\cite{fp97}).\n\n\\end{table}\n\n\n\\subsection{Near infrared extinction in the Galactic plane}\\label{ir_ext}\n\nWe now consider the possibility  of explaining the flat extinction\nacross the $3 - 8\\,\\mu{\\rm m}$ wavelength range observed for\nseveral lines of sight. This flattening was first measured by\nLutz et al.~(\\cite{luetal96}) toward the Galactic center\nwith {\\it ISO}, using hydrogen recombination lines.\nLater Lutz~(\\cite{lutz99}) confirmed the effect using more\nrecombination lines. Recently, Indebetouw et al.~(\\cite{inetal05})\nfound a similar flat extinction along two lines of sight:\n$l=42\\degr$ and $l=284\\degr$. The extinction was obtained at seven\nwavelengths ($1.2 - 8\\,\\mu{\\rm m}$) by combining images from the {\\it Spitzer\nSpace Telescope} with the 2MASS point-source catalog.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f6.eps}}\n\\caption\n{Observed and calculated extinction in the near-IR part\nof spectrum. The observations correspond to the average extinction\nfor two lines of sight along the Galactic plane\n(Indebetouw et al.~\\cite{inetal05})\ntransformed into magnitudes of extinction per kpc.\nThe theoretical extinction was calculated for component~(I)\nof the model used for  $\\zeta$ Oph (${\\cal P}=0.9$, short dashed curve).\nTwo other curves correspond to the same component but with\nanother particle porosity.\n{\\rm The dot-dashed curve is the approximation of Cardelli et al. (\\cite{ccm89})\nwith $R_{\\rm V}=3.1$.}\n}\n\\label{gc}\\end{center}\\end{figure}\nWe used the average extinction given in Table~1 of\nIndebetouw et al.~(\\cite{inetal05}) and transformed\nit into magnitudes of extinction per kpc using\nthe measured value  $A_{\\rm K}\/D = 0\\fm15 \\pm 0\\fm1\\,{\\rm kpc}^{-1}$\n(Indebetouw et al.~\\cite{inetal05}).\nThe observations are plotted in Fig.~\\ref{gc} together with\nthree theoretical curves.\nBecause we have  little information about the UV-visual extinction and\ngas-phase abundances in these directions, we (rather arbitrarily)\napplied the model used for\n$\\zeta$ Oph (porous component (I): Be1\/pyroxene, porosity 90\\%; see\nSect.~\\ref{st_ext}). This model (short dashed curve on Fig.~\\ref{gc})\nwell explains the flat extinction at $\\lambda > 3\\,\\mu{\\rm m}$$^($\\footnote{Note\nthat porous particles from magnetite (component (III); see\nSect.~\\ref{st_ext}) cannot fit well the extinction at these wavelengths\nbecause of a bump at $\\lambda \\approx 2\\,\\mu{\\rm m}$.}$^)$\nbut the extinction in the J and H  bands is  too small.\nCompact particles (long-dashed curve in Fig.~\\ref{gc}) produce\neven larger extinction at these bands than the observed one.\nHowever, the extinction from such particles at longer wavelengths\ndecreases rapidly. Our preliminary analysis shows that\nparticles with a porosity of\nabout 0.6 (solid curve on Fig.~\\ref{gc})\ncan be chosen as an appropriate model.\nEvidently, a similar curve can be obtained as a combination\nof compact and very porous particles.\nQuite close extinction can be found with\nthe CCM approximation and the standard value $R_{\\rm V}=3.1$.\nWe arbitrarily extrapolated this approximation to long wavelengths\n($\\lambda^{-1} < 0.3 \\,\\mu{\\rm m}^{-1}$) where it gives too small extinction.\n\n Extinction produced by porous grains was also rather flat\nbetween 1.0 and 2.2~$\\mu{\\rm m}$ (for example,\n$A(\\lambda) \\propto \\lambda^{-1.3}$ for ${\\cal P}=0.6$) as was detected\nfor several ultracompact HII regions with $A_{\\rm V} \\ga 15^{\\rm m}$\n(Moore et al. \\cite{metal05}).\n\nIn order to calculate the dust-phase  abundances\nfor models presented in Fig.~\\ref{gc}\nwe estimated the total hydrogen column density\nusing Eqs.~(3.26), (3.27) and (3.22) from Voshchinnikov~(\\cite{v02}).\nFirst, we found  the column density of\natomic hydrogen $N({\\rm HI})$ from the extinction at J band and then\ntransformed $N({\\rm HI})$ into a total hydrogen column density\n$N({\\rm H})$ using the ratio of total to selective extinction\n$R_{\\rm V}=3.1$. A value of\n$N({\\rm H})\/D=2.79 \\, 10^{21}\\, {\\rm cm}^{-2}{\\rm kpc}^{-1}$\nwas obtained.\nThe calculated abundances are given in Table~\\ref{igc}, which also\ncontains the visual extinction calculated for three\nmodels. Note that the model of grains with porosity ${\\cal P}=0.6$\ngives the largest contribution to $A_{\\rm V}$ in comparison with\ntwo other models.\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption[]{Dust-phase  abundances (in ppm)\nand visual extinction for three models presented\nin Fig.~\\ref{gc}.}\\label{igc}\n\\begin{tabular}{cccc}\n\\hline\n\\noalign{\\smallskip}\nElement & ${\\cal P} = 0$ & ${\\cal P} = 0.6$ & ${\\cal P} = 0.9$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n~C&      96  &  84  &   62   \\\\\n~O&      53  &  46  &   34   \\\\\nMg&      ~8.9& ~7.7 &  ~5.7   \\\\\nSi&      18  &  16  &   11    \\\\\nFe&    ~8.9  & ~7.7 &   ~5.7  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$A_{\\rm V}$ &   0\\fm60  & 0\\fm79 &  0\\fm61  \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\\end{center}\n\\end{table}\n\nThere are useful observational data\nfor foreground stars in the field $l=284\\degr$.\nFor stars HD~90273,  HD~93205 and HD~93222,\nWegner~(\\cite{ww02}) and Barbaro et al.~(\\cite{betal04})\nestimate  the values of $R_{\\rm V}$ which lie between 3.4 and 4.0.\nFor HD~90273 Barbaro et al.~(\\cite{betal04}) also find\nan anomalously high gas to dust ratio\n$N({\\rm H})\/E({\\rm B}-{\\rm V})=\n1.04\\,10^{22}\\,{\\rm atoms}\\,{\\rm cm}^{-2}\\,{\\rm mag.}^{-1}$\nEnlargement of $R_{\\rm V}$ increases the required dust-phase\nabundances while the decrease of the gas to dust ratio  reduces them.\nAndr\\'e et al.~(\\cite{and03}) measured the  interstellar gas-phase\noxygen abundances along the sight lines toward 5 early-type stars\nwith  $l=285\\fdg3-287\\fdg7$ and  $b=-5\\fdg5~-~+0\\fdg1$.\nThe values of $[{\\rm O}\/{\\rm H}]_{\\rm g} = 443$~ppm vary from\n 356~ppm to 512~ppm, the average value being 443~ppm.\nThis gives for the mean dust-phase\nabundance $[{\\rm O}\/{\\rm H}]_{\\rm d} = 14$~ppm.\nHowever, the extinction curves for HD~93205 and HD~93222\npublished by Wegner~(\\cite{ww02}) have strong UV bumps and flat\nextinction in the far-UV. This means that extinction can be mainly produced\nby carbonaceous grains.\nEvidently, the most reasonable way to solve the problem of\nabundances  is a re-examination of the reference cosmic\nabnudances and a detailed study of their local values.\n\n\\section{Infrared radiation}\\label{irr}\n\\subsection{Dust temperature}\\label{temp}\n\n\n\\begin{figure}\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f7.eps}}\n\\caption{Size dependence  of the temperature\nfor spherical particles.\nThe particles are located at  a distance of\n10$^4\\,R_\\star$ from a star with an effective temperature\n$T_\\star=2500$\\,K.\nUpper panel: calculations based on the EMT-Mie theory.\nLower panel: calculations based on the layered-sphere theory.\n}\\label{td}\\end{center}\n\\end{figure}\nThe commonly used equilibrium temperature\nof cosmic grains is derived from a balance {\\rm between} the energy gain\ndue to absorption of the UV and visual stellar photons\nand  the energy loss due to re-emission of IR photons.\nThe temperature of porous and compact particles\nof different size and porosity is shown in\nFigs.~\\ref{td} and \\ref{td-ppp} as a function of particle\nsize and porosity, respectively.\nThe results were calculated for particles located at a distance of\n10$^4\\,R_\\star$ from a star with an effective temperature\n$T_\\star=2500$\\,K. In the case of the layered spheres,\nan increase of the vacuum fraction\ncauses a decrease of the grain temperature if the  amount of the\nsolid material is kept constant. This behaviour holds for particles of all\nsizes as well as for particles located closer to the star or farther away\nand for other values of $T_\\star$. If the  EMT-Mie theory is applied,\nthe temperature  drops when the porosity grows up to $\\sim 0.7$ and then\nstarts to increase (see Fig.~\\ref{td}, upper panel and Fig.~\\ref{td-ppp}).\nSuch a behaviour corresponds to the results\nof Greenberg \\& Hage~(\\cite{grha91}) who found an increase of\ntemperature for large grain porosity (see Fig.~4 in their paper).\n\n\n\\begin{figure}[htb]\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f8.eps}}\n\\caption{Dependence  of the dust temperature on particle porosity.\nSolid lines: calculations based on the EMT-Mie theory.\ndashed lines: calculations based on the layered-sphere theory.\nOther parameters are the same as in Fig.~\\ref{td}.\n}\\label{td-ppp}\\end{center}\\end{figure}\nAs it follows from Figs.~\\ref{td} and \\ref{td-ppp},\nthe difference in the temperature of very porous grains\ncalculated using the two models can reach $\\sim 6$\\,K or  $\\sim 15$\\%\nwhile the temperature of compact (${\\cal P} = 0$)\ncomposite grains differs by less than 1\\%.\nNote that the relative difference in temperatures of $\\sim 15$\\%\nbetween particles of the two types is kept for other stellar temperatures\n(e.g., in the case of the Sun or an interstellar radiation field).\n\n\nThe intermediate porosity of grains leads to a shift of the\npeak position of IR emission to larger wavelengths\nin comparison with compact particles. This occurs independently of\nparticle structure (small or various size inclusions).\nHowever, very porous grains with Rayleigh\/non-Rayleigh inclusions\nare expected to be systematically cooler than\nparticles  with Rayleigh inclusions. Such a difference can be of\ngreat importance at a lower temperature regime because it can influence\nthe growth\/destruction of mantles on grains in molecular clouds.\n\n                               \n\n\\subsection{Infrared features}\\label{ir_b}\n\nIt is well known that the shape of the IR dust features is a good indicator\nof  the particle size and chemical composition.  With an increase of the size,\na  feature becomes wider and eventually fades away.\nFor example, in the case of compact spherical grains of astrosil,\nthe 10~$\\mu{\\rm m}$ and  18~$\\mu{\\rm m}$  features disappear when the grain radius\nexceeds $\\sim 2-3 \\,\\mu{\\rm m}$.\nObserved differences in small scale structure of the features\nare usually attributed to variations of  the composition\n(e.g., changes of the ratio of magnesium\nto iron in silicates) or  material state (amorphous\/crystalline).\n\n\\begin{figure}\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f9.eps}}\n\\caption{Wavelength dependence  of the absorption efficiency factors\nfor spherical particles\nof radius $r_{\\rm s, \\, compact}=0.1 \\,\\mu{\\rm m}$.\nUpper panel: calculations based on the EMT-Mie theory.\nLower panel: calculations based on the layered-sphere theory.\n}\\label{abs01}\\end{center}\n\\end{figure}\nIn Fig.~\\ref{abs01} we compare the wavelength dependence of\nthe absorption efficiency factors for particles of the same mass but different\nstructure.\n The upper panel shows  results obtained with the EMT-Mie model\nfor particles with Rayleigh inclusions.\nIt can be seen that the central position and the width of the dust features\ndoes not really change.\nLarger changes occur for the layered-sphere model\n(Fig.~\\ref{abs01}, lower panel).\n In this case a growth of ${\\cal P}$ causes a shift of the center\nof the feature to longer wavelengths and its broadening.\n For particles with ${\\cal P}=0.9$, the 10~$\\mu{\\rm m}$  feature transforms into\na plateau while the 18~$\\mu{\\rm m}$  feature disappears.\n\n\\begin{figure}\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f10.eps}}\n\\caption[]\n{Emission in the disc around the star ${\\beta}~\\rm Pictoris$  in the region of silicate\n10\\,$\\mu{\\rm m}$ band.\nStars and squares are the observations of Knacke et al.~(\\cite{kn93})\nand of Telesco \\& Knacke~(\\cite{tk91}). The curves present the results\nof calculations for particles of radius $r_{\\rm s, \\, compact}=0.1 \\,\\mu{\\rm m}$\nas shown in Fig.~\\ref{abs01} but normalized at $\\lambda=9.6\\,\\mu{\\rm m}$.\n}\\label{bet}\\end{center}\n\\end{figure}\nWe plotted in Fig.~\\ref{bet} our data from Fig.~\\ref{abs01} in a normalized\nmanner together with observations of ${\\beta}~\\rm Pictoris$ made by\nKnacke et al.~(\\cite{kn93}) and Telesco \\& Knacke~(\\cite{tk91}).\nAs follows from Fig.~\\ref{bet}, for given optical constants of the silicate\nthe observed shape of the 10~$\\mu{\\rm m}$ feature\nis better reproduced  by either compact or porous particles\nwith small size inclusions of materials.\nNote that in the case of  ${\\beta}~\\rm Pictoris$ the EMT-Mie calculations\nwere earlier used by  Li \\& Greenberg~(\\cite{li:gre98}) for the\nexplanation of the 10~$\\mu{\\rm m}$ emission feature and by\nVoshchinnikov \\& Kr\\\"ugel~(\\cite{vk99}) for\nthe interpretation of the positional\nand wavelength dependence of polarization.\nThe best fit was obtained for very\nporous particles: ${\\cal P} \\approx 0.95$ and 0.76, respectively.\n\n\n\\begin{figure}\\begin{center}\n\\resizebox{\\hsize}{!}{\\includegraphics{3371f11.eps}}\n\\caption{Wavelength dependence  of the normalized absorption efficiency factors\nfor spherical particles of radius $r_{\\rm s, \\, compact}=2 \\,\\mu{\\rm m}$.\nUpper panel: calculations based on the EMT-Mie theory.\nLower panel: calculations based on the layered-sphere theory.\n}\\label{abs2}\\end{center}\n\\end{figure}\nFigure~\\ref{abs2} shows  the normalized absorption efficiency factors\nfor spherical particles of radius $r_{\\rm s, \\, compact}=2 \\,\\mu{\\rm m}$.\nFor compact grains the 10~$\\mu{\\rm m}$ feature almost disappears.\nWhen the porosity increases the strength of the feature grows\nin the case of the Bruggeman-Mie calculations.\nThis tendency coincides with the results\nshown in Fig.~7 of Hage \\& Greenberg~(\\cite{hagr90}) who found\nthat the higher the porosity, the sharper the silicate emission {\\rm became}.\nFor the case of layered spheres the feature becomes only slightly\nstronger but its peak shifts to longer wavelengths.\n\n\nA standard ``compact'' approach to the modelling of\nthe 10~$\\mu{\\rm m}$  feature was used  by\nvan Boekel et al.~(\\cite{vb03}, \\cite{vb05})\nand Przygodda et al. (\\cite{pr03}) who considered the flattening of the 10~$\\mu{\\rm m}$  feature\nas an evidence of grain growth in the discs around Herbig Ae\/Be\nstars and T Tauri stars, respectively.\nOur investigations show that the variations\nof the shape of the feature and its position and strength can\nalso be attributed to the change of porosity and relative amount\nof carbon in composite grains of small sizes.\n\n\n\\subsection{Dust opacities}\\label{opa}\n\n\\begin{table*}[htb]\n\\caption[]{Mass absorption coefficients at $\\lambda = 1$\\,mm of\ncompact and porous spheres consisting of\nAC1$^\\ast$ and (or) astrosil$^{\\ast\\ast}$.}    \\label{t1}\n\\begin{center}\\begin{tabular}{cccccccccc}\n\\hline\n\\noalign{\\smallskip}\n&\\multicolumn{3}{c}{AC1 + astrosil}&\\multicolumn{3}{c}{astrosil}\n&\\multicolumn{3}{c}{AC1 } \\\\\n\\noalign{\\smallskip}  \\cline{2-10} \\noalign{\\smallskip}\n${\\cal P}$ &  $\\rho_{\\rm d}$&\\multicolumn{2}{c}{$\\kappa$, cm$^2$\/g}\n&  $\\rho_{\\rm d}$ &\\multicolumn{2}{c}{$\\kappa$, cm$^2$\/g}\n&  $\\rho_{\\rm d}$ &\\multicolumn{2}{c}{$\\kappa$, cm$^2$\/g} \\\\\n\\noalign{\\smallskip}  \\cline{2-10} \\noalign{\\smallskip}\n&&Brugg.--Mie & lay. spheres\n&&Brugg.--Mie & lay. spheres\n&&Brugg.--Mie & lay. spheres \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n 0.00 & ~2.58   &   1.58 &   1.58 & ~3.30   &   0.310&   0.310 & ~1.85  &   4.37 &   ~4.37\\\\\n 0.10 & ~2.32   &   1.89 &   1.55 & ~2.97   &   0.371&   0.334 & ~1.66  &   5.13 &   ~4.60\\\\\n 0.30 & ~1.80   &   2.75 &   1.87 & ~2.31   &   0.548&   0.446 & ~1.30  &   7.09 &   ~5.77\\\\\n 0.50 & ~1.29   &   3.83 &   2.57 & ~1.65   &   0.778&   0.646 & 0.925  &   9.22 &   ~7.88\\\\\n 0.70 & 0.772   &   3.94 &   4.04 & 0.990   &   0.794&  ~1.05  & 0.555  &   9.14 &  11.9 \\\\\n 0.90 & 0.258   &   2.45 &   8.12 & 0.330   &   0.431&  ~2.20  & 0.185  &   5.94 &  21.8 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\end{tabular}\\end{center}\n $^{\\ast}$ $m(\\lambda=1 {\\rm mm})=2.93+0.276i$, $\\rho_{\\rm d}=1.85$\\,g\/cm$^3$\n\n $^{\\ast\\ast}$ $m(\\lambda=1 {\\rm mm})=3.43+0.050i$, $\\rho_{\\rm d}=3.3$\\,g\/cm$^3$\n\\end{table*}\n\nThe dust opacity or\nthe mass absorption coefficient of a grain material $\\kappa(\\lambda)$\nenters directly in the expression for\nthe dust mass of an object $M_{\\rm d}$ which is determined from\noptically thin millimeter emission\n\\begin{equation}\nM_{\\rm d} =  \\frac{F_{\\rm mm}(\\lambda) D^2}{\\kappa(\\lambda) B_\\lambda(T_{\\rm d})}.\n        \\label{m}\n\\end{equation}\nHere, $F_{\\rm mm}(\\lambda)$ is  the observed flux,\n$D$ the distance to the object, $B_\\lambda(T_{\\rm d})$ the Planck function,\n$T_{\\rm d}$ the dust temperature.\nThe mass absorption coefficient  $\\kappa(\\lambda)$ depends on\nthe particle volume $V_{\\rm total}$,  the material density $\\rho_{\\rm d}$\nand the extinction cross-section $C_{\\rm ext}$ as follows:\n\\begin{equation}\n\\kappa(\\lambda) = \\frac{C_{\\rm ext}}{\\rho_{\\rm d} V_{\\rm total}}  \\approx\n  \\frac{3}{\\rho_{\\rm d}} \\, \\left(\\frac{2 \\pi}{\\lambda}\\right) \\,\n {\\rm Im} \\left\\{\\frac{\\varepsilon_{\\rm eff}-1}{\\varepsilon_{\\rm eff}+2} \\right\\}.\n        \\label{kap}\n\\end{equation}\nAt long wavelengths the scattering can be neglected\n{\\rm ($C_{\\rm ext} \\approx C_{\\rm abs}$)} and\n$C_{\\rm abs}$ {\\rm can be evaluated} in the Rayleigh approximation.\nThen the mass absorption coefficient does not depend on the particle size\nas shown in  the right part of Eq.~(\\ref{kap}).\nThe effective dielectric permittivity $\\varepsilon_{\\rm eff}$\nin Eq.~(\\ref{kap}) can be found from the Bruggeman rule (see Eq.~(\\ref{bru})) or\nthe layered-sphere rule of the EMT (see Eqs.~(7), (8) in\nVoshchinnikov et al. \\cite{vih05}).\n\n\nExtensive  studies of the mass absorption  coefficient dependence\non the material properties and grain shape are\nsummarized by Henning~(\\cite{h96}) who,\nin particular, {\\rm notes} that the opacities at 1~mm are considerably\nlarger for non-spherical particles {\\rm than for} spheres\n(see also Ossenkopf \\& Henning \\cite{oh94}).\nWe find that a similar effect  (an increase of opacity in comparison\nwith compact spheres) is\nproduced by inclusion of a large fraction of vacuum\ninto the particles. This follows  from Table~\\ref{t1}\nwhere the opacities at $\\lambda = 1$\\,mm are  presented.\nThis Table contains the results for particles consisting of three\nmaterials (AC1, astrosil and vacuum) or\ntwo materials (AC1 or astrosil and vacuum).\nIn the first case,\nthe volume fractions of AC1 and astrosil are equal\n($V_{\\rm AC1} \/V_{\\rm total} = V_{\\rm astrosil} \/V_{\\rm total} =\n1\/2 \\, (1 - {\\cal P})$) while in the second case the volume fraction\nof solid material is $1 - {\\cal P}$.\nIt can be seen that the values of $\\kappa$ are generally larger for particles\nwith a larger fraction of vacuum. This is related\nto the decrease of the particle density $\\rho_{\\rm d}$ which is calculated\nas the volume-average quantity.\n As the mass of dust in an object is proportional to $\\rho_{\\rm d}$\n(see Eqs.~(\\ref{m}) and (\\ref{kap})), the assumption of porous grains\n can lead to considerably smaller mass estimates.\nNote that the opacities are larger for more absorbing\ncarbon particles. A similar effect was  noted by\nQuinten et al.~(\\cite{qkhm02}) who theoretically studied the wavelength\ndependence of extinction of different carbonaceous particles.\nThey also showed that the far IR extinction was larger\nfor clusters of spheres and spheroids than for compact spheres.\nA very large enhancement of the submm opacities was  found\nby Ossenkopf \\& Henning~(\\cite{oh94}) in the case of pure\ncarbon aggregates or carbon on silicate grains.\n\nUsing Eq.~(\\ref{m}) and data from Table~\\ref{t1} shows\nhow the particle porosity and structure\ncan influence estimates  of dust mass in an object.\nThe mass ratio estimates can be found in the Rayleigh--Jeans approximation\n$$\n\\frac{M_{\\rm d} ({\\rm \\mbox{compact}})}{M_{\\rm d} ({\\rm porous})}\n=  \\frac{\\kappa_{\\rm porous}(\\lambda) \\, T_{\\rm d, porous}}\n    {\\kappa_{\\rm compact}(\\lambda) \\, T_{\\rm d, compact}}.\n$$\nWith grain temperatures from Fig.~\\ref{td-ppp}\nand the values of $\\kappa$ for composite grains and EMT-Mie theory\n(the third column in Table~\\ref{t1}) we can find that the ratio\n$M_{\\rm d} ({\\rm \\mbox{compact}})\/M_{\\rm d} ({\\cal P} = 0.9)$\nlies between $\\sim 1.3$ and $\\sim 1.5$. If the layered-sphere model\nis used (the forth column in Table~\\ref{t1}) the ratio increases to\n$3.8 - 4.3$. This means that the calculated mass of an object can be reduced\nif compact grains are replaced by porous ones.\n\nThe ratio of dust masses calculated for two grain models is\n$$\n\\frac{M_{\\rm d} ({\\rm \\mbox{EMT-Mie}})}{M_{\\rm d} ({\\rm layered \\, sphere})}\n=  \\frac{\\kappa_{\\rm lay \\, sphere}(\\lambda) \\, T_{\\rm d, lay \\, sphere}}\n    {\\kappa_{\\rm EMT-Mie}(\\lambda) \\, T_{\\rm d, EMT-Mie}} \\approx 2.8.\n$$\nThe numerical value was obtained for particles\nconsisting of AC1, astrosil and vacuum with ${\\cal P} = 0.9$\nand the temperature ratio\n$T_{\\rm d, lay \\, sphere}\/T_{\\rm d, EMT-Mie}=0.85$\nas discussed in Sect.~\\ref{temp}.\nIf we consider  particles of the same porosity but consisting of two\nmaterials, the ratio of masses will\nbe even larger (3.1 for AC1 and 4.3 for astrosil).\nThus,  one can overestimate the mass of an object\nby  a factor of 3 or more  if the EMT-Mie model is applied,\nsince real dust grains in molecular cloud cores\nshould be very porous and should have non-Rayleigh inclusions.\nAnother case when the effect can be important is in circumstellar\ndiscs, e.g. Takeushi et al.~(\\cite{tcl05}) used $\\kappa = 0.3\\,{\\rm cm^2\/g}$\nat $\\lambda = 1$\\,mm for highly porous silicate grains,\nwhich is a good approximation only for particles with small size\ninclusions (see Table~\\ref{t1}).\n\n\\section{Concluding remarks}\\label{concl}\n\nWe have considered how the porosity of composite cosmic dust grains\ncan affect their optical properties important for interpretation of\nobservations of interstellar, circumstellar and cometary dust.\n Two models of particle structure were used.\n Particles of the first kind had well-mixed inclusions\nsmall in comparison with the wavelength,\nwhile those of the second kind consisted of very thin,\ncyclically repeating layers.\n Earlier we  showed that the optical properties of such layered particles\nare close to those of particles with small and large inclusions\n(see Voshchinnikov et al. \\cite{vih05}).\n As effective medium theories give reliable results\nfor particles with small inclusions,\ntwo very different particle structure models can be\nsimply realized and extensive computations can be performed.\n\nFor both models, we studied\nhow an increase of the volume fraction of vacuum could change\nthe extinction efficiencies at different wavelengths,\ntemperature of dust grains, profiles of the IR silicate bands and\ndust millimeter opacities.\n It is found that the models begin to differ\nessentially when the porosity exceeds $\\sim 0.5$.\n This difference appears as lower temperatures (Sect.~\\ref{temp}),\nshifted central peaks of the silicate bands\n(Sect.~\\ref{ir_b}) and larger millimeter opacities\n(Sect.~\\ref{opa})\nfor layered particle model in comparison with that based\non EMT calculations.\n The latter model also requires larger dust-phase abundances\nthan the layered model (Sect.~\\ref{st_ext})\nto produce the same interstellar extinction.\n\nThe assumption that interstellar particles have only small size inclusions\nlooks to some extent artificial\n(excluding, of course, the case of special laboratory samples).\n Therefore, we believe that the layered sphere model well describing\nlight scattering by very porous quasispherical particles with\ninclusions of different sizes\nshould find wide applications in interpretation of different phenomena.\n In particular, this model has good perspectives\nto explanation of the flat interstellar extinction observed\nin the near-IR part of spectrum (Sect.~\\ref{ir_ext}) and\nvariations of the shape of the silicate feature detected in spectra\nof T Tau and Herbig Ae\/Be stars\n(the results will be in a subsequent next paper).\n\n\n\\acknowledgements{\nWe are grateful to Walter Wegner for the possibility to use unpublished data\nand to Bruce Draine for  comments on an earlier version of the paper.\nWe are also grateful to the referee Michael Wolff and scientific editor\nAnthony Jones for useful suggestions.\nNVV acknowledges the hospitality of the Max-Planck-Institut f\\\"ur Astronomie\nwhere this work was finished.\nThe work was partly supported by grant 1088.2003.2 of the President of the\nRussian Federation for leading scientific schools.\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe properties and dynamics of physical systems are closely tied to their symmetries.\nOften these symmetries are known from fundamental principles. There are also, however, systems with unknown or emergent symmetries.\nDiscovering and characterizing these symmetries is an essential component of physics research.\n\n\nBeyond their inherent interest, symmetries are also practically useful for increasing the statistical power of datasets for various analysis goals.\nFor example, a dataset can be augmented with pseudodata generated by applying symmetry transformations to existing data, thereby creating a larger training sample for machine learning tasks.\nNeural network architectures can be constructed to respect symmetries (e.g.~convolutional neural networks and translation symmetries~\\cite{6795724}), in order to improve generalization and reduce the number of model parameters.\nFurthermore, symmetries can significantly increase the size of a useful synthetic dataset created from a generative model trained on a limited set of examples~\\cite{2008.06545}.\n\n\nDeep learning is a powerful tool for identifying patterns in high-dimensional data and is therefore a promising technique for symmetry discovery.\nA variety of deep learning methods have been proposed for symmetry discovery and related tasks.\nNeural networks can parametrize the equations of motion for physical systems, which can have conserved quantities resulting from symmetries~\\cite{greydanus2019hamiltonian,cranmer2020lagrangian}.\nGeneric neural networks targeting classification tasks can encode symmetries in their hidden layers~\\cite{Barenboim:2021vzh,Krippendorf:2020gny}. %\nThis possibility can be used to actively learn symmetries by encoding a shared equivariance in hidden layers across learning tasks~\\cite{zhou2021metalearning}.\nDirectly learning symmetries can be framed as an inference problem given access to parametric symmetry transformations of the same dataset~\\cite{benton2020learning}.\nA given symmetry can be identified in data if a classifier is unable to distinguish a dataset from its symmetric counterpart~\\cite{Tombs:2021wae,Lester:2021kur,Lester:2021aks} (similar to anomaly detection methods comparing data to a reference~\\cite{Collins:2018epr,Collins:2019jip,DAgnolo:2018cun}).\nAnother class of targeted approaches can be found in the domain of automatic data augmentation.\nIf a dataset can be augmented without changing its statistical properties, then one has learned a symmetry. Significant advances in this area have used reinforcement learning~\\cite{cubuk2019autoaugment,lim2019fast}. \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Images\/Plots\/SchematicSymmetry2.pdf}\n    \\caption{\n    A schematic diagram of (top) the training setup for a usual GAN and (bottom) the SymmetryGAN variation discussed in this paper for automatically discovering symmetries.\n    %\n    Here, $g$ is the generator and $d$ is the discriminator.\n    %\n    Not represented here is the incorporation of the inertial reference dataset.\n    %\n    In our numerical examples, this is accomplished by directly imposing constraints on $g$.\n    }\n    \\label{fig:schematic}\n\\end{figure}\n\n\nAn alternative symmetry discovery approach that is flexible, fully differentiable, and simple is based on generative models~\\cite{hataya2019faster,antoniou2018data}.\nUsually, a generative model is a function that maps random numbers to structured data.\nFor example, a deep generative surrogate model can be trained such that the resulting probability density matches that of a target dataset.\nFor symmetry discovery, by contrast, the random numbers are replaced with the target dataset itself.\nIn this way, a well-trained generator designed to confound an adversary will implement a symmetry transformation.  \nWe call this generative model framework for symmetry discovery \\emph{SymmetryGAN}, since it has the same basic training strategy as a generative adversarial network (GAN)~\\cite{Goodfellow:2014upx}, as shown in \\Fig{schematic}.\n\n\nIn this paper, we extend the SymmetryGAN approach and introduce it to the physics community.\nIn particular, we build a rigorous statistical framework for describing the symmetries of a dataset and construct a learning paradigm for automatically detecting generic symmetries.\nThe key idea is that symmetries of a target dataset have to be defined with respect to an \\emph{inertial} reference dataset, analogous to inertial frames in classical mechanics.\nOur deep learning setup is simpler than existing approaches and we develop an analytic understanding of the algorithm's performance in simple cases.\nThis in turn allows us to understand the dynamics of the machine learning as it trains from a random initialization to an element of the symmetry group.\n\n\n\n This rest of this paper is organized as follows.\n %\n In \\Sec{stats}, we build a rigorous statistical framework for discovering the symmetries of a dataset, contrasting it with discovering the symmetries of an individual data element.\n %\n Our machine learning approach with an inertial restriction is introduced in \\Sec{MLa} and the deep learning implementation is described in \\Sec{ML}.\n %\n Empirical studies of simple Gaussian examples, including both analytic and numerical results, are presented in \\Sec{results}.\n %\n We then apply our method to a high energy physics dataset in \\Sec{hepexample}.\n %\n  In \\Sec{inference}, we discuss possible ways to go beyond symmetry discovery and towards symmetry inference, with further studies in \\App{symmetry_discovery_map}.\n %\n Our conclusions and outlook are in \\Sec{conclusions}.\n\n\n\n\\section{Statistics of Symmetries}\n\\label{sec:stats}\n\nWhat is a symmetry?\nLet $X$ be a random variable on an open set $\\O\\subseteq\\mathbb{R}^n$, and let $x$ be an instantiation of $X$.\nWhen we refer to the symmetry of an individual data element $x \\in X$, we usually mean a transformation $h:\\O\\rightarrow\\O$ such that:\n\\begin{equation}\n    h(x) = x,\n\\end{equation}\ni.e.\\ $x$ is invariant to the transformation $h$.\nMore generally, we can consider functions of individual data elements, $f:\\O\\subseteq\\mathbb{R}^n\\rightarrow\\O'\\subseteq\\mathbb{R}^m$.\nIn that case, the function is symmetric if\n\\begin{equation}\n\\label{eq:functionsymmetry}\n    f(h(x)) = f(x),\n\\end{equation}\ni.e. the output of $f$ is invariant to the transformation $h$ acting on $x$.\nOne can also consider equivariances, where the output of $f$ has well-defined transformation properties under the symmetry~\\cite{Dolan:2020qkr,serviansky2020set2graph,Bogatskiy:2020tje,Shimmin:2021pkm}.\nWhile symmetries acting on individual data elements are interesting, they are \\emph{not} the focus of this paper.\n\n\nWe are interested in the symmetries of a dataset as a whole, treated as a statistical distribution.\nLet $X$ be governed by the probability density function (PDF) $p$.\nNaively, a symmetry of the dataset $X$ is a map $g:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n$ such that $g$ preserves the PDF:\n\\begin{equation}\n\\label{eq:naivesymmetry}\np(X = x) = p(X = g(x)) \\, |g'(x)|,\n\\end{equation}\nwhere $|g'(x)|$ is the Jacobian determinant of $g$.\nWhile it is necessary that any candidate symmetry preserves the probability density, it is not sufficient, at least not in the usual way that physicists think about symmetries.\n\n\nConsider the simple case of $n=1$.\nLet $F$ be the cumulative distribution function (CDF) of $X$.\n$F(X)$ is itself a random variable satisfying \n\\begin{equation}\n    F(X)\\sim\\mathcal{U}[0,1],\n\\end{equation}\nwhere $\\mathcal{U}(\\O)$ is the uniform random variable on $\\O$.\nConversely, $F^{-1}(\\mathcal{U}[0,1])$ is a random variable governed by the PDF $p$ (for technical details, see~\\Ref{10.2307\/2132726}).\nThe uniform distribution on the interval $[0,1]$ has many PDF-preserving maps, such as the quantile inversion map:\n\\begin{equation}\n\\widetilde{g}(x)=1-x.\n\\end{equation}\nThis map has the additional property that $\\widetilde{g}^2(x)=x$, so it appears to represent a $\\mathbb{Z}_2$ (i.e.~parity) symmetry.\nUsing the CDF map from above, every probability density $p$ admits a $\\mathbb{Z}_2$ PDF-preserving map:\n\\begin{equation}\n\\label{eq:PDFpreserveZ2}\n    g=F^{-1}\\circ\\widetilde{g}\\circ F.\n\\end{equation}\n\n\nIf we were to accept \\Eq{naivesymmetry} as the definition of a symmetry, then \\emph{all} one-dimensional random variables would have a $\\mathbb{Z}_2$ symmetry, namely the one in \\Eq{PDFpreserveZ2}.\nWhile true in a technical sense, this is not what physicists (or, to our knowledge, any domain experts) think of as a symmetry of a dataset.\nThe precise definition of a symmetry must therefore be stricter than simply PDF-preserving.\nIn particular, while this $\\mathbb{Z}_2$ PDF-preserving map applies to every one-dimensional random variable, it requires a different map for each such variable.\nWhen we usually think about symmetries, we imagine common maps that can be applied to a variety of physical systems that share the same underlying symmetry structure.\n\n\n\n\nThis line of thinking suggests a sharper definition of a symmetry that makes use of a reference distribution.\nConsider two probability densities\n\\begin{equation}\n\\label{eq:twoPDFs}\np:\\mathbb{R}^n\\rightarrow\\mathbb{R}, \\quad p_I:\\mathbb{R}^n\\rightarrow\\mathbb{R}.\n\\end{equation}\nA map $g:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n$ is defined to be a symmetry of $p$ \\emph{relative} to $p_I$ if it is PDF-preserving for both $p$ and $p_I$:\n\\begin{equation}\n\\label{eq:improvedsymmetry}\np(x) = p(g(x)) \\, |g'(x)|, \\quad p_I(x) = p_I(g(x)) \\, |g'(x)|.\n\\end{equation}\nThe reference or \\textit{inertial} density $p_I$ is the analogue of an inertial reference frame in classical mechanics.\nThis new definition of a symmetry will typically exclude quantile maps, like $\\widetilde{g}$ above, because the $\\widetilde{g}$ that works for one random variable will typically not work for another (e.g.\\ Gaussian and exponential random variables).\n\n\nWhile this new definition solves the problem of ``fake'' symmetries, it also introduces a dependence on the inertial distribution.  \nJust as with inertial reference frames, however, there is often a canonical choice for $p_I$ which reduces the number of possibilities in practice.\nA natural choice for many physics datasets is to pick the uniform distribution on $\\mathbb{R}^{n}$, where $n$ is the dimension of the dataset.\nThis not a proper (i.e.\\ normalizable) probability density, though, so we discuss techniques below to use it as the inertial distribution nonetheless.\n\n\nFinally, it is instructive to relate the definitions of symmetries for datasets and functions.\nGiven the two PDFs in \\Eq{twoPDFs}, we can construct the likelihood ratio\n\\begin{equation}\n    \\ell(x) \\equiv \\frac{p(x)}{p_I(x)}.\n\\end{equation}\nApplying the symmetry map $g$ as in \\Eq{improvedsymmetry}, the likelihood ratio transforms as:\n\\begin{equation}\n    \\ell(g(x)) = \\frac{p(g(x))}{p_I(g(x))} = \\frac{p(x)}{p_I(x)} = \\ell(x),\n\\end{equation}\nwhere the Jacobian factor $|g'(x)|$ cancels between the numerator and denominator.\nTherefore the likelihood ratio, which is an ordinary function, is symmetric by the definition in \\Eq{functionsymmetry}.\nThis cancelling of the Jacobian factor is an intuitive way to understand why an inertial reference density is necessary to define the symmetry of a dataset.\n\n\n\n\\section{Machine Learning with Inertial Restrictions}\n\\label{sec:MLa}\n\nThe SymmetryGAN paradigm for discovering symmetries in a dataset involves simultaneously learning two functions:\n\\begin{align}\n    g:\\mathbb{R}^n &\\rightarrow\\mathbb{R}^n,\\\\\n    d:\\mathbb{R}^n &\\rightarrow[0,1].\n\\end{align}\nThe function $g$ is a \\emph{generator} that represents the symmetry map.%\n\\footnote{Here, we are using the machine learning meaning of a ``generator'', which differs from the generator of a symmetry group, though they are closely related.}\nThe function $d$ is a \\emph{discriminator} that tries to distinguish the input data $\\{x_i\\}$ from the transformed data $\\{g(x_i)\\}$.\nWhen the discriminator cannot distinguish the original data from the transformed data, then $g$ will be a symmetry.\nThe technical details of this approach are provided in \\Sec{ML} using the framework of adversarial networks.\n\n\nAs described in \\Sec{stats}, it is not sufficient to require that $g$ preserves the PDF of the input data; it also has to preserve the PDF of the inertial density.\nThere are several methods to implement an inertial restriction into the machine learning strategy.\n\\begin{itemize}\n    \\item \\emph{Simultaneous discrimination:}\n    %\n    In this method, the discriminator $d$ is applied both to the input dataset and to data drawn from the inertial density $p_I$.\n    %\n    The training procedure penalizes any map $g$ that does not fool $d$ for both datasets.\n    %\n    In practice, it might be advantageous to use two separate discriminators $d$ and $d_I$ for this approach.\n    %\n    \\item \\emph{Two stage selection:}\n    %\n    Here, one first identifies all PDF-preserving maps $g$.\n    %\n    Then one \\textit{post hoc} selects the ones that also preserve the inertial density.\n    %\n    \\item \\emph{Upfront restriction:}\n    %\n    If the PDF-preserving maps of $p_I$ are already known, then one could restrict the set of maps $g$ at the outset.\n    %\n    This allows one to perform an unconstrained optimization on the restricted search space.\n    %\n\\end{itemize}\n\n\n\nEach of these methods has advantages and disadvantages.\nThe first two options require sampling from the inertial density $p_I$.\nThis is advantageous in cases where the symmetries of the inertial density are not known analytically.\nWhen $p_I$ is uniform on $\\mathbb{R}^n$ or another unbounded domain, though, these approaches are not feasible.%\n\\footnote{One could try to leverage approximate strategies, such as cutting off the support for $p_I$ a few standard deviations away from the mean of $p$.  Still, one can run into edge effects if there is a mismatch between the domain and range of $g$.}\nThe second option is computationally wasteful, as the space of PDF-preserving maps is generally much larger than the space of symmetry maps.\nWe focus on the third option: restricting the set of functions $g$ to be automatically PDF-preserving for $p_I$.\nThis in turn requires a way to parametrize all such $g$, or at least a large subset of them.\n\nFor all of the studies in this paper, we further focus on the case where the inertial distribution $p_I$ is uniform on $\\mathbb{R}^n$.\nFor any open set $\\O\\subseteq\\mathbb{R}^n$, a differentiable function $g:\\O\\to \\O$ preserves the PDF of the uniform distribution $\\mathcal{U}(\\O)$ if and only if $g$ is an equiareal map.%\n\\footnote{By carefully taking suitable limits, these ideas go through even if $\\mathcal{U}(\\O)$ is an improper prior. The important takeaway is that uniform distributions are preserved by equiareal maps.}\nTo see this, note that the PDF of $X\\sim\\mathcal{U}(\\O)$ is $p(X = x) = 1\/\\operatorname{Vol}(\\O)$.\nHence, the PDF-preserving condition $p = p\\circ g\\cdot |g'|$ is met if and only if $|g'| = 1$.\nA map is equiareal if and only if its Jacobian determinant is $1$, which proves our claim.\nTherefore, our search space to discover symmetries of physics datasets will be the space of equiareal maps of appropriate dimension.\nOf course, there are interesting physics symmetries that do not preserve uniform distributions on $\\mathbb{R}^n$; these would require an alternative approach.\n\n\nThe set of equiareal maps for $n > 1$ is not well characterized.\nFor example, even for $n = 2$, not all equiareal maps are linear.\nA simple example of a non-linear area-preserving map is the H\\'{e}non map~\\cite{10.2307\/43635985}: $g(x,y)=(x,y-x^2)$.\nThis makes the space of equiareal maps difficult to directly encode into the learning.\nWhile the general set of equiareal maps is difficult to parametrize, the set of area preserving linear maps on $\\mathbb R^n$ is well understood:\n\\begin{multline*}\n    \\mathbb{A} SL^\\pm_n(\\mathbb{R}) = \\{g: \\mathbb{R}^n\\to\\mathbb{R}^n \\; | \\; g(x) = Mx + V,\\\\ M\\in \\mathbb{R}^{n\\times n}, \\det M = \\pm 1, V\\in \\mathbb{R}^n\\}.\n\\end{multline*}\nThis is a subgroup of the general affine group $\\operatorname{Aff}_n(\\mathbb{R})$, and it can be characterized as a topological group of dimension $n(n+1) - 1$.\nThese maps even have complete parametrizations such as the \\textit{Iwasawa decomposition}~\\cite{10.2307\/1969548} which significantly aid the symmetry discovery process.\n\n\nNot all symmetries are linear, however, and if one chooses $\\mathbb{A} SL^\\pm_n(\\mathbb{R})$ as the search space, one cannot discover non-linear maps.\nEven so, the subset of symmetries discoverable within $\\mathbb{A} SL^\\pm_n(\\mathbb{R})$ is rich enough, and the benefits of having a known parameterization valuable enough, that we focus on linear symmetries in this paper and leave the study of non-linear symmetries to future work.\n\n\n\\section{Deep Learning Implementation}\n\\label{sec:ML}\n\n\nTo implement the SymmetryGAN procedure, we modify the learning setup of a GAN~\\cite{Goodfellow:2014upx}.\nFor a typical GAN, a generator function $g$ surjects a latent space onto a data space.%\n\\footnote{While all the GANs discussed here are (approximately) bijective, GANs in general need not be. Symmetry discovery requires the generator to be bijective, so one may want to leverage nomalizing flows~\\cite{10.5555\/3045118.3045281,Kobyzev2020} in future work.}\nThen, a discriminator distinguishes generated examples from target examples.\n\n\nFor a SymmetryGAN, the latent probability density is the \\emph{same} as the target probability density, as illustrated in \\Fig{schematic}.\nThe generator $g$ and discriminator $d$ are parametrized as neural networks.\nThey are then trained simultaneously to optimize the binary cross entropy loss functional:\n\\begin{align}\n\\label{eq:numericloss}\n    L[g,d]=-\\frac1N\\sum_{x\\in\\{x_i\\}_{i=1}^N}\\Big[\\log\\big(d(x)\\big) + \\log\\big(1-d(g(x))\\big)\\Big]\\,.\n\\end{align}\nThis differs from the usual binary cross entropy in that the same samples appear in the first and second terms.\nA similar structure appears in neural resampling~\\cite{Nachman:2020fff} and in step 2 of the \\textsc{OmniFold} algorithm~\\cite{Andreassen:2019cjw}.\nFollowing \\Sec{MLa}, we assume that the generator $g$ already preserves the inertial distribution. \n\n\nThe behavior of \\Eq{numericloss} can be understood analytically by considering the limit of infinite data:\n\\begin{align}\\nonumber\n    L[g,d]&=-\\int \\Big[\\log\\big(d(x)\\big) \\, p(x)\\\\\\label{eq:loss}\n    &\\qquad+\\log\\big(1-d(g(x))\\big)\\, p(g(x))\\, |g'(x)|\\Big]\\dd x\\,,\n\\end{align}\nwhere the Jacobian factor $|g'(x)|$ is now made manifest.\nFor a fixed $g$, the optimal $d$ is the usual result from binary classification (see e.g.\\ \\Ref{hastie01statisticallearning,sugiyama_suzuki_kanamori_2012}):\n\\begin{align}\n\\label{eq:optimalf}\n    d_*=\\frac{p(x)}{p(x)+p(g(x)) \\, |g'(x)|}\\,,\n\\end{align}\nwhich is the ratio of the probability density of the first term in \\Eq{loss} to the sum of the densities of both terms.\nInserting $d_*$ into \\Eq{loss} and optimizing using the Euler-Lagrange equation:\n\\begin{equation}\n\\frac{\\delta L[g,g']}{\\delta g}=\\frac{\\partial L}{\\partial g}-\\frac{\\dd}{\\dd x} \\frac{\\partial L}{\\partial g'} = 0,\n\\end{equation}\none can show that the optimal $g$ satisfies\n\\begin{equation}\np(x)=p(g_*(x)) \\, |g_{*}'(x)|,\n\\end{equation}\ni.e.\\ $g$ is PDF-preserving as in \\Eq{naivesymmetry}.\nFor such a $g$, we have that $d_* = \\frac12$, the loss is maximized at a value of $2 \\log 2$, and the discriminator is maximally confounded. The generator tries to maximize the loss with respect to $g$ and the discriminator tries to minimize the loss with respect to $d$.\n\n\nThe SymmetryGAN approach has the potential to find any symmetry representable by $g(x)$.\nTo target a particular symmetry subgroup, $G \\leq \\mathbb{A} SL_n^\\pm(\\mathbb{R})$, we can add a term to the loss function.\nFor example, to discover a cyclic symmetry group, $G = \\mathbb{Z}_q, q\\in\\mathbb N$, the loss function can be augmented with a mean squared error term:\n\\begin{align}\n\\label{eq:cyclicloss}\n    L[g,d] = L_\\text{BCE}[g,d]-\\frac\\alpha N\\sum_{x\\in\\{x_i\\}_{i=1}^N}(g^q(x) - x)^2,\n\\end{align}\n %\n %\nwhere $L_\\text{BCE}$ is the binary cross entropy loss in \\Eq{numericloss}, $g^q$ is $g$ composed with itself $q$ times, and $\\alpha>0$ is a weighting hyperparameter.\nA SymmetryGAN with this loss function will discover the largest subgroup of $G$ that is a symmetry of the dataset.\n\n\n\\section{Empirical Gaussian Experiments}\n\\label{sec:results}\n\nIn this section, we study the SymmetryGAN approach both analytically and numerically in a variety of simple Gaussian examples.\nFor the empirical studies here and in \\Sec{hepexample}, all neural networks are implemented using \\textsc{Keras}~\\cite{keras} with the \\textsc{Tensorflow} backend~\\cite{tensorflow} and optimized with \\textsc{Adam}~\\cite{adam}.\nThe generator function $g$ is parametrized as a linear function, with constraints that vary by example and are described further below.\nThe discriminator function $d$ is parametrized with two hidden layers, using 25 nodes per layer.\nRectified Linear (ReLU) activation functions are used for the intermediate layers and a sigmoid function is used for the last layer.\nFor the empirical studies, $128$ events are generated for each example.\n\n\\subsection{One-Dimensional Gaussian}\n\\label{sec:1d_example}\n\nOur first example involves data drawn from a one-dimensional Gaussian distribution with a $\\mathbb{Z}_2$ reflection symmetry.\nData are distributed according to the probability distribution $\\mathcal N(0.5, 1.0),$ i.e.\\ a Gaussian with $\\mu = 0.5$ and $\\sigma^2 = 1.0$.\nThis distribution has precisely two symmetries, both linear:\n\\begin{equation}\n\\label{eq:1D_minima}\ng(x) = x, \\qquad g(x) = 1-x.\n\\end{equation}\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Images\/Plots\/Z2analytic.pdf}\n    \\caption{\n    %\n    The analytic loss landscape in the slope ($c$) vs.\\ intercept ($b$) space for the one-dimensional Gaussian example.\n    %\n    The two maxima are indicated by stars.}\n    \\label{fig:Z2analytic}\n\\end{figure}\n\n\nImplicitly, we are taking the inertial distribution to be uniform on $\\mathbb{R}$.\nAs stated earlier, the PDF-preserving maps of $\\mathcal{U}(\\mathbb{R})$ are equireal.\nIn one dimension, the only equireal maps are linear.\nLinear maps in one dimension are defined by two numbers, so the generator function can be parametrized as\n\\begin{equation}\n    \\label{eq:linear_form}\n    g(x) = b + c \\, x. \n\\end{equation}\nIn \\Fig{Z2analytic}, we show the analytically computed loss from \\Eq{loss} as a function of $b$ and $c$.\nIn this figure, the discriminator $d$ is taken to be the analytic optimum in \\Eq{optimalf}.\nThere are two maxima in the loss landscape, one corresponding to each of the linear symmetries from \\Eq{1D_minima}.\nHere, and in most subsequent examples below, we have shifted the output such that maximum loss value is $0$.\n\n\nAnother interesting feature of the loss landscape is the deep minimum at $c=0$ that divides the space into two parts.\nThis gives rise to the prediction that, under gradient descent, the neural network will find $g(x)= 1-x$ when $c$ is initialized negative and find $g(x) = x$ when $c$ is initialized positive.\nIn the edge case when $c$ is initialized to precisely zero, the generator is degenerate and no longer even bijective and the outcome is indeterminate, but the likelihood of sampling $c$ to be precisely zero is, of course, zero.\nFor the rest of the paper, we ignore such edge cases.\nThere are no such features in the loss landscape as a function of  $b$, suggesting that there should be little dependence on the initial value of $b$.\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/b_fc_f.pdf}\n    \\label{fig:Z2numeric_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/c_ic_f.pdf}\n    \\label{fig:Z2numeric_ii}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/b_ib_f.pdf}\n    \\label{fig:Z2numeric_iii}}\n    \\caption{\n    %\n    The empirical symmetry discovery process for the one-dimensional Gaussian example.\n    %\n    The initial parameters have a subscript $i$ and the final parameters have a subscript $f$.\n    %\n    (i) Final slope ($c_f)$ vs.\\ final intercept ($b_f$), showing that the network finds the two maxima.\n    %\n    (ii) Final slope ($c_f)$ vs.\\ initial slope ($c_i$), showing the phase transition at $c_i = 0$.\n    %\n    (iii) Final intercept ($b_f)$ vs.\\ initial intercept ($b_i$), showing the independence on $b_i$.}\n    \\label{fig:Z2numeric}\n\\end{figure*}\n\n\n    These predictions are tested empirically in \\Fig{Z2numeric}, where the initialized parameters are $(b_i,c_i)\\sim \\mathcal{U}([-5, 5]^2)$ and the learned parameters are $(b_f,c_f)$.\n    %\n    In \\Fig{Z2numeric_i}, there are distinct clusters at $(b_f, c_f) = (0, 1)$ and $(1, -1)$,\n    showing that the SymmetryGAN correctly finds both symmetries of the distribution and nothing else.\n    %\n    In \\Fig{Z2numeric_ii}, there is a demonstration of the loss barrier in slope space; if the initial slope is positive, the final slope is $+1$, whereas if the initial slope is negative, the final slope is $-1$.\n    %\n    Finally, \\Fig{Z2numeric_iii} shows the absence of a loss barrier in intercept space; the final intercepts are scattered between $0$ and $1$ independent of the initialized intercept. \n    %\n    We discuss further the \\emph{symmetry discovery map} from initialized to learned parameters in \\Sec{symmetry_discovery_map} and \\App{symmetry_discovery_map}.\n    \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Images\/Plots\/restrictedZ2.pdf}\n    \\caption{\n    The analytic loss landscape for the restricted generator $g(x) = 1 + cx$, with two local maxima at $c = -1$ and $c = 0.5$.}\n    \\label{fig:Z2restricted}\n\\end{figure}\n    \nIn the above example, the parameterization of $g$ was sufficiently flexible that the SymmetryGAN could find both symmetries and the loss landscape had no other maxima.\nIf the space is incompletely parameterized, though, then local maxima can manifest as false symmetries.\nFor example, suppose instead of a two parameter $g$ as above, $g$ were parameterized as $g(x) = 1 + c\\, x$.\nThe corresponding analytic loss landscape is shown in \\Fig{Z2restricted}.\nA SymmetryGAN initialized with a negative slope correctly finds the only symmetry of this form, $g(x) = 1 - x$, but a neural network initialised with positive slope is unable to cross over the loss barrier at $c = 0$ and instead settles at the locally loss maximizing $g(x) = 1 + 0.5 \\,x$.\nWhile our investigations of $\\mathbb{A} SL_n^\\pm(\\mathbb{R})$ suggest that this does not happen with the full parametrization, the topology of the set of equiareal maps is not known and therefore obstructions like the one illustrated here are possible.\nIt is always possible to check if a solution is a symmetry, however.\nSpecifically, one can apply the learned function to the data and train a \\textit{post hoc} discriminator to ensure that its performance is equivalent to random guessing.\nFor an analytic symmetry, we know that at the point of loss maximisation  $p = p\\circ g\\cdot |g'|$, and consequently $d = \\frac{p}{p + p\\circ g\\cdot |g'|} = \\frac12$.\nHence, at the global (symmetry) maxima, $L = - \\frac1N\\sum_{x_i}\\qty[\\log d + \\log (d\\circ g)] = 2\\log2$.\nOn the other hand, there is no way for the neural network to get stuck at non-symmetry local maxima with $L = 2 \\log 2$.\nHence, the true symmetries can be distinguished from local optima by checking the value of the loss.\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/SO2symmAnalytic.pdf} \\label{fig:SO2_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/4O2asymmAnalytic.pdf}\n    \\label{fig:SO2_iii}}\n    %\n    \\caption{\n    The analytic loss landscapes overlaid with empirically discovered symmetries for the two-dimensional Gaussian examples with the generator restriction in \\Eq{rotation}.\n    %\n    (i) The Gaussian $N_{1,1}$ with uniform covariance, which has loss maxima on the unit circle $c^2 + s^2 = 1$.\n    %\n(ii) The Gaussian $N_{1,2}$ whose covariance matrix has non-equal diagonal elements, which only has symmetries at $c = \\pm 1$ and $s = 0$.\n    }\n    \\label{fig:SO2}\n\\end{figure*}\n\n\\subsection{Two-Dimensional Gaussian}\n\\label{sec:2d_example}\n\nNext, we consider cases of two-dimensional Gaussian random variables.\nThese examples offer much richer symmetry groups for study as well as a greater scope for variations.\nWe take the inertial distribution to be uniform on $\\mathbb{R}^2$.\n\n\nWe start with the standard normal distribution in two dimensions,\n\\begin{equation}\nN_{1,1}\\equiv\\mathcal{N}\\qty(\\vec{0},\\mathbbm{1}_2),\n\\end{equation}\nwhere $\\mathbbm{1}_n$ is the $n\\times n$ identity matrix.\nThis distribution has as its linear symmetries all rotations about the origin and all reflections about lines through the origin, which constitute the group $O(2)$.\nFor further exploration, we consider a two-dimensional Gaussian with covariance not proportional to the identity,\n\\begin{equation}\nN_{1,2} \\equiv \\mathcal{N}\\qty(\\vec{0},\\mqty[1&0\\\\0&2]).\n\\end{equation}\nThe symmetry group of this distribution is quite complicated and described below.\nAmong other features, it contains the Klein 4--group, $V_4 = \\qty{\\mathbbm{1}, -\\mathbbm{1}, \\sigma_3, -\\sigma_3}$, for Pauli matrix $\\sigma_3$.\n\n\n\nThe linear search space that preserves $\\mathbb{R}^2$, the general affine group in two dimensions, $\\operatorname{Aff}_2(\\mathbb{R}) = \\mathbb{A} GL_2(\\mathbb{R})$, has six real parameters.\nBefore exploring the entire space, we first examine the subspace:\n\\begin{align}\n\\label{eq:rotation}\ng(X) = \\mqty[c&s\\\\-s&c]\\,X,\n\\end{align}\nfor $c, s\\in \\mathbb{R}^\\times$, where $\\mathbb{R}^\\times$ is the set of non-zero real numbers.\nWhile this is only a rotation if $c^2+s^2=1$, we want to test if a SymmetryGAN can discover this relationship starting from this more general representation.\nThe symmetries represented by \\Eq{rotation} are a subgroup of $GL_2(\\mathbb{R})$: $SO(2)\\times \\mathbb{R}^+ = \\ev{\\theta, r|\\theta\\in [0, 2\\pi), r\\in\\mathbb{R}^+}$, where $\\mathbb{R}^+$ is the set of positive real numbers.\nFor the $ N_{1,1}$ Gaussian, this means looking for the $r = 1$ subgroup, which is indicated by the red circle in the loss landscape in \\Fig{SO2_i}.\nTo test the SymmetryGAN, we sample the parameters $c$ and $s$ uniformly at random from $[-1, 1]^2$, and the learned $c$ and $s$ values correspond to the expected $SO(2)$ unit circle, also shown in \\Fig{SO2_i}.\nWe repeat this exercise for the $ N_{1,2}$ Gaussian in \\Fig{SO2_iii}, where the SymmetryGAN discovers the $\\mathbb{Z}_2$ subgroup of $V_4$ generated by a rotation by $\\pi$.\n\n\n\\begin{figure*}[p]\n    \\centering\n   \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/C2MSE.pdf}\n       \\label{fig:MSE_i}\n       }\n   $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/C3MSE.pdf}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/C7MSE.pdf}}\\\\\n\n    \\caption{The analytic loss landscapes overlaid with empirically discovered symmetries for the $N_{1,1}$ example with a cyclic-enforcing term added to the loss, to be compared to \\Fig{SO2_i}.\n    %\n    The cases studied are (i) $\\mathbb{Z}_2$, (ii) $\\mathbb{Z}_3$, and (iii) $\\mathbb{Z}_7$.}\n    \\label{fig:MSE}\n\\end{figure*}\n\n\\begin{figure*}[p]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/6O2d-tsymm.pdf}} $\\quad$\n     \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/6GL2symmRU.pdf}} $\\quad$\n  \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/6GL2symmAB.pdf}}\\\\\n  \\caption{\n    %\n    Slices through the analytic loss landscape together with empirically discovered symmetries for $\\mathcal N_{1,1}$ with the full $\\mathbb{A} GL_2(\\mathbb{R})$ search space.\n    %\n    (i) The determinant-rotation angle space.  The maxima are indicated by vertical red lines.\n    %\n    (ii) The dilatation-shear space.  The maximum is indicated by a red star.\n    %\n    (iii) The affine translation space.  The maximum is indicated by a red star at the origin.}\n    \\label{fig:AGL2symm}\n\\end{figure*}\n\n\\begin{figure*}[p]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/6GL2asymmDU.pdf}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/6GL2asymmTR.pdf}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.31\\textwidth]{Images\/Plots\/6GL2asymmAB.pdf}}\\\\\n    \n    \\caption{\n    %\n    Similar to \\Fig{AGL2symm} but for the $N_{1,2}$ distribution.\n    %\n    (i) The determinant-sheer space.  The maxima are indicated by two red stars.\n    %\n    (ii) The dilatation-rotation angle space.  The maxima are indicated by four red stars.\n    %\n    (iii) The affine translation space.  The maximum is indicated by a red star at the origin.}\n    \\label{fig:AGL2asymm}\n\\end{figure*}\n\n\n\nThis two-dimensional example allows us to test the approach in \\Eq{cyclicloss} for finding $\\mathbb{Z}_q$ subgroups of the full symmetry group.\nRestricting our attention to the $N_{1, 1}$ example and the $SO(2)\\times \\mathbb{R}^+$ subgroup in \\Eq{rotation}, we add the cyclic-enforcing mean squared error term to the loss with $\\alpha=0.1$.\nResults are shown in \\Fig{MSE} for $q = 2$, $3$, and $7$, where the analytic loss optima and empirically found symmetries are broken into discretely many solutions, with the number corresponding to the $q^{\\textrm{th}}$ roots of unity, as expected.\n\nWe now consider the general affine group, $\\operatorname{Aff}_2(\\mathbb{R})$.\nIn two dimensions, the elements of this group can be represented as a matrix with 6 parameters:\n\\begin{itemize}\n    \\item $d\\in\\mathbb{R}^\\times$, the determinant;\n    \\item $\\theta\\in [0, 2\\pi)$, the angle of rotation;\n    \\item $r\\in \\mathbb{R}^+$, the dilatation;\n    \\item $u\\in \\mathbb{R}$, the shear in the $x$-direction; and\n    \\item $(a, b)\\in \\mathbb{R}^2$ the overall affine shift.\n\\end{itemize}\nBy Iwasawa's decomposition~\\cite{10.2307\/1969548}, the full transformation can be written as\n\\begin{align}\n&g(X) =\\sqrt{|d|}\\,\\mqty[1&0\\\\0&-1]^{\\delta}\\mqty[c_\\theta&s_\\theta\\\\-s_\\theta&c_\\theta]\\mqty[r&0\\\\0&\\frac{1}{r}]\\mqty[1&u\\\\0&1]\\,X+\\mqty[a\\\\b]\\,,\n\\end{align}\nwhere $\\delta=\\frac{1 - \\operatorname{sgn}(d)}{2}$ and $c_\\theta=\\cos(\\theta)$ and $s_\\theta=\\sin(\\theta)$.\n\n\n\nFor the distribution $N_{1,1}$, the symmetry group is $O(2)$, described by the parameters $d = \\pm 1, \\theta\\in[0, 2\\pi), r = 1$, and $u = a = b = 0$.\nVisualizing this space is difficult, but multiple slices through the analytic loss landscape are presented in \\Fig{AGL2symm}.\nThe neural network is trained over all six parameters of the Iwasawa decomposition of $\\operatorname{Aff}_2(\\mathbb{R})$.\nThe empirically discovered symmetries, shown as yellow dots in \\Fig{AGL2symm}, are two-parameter slices of the discovered symmetry group, where slices are chosen such that the parameters not under study are closest to $d = r = 1$, $\\theta = a = b = 0$.\nThe empirical data agree well with the predictions.\n\n\n\n\n\nThe same analysis of $N_{1,2}$ is more complex because the corresponding symmetry group is more complicated than for $N_{1,1}$.\nWhen $r = 1$ and $u = 0$, the symmetries are the $V_4$ we saw earlier ($\\theta = 0,\\pi$ and $d = \\pm 1$).\nBy varying $r$ and $u$, however, one can in fact undo the symmetry-breaking induced by the non-identity covariance, thereby restoring the rotational symmetry.\nFor example, when $r = \\sqrt{2}$, $N_{1,2}$ is transformed into a Gaussian with covariance $\\mathrm{diag}[2,1]$, therefore $r = \\sqrt 2$ and $\\theta = \\frac\\pi2, \\frac{3\\pi}2$ constitutes a symmetry.\nIt is difficult to describe the whole symmetry group in closed form, or even to visualise it because it does not live in any single planar slice of $\\mathbb{A} GL_2(\\mathbb{R})$.\nAs shown for various parameter slices in \\Fig{AGL2asymm}, though, the empirical results agree well with the analytic predictions.\n\n\n\\begin{figure*}[p]\n    \\centering\n   \\subfloat[]{\n        \\includegraphics[height=0.4\\textwidth]{Images\/Plots\/1d_bimodalplot.pdf}\n        \\label{fig:otherdistributions_1Di}\n    }\n    \\subfloat[]{\n        \\includegraphics[height=0.4\\textwidth]{Images\/Plots\/1d_bimodalsymm.pdf}\n         \\label{fig:otherdistributions_1Dii}\n    }\n    \\caption{\n    Empirical distribution (i) and empirically discovered symmetries overlaid on the analytic loss landscape (ii) for a one-dimensional bimodal distribution inspired by \\Ref{fisher2018boltzmann}.\n    }\n    \\label{fig:otherdistributions_1D}\n\\end{figure*}\n\n\\begin{figure*}\n    \\subfloat[]{\\includegraphics[height=0.27\\textwidth]{Images\/Plots\/2d_circularplot.pdf}}\\subfloat[]{ \\includegraphics[height=0.28\\textwidth]{Images\/Plots\/2doctagonalrotations.pdf}}\\subfloat[]{ \\includegraphics[height=0.28\\textwidth]{Images\/Plots\/2doctagonalreflections.pdf}}\\\\\n    \\subfloat[]{\\includegraphics[height=0.27\\textwidth]{Images\/Plots\/2d_squareplot.pdf}}\\subfloat[]{\\includegraphics[height=0.28\\textwidth]{Images\/Plots\/2dsquarerotations.pdf}}\\subfloat[]{\\includegraphics[height=0.28\\textwidth]{Images\/Plots\/2dsquarereflections.pdf}}\n    \\caption{\n    Empirical distributions (left column) and empirically discovered rotations (middle column) and reflections (right column) overlaid on the analytic loss landscape for two two-dimensional Gaussian mixture models inspired by \\Ref{fisher2018boltzmann}.\n    %\n    The studied examples are (i,ii,iii) a two-dimensional octagonal distribution, and (iv,v,vi) a two-dimensional 5$\\times$5 distribution. Note that antipodal points on (iii) and (vi) represent the same reflection}\n    \\label{fig:otherdistributions_2D}\n\\end{figure*}\n\n\n\\subsection{Gaussian Mixtures}\n\n\nAs our last set of simple examples, we apply the SymmetryGAN approach to three Gaussian mixture models, inspired by the examples in \\Ref{fisher2018boltzmann}.\nThe first is a one-dimensional bimodal probability distribution:\n\\begin{align}\np(x) = \\frac12\\mathcal N (-1, 1) + \\frac12 \\mathcal N (1, 1),\n\\end{align}\nwhich respects the $\\mathbb{Z}_2$ symmetry group $g(x) = \\pm x$.\nThe empirical distribution for this example is shown in \\Fig{otherdistributions_1Di}.\nApplying SymmetryGAN starting from the generator for linear transformations in \\Eq{linear_form}, it finds the predicted symmetries with great accuracy, as shown in \\Fig{otherdistributions_1Dii}.\n\n\n\n\n\nWe next consider two two-dimensional Gaussian mixtures.\nThe octagonal distribution,\n\\begin{align}\np(x) = \\frac18\\sum_{i = 1}^8\\mathcal N\\left(\\cos\\frac{2\\pi i}{8}, 0.1\\right)\\times \\mathcal N\\left(\\sin\\frac{2\\pi i}{8}, 0.1\\right),\n\\end{align}\nhas the dihedral symmetry group of an octagon $D_{8}$.\nThe two-dimensional $5\\times 5$ square distribution,\n\\begin{align}\np(x) = \\frac1{25} \\sum_{i = 1}^5\\sum_{j = 1}^5\\mathcal N (i - 2 , 0.1)\\times\\mathcal N (j-2, 0.1),\n\\end{align}\nhas the symmetry group of a square $D_4$.\nWe use the generator \\begin{equation}\n    \\label{eq:O2}\n    g(X) = \\mqty[c&s\\\\-s&(-1)^\\delta c]X,\n\\end{equation}\nwhich can discover the the entire symmetry subgroup (rotations and reflections) in $O(2)$.\nData sampled from these distributions are shown in the left column of \\Fig{otherdistributions_2D}.\nIn the middle and right columns of \\Fig{otherdistributions_2D}, we see that SymmetryGAN finds the expected rotations and reflections, respectively.\n\n\n\\section{Particle Physics Example}\n\\label{sec:hepexample}\n\nWe now turn to an application of SymmetryGANs in particle physics.\nHere, we are interested to learn if this approach can recover well-known azimuthal symmetries in collider physics and possibly identify symmetries that are not immediately obvious.\n\n\\subsection{Dataset and Preprocessing}\n\nThis case study is based on dijet events.\nJets are collimated sprays of particles produced from the fragmentation of quarks and gluons, and pairs of jets are one of the most common configurations encountered at the LHC.\nWith a suitable jet clustering algorithm, each jet has a well-defined momentum, and we can search for symmetries of the jet momentum distributions.\n\nThe dataset we use is the background dijet sample from the LHC Olympics anomaly detection challenge~\\cite{gregor_kasieczka_2019_4536377,Kasieczka:2021xcg}.\nThese events are generated using \\texttt{Pythia} 8.219~\\cite{Sjostrand:2006za,Sjostrand:2007gs} with detector simulation provided by \\texttt{Delphes} 3.4.1~\\cite{deFavereau:2013fsa,Mertens:2015kba,Selvaggi:2014mya}\nThe reconstructed particle-like objects in each event are clustered into\n$R=1$ anti-$k_T$~\\cite{Cacciari:2008gp} jets using \\texttt{FastJet} 3.3.0~\\cite{Cacciari:2011ma,Cacciari:2005hq}.\nAll events are required to satisfy a single $p_T>1.2$~TeV jet trigger, and our analysis is based on the leading two jets in each event, where leading refers to the ones with largest transverse momenta ($p_T^2 = {p_x^2 + p_y^2}$).\n\n\nEach event is represented as a four-dimensional vector:\n\\begin{equation}\nX = (p_{1x},p_{1y},p_{2x},p_{2y}),\n\\end{equation}\nwhere $p_1$ refers to the momentum of the leading jet, $p_2$ represents the momentum of the subleading jet, and $x$ and $y$ are the Cartesian coordinates in the transverse plane.\nWe focus on the transverse plane because the jets are typically back-to-back in this plane as a result of momentum conservation.\nThe longitudinal momentum of the parton-parton interaction is not known and so there is no corresponding conservation law for $p_z$.%\n\\footnote{In principle, we could use SymmetryGAN to confirm the absence of a symmetry in $p_z$.}\n\nSince we have a four-dimensional input space, a natural search space for symmetries is $SO(4)$, the group of all rotations on $\\mathbb{R}^4$.  Before exploring the whole candidate symmetry space, we first consider an $SO(2)\\times SO(2)$ subspace where the two leading jets are independently rotated.\n\n\\subsection{$SO(2) \\times SO(2)$ Subspace}\n\n\n\\begin{figure*}[p]\n    \\centering\n    \\subfloat[]{\\includegraphics[height=0.45\\textwidth]{Images\/Plots\/LHCO.pdf}\n    \\label{fig:LHCO_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[height=0.45\\textwidth]{Images\/Plots\/LHCO_avg.pdf}\n    \\label{fig:LHCO_ii}}\n    \\caption{\n    %\n    (i) Empirically discovered symmetries in the LHC Olympics dijet dataset.\n    %\n    The final values of $\\theta_1$ and $\\theta_2$ from the SymmetryGAN are plotted over the line $\\theta_1 = \\theta_2$.\n    %\n    (ii) The map between initial and final symmetry parameters.\n    %\n    The final rotation angle is the average of the initialized rotation angles, offset by $\\pi$ if the angle between the initialized angles is reflex. \n    }\n    \\label{fig:LHCO}\n\\end{figure*}  \n\nBecause of momentum conservation, we expect that only those rotations that simultaneously rotate both jets by the same angle will be symmetries.\nWe start from a generic $SO(2)\\times SO(2)$ group element:\n\\begin{equation}\n    g_{\\theta_1, \\theta_2}\\mqty[p_{1x}\\\\ p_{1y}\\\\p_{2x}\\\\p_{2y}] = \\mqty[\\cos\\theta_1&\\sin\\theta_1&0&0\\\\-\\sin\\theta_1&\\cos\\theta_1&0&0\\\\0&0&\\cos\\theta_2&\\sin\\theta_2\\\\0&0&-\\sin\\theta_2&\\cos\\theta_2]\\mqty[p_{1x}\\\\ p_{1y}\\\\p_{2x}\\\\p_{2y}],\n\\end{equation}\nwhere $(\\theta_1, \\theta_2) \\in [0, 2\\pi)^2$.\nWe expect the symmetries to correspond to the subgroup $\\qty{g_{\\theta_1, \\theta_2}|\\theta_1 = \\theta_2}\\cong SO(2)$.\nThis prediction is borne out in \\Fig{LHCO_i}.\n\n\nWe can also study the training dynamics of the SymmetryGAN.\nMore information about this procedure is given in \\App{symmetry_discovery_map}, but the idea is to find a symmetry discovery map $\\Omega: SO(2)\\times SO(2) \\to SO(2)$, $(\\theta_{1i}, \\theta_{2i})\\mapsto\\theta_{f},$ that describes how the initial parameters map to the learned ones.\nWe propose the map given by\n\\begin{equation}\n\\begin{split}\n    \\Omega(\\theta_1, \\theta_2) &= \\begin{cases}\\frac{\\theta_1 + \\theta_2}{2}& |\\theta_1 - \\theta_2| < \\pi\\,,\\\\\n    \\frac{\\theta_1 + \\theta_2}{2} - \\pi & |\\theta_1 - \\theta_2| > \\pi\\,,\n    \\end{cases}\n    \\end{split}\n\\end{equation}\nwhere there is only one output angle even though the output space is two-dimensional.\nThis map posits that the final angle will bisect the smaller angle between $\\theta_1$ and $\\theta_2$, which is validated by the empirical results shown in \\Fig{LHCO_ii}.\n\n\n\\subsection{$SO(4)$ Search Space}\n\nWe now turn to the four-dimensional rotation group.\n$SO(4)$ is a six parameter group, specified by $\\qty{\\theta_i}_{i=1}^6,$ which parametrize the six independent rotations:\n\\begin{align}\nR_1\\colon p_{1x}&\\leadsto p_{1y},&\nR_2\\colon p_{1x}&\\leadsto p_{2x},\\\\\nR_3\\colon p_{1x}&\\leadsto p_{2y},&\nR_4\\colon p_{1y}&\\leadsto p_{2x},\\\\\nR_5\\colon p_{1y}&\\leadsto p_{2y},&\nR_6\\colon p_{2x}&\\leadsto p_{2y},\n\\end{align}\nwhere the notation $R: a \\leadsto b$ means\n\\begin{align}\n    R(a) &= a\\cos\\theta + b\\sin\\theta,\\\\\n    R(b) &= b\\cos\\theta - a\\sin\\theta.\n\\end{align}\nOne way to describe a generic generator $g_{\\vb*\\theta}$ is\nby\n\\begin{equation}\n    g_{\\vb*\\theta}(X) = R_1 R_2 R_3 R_4 R_5 R_6 \\, X.\n\\end{equation}\n\n\\begin{figure*}[p]\n    \\centering\n        \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/LHCO_Comparison1.pdf} \\label{fig:LHCOComparison_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/LHCO_Comparison2.pdf}\n    \\label{fig:LHCOComparison_ii}}\n    \\caption{\n    %\n    Two dimensional projection of (i) the original LHC Olympics dijet dataset and (ii) its transformation by one of the generators discovered by the SymmetryGAN.\n    %\n    Here, we plot the momenta of the two leading jets in the transverse plane.}\n    \\label{fig:LHCO_Comparison}\n\\end{figure*}\n\nIt is not easy to visualize a six-dimensional space, and the symmetries discovered by SymmetryGAN\ndo not lie in any single $2$-plane or even $3$-plane.\nTherefore, we need alternative methods to verify that the maps discovered by the neural network are indeed symmetries.\n\n\nOne verification strategy is to visually inspect $X$ and $g_{\\vb*\\theta}(X)$ to see if the spectra look the same.\nIn \\Fig{LHCO_Comparison}, we show a projection of the distribution of $X$ and one instance of $g_{\\vb*\\theta}(X)$, which suggests that the found $g_{\\vb*\\theta}$ is indeed a symmetry.\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/KL_rand1.pdf} \\label{fig:KLrand_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/KL_rand2.pdf}\n    \\label{fig:KLrand_ii}}\n    %\n    \\caption{\n    %\n    %\nAn example of the jet azimuthal angle distributions, (i)$\\widetilde{\\phi}_{1}$ and (ii)$\\widetilde{\\phi}_{2}$, of the LHC Olympics dijet data rotated by a randomly selected rotation in $SO(4)$.\nThe distribution is not uniform, so a random rotation is not a symmetry.}\n    \\label{fig:KL_rand}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/KL_symm1.pdf} \\label{fig:KLsymm_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/KL_symm2.pdf}\n    \\label{fig:KLsymm_ii}}\n    %\n    \\caption{\n        %\n    %\n    The same as \\Fig{KL_rand}, but for a symmetry in $SO(4)$.\n    %\n    The distribution is uniform, so this rotation is a candidate symmetry.\n    }\n    \\label{fig:KL_symm}\n\\end{figure*}\n\nAnother verification strategy is to test if the discovered symmetries preserve special projections of the dataset.\nEach of the two jets has an azimuthal angle $\\phi_{j} = \\text{arctan2}(p_{jy}, p_{jx})$ for $j = 1,2$ that is uniformly distributed over $[-\\pi, \\pi)$, where $\\text{arctan2}$ is the two argument arctangent function, which returns the principal value of the polar angle $\\theta\\in (-\\pi, \\pi]$.\nSymbolically, the data can be represented as\n\\begin{equation}\n    X = \\mqty[p_{1x}\\\\ p_{1y}\\\\p_{2x}\\\\p_{2y}] = \\mqty[p_{1T}\\cos\\phi_{i1}\\\\p_{1T}\\sin\\phi_{1}\\\\p_{2T}\\cos\\phi_{2}\\\\p_{2T}\\sin\\phi_{2}],\\qquad \\phi_{j}\\sim\\mathcal{U}[-\\pi, \\pi)\\,,\n\\end{equation}\nwhere $p_{jT}$ is the transverse momentum of each jet (which is approximately the same for both jets since they are roughly back to back).\nIf one applies an arbitrary rotation, there is no reason the new azimuthal angles,\n\\begin{align}\n    \\widetilde{\\phi}_{1} &= \\text{arctan2}(g_{\\vb*\\theta}(X)_2, g_{\\vb*\\theta}(X)_1), \\\\\n    \\widetilde{\\phi}_{2} &=  \\text{arctan2}(g_{\\vb*\\theta}(X)_4, g_{\\vb*\\theta}(X)_3),\n\\end{align}\nshould be uniformly distributed anymore, as \\Fig{KL_rand} demonstrates.\nIf one of the symmetry rotations discovered by the neural network is applied to $X$, however, $\\widetilde{\\phi}_{j}$ must remain uniformly distributed, as shown in \\Fig{KL_symm}.\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/KL_Div1.pdf} \\label{fig:KLdiv_i}}\n    $\\quad$\n    \\subfloat[]{\\includegraphics[width=0.45\\textwidth]{Images\/Plots\/KL_DIv2.pdf}\n    \\label{fig:KLdiv_ii}}\n    %\n    \\caption{\n        %\n    %\n    The KL divergence between the jet azimuthal angle distribution before and after a random rotation or a symmetry rotation, for the (i) leading jet and (ii) subleading jet.\n    %\n    The KL divergence between two samples drawn from $\\mathcal U[-\\pi, \\pi)$ is shown for comparison.\n    }\n    \\label{fig:KL_div}\n\\end{figure*}\n\nThis effect can be quantified by computing the Kullback-Leibler (KL) divergence of the two $\\widetilde{\\phi}_{j}$ distributions against that of $\\phi_{j}$.\nIn \\Fig{KL_div}, we see that the KL divergence of the symmetries is much smaller than the KL divergence of the random rotations.\nAlso plotted on the same figure is the KL divergence of two samples drawn from $\\mathcal{U}[-\\pi, \\pi)$, which represents the irreducible effect from considering a finite dataset.\nThis would be the KL divergence of $\\widetilde{\\phi}_{j}$ obtained from applying an ideal analytic symmetry to $X$, against $\\phi_{j}$.\nIt is instructive to consider the means of the histograms.\nThe KL divergence of randomly selected elements of $SO(4)$ has means of $0.37$ ($0.34$) for the leading (subleading) jet, while the KL divergence of symmetries in $SO(4)$ has respective means $0.0058$ ($0.0090$).\nThe irreducible statistical noise has a mean of $0.0010$.\n\nClearly, the symmetries reconstruct the distribution much better than randomly selected elements of $SO(4)$, and are in fact quite close to the irreducible KL divergence due to finite sample size.\nNote that the x-axis of \\Fig{KL_div} is logarithmic, which magnifies the region near zero, so the difference between the symmetry histogram and the statistical noise histogram is smaller than it might appear.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{Images\/Plots\/LHCOLosses.pdf}\n    \\caption{\n        %\n    %\n    The loss of random rotations in $SO(4)$ compared to the loss of rotations learned by SymmetryGAN, overlaid with the analytic loss of a symmetry, $2\\log2$.}\n    \\label{fig:LHCOLosses}\n\\end{figure}\n\nA final method to independently verify that the rotations SymmetryGAN finds are symmetries of the LHC Olympics data is by computing the loss function.\nAs discussed at the end \\Sec{1d_example}, when $g$ represents a symmetry and $d$ is an ideal discriminator, the binary cross entropy loss is $2\\log2$.\nBy training a \\textit{post hoc} classifier, we can therefore compute the loss of a specific symmetry generator.%\n\\footnote{In principle, one could look at the value of the loss after training the discriminator.  In practice, a post-hoc classifier yields more reliable behavior; see related discussion in \\Ref{Diefenbacher:2020rna}.}\nIn \\Fig{LHCOLosses}, we compare the loss of randomly sampled rotations from $SO(4)$ to the loss of rotations discovered by SymmetryGAN.\nThe latter is quite close to the analytic optimum, $2\\log 2$.\n\n\nFrom these tests, we conclude that SymmetryGAN has discovered symmetries of the LHC Olympics dataset.\nAs discussed further in \\Sec{inference} below, though, discovering symmetries is different from inferring the structure of found subgroup from the six-dimensional search space.\nMimicking the study from \\Fig{MSE_i}, we can study its $\\mathbb{Z}_2$ subgroups, through the loss function in \\Eq{cyclicloss} with $q=2$.\nThe backbone of this subgroup is expected to be the reflections $p_{1k}\\leftrightarrow p_{2k}$ (because both jets have approximately the same momenta) and $p_{jx}\\leftrightarrow p_{jy}$ (because $\\sin\\phi$ and $\\cos\\phi$ look the same upon drawing sufficiently many samples of $\\phi$).\nThe learning process reveals a much larger group, though.\nThere is in fact a continuous group of $\\mathbb{Z}_2$ symmetries, which combine an overall azimuthal rotation and one of the aforementioned backbone reflections.\nIn retrospect, these $\\mathbb{Z}_2$ symmetries should have been expected, since they are compositions of well-known symmetry transformations.\nThis example highlights the need to go beyond symmetry discovery and towards symmetry inference.\n\n\n\\section{Towards Symmetry Inference}\n\\label{sec:inference}\n\nThe examples in \\Secs{results}{hepexample} highlight the potential power of SymmetryGAN for discovering symmetries using deep learning.\nDespite the many maps discovered by the neural network, though, it is difficult to infer, for example, the precise Lie subgroup of $SO(4)$ respected by the LHC Olympics data.\nThis highlights a limitation of this  approach and the distinction between ``symmetry discovery'' and ``symmetry inference''.\nThough SymmetryGAN can identify points on the Lie group manifold, there is no simple way to infer precisely which Lie group has been discovered.\nIn this section, we mention three potential methods to assist in the process of symmetry inference.\n\n\\subsection{Finding Discrete Subgroups}\n\\label{sec:discrete_subgroups}\n\nOne way to better understand the structure of the learned symmetries is to look for discrete subgroups.\nAs already shown in \\Fig{MSE} and mentioned in the particle physics case, we can identify discrete $\\mathbb{Z}_q$ symmetry transformations by augmenting the loss with \\Eq{cyclicloss}.\nBy forcing the symmetries to take a particular form, we can infer the presence (or absence) of such a subgroup.\n\n\nIt is interesting to consider possible modifications to \\Eq{cyclicloss} to handle non-Abelian discrete symmetries.\nThe goal would be to learn multiple symmetries simultaneously that satisfy known group theoretic relations.\nFor example in the Abelian case, a loss term like \n\\begin{equation}\n\\label{eq:abeliansymm}\n    -\\frac\\alpha N\\sum_{x\\in\\{x_i\\}_{i=1}^N}(g_1 \\circ g_2(x) - g_2 \\circ g_1(x))^2\n\\end{equation}\ncould be used to identify any two symmetries $g_1$ and $g_2$ that commute.\nWe leave a study of these possibilities to future work.\n\n\n\\subsection{Group Composition}\n\nBy running SymmetryGAN a few times, one may discover a few points on the symmetry manifold.\nBy composing these discovered symmetries together, one can rapidly increase the number of known points on the manifold because the discovered symmetries are elements of a group, by construction, so their composition is still an element of the group.\n\n\nThis notion is quite powerful.\nThe ergodicity of the orbits of group elements is a richly studied and complex area of mathematics (see e.g.~\\Ref{bams\/1183548783}).\nMany groups of physical interest are locally connected, compact, and have additional structure.\nIn that context, it is likely that the full symmetry group is generated by $\\qty{r_1,\\dots, r_\\nu}$, where $r_i$ is randomly drawn from the group and $\\nu$ is the product of the representation dimension and the number of connected components.\n\nFor example, consider the group $U(1)\\cong SO(2)$, which has $\\nu=1$.\nAlmost any element of $U(1), e^{i\\theta}$, has rotation angle which is an irrational multiple of $\\pi, \\frac\\theta\\pi\\in\\mathbb{R}\\setminus\\mathbb{Q}$.\nWe can therefore approximate any element $e^{i\\phi}\\in U(1)$ by repeated applications of $e^{i\\theta}$:\n\\begin{equation}\n    \\forall e^{i\\phi}\\in U(1)\\;\\forall\\epsilon > 0\\;\\exists n\\in \\mathbb{N}\\; \\norm{e^{i\\phi} - e^{in\\theta}} < \\epsilon\\,.\n\\end{equation}\nIn other words, the subgroup generated by $e^{i\\theta}$ is dense in $U(1)$.\n\n\nIn practice, the symmetries discovered by SymmetryGAN will be not exact due to numerical considerations.\nSince the network learns approximate symmetries with some associated error, each composition compounds this error.\nThus, there are practical limits on the number of compositions that can be carried out with numeric data.\n\n\n\\subsection{The Symmetry Discovery Map}\n\\label{sec:symmetry_discovery_map}\n\nSo far, we have initialized a SymmetryGAN with uniformly distributed values of certain parameters, and then trained it to return the values of those parameters that constitute a symmetry.\nWe can define a \\textit{symmetry discovery map}, which connects the initialized parameters of $g$ to the parameters of the learned function:\n\\begin{equation}\n\\Omega: \\mathbb{R}^k \\to \\mathbb{R}^k,\n\\end{equation}\nwhere $k$ is the dimension of the parameter space.\nThis is a powerful object not only for characterizing the learning dynamics but also to assist in the process of symmetry discovery and inference.\n\n\nThere are at least two distinct reasons why knowledge of this symmetry discovery map is useful.\nFirst, the map is of theoretical interest.\nWe discussed in \\Sec{1d_example} the importance of understanding the topology of the symmetry group. \nThe symmetry discovery map induces a \\emph{deformation retract} from the search space to the symmetry space.\nEvery deformation retract is a homotopy equivalence, and by the Eilenberg-Steenrod axiom of homotopy equivalence~\\cite{Eilenberg117}, the homology groups of the symmetry group can be constructed from the homology groups of the search space.\nEven in low dimensions, the topology of the symmetry group can be non-trivial (cf.~\\Sec{2d_example} for an example in 2D).\nThe topology of $GL_n(\\mathbb{R})$, however, has been studied for over half a century, and the homotopy and homology groups of several non-trivial subgroups of $\\operatorname{Aff}_n(\\mathbb{R})$ have been fully determined~\\cite{SCHLICHTING20171}.\nHence, if the symmetry discovery map were known, one could leverage the full scope of algebraic topology and the known results for the linear groups to understand the topology of the symmetry group.\n\n\nSecond, this map has practical value.\nEvery time a SymmetryGAN is trained, it must relearn how to move the initialized values of $g$ to the final values.\nIntuitively, nearby initial values should map to nearby final values, so learning the symmetry discovery map should enable a more efficient exploration of the symmetry group.\nIn practice, this can be accomplished by augmenting the loss function in \\Eq{numericloss}.\nLet $g(x|c)$ be the symmetry generator, with the parameters $c$ made explicit.\nLet $\\Omega(c)$ be a neural network representing the symmetry discovery map.\nSampling parameters from the space of parameters $\\mathbb{R}^k$ and data points from $X$, we can optimize the following loss:\n\\begin{align}\n\\label{eq:numericloss_parameters}\n    L[\\Omega,d]&=-\\sum_{c \\in \\{c_a\\}} \\sum_{x\\in\\{x_i\\}}  \\Big[\\log\\big(d(x)\\big) \\\\\n    & \\nonumber \\qquad \\qquad \\qquad \\qquad + \\log\\big(1-d(g(x|\\Omega(c)))\\big)\\Big]\\,.\n\\end{align}\nNote that this loss is now a functional of $\\Omega$ instead of $g$.\nIf $\\Omega(c)$ can be initialized to the identity function, then gradient descent acting on $\\Omega(c)$ is (asymptotically) the same as gradient descent acting on the original parameters.\nThus, as long as $\\Omega(c)$ has a sufficiently flexible parametrization, the learned $\\Omega(c)$ will be a good approximation to the symmetry discovery map learned by the original SymmetryGAN.\n\n\nWe defer a full exploration of the symmetry discovery map to future work.\nPreliminary analytic and numerical studies of the symmetry discovery map are shown in \\App{symmetry_discovery_map}.\n\n\n\\section{Conclusions and Outlook}\n\\label{sec:conclusions}\n\n\nIn this paper, we provided a rigorous statistical definition of the term ``symmetry'' in the context of probability densities.\nThis is highly relevant for the field of high energy collider physics where the key objects of study are scattering cross sections.\nWe proposed SymmetryGAN as a novel, flexible, and fully differentiable deep learning approach to symmetry discovery.\nSymmetryGAN showed promising results when applied to Gaussian datasets as well as to dijet events from the LHC, conforming with our analytic predictions and providing new insights in some cases.\n\n\nA key takeaway lesson is that the symmetry of a probability density only makes sense when compared to an inertial density.\nFor our studies, we focused exclusively on the inertial density corresponding to the uniform distribution on (an open subset of) $\\mathbb{R}^n$, since Euclidean symmetries are ubiquitous in physics.\nFurthermore, we only considered area preserving linear maps on $\\mathbb{R}^n$, a simple yet rich group of symmetries that maintain this inertial density.\nMoving forward, there are many opportunities to further develop the concepts introduced in this paper.\nAs a straightforward extension, non-linear equiareal maps over $\\mathbb{R}^n$ could be added to the linear parameterizations we explored, as could Lorentz-like symmetries.\nIn more complex cases where there is no obvious notion of an inertial density, one could study the relative symmetries between two different datasets.\nIt would also be interesting to discover approximate symmetries and rigorously quantify the degree of symmetry breaking.\nThis is relevant in cases where the complete symmetry group is obscured by experimental acceptances and efficiencies.\n\n\nA key open question is how to go beyond symmetry discovery and towards symmetry inference.\nWe showed how one can introduce loss function modifications to emphasize the discovery of discrete subgroups.\nOne could imagine extending this strategy to continuous subgroups to gain a better handle on group theoretic structures.\nThe symmetry discovery map is a potentially powerful tool for symmetry inference, since it in principle allows the entire symmetry group to be discovered in a single training.\nIn practice, though, we found learning the symmetry discovery map to be particularly challenging.\nWe hope future algorithmic and implementation developments will enable more effective strategies for symmetry discovery and inference, in particle physics and beyond.\n\nThe code for this paper can be found in this \\href{https:\/\/github.com\/hep-lbdl\/symmetrydiscovery}{\\underline{GitHub repository}}.\n\n\\section*{Acknowledgments}\n\nKD and BN are supported by the U.S. Department of Energy, Office of Science under contract DE-AC02-05CH11231.\nJT is supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, \\url{http:\/\/iaifi.org\/}), and by the U.S. DOE Office of High Energy Physics under grant number DE-SC0012567.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\tA quantum-chromodynamic potential model was proposed by us\\cite{GRR}\nin 1982, which not only yielded results for the $c\\bar{c}$ and $b\\bar{b}$\nenergy levels and their spin splittings in good agreement with the existing\nexperimental data but its predictions were also confirmed by\nlater experiments at the Cornell Electron Storage Ring\\cite{CESR}.\nAn essential feature of our model was the inclusion of the\none-loop radiative corrections to the quark-antiquark potential, which\nhad been derived by us in an earlier investigation\\cite{GR}. Subsequently,\nthe model was improved by using relativisitic kinematics\\cite{GRR2}\nand a nonsingular form of the quarkonium potential\\cite{GRS}.\nAs shown by us, in addition to the energy levels of $c\\bar{c}$ and $b\\bar{b}$,\nour model also yields results in good agreement with the experimental data\nfor the leptonic and E1 transition widths.  It was further shown by\nZhang, Sebastian, and Grotch\\cite{zhang} that the M1 transition\nwidths for $c\\bar{c}$ and $b\\bar{b}$ obtained from our model\nare in better agreement with the experimental data than those predicted\nusing other potential models.\n\n\tRecently the mass of the ${}^1P_1$ state of charmonium has\nbeen determined by the E760 collaboration\\cite{E760} in $p\\bar{p}$\nannihilations at Fermilab, and the splitting between the center of\ngravity of the ${}^3P_J$ states and the ${}^1P_1$ state, denoted as\n$\\Delta M_P$, is found to be approximately $-0.9$~MeV.  This experimental\nresult has created much interest since it provides a new test for the\npotential models for heavy quarkonia.\n\n\tIf the spin-dependent forces in the quarkonium potential could be\ntreated perturbatively, the $\\Delta M_P$ splitting would arise solely\nfrom the spin-spin (color hyperfine) interaction.  However, the spin-dependent\nforces are known to be quite large and, as observed by Lichtenberg and Potting\n\\cite{lichten}, the contributions of the spin-orbit and tensor interactions to\n$\\Delta M_P$ cannot be ignored in a nonperturbative treatment.  We shall\nanalyze this complex situation with the use of our model which\navoids the use of an illegitimate perturbative treatment, and provide an\nexplanation for the observed splittings of the charmonium P~states.\n\n\tSeveral authors\\cite{halzen,chen,grotch} have recently shown that\na theoretical value for $\\Delta M_P$ in close agreement with the\nexperimental value can be readily obtained from the spin-spin\ninteraction terms in the quarkonium potential.  However,\nsince they have employed an illegitimate perturbative treatment, the\nsignificance of this simple interpretation remains an open question.\n\n\tOnly a quarkonium model which is in good overall agreement with\nthe experimental data can be taken seriously. Our model for heavy quarkonia\nsatisfies this requirement.\n\n\\section{$\\symbol{'143}\\bar{\\symbol{'143}}$ SPECTRUM}\n\n\tOur model is based on the Hamiltonian\n\\begin{equation}\nH=H_0+V_p+V_c , \\label{hamiltonian}\n\\end{equation}\nwhere\n\\begin{equation}\nH_0=2(m^2+{\\bf p}^2)^{1\/2}\n\\end{equation}\nis the relativistic kinetic energy term, and $V_p$ and $V_c$ are nonsingular\nquasistatic perturbative and confining potentials, which are\ngiven in the Appendix.\nWe found a trial wave function introduced by Jacobs, Olsson, and Suchyta\n\\cite{jacobs} particularly suitable for obtaining the quarkonium energy levels\nand wave functions.\n\n\tOur results for the energy-level splittings as well as the individual\nenergy levels of $c\\bar{c}$ are given in Tables~\\ref{splittings}\nand~\\ref{levels}.\nFor experimental data we have generally relied on the Particle Data Group\n\\cite{PDG}, but for the $\\eta_c$ state we have used the new result announced by\nthe E760 collaboration \\cite{appel}.  The two sets\nof theoretical results in these tables correspond to the scalar-exchange and\nthe scalar-vector-exchange forms of the confining\npotential, given by\n\\begin{mathletters}\n\\begin{equation}\nV_c=V_S ,\n\\end{equation}\nand\n\\begin{equation}\nV_c=(1-B)V_S+BV_V ,\n\\label{vconfine}\n\\end{equation}\n\\end{mathletters}\nrespectively.  The results obtained with the scalar-exchange confining\npotential are\nunsatisfactory, while the scalar-vector-exchange results\nare in surprisingly close agreement with the\nexperimental data, including the observed mass of the ${}^1P_1$ state\nand the $\\Delta M_P$ splitting.  The scalar-vector mixing parameter $B$\nis found to be about $\\frac{1}{4}$.\n\n\tIn Table~\\ref{mp}, we display the contributions to $\\Delta M_P$ from\nthe various types of terms in the Hamiltonian (\\ref{hamiltonian}) with\nthe confining potential (\\ref{vconfine}).\nThe table shows comparable contributions to $\\Delta M_P$\nfrom several sources, which brings out the complexity of this splitting when\nspin-dependent potential terms are included in the unperturbed Hamiltonian.\nThe $\\Delta M_P$ splitting, therefore, does not provide a direct test of the\nspin-spin interaction in heavy quarkonia.\n\n\tIn Tables~IV and~V, we give the results for the leptonic and E1 transition\nwidths corresponding to the scalar-vector-exchange confining potential by using\nthe formulae\n\\begin{equation}\n\\Gamma_{ee}({}^3S_1 \\to e^+e^-) = \\frac{16\\pi\\alpha^2\ne_Q^2}{M^2(Q\\overline{Q})}\n\t\\left| \\Psi(0)\\right|^2 \\left( 1-\\frac{16\\alpha_s}{3\\pi}\\right),\n\\end{equation}\nand\n\\begin{eqnarray}\n\\Gamma_{E1}({}^3S_1 \\to {}^3P_J) &=& \\frac{4}{9}\\frac{2J+1}{3}\\alpha e_Q^2\n\tk_J^3 |r_{fi}|^2,\\nonumber \\\\\n\\Gamma_{E1}({}^3P_J \\to {}^3S_1) &=&\\frac{4}{9}\\alpha e_Q^2\n\tk_J^3 |r_{fi}|^2,\\\\\n\\Gamma_{E1}({}^1P_1 \\to {}^1S_0) &=&\\frac{4}{9}\\alpha e_Q^2\n\tk_J^3 |r_{fi}|^2.\\nonumber\n\\end{eqnarray}\nThe photon energies for the E1 transition widths have been\nobtained from the energy difference of the initial and the\nfinal $c\\bar{c}$ states by taking into account the recoil correction.\nOur results are\nin good agreement with the available experimental data \\cite{PDG}, and\nour prediction for the E1 transition width of $1{}^1P_1\\to 1{}^1S_0$ is\n341.8 keV.\n\n\\section{CONCLUSION}\n\tWe conclude with explanatory remarks concerning some features of our\nquarkonium potential.\n\n\\subsection{Renormalization scheme}\n\n\tWe have used the Gupta-Radford (GR) renormalization scheme \\cite{GR2}\nfor the one-loop radiative corrections to the quarkonium potential\nrather than the modified minimal-subtraction ($\\overline{\\rm MS}$)\nscheme.  The GR scheme is a simplified momentum-space subtraction\nscheme, and the parameter $\\mu$ can be interpreted as representing the\nmomentum scale of the physical process.\nThis scheme also has the desirable feature that\nit satisfies the decoupling theorem\\cite{appelquist}.  On the other\nhand, in the $\\overline{\\rm MS}$ scheme $\\mu$ appears as a mathematical\nparameter, and in this scheme decoupling-theorem-violating terms are\nsimply ignored.\n\n\tThe one-loop radiative corrections in the GR scheme can be converted\ninto those in the $\\overline{\\rm MS}$ scheme by means of the relation\n\\cite{GR2}\n\\begin{equation}\n\\alpha_s=\\bar{\\alpha_s}\\left[ 1+\\frac{\\bar{\\alpha_s}}{4\\pi}\\left(\n\t\\frac{49}{3}-\\frac{10}{9}n_l+\\frac{2}{3}\\sum_{n_h}\n\t\\ln\\frac{m^2}{\\mu^2} \\right)\\right]\\, , \\label{renorm}\n\\end{equation}\nwhere $\\bar{\\alpha_s}$ refers to the $\\overline{\\rm MS}$ scheme, and $n_l$\nand $n_h$ are the numbers of light and heavy quark flavors.  If we drop\nthe decoupling-theorem-violating terms that appear in the $\\overline{\\rm MS}$\nscheme, we can put $n_l=n_f$ and $n_h=0$, and (\\ref{renorm}) reduces to\n\\begin{equation}\n\\alpha_s=\\bar{\\alpha_s}\\left[ 1+\\frac{\\bar{\\alpha_s}}{4\\pi}\\left(\n\t\\frac{49}{3}-\\frac{10}{9}n_f \\right)\\right]\\, .\n\\end{equation}\n\n\\subsection{Quasistatic potential}\n\n\tIn an earlier investigation \\cite{GRR2}, we arrived at the surprising\nconclusion that while the quasistatic form of the quarkonium potential yields\nresults in good agreement with the experimental data, this is not the case\nfor the momentum-dependent form.  This conclusion has also been confirmed by\nthe\nrecent investigations of Gara {\\it et al.} \\cite{gara} and Lucha {\\it et al.}\n\\cite{lucha}.\n\n\tIt appears to us that the success of the quasistatic potential is related\nto the phenomenon of quark confinement.  Since a rigorous treatment of\nquark confinement does not exist at this time, we shall only offer a\nplausible argument. It was argued earlier \\cite{GR3} with the use\nof a renormalization-group-improved quantum-chromodynamic treatment\nthat quark confinement can be understood as a consequence of the fact\nthat quarks and antiquarks are unable to exchange low-momentum gluons.\nMoreover, since for the quark-antiquark scattering in the center-of-mass\nsystem\n\\begin{equation}\n{\\bf p}^2=\\frac{1}{4}{\\bf k}^2+\\frac{1}{4}{\\bf s}^2,\n\\end{equation}\nwhere\n\\begin{equation}\n{\\bf k}={\\bf p'}-{\\bf p},\\qquad {\\bf s}={\\bf p'}+{\\bf p},\n\\end{equation}\nit follows that if ${\\bf k}^2$ is allowed to take only large values,\n${\\bf s}^2$ can be treated as small.  This may be regarded as a justification\nfor the quasistatic approximation in which terms of second and higher\norders in ${\\bf s}$ are ignored.\n\n\tOur quarkonium perturbative and confining potentials are not\nonly quasistatic but also nonsingular.  In the momentum space, these\npotentials are obtained by first expanding in powers of\n${\\bf p}^2\/(m^2+{\\bf p}^2)$, and then approximating ${\\bf p}^2$ as $\\frac{1}{4}{\\bf k}^2$.\nThe perturbative potential in powers of ${\\bf p}^2\/(m^2+{\\bf p}^2)$ includes, among\nothers, terms of the form\n\\begin{equation}\nf({\\bf p}^2)=\\frac{a+b\\;{\\bf S}_1\\cdot{\\bf S}_2}{m^2+{\\bf p}^2}\\ ,\n\\label{zero}\n\\end{equation}\nwhich becomes in the quasistatic approximation\n\\begin{equation}\nf({\\bf k}^2)=\\frac{a+b\\;{\\bf S}_1\\cdot{\\bf S}_2}{m^2+\\frac{1}{4}{\\bf k}^2}\\ .\n\\label{quasizero}\n\\end{equation}\nIt has been observed by Grotch, Sebastian, and Zhang \\cite{grotch} that while\nthe\ncontribution of $f({\\bf p}^2)$ vanishes for the P states due to the\nvanishing of the wave function at the origin, $f({\\bf k}^2)$ yields a small\nbut nonvanishing contribution for these states.  Consequently, for P and higher\nangular-momentum states it would be more accurate to drop\nterms of the form (\\ref{zero}) than to convert them into the approximate form\n(\\ref{quasizero}).  We agree with the observation of Grotch {\\it et al.}\nAccordingly, in the treatment of states with $l\\neq 0$ we shall\ndrop terms of the form (\\ref{quasizero}) in the momentum-space potentials\nand the corresponding terms of the form\n\\begin{equation}\nf({\\bf r})=\\frac{a+b\\;{\\bf S}_1\\cdot{\\bf S}_2}{\\pi r}e^{-2mr}\n\\label{coordzero}\n\\end{equation}\nin the coordinate-space potentials.\n\n\\subsection{Confining potential}\n\n\tIn our theoretical treatment, our aim has been to avoid phenomenology\nexcept in the choice of the long-range confining potential, which\ncannot be derived sufficiently accurately by any known theoretical\ntechnique.  It is indeed remarkable that the results obtained from our\nfield-theoretical perturbative potential supplemented with a phenomenological\nconfining potential are in\nexcellent over-all agreement with the experimental data including the\n$\\Delta M_P$ splitting. It should be noted that we have neglected\neffect of coupling of the energy levels to virtual decay\nchannels and possibly other small effects.\nSuch effects presumably have also been taken into account in\nour phenomenological confining potential.\n\n\\acknowledgments\n\tThis work was supported in part by the U.S. Department of Energy\nunder Grant No. DE-FG02-85ER40209 and the National Science Foundation\nunder Grant No. PHY-93-07980. W.~W.~R. would like to acknowledge conversations\nwith R.~Lewis and G.~A.~Smith regarding the results of the E760\ncollaboration.\n\n\\cleardoublepage\n\\widetext\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nThe 20th century has seen the development of two pillars of modern theoretical physics: quantum field theory (QFT) and general relativity (GR). The standard model of particle physics, which successfully describes the quantum properties of the strong and electroweak interactions, is based on the former framework. However, its na\\\"{i}ve application to the latter yields a QFT of gravity prone to (perturbatively) non-renormalizable ultraviolet (UV) divergences~\\cite{tHooft:1974toh,Goroff:1985th,Goroff:1985sz}.\n\nDespite remarkable progress in a number of directions, the difficulties of formulating a complete theory of quantum gravity have led a considerable portion of the community to shift the focus on general, possibly model-independent lessons that could shed light on the nature of gravity at all scales. Many of these proposals, commonly dubbed ``swampland conjectures''~\\cite{Vafa:2005ui} in the context of string theory, rest on considerations on black-hole physics, which often can lie entirely in the semi-classical regime where one expects low-energy effective field theory (EFT) to be a reliable description.\n\nOn the other hand, some aspects of these conjectures arose from and are tied to string theory and, in particular, its spacetime-supersymmetric incarnations. These settings are much better understood, since quantum corrections are often under quantitative control~\\cite{Berglund:2005dm, Gonzalo:2018guu, Marchesano:2019ifh, Blumenhagen:2019vgj, Baume:2019sry, Palti:2020qlc, Marchesano:2021gyv, Lee:2018urn, Lee:2018spm, Lee:2019tst, Lee:2019xtm} and can sometimes even be computed exactly~\\cite{Klaewer:2020lfg, Klaewer:2021vkr}. In certain settings, such as $\\mathcal{N} = 2$ Calabi-Yau flux compactifications, one can even classify general families of models at once~\\cite{Grimm:2018ohb, Grimm:2019wtx, Grimm:2019ixq, Gendler:2020dfp, Grimm:2020ouv, Bastian:2021eom}. In some string models with high-energy supersymmetry breaking, a variety of swampland proposals were verified~\\cite{Basile:2020mpt, Basile:2021mkd}, although a construction of realistic and (meta-) stable vacua is still an open problem~\\cite{Kachru:2003aw, Balasubramanian:2005zx, Koerber:2007xk, Danielsson:2009ff, Moritz:2017xto, Kallosh:2018nrk, Bena:2018fqc, Gautason:2018gln, Cordova:2018dbb, Blaback:2019zig, Hamada:2019ack, Gautason:2019jwq, Cribiori:2019clo, Andriot:2019wrs, Shukla:2019dqd, Shukla:2019akv, Cordova:2019cvf, Andriot:2020wpp, Andriot:2020vlg,Farakos:2020wfc,Gao:2020xqh,Bena:2020qpa,Bena:2020xrh,Crino:2020qwk,Dine:2020vmr,Basiouris:2020jgp,Cribiori:2020use,Hebecker:2020ejb,Andriot:2021rdy,DeLuca:2021pej,Cicoli:2021dhg,Cribiori:2021djm}. At any rate, it is paramount to understand the consequences of general consistency conditions outside of the specific contexts arising from (supersymmetric) string compactifications, although parametric control is most likely going to be problematic~\\cite{Dine:1985he,Bena:2018fqc,Gao:2020xqh,Bena:2020xrh,Dine:2020vmr} due to unknown or uncalculable corrections. To wit, efforts to test swampland proposals have almost entirely focused on supersymmetric settings, and in particular on stringy constructions. In order to shed light on whether they only encode a ``string lamppost principle''~\\cite{Montero:2020icj} or they hold in more generality, it is important to extend their exploration to a broader class of quantum gravity models. On the other hand, the more stringent and well-grounded swampland proposals, such as the ``no global symmetries''~\\cite{Misner:1957mt, Polchinski:2003bq, Banks:2010zn, Harlow:2018tng} and weak gravity~\\cite{Arkani-Hamed:2006emk} conjectures, could help guide the search for asymptotic safety, which is currently faced with the daunting prospect of navigating ever-larger theory spaces~\\cite{Benedetti:2010nr, Knorr:2021slg}.\n\nA concrete problem that can be phenomenologically relevant in the near future is constraining the possible values of the Wilson coefficients of a curvature expansion of the gravitational EFT. In particular, one can expect that generic detectable leading-order effects of quantum gravity be encoded in the coefficients of the quadratic curvature invariants, which we shall discuss in detail in the following, or in some specific non-local form factors~\\cite{Belgacem:2017cqo,Knorr:2018kog}. Some efforts in this direction have been made using the S-matrix bootstrap~\\cite{Guerrieri:2021ivu, Caron-Huot:2021rmr}, finding compelling agreement with the parameter space allowed by string theory. In the EFT framework, the problem has also been investigated via positivity bounds~\\cite{deRham:2017zjm, deRham:2017xox, deRham:2018qqo, DeRham:2018bgz, Alberte:2019lnd, Alberte:2019xfh, Alberte:2019zhd, Alberte:2020bdz, Alberte:2020jsk, deRham:2021fpu}.\n\nIn this paper we approach this issue from a novel direction, studying the constraints coming from swampland conjectures together with the consistency conditions required by the existence of a UV fixed point of the gravitational renormalization group (RG) flow\\footnote{See also~\\cite{deAlwis:2019aud} for related discussions on the weak gravity conjecture in the context of asymptotically safe gravity.}. The latter scenario has been termed ``asymptotic safety'', in analogy with asymptotic freedom as a particular case. This idea, originally due to Weinberg~\\cite{1976W}, would imply that the Wilson coefficients of the IR effective action stem from a UV-complete RG trajectory, which in turn would be determined by a finite number of relevant deformations from the fixed point. Recently, this area of research has witnessed considerable development of the theoretical framework to investigate RG flows beyond perturbation theory~\\cite{Dupuis:2020fhh} and finding evidence for the existence of the Reuter fixed point in a variety of different approximation schemes~\\cite{Souma:1999at,Lauscher:2002sq,Litim:2003vp,Codello:2006in,Machado:2007ea,Benedetti:2009rx,Dietz:2012ic,Dona:2013qba,Eichhorn:2013xr,Dona:2014pla,Christiansen:2014raa,Falls:2014tra,Christiansen:2015rva,Meibohm:2015twa,Oda:2015sma,Dona:2015tnf,Biemans:2016rvp,Eichhorn:2016esv, Dietz:2016gzg, Falls:2016msz,Gies:2016con,Biemans:2017zca,Christiansen:2017cxa,Hamada:2017rvn,Platania:2017djo,Falls:2017lst,deBrito:2018jxt,Eichhorn:2018nda,Eichhorn:2019yzm,deBrito:2019umw,Knorr:2021slg} (see also~\\cite{Donoghue:2019clr,Bonanno:2020bil} for critical assessments on the status of the field and its open questions). Possible implications of asymptotically safe gravity in astrophysics and cosmology (see~\\cite{Bonanno:2017pkg,Platania:2020lqb} for reviews) have been investigated using simplified models~\\cite{Bonanno:2006eu,Falls:2012nd,torres15,Koch:2015nva,Bonanno:2015fga,Bonanno:2016rpx,Kofinas:2016lcz,Falls:2016wsa,Bonanno:2016dyv,Bonanno:2017gji,Bonanno:2017kta,Bonanno:2017zen,Bonanno:2018gck,Liu:2018hno,Majhi:2018uao,Anagnostopoulos:2018jdq,Adeifeoba:2018ydh,Pawlowski:2018swz,Gubitosi:2018gsl,Platania:2019qvo,Platania:2019kyx,Bonanno:2019ilz,Held:2019xde} and more elaborate computations~\\cite{Bosma:2019aiu}, leading to the tentative conclusions that black-hole and cosmological singularities could be resolved by quantum effects, and that the nearly scale-invariant  cosmological power spectrum could arise naturally from a nearly scale-invariant  asymptotically safe regime.\n\nIn this paper we shall propose a concrete method to extract the allowed region of IR parameters from the RG flow of asymptotically safe trajectories. \nIn particular, we shall focus on the simpler case of the one-loop approximation in quadratic gravity~\\cite{Codello:2006in,Niedermaier:2009zz, Niedermaier:2010zz} in order to test our construction and provide a proof of principle of our idea. We will show that the IR limit of asymptotically safe trajectories falls inside the region allowed by the weak gravity conjecture and electromagnetic duality, and display a non-trivial intersection with the one allowed by the de Sitter and trans-Planckian censorship bounds.\n\nThe contents of this paper are organized as follows. In sect.~\\ref{sec:swampland} we provide a brief overview of swampland conjectures, focusing on the weak gravity conjecture, the de Sitter conjecture and the trans-Planckian censorship conjecture, since they entail the most relevant bounds for our subsequent analysis. In sect.~\\ref{sec:one-loop} we describe in detail the one-loop approxima\\-tion to quadratic gravity that we employ as testing grounds, and our method of extracting the physical IR Wilson coefficients. The resulting effective action turns out to contain non-local form factors. In sect.~\\ref{sec:results} we collect our results: in sect.~\\ref{sec:IR_space} we present the allowed region of parameter space that we found, which spans a plane in the three-dimensional space of dimensionless IR parameters, and in sect.~\\ref{sec:wgc_constraints} and sect.~\\ref{sec:dsc_constraints} we study the constraints stemming from the swampland conjectures discussed in sect.~\\ref{sec:wgc_intro} and sect.~\\ref{sec:dsc_tcc_intro} respectively. In sect.~\\ref{sec:intersections} we discuss and display the intersection of all regions. We conclude with a summary and some perspectives in sect.~\\ref{sec:conclusions}.\n\n\\section{An overview of swampland conjectures}\\label{sec:swampland}\n\nAs we have anticipated in the introduction, swampland conjectures are proposals that ought to rule out EFTs of gravity that do not admit UV completions~\\cite{Vafa:2005ui}. These conjectures are generally motivated in part by purely low-energy considerations, stemming from black-hole physics or inflation, but they also arise from detailed investigations of string-theoretic settings, where generally one has more control over corrections and patterns can be corrobo\\-rated across families of EFTs. The latter approach has led some to describe a ``lamppost'' effect~\\cite{Montero:2020icj}, whereby only settings that are somewhat under control can be investigated and thus it is unclear to which extent the resulting conclusions can be generalized. Furthermore, while at least minimal supersymmetry is generally unbroken in order to retain computational control, recent considerations~\\cite{Cribiori:2021gbf,Castellano:2021yye} point to a tension between low-energy supersymme\\-try breaking\\footnote{Nevertheless, scenarios with high-energy supersymmetry breaking have been investigated in the context of the swampland~\\cite{Bonnefoy:2018tcp,Basile:2020mpt,Basile:2021mkd}. See~\\cite{Mourad:2017rrl, Basile:2021vxh, Mourad:2021lma} for recent reviews.} and the consistency of the EFT. As we have discussed in the preceding section, one of the motivations behind this work is indeed to go beyond the usual settings, seeking lessons for other approaches to quantum gravity.\n\nSince its inception, the swampland program aims to describe the boundary between the landscape of consistent gravitational EFTs with a growing number of proposed criteria\\footnote{See~\\cite{Palti:2019pca, vanBeest:2021lhn} for reviews.}, numerous relations among which~\\cite{Andriot:2020lea, Lanza:2020qmt} point to a deeper underlying principle. In particular, connections between the distance conjecture~\\cite{Ooguri:2006in, Ooguri:2018wrx} and string dualities suggest that an organizing principle for these consistency criteria in the IR be related to a non-perturbative UV formulation of quantum gravity. Furthermore, as we shall see in the following, swampland considerations have provided intriguing clues toward a number of phenomenological puzzles~\\cite{Grana:2021zvf}. \n\nIn this paper we shall focus on some conjectures which can provide bounds for the Wilson coefficients of the gravitational EFT. In particular,\n\n\\begin{itemize}\n    \\item The \\emph{weak gravity conjecture} (WGC)~\\cite{Arkani-Hamed:2006emk} relates the mass and charge of light states and black holes;\n    \\item The \\emph{de Sitter conjecture} (dSC)~\\cite{Obied:2018sgi}, along with its refined versions~\\cite{Ooguri:2018wrx,Garg:2018reu,Andriot:2018mav}, constrains the behavior of scalar potentials and their derivatives, leading to an obstruction to the existence of de Sitter vacua that is $\\mathcal{O}(1)$ in Planck units;\n    \\item The \\emph{trans-Planckian censorship conjecture} (TCC)~\\cite{Bedroya:2019snp, Brandenberger:2021pzy} constrains sub-Planckian cosmological perturbations to remain sub-Planckian across inflation, and leads to bounds on the lifetime of metastable de Sitter configurations as well as on the $\\mathcal{O}(1)$ parameter that appears in the dSC, at least in asymptotic regions of field space.\n\\end{itemize}\n\nIn light of the latter consideration, for the purposes of this paper in the following we shall investigate the consequences of the TCC on Starobinsky-like inflationary potentials as a special case of the dSC. Indeed, we shall restrict ourselves to the asymptotic region of field space corresponding to small curvatures in Planck units, where the TCC could provide a dSC bound with a specific $\\mathcal{O}(1)$, as we shall see below.\n\n\\subsection{Weak gravity conjecture and black holes}\\label{sec:wgc_intro}\n\nLet us begin reviewing some features of the (electric) WGC, referring the reader to~\\cite{Palti:2019pca,  vanBeest:2021lhn} for more details. In its most basic form, it states that in a consistent EFT of gravity coupled to a $U(1)$ gauge field there exists a state whose mass $m$ is lower than its charge $q$ in Planck units. In four dimensions, the bound for charged particles reads\n\\begin{eqaed}\\label{eq:wgc_basic}\n    \\frac{m}{M_\\text{Pl}} \\leq \\mathcal{O}(1) \\, q \\, ,\n\\end{eqaed}\nwhere the model-dependent $\\mathcal{O}(1)$ constant is $\\frac{1}{\\sqrt{2}}$ in Einstein-Maxwell theory.\n\nAmong various motivations and evidence gathered in the literature, the WGC is grounded in black-hole physics from the requirement that charged, extremal black holes be able to decay, lest protected by a symmetry (such as supersymmetry, in the case of BPS-saturated states). The rationale behind this lies in avoiding remnants while keeping the black hole from violating the extremality bound, since a violation of either would presumably lead to consistency issues potentially within the EFT regime~\\cite{Giddings:1992hh, Susskind:1995da, Arkani-Hamed:2006emk}. For charged black holes of mass $M$ and charge $Q$, this requirement translates into\n\\begin{eqaed}\\label{eq:wgc_bhs}\n    \\frac{M}{Q} \\geq \\left(\\frac{M}{Q}\\right)_\\text{extremal} \\, ,\n\\end{eqaed}\nwhere the latter is generally an $\\mathcal{O}(1)$ constant\\footnote{For Einstein-Maxwell theory, the extremality bound reads $\\frac{M}{Q} \\geq \\sqrt{2} \\, M_\\text{Pl}$.}. However, higher-curvature corrections could potentially spoil this condition even for macroscopic black holes, provided they are sufficiently close to extremality. Writing the leading quartic corrections according to~\\cite{Kats:2006xp}\n\\begin{eqaed}\\label{eq:eft_corr}\n    \\Delta \\mathcal{L} = c_1 \\, R^2 + c_2 \\, R_{\\mu \\nu}R^{\\mu \\nu} + c_3 \\, R_{\\mu \\nu \\rho \\sigma} R^{\\mu \\nu \\rho \\sigma} \\, ,\n\\end{eqaed}\nthe resulting bounds for the corresponding Wilson coefficients $c_i$ comprise a family of inequalities for linear combinations of the $c_i$, parametrized by the extremality parameter of the black hole~\\cite{Arkani-Hamed:2006emk,Kats:2006xp,Cheung:2018cwt,Charles:2017dbr,Hamada:2018dde,Charles:2019qqt}. The extremality bound in general now takes the form\n\\begin{eqaed}\\label{eq:wgc_bhs_hd}\n    \\frac{M}{Q} \\geq \\left(\\frac{M}{Q}\\right)_\\text{extremal} \\left(1 - \\frac{\\Delta}{M^2} \\right) \\, ,\n\\end{eqaed}\nwhere the linear combination $\\Delta$ of Wilson coefficients is to be non-negative in order for the WGC to hold, and is proportional to the coefficient $c_2 + 4c_3$ of the Weyl-squared term~\\cite{Kats:2006xp,Charles:2019qqt}.\n\nThe leading order contributions to $\\Delta$ comprise not only the Wilson coefficients in the effective action of eq.~\\eqref{eq:eft_corr}, but also the Wilson coefficients that involve the $U(1)$ gauge field. It has been recently shown~\\cite{Cano:2021tfs} that, \\emph{assuming invariance under electromagnetic duality}, higher-curvature corrections up to sextic order can be written in terms of purely gravitational terms, up to field redefinitions. Let us stress that our aim is to intersect swampland bounds with the constraints provided by asymptotic safety, and the technical obstacles to compute its consequences for quartic electromagnetic couplings in gravity, which would entail involved FRG computations along the lines of~\\cite{Knorr:2021slg}, compel us to focus on the duality-invariant scenario of~\\cite{Cano:2021tfs}, which at any rate appears intriguing on its own\\footnote{Another instance of the interplay between duality and the WGC has been studied in~\\cite{Loges:2020trf}.}. Moreover, the electromagnetic couplings do not run under the RG flow at one loop because of tree-level duality~\\cite{Charles:2017dbr,Charles:2019qqt}. This has been used to argue that the low-energy behavior of the correction $\\Delta$ to the extremality bound is dominated by the Weyl anomaly coefficients that drive the running of the $c_i$, and in particular of the Weyl-squared Wilson coefficient $c_2 + 4c_3$, as we shall see in the following. However, in the present case we would like to constrain the physical parameters built out of the Wilson coefficients and the Planck scale, in order to compare the resulting bounds with the constraints of asymptotic safety. We shall describe the procedure in detail in the following section.\n\n\\subsection{de Sitter and trans-Planckian censorship}\\label{sec:dsc_tcc_intro}\n\nLet us now move on to discuss the dSC and the TCC. The former quantifies an obstruction to the existence of de Sitter vacua, in the form of a bound for the (field-space gradient of the) scalar potential $V(\\phi)$. Indeed, since in this setting de Sitter vacua would arise as positive-energy critical points of $V$, a natural bound that would prevent these takes the form\n\\begin{eqaed}\\label{eq:dsc_basic}\n    M_\\text{Pl}\\, \\abs{\\nabla V} \\geq c V \n\\end{eqaed}\nfor field ranges\n\\begin{eqaed}\\label{eq:field_range}\n    \\Delta \\phi \\lesssim f \\, M_\\text{Pl} \\, ,\n\\end{eqaed}\nwhere $c \\, , \\, f > 0$ are (\\emph{a priori} model-dependent) constants. Within the EFT framework, their natural values are $\\mathcal{O}(1)$, indicating that the obstruction is tied to the expected cutoff of the EFT. However, one is readily confronted with a tension between the bound in eq.~\\eqref{eq:dsc_basic} and slow-roll inflation~\\cite{Garg:2018reu, Kinney:2018nny}\\footnote{See also~\\cite{Rudelius:2019cfh,Rudelius:2021oaz,Chojnacki:2021fag,Jonas:2021xkx} for discussions on eternal inflation and the swampland.}, leading to refinements involving the Hessian matrix of the potential~\\cite{Garg:2018reu,Ooguri:2018wrx}. In particular, whenever the bound of eq.~\\eqref{eq:dsc_basic} would be violated, the matrix\n\\begin{eqaed}\\label{eq:ref_dsc}\n    M_\\text{Pl}^2\\, \\text{Hess}(V) + \\, c' \\, V\n\\end{eqaed}\nwould be negative semidefinite, with $c' > 0$ another $\\mathcal{O}(1)$ constant. Further refinements were proposed in~\\cite{Andriot:2018mav}, but in our setting we shall find that the first bound of eq.~\\eqref{eq:dsc_basic} is sufficient, since it encompasses eq.~\\eqref{eq:ref_dsc} in the regions of parameter space that we are concerned with.\n\nOn the other hand, the TCC surmises that sub-Planckian quantum fluctuations in the early universe at initial time $t_i$ never grow macroscopic at a final time $t_f$. In particular, they ought to never cross the Hubble horizon and freeze. This requirement can be formulated, in terms of the scale factor $a(t)$ and the corresponding Hubble parameter $H$, by~\\cite{Bedroya:2019snp, Brandenberger:2021pzy}\n\\begin{eqaed}\\label{eq:tcc_basic}\n    \\frac{a(t_f)}{a(t_i)} \\lesssim \\frac{M_\\text{Pl}}{H(t_f)} \\, ,\n\\end{eqaed}\nagain up to an $\\mathcal{O}(1)$ constant. An intriguing consequence of eq.~\\eqref{eq:tcc_basic} is that de Sitter configurations are not prohibited, but they are metastable with a lifetime $T$ bounded by\n\\begin{eqaed}\\label{eq:tcc_life}\n    T \\lesssim \\frac{1}{H} \\, \\log \\frac{M_\\text{Pl}}{H} \\, ,\n\\end{eqaed}\nof the order of a trillion years. This results points to a possible resolution of the coincidence problem in this setting.\n\nThe most relevant consequence of the TCC for the purposes of this paper is that, in the presence of a scalar potential, it leads to a bound of the form of eq.~\\eqref{eq:dsc_basic} with\n\\begin{eqaed}\\label{eq:tcc_c1}\n    c = \\frac{2}{\\sqrt{(d-1)(d-2)}}\n\\end{eqaed}\nin $d$ spacetime dimensions, at least in asymptotic regimes of field space. In the present setting, the scalar potential arises from the quadratic curvature terms, and the corresponding asymptotic regime for gravitational field fluctuations is that of small curvatures~\\cite{Lust:2019zwm}. This regime is mapped to a neighbourhood of the origin in the inflaton description. For generic curvatures, one expects that both the purely gravitational description and the inflaton description be modified, including the geometry of field space. Nevertheless, since our current setup does not allow for precise quantitative bounds, we shall henceforth take eq.~\\eqref{eq:tcc_c1} simply as a reference point around which to study the more general bound of eq.~\\eqref{eq:dsc_basic}. Let us also remark that this value appears in a number of related swampland bounds~\\cite{Andriot:2020lea} and is well-behaved under dimensional reduction~\\cite{Rudelius:2021oaz}, and thus it may play a more prominent role in the story. At any rate, it would be interesting to explore the more direct implications of eq.~\\eqref{eq:tcc_basic} studying cosmological solutions or exploring the considerations of~\\cite{Bedroya:2019tba, Brandenberger:2021pzy} within our setup.\n\n\\section{One-loop RG flow in quadratic gravity}\\label{sec:one-loop}\n\nLet us now discuss the concrete setting in which we shall compute the possible values of the Wilson coefficients of the effective gravitational action. In this work we focus on the quadratic truncation\\footnote{Let us remark that here ``quadratic'' refers to the order in the curvatures. In terms of derivatives, the action in eq.~\\eqref{eq:quadratic_lagrangian} is quartic.}, in the one-loop approximation. In Euclidean signature, the Lagrangian pertaining to the full quadratic truncation reads\n\\begin{eqaed}\\label{eq:quadratic_lagrangian}\n    \\mathcal{L}=\\frac{2\\Lambda-R}{16\\pi G}+\\frac{1}{2\\lambda}\\,C^2-\\frac{\\omega}{3\\lambda}R^2+\\frac{\\theta}{\\lambda}E \\, ,\n\\end{eqaed}\nwhere $C^2 \\equiv C_{\\mu\\nu\\rho\\sigma}C^{\\mu\\nu\\rho\\sigma}$ is the square of the Weyl tensor, $E$ is the Gauss-Bonnet density and and the Wilson coefficients\n\\begin{eqaed}\\label{eq:wilson_coeffs}\n    g_C \\equiv \\frac{1}{2\\lambda} \\, , \\qquad\n    g_R \\equiv - \\, \\frac{\\omega}{3\\lambda}\n\\end{eqaed}\ncan be related to the $c_i$ coefficients in eq.~\\eqref{eq:eft_corr}. Indeed, since\n\\begin{eqaed}\\label{eq:weyl_gb}\n    C^2&=R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}-2R_{\\mu\\nu}R^{\\mu\\nu}+\\frac{R^2}{3}\\,,\\\\\n    E&=R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}-4R_{\\mu\\nu}R^{\\mu\\nu}+R^2\\,,\n\\end{eqaed}\nthe $c_i$ are related to the couplings in eq.~\\eqref{eq:quadratic_lagrangian} according to\n\\begin{eqaed}\\label{eq:couplings_relation}\n    c_1&=\\frac{1}{6\\lambda}-\\frac{\\omega}{3\\lambda}+\\frac{\\theta}{\\lambda} \\, ,\\\\\n    c_2&=-\\frac{1}{\\lambda}-\\frac{4\\theta}{\\lambda} \\,,\\\\\n    c_3&=\\frac{1}{2\\lambda}+\\frac{\\theta}{\\lambda} \\,.\n\\end{eqaed}\nWhile this setup holds in general spacetime dimensions $d$, we now restrict to $d = 4$. The one-loop beta functions of the couplings of eq.~\\eqref{eq:quadratic_lagrangian} read~\\cite{Codello:2006in,Niedermaier:2009zz, Niedermaier:2010zz}\n\\begin{eqaed}\\label{eq:betas}\n\\beta_{\\widetilde{\\Lambda}} & =-\\,2 \\widetilde{\\Lambda}+\\frac{1}{(4 \\pi)^{2}}\\left[\\frac{1+20 \\omega^{2}}{256 \\pi \\widetilde{G} \\omega^{2}} \\lambda^{2}+\\frac{1+86 \\omega+40 \\omega^{2}}{12 \\omega} \\lambda \\widetilde{\\Lambda}\\right] \\\\\n& \\quad -\\frac{1+10 \\omega^{2}}{64 \\pi^{2} \\omega} \\lambda +\\frac{2 \\widetilde{G}}{\\pi}-\\frac{83+70 \\omega+8 \\omega^{2}}{18 \\pi} \\widetilde{G} \\widetilde{\\Lambda} \\, , \\\\\n\\beta_{\\widetilde{G}} & =2 \\widetilde{G}-\\frac{1}{(4 \\pi)^{2}} \\frac{3+26 \\omega-40 \\omega^{2}}{12 \\omega} \\lambda \\widetilde{G}-\\frac{83+70 \\omega+8 \\omega^{2}}{18 \\pi} \\widetilde{G}^{2} \\, ,\n\\\\\n\\beta_{\\lambda} & =-\\frac{1}{(4 \\pi)^{2}} \\frac{133}{10} \\lambda^{2} \\, , \\\\\n\\beta_{\\omega} & =-\\frac{1}{(4 \\pi)^{2}} \\frac{25+1098 \\,\\omega+200 \\,\\omega^{2}}{60} \\lambda \\, , \\\\\n\\beta_{\\theta} & =\\frac{1}{(4 \\pi)^{2}} \\frac{7\\,(56-171\\, \\theta)}{90} \\lambda\n\\end{eqaed}\nwhere $\\widetilde{G}_k=G_k\\,k^2$ and $\\widetilde{\\Lambda}_k=\\Lambda_k\\,k^{-2}$\nare the dimensionless Newton coupling and cosmological constant respectively, and we have suppressed the subscript $k$ in eq.~\\eqref{eq:betas} for the sake of clarity.\n\nIn our setting, the flow of the (classically) marginal couplings $\\lambda$, $\\omega$ and $\\theta$ is decoupled from that of the Einstein-Hilbert couplings. Out of the UV fixed points\n\\begin{eqaed}\\label{eq:fixed_points}\n    \\lambda_\\ast = 0 \\, , \\qquad \\omega_\\ast = \\omega_\\pm \\equiv \\frac{-549 \\pm 7\\sqrt{6049}}{200} \\, , \\qquad \\theta_\\ast = \\frac{56}{171} \\, ,\n\\end{eqaed}\nUV completeness selects $\\omega_\\ast = \\omega_+ \\approx - \\, 0.023$~\\cite{Codello:2008vh}, which the solutions approach as the RG time\\footnote{Note that, since we are interested in the IR regime, our convention for the RG time is such that $t \\to +\\infty$ in the IR.} $t \\equiv \\log\\frac{k_0}{k} \\to -\\infty$, as is apparent from fig.~\\ref{fig:flowlambdaomega}. Let us remark that this fixed point is asymptotically safe, \\emph{i.e.} at least one coupling is not asymptotically free~\\cite{Codello:2008vh, Niedermaier:2009zz, Niedermaier:2010zz, Groh:2011vn}. Indeed, the critical exponents of $G$ and $\\Lambda$ are 2 and 4\\footnote{Although 2 and 4 are not the canonical mass dimensions of $G$ and $\\Lambda$, they are the canonical dimension of the couplings $1\/G$ and $\\Lambda\/G$ that multiply the operators $\\sqrt{-g}$ and $\\sqrt{-g} R$. This occurs because the transformation between these couplings is non-singular, as explained in~\\cite{Percacci:2007sz}. On the other hand, at the Gaussian fixed point the transformation between the couplings is singular, and the dimensions change accordingly.}, while in the IR they become the canonical -2 and 2 respectively~\\cite{Litim:2012vz}. The fact that all couplings are attracted to the fixed point in the UV~\\cite{Codello:2008vh} is instead an artifact of the one-loop approximation. Indeed, more sophisticated FRG computations yield a fixed point with a three-dimensional critical surface~\\cite{Benedetti:2009rx}.\n\nThe flow can be solved analytically in terms of the deformations $\\delta \\lambda \\, , \\, \\delta \\omega$ from the UV fixed point, and yields the closed-form solution\n\\begin{eqaed}\\label{eq:marginal_flow}\n    \\lambda(t) & = \\frac{\\delta \\lambda}{1 - \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t} \\,,\n    \\\\\n    \\omega(t) & = \\frac{\\omega_- - \\, \\omega_+ \\, \\left(1 + \\frac{\\Delta}{\\delta \\omega}\\right) \\left(1 - \\, \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t \\right)^{\\frac{7\\sqrt{6049}}{399}}}{1 - \\, \\left(1 + \\frac{\\Delta}{\\delta \\omega}\\right) \\left(1 - \\, \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t \\right)^{\\frac{7\\sqrt{6049}}{399}}} \\,,\n    \\\\\n    \\theta(t) & = \\frac{56}{171} + \\frac{\\delta \\theta}{1 - \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t} \\,,\n\\end{eqaed}\nwhere $\\Delta \\equiv \\omega_+ - \\omega_-$. The vector field generating this flow is displayed in fig.~\\ref{fig:flowlambdaomega} in the $(\\omega,\\lambda)$ subspace and in fig.~\\ref{fig:flowMC} with various 3D plots. Let us observe that the UV completeness of the trajectory requires $\\delta \\lambda > 0$, and that the IR flow ends at $t = t_\\text{IR} \\equiv \\frac{160\\pi^2}{133 \\delta \\lambda}$. However, since $\\delta \\lambda \\ll 1$ this RG time is parametrically large, and one can reliably extract the perturbative IR behavior for the Wilson coefficients. Furthermore, reaching a physical IR regime with $\\widetilde{G} \\to 0^+$ requires that $\\delta \\widetilde{G} \\, , \\, \\delta \\omega < 0$, in order that their flows remain between the UV and IR fixed-point values avoiding runaway. The flow of the relevant deformations from the fixed point is shown in Fig.~\\ref{fig:flow-reldeformations}.\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.7]{\"Images\/FlowLO\".pdf}\n\\caption{Flow of one-loop quadratic gravity in the $(\\omega,\\lambda)$ subspace. The arrows point toward the IR. In this setting, two non-trivial fixed points are present, and the Reuter fixed point is the one with the smaller absolute value~\\cite{Codello:2008vh}. \\label{fig:flowlambdaomega}}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.4]{\"Images\/Flow3D\".pdf}\\\\\\belowbaseline[0pt]{\n\\includegraphics[scale=0.45]{\"Images\/3Dplot-marg-4\".pdf}}~$\\qquad$\\belowbaseline[0pt]{\\includegraphics[scale=0.47]{\"Images\/3Dplot-marg-2\".pdf}}\n\\caption{Flow of the classically marginal couplings $(\\omega,\\lambda,\\theta)$ in one-loop quadratic gravity. The arrows point towards the IR, and different viewpoints are shown to better visualize the flow. The color coding of the arrows is identical to that of fig.~\\ref{fig:flowlambdaomega}.\\label{fig:flowMC}}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.68]{\"Images\/FlowRDomlam\".pdf}\\includegraphics[scale=0.68]{\"Images\/FlowRDglambda\".pdf}\n\\caption{Flow of the relevant deformations from the fixed point. The left panel depicts the flow in the $(\\omega,\\lambda)$ subspace. The right panel depicts the flow in the $(G, \\Lambda)$ subspace where the classically marginal couplings have been set to the UV fixed point. The arrows indicate the the flow from the UV to the IR.\\label{fig:flow-reldeformations}}\n\\end{figure}\n\nSubstituting the expressions of eq.~\\eqref{eq:marginal_flow} in eq.~\\eqref{eq:betas}, one can then solve the remaining flow equations numerically varying the initial conditions, or, equivalently, the deformations $\\delta \\omega, \\delta \\lambda, \\delta \\widetilde{G} , \\delta \\widetilde{\\Lambda}$ from the UV fixed point. The RG flow then drives the running couplings to the weakly coupled IR, where the running couplings $g_C$ and $g_R$, defined in eq.~\\eqref{eq:wilson_coeffs}, behave logarithmically (linearly in $t$) as $t \\to t_\\text{IR}^-$. This result is consistent with perturbative computations, and the resulting asymptotic expressions read\n\\begin{eqaed}\\label{eq:log_running}\n    g_C(t) & \\sim \\frac{1}{2\\delta \\lambda} - \\frac{133}{320\\pi^2} \\, t \\, , \\\\\n    g_R(t) & \\sim - \\, \\frac{\\omega_-}{3\\delta \\lambda} + \\frac{133}{480\\pi^2} \\, \\omega_- \\, t \\, .\n\\end{eqaed}\nIn order to extract the physical IR parameters, we shall identify the (square of the) RG scale $k^2$ with the covariant Laplacian\/d'Alembertian $\\Box$. In order to eliminate the arbitrary reference scale $k_0$ that defines the initial condition for the RG flow, one can express every quantity in units of the IR Planck mass\\footnote{Notice that our convention for the Planck mass differs from the more widespread ``reduced'' Planck mass $\\widehat{M}_\\text{Pl}^{-2} = 8\\pi G$.} $M_\\text{Pl}^{-2} = G$. To this end, since $e^{2t} \\, \\widetilde{G}(t) \\to \\widetilde{G}_0$ tends to a constant in the IR, one can evaluate the running Wilson coefficients of eq.~\\eqref{eq:log_running} replacing $t \\to - \\, \\frac{1}{2} \\log \\widetilde{G}(t)$, so that\n\\begin{eqaed}\\label{eq:t_planck_sub}\n    \\log \\frac{M_\\text{Pl}}{k} & = \\log \\frac{M_\\text{Pl}}{k_0} + t \\\\\n    & \\sim - \\, \\frac{1}{2} \\log \\widetilde{G}(t) \\\\\n    &  \\sim - \\, \\frac{1}{2} \\log \\widetilde{G}_0 + t \\, .\n\\end{eqaed}\nSince $\\widetilde{G}_0$ can be extracted from the numerical solution of eqs.~\\eqref{eq:betas}, identifying \n\\begin{eqaed}\\label{eq:form_factor_sub}\n    \\log \\frac{k^2}{M_\\text{Pl}^2} \\longrightarrow \\log \\frac{\\Box}{M_\\text{Pl}^2} \\, ,\n\\end{eqaed}\naccording to the preceding considerations, one can reconstruct an effective action of the form\n\\begin{eqaed}\\label{eq:form_factor_eft}\n    \\Gamma = \\int d^4x \\sqrt{g} \\, \\left(\\frac{2\\Lambda - R}{16\\pi G} + g_C \\, C^2 + g_R \\, R^2 + b_C \\, C \\log \\frac{\\Box}{M_\\text{Pl}^2} \\, C + b_R \\, R \\log \\frac{\\Box}{M_\\text{Pl}^2} \\, R \\right) \\, , \n\\end{eqaed}\nwhere Weyl-tensor contraction is understood. The appearance of non-local form factors resonates with the considerations in~\\cite{Knorr:2019atm, Draper:2020bop, Draper:2020knh}. While we shall neglect them in the following, the presence of form factors of this type seems largely consistent with preceding results~\\cite{Riegert:1984kt, Deser:1996na, Erdmenger:1996yc, Erdmenger:1997gy, Deser:1999zv, Bautista:2017enk} (see also~\\cite{Donoghue:2015nba} for a discussion of logarithmic form factors). Note that, despite their behavior at low energies, one expects a resummation of such non-local form factors to yield a result that is both well-defined and subleading in the IR compared to the local terms~\\cite{Draper:2020bop}\\footnote{One exception could be a non-local form factor of the type $\\sim 1\/\\Box$, as discussed in~\\cite{Belgacem:2017cqo,Knorr:2018kog}.}. Furthermore, they do not contribute to the scalar potential that we shall discuss in Section~\\ref{sec:dsc_constraints} in the context of the dSC. Notwithstanding the importance of form factors in establishing a non-local behavior of gravity, we would like to understand which values of the three dimensionless combinations\n\\begin{eqaed}\\label{eq:IR_params}\n    G\\Lambda \\, , \\quad g_C \\, , \\quad g_R\n\\end{eqaed}\nare allowed starting from any initial condition, \\emph{i.e.}, any perturbation of the asymptotically safe UV fixed point along UV-attractive directions. To this end, we have evaluated numerically these combinations in the IR, implementing the substitution of eq.~\\eqref{eq:t_planck_sub}. The following plots highlighting the swampland constraints, the IR limits of asymptotically safe RG trajectories, as well as the final intersection between the allowed regions, will pertain to the $(G\\Lambda,g_C,g_R)$ theory space.\n\nTo conclude this section, let us collect a few words of caution regarding the one-loop approximation. In general, in the context of gravity, one expects it to be only reliable in the IR, despite the appearance of a UV fixed point outside of the perturbative regime. The methods of the functional RG have been employed, both in earlier~\\cite{Benedetti:2009rx,Benedetti:2009gn} and recent~\\cite{Knorr:2021slg} efforts, to obtain non-perturbative flow equations in the quadratic truncation, but applying our method to extract the allowed region of parameter space in the IR entails highly involved and unstable numerical analysis. In order to circumvent these obstacles, and address the problem in a more quantitative fashion, a natural first step would entail performing novel FRG computations. The simplest relevant setting would include the most general quadratic truncation coupling the electromagnetic field to gravity, which, while daunting, appears feasible via the methods that have been very recently introduced in~\\cite{Knorr:2021slg} to study the purely gravitational sector. In light of these (and other related) issues, in this work we have focused on the one-loop approximation as a proof of principle, with the hope of uncovering some instructive general lessons from the results that we are now about to present.\n\nFinally, let us stress that, although quadratic truncations of the gravitational action are typically associated with a loss of physical unitarity~\\cite{Stelle:1977ry}, the Stelle ghost could be a truncation artifact~\\cite{Platania:2020knd}. Integrating out quantum fluctuations could lead to well-behaved, unitary scattering amplitudes~\\cite{Draper:2020bop,Draper:2020knh}, as explicit computations seem to suggest~\\cite{Bonanno:2021squ}. This issue was also discussed within the setting explored in this paper in~\\cite{Niedermaier:2009zz, Niedermaier:2010zz}.\n\n\\section{Results}\\label{sec:results}\n\nLet us now describe in detail our results on the allowed values of physical IR parameters that we have obtained from the calculations outlined in the preceding section, along with the swampland constraints that we have discussed in sect.~\\ref{sec:swampland}.\n\n\\subsection{Infrared limit of asymptotically safe RG trajectories in one-loop quadratic gravity}\\label{sec:IR_space}\n\nIn order to uncover the space of physical parameters appearing in the (local sector of the) effective action of eq.~\\eqref{eq:form_factor_eft}, we have sampled the space of allowed deformations $(\\delta \\omega, \\delta \\lambda, \\delta \\widetilde{G} , \\delta \\widetilde{\\Lambda})$ from the UV fixed point, and extracted the resulting IR values of the parameters in eq.~\\eqref{eq:IR_params} evaluating the flow of $\\widetilde{G}$ and $\\widetilde{\\Lambda}$ for a suitably large RG time $t \\approx 30$, exploiting the rapid convergence of the combination in eq.~\\eqref{eq:t_planck_sub}. The resulting values for $\\widetilde{G}\\widetilde{\\Lambda} = G\\Lambda$, or equivalently $\\Lambda\/M_\\text{Pl}^2$, span a wide range of values, of the order of $10^5$ for the region of initial deformations that we have explored. Moreover, the closed-form flow that one obtains setting the classically marginal couplings to their fixed-point values spans the whole real axis~\\cite{Codello:2008vh}. We are thus led to conclude that the allowed (IR) values of $G\\Lambda$ are unrestricted. On the other hand, the values of $g_C$ and $g_R$ appear to lie on the line\n\\begin{eqaed}\\label{eq:marginal_line}\n    g_R \\approx - \\, 0.74655 + 3.64447 \\, g_C \\, ,\n\\end{eqaed}\nas depicted in fig.~\\ref{fig:IR2d-fit} and fig.~\\ref{fig:IR3d}. This result appears to be very robust upon increase of the sample size, and in particular for $10^6$ points the covariance matrix of the fit is of the order $\\mathcal{O}(10^{-8})$. Let us observe that, neglecting the intercept term, eq.~\\eqref{eq:marginal_line} follows from eq.~\\eqref{eq:log_running} as $t \\to t_\\text{IR}^-$, whereby $g_R \\sim - \\, 2\\omega_-\/3 \\, g_C$. Since we instead evaluate the IR couplings at a fixed, albeit sufficiently large, RG time, it is tempting to speculate that the intercept term in eq.~\\eqref{eq:marginal_line} is a correction arising from RG trajectories that approach the IR more slowly. Therefore, at least within the scope of our approximations, the presence of a UV fixed point appears to constrain the allowed physical coefficients in eq.~\\eqref{eq:IR_params} to a specific plane, and we shall now compare this result to the constraints arising from the swampland conjectures that we have discussed in sect.~\\ref{sec:swampland}. \n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.5]{\"Images\/LineEqs\".pdf}\n\\caption{The line of equation $g_R = - \\, 0.74655 + 3.64447 \\, g_C$ fitting the IR values obtained from the flow of eq.~\\eqref{eq:betas} sampling UV initial conditions. The covariance matrix evaluates to $\\mathcal{O}(10^{-8})$ with~$10^6$ data points.}\\label{fig:IR2d-fit}\n\\end{figure}\n\n\\begin{figure}[t!]\n$\\hspace{-0.2cm}$\\includegraphics[scale=0.53]{\"Images\/Plot3D-1\".pdf}$\\hspace{-0.5cm}$\\includegraphics[scale=0.53]{\"Images\/Plot3D-2\".pdf}\\\\\n\\centering\\includegraphics[scale=0.55]{\"Images\/Plot3D-3\".pdf}\n\\caption{Plots depicting the IR endpoints of asymptotically safe RG trajectories. The points lie on the plane of equation $g_R = - \\, 0.74655 + 3.64447 \\, g_C$. The values of $G\\Lambda$ span a vast range, which, together with the closed-form flow of~\\cite{Codello:2008vh} depicted in fig.~\\ref{fig:flow-reldeformations}, leads us to infer that they are unrestricted.}\\label{fig:IR3d}\n\\end{figure}\n\n\\subsection{Constraints on quadratic gravity from WGC}\\label{sec:wgc_constraints}\n\nAs we have discussed in sect.~\\ref{sec:wgc_intro}, the WGC entails positivity bounds for the Wilson coefficients of the higher-derivative corrections to Einstein gravity. Since these bounds involve charged particles and black holes, higher-derivative couplings of a $U(1)$ gauge field ought to be included, although the resulting RG flow is extremely involved technically and has not been computed hitherto. On the other hand, the considerations of~\\cite{Charles:2017dbr, Charles:2019qqt, Cano:2021tfs}, based on electromagnetic duality, show that one can still make use of our results to constrain higher-derivative corrections in a \\emph{duality-invariant} scenario using the WGC. To this end, expressing the higher-curvature in terms of the $c_i$ coefficients of eq.~\\eqref{eq:eft_corr}, the (family of) positivity bound(s) of~\\cite{Cheung:2018cwt} reads\n\\begin{eqaed}\\label{eq:positivity_bound}\n    (1-\\xi)^2\\,c_0+20\\xi c_3-5\\xi(1-\\xi)(2c_3)>0\n\\end{eqaed}\nwhere $\\xi\\equiv\\sqrt{1-Q^2\/M^2}$ is the extremality parameter of Reissner-Nordstr\\\"{o}m black holes with mass $M$ and charge $Q$, $0<\\xi<1\/2$ for black holes with positive specific heat and\n\\begin{eqaed}\\label{eq:c0_coeff}\n    c_0 \\equiv c_2 + 4 \\, c_3 \\, .\n\\end{eqaed}\nIn terms of the couplings in eq.~\\eqref{eq:quadratic_lagrangian}, the bound of eq.~\\eqref{eq:positivity_bound} takes the simpler form\n\\begin{eqaed}\\label{eq:simpler_wgc_bound}\n    \\frac{1}{\\lambda}(10\\, \\theta \\, (\\xi +1) \\xi +6 \\xi ^2+3 \\xi +1)>0 \\, ,\n\\end{eqaed}\nwhich holds for~$\\lambda>0$ provided that~$\\xi>0$ (which is always satisfied by the extremality parameter) and that~$\\theta>0$. As we have discussed in sect.~\\ref{sec:one-loop}, the latter condition is fulfilled  if $\\delta \\theta > 0$, since $\\delta \\lambda > 0$ in eq.~\\eqref{eq:marginal_flow}. Hence, the (duality-invariant) WGC constrains~$\\lambda > 0$, which is included by the analysis of the preceding section and does not entail additional conditions. Let us observe that, although $\\theta$ encodes the coupling of the Gauss-Bonnet invariant, it contributes to the entropy of a black hole~\\cite{Myers:1988ze,Myers:1998gt,Clunan:2004tb} even in four dimensions, where it is topological. It does not contribute in the limit $\\xi \\ll 1$, since the resulting bound also describes the positivity of the extremality ratio~\\cite{Kats:2006xp,Charles:2019qqt}.\n\n\\subsection{Constraints on quadratic gravity from dS and TC conjectures}\\label{sec:dsc_constraints}\n\nLet us now discuss the constraints arising from the dSC and the TCC. As we have anticipated in sect.~\\ref{sec:dsc_tcc_intro}, we shall focus on the bounds that the dSC and the TCC entail for the scalar potential that arises from the higher-derivative corrections of eq.~\\eqref{eq:form_factor_eft}. In order to extract the potential proper, we shall concern ourselves with the local sector of the theory, neglecting the form factors and the Weyl term, which vanishes on cosmological backgrounds.\n\nFollowing the standard procedure to obtain inflaton potentials from $F(R)$ Lagrangians (see, \\emph{e.g.},~\\cite{inflationaris,Platania:2019qvo}), one begins from $R^2$ gravity with a cosmological constant,\n\\begin{eqaed}\\label{eq:our_f}\n    F(R)=\\frac{1}{16\\pi G}\\left(R-2\\Lambda + \\frac{R^2}{6m^2}\\right) \\, ,\n\\end{eqaed}\nwhere the coupling $g_R$ is related to the inflaton mass according to\n\\begin{eqaed}\\label{eq:grm2}\n    g_R = - \\, \\frac{M_\\text{Pl}^2}{(8\\pi)\\cdot 12m^2} \\, .\n\\end{eqaed}\nOne then arrives at the inflaton potential\n\\begin{eqaed}\\label{eq:scalar_pot}\n    V(\\phi)=\\frac{M_\\text{Pl}^2}{8\\pi}\\,e^{-2\\sqrt{\\frac{2}{3}}\\frac{\\phi}{M_\\text{Pl}}} \\left(\\frac{3 m^2}{4}\\left(e^{\\sqrt{\\frac{2}{3}}\\frac{\\phi}{M_\\text{Pl}}}-1\\right)^2+\\Lambda \\right) \\, .\n\\end{eqaed}\nIn order to retain compatibility with the EFT, we shall consider field values in eq.~\\eqref{eq:field_range} around $\\phi \\ll M_\\text{Pl}$, since it corresponds to small curvatures. Indeed, the procedure to obtain inflationary potentials from quadratic gravity yields $\\phi = \\sqrt{\\frac{3}{2}} \\, M_\\text{Pl} \\log\\left( 1 + \\mathcal{O}(M_\\text{Pl}^{-2})\\right)$~\\cite{inflationaris,Platania:2019qvo}. One can then study dSC and TCC constraints of eqs.~\\eqref{eq:dsc_basic},~\\eqref{eq:ref_dsc} and~\\eqref{eq:tcc_basic} numerically varying the $\\mathcal{O}(1)$ constants $c$ and $f$, imposing that the bounds be satisfied for all $\\phi$ in the range allowed by eq.~\\eqref{eq:field_range}. The resulting regions are highlighted in fig.~\\ref{fig:TCC1} and in fig.~\\ref{fig:TCC2}, where each panel corresponds to a particular value of $f$ and consists of two plots which display the bounds in the $(m^2,\\Lambda)$ (left panels) and $(g_R,G\\Lambda)$ (right panels) planes. Due to the inverse relation in eq.~\\eqref{eq:grm2} between $m^2\/M_{\\text{Pl}}^2$ and $g_R$, the linear bounds in the~$(m^2, \\Lambda)$ plane translate into hyperbolas in the~$(g_R,G\\Lambda)$ plane. Note that whether the dimensionless minimum $\\phi_{\\mathrm{min}}\/M_{\\mathrm{Pl}}$, which exists for $\\Lambda \\,\/{m^2} >-3\/4$, falls inside the interval $(-f,+f)$ depends on the ratio $\\Lambda\/m^2$. In particular, the minimum falls outside the interval for $f<|\\phi_\\mathrm{min}\/M_\\mathrm{Pl}|$. Consequently, even if $V(\\phi_\\text{min})$ were to be positive for some $\\Lambda$ and $m^2$, this would not necessarily violate the dSC\/TCC bounds for fixed values of $f$ and $c$. Consequently, the bounds displayed in fig.~\\ref{fig:TCC1} and in fig.~\\ref{fig:TCC2} are non-trivially affected by eq.~\\eqref{eq:field_range}, and by the specific values of $f$ and $c$. For instance, it is worth noticing that smaller values of $f$ entail smaller field intervals where the dSC\/TCC bounds are to be satisfied. Thus, the bounds are less stringent, and the allowed region bigger. Similarly, the bound is more restrictive for higher values of $c$. In particular, fig.~\\ref{fig:TCC2} depicts the bounds derived from $c = \\sqrt{2\/3}$, the value pertaining to the TCC. While our analysis cannot probe the TCC in the large-excursion regime, where it differs substantially from the dSC, the additional considerations of~\\cite{Andriot:2020lea, Rudelius:2021oaz} point to a deeper role of this value of $c$ which could manifest itself, in the low-curvature regime at stake, in further investigations of swampland bounds and\/or dimensional reduction.\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCf01-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCf01-gr\".pdf}\\\\\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCf05-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCf05-gr\".pdf}\\\\\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCf1-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCf1-gr\".pdf}\n\\caption{dSC\/TCC constraints for $f=0.1$ (top panel), $f=0.5$ (central panel) and $f=1$ (bottom panel), and various values of $c$. Due to the inverse relation in eq.~\\eqref{eq:grm2} between the inflaton mass in Planck units $m^2\/M_{\\text{Pl}}^2$ and the coupling $g_R$, the linear bounds in the~$(m^2, \\Lambda)$ plane translate into hyperbolas in the~$(g_R,G\\Lambda)$ plane. \\label{fig:TCC1}}\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCc23-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCc23-gr\".pdf}\n\\caption{TCC constraints, corresponding to $c=\\sqrt{2\/3}$, for various values of $f$. The bounds are not qualitatively different from to the dSC bounds displayed in fig.~\\ref{fig:TCC1}.\\label{fig:TCC2}}\n\\end{figure}\n\n\\subsection{Intersections of allowed regions: compatibility of asymptotic safety with dS, TC and WG conjectures}\\label{sec:intersections}\n\nWe are now ready to collect the results that we have discussed in the preceding sections, and to visualize the intersection between the different allowed regions. Within (an extrapolation of the) one-loop approximation, asymptotic safety of the RG flow constrains the physical IR parameters of eq.~\\eqref{eq:IR_params} to lie on the plane of eq.~\\eqref{eq:marginal_line}. On the other hand, while the (duality-invariant) WGC does not entail any additional constraint, the dSC\/TCC conditions for the inflaton potential place constraints on the cosmological constant and the inflaton mass. One can plot the intersections for any values of the $\\mathcal{O}(1)$ constants $c$ and $f$, and fig.~\\ref{fig:intersections3D}, the main result of our work, refers to the representative choice $c=f=1$. Albeit difficult to visualize, there is in general a region of the asymptotically safe plane that appears compatible with the swampland constraints that we have investigated. One can straightforwardly verify that the same conclusion is reached for different values of $f$ and $c$. Our findings, based on the quadratic one-loop approximation, thus point at a non-trivial compatibility between the conditions for UV-completeness dictated by asymptotic safety and some of the most relevant swampland conjectures. Consequently, it also points at the possibility, partially supported by~\\cite{Basile:2020dzh,Basile:2021krk}, of a connection between the frameworks of asymptotic safety and string theory~\\cite{deAlwis:2019aud}. Within this picture, field-theoretical asymptotic safety would serve as a ``pivot'' for the RG flow from string theory to low-energy gravity, in the sense that below a certain scale the flow of string theory toward the IR closely approaches a field-theoretical trajectory controlled by a UV fixed point.\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.5]{\"Images\/Intersections-3D\".pdf}\\includegraphics[scale=0.5]{\"Images\/Intersections-3D-2\".pdf}\\\\\\includegraphics[scale=0.5]{\"Images\/Intersections-3D-3\".pdf}\n\\caption{Intersections between the regions allowed by asymptotic safety, the WGC and the dSC\/TCC for the representative values $c=f=1$. The WGC bound corresponds to the yellow region, while the dSC\/TCC bound corresponds to the blue region. The green region depicts the space of IR parameters spanned by asymptotically safe trajectories, which lies within the region allowed by the WGC.}\\label{fig:intersections3D}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this paper we have analyzed the intersection of consistency conditions for Wilson coefficients of gravitational EFTs, combining the constraints of asymptotic safety and swampland conjectures. In particular, in sect.~\\ref{sec:one-loop} we have employed a systematic method to extract the  hypersurface of allowed IR parameters stemming from UV-complete RG trajectories in gravitational theories by randomly sampling its relevant deformations, and we have applied this technique to the flow equations stemming from one-loop quadratic gravity~\\cite{Codello:2008vh}. Despite expecting that this approximation be reliable in the IR, the resulting RG flow exhibits a UV non-Gaussian fixed point, consistently with more refined functional RG computations~\\cite{Percacci:2017fkn, Reuter:2019byg, Pawlowski:2020qer}. However, the dimension of its critical surface is larger than what is suggested by the functional RG~\\cite{Benedetti:2009rx,Benedetti:2009gn,Falls:2020qhj,Knorr:2021slg}. As we have discussed in detail in sect.~\\ref{sec:results}, our findings suggest that the requirement that the RG flow be asymptotically safe constrains the physical parameters to lie on a plane, which we have determined, within our one-loop framework, to a precision of order $\\mathcal{O}(10^{-8})$. The values of the cosmological constant in Planck units seems not to be restricted by these considerations, while the classically marginal couplings lie on a line. \n\nIn sect.~\\ref{sec:wgc_constraints} and sect.~\\ref{sec:dsc_constraints} we have investigated the constraints on the Wilson coefficients arising from the weak gravity conjecture (WGC), the de Sitter conjecture (dSC) and the trans-Planckian censorship conjecture (TCC). In particular, the WGC does not entail additional bounds and is compatible with the UV-complete RG trajectories, while the dSC\/TCC bounds are more restrictive. To wit, the Starobinsky-like scalar potential stemming from the (local sector of the) effective action involves the ratio of the cosmological constant to the (squared) inflaton mass, and therefore the corresponding bounds place constraints on the dimensionless ratios $\\Lambda\/M_\\text{Pl}^2$ and $m\/M_\\text{Pl}$. These bounds depend on some dimensionless $\\mathcal{O}(1)$ constants, which we have varied to some extent in our analysis, and generally trace out a region in the plane allowed by asymptotic safety. While we expect that the qualitative results be unaffected by improving the truncation scheme, at least to some extent, it would be interesting to investigate the quantitative deviations in this respect.\n\nWhile in this work we have focused on the local sector of the effective action, our computation also yields the coefficients of non-local logarithmic form factors, akin to those arising from non-local heat kernel computations~\\cite{Barvinsky:1987uw, Barvinsky:1990up, Barvinsky:1990uq, Barvinsky:1993en, Avramidi:1990ap, Codello:2012kq}. It would be interesting to explore their consequences and their role within asymptotically safe gravity~\\cite{Knorr:2019atm, Draper:2020bop, Draper:2020knh} and their connection to massive matter fields~\\cite{Ohta:2020bsc}.\n\nAll in all, our results suggest that swampland constraints can be compatible with restrictions coming from UV completeness of the RG flow, but in a non-trivial fashion: the allowed parameter space is restricted to a non-trivial intersection. In retrospect, one could have expected this result on the grounds that some swampland criteria purport to be necessary conditions for UV completeness that cannot be derived from  purely field-theoretical considerations, and thus they could constrain further the parameter space compatible with asymptotic safety. On the other hand, the one-loop approximation that we have studied already features the appearance of non-local form factors. In general, non-locality at the level of the effective action is a feature of any standard (local) QFT, and thus it is in principle unrelated to possible fundamental non-localities in the bare (fixed-point) action. Precisely how the notion of (non-)locality is realized in quantum gravity is an open and intriguing question, partly related to the problem of observables~\\cite{Donnelly:2015hta, Rejzner:2016yuy, Donnelly:2016rvo, Klitgaard:2017ebu, Rudelius:2021azq}. However, a number of semi-classical considerations~\\cite{Giddings:1992hh, Susskind:1993if, Almheiri:2012rt, Giddings:2012gc, Dvali:2014ila, Keltner:2015xda, Mann:2015luq} point to the breaking of the familiar concept of locality microscopically. Whether asymptotically safe gravity is realized by a bare action polynomial in derivatives (and thus ``local'' in some sense) is not established yet. Should fundamental non-locality turn out to emerge as a feature of asymptotic safety, this would strengthen its potential connections with the frameworks of non-local gravity~\\cite{Modesto:2011kw, Modesto:2017sdr, Buoninfante:2018mre, Buoninfante:2018xiw} and string theory~\\cite{Giddings:2006vu}.\n\nDue to the nature of our approximations, this work constitutes only a first step toward determining whether the asymptotic safety scenario is compatible with the peculiar behavior and UV\/IR mixing that gravity could exhibit already at the semi-classical level due to black holes (see~\\cite{Buoninfante:2021ijy} for a very recent discussion on their validity and limitations), or with general indications from string theory. In particular, a possible connection between asymptotically safe gravity and string theory has been conjectured in~\\cite{deAlwis:2019aud}, and it is tempting to speculate that it could explain our findings. Computations combining the functional renormalization group techniques~\\cite{Dupuis:2020fhh} with symmetries of string theory~\\cite{Veneziano:1991ek, Meissner:1991zj, Meissner:1996sa, Hohm:2015doa, Hohm:2019ccp, Hohm:2019jgu} have provided preliminary evidence in favour of this scenario~\\cite{Basile:2020dzh,Basile:2021krk}. This possibility extends to the more general notion of ``effective asymptotic safety''~\\cite{Held:2020kze}, and swampland bounds could further constrain which RG flows controlled by the ``effective'' fixed point are closely approached by the RG flow arising from the proper UV completion in the IR.\n\nMost prominently, the absence of continuous global symmetries\\footnote{The fate of global discrete symmetries has been investigated in the context of asymptotically safe gravity in~\\cite{Eichhorn:2020sbo, Ali:2020znq}.} is supported by a variety of arguments from black-hole physics, string theory and holography~\\cite{Misner:1957mt, Banks:1988yz, Kallosh:1995hi, Polchinski:2003bq, Banks:2010zn, Harlow:2018tng, McNamara:2019rup}, and it would be interesting to explore this foundational issue further in the direction that we have outlined in this paper.\n\n\\section*{Acknowledgements}\n\nThe authors would like to thank F. Saueressig for insighful discussions and B. Knorr for feedback on the manuscript. The authors thank also B. Holdom for spotting a typo.\n\nThe work of I.B. is supported by the Fonds de la Recherche Scientifique - FNRS under Grants No. F.4503.20 (\"HighSpinSymm\") and T.0022.19 (\"Fundamental issues in extended gravitational theories\"). A.P. acknowledges support by Perimeter Institute for Theoretical Physics.  Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities.\n\n\\bibliographystyle{JHEP}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\\protect\\setcounter{equation}{0}\n\n\n\\subsection{}\nThe quantum systems studied in this paper are obtained by coupling a\ncertain number (finite or infinite) of $N$-body systems. A\n(standard) $N$-body system consists of a fixed number $N$ of\nparticles which interact through $k$-body forces which preserve $N$\n(arbitrary $1\\leq k \\leq N$). The many-body type interactions\ninclude forces which allow the system to make transitions between\nstates with different numbers of particles. These transitions are\nrealized by creation-annihilation processes as in quantum field\ntheory.\n\nThe Hamiltonians we want to analyze are rather complex objects and\nstandard Hilbert space techniques seem to us inefficient in this\nsituation. Our approach is based on the observation that the\n$C^*$-algebra $\\mathscr{C}$ generated by a class of physically interesting\nHamiltonians often has a quite simple structure which allows one to\ndescribe its quotient with respect to the ideal of compact operators\nin rather explicit terms \\cite{GI1,GI2}.  From this one can deduce\ncertain important spectral properties of the Hamiltonians.  We refer\nto $\\mathscr{C}$ as the \\emph{Hamiltonian algebra} (or $C^*$-algebra of\nHamiltonians) of the system. \n\nThe main difficulty in this algebraic approach is to isolate the\ncorrect $C^*$-algebra. This is especially problematic in the present\nsituations since it is not a priori clear how to define the\ncouplings between the various $N$-body systems but in very special\nsituations.  It is rather remarkable that the $C^*$-algebra\ngenerated by a small class of elementary and natural Hamiltonians\nwill finally prove to be a fruitful choice.  These elementary\nHamiltonians are analogs of the Pauli-Fierz Hamiltonians.\n\n\nThe purpose of the preliminary Section \\ref{s:euclid} is to present\nthis approach in the simplest but physically important case when the\nconfiguration spaces of the $N$-body systems are Euclidean\nspaces. We start with a fundamental example, the standard $N$-body\ncase. Then we describe the many-body formalism in the Euclidean case\nand we state our main results on the spectral analysis of the\ncorresponding Hamiltonians.\n\nThere is one substantial simplification in the Euclidean case: each\nsubspace has a canonical supplement, the subspace orthogonal to\nit. This plays a role in the way we present the framework in Section\n\\ref{s:euclid}.  However, the main constructions and results do not\ndepend on the existence of a supplement but to see this requires\nmore sophisticated tools from the theory of crossed product\n$C^*$-algebras and Hilbert $C^*$-modules which are not apparent in\nthis introductory part.  In the rest of the paper we consider\nmany-body type couplings of systems whose configuration space is an\narbitrary abelian locally compact group.  One of the simplest\nnontrivial physically interesting cases covered by this framework is\nthat when the configuration spaces of the $N$-body systems are\ndiscrete groups, e.g.  discretizations $\\mathbb{Z}^D$ of $\\mathbb{R}^D$.\n\n\\subsection{}\nWe summarize now the content of the paper.  Section \\ref{s:euclid}\nstarts with a short presentation of the standard \\mbox{$N$-body}\nformalism, the rest of the section being devoted to a rather\ndetailed description of our framework and main results in the case\nwhen the configuration spaces of the $N$-body subsystems are\nEuclidean spaces. These results are proven in a more general and\nnatural setting in the rest of the paper.  In Section \\ref{s:grad}\nwe recall some facts concerning $C^*$-algebras graded by a\nsemilattice $\\mathcal{S}$ (we take here into account the results of Athina\nMageira's thesis \\cite{Ma}) and then we present some results on\n$\\mathcal{S}$-graded Hilbert $C^*$-modules. This notion, due to Georges\nSkandalis \\cite{Sk}, proved to be very natural and useful in our\ncontext: thanks to it many results can be expressed in a simple and\nsystematic way thus giving a new and interesting perspective to the\nsubject (this is discussed in more detail in \\cite{DG4}). The heart\nof the paper is Section \\ref{s:grass}, where we define the many-body\nHamiltonian algebra $\\mathscr{C}$ in a general setting and prove that it is\nnaturally graded by a certain semilattice $\\mathcal{S}$. In Section\n\\ref{s:id} we give alternative descriptions of the components of\n$\\mathscr{C}$ which are important for the affiliation criteria presented in\nSection \\ref{s:af}, where we point out a large class of self-adjoint\noperators affiliated to the many-body algebra.  The $\\mathcal{S}$-graded\nstructure of $\\mathscr{C}$ gives then an HVZ type description of the\nessential spectrum for all these operators. The main result of\nSection \\ref{s:mou} is the proof of the Mourre estimate for\nnonrelativistic many-body Hamiltonians. Finally, an Appendix is\ndevoted to the question of generation of some classes of\n$C^*$-algebras by \"elementary\" Hamiltonians.\n\n\n\\subsection{Notations} \n\\label{ss:inotations}\nWe recall some notations and terminology.  If $\\mathcal{E},\\mathcal{F}$ are normed\nspaces then $L(\\mathcal{E},\\mathcal{F})$ is the space of bounded operators\n$\\mathcal{E}\\rightarrow\\mathcal{F}$ and $K(\\mathcal{E},\\mathcal{F})$ the subspace consisting of compact\noperators. If $\\mathcal{G}$ is a third normed space and $(e,f)\\mapsto ef$ is\na bilinear map $\\mathcal{E}\\times\\mathcal{F}\\to\\mathcal{G}$ then $\\mathcal{E}\\mathcal{F}$ is the linear\nsubspace of $\\mathcal{G}$ generated by the elements $ef$ with\n$e\\in\\mathcal{E},f\\in\\mathcal{F}$ and $\\mathcal{E}\\cdot\\mathcal{F}$ is its closure. If $\\mathcal{E}=\\mathcal{F}$\nthen we set $\\mathcal{E}^2=\\mathcal{E}\\cdot\\mathcal{E}$.  Two unusual abbreviations are\nconvenient: by \\emph{lspan} and \\emph{clspan} we mean ``linear\nspan'' and ``closed linear span'' respectively.  If $\\mathcal{A}_i$ are\nsubspaces of a normed space then $\\sum^\\mathrm{c}_i\\mathcal{A}_i$ is the clspan of\n$\\cup_i\\mathcal{A}_i$.  If $X$ is a locally compact topological space then\n$\\cc_{\\mathrm{o}}(X)$ is the space of continuous complex functions which tend to\nzero at infinity and $\\cc_{\\mathrm{c}}(X)$ the subspace of functions with compact\nsupport.\n\nBy \\emph{ideal} in a $C^*$-algebra we mean a closed self-adjoint\nideal. A $*$-homomorphism between two \\mbox{$C^*$-algebras} will be\ncalled \\emph{morphism}.  We write $\\mathscr{A}\\simeq\\mathscr{B}$ if the\n$C^*$-algebras $\\mathscr{A},\\mathscr{B}$ are isomorphic and $\\mathscr{A}\\cong\\mathscr{B}$ if they\nare canonically isomorphic (the isomorphism should be clear from the\ncontext).\n\n\nA self-adjoint operator $H$ on a Hilbert space $\\mathcal{H}$ is\n\\emph{affiliated} to a $C^*$-algebra $\\mathscr{A}$ of operators on $\\mathcal{H}$ if\n$(H+i)^{-1}\\in\\mathscr{A}$; then $\\varphi(H)\\in\\mathscr{A}$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$. If $\\mathscr{A}$ is the closed linear span of the\nelements $\\varphi(H)A$ with $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$ and $A\\in\\mathscr{A}$, we\nsay that $H$ is \\emph{strictly affiliated to $\\mathscr{A}$}.  The\n$C^*$-algebra generated by a set $\\mathscr{E}$ of self-adjoint operators is\nthe smallest $C^*$-algebra such that each $H\\in\\mathscr{E}$ is affiliated to\nit.\n\nWe now recall the definition of $\\mathcal{S}$-graded $C^*$-algebras\nfollowing \\cite{Ma2}. Here $\\mathcal{S}$ is a \\emph{semilattice}, i.e. a set\nequipped with an order relation $\\leq$ such that the lower bound\n$\\sigma\\wedge\\tau$ of each couple of elements $\\sigma,\\tau$ exists.\nWe say that a subset $\\mathcal{T}$ of $\\mathcal{S}$ is a \\emph{sub-semilattice} of\n$\\mathcal{S}$ if $\\sigma,\\tau\\in\\mathcal{T}\\Rightarrow\\sigma\\wedge\\tau\\in\\mathcal{T}$.  The\nset $\\mathscr{S}$ of all closed subgroups of a locally compact abelian group\nis a semilattice for the order relation given by set inclusion.  The\nsemilattices which are of main interest for us are (inductive limits\nof) sub-semilattices of $\\mathscr{S}$.\n\nA $C^*$-algebra $\\mathscr{A}$ is called \\emph{$\\mathcal{S}$-graded} if a linearly\nindependent family $\\{\\mathscr{A}(\\sigma)\\}_{\\sigma\\in\\mathcal{S}}$ of\n$C^*$-subalgebras of $\\mathscr{A}$ has been given such that\n$\\sum^\\mathrm{c}_{\\sigma\\in\\mathcal{S}}\\mathscr{A}(\\sigma)=\\mathscr{A}$ and\n$\\mathscr{A}(\\sigma)\\mathscr{A}(\\tau)\\subset\\mathscr{A}(\\sigma\\wedge\\tau)$ for all\n$\\sigma,\\tau$. The algebras $\\mathscr{A}(\\sigma)$ are the \\emph{components\n  of $\\mathscr{A}$}. It is useful to note that some of the algebras\n$\\mathscr{A}(\\sigma)$ could be zero. If $\\mathcal{T}$ is a sub-semilattice of\n$\\mathcal{S}$ and $\\mathscr{A}(\\sigma)=\\{0\\}$ for $\\sigma\\notin\\mathcal{T}$ we say that $\\mathscr{A}$\nis \\emph{supported} by $\\mathcal{T}$; then $\\mathscr{A}$ is in fact\n$\\mathcal{T}$-graded. Reciprocally, any $\\mathcal{T}$-graded $C^*$-algebra becomes\n$\\mathcal{S}$-graded if we set $\\mathscr{A}(\\sigma)=\\{0\\}$ for $\\sigma\\notin\\mathcal{T}$.\n\n\n\n\n\\subsection{Note}\nThe preprint \\cite{DG4} is a preliminary version of this paper. We\ndecided to change the title because the differences between the two\nversions are rather important: the preliminaries concerning the\ntheory of Hilbert $C^*$-modules and the role of the imprimitivity\nalgebra of a Hilbert $C^*$-module in the spectral analysis of\nmany-body systems are now reduced to a minimum; on the other hand,\nthe Euclidean case and the spectral theory of the corresponding\nHamiltonians are treated in more detail.\n\n\\vspace{3mm}\n\n\\begin{acknowledgement}{\\rm\nThe authors thank Georges Skandalis for very helpful suggestions and\nremarks.}\n\\end{acknowledgement}\n\n\n\\section{Euclidean framework: main results}\n\\label{s:euclid}\n\\protect\\setcounter{equation}{0}\n\n\\subsection{The Hamiltonian algebra of a standard $N$-body system}\n\\label{ss:inb}\n\nConsider a system of $N$ particles moving in the physical space\n$\\mathbb{R}^d$. In the nonrelativistic case the Hamiltonian is of the form\n\\begin{equation}\\label{eq:inonrel}\nH={\\textstyle\\sum_{j=1}^N P_j^2\/2m_j + \\sum_{j=1}^N V_j(x_j)+\n\\sum_{j<k} V_{jk}(x_j-x_k)}\n\\end{equation}\nwhere $m_1,\\dots,m_N$ are the masses of the particles,\n$x_1,\\dots,x_N\\in\\mathbb{R}^d$ their positions, and $P_j=-i\\nabla_{x_j}$\ntheir momenta. In the simplest situation the potentials $V_j,V_{jk}$\nare real continuous functions with compact support on $\\mathbb{R}^d$. The\nstate space of the system is the Hilbert space $L^2(X)$ with\n$X=(\\mathbb{R}^d)^N$.\n\nLet $P=(P_1,\\dots,P_N)$, this is a set of commuting self-adjoint\noperators on $L^2(X)$ and so $h(P)$ is a well defined self-adjoint\noperator for any real Borel function $h$ on $X$. In what follows we\nreplace the kinetic energy part $\\sum_j P_j^2\/2m_j$ in\n\\eqref{eq:inonrel} by an operator $h(P)$ with $h$ continuous and\ndivergent at infinity. Denote $x=(x_1,\\dots,x_N)$ the points of $X$\nand let us consider the linear subspaces of $X$ defined as follows:\n$X_j=\\{x\\in X\\mid x_j=0\\}$ if $1\\le j \\le N$ and $X_{jk}=\\{x\\in\nX\\mid x_j=x_k\\}$ for $j<k$.  Let $\\pi_j$ and $\\pi_{jk}$ be the\nnatural maps of $X$ onto the quotient spaces $X\/X_j$ and $X\/X_{jk}$\nrespectively (the Euclidean structure of $X$ allows us to identify\nthese abstract spaces with the subspaces of $X$ orthogonal to $X_j$\nand $X_{jk}$, but this is irrelevant here). Then $H$ may be written\nin the form\n\\begin{equation}\\label{eq:inonr}\n  H=h(P) + {\\textstyle\\sum_j v_j\\circ\\pi_j(x)+\n    \\sum_{j<k} v_{jk}\\circ\\pi_{jk}(x)}\n\\end{equation}  \nfor some real functions $v_j\\in\\cc_{\\mathrm{c}}(X\/X_j)$ and\n$v_{jk}\\in\\cc_{\\mathrm{c}}(X\/X_{jk})$. Thus Hamiltonians of the form\n\\eqref{eq:inonr} are natural objects in the $N$-body problem.  Note\nthat there should be no privileged origin in the momentum space, so\nif we accept $h(P)$ as an admissible kinetic energy operator then\n$h(P+p)$ should also be admissible for any\n$p=(p_1,\\dots,p_N)\\in(\\mathbb{R}^d)^N$.\n\nLet $\\mathscr{S}\\equiv\\mathscr{S}(X)$ be the set of linear subspaces of $X$ equipped\nwith the order relation $Y\\leq Z \\Leftrightarrow Y\\subset Z$. Then\n$\\mathscr{S}$ is a semilattice with $Y\\wedge Z =Y\\cap Z$.  If $Y\\in\\mathscr{S}$ then\nwe realize $\\cc_{\\mathrm{o}}(X\/Y)$ as a $C^*$-algebra of operators on $L^2(X)$ by\nassociating to $v$ the operator of multiplication by $v\\circ\\pi_Y$,\nwhere $\\pi_Y:X\\to X\/Y$ is the canonical surjection. The following\nfact is easy to prove: \\emph{if $\\mathcal{S}\\subset\\mathscr{S}$ is finite then\n$\\mathcal{C}(\\mathcal{S}):=\\sum_{Y\\in\\mathcal{S}} \\cc_{\\mathrm{o}}(X\/Y)$ is a direct topological sum and\nis an algebra if and only if $\\mathcal{S}$ is a sub-semilattice of $\\mathscr{S}$; in\nthis case, $\\mathcal{C}(\\mathcal{S})$ is an $\\mathcal{S}$-graded $C^*$-algebra}.\n\nIn the next proposition $\\mathcal{S}$ is the semilattice of subspaces of $X$\ngenerated by the $X_j$ and $X_{jk}$, i.e. the set of subspaces of\n$X$ obtained by taking arbitrary intersections of subspaces of the\nform $X_j$ and $X_{jk}$.  We denote $\\mathscr{T}_X$ the $C^*$-algebra of\noperators on $L^2(X)$ of the from $\\varphi(P)$ with\n$\\varphi\\in\\cc_{\\mathrm{o}}(X)$.\n\n\\begin{proposition}\\label{pr:mad}\n  Let $h:X\\to\\mathbb{R}$ be continuous with $h(x)\\to\\infty$ if\n  $x\\to\\infty$ and let $H_p$ be the self-adjoint operator\n  \\eqref{eq:inonr} with $h(P)$ replaced by $h(P+p)$. Then the\n  $C^*$-algebra generated by the operators $H_p$ when $p$ runs over\n  $(\\mathbb{R}^d)^N$ and $ v_j$ and $ v_{jk}$ run over the set of real\n  functions in $\\cc_{\\mathrm{c}}(X\/X_j)$ and $\\cc_{\\mathrm{c}}(X\/X_{jk})$ respectively is\n  $\\mathscr{C}=\\mathcal{C}(\\mathcal{S})\\cdot\\mathscr{T}_X=\\sum_{Y\\in\\mathcal{S}}\n  \\cc_{\\mathrm{o}}(X\/Y)\\cdot\\mathscr{T}_X$. Moreover, $\\mathscr{C}$ is $\\mathcal{S}$-graded by this\n  decomposition.\n\\end{proposition}\n\n \nThis has been proved in \\cite{DG1}. More general results of this\nnature are presented in Appendix \\ref{s:appb}.  Observe that we decided\nto fix the function $h$ which represents the kinetic energy but not\nthe potentials $v_j,v_{jk}$. However, as a consequence of\nProposition \\ref{pr:mad}, if we allow $h$ to vary we get the same\nalgebra.\n\nProposition \\ref{pr:mad} provides a basic example of ``Hamiltonian\nalgebra''.  We mention that $\\mathscr{C}$ is the crossed product of the\n$C^*$-algebra $\\mathcal{C}(\\mathcal{S})$ by the natural action of the additive group\n$X$, so it is a natural mathematical object.  We shall see in a more\ngeneral context that the set of self-adjoint operators affiliated to\nit is much larger than expected (cf. Theorem \\ref{th:iaff} for\nexample).\n\nIf we are in the nonrelativistic case and $V_j=0$ for all $j$ then\nthe center of mass of the system moves freely and it is more\nconvenient to eliminate it and to take as origin of the reference\nsystem the center of mass of the $N$ particles. Then the\nconfiguration space $X$ is the set of points\n$x=(x_1,\\dots,x_N)\\in(\\mathbb{R}^d)^N$ such that $\\sum_km_k x_k=0$.\nProposition \\ref{pr:mad}  remains valid if $\\mathcal{S}$ is conveniently\ndefined, see \\S \\ref{ss:cexample}.\n\nThe following ``generalized'' class of $N$-body systems is suggested\nby results from \\cite{Ma,Ma3}.\n\n\\begin{definition}\\label{df:nb}\n  An \\emph{$N$-body structure on a locally compact abelian group\n    $X$} is a set $\\mathcal{S}_X$ of closed subgroups such that\n  $X\\in\\mathcal{S}_X$ and such that for all $Y,Z\\in\\mathcal{S}_X$ the following\n  three conditions are satisfied: (i)~ $Y\\cap Z \\in \\mathcal{S}$; (ii) the\n  subgroups $Y,Z$ of $X$ are compatible; (iii) if $Y\\supsetneq Z$\n  then $Y\/Z$ is not compact.\n\\end{definition}\n$X$ must be thought as configuration space of the system. The notion\nof compatible subgroups is defined in Subsection \\ref{ss:compat} (if\n$X$ is a $\\sigma$-compact topological space this means that $Y+Z$ is\na closed subgroup).  We shall see that the Hamiltonian algebra\nassociated to such an $N$-body system is an interesting object:\n\\begin{equation}\\label{eq:gnbh}\n\\mathscr{C}_X(\\mathcal{S}_X):= \\mathcal{C}(\\mathcal{S}_X)\\cdot\\mathscr{T}_X \\cong \\mathcal{C}(\\mathcal{S}_X)\\rtimes X\n\\quad\\text{where}\\quad  \\mathcal{C}(\\mathcal{S}_X)=\n\\textstyle{\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}_X}} \\cc_{\\mathrm{o}}(X\/Y).\n\\end{equation} \n\nHere $\\mathscr{T}_X\\cong\\cc_{\\mathrm{o}}(X^*)$ is the group $C^*$-algebra of $X$ and\n$\\rtimes$ means crossed product.\n\n\\begin{example}\\label{ex:}\n  This framework covers an interesting extension of the standard\n  $N$-body setting. Assume that $X$ is a finite dimensional real\n  vector space.  In the standard framework the semilattice $\\mathcal{S}$\n  consists of linear subspaces of $X$ but here we allow them to be\n  closed additive subgroups. The closed additive\n  subgroups of $X$ are of the form $Y=E+L$ where $E$ is a vector\n  subspace of $X$ and $L$ is a lattice in a vector subspace $F$ of\n  $X$ such that $E\\cap F=\\{0\\}$.  More precisely, $L=\\sum_k\\mathbb{Z} f_k$\n  where $\\{f_k\\}$ is a basis in $F$. Thus $F\/L$ is a torus and if\n  $G$ is a third vector subspace such that $X=E\\oplus F\\oplus G$\n  then the space $X\/Y\\simeq (F\/L)\\oplus G$ is a cylinder with $F\/L$\n  as basis.\n\\end{example}\n\n\n\\subsection{The Euclidean many-body algebra}\n\\label{ss:sintro}\n\nWe introduce here an abstract framework which allows us to study\ncouplings between several $N$-body systems of the type considered\nabove. A concrete and physically interesting example may be found in\n\\S\\ref{ss:cexample}.\n\nLet $\\mathcal{X}$ be a real prehilbert space. Let $\\mathscr{S}(\\mathcal{X})$ be the set of\nfinite dimensional subspaces of $\\mathcal{X}$ equipped with the order\nrelation given by set inclusion. This is a lattice with $X\\wedge\nY=X\\cap Y$ and $X\\vee Y=X+Y$, but only the semilattice structure is\nrelevant for what follows. \n\n\nEach finite dimensional subspace\n$X\\subset \\mathcal{X}$ is equipped with the Euclidean structure induced by\n$\\mathcal{X}$ hence the Hilbert space $\\mathcal{H}_X= L^2(X)$ and the $C^*$-algebras\n$\\mathscr{L}_X=L(\\mathcal{H}_X)$ and $\\mathscr{K}_X=K(\\mathcal{H}_X)$ are well defined.  The group\nalgebra $\\mathscr{T}_X$ is defined as the closure in $\\mathscr{L}_X$ of the set of\noperators of convolution with functions of class $\\cc_{\\mathrm{c}}(X)$.  If\n$O=\\{0\\}$ then $\\mathcal{H}_O=\\mathbb{C}$ and $\\mathscr{L}_O=\\mathscr{K}_O=\\mathscr{T}_O=\\mathbb{C}$ by\nconvention. \n\nWe denote $h(P)$ the operator on $\\mathcal{H}_X$ given by\n$\\mathcal{F}_X^{-1}M_h\\mathcal{F}_X$, where $\\mathcal{F}_X$ is the Fourier transformation\nand $M_h$ is the operator of multiplication by the function\n$h:X\\to\\mathbb{C}$. Then $\\mathscr{T}_X=\\{\\psi(P)\\mid\\psi\\in\\cc_{\\mathrm{o}}(X)\\}$. We use the\nnotation $P=P_X$  if the space $X$ has to be specified. \n\nIf $X,Y$ are finite dimensional subspaces of $\\mathcal{X}$ we set\n$\\mathscr{L}_{XY}=L(\\mathcal{H}_Y,\\mathcal{H}_X)$ and $\\mathscr{K}_{XY}=K(\\mathcal{H}_Y,\\mathcal{H}_X)$.  We define\na closed subspace $\\mathscr{T}_{XY}\\subset\\mathscr{L}_{XY}$ as follows. If\n$\\varphi\\in\\cc_{\\mathrm{c}}(X+Y)$ then one may easily check that\n$(T_{XY}(\\varphi)f)(x)=\\int_Y\\varphi(x-y)f(y) \\text{d} y$ defines a\ncontinuous operator $\\mathcal{H}_Y\\to\\mathcal{H}_X$. Let\n\\begin{equation}\\label{eq:txy}\n\\mathscr{T}_{XY}= \\text{ norm closure of the set of operators} \\hspace{2mm}\nT_{XY}(\\varphi)  \\hspace{2mm} \\text{with} \\hspace{2mm} \n\\varphi\\in\\cc_{\\mathrm{c}}(X+Y).\n\\end{equation}\nClearly $\\mathscr{T}_{XX}=\\mathscr{T}_X$. The space $\\mathscr{T}_{XY}$ is a ``concrete''\nrealization of the Hilbert $C^*$-module introduced by Philip Green\nto show the Morita equivalence of the crossed products\n$\\cc_{\\mathrm{o}}(Z\/Y)\\rtimes X$ and $\\cc_{\\mathrm{o}}(Z\/X)\\rtimes Y$ where $Z=X+Y$ (this has\nbeen noticed by Georges Skandalis, see Remark \\ref{re:rief} for\nmore details).\n\n\nNow we fix a sub-semilattice $\\mathcal{S}\\subset\\mathscr{S}$, i.e. we assume that\n$X\\cap Y\\in\\mathcal{S}$ if $X,Y\\in\\mathcal{S}$.  This set completely determines the\nmany-body system and the class of Hamiltonians that we intend to\nstudy. For each $X\\in\\mathcal{S}$ the Hilbert space $\\mathcal{H}_X$ is thought as\nthe state space of an $N$-body system with $X$ as configuration\nspace.  We define the state space of the many-body system as the\nHilbertian direct sum\n\\begin{equation}\\label{eq:hs}\n\\mathcal{H}\\equiv\\mathcal{H}_\\mathcal{S}=\\oplus_{X\\in\\mathcal{S}} \\mathcal{H}_X.\n\\end{equation}\nWe have a natural embedding $\\mathscr{L}_{XY}\\subset L(\\mathcal{H})$ for all\n$X,Y\\in\\mathcal{S}$. Let $\\mathscr{L}\\equiv\\mathscr{L}_\\mathcal{S}$ be the closed linear span of the\nsubspaces $\\mathscr{L}_{XY}$. Clearly $\\mathscr{L}$ is a $C^*$-subalgebra of\n$L(\\mathcal{H})$ which is equal to $L(\\mathcal{H})$ if and only if $\\mathcal{S}$ is finite.\nWe will be interested in subspaces $\\mathscr{R}$ of $\\mathscr{L}$ constructed as\nfollows: for each couple $X,Y$ we are given a closed subspace\n$\\mathscr{R}_{XY}\\subset\\mathscr{L}_{XY}$ and $\\mathscr{R}\\equiv(\\mathscr{R}_{XY})_{X,Y\\in\\mathcal{S}}=\n{\\textstyle\\sum^\\mathrm{c}_{X,Y\\in\\mathcal{S}}}\\mathscr{R}_{XY}$ where $\\sum^\\mathrm{c}$ means\nclosure of the sum. Note that\n$\\mathscr{K}\\equiv\\mathscr{K}_\\mathcal{S}=(\\mathscr{K}_{XY})_{X,Y\\in\\mathcal{S}}=K(\\mathcal{H})$.\n\n\\begin{theorem}\\label{th:C}\n  Let $\\mathscr{T}\\equiv\\mathscr{T}_\\mathcal{S}=(\\mathscr{T}_{XY})_{X,Y\\in\\mathcal{S}}$. Then $\\mathscr{T}$ is a\n  closed self-adjoint subspace of $\\mathscr{L}$ and $\\mathscr{C}\\equiv\\mathscr{C}_\\mathcal{S}=\\mathscr{T}^2$\n  is a non-degenerate $C^*$-algebra of operators on $\\mathcal{H}$.\n\\end{theorem}\n\nWe say that \\emph{$\\mathscr{C}$ is the Hamiltonian algebra of the many-body\n  system $\\mathcal{S}$}.  This terminology will be justified later on: we\nshall see that physically interesting many-body Hamiltonians are\nself-adjoint operators affiliated to $\\mathscr{C}$. Moreover, in a quite\nprecise way, $\\mathscr{C}$ is the smallest $C^*$-algebra with this\nproperty. For the purposes of this paper \\emph{we define a\nmany-body Hamiltonian as a self-adjoint operator affiliated to\n$\\mathscr{C}$}.\n\nWe now equip $\\mathscr{C}$ with an $\\mathcal{S}$-graded $C^*$-algebras\nstructure. This structure will play a central role in the spectral\nanalysis of self-adjoint operators affiliated to $\\mathscr{C}$.  We often\nsay ``graded'' instead of $\\mathcal{S}$-graded.\n\nTo define the grading we need new objects.  If $Y\\not\\subset X$ we\nset $\\mathcal{C}_X(Y)=\\{0\\}$. If $Y\\subset X$ then we define $\\mathcal{C}_X(Y)$ as\nthe set of continuous functions on $X$ which are invariant under\ntranslations in the $Y$ directions and tend to zero in the $Y^\\perp$\ndirections.  This is a $C^*$-algebra of bounded uniformly continuous\nfunctions on $X$ canonically isomorphic with $\\cc_{\\mathrm{o}}(X\/Y)$ where $X\/Y$\nis the orthogonal of $Y$ in $X$. Thus\n\\begin{equation}\\label{eq:2cxy}\n\\mathcal{C}_X(Y)\\cong\\cc_{\\mathrm{o}}(X\/Y) \\text{ if } Y\\subset X \\quad\\text{and}\\quad\n\\mathcal{C}_X(Y)=\\{0\\} \\text{ if } Y\\not\\subset X.\n\\end{equation}\nLet $\\mathcal{C}_X\\equiv\\mathcal{C}_X(\\mathcal{S}):=\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}}\\mathcal{C}_X(Y)$, this is a\n$C^*$-algebra of bounded uniformly continuous functions on $X$.  We\nembed it in $\\mathscr{L}_X$ by identifying a function $\\varphi$ with the\noperator on $\\mathcal{H}_X$ of multiplication by $\\varphi$. Then\n\\begin{equation}\\label{eq:imp0}\n\\mathcal{C}\\equiv\\mathcal{C}_\\mathcal{S}=\\oplus_{X\\in\\mathcal{S}}\\mathcal{C}_X\n\\end{equation}\nis a $C^*$-algebra of operators on $\\mathcal{H}$ included in $\\mathscr{L}$.  For\neach $Z\\in\\mathcal{S}$ we define a $C^*$-subalgebra of $\\mathcal{C}$ by\n\\begin{equation}\\label{eq:impz}\n\\mathcal{C}(Z)\\equiv\\mathcal{C}_\\mathcal{S}(Z)=\\oplus_X\\mathcal{C}_X(Z)=\\oplus_{X\\supset Z}\\mathcal{C}_X(Z).\n\\end{equation}\nIt is easy to see that the family $\\{\\mathcal{C}(Z)\\}_{Z\\in\\mathcal{S}}$ defines a\ngraded $C^*$-algebra structure on $\\mathcal{C}$.\n\n\\begin{theorem}\\label{th:CG}\n  We have $\\mathscr{C}=\\mathscr{T}\\cdot\\mathcal{C}=\\mathcal{C}\\cdot\\mathscr{T}$.  For each $Z\\in\\mathcal{S}$ the\n  space $\\mathscr{C}(Z)=\\mathscr{T}\\cdot\\mathcal{C}(Z)=\\mathcal{C}(Z)\\cdot\\mathscr{T}$ is a $C^*$-subalgebra\n  of $\\mathscr{C}$ and the family $\\{\\mathscr{C}(Z)\\}_{Z\\in\\mathcal{S}}$ defines a graded\n  $C^*$-algebra structure on $\\mathscr{C}$.\n\\end{theorem}\n\nIn particular we get $\\mathscr{C}=(\\mathscr{C}_{XY})_{X,Y\\in\\mathcal{S}}$ with (the second\nand fourth equalities are not obvious):\n\\begin{align}\n\\mathscr{C}_{XY} \n&=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y \n={\\textstyle\\sum^\\mathrm{c}_{Z\\subset X\\cap Y}}\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Z)\n\\label{eq:icxy1} \\\\\n&=\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}= {\\textstyle\\sum^\\mathrm{c}_{Z\\subset X\\cap\n    Y}}\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}. \\label{eq:icxy2}\n\\end{align}\nThe $C^*$-algebras $\\mathcal{C}(Z)$ and $\\mathscr{C}(Z)$ ``live'' in the closed\nsubspace $\\mathcal{H}_{\\geq Z}=\\oplus_{X\\supset Z}\\mathcal{H}_X$ of $\\mathcal{H}$.  More\nprecisely, they leave invariant $\\mathcal{H}_{\\geq Z}$ and their restriction\nto its orthogonal subspace is zero. Moreover, if we denote\n$\\mathscr{L}_{\\geq Z}=(\\mathscr{L}_{X,Y})_{X,Y\\supset Z}\\subset L(\\mathcal{H}_{\\geq\n  Z})$ then clearly $\\mathcal{C}(Z)$ and $\\mathscr{C}(Z)$ are subalgebras of\n$\\mathscr{L}_{\\geq Z}$.\n\n\n\\begin{remark}\\label{re:nbody}{\\rm \nThe diagonal element $\\mathscr{C}_{XX}\\equiv\\mathscr{C}_X$ of the ``matrix'' $\\mathscr{C}$\nis given  by\n\\begin{equation}\\label{eq:NB}\n\\mathscr{C}_X=\\mathcal{C}_X\\cdot\\mathscr{T}_X=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\subset X}\\cc_{\\mathrm{o}}(X\/Z)\\cdot\\mathscr{T}_X}\n\\subset\\mathscr{L}_X.\n\\end{equation}\nThis $C^*$-algebra is \\emph{the Hamiltonian algebra of the\n  (generalized) $N$-body system} associated to the semi-lattice\n$\\mathcal{S}_X=\\{Z\\in\\mathcal{S}\\mid Z\\subset X\\}$ of subspaces of $X$,\ncf. \\eqref{eq:gnbh}. The non-diagonal elements $\\mathscr{C}_{XY}$ are\nHilbert $C^*$-bimodules which define the coupling between the\n$N$-body type systems $X$ and $Y$.  }\\end{remark}\n\n\\begin{remark}\\label{re:bigc}{\\rm Note that if we take $\\mathcal{S}$ equal\n    to the set of all finite dimensional subspaces of $\\mathcal{X}$ then we\n    get a graded $C^*$-algebra $\\mathscr{C}$ canonically associated to the\n    prehilbert space $\\mathcal{X}$. According to a remark in\n    \\S\\ref{ss:inotations}, any other choice of $\\mathcal{S}$ would give us a\n    graded $C^*$-subalgebra of this one.  }\\end{remark}\n\n\n\\begin{remark}\\label{re:sym}\n  The algebra $\\mathscr{C}$ is not adapted to symmetry considerations, in\n  particular in applications to physical systems consisting of\n  particles one has to assume them distinguishable. The Hamiltonian\n  algebra for systems of identical particles interacting through\n  field type forces (both bosonic and fermionic case) is constructed\n  in \\cite{Geo}. We mention that a quantum field model without\n  symmetry considerations corresponds to the case when $\\mathcal{S}$ is a\n  distributive relatively ortho-complemented lattice.\n\\end{remark}\n\n\n\\subsection{Particle systems with conserved total mass}\n\\label{ss:cexample} \n\n\nWe give now a physically interesting example of the preceding abstract\nconstruction.  We shall describe the many-body system associated to\n$N$ ``elementary particles'' of masses $m_1,\\dots,m_N$ moving in the\nphysical space $\\mathbb{R}^d$ without external fields. We shall get\n\\emph{a system in which the the total mass is conserved but not the\n  number of particles.}\n\nWe go back the framework of \\S \\ref{ss:inb} but assume that \nthe particles interact only through 2-body forces. Then\n\\begin{equation}\\label{eq:nonrel}\nH={\\textstyle\\sum_{j=1}^N P_j^2\/2m_j +\\sum_{j<k} V_{jk}(x_j-x_k)}.\n\\end{equation}\nand the center of mass of the system moves freely so it is\nconvenient to eliminate it. This is a standard procedure that we\nsketch now, cf.  \\cite{ABG,DeG1} for a detailed discussion of the\nformalism.  We take as origin of the reference system the center of\nmass of the $N$ particles so the configuration space $X$ is the set\nof points $x=(x_1,\\dots,x_N)\\in(\\mathbb{R}^d)^N$ such that $\\sum_km_k\nx_k=0$. We equip $X$ with the scalar product\n$\\braket{x}{y}=\\sum_{k=1}^N2m_k x_k y_k$. The advantage is that the\nreduced Hamiltonian, the operator acting in $L^2(X)$ naturally\nassociated to the expression \\eqref{eq:nonrel}, is\n$\\Delta_X+\\sum_{j<k}V_{jk}(x_j-x_k)$ where $\\Delta_X$ is the\nLaplacian associated to this scalar product. We denote by the same\nsymbol $H$ this reduced operator.\n\nThe first step is to describe the $C^*$-algebra generated by these\nHamiltonians, i.e. to get the analog of Proposition \\ref{pr:mad} in\nthe present context. Thus we have to describe the semilattice of\nsubspaces of $X$ generated by the $X_{(jk)}:=X_{jk}\\cap X$. We give\nthe result below and refer to the Appendix \\S\\ref{ss:anbody} for\nproofs.\n\nA partition $\\sigma$ of the set $\\{1,\\dots,N\\}$ is also called\ncluster decomposition. Then the sets of the partition are called\nclusters. A cluster $a\\in\\sigma$ is thought as a ``composite\nparticle'' of mass $m_a=\\sum_{k\\in a}m_k$.  Let $|\\sigma|$ be the\nnumber of clusters of $\\sigma$. We interpret $\\sigma$ as a system of\n$|\\sigma|$ particles with masses $m_a$ hence its configuration space\nshould be the set of $x=(x_a)_{a\\in\\sigma}\\in(\\mathbb{R}^d)^{|\\sigma|}$\nsuch that $\\sum_a m_a x_a=0$ equipped with a scalar product similar\nto that defined above.\n\nLet $X_\\sigma$ be the set of $x\\in X$ such that $x_i=x_j$ if $i,j$\nbelong to the same cluster and let us equip $X_\\sigma$ with the\nscalar product induced by $X$. Then there is an obvious isometric\nidentification of $X_\\sigma$ with the configuration space of the\nsystem $\\sigma$ as defined above. The advantage now is that all the\nspaces $X_\\sigma$ are isometrically embedded in the same $X$.  We\nequip the set $\\mathfrak{S}$ of partitions with the order relation:\n$\\sigma\\leq\\tau$ if and only if ``$\\tau$ is finer than $\\sigma$''\n(this is opposite to the usual convention). Then $\\sigma\\leq\\tau$ is\nequivalent to $X_\\sigma\\subset X_\\tau$ and $X_\\sigma\\cap\nX_\\tau=X_{\\sigma\\wedge\\tau}$.  Thus we see that $\\mathfrak{S}$ is\nisomorphic as semilattice with the set $\\mathcal{S}=\\{X_\\sigma\\mid\n\\sigma\\in\\mathfrak{S}\\}$ of subspaces of $X$ with inclusion as order\nrelation. Now it is easy to check that \\emph{$\\mathcal{S}$ coincides with\n  the semilattice of subspaces of $X$ generated by the $X_{(jk)}$}.\n\nWe abbreviate $\\mathcal{H}_\\sigma=\\mathcal{H}_{X_\\sigma}=L^2(X_\\sigma)$.  According\nto the identifications made above, this is the state space of a\nsystem of $|\\sigma|$ particles with masses $(m_a)_{a\\in\\sigma}$.\n\nNow we may apply our construction to $\\mathcal{S}$. We get a system whose\nstate space is $\\mathcal{H}=\\oplus_\\sigma \\mathcal{H}_{\\sigma}$. If the system is in\na state $u\\in\\mathcal{H}_{\\sigma}$ then it consists of $|\\sigma|$ particles\nof masses $m_a$. Note that $\\min\\mathfrak{S}$ is the partition\nconsisting of only one cluster $\\{1,\\dots,N\\}$ with mass\n$M=m_1+\\dots+m_N$. Since there are no external fields and we decided\nto eliminate the motion of the center of mass, this system must be\nthe vacuum. And its state space is indeed\n$\\mathcal{H}_{\\min\\mathfrak{S}}=\\mathbb{C}$. The algebra $\\mathscr{C}$ in this case\npredicts usual inter-cluster interactions associated, for examples,\nto potentials defined on $X^\\sigma=X\/X_\\sigma$, but also\ninteractions which force the system to make a transition from a\n``phase'' $\\sigma$ to a ``phase'' $\\tau$. In other terms, the system\nof $|\\sigma|$ particles with masses $(m_a)_{a\\in\\sigma}$ is\ntransformed into a system of $|\\tau|$ particles with masses\n$(m_b)_{b\\in\\tau}$. Thus the number of particles varies from $1$ to\n$N$ but the total mass is constant and equal to $M$.\n\n\n\n\n\\subsection{Natural morphisms and essential spectrum}\n\\label{ss:specan} \n\nWe return to the general case.  Sub-semilattices $\\mathcal{T}$ of $\\mathcal{S}$\ndefine many-body type subsystems (this is discussed in more detail\nin\\S\\ref{ss:subsys}). The spectral properties of the total many-body\nHamiltonian are described in terms of a special class of such\nsubsystems.\n\n\nEach $X\\in\\mathcal{S}$ determines a new many-body system $\\mathcal{S}_{\\geq\n  X}=\\{Y\\in\\mathcal{S}\\mid Y\\supset X\\}$ whose state space is $\\mathcal{H}_{\\geq\n  X}$. Let $\\mathscr{C}_{\\ge X}$ be the corresponding Hamiltonian algebra\n$\\mathscr{C}_{\\mathcal{S}_{\\geq X}}$. It is easy to see that $\\mathscr{C}_{\\geq\n  X}=\\sum^\\mathrm{c}_{Y\\supset X}\\mathscr{C}(Y)$.  Thus $\\mathscr{C}_{\\geq X}$ is a\n$C^*$-subalgebra of $\\mathscr{C}$ which lives and is non-degenerate on the\nsubspace $\\mathcal{H}_{\\geq X}$ of $\\mathcal{H}$. We mention one fact:\nif $\\Pi_{\\ge X}$ is the orthogonal projection $\\mathcal{H}\\to\\mathcal{H}_{\\ge X}$,\nthen $\\Pi_{\\ge X}\\mathscr{C}\\Pi_{\\ge X}$ is an $\\mathcal{S}$-graded\n$C^*$-subalgebra of $\\mathscr{C}_\\mathcal{S}$ and we have $\\mathscr{C}_{\\geq\n  X}\\subset\\Pi_{\\ge X}\\mathscr{C}\\Pi_{\\ge X}$ strictly in general.\n\n\nThen the general theory of graded\n$C^*$-algebras implies that \\emph{there is a unique linear\n  continuous projection $\\mathscr{P}_{\\geq X}:\\mathscr{C}\\to\\mathscr{C}_{\\geq X}$ such that\n  $\\mathscr{P}_{\\geq X}(T)=0$ if $T\\in\\mathscr{C}(Y)$ with $Y\\not\\supset X$ and this\n  projection is a morphism}. These are the \\emph{natural morphisms}\n  of the graded algebra $\\mathscr{C}$.\n\n\nThis extends to unbounded operators as follows: if $H$ is a\nself-adjoint operator on $\\mathcal{H}$ strictly affiliated to $\\mathscr{C}$ then\nthere is a unique self-adjoint operator $H_{\\ge X}\\equiv\\mathscr{P}_{\\ge\n  X}(H)$ on $\\mathcal{H}_{\\ge X}$ such that $\\mathscr{P}_{\\ge\n  X}(\\varphi(H))=\\varphi(H_{\\ge X})$ for all $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$.\nIf $H$ is only affiliated to $\\mathscr{C}$ then $H_{\\ge X}$ could be not\ndensely defined.\n\n\n\nAssume that the semilattice $\\mathcal{S}$ has a smallest element\n$\\min\\mathcal{S}$. Then $X\\in\\mathcal{S}$ is called \\emph{atom} if the only element\nof $\\mathcal{S}$ strictly included in $X$ is $\\min\\mathcal{S}$.  Let $\\mathcal{P}(\\mathcal{S})$ be\nthe set of atoms of $\\mathcal{S}$. We say that $\\mathcal{S}$ is \\emph{atomic} if\neach of its elements distinct from $\\min\\mathcal{S}$ contains an atom.  The\nfollowing HVZ type theorem is an immediate consequence Theorem\n\\ref{th:ga}. The symbol $\\overline{\\cup}$ means ``closure of\nunion''.\n\n\\begin{theorem}\\label{th:imp4}\nIf $H$ is a self-adjoint operator on $\\mathcal{H}$ strictly affiliated to\n$\\mathscr{C}$ then for each $X\\in\\mathcal{S}$ there is a unique self-adjoint\noperator $H_{\\geq X}\\equiv\\mathscr{P}_{\\geq X}(H)$ on $\\mathcal{H}_{\\geq X}$ such\nthat $\\mathscr{P}_{\\geq X} (\\varphi(H))=\\varphi(H_{\\geq X})$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$. The operator $H_{\\geq X}$ is strictly\naffiliated to $\\mathscr{C}_{\\geq X}$.  If $O\\in\\mathcal{S}$ and $\\mathcal{S}$ is atomic then\n\\begin{equation}\\label{eq:esi}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{X\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq X}).\n\\end{equation} \n\\end{theorem}\n\nThe theorem remains valid for operators which are only affiliated to\n$\\mathscr{C}$ but then we must allow them to be non-densely defined.\n\n\n\n\\subsection{Subsystems and subhamiltonians}\n\\label{ss:subsys}\n\n\n\nIf $\\mathcal{T}$ is an arbitrary subset of $\\mathcal{S}$ then the Hilbert space\n$\\mathcal{H}_\\mathcal{T}=\\oplus_{X\\in\\mathcal{T}}\\mathcal{H}_X$ is well defined and naturally\nembedded as a closed subspace of $\\mathcal{H}_\\mathcal{S}$. Let $\\Pi_\\mathcal{T}$ be the\northogonal projection of $\\mathcal{H}$ onto $\\mathcal{H}_\\mathcal{T}$. Note that the\ndefinition of $\\mathscr{T}_\\mathcal{S}$ makes sense for any set $\\mathcal{S}$ of finite\ndimensional subspaces , in particular we may replace $\\mathcal{S}$ by\n$\\mathcal{T}$. Then we have:\n\\begin{align}\n& \\mathscr{T}_\\mathcal{T} =\\Pi_\\mathcal{T}\\mathscr{T}_\\mathcal{S}\\Pi_\\mathcal{T}, \n\\quad\n\\mathcal{C}_\\mathcal{S}^\\mathcal{T} := \\Pi_\\mathcal{T}\\mathcal{C}_\\mathcal{S}\\Pi_\\mathcal{T}= \\oplus_{X\\in\\mathcal{T}}\\mathcal{C}_X(\\mathcal{S}), \n\\label{eq:subs1}\\\\\n& \\mathscr{C}_\\mathcal{S}^\\mathcal{T}:= \\Pi_\\mathcal{T}\\mathscr{C}_\\mathcal{S}\\Pi_\\mathcal{T}\n=\\Pi_\\mathcal{T}\\mathscr{T}_\\mathcal{S}\\cdot\\mathcal{C}_\\mathcal{S}\\Pi_\\mathcal{T}=\n\\mathscr{T}_\\mathcal{T}\\cdot\\mathcal{C}_\\mathcal{S}^\\mathcal{T} \\label{eq:subs2}\n\\end{align}\nFrom this we easily get that $\\mathscr{C}_\\mathcal{S}^\\mathcal{T}=\n{\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY}$ is an $\\mathcal{S}$-graded\n$C^*$-subalgebra of $\\mathscr{C}$ supported by the ideal generated by $\\mathcal{T}$\nin $\\mathcal{S}$ (an ideal is a subset $\\mathcal{J}$ of $\\mathcal{S}$ such that $X\\subset\nY\\in\\mathcal{J}\\Rightarrow X\\in\\mathcal{J}$).  The operators affiliated to\n$\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$ are affiliated to $\\mathscr{C}_\\mathcal{S}$, so are many-body\nHamiltonians in our sense. \n\nThe case when $\\mathcal{T}$ is a sub-semilattice of $\\mathcal{S}$ is interesting.\nIndeed, then $\\mathcal{T}$ defines a many-body system whose Hamiltonian\nalgebra $\\mathscr{C}_\\mathcal{T}$ is a $\\mathcal{T}$-graded $C^*$-algebra of operators on\nthe Hilbert space $\\mathcal{H}_\\mathcal{T}$. We emphasize that \\emph{this algebra\n  does not coincide with $\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$}.  We always have\n$\\mathscr{C}_\\mathcal{T}\\subset\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$ but \\emph{the inclusion is strict unless\n  $\\mathcal{T}$ is an ideal of $\\mathcal{S}$}.  This is clear because\n$\\mathcal{C}_\\mathcal{T}\\subset\\mathcal{C}_\\mathcal{S}^\\mathcal{T}$ strictly in general.\n\nThe simplest sub-semilattices are the chains (totally ordered\nsubsets). Then $\\mathcal{H}_\\mathcal{T}$ has a structure analogous to a Fock\nspace. Nonrelativistic Hamiltonians affiliated to such\n$C^*$-algebras $\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$ have been studied before in \\cite{SSZ}.\nIf we take $\\mathcal{T}=\\{X\\}$ for an arbitrary $X\\in\\mathcal{S}$ then the\nassociated subsystem has $\\mathcal{H}_X$ as state space and its Hamiltonian\nalgebra is just $\\mathscr{C}_X=\\mathcal{C}_X(\\mathcal{S})\\cdot\\mathscr{T}_X$, the Hamiltonian\nalgebra of the (generalized) $N$-body system determined by the\nsemi-lattice $\\mathcal{S}_X$. We refer to \\S\\ref{ss:T} and to the Example\n\\ref{ex:fried} for other simple but instructive examples of\nsubsystems.\n\n\n\n\\subsection{Intrinsic descriptions}\n\\label{ss:id}\n\nWe give two explicit descriptions of $\\mathscr{C}(Z)$ as an algebra of\noperators on $\\mathcal{H}_{\\geq Z}$. Since $\\mathscr{C}$ is the closure of the sum\nof the algebras $\\mathscr{C}(Z)$, these descriptions allow one to check\nrather easily whether a self-adjoint operator is affiliated to $\\mathscr{C}$\nor not. Both theorems are consequences of more general results in\nSection \\ref{s:id}.\n\nFor any vector $a\\in\\mathcal{X}$ and any finite dimensional subspace $X$ of\n$\\mathcal{X}$ we define two unitary operators in $\\mathcal{H}_X$ by\n$(U_af)(x)=f(x+a_X)$ and $(V_a f)(x)=\\mathrm{e}^{i\\braket{a}{x}} f(x)$\nwhere $a_X$ is the orthogonal projection of $x$ on $X$. Then\n$\\{U_a\\}_{a\\in\\mathcal{X}}$ and $\\{V_a\\}_{a\\in\\mathcal{X}}$ are strongly continuous\nrepresentations of (the additive group) $\\mathcal{X}$ on $\\mathcal{H}_X$ such that\n$U_a=1 \\Leftrightarrow V_a=1 \\Leftrightarrow a\\perp X$. The direct\nsum over $X\\in\\mathcal{S}$ of these representations give representations of\n$\\mathcal{X}$ on $\\mathcal{H}$ for which we use the same notations.\n\n\n\\begin{theorem}\\label{th:CZU}\n  $\\mathscr{C}(Z)$ is the set of $T\\in\\mathscr{L}_{\\geq Z}$ such that\n\\begin{compactenum}\n\\item[{\\rm(i)}] $U_a^*T U_a=T$ for all $a\\in Z$ and $\\|T(V_a-1)\\|\\to\n  0$ if $a\\to 0$ in $Z^\\perp$ ,\n\\item[{\\rm(ii)}] $\\|T(U_a-1)\\|\\to 0$ and $\\|V^*_a T V_a-T\\|\\to 0$ if\n  $a\\to 0$ in $\\mathcal{X}$.\n\\end{compactenum}\n\\end{theorem}\n\n\nLet $\\mathcal{S}\/Z$ be the set of subspaces of $\\mathcal{X}\/Z=Z^\\perp$ of the form\n$X\/Z$ with $X\\in\\mathcal{S}, X\\supset Z$. Clearly $\\mathcal{S}\/Z$ is a semilattice\nof finite dimensional subspaces of $\\mathcal{X}\/Z$ so the Hilbert space\n$\\mathcal{H}_{\\mathcal{S}\/Z}$ and the corresponding algebra of compact operators\n$\\mathscr{K}_{\\mathcal{S}\/Z}$ are well defined.  If $X\\supset Z$ then $X=Z\\oplus\nX\/Z$ so we have a canonical factorization\n$\\mathcal{H}_X=\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}$. Thus $\\mathcal{H}_{\\geq\n  Z}=\\mathcal{H}_Z\\otimes\\mathcal{H}_{\\mathcal{S}\/Z}$.\n\n\\begin{theorem}\\label{th:CZ}\n$\\mathscr{C}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{\\mathcal{S}\/Z}$ relatively to the factorization\n$\\mathcal{H}_{\\geq Z}=\\mathcal{H}_Z\\otimes\\mathcal{H}_{\\mathcal{S}\/Z}$.\n\\end{theorem}\n\n\\subsection{Factorization properties}\n\\label{ss:fact}\n\nFor $Z\\subset X\\cap Y$ we have $X=Z\\oplus (X\/Z)$ and $Y=Y\\oplus\n(Y\/Z)$ hence we have canonical factorizations\n\\begin{equation}\\label{eq:xyzint}\n\\mathcal{H}_X=\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}\n\\quad\\text{and}\\quad \n\\mathcal{H}_Y=\\mathcal{H}_Z\\otimes\\mathcal{H}_{Y\/Z}.\n\\end{equation}\nRelatively to these factorizations, we get from Theorem \\ref{th:CZ}:\n\\begin{equation}\\label{eq:CZ}\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{X\/Z,Y\/Z} \\cong \\cc_{\\mathrm{o}}(Z^*;\\mathscr{K}_{X\/Z,Y\/Z}).\n\\end{equation}\nThe tensor product (and those below) is in the category of Hilbert\nmodules, cf.  \\S\\ref{ss:ha}. We have written $Z^*$ above in spite of\nthe canonical Euclidean isomorphism $Z^*\\cong Z$ in order to stress\nthat we consider functions of momentum not of position. For any\n$X,Y$ we set\n\\begin{equation}\\label{eq:xony}\nX\/Y=X\/(X\\cap Y)= X\\ominus (X\\cap Y)\n\\end{equation}\nand so we have \n\\begin{equation}\\label{eq:xyonz}\nX\/Z=X\/(X\\cap Y)\\oplus (X\\cap Y)\/Z = X\/Y\\oplus (X\\cap Y)\/Z \n\\end{equation}\nand similarly for $Y\/Z$. Then from \\eqref{eq:CZ} and\n\\eqref{eq:comtens} we get the finer factorization:\n\\begin{equation}\\label{eq:xyzeptens}\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{(X\\cap Y)\/Z}\n\\otimes \\mathscr{K}_{X\/Y,Y\/X}. \n\\end{equation}\nIn particular, we get\n\\begin{equation}\\label{eq:xyzeint}\n\\mathscr{C}_{XY}=\\mathscr{C}_{X\\cap Y}\\otimes \\mathscr{K}_{X\/Y,Y\/X}\n\\end{equation}\nrelatively to the tensor factorizations \n\\begin{equation}\\label{eq:xyeht}\n\\mathcal{H}_X=\\mathcal{H}_{X\\cap Y}\\otimes\\mathcal{H}_{X\/Y} \\quad\\text{and}\\quad \n\\mathcal{H}_Y=\\mathcal{H}_{X\\cap Y}\\otimes\\mathcal{H}_{Y\/X}.\n\\end{equation}\nSince $\\mathscr{K}_{X\/Y,O}\\cong\\mathcal{H}_{X\/Y}$\nin the special case $Z\\subset Y\\subset X$ we have \n\\begin{equation}\\label{eq:xby}\n\\mathscr{C}_{XY}=\\mathscr{C}_{Y}\\otimes\\mathcal{H}_{X\/Y}\n\\quad\\text{and}\\quad\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{Y\/Z} \\otimes \\mathcal{H}_{X\/Y}.\n\\end{equation}\n\n\n\\begin{example}\\label{ex:fried}\n  These factorizations give us the possibility of expressing quite\n  explicitly the Hamiltonian algebra of some subsystems. We refer to\n  \\S\\ref{ss:T} for more general situations and consider here\n  sub-semilattices of the form $\\mathcal{T}=\\{X,Y\\}$ with $X\\supset Y$.\n  This is a toy model, an $N$-body system coupled to one of its\n  subsystems, and can be nicely formulated in a purely abstract\n  setting, cf. Proposition \\ref{pr:toymodel}.  We have\n  $\\mathcal{H}_\\mathcal{T}=\\mathcal{H}_X\\oplus\\mathcal{H}_Y$ with $\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}$.\n  From \\eqref{eq:xby} we have\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}=\n\\begin{pmatrix}\n\\mathscr{C}_X & \\mathscr{C}_{Y}\\otimes\\mathcal{H}_{X\/Y}\\\\\n\\mathscr{C}_{Y}\\otimes\\mathcal{H}^*_{X\/Y} & \\mathscr{C}_Y\n\\end{pmatrix}.\n\\]\nwhere $\\mathcal{H}^*_{X\/Y}$ has a natural meaning (see\n\\S\\ref{ss:affil}). The grading is defined for $Z\\in\\mathcal{S}_X$\nby\n\\begin{compactenum}\n\\item\nIf  $Z\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(Z)=\n\\begin{pmatrix}\n\\mathscr{C}_X(Z) & \\mathscr{C}_{Y}(Z)\\otimes\\mathcal{H}_{X\/Y}\\\\\n\\mathscr{C}_{Y}(Z)\\otimes\\mathcal{H}^*_{X\/Y} & \\mathscr{C}_Y(Z)\n\\end{pmatrix}.\n\\]\n\\item   \\label{p:i2ex}\nIf  $Z\\not\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(Z)=\n\\begin{pmatrix}\n\\mathscr{C}_X(Z) & 0\\\\\n0 & 0\n\\end{pmatrix}.\n\\]\n\\end{compactenum}\nIf $\\mathcal{T}=\\{X,O\\}$ we get a version of the Friedrichs model: an\n$N$-body system coupled to the vacuum. The case when $\\mathcal{T}$ is an\narbitrary chain (a totally ordered subset of $\\mathcal{S}$) is very similar.\nThe case $\\mathcal{T}=\\{X,Y\\}$ with not comparable $X,Y$ is more complicated\nand is treated in \\S\\ref{ss:T} in a more general setting.\n\\end{example}\n\n\n\n\\subsection{Examples of many-body Hamiltonians}\n\\label{ss:ex}\n\n\nHere we use Theorems \\ref{th:CZ} and \\ref{th:CZU} to construct\nself-adjoint operators strictly affiliated to $\\mathscr{C}$. For simplicity,\n\\emph{in this and the next subsections $\\mathcal{S}$ is assumed finite}. If\n$\\mathcal{S}$ is infinite then an assumption of the same nature as the\nnon-zero mass condition in quantum field theory models is needed to\nensure that the kinetic energy operator $K$ is affiliated to $\\mathscr{C}$.\n\nThe Hamiltonians will be of the form $H=K+I$ where the self-adjoint\noperator $K$ is the kinetic energy and $I$ is an\ninteraction term bounded in form sense by $K$. More precisely, $I$\nis a symmetric sesquilinear form on the domain of $|K|^{1\/2}$ which\nis continuous, i.e. satisfies\n\\begin{equation}\\label{eq:formbound}\n\\pm I\\leq \\mu|K+ia| \\hspace{2mm}\\text{for some real numbers } \\mu, a.\n\\end{equation}\n$H$ and $K$ are matrices of operators, e.g. $H=(H_{XY})_{X,Y\\in\\mathcal{S}}$\nwhere $H_{XY}$ is defined on a subspace of $\\mathcal{H}_Y$ and has values in\n$\\mathcal{H}_X$ and the relation $H_{XY}^*=H_{YX}$ holds at least\nformally. By construction $K$ is given by a diagonal matrix, so\n$K_{XY}=0$ if $X\\neq Y$, and we set $K_X=K_{XX}$. The interaction\nwill be a matrix of sesquilinear forms. Then $H_{XX}=K_{X}+I_{XX}$\nwill be an $N$-body type Hamiltonian, i.e. a self-adjoint operator\naffiliated to $\\mathscr{C}_X$, cf. Remark \\ref{re:nbody}. The non-diagonal\nelements $H_{XY}=I_{XY}$ define the interaction between the systems\n$X$ and $Y$.  We give now a rigorous construction of such\nHamiltonians.\n\n\n{\\bf(a)} For each $X$ we choose a kinetic energy operator\n$K_X=h_X(P)$ for the system having $X$ as configuration space. The\nfunction $h_X:X\\to\\mathbb{R}$ is continuous and such that\n$h_X(x)\\to\\infty$ if $x\\to\\infty$.  We stress that there are no\nrelations between the kinetic energies of the systems corresponding\nto different $X$. Denote $\\mathcal{G}^2_X$ the domain of $K_X$ equipped with\nthe graph norm and let $\\mathcal{G}^s_X$ ($s\\in\\mathbb{R}$) be the scale of\nHilbert spaces associated to it, e.g.  $\\mathcal{G}^0_X=\\mathcal{H}_X$,\n$\\mathcal{G}^1_X=D(K_X^{1\/2})$ is the form domain of $K_X$, and $\\mathcal{G}^{-1}_X$\nits adjoint space.\n\n\n{\\bf(b)} The total kinetic energy of the system is by definition\n$K=\\oplus_X K_X$. We call this a \\emph{standard kinetic energy\n  operator}.  Then the spaces $\\mathcal{G}^s$ of the scale determined by the\ndomain $\\mathcal{G}^2$ of $K$ can be identified with direct sums\n$\\mathcal{G}^s\\simeq\\oplus_X\\mathcal{G}^s_X$. In particular this holds for the form\ndomain $\\mathcal{G}^1=D(K^{1\/2})$ and for its adjoint space $\\mathcal{G}^{-1}$.\nNote that we may also introduce the operators $K_{\\geq\n  X}=\\oplus_{Y\\supset X} K_Y$ and the associated spaces $\\mathcal{G}^s_{\\ge\n  X}$. If $s>0$ we have $\\mathcal{G}^s_{\\ge X}=\\mathcal{G}^s\\cap\\mathcal{H}_{\\ge X}$.\n\n{\\bf(c)} The simplest type of interactions that we may consider are\ngiven by symmetric elements $I$ of the multiplier algebra of\n$\\mathscr{C}$. Then $H=K+I$ is strictly affiliated to $\\mathscr{C}$ and $\\mathscr{P}_{\\geq\n  X}(H)=K_{\\geq X}+\\mathscr{P}_{\\geq X} (I)$ where $\\mathscr{P}_{\\geq X}$ is\nextended to the multiplier algebras as explained in\n\\cite[p. 18]{La}.\n\n{\\bf(d)} In order to cover singular interactions (form bounded but\nnot necessarily operator bounded by $K$) we assume that the\nfunctions $h_X$ are equivalent to regular weights. This is a quite\nweak assumption, cf. page \\pageref{p:regw}.  For example, it\nsuffices that $c'|x|^{\\alpha}\\leq h_X(x)\\leq c''|x|^{\\alpha}$ for\nlarge $x$ where $c',c'',\\alpha>0$ are numbers depending on $X$.\nThen $U_a,V_a$ induce continuous operators in each of the spaces\n$\\mathcal{G}^s_X$, $\\mathcal{G}^s$, $\\mathcal{G}^s_{\\ge X}$. \n\n\n{\\bf(e)} The interaction will be of the form $I=\\sum_{Z\\in\\mathcal{S}} I(Z)$\nwhere the $I(Z)$ are continuous symmetric sesquilinear forms on\n$\\mathcal{G}^1$ such that $ I(Z) \\geq -\\mu_Z K -\\nu$ for some positive\nnumbers $\\mu_Z$ and $\\nu$ with $\\sum_Z\\mu_Z<1$. Then the form sum\n$K+I$ defines a self-adjoint operator $H$ on $\\mathcal{H}$.\n\n{\\bf(f)} We identify $I(Z)$ with a symmetric operator\n$\\mathcal{G}^1\\to\\mathcal{G}^{-1}$ and we assume that $I(Z)$ is supported by the\nsubspace $\\mathcal{H}_{\\ge Z}$.  In other terms, $I(Z)$ is the sesquilinear\nform on $\\mathcal{G}^1$ associated to an operator $I(Z):\\mathcal{G}^1_{\\ge\n  Z}\\to\\mathcal{G}^{-1}_{\\ge Z}$.  Moreover, we assume that this last\noperator satisfies\n\\begin{equation}\\label{eq:iaff}\nU_a I(Z)=I(Z) U_a \\text{ if } a\\in Z, \\ \nI(Z)(V_a-1)\\to 0 \\text{ if } a\\to 0 \\text{ in } Z^\\perp, \\ \nV^*_a I(Z) V_a\\to I(Z) \\text{ if } a\\to 0\n\\end{equation}\nwhere the limits hold in norm in $L(\\mathcal{G}^2_{\\ge Z},\\mathcal{G}^{-1}_{\\ge Z})$.\n\nNote that the first part of condition {\\bf(f)}, saying that $I(Z)$\nis supported by $\\mathcal{H}_{\\ge Z}$, is equivalent to an estimate of the\nform $\\pm I(Z)\\le\\mu K_{\\geq Z}+ \\nu\\Pi_{\\geq Z}$ for some positive\nnumbers $\\mu,\\nu$. See also Remark \\ref{re:iaff}.\n\n\\begin{theorem}\\label{th:iaff}\n  The Hamiltonian $H$ is a self-adjoint operator strictly affiliated\n  to $\\mathscr{C}$, we have   \n  $H_{\\geq X}=K_{\\geq X}+\\sum_{Z\\geq X} I(Z)$, \n  and $\\mathrm{Sp_{ess}}(H)=\\textstyle{\\bigcup}_{X\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq X})$.\n\\end{theorem}\n\n\\begin{remark}\\label{re:pauli}\n  We required the $h_X$ to be bounded from below only for the\n  simplicity of the statements. Moreover, a simple extension of the\n  formalism allows one to treat particles with arbitrary\n  spin. Indeed, if $E$ is a complex Hilbert then Theorem \\ref{th:CG}\n  remains true if $\\mathscr{C}$ is replaced by $\\mathscr{C}^E=\\mathscr{C}\\otimes K(E)$ and\n  the $\\mathscr{C}(Z)$ by $\\mathscr{C}(Z)\\otimes K(E)$. If $E$ is the spin space\n  then it is finite dimensional and one obtains $\\mathscr{C}^E$ exactly as\n  above by replacing the $\\mathcal{H}(X)$ by $\\mathcal{H}(X)\\otimes\n  E=L^2(X;E)$. Then one may consider instead of scalar kinetic\n  energy functions $h$ self-adjoint operator valued functions\n  $h:X^*\\to L(E)$. For example, we may take as one particle kinetic\n  energy operators the Pauli or Dirac Hamiltonians.\n\\end{remark}\n\n\n\\begin{remark}\\label{re:iaff}\nWe give here a second, more explicit version of condition\n{\\bf(f)}. Since \n$I(Z)$ is a continuous symmetric operator $\\mathcal{G}^1\\to\\mathcal{G}^{-1}$ we may\nrepresent it as a matrix $I(Z)=(I_{XY}(Z))_{X,Y\\in\\mathcal{S}}$ of\ncontinuous operators $I_{XY}(Z):\\mathcal{G}^1_Y\\to\\mathcal{G}^{-1}_X$ with\n$I_{XY}(Z)^*=I_{YX}(Z)$. We take $I_{XY}(Z)=0$ if $Z\\not\\subset\nX\\cap Y$ and if $Z\\subset X\\cap Y$ we assume\n$V^*_a I_{XY}(Z) V_a\\to I_{XY}(Z) \\text{ if } a\\to 0 \\text{ in }\nX+Y$ and \n\\begin{equation}\\label{eq:iiaff}\nU_a I_{XY}(Z)=I_{XY}(Z) U_a \\text{ if }a\\in Z,  \\quad\nI_{XY}(Z)(V_a-1)\\to 0 \\text{ if } a\\to 0 \\text{ in } Y\/Z.\n\\end{equation}\nThe limits should hold in norm in $L(\\mathcal{G}^2_Y,\\mathcal{G}^{-1}_X)$.\n\\end{remark}\n\nThe operators $I_{XY}(Z)$satisfying \\eqref{eq:iiaff} are described\nin more detail in Proposition \\ref{pr:zxy}. In the next example we\nconsider the simplest situation which is useful in the\nnonrelativistic case.\n\nIf $E$ is an Euclidean space and $s$ is a real number let $\\mathcal{H}^s_E$\nbe the Sobolev space defined by the norm\n\\[\n\\|u\\|_{\\mathcal{H}^s} = \\|(1+\\Delta_E)^{s\/2}u\\|\n\\] \nwhere $\\Delta_E$ is the (positive) Laplacian associated to the\nEuclidean space $E$. The space $\\mathcal{H}^s_E$ is equipped with two\ncontinuous representations of $E$, a unitary one induced by\n$\\{U_x\\}_{x\\in E}$ and a non-unitary one induced by $\\{V_x\\}_{x\\in\n  E}$.  If $E=O:=\\{0\\}$ we define $\\mathcal{H}_E^s=\\mathbb{C}$.\n\n\\begin{definition}\\label{df:small}\n  If $E,F$ are Euclidean spaces and $T:\\mathcal{H}^s_E\\to\\mathcal{H}^t_F$ is a\n  linear map, we say that \\emph{$T$ is small at infinity} if there\n  is $\\varepsilon>0$ such that when viewed as a map\n  $\\mathcal{H}^{s+\\varepsilon}_E\\to\\mathcal{H}^{t}_F$ the operator $T$ is compact.\n\\end{definition}\nBy the closed graph theorem $T$ is continuous and the compactness\nproperty holds for all $\\varepsilon>0$.  If $E=O$ or $F=O$ then we\nconsider that all the operators $T:\\mathcal{H}^s_E\\to\\mathcal{H}^t_F$ are small at\ninfinity.\n\n\\begin{example}\\label{ex:zxy}\n  Due to assumption {\\bf(d)} the form domains of $K_X$ and $K_Y$ are\n  Sobolev spaces, for example $\\mathcal{G}^1_X=\\mathcal{H}^s_X$ and\n  $\\mathcal{G}^1_Y=\\mathcal{H}^t_Y$. Let $I_{XY}^Z:\\mathcal{H}^t_{Y\/Z}\\to\\mathcal{H}^{-s}_{X\/Z}$ be\n  a linear small at infinity map. Then we may take\n  $I_{XY}(Z)=1_Z\\otimes I_{XY}^Z$ relatively to the tensor\n  factorizations \\eqref{eq:xyzint}.\n\\end{example}\n\n\nWe make now some comments to clarify the conditions {\\bf(a)} -\n{\\bf(f)}.  Assume, more generally, that $\\mathscr{C}$ is a $C^*$-algebra of\noperators on a Hilbert space $\\mathcal{H}$ and that $K$ is a self-adjoint\noperator on $\\mathcal{H}$ affiliated to $\\mathscr{C}$. Let $I$ be a continuous\nsymmetric sesquilinear form on the domain of $|K|^{1\/2}$. Then for\nsmall real $\\nu$ the form sum $K+\\nu I$ is a self-adjoint operator\n$H_\\nu$. If $H_\\nu$ is affiliated to $\\mathscr{C}$ for small $\\nu$, and\nsince the derivative with respect to $\\nu$ at zero of\n$(H_\\nu+i)^{-1}$ exists in norm, we get\n$(K+i)^{-1}I(K+i)^{-1}\\in\\mathscr{C}$. This clearly implies\n$\\jap{K}^{-2}I\\jap{K}^{-2}\\in\\mathscr{C}$.  Since\n$\\jap{K}^{-1\/2}I\\jap{K}^{-1\/2}$ is a bounded operator, the map\n$z\\mapsto\\jap{K}^{-z}I\\jap{K}^{-z}$ is holomorphic on $\\Re{z}>1\/2$\nhence we get\n\\begin{equation}\\label{eq:ent}\n\\jap{K}^{-\\alpha}I\\jap{K}^{-\\alpha}\\in\\mathscr{C} \\text{ if } \\alpha>1\/2. \n\\end{equation}\nReciprocally, if $K$ is strictly affiliated to $\\mathscr{C}$ (and $K$ as\ndefined at (b) has this property) then Theorem 2.8 from \\cite{DG3}\nsays that $\\jap{K}^{-1\/2}I\\jap{K}^{-\\alpha}\\in\\mathscr{C}$ suffices to\nensure that $H=K+I$ is strictly affiliated to $\\mathscr{C}$ under a quite\ngeneral condition needed to make this operator well defined (this is\nthe role of assumption (e) above).  Condition (f) is formulated such\nas to imply $\\jap{K}^{-1\/2}I\\jap{K}^{-1}\\in\\mathscr{C}$. To simplify the\nstatement we added condition (d) which implies that the spaces\n$\\mathcal{G}^s$ are stable under the group $V_a$. Formally\n\\[\n(\\jap{K}^{-1\/2}I\\jap{K}^{-1})_{XY}=\n\\jap{K_X}^{-1\/2}I_{XY}\\jap{K_Y}^{-1}.\n\\]\nSo this should belong to $\\mathscr{C}_{XY}=\\sum_{Z\\subset X\\cap\n  Y}\\mathscr{C}_{XY}(Z)$. Thus $I_{XY}$ must be a sum of terms $I_{XY}(Z)$\nwith\n\\[\n\\jap{K_X}^{-1\/2}I_{XY}(Z)\\jap{K_Y}^{-1}\\in\\mathscr{C}_{XY}(Z). \n\\]\nConditions (d) and (f) are formulated such as this to hold,\ncf. Remark \\ref{re:iaff} and Theorem \\ref{th:CZU}.\n\n\n\n\\subsection{Pauli-Fierz   Hamiltonians}\n\\label{ss:affil} \n\nThe next result is an a priori argument which supports our\ninterpretation of $\\mathscr{C}$ as Hamiltonian algebra of a many-body\nsystem: we show that $\\mathscr{C}$ is the $C^*$-algebra generated by a\nsimple class of Hamiltonians which have a natural quantum field\ntheoretic interpretation. For simplicity we state this only for\nfinite $\\mathcal{S}$.\n\nFor each couple $X,Y\\in\\mathcal{S}$ such that $X\\supset Y$ we have\n$\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}$.  Then we define\n$\\Phi_{XY}\\subset\\mathscr{L}_{XY}$ as the closed linear subspace consisting\nof ``creation operators'' associated to states from $\\mathcal{H}_{X\/Y}$,\ni.e. operators $a^*(\\theta):\\mathcal{H}_Y\\to\\mathcal{H}_X$ with $\\theta\\in\\mathcal{H}_{X\/Y}$\nwhich act as $u\\mapsto u\\otimes\\theta$.  We set\n$\\Phi_{YX}=\\Phi_{XY}^*\\subset\\mathscr{L}_{YX}$, this is the space of\n``annihilation operators'' $a(\\theta)=a^*(\\theta)^*$ defined by\n$\\mathcal{H}_{X\/Y}$.  This defines $\\Phi_{XY}$ when $X,Y$ are comparable,\ni.e. $X\\supset Y$ or $X\\subset Y$, which we abbreviate by $X\\sim\nY$. If $X\\not\\sim Y$ then we take $\\Phi_{XY}=0$. Note that\n$\\Phi_{XX}=\\mathbb{C} 1_X$, where $1_X$ is the identity operator on\n$\\mathcal{H}_X$.  We have\n\\begin{equation}\\label{eq:Phi}\n\\mathscr{T}_X\\cdot\\Phi_{XY}=\\Phi_{XY}\\cdot \\mathscr{T}_Y=\\mathscr{T}_{XY} \\quad\n\\text{if } X\\sim Y.\n\\end{equation}\n\nNow let $\\Phi=(\\Phi_{XY})_{X,Y\\in\\mathcal{S}}\\subset\\mathscr{L}$. This is\na closed self-adjoint linear space of bounded operators on $\\mathcal{H}$.  A\nsymmetric element $\\phi\\in\\Phi$ will be called \\emph{field\n  operator}. Giving such a $\\phi$ is equivalent to giving a family\n$\\theta=(\\theta_{XY})_{X\\supset Y}$ of elements\n$\\theta_{XY}\\in\\mathcal{H}_{X\/Y}$, the components of the operator\n$\\phi\\equiv\\phi(\\theta)$ being given by:\n$\\phi_{XY}=a^*(\\theta_{XY})$ if $X\\supset Y$,\n$\\phi_{XY}=a(\\theta_{YX})$ if $X\\subset Y$, and $\\phi_{XY}=0$ if\n$X\\not\\sim Y$. \n\nThe operators of the form $K+\\phi$, where $K$ is a standard kinetic\nenergy operator and $\\phi\\in\\Phi$ is a field operator, will be\ncalled \\emph{Pauli-Fierz Hamiltonians}.\n\n\\begin{theorem}\\label{th:motiv}\n  If $\\mathcal{S}$ is finite then $\\mathscr{C}$ is the $C^*$-algebra\n  generated by the Pauli-Fierz Hamiltonians.\n\\end{theorem}\n\nThus $\\mathscr{C}$ is generated by a class of Hamiltonians involving only\nelementary field type interactions. On the other hand, we have seen\nbefore that the class of Hamiltonians affiliated to $\\mathscr{C}$ is very\nlarge and covers \\mbox{$N$-body} systems interacting between\nthemselves with field type interactions. We emphasize that the\n$k$-body type interactions \\emph{inside} each of the $N$-body\nsubsystems are generated by pure field interactions.\n\n\n\\subsection{Nonrelativistic Hamiltonians and Mourre\n  estimate}\n\\label{ss:mouint}  \n\nWe prove the Mourre estimate only for nonrelativistic many-body systems. \nThere are serious difficulties when\nthe kinetic energy is not a quadratic form even in the much simpler\ncase of $N$-body Hamiltonians, but see \\cite{De1,Ger1,DG2} for some\npartial results which could be extended to our setting. Note that\nthe quantum field case is much easier from this point of view\nbecause of the special nature of the interactions\n\\cite{DeG2,Ger2,Geo}. \n\nLet $\\mathcal{S}$ be a finite semilattice of subspaces of $\\mathcal{X}$.  Recall\nthat for $X\\in\\mathcal{S}$ we denote $\\mathcal{S}\/X$ the set of subspaces $Y\/X=Y\\cap\nX^\\perp$ with $Y\\in\\mathcal{S}_{\\ge X}$. This is a finite semilattice of\nsubspaces of $\\mathcal{X}$ which contains $O$. Hence the Hilbert space\n$\\mathcal{H}_{\\mathcal{S}\/X}$ and the $C^*$-algebra $\\mathscr{C}_{\\mathcal{S}\/X}$ are well defined\nby our general rules and (cf. \\S\\ref{ss:fact}):\n\\begin{equation}\\label{eq:f}\n\\mathcal{H}_{\\geq X}=\\mathcal{H}_X\\otimes\\mathcal{H}_{\\mathcal{S}\/X}\\quad \\text{and} \\quad\n\\mathscr{C}_{\\geq X}=\\mathscr{T}_X\\otimes\\mathscr{C}_{\\mathcal{S}\/X} .\n\\end{equation}\nDenote $\\Delta_X$ the (positive) Laplacian associated to the Euclidean\nspace $X$ with the convention $\\Delta_O=0$. We have\n$\\Delta_X=h_X(P)$ with $h_X(x)=\\|x\\|^2$.  We set\n$\\Delta\\equiv\\Delta_\\mathcal{S}=\\oplus_X \\Delta_X$ and define $\\Delta_{\\geq X}$\nsimilarly. If $Y\\supset X$ then $\\Delta_Y=\\Delta_X\\otimes 1\n+1\\otimes\\Delta_{Y\/X}$ hence $\\Delta_{\\geq X}=\\Delta_X\\otimes 1\n+1\\otimes\\Delta_{\\mathcal{S}\/X}$.  The domain and form domain of the\noperator $\\Delta_\\mathcal{S}$ are given by $\\mathcal{H}_\\mathcal{S}^2$ and $\\mathcal{H}_\\mathcal{S}^1$ where\n$\\mathcal{H}_\\mathcal{S}^s\\equiv\\mathcal{H}^s=\\oplus_X \\mathcal{H}^s(X)$ for any real\n$s$.\n\n \nWe define nonrelativistic many-body Hamiltonian by extending to the\npresent setting \\cite[Def. 9.1]{ABG}. We consider only strictly\naffiliated operators to avoid working with not densely defined\noperators. Note that the general case of affiliated operators covers\ninteresting physical situations (hard-core interactions).\n\n\n\\begin{definition}\\label{df:NR}\n\\emph{A nonrelativistic many-body Hamiltonian of type $\\mathcal{S}$} is a\nbounded from below self-adjoint operator $H=H_\\mathcal{S}$ on $\\mathcal{H}=\\mathcal{H}_\\mathcal{S}$\nwhich is strictly affiliated to $\\mathscr{C}=\\mathscr{C}_\\mathcal{S}$ and has the following\nproperty: for each $X\\in\\mathcal{S}$ there is a bounded from below\nself-adjoint operator $H_{\\mathcal{S}\/X}$ on $\\mathcal{H}_{\\geq X}$\nsuch that \n\\begin{equation}\\label{eq:NR}\n\\mathscr{P}_{\\geq X}(H)\\equiv H_{\\geq X}=\\Delta_X\\otimes1+ 1\\otimes H_{\\mathcal{S}\/X}\n\\end{equation}\nrelatively to the tensor factorization $\\mathcal{H}_{\\geq\n  X}=\\mathcal{H}_X\\otimes\\mathcal{H}_{\\mathcal{S}\/X}$. \n\\end{definition}\n\nThen \\emph{each $H_{\\mathcal{S}\/X}$ is a nonrelativistic many-body\n  Hamiltonian of type $\\mathcal{S}\/X$}. Indeed, the argument from \\cite[p.\\\n415]{ABG} extends in a straightforward way to the present situation.\n\n\\begin{remark}\\label{re:maxs}\n  If $X$ is a maximal element in $\\mathcal{S}$ then $\\mathcal{S}\/X=\\{O\\}$ hence\n  $\\mathcal{H}_{\\mathcal{S}\/X}=\\mathcal{H}_O=\\mathbb{C}$ and $H_O$ will necessarily be a real\n  number.  Then we get $\\mathcal{H}_{\\geq X}=\\mathcal{H}_X$, $\\mathscr{C}_{\\geq X}=\\mathscr{T}_X$,\n  and $H_{\\ge X}=\\Delta_X +H_O$ on $\\mathcal{H}_X$.\n\\end{remark}\n\n\\begin{remark}\\label{re:mins}\n  Since $\\mathcal{S}$ is a finite semilattice, it has a least element\n  $\\min\\mathcal{S}$. If $\\mathcal{S}_o=\\mathcal{S}\/{\\min\\mathcal{S}}$, we get\n\\begin{equation}\\label{eq:min}\n\\mathcal{H}_{\\mathcal{S}}=\\mathcal{H}_{\\min\\mathcal{S}}\\otimes\\mathcal{H}_{\\mathcal{S}_o}, \\quad\n\\mathscr{C}_{\\mathcal{S}}=\\mathscr{T}_X\\otimes \\mathscr{C}_{\\mathcal{S}_o}, \\quad  \nH_{\\mathcal{S}}=\\Delta_{\\min\\mathcal{S}}\\otimes 1 +  1\\otimes H_{\\mathcal{S}_o}.\n\\end{equation}\n\\end{remark}\n\nNow we give an HVZ type description of the essential spectrum of a\nnonrelativistic many-body Hamiltonian. For a more detailed\nstatement, see the proof.\n\n\\begin{theorem}\\label{th:nrhvz}\n  Denote $\\tau_X=\\inf H_{\\mathcal{S}\/X}$ the bottom of the spectrum of\n  $H_{\\mathcal{S}\/X}$. Then\n\\begin{equation}\\label{eq:hvzunif}\n  \\mathrm{Sp_{ess}}(H)=[\\tau,\\infty[ \\hspace{2mm}\\text{with}\\hspace{2mm}\n  \\tau=\\min \\{ \\tau_X \\mid X \\text{ is minimal in } \\mathcal{S}\\setminus\n  \\{O\\} \\}. \n\\end{equation}\n\\end{theorem}\n\\proof\n  From \\eqref{eq:NR} we get\n\\begin{equation}\\label{eq:speint}\n\\mathrm{Sp}(H_{\\geq X})=[0,\\infty[\\ +\\, \\mathrm{Sp}(H_{\\mathcal{S}\/X})=\n[\\tau_X,\\infty[ \\quad\n\\text{if } X\\neq O.\n\\end{equation}\nIn particular, if $O\\notin\\mathcal{S}$ then by taking $X=\\min\\mathcal{S}$\nin \\eqref{eq:min} we get \n\\begin{equation}\\label{eq:mino}\n\\mathrm{Sp}(H)=\\mathrm{Sp_{ess}}(H)=[\\inf H_{\\mathcal{S}_o},\\infty[ .\n\\end{equation}\nIf $O\\in\\mathcal{S}$ then Theorem \\ref{th:imp4} implies \n\\begin{equation}\\label{eq:hvznrint}\n\\mathrm{Sp_{ess}}(H)=[\\tau,\\infty[ \\hspace{2mm}\\text{with}\\hspace{2mm}\n\\tau=\\min_{X\\in\\mathcal{P}(\\mathcal{S})} \\tau_X.\n\\end{equation}\nThe relation \\eqref{eq:hvzunif} expresses \\eqref{eq:mino} and\n\\eqref{eq:hvznrint} in a unified way. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nFor $X\\in\\mathcal{S}$ we consider the dilation group $W_\\tau=\\mathrm{e}^{i\\tau D}$\ndefined on $\\mathcal{H}_X$ by (set $n=\\dim X$):\n\\begin{equation}\\label{eq:fed}\n(W_\\tau u)(x)=\\mathrm{e}^{n\\tau\/4}u(\\mathrm{e}^{\\tau\/2}x), \n\\quad \n2iD=x\\cdot\\nabla_x+n\/2= \\nabla_x\\cdot x-n\/2. \n\\end{equation}\nLet $D_O=0$. We keep the same notation for the unitary operator\n$\\oplus_X W_\\tau$ on the direct sum $\\mathcal{H}=\\oplus_X\\mathcal{H}_X$ and we do\nnot indicate explicitly the dependence on $X$ or $\\mathcal{S}$ of $W_\\tau$\nand $D$ unless this is really needed. Note that $D$ has\nfactorization properties similar to that of the Laplacian,\ne.g. $D_{\\geq X}=D_X\\otimes 1 +1\\otimes D_{\\mathcal{S}\/X}$.\n\n\nWe refer to Subsection \\ref{ss:mest} for terminology related to the\nMourre estimate. We take $D$ as conjugate operator and we denote by\n$\\widehat\\rho_H(\\lambda)$ the best constant (which could be infinite)\nin the Mourre estimate at point $\\lambda$. The \\emph{threshold set}\n$\\tau(H)$ of $H$ with respect to $D$ is the set where\n$\\widehat\\rho_H(\\lambda)\\leq0$.  If $A$ is a real set then we define\n$N_A:\\mathbb{R}\\to[-\\infty,\\infty[$ by $N_A(\\lambda)=\\sup\\{ x\\in A \\mid\nx\\leq\\lambda\\}$ with the convention $\\sup\\emptyset=-\\infty$.  Denote\n$\\mathrm{ev}(T)$ the set of eigenvalues of an operator $T$.\n\n\\begin{theorem}\\label{th:thrintr}\n  Let $H=H_\\mathcal{S}$ be a nonrelativistic many-body Hamiltonian of type\n  $\\mathcal{S}$ and of class $C^1_\\mathrm{u}(D)$. Then $\\tau(H)$ is a closed\n  \\emph{countable} real set given by\n\\begin{equation}\\label{eq:thrintr}\n\\tau(H)=\\textstyle{\\bigcup}_{X\\neq O}\\mathrm{ev}(H_{\\mathcal{S}\/X}).\n\\end{equation}\nThe eigenvalues of $H$ which do not belong to $\\tau(H)$ are of\nfinite multiplicity and may accumulate only to points from\n$\\tau(H)$. We have\n$\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ for all real\n$\\lambda$.\n\\end{theorem}\n\nWe emphasize that if $O\\notin\\mathcal{S}$ the threshold set\n\\begin{equation}\\label{eq:thrinto}\n\\tau(H)=\\textstyle{\\bigcup}_{X\\in\\mathcal{S}} \\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\end{equation}\nis very rich although the spectrum of\n$H=\\Delta_{\\min\\mathcal{S}}\\otimes1+1\\otimes H_{\\mathcal{S}_o}$ is purely absolutely\ncontinuous.\n\n\\begin{remark}\\label{re:NM}\nWe thus see that there is no difference between nonrelativistic\n$N$-body and many-body Hamiltonians from the point of view of their\nchannel structure.  The formulas which give the essential spectrum\nand the threshold set relevant in the Mourre estimate are identical,\ncf. \\eqref{eq:hvznrint} and \\eqref{eq:thrintr}. This is due to\nthe fact that both Hamiltonian algebras are graded by the same\nsemilattice $\\mathcal{S}$.\n\\end{remark}\n\n\n\\subsection{Examples of nonrelativistic many-body Hamiltonians}\n\\label{ss:examples}\n\nLet $H=K+I$ with kinetic energy  $K=\\Delta$. Hence\n$\\mathcal{G}^1=\\mathcal{H}^1=\\oplus_X \\mathcal{H}^1_X$ and $\\mathcal{G}^{-1}=\\mathcal{H}^{-1}=\\oplus_X\n\\mathcal{H}^{-1}_X$ with the notations of \\S\\ref{ss:ex}.  The interaction\nterm is an operator $I:\\mathcal{H}^1\\to\\mathcal{H}^{-1}$ given by a sum\n$I=\\sum_{Z\\in\\mathcal{S}} I(Z)$ where each $I(Z)$ is defined with the help\nof the tensor factorization $\\mathcal{H}_{\\ge Z}=\\mathcal{H}_Z\\otimes\\mathcal{H}_{\\mathcal{S}\/Z}$.\n\n\\begin{proposition}\\label{pr:exnr}\n  Let $I^Z:\\mathcal{H}^1_{\\mathcal{S}\/Z}\\to\\mathcal{H}^{-1}_{\\mathcal{S}\/Z}$ be symmetric and small\n  at infinity and let $I(Z):=1_Z\\otimes I^Z$ which is naturally\n  defined as a symmetric operator $\\mathcal{H}^1\\to\\mathcal{H}^{-1}$. Assume that\n  $I(Z)\\geq -\\mu_Z\\Delta-\\nu$ for some numbers $\\mu_Z,\\nu\\ge0$\n  with $\\sum\\mu_Z<1$. Then $H=\\Delta+I$ defined in the quadratic\n  form sense is a nonrelativistic many-body Hamiltonian of type\n  $\\mathcal{S}$ and we have $H_{\\geq X}=\\Delta_{\\geq X}+\\sum_{Z\\supset X}\n  I(Z)$.\n\\end{proposition}\n\n\nThe first condition on $I^Z$ can be stated in terms of its\ncoefficients as follows: if $Z\\subset X\\cap Y$ then the operator\n$I_{XY}^Z:\\mathcal{H}^1_{Y\/Z}\\to\\mathcal{H}^{-1}_{X\/Z}$ is small at infinity and\nsuch that $(I_{XY}^Z)^*=I_{YX}^Z$.  On the other hand, note that if\nthe operators $I^Z:\\mathcal{H}^1_{\\mathcal{S}\/Z}\\to\\mathcal{H}^{-1}_{\\mathcal{S}\/Z}$ are compact\nthen they are small at infinity and for any $\\mu>0$ there is a\nnumber $\\nu$ such that $\\pm I(Z)\\leq \\mu\\Delta_\\mathcal{S}+\\nu$ for all\n$Z$. The more general smallness at infinity condition covers second\norder perturbations of $\\Delta_\\mathcal{S}$.\n\n\n\nIn the next proposition we give examples of nonrelativistic\noperators of class $C^1_\\mathrm{u}(D)$. The operator $H$ is constructed as\nin Proposition \\ref{pr:exnr} but we consider only interactions which\nare relatively bounded in \\emph{operator} sense with respect to the\nkinetic energy such as to force the domain of $H$ to be equal to the\ndomain of $\\Delta$, hence to $\\mathcal{H}^2=\\oplus_X\\mathcal{H}^2_X$.  Since\nthis space is stable under the action of the operators $W_\\tau$,\nwe shall get a simple condition for $H$ to be of class\n$C^1_\\mathrm{u}(D)$. \n\n\n\\begin{proposition}\\label{pr:nrm}\n  For each $Z\\in\\mathcal{S}$ assume that $I^Z:\\mathcal{H}^2_{\\mathcal{S}\/Z}\\to \\mathcal{H}_{\\mathcal{S}\/Z}$\n  is compact and symmetric as operator on $\\mathcal{H}_{\\mathcal{S}\/Z}$ and that\n  $[D,I^Z]: \\mathcal{H}^2_{\\mathcal{S}\/Z}\\to\\mathcal{H}^{-2}_{\\mathcal{S}\/Z}$ is compact.  Then the\n  conditions of Proposition \\ref{pr:exnr} are fulfilled and each\n  operator $I(Z):\\mathcal{H}^2\\to\\mathcal{H}$ is $\\Delta$-bounded with relative\n  bound zero.  The operator $H$ is self-adjoint on $\\mathcal{H}^2$ and of\n  class $C^1_\\mathrm{u}(D)$.\n\\end{proposition}\n\nSo for the coefficients $I^Z_{XY}$ we ask $I^Z_{XY}=0$ if\n$Z\\not\\subset X\\cap Y$ and if $Z\\subset X\\cap Y$ then\n$(I^{Z}_{XY})^*\\supset I^Z_{YX}$ and\n\\begin{equation}\\label{eq:3d}\nI^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{X\/Z} \\text{ and } \n[D,I^Z_{XY}]: \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\text{  are  compact\n  operators.} \n\\end{equation}\nThe expression $[D,I^Z_{XY}]=D_{X\/Z}I^Z_{XY}-I^Z_{XY}D_{Y\/Z}$ is not\nreally a commutator. Indeed, if we denote $E=(X\\cap Y)\/Z$, so\n$Y\/Z=E\\oplus(Y\/X)$ and $X\/Z=E\\oplus(X\/Y)$, then $\\mathcal{H}_{X\/Z}\n=\\mathcal{H}_E\\otimes\\mathcal{H}_{X\/Y}$ and $\\mathcal{H}_{Y\/Z} =\\mathcal{H}_E\\otimes\\mathcal{H}_{Y\/X}$.\nHence the relation $D_{X\/Z}=D_E\\otimes 1 + 1\\otimes D_{X\/Y}$ and a\nsimilar one for $Y\/Z$ give\n\\begin{equation*}\n[D,I^Z_{XY}] =[D_E,I^Z_{XY}] +D_{X\/Y}I^Z_{XY} -I^Z_{XY}D_{Y\/X}.\n\\end{equation*}\nThe first term above is a commutator and so is of a different nature\nthan the next two. Since $I^Z_{XY}D_{Y\/X}$ is a restriction of\n$(D_{Y\/X}I^Z_{YX})^*$ it is clear that the second part of condition\n\\eqref{eq:3d} follows from:\n\\begin{equation}\\label{eq:2d}\n[D_E,I^Z_{XY}] \\text{ and } D_{X\/Y}I^Z_{XY} \\text{ are compact\n  operators } \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\text{ for all } X,Y,Z.\n\\end{equation}\nWe consider some simple examples of operators $I_{XY}^Z$ to clarify\nthe difference with respect to the $N$-body situation (see\n\\S\\ref{ss:scs} for details and generalizations). If $E,F$ are\nEuclidean spaces we denote\n\\begin{equation}\\label{eq:ikef}\n\\mathscr{K}^2_{FE} =K(\\mathcal{H}^2_E,\\mathcal{H}_F) \\quad \\text{and} \\quad\n\\mathscr{K}^{2}_E=\\mathscr{K}^2_{E,E}=K(\\mathcal{H}^2_E,\\mathcal{H}_E). \n\\end{equation}\nDenote $X \\boxplus Y = X\/Y\\oplus Y\/X$ and embed $L^2(X \\boxplus\nY)\\subset \\mathscr{K}_{X\/Y,Y\/X}$ by identifying a Hilbert-Schmidt operator\nwith its kernel. Then\n\\begin{equation*}\nL^2(X \\boxplus Y;\\mathscr{K}^2_E) \\subset \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X}\n\\subset \\mathscr{K}^2_{X\/Z,Y\/Z}.\n\\end{equation*}\nThus $I_{XY}^Z \\in L^2(X \\boxplus Y;\\mathscr{K}^2_E)$ is a simple example of\noperator satisfying the first part of condition \\eqref{eq:3d}. Such\nan $I_{XY}^Z$ acts as follows: if $u\\in\\mathcal{H}^2_{Y\/Z}\\subset\nL^2(Y\/X;\\mathcal{H}^2_E)$ then\n\\[\nI_{XY}^Z u\\in \\mathcal{H}_{X\/Z} = L^2(X\/Y;\\mathcal{H}_E) \\quad\\text{is given by}\\quad\n(I_{XY}^Z u)(x')={\\textstyle\\int_{Y\/X}} I_{XY}^Z(x',y')u(y') \\text{d} y'.\n\\]\nNow we consider \\eqref{eq:2d}.\nSince $(x',y')\\mapsto[D_E,I^Z_{XY}(x',y')]$ is the kernel of the\noperator $[D_E,I^Z_{XY}]$, if\n\\[\n[D_E,I^Z_{XY}]\\in L^2(X \\boxplus Y;K(\\mathcal{H}^2_E,\\mathcal{H}^{-2}_E)\n\\]\nthen $[D_E,I^Z_{XY}]$ is a compact operator\n$\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}$. For the term $D_{X\/Y}I^Z_{XY}$ it\nsuffices to require the compactness of the operator\n\\[\nD_{X\/Y}I^Z_{XY} = 1_E\\otimes D_{X\/Y} \\cdot I^Z_{XY}: \\mathcal{H}^2_{Y\/Z}\\to\n\\mathcal{H}_E\\otimes\\mathcal{H}^{-2}_{X\/Y}.\n\\]\nFrom \\eqref{eq:fed} we see that this is a condition on the kernel\n$x'\\cdot\\nabla_{x'}I_{XY}^Z(x',y')$. For example, it suffices that\nthe operator $\\jap{Q_{X\/Y}}I^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{X\/Z}$ be\ncompact, which is a short range assumption. In summary:\n\n\\begin{example}\\label{ex:hschmidt}\n  For each $Z\\subset X\\cap Y$ let $I^Z_{XY} \\in L^2(X \\boxplus\n  Y;\\mathscr{K}^2_E)$ such that the adjoint of $I^Z_{XY}(x',y')$ is an\n  extension of $I^Z_{YX}(y',x')$. Assume that kernel\n  $[D_E,I^Z_{XY}(x',y')]$ belongs to $L^2(X \\boxplus\n  Y;K(\\mathcal{H}^2_E,\\mathcal{H}^{-2}_E)$ and that the kernel\n  $x'\\cdot\\nabla_{x'}I_{XY}^Z(x',y')$ defines a compact operator\n  $\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}$. Then \\eqref{eq:3d} is fulfilled.\n\\end{example}\n\n\\begin{example}\\label{ex:anc}\n  Here we consider the particular case $Y\\subset X$ to see the\n  structure of a generalized creation operator which appears in this\n  context. For each $Z\\subset Y$ let $I^Z_{XY}\\in\n  \\mathscr{K}^2_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}$, where the tensor product is a kind\n  of weak version of $L^2(X\/Y; \\mathscr{K}^2_{Y\/Z})$ discussed in\n  \\S\\ref{ss:ha}. Furthermore, assume that $[D_{Y\/Z},I^Z_{XY}]\\in\n  K(\\mathcal{H}^2(Y\/Z),\\mathcal{H}^{-2}_{Y\/Z})\\otimes\\mathcal{H}_{X\/Y}$ and\n  $D_{X\/Y}I^Z_{XY}\\in \\mathscr{K}^2_{Y\/Z}\\otimes\\mathcal{H}^{-2}_{X\/Y}$. Then\n  \\eqref{eq:3d} holds.\n\\end{example}\n\n\n\\subsection{Boundary values of the resolvent}\n\\label{ss:bvr}\n\nTheorem \\ref{th:thrintr} has important consequences in the spectral\ntheory of the operator $H$: we shall use it together with\n\\cite[Theorem 7.4.1]{ABG} to show that $H$ has no singular\ncontinuous spectrum and to prove the existence of the boundary\nvalues of its resolvent in the class of weighted $L^2$ spaces that\nwe define now. Let $\\mathcal{H}_{s,p}=\\oplus_X L^2_{s,p}(X)$ where the\n$L^2_{s,p}(X)$ are the Besov spaces associated to the position\nobservable on $X$ (these are obtained from the usual Besov spaces\nassociated to $L^2(X)$ by a Fourier transformation).  Note that\n$\\mathcal{H}_{s}=\\mathcal{H}_{s,2}$ is the Fourier transform of the Sobolev space\n$\\mathcal{H}^s$.  Let $\\mathbb{C}_+$ be the open upper half plane and\n$\\mathbb{C}^H_+=\\mathbb{C}_+\\cup(\\mathbb{R}\\setminus\\tau(H))$. If we replace the upper\nhalf plane by the lower one we similarly get the sets $\\mathbb{C}_-$ and\n$\\mathbb{C}^H_-$. We define two holomorphic maps $R_\\pm:\\mathbb{C}_\\pm\\to\nL(\\mathcal{H})$ by $R_\\pm(z)=(H-z)^{-1}$ and note that we have continuous\nembeddings\n\\[\nL(\\mathcal{H})\\subset L(\\mathcal{H}_{1\/2,1},\\mathcal{H}_{-1\/2,\\infty}) \\subset\nL(\\mathcal{H}_{s},\\mathcal{H}_{-s}) \\quad\\text{if } s>1\/2\n\\]\nso we may consider $R_\\pm$ as maps with values in\n$L(\\mathcal{H}_{1\/2,1},\\mathcal{H}_{-1\/2,\\infty})$.\n\n\\begin{theorem}\\label{th:c11}\nIf $H$ is of class $C^{1,1}(D)$ then its singular continuous\nspectrum is empty and the holomorphic maps $R_\\pm:\\mathbb{C}_\\pm\\to\nL(\\mathcal{H}_{1\/2,1},\\mathcal{H}_{-1\/2,\\infty})$ extend to weak$^*$ continuous\nfunctions $\\bar{R}_\\pm$ on $\\mathbb{C}^H_\\pm$.  The maps\n$\\bar{R}_\\pm:\\mathbb{C}^H_\\pm\\to L(\\mathcal{H}_{s},\\mathcal{H}_{-s})$ are norm continuous\nif $s>1\/2$.\n\\end{theorem}\n\nThis result is optimal both with regard to the regularity of the\nHamiltonian relatively to the conjugate operator $D$ and to the\nBesov spaces in which we establish the existence of the boundary\nvalues of the resolvent. The class $C^{1,1}(D)$ will be discussed\nand its optimality will be made precise in \\S\\ref{ss:scsc} but we\ngive some examples below.\n\nWe state first the simplest sufficient condition: \\emph{assume that\n  $H$ is as in Proposition \\ref{pr:exnr} and that its domain is\n  equal to $\\mathcal{H}^2$;if $\\,[D,[D,I^Z]]\\in\n  L(\\mathcal{H}^2_{\\mathcal{S}\/Z},\\mathcal{H}^{-2}_{\\mathcal{S}\/Z})$ for all $Z$ then $H$ is of\n  class $C^{1,1}(D)$}. This follows from Theorem 6.3.4 in\n\\cite{ABG}. The condition on $\\,[D,[D,I^Z]]$ can easily be written\nin terms of the coefficients $I^Z_{XY}$ by arguments similar to\nthose of \\S\\ref{ss:examples}.  Refinements allow the addition of\nlong range and short range interactions as in \\cite[\\S 9.4.2]{ABG}.\n\nLet $\\xi:\\mathbb{R}\\to\\mathbb{R}$ be of class $C^\\infty$ and such that\n$\\xi(\\lambda)=0$ if $\\lambda\\le 1$ and $\\xi(\\lambda)=1$ if\n$\\lambda\\ge 2$. For each Euclidean space $X$ and real $r\\ge 1$ we\ndenote $\\xi^r_X$ the operator of multiplication by the function\n$x\\mapsto\\xi(|x|\/r)$ on any Sobolev space over $X$. Then we define\n$\\xi^r_\\mathcal{S}=\\oplus_{X\\in\\mathcal{S}}\\xi^r_X$ considered as operator on\n$\\mathcal{H}^s_\\mathcal{S}$ for any real $s$.\n\n\\begin{definition}\\label{df:slr}\n  Let $T:\\mathcal{H}^2_\\mathcal{S}\\to\\mathcal{H}_\\mathcal{S}$ be a symmetric operator. We say that\n  $T$ is a \\emph{long range interaction} if $[D,T]\\in\n  L(\\mathcal{H}^2_\\mathcal{S},\\mathcal{H}^{-1}_\\mathcal{S})$ and $\\int_1^\\infty \\|\\xi^r_\\mathcal{S}\n  [D,T]\\|_{\\mathcal{H}^2_\\mathcal{S}\\to\\mathcal{H}^{-1}_\\mathcal{S}} \\text{d} r\/r <\\infty$.  We say that\n  $T$ is a \\emph{short range interaction} if $\\int_1^\\infty\n  \\|\\xi^r_\\mathcal{S} [D,T]\\|_{\\mathcal{H}^2_\\mathcal{S}\\to\\mathcal{H}_\\mathcal{S}} \\text{d} r <\\infty$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{th:BVR}\n  Assume that $H=\\Delta_\\mathcal{S} + \\sum_{Z\\in\\mathcal{S}} 1_Z\\otimes I^Z$ where\n  each $I^z:\\mathcal{H}^2_{\\mathcal{S}\/Z}\\to\\mathcal{H}_{\\mathcal{S}\/Z}$ is symmetric, compact, and\n  is the sum of a long range and a short range interaction. Then $H$\n  is a nonrelativistic many-body Hamiltonian of class $C^{1,1}(D)$,\n  hence the conclusions of Theorem \\ref{th:c11} are true.\n\\end{theorem}\n\nScattering channels may be defined in a natural way in the context\nof the theorem.  If the long range interactions are absent we expect\nthat asymptotic completeness holds.\n\n\n\n\n\n\\section{Graded Hilbert $C^*$-modules}\n\\label{s:grad}\n\\protect\\setcounter{equation}{0}\n\n\\subsection{Graded $\\boldsymbol{C^*}$-algebras}\n\\label{ss:grca}\n\nThe natural framework for the systems considered in this paper is\nthat of $C^*$-algebras graded by semilattices. We refer to\n\\cite{Ma2,Ma3} for a detailed study of this class of algebras.\n\nLet $\\mathcal{S}$ be a semilattice and $\\mathscr{A}$ a graded $C^*$-algebra.\nFollowing \\cite{Ma2} we say that $\\mathscr{B}\\subset\\mathscr{A}$ is a \\emph{graded\n  $C^*$-subalgebra} if $\\mathscr{B}$ is a $C^*$-subalgebra of $\\mathscr{A}$ equal to\n$\\sum^\\mathrm{c}_\\sigma\\mathscr{B}\\cap\\mathscr{A}(\\sigma)$. Then $\\mathscr{B}$ has a natural\ngraded $C^*$-algebra structure: $\\mathscr{B}(\\sigma)=\\mathscr{B}\\cap\\mathscr{A}(\\sigma)$.\nIf $\\mathscr{B}$ is also an ideal of $\\mathscr{A}$ then $\\mathscr{B}$ is a \\emph{graded\n  ideal}.\n\nA subset $\\mathcal{T}$ of a semilattice $\\mathcal{S}$ is a \\emph{sub-semilattice of\n  $\\mathcal{S}$} if $\\sigma,\\tau\\in\\mathcal{T} \\Rightarrow\n\\sigma\\wedge\\tau\\in\\mathcal{T}$. We say that $\\mathcal{T}$ is an \\emph{ideal of\n  $\\mathcal{S}$} if $\\sigma\\leq\\tau\\in\\mathcal{T} \\Rightarrow \\sigma\\in\\mathcal{T}$. If\n$\\mathscr{A}$ is an $\\mathcal{S}$-graded $C^*$-algebra and $\\mathcal{T}\\subset\\mathcal{S}$ let\n$\\mathscr{A}(\\mathcal{T})=\\sum^\\mathrm{c}_{\\sigma\\in\\mathcal{T}}\\mathscr{A}(\\sigma)$ (if $\\mathcal{T}$ is finite\nthe sum is already closed).  If $\\mathcal{T}$ is a sub-semilattice or an\nideal then clearly $\\mathscr{A}(\\mathcal{T})$ is a $C^*$-subalgebra or an ideal of\n$\\mathscr{A}$ respectively.\n\nWe say that $\\mathscr{A}$ is \\emph{supported by a sub-semilattice $\\mathcal{T}$} if\n$\\mathscr{A}=\\mathscr{A}(\\mathcal{T})$, i.e. $\\mathscr{A}(\\sigma)=\\{0\\}$ for $\\sigma\\notin\\mathcal{T}$. Then\n$\\mathscr{A}$ is also $\\mathcal{T}$-graded.  The smallest sub-semilattice with this\nproperty will be called \\emph{support of $\\mathscr{A}$}.  If $\\mathcal{T}$ is a\nsub-semilattice of $\\mathcal{S}$ and $\\mathscr{A}$ is a $\\mathcal{T}$-graded algebra then\n$\\mathscr{A}$ is $\\mathcal{S}$-graded: set $\\mathscr{A}(\\sigma)=\\{0\\}$ for\n$\\sigma\\in\\mathcal{S}\\setminus\\mathcal{T}$.\n\nThe next result is obvious if $\\mathcal{S}$ is finite. For the general case,\nsee the proof of Proposition 3.3 in \\cite{DG3}.\n\n\n\\begin{proposition}\\label{pr:gsalg}\n  Let $\\mathcal{T}$ be a sub-semilattice of $\\mathcal{S}$ such that\n  $\\mathcal{T}'=\\mathcal{S}\\setminus\\mathcal{T}$ is an ideal. Then $\\mathscr{A}(\\mathcal{T})$ is a\n  $C^*$-subalgebra of $\\mathscr{A}$, $\\mathscr{A}(\\mathcal{T}')$ is an ideal of $\\mathscr{A}$, and\n  $\\mathscr{A}=\\mathscr{A}(\\mathcal{T})+\\mathscr{A}(\\mathcal{T}')$ with $\\mathscr{A}(\\mathcal{T})\\cap\\mathscr{A}(\\mathcal{T}')=\\{0\\}$.\n  In particular, the natural linear projection\n  $\\mathscr{P}(\\mathcal{T}):\\mathscr{A}\\to\\mathscr{A}(\\mathcal{T})$ is a morphism.\n\\end{proposition}\n\nIf $\\mathcal{T}$ is a sub-semilattice then $\\mathcal{T}'$ is an ideal if and only if\n$\\mathcal{T}$ is a filter\n(i.e. $\\sigma\\ge\\tau\\in\\mathcal{T}\\Rightarrow\\sigma\\in\\mathcal{T}$). Thus if $\\mathcal{S}$\nis finite then the only sub-semilattices which have this property\nare the $\\mathcal{S}_{\\ge\\sigma}$ introduced below.\n\nThe simplest sub-semilattices are the chains (totally ordered\nsubsets). If $\\sigma\\in \\mathcal{S}$ and\n\\begin{equation}\\label{eq:Aa}\n\\mathcal{S}_{\\geq\\sigma}=\\{\\tau\\in \\mathcal{S}\\mid \\tau\\geq\\sigma\\},\n\\quad%\n\\mathcal{S}_{\\not\\geq\\sigma}=\n\\mathcal{S}'_{\\geq\\sigma}=\\{\\tau\\in\\mathcal{S}\\mid\\tau\\not\\geq\\sigma\\},\n\\quad\n\\mathcal{S}_{\\leq\\sigma}=\\{\\tau\\in \\mathcal{S}\\mid \\tau\\leq\\sigma\\}\n\\end{equation}\nthen $\\mathcal{S}_{\\geq\\sigma}$ is a sub-semilattice and\n$\\mathcal{S}_{\\not\\geq\\sigma}$ and $\\mathcal{S}_{\\leq\\sigma}$ are ideals. So\n$\\mathscr{A}_{\\ge\\sigma}\\equiv\\mathscr{A}(\\mathcal{S}_{\\geq\\sigma})$ is a graded\n$C^*$-subalgebra of $\\mathscr{A}$ supported by $\\mathcal{S}_{\\geq\\sigma}$ and\n$\\mathscr{A}(\\mathcal{S}_{\\not\\geq\\sigma})$ is a graded ideal supported by\n$\\mathcal{S}_{\\not\\geq\\sigma}$ such that\n\\begin{equation}\\label{eq:dsum}\n\\mathscr{A}=\\mathscr{A}_{\\ge\\sigma}+\\mathscr{A}(\\mathcal{S}_{\\not\\geq\\sigma})\n\\quad\\text{with} \\quad\n\\mathscr{A}_{\\ge\\sigma}\\cap\\mathscr{A}(\\mathcal{S}_{\\not\\geq\\sigma})=\\{0\\}. \n\\end{equation}\nThe projection morphism $\\mathscr{P}_{\\ge\\sigma}:\\mathscr{A}\\to\\mathscr{A}_{\\geq\\sigma}$\ndefined by \\eqref{eq:dsum} is the unique linear continuous map\n$\\mathscr{P}_{\\geq\\sigma}:\\mathscr{A}\\rightarrow\\mathscr{A}$ such that $\\mathscr{P}_{\\geq\\sigma}A=A$ if\n$A\\in\\mathscr{A}(\\tau)$ for some $\\tau\\geq\\sigma$ and $\\mathscr{P}_{\\geq\\sigma}A=0$\notherwise.\n\n$\\mathcal{S}$ is called \\emph{atomic} if it has a smallest element\n$o\\equiv\\min \\mathcal{S}$ and if each $\\sigma\\neq o$ is minorated by an\natom.  We denote by $\\mathcal{P}(\\mathcal{S})$ the set of atoms of $\\mathcal{S}$.  If $\\mathcal{T}$\nis an ideal of $\\mathcal{S}$ and $\\mathcal{S}$ is atomic then $\\mathcal{T}$ is atomic, we\nhave $\\min\\mathcal{T}=\\min \\mathcal{S}$, and $\\mathcal{P}(\\mathcal{T})=\\mathcal{P}(\\mathcal{S})\\cap\\mathcal{T}$.  This next\nresult is also easy to prove \\cite{DG3}.\n\n\\begin{theorem}\\label{th:ga}  \n  If $\\mathcal{S}$ is atomic then $\\mathscr{P}\n  A=(\\mathscr{P}_{\\geq\\alpha}A)_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}$ defines a morphism\n  $\\mathscr{P}:\\mathscr{A}\\to\\prod_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}\\mathscr{A}_{\\geq\\alpha}$ with\n  $\\mathscr{A}(o)$ as kernel. This gives us a canonical embedding\n\\begin{equation}\\label{eq:quot}\n\\mathscr{A}\/\\mathscr{A}(o)\\subset\\displaystyle\\mbox{$\\textstyle\\prod$}_{\\substack{i}}_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}\\mathscr{A}_{\\geq\\alpha}.\n\\end{equation}\n\\end{theorem}\n\nWe call this ``theorem'' because it has important consequences in\nthe spectral theory of many-body Hamiltonians: it allows us to\ncompute their essential spectrum and to prove the Mourre estimate.\n\nWe assume that $\\mathcal{S}$ is atomic so that $\\mathscr{A}$ comes equipped with a\nremarkable ideal $\\mathscr{A}(o)$. Then for $A\\in\\mathscr{A}$ we define its\n\\emph{essential spectrum} (relatively to $\\mathscr{A}(o)$) by the formula\n\\begin{equation}\\label{eq:eso}\n\\mathrm{Sp_{ess}}(A)\\equiv\\mathrm{Sp}(\\mathscr{P} A).\n\\end{equation}\nIn our concrete examples $\\mathscr{A}$ is represented on a Hilbert space\n$\\mathcal{H}$ and $\\mathscr{A}(o)= K(\\mathcal{H})$, so we get the usual Hilbertian notion of\nessential spectrum. \n\nIn order to extend this to unbounded operators it is convenient to\ndefine an \\emph{observable affiliated to $\\mathscr{A}$} as a morphism\n$H:\\cc_{\\mathrm{o}}(\\mathbb{R})\\to\\mathscr{A}$.  We set $\\varphi(H)\\equiv H(\\varphi)$.  If\n$\\mathscr{A}$ is realized on  $\\mathcal{H}$ then a self-adjoint\noperator on $\\mathcal{H}$ such that $(H+i)^{-1}\\in\\mathscr{A}$ is said to be\naffiliated to $\\mathscr{A}$; then $H(\\varphi)=\\varphi(H)$ defines an\nobservable affiliated to $\\mathscr{A}$ (see Appendix A in \\cite{DG3} for a\nprecise description of the relation between observables and\nself-adjoint operators affiliated to $\\mathscr{A}$). The spectrum of an\nobservable is by definition the support of the morphism $H$:\n\\begin{equation}\\label{eq:sp}\n\\mathrm{Sp}(H)=\\{\\lambda\\in\\mathbb{R} \\mid\n\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R}),\\varphi(\\lambda)\\neq 0 \\Rightarrow\n\\varphi(H)\\neq0\\}. \n\\end{equation}\nNow note that $\\mathscr{P} H\\equiv\\mathscr{P}\\circ H$ is an observable affiliated to\nthe quotient algebra $\\mathscr{A}\/\\mathscr{A}(o)$ so we may define the essential\nspectrum of $H$ as the spectrum of $\\mathscr{P} H$. Explicitly, we get:\n\\begin{equation}\\label{eq:es1}\n\\mathrm{Sp_{ess}}(H)=\\{\\lambda\\in\\mathbb{R} \\mid \n\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R}),\\varphi(\\lambda)\\neq 0 \\Rightarrow\n\\varphi(H)\\notin \\mathscr{A}(o)\\}. \n\\end{equation}\nNow the first assertion of the next theorem follows immediately from\nTheorem \\ref{th:ga}. For the second assertion, see the proof of\nTheorem 2.10 in \\cite{DG2}. By $\\overline{\\cup}$ we denote the\nclosure of the union.\n\n\\begin{theorem}\\label{th:gas}\nLet $\\mathcal{S}$ be atomic. If $H$ is an observable affiliated to $\\mathscr{A}$\nthen $H_{\\geq\\alpha}=\\mathscr{P}_{\\geq\\alpha}H$ is an observable affiliated\nto $\\mathscr{A}_{\\geq\\alpha}$ and we have:\n\\begin{equation}\\label{eq:es2}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq\\alpha}).\n\\end{equation}\nIf for each $A\\in\\mathscr{A}$ the set of $\\mathscr{P}_{\\geq\\alpha}A$ with\n$\\alpha\\in\\mathcal{P}(\\mathcal{S})$ is compact in $\\mathscr{A}$ then the union in\n\\eqref{eq:es2} is closed.\n\\end{theorem}\n\n\\subsection{Hilbert $C^*$-modules}\n\\label{ss:preh}\n\nSome basic knowledge of the theory of Hilbert\n$C^*$-modules is useful but not indispensable for understanding our\nconstructions. We translate here the necessary facts in a purely\nHilbert space language.  Our main reference for the general theory\nof Hilbert $C^*$-modules is \\cite{La} but see also \\cite{Bl,RW}.\nThe examples of interest in this paper are the ``concrete'' Hilbert\n$C^*$-modules described below as Hilbert $C^*$-submodules of\n$L(\\mathcal{E},\\mathcal{F})$. We recall, however, the general definition.\n\n\nIf $\\mathscr{A}$ is a $C^*$-algebra then a \\emph{Banach $\\mathscr{A}$-module} is a\nBanach space $\\mathscr{M}$ equipped with a continuous bilinear map\n$\\mathscr{A}\\times\\mathscr{M}\\ni(A,M)\\mapsto MA\\in\\mathscr{M}$ such that $(MA)B=M(AB)$.  We\ndenote $\\mathscr{M}\\cdot\\mathscr{A}$ the clspan of the elements $MA$ with $A\\in\\mathscr{A}$\nand $M\\in\\mathscr{M}$.  By the Cohen-Hewitt theorem \\cite{FD} for each\n$N\\in\\mathscr{M}\\cdot\\mathscr{A}$ there are $A\\in\\mathscr{A}$ and $M\\in\\mathscr{M}$ such that\n$N=MA$, in particular $\\mathscr{M}\\cdot\\mathscr{A}=\\mathscr{M}\\mathscr{A}$.  Note that by module we\nmean ``right module'' but the Cohen-Hewitt theorem is also valid for\nleft Banach modules.\n\nLet $\\mathscr{A}$ be a $C^*$-algebra. A (right) \\emph{Hilbert $\\mathscr{A}$-module}\nis a Banach $\\mathscr{A}$-module $\\mathscr{M}$ equipped with an $\\mathscr{A}$-valued\nsesquilinear map\n$\\braket{\\cdot}{\\cdot}\\equiv\\braket{\\cdot}{\\cdot}_\\mathscr{A}$ which is\npositive (i.e. $\\braket{M}{M}\\geq0$) $\\mathscr{A}$-sesquilinear\n(i.e. $\\braket{M}{NA}=\\braket{M}{N}A$) and such that\n$\\|M\\|\\equiv\\|\\braket{M}{M}\\|^{1\/2}$. Then $\\mathscr{M}=\\mathscr{M}\\mathscr{A}$.  The clspan\nof the elements $\\braket{M}{M}$ is an ideal of $\\mathscr{A}$ denoted\n$\\braket{\\mathscr{M}}{\\mathscr{M}}$. One says that $\\mathscr{M}$ is \\emph{full} if\n$\\braket{\\mathscr{M}}{\\mathscr{M}}=\\mathscr{A}$.  If $\\mathscr{A}$ is an ideal of a $C^*$-algebra\n$\\mathscr{C}$ then $\\mathscr{M}$ is equipped with an obvious structure of Hilbert\n$\\mathscr{C}$-module. Left Hilbert $\\mathscr{A}$-modules are defined similarly. \n\nIf $\\mathscr{M},\\mathscr{N}$ are Hilbert $\\mathscr{A}$-modules and $(M,N)\\in\\mathscr{M}\\times\\mathscr{N}$\nthen $M'\\mapsto N\\braket{M}{M'}$ is a linear continuous map\n$\\mathscr{M}\\to\\mathscr{N}$ denoted $\\ket{N}\\bra{M}$ or $NM^*$.  The closed linear\nsubspace of $L(\\mathscr{M},\\mathscr{N})$ generated by these elements is denoted\n$\\mathcal{K}(\\mathscr{M},\\mathscr{N})$. There is a unique antilinear isometric map $T\\mapsto\nT^*$ of $\\mathcal{K}(\\mathscr{M},\\mathscr{N})$ onto $\\mathcal{K}(\\mathscr{N},\\mathscr{M})$ which sends\n$\\ket{N}\\bra{M}$ into $\\ket{M}\\bra{N}$. The space\n$\\mathcal{K}(\\mathscr{M})\\equiv\\mathcal{K}(\\mathscr{M},\\mathscr{M})$ is a $C^*$-algebra called\n\\emph{imprimitivity algebra} of the Hilbert $\\mathscr{A}$-module $\\mathscr{M}$.\n\nAssume that $\\mathscr{N}$ is a closed subspace of a Hilbert $\\mathscr{A}$-module $\\mathscr{M}$\nand let $\\braket{\\mathscr{N}}{\\mathscr{N}}$ be the clspan of the elements\n$\\braket{N}{N}$ in $\\mathscr{A}$. If $\\mathscr{N}$ is an $\\mathscr{A}$-submodule of $\\mathscr{M}$ then\nit inherits an obvious Hilbert $\\mathscr{A}$-module structure from $\\mathscr{M}$.  If\n$\\mathscr{N}$ is not an $\\mathscr{A}$-submodule of $\\mathscr{M}$ it may happen that there is a\n$C^*$-subalgebra $\\mathscr{B}\\subset\\mathscr{A}$ such that $\\mathscr{N}\\mathscr{B}\\subset\\mathscr{N}$ and\n$\\braket{\\mathscr{N}}{\\mathscr{N}}\\subset\\mathscr{B}$. Then clearly we get a Hilbert\n$\\mathscr{B}$-module structure on $\\mathscr{N}$. On the other hand, it is clear that\nsuch a $\\mathscr{B}$ exists if and only if $\\mathscr{N}\\braket{\\mathscr{N}}{\\mathscr{N}}\\subset\\mathscr{N}$\nand then $\\braket{\\mathscr{N}}{\\mathscr{N}}$ is a $C^*$-subalgebra of $\\mathscr{A}$. Under\nthese conditions we say that \\emph{$\\mathscr{N}$ is a Hilbert $C^*$-submodule}\nof the Hilbert $\\mathscr{A}$-module $\\mathscr{M}$. Then $\\mathscr{N}$ inherits a Hilbert\n$\\braket{\\mathscr{N}}{\\mathscr{N}}$-module structure and this defines the\n$C^*$-algebra $\\mathcal{K}(\\mathscr{N})$. Moreover, if $\\mathscr{B}$ is as above then\n$\\mathcal{K}(\\mathscr{N})=\\mathcal{K}_\\mathscr{B}(\\mathscr{N})$.\n\nIf $\\mathscr{N}$ is a closed subspace of a Hilbert $\\mathscr{A}$-module $\\mathscr{M}$ then\nlet $\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$ be the closed subspace of $\\mathcal{K}(\\mathscr{M})$ generated by\nthe elements $NN^*$ with $N\\in\\mathscr{N}$. It is easy to prove that\n\\emph{if $\\mathscr{N}$ is a Hilbert $C^*$-submodule of $\\mathscr{M}$ then\n$\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$ is a $C^*$-subalgebra of $\\mathcal{K}(\\mathscr{M})$ and the map\n$T\\mapsto T|_\\mathscr{N}$ sends $\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$ onto $\\mathcal{K}(\\mathscr{N})$ and is an\nisomorphism of $C^*$-algebras}. Then we identify $\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$\nwith $\\mathcal{K}(\\mathscr{N})$.\n\nIf $\\mathcal{E},\\mathcal{F}$ are Hilbert spaces then we equip $L(\\mathcal{E},\\mathcal{F})$ with the\nHilbert $L(\\mathcal{E})$-module structure defined as follows: the\n$C^*$-algebra $L(\\mathcal{E})$ acts to the right by composition and we take\n$\\braket{M}{N}=M^*N$ as inner product, where $M^*$ is the usual\nadjoint of the operator $M$. Note that $L(\\mathcal{E},\\mathcal{F})$ is also equipped\nwith a natural left Hilbert $L(\\mathcal{F})$-module structure: this time the\ninner product is $MN^*$.\n\nIf $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$ is a linear subspace then $\\mathscr{M}^*\\subset\nL(\\mathcal{F},\\mathcal{E})$ is the set of adjoint operators $M^*$ with $M\\in\\mathscr{M}$.\nClearly $\\mathscr{M}_1\\subset\\mathscr{M}_2\\Rightarrow\\mathscr{M}_1^*\\subset\\mathscr{M}_2^*$. If\n$\\mathcal{G}$ is a third Hilbert spaces and $\\mathscr{N}\\subset L(\\mathcal{F},\\mathcal{G})$ is a\nlinear subspace then $(\\mathscr{N}\\cdot\\mathscr{M})^*=\\mathscr{M}^*\\cdot\\mathscr{N}^*$.  In\nparticular, if $\\mathcal{E}=\\mathcal{F}=\\mathcal{G}$, $\\mathscr{M}=\\mathscr{M}^*$, and $\\mathscr{N}=\\mathscr{N}^*$ then\n$\\mathscr{M}\\cdot\\mathscr{N}\\subset\\mathscr{N}\\cdot\\mathscr{M}$ is equivalent to\n$\\mathscr{M}\\cdot\\mathscr{N}=\\mathscr{N}\\cdot\\mathscr{M}$.\n\n\nNow let $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$ be a closed linear subspace. Then\n\\emph{$\\mathscr{M}$ is a Hilbert $C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$ if and only\n  if $\\mathscr{M}\\mr^*\\mathscr{M}\\subset\\mathscr{M}$}.\n\nThese are the ``concrete'' Hilbert $C^*$-modules we are interested\nin. It is clear that $\\mathscr{M}^*$ will be a Hilbert $C^*$-submodule of\n$L(\\mathcal{F},\\mathcal{E})$. We mention that $\\mathscr{M}^*$ is canonically identified with\nthe left Hilbert $\\mathscr{A}$-module $\\mathcal{K}(\\mathscr{M},\\mathscr{A})$ dual to $\\mathscr{M}$. \n\n\\begin{proposition}\\label{pr:ss}\nLet $\\mathcal{E},\\mathcal{F}$ be Hilbert spaces and let $\\mathscr{M}$ be a Hilbert\n$C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$. Then $\\mathscr{A}\\equiv\\mathscr{M}^*\\cdot\\mathscr{M}$ and\n$\\mathscr{B}\\equiv\\mathscr{M}\\cdot\\mathscr{M}^*$ are $C^*$-algebras of operators on $\\mathcal{E}$\nand $\\mathcal{F}$ respectively and $\\mathscr{M}$ is equipped with a canonical\nstructure of $(\\mathscr{B},\\mathscr{A})$ imprimitivity bimodule.\n\\end{proposition}\n\nFor the needs of this paper the last assertion of the proposition\ncould be interpreted as a definition.\n\n\\begin{proposition}\\label{pr:clsubmod}\nLet $\\mathscr{N}$ be a $C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$ such that\n$\\mathscr{N}\\subset\\mathscr{M}$ and $\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{M}^*\\cdot\\mathscr{M}$,\n$\\mathscr{N}\\cdot\\mathscr{N}^*=\\mathscr{M}\\cdot\\mathscr{M}^*$. Then $\\mathscr{N}=\\mathscr{M}$.\n\\end{proposition}\n\\proof If $M\\in\\mathscr{M}$ and $N\\in\\mathscr{N}$ then $MN^*\\in\\mathscr{B}=\\mathscr{N}\\cdot\\mathscr{N}^*$ and\n$\\mathscr{N}\\rn^*\\mathscr{N}\\subset\\mathscr{N}$ hence $MN^*N\\in\\mathscr{N}$. Since $\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{A}$\nwe get $MA\\in\\mathscr{N}$ for all $A\\in\\mathscr{A}$. Let $A_i$ be an approximate\nidentity for the $C^*$-algebra $\\mathscr{A}$. Since one can factorize $M=M'A'$\nwith $M'\\in\\mathscr{M}$ and $A'\\in\\mathscr{A}$ the sequence $MA_i=M'A'A_i$ converges\nto $ M'A'=M$ in norm. Thus $M\\in\\mathscr{N}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{proposition}\\label{pr:2ss}\nLet $\\mathcal{E},\\mathcal{F},\\mathcal{H}$ be Hilbert spaces and let $\\mathscr{M}\\subset L(\\mathcal{H},\\mathcal{E})$\nand $\\mathscr{N}\\subset L(\\mathcal{H},\\mathcal{F})$ be Hilbert $C^*$-submodules. Let $\\mathscr{A}$ be\na $C^*$-algebra of operators on $\\mathcal{H}$ such that $\\mathscr{M}^*\\cdot\\mathscr{M}$ and\n$\\mathscr{N}^*\\cdot\\mathscr{N}$ are ideals of $\\mathscr{A}$ and let us view $\\mathscr{M}$ and $\\mathscr{N}$ as\nHilbert $\\mathscr{A}$-modules. Then $\\mathcal{K}(\\mathscr{M},\\mathscr{N})\\cong\\mathscr{N}\\cdot\\mathscr{M}^*$ the\nisometric isomorphism being determined by the condition\n$\\ket{N}\\bra{M}=NM^*$. \n\\end{proposition}\n\n\\subsection{Graded Hilbert $C^*$-modules}\n\\label{ss:gf}\n\nThis is due to Georges Skandalis \\cite{Sk} (see also Remark\n\\ref{re:squant}).\n\n\\begin{definition}\\label{df:grm}\nLet $\\mathcal{S}$ be a semilattice and $\\mathscr{A}$ an $\\mathcal{S}$-graded\n$C^*$-algebra. A Hilbert $\\mathscr{A}$-module $\\mathscr{M}$ is an \\emph{$\\mathcal{S}$-graded\n  Hilbert $\\mathscr{A}$-module} if a linearly independent family\n$\\{\\mathscr{M}(\\sigma)\\}_{\\sigma\\in \\mathcal{S}}$ of closed subspaces of $\\mathscr{M}$ is\ngiven such that $\\sum_\\sigma\\mathscr{M}(\\sigma)$ is dense in $\\mathscr{M}$ and:\n\\begin{equation}\\label{eq:grm}\n\\mathscr{M}(\\sigma)\\mathscr{A}(\\tau)\\subset\\mathscr{M}(\\sigma\\wedge\\tau) \n\\hspace{2mm}\\text{and}\\hspace{2mm}\n\\braket{\\mathscr{M}(\\sigma)}{\\mathscr{M}(\\tau)}\\subset\\mathscr{A}(\\sigma\\wedge\\tau)\n\\hspace{2mm} \\text{for all } \\sigma,\\tau\\in \\mathcal{S}.\n\\end{equation}\n\\end{definition}\nNote that $\\mathscr{A}$ equipped with its canonical Hilbert $\\mathscr{A}$-module\nstructure is an $\\mathcal{S}$-graded Hilbert \\mbox{$\\mathscr{A}$-module}. \n\\eqref{eq:grm} implies that each $\\mathscr{M}(\\sigma)$ is a\nHilbert $\\mathscr{A}(\\sigma)$-module and if $\\sigma\\leq\\tau$ then\n$\\mathscr{M}(\\sigma)$ is an $\\mathscr{A}(\\tau)$-module.\n\n\nFrom \\eqref{eq:grm} we also see that \\emph{the imprimitivity algebra\n  $\\mathcal{K}(\\mathscr{M}(\\sigma))$ of the Hilbert $\\mathscr{A}(\\sigma)$-module\n  $\\mathscr{M}(\\sigma)$ is naturally identified with the clspan in\n  $\\mathcal{K}(\\mathscr{M})$ of the elements $MM^*$ with $M\\in\\mathscr{M}(\\sigma)$}. Thus\n$\\mathcal{K}(\\mathscr{M}(\\sigma))$ is identified with a $C^*$-subalgebra of\n$\\mathcal{K}(\\mathscr{M})$. We use this identification below.\n\n\\begin{theorem}\\label{th:kghm}\nIf $\\mathscr{M}$ is a graded Hilbert $\\mathscr{A}$-module then $\\mathcal{K}(\\mathscr{M})$ becomes a\ngraded $C^*$-algebra if we define\n$\\mathcal{K}(\\mathscr{M})(\\sigma)=\\mathcal{K}(\\mathscr{M}(\\sigma))$. If $M\\in\\mathscr{M}(\\sigma)$ and\n$N\\in\\mathscr{M}(\\tau)$ then there are elements $M'$ and $N'$ in\n$\\mathscr{M}(\\sigma\\wedge\\tau)$ such that $MN^*=M'N'^*$;\nin particular $MN^*\\in\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$.\n\\end{theorem}\n\\proof As explained before, $\\mathcal{K}(\\mathscr{M})(\\sigma)$ are $C^*$-subalgebras\nof $\\mathcal{K}(\\mathscr{M})$. To show that they are linearly independent, let\n$T(\\sigma)\\in\\mathcal{K}(\\mathscr{M})(\\sigma)$ such that $T(\\sigma)=0$ but for a\nfinite number of $\\sigma$ and assume $\\sum_\\sigma T(\\sigma)=0$. Then\nfor each $M\\in\\mathscr{M}$ we have $\\sum_\\sigma T(\\sigma)M=0$. Note that the\nrange of $T(\\sigma)$ is included in $\\mathscr{M}(\\sigma)$. Since the linear\nspaces $\\mathscr{M}(\\sigma)$ are linearly independent we get $T(\\sigma)M=0$\nfor all $\\sigma$ and $M$ hence $T(\\sigma)=0$ for all $\\sigma$.\n\nWe now prove the second assertion of the proposition. Since\n$\\mathscr{M}(\\sigma)$ is a Hilbert $\\mathscr{A}(\\sigma)$-module there are\n$M_1\\in\\mathscr{M}(\\sigma)$ and $S\\in\\mathscr{A}(\\sigma)$ such that $M=M_1S$, cf. the\nCohen-Hewitt theorem or Lemma 4.4 in \\cite{La}. Similarly, $N=N_1T$\nwith $N_1\\in\\mathscr{M}(\\tau)$ and $T\\in\\mathscr{A}(\\tau)$. Then $MN^*=M_1(S\nT^*)N_1^*$ and $S T^*\\in \\mathscr{A}(\\sigma\\wedge\\tau)$ so we may factorize it\nas $S T^*=UV^*$ with $U,V\\in \\mathscr{A}(\\sigma\\wedge\\tau)$, hence\n$MN^*=(M_1U)(N_1V)^*$.  By using \\eqref{eq:grm} we see that $M'=M_1U$\nand $N'=N_1V$ belong to $\\mathscr{M}(\\sigma\\wedge\\tau)$.  In particular, we\nhave $MN^*\\in\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$ if $M\\in\\mathscr{M}(\\sigma)$ and\n$N\\in\\mathscr{M}(\\tau)$.\n\nObserve that the assertion we just proved implies that\n$\\sum_\\sigma\\mathcal{K}(\\mathscr{M})(\\sigma)$ is dense in $\\mathcal{K}(\\mathscr{M})$.  \nIt remains to see that\n$\\mathcal{K}(\\mathscr{M})(\\sigma)\\mathcal{K}(\\mathscr{M})(\\tau)\\subset\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$.\nFor this it suffices that $M\\braket{M}{N}N^*$ be in\n$\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$ if $M\\in\\mathscr{M}(\\sigma)$ and\n$N\\in\\mathscr{M}(\\tau)$. Since $\\braket{M}{N}\\in\\mathscr{A}(\\sigma\\wedge\\tau)$ we\nmay write $\\braket{M}{N}=S T^*$ with $S,T\\in\\mathscr{A}(\\sigma\\wedge\\tau)$\nso $M\\braket{M}{N}N^*=(MS)(NT)^*\\in\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$ by\n\\eqref{eq:grm}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip \n\n\nWe recall that the direct sum of a family $\\{\\mathscr{M}_i\\}$ of Hilbert\n$\\mathscr{A}$-modules is defined as follows: $\\oplus_i\\mathscr{M}_i$ is the space of\nelements $(M_i)_i\\in\\prod_i\\mathscr{M}_i$ such that the series\n$\\sum_i\\braket{M_i}{M_i}$ converges in $\\mathscr{A}$ equipped with the natural\n$\\mathscr{A}$-module structure and with the $\\mathscr{A}$-valued inner product defined\nby\n\\begin{equation}\\label{eq:sum}\n\\braket{(M_i)_i}{(N_i)_i} \n=\\textstyle\\sum_i\\braket{M_i}{N_i}.\n\\end{equation} \nThe algebraic direct sum of the $\\mathscr{A}$-modules $\\mathscr{M}_i$ is dense in\n$\\oplus_i\\mathscr{M}_i$.\n\n\nIt is easy to check that if each $\\mathscr{M}_i$ is graded and if we set\n$\\mathscr{M}(\\sigma)=\\oplus_i\\mathscr{M}_i(\\sigma)$ then $\\mathscr{M}$ becomes a graded\nHilbert $\\mathscr{A}$-module.  For example, if $\\mathscr{N}$ is a graded Hilbert\n$\\mathscr{A}$-module then $\\mathscr{N}\\oplus\\mathscr{A}$ is a graded Hilbert $\\mathscr{A}$-module and\nso the \\emph{linking algebra $\\mathcal{K}(\\mathscr{N}\\oplus\\mathscr{A})$ is equipped with a\ngraded algebra structure}. We recall \\cite[p. 50-52]{RW} that we\nhave a natural identification\n\\begin{equation}\\label{eq:link}\n\\mathcal{K}(\\mathscr{N}\\oplus\\mathscr{A})=\n\\begin{pmatrix}\n\\mathcal{K}(\\mathscr{N})& \\mathscr{N}\\\\\n\\mathscr{N}^*& \\mathscr{A}\n\\end{pmatrix}\n\\end{equation}\nand by Theorem \\ref{th:kghm} this is a graded algebra whose\n$\\sigma$-component is equal to\n\\begin{equation}\\label{eq:links}\n\\mathcal{K}(\\mathscr{N}(\\sigma)\\oplus\\mathscr{A}(\\sigma))=\n\\begin{pmatrix}\n\\mathcal{K}(\\mathscr{N}(\\sigma))& \\mathscr{N}(\\sigma)\\\\\n\\mathscr{N}(\\sigma)^*& \\mathscr{A}(\\sigma)\n\\end{pmatrix}.\n\\end{equation}\nIf $\\mathscr{N}$ is a $C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$ and if we set\n$\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{A},\\mathscr{N}\\cdot\\mathscr{N}^*=\\mathscr{B}$ then the linking algebra\n$\\begin{pmatrix}\\mathscr{B}& \\mathscr{M}\\\\\\mathscr{M}^*& \\mathscr{A}\\end{pmatrix}$ of $\\mathscr{M}$ is a\n$C^*$-algebra of operators on $\\mathcal{F}\\oplus\\mathcal{E}$.\n\nSome of the graded Hilbert $C^*$-modules which we shall use later on\nwill be constructed as follows.\n\n\\begin{proposition}\\label{pr:rhm}\nLet $\\mathcal{E},\\mathcal{F}$ be Hilbert spaces and let $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$ be a\nHilbert $C^*$-submodule, so that $\\mathscr{A}\\equiv\\mathscr{M}^*\\cdot\\mathscr{M}\\subset\nL(\\mathcal{E})$ is a $C^*$-algebra and $\\mathscr{M}$ is a full Hilbert\n$\\mathscr{A}$-module. Let $\\mathcal{C}$ be a $C^*$-algebra of operators on $\\mathcal{E}$\ngraded by the family of $C^*$-subalgebras\n$\\{\\mathcal{C}(\\sigma)\\}_{\\sigma\\in\\mathcal{S}}$. Assume that we have\n\\begin{equation}\\label{eq:tas}\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)=\\mathcal{C}(\\sigma)\\cdot\\mathscr{A}\n\\equiv\\mathscr{C}(\\sigma)\n\\hspace{2mm} \\text{for all } \\sigma\\in\\mathcal{S}\n\\end{equation}\nand that the family $\\{\\mathscr{C}(\\sigma)\\}$ of subspaces of $L(\\mathcal{F})$ is\nlinearly independent. Then the $\\mathscr{C}(\\sigma)$ are $C^*$-algebras of\noperators on $\\mathcal{E}$ and $\\mathscr{C}=\\sum^\\mathrm{c}_\\sigma\\mathscr{C}(\\sigma)$ is a\n$C^*$-algebra graded by the family $\\{\\mathscr{C}(\\sigma)\\}$. If\n$\\mathscr{N}(\\sigma)\\equiv\\mathscr{M}\\cdot\\mathcal{C}(\\sigma)$ then\n$\\mathscr{N}=\\sum^\\mathrm{c}_\\sigma\\mathscr{N}(\\sigma)$ is a full Hilbert $\\mathscr{C}$-module\ngraded by $\\{\\mathscr{N}(\\sigma)\\}$.\n\\end{proposition}\n\\proof\nWe have\n$$\n\\mathscr{C}(\\sigma)\\cdot\\mathscr{C}(\\tau)=\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\tau)\n=\\mathscr{A}\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathcal{C}(\\tau)\\subset\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\\mathscr{C}(\\sigma\\wedge\\tau).\n$$ \nThis proves that the $\\mathscr{C}(\\sigma)$ are $C^*$-algebras and that\n$\\mathscr{C}$ is $\\mathcal{S}$-graded. Then:\n$$\n\\mathscr{N}(\\sigma)\\cdot\\mathscr{C}(\\tau)=\\mathscr{M}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathcal{C}(\\tau)\\cdot\\mathscr{A}\n\\subset\\mathscr{M}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)\\cdot\\mathscr{A}=\n\\mathscr{M}\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\n\\mathscr{M}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\\mathscr{N}(\\sigma\\wedge\\tau)\n$$\nand\n$$\n\\mathscr{N}(\\sigma)^*\\cdot\\mathscr{N}(\\tau)=\n\\mathcal{C}(\\sigma)\\cdot\\mathscr{M}^*\\cdot\\mathscr{M}\\cdot\\mathcal{C}(\\tau)=\n\\mathcal{C}(\\sigma)\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\tau)=\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathcal{C}(\\tau)\\subset\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\\mathscr{C}(\\sigma\\wedge\\tau).\n$$\nObserve that this computation also gives\n$\\mathscr{N}(\\sigma)^*\\cdot\\mathscr{N}(\\sigma)=\\mathscr{C}(\\sigma)$.  Then\n$$\n\\big({\\textstyle\\sum_\\sigma}\\mathscr{N}(\\sigma)^*\\big)\n\\big({\\textstyle\\sum_\\sigma}\\mathscr{N}(\\sigma)\\big)=\n{\\textstyle\\sum_{\\sigma,\\tau}}\\mathscr{N}(\\sigma)^*\\mathscr{N}(\\tau)\\subset\n{\\textstyle\\sum_{\\sigma,\\tau}}\\mathscr{C}(\\sigma\\wedge\\tau)\\subset\n{\\textstyle\\sum_{\\sigma}}\\mathscr{C}(\\sigma)\n$$ and by the preceding remark we get $\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{C}$ so $\\mathscr{N}$\nis a full Hilbert $\\mathscr{C}$-module. To show the grading property it\nsuffices to prove that the family of subspaces $\\mathscr{N}(\\sigma)$ is\nlinearly independent. Assume that $\\sum N(\\sigma)=0$ with\n$N(\\sigma)\\in\\mathscr{N}(\\sigma)$ and $N(\\sigma)=0$ for all but a finite\nnumber of $\\sigma$. Assuming that there are non-zero elements in\nthis sum, let $\\tau$ be a maximal element of the set of $\\sigma$\nsuch that $N(\\sigma)\\neq0$. From\n$\\sum_{\\sigma_1,\\sigma_2}N(\\sigma_1)^*N(\\sigma_2)=0$ and since\n$N(\\sigma_1)^*N(\\sigma_2)\\in\\mathscr{C}(\\sigma_1\\wedge\\sigma_2)$ we get\n$\\sum_{\\sigma_1\\wedge\\sigma_2=\\sigma}N(\\sigma_1)^*N(\\sigma_2)=0$ for\neach $\\sigma$. Take here $\\sigma=\\tau$ and observe that if\n$\\sigma_1\\wedge\\sigma_2=\\tau$ and $\\sigma_1>\\tau$ or $\\sigma_2>\\tau$\nthen $N(\\sigma_1)^*N(\\sigma_2)=0$. Thus $N(\\tau)^*N(\\tau)=0$ so\n$N(\\tau)=0$. But this contradicts the choice of $\\tau$, so\n$N(\\sigma)=0$ for all $\\sigma$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Tensor products}\n\\label{ss:ha}\n\nIn this subsection we collect some facts concerning tensor products\nwhich are useful in what follows.  We recall the definition of the\ntensor product of a Hilbert space $\\mathcal{E}$ and a \\mbox{$C^*$-algebra}\n$\\mathscr{A}$ in the category of Hilbert $C^*$-modules, cf. \\cite{La}. We\nequip the algebraic tensor product $\\mathcal{E}\\odot\\mathscr{A}$ with the obvious\nright $\\mathscr{A}$-module structure and with the $\\mathscr{A}$-valued sesquilinear\nmap given by\n\\begin{equation}\\label{eq:hoa}\n\\braket{{\\textstyle\\sum_{u\\in\\mathcal{E}}}u\\otimes\n  A_u}{{\\textstyle\\sum_{v\\in\\mathcal{E}}}v\\otimes B_v} \n={\\textstyle\\sum_{u,v}}\\braket{u}{v}A^*_uB_v\n\\end{equation}\nwhere $A_u=B_u=0$ outside a finite set. Then the completion of\n$\\mathcal{E}\\odot\\mathscr{A}$ for the norm $\\|M\\|:=\\|\\braket{M}{M}\\|^{1\/2}$ is a\nfull Hilbert $\\mathscr{A}$-module denoted $\\mathcal{E}\\otimes\\mathscr{A}$. Clearly its\nimprimitivity algebra is\n\\begin{equation}\\label{eq:cha}\n\\mathcal{K}(\\mathcal{E}\\otimes\\mathscr{A})=K(\\mathcal{E})\\otimes\\mathscr{A}.\n\\end{equation}  \nIf $\\mathscr{A}$ is $\\mathcal{S}$-graded then $\\mathcal{E}\\otimes\\mathscr{A}$ is equipped with an\nobvious structure of $\\mathcal{S}$-graded Hilbert $\\mathscr{A}$-module.\n\nIf $\\mathscr{A}$ is realized on a Hilbert space $\\mathcal{F}$ then one has a natural\nisometric embedding $\\mathcal{E}\\otimes\\mathscr{A} \\subset L(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$.\nIndeed, there is a unique linear map $\\mathcal{E}\\otimes\\mathscr{A}\\to\nL(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$ which associates to $u\\otimes A$ the function\n$f\\mapsto u\\otimes(Af)$ and due to \\eqref{eq:hoa} this map is an\nisometry.  Thus the Hilbert $\\mathscr{A}$-module $\\mathcal{E}\\otimes\\mathscr{A}$ is realized\nas a Hilbert $C^*$-submodule of $L(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$, the dual\nmodule is realized as the set of adjoint operators\n$(\\mathcal{E}\\otimes\\mathscr{A})^*\\subset L(\\mathcal{E}\\otimes\\mathcal{F},\\mathcal{E})$, and one clearly has\n\\begin{equation}\\label{eq:etens}\n(\\mathcal{E}\\otimes\\mathscr{A})^*\\cdot (\\mathcal{E}\\otimes\\mathscr{A})=\\mathscr{A}, \\hspace{2mm}\n(\\mathcal{E}\\otimes\\mathscr{A})\\cdot(\\mathcal{E}\\otimes\\mathscr{A})^*=K(\\mathcal{E})\\otimes\\mathscr{A}.\n\\end{equation} \n\n\nIf $X$ is a locally compact space equipped with a Radon measure then\n$L^2(X)\\otimes\\mathscr{A}$ is the completion of $\\cc_{\\mathrm{c}}(X;\\mathscr{A})$ for the norm\n$\\|\\int_X F(x)^*F(x) \\text{d} x\\|^{1\/2}$.  Note that $L^2(X;\\mathscr{A})\\subset\nL^2(X)\\otimes\\mathscr{A}$ strictly in general, cf.\\ the example below.  If\n$\\mathscr{A}\\subset L(\\mathcal{F})$ then the norm on $L^2(X)\\otimes\\mathscr{A}$ is\n\\begin{equation}\\label{eq:L2a}\n\\|{\\textstyle\\int_X} F(x)^*F(x) \\text{d} x\\|^2=\n{\\textstyle\\sup_{f\\in\\mathcal{F},\\|f\\|=1}} {\\textstyle\\int_X} \n\\|F(x)f\\|^2 \\text{d} x.\n\\end{equation}\nIf $Y$ is a locally compact space then\n$\\mathcal{E}\\otimes\\cc_{\\mathrm{o}}(Y)\\cong\\cc_{\\mathrm{o}}(Y;\\mathcal{E})$.  Hence $L^2(X)\\otimes\\cc_{\\mathrm{o}}(Y)$ is\nthe completion of $\\cc_{\\mathrm{c}}(X\\times Y)$ for the norm $\\sup_{y\\in\n  Y}(\\int_X |F(x,y)|^2 \\text{d} x)^{1\/2}$. Assume that $X=Y$ is a\nlocally compact abelian group and let $f\\in L^\\infty(X)$ with\ncompact support and $g\\in L^2(X)$. It is easy to check that\n$F(x,y)=f(x)g(x+y)$ is an element of\n$\\cc_{\\mathrm{o}}(X;L^2(X))=L^2(X)\\otimes\\cc_{\\mathrm{o}}(X)$ but if $F(x,\\cdot)=f(x)U_x g$ is\nnot zero then it does not belong to $\\cc_{\\mathrm{o}}(X)$ and is not even a bounded\nfunction if $g$ is not. Thus the elements of $L^2(X)\\otimes\\mathscr{A}$ can\nnot be realized as bounded operator valued (equivalence classes of)\nfunctions on $X$.\n\n\n\nMore generally, if $\\mathcal{F}'$, $\\mathcal{F}''$ are Hilbert spaces and\n$\\mathscr{M}\\subset L(\\mathcal{F}',\\mathcal{F}'')$ is a closed subspace then we define\n$L^2(X)\\otimes\\mathscr{M}$ as the completion of the space $\\cc_{\\mathrm{c}}(X;\\mathscr{M})$ for a\nnorm similar to \\eqref{eq:L2a}. We clearly have\n$L^2(X)\\otimes\\mathscr{M}\\subset L(\\mathcal{F}',L^2(X)\\otimes\\mathcal{F}'')$ isometrically\nand $L^2(X;\\mathscr{M})\\subset L^2(X)\\otimes\\mathscr{M}$ continuously.\n\n\n\nIf $\\mathcal{E},\\mathcal{F},\\mathcal{G},\\mathcal{H}$ are Hilbert spaces and $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$\nand $\\mathscr{N}\\subset L(\\mathcal{G},\\mathcal{H})$ are closed linear subspaces then we\ndenote $\\mathscr{M}\\otimes\\mathscr{N}$ the closure in\n$L(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H})$ of the algebraic tensor product of\n$\\mathscr{M}$ and $\\mathscr{N}$. Now suppose that $\\mathscr{M}$ is a $C^*$-submodule of\n$L(\\mathcal{E},\\mathcal{F})$ and that $\\mathscr{N}$ is a $C^*$-submodule of $L(\\mathcal{G},\\mathcal{H})$ and\nlet $\\mathscr{A}=\\mathscr{M}^*\\cdot\\mathscr{M}$ and $\\mathscr{B}=\\mathscr{N}^*\\cdot\\mathscr{N}$. Then $\\mathscr{M}$ is a\nHilbert $\\mathscr{A}$-module and $\\mathscr{N}$ is a Hilbert $\\mathscr{B}$-module hence the\nexterior tensor product, denoted temporarily\n$\\mathscr{M}\\otimes_{\\text{ext}}\\mathscr{N}$, is well defined in the category of\nHilbert $C^*$-modules \\cite{La} and is a Hilbert\n$\\mathscr{A}\\otimes\\mathscr{B}$-module. On the other hand, it is easy to check that\n$(\\mathscr{M}\\otimes\\mathscr{N})^*=\\mathscr{M}^*\\otimes\\mathscr{N}^*$ and then that $\\mathscr{M}\\otimes\\mathscr{N}$\nis a Hilbert $C^*$-submodule of $L(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H})$\nsuch that\n$(\\mathscr{M}\\otimes\\mathscr{N})^*\\cdot(\\mathscr{M}\\otimes\\mathscr{N})=\\mathscr{A}\\otimes\\mathscr{B}$. Finally, it\nis clear that $L(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H})$ and\n$\\mathscr{M}\\otimes_{\\text{ext}}\\mathscr{N}$ induce the same $\\mathscr{A}\\otimes\\mathscr{B}$-valued\ninner product on the algebraic tensor product of $\\mathscr{M}$ and $\\mathscr{N}$.\nThus we we get a canonical isometric isomorphism\n$\\mathscr{M}\\otimes_{\\text{ext}}\\mathscr{N}=\\mathscr{M}\\otimes\\mathscr{N}$.\n\n\nAs an application we give now an abstract version of the \"toy\nmodels\" described in Example \\ref{ex:fried}. Let $\\mathcal{E},\\mathcal{F}$ be\nHilbert spaces and let us define $\\mathcal{H}=(\\mathcal{E}\\otimes\\mathcal{F})\\oplus\\mathcal{F}$.\nLet $\\mathscr{A}$ and $\\mathscr{B}$ be $C^*$-algebras of operators on $\\mathcal{F}$ and\n$\\mathcal{E}\\otimes\\mathcal{F}$ respectively. We embed $\\mathcal{E}\\otimes\\mathscr{A} \\subset\nL(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$ as above. We simplify notation and denote\n$\\mathcal{E}^*\\otimes\\mathscr{A}:=(\\mathcal{E}\\otimes\\mathscr{A})^*\\subset L(\\mathcal{E}\\otimes\\mathcal{F},\\mathcal{F})$ the\ndual module.\n\n\\begin{proposition}\\label{pr:toymodel}\n  Let $\\mathcal{S}$ be a semilattice and $\\mathcal{T}$ an ideal of $\\mathcal{S}$. Assume\n  that the $C^*$-algebras $\\mathscr{A}$ and $\\mathscr{B}$ are $\\mathcal{S}$-graded and that\n  we have $\\mathscr{A}(\\sigma)=\\{0\\}$ if $\\sigma\\notin\\mathcal{T}$ and $\\mathscr{B}(\\tau)=\n  K(\\mathcal{E})\\otimes \\mathscr{A}(\\tau)$ for $\\tau\\in\\mathcal{T}$.  Then\n\\begin{equation}\\label{e:toy}\n\\mathscr{C}=\n\\begin{pmatrix}\n\\mathscr{B} & \\mathcal{E}\\otimes\\mathscr{A}\\\\\n\\mathcal{E}^*\\otimes\\mathscr{A} & \\mathscr{A}\n\\end{pmatrix}.\n\\end{equation}\nis an $\\mathcal{S}$-graded $C^*$-algebra if we define its components as\nfollows:\n\\begin{equation}\\label{e:gtoy}\n\\mathscr{C}(\\sigma)=\n\\begin{pmatrix}\n\\mathscr{B}(\\sigma) & \\mathcal{E}\\otimes\\mathscr{A}(\\sigma)\\\\\n\\mathcal{E}^*\\otimes\\mathscr{A}(\\sigma) & \\mathscr{A}(\\sigma)\n\\end{pmatrix} \\quad \\text{for all} \\quad \\sigma\\in\\mathcal{S}.\n\\end{equation}\n\\end{proposition}\n\\proof\nObserve that if we set $\\mathcal{T}'=\\mathcal{S}\\setminus\\mathcal{T}$ then\n\\begin{equation}\\label{e:imptoy}\n\\mathscr{C}=\n\\begin{pmatrix}\nK(\\mathcal{E})\\otimes\\mathscr{A} & \\mathcal{E}\\otimes\\mathscr{A}\\\\\n\\mathcal{E}^*\\otimes\\mathscr{A} & \\mathscr{A}\n\\end{pmatrix} +\n\\begin{pmatrix}\n\\mathscr{B}(\\mathcal{T}') &  0 \\\\\n0 & 0\n\\end{pmatrix} =\n\\mathcal{K}(\\mathscr{N}\\oplus\\mathscr{A}) +\n\\begin{pmatrix}\n\\mathscr{B}(\\mathcal{T}') &  0 \\\\\n0 & 0\n\\end{pmatrix}\n\\end{equation}\nwhere $\\mathscr{N}=\\mathcal{E}\\otimes\\mathscr{A}$ is an $\\mathcal{S}$-graded Hilbert $\\mathscr{A}$-module,\ncf. \\eqref{eq:link} and \\eqref{eq:cha}. It is easy to see that the\nfamily $\\{\\mathscr{C}(\\sigma)\\}$ is linearly independent and that $\\mathscr{C}$ is\nthe closure of its sum. By taking into account \\eqref{eq:links} we\nsee that it suffices to show that\n$\\mathscr{C}(\\sigma)\\mathscr{C}(\\tau)\\subset\\mathscr{C}(\\sigma\\wedge\\tau)$ if\n$\\sigma\\in\\mathcal{T}'$ and $\\tau\\in\\mathcal{T}$. After computing the coefficients\nof the matrices we see that it suffices to check that\n$\\mathscr{B}(\\sigma)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau)\n\\subset\\mathcal{E}\\otimes\\mathscr{A}(\\sigma\\wedge\\tau)$. \nBut:\n\\begin{align*}\n\\mathscr{B}(\\sigma)\\cdot\\mathcal{E}\\otimes\\mathscr{A}(\\tau) & =\n\\mathscr{B}(\\sigma)\\cdot K(\\mathcal{E})\\otimes\\mathscr{A}(\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau) = \n\\mathscr{B}(\\sigma)\\cdot\\mathscr{B}(\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau) \\\\\n& \\subset \\mathscr{B}(\\sigma\\wedge\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau)\n=K(\\mathcal{E})\\otimes\\mathscr{A}(\\sigma\\wedge\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau)\n\\subset \\mathcal{E}\\otimes\\mathscr{A}(\\sigma\\wedge\\tau)\n\\end{align*}\nwhich finishes the proof.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nThe extension to an increasing family of ideals\n$\\mathcal{T}_1\\subset\\mathcal{T}_2\\dots\\subset\\mathcal{S}$ is straightforward. \n\n\n\\section{The many-body $C^*$-algebra }\n\\label{s:grass}\n\\protect\\setcounter{equation}{0}\n\nIn this section we introduce the many-body $C^*$-algebra and\ndescribe its main properties (in particular, we prove the theorems\n\\ref{th:C} and \\ref{th:CG}). Subsection \\ref{ss:hilbert} contains\nsome preparatory material on concrete realizations of Hilbert\n$C^*$-modules which implement the Morita equivalence between some\ncrossed products.\n\n\n\\subsection{Notations}\n\\label{ss:group}\n\n\nLet $X$ be a locally compact abelian group with operation denoted\nadditively equipped with a Haar measures $\\text{d} x$. We abbreviate\nthis by saying that \\emph{$X$ is an lca group}.  We set $\\mathscr{L}_X\\equiv\nL(L^2(X))$ and $\\mathscr{K}_X\\equiv K(L^2(X))$ and note that these are\n$C^*$-algebras independent of the choice of the measure on $X$.  If\n$Y$ is a second lca group we shall use the abbreviations\n\\begin{equation}\\label{eq:lkxy}\n\\mathscr{L}_{XY}=L(L^2(Y),L^2(X)) \\quad\\text{and}\\quad\n\\mathscr{K}_{XY}=K(L^2(Y),L^2(X)). \n\\end{equation}\nWe denote by $\\varphi(Q)$ the operator in $L^2(X)$ of multiplication\nby a function $\\varphi$ and if $X$ has to be explicitly specified we\nset $Q=Q_X$.  The bounded uniformly continuous functions on $X$ form\na $C^*$-algebra $\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ which contains the algebras $\\cc_{\\mathrm{c}}(X)$ and\n$\\cc_{\\mathrm{o}}(X)$.  The map $\\varphi\\mapsto\\varphi(Q)$ is an embedding\n$\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)\\subset\\mathscr{L}_X$.\n\nThe group $\\mathcal{C}^*$-algebra $\\mathscr{T}_X$ of $X$ is the closed linear\nsubspace of $\\mathscr{L}_X$ generated by the convolution operators of the\nform $(\\varphi*f)(x)=\\int_X \\varphi(x-y)f(y)\\text{d} y$ with\n$\\varphi\\in\\cc_{\\mathrm{c}}(X)$.  Observe that $f\\mapsto\\varphi*f$ is equal to\n$\\int_X\\varphi(-a)U_a\\,\\text{d} a$ where $U_a$ is the unitary\ntranslation operator on $L^2(X)$ defined by $(U_af)(x)=f(x+a)$.\n\nLet $X^*$ be the group dual to $X$ with operation denoted\nadditively\\symbolfootnote[2]{\\ Then $(k+p)(x)=k(x)p(x)$, $0(x)=1$,\n  and the element $-k$ of $X^*$ represents the function\n  $\\bar{k}$. In order to avoid such strange looking expressions one\n  might use the notation $k(x)=[x,k]$.  }.  If $k\\in X^*$ we define\na unitary operator $V_k$ on $L^2(X)$ by $(V_ku)(x)=k(x) u(x)$.  The\nFourier transform of an integrable measure $\\mu$ on $X$ is defined\nby $(F\\mu)(k)=\\int \\bar{k}(x)\\mu(\\text{d} x)$.  Then $F$ induces a\nbijective map $L^2(X)\\rightarrow L^2(X^*)$ hence a canonical isomorphism\n$S\\mapsto F^{-1}S F$ of $\\mathscr{L}_{X^*}$ onto $\\mathscr{L}_X$.  If $\\psi$ is a\nfunction on $X^*$ we set $\\psi(P)\\equiv\\psi(P_X)=F^{-1}M_\\psi F$,\nwhere $M_\\psi=\\psi(Q_{X^*})$ is the operator of multiplication by\n$\\psi$ on $L^2(X^*)$.  The map $\\psi\\mapsto\\psi(P)$ gives an\nisomorphism $\\cc_{\\mathrm{o}}(X^*)\\cong \\mathscr{T}_X$.\n\n\nIf $Y\\subset X$ is a closed subgroup then $\\pi_Y:X\\to X\/Y$ is the\ncanonical surjection.  We embed $\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ with the\nhelp of the injective morphism $\\varphi\\mapsto\\varphi\\circ\\pi_Y$. So\n$\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)$ is identified with the set of functions\n$\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ such that $\\varphi(x+y)=\\varphi(x)$ for all\n$x\\in X$ and $y\\in Y$.\n\nIn particular, $\\cc_{\\mathrm{o}}(X\/Y)$ is identified with the set of continuous\nfunctions $\\varphi$ on $X$ such that $\\varphi(x+y)=\\varphi(x)$ for\nall $x\\in X$ and $y\\in Y$ and such that for each $\\varepsilon>0$\nthere is a compact $K\\subset X$ such that $|\\varphi(x)|<\\varepsilon$\nif $x\\notin K+Y$. By $x\/Y\\to\\infty$ we mean $\\pi_Y(x)\\to\\infty$, so\nthe last condition is equivalent to $\\varphi(x)\\to0$ if\n$x\/Y\\to\\infty$.  For coherence with later notations we set\n\\begin{equation}\\label{eq:coxy}\n\\mathcal{C}_X(Y)=\\cc_{\\mathrm{o}}(X\/Y)\n\\end{equation}\nObserve that to an element $y\\in Y$ we may associate a translation\noperator $U_y$ in $L^2(X)$ and another translation operator in\n$L^2(Y)$. However, in order not to overcharge the writing we shall\ndenote the second operator also by $U_y$.  The restriction map\n$k\\mapsto k|_Y$ is a continuous surjective group morphism $X^*\\to\nY^*$ with kernel equal to $Y^\\perp=\\{k\\in X^*\\mid\nk(y)=1\\hspace{1mm}\\forall y\\in Y\\}$ which defines the canonical\nidentification $Y^*\\cong X^*\/Y^\\perp$. We denote by the same symbol\n$V_k$ the operator of multiplication by the character $k\\in X^*$ in\n$L^2(X)$ and by the character $k|_Y\\in Y^*$ in $L^2(Y)$.\n\nWe shall write $X=Y\\oplus Z$ if $X$ is the direct sum of the two\nclosed subgroups $Y,Z$ equipped with compatible Haar measures, in\nthe sense that $\\text{d} x=\\text{d} y\\otimes \\text{d} z$.  Then\n$L^2(X)=L^2(Y)\\otimes L^2(Z)$ as Hilbert spaces and\n$\\mathscr{K}_X=\\mathscr{K}_Y\\otimes \\mathscr{K}_Z$ and $\\mathcal{C}_X(Y)=1\\otimes\\cc_{\\mathrm{o}}(Z)$ as\n$C^*$-algebras.\n\nLet $O=\\{0\\}$ be the trivial group equipped with the Haar measure of\ntotal mass $1$. Then $L^2(O)=\\mathbb{C}$.\n\n\\subsection{Crossed products}\n\\label{ss:nbcrp}\n\nLet $X$ be a locally compact abelian group. A $C^*$-subalgebra\n$\\mathcal{A}\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ stable under translations will be called\n\\emph{$X$-algebra}.  The \\emph{crossed product of\n  $\\mathcal{A}$ by the action of $X$} is an abstractly defined $C^*$-algebra\n$\\mathcal{A}\\rtimes X$ canonically identified with the $C^*$-algebra of\noperators on $L^2(X)$ given by\n\\begin{equation}\\label{eq:crp}\n\\mathcal{A}\\rtimes X\\equiv\\mathcal{A}\\cdot \\mathscr{T}_X=\\mathscr{T}_X\\cdot\\mathcal{A}.\n\\end{equation}\nCrossed products of the form $\\mathcal{C}_X(Y)\\rtimes X$ where $Y$ is a\nclosed subgroup of $X$ play an important role in the many-body\nproblem. To simplify notations we set\n\\begin{equation}\\label{eq:Cxy}\n\\mathscr{C}_X(Y)=\\mathcal{C}_X(Y)\\rtimes X=\\mathcal{C}_X(Y)\\cdot\\mathscr{T}_X=\n\\mathscr{T}_X\\cdot\\mathcal{C}_X(Y).\n\\end{equation}\nIf $X=Y\\oplus Z$ and if we identify $L^2(X)=L^2(Y)\\otimes L^2(Z)$ then\n$\\mathscr{T}_X=\\mathscr{T}_Y\\otimes\\mathscr{T}_Z$ hence\n\\begin{equation}\\label{eq:crxyz}\n\\mathscr{C}_X(Y)=\\mathscr{T}_Y\\otimes \\mathscr{K}_Z.\n\\end{equation}\nA useful ``symmetric'' description of $\\mathscr{C}_X(Y)$ is contained in the\nnext lemma. Let $Y^{(2)}$ be the closed subgroup of $X^2\\equiv X\\oplus\nX$ consisting of elements of the form $(y,y)$ with $y\\in Y$.\n\n\\begin{lemma}\\label{lm:sym}\n$\\mathscr{C}_X(Y)$ is the closure of the\nset of integral operators with kernels $\\theta\\in\\cc_{\\mathrm{c}}(X^2\/Y^{(2)})$.\n\\end{lemma}\n\\proof Let $\\mathscr{C}$ be the norm closure of the set of integral\noperators with kernels $\\theta\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X^2)$ having the properties:\n(1) $\\theta(x+y,x'+y)=\\theta(x,x')$ for all $x,x'\\in X$ and $y\\in\nY$; (2) $\\mbox{\\rm supp\\! }\\theta\\subset K_\\theta+Y$ for some compact\n$K_\\theta\\subset X^2$. We show $\\mathscr{C}=\\mathscr{C}_X(Y)$.  Observe that the map\nin $X^2$ defined by $(x,x')\\mapsto(x-x',x')$ is a topological group\nisomorphism with inverse $(x_1,x_2)\\mapsto(x_1+x_2,x_2)$ and sends\nthe subgroup $Y^{(2)}$ onto the subgroup $\\{0\\}\\oplus Y$. This map\ninduces an isomorphism $X^2\/Y^{(2)}\\simeq X\\oplus(X\/Y)$. Thus any\n$\\theta\\in\\cc_{\\mathrm{c}}(X^2\/Y^{(2)})$ is of the form\n$\\theta(x,x')=\\widetilde\\theta(x-x',x')$ for some\n$\\widetilde\\theta\\in\\cc_{\\mathrm{c}}(X\\oplus(X\/Y))$. Thus $\\mathscr{C}$ is the closure in\n$\\mathscr{L}_X$ of the set of operators of the form\n$(Tu)(x)=\\int_X\\widetilde\\theta(x-x',x')u(x') \\text{d} x'$. Since we may\napproximate $\\widetilde\\theta$ with linear combinations of functions of\nthe form $a\\otimes b$ with $a\\in\\cc_{\\mathrm{c}}(X), b\\in\\cc_{\\mathrm{c}}(X\/Y)$ we see that\n$\\mathscr{C}$ is the clspan of the set of operators of the form\n$(Tu)(x)=\\int_X a(x-x')b(x')u(x') \\text{d} x'$. But this clspan is\n$\\mathscr{T}_X\\cdot\\mathcal{C}_X(Y)=\\mathscr{C}_X(Y)$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Compatible subgroups}\n\\label{ss:compat}\n\nIf $X,Y$ is an arbitrary pair of lca groups then $X\\oplus Y$ is the\nset $X\\times Y$ equipped with the product topology and group\nstructure. If $X,Y$ are closed subgroups of an lca group $G$ and if\nthe map $Y\\oplus Z\\to Y+Z$ defined by $(y,z)\\mapsto y+z$ is open, we\nsay that they are \\emph{compatible subgroups of $G$}.  In this case\n$Y+Z$ is a closed subgroup of $X$.\n\n\n\\begin{remark}\\label{re:DP}{\\rm\nIf $G$ is $\\sigma$-compact then $X,Y$ are compatible if and only if\n$X+Y$ is closed. Indeed, a continuous surjective morphism between\ntwo locally compact $\\sigma$-compact groups is open and a subgroup\n$H$ of a locally compact group $G$ is closed if and only if $H$ is\nlocally compact for the induced topology, see Theorems 5.11 and 5.29\nin \\cite{HR}.  We thank Lo\\\"ic Dubois and Benoit Pausader for\nenlightening discussions on this matter.\n}\\end{remark}\n\n\nThe importance of the compatibility condition in the context of\ngraded $C^*$-algebras has been pointed out in \\cite[Lemma 6.1.1]{Ma}\nand one may find there several descriptions of this condition (see\nalso Lemma 3.1 from \\cite{Ma3}). We quote two of them. Let $X\/Y$ be\nthe image of $X$ in $G\/Y$ considered as a subgroup of $G\/Y$ equipped\nwith the induced topology. The group $X\/(X\\cap Y)$ is equipped with\nthe locally compact quotient topology and we have a natural map\n$X\/(X\\cap Y)\\to X\/Y$ which is a bijective continuous group morphism.\nThen $X,Y$ are compatible if and only if the following equivalent\nconditions are satisfied:\n\\begin{align}\n& \\text{the natural map} \\hspace{2mm} X\/(X\\cap Y)\\to X\/Y \\hspace{2mm}\n\\text{is a homeomorphism}, \\label{eq:ma1} \\\\\n&\n\\text{the natural map }\nG\/(X\\cap Y)\\to G\/X\\times G\/Y \\hspace{1mm} \\text{is closed}.\n\\label{eq:ma2}\n\\end{align}\n\nIf $\\mathcal{A}$ is a $G$-algebra let $\\mathcal{A}|_X$ be the set of restrictions\nto $X$ of the functions from $\\mathcal{A}$. This is an $X$-algebra.\n\n\\begin{lemma}\\label{lm:reg}\nIf $X,Y$ are compatible subgroups of $G$ then\n\\begin{align}\n& \\mathcal{C}_G(X)\\cdot\\mathcal{C}_G(Y) = \\mathcal{C}_G(X\\cap Y)  \\label{eq:reg1}\\\\\n& \\mathcal{C}_G(Y)|_X =  \\mathcal{C}_X(X\\cap Y).\n\\label{eq:reg2}\n\\end{align}\nThe second relation remains valid for the subalgebras $\\cc_{\\mathrm{c}}$.\n\\end{lemma}\n\\proof The fact that the inclusion $\\subset$ in \\eqref{eq:reg1} is\nequivalent to the compatibility of $X$ and $Y$ is shown in Lemma\n6.1.1 from \\cite{Ma}, so we only have to prove that the equality\nholds. Let $E = (G\/X) \\times (G\/Y)$. If $\\varphi \\in \\cc_{\\mathrm{o}}(G\/X)$ and\n$\\psi \\in \\cc_{\\mathrm{o}}(G\/Y)$ then $\\varphi \\otimes \\psi$ denotes the function\n$(s, t) \\longmapsto \\varphi(s) \\psi(t)$, which belongs to\n$\\cc_{\\mathrm{o}}(E)$. The subspace generated by the functions of the form\n$\\varphi \\otimes \\psi$ is dense in $\\cc_{\\mathrm{o}}(E)$ by the Stone-Weierstrass\ntheorem. If $F$ is a closed subset of $E$ then, by the Tietze\nextension theorem, each function in $\\cc_{\\mathrm{c}}(F)$ extends to a function\nin $\\cc_{\\mathrm{c}}(E)$, so the restrictions $(\\varphi \\otimes \\psi)|_F$\ngenerate a dense linear subspace of $\\cc_{\\mathrm{o}}(F)$.  Let us denote by\n$\\pi$ the map $x \\mapsto (\\pi_X(x), \\pi_Y(x))$, so $\\pi$ is a group\nmorphism from $G$ to $E$ with kernel $V=X\\cap Y$.  Then by\n\\eqref{eq:ma2} the range $F$ of $\\pi$ is closed and the quotient map\n$\\widetilde\\pi : G\/V \\to F$ is a continuous and closed bijection, hence\nis a homeomorphism. So $\\theta \\mapsto \\theta \\circ \\tilde \\pi$ is\nan isometric isomorphism of $\\cc_{\\mathrm{o}}(F)$ onto $\\cc_{\\mathrm{o}}(G\/V)$. Hence for\n$\\varphi \\in \\cc_{\\mathrm{o}}(G\/X)$ and $\\psi \\in \\cc_{\\mathrm{o}}(G\/Y)$ the function $\\theta\n= (\\varphi \\otimes \\psi) \\circ \\tilde \\pi$ belongs to $\\cc_{\\mathrm{o}}(G\/V)$, it\nhas the property $\\theta \\circ \\pi_V = \\varphi \\circ \\pi_X \\cdot\n\\psi \\circ \\pi_Y$, and the functions of this form generate a dense\nlinear subspace of $\\cc_{\\mathrm{o}}(G\/V)$.\n\nNow we prove \\eqref{eq:reg2}. Recall that we identify $\\mathcal{C}_G(Y)$\nwith a subset of $\\cc_{\\mathrm{b}}^{\\mathrm{u}}(G)$ by using $\\varphi\\mapsto\\varphi\\circ\\pi_Y$\nso in terms of $\\varphi$ the restriction map which defines\n$\\mathcal{C}_G(Y)|_X$ is just $\\varphi\\mapsto\\varphi|_{X\/Y}$. Thus we have a\ncanonical embedding $\\mathcal{C}_G(Y)|_X\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)$ for an arbitrary\npair $X,Y$ . Then the continuous bijective group morphism\n$\\theta:X\/(X\\cap Y)\\to X\/Y$ allows us to embed\n$\\mathcal{C}_G(Y)|_X\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/(X\\cap Y))$. That the range of this map is\nnot $\\mathcal{C}_X(X\\cap Y)$ in general is clear from the example $G=\\mathbb{R},\nX=\\pi\\mathbb{Z},Y=\\mathbb{Z}$. But if $X,Y$ are compatible then $X\/Y$ is closed\nin $G\/Y$, so $\\mathcal{C}_G(Y)|_X=\\cc_{\\mathrm{o}}(X\/Y)$ by the Tietze extension theorem,\nand $\\theta$ is a homeomorphism, hence we get \\eqref{eq:reg2}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\\begin{lemma}\\label{lm:double}\nIf $X,Y$ are compatible subgroups of $G$ then $X^2=X\\oplus X$ and\n$Y^{(2)}=\\{(y,y)\\mid y\\in Y\\}$ is a compatible pair of closed\nsubgroups of $G^2=G\\oplus G$.\n\\end{lemma}\n\\proof Let $D=X^2\\cap Y^{(2)}=\\{(x,x)\\mid x\\in X\\cap Y\\}$. Due to to\n\\eqref{eq:ma1} it suffices to show that the natural map\n$Y^{(2)}\/D\\to Y^{(2)}\/X^2$ is a homeomorphism.  Here $Y^{(2)}\/X^2$\nis the image of $Y^{(2)}$ in $G^2\/X^2\\cong (G\/X)\\oplus(G\/X)$, more\nprecisely it is the subset of pairs $(a,a)$ with $a=\\pi_X(z)$ and\n$z\\in Y$, equipped with the topology induced by\n$(G\/X)\\oplus(G\/X)$. Thus the natural map $Y\/X\\to Y^{(2)}\/X^2$ is a\nhomeomorphism. On the other hand, the natural map $Y\/(X\\cap Y)\\to\nY^{(2)}\/D$ is clearly a homeomorphism.  To finish the proof note\nthat $Y\/(X\\cap Y)\\to Y\/X$ is a homeomorphism because $X,Y$ is a\nregular pair.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{lemma}\\label{lm:regp}\n  Let $X,Y$ be compatible subgroups of an lca group $G$ and let\n  $X^\\perp,Y^\\perp$ be their orthogonals in $G^*$.  Then $(X\\cap\n  Y)^\\perp=X^\\perp+Y^\\perp$ and the closed subgroups\n  $X^\\perp,Y^\\perp$ of $G^*$ are compatible.\n\\end{lemma}\n\\proof $X+Y$ is closed and, since\n$(x,y)\\mapsto(x,-y)$ is a homeomorphism, the map $S:X\\oplus Y\\to\nX+Y$ defined by $S(x,y)=x+y$ is an open surjective morphism. Then\nfrom the Theorem 9.5, Chapter 2 of \\cite{Gu} it follows that the\nadjoint map $S^*$ is a homeomorphism between $(X+Y)^*$ and its\nrange. In particular its range is a locally compact subgroup for the\ntopology induced by $X^*\\oplus Y^*$ hence is a closed subgroup of\n$X^*\\oplus Y^*$, see Remark \\ref{re:DP}.  We\nhave $(X+Y)^\\perp=X^\\perp\\cap Y^\\perp$, cf. 23.29 in \\cite{HR}. Thus\nfrom $X^*\\cong G^*\/X^\\perp$ and similar representations for $Y^*$\nand $(X+Y)^*$ we see that\n$$\nS^*:G^*\/(X^\\perp \\cap Y^\\perp)\\to G^*\/X^\\perp\\oplus G^*\/Y^\\perp\n$$ is a closed map. But $S^*$ is clearly the natural map involved in\n\\eqref{eq:ma2}, hence the pair $X^\\perp,Y^\\perp$ is\nregular. Finally, note that $(X\\cap Y)^\\perp$ is always equal to the\nclosure of the subgroup $X^\\perp+Y^\\perp$, cf. 23.29 and 24.10 in\n\\cite{HR}, and in our case $X^\\perp+Y^\\perp$ is closed.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Green Hilbert $C^*$-modules} \n\\label{ss:hilbert} \n\nLet $X,Y$ be a compatible pair of closed subgroups of a locally\ncompact abelian group $G$. Then the subgroup $X+Y$ of $G$ generated\nby $X\\cup Y$ is also closed. If we identify $X\\cap Y$ with the\nclosed subgroup $D$ of $X\\oplus Y$ consisting of the elements of the\nform $(z,z)$ with $z\\in X\\cap Y$ then the quotient group $X\\uplus Y\n\\equiv (X\\oplus Y)\/(X\\cap Y)$ is locally compact and the map\n\\begin{equation}\\label{eq:nat}\n\\phi:X\\oplus Y \\to X+Y \\hspace{2mm}\\text{defined by}\\hspace{2mm}\n \\phi(x,y)=x-y\n\\end{equation}\nis an open continuous surjective group morphism $X\\oplus Y\\to X+Y$\nwith $X\\cap Y$ as kernel.  Hence the group morphism\n$\\phi^\\circ:X\\uplus Y\\to X+Y$ induced by $\\phi$ is a homeomorphism.\n\n\nSince $\\cc_{\\mathrm{c}}(X\\uplus Y)\\subset \\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\\oplus Y)$ the elements\n$\\theta\\in\\cc_{\\mathrm{c}}(X\\uplus Y)$ are functions $\\theta:X\\times Y\\to\\mathbb{C}$\nand we may think of them as kernels of integral operators.\n\n\\begin{lemma}\\label{lm:bound}\n  If $\\theta\\in\\cc_{\\mathrm{c}}(X\\uplus Y)$ then $(T_\\theta\n  u)(x)=\\int_Y\\theta(x,y)u(y) \\text{d} y$ defines an operator in\n  $\\mathscr{L}_{XY}$ with norm $\\|T_\\theta\\|\\leq C\\sup|\\theta|$ where $C$\n  depends only on a compact which contains the support of $\\theta$.\n\\end{lemma}\n\\proof \nBy the Schur test\n$$\n\\|T_\\theta\\|^2\\leq \n{\\textstyle\\sup_{x\\in X}}\\int_Y|\\theta(x,y)\\text{d}  y \\cdot\n{\\textstyle\\sup_{y\\in Y}}\\int_X|\\theta(x,y)\\text{d} x.\n$$ \nLet $K\\subset X$ and $L\\subset Y$ be compact sets such that\n$(K\\times L) + D$ contains the support of $\\theta$.\nThus if $\\theta(x,y)\\neq0$ then $x\\in z+K$ and $y\\in z+L$ for some \n$z\\in X\\cap Y$ hence $ \\int_Y|\\theta(x,y)\\text{d} y \\leq\n\\sup|\\theta| \\lambda_Y(L).  $ Similarly $\\int_X|\\theta(x,y)\\text{d} x\n\\leq \\sup|\\theta| \\lambda_X(K)$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{definition}\\label{df:ryz}\n$\\mathscr{T}_{XY}$ is the norm closure in $\\mathscr{L}_{XY}$ of the set of operators\n$T_\\theta$ as in Lemma \\ref{lm:bound}.\n\\end{definition}\n\n\n\n\\begin{remark}\\label{re:rief}\n  If $X\\supset Y$ then $\\mathscr{T}_{XY}$ is a ``concrete'' realization of\n  the Hilbert $C^*$-module introduced by Rieffel in \\cite{Ri} which\n  implements the Morita equivalence between the group $C^*$-algebra\n  $\\mathcal{C}^*(Y)$ and the crossed product $\\cc_{\\mathrm{o}}(X\/Y)\\rtimes X$. More\n  precisely, $\\mathscr{T}_{XY}$ is a Hilbert $\\mathcal{C}^*(Y)$-module and its\n  imprimitivity algebra is canonically isomorphic with\n  $\\cc_{\\mathrm{o}}(X\/Y)\\rtimes X$.  If $X,Y$ is an arbitrary couple of\n  compatible subgroups of $G$ then we defined $\\mathscr{T}_{XY}$ such that\n  $\\mathscr{T}_{XY}=\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{GY}$. On the other hand, from\n  \\eqref{eq:factor} we get $\\mathscr{T}_{XY}=\\mathscr{T}_{XE}\\cdot\\mathscr{T}_{EY}$ with\n  $E=X\\cap Y$, hence $\\mathscr{T}_{XY}$ is naturally a Hilbert\n  $(\\cc_{\\mathrm{o}}(X\/E)\\rtimes X,\\cc_{\\mathrm{o}}(Y\/E)\\rtimes Y)$ imprimitivity bimodule.\n  It has been noticed by Georges Skandalis that $\\mathscr{T}_{XY}$ is in\n  fact a ``concrete'' realization of a Hilbert $C^*$-module\n  introduced by Green to show the Morita equivalence of the\n  $C^*$-algebras $\\cc_{\\mathrm{o}}(Z\/Y)\\rtimes X$ and $\\cc_{\\mathrm{o}}(Z\/X)\\rtimes Y$ where\n  we take $Z=X+Y$, cf. \\cite[Example 4.13]{Wi}.\n\\end{remark}\n\n\nWe give now an alternative definition of $\\mathscr{T}_{XY}$. If\n$\\varphi\\in\\cc_{\\mathrm{c}}(G)$ we define $T_{XY}(\\varphi):\\cc_{\\mathrm{c}}(Y)\\to\\cc_{\\mathrm{c}}(X)$ by\n\\begin{equation}\\label{eq:ryz}\n(T_{XY}(\\varphi)u)(x)=\\int_Y\\varphi(x-y)u(y)\\text{d} y.\n\\end{equation}\nThis operator depends only the restriction $\\varphi|_{X+Y}$ hence,\nby the Tietze extension theorem, we could take $\\varphi\\in\\cc_{\\mathrm{c}}(Z)$\ninstead of $\\varphi\\in\\cc_{\\mathrm{c}}(G)$, where $Z$ is any closed subgroup of\n$G$ containing $X\\cup Y$.\n\n\\begin{proposition}\\label{pr:def2}\n$T_{XY}(\\varphi)$ extends to a bounded operator $L^2(Y)\\to L^2(X)$,\nalso denoted $T_{XY}(\\varphi)$, and for each compact $K\\subset G$\nthere is a constant $C$ such that if $\\mbox{\\rm supp\\! }\\varphi\\subset K$\n\\begin{equation}\\label{eq:nyz}\n\\|T_{XY}(\\varphi)\\|\\leq C \\sup\\nolimits_{x\\in G}|\\varphi(x)|.\n\\end{equation}\nThe adjoint operator is given by $T_{XY}(\\varphi)^*=\nT_{YX}(\\varphi^*)$ where $\\varphi^*(x)=\\bar\\varphi(-x)$.  The space\n$\\mathscr{T}_{XY}$ coincides with the closure in $\\mathscr{L}_{XY}$ of the set of\noperators of the from $T_{XY}(\\varphi)$.\n\\end{proposition}\n\\proof The set $X+Y$ is closed in $G$ hence the restriction map\n$\\cc_{\\mathrm{c}}(G)\\to\\cc_{\\mathrm{c}}(X+Y)$ is surjective. On the other hand, the map\n$\\phi^\\circ:X\\uplus Y\\to X+Y$, defined after \\eqref{eq:nat}, is a\nhomeomorphism so it induces an isomorphism\n$\\varphi\\to\\varphi\\circ\\phi^\\circ$ of $\\cc_{\\mathrm{c}}(X+Y)$ onto $\\cc_{\\mathrm{c}}(X\\uplus\nY)$. Clearly $T_{XY}(\\varphi)=T_\\theta$ if $\\theta=\\varphi\\circ\\phi$,\nso the proposition follows from Lemma \\ref{lm:bound}.  \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nWe discuss now some properties of the spaces $\\mathscr{T}_{XY}$.\nWe set $\\mathscr{T}_{XY}^*\\equiv(\\mathscr{T}_{XY})^*\\subset\\mathscr{L}_{YX}$. \n\n\\begin{proposition}\\label{pr:nyza}\nWe have  $\\mathscr{T}_{XX}=\\mathscr{T}_X$ and:\n\\begin{align}\n& \\mathscr{T}_{XY}^* =\\mathscr{T}_{YX} \\label{eq:rad} \\\\\n& \\mathscr{T}_{XY}   =\\mathscr{T}_{XY}\\cdot \\mathscr{T}_Y=\\mathscr{T}_X\\cdot\\mathscr{T}_{XY}\n\\label{eq:cyzc}  \\\\\n& \\mathcal{A}|_X\\cdot\\mathscr{T}_{XY} =\\mathscr{T}_{XY}\\cdot\\mathcal{A}|_Y \\label{eq:ayza}\n\\end{align}\nwhere $\\mathcal{A}$ is an arbitrary  $G$-algebra.\n\\end{proposition}\n\\proof The relations $\\mathscr{T}_{XX}=\\mathscr{T}_X$ and \\eqref{eq:rad} are\nobvious.  Now we prove the first equality in \\eqref{eq:cyzc} (then\nthe second one follows by taking adjoints). If $C(\\eta)$ is the\noperator of convolution in $L^2(Y)$ with $\\eta\\in \\cc_{\\mathrm{c}}(Y)$ then a\nshort computation gives\n\\begin{equation}\\label{eq:yzc}\nT_{XY}(\\varphi)C(\\eta)=T_{XY}(T_{G Y}(\\varphi)\\eta)\n\\end{equation}\nfor $\\varphi\\in\\cc_{\\mathrm{c}}(G)$. Since $T_{G Y}(\\varphi)\\eta\\in\\cc_{\\mathrm{c}}(G)$ we get\n$T_{XY}(\\varphi)C(\\eta)\\in\\mathscr{T}_{GX}$, so $\\mathscr{T}_{XY}\\cdot\n\\mathscr{T}_Y\\subset\\mathscr{T}_{XY}$. The converse follows by a standard\napproximation argument.\n\nLet $\\varphi\\in\\cc_{\\mathrm{c}}(G)$ and $\\theta\\in\\mathcal{A}$. We shall denote by\n$\\theta(Q_X)$ the operator of multiplication by $\\theta|_X$ in\n$L^2(X)$ and by $\\theta(Q_Y)$ that of multiplication by $\\theta|_Y$\nin $L^2(Y)$.  Choose some $\\varepsilon>0$ and let $V$ be a compact\nneighborhood of the origin in $G$ such that\n$|\\theta(z)-\\theta(z')|<\\varepsilon$ if $z-z'\\in V$. There are\nfunctions $\\alpha_k\\in\\cc_{\\mathrm{c}}(G)$ with $0\\leq\\alpha_k\\leq1$ such that\n$\\sum_k\\alpha_k=1$ on the support of $\\varphi$ and\n$\\mbox{\\rm supp\\! }\\alpha_k\\subset z_k+V$ for some points $z_k$.  Below we shall\nprove:\n\\begin{equation}\\label{eq:byza}\n\\|T_{XY}(\\varphi)\\theta(Q_Y)- \n{\\textstyle\\sum_k}\\theta(Q_X-z_k)T_{XY}(\\varphi\\alpha_k)\\| \n\\leq\n\\varepsilon\\|T_{XY}(|\\varphi|)\\|.\n\\end{equation}\nThis implies $\\mathscr{T}_{XY}\\cdot\\mathcal{A}|_Y\\subset\\mathcal{A}|_X\\cdot\\mathscr{T}_{XY}$. If we\ntake adjoints, use \\eqref{eq:rad} and interchange $X$ and $Y$ in the\nfinal relation, we obtain $\\mathcal{A}|_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{A}|_Y$\nhence the proposition is proved.  For $u\\in\\cc_{\\mathrm{c}}(X)$ we have:\n\\begin{align*}\n(T_{XY}(\\varphi)\\theta(Q_Y)u)(x) &=\n\\int_Y\\varphi(x-y)\\theta(y)u(y)\\text{d} y \n=\\sum_k\\int_Y\\varphi(x-y)\\alpha_k(x-y)\\theta(y)u(y)\\text{d} y \\\\\n&=\n\\sum_k\\int_Y\\varphi(x-y)\\alpha_k(x-y)\\theta(x-z_k)u(y)\\text{d} y\n+(Ru)(x)\\\\ \n&= \n\\sum_k\\left(\\theta(Q_X-z_k)T_{XY}(\\varphi\\alpha_k)u\\right)(x) \n+(Ru)(x).\n\\end{align*}\nWe can estimate the remainder as follows\n$$\n|(Ru)(x)|=\\left|\\sum_k\\int_Y\\varphi(x-y)\\alpha_k(x-y)\n[\\theta(y)-\\theta(x-z_k)]u(y)\\text{d} y \\right|\\leq\n\\varepsilon\\int_Y|\\varphi(x-y)u(y)|\\text{d} y.\n$$ \nbecause $x-z_k-y\\in V$.  This proves \\eqref{eq:byza}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\n\\begin{proposition}\\label{pr:ryz}\n $\\mathscr{T}_{XY}$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ and\n\\begin{equation}\\label{eq:hyz}\n\\mathscr{T}_{XY}^*\\cdot\\mathscr{T}_{XY}=\\mathscr{C}_Y(X\\cap Y), \\hspace{2mm}\n\\mathscr{T}_{XY}\\cdot\\mathscr{T}_{XY}^*=\\mathscr{C}_X(X\\cap Y).\n\\end{equation}\nThus $\\mathscr{T}_{XY}$ is a $(\\mathscr{C}_X(X\\cap Y),\\mathscr{C}_Y(X\\cap Y))$ imprimitivity\nbimodule.\n\\end{proposition}\n\\proof Due to \\eqref{eq:rad}, to prove the first relation in\n\\eqref{eq:hyz} we have to compute the clspan $\\mathscr{C}$ of the operators\n$T_{XY}(\\varphi)T_{YX}(\\psi)$ with $\\varphi,\\psi$ in $\\cc_{\\mathrm{c}}(G)$.  We\nrecall the notation $G^2=G\\oplus G$, this is a locally compact\nabelian group and $X^2=X\\oplus X$ is a closed subgroup. Let us\nchoose functions $\\varphi_k,\\psi_k\\in\\cc_{\\mathrm{c}}(G)$ and let\n$\\Phi=\\sum_k\\varphi_k\\otimes\\psi_k\\in\\cc_{\\mathrm{c}}(G^2)$. If\n$\\psi_k^\\dag(x)=\\psi_k(-x)$, then $\\sum_k\nT_{XY}(\\varphi_k)T_{YX}(\\psi_k^\\dag)$ is an integral operator on\n$L^2(X)$ with kernel $\\theta_X=\\theta|_{X^2}$ where $\\theta:G^2\\to\n\\mathbb{C}$ is given by\n$$ \n\\theta(x,x')= \\int_Y\\Phi(x+y,x'+y)\\text{d} y.\n$$ Since the set of decomposable functions is dense in $\\cc_{\\mathrm{c}}(G^2)$ in\nthe inductive limit topology, an easy approximation argument shows\nthat $\\mathscr{C}$ contains all integral operators with kernels of the same\nform as $\\theta_X$ but with arbitrary $\\Phi\\in\\cc_{\\mathrm{c}}(G^2)$.  Let\n$Y^{(2)}$ be the closed subgroup of $G^2$ consisting\nof the elements $(y,y)$ with $y\\in Y$. Then $K=\\mbox{\\rm supp\\! }\\Phi\\subset G^2$\nis a compact, $\\theta$ is zero outside $K+Y^{(2)}$, and\n$\\theta(a+b)=\\theta(a)$ for all $a\\in G^2,b\\in Y^{(2)}$. Thus\n$\\theta\\in\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})$, with the usual identification\n$\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(G^2)$. From Proposition 2.48 in\n\\cite{Fo} it follows that reciprocally, any function $\\theta$ in\n$\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})$ can be represented in terms of some $\\Phi$ in\n$\\cc_{\\mathrm{c}}(G^2)$ as above. Thus $\\mathscr{C}$ is the closure of the set of\nintegral operators on $L^2(X)$ with kernels of the form $\\theta_X$\nwith $\\theta\\in\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})$. According to Lemma\n\\ref{lm:double}, the pair of subgroups $X^2,Y^{(2)}$ is regular, so\nwe may apply Lemma \\ref{lm:reg} to get\n$\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})|_{X^2}=\\cc_{\\mathrm{c}}(X^2\/D)$ where $D=X^2\\cap\nY^{(2)}=\\{(x,x)\\mid x\\in X\\cap Y\\}$.  But by Lemma \\ref{lm:sym} the\nnorm closure in $\\mathscr{L}_X$ of the set of integral operators with\nkernel in $\\cc_{\\mathrm{c}}(X^2\/D)$ is $\\mathscr{C}_X\/(X\\cap Y)$. This proves\n\\eqref{eq:hyz}.\n\nIt remains to prove that $\\mathscr{T}_{XY}$ is a Hilbert $C^*$-submodule of\n$\\mathscr{L}_{XY}$, i.e. that we have\n\\begin{equation}\\label{eq:hyz1}\n\\mathscr{T}_{XY}\\cdot\\mathscr{T}_{XY}^*\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}.\n\\end{equation}\nThe first identity in \\eqref{eq:hyz} and \\eqref{eq:cyzc} imply\n\\begin{equation*}\n\\mathscr{T}_{XY}\\cdot\\mathscr{T}_{XY}^*\\cdot\\mathscr{T}_{XY} = \n\\mathscr{T}_{XY}\\cdot \\mathscr{T}_Y\\cdot\\mathcal{C}_Y(X\\cap Y)=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y).\n\\end{equation*}\nFrom Lemma \\ref{lm:reg} we get \n$$\n\\mathcal{C}_Y(X\\cap Y)=\\mathcal{C}_G(X\\cap Y)|_Y =\n\\mathcal{C}_G(X)|_Y \\cdot \\mathcal{C}_G(Y)|_Y = \\mathcal{C}_G(X)|_Y \n$$ because $\\mathcal{C}_G(Y)|_Y=\\mathbb{C}$.  Then by using Proposition\n\\ref{pr:nyza} we obtain\n$$\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y) = \\mathscr{T}_{XY}\\cdot\\mathcal{C}_G(X)|_Y =\n\\mathcal{C}_G(X)|_X \\cdot\\mathscr{T}_{XY} =\\mathscr{T}_{XY}\n$$\nbecause $\\mathcal{C}_G(X)|_X=\\mathbb{C}$. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{corollary}\\label{co:txy}\nWe have\n\\begin{align}\n\\mathscr{T}_{XY} &= \\mathscr{T}_{XY}\\mathscr{T}_Y=\\mathscr{T}_{XY}\\mathcal{C}_Y(X\\cap Y) \n\\label{eq:txy1}\n\\\\\n&= \\mathscr{T}_X\\mathscr{T}_{XY}=\\mathcal{C}_X(X\\cap Y)\\mathscr{T}_{XY}.\n\\label{eq:txy2}\n\\end{align}\n\\end{corollary}\n\\proof If $\\mathscr{M}$ is a Hilbert $\\mathscr{A}$-module then $\\mathscr{M}=\\mathscr{M}\\mathscr{A}$ hence\n$\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\mathscr{C}_Y(X\\cap Y)$ by Proposition \\ref{pr:ryz}. The\nspace $\\mathscr{C}_Y(X\\cap Y)$ is a $\\mathscr{T}_Y$-bimodule and $\\mathscr{C}_Y(X\\cap\nY)=\\mathscr{C}_Y(X\\cap Y)\\cdot \\mathscr{T}_Y$ by \\eqref{eq:Cxy} hence we get\n$\\mathscr{C}_Y(X\\cap Y)=\\mathscr{C}_Y(X\\cap Y)\\mathscr{T}_Y$ by the Cohen-Hewitt theorem.\nThis proves the first equality in \\eqref{eq:txy1} and the other ones\nare proved similarly.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nIf $\\mathcal{G}$ is a set of closed subgroups of $G$ then the\n\\emph{semilattice generated by $\\mathcal{G}$} is the set of finite\nintersections of elements of $\\mathcal{G}$.\n\n\\begin{proposition}\\label{pr:product}\nLet $X,Y,Z$ be closed subgroups of $G$ such that any two subgroups\nfrom the semilattice generated by the family $\\{X,Y,Z\\}$ are\ncompatible. Then:\n\\begin{align}\\label{eq:product}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY} &=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap Z)= \\mathcal{C}_X(X\\cap Z)\\cdot\\mathscr{T}_{XY} \\\\\n&= \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)= \n\\mathcal{C}_X(X\\cap Y\\cap Z)\\cdot\\mathscr{T}_{XY}.\n\\end{align} \nIn particular, if $Z\\supset X\\cap Y$ then\n\\begin{equation}\\label{eq:factor}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}.\n\\end{equation}\n\\end{proposition}\n\\proof We first prove \\eqref{eq:factor} in the particular case\n$Z=G$. As in the proof of Proposition \\ref{pr:ryz} we see that\n$\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{G Y}$ is the the closure in $\\mathscr{L}_{XY}$ of the\nset of integral operators with kernels \n$\\theta_{XY}=\\theta|_{X\\times Y}$ where $\\theta:G^2\\to \\mathbb{C}$ is\ngiven by \n$$ \n\\theta(x,y)= \\int_G\\sum_k\\varphi_k(x-z)\\psi_k(z-y)\\text{d} z=\n\\int_G\\sum_k\\varphi_k(x-y-z)\\psi_k(z)\\text{d} z\\equiv\\xi(x-y)\n$$ where $\\varphi_k,\\psi_k\\in\\cc_{\\mathrm{c}}(G)$ and $\\xi=\\sum_k\\varphi_k*\\psi_k$\nconvolution product on $G$. Since $\\cc_{\\mathrm{c}}(G)*\\cc_{\\mathrm{c}}(G)$ is dense in\n$\\cc_{\\mathrm{c}}(G)$ in the inductive limit topology, the space\n$\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{G Y}$ is the the closure of the set of integral\noperators with kernels $\\theta(x,y)=\\xi(x-y)$ with $\\xi\\in\\cc_{\\mathrm{c}}(G)$.\nBy Proposition \\ref{pr:def2} this is  $\\mathscr{T}_{XY}$. \n\nNow we prove \\eqref{eq:product}. From \\eqref{eq:factor} with $Z=G$\nand \\eqref{eq:hyz} we get:\n\\begin{equation*}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY} =\n\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{G Z}\\cdot\\mathscr{T}_{ZG}\\cdot\\mathscr{T}_{G Y}\n= \\mathscr{T}_{XG}\\cdot\\mathcal{C}_G(Z)\\cdot \\mathscr{T}_G\\cdot\\mathscr{T}_{GY}.\n\\end{equation*}\nThen from Proposition \\eqref{pr:nyza} and Lemma \\ref{lm:reg} we get:\n$$\n\\mathcal{C}_G(Z)\\cdot \\mathscr{T}_G\\cdot\\mathscr{T}_{GY}=\\mathcal{C}_G(Z)\\cdot\\mathscr{T}_{G Y}=\n\\mathscr{T}_{G Y}\\cdot\\mathcal{C}_G(Z)|_Y= \\mathscr{T}_{G Y}\\cdot\\mathcal{C}_Y(Y\\cap Z).\n$$ \nWe obtain \\eqref{eq:product} by using once again\n\\eqref{eq:factor} with $Z=G$ and taking adjoints. On the other hand,\nthe relation $\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y)$ holds because\nof \\eqref{eq:txy1}, so we have\n$$\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap Z)=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y)\\cdot\\mathcal{C}_Y(Y\\cap Z)=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\n$$ where we also used \\eqref{eq:reg1} and the fact that $X\\cap Y$,\n$Z\\cap Y$ are compatible. Finally, to get \\eqref{eq:factor} for\n$Z\\supset X\\cap Y$ we use once again \\eqref{eq:hyz}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nThe object of main interest for us is introduced in the next\ndefinition. \n\n\\smallskip\n\n\\begin{definition}\\label{df:nxyz}\nIf $X,Y$ are compatible subgroups and $Z$ is a closed subgroup of\n$X\\cap Y$ then we set\n\\begin{equation}\\label{eq:nxyz}\n\\mathscr{C}_{XY}(Z):=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Z)=\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}.\n\\end{equation} \n\\end{definition}\nThe equality above follows from \\eqref{eq:ayza} with $\\mathcal{A}=\\mathcal{C}_G(Z)$.\nWe clearly have $\\mathscr{C}_{XY}(X\\cap Y)=\\mathscr{T}_{XY}$ and\n$\\mathscr{C}_{XX}(Y)=\\mathscr{C}_X(Y)$ if $X\\supset Y$. Moreover\n\\begin{equation}\\label{eq:nadj}\n\\mathscr{C}_{XY}^*(Z):=\\mathscr{C}_{XY}(Z)^*=\\mathscr{C}_{YX}(Z)\n\\end{equation}\nbecause of \\eqref{eq:rad}.\n\n\n\\begin{theorem}\\label{th:nxyz}\n$\\mathscr{C}_{XY}(Z)$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ such that\n\\begin{equation}\\label{eq:nnz}\n\\mathscr{C}_{XY}^*(Z)\\cdot\\mathscr{C}_{XY}(Z)=\\mathscr{C}_Y(Z)\n\\hspace{2mm}\\text{and}\\hspace{2mm}\n\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{XY}^*(Z)=\\mathscr{C}_X(Z).\n\\end{equation} \nIn particular, $\\mathscr{C}_{XY}(Z)$ is a\n$(\\mathscr{C}_X(Z),\\mathscr{C}_Y(Z))$ imprimitivity bimodule.\n\\end{theorem}\n\\proof \nBy using \\eqref{eq:nadj}, the definition \\eqref{eq:nxyz}, and\n\\eqref{eq:reg1} we get\n\\begin{align*}\n\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{YX}(Z) &=\n\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}\\cdot \\mathscr{T}_{YX}\\cdot\\mathcal{C}_X(Z)\\\\\n&=\n\\mathcal{C}_X(Z)\\cdot\\mathcal{C}_X(X\\cap Y)\\cdot \\mathscr{T}_X\\cdot\\mathcal{C}_X(Z)\\\\\n&=\n\\mathcal{C}_X(Z)\\cdot \\mathscr{T}_X\\cdot\\mathcal{C}_X(Z)= \\mathcal{C}_X(Z)\\cdot \\mathscr{T}_X\n\\end{align*}\nwhich proves the second equality in \\eqref{eq:nnz}.  The first one\nfollows by interchanging $X$ and $Y$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\n\n\\subsection{Many-body systems}\n\\label{ss:gaz} \n\nHere we give a formal definition of the notion of ``many-body\nsystem'' then define and discuss the Hamiltonian algebra associated\nto it.\n\n\nLet $\\mathscr{S}$ be a set of locally compact abelian groups with the\nfollowing property: for any $X,Y\\in\\mathscr{S}$ there is $Z\\in\\mathscr{S}$ such that\n$X$ and $Y$ are compatible subgroups of $Z$. Note that this implies\nthe following: if $Y\\subset X$ then the topology and the group\nstructure of $Y$ coincide with those induced by $X$.\n\nIf $\\mathscr{S}$ is a set of $\\sigma$-compact locally compact abelian groups\nthen the compatibility assumption is equivalent to the following\nmore explicit condition: for any $X,Y\\in\\mathscr{S}$ there is $Z\\in\\mathscr{S}$ such\nthat $X$ and $Y$ are closed subgroups of $Z$ and $X+Y$ is closed in\n$Z$. \n\n\n\\begin{definition}\\label{df:mb}\nA \\emph{many-body system} is a couple $(\\mathcal{S},\\lambda)$ where:\n\\begin{compactenum}\n\\item[(i)] $\\mathcal{S}\\subset\\mathscr{S}$ is a subset such that \n$X,Y\\in\\mathcal{S}\\Rightarrow X\\cap Y\\in\\mathcal{S}$ and if\n$X\\supsetneq Y$ then $X\/Y$ is not compact,\n\\item[(ii)] \n$\\lambda$ is a map $X\\mapsto\\lambda_X$ which associates a Haar\nmeasures $\\lambda_X$ on $X$ to each $X\\in\\mathcal{S}$. \n\\end{compactenum}\n\\end{definition}\n\nWe identify $\\mathcal{S}=(\\mathcal{S},\\lambda)$ so the choice of Haar measures is\nimplicit. Note that the Hilbert space $\\mathcal{H}_\\mathcal{S}$ and the\n$C^*$-algebra $\\mathscr{C}_\\mathcal{S}$ that we introduce below depend on $\\lambda$\nbut different choices give isomorphic objects.  Each $X\\in\\mathcal{S}$ is\nequipped with a Haar measure so the Hilbert spaces $\\mathcal{H}_X= L^2(X)$\nare well defined. If $Y\\subset X$ are in $\\mathcal{S}$ then $X\/Y$ is\nequipped with the quotient measure so $\\mathcal{H}_{X\/Y}= L^2(X\/Y)$ is well\ndefined.\n\n\n{\\bf Example:} Let $\\mathscr{S}$ the set of all finite dimensional vector\nsubspaces of a vector space over an infinite locally compact field\nand let $\\mathcal{S}$ be any subset of $\\mathscr{S}$ such that $X,Y\\in\\mathcal{S}\\Rightarrow\nX\\cap Y \\in \\mathcal{S}$. \n\n\nFor each $X\\in\\mathcal{S}$ let $\\mathcal{S}_X$ be the set of $Y\\in\\mathcal{S}$ such that\n$Y\\subset X$.  This is an $N$-body system with $X$ as configuration\nspace in the sense of Definition \\ref{df:nb}.  Then by Lemma\n\\ref{lm:reg} the space\n\\begin{equation}\\label{eq:sax}\n\\mathcal{C}_X := {\\textstyle\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}_X}}\\mathcal{C}_X(Y)\n\\end{equation}\nis an $X$-algebra so the crossed product $\\mathcal{C}_X\\rtimes X$ is well\ndefined and we clearly have\n\\begin{equation}\\label{eq:crsax}\n\\mathscr{C}_X := \\mathcal{C}_X\\rtimes X \\equiv \\mathcal{C}_X\\cdot\\mathscr{T}_X =\n{\\textstyle\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}_X}}\\mathscr{C}_X(Y).\n\\end{equation}\nThe $C^*$-algebra $\\mathscr{C}_X$ is realized on the Hilbert space $\\mathcal{H}_X$\nand we think of it as the Hamiltonian algebra of the $N$-body system\ndetermined by $\\mathcal{S}_X$.\n\n\\begin{theorem}\\label{th:grsax}\nThe $C^*$-algebras $\\mathcal{C}_X$ and $\\mathscr{C}_X$ are $\\mathcal{S}_X$-graded by \nthe decompositions \\eqref{eq:sax} and \\eqref{eq:crsax}. \n\\end{theorem}\n\nThe theorem is a particular case of results due to A. Mageira,\ncf. Propositions 6.1.2, 6.1.3 and 4.2.1 in \\cite{Ma} (or see\n\\cite{Ma3}). We mention that the results in \\cite{Ma,Ma3} are much\ndeeper since the groups are allowed to be noncommutative and the\ntreatment is so that the second part of condition (i) is not\nneeded. The case when $\\mathcal{S}$ consists of linear subspaces of a finite\ndimensional real vector space has been considered in \\cite{BG1,DG1}\nand the corresponding version of Theorem \\ref{th:grsax} is proved\nthere by elementary means.\n\n\n\\begin{definition}\\label{df:main}\nIf $X,Y\\in\\mathcal{S}$ then\n$\\mathscr{C}_{XY} := \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y=\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}$.\n\\end{definition}\n\nIn particular $\\mathscr{C}_{XX}=\\mathscr{C}_X$ is a $C^*$-algebra of operators on\n$\\mathcal{H}_X$.  For $X\\neq Y$ the space $\\mathscr{C}_{XY}$ is a closed linear\nspace of operators $\\mathcal{H}_Y\\to \\mathcal{H}_X$ canonically associated to the\nsemilattice of groups $\\mathcal{S}_{X\\cap Y}$, cf. \\eqref{eq:main}. We call\nthese spaces \\emph{coupling modules} because they are Hilbert\n$C^*$-modules and determine the way the systems corresponding to $X$\nand $Y$ are allowed to interact.\n\n\n\nFor each pair $X,Y\\in\\mathcal{S}$ with $X\\supset Y$ we set \n\\begin{equation}\\label{eq:saX}\n\\mathcal{C}^Y_X :=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_Y}}\\mathcal{C}_X(Z).\n\\end{equation}\nThis is also an $X$-algebra so we may define $\\mathscr{C}^Y_X=\\mathcal{C}^Y_X\\rtimes\nX$ and we have\n\\begin{equation}\\label{eq:saX1}\n\\mathscr{C}^Y_X := \\mathcal{C}^Y_X\\rtimes X=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_Y}}\\mathscr{C}_X(Z).\n\\end{equation}\nIf $X=Y\\oplus Z$ then $\\mathcal{C}_X^Y\\simeq\\mathcal{C}_Y\\otimes 1$ and \n$\\mathscr{C}_X^Y\\simeq\\mathscr{C}_Y\\otimes \\mathscr{T}_Z$. \n\n\\begin{lemma}\\label{lm:xyprod}\nLet $X\\in\\mathcal{S}$ and $Y\\in\\mathcal{S}_X$. Then\n\\begin{equation}\\label{eq:xy1}\n\\mathcal{C}_X^{Y}=\\mathcal{C}_X(Y)\\cdot\\mathcal{C}_X \\hspace{2mm}\\text{and}\\hspace{2mm}\n\\mathscr{C}_X^{Y}=\\mathcal{C}_X(Y)\\cdot\\mathscr{C}_X=\\mathscr{C}_X\\cdot\\mathcal{C}_X(Y).\n\\end{equation}\nMoreover, for all  $Y,Z\\in\\mathcal{S}_X$ we have\n\\begin{equation}\\label{eq:xy2}\n\\mathcal{C}_X^Y\\cdot\\mathcal{C}_X^Z=\\mathcal{C}_X^{Y\\cap Z} \n\\hspace{2mm}\\text{and}\\hspace{2mm} \n\\mathscr{C}_X^Y\\cdot\\mathscr{C}_X^Z=\\mathscr{C}_X^{Y\\cap Z}.\n\\end{equation}\n\\end{lemma}\n\\proof The abelian case follows from \\eqref{eq:reg1} and a\nstraightforward computation. For the crossed product algebras we use\n$\\mathcal{C}_X(Y)\\cdot\\mathscr{C}_X=\\mathcal{C}_X(Y)\\cdot\\mathcal{C}_X\\cdot \\mathscr{T}_X$ and the first\nrelation in \\eqref{eq:xy1} for example.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{lemma}\\label{lm:nxy}\nFor arbitrary $X,Y\\in\\mathcal{S}$ we have\n\\begin{equation}\\label{eq:main}\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y\n=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}=\n\\mathcal{C}_X^{X\\cap Y}\\cdot\\mathscr{T}_{XY}.\n\\end{equation}\n\\end{lemma}\n\\proof\nIf $G\\in\\mathscr{S}$ contains $X\\cup Y$ then clearly\n$$\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_X} }\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_X} }\\mathcal{C}_G(Z)|_X\\cdot\\mathscr{T}_{XY}.\n$$\nFrom \\eqref{eq:ayza} and \\eqref{eq:reg2} we get\n$$\n\\mathcal{C}_G(Z)|_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap Z).\n$$ \nSince $Y\\cap Z$ runs over $\\mathcal{S}_{X\\cap Y}$ when $Z$ runs over\n$\\mathcal{S}_X$ we obtain $\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}$.\nSimilarly $\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y=\\mathcal{C}_X^{X\\cap Y}\\cdot\\mathscr{T}_{XY}$. \nOn the other hand $\\mathcal{C}_X^{X\\cap Y}=\\mathcal{C}_G^{X\\cap Y}|_X$ and similarly\nwith $X,Y$ interchanged, hence \n$\\mathcal{C}_X^{X\\cap Y}\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}$\nbecause of \\eqref{eq:ayza}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\\begin{proposition}\\label{pr:mxyz}\nLet $X,Y,Z\\in\\mathcal{S}$. Then $\\mathscr{C}_{XY}^*=\\mathscr{C}_{YX}$ and\n\\begin{equation}\\label{eq:mxyz}\n\\mathscr{C}_{XZ}\\cdot\\mathscr{C}_{ZY}=\\mathscr{C}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y\\cap Z}=\n\\mathcal{C}_X^{X\\cap Y\\cap Z}\\cdot\\mathscr{C}_{XY} \\subset \\mathscr{C}_{XY}. \n\\end{equation}\nIn particular $\\mathscr{C}_{XZ}\\cdot\\mathscr{C}_{ZY}=\\mathscr{C}_{XY}$ if $Z\\supset X\\cap Y$.\n\\end{proposition}\n\\proof \nThe first assertion follows from \\eqref{eq:rad}.  From the\nDefinition \\ref{df:main} and Proposition \\ref{pr:product} we then get\n\\begin{align*}\n\\mathscr{C}_{XZ}\\cdot\\mathscr{C}_{ZY} &=\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}\\cdot\\mathcal{C}_Y =\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y \\\\ &=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y =\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y.\n\\end{align*}\nBut $\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y=\\mathcal{C}_Y^{X\\cap Y\\cap Z}$ by Lemma\n\\ref{lm:xyprod}. For the last inclusion in \\eqref{eq:mxyz} we use\nthe obvious relation $\\mathcal{C}_Y^{X\\cap Y\\cap Z}\\cdot\\mathcal{C}_Y\\subset\\mathcal{C}_Y$.\nThe last assertion of the proposition follows from \\eqref{eq:main}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nThe following theorem is a consequence of the results obtained so\nfar.\n\n\\begin{theorem}\\label{th:nmod}\n$\\mathscr{C}_{XY}$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ such that \n\\begin{equation}\\label{eq:nmod}\n\\mathscr{C}_{XY}^*\\cdot\\mathscr{C}_{XY}=\\mathscr{C}_Y^{X\\cap Y} \\text{ and }\n\\mathscr{C}_{XY}\\cdot\\mathscr{C}_{XY}^*=\\mathscr{C}_X^{X\\cap Y}.\n\\end{equation}\nIn particular, $\\mathscr{C}_{XY}$ is a\n$(\\mathscr{C}_X^{X\\cap Y},\\mathscr{C}_Y^{X\\cap Y})$ imprimitivity bimodule. \n\\end{theorem}\n\nWe recall the conventions\n\\begin{align}\n& X,Y\\in\\mathcal{S} \\text{ and } Y\\not\\subset X \\Rightarrow \n\\mathcal{C}_X(Y)= \\mathscr{C}_X(Y)=\\{0\\}, \\label{eq:convn} \\\\\n& X,Y,Z\\in\\mathcal{S} \\text{ and } Z\\not\\subset X\\cap Y \\Rightarrow\n\\mathscr{C}_{XY}(Z)=\\{0\\}.\\label{eq:convn1}\n\\end{align}\nFrom now on by ``graded'' we mean $\\mathcal{S}$-graded.  Then\n$\\mathscr{C}_X=\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}}\\mathscr{C}_X(Y)$ is a graded\n$C^*$-algebras supported by the ideal $\\mathcal{S}_X$ of $\\mathcal{S}$, in\nparticular it is a graded ideal in $\\mathscr{C}_X$.  With the notations of\nSubsection \\ref{ss:grca} the algebra $\\mathscr{C}^Y_X=\\mathscr{C}_X(\\mathcal{S}_Y)$ is a\ngraded ideal of $\\mathscr{C}_X$ supported by $\\mathcal{S}_Y$. Similarly for $\\mathcal{C}_X$\nand $\\mathcal{C}_X^Y$.\n\nSince $\\mathscr{C}_X^{X\\cap Y}$ and $\\mathscr{C}_Y^{X\\cap Y}$ are ideals in $\\mathscr{C}_X$\nand $\\mathscr{C}_Y$ respectively, Theorem \\ref{th:nmod} allows us to equip\n$\\mathscr{C}_{XY}$ with (right) Hilbert $\\mathscr{C}_Y$-module and left Hilbert\n$\\mathscr{C}_X$-module structures (which are not full in general).\n\n\\begin{theorem}\\label{th:nmain}\nThe Hilbert $\\mathscr{C}_Y$-module $\\mathscr{C}_{XY}$ is graded by the family of\n$C^*$-submodules  $\\{\\mathscr{C}_{XY}(Z)\\}_{Z\\in\\mathcal{S}}$.\n\\end{theorem}\n\\proof We use Proposition \\ref{pr:rhm} with $\\mathscr{M}=\\mathscr{T}_{XY}$ and\n$\\mathcal{C}_Y(Z)$ as algebras $\\mathcal{C}(\\sigma)$. Then $\\mathscr{A}=\\mathscr{C}_Y(X\\cap Y)$ by\n\\eqref{eq:hyz} hence $\\mathscr{A}\\cdot\\mathcal{C}_Y(Z)=\\mathscr{C}_Y(Z)$ and the conditions\nof the proposition are satisfied.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\\begin{remark}\\label{re:precise}\nThe following more precise statement is a consequence of the Theorem\n\\ref{th:nmain}: the Hilbert $\\mathscr{C}_Y^{X\\cap Y}$-module $\\mathscr{C}_{XY}$ is\n$\\mathcal{S}_{X\\cap Y}$-graded by the family of $C^*$-submodules\n$\\{\\mathscr{C}_{XY}(Z)\\}_{Z\\in\\mathcal{S}_{X\\cap Y}}$.\n\\end{remark}\n\nFinally, we may construct the $C^*$-algebra $\\mathscr{C}$ which is of main\ninterest for us, the many-body Hamiltonian algebra. We shall\ndescribe it as an algebra of operators on the Hilbert space\n\\begin{equation}\\label{eq:bigh}\n\\mathcal{H}\\equiv\\mathcal{H}_\\mathcal{S}={\\textstyle\\oplus_{X\\in\\mathcal{S}}} \\mathcal{H}_X\n\\end{equation}\nwhich is a kind of Boltzmann-Fock space (without symmetrization or\nanti-symmetrization) determined by the semilattice $\\mathcal{S}$. Note that\nif the zero group $O=\\{0\\}$ belongs to $\\mathcal{S}$ then $\\mathcal{H}$ contains\n$\\mathcal{H}_O=\\mathbb{C}$ as a subspace, this is the vacuum sector.  Let\n$\\Pi_{X}$ be the orthogonal projection of $\\mathcal{H}$ onto $\\mathcal{H}_X$ and let\nus think of its adjoint $\\Pi_{X}^*$ as the natural embedding\n$\\mathcal{H}_X\\subset\\mathcal{H}$. Then for any pair $X,Y\\in\\mathcal{S}$ we identify\n\\begin{equation}\\label{eq:identc}\n\\mathscr{C}_{XY}\\equiv\\Pi^*_{X}\\mathscr{C}_{XY}\\Pi_{Y} \\subset L(\\mathcal{H}).\n\\end{equation}\nThus we realize $\\{\\mathscr{C}_{XY}\\}_{X,Y\\in\\mathcal{S}}$ as a linearly independent\nfamily of closed subspaces of $L(\\mathcal{H})$ such that\n$\\mathscr{C}_{XY}^*=\\mathscr{C}_{YX}$ and $\\mathscr{C}_{XZ}\\mathscr{C}_{Z'Y}\\subset\\mathscr{C}_{XY}$ for all\n$X,Y,Z,Z'\\in\\mathcal{S}$. Then by what we proved before, especially\nProposition \\ref{pr:mxyz}, the space\n$\\sum\\nolimits_{X,Y\\in\\mathcal{S}}\\mathscr{C}_{XY}$ is a $*$-subalgebra of $L(\\mathcal{H})$\nhence its closure\n\\begin{equation}\\label{eq:bigco}\n\\mathscr{C}\\equiv\\mathscr{C}_\\mathcal{S}= {\\textstyle\\sum^\\mathrm{c}_{X,Y\\in\\mathcal{S}}}\\mathscr{C}_{XY}.\n\\end{equation}\nis a $C^*$-algebra of operators on $\\mathcal{H}$.  Note that one may view\n$\\mathscr{C}$ as a matrix $(\\mathscr{C}_{XY})_{X,Y\\in\\mathcal{S}}$.  \n\nIn a similar way one may associate to the spaces $\\mathscr{T}_{XY}$ a\nclosed self-adjoint subspace $\\mathscr{T}\\subset L(\\mathcal{H})$.  It is also useful\nto define a new subspace $\\mathscr{T}^\\circ\\subset L(\\mathcal{H})$ by\n$\\mathscr{T}^\\circ_{XY}=\\mathscr{T}_{XY}$ if $X\\sim Y$ and $\\mathscr{T}^\\circ=\\{0\\}$ if\n$X\\not\\sim Y$. Here $X\\sim Y$ means $X\\subset Y$ or $Y\\subset X$\n. Clearly $\\mathscr{T}^\\circ$ is a closed self-adjoint linear subspace of\n$\\mathscr{T}$.  Finally, let $\\mathcal{C}$ be the diagonal $C^*$-algebra\n$\\mathcal{C}\\equiv\\oplus_X\\mathcal{C}_X$ of operators on $\\mathcal{H}$.\n\n\\begin{theorem}\\label{th:tc}\nWe have $\\mathscr{C}=\\mathscr{T}\\cdot\\mathcal{C}=\\mathcal{C}\\cdot\\mathscr{T}=\\mathscr{T}\\cdot\\mathscr{T}=\n\\mathscr{T}^\\circ\\cdot\\mathscr{T}^\\circ$.\n\\end{theorem}\n\\proof The first two equalities are an immediate consequence of the\nDefinition \\ref{df:main}. To prove the third equality we use\nProposition \\ref{pr:product}, more precisely the relation\n\\[\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)=\n\\mathscr{C}_{XY}(X\\cap Y\\cap Z)\n\\]\nwhich holds for any $X,Y,Z$. Then\n\\[\n{\\textstyle\\sum^\\mathrm{c}_Z}\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\n{\\textstyle\\sum^\\mathrm{c}_Z}\\mathscr{C}_{XY}(X\\cap Y\\cap Z)=\n{\\textstyle\\sum^\\mathrm{c}_Z}\\mathscr{C}_{XY}(Z)=\\mathscr{C}_{XY}\n\\]\nwhich is equivalent to $\\mathscr{T}\\cdot\\mathscr{T}=\\mathscr{C}$. Now we prove the last\nequality in the proposition. We have\n\\[\n{\\textstyle\\sum^\\mathrm{c}_Z} \\mathscr{T}^\\circ_{XZ}\\cdot\\mathscr{T}^\\circ_{ZY}= \n\\text{ closure of the sum }\n{\\textstyle\\sum_{\\substack{Z\\sim X\\\\ Z\\sim Y}}}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}. \n\\]\nIn the last sum we have four possibilities: $Z\\supset X\\cup Y$,\n$X\\supset Z\\supset Y$, $Y\\supset Z\\supset X$, and $Z\\subset X\\cap\nY$. In the first three cases we have $Z\\supset X\\cap Y$ hence\n$\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}$ by \\eqref{eq:factor}. In the last\ncase we have $\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Z)$ by\n\\eqref{eq:product}. This proves $\\mathscr{T}^\\circ\\cdot\\mathscr{T}^\\circ=\\mathscr{C}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nFinally, we are able to equip $\\mathscr{C}$ with an $\\mathcal{S}$-graded\n$C^*$-algebra structure.\n\n\\begin{theorem}\\label{th:cgrad}\nFor each $Z\\in\\mathcal{S}$ the space $\\mathscr{C}(Z):=\n\\sum^\\mathrm{c}_{X,Y\\in\\mathcal{S}}\\mathscr{C}_{XY}(Z)$ is a $C^*$-subalgebra of\n$\\mathscr{C}$. The family $\\{\\mathscr{C}(Z)\\}_{Z\\in\\mathcal{S}}$ defines a graded\n$C^*$-algebra structure on $\\mathscr{C}$.\n\\end{theorem}\n\\proof \nWe first prove the following relation:\n\\begin{equation}\\label{eq:xyzef}\n\\mathscr{C}_{XZ}(E)\\cdot\\mathscr{C}_{ZY}(F)=\\mathscr{C}_{XY}(E\\cap F)\n\\quad \\text{if } X,Y,Z\\in\\mathcal{S} \\text{ and } E\\subset{X\\cap Z},\nF\\subset{Y\\cap Z}.\n\\end{equation}\nFrom Definition \\ref{df:nxyz}, Proposition \\ref{pr:product},\nrelations \\eqref{eq:reg1} and \\eqref{eq:ayza}, and \n$F\\subset Y\\cap Z$, we get\n\\begin{align*}\n\\mathscr{C}_{XZ}(E)\\cdot\\mathscr{C}_{ZY}(F) &=\n\\mathcal{C}_X(E)\\cdot\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}\\cdot\\mathcal{C}_Y(F) \\\\\n&= \\mathcal{C}_X(E)\\cdot\\mathscr{T}_{XY}\\cdot\\mathcal{C}_{Y}(Y\\cap Z)\\cdot\\mathcal{C}_Y(F) \\\\\n&= \\mathcal{C}_X(E)\\cdot\\mathscr{T}_{XY}\\cdot\\mathcal{C}_{Y}(F) \\\\\n&= \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap E)\\cdot\\mathcal{C}_{Y}(F) \\\\\n&= \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap E\\cap F).\n\\end{align*}\nAt the next to last step we used $\\mathcal{C}_X(E)=\\mathcal{C}_G(E)|_X$ for some\n$G\\in\\mathscr{S}$ containing both $X$ and $Y$ and then \\eqref{eq:ayza},\n\\eqref{eq:reg2}. Finally, we use $\\mathcal{C}_Y(Y\\cap E\\cap F)=\\mathcal{C}_Y(E\\cap\nF)$ and the Definition \\ref{df:nxyz}. This proves \\eqref{eq:xyzef}.\nDue to the conventions \\eqref{eq:convn}, \\eqref{eq:convn1} we now\nget from \\eqref{eq:xyzef} for $E,F\\in\\mathcal{S}$\n\\[\n{\\textstyle\\sum_{Z\\in\\mathcal{S}}}\\mathscr{C}_{XZ}(E)\\cdot\\mathscr{C}_{ZY}(F)=\n\\mathscr{C}_{XY}(E\\cap F).\n\\]\nThus $\\mathscr{C}(E)\\mathscr{C}(F)\\subset\\mathscr{C}(E\\cap F)$, in particular $\\mathscr{C}(E)$ is a\n$C^*$-algebra. It remains to be shown that the family of\n$C^*$-algebras $\\{\\mathscr{C}(E)\\}_{E\\in\\mathcal{S}}$ is linearly independent. Let\n$A(E)\\in\\mathscr{C}(E)$ such that $A(E)=0$ but for a finite number of $E$\nand assume that $\\sum_E A(E)=0$. Then for all $X,Y\\in\\mathcal{S}$\nwe have $\\sum_E \\Pi_X A(E) \\Pi_Y^* =0$. Clearly \n$\\Pi_X A(E) \\Pi_Y^*\\in\\mathscr{C}_{XY}(E)$ hence from Theorem \\ref{th:nmain}\nwe get $\\Pi_X A(E) \\Pi_Y^*=0$ for all $X,Y$ so $A(E)=0$ for all $E$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Subsystems}\n\\label{ss:T} \n\nWe now point out some interesting subalgebras of\n$\\mathscr{C}$. If $\\mathcal{T}\\subset\\mathcal{S}$ is any subset let\n\\begin{equation}\\label{eq:t}\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}\\equiv{\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY} \\quad\n\\text{and} \\quad \\mathcal{H}_\\mathcal{T}\\equiv\\oplus_{X\\in\\mathcal{T}}\\mathcal{H}_X.\n\\end{equation}\nNote that the sum defining $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is already closed if $\\mathcal{T}$ is\nfinite and that $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is a $C^*$-algebra which lives on the\nsubspace $\\mathcal{H}_\\mathcal{T}$ of $\\mathcal{H}$. In fact, if $\\Pi_\\mathcal{T}$ is the orthogonal\nprojection of $\\mathcal{H}$ onto $\\mathcal{H}_\\mathcal{T}$ then\n\\begin{equation}\\label{eq:tt}\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}=\\Pi_\\mathcal{T}\\mathscr{C}_\\mathcal{S}\\Pi_\\mathcal{T}\n\\end{equation}\nand this is a $C^*$-algebra because $\\mathscr{C}\\Pi_\\mathcal{T}\\mathscr{C}\\subset\\mathscr{C}$ by\nProposition \\ref{pr:mxyz}. \nIt is easy to check that $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is a graded $C^*$-subalgebra of\n$\\mathscr{C}$ supported by the ideal $\\textstyle{\\bigcup}_{X\\in\\mathcal{T}}\\mathcal{S}_X$ generated by\n$\\mathcal{T}$ in $\\mathcal{S}$. Indeed, we have\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}\\,\\textstyle{\\bigcap}\\,\\mathscr{C}(E)=\n\\left({\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY}\\right) \\textstyle{\\bigcap}\n\\left({\\textstyle\\sum_{X,Y\\in\\mathcal{S}}^\\mathrm{c}}\\mathscr{C}_{XY}(E)\\right) =\n{\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY}(E).\n\\]\nIt is clear that $\\mathscr{C}$ is the inductive limit of the increasing\nfamily of $C^*$-algebras $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ with finite $\\mathcal{T}$.\n\nIf $\\mathcal{T}=\\{X\\}$ then $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is just $\\mathscr{C}_X$. If\n$\\mathcal{T}=\\{X,Y\\}$ with distinct $X,Y$ we get a simple but nontrivial\nsituation. Indeed, we shall have $\\mathcal{H}_\\mathcal{T}=\\mathcal{H}_X\\oplus\\mathcal{H}_Y$ and\n$\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ may be thought as a matrix\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}=\n\\begin{pmatrix}\n\\mathscr{C}_X & \\mathscr{C}_{XY}\\\\\n\\mathscr{C}_{YX} & \\mathscr{C}_Y\n\\end{pmatrix}.\n\\]\nThe grading is now explicitly defined as follows: \n\\begin{compactenum}\n\\item\nIf  $E\\subset X\\cap Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(E)=\n\\begin{pmatrix}\n\\mathscr{C}_X(E) & \\mathscr{C}_{XY}(E)\\\\\n\\mathscr{C}_{YX}(E) & \\mathscr{C}_Y(E)\n\\end{pmatrix}.\n\\]\n\\item   \\label{p:2ex}\nIf  $E\\subset X$ and $E\\not\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(E)=\n\\begin{pmatrix}\n\\mathscr{C}_X(E) & 0\\\\\n0 & 0\n\\end{pmatrix}.\n\\]\n\\item\nIf  $E\\not\\subset X$ and $E\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(E)=\n\\begin{pmatrix}\n0 & 0\\\\\n0 & \\mathscr{C}_Y(E)\n\\end{pmatrix}.\n\\]\n\\end{compactenum}\n\n\nThe case when $\\mathcal{T}$ is of the form $\\mathcal{S}_X$ for some $X\\in\\mathcal{S}$ is\nespecially interesting.  We denote $\\mathscr{C}_X^\\#\\equiv\\mathscr{C}_{\\mathcal{S}_X}$ and\nwe say that the $\\mathcal{S}_X$-graded $C^*$-algebra is the\n\\emph{unfolding} of the algebra $\\mathscr{C}_X$.  More explicitly\n\\begin{equation}\\label{eq:xc}\n\\mathscr{C}_X^\\#\\equiv{\\textstyle\\sum^\\mathrm{c}_{Y,Z\\in\\mathcal{S}_X}}\\mathscr{C}_{YZ}.\n\\end{equation}\nThe self-adjoint operators affiliated to $\\mathscr{C}_X$ live on the Hilbert\nspace $\\mathcal{H}_X$ and are (an abstract version of) Hamiltonians of an\n$N$-particle system $\\mathscr{S}$ with a fixed $N$ (the configuration space\nis $X$ and $N$ is the number of levels of the semilattice\n$\\mathcal{S}_X$). The unfolding $\\mathscr{C}_X^\\#$ lives on the ``Boltzmann-Fock\nspace'' $\\mathcal{H}_{\\mathcal{S}_X}$ and is obtained by adding interactions which\ncouple the subsystems of $\\mathcal{S}$ which have the groups $Y\\in\\mathcal{S}_X$ as\nconfiguration spaces and $\\mathscr{C}_Y$ as Hamiltonian algebras.\n\nClearly $\\mathscr{C}_X^\\#\\subset\\mathscr{C}_Y^\\#$ if $X\\subset Y$ and $\\mathscr{C}$ is the\ninductive limit of the algebras $\\mathscr{C}_X^\\#$.  Below we give an\ninteresting alternative description of $\\mathscr{C}_X^\\#$.\n\n\\begin{theorem}\\label{th:mor}\nLet $\\mathscr{N}_X=\\oplus_{Y\\in\\mathcal{S}_X}\\mathscr{C}_{YX}$ be the direct sum of the\nHilbert $\\mathscr{C}_X$-modules $\\mathscr{C}_{YX}$ equipped with the direct sum graded\nstructure. Then $\\mathcal{K}(\\mathscr{N}_X) \\cong \\mathscr{C}_X^\\#$ the isomorphism being such\nthat the graded structure on $\\mathcal{K}(\\mathscr{N}_X)$ defined in Theorem\n\\ref{th:kghm} is transported into that of $\\mathscr{C}_X^\\#$.  In other terms,\n$\\mathscr{C}_X^\\#$ is the imprimitivity algebra of the full Hilbert\n$\\mathscr{C}_X$-module $\\mathscr{N}_X$ and $\\mathscr{C}_X$ and $\\mathscr{C}_X^\\#$ are Morita\nequivalent.\n\\end{theorem}\n\\proof If $Y\\subset X$ then $\\mathscr{C}^*_{YX}\\cdot\\mathscr{C}_{YX}=\\mathscr{C}_X^Y$ and\n$\\mathscr{C}_{YX}$ is a full Hilbert $\\mathscr{C}_X^Y$-module. Since the $\\mathscr{C}_X^Y$\nare ideals in $\\mathscr{C}_X$ and their sum over $Y\\in\\mathcal{S}_X$ is equal to\n$\\mathscr{C}_X$ we see that $\\mathscr{N}_X$ becomes a full Hilbert graded\n$\\mathscr{C}_X$-module supported by $\\mathcal{S}_X$, cf.  Section \\ref{s:grad}. By\nTheorem \\ref{th:kghm} the imprimitivity $C^*$-algebra $\\mathcal{K}(\\mathscr{N}_X)$\nis equipped with a canonical $\\mathcal{S}_X$-graded structure.\n\nWe shall make a comment on $\\mathcal{K}(\\mathscr{M})$ in the more general the case\nwhen $\\mathscr{M}=\\oplus_i\\mathscr{M}_i$ is a direct sum of Hilbert $\\mathscr{A}$-modules\n$\\mathscr{M}_i$, cf. \\S\\ref{ss:gf}. First, it is clear that we have \n\\[\n\\mathcal{K}(\\mathscr{M})={\\textstyle\\sum^\\mathrm{c}_{ij}}\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i)\\cong\n(\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i))_{ij}.\n\\]\nNow assume that $\\mathcal{E},\\mathcal{E}_i$ are Hilbert spaces such that $\\mathscr{A}$ is a\n$C^*$-algebra of operators on $\\mathcal{E}$ and $\\mathscr{M}_i$ is a Hilbert\n$C^*$-submodule of $L(\\mathcal{E},\\mathcal{E}_i)$ such that\n$\\mathscr{A}_i\\equiv\\mathscr{M}_i^*\\cdot\\mathscr{M}_i$ is an ideal of $\\mathscr{A}$. \nThen by Proposition \\ref{pr:2ss} we have\n$\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i)\\cong\\mathscr{M}_i\\cdot\\mathscr{M}_j^*\\subset L(\\mathcal{E}_j,\\mathcal{E}_i)$. \n\nIn our case we take \n\\[\ni=Y\\in\\mathcal{S}_X,\\quad \\mathscr{M}_i=\\mathscr{C}_{YX}, \\quad \\mathscr{A}=\\mathscr{C}_X,\\quad\n\\mathcal{E}=\\mathcal{H}_X,\\quad  \\mathcal{E}_i=\\mathcal{H}_Y,\\quad   \\mathscr{A}_i=\\mathscr{C}_X^Y.\n\\]\nThen we get\n\\[\n\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i)\\equiv\\mathcal{K}(\\mathscr{C}_{ZX},\\mathscr{C}_{YX})\\cong\n\\mathscr{C}_{YX}\\cdot\\mathscr{C}_{ZX}^*=\\mathscr{C}_{YX}\\cdot\\mathscr{C}_{XZ}=\\mathscr{C}_{YZ}\n\\]\nby Proposition \\ref{pr:mxyz}.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\\begin{remark}\\label{re:squant}\n  We understood the role in our work of the imprimitivity algebra of\n  a Hilbert $C^*$-module thanks to a discussion with Georges\n  Skandalis: he recognized (a particular case of) the main\n  $C^*$-algebra $\\mathscr{C}$ we have constructed as the imprimitivity\n  algebra of a certain Hilbert $C^*$-module. Theorem \\ref{th:mor} is\n  a reformulation of his observation and of his abstract\n  construction of graded Hilbert $C^*$-modules in the present\n  framework (at the time of the discussion our definition of $\\mathscr{C}$\n  was rather different because we were working in a tensor product\n  formalism).  More generally, if $\\mathscr{M}$ is a full Hilbert\n  $\\mathscr{A}$-module then the imprimitivity $C^*$-algebra $\\mathcal{K}(\\mathscr{M})$ could\n  also be interpreted as Hamiltonian algebra of a system related in\n  some natural way to the initial one.  For example, this is a\n  natural method of ``second quantizing'' \\mbox{$N$-body} systems,\n  i.e.  introducing interactions which couple subsystems\n  corresponding to different cluster decompositions of the $N$-body\n  systems.  This is clear in the physical $N$-body situation\n  discussed in \\S\\ref{ss:cexample}\n\\end{remark}\n\n\n\\section{An intrinsic description}\n\\label{s:id}\n\\protect\\setcounter{equation}{0}\n\nWe begin with some preliminary facts on crossed products.  Let $X$\nbe a locally compact abelian group.  The next result, due to\nLandstad \\cite{Ld}, gives an ``intrinsic'' characterization of\ncrossed products of \\mbox{$X$-algebras} by the action of $X$.  We\nfollow the presentation from \\cite[Theorem 3.7]{GI4} which takes\nadvantage of the fact that $X$ is abelian.\n\n\\begin{theorem}\\label{th:land}\nA $C^*$-algebra $\\mathscr{A}\\subset \\mathscr{L}_X$ is a crossed product\nif and only for each $A\\in\\mathscr{A}$ we have:\n\\begin{itemize}\n\\vspace{-2mm} \n\\item\nif $k\\in X^*$ then $V_k^*AV_k\\in\\mathscr{A}$ and\n$\\lim_{k\\rarrow0}\\|V_k^*AV_k-A\\|=0$,\n\\item\nif $x\\in X$ then $U_xA\\in\\mathscr{A}$ and $\\lim_{x\\rarrow0}\\|(U_x-1)A\\|=0$.\n\\vspace{-2mm}\n\\end{itemize} \nIn this case one has $\\mathscr{A}=\\mathcal{A}\\rtimes X$ for a unique $X$-algebra\n$\\mathcal{A}\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$  and this algebra is given by\n\\begin{equation}\\label{eq:land}\n\\mathcal{A} =\\{\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)\\mid \n\\varphi(Q)S \\in {\\mathscr{A}} \\hspace{1mm}\\text{and}\\hspace{1mm}\n\\bar\\varphi(Q)S \\in {\\mathscr{A}}\n\\hspace{1mm}\\text{for all}\\hspace{1mm} S \\in \\mathscr{T}_X\\}.\n\\end{equation} \n\\end{theorem}\nNote that the second condition above is equivalent\nto $\\mathscr{T}_X\\cdot\\mathscr{A}=\\mathscr{A}$, cf. Lemma \\ref{lm:help}.\n\n\nThe following consequence of Landstad's theorem is an intrinsic \ndescription of $\\mathscr{C}_X(Y)$.\n\n\\begin{theorem}\\label{th:cxy} \n$\\mathscr{C}_X(Y)$ is the set of $A\\in \\mathscr{L}_X$ such that $U_y^*AU_y=A$ for all\n$y\\in Y$ and:\n\\begin{enumerate}\n\\item[{\\rm(1)}]\n$\\|U_x^*AU_x-A\\|\\to 0$ if $x\\to 0$ in $X$ and \n$\\|V_k^*AV_k-A\\|\\to 0$ if $k\\to 0$ in $X^*$,\n\\item[{\\rm(2)}]\n$\\|(U_x-1)A\\|\\to 0$ if $x\\to 0$ in $X$ and $\\|(V_k-1)A\\|\\to 0$ if\n$k\\to 0$ in $Y^\\perp$. \n\\end{enumerate}   \n\\end{theorem}\n\nBy ``$k\\to 0$ in $Y^\\perp$'' we mean: $k\\in Y^\\perp$ and $k\\to 0$.\nNote that the second condition above is equivalent to:\n\\begin{equation}\\label{eq:cxy}\n\\text{there are } \\theta\\in\\mathscr{T}_X,\\ \\psi\\in\\mathcal{C}_X(Y)\n\\text{ and }  B,C\\in\\mathscr{L}_X \\text{ such that } \nA=\\theta(P)B=\\psi(Q)C.\n\\end{equation}\nFor the proof, use $Y^\\perp\\cong (X\/Y)^*$ and apply Lemma\n\\ref{lm:help}. In particular, the last factorization shows that for\neach $\\varepsilon>$ there is a compact set $M\\subset X$ such that\n$\\|\\cchi_V(Q)A\\|<\\varepsilon$, where $V=X\\setminus(M+Y)$.\n\n\\noindent{\\bf Proof of Theorem \\ref{th:cxy}:} Let $\\mathscr{A}\\subset \\mathscr{L}_X$\nbe the set of operators $A$ satisfying the conditions from the\nstatement of the theorem.  We first prove that $\\mathscr{A}$ satisfies the\ntwo conditions of Theorem \\ref{th:land}. Let $A\\in\\mathscr{A}$.  We have to\nshow that $A_p\\equiv V_p^*AV_p\\in\\mathscr{A}$ and $\\|V_p^*AV_p-A\\|\\to0$ as\n$p\\to0$.  From the commutation relations $U_xV_p=p(x)V_pU_x$ we get\n$\\|(U_x-1)A_p\\|=\\|(U_x-p(x))A\\|\\to0$ if $x\\to0$ and the second part\nof condition 1 of the theorem is obviously satisfied by $A_p$. Then\nfor $y\\in Y$\n$$\nU_y^*A_pU_y=U_y^*V_p^*AV_pU_y=V_p^*U_y^*AU_yV_p=V_p^*AV_p=A_p.\n$$ Condition 2 is clear so we have $A_p\\in\\mathscr{A}$ and the fact that\n$\\|V_p^*AV_p-A\\|\\to0$ as $p\\to0$ is obvious. That $A$ satisfies the\nsecond Landstad condition, namely that for each $a\\in X$ we have\n$U_aA\\in\\mathscr{A}$ and $\\|(U_a-1)A\\|\\to0$ as $a\\to0$, is also clear because\n$\\|[U_a,V_k]\\|\\to0$ as $k\\to0$.\n\nNow we have to find the algebra $\\mathcal{A}$ defined by \\eqref{eq:land}.\nAssume that $\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ satisfies $\\varphi(Q)S\\in\\mathscr{A}$ for all\n$S\\in \\mathscr{T}_X$. Since $U_y^*\\varphi(Q)U_y=\\varphi(Q-y)$ we get\n$(\\varphi(Q)-\\varphi(Q-y))S=0$ for all such $S$ and all $y\\in Y$,\nhence $\\varphi(Q)-\\varphi(Q-y)=0$ which means $\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)$.\nWe shall prove that $\\varphi\\in\\mathcal{C}_X(Y)$ by reductio ad absurdum. \n\nIf $\\varphi\\notin\\mathcal{C}_X(Y)$ then there is $\\mu>0$ and there is a\nsequence of points $x_n\\in X$ such that $x_n\/Y\\to\\infty$ and\n$|\\varphi(x_n)|>2\\mu$. From the uniform continuity of $\\varphi$ we\nsee that there is a compact neighborhood $K$ of zero in $X$ such\nthat $|\\varphi|>\\mu$ on $\\bigcup_n(x_n+K)$.  Let $K'$ be a compact\nneighborhood of zero such that $K'+K'\\subset K$ and let us choose\ntwo positive not zero functions $\\psi,f\\in\\cc_{\\mathrm{c}}(K')$. We define $S\\in\n\\mathscr{T}_X$ by $Su=\\psi*u$ and recall that $\\mbox{\\rm supp\\! } Su\\subset\\mbox{\\rm supp\\! }\n\\psi+\\mbox{\\rm supp\\! } u$. Thus $\\mbox{\\rm supp\\! } SU_{x_n}^*f\\subset K'+x_n+K'\\subset\nx_n+K$. Now let $V$ be as in the remarks after \\eqref{eq:cxy}. Since\n$\\pi_Y(x_n)\\to\\infty$ we have $x_n+K\\subset V$ for $n$ large enough,\nhence\n$$\n\\|\\cchi_V(Q)\\varphi(Q)SU_{x_n}^*f\\|\\geq\\mu\\|SU_{x_n}^*f\\|=\n\\mu\\|Sf\\| >0.\n$$\nOn the other hand, for each $\\varepsilon>0$ one can choose $V$ such\nthat $\\|\\cchi_V(Q)\\varphi(Q)S\\|<\\varepsilon$.  Then we shall have\n$\\|\\cchi_V(Q)\\varphi(Q)SU_{x_n}^*f\\|\\leq\\varepsilon\\|f\\|$ so\n$\\mu\\|Sf\\|\\leq\\varepsilon\\|f\\|$ for all $\\varepsilon>0$ which is\nabsurd.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nWe now give a similar characterization of $\\mathscr{C}_{XY}(Z)$ where $X,Y$\nis a compatible pair of closed subgroups of an lca group $G$.\n\n\\begin{theorem}\\label{th:yzintr}\n  $\\mathscr{C}_{XY}(Z)$ is the set of $T\\in\\mathscr{L}_{XY}$ satisfying the\n  following conditions:\n\\begin{enumerate} \\vspace{-2mm}\n\\item[{\\rm(1)}] $U_z^*T U_z=T$ if $z\\in Z$ and $\\|V^*_k T V_k-T\\|\\to\n  0$ if $k\\to 0$ in $(X+Y)^*$\n\\item[{\\rm(2)}]\n$\\|(U_x-1)T\\|\\to 0$ if $x\\to 0$ in $X$ and \n$\\|T(U_y-1)\\|\\to 0$ if $y\\to 0$ in $Y$, \n\\item[{\\rm(3)}]\n$\\|(V_k-1)T\\|\\to 0$ if $k\\to 0$ in $(X\/Z)^*$ and \n$\\|T(V_k-1)\\|\\to 0$ if $k\\to 0$ in $(Y\/Z)^*$.\n\\end{enumerate}\n\\end{theorem}\n\nBefore the proof we make some preliminary comments. We think of\n$X+Y$ as a closed subgroup of $G\\in\\mathscr{S}$ which contains $X$ and $Y$\nas closed subgroups.  Each character $k\\in(X+Y)^*$ defines by\nrestriction a character $k|_X\\in X^*$ and the map $k\\mapsto k|_X$ is\na continuous open surjection. And similarly if $X$ is replaced by\n$Y$. In (1) the operator $V_k$ acts in $L^2(X)$ as multiplication by\n$k|_X$ and in $L^2(Y)$ as multiplication by $k|_Y$. In the first\npart of (3) we take $k\\in X^*$ and identify $(X\/Z)^*$ with the\northogonal of $Z$ in $X^*$ and similarly for the second part.\n\nAssumptions (2) and (3) of Theorem \\ref{th:yzintr} are decay\nconditions in certain directions in $P$ and $Q$ space. Indeed, by\nLemma \\ref{lm:help} condition (2) is equivalent to:\n\\begin{equation}\\label{eq:cond1}\n\\text{there are } S_1\\in \\mathscr{T}_X, S_2\\in \\mathscr{T}_Y \\text{ and }\nR_1,R_2\\in\\mathscr{L}_{XY} \\text{ such that } T=S_1R_1=R_2S_2.\n\\end{equation}\nRecall that $\\mathscr{T}_X\\cong\\cc_{\\mathrm{o}}(X^*)$ for example. Then\ncondition (3) is equivalent to:\n\\begin{equation}\\label{eq:cond2}\n\\text{there are } S_1\\in \\mathcal{C}_X(Z), S_2\\in \\mathcal{C}_Y(Z) \\text{ and }\nR_1,R_2\\in\\mathscr{L}_{XY} \\text{ such that } T=S_1R_1=R_2S_2.\n\\end{equation}\n\n\n\\noindent{\\bf Proof of Theorem \\ref{th:yzintr}:} The set $\\mathscr{C}$ of all\nthe operators satisfying the conditions of the theorem is clearly a\nclosed subspace of $\\mathscr{L}_{XY}$.  We have $\\mathscr{C}_{X,Y}(Z)\\subset\\mathscr{C}$\nbecause \\eqref{eq:cond1}, \\eqref{eq:cond2} are satisfied by any\n$T\\in\\mathscr{C}_{XY}(Z)$ as a consequence of Theorem \\ref{th:nxyz}.  Then we\nget:\n$$\n\\mathscr{C}_Y(Z)= \n\\mathscr{C}_{XY}^*(Z)\\cdot\\mathscr{C}_{XY}(Z)\\subset \\mathscr{C}^*\\cdot\\mathscr{C}, \n\\hspace{1mm}\n\\mathscr{C}_X(Z)=\n\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{XY}^*(Z)\\subset \\mathscr{C}\\cdot\\mathscr{C}^*.\n$$ \nWe prove that equality holds in both these relations. We show, for\nexample, that $A\\equiv TT^*$ belongs to $\\mathscr{C}_X(Z)$ if $T\\in\\mathscr{C}$ and\nfor this we shall use Theorem \\ref{th:cxy} with $Y$ replaced by\n$Z$. That $U_z^*AU_z=A$ for $z\\in Z$ is clear. From \\eqref{eq:cond1}\nwe get $A=S_1R_1R_1^*S_1^*$ with $S_1\\in\\mathscr{T}_X$ hence\n$\\|(U_x-1)A\\|\\to 0$ and $\\|A(U_x-1)\\|\\to 0$ as $x\\to0$ in $X$ are\nobvious and imply $\\|U_x^*AU_x-A\\|\\to 0$. Then \\eqref{eq:cond2}\nimplies $A=\\psi(Q)C$ with $\\psi\\in\\mathcal{C}_X(Z)$ and bounded $C$ hence\n\\eqref{eq:cxy} is satisfied.\n\nThat $\\mathscr{C}\\rc_Y(Z)\\subset\\mathscr{C}$ is easily proven because $T=SA$ has the\nproperties \\eqref{eq:cond1} and \\eqref{eq:cond2} if $S$ belongs to\n$\\mathscr{C}$ and $A$ to $\\mathscr{C}_Y(Z)$, cf. Theorem \\ref{th:cxy}. From what we\nhave shown above we get $\\mathscr{C}\\rc^*\\mathscr{C}\\subset\\mathscr{C}\\rc_Y(Z)\\subset\\mathscr{C}$ so\n$\\mathscr{C}$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$. On the other hand,\n$\\mathscr{C}_{XY}(Z)$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ such that\n$\\mathscr{C}_{XY}^*(Z)\\cdot\\mathscr{C}_{XY}(Z)=\\mathscr{C}^*\\cdot\\mathscr{C}$ and\n$\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{XY}^*(Z)=\\mathscr{C}\\cdot\\mathscr{C}^*$. Since\n$\\mathscr{C}_{XY}(Z)\\subset\\mathscr{C}$ we get $\\mathscr{C}=\\mathscr{C}_{XY}(Z)$ from Proposition\n\\ref{pr:clsubmod}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\nIf $Z=X\\cap Y$ then Theorem \\ref{th:yzintr} gives an intrinsic\ndescription of the space $\\mathscr{T}_{XY}$. For example:\n\n\\begin{corollary}\\label{co:txyintr}\nIf $X\\supset Y$ then $\\mathscr{T}_{XY}$ is the set of $T\\in\\mathscr{L}_{XY}$\nsatisfying $U_y^*T U_y=T$    if $y\\in Y$ and such that:\n$U_xT\\to T$ if $x\\to 0$ in $X$, \n$V^*_k T V_k\\to T$ if $k\\to 0$ in $X^*$ and \n$V_kT\\to T$ if $k\\to 0$ in $Y^\\perp$.\n\\end{corollary}\n\n\nIn the rest of this section we describe the structure of the objects\nintroduced in Section \\ref{s:grass} when the subgroups are\ncomplemented, e.g. if $\\mathcal{S}$ consists of finite dimensional vector\nspaces.\n\n\nWe say that \\emph{$Z$ is complemented in $X$} if $X=Z\\oplus E$ for\nsome closed subgroup $E$ of $X$. If $X,Z$ are equipped with Haar\nmeasures then $X\/Z$ is equipped with the quotient Haar measure and we\nhave $E\\simeq X\/Z$.  If $Z$ is complemented in $X$ and $Y$ then\n$\\mathscr{C}_{XY}(Z)$ can be expressed as a tensor product.\n\n\\begin{proposition}\\label{pr:def3}\nIf $Z$ is complemented in $X$ and $Y$ then\n\\begin{equation}\\label{eq:ryzsum}\n\\mathscr{C}_{XY}(Z)\\simeq \\mathscr{T}_Z\\otimes \\mathscr{K}_{X\/Z,Y\/Z}.\n\\end{equation}\nIf $Y\\subset X$ then $\\mathscr{T}_{XY}\\simeq \\mathscr{T}_Y\\otimes L^2(X\/Y)$ tensor\nproduct of Hilbert $C^*$-modules.\n\\end{proposition} \n\\proof Note first that the tensor product in \\eqref{eq:ryzsum} is\ninterpreted as the exterior tensor product of the Hilbert\n$C^*$-modules $\\mathscr{T}_Z$ and $\\mathscr{K}_{X\/Z,Y\/Z}$. Let $X=Z\\oplus E$ and\n$Y=Z\\oplus F$ for some closed subgroups $E,F$. Then, as explained in\n\\S\\ref{ss:ha}, we may also view the tensor product as the norm closure\nin the space of continuous operators from $L^2(Y)\\simeq L^2(Z)\\otimes\nL^2(F)$ to $L^2(X)\\simeq L^2(Z)\\otimes L^2(E)$ of the linear space\ngenerated by the operators of the form $T\\otimes K$ with $T\\in\n\\mathscr{T}_Z$ and $K\\in \\mathscr{K}_{EF}$.\n\n\n\n\nWe now show that under the conditions of the proposition $X+Y\\simeq\nZ\\oplus E\\oplus F$ algebraically and topologically. The natural map\n$\\theta:Z\\oplus E\\oplus F\\to Z+E+F=X+Y$ is a continuous bijective\nmorphism, we have to prove that it is open.  Since $X,Y$ are\ncompatible, the  map \\eqref{eq:nat} is a continuous open\nsurjection.  If we represent $X\\oplus Y\\simeq Z\\oplus Z\\oplus E\\oplus\nF$ then this map becomes $\\phi(a,b,c,d)=(a-b)+c+d$.  Let\n$\\psi=\\xi\\oplus{\\rm id}_E\\oplus{\\rm id}_F$ where $\\xi:Z\\oplus Z\\to Z$\nis given by $\\xi(a,b)=a-b$. Then $\\xi$ is continuous surjective and\nopen because if $U$ is an open neighborhood of zero in $Z$ then\n$U-U$ is also an open neighborhood of zero. Thus\n$\\psi:(Z\\oplus Z)\\oplus E\\oplus F \\to Z\\oplus E\\oplus F$ is a\ncontinuous open surjection and $\\phi=\\theta\\circ\\psi$. So if $V$ is\nopen in $Z\\oplus E\\oplus F$ then there is an open \n$U\\subset Z\\oplus Z\\oplus E\\oplus F$ such that $V=\\psi(U)$ and then\n$\\theta(V)=\\theta\\circ\\psi(U)=\\phi(U)$ is open in $Z+E+F$.\n\nThus we may identify $L^2(Y)\\simeq L^2(Z)\\otimes L^2(F)$ and\n$L^2(X)\\simeq L^2(Z)\\otimes L^2(E)$ and we must describe the norm\nclosure of the set of operators $T_{XY}(\\varphi)\\psi(Q)$ with\n$\\varphi\\in\\cc_{\\mathrm{c}}(X+Y)$ (cf. the remark after \\eqref{eq:ryz} and the fact\nthat $X+Y$ is closed) and $\\psi\\in\\cc_{\\mathrm{o}}(Y\/Z)$. Since $X+Y\\simeq Z\\oplus\nE\\oplus F$ and $Y=Z\\oplus F$ it suffices to describe the clspan of the\noperators $T_{XY}(\\varphi)\\psi(Q)$ with\n$\\varphi=\\varphi_Z\\otimes\\varphi_E\\otimes\\varphi_F$ and\n$\\varphi_Z,\\varphi_E,\\varphi_F$ continuous functions with compact\nsupport on $Z,E,F$ respectively and $\\psi=1\\otimes\\eta$ where $1$ is\nthe function identically equal to $1$ on $Z$ and\n$\\eta\\in\\cc_{\\mathrm{o}}(F)$. Then, if $x=(a,c)\\in Z\\times E$ and $y=(b,d)\\in\nZ\\times F$, we get:\n$$\n(T_{XY}(\\varphi)\\psi(Q)u)(a,c)=\\int_{Z\\times F} \\varphi_Z(a-b)\n\\varphi_E(c)\n\\varphi_F(d)  \\eta(d) u(b,d) \\text{d} b \\text{d} d. \n$$ But this is just\n$C(\\varphi_Z)\\otimes\\ket{\\varphi_E}\\bra{\\bar\\eta\\bar\\varphi_F}$ where\n$\\ket{\\varphi_E}\\bra{\\bar\\eta\\bar\\varphi_F}$ is a rank one operator\n$L^2(F)\\to L^2(E)$ and $C(\\varphi_Z)$ is the operator of convolution\nby $\\varphi_Z$ on $L^2(Z)$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nIf $X\\cap Y$ is complemented in $X$ and $Y$ then $\\mathscr{C}_{XY}$ can be\nexpressed (non canonically) as a tensor product.\n\n\\begin{proposition}\\label{pr:xytens}\nIf $X\\cap Y$ is complemented in $X$ and $Y$ then\n\\[\n\\mathscr{C}_{XY}\\simeq \\mathscr{C}_{X\\cap Y} \\otimes \\mathscr{K}_{X\/Y,Y\/X}.\n\\] \nIn particular,\nif $X\\supset Y$ then $\\mathscr{C}_{XY}\\simeq \\mathscr{C}_{Y} \\otimes \\mathcal{H}_{X\/Y}$.\n\\end{proposition}\n\\proof If $X=(X\\cap Y)\\oplus E$ and $Y=(X\\cap Y)\\oplus F$ then we have\nto show that $\\mathscr{C}_{XY}\\simeq \\mathscr{C}_{X\\cap Y} \\otimes \\mathscr{K}_{EF}$ where the\ntensor product may be interpreted either as the exterior tensor\nproduct of the Hilbert $C^*$-modules $\\mathscr{C}_{X\\cap Y}$ and $\\mathscr{K}_{EF}$ or\nas the norm closure in the space of continuous operators from\n$L^2(Y)\\simeq L^2(X\\cap Y)\\otimes L^2(F)$ to $L^2(X)\\simeq L^2(X\\cap\nY)\\otimes L^2(E)$ of the algebraic tensor product of $\\mathscr{C}_{X\\cap Y}$\nand $\\mathscr{K}_{EF}$.  From Proposition \\ref{pr:def3} with $Z=X\\cap Y$ we\nget $\\mathscr{T}_{XY}\\simeq\\mathscr{T}_{X\\cap Y}\\otimes\\mathscr{K}_{EF}$. The relations\n\\eqref{eq:main} and the Definition \\ref{df:main} imply\n$\\mathscr{C}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}$ and we clearly have\n\\[\n\\mathcal{C}_Y^{X\\cap Y}={\\textstyle\\sum_{Z\\subset X\\cap Y}^\\mathrm{c}} \\mathcal{C}_Y(Z)\n\\simeq {\\textstyle\\sum_{Z\\subset X\\cap Y}^\\mathrm{c}} \n\\mathcal{C}_{X\\cap Y}(Z)\\otimes \\cc_{\\mathrm{o}}(F)\n\\simeq \\mathcal{C}_{X\\cap Y}\\otimes \\cc_{\\mathrm{o}}(F).\n\\]     \nThen we get\n\\[\n\\mathscr{C}_{XY}\\simeq \\mathscr{T}_{X\\cap Y}\\otimes\\mathscr{K}_{EF}\\cdot \n\\mathcal{C}_{X\\cap Y}\\otimes \\cc_{\\mathrm{o}}(F)=\n\\big(\\mathscr{T}_{X\\cap Y}\\cdot\\mathcal{C}_{X\\cap Y} \\big)\\otimes\n\\big(\\mathscr{K}_{EF}\\cdot\\cc_{\\mathrm{o}}(F)\\big)\n\\]\nand this is $\\mathscr{C}_{X\\cap Y} \\otimes \\mathscr{K}_{EF}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nIf $Z$ is complemented in $X$ and $Y$ then Theorem \\ref{th:yzintr}\ncan be improved. We shall describe this improvement only in the\nEuclidean case which will be useful in our treatment of\nnonrelativistic Hamiltonians. Thus below we assume that $X,Y$ are\nsubspaces of an Euclidean space (see \\S\\ref{ss:mouint} for\nnotations). Note that $V_k$ is the operator of multiplication by the\nfunction $x\\mapsto\\mathrm{e}^{i\\braket{x}{k}}$ where the scalar product\n$\\braket{x}{k}$ is well defined for any $x,k$ in the ambient space\n$\\mathcal{X}$. \n\n\\begin{theorem}\\label{th:xyzeintr}\n$\\mathscr{C}_{XY}(Z)$ is the set of $T\\in\\mathscr{L}_{XY}$ satisfying:\n\\begin{enumerate} \n\\item[{\\rm(1)}] \n$U_z^*T U_z=T$ for $z\\in Z$ and\n$\\|V^*_z T V_z-T\\|\\to 0$ if $z\\to 0$ in $Z$,\n\\item[{\\rm(2)}] \n$\\|(U_x-1)T\\|\\to 0$ if $x\\to 0$ in $X$ and $\\|(V_k-1)T\\|\\to 0$ if\n$k\\to 0$ in $X\/Z$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\\label{re:xyzeintr}\nCondition 2 may be replaced by\n\\begin{compactenum}\n\\item[{\\rm(2$'$)}]\n$\\|T(U_y-1)\\|\\to 0$ if $y\\to 0$ in $Y$ and $\\|T(V_k-1)\\|\\to 0$ if\n$k\\to 0$ in $Y\/Z$.\n\\end{compactenum}\nThis will be clear from the next proof.\n\\end{remark}\n\\proof Let $\\mathcal{F}\\equiv\\mathcal{F}_Z$ be the Fourier transformation in the\nspace $Z$, this is a unitary operator in the space $L^2(Z)$ which\ninterchanges the position and momentum observables $Q_Z,P_Z$. We\ndenote also by $\\mathcal{F}$ the operators $\\mathcal{F}\\otimes1_{\\mathcal{H}_{X\/Z}}$ and\n$\\mathcal{F}\\otimes1_{\\mathcal{H}_{Y\/Z}}$ which are unitary operators in the spaces\n$\\mathcal{H}_X$ and $\\mathcal{H}_Y$ due to \\eqref{eq:xyzint}.  If $S=\\mathcal{F} T\n\\mathcal{F}^{-1}$ then $S$ satisfies the following conditions:\n\\begin{enumerate} \n\\item[(i)]\n$V_z^*S V_z=S$ for $z\\in Z$, $\\|(V_z-1)S\\|\\to 0$ if $z\\to 0$ in $Z$,\nand $\\|U_z S U^*_z-S\\|\\to 0$ if $z\\to 0$ in $Z$;\n\\item[(ii)] \n$\\|(U_x-1)S\\|\\to 0$ and $\\|(V_x-1)S\\|\\to 0$ if $x\\to 0$ in $X\/Z$.\n\\end{enumerate}\nFor the proof, observe that the first part of condition (2) may be\nwritten as the conjunction of the two relations $\\|(U_z-1)T\\|\\to 0$\nif $z\\to 0$ in $Z$ and $\\|(U_x-1)T\\|\\to 0$ if $x\\to 0$ in $X\/Z$.  We\nshall work in the representations\n\\begin{equation}\\label{eq:fiber}\n\\mathcal{H}_X= L^2(Z;\\mathcal{H}_{X\/Z}) \\quad \\text{and} \\quad\n\\mathcal{H}_Y= L^2(Z;\\mathcal{H}_{Y\/Z}).\n\\end{equation} \nFrom the relation $V_z^*S V_z=S$ for all $z\\in Z$ it follows that\nthere is a bounded weakly measurable function\n$S(\\cdot):Z\\to\\mathscr{L}_{X\/Z,Y\/Z}$ such that in the representations\n\\eqref{eq:fiber} $S$ is the operator of multiplication by\n$S(\\cdot)$. Then $\\|U_z S U^*_z-S\\|\\to 0$ if $z\\to 0$ in $Z$ means\nthat the function $S(\\cdot)$ is uniformly continuous. And clearly\n$\\|(V_z-1)S\\|\\to 0$ if $z\\to 0$ in $Z$ is equivalent to the fact\nthat $S(\\cdot)$ tends to zero at infinity. Thus we see that\n$S(\\cdot)\\in\\cc_{\\mathrm{o}}(Z;\\mathscr{L}_{X\/Z,Y\/Z})$.\nThe condition (ii) can now be written\n\\[\n\\sup_{z\\in Z}\\big(\\|(U_x-1)S(z)\\|+ \\|(V_x-1)S(z)\\|\\big)\\to 0\n\\quad \\text{if } x\\to 0 \\text{ in } X\/Z.\n\\]\nFrom the Riesz-Kolmogorov theorem it follows that each $S(z)$ is a\ncompact operator. Thus we have $S(\\cdot)\\in\\cc_{\\mathrm{o}}(Z;\\mathscr{K}_{X\/Z,Y\/Z})$\nwhich implies $T\\in\\mathscr{C}_{XY}(Z)$ by Proposition \\ref{pr:def3}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{remark}\\label{re:half}\nSince $S(\\cdot)$ is continuous and tends to zero at infinity, for\neach $\\varepsilon>0$ there are points $z_1,\\dots,z_n\\in Z$ and\ncomplex functions $\\varphi_1,\\dots,\\varphi_n\\in\\cc_{\\mathrm{c}}(Z)$ such that\n\\[\n\\|S(z)-{\\textstyle\\sum_k}\\varphi_k(z)S(z_k)\\|\\leq\\varepsilon\\quad\n\\forall z\\in Z.\n\\]\nThe operators $S(z_k)$ being compact, applying once again the\nRiesz-Kolmogorov theorem we get\n\\[\n\\sup_{z\\in Z}\\big(\\|S(z)(U_y-1)\\|+ \\|S(z)(V_y-1)\\|\\big)\\to 0 \n\\quad \\text{if } y\\to 0 \\text{ in } Y\/Z.\n\\]\nThis explains why the second parts of conditions (2) and (3) of\nTheorem \\ref{th:yzintr} is not needed. \n\\end{remark}\n\n\n\n\n\\section{Affiliated operators}\n\\label{s:af}\n\\protect\\setcounter{equation}{0}\n\nIn this section we give examples of self-adjoint operators\naffiliated to the algebra $\\mathscr{C}$ constructed in Section \\ref{s:grass}\nand then we give a formula for their essential spectrum. We refer to\n\\S\\ref{ss:grca} for terminology and basic results related to the\nnotion of affiliation that we use and to \\cite{ABG,GI1,DG3} for\ndetails.\n\nWe recall that a self-adjoint operator $H$ on a Hilbert space $\\mathcal{H}$\nis \\emph{strictly affiliated} to a $C^*$-algebra of operators $\\mathscr{A}$\non $\\mathcal{H}$ if $(H+i)^{-1}\\in\\mathscr{A}$ (then $\\varphi(H)\\in\\mathscr{A}$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$) and if $\\mathscr{A}$ is the clspan of the elements\n$\\varphi(H)A$ with $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$ and $A\\in\\mathscr{A}$. This class\nof operators has the advantage that each time $\\mathscr{A}$ is\nnon-degenerately represented on a Hilbert space $\\mathcal{H}'$ with the help\nof a morphism $\\mathscr{P}:\\mathscr{A}\\to L(\\mathcal{H}')$, the observable $\\mathscr{P} H$ is\nrepresented by a usual (densely defined) self-adjoint operator on\n$\\mathcal{H}'$.\n\nThe diagonal algebra\n\\begin{equation}\\label{eq:d}\n\\mathscr{T}_{\\text{d}}\\equiv(\\mathscr{T}_{\\mathcal{S}})_\\text{d}=\\oplus_{X\\in\\mathcal{S}} \\mathscr{T}_X\n\\end{equation}\nhas a simple physical interpretation: this is the $C^*$-algebra\ngenerated by the kinetic energy operators.  Since\n$\\mathscr{C}_{XX}=\\mathscr{C}_X\\supset\\mathscr{C}_{X}(X)= \\mathscr{T}_X$ we see that $\\mathscr{T}_\\text{d}$\nis a $C^*$-subalgebra of $\\mathscr{C}$. From \\eqref{eq:nxyz},\n\\eqref{eq:txy1}, \\eqref{eq:txy2} and the Cohen-Hewitt theorem we get\n\\begin{equation}\\label{eq:ed}\n\\mathscr{C}(Z)\\mathscr{T}_\\text{d}=\\mathscr{T}_\\text{d}\\mathscr{C}(Z)=\\mathscr{C}(Z)\\quad \\forall Z\\in\\mathcal{S} \\quad \n\\text{and}\\hspace{2mm} \\mathscr{C} \\mathscr{T}_\\text{d}=\\mathscr{T}_\\text{d}\\mathscr{C}=\\mathscr{C}.\n\\end{equation} \nIn other terms, $\\mathscr{T}_\\text{d}$ acts\nnon-degenerately\\symbolfootnote[2]{\\ Note that if $\\mathcal{S}$ has a\n  largest element $\\mathcal{X}$ then the algebra $\\mathscr{C}(\\mathcal{X})$ acts on each\n  $\\mathscr{C}(Z)$ but this action is degenerate.} \non each $\\mathscr{C}(Z)$ and on $\\mathscr{C}$. It follows that a self-adjoint\noperator strictly affiliated to $\\mathscr{T}_\\text{d}$ is also strictly\naffiliated to $\\mathscr{C}$. \n\n\nFor each $X\\in\\mathcal{S}$ let $h_X:X^*\\to\\mathbb{R}$ be a continuous function\nsuch that $|h_X(k)|\\to\\infty$ if $k\\to\\infty$ in $X^*$. Then the\nself-adjoint operator $K_X\\equiv h_X(P)$ on $\\mathcal{H}_X$ is strictly\naffiliated to $\\mathscr{T}_X$ and the norm of $(K_X+i)^{-1}$ is equal to\n$\\sup_k(h^2_X(k)+1)^{-1\/2}$. Let $K\\equiv\\bigoplus_{X\\in\\mathcal{S}}K_X$,\nthis is a self-adjoint operator $\\mathcal{H}$. Clearly $K$ is affiliated to \n$\\mathscr{T}_\\text{d}$ if and only if \n\\begin{equation}\\label{eq:kin}\n\\lim_{X\\to\\infty}\\sup\\nolimits_{k}(h^2_X(k)+1)^{-1\/2}=0\n\\end{equation}\nand then $K$ is strictly affiliated to $\\mathscr{T}_\\text{d}$ (the set $\\mathcal{S}$ is\nequipped with the discrete topology). If the functions $h_X$ are\npositive this means that $\\min h_X$ tends to infinity when\n$X\\to\\infty$. One could avoid such a condition by considering an\nalgebra larger then $\\mathscr{C}$ such as to contain \n$\\prod_{X\\in\\mathcal{S}} \\mathscr{T}_X$, but we shall not develop this here.\n\nNow let $H=K+I$ with $I\\in\\mathscr{C}$ (or in the multiplier algebra) a\nsymmetric element. Then\n\\begin{equation}\\label{eq:res}\n(\\lambda-H)^{-1}=(\\lambda-K)^{-1}\\left(1-I(\\lambda-K)^{-1}\\right)^{-1}\n\\end{equation}\nif $\\lambda\\notin\\mathrm{Sp}(H)\\cup\\mathrm{Sp}(K)$ . Thus $H$ is strictly affiliated\nto $\\mathscr{C}$.  We interpret $H$ as the Hamiltonian of our system of\nparticles when the kinetic energy is $K$ and the interactions\nbetween particles are described by $I$. Even in the simple case\n$I\\in\\mathscr{C}$ these interactions are of a very general nature being a\nmixture of $N$-body and quantum field type interactions (which\ninvolve creation and annihilation operators so the number of\nparticles is not preserved).\n\nWe shall now use Theorem \\ref{th:gas} in order to compute the\nessential spectrum of an operator like $H$. The case of unbounded\ninteractions will be treated later on. Let $\\mathscr{C}_{\\geq E}$ be the\n$C^*$-subalgebra of $\\mathscr{C}$ determined by $E\\in\\mathcal{S}$ according to the\nrules of $\\S\\ref{ss:grca}$. More explicitly, we set\n\\begin{equation}\\label{eq:geqe}\n\\mathscr{C}_{\\geq E}={\\textstyle\\sum^\\mathrm{c}_{F\\supset E}\\mathscr{C}(F)}\\cong \n\\big({\\textstyle\\sum^\\mathrm{c}_{F\\supset E}}\n\\mathscr{C}_{XY}(F)\\big)_{X\\cap Y\\supset E}\n\\end{equation}\nand note that $\\mathscr{C}_{\\geq E}$ lives on the subspace $\\mathcal{H}_{\\geq\n  E}=\\bigoplus_{X\\supset E}\\mathcal{H}_X$ of $\\mathcal{H}$. Since in the second sum\nfrom \\eqref{eq:geqe} the group $F$ is such that $E\\subset F\\subset\nX\\cap Y$ the algebra $\\mathscr{C}_{\\geq E}$ is strictly included in the\nalgebra $\\mathscr{C}_\\mathcal{T}$ obtained by taking $\\mathcal{T}=\\{F\\in\\mathcal{S} \\mid F\\supset\nE\\}$ in \\eqref{eq:t}. \n\nLet $\\mathscr{P}_{\\geq E}$ be the canonical idempotent morphism of $\\mathscr{C}$\nonto $\\mathscr{C}_{\\geq E}$ introduced in \\S\\ref{ss:grca}. We consider\nthe self-adjoint operator on the Hilbert space $\\mathcal{H}_{\\geq E}$\ndefined as follows:\n\\begin{equation}\\label{eq:post}\n H_{\\geq E}=K_{\\geq E}+I_{\\geq E} \\quad \\text{where} \\quad \nK_{\\geq E}= \\oplus_{X\\geq E} K_X \\hspace{2mm} \\text{and} \n\\hspace{2mm} I_{\\geq E}=\\mathscr{P}_{\\geq E}I.\n\\end{equation}\nThen $H_{\\geq E}$ is strictly affiliated to $\\mathscr{C}_{\\geq E}$ and it\nfollows easily from \\eqref{eq:res} that\n\\begin{equation}\\label{eq:pre}\n\\mathscr{P}_{\\geq E}\\varphi(H)=\\varphi(H_{\\geq E}) \\quad \\forall\n\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R}).\n\\end{equation} \nNow let us assume that the group $O=\\{0\\}$ belongs to $\\mathcal{S}$. Then we\nhave\n\\begin{equation}\\label{eq:O}\n\\mathscr{C}(O)=K(\\mathcal{H}).\n\\end{equation} \nIndeed, from \\eqref{eq:nxyz} we get\n$\\mathscr{C}_{XY}(O)=\\mathscr{T}_{XY}\\cdot\\cc_{\\mathrm{o}}(Y)=\\mathscr{K}_{XY}$ which implies the\npreceding relation. If we also assume that $\\mathcal{S}$ is atomic and we\ndenote $\\mathcal{P}(\\mathcal{S})$ its set of atoms, then from Theorem \\ref{th:ga} we\nget a canonical embedding\n\\begin{equation}\\label{eq:quotc}\n\\mathscr{C}\/K(\\mathcal{H})\\subset\\displaystyle\\mbox{$\\textstyle\\prod$}_{\\substack{i}}\\nolimits_{E\\in\\mathcal{P}(\\mathcal{S})}\\mathscr{C}_{\\geq E}\n\\end{equation} \ndefined by the morphism $\\mathscr{P}\\equiv(\\mathscr{P}_{\\geq E})_{E\\in\\mathcal{P}(\\mathcal{S})}$.\nThen from \\eqref{eq:es2} we obtain:\n\\begin{equation}\\label{eq:ess1}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{E\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq E}).\n\\end{equation}\nOur next purpose is to prove a similar formula for a certain class\nof unbounded interactions $I$.\n\nLet $\\mathcal{G}\\equiv\\mathcal{G}_\\mathcal{S}=D(|K|^{1\/2})$ be the form domain of $K$\nequipped with the graph topology. Then $\\mathcal{G}\\subset\\mathcal{H}$ continuously\nand densely so after the Riesz identification of $\\mathcal{H}$ with its\nadjoint space $\\mathcal{H}^*$ we get the usual scale\n$\\mathcal{G}\\subset\\mathcal{H}\\subset\\mathcal{G}^*$ with continuous and dense embeddings.\nLet us denote\n\\begin{equation}\\label{eq:jap}\n\\jap{K} =|K+i|=\\sqrt{K^2+1}.\n\\end{equation}\nThen $\\jap{K}^{1\/2}$ is a self-adjoint operator on $\\mathcal{H}$ with domain\n$\\mathcal{G}$ and $\\jap{K}$ induces an isomorphism $\\mathcal{G}\\to\\mathcal{G}^*$.  The\nfollowing result is a straightforward consequence of Theorem 2.8 and\nLemma 2.9 from \\cite{DG3}. \n\n\\begin{theorem}\\label{th:af}\nLet $I:\\mathcal{G}\\to\\mathcal{G}^*$ be a continuous symmetric operator and let us\nassume that there are real numbers $\\mu,a$ with $0<\\mu<1$ such that\none of the following conditions is satisfied:\n\\begin{compactenum}[(i)]\n\\item\n$\\pm I \\leq\\mu|K+ia|,$\n\\item\n$K$ is bounded from below and $ I \\geq -\\mu|K+ia|.$\n\\end{compactenum}\nLet $H=K+I$ be the form sum of $K$ and $I$, so $H$ has as domain the\nset  of $u\\in\\mathcal{G}$ such that $Ku+Iu\\in\\mathcal{H}$ and acts as $Hu=Ku+Iu$. \nThen $H$ is a self-adjoint operator on $\\mathcal{H}$. If there is\n$\\alpha>1\/2$ such that \n$\\langle K\\rangle^{-1\/2}I\\langle K\\rangle^{-\\alpha}\\in\\mathscr{C}$ then $H$\nis strictly affiliated to $\\mathscr{C}$. \nIf $O\\in\\mathcal{S}$ and the semilattice $\\mathcal{S}$ is atomic then\n\\begin{equation}\\label{eq:ess2}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{E\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq E}).\n\\end{equation}\n\\end{theorem}\n\nThe last assertion of the theorem follows immediately from Theorem\n\\ref{th:gas} and is a general version of the HVZ theorem.  In order\nto have a more explicit description of the observables $H_{\\geq\n  E}\\equiv\\mathscr{P}_{\\geq E}H$ we now prove an analog of Theorem 3.5 from\n\\cite{DG3}. We cannot use that theorem in our context for three\nreasons: first we did not suppose that $\\mathcal{S}$ has a maximal element,\nthen even if $\\mathcal{S}$ has a maximal element $\\mathcal{X}$ the action of the\ncorresponding algebra $\\mathscr{C}(\\mathcal{X})$ on the algebras $\\mathscr{C}(E)$ is\ndegenerate, and finally our ``free'' operator $K$ is not affiliated\nto $\\mathscr{C}(\\mathcal{X})$.\n\n\\begin{theorem}\\label{th:afi}\nFor each $E\\in\\mathcal{S}$ let $I(E)\\in L(\\mathcal{G},\\mathcal{G}^*)$ be a symmetric\noperator such that:\n\\begin{compactenum}[(i)]\n\\item\n$\\jap{K}^{-1\/2}I(E)\\jap{K}^{-\\alpha}\\in\\mathscr{C}(E)$ for some\n$\\alpha> 1\/2$ independent of $E$,\n\\item\nthere are real positive numbers $\\mu_E,a$ such that either $\\pm\nI(E) \\leq\\mu_E|K+ia|$ for all $E$ or $K$ is bounded from below and\n$ I(E) \\geq -\\mu_E|K+ia|$ for all $E$,\n\\item\nwe have $\\sum_E\\mu_E\\equiv\\mu<1$ and the series $\\sum_E I(E)\\equiv I$\nis norm summable in $L(\\mathcal{G},\\mathcal{G}^*)$.\n\\end{compactenum}\nLet us set $I_{\\geq E}=\\sum_{F\\geq E}I(F)$. Define the self-adjoint\noperator $H=K+I$ on $\\mathcal{H}$ as in Theorem \\ref{th:af} and define\nsimilarly the self-adjoint operator $H_{\\geq E}=K_{\\geq E}+I_{\\geq\n  E}$ on $\\mathcal{H}_{\\geq E}$.  Then the operator $H$ is strictly\naffiliated to $\\mathscr{C}$, the operator $H_{\\geq E}$ is strictly\naffiliated to $\\mathscr{C}_{\\geq E}$, and we have $\\mathscr{P}_{\\geq E}H=H_{\\geq E}$.\n\\end{theorem}\n\\proof  We shall consider only the case when $\\pm I(E)\n\\leq\\mu_E|K+ia|$ for all $E$. The more singular situation when $K$\nis bounded from below but there is no restriction on the positive\npart of the operators $I(E)$ (besides summability) is more difficult\nbut the main idea has been explained in \\cite{DG3}.\n\nWe first make some comments to clarify the definition of the\noperators $H$ and $H_{\\geq E}$.  Observe that our assumptions imply\n$\\pm I\\leq\\mu|K+ia|$ hence if we set\n\\[\n\\Lambda\\equiv|K+ia|^{-1\/2}=(K^2+a^2)^{-1\/4}\\in \\mathscr{T}_\\text{d}\n\\]\nthen  we obtain\n\\[\n\\pm\\braket{u}{Iu}\\leq\\mu\\braket{u}{|K+ia|u}=\n\\mu\\| |K+ia|^{1\/2}u\\| = \\mu\\| \\Lambda^{-1} u\\|\n\\] \nwhich is equivalent to $\\pm\\Lambda I\\Lambda\\leq\\mu$ or $\\|\\Lambda\nI\\Lambda\\|\\leq\\mu$. In particular we may use Theorem \\ref{th:af}\nin order to define the self-adjoint operator $H$. Moreover, we have\n\\[\n\\jap{K}^{-1\/2}I\\jap{K}^{-\\alpha}={\\textstyle\\sum_E}\n\\jap{K}^{-1\/2}I(E)\\jap{K}^{-\\alpha}\\in\\mathscr{C}\n\\]\nbecause the series is norm summable in $L(\\mathcal{H})$. Thus $H$ is\nstrictly affiliated to $\\mathscr{C}$.  \n\nIn order to define $H_{\\geq E}$ we first make a remark on \n$I_{\\geq E}$.  If we set $\\mathcal{G}_X=D(|K_X|^{-1\/2})$ and if we equip\n$\\mathcal{G}$ and $\\mathcal{G}_X$ with the norms  \\label{p:formd}\n\\[\n\\|u\\|_\\mathcal{G}=\\|\\jap{K}^{1\/2}u\\|_\\mathcal{H} \\quad \\text{and} \\quad\n\\|u\\|_{\\mathcal{G}_X}=\\|\\jap{K_X}^{1\/2}u\\|_{\\mathcal{H}_X}\n\\]\nrespectively then clearly $\\mathcal{G}=\\oplus_X\\mathcal{G}_X$ and\n$\\mathcal{G}^*=\\oplus_X\\mathcal{G}^*_X $ where the sums are Hilbertian direct sums\nand $\\mathcal{G}^*$ and $\\mathcal{G}_X^*$ are equipped with the dual norms.  Then\neach $I(F)$ may be represented as a matrix\n$I(F)=(I_{XY}(F))_{X,Y\\in\\mathcal{S}}$ of continuous operators\n$I_{XY}(E):\\mathcal{G}_Y\\to\\mathcal{G}^*_X$. Clearly\n\\[\n\\jap{K}^{-1\/2}I(F)\\jap{K}^{-\\alpha}=\n\\left(\\jap{K_X}^{-1\/2}I_{XY}(F)\\jap{K_Y}^{-\\alpha}\\right)_{X,Y\\in\\mathcal{S}}\n\\] \nand since by assumption (i) this belongs to $\\mathscr{C}(F)$ we see that\n$I_{XY}(F)=0$ if $X\\not\\supset F$ or $Y\\not\\supset F$. Now fix $E$\nand let $F\\supset E$. Then, when viewed as a sesquilinear form,\n$I(F)$ is supported by the subspace $\\mathcal{H}_{\\geq E}$ and has domain\n$\\mathcal{G}_{\\geq E}= D(|K_{\\geq E}|^{1\/2}$.  It follows that $I_{\\geq E}$ \nis a sesquilinear form with domain $\\mathcal{G}_{\\geq E}$ supported by the\nsubspace $\\mathcal{H}_{\\geq E}$ and may be thought as an element of\n$L(\\mathcal{G}_{\\geq E},\\mathcal{G}^*_{\\geq E})$ such that\n$\\pm I_{\\geq E}\\leq \\mu |K_{\\geq E}+ia|$ because \n$\\sum_{F\\supset E}\\mu_F\\leq \\mu$. To conclude, we may now define \n$H_{\\geq E}=K_{\\geq E}+I_{\\geq E}$ exactly as in the case of $H$ and\nget a self-adjoint operator on $\\mathcal{H}_{\\geq E}$ strictly affiliated to\n$\\mathscr{C}_{\\geq E}$. Note that this argument also gives\n\\begin{equation}\\label{eq:ek}\n\\jap{K}^{-1\/2} I(F) \\jap{K}^{-1\/2}=\n\\jap{K_{\\geq E}}^{-1\/2} I(F) \\jap{K_{\\geq E}}^{-1\/2}.\n\\end{equation}\nIt remains to be shown that $\\mathscr{P}_{\\geq E}H=H_{\\geq E}$. If we set\n$R\\equiv(ia-H)^{-1}$ and $R_{\\geq E}\\equiv(ia-H_{\\geq E})^{-1}$ then\nthis is equivalent to $\\mathscr{P}_{\\geq E}R=R_{\\geq E}$.  Let us set\n\\[\nU=|ia-K|(ia-K)^{-1}=\\Lambda^{-2}(ia-K)^{-1}, \\quad\nJ=\\Lambda I\\Lambda U.\n\\]\nThen $U$ is a unitary operator and $\\|J\\|<1$, so we get a norm\nconvergent series expansion\n\\[\nR=(ia-K-I)^{-1}=\n\\Lambda U(1-\\Lambda I\\Lambda U)^{-1}\\Lambda =\n{\\textstyle\\sum_{n\\geq0}}\\Lambda U J^n\\Lambda\n\\]\nwhich implies\n\\[\n\\mathscr{P}_{\\geq E} (R)=\n{\\textstyle\\sum_{n\\geq0}}\n\\mathscr{P}_{\\geq E}\\big(\\Lambda U J^{n}\\Lambda\\big)\n\\]\nthe series being norm convergent. Thus it suffices to prove that for\neach $n\\geq0$ \n\\begin{equation}\\label{eq:ekk}\n\\mathscr{P}_{\\geq E}\\big(\\Lambda U J^{n}\\Lambda\\big)=\n\\Lambda_{\\geq E} (J_{\\geq E})^{n}\\Lambda_{\\geq E}\n\\end{equation}\nwhere $J_{\\geq E}=\\Lambda_{\\geq E} I_{\\geq E}\\Lambda_{\\geq E}\nU_{\\geq E}$.  Here $\\Lambda_{\\geq E}$ and $U_{\\geq E}$ are\nassociated to $K_{\\geq E}$ in the same way $\\Lambda$ and $K$ are\nassociated to $K$. For $n=0$ this is obvious because \n$\\mathscr{P}_{\\geq E}K=K_{\\geq E}$. If $n=1$ this is easy because\n\\begin{align}\\label{eq:e}\n\\Lambda U J\\Lambda &= \\Lambda U \\Lambda I \\Lambda U\\Lambda=\n(ia-K)^{-1} I (ia-K)^{-1} \\\\\n&=\n[(ia-K)^{-1}\\jap{K}^{1\/2}] \\cdot \n[\\jap{K}^{-1\/2} I \\jap{K}^{-\\alpha}] \\cdot\n[\\jap{K}^{\\alpha} (ia-K)^{-1}] \\nonumber\n\\end{align}\nand it suffices to note that \n$\\mathscr{P}_{\\geq E}(\\jap{K}^{-1\/2} I(F) \\jap{K}^{-\\alpha})=0$ if\n$F\\not\\supset E$ and to use \\eqref{eq:ek} for $F\\supset E$. \n\nTo treat the general case we make some preliminary remarks. \nIf $J(F)=\\Lambda I(F) \\Lambda U$ then $J=\\sum_F J(F)$ where the\nconvergence holds in norm on $\\mathcal{H}$ because of the condition (iii). \nThen we have a norm convergent expansion\n\\[\n\\Lambda U J^n \\Lambda ={\\textstyle\\sum_{F_1,\\dots,F_n\\in\\mathcal{S}}}\n\\Lambda U J(F_1)\\dots J(F_n) \\Lambda.\n\\]\nAssume that we have shown $\\Lambda U J(F_1)\\dots\nJ(F_n)\\Lambda\\in\\mathscr{C}(F_1\\cap\\dots\\cap F_n)$.  Then we get\n\\begin{equation}\\label{eq:ekkk}\n\\mathscr{P}_{\\geq E}(\\Lambda U J^n \\Lambda)=\n{\\textstyle\\sum_{F_1\\geq E,\\dots,F_n\\geq E}}\n\\Lambda U J(F_1)\\dots J(F_n) \\Lambda\n\\end{equation}\nbecause if one $F_k$ does not contain $E$ then the intersection\n$F_1\\cap\\dots\\cap F_n$ does not contain $E$ hence $\\mathscr{P}_{\\geq E}$\napplied to the corresponding term gives $0$. Because of\n\\eqref{eq:ek} we have $J(F)=\\Lambda_{\\geq E} I(F) \\Lambda_{\\geq E}\nU_{\\geq E}$ if $F\\supset E$ and we may replace everywhere in the\nright hand side of \\eqref{eq:ekkk} $\\Lambda$ and $U$ by\n$\\Lambda_{\\geq E}$ and $U_{\\geq E}$. This clearly proves\n\\eqref{eq:ekk}. \n\nNow we prove the stronger fact \n$\\Lambda U J(F_1)\\dots J(F_n)\\in\\mathscr{C}(F_1\\cap\\dots\\cap F_n)$.  \nIf $n=1$ this follows from a slight modification of\n\\eqref{eq:e}: the last factor on the right hand side of \\eqref{eq:e}\nis missing but is not needed. Assume that the assertion holds for\nsome   $n$. Since $K$ is strictly affiliated to $\\mathscr{T}_\\text{d}$ and\n$\\mathscr{T}_\\text{d}$ acts non-degenerately on each $\\mathscr{C}(F)$ we may use the\nCohen-Hewitt theorem to deduce that there is $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$\nsuch that\n$\\Lambda U J(F_1)\\dots J(F_n)=T\\varphi(K)$ \nfor some $T\\in\\mathscr{C}(F_1\\cap\\dots\\cap F_n)$. \nThen\n\\[\n\\Lambda U J(F_1)\\dots J(F_n)J(F_{n+1})=T\\varphi(K)J(F_{n+1})\n\\]\nhence it suffices to prove that $\\varphi(K)J(F)\\in\\mathscr{C}(F)$ for any\n$F\\in\\mathcal{S}$ and any $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$. But the set of $\\varphi$\nwhich have this property is a closed subspace of $\\cc_{\\mathrm{o}}(\\mathbb{R})$ which\nclearly contains the functions $\\varphi(\\lambda)=(\\lambda -z)^{-1}$\nif $z$ is not real hence is equal to $\\cc_{\\mathrm{o}}(\\mathbb{R})$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\\begin{remark}\\label{re:alpha}\nChoosing $\\alpha>1\/2$ allows one to consider perturbations of $K$\nwhich are of the same order as $K$, e.g. in the $N$-body situations\none may add to the Laplacian $\\Delta$ on operator like $\\nabla^*\nM\\nabla$ where the function $M$ is bounded measurable and has the\nstructure of an $N$-body type potential, cf. \\cite{DG3,DerI}.  \n\\end{remark}\n\n\nThe only assumption of Theorem \\ref{th:afi} which is really relevant\nis $\\jap{K}^{-1\/2}I(E)\\jap{K}^{-\\alpha}\\in\\mathscr{C}(E)$.  We shall give\nbelow more explicit conditions which imply it.  If we change\nnotation $E\\to Z$ and use the formalism introduced in the proof of\nTheorem \\ref{th:afi} we have\n\\begin{equation}\\label{eq:xye}\nI(Z)=(I_{XY}(Z))_{X,Y\\in\\mathcal{S}} \\quad \\text{with} \\quad\nI_{XY}(Z):\\mathcal{G}_Y\\to\\mathcal{G}^*_X \\text{ continuous}.\n\\end{equation}\nWe are interested in conditions on $I_{XY}(Z)$ which imply\n\\begin{equation}\\label{eq:xye1}\n\\jap{K_X}^{-1\/2}I_{XY}(Z)\\jap{K_X}^{-\\alpha} \\in \\mathscr{C}_{XY}(Z).\n\\end{equation}\nFor this we shall use Theorem \\ref{th:yzintr} which gives a simple\nintrinsic characterization of $\\mathscr{C}_{XY}(Z)$.\n\nThe construction which follows is interesting only if $X$ is not a\ndiscrete group, otherwise $X^*$ is compact and many conditions are\ntrivially satisfied. We shall use weights only in order to avoid\nimposing on the functions $h_X$ regularity conditions stronger than\ncontinuity.\n\nA positive function $w$ on $X^*$ is a \\emph{weight} if\n$\\lim_{k\\to\\infty} w(k)=\\infty$ and $w(k+p)\\leq\\omega(k)w(p)$ for\nsome function $\\omega$ on $X^*$ and all $k,p$. We say that $w$ is\n\\emph{regular} if one may choose $\\omega$ such that\n$\\lim_{k\\to0}\\omega(k)=1$.  The example one should have in mind when\n$X$ is an Euclidean space is $w(k)=\\jap{k}^s$ for some $s>0$. Note\nthat we have $\\omega(-k)^{-1}\\leq w(k+p)w(p)^{-1} \\leq \\omega(k)$\nhence if $w$ is a regular weight then \\label{p:regw}\n\\begin{equation}\\label{eq:regw}\n\\theta(k)\\equiv \\sup_{p\\in X^*}\\frac{|w(k+p)-w(p)|}{w(p)} \n\\Longrightarrow\n\\lim_{k\\to0}\\theta(k)=0.\n\\end{equation}\nIt is clear that if $w$ is a regular weight and $\\sigma\\geq 0$ is a\nreal number then $w^\\sigma$ is also a regular weight.\n\nWe say that two functions $f,g$ defined on a neighborhood of \ninfinity of $X^*$ are \\emph{equivalent} and we write $f\\sim g$ if\nthere are numbers $a,b$ such that $a|f(k)|\\leq|g(k)|\\leq\nb|f(k)|$. Then $|f|^\\sigma\\sim|g|^\\sigma$ for all $\\sigma>0$.\n\n\nWe denote $\\mathcal{G}^\\sigma_X=D(|K_X|^{\\sigma\/2})$ and\n$\\mathcal{G}^{-\\sigma}_X\\equiv(\\mathcal{G}^{\\sigma}_X)^*$ with $\\sigma \\geq 1$.  In\nparticular $\\mathcal{G}^1_X=\\mathcal{G}_X$ and $\\mathcal{G}^{-1}_X=\\mathcal{G}^*_X$.\n\n\\begin{proposition}\\label{pr:tex}\n  Assume that $h_X,h_Y$ are equivalent to regular weights. Let\n  $Z\\subset X\\cap Y$ and let $I_{XY}(Z)$ be a continuous map\n  $\\mathcal{G}_Y\\to\\mathcal{G}^*_X$ such that\n\\begin{enumerate}\n\\item\n$U_z I_{XY}(Z)=I_{XY}(Z) U_z$ if $z\\in Z$ and\n$V^*_k I_{XY}(Z) V_k\\to I_{XY}(Z)$ if $k\\to 0$ in $(X+Y)^*$,\n\\item \n$(U_x-1)I_{XY}(Z)\\to 0$ if $x\\to 0$ in $X$ and\n$(V_k-1) I_{XY}(Z)\\to 0$ if $k\\to 0$ in $(X\/Z)^*$,\n\\end{enumerate}\nwhere the limits hold in norm in $L(\\mathcal{G}^{\\sigma}_Y,\\mathcal{G}^{-1}_X)$ for\nsome $\\sigma\\geq1$. Then \\eqref{eq:xye1} holds with\n$\\alpha=\\sigma\/2$.\n\\end{proposition}\n\\proof We begin with some general comments on weights. Let $w$ be a\nregular weight and let $\\mathcal{G}_X$ be the domain of the operator $w(P)$\nin $\\mathcal{H}_X$ equipped with the norm $\\|w(P)u\\|$. Then $\\mathcal{G}_X$ is a\nHilbert space and if $\\mathcal{G}^*_X$ is its adjoint space then we get a\nscale of Hilbert spaces $\\mathcal{G}_X\\subset\\mathcal{H}_X\\subset\\mathcal{G}^*_X$ with\ncontinuous and dense embeddings. Since $U_x$ commutes with $w(P)$ it\nis clear that $\\{U_x\\}_{x\\in X}$ induces strongly continuous unitary\nrepresentation of $X$ on $\\mathcal{G}_X$ and $\\mathcal{G}^*_X$.  Then\n\\[\n\\|V_k u\\|_{\\mathcal{G}_X}=\\|w(k+P)u\\|\\leq\\omega(k)\\|u\\|_{\\mathcal{G}_X}\n\\]\nfrom which it follows that $\\{V_k\\}_{k\\in X^*}$ induces by\nrestriction and extension strongly continuous representations of\n$X^*$ in $\\mathcal{G}_X$ and $\\mathcal{G}^*_X$. Moreover, as operators on $\\mathcal{H}_X$\nwe have \\label{p:RK}\n\\begin{align}  \n|V_k^*w(P)^{-1}V_k-w(P)^{-1}| \n&=|w(k+P)^{-1}-w(P)^{-1}| \n= |w(k+P)^{-1}(w(P)-w(k+P))w(P)^{-1}|  \\nonumber \\\\\n& \\leq \\omega(-k)|(w(P)-w(k+P))w(P)^{-2}|\n\\leq \\omega(-k)\\theta(k) w(P)^{-1}. \\label{eq:refw} \n\\end{align}\nNow let $w_X,w_Y$ be regular weights equivalent to\n$|h_X|^{1\/2},|h_Y|^{1\/2}$ and let us set $S=I_{XY}(Z)$. Then\n\\[\n\\jap{K_X}^{-1\/2}S\\jap{K_Y}^{-\\alpha}=\n\\jap{K_X}^{-1\/2}w_X(P)\\cdot \nw_X(P)S w_Y(P)^{-2\\alpha} \\cdot\nw_Y(P)^{2\\alpha}\\jap{K_Y}^{-\\alpha}\n\\]\nand $\\jap{h_X}^{-1\/2}w_X$, $\\jap{h_Y}^{-\\alpha}w_Y^{2\\alpha}$ and\ntheir inverses are bounded continuous functions on $X,Y$. Since\n$\\mathscr{C}_{XY}(Z)$ is a non-degenerate left $\\mathscr{T}_X$-module and right\n$\\mathscr{T}_Y$-module we may use the Cohen-Hewitt theorem to deduce that\n\\eqref{eq:xye1} is equivalent to\n\\begin{equation}\\label{eq:xye2}\nw_X(P)^{-1}I_{XY}(Z) w_Y(P)^{-\\sigma} \\in \\mathscr{C}_{XY}(Z)\n\\end{equation}\nwhere $\\sigma=2\\alpha$.  To simplify notations we set $W_X=w_X(P),\nW_Y=w^\\sigma_Y(P)$.  We also omit the index $X$ or $Y$ for the\noperators $W_X,W_Y$ since their value is obvious from the\ncontext. In order to show $W^{-1}SW^{-1}\\in \\mathscr{C}_{XY}(Z)$ we check\nthe conditions of Theorem \\ref{th:yzintr} with $T=W^{-1}SW^{-1}$.\nThe first part of condition (2) of the theorem is verified by the\nhypothesis (2) of the present proposition.  We may assume $\\sigma>1$\nand then hence the second part of condition (2) of the theorem\nfollows from\n\\[\n\\|T(U_y-1)\\|\\leq \\|W^{-1}I_{XY}(Z)w_Y^{-1}(P)\\|\\to 0 \n\\|(U_y-1)w_Y^{1-\\sigma}(P)\\| \\quad \\text{if } y\\to0.\n\\]\nTo check the second part of condition (1) of the theorem set\n$W_k=V_k^*WV_k$ and $S_k=V_k^* SV_k$ and write\n\\begin{align*}\nV_k^*TV_k-T \n&= W_k^{-1} S_k W_k^{-1}-W^{-1}SW^{-1}\\\\\n&= (W_k^{-1}-W^{-1})S_kW_k^{-1} + W^{-1}(S_k-S)W_k^{-1}\n+W^{-1}S(W_k^{-1} -W^{-1}).\n\\end{align*}\nNow if we use \\eqref{eq:refw} and set $\\xi(k)=\\omega(-k)\\theta(k)$\nwe get\n\\begin{align*}\n\\|V_k^*TV_k-T\\| &\\leq \n\\xi(k)\\|W^{-1}S_kW_k^{-1}\\| + \\|W^{-1}(S_k-S)W^{-1}\\|\\|W W_k^{-1}\\|\n+\\xi(k)\\|W^{-1}SW^{-1}\\|\n\\end{align*}\nwhich clearly tends to zero if $k\\to0$. Condition\n(3) of Theorem \\ref{th:yzintr} follows by a similar argument.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nNow let $H$ be defined according to the algorithm of \\S\\ref{ss:ex}.\nThen condition (i) of Theorem \\ref{th:afi} will be satisfied for all\n$\\alpha >1\/2$. Indeed, from Proposition \\ref{pr:tex} we get\n$\\jap{K}^{-1\/2}\\Pi_\\mathcal{T} I(Z)\\Pi_\\mathcal{T}\\jap{K}^{-\\alpha}\\in\\mathscr{C}(Z)$ for\nany finite $\\mathcal{T}$ and this operator converges in norm to\n$\\jap{K}^{-1\/2} I(Z)\\jap{K}^{-\\alpha}$.  Thus all conditions of\nTheorem \\ref{th:afi} are fulfilled by the Hamiltonian $H=K+I$ and so\n$H$ is strictly affiliated to $\\mathscr{C}$.  \\label{p:algor}\n\n\n\n\\section{The Mourre estimate} \n\\label{s:mou}\n\\protect\\setcounter{equation}{0}\n\n\n\n\\subsection{Proof of the Mourre estimate}\n\\label{ss:mest}\n\nFrom now on we work in the framework of the second part of Section\n\\ref{s:euclid}, so we assume that $\\mathcal{S}$ is a \\emph{finite}\nsemilattice of finite dimensional subspaces of an Euclidean\nspace. In this subsection we prove the Mourre estimate for\nnonrelativistic Hamiltonians.  The strategy of the proof is that\nintroduced in \\cite{BG2} and further developed in \\cite{ABG,DG2}\n(graded $C^*$-algebras over infinite semilattices and dispersive\nHamiltonians are considered in Section 5 from \\cite{DG2}).  We\nchoose the generator $D$ of the dilation group $W_\\tau$ in $\\mathcal{H}$ as\nconjugate operator for reasons explained below. For special types of\ninteractions, similar to those occurring in quantum field models,\nwhich are allowed by our formalism, better choices can be made, but\nat a technical level there is nothing new in that with respect to\n\\cite{Geo} (these special interactions correspond to distributive\nsemilattices $\\mathcal{S}$).\n\n\n\n\nThe dilations implement a group of automorphisms of the\n$C^*$-algebra $\\mathscr{C}$ which is compatible with the grading, i.e. it\nleaves invariant each component $\\mathscr{C}(Z)$ of $\\mathscr{C}$.  In fact, it is\nclear that $W_\\tau^*\\mathscr{C}_{XY}(Z)W_\\tau=\\mathscr{C}_{XY}(Z)$ for all $X,Y,Z$\nhence $W_\\tau^*\\mathscr{C}(Z)W_\\tau=\\mathscr{C}(Z)$.  This fact plays a fundamental\nrole in the proof of the Mourre estimate for operators affiliated to\n$\\mathscr{C}$ and explains the choice of $D$ as conjugate operator.\nMoreover, for each $T\\in\\mathscr{C}$ the map $\\tau\\mapsto W^*_\\tau T W_\\tau$\nis norm continuous. We can compute explicitly the function\n$\\widehat\\rho_H$ thanks to the relation\n\\begin{equation}\\label{eq:dlap}\nW^*_\\tau \\Delta_X W_\\tau = \\mathrm{e}^\\tau\\Delta_X \\quad \\text{or}\\quad\n[\\Delta_X, i D]=\\Delta_X \n\\end{equation}\nWe say that a self-adjoint operator $H$ \\emph{is of class $C^1(D)$}\nor \\emph{of class $C^1_\\mathrm{u}(D)$} if $W^*_\\tau RW_\\tau$ as a function\nof $\\tau$ is of class $C^1$ strongly or in norm respectively.  Here\n$R=(H-z)^{-1}$ for some $z$ outside the spectrum of $H$.  The formal\nrelation\n\\begin{equation}\\label{eq:dres}\n[D,R]= R[H,D] R \n\\end{equation}\ncan be given a rigorous meaning as follows. If $H$ is of class\n$C^1(D)$ then the intersection $\\mathscr{D}$ of the domains of the operators\n$H$ and $D$ is dense in $D(H)$ and the sesquilinear form with domain\n$\\mathscr{D}$ associated to the formal expression $HD-DH$ is continuous for\nthe topology of $D(H)$ so extends uniquely to a continuous\nsesquilinear form on the domain of $H$ which is denoted\n$[H,D]$. This defines the right hand side of \\eqref{eq:dres}.  The\nleft hand side can be defined for example as \n$i\\frac{d}{d\\tau}W_\\tau^*RW_\\tau|_{\\tau=0}$. \n\nFor Hamiltonians as those considered here it is easy to decide that\n$H$ is of class $C^1(D)$ in terms of properties of the commutator\n$[H, D]$.  Moreover, the following is easy to prove: \\emph{if $H$\n  is affiliated to $\\mathscr{C}$ then $H$ is of class $C^1_\\mathrm{u}(D)$ if and\n  only if $H$ is of class $C^1(D)$ and $[R,D]\\in\\mathscr{C}$}.\n\nLet $H$ be of class $C^1(D)$ and $\\lambda\\in\\mathbb{R}$. Then for each\n$\\theta\\in\\cc_{\\mathrm{c}}(\\mathbb{R})$ with $\\theta(\\lambda)\\neq0$ one may find a real\nnumber $a$ and a compact operator $K$ such that \n\\begin{equation}\\label{eq:must}\n\\theta(H)^*[H,iD]\\theta(H)\\geq a|\\theta(H)|^2+K.\n\\end{equation}\n\n\\begin{definition}\\label{df:must}\nThe upper bound $\\widehat\\rho_H(\\lambda)$ of the numbers $a$ for which\nsuch an estimate holds is \\emph{the best constant in the Mourre\n  estimate for $H$ at $\\lambda$}.  The \\emph{threshold set} of $H$\n(relative to $D$) is the closed real set\n\\begin{equation}\\label{eq:thr0}\n\\tau(H)=\\{\\lambda \\mid \\widehat\\rho_H(\\lambda)\\leq0\\}\n\\end{equation}\nOne says that $D$ is \\emph{conjugate to} $H$ at $\\lambda$ if \n$\\widehat\\rho_H(\\lambda)>0$. \n\\end{definition}\n\nThe set $\\tau(H)$ is closed because the function\n$\\widehat\\rho_H:\\mathbb{R}\\to]-\\infty,\\infty]$ is lower semicontinuous.\n\n\nTo each closed real set $A$ we associate the function\n$N_A:\\mathbb{R}\\to[-\\infty,\\infty[$ defined by\n\\begin{equation}\\label{eq:na}\nN_A(\\lambda)=\\sup\\{ x\\in A \\mid x\\leq\\lambda\\}.\n\\end{equation}\nWe make the convention $\\sup\\emptyset=-\\infty$. Thus $N_A$ may take\nthe value $-\\infty$ if and only if $A$ is bounded from below and then\n$N_A(\\lambda)=-\\infty$ if and only if $\\lambda<\\min A$. The function\n$N_A$ is further discussed during the proof of Lemma \\ref{lm:nab}.\n\n\nNonrelativistic many-body Hamiltonians have been introduced in\nDefinition \\ref{df:NR}.  Let $\\mathrm{ev}(T)$ be the set of\neigenvalues of an operator $T$.\n\n\\begin{theorem}\\label{th:thr}\n  Let $\\mathcal{S}$ be finite and let $H=H_\\mathcal{S}$ be a nonrelativistic\n  many-body Hamiltonian of class $C^1_\\mathrm{u}(D)$. Then\n  $\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ for all real\n  $\\lambda$ and $\\tau(H)$ is a closed \\emph{countable} real set\n  given by\n\\begin{equation}\\label{eq:thr}\n\\tau(H)=\\textstyle{\\bigcup}_{X\\neq O}\\mathrm{ev}(H_{\\mathcal{S}\/X}).\n\\end{equation} \n\\end{theorem}\n\\proof We first treat the case $O\\in\\mathcal{S}$. We need some facts which\nare discussed in detail in Sections 7.2, 8.3 and 8.4 from \\cite{ABG}\n(see pages 51--61 in \\cite{BG2} for a shorter presentation).\n\n(i) For each real $\\lambda$ let $\\rho_H(\\lambda)$ be the upper bound\nof the numbers $a$ for which an estimate like \\eqref{eq:must} but\nwith $K=0$ holds. This defines a lower semicontinuous function\n$\\rho_H:\\mathbb{R}\\to ]-\\infty,\\infty] $ hence the set\n$\\varkappa(H)=\\{\\lambda \\mid \\rho_H(\\lambda)\\leq 0\\}$ is a closed\nreal set called \\emph{critical set} of $H$ (relative to $D$). We\nclearly have $\\rho_H\\leq\\widehat\\rho_H$ and so\n$\\tau(H)\\subset\\varkappa(H)$.\n\n(ii) Let $\\mu(H)$ be the set of eigenvalues of $H$ such that\n$\\widehat\\rho_H(\\lambda)>0$. Then $\\mu(H)$ is a discrete subset of\n$\\mathrm{ev}(H)$ consisting of eigenvalues of finite\nmultiplicity. This is essentially the virial theorem.\n\n(iii) There is a simple and rather unexpected relation between the\nfunctions $\\rho_H$ and $\\widehat\\rho_H$: they are ``almost'' equal. In\nfact, $\\rho_H(\\lambda)=0$ if $\\lambda\\in\\mu(H)$ and\n$\\rho_H(\\lambda)=\\widehat\\rho_H(\\lambda)$ otherwise. In particular\n\\begin{equation}\\label{eq:tev}\n\\varkappa(H)=\\tau(H)\\cup \\mathrm{ev}(H)=\\tau(H)\\sqcup\\mu(H)\n\\end{equation}\nwhere $\\sqcup$ denotes disjoint union. \n\n(iv) This step is easy but rather abstract and the $C^*$-algebra\nsetting really comes into play. We assume that $H$ is affiliated to\nour algebra $\\mathscr{C}$. The preceding arguments did not require more than\nthe $C^1(D)$ class. Now we require $H$ to be of class $C^1_\\mathrm{u}(D)$.\nThen the operators $H_{\\geq X}$ are also of class $C^1_\\mathrm{u}(D)$ and\nwe have the important relation (Theorem 8.4.3 in \\cite{ABG} or\nTheorem 4.4 in \\cite{BG2})\n\\[\n\\widehat\\rho_H=\\min_{X\\in\\mathcal{P}(\\mathcal{S})}\\rho_{H_{\\geq X}}.\n\\]\nTo simplify notations we adopt the abbreviations $\\rho_{H_{\\geq\n    X}}=\\rho_{\\geq X}$ and instead of $X\\in\\mathcal{P}(\\mathcal{S})$ we write\n$X\\gtrdot O$, which should be read ``$X$ covers $O$''. For coherence\nwith later notations we also set $\\widehat\\rho_H=\\widehat\\rho_\\mathcal{S}$.  So\n\\eqref{eq:mustq} may be written\n\\begin{equation}\\label{eq:mustq}\n\\widehat\\rho_\\mathcal{S}=\\min_{X\\gtrdot O}\\rho_{\\geq X}.\n\\end{equation}\n\n(v) From \\eqref{eq:dlap} and \\eqref{eq:NR} we get\n\\[\nH_{\\geq X}=\\Delta_X\\otimes1+ 1\\otimes H_{\\mathcal{S}\/X}, \\quad\n[H_{\\geq X},iD]=\\Delta_X\\otimes1+ 1\\otimes [D,i H_{\\mathcal{S}\/X}].\n\\]\nRecall that we denote $D$ the generator of the dilation group\nindependently of the space in which it acts. We note that the formal\nargument which gives the second relation above can easily be made\nrigorous but this does not matter here. Indeed, since $H_{\\geq X}$\nis of class $C^1_\\mathrm{u}(D)$ and by using the first relation above, one\ncan easily show that $H_{\\mathcal{S}\/X}$ is also of class $C^1_\\mathrm{u}(D)$ (see\nthe proof of Lemma 9.4.3 in \\cite{ABG}). Now we may use Theorem\n8.3.6 from \\cite{ABG} to get\n\\[\n\\rho_{\\geq X}(\\lambda)=\\inf_{\\lambda_1+\\lambda_2=\\lambda}\n\\big(\\rho_{\\Delta_X}(\\lambda_1) + \\rho_{\\mathcal{S}\/X}(\\lambda_2) \\big)\n\\]\nwhere $\\rho_{\\mathcal{S}\/X}=\\rho_{H_{\\mathcal{S}\/X}}$. But clearly if $X\\neq O$ we\nhave $\\rho_{\\Delta_X}(\\lambda)=\\infty$ if $\\lambda<0$ and \n$\\rho_{\\Delta_X}(\\lambda)=\\lambda$ if $\\lambda\\geq0$. Thus we get\n\\begin{equation}\\label{eq:mustt}\n\\rho_{\\geq X}(\\lambda)=\\inf_{\\mu\\leq\\lambda}\n\\big(\\lambda-\\mu + \\rho_{\\mathcal{S}\/X}(\\mu) \\big)\n=\\lambda-  \\sup_{\\mu\\leq\\lambda}\\big(\\mu - \\rho_{\\mathcal{S}\/X}(\\mu) \\big).\n\\end{equation}\n\n(vi) Now from \\eqref{eq:mustq} and \\eqref{eq:mustt} we get\n\\begin{equation}\\label{eq:musr}\n\\lambda-\\widehat\\rho_\\mathcal{S}(\\lambda)=\n\\max_{X\\gtrdot O}\\sup_{\\mu\\leq\\lambda}\n\\big(\\mu - \\rho_{\\mathcal{S}\/X}(\\mu)\\big).\n\\end{equation}\nFinally, we are able to prove the formula\n$\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ by induction\nover the semilattice $\\mathcal{S}$. In other terms, we assume that the\nformula is correct if $H$ is replaced by $H_{\\mathcal{S}\/X}$ for all $X\\neq\nO$ and we prove it for $H=H_{\\mathcal{S}\/O}$. So we have to show that the\nright hand side of \\eqref{eq:musr} is equal to\n$N_{\\tau(H)}(\\lambda)$.\n\nAccording to step (iii) above we have $\\rho_{\\mathcal{S}\/X}(\\mu)=0$ if\n$\\mu\\in\\mu(H_{\\mathcal{S}\/X})$ and\n$\\rho_{\\mathcal{S}\/X}(\\mu)=\\widehat\\rho_{\\mathcal{S}\/X}(\\mu)$ otherwise. Since by the\nexplicit expression of $\\widehat\\rho_{\\mathcal{S}\/X}$ this is a positive\nfunction and since $\\rho_H(\\lambda)\\leq0$ is always true if\n$\\lambda$ is an eigenvalue, we get $\\mu-\\rho_{\\mathcal{S}\/X}(\\mu)=\\mu$ if\n$\\mu\\in\\mathrm{ev}(H_{\\mathcal{S}\/X})$ and\n\\[\n\\mu-\\rho_{\\mathcal{S}\/X}(\\mu)=\\mu-\\widehat\\rho_{\\mathcal{S}\/X}(\\mu)=\nN_{\\tau(H_{\\mathcal{S}\/X})}(\\mu)\n\\]\notherwise. From the first part of Lemma \\ref{lm:nab} below we get\n\\[\n\\sup_{\\mu\\leq\\lambda}\\big(\\mu - \\rho_{\\mathcal{S}\/X}(\\mu)\\big)=\nN_{\\mathrm{ev}(H_{\\mathcal{S}\/X}) \\cup \\tau(H_{\\mathcal{S}\/X})}.\n\\]\nIf we use the second part of Lemma \\ref{lm:nab} then we see that\n\\[\n\\max_{X\\gtrdot O}\\sup_{\\mu\\leq\\lambda}\\big(\\mu -\n\\rho_{\\mathcal{S}\/X}(\\mu)\\big)=\n\\max_{X\\gtrdot O}N_{\\mathrm{ev}(H_{\\mathcal{S}\/X}) \\cup \\tau(H_{\\mathcal{S}\/X})}\n\\]\nis the $N$ function of the set \n\\[\n\\bigcup_{X\\gtrdot O}\\big(\\mathrm{ev}(H_{\\mathcal{S}\/X}) \n\\cup \\tau(H_{\\mathcal{S}\/X})\\big)=\n\\bigcup_{X\\gtrdot O}\\left(\\mathrm{ev}(H_{\\mathcal{S}\/X}) \n\\bigcup \\bigcup_{Y>X}\\mathrm{ev}(H_{\\mathcal{S}\/Y})\\right)=\n\\bigcup_{X>O}\\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\]\nwhich finishes the proof of\n$\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ hence the proof\nof Theorem \\ref{th:thr} in the case $O\\in\\mathcal{S}$.\n\nNo assume $O\\notin\\mathcal{S}$ and let $E=\\min\\mathcal{S}$. Then $O\\in\\mathcal{S}\/E$ so we may\nuse the preceding result for $H_{\\mathcal{S}\/E}$. Moreover, we have\n$H=\\Delta_E\\otimes 1 + 1\\otimes H_{\\mathcal{S}\/E}$. Thus\n$\\mathrm{ev}(H)=\\emptyset$, $\\widehat\\rho_H=\\rho_H$, and we may use a\nrelation similar to \\eqref{eq:mustt} to get\n\\[\n\\lambda-\\widehat\\rho_H(\\lambda)=\n\\sup_{\\mu\\leq\\lambda}(\\mu-\\rho_{\\mathcal{S}\/E}(\\mu)).\n\\]\nBy what we have shown before we have\n$\\mu-\\rho_{\\mathcal{S}\/E}(\\mu)=N_{\\tau(H_{\\mathcal{S}\/E})}(\\mu)$ if\n$\\mu\\notin\\mu(H_{\\mathcal{S}\/E})$ and otherwise $\\mu-\\rho_{\\mathcal{S}\/E}(\\mu)=\\mu$.\nFrom Lemma \\ref{lm:nab} we get\n$\\lambda-\\widehat\\rho_H(\\lambda)=N_{\\tau(H_{\\mathcal{S}\/E})\\cup\\mu(H_{\\mathcal{S}\/E})}$.\nBut from \\eqref{eq:tev} we get $\\tau(H_{\\mathcal{S}\/E})\\cup\\mu(H_{\\mathcal{S}\/E})=\n\\tau(H_{\\mathcal{S}\/E})\\cup\\mathrm{ev}(H_{\\mathcal{S}\/E})$. From\n\\eqref{eq:thr} we get\n\\[\n\\tau(H_{\\mathcal{S}\/E})=\\textstyle{\\bigcup}_{Y\\in\\mathcal{S}\/E, Y\\neq O}\\mathrm{ev}(H_{(\\mathcal{S}\/E)\/Y})\n=\\textstyle{\\bigcup}_{X\\in\\mathcal{S}, X\\neq E} \\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\] \nbecause if we write $Y=X\/E$ with $X\\in\\mathcal{S}, X\\neq E$ then\n$(\\mathcal{S}\/E)\/(X\/E)=\\mathcal{S}\/X$. Finally,\n\\[\n\\tau(H_{\\mathcal{S}\/E})\\,\\textstyle{\\bigcup}\\,\\mathrm{ev}(H_{\\mathcal{S}\/E})=\n\\textstyle{\\bigcup}_{X\\in\\mathcal{S}} \\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\]\nwhich proves the Theorem in the case $O\\notin\\mathcal{S}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\nIt remains to show the following fact which was used above.\n\n\\begin{lemma}\\label{lm:nab}\nIf $A$ and $A\\cup B$ are closed and if $M$ is the function given by\n$M(\\mu)=N_A(\\mu)$ for $\\mu\\notin B$ and $M(\\mu)=\\mu$ for $\\mu\\in B$\nthen $\\sup_{\\mu\\leq\\lambda}M(\\mu)=N_{A\\cup B}(\\lambda)$.  If $A,B$\nare closed then $\\sup(N_A,N_B)=N_{A\\cup B}$.\n\\end{lemma}\n\\proof The last assertion of the lemma is easy to check, we prove\nthe first one.  Observe first that the function $N_A$ has the\nfollowing properties:\n\\begin{compactenum}\n\\item[(i)]\n$N_A$ is increasing and right-continuous,\n\\item[(ii)]\n$N_A(\\lambda)=\\lambda$ if $\\lambda\\in A$,\n\\item[(iii)] $N_A$ is locally constant and $N_A(\\lambda)<\\lambda$ on\n  $A^\\mathrm{c}\\equiv \\mathbb{R}\\setminus A$.\n\\end{compactenum} \nIndeed, the first assertion in (i) and assertion (ii) are obvious.\nThe second part of (i) follows from the more precise and easy to prove\nfact \n\\begin{equation}\\label{eq:nae}\nN_A(\\lambda+\\varepsilon)\\leq N_A(\\lambda)+\\varepsilon \\quad\n\\text{for all real } \\lambda \\text{ and } \\varepsilon>0.\n\\end{equation}\nA connected component of the open set $A^\\mathrm{c}$ is necessarily an\nopen interval of one of the forms $]-\\infty,y[$ or $]x,y[$ or\n$]x,\\infty[$ with $x,y\\in A$. On the first interval (if such an\ninterval appears) $N_A$ is equal to $-\\infty$ and on the second or\nthe third one it is clearly constant and equal to $N_A(x)$. We also\nnote that the function $N_A$ is characterized by the properties\n(i)--(iii).\n\nThus, if we denote $N(\\lambda)=\\sup_{\\mu\\leq\\lambda}M(\\mu)$, then it\nwill suffices to show that the function $N$ satisfies the conditions\n(i)--(iii) with $A$ replace by $A\\cup B$. Observe that $M(\\mu)\\leq\\mu$\nand the equality holds if and only if $\\mu\\in A\\cup B$.  Thus $N$ is\nincreasing, $N(\\lambda)\\leq\\lambda$, and $N(\\lambda)=\\lambda$ if\n$\\lambda\\in A\\cup B$.\n\nNow assume that $\\lambda$ belongs to a bounded connected component\n$]x,y[ $ of $A\\cup B$ (the unbounded case is easier to treat). If\n$x<\\mu<y$ then $\\mu\\notin B$ so $M(\\mu)=N_A(\\mu)$ and $]x,y[$ is\nincluded in a connected component of $A^\\mathrm{c}$ hence $M(\\mu)=N_A(x)$.\nThen $N(\\lambda)= \\max(\\sup_{\\nu\\leq x}M(\\nu),N_A(x))$ hence $N$ is \nconstant on $]x,y[$. Here we have $M(\\nu)\\leq\\nu\\leq x$ so if $x\\in A$\nthen $N_A(x)=x$ and we get $N(\\lambda)=x$. If $x\\in B\\setminus A$ then\n$M(x)=x$ so $\\sup_{\\nu\\leq x}M(\\nu)=x$ and $N_A(x)<x$ hence \n$N(\\lambda)=x$. Since $x\\in A\\cup B$ one of these two cases is \ncertainly realized and the same argument gives $N(x)=x$. Thus the \nvalue of $N$ on $]x,y[$ is $N(x)$ so $N$ is right continuous on \n$[x,y[$.  Thus we proved that $N$ is locally constant and right \ncontinuous on the complement of $A\\cup B$ and also that \n$N(\\lambda)<\\lambda$ there.\n\nIt remains to be shown that $N$ is right continuous at each point of\n$\\lambda\\in A\\cup B$. We show that \\eqref{eq:nae} hold with $N_A$\nreplaced by $N$. If $\\mu\\leq\\lambda$ then \n$M(\\mu)\\leq\\mu\\leq\\lambda=M(\\lambda)$ hence we have \n\\[\nN(\\lambda+\\varepsilon)=\n\\sup_{\\lambda\\leq\\mu\\leq\\lambda+\\varepsilon}M(\\mu). \n\\]\nBut $M(\\mu)$ above is either $N_A(\\mu)$ either $\\mu$. In the second\ncase $\\mu\\leq\\lambda+\\varepsilon$ and  in the first case\n\\[\nN_A(\\mu)\\leq N_A(\\lambda+\\varepsilon)\\leq N_A(\\lambda)+\\varepsilon\n\\leq \\lambda +\\varepsilon. \n\\]\nThus we certainly have $N(\\lambda+\\varepsilon)\\leq\\lambda+\\varepsilon$\nand $\\lambda=N(\\lambda)$ because $\\lambda\\in A\\cup B$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\\subsection{A general class of interactions}\n\\label{ss:euc}\n\n\nThe rest of this section is devoted to some technical questions. Our\nmain purpose is to clarify the structure of the interactions in the\nEuclidean case.\n\nThe following compactness criterion will be useful. This is a\nconsequence of the Riesz-Kolmogorov theorem and of the argument page\n\\pageref{p:RK} involving the regularity of the weight. Let $E,F$ be\narbitrary Euclidean space and $s,t\\in\\mathbb{R}$.\n\n\\begin{proposition}\\label{pr:stsob}\n  An operator $T\\in L(\\mathcal{H}^s_E,\\mathcal{H}^t_F)$ is compact if and only if\n  one of the next two equivalent conditions is satisfied,\n  where $\\|\\cdot\\|$ is the norm in $L(\\mathcal{H}^s_E,\\mathcal{H}^t_F)$:\n\\begin{compactenum}\n\\item[{\\rm(i)}]\n$\\|(U_x-1)T\\| + \\|(V_x-1)T\\| \\to 0 \\quad \n\\text{if } x\\to 0 \\text{ in }F$,\n\\item[{\\rm(ii)}]\n$\\|T(U_x-1)\\| + \\|T(V_x-1)\\| \\to 0 \\quad \n\\text{if } x\\to 0 \\text{ in }E$.\n\\end{compactenum}\n\\end{proposition}\n\nWe denote $L^\\circ(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$ the set of small at infinity\noperators, cf.  Definition \\ref{df:small}. Clearly\n\\emph{$L^\\circ(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$ is a closed subspace of\n  $L(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$}.\n\n\\begin{corollary}\\label{co:small}\n  An operator $T\\in L(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$ is small at infinity if\n  and only if $\\lim_{k\\to0}T(V_k-1)= 0$ in norm in\n  $L(\\mathcal{H}^{s+\\varepsilon}_E,\\mathcal{H}^{t}_F)$ for some $\\varepsilon>0$.\n  Then this holds for all $\\varepsilon>0$.\n\\end{corollary}\n\nIndeed, the first part of condition (ii) of Proposition\n\\ref{pr:stsob} ($s$ replaced by $s+\\varepsilon$) is\nautomatically satisfied.\n\nWe now give a Sobolev space version of Proposition \\ref{pr:tex}\nwhich uses the weights $\\jap{\\cdot}^s$ and is convenient in\napplications.  By using Theorem \\ref{th:xyzeintr} instead of Theorem\n\\ref{th:yzintr} in the proof of Proposition \\ref{pr:tex} we get:\n\n\\begin{proposition}\\label{pr:etex}\nLet $s,t>0$ and $Z\\subset X\\cap Y$. Let \n$I_{XY}(Z)\\in  L(\\mathcal{H}^t_Y,\\mathcal{H}^{-s}_X)$\nsuch that the following relations hold in norm in\n$ L(\\mathcal{H}^{t+\\varepsilon}_Y,\\mathcal{H}^{-s}_X)$ for some $\\varepsilon > 0$:\n\\begin{enumerate}\n\\item[{\\rm(1)}]\n$U_z I_{XY}(Z)=I_{XY}(Z) U_z$ if $z\\in Z$ and\n$V^*_z I_{XY}(Z) V_z\\to I_{XY}(Z)$ if $z\\to 0$ in $Z$,\n\\item[{\\rm(2)}] \n$I_{XY}(Z)(V_y-1)\\to 0$ if $y\\to 0$ in $Y\/Z$.\n\\end{enumerate}\nIf $h_X,h_Y$ are continuous real functions on $X,Y$ such that\n$h_X(x)\\sim\\jap{x}^{2s}$ and $h_Y(y)\\sim\\jap{y}^{2t}$ and if we set\n$K_X=h_X(P), K_Y=h_Y(P)$ then\n$\\jap{K_X}^{-1\/2}I_{XY}(Z)\\jap{K_Y}^{-\\alpha}\\in\\mathscr{C}_{XY}(Z)$ if\n$\\alpha>1\/2$.\n\\end{proposition}\n\nOur next purpose is to discuss in more detail the structure of the\noperators $I_{XY}(Z)$ from Proposition \\ref{pr:etex}. For this we\nmake a Fourier transformation $\\mathcal{F}_Z$ in the $Z$ variable as in the\nproof of Theorem \\ref{th:xyzeintr}.\n\nWe fix $X,Y,Z$ with $Z\\subset X\\cap Y$, use the tensor\nfactorizations \\eqref{eq:xyzint} and make identifications like\n$\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}= L^2(Z;\\mathcal{H}_{X\/Z})$. Thus\n$\\mathcal{H}_X=\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}$ and $\\Delta_X=\\Delta_Z\\otimes 1 +\n1\\otimes \\Delta_{X\/Z}$ hence if $s\\geq0$\n\\begin{equation}\\label{eq:stens}\n\\mathcal{H}^s(X)=\\mathcal{H}(Z;\\mathcal{H}^s(X\/Z))\\cap \\mathcal{H}^s(Z;\\mathcal{H}_{X\/Z})=\n\\big(\\mathcal{H}_Z\\otimes\\mathcal{H}^s(X\/Z)\\big)\\cap \n\\big(\\mathcal{H}^s(Z)\\otimes\\mathcal{H}_{X\/Z}\\big)\n\\end{equation}\nwhere our notations are extended to  vector-valued Sobolev spaces. \nClearly\n\\begin{equation}\\label{eq:lap}\n\\mathcal{F}_Z \\jap{P_X}^s \\mathcal{F}_Z^{-1} = \n\\int_Z^\\oplus (1+|k|^2+|P_{X\/Z}|^2)^{s\/2} \\text{d} k.\n\\end{equation}\nThen from \\eqref{eq:CZ} and $\\mathscr{T}_Z=\\mathcal{F}_Z^{-1}\\cc_{\\mathrm{o}}(Z)\\mathcal{F}_Z$ we get\n\\begin{equation*}\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{X\/Z,Y\/Z}=\n\\mathcal{F}_Z^{-1}\\cc_{\\mathrm{o}}(Z;\\mathscr{K}_{X\/Z,Y\/Z})\\mathcal{F}_Z.\n\\end{equation*}\nTo each weakly measurable map $I_{XY}^Z:Z\\to\nL(\\mathcal{H}^t_{Y\/Z},\\mathcal{H}^{-s}_{X\/Z})$ such that\n\\begin{equation}\\label{eq:Iest}\n\\sup\\nolimits_k\n\\|(1+|k|+|P_{X\/Z}|)^{-s}I_{XY}^Z(k)(1+|k|+|P_{Y\/Z}|)^{-t}\\| <\\infty.\n\\end{equation} \nwe associate a continuous operator \n$I_{XY}(Z):\\mathcal{H}^t_Y\\to\\mathcal{H}^{-s}_X$ by the relation\n\\begin{equation}\\label{eq:Ixyz}\n\\mathcal{F}_Z I_{XY}(Z) \\mathcal{F}_Z^{-1} \\equiv \\int_Z^\\oplus I_{XY}^Z(k) \\text{d} k.\n\\end{equation}\nThe following fact is known: a continuous operator\n$T:\\mathcal{H}^t_Y\\to\\mathcal{H}^{-s}_X$ is of the preceding form if and only if\n$U_aT=TUa$ for all $a\\in Z$. From the preceding results we get\n(notations are as in Remark \\ref{re:iaff}):\n\n\\begin{proposition}\\label{pr:zxy}\n  Let $X,Y,Z\\in\\mathcal{S}$ with $Z\\subset X\\cap Y$ and assume that\n  $\\mathcal{G}^1_X=\\mathcal{H}^s_X$ and $\\mathcal{G}^1_Y=\\mathcal{H}^t_Y$. An operator\n  $I_{XY}(Z):\\mathcal{H}^t_Y\\to\\mathcal{H}^{-s}_X$ satisfies the conditions of\n  Remark \\ref{re:iaff} if and only if it is of the form\n  \\eqref{eq:Ixyz} with a norm continuous function $I_{XY}^Z:Z\\to\n  L^\\circ(\\mathcal{H}^t_{Y\/Z},\\mathcal{H}^{-s}_{X\/Z})$ satisfying \\eqref{eq:Iest}.\n\\end{proposition}\n\n\n\\subsection{Auxiliary results}\n\\label{ss:lemma}\n\nIn this subsection we collect some useful technical results.  Let\n$\\mathcal{E},\\mathcal{F},\\mathcal{G},\\mathcal{H}$ be Hilbert spaces. Note that we have a canonical\nidentification (tensor products are discussed in \\S\\ref{ss:ha})\n\\begin{equation}\\label{eq:comtens}\nK(\\mathcal{E},\\mathcal{F})\\otimes K(\\mathcal{G},\\mathcal{H})\\cong K(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H}),\n\\hspace{2mm}\\text{in particular}\\hspace{2mm}\nK(\\mathcal{E},\\mathcal{F}\\otimes\\mathcal{H})\\cong K(\\mathcal{E},\\mathcal{F})\\otimes\\mathcal{H}.\n\\end{equation}\nAssume that we have continuous injective embeddings $\\mathcal{E}\\subset\\mathcal{G}$\nand $\\mathcal{F}\\subset\\mathcal{G}$. We equip $\\mathcal{E}\\cap\\mathcal{F}$ with the intersection\ntopology defined by the norm $(\\|g\\|_\\mathcal{E}^2+\\|g\\|_\\mathcal{F}^2)^{1\/2}$, hence\n$\\mathcal{E}\\cap\\mathcal{F}$ becomes a Hilbert space continuously embedded in $\\mathcal{G}$.\n\n\\begin{lemma}\\label{lm:efgh}\n  The map $K(\\mathcal{E},\\mathcal{H})\\times K(\\mathcal{F},\\mathcal{H}) \\to K(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})$ which\n  associates to $S\\in K(\\mathcal{E},\\mathcal{H})$ and $T\\in K(\\mathcal{F},\\mathcal{H})$ the operator\n  $S|_{\\mathcal{E}\\cap\\mathcal{F}}+T|_{\\mathcal{E}\\cap\\mathcal{F}} \\in K(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})$ is\n  surjective. Thus, slightly formally,\n\\begin{equation}\\label{eq:efgh}\nK(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})=K(\\mathcal{E},\\mathcal{H}) + K(\\mathcal{F},\\mathcal{H}).\n\\end{equation}\n\\end{lemma}\n\\proof It is clear that the map is well defined. Let $R\\in\nK(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})$, we have to show that there are $S,T$ as in the\nstatement of the proposition such that $R=\nS|_{\\mathcal{E}\\cap\\mathcal{F}}+T|_{\\mathcal{E}\\cap\\mathcal{F}}$.  Observe that the norm on\n$\\mathcal{E}\\cap\\mathcal{F}$ has been chosen such that the linear map\n$g\\mapsto(g,g)\\in\\mathcal{E}\\oplus\\mathcal{F}$ be an isometry with range a closed\nlinear subspace $\\mathcal{I}$. Consider $R$ as a linear map $\\mathcal{I}\\to\\mathcal{H}$ and\nextend it to the orthogonal of $\\mathcal{I}$ by zero. The so defined map\n$\\widetilde R:\\mathcal{I}\\to\\mathcal{H}$ is clearly compact. Let $S,T$ be defined by\n$Se=\\widetilde R(e,0)$ and $Tf=\\widetilde R(0,f)$. Clearly $S\\in\nK(\\mathcal{E},\\mathcal{H})$ and $T\\in K(\\mathcal{F},\\mathcal{H})$ and if $g\\in\\mathcal{E}\\cap\\mathcal{F}$ then\n\\[\nSg+Tg=\\widetilde R(g,0)+\\widetilde R(0,g)=\\widetilde R(g,g)=Rg\n\\]\nwhich proves the lemma.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nWe give some applications. If $E,F$ are Euclidean spaces and $s>0$\nis real then\n\\begin{equation}\\label{eq:EFs}\n\\mathcal{H}^s_{E\\oplus F}=\\big(\\mathcal{H}^s_E\\otimes\\mathcal{H}_F\\big)\\cap\n\\big(\\mathcal{H}_E\\otimes\\mathcal{H}^s_F\\big)\n\\end{equation}\nhence Lemma \\ref{lm:efgh} gives for an arbitrary Hilbert space $\\mathcal{H}$\n\\begin{equation}\\label{eq:EFH}\nK(\\mathcal{H}^s_{E\\oplus F},\\mathcal{H})=\nK(\\mathcal{H}^s_E\\otimes\\mathcal{H}_F,\\mathcal{H}) + K(\\mathcal{H}_E\\otimes\\mathcal{H}^s_F,\\mathcal{H}).\n\\end{equation}\nIf $\\mathcal{H}$ itself is a tensor product $\\mathcal{H}=\\mathcal{H}'\\otimes\\mathcal{H}''$ then we\ncan combine this with  \\eqref{eq:comtens} and get\n\\begin{equation}\\label{eq:EFHEF}\nK(\\mathcal{H}^s_{E\\oplus F},\\mathcal{H}'\\otimes\\mathcal{H}'')  =\nK(\\mathcal{H}^s_E,\\mathcal{H}')\\otimes K(\\mathcal{H}_F,\\mathcal{H}'')\n + K(\\mathcal{H}_E,\\mathcal{H}') \\otimes K(\\mathcal{H}^s_F,\\mathcal{H}'').\n\\end{equation}\nConsider now a triplet $X,Y,Z$ such that $Z\\subset X\\cap Y$ and\ndenote\n\\begin{equation}\\label{eq:xyze}\nE=(X\\cap Y)\/Z \\hspace{2mm}\\text{and}\\hspace{2mm} \nX \\boxplus Y = X\/Y\\times Y\/X.\n\\end{equation}\nThen $Y\/Z=E\\oplus(Y\/X) \\text{ and } X\/Z=E\\oplus(X\/Y)$ hence by using\n\\eqref{eq:EFHEF} we get for example \n\\begin{align}\n\\mathcal{H}_{Y\/Z} &=\\mathcal{H}_E\\otimes\\mathcal{H}_{Y\/X} \\text{ and \\ }\n\\mathcal{H}_{X\/Z} =\\mathcal{H}_E\\otimes\\mathcal{H}_{X\/Y} \\label{eq:new} \\\\\n\\mathcal{H}^2_{Y\/Z} &=\\big(\\mathcal{H}^2_E\\otimes\\mathcal{H}_{Y\/X}\\big)\\cap \n\\big(\\mathcal{H}_E\\otimes\\mathcal{H}^2_{Y\/X}\\big) \\label{eq:klea} \\\\\n\\mathcal{H}^{-2}_{X\/Z} &= \\mathcal{H}^{-2}_E\\otimes\\mathcal{H}_{X\/Y}\n+\\mathcal{H}_E\\otimes\\mathcal{H}^{-2}_{X\/Y}. \\label{eq:klean}\n\\end{align}\nBy using once again \\eqref{eq:EFHEF} and the notations introduced in\n\\eqref{eq:ikef}, we get\n\\begin{equation}\\label{eq:kde3}\n\\mathscr{K}^2_{X\/Z,Y\/Z} = \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X} + \n\\mathscr{K}_E\\otimes\\mathscr{K}^2_{X\/Y,Y\/X}.\n\\end{equation}\nWe identify a Hilbert-Schmidt operator with its kernel, so $L^2(X\n\\boxplus Y)\\subset \\mathscr{K}_{X\/Y,Y\/X}$ is the subspace of Hilbert-Schmidt\noperators. The we have a strict inclusion\n\\begin{equation} \\label{eq:kde9} \nL^2(X \\boxplus Y;\\mathscr{K}^2_E) \\subset \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X}\n\\end{equation}\n\n\n\\subsection{First order regularity conditions}\n\\label{ss:scs} \n\nIn the next two subsections we consider interactions as in\nProposition \\ref{pr:nrm} and give explicit conditions on the\n$I_{XY}^Z$ such that $H$ be of class $C^1_\\mathrm{u}(D)$. We recall that\nthe assumptions of Proposition \\ref{pr:nrm} can be stated as\nfollows: for all $Z\\subset X\\cap Y$\n\\begin{align}\n& I^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{X\/Z} \n\\hspace{2mm} \\text{is compact and satisfies}\\hspace{2mm} \n(I^{Z}_{XY})^*\\supset I^Z_{YX}, \\label{eq:A}\\\\\n& [D,I^Z_{XY}]:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\hspace{2mm}\\text{is\ncompact}. \\label{eq:B}\n\\end{align}\nIf \\eqref{eq:A} is satisfied then by duality and interpolation we\nget\n\\begin{equation}\\label{eq:interpol}\nI^Z_{XY}:\\mathcal{H}^\\theta_{Y\/Z}\\to\\mathcal{H}^{\\theta-2}_{X\/Z} \\quad \n\\text{is a compact operator for all } 0\\leq \\theta \\leq 2,\n\\end{equation} \nin particular the operator $[D,I^Z_{XY}]\\equiv\nD_{X\/Z}I^Z_{XY}-I^Z_{XY}D_{Y\/Z}$ restricted to the space of\nfunctions in $\\mathcal{H}^2_{Y\/Z}$ with compact support has values in the\nspace of functions locally in $\\mathcal{H}^{-1}_{X\/Z}$.  We use, for\nexample, the relation $D_{X\/Z}=D_E\\otimes 1 + 1\\otimes D_{X\/Y}$\nrelatively to \\eqref{eq:new} to decompose this operator as follows:\n\\begin{align}\n[D,I^Z_{XY}] \n&=(D_E+D_{X\/Y})I^Z_{XY}-I^Z_{XY}(D_E+D_{Y\/X}) \\nonumber \\\\\n&=[D_E,I^Z_{XY}] +D_{X\/Y}I^Z_{XY} -I^Z_{XY}D_{Y\/X}. \\label{eq:dec}\n\\end{align}\nSince $I^Z_{XY}D_{Y\/X}\\subset (D_{Y\/X}I^Z_{YX})^*$ if \\eqref{eq:A}\nis satisfied then condition \\eqref{eq:B} follows from:\n\\begin{equation}\\label{eq:dii}\n[D_E,I^Z_{XY}] \\text{ and } D_{X\/Y}I^Z_{XY} \\text{ are compact\n  operators } \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\text{ for all } X,Y,Z.\n\\end{equation}\nAccording to \\eqref{eq:kde3} the first part of condition \n\\eqref{eq:A} is equivalent to\n\\begin{equation}\\label{eq:kde4}\nI_{XY}^Z=J+J' \\text{ for some }\nJ\\in \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X} \\text{ and }\nJ'\\in\\mathscr{K}_E\\otimes\\mathscr{K}^2_{X\/Y,Y\/X}. \n\\end{equation}\nAs a particular case, from \\eqref{eq:kde9} we obtain the example\ndiscussed in \\S\\ref{ss:examples}. The compactness conditions\n\\eqref{eq:dii} are conditions on the kernels $[D_E,I^Z_{XY}(x',y')]$\nand $x'\\cdot\\nabla_{x'}I_{XY}^Z(x',y')$ of the operators\n$[D_E,I^Z_{XY}]$ and $D_{X\/Y}I^Z_{XY}$.  Note that a condition on\n$I^Z_{XY}D_{Y\/X}$ is a requirement on the kernel\n$y'\\cdot\\nabla_{y'}I_{XY}^Z(x',y')$.\n\n\n\\subsection{Creation-annihilation type interactions}\n\\label{ss:xsupy} \n\nTo see the relation with the\ncreation-annihilation type interactions characteristic to quantum\nfield models we consider now the case when\n$Y\\subset X$ strictly. Then\n\\begin{equation}\\label{eq:ysux}\n\\mathscr{C}_{XY}=\\mathscr{C}_Y\\otimes\\mathcal{H}_{X\/Y}, \\quad \n\\mathscr{C}_{XY}(Z)=\\mathscr{C}_Y(Z)\\otimes\\mathcal{H}_{X\/Y}, \\quad\n\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}\n\\end{equation}\nwhere the first two tensor product have to be interpreted as\nexplained in \\S\\ref{ss:ha}. In particular we have\n\\begin{equation}\\label{eq:sux}\nL^2(X\/Y;\\mathscr{C}_Y)\\subset\\mathscr{C}_{XY} \\quad \\text{and} \\quad\nL^2(X\/Y;\\mathscr{C}_Y(Z))\\subset\\mathscr{C}_{XY}(Z)\n\\quad \\text{strictly}. \n\\end{equation}\nIf $Z\\subset Y$ then $X=Z\\oplus(Y\/Z)\\oplus(X\/Y)$\nand $X\/Z=(Y\/Z)\\oplus(X\/Y)$ hence $\\mathcal{H}_{X\/Z}=\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}$\nand thus the operator $I^Z_{XY}$ is\njust a compact operator\n\\begin{equation}\\label{eq:ixyo}\nI^Z_{XY} : \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}.\n\\end{equation}\nIf $\\mathcal{E},\\mathcal{F},\\mathcal{G}$ are Hilbert spaces then $K(\\mathcal{E},\\mathcal{F}\\otimes\\mathcal{G})\\cong\nK(\\mathcal{E},\\mathcal{F})\\otimes\\mathcal{G}$. Hence \\eqref{eq:ixyo} means\n\\begin{equation}\\label{eq:ixyoo}\nI^Z_{XY} \\in \\mathscr{K}^2_{Y\/Z}\\otimes \\mathcal{H}_{X\/Y}.\n\\end{equation}\nLet $\\mathscr{I}_{XY}= {\\textstyle\\sum_{Z\\subset X\\cap Y}} 1_Z\\otimes\n\\mathscr{K}^2_{X\/Z,Y\/Z}$, where the sum is direct and closed in\n$\\mathscr{K}^2_{XY}$.  A usual nonrelativistic $N$-body Hamiltonian\nassociated to the semilattice $\\mathcal{S}_X$ of subspaces of $X$ is of the\nform $\\Delta_X+I_X$ with $I_X\\in\\mathscr{I}_{X}\\equiv\\mathscr{I}_{XX}$. Thus the\ninteraction which couples the $X$ and $Y$ systems is of the form\n\\begin{equation}\\label{eq:xyinter}\nI_{XY}={\\textstyle\\sum_{Z\\in\\mathcal{S}_Y}} \n1_Z\\otimes I^Z_{XY} \\in \\mathscr{I}_{Y}\\otimes \\mathcal{H}_{X\/Y}. \n\\end{equation}\nIn particular we may take $I_{XY}\\in L^2(X\/Y;\\mathscr{I}_Y)$, but we stress\nthat the space $\\mathscr{I}_{Y}\\otimes \\mathcal{H}_{X\/Y}$ is much larger (see\n\\S\\ref{ss:ha}).  More explicitly, a square integrable function\n$I_{XY}:X\/Y\\to\\mathscr{I}_Y$ determines an operator $I_{XY}:\\mathcal{H}^2_Y\\to\\mathcal{H}_X$\nby the following rule: it associates to $u\\in\\mathcal{H}^2(Y)$ the function\n$y'\\mapsto I_{XY}(y')u$ which belongs to $L^2(X\/Y;\\mathcal{H}_{X\/Y})=\\mathcal{H}_X$.\nWe may also write\n\\begin{equation}\\label{eq:IV}\n(I_{XY}u)(x)=(I_{XY}(y')u)(y) \\quad \\text{where }\nx\\in X=Y\\oplus X\/Y \\text{ is written as } x=(y,y').\n\\end{equation}\nWe say that the operator valued function $I_{XY}$ is the kernel of\nthe operator $I_{XY}$. The adjoint $I_{YX}=I_{XY}^*$ is an integral\noperator in the $y'$ variable (like an annihilation\noperator). Indeed, if $v\\in\\mathcal{H}_X$ is thought as a map $y'\\mapsto\nv(y')\\in\\mathcal{H}_Y$ then we have $I_{YX}v=\\int_{X\/Y}\nI^*_{XY}(y')v(y')\\text{d} y'$ at least formally.\n\n\nThe particular case when the function $I_{XY}$ is factorizable\nclarifies the connection with the quantum field type interactions:\nlet $I_{XY}$ be a finite sum $I_{XY}=\\sum_i V_Y^i\\otimes\\phi_i$\nwhere $V^i_Y\\in\\mathscr{I}_Y$ and $\\phi_i\\in \\mathcal{H}_{X\/Y}$, then\n\\begin{equation}\\label{eq:qft}\nI_{XY}u={\\textstyle\\sum_i} (V_Y^i u)\\otimes\\phi_i \\quad\n\\text{as an operator }  \nI_{XY}:\\mathcal{H}^2_Y\\to\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}.\n\\end{equation}\nThis is a sum of $N$-body type interactions $V^i_Y$ tensorized with\noperators which create particles in states~ $\\phi_i$.\n\nThe conditions on the ``commutator'' $[D,I_{XY}]$ may be written\nin terms of the kernel of $I_{XY}$. The\nrelation \\eqref{eq:dec}  becomes $\n[D,I_{XY}]=[D_Y,I_{XY}]+D_{X\/Y}I_{XY}$. The operator $D_Y$ acts only\non the variable $y$ and $D_{X\/Y}$ acts only on the variable\n$y'$. Thus $[D_Y,I_{XY}]$ and $D_{X\/Y}I_{XY}$ are operators of the\nsame nature as $I_{XY}$ but more singular: the kernel of\n$[D_Y,I_{XY}]$ is the function $y'\\mapsto [D_Y,I_{XY}(y')]$ and that\nof $2iD_{X\/Y}I_{XY}$ is the function $ y'\\mapsto\n(y'\\cdot\\nabla_{y'}+n\/2) I_{XY}(y')$.  Thus, to get\n\\eqref{eq:B} it suffices to\nrequire two conditions on the kernel $I_{XY}$, one on\n$[D_Y,I_{XY}(y')]$ and a second one on\n$y'\\cdot\\nabla_{y'}I_{XY}(y')$.\n\nIf we decompose $I_{XY}$ as in \\eqref{eq:xyinter} with\n$I^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}$ compact then the\n(formal) kernel of $I^Z_{XY}$ is a $\\mathscr{K}^2_{Y\/Z}$ valued map on\n$X\/Y$.  We require that $[D_{Y\/Z},I^Z_{XY}]$ and $D_{X\/Y}I^Z_{XY}$\nbe compact operators $\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}$. From\n\\eqref{eq:stens} and $X\/Z=(Y\/Z)\\oplus(X\/Y)$ we get\n\\begin{equation*}\n\\mathcal{H}^2_{X\/Z} = \\big(\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}^2_{X\/Y}\\big)\\cap\n\\big(\\mathcal{H}^2_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}\\big), \\quad\n\\mathcal{H}^{-2}_{X\/Z} = \\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}^{-2}_{X\/Y} +\n\\mathcal{H}^{-2}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}\n\\end{equation*}  \nwhich are helpful in checking these compactness requirements.\n\n\n\n\\subsection{Besov regularity classes}\n\\label{ss:scsc} \n\nWe recall some facts concerning the Besov type regularity class $\nC^{1,1}(D)$; we refer to \\cite{ABG} for details on these\nmatters. Since the conjugate operator $D$ is fixed we shall not\nindicate it in the notation from now on. An operator $T\\in L(\\mathcal{H})$\nis of class $C^{1,1}$ if\n\\begin{equation}\\label{eq:c11}\n\\int_0^1\\|W^*_{2\\varepsilon}T W_{2\\varepsilon}-\n2W^*_{\\varepsilon}T W_{\\varepsilon} +T\\|\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} \\equiv\n\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2\nT\\|\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} <\\infty\n\\end{equation}\nwhere $\\mathcal{W}_\\varepsilon$ is the automorphism of $L(\\mathcal{H})$ defined by\n$\\mathcal{W}_\\varepsilon T=W_\\varepsilon^*T W_\\varepsilon$. The condition\n\\eqref{eq:c11} implies that $T$ is of class $C^1_\\mathrm{u}$ and is\njust slightly more than this. Indeed, $T$ is of class $C^1$ or\n$C^1_\\mathrm{u}$ if and only if the limit\n\\[\n\\lim_{\\tau\\to 0}\\int_\\tau^1 (\\mathcal{W}_\\varepsilon-1)^2 T\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2}\n\\] \nexists strongly or in norm respectively. The following subclass of\n$C^{1,1}$ is useful in applications:  $T$ is called of class\n$C^{1+}$ if $T$ is of class $C^1$, so the commutator $[D,T]$ is a\nbounded operator, and\n\\begin{equation}\\label{eq:din}\n\\int_0^1\\|W_\\varepsilon^*[D,T]W_\\varepsilon-[D,T]\\|\n\\frac{\\text{d}\\varepsilon}{\\varepsilon} <\\infty.\n\\end{equation}\nThen $C^{1+}\\subset C^{1,1}$. The class most frequently used in the\ncontext of the Mourre theorem is $C^2$: this is the set of $T\\in\nC^1$ such that $[D,T]\\in C^1$. But $[D,T]\\in C^1$ if and only if\n\\[\n\\|W_\\varepsilon^*[D,T] W_\\varepsilon-[D,T]\\| \\leq \nC |\\varepsilon| \\quad \\text{for some constant $C$ and all real }\n\\varepsilon \n\\]\nhence this condition is much stronger then the Dini type condition\n\\eqref{eq:din}.  A self-adjoint operator $H$ is of class $C^{1,1}$,\n$C^{1+}$ or $C^2$ if its resolvent is of class $C^{1,1}$, $C^{1+}$\nor $C^2$ respectively. \n\nWe now consider a Hamiltonian as in Proposition \\ref{pr:nrm} and\ndiscuss conditions which ensure that $H$ is of class $C^{1,1}$. An\nimportant point is that the domain $\\mathcal{H}^2$ of $H$ is stable under\nthe dilation group $W_\\tau$.  Then Theorem 6.3.4 from \\cite{ABG}\nimplies that $H$ is of class $C^{1,1}$ if and only if\n\\begin{equation}\\label{eq:c11h}\n\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2H\\|_{\\mathcal{H}^2\\to\\mathcal{H}^{-2}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} <\\infty.\n\\end{equation}\nAs above $\\mathcal{W}_{\\varepsilon} H=W^*_{\\varepsilon}H W_{\\varepsilon}$\nhence $ (\\mathcal{W}_\\varepsilon-1)^2H=W^*_{2\\varepsilon}H W_{2\\varepsilon}-\n2W^*_{\\varepsilon}H W_{\\varepsilon} +H$.  We have $H=\\Delta +I$ and\ndue to \\eqref{eq:dlap} the relation \\eqref{eq:c11h} is trivially\nverified by the kinetic part $\\Delta$ of $H$ hence we are only\ninterested in conditions on $I$ which ensure that \\eqref{eq:c11h} is\nsatisfied with $H$ replaced by $I$. If this is the case, by a slight\nabuse of language we say that $I$ is of class $C^{1,1}$. In terms of\nthe coefficients $I_{XY}$, this means\n\\begin{equation}\\label{eq:c11xyz}\n\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2I_{XY}^Z\\|_{\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} <\\infty\n\\quad\\text{for all } X,Y,Z.\n\\end{equation}\nWe recall one fact (see \\cite[Ch. 5]{ABG}). Let $I:\\mathcal{H}^2\\to\\mathcal{H}^{-2}$\nbe an arbitrary linear continuous operator. Then\n$[D,I]:\\mathcal{H}^2_\\mathrm{c}\\to\\mathcal{H}^{-3}_{\\mathrm{loc}}$ is well defined and $I$\nis of class $C^1$ (in an obvious sense) if and only if this operator\nis the restriction of a continuous map $\\mathcal{H}^2\\to\\mathcal{H}^{-2}$, which\nwill be denoted also $[D,I]$. We say that $I$ is of class $C^{1+}$\nif this condition is satisfied and \n\\begin{equation}\\label{eq:dini}\n\\int_0^1\\|W_\\varepsilon^*[D,I]W_\\varepsilon-[D,I]\\|_{\\mathcal{H}^2\\to\\mathcal{H}^{-2}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon} <\\infty.\n\\end{equation}\nAs before, if $I$ is of class $C^{1+}$ then it is of class\n$C^{1,1}$. In terms of the coefficients $I_{XY}^Z$ this means\n\\begin{equation}\\label{eq:dinix}\n\\int_0^1\\|W_\\varepsilon^*[D,I_{XY}^Z]W_\\varepsilon-[D,I_{XY}^Z]\n\\|_{\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon} <\\infty.\n\\end{equation}\nSuch a condition should be imposed on each of the three terms in the\ndecomposition \\eqref{eq:dec} separately.\n\nThe techniques developed in \\S 7.5.3 and on pages 425--429 from\n\\cite{ABG} can be used to get more concrete conditions.  The only\nnew fact with respect to the $N$-body situation as treated there is\nthat $\\mathcal{W}_\\tau$ when considered as an operator on $\\mathscr{L}_{X\/Z,Y,Z}$\nfactorizes in a product of three commuting operators. Indeed, if we\nwrite $\\mathcal{H}_{Y\/Z}=\\mathcal{H}_E\\otimes\\mathcal{H}_{Y\/X}$ and\n$\\mathcal{H}_{X\/Z}=\\mathcal{H}_E\\otimes\\mathcal{H}_{X\/Y}$ then we get\n$\\mathcal{W}_\\tau(T)=W^{X\/Y}_{-\\tau}\\mathcal{W}^E_\\tau(T)W^{Y\/X}_\\tau$ where this\ntime we indicated by an upper index the space to which the operator\nis related and, for example, we identified $W^{Y\/X}_\\tau=1_E\\otimes\nW^{Y\/X}_\\tau$. To check the $C^{1,1}$ property in this context\none may use:\n\n\n\\begin{proposition}\\label{pr:inter}\nIf $T\\in\\mathscr{L}:= L(\\mathcal{H}^2_{Y\/Z},\\mathcal{H}^{-2}_{X\/Z})$ then \n$\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2T\\|_\\mathscr{L}\n\\text{d}\\varepsilon\/\\varepsilon^2<\\infty$ follows from\n\\begin{equation}\\label{eq:inter}\n\\int_0^1\\left(\n\\|(W^{X\/Y}_{\\varepsilon}-1)^2T\\|_\\mathscr{L}+ \n\\|(\\mathcal{W}^E_\\varepsilon-1)^2T\\|_\\mathscr{L}+\n\\|T(W^{Y\/X}_{\\varepsilon}-1)^2\\|_\\mathscr{L}\n\\right)\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} < \\infty.\n\\end{equation}\n\\end{proposition}\n\\proof We shall interpret $\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2T\\|_\\mathscr{L}\n\\text{d}\\varepsilon\/\\varepsilon^2<\\infty$ in terms of real interpolation\ntheory. Let $L_\\tau$ be the operator of left multiplication by\n$W^{X\/Y}_{-\\tau}$ and $N_\\tau$ the operator of right multiplication\nby $W^{Y\/X}_{\\tau}$ on $\\mathscr{L}_{X\/Z,Y\/Z}$. If we also set\n$M_\\tau=\\mathcal{W}^E_\\tau$ then we get three commuting operators\n$L_\\tau,M_\\tau,N\\tau$ on $\\mathscr{L}_{X\/Z,Y\/Z}$ such that $\\mathcal{W}_\\tau=L_\\tau\nM_\\tau N_\\tau$.  Then it is easy to check a Dini type condition like\n\\eqref{eq:dinix} by using\n\\begin{equation}\\label{eq:tmp}\n\\mathcal{W}_\\tau-1=(L_\\tau-1)M_\\tau N_\\tau+(M_\\tau -1)N_\\tau+(N_\\tau-1). \n\\end{equation}\nOn the other hand, observe that $\\mathcal{W}_\\tau,L_\\tau,M_\\tau,N_\\tau$ are\none parameter groups of operators on the Banach space $\\mathscr{L}$.  These\ngroups are not continuous in the ordinary sense but this does not\nreally matter, in fact we are in the setting of \\cite[Ch. 5]{ABG}.\nThe main point is that the integral\n$\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2T\\|_\\mathscr{L}\\text{d}\\varepsilon\/\\varepsilon^2$\nis finite if and only if\n$\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^6T\\|_\\mathscr{L}\\text{d}\\varepsilon\/\\varepsilon^2$\nis finite (see Theorem 3.4.6 in \\cite{ABG}; this is where real\ninterpolation comes into play).  Now by taking the sixth power of\n\\eqref{eq:tmp} and developing the right hand side we easily get the\nresult, cf. the formula on top of page 132 of \\cite{ABG}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nThe proof of Theorem \\ref{th:BVR} is based on an extension of\nPropositions 9.4.11 and 9.4.12 from \\cite{ABG} to the present\ncontext. Since the argument is very similar, we do not enter into\ndetails. We mention only that the operator $D$ can be written as\n$4D=P \\cdot Q+Q \\cdot P$ where $P=\\oplus_X P_X$ and $Q=\\oplus_X Q_X$\nare suitably interpreted. The proofs in \\cite{ABG} depend only on\nthis structure.\n\n\n \n\n\\section{Appendix: Hamiltonian algebras} \n\\label{s:appb}\n\n\nWe prove here some results on $C^*$-algebras generated by certain\nclasses of ``elementary'' Hamiltonians.\n\n\\subsection{}\\label{ss:a1}\n\nLet $X$ be a locally compact abelian group and let\n$\\{U_x\\}_{x\\in X}$ be a strongly continuous unitary representation\nof $X$ on a Hilbert space $\\mathcal{H}$. Then one can associate to it a\nBorel regular spectral measure $E$ on $X^*$ with values projectors\non $\\mathcal{H}$ such that $U_x=\\int_{X^*}k(x)E(\\text{d} k)$ and this allows us\nto define for each Borel function $\\psi:X^*\\to\\mathbb{C}$ a normal\noperator on $\\mathcal{H}$ by the formula $\\psi(P)=\\int_{X^*} \\psi(k)E(\\text{d}\nk)$.  The set $\\mathscr{T}_X(\\mathcal{H})$ of all the operators $\\psi(P)$ with\n$\\psi\\in\\cc_{\\mathrm{o}}(X^*)$ is clearly a non-degenerate $C^*$-algebra of\noperators on $\\mathcal{H}$. The following result, which will be useful in\nseveral contexts, is an easy consequence of the Cohen-Hewitt\nfactorization theorem, see Lemma 3.8 from \\cite{GI4}. Consider an\noperator $A\\in L(\\mathcal{H})$.\n\n\\begin{lemma}\\label{lm:help}\n$\\displaystyle{\\lim_{x\\to0}}\\|(U_x-1)A\\|=0$ if and\nonly if $A=\\psi(P)B$ for some $\\psi\\in\\cc_{\\mathrm{o}}(X^*)$ and $B\\in L(\\mathcal{H})$.\n\\end{lemma}\n\nWe say that an operator $S\\in L(\\mathcal{H})$ is of class $C^0(P)$ if the\nmap $x\\mapsto U_xSU_x^*$ is norm continuous.\n\n\\begin{lemma}\\label{lm:cop}\nLet $S\\in L(\\mathcal{H})$ be of class $C^0(P)$ and let $T\\in\n\\mathscr{T}_X(\\mathcal{H})$. Then for each $\\varepsilon>0$ there is $Y\\subset X$\nfinite and there are operators $T_y\\in \\mathscr{T}_X(\\mathcal{H})$ such that\n$\\|ST-\\sum_{y\\in Y} T_y U_{y}SU_{y}^*\\|<\\varepsilon$.\n\\end{lemma}\n\\proof It suffices to assume that $T=\\psi(P)$ where $\\psi$ has a\nFourier transform integrable on $X$, so that $T=\\int_X U_x\n\\widehat\\psi(x) \\text{d} x$, and then to use a partition of unity on $X$\nand the uniform continuity of the map $x\\mapsto U_xSU_x^*$ (see the\nproof of Lemma 2.1 in \\cite{DG1}).  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nWe say that a subset $\\mathcal{B}$ of $L(\\mathcal{H})$ is $X$-stable if\n$U_xSU_x^*\\in\\mathcal{B}$ whenever $S\\in\\mathcal{B}$ and $x\\in X$. From Lemma\n\\ref{lm:cop} we see that if $\\mathcal{B}$ is an $X$-stable real linear space\nof operators of class $C^0(P)$ then\n\\[\n\\mathcal{B}\\cdot \\mathscr{T}_X(\\mathcal{H})= \\mathscr{T}_X(\\mathcal{H})\\cdot\\mathcal{B}.\n\\] \nSince the $C^*$-algebra $\\mathcal{A}$ generated by $\\mathcal{B}$ is also $X$-stable\nand consists of operators of class $C^0(P)$\n\\begin{equation}\\label{eq:cop}\n\\mathscr{A}\\equiv\\mathcal{A}\\cdot \\mathscr{T}_X(\\mathcal{H})= \\mathscr{T}_X(\\mathcal{H})\\cdot\\mathcal{A}\n\\end{equation} \nis a $C^*$-algebra. The operators $U_x$ implement a norm continuous\naction of $X$ by automorphisms of the algebra $\\mathcal{A}$ so the\n$C^*$-algebra crossed product $\\mathcal{A}\\rtimes X$ is well defined and the\nalgebra $\\mathscr{A}$ is a quotient of this crossed product.\n\nA function $h$ on $X^*$ is called \\emph{$p$-periodic} for some\nnon-zero $p\\in X^*$ if $h(k+p)=h(k)$ for all $k\\in X^*$.\n\n\\begin{proposition}\\label{pr:cop}\nLet $\\mathcal{V}$ be an $X$-stable set of symmetric bounded operators of\nclass $C^0(P)$ and such that $\\lambda\\mathcal{V}\\subset\\mathcal{V}$ if\n$\\lambda\\in\\mathbb{R}$. Denote $\\mathcal{A}$ the $C^*$-algebra generated by $\\mathcal{V}$\nand define $\\mathscr{A}$ by \\eqref{eq:cop}.  Let $h:X^*\\to\\mathbb{R}$ be\ncontinuous, not $p$-periodic if $p\\neq0$, and such that\n$|h(k)|\\to\\infty$ as $k\\to\\infty$.  Then $\\mathscr{A}$ is the $C^*$-algebra\ngenerated by the self-adjoint operators of the form $h(P+k)+V$ with\n$k\\in X^*$ and $V\\in\\mathcal{V}$.\n\\end{proposition}\n\\proof Denote $K=h(P+k)$ and let $R_\\lambda=(z-K-\\lambda V)^{-1}$\nwith $z$ not real and $\\lambda$ real. Let $\\mathscr{C}$ be the $C^*$-algebra\ngenerated by such operators (with varying $k$ and $V$). By taking\n$V=0$ we see that $\\mathscr{C}$ will contain the $C^*$-algebra generated by\nthe operators $R_0$. By the Stone-Weierstrass theorem this algebra\nis $\\mathscr{T}_X(\\mathcal{H})$ because the set of functions $p\\to(z-h(p+k))^{-1}$\nwhere $k$ runs over $X^*$ separates the points of $X^*$. The\nderivative with respect to $\\lambda$ at $\\lambda=0$ of $R_\\lambda$\nexists in norm and is equal to $R_0VR_0$, so $R_0VR_0\\in\\mathscr{C}$. Since\n$\\mathscr{T}_X\\subset\\mathscr{C}$ we get $\\phi(P)V\\psi(P)\\in\\mathscr{C}$ for all\n$\\phi,\\psi\\in\\cc_{\\mathrm{o}}(X^*)$ and all $V\\in\\mathcal{V}$. Since $V$ is of class\n$C^0(P)$ we have $(U_x-1)V\\psi(P)\\sim V(U_x-1)\\psi(P)\\to0$ in norm\nas $x\\to0$ from which we get $\\phi(P)V\\psi(P)\\to S\\psi(P)$ in norm\nas $\\phi\\to1$ conveniently. Thus $V\\psi(P)\\in\\mathscr{C}$ for $V,\\psi$ as\nabove. This implies $V_1\\cdots V_n\\psi(P)\\in\\mathscr{C}$ for all\n$V_1,\\dots,V_n\\in\\mathcal{V}$. Indeed, assuming $n=2$ for simplicity, we\nwrite $\\psi=\\psi_1\\psi_2$ with $\\psi_i\\in\\cc_{\\mathrm{o}}(X^*)$ and then Lemma\n\\ref{lm:cop} allows us to approximate $V_2\\psi_1(P)$ in norm with\nlinear combinations of operators of the form $\\phi(P)V^x_2$ where\nthe $V^x_2$ are translates of $V_2$.  Since $\\mathscr{C}$ is an algebra we\nget $V_1\\phi(P) V^x_2\\psi_2(P)\\in\\mathscr{C}$ hence passing to the limit we\nget $V_1V_2\\psi(P)\\in\\mathscr{C}$. Thus we proved $\\mathscr{A}\\subset\\mathscr{C}$. The\nconverse inclusion follows from a series expansion of $R_\\lambda$ in\npowers of $V$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nThe next two corollaries follow easily from Proposition\n\\ref{pr:cop}. We take $\\mathcal{H}=L^2(X)$ which is equipped with the usual\nrepresentations $U_x,V_k$ of $X$ and $X^*$ respectively. Let\n$W_\\xi=U_xV_k$ with $\\xi=(x,k)$ be the phase space translation\noperator, so that $\\{W_\\xi\\}$ is a projective representation of the\nphase space $\\Xi=X\\oplus X^*$.  Fix some classical kinetic energy\nfunction $h$ as in the statement of Proposition \\ref{pr:cop} and let\nthe classical potential $v:X\\to\\mathbb{R}$ be a bounded uniformly\ncontinuous function. Then the quantum Hamiltonian will be\n$H=h(P)+v(Q)\\equiv K+V$. Since the origins in the configuration and\nmomentum spaces $X$ and $X^*$ have no special physical meaning one\nmay argue \\cite{Be1,Be2} that $W_\\xi H W^*_\\xi=h(P-k)+v(Q+x)$ is a\nHamiltonian as good as $H$ for the description of the evolution of\nthe system. It is not clear to us whether the algebra generated by\nsuch Hamiltonians (with $h$ and $v$ fixed) is in a natural way a\ncrossed product. On the other hand, it is natural to say that the\ncoupling constant in front of the potential is also a variable of\nthe system and so the Hamiltonians $H_\\lambda=K+\\lambda V$ with any\nreal $\\lambda$ are as relevant as $H$. Then we may apply Proposition\n\\ref{pr:cop} with $\\mathcal{V}$ equal to the set of operators of the form\n$\\lambda v(Q+x)$. Thus:\n\n\n\\begin{corollary}\\label{co:cop1}\nLet $v\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ real and let $\\mathcal{A}$ be the $C^*$-subalgebra of\n$\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ generated by the translates of $v$.  Let $h:X^*\\to\\mathbb{R}$ be\ncontinuous, not $p$-periodic if $p\\neq0$, and such that\n$|h(k)|\\to\\infty$ as $k\\to\\infty$.  Then the $C^*$-algebra generated\nby the self-adjoint operators of the form $W_\\xi H_\\lambda W^*_\\xi$\nwith $\\xi\\in\\Xi$ and real $\\lambda$ is the crossed product\n$\\mathcal{A}\\rtimes X$.\n\\end{corollary}\n\nNow let $\\mathcal{T}$ be a set of closed subgroups of $X$ such that the\nsemilattice $\\mathcal{S}$ generated by it (i.e. the set of finite\nintersections of elements of $\\mathcal{T}$) consists of pairwise compatible\nsubgroups. Set $\\mathcal{C}_X(\\mathcal{S})=\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}} \\mathcal{C}_X(Y)$. From\n\\eqref{eq:reg1} it follows that this is the $C^*$-algebra generated\nby  $\\sum_{Y\\in\\mathcal{T}} \\mathcal{C}_X(Y)$.\n\n\\begin{corollary}\\label{co:cop2}\nLet $h$ be as in Corollary \\ref{co:cop1}. Then the $C^*$-algebra\ngenerated by the self-adjoint operators of the form $h(P+k)+v(Q)$\nwith $k\\in X^*$ and $v\\in\\sum_{Y\\in\\mathcal{T}} \\mathcal{C}_X(Y)$ is the\ncrossed product $\\mathcal{C}_X(\\mathcal{S})\\rtimes X$.\n\\end{corollary}\n\n\n\\begin{remark}\\label{re:cop}\nProposition \\ref{pr:cop} and Corollaries \\ref{co:cop1} and\n\\ref{co:cop2} remain true and are easier to prove if we consider the\n$C^*$-algebra generated by the operators $h(P)+V$ with all\n$h:X^*\\to\\mathbb{R}$ continuous and such that $|h(k)|\\to\\infty$ as\n$k\\to\\infty$. If in Proposition \\ref{pr:cop} we take $\\mathcal{H}=L^2(X;E)$\nwith $E$ a finite dimensional Hilbert space (describing the spin\ndegrees of freedom) then the operators $H_0=h(P)$ with $h:X\\to L(E)$\na continuous symmetric operator valued function such that\n$\\|(h(k)+i)^{-1}\\|\\to 0$ as $k\\to\\infty$ are affiliated to $\\mathscr{A}$\nhence also their perturbations $H_0+V$ where $V$ satisfies the\ncriteria from \\cite{DG3}, for example.\n\\end{remark}\n\n\n\\subsection{}\n\\label{ss:anbody} \n\nWe consider the framework of \\S\\ref{ss:cexample} and use Corollary\n\\ref{co:cop2} to prove that the Hamiltonian algebra of a\nnonrelativistic $N$-body system is generated in a natural way by the\noperators of the form \\eqref{eq:nonrel}. To state a precise result\nit suffices to consider the reduced Hamiltonians (for which we keep\nthe notation $H$).\n\nLet $\\mathfrak{S}_2$ be the set of cluster decompositions which\ncontain only one nontrivial cluster which consists of exactly two\nelements. This cluster is of the form $\\{j,k\\}$ for a unique pair of\nnumbers $1\\leq j < k \\leq N$ and we denote by $(jk)$ the\ncorresponding cluster decomposition. The map $x\\mapsto x_j-x_k$\nsends $X$ onto $\\mathbb{R}^d$ and has $X_{(jk)}$ as kernel hence\n$V_{jk}(x_j-x_k)=V_{(jk)}\\circ\\pi_{(jk)}(x)$ where\n$V_{(jk)}:X\/X_{(jk)}\\to\\mathbb{R}$ is continuous with compact support and\n$\\pi_{(jk)}:X\\to X\/X_{(jk)}$ is the canonical surjection.\n\nThus the reduced Hamiltonians corresponding to \\eqref{eq:nonrel} are\nthe operators on $\\mathcal{H}_X$ of the form\n\\begin{equation}\\label{eq:nonre}\n\\Delta_X+\n{\\textstyle{\\sum_{\\sigma\\in\\mathfrak{S}_2}}} V_\\sigma\\circ\\pi_\\sigma\n\\hspace{2mm}\\text{with}\\hspace{2mm} V_\\sigma:X\/X_\\sigma\\to\\mathbb{R}\n\\text{ continuous with compact support}.\n\\end{equation} \nThese operators must be affiliated to the Hamiltonian algebra of the\n$N$-body system. On the other hand, if a Hamiltonian $h(P)+V$ is\nconsidered as physically admissible then $h(P+k)+V$ should be\nadmissible too because the zero momentum $k=0$ should not play a\nspecial role. In other terms, translations in momentum space should\nleave invariant the set of admissible Hamiltonians. Hence it is\nnatural to consider \\emph{the smallest $C^*$-algebra $\\mathscr{C}_X(\\mathcal{S})$\n  such that the operators \\eqref{eq:nonre} are affiliated to it and\n  which is stable under translations in momentum space. But this\n  algebra is exactly the crossed product}\n\\[\n\\mathscr{C}_X=\\mathcal{C}_X\\rtimes X=\\mathcal{C}_X\\cdot \\mathscr{T}_X \n\\hspace{2mm}\\text{with}\\hspace{2mm}\n\\mathcal{C}_X={\\textstyle\\sum_\\sigma}\\mathscr{C}_X(X_\\sigma).\n\\]\nIndeed, it is clear that the semilattice generated by\n$\\mathfrak{S}_2$ is $\\mathfrak{S}$ so we may apply Corollary\n\\ref{co:cop2}.\n\n\n\\subsection{}\n\\label{ss:amotiv} \n\nHere we prove Theorem \\ref{th:motiv}.\n\nLet $\\mathscr{C}'$ be the $C^*$-algebra generated by the operators of the\nform $(z-K-\\phi)^{-1}$ where $z$ is a not real number, $K$ is a\nstandard kinetic energy operator, and $\\phi$ is a symmetric field\noperator. With the notation \\eqref{eq:d} we easily get\n$\\mathscr{T}_\\text{d}\\subset\\mathscr{C}'$.  If $\\lambda\\in\\mathbb{R}$ then $\\lambda\\phi$ is\nalso a field operator so $(z-K-\\lambda\\phi)^{-1}\\in\\mathscr{C}'$. By taking\nthe derivative with respect to $\\lambda$ at $\\lambda=0$ of this\noperator we get $(z-K)^{-1}\\phi (z-K)^{-1}\\in\\mathscr{C}$. Since\n$(z-K)^{-1}=\\oplus_X(z-h_X(P))^{-1}$ (recall that $P$ is the\nmomentum observable independently of the group $X$) and since\n$\\mathscr{T}_\\text{d}\\subset\\mathscr{C}'$ we get $S\\phi(\\theta) T\\in\\mathscr{C}'$ for all\n$S,T\\in \\mathscr{T}_\\text{d}$ and $\\theta=(\\theta_{XY})_{X\\supset Y}$.\n\nLet $\\mathscr{C}'_{XY}=\\Pi_X\\mathscr{C}'\\Pi_Y\\subset\\mathscr{L}_{XY}$ be the components of\nthe algebra $\\mathscr{C}'$ and let us fix $X\\supset Y$. Then we get\n$\\varphi(P) a^*(u) \\psi(P)\\in\\mathscr{C}'_{XY}$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(X^*)$, $\\psi\\in\\cc_{\\mathrm{o}}(Y^*)$, and $u\\in\\mathcal{H}_{X\/Y}$.  The\nclspan of the operators $a^*(u) \\psi(P)$ is $\\mathscr{T}_{XY}$, see\nProposition \\ref{pr:def3} and the comments after \\eqref{eq:L2a}, and\nfrom \\eqref{eq:cyzc} we have $\\mathscr{T}_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}$.  Thus\nthe clspan of the operators $\\varphi(P) a^*(u) \\psi(P)$ is\n$\\mathscr{T}_{XY}$ for each $X\\supset Y$ and then we get\n$\\mathscr{T}_{XY}\\subset\\mathscr{C}'_{XY}$.  By taking adjoints we get\n$\\mathscr{T}_{XY}\\subset\\mathscr{C}'_{XY}$ if $X\\sim Y$.\n\nNow recall that the subspace $\\mathscr{T}^\\circ\\subset L(\\mathcal{H})$ defined by\n$\\mathscr{T}^\\circ_{XY}=\\mathscr{T}_{XY}$ if $X\\sim Y$ and $\\mathscr{T}^\\circ=\\{0\\}$ if\n$X\\not\\sim Y$ is a closed self-adjoint linear subspace of $\\mathscr{T}$ and\nthat $\\mathscr{T}^\\circ\\cdot\\mathscr{T}^\\circ=\\mathscr{C}$, cf. Theorem \\ref{th:tc}. By\nwhat we proved before we have $\\mathscr{T}^\\circ\\subset\\mathscr{C}'$ hence\n$\\mathscr{C}\\subset\\mathscr{C}'$. The converse inclusions is easy to prove. This\nfinishes the proof of Theorem \\ref{th:motiv}. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{S1}\n\\setcounter{equation}{0}\nThe Laguerre unitary ensemble ($ {\\rm LUE}_N $) refers to the eigenvalue probability\ndensity function (p.d.f.)\n\\begin{equation}\n \\frac{1}{C_{N,a}}\n \\prod^{N}_{l=1}\\lambda_l^ae^{-\\lambda_l}\\prod_{1 \\leq j<k \\leq N}(\\lambda_k-\\lambda_j)^2 , \n\\label{LUE_pdf}\n\\end{equation}\nwhere\n\\begin{equation}\n  C_{N,a} := \\prod^{N-1}_{j=0}\\Gamma(j+1)\\Gamma(j+a+1) ,\n\\label{LUE_norm}\n\\end{equation}\nwith support on $ \\lambda_l \\in [0,\\infty) $. For $ a \\in \\mathbb Z_{\\geq 0} $, this\neigenvalue p.d.f. is realised by non-negative matrices $ X^{\\dagger}X $ where \n$ X $ is an $ M\\times N $ complex Gaussian matrix and $ a=M-N $. In our work\n\\cite{FW_2002a} the average over (\\ref{LUE_pdf})\n\\begin{equation}\n  \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) := \\frac{C_{N,a}}{C_{N,a+\\mu}}\n  \\Big\\langle \\prod^N_{l=1}(1-\\xi\\chi^{(l)}_{(0,s)})(\\lambda_l-s)^{\\mu} \\Big\\rangle_{{\\rm LUE}_N} ,\n\\label{LUE_E}\n\\end{equation}\nwhere $ \\chi^{(l)}_{J}=1 $ for $ \\lambda_l \\in J $, $ \\chi^{(l)}_{J}=0 $ otherwise,\nand the normalisation is chosen so that \n\\begin{equation}\n  \\left.\\tilde{E}_{2,N}((0,s);a,\\mu;\\xi)\\right|_{s=0} := 1 ,\n\\label{LUE_Enorm}\n\\end{equation}\nwas characterised as a $ \\tau $-function for the Painlev\\'e V system. As a consequence,\nit was shown that \n\\begin{equation}\n W_N(s;a,\\mu;\\xi) \n  := s\\frac{d}{ds}\\log\\left( s^{-N\\mu}\\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) \\right) ,\n\\label{LUE_sigma}\n\\end{equation}\nsatisfies the Jimbo-Miwa-Okamoto $ \\sigma $-form of the Painlev\\'e V equation\n\\begin{equation}\n  (s\\sigma''_V)^2 - \\left[ \\sigma_V - s\\sigma'_V + 2(\\sigma'_V)^2 +\n  (\\sum^3_{j=0}\\nu_j) \\sigma'_V \\right]^2\n  + 4 \\prod^3_{j=0}(\\nu_j + \\sigma'_V) = 0 ,\n\\label{PV_sigma}\n\\end{equation}\nwith\n\\begin{equation}\n  \\nu_0 = 0, \\quad \\nu_1=-\\mu, \\quad \\nu_2 = N+a, \\quad \\nu_3 = N, \\quad\n  \\sum^3_{j=0}\\nu_j = 2N+a-\\mu .\n\\label{LUE_Vparam}\n\\end{equation}\n\nFor this to uniquely characterise $ W_N $, a boundary condition must be specified. \nHowever in \\cite{FW_2002a} only in the cases $ \\mu=0 $ and $ \\mu=2 $ were a\nboundary condition specified for general $ \\xi $.\n\nAlso considered in \\cite{FW_2002a} was the hard edge limiting average\n\\begin{equation}\n  \\tilde{E}^{\\rm hard}_N(t;a,\\mu;\\xi) :=\n  \\lim_{N \\to \\infty} \\Big(\n  \\frac{C_{N,a}}{C_{N,a+\\mu}}\\tilde{E}_{2,N}((0,\\frac{t}{4N});a,\\mu;\\xi) \\Big) .\n\\label{HE_Edefn}\n\\end{equation}\nIt was shown that\n\\begin{equation}\n  \\tilde{E}^{\\rm hard}_2(t;a,\\mu;\\xi) = \\exp\\int_{0}^t u^{h}(s;a,\\mu;\\xi)\\,\\frac{ds}{s} ,\n\\label{HE_E}\n\\end{equation}\nwhere setting\n\\begin{equation}\n  u^{h}(s;a,\\mu;\\xi) =-(\\sigma_{{\\rm III}'}(s)+\\frac{\\mu(\\mu+a)}{2}) ,\n\\label{HE_E.2}\n\\end{equation}\nthe function $ \\sigma_{{\\rm III}'}(s) $ satisfies the Jimbo-Miwa-Okamoto $ \\sigma $-form\nof the Painlev\\'e ${\\rm III^{\\prime}}\\;$ equation\n\\begin{equation}\n  (s\\sigma''_{{\\rm III}'})^2 - v_1v_2(\\sigma'_{{\\rm III}'})^2\n  + \\sigma'_{{\\rm III}'}(4\\sigma'_{{\\rm III}'}-1)(\\sigma_{{\\rm III}'}-s\\sigma'_{{\\rm III}'})\n  - \\frac{1}{4^3}(v_1-v_2)^2 = 0 ,\n\\label{PIII_sigma}\n\\end{equation}\nwith parameters\n\\begin{equation}\n  v_1 = a+\\mu, \\qquad v_2 = a-\\mu .\n\\label{HE_IIIparam}\n\\end{equation}\nAgain only in the cases $ \\mu=0 $ and $ \\mu=2 $ were boundary conditions specified for \ngeneral $ \\xi $.\n\nThe aim of this work is to specify the boundary conditions relevant to both \n(\\ref{LUE_sigma}) and (\\ref{PV_sigma}) for general values of the parameters. In the \ncase of (\\ref{LUE_sigma}) this is done by writing the average (\\ref{LUE_E}) in\nits equivalent determinant form and evaluating the matrix elements in terms of\ncertain $ {}_1F_1 $ hypergeometric functions. With the small $ s $ behaviour \nof the matrix elements determined, it turns out that the determinant is such that\nits corresponding small $ s $ behaviour can readily be deduced. The small $ s $\nasymptotic form of (\\ref{LUE_E}) then follows immediately. Scaling this \nasymptotic form as required by (\\ref{HE_Edefn}) then gives the small $ t $\nbehaviour of $ \\tilde{E}^{\\rm hard}_N(t;a,\\mu;\\xi) $ and the small $ s $ behaviour\nof $ u^{h}(s;a,\\mu;\\xi) $.\n\nThe general small $ s $ asymptotic form of the permitted solutions of \n(\\ref{PV_sigma}) and (\\ref{PIII_sigma}) have been given by Jimbo \\cite{Ji_1982}.\nAs part of this study the boundary conditions found here are compared against\nthese general forms. It is found that in both cases only one of the two branches\npermitted by the general solution is present in our random matrix problem.\n\nAs an application of our results we specify the rapid computation of the power\nseries expansion of a certain Hankel determinant of Bessel functions. The latter\nis known form our work \\cite{FW_2002a} to be a special case of \n$ \\tilde{E}^{\\rm hard}_N(t;a,\\mu;\\xi) $. The coefficients in the power series appear\nin an asymptotic formula obtained recently by Conrey, Rubinstein and Snaith \n\\cite{CRS_2005} for the integer moments of the derivative of the characteristic\npolynomial of a unitary random matrix. This in turn has application to the study\nof the derivative of the Riemann zeta function on the critical line.\n \n\\section{Small $ s $ Expansion of $ \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) $}\\label{S2}\n\\setcounter{equation}{0}\nA standard result in random matrix theory, which in fact goes back to an identity\nof Heine (see \\cite{ops_Sz}) expresses the random matrix average (\\ref{LUE_E}) as\na determinant\n\\begin{equation}\n \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) \n  = \\frac{N!C_{N,a}}{C_{N,a+\\mu}}\\det[w_{j+k}]_{j,k=0,\\ldots,N-1} , \n\\label{LUE_Hdet}\n\\end{equation}\nwhere\n\\begin{equation}\n  w_{n} \n  := \\int^{\\infty}_0 d\\lambda\\; (1-\\xi\\chi_{(0,s)})(\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda} .\n\\label{LUE_Hsym}\n\\end{equation}\nUnless $ \\mu $ is a non-negative integer (\\ref{LUE_Hsym}) is not well defined for\n$ s $ real and positive, which is the domain of interest. To remedy this, we note \nthat simple manipulation gives\n\\begin{equation}\n  w_{n} \n  := \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n     + (1-\\xi)\\int^{s}_0 d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda} ,\n\\label{LUE_Hsym.2}\n\\end{equation}\nand in the second integral of this expression write \n$ (\\lambda-s)^{\\mu} = e^{\\mu\\log(\\lambda-s)} $ with \n$ -\\pi < \\arg\\log(\\lambda-s) \\leq \\pi $. We then obtain\n\\begin{equation}\n  w_{n} \n  := \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n     + (1-\\xi)e^{\\pi i\\mu}\\int^{s}_0 d\\lambda\\; (s-\\lambda)^{\\mu}\\lambda^{n+a}e^{-\\lambda} ,\n\\label{LUE_Hsym.3}\n\\end{equation}\nwhich is well defined for $ \\Re(\\mu) > -1 $ and $ \\Re(a) > -1 $ for \n$ s>0 $ with the additional constraint $ \\Re(\\mu+a) > -1 $ at $ s=0 $.\n\nWe seek the leading terms in the small $ s$ expansion of (\\ref{LUE_Hsym.3}). \nThese can be read off from an explicit evaluation in terms of the $ {}_1F_1 $\nconfluent hypergeometric function \\cite{WW_1965}. \n\\begin{proposition}\nSubject to the conditions $ \\Re(\\mu) > -1 $, $ \\Re(a) > -1 $,\n$ \\Re(\\mu+a) > -1 $ and $ \\mu+a\\notin \\mathbb Z_{\\geq 0} $ we have\n\\begin{equation}\n  w_{n} = a_n(s) + s^{n+\\mu+a+1}b_n(s) ,\n\\label{LUE_Hsym.4}\n\\end{equation}\nwhere $ a_n(s), b_n(s) $ are analytic about $ s=0 $ and given explicitly by\n\\begin{equation}\n\\begin{split}\n  a_n(s) &=\n  \\Gamma(\\mu+n+a+1)e^{-s}{}_1F_1(-a-n;-\\mu-a-n;s) ,\n  \\\\\n  b_n(s) &=\n  \\frac{\\Gamma(\\mu+1)\\Gamma(n+a+1)}{\\Gamma(\\mu+n+a+2)}\n  \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right)\n  \\\\\n         & \\qquad\\times\n  e^{-s}{}_1F_1(\\mu+1;\\mu+a+n+2;s) .\n\\end{split}\n\\label{Hsym_1F1}\n\\end{equation}\nIn particular, under the above conditions,\n\\begin{equation}\n  w_{n} \\mathop{\\sim}\\limits_{s \\to 0} a_n(0)+sa'_n(0)+s^{n+\\mu+a+1}b_n(0) ,\n\\label{LUE_Hsym.5}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n  a_n(0)  &= \\Gamma(\\mu+n+a+1) ,\n  \\\\\n  a'_n(0) &= -\\mu\\Gamma(\\mu+n+a) ,\n  \\\\\n  b_n(0) &=\n  \\frac{\\Gamma(\\mu+1)\\Gamma(n+a+1)}{\\Gamma(\\mu+n+a+2)}\n  \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) .\n\\end{split}\n\\label{Hsym_1F1_exp}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe results (\\ref{LUE_Hsym.5}) and (\\ref{Hsym_1F1_exp}) are immediate corollaries\nof (\\ref{LUE_Hsym.4}) and (\\ref{Hsym_1F1}) and the fact that\n\\begin{equation*}\n  {}_1F_1(\\gamma;\\alpha;s) = 1+\\frac{\\gamma}{\\alpha}s+{\\rm O}(s^2) .\n\\end{equation*}\nTo derive (\\ref{LUE_Hsym.4}), we note that simple manipulation shows\n\\begin{equation*}\n  \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n = s^{a+n}e^{-s}\\int^{\\infty}_0 d\\lambda\\; (1+\\lambda\/s)^{n+a}\\lambda^{\\mu}e^{-\\lambda} .\n\\end{equation*}\nBut with \n\\begin{equation*}\n  W_{k,m}(z) = \\frac{z^ke^{-z\/2}}{\\Gamma(1\/2-k+m)}\n   \\int^{\\infty}_0 dt\\; (1+t\/z)^{k-1\/2+m}t^{-k-1\/2+m}e^{-t} ,\n\\end{equation*}\nspecifying the Whittaker function, it is known that \\cite{WW_1965}\n\\begin{equation*}\n  W_{k,m}(z)\n  = \\frac{\\Gamma(-2m)}{\\Gamma(1\/2-k-m)}M_{k,m}(z)+\\frac{\\Gamma(2m)}{\\Gamma(1\/2-k+m)}M_{k,-m}(z)\n\\end{equation*}\nwhere\n\\begin{equation*}\n  M_{k,m}(z) = z^{m+1\/2}e^{-z\/2}{}_1F_1(1\/2-k+m;2m+1;z) .\n\\end{equation*}\nConsequently\n\\begin{multline}\n  \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n =  \\Gamma(\\mu+a+n+1)e^{-s}{}_1F_1(-a-n;-\\mu-a-n;s) \n \\\\\n  + \\frac{\\Gamma(\\mu+1)\\Gamma(-\\mu-a-n-1)}{\\Gamma(-a-n)}\n    s^{\\mu+a+n+1}e^{-s}{}_1F_1(\\mu+1;\\mu+a+n+2;s) .\n\\label{ws6}\n\\end{multline}\nThe left-hand side of (\\ref{ws6}) exists for $ \\Re(\\mu)>-1 $ if $ s>0 $ and\n$ \\Re(\\mu+a)>-1 $ if $ s=0 $, whereas the right-hand side is valid in \nthis parameter domain except for $ \\mu+a+n\\in\\mathbb Z_{\\geq 0} $, and in this case the individual\nterms have a simple pole at $ a+n\\notin\\mathbb Z_{\\geq 0} $ or are undefined when $ a+n\\in\\mathbb Z_{\\geq 0} $. \nNeedless to say the sum of the terms on the right-hand side has the same analytic\ncharacter as the left-hand side.\n\nRegarding the second integral in (\\ref{LUE_Hsym.3}), we first note that a simple \nchange of variables gives\n\\begin{equation*}\n  \\int^{s}_0 d\\lambda\\; (s-\\lambda)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n  = s^{n+1+a+\\mu}e^{-s}\\int^{1}_0 dx\\; (1-x)^{n+a}x^{\\mu}e^{sx} .\n\\end{equation*}\nBut the integral on the right hand side is the Euler integral representation of\nthe $ {}_1F_1 $ function, which shows\n\\begin{multline}\n \\int^{s}_0 d\\lambda\\; (s-\\lambda)^{\\mu}\\lambda^{n+a}e^{-\\lambda} \n \\\\\n  = \\frac{\\Gamma(\\mu+1)\\Gamma(a+n+1)}{\\Gamma(\\mu+a+n+2)}\n    s^{\\mu+a+n+1}e^{-s}{}_1F_1(\\mu+1;\\mu+a+n+2;s) .\n\\label{ws7}\n\\end{multline}\nThis latter relation is valid for $ \\Re(\\mu)>-1 $ and $ \\Re(a)>-1 $ when\n$ s>0 $.\nSubstituting (\\ref{ws6}) and (\\ref{ws7}) in (\\ref{LUE_Hsym.3}) and using the \nappropriate gamma function identities gives (\\ref{LUE_Hsym.4}), (\\ref{Hsym_1F1}).\n\\end{proof}\n\nWhen $ \\mu+a \\in\\mathbb Z_{\\geq 0} $ we have to consider two exceptional cases where one of the \nhypergeometric functions are not defined - the first when $ a+n \\in\\mathbb Z_{\\geq 0} $ \nfor which the hypergeometric function is indeterminate, and the second when\n$ a+n \\notin\\mathbb Z_{\\geq 0} $ and the hypergeometric function has a simple pole.\nThese two cases can be recovered by taking suitable limits and we just state the\nfinal results.\n\n\\begin{proposition}\nWhen $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and $ a+n=k \\in\\mathbb Z_{\\geq 0} $ with $ n+j\\geq k $ \nwe have\n\\begin{multline}\n w_n = k!e^{-s}\\Bigg\\{  \\sum^k_{l=0}\\frac{(n+j-l)!}{(k-l)!l!}s^l\n \\\\\n  + (-1)^{n+j+k}(1-\\xi)\\frac{(n+j-k)!}{(n+j+1)!}s^{n+j+1}\n    {}_1F_1(n+j+1-k;n+j+2;s) \\Bigg\\} ,\n\\label{Hsym_indeterm}\n\\end{multline}\nand to leading order in small $ s $ we have\n\\begin{equation}\n  w_{n} \\mathop{\\sim}\\limits_{s \\to 0} \n  (n+j)! - (n+j-k)(n+j-1)!s + (-1)^{n+j+k}(1-\\xi)\\frac{(n+j-k)!k!}{(n+j+1)!}s^{n+j+1} .\n\\label{Exp_indeterm}\n\\end{equation}\n\\end{proposition}\nNote that the condition $ n+j\\geq k $ is the same as $ \\mu \\geq 0 $, which falls\nwithin the domain of interest. The key difference of (\\ref{Exp_indeterm}) with\n(\\ref{LUE_Hsym.5}) and (\\ref{Hsym_1F1_exp}) is that the non-analytic term is\nnow polynomial and the second part of this term is absent having been cancelled by\na counterbalancing term.\n\n\\begin{proposition}\nWhen $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and $ a+n \\notin\\mathbb Z_{\\geq 0} $ we have \n\\begin{multline}\n w_n = e^{-s}\\Bigg\\{ \\sum^{n+j}_{l=0}\\frac{(-a-n)_l(n+j-l)!}{l!}(-s)^l \n \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+n+1)}{(n+j+1)!}(1-\\xi)e^{i\\pi\\mu}s^{n+j+1}\n   {}_1F_1(\\mu+1;n+j+2;s)\n \\\\\n  + (-1)^j\\frac{\\sin\\pi a}{\\pi}\\frac{\\Gamma(\\mu+1)\\Gamma(a+n+1)}{(n+j+1)!}s^{n+j+1}\n \\\\\n \\times\n    \\sum^{\\infty}_{l=0}\\left[\\psi(l+1)+\\psi(n+j+l+2)-\\psi(\\mu+l+1)-\\log s \\right]\n          \\frac{(\\mu+1)_l}{(n+j+2)_l}\\frac{s^l}{l!} \\Bigg\\} ,\n\\label{Hsym_pole}\n\\end{multline}\nand its leading order behaviour for small $ s $ is\n\\begin{multline}\n  w_{n} \\mathop{\\sim}\\limits_{s \\to 0} \n  (n+j)!+(a-j)(n+j-1)!s \n  \\\\\n  +\\frac{(a-j)_{n+j+1}}{(n+j+1)!}s^{n+j+1}\n   \\left[\\frac{\\pi e^{-i\\pi a}}{\\sin\\pi a}(1-\\xi)+\\psi(1)+\\psi(n+j+2)-\\psi(\\mu+1)-\\log s \\right] .\n\\label{Exp_pole}\n\\end{multline}\n\\end{proposition}\nThe expansion (\\ref{Exp_pole}) differs significantly from (\\ref{LUE_Hsym.5}) and \n(\\ref{Hsym_1F1_exp}) because of the presence of logarithmic terms which now \nreplace the non-analytic contributions of the generic case.\n\n\\begin{corollary}\nUnder generic conditions on $ \\mu+a $ we have\n\\begin{multline}\n  \\det[w_{j+k}]_{j,k=0,\\ldots,N-1} \n  \\\\\n  = \\det[\\Gamma(\\mu+a+1+j+k)]_{j,k=0,\\ldots,N-1}\n  \\\\\n    - \\mu s\\det[\\Gamma(\\mu+a+j) \\;\\; \\Gamma(\\mu+a+1+j+k)]_{{j=0,\\ldots,N-1}\\atop\n                                                         {k=1,\\ldots,N-1}}\n    + {\\rm O}(s^2)\n  \\\\\n    + s^{\\mu+a+1}b_0(0)\\det[\\Gamma(\\mu+a+3+j+k)]_{j,k=0,\\ldots,N-2}\n      \\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\\n    + {\\rm O}(s^{2(\\mu+a+1)}) .\n\\label{Hdet_exp}\n\\end{multline}\n\\end{corollary}\n\\begin{proof}\nAccording to (\\ref{LUE_Hsym.5})\n\\begin{multline*}\n  \\det[w_{j+k}]_{j,k=0,\\ldots,N-1} \n  \\\\\n  \\mathop{\\sim}\\limits_{s \\to 0}\n  \\det[a_{j+k}(0)+sa'_{j+k}(0)+s^{\\mu+a+1+j+k}b_{j+k}(0)]_{j,k=0,\\ldots,N-1}\n  \\\\\n  \\mathop{\\sim}\\limits_{s \\to 0}\n  \\det[a_{j+k}(0)]_{j,k=0,\\ldots,N-1}+s[s]\\det[a_{j+k}(0)+sa'_{j+k}(0)]_{j,k=0,\\ldots,N-1}\n  \\\\\n  + s^{\\mu+a+1}b_0(0)\\det[a_{j+k+2}(0)]_{j,k=0,\\ldots,N-2} ,\n\\end{multline*}\nwhere $ [s]P(s) $ denotes the coefficient of $ s $ in $ P(s) $. Recalling the \nexplicit formula for $ a_n(0) $ as given in (\\ref{Hsym_1F1_exp}) we obtain the \nconstant term and the term proportional to $ s^{\\mu+a+1} $ in (\\ref{Hdet_exp}).\nIt remains to compute the coefficient of $ s $, which according to \n(\\ref{Hsym_1F1_exp}) has the explicit form\n\\begin{equation}\n   [s]\\det[\\Gamma(\\mu+a+1+j+k)-\\mu s\\Gamma(\\mu+a+j+k)]_{j,k=0,\\ldots,N-1} .\n\\label{ws9}\n\\end{equation}\nUsing the linearity formula\n\\begin{equation*}\n   \\det[{\\bf a}_1 \\cdots {\\bf a}_j+{\\bf b}_j \\cdots {\\bf a}_n]\n   = \\det[{\\bf a}_1 \\cdots {\\bf a}_j \\cdots {\\bf a}_n]\n    +\\det[{\\bf a}_1 \\cdots {\\bf b}_j \\cdots {\\bf a}_n] ,\n\\end{equation*}\nwhere the $ \\bf{a} $'s and $ \\bf{b} $'s are column vectors, on each column of\nthe determinant we see that of the terms proportional to $ s $ only the one\nobtained from expanding the first column in non-zero (all the rest result in \ntwo identical columns), and the determinant given by (\\ref{Hdet_exp}) results.\n\\end{proof}\n\nIt remains to evaluate the determinants. For this task we make use of the identity\n\\cite{Nd_2004}\n\\begin{equation*}\n   \\det[\\Gamma(z_k+j)]_{j,k=0,\\ldots,n-1}\n   = \\prod^{n-1}_{j=0}\\Gamma(z_j)\\prod_{0\\leq j<k\\leq n-1}(z_k-z_j) .\n\\end{equation*}\nAfter straightforward manipulations, gamma function evaluations of all the \ndeterminants in (\\ref{Hdet_exp}) can be obtained. Substituting in (\\ref{LUE_Hdet}),\nand recalling that the normalisation is such that at $ s=0 $ $ \\tilde{E}_N $ \nis equal to unity, we obtain the sought small $ s $ expansion of $ \\tilde{E}_N $\nand thus $ W_N $ valid for general values of the parameters.\n\n\\begin{proposition}\\label{LUE_exp}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ and $ \\mu+a\\notin\\mathbb Z_{\\geq 0} $ we have \n\\begin{multline}\n  \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) = 1-\\frac{\\mu N}{\\mu+a}s+{\\rm O}(s^2)\n  \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)\\Gamma(\\mu+a+N+1)}{\\Gamma^2(\\mu+a+2)\\Gamma(\\mu+a+1)\\Gamma(N)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   s^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) ,\n\\label{LUE_Eexp}\n\\end{multline}\nand consequently\n\\begin{multline}\n  W_{N}(s;a,\\mu;\\xi) = -N\\mu-\\frac{\\mu N}{\\mu+a}s+{\\rm O}(s^2)\n  \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)\\Gamma(\\mu+a+N+1)}{\\Gamma(\\mu+a+2)\\Gamma^2(\\mu+a+1)\\Gamma(N)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   s^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) .\n\\label{LUE_Wexp}\n\\end{multline}\n\\end{proposition}\nIn the first exceptional case $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and $ a=k \\in\\mathbb Z_{\\geq 0} $ \nwith $ j\\geq k $ one can still use (\\ref{LUE_Eexp}) but omitting the term involving\nthe ratio of sines, in the case $ j=0 $, or the whole term if $ j>0 $.\nThe situation of the other exceptional case $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and \n$ a \\notin\\mathbb Z_{\\geq 0} $ is more complicated and more so for larger $ j $, and we \nonly give the examples of $ j=0,1 $.\n\\begin{proposition}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ with $ \\mu+a=0 $ we have \n\\begin{multline}\n\\tilde{E}_{2,N}((0,s);a,\\mu=-a;\\xi) = 1\n  \\\\\n  +\\Big\\{-1 + \\frac{\\pi a}{\\sin\\pi a}e^{-i\\pi a}(1-\\xi)\n   +a\\left[2\\psi(2)+\\psi(1)-\\psi(1-a)-\\psi(N+1)-\\log s \\right] \\Big\\}Ns\n  \\\\ + {\\rm o}(s) .\n\\label{LUE_Eexp_pole0}\n\\end{multline}\nFor $ \\mu+a=1 $ we have\n\\begin{multline}\n\\tilde{E}_{2,N}((0,s);a,\\mu=1-a;\\xi) = 1+(a-1)Ns\n  \\\\\n  +\\frac{a(a-1)}{4}\\Big\\{\\frac{\\pi}{\\sin\\pi a}e^{-i\\pi a}(1-\\xi)\n   +2\\psi(3)+\\psi(2)-\\psi(2-a)-\\psi(N+2)-\\log s \\Big\\}(N+1)Ns^2\n  \\\\ + {\\rm o}(s^2) .\n\\label{LUE_Eexp_pole1}\n\\end{multline}\n\\end{proposition}\n\n\\section{Comparison with the Jimbo solution}\\label{S3}\n\\setcounter{equation}{0}\nThe small $ s$ expansion of the most general solution permitted by (\\ref{PV_sigma}),\nor more precisely its corresponding $ \\tau $-function (see (\\ref{PV_tau}) below) has \nbeen determined by Jimbo \\cite{Ji_1982}. However in \\cite{Ji_1982} the equation\n(\\ref{PV_sigma}) is not treated directly. Instead the discussion is based on\nthe equation\n\\begin{multline}\n  (s\\zeta'')^2 - \\left[ \\zeta - s\\zeta' + 2(\\zeta')^2 -\n  (2\\theta_0+\\theta_{\\infty}) \\zeta' \\right]^2 \\\\\n  + 4\\zeta'(\\zeta'-\\theta_0)(\\zeta'-\\frac{1}{2}(\\theta_0-\\theta_s+\\theta_{\\infty}))\n    (\\zeta'-\\frac{1}{2}(\\theta_0+\\theta_s+\\theta_{\\infty}))\n  = 0 ,\n\\label{PV_zeta}\n\\end{multline}\nand the small $ s $ behaviour of the corresponding $ \\tau $-function $ \\tau_V(s) $,\nspecified by the the requirement that\n\\begin{equation}\n  \\zeta(s) = s\\frac{d}{ds}\\log\\tau_V(s)+\\frac{1}{2}(\\theta_0+\\theta_{\\infty})s\n          +\\frac{1}{4}[(\\theta_0+\\theta_{\\infty})^2-\\theta_s^2] , \n\\label{PV_tau}\n\\end{equation}\nwas determined.\n\nComparison of (\\ref{PV_zeta}), (\\ref{PV_tau}) with (\\ref{PV_sigma}), \n(\\ref{LUE_sigma}) shows that for the parameters (\\ref{LUE_Vparam})\n\\begin{equation}\n  \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) = s^{N^2+N(a+\\mu)}e^{-(N+a\/2)s}\\tau_V(s) ,\n\\label{LUE_Vtau}\n\\end{equation}\nwhile in general\n\\begin{equation}\n  \\theta_0 = -\\nu_1, \\quad \\theta_s = \\nu_2-\\nu_3, \n  \\quad \\theta_{\\infty} = \\nu_1-\\nu_2-\\nu_3 .\n\\label{Vnu_theta}\n\\end{equation}\nNote that for the parameters (\\ref{LUE_Vparam}) we thus thus have\n\\begin{equation}\n  \\theta_0 = \\mu, \\quad \\theta_s = a, \\quad \\theta_{\\infty} = -2N-a-\\mu .\n\\label{LUE_theta}\n\\end{equation}\n\nThe relevant result from \\cite{Ji_1982} can now be presented. It states that the\nmost general small $ s $ behaviour of $ \\tau_V(s) $ permitted by the equation\n(\\ref{PV_zeta}) is\n\\begin{multline}\n  \\tau_V(s) = Cs^{(\\sigma^2-\\theta^2_{\\infty})\/4}\n       \\Bigg\\{ 1\n               - \\frac{\\theta_{\\infty}(\\theta^2_s-\\theta^2_0+\\sigma^2)}\n                     {4\\sigma^2}s \\\\\n   + u\n      \\frac{\\Gamma^2(-\\sigma)}{\\Gamma^2(2+\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle\\theta_s+\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_s-\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_{\\infty}+\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle\\theta_s+\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_s-\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_{\\infty}-\\sigma}{\\displaystyle 2})}\n      s^{1+\\sigma}\n   \\\\\n   + \\frac{1}{u}\n      \\frac{\\Gamma^2(\\sigma)}{\\Gamma^2(2-\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle\\theta_s+\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_s-\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_{\\infty}-\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle\\theta_s+\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_s-\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_{\\infty}+\\sigma}{\\displaystyle 2})}\n      s^{1-\\sigma}\n   \\\\\n   + {\\rm O}(|s|^{2(1-\\Re(\\sigma))}) \\Bigg\\} ,\n\\label{PV_sExp_0}\n\\end{multline}\nwhere $ C $ is a normalisation constant, while $ u $ and $ \\sigma $ are arbitrary\nparameters. The above result was derived subject to the conditions \n$ \\theta_0, \\theta_s \\notin\\mathbb Z $,\n$ \\frac{1}{2}(\\theta_{\\infty}\\pm\\sigma) \\notin\\mathbb Z $,\n$ \\frac{1}{2}(\\theta_{s}\\pm\\theta_0\\pm\\sigma) \\notin\\mathbb Z $ and\nthat $ 0 < \\Re(\\sigma) < 1 $ (a distinct solution was presented for $ \\sigma=0 $).\nThese conditions therefore strictly apply only to the generic or transcendental solutions of\nthe fifth Painlev\\'e equation. \nFor generic parameter values the terms given in (\\ref{PV_sExp_0}) uniquely \nspecify all the subsequent terms in the convergent Puisuex-type expansion for \n$ \\zeta(s) $ about $ s=0 $\n\\begin{equation}\n   \\zeta(s) = \n   \\sum^{\\infty}_{j=0}\\sum_{|k|\\leq j}c_{j,k}s^{j+k\\sigma} ,\n\\label{PV_puisuex}\n\\end{equation}\ni.e. with any two of $ c_{1,0},c_{1,1} $ or $ c_{1,-1} $ given. \n\nTo relate this to $ \\tilde{E}_{2,N} $, we see from (\\ref{LUE_Vtau}) and \n(\\ref{LUE_theta}) that we require $ \\sigma^2=(a+\\mu)^2 $ and thus we can choose\n\\begin{equation}\n  \\sigma=a+\\mu .\n\\label{LUE_const}\n\\end{equation}\nThis relation, $ \\sigma=\\theta_0+\\theta_{s} $, is a violation of one of the strict \nconditions given above and is in fact a sufficient condition for a classical solution,\nalong with the necessary condition $ \\theta_0+\\theta_s+\\theta_{\\infty}=-2N\\in\\mathbb Z $,\nwhich is the type of solution that we are dealing with here. \nHowever we conjecture that Jimbo's conditions\ncan be relaxed to accommodate such solutions and the corresponding formulae \n(or limiting forms if necessary) still hold. \nWith this choice of $ \\sigma $ the coefficient of $ s^{1-\\sigma} $ in (\\ref{PV_sExp_0})\ncontains a factor of\n\\begin{equation*}\n  \\frac{1}{\\Gamma(\\frac{\\displaystyle\\theta_{\\infty}+\\sigma}{\\displaystyle 2})}\n  = \\frac{1}{\\Gamma(-N)}\n\\end{equation*}\nand thus vanishes. Simplifying the other terms gives\n\\begin{multline*}\n  \\tau_V(s) \\sim Cs^{-N^2-N(a+\\mu)}\n       \\Bigg\\{ 1 + \\frac{(2N+a+\\mu)a}{2(a+\\mu)}s \\\\\n   + u \\frac{\\sin\\pi\\mu}{\\sin\\pi(a+\\mu)}\n       \\frac{\\Gamma(a+1)\\Gamma(\\mu+1)\\Gamma(N+1+a+\\mu)}\n            {\\Gamma^2(2+a+\\mu)\\Gamma(1+a+\\mu)\\Gamma(N)}\n      s^{1+a+\\mu} \\Bigg\\} .\n\\end{multline*}\nSubstituting in (\\ref{LUE_Vtau}) we see that this is in precise agreement with \n(\\ref{LUE_Eexp}) provided we choose\n\\begin{equation}\n  u\\frac{\\sin\\pi\\mu}{\\sin\\pi(a+\\mu)} = (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(a+\\mu)}\n\\label{LUE_u}\n\\end{equation}\n\n\\section{The hard edge limit}\\label{S4}\n\\setcounter{equation}{0}\nThe hard edge limit is defined by (\\ref{HE_Edefn}). However, only in the cases\n$ \\mu=0 $, $ \\mu=2 $ do we know how to prove its existence for general $ \\xi $\n(in the case $ \\mu=0 $ $ \\tilde{E}_{2,N} $ can be written as a Fredholm determinant,\nwhile the case $ \\mu=2 $ is related to this via differentiation). However a log-gas\nviewpoint (\\cite{rmt_Fo}) indicates that the limit will be well defined, and\nmoreover we expect that it can be taken term-by-term in the small $ s $ expansion\nof $ \\tilde{E}_{2,N} $. In this section we will show that taking the hard edge \nlimit of the small $ s $ expansion (\\ref{LUE_Eexp}) give rise to an initial \ncondition for the solution of (\\ref{PIII_sigma}) consistent with that allowed\nby Jimbo's theory of the small $ s $ expansion of the Painlev\\'e ${\\rm III^{\\prime}}\\;$ equation. From a \npractical perspective this specifies $ \\tilde{E}^{\\rm hard}_{2} $ for general\nvalues of the parameters according to (\\ref{HE_E}), while from a theoretical\nviewpoint it lends weight to the belief that (\\ref{HE_E}) is indeed the correct\nlimiting evaluation for general values of the parameters.\n\nUnder the assumption that the hard edge limit can be taken term-by-term in the\nsmall $ s $ expansion of Proposition (\\ref{LUE_exp}) is immediate.\n\n\\begin{corollary}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ and $ \\mu+a\\notin\\mathbb Z_{\\geq 0} $ \nwe have\n\\begin{multline}\n  \\tilde{E}^{\\rm hard}_{2}(s;a,\\mu;\\xi) = 1-\\frac{\\mu}{4(a+\\mu)}s+{\\rm O}(s^2)\n  \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)}{\\Gamma^2(\\mu+a+2)\\Gamma(\\mu+a+1)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   \\left(\\frac{s}{4}\\right)^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) ,\n\\label{HE_Eexp}\n\\end{multline}\nand consequently the $ \\sigma $-function $ \\sigma_{\\rm III'}(s) $ in (\\ref{HE_E.2})\nhas the small $ s $ expansion\n\\begin{multline}\n  \\sigma_{\\rm III'}(s) = -\\frac{\\mu(\\mu+a)}{2}+\\frac{\\mu}{4(\\mu+a)}s+{\\rm O}(s^2)\n  \\\\\n  -\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)}{\\Gamma(\\mu+a+2)\\Gamma^2(\\mu+a+1)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   \\left(\\frac{s}{4}\\right)^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) .\n\\label{HE_Sexp}\n\\end{multline}\n\\end{corollary}\n\nSome examples of exceptional cases not covered by the preceding corollary\nare the following. They are obtained by taking the hard edge limit of\n(\\ref{LUE_Eexp_pole0}) and (\\ref{LUE_Eexp_pole1}). \n\\begin{corollary}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ and $ \\mu+a=0 $ we have \n\\begin{multline}\n\\tilde{E}^{\\rm hard}_{2}(s;a,\\mu=-a;\\xi) = 1\n  \\\\\n  +\\Big\\{\n   -1+\\frac{\\pi a}{\\sin\\pi a}e^{-\\pi ia}(1-\\xi)\n     +a[2\\psi(2)+\\psi(1)-\\psi(1-a)-\\log(s\/4)] \\Big\\}\\frac{s}{4}\n  \\\\ +{\\rm o}(s) ,\n\\label{HE_Eexp0}\n\\end{multline}\nwhilst for $ \\mu+a=1 $ we have\n\\begin{multline}\n\\tilde{E}^{\\rm hard}_{2}(s;a,\\mu=1-a;\\xi) = 1+(a-1)\\frac{s}{4}\n  \\\\\n  +\\frac{a(a-1)}{4} \\Big\\{\n   \\frac{\\pi}{\\sin\\pi a}e^{-\\pi ia}(1-\\xi)\n     +2\\psi(3)+\\psi(2)-\\psi(2-a)-\\log(s\/4) \\Big\\}\\left(\\frac{s}{4}\\right)^2\n  \\\\ +{\\rm o}(s^2) .\n\\label{HE_Eexp1}\n\\end{multline}\n\\end{corollary}\n\nTo compare these results to the small independent variable expansions given by \nJimbo in the theory of ${\\rm III^{\\prime}}\\;$, we must first undertake some preliminary\ncalculations as the equation (\\ref{PIII_sigma}) is not directly studied in \n\\cite{Ji_1982}. Rather the equation studied is\n\\begin{equation}\n  (t\\zeta'')^2 = 4\\zeta'(\\zeta' -1)(\\zeta - t\\zeta')\n  + \\left( \\frac{v_1+v_2}{2}-v_1\\zeta'\\right)^2 ,\n\\label{PIII_zeta}\n\\end{equation}\nwhere we have identified $ \\theta_0=-v_2 $, $ \\theta_{\\infty}=-v_1 $ \n($ \\theta_0, \\theta_{\\infty} $ are the parameters appearing in \\cite{Ji_1982}).\nIn terms of $ \\zeta(t) $ the $ \\tau $-function $ \\tau_{\\rm III'}(t) $ is specified\nby the requirement that\n\\begin{equation}\n  \\zeta(t) = t\\frac{d}{dt}\\log\\tau_{\\rm III'}(t)+\\frac{v_2^2-v_1^2}{4}+t ,\n\\label{PIII_zeta_tau}\n\\end{equation}\nand it is the small $ t $ expansion of $ \\tau_{\\rm III'}(t) $ presented in \n\\cite{Ji_1982}. Comparison of (\\ref{PIII_zeta}) and (\\ref{PIII_sigma}) shows that\n\\begin{equation}\n  \\zeta(t) = -\\sigma_{\\rm III'}(s)+\\frac{v_1(v_2-v_1)}{4}+\\frac{s}{4} ,\n  \\qquad t = \\frac{s}{4} .\n\\label{HE_zeta}\n\\end{equation}\nRecalling (\\ref{HE_E.2}), (\\ref{HE_E}), (\\ref{HE_zeta}) and (\\ref{PIII_zeta_tau}) \nwe see\n\\begin{equation}\n \\tilde{E}^{\\rm hard}_{2}(s;a,\\mu;\\xi) = t^{(v_2^2-v_1^2)\/4}\\tau_{\\rm III'}(t) .\n\\label{HE_tau}\n\\end{equation}\n\nIn \\cite{Ji_1982} the most general small $ t $ expansion of $ \\tau_{\\rm III'}(t) $\nas permitted by (\\ref{PIII_zeta}) is presented. It reads\n\\begin{multline}\n  \\tau_{\\rm III'}(t) = Ct^{(\\sigma^2-v^2_2)\/4}\n       \\Bigg\\{ 1 + \\frac{v_1v_2-\\sigma^2}{2\\sigma^2}t\n   \\\\\n   - u\n      \\frac{\\Gamma^2(-\\sigma)}{\\Gamma^2(2+\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle v_2+\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle v_1+\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle v_2-\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n      t^{1+\\sigma}\n   \\\\\n   - \\frac{1}{u}\n      \\frac{\\Gamma^2(\\sigma)}{\\Gamma^2(2-\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle v_2-\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle v_2+\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle v_1+\\sigma}{\\displaystyle 2})}\n      t^{1-\\sigma}\n   \\\\\n   + {\\rm O}(|t|^{2(1-\\Re(\\sigma))}) \\Bigg\\} ,\n\\label{PIII_tExp_0}\n\\end{multline}\nwhere as in (\\ref{PV_sExp_0}) $ C $ is a normalisation, while $ u $ and $ \\sigma $\nare arbitrary parameters. This result was established under the assumptions that\n$ \\frac{1}{2}(v_1\\pm\\sigma) \\notin\\mathbb Z $ and $ \\frac{1}{2}(v_2\\pm\\sigma) \\notin\\mathbb Z $\nalong with $ 0 < \\Re\\sigma < 1 $ (for $ \\sigma=0 $ a distinct solution is given).\n\nTo see that this structure is consistent with (\\ref{HE_Eexp}) and (\\ref{HE_tau}),\nrecalling (\\ref{HE_IIIparam}) we see that for the right hand side of \n(\\ref{HE_tau}) to tend to $ 1 $ as $ t $ tends to zero we must have $ C=1 $ and\n$ \\sigma=\\pm v_1 $. Again this is a violation of first condition given above but we\nconjecture that the formulae have meaning under the following limiting procedure and\nare correct. Choosing the positive sign for definiteness, and then writing \n\\begin{equation*}\n   \\frac{u}{\\Gamma(\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n  = \\frac{u(v_1-\\sigma)}{2\\Gamma(1+\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n\\end{equation*}\nwe see that requiring \n\\begin{equation*}\n  \\frac{u}{2}(v_1-\\sigma) \\to \\tilde{u}\\frac{\\sin\\pi v_1}{\\pi}\n  \\quad {\\rm as} \\quad \\sigma \\to v_1 ,\n\\end{equation*}\n(\\ref{PIII_tExp_0}) reads\n\\begin{multline}\n  \\tau_{\\rm III'}(t) \\sim t^{(v_1^2-v^2_2)\/4}\n       \\Bigg\\{ 1 + \\frac{v_1v_2-v_1^2}{2v_1^2}t\n   \\\\\n   + \\tilde{u}\n      \\frac{\\sin\\pi(v_1-v_2)\/2}{\\sin\\pi v_1}\n      \\frac{\\Gamma(1-\\frac{\\displaystyle v_2-v_1}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle v_2+v_1}{\\displaystyle 2})}\n           {\\Gamma^2(2+v_1)\\Gamma(1+v_1)}\n      \\left( \\frac{t}{4}\\right)^{1+v_1} \\Bigg\\} .\n\\label{PIII_tExp_0.2}\n\\end{multline}\nRecalling again (\\ref{HE_IIIparam}) and (\\ref{HE_tau}) we see that this agrees\nwith (\\ref{HE_Eexp}) provided\n\\begin{equation}\n  \\tilde{u}\\frac{\\sin\\pi\\mu}{\\sin\\pi(a+\\mu)}\n  = (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(a+\\mu)} ,\n\\label{HE_u}\n\\end{equation}\n(cf. (\\ref{LUE_u})).\n\n\\section{Application}\\label{S5}\n\\setcounter{equation}{0}\nIn a recent work relating to the application of random matrix theory to the\nstudy of moments of the derivative of the Riemann zeta-function, Conrey,\nRubinstein and Snaith \\cite{CRS_2005} obtained two asymptotic expressions\nassociated with the derivative of characteristic polynomials for random unitary\nmatrices. With $ U $ a Haar distributed element of the unitary group $ U(N) $,\nand $ e^{i\\theta_1},\\ldots,e^{i\\theta_N} $ its eigenvalues, let\n\\begin{equation}\n  \\Lambda_A(s) = \\prod^N_{j=1}(1-se^{-i\\theta_j}) ,\n\\label{UcharP}\n\\end{equation}\nand\n\\begin{equation}\n  {\\mathcal Z}_A(s) = e^{-\\pi iN\/2}e^{i\\sum^N_{n=1}\\theta_n\/2}s^{-N\/2}\\Lambda_A(s) ,\n\\label{UcharZ}\n\\end{equation}\n(note that $ {\\mathcal Z}_A(e^{i\\theta}) $ is real for $ \\theta $ real). In terms\nof this notation, the two results from \\cite{CRS_2005} are\n\\begin{equation}\n  \\langle |\\Lambda'_A(1)|^{2k} \\rangle_{A\\in U(N)} \\mathop{\\sim}\\limits_{N \\to \\infty}\n  b_k N^{k^2+2k} ,\n\\label{LcharM.1}\n\\end{equation}\nwhere\n\\begin{multline}\n b_k = (-1)^{k(k+1)\/2} \\sum^{k}_{h=0}{{k}\\choose{h}}(k+h)!\n \\\\\n \\times\n [x^{k+h}]\\left( e^{-x}x^{-k^2\/2}\n          \\det[I_{\\alpha+\\beta-1}(2\\sqrt{x})]_{\\alpha,\\beta=1,\\ldots,k}\\right) ,\n\\label{LcharM.1aux}\n\\end{multline}\nand\n\\begin{equation}\n  \\langle |{\\mathcal Z}'_A(1)|^{2k} \\rangle_{A\\in U(N)} \\mathop{\\sim}\\limits_{N \\to \\infty}\n  b'_k N^{k^2+2k} ,\n\\label{ZcharM.1}\n\\end{equation}\nwhere\n\\begin{equation}\n b'_k = (-1)^{k(k+1)\/2} (2k)!\n [x^{2k}]\\left( e^{-x\/2}x^{-k^2\/2}\n         \\det[I_{\\alpha+\\beta-1}(2\\sqrt{x})]_{\\alpha,\\beta=1,\\ldots,k}\\right) .\n\\label{ZcharM.1aux}\n\\end{equation}\nIn (\\ref{LcharM.1aux}) and (\\ref{ZcharM.1aux}) the notation $ [x^p]f(x) $ denotes \nthe coefficient of $ x^p $ in $ f(x) $.\n\nThe relevance of these formulae to the present study is that the determinant \ntherein can be identified in terms of $ \\tilde{E}^{\\rm hard}_{2} $. Thus, we have\nshown in a previous study \\cite{FW_2002a} that for $ a \\in \\mathbb Z_{\\geq 0} $\n\\begin{equation}\n \\tilde{E}^{\\rm hard}_{2}(s;a,\\mu;\\xi=1)\n  = A(a,\\mu)\\left(\\frac{2}{\\sqrt{s}}\\right)^{a\\mu}e^{-s\/4}\n                           \\det[I_{\\mu+\\alpha-\\beta}(\\sqrt{s})]_{\\alpha,\\beta=1,\\ldots,a} .\n\\label{HE_bessel}\n\\end{equation}\nwhere\n\\begin{equation}\n  A(a,\\mu) = a!\\prod^a_{j=1}\\frac{(j+\\mu-1)!}{j!} .\n\\label{.1}\n\\end{equation}\nInterchanging row $ \\beta $ by row $ a-\\beta+1 $ ($ \\beta=1,\\ldots,a $ in order) \nwe see from this that\n\\begin{equation}\n\\begin{split}\n  b_k  & = \\frac{(-1)^k}{A(k,k)}\\sum^k_{h=0} {{k}\\choose{h}}(k+h)!\n          [x^{k+h}]\\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1)\n  \\\\\n  b'_k & = \\frac{(-1)^k}{A(k,k)}(2k)!\n          [x^{2k}]\\left(  e^{x\/2}\\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) \\right)\n\\end{split}\n\\label{:a}\n\\end{equation}\nNote that the Painlev\\'e ${\\rm III^{\\prime}}\\;$ parameters appearing in this solution are \n$ \\mu=a=k\\in\\mathbb N $ and $ \\mu+a=2k\\in2\\mathbb N $ and thus we are dealing with the exceptional \ncase of indeterminacy referred to in Section 2. However as was noted there the \ngeneric formulae still hold with to the modifications discussed and in particular \nthe $\\sigma$-function has a small argument expansion of a purely analytic form. \n\nFrom (\\cite{FW_2003b}) it is known that the determinants in (\\ref{LcharM.1aux})\nand (\\ref{ZcharM.1aux}) can also be expressed as a particular generalised \nhypergeometric function. Such an observation implies, for instance, that\n\\begin{equation}\n x^{-k^2\/2}\\det[I_{\\alpha+\\beta-1}(2\\sqrt{x})]_{\\alpha,\\beta=1,\\ldots,k} \n   = \\prod^k_{j=1}\\frac{j!}{\\Gamma(j+k)}\n     {{}^{\\vphantom{(1)}}_0}F^{(1)}_1(;2k;x_1,\\ldots,x_k)|_{x_j=x} ,\n\\label{.b}\n\\end{equation} \nwhere $ {{}^{\\vphantom{(1)}}_0}F^{(1)}_1(;c;x_1,\\ldots,x_k) $ has a series\ndevelopment about $ x_1,\\ldots,x_k=0 $ with an explicitly given coefficient for an\narbitrary term.\nHowever this is not a practical or efficient way to compute the coefficients \nrequired in (\\ref{LcharM.1aux}) or (\\ref{ZcharM.1aux}) for moderate or large $ k $\nas it involves the hook lengths of Young diagrams associated with the partitions\nof $ k $. \n\nAccording to (\\ref{HE_E}), (\\ref{PIII_sigma}) and (\\ref{HE_Sexp})\n\\begin{equation}\n \\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) \n = \\exp\\left( -\\int^{4x}_0\\frac{ds}{s}\\;(\\sigma_{\\rm III'}(s)+k^2)\\right) ,\n\\label{Dzeta_tau}\n\\end{equation}\nwhere $ \\sigma_{\\rm III'}(s) $ satisfies the particular $ \\sigma $-Painlev\\'e \n${\\rm III^{\\prime}}\\;$ equation\n\\begin{equation}\n  (s\\sigma''_{{\\rm III}'})^2 \n  + \\sigma'_{{\\rm III}'}(4\\sigma'_{{\\rm III}'}-1)(\\sigma_{{\\rm III}'}-s\\sigma'_{{\\rm III}'})\n  - \\frac{k^2}{16} = 0 ,\n\\label{Dzeta_PIIIsigma}\n\\end{equation}\nsubject to the boundary condition\n\\begin{equation}\n  \\sigma_{\\rm III'}(s) \\mathop{\\sim}\\limits_{s \\to 0} -k^2+\\frac{s}{8}+{\\rm O}(s^2),\n  \\quad k\\in\\mathbb N .\n\\label{Dzeta_PIIIBC}\n\\end{equation}\nSubstituting \n\\begin{equation}\n  \\sigma_{\\rm III'}(s) = \\eta(s)+\\frac{s}{8} ,\n\\label{Dzeta_xfm}\n\\end{equation}\n(\\ref{Dzeta_PIIIsigma}) reads\n\\begin{equation}\n  (s\\eta'')^2 + 4((\\eta')^2-\\frac{1}{64})(\\eta-s\\eta') - \\frac{k^2}{4^2} = 0 .\n\\label{Dzeta_PIII.2}\n\\end{equation}\nWe see immediately that $ \\eta(s) $ can be expanded in an even function of $ s $ about\n$ s=0 $,\n\\begin{equation}\n  \\eta(s) = \\sum^{\\infty}_{n=0}c_ns^{2n}, \\qquad c_0=-k^2,\n  \\quad k\\in\\mathbb N .\n\\label{Dzeta_Exp}\n\\end{equation}\nMoreover the coefficients can be computed by a recurrence relation.\n\n\\begin{proposition}\\label{Dzeta_recur}\nSubstituting (\\ref{Dzeta_Exp}) in (\\ref{Dzeta_PIII.2}) shows\n\\begin{equation}\n  c_1 = \\frac{1}{64(4k^2-1)} ,\n\\label{Dzeta_c1}\n\\end{equation}\nwhile for $ p \\geq 2 $\n\\begin{multline}\n c_p = \\frac{1}{2c_1p(2p-1)+(2p-1)\/64-8pk^2c_1}\n \\\\\n \\times\\Bigg( 4k^2\\sum^{p-2}_{l=1}(l+1)(p-l)c_{l+1}c_{p-l}\n \\\\\n             -\\sum^{p-2}_{l=1}(l+1)(p-l)(2l+1)(2p-2l+1)c_{l+1}c_{p-l}\n \\\\\n             -\\sum^{p-1}_{l=1}(1-2l)c_{l}A_{p-l-1} \\Bigg) ,\n\\label{Dzeta_cp}\n\\end{multline}\nwhere\n\\begin{equation}\n  A_q = \\sum^{q}_{l=0}(l+1)(q-l+1)c_{l+1}c_{q-l+1} .\n\\label{Dzeta_aux}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWith $ h_l := (l+1)(2l+1)c_{l+1} $ we see\n\\begin{equation}\n  (s\\eta'')^2 = 4\\sum^{\\infty}_{p=1}H_{p-1}s^{2p}, \\qquad\n  H_p = \\sum^{p}_{l=0}h_lh_{p-l} ,\n\\label{Dzeta_aux1}\n\\end{equation}\nand similarly with $ a_l := (l+1)c_{l+1} $ we have\n\\begin{equation*}\n  (s\\eta')^2 = 4s^2\\sum^{\\infty}_{p=0}A_{p}s^{2p}, \\qquad\n  A_p = \\sum^{p}_{l=0}a_la_{p-l} .\n\\end{equation*}\nIt follows from this latter result that\n\\begin{equation}\n  \\left( (\\eta')^2-\\frac{1}{64} \\right)(\\eta-s\\eta')  = \\sum^{\\infty}_{p=0}G_{p}s^{2p} ,\n\\label{Dzeta_aux2}\n\\end{equation}\nwhere\n\\begin{equation*}\n  G_p = \\sum^{p}_{l=0}(1-2l)c_lb_{p-l}, \\quad\n  b_0 = -\\frac{1}{64}, \\quad b_p = 4A_{p-1} \\quad (p\\geq 1) .\n\\end{equation*}\nSubstituting (\\ref{Dzeta_aux1}) and (\\ref{Dzeta_aux2}) in (\\ref{Dzeta_PIII.2})\nand equating like coefficients of $ s^{2p} $ to zero shows that for $ p\\geq 1 $\n\\begin{equation*}\n  H_{p-1}+G_{p} = 0 .\n\\end{equation*}\nThis for $ p=1 $ implies (\\ref{Dzeta_c1}), and for $ p>1 $ implies (\\ref{Dzeta_cp}).\n\\end{proof}\n\nUsing Proposition \\ref{Dzeta_recur} it is straightforward to calculate, via \ncomputer algebra, the first $ k $ coefficients in (\\ref{Dzeta_Exp}) for any \nparticular value of $ k $. Furthermore use of computer algebra gives the \npower series up to $ x^{2k} $ of \n\\begin{equation*}\n  \\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) \\quad{\\rm and}\\quad\n  e^{x\/2}\\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) ,\n\\end{equation*}\naccording to (\\ref{Dzeta_tau}). From these power series the formulae \n(\\ref{:a}) are used to compute $ b_k $ and $ b'_k $. In \\cite{CRS_2005} the\nfirst 15 values of both $ b_k $ and $ b'_k $ were tabulated. This can be rapidly\nextended using the present method. However the resulting rational numbers \nquickly become unwieldy to record. Let us then be content by presenting just the\n16th member of the sequences,\n\\begin{equation*}\n  b_{16} = \\frac{\\scriptstyle \n307\n\\cdot\n23581\n\\cdot\n92867\n\\cdot\n760550281759\n             }\n             {\\scriptstyle \n2^{272}\n\\cdot\n3^{130}\n\\cdot\n5^{66}\n\\cdot\n7^{42}\n\\cdot\n11^{24}\n\\cdot\n13^{21}\n\\cdot\n17^{16}\n\\cdot\n19^{14}\n\\cdot\n23^{10}\n\\cdot\n29^{6}\n\\cdot\n31^{5}\n\\cdot\n37^{3}\n\\cdot\n41^{2}\n\\cdot\n43^{2}\n\\cdot\n47\n\\cdot\n53\n\\cdot\n59\n\\cdot\n61\n             } ,\n\\end{equation*}\n\\begin{equation*}\n  b'_{16} = \\frac{\\scriptstyle \n4148297603\n\\cdot\n7623077808870586151748455369217213506671334530597\n              }\n              {\\scriptstyle\n2^{264}\n\\cdot\n3^{133}\n\\cdot\n5^{66}\n\\cdot\n7^{42}\n\\cdot\n11^{25}\n\\cdot\n13^{21}\n\\cdot\n17^{16}\n\\cdot\n19^{14}\n\\cdot\n23^{11}\n\\cdot\n29^{7}\n\\cdot\n31^{6}\n\\cdot\n37^{3}\n\\cdot\n41^{2}\n\\cdot\n43^{2}\n\\cdot\n47\n\\cdot\n53\n\\cdot\n59\n\\cdot\n61\n              } .\n\\end{equation*}\n\n\\section{Acknowledgements}\nThis work was supported by the Australian Research Council. PJF thanks M. Rubinstein\nfor relating the results of \\cite{CRS_2005} before publication, and for the \norganisers of the Newton Institute program `Random matrix approaches in number theory'\nheld in the first half of 2004 for making this possible. NSW wishes to thank the \norganisers of the CRM program `Random Matrices, Random Processes and Integrable Systems'\nheld in Montreal 2005 for the opportunity to attend the program and in particular the \nhospitality of John Harnad during his stay.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nIn the field of machine learning, when there is a large number of features, determining the smallest subset that exhibits the strongest effect often decreases model complexity and increases prediction accuracy.\nThis process is called feature selection.\\footnote{Feature selection is also known as variable selection, attribute selection, and variable subset selection.} There are two well-known types of feature selection methods: filters and wrappers.\nFilters select features based on their relevance to the response variable independently of the prediction model.\nWrappers select features that increase the prediction accuracy of the model.\nHowever, as with any design decision during the construction of a prediction model, one needs to evaluate different feature selection methods in order to choose one, and above all to assess whether it is needed or not.\n\nWhile there has been extensive research on the impact of feature selection on prediction models in different domains, our investigation reveals that it is a rarely studied topic in the domain of bug prediction.\nFew studies explore how feature selection affects the accuracy of classifying software entities into buggy or clean~\\cite{Shiv13a}\\cite{Gao2011a}\\cite{Cata09b}\\cite{Kris11a}\\cite{Wang12a}\\cite{Khos10a}\\cite{Khos14a}\\cite{Ghot17a}, but to the best of our knowledge no dedicated study exists on the impact of feature selection on the accuracy of predicting the number of bugs.\nAs a result of this research gap, researchers often overlook feature selection and provide their prediction models with all the metrics they have on a software project or in a dataset.\nWe argue that feature selection is a mandatory step in the bug prediction pipeline and its application might alter previous findings in the literature, especially when it comes to comparing different machine learning models or different software metrics.\n\nIn this paper we treat bug prediction as a regression problem where a bug predictor predicts the number of bugs in software entities as opposed to classifying software entities as buggy or clean.\nWe investigate the impact of filter and wrapper feature selection methods on the prediction accuracy of five machine learning models: K-Nearest Neighbour, Linear Regression, Multilayer Perceptron, Random Forest, and Support Vector Machine.\nMore specifically, we carry out an empirical study on five open source Java projects: Eclipse JDT Core, Eclipse PDE UI, Equinox, Lucene, and Mylyn to answer the following research questions:\n\\vspace{0.2cm}\n\\\\\\emph{RQ1: How does feature selection impact the prediction accuracy?}\nOur results show that applying correlation-based feature selection (CFS) improves the prediction accuracy in 32\\% of the experiments, degrades it in 24\\%, and keeps it unchanged in the rest.\nOn the other hand, applying the wrapper feature selection method improves prediction accuracy by up to 33\\% in 76\\% of the experiments and never degrades it in any experiment. However, the impact of feature selection varies depending on the underlying machine learning model as different models vary in their sensitivity to noisy, redundant, and correlated features in the data.\nWe observe zero to negligible effect in the case of Random Forest models.\n\\vspace{0.2cm}\n\\\\\\emph{RQ2: Are wrapper feature selection methods better than filters?}\nWrapper feature selection methods are consistently either better than or similar to CFS.\nApplying wrapper feature selection eliminates noisy and redundant features and keeps only relevant features for that specific project, increasing the prediction accuracy of the machine learning model.\n\\vspace{0.2cm}\n\\\\\\emph{RQ3: Do different methods choose different feature subsets?}\nWe realize there is no optimal feature subset that works for every project and feature selection should be applied separately for each new project.\nWe find that not only different methods choose different feature subsets on the same projects, but also the same feature selection method chooses different feature subsets for different projects.\nInterestingly however, all selected feature subsets include a mix of change and source code metrics.\n\\vspace{0.2cm}\n\nIn summary, this paper makes the following contributions:\n\\begin{enumerate}\n\\item A detailed comparison between filter and wrapper feature selection methods in the context of bug prediction as a regression problem.\n\\item A detailed analysis on the impact of feature selection on five widely-used machine learning models in the literature.\n\\item A comparison between the selected features by different methods.\n\\end{enumerate}\n\nThe rest of this paper is organized as follows: In \\autoref{background}, we give a technical background about feature selection in machine learning.\nWe motivate our work in \\autoref{motivation}, and show how we are the first to study wrapper feature selection methods when predicting the number of bugs.\nIn \\autoref{empirical}, we explain the experimental setup, discuss the results of our empirical study, and elaborate on the threats to validity of the results.\nFinally, we discuss the related work in \\autoref{relatedWork} showing how our findings are similar or different from the state of the art, then conclude this paper in \\autoref{conclusions}.\n\n\n\n\\begin{figure}\n\\center{\\includegraphics[width=1.0\\linewidth]{.\/img\/biasvariance.pdf}}\n\\caption{The relationship between model complexity and model error \\cite{BiasVariance}.}\n\\label{fig:complexity}\n\\end{figure}\n\n\\section{Technical Background}\n\\label{background}\nTrained on bug data and software metrics, a bug predictor is a machine learning model that predicts defective software entities using software metrics.\nThe software metrics are called the independent variables or the features.\nThe prediction itself is called the response variable or the dependent variable.\nIf the response variable is the absence\/presence of bugs then bug prediction becomes a classification problem and the machine learning model is called a classifier.\nIf the response variable is the number of bugs in a software entity then bug prediction is a regression problem and the model is called a regressor.\n\nFeature selection is an essential part in any machine learning process.\nIt aims at removing irrelevant and correlated features to achieve better accuracy, build faster models with stable performance, and reduce the cost of collecting features for later models.\nModel error is known to be increased by both noise \\cite{Atla11a} and feature multicollinearity \\cite{Alle97c}.\nDifferent feature selection algorithms eliminate this problem in different ways.\nFor instance, correlation based filter selection chooses features with high correlation with the response variable and low correlation with each other.\n\nAlso when we build a prediction model, we often favour less complex models over more complex ones due to the known relationship between model complexity and model error, as shown in Figure \\autoref{fig:complexity}.\nFeature selection algorithms try to reduce model complexity down to the sweet spot where the total error is minimal.\nThis point is called the optimum model complexity.\nModel error is computed via the mean squared error (MSE) as: \n\\\\$MSE = \\frac{1}{N} \\sum_{i=1}^{N} (\\hat{Y}_i - Y_i)^2$ \\\\where $\\hat{Y}_i$ is the predicted value and $Y_i$ is the actual value.\nMSE can be decomposed into model bias and model variance as: \\\\$MSE = Bias^2 + Variance + Irreducible Error $ \\cite{Hast05a}\n\nBias is the difference between the average prediction of our model to the true unknown value we are trying to predict.\nVariance is the variability of a model prediction for a given data point.\nAs can be seen in Figure \\autoref{fig:complexity}, reducing model complexity increases the bias but decreases the variance.\nFeature selection sacrifices a little bit of bias in order to reduce variance and, consequently, the overall MSE.\n\nEvery feature selection method consists of two parts: a search strategy and a scoring function.\nThe search strategy guides the addition or removal of features to the subset at hand and the scoring function evaluates the performance of that subset.\nThis process is repeated until no further improvement is observed.\n\n\\begin{table}\n\\scriptsize\n\\caption{The CK Metrics Suite \\cite{Chid94a} and other object-oriented metrics included as the source code metrics in the bug prediction dataset \\cite{DAmb10c}}\n\\begin{center}\n\\begin{tabular}{ll} \n{Metric Name}\t\t& {Description}\\\\ \\hline\nCBO   & Coupling Between Objects \\\\[0.05cm] \nDIT   & Depth of Inheritance Tree \\\\[0.05cm] \nFanIn   & Number of classes that reference the class \\\\[0.05cm] \nFanOut   & Number of classes referenced by the class \\\\[0.05cm] \nLCOM   & Lack of Cohesion in Methods \\\\[0.05cm] \nNOC   & Number Of Children \\\\[0.05cm] \nNOA   & Number Of Attributes in the class \\\\[0.05cm] \nNOIA   & Number Of Inherited Attributes in the class \\\\[0.05cm] \nLOC   & Number of lines of code \\\\[0.05cm] \nNOM   & Number Of Methods \\\\[0.05cm] \nNOIM   & Number of Inherited Methods \\\\[0.05cm] \nNOPRA   & Number Of PRivate Atributes \\\\[0.05cm] \nNOPRM   & Number Of PRivate Methods \\\\[0.05cm] \nNOPA   & Number Of Public Atributes \\\\[0.05cm] \nNOPM   & Number Of Public Methods \\\\[0.05cm] \nRFC   & Response For Class \\\\[0.05cm] \nWMC   & Weighted Method Count  \\\\\\hline\n\n\\label{tbl:sourceMetrics}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\scriptsize\n\\caption{The change metrics proposed by Moser \\emph{et al.}\\xspace \\cite{Mose08a} included in the bug prediction dataset \\cite{DAmb10c}}\n\\label{tbl:changeMetrics}\n\\begin{center}\n\\begin{tabularx}{0.48\\textwidth}{lX} \n{Metric Name}\t\t& {Description}\\\\ \\hline\nREVISIONS   & Number of reversions \\\\[0.05cm] \nBUGFIXES   & Number of bug fixes \\\\[0.05cm] \nREFACTORINGS   & Number Of Refactorings \\\\[0.05cm] \nAUTHORS   & Number of distinct authors that checked a file into the repository \\\\[0.05cm] \nLOC\\_ADDED   & Sum over all revisions of the lines of code added to a file \\\\[0.05cm] \nMAX\\_LOC\\_ADDED   & Maximum number of lines of code added for all revisions \\\\[0.05cm] \nAVE\\_LOC\\_ADDED   & Average lines of code added per revision \\\\[0.05cm] \nLOC\\_DELETED   & Sum over all revisions of the lines ofcode deleted from a file \\\\[0.05cm] \nMAX\\_LOC\\_DELETED   & Maximum number of lines of code deleted for all revisions \\\\[0.05cm] \nAVE\\_LOC\\_DELETED   & Average lines of code deleted per revision \\\\[0.05cm] \nCODECHURN   & Sum of (added lines of code - deleted lines of code) over all revisions \\\\[0.05cm] \nMAX\\_CODECHURN   & Maximum CODECHURN for all revisions \\\\[0.05cm] \nAVE\\_CODECHURN   & Average CODECHURN for all revisions \\\\[0.05cm] \nAGE   & Age of a file in weeks (counting backwards from a specific release) \\\\[0.05cm] \nWEIGHTED\\_AGE   & Sum over age of a file in weeks times number of lines added during that week normalized by the total number of lines added to that file \\\\[0.05cm] \\hline\n\n\\end{tabularx}\n\\end{center}\n\\end{table}\n\n\n\\section{Motivation}\n\\label{motivation}\n\nIn this section, we shortly discuss the importance of predicting the number of bugs in software entities.\nThen, we highlight the impact of feature selection on bug prediction and particularly motivate the need for studying the wrapper methods.\n\n\\subsection{Regression vs Classification}\nMost of the previous research treats bug prediction as a classification problem where software entities are classified as either buggy or clean, and there have been several studies on the impact of feature selection on defect classification models.\nOn the other hand, bug prediction as a regression problem is not well-studied, and the effect of feature selection on predicting the number of bugs is not well-understood.\n\nSoftware bugs are not evenly distributed and tend to cluster \\cite{Ostr04a}, and some software entities commonly have larger numbers of bugs compared to others.\nPredicting the number of bugs in each entity provides valuable insights about the quality of these software entities~\\cite{Osma16c}, which helps in prioritizing software entities to increase the efficiency of related development tasks such as testing and code reviewing~\\cite{Khos03a}.\nThis is an important quality of a bug predictor especially for cost-aware bug prediction~\\cite{Mend09b}\\cite{Aris10a}\\cite{Kame10a}\\cite{Koba11a}\\cite{Hata12a}.\nIn fact, predicting the number of bugs in software entities and then ordering these entities based on bug density is the most cost-effective option \\cite{Osma17f}.\n\n\\subsection{Dimensionality Reduction}\n\\label{reduction}\n\nWhen the dimensionality of data increases, distances grow more and alike between the vectors and it becomes harder to detect patterns in data~\\cite{Bell60a}.\nFeature selection not only eliminates the confounding effects of noise and feature multicollinearity, but also reduces the dimensionality of the data to improve accuracy.\nHowever, feature selection does not seem to be considered as important as it should be in the field of bug prediction.\nFor instance, only 25 out of the 64 studied techniques in a recent research apply feature selection before training a machine learning model~\\cite{Malh15a}.\nOnly 2 out of the 25 are applied to bug prediction as a regression problem.\n\n\\subsection{Filters vs Wrappers}\n\\label{filtersWrappers}\nFeature selection methods are of two types: wrappers and filters \\cite{Koha97a}.\nWith wrappers, the scoring function is the accuracy of the prediction model itself.\nWrappers look for the feature subset that works best with a specific machine learning model.\nThey are called wrappers because the machine learning algorithm is wrapped into the selection procedure.\nWith filters (\\emph{e.g.},\\xspace CFS, InfoGain, PCA), the scoring function is independent of the machine learning model.\nThey are called filters because the attribute set is filtered before the training phase.\nGenerally, filters are faster than wrappers but less powerful because wrappers address the fact that different learning algorithms can achieve best performance with different feature subsets.\nIn this paper we aim at finding whether there is actually a difference between filters and wrappers in bug prediction, and then quantifying this difference.\n\nWrappers are known to be computationally expensive.\nThey become a bottleneck when the size of a dataset (features + data items) becomes large.\nHowever, this rarely happens in the bug prediction and bug prediction datasets tend to be relatively small.\nThis means that although wrappers are more resource intensive, they are easily applicable to bug prediction.\nNevertheless, our literature research yielded relatively few works that use wrappers for predicting number of bugs.\n\n\n\\section{Empirical Study}\n\\label{empirical}\n\nIn this section, we investigate the effect of feature selection on the accuracy of predicting the number of bugs in Java classes.\nSpecifically, we compare five widely-used machine learning models applied to five open source Java projects to answer the following research questions:\n\\\\\\emph{RQ1: How does feature selection impact the prediction accuracy?}\n\\\\\\emph{RQ2: Are wrapper feature selection methods better than filters?}\n\\\\\\emph{RQ3: Do different methods choose different feature subsets?}\n\n\\begin{table*}[h!]\n\\renewcommand{\\arraystretch}{1.0}\n\\footnotesize\n\\caption{Details about the systems in the studied dataset, as reported by D'Ambros \\emph{et al.}\\xspace~\\cite{DAmb10c}}\n\\begin{center}\n\\begin{tabular}{llrrrr} \n\t\t\t\t&\t\t&\t\t\t\t&\t\t\t&\t\t\t\t&\\% classes with more \\\\\nSystem \t\t\t&Release\t&KLOC\t\t\t&\\#Classes  \t& \\% Buggy\t\t&than one bug\t\\\\ \\hline\n\nEclipse JDT Core\t&3.4\t\t&$\\approx 224 $\t&997\t\t\t& $\\approx 20\\%$\t& $\\approx 7\\%$\\\\ \n\t\t\t\nEclipse PDE UI\t\t&3.4.1\t&$\\approx 40 $\t\t&1,497\t\t& $\\approx 14\\%$\t & $\\approx 5\\%$  \\\\ \n\t\t\t\nEquinox\t\t\t&3.4\t\t&$\\approx 39 $\t\t&324\t\t\t& $\\approx 40\\%$   \t& $\\approx 15\\%$ \\\\ \n\t\t\t\nMylyn\t\t\t&3.41\t&$\\approx 156$\t&1,862\t\t& $\\approx 13\\%$\t& $\\approx 4\\%$    \\\\ \n\t\t\t\nLucene\t\t\t&2.4.0\t&$\\approx 146$\t&691\t\t\t& $\\approx 9\\%$  \t& $\\approx 3\\%$\\\\ \\hline\n\\label{tbl:dataset}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\subsection{Experimental Setup}\n\\subsubsection*{Dataset}\nWe adopt the ``Bug Prediction Dataset\" provided by D'Ambros \\emph{et al.}\\xspace \\cite{DAmb10c} which serves as a benchmark for bug prediction studies.\nWe choose this dataset because it is the only dataset that contains both source code and change metrics at the class level, in total 32 metrics listed in~\\autoref{tbl:sourceMetrics} and \\autoref{tbl:changeMetrics}; and also provides the number of post-release bugs as the response variable for five large open source Java systems listed in \\autoref{tbl:dataset}.\nThe other dataset that has the number of bugs as a response variable comes from the PROMISE repository, but contains only 21 source code metrics \\cite{Jure10a}.\n \n\\subsubsection*{Prediction Models}\nWe use Multi-Layer Perceptron (MLP), Random Forest (RF), Support Vector Machine (SVM), Linear Regression (LR), and an implementation of the k-nearest neighbour algorithm called IBK.\nEach model represents a different category of statistical and machine learning models that is widely used in the bug prediction research~\\cite{Malh15a}.\n\nWe use the correlation-based feature selection (CFS) method \\cite{Hall00a}, the best \\cite{Chal08b}\\cite{Ghot17a} and the most commonly-used filter method in the literature \\cite{Malh15a}.\nFor the wrapper feature selection method we use the corresponding wrapper applicable to each prediction model.\nIn other words, we use MLP wrapper for MLP, RF wrapper for RF, SVM wrapper for SVM, LR wrapper for LR, and IBK wrapper for IBK.\nEvery feature selection method also needs a search algorithm.\nWe use the \\emph{Best First} search algorithm which searches the space of feature subsets using a greedy hill-climbing procedure with a backtracking facility.\nWe use this search algorithm because it returns the results in a reasonable amount of time while being exhaustive to a certain degree.\n \nWe use the Weka data mining tool \\cite{Hall09} to build prediction models for each project in the dataset.\nFollowing an empirical method similar to that of Hall and Holmes \\cite{Hall03a}, we apply each prediction model to three feature sets: the full set, the subset chosen by CFS, and the subset chosen by the wrapper.\nThe prediction model is built and evaluated following the 10-fold cross validation procedure.\nThe wrapper feature selection is applied using a 5-fold cross validation on the training set of each fold, then the best feature set is used.\nThe CFS algorithm is applied to the whole training set of each fold.\nThen the whole process is repeated 30 times.\nWe evaluate the predictions by means of the root mean squared error (RMSE).\nIn total, we have 25 experiments.\nEach experiment corresponds to a specific project and a specific prediction model trained on the three feature sets.\n\nWe use the default hyperparameter (\\emph{i.e.},\\xspace configuration) values of Weka 3.8.0 for the used machine learning models.\nAlthough hyperparameters can be tuned \\cite{Tant16a}\\cite{Osma17a}, we do not perform this optimization because we want to isolate the effect of feature selection.\nBesides, Linear Regression does not have hyperparameters and the gained improvement of optimizing SVM and RF is negligible \\cite{Tant16a}\\cite{Osma17a}.\n\n\n\n\\subsection{Results}\n\\autoref{fig:boxplots} shows standard box plots for the different RMSE values obtained by the different feature sets per prediction model per project.\nEach box shows 50\\% of that specific population.\\footnote{By population we mean the RMSE values of a specific experiment with a specific feature set.\nEach population consists of $10 \\times 50 = 500$ data items (10-fold cross validation done 50 times)} We can see that the wrapper populations are almost always lower than the full set ones, have smaller boxes, and have fewer outliers.\nThis means that applying the wrapper gives better and more consistent predictions.\nOn the other hand, we cannot make any observations about applying CFS because the difference between the CFS populations and the full set populations are not apparent.\n\n\\begin{figure*}\n\\center{\\includegraphics[width=1.0\\linewidth]{.\/img\/boxplots.pdf}}\n\\caption{Boxplots of all the experiments in our empirical study.\nThe y-axis represents the root mean squared error (RMSE).\nFor each project\/model, we examine three feature sets: the full set, the subset chosen by the CFS filter, and the subset chosen by the wrapper corresponding to the model.}\n\\label{fig:boxplots}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\center{\n\\includegraphics[width=1.0\\linewidth]{.\/img\/effect_size.pdf}}\n\\caption{This figure shows the bar plots of the effect size of the Dunn post-hoc analysis, which is carried out at the 95\\% confidence interval.\nThe x-axis indicates the pairwise comparison and the y-axis indicates the effect size.\nThe bars are color-coded.\nIf the bar is red, this means that the difference is not statistically significant.\nGrey means that there is a statistical significant difference, but the effect is negligible.\nBlue, golden, and green indicate a small, medium, and large statistically significant effect, respectively.}\n\\label{fig:effectSize}\n\\end{figure*}\n\n\nWhile box plots are usually good to get an overview of the different populations and how they compare to each other, they do not provide any statistical evidence.\nTo get more concrete insights, we follow the two-stage statistical test: Kruskal-Wallis + Dunn post-hoc analysis, both at the 95\\% confidence interval.\nWe apply the Kruskal-Wallis test on the results to determine whether different feature subsets have different prediction accuracies (\\emph{i.e.},\\xspace different RMSE).\nOnly when this test indicates that the populations are different, can we quantify such differences with a post-hoc analysis.\nWe perform Dunn post-hoc pairwise comparisons and analyze the effect size between each two populations.\n\\autoref{fig:effectSize} shows on the y-axis the detailed effect size between the two compared RMSE populations on the x-axis.\nIn this plot, there are two possible scenarios:\n\\begin{enumerate}\n\\item The Kruskal-Wallis test indicates that there is no statistical difference between the populations.\nThen all the bars are red to show that there is no effect between any two populations.\n\\item The Kruskal-Wallis test indicates a statistically significant difference between the populations.\nThen the color of the bars encode the pairwise effect size.\nRed means no difference and the two populations are equivalent.\nGrey means that there is a significant difference but can be ignored due to the negligible effect size.\nBlue, golden, and green mean small, medium, and large effect size respectively.\n\\end{enumerate}\n\nTo see how feature selection methods impact the prediction accuracy (\\emph{RQ1}), we compare the RMSE values obtained by applying CFS and wrappers with those obtained by the full feature set.\nWe observe that the RMSE value obtained by the CFS feature subset is statistically lower than the full set in 8 experiments (32\\%),\\footnote{negative non-red effect size in \\autoref{fig:effectSize}} statistically higher in other 6 experiments (24\\%),\\footnote{positive non-red effect size in \\autoref{fig:effectSize}} and statistically equivalent in 11 experiments (44\\%).\\footnote{red effect size in \\autoref{fig:effectSize}} Although CFS can decrease the RMSE by 24\\% on average (MLP with Mylyn), it can increase it by up to 24\\% (SVM with Lucene).\nWe also notice that applying CFS is not consistent within experiments using the same model.\nIt does not always improve, or always degrade, or always retain the performance of any model throughout the experiments.\nWe conclude that CFS is unreliable and gives unstable results.\nFurthermore, even when CFS reduces the RMSE, the effect size is at most small.\n\nOn the other hand, the RMSE value of the wrapper feature subset is statistically lower than that of the full set in 19 experiments (76\\%)  and statistically equivalent in the rest.\nApplying the wrapper feature selection method can decrease RMSE of a model by up to 33\\% (MLP with Eclipse JDT).\nWe also observe that the impact of the wrapper feature selection method on the accuracy is different from one model to another.\nIt has a non-negligible improvement on the prediction accuracy of IBK, LR, MLP, RF, and SVM in 80\\%, 60\\%, 100\\%, 20\\%, and 20\\% of the experiments, respectively.\nThis is due to the fact that different machine learning models are different in their robustness against noise and multicollinearity.\nMLP, IBK, and LR were improved significantly almost always in our experiments.\nOn the other hand, SVM and RF were not improved as often, because they are known to be resistant to noise, especially when the number of features is not too high.\nRF is an ensemble of decision trees created by using bootstrap samples of the training data and random feature selection in tree induction \\cite{Brei01a}.\nThis gives RF the ability to work well with high-dimensional data and sift the noise away.\nThe SVM algorithm is also designed to operate in a high-dimensional feature space and can automatically select relevant features \\cite{Hsu03a}.\nIn fact, this might be the reason behind the proven record of Random Forest and Support Vector Machine in bug prediction \\cite{Guo04a}\\cite{Elis08a}.\n\n\n\\begin{table}[]\n\\renewcommand{\\arraystretch}{1.2}\n\\normalsize\n\\caption{The level of agreement between different feature selection methods in each project}\n\\begin{center}\n\\begin{tabular}{lll} \nProject \t\t\t& $k$ \t&Agreement \\\\ \\hline\nEclipse JDT Core\t&0.18\t& Slight\\\\ \n\t\t\t\nEclipse PDE UI\t\t&0.17\t & Slight   \\\\ \n\t\t\t\nEquinox\t\t\t&0.40\t  & Fair  \\\\ \n\t\t\t\nMylyn\t\t\t&0.08\t & Slight   \\\\ \n\t\t\t\nLucene\t\t\t&0.18\t& Slight  \\\\ \\hline\n\\label{tbl:kappaOfProjects}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[]\n\\renewcommand{\\arraystretch}{1.2}\n\\normalsize\n\\caption{The level of agreement between the feature subsets selected by each method over all projects}\n\\begin{center}\n\\begin{tabular}{lll} \nFeature Selection Method \t\t\t& $k$ \t&Agreement \\\\ \\hline\nCFS\t\t\t\t\t\t\t\t&0.23\t& Fair\\\\ \n\t\t\t\nIBK Wrapper\t\t\t\t\t\t&0.26\t & Fair   \\\\ \n\t\t\t\nLR Wrapper\t\t\t\t\t\t&0.16\t  & Slight  \\\\ \n\t\t\t\nMLP Wrapper\t\t\t\t\t\t&0.04\t & Slight   \\\\ \n\nRF Wrapper\t\t\t\t\t\t&0.04\t & Slight   \\\\ \n\t\t\t\nSVM Wrapper\t\t\t\t\t\t&-0.01\t& Poor  \\\\ \\hline\n\n\\label{tbl:kappaOfMethods}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe wrapper method is statistically better than CFS in 18 experiments, statistically equivalent in 6 experiments, and worse in one experiment, but with a negligible effect size.\nThese results along with the fact that CFS sometimes increases the RMSE, clearly show that the wrapper selection method is a better choice than CFS (\\emph{RQ2}).\n\n\\begin{figure*}[]\n     \\begin{center}\n        \\subfigure[]{%\n            \\label{fig:selectedFeaturesGrid}\n            \\includegraphics[width=0.9\\textwidth]{.\/img\/selectedFeaturesGrid2.pdf}}\n\n        \\subfigure[]{%\n            \\label{fig:selectedFeaturesCounts}\n            \\includegraphics[width=0.9\\textwidth]{.\/img\/selectedFeaturesCounts2.pdf}}\n    \\end{center}\n\\end{figure*}\n\\begin{figure*}[]\n     \\begin{center}\n\n \\subfigure[]{%\n            \\label{fig:selectedFeaturesPerProject}\n            \\includegraphics[width=1\\textwidth]{.\/img\/selectedFeaturesPerProject.pdf}}\n\n    \\end{center}\n    \\caption{Subfigure (a) shows the features selected by each method using the whole data of each project.\nSubfigure (b) shows the number of times each feature is selected out of the 30 (1 CFS feature set + 5 wrapper sets per project).\nThe more times a feature is selected the more important it is for making accurate predictions.\nSubfigure (c) shows how different selection methods vary in the number of selected features.\nDetails about the features (metrics) are in \\autoref{tbl:sourceMetrics} and \\autoref{tbl:changeMetrics}}\n   \\label{fig:selectedFeatures}\n\\end{figure*}\n\n\n\\autoref{fig:selectedFeatures} shows the details about the features selected by each method using the whole data of each project in the dataset.\nTo answer the third research question (\\emph{RQ3}), we use the Fleiss' kappa statistical measure \\cite{Flei71a} to evaluate the level of agreement between the different feature selection methods for each project and the level of agreement of each feature selection method over the different projects. The Fleiss' kappa value, called $k$, is interpreted as follows:\n\\\\$k\\leq0 \\implies$ poor agreement\n\\\\$0.01 <k \\leq 0.20 \\implies$ Slight agreement\n\\\\$0.21 <k \\leq 0.40 \\implies$ Fair agreement\n\\\\$0.41 <k \\leq 0.60 \\implies$ Moderate agreement\n\\\\$0.61 <k \\leq 0.80 \\implies$ Substantial agreement\n\\\\$0.81 <k \\leq 1.00 \\implies$ Almost perfect agreement\n\n\nFigure \\autoref{fig:selectedFeaturesGrid} shows that different methods choose different features in each project.\nThe level of agreement between the different methods is \\emph{slight} in four projects and \\emph{fair} in only one, as detailed in \\autoref{tbl:kappaOfProjects}.\nAlso the same method chooses different features in different projects.\n\\autoref{tbl:kappaOfMethods} shows that the level of agreement between the feature subsets selected by the same method in different projects is at most \\emph{fair}.\nHowever, there exists some agreement on some features.\nFigure \\autoref{fig:selectedFeaturesCounts} shows that \\emph{REVISIONS, FanOut, NOIM,} and \\emph{FanIn} are chosen most often.\n\\emph{REVISIONS} in particular is chosen by all methods almost all the time.\nIt is selected in 28 out of 30 feature subsets,\\footnote{For each one of the 5 projects in the dataset, there are 6 feature subsets: 1 CFS subset and 5 wrapper subsets.} meaning that it has a high predictive power.\nOn the other hand, \\emph{RFC, MAX\\_LOC\\_DELETED, NOM}, and \\emph{LCOM} are picked the least number of times, which means they have little to no predictive power.\n\nThe number of discarded features varies from 10 to 28 features out of 32 in total, as detailed in Figure \\autoref{fig:selectedFeaturesPerProject}.\nThis means that most of the features can be removed while enhancing (in 48\\% of the experiments), or at least retaining (in 52\\% of the experiments), the prediction accuracy.\n\nAnother important observation is that no feature subset contains only source code metrics or only change metrics, but always a mix of both.\nThis means that no category of metrics (\\emph{i.e.},\\xspace change and source code) alone is good for predicting the number of bugs, but they should be combined to achieve better performance.\nPrevious studies show that change metrics are better than source code metrics in bug prediction and combining the two sets either does not bring any benefit \\cite{Mose08a} or hinders the performance \\cite{Aris10a}.\nHowever, these studies either did not employ feature selection at all \\cite{Mose08a} or employed only CFS \\cite{Aris10a}.\n\n\n\\subsection{Threats to Validity}\n\nAlthough we use a well-known benchmark as our dataset, the quality of our results is very dependent on the quality of that dataset.\nAlso our dependence on WEKA for building the machine learning models, makes the quality of the models dependent solely on the quality of the WEKA implementation itself.\nSince we are dealing with machine learning, the risk of overfitting always exists.\nIn general, we mitigate this risk by repeating each feature selection\/training\/testing process 50 times.\nAlso for the wrapper method, we apply the train-test-validate (TTV) approach to mitigate any overfitting bias in favour of wrappers.\\footnote{The train-test-validate is not applicable to CFS due to how CFS operates} \n\nIn our study, the dataset contains metrics only from open source software systems.\nIt is hard to generalize the results to all systems due to the differences between industrial and open source projects.\nAlso all the systems in the dataset are developed in Java and the findings might not generalize to systems developed in other languages.\nAnother important threat to external validity comes from the chosen prediction models.\nAlthough we try to pick a representative set of prediction models, our findings might not generalize to other machine learning models such as hybrid and genetic ones.\n\nMachine learning methods depend on the underlying data.\nAlthough our findings are statistically strong, they may not generalize to all open source Java systems.\nPractitioners and researchers are encouraged to run comparative studies, similar to ours, before making decisions regarding the machine learning techniques employed for bug prediction.\n\n\\section{Related Work}\n\\label{relatedWork}\n\nIn a recent systematic literature review of the machine learning techniques in the bug prediction field~\\cite{Malh15a}, Most studies (60\\%) do not apply feature selection at all.\nOur investigation shows that researchers in bug prediction often undermine the importance of feature selection.\n\nAmong the studies that apply feature selection, CFS \\cite{Hall00a} is the most commonly used \\cite{Malh15a}.\nMany studies employ the CFS filter technique before training machine learning models \\cite{Elis08a}\\cite{Kaur08a}\\cite{Aris10a} \\cite{DeCa10a}\\cite{Malh11a}\\cite{Zimm09a}\\cite{Twal11a}\\cite{Okut14a}.\nWe also apply CFS as the baseline filter technique and compare it to wrapper techniques to show that wrappers outperform this filter in most cases.\n\nMost studies that apply feature selection actually apply filter techniques like principal component analysis (PCA) \\cite{Khos06a}\\cite{DAmb10c}\\cite{Naga06a}\\cite{Kanm07a}\\cite{Sing10a}, consistency based selection (CBS) \\cite{Zimm09a}, and InfoGain \\cite{Menz07a}\\cite{Turh07a}\\cite{Sing09a}\\cite{Misi11a}.\nVery few studies apply wrapper feature selection techniques \\cite{Cahi13a}\\cite{Gond08a} when classifying software entities as buggy or clean (classification).\nHowever, to the best of our knowledge, there is no study that applies wrapper techniques when predicting the number of bugs in software entities (regression).\nIn our study we compare the CFS filter with different wrappers while treating bug prediction as a regression problem.\n\nShivaji \\emph{et al.}\\xspace \\cite{Shiv13a} study the impact of multiple feature selection techniques (filters and wrappers) on the performance of Na\\\"ive Bayes and Support Vector Machines when classifying code changes as buggy or clean.\nThey report a significant enhancement in the accuracy measures of both classifiers when feature selection is applied.\nHowever, Shivaji \\emph{et al.}\\xspace consider \\emph{``everything in the source code separated by whitespace or a semicolon\"} as a feature \\cite{Shiv13a}.\nThis includes variable names, method names, keywords, comment words, \\emph{etc.}\\xspace\nThey end up with a staggering number of features ranging from 6,127 to 41,942 features.\nThe vast majority of features in the initial feature set are bound to be irrelevant, hence the results of their study are exaggerated and cannot be generalized.\n\nChallagulla \\emph{et al.}\\xspace \\cite{Chal08b} report that performing principal component analysis before training the models does not result in a significant improvement of the performance of the bug predictors, while correlation-based feature selection (CFS) and consistency-based subset evaluation (CBS) actually decrease the prediction error of the machine learning models.\nWe actually report different results when applying CFS to regressors.\n\nKhoshgoftaar \\emph{et al.}\\xspace \\cite{Khos10a} combine filter feature selection and data sampling to achieve better classification accuracy on the package level.\nThey report an improvement of around 2\\% to the area under the receiver operator characteristic curve (AUC-ROC) when the filters are applied to sampled data rather than the whole data.\nThen Khoshgoftaar \\emph{et al.}\\xspace \\cite{Khos14a} improve the performance by repeatedly applying the sampling and filtering several times then aggregating the results.\n\nIn their defect-proneness prediction framework, Song \\emph{et al.}\\xspace \\cite{Song11a} included a special step for feature selection.\nHowever, their framework treats bug prediction as a classification problem.\nIn this paper we study the effect of wrapper feature selection methods on bug prediction as a regression problem.\n\nGao \\emph{et al.}\\xspace \\cite{Gao2011a} studied seven filter feature selection techniques.\nThey report that the studied classification models were either improved or remained unchanged while 85\\% of the original features were eliminated.\nKrishnan \\emph{et al.}\\xspace \\cite{Kris11a} analyze whether change metrics remain good predictors during the evolution of Eclipse.\nIn their analysis, they use J48 decision tree as the machine learning algorithm and refer to the top five features as the set of prominent predictors, then study the consistency of this set over the consecutive versions of Eclipse.\nThey report that there is a small subset of change metrics that is consistently good at classifying software entities as buggy or clean across products and revisions.\nWang \\emph{et al.}\\xspace \\cite{Wang12a} study the effect of removing redundant or useless features from the PROMISE dataset.\\footnote{\\url{http:\/\/openscience.us\/repo\/}}\nThey report that feature selection improves classification accuracy.\nCatal and Diri \\cite{Cata09b} explore which machine learning algorithm performs best before and after applying feature reduction.\nIn these studies, researchers treat bug prediction as a classification problem while we study the effect of feature selection on bug prediction as a regression problem.\n\nTurhan and Bener \\cite{Turh09a} argue that Na\\\"ive Bayes assumes the independence and the equal importance of features.\nThese assumptions are not true in the context of bug prediction.\nYet, Na\\\"ive Bayes is one of the best classifiers \\cite{Menz07a}.\nThey empirically show that the independence assumption is not harmful for defect prediction using Na\\\"ive Bayes and assigning weights to features increases the performance and removes the need for feature selection.\nThey conclude that \\emph{``either weighted Naive Bayes or pre-processing data with PCA may produce better results to locate software bugs in source code\"}\\cite{Turh09a}.\n\nWe confirm that most of the existing literature treats bug prediction as a classification problem, and that studying the number of bugs is neglected in the field.\nFor instance, Ostrand \\emph{et al.}\\xspace \\cite{Ostr05a} use negative binomial regression (NBR) to predict the number of bugs in software modules.\nThey report that NBR fits the bug prediction problem and demonstrates high accuracy.\nJanes \\emph{et al.}\\xspace \\cite{Jane06a} compare three count models to predict the number of bugs and find out that zero-inflated NBR performs better than Poisson regression and NBR.\nRathore and Kumar \\cite{Rath16a} investigate six different fault prediction models for predicting the number of bugs and show that count models (\\emph{i.e.},\\xspace NBR and Zero-Inflated Poisson regression) underperform compared to linear regression, decision tree regression, genetic programming and multilayer perceptron.\nGraves \\emph{et al.}\\xspace \\cite{Grav00a} build a generalized linear regression model to predict the number of bugs based on the various change metrics.\nGao and Khoshgoftaar \\cite{Gao07a} compare the performance of several count models (\\emph{e.g.},\\xspace Poisson regression) and show that Hurdle Poisson regression has the most accurate predictions.\nNevertheless, these studies do not apply feature selection and their results should be reassessed in the light of our findings.\n\n\n\\section{Conclusions}\n\\label{conclusions}\nGeneralizing bug prediction findings is hard.\nSoftware projects have different teams, cultures, frameworks, and architectures.\nConsequently, software metrics have different correlations with the number of bugs in different projects.\nThese correlations can be captured differently by distinct prediction models.\nWe argue that wrapper feature selection methods fit this problem best because they not only choose features relevant to the response variable but also to the prediction model itself.\nIndeed, our results show that wrapper feature selection is always better than CFS and improves the performance of a model (by up to 47\\%) while eliminating most of the features (up to 87\\%).\nOur results also reveal that there is little agreement on the selected features.\nEven the same method chooses different features in different projects.\nWe cannot generalize what feature subset to use, but we can recommend combining both change and source code metrics and letting the wrapper feature selection method choose the right subset.\n\n\nIn the future, we plan to investigate how the selected feature subset changes with the evolution of a certain project from the size and selection perspectives.\nLastly, while carrying out this research we realized that datasets providing the number of bugs as the response variable are scarce, which could be a hurdle to studies predicting the number of bugs in software systems.\nWe therefore encourage the community to publish such defect datasets more frequently.\n\\section*{Acknowledgments}\nThe authors gratefully acknowledge the funding of the Swiss National Science Foundations for the project ``Agile Software Analysis\" (SNF project No. 200020\\_162352, Jan 1, 2016 - Dec. 30, 2018).\\footnote{\\url{http:\/\/p3.snf.ch\/Project-162352}}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn perturbative quantum field theory, the traditional method to compute\ncross sections and distributions for a physical process involves generating\ntree and loop amplitudes via Feynman diagrams and interfering them.\nSimplifications of the expressions are performed at the analytical level.\nHere an essential part is the reduction of the typically dimensionally regularized\nloop integrals~\\cite{'tHooft:1972fi} to a small number of standard integrals.\nThis step can be performed at the amplitude level for tensor integrals or,\nafter contraction of Lorentz indices, at the level of interferences for scalar\nintegrals. Considering the case of scalar integrals,  \nintegration by parts (IBP) identities~\\cite{Tkachov:1981wb,Chetyrkin:1981qh}\nand Lorentz invariance (LI) identities~\\cite{Gehrmann:1999as} may be used\nfor a systematic reduction to a set of independent integrals, called master\nintegrals.\nThe standard reduction algorithm by Laporta~\\cite{laporta} defines an\nordering for Feynman integrals, generates identities and solves the\nresulting system of linear equations.\nAlternative methods to exploit IBP and LI identities for reductions have been\nproposed~\\cite{Smirnov:2005ky,Smirnov:2006tz,Gluza:2010ws,Schabinger:2011dz},\nsee also \\cite{Lee:2008tj,Grozin:2011mt} and references therein.\nPublic implementations of different reduction algorithms are available with the\ncomputer programs AIR~\\cite{Anastasiou:2004vj}, FIRE~\\cite{Smirnov:2008iw}\nand the first version of \\reduze~\\cite{Studerus:2009ye}.\n\nThis article presents the new public reduction program \\reduze\\;2.\nIt is written in {\\tt C++} and represents a major rewrite and extension of its\npredecessor \\reduze. In the following, the name \\reduze\\ refers to the new\nversion presented here.\n\nIn \\reduze, integrals are indexed by integral families (``auxiliary topologies'')\nand sectors (``topologies'') therein.\nFor the reduction, the program implements a fully distributed variant of\nLaporta's algorithm using the Message Passing Interface (MPI).\nIn this way, not only different sectors can be reduced in parallel, but also\nthe integrals of a single sector can be reduced in a distributed computation.\n\nThe program allows to utilize multiple integral families within a calculation.\nSpecial emphasis has been placed on finding relations between sectors\nof the same or different integral families and employing them\nto eliminate integrals.\nBesides a straightforward combinatorial matcher, the program implements\ngraph and matroid theory based algorithms to compute such relations,\ntaking into account possible crossings of external momenta.\nSimilar to the program \\diana~\\cite{Tentyukov:1999yq}, \\reduze\\ may be used\nto shift loop momenta of Feynman diagrams generated by a program like\n\\qgraf~\\cite{qgraf} or \\feynarts~\\cite{Hahn:2000kx} to match sectors of\nintegral families.\n\nOther features include the generation of differential equations for Feynman\nintegrals and the computation of (bare) amplitude interferences up to master\nintegrals, starting from Feynman diagrams generated by \\qgraf.\nFor storing intermediate results of a reduction, optionally,\nthe transitional open source database \\berkeleydb~\\cite{berkeley}\ncan be used.\n\nFor the normalization of algebraic coefficients\nin the identities, one can choose between \\ginac~\\cite{Bauer:2000cp}\nand \\fermat~\\cite{fermat}.\nReduction identities and other results can be exported to\n\\form~\\cite{form}, \\mathematica~\\cite{mathematica}, and \\maple~\\cite{maple}\nformat.\nConfiguration and job files use the \\yaml\\ format~\\cite{yaml} and are parsed\nwith the {\\tt yaml-cpp} parser~\\cite{yamlcpp}.\n\n\\reduze\\;2 was used to calculate the two-loop leading color corrections\nto heavy-quark pair production in the gluon fusion channel~\\cite{Bonciani:2010mn}.\nLast but not least, \\reduze\\ is published as open source under the\n{\\tt GNU} General Public License (GPL) v3 and has no mandatory dependencies\non proprietary software.\n\n\n\\section{Basic concepts and notations}\n\n\\subsection{Integral families, sectors, and integrals}\n\\label{sec:fam_sec_int}\n\nA \\emph{propagator} $P$ is defined as the expression $1\/(q^2 - a)$\nwhere $q$ is a four-momentum and $a$ is constant.\nThe momentum $q$ of a propagator (defined up to a minus sign) is a\nlinear combination of loop momenta $k_i$ and external momenta $p_i$,\nand $q^2$ is a scalar product in Minkowski space with the metric convention\n$g = \\mbox{diag}(1, -1, -1, -1)$.\nIn \\reduze, also generalized propagators $1\/(q l - m^2)$\nwith the scalar product of two different momenta $q$ and $l$ are available\nto support more general irreducible numerators.\n\nAn $l$-loop \\emph{integral family} (or ``auxiliary topology'') $F$ is an\nordered set $\\{ P_1, \\ldots, P_n \\}$\nof propagators $P_i$, $i=1,\\ldots,n$, which is minimal and complete in the sense,\nthat any scalar product of a loop momentum $k_i$ with a loop momentum $k_j$\nor an external momentum $p_j$ can be uniquely expressed as a linear combination\nof inverse propagators and kinematic invariants.\nDenoting the number of independent\nexternal momenta by $m$, an integral family must contain exactly\n$l\\,(l+1)\/2+l\\,m$ propagators, where the first term counts the scalar products\nbetween loop momenta only and the second term the products involving both loop and\nexternal momenta.\nA new feature of \\reduze\\  is its ability to handle several integral families\nsimultaneously.\n\nA selection of $t$ propagators of an integral family defines a \\emph{sector} of\nthis family.\nAssuming a sector has the propagators $P_{j_1},\\ldots, P_{j_t}$ with\n$\\{j_1,\\ldots,j_t\\} \\subset \\{1,\\ldots,n\\}$, then its identification number\nis defined as\n\\begin{equation}\n ID = \\sum_{k=1}^{t} 2^{j_k-1} \\, .\n\\end{equation}\nThere are in general $\\binom{n}{t}$ different $t$-propagator sectors and\n$\\sum_{t=0}^{n} \\binom{n}{t} = 2^n$ sectors are contained in an integral\nfamily. Their identification numbers fulfill $0 \\leq ID \\leq 2^n - 1$ .\n\nA sector whose propagators form a subset of the propagators of another sector\nof the same integral family is a \\emph{subsector} of the other sector.\n\nThe purpose of an integral family is to index scalar loop integrals.\nTo every $t$-propagator sector with propagators $P_{j_1}$, $\\ldots$, $P_{j_t}$\nbelongs a infinite set of $d$-dimensionally regularized $l$-loop\nintegrals~\\cite{'tHooft:1972fi} which all share the same propagators.\nThese integrals have the generic form\n\\begin{equation}\\label{eq:dim_reg_int}\nI = \\int \\mbox{d}^d k_1 \\ldots \\int \\mbox{d}^d k_l \\,\nP_{j_1}^{r_1} \\ldots P_{j_t}^{r_t}\n     P_{j_{t+1}}^{-s_1} \\ldots P_{j_n}^{-s_{n-t}}\n\\end{equation}\nwith integer exponents $r_i \\geq 1$ and $s_i \\geq 0$.\nIn \\reduze\\  such an integral is represented by\n\\begin{equation}\\label{eq:int_reduze_format}\n\\mbox{I}(F, t, ID, r, s,\\{v_1,\\ldots,v_{n}\\})\n\\end{equation}\nwhere $F$ denotes the integral family,\n$r=\\sum_{i=1}^t r_i \\geq t$, $s=\\sum_{i=1}^{n-t} s_i \\geq 0$ and\n$v_i$ is the exponent of propagator $P_i$.\nPositive $v_i$ denote powers of regular propagators (non-trivial denominator),\nnegative $v_i$ denote powers of inverse propagators (non-trivial numerator),\nand zero means absence of a propagator.\nThe numbers $t$, $r$, $s$ as well as the identification number $ID$ of the\nsector, to which the integral belongs, can be calculated from the vector $v$.\n\nConsider a $t$-propagator sector of a $n$-propagator integral family.\nThe number of integrals that one can build for certain values of $r$ and $s$ is given by\n$\n{\\mathcal N}(n,t,r,s) = \\binom{r-1}{t-1} \\binom{s+n-t-1}{n-t-1} \\, .\n$\nThe two binomial factors count all possible ways to arrange the exponents of\nthe propagators in the denominator and numerator, respectively.\n\nThe integral with $r=t$ and $s=0$ of some sector is called \n\\emph{corner integral} of this sector.\n\n\n\n\\subsection{Integration by parts (IBP) identities}\nIn dimensional regularization~\\cite{'tHooft:1972fi} the integral over a total derivative\nis zero. Let ${\\mathbf I'}$ be the integrand of an integral of the form (\\ref{eq:dim_reg_int}).\nThen, working out the differentiation in\n\\begin{equation}\n \\int \\mbox{d}^d k_{i} \\, \\frac{\\partial}{\\partial k_{i}^{\\mu}} \\, \\big[ q^{\\mu} \\,\n {\\mathbf I'}(p_1, \\ldots, p_{m}, k_1, \\ldots, k_l) \\big] = 0\n\\end{equation}\n\nleads to the integration by parts (IBP) identities~\\cite{Tkachov:1981wb,Chetyrkin:1981qh}.\nThe momentum $q$ is an arbitrary loop or external momentum. The index $\\mu$ is summed over\nbut the index $i$ is not. If there are $l$ loop momenta and $m$ independent external\nmomenta one can therefore build $l\\, (l+m)$ equations from one integral, the\n\\emph{seed integral}.\n\n\n\\subsection{Lorentz invariance (LI) identities}\nOne can also use the Lorentz invariance of the integrals~\\cite{Gehrmann:1999as}. Taking\nan integral ${\\mathbf I}(p_{1},\\ldots ,p_{m})$ the following equation holds\n\\begin{equation}\n\\sum_{n=1}^m \\Big( p_{n}^{\\nu} \\frac{\\partial}{\\partial p_{n \\mu}}\n    - p_{n}^{\\mu} \\frac{\\partial}{\\partial p_{n \\nu}} \\Big)\\, {\\mathbf I}\n    (p_{1},\\ldots ,p_{m}) = 0 \\, .\n\\end{equation}\nThe derivatives can be shifted directly to the integrand of the integral ${\\mathbf I}$.\nThis equation can be contracted with all possible antisymmetric combinations of the\nexternal momenta, e.g.\\ $p_{1 \\mu} p_{2 \\nu} - p_{1 \\nu} p_{2 \\mu}$, which leads to\n$m\\,(m-1)\/2$ equations where $m$ denotes the number of independent external momenta.\nAs it was shown in~\\cite{Lee:2008tj} the LIs do not give new linearly independent\nequations in addition to the IBPs. However, they can accelerate the convergence in a\nreduction, since in general an LI identity generated from one seed integral cannot be\nreproduced with the IBP identities generated from the same seed integral alone.\n\\reduze\\  offers the possibility to use the LIs.\n\n\n\\subsection{Zero sectors}\n\nIt is possible that a whole sector is zero which means that all integrals belonging\nto this sector are zero. A sector of an $l$-loop integral family is trivially\nzero if it does not allow for a selection of $l$ propagator momenta which are\nindependent with respect to the $l$ loop momenta.\nThe graph based methods in \\reduze, see section \\ref{sec:physical_secs},\nwill automatically detect these cases.\nAs a second method, a sector is set to zero if the reduced IBP identities\ngenerated from the seed integrals of this sector with $r=t$ and\n$s=0, 1$ show that its corner integral is zero.\n\n\\subsection{Sector relations} \\label{sec:sec_rel}\n\nGiven a scalar loop integral as well as one or several integral families,\nsuppose it is possible to map the integral to a linear combination of indexed\nintegrals of type \\eqref{eq:int_reduze_format}.\nIn general, such a map is not unique.\nAmbiguities may arise if sectors from different integral families have the same\nset of propagators or if a transformation of loop and external momenta in\n\\eqref{eq:dim_reg_int} leads to a different linear combination of type\n\\eqref{eq:int_reduze_format}.\nFor the corner integral of some sector $S$ written in the form\n\\eqref{eq:dim_reg_int},\nconsider the transformation of integration variables\n\\begin{align}\\label{eq:shift_trafo}\nk_i &\\to \\sum_{j=1}^{l} M_{ij} k_j + \\sum_{j=1}^{m} N_{ij} p_j\n\\end{align}\nwith $|\\det{M}|=1$.\nIf the new integrand factors can be identified with propagators of a\nsector $S'$, the ``shift'' transformation \\eqref{eq:shift_trafo} defines a\n\\emph{sector relation} between $S$ and $S'$.\nIn this case, any integral in the sector $S$ can be expressed as a linear\ncombination of integrals in the sector $S'$ and subsectors of $S'$.\nIf $S$ and $S'$ are different, such a relation can be used to eliminate\none of the two sectors completely.\nThe case where $S$ and $S'$ are identical is discussed in\nsection~\\ref{sec:sec_symm}.\n\n\\reduze\\  is able to automatically detect sector relations or handle relations\nsupplied by the user.\nSector relations will be used to eliminate redundant sectors or integrals,\nusually at the earliest possible stage.\nAs a particularly useful special case, a shift \\eqref{eq:shift_trafo} might\nmap each propagator of an integral family to another propagator of the same\nintegral family.\nThis leads to a one--to--one mapping between integrals as well as sectors\nof the integral family.\nSuch a relation can be entered for the full integral family via permutations\nof propagators and allows for particularly efficient removal of redundancies.\n\nAs a generalization of the above described concepts,\n\\reduze\\  also allows for crossings of\nexternal momenta in addition to \\eqref{eq:shift_trafo}.\nIf the involved crossing leaves the kinematic invariants\nunchanged, the corresponding relations can be directly exploited for\nrelations between sectors as described before.\nIn the general case, \\reduze\\  also handles relations between sectors of\nintegral families defined with crossed kinematics.\n\n\\subsection{Sector symmetries}\n\\label{sec:sec_symm}\n\nSpecial shifts of the loop momenta as in \\eqref{eq:shift_trafo} which\ntransform a sector to itself are called \\emph{sector symmetries}.\nThese shifts are also allowed to contain a permutation of the external\nmomenta as long as it does not change the kinematic invariants.\nSector symmetries may be used to express integrals in terms of other\nintegrals in the same sector and its subsectors.\nThese relations may provide information complementary to the IBP and LI\nidentities and can be used in the reduction to find a minimal number of\nmaster integrals.\n\\reduze\\ is capable of automatically determining sector symmetries\nor handling user supplied rules, and offers to exploit them for reductions.\n\n\n\\section{Graph and matroid based algorithms for sectors}\n\n\\subsection{Physical sectors}\n\\label{sec:physical_secs}\n\nA physical sector is a sector whose propagators correspond to edges in\na graph such that momentum is conserved. The construction of a graph from\na sector, i.e. from the momenta of a set of propagators with $l$-loop momenta,\ncan be done by choosing $l$ propagators which have independent momenta with\nrespect to the loop momenta and identifying them as edges in a graph with\nboth ends glued together in a single root vertex. External edges are also\nattached to this root vertex with one of their ends.\nSubsequently, for each of the remaining propagators, a vertex of the graph\n(first the root vertex) is cleaved into two vertices, and a new edge is inserted\nbetween these vertices, such that the edge's momentum (determined by momentum\nconservation) exactly matches the propagator's momentum.\n\nWith this procedure \\reduze\\ automatically constructs graphs for sectors\nwhere this is possible and thus identifies physical sectors.\nThe possibility of having graph representations for sectors gives\naccess to fast algorithms for identifying isomorphic graphs and\nfinding sector relations and sector symmetries.\n\n\n\\subsection{Sector relations}\n\nIf graphs constructed for two different physical sectors are isomorphic,\na shift of the form \\eqref{eq:shift_trafo} between the two sets of loop momenta\ncan be derived by identifying the edges of the two graphs together\nwith their oriented momentum flow labeling.\n\\reduze\\ offers the possibility to find relations between physical sectors\nby this strategy, allowing also crossings of external legs for the graph\nisomorphism but restricting to cases with $|\\det{M}|=1$.\nFor the graph of each physical sector a standard form of its adjacencies,\na \\emph{canonical label}, is computed with the algorithm\n\\cite{McKay1981}.\nWe distinguish different masses by replacing massive edges with a chain\nof several edges, where the length of such a chain labels a mass uniquely.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{matroid1}\\hspace{20mm}\n\\includegraphics[width=0.25\\textwidth]{matroid2}\n\\end{center}\n\\caption{\\label{fig:twists}\nTwo non-isomorphic graphs which are related by a twist.\nThe matroids of these graphs are isomorphic.\n}\n\\end{figure}\n\nGraph isomorphism takes into account the ambiguities in labeling the\n\\emph{nodes} of a graph.\nWhile isomorphic graphs can be described by the same propagators (possibly\nwith a crossing of external momenta), the inverse is not true.\nConsider for instance the two vacuum graphs in figure~\\ref{fig:twists}.\nThe two graphs are non-isomorphic but can be described by the same\npropagators, i.e.\\ by the same sector.\nHere, a more appropriate object to consider is not the graph of the sector,\nbut the associated matroid.\nMatroids are based on the notion of a set of linearly independent sets\nand may be considered as generalizations of graphs.\nFor a graph an associated graph matroid (or cycle matroid) can be defined via\nits edges. Definitions and fundamental properties are given in the review\n\\cite{Bogner:2010kv} and references therein.\nEssential for us is the following chain of statements.\nThe relevant properties of a vacuum graph where all edges share a\ncommon mass is encoded in the first Symanzik polynomial\n($\\mathcal{U}$ polynomial).\nFor brevity of the argument let us furthermore restrict to biconnected graphs.\nThe generalization to arbitrary vacuum components with different masses is\nrather straightforward.\nAn immediate combinatorial approach to isomorphisms of the\nSymanzik polynomials, which is not restricted to vacuum graphs, was presented in\n\\cite{Pak:2011xt}.\nHere, we note that the first Symanzik polynomials\nof two graphs are equal up to a permutation of edge variables exactly if\ntheir matroids are isomorphic, see \\cite{Bogner:2010kv} and references therein.\nTwo matroids of biconnected vacuum graphs are isomorphic exactly if the\ntwo graphs are isomorphic up to a series of \\emph{twists}, a statement known\nas Whitney's theorem \\cite{whitney}.\nA twist operation starts by breaking a graph into two graphs such that\nidentification of \\emph{separation pairs} of nodes in both graphs restore\nthe original graph.\nAs the second step of the twist, the separation pairs are identified\nwith flipped orientation.\nIn figure~\\ref{fig:twists} the graphs are related by a twist around\nthe left-most and right-most nodes and thus have the same matroid.\n\nThese statements can be turned into an algorithm to detect shifts \\eqref{eq:shift_trafo}\nbetween vacuum sectors, which is sketched in the following and implemented in\n\\reduze.\nWe extend the graph isomorphism based shift detection by modifying the\ngenerated graphs with twists such that their canonical labels are minimized.\nA graph of a physical sector is decomposed into biconnected components with the\nalgorithm \\cite{HopcroftT71}.\nPossible separation pairs of biconnected components are identified via a\ndecomposition into \\emph{triconnected components}.\nWe implemented the algorithm \\cite{HopcroftT73, GutwengerMutzel2001}\nfor this purpose.\nIn order to generate at least one representative for each graph isomorphism\nclass, it is necessary to perform twists around separation pairs\nas specified by \\emph{virtual edges} as well as twists which correspond to\npermutations of edges within polygon components of the decomposition.\nWhile twisting we track the edges including their orientations to be able to\nidentify propagator momenta and different masses.\nGraphs with external legs are handled by intermediately joining their external\nnodes into one node and restricting to those twists, which keep the external\nlegs joined into one node with their original orientation.\n\nAlternatively, \\reduze\\ also offers a procedure to find all sector relations,\nwhich tries to identify sets of propagators in a straightforward approach\nbased on linear algebra and combinatorics.\nWhile this procedure finds all shifts between arbitrary sectors,\nincluding sectors not corresponding to a graph,\nit is usually much less efficient than the graph based methods.\n\n\n\\subsection{Sector symmetries}\n\\label{sec:sym_rel}\n\nSector symmetries are shifts of the form \\eqref{eq:shift_trafo} with\nmap a sector onto itself.\nAdditional permutations of the external momenta are permitted as long as the\nkinematic invariants are unchanged.\nDifferent such sector symmetries for physical sectors can be found by the\nunderlying symmetries of the graph as vertex permutations from the automorphism\ngroup and permutations of multi-edges. The automorphism group of a graph\nconsists of all permutations of the vertices which leave the canonical label\nof the graph unchanged. These transformations are calculated\nin \\reduze\\ with the algorithm \\cite{McKay1981} and are used to derive the\nmapping of the edges between pairs of vertices and the associated shift of\nthe loop momenta. In the case where there is more than one edge between\ntwo vertices also permutations of the edges with the same mass are considered.\n\nAlternatively, a complete set of sector symmetries can be calculated\nby \\reduze\\ using a combinatorial propagator matching approach.\n\n\n\\subsection{Matching of diagrams to sectors}\n\\label{sec:match_dias}\n\nAs discussed above, the assignment of momenta to propagators of Feynman\ndiagrams is ambiguous.\nIf diagrams are generated with a program like \\qgraf\\ \\cite{qgraf}, typically\nthe assigned loop momenta have to be shifted in order to index the involved\nloop integrals via integral families.\n\\reduze\\ can compute these shifts for diagram files generated with \\qgraf.\n\\reduze\\ can also handle permutations of external momenta and find matchings of\ndiagrams to crossed sectors.\n\n\n\\section{Distributed reduction algorithm}\n\n\\subsection{Load balanced system solving}\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}[b]{0.5\\textwidth}\n{\\small\n\\fcolorbox{framecolor}{shadecolor}{\n\\parbox{52mm}{\n\\mbox{}\\vspace*{-5mm}\n\\begin{align*}\n \\textcolor{emphcolor}{\\boldsymbol{I_5}} + c_{14} \\boldsymbol{I_4} + c_{13} \\boldsymbol{I_3} \\qquad\\quad\\,\\, \\qquad\\quad\\,\\,&= 0 \\\\\n \\textcolor{emphcolor}{\\boldsymbol{I_5}} + c_{24} \\boldsymbol{I_4} \\qquad\\quad\\,\\,  + c_{22} \\boldsymbol{I_2} \\qquad\\quad\\,\\,&= 0\\\\\n \\textcolor{emphcolor}{\\boldsymbol{I_5}} \\qquad\\quad\\,\\, + c_{33} \\boldsymbol{I_3} + c_{32} \\boldsymbol{I_2} \\qquad\\quad\\,\\,&= 0\n\\end{align*}\n\\mbox{}\\vspace*{-7mm}\n}}\n\\\\[1mm]\n\\fcolorbox{framecolor}{shadecolor}{\n\\parbox{58mm}{\n\\mbox{}\\vspace*{-5mm}\n\\begin{align*}\n\\qquad\\quad\\,\\,  \\textcolor{emphcolor}{\\boldsymbol{I_3}} + c_{42} \\boldsymbol{I_2} \\qquad\\quad\\,\\,  &= 0 \\\\\n\\qquad\\quad\\,\\,\\qquad\\quad\\,\\, \\textcolor{emphcolor}{\\boldsymbol{I_3}}\\qquad\\quad\\,\\,  + c_{51} \\boldsymbol{I_1} &= 0\n\\end{align*}\n\\mbox{}\\vspace*{-7mm}\n}}\n\\\\[1mm]\n\\fcolorbox{framecolor}{shadecolor}{\n\\parbox{58mm}{\n\\mbox{}\\vspace*{-5mm}\n\\begin{align*}\n\\qquad\\quad\\,\\,\\qquad\\quad\\,\\,\\qquad\\quad\\,\\, \\textcolor{emphcolor}{\\boldsymbol{I_2}} + c_{61} \\boldsymbol{I_1} &= 0\n\\end{align*}\n\\mbox{}\\vspace*{-7mm}\n}}\n}\\end{minipage}\n\\hspace*{5mm}\n\\includegraphics[width=0.35\\textwidth]{manager-worker-topo}\n\\end{center}\n\\caption{\\label{fig:blocks}\n\\emph{Left:} Three blocks of equations for loop integrals $I_i$\nwith coefficients $c_{ij}$ depending on kinematic invariants\nand the space--time dimension.\nEach block (shaded rectangle) contains equations\nfor the same leading integral (bold face $I_i$).\n\\emph{Right:} communication topology for MPI processes involved in a\ndistributed reduction.\n}\n\\end{figure}\n\n\\reduze\\ computes reductions for Feynman integrals by generating identities\nfor a range of seed integrals and reducing this linear system of equations.\nThe seed integrals are usually chosen for ranges in the propagator\nexponent sums $r$ and $s$.\nIn a typical application, reductions for integrals from several sectors are\nneeded.\nMoreover, a full reduction of a specific sector requires in general\nalso the reduction of subsector integrals.\n\\reduze\\  proceeds bottom-up: the reduction of a sector is started only after\nall subsector results are available.\nSectors which are no subsectors of each other can be reduced independently such\nthat these tasks are easily parallelized.\nThis kind of parallelization is available in the first version of \\reduze\\ \nvia a shell script, which launches programs for different sectors.\nIn version\\ 2 of \\reduze, two levels of parallelization are implemented via the\nmessage--passing--interface (MPI) standard.\nThe reduction of a sector becomes a \\emph{job} and\nseveral independent such reduction jobs can be executed in parallel.\nThe job system is described in more detail in the next section.\nOn top of this first level of parallelization, this version of \\reduze\\  implements\na distributed reduction algorithm for the reduction of a single sector.\nSince in this case the parallelization is less obvious we give some details about\nour implementation in the following subsection.\n\nIn \\reduze, a total ordering is defined for indexed integrals of the type\n\\eqref{eq:int_reduze_format}. The ordering defines integrals of a sector to be more\ncomplicated than integrals of its subsectors.\nIn the following, terms like ``leading'' or ``lower'' integral refer\nto this ordering.\nTo choose specific integrals as master\nintegrals the user may adjust the ordering, possibly at a later stage.\nIn order to reduce integrals of a given sector, IBP and LI identities are\ngenerated for a specified range of seed integrals.\nThis results in a sparse homogeneous linear system of equations for the indexed\nintegrals where the coefficients are rational functions of the kinematic\ninvariants and the space--time dimension.\nThe equations are sorted into blocks\nof equations with the same leading integral,\nsee left panel of figure \\ref{fig:blocks}.\n\nThe blocks are reduced bottom up, starting from the block with the lowest\nleading integral.\nFor each block, integrals are reduced, i.e. replaced by linear combinations of\nlower integrals, according to the results from lower blocks and subsectors\n(``back substitution'') if possible.\nIf a block contains several equations, one is kept and used to eliminate\nthe leading integral from all other equations in the block (``forward\nelimination''), which are subsequently solved for the new leading integral.\nThe coefficients of the integrals are normalized such that zeros are detected\nand numerator and denominator are coprime.\nThis requires multivariate polynomial greatest--common--divisor (GCD)\ncomputations which typically present the most time consuming part of the\nfull calculation.\nThe result of this reduction of a block is one equation for the block and\npossibly further equations with lower leading integrals to be inserted\ninto lower blocks.\nThe next block to be selected is the lowest block which contains more than\none equation or involves integrals which can be reduced.\n\n\\reduze\\  offers both, a purely serial reduction for one sector as well as\na distributed execution.\nIn the serial version, the above steps are performed in deterministic order\non one core.\nThe distributed version employs a star topology of MPI processes with one\nmanager and one or more workers, see right panel of figure~\\ref{fig:blocks}.\nThe workers perform the actual reduction steps for a block, while\nthe manager keeps track of the complete system of equations and balances the\nwork between the workers.\nMore specifically, an idle worker contacts the manager to request work.\nThe manager looks up the next block to process and sends its equations together\nwith equations needed for back substitutions to the worker.\nThe worker reduces the block and sends the result to the manager.\n\nOur motivation for this distributed algorithm is as follows.\nExperiments show that in typical applications the time needed for the reduction\nof one block can easily differ by more than 6 orders of magnitude.\nMoreover, the exact order of the individual reduction steps significantly\ndetermines the execution time for the full reduction and a bottom-up order\ntypically shows the best performance.\nBoth issues are directly addressed by the dynamical load balancing mechanism\npresented above, at least for a not--too--large number of worker processes.\nHow well this works in practice is quantified in the following subsection.\n\n\\reduze\\ allows to choose between \\ginac\\ \\cite{Bauer:2000cp}\nand \\fermat\\ \\cite{fermat} for the GCD computation\nneeded to normalize coefficients.\nDuring reduction, the equations are stored either in RAM or optionally\nin a transactional database as implemented by the open source Berkeley DB\n\\cite{berkeley}.\nWith transactions turned on, an aborted reduction of a single sector may be\nresumed at a later time; recovering completed reductions for sectors is\navailable in any case.\n\\reduze\\ supports different ways to split a calculation into several runs,\nthis is described in the tutorial provided with the \\reduze\\ distribution.\nReductions for crossed integrals are automatically be obtained via reduction\nresults from its uncrossed counterpart in order to save computation time\nand disk space.\n\n\\subsection{Performance results for single sectors}\n\\label{sec:perf_single}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.5\\textwidth]{perfsingle}}\n\\caption{\\label{fig:perf_single}\nExecution time for the reduction of single sectors in dependence\nof the number of worker processes employed for the reduction.\n}\n\\end{figure}\n\nPerformance results for the reduction of sectors for two--loop contributions\nto heavy--quark pair production are shown in figure~\\ref{fig:perf_single}.\nThese sectors have $t=4$ or $5$, respectively,\nand reductions were computed for integrals with $r=t\\ldots 7$ and\n$s=0\\ldots3$ or 4, respectively.\nWe used a computer with 48 CPU cores operating at $2.1$~GHz and started\n\\reduze\\ with $n_{\\text{workers}}+2$ MPI processes (one job center process\nand one manager process should be overbooked for a small number of available\ncores).\n\nIn general, we observe that the scaling with the number of processes is\nproblem specific and depends on the configuration of \\reduze, such as the\nchosen computer algebra system.\nThe upper two curves in the figure show an example with a good scaling for up\nto 22 workers.\nIndeed, we find examples where the scaling is good up to 48 workers.\nWe think this good scaling behavior is noteworthy, given the fact that it\ndescribes the distributed computation of a not too loosely coupled system.\nAs expected, we observe that beyond some number of worker\nprocesses the run time decreases less and less with additional workers\nand finally increases for even larger number of workers.\nContributions to this behavior is expected from serial parts\nof the code, communication overhead, but potentially also from\na ``less ideal'' order of evaluation when solving the system of equations\nwith a larger number of workers. \nThe lowest curve was obtained for an example of a reduction of a comparably\nsimple system of equations, where the onset of such a behavior is clearly\nvisible.\nIt is also not difficult to find examples with worse scaling,\nwhere a minimal runtime is obtained for only a few worker processes.\n\nUsing \\fermat\\ instead of \\ginac\\ for the GCD computations can easily result\nin a speed-up by an order of magnitude, see the two blue curves for\nsector~B in the figure.\nFor the displayed \\fermat\\ benchmarks, the system to be reduced was kept in RAM,\nwhile for the \\ginac\\ benchmarks a database was used.\nOur tests indicate that for examples of this type, the performance penalty for\nusing a database is considerably less than the differences due to the two\nalgebra systems considered here.\nNevertheless, also this option is relevant for optimal performance,\nespecially for a larger number of workers.\nTypically, if the coefficients in the equations become more involved, the\nreduction benefits considerably from a larger amount of workers compared\nto cases with compact coefficients.\nThe performance in a real application example, involving the reduction of\nseveral sectors, is discussed at the end of the next section.\n\n\n\\section{Job system}\n\n\\subsection{Load balanced execution of jobs}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.8\\textwidth]{jobcenter}}\n\\caption{\\label{fig:jobcenter}\nDynamical load balancing in \\reduze}\n\\end{figure}\n\nIn \\reduze, a \\emph{job} represents a sequence of computations which\ncan be performed once its dependencies, specified via the presence of input\nfiles, are fulfilled.\nMost jobs in \\reduze\\ are \\emph{serial jobs}: they are executed on a single\ncore but possibly in parallel to other jobs.\nParallel execution of different jobs represents the top layer parallelization\nmechanism of \\reduze\\ which is automatically available for any job type added\nto \\reduze.\nReduction of identities is an example for a \\emph{distributed job}.\nSuch a job can be executed by several processes in parallel: one process,\nthe \\emph{manager}, is responsible for the full execution of the job,\nother processes, the \\emph{workers}, help for some time with the execution.\nIn order to employ this second layer parallelization, a dedicated distributed\nalgorithm needs to be implemented for the job.\nFor each run of \\reduze, the user specifies a sequence of such jobs, which\nare inserted into a job queue. A job may generate additional auxiliary jobs\nautomatically. The job queue is responsible for resolving the dependencies\nbetween the jobs and determining the next job to be executed.\n\nIf \\reduze\\ is started with several MPI processes, one process will act as a\n\\emph{job center} and dynamically balance work between the other processes,\nthe \\emph{clients} of the job center, see figure~\\ref{fig:jobcenter}.\nThe job center schedules jobs using the job queue and assigns them to clients.\nAn idle client contacts the job center and requests work.\nThe job center responds by assigning a job to the client, either to be executed\nas a manager or as a worker.\nThe client changes its role accordingly.\nAs a worker, it contacts the responsible manager of the job and helps with the job\nexecution.\nAs a manager, it can either execute the job by itself or\nregister as an \\emph{employer} at the job center to request workers.\n\nIn order to optimize the efficiency of the parallelization,\nthe job center periodically collects performance measurements from the employers\nand estimates an optimal distribution of workers based on it.\nAccording to this estimate, workers will effectively be reassigned to other\nemployers by requesting release of workers from ``inefficient'' employers and\nassigning idle customer to ``efficient'' employers.\nThe basic idea is to avoid low efficiencies due to overloaded managers by\nassigning workers preferably to managers which idle a high percentage of their\nCPU time.\n\n\\subsection{Performance results for multiple sectors}\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}[b]{0.21\\textwidth}\n \\includegraphics[width=\\textwidth]{A_446}\\\\[2ex]\n \\includegraphics[width=\\textwidth]{A_247}\\\\[2ex]\n \\includegraphics[width=\\textwidth]{A_431}\\\\[2ex]\n \\includegraphics[width=\\textwidth]{A_191}\n\\vspace{2ex}\n\\end{minipage}\n\\hspace{10ex}\n\\includegraphics[width=0.5\\textwidth]{perfmulti}\n\\end{center}\n\\caption{\\label{fig:perf_multi}\nRuntime decrease for the reduction of multiple sectors using a large number\nof processor cores.\nThe reductions cover the sectors of the depicted graphs and all of their\nsubsectors.\nIn the graphs, thick lines represent massive propagators.\n}\n\\end{figure}\n\nPerformance results for the reduction of a selection of sectors are shown in\nfigure~\\ref{fig:perf_multi}.\nThese sectors are encountered in the calculation of two-loop corrections to\nheavy-quark pair production.\nReductions are calculated for integrals of the depicted sectors and all of\ntheir subsectors, where $r=t\\ldots 7$ and $s=0\\ldots 4$.\nThe benchmarks for the solid curve were performed on a computer with 48 CPU\ncores operating at $2.1$~GHz using \\fermat\\ and keeping all equations in RAM.\nThe measurements for the dashed curve were obtained using the ``Schr\\\"odinger''\ncluster of the University of Zurich with $2.8$~GHz cores; these runs were\nconfigured to use \\ginac\\ and a database.\n\nThe figure shows for this realistic application example that the calculation\nbenefits considerably from up to 96 cores, if available.\nLet us stress again, that the scaling behavior is problem specific and\nmay be worse for other types of applications.\nIn the present example,\nthe run--time decrease due to additional cores is quite close to an optimal\n$1\/n_{\\text{cores}}$ behavior for smaller numbers of cores.\n\n\n\\section{Other features}\n\n\\subsection{Differential equations for master integrals}\nOne method to solve master integrals consists\nof deriving differential equations by taking derivatives in the kinematic\ninvariants, replacing new integrals with the reduction results and solving\nthe differential equations by integration.\nIn particular for sectors with several independent integrals,\na change of basis for the master integrals may be required.\n\\reduze\\ offers the possibility to derive differential equations for Feynman\nintegrals and reduce these equations for some user choice of master integrals.\n\n\\subsection{Interference terms}\nStarting from diagrams generated by \\qgraf, \\reduze\\ can compute scalar\ninterferences of (bare) QED or QCD amplitudes in dimensional regularization.\nThis includes insertion of user--defined Feynman rules, contraction of Lorentz\nvector indices, performing Dirac traces, and evaluating color structures.\nIn case of an interference of a tree-level diagram with a diagram that\ncould be matched to a sector of an integral family (section \\ref{sec:match_dias})\nalso the occurring integrals, which belong to the matched sector and its subsectors,\nare indexed by the corresponding integral family.\nIn a further step, these integrals can be reduced to master integrals and\nsubstituted correspondingly in the interference terms.\nEach computation of an interference of two diagrams is treated as a job, and\nwhen MPI is used, these jobs can be performed in parallel.\nSpecific examples are distributed with the \\reduze\\ package.\n\n\n\\section{Usage}\n\nThe package \\reduze\\ can be downloaded from the web page\n \\url{http:\/\/projects.hepforge.org\/reduze}\\,.\nThe distribution contains a tutorial with detailed description of\ninstallation and usage as well as several example files.\n\n\n\\acknowledgments{\nWe thank York Schr\\\"oder for many useful discussions and comments on the\ndraft.\nWe are grateful to the CEDAR collaboration for hosting our project web page\non HepForge~\\cite{Buckley:2006np}.\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG SCHR 993\/2-1)\nand the Schweizer Nationalfonds (Grant 200020\\_124773\/1).}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe main problem we will consider here is to give a lower bound for the dimension of the union of a collection of lines in terms of the dimension of the collection of lines, without imposing a structural hypothesis on the collection (in contrast to the Kakeya problem where one assumes that the lines are direction-separated, or perhaps satisfy the weaker ``Wolff axiom''). \n\nSpecifically, we are motivated by the following conjecture of D. Oberlin ($\\mathop{\\mathrm{hdim}}$ denotes Hausdorff dimension).\n\\begin{conjecture}\nSuppose $d \\geq 1$ is an integer, that $0 \\leq \\beta \\leq 1,$ and that ${\\bf L}$ is a collection of lines in $\\rea^n$ with $\\mathop{\\mathrm{hdim}}({\\bf L}) \\geq 2(d-1) + \\beta.$ Then\n\\begin{equation} \\label{dimrange}\n\\mathop{\\mathrm{hdim}}(\\bigcup_{L \\in {\\bf L}}L) \\geq d + \\beta.\n\\end{equation}\n\\end{conjecture}\nThe bound \\eqref{dimrange}, if true, would be sharp, as one can see by taking ${\\bf L}$ to be the set of lines contained in the $d$-planes belonging to  a $\\beta$-dimensional family of $d$-planes. (Furthermore, there is nothing to be gained by taking $1 < \\beta \\leq 2$ since the dimension of the set of lines contained in a $d+1$-plane is $2(d-1) + 2$.)\n\nStandard Fourier-analytic methods show that \\eqref{dimrange} holds for $d=1$, but the conjecture is open for $d > 1.$ As a model problem, one may consider an analogous question where $\\rea^n$ is replaced by a vector space over a finite-field. Our main result is that the corresponding conjecture holds for all $d$ ($|\\cdot|$ denotes cardinality).\n\n\\begin{theorem} \\label{firstlinethm}\nSuppose $d \\geq 1$ is an integer, $F$ is a finite field, $0 \\leq \\beta \\leq 1,$ and that ${\\bf L}$ is a collection of lines in $F^n$ with $|{\\bf L}| \\geq |F|^{2(d-1) + \\beta}.$ Then\n\\begin{equation} \\label{dimrangeff} \n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{d + \\beta}\n\\end{equation}\nwhere the implicit constant may depend on $d$, but is independent\\footnote{The constant is also independent of $\\beta$ and $n$, but this is only of secondary interest.} of $F$.\n\\end{theorem}\n\nReusing the examples above, one sees that \\eqref{dimrangeff} is sharp, up to the loss in the implicit constant, and that there is nothing to be gained by taking $1 < \\beta \\leq 2.$\n\nThe main tool we use in the proof of \\eqref{dimrangeff} is an iterated version of Wolff's hairbrush argument \\cite{wolff95ibk}. For comparison, we state the finite-field version of his result\\footnote{Wolff's main interest in this method was likely its use towards a partial resolution of the Kakeya conjecture (up to a negligible constant, any direction separated collection of lines satisfies the Wolff axiom). To that end, it has been superceded by Dvir's theorem \\cite{dvir09osk} (see also \\cite{ellenberg09ksm}), whose proof makes stronger use of the direction-separation hypotheses and does not seem to be applicable to the present question.} (see \\cite{wolff99rwc},\\cite{mockenhaupt04rkp}), starting with the following definition. A set of lines ${\\bf L}$ in $F^n$ satisfies the \\emph{Wolff axiom} if for every two-plane $R \\subset F^n$ \n\\[\n|\\{L \\in {\\bf L} : L \\subset R\\}| < |F|.\n\\]\n\n\\begin{theorem}[Wolff]\nSuppose that $\\alpha \\geq 1$, $F$ is a finite field, and ${\\bf L}$ is a collection of lines in $F^n$ with $|{\\bf L}| \\geq |F|^{\\alpha}.$ If ${\\bf L}$ satisfies the Wolff axiom then\n\\begin{equation} \\label{wolffexponent}\n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{\\frac{\\alpha + 3}{2}}\n\\end{equation}\nwhere the implicit constant is independent of $F$.\n\\end{theorem}\n\nAn immediate consequence of Theorem \\ref{firstlinethm} is that, for odd integers $\\alpha$, Wolff's theorem holds even for collections of lines that do not satisfy the Wolff axiom. \n\nThe proof of Theorem \\ref{firstlinethm} also shows that the Wolff axiom can be relaxed for general values of $\\alpha.$ We say that a set of lines ${\\bf L}$ in $F^n$ satisfies the \\emph{$d$-plane Wolff axiom} if for every $d$-plane $R \\subset F^n$ \n\\begin{equation} \\label{dpwa}\n|\\{L \\in {\\bf L} : L \\subset R\\}| < |F|^{2d-3}.\n\\end{equation}\nIf $R$ is a $d$-plane then there are approximately $|F|^{3(d-2)}$ 2-planes $S$ contained in $R$ and for each line $L \\subset R$ there are approximately $|F|^{d-2}$ 2-planes $S$ with $L \\subset S \\subset R.$ Thus, the $d$-plane Wolff axiom asserts that for every $d$-plane $R$ the standard Wolff axiom holds ``on average'' for two-planes $S \\subset R.$ In particular, the d-plane Wolff axiom is weaker than the standard Wolff axiom when $d > 2$ (assuming one is willing to adjust the axioms by a constant factor, which would make no impact on the validity of the stated theorems). \n\n\\begin{theorem} \\label{wolffrelaxed}\nSuppose that $d > 2$ is an integer, $2(d-1) - 1 < \\alpha < 2(d-1) + 1$, $F$ is a finite field, and ${\\bf L}$ is a collection of lines in $F^n$ with $|{\\bf L}| \\geq |F|^{\\alpha}.$ If ${\\bf L}$ satisfies the $d$-plane Wolff axiom \\eqref{dpwa} then\n\\begin{equation} \\label{wolffexponent2}\n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{\\frac{\\alpha + 3}{2}}\n\\end{equation}\nwhere the implicit constant is independent of $F$.\n\\end{theorem}\n\nBounds of the form \\eqref{wolffexponent} do not seem to be sharp; at least, they can be slightly strengthened in the case when $F = \\BBZ_p$, $\\alpha = 2$, and $n=3$, see \\cite{bourgain04spe}. \n\nSince \\eqref{wolffexponent2} improves on \\eqref{dimrangeff} when $\\beta < 1$, one can use the $d$-plane Wolff axiom to extract structural information about quasi-extremizers (cf. \\cite{christ06qer}) of \\eqref{dimrangeff}.\n\n\\begin{theorem} \\label{qethm}\nSuppose $d \\geq 1$ is an integer, and that $0 \\leq \\beta < 1$. Then, for every $C$ there exist $M$ and $c > 0$ such that if $F$ is a finite field with $|F| \\geq M$ and ${\\bf L}$ is a collection of lines in $F^n$ with $|L| \\geq |F|^{2(d-1) + \\beta}$ satisfying\n\\begin{equation} \\label{qebound}\n|{\\bf P}| \\leq C  |F|^{d + \\beta}\n\\end{equation}\nwhere ${\\bf P} := \\bigcup_{L \\in {\\bf L}}L,$\nthen there are $d$-planes $R_1, \\ldots, R_N$ with $N \\geq c |F|^\\beta$ such that \n\\[\n|{\\bf P} \\cap \\bigcup_{j} R_j| \\geq c |{\\bf P}|\n\\]\nand for each $j$\n\\begin{equation} \\label{denseindplane}\n|{\\bf P} \\cap R_j| \\geq c |R_j|.\n\\end{equation}\n\\end{theorem}\n\nOne can also prove a version of the statement above for $-1 < \\beta < 0,$ but we omit the details. \n\nBy adding two additional layers of recursion, the method of Theorem \\ref{firstlinethm} can be adapted to treat unions of $k$-planes. \n\n\\begin{theorem} \\label{upthm1}\nSuppose $d \\geq k > 0$ are integers, that $0 \\leq \\beta \\leq 1$ and that ${\\bf L}$ is a collection of $k$-planes in $F^n$ with \n\\begin{equation} \\label{upthm1hyp}\n|{\\bf L}| \\geq  |F|^{(k+1)(d-k)+\\beta}.\n\\end{equation}\nThen \n\\begin{equation} \\label{unionplanesbound}\n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{d + \\beta}.\n\\end{equation}\n\\end{theorem}\n\nOur proof requires simultaneous treatment of the following more general result.\n\n\\begin{theorem} \\label{upthm2}\nSuppose $d \\geq k > k' \\geq 0$ are integers, that $0 \\leq \\beta \\leq k' + 1$ and that ${\\bf L}$ is a collection of $k$-planes in $F^n$ with \n\\[\n|{\\bf L}| \\geq  |F|^{(k+1)(d-k)+\\beta}.\n\\]\nLetting ${\\bf P}_L = \\{k'\\text{-planes\\ }P : P \\subset L\\}$ we have\n\\begin{equation} \n|\\bigcup_{L \\in {\\bf L}}{\\bf P}_L| \\gtrsim |F|^{(k' + 1)(d - k') + \\beta}.\n\\end{equation}\n\\end{theorem}\n\nTheorems \\ref{upthm1} and \\ref{upthm2} are sharp in the same sense as Theorem \\ref{firstlinethm}. For a $k$-plane analog of Wolff's theorem see \\cite{bueti06ibk}. It may be possible to modify the proof of \\eqref{unionplanesbound} to obtain a $k$-plane analog of Theorem \\ref{wolffrelaxed}, but we do not pursue the details here.\n\nThe outline of this article is as follows: Section \\ref{prsection} contains some technical machinery, Section \\ref{linesection} contains the proofs of Theorems \\ref{firstlinethm}, \\ref{wolffrelaxed}, and \\ref{qethm},  and Section \\ref{planesection} contains the proofs of Theorems \\ref{upthm1} and \\ref{upthm2}.\n\n\n\\section{Preliminaries} \\label{prsection}\n\nWe start by roughly describing the approach of \\cite{wolff95ibk}. If a union of lines is small, then there must exist a ``hairbrush'' of many lines intersecting one common line. The ambient space can then be foliated into two-dimensional planes containing the common line, and a classical bound can be applied to estimate the union of lines contained in each two-plane.  \n\nIn the present situation, we instead consider a hairbrush of many lines or $k$-planes intersecting a common $m$-dimensional plane. The following lemma is used to determine the appropriate choice of $m$.\n\n\\begin{lemma} \\label{pslemma}\nLet ${\\bf L}$ be a collection of $k$-planes in $F^n$ and suppose $d$ is a nonnegative integer with $k \\leq d \\leq n$. There is an $m$ with $k \\leq m \\leq d$, a collection of $m$-planes $R_1, \\ldots, R_N$, and collections of $k$-planes ${\\bf L}_{R_1}, \\ldots, {\\bf L}_{R_N}$ such that\n\\begin{enumerate}[(a)]\n\\item the ${\\bf L}_{R_j}$ are pairwise disjoint subsets of ${\\bf L}$ with $L \\subset R_j$ for $L \\in {\\bf L}_{R_j}$;\n\\item if $m > k$ then $|{\\bf L}_{R_j}| \\geq  |F|^{(k+1)(m - 1 - k) + k}$;\n\\item if $m=k$ then $|{\\bf L}_{R_j}| = 1$;\n\\item \\label{epcondition} letting ${\\bf L}^m = \\bigcup_{j}{\\bf L}_{R_j}$, we have $|{\\bf L}^m| \\geq 2^{-(d-m+1)}|{\\bf L}|$;\n\\item \\label{genwolffcondition} if $m < m' \\leq d$ then for every $m'$-plane $S$, $|\\{L \\in {\\bf L}^m : L \\subset S\\}| < |F|^{(k+1)(m' - 1 - k) + k} .$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSet ${\\bf L}^{*,d+1} := {\\bf L}.$ For $k \\leq m \\leq d$, suppose that ${\\bf L}^{*,m+1} \\subset {\\bf L}$ has been chosen so that $|{\\bf L}^{*,m+1}| \\geq 2^{-(d-m)} |{\\bf L}|$. Starting at $j = 1$, suppose $m$-planes $R^m_{j'}$ and collections of $k$-planes ${\\bf L}_{R^m_{j'}}$ have been selected for all integers $0 < j' < j$.\n\nIf there is an $m$-plane $R$ so that \n\\[\n|\\{L \\in {\\bf L}^{*,m+1} \\setminus \\bigcup_{j'  < j} {\\bf L}_{R^m_{j'}} : L \\subset R\\}| \\geq |F|^{(k+1)(m-1-k) + k}\n\\]\nthen let $R^m_j$ be such an $m$-plane, let\n\\[\n{\\bf L}_{R^m_j} = \\{L \\in {\\bf L}^{*,m+1} \\setminus \\bigcup_{j'  < j} {\\bf L}_{R^m_{j'}} : L \\subset R^m_j\\}\n\\]\nand continue the process with $j+1.$ If there is no such plane then terminate the process and set \n\\[\n{\\bf L}^{*,m} = {\\bf L}^{*,m+1} \\setminus \\bigcup_{j'  < j} {\\bf L}_{R^m_{j'}}.\n\\]\nIf $|{\\bf L}^{*,m}| < |{\\bf L}^{*,m+1}|\/2$ then property (\\ref{epcondition}) is satisfied and we terminate the process. Otherwise, continue with $m-1.$\n\nIf the process reaches the stage $m=k,$ then let $R^{k}_1, \\ldots, R^k_N$ be some enumeration of ${\\bf L}^{*,k+1}$ and ${\\bf L}_{R^{k}_j} = \\{R^k_j\\}$ and we are finished.\n\\end{proof}\n\nNext, we describe in detail the foliation of the ambient space.\n\n\\begin{lemma} \\label{folprop}\nSuppose that $S$ is an $m$-plane in $F^n$, that $0 \\leq q \\leq k-1,$ and that $m + k-q \\leq n.$ Then we can find $(m+k-q)$-planes $T_1, \\ldots, T_N$ with $S \\subset T_i$ for all $i$ such that for all $(k-1)$-planes $P$ and $k$-planes $L$ satisfying\n\\begin{enumerate}[(a)]\n\\item $P \\subset L$\n\\item $L \\cap S$ is a $q$-plane\n\\item $P \\cap S$ is a $(q-1)$-plane if $q > 0$ and $P \\cap S = \\emptyset$ if $q = 0$\n\\end{enumerate}\nwe have $L \\subset T_i$ for some $i$ and $P \\not\\subset T_{i'}$ for $i' \\neq i.$\n\\end{lemma}\n\n\\begin{proof}\nTo find the $T_i$, write $S=x + \\spa(e_1, \\ldots, e_m)$ and \n\\[\nF^n = \\spa(e_1, \\ldots, e_m, f_1, \\ldots f_{n-m}).\n\\]\nThen write $T_i = S + V_i$ where, as $i$ varies, $V_i$ ranges over all $(k-q)$-dimensional subspaces of $\\spa(f_1, \\ldots, f_{n-m}).$\n\nFix some $P,L$ satisfying the hypotheses. One can check that there is an $i$ such that $L \\subset T_i.$ For any $i' \\neq i$ we have that $T_i \\cap T_{i'}$ contains $S$ and is, at most, an $(m +  k-q - 1)$-plane. \n\nFirst consider the case $q > 0.$ Choose $y \\in P \\cap S$ and write \n\\begin{align*}\nP \\cap S &= y + \\spa(g_1, \\ldots, g_{q-1}),\\\\\nP &= y + \\spa(g_1, \\ldots, g_{q-1}, h_1, \\ldots, h_{k-q}),\\\\\n\\intertext{and}\nS &= y + \\spa(g_1, \\ldots, g_{q-1}, h'_{1}, \\ldots, h'_{m + 1 -q}).\n\\end{align*}\nClearly, \n\\[\nW := \\{g_1, \\ldots, g_{q-1}, h_1, \\ldots, h_{k-q}, h'_{1}, \\ldots, h'_{m + 1 -q } \\}\n\\]\nis linearly independent and, since $S \\subset T_i \\cap T_{i'}$, $P \\subset T_i \\cap T_{i'}$ would imply $y + \\spa(W) \\subset T_i \\cap T_{i'}$ contradicting the dimension estimate on the latter set.  \n\nFor the case $q=0$ write \n\\begin{align*}\nP &= z + \\spa(h_1, \\ldots, h_{k-1})\\\\\n\\intertext{and}\nS &= y + \\spa(h'_1, \\ldots, h'_m).\n\\end{align*}\nAgain using the dimension estimate on $T_i \\cap T_{i'}$, we see that if $P \\subset T_{i} \\cap T_{i'}$ then, since $P \\cap S = \\emptyset,$ we have $v \\in \\spa(h_1', \\ldots, h'_m)$ for some $0 \\neq v \\in \\spa(h_1, \\ldots, h_{k-1}).$ But, since $L \\cap S \\neq \\emptyset$, this implies $\\dim(L \\cap S) > 0$, contradicting the assumption that $\\dim(L \\cap S) = 0.$\n\\end{proof}\n\nTo estimate the union of lines or $k$-planes contained in each leaf of the foliation, we will appeal to recursion. However,  at the root we still use the classical method:\n \n\\begin{lemma} \\label{classmethod}\nSuppose that ${\\bf L}$ is a collection of $m$-planes, ${\\bf P}$ is a collection of $(k-1)$-planes such that for every $L \\in {\\bf L}$\n\\[\n|\\{P \\in {\\bf P} : P \\subset L\\}| \\geq M,\n\\]\nand \n\\begin{equation} \\label{mustpruneeq}\n|{\\bf L}| |F|^{k(m-k)} \\leq M.\n\\end{equation}\nThen\n\\[\n|{\\bf P}| \\gtrsim |{\\bf L}| M.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nFor each $L \\in {\\bf L}$, let ${\\bf P}_L$ be a subset of $\\{P \\in {\\bf P} : P \\subset L\\}$ with $M \\leq {\\bf P}_L \\leq 2M.$ Set\n\\[\n{U} = \\{(P,L,L')  : (L,L') \\in {\\bf L}^2, P \\in {\\bf P}_L \\cap {\\bf P}_{L'} \\}.\n\\]\nAn application of Cauchy-Schwarz gives \n\\[\n|{U}| \\geq \\frac{M^2|{\\bf L}|^2}{|{\\bf P}|}.\n\\]\nAny two distinct $m$-planes intersect in, at most, an $(m-1)$-plane. Since, by  \\eqref{plcount} below, an $(m-1)$-plane contains $\\lesssim |F|^{k(m-k)}$ $(k-1)$-planes, we have \n\\begin{align*}\n|{U}| &\\leq C |{\\bf L}|^2 |F|^{k(m-k)} + |\\{(P,L) : L \\in {\\bf L}, P \\in {\\bf P}_L\\}| \\\\\n&\\leq C|{\\bf L}|^2|F|^{k(m-k)} + |{\\bf L}| 2 M \\\\\n&\\lesssim |{\\bf L}|M.\n\\end{align*}\n\\end{proof}\n\nWe finish the section with three standard estimates for collections of planes.  \n\n\\begin{lemma} \\label{grascount}\nFor integers $0 \\leq k \\leq d$ we have\n\\begin{equation} \\label{sscount}\n|G(d,k)| \\approx |F|^{k(d-k)}\n\\end{equation}\nand\n\\begin{equation} \\label{plcount}\n|G'(d,k)| \\approx |F|^{(k+1)(d-k)}\n\\end{equation}\nwhere $G(d,k)$ is the set of $k$-dimensional subspaces of $F^d$ and $G'(d,k)$ is the set of $k$-planes in $F^d.$ \n\\end{lemma}\n\n\\begin{proof}\nA generic choice of $k$ vectors in $F^d$ is linearly independent, and there are approximately $|F|^{kd}$ such choices. By the same logic, each $k$-plane has approximately $F^{k^2}$ choices of basis, and hence we have \\eqref{sscount}. \n\nGiven a $k$-dimensional subspace $P$ with basis $e_1, \\ldots, e_k$, choose \\\\ $f_1, \\ldots, f_{d-k}$ so that $F^d = \\spa(e_1, \\ldots, e_k, f_1, \\ldots, f_{d-k})$. Then there is a one-to-one correspondence between linear combinations of $f_1, \\ldots, f_{d-k}$ and distinct translates of $P$, giving \\eqref{plcount}.\n\\end{proof}\n\n\n\\begin{lemma} \\label{sccprop}\nSuppose that $S$ is an $l$-plane in $F^m.$ Then for $l < l' \\leq m$\n\\begin{equation} \\label{scontainedcount}\n|\\{P \\subset F^m : P \\text{\\ is a\\ } l'\\text{-plane\\ and\\ } S \\subset P\\}| \\lesssim |F|^{(l' - l)(m - l')}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWrite $S = x + \\spa(e_1, \\ldots, e_l)$ and \n\\[\nF^m = \\spa(e_1, \\ldots, e_l, f_1, \\ldots, f_{m-l}).\n\\] \nThen there is a one-to-one correspondence between $l'$-planes $P \\supset S$ and $(l'-l)$-dimensional subspaces of $\\spa(f_1, \\ldots, f_{m-l})$, so \\eqref{scontainedcount} follows from \\eqref{sscount}.\n\\end{proof}\n\n\\begin{lemma} \\label{eiprop}\nSuppose $S$ is an $l$-plane in $F^k.$ Then\n\\[\n|\\{P \\subset F^k : P \\text{\\ is a\\ } (k-1)\\text{-plane\\ and\\ } P \\cap S = \\emptyset\\}| \\lesssim |F|^{k-l}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWrite $S = x + \\spa(e_1, \\ldots, e_l)$, and fix $P = y + \\spa(f_1, \\ldots, f_{k-1}).$ If $P \\cap S = \\emptyset$, we must have $e_j \\in \\spa(f_1, \\ldots, f_{k-1})$ for $j = 1, \\ldots, l.$ Thus, $P$ is a translate of a $(k-1)$-plane containing $S$. Since, by Lemma \\ref{sccprop}, there are $\\lesssim |F|^{k-l-1}$ $(k-1)$-planes containing $S$, we have at most $|F|^{k-l}$ possible planes $P$. \n\\end{proof}\n\n\n\\section{Unions of lines} \\label{linesection}\nTheorems \\ref{firstlinethm} and \\ref{wolffrelaxed} follow immediately from:\n\\begin{proposition} \\label{linetheorem}\nSuppose $d \\geq 1$ is an integer, that $0 < \\gamma,\\lambda \\leq 1$, that $\\max(1 - d,-1) \\leq \\beta \\leq 1,$ that ${\\bf L}$ is a collection of lines in $F^n$ with \n\\[\n|{\\bf L}| \\geq \\gamma |F|^{2(d-1)+\\beta},\n\\]\nand that ${\\bf P}$ is a collection of points in $F^n$ satisfying \n\\[\n|\\{P \\in {\\bf P} : P \\in L\\}| \\geq \\lambda |F|\n\\]\nfor every $L \\in {\\bf L}.$ Then\n\\begin{equation} \\label{nondpwaresult}\n|{\\bf P}| \\gtrsim  |F|^{d +\\max(0,\\beta)}.\n\\end{equation}\nwhere the implicit constant may depend on $d,\\gamma,\\lambda.$ Furthermore, if $d \\geq 2$ and ${\\bf L}$ satisfies the $d$-plane Wolff axiom \\eqref{dpwa} then we have\n\\begin{equation} \\label{dpwaresult}\n|{\\bf P}| \\gtrsim |F|^{d + \\frac{\\beta + 1}{2}}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem \\ref{qethm} assuming Proposition \\ref{linetheorem}]\nSuppose that ${\\bf L}$ satisfies \\eqref{qebound} and that $|F| \\geq M$ where $M$ is large and to be determined later.\n\n For $d$-planes $R$ let ${\\bf L}_R = \\{L \\in {\\bf L} : L \\subset R\\}$. Choose $d$-planes $R_1, \\ldots, R_N$ so that for each $j$, $|{\\bf L}_{R_j}| \\geq |F|^{2d-3}$ and such that for $R \\not\\in \\{R_j\\}_{j=1}^N$ we have $|{\\bf L}_{R}| < |F|^{2d-3}. $\n\nSetting ${\\bf L}' = \\bigcup_{j}{\\bf L}_{R_j}$, ${\\bf L}'' = {\\bf L} \\setminus {\\bf L}'$ and ${\\bf P}'' = \\bigcup_{L \\in {\\bf L}''}L$, we must have $|{\\bf L}''| < \\frac{1}{2}|{\\bf L}'|$ or \nelse we would have\n\\[\n|{\\bf P}| \\geq |{\\bf P}''| \\geq c' |F|^{d + \\frac{\\beta + 1}{2}} > C |F|^{d+\\beta}\n\\]\nwhere $c'$ is the implicit constant from \\eqref{dpwaresult} and $M$ is chosen large enough to overwhelm $\\frac{C}{c'}$ (In the second inequality above we have used the fact that ${\\bf L}''$ satisfies \\eqref{dpwa} to obtain \\eqref{dpwaresult} from Proposition \\ref{linetheorem}.) \n\nThus, $|{\\bf L}'| \\geq \\frac{1}{2}|{\\bf L}|$ and, by \\eqref{plcount} with $k=1$, $N \\geq c |F|^{\\beta}.$ Letting ${\\bf P}' = \\bigcup_{L \\in {\\bf L}'}L$ we have\n\\[\n|{\\bf P}'| \\geq c'' |F|^{d + \\beta} \\geq c C|F|^{d + \\beta} \\geq c |{\\bf P}|\n\\]\nwhere $c''$ is the implicit constant from \\eqref{nondpwaresult}, $c$ is chosen small enough to underwhelm $\\frac{c''}{C},$ and the last inequality follows from \\eqref{qebound}.\n\nA final application of $\\eqref{nondpwaresult}$ with $d' = d-1$ and $\\beta' = 1$ then gives \\eqref{denseindplane} since $|{\\bf L}_{R_j}| \\geq |F|^{2(d-1-1)+1}.$\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{linetheorem}]\nWe use induction on $d$. When $d=1$ the result follows from an application of Lemma \\ref{classmethod} (with $m=1$, $k=1$, and ${\\bf L}$ replaced by a subset of itself with cardinality $\\approx \\min(\\gamma,\\lambda) |F|^\\beta$ ), so we will prove it for $d > 1$, working under the assumption that it has already been proven for $1 \\leq d' < d.$\n\nAfter possibly deleting lines, we may assume\n\\begin{equation} \\label{refinedkk1}\n|{\\bf L}| < 2\\gamma |F|^{2(d-1)+\\beta}.\n\\end{equation}\nApplying Lemma \\ref{pslemma} to ${\\bf L}$ we obtain $m$-planes $R_1, \\ldots, R_N.$ Note that if $L$ satisfies the $d$-plane Wolff axiom \\eqref{dpwa} then we must have $m < d.$ \n\n\\ \n\n\\noindent{\\bf Case 1, $(m=d)$:}\\\\ \nLet ${\\bf P}_{R_j} = \\{P \\in {\\bf P} : P \\in R_j\\}.$ Applying the case $d' = d-1$ of the proposition to the ${\\bf L}_{R_j}$, we deduce that $|{\\bf P}_{R_j}| \\gtrsim |F|^{d}$ for each $j$. Since, by \\eqref{plcount}, $|{\\bf L}_{R_j}| \\lesssim |F|^{2(d-1)}$, we have $N \\gtrsim |F|^{\\max(0,\\beta)}.$ Then using Lemma \\ref{classmethod} (possibly applying it to a subset of $\\{R_j\\}_{j=1}^N$ in order to satisfy \\eqref{mustpruneeq}) to estimate $|\\bigcup_{j}{\\bf P}_{R_j}|$ shows that \n\\[\n|{\\bf P}| \\gtrsim  |F|^{d + \\max(0,\\beta)}\n\\]\nas desired.\n \n\\ \n\n\\noindent{\\bf Case 2, $(m < d)$:}\\\\\nWe construct sets of lines and points using a standard ``popularity'' argument. Fix some large $C$ to be determined later and let \n\\[\n{\\bf P}^{\\sharp} = \\{P \\in {\\bf P} : |\\{L \\in {\\bf L} : P \\in L\\}| \\geq C |F|^{d-1 - \\frac{1-\\beta}{2}}\\}\n\\]\nand\n\\[\n{\\bf L}^{\\sharp} = \\{L \\in {\\bf L} : |\\{P \\in {\\bf P}^{\\sharp} : P \\in L\\}| \\geq \\frac{1}{4} \\lambda |F|\\}.\n\\]\nLetting ${\\bf L}^m$ be as in Lemma \\ref{pslemma}, we then have either \n\\begin{align} \\label{firstpossibilityk1}\n|{\\bf P}| &\\geq \\frac{1}{2}\\lambda |F|  |{\\bf L}^m|\/(C|F|^{(d-1) - \\frac{1-\\beta}{2}})\n\\end{align}\n(in which case we are finished since the right hand side above is $\\gtrsim |F|^{d + \\frac{\\beta+1}{2}}$) or\n\\begin{equation} \\label{secondpossibilityk1}\n|{\\bf L}^{\\sharp}| \\geq \\frac{1}{8} |{\\bf L}^m|\n\\end{equation}\n\nIndeed, suppose that \\eqref{firstpossibilityk1} does not hold. Set\n\\[\nI = \\{(P,L) : P \\in {\\bf P}_L, L \\in {\\bf L}^{m}\\}\n\\]\nwhere, for each $L$, ${\\bf P}_L$ is a subset of ${\\bf P} \\cap L$ with $\\lambda |F| \\leq |{\\bf P}_L|  < 2 \\lambda |F|.$ \nThen letting \n\\[\nI' = \\{(P,L) : P \\in {\\bf P}_L \\setminus {\\bf P}^{\\sharp} , L \\in {\\bf L}^{m}\\}\n\\]\nwe have\n\\[\n|I'| < C |F|^{d-1 - \\frac{1-\\beta}{2}} |{\\bf P}| < \\frac{1}{2} |I|\n\\]\nand so\n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp}, L \\in {\\bf L}^{m}\\}| \\geq \\frac{1}{2} \\lambda |F| |{\\bf L}^{m}|\n\\]\ngiving \n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp}, L \\in {\\bf L}^{\\sharp}\\}| \\geq \\frac{1}{4} \\lambda |F||{\\bf L}^{m}|\n\\]\nthus leading (by the upper bound on $|{\\bf P}_L|$) to \\eqref{secondpossibilityk1} as claimed.\n\nLet ${\\bf L}^{\\sharp}_{R_j} = {\\bf L}_{R_j} \\cap {\\bf L}^{\\sharp}$, ${\\bf P}^{\\sharp}_{R_j} = R_j \\cap {\\bf P}^{\\sharp}$, \n\\[\n{\\bf L}'_{R_j} = \\{L \\in {\\bf L}^m : |L \\cap R_j| = 1\\},\n\\]\nand\n\\[\n{\\bf P}'_{R_j} = {\\bf P} \\setminus R_j.\n\\]\nFix $j$ so that $|{\\bf L}^{\\sharp}_{R_j}| \\gtrsim |{\\bf L}_{R_j}|,$ and recall $|{\\bf L}_{R_j}| \\geq |F|^{2m-3}$ by Lemma \\ref{pslemma}.\nApplying the previously known case $d' = m-1$ of the proposition (or using the trivial estimate if $m=1$) we have\n\\begin{equation} \\label{lequalszeroPk1}\n|{\\bf P}^{\\sharp}_{R_j}| \\gtrsim |F|^{m}.\n\\end{equation}\n\nFor each point $P \\in {\\bf P}^{\\sharp}_{R_j}$ there are $\\geq C |F|^{d-1 -  \\frac{1-\\beta}{2}}$ lines from ${\\bf L}^{m}$ intersecting $P$. Since, by Lemma \\ref{sccprop}, $\\lesssim |F|^{m-1}$ of these lines are contained in $R_j$, we have\n\\[\n|\\{L \\in {\\bf L}'_{R_j} : P \\in L\\}| \\geq \\frac{1}{2} |F|^{d - 1 - \\frac{1-\\beta}{2}}\n\\]\nprovided that $C$ is chosen sufficiently large. \nThus, \n\\begin{align*}\n|{\\bf L}'_{R_j}| &\\gtrsim |{\\bf P}^{\\sharp}_{R_j}| |F|^{d - 1 - \\frac{1 - \\beta}{2}} \\\\\n&\\gtrsim |F|^{m + d-1 - \\frac{1-\\beta}{2}}.\n\\end{align*}\nNote\\footnote{We may assume throughout that $|F|$ is sufficiently large relative to certain parameters (for instance $\\lambda$) since the implicit constants may be chosen so that the conclusion holds trivially for small $|F|$.} that for each $L \\in {\\bf L}'_{R_j}$, $|L \\cap {\\bf P}'_{R_j}| \\geq \\lambda |F| - 1 \\gtrsim |F|.$\n\n\nApplying Lemma \\ref{folprop}, we write $F^n$ as the union of $(m + 1)$-planes $T_i$ containing $R_j$. Let\n\\begin{align*}\n{\\bf L}_i &= \\{L \\in {\\bf L}'_{R_j} : L \\subset T_i\\} \\\\ \n{\\bf P}_i &= \\{P \\in {\\bf P}'_{R_j} : P \\in T_i\\}.\n\\end{align*}\nThen \n\\begin{align*}\n|{\\bf P}| &\\geq \\sum_i |{\\bf P}_i| \\\\ \n&\\gtrsim \\sum_i |{\\bf L}_i|\/|F|^{m-2} \\\\ \n&\\geq |{\\bf L}'_{R_j}|\/|F|^{m - 2}\\\\ \n&\\gtrsim |F|^{d +\\frac{\\beta+1}{2}}\n\\end{align*}\nwhere, for the second inequality, we used the fact (which follows from Lemma \\ref{pslemma}) that $|{\\bf L}_i| < |F|^{2(m -1) + 1} $ to see that $|{\\bf L}_i| = |F|^{2(d' - 1) + \\beta'}$ for some $d' \\leq m$ and so we can estimate $|{\\bf P}_i|$ using the previously known case $d'$ of \\eqref{nondpwaresult}.\n\\end{proof}\n\n\\section{Unions of Planes} \\label{planesection}\nTheorem \\ref{upthm2} is obtained by induction from the hyperplane case:\n\\begin{proposition} \\label{maintheoremcd1}\nSuppose $d,k > 0$ are integers, that $0 < \\gamma,\\lambda \\leq 1$, that  $d \\geq k$, that $0 \\leq \\beta \\leq k$, that ${\\bf L}$ is a collection of $k$-planes in $F^n$ with \n\\[\n|{\\bf L}| \\geq \\gamma |F|^{(k+1)(d-k)+\\beta},\n\\]\nand that ${\\bf P}$ is a collection of $(k-1)$-planes in $F^n$ satisfying \n\\[\n|\\{P \\in {\\bf P} : P \\subset L\\}| \\geq \\lambda |F|^{k}\n\\]\n for every $L \\in {\\bf L}.$ Then\n\\[\n|{\\bf P}| \\gtrsim  |F|^{k(d - k + 1)+\\beta}\n\\]\nwhere the implicit constant may depend on $d,\\gamma,\\lambda, k.$\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem \\ref{upthm2} assuming Proposition \\ref{maintheoremcd1}]\nTheorem \\ref{upthm2} follows directly from Proposition \\ref{maintheoremcd1} when $k-k' = 1$. Fix $k_0 \\geq 1$ and assume that the theorem holds for all $k-k' = k_0,$ and fix some $k,k'$ with $k-k' = k_0 + 1.$ \n\nFor $L \\in {\\bf L}$ satisfying \\eqref{upthm1hyp} let \n\\[\n{\\bf P}'_L = \\{(k'+1)\\text{-planes\\ }P' : P' \\subset L\\}\n\\] \nand for any $(k'+1)$-plane $P'$ let \n\\[\n{\\bf P}''_{P'} = \\{k'\\text{-planes\\ }P : P \\subset P'\\}.\n\\]\nApplying the previously known case of the theorem, we have\n\\[\n|\\bigcup_{L \\in {\\bf L}}{\\bf P}'_L| \\gtrsim |F|^{(k' + 2)(d - (k' + 1)) + \\beta}\n\\]\nand thus a second application of Proposition \\ref{maintheoremcd1} gives\n\\[\n|\\bigcup_{L \\in {\\bf L}}{\\bf P}_L| = |\\bigcup_{L} \\bigcup_{P' \\in {\\bf P}'_L} {\\bf P}''_{P'}| \\gtrsim |F|^{(k' + 1)(d - k') + \\beta}.\n\\]\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{maintheoremcd1}]\nWe use induction on $d$. When $d=k$ the result follows from an application of Lemma \\ref{classmethod} (with $m=k$). So we will prove it for $d > k$, working under the assumption that it has already been proven for $k \\leq d' < d.$\n\nAfter possibly deleting planes, we may assume\n\\begin{equation} \\label{refinedk}\n|{\\bf L}| < 2\\gamma |F|^{(k+1)(d-k)+\\beta}.\n\\end{equation}\nApplying Lemma \\ref{pslemma} to ${\\bf L}$ we obtain $m$-planes $R_1, \\ldots, R_N.$\n\n\\ \n\n\\noindent{\\bf Case 1, $(m=d)$:}\\\\ \nLet ${\\bf P}_{R_j} = \\{P \\in {\\bf P} : P \\subset R_j\\}.$ Applying the case $d' = d-1$ of the theorem to the ${\\bf L}_{R_j}$, we deduce that $|{\\bf P}_{R_j}| \\gtrsim |F|^{k(d-k+1)}$ for each $j$. Since, by \\eqref{plcount}, $|{\\bf L}_{R_j}| \\lesssim |F|^{(k+1)(d-k)}$, we have $N \\gtrsim |F|^{\\beta}.$ Using Lemma \\ref{classmethod} to estimate $|\\bigcup_{j}{\\bf P}_{R_j}|,$ we conclude\n\\[\n|{\\bf P}| \\gtrsim  |F|^{k(d-k+1)+\\beta}\n\\]\nas desired.\n \n\\ \n\n\\noindent{\\bf Case 2, $(m < d)$:}\\\\\nWe construct sets of $k$-planes and $(k-1)$-planes using a standard ``iterated-popularity'' argument. Fix some large $C$ to be determined later. Let ${\\bf L}^{\\sharp,0} = {\\bf L}^m$, ${\\bf P}^{\\sharp,0} = {\\bf P}$ and for $1 \\leq q \\leq k$ let\n\\[\n{\\bf P}^{\\sharp,q} = \\{P \\in {\\bf P}^{\\sharp,q-1} : |L \\in {\\bf L}^{\\sharp, q-1} : P \\subset L| \\geq C|F|^{d-k}\\}\n\\]\nand\n\\[\n{\\bf L}^{\\sharp,q} = \\{L \\in {\\bf L}^{\\sharp,q-1} : |P \\in {\\bf P}^{\\sharp,q} : P \\subset L| \\geq 2^{-2q} \\lambda |F|^k\\}.\n\\]\nThen either \n\\begin{align} \\label{firstpossibility}\n|{\\bf P}| &\\geq 2^{-5k}\\lambda |F|^k  |{\\bf L}^m|\/(C|F|^{d-k})\n\\end{align}\n(in which case we are finished since the right hand side above is $\\gtrsim |F|^{k(d-k+1) + \\beta}$) or for each $q \\leq k$\n\\begin{equation} \\label{secondpossibility}\n|{\\bf L}^{\\sharp,q}| \\geq 2^{-3q} |{\\bf L}^m|.\n\\end{equation}\n\nIndeed, suppose that \\eqref{firstpossibility} does not hold and that \\eqref{secondpossibility} holds for $0 \\leq q \\leq q_0 < k$. Set \n\\[\nI = \\{(P,L) : P \\in {\\bf P}_L, L \\in {\\bf L}^{\\sharp,q_0}\\}\n\\]\nwhere, for each $L$, ${\\bf P}_L$ is a subset of ${\\bf P}^{\\sharp,q_0}$ with $P \\subset L$ for every $P \\in {\\bf P}_L$ and $2^{-2q_0} \\lambda |F|^k \\leq |{\\bf P}_L|  < 2^{-(2q_0-1)} \\lambda |F|^k.$ Note\n\\[\n|I| \\geq 2^{-2q_0} \\lambda |F|^k |{\\bf L}^{\\sharp,q_0}| \\geq 2^{-5q_0} \\lambda |F|^k |{\\bf L}^m|.\n\\]\nThen letting \n\\[\nI' = \\{(P,L) : P \\in {\\bf P}_L \\setminus {\\bf P}^{\\sharp,q_0 + 1} , L \\in {\\bf L}^{\\sharp,q_0}\\}\n\\]\nwe have\n\\[\n|I'| < C|F|^{d-k} |{\\bf P}| \\leq 2^{-5(k - q_0)} |I| \\leq \\frac{1}{2} |I|\n\\]\nand so\n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp,q_0 + 1}, L \\in {\\bf L}^{\\sharp,q_0}\\}| \\geq \\frac{1}{2} 2^{-2q_0} \\lambda |F|^k |{\\bf L}^{\\sharp,q_0}|.\n\\]\nThis gives\n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp,q_0 + 1}, L \\in {\\bf L}^{\\sharp,q_0+1}\\}| \\geq \\frac{1}{4} 2^{-2q_0} \\lambda |F|^k |{\\bf L}^{\\sharp,q_0}|\n\\]\nthus leading (by the upper bound on $|{\\bf P}_L|$) to \n\\[\n|{\\bf L}^{\\sharp,q_0 + 1}| \\geq \\frac{1}{8} |{\\bf L}^{\\sharp,q_0}|\n\\]\nas claimed.\n\nFor each $1 < q \\leq k$ let \n\\begin{align*}\n{\\bf L}^{\\sharp,q}_{R_j} &= \\{L \\in {\\bf L}^{\\sharp,q} : L \\cap R_j \\text{\\ is a }q\\text{-plane}\\} \\\\\n{\\bf P}^{\\sharp,q}_{R_j} &= \\{P \\in {\\bf P}^{\\sharp,q} : P \\cap R_j \\text{\\ is a }(q-1)\\text{-plane}\\}.\n\\end{align*}\nDefine ${\\bf L}^{\\sharp,0}_{R_j}$ as above and let ${\\bf P}^{\\sharp,0}_{R_j} = \\{P \\in {\\bf P}^{\\sharp,0} : P \\cap R_j = \\emptyset\\}.$\n\nLetting ${\\bf L}_{R_j}$ be as in Lemma 2.1, fix $j$ so that $|{\\bf L}^{\\sharp,k}_{R_j}| \\gtrsim |{\\bf L}_{R_j}|.$\nApplying the previously known case $d' = m-1$ of the theorem (or using the trivial estimate if $m=k$) we have\n\\begin{equation} \\label{lequalszeroP}\n|{\\bf P}^{\\sharp,k}_{R_j}| \\gtrsim |F|^{k(m-k+1)}.\n\\end{equation}\n\nSuppose for some $1 \\leq q \\leq k$ that  \n\\begin{equation} \\label{equationA}\n|{\\bf P}^{\\sharp,q}_{R_j}| \\gtrsim |F|^{\\omega}.\n\\end{equation}\nOur immediate goal is to see that under certain conditions, \\eqref{equationA} implies \\eqref{equationB} and \\eqref{equationstar} below. For each $P \\in {\\bf P}^{\\sharp,q}_{R_j}$ there are $\\geq C |F|^{d-k}$ $k$-planes from ${\\bf L}^{\\sharp,q-1}$ containing $P$. Since, by Lemma \\ref{sccprop}, $\\lesssim |F|^{m-q}$ of these $k$-planes intersect $R_j$ in $q$-planes (and there is no possibility that for a $k$-plane $L \\supset P$ we have $\\dim(L \\cap R_j) > q$, since $\\dim(L) = \\dim(P) + 1$), we have\n\\[\n|\\{L \\in {\\bf L}^{\\sharp,q-1}_{R_j} : P \\subset L\\}| \\geq \\frac{1}{2} C |F|^{d - k}\n\\]\nprovided that $C$ is chosen sufficiently large and \n\\begin{equation} \\label{firstiterationrequirement}\nm + k-q \\leq d . \n\\end{equation}\nThus, using the fact (which follows from Lemma \\ref{sccprop}) that for each $(q-1)$-plane $S \\subset L$ there are at most $|F|^{k-q}$ $(k-1)$-planes $P$ with $S \\subset P \\subset L$, we have\n\\begin{equation} \\label{equationB}\n|{\\bf L}^{\\sharp,q-1}_{R_j}| \\gtrsim C |F|^{\\omega + (d-k) - (k-q)}\n\\end{equation}\nassuming \\eqref{firstiterationrequirement}. \n\nIt follows from the definition of ${\\bf L}^{\\sharp,q-1}$ that for each $L \\in {\\bf L}^{\\sharp,q-1}_{R_j}$\n\\[\n|\\{P \\in {\\bf P}^{\\sharp,q-1}: P \\subset L\\}| \\gtrsim |F|^k.\n\\]\nThus, using Lemma \\ref{sccprop} to obtain\n\\[\n|\\{P \\subset L : (L \\cap R_j) \\subset P\\}| \\lesssim |F|^{k-q} \\ll |F|^k\n\\]\nand (for $q > 1$) Lemma \\ref{eiprop} to obtain\n\\begin{equation} \\label{eipropapp}\n|\\{P \\subset L : (L \\cap R_j) \\cap P = \\emptyset\\}| \\lesssim |F|^{k-q+1} \\ll |F|^k\n\\end{equation}\nit follows that\n\\begin{equation} \\label{rieqn}\n|\\{P \\in {\\bf P}^{\\sharp,q-1}_{R_j}: P \\subset L\\}| \\gtrsim |F|^k.\n\\end{equation}\nEstimate \\eqref{rieqn} also holds when $q=1$, since the special definition of ${\\bf P}^{\\sharp,0}_{R_j}$ means that we do not need to use \\eqref{eipropapp}.\n\nApplying Lemma \\ref{folprop}\ngives $(m + k -q  + 1)$-planes $T_i$ containing $R_j$. Let\n\\begin{align*}\n{\\bf L}_i &= \\{L \\in {\\bf L}^{\\sharp,q-1}_{R_j} : L \\subset T_i\\} \\\\ \n{\\bf P}_i &= \\bigcup_{L \\in {\\bf L}_i}\\{P \\in {\\bf P}^{\\sharp,q-1}_{R_j}: P \\subset L\\}.\n\\end{align*}\nThen assuming \n\\begin{equation} \\label{seconditerationrequirement}\nm + k -q + 1\\leq d\n\\end{equation}\nwe have\n\\begin{align}\n\\nonumber |{\\bf P}^{\\sharp,q-1}_{R_j}| &\\geq \\sum_i |{\\bf P}_i| \\\\ \n\\nonumber&\\gtrsim \\sum_i |{\\bf L}_i|\/|F|^{m-q-k} \\\\ \n\\nonumber&\\geq |{\\bf L}^{\\sharp,q-1}_{R_j}|\/|F|^{m-q - k}\\\\ \n\\label{equationstar}&\\gtrsim C |F|^{\\omega + d - (m+k-q) + q}\n\\end{align}\nwhere, for the second inequality, we used the fact (which follows from Lemma \\ref{pslemma}) that $|{\\bf L}_i| < |F|^{(k+1)(m -q) + k} $ to see that for some $d' \\leq m+k-q$ we can estimate $|{\\bf P}_i|$ using the previously known case $d'$ of the theorem.\n\nStarting with \\eqref{lequalszeroP} and iterating the fact that \\eqref{equationA} implies \\eqref{equationB} and \\eqref{equationstar}, we obtain for $0 \\leq q \\leq k$\n\\[\n|{\\bf P}^{\\sharp,q}_{R_j}| \\gtrsim C^{k-q} |F|^{k + q(m-k) + (k-q)d - (k-q)(k-q-1)}\n\\]\nif $m + k-q \\leq d$ and\n\\[\n|{\\bf L}^{\\sharp,q}_{R_j}| \\gtrsim C^{k-q} |F|^{k + (q+1)(m-k) + (k-q-1)d - (k-q-1)(k-q-2) + (d - k) - (k-q-1)} \n\\]\nif $m + k-q - 1 \\leq d.$\n\nSo if $m+k \\leq d$ we have\n\\[\n|{\\bf P}| \\geq |{\\bf P}^{\\sharp,0}_{R_j}| \\gtrsim |F|^{k(d - k + 1) + k}\n\\]\nas desired, and otherwise we have\n\\begin{equation} \\label{toomanykplanes}\n|{\\bf L}| \\geq |{\\bf L}^{\\sharp,k-(d + 1 - m)}_{R_j}| \\gtrsim C^{d+1-m} |F|^{(k+1)(d-k) + k}.\n\\end{equation}\nChoosing $C$ sufficiently large depending on the relevant implicit constants (none of which depended on $C$), we see that \\eqref{toomanykplanes} contradicts the assumption \\eqref{refinedk} and so we must have \\eqref{firstpossibility}.\n\\end{proof}\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent\nExplosive nuclear burning in astrophysical environments produces\nunstable nuclei which again can be targets for subsequent reactions.\nMost of these nuclei are not accessible in terrestrial laboratories or not fully\nexplored by experiments, yet.\n\nApproximately half of all stable nuclei observed in nature in the heavy\nelement region $A>60$ were produced in the so-called r-process (i.e.,\nrapid neutron capture process), which is believed to occur\nin type II supernova explosions (see e.g., \\cite{cowan,ohurev}).\nAn environment with a high neutron density is the prerequisite for such an\nr-process, in which heavier elements are built up from seed elements by\nconsecutive neutron captures and $\\beta$-decays. Because of the abundant\nneutrons, a multitude of neutron captures ($\\simeq 15-35$)\nmay occur until the $\\beta$-decay\nhalf-life becomes shorter than the half-life against neutron capture. Thus,\nthe r-process path along which reactions take place, is pushed off the\nregion of stability towards neutron-rich unstable nuclei. The location of\nthe path has consequences for the resulting nuclear abundances, calculated in\nastrophysical models~\\cite{friedel,chen96}.\n\nFor most of the required neutron capture cross sections the statistical\nmodel (compound nucleus (CN) mechanism, Hauser-Feshbach approach)\ncan be applied. This\nmodel employs a statistical average over resonances, for which one has to\nknow level densities but not necessarily exact excitation energies and level\nspin assignments. However, the criterion for the applicability of that model\nis a sufficiently high level density. Especially for some light nuclei it\nhas been known for years that the statistical model cannot be applied and that\nthe direct capture (DC) mechanism dominates the cross sections. Nevertheless,\nit has only been realized recently that also for intermediate and heavy \nnuclei the\ndirect mechanism can become important near shell closures and for neutron-rich\nisotopes when the level density becomes too low for the CN\nmechanism. When approaching the drip-line, neutron separation energies\ndecrease and the nuclei become less deformed, both leading to a smaller\nlevel density at the relevant projectile energy. This relevant energy is\ndetermined by the peak\n$E=kT$ of the Maxwell-Boltzmann velocity distribution of the neutron gas.\nIf a segment of the r-process path at a given element lies close enough\nto the drip-line, the statistical model will not be applicable anymore and\nthe DC reactions will dominate~\\cite{mat83,gor97}.\n\nThe relation between DC and CN mechanisms has already been studied\nfor neutron capture by light\nand intermediate target nuclei~\\cite{ohurev,bal,gru,mei,meis,bee,kra}.\nInvestigations of the dependence of the level density on charge and mass number\nand a discussion of the applicability of the statistical model have been\ngiven elsewhere~\\cite{tfkl1}. In this paper we want to investigate\ndirect neutron capture on neutron-rich Sn and Pb isotopes with the emphasis\non discussing the difficulties, the level of reliability as well as\nthe predictive power of\ntheoretical calculations.\n\nThe main problem for the DC predictions is that neutron separation\nenergies and level properties (excitation energies, spins, parities)\nhave to be\nknown accurately, contrary to a statistical calculation in which it is\nsufficient to know the level density. As\nin the foreseeable future one can not expect\nany experimental\ninformation for the majority of nuclei close to the drip-line,\none has to turn to theory for providing the input for the DC\ncalculations. At the moment, there are several microscopic and\nmacroscopic-microscopic descriptions competing in the quest for predicting\nnuclear properties far off stability. For the first time, in this work we\nwant to investigate the difference in the level structure between several\nmodels and its impact on predicted neutron capture cross sections. The\ncompared models are a Hartree-Fock-Bogoliubov (HFB) model with the Skyrme SkP\nforce~\\cite{doba1,doba}, a relativistic mean field theory (RMFT) with the \nparameter set NLSH~\\cite{sharma1,sharma2}, and the macroscopic-microscopic\nfinite-range droplet model FRDM (1992) which was also used in\ncalculations of nuclear ground-state masses and \ndeformations~\\cite{moeller,moell2} and in calculations of quantities of\nastrophysical interest~\\cite{moellkra}.\n\nIn Section~\\ref{secDC} we very briefly introduce the method of the DC\ncalculation and Section~\\ref{secMic} gives an overview of the utilized\nmicroscopic models. For $^{208}$Pb, the DC results can directly be compared to\nexperimental values. This is described in Section~\\ref{secExp}.\nIn the following Sections~\\ref{secPb} and ~\\ref{secSn} we present our results\nfor the heavy Pb and Sn isotopes. Possible astrophysical signatures and\nremaining uncertainties are discussed in Section~\\ref{SecDis}.\nThe paper is concluded by the summary section~\\ref{summary}.\n\n\n\n\\section{Direct Capture and Folding Procedure}\n\\label{secDC}\n\\noindent\nThe theoretical cross section $\\sigma^{\\mathrm{th}}$ is derived from the\nDC cross section $\\sigma^{\\mathrm{DC}}$ given by~\\cite{kra,kim}\n\\begin{equation}\n\\sigma^{\\mathrm{th}}=\\sum_{i}C_i^2S_i\\sigma^{\\mathrm{DC}}_i \\quad .\n\\end{equation}\nThe sum extends over all possible final states (ground state and\nexcited states) in the residual nucleus. The isospin Clebsch-Gordan\ncoefficients and spectroscopic factors are denoted by $C_i$ and $S_i$,\nrespectively. The DC cross sections $\\sigma^{\\mathrm{DC}}_i$ are\nessentially determined by the overlap of the scattering wave function in\nthe entrance channel, the bound-state wave function in the exit channel,\nand the multipole transition operator. For the computation of the DC\ncross section we used the direct capture code TEDCA~\\cite{TEDCA}, which\nincludes E1, M1 and E2 transitions.\n\nFor determining the nucleon-nucleus potential the folding procedure was\nemployed, a method already successfully applied in the description of\nmany systems. In this approach the nuclear target\ndensity $\\rho_{\\mathrm{T}}$ is folded\nwith an energy and density dependent nucleon-nucleon interaction\n$v_{\\mathrm{eff}}$~\\cite{kob}:\n\\begin{equation}\nV(R)=\\lambda V_{\\mathrm{F}}(R)=\\lambda \\int \\rho_{\\mathrm{T}}\n(\\bbox{r})v_{\\mathrm{eff}}(E,\\rho_{\\mathrm{T}},\n\\vert\\bbox{R} - \\bbox{r} \\vert )d\\bbox{r} \\quad ,\n\\end{equation}\nwith $\\bbox{R}$ being the separation of the centers of mass of the two\ncolliding nuclei. The normalization factor $\\lambda$ accounts for\neffects of antisymmetrization and is close to unity. The nuclear density\n$\\rho_{\\mathrm{T}}$ can be derived from experimental charge\ndistributions or from theory. The potential\nobtained in this way ensures the correct behavior of the wave functions\nin the nuclear exterior. At the low energies considered in astrophysical\nevents the imaginary parts of the optical potentials are small.\n\nIn connection with the results presented below it is useful to recapitulate\nthe sensitivity of the DC calculations to various elements of the\ndescription. In ascending importance,\nin the present context the DC is sensitive to the optical potential and\ndensity distribution, respectively, the reaction $Q$-value, and the spin and\nparity of a level.\n\nFor the accuracy attempted here, there is almost no\ndifference in the results obtained by employing the optical potentials\nderived from the density distributions of the different models while leaving\nall other properties unchanged.\n\nA stronger dependence is seen when examining changes in the $Q$-value. An \nincrease\nin the $Q$-value will give a non-linear increase in the resulting cross section.\nAs the $Q$-value is computed as the difference in the binding energies of\ntarget and residual nucleus (i.e., the neutron separation energy) minus the\nexcitation energy of the level into which the neutron is captured\n\\begin{equation}\nQ_i=(B_{\\mathrm{T}}-B_{\\mathrm{R}})-E_i=S_{\\mathrm{n}}-E_i\\quad,\n\\end{equation}\nthe cross section will be sensitive to the masses (separation energies) derived\nin the different microscopic models as well as the level structure (excitation\nenergies) given in these models.\n\nThe by far strongest sensitivity is that to spins and parities of the\ninvolved initial and final states. In\norder to comply with the electromagnetic selection rules, a state has to\nhave the proper parity to contribute to the cross section significantly.\nThe dominant contribution to the DC cross section will stem from an E1\ntransition. In this case, parity has to change.\nConsequently, the capture of an incoming neutron $p$-wave will be important \nfor the Pb isotopes, whereas\n$s$-wave capture is dominating in the Sn cases.\nFurthermore, significant contributions only arise from low spin states\nlike 1\/2 and 3\/2 states, whereas the capture to levels with higher spins is \nstrongly suppressed.\nIn this respect, it will prove to be important that the different\nmicroscopic models make different predictions on which states are\nneutron-bound and which are not, since DC can only populate bound states.\n\n\\section{The Microscopic Input}\n\\label{secMic}\n\\noindent\nThe energy levels, masses, and nuclear density distributions\nneeded as input for the DC calculation were\ntaken from three different approaches. The first one was the\nRMFT which has turned out to be a successful tool\nfor the description of many nuclear properties~\\cite{gam}. The RMFT\ndescribes the nucleus as a system of Dirac nucleons interacting via\nvarious meson fields. There are six parameters which are usually\nobtained by fits to finite nuclear properties. For our calculations we\nhave used the parameter set NLSH~\\cite{sharma1,sharma2}.\n\nThe second method was FRDM (1992), which is a macroscopic-microscopic\nmodel based on the finite-range droplet macroscopic model and a\nfolded-Yukawa single-particle potential~\\cite{moeller}.\nFor pairing, the\nLipkin-Nogami pairing model~\\cite{yuk} is employed. This model proved to\nbe very successful in reproducing ground state spins along magic\nnumbers~\\cite{moell1} and\nhas been used in QRPA calculations of $\\beta$-decay half\nlives~\\cite{moell1,moellkra} and\nfor nuclear mass determinations~\\cite{moell2}.\n\nFinally, we also utilized the self-consistent mean field HFB\nmodel~\\cite{doba1,doba} in which the nuclear states are calculated by a one-step\nvariational procedure minimizing the total energy with respect\nto the occupation factors and the single-particle wave functions simultaneously.\n\nTo be able to compare the predictions from all of the models the nuclei were\nconsidered to be spherically symmetric. The limitations of such a\nrestriction\nare discussed in Section \\ref{SecDis}.\n\n\\newcommand{$^{208}$Pb(n,$\\gamma$)$^{209}$Pb}{$^{208}$Pb(n,$\\gamma$)$^{209}$Pb}\n\\section{Comparison with Experiments for the {\\protect$^{208}$Pb(n,$\\gamma$)$^{209}$Pb} reaction}\n\\label{secExp}\n\n\\noindent\nRecently, it became possible to extract the non-resonant part of the\nexperimental capture cross section for the\n$^{208}$Pb(n,$\\gamma$)$^{209}$Pb reaction~\\cite{corvi}. In that work, high\nresolution neutron capture measurements were carried out in order to\ndetermine twelve resonances in the range 1--400 keV. From these values the\nresonant Maxwellian-averaged cross section $<\\sigma>^{\\mathrm{R}}_{30\n\\mathrm{keV}}$=0.221(27)\\,mb was calculated.\nMeasurements of the total cross section using neutron\nactivation~\\cite{macklin,ratzel} are also available at 30 keV,\nyielding the value\n$<\\sigma>^{\\mathrm{t}}_{30\\mathrm{keV}}$=0.36(3)\\,mb. By a simple\nsubtraction of the\nresonant part from the total cross section the value of\n$<\\sigma>^{\\mathrm{NR}}_{30\\mathrm{keV}}$=0.14(4)\\,mb can be deduced\nfor the\nnon-resonant capture cross section.\n\nUsing the experimentally known density distributions~\\cite{devries},\nmasses~\\cite{audi} and energy levels~\\cite{nucldatasheets}, we\ncalculated the non-resonant contribution in the DC model. The strength\nparameter $\\lambda$ of the folding potential in the neutron channel was\nfitted to experimental scattering data at low energies~\\cite{scdata}.\nThe value of $\\lambda$ for the bound state is fixed by the requirement\nof correct reproduction of the binding energies.\nThe spectroscopic factors for the relevant low lying states of $^{209}$Pb\nare close to unity as can be inferred from\ndifferent $^{208}$Pb(d,p)$^{209}$Pb reaction data~\\cite{nucldatasheets}.\nFor the Maxwellian-averaged non--resonant DC cross section we obtained\n$<\\sigma>^{\\mathrm{DC}}_{30\\mathrm{keV}}=0.135$ mb, which is in excellent\nagreement with experiment. The by far highest contributions to the DC\ncross section come from the E1 $p$-wave capture to the low spin states\n$J^{\\pi}=1\/2^+, 3\/2^+, 5\/2^+$. Capture to the other states is negligible.\n\nIn order to test the different microscopic approaches we also calculated\nnon-resonant DC on $^{208}$Pb by consistently\ntaking the input (energy levels, masses\nand nuclear\ndensities) from the models described above.\nAgain, the strength parameter $\\lambda$ of the folding potential in the\nentrance channel was adjusted to the elastic scattering data for each\nof the models. The calculations for the neutron capture cross\nsections yield  0.0289 mb, 0.0508 mb, and 0.0135 mb\nfor RMFT, FRDM, and HFB, respectively.\nHence, each of the models gives a smaller value for\nthe Maxwellian-averaged 30 keV capture cross section than the\ncalculation using experimental input data. The differences are due to the\nneutron separation energies and level schemes of the relevant states in\n$^{209}$Pb (see\nFig.~\\ref{209}) in the microscopic models, leading to different $Q$-values\nfor capture to the\nexcited states ($J^{\\pi}=1\/2^+,3\/2^+,5\/2^+$).\nIt should be noted that in Fig.~\\ref{209} only those theoretical levels\nare shown which contribute to the cross section, i.e. only particle states.\nCapture into hole states is strongly suppressed by the fact that a re-ordering\nprocess would be required in the final nucleus (see e.g.~\\cite{tomenam} for\na similar case). This would be reflected in\nextremely small spectroscopic\nfactors. Therefore, the DC to such states is negligible.\n\n\\section{Results for Neutron-rich Pb Isotopes}\n\\label{secPb}\n\\noindent\nWe also investigated the model dependence of neutron capture on the\nneutron-rich even-even isotopes $^{210-238}$Pb. For these isotopes\nexperimental data are only available near the region of stability. For\nmore neutron-rich nuclei one has to rely solely on input parameters from\nmicroscopic models. In this and the following section we compare cross\nsections calculated with the nuclear properties predicted by different\nnuclear-structure models. Therefore, we consider nuclear cross sections\ninstead of Maxwellian-averaged ones as in the previous section.\n\nHaving obtained the relevant spins and calculated the $Q$-values from\nthe masses as\ndiscussed above, we still had to determine the scattering potentials\nwith their respective strength parameters\n(see Eq.~2). As a first\nstep, the folding potentials were calculated, using the density\ndistributions taken from the three different nuclear-structure models\n(HFB, RMFT, FRDM).\nIn the potentials for each of the isotopes a factor $\\lambda$ was chosen\ngiving the\nsame volume integral as for the fitted $^{208}$Pb+n potential, which was\nobtained as described in the previous section. This is justified because\nit is known that the\nvolume integrals only change very slowly when adding neutrons to a\nnucleus~\\cite{satchler}.\nFor the bound\nstate potentials $\\lambda$ is fixed by the requirement of correct\nreproduction of the binding energies. The spectroscopic factors were\nassumed to be unity for all transitions considered.\n\nThe results of our calculations are summarized in Fig.~\\ref{vgl}. For\ncomparison, the levels from all of the models\nfor $^{219}$Pb, $^{229}$Pb, and $^{239}$Pb are shown in\nFigs.~\\ref{level1}--\\ref{level3}.\nThe most\nstriking feature in Fig.~\\ref{vgl} is the sudden drop over several\norders of magnitude in\nthe cross sections calculated with the RMFT levels in the mass range\n$A=212-220$. This is due to the\nlack of low spin levels which are cut off by the decreasing neutron\nseparation energy.  Only after the\n1i$_{11\/2}$ orbital\n(which forms the state at lowest energy in the RMFT) has been filled\ncompletely at\n$^{222}$Pb the\ncross section is increasing because low spin states become\navailable again. A similar gap is seen for $A=230-232$, and it is\nexpected that those gaps will repeatedly appear when approaching the\ndrip-line.\nSince in some cases there are unbound low spin states\nclose to the threshold a small shift in the level\nenergies could already close such a gap. However, note that the level spacing\nin the\nRMFT has the tendency to increase towards neutron rich\nnuclei~\\cite{sharma3},\ncontrary to the FRDM and the HFB prediction.\n\nThe values resulting from the FRDM exhibit a smoother and\nalmost constant\nbehavior in the considered mass range.\nOnly a slight dip is visible for $^{220}$Pb(n,$\\gamma$) since\nthe previously accessible 1\/2$^+$ and 3\/2$^+$ states have become\nunbound in $^{221}$Pb. The 2g$_{9\/2}$ orbital is at lower energy than the\n11\/2$^+$ level in this model.\nBeyond $^{223}$Pb it has been filled and\nat least one of the low spin states can be populated again. The known\nground state spins for the lighter isotopes are also reproduced\ncorrectly. For higher mass numbers the cross sections are similar to the\nones obtained in the HFB model.\n\nFor mass numbers below $A=232$, the HFB capture cross sections\nare always larger than those obtained in the other models.\nAlthough the neutron separation energies are also decreasing, the $Q$-values\nfor the capture to the low spin states \nbecome even larger, because the states are moving towards lower\nexcitation energies.\nIn general, the HFB cross sections of the investigated capture reactions\nexhibit a very smooth behavior with increasing neutron number.\n\n\\section{Results for Sn Isotopes}\n\\label{secSn}\n\\noindent\nProceeding in the same manner as for the Pb isotopes (Sec.\\ \\ref{secPb}),\nwe extended our investigation to the Sn nuclei.\nHere, the situation is different in\ntwo ways: Firstly, the drip-line lies at relatively much lower\nneutron numbers and the r-process path is not so\nfar off stability, and secondly, there are more experimental data available\nalso for the unstable nuclei close to or in the r-process path, which makes\na test of theoretical models possible.\n\nAgain, we took the nuclear properties and density distributions from the\nabove described models. The strengths of the scattering potentials were\nadjusted to reproduce the same value of the volume integral of 425 MeVfm$^3$\nas determined from the experimental elastic scattering data on the stable\nSn isotopes~\\cite{bal}.\nWe calculated the capture cross sections from\nthe stable isotope $^{124}$Sn out to the r-process path which\nis predicted at a neutron separation energy of about 2 MeV~\\cite{friedel}. \nAs the\nmodels make different predictions about masses and separation energies,\nthe r-process path is located at different mass numbers: $A\\simeq 135$ for\nRMFT and FRDM and $A \\simeq 145$ in the case of HFB.\nContrary to the Pb isotopes for which the $p$-wave capture is the main \ncontribution\nallowed by the electromagnetic selection rules, the Sn cross sections\nare dominated by the $s$-wave captures, due to the negative parities of the \nfinal states.\n\nThe level schemes of the $^{125}$Sn, $^{133}$Sn, and $^{141}$Sn\nnuclei are shown in\nFigs.~\\ref{sn1}--\\ref{sn3}, and the resulting cross sections\nfor all considered nuclei and models\nare combined in Fig.~\\ref{snfig}.\nSimilarly as in the Pb case,\nthe dependence of the cross sections on the mass number can be understood\nby considering the excitation energies of the low-spin states relative to the\nneutron separation energy predicted in various models\n(Figs.~\\ref{spinhfb}--\\ref{spinfrdm}). The 3\/2$^-$ state is bound in the\nFRDM already at low mass number, whereas it becomes bound only at $A=131$\nand $A=133$ in HFB and RMFT, respectively. Therefore, the FRDM cross sections\nare larger than the ones from HFB and RMFT for $A<133$. The drop in the\nFRDM cross sections beyond the $N=82$ shell is due to the fact that the\n1\/2$^-$ and 3\/2$^-$ states slowly become unbound (see Fig.~\\ref{spinfrdm}).\nIn the HFB model the two low-spin states move down in energy faster than\nthe neutron separation energy, thus providing an increasing $Q$-value and\nslightly increasing cross sections (Fig.~\\ref{spinhfb}).\nA similar trend can be found in the\nlevels from RMFT, although with a less pronounced increase of the $Q$-value\n(Fig.~\\ref{spinrmf}).\n\nThere are no data available concerning the pure DC contribution\nto the cross sections for the neutron-rich Sn isotopes. However, there is\nexperimental information regarding masses and level schemes. This can be\ncompared to theory (see Fig.~\\ref{sn2}).\nFor the experimentally\nknown isotope $^{133}$Sn we calculated DC by taking the experimentally known\nmasses and levels~\\cite{hoff} as input for the DC-calculation,\nthus arriving at a pseudo-experimental value for\nthe cross section which can be compared to the purely theoretical predictions.\nThe resulting value is marked by a cross in Fig.~\\ref{snfig}.\nNeutron capture on $^{132}$Sn is particularly\ninteresting because $^{133}$Sn is predicted to be already very close to\nthe r-process path by the two models RMFT and FRDM.\nAs it turns out, however, the resulting cross sections show the closest\nagreement among the investigated nuclei for this case.\nAll of the considered models predict the same ground state spin, a bound\n3\/2$^-$ state and a (barely) unbound 1\/2$^-$ state (cf.,\nFigs.~\\ref{spinhfb}--\\ref{spinfrdm}, and Fig.~\\ref{sn2};\nnote that the mass ranges in the plots are different). However, the resulting\n$Q$-value is largest in the RMFT, yielding the highest cross section. The cross\nsections from the HFB and FRDM levels are smaller by about a factor of 2\nbecause of the less strongly bound 3\/2$^-$ state. The additional 5\/2$^-$\nstate found in HFB gives only a small contribution to the total cross section\nand cannot compensate for the comparatively low $Q$-value of the capture to\nthe 3\/2$^-$ level. Nevertheless, compared to the large discrepancies\nregarding other nuclei, there is good agreement in the resulting cross\nsections. Therefore, this\nnucleus may be a bad choice to select between the different models, but it\nis reassuring in the astrophysics context that the cross sections agree so\nwell.\n\n\\section{Discussion}\n\\label{SecDis}\n\\noindent\nIn systematic r-process studies~\\cite{friedel} it was found that\nthe r-process path is touching nuclei with neutron separation energies\naround 2.5--1.7\\,MeV in the Sn region and\n$S_{\\mathrm{n}}\\simeq 1.5-0.9$\\,MeV in the\nPb region~\\cite{friedel}.\nIn our calculations for Pb (including $^{239}$Pb+n) we cover the\nastrophysically relevant mass region, with the possible exception of the\nHFB model. The neutron separation energies in the HFB model decrease\nmuch slower with increasing mass number than in the other models\n(cf., Fig.\\ \\ref{level3}), thus\nnot only leading to a drip-line at higher mass but also pushing the\nr-process path further out. However, the most extreme path location\nmight still be further out by not more than two or three isotopes from\n$^{240}$Pb, and therefore it is possible to extrapolate the trend seen in the\nHFB calculation at lower mass numbers.\nIt should be kept in mind, however,\nthat the location of the r-process path is determined by the ratio\nbetween neutron capture half-life and $\\beta$-decay half-life.\n\nIn the following we briefly discuss the possible astrophysical\nconsequences of the effects\nfound in the cross section behavior given by the different models.\nComplete r-process network calculations, which take into account all\npossible reaction links and do not postulate an a-priori $\\beta$-flow\nequilibrium, require a large number of astrophysical and nuclear-physics\ninput parameters (for a detailed discussion, see e.g.~\\cite{cowan}). In such\na non-equilibrium scenario, the location of the r-process path as well as\nthe time-scale of the r-matter flow is mainly determined by the neutron\ndensity as astrophysical quantity, and by the nuclear-physics parameters:\nthe neutron separation energy $S_{\\mathrm{n}}$ and the capture cross sections\n$\\sigma_{\\mathrm{n}}$. With this, details of the r-process are depending\non the specific nuclear models used. In the following discussion\nwe will consider as a first estimate\nonly the r-process paths found in detailed studies making use of FRDM\nmasses~\\cite{friedel} and vary the capture cross sections according to our\nfindings for the different microscopic inputs.\n\nIn the mass region beyond the $A\\simeq195$ r-abundance peak, neutron\ndensities of $n_{\\mathrm{n}}\\simeq10^{25}-10^{27}$\\,cm$^{-3}$ are required\nto produce sizeable amounts of $Z\\simeq80-84$, $A\\simeq230-250$ r-process\nisotopes very far from $\\beta$-stability. After successive $\\beta^-$- and\n$\\alpha$-decays they will form the long-lived r-chronometers $^{232}$Th and\n$^{235,238}$U, and the major part of the r-abundances of $^{206-208}$Pb and\n$^{209}$Bi (see, e.g.~\\cite{pfeiff97}). When regarding the\n$\\sigma_{\\mathrm{n}}$ cross sections for Pb from FRDM and HFB\n(see Fig.~\\ref{vgl}), very similar results are expected for the $^{230-238}$Pb\nprogenitor isotopes. Thus, also similar initial r-abundances for\n$^{232}$Th and $^{235,238}$U will result. However, when using the RMFT\ncross sections, a considerable hindrance of the nuclear flow around\n$A\\simeq130$ may occur which consequently would change the Th\/U abundance\nratios. These neutron capture cross sections which are 5 or more orders of\nmagnitude smaller than the ones given by FRDM and HFB levels would\nincrease the life-time of a nucleus against neutron-capture by the same\norder of magnitude and thus even prevent the flow to heavier elements\nwithin the time-scales given by the astrophysical environment.\n\nIn the case of the Sn isotopes, the situation is quite different from the\nPb region. The range of astrophysically realistic $n_{\\mathrm{n}}$-conditions\nfor producing the $A\\simeq130$ r-abundances is lower, with\n$n_{\\mathrm{n}}\\simeq10^{22}-5\\times10^{24}$\\,cm$^{-3}$. Hence, the r-process\npath is much closer to $\\beta$-stability, involving the progenitor isotopes\n$^{134,136,138}$Sn only a few neutrons beyond the doubly magic nucleus\n$^{132}_{50}$Sn$_{82}$. For these isotopes the Hauser-Feshbach (HF)\ncross sections used so far~\\cite{cowan} are of the order of 10$^{-4}$ to\n$5\\times10^{-5}$\\,barn. According to a recent investigation~\\cite{tfkl1},\nthe statistical model cannot be applied in that region and will overestimate\nthe capture cross sections. However, even if we use the experimental levels to\ncalculate a Breit-Wigner resonant cross section for\n$^{132}$Sn(n,$\\gamma$)$^{133}$Sn, we find it to be a factor of about 6 lower\nthan the HF cross sections.\nOur present calculations would add another DC\ncontribution of about the same magnitude as given by\nHF (see Fig.~\\ref{snfig}), which has so\nfar not been taken into account. As a consequence of the larger total cross\nsection, the r-matter flow to heavier elements would be facilitated, thus\navoiding the formation of a pronounced $A\\simeq134-138$ ``satellite peak'' in\nthe r-abundance curve sometimes observed in steady-flow calculations\n(see, e.g.\\ Fig.\\ 2 in \\cite{chen96}, or Fig.\\ 5 in \\cite{friedel}).\nSuch a signature is only indicated in the heavy-mass wing of the\n$A\\simeq130$ $N_{r,\\odot}$-peak. It is interesting to note in this context\nthat the HFB model, which exhibits the weakest $N=82$ shell closure and\nwith this also the weakest ``bottle-neck'' for the r-matter transit in this\nregion (for a detailed discussion, see e.g.~\\cite{klk97}), yields the\nhighest DC cross sections for the $A\\geq134$ Sn isotopes.\n\nSince we assumed spherical nuclei in order to be able to compare the\ndifferent microscopic models, deformation effects were not taken into\naccount which lead to level splitting and thus can increase the number\nof accessible levels. When considering deformation our results could be\nmodified in two ways: Firstly, the number of bound low-spin levels could\nbe increased, leading to larger DC cross sections; secondly,\ndue to a possibly larger number of levels at and above the neutron\nseparation energy, the compound reaction mechanism could be further\nenhanced and clearly dominate the resulting cross sections. However,\nas can be seen from level density~\\cite{ohurev,tfkl1} and\ndeformation (e.g., \\cite{moell2}) studies, deformation of Pb isotopes\nsets in at a mass number of about\n$A\\simeq 220$ and decreases already for masses beyond $A\\simeq 230$.\nCloser to the drip-line, the nuclei show low level densities again,\nnot only due to low neutron separation energies but also because of\nsphericity. Lead isotopes in the r-process path (especially for components\nwith low $S_{\\mathrm{n}}$) will therefore already\nhave reduced deformation and the DC -- being sensitive to the\nlevel structure -- will give an important contribution to the total\ncapture cross sections. Concerning Sn, a theoretical study of the ratio of\nDC over CN contributions for Sn isotopes~\\cite{bal} shows that CN dominates\nup to a mass number $A \\simeq 130$. Moreover, deformation is predicted to set\nin only at $A\\simeq140$ for Sn~\\cite{moellkra}.\nThis is supported by level density\nconsiderations~\\cite{tfkl1}, showing that the level density is too low\nin this region to apply the statistical model. Therefore, depending on the\nmodel, the r-process path lies at the border of or already well inside the region\nwhere the DC is non-negligible and dominating.\n\nAnother source of uncertainty is the assumption of pure single-particle\nstates, i.e., setting the spectroscopic factors to unity. This has been\nshown to be a good approximation for Pb isotopes close to stability and\nit is expected to hold for neutron-rich Pb isotopes. However, a range of\n0.01--1.0 for the spectroscopic factors could be realistic. This will play\nonly a minor role in the present comparison of different microscopic\nmodels, as the differences in the models may be only slightly enhanced\nwhen considering different theoretical spectroscopic factors. Nevertheless,\nit will be important in quantitative calculations of abundances, invoking\ncomplicated reaction networks.\n\n\\section{Summary}\n\\label{summary}\n\\noindent\nWe have shown that theoretical capture cross sections can depend\nsensitively on the microscopic models utilized to determine the\nnecessary input parameters.\nBecause of low level densities, the\ncompound nucleus model will not be applicable in those cases.\nDrops over several orders of magnitude in the cross sections -- as found\nwith the RMFT for Pb -- would change the position of the r-process path\nand possibly influence\nthe formation of heavy chronometer elements,\nwhereas the enhanced capture rates on Sn\ncould have direct effects in the final r-process abundance distribution.\nDeformation effects and the compound nucleus reaction mechanism\nmay still be of importance for the Pb isotopes and further investigations\nare needed. Nevertheless, the DC will be of major importance in\nthe Sn region. This region is also interesting for future experimental\ninvestigations of $S_{\\mathrm{n}}$, neutron single-particle levels and\n(d,p)-reactions studying spectroscopic factors.\nThere is also a need for improved microscopic nuclear-structure models\nwhich can also be compared in an astrophysical context following the\nsuccessful tradition of the interplay between nuclear physics and\nastrophysics.\n\n\n\n\\acknowledgements\n\\noindent\nThis work was supported in part by the Austrian Science Foundation\n(project S7307--AST) and by the Polish Committee for\nScientific Research.\nTR acknowledges support by an APART fellowship from\nthe Austrian Academy of Sciences.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\noindent The strategies to search for a dark matter (DM) component in the Universe are nowadays extremely varied, targeting \nmany possible gravitational and non-gravitational properties such as the DM mass or standard model (SM) \ncouplings~\\cite{Bertone:2004pz}. In astrophysical, cosmological, and laboratory settings, this broadband approach has yet to \nconclusively reveal any non-gravitational signatures. However, via both indirect and direct searches, the very wide DM \nmodel space has been significantly restricted. The focus of this article concerns the reach of the generic \nclass of experiments aiming to directly detect DM through a possible DM-nucleon coupling~\\cite{Goodman:1984dc}, \nknown as direct detection facilities. Currently, world-leading examples of this setup include \ne.g.~LUX-ZEPLIN (LZ)~\\cite{LZ:2022ufs}, PandaX-4T~\\cite{PandaX-4T:2021bab}, and Xenon-1T~\\cite{XENON:2018voc},\nwhich set the strongest limits in the DM mass \n$m_\\chi$ vs.~spin-independent nuclear coupling $\\sigma_{\\mathrm{SI}}$ parameter space.\n\nThe sensitivity of a given direct detection experiment is controlled by a number of factors. Firstly, the event rate $\\Gamma_N$\nscales with the number of DM particles that have a sufficiently large kinetic energy. Specifically, the DM energy \nmust be large enough \nto induce a nuclear recoil that can trigger a signal above the detector threshold. Secondly, the rate \nalso scales linearly with the DM-nucleon cross section $\\mathrm{d}\\sigma_{\\chi N} \/ \\mathrm{d} T_N$, \nat least in the above examples, where $T_N$ is the nuclear recoil energy. Thirdly, as in any count-based experiment, \nthis signal rate should be compared to some \nbackground event rate to derive a statistically significant detection threshold. Notably, in direct detection facilities, the background \nrates are typically extremely low as necessitated by the small expected signal rates, although there are some important \nexceptions, such as a dedicated CRESST surface run~\\cite{CRESST:2017ues}. \n\nThe standard target for these experiments is the DM in the Galactic halo, which has characteristic velocities of the order\n$v_\\chi \\sim 10^{-3}c$ and in any case cannot exceed the Galactic escape velocity \n$v_\\mathrm{esc} \\sim 540 \\, \\mathrm{km\/s}$~\\cite{Evans:2000gr,Evans:2005tn}. For a given DM mass \n$m_\\chi$, there is hence unavoidably a maximum DM kinetic energy available to excite nuclear recoil signals of the order \n$T_N \\sim m_\\chi^2 v_\\mathrm{esc}^2 \/ m_N$. For some DM mass $m_\\chi^\\mathrm{min}$\nthis must fall below the detectable threshold, and the experimental sensitivity drops to zero. For experiments such as \nXenon, PandaX and LZ, it is well-known that this cut-off\nlies around the GeV-scale, corresponding to a detectable threshold in the keV range. As such, even though these detectors \nhave impressive reach in interaction cross-section -- currently down to the level of \n$\\sigma_\\mathrm{SI} \\sim 10^{-47}\\,\\mathrm{cm}^2$~\\cite{XENON:2018voc,PandaX-4T:2021bab,LZ:2022ufs}, and even \napproaching the neutrino floor with ongoing searches~\\cite{Strigari:2009bq,OHare:2021utq} -- there is ample motivation \n(and hence, in fact, both experimental and theoretical activity)\nfor methods to probe the sub-GeV mass range~\\cite{Knapen:2017xzo,Essig:2022dfa}. \nThis describes the first ``window\" in which DM can hide -- it could just be that DM has \na small mass out of the reach of direct detection experiments. \nThere is yet another window at {\\it large} values of the cross-section \n$\\sigma_{\\mathrm{SI}}$, however, which will be a key focus of this article. This arises due to the fact that if DM interacts \\textit{too} \nstrongly, then it can actually be the case that DM particles are unable to reach the detectors due to the attenuation of the flux \nin the atmosphere or the rock overburden~\\cite{Starkman:1990nj,Zaharijas:2004jv, Mack:2007xj}. \nThis typically becomes the main prohibitive factor for cross \nsections at the level of $\\sigma_{\\mathrm{SI}} \\gtrsim 10^{-28}\\,\\mathrm{cm}^2$~\\cite{Emken:2018run}.\n\nThere have been a number of promising experimental proposals to probe these two open windows. Attempts \nto extend the sensitivity to DM-nucleus interactions into the sub-GeV realm include \nsearches for Migdal electrons~\\cite{Ibe:2017yqa,XENON:2019zpr} or bremsstrahlung photons~\\cite{Kouvaris:2016afs}, \naccompanied by an intense low-threshold direct detection program in the development of novel detector \nconcepts (for a recent review, see Ref.~\\cite{Essig:2022dfa}).\nCross sections sufficiently large for DM to scatter inside the Earth before reaching underground detectors,\non the other hand,\ncan be probed by surface runs of conventional direct detection experiments (like the one performed by the CRESST \ncollaboration~\\cite{CRESST:2017ues}), or by targeting the expected diurnal modulation in the\nsignal in this case~\\cite{Collar:1992qc,Collar:1993ss}. As far as this work is \nconcerned, however, we will be interested in the role played by the irreducible astrophysical flux of\nhighly boosted DM that originates from cosmic ray collisions with DM particles in the Galactic halo (CRDM). \nThis was pointed out only relatively recently~\\cite{Bringmann:2018cvk,Cappiello:2018hsu}, and subverts the issue\nof a loss in sensitivity by noting that a sub-dominant component of DM with velocities well above those in the Galactic halo can \nproduce a detectable signal even if it is very light, i.e.~for DM masses (well) below $1 \\, \\mathrm{GeV}$. \nThe sub-dominant nature of the flux \nnaturally introduces a trade-off with the interaction rates that can be probed, quantitatively resulting in limits\nat the level of $\\sigma_\\mathrm{SI} \\sim 10^{-31}\\,\\mathrm{cm}^2$~\\cite{Bringmann:2018cvk}.\nInterestingly, CRDM does not only probe previously open parameter space at small DM masses but also results\nin bounds extending into the relevant regime of the second open window described above. \nAfter this initial work pointed out the advantages of \nconsidering such a boosting mechanism, a large number of further analyses have addressed various aspects of the \nproduction~\\cite{Alvey:2019zaa,DeRocco:2019jti,Dent:2019krz,Wang:2019jtk,Zhang:2020nis,Plestid:2020kdm,Su:2020zny,Cho:2020mnc,Guo:2020oum,Xia:2020apm,Dent:2020syp,Dent:2020qev,Emken:2021lgc,Das:2021lcr,Bell:2021xff,An:2021qdl,Feng:2021hyz,Wang:2021jic,Granelli:2022ysi,Xia:2022tid,Bardhan:2022ywd}, \nattenuation~\\cite{Bondarenko:2019vrb,McKeen:2022poo}, and \ndetection~\\cite{Ema:2018bih, Cappiello:2019qsw,Berger:2019ttc,Kim:2020ipj,Guo:2020drq,DeRoeck:2020ntj,Ge:2020yuf,Cao:2020bwd,Jho:2020sku,Lei:2020mii,Harnik:2020ugb,Ema:2020ulo,Bramante:2021dyx,Emken:2021vmf,PandaX-II:2021kai} \nof astrophysically boosted DM. \nFor a recent comprehensive \\mbox{(re-)analysis} of all of these aspects see, e.g.~Xia~{\\it et al.}~\\cite{Xia:2021vbz},\nwho stressed in particular that form-factor suppressed attenuation in the overburden seemingly allows us to exclude\ncross sections much larger than $\\sigma_\\mathrm{SI} \\sim 10^{-28}\\,\\mathrm{cm}^2$.\n\nThis article builds on this literature in three important ways: firstly, we point out that when DM acquires such large energies, \ninelastic scattering in the rock overburden above detectors such as Xenon-1T will at some point become the dominant\nattenuation mechanism. \nAs such, to avoid being over-optimistic in terms of how much parameter space is excluded, we show how \nto include this physical effect in a self-consistent manner and\nderive the resulting bounds. Secondly, we broaden the applicability of these limits to models that are more realistic for DM \nwith sub-GeV masses, moving beyond simplified contact interactions to\ninteractions mediated by vector or scalar mediators, or DM that has some internal structure. Finally, we argue that with \nthese improvements, and when taking into account fully complementary constraints from cosmology,\nthere is generically no remaining open parameter space left unconstrained for nuclear cross sections exceeding \n$10^{-30}\\,\\mathrm{cm}^2$, \nfor DM masses in the entire MeV to GeV range. We demonstrate that possible loopholes to this statement -- still allowing \nan open window at larger cross sections -- require a combination\nof {\\it (i)} questioning the principal ability of CRESST to probe DM masses down to the published limit of \n$m_\\chi=140$\\,MeV~\\cite{CRESST:2017ues}  and \n{\\it (ii)} choosing a rather narrow range of mediator masses $m_\\phi\\sim 30$\\,MeV (or finite DM extent $r_\\chi\\sim10$\\,fm).\nFor our numerical analysis throughout the article, we use the package {\\sf DarkSUSY}~\\cite{Bringmann:2018lay}. The improved CRDM \ntreatment reported in this work, including also updated cosmic ray fluxes and a more sophisticated use of form factors in the \nattenuation part, will be included in the next public release of the code.\n\n\\noindent The rest of the article is organized as follows: we start in section~\\ref{sec:crdm} by briefly reviewing the production \nof CRDM and the attenuation of the subsequent flux on its way to the detector, establishing our notation and\nsetting up the basic formalism that our analysis relies on. In the next two sections, we discuss in more detail how to model nuclear \nform factors (section~\\ref{sec:form_factors}) and the impact of inelastic scattering (section~\\ref{sec:inel}) on the attenuation \nof the flux. In section~\\ref{sec:m2}, we consider a number of \ngeneric options for the $Q^2$- and $s$-dependence of the scattering amplitude that are more realistic than assuming a constant \ncross-section. We complement this in section~\\ref{sec:hexaquark} with the analysis of a specific example, namely a\nbaryonic DM candidate that has been argued to evade traditional direct detection bounds despite its relatively\nstrong interactions with nuclei. We conclude and summarise our results in section~\\ref{sec:conclusions}.\n\n\n\n\n\\section{Cosmic-ray upscattering of dark matter}\n\\label{sec:crdm}\n\nWe describe here, in turn, how initially non-relativistic DM particles in the Galactic halo are up-scattered by cosmic rays (CRs),\nhow the flux of these relativistic CRDM particles is attenuated before reaching detectors at Earth, and\nhow to compute the resulting elastic scattering rate in direct detection experiments.\n\n\\medskip\n\\noindent\\textbf{Production:} The basic mechanism that we consider is the elastic scattering of CR nuclei $N$,  \nwith a flux of ${{d\\Phi_N}}\/{dT_N}$, \non non-relativistic DM particles $\\chi$ in the Galactic halo. For a DM mass $m_\\chi$ and \ndensity profile $\\rho_\\chi(\\mathbf{r})$, this induces a relativistic CRDM flux incident on Earth \nof~\\cite{Bringmann:2018cvk,Bondarenko:2019vrb} \n\\begin{eqnarray}\n\\frac{d\\Phi_{\\chi}}{dT_\\chi}&=&\\int\\frac{d\\Omega}{4\\pi} \\int_{\\rm l.o.s.} \\!\\!\\!\\!\\!\\!d\\ell \\, \\frac{\\rho_\\chi}{m_\\chi} \n\\sum_N\n\\int_{T_N^\\mathrm{min}}^\\infty d T_N\\, \\frac{d \\sigma_{\\chi N} }{dT_\\chi} \\frac{{d\\Phi_N}}{dT_N}\\\\\n&\\equiv&  \nD_\\mathrm{eff} \\frac{\\rho_\\chi^\\mathrm{local}}{m_\\chi}  \n\\sum_N\n\\int_{T_N^\\mathrm{min}}^\\infty d T_N\\, \\frac{d \\sigma_{\\chi N} }{dT_\\chi} \\frac{{d\\Phi^\\mathrm{LIS}_N}}{dT_N}\n\\label{eq:chiCR}\n\\,.\n\\end{eqnarray}\nHere $\\mathbf{r}$ denotes the Galactic position, and \n${d \\sigma_{\\chi N} }\/{dT_\\chi}$ is the differential elastic scattering cross section\nfor accelerating a DM particle to a kinetic recoil energy $T_\\chi$. \nFor DM particles initially at rest, this requires a minimal CR energy $T_N^\\mathrm{min}$ of\n\\begin{equation}\n\\label{eq:Tmin}\nT_N^\\mathrm{min}=\n\\left\\{\n\\begin{array}{ll}\n\\left( \\frac{T_\\chi}{2}  - m_N\\right) \\left[\n1-\\sqrt{1+\\frac{2 T_\\chi}{m_\\chi}\\frac{\\left(m_N + m_\\chi\\right)^2}{\\left(2m_N - {T_\\chi}\\right)^{2}}}\n\\right] & \\quad\\mathrm{for~}T_\\chi<2m_N\\\\\n\\sqrt{\\frac{m_N}{m_\\chi}} \\left(m_N + m_\\chi\\right) & \\quad\\mathrm{for~}T_\\chi=2m_N\\\\\n\\left( \\frac{T_\\chi}{2}  - m_N\\right) \\left[\n1+\\sqrt{1+\\frac{2 T_\\chi}{m_\\chi}\\frac{\\left(m_N + m_\\chi\\right)^2}{\\left(2m_N - {T_\\chi}\\right)^{2}}}\n\\right] & \\quad\\mathrm{for~}T_\\chi>2m_N\n\\end{array}\n\\right. \\,.\n\\end{equation}\nFurthermore, in the second line of Eq.~(\\ref{eq:chiCR}), we have introduced an effective distance $D_{\\rm eff}$ that allows us to \nexpress the CRDM flux in the solar system in terms of the relatively well measured {\\it local} interstellar CR flux, \n${{d\\Phi_N}^{\\rm LIS}}\/{dT_N}$, and the {\\it local} DM density, for which we adopt \n$\\rho^\\mathrm{local}_\\chi=0.3\\,\\mathrm{GeV}\/\\mathrm{cm}^3$~\\cite{Read:2014qva} (noting that our final limits are \nindependent of this choice). The advantage of this parametersation is that uncertainties deriving from the integration \nover the volume relevant for CRDM production, $\\int d\\Omega \\int \\!d\\ell$,  are captured in a single \nphenomenological parameter $D_\\mathrm{eff}$. Indeed, despite the complicated underlying physics, this parameter is \nsurprisingly well constrained,\nwith uncertainties dominated by the vertical extent of the confinement zone of Galactic CRs. \nIn what follows, we will use a fiducial value of $D_{\\rm eff}=10$\\,kpc.\\footnote{%\nWhen assuming  an Einasto profile~\\cite{Einasto:2009zd} for the DM density, and a cylindric CR diffusion model \ntuned with {\\sf GalProp}~\\cite{Strong:1998pw} to describe the observed flux of light CR nuclei,   \na more detailed analysis reveals that $D_{\\rm eff}$ varies between $\\sim9$\\,kpc and \n$\\sim11$\\,kpc for DM recoil energies above 1\\,MeV~\\cite{Xia:2021vbz} . \n}\nWe note that our final limits only depend logarithmically on this quantity, for large interaction rates,\nor scale as $D_{\\rm eff}^{-1\/2}$ when attenuation in the soil or atmosphere is inefficient, respectively.\n\nWhen computing the CRDM flux in Eq.~(\\ref{eq:chiCR}), we take into account the four most abundant\nCR species, $N=\\{p,{\\rm He},{\\rm C}, {\\rm O}\\}$, for which high-quality determinations of the local \ninterstellar fluxes exist~\\cite{Boschini:2018baj}.  The fluxes of heavier nuclei are subject to significant \nuncertainties for the energies of interest to us, see e.g.~the discussion in Ref.~\\cite{Boschini:2020jty}, not least due to \napparent discrepancies between AMS-02 data~\\cite{AMS:2018tbl,AMS:2018cen,AMS:2020cai} and \nearlier measurements. We also note that the CRDM flux contribution from these heavier elements is \nstrongly form-factor suppressed at large $T_\\chi$, see section \\ref{sec:form_factors}, and hence \nanyway not relevant for constraining DM with masses $m_\\chi\\gtrsim0.1$\\,GeV.\n\n\\medskip\n\\noindent\\textbf{Attenuation:} On its way to the detector, the CRDM flux given by Eq.~(\\ref{eq:chiCR}) is attenuated due to \nscattering of the CRDM particles with nuclei in the atmosphere and soil (overburden) above the experimental\nlocation. This effect can be well modelled by the energy loss equation\n\\begin{equation}\n\\label{eq:eloss}\n\\frac{dT_\\chi^z}{dz}=-\\sum_N n_N\\int_0^{\\omega_\\chi^\\mathrm{max}}\\!\\!\\!d\\omega_\\chi\\,\\frac{d \\sigma_{\\chi N}}{d\\omega_\\chi} \\omega_\\chi\\,,\n\\end{equation}\nwhich can be used to relate the average kinetic energy at depth $z$, $T_\\chi^z$, to an initial energy \n$T_\\chi$ at the top of the atmosphere.\nHere, the sum runs over the nuclei $N$ in the overburden,\ni.e.~no longer over the CR species, and $\\omega_\\chi$ is the {\\it energy loss} of a DM particle\nin a single collision. \nFor elastic scattering, $\\omega_\\chi$ is equal to the nuclear recoil energy $T_N$.\nIn that case, the maximal energy loss of a DM particle with initial kinetic energy $T_\\chi^z$\nis given by\n\\begin{equation}\n\\label{eq:tmax}\n\\omega_\\chi^\\mathrm{max}=T_N^\\mathrm{max}=\\frac{2m_N}{s}\\left[\\left(T_\\chi^z\\right)^2+2m_\\chi T_\\chi^z\\right],\n\\end{equation} \nwhere \n\\begin{equation}\n\\label{eq:sdef}\ns=(m_N+m_\\chi)^2+2m_N T_\\chi^z\n\\end{equation}\nis the (squared) CMS energy of the process. For inelastic scattering on the other hand, which we will discuss in more detail \nin section \\ref{sec:inel}, the energy loss can in principle be as high as $\\omega_\\chi^\\mathrm{max}=T_\\chi^z$.\nFor the purpose of this work we will mostly be interested in the Xenon-1T\ndetector, located at a depth of $z=1.4\\, \\text{km}$  in the Gran Sasso laboratory. In this case the \nlimestone overburden has a density of 2.71 g\/cm$^3$~\\cite{Miramonti:2005xq},\nmostly consisting of an admixture of CaCO$_3$ and MgCO$_3$, and attenuation in the\natmosphere can be neglected; in terms of weight percentages\nthe dominant elements are O (47.91\\%), Ca (30.29\\%), C (11.88\\%), Mg (5.58\\%), Si (1.27\\%),\nAl (1.03\\%) and K (1.03\\%)~\\cite{Wulandari:2003cr}. We note that Eq.~(\\ref{eq:eloss}) only provides an approximate \ndescription of the stopping \neffect of the overburden, which is nonetheless sufficiently accurate for our purposes. For a detailed comparison of this \napproach with Monte Carlo simulations of individual particle trajectories, see \nRefs.~\\cite{Emken:2017qmp,Emken:2018run,Mahdawi:2018euy,Emken:2019hgy,Xia:2021vbz}\n\n\\medskip\n\\noindent\\textbf{Detection:} The elastic scattering rate of relativistic CRDM particles arriving at underground detectors \nlike the Xenon-1T experiment is given by\n\\begin{equation}\n\\label{eq:gammarate}\n {\\frac{d\\Gamma_N}{d T_{N}}=\n \\int_{T_\\chi^{\\rm min}}^\\infty \\!\\!dT_\\chi\\ \n \\frac{d \\sigma_{\\chi N}}{dT_N} \\frac{d\\Phi_\\chi}{dT_\\chi}} \\,.\n\\end{equation}\nNote that the above integral is over the energy of the DM particles \\emph{before} entering the atmosphere. \nOn the other hand, the elastic scattering cross section ${d \\sigma_{\\chi N}}\/{dT_N} $ must still be evaluated at the actual \nDM energy, $T_\\chi^z$, at the detector location, which requires numerically solving Eq.~(\\ref{eq:eloss}) \nfor $T_\\chi^z(T_\\chi)$. The lower bound on the integral then represents the minimal {\\it initial} CRDM energy \nthat is needed to induce a nuclear recoil of energy $T_N$ {\\it at depth $z$}, i.e.\n$T_\\chi^{\\rm min}=T_\\chi(T_\\chi^{z, \\mathrm{min}})$. This can be obtained by inverting the solution of Eq.~(\\ref{eq:eloss}),\nwhere $T_\\chi^{z, \\mathrm{min}}$ is given by the right-hand side of Eq.~(\\ref{eq:Tmin}) under the replacement\n$(T_\\chi,m_\\chi,m_N)\\to(T_N,m_N,m_\\chi)$.\nIn general, the elastic nuclear scattering cross section \n${d \\sigma_{\\chi N}}\/{dT_N} $ \nis a function of both $s$ and the (spatial) momentum transfer, \n\\begin{equation}\n\\label{eq:q2}\nQ^2=2m_N T_N\\,.\n\\end{equation}\nIf the dependence on $s$ can be neglected or the (dominant) dependence on $Q^2$ factorizes -- as in the case of\nstandard form factors -- then the rate in the detector given in Eq.~(\\ref{eq:gammarate}) will have an {\\it identical}\n$Q^2$-dependence as compared to the corresponding rate expected from the standard population of \nnon-relativistic halo DM. As pointed out in Ref.~\\cite{Bringmann:2018cvk}, this salient feature makes it possible to \ndirectly re-interpret published limits on the \nlatter (conventionally expressed as limits on the scattering cross section with protons) into limits on the \nformer. Otherwise, for an accurate determination of the expected count rate in \na given analysis window, one would in principle have to also model the detector response in the \nevaluation of Eq.~(\\ref{eq:gammarate}) and then infer limits based on the full detector likelihood \n(e.g.~with a tool like {\\sf DDCalc}~\\cite{GAMBITDarkMatterWorkgroup:2017fax,GAMBIT:2018eea}).\n\n\\section{Nuclear form factors}\n\\label{sec:form_factors}\n\nThe target nuclei used in direct detection experiments \nare typically larger than the de Broglie wavelength of DM with standard Galactic velocities, \nat least for heavy nuclei, implying that the incoming DM particles only `see' part of the nucleus. \nSince the elastic scattering process is fundamentally induced by a coupling between DM and the \nconstituents of these nuclei, this means that it should be suppressed by a \nnuclear form factor, $G^2(Q^2)$, compared to the naive expectation that the nuclear cross section \nis merely a coherent sum of the cross sections of all the constituents (for recent pedagogic \naccounts of conventional direct DM searches, see e.g.~Refs.~\\cite{DelNobile:2021icc, Cooley:2021rws}).\\footnote{%\nWe focus here on spin-independent elastic scattering.  For {\\it spin-dependent} scattering,  \nthe sum would not be coherent and hence generally result in much smaller cross sections.\nThis prevents standard DM from being stopped in the overburden before reaching the experimental\nlocation -- unless the scattering cross section {\\it per nucleon} is so large that it becomes incompatible with other\nastrophysical constraints.\nA detailed treatment of attenuation in the Earth's crust is, hence, less relevant in this case. \n}\nFor CRDM, this effect is amplified, given the smaller de Broglie wavelengths\nassociated to the faster moving upscattered DM particles. \n\nThese nuclear form factors are essentially Fourier transforms of the number density of nucleons inside\nthe nucleus, usually approximated by the experimentally easier accessible charge density. A common\nparameterization is the one suggested by Helm~\\cite{Helm:1956zz}, which is based on modelling \nthe nucleus as a hard sphere with a Gaussian smearing (in configuration space).  For heavy nuclei we follow instead a slightly\nmore accurate approach and implement model-independent form factors~\\cite{Duda:2006uk}, based\non elastic electron scattering data. Concretely, we implement their Fourier-Bessel (FB) expansion approach,\nwith parameters taken from Ref.~\\cite{DeVries:1987atn}. For nuclei where the FB parameters\nare not available, notably Mg and K, we use model-independent Sum of Gaussians (SOG) form factors instead. \n\nFor $Q^2\\gg (0.1\\,\\mathrm{GeV})^2$ one starts\nto resolve the inner structure of the nucleons themselves, which we discuss in more detail in section \\ref{sec:inel}. \nLet us however briefly mention \nthat in the case of He, this effect is already largely captured by the above description in that we take the \nSOG form factors from Ref.~\\cite{DeVries:1987atn} (thus improving on the simple dipole prescription used, e.g.,\nin Ref.~\\cite{Bringmann:2018cvk}). For the proton, we adopt the usual dipole {\\it nucleon} form factor,\nnoting that the {\\it nuclear} form factor would formally equal unity,\n\\begin{equation}\n\\label{eq:Gp}\nG_p^2(Q^2)=\\left(1+ Q^2\/\\Lambda_p^2 \\right)^{-4}\\,,\n\\end{equation}\nwith $\\Lambda_p=0.843$\\,GeV. This provides a very good fit to experimental data up to momentum\ntransfers of at least $Q^2\\sim1$\\,GeV$^2$, with an agreement of better than 10\\% for \n$Q^2\\leq10$\\,GeV$^2$~\\cite{Perdrisat:2006hj,Punjabi:2015bba}. \nWe note that our final results are highly insensitive to such large momenta.\n\n\nIn the rest of the section, we will briefly describe the impact of nuclear form factors on \nthe CRDM flux and the attenuation of this flux on its way to the detector.\nIn both cases the effect is sizeable, motivating the need for a precise modelling of $G^2(Q^2)$.\n\n\n\\subsection{Impact on production}\n\\label{sec:FF_prod}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_one_wl}\n\\caption{{\\it Left panel.} Expected CRDM fluxes for DM masses $m_\\chi=0.001, 0.01, 0.1,1,10$\\,GeV, \nfrom top to bottom, assuming a constant spin-independent scattering cross section of \n$\\sigma_{\\rm SI}^{p,n}=10^{-30}\\,\\mathrm{cm}^2$ (solid lines). The effect of inelastic scattering is neglected.\nDashed lines show the CRDM fluxes that would result when not taking into account \nthe effect of form factors. \n{\\it Right panel.} Black lines indicate the individual contributions to the CRDM flux from scattering on CR $p$, He, C and O,\nfor the example of $m_\\chi=100$\\,MeV. Other lines (highlighted only for the  $m_\\chi=100$\\,MeV case) show the\ntotal flux, as in the left panel.\n}\n\\label{fig:flux_ff}\n\\end{center}\n\\end{figure}\n\n\nThe solid lines in Fig.~\\ref{fig:flux_ff} show the expected CRDM flux before attenuation, cf.~Eq.~(\\ref{eq:chiCR}),\nfor a range of DM masses. For the purpose of this figure, we have assumed a constant elastic \nscattering cross section $\\sigma_{\\rm SI}^p=\\sigma_{\\rm SI}^n$ on nucleons, i.e.~a nuclear cross section\ngiven by\n\\begin{equation}\n\\frac{d \\sigma_{\\chi N}}{dT_\\chi} = \\mathcal{C}^2\n\\times \\frac{\\sigma_{\\rm SI}^p}{T_\\chi^{\\rm max}} \\times G^2(2T_\\chi m_\\chi)\\,.\n\\label{eq:siconst}\n\\end{equation}\nHere, \n\\begin{equation}\n\\label{eq:c_coh}\n\\mathcal{C}^2= A^2\\frac{\\mu_{\\chi N}^2}{\\mu_{\\chi p}^2}\n\\end{equation}\ndescribes the usual coherent enhancement, in this case proportional to the square of the atomic number $A$ \nof nucleus $N$. In the rest of the expression, $\\mu_{\\chi N}$ ($\\mu_{\\chi p}$) is the reduced mass of the \nDM\/nucleus (DM\/nucleon) system and\nthe maximal DM energy $T_\\chi^{\\rm max}$ that can result from a CR nucleus with energy $T_N$ \nis given by the right-hand side of Eq.~(\\ref{eq:tmax}) after replacing $T_\\chi^z \\to T_N$ and $m_\\chi\\leftrightarrow m_N$.\n\nIn the left panel of the figure, we show that neglecting nuclear form factors (dashed lines) would lead to a \nsignificant overestimate of the CRDM flux at high energies. For $m_\\chi\\gtrsim0.1$\\,GeV, the form factor\nsuppression even becomes the dominant effect to determine the overall normalization of the flux,\nwhile for lower DM masses, the peak of the distribution is entirely determined by the fact that the \nCR flux itself peaks at GeV energies. This suppression in the flux leads to a rapid deterioration \nof CRDM limits.\nModelling form factors correctly is thus particularly important for the highest DM masses that can be\nprobed by cosmic-ray upscattering, i.e.~for $m_\\chi \\sim 1 - 10 \\, \\mathrm{GeV}$.\n\nIn the right panel of Fig.~\\ref{fig:flux_ff}, the contributions from the individual CR nuclei to the  \nCRDM flux are shown. At low energies the dominant contribution is always from Helium, closely followed by the one from protons. \nThe high-energy part of the CRDM flux, on the other hand, \nis almost exclusively due to CR protons because the contribution from heavier CR nuclei is \nheavily form-factor suppressed. In addition, for $m_\\chi\\gtrsim1$\\,GeV, the \npeak amplitude of the CRDM flux -- which typically has the most constraining\npower in direct detection experiments -- is almost exclusively  determined by CR $p$ and He nuclei\n (see also Fig.~\\ref{fig:attenuation_ff} below to better gauge the relevant range of energies {\\it after} attenuation\n in the overburden).\nFor lower DM masses, on the other hand, including further high-$Z$ CR species than those taken into account \nhere could in principle increase the relevant part of the CRDM flux by up to $\\sim50$\\,\\%~\\cite{Xia:2021vbz}. \nIn what follows, we conservatively neglect these contributions, in view of both the larger uncertainties in the underlying\nCR fluxes and the fact that we are mainly interested in DM masses around the GeV scale.\n\n\n\\subsection{Impact on attenuation}\n\\label{sec:FF_att}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figures_draft\/figure_two_wl.pdf}\n\\caption{%\nMinimal kinetic energy $T_\\chi$ that a DM particle must have at the surface of the Earth ($z=0$) in order \nto trigger a signal in the Xenon-1T experiment,  as a function\nof a (constant) spin-independent scattering cross section $\\sigma_{\\rm SI}^{p,n}$.\n Different colors correspond to different DM masses, \n as  in Fig.~\\ref{fig:flux_ff}.\n Dash-dotted lines show the kinetic energies that would be necessary when computing the attenuation in the \nzero momentum transfer limit. Dashed lines illustrate the effect of adding\nthe expected form factor suppression, cf.~section \\ref{sec:form_factors}, while solid \nlines show the result of our full treatment, including also inelastic scattering events \n(discussed in section \\ref{sec:inel}).\n}\n\\label{fig:attenuation_ff}\n\\end{center}\n\\end{figure}\n\n\nWe now turn our attention to assessing the effect that the form factor suppression has on the attenuation of DM\nparticles on their way to the detector in a direct detection experiment. For concreteness we will again focus\non the case of Xenon-1T, where Xe nuclei recoiling with an energy of at least\n$T_{\\rm Xe}=4.9$\\,keV trigger a detectable signal~\\cite{XENON:2018voc}. In \nFig.~\\ref{fig:attenuation_ff}, we show the minimal initial DM energy that is required to kinematically allow\nfor this, after penetrating through the Gran Sasso rock. In practice this is done by numerically solving Eq.~(\\ref{eq:eloss}) with\n{\\sf DarkSUSY}. Dash-dotted lines indicate the result when conservatively\nassuming that the stopping power in the overburden is as efficient as in the zero-momentum transfer\nlimit (as in Ref.~\\cite{Bringmann:2018cvk}), while dashed lines show the effect of adding the additional\nform factor suppression for high $Q^2$ (as in Refs.~\\cite{Bell:2021xff,Xia:2021vbz}). \nSolid lines, finally, demonstrate the effect of also adding the attenuation power of inelastic scattering events,\nas described in detail below in Section \\ref{sec:inel}.\n\nFor small cross sections, attenuation is inefficient and, as expected, the three approaches give the \nsame answer. In this limit, the difference in the required DM energy is entirely due to the well-known\nkinematic effect, cf.~Eq.~(\\ref{eq:Tmin}), that lighter particles require a higher energy to induce a \ngiven recoil of much heavier particles\n(up to a minimum energy of $T_\\chi\\geq\\sqrt{m_{\\rm Xe}T_{\\rm Xe}\/2} =17.3$\\,MeV in the limiting case where $m_\\chi\\to0$).\nCorrespondingly, this also means that the CRDM fluxes cannot actually be probed by Xenon-1T \nfor the entire range of $T_\\chi$ shown in Fig.~\\ref{fig:flux_ff};\nunless $m_\\chi\\lesssim10$\\,MeV, however, the lowest detectable energy is always smaller\n than the energy at which the CRDM flux peaks.\n \nFor large cross sections, on the other hand, Fig.~\\ref{fig:attenuation_ff} shows a \npronounced difference between the three \napproaches: while in the case of a constant cross section (dash-dotted lines) the energy loss equation\nresults in an exponential attenuation, adding form factors (dashed lines) implies that the required initial \nDM energy only rises as the square root of the scattering cross section in the $Q^2=0$ limit.\nIn fact, we note that this is exactly the behaviour one would expect from Eq.~(\\ref{eq:eloss}) for a \ncross section that falls off very rapidly at large momentum transfers. \nComparing again to Fig.~\\ref{fig:flux_ff},\nthis correspondingly enlarged range of kinetic energies that becomes kinematically accessible to Xenon-1T will \ninevitably lead to significantly larger rates in the detector -- which, indeed, is exactly the conclusion reached in\nRefs.~\\cite{Bell:2021xff,Xia:2021vbz}. However, such a strong \nsuppression of the physical stopping power of the Gran Sasso rock for a relativistic particle is highly \nunphysical. As we discuss in the next section, this is simply because the DM particles will start to scatter off the constituent \nnucleons themselves, albeit not \ncoherently across the whole nucleus. Adding this effect (solid lines), \nresults again in exponential attenuation in the overburden -- though only at significantly larger cross sections \nthan what would be expected when adopting a constant cross section for simplicity. \n\n\n\\section{Inelastic Scattering}\n\\label{sec:inel}\n\nOur discussion so far has largely neglected the impact of inelastic scattering events of relativistic DM particles incident on nuclei \nat rest, or {\\it vice versa}. Physically, the inclusion of inelastic scattering processes is non-negotiable and should be considered in \na full treatment. This is because, whilst the form factor suppression described above is the relevant feature in the transition from \ncoherently scattering off the whole nucleus to only parts of it, once the DM or nucleus transfers a sufficiently large amount \nof energy $\\omega$, the scattering will probe individual nucleon-, or even quark-level processes. The result is an additional \ncontribution to the total scattering cross-section that can easily dominate \nin the large energy transfer regime. \nAs far as CRDM limits are concerned, the most important effect \nthat the inclusion of inelastic scattering modifies is the attenuation of the flux through the Earth or atmosphere.\nNot including it, therefore, will lead to an overly optimistic estimate as to the amount of parameter space that is ruled out via this \nmechanism.\\footnote{In order to keep our results conservative, we neglect the effect of inelastic scattering on CRDM {\\it \nproduction} in our analysis. We leave the study of this additional contribution of the flux to future work, noting that we \nexpect it to mostly improve limits for larger DM masses (where the form factor suppression nominally leads \nto a significant reduction of the CRDM flux, see Fig.~\\ref{fig:flux_ff}).}\nLet us note that inelastic scattering of {\\it non-relativistic} DM, resulting in the excitation of low-lying states in the target nuclei,  \nwas previously both studied \ntheoretically~\\cite{Baudis:2013bba, McCabe:2015eia, Kouvaris:2016afs, Hoferichter:2018acd}\nand searched for experimentally~\\cite{XMASS-I:2014lnb,XENON:2017kwv,Lehnert:2019tuw,XENON:2020fgj}. \nHere we concentrate on different types of inelastic processes that are only accessible to nuclei scattering off \nhigh-energy DM particles.\n\nThe rest of this section is organised as follows: firstly we give a qualitative description of the most important \ninelastic scattering processes, such as the excitation of hadronic resonances or quasi-elastic scattering \noff individual nucleons. Secondly, we explain how we obtain a quantitative estimate of these complicated nuclear interactions \nby making a direct analogy to the case of neutrino-nucleus scattering. In this regard, we make use of the public code \n\\texttt{GiBUU}~\\cite{Buss:2011mx,gibuuweb}.\nFinally, we will explain how to build this into the formalism described in section~\\ref{sec:crdm} in terms of the DM energy loss, see \nEq.~\\eqref{eq:eloss}.\n\n\\subsection{Scattering processes and associated energy scales}\nThere are a number of relevant contributions to scattering cross-sections on nuclei that are associated to certain \ncharacteristic energies or nuclear length scales. In the highly non-relativistic limit, as described above,\ncoherently enhanced elastic scattering dominates. At somewhat higher energies, more specifically momentum transfers\ncorresponding to (inverse) length scales smaller than the size of the nucleus,\nthe elastic scattering becomes form factor suppressed -- a description which physically assumes a smooth distribution of \nscattering centres throughout the nucleus. The main characteristic of elastic scattering in both of these regimes is that \nthe energy loss of the incident DM particle is uniquely related to the momentum transfer by $\\omega=Q^2\/(2m_N)$.\n\nThis relation no longer holds for inelastic scattering processes, which are expected to become relevant at even higher energies. \nFor our purposes, these inelastic processes can be broadly split up into three scattering regimes, depending\non the energy that is transferred (see also Fig.~\\ref{fig:mchi_1GeV} below, as well as a review~\\cite{Formaggio:2012cpf} \nfor the discussion of the analogous situation in the case of neutrino-nucleus scattering):\n       \n\\begin{itemize}\n      \\item \\textbf{Quasi-Elastic Scattering} \\textbf{(}$\\mathbf{\\omega \\gtrsim 10^{-2}}$\\,\\textbf{GeV):} At suitably large \n\tenergy transfers, the form factor suppression cannot be totally physical. This is because  \n\tthe incident DM particles will probe directly the constituent nucleons, which are \n\tinherently not smoothly distributed. \\emph{Quasi-elastic scattering} (QE) dominates for \n\t$10^{-2}\\,\\mathrm{GeV} \\lesssim \\omega \\lesssim 1 \\, \\mathrm{GeV}$, and describes this situation, \n\ti.e.~where the dominant scattering is directly off {\\it individual} protons (and neutrons) inside the nucleus, \n\t$\\chi\\, p (n) \\rightarrow \\chi\\, p (n)$.\n\n\t\\item \\textbf{Excitation of Hadronic Resonances} \\textbf{(}$\\mathbf{\\omega \\gtrsim 0.2}$\\,\\textbf{GeV):} At higher energies \n\tstill, DM-nucleon scattering can excite nuclear resonances such as \n\t$\\chi \\, p \\rightarrow \\chi \\, (\\Delta \\rightarrow p \\pi^0)$ etc., leading to a wide variety of hadronic final states. Often, the contribution due to the lowest lying \n\t$\\Delta$ resonances (DR) is distinguished from contributions from higher resonances (HR) since the former can  \n\tbe well resolved and starts playing role at considerably smaller transferred energies. \n\tIn a complicated nucleus such as ${^{16}}\\mathrm{O}$, both the QE and resonance contributions to the scattering cross-section must be resolved numerically, \n\ttaking into account effects such as the nuclear potential and spin statistics. \n\n\t\\item \\textbf{Deep Inelastic Scattering} \\textbf{(}$\\mathbf{\\omega \\gtrsim 1}$ \\textbf{GeV):} Most DM couplings to \n\tnuclei and nucleons result from more fundamental couplings to quarks or gluons. \n\tAs such, once the energy transfer is large enough to probe the inner structure of the nucleons \n\t($\\omega \\gtrsim 1 \\, \\mathrm{GeV}$), then \\emph{deep inelastic scattering} (DIS) of DM with partons inside the nucleons\n\tcan occur. Again, this should be \n\tresolved numerically to give an accurate estimate of the impact at the level of the scattering cross-section.\n\\end{itemize}\n\n\n\\subsection{Computation of the inelastic cross-section for neutrinos}\n\\label{sec_gibuu}\n\nDue to the complicated nuclear structure of the relevant atomic targets in the Earth, \nor in the composition of cosmic rays, \nit is typically not possible to analytically compute all the contributions to DM-nucleus scattering \ndescribed above. Instead, to estimate their impact on our \nconclusions and limits, we will make a direct connection with the physics of neutrino-nucleus scattering for which numerical codes \n-- such as \\texttt{GiBUU}~\\cite{Buss:2011mx} -- are capable of generating the relevant differential cross-sections.\n\nIn more detail, we draw the analogy between neutral current neutrino-nucleon scattering via processes such as \n$\\nu \\, p \\rightarrow \\nu \\, p$ and DM-nucleon scattering. Numerically modelling the neutral current quasi-elastic scattering, \nresonances and deep inelastic scattering as a function of the energy transferred to the nucleus, $\\omega$, allows us to \nunderstand the relative importance of these processes as a function of the incoming neutrino energy\n(or DM kinetic energy $T_\\chi$). Of course, since \nthese codes are tuned for neutrino physics, simply outputting the differential cross-sections such as \n$\\mathrm{d}\\sigma_{\\nu N} \/ \\mathrm{d}\\omega$ is not sufficient. To map the results onto DM, see \nsection \\ref{sec:map_dm} below for further details, we should re-scale the results so \nas to respect both the relative interaction strengths and model dependences such as e.g. the mediator mass. In general, we \nexpect this approach to provide a good estimate of the DM-nucleus cross section (at least) for contact interactions and  \nscattering processes dominated by mediators in the $t$-channel.\n\n\nAt the level of implementation, we choose the settings in the \\texttt{GiBUU} code described in Tab.~\\ref{tab:gibuu}. \nSince we are interested in quantifying the effect of inelastic scattering on the attenuation of the CRDM flux as it passes \nthrough the Earth, \nwe mostly focus on the total inelastic scattering cross section, i.e.~the sum over all the processes described in the \nprevious section. We numerically calculate this for the most abundant nuclei in the Gran Sasso rock, \n$N = \\{\\mathrm{O}, \\mathrm{Ca}, \\mathrm{C}, \\mathrm{Mg}, \\mathrm{Si}, \\mathrm{Al}, \\mathrm{K}\\}$.\nFundamentally, inelastic cross-sections are expressed in terms of double-differential cross-sections \nlike $\\mathrm{d}^2 \\sigma_{\\nu N} \/ \\mathrm{d}Q^2 \\mathrm{d}\\omega$, since for inelastic scattering $Q^2$ and $\\omega$ are \nindependent variables. \nFor integrating the energy loss equation, Eq.~\\eqref{eq:eloss}, however, it suffices to compute\n\\begin{equation}\n\\frac{\\mathrm{d} \\sigma_{\\nu N}}{ \\mathrm{d}\\omega} \\equiv \\int_{Q^2} \\frac{\\mathrm{d}^2 \\sigma_{\\nu N}}{ \\mathrm{d}Q^2 \\,\\mathrm{d}\\omega} \\,\\mathrm{d}Q^2\\,.\n\\end{equation}\nOn the other hand, the full information about the $Q^2$-dependence of \n$\\mathrm{d}^2 \\sigma_{\\nu N} \/ \\mathrm{d}Q^2 \\mathrm{d}\\omega$ provided by \\texttt{GiBUU} still remains a highly\nuseful input to our analysis. This is because  the double-differential cross \nsections of the individual inelastic processes turn out to sharply peak at values of $Q^2$ that have simple relations to $\\omega$.\nFor example, the peak position for the QE contribution corresponds to the `elastic' relation~\\eqref{eq:q2} for nucleons. \nAs described below, this information will be used \nfor setting realistic reference values of $Q^2$ to capture the model-dependence of the DM cross sections.\n\n\n\\begin{table}[]\\label{tab:gibuu}\n\t\\resizebox{\\textwidth}{!}{\n\t\\begin{tabular}{@{}llllll@{}}\n\t\\toprule\n\t\\multicolumn{2}{l}{\\hspace{-0.2cm}\\textbf{\\&neutrino\\_induced}} & \\multicolumn{2}{l}{\\textbf{\\&input}}               & \\multicolumn{2}{l}{\\textbf{\\&nl\\_dSigmadElepton}} \\\\\n\t\\texttt{process\\_ID}               & 3           & \\texttt{eventtype}                     & 5                & \\texttt{enu}                  & $T_\\chi$             \\\\\n\t\\texttt{flavor\\_ID}                & 2           & \\texttt{numEnsembles}                  & 100              & \\texttt{elepton}              & $0.005 T_\\chi$       \\\\\n\t\\texttt{nuXsectionMode}            & 2           & \\texttt{numTimeSteps}                  & 0                & \\texttt{delta\\_elepton}       & $\\Delta E_\\ell$      \\\\\n\t\\texttt{nuExp}                     & 0           & \\texttt{num\\_Energies}                 & 50               & \\multicolumn{2}{l}{\\textbf{\\&target}}                \\\\\n\t\\texttt{includeQE}                 & T\/F         & \\texttt{num\\_runs\\_sameEnergy}         & 1                & \\texttt{Target\\_A}            & $A$                  \\\\\n\t\\texttt{includeDELTA}              & T\/F         & \\texttt{delta\\_T}                      & 0.2              & \\texttt{Target\\_Z}            & $Z$                  \\\\\n\t\\texttt{includeRES}                & T\/F         & \\texttt{localEnsemble}                 & T                & \\multicolumn{2}{l}{\\textbf{\\&initDensity}}           \\\\\n\t\\texttt{path\\_To\\_Input}           & \\texttt{\/path\/to\/buuinput} & \\texttt{include1pi}     & F                & \\texttt{densitySwitch}        & 2                    \\\\\n\t\\texttt{includeDIS}                & T\/F         & \\multicolumn{2}{l}{\\textbf{\\&neutrinoAnalysis}}           & \\multicolumn{2}{l}{\\textbf{\\&initPauli}}             \\\\\n\t\\texttt{2p2hQE}                    & F           & \\texttt{XSection\\_analysis}            & T                & \\texttt{pauliSwitch}          & 2                    \\\\\n\t\\texttt{include2p2hDelta}          & F           & \\texttt{detailed\\_diff\\_output}        & F                &  \t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\n    \\texttt{include2pi}                & F           &                                        &                  &  \t\t\t\t\t\t\t\t\t\t\t\t\t\\\\ \\bottomrule\n\t\\end{tabular}\n\t}\n\t\\caption{Settings choices for running \\texttt{GiBUU} to study neutral current neutrino scattering. \n\tWe also enforced a logarithmic binning in the outgoing lepton energy, by changing the variable assignment \n\tof \\texttt{dElepton} from $E_\\ell \\rightarrow E_\\ell + \\Delta E_\\ell$ to $E_\\ell \\rightarrow (1 + \\Delta E_\\ell) E_\\ell$.\n\t}\n\t\\end{table}\n\n\n\\subsection{Mapping to the dark matter case}\n\\label{sec:map_dm}\n\nHaving described the technical details of how we obtain the neutrino-nucleus inelastic cross-sections using \\texttt{GiBUU}, we \nnow turn our attention to the mapping of these quantities onto DM models. \nThis is a necessary step for two broad reasons: \\emph{(a)} the interaction strength governing the DM-nucleus interactions is \ntypically very different from the neutrino-nucleus SM value, and \\emph{(b)} the way the interaction proceeds via e.g.~a contact \ninteraction or mediator exchange can lead to substantially different kinematics and non-trivial $Q^2$- or $s$-dependences.\n\nThe total scattering cross-section $\\mathrm{d}\\sigma_{\\chi N}\/\\mathrm{d}\\omega$ consists of the coherent elastic scattering contribution that we compute analytically for each of the models considered in this work, and the inelastic scattering cross section that we want to estimate based on the \\texttt{GiBUU} output:\n\\begin{align}\\nonumber\n\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega} &= \\left.\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{el}}+\\left.\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{inel}}\\\\ \\label{eq:rescaling}\n&\\equiv  \n\t\\left.\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{el}, Q^2=2\\omega m_N} + \\sum_{i} \\left.\\frac{\\mathrm{d}\\sigma_{\\mathrm{SI}} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{el}, Q^2=Q_{i,\\mathrm{ref}}^2} \n\\times I_{\\chi,i}(T_\\chi,\\omega)\\,.\n\\end{align}\nHere $\\left.\\mathrm{d}\\sigma_{\\mathrm{SI}} \/ \\mathrm{d}\\omega\\right|_{\\mathrm{el}}$ is the differential\nDM-nucleon elastic cross section, excluding nucleon form factors such as the one given in \nEq.~(\\ref{eq:Gp}). \nThe sum runs over the various individual processes, $i\\in$(QE, DR, HR, DIS),\nwhich all have characteristic reference values of $Q^2=Q^2_{i, \\mathrm{ref}}(\\omega)$ where\nthe respective inelastic cross section peaks. In the second step above, we thus choose to rescale the inelastic scattering \nevents to the elastic scattering off a point-like nucleon. \nThis rescaling is motivated by the fact that for inelastic contributions like QE, the underlying process is much better \ndescribed by scattering on individual nucleons than on the entire nucleus.\nThe factor\n\\begin{equation}\nI_{\\chi,i}(T_\\chi,\\omega) \\equiv \\frac{\\mathrm{d}\\sigma^i_{\\chi N} \/\\mathrm{d}\\omega \\big|_{\\mathrm{inel}}}\n\t{\\mathrm{d}{\\sigma}_{\\mathrm{SI}} \/\\mathrm{d}\\omega \\big|_{\\mathrm{el},Q^2=Q^2_{i, \\mathrm{ref}}}}\n\\end{equation}\nthus quantifies the ratio of the inelastic scattering process on a nucleus to the elastic scattering on an individual\nnucleon.\n\nWe now make the simplifying assumption that this ratio is to a certain degree model-independent,\nbased on the expectation that DM should probe the inner structure of nucleons in a similar way as neutrinos do\nwhen only neutral current interactions are involved. Physically, indeed, this closely resembles the situation both for \ncontact interactions and $t$-channel mediators.\nThe model dependence thus dominantly comes from the structure of the term \n$\\left.\\mathrm{d}\\sigma_{\\mathrm{SI}} \/ \\mathrm{d}\\omega\\right|_{\\mathrm{el}}$, and we approximate\n\\begin{equation}\n\\label{eq:I2}\nI_{\\chi,i}(T_\\chi,\\omega) \\approx I_{\\nu,i}(E_\\nu,\\omega)\\equiv\\frac{\\left.\\mathrm{d}\\sigma^i_{\\nu N} \/\\mathrm{d}\\omega \\right|_{\\mathrm{inel}}}\n\t{\\mathrm{d}{\\sigma}^i_{\\nu,\\mathrm{SI}} \/\\mathrm{d}\\omega \\big|_{\\mathrm{el}}}\\,.\n\\end{equation}\nHere, the inelastic neutrino-nucleus cross section \n$\\left.\\mathrm{d}\\sigma_{\\nu N}^i\/\\mathrm{d}\\omega\\right|_{\\mathrm{inel}}(E_\\nu,\\omega)$ \ncan be obtained using the \\texttt{GiBUU} code, as described in section \\ref{sec_gibuu}, and we evaluate it \nat the incoming DM kinetic energy, $E_\\nu = T_\\chi$.\n On the other hand, a possible estimate for the denominator -- the elastic neutral current neutrino-nucleon cross \nsection without the form factor -- is the average of the proton and neutron cross sections in the $\\omega \\rightarrow 0$ \nlimit~\\cite{Formaggio:2012cpf}:\n\\begin{equation}\n\\label{eq:nuel_simp}\n\\left.\\frac{\\mathrm{d}\\sigma^i_{\\nu,\\mathrm{SI}}}{\\mathrm{d}\\omega} \\right|_{\\mathrm{el}} \n= \\frac{1}{2} \\sum_{j=n,p} \\frac{m_j G_F^2}{4\\pi}\\left[(g_A\\tau_3^j - \\Delta_S)^2 + (\\tau_3^j-2(1+\\tau_3^j)\\sin^2\\theta_W)^2\\right].\n\\end{equation}\nHere $\\tau_3^p=1$ and $\\tau_3^n=-1$, $\\theta_W$ is the weak mixing angle and $G_F$ is the Fermi constant.\nThe axial vector and strange quark contributions are encoded in the parameters \n$\\Delta_S\\approx -0.15$ (see, e.g., Ref.~\\cite{Alberico:1997vh} for a discussion) and  \n$g_A= 1.267$~\\cite{ParticleDataGroup:2008zun}, respectively. Numerically the square bracket evaluates\nto a factor of $\\sim\\!2.24\\,(2.01)$ for neutrons (protons). \nLet us stress, however, that this formula is valid only for energies \nrelevant for inelastic scattering, $0.1\\,\\mathrm{GeV}\\lesssim E_\\nu\\lesssim10$\\,GeV. \nAt much smaller energies, only the valence quarks contribute to the scattering, and we would instead have\n\\begin{equation}\n\\label{eq:nuel_simpNR}\n\\left.\\frac{\\mathrm{d}\\sigma^i_{\\nu,\\mathrm{SI}}}{\\mathrm{d}\\omega} \\right|_{\\mathrm{el}} \n= \\frac{m_n G_F^2}{4\\pi}\n\\end{equation}\nfor neutrons, while the scattering on protons is strongly suppressed by a factor of \n$Q_W^2=(1-4\\sin^2\\theta_W)^2\\approx0.012$.\n\n \nIt is worth noting that in principle, we could improve the assumption made in Eq.~(\\ref{eq:I2})\nfor the quasi-elastic process, because there is a \nwell-controlled understanding of the analytic QE cross-section via the Llewellyn-Smith formalism (see section~V \nof~Ref.~\\cite{Formaggio:2012cpf}). For clarity, we choose to take a consistent prescription across all inelastic processes, \nand we have checked that including the full QE cross-section would only introduce an \nadditional $\\mathcal{O}(1)$ factor in the DM QE cross-section.\nFor the numerical implementation in {\\sf DarkSUSY}, we pre-tabulate $I_{\\nu,i}$ from $T_\\chi=0.01$\\,GeV up to energies of $T_\\chi=10$\\,GeV,\nwith $200$ ($101$) equally log-spaced bins in $T_\\chi$ ($\\omega$)\nand a normalization as given by Eq.~(\\ref{eq:nuel_simp}),  and then interpolate between \nthese values.\\footnote\nFor significantly higher energies,  \\texttt{GiBUU} is no longer numerically stable. Furthermore, \nthe underlying equations that describe the interaction processes begin to fall outside their ranges of validity\nas the $Z$ boson mass starts to get resolved.\nAt higher energies, where anyway only the DIS contribution is non-negligible,  a reasonable estimate \ncan still be obtained by a simple extrapolation \n$I_{\\nu,i}(T_\\chi,\\omega)\\to I_{\\nu,i}(T_\\chi^{\\rm ref},\\omega^{\\rm ref})$, \nwith $\\omega^{\\rm ref} =\\omega\\,(T_\\chi^{\\rm ref}\/T_\\chi)^{0.25}$,  beyond some reference energy\n$T_\\chi^{\\rm ref}\\approx10$\\,GeV\\@. By running  \\texttt{GiBUU} up to $E_\\nu\\sim30$\\,GeV, we checked that\nthis prescription traces the peak location (in $\\omega$) of the DIS contribution very well, \nindependently of the exact choice of $T_\\chi^{\\rm ref}$. We also confirmed that \nthe peak value of $I$ becomes roughly constant for such large energies. \nOn the other hand, higher-order inelastic processes are expected to become increasingly important \nat very large energies, not covered in \\texttt{GiBUU}.\nWe therefore only add the above extrapolation as an {\\it option} in {\\sf DarkSUSY}, and instead completely cut the incoming CRDM\nflux at $10$\\,GeV in the default implementation. As a result, our bounds on the interaction strength\nmay be overly conservative for small DM masses $m_\\chi\\lesssim0.1$\\,GeV.\n}\n\nWe also must choose the reference values for the transferred momentum $Q^2_{i, \\mathrm{ref}}$, which allows us to account \nfor e.g.~mediators that may be much lighter than the electroweak scale. \nImportantly, each process (quasi-elastic, \n$\\Delta$-resonance,...) is expected to have a different characteristic $Q^2$-$\\omega$ dependence that takes into account the \nrelevant binding energies and kinematic scaling. For example, in the case of elastic scattering, the \nrelation $Q^2 = 2 m_N \\omega$ holds, whilst for quasi-elastic processes, the relevant scattering component is a nucleon such \nthat the cross-section is peaked around $Q^2 \\sim 2 \\,{\\overline{ m}}\\,\\omega$, where ${\\overline m} \\equiv (m_n + m_p)\/2$. \nThe resonance of a particle with mass $m_\\mathrm{res}$ can be accounted for by noting that part of the transferred\nkinetic energy is used to excite the resonance, such that the cross-section peaks around \n$Q^2 \\sim 2\\, {\\overline m}\\, (\\omega - (m_\\mathrm{res} - \\overline{m}))$. \nWe have confirmed these expectations numerically by comparing directly to the \ndoubly-differential cross-section extracted from \\texttt{GiBUU}.\nFrom this numerical comparison we further extract that $Q^2 \\sim 0.6\\,\\overline{m}\\,(\\omega \\!-\\! \\omega_{\\rm DIS})$,\nwith $\\omega_{\\rm DIS}=1.0$\\,GeV, constitutes a very\ngood fit to the peak location of the DIS cross-section.\nIn summary, we take the following reference values across the four inelastic processes:\n\\begin{align}\n\tQ^2_{\\mathrm{QE}, \\mathrm{ref}} = 2\\, {\\overline m} \\omega \\, \\, , \\,\\,\\,\\, &Q^2_{\\Delta, \\mathrm{ref}} = 2\\, {\\overline m}\\, (\\omega - \\Delta m_\\Delta)  \\nonumber \\\\\n\tQ^2_{\\mathrm{res}, \\mathrm{ref}} = 2\\, {\\overline m}\\, (\\omega - \\Delta m_{\\mathrm{res}}) \\, \\, , \\,\\,\\,\\, &Q^2_{\\mathrm{DIS}, \\mathrm{ref}} = 0.6\\, {\\overline m}\\, (\\omega - \\omega_{\\rm DIS})\\,.\n\\end{align}\nHere, $\\Delta m_\\Delta = 0.29 \\, \\mathrm{GeV}$ is the mass difference between the $\\Delta$ baryon and an average nucleon, \nand $\\Delta m_\\mathrm{res} = 0.40 \\, \\mathrm{GeV}$ is an estimate for the corresponding average mass difference of the higher \nresonances (we checked that our final limits are insensitive to the exact value taken here).\n\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\\includegraphics[width=0.9\\textwidth]{figures_draft\/figure_three_wl}\n\t\\caption{Comparison between the elastic (green, lower energies) and inelastic (blue, higher energies) contributions to the \n\tDM-nucleus differential cross section $\\mathrm{d}\\sigma_{\\chi N}\/\\mathrm{d}\\omega$, where $\\omega$ is the \n\tDM energy loss. This figure shows these contributions \tfor a constant isospin-conserving DM-nucleus cross section, with \n\t$m_\\chi = 1\\, \\mathrm{GeV}$ and $N = {^{16}}\\mathrm{O}$. The small colorbar on \n\tthe inset of the plots, along with the stated numerical ratio, indicates the balance between elastic and inelastic\n\tscattering in terms of the contribution to the integrated cross section \n\t$\\sigma_{\\chi N}^\\mathrm{tot}$.\n\t}\n\t\\label{fig:mchi_1GeV}\n\t\\end{center}\n\\end{figure}\n\n\nTo illustrate this procedure concretely, we consider the simple case of a contact interaction where, cf.~Eq.~(\\ref{eq:siconst}), \n$\\left.\\mathrm{d}\\sigma_{{\\rm SI}}\/\\mathrm{d}\\omega\\right|_{\\mathrm{el.}} = \\sigma_{\\mathrm{SI}} \/ \\omega^\\mathrm{max}$ and \n$\\omega^\\mathrm{max} = 2\\, {\\overline m} (T_\\chi^2 + 2 \\chi T_\\chi) \/ (({\\overline m} + m_\\chi)^2 + 2 {\\overline m} T_\\chi)$. The results for the \nrescaled inelastic cross-section (blue) are shown in Fig.~\\ref{fig:mchi_1GeV} for a DM mass $m_\\chi = 1\\,\\mathrm{GeV}$ incident \non a $^{16}\\mathrm{O}$ nucleus. In this figure, we also compare to the coherent elastic contribution (green) and highlight the \nbalance between the relative contributions to the total (integrated) cross-section $\\sigma^\\mathrm{tot}_{\\chi N}$. In particular, we \nsee that above kinetic energies $T_\\chi \\gtrsim 0.2\\,\\mathrm{GeV}$, the inelastic contribution dominates, clearly \nmotivating the necessity of its  inclusion. This is consistent with the picture previously encountered in Fig.~\\ref{fig:attenuation_ff}, \nwhere we could see the impact of inelastic scattering on the energy loss.  More concretely, \nthe result lies in some intermediate regime between the $G(Q^2) = 1$ and $G(Q^2) \\neq 1$ cases, the former\/latter leading \nto conservative\/overly optimistic limits respectively. \nIn the next section we will derive the relevant CRDM limits in the \n$\\sigma_{\\mathrm{SI}}-m_\\chi$ plane for a number of models to make this point quantitatively.\n\n\nLet us conclude this section by briefly returning to the implicit assumption of isospin-conserving DM interactions that\nwe made above, with \n$\\sigma_{\\mathrm{SI}}=\\sigma^p_{\\mathrm{SI}}=\\sigma^n_{\\mathrm{SI}}$.\nInterestingly, neutral-current induced\ninelastic scatterings between neutrinos and nucleons hardly distinguish between protons and \nneutrons~\\cite{Formaggio:2012cpf}, such that the factor $I_{\\chi,i}\\approx I_{\\nu,i}$  indeed becomes, by construction,\nlargely independent of the nucleon nature. Naively, one would thus conclude that\nisospin-violating DM couplings can easily be incorporated in our treatment of inelastic scattering by replacing\n$\\sigma_{\\mathrm{SI}} \\to (1 \/ A) \\times (Z \\sigma^p_{\\mathrm{SI}} + (A - Z) \\sigma^n_{\\mathrm{SI}})$  in Eq.~(\\ref{eq:rescaling}).\nWhen doing so, however, it is important to keep in mind that the nucleon cross sections should be evaluated at \nenergies that are relevant for inelastic scattering, not  in the highly non-relativistic limit.\nAt these high energies, isospin symmetry is typically largely restored because the nucleon couplings are no longer \nexclusively determined by the valence quarks, and instead receive corrections from a large number of sea quarks \n(and, in principle, gluons).\nAs pointed out above, the example of neutrino scattering illustrates this effect very clearly: even though isospin is almost\nmaximally violated at low energies, the effective neutrino couplings to neutrons and protons agree within\n$\\sim5$\\,\\% at energies around 0.1\\,GeV, cf.~Eqs.~(\\ref{eq:nuel_simp}) and (\\ref{eq:nuel_simpNR}). In practice, however,\na possible complication often arises in that the nucleon couplings $g_n$ and $g_p$ are only provided in the highly\nnon-relativistic limit. \nIn that case, an educated guess for $\\sigma_{\\rm SI}$ in the second term of Eq.~(\\ref{eq:rescaling})\nis to anyway take the leading order (Born) expression -- but to adopt (effective)\nvalues for {\\it both} nucleon couplings that correspond to\nthe maximum of $\\left|g_p\\right|$ and $\\left|g_n\\right|$ in the non-relativistic limit. \nThis induces a model-dependent uncertainty \nin the normalization of the inelastic contribution that can in principle only be avoided by fully implementing the concrete\ninteraction model in a code like \\texttt{GiBUU}. On the other hand, the neutrino example illustrates that this error should\ngenerally not be expected to be larger than a factor of $\\sim$\\,2, implying that for most applications such a \nmore sophisticated treatment is not warranted.\n\n\n\\section{Contact interactions and beyond}\n\\label{sec:m2}\n\nIn sections \\ref{sec:form_factors} and \\ref{sec:inel} we have discussed in detail the \n$Q^2$-dependence that arises due to both form factor suppression and inelastic\nscattering, as well as the impact this has on the production and attenuation of the CRDM flux.\nThis does not yet take into account, however, the possible angular and energy dependence of the\nelastic scattering cross section itself. In fact, for \\mbox{(sub-)GeV} DM, a significant dependence of this type is \nactually expected in view of null searches for new light particles at colliders. For example, it has been \ndemonstrated in a recent global analysis~\\cite{GAMBIT:2021rlp} that it is impossible to satisfy all relevant \nconstraints simultaneously (even well above GeV DM masses) \nand at the same time maintain the validity of an effective field theory description at LHC energies.\n\n\nOf course, this necessarily introduces a model-dependent element to the discussion, and in this section, the\naim will be to analyse the most generic situations that can appear when considering models beyond simple contact \ninteractions.  Concretely, in section~\\ref{sec:scalar} we will study the case of a light scalar mediator, a light vector\nmediator in section~\\ref{sec:vector}, and the scenario where DM particles have a finite extent\nin section~\\ref{sec:puffy}. In all these cases, we will re-interpret the published Xenon-1T limits and assess \nwhether there is a remaining unconstrained window of large scattering cross sections for GeV-scale DM. \nJust before this, however, in section~\\ref{sec:const} we will briefly revisit the (physically less motivated) case of \na constant cross-section, which can be viewed as the highly non-relativistic limit of a contact interaction. \nThis will allow us to illustrate how the resulting CRDM constraints compare with \nestablished bounds from both surface and astrophysical experiments, as well as provide a more direct comparison with the \nexisting literature.\n\n\n\n\\subsection{Constant cross section}\n\\label{sec:const}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_four_wl}\n\\caption{%\n{\\it Left panel.} Limits on a constant spin-independent DM-nucleon scattering\ncross section as a function of the DM mass, \nbased on a re-interpretation of Xenon-1T limits on non-relativistic DM~\\cite{XENON:2018voc}\nfor the CRDM component studied in this work (solid lines).\nDash-dotted lines show the excluded region that results when assuming a constant cross\nsection in the attenuation part (as in Ref.~\\cite{Bringmann:2018cvk}).\nDashed lines show the effects of adding form factors in the attenuation part, but no \ninelastic scattering, resulting in limits similar to those derived in Ref.~\\cite{Xia:2021vbz}.\nFor the latter case, for comparison, we also show the effect of artificially cutting the incoming CRDM flux \nat the indicated energies.\\\\ \n{\\it Right panel.} Updated CRDM limits (coinciding with the solid lines from the left panel) in comparison \nto limits from the Lyman-$\\alpha$ forest~\\cite{Rogers:2021byl}, the Milky Way satellite \npopulation~\\cite{Maamari:2020aqz}, gas clouds in the Galactic Centre \nregion~\\cite{Bhoonah:2018gjb}, the XQC experiment~\\cite{McCammon:2002gb,Mahdawi:2018euy}, and \na recently analysed storage dewar experiment~\\cite{Neufeld:2019xes,Xu:2021lmg}.\nWe also show upper limits on the cross section as published by the CRESST \ncollaboration~\\cite{CRESST:2017ues} (solid green lines), based on a surface run of their \nexperiment, along with the maximal cross section where \nattenuation does not prevent DM from leaving a signal in the detector~\\cite{Emken:2018run}.\nAlternative limits are indicated by green dashed~\\cite{Mahdawi:2018euy} \nand dash-dotted lines~\\cite{Xu:2020qjk}, based on the assumption of a thermalization efficiency \nof $\\epsilon_{\\rm th}=2$\\,\\% and $\\epsilon_{\\rm th}=1$\\,\\%, respectively, which is \nsignificantly worse than the one adopted in the CRESST analysis.\n}\n\\label{fig:constraints_constant}\n\\end{center}\n\\end{figure}\n\nFor the discussion of a constant cross section, we will again consider the case of spin-independent scattering with isospin \nconserving nucleon couplings, as described by Eq.~(\\ref{eq:siconst}). In the left panel of Fig.~\\ref{fig:constraints_constant}, we \nshow our improved constraints from a re-interpretation of the Xenon-1T limits in this case.\nBroadly, these updated and refined CRDM limits cover the mass range up to \n$m_\\chi \\lesssim 10 \\,\\mathrm{GeV}$ for cross-sections \n$10^{-31}\\,\\mathrm{cm}^2 \\lesssim \\sigma_{\\mathrm{SI}} \\lesssim 2 \\times 10^{-28}\\,\\mathrm{cm}^2$.\n\nFor comparison, we also indicate (with dash-dotted lines) the limits that result when neglecting both form-factor\ndependence of the cross section and inelastic scatterings in the attenuation part. As expected, this leads to a shape of the \nexcluded region very similar to that originally derived in Ref.~\\cite{Bringmann:2018cvk}, where the same simplifying\nassumptions were made. As a result of our improved treatment of CR fluxes and form factors, \nhowever, the limits indicated with dash-dotted lines are overall  slightly more stringent than what is reported in that analysis.\nWe find that for very light DM, with $m_\\chi\\lesssim10$\\,MeV, this simplistic treatment actually leads to rather\nrealistic limits, the reason being that for highly relativistic particles the typical momentum transfer is always so large\nthat efficient inelastic scattering becomes relevant. For heavier DM masses, on the other hand, this treatment clearly \noverestimates the stopping power because it neglects the form factor suppression  relevant for semi-relativistic DM\nscattering on nuclei.\n\nDashed lines furthermore show the effect of adding the form factor suppression during the attenuation in the soil, as done in \nRef.~\\cite{Xia:2021vbz}, but still not including inelastic scattering. Clearly, this vastly underestimates the actual\nattenuation taking place and therefore appears to exclude very large cross sections.\\footnote{%\nCompared to Ref.~\\cite{Xia:2021vbz}, we also find that the excluded region extends to somewhat larger DM masses,\nmostly as a result of our updated treatment of elastic form factors.\nOn the other hand, we recall that our attenuation prescription is based on the analytical energy loss treatment outlined in \nsection~\\ref{sec:crdm}, rather than a full Monte Carlo simulation. This likely overestimates the maximally excluded DM mass,\nbut only by less than a factor of 2~\\cite{Xia:2021vbz}.\n} \nIn order to gain a better intuitive understanding for the shape and strength of our final limits, finally, we also indicate the \neffect of neglecting inelastic scattering and instead artificially cutting the CRDM flux (prior to entering the soil) above \nsome given energy.  The resulting upper limit on the cross section that can be probed in this fiducial setup strongly suggests that \ninelastic scattering events very efficiently stop the incident CRDM flux in the overburden as soon as they become\nrelevant compared to elastic scattering events. From Fig.~\\ref{fig:constraints_constant}, and well in accordance with the\nexpectations from section \\ref{sec:inel}, this happens at CRDM energies $T_\\chi\\gtrsim0.2$\\,GeV.\n\nIn the right panel of Fig.~\\ref{fig:constraints_constant} we show our improved constraints \nfrom a re-interpretation of the Xenon-1T limits in comparison with complementary limits from direct\nprobes of the DM-nucleon scattering cross section.\nAt small DM masses the dominant\nconstraint results from analysing the distribution of large-scale structures as traced by the Lyman-$\\alpha$ \nforest. This is based on the fact that protons scattering too strongly off DM \nwould accelerate the latter and thereby suppress the matter power spectrum at sub-Mpc scales. \nSuch limits have recently been significantly tightened~\\cite{Rogers:2021byl},  utilizing\nstate-of-the-art cosmological hydrodynamical simulations of the intergalactic medium at redshifts \n$2\\lesssim z\\lesssim6$. Similar bounds from the CMB (not shown here) are generally weaker by up to three \norders of magnitude~\\cite{Rogers:2021byl, Planck:2015bpv, Xu:2018efh}, while\nthe Milky Way satellite population~\\cite{Maamari:2020aqz} -- as inferred from the Dark Energy Survey and \nPanSTARRS-1~\\cite{DES:2019vzn} -- places bounds that are roughly one order of magnitude weaker.\nBeyond cosmological bounds, cold gas clouds near the Galactic Center provide an interesting complementary testbed, \nin particular at high DM masses, where halo DM particles scattering too efficiently on the much {\\it colder} baryon population \nwould heat up the latter~\\cite{Bhoonah:2018wmw}. Here we show updated \nconstraints~\\cite{Bhoonah:2018gjb} based on \nthe cloud G357.8-4.7-55, noting that these constraints might be improved by more than one order\nof magnitude if G1.4-1.8+87 is indeed as cold as $T\\leq22$\\,K (as reported in \nRefs.~\\cite{McClure-Griffiths:2013awa,DiTeodoro:2018ybg} but disputed in Ref.~\\cite{Farrar:2019qrv}).\nWe also display the limits~\\cite{Mahdawi:2018euy} that result from the ten minutes' flight of the X-ray Calorimetry \nRocket (XQC)~\\cite{McCammon:2002gb}, based on the observation that ambient DM particles scattering\noff the silicon nuclei in the quantum calorimeter would deposit (part of) their energy in the \nprocess~\\cite{Wandelt:2000ad,Zaharijas:2004jv,Erickcek:2007jv}. \nIn deriving these XQC limits, one must take into account that the recoil energy of a silicon nucleus \npotentially thermalizes much less efficiently in the calorimeter than the $e^\\pm$ pairs produced from \nan incoming X-ray photon, such that a nuclear recoil energy $T_N$ will leave a signal equivalent  \nto a photon with a reduced `thermal' recoil energy $T_T=\\epsilon_{\\rm th} T_N$. Concretely, the limits shown in the \nplot are based on the very conservative assumption of a thermalization efficiency factor of $\\epsilon_{\\rm th} = 0.02$.\\footnote{%\nWhen the scattering is mediated by a Yukawa-like interaction, a perturbative description of the scattering\nprocess may no longer be adequate. In that case the constraints shown here, in particular for XQC, receive \ncorrections due to non-perturbative effects leading to resonances or anti-resonances in the scattering cross \nsection~\\cite{Xu:2020qjk}. Here, we will not consider this possibility further, noting that a variation of the relatively \nuncertain value of $\\epsilon_{\\rm th}$ anyway has a larger impact on the XQC limits~\\cite{Mahdawi:2018euy}.\n}\n\nFurthermore, in order to directly probe sub-GeV DM with very large cross sections, the CRESST collaboration has performed\na dedicated surface run of their experiment~\\cite{CRESST:2017ues}, deliberately avoiding the shielding \nof the Gran Sasso rock used in the standard run~\\cite{CRESST:2015txj}. The result of this search is the exclusion region \nindicated by the solid green line in Fig.~\\ref{fig:constraints_constant}. Here, upper bounds on the cross section correspond to \nthe published limits, obtained under the assumption that any attenuation in the overburden can be neglected. \nModelling the effect of attenuation with detailed numerical simulations also results in  \nthe exclusion region limited from above~\\cite{Emken:2018run}, coming from the fact that one must have a sufficiently large flux of \nDM particles at the detector location.\nIn a series of papers, Farrar~\\textit{et al.}~have claimed  \nthat the CRESST thermalization efficiency adopted in the official analysis is too optimistic~\\cite{Mahdawi:2018euy,Wadekar:2019mpc,Xu:2020qjk,Xu:2021lmg}, challenging the general ability of the experiment to probe sub-GeV DM.\nWe indicate the resulting alternatives to the published CRESST limits in the same figure, albeit noting that the underlying assumption \nof an efficiency as low as $\\epsilon_{\\rm th}\\sim1$\\,\\% is not supported by data or simulations.\nFor example, no indication for such a dramatic loss of efficiency at low energies is observed for neutrons from an AmBe \nneutron calibration source~\\cite{florian}.\n\nTo summarise, Fig.~\\ref{fig:constraints_constant} illustrates the fact that the existence of the CRDM component provides an \nimportant probe of strongly interacting light DM. In particular, below $m_\\chi\\lesssim100$\\,MeV, it restricts parameter space that \nis otherwise either unconstrained or only testable with \ncosmological probes (which -- at least to some degree -- are \nsubject to modelling caveats regarding the Lyman-$\\alpha$ forest and the non-linear evolution of density perturbations \nat small scales; see, e.g., Refs.~\\cite{Hui:2016ltb,Irsic:2017ixq}). The CRDM component also leads to\nhighly relevant complementary constraints up to DM masses of a few GeV, especially when noting that  \nthese constraints are independent of the thermalization efficiency discussion above.\n\n\n\n\\subsection{Scalar mediators}\n\\label{sec:scalar}\n\nAs our first example beyond a constant scattering cross section we consider the case where \na new light scalar particle $\\phi$ mediates the interaction between DM and nucleons.  \nWe thus consider the interaction Lagrangian \n\\begin{equation}\n\\mathcal{L}_{\\rm int}= - g_\\chi \\phi\\overline\\chi\\chi - g_p \\phi \\overline p p- g_n \\phi \\overline n n\\,,\n\\end{equation}\nand assume, for simplicity, isospin conservation ($g_p = g_n $).\nAt the level of the effective nuclear interaction Lagrangian, the dominant interaction terms with scalar ($N_0$) and fermionic ($N_{1\/2}$) \n{\\it nuclei} are thus given by\\footnote{%\nWhile the dominant cosmic-ray nuclei are either scalar or spin $1\/2$ particles, some heavier nuclei in the overburden\nhave higher spins. For simplicity we treat those nuclei as scalars when determining their contribution to the energy loss,\nas described by Eq.~(\\ref{eq:eloss}), noting that this induces a neglible error in the estimated elastic scattering cross section,\nof the order of $Q^2\/m_N^2\\ll1$. Moreover, nuclei with higher spins make up only about 2\\% of the total mass in the overburden.\n}\n\\begin{equation}\n\\label{eq:leff_scalar}\n\\mathcal{L}_{\\rm int}= -g_N\\left( 2m_N N_0N_0+\\overline N_{1\/2}N_{1\/2}\\right)\\,.\n\\end{equation}\nHere, the dimensionful coupling to scalar nuclei has been normalized such that both terms in the above expression result\nin the same scattering cross section in the highly non-relativistic limit. In addition, the coupling to individual nucleons is coherently \nenhanced across the nucleus, resulting in an effective coupling to both scalar and fermionic nuclei given by\n\\begin{equation}\n\\label{eq:gN_coh}\ng_N^2 =A^2 \\, g_p^2 \\times G_N^2(Q^2)\\,,\n\\end{equation}\nwhere $G_N$ is the same form-factor as in the case of a `constant' cross section.\nFor the resulting elastic scattering cross section for DM incident on nuclei at rest we find\n\\begin{equation}\n\\label{diffsig_full_scalar}\n\\frac{d\\sigma_{\\chi N}}{d T_N}=\\frac{\\mathcal{C}^2\\sigma_{\\rm SI}^\\mathrm{NR}}{T_N^\\mathrm{max}}\n\\frac{m_\\phi^4}{(Q^2+m_\\phi^2)^2}\n\\frac{m_N^2\\left(Q^2+4m_\\chi^2 \\right)}{4 s\\,{\\mu_{\\chi N}^2}}\n\\times\\left\\{\n\\begin{array}{ll}\n1& ~~\\mathrm{for~scalar~}N\\\\\n1+\\frac{Q^2}{4m_N^2} &  ~~\\mathrm{for~fermionic~}N\n\\end{array}\n\\right\\}\n\\times G_N^2(Q^2)\\,,\n\\end{equation}\nwhere $\\mu_{\\chi p}$ is the reduced mass of the DM\/nucleon system and  \n\\begin{equation}\n\\label{eq:sig0_scalar}\n\\sigma_{\\rm SI}^\\mathrm{NR} = \\frac{g_\\chi^2 g_p^2 \\mu_{\\chi p}^2}{\\pi m_\\phi^4}\n\\end{equation}\nis the spin-independent scattering cross section {\\it per nucleon} in the ultra non-relativistic limit.\nFor reference, the kinematic quantities $T_N^\\mathrm{max}$, $s$ and $Q^2$ are given by \nEqs.~(\\ref{eq:tmax}), (\\ref{eq:sdef}) and (\\ref{eq:q2}), respectively. For the production part of the process, \nwhere CR nuclei collide with DM at rest, one\nsimply has to exchange $T_N\\leftrightarrow T_\\chi$ and $m_\\chi\\leftrightarrow m_N$ in \nthese expressions for kinematic variables -- but not in the rest of Eq.~(\\ref{diffsig_full_scalar}) \n-- in order to obtain ${d\\sigma_{\\chi N}}\/{d T_\\chi}$.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_five_wl}\n\\caption{{\\it Left panel.} Solid lines show the CRDM flux before attenuation for a constant interaction\ncross section, as in Fig.~\\ref{fig:flux_ff}, for DM masses  $m_\\chi=10$\\,MeV and  $m_\\chi=1$\\,GeV. \nFor comparison we also indicate the corresponding CRDM flux for a {\\it scalar} mediator, cf.~Eq.~(\\ref{diffsig_full_scalar}), \nwith mass  $m_\\phi=100$\\,MeV (dash-dotted), $m_\\phi=10$\\,MeV \n(dashed) and $m_\\phi=1$\\,MeV (dotted), for a cross section (in the non-relativistic limit) of  \n$\\sigma_{\\rm SI}^{\\rm NR}=10^{-30}\\,\\mathrm{cm}^2$. \n{\\it Right panel.} \nMinimal kinetic energy $T_\\chi$ that a DM particle must have, prior to attenuation, in order \nto trigger a signal in the Xenon-1T experiment. Line styles and colors match those of the left\npanel. In particular, solid lines show the case of a constant spin-independent scattering cross \nsection and are identical to those displayed  in Fig.~\\ref{fig:attenuation_ff}.\n}\n\\label{fig:results_scalar}\n\\end{center}\n\\end{figure}\n\nIn the left panel of Fig.~\\ref{fig:results_scalar} we show the resulting CRDM fluxes for this\nmodel. For small kinetic energies these fluxes are, as expected, identical to those shown in Fig.~\\ref{fig:flux_ff} \nfor the case of a constant cross section. This is the regime where $Q^2=2m_\\chi T_\\chi$ is smaller\nthan the masses of both the mediator and CR nuclei, such that Eq.~(\\ref{diffsig_full_scalar}) reduces to \nEq.~(\\ref{eq:siconst}). For $Q^2\\gtrsim m_\\phi^2$, on the other hand, the presence of a light mediator \nclearly suppresses the fluxes. Note that the matrix element also contains a factor of $(Q^2+4m_\\chi^2)$, \nwhich additionally leads\nto a flux {\\it enhancement} for fully relativistic DM particles, $T_\\chi\\gtrsim 2 m_\\chi$. \nIn the figure, this latter effect is clearly visible for the case of $m_\\chi=10$\\,MeV and a heavy mediator.\nIn general, the appearance of such model-dependent features demonstrates the need to use the full matrix \nelement for the relativistic cross section. This is in contrast to the non-relativistic case, where a model-independent rescaling \nof the cross section by a factor of $(1+Q^2\/m_\\phi^2)^{-2}$ is usually sufficient to model the effect of a light mediator\n(see, e.g., Refs.~\\cite{Chang:2009yt,Fornengo:2011sz,Kaplinghat:2013yxa}).\n\nIn the right panel of Fig.~\\ref{fig:results_scalar}, we explore the minimal CRDM energy $T_\\chi$ that\nis needed to induce a detectable nuclear recoil. Compared to the situation of a constant scattering cross section (depicted by the \nsolid lines for easy comparison), the attenuation is as expected rather strongly suppressed when light scalar mediators are \npresent (with the exception of the\ncase with $m_\\chi=10$\\,MeV and $m_\\phi=100$\\,MeV, where the cross section is enhanced \ndue to the $(Q^2+4m_\\chi^2)$ factor in the squared matrix element). In order to understand the qualitative behaviour of \n$T_\\chi^{\\rm min} (z=0)$ better, we recall from the discussion of Fig.~\\ref{fig:attenuation_ff}\nthat there are two generic scaling regimes for solutions of the energy loss equation. Firstly, for cross-sections with no \n-- or only a mild --  dependence on the momentum transfer, $T_{\\chi}^{\\rm min}(z=0)$ grows exponentially \nwith increasing $\\sigma_{\\rm SI}^{\\rm NR}$. Secondly, in the presence of an effective cutoff in the cross section (like when form \nfactors or light mediators are introduced), $T_{\\chi}^{\\rm min}(z=0) \\propto \\sqrt{\\sigma_{\\rm SI}^{\\rm NR}}$ for large energies\n$T_\\chi$. These different regimes are clearly visible in the figure. \nFor the green dot-dashed curve ($m_\\chi=1$\\,GeV, $m_\\phi=100$\\,MeV), for example,\none observes as expected an initial steep rise at the smallest DM energies -- until the form factor and mediator suppression \nof the cross section cause a scaling with $\\sqrt{\\sigma_{\\rm SI}^{\\rm NR}}$ for kinetic energies above a few MeV. \nAt roughly $T_\\chi\\gtrsim0.1$\\,GeV, inelastic scattering kicks in, leading again to an exponential suppression of the flux. \nFor even higher energies, finally, the scattering cross section falls off so rapidly that the required initial DM energy once\nagain only grows as $\\sqrt{\\sigma_{\\rm SI}^{\\rm NR}}$.\n\nTurning our attention to the resulting CRDM limits, it is worth stressing here that $\\sigma_{\\rm SI}^\\mathrm{NR}$, as introduced in Eq.~(\\ref{eq:sig0_scalar}),\nis a somewhat artificial object that only describes the cross section for physical processes\nrestricted to $Q^2\\lesssim m_\\phi^2$.\nIn a direct detection experiment like Xenon-1T this is necessarily violated for $m_\\phi\\lesssim \\sqrt{2m_N T_N^{\\rm thr}}\\sim35$\\,MeV, \ngiven that  $T_N^{\\rm thr}=4.9$\\,keV is the  minimal recoil \nenergy needed to generate a signal. A natural consequence of this is that making a straight-forward comparison \nto the $\\sigma_{\\rm SI}$ appearing in the `constant cross section' case discussed in \nsection \\ref{sec:const} is challenging. Instead, the best we can achieve in terms of a meaningful \ncomparison is to define a {\\it reference cross section}\n\\begin{equation}\n\\label{eq:Qref}\n \\tilde \\sigma_{\\rm Xe,SI}^p \\equiv \\sigma_{\\rm SI}^\\mathrm{NR}\\times \\frac{m_\\phi^4}{(Q_{\\rm Xe,ref}^2+m_\\phi^2)^2}\n\\frac{Q^2_{\\rm Xe,ref}+4m_\\chi^2}{4m_\\chi^2}\\,,\n\\end{equation}\nwhere $Q_{\\rm Xe,ref}\\sim35$\\,MeV. It follows from Eq.~(\\ref{diffsig_full_scalar}) and Eq.~(\\ref{eq:siconst}), \nand the fact that $s\\approx (m_\\chi+m_N)^2$ for the energies of interest here, that $ \\tilde \\sigma_{\\rm Xe,SI}^p$ \nshould be interpreted as the effective CRDM cross section per nucleon that is dominantly seen in the Xenon-1T \nanalysis window.\nIt is thus this quantity, not the $\\sigma_{\\rm SI}^\\mathrm{NR}$ from Eq.~(\\ref{eq:sig0_scalar}), that should\nbe compared to the published Xenon-1T limits on the DM-nucleon cross section.\n\nThis also allows us to address the question of how the limits on the DM-nucleon coupling coming from the CRDM component\ncompare to the complementary constraints introduced in section \\ref{sec:const} (cf.~the right\npanel of Fig.~\\ref{fig:constraints_constant}). In order to do so, one first needs to\nrealize that all of those limits are derived under the assumption of non-relativistic\nDM and a constant cross section. In reality, however, they probe very different physical environments and typical \nmomentum transfers. In order to allow for a direct comparison, therefore, they also need to be re-scaled to a common \nreference cross section.\nAssuming that the DM energies in Eq.~(\\ref{diffsig_full_scalar}) are non-relativistic, a reported limit on the DM-nucleon \ncross-section $\\sigma_{\\rm SI}^p$ from an experiment probing typical momentum transfers of the order $Q^2_{\\rm ref}$ would \ncorrespond to a cross section of\n\\begin{equation}\n\\label{eq:rescale_scalar}\n \\tilde \\sigma_{\\rm Xe,SI}^p = \\sigma_{\\rm SI}^p\\times \n \\left(\\frac{Q^2_{\\rm ref}+m_\\phi^2}{Q^2_{\\rm Xe,ref}+m_\\phi^2}\\right)^2\n\\frac{Q^2_{\\rm Xe,ref}+4m_\\chi^2}{Q^2_{\\rm ref}+4m_\\chi^2}\n\\end{equation}\nin the Xenon-1T detector. As an example, consider the CRESST surface run~\\cite{CRESST:2017ues}, where a threshold energy of $\\sim20$\\,eV\nfor the sapphire \ndetector would imply $Q^2_{\\rm ref}\\sim (0.98\\,\\mathrm{MeV})^2\/\\epsilon_{\\rm th}$.  Similarly, a thermal recoil energy of of $29$\\,eV\nin XQC corresponds to $Q^2_{\\rm ref}\\sim (8.7\\,\\mathrm{MeV})^2$ for the nuclear recoil on Si nuclei\n(assuming $\\epsilon_{\\rm th} = 0.02$ as for the unscaled limits).\nTurning to cosmological limits, a baryon velocity of $v_b^{\\rm rms}\\sim33$km\/s at the times relevant for the emission\nof Lyman-$\\alpha$ photons~\\cite{Silk:1967kq} implies typical momentum transfers from the Helium nuclei to DM of \n$Q_{\\rm ref}^2\\sim4 \\mu_{\\chi{\\rm He}}^2\\times10^{-8}$. This means that, for the range of DM and mediator masses considered \nhere, the cross section at these times becomes roughly constant and we can approximate $Q_{\\rm ref}^2\\approx0$ \nin Eq.~(\\ref{eq:rescale_scalar}). The same goes for the constraints stemming from the MW satellite abundance,\nwhich are sensitive to even lower redshifts and thus smaller momentum transfers~\\cite{Nadler:2019zrb,Maamari:2020aqz}.\n\nIn Fig.~\\ref{fig:limits_scalar} we show a subset of these correspondingly rescaled constraints\\footnote{%\nUpper bounds on the excluded cross section, due to attenuation effects, cannot simply be rescaled as\nin Eq.~(\\ref{eq:rescale_scalar}). \nFor the sake of Fig.~\\ref{fig:limits_scalar}, we instead adopt a rather simplistic \napproach~\\cite{Davis:2017noy,Kouvaris:2014lpa,Emken:2017erx,Emken:2018run} to estimate these \nlimits by requiring that the most energetic halo DM particles, with an assumed velocity $v_{\\rm max}$,\ncan trigger nuclear recoils above the CRESST threshold of 19.7\\,eV\/$\\epsilon_{\\rm th}$ after attenuation in the \nEarth's atmosphere. For the average density and distribution of elements in the atmosphere, we follow Ref.~\\cite{USatm}.\nBy treating $v_{\\rm max}$ and the effective height of the atmosphere, $h_a$, \nas free parameters, we can then rather accurately fit the results of more detailed \ncalculations~\\cite{Emken:2018run,Mahdawi:2018euy} for the case of a constant cross section\n-- with numerical values in reasonable agreement with the physical expectation in such a heuristic approach. \nFinally, we adopt those values of $v_{\\rm max}$ and $h_a$ to derive the corresponding limits for the case of a scalar mediator,\nas displayed in Fig.~\\ref{fig:limits_scalar}.\n}\n -- for mediator masses \n$m_\\phi=1$\\,MeV, 10\\,MeV, 100\\,MeV and 1\\,GeV -- along with the full CRDM constraints derived here.\nWe also indicate, for comparison, with dotted black lines where non-perturbative couplings would be needed in this model to \nrealize the stated cross section. This line is only visible for the case of $m_\\phi=1$\\,GeV,\nwhich demonstrates that it is generically challenging to realize large cross sections without invoking light mediators.\nThe presence of an abundant species with a mass below a few MeV, furthermore, would affect how light elements are\nproduced during big bang nucleosynthesis (BBN). For a 1\\,MeV particle with one degree of freedom, like $\\phi$,  \nthis can be formulated as a constraint of $\\tau>0.43$\\,s~\\cite{Depta:2020zbh} on the lifetime of such a particle.\nPhysically, this constraint derives from freeze-in production of $\\phi$ via the inverse decay process. \nSince $\\phi\\to\\gamma\\gamma$ (apart from $\\phi\\to\\bar \\nu\\nu$) is the only kinematically possible SM decay channel, \nthe translation of this bound to a constraint on the SM coupling $g_p$ is somewhat model-dependent.\nFor concreteness we consider the Higgs portal model, where $\\tau>1$\\,s at $m_\\phi=1$\\,MeV corresponds to \na squared mixing angle\n$\\sin^2\\theta=(8.62\\times10^2g_p)^2>3.8\\times10^{-4}$~\\cite{Krnjaic:2015mbs}. The area above the dashed\nline in the top left panel of Fig.~\\ref{fig:limits_scalar} requires either a {\\it larger} value of $g_p$ than what is given by this\nbound, or a non-perturbative coupling $g_\\chi^2>4\\pi$. This confirms the generic expectation that for very light particles \nBBN constraints are more stringent than those stemming from the CRDM \ncomponent~\\cite{Krnjaic:2019dzc,Bondarenko:2019vrb}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{figures_draft\/figure_six_wl.pdf}\n\\caption{%\nLimits on the DM-nucleon scattering cross section evaluated at a reference momentum transfer of \n$Q_{\\rm Xe,ref}=35$\\,MeV, as a function of the DM mass $m_\\chi$. From top left to bottom right,\nthe panels show the case of a {\\it scalar mediator} with mass $m_\\phi=1$\\,MeV, 10\\,MeV, 100\\,MeV and 1\\,GeV.\nSolid purple lines show the updated CRDM limits studied in this work. We further\nshow limits from the Lyman-$\\alpha$ forest~\\cite{Rogers:2021byl}, the XQC \nexperiment~\\cite{McCammon:2002gb,Mahdawi:2018euy}, the CRESST surface \nrun~\\cite{CRESST:2017ues, Emken:2018run} and an alternative analysis of the CRESST \nlimits~\\cite{Mahdawi:2018euy}. All these limits are rescaled to match the situation of a light mediator,\nas explained in the text. The parameter region above the dotted black line in the bottom right panel\nrequires non-perturbative couplings, while the area above the dotted line in the top left panel is excluded\nby BBN.\n}\n\\label{fig:limits_scalar}\n\\end{center}\n\\end{figure}\n\n\nOur results demonstrate that in the presence of light mediators the largest DM mass that can be constrained \ndue to CR upscattering is reduced from about 10\\,GeV, cf.~Fig.~\\ref{fig:constraints_constant},\nto just above 1\\,GeV (for $m_\\phi\\sim1$\\,MeV). This is a direct consequence of the suppressed CRDM production\nrate discussed above. On the other hand, the reduction of the cross section also implies a smaller attenuation\neffect, thus closing parameter space at larger cross sections. More importantly, complementary constraints\nfrom cosmology and dedicated surface experiments become {\\it more stringent} in the presence of light mediators, \nonce they are translated to a common reference cross section. To put this in context, let us first recall that in the constant \ncross section case, Fig.~\\ref{fig:constraints_constant} tells us that cross sections \n$\\sigma_{\\rm SI}\\gtrsim 2\\cdot10^{-31}\\,{\\rm cm}^2$  are safely excluded across the entire DM mass range \n(or $\\sigma_{\\rm SI}\\gtrsim 6\\cdot10^{-31}\\,{\\rm cm}^2$ when assuming that the thermalization efficiency of \nCRESST is indeed as low as 2\\,\\%). From Fig.~\\ref{fig:limits_scalar} we infer that these limits can be somewhat \nweakened for sub-GeV DM, when considering light meditators in the mass range \n$10\\,{\\rm MeV}\\lesssim m_\\phi\\lesssim 100\\,{\\rm MeV}$ (as we will see further down, the situation of a vector mediator\nis not appreciably different from that of the scalar mediator shown here). Concretely, the upper bound on the \ncross section now becomes $\\tilde \\sigma_{\\rm SI}\\lesssim 3\\cdot10^{-31}\\,{\\rm cm}^2$, independently of the DM\n{\\it and} mediator mass. For a 2\\,\\% thermalization efficiency of CRESST~\\cite{Mahdawi:2018euy} and a narrow \nrange of mediator masses, $10\\,{\\rm MeV}\\lesssim m_\\phi\\ll100\\,{\\rm MeV}$,\na small window opens up above the maximal cross section that can be probed with CRESST. \nThe reason is the gap between Lyman-$\\alpha$ bounds and the weakened CRESST limits from Ref.~\\cite{Mahdawi:2018euy}\nthat is visible in the figure, for $m_\\phi\\gtrsim10\\,{\\rm MeV}$, and which is closed by the CRDM limits only for mediator\nmasses of $m_\\phi\\gtrsim 30$\\,MeV. Nominally, for $m_\\chi\\sim2$\\,GeV and $m_\\phi\\sim 30$\\,MeV, \nthis would allow for cross sections as large as $\\tilde \\sigma_{\\rm SI}\\sim 4\\cdot10^{-29}\\,{\\rm cm}^2$.\nIn either case, the conclusion remains that CRDM leads to highly complementary limits, and that this \nrelativistic component of the DM flux is in fact crucial for excluding the possibility of very large DM-nucleon\ninteractions.\n\n\n\\subsection{Vector mediators}\n\\label{sec:vector}\n\nWe next consider the general case of a massive vector mediator $V$, with interactions given by\n\\begin{equation}\n\\mathcal{L}=   V_\\mu \\left(g_\\chi \\overline{\\chi}\\gamma^\\mu \\chi + g_{p}\\overline{p}\\gamma^\\mu p + g_{n}\\overline{n}\\gamma^\\mu n\\right)\\,.\n\\end{equation}\nWe will again assume  $g_n=g_p$  for simplicity, noting that smaller values of the ratio $g_n\/g_p$ can lead to\nsignificantly smaller cross sections (see, e.g., Refs.~\\cite{Frandsen:2011cg,Kaplinghat:2013yxa}); in our context this would\nmostly imply that the attenuation in the overburden becomes less relevant, leading to more stringent constraints.\nIn analogy to Eq.~(\\ref{eq:leff_scalar}), this implies the following dominant interaction terms with scalar and fermionic nuclei, respectively:\n\\begin{equation}\n\\label{eq:leff_vector}\n\\mathcal{L}_{\\rm int}= -g_N V_\\mu\\left( i N_0^*{\\mathop{\\partial^\\mu}^{\\leftrightarrow}} N_0+\\overline N_{1\/2}\\gamma^\\mu N_{1\/2}\\right),\n\\end{equation}\nwhere the effective mediator coupling to nuclei, $g_N$, is again given by the coherent enhancement stated in Eq.~(\\ref{eq:gN_coh}).\nFor the elastic scattering cross section on nuclei we find\n\\begin{eqnarray}\n\\label{diffsig_full_vector}\n\\frac{d\\sigma_{\\chi N}}{d T_N}&=&\\frac{\\mathcal{C}^2 \\sigma_{\\rm SI}^\\mathrm{NR}}{T_N^\\mathrm{max}}\n\\frac{m_A^4}{(Q^2+m_A^2)^2}\n\\times G_N^2(Q^2)\\\\\n&&\n\\times\\frac{1}{4s \\mu_{\\chi N}^2}\n\\left\\{\n\\begin{array}{ll}\nm_\\chi^2Q^2-Q^2s+(s-m_N^2-m_\\chi^2)^2& ~~\\mathrm{for~scalar~}N\\\\\n\\frac12 Q^4 -Q^2s+(s-m_N^2-m_\\chi^2)^2&  ~~\\mathrm{for~fermionic~}N\n\\end{array}\n\\right..\\nonumber\n\\end{eqnarray}\nHere, the cross section in the ultra-nonrelativistic limit,\n\\begin{equation}\n\\label{eq:sig0_vector}\n\\sigma_{\\rm SI}^\\mathrm{NR} = \\frac{g_\\chi^2 g_p^2 \\mu_{\\chi p}^2}{\\pi m_A^4}\\,,\n\\end{equation}\ni.e.~for $Q^2\\to0$ and $s\\to(m_N+m_\\chi)^2$, agrees exactly with the result obtained for the scalar case, as expected.\nFor large energies and momentum transfers, on the other hand, the behaviour\nis different.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_seven_wl}\n\\caption{%\n{\\it Left panel.} \nMinimal kinetic energy $T_\\chi$ that a DM particle must have, prior to attenuation, in order \nto trigger a signal in the Xenon-1T experiment for DM nucleus interactions via a {\\it vector mediator},\nas a function of the spin-independent DM-nucleon scattering cross section in the highly non-relativistic limit, \n$\\sigma_{\\rm SI}^\\mathrm{NR}$. Yellow (green) lines indicate a DM mass\n$m_\\chi=10$\\,MeV ($m_\\chi=1$\\,GeV), and different line styles correspond to  mediator masses \n$m_A=1,10,100$\\,MeV as indicated.\nSolid lines show the case of a constant spin-independent scattering cross \nsection and are identical to those displayed  in Fig.~\\ref{fig:attenuation_ff}.\n{\\it Right panel.} \nConstraints on $\\sigma_{\\rm SI}^\\mathrm{NR}$ as a function of the DM mass $m_\\chi$.  Solid purple lines \nrefer to the case of a constant cross section, as in Fig.~\\ref{fig:constraints_constant}, while other line \nstyles show the case where the interaction is mediated by a light scalar (red) or vector (green) particle \nwith mass $m_{\\rm med}=10$\\,MeV and $1$\\,GeV, respectively.}\n\\label{fig:limits_vector}\n\\end{center}\n\\end{figure}\n\nThe resulting CRDM fluxes are nonetheless so similar to the scalar case shown in the left panel of Fig.~\\ref{fig:results_scalar} \nthat we refrain from plotting them separately.  Differences do exist, however, for the stopping power in the overburden.\nIn the left panel of Fig.~\\ref{fig:limits_vector} we therefore show the minimal initial kinetic energy needed by a CRDM\nparticle to induce detectable nuclear recoils in Xenon-1T. Compared to the scalar case, cf.~the right panel of \nFig.~\\ref{fig:results_scalar},  the attenuation is more efficient for highly relativistic DM particles due to the $s$-dependence \nof the terms in the second line of Eq.~(\\ref{diffsig_full_vector}). As before, the effect of these model-dependent terms\nfrom the scattering amplitude \nis most visible for highly relativistic particles, with small $m_\\chi$, and large mediator masses, where \nthe suppression due to the factor $(1+Q^2\/m_A^2)^{-2}$ is less significant.\n\nIn the right panel of  Fig.~\\ref{fig:limits_vector} we compare the  final exclusion regions for the \nsituations considered so far, i.e.~for a contact interaction, scalar mediators and vector mediators, respectively. \nFor the sake of comparison in one single figure, \nwe plot here the cross section in the ultra-nonrelativistic limit. For an interpretation of \nthese limits in comparison to complementary constraints on DM-nucleon interactions \nwe thus refer to the discussion of Fig.~\\ref{fig:limits_scalar},\nnoting that the rescaling prescriptions for vector and scalar mediators are qualitatively the same.\nThe first thing to take away from Fig.~\\ref{fig:limits_vector} is that, as expected, the exclusion regions \nfor heavy mediators resemble those obtained for the constant cross section case. The figure\nfurther demonstrates that the only significant difference between scalar and vector mediators appears  \nat smaller mediator masses, where the former are somewhat less efficiently stopped in the overburden. \nIt is worth noting, however, that this region of parameter space where the vector and scalar cases differ substantially\nis nonetheless excluded by Lyman-$\\alpha$ bounds. \nThe general discussion and conclusions from the scalar mediator case \nexplored in the previous subsection thus also applies to interactions mediated by vector particles.\n\n\n\n\n\\subsection{Finite-size dark matter}\n\\label{sec:puffy}\n\nAs a final generic example of a $Q^2$-suppressed cross section let us consider the situation \nwhere the DM particle itself has a finite size that is larger than its Compton wavelength. \nThe corresponding scattering cross section then takes the same form as in the point-like case,\nmultiplied by another factor $\\left|G_\\chi(Q^2)\\right|^2$ that reflects the spatial extent of \n$\\chi$~\\cite{Feldstein:2009tr,Laha:2013gva,Chu:2018faw} (for concrete models see also, e.g., \nRefs.~\\cite{Nussinov:1985xr,Chivukula:1989qb,Cline:2013zca,Krnjaic:2014xza,Wise:2014ola,Hardy:2015boa,Coskuner:2018are,Contino:2018crt}).\nSpecifically, just as for nuclear form factors, we have \n\\begin{equation}\nG_\\chi(Q^2)=\\int d^3 x\\,e^{i\\mathbf{q}\\cdot\\mathbf{x}}\\rho_\\chi(\\mathbf{x})\\,,\n\\end{equation}\nwhere $\\rho_\\chi(\\mathbf{x})$ is the distribution of the effective charge density that the interaction couples to.\nFor simplicity we will  choose a dipole form factor of the form\\footnote{%\nThe exact choice of the form factor does not significantly affect our results, as long as $G_\\chi(Q^2)<G_\\chi(0)=1$.\nAn interesting, qualitatively different situation occurs when $G_\\chi(0)=0$, i.e.~for a form factor that {\\it grows} \nwith $Q^2$. This is, e.g., realized if the scattering is mediated by a dark $U(1)'$ under which $\\chi$ is \nneutral~\\cite{Feldstein:2009tr,Chu:2018faw}. We will not consider this class of models in this work.\n}\n\\begin{equation} \\label{eq:dipole}\nG_\\chi(Q^2)=\\left(1+\\frac{r_\\chi^2}{12}Q^2\\right)^{-2}\\,,\n\\end{equation}\nwith $r_\\chi$ being the r.m.s.~radius of the DM particle, $r_\\chi^2=\\int d^3 x\\,\\mathbf{x}^2 \\rho_\\chi(\\mathbf{x})$.\nWe then multiply $G^2_\\chi(Q^2)$ with Eq.~(\\ref{eq:siconst}) in order to obtain \n${d\\sigma_{\\chi N}}\/{d T_N}$, thus describing an effective scalar interaction with the usual coherent\nenhancement inside the  nucleus -- but where each of the nucleons only `sees' some fraction of the entire DM \nparticle.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_eight_wl}\n\\caption{{\\it Left panel.} \nSolid lines show the CRDM flux before attenuation for a constant interaction\ncross section, as in Fig.~\\ref{fig:flux_ff}, for DM masses  $m_\\chi=10$\\,MeV and  $m_\\chi=1$\\,GeV. \nFor comparison we indicate the corresponding CRDM flux for {\\it finite size} DM,  \nwith  $r_\\chi =1$\\,fm (dotted) and $r_\\chi=10$\\,fm (dashed), for a cross section of  \n$\\sigma_{\\rm SI}^{\\rm NR}=10^{-30}\\,\\mathrm{cm}^2$. \n{\\it Right panel.} Limits on the spin-independent DM-nucleon cross section, with line styles and colors matching \nthose of the left panel. In particular, solid lines show the case of a constant scattering cross \nsection and are identical to those displayed  in the left panel of Fig.~\\ref{fig:constraints_constant}.\n\\label{fig:results_puffy}\n}\n\\end{center}\n\\end{figure}\n\nIn a very similar fashion to what happens in the presence of a light mediator $\\phi$, such a cross section features a sharp \nsuppression for momentum transfers exceeding a `mass' scale $m_\\phi\\sim \\sqrt{12}\/r_\\chi$. Sharper than in that case, in fact, as \nthe suppression scales with \na power of $Q^{-8}$ rather than just $Q^{-4}$. This is clearly visible in the left panel of Fig.~\\ref{fig:results_puffy},\nwhere we plot the expected CRDM flux for DM with a finite size, for various values of $m_\\chi$ and $r_\\chi$. \nFor example, for $r_\\chi=10$\\,fm, we have \n$\\sqrt{12}\/r_\\chi\\sim68$\\,MeV and the cutoff indeed appears at only slightly smaller values of $T_\\chi$ than in the case\nof the $100$\\,MeV mediator displayed in Fig.~\\ref{fig:results_scalar} (for $m_\\chi=1$\\,GeV). The slope above the cutoff, \nhowever, is twice as steep -- as expected from the $Q^{-8}$ suppression.\n\nIn the right panel of Fig.~\\ref{fig:results_puffy} we show how the constraints on a constant DM-nucleon cross section\nweaken when considering the situation where the DM particles themselves have a finite extent. Concretely,\nfor a DM radius of $r_\\chi=1$\\,fm ($r_\\chi=10$\\,fm) the maximal DM mass that can be probed decreases from\n$\\sim10$\\,GeV to about $4.5$\\,GeV ($1.1$\\,GeV). The reduced CRDM flux for extended DM, cf.~the left panel\nof the figure, also \nvisibly weakens the lower bound on the exclusion region. At the same time, attenuation is also less efficient\nfor a given cross section in the non-relativistic limit (inelastic scattering still effectively cuts off the incoming CRDM\nflux above $\\sim$0.2\\,GeV, explaining e.g.~the upper, almost horizontal boundary of the exclusion region in the\n$r_\\chi=10$\\,fm case).\nFor $r_\\chi\\gtrsim1$\\,fm, this starts to significantly enlarge the excluded region to higher cross sections.\nOn the other hand, it should be noted that for composite DM particles the interaction\ncross section may not actually continue to drop as $Q^{-8}$ for very large momentum transfers,\nas would be implied by Eq.~(\\ref{eq:dipole}). At some point, instead, inelastic scattering events on the DM constituents \nwill take over, in analogy to what we discussed for nuclei in section \\ref{sec:inel}. This is particularly relevant \nif the DM constituents are themselves finite in size, in which case the upper boundaries of the\nexclusion regions shown in Fig.~\\ref{fig:results_puffy} would be overly optimistic for very large $r_\\chi$.\n\nSimilar to the discussion in section \\ref{sec:scalar}, a comparison of the limits shown in Fig.~\\ref{fig:results_puffy}  \nwith complementary limits requires a re-scaling of $\\sigma_{\\rm SI}$ to a common reference cross section. Due to the\nstrong form factor suppression, this rescaling has an even larger effect than in the light mediator case; concretely,\ninstead of Eq.~(\\ref{eq:rescale_scalar}), the rescaling of reported limits, $\\sigma_{\\rm SI}^p$, to those relevant \nfor the Xenon-1T detector now takes the form\n\\begin{equation}\n \\tilde \\sigma_{\\rm Xe,SI}^p = \\sigma_{\\rm SI}^p\\times \n \\left(\\frac{Q^2_{\\rm ref}+12\/r_\\chi^2}{Q^2_{\\rm Xe,ref}+12\/r_\\chi^2}\\right)^4\\,.\n\\end{equation}\nQualitatively, however, this does not change the lesson learned in the light mediator case: while limits from the CRDM component\ncan be weakened by increasing $r_\\chi$, this will inevitably strengthen complementary bounds from cosmology. As a result,\nwe find once again an absolute upper bound on the cross section of about  \n$\\tilde \\sigma_{\\rm SI}\\sim 3\\cdot10^{-31}\\,{\\rm cm}^2$, independently of the DM mass\n{\\it and} size. Also in this case there is a small loophole to this statement if one is willing to assume that the thermalization\nefficiency of CRESST is as small as 2\\,\\%: when tuning the size of the DM particles to $r_\\chi\\simeq10$\\,fm,\nwe find that cross sections two orders of magnitude larger may in that case be viable for DM masses in a narrow range \nbetween around 1\\,GeV and 2\\,GeV.\n\n\n\n\\section{Hexaquarks: a viable baryonic dark matter candidate?}\n\\label{sec:hexaquark}\n\nIn section \\ref{sec:m2} we discussed various generic situations where the amplitude for elastic scattering \nshows a significant dependence on the momentum transfer, and how this impacts the conclusions about \nwhether a window of large scattering cross sections remains open or not. In this section we complement\nthose more model-independent considerations by taking a closer look at a specific DM candidate\nin the GeV range, with relatively large nuclear interactions.\nConcretely, it has been conjectured that a neutral (color-flavor-spin-singlet) bound state of six light quarks \n$uuddss$ may exist, and provide a plausible DM candidate that would evade all current\nconstraints despite its baryonic nature~\\cite{Farrar:2002ic,Zaharijas:2004jv,Farrar:2017eqq, Farrar:2020zeo,Farrar:2022mih}. \nIn particular, this \\emph{sexaquark} $S$ (to be distinguished from a generic 6-quark state, often referred to as {\\it hexaquark})\nwould form early enough to behave like standard cold DM during both big bang nucleosynthesis and recombination.\nIt would thus not be in conflict with the independent, and rather precise, measurements~\\cite{Aver:2015iza,Planck:2018vyg} \nof the cosmological baryon density during these epochs.\n\nCompared to the H-dibaryon that was suggested earlier~\\cite{Jaffe:1976yi} and thoroughly studied both theoretically  and experimentally (see Refs.~\\cite{Sakai:1999qm,Clement:2016vnl} for reviews), furthermore, the $S$ should be much \nmore tightly bound, leading to weaker interactions with ordinary baryons and thus evading direct searches.  \nSuch a particle would be absolutely \nstable for  $m_S<m_D + m_e\\simeq 1.88$\\,GeV, and decay with a lifetime exceeding the age of the \nUniverse for $m_S\\lesssim 2\\,$GeV~\\cite{Farrar:2020zeo}. Determining its expected mass exactly, however,\nis challenging; lattice simulations, for example, remain somewhat inconclusive (see, e.g., \nRefs.~\\cite{NPLQCD:2010ocs,Inoue:2010es,NPLQCD:2012mex,Francis:2018qch} where the results for binding energies \nof the H-dibaryon state range from $\\sim$17\\,MeV to $\\sim$75\\,MeV relying, however, on unrealistically large quark masses).\nEven if the sexaquark is stable on cosmological timescales, its relic abundance would generally be much smaller than\nthe observed DM abundance if one assumes that its interactions in the early universe are of the order of the \nstrong force~\\cite{Kolb:2018bxv,Gross:2018ivp}. If instead, one postulates much weaker interactions due to the assumed\ncompactness of the sexaquark, thermal equilibrium with the SM heat bath would\nnot be possible to maintain after the QCD phase transition and the correct DM abundance might be \nachieved -- in a region of parameter space claimed to evade all existing constraints~\\cite{Farrar:2020zeo}.\n\nMotivated by this intriguing possibility, for simplicity we will adopt the description of sexaquark interactions \nfrom Ref.~\\cite{Farrar:2020zeo}, i.e.~we model the interaction with nucleons by the exchange \nof a vector meson. \nIn particular, the relevant interaction terms with the flavour-neutral mixture of $\\phi$ and $\\omega$, denoted by $V$, are given by\n\\begin{equation}\n\\label{eq:laghexaquarks}\n\\mathcal{L}= V_\\mu \\left( i g_{S} S^\\dagger {\\mathop{\\partial^\\mu}^{\\leftrightarrow}} S  + g_{p}\\overline{p}\\gamma^\\mu p + g_{n}\\overline{n}\\gamma^\\mu n\\right),\n\\end{equation}\nand we adopt the value $m_V = 1\\,$GeV used in Ref.~\\cite{Farrar:2020zeo} for our calculations.\nThe value of $g_n=g_p\\sim 2.6\\sqrt{4\\pi}$ \ncan be inferred from the literature on the one-boson-exchange model~\\cite{Maessen:1989sx} although $\\mathcal{O}(1)$ \nuncertainties can be expected here.\\footnote{%\nIn particular, we note that modern analyses of low-energy baryon-baryon scattering consider processes beyond single \nmeson exchange~\\cite{Nagels:2015lfa}, and that baryon-baryon interactions can also be treated within the more \nsystematic approach \nof chiral perturbation theory \\cite{Weinberg:1990rz}. However, given the significant uncertainties on the sexaquark \ncouplings we consider the one-boson-exchange approximation to be sufficient for our purposes.\n}\nThe coupling $g_{S}$ is largely unknown, though simple scaling arguments suggest that\n\\begin{equation}\n\\alpha_{\\mathrm{SN}} \\equiv \\frac{g_{S}g_{p}}{4\\pi}\n\\end{equation}\nis very roughly of the order of $\\sim0.1$~\\cite{Farrar:2020zeo}. Following that reference, we will treat $\\alpha_{\\mathrm{SN}}$\nas a free parameter that we will generously vary in the interval $(10^{-3},10)$. \nImportantly however -- at least in this parameter range -- the DM relic abundance \nis independent of $\\alpha_{\\mathrm{SN}}$. Instead, the final abundance of $S$ is set by an independent coupling constant $\\tilde{g}$~\\cite{Farrar:2020zeo} \nthat describes the (much weaker) sexaquark-breaking interactions within the effective description. This coupling does not directly enter the analysis presented here.\n\nWe treat the interaction of $V$ with nuclei similarly to that in section~\\ref{sec:vector}, i.e.~we describe it by the effective \nLagrangian~\\eqref{eq:leff_vector} with the coherently enhanced, effective coupling $g_N$ given by Eq.~\\eqref{eq:gN_coh}. \nFor the elastic scattering cross section on nuclei we thus find\n\\begin{eqnarray}\n\\label{diffsig_full_hex}\n\\frac{d\\sigma_{S N}}{d T_N}&=&\\frac{\\mathcal{C}^2 \\sigma_{\\rm SI}^\\mathrm{NR}}{T_N^\\mathrm{max}}\n\\frac{m_V^4}{(Q^2+m_V^2)^2}\n\\times G_N^2(Q^2) G_V^2(Q^2)\\\\\n&&\n\\times\\frac{1}{4s \\mu_{S N}^2}\n\\left\\{\n\\begin{array}{ll}\n(s-\\tfrac{1}{2}Q^2-m_N^2-m_\\chi^2)^2& ~~\\mathrm{for~scalar~}N\\\\\nm_N^2Q^2-Q^2s+(s-m_N^2-m_S^2)^2&  ~~\\mathrm{for~fermionic~}N\n\\end{array}\n\\right..\\nonumber\n\\end{eqnarray}\nHere, \n\\begin{equation}\n\\sigma_{\\rm SI}^{\\rm NR} = \\frac{16 \\pi \\alpha_{\\mathrm{SN}}^2 \\mu_{S p}^2}{m_V^4}\n\\end{equation}\nis the scattering cross section on nucleons in the non-relativistic limit and $\\mu_{S p}$ ($\\mu_{S N}$) is the reduced mass of \nthe sexaquark-nucleon (nucleus) system. \n\nCompared to the treatment in section~\\ref{sec:vector}, we introduce an additional form factor $G_V$ related to the cutoff in the \none-boson-exchange models. In this context, exponential cutoffs\n\\begin{equation}\nG_V(Q^2) = e^{-\\frac{Q^2}{\\Lambda_V^2}}\n\\end{equation}\nare mostly used and the cutoff mass $\\Lambda_V$ is fitted to data (and can in principle differ for different meson exchange \nchannels). For example, within the fit to data taking into account hyperon-nucleon interactions \\cite{Maessen:1989sx}, these cutoff \nmasses were found to range between 820\\,MeV and 1270\\,MeV. Since yet lower cutoff masses appear in related literature (e.g., \ndown to 590\\,MeV in \\cite{Stoks:1996yj}), we generously vary $\\Lambda_V$ between 500 and 1500\\,MeV.\nWe note that for $\\Lambda_V\\gtrsim1500\\,$MeV, CRDM limits become in fact independent of the cutoff scale.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{figures_draft\/figure_nine_wl}\n\\caption{Effective sexaquark coupling $\\alpha_{\\mathrm{SN}}$ vs.~sexaquark mass $m_S$.\nThe purple region shows the parameter range that is excluded by the analysis in this work, assuming that sexaquarks\nmake up all of the cosmologically observed DM;\ndifferent line styles correspond, as indicated, to cutoff masses $\\Lambda_V\/{\\rm GeV}\\in\\{0.5, 1,1.5\\}$\nin the one-boson exchange approximation. All other constraints are, for easier comparison, directly\nreproduced from Fig.~10 of Ref.~\\cite{Farrar:2020zeo}, conservatively assuming an attractive Yukawa force between $S$\nand nuclei. The thin vertical stripe corresponds to the mass range\nwhere, according to that analysis, the sexaquark would be a viable DM candidate without being in conflict\nwith other particle physics observation, in particular the stability of deuterons based on SNO data~\\cite{Bellerive:2016byv}. \nThe upper end of that mass range may increase from 1890\\,MeV to up to 2054\\,MeV if sexaquark DM does not \naccumulate in the Earth at the level claimed in Ref.~\\cite{Neufeld:2018slx}.\n\\label{fig:results_sexaquark}\n}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:results_sexaquark} we show the parameter space in the $\\alpha_{\\mathrm{SN}}$ vs.~$m_S$ plane where sexaquark DM\nis excluded because of the irreducible CRDM component. For a better direct comparison, we also indicate the preferred\nmass range according to Ref.~\\cite{Farrar:2020zeo}, along with the complementary limits presented in that analysis. \nFrom this figure, it is clear that our new limits close a significant part of the viable parameter region where sexaquarks could be\nthe dominant DM component -- even without taking into account the CRESST results. \nIn particular, we note that the Lyman-$\\alpha$ limits~\\cite{Rogers:2021byl}  \nshown in figures~\\ref{fig:constraints_constant} and \\ref{fig:limits_scalar} were presented subsequent to the analysis of\nRef.~\\cite{Farrar:2020zeo} and are significantly stronger than the CMB limits indicated in Fig.~\\ref{fig:results_sexaquark}.\nThe apparently open window at $\\alpha_{\\mathrm{SN}}\\sim0.3$ is thus also robustly excluded.\nOn the other hand, a small open window remains for $\\alpha_{\\mathrm{SN}}\\lesssim 4\\cdot10^{-3}$. While not being\nin conflict with the DM abundance, as explained above, we recall that \nsuch values of  $\\alpha_{\\mathrm{SN}}$ are somewhat smaller than intrinsically expected.\n\nLet us, finally, briefly comment on the fact that the DM-nucleon scattering cross section can, strictly speaking,\nonly be calculated perturbatively in the Born limit, $\\alpha_{\\mathrm{SN}} \\mu_{\\chi N}\\lesssim m_V$. Outside this regime,\nnon-relativistic scattering in a Yukawa potential exhibits parametric resonances where the scattering amplitude\nis significantly enhanced or suppressed. This non-perturbative \neffect is well-known from the self-scattering of cold DM in the presence\nof light mediators~\\cite{Tulin:2013teo}, and it is the origin of the resonant structure in the complementary limits from \nRef.~\\cite{Farrar:2020zeo} that is visible in Fig.~\\ref{fig:results_sexaquark}. For our CRDM limits, on the other hand, \nthis additional complication does not arise because such non-perturbative corrections are \nlargely irrelevant for relativistic scattering; in fact, \nalready for the typical velocities during the freeze-out process of thermally produced DM, $v_\\chi\\sim0.3$, the impact\nis strongly suppressed~\\cite{Tulin:2013teo}. The CRDM limits are thus also robust w.r.t.~underlying model assumptions \nsuch as whether the force mediated by the Yukawa potential is attractive or repulsive.\n\n\n\n\n\\section{Summary and Conclusions}\n\\label{sec:conclusions}\n\nFor sizeable elastic scattering rates between DM and nuclei there is an irreducible relativistic\ncomponent of the flux of DM particles arriving at Earth. This extends the sensitivity of conventional\ndirect detection experiments both to sub-GeV masses and to scattering cross sections above the limit\nset by a too efficient attenuation of the DM flux on the way to the detector. While such large scattering cross \nsections are also constrained by complementary probes from astrophysics and cosmology, it has repeatedly \nbeen pointed out that there might be an open window of relatively strongly interacting DM with a mass in the\nballpark of $\\sim1$\\,GeV.\n\nWe find that the CRDM component in the DM flux generically closes this window, under rather minimal \nassumptions. In order to arrive at this conclusion, we included in our analysis a detailed treatment\nof the inelastic scattering of DM off nuclei (section \\ref{sec:inel}). We demonstrate that this provides \nan important additional stopping channel for CRDM particles on their way to direct detection facilities\n-- unlike for non-relativistic DM, where only elastic scattering is relevant.\nWe also investigated the extent to which a possible energy or momentum-transfer dependence of the cross section\ncould weaken our general conclusions. \nFor this purpose, we considered {\\it i)} a class of simplified models where the scattering with nuclei is mediated by a light\nscalar (section \\ref{sec:scalar}) or vector (section \\ref{sec:vector})\nparticle, as well as {\\it ii)} situations where DM particles cannot be described as being point-like (section \\ref{sec:puffy}). \nIn all these cases, the additional momentum-transfer dependence indeed weakens the limits from\ndirect detection -- which however is compensated for by a corresponding strengthening of complementary\nlimits, in particular from cosmology. In combination, these limits stringently constrain the \npossibility of cross sections larger than a few times $10^{-31}\\,{\\rm cm}^2$, over a wide range of DM masses. Interestingly, this is\nlargely independent of underlying modelling assumptions such as the mass of new mediator particles or\nthe DM particles' radius.\n\nFinally, an exotic QCD bound state that is produced well before BBN, has repeatedly been put forward as \na potential DM candidate. While it is theoretically unclear whether such states could actually exist, \nadding to significant experimental constraints, it is certainly an intriguing\nidea to have a `baryonic' DM candidate that would in fact evade the strong\nevidence from BBN and CMB against this possibility. However, cosmic-ray upscattering\nof such particles leads to stringent new constraints that have not previously been pointed out\nin this context. For the concrete case of stable sexaquark DM, as discussed in \nsection~\\ref{sec:hexaquark}, we find that the parameter space giving the correct cosmological\nabundance is strongly pressured.\n\nFor the analysis performed in this work we used the numerical tool {\\sf DarkSUSY}~\\cite{Bringmann:2018lay} \nto compute CRDM fluxes and limits. In doing so we significantly expanded the general numerical routines\nprovided therein, adding in particular inelastic scattering, the contribution \nfrom CRs beyond $p$ and He, and an updated treatment of nuclear form factors in the context of\nCRDM attenuation.   These updates\nwill be included in the next public release\nof the code.\n\n\n\n\\acknowledgments\nWe thank Timon Emken and Florian Reindl for enlightening discussions about the thermalization efficiency of the\nCRESST experiment, and Felix Kahlhoefer for insightful comments on how to map nucleon to nuclear cross sections. We further \nthank Assumpta Pare\\~{n}o, Gilberto Colangelo and Urs Wiedemann for comments related to the hexaquark state. TB warmly \nthanks the Albert Einstein Institute in Bern, and the CERN Theoretical Physics Department, \nfor support and hospitality during the preparation of this manuscript. JA is supported through the research program ``The Hidden \nUniverse of Weakly Interacting Particles\" with project number 680.92.18.03 (NWO Vrije Programma), which is partly financed by \nthe Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Dutch Research Council). HK is supported by the ToppForsk-UiS Grant No. PR-10614.\n\n\n\\bibliographystyle{JHEP_improved}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe high-energy cosmic ray, $\\gamma$-ray and neutrino emission of the Universe are fascinating phenomena based on poorly understood non-thermal processes in astrophysical environments. Whereas high-energy $\\gamma$-rays can be produced via leptonic processes like inverse Compton scattering or bremsstrahlung, the production of neutrinos require interactions of high energy cosmic rays with radiation or matter. Hence, cosmic neutrinos are the unambiguous tracers of cosmic ray interactions in our Universe and neutrino astronomy promises to unravel their sources.\n\nHigh energy astrophysical neutrinos become visible in neutrino observatories once they interact in the detector vicinity via charged and neutral current interactions. The high energy secondary particles produced in these interactions are detected in transparent media like water or ice via Cherenkov light emission. The cross sections of these processes are very low. For instance, at PeV energies the neutrino interacts with nucleons with a cross section of about 1 nbarn. Hence, for a detector density of the order of $N_A\\,{\\rm cm}^{-3}$, we expect only a fraction of about $10^{-5}$ to interact in 1 km of the medium. These low event rates and the large background from atmospheric cosmic ray interactions are the experimental challenges for neutrino astronomy.\n\nThe IceCube Collaboration~\\cite{Aartsen:2013jdh,Aartsen:2014bea} has recently reported the detection of a cosmic flux of high energy neutrinos with a significance of $5.7\\sigma$. The flux is consistent with an $E^{-\\gamma}$ power spectrum with spectral index $\\gamma\\simeq2.3\\pm0.3$ and equal distribution between flavor and isotropic arrival direction of neutrinos. The best fit $E^{-2}$-flux is given as\n\\begin{equation}\\label{eq:ICnu}\nE_\\nu^2J^{\\rm IC}_\\nu(E_\\nu) \\simeq (0.95\\pm0.3)\\times10^{-8}\\,\\frac{{\\rm GeV}}{{\\rm cm}^{2}\\,{\\rm s}\\,{\\rm sr}}\n\\end{equation}\nThe origin of these neutrinos is unknown. A statistically weak cluster of events near the Galactic Centre has motivated speculations about a Galactic origin of the signal. These scenarios include the diffuse neutrino emission of Galactic CRs~\\cite{Ahlers:2013xia,Kachelriess:2014oma}, the joint emission of Galactic PeVatrons~\\cite{Fox:2013oza,Gonzalez-Garcia:2013iha} or extended Galactic structures like the Fermi Bubbles~\\cite{Razzaque:2013uoa,Ahlers:2013xia,Lunardini:2013gva} or the Galactic Halo~\\cite{Taylor:2014hya}. A possible association with the sub-TeV diffuse Galactic $\\gamma$-ray emission~\\cite{Neronov:2013lza} and constraints from the non-observation from diffuse Galactic PeV $\\gamma$-rays~\\cite{Gupta:2013xfa,Ahlers:2013xia} have also been mentioned. More radical suggestions include PeV dark matter decay scenarios~\\cite{Feldstein:2013kka,Esmaili:2013gha,Bai:2013nga}. The extension of the events to large Galactic latitudes and the absence of significant event clusters suggest that a significant contribution of the signal originates in extragalactic sources. Possible source candidates include galaxies with intense star-formation~\\cite{Loeb:2006tw,Murase:2013rfa,He:2013cqa}, cores of active galactic nuclei~\\cite{Stecker1991,Stecker:2013fxa}, low-power $\\gamma$-ray bursts~\\cite{Waxman:1997ti,Murase:2013ffa}, intergalactic shocks and active galaxies embedded in structured regions~\\cite{Berezinsky:1996wx,Murase:2013rfa}. \n\nThe search for transient and continuous neutrino sources has so far been unsuccessful providing upper neutrino flux limits on individual Galactic and extragalactic source candidates~\\cite{IceCube:2011ai,AdrianMartinez:2012rp,Abbasi:2012zw,Aartsen:2013uuv,Adrian-Martinez:2014wzf,Adrian-Martinez:2013dsk}. If the IceCube observation is a superposition of individual (possibly extended) sources, in contrast to a truely diffuse emission, these searches should eventually uncover individual sources or at least place upper limits on possible source candidates. However, it is important to keep in mind that the interaction rate of a neutrino is so low that it travels basically unattenuated through the tenuous matter and radiation backgrounds of the Universe over cosmic distances. The un-resolved extragalactic (or quasi-diffuse) flux is hence a superposition of many sources ranging from relatively recent and local objects to old and distant sources as far as the Hubble horizon. This makes the identification of individual point-sources contributing to the IceCube flux challenging. \n\nIn this paper we investigate the necessary performance of a neutrino observatory for the detection of neutrino point-sources in the form of event clusters and in association with close-by sources or source catalogues. We will start in Section~\\ref{sec2} with a derivation of the contribution of individual neutrino sources to a quasi-diffuse flux. In Section~\\ref{sec3} we derive simple estimates of the required event numbers and rates of the quasi-diffuse emission for the identification of sources assuming a spatially homogeneous local distribution. For continuously emitting neutrino sources an important aspect of the analysis is the background of atmospheric neutrinos and the quasi-diffuse neutrino signal itself. In Section~\\ref{sec4} we include these contributions quantitatively via a significance tests introducing simple test statistics for event clusters and source associations. We compare these results with specific source scenarios in Section~\\ref{sec5} and conclude in Section~\\ref{sec6}. In the following we work in Heaviside-Lorentz units and make use of the abbreviation $A_x = A\/(10^xu)$, where $u$ is the (canonical) unit of the quantity $A$. \n\n\\section{Neutrino Point-Sources}\\label{sec2}\n\nHigh-energy neutrinos are produced by the decay of charged pions from hadronic interactions of CRs with radiation ($p\\gamma$) and matter ($pp$). The same mechanism produces also high-energy $\\gamma$-rays from the formation of neutral pions. On production, the emission rates of neutrinos (summed over neutrinos and anti-neutrinos with flavor $\\nu_\\alpha$) and $\\gamma$-rays are related to the emission rate of CR nucleons ($N$) as\n\\begin{equation}\\label{eq:MM}\n\\frac{1}{3}\\sum_{\\alpha}E_\\nu Q_{\\nu_\\alpha} \\simeq \\frac{K_\\pi}{2}  E_\\gamma Q_\\gamma\\simeq \\frac{f_\\pi}{\\kappa}\\frac{K_\\pi}{1+K_\\pi}E_NQ_N\\,,\n\\end{equation}\nwhere the (average) energies are related as $E_\\nu\\simeq E_\\gamma\/2\\simeq E_N\/20$. We will assume in the following that energy losses of the charged pions and muons prior to their decay are unimportant. In this case the relation (\\ref{eq:MM}) has to be considered as lower bound on the total CR energy of the sources. The parameter $K_\\pi\\simeq1$ ($K_\\pi\\simeq2$) denotes the average ratio between charged and neutral pions in $p\\gamma$ ($pp$) interactions and the pion production efficiency, that is related to the optical depth $\\tau$ for hadronic interactions with inelasticity $\\kappa$ as $f_\\pi \\simeq 1-\\exp(-\\kappa\\tau)$.\n\nThe IceCube flux between 60~TeV to 2~PeV corresponds to $\\gamma$-rays between 100~TeV and 4~PeV or CR nucleons between 1~PeV and 40~PeV. Cosmic rays at these energies are deflected in cosmic magnetic fields and can only be used for point-source associations at very high energies. For instance, ultra-high energy (UHE) CR protons beyond 60~EeV are expected to originate within about 200~Mpc due to the strong absorption in the cosmic microwave background (CMB). Even at these extreme energies the deflection via turbulent Galactic magnetic field can be of the order of $0.5^\\circ$~\\cite{Giacinti:2011uj}. The contribution of extragalactic magnetic fields is more uncertain and can in principle be larger~\\cite{Dolag:2004kp}.\n\nGamma-rays with PeV energies experience no electro-magnetic deflections, but have a short absorption length of about 10~kpc due to $e^+e^-$ production via scattering off of the photons of the CMB. At about 100~TeV interactions with the extragalactic background light limit the propagation distance to Mpc scales. Direct observation of $\\gamma$-ray emission in association with the IceCube flux is hence not feasible, unless there is a significant Galactic contribution~\\cite{Ahlers:2013xia}. However, the sub-TeV extension of the IceCube signal (expected for a $pp$ origin of the signal) as well as the sub-TeV contribution of cascaded $\\gamma$-rays via inverse-Compton scattering of the high-energy $e^+e^-$ can be visible as an (extended) point-source TeV $\\gamma$-ray emission and provide an additional constraint for the contribution of close-by sources~\\cite{Murase:2013rfa}.\n\nNeutrinos on the other hand have negligible interactions during propagation and are ideal point-source messengers.\nThe (quasi-)diffuse flux of neutrinos $J$ (in units of ${\\rm GeV}^{-1} {\\rm s}^{-1} {\\rm cm}^{-2} {\\rm s}^{-1}$) originating in multiple cosmic sources is simply given by\n\\begin{equation}\nJ_\\nu(E_\\nu) = \\frac{1}{4\\pi}\\int_0^\\infty\\frac{{\\rm d}z}{H(z)}\\mathcal{L}_\\nu(z,(1+z)E_\\nu)\\,,\n\\end{equation}\nwhere $H$ is the red-shift dependent Hubble expansion rate and $\\mathcal{L}$ is the spectral emission rate density. In the case of a continuous neutrino emission we can decompose this into $\\mathcal{L}(z,E) = \\mathcal{H}(z)Q_\\nu(E)$ where $\\mathcal{H}$ is the source density and $Q_\\nu$ is the emission rate per source. In the following we will assume evolution following the star-formation rate (SFR)~\\cite{Hopkins:2006bw,Yuksel:2008cu} $\\mathcal{H}_{\\rm SFR}(z) \\propto (1+z)^{n_i}$ with $n_i=3.4$ for $z<1$,  $n_i=-0.3$ for $1<z<4$ and  $n_i=-3.5$ otherwise. The red-shift dependence of the source distribution can be parametrized by the energy dependent quantity\n\\begin{equation}\n\\xi_z(E) = \\int_0^\\infty{\\rm d}z\\frac{H_0}{H(z)}\\frac{\\mathcal{L}_\\nu(z,(1+z)E)}{\\mathcal{L}_\\nu(0,E)}\\,.\n\\end{equation}\nFor the special case of power-law spectra $\\mathcal{L}_\\nu(E)\\propto E^{-\\gamma}$ this quantity is energy independent and we will assume the case $\\gamma=2$ in the following corresponding to the fit (\\ref{eq:ICnu}). In this case we have $\\xi_z\\simeq2.4$ assuming evolution with SFR of the CR sources. For a source distribution with no evolution in the local ($z<2$) Universe this reduces to $\\xi_z\\simeq0.5$. \n\nFrom this we can estimate the contribution of an individual continuously emitting point sources. For a distance $d = d_1 10$~Mpc the mean neutrino flux is given as\n\\begin{align}\\label{eq:JPS}\nE_\\nu^2J_\\nu \n\\simeq \\frac{(0.9\\pm0.3)\\times10^{-12}}{\\xi_{z, 2.4}\\mathcal{H}_{0, -5}d_1^{2}} \\frac{\\rm TeV}{{\\rm cm}^{2}\\,{\\rm s}}\\,,\n\\end{align}\nwhere $\\mathcal{H}_0 = \\mathcal{H}_{0, -5} 10^{-5} {\\rm Mpc}^{-3}$ is the local source density. For this choice of parameters the contribution of a source is consistent with upper limits of neutrino point sources in an un-binned search~\\cite{Aartsen:2013uuv}. In the case of transient sources we write instead $\\mathcal{L}(z,E) = {\\dot{\\mathcal{H}}}(z){\\rm d}N\/{\\rm d}E(E)$ with transient rate density ${\\dot{\\mathcal{H}}}(z)$ and spectrum ${\\rm d}N\/{\\rm d}E$ of an individual flare. In this case the mean neutrino fluence $F$ from an individual transient can be expressed as \n\\begin{align}\\label{eq:FPS}\nE_\\nu^2F_\\nu \n\\simeq \\frac{0.3\\pm0.1}{\\xi_{z, 2.4}{\\dot{\\mathcal{H}}}_{0, -6}d_1^{2}} \\frac{\\rm GeV}{{\\rm cm}^{2}}\\,,\n\\end{align}\nwhere ${\\dot{\\mathcal{H}}}_0 = {\\dot{\\mathcal{H}}}_{0, -6}10^{-6} {\\rm Mpc}^{-3}{\\rm yr}^{-1}$ is the local flaring\/burst density rate.\nThe sensitivity of IceCube for triggered transient sources lies at about $0.1{\\rm GeV}\/{\\rm cm}^2$ depending on zenith angle and emission time scale~\\cite{IceCube:2011ai,Abbasi:2012zw}.\n\nThe previous estimates depend on the distance of the source. In the case of a large number of sources we can express the probability that the closest source contributes with an expected number of events $n$ as \\begin{equation}\\label{eq:ploc}\np_1(n) \\simeq \\frac{3}{2}\\frac{1}{ n}\\left(\\frac{ n(\\hat{r})}{ n}\\right)^\\frac{3}{2}e^{-\\left(\\frac{ n(\\hat{r})}{ n}\\right)^\\frac{3}{2}}\n\\end{equation}\nwhere the distance $\\hat{r}$ is defined via $\\mathcal{H}_0\\hat{r}^3 \\Delta \\Omega\/3=1$, {\\it i.e.}~the radius of a sphere where we expect one source in the experimental field of view (FoV) $\\Delta \\Omega$. The general probability distribution of the $k$th-closest source is given in Appendix~\\ref{app1}.\n\n\\begin{figure}[t]\\centering\n\\includegraphics[width=\\linewidth]{PSsens5yr.pdf}\n\\caption[]{Point sources sensitivity for continuous (top) and transient (bottom) sources. The shaded areas show the $10\\%$ quantiles around the median expectation from the closest sources of the ensemble. The dotted horizontal line show the IceCube sensitivity after five years estimated for a muon energy threshold of 10~TeV (see main text).}\\label{fig1}\n\\end{figure}\n\nIn Figure~\\ref{fig1} we show the contribution of the closest continuous or transient source in terms of the density of the underlying source population and as $10\\%$ quantiles around the median (solid lines) according to Eq.~(\\ref{eq:ploc}) and Eqs.~(\\ref{eq:JPS}) and (\\ref{eq:FPS}), respectively. We assume an observation time $T_{\\rm live}=5$~yrs for the total number of transient sources. The dotted horizontal lines show the estimated sensitivity of IceCube to continuous and transient sources for a detector live-time of five years assuming an event rate of $\\dot{N}\\sim 50~{\\rm yr}^{-1}$ above a muon energy threshold of 10~TeV and low background~\\cite{Karle2014}. The estimated sensitivity (90\\% C.L.) shown in Fig.~\\ref{fig1} is then simply $J \\simeq f_{\\rm sky}4\\pi J^{\\rm IC}_\\nu\\times 2.3\/(T_{\\rm live}\\dot{N})$ for event associations with continuous sources and $F \\simeq f_{\\rm sky}4\\pi  J^{\\rm IC}_\\nu\\times 2.3\/\\dot{N}$ for events triggered by transient sources, where we introduced the fractional sky coverage $f_{\\rm sky}=\\Delta\\Omega\/(4\\pi)$. We will provide a more precise estimate of the detector sensitivity including backgrounds in the following sections.\n\nThese results already indicate that the non-observation of individual neutrino sources (and in particular the closest one) is consistent with the hypothesis of an extragalactic origin of the recent IceCube observation (\\ref{eq:ICnu}) for sufficiently large source densities and\/or rates. On the other hand, the identification of individual neutrino sources with the continued observation with IceCube over the next years will be challenging unless the source distribution is sufficiently sparse and\/or rare. In the next sections we will make this statement more quantitative and discuss the required event numbers and search strategies for an identification of the sources with IceCube or next-generation neutrino observatories.\n\n\\section{Point Source Statistics}\\label{sec3}\n\nA model-independent identification of neutrino sources can be the detection of spatial or temporal clusters with total number of $m$ neutrino events or more. The required value for $m$ needed for a significant detection depends on the density of sources as well as the expected number of signal and background events. The neutrino event clusters will be most likely associated with local neutrino sources and we can hence simplify the discussion by considering Euclidean space, where we neglect redshift scaling of energy and comoving volume. The contribution from a single local source at distance $r\\leq H_0^{-1}$ can be expressed as\n\\begin{equation}\\label{eq:nr}\n n(r) \n\\simeq \\frac{H_0}{f_{\\rm sky}4\\pi r^2\\xi_z}\\times\\begin{cases} N\/\\mathcal{H}_0& \\text{(continuous)}\\\\ \\dot{ N}\/{\\dot{\\mathcal{H}}}_0& \\text{(transient)}\\end{cases}\n\\end{equation}\nIn the following we will derive results in terms of the expected number of events $ N$ or $\\dot{ N} =  N\/T$ of all neutrino sources. \n\nAs a back-of-the-envelope estimate of the required total event numbers for the observation of event multiplets we can consider the contribution of the closest source of the ensemble. For a fractional sky coverage $f_{\\rm sky}$ and local source density $\\mathcal{H}_0$ we expect one source in the FoV within a sphere of volume $V_1 = 1\/(f_{\\rm sky}\\mathcal{H}_0)$. The total number of events that we expect from this volume is given by the integral of Eq.~(\\ref{eq:nr}) over $V_1$ and yields $m =  N(V_1\/V_H)^\\frac{1}{3}\/\\xi_z$ where we introduce the Hubble volume $V_H = 4\\pi\/(3H_0^3)$. Note, that $V_H\/V_1$ correspond to the effective number of sources in the FoV. In the case of continuous sources we arrive then at an expected total event number for $m$ local events of\\begin{equation}\\label{eq:Ncont}\n N_{\\rm cont} \\simeq 740\\,\\left(\\frac{m}{2}\\right)\\,\\xi_{z, 2.4}\\,\\left(f_{\\rm sky}\\,\\mathcal{H}_{0,-5}\\right)^{\\frac{1}{3}}\\,.\n\\end{equation}\nIn the case of transient sources we have to take into account that the number of sources is increasing with observation time $T_{\\rm live} =  N\/\\dot{ N}$. Solving in terms of the total observation rate $\\dot{ N}$ we arrive at \n\\begin{equation}\\label{eq:Ntrans}\n N_{\\rm trans} \\simeq 637\\left(\\frac{m}{2}\\right)^\\frac{3}{2}\\,\\xi_{z, 2.4}^\\frac{3}{2}\\,\\left(f_{\\rm sky}{{\\dot{\\mathcal{H}}}_{0,-6}}\/{\\dot{ N}_2}\\right)^\\frac{1}{2}\\,,\n\\end{equation}\nwith an event rate $\\dot{ N} = 100\\dot{ N}_2\/{\\rm yr}$. Note, that the event clusters in the transient case should also show a strong temporal coincidence within the burst or flaring time-scale of the source. This fact is important for the comparison with background events due to random clusters of atmospheric neutrinos and muons.\n\nInstead of searching for auto-correlations of events we can also look for cross-correlations with catalogues of candidate sources. For simplicity, let's assume that the catalogue is locally complete up to a distance $r_{\\rm cat}$ containing $C = f_{\\rm sky}\\mathcal{H}_0V_{\\rm cat}$ local neutrino sources in the FoV. Following the same line of arguments as in the case of event clustering we can estimate that $m =  N(V_{\\rm cat}\/V_H)^\\frac{1}{3}\/\\xi_z$ events are expected to correlate with sources of the catalogue. Note that this expression is only valid for $V_{\\rm cat}<V_H$; for complete catalogues we simply have $N(m) = m$. For transient sources we assume that the catalogue itself grows in time and remains complete within a distance $r_{\\rm cat}$. In this case the expected total number of events for $m$ associations with sources of the catalogue is\n\\begin{equation}\\label{eq:Nass}\n N_{\\rm ass} \\simeq 107\\, m\\,\\xi_{z, 2.4}r_{\\rm cat, 2}^{-1}f_{\\rm sky}^{-\\frac{1}{3}}\\,.\n\\end{equation}\nThe result is the same for continuous and transient sources if we assume that the catalogue grows in time for transient sources. In the case of extended source catalogues with $r_{\\rm cat}\\sim 1\/H_0$ already a few signal events can be sufficient for a point-source association. This requires sources with powerful electromagnetic emission which is typical for transient sources like GRBs or AGN flares. \n\nA crucial aspect for a statistically significant detection of local neutrino sources are the backgrounds of distant neutrino sources as well as atmospheric showers. For transient sources the time-stamp of the signal can lead to a significant reduction of backgrounds. In this case Eqs.~(\\ref{eq:Ntrans}) and (\\ref{eq:Nass}) already indicate the number of events required for source identifications. However, continuous sources require a more careful statistical discussion. The previous estimates suggest that the required number of observed signal events $N$ has to reach at least a level of 100 before we can expect to observe an association of events with local continuous sources. \n\nThe classical muon neutrino search has a sky coverage of about $f_{\\rm sky}\\lesssim 0.5$. For the following analysis we will assume that a suitable application of low level event filters has already reduced the background to a level $S\/B$. The significance of spatial clustering or association of events depends on the experimental angular resolution $\\Delta\\theta$. This defines the effective number of bins in the sky as $n_{\\rm bin} \\simeq 2f_{\\rm sky}\/(1-\\cos(\\Delta\\theta))$. For instance, we have $n_{\\rm bin}\\simeq6600$ for $\\Delta\\theta\\simeq 1^\\circ$ and $f_{\\rm sky}=0.5$, but only $n_{\\rm bin}\\simeq66$ for $\\Delta\\theta\\simeq 10^\\circ$. The expected number of evens from a random distribution of $N_{\\rm bg}$ background events is $m\\simeq {N}_{\\rm bg}\/n_{\\rm bin}$ per bin. Hence, for poorly resolved cascade events with $\\gtrsim10^\\circ$ resolution even a low background contribution of $S\/B\\simeq 1$ already produces random event clusters and association.\n\nIn the case of transient sources with burst or flaring time window $\\Delta T\\ll T_{\\rm live}$ we can make use of the fact that the CR background is continuous. We can account for this by redefining the effective number of bins as $n_{\\rm bin}\\to T_{\\rm live}\/\\Delta Tn_{\\rm bin}$. In general, $T_{\\rm live}\/\\Delta T$ is expected to be very large and depends on the specific source. The background cluster probability in the transient case is expected to be very low. \n\n\\section{Significance Test}\\label{sec4}\n\nIn order to test the statistical significance of neutrino point-sources we introduce two simple test statistics (TS), {\\it i)}  ${\\rm TS}_1 = \\max\\lbrace {\\bf k}\\rbrace$ for cluster tests and {\\it ii)} ${\\rm TS}_2 = \\sum_{N_s} k_i$ for a source association with $N_s$ closest sources. If ${\\bf k}$ is the experimental result and ${\\bf k}_0$ a possible background distribution we define the ensemble-averaged $p$-value as\n\\begin{equation}\np= \\int\\prod_{i=1}^{N_s}{\\rm d}n_i\\,p_{\\rm PS}(n_i)\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\sum_{{\\rm TS}({\\bf k}_0)\\geq{\\rm TS}({\\bf k})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!P({\\bf k})P_0({\\bf k}_0)\\,,\n\\end{equation}\nwhere $p_{\\rm PS}(n)$ is the probability (\\ref{eq:pPS}) for an individual source out of $N_s$ sources to contribute with an expectation value $n$. The event probability distributions for signal $P$ and background $P_0$ are products of Poisson distributions for the content of the individual bins (see Appendix~\\ref{app1}). \n\nOne advantage of these test statistics is that the $p$-value reduces to simple analytic formulae in the background-free case, which is typically the case for transient sources. In this case we simply have $p({\\rm TS}_1) \\simeq 1-P_{\\rm cl}(2)$ with Eq.~(\\ref{eq:Pcl}) and $p({\\rm TS}_2) \\simeq 1-P_{\\rm ass}(1)$ with Eq.~(\\ref{eq:Pass}) derived in Appendix~\\ref{app2}. In Appendix~\\ref{app2} we also show that in the background-free case the required total signal event numbers at the significance level $p=0.1$ are the same as in Eq.~(\\ref{eq:Ncont}) or (\\ref{eq:Ntrans}) with the replacement $m\\to M\\simeq 3.57$. From this we can estimate that IceCube is sensitive to multiplets of rare transient sources $\\dot{\\mathcal{H}}_0\\lesssim10^{-8}\\,{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$ after five years of observation.\n\nIn the case of continuous sources the background probability is not negligible. In the following we discuss two experimental scenarios parametrized by the angular resolution $\\Delta\\theta$, partial sky fraction of the FoV $f_{\\rm sky}$ and the signal-to-background ratio $S\/B$ of the observation. In the present IceCube 86-string configuration we expect about 40-60 signal events per year in the Northern Hemisphere above a muon energy threshold of 10~TeV, depending on the spectral index of the observation~\\cite{Karle2014}. The atmospheric muon neutrino background is at the level of about 500 events per year. The angular resolution of these events at about 10~TeV is of the order of $0.6^\\circ$. We will hence use the combination {\\it a)} $f_{\\rm sky}=0.5$, $S\/B = 0.1$ and $\\Delta\\theta=0.6^\\circ$ as a first detector performance benchmark. At muon energies above about 100 TeV the signal-to-background ratio in the Northern Hemisphere increases to about $S\/B=1$ with an increased angular resolution. However, due to neutrino absorptions inside the Earth the effective FoV is reduced. In this case we use as a second performance benchmark {\\it b)} $f_{\\rm sky}=0.25$, $S\/B = 1$ and $\\Delta\\theta=0.3^\\circ$.\n\nIn Figure~\\ref{fig2} we show results of the averaged $p$-value for different astrophysical scenarios of continuous neutrino sources and detector parameters. As usual the sensitivity is defined as a 90\\% confidence level (C.L.) of the signal hypothesis or, equivalently, $p=0.1$. The upper panel of Fig.~\\ref{fig2} shows the results for ${\\rm TS}_1$ for the search of event clusters for the benchmark point {\\it a)} (left panel) and {\\it b)} (right panel) . Even with high $S\/B$ and good angular resolution the significance of event clusters requires event numbers of $10^3$ in the most favorable astrophysical scenario of low source densities of $10^{-6}~{\\rm Mpc}^{-3}$. Hence, a model-independent identification of extragalactic neutrino sources is challenging even with a detector with ten times the event rates of the present IceCube configuration. Note that the oscillatory pattern in the $p$-values is due to the averaging over ensembles and experimental realizations. For an actual realization with maximum multiplet $k_{\\rm max}$ the $p$-value is simply $p=1-P_{\\rm bg}(k_{\\rm max})$ which is continuously decreasing with $N$. \n\nThe situation for continuous sources improves in the case of associations of neutrino events with candidate source catalogues tested by the test statistic ${\\rm TS}_2$. We show results in the bottom panels of Fig.~\\ref{fig2} for the benchmark points {\\it a)} (left panel) and {\\it b)} (right panel), respectively. The dashed lines show the results of association with the closest source of the ensemble following the distribution (\\ref{eq:ploc}). The detector performance used in the middle panel corresponds to the case shown in Fig.~\\ref{fig1}. For instance, the optimistic source density of $10^{-6}~{\\rm Mpc}^{-3}$ requires 1000 event for $p=0.1$. With a present IceCube rate of 50 signal events per year this requires an experimental live-time of $T_{\\rm live}\\simeq 20$~yrs, consistent with the sensitivity level indicated in Fig.~\\ref{fig1}. This provides an {\\it a posteriori} justification of the background-free estimate.\n\nThe solid lines in the lower panels of Fig.~\\ref{fig2} show the results of ${\\rm TS}_2$ in the case of an association of events with the 100 closest sources. In the case of the detector performance {\\it b)} with a high purity of events of $S\/B\\simeq1$ and good angular resolution the observation of $50$--$500$ events is required for source densities of $10^{-6}$--$10^{-4}\\,{\\rm Mpc}^{-3}$ and a moderate source evolution following the star-formation rate ($\\xi_z\\simeq2.4$). For a detector with  $\\dot{N}\\simeq 50~{\\rm yr}^{-1}$ corresponding to five times IceCube's event rates above muon energies of 100~TeV this would require 1--10 years of observation. \n\n\\begin{figure*}[t]\\centering\n\\includegraphics[width=\\linewidth]{TS12.pdf}\n\\caption[]{Results of a significance test of continuous point-sources. \n{\\bf Top Panels:} The results for the test statistic ${\\rm TS}_1$ looking for significance of event clusters in a map. The four different scenarios are parametrized in terms of an increasing expected total number of signal events $ N$ and the fixed combination of local source density $\\mathcal{H}_0$ and source evolution $\\xi_z$. The dashed horizontal line indicates the sensitivity threshold of $1-p=90\\%$. We show results for a benchmark detector performance {\\it a)} $S\/B=0.1$, $f_{\\rm sky}=0.5$ and $\\Delta\\theta =0.6^\\circ$ (left panel) and {\\it b)} $S\/B=1$, $f_{\\rm sky}=0.25$ and $\\Delta\\theta =0.3^\\circ$ (right panel), corresponding to muon energy thresholds of about $10$~TeV and $100$~TeV, respectively~\\cite{Karle2014}. As a reference value, the signal rate of the full IceCube detector in the Northern Hemisphere is about $\\dot N \\simeq 50\\, {\\rm yr}^{-1}$ above muon energies of 10~TeV and $\\dot N \\simeq 10\\, {\\rm yr}^{-1}$ above 100~TeV. \n{\\bf Lower Panels:} Same as top panels but now showing the results for the test statistic ${\\rm TS}_2$ looking for association with 100 closest members of the ensemble (solid line) and the closest source (dashed line).}\\label{fig2}\n\\end{figure*}\n\n\\section{Source Candidates}\\label{sec5}\n\nFor a given source density $\\mathcal{H}$ or rate density ${\\dot{\\mathcal{H}}}$ of neutrino sources we can derive a lower limit on the energy budget of the individual sources. The neutrino emission extends in the energy range from $60$~TeV to $2$PeV and hence, conservatively, the corresponding range of the underlying CR population ranges from $E_-\\simeq m_p$ to $E_+\\simeq 40$~PeV. The mean integrated emission rate of continuous point sources can then be estimated as\n\\begin{equation}\\label{eq:Qnu}\n\\int\\limits_{E_-}^{E_+}{\\rm d} EEQ_N \n\\simeq8.9\\times10^{42}\\frac{\\mathcal{R}_{17.5}}{f_\\pi\\xi_{z,2.4}\\mathcal{H}_{0, -5}}\\frac{\\rm erg}{\\rm s}\\,,\n\\end{equation}\nwhere we assumed $K_\\pi=2$ ($pp$) and an energy independent pion fraction efficiency $f_\\pi$ between $E_-$ and $E_+$. We also assume a spectral index $\\gamma=2$ corresponding to a bolometric correction factor $\\mathcal{R} = \\ln(E_+\/E_-) = 17.5\\mathcal{R}_{17.5}$. In the case of transient sources the emission spectra ${\\rm d}N\/{\\rm d}E$ of neutrinos, $\\gamma$-rays and CRs are related via the analogue of Eq.~(\\ref{eq:MM}). The mean emission of an individual point source corresponding to the IceCube observation can then be estimated as\n\\begin{equation}\\label{eq:Nnu}\n\\int\\limits_{E_-}^{E_+}{\\rm d}EE\\frac{{\\rm d}N_N}{{\\rm d}E}\n\\simeq2.8\\times10^{51}\\frac{\\mathcal{R}_{17.5}}{f_\\pi\\xi_{z,2.4}{\\dot{\\mathcal{H}}}_{0, -6}}{\\rm erg}\\,.\n\\end{equation}\nNote, that for UHE CR sources reaching energies of the order of $E_+\\simeq10^{12}$~GeV we have $\\mathcal{R} \\simeq 27.6$. We will discuss in the following specific source scenarios in terms of the required source energy budget, density, rate and evolution.\n\n\\subsubsection{Active Galactic Nuclei}\n\nThe population of hard x-ray emitting AGN has a peak luminosity at about $10^{43}$--$10^{44}$~erg\/s with a source density of $\\mathcal{H}_{0}\\simeq 10^{-5}$--$10^{-4}{\\rm Mpc}^{-3}$\\cite{Barger:2004is,Tueller:2007rk}. If we assume that CRs are accelerated to a comparable power we can see that the requirements of Eq.~(\\ref{eq:Qnu}) are fulfilled with high pion production efficiencies. We assume that the neutrino luminosity is proportional to the x-ray luminosity and follow the model of~\\cite{Barger:2004is,Aird:2009sg}. In this case the red-shift evolution factor is given by $\\xi_z \\simeq 3.6$ and $\\mathcal{H}_0\\simeq 10^{-5}{\\rm Mpc}^{-3}$. \n\nFor instance, the close-by radio galaxy Cen A at distance of about 4~Mpc has presently an upper limit for muon neutrinos that is about 40 times higher then Eq.~(\\ref{eq:JPS})~\\cite{Aartsen:2013uuv}. However, note that the TeV $\\gamma$-ray emission is about $2.5\\times10^{-13}{\\rm TeV}^{-1}{\\rm cm}^{-2}{\\rm s}^{-1}$~\\cite{Aharonian:2009xn}, which is almost two orders of magnitude lower than the hadronic $\\gamma$-ray emission expected from the relations (\\ref{eq:MM}). Another close radio galaxy M87 at 16~Mpc has a stronger upper limit on continuous neutrino emission that is only a factor 5 higher than the prediction (\\ref{eq:JPS}). The observed TeV $\\gamma$-ray emission is of the order of $6\\times10^{-12}{\\rm TeV}^{-1}{\\rm cm}^{-2}{\\rm s}^{-1}$~\\cite{Aleksic:2012uma}, which is one order of magnitude lower than the estimate (\\ref{eq:MM}).\n\nIn the unified AGN model the population of blazars is a fraction of those radio-load AGNs where the jet emission is aligned with the observation axis. For instance, the isotropic equivalent density of flat-spectrum radio quasars peaks at a jet luminosity of $10^{47}$~erg\/s with $\\mathcal{H}_{0}\\simeq 10^{-9}{\\rm Mpc}^{-3}$~\\cite{Ajello:2011zi}. We assume the same evolution as the underlying AGN source population giving $\\xi_z\\simeq3.6$. Two of the closest blazars are Mrk 421 ($\\sim$130~Mpc) and Mrk 501 ($\\sim$140~Mpc). In this case the neutrino upper limits are one order of magnitude stronger than the average emission predicted by Eq.~(\\ref{eq:JPS}) and disfavor a blazar origin of the IceCube observation.\n\nGiant AGN flares with an energy release of the order of $10^{51}$~erg have been speculated as a possible source of UHE CRs~\\cite{Farrar:2008ex}. Their event rate is estimated to be of the order of ${\\dot{\\mathcal{H}}}_{0}\\simeq 10^{-6}~{\\rm Mpc}^{-3}{\\rm yr}^{-1}$ consistent with Eq.~(\\ref{eq:Nnu}) for high pion production efficiencies. Note that the present IceCube configuration would only be sensitive to these flares via triggering on known close-by sources. On the other hand a detector with five times the effective area would be sensitive to this source class model-independently via a significant spatial and temporal clustering of events. \n\n\\subsubsection{Starburst Galaxies}\n\nStarburst galaxies show a high star formation rate of $10^{-3} M_\\odot {\\rm Mpc}^{-3} {\\rm yr}^{-1}$. Assuming that CRs are accelerated in supernova shocks with a total CR energy of $10^{50}$~erg at a rate of about $0.1\\,{\\rm yr}^{-1}$~\\cite{Lacki:2013ata} we can estimate a total power of $3\\times10^{41}{\\rm erg}\/{\\rm s}$ per source. The redshift evolution of starburst galaxies follows the average SFR at high redshift ($z\\gtrsim1$). However, the local contributions is dominated by normal galaxies and starburst only contribute at a level of 10\\%. We account for relative evolution of the starburst density by a factor ${\\rm min}(0.1+0.9z,1)\/0.1$ suggested in Ref.~\\cite{Thompson:2006np} resulting in an evolution parameter $\\xi\\simeq18$. Hence, we can estimate the required average CR power of starburst galaxies to match the IceCube flux as $10^{41}\/f_\\pi$~erg\/s and hence $f_\\pi\\simeq 1$, {\\it i.e.}~CR {\\it calorimetry}~\\cite{Loeb:2006tw}. \n\nThe expected neutrino flux (\\ref{eq:JPS}) on starburst galaxy M82 at about 3.5~Mpc is about 40 times lower than present upper limit~\\cite{Aartsen:2013uuv}. The starburst NGC 253 at about 2.5~Mpc lies in the the Southern Hemisphere and point-source neutrino limits from IceCube are weaker. Intriguingly, the expected photon point-source flux from Eq.~(\\ref{eq:MM}) is comparable to that observed in TeV $\\gamma$-ray emission for M82 and NGC 253~\\cite{Karlsson:2009hd,Abramowski:2012xy}. This agrees with the result of Ref.~\\cite{Lacki:2010vs} based on proton calorimetry.\n\nNote that the large evolution factor $\\xi_z\\simeq18$ in this scenario produces a large background consisting of distant neutrino emitters. The detection of individual neutrino sources is hence challenging. The corresponding results of the significance tests are shown as green lines in Fig.~\\ref{fig2}. For all scenarios the required signal event number exceed $10^4$.\n \n\\subsubsection{Gamma-Ray Bursts}\n\nLong duration GRBs following the collapse of massive stars occur with an (isotropic equivalent) rate density of ${\\dot{\\mathcal{H}}}_{0}\\simeq 10^{-9} {\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$~\\cite{Wanderman:2009es} and energy of $M_\\odot \\simeq 2\\times10^{54}$~erg. Following the GRB evolution model of Ref.~\\cite{Wanderman:2009es} gives an evolution parameter of $\\xi_z\\simeq1.9$. Again, this is consistent with the required power (\\ref{eq:Nnu}) if the pion production efficiency is high. However, in the case of GRBs there are strong bounds on the neutrino emission in coincidence with the $\\gamma$-ray display~\\cite{Abbasi:2012zw} which are a factor 5 lower than the observed diffuse flux (\\ref{eq:ICnu}). \n\nIt has been speculated that low-power GRBs that are unobservable via their burst might be more efficient neutrino factories~\\cite{Murase:2013ffa}. In this case the strong IceCube bounds don't apply~\\cite{Abbasi:2012zw}. However, even for IceCube's moderate signal event rates of the order of $50~{\\rm yr}^{-1}$ above a muon energy threshold of 10~TeV event clusters in time and space are expected to appear already within one year of observation ($m\\to M\\simeq3.57$ for 90\\% C.L.~of doublets in Eq.~(\\ref{eq:Ntrans})). Hence, already the present neutrino data can constrain this scenario model-independently. The triggered search on known GRBs is even more sensitive~\\cite{Abbasi:2012zw}.\n\n\\subsubsection{UHE CR Calorimeters}\n\nThe required emission rate density of UHE CR proton sources can be estimated as $E^2_pQ_p(E_p) \\simeq (1-2)\\times10^{44}\\,{\\rm erg}\\,{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$~\\cite{Ahlers:2012rz}. If we assume that the emission spectrum $Q_p$ extends to lower energies the corresponding neutrino flux can be determined via Eq.~(\\ref{eq:MM}) and can be expressed as\nNeglecting a redshift dependence of $f_\\pi$ this translates into an $E^{-2}$ flux ($K_\\pi=2$) of \n\\begin{equation}\nE_\\nu^2J_\\nu(E_\\nu) \\simeq \\xi_{z,2.4} f_\\pi(3-6)\\times10^{-8}\\,\\frac{\\rm GeV}{{\\rm cm}^{2}\\,{\\rm s}\\,{\\rm sr}}\\,,\\label{eq:UHECRnu}\n\\end{equation}\nThe similarity of Eqs.~(\\ref{eq:ICnu}) and (\\ref{eq:UHECRnu}) suggests that the IceCube flux is related to the sources of UHE CRs. The limit $f_\\pi\\gtrsim1$ in Eq.~(\\ref{eq:UHECRnu}) corresponds to the Waxman-Bahcall bound~\\cite{Waxman:1998yy,Bahcall:1999yr}. Interestingly, the observation (\\ref{eq:ICnu}) is close to this bounds requiring high pion production efficiencies. \n\nOn the other hand, the energy loss due to pion production during acceleration has to be sufficiently low for UHE CR sources in order to compete with the energy gain per acceleration cycle. A natural solution to this fine-tuning problem occurs if the acceleration site is distinct from the CR calorimeter. Such a scenario could be provided in starburst galaxies, where the sub-PeV neutrino production happens during CR propagation in the dusty starburst environment after they have been released from transient UHE CR sources~\\cite{Katz:2013ooa}. Other scenarios consider massive galaxy clusters with local densities of the order of $\\mathcal{H}_0\\simeq 10^{-6}$--$10^{-5}{\\rm Mpc}^{-3}$ as CR calorimeters~\\cite{Berezinsky:1996wx,Murase:2008yt}. Note that the neutrino emission from CR propagation in CR calorimeters is expected to be continuous even if the sources of UHE CRs are transients, such as flaring AGNs or GRBs. \n\n\\section{Conclusion}\\label{sec6}\n\nIn this work we have studied the probability of identifying extra-galactic neutrino sources as the likely origin of the astrophysical neutrino flux observed by the IceCube Collaboration. We have derived the expected flux of individual continuous and transient sources based on the IceCube observation and depending on source density and rate. Our analysis takes into account the ensemble variation of the source distribution, their cosmic evolution and the Poisson statistics of weak nearby sources. In order to account for the background of atmospheric neutrinos and distant neutrino sources we introduced test statistics for significance tests of event clusters and source associations.\n\nOur findings are as follows. A model-independent observation of neutrino event clusters from a continuously emitting source population is challenging due to large atmospheric backgrounds unless the experimental angular resolution can be improved significantly. For transient sources these backgrounds are in general much lower due to the correlation of signal events in space and time. We estimate that the present IceCube detector is sensitive to rare $\\dot{\\mathcal{H}}_0\\lesssim10^{-8}~{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$ source classes within five years of operation. A next generation telescope with a five times extended effective muon area would be sensitive to multiplets from transient sources with $\\dot{\\mathcal{H}}_0\\lesssim10^{-6}~{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$. \n\nThe sensitivity to the underlying source population can be increased in model-dependent associations of events with source catalogues. We estimate that the IceCube detector is already sensitive to sparse continuous sources with $\\mathcal{H}_0\\lesssim10^{-6}~{\\rm Mpc}^{-3}$ as well as transient sources at the level of $\\dot{\\mathcal{H}}_0\\lesssim10^{-5}~{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$ via the association of events with the 100 closest sources of the ensemble. Again, a next-generation detector with 5 times the effective area as IceCube and otherwise identical performance in terms of angular resolution and FoV would improve the sensitive of these searches by about two orders of magnitude. A few final remarks are in order. \n\n{\\it i)} In this analysis we have only considered the scenario that the IceCube observation has an extragalactic origin. However, a (partial) Galactic origin of the observation is not yet excluded on statistical grounds. The presence of Galactic contributions will most likely emerge as a large scale anisotropy of the signal which has not been discussed in this work. \n\n{\\it ii)} We have expressed our results via ensemble-averaged $p$-values. The shaded regions shown in Fig.~\\ref{fig1} indicate the $\\pm40\\%$ fluctuation around the median. This {\\it cosmic variance} of the closest source can hence increase or decrease the prospects of detecting the closest source, depending on the sign of the fluctuation.\n\n{\\it iii)} We assumed that the individual sources of the ensemble can be approximated by a universal (average) luminosity. It is straightforward to include a non-trivial luminosity function or other source nuisance parameters in the definition of the point source distribution (\\ref{eq:pPS}) and related quantities. \n\n{\\it iv)} Future neutrino observatories may employ an extended surface veto for atmospheric events that would reduce the contribution of atmospheric neutrinos~\\cite{Schonert:2008is,Karle2014}. This can increase the signal-to-background ratio at lower muon energy threshold and improve the detection prospects estimated in this analysis. \n\n{\\it v)} The identification of sources via the association of events with source catalogues is model-dependent and the statistical results need to be corrected by trials factors. However, complementary to the identification of individual neutrino sources the study of spectral properties of the quasi-diffuse neutrino flux will help to narrow down valid astrophysical scenarios and should hence limit the number of trials.\n\n{\\it vi)} The identification of extra-galactic neutrino point-sources from a quasi-diffuse flux has also been discussed recently in Refs.~\\cite{Fargion:2014mda,Anchordoqui:2014yva}. In this analysis we have carefully studied the effect of ensemble and statistical variations and accounted for backgrounds via a significance test. This makes our estimates more robust and less optimistic than Refs.~\\cite{Fargion:2014mda,Anchordoqui:2014yva}.\n\n\\begin{acknowledgements}\nWe would like to thank the IceCube Collaboration for many fruitful discussions, in particular Jake Feintzeig, Gary Hill, Albrecht Karle, Claudio Kopper and Chris Weaver. We acknowledge support by the U.S. National Science Foundation (NSF) under grants OPP-0236449 and PHY-0236449.\n\\end{acknowledgements}\n\n\\begin{appendix}\n\n\\section{Source Distribution}\\label{app1}\n\nThe probability distribution of single source within $r<R$ is $p(r) = 3r^2\/{R^3}$. We can express this in terms of the expected events from a point-source using ${\\rm d} n\/{\\rm d}r = - 2 n\/r$ as\n\\begin{equation}\\label{eq:pPS}\np_{\\rm PS}(n,n(R)) = \\Theta(n-n(R))\\frac{3}{2}\\frac{1}{ n}\\left(\\frac{ n(R)}{ n}\\right)^{3\/2}\\,,\n\\end{equation}\nwith $n(R)$ defined by Eq.~(\\ref{eq:nr}). The expected event distribution for the $k$th-closest of the $N_s$ sources of the ensemble is given by\n\\begin{multline}\np_k(n) = \\Theta(n-n(R))\\frac{N_s!}{(k-1)!(N_s-k)!}\\\\\\times\\frac{3}{2}\\frac{1}{n}\\left(\\frac{n(R)}{n}\\right)^{\\frac{3k}{2}}\\left[1-\\left(\\frac{n(R)}{n}\\right)^{\\frac{3}{2}}\\right]^{N_s-k}\\,.\n\\end{multline}\nFor $N_s\\gg k$ we can approximate the last term as\n\\begin{equation}\n\\left[1-\\left(\\frac{n(R)}{n}\\right)^{\\frac{3}{2}}\\right]^{N_s-k}\n\\simeq e^{-N_s\\left(\\frac{n(R)}{n}\\right)^{\\frac{3}{2}}}\\,,\n\\end{equation}\nand using $(N_s-k) n(R)^{3\/2} \\simeq n(\\hat{r})^{3\/2}$ we arrive at\n\\begin{equation}\np_k(n) \\simeq \\frac{3}{2(k-1)!}\\frac{1}{n}\\left(\\frac{n(\\hat{r})}{n}\\right)^{\\frac{3k}{2}}e^{-\\left(\\frac{ n(\\hat{r})}{ n}\\right)^\\frac{3}{2}}\\,.\n\\end{equation}\nFor $k=1$ this agrees with Eq.~(\\ref{eq:ploc}) for the closest source.\n\n\\section{Signal Probability}\\label{app2}\n\nIn general, the probability of the events ${\\bf k}$ is given by the Poisson distribution of events in the individual bins with expected signal events $\\lambda_i$ and background $\\lambda_{\\rm bg}$,\n\\begin{equation}\nP({\\bf k}) = \\prod_{i=1}^{n_{\\rm bin}}\\frac{(\\lambda_i+\\lambda_{\\rm bg})^{k_i}}{k_i!}e^{-\\lambda_{\\rm bg}-\\lambda_i}\\,,\n\\end{equation}\nwhere $\\sum_i (\\lambda_i+\\lambda_{\\rm bg}) = N_{\\rm tot}$. The total number of events can be expressed via the expected number of all signal events and the signal-to-background ratio $S\/B$ as $N_{\\rm tot}= (1+(S\/B)^{-1})N$. The background probability is simply\n\\begin{equation}\nP_0({\\bf k}) = \\prod_{i=1}^{n_{\\rm bin}}\\frac{\\lambda_0^{k_i}}{k_i!}e^{-\\lambda_0}\\,,\n\\end{equation}\nwith $n_{\\rm bin}\\lambda_0 = N_{\\rm tot}$.\n\nIn the case of the test statistics ${\\rm TS}_{1\/2}$ for event clusters and source associations we can derive simple probabilities in the background free case. The probability of observing less than $m$ events from $ n$ expected events can be expressed via incomplete $\\Gamma$-functions as\n\\begin{equation}\n\\sum_{k=0}^{m-1}  \\frac{ n^k}{k!}e^{- n} = \\frac{\\Gamma(m, n)}{\\Gamma(m)}\\,.\n\\end{equation}\nHence, the probability of observing less than $m$ events from a single source within distance $R$ is given as\n\\begin{equation}\n P_{\\rm PS}(m) = \\int\\limits_{ n(R)}^\\infty {\\rm d}n\\, p_{\\rm PS}(n,n(R))\\frac{\\Gamma(m, n)}{\\Gamma(m)}\\,.\n\\end{equation}\nThis can be expressed as\n\\begin{equation}\n P_{\\rm PS}(m) = \\frac{\\Gamma\\left(m, n(R)\\right)-(n(R))^{3\/2}\\Gamma\\left(m-\\frac{3}{2}, n(R)\\right)}{(m-1)!}\\,.\n\\end{equation}\nThe probability of observing at least one cluster of $m$ or more neutrinos from a local source is given by the expression $P_{\\rm cl} = 1 -  P_{\\rm PS}^{N_s}$.\nIn the limit $n(R)\\ll 1$ and for $m\\geq2$ this can be approximated as\n\\begin{equation}\\label{eq:Pcl}\nP_{\\rm cl}\\simeq1-\\exp\\left(-\\frac{\\Gamma(m-\\frac{3}{2})}{3^{\\frac{3}{2}}\\Gamma(m)}\\left(\\frac{N}{\\xi_z}\\right)^\\frac{3}{2}\\left(\\frac{V_1}{V_H}\\right)^\\frac{1}{2}\\right)\n\\,.\n\\end{equation}\nHence, this result is independent of the auxiliary distance $R$ as expected. For a significance level $P$ the required total signal event number is the same as in Eqs.~(\\ref{eq:Ncont}) or (\\ref{eq:Ntrans}) with the replacement\n\\begin{equation}\\label{eq:mM}\nm\\to M=\\left(\\frac{3^\\frac{3}{2}\\Gamma(m)}{\\Gamma(m-\\frac{3}{2})}\\ln\\frac{1}{1-P}\\right)^\\frac{2}{3}\\,.\n\\end{equation}\n\nFor the case of local source associations we can write the probability of observing $m$ or more events as\n\\begin{multline}\\label{eq:Pass}\nP_{\\rm ass}(m) = 1-\\int{\\rm d}n_Cp_C(n_C)\\prod_{i=1}^{C-1}{\\rm d}n_i p_{\\rm PS}(n_i,n_C)\\\\\\times\\frac{\\Gamma(m,\\sum_{i=1}^Cn_i)}{\\Gamma(m)} \\,.\n\\end{multline}\nFor large number of local sources $C$ we can approximate this by the average number of expected events as\n\\begin{equation}\nP_{\\rm ass}(m) \\simeq 1-\\frac{\\Gamma(m,N(V_C\/V_H)^\\frac{1}{3}\/\\xi_z)}{\\Gamma(m)}\\,.\n\\end{equation}\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nLoaded granular materials made of highly deformable particles are ubiquitous in nature and industry from cell monolayer growth \\cite{angelini2011_pnas,martin2004_dev} to foam and emulsion compression \\cite{bolton1990_prl,katgert2010_epl,brujic2007_prl} and even metal or plastic powder sintering \\cite{kim1987_ijp,bares2014_prl}. In some cases, because of impurities or for technical and recycling purposes, rigid, undeformable particles are mixed among soft ones \\cite{tsoi2011_geo,khatami2020_gi}. This is the case for example in sand-rubber granular layers used as low-cost seismic \\md{isolations} \\cite{tsiavos2019_sdee}. \n\nThese kinds of mixed soft-rigid granular systems display singular behaviors different from purely rigid or \\md{purely} soft systems. Among other purposes, modifying the mixing ratio and the particle nature permits to tune the mechanical properties of a granular medium. In order to optimize their industrial applications and to control \\md{their} behavior in \\md{natural phenomena}, it is needed to know the connection between their macroscopic behaviors and their microscale composition and organisation. Because granular matter is naturally \\md{multiscale}, \\md{a} clear understanding of its global behavior necessitates to figure out and to measure what happens at the local\/grain scale and at all the intermediate scales up-to the global one. Due to the already complex process of measuring local mechanical properties of each grain for fewly deformed particles \\cite{zadeh2019_gm} the challenge is even more important of measuring them for soft highly deformed particles. \n\nVery recently this challenge has been overcome \\cite{vu2019_pre,vu2019_pre_2} using an experimental method coupling Digital Image Correlation (DIC) and very \\md{acurate} imaging. In this paper, we extend this method to the so called case of mixtures of soft and rigid particles. We show how this experimental set-up is \\md{modified} to compress \\md{these} mixtures and how the image post-processing is modified accordingly. A selection of resulting experimental data from the inside of the particle up-to the system size is presented. This allows us to understand better the mechanical behavior of these materials and will permit to improve their compression models \\cite{cardenas2020_pre}.\n\n\\section{Experimental method} \\label{method}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig1.png}\n\\caption{Experimental set-up. Top: 3D \\md{drawing} of the compression device and particle packing laying on a photographic flatbed scanner \\cite{scanner}. Bottom-left: squeezed soft silicone particle \\cite{silicone,vu2019_pre_2}. The top shiny aspect is due to a layer of thin metallic glitter \\cite{silver_glitter}. Bottom-right: rigid PVC particle. The top shiny aspect is due to silver paint. \\md{During experiments particles are used with their shiny part in contact with the scanner.}}\n\\label{fig_1}\n\\end{figure}\n\nIn Fig. \\ref{fig_1} we show the experimental set-up already presented in \\cite{vu2019_pre_2}. It is composed of a bidimensional piston of initial dimension $270\\times202$ mm$^2$ \\md{which permits to compress uniaxially the system}. A stepper motor rotates a screw which \\md{translate the moving edge of the piston inward in an accurate manner ($25$ $\\mu$m per motor step)}. Two force sensors are attached to this latter, they record the global stress evolution ($\\sigma$) when compressing a mixed granular system. The global stress is measured continuously with a frequency of $100$ Hz while the system is compressed stepwisely. At each step the piston moves of $0.5$ mm at a speed of $2$ mm\/min to make \\md{it} sure to stay in the quasistatic regime. Then, the system rests during $1$ min to relax and is imaged from below with a flatbed \\cite{scanner} with a resolution of $2400$ dpi ($10.6$ $\\mu$m\/px).\n\nThe granular \\md{systems are} composed of soft hyperelastic cylinders made of silicone \\cite{silicone,vu2019_pre_2} and of rigid \\md{cylinders} made of PVC (see Fig. \\ref{fig_1}). Their Young moduli are $0.45$ MPa \\cite{vu2019_em} and $1.2$ GPa, respectively. Both soft and rigid granular packings are bidisperse cylinders of diameters $20$ mm and $30$ mm and of height $15$ mm. For the \\md{experiments} presented in this paper, $96$ particles are used, $1\/2$ of them being soft and $2\/3$ of them being small. This last ratio is equally spread among soft and rigid particles, this avoids phase crystallization. \\md{The ratio of soft to rigid particles, $\\kappa = \\rm{n}_{\\rm{soft}}\/\\rm{n}_{\\rm{total}}$ is varied from $0.2$ to $0.8$.} To avoid basal friction the scanner glass is coated with oil which also lubricate the contacts making particles almost frictionless. \\md{Namely the measured friction coefficient is $0.05 \\pm 0.02$.}\n\nTo measure displacement fields on the particle bottom faces, the DIC method requires a thin random pattern with a high optical contrast. In the case of soft particles, this is obtained by coating the silicone with a very thin layer of \\md{shiny} glitter \\cite{silver_glitter}. For rigid particles, PVC is simply coated with \\md{silver metallic} paint. In both cases, the glitter characteristic size is around $25$ $\\mu$m. \\md{This} creates a random pattern with a correlation length of \\md{about} $50$ $\\mu$m on the bottom face of each particle.\n\n\\md{For each experiment a set of about $100$} pictures is obtained with this experimental set-up by compressing the system over $15$\\%. Pictures are then post-processed with an algorithm modified from \\cite{vu2019_pre_2, DIC_code}. First, particles are detected individually by thresholding the initial\/undeformed picture. Each particule is then tracked in the full set of pictures by using image correlation. Then, from the measurement of the particle rotation and translation, subsets of images are extracted following each particle and correcting its solid rigid motion. For soft particles a DIC algorithm aimed for large \\md{deformations} already presented in \\cite{vu2019_em} is used to obtain the displacement field ($\\bm{u}$) inside each particle. For the rigid particles a more classical DIC algorithm is used. This latter correlates all images with the initial one as classically done for systems \\md{in} the small deformation assumption. In both cases the correlation cell size is self-adjusted to take random pattern inhomogeneities into account as already presented in \\cite{vu2019_pre_2}. \n\nThen, for soft particles the deformation gradient tensor is computed from $\\bm{u}$ \\cite{taber2004_book}:\n\\begin{equation}\n\t\\bm{\\bm{F}}=\\bm{\\nabla}\\bm{u}+\\bm{\\bm{I}}\n\\end{equation}\nwith $\\bm{\\bm{I}}$ being  the  second  order  identity  tensor. The right Cauchy-Green strain tensor is finally derived \\cite{taber2004_book}:\n\\begin{equation}\n\t\\bm{\\bm{C}}=\\bm{\\bm{F}}^T\\bm{\\bm{F}}\n\\end{equation}\nFor rigid particles, the infinitesimal strain tensor is just directly computed ($\\bm{\\bm{\\varepsilon}}=1\/2((\\bm{\\nabla}\\bm{u})^T+\\bm{\\nabla}\\bm{u})$). From $\\bm{u}$ the evolution of the exact particle boundaries is also deduced which permits to get all geometric \\md{observables} for both particles and voids, but also contacts, at each compression step. \\md{Contacts are detected by considering first the proximity of each point of the particles boundaries with the boundaries of neighbor particles. Then, for each point closer than a certain threshold the von Mises strain around this point is measured and the contact is assumed to be real if this value is above a certain threshold as detailed in \\cite{vu2019_pre_2}.} For the sake of conciseness only a meaningful selection of \\md{the different} mechanical and geometrical observables are presented here.\n\n\n\\section{Results and discussions} \\label{result}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig2.png}\n\\caption{Global observables. Variation of the global stress ($\\sigma$), coordination ($Z$), specific contact length ($l$), average value of the von Mises of the right Cauchy-Green strain tensor ($|\\bm{\\bm{C}}|$) of soft particles, particle asphericity ($a$) and void area ($A$) as a function of the packing fraction ($\\phi$) for a compression experiment made with $50$\\% of rigid particles ($\\kappa = 0.5$).}\n\\label{fig_2}\n\\end{figure}\n\nAs presented in Fig. \\ref{fig_2}, \\md{a granular system with $\\kappa = 0.5$ is compressed for a packing fraction ($\\phi$) going from $0.795$ to $0.915$}. While loading, the global stress first stays close to $0$ MPa and then increases progressively from the jamming point \\cite{liu1998_nat}. This occurs for $\\phi$ close to \\md{$\\phi_j = 0.84$}, as already observed in many other experimental studies \\cite{behringer2018_rpp}. It also corresponds with a coordination value slightly above $3$ as predicted theoretically \\cite{van2009_jpcm} since particles are slightly frictional. In Fig. \\ref{fig_2} we also plot the evolution of the specific contact length, which is the average of \\md{the ratio of the particle perimeter in contact with another particle or with the system boundaries ($l_{\\rm contact}$) over the particle perimeter: $l = \\left< l_{\\rm contact}\/\\rm{perimeter}\\right>_{\\rm particles}$.} This quantity increases linearly with $\\phi$, which is not in agreement with the Hertz's contact law in 2D \\cite{johnson_book}. This last point is not surprising since soft particles are far from the small deformation assumption \\md{in our study}. \n\nFigure \\ref{fig_2} also shows that the average value of the von Mises strain ($|\\bm{\\bm{C}}|$) in soft particles decreases parabolically all along the compression while the average soft particle asphericity first gently increases and then sharply rises after $\\phi=0.88$. This quantity is computed as the ratio between the actual particle area and the area deduced from the perimeter assuming the particle would be circular. The packing fraction $\\phi=0.88$ also corresponds with the maximum value of the total void area which first increases, since void clusters form, and then decreases, since particles are squeezed and reduce these \\md{cluster} areas, as already observed for fully soft systems \\cite{vu2019_pre_2}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig3.png}\n\\caption{Local observables. Field of the first invariant of the right Cauchy-Green strain tensor ($\\bm{\\bm{C}}$) for soft particles and of the infinitesimal strain tensor ($\\bm{\\bm{\\varepsilon}}$) for rigid particles. These fields are computed for a compression experiment made with $50$\\% of rigid particles at a packing fraction $\\phi = 0.89$ far above the jamming point.}\n\\label{fig_3}\n\\end{figure}\n\nIn Fig. \\ref{fig_3}, the first invariant of the strain fields measured in each particle is shown for $\\phi = 0.89$. For soft particles the right Cauchy-Green strain tensor ($\\bm{\\bm{C}}$) is used while it is the infinitesimal strain tensor ($\\bm{\\bm{\\varepsilon}}$) \\md{which is used} for rigid particles. The field for soft particles distinctly shows high strain chains with clear areas near the contact points where the deformation is maximum. On the contrary the patterns observed on the rigid particles suggest that the imaging method is not accurate enough to \\md{highlight} grain deformations. This is not surprising since the level of compression is rather low and the \\md{ratio of soft to rigid} particle stiffness covers several orders of magnitude: \\md{deformation is almost exclusively carried by soft particles.} The lines that appear on the rigid particles actually correspond with the small speed inhomogeneities of the \\md{imaging method}.\n \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig4.png}\n\\caption{Statistical observation. \\md{Main panel}: Evolution of the PDF of the value of the von Mises of the right Cauchy-Green strain tensor ($\\bm{C}$) measured in soft particles as a function of the packing fraction for a compression experiment made with $50$\\% of rigid particles. Scale is semi-log. \\md{Inset: Evolution of standard deviation of $\\bm{C}$ for different particle mixes ($\\kappa$) as function of the distance to the jamming point.}}\n\\label{fig_4}\n\\end{figure}\n\nIn Fig. \\ref{fig_4}, we present the evolution of the strain field statistics. More exactly, we plot the probability density functions (PDF) of the von Mises values of the right Cauchy-Green strain tensor ($|\\bm{\\bm{C}}|$) in soft particles for different packing fractions. For gently squeezed states, the PDF is a Gaussian centered around $1$ which corresponds with the fact that the particles are mainly undeformed. While the system is loaded, the PDF gets broader and broader and the average value slightly shifts to the left which corresponds with the global compression of the system. For densely packed states, the PDFs display an exponential tail with a coefficient decreasing with the packing fraction. This exponential tail is reminiscent of what is observed for force values in packing of rigid granular systems \\cite{radjai1996_prl} and remains unchanged from observations made on purely soft systems \\cite{vu2019_pre_2}. \\md{In the inset of Fig. \\ref{fig_4}, the evolution of the standard deviation of $\\bm{C}$ is shown for different ratios between soft and rigid particles. When the system is mainly composed of rigid particles ($\\kappa = 0.2$), this value rapidly increases right after the jamming point. In this case lot of particles undergo large deformation and display more extreme strain values. On the contrary, when the system is composed of more soft particles ($\\kappa = 0.8$) the deformation is shared over more particles and less extreme strain values are observed so the standard deviation increases less rapidly.}\n\n\\section{Conclusion} \\label{conclusion}\n\nIn summary, we have developed an experimental tool that \\md{can} permit to measure all mechanical and geometrical observables from \\md{the displacement field at} the particle scale for bidimensional granular systems under uniaxial compression, whatever the particle mechanical properties. \\md{We believe this measurement method at the \\textit{sub-micro scale} will permits to model granular matter in a more efficient way to predict its behavior.} It is applied here to the case of a mixed granular \\md{media} made of soft squeezable particles and rigid ones. The system is loaded step by step and imaged very accurately, while \\md{the} global force applied to it is recorded continuously. By mean of DIC, the evolution of mechanical fields inside each particle are measured which permits to deduce observables as diverse as packing fraction, coordination, contact length, particle asphericity, void area and strain fields to name a few.\n\nMore specifically we have shown that our system crosses the jamming transition for a packing fraction and a coordination value close from what has been observed in the case of purely rigid systems. This point validate our experimental approach. \\md{We have also shown that the average von Mises strain in soft particles decreases with the exact same tendency as the global stress increases in the system. Since rigid particle deformation is negligible, this suggests that rigid particles do not carry more load than the soft ones. The loading seems to be balanced between soft and rigid particles.} The stress vs. packing fraction curve is also similar to what has been observed in a recent numerical study \\cite{cardenas2020_pre}. \n\nDeep in the jammed state the PDF of the von Mises strain tensor displays an exponential tail as already observed for purely soft systems and similarly to what is observed for the \\md{intergranular} force PDF \\md{of} purely rigid systems. \\md{Also, the standard deviation of this quantity increases more rapidly for systems composed of more rigid particles. This is explained by the fact that deformation is mainly carried by soft particles and that they are individually more deformed when they are less for a given deformation level.}\n\nBeyond the study of mixed soft-rigid granular systems, we believe that this experimental tool offers an opportunity to catch experimentally the behavior of breakable and polydisperse (in size, shape and material) \\md{particles} under different loading geometries. \n\n{\\small\nGille Camp, St\\'{e}phan Devic and R\\'{e}my Mozul are greatly thanked for their technical support. We warmly thank Vincent Huon for for fruitful discussions.\n}\n\\newline\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and observations}\n\nThis paper is a sequel to the paper by Hutchings, Maddox, Cutri, and\nNelson (2003: paper 1), which presented optical imaging of QSOs of redshift\n0.3 and lower, selected from the Two Micron All Sky Survey (2MASS). HST\nimaging of a different sample of 2MASS QSOs is given by Marble et al (2003).\nIn this paper we present optical imaging of a sample of 2MASS QSOs at \nredshifts above 0.3. The sample of QSOs is the same as in paper 1, but \nwith the higher redshift bin.\n\nThe QSOs are identified by spectroscopy after selection by colours\nJ-K larger than 2.0, and are classified as types 1, 2 or intermediate,\nbased on the ratio of narrow to broad line emission. These do not include\nany previously known AGN. 2MASS QSO selection is described in Cutri et al.\n(2002).  The present study includes objects between\n-30$^o$ and 60$^o$ declination, and thus suitable for observation with the \nCFHT. Paper I included data on 76 of 243 eligible objects, and this paper\nincludes data on 21 of 64 eligible\nobjects. All but 2 of the eligible objects have redshifts below 0.7.\n\n\n\nThe observations were taken as service-observing `snapshots' with\nthe Megaprime camera and r-band filter, with exposures of 340 secs. The pixel\nsampling is 0.187 arcsec. (The paper 1 data were taken with R band, and 0.206\narcsec sampling, and 200 sec exposures, with the CFHT 12K camera. The CFHT\nwebsite gives details of the different filters.) All\nobjects had two independent images, 5 had four images, and two had 8 images.\nThe data were obtained over the period April through June 2004, in seeing\nconditions that varied from 0.6 to 1.3 arcsec FWHM. Table 1 lists the\nobjects observed, and various of their properties, from the 2MASS database\nand also measures from the present work.\n\nAs the selection of observed objects was dictated mainly by scheduling,\nwe show in Figure 1 the distribution of observed objects within the whole\nsample of 64 high redshift objects, and also include the lower redshift\nsample from paper 1.\nThe observed subsample is slightly skewed to higher luminosity, but there\nis K-band luminosity overlap of about 15 new objects with 21 of the highest\nluminosity objects from the lower redshift sample. In terms of J-K colour,\nthe observed subset is slightly\nskewed to redder objects than the average by about 0.5 magnitudes.\nThe spectral classifications among the new sample\nare representative of the 2MASS QSOs in this redshift range.\n\n\\section{Data processing and measurements}\n\nAfter removal of the instrumental signature, subimages were generated of\neach QSO and 3 PSF stars from the same CCD and observation, as near as\npossible to the\nQSO. The two (or more) images of each field were combined, to help remove\ncosmic rays and to increase the signal. In a couple of cases, some of\nthe images had very different FWHM as they were taken on different nights,\nand such images were not combined.\n\nPSF stars were chosen to have more signal than the QSOs, and to be\nfree of nearby companions and image flaws. Any other stars near the PSF\nstars were edited out of the images before generating the profiles, to\nget the most representative PSF profiles. The images and luminosity\nprofiles were compared among the PSF stars, and generally the cleanest\none was picked as the PSF to be used. Occasionally\ntwo PSF stars were co-added, to increase the signal, after checking that\nthe profiles agreed.\n\nMeasurements were made of every individual image, as well as the\ncombined images, to give independent values of the fluxes and PSF-removal.\nThe measurements made were the flux of the QSO and the PSF stars, and the\nimage FWHM from profiles. Profiles were generated by the IRAF task ellipse,\nafter careful sky removal. Companions near the QSOs were included in\ntheir profiles,\nalthough in some cases, well separated companions were edited out to better\nstudy the extended flux centered on the QSO. Figure 2 shows profiles\nfrom one object as an example.\n\nThe host galaxy fluxes were estimated in two ways. First, the flux difference\nwas measured between the QSO and PSF profiles, after scaling the PSF\nprofile to the same peak value as the QSO. This yields a measure\nof the resolved flux which is a lower limit to the QSO host flux. Second,\ndifferently scaled PSF images were subtracted from the QSO to yield a\nresulting\nprofile that increases monotonically to the nucleus. The value for\nthe profile that just turns over near the nucleus yields an upper\nlimit for the host galaxy flux. The best value was taken to be the\nintermediate case where the difference flux follows a smooth profile for\nits inner part. While this was subjective, and depends in principle\non whether the difference follows an exponential or de Vaucouleurs\nprofile, or combination, the QSO nuclear flux was generally small\nenough that the spread in these difference fluxes was small. Error\nbars were taken for each host galaxy flux as the upper and lower\nvalues described above.\n\nThe same procedure was repeated for the co-added images for each QSO.\nTable 1 shows the final adopted values for the ratio of nuclear to\nresolved flux for each QSO. The agreement between the individual and\nthe combined images was in all cases within 20\\%, and the ranges as\ndescribed above have similar values. Thus, in Table 1 we show the\nmean values (average discarding outliers) for the nuclear to flux ratio, \nand we assign an\noverall uncertainty of 30\\% in those values. Note that the resolved flux\ngiven does not include any separate close companions within the\nradius of 10 arcsec: where these are present they were not included.\nThe resolved flux does include any features that appear to be part of\nor connected to of the host galaxy, even though they introduce irregularities \ninto the azimuthally averaged profile. Many of the profiles are\nirregular, which means that fit to either standard model is\nsomewhat arbitrary. We have noted where the profiles do have some\nregion of good fit to one or other model. There are ten with some\nspheroidal component and four with a good exponential. Four others\nhave so many companions that no fit is good, and another three are clean\nbut do not have clean regions of good fit. In several cases, the host\ngalaxy appears to have both bulge and disk components to the azimuthally\naveraged profile. \n\nThe morphology of the QSO images, both raw and PSF-subtracted, was\ninspected to make an estimate of the degree of tidal disturbance.\nAs in paper 1, this value takes into account the presence of connecting\nbridges to companions, single asymmetric arms or extensions, radial\njet-like features, warps, and other asymmetries. This is quantified as\nan interaction strength index from 0 to 3, as in paper 1. Two objects have\narchival HST images from Marble et al 2003 (the last and third last ones \nin Table 1). In the first of these, no interaction is seen in either\ntelescope image. In the second, our CFHT image has some asymmetry (interaction\nindex 1), but the HST image shows considerable structure that clearly\nindicates a significant disturbance on small scales (see Figure 3). \nThus, our interaction indices may well be conservative. Figure 3 shows \nexamples of the three levels of interaction, plus an HST image for comparison.\n\n\\section{Results}\n\n   All the QSOs in the sample are resolved, most of them easily, with \nratio of nuclear to\nresolved flux ranging from 0.17 to 25. We discuss the properties\nof the resolved flux in the subsections below, but begin with a look\nat possible biases and selection effects within the dataset.\n\n\\subsection{Biases and systematics}\n\n   Since there is a range of image quality in the sample (from 0.6 to\n1.3 arcsec), we looked for measured quantities that may depend on it.\nThe image quality shows no correlation with object redshift.\nThe interaction index shows no envelope or correlation. The ratio of\nnuclear to resolved flux also does not show any trend, and certainly not\nin the expected sense of lower ratios for better seeing. The dynamic range\nof `useful' QSO image does show an upper envelope where the maximum range is\nsmaller for larger FWHM images. This is largely the effect of reducing the\npeak central signal with poor image resolution, and the limiting signal is \nnot affected. Overall, for the sample and measurements we discuss, the image\nquality is not very important and is not introducing any significant bias\nto the results.\n\n   Another observational variable is the total exposure time. While most\nobjects have 680 secs integration, three have twice that, and two have\nfour times that. The interaction index, and useable dynamic range are\nnot correlated with the exposure, and the two best-resolved objects\nhave the minimum exposure. Thus, exposure time is not biassing the\noverall results of the work.\n\n  The mix of spectral types is not correlated with object redshift. The\nmagnitude of the QSO is not correlated with the dynamic range of the\nimages. The least resolved objects are not the highest redshift ones, so\nthere is no observational bias obvious. The median ratio of nuclear to\nresolved flux is 4, and values near this are spread evenly across the\nredshift range. There is no systematic colour change with redshift, or\nchange of scale length with redshift, so the mix of objects appears to\nbe unrelated to redshift.\n\n\\subsection{Nucleus to host ratio}\n\n   Figure 4 shows the ratio of nuclear to resolved flux with spectral\nclass. The ratio increases with the dominance of the broad emission\nline components, which is as expected for the general model of\ncentral obscuration by a torus for type 2 objects.\n\n   The lowest dynamic range observations have the lowest nuclear light\ndomination. This means that the resolved flux is in the inner parts\nof the host galaxies, and that low nuclear domination is caused by\nobscuration within\nthe central host galaxy. However, overall QSO colour is loosely\ncorrelated with nuclear domination, so the obscuration is connected with\nreddening of the nuclear light - a separate phenomenon from the\nobscuration of line emission by the torus.\n\nThe result that obscuration is connected with the diminishing\nof the nuclear light in the optical is supported by the findings\nof Francis, Nelson and Cutri (2004) who found that a number of\nnear-ir flux-selected AGN were missed in SDSS because their optical colors\nwere indistinguishable from normal galaxies.  They speculated this\nwas because the nuclei were preferentially reddened and thus better\nvisible in the near-ir.\n\n   Scale length of the host galaxy is not correlated with nuclear domination,\nso the nuclear region is obscured without affecting the outer parts of the\nhost galaxy. Scale length is also not related to the spectral class or the\ncolour of the QSO. Further mention is made of the scale length below.\n\n\\subsection{Interaction status}\n\n  The level of interaction seen is not strongly related with the QSO colour,\nbut there is a trend towards redder colour for more strongly interacting\nsystems. There is a clear dependence and upper envelope with the nuclear\nfraction, in the sense that the more interacting objects have more \nobscured nuclei (see Figure 5).\n\n   It is also clear that interactions are less obvious in higher redshift\nobjects, as expected as the signal to noise and the angular scales decrease.\nWe note that the object with HST imaging, given interaction class 1 from\nthe CFHT data, is clearly interacting in the HST images, and would have\nindex 2. Thus, the CFHT interaction indices should be regarded as lower limits\noverall, and particularly for the higher redshift objects.\n\n   The host galaxy scale length shows a trend where they are larger\nfor weaker interacting systems, although there is considerable spread.\nThis will reflect the presence and decay of disks, tidal arms, and\neventual increase in spheroid structure during a major interaction.\n\n   Comparison with the z$<$0.3 sample in paper 1 is of interest. Figure 6\nshows the fractions with interaction index with redshift in different\nQSO samples,\nwhere the 2MASS lower redshift sample is large enough to split at\nz=0.2. There is a\nsystematic shift towards lower interaction index with increasing redshift.\nEven allowing for lowering the index by one grade point for 1\/2 of them\n(which would allow for missed signatures seen only with higher resolution,\nas is Figure 3),\nwe find the interactions are stronger at lower redshifts. Since the\nhigher redshift objects are more luminous, it may be that the more interacting\nand hence more obscured objects are not detected in the higher redshift\nbins because of flux limits. The ratio of counts of objects above and below\nz=0.3 (64 to 243, plus lack of LINERS at z$>$0.3), shows the 2MASS flux\nlimitation clearly.\n\n\n\\subsection{Radio flux}\n\n  Six of the sample of 21 appear in the FIRST radio catalogue (although\n2 of them\nare too far south to appear in the FIRST catalogue: see Becker et al 1995).\nThey are\nall unresolved, and faint - see Table 1 for the fluxes. The mean redshift\nof the radio sources is the same as that for the others ($\\sim$0.40).\nThere is no correlation between radio flux and colour. The mean ratio\nof nucleus to host for the radio sources is 2.5, compared with 7.2 for\nthe others (medians 1.3 and 3.2). The spectroscopic types have the same\n(full) spread for radio and radio-quiet objects. The radio sources\nhave an average interaction index of 2 while the others have value 1,\nand this difference is significant at the 95\\% level, if the\ndistributions\nare gaussian. This suggests that\nnuclear radio sources occur in recently activated nuclei where the\nsigns of interaction, and dust obscuration are higher. This is similar\nto the result on IRAS galaxies investigated by Neff and Hutchings\n(1992).\n\n\\subsection{Asymmetries and and profiles}\n\n   The ellipticities of the contours in this sample are lower than\nthose in paper 1, and follow the same upper limit which decreases with\nincreasing redshift. This is largely the effect of the diminishing scale and\nsurface brightness with redshift, but the values are less determinate as the \npresence of line of sight companions increases too.\n\n   Profiles were classified as good fits to R$^{1\/4}$ law or\nexponentials\nif they had significant linear sections in the relevant plots. Many\nobjects have nearby companions or asymmetries which made these simple\nclassifications impossible. However, there were 3 objects with good\nexponential\nprofile sections and 9 with good bulge components. Their mean redshifts\nare 0.38 and 0.43, respectively, so that the outer exponential tails may\nsimply be less detectable in the fainter higher redshift objects.\n\n   It is of interest to compare the profiles of an object with HST imaging\n(1715+281).\nThe HST image shows strong asymmetry of the resolved flux and is traceable to\na radius of 2.4\", while the CFHT image shows less of the structure but\nthe same asymmetry out to 7\" (see Figure 2). \nThe profiles agree and both indicate\nthat r$^{1\/4}$ fits quite well, while an exponential does not. Thus, there\nis good agreement in this one case of overlap with HST data.\n\n\\subsection{Scale lengths}\n\n   The scale lengths given in Table 1 are derived from the slopes of the\nradius-magnitude plots for the images, outside the unresolved cores. They\ndo not include any signficant companions, and are converted to Kpc for\na Hubble constant of 75. Cases where the slope cannot be measured\nwith any reliability are left blank. Scale length is not correlated\nwith the dynamic range of the images, although the latter range only\nby about 25\\%. It is also not correlated with the image quality - as\nexpected\nsince the profiles are azimuthally averaged over values far greater than\nthe image quality.\n\n  The scale length is larger for the more interacting systems. This\nis a quantitative measure of the extended arms and asymmetries that\nlead to the higher interaction indices. In general the scale length values\nare comparable with large nearby galaxies.\n\n\n\\section{Discussion}\n\n    In paper 1 we noted a number of differences between the 2MASS low\nredshift QSOs and standard blue QSOs. Generally speaking, the 2MASS obects\nare redder and have more obscured nuclei, and the host galaxies show a far\ngreater proportion of tidal interactions.\n\n   We note that the 2MASS sample requires detection in all three of the\nJHK passbands, so that there is a bias against highly reddened objects.\nThe J-K colours average at 2.2 mag in both the lower and higher\nredshift subsamples, but the B-R colours are different, as seen in Table 2.\nThe selection will also lose the less luminous objects as the redshift\nincreases. While this is clearly true from Figure 1, it is interesting to \nnote the large fraction of more luminous objects that appear at higher\nredshifts. This increase is just what is expected from the 6-fold increase \nin volume of space sampled, so there may not be significant selection effects\nbetween the present sample and the lower redshift objects in paper 1.\n\n\n    In most of the aspects considered, the higher redshift 2MASS objects are\nmore similar to `normal' blue-selected QSOs, but they are intermediate between\nthem and the low redshift 2MASS QSOs. Table 2 shows some key comparison\nnumbers, based on paper 1 and the optically selected sample of Hutchings\nand Neff (1991).\n\n    The higher redshift sample of this paper correspond to the highest\nluminosity objects in paper 1. The subset of the paper 1 objects that\nmatch this K-band luminosity are also shown in Table 2, and they have higher\nnucleus to host ratio and bluer colour than the full paper 1 sample,\nbut not as high or blue as the sample in this paper. Thus, luminosity\nis a relevant parameter as well as redshift. The single high redshift object\n(at z=2.37) has very high luminosity (although optical emisssion lines \nshifted into the 2MASS bandpasses probably contribute), and is hardly\nresolved in our data. Thus, there is probably nothing very remarkable\nin this object from the data presented here.\n\n   The higher redshift (and luminosity) sample of this paper is\ngenerally similar to normal blue QSOs, but still have a higher fraction of\ninteraction evidence, although the fraction of highly interacting objects\nis similar to that for blue QSOs. This is seen in both morphology and\nluminosity profiles. It seems\nlikely that the higher luminosity QSOs blow away the circumnuclear dust\nfaster, but are still relatively young AGN. The unresolved nature of the \nradio sources is consistent with this idea.\n\n   The sample does not reach the faint luminosities of the low redshift\nobjects in paper 1, so the evolution of these sources cannot be\ntraced until we have a deeper NIR QSO survey. We note that the Spitzer\ntelescope has reported finding a large population of lower luminosity\nAGN in their deeper survey. It will be important to follow those up\nwith high resolution imaging. \n\nThis publication makes use of data products from the Two Micron All Sky\nSurvey, which is a joint project of the University of Massachusetts and the\nInfrared Processing and Analysis Center\/California Institute of Technology,\nfunded by the National Aeronautics and Space Administration and the\nNational Science Foundation.\n\n\\newpage\n\\scriptsize\n\\begin{deluxetable}{lrllrrccrlrc}\n\\tablenum{1}\n\\tablecaption{2MASS QSO sample}\n\\tablehead{\\colhead{RA} &\\colhead{Dec}\n&\\colhead{z} &\\colhead{Typ} &\\colhead{B, R, J, H, K} &\\colhead{N\/H}\n&\\colhead{Int} &\\colhead{Sc L} &\\colhead{Range} &\\colhead{20cm}\n&\\colhead{IQ} &\\colhead{M$_K$}\\\\\n&&&\\colhead{\\tablenotemark{a}} &&\\colhead{\\tablenotemark{b}}\n&\\colhead{\\tablenotemark{c}} &\\colhead{Kpc\\tablenotemark{d}}\n&\\colhead{mag\\tablenotemark{e}} &\\colhead{mJy} &\\colhead{\"\\tablenotemark{f}}}\n\\startdata\n11 03 12.93  & 41 41 54.9  &0.403 &1.2\n&16.6,16.1,15.2,14.2,13.0 &16.6&1. &-- &9.5&-- &1.1 &-29.1\\\\\n13 32 31.17  & 03 59 28.0  &0.346& 1.\n&17.4,17.3,16.0,14.8,13.4 & 4.3& 1.&--& 11.3&1.5\n& 0.9 &-28.3\\\\\n13 45 17.89  & -08 29 57.3 &0.473& 1.\n& --,  ~~~~--, 15.6,14.2,12.8\n& 3.9& 2.&7.7 &9.6&-- & 0.8 &-29.6\\\\\n14 32 04.62  & 39 44 38.9 &0.349&1.5\n&16.4,16.3,16.0,15.4,14.4\n& 5.4& 1. &4.4 &8.5&-- & 0.8 &-27.3\\\\\n14 35 15.66  & 02 32 21.7 &0.305&1.2\n&16.6,16.7,15.6,14.4,12.9\n& 12.& 1.&--& 11.5&-- & 0.9 &-28.5\\\\\n14 38 27.94  & -11 22 49.5 &0.401&1.5\n&19.6,18.7,16.3,15.1,13.6\n&1.2& 1.&8.6 & 9.0 &-- & 1.1 &-28.4\\\\\n14 41 18.87  & -11 31 47.5 & 0.330 &1.2\n&16.6,16.8,16.2,15.2,14.0\n& 2.5& 1.&4.8 & 10.5&-- &1.0 &-27.6\\\\\n14 42 02.95  & 14 55 39.3 &0.307&1.2\n&17.6,17.0,16.2,15.2,14.2\n& 1.8& 3. &5.9 & 10.4& 104. & 0.9 &-27.2\\\\\n14 50 00.90  & 14 29 48.7 &0.358&1.2\n&17.7,17.1,16.4,15.6,14.4\n& 12.& 0.&--&12.0 &-- & 0.7 &-27.4\\\\\n15 00 13.40  & 12 36 45.2 &0.407& 1.9\n&19.5,18.9,16.8,15.9,14.6\n& 0.3& 3. &4.7 &8.1&-- & 0.8 &-27.5\\\\\n15 01 50.51  & 49 33 38.2 &0.337&1.9\n&18.1,17.1,16.4,14.9,13.5\n&0.26& 3.&4.8 & 8.0 &3.1 & 0.8 &-28.2\\\\\n15 31 07.19  & 12 08 14.4 &0.542&1.5\n&19.8,18.7,17.1,15.7,14.6\n& 3.2& 2.&6.7& 10.2&-- &1.0 &-28.1\\\\\n15 36 44.92  & 14 12 29.4 &0.399&1.2\n&16.7,17.3,16.1,15.4,14.0\n& 4.8& 2. &6.6 & 10.5&-- & 0.9 &-28.0\\\\\n15 40 19.57  & -02 05 05.3 &0.319& 1.\n& 16.0,15.8,15.3,14.3,13.2\n&6.& 2.&4.7 &9.3&4.7 &1.0 &-28.3\\\\\n15 49 38.73  & 12 45 09.2 & 2.37& 1.9\n&18.7,17.3,15.8,14.5,13.5\n& 25.& 0. &-- &12.5 &-- & 0.7 &-33.0\\\\\n15 50 59.30  & 21 28 08.8 &0.373& 1.\n&17.3,16.8,16.3,15.7,14.3\n& 8.8& 0.&--& 12.8&-- & 0.6 &-27.6\\\\\n16 18 09.74  & 35 02 08.9 &0.446& 1.9\n&18.8,18.2,16.8,15.4,14.1\n& 2.6& 2.&6.9&10.0 &14. & 0.9 &-28.2\\\\\n16 44 20.14  & 56 36 44.6 &0.329&1.5\n&19.8,18.2,17.2,15.9,14.6\n&0.17& 1.&4.7 &9.5&-- & 1.3 &-27.0\\\\\n17 00 02.99  & 21 18 23.3  &0.596 &1.5\n& --,  ~~~~--, 17.4,15.9,14.9\n&1.4 & 0.&3.8 &8.3&-- & 1.1 &-28.1\\\\\n17 00 56.01  & 24 39 28.2 &0.509&1.5\n&16.8,16.8,16.0,15.3,14.3\n& 10.& 1. &8.7 & 10.3&-- & 0.7 &-28.3\\\\\n17 15 59.77  & 28 07 16.9  &0.524 &1.8\n& --,  ~~~~--, 17.2,15.8,14.6\n&0.32& 1. &4.4 &7.8&1.6 & 0.8 &-28.0\\\\\n\\enddata\n\\tablenotetext{a}{ The range from 1 to 2 reflects the ratio of narrow to\nbroad emission lines.}\n\\tablenotetext{b}{ The ratio of nuclear to host flux in r-band, from\nthis work. Uncertainties are estimated to be 30\\%.}\n\\tablenotetext{c}{ Strength of host galaxy interaction based on observed\nmorphological features, as in paper 1.}\n\\tablenotetext{d}{ 1\/e profile flux drop}\n\\tablenotetext{e}{ Total dynamic range of profile in magnitudes}\n\\tablenotetext{f}{FWHM of stellar images in arcsec}\n\n\\end{deluxetable}\n\n\\newpage\n\\normalsize\n\\begin{deluxetable}{lcccc}\n\\tablenum{2}\n\\tablecaption{2MASS QSO sample comparisons}\n\\tablehead{\\colhead{Property} &\\multicolumn{3}{c}{2MASS QSOs}\n&\\colhead{Blue QSOs}\\\\\n&\\colhead{z$<$0.3} &\\colhead{z$<$0.3,luminous} &\\colhead{z$>$0.3} }\n\\startdata\nAverage z &0.2 &0.2 &0.4 &0.2\\\\\nSample size &76 &12 &21 &28\\\\\n\\\\\nB-R &1.1 &0.8 &0.5 &0.3\\\\\nNuc\/Host &0.5 &1.2 &6.0 &5.0\\\\\nInt strong &33\\% &42\\%&14\\% &11\\%\\\\\nInt (any) &75\\% &58\\% &81\\% &39\\%\\\\\nExp profile &16\\% &-- &14\\% &18\\%\\\\\nBulge profile &20\\% &-- &48\\% &54\\%\\\\\nMessy profile &64\\% &-- &38\\% &28\\%\\\\\nRadio detected &40\\% &50\\% &29\\% &43\\%\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\newpage\n\\centerline{References}\n\nBecker, R., White R.L., and Helfand D.J., 1995, ApJ, 450, 559\n\nCutri,R.M. et al, 2002, ASP Conf Ser vol 284, 127\n\nFrancis P.J., Nelson B.O., and Cutri, R.M., 2004, AJ, 127, 646\n\nHutchings J.B., Maddox N., Cutri R.M., Nelson B.O., 2003, AJ, 126, 63\n(paper 1)\n\nHutchings J.B. and Neff S.G., 1992, AJ, 101, 434\n\nMarble A.R. et al., 2003, ApJ, 590, 707\n\nNeff S.G. and Hutchings J.B., 1992, AJ, 103, 1992\n\n\n\\newpage\n\\normalsize\n\\centerline{Captions to figures}\n\n1. Sample selection in this paper (z$>$0.3) and paper 1, from the full\n2MASS sample observable from CFHT. Filled circles are the objects observed. \nTop: dashed lines are contours of constant luminosity. Bottom: spectral\ntypes range from broad to narrow emission lines, with intermediate\nvalues. Dashes are the full sample and dots are those observed.\nThe paper 1 observed sample is grouped as types 1 and 2 only, as given\nin that paper.\n\n2. Azumuthally averaged profiles of QSO 1715+281, with the CFHT PSF \nand also an HST image (with slightly different filter).\n\n3. Images of representative objects from the sample, in decreasing levels\nof interaction - top left (1500+12) level 3, top right (1442+14) level 2, \nbottom (1715+28) level 1. The lower right is the HST image of the object \nat lower left. The images are 11 arcsec on a side except for the HST which\nis 6 arcsec.\n\n4. Correlations with nuclear light fraction. Top: the type 1 objects\nhave higher nuclear flux, consistent with the orientation expectation\nfor broad-line objects. Bottom: With the high value exception, there\nis correlation with overall QSO colour, consistent with the nuclear\nfraction being reduced by dust reddening. In the lower panel, open dots \nare B-K and filled dots are 3(B-R). The line is the linear fit \nthrough all except the top right object.\n\n5. Correlations with interaction level of host galaxies. The interaction\nscale was estmated on a 5-point scale but is reduced to 3 here to match\nthe values from paper 1. Top: high\nlevels of interaction are not seen in the highest redshift objects,\npresumably because of signal level and angular scale. Bottom: Highly\ninteracting hosts have low flux nuclei, presumably because they are\nobscured by dust.\n\n6. Fraction of hosts in different interaction levels in three redshift\nbins with about equal sample numbers. There is a systematic change with\nredshift whereby lower redshift objects are more interacting. This may\nnot all be due to detectability changes with redshift.\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\n\n\\subsection{Background and Motivation}\n\nAs one studies \\emph{differential structures} on a manifold such as Milnor \\cite{milnor},   \\emph{affine structures} on a scheme, taken as counterparts,  will be introduced and discussed in this paper. Here, we will obtain several properties on affine structures in a rigour and systematic manner. These results  in fact will have been applied to the\ndiscussions on the complex analytical space of an algebraic scheme over a number field.\n\n\\subsubsection{An affine covering of a scheme}\n\nAs well-known, a scheme (or a projective scheme, respectively) is defined to be a\nringed space that can be covered by a family of affine (or\nprojective, respectively) schemes, called \\emph{an affine covering} of the scheme $X$.  An affine scheme is the spectrum of a\ncommutative ring equipped with the sheaf in an evident manner (\\cite{ega,hartshorne}). In the paper it will be seen that such a family of affine schemes determines a unique \\emph{affine\nstructure} on the scheme.\n\n\\subsubsection{Each affine covering produces a complex analytical space}\n\nFixed an algebraic scheme $X$ over a number field $K$. Let $\\{A_{\\alpha}\\}_{\\alpha \\in\\Gamma}$\nbe a family of finitely generated algebras over $K$ such that their spectra\n$Spec{A_{\\alpha}}$ cover $X$, i.e., $$\\bigcup_{\\alpha} Spec{A_{\\alpha}}\\supseteq X$$ holds. Here, each $A_{\\alpha}$ is isomorphic to the\nquotient of some polynomial ring $K[t_{1},t_{2},\\cdots,t_{n_{\\alpha}}]$  by a\nfinite number of polynomials $$f_{1},f_{2},\\cdots,f_{r_{\\alpha}}$$ over $K$ in the variables.\n\nBy Serre's\n\\emph{GAGA} (\\cite{sga1,serre-gaga}), each open subscheme $Spec{A_{\\alpha}}$ has an analytical space\n$X^{an}_{\\alpha}$ that is defined by the common zeros of the polynomials\n$$f_{1},f_{2},\\cdots,f_{r_{\\alpha}}$$ mentioned above. Gluing these analytical spaces\n$X^{an}_{\\alpha}$, we will obtain an analytical space $X^{an}$, called the\n\\emph{complex analytical space} of $X$. It has been seen that such a process has several functorial\nproperties with respect to $X$ (\\cite{sga1,serre-gaga}).\n\nHence, every affine covering of $X$ produces a complex analytical space $X^{an}$ of $X$.\n\nHowever, in general, an algebraic scheme $X$ can have many affine coverings. Then what about the complex analytical spaces of $X$ produced by different affine coverings of $X$?\n\n\\subsubsection{Different affine coverings can produce different complex analytical spaces}\n\nLet's take an example raised by Serre (\\cite{serre-example}):\n\\begin{quotation}\n\\emph{Let $V$ be the\nnonsingular projective variety over a number field $K$ as defined in \\emph{\\cite{serre-example}}. Suppose that\n$V_{\\phi}$ is a conjugate variety of $V$ defined by an isomorphism $\\phi$.\nThen there is such an isomorphism $\\phi$ that the complex analytic spaces $V^{an}$ and $V^{an}_{\\phi}$ are not of\nthe same homotopy type.}\n\\end{quotation}\n\nDenote by $A$ and $A_{\\phi}$\nthe homogeneous coordinate rings of $V$ and $V_{\\phi}$, respectively. Then\nwe have projective schemes $$X=Spec{A}; \\, X_{\\phi}=Spec{A_{\\phi}}$$\nwith complex analytical spaces $$X^{an}=V^{an}; \\, X_{\\phi}^{an}=V^{an}_{\\phi}$$ respectively.\n\nThis shows that different affine coverings of the same scheme can produce different complex analytical spaces.\n\nFor this phenomenon,  there are also some more examples arising from abelian varieties and Shimura varieties (\\cite{deligne-milne,milne-suh}).\n\nNow we come to a conclusion that there do exist evidences, such as related examples in \\cite{deligne-milne,sga1,milne-suh,serre-example}, that different affine\ncoverings can produce different complex analytical spaces for a fixed algebraic scheme.\n\nWhy there does exist such a phenomena? It is an interesting problem. Such related topics will be\ndiscussed in our subsequent papers.\n\n\\subsubsection{Several problems}\n\nIn the above it has been seen that different affine coverings of an algebraic scheme can produce different complex analytical space. Then it is natural for one to have several questions (similarly for projective schemes)\nsuch as the following:\n\\begin{itemize}\n\\item    \\emph{How can we give a restrictive definition for such a family of affine schemes to\n    patch a scheme?}\n\n\\item    \\emph{Does there exist another family of affine schemes covering\n    the fixed scheme and making it into a scheme?}\n\n\\item    \\emph{Given another family of affine schemes which cover the fixed scheme.\n    Will we obtain the same scheme?}\n\n\\item    \\emph{Given a ringed space. How many families of affine\n    schemes do patch it? How many schemes do there exist on the same underlying\n    topological space?}\n\n\\item    \\emph{In particular, given an algebraic scheme $X$ over a number field. There can be\n    many families of affine schemes covering $X$. Each such a family produces an analytical\n    space $X^{an}$ of $X$. When are these analytical spaces $X^{an}$ either  diffeomorphic to each other or of\n    the same homotopy types?}\n\\end{itemize}\n\nSeveral questions related to the above will be in part discussed in the paper.\n\nAt the same time, affine structures have also been encountered by us during the discussions on a type of Galois covers of algebraic and arithmetic schemes, where such a scheme is said to be \\emph{Galois closed} if it has only one affine\nstructures.\n\nThe Galois closed schemes have several nice properties with applications to\nclass fields, for example, their Galois groups of rational fields are\nisomorphic to their groups of automorphisms (for instance, see \\cite{an}).\n\n\n\\subsection{Techniques in the Paper}\n\nAs a counterpart, an affine structure on a scheme behaves exactly like a differential\nstructure on a manifold.\n\n\\subsubsection{A smooth manifold can have many differential structures}\n\nIn a classical way, a differential manifold is a\ntopological space covered by a family of open subsets in some Euclidean space,\nwhich is obtained by glueing such a family of open sets as patches. Under some\ntechnical conditions, such a family of open sets is called a \\emph{differential\nstructure} on the manifold.\n\nNowadays there have been many well-known facts about\nmanifolds and their differential structures:\n\\begin{itemize}\n\n\\item    \\emph{There exist differential manifolds which have many\ndifferential structures on them that are not diffeomorphic to each\nother} (\\cite{milnor}).\n\n\\item     \\emph{There exist topological spaces that have no differential structures on\nthem}.\n\n\\item     \\emph{There exist  differential\nmanifolds will be of different properties if we establish different differential\nstructures on the (same) underlying spaces}.\n\\end{itemize}\n\n\\subsubsection{Affine structure v.s. differential structure}\n\nMany approaches and skills in topology can be applied here to schemes. The techniques in the\npresent paper, which we borrowed from differential topology, is thus not new to\nsome certain degree.\n\n\\subsubsection{Known related results on affine coverings}\n\nFor affine coverings, there have been several informal\ndiscussions, for example, see \\cite{hartshorne,shafarevich}, on how to patch a scheme, that is, how to glue a given\nfamily of affine schemes into a scheme; for a more abstract case of categories,\nthere have been fibered categories and groupoids (\\cite{sga3-1,sga1}) which can be\napplied to discuss such coverings.\n\nHowever, all those discussions involved in \\cite{sga3-1,sga1,hartshorne,shafarevich} deal only with \\emph{coverings},\nthat is, a family of abstract objects over a fixed object.\n\n\\subsubsection{Further results obtained in the paper}\n\nIn the paper here we will have further discussions on\nsuch affine coverings and several results will be obtained. We will introduce and discuss affine\nstructures in a rigor and systematic manner, where an affine structure is an affine covering that is taken as maximal families of objects  covering a given object and satisfying the certain properties.\n\nIn fact, the affine structures will afford a platform to\nus to discuss the problems mentioned above \\S 1.1. In particular, an\nalgebraic scheme over $\\mathbb{C}$ can have a unique associative analytical\nspace if there exists only one affine structure on its underlying space.\n\nFurthermore, in a rigor manner, a scheme is a locally ringed space with a\nspecified affine structure on it; it follows that in such a case an algebraic\nscheme over a number field can be associated exactly with a unique complex\nanalytical space.\n\nThe main results obtained in the paper are that by the whole of affine structures on a space,\nit will be seen whether\ntwo spaces are homeomorphic  and whether two schemes are isomorphic. In other words, the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively.\n\nSuch results can be applied to complex analytical spaces of algebraic schemes and arithmetic schemes.\n\n\\subsection{Outline of the Paper}\n\nAt last we give an outline of the paper.\nIn \\S 2 we will use \\emph{pseudogroups $\\Gamma$ of affine transformations} to\ndefine an \\emph{affine $\\Gamma-$atlas} on a topological space, which consists of a\nfamily of \\emph{affine charts}. An \\emph{affine $\\Gamma-$structure} on a space is an affine\n$\\Gamma-$atlas which is maximal. Our discussion can be regarded as an\nalgebraic version of differential structures (\\cite{str,kobayashi}).\n\nIn \\S 3 an affine $\\Gamma-$structure on a space  is said to be\n\\emph{admissible} if there is a sheaf on the space such that they are coincide with\neach other on each affine chart. Here, such a sheaf is called an \\emph{extension} of\nthe given affine structure.\n\nAn affine structure which is not admissible will be\nof no practical use.\n\nGiven a  scheme $(X,\\mathcal{O}_{X})$ in the usual manner (\\cite{ega,hartshorne}). In \\S 4\nwe will discuss the special types of affine structures on the space $X$, called\nthe \\emph{canonical} and the \\emph{relative canonical} affine structures in the scheme\n$(X,\\mathcal{O}_{X})$ respectively. Their extensions are called the \\emph{associate\nschemes} of $(X,\\mathcal{O}_{X})$.\n\nEvery scheme has an associate scheme. In\nparticular, a scheme itself is an associate scheme of it. As schemes, a fixed\nscheme and their associate schemes are isomorphic with each other.\n\nNow put\n\\begin{itemize}\n\\item \\emph{$\\mathbb{A}\\left( X\\right)\\triangleq$  the set of all admissible affine structures\non a topological space $X$;}\n\n\\item \\emph{$ \\mathbb{A}_{0}\\left(\nX,\\mathcal{O}_{X}\\right) \\triangleq$  the set of all the relative\ncanonical affine structures in a scheme} $\\left( X,\\mathcal{O}_{X}\\right)$.\n\\end{itemize}\n\nUsing the set of affine structures on a space, in \\S 5 we will give the statements of the two main\ntheorems in the paper:\n\n\\textbf{Theorem 5.1.} \\emph{Let $X$ and $Y$ be two\ntopological spaces such that either $\\mathbb{A}\\left(\nX\\right)\\not=\\emptyset$ or $\\mathbb{A}\\left( Y\\right)\\not=\\emptyset$\nholds. Then $X$ and $Y$ are homeomorphic if and only if there is $$ \\mathbb{A}\\left(\nX\\right) =\\mathbb{A}\\left( Y\\right).$$}\n\n\\textbf{Theorem 5.2.} \\emph{Any two schemes $\\left(\nX,\\mathcal{O}_{X}\\right) $ and $ \\left( Y, \\mathcal{O}_{Y}\\right) $ are\nisomorphic if and only if we have $$ \\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right)\n\\cong \\mathbb{A}_{0}\\left( Y, \\mathcal{O}_{Y}\\right) .$$}\n\n\nFrom the two main theorems it will be seen that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively. Such results will be applied to complex analytical spaces of algebraic schemes and arithmetic schemes. The two theorems will be proved in \\S 7 and \\S 8, respectively.\n\n\nAs a conclusion, in \\S 6 we will give several concluding remarks. Particularly, we will come to a conclusion that to be\nprecisely defined, a scheme should be a locally ringed space together with a\ngiven admissible affine structure on it if the affine structures are in action in a\nparticular case.\n\n\\bigskip\n\n\\textbf{Acknowledgment}\nThe author would like to express his sincere\ngratitude to Professor Li Banghe for his advice and instructions\non algebraic geometry and topology.\n\n\n\n\n\n\\section{Definitions for Affine Structures}\n\nIn this section we will introduce affine structures on a space in a evident manner as one does for differential structures on a space (for instance, see \\cite{str,kobayashi}).\n\n\\subsection{Pseudogroup of Affine Transformations}\n\nLet $\\mathfrak{Comm}$ be the category of commutative rings with identities,\nand $\\mathfrak{Comm}\/k$ the category of finitely generated algebras over a\nfield $k.$ Here, a pseudogroup (or groupoid) is a small category in which every\nmorphism is invertible (\\cite{maclane}).\n\n\\begin{definition}\nA \\textbf{pseudogroup $\\Gamma$ of affine transformations}, as a subcategory of $\\mathfrak{Comm}$, is a pseudogroup of isomorphisms between commutative rings satisfying the\nconditions 1-5:\n\\begin{enumerate}\n\\item Each $\\sigma\\in\\Gamma$ is an isomorphism from\na ring $dom\\left( \\sigma\\right) $ onto a ring $rang\\left( \\sigma\\right)$\ncontained in $\\Gamma$, called the \\textbf{domain} and\n\\textbf{range} of $\\sigma$, respectively.\n\n\\item If $\\sigma\\in\\Gamma$,  the inverse\n$\\sigma^{-1}$ is contained in $\\Gamma.$\n\n\\item The identity map $id_{A}$ on $A$ is contained in \n\\Gamma$ if there is some \n\\delta\\in\\Gamma$ with $dom\\left( \\delta\\right) =A.$\n\n\\item If $\\sigma\\in\\Gamma$,  the isomorphism\ninduced by $\\sigma$ defined on the localization $dom\\left(\n\\sigma\\right)_{f}$ of the ring $dom\\left( \\sigma\\right)$ at any\n $0\\not= f\\in dom\\left( \\sigma\\right) $ is contained in $\\Gamma.$\n\n\\item Given any $\\sigma,\\delta\\in\\Gamma$. Then\nthe isomorphism factorized by $dom\\left( \\tau\\right) $ from\n$dom\\left( \\sigma\\right) _{f}$ onto $rang\\left( \\delta\\right) _{g}$\nis contained in $\\Gamma$ if for some\n$\\tau\\in\\Gamma$ there are isomorphisms $dom\\left( \\tau\\right) \\cong\ndom\\left( \\sigma\\right) _{f}$ and $dom\\left( \\tau\\right) \\cong\nrang\\left( \\delta\\right) _{g}$ with $0\\not= f\\in dom\\left(\n\\sigma\\right) $ and $0\\not= g\\in rang\\left( \\delta\\right) .$\n\\end{enumerate}\n\nSuch a pseudogroup $\\Gamma$ is said to be a \\textbf{pseudogroup of } $k-$\\textbf{affine transformation} if $\\Gamma$\nis contained in the category $\\mathfrak{Comm}{\/k}$, or equivalently,\nif each isomorphism in $\\Gamma$ is an isomorphism of finitely\ngenerated algebras over a field $k.$\n\\end{definition}\n\n\n\\subsection{Affine Charts and Affine Atlas}\n\nFor a topological space, we give the notions of affine charts and affine atlas.\n\n\\begin{definition}\nLet $X$ be a topological space and  $\\Gamma$ a\npseudogroup of affine transformations. Then an \\textbf{affine }$\\Gamma-$\\textbf{atlas}\n $\\mathcal{A}\\left( X,\\Gamma\\right) $ on $X$ is a collection of pairs $\\left( U_{j},\\varphi_{j}\\right)\n$ with $j\\in\\Delta$, called \\textbf{affine charts}, satisfying the conditions\n1-3:\n\\begin{enumerate}\n\\item {For every pair} $\\left( U_{j},\\varphi\n_{j}\\right) \\in\\mathcal{A}\\left( X,\\Gamma\\right) ,${\n}$U_{j}${ is an open subset of }$X${ and\n}$\\varphi_{j}${ is an homeomorphism of }$U_{j}${ onto\n}$Spec\\left( A_{j}\\right) ,${ where }$A_{j}${ is a\ncommutative ring contained in $\\Gamma$.}\n\n\\item $\\bigcup_{j\\in\\Delta} U_{j}\\supseteq X${ is an open covering\nof }$X.$\n\n\\item {Given any} $\\left( U_{i},\\varphi\n_{i}\\right) ,\\left( U_{j},\\varphi_{j}\\right) \\in\\mathcal{A}\\left(\nX,\\Gamma\\right) $ {with} $U_{i}\\cap U_{j}\\not\n=\\varnothing${. There exists a pair }$\\left(\nW_{ij},\\varphi_{ij}\\right) \\in\\mathcal{A}\\left( X,\\Gamma\\right) $\n{such that }$W_{ij}\\subseteq U_{i}\\cap U_{j}${ and that the\nisomorphism from the localization }$\\left( A_{j}\\right)\n_{f_{j}}${ onto the localization }$\\left( A_{i}\\right)\n_{f_{i}}${ that is induced by the restriction\n}$$\\varphi_{j}\\circ\\varphi_{i}^{-1}\\mid_{W_{ij}}:\\varphi_{i}(W_{ij})\n\\rightarrow\\varphi_{j}(W_{ij})$$\n{is also contained in }$\\Gamma$. {Here} $A_{i}$ and $A_{j}$\n{are commutative rings contained in $\\Gamma$ such that}\n$$\\varphi _{i}\\left( U_{i}\\right) =SpecA_{i}\\,  and \\,\n\\varphi_{j}\\left( U_{j}\\right) =SpecA_{j}$$ {hold and that there\nare homeomorphisms}\n$$\\varphi_{i}\\left( W_{ij}\\right) \\cong Spec\\left( A_{i}\\right)\n_{f_{i}}{ and }\\varphi_{j}\\left( W_{ij}\\right) \\cong Spec\\left(\nA_{j}\\right) _{f_{j}}$${for some} $f_{i}\\in A_{i}$ \\, and \\, \nf_{j}\\in A_{j}.$\n\\end{enumerate}\n\nMoreover, $\\mathcal{A\n\\left( X,\\Gamma\\right) ${ is said to be a $k-$\\textbf{affine }$\\Gamma- $\\textbf{atlas}}{ on }$X$ {if\n}$\\Gamma$ is a subcategory of $\\mathfrak{Comm}\/k.$\n\n{An affine }$\\Gamma-${atlas }$\\mathcal{A}\\left( X,\\Gamma\\right) \n{ on }$X${ is said to be \\textbf{complete} (or\n\\textbf{maximal}) if it can not be contained properly in any other\naffine }$\\Gamma-${atlas of }$X.$\n\\end{definition}\n\n\\begin{remark}\n{The above construction in Definition 2.2 is\nwell-defined since the open covering }$\\{U_{j}\\}$ {such that}\n$\\left( U_{j},\\varphi _{j}\\right) \\in\\mathcal{A}\\left(\nX,\\Gamma\\right) ${ is a base for the topology on }$X.$\n\n{Let }$\\mathcal{A}\\left( X,\\Gamma\\right) $ \nand }$\\mathcal{A}\\left( X,\\Gamma^{\\prime}\\right) $ {be atlases\non a\nspace }$X.$ {Then }$\\Gamma\\supseteq\\Gamma^{\\prime}${ holds if }\n\\mathcal{A}\\left( X,\\Gamma\\right) \\supseteq\\mathcal{A}\\left(\nX,\\Gamma^{\\prime}\\right) .$\n\\end{remark}\n\n\\subsection{Affine Structures}\n\nAs one has differential structures on a manifold, here we have affine structures on a space such as the following.\n\n\\begin{definition}\n{Let }$X${ be a topological space and }$\\Gamma${ a\npseudogroup of affine transformations. Then two affine }$\\Gamma-$\natlases }$\\mathcal{A}$ {and} $\\mathcal{A}^{\\prime}${ on }$X$\n\\ are said to be $\\Gamma-$\\textbf{compatible}} {if the\n condition below is satisfied:}\n\\begin{quotation}\nFor any $\\left( U,\\varphi\\right) \\in\\mathcal{A}$ {and }\n\\left( U^{\\prime},\\varphi^{\\prime}\\right) \\in\\mathcal{A}^{\\prime}$\n{with} $U\\cap U^{\\prime}\\not =\\varnothing$  {there\nexists an affine chart }$\\left( W,\\varphi^{\\prime\\prime}\\right) \\in\\mathcal{\n}\\bigcap\\mathcal{A}^{\\prime}$ {such that }$W\\subseteq U\\cap U^{\\prime}\n{ and that the isomorphism from the localization }$\\left( A\\right)\n_{f}${ onto the localization }$\\left( A^{\\prime}\\right)\n_{f^{\\prime}}${\ninduced by the restriction }$\\varphi^{\\prime}\\circ\\varphi^{-1}\\mid _{W}\n{ is also contained in }$\\Gamma$. {Here }$A$ and $A^{\\prime}$\n{are commutative rings contained in $\\Gamma$ such that}\n$\\varphi\\left( U\\right) =SpecA {\\, and \\, } \\varphi^{\\prime}\\left(\nU^{\\prime}\\right) =SpecA^{\\prime}$ {hold and that there are homeomorphisms}\n$\\varphi\\left( W\\right) \\cong Spec\\left( A\\right)_{f}{\\, and \\,\n}\\varphi^{\\prime}\\left( W\\right) \\cong Spec\\left( A^{\\prime}\\right)\n_{f^{\\prime}}$ {for some} $f\\in A$ {and} $f^{\\prime}\\in\nA^{\\prime}.$\n\\end{quotation}\n\\end{definition}\n\n\\begin{proposition}\n{Let }$X${ be a topological space and let }$\\Gamma${ be a\npseudogroup of affine transformations. Then for any given affine }$\\Gamma-\n{atlas $\\mathcal{A}$ on }$X,${ there is a unique complete affine \n$\\Gamma-${atlas }$\\mathcal{A}_{m}${ on }$X${ such\nthat}\n\\begin{itemize}\n\\item $\\mathcal{A\\subseteq A}_{m};$\n\n\\item $\\mathcal{A}$ {and }$\\mathcal{A}_{m}$ \nare }$\\Gamma-${compatible.}\n\\end{itemize}\n\\end{proposition}\n\n{In such a case, we will say that $\\mathcal{A}$ is a \\textbf{base} for\n}$\\mathcal{A}_{m}$ {and} $\\mathcal{A}_{m}${ is the\n\\textbf{complete affine }$\\Gamma-$\\textbf{atlas}}{ determined\nby $\\mathcal{A}.$}\n\n\n\\begin{proof}\nProve the existence. Let $\\Sigma$ be the collection of affine $\\Gamma-\natlases $\\mathcal{A}_{\\alpha}$ on $X$ such that\n$\\mathcal{A}\\subseteq\\mathcal{A}_{\\alpha}$ and that\n$\\mathcal{A}$ and $\\mathcal{A}_{\\alpha}$ are $\\Gamma-$compatible.\n\nThen $\\Sigma$ is a partially ordered set together with the inclusions of sets\n$\\mathcal\nA}_{\\alpha}\\subseteq\\mathcal{A}_{\\beta}$ for any $\\mathcal{A}_{\\alpha}\n\\mathcal{A}_{\\beta}\\in\\Sigma.$ It is clear that every totally\nordered subset of $\\Sigma$ has a upper bound in $\\Sigma$. By Zorn's\nLemma, $\\Sigma$ has maximal elements.\n\nProve the uniqueness. Let $\\mathcal{A}_{m}$ and $\\mathcal{A}_{m}^{\\prime}$\nbe two maximal elements of $\\Sigma.$ Then we must have $\\mathcal{A}_{m}\n\\mathcal{A}_{m}^{\\prime}.$ Otherwise, hypothesize $\\mathcal{A}_{m}\\not \n\\mathcal{A}_{m}^{\\prime}.$ It is seen that $\\mathcal{A}_{m}$ and $\\mathcal{A\n_{m}^{\\prime}$ are $\\Gamma-$compatible since they are $\\Gamma\n-$compatible respectively\nwith $\\mathcal{A}$; then the union $\\mathcal{A}_{m}\\cup\\mathcal\nA}_{m}^{\\prime}$ is contained in $\\Sigma,$ where we will obtain a\ncontradiction.\n\\end{proof}\n\n\\begin{definition}\n{Let }$X${ be a topological space. An \\textbf{affine }}\n\\Gamma -${\\textbf{structure} on }$X$ {is a complete affine }\n\\Gamma -${atlas }$\\mathcal{A}\\left( \\Gamma \\right) $ {on\n}$X,$ {where $\\Gamma $ is a given pseudogroup of affine\ntransformations.}\n\\end{definition}\n\nLikewise, we define a {$k-$\\textbf{affine\n}$\\Gamma-$\\textbf{structure}} if $\\Gamma\\subseteq\\mathfrak{Comm}\/k.$\n\n\n\n\n\n\\section{Admissible Affine Structures}\n\n\n\nBy Proposition 2.1 it is seen that an affine atlas on a topological space\n$X$ determines a unique affine structure on it. From this view of point, we\nsometimes identify an affine atlas on $X$ with its determined complete\naffine structure on $X.$ In this section will discuss admissible affine structures on a space. On a given space, only admissible affine structures are interesting and are of the practical uses.\n\n\\begin{definition}\n{Let $\\mathcal{A}$}$\\left( \\Gamma \\right) ${ be an affine }\n\\Gamma -${structure on a topological space }$X${. Suppose\nthat there exists a locally ringed space }$\\left(\nX,\\mathcal{F}\\right) $ such that\n$\\varphi _{\\alpha \\ast }\\mathcal{F}\\mid\n_{U_{\\alpha }}\\left( SpecA_{\\alpha }\\right) =A_{\\alpha }$ holds for each $\\left( U_{\\alpha },\\varphi\n_{\\alpha }\\right) \\in ${$\\mathcal{A}$}$\\left( \\Gamma \\right) $,\n{where} $A_{\\alpha }$ {is a commutative ring contained in\n$\\Gamma$ with} $\\varphi _{\\alpha }\\left( U_{\\alpha }\\right)\n=SpecA_{\\alpha }.$\n\n{Then }$\\mathcal{A}\\left( \\Gamma\n\\right) ${ is said to be an \\textbf{admissible affine structure} on }\nX$ {and }$\\left( X,\\mathcal{F}\\right) $ {is said to be an\n\\textbf{extension}} {of the affine }$\\Gamma -${structure }$\\mathcal\nA}\\left( \\Gamma \\right) .$\n\\end{definition}\n\n\\begin{proposition}\n{All extensions of an admissible affine\nstructure on a topological space are schemes which are isomorphic with each other.}\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{A}$ be an admissible affine structure on a topological space $X$.\nIt is evident that each extension of $\\mathcal{A}$ on $X$ is a scheme.\n\nNow fixed any extensions $\\left( X,\\mathcal{F}\\right) $ and $\n\\left( X,\\mathcal{G}\\right) $  of $\\mathcal{A}$ on $X$.\nWe prove $\\mathcal{F}\\cong\\mathcal{G}.$\n\nIn deed, let $U_{\\alpha }$ be an open subset of $X$ contained in\n$\\mathcal{A}$. From the assumption we have $$\\Gamma \\left(\n\\mathcal{F},U_{\\alpha }\\right) =\\Gamma \\left( \\mathcal{G},\nU_{\\alpha }\\right) .$$\n\nTake any open subset $U$ of $X$. We have $$U=\\bigcup_{\\alpha\n}U_{\\alpha }$$ with $U_{\\alpha }\\in \\mathcal{A}$.\nDefine a map $$\\phi:\\Gamma \\left(\n\\mathcal{F},U\\right) =\\Gamma \\left( \\mathcal{G},\nU\\right),\\\\ t\\mapsto \\phi (t)$$\nwhere $\\phi (t)\\in \\Gamma \\left( \\mathcal{G},\nU\\right)$ is the section on $U$ determined by $$t\\mid _{U_{\\alpha}}=\n\\phi (t)\\mid _{U_{\\alpha}}.$$\nThen $\\phi$ is an isomorphism for every open subset $U$ of $X$.\n\nBy $\\phi$\nwe obtain an\nisomorphism $\\mathcal{F}_{x}\\cong \\mathcal{G}_{x}$ at every $x\\in X$ and\nhence $\\mathcal{F}\\cong\\mathcal{G}$ holds.\n\\end{proof}\n\n\\begin{corollary}\nFor affine structures, there are the following statements:\n\\begin{enumerate}\n\\item {Let }$\\left( X,\\mathcal{F}\\right) $ {be an\nextension of the affine }$\\Gamma -${structure }$\\mathcal{A}\\left(\n\\Gamma \\right) $ {on a space }$X.$ {Then we have}\n$$\n\\left( U,\\mathcal{F}\\mid _{U}\\right) \\cong \\left( SpecA,\\widetilde{A\n\\right) \\text{ {and }}\\mathcal{F}\\mid _{U}=\\varphi _{\\ast }^{-1\n\\widetilde{A}\n$$\n{for every affine chart }$\\left( U,\\varphi \\right) \\in ${$\\mathcal\nA}$}$\\left( \\Gamma \\right) $ {with }$\\varphi \\left( U\\right)\n=Spec\\left( A\\right) ,${ where }$A$\n{is a commutative ring contained in }\n\\Gamma .$\n\n\\item {An affine structure }$\\mathcal{A}${\non a space }$X${ is admissible if and only if\n}$\\mathcal{A}${ can be\nextended to be a sheaf }$\\mathcal{F}${ on }$X$ {such that }\n\\left( X,\\mathcal{F}\\right) $ {is a locally ringed space.}\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nIt is immediate from Definition 3.1 and Proposition 3.1.\n\\end{proof}\n\n\n\\section{Canonical Affine Structures}\n\nIn this section it will be seen that for a given scheme there\ncan be many different admissible affine structures on the underlying\ntopological space of the scheme. That is, a given scheme can have many associate schemes. All associate schemes of a given scheme are isomorphic as schemes but have different affine structures.\n\n\\subsection{Canonical Pseudogroups of Affine Transformations}\n\nTo start with, let's  consider an example.\n\n\\begin{example}\n\\textbf{(Different Affine Structures)} {Let} $k$ {be a field.}\n\\begin{enumerate}\n\\item {Put} $$\\Gamma _{1}=\\{\\text{{the\nidentity }}1_{k}:k\\rightarrow k\\};$$\n\\begin{equation*}\n\\begin{array}{c}\n\\Gamma _{2}=\\{\\text{{the identity }}1_{k}:k\\rightarrow k\\}\\bigcup \\{\\text{{a field isomorphism\n}}\\sigma\n:k\\rightarrow k^{\\prime }\\}\\\\\n\n \\bigcup \\{\\text{{the inverse }}\\sigma\n^{-1}:k^{\\prime }\\rightarrow\nk\\}.\\\\\n\\end{array}\n\\end{equation*}\n{Then }$\\Gamma _{1}${ and }$\\Gamma _{2}${ are both\npseudogroups of\\ affine transformations.}\n\n\\item Let\n$$\\mathcal{A}\\left( \\Gamma _{1}\\right) =\\{\\left( U,\\varphi\n\\right) \\};$$ $$ \\mathcal{A}\\left( \\Gamma _{2}\\right)\n=\\{\\left( U,\\varphi \\right) ,\\left( V,\\eta \\right) \\},$$ {where\n}$$U=V=Spec(k),\\text{ }\\varphi \\left( U\\right) =Spec(k),\\text{ and }\\eta \\left( V\\right) =Spec(k^{\\prime }).$$\n{Then }$Spec(k)${ is an extension of the affine structure }\n\\mathcal{A}\\left( \\Gamma _{1}\\right) $.\n{In general, it is not true that\n}$Speck${ is an extension of }$\\mathcal{A}\\left( \\Gamma\n_{2}\\right) .$\n{For example, let }$\\sqrt[3]{2},\\xi ,\\overline{\\xi }$ {be\nthe roots of the equation }$X^{3}-2=0$ {in }$\\mathbb{C}.$\n{Consider }$k=\\mathbb{Q}\\left( \\sqrt[3]{2}\\right); k^{\\prime } =\\mathbb{Q}\\left( \\xi \\right) .$\n\\end{enumerate}\n\\end{example}\n\n\\begin{definition}\n{Let }$\\left( X,\\mathcal{O}_{X}\\right) $ be a scheme.\n{Denote by $\\Gamma_{ 0}$ (respectively, $\\Gamma^{max}$)\nthe union of\nthe set of some (respectively, all) identities of commutative rings}\n$$id_{A_{\\alpha }}:A_{\\alpha }\\rightarrow A_{\\alpha }$$ {and\nthe set of some (respectively, all) isomorphisms of commutative rings}\n$$\\sigma _{\\alpha \\beta }:\\left( A_{\\alpha }\\right) _{f_{\\alpha\n}}\\rightarrow \\left( A_{\\beta }\\right) _{f_{\\beta }},$$\n{satisfying the conditions 1-2:}\n\\begin{enumerate}\n\\item {Each} $ A_{\\alpha },A_{\\beta },A_{\\gamma }\\in Comm$\n{are commutative rings such that there are affine open subsets}\n$$U_{\\alpha },U_{\\beta }, { and \\, }U_{\\gamma }\\subseteq U_{\\alpha\n}\\cap U_{\\beta }$${of} $X$ {satisfying the conditions}\n$$\\varphi _{\\alpha }\\left( U_{\\alpha }\\right) =SpecA_{\\alpha\n}, \\varphi _{\\beta }\\left( U_{\\beta }\\right) =SpecA_{\\beta\n},{ and \\,} \\varphi _{\\gamma }\\left( U_{\\gamma }\\right)\n=SpecA_{\\gamma }.$$\n\n\\item {Each} $\\sigma _{\\alpha \\beta }:\\left( A_{\\alpha\n}\\right) _{f_{\\alpha }}\\rightarrow \\left( A_{\\beta }\\right)\n_{f_{\\beta }}${is induced from the homeomorphism}\n$$\\varphi _{\\alpha }\\circ \\varphi _{\\beta }^{-1}\\mid _{U_{\\gamma}}:\n\\varphi _{\\beta }({U_{\\gamma}})\\rightarrow\\varphi _{\\alpha}({U_{\\gamma}})$$\n{such that}\n$$\n\\varphi _{\\alpha }\\left( U_{\\gamma }\\right) \\cong Spec\\left( A_{\\alpha\n}\\right) _{f_{\\alpha }}\\text{{ and }}\\varphi _{\\beta }\\left( U_{\\gamma\n}\\right) \\cong Spec\\left( A_{\\beta }\\right) _{f_{\\beta }}\n$$\n{hold for some} $f_{\\alpha }\\in A_{\\alpha }$ {and\n}$f_{\\beta }\\in A_{\\beta }.$\n\\end{enumerate}\n\n{Then the pseudogroup generated by\n$\\Gamma_{0}$ in $\\mathfrak{Comm}$, denoted by\n$\\Gamma_{X,\\mathcal{O}_{X}}$, which is the smallest pseudogroup\ncontaining  $\\Gamma_{0}$ in $\\mathfrak{Comm}$, is called \\textbf{a\npseudogroup of affine transformations} in $\\left(\nX,\\mathcal{O}_{X}\\right) $.}\n\n{The pseudogroup generated by\n$\\Gamma^{max}$ in $\\mathfrak{Comm}$, denoted by\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$, is called \\textbf{the maximal\npseudogroup of affine transformations} in $\\left(\nX,\\mathcal{O}_{X}\\right) $.}\n\n{For any given $\\Gamma_{X,\\mathcal{O}_{X}}$, define} $$\\mathcal{A}^{\\ast }\n\\left( \\Gamma_{ X,\\mathcal{O}_{X}}\n\\right) =\\{\\left( U_{\\alpha },\\varphi _{\\alpha }\\right) :\\varphi _{\\alpha\n}\\left( U_{\\alpha }\\right) =SpecA_{\\alpha } \\text{ and }A_{\\alpha }\\in\n\\Gamma_{ X,\\mathcal{O}_{X}} \\}$${where each $U_{\\alpha}$ is an\naffine open subset in the scheme $X$.}\n\\end{definition}\n\n\\begin{definition}\n{Let }$\\left( X,\\mathcal{O}_{X}\\right) $ be a scheme.\n{Given such a pseudogroup $\\Gamma_{\nX,\\mathcal{O}_{X}}$ in $\\left(\nX,\\mathcal{O}_{X}\\right)$. Suppose that $\\mathcal{A}^{\\ast }\\left(\n\\Gamma_{ X,\\mathcal{O}_{X}}\n\\right)$ is an affine $\\Gamma_{ X,\\mathcal{O}_{X}}\n-$atlas on the space $X$.} {Then $\\Gamma_{\nX,\\mathcal{O}_{X}}$ is said to be a \\textbf{canonical pseudogroup of affine transformations}\nin\nthe scheme $\\left( X,\\mathcal{O}_{X}\\right)$ and $\\mathcal{A}^{\\ast }\\left(\n\\Gamma_{ X,\\mathcal{O}_{X}}\n\\right)$ is said to be an \\textbf{affine atlas} in the scheme $(X,\\mathcal{O}_{X})$.}\n\\end{definition}\n\nIt is immediate that $\\Gamma_{X,\\mathcal{O}_{X}}$ is a sub-pseudogroup of $\\Gamma^{max}_{X,\n\\mathcal{O}_{X}}$.\nThere can be many canonical pseudogroups of affine transformations in the scheme\n$(X,\\mathcal{O}_{X})$. By Zorn's Lemma it is seen that\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$ is maximal among these pseudogroups.\n\n\nTake an example. Let $X=Spec(\\mathbb{Z})$ and $Y$ be the disjoint union of $X$.\nThen there are three canonical pseudogroups of affine transformations in the\nscheme $Y$,\nwhich are generated respectively by $\\mathbb{Z}$ and its localisations, by\n$\\mathbb{Z}\\oplus\\mathbb{Z}$\nand its localisations, and by $\\mathbb{Z}$ and $\\mathbb{Z}\\oplus\\mathbb{Z}$\nand their localisations.\n\n\\subsection{Canonical Affine Structures}\n\nIn general, the underlying space of a scheme can have many affine structures on it.\n\n\\begin{definition}\n{Let $\\Gamma$ be a canonical\npseudogroup of affine transformations in a scheme} $\\left(\nX,\\mathcal{O}_{X}\\right) .$\n\n{An affine $\\Gamma-$atlas $\\mathcal{A}$ on the space $X$ is\nsaid to be \\textbf{a canonical affine structure} in the scheme $(X,\\mathcal{O}_{X})$\nif $\\mathcal{A}$\nis the affine $\\Gamma-$structure on $X$ determined by the\naffine $\\Gamma-$atlas $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right)$.}\n\n{An affine $\\Gamma-$atlas $\\mathcal{A}$ on the space $X$\nis said to be \\textbf{a relative canonical affine structure} in the scheme\n$(X,\\mathcal{O}_{X})$ if\n$\\mathcal{A}$ is maximal among all the\naffine $\\Gamma-$atlases in $(X,\\mathcal{O}_{X})$ which contain the\naffine $\\Gamma-$atlas $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right)$ and are\n$\\Gamma-$compatible.}\n\n{A scheme is said to have \\textbf{a unique (}respectively,\n\\textbf{relative}\\textbf{)\ncanonical affine structure} if there exists only one (respectively, relative)\ncanonical affine\nstructure in it.}\n\\end{definition}\n\n\\begin{proposition}\n{Let }$\\Gamma ${ be the maximal pseudogroup of affine\ntransformations in a scheme} $\\left( X,\\mathcal{O}_{X}\\right) .$ {Then }\n\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) ${ is a relative canonical affine $\\Gamma\n-$structure in} $\\left(\nX,\\mathcal{O}_{X}\\right) .$\n\\end{proposition}\n\n\\begin{proof}\nProve $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ is a $\\Gamma -$atlas\nof the space $X.$ In fact, it is clear that  $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ is $\\Gamma -$compatible. Hence, it suffices to prove that\n$\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ affords us a base for the topology on\nthe space $X.$\n\nFixed any point $x\\in X$. Take any affine open subset $U\\ni x$ in $\\left( X\n\\mathcal{O}_{X}\\right) $ such that there is an isomorphism $$\\left( \\varphi ,\n\\widetilde{\\varphi \n\\right) :\\left( U,\\mathcal{O}_{X}\\mid _{U}\\right) \\cong \\left( SpecA\n\\widetilde{A}\\right) $$ where $A\\in \\mathfrak{Comm}.$\n\nThere is some $f\\in A$ such that $A_{f}$ is contained in $\\Gamma$.\nHypothesize that $A_{f}\\not\\in \\Gamma $ holds for any $f\\in A.$ Then\n$\\{\\left( U,\\varphi \\right) \\}$and $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ are\n$\\Gamma -$compatible; it follows that $\\{id_{A}\\}\\bigcup\\Gamma$ is a\npseudogroup of affine transformations in $(X,\\mathcal{O}_{X})$,\nwhere $id_{A}:A\\rightarrow A$ is the identity map; hence, we have\n$$\\Gamma \\subsetneqq \\{id_{A}\\}\\bigcup\\Gamma,$$ which will be in\ncontradiction with the assumption.\n\nNow take any $f \\in A$ such that $A_{f}\\in \\Gamma $. Let $SpecA$ be\nirreducible without loss of generality. We have $$Spec(A_{f})\\cong\nD(f)\\subseteq SpecA.$$ Then $\\{\\left(\nU,\\varphi \\right) \\}$ and $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ are $\\Gamma -\ncompatible.\n\nAs $U$ is an affine open subset in $X$, we have $\\left(\nU,\\varphi \\right)\\in \\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $; as $\\Gamma $ is\nmaximal in $\\left( X,\\mathcal{O}_{X}\\right) ,$ it is seen that $A$ is\ncontained in $\\Gamma .$\n\nThis proves that for any $x\\in X$ there is an affine chart $\\left(\nU,\\varphi \\right) \\in \\mathcal{A}^{\\ast }\\left( \\Gamma \\right)$ such that\n$x\\in U$.\n\\end{proof}\n\n\\begin{remark}\nLet $\\left( X,\\mathcal{O}_{X}\\right) $ be a scheme. Then\n$\\mathcal{A}^{\\ast }\\left( \\Gamma_{X,\\mathcal{O}_{X}} \\right)\n\\subseteqq \\mathcal{A}^{\\ast }\\left( \\Gamma^{m}_{X,\\mathcal{O}_{X}}\n\\right).$\nIn particular, each relative canonical affine\n$\\Gamma_{X,\\mathcal{O}_{X}}-$structure\nin $(X,\\mathcal{O}_{X})$ is contained in $\\mathcal{A}^{\\ast }\\left( \\Gamma^{m}_{X,\n\\mathcal{O}_{X}} \\right)$.\nIn general, there can be different relative canonical affine structures in a scheme.\n\\end{remark}\n\nFurthermore, we have the following conclusions.\n\n\\begin{proposition}\nLet $\\left( X,\\mathcal{O}_{X}\\right)$ be a scheme. There are the following statements.\n\\begin{enumerate}\n\\item {Let $\\Gamma$ be a canonical pseudogroup\nof affine transformations in $\\left( X,\\mathcal{O}_{X}\\right)$. Then there is a\nunique (respectively, relative)\ncanonical affine $\\Gamma-$structure in $\\left( X,\\mathcal{O}_{X}\\right)$.}\n\n\\quad Furthermore, given any affine open subset $U$ in $\\left( X,\\mathcal{O}_{X}\\right)$. Then\n$U$ is contained in the canonical affine $\\Gamma-$structure in $\\left( X,\\mathcal{O}_{X}\\right)$\nif and only if $U$ is contained in the relative canonical affine $\\Gamma-$structure\nin $\\left( X,\\mathcal{O}_{X}\\right)$.\n\n\\item {The scheme $\\left( X,\\mathcal{O}_{X}\\right)$ has a unique affine structure\nif and only if $\\left( X,\\mathcal{O}_{X}\\right)$\nhas a unique relative affine structure.}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n$\\emph{1}$. It is immediate from definition.\n\n$\\emph{2}$. Assume that $\\left( X,\\mathcal{O}_{X}\\right)$ has a unique affine structure.\nHypothesize that $\\mathcal{A}(\\Gamma _{1})$\nand $\\mathcal{A}(\\Gamma _{2})$ are two distinct relative canonical affine structures\nin $\\left( X,\\mathcal{O}_{X}\\right)$ together\nwith the canonical\npseudogroups $\\Gamma_{1}$ and $\\Gamma_{2}$ respectively.\n\nFrom $\\Gamma_{1}$ and $\\Gamma_{2}$ we obtain two canonical affine structures\n$\\mathcal{B}(\\Gamma _{1})$\nand $\\mathcal{B}(\\Gamma _{2})$ in $(X,\\mathcal{O}_{X})$. Then $\\mathcal{B}(\\Gamma _{1})$\nand $\\mathcal{B}(\\Gamma _{2})$ are neither $\\Gamma_{1}-$compatible nor\n$\\Gamma_{2}-$compatible. Otherwise, if they are $\\Gamma_{1}-$compatible, by $(i)$\nit will be seen that\n$\\mathcal{A}(\\Gamma _{1})$\nand $\\mathcal{A}(\\Gamma _{2})$ are $\\Gamma_{1}-$compatible.\n\nHence, there are two distinct canonical affine structures in $\\left( X,\\mathcal{O}_{X}\n\\right)$,\nwhich is in contradiction to the assumption.\n\nConversely, assume that $\\left( X,\\mathcal{O}_{X}\\right)$ has a unique relative affine\nstructure.\nIf $(X,\\mathcal{O}_{X})$ has two distinct canonical affine structures $\\mathcal{B}\n(\\Gamma _{1})$\nand $\\mathcal{B}(\\Gamma _{2})$, by (1) we will obtain two relative\ncanonical affine structures in $\\left( X,\\mathcal{O}_{X}\\right)$ which are neither\n$\\Gamma_{1}-$compatible nor\n$\\Gamma_{2}-$compatible in virtue of the property of a base for the topology on $X$,\nwhere there will be a contradiction.\n\\end{proof}\n\n\\subsection{Associate Schemes}\n\nIn the following it will be seen that a scheme can have many associate schemes.\n\n\\begin{proposition}\nAll (respectively,\nrelative) canonical affine structures in a scheme $\\left( X,\\mathcal{O}_{X}\\right)$ are admissible; moreover,\ntheir extensions are all isomorphic to $\\left( X,\\mathcal{O}_{X}\\right)$ as schemes.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $ be a (respectively, relative)\ncanonical\naffine structure on $X.$ Take any $\\left( U_{\\alpha },\\varphi _{\\alpha\n}\\right) \\in \\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) .$ There is\nthe isomorphism\n\\begin{equation*}\n\\left( \\tau _{\\alpha },\\widetilde{\\tau _{\\alpha }}\\right) :\\left( U_{\\alpha\n},\\mathcal{O}_{X}\\mid _{U_{\\alpha }}\\right) \\cong \\left( SpecA_{\\alpha }\n\\widetilde{A_{\\alpha }}\\right)\n\\end{equation*}\nwhere $$\\varphi _{\\alpha }\\left( U_{\\alpha }\\right) =\\tau _{\\alpha }\\left(\nU_{\\alpha }\\right) ;$$ $$\\widetilde{\\tau _{\\alpha }}\\left( \\widetilde\nA_{\\alpha }}\\right) =\\tau _{\\alpha \\ast }\\mathcal{O}_{X}\\mid _{U_{\\alpha }}.$$\n\nThis proves that the scheme $(X,\\mathcal{O}_{X})$ is at least an extension of\n$\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $.\nIt follows that $\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $ is admissible.\n\nTake any extension $(X,\\mathcal{F})$ of $\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $.\nBy gluing sections, it is seen that $(X,\\mathcal{F})$ and $\\left( X,\\mathcal{O}_{X}\\right)$ are\nisomorphic schemes.\n\\end{proof}\n\n\\begin{definition}\nAn \\textbf{associate\nscheme} of a given scheme $\\left( X,\\mathcal{O}_{X}\\right) $ is an extension on the space $X$ of a\ncanonical affine structure or a relative canonical affine structure in $\\left( X,\\mathcal{O\n_{X}\\right) $.\n\\end{definition}\n\n\\begin{remark} By Proposition 4.3 we have the following statements:\n\n$1.$ Every scheme has an associate scheme.\nIn particular, a scheme  is an associate scheme of itself.\n\n$2.$ All associate schemes of a given scheme are\nisomorphic as schemes.\n\\end{remark}\n\n\n\\section{Statements of the Main Theorems}\n\nIn the paper there are two main theorems that will be stated in the following.\n\n\\subsection{Definitions and Notations}\n\nLet's fix notations and terminology.\n\nFor a topological space $X$,\nput\n\\begin{itemize}\n\\item $\\mathbb{A}\\left( X\\right)\\triangleq$  \\emph{the set of all admissible affine structures\non the space} $X$.\n\\end{itemize}\nFor a scheme $\\left( X,\\mathcal{O}_{X}\\right)$, set\n\\begin{itemize}\n\\item $ \\mathbb{A}_{0}\\left(\nX,\\mathcal{O}_{X}\\right) \\triangleq$  \\emph{the set of all the relative\ncanonical affine structures in the scheme} $\\left( X,\\mathcal{O}_{X}\\right)$.\n\\end{itemize}\n\nLikewise, we can define $ \\mathbb{A}\\left( X;k\\right)$ and $\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X};k\\right) $ for $k-$affine\nstructures.\n\n\n\\begin{definition}\n{For two spaces $X$ and $Y$, we say\n$$\n\\mathbb{A}\\left( X\\right) \\subseteq\\mathbb{A}\\left( Y\\right)\n$$\nif the below condition is satisfied:}\n\\begin{quotation}\nGiven any affine chart $\\left( U_{\\alpha},\\varphi_{\\alpha}\\right) $\ncontained in an affine structure\n$\\mathcal{A}\\left( X\\right) $ belonging to $\\mathbb{A}\\left( X\\right)$. There\nis an affine chart $\\left( V_{\\alpha},\\psi_{\\alpha}\\right) $ contained in an\naffine structure $\\mathcal{A}\\left( Y\\right) $ belonging to $\\mathbb{A\n\\left( Y\\right) $ such that $B_{\\alpha}=A_{\\alpha}$. Here\n$\nA_{\\alpha},B_{\\alpha}\\in\\mathfrak{Comm}$, $\\varphi_{a}\\left( U_{\\alpha}\\right)\n=SpecA_{\\alpha}$, and $ \\psi_{\\alpha\n}\\left( V_{\\alpha}\\right) =SpecB_{\\alpha}.$\n\\end{quotation}\n\\end{definition}\n\n\\begin{definition}\nFor two spaces $X$ and $Y$, we say\n$$\n\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)\n$$\nif there are relations\n\\begin{equation*}\n\\mathbb{A}\\left( X\\right) \\subseteq\\mathbb{A}\\left( Y\\right) \\text{ and \n\\mathbb{A}\\left( X\\right) \\supseteq\\mathbb{A}\\left( Y\\right).\n\\end{equation*}\n\\end{definition}\n\nLikewise, replacing admissible by relative canonical, we define $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) =\\mathbb{A}_{0}\\left( Y,\n\\mathcal{O}_{Y}\\right) $$\nfor two schemes $\\left( X,\\mathcal{O}_{X}\\right) $ and $\\left( Y,\\mathcal{O\n_{Y}\\right) .$\n\n\\begin{definition}\nFor two spaces $X$ and $Y$, we say $$ \\mathbb{A}\\left( X\\right)\n\\trianglelefteq\\mathbb{A}\\left( Y\\right) $$  if the below conditions 1-3 are\nsatisfied:\n\\begin{enumerate}\n\\item \\textbf{(Local Isomorphism)} Given any affine chart $\\left( U_{\\alpha\n},\\varphi_{\\alpha}\\right) $ contained in an affine structure $\\mathcal{A}\n\\left( X\\right) $ belonging to $\\mathbb{A}\\left( X\\right)$.\n\n\\quad Then there is an affine chart\n$\\left( V_{\\alpha},\\psi_{\\alpha}\\right) $ contained in an\naffine structure $\\mathcal{A}\\left( Y\\right) $ belonging to $\\mathbb{A}\n\\left( Y\\right) $ such that $A_{\\alpha}$ and $B_{\\alpha}$ are isomorphic\nrings. Here $ A_{\\alpha},B_{\\alpha}\\in\\mathfrak{Comm}$,\n$\\varphi_{a}\\left( U_{\\alpha}\\right) =SpecA_{\\alpha}$, and $\\psi_{\\alpha\n}\\left( V_{\\alpha}\\right) =SpecB_{\\alpha}. $\n\n\\item \\textbf{(Covering)} Let $\\{\\left( U_{\\alpha },\\varphi_{\\alpha}\\right)\n\\}_{\\alpha\\in\\Gamma}$ be a family of affine charts $\\left( U_{\\alpha\n},\\varphi_{\\alpha}\\right) $ contained in some affine structures\n$\\mathcal{A}(\\Gamma_{\\alpha})$ belonging to $\\mathbb{A}\\left( X\\right)$\nsuch that $\\varphi_{a}\\left( U_{\\alpha}\\right)$ $ =SpecA_{\\alpha}$ and\n$\\bigcup_{\\alpha\\in\\Gamma}U_{\\alpha }\\supseteq X$.\n\n\\quad Then\n$\\bigcup_{i,\\alpha}V_{i,\\alpha }\\supseteq Y$ holds, where $V_{i,\\alpha }$\nruns through all the affine charts $(V_{i,\\alpha },\\psi_{i,\\alpha})$ contained in\nany affine structures $\\mathcal{A}(\\Gamma_{i,\\alpha})$ belonging to $\\mathbb{A\n\\left( Y\\right) $ such that $B_{i,\\alpha }\\cong A_{\\alpha}$ and\n$\\psi_{i,\\alpha}(B_{i,\\alpha })=SpecB_{i,\\alpha }$.\n\n\\item \\textbf{(Filtering)} Let $\\left( U_{\\alpha },\\varphi_{\\alpha}\\right)$ and\n$(U_{\\beta},\\varphi_{\\beta})$ be two affine charts contained in some affine\nstructures $\\mathcal{A}(\\Gamma_{\\alpha})$ and\n$\\mathcal{A}(\\Gamma_{\\beta})$ belonging to $\\mathbb{A}\\left( X\\right)$\nrespectively. Given any $x_{\\alpha}\\in SpecA_{\\alpha}$ and $x_{\\beta}\\in\nSpecA_{\\beta}$ with\n$\\varphi^{-1}_{\\alpha}(x_{\\alpha})=\\varphi^{-1}_{\\beta}(x_{\\beta})$, where\n$\\varphi_{\\alpha}\\left( U_{\\alpha}\\right) =SpecA_{\\alpha}$ and\n$\\varphi_{\\beta}\\left( U_{\\beta}\\right) =SpecA_{\\beta}$.\n\n\\quad Then there exist\naffine charts $\\left( V_{\\alpha },\\psi_{\\alpha}\\right)$ and\n$(V_{\\beta},\\psi_{\\beta})$ respectively contained in some affine structures\n$\\mathcal{A}^{\\prime}(\\Gamma^{\\prime}_{\\alpha})$ and\n$\\mathcal{A}^{\\prime}(\\Gamma^{\\prime}_{\\beta})$ belonging to\n$\\mathbb{A}\\left( Y\\right)$ such that\n$\\psi^{-1}_{\\alpha}\\circ\\sigma_{\\alpha}(x_{\\alpha})=\\psi^{-1}_{\\beta}\\circ\n\\sigma_{\\beta}(x_{\\beta})$\nholds and that there are ring isomorphisms $\\delta\n_{\\alpha}:B_{\\alpha}\\cong A_{\\alpha}\\text{ and }\\delta_{\\beta\n}:B_{\\beta}\\cong A_{\\beta},$\n where $\\psi_{\\alpha}(V_{\\alpha})=SpecB_{\\alpha}$,  $\\psi_{\\beta}(V_{\\beta})=SpecB_{\\beta}$,\n and $\\sigma\n_{\\alpha}:SpecA_{\\alpha}\\rightarrow SpecB_{\\alpha}\\text{ and }\\sigma_{\\beta\n}:SpecA_{\\beta}\\rightarrow SpecB_{\\beta}$ are the isomorphisms induced\nfrom $\\delta _{\\alpha}$ and $\\delta_{\\beta }$, respectively.\n\\end{enumerate}\n\\end{definition}\n\n\n\nLikewise, replacing admissible by relative canonical, we define $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\unlhd\\mathbb{A}_{0}\\left( Y\n\\mathcal{O}_{Y}\\right) $$\nfor two schemes $\\left( X,\\mathcal{O}_{X}\\right) $ and $\\left( Y,\\mathcal{O\n_{Y}\\right) .$\n\n\nSuch an isomorphism $\\delta_{\\alpha}:A_{\\alpha}\\cong B_{\\alpha}$ is\ncalled a \\emph{deck transformation} from $X$ into $Y.$\n\n\\begin{definition}\nFor two spaces $X$ and $Y$, we say\n$$\n\\mathbb{A}\\left( X\\right) \\cong\\mathbb{A}\\left( Y\\right)\n$$\nif there are relations $$\\mathbb{A}\\left( X\\right) \\trianglelefteq\\mathbb{A}\\left(\nY\\right) \\text{ and }\\mathbb{A}\\left( X\\right) \\trianglerighteq\\mathbb{A}\\left(\nY\\right) .$$\n\\end{definition}\n\nLikewise, replacing admissible by relative canonical, we define $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y\n\\mathcal{O}_{Y}\\right) $$\nfor two schemes $\\left( X,\\mathcal{O}_{X}\\right) $ and $\\left( Y,\\mathcal{O\n_{Y}\\right) .$\n\n\n\\subsection{Statements of the Main Theorems}\n\nNow we give the statements of the two main theorems of the present paper.\n\n\\begin{theorem}\nLet $X$ and $Y$ be two topological spaces such\nthat either $\\mathbb{A}\\left( X\\right)\\not=\\emptyset$ or $\\mathbb{A}\\left(\nY\\right)\\not=\\emptyset$ holds. Then $X$ and $Y$ are homeomorphic if and\nonly if there is $$ \\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right).$$\n\\end{theorem}\n\n\\begin{theorem}\nAny two schemes $\\left( X,\\mathcal{O}_{X}\\right)\n$ and $ \\left( Y, \\mathcal{O}_{Y}\\right) $ are isomorphic if and only if we have $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y,\n\\mathcal{O}_{Y}\\right) .$$\n\\end{theorem}\n\nWe will prove Theorems 5.1 and 5.2 in \\S 7 and \\S 8, respectively.\n\n\\begin{remark}\nFrom the two main theorems above it is seen that the whole of affine structures on a space and the underlying space of a scheme, as local data of the space, encode the global data of the space and the scheme, in particular, the global topology of the space and the scheme, respectively.\n\\end{remark}\n\n\\begin{remark}\nIn Theorem 5.2 the condition $$ \\mathbb{A}_{0}\\left(\nX,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y,\n\\mathcal{O}_{Y}\\right)$$ can not be replaced by $$ \\mathbb{A}\\left(\nX,\\mathcal{O}_{X}\\right) = \\mathbb{A}\\left( Y, \\mathcal{O}_{Y}\\right).$$\nFor example, consider $X=Spec({\\mathbb{Q}})$ and\n$Y=Spec({\\mathbb{Q}}(\\sqrt{2}))$.\n\\end{remark}\n\n\n\n\\section{Concluding Remarks}\n\nFrom Remark 4.2 and Theorem 5.2 we have the following proposition, a comparison between two schemes of the same underlying space.\n\n\\begin{proposition}\n{Let $\\left( X,\\mathcal{O}_{X}\\right) $ and $ \\left(\nX,\\mathcal{O}^{\\prime}_{X}\\right)$ be two schemes. The\nfollowing statements are equivalent.}\n\\begin{itemize}\n\\item $\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong\n\\mathbb{A}_{0}\\left(X, \\mathcal{O}^{\\prime}_{X}\\right) $ holds.\n\n\\item $\\left( X,\\mathcal{O}_{X}\\right) $ and $ \\left(\nX,\\mathcal{O}^{\\prime}_{X}\\right)$ are isomorphic schemes.\n\n\\item There is an isomorphism $ \\left(\nX,\\mathcal{O}_{X}^{\\symbol{94}}\\right) \\cong \\left(\nX,\\mathcal{O}_{X}^{\\prime \\symbol{94}}\\right)$ for any associate\nschemes $\\left( X,\\mathcal{O}_{X}^{\\symbol{94}}\\right)$ of $\\left(\nX,\\mathcal{O}_{X}\\right) $ and $\\left( X,\\mathcal{O}_{X}^{\\prime\n\\symbol{94}}\\right) $ of and $\\left( X,\\mathcal{O}_{X}^{\\prime\n}\\right) $.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{example}\n{Let $K\/k$ be a Galois extension. Then $SpecK$ has a unique\nassociate scheme and there exists a unique admissible $k-$affine\nstructure in the scheme $SpecK$.}\n\\end{example}\n\n\\begin{definition}\n{A scheme $\\left( X,\\mathcal{O}_{X}\\right)$ is said to\n\\textbf{have a property} $P$ \\textbf{for an admissible affine\nstructures} $\\mathcal{A}$ on $X$ if as a scheme any extension\n$\\left( X,\\mathcal{O}_{ \\mathcal{A}\\left( \\Gamma \\right) }\\right)$\nof $\\mathcal{A}$ has that property $P$.}\n\\end{definition}\n\n\\begin{remark}\nLet $\\left( X,\\mathcal{O}_{\\mathcal{X}}\\right) $ be a scheme. There are the following conclusions.\n\\begin{itemize}\n\\item Fixed an associate scheme\n$(X,\\mathcal{O}_{\\mathcal{A}})$ of\n$(X,\\mathcal{O}_{X})$. In general, it is not true that $\\left( X,\\mathcal{O}_{\\mathcal{A\n}\\right)=(X,\\mathcal{O}_{X}) $ although they are isomorphic.\n\n\\item {There can be a scheme }$\\left( X,\\mathcal{O}_{X}\\right) $\n and an admissible affine structure $\\mathcal{A}$ on\nthe space $X$ such that there is some property $P$\nthat $\\left(\nX,\\mathcal{O}_{X}\\right) $ holds but an extension $\\left( X,\\mathcal{O}_{\\mathcal{A}}\\right) $ of $\\mathcal{A}$ does not hold.\n\n\\item One says that a scheme $\\left( X,\\mathcal{O}\n_{X}\\right) $ has a property $P$. But it is not specified that the\nproperty $P$ holds for some certain or all the admissible affine\nstructures on the space $X$.\n\n\\quad This situation is very similar to that in differential\ntopology. As usual, a differential manifold is said to have some\nproperty if the property holds for all the differential structures\nuntil such a structure is especially specified.\n\n\\quad It has been known that\nthere is some property $P$ on some manifold $X$ which does not hold for any\nother differential structures on $X$.\n\\end{itemize}\n\\end{remark}\n\n\\begin{remark}\nIn a precise and rigour manner, a scheme is defined to be a ringed space\ntogether with a specified admissible affine structure on it if the affine structures\nare in action in a particular case.\n\\end{remark}\n\n\\begin{remark}\n{Let $\\left( X,\\mathcal{O}_{X}\\right) $ be a\nscheme. For any $\\mathcal{A},\\mathcal{B}\\in \\mathbb{A\n\\left( X\\right) $, we say $$\\mathcal{A}\\sim \\mathcal{B}$$ if and only\nif there is an isomorphism $\\left(\nX,\\mathcal{O}_{\\mathcal{A}}\\right) \\cong \\left(\nX,\\mathcal{O}_{\\mathcal{B}}\\right) .$}\n\n{Then the quotient set $$\\mathbb{A\n\\left( X\\right) \/\\sim $$ is the whole of the schemes on the space $X$\nupon isomorphisms.}\n\\end{remark}\n\n\n\n\n\n\\section{Proof of the First Main Theorem}\n\n\nIn this section we give the proof of Theorems 5.1.\n\n\\begin{proof}\n\\textbf{(Proof of Theorem 5.1)}\nAs $\\mathbb{A}\\left( X\\right)\\not=\\emptyset$ and $\\mathbb{A}\\left(\nY\\right)\\not=\\emptyset$, we can choose two admissible affine structures\non $X$ and $Y$ respectively. Fixed their extensions $\\left(\nX,\\mathcal{O}_{X}\\right) $ and $\\left( Y, \\mathcal{O}_{Y}\\right)$, which are\nschemes.\n\n\n\n$\\implies$. Prove that $\\mathbb{A}\\left( X\\right) \\subseteq\n\\mathbb{A}\\left( Y\\right)$ holds.\n\nIn deed, take any affine chart $\\left( U_{\\alpha },\\varphi _{\\alpha }\\right)\n$ contained in an affine structure $\\mathcal{A}\\left( X\\right) $\nbelonging to $\\mathbb{A}\\left( X\\right) ,$ where $\\varphi _{a}\\left(\nU_{\\alpha }\\right) =SpecA_{\\alpha }$ and $A_{\\alpha }\\in\n\\mathfrak{Comm}.$\n\nThen $\\left( \\tau \\left( U\\right) ,\\varphi _{\\alpha }\\circ\n\\tau ^{-1}\\right) $ is an affine chart contained in an affine structure $$\n\\mathcal{A}\\left( \\tau \\left( X\\right) \\right) =\\mathcal{A}\\left( Y\\right) $$\nbelonging to $\\mathbb{A}\\left( Y\\right) .$ Hence, we have\n$$\\mathbb{A}\\left( X\\right) \\subseteq \\mathbb{A}\\left( Y\\right) .$$\n\nSimilarly, we have $$\\mathbb{A}\\left( X\\right) \\supseteq \\mathbb{A}\\left(\nY\\right).$$ So $$\\mathbb{ A}\\left( X\\right) = \\mathbb{A}\\left( Y\\right).$$\n\n\n$\\impliedby$. Let $\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)\n$. We will prove that there exists a homeomorphism\n$$\\tau:X\\longrightarrow Y.$$ We will proceed in several steps.\n\n\n\n$(i)$ Take an affine chart $\\left( U_{\\alpha},\\varphi_{\\alpha}\\right) $ contained\nin an affine structure\n$\\mathcal{A}\\left( X\\right) $ belonging to $\\mathbb{A}\\left( X\\right)$, where\n$\nA_{\\alpha}\\in\\mathfrak{Comm}$ is a commutative ring and $$\\varphi_{a}\\left( U_{\\alpha}\n\\right) =SpecA_{\\alpha}\n.$$ Then there\nis an affine chart $\\left( V_{\\alpha},\\psi_{\\alpha}\\right) $ contained in an\naffine structure $\\mathcal{A}\\left( Y\\right) $ belonging to $\\mathbb{A\n\\left( Y\\right) $ such that $$ \\psi_{\\alpha\n}\\left( V_{\\alpha}\\right) =SpecA_{\\alpha}.$$\n\nThe converse is true since we have $$\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)\n.$$\n\nLet $\\Sigma$ be the disjoint union of all such open sets $SpecA_{\\alpha}$. Take any\npoints $x,y\\in\\Sigma$.\n\nWe say $$x\\thicksim_{X} y$$ if there exist admissible affine\nstructures $\\mathcal{A}(\\Gamma_{\\alpha})$ and $\\mathcal{A}(\\Gamma_{\\beta})$\ncontained in $\\mathbb{A}\\left( X\\right)$ satisfying the condition:\n\\begin{quotation}\n\\emph{There are affine charts $(U_{\\alpha},\\varphi _{\\alpha })\\in\n\\mathcal{A}(\\Gamma_{\\alpha}),(U_{\\beta},\\varphi _{\\beta })\\in\n\\mathcal{A}(\\Gamma_{\\beta})$ such that $\\varphi _{\\alpha}^{-1}(x)=\n\\varphi _{\\beta }^{-1}(y)$, where $x\\in SpecA_{\\alpha}=\n\\varphi_{\\alpha}(U_{\\alpha})$ and $y\\in\nSpecA_{\\beta}=\\varphi_{\\beta}(U_{\\beta}).$}\n\\end{quotation}\n\nLikewise, we say $$x\\thicksim_{Y} y$$ if there exist admissible affine\nstructures $\\mathcal{A}(\\Gamma_{\\alpha})$ and $\\mathcal{A}(\\Gamma_{\\beta})$\ncontained in $\\mathbb{A}\\left( Y\\right)$ satisfying the condition:\n\\begin{quotation}\n\\emph{There are affine charts $(V_{\\alpha},\\psi _{\\alpha })\\in\n\\mathcal{A}(\\Gamma_{\\alpha}),(V_{\\beta},\\psi_{\\beta })\\in\n\\mathcal{A}(\\Gamma_{\\beta})$ such that $\\psi _{\\alpha}^{-1}(x)= \\psi\n_{\\beta }^{-1}(y)$, where $x\\in SpecA_{\\alpha}=\n\\psi_{\\alpha}(V_{\\alpha})$ and $y\\in\nSpecA_{\\beta}=\\psi_{\\beta}(V_{\\beta}).$}\n\\end{quotation}\n\n\n\n$(ii)$ Let $\\Sigma_{X}$ be the quotient of the set $\\Sigma$ by relation $\\thicksim_{X}$,\nand let\n$$\\pi_{X}:\\Sigma\\longrightarrow \\Sigma_{X}$$ be the canonical map.\n\nProve that there is a bijection $\\rho_{X}$ from the set $\\Sigma_{X}$ onto the set $X$.\n\nIn deed, we have a mapping\n$$\\rho:\\Sigma \\longrightarrow X$$ given by $$z\\longmapsto \\varphi_{\\alpha}^{-1}(z)$$ where\n$(U_{\\alpha},\\varphi\n_{\\alpha })$ is the affine chart contained in an affine structure belonging to\n$\\mathbb{A}(X)$ such that\n$$z\\in SpecA_{\\alpha}=\\varphi_{\\alpha }(U_{\\alpha}).$$ Then we have a map$$\\rho_{X}:\\Sigma_{X}\n\\longrightarrow X,\n\\pi_{X}(z)\\longmapsto \\rho (z).$$\n\nEvidently, $\\rho_{X}$ is a surjection. From the definition\nfor $\\thicksim_{X}$, it is easily seen\nthat $\\rho_{X}$ is an injection. This proves $\\rho_{X}$ is a bijection.\n\nHence, $\\Sigma_{X}$ is a topological space together with the topology on the space $X$.\n\nSimilarly, let $\\Sigma_{Y}$ be the quotient of the set $\\Sigma$ by relation $\\thicksim_{Y}$,\nand let\n$$\\pi_{Y}:\\Sigma\\longrightarrow \\Sigma_{Y}$$ be the canonical map.\nAs $$\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right),$$\nit is clear that there is a bijection $\\rho_{Y}$ from the set $\\Sigma_{Y}$ onto the set $Y$.\nThen\n$\\Sigma_{Y}$ is a topological space together with the topology on the space $Y$.\n\n\n\n$(iii)$ Take any $x,y\\in\\Sigma$. Prove that $$x\\thicksim_{X} y$$ holds if and only if\n$$x\\thicksim_{Y} y$$ holds.\n\nIn deed, let $x\\thicksim_{X} y$, that is, we have $$\\varphi\n_{\\alpha}^{-1}(x)= \\varphi _{\\beta }^{-1}(y)$$ for some affine charts\n$$(U_{\\alpha},\\varphi _{\\alpha })\\in\n\\mathcal{A}(\\Gamma_{\\alpha})\\text{ and }(U_{\\beta},\\varphi _{\\beta })\\in\n\\mathcal{A}(\\Gamma_{\\beta})$$ such that $$x\\in\nSpecA_{\\alpha}=\\varphi_{\\alpha}(U_{\\alpha})\\text{ and }y\\in\nSpecA_{\\beta}=\\varphi_{\\beta}(U_{\\beta}).$$\n\nLet $\\Gamma^{max}_{X,\\mathcal{O}_{X}}$be the maximal\npseudogroup of affine transformations in the scheme $\\left(X,\\mathcal{O}_{X}\\right) $.\n\nWe choose the above open sets\n$U_{\\alpha}$ and $U_{\\beta}$ to be affine open subsets of the scheme $\\left(X,\n\\mathcal{O}_{X}\\right) $. That is, $U_{\\alpha}$ and $U_{\\beta}$\nare contained in the pseudogroup $\\Gamma^{max}_{X,\\mathcal{O}_{X}}$.\n\nThen there is an affine open subset $U_{\\alpha \\beta}$ of $\\left(X,\n\\mathcal{O}_{X}\\right) $ contained in\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$\nsuch that $$\\varphi _{\\alpha}^{-1}(x)\\in U_{\\alpha \\beta} \\subseteq U_{\\alpha}\n\\bigcap U_{\\beta};$$\n $$\\varphi_{\\alpha}(U_{\\alpha \\beta})=Spec(A_{\\alpha})_{f_{\\alpha}};$$\n $$\\varphi_{\\beta}\n(U_{\\alpha \\beta})=Spec(A_{\\beta})_{f_{\\beta}}$$\nfor some $f_{\\alpha}\\in A_{\\alpha}$ and $f_{\\beta} \\in A_{\\beta}$, where the isomorphism\n$\\sigma_{\\alpha \\beta}$ from\n$(A_{\\alpha})_{f_{\\alpha}}$ onto $(A_{\\beta})_{f_{\\beta}}$ is contained in\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$.\n\nAs $$\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right),$$ we have affine charts\n$(V_{\\alpha},\\psi_{\\alpha})$\nand $(V_{\\beta},\\psi_{\\beta})$ respectively contained in some affine structures\nbelonging to $\\mathbb{A}(Y)$, where\n$$V_{\\alpha}=\\psi_{\\alpha}^{-1}(SpecA_{\\alpha})\\text{ and\n}V_{\\beta}=\\psi_{\\beta}^{-1}(SpecA_{\\beta}).$$\n\nSet $$V_{\\alpha \\beta}=\\psi_{\\alpha}^{-1}(Spec(A_{\\alpha})_{f_{\\alpha}});$$\n$$V_{\\beta \\alpha}=\\psi_{\\beta}^{-1}(Spec(A_{\\beta})_{f_{\\beta}}).$$\nDenote by $\\psi_{\\beta \\alpha}$ the homeomorphism of $V_{\\beta \\alpha}$ onto\n$V_{\\alpha \\beta}$ which is induced\nfrom $\\sigma_{\\alpha \\beta}$.\n\nIt is easily seen that $(V_{\\alpha \\beta},\\psi_{\\beta}\\circ\\psi_{\\beta \\alpha}^{-1})$\nis an affine chart contained in\nsome admissible affine structure belonging to $\\mathbb{A}(Y)$. In fact, fixed any\nadmissible affine $\\Gamma_{0}-$structure $\\mathcal{A}(\\Gamma_{0})$ on the space $Y$ which\ncontains the affine chart $(V_{\\alpha},\\psi_{\\alpha})$. Let $\\Gamma_{1}$ be the pseudogroup\nof affine transformations\nin $\\frak{comm}$ generated by the union of $\\Gamma_{0}$ and the set of the identity on\n$(A_{\\beta})_{f_{\\beta}}$ and all\nthe possible isomorphisms between the localisations of the rings. Then\n$\\{(V_{\\alpha \\beta},\\psi_{\\beta}\\circ\\psi_{\\beta \\alpha}^{-1})\\}$\nand $\\mathcal{A}(\\Gamma_{1})$ are $\\Gamma_{1}-$compatible. Hence,\n$$(V_{\\alpha \\beta},\\psi_{\\beta}\\circ\\psi_{\\beta \\alpha}^{-1})\\in \\mathcal{A}(\\Gamma_{1}).$$\n\nConsider $$y\\in V_{\\beta\n\\alpha};$$ $$x\\in Spec(A_{\\alpha})_{f_{\\alpha}}\\subseteq SpecA_{\\alpha}.$$ It is evident that $$\\psi_{\\alpha}^{-1}(x)=\n\\psi_{\\beta \\alpha}\\circ \\psi_{\\beta}^{-1}(y)$$ holds since $$\\psi_{\\alpha}^{-1}\n(Spec(A_{\\alpha})_{f_{\\alpha}})\n=\\psi_{\\beta \\alpha}\\circ \\psi_{\\beta}^{-1}(Spec(A_{\\beta})_{f_{\\beta}})=V_{\\alpha \\beta}.$$\nThis proves $$x\\thicksim_{Y}y$$ holds.\n\nIn a similar manner, it is seen that the converse is true.\n\n\n\n$(iv)$ The map from $\\Sigma_{X}$ into $\\Sigma_{Y}$ defined by $$\\pi_{X}(z)\\longmapsto\n\\pi_{Y}(z)$$ for $z\\in \\Sigma$\ngives us a bijection $$\\tau:X\\longrightarrow Y,$$ which is well-defined from $(iii)$.\n\nAll the open sets $SpecA_{\\alpha}$ determine a topology on the set\n$\\Sigma$ in such a manner:\n\\begin{quotation}\n\\emph{A subset $W$ of $\\Sigma$ is open if and only if $\\pi_{X}(W)$ is open\nin $\\Sigma_{X}$.}\n\\end{quotation}\n\nIt follows that $\\Sigma_{X}$ is the quotient space of $\\Sigma$ by $\\pi_{X}$.\nAs $\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)$, $\\Sigma_{Y}$ is\nthe quotient space of $\\Sigma$ by $\\pi_{Y}$. Hence, $$\\tau:X\\longrightarrow\nY$$ is a homeomorphism.\n\nThis completes the proof.\n\\end{proof}\n\n\n\n\\section{Proof of the Second Main Theorem}\n\n\n\nIn this section we give the proof of Theorem 5.2.\n\n\\begin{proof}\n\\textbf{(Proof of Theorem 5.2)}\n$\\implies$. Let $$\\tau:\\left( X,\\mathcal{O}_{X}\\right) \\cong \\left( Y,\n\\mathcal{O}_{Y}\\right)$$ be an isomorphism.\n\nAs $$\\tau _{\\ast }\\mathcal{O}_{X}\\cong \\mathcal{O}_{Y},$$ we have\n\\begin{equation*}\n\\begin{array}{l}\n\\left( \\varphi _{\\alpha }^{-1}\\left( SpecA_{\\alpha }\\right) ,\\left(\n\\varphi _{\\alpha }^{-1}\\right) _{\\ast }\\widetilde{A_{\\alpha }}\\right)\\\\\n\n\\cong\\left( U_{\\alpha },\\mathcal{O}_{X}\\mid _{U_{\\alpha }}\\right) \\\\\n\n\\cong \\left( \\tau \\left( U_{\\alpha }\\right) ,\\tau _{\\ast }\\mathcal{O}\n_{X}\\mid _{U_{\\alpha }}\\right) \\\\\n\n\\cong \\left( \\tau \\left( U_{\\alpha }\\right) ,\\mathcal{O}_{Y}\\mid\n_{\\tau \\left( U_{\\alpha }\\right) }\\right) \\\\\n\n\\cong\\left( \\psi _{\\alpha }^{-1}\\left( SpecB_{\\alpha }\\right) ,\\left( \\psi\n_{\\alpha }^{-1}\\right) _{\\ast }\\widetilde{B_{\\alpha }}\\right)\n\\end{array}\n\\end{equation*}\nfor any affine open set $U_{\\alpha}$ of $X$ such that $$\\varphi _{\\alpha }\n\\left( U_{\\alpha }\\right) =SpecA_{\\alpha }$$ and $\\left( U_{\\alpha },\\varphi\n_{\\alpha }\\right) $ is contained in some canonical affine structure\nbelonging\nto $\\mathbb\nA}_{0}\\left( X,\\mathcal{O}_{X}\\right)$.\n\nThen $\\left( \\tau \\left( U_{\\alpha }\\right) ,\\varphi _{\\alpha }\\right) $ is an affine\nchart with $$\\psi _{\\alpha }\\left( \\tau \\left( U_{\\alpha }\\right) \\right)\n=SpecB_{\\alpha },$$ which is contained in some canonical affine structure\nbelonging to $\\mathbb{A}_{0}\\left( Y,\\mathcal{O}_{Y}\\right) $.\n\nBy the\nisomorphism $\\tau$ it is easily seen that the conditions $\\emph{1}$-$\\emph{2}$ in\nDefinition 5.3 are satisfied.\n\nNow let $\\Gamma_{X}$ and $\\Gamma_{Y}$ be the maximal pseudogroups\nof affine transformations in the schemes $\\left( X,\\mathcal{O}_{X}\\right)$\nand $\\left( Y,\\mathcal{O}_{Y}\\right)$, respectively. Via the isomorphism\n$\\tau$, every affine chart in $\\mathcal{A}^{*}(\\Gamma_{X})$ is an affine chart\nin the $\\mathcal{A}^{*}(\\Gamma_{Y})$; the converse is true.\n\nUsing the same\nprocedure in proving Theorem 5.1, we can prove such a claim that\n\\textquotedblleft $x\\thicksim _{X}y$\\textquotedblright\\ for $X$ holds if and\nonly if \\textquotedblleft $x\\thicksim _{Y}y$\\textquotedblright\\ for $Y$\nholds.\n\nIt follows that the condition $\\emph{3}$ in Definition 5.3 is satisfied. Hence, we\nhave\n$$\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\trianglelefteq \\mathbb{A\n_{0}\\left( Y,\\mathcal{O}_{Y}\\right).$$\n\nSimilarly, we prove $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\trianglerighteq \\mathbb{A\n_{0}\\left( Y,\\mathcal{O}_{Y}\\right).$$ This proves $$\\mathbb{A}_{0}\n\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y\n\\mathcal{O}_{Y}\\right).$$\n\n\n\n$\\impliedby$. Put $$\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A\n_{0}\\left( Y,\\mathcal{O}_{Y}\\right) .$$ We prove that there exists an\nisomorphism from $\\left( X,\\mathcal{O}_{X}\\right) $ onto $\\left(\nY,\\mathcal{O}_{Y}\\right) $.\n\nIn the following we will proceed in several steps in a manner similar to the procedure in proving\nTheorem 5.1.\n\n\n\n$(i)$ Let $\\Gamma_{Y}$ be the maximal pseudogroup of affine\ntransformations in the scheme $\\left(Y,\\mathcal{O}_{Y}\\right) $. We obtain a\nrelative canonical affine $\\Gamma_{Y}-$structure $$\\mathcal{A}\\left(\n\\Gamma_{Y}\\right)\\triangleq\\mathcal{A}^{*}\\left( \\Gamma_{Y}\\right) $$ on\n$Y$. Then $V_{\\alpha}$ is an affine open set in the scheme\n$(Y,\\mathcal{O}_{Y})$ for every $(V_{\\alpha},\\psi_{\\alpha})$ contained in $\\mathcal{A}\\left(\n\\Gamma_{Y}\\right)$.\n\nFor each $(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) $\nwe put\n$$\n\\psi_{\\alpha}(V_{\\alpha})=SpecB_{\\alpha}\n$$\nwhere $B_{\\alpha}$ is a commutative ring contained in the pseudogroup\n$\\Gamma_{Y}$.\n\nFrom Definition 4.1 we have $$\\mathcal{A}\\left( \\Gamma_{Y}\\right) \\supseteq\n\\mathcal{A}^{*}\\left( \\Gamma_{Y}\\right);$$ then\n$$\\bigcup_{(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) }\nV_{\\alpha} \\supseteq Y.$$\n\nLet $\\Sigma$ be the disjoint union of all the open sets $SpecB_{\\alpha}$\nsuch that $$\\psi_{\\alpha}(V_{\\alpha})=SpecB_{\\alpha}\\text{ and\n}(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) .$$ Take\nany points $x,y\\in\\Sigma$.\n\nWe say $$x\\thicksim_{Y} y$$ if there are affine charts $(V_{\\alpha},\\psi _{\\alpha\n}),(V_{\\beta},\\psi_{\\beta })\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) $ such that\n$$\\psi _{\\alpha}^{-1}(x)= \\psi _{\\beta }^{-1}(y),$$ where $$x\\in SpecB_{\\alpha}=\n\\psi_{\\alpha}(V_{\\alpha});$$ $$y\\in SpecB_{\\beta}=\\psi_{\\beta}(V_{\\beta}).$$\n\n\n\n$(ii)$ For each $(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left(\n\\Gamma_{Y}\\right) $, define\n$$\\{(U_{i,\\alpha},\\varphi_{i,\\alpha})\\}_{i\\in I_{\\alpha}}$$ to be the\nset of all the affine charts contained in each relative canonical\naffine structures in the scheme $(X,\\mathcal{O}_{X})$ such that\n$$\\varphi_{i,\\alpha}(U_{i,\\alpha})=SpecA_{i,\\alpha}$$ and that there\nis an isomorphism $$\\delta_{i,\\alpha}: A_{i,\\alpha}\\cong B_{\\alpha}.$$\n\nDenote by $\\Delta_{X}$ the set of all such affine charts\n$(U_{i,\\alpha},\\varphi_{i,\\alpha})$, where $i\\in I_{\\alpha}$ and\n$(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) $.\n\nAs $$\\mathbb{A}_{0}\\left( Y,\\mathcal{O}_{Y}\\right) \\unlhd\\mathbb{A}_{0}\\left(\nX, \\mathcal{O}_{X}\\right) ,$$ we have\n$$\\bigcup_{(U_{i,\\alpha},\\varphi_{i,\\alpha})\\in \\Delta_{X}} U_{i,\\alpha} \\supseteq X.$$\n\nLet $\\Sigma^{*}$ be the disjoint union of all the open sets\n$SpecA_{i,\\alpha}$ such that $$A_{i,\\alpha}\\cong B_{\\alpha}\\text{\nand }(U_{i,\\alpha},\\varphi_{i,\\alpha})\\in \\Delta_{X}.$$ Take any\n$x,y\\in \\Sigma^{*}$.\n\nWe say $$x\\thicksim_{X} y$$ if there are affine charts\n$(U_{i,\\alpha},\\varphi _{i,\\alpha }),(U_{j,\\beta},\\varphi _{j,\\beta\n})\\in \\Delta_{X}$ such that $$\\varphi _{i,\\alpha}^{-1}(x)= \\varphi\n_{j,\\beta }^{-1}(y)$$ holds, where $$x\\in SpecA_{i,\\alpha}=\n\\varphi_{i,\\alpha}(U_{i,\\alpha});$$ $$y\\in\nSpecA_{j,\\beta}=\\varphi_{j,\\beta}(U_{j,\\beta}).$$\n\nWe say $$x\\thicksim_{\\Sigma} y$$ if there are affine charts\n$(U_{i,\\alpha},\\varphi _{i,\\alpha }),(U_{j,\\beta},\\varphi _{j,\\beta })\\in\n\\Delta_{X}$ such that $$\\sigma _{i,\\alpha}^{-1}(x)=\\sigma_{j,\\beta\n}^{-1}(y)$$ holds, where $$x\\in SpecA_{i,\\alpha}= \\varphi_{i,\\alpha}(U_{i,\\alpha}), \\\\\ny\\in SpecA_{j,\\beta}=\\varphi_{j,\\beta}(U_{j,\\beta}),$$ and\n$$\\sigma\n_{i,\\alpha}:SpecB_{\\alpha}\\rightarrow SpecA_{i,\\alpha}\\text{ and\n}\\sigma_{j,\\beta }:SpecB_{\\beta}\\rightarrow SpecA_{j,\\beta}$$ are the scheme\nisomorphisms induced from the ring isomorphisms $$\\delta\n_{i,\\alpha}:A_{i,\\alpha}\\cong B_{\\alpha}\\text{ and }\\delta_{j,\\beta\n}:A_{j,\\beta}\\cong B_{\\beta}$$ respectively.\n\n\n\n$(iii)$ Let $\\Sigma^{*}_{X}$ be the quotient of the set $\\Sigma^{*}$ by\n$\\thicksim_{X}$, and let\n$$\\pi_{X}:\\Sigma^{*}\\longrightarrow \\Sigma^{*}_{X}$$ be the canonical map.\nWe have got schemes $\\Sigma^{*}$  and $\\Sigma^{*}_{X}$ in an evident\nmanner. It is clear that $\\pi_{X}$ is a morphism of the schemes.\n\nProve that there is an isomorphism $\\rho_{X}$ from the scheme\n$\\Sigma^{*}_{X}$ onto the scheme $X$.\n\nIn deed, we have a mapping $$\\rho:\\Sigma^{*} \\longrightarrow X$$ given by\n$$z\\longmapsto \\varphi_{i,\\alpha}^{-1}(z),$$ where $(U_{i,\\alpha},\\varphi\n_{i,\\alpha })\\in\\Delta_{X}$ such that $$z\\in SpecA_{i,\\alpha}=\\varphi_{i,\\alpha\n}(U_{i,\\alpha}).$$ Then we have a mapping $$\\rho_{X}:\\Sigma^{*}_{X} \\longrightarrow X,\n\\pi_{X}(z)\\longmapsto \\rho (z).$$\n\nEvidently, $\\rho_{X}$ is a surjection. From the definition for $\\thicksim_{X}$,\nit is  seen that $\\rho_{X}$ is an injection. Hence, $\\rho_{X}$ is a\nhomeomorphism from the space $\\Sigma^{*}_{X}$ onto the space $X$. By\nthe construction, it is seen that $\\rho_{X}$ is an isomorphism of the\nschemes.\n\nSimilarly, let $\\Sigma_{Y}$ be the quotient of the set $\\Sigma$ by\n$\\thicksim_{Y}$, and let\n$$\\pi_{Y}:\\Sigma\\longrightarrow \\Sigma_{Y}$$ be the canonical map.\n\nThen $\\Sigma$ and $\\Sigma_{Y}$ are schemes, and $\\pi_{Y}$ is a scheme\nmorphism. There is an isomorphism $\\rho_{Y}$ from the scheme\n$\\Sigma_{Y}$ onto the scheme $Y$.\n\nLet $\\Sigma^{*}_{\\Sigma}$ be the quotient of the set $\\Sigma^{*}$ by\n$\\thicksim_{\\Sigma}$, and let\n$$\\pi_{\\Sigma}:\\Sigma^{*}\\longrightarrow \\Sigma^{*}_{\\Sigma}$$ be the canonical map.\n\nThen $\\Sigma^{*}_{\\Sigma}$ is a scheme and $\\pi_{\\Sigma}$ is a morphism.\nThere is an isomorphism $\\rho_{\\Sigma}$ from the scheme\n$\\Sigma^{*}_{\\Sigma}$ onto the scheme $\\Sigma$.\n\n\n\n$(iv)$ Take any $x,y\\in \\Sigma^{*}$. We prove\n$$\\rho_{X}\\circ\\pi_{X}(x)=\\rho_{X}\\circ\\pi_{X}(y)$$ if and only if\n$$\\rho_{Y}\\circ\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(x)=\\rho_{Y}\\circ\n\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(y).$$\n\nIn deed, let $$\\rho_{X}\\circ\\pi_{X}(x)=\\rho_{X}\\circ\\pi_{X}(y).$$ We have\n$$\\varphi _{i,\\alpha}^{-1}(x)= \\varphi _{j,\\beta }^{-1}(y)$$ for some affine charts\n$$(U_{i,\\alpha},\\varphi _{i,\\alpha }), (U_{j,\\beta},\\varphi _{j,\\beta })\\in\n\\Delta_{X}$$ such that $$x\\in\nSpecA_{i,\\alpha}=\\varphi_{i,\\alpha}(U_{i,\\alpha});$$ $$y\\in\nSpecA_{j,\\beta}=\\varphi_{j,\\beta}(U_{j,\\beta}).$$\n\nAs $$\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\unlhd\\mathbb{A}_{0}\\left(\nY, \\mathcal{O}_{Y}\\right) ,$$ we have affine charts $(V_{i,\\alpha},\\psi _{i,\\alpha\n})$ and $ (V_{j,\\beta},\\psi_{j,\\beta })$ contained in\n$\\mathcal{A}(\\Gamma_{Y})$ such that\n$$\\psi^{-1}_{i,\\alpha}\\circ\\sigma\n_{i,\\alpha}^{-1}(x)=\\psi^{-1}_{j,\\beta }\\circ\\sigma_{j,\\beta }^{-1}(y)$$\nwhere\n$$\\psi _{i,\\alpha }(V_{i,\\alpha})=SpecB_{i,\\alpha},\\, \\psi_{j,\\beta\n}(V_{j,\\beta})=SpecB_{j,\\beta}$$ and\n$$\\sigma\n_{i,\\alpha}:SpecB_{i,\\alpha}\\rightarrow SpecA_{i,\\alpha}\\text{ and\n}\\sigma_{j,\\beta }:SpecB_{j,\\beta}\\rightarrow SpecA_{j,\\beta}$$\nare the isomorphisms induced from\n the ring isomorphisms\n $$\\delta\n_{i,\\alpha}:A_{i,\\alpha}\\cong B_{i,\\alpha}\\text{ and }\\delta_{j,\\beta\n}:A_{j,\\beta}\\cong B_{j,\\beta}$$ respectively.\n\nHence, we have\n$$\\rho_{Y}\\circ\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(x)=\\rho_{Y}\\circ\\pi_{Y}\n\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(y).$$\n\nIn a similar manner, it is seen  that the converse is true.\n\n\n\n$(v)$ Define a map $\\tau:X\\longrightarrow Y$ by\n$$\\tau(\\rho_{X}\\circ\\pi_{X}(z))=\\rho_{Y}\\circ\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(z)$$\nfor every $z\\in \\Sigma^{*}$.\n\nBy $(iv)$ it is seen that $\\tau$ is well-defined. It is immediate that $\\tau$ is the\ndesired scheme isomorphism from $\\left( X,\\mathcal{O}_{X}\\right) $ onto $\n\\left( Y,\\mathcal{O}_{Y}\\right)$.\n\nThis completes the proof.\n\\end{proof}\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\nInference of the evolutionary histories of species, commonly represented by a species tree, is complicated by the divergent evolutionary history of different parts of the genome. Different loci on the genome can have different histories from the underlying species tree (and each other) due to processes such as incomplete lineage sorting (ILS), gene duplication and loss, and horizontal gene transfer. The multispecies coalescent is a commonly used model for performing inference on species and gene trees in the presence of ILS. This paper introduces Lily-T and Lily-Q, two new methods for species tree inference under the multispecies coalescent. We then compare them to two frequently used methods, SVDQuartets and ASTRAL, using simulated and empirical data. Both methods generally showed improvement over SVDQuartets, and Lily-Q was superior to Lily-T for most simulation settings. The comparison to ASTRAL was more mixed -- Lily-Q tended to be better than ASTRAL when the length of recombination-free loci was short, when the coalescent population parameter $\\theta$ was small, or when the internal branch lengths were longer.\n\n\\section{Introduction}\nThe phylogenetic inference problem is concerned with using data, including but not limited to DNA sequences, to understand the evolutionary history of a collection of species. Consider the collection of mammals shown in figure \\ref{exampleTree}. We are concerned with three aspects shown in the figure. First, is the unlabeled topology correct? In other words, for each node, how many descendants are there on the left and right branches? Second, is the labeled topology correct, or does it need to be permuted? Third, when do the speciation events occur?\n\n\\begin{figure}\n            \\centering\n        \\includegraphics[width=0.8\\linewidth]{\"BigTree\".png}\n    \\caption{Example phylogenetic tree featuring six mammalian species.}\n    \\label{exampleTree}\n\\end{figure}\n\nWe usually begin with the assumption that when a speciation event occurs each ancestral species divides into exactly two daughter species. Thus, the evolutionary history can be represented as a binary tree known as a species tree.\n\\begin{definition}\nA \\textit{species tree} is an acyclic graph $S=(V(S), E(S), \\boldsymbol{\\tau}_S)$ where $V(S)$ is the vertex set of $S$, $E(S)$ is the edge set of $S$, and $\\boldsymbol{\\tau}_S$ is a set of branch lengths. \n\\end{definition}\n\\label{Species tree}\nBiologically, internal nodes represent speciation events while the leaves represent extant species. We call this leaf set $L_S$. The leaves will be represented by lower case letters $a, b, c,$ etc., and internal nodes by letters later in the alphabet. Internal branches represent ancestral species. If we know the common ancestor of all the species under consideration, then the tree is \\textit{rooted}, and the tree becomes a directed graph from the root outward. If the root is unknown, then we only know the direction of the branches that connect to external nodes. The \\textit{degree} of a node is the number of other nodes a node is connected to. For a species tree, the leaves are of degree one, and due to the binary tree assumption, all internal nodes are of degree three (except the root, if known, which is of degree two).\n\nA common and useful assumption, known as the \\textit{molecular clock}, is that the mutation rate is constant over time (or more precisely, if $\\lambda$ represents the mutation rate there is a common $\\lambda(t)$ for all branches at time $t$). In that case, the rooted tree is \\textit{ultrametric}, meaning that each leaf will be equidistant from the root. For ultrametric trees, an equivalent parameterization to the set of branch lengths $\\boldsymbol{\\tau}_S$ is the set of node times $\\boldsymbol{\\tau}_{S_{nodes}}$. Because we assume the molecular clock, we will simplify the notation so that $\\boldsymbol{\\tau}$ represents the node times for the remainder of the paper.\n\nLet $\\mathcal{S}^{(n)}$ refer to the set of all possible trees with $n$ taxa. The superscript is in parentheses to highlight that it refers to the number of taxa rather than as an exponent. Then, for example, $\\mathcal{S}^{(4)}_1$ and $\\mathcal{S}^{(4)}_2$ can indicate the first and second 4-taxon species trees from figure \\ref{quartets}. If the labeling is unimportant, we can also use $\\mathcal{S}^{(4)}_{u1}$ and $\\mathcal{S}^{(4)}_{u2}$ to indicate the two \\textit{unlabeled} 4-taxon topologies. \n\n    \\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{\"rootedQ\".png}\n    \\caption{The fifteen rooted quartets. We can label these $\\mathcal{S}^{(4)}_1, \\mathcal{S}^{(4)}_2 \\ldots \\mathcal{S}^{(4)}_{15}$. If we are unconcerned with the labels, the first column is $\\mathcal{S}^{(4)}_{u1}$ and the other columns $\\mathcal{S}^{(4)}_{u2}$. Note that each row corresponds to one of the three unrooted quartets.}\n    \\label{quartets}\n\\end{figure} \n\nSpecies tree inference is complicated by the possibility for divergence between the evolutionary history of species and individual elements of their genomes. Causes for this divergence include incomplete lineage sorting (ILS), gene duplication and loss (GDL) and horizontal gene transfer (HGT). ILS is commonly modeled by the coalescent process \\cite{kingman1982}. The history of individual loci on the genome is represented by a gene tree.\n\n\\begin{definition}\nA \\textit{gene tree} is a network $G=(V(G), E(G), \\boldsymbol{t}_G)$ where $V(G)$ is the vertex set of $G$, $E(G)$ is the edge set of $G$, and $\\boldsymbol{t}_G$ is a set of branch lengths. When the molecular clock is assumed, $\\boldsymbol{t}_G$ can equivalently represent node times as with the species tree above and the notation simplified to $\\boldsymbol{t}$. These node times are subject to the constraint that the coalescent events in question must occur prior to the divergence time of the species in question.\n\\end{definition}\n\nThe gene tree is embedded within the species tree (see figure \\ref{coalescent}) and usually both trees have the same leaf set. Exceptions can occur if the gene has not been sampled for all species under consideration or if there are multiple sampled individuals per species. Thus, unless otherwise noted, we will drop the subscript and just refer to the set of species under consideration as $L$. When we need to distinguish the leaves of $G$ from the leaves of $S$ we use capital letters $A, B, C, \\ldots$. Care should be taken, however, to distinguish when $A$ or $C$ are used to denote members of the leaf set and when they are used as abbreviations for nucleotides. Internal nodes represent coalescent events, which identify the most recent common ancestor of two gene lineages. \n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[h]{.48\\linewidth}\n   \\centering\n\\includegraphics[width=.9\\linewidth]{\"sym2\".png}\n\\caption{Symmetric species tree}\n\\end{subfigure}\n\\begin{subfigure}[h]{.44\\linewidth}\n\\centering\n\\includegraphics[width=.9\\linewidth]{\"asym2\".png}\n\\caption{Asymmetric species tree}\n\\end{subfigure}\n\\caption{Species trees with four taxa under the coalescent process. The green lines show example gene trees evolving within the underlying species tree.}\n\\label{coalescent}\n\\end{figure}\n\nIt is worth noting here that our definition of a gene, as is common in the phylogenetic literature, refers to a recombination-free region of the genome. Thus, the common assumption is that no recombination occurs within a gene, and all sites in a gene share a common evolutionary history, while sites on separate genes are independent conditional only on the underlying species tree. This differs from the biological definition of a gene as a segment of DNA coding for a polypeptide. There is, of course, no underlying reason why a biological gene can or should share a common history in its entirety. To avoid this confusion, \\textit{locus} is also sometimes used to describe a recombination-free region, however, it is still more common to refer to ``gene trees\" rather than ``locus trees\".\n\nA number of different approaches have been taken with regard to species tree inference in the presence of ILS. The first is essentially to ignore the problem: perform gene tree inference on concatenated data using methods such as RAxML \\cite{raml} or FastTree \\cite{FastTree}, treating all sites as if they share a single, common evolutionary history. This can be fast and accurate for estimating $S$. But, there are some concerns: concatenation has been shown to be statistically inconsistent for some values of $(S,\\boldsymbol{\\tau})$ \\cite{degnan05,roch15}, and speciation time estimates are biased since the coalescent event must naturally occur before the speciation time. Another approach is the use of summary statistics that first estimate the gene trees independently for each gene, and then use the gene tree estimates as inputs for species tree estimates. Examples of this approach include STEM \\cite{kubatko09}, ASTRAL \\cite{mirarab14,mirarab15,zhang18}, and MP-EST \\cite{MPEST}. These methods can be computationally efficient, however they depend on the accuracy of the gene tree estimates that are used as inputs as well as proper delineation of recombination-free segments of the genome \\cite{gatesy14,springer16}. A third approach uses the full data to coestimate the species tree and each of the gene trees, generally using Markov chain Monte Carlo (MCMC) methods. Examples of this approach include BEST \\cite{BEST}, *BEAST \\cite{BEAST,SB2},  and BPP \\cite{BPP}. These methods can be quite accurate but are very computationally intensive when the number of loci and\/or leaf set is large. Assessment of convergence is also a challenge, especially due to the multi-modal nature of the likelihood in the tree space \\cite{salter01}.\n\nA fourth approach, and the one we take in this paper, is to treat the gene trees as a nuisance parameter that can be integrated over. A previous example of this approach is SVDQuartets \\cite{chifman14}, which uses a rank-based methodology to infer the proper unrooted species tree for each quartet of species under consideration, and then uses an assembly algorithm to infer the final n-taxon species tree estimate taking the set of unrooted quartet trees as input. The theory behind the method assumes unlinked coalescent-independent sites (CIS) data such that the gene tree underlying each site can be treated as a random draw from the distribution of all possible gene trees given the species tree, however Wascher and Kubatko (2020)\\nocite{wascher20} recently proved that SVDQuartets is statistically consistent for multilocus data as well.\n\nOur method, \\textbf{(Li)}kelihood-based assemb\\textbf{(ly)} (\\textbf{Lily}), also assumes unlinked CIS data. From \\cite{chifman15} we have the site pattern probabilities given a species tree topology and branching times. Then, a prior on these topologies and branching times is assumed, and posterior probabilities for each set of rooted triplets (Lily-T) or unrooted quartets (Lily-Q) are calculated. These posterior probabilities are then used as weights in an assembly algorithm to infer the final n-taxon estimated topology $\\hat{S}$. The details of the procedures are described in the next section.\n\n\\section{Method}\nThe outline for the Lily-T and Lily-Q procedures are laid out in algorithms \\ref{tProd} and \\ref{Qprod}. For each triplet (Lily-T) or quartet (Lily-Q) of species, first the site pattern frequencies are found. Then, the likelihood for each rooted topology is calculated. Using Bayes's Theorem, the posterior probability of each rooted triplet or unrooted quartet is then determined. Finally, given these posteriors as inputs, the n-taxon topology is estimated using supertree assembly methods. Each step is discussed in detail in the following sections.\n\\begin{algorithm}\n\\SetAlgoLined\n  \\For{$l\\gets1$ \\KwTo ${n \\choose 3}$}{\n    Find site pattern frequencies $\\boldsymbol{D}_{JC}$ for the $l$\\textsuperscript{th} triplet of species (section \\ref{dataSec})\\;\n    Find the site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau)}$ for each of the three rooted triplet topologies from \\cite{chifman15} (section \\ref{method1} and appendix \\ref{tripletApp})\\;\n    Integrate over $\\boldsymbol{\\tau}$ using $\\theta=0.003$ and $\\beta$ as estimated from equation \\ref{distEst} to find $\\boldsymbol{\\delta}|S$ and then $L(S|\\boldsymbol{D}_{JC})$ for each rooted triplet (sections \\ref{derivation} and \\ref{robustness})\\;\n    From Bayes's Theorem find the posterior probability of each of the three rooted triplets (section \\ref{derivation})\n    }\n    Using the $3{n \\choose 3}$ posterior probabilities as input, estimate the n-taxon rooted topology using the Triplet MaxCut algorithm \\cite{sevillya16} (section \\ref{assembly})\\;\n \\caption{Lily-T procedure}\n \\label{tProd}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\SetAlgoLined\n  \\For{$l\\gets1$ \\KwTo ${n \\choose 4}$}{\n    Find site pattern frequencies $\\boldsymbol{D}_{JC}$ for the $l$\\textsuperscript{th} quartet of species (section \\ref{dataSec})\\;\n    Find the site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau)}$ for each of the fifteen rooted quartet topologies from \\cite{chifman15} (section \\ref{method1})\\;\n    Integrate over $\\boldsymbol{\\tau}$ using $\\theta=0.003$ and $\\beta$ as estimated from equation \\ref{distEst} to find $\\boldsymbol{\\delta}|S$ and then $L(S|\\boldsymbol{D}_{JC})$ for each rooted quartet (sections \\ref{derivation} and \\ref{robustness})\\;\n    From Bayes's Theorem find the posterior probability of each of the fifteen rooted quartets (section \\ref{derivation}) \\;\n    Calculate the posterior probability of the three unrooted quartet topologies as the sum of the corresponding rooted quartets (see figure \\ref{rooting})\\;\n    }\n    Given the $3{n \\choose 4}$ posterior probabilities as input, estimate the n-taxon rooted topology using the Weighted QMC algorithm \\cite{avni14} (section \\ref{assembly})\\;\n \\caption{Lily-Q procedure}\n \\label{Qprod}\n\\end{algorithm}\n\n\\subsection{Data structure}\n\\label{dataSec}\nOur data structure and data reduction method for Lily-Q are summarized in figure \\ref{data_map}. We begin by assuming a matrix of aligned sequence data $\\textbf{D}_{raw}$ where the rows represent the species under consideration and the columns each represent an aligned site. We assume in the sequel that this alignment has been performed without error. Even with a correct alignment, there may also be sites present in one sequence and not another, either due to the data truly being missing or because of an insertion or deletion. Thus, $D_{ij} \\in \\{A,C,G,T,-\\}$.  Here, $i$ is an index for the taxon, $j$ is the index for the site, the letters represent the four nucleotides, and the dash represents missing data.\n\nIf the data are iid, or if the possibility of varying rates across sites is treated as a random effect that can be integrated over, then the columns are exchangeable. We begin with one subset of four of the $n$ taxa. Let $\\delta_{TCCG}$ represent the probability that at a certain site the first species has A, the next two have C, and the fourth species has G, i.e., $\\delta_{TCCG}=P(i_A=T, i_B=C, i_C=C, i_D=G)$ where $i_x$ is the state for species $x$. As discussed before, the abbreviation C is overloaded so care should be taken to note that $i_C=C$ means the third species has cytosine at this site. Under this assumption of exchangeability, $\\delta_{TCCG}$ will be the same at every site. Then, the number of sites where the first taxon has character $i_A$, the second has character $i_B$, etc., which we label $d_{i_A, i_B, \\ldots i_n}$ will follow a binomial distribution with probability $\\delta_{i_A,i_B, \\ldots i_n}$, and the joint probability of all possible site patterns is a multinomial distribution. Since there are $4^n$ possibilities (or $5^n$ with missing data), the numbers of sites that follow each pattern $d_{i_A, i_B, \\ldots i_n}$ is a sufficient statistic. Then we can map down $\\boldsymbol{D}_{raw}$ down to a $4^n \\times 1$ vector $\\boldsymbol{D}_{ind} \\sim Multinom(J,\\boldsymbol{\\delta}_{ind})$, where each element of $\\boldsymbol{\\delta}$ represents one of the site pattern probabilities and $J$ is the total number of sites. In the sequel, we will only consider sites where all four species in the quartet have a nucleotide present -- i.e., sites where $D_{ij} \\in \\{A,C,G,T\\}$, $i=A,B,C,D$, which will not affect inference if sites are missing at random. \n\nIf we further assume the Jukes-Cantor (JC69) substitution model \\cite{jukes69}, all nucleotides have the same limiting frequency of 1\/4 and all substitution rates between nucleotides are the same. As a result, for determining probabilities, if we use the JC69 model we don't need to keep track of what nucleotides are present where; we only need to note whether or not they are the same. For example with two taxa, $P(i_A=C, i_B=C)=P(i_A=G, i_B=G)$, as is the probability of any two different nucleotides at the same site. So, we can call two identical nucleotides at the same site an XX pattern regardless of whether they represent AA, CC, GG, or TT. Similarly, XY represents the case where the two nucleotides are different. Again, the distribution of the site pattern frequencies follows a multinomial distribution, but we need to keep track of fewer cases. The number of different cases required for $n$ taxa is $\\sum_{i=1}^4 S_{n,i}$ where $S_{n,i}$ is the Sterling number of the second kind. Most relevant for our later discussion, for a four-taxon tree, we can map $\\boldsymbol{D}_{ind}$ down to a $15 \\times 1$ vector $\\boldsymbol{D}_{JC}$. We repeat this mapping for each quartet of species, creating an ${n \\choose 4}$ set of site pattern frequency vectors $\\boldsymbol{D}_{JC}$.\n\nThis data mapping process is similar for Lily-T, except that it is repeated of each of the ${n \\choose 3}$ set of triplets. $\\boldsymbol{D}_{ind}$ then is a $4^3 \\times 1$ vector and  $\\boldsymbol{D}_{JC}$ is a $5 \\times 1$ vector.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{\"DataSetup\".png}\n    \\caption{Data reduction (Lily-Q): For each set of four species, the raw aligned sequence data can be reduced down to $\\boldsymbol{D}_{ind}$ and then $\\boldsymbol{D}_{JC}$ under the Jukes-Cantor assumptions.}\n    \\label{data_map}\n\\end{figure}\n\n\\subsection{Derivation of $L((S,\\boldsymbol{\\tau)|\\boldsymbol{D}_{JC}})$ for 3- or 4-taxon trees}\n\\label{method1}\nChifman and Kubatko (2015)\\nocite{chifman15} derived the site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau})$ for a 4-taxon tree where $\\delta_k|(S,\\boldsymbol{\\tau})$  is the probability of the $k$\\textsuperscript{th} site pattern, $k \\in \\{XXXX, XXXY, \\ldots XYZW\\}$, occurring at a given site given the species tree topology and branching times under the JC69 model and the molecular clock. Then since $\\boldsymbol{D}_{JC} \\sim Multinom(J, \\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}))$ the likelihood is given by \n\n\\begin{equation}\n    L((S,\\boldsymbol{\\tau})|\\boldsymbol{D}_{JC}) \\propto \\prod_{k=1}^{K} [\\delta_k|(S,\\boldsymbol{\\tau})]^{d_k} \n    \\label{lik1}\n\\end{equation}\n\nTo find $P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(S,\\boldsymbol{\\tau}))$ we first recognize that this can be factored into two processes: the coalescent process and the substitution process:\n$$ P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(S,\\boldsymbol{\\tau}))=P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(G,\\boldsymbol{t}),(S, \\boldsymbol{\\tau}))f((G,\\boldsymbol{t})|(S, \\boldsymbol{\\tau})),  $$\n$$=P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(G,\\boldsymbol{t}))f((G,\\boldsymbol{t})|(S, \\boldsymbol{\\tau})).  $$\nwhere the second equality is true under the assumption of neutral selection, whereby the substitution process and coalescent process are independent and then the first term depends directly only on the gene tree \\cite{whidden15}.\n\nWe begin by noting that:\n$$\\delta_{XXXY}|((a,(b,(c,d))),\\boldsymbol{\\tau}))=\\delta_{XYXX}|((a,(d,(c,b))),\\boldsymbol{\\tau})).$$ A similar argument can be made for all fifteen site pattern frequencies and all fifteen rooted 4-taxon topologies. Thus, we need only derive the site pattern probabilities for the two topologies shown in figure \\ref{coalescent}, $((a,b),(c,d))$ and $(a,(b,(c,d)))$ and the site pattern probabilities for the other 13 topologies is a permutation of one of these two cases. The likelihood of any other 4-taxon tree is then given by equation \\ref{lik1} with the data permuted as necessary by the permutation function $\\sigma(\\cdot)$:\n\\begin{equation}\nL((S,\\boldsymbol{\\tau})|\\boldsymbol{D}_{JC}) \\propto \\prod_{k=1}^{K} [\\delta_k|(S,\\boldsymbol{\\tau})]^{\\sigma(d_k)} \n\\label{lik2}\n\\end{equation}\n\n$\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau})$ for the lone unlabeled 3-taxon tree topology can be found be marginalizing over the fourth taxa in figure \\ref{coalescent}. The details are shown in the Appendix.\n\n\\subsection{Derivation of $P(S|\\boldsymbol{D}_{JC})$ for 3- or 4-taxon trees}\n\\label{derivation}\nFrom the Law of Total Probability, it is immediate that: \n$$ \\boldsymbol{\\delta}|S = \\int_{\\boldsymbol{\\tau}} \\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}) f(\\boldsymbol{\\tau}) d\\boldsymbol{\\tau}$$\nA wide variety of forms for the density $f(\\boldsymbol{\\tau})$ can be chosen. We choose priors to be uninformative and to allow for analytic solutions to $\\boldsymbol{\\delta}|S$. Refer to figure \\ref{coalescent} for a description of $\\boldsymbol{\\tau}$ and note that the 3-taxon case can be viewed as the asymmetric case with taxon $a$ removed. For a 3-taxon tree, we choose $\\tau_2 \\sim Exp(\\beta)$ and $\\tau_1|\\tau_2 \\sim U(0,\\tau_2)$. For the 4-taxon symmetric tree, we choose $\\tau_3 \\sim Exp(\\beta)$ and $\\tau_1|\\tau_3,\\tau_2|\\tau_3$ independently $\\sim U(0,\\tau_3)$ For the 4-taxon asymmetric tree, we choose $\\tau_3 \\sim Exp(\\beta)$ and $(\\frac{\\tau_1}{\\tau_3}, \\frac{\\tau_2}{\\tau_3}) \\sim Dirichlet(1,1,1)$. Choosing a prior on the root age gives us a prior on the age of the tree as a whole; setting exponential priors on each branch length leads to the total age of the tree being dependent on the degree of tree symmetry. Given these priors on the branching times, $\\boldsymbol{\\delta}|S$ is calculated as derived in the appendix and the likelihood of any tree can be found as before by taking these site pattern probabilities and applying the standard multinomial likelihood to a permutation of the data:\n$$L(S|\\boldsymbol{D}_{JC}) \\propto \\prod_{k=1}^{K} [\\delta_k|S]^{\\sigma(d_k)} $$\n\nAn unfortunate side effect of this choice of prior is that for four-taxon trees (by a simple application of the law of conditional expectation) $E(\\tau_1)=\\frac{1}{2}\\tau_3$ for the symmetric topology and $E(\\tau_1)=\\frac{1}{3}\\tau_3$ for the asymmetric topology. In other words, if (C,D) is a cherry, inferring an asymmetric topology automatically implies the prior assumption that this divergence time between species C and D occurs more recently. This does not appear to be a problem at first glance, since we are not concerned with inferring species divergence times here, and are treating $\\boldsymbol{\\tau} $ as a nuisance parameter. But, it turns out that this makes accurate inference of the root location of a 4-taxon tree impossible. Given our choice of model, if the true tree is symmetric -- for example $((a,b),(c,d))$ -- then various pairs of rooted asymmetric trees have the same likelihood with speciation times integrated out: $ L[(a,(b,(c,d)))]=L[(b,(a,(c,d)))]$, $ L[(c,(d,(a,b)))]=L[(d,(c,(a,b)))]$, etc. Worse, for some true values of $\\tau_1, \\tau_2, \\tau_3$, $ L[(a,(b,(c,d)))]>L[((a,b),(c,d))]$. These inference errors all concern the location of the root rather than the unrooted topology. As a result, we limit our inference on 4-taxon trees to unrooted topologies. \n\nFor the prior on topologies, we assume the tree generation follows a Yule model: there is a constant rate of species divergence over time, and the rate of species divergence is equal for all branches. This is meant to be as uninformative as possible. For 3-taxon trees, this intuitively assigns a 1\/3 probability to each of the 3 rooted 3-taxon trees. For 4-taxon trees, each of the three symmetric topologies has a 1\/9 prior probability and each of the twelve asymmetric topologies has a 1\/18 prior probability. Interestingly, while any individual symmetric topology is twice as probable as any individual asymmetric topology, in the prior it is twice as probable that the unlabeled topology will be asymmetric rather than symmetric. The details of the prior calculation are given by \\cite{harding71}.\n\nTaking both the prior on the topologies and each topology's likelihood, it is a simple application of Bayes's Theorem in the 3-taxon case to show:\n$$ P(S=s|\\textbf{D})=\\frac{L(S|\\textbf{D})P(S=s)}{\\sum_{i=1}^3 L(S_i|\\textbf{D})P(S_i=s_i)} $$ \nFor the 4-taxon case, the summation is performed over 15 rather than 3 topologies, and the final probability of the unrooted topology is the sum of the five rooted topologies compatible with it (see figure \\ref{rooting}).\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{\"rooting\".png}\n    \\caption{Each unrooted 4-taxon tree corresponds to five different rooted trees, arising from placing the root on one of the five branches in the unrooted tree.}\n    \\label{rooting}\n\\end{figure}\n\nFor the 3-taxon case, we can show that given a sufficient number of unlinked sites under the molecular clock, we can infer the correct topology with probability 1. For the unrooted 4-taxon case, we have demonstrated this in simulation studies for a wide variety of true trees, but it remains unproven.\n\\begin{theorem}\nGiven CIS data for three taxa evolving under the multispecies coalescent with the JC69 model and the molecular clock, if $\\hat{S}$ is the maximum a posteriori tree, $P(\\hat{S}=S) \\rightarrow 1$ as the number of sites $J \\rightarrow \\infty$ for any prior for which $0<\\tau_1<\\tau_2$ holds with probability 1.\n\\end{theorem}\nThen, using a Bonferroni adjustment, we can ensure that given a sufficient number of sites, $P(\\hat{S}=S) > 1- \\epsilon$ \\textit{uniformly} for all the triplets. It then follows that given proposition four from \\cite{steel93}, we can estimate the n-taxon topology with probability $>1-\\epsilon$.\n\n\\subsection{Estimating the n-taxon species tree}\n\\label{assembly}\nThere is no theoretical impediment to prevent extending this procedure to infer full posterior probabilities for 5-, 6-, or even n-taxon trees. But, to see the practical difficulties, consider the challenges to extend this result to just five taxa. First, for 5 taxa $\\boldsymbol{\\delta}_{JC}$ is now a $51 \\times 1$ vector for each of three unlabeled topologies. Then, a $51 \\times 105$ permutation matrix is needed to find the site pattern probabilities for all 105 5-taxon labeled gene tree topologies. A general recursive formula for the number of unordered gene histories embedded in an n-taxon species tree is provided by \\cite{rosenberg07}. We calculated this total to be 379, 313, and 208 for $S_1$, $S_2$, and $S_3$ from figure \\ref{topologies}, respectively. The probabilities of these histories need to be calculated and then integrated over to find the site pattern probabilities of the three unlabeled species tree topologies, then the same $51 \\times 105$ permutation matrix must again be applied to get the full site pattern probabilities for all species tree topologies. As a result, the full set of site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau})$ remains to be calculated for 5 taxa, and an even more complex process would be needed to extend to six or more taxa.\n\nInstead, to infer the n-taxon topology, we use an assembly algorithm that takes the posterior probabilities of the rooted triplets or unrooted quartets as inputs to infer the final n-taxon species tree estimate. We repeat the process of sections \\ref{method1} and \\ref{derivation} for each of the ${n \\choose 3}$ sets of triplets (Lily-T) or ${n \\choose 4}$ quartets (Lily-Q). Then we have a set of $3{n \\choose 3}$ posterior probabilities for each possible rooted triplet or $3{n \\choose 4}$ posterior probabilities for each possible unrooted quartet from the leaf set. These probabilities are then used in an assembly algorithm to infer the estimated n-taxon species tree $\\hat{S}$. There are many options for assembly methods. The methods we choose allow us to use the posterior probabilities as weights in the subtree inputs. Both SVDQuartets and ASTRAL are unweighted methods and thus treat all inputs equally regardless of the inference uncertainty (see figure \\ref{weighting}). For Lily-T, we chose the Triplet MaxCut (TMC) method of Sevillya et al. (2016)\\nocite{sevillya16} due to its speed, accuracy, ease of implementation, and its ability to work with rooted trees as inputs, and similarily chose the Weighted Quartet Max Cut (weighted QMC) algorithm of Avni et al. (2015)\\nocite{avni14} for Lily-Q. Further, the implementation of these algorithms are very similar to the algorithm used by SVDQuartets, and so using it reduces, but does not eliminate, one source of variation between the two methods.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{\"weighting\".png}\n    \\caption{Unweighted tree inputs: an unweighted method treats the two cases the same, even though the right side contains far more information about the true topology.}\n    \\label{weighting}\n\\end{figure}\n\nTriplet MaxCut works as follows: first we obtain posterior probabilities on all ${n \\choose 3}$ subsets from the leaf set. When we have multiple individuals from a species, posterior probabilities for each combination of individuals from each triplet of species are calculated. Then two symmetric $n \\times n$ matrices are formed, $\\textbf{G}$ and $\\textbf{B}$, called the ``good\" and ``bad\" matrices with $i,j=1,2,3 \\ldots n$ corresponding to the taxa in alphabetical order: $a, b, c,$ etc. For each triplet input, we have a set of two taxa that form a cherry and a third that is not part of the cherry, as well as a weighting for the triplet. For each of the ${3 \\choose 2}$ pairs in the triplet, if taxa $i$ and $j$ form a cherry, the weight of the triplet is added to $B_{ij}$ and $B_{ji}$ and if $i$ and $j$ do not form a cherry, the weight of the triplet is added to $G_{ij}$ and $G_{ji}$. As an example, consider the triplet $(a,(c,d))$ with weight 0.5. We would add 0.5 to $B_{34}$, $B_{43}$, $G_{13}$, $G_{31}$, $G_{14}$, and $G_{41}$. This process is repeated for all sets of input triplets. Thus, large entries in $G_{ij}$ indicate taxa $i$ and $j$ should, all else equal, be separated, and large entries in $B_{ij}$ indicate taxa $i$ and $j$ should, all else equal, be grouped together in the final tree S.\n\nNext, for the set of $n$ taxa $L=\\{1,2,3 \\ldots\\ n\\}$ we obtain all subsets of the taxa of cardinality $\\leq \\floor*{\\frac{n}{2}}$. For each subset, we can arbitrarily label the taxa in the subset $L_1$ and call the remainder $L_2=L\\setminus L_1$, resulting in a bipartitioning of $L$. Then, for each bipartition, we score the bipartition by the ratio $ \\frac{\\sum_{i \\in L_1} \\sum _{j \\in L_2} G_{ij}}{\\sum_{i \\in L_1} \\sum _{j \\in L_2} B_{ij}}$. Note that this score can be undefined if there are entries in $\\boldsymbol{B}=0$. To avoid this problem, a very small number such as $10^{-200}$ can be added to each element of $\\boldsymbol{B}$. The bipartition with the highest score is accepted. If the cardinality of either $L_1$ or $L_2$ is greater than two, the process is repeated recursively on $L_1$ and\/or $L_2$ as needed. In essence, each step of the procedure creates a node and assigns taxa to the left and right branches of the node until the final tree is resolved.\n\nThe operation of the weighted QMC is largely similar, with posteriors first being calculated for all ${n \\choose 4}$ subsets of quartets from the leaf set to use as inputs in the assembly algorithm. The major difference in the assembly is that care must be taken if one of $L_1$ or $L_2$ is a set of 3 elements. Then the set is augmented with an additional artificial taxon and the procedure is performed on this augmented set to evaluate which two elements of the set constitute a cherry.\n\n\\subsection{Uncertainty quantification}\nAn immediate concern with using an assembly procedure is that one of the key advantages of our likelihood-based approach -- the ability to produce posterior probabilities -- is lost. The assembly procedures we chose are based on heuristics that make sense -- grouping species together that have a high probability of being cherries. But, unfortunately we have no distributional results to assess. The output tree is a point estimate only, and while we can generate simulation data to say that it is reasonably accurate, once we apply it to real data we no longer have a measure of uncertainty. The difficulty lies in the same factors that led us to pursue the assembly procedure in the first place -- the inability to calculate a joint $n$-taxon set of site pattern probabilities.\n\nA standard method for measuring uncertainties of estimated phylogenies is the bootstrapping method of Efron (1979) \\cite{efron79}. We implemented a nonparametric bootstrap: for CIS data we resampled with replacement from all sites, and for multi-locus data we resampled the genes with replacement and then for each gene we resampled with replacement the sites within each gene. An advantage of the bootstrap is that it can give unbiased estimates of uncertainty without any distributional assumptions. But, a number of cautions are in order. First, we only have asymptotic guarantees about the approximation of $\\frac{\\boldsymbol{d}_k}{J}$ to $\\delta_k$ and we have no finite-sample knowledge of how good the approximation is. As a result, we don't know how much uncertainty we have about our uncertainty. Second, without parallelization, the time required to estimate the bootstrap samples increases linearly with the number of bootstrap samples taken. Last, bootstrap support values are \\textit{not} probabilities, and should not be treated as such. That said, we can compare the bootstrap support to the actual proportion of times we infer the correct tree to see if the bootstrap support reasonably ``mimics\" the true probability of being correct for reasonable parameter values.\n\n\\section{Implementation}\n\\label{implement}\nWe have written source code in C++ that implements Lily-T and Lily-Q, as well as programs that summarize the distances from the true tree for our simulation runs. The programs take as inputs alignments in PHYLIP format and output either the final output tree (Lily-T) or a properly formatted input file for the weighted QMC program (Lily-Q). We also wrote programs for calculating the number of gene tree histories in section \\ref{assembly} and a Perl wrapper for executing the simulation runs. These programs, as well as a file summarizing the results are available at \\url{https:\/\/github.com\/richards-1227\/Lily}. \n\n\\section{Results}\nData were simulated using the \\textit{ms} \\cite{MS} and Seq-Gen \\cite{seq-gen} programs in C++ using a Unix HPC platform. The \\textit{ms} program takes the node times (in coalescent units) and population parameter $\\theta$ as input and simulates a set of gene trees under the multispecies coalescent model. Seq-Gen then takes these gene trees and the mutation model parameters as input and generates the aligned sequence data $\\boldsymbol{D}_{raw}$. Caution should be taken in working with these programs for diploid organisms -- to get the correct simulated probabilities, you enter one-half the node times (in coalescent units) and $2\\theta$ rather than their actual values. 1,000,000 CIS were simulated at various settings of $\\boldsymbol{\\tau}$ and $\\theta$ and we compared the observed values of $\\boldsymbol{d}_{JC}$ to $J\\boldsymbol{\\delta}_{JC}|(S,\\boldsymbol{\\tau})$. Chi-squared goodness of fit tests were performed and there was no evidence that the simulated frequencies differed from that expected (only one individual test p-value was below 0.05 (0.006), and it was no longer significant after making a Bonferroni adjustment).\n\n\\subsection{Testing robustness to model assumptions and prior specification}\n\\label{robustness}\nWe next relaxed various assumptions of the substitution model to verify that the true tree was still estimated with high probability. One thousand runs were performed with 25,000 CIS generated with the Jukes-Cantor assumptions being progressively relaxed, first allowing for different stationary probabilities, then different relative substitution rates, then between-site rate heterogeneity, and then allowing for invariant sites. The settings are summarized in table \\ref{JCsettings}.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n    \\hline\n       &   & \\multicolumn{5}{|c|}{Relative rates vs. A-C} &  &  & \\\\\n      \\hline\n      Setting & CG content  & A-G & A-T  & C-G  & C-T  & G-T  & $\\alpha$ & \\% Invariable &  $P(S|\\boldsymbol{d})$\\\\\n       \\hline\n        1 & 25\\% & 1 & 1 & 1 & 1 & 1 & N\/A & 0 & 0.983      \\\\ \n               \\hline\n        2 & 35\\% & 1 & 1 & 1 & 1 & 1 & N\/A & 0   & 0.971\\\\ \n        \\hline\n                3 & 45\\% & 3 & 1 & 1 & 3 & 1 & N\/A & 0 & 0.951 \\\\ \n        \\hline\n                4 & 35\\% & 5 & 1 & 1 & 3 & 1 & N\/A & 0  & 0.951\\\\ \n        \\hline\n                5 & 35\\% & 5 & 0.75 & 0.5 & 3 & 0.25 & N\/A & 0 & 0.971 \\\\ \n        \\hline\n                6 & 35\\% & 2 & 0.75 & 0.5 & 1.5 & 0.25 & 3 & 0 & 0.974  \\\\ \n        \\hline\n                7 & 35\\% & 2 & 0.75 & 0.5 & 1.5 & 0.25 & 9 & 0  & 0.960\\\\ \n        \\hline\n                8 & 35\\% & 2 & 0.75 & 0.5 & 1.5 & 0.25 & 3 & 0.2  & 0.960 \\\\ \n        \\hline\n    \\end{tabular}\n    \\caption{Initial simulation test settings. The final column gives the average posterior probabilities assigned to the true tree as the JC69 assumptions are progressively relaxed.}\n    \\label{JCsettings}\n\\end{table}\n\nThe results shown in table \\ref{JCsettings} are typical and are presented for three taxa and each of the eight substitution model settings using $\\tau_1=5.2$, $\\tau_2=6.0$, $\\theta=0.01$, and $\\beta=0.1$. There does not appear to be a large impact from relaxing the JC69 assumptions, and for the remainder of our simulations all the data were generated under simulation setting 8, so performance of our methods is measured against a deliberately misspecified model.\n\nHere we should note that the site pattern probabilities $\\boldsymbol{\\delta}$ are also conditional on $\\theta$ as well as $(S,\\boldsymbol{\\tau})$, but keeping with the notation of \\cite{chifman15} we have ignored this conditioning. For the remainder of our work, we used $\\hat{\\theta}=0.003$ as an input to both Lily-T and Lily-Q, which we justify as follows. Most empirical values of $\\theta$ fall between 0.001 and 0.01 (see \\cite{kubatko07,rannala03,jennings05,kopp05} for estimates on species ranging from primates to finches). So, we simulated data using a true $\\theta=\\{0.0003, 0.001, 0.003, 0.01, 0.03\\}$, extending beyond the empirical range, and calculated posteriors using $\\hat{\\theta}=0.01$ and $\\hat{\\theta}=0.001$. The results for Lily-T and Lily-Q are shown in figure \\ref{ThetaEffect}. Since there is no apparent difference in performance for assuming a $\\theta$ different from the actual data-generating $\\theta$, we simply use a $\\hat{\\theta}=0.003$ as it is in the middle of the empirical range on the log scale.\n\n\\begin{figure}\n    \\centering\n   \\begin{subfigure}[b]{0.43\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"theta3\".png}\n   \\caption{$\\hat{\\theta}=0.001$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.43\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"theta4\".png}\n   \\caption{$\\hat{\\theta}=0.01$}\n\\end{subfigure}\n    \\caption{Mean posterior probability assigned by Lily-Q to the true tree for varying numbers of sites and population parameter $\\theta$ for a symmetric 4-taxon  with branch lengths of 0.5 coalescent units. The left panel (a) uses $\\hat{\\theta}=0.001$ and the right panel (b) uses $\\hat{\\theta}=0.01$.}\n    \\label{ThetaEffect}\n\\end{figure}\n\nWe cannot, however, simply assume a value for the root age hyperparameter $\\beta$. Figure \\ref{betaEffect} shows the posterior assigned to the correct tree, using a $\\beta$ where $1\/\\beta$ varies from 2 orders of magnitude below to 2 orders of magnitude above the actual root age. There is very little loss in accuracy for overestimating the root age (very small values of $\\beta$) except with very few sites (see the $J=1000$ column). If the root age is underestimated, however, there can be a large loss of accuracy, even with a larger number of sites. This is to be expected given the asymmetric nature of the exponential prior -- a small $\\beta$ flattens the prior and can still have adequate prior mass near the true value, while a large $\\beta$ puts most prior mass near zero. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{\"BetaEffect\".png}\n    \\caption{Mean posterior probability assigned by Lily-T to the true tree for varying numbers of sites and root age hyperparameter $\\beta$ for a 3-taxon tree with branch lengths of 0.5 coalescent units.}\n    \\label{betaEffect}\n\\end{figure}\n\nWe take an empirical Bayes approach to choosing $\\beta$. From the JC69 model we can infer the well-known pairwise distance estimate for each pair of species\n$$\\hat{\\lambda} \\hat{t}=\\frac{-3log(1-\\frac{4\\hat{p}}{3})}{4} $$\nwhere $t$ is the time measured in years, $\\lambda$ is the per-year mutation rate, and $\\hat{p}$ is the fraction of discordant sites between the two species. Then, assuming a value of 0.003 for $\\theta$ and generation time $g$, we have:\n        $$0.003=\\theta=4N_e\\mu=4\\lambda N_e g $$\nAfter a quick algebraic manipulation we can estimate time in coalescent units ($2N_e$ generations) as:\n  \\begin{equation}\n      \\frac{\\hat{t}}{2N_eg}=\\frac{-6log(1-\\frac{4\\hat{p}}{3})}{0.012} \n      \\label{distEst}\n  \\end{equation}      \nFor each set of three or four species, we will have either three or six of these pairwise divergence estimates from equation \\ref{distEst}. Lily-T and Lily-Q use the mean of these estimates as the value for the hyperparameter $\\beta$. It is worth noting that because these estimates use concatenated data, they are biased upwards with respect to the species tree root time since they estimate the time of coalescent events and any coalescent event must occur prior to species divergence. But, here, that bias is useful as it helps reduce the risk of underestimating the root age, which has a greater performance impact than overestimation. \n\n\\subsection{Application to simulated data}\nData were simulated for 5, 8, and 12-taxon trees for a total of 12 different topologies ranging from fully symmetric to fully asymmetric. The full set of topologies simulated is displayed in figure \\ref{topologies}. For each topology, we used four different values of the population parameter: $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$. These were chosen to extend slightly above and below the empirical range of 0.001 to 0.01 and the values are linear on the log scale. We used four different settings for the minimum branch length (MBL): 0.2, 0.5, 1.0, and 2.0 coalescent units. Multi-locus data were simulated with 10, 50, and 500 genes and 50, 200, and 500 sites per gene. For CIS data, we simulated either 5,000 or 50,000 sites. 100 runs were performed at each combination of settings.\n\\begin{comment}\n\\begin{table}[]\n    \\centering\n    Five taxa:\n    \\vskip 1mm\n    \\begin{tabular}{c c c }\n       $(a,(b,(c,(d,e))))$  & $(a,((b,c),(d,e)))$  & $((a,b),(c,(d,e)))$ \n            \\end{tabular}\n          \\vskip 1mm\n                Eight taxa:\n    \\vskip 1mm\n       \\begin{tabular}{c c }\n      $(a,(b,(c,(d,(e,(f,(g,h)))))))$   & $((a,b),((c,d),(e,(f,(g,h)))))$ \\\\ $(((a,b),c),((d,e),(f,(g,h))))$   &  $(((a,b),(c,d)),((e,f),(g,h)))$ \\\\ \n             \\end{tabular}\n          \\vskip 1mm\n                Twelve taxa:\n    \\vskip 1.5mm\n       \\begin{tabular}{c c }\n      $(a,(b,(c,(d,(e,(f,(g,(h,(i,(j,(k,l)))))))))))$ & $((a,b),((c,(d,(e,f))),((g,h),((i,j),(k,l)))))$ \\\\ $((a,(b,c)),((d,(e,f)),((g,(h,i)),(j,(k,l)))))$  & $(((a,b),(c,d)),(((e,f),(g,h)),((i,j),(k,l))))$ \n                  \\end{tabular}\n       \\begin{tabular}{c } $(((a,(b,c)),(d,(e,f))),((g,(h,i)),(j,(k,l))))$   \n    \\end{tabular}\n    \\caption{Species tree topologies used in simulation study.}\n    \\label{top_used}\n\\end{table}\n\\end{comment}\n\nFocusing on the minimum branch length created a number of side effects. Comparing trees $S_4$ and $S_5$ in figure \\ref{topologies} we note that for the same MBL, the root for the asymmetric tree $S_5$ will occur much further in the past than for the symmetric tree $S_4$. Second, for trees that are neither fully symmetric nor asymmetric such as $S_{12}$, some internal branches naturally have to be longer than the MBL, in which case the internal nodes were chosen to be equally spaced. \n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}{.47\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{\"topology1\".png}\n\\end{subfigure}\n\\begin{subfigure}{.47\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{\"topology2\".png}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.47\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{\"topology3\".png}\n\\end{subfigure}\n    \\caption{The twelve topologies used in the simulation study. 5 taxa: $S_1$ through $S_3$, 8 taxa: $S_4$ through $S_7$, 12 taxa: $S_8$ though $S_{12}$. The red bar indicates scale of the MBL, which was set at 0.2, 0.5, 1.0, and 2.0 coalescent units.}\n    \\label{topologies}\n\\end{figure}\n\nIn order to evaluate how well each method performed, we first need a proper metric of distance between trees. The most common is the Robinson Folds (RF) distance from \\cite{RFdist}. They define the distance between two trees $T_1$ and $T_2$ as the sum of the number of bipartitions in $T_1$ not contained in $T_2$ and vice versa. The presence of a bipartition is symmetric, since the number of bipartitions is equal to the number of internal branches, which is in turn equal to $n-2$ for a rooted tree and $n-3$ for an unrooted tree. Thus, if there is a bipartition of $T_1$ not in $T_2$, there must be a bipartition in $T_2$ not in $T_1$. As a result, the RF metric can take on any even value from zero to $2(n-2)$ (or $2(n-3)$ if unrooted). Since the metric is also a function of whether or not the two trees are rooted, to compare Lily-T to the other methods (which produce unrooted trees), we must first unroot both the estimated and true tree. For multi-locus data, we compared SVDQuartets (as implemented in PAUP* \\cite{paup}), ASTRAL (using FastTree for gene tree estimation as per the example in \\cite{chou15}), Lily-T, and Lily-Q for each run and calculated the mean RF distance and standard error for the 100 runs. For CIS data, we excluded ASTRAL from comparison since gene tree cannot be estimated for a single site.\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF5pt2\".png}\n   \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF5pt5\".png}\n   \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF5_1\".png}\n   \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF5_2\".png}\n   \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for true tree $(a,(b,(c,(d,e))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL is displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8bpt2\".png}\n   \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8bpt5\".png}\n   \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8b1\".png}\n   \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8b2\".png}\n   \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for true tree $(((a,b),c),((d,e),(f,(g,h))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL is displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF6}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8bpt2s\".png}\n   \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8bpt5s\".png}\n   \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8b1s\".png}\n   \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF8b2s\".png}\n   \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-T vs. SVDQuartets for true tree $(((a,b),c),((d,e),(f,(g,h))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- dark, 50 -- medium, 500 -- light. SVDQuartets displayed in grey shading and Lily-T in green.}\n\\label{simulation_RF6s}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF12dpt2\".png}\n   \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF12dpt5\".png}\n   \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF12d1\".png}\n   \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RF12d2\".png}\n   \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for true tree $(((a,(b,c)),(d,(e,f))),((g,(h,i)),(j,(k,l))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL is displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF12}\n\\end{figure}\n\nA representative selection of the simulation results are shown in figures \\ref{simulation_RF1} through \\ref{simulation_RF12} (the full plots are in the supplementary material.) One complication is that for very small number of sites and a small value of $\\theta$, there may be cases where there is no divergence between any set of three species. The proper inference in this case would be to return a polytomous tree. Each method, however, treats this case differently. SVDQuartets will either return a polytomous tree or an error. Lily-Q infers an 1\/3 probability for each of the three possible unrooted quartets involving three or more zero-divergence species, but the assembly procedure will then produce an error message. Lily-T will do the same, but will generate an error only after consuming much time and memory. ASTRAL will infer an apparently random binary tree, but that is due to FastTree incorrectly inferring binary trees for the gene tree inputs. As a result, for those settings where this occurred, we display the Lily-Q vs. ASTRAL comparison for first 100 valid runs without any set of three zero-distance taxa. We also did not use Lily-T or SVDQuartets on the 10 genes and 50 sites-per-gene setting because SVDQuartets invariably produced errors and Lily-T produced errors while materially slowing down the simulation process. Thus, the plots for Lily-T and SVDQuartets do not show results for some settings. \n\nA number of results become evident from the RF plots. First, all estimation is better with a larger $\\theta$, as there is more mutation along each branch allowing us to pick up more of the phylogenetic signal. A lone exception to this trend is that estimation gets worse for SVDQuartets for $S_8$ with $\\theta=0.0216$ and a minimum branch length of 2.0 coalescent units. It is worth noting that for these settings the root is 22.0 coalescent units in the past and the sequences may be nearing saturation and so SVDQuartets may have greater difficulty resolving the phylogeny near saturation. Second, with enough data, all methods perform well. Third, by comparing the 50 genes\/500 sites per gene and the 500 genes\/50 sites per gene case, we can see that all methods do better with more genes even when the total number of sites is held constant. At smaller sample sizes, SVDQuartets is generally outperformed by all other methods, and Lily-T is in turn outperformed by Lily-Q under most simulation settings.\n\nTherefore, we focused on the comparison between Lily-Q and ASTRAL. For MBL of 1.0 or 2.0 coalescent units, Lily-Q generally performs no worse, and in many cases, better than ASTRAL. For an MBL of 0.5 coalescent units, ASTRAL does better for 200 or 500 sites per gene and $\\theta$ of 0.0072 or 0.0216, and this effect appears to be stronger with fewer taxa. Lily-Q does as well or better for almost all cases with 50 sites per gene and 10 or 50 genes. For 0.2 CI, Lily-Q does better with fewer sites per gene and with smaller values of $\\theta$ and ASTRAL performs better with more sites per gene and larger values of $\\theta$. The comparison also seems somewhat dependent on topology, but without any clear pattern. ASTRAL outperforms Lily-T for most settings, except when the branch lengths are large and $\\theta$ is smaller. \n\nThese results match our expectations, as ASTRAL depends on accurate gene tree inputs and those gene trees are easier to resolve when the genes are long (many sites per gene) and there is more mutation along each branch (higher $\\theta$). One oddity is that for Lily-Q, with 10 genes, sometimes the RF distance was higher going from 200 to 500 sites per gene. But, the standard errors almost always overlapped so this appears to be random noise arising out of the fact that, with only 10 genes, if the gene trees differ from the species tree, increasing the number of sites only allow better inference of the mismatched gene trees rather than the underlying species tree.\n\n\\begin{comment}\nTables \\ref{det_result1} through \\ref{det_result4} show the results for a single topology $((a,b),((c,d),(e,(f,(g,h)))))$, with a minimum branch length of 0.5 coalescent units. A number of results become evident. First, all estimation is better with a larger $\\theta$, as there is more mutation along each branch allowing us to pick up more of the phylogenetic signal. Second, with enough data, all methods perform well. Third, by comparing the 50 genes\/500 sites per gene and the 500 genes\/50 sites per gene case, we can see that all methods do better with more genes even when the total number of sites is held constant. At smaller sample sizes, SVDQuartets is generally outperformed by all other methods, and Lily-T is outperformed by Lily-Q and ASTRAL. Therefore, we focused on the comparison between Lily-Q and ASTRAL and calculated a (pooled variance) T-statistic for the difference between them. As expected, ASTRAL performs better under conditions where gene tree estimation is more accurate -- when both the gene length and $\\theta$ is large. Lily-Q, on the other hand, tends to perform better when the gene lengths are shorter or $\\theta$ is smaller. \n\nOur program in C++ for calculating RF distances was written assuming binary trees. In some cases for Lily-T and especially SVDQuartets, the assembly algorithm did not output a binary tree in all 100 runs and in those cases we did not calculate the mean RF distance for comparison (represented as dashes in the tables).\n\n\\begin{table}[]\n    \\centering\n    \\begin{scriptsize}\n    \\begin{tabular}{|c|c|c|c|c|c|c|}\n    \\hline\n       \\textbf{Genes}  & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n    \\hline\n       500  & 500  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 200  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 50  & 0.56(0.09)  & 0.2(0.06)  &  0.02(0.02)  & 0.22(0.07)  & 2.80\\\\\n         \\hline\n                50  & 500  & 0.68(0.10)  & 0.32(0.08)  &  0.16(0.05)  & 0.30(0.08)  & 1.49\\\\\n         \\hline\n                50  & 200  & 2.14(0.24)  & 1.22(0.14)  &  0.74(0.11)  & 1.30(0.15)  & 2.95\\\\\n         \\hline\n                50  & 50  & --  & 3.86(0.27)  &  2.52(0.21)  & 3.34(0.20)  & 2.81\\\\\n         \\hline\n                10  & 500  & --  & 2.64(0.23)  &  2.06(0.19)  & 2.8(0.21)  & 2.64\\\\\n         \\hline\n                10  & 200  & --  & 4.52(0.27) &  3.06(0.20)  & 4.32(0.21)  & 4.30\\\\\n         \\hline\n                10  & 50  & --  & --  &  6.06(0.22)  & 6.94(0.21)  & 2.89\\\\\n                 \\hline\n                25000  & 1  & 0.56(0.10)  & 0.12(0.05)  &  0.02(0.02)  & --  & -- \\\\\n                         \\hline\n                5000  & 1  & --  & 1.84(0.18)  &  1.00(0.14)  & --  & -- \\\\\n         \\hline\n    \\end{tabular}\n        \\end{scriptsize}\n    \\caption{Mean RF distance (with standard error) from true tree  $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0008$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n    \\label{det_result1}\n\\end{table}\n\\begin{table}[]\n    \\centering\n    \\begin{scriptsize}\n        \\begin{tabular}{|c|c|c|c|c|c|c|}\n    \\hline\n       \\textbf{Genes}  & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n    \\hline\n       500  & 500  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 200  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 50  & 0.06(0.03)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                50  & 500  & 0.32(0.07)  & 0.12(0.05)  &  0.02(0.02)  & 0.12(0.05)  & 1.94\\\\\n         \\hline\n                50  & 200  & 0.48(0.09)  & 0.26(0.07)  &  0.08(0.04)  & 0.34(0.08)  & 3.07\\\\\n         \\hline\n                50  & 50  & 3.18(0.32)  & 1.48(0.17)  &  0.80(0.11)  & 1.60(0.15)  & 4.31\\\\\n         \\hline\n                10  & 500  & 1.96(0.17)  & 1.58(0.16)  &  1.44(0.15)  & 1.42(0.14)  & -0.10\\\\\n         \\hline\n                10  & 200  & --  & 2.62(0.20) &  1.84(0.17)  & 2.24(0.19)  & 1.59\\\\\n         \\hline\n                10  & 50  & --  & --  &  3.58(0.26)  & 5.14(0.27)  & 4.18\\\\\n                                 \\hline\n                25000  & 1  & 0.04(0.03)  & 0(0)  &  0(0)  & --  & -- \\\\\n                         \\hline\n                5000  & 1  & 0.76(0.11)  & 0.44(0.08)  &  0.14(0.05)  & --  & -- \\\\\n         \\hline\n    \\end{tabular}\n        \\end{scriptsize}\n    \\caption{Mean RF distance (with standard error) from true tree  $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0024$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n\\end{table}\n\\begin{table}[]\n    \\centering\n    \\begin{scriptsize}\n        \\begin{tabular}{|c|c|c|c|c|c|c|}\n    \\hline\n       \\textbf{Genes}  & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n    \\hline\n       500  & 500  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 200  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 50  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                50  & 500  & 0.28(0.07)  & 0.04(0.03)  &  0.08(0.04)  & 0(0)  & -2.04\\\\\n         \\hline\n                50  & 200  & 0.28(0.07)  & 0.04(0.03)  &  0.06(0.03)  & 0.08(0.04)  & 0.38\\\\\n         \\hline\n                50  & 50  & 1.00(0.11)  & 0.56(0.10)  &  0.34(0.08)  & 0.50(0.10)  & 1.31\\\\\n         \\hline\n                10  & 500  & 1.14(0.14)  & 0.68(0.11)  &  0.64(0.11)  & 0.54(0.09)  & -0.71\\\\\n         \\hline\n                10  & 200  & 1.70(0.15)  & 1.48(0.15) &  1.30(0.13)  & 1.48(0.14)  & 0.84\\\\\n         \\hline\n                10  & 50  & --  & --  &  2.40(0.18)  & 2.98(0.22)  & 2.04\\\\\n                                 \\hline\n                25000  & 1  & 0(0)  & 0(0)  &  0(0)  & --  & -- \\\\\n                         \\hline\n                5000  & 1  & 0.36(0.08)  & 0.06(0.03)  &  0(0)  & --  & -- \\\\\n         \\hline\n    \\end{tabular}\n       \\end{scriptsize}\n    \\caption{Mean RF distance (with standard error) from true tree  $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0072$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n    \\begin{scriptsize}\n        \\begin{tabular}{|c|c|c|c|c|c|c|}\n    \\hline\n       \\textbf{Genes}  & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n    \\hline\n       500  & 500  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 200  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                500  & 50  & 0(0)  & 0(0)  &  0(0)  & 0(0)  & 0\\\\\n         \\hline\n                50  & 500  & 0.06(0.03)  & 0(0)  &  0.02(0.01)  & 0(0)  & -1.00\\\\\n         \\hline\n                50  & 200  & 0.28(0.07)  & 0.04(0.03)  &  0.02(0.02)  & 0(0)  & -1.00\\\\\n         \\hline\n                50  & 50  & 0.38(0.08)  & 0.22(0.06)  &  0.16(0.05)  & 0.2(0.06)  & 0.49\\\\\n         \\hline\n                10  & 500  & 1.82(0.17)  & 1.18(0.15)  &  1.08(0.15)  & 0.94(0.12)  & -0.73\\\\\n         \\hline\n                10  & 200  & 1.32(0.15)  & 0.84(0.12) &  0.6(0.10)  & 0.70(0.11)  & 0.68\\\\\n         \\hline\n                10  & 50  & --  & --  &  1.54(0.17)  & 1.80(0.17)  & 1.10\\\\\n                                 \\hline\n                25000  & 1  & 0(0)  & 0(0)  &  0(0)  & --  & -- \\\\\n                         \\hline\n                5000  & 1  & 0(0)  & 0(0)  &  0(0)  & --  & -- \\\\\n         \\hline\n    \\end{tabular}\n    \\end{scriptsize}\n    \\caption{Mean RF distance (with standard error) from true tree  $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0216$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n    \\label{det_result4}\n\\end{table}\n\nFigures \\ref{simulation_RF1} through \\ref{simulation_RF6} show a graphical comparison of Lily-Q vs. ASTRAL for all cases with an MBL of 0.5 coalescent units. These show the normalized RF distance (RF distance divided by the maximum value of $2(n-3)$). The direction of results are largely similar for other branch lengths, but tend to be larger in magnitude for MBL of 0.2 coalescent units and smaller for higher MBL. The full results is available online (see section \\ref{implement}, and a complete table of cases where there is a significant difference between Lily-Q and ASTRAL (defined as a T-statistic with an absolute value greater than 2) is in the supplemental material. With 500 genes, there is almost no difference between methods, although Lily-Q appears to do slightly better with 50 sites per gene. With 50 genes, at higher $\\theta$ values, there is little error, except again a slight outperformance by Lily-Q with 50 sites per gene. At smaller $\\theta$ values, with 50 genes, Lily-Q appears to outperform especially with smaller numbers of sites per gene. With 10 genes, Lily-Q again outperforms at small values of $\\theta$ even when the number of sites per gene large. For higher values of $\\theta$ Lily-Q has lower RF distance for few sites per gene and ASTRAL does better for many sites per gene.\n\nOne oddity is that for Lily-Q, with 10 genes, sometimes the RF distance was higher going from 200 to 500 sites per gene. But, the standard errors almost always overlapped so this appears to be random noise arising out of the fact that, with only 10 genes, if the gene trees differ from the species tree, increasing the number of sites only allow better inference of the mismatched gene trees rather than the underlying species tree.\n\nFinally, it is worth noting that the RF metric is heavily left-skewed even with  moderate numbers of taxa \\cite{steel93}. As a result, even with only 10 sites and 50 sites per gene, the average estimated tree for both methods is closer to the true tree than the vast majority of the tree space. \n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot1\".png}\n   \\caption{$(a,(b,(c,(d,e))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot2\".png}\n   \\caption{$(a,((b,c),(d,e)))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot3\".png}\n   \\caption{$((a,b),(c,(d,e)))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot4\".png}\n   \\caption{$(a,(b,(c,(d,(e,(f,(g,h)))))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot5\".png}\n   \\caption{$((a,b),((c,d),(e,(f,(g,h)))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot6\".png}\n   \\caption{$(((a,b),c),((d,e),(f,(g,h))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF3}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot7\".png}\n   \\caption{$(((a,b),(c,d)),((e,f),(g,h)))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot8\".png}\n   \\caption{$(a,(b,(c,(d,(e,(f,(g,(h,(i,(j,(k,l)))))))))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot9\".png}\n   \\caption{$((a,b),((c,(d,(e,f))),((g,h),((i,j),(k,l)))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot10\".png}\n   \\caption{$((a,(b,c)),((d,(e,f)),((g,(h,i)),(j,(k,l)))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF5}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot11\".png}\n   \\caption{$(((a,b),(c,d)),(((e,f),(g,h)),((i,j),(k,l))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n   \\includegraphics[width=\\linewidth]{\"RFplot12\".png}\n   \\caption{$(((a,(b,c)),(d,(e,f))),((g,(h,i)),(j,(k,l))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF6}\n\\end{figure}\n\n\\end{comment}\n\n\\subsection{Bootstrapping results}\nFor twenty different combinations of topology, $\\theta$, MBL, gene length, and number of genes, we drew 100 bootstrap samples by resampling both the genes and then resampling the sites within each gene. For both Lily-T and Lily-Q, we compared the bootstrap support for the estimated tree to the RF distance from the true tree. Figure \\ref{RFboot5} shows the bootstrap support given the RF distance between the estimated tree and the true tree. We can see that the farther the estimated tree is from the true tree, the lower the bootstrap support tends to be. Tables \\ref{boot_table5} to \\ref{boot_table12Q} in turn show the positive and negative predictive value of high or low bootstrap support. What we see is that we have very good positive predictive value -- for Lily-Q only 5 of 697 trees ($<$1\\%) with bootstrap support greater than 0.7 were incorrect, and all of them had an RF distance of 2. With lower bootstrap support, however, more trees tend to be further from the true tree but the estimated tree may still be close to the correct tree -- for Lily-Q 356 of 830 estimated trees (43\\%) with bootstrap support less than 0.4 were in fact correct.\n\nThese simulation studies suggest that the bootstrap support may be useful as a guide to triaging larger phylogenetic questions -- if the support is high, the faster results that can be obtained from Lily-T or Lily-Q can be accepted with confidence. Conversely, if the support is low, it serves as an indication that a more computationally intensive MCMC-based method may be necessary for accurate results.  \n\n\\begin{figure}\n    \\centering\n   \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"taxa5_boxplot\".png}\n     \\caption{Lily-T (5 taxa)}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"Quart5taxa_boxplot\".png}\n     \\caption{Lily-Q (5 taxa)}\n   \\end{subfigure}\n   \\newline\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"taxa8_boxplot\".png}\n     \\caption{Lily-T (8 taxa)}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"Quart8taxa_boxplot\".png}\n     \\caption{Lily-Q (8 taxa)}\n   \\end{subfigure}\n   \\newline\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"taxa12_boxplot\".png}\n     \\caption{Lily-T (12 taxa)}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"Quart12taxa_boxplot\".png}\n     \\caption{Lily-Q (12 taxa)}\n   \\end{subfigure}\n    \\caption{Bootstrap support for estimated trees grouped by RF distance}\n    \\label{RFboot5}\n\\end{figure}\n\n\n\\begin{table}[]\n    \\centering\n    \\begin{minipage}{0.59\\linewidth}\n        \\begin{tabular}{|c|c|c|c|c|}\n        \\hline\n        \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} \\\\\n         \\hline\n       0  & 0  & 0  & 0  & 1\\\\\n        \\hline\n       (0, 0.4]  & 121  & 135  & 109  & 34\\\\\n        \\hline\n       (0.4, 0.7]  & 107  & 22  & 1  & 0\\\\\n        \\hline\n       (0.7, 0.9]  & 62  & 0  & 0  & 0\\\\\n        \\hline\n       (0.9, 1]  & 108  & 0  & 0  & 0 \\\\\n        \\hline\n    \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}{0.39\\linewidth}\n        \\begin{tabular}{|c|c|c|c|}\n         \\hline\n        \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4}  \\\\\n         \\hline\n       0  & 0  & 0  & 0  \\\\\n        \\hline\n       (0, 0.4]  & 46  & 88  & 62  \\\\\n        \\hline\n       (0.4, 0.7]  & 114  & 49  & 12  \\\\\n        \\hline\n       (0.7, 0.9]  & 153  & 5  & 0  \\\\\n        \\hline\n       (0.9, 1]  & 171  & 0  & 0  \\\\\n        \\hline\n    \\end{tabular}\n    \\end{minipage}\n    \\caption{RF distances of estimated 5-taxon trees by bootstrap support level for Lily-T (left) and Lily-Q (right).}\n    \\label{boot_table5}\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n        \\begin{tabular}{|c|c|c|c|c|c|c|}\n         \\hline\n        \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} & \\textbf{RF 10}  \\\\\n         \\hline\n       0  & 1  & 1  & 4  & 1 & 1 & 1 \\\\\n        \\hline\n       (0, 0.2]  & 129  & 117  & 72 & 19 & 3 & 1  \\\\\n        \\hline\n       (0.2, 0.9] & 205  & 41  & 3 & 0 & 0 & 0  \\\\\n        \\hline\n       (0.9, 1] & 100  & 0  & 0 & 0 & 0 & 0   \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{RF distances of 8-taxon trees estimated by Lily-T for a given bootstrap support level}\n    \\label{boot_table8T}\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n        \\begin{tabular}{|c|c|c|c|c|c|}\n         \\hline\n        \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8}  \\\\\n         \\hline\n       (0, 0.4]  & 143  & 113  & 47 & 14 & 3  \\\\\n        \\hline\n       (0.4, 0.7]  & 149  & 13  & 0 & 0 & 0   \\\\\n        \\hline\n       (0.7, 0.9] & 87  & 0  & 0 & 0 & 0   \\\\\n        \\hline\n       (0.9, 1] & 131  & 0  & 0 & 0 & 0    \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{RF distances of 8-taxon trees estimated by Lily-Q for a given bootstrap support level}\n    \\label{boot_table8Q}\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n        \\begin{tabular}{|c|c|c|c|c|c|c|}\n         \\hline\n        \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} & \\textbf{RF 10}  \\\\\n         \\hline\n       0  & 3  & 5  & 3  & 2 & 0 & 1 \\\\\n        \\hline\n        (0, 0.2]  & 131  & 108  & 55  & 27 & 6 & 1 \\\\\n        \\hline\n       (0.2, 0.4]  & 68  & 25  & 3 & 0 & 0 & 0  \\\\\n        \\hline\n       (0.4, 0.7] & 95  & 4  & 0 & 0 & 0 & 0  \\\\\n        \\hline\n       (0.7, 1] & 63  & 0  & 0 & 0 & 0 & 0   \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{RF distances of 12-taxon trees estimated by Lily-T for a given bootstrap support level}\n    \\label{boot_table12T}\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n        \\begin{tabular}{|c|c|c|c|c|c|c|}\n         \\hline\n        \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} & \\textbf{RF 10}  \\\\\n         \\hline\n       0  & 1  & 2  & 1  & 1 & 0 & 0 \\\\\n        \\hline\n        (0, 0.2]  & 75  & 52  & 36  & 8 & 6 & 1 \\\\\n        \\hline\n       (0.2, 0.4]  & 91  & 36  & 4 & 0 & 0 & 0  \\\\\n        \\hline\n       (0.4, 0.7] & 129  & 7  & 0 & 0 & 0 & 0  \\\\\n        \\hline\n      (0.7, 0.9] & 88  & 0  & 0 & 0 & 0 & 0  \\\\\n        \\hline\n       (0.9, 1] & 62  & 0  & 0 & 0 & 0 & 0   \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{RF distances of 12-taxon trees estimated by Lily-Q for a given bootstrap support level}\n    \\label{boot_table12Q}\n\\end{table}\n\n\\subsection{Application to empirical sequence data}\nWe applied Lily-T and Lily-Q to four empirical datasets. For all the datasets we estimated the topology and then drew 100 bootstrap samples and calculated the bootstrap support at each node. The first dataset is a somewhat toy example: a segment of 888 sites over five genes coming from the mitochondria of nine primates \\cite{yang97}. Since mitochondria are passed down solely from the mother, we performed the bootstrap as if the data were from a single 888-site gene since all sites share a common lineage. The results are displayed in figure \\ref{primate_out}. Both Lily-T and Lily-Q inferred the same tree as \\cite{yang97}. This tree is different than the primate consensus from \\cite{chou15, gatesy17} in that infers that tarsiers and lemurs are a cherry rather than that lemurs are an outgroup, likely due to the small number of sites from a single lineage. Lily-Q had higher bootstrap support than Lily-T for most nodes, especially for the (human, chimp) clade. Run time for both methods was a matter of seconds, including bootstrapping.\n\n\\begin{figure}\n    \\centering\n   \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.4]{\"priT\".png}\n     \\caption{Lily-T}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.4]{\"priQ\".png}\n     \\caption{Lily-Q}\n   \\end{subfigure}\n    \\caption{Estimates for 9-taxon primate tree with bootstrap support for each clade. Lily-Q display assumes root location is known.}\n    \\label{primate_out}\n\\end{figure}\n\nThe second dataset consisted of approximately 127,000 sites over 106 genes from eight yeast species: \\textit{S.cerevisiae} (Scer), \\textit{S. paradoxus} (Spar), \\textit{S. mikatae} (Smik), \\textit{S. kudriavzevii} (Skud), \\textit{S. bayanus} (Sbay), \\textit{S. castellii} (Scas), \\textit{S. kluyveri} (Sklu), and the outgroup \\textit{C. albicans} (Calb) \\cite{rokas03, wen18}. Both methods matched the results from \\cite{wen18}. Estimation took approximately 4 min including bootstrapping on a MacBook Air. All but one node had 100\\% bootstrap support. The exception was some uncertainty over whether \\textit{Skud} and \\textit{Sbay} represent a clade or if \\textit{Sbay} is an outgroup to all of \\textit{Scer}, \\textit{Spar}, and \\textit{Smik} (see figure \\ref{yeast_out}).\n\n\\begin{figure}\n    \\centering\n   \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.4]{\"YeastT\".png}\n     \\caption{Lily-T}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.4]{\"YeastQ\".png}\n     \\caption{Lily-Q}\n   \\end{subfigure}\n    \\caption{Estimates for 8-taxon yeast tree with bootstrap support for each clade. Lily-Q display assumes root location is known.}\n    \\label{yeast_out}\n\\end{figure}\n\nNext, we applied Lily-T and Lily-Q to a dataset of 52 individuals from seven North American snake species consisting of aligned sequences from 18 nuclear and 1 mitochondrial genes with 190-850 characters per gene. Runtime was around a half hour on a Unix HPC platform (the increase in run time was due to the many different combination of individuals within each subset of three or four species). This is much less than the runtimes reported by \\cite{chifman14}: $<$1 day for SVDQuartets and $\\approx$10 days for *BEAST, although we acknowledge runtimes for these methods have improved in the time since those results were published. Our results are shown in figure \\ref{snake_out}. Bootstrap support was over 90\\% for all clades except the (S.m.miliarius, S.m.barbouri) clade. For that clade, bootstrap support was 34\\% for Lily-T and 67\\% for Lily-Q. SVDQuartets also exhibited low bootstrap support for this clade, so the weak support may be a function of the weak phylogenetic signal present in the data.\n\n\\begin{figure}\n    \\centering\n   \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.4]{\"snakeT\".png}\n     \\caption{Lily-T}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.4]{\"snakeQ\".png}\n     \\caption{Lily-Q}\n   \\end{subfigure}\n    \\caption{Estimates for 7-taxon snake tree with bootstrap support for each clade. (Species names abbreviated for clarity. Full names in \\cite{chifman14}). Lily-Q display assumes root location is known.}\n    \\label{snake_out}\n\\end{figure}\n\nFinally, we estimated the topology for four mosquito species using aligned sequences consisting of over 25 million sites from around 80,000 different genes. For Lily-T, we ran 100 bootstrap samples and there was 100\\% bootstrap support for each node, with total computation time under an hour using a Unix HPC platform even with this large dataset. Because we did not have the delineation of the different genes, we resampled the sites for bootstrapping in a single stage. The estimated species tree is shown in figure \\ref{mos_out} and matched the results from \\cite{tha18}. Since there were only four taxa, Lily-Q could calculate a posterior probability of 100\\% using a single run which took around 15 seconds. \n\n\\begin{figure}\n    \\centering\n   \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"mosT\".png}\n     \\caption{Lily-T}\n   \\end{subfigure}\n      \\begin{subfigure}{0.49\\linewidth} \\centering\n     \\includegraphics[scale=0.3]{\"mosQ\".png}\n     \\caption{Lily-Q}\n   \\end{subfigure}\n    \\caption{Estimates for 4-taxon yeast mosquito with bootstrap support (Lily-T). Lily-Q display assumes root location is known (posterior probability of 100\\% not shown).}\n    \\label{mos_out}\n\\end{figure}\n\n\\section{Conclusions and further work}\nLily-T and Lily-Q are two new methods for fast, accurate species tree estimation under the multispecies coalescent model along with the assumption of the molecular clock. The methods are insensitive to the value of the coalescent population parameter $\\theta$, and sensitivity to the prior on branching times can be controlled through our method for estimating root age from the data. Lily-Q is more accurate than Lily-T, but Lily-T does have certain advantages in that it can estimate the root location, and there is a theoretical guarantee as the number of unlinked sites goes to infinity.\n\nThe comparison between Lily-Q and ASTRAL follows our expectations in that ASTRAL is able to perform well under conditions when estimation of the individual genes is likely to be accurate -- when $\\theta$ is large and when there are a large number of sites per gene. It is worth noting that our comparisons are of a correctly specified ASTRAL model against an incorrectly specified Lily-Q model: we assume that we have delineated the genes correctly and that sites within each gene have a common history whereas Lily-Q assumes that all sites are unlinked. Even with this disadvantage, Lily-Q outperformed ASTRAL for many parameter settings. There is also reason to believe that the number of sites per gene needed for ASTRAL to do better than Lily-Q may not be realistic -- \\cite{hobolth11} estimated 75\\% of loci in a primate dataset have a recombination-free length of between 17 and 93bp and \\cite{springer16} reported a loci length of around 12bp for a large mammalian dataset. An avenue for further research would be to test Lily-Q against ASTRAL when the genes are not properly delimited to measure how much this degrades ASTRAL performance.\n\nWe also did not compare directly to any concatenation methods. These comparisons may indicate the limits of concatenation methods, especially for the highly asymmetric trees with short MBL. One comparison in particular that may be of interest would be to compare Lily-T to SMRT-ML \\cite{SMRTML}, which estimates rooted triplets using concatenated data. We should highlight that, unlike concatentation or coestimation methods, we can only estimate $S$, and not any other parameters of interest such as $\\boldsymbol{\\tau}$, although other methods for estimating $\\boldsymbol{\\tau}$ incorporating ILS such as that of \\cite{peng20} may be able to be used in conjunction with our work.\n\nWhile both the Lily-T and Lily-Q assembly steps generate errors in the face of three or more identical sequences, there are ways to work around this problem. For example, consider the case with six taxa where taxa $a$, $b$, and $c$ have identical sequences. As a first step $a$ and $b$ could be simply removed from consideration, inference applied to the remaining taxa (perhaps inferring the split $(c,d,(e,f))$, which results in the final inferred tree in figure \\ref{polytomy}. While such a workaround would be unwieldy for the large-scale simulation study performed here, it could be reasonably implemented on empirical data.\n\n\\begin{figure}\n            \\centering\n        \\includegraphics[width=0.5\\linewidth]{\"polytomy\".png}\n    \\caption{Handling identical sequences: After first treating the identical sequences as coming from a single species, the rest of the tree is resolved as $(c,d,(e,f))$. Then $c$ is replaced by the polytomy $(a,b,c)$.}\n    \\label{polytomy}\n\\end{figure}\n\nWe have presented some data showing that high bootstrap support is indicative of high tree estimation accuracy, but we wish to highlight that we need to explore this claim further in future work. First, to minimize simulation time, we only performed the bootstrap analysis on twenty different parameter settings. Second, we would like to perform further simulation studies to investigate whether bootstrap support for a particular clade has predictive value for the clade being present in the true tree.\n\nFinally, this work has all been done under the molecular clock assumption, and we have some early signs that the methods are in fact quite sensitive to this assumption. To see why, consider a true tree $((a,b),c)$ with the branch leading to $a$ having a much larger mutation rate than other branches. This would lead to a larger number of YXX patterns than under equal rates, which if the divergence in rates was large enough, would lead Lily-T to infer $(a,(b,c))$ rather than the true tree. We hope to investigate this in greater detail in future work. Correcting for sensitivity to violations of the clock would require deriving $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau},\\boldsymbol{\\gamma}) $ where $\\boldsymbol{\\gamma}$ is the vector of relative rates along different branches of the tree. Then we would need to either estimate $\\hat{\\boldsymbol{\\gamma}}$ from the data or apply a prior to $\\boldsymbol{\\gamma}$ and integrate over it to get $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}) $. Whether these steps can be performed without substantially slowing down the methodology remains an open question.\n\n\\section{Appendix}\n\n\\subsection{Calculation of $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}))$ for 3-taxon trees}\n\\label{tripletApp}\nThe fifteen site pattern probabilities derived in \\cite{chifman15} can be mapped down to the five site patterns for three taxa by summing over the fourth taxon we are not interested in to get the following relationships (dropping the conditioning on $(S, \\boldsymbol{\\tau})$ for clarity):\n$$p_0=\\delta_{XXX}=\\delta_{XXXX}+3\\delta_{YXXX}$$\n$$p_1=\\delta_{YXX}=\\delta_{XYXX}+\\delta_{XXYY}+2\\delta_{YZXX}$$\n$$p_2=\\delta_{XXY}=\\delta_{XYX}=\\delta_{XXXY}+\\delta_{XYYX}+2\\delta_{YXXZ}=\\delta_{XXYX}+\\delta_{XYXY}+2\\delta_{YXZX}$$\n$$p_3=\\delta_{XYZ}=\\delta_{XXYZ}+\\delta_{XYXZ}+\\delta_{XYZX}+\\delta_{XYZW}$$\nThese probabilities take the form:\n\\begin{eqnarray}\n\\label{3taxaProb1}\n  \\left\\{\n  \\begin{aligned}\n& p_0=c_0+3c_1+6c_2+12c_3\\\\\n& p_1=c_0+3c_1-2c_2-4c_3\\\\\n& p_2=c_0-c_1+2c_2-4c_3\\\\\n& p_3=c_0-c_1-2c_2+4c_3\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nwhere (measuring $\\tau_1$ and $\\tau_2$ in coalescent units):\n\\begin{eqnarray}\n\\label{3taxaProb2}\n  \\left\\{\n  \\begin{aligned}\n& c_0=1\/64\\\\\n& c_1=\\frac{e^{-8\\tau_1\\theta\/3}}{64(1+\\frac{8\\theta}{3})}\\\\\n& c_2=\\frac{e^{-8\\tau_2\\theta\/3}}{64(1+\\frac{8\\theta}{3})}\\\\\n& c_3=\\frac{e^{-4\\tau_1\\theta\/3}e^{-8\\tau_2\\theta\/3}}{128(1+\\frac{8\\theta}{3})(1+\\frac{4\\theta}{3})}\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nIt is easily verified that:\n\\begin{equation}\n\\label{3taxaProb3}\n    4p_0+12p_1+24p_2+24p_3=1\n\\end{equation}\nThe coefficients in equation \\ref{3taxaProb3} arise as follows. $X$ can represent any of $A$, $C$, $G$, or $T$. $Y$ can represent any of the remaining three characters, and $Z$ any of the remaining two. Finally, the coefficient of $p_2$ is doubled to account for both $\\delta_{XYX}$ and $\\delta_{XXY}$.\n\n\\subsection{Calculation of $\\boldsymbol{\\delta}|S$ for 3-taxon trees}\nOne can see from equation \\ref{3taxaProb1} that $\\delta_k|(S,\\boldsymbol{\\tau})$ is a linear combination of terms, so $\\delta_k|S$ is also linear combination of terms where each integral is of one of two forms (since the prior on $\\boldsymbol{\\tau}$ is $f(\\boldsymbol{\\tau}) = \\frac{1}{\\tau_2}\\beta e^{-\\beta\\tau_2}$): \n\n\\textbf{Form one:}\n\\begin{equation}\n\\label{post1}\n    \\int_0^{\\infty} \\int_0^{\\tau_2} \\frac{c}{\\tau_2}e^{-a\\tau_2} d\\tau_1 d\\tau_2=\\int_0^{\\infty} ce^{-a\\tau_2}d\\tau_2=\\frac{c}{a}\n\\end{equation}\n\n\\textbf{Form two:}\n$$\\int_0^{\\infty} \\int_0^{\\tau_2} \\frac{c}{\\tau_2}e^{-b\\tau_1}e^{-a\\tau_2} d\\tau_1 d\\tau_2$$\n$$=\\int_0^{\\infty}\\frac{c}{b\\tau_2} (1-e^{-b\\tau_2}) e^{-a\\tau_2}d\\tau_2$$\n\\begin{equation}\n\\label{post2}\n    =\\frac{c}{b}log(\\frac{a+b}{a})\n\\end{equation}\nTaken together, $\\delta_k|S$ has the linear form of equation \\ref{3taxaProb1} with the following terms replacing those of equation \\ref{3taxaProb2}: \n\\begin{eqnarray}\n\\label{3taxaProb4}\n  \\left\\{\n  \\begin{aligned}\n& c_0=1\/64\\\\\n& c_1=\\frac{\\beta}{64(1+\\frac{8\\theta}{3})(\\frac{8\\theta}{3})}log(\\frac{\\beta+\\frac{8\\theta}{3}}{\\beta})\\\\\n& c_2=\\frac{\\beta}{64(1+\\frac{8\\theta}{3})(\\beta+\\frac{8\\theta}{3})}\\\\\n& c_3=\\frac{\\beta}{64(1+\\frac{8\\theta}{3})(2+\\frac{8\\theta}{3})(\\frac{4\\theta}{3})}log(\\frac{\\beta+4\\theta}{\\beta+\\frac{8\\theta}{3}})\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nThese results were verified by Monte Carlo integration with 10,000 draws from the prior on $\\boldsymbol{\\tau}$ with $\\beta=0.1$.\n\n\\subsection{Calculation of $\\boldsymbol{\\delta}|S$ for 4-taxon trees}\n\nFrom Chifman and Kubatko (2015), $P(\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}))$ is the inner product $\\textbf{c}^{T}\\textbf{b}$ where $\\textbf{c}$ is a vector of constants (with respect to the branching times) and $\\textbf{b}$ is a vector of functions of the branching times. These two vectors (where $\\alpha=4\/3$ is a constant that comes from the mutation rate when time is measured in coalescent units) are:\n$$\\boldsymbol{b}_{sym}=\\{1,e^{-2\\alpha\\theta\\tau_1},e^{-2\\alpha\\theta\\tau_2}, e^{-2\\alpha\\theta(\\tau_1+\\tau_2)}, e^{-2\\alpha\\theta\\tau_3},\n           e^{-\\alpha\\theta(\\tau_1+2\\tau3)},  e^{-\\alpha\\theta(\\tau_2+2\\tau3)},$$  $$e^{-\\alpha\\theta(\\tau_1+\\tau_2+2\\tau3)},\n            e^{2\\alpha(\\tau_1+\\tau_2)-4\\theta\\tau_3(\\alpha+\\frac{1}{2\\theta})}\n            \\} $$\n$$\\boldsymbol{b}_{asymm}=\\{1,e^{-2\\alpha\\theta\\tau_1},e^{-2\\alpha\\theta\\tau_2}, e^{-\\alpha\\theta(\\tau_1+2\\tau_2)}, e^{-2\\alpha\\theta\\tau_3},\n           e^{-\\alpha\\theta(\\tau_1+2\\tau3)}, e^{-2\\alpha\\theta(\\tau_1+\\tau3)},$$ $$ e^{-\\alpha\\theta(\\tau_2+2\\tau3)},  e^{-\\alpha\\theta(\\tau_1+\\tau_2+2\\tau3)},\n            e^{2(\\tau_1-\\tau_2)-2\\theta\\alpha(\\tau_2+\\tau3)}\n            \\} $$\nSince the priors are $f(\\boldsymbol{\\tau}) = \\frac{1}{\\tau_3^2}\\beta e^{-\\beta\\tau_3}$ for the symmetric topology and $f(\\boldsymbol{\\tau}) = \\frac{2}{\\tau_3^2}\\beta e^{-\\beta\\tau_3}$ for the asymmetric topology, the integrals required to integrate out the branching times take on only eight different forms (four for each topology).\n\\subsubsection{Symmetric topology}\nForm One:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} c e^{-f\\tau_3}d\\tau_3= \\frac{c}{f}$$\nForm two:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\frac{c}{a\\tau_3}(1-e^{-a\\tau_3}) e^{-f\\tau_3}d\\tau_3= \\frac{c}{a}(log(a+f)-log(f))$$\nAfter applying integration by parts.\\\\\nForm three\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\frac{c}{b\\tau_3}(1-e^{-b\\tau_3}) e^{-f\\tau_3}d\\tau_3= \\frac{c}{b}(log(b+f)-log(f))$$\nForm four:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2}  e^{-a\\tau_1} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3= \\int_0^{\\infty} \\frac{c}{ab\\tau_3^2} (1-e^{-a\\tau_3})(1-e^{-b\\tau_3})e^{-f\\tau_3} d\\tau_3$$\nUse integration by parts to obtain (the first term vanishes):\n$$=\\frac{c}{ab}\\int_0^{\\infty}\\frac{1}{\\tau_3}(-fe^{-f\\tau_3}+(a+f)e^{-(a+f)\\tau_3}+(b+f)e^{-(b+f)\\tau_3}-(a+b+f)e^{-(a+b+f)\\tau_3}) d\\tau_3$$\nAfter applying integration by parts a second time (again, the first term vanishes):\n$$=\\frac{c}{ab}(flog(f)-(a+f)log(a+f)-(b+f)log(b+f)+(a+b+f)log(a+b+f))$$\n\\subsubsection{Asymmetric topology}\nForm One:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\frac{c}{2} e^{-f\\tau_3}d\\tau_3= \\frac{c}{2f}$$\nForm Two:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3 =\\int_0^{\\infty} \\int_0^{\\tau_3} \\frac{c}{a\\tau_3^2} (1-e^{-a\\tau_2})e^{-f\\tau_3} d\\tau_2 d\\tau_3$$\n$$=\\int_0^{\\infty} \\frac{c}{a^2\\tau_3^2}(a\\tau_3e^{-f\\tau_3}-e^{-f\\tau_3}+e^{-(a+f)\\tau_3}) d\\tau_3$$\n$$= \\int_0^{\\infty} \\frac{c}{a\\tau_3}e^{-f\\tau_3} d\\tau_3 + \\int_0^{\\infty} \\frac{c}{a^2\\tau_3^2} (e^{-(a+f)\\tau_3}-e^{-f\\tau_3})$$\nPerform integration by parts once on the second integral to obtain:\n$$= \\int_0^{\\infty} \\frac{c}{a\\tau_3}e^{-f\\tau_3} d\\tau_3 + \\int_0^{\\infty} \\frac{c}{a^2\\tau_3} (fe^{-f\\tau_3}-(a+f)e^{-(a+f)\\tau_3}) - \\frac{c}{a}$$\nGrouping terms:\n$$= \\frac{c(a+f)}{a^2} \\int_0^{\\infty} \\frac{1}{\\tau_3} (e^{-f\\tau_3}-e^{-(a+f)\\tau_3}) - \\frac{c}{a}$$\nAfter performing integration by parts again:\n$$=\\frac{c(a+f)}{a^2}(log(a+f)-log(f))-\\frac{c}{a} $$\nForm three:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\int_0^{\\tau_3} \\frac{c\\tau_2}{\\tau_3^2}e^{-b\\tau_2}e^{-f\\tau_3} d\\tau_2d\\tau_3$$\n$$=\\int_0^{\\infty} \\frac{c}{b^2\\tau_3^2}(e^{-d\\tau_3}-e^{-(b+d)\\tau_3}-b\\tau_3e^{-(b+d)\\tau_3}) d\\tau_3$$\nUsing integration by parts gives:\n$$=\\frac{c}{b}+\\int_0^{\\infty} \\frac{cf}{b^2\\tau_3} (e^{-(b+f)\\tau_3}-e^{-f\\tau_3})$$\nAfter applying integration by parts a second time:\n$$=\\frac{c}{b}+\\frac{cf}{b^2}(log(f)-log(b+f)) $$\nForm four: \n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\int_0^{\\tau_3} \\frac{c}{a\\tau_3^2} (1-e^{-a\\tau_2}) e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_2 d\\tau_3$$\n$$=\\int_0^{\\infty} \\frac{c}{ab(a+b)\\tau_3^2} (ae^{-f\\tau_3}-(a+b)e^{-(b+f)\\tau_3}+be^{-(a+b+f)\\tau_3}) d\\tau_3$$\nApplying integration by parts (the first term vanishes) gives:\n$$= \\int_0^{\\infty} \\frac{c}{ab(a+b)\\tau_3} (-afe^{-f\\tau_3}+(a+f)(b+f)e^{(b+f)\\tau_3}-b(a+b+f)e^{-a+b+f)\\tau_3}) d\\tau_3$$\nAfter integrating by parts a second time (again the first term vanishes):\n$$=\\frac{c}{ab(a+b)}(aflog(f)-(a+b)(b+f)log(b+f)+b(a+b+f)log(a+b+f))$$\n\\subsubsection{Example}\nAs an example, we will show the calculation of the second term of $\\textbf{c}^T\\textbf{d}$ for an asymmetric topology. From above, $b_2=e^{-2\\alpha\\theta\\tau_1}$ and so the integral is:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{2\\beta c_2}{\\tau_3^2} e^{-2\\alpha\\theta\\tau_1} e^{-\\beta\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3 $$\nThis is of form two where $c=2\\beta c_2$, $a=2\\alpha\\theta$, and $f=\\beta$.\nThen, from above,\n$$d_2=\\frac{2\\beta c_2(2\\alpha\\theta+\\beta)}{(2\\alpha\\theta)^2}(log(2\\alpha\\theta+\\beta)-log(\\beta))-\\frac{2\\beta c_2}{2\\alpha\\theta}$$\n\\subsubsection{Final results}\n$$d_{sym,1}=1 $$\n$$d_{sym,2}=d_{sym,3}=\\beta\\frac{log(2\\alpha\\theta+\\beta)-log(\\beta)}{2\\alpha\\theta}$$\n$$d_{sym,4}= \\beta\\frac{\\beta log(\\beta)-2(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)+(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta)}{4\\alpha^2\\theta^2}$$\n$$d_{sym,5}= \\beta\\frac{1}{2\\alpha\\theta+\\beta}$$\n$$d_{sym,6}= d_{sym,7}=\\beta\\frac{log(3\\alpha\\theta+\\beta)-log(2\\alpha\\theta+\\beta)}{\\alpha\\theta}$$\n$$d_{sym,8}= \\beta\\frac{(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)\n-2(3\\alpha\\theta+\\beta)log(3\\alpha\\theta+\\beta)\n+(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta)}{\\alpha^2\\theta^2}$$\n$$d_{sym,9}=\\beta(\\beta+4\\theta(\\alpha+\\frac{1}{2\\theta}))log((\\beta+4\\theta(\\alpha+\\frac{1}{2\\theta}))\n-2(\\beta-1+4\\theta(\\alpha+\\frac{1}{2\\theta}))log((\\beta-1+4\\theta(\\alpha+\\frac{1}{2\\theta}))$$\n$$+(\\beta-2+4\\theta(\\alpha+\\frac{1}{2\\theta}))log((\\beta-2+4\\theta(\\alpha+\\frac{1}{2\\theta}))$$\n$$d_{asymm,1}=1 $$\n$$d_{asymm,2}= 2\\beta\\frac{2\\alpha\\theta+\\beta}{4\\alpha^2\\theta^2}(log(2\\alpha\\theta+\\beta)-log(\\beta))-\\frac{2\\beta}{2\\alpha\\theta}$$\n$$d_{asymm,3}= 2\\beta\\frac{\\beta}{4\\alpha^2\\theta^2}(log(\\beta)-log(2\\alpha\\theta+\\beta))+\\frac{2\\beta}{2\\alpha\\theta}$$\n$$d_{asymm,4}= \\frac{2\\beta}{6\\alpha^3\\theta^3}(\\alpha\\theta\\beta log(\\beta)-(3\\alpha\\theta)(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)+(2\\alpha\\theta)(3\\alpha\\theta+\\beta)log(3\\alpha\\theta+\\beta))$$\n$$d_{asymm,5}=\\frac{2\\beta}{4\\alpha\\theta+2\\beta}$$\n$$d_{asymm,6}=2\\beta\\frac{3\\alpha\\theta+\\beta}{\\alpha^2\\theta^2}(log(3\\alpha\\theta+\\beta)-log(2\\alpha\\theta+\\beta))-\\frac{2\\beta}{\\alpha\\theta}$$\n$$d_{asymm,7}=2\\beta\\frac{4\\alpha\\theta+\\beta}{4\\alpha^2\\theta^2}(log(4\\alpha\\theta+\\beta)-log(2\\alpha\\theta+\\beta))-\\frac{2\\beta}{2\\alpha\\theta}$$\n$$d_{asymm,8}=2\\beta\\frac{2\\alpha\\theta+\\beta}{\\alpha^2\\theta^2}(log(2\\alpha\\theta+\\beta)-log(3\\alpha\\theta+\\beta))+\\frac{2\\beta}{\\alpha\\theta}$$\n$$d_{asymm,9}=\\frac{2\\beta}{2\\alpha^3\\theta^3}(\\alpha\\theta(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)-(2\\alpha\\theta)(3\\alpha\\theta+\\beta)log(3\\alpha\\theta+\\beta)+(\\alpha\\theta)(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta))$$\n$$d_{asymm,10}=\\frac{-2\\beta}{(2\\alpha\\theta)(1+2\\alpha\\theta)}(-(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)-(2\\alpha\\theta)(1+4\\alpha\\theta+\\beta)log(1+4\\alpha\\theta+\\beta)$$\n$$+(1+2\\alpha\\theta)(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta)) $$\n\n\\subsection{Proof of Theorem 1}\n\\label{mainTheorem}\n\n\\begin{proof}\nWithout loss of generality, let $S_0=(a,(b,c))$ be the true tree and $\\boldsymbol{\\tau}_0$ be the true value of $\\boldsymbol{\\tau}$. We want to show that $\\forall \\epsilon>0 \\, \\, \\exists J_0 : \\forall J>J_0 \\, \\, P(\\hat{S}=S)>1-\\epsilon$. First, note that for all values of $(\\boldsymbol{\\tau}, \\theta)$, we have $\\delta_{XXY}=\\delta_{XYX}$. \n\nSecond, we note using the results of equation \\ref{3taxaProb1}\n\\begin{equation}\n    \\delta_{YXX}|((a,(b,c)),\\boldsymbol{\\tau}) > \\delta_{XYX}|((a,(b,c)),\\boldsymbol{\\tau})\n    \\label{expectation}\n\\end{equation}\n$$\\leftrightarrow c_0+3c_1-2c_2-4c_3 > c_0-c_1+2c_2-4c_3$$\n$$\\leftrightarrow \\frac{e^{-2m\\tau_1\\theta}}{64(1+2m\\theta)}=c1>c2=\\frac{e^{-2m\\tau_2\\theta}}{64(1+2m\\theta)}$$\n$$\\leftrightarrow \\tau_1<\\tau_2$$\nwhich holds w.p.1 by assumption.\n\nNext, note \n$$\\int_{\\boldsymbol{\\tau}}\\delta_{YXX}|((a,(b,c)),\\boldsymbol{\\tau})f(\\boldsymbol{\\tau})d(\\boldsymbol{\\tau})=\\delta_{YXX}|(a,(b,c))$$ \n$$> \\delta_{XYX}|(a,(b,c))=\\int_{\\boldsymbol{\\tau}}\\delta_{XYX}|((a,(b,c)),\\boldsymbol{\\tau})f(\\boldsymbol{\\tau})d(\\boldsymbol{\\tau})$$\nby properties of expectations from equation \\ref{expectation} and since $f(\\boldsymbol{\\tau})>0$ a.e. under the prior. \n\nSince each topology $(a,(b,c))$, $(b,(a,c))$, and $(c,(a,b))$ has a prior probability of 1\/3, the maximum posterior topology will be the maximum likelihood topology. Recall from section \\ref{derivation} that we can permute any site pattern probability given one topology to find a site pattern probability given another topology -- e.g.,   $\\delta_{YXX}|(a,(b,c))=\\delta_{XYX}|(b,(a,c))$.\nSo, the likelihood of trees $(a,(b,c))$ and $(b,(a,c))$ given the data are:\n$$L((a,(b,c))|\\boldsymbol{d}) =c_{\\boldsymbol{d}} (p_{xxx})^{d_{xxx}}(p_{xxy})^{d_{xxy}}(p_{xyx})^{d_{xyx}}(p_{yxx})^{d_{yxx}}(p_{xyz})^{d_{xyz}} $$\n$$L((b,(a,c))|\\boldsymbol{d}) =c_{\\boldsymbol{d}} (p_{xxx})^{d_{xxx}}(p_{xxy})^{d_{xxy}}(p_{xyx})^{d_{yxx}}(p_{yxx})^{d_{xyx}}(p_{xyz})^{d_{xyz}} $$\nBy cancelling terms in common, one can see that the likelihood ratio comes down to permuting the number of sites that follow the XYX and YXX patterns:\n$$\\frac{L((a,(b,c))|\\boldsymbol{d})}{L((b,(a,c))|\\boldsymbol{d})}= (p_{xyx})^{d_{xyx}-d_{yxx}}(p_{yxx})^{d_{yxx}-d_{xyx}}$$\nTogether $p_{yxx}>p_{xyx}$ and $d_{yxx}>d_{xyx}$ imply that  $\\frac{L((a,(b,c))|\\boldsymbol{d})}{L((b,(a,c))|\\boldsymbol{d})}>1$, which implies that $\\hat{S}=S$. As we have shown that $p_{yxx}>p_{xyx}$, it is sufficient to show that $d_{yxx} \\rightarrow p_{yxx}$ and $d_{xyx} \\rightarrow p_{xyx}$ for sufficiently large number of sites.\n\nLet $p_{yxx,0}=\\delta_{yxx}|((a,(b,c)),\\boldsymbol{\\tau}_0)$ and $p_{xyx,0}=\\delta_{xyx}|((a,(b,c)),\\boldsymbol{\\tau}_0)$. From the SLLN, we have \n$$\\forall \\epsilon_1,\\delta_1>0 \\, \\, \\exists J_1 : \\forall J>J_1 \\, \\, P(\\frac{d_{yxx}}{J}>p_{yxx,0}-\\delta_1)>1-\\epsilon_1$$\n$$\\forall \\epsilon_2,\\delta_2>0 \\, \\, \\exists J_2 : \\forall J>J_2 \\, \\, P(\\frac{d_{xyx}}{J}<p_{xyx,0}+\\delta_2)>1-\\epsilon_2$$\nChoose $\\delta_1,\\delta_2$ such that $\\delta_1+\\delta_2<p_{yxx,0}-p_{xyx,0}$,  $\\epsilon_1=\\epsilon_2=\\epsilon\/2$, and $J_0=max\\{J_1,J_2\\}$ by applying the Bonferroni inequality we have that  $P(\\frac{d_yxx}{J}>p_{yxx,0}-\\delta_1, p_{xyx,0}+\\delta_2>\\frac{d_{xyx}}{J})>1-\\epsilon$ and the result holds.\n\n\\end{proof}\n\n\n\n\\def\\BibTeX{{\\rm B\\kern-.05em{\\sc i\\kern-.025em b}\\kern-.08em\n    T\\kern-.1667em\\lower.7ex\\hbox{E}\\kern-.125emX}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the field-antifield formalism \\cite{BV,BV1,BV2}, the concept of\ndeformations based on a nilpotent higher-order operator $\\Delta$ was\ndeveloped in a series of articles\n\\cite{BT3,Ak,AD,BDA,BBD1,BBD3,BM1,BM2,BM3}. Such deformations\nusually modify the Jacobi identity with BRST exact terms. In\ncontrast to that, one can, with no assumptions {\\it a priori} on\nunderlying $\\Delta$ operator, consider \"local\" deformations of the\nantibracket with a Boson deformation parameter, such that the Jacobi\nidentity holds strongly.\n\nHistorically,  the deformed antibracket has been studied in the\narticles \\cite{LSh,KoT,KoT1,KoT2}. In Ref. \\cite{BB2}, the deformed\n$\\Delta$-operator has been found that differentiates the deformed\nantibracket, and the first attempt has been made to understand\nactually possible role of the deformed antibracket and\n$\\Delta$-operator  in the construction of the $W-X$ version\n\\cite{BT1,BT2,BT3,BMS,BT4,BBD1,BBD2,BB3} of the  Lagrange deformed\nfield-antifield Batalin-Vilkovisky (BV)-formalism \\cite{BB2}. If one\nbelieves that the deformed BV - formalism still describes\ngauge-invariant field systems, then there appears a difficult\nproblem of how the deformation can coexist with the usual\ngauge-fixing mechanism. Or, in other words, if that is possible to\nprovide for a  proper solution to the deformed classical\/quantum\nmaster equation.  In the present article, we will  try to pay\nattention enough to seek for a possible way to resolve the mentioned\nproblem.\n\nIn principle, our present consideration is based essentially on the logic\nand mathematics of the article \\cite{BB2} of Batalin and Bering.\nRegrettably, these authors did not\naccomplish their task of construction of the nontrivially deformed field-antifield\nformalism\nbased on the deformed antibracket and $\\Delta$ operator. The main idea was to extend the\noriginal antisymplectic phase space with a single extra field-antifield pair just\ncontrolling the scale of deformation. Then, one defines a trivial deformation\nwithin the extended phase space,\nand then one should reduce effectively the scale of trivial deformation in\nsuch a way that the latter becomes nontrivial in a consistent manner. That idea seems\npromising, the same as before. However, there remains a difficult unresolved problem of\nconsistent coexistence between the  properness principle and non-triviality of\nthe deformation. We would like to try again to attack that problem.\n\n\n\\section{Extended $\\Delta$-Operator}\n\nWe begin with the standard odd Laplacian operator,\n\\beq \\label{i2}\n\\Delta = \\frac{1}{2}(-1)^{\\varepsilon_{A}} \\pa_{A} E^{AB} \\pa_{B},\\quad\n\\pa_A= \\frac{\\pa}{\\pa Z^A}, \\quad\n\\varepsilon(\\Delta) = 1,  \\quad \\Delta^{2} = 0,  \n\\eeq where $Z^{A}$ , $\\varepsilon_{A} = \\varepsilon( Z^{A})$, are\noriginal  Darboux coordinates of the field-antifield formalism, and\n$E^{AB}$ is a constant invertible antisymplectic metric with the\nusual  statistics and dual antisymmetry properties,\n\\beq\n\\label{2.2}\n\\varepsilon( E^{AB} ) = \\varepsilon_{A} + \\varepsilon_{B} + 1, \\quad\nE^{AB} = - E^{BA}(-1)^{ (\\varepsilon_{A} + 1) (\\varepsilon _{B} +\n1)}.\n\\eeq\nIn what follows below, typical functions, depending only\non the $Z^{A}$ variables of the original sector, will be denoted in\nsmall letters. Now, let us extend the original $Z$-sector by\nincluding two new variables, a Boson $t$ and a Fermion $\\theta$, to\nextend the original odd Laplacian, to become \\cite{BB2} (see also\nAppendix A)\n\\beq\n\\label{i1} \\Delta_{\\tau} = t^{2} \\Delta + N_{\\tau}\n\\pa_{\\theta}, \\quad\n \\pa_{\\theta}=\\frac{\\pa}{\\pa \\theta}, \\quad\n \\varepsilon(\\Delta_{\\tau}) = 1,\\quad \\Delta^2_{\\tau} = 0,\n\\eeq\nwhere\n\\beq\n\\label{i3}\nN_{\\tau} = N + t \\pa_{t}, \\quad N = N^{A} \\pa_{A}.      \n\\eeq\nIn what follows below, typical functions depending on the full set of variables,\n$Z^{A}, t, \\theta$, will be denoted in capital letters.\nNilpotency of $\\Delta_{\\tau}$ requires\n\\beq\n\\label{i4}\n[ \\Delta, N ] = 2 \\Delta,      \n\\eeq\nor, in more detail,\n\\beq\n\\nonumber\n&&[ \\Delta, N ] = [ \\Delta, N^{A} ] \\pa_{A} = \\left[ ( \\Delta N^{A} ) +\n{\\rm ad}( N^{A} ) (-1)^{\\varepsilon_{A}} \\right]\\pa_{A} = \\\\\n\\nonumber\n&&=( \\Delta N^{A} ) \\pa_{A} +\n\\frac{1}{2} \\left[ ( N^{A}, Z^{B} ) -\n( A \\leftrightarrow B ) (-1)^{ (\\varepsilon_{A} +1) (\\varepsilon_{B} +1) } \\right]\n(-1)^{\\varepsilon_{A}} \\pa_{B}\\pa_{A} =\\\\\n\\label{i5}\n&&=2\\Delta  =  E^{AB} (-1)^{\\varepsilon_{A}} \\pa_{B} \\pa_{A},     \n\\eeq\nwhich, in turn, implies\n\\beq\n\\label{i6}\n\\Delta N^{A} = 0,        \n\\eeq\nand\n\\beq\n\\label{i7}\nE^{AB} =\\frac{1}{2} \\left[ ( N^{A}, Z^{B} ) - ( A \\leftrightarrow B )\n (-1)^{ ( \\varepsilon_{A} +1 ) ( \\varepsilon_{B} + 1 ) } \\right].    \n\\eeq\nHere and below the notation\n\\beq\n\\label{ad}\n{\\rm ad}(F)(...)=(F,(...))\n\\eeq\nfor the left adjoint of the antibracket is used.\n\nIt follows immediately from (\\ref{i7}) that\n\\beq\n\\label{i10}\n( N^{A}, N^{B} ) =\\frac{1}{2}\\left[ N^{A}\\overleftarrow{\\pa}_{C} ( N^{C}, N^{B} ) -\n(A\\leftrightarrow B )(-1)^{(\\varepsilon_{A} + 1 )( \\varepsilon_{B} + 1)} \\right].\n\\eeq Here, in (\\ref{i5}), (\\ref{i7}), (\\ref{ad}), (\\ref{i10}), the usual\nantibracket, generated by the operator $\\Delta$, is used\n\\beq\n\\label{i8} ( f, g ) = (-1)^{\\varepsilon(f)}[ [ \\Delta, f ], g ]\n\\cdot 1=f \\overleftarrow{\\pa}_{A} E^{AB} \\overrightarrow{\\pa}_{B} g .     \n\\eeq\nThus, the coefficients $N^{A}$ of the vector field $N$ should satisfy Eqs.\n(\\ref{i6}) and (\\ref{i7}).\nOf course,  the simplest solution is obvious,\n\\beq\n\\label{i9}\nN^{A} = Z^{A},   \\quad  N = Z^{A}\\pa_{A}. \n\\eeq\nThat was exactly the simplest ansatz  used in the article \\cite{BB2} from the very\nbeginning. Here, in the present article, we do not restrict ourselves with any {\\it a priori}\nchoice of a special solution to Eqb. (\\ref{i6}) and (\\ref{i7}). Only these equations\nthemselves will be used in our further reasoning, nothing else.\nIt can be shown that the general solution to Eq. (\\ref{i7}) is\n\\beq\n\\label{i11}\nN^{A} = Z^{A} + 2( F, Z^{A} ), \\quad   \\varepsilon( F ) = 1      \n\\eeq\nwith $F$ being arbitrary Fermion.\nIn that case, Eqs. (\\ref{i6}) and (\\ref{i7}) are satisfied as follows:\n\\beq\n\\label{i12}\nE^{AB} =\\frac{1}{2}\\left[ 2 E^{AB} + 2(F, E^{AB} ) \\right] = E^{AB},\n\\eeq\n\\beq\n\\label{i13}\n\\Delta N^{A} = 2(\\Delta F, Z^{A} ) = 0,  \n\\eeq\n\\beq\n\\label{i14}\n\\Delta F = {\\rm const}( Z ).      \n\\eeq\nThus, we find\n\\beq\n\\label{i15}\nN = Z^{A}\\pa_{A} + 2\\;{\\rm ad}( F ).  \n\\eeq\n\nOne can rewrite Eq. (\\ref{i7}) in its natural form\n\\beq\n\\label{i16}\n\\delta^{A}_{D} - \\frac{1}{2} N^{A} \\overleftarrow{\\pa}_{D} =\n\\frac{1}{2} E^{AB} (\\overrightarrow{\\pa}_{B} N^{C } ) E_{CD},       \n\\eeq\nwhere $E_{AB}$ is the inverse to $E^{AB}$,\n\\beq\n\\label{i17}\nE^{AB} E_{BC} = \\delta^{A}_{C}.   \n\\eeq\nNow, the super trace of (\\ref{i16}) yields\n\\beq\n\\label{i18}\n\\sTr I = \\delta^{A}_{A} (-1)^{\\varepsilon_{A}} = 0,  \n\\eeq\nthat is fulfilled identically due to equal  number of Bosons and Fermions\namong the variables $Z^{A}$.\nThe supertrace imposes no restrictions for the divergence of vector field $N$,\n\\beq\n\\label{i19}\n\\div N = \\pa_{A} N^{A} (-1)^{\\varepsilon_{A}},      \n\\eeq\nNote that $E^{AB}$ and $E_{AB}$ on the right-hand side of\n(\\ref{i16}) do enter in the form of a similarity transformation (\nthat is just the meaning  of the naturalness of (\\ref{i16}) ) and,\ntherefore, they cancel each other when taking the supertrace\nor superdeterminant. Notice also, that (\\ref{i16}) is a \"bridge\"\nbetween the initial equation (\\ref{i7}) and its \"dual\" form, \\beq\n\\label{i20} E_{AB} = \\frac{1}{2} \\left[ E_{AC} ( N^{C}\n\\overleftarrow{\\pa}_{B} ) -\n( A \\leftrightarrow B ) (-1)^{ \\varepsilon_{A} \\varepsilon_{B} } \\right].  \n\\eeq\nIn turn, by substituting\n\\beq\n\\label{i20.7}\nN^{A} = - 2 E^{AB} V_{B}\n = 2 V_{B}  E^{BA}, \n\\eeq\n\\beq\n\\varepsilon( V_{A} ) = \\varepsilon_{A} + 1,  \n\\eeq into (\\ref{i20}), the latter takes the form\n\\beq\n\\label{i20.8}\nE_{AB}  = \\pa_{A} V_{B} -\n\\pa_{B} V_{A} (-1)^{\\varepsilon_{A}\\varepsilon_{B}},\n\\eeq\nwhich tells us that $V_{A}$ is just the antisymplectic potential,\ngenerating a constant invertible\nmetric in its covariant components $E_{AB}$. Thereby, one realizes\nthat the arbitrariness in $N^{A}$\nis generated by the natural geometric arbitrariness in the choice of\nthe antisymplectic potential.\nIt can be shown that the general solution to Eq. (\\ref{i20.8}) is\n\\beq\n\\label{i.V}\nV_{B}  = \\frac{1}{2} Z^{A} E_{AB} + \\pa_{B} F,\n\\eeq\nwith $F$ being arbitrary Fermion, so that (\\ref{i.V})\nis consistent with (\\ref{i20.8}), (\\ref{i20.7}).\nOf course, it follows from (\\ref{i6}), (\\ref{i20.7}), that  the antisymplectic potential\n$V_{A}$ should satisfy the condition\n\\beq\n\\label{i20.9}\n\\Delta V_{A} = 0,   \n\\eeq\nwhich is consistent with (\\ref{i14}).\nThe relation (\\ref{i20.8}) is invariant under the shift,\n\\beq\n\\label{i2.31}\nV_{A} = V'_{A} + \\pa_{A} F' ,\\quad  \\varepsilon( F' ) = 1.\n\\eeq\nOn the other hand, we have,\n\\beq\n\\label{i2.32}\n\\Delta V_{A}=\\Delta V'_{A}+\\pa_{A} \\Delta F'(-1)^{\\varepsilon_A} = 0,   \n\\eeq\n\\beq\n\\label{i2.33}\nN^{A} = N^{'A}  - 2 E^{AB} \\pa_{B} F', \n\\eeq\n\\beq\n\\label{i2.34}\n\\Delta N^{A} = \\Delta N^{'A}  +\n2 E^{AB} \\pa_{B} \\Delta F' (-1)^{\\varepsilon_{A}} = 0, \n\\eeq\n\\beq\n\\label{i2.35}\nN = N' + 2 {\\rm ad}( F' ) , \n\\eeq\n\\beq\n\\label{i2.36}\n\\div N = \\div N' - 4 \\Delta F' .   \n\\eeq\nSo, if one chooses in (\\ref{i2.31}) - (\\ref{i2.35}) for the $F'$ to satisfy the relation\n\\beq\n\\label{i2.37}\n\\div N' = \\div N + 4 \\Delta F' = 0,  \n\\eeq\nthen the new value (\\ref{i2.37}) of the $\\div N'$ is zero. Thereby, the new\noperator\n\\beq\n\\label{i2.38}\nN' = N - 2 {\\rm ad}(F') = - N'^{T},  \n\\eeq is an antisymmetric one. The transposed operation is defined\nvia\n\\beq\n\\int [dZ] (A^TF)G=(-1)^{\\varepsilon(A)\\varepsilon(F)}\\int [dZ]\nF(AG). \\eeq So far, the condition (\\ref{i2.37}) seems to be the only\nrestriction on $F'$. However, let us consider the commutator \\beq\n\\label{i2.39}\n[ \\Delta, N' ]=[ \\Delta,N - 2 {\\rm ad}( F' ) ] = 2 \\Delta -2 {\\rm ad}( \\Delta F' ).\n\\eeq\nSo, if we would like for the new operator $N'$ to maintain the relation\n\\beq\n\\label{i2.40}\n[ \\Delta, N' ] = 2 \\Delta,   \n\\eeq\nthen there should be\n\\beq\n\\label{i2.41}\n\\Delta F' = {\\rm const} (Z).   \n\\eeq\nDue to the latter, it follows from (\\ref{i2.34})\n\\beq\n\\label{i2.42}\n\\Delta N^{A} = \\Delta N'^{A} = 0.   \n\\eeq\nIn turn, due to (\\ref{i2.41}), it follows from (\\ref{i2.37}) that\n\\beq\n\\label{i2.43}\n\\div N = {\\rm const} (Z).   \n\\eeq\nThus, we see that the deviation from zero allowed for $\\div N$ is not so\narbitrary. That is because  the condition (\\ref{i2.42}) is rather restrictive.\nWe see that the new antisymmetric $N'$ does maintain all the basic\nconditions (\\ref{i2.40}) and (\\ref{i2.42}), provided the condition (\\ref{i2.43}) holds.\nIn what follows below, we do mean that our $N'$-operator is chosen just in its\nantisymmetrical form (\\ref{i2.38}), \"from the very beginning\".  For brevity,\nin all further formulae we omit the prime of $N'$.\n\nThere exists a crucially important consequence of (\\ref{i7}),\nthat the operator $( N - 2 )$ does differentiate the antibracket,\n\\beq\n\\label{i26}\n( N - 2 ) ( f, g ) = ( ( N - 2 ) f, g ) + ( f, ( N - 2 ) g ).   \n\\eeq\nThat goes as follows,\n\\beq\n\\nonumber\n&&N ( f, g ) = ( N f, g ) + ( f, N g) - f \\overleftarrow{\\pa}_{A} \\left[ (\nN^{A}\\overleftarrow{\\pa}_{C} ) E^{CB} -\n( A \\leftrightarrow B ) (-1)^{ ( \\varepsilon_{A} + 1 ) ( \\varepsilon_{B} + 1\n) } \\right]\\overrightarrow{\\pa} _{B} g  =\\\\\n\\label{i27}\n&&=( N f, g ) + ( f, N g ) - 2 ( f, g ),   \n\\eeq\nwhich is equivalent to (\\ref{i26}).\n\n\\section{Extended Antibracket}\n\n\nNow, let us consider the antibracket generated by the extended operator\n$\\Delta_{\\tau}$ \\cite{BB2},\n\\beq\n\\label{i28}\n( F, G )_{\\tau} = (-1)^{\\varepsilon(F)}[ [ \\Delta_{\\tau}, F], G ]\\cdot 1=t^{2} ( F, G ) +\n( N_{\\tau} F ) \\pa_{\\theta} G - F \\overleftarrow{\\pa}_{\\theta} N_{\\tau} G,     \n\\eeq\nwhere $( F, G )$ is the ( usual ) antibracket (\\ref{i8}) in the original $Z^{A}$ -sector,\nalthough functions $F, G$ themselves,\nstanding for $f, g$, respectively, do depend on $t, \\theta $ as well. $N_{\\tau}$ is\ndefined in (\\ref{i3}).  Due to (\\ref{i26}), the\noperator $( N_{\\tau} - 2 )$ does differentiate the usual antibracket $( F, G )$,\nas well. In fact, we will use the relation equivalent to that,\n\\beq\n\\label{i30}\nN_{\\tau} ( F, G ) = ( N_{\\tau} F, G ) + ( F, N_{\\tau} G ) - 2 ( F, G).\n\\eeq\nOne can state that the extended antibracket (\\ref{i28}) does satisfy the strong\nJacobi identity, provided  the usual\nantibracket has that property. In particular, one assumes that the strong\nJacobi identity holds for any Boson $B$,\n\\beq\n\\label{i31}\n( ( B, B ) , B ) = 0, \\quad     \\varepsilon( B ) = 0.\n\\eeq In its general form, the Jacobi identity can be reproduced from\n(\\ref{i31}) via the differential polarization procedure. To do this,\none has to choose a specific form for $B$\n\\cite{B},\n\\beq\n\\label{i32}\nB = \\sum_{i = 1}^{3} m_{i} n_{i},  \\quad  n_{1} = F,\\;  n_{2} = G, \\; n_{3} =H.\n\\eeq\nThen, the operator\n\\beq\n\\label{i33}\n\\pa_{1} \\pa_{2} \\pa_{3} (-1)^{ ( \\varepsilon_{1} + 1 ) ( \\varepsilon_{3}\n+ 1 ) + \\varepsilon_{2} },   \n\\eeq\nshould be applied to (\\ref{i31}), where $\\pa_{i}$ are partial $m_{i}$ -derivatives,\n$\\varepsilon_{i}$ are Grassmann parities,\n\\beq\n\\label{i34}\n\\varepsilon_{i} = \\varepsilon( n_{i} ) = \\varepsilon( m_{i} ).  \n\\eeq\n\nIt is our task now, to prove that the extended antibracket (\\ref{i28}) satisfies\nthe compact form of the strong Jacobi identities,\n\\beq\n\\label{i35}\n( ( B, B )_{\\tau}, B )_{\\tau} = 0,   \\quad     \\varepsilon( B ) = 0, \n\\eeq\nprovided  similar compact form (\\ref{i31}) holds. We have\n\\beq\n\\label{i36}\n( B, B )_{\\tau} = t^{2} ( B, B ) + 2 ( N_{\\tau} B ) ( \\pa_{\\theta} B).\n\\eeq\nBy substituting that in (\\ref{i35}), one gets\n\\beq\n\\nonumber\n&&(( B, B)_{\\tau}, B )_{\\tau} = t^{2}( t^{2}( B, B )+2(N_{\\tau}B)(\\pa_{\\theta}B),B)+\n( N_{\\tau} ( t^{2} ( B, B ) +\\\\\n\\label{i37}\n&&+\n2( N_{\\tau}B)(\\pa_{\\theta} B ) ))( \\pa_{\\theta} B ) -\n( \\pa_{\\theta} ( t^{2} ( B, B ) + 2 ( N_{\\tau} B ) ( \\pa_{\\theta} B )\n) ) ( N_{\\tau} B ),   \n\\eeq\nwhere the relations (\\ref{i30}), (\\ref{i31}) will be used, together with\n\\beq\n\\label{i38}\n( \\pa_{\\theta} B )^{2} = 0    \n\\eeq\nThus, the right-hand side of (\\ref{i37}) takes the form\n\\beq\n\\nonumber\n&&( 2 t^{2} ( B, B ) + 2 t^{2} ( N_{\\tau} B, B ) - 2 t^{2} ( B, B ) )\n( \\pa_{\\theta} B ) +\n2 t^{2} ( N_{\\tau} B ) ( \\pa_{\\theta} B, B )-\\\\\n\\nonumber\n &&- 2 t^{2} ( N_{\\tau}\nB, B ) ( \\pa_{\\theta} B ) -\n2 t^{2} ( \\pa_{\\theta} B, B ) ( N_{\\tau} B ) + \\\\\n\\label{i39}\n&&2 ( N_{\\tau} B ) (\nN_{\\tau} \\pa_{\\theta} B ) ( \\pa_{\\theta} B ) -\n2 ( \\pa_{\\theta} N_{\\tau} B ) ( \\pa_{\\theta} B ) ( N_{\\tau} B ) = 0.  \n\\eeq\nHere, the first and third, the second and fifth, the fourth and sixth, the seventh and\neight terms,  compensate each other in\nevery pair mentioned.  Finally, the strong Jacobi identity (\\ref{i35}) for the\nextended antibracket (\\ref{i28}) is proven.\n\n\\section{ Nontrivial Deformation in the Sector of Original Variables}\n\nIn turn, let us study the role of the operator $N$ in construction of a\nnontrivially deformed antibracket in the original\n$Z^{A}$-sector. So, let $\\kappa$ be a deformation parameter. Consider the\noperator\n\\beq\n\\label{i40}\nK = K(N) = \\kappa ( N - 2 ),\\quad       K(N + 2) = \\kappa N.\n\\eeq\nWe have\n\\beq\n\\label{i41}\nK(N + 2) - K(N) = 2 \\kappa,    \n\\eeq\n\\beq\n\\label{i42}\nK(N) ( f g )=( K(N + 2) f ) g + f( K(N) g)=( K(N) f ) g+f( K(N + 2)g ),     \n\\eeq\n\\beq\n\\label{i43}\nK ( f, g ) = ( K f, g ) + ( f, K g ).    \n\\eeq\nBy using the well-known Witten formula for the usual antibracket,\n\\beq\n\\label{i44}\n( f, g) = \\Delta( f g ) (-1)^{\\varepsilon(f)} - [ f (\\Delta g) + (\\Delta f )\ng (-1)^{\\varepsilon(f)} ],    \n\\eeq\nwe define the deformed antibracket by the relation \\cite{BB2},\n\\beq\n\\label{i45}\n( f, g )_{*} = \\Delta ( f g ) (-1)^{\\varepsilon(f)} -\n( 1 - K ) [ f (\\Delta_{*} g) + ( \\Delta_{*} f ) g (-1)^{\\varepsilon(f)} ] ,\n\\eeq\nwhere\n\\beq\n\\label{i46}\n\\Delta_{*} = \\Delta (1 - K )^{-1} = ( 1 - K( N + 2 ) )^{-1} \\Delta.\n\\eeq\nBy using (\\ref{i42}) and (\\ref{i44}), the relation (\\ref{i45}) can be rewritten in the form\n\\beq\n\\label{i47}\n( f, g )_{*} = ( f, g ) + ( K f ) ( \\Delta_{*} g ) + ( \\Delta_{*} f ) ( K g\n) (-1)^{\\varepsilon(f)}.     \n\\eeq\nUsually, the deformation of the antibracket is defined by that formula.\nIt follows from (\\ref{i45})\n\\beq\n\\label{i48}\n\\Delta_{*} ( f, g )_{*} = - \\Delta \\left[ f ( \\Delta_{*} g) + ( \\Delta_{*} f) g\n(-1)^{\\varepsilon(f)} \\right].   \n\\eeq\n\\beq\n\\label{i49}\n( ( \\Delta_{*} f ), g )_{*} = \\Delta ( ( \\Delta_{*} f ) g)\n(-1)^{\\varepsilon(f) + 1} -\n( 1 - K ) [ ( \\Delta_{*} f ) ( \\Delta_{*} g ) ],    \n\\eeq\n\\beq\n( f, ( \\Delta_{*} g ) )_{*} = \\Delta ( f ( \\Delta_{*} g ) )\n(-1)^{\\varepsilon( f )} -\n( 1 - K ) [ ( \\Delta_{*} f ) ( \\Delta_{*} g ) ] (-1)^{\\varepsilon( f )} ,\n\\eeq\n\\beq\n\\label{i50}\n( ( \\Delta_{*} f ), g )_{*} - ( f, ( \\Delta_{*} g) )_{*}\n(-1)^{\\varepsilon(f)}  =\n\\Delta \\left[ - ( \\Delta_{*} f ) g (-1)^{\\varepsilon(f)} - f ( \\Delta_{*} g )\n\\right] = \\Delta_{*} ( f, g )_{*}.    \n\\eeq\nThe latter means that the deformed operator (\\ref{i46}) does differentiate the\ndeformed antibracket (\\ref{i45}), or (\\ref{i47}) \\cite{BB2}.\n\nFinally, one can state that the nontrivially deformed antibracket (\\ref{i45}), or\n(\\ref{i47}), does satisfy the strong\nJacobi identity. Indeed, we have\n\\beq\n\\nonumber\n&&( ( B, B )_{*}, B )_{*} = ( ( B, B ) + 2 ( K B ) ( \\Delta_{*} B ), B ) +\n( K ( ( B, B ) + 2 ( K B ) ( \\Delta_{*} B ) ) ) ( \\Delta_{*} B ) -\\\\\n\\nonumber\n&&-\n2 ( ( \\Delta_{*} B ), B )_{*}  ( K B ) = 2 ( K B ) ( ( \\Delta_{*} B ) , B\n) - 2 ( ( K B ), B ) ( \\Delta_{*} B ) +\n2 ( ( K B ), B ) ( \\Delta_{*} B )+\\\\\n\\label{i51}\n&& + 2 ( K ( ( K B ) ( \\Delta_{*} B ) ) ) (\n\\Delta_{*} B )  -\n2 ( ( \\Delta_{*} B ), B ) ( K B ) - 2 ( K \\Delta_{*} B ) ( \\Delta_{*} B )\n( K B ) = 0.           \n\\eeq\nHere, on the right-hand side of the last equality, the first and  fifth, the\nsecond and third, the fourth and\nsixth terms do compensate each other in every pair mentioned.  Thus, the\nstrong Jacobi identity for the\nnontrivially deformed antibracket (\\ref{i45}), or (\\ref{i47}), in the original $Z^{A}$\n-sector is proven.\n\n\n\\section*{5\\;\\; Trivial $\\tau$-Extended Deformation\n\\footnote{This  section represents the main  results and formulae of\nRef. \\cite{BB2}, related in general  to  trivial deformations\nin $\\tau$ -extended phase space.}}\n\\setcounter{section}{5}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\setcounter{equation}{0}\n\nNow, we have to consider a trivial $\\tau$ extended deformation in the extended\nphase space including\nthe variables $t$ and $\\theta$. Let us introduce the operator\n\\beq\n\\label{i52}\n K_{\\tau} = \\kappa N_{\\tau},     \\quad    [ K_{\\tau}, \\Delta_{\\tau} ]= 0,\n\\eeq\nand define trivially deformed  extended operator\n\\beq\n\\label{i53}\n\\Delta_{\\tau *} = \\Delta_{\\tau} ( 1 - K_{\\tau} )^{-1},\n\\quad \\Delta^2_{\\tau *} = 0.     \n\\eeq\nNow, introduce the operator\n\\beq\n\\label{i54}\nT = 1 + \\kappa \\theta \\Delta_{\\tau *} ,             \n\\eeq\nand its inverse\n\\beq\n\\label{i55}\nT^{-1} = 1 - \\kappa \\theta \\Delta_{\\tau}.                \n\\eeq\nThen, one finds that\n\\beq\n\\label{i56}\n\\Delta_{\\tau *}  = T^{-1} \\Delta_{\\tau} T.     \n\\eeq\nAlso, it follows that the operator T does satisfy the equations (see also App. D)\n\\beq\n\\label{5.6}\n[ \\Delta_{\\tau}, T ] = \\Delta_{\\tau} T K_{\\tau}, \\quad [ T, K_{\\tau} ] =0.\n\\eeq\nTogether with (\\ref{i1}) and (\\ref{i52}), these equations constitute what we\ncall \"T-algebra\".\n\nNext, define a trivially deformed extended antibracket,\n\\beq\n\\nonumber\n&&( F, G )_{\\tau *} = T^{-1} ( ( T F ), ( T G ) )_{\\tau} = \\\\\n\\nonumber\n&&=( F, G\n)_{\\tau} +\n( K_{\\tau} F ) ( \\Delta_{\\tau *} G ) + ( \\Delta_{\\tau *} F )\n(K_{\\tau} G ) (-1)^{\\varepsilon(F)} =\\\\\n\\label{i57}\n&&=\\Delta_{\\tau} ( F G ) (-1)^{\\varepsilon(F)} - ( 1 - K_{\\tau} )\\left [\nF ( \\Delta_{\\tau *} G ) +\n( \\Delta_{\\tau *} F ) G (-1)^{\\varepsilon(F)} \\right].\n\\eeq\nThe latter formula allows for a natural rewriting in terms of the $*$-\nmodified double-commutator formula generalizing (\\ref{i8}) and (\\ref{i28}),\n\\beq\n\\label{i57.3}\n( F, G )_{\\tau *}  =  (-1)^{\\varepsilon(F)} [ [ \\Delta_{\\tau *},\n( T F)_{*} ], ( T G )_{*} ]\\cdot 1 ,   \n\\eeq\nwhere we have used (\\ref{i28}) and (\\ref{i56}), and\n\\beq\n\\label{i57.4}\n( T F )_{*}  =  T^{-1} ( T F ) T,  \\quad   ( T G )_{*}  =  T^{-1} ( T G ) T.  \n\\eeq\nHere in (\\ref{i57.4}), on the right-hand sides, the operator $T$  in the middle\nfactors applies only to the function\nstanding to the right within the respective round bracket.\n\nNotice that within the class of functions,\n\\beq\n\\label{i57.1}\n F = t^{-2} f, \\quad  G = t^{-2} g,   \n\\eeq\nthe trivially deformed $\\tau$ -extended operator (\\ref{i56}) and antibracket (\\ref{i57})\nreduces, respectively,  to the\nnon-trivially deformed operator (\\ref{i46}) and antibracket (\\ref{i45})\nin the original sector,\n\\beq\n\\label{i57.2}\n\\Delta_{\\tau *}F=\\Delta_{*}f,\\quad ( F, G )_{\\tau *} = t^{-2} ( f, g )_{*} .  \n\\eeq\n\nBy construction, the trivially deformed extended antibracket (\\ref{i57}) does\nsatisfy the strong Jacobi identity.\nIn turn, define a trivial associative and commutative  star-product\n\\beq\n\\label{i58}\n( F * G ) = T^{-1} ( ( T F ) ( T G ) ) = F G - \\kappa \\theta ( F, G\n)_{ \\tau *} (-1)^{\\varepsilon(F)}.    \n\\eeq\nIt is worthy to mention here that the operators (\\ref{i57.4}) apply to a function\nas to yield the left adjoint of the symbol multiplication (\\ref{i58}),\n\\beq\n\\label{i58.1}\n( (T F)_{*} G ) =  F * G,\\quad (T F)_{*} = F -\n\\kappa\\!\\; \\theta\\!\\; {\\rm ad}_{\\tau_{*}} ( F ) (-1)^{\\varepsilon( F )}. \n\\eeq\n\n Due to (\\ref{i57.2}), within the class of functions (\\ref{i57.1}),\n the star-product (\\ref{i58}) reduces as follows\n\\beq\n\\label{5.13}\nF * G = ( t^{-2} f ) ( t^{-2} g ) - \\kappa \\theta t^{-2} ( f, g )_{*}\n(-1)^{\\varepsilon(f)}.   \n\\eeq\nThen, with respect to the star-product (\\ref{i58}), we have the\ntrivially  deformed extended Witten formula \\footnote{The same as in\nthe undeformed case, the deformed extended Witten formula\n(\\ref{i59}) follows directly from the double-commutator formula\n(\\ref{i57.3}) with (\\ref{i57.4}), (\\ref{i58}), (\\ref{i58.1}) taken\ninto account. } \\beq \\label{i59} ( F, G )_{\\tau *} = \\Delta_{\\tau *}\n( F * G ) (-1)^{\\varepsilon(F)} - F\n* ( \\Delta_{\\tau *} G ) -\n( \\Delta_{\\tau *} F ) * G (-1)^{\\varepsilon(F)},    \n\\eeq\nthe Leibnitz rule,\n\\beq\n\\label{i60}\n( ( F * G ), H )_{\\tau *} =  F * ( G, H )_{\\tau *} +\nG * ( F, H )_{\\tau *} (-1)^{\\varepsilon(F) \\varepsilon(G)},\n\\eeq\nthe Getzler identity \\cite{G} providing for the absence of higher antibrackets  in\nthe BV -algebra\n\\beq\n\\nonumber\n&&\\Delta_{\\tau *} ( F * G * H ) - \\Delta_{\\tau *} ( F * G ) * H -\nF * \\Delta_{\\tau *} ( G * H ) (-1)^{\\varepsilon(F)} -\\\\\n\\nonumber\n&&- \\Delta_{\\tau *} ( F * H ) * G (-1)^{\\varepsilon(G)\n\\varepsilon(H)}+( \\Delta_{\\tau *} F ) * G * H-\\\\\n\\label{i61}\n&&- F * ( \\Delta_{\\tau *} G )* H\n(-1)^{\\varepsilon_(F)} +\nF * G * ( \\Delta_{\\tau *} H ) (-1)^{\\varepsilon(F) +\n\\varepsilon(G)} = 0.    \n\\eeq\nThe star exponential  is defined as (see also App. C)\n\\beq\n\\nonumber\n&& \\exp_{*}\\{ B \\}  =\n1 + B + \\frac{1}{2} B * B + \\frac{1}{3!} B * B * B + ... =\\\\\n\\label{i62}\n&&=T^{-1} \\exp\\{( T B )\\} =\n\\exp\\left\\{ B -\\frac{1}{2} \\kappa \\theta ( B, B )_{\\tau *} \\right\\}.   \n\\eeq\nThe latter satisfies\n\\beq\n\\label{i63.1}\n&&\\exp_{*}\\{ - B \\}  *  \\exp_{*}\\{ B \\}  = 1, \\\\\n\\label{i63.2}\n&&\\exp_{*}\\{ - B \\}  * (\n\\Delta_{\\tau *} \\exp_{*}\\{ B \\} ) =\n( \\Delta_{\\tau *} B ) + \\frac{1}{2} ( B, B )_{\\tau *}, \\\\\n\\label{i63.3}\n&&\\exp_{*}\\{ B + B'\\} =\\exp_{*}\\{ B \\}  * \\exp_{*}\\{ B'\\}, \\\\\n\\label{i63.4}\n&&\\delta  \\exp_{*}\\{ B \\}  = \\exp_{*}\\{ B \\} * \\delta B,\\quad\n\\varepsilon(B) = \\varepsilon(B') = 0.             \n\\eeq\nThe trivially deformed extended  quantum master equation has the form\n\\beq\n\\label{i64}\n\\Delta_{\\tau *} \\exp_{*}\\left\\{ \\frac{i}{\\hbar}\\; W \\right\\}  = 0,\n\\eeq\nor, equivalently,\n\\beq\n\\label{i65}\n\\frac{1}{2} ( W, W )_{\\tau *} = i \\hbar \\Delta_{\\tau *} W.    \n\\eeq\nOne has to seek for a solution to that equation in the form\n\\beq\n\\label{i5.21}\nW = \\sum_{k = - 2}^{\\infty} W_{ (k|0) } t^{k} + \\theta \\sum_{k = 1}^{\\infty}\nW_{ (k|1) } t^{k},   \n\\eeq\nwhere the component $W_{ (-2|0) } = S$  is  identified with the classical\nnontrivially  deformed proper action, (see also App. B)\n\\beq\n\\label{i5.22}\n( S, S )_{*} = 0.    \n\\eeq\nThe detailed form of the equations for coefficients in (\\ref{i5.21}), together with\nthe corresponding formal techniques, can be found in Ref. \\cite{BB2}.\n\nThe trivially deformed  path integral with  a measure $d\\mu$  in the extended\nphase space is defined as\n\\beq\n\\label{i66}\n\\mathcal{Z} = \\int d\\mu \\exp_{* ( \\kappa)}\\left\\{\\frac{i}{ \\hbar}\\; W \\right\\}\n \\exp_{* ( -\\kappa)}\\left\\{\\frac{i}{\\hbar }\\; X \\right\\}   =\n\\int d\\mu \\exp\\left\\{\\frac{i}{\\hbar }\\; A \\right\\},\n\\eeq\nwhere\n\\beq\n\\label{5.27}\nA = T_{ (\\kappa)} W + T_{( - \\kappa)}X,        \n\\eeq\n\\beq\n\\Delta_{\\tau * ( \\kappa ) }\\left(\\exp_{*(\\kappa )}\\left\\{\\frac{ i}{ \\hbar}\\; W \\right\\}\n\\right) = 0,\\quad\n\\Delta_{\\tau * (- \\kappa )}\\left(\\exp_{*( - \\kappa )} \\left\\{\\frac{i}{\\hbar}\\; X \\right\\}\n\\right)=  0.\n\\eeq\nHere, $X$ satisfies the same equation as $W$ does, but with the formal replacement\n$\\kappa \\rightarrow - \\kappa$.\nThis replacement just provides for right transposition properties when\nintegrating by part.\n\nWe proceed in (\\ref{i66}) with the following integration measure\n\\beq\n\\label{i5.25}\nd \\mu = t^{-1} d t d\\theta d \\lambda_{\\theta} [ d Z ] [ d \\lambda ].   \n\\eeq\nThe transposed operator $A^{T}$ of the operator $A$ is defined via\n\\beq\n\\label{i5.26}\n\\int d\\mu ( A^{T} F ) G  =   (-1)^{\\varepsilon(A) \\varepsilon(F)}  \\int d\n\\mu  F  ( A G).      \n\\eeq\nOur main transposed operators are\n\\beq\n\\label{i5.27}\n\\Delta^{T} = \\Delta,  \\quad N^{T} = - N, \\quad\n\\Delta^{T}_{\\tau} = \\Delta_{\\tau},\\quad N^{T}_{\\tau} = - N_{\\tau},\\quad\n\\Delta^{T}_{\\tau *( \\kappa )} =  \\Delta_{\\tau * (- \\kappa )}.  \n\\eeq\nLet us make in (\\ref{i66}) the variation of the form\n\\beq\n\\label{i5.28}\n\\delta \\exp_{*(-\\kappa)}\\left\\{\\frac{ i}{\\hbar }\\; X \\right\\} =\n\\Delta_{\\tau *(-\\kappa)} \\left( \\exp_{*(-\\kappa)}\\left\\{\\frac{i}{\\hbar\n}\\; X \\right\\} *_{(-\\kappa)} \\delta \\Psi \\right),   \n\\eeq\nwith arbitrary infinitesimal Fermion $\\delta \\Psi$. The (\\ref{i5.28}) is consistent\nwith (\\ref{i63.4}) due to the quantum master equation for the  $X$.\nThen, we deduce that the path\nintegral is independent of the gauge-fixing action $X$,\n\\beq\n\\nonumber\n&&\\delta_{X} \\mathcal{Z} = \\int d \\mu \\exp_{*( \\kappa )}\\left\\{\\frac{i}{\\hbar }\\;W \\right\\}\n\\delta \\exp_{* ( -\\kappa ) }\\left\\{\\frac{i}{\\hbar }\\; X \\right\\} =\\\\\n\\nonumber\n&&=\\int d \\mu \\exp_{* ( \\kappa ) }\\left\\{\\frac{i}{\\hbar}\\; W \\right\\}\\Delta_{\\tau *( -\n\\kappa )} \\left( \\exp_{ * ( - \\kappa ) }\\left\\{\\frac{i}{\\hbar }\\; X \\right\\}\n*_{ ( - \\kappa ) } \\delta \\Psi \\right) =\\\\\n\\label{i5.29}\n&&=\\int d \\mu \\left( \\Delta_{\\tau * ( \\kappa )}\n\\exp_{ * ( \\kappa ) }\\left\\{\\frac{i}{\\hbar }\\; W \\right\\} \\right)\n\\left(\\exp_{* ( - \\kappa) }\n\\left\\{\\frac{i}{\\hbar }\\; X \\right\\} *_{ ( - \\kappa ) } \\delta \\Psi \\right)\n= 0.     \n\\eeq\n\nIt follows from (\\ref{i56}), (\\ref{i62}) and (\\ref{i64}) that\n\\beq\n\\Delta_{\\tau}  \\exp\\left\\{\\frac{i}{\\hbar }\\; T_{ ( \\kappa) } W  \\right\\} = 0.\n\\eeq\nFor similar reasons, it follows that\n\\beq\n\\Delta_{\\tau}\\exp\\left\\{\\frac{i}{\\hbar}\\;T_{( - \\kappa ) } X \\right\\} = 0.\n\\eeq\nThese equations tell us that in terms of the barred actions,\n\\beq\n\\bar{W} = T_{ (\\kappa) } W,\\quad \\bar{X} = T_{ ( -\\kappa ) } X,\n\\eeq\nthe path integral (\\ref{i66}) is just the standard $W - X$ version of the\nfield-antifield formalism. From the latter point of view, it is\nwell-known that the path integral (\\ref{i66}) is stable  under the\ngauge variation\n\\beq\n\\delta{ \\bar{X} } = \\sigma_{\\tau}(\\bar{X}) \\bar{\\Psi },\n\\eeq\nwhere\n\\beq\n\\sigma_{\\tau}( \\bar{X} ) =-i\\hbar \\;\\Delta_{\\tau}+{\\rm ad}_{\\tau}( \\bar{X} ),\n\\eeq\nso that\n\\beq\n\\delta X = T^{-1}_{( - \\kappa ) } \\delta \\bar{X}.\n\\eeq\n If one identifies $\\bar{ \\Psi } =T_{ ( - \\kappa ) } \\Psi$ , then\n\\beq\n\\delta X = \\sigma_{ \\tau * ( -\\kappa ) }( X ) \\Psi ,\n\\eeq\nwhere\n\\beq\n\\sigma_{ \\tau * ( - \\kappa ) }( X )\n= -i\\hbar\\; \\Delta_{\\tau *( - \\kappa )}+{\\rm ad}_{\\tau *( -\\kappa)}( X ),\n\\eeq\nwhich is exactly the variation of $X$ generated by (\\ref{i5.28}).\n\n\n\\section{ Gauge-Fixing in the Classical Extended Nondeformed\\\\ Master\nEquation}\n\n\nHere we study, if the standard gauge-fixing procedure is capable to\neliminate the extra variable $t$, as applied\nto the classical  $\\tau$ -extended nondeformed master equation,\n\\beq\n\\label{i6.1}\n( S, S )_{\\tau} = t^{2} ( S, S ) + 2 ( N_{\\tau} S ) ( \\pa_{\\theta} S ) = 0.\n\\eeq\nwhere we restrict ourselves to the simplest choice for $N_{\\tau}$,\n\\beq\n\\label{i6.2}\nN_{\\tau} = Z^{A} \\pa_{A} + t \\pa_{t}.   \n\\eeq\nWe proceed with the following ansatz for S,\n\\beq\n\\label{i6.3}\nS  = S ( Z, t, \\theta ) = S ( t^{-1} Z, \\theta  ),       \n\\eeq\nwhich implies\n\\beq\n\\label{i6.4}\nN_{\\tau} S = 0,    \n\\eeq\nso that Eq.  (\\ref{i6.1}) takes the usual form of the classical master\nequation,\n\\beq\n\\label{i6.5}\n( S, S ) = 0.      \n\\eeq\nLet $\\mathcal{S}( \\phi )$ be an original action of original fields $\\phi^{i}$ ,\nand $R^{i}_{\\alpha}( \\phi )$ do satisfy the Noether identities,\n\\beq\n\\label{i6.6}\n\\mathcal{S} \\overleftarrow{\\pa}_{i} R^{i}_{\\alpha} = 0,\n\\quad \\pa_i=\\frac{\\pa}{\\pa \\phi^i}.    \n\\eeq\nFor the sake of simplicity, let the generators $R^{i}_{\\alpha}$ be\nlinearly independent, so that the theory is irreducible.  Let us expand\nthe ansatz (\\ref{i6.3}) in powers of antifields,\n\\beq\n\\nonumber\n&&S = \\mathcal{S}(t^{-1}\\phi) + t^{-1}\\phi^*_{i} R^{i}_{\\alpha}(t^{-1}\\phi)\nt^{-1} C^{\\alpha} + \\theta t^{-1} C^{t} +\n\\frac{1}{2}t^{-1} C^*_{\\gamma} U^{\\gamma}_{\\alpha \\beta}(t^{-1}\\phi) t^{-1}\nC^{\\beta} t^{-1} C^{\\alpha} (-1)^{\\varepsilon_{\\alpha}} +\\\\\n\\label{i6.7}\n&&+\nt^{-1} {\\bar C}^{*\\alpha} t^{-1} B_{\\alpha} + t^{-1} {\\bar C}^{*t} t^{-1}\nB_{t} + ... ,      \n\\eeq\nwhere the terms presented explicitly are enough for a rank one theory, while\nellipses mean terms nonlinear in antifields.\n\nNow, let us split $Z^{A}$ into fields and antifields,\n\\beq\n\\label{i6.8}\nZ^{A} = \\{\\Phi^{a}, \\Phi^*_{a}\\}. \n\\eeq\nThen the gauge-fixing Fermion allowed has the form\n\\beq\n\\label{i6.9}\n\\Psi= \\Psi( \\Phi, t ) = \\Psi( t^{-1} \\Phi, \\ln t ),      \n\\eeq\nso that the antifields $\\Phi^*_{a}$ should be eliminated in (\\ref{i6.7}) by the\nconditions\n\\beq\n\\label{i6.10}\n\\Phi^*_{a} = t ^{2} \\Psi \\overleftarrow{\\pa}_{a}, \\quad \\pa_a=\\frac{\\pa}{\\pa \\Phi^a},    \n\\eeq\n\\beq\n\\label{i6.11}\n\\theta = N_{\\tau} \\Psi.       \n\\eeq\nThese conditions do correspond to the following ansatz for the gauge-fixing\nmaster action X,\n\\beq\n\\label{i6.12.1}\nX = \\left( t^{-1} \\Phi^*_{a} - t \\Psi \\overleftarrow{\\pa}_{a} \\right) \\lambda^{a} + (\n\\theta - N_{\\tau} \\Psi ) \\lambda^{\\theta},     \n\\eeq\nwhere $\\lambda^{a}$ and $\\lambda^{\\theta}$ is the corresponding Lagrange\nmultiplier.\nIn fact, it is enough for our purposes to use the ansatz\n\\beq\n\\label{i6.12}\n\\Psi = t^{-1} \\bar{C}_{\\alpha} \\chi^{\\alpha}( t^{-1} \\phi ) + t^{-1}\n\\bar{C}_{t} \\chi^{t}( \\ln t ),   \n\\eeq\nwith the following identification of fields\n\\beq\n\\label{i6.13}\n\\Phi^{a} = \\{\\phi^{i}, B_{\\alpha}, C^{\\alpha}, \\bar{C}_{\\alpha}, B_{t}, C^{t},\n\\bar{C}_{t}\\}.   \n\\eeq\n\nThus, we arrive at the following complete gauge-fixed action\n\\beq\n\\nonumber\n&&S_{gauge-fixed} = \\mathcal{S}( t^{-1}\\phi ) +\n t^{-1} \\bar{C}_{\\alpha} \\chi( t^{-1} \\phi ) \\overleftarrow{\\pa}_{i}\nR^{i}_{\\alpha}( t^{-1} \\phi ) C^{\\alpha} +\nt^{-2} \\bar{C}_{t}( N_{\\tau} \\chi^{t} ) C^{t}+ \\\\\n\\label{i6.14}\n&&+ \\chi^{\\alpha}( t^{-1} \\phi\n) t^{-1} B_{\\alpha}  + \\chi^{t}( \\ln t ) t^{-1} B_{t} + ... .  \n\\eeq\nThat action has, in its terms presented explicitly, the standard structure\nof the Faddeev-Popov action, both\nin the sector of the usual gauge $\\chi^{\\alpha}$ and of the extra gauge\n$\\chi^{t}$.  By choosing $\\chi^{t} = \\ln t$,\none removes the $t$-integration at the value $t = 1$. Thereby, it is shown that\nthe extra variable t is eliminated actually via the standard gauge-fixing procedure.\n\n\n\\section{ Gauge-Fixing in the Extended Trivially Deformed Classical\/Quantum Master\nEquation}\n\n\nLet us consider the  extended trivially deformed classical master equation,\n\\beq\n\\label{i7.1}\n( S, S )_{\\tau{*}} = 0,    \n\\eeq\nor in more detail,\n\\beq\n\\label{i7.2}\nt^{2} (S, S ) + 2 ( N_{\\tau} S )( \\pa_{\\theta} S ) +\n2 \\kappa ( N_{\\tau} S ) ( ( t^{2} \\Delta + N_{\\tau} \\pa_{\\theta} )\n( 1 - \\kappa N_{\\tau} )^{-1} S ) = 0.    \n\\eeq\nThe same as in  Sec. 6, we have chosen $N _{\\tau}$ in the\nsimplest form (\\ref{i6.2}).\nIn contrast to the previous section, we are not allowed to require for the\noperator $N_{\\tau}$\nto annihilate the $S$, as the deformation by itself would be eliminated\nimmediately in this way.\nHowever, if we do believe that a solution for $S$ does exist, we can try to\nrequire for $N_{\\tau}$ to\nannihilate the gauge-fixing part of $S$, at least. When doing that, we should\nprovide for the form\nof Eqs. (\\ref{i7.1}), or (\\ref{i7.2}), to be respected. Let us seek for $S$ in the\nform\n\\beq\n\\label{i7.3}\nS = S_{min} + t^{-2} ( \\bar{C}^{*{\\alpha}} B_{\\alpha} + \\bar{C}^{*{t}} B_{t} ),  \n\\eeq\nwhere the minimal action $S_{min}$ depends on the minimal,  gauge-algebra\ngenerating, set of variables, only,\n\\beq\n\\label{i7.4}\nS_{min} = S_{min}( \\phi, \\phi^*; t, \\theta; C, C^*; C^{t}, C^*_{t} ).  \n\\eeq\nBy construction, the second and third term in (\\ref{i7.3}), those are just the\ngauge-fixing parts of $S$ , are\ncertainly annihilated by the $N_{\\tau}$, while the $S_{min}$ is not. In this\nway, one can see that\nthe $S_{min}$ by itself does satisfy exactly Eqs. (\\ref{i7.1}),\nor (\\ref{i7.2}). If\none chooses the gauge\nFermion $\\Psi$ in the simplest form (\\ref{i6.12}), then the antifields should be\neliminated by the conditions\n(\\ref{i6.10}), see also (\\ref{i6.13}).  In turn, the second and third term\nin (\\ref{i7.3})\ntake exactly the form of fourth\nand fifth term in (\\ref{i6.14}), respectively. Thereby, it is shown that the extra\nvariable $t$  is eliminated\nactually via the standard gauge-fixing procedure.\n\nIn the same way, one can consider the extended trivially deformed quantum\nmaster equation, (\\ref{i65}),\n\\beq\n\\label{7.5}\nt^{2} ( W, W ) + 2 ( N_{\\tau} W ) ( \\pa_{\\theta} W ) +\n2( (\\kappa N_{\\tau} W)  - i \\hbar) (( t^{2} \\Delta + N_{\\tau}\n\\pa_{\\theta} ) ( 1 - \\kappa N_{\\tau} )^{-1} W ) =  0.    \n\\eeq\nAs Eq. (\\ref{i7.2}) is a classical limit to the quantum equation (\\ref{7.5}),\nall the above reasoning, as well as the final statement remains the same.\n\nFinally, let us notice the following. In Secs. 6 and 7,  we have used the\nzero mode $t^{-1} Z^{A}$ of the\noperator (\\ref{i6.2}).  Here, we mention in short how to deal with the general\noperator (\\ref{i3}), where $N^{A}$\nis defined by (\\ref{i20.7}) - (\\ref{i20.9}). Let $\\bar{Z}^{A}$\nbe the zero mode of the\noperator (\\ref{i3}). Then we have\nformally,\n\\beq\n\\label{i7.5}\n\\bar{Z}^{A} = \\exp\\{ - ( \\ln t ) N \\} Z^{A},  \\quad    N = N^{A} \\pa_{A}.\n\\eeq\nIt follows from (\\ref{i26}) that\n\\beq\n\\label{i7.6.1}\n( \\bar{Z}^{A}, \\bar{Z}^{B} ) = t^{-2} E^{AB}, \\quad\n( \\bar{Z}^{A}, Z^{C} ) E_{CD} ( Z^{D}, \\bar{Z}^{B} )=( \\bar{Z}^{A},\\bar{Z}^{B} ). \n\\eeq\nIn  turn,  it  follows  from  (\\ref{i7.6.1})  that  the general  solution for  the\nleft  off-diagonal  block  has  the  form,\n\\beq\n\\label{78}\n( \\bar{Z}^{A}, Z^{B} )  =  t^{-1} S^{A}_{\\;\\;C}( t ) E^{CB}, \\quad\n\\bar{Z}^{A}  =  t^{-1} S^{A}_{\\;\\;B}( t ) Z^{B},\n\\eeq\nwhere $S^{A}_{\\;\\;B}(t)={\\rm const}(Z)$ is a $t$-dependent antisymplectic matrix,\n\\beq\n\\label{7.9}\nS^{A}_{\\;\\;C}( t ) E^{CD} S^{B}_{\\;\\;D}( t ) (-1)^{\\varepsilon_{D} ( \\varepsilon_{B}\n+ 1 ) }  =  E^{AB} ,    \n\\eeq\nsuch that\n\\beq\n\\label{7.10}\nS^{A}_{\\;\\;B}( t = 1 ) = \\delta^{A}_{\\;\\;B}.       \n\\eeq\nOne can get the general solution for the right off-diagonal block via the\nsupertransposition  in (\\ref{78}).\n\nNow, let us consider some explicit formulae concerning the modified\n$N$-operator. First, let us choose the quadratic Fermion $F$ entering (\\ref{i11})\nthat meets the condition (\\ref{i14}),\n\\beq\n\\label{7.11}\n2 F = Z^{A} F_{AB} Z^{B},  \\quad   \\varepsilon (F) = 1,    \n\\eeq\n\\beq\n\\label{7.12}\n\\varepsilon( F_{AB} )  = \\varepsilon_{A}+\\varepsilon_{B}+1,  \\quad\nF_{AB} = F_{BA} (-1)^{\\varepsilon_{A} \\varepsilon_{B}} = {\\rm const}(Z).  \n\\eeq\nWe have\n\\beq\n2\\Delta F = E^{AB} F_{BA} (-1)^{ \\varepsilon_{A} } = {\\rm const}( Z ),\n\\eeq\n\\beq\n\\label{7.14}\nN^{A} = Z^{A} + 2( F, Z^{A} ) = Z^{B} ( \\delta_{B}^{\\;\\;A} + 2F_{BC} E^{CA} ) =\n(\\delta^{A}_{\\;\\;B} - 2 E^{AC} F_{CB} ) Z^{B}.    \n\\eeq\nOn the other hand,  as the second in (\\ref{78})  is the zero mode of $N_{\\tau}$,\nwe have another expression for $N^{A}$,\n\\beq\n\\label{7.15}\nN^{A} = - ( S^{-1} )^{A}_{\\;\\;C}\\; t^{2} \\pa_{t} ( t^{-1} S^{C}_{\\;\\;B} ) Z^{B}.\n\\eeq\nIt follows from  (\\ref{7.14}) and (\\ref{7.15}) that the Lie equation holds\n\\beq\n\\label{7.16}\nt \\pa_{t} S^{A}_{\\;\\;B} = 2 S^{A}_{\\;\\;C} E^{CD} F_{DB}, \\quad\nS^{A}_{\\;\\;B}( t = 1 ) =\\delta^{A}_{\\;\\;B},     \n\\eeq\nwhose formal matrix solution is\n\\beq\n\\label{7.17}\nS =  S( t ) = \\exp\\{ 2\\ln( t ) E F \\}.     \n\\eeq\nBy $t$-differentiating the formula (\\ref{7.9}), and then using the equation (\\ref{7.16}),\none confirms that  the matrix (\\ref{7.17})\nby  itself  does satisfies exactly  the antisymplicticity equation (\\ref{7.9}).\nThus, we see that the exponential (\\ref{7.17})\nprovides for the  $t$-parametrization of a family of antisymplectic matrices,\nwith $( EF )$ being a generator.\n\n\nIf one splits the full set  $Z^{A}$ into minimal sector $Z_{min}$ in (\\ref{i7.4}),\nexcept  for $\\{ t, \\theta \\}$, and\nthe rest, $Z_{aux}$, $\\{Z\\} =\\{ Z_{min}\\}\\oplus \\{Z_{aux}\\}$, then, by choosing in (\\ref{i.V})\n$F=F(Z_{min})$,  one has\n\\beq\n\\label{i7.6.2}\n\\bar{Z}_{aux} = t^{-1} Z_{aux}.\n\\eeq\nIn terms of (\\ref{i7.5}), the formula (\\ref{i6.3}) and (\\ref{i6.9}) takes the form,\n\\beq\n\\label{i7.6}\nS = S( \\bar{Z}, \\theta ),   \n\\eeq\nand\n\\beq\n\\label{i7.7}\n\\Psi = \\Psi( \\bar{\\Phi}, \\ln t ),   \n\\eeq respectively.\nIn turn, the formula (\\ref{i7.3}) in terms of\n(\\ref{i7.5}) preserves its\nform\n\\beq\n\\label{i7.8}\n S = S_{min} +\n(\\overline{\\bar{C}^{*\\alpha}B_{\\alpha}}   +\n\\overline{\\bar{C}^{*t}B_t } )=\nS_{min} + t^{-2} ( \\bar{C}^{*{\\alpha}} B_{\\alpha} + \\bar{C}^{*{t}} B_{t} ),    \n\\eeq\nwhere $S_{min}$ is given by (\\ref{i7.4}).\nFor the particular case $N^{A} = Z^{A}$, we reproduce from (\\ref{i7.5})\n$\\bar{Z}^{A} = t ^{-1} Z^{A}$.\n\nDue to (\\ref{i7.6.2}) and (\\ref{i7.8}),  one has\n\\beq\n\\label{i7.11}\n( S_{min}, S_{aux} ) = 0,  \\;\\;  ( S_{aux}, S_{aux} ) = 0,  \\;\\; N_{\\tau} S_{aux} =\n0, \\;\\;  \\pa_{\\theta} S_{aux} = 0,\\;\\; S_{aux} = S - S_{min},   \n\\eeq\ntogether with\n\\beq\n\\label{i7.12}\n\\Delta S_{aux} = 0.   \n\\eeq\nThe relations (\\ref{i7.11}) and (\\ref{i7.12}) allow one to preserve the form of the equation\n(\\ref{i7.2}) for the minimal action (\\ref{i7.4}) in the general case of (\\ref{i7.5}).\n\n\n\n\\section{ Generalized Darboux Coordinates \\cite{BB2}}\n\n\n\nThe $\\tau$ -extended trivially deformed classical\/quantum master\nequation takes its simplest form  in the so-called generalized\nDarboux coordinates,\n\\beq\n\\label{i8.1}\n\\tau_{0}:  Z^{A}_{0} = t^{-1} Z^{A},\\quad t_{0} = \\ln t,\\quad  t^*_{0} = \\theta,\n\\eeq\nwith the following integration measure,\n\\beq\nd\\mu=d t_{0} d t^*_{0} d\\lambda_{t^*_{0}} [ d Z_{0} ] [ d \\lambda ].\n\\eeq\nWe have already used these coordinates  partially when discussing\nthe gauge-fixing\nprocedure. In the case of the master equation, we have, with the use\nof (\\ref{i8.1}),\n\\beq\n\\label{i8.2}\n( W, W )_{\\tau_{0}{*}} = 2  i  \\hbar \\Delta_{\\tau_{0}{*}} W,   \n\\eeq\nwhere\n\\beq\n\\label{i8.3}\n( F, G )_{\\tau_{0}{*}} = ( F, G )_{\\tau_{0}} + ( K_{\\tau_{0}} F ) (\n\\Delta_{\\tau_{0}{*}} G ) +\n( \\Delta_{\\tau_{0}{*}} F ) ( K_{\\tau_{0}} G ) (-1)^{\n\\varepsilon_{F} },\n\\eeq\n\\beq\n\\label{i8.4}\n( F, G )_{\\tau_{0}} = F [ \\overleftarrow{\\pa}_{A 0} E^{AB}\n\\overrightarrow{\\pa}_{B 0} +\n\\overleftarrow{\\pa}_{t_{0}} \\overrightarrow{\\pa}_{t^*_{0}} -\n\\overleftarrow{\\pa}_{t^*_{0}} \\overrightarrow{\\pa}_{t_{0}} ] G,\\quad   \\pa_{A 0} =\n\\frac{\\pa}{\\pa Z^{A}_{0}},     \n\\eeq\n\\beq\n\\label{i8.5}\nK_{\\tau_{0}} = \\kappa \\pa_{t_{0}},  \n\\eeq\n\\beq\n\\label{i8.6}\n\\Delta_{\\tau_{0}{*}} = \\Delta_{\\tau_{0}} ( 1 - K_{\\tau_{0}} )^{-1},\n\\eeq\n\\beq\n\\label{i8.7}\n\\Delta_{\\tau_{0}} = \\frac{1}{ 2 } (-1)^{\\varepsilon_{A}} \\pa_{A 0} E^{AB}\n\\pa_{B 0} + \\pa_{t_{0}} \\pa_{t^*_{0}}.   \n\\eeq\nWe have the two main simplifications here. The first is the absence\nof the $t_{0}$ -dependent factors in the square bracket in (\\ref{i8.4}).  The second\nis a very simple form of the  operator (\\ref{i8.5}). The latter reduces to the $t_{0}$\n-derivative.\n\nThe antifields are eliminated by the conditions\n\\beq\n\\label{i8.8}\n\\Phi^*_{a 0} = \\Psi\\overleftarrow{\\pa}_{a 0},  \\quad   t^*_{0} =\n\\Psi\\overleftarrow{\\pa}_{t_{0}},  \n\\eeq\nwhere\n\\beq\n\\label{i8.9}\nZ^{A}_{0} = \\{\\Phi^{a}_{0},  \\Phi^*_{a 0}\\},\\quad\n\\pa_{a 0} = \\frac{\\pa}{\\pa \\Phi^{a}_{0}}.        \n\\eeq\n\nFinally, let us consider in short what happens  if one ignores the\ngauge-fixing mechanism as to eliminate the extra variable $t$.\nIn that case, one assumes the $W$ to be $\\theta$ -independent,\n\\beq\n\\label{i8.10}\n\\pa_{\\theta} W = \\pa_{t^*_{0}} W = 0.    \n\\eeq\nUnder the assumption (\\ref{i8.10}),  the path integral  (\\ref{i66}) becomes as\nrepresented in coordinates (\\ref{i8.1}),\n\\beq\n\\label{i8.11}\n\\mathcal{Z}= \\int d t_{0}\\int d \\Phi_{0}\\exp \\left\\{\\frac{i}{\\hbar}A \\right\\},\n\\eeq\n\\beq\n\\label{i8.12}\nA = W( \\Phi, \\Phi^*, t, \\kappa, \\hbar ) = W \\left(\\exp\\{ t _{0} \\} \\Phi_{0},\n\\exp\\{ t_{0} \\} \\left( \\Psi( \\Phi_{0} )\\frac{\\overleftarrow{\\pa}}{\\pa\\Phi_{0}} \\right),\n\\exp\\{t_{0}\\},  \\kappa, \\hbar \\right),      \n\\eeq\n\\beq\n\\label{i8.13}\n( W, W )_{0} + 2 \\left( \\kappa ( \\pa_{t_{0}} W )  -  i \\hbar  )\n( \\Delta_{0} ( 1 - \\kappa \\pa_{ t_{0} } )^{-1} W \\right) = 0,    \n\\eeq\n\\beq\n\\label{i8.14}\n( F, G )_{0} = F \\overleftarrow{\\pa}_{A 0} E^{AB}\\overrightarrow{\\pa}_{B 0} G,\n\\eeq\n\\beq\n\\label{i8.15}\n\\Delta_{0} =\\frac{1}{2 } (-1)^{\\varepsilon_{A}} \\pa_{A 0} E^{AB} \\pa_{B 0}.\n\\eeq\n\nIn Ref. \\cite{BB2}, it was suggested that the extra variable $t_{0}$, remaining in the\npath integral (\\ref{i8.11}), plays the role of the Schwinger proper time\nin the field-antifield formalism. It\nseems rather plausible that the variable $t$ has a non-perturbative status. If\none rescales in (\\ref{i8.12}):  $W \\rightarrow \\exp\\{ - 2 t_{0} \\} W$,\nthen in (\\ref{i8.12}), (\\ref{i8.13}) one should\nsubstitute: $\\hbar \\rightarrow \\exp\\{ 2 t_{0} \\} \\hbar$,\n$ \\pa_{t_{0}}\\rightarrow \\pa_{t_0} - 2$.\n\n\n\n\\section*{Acknowledgments}\n\\noindent\n I. A. Batalin would like  to thank Klaus Bering of Masaryk\nUniversity for interesting discussions. The work of I. A. Batalin is\nsupported in part by the RFBR grants 14-01-00489 and 14-02-01171.\n The work of P. M. Lavrov is supported in part by the Presidential grant\n 88.2014.2 for LRSS and by the RFBR grant 15-02-03594.\n\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe recent paper \\cite{BCMNO} characterises the automorphisms of the Higman-Thompson groups $G_{n,r}$ as a subgroup of the rational group $\\T{R}_{n,r}$ consisting of those elements which have the additional property of being bi-synchronizing. In this article, we extend the arguments of \\cite{BCMNO} and characterise the automorphism group of $T_{n,r}$, generalisations of Thompson's group $T$, as a subgroup of $\\aut{G}_{n,r}$.\n\n The automorphism group  of $T_{2,1}$, or $T$, as well as the automorphism group of Thompson's group $F$, were studied in the paper \\cite{MBrin2}, which also demonstrates that $\\out{T_2}$ is isomorphic to the cyclic group of order $2$. The paper \\cite{JBurilloSClearyF} studies the metric properties of $\\aut{F}$ and gives a presentation for this group, whilst  the follow up paper \\cite{MBrinFGuzman} studies the automorphisms of the groups $F_{n}$ and $T_{n,n-1}$, generalisations of Thompson's group $F$ and $T$ (the approach there gives no information about $\\aut{T_{n,r}}$ when $r \\ne n-1$). The  paper \\cite{MBrinFGuzman}, amongst other things, demonstrates that $\\out{T_{n,n-1}}$  for $n\\ge3$ is infinite and contains an isomorphic copy of Thompson's group $F$. \n\nIn this paper we extend the results of \\cite{MBrinFGuzman} to the groups $\\Tnr$ for $1 \\le r < n-1$. Firstly, we prove the following,\n\n\\begin{Theorem}\nThe group $\\aut{T_{n,r}}$ consists of those elements of $ \\Rnr$ which can be represented by bi-synchronizing transducers such that the induced homeomorphisms on Cantor space respects the cyclic ordering. \n\\end{Theorem}\n\nOne immediately deduces the following corollary:\n\\begin{corollary}\n$\\aut{\\Tnr} < \\aut{G_{n,r}}$.\n\\end{corollary}\n\nWe now turn our attention to the quotient group $\\out{\\Tnr} = \\aut{\\Tnr}\/ \\inn{\\Tnr}$. The results below are equally true in the $\\Gnr$ context and so to avoid repetition we shall sometimes use the symbol $X$ to represent either $T$ or $G$, analogously the symbol $\\T{X}$ will represent either $\\T{TO}$ or $\\T{O}$.\n \n It is a result in the paper \\cite{BCMNO} that for $1 \\le r < n$ the group $\\out{\\Gnr}$ is isomorphic to a subgroup $\\Onr$ of a group $\\On$ consisting of non-initial, bi-synchronizing transducers. Furthermore, that paper also shows that $\\On = \\cup_{1\\le r \\le n-1} \\Onr$. More specifically, it is demonstrated in \\cite{BCMNO} that for $1 \\le i \\le j \\le n-1$ such that $i$ divides $j$ in the additive group $\\Z_{n-1}$,   $\\Ons{i} \\le \\Ons{j}$. As a consequence of this, one may deduce that for all $1 \\le i \\le n-1$ $\\Ons{1}\\le \\Ons{i}$, $\\Ons{i} \\le \\Ons{n-1}$  (and so $\\On = \\Ons{n-1}$) and $\\Ons{i} = \\Ons{d}$ for $d = \\gcd(n-1, i)$.  We extend these result to the group $\\out{\\Tnr}$. That is, we have:\n\n\\begin{Theorem}\nFor all $1 \\le r \\le n-1$, the group $\\out{\\Tnr}$ is isomorphic to a subgroup $\\TOnr$ of $\\Onr$. Moreover for all $1 \\le i \\le j \\le n-1$ such that $i$ divides $j$ in the additive group $\\Z_{n-1}$, we have   $\\TOns{i} \\le \\TOns{j}$. \n\\end{Theorem} \n\n\\begin{corollary}\\label{Corollary:OniequalsOnjifgcdequal}\nFor $1 \\le 1 \\le n-1$, we have $\\TOns{1} \\le \\TOns{i}$, $\\TOns{i} \\le \\TOns{n-1}$  and $\\TOns{i} = \\TOns{d}$ for $d = \\gcd(n-1, i)$.\n\\end{corollary}\n\nWe shall subsequently  use the symbol $\\TOn$ for the group $\\TOns{n-1}$.\n\nThe last phrase of Corollary~\\ref{Corollary:OniequalsOnjifgcdequal} is perhaps to be expected, certainly when $X = G$, as  results of Higman (\\cite{GHigman}), Pardo (\\cite{EPardo}), and Dicks and Mart{\\' \\i}nez-P{\\' e}rez (\\cite{DicksPerez})  demonstrates that $\\Gnr \\cong \\Gmr{m}{s}$ if and only if $n=m$ and $\\gcd(n-1, r) = \\gcd(n-1,s)$. In fact it is a question in \\cite{BCMNO} whether or not $\\Onr \\cong \\Ons{s}$ if and only if $\\gcd(n-1,r) = \\gcd(n-1,s)$. We show that this question has a negative answer in both the $\\Gnr$ and $\\Tnr$ context. That is we prove the following:\n\n\\begin{Theorem}\nThere is a number $n \\in \\N$, $n >2$, and $1 \\le r, s \\le n-1$ such that, for $X = \\T{TO}, \\T{O}$, $\\XOnr = \\XOns{s}$ but $\\gcd(n-1, r) \\ne \\gcd(n-1, s)$.\n\\end{Theorem}\n\n\nOur next result demonstrates that the groups $\\XOns{r}$ for all $1 \\le r \\le n-1$ are normal subgroups of $\\XOn$.\n\n\\begin{Theorem}\nFor all $1 \\le r \\le n-1$ we have $\\XOns{r} \\unlhd \\XOn$.\n\\end{Theorem}\n\n \n  \nWe next extend a result of Brin and Guzm{\\'a}n \\cite{MBrinFGuzman} for $\\TOns{n-1}$, and show that for $n \\ge 3$, $\\XOns{1}$ contains an isomorphic copy of R. Thompson's group $F$:\n\n\\begin{Theorem}\nLet $n \\ge 3$, then $\\out{X_{n,1}}$ contains a subgroup isomorphic to R. Thompson's group $F$. \n\\end{Theorem}\n\n From this and Corollary~\\ref{Corollary:OniequalsOnjifgcdequal}, we have the following:\n \\begin{corollary}\n For all $1 \\le r\\le n-1$, $\\XOns{r}$ contains  a subgroup isomorphic to R. Thompson's group $F$.\n \\end{corollary}\n  \n  \n  We further demonstrate (Section~\\ref{Section:nestingproperties2}), in the case $r=4$, that the set  $\\TOms{4}{3}\\backslash \\TOms{4}{1}$ is non-empty. Notice that since $3$ is prime, for $ 1 \\le r <3$, $\\TOms{4}{r} = \\TOms{4}{1}$, thus this result indicates that, in general, the group $\\TOnr$ might depend on $r$.\n\n\n\n We also investigate in Section~\\ref{Section:nestingproperties2} the nesting properties of the groups $\\XOnr$ of $\\XOn$ for $1 \\le r \\le n-1$. We show that these groups from a lattice with the `meet' of $\\XOnr$ and $\\XOns{s}$ being the intersection of the two groups and the `join' of $\\XOnr$ and $\\XOns{s}$ being the smallest $t$, $1 \\le t \\le n-1$ such  that $\\XOnr$ and $\\XOns{s}$ are subgroups of $\\XOns{t}$. We do not know if it is in fact the case that $\\XOns{t} = \\gen{\\XOns{s}, \\XOns{r}}$ (see Question~\\ref{Question:canjoinbereplacedwithubgroupgenerated}). To each element of $\\XOn$ we associate a numerical invariant which yields a group homomorphism from $\\XOn$ to the group of units of $\\Z_{n-1}$ (notice that for $n=2$,  $\\Z_{n-1}$ is equal to its group of units), with kernel $\\XOns{1}$. That is, we prove the following:\n \n \\begin{Theorem}\\label{Theorem:homfromoutintounitsofZn}\n There is  homomorphism from $\\XOn$ to the group of units of $\\Z_{n-1}$ with kernel $\\XOns{1}$. \n \\end{Theorem}\n \nWe should point out that the existence of this homomorphism is already a consequence of the fact that the  dimension group (see \\cite{KriegerW1980} for the definition of the dimension group) of $\\XOnr$ is equal to the additive group $\\Z_{n-1}$ and automorphisms of $\\XOnr$ yield automorphisms of the dimension group of $\\XOnr$. The author is grateful to  Prof. Nekrashevych for drawing these facts to his attention. Our proof of Theorem~\\ref{Theorem:homfromoutintounitsofZn} however does not rely on these observations, instead we explicitly construct the homomorphism by making use of the synchronizing condition to associate a numerical invariant to every element of $\\XOnr$. Our approach gives a means of resolving this question as we  reduce it to one of constructing transducers which have certain properties. In particular, Theorem~\\ref{Theorem:homfromoutintounitsofZn} demonstrates that the homomorphism from $\\XOns{1}$ into the group of units of $\\Z_{n-1}$ is the trivial homomorphism. Though we are unable show in general that the homomorphism of Theorem~\\ref{Theorem:homfromoutintounitsofZn} is surjective, we show that under the assumption that it is surjective, then  for $1 \\le s,r,t \\le n-1$ and $t$ the smallest element of $\\Z_{n}$ such that $\\XOms{n}{t}$ contains $\\XOms{n}{s}$ and $\\XOms{n}{r}$, $\\XOms{n}{t}= \\gen{\\XOms{n}{s}, \\XOms{n}{r}}$. The Theorem~\\ref{Theorem:surjectivityofrsigpartial} below, proven in Section~\\ref{Section:onsurjectivityofrsig}, indicates that the homomorphism of Theorem~\\ref{Theorem:homfromoutintounitsofZn} from $\\On$ to the group of units of $\\Z_{n-1}$ is surjective in many cases, indeed, elementary results in number theory indicate that there are infinitely many numbers $n$ which satisfying the hypothesis of the theorem. We do not know if the restriction to  $\\TOn$ is also surjective.\n\n\n\\begin{Theorem}\\label{Theorem:surjectivityofrsigpartial}\nIf the divisiors of $n$ generate the group of units of $\\Z_{n-1}$ then the homomorphism of Theorem~\\ref{Theorem:rsigisahomomorphism} defined on $\\On$ is unto the group of units of $\\Z_{n-1}$.\n\\end{Theorem}\n\n \n \nWe conclude in Section~\\ref{Section:RftyTn} with the following result:\n\\begin{Theorem}\nThe groups $\\Tnr$ have the $\\rfty$ property.\n\\end{Theorem}\nWe recall that a group $G$ is said to have the $R_{\\infty}$ property if for every automorphism $\\varphi$ of $G$, the equivalence relation defined on $G$ by, for $x,y \\in G$, $x$ is equivalent to $y$ if there is an element $h$ in $G$ such that $h^{-1}x(h)\\varphi = y$,  has infinitely many equivalence classes. The question of which groups have the $\\rfty$ property has received a lot of attention. This class of groups has been shown to  include, for instance, all non-elementary Gromov hyperbolic groups (\\cite{Fel'shtyn2004}, \\cite{GLevittMLustig}), Baumslag-Solitar groups $BS(m,n)$ for $m,n \\in \\Z\\backslash\\{0\\}, (m,n) \\ne (1,1)$ (\\cite{AFelshtynDGonclavesBSG}), lamplighter groups $\\Z_{n} \\wr \\Z$ where $2|n$ or $3|n$ (\\cite{DGonclavesPWong}), the first Grigorchuk  and the Gupta-Sidki groups (\\cite{AFelshtynLYuriyandETroitsky}), and R. Thompson's group $F$ (\\cite{BlkFelAlxGon}, \\cite{BMV}). It was shown independently by Burillo, Matucci, and Ventura (\\cite{BMV}), and  Gon{\\c c}alves and Sankaran (\\cite{DGonclavesPSankaran}) that $R$. Thompson's group $T$ is in this class of groups. We thus extend this result to the family of groups $T_{n,r}$.\n\nInterlaced through out the text are several open questions. In work in preparation we apply the techniques in the current paper and in the paper \\cite{BCMNO} to the generalizations of Thompson's group $F$. \n\\section*{Acknowledgements}\nThe author is grateful to Collin Bleak and Matthew Brin for helpful discussions and their comments on early drafts of this article. This work was partially supported by Leverhulme Trust Research Project Grant RPG-2017-159.\n\\section{Some preliminaries and the groups \\texorpdfstring{$G_{n,r} \\mbox{ and } T_{n,r}$}{Lg}}\\label{Section:Preliminaries}\nWe begin by setting up some notation.\n\nLet $X$ be a topological space, we shall denote by $H(X)$ the group of homeomorphisms of $X$, and for $G \\le H(X)$, $N_{H(X)}(G)$ shall denote the normaliser of $G$ in $H(X)$. Let $S_r:= \\mathbb{R}\/r \\Z$ for $r \\in \\mathbb{R}$, the circle of  length $r$ and for $n \\in \\N \\backslash \\{0\\}$, let $\\Z[1\/n] := \\{ a\/n : a \\in \\Z\\}$, the  $n$-adic rationals.  Though we will mainly think of $T_{n,r}$ as a subgroup of $G_{n,r}$, we shall at times consider it as a subgroup of $H(S_r)$ and we shall make it apparent at such times that we are doing so. We establish some further notation required to define the groups $G_{n,r}$ and $\\Tnr$.\n\nLet $\\dotr:= \\{\\dot{0},\\dot{1},\\ldots, \\dot{r-1}\\}$, and let $X_n := \\{0,1,\\ldots, n-1\\}$. We shall take the ordering $\\dot{0} < \\dot{1} < \\ldots < \\dot{r-1}$ and $0< 1 < \\ldots < n-1$ on $\\dotr$ and $\\Xn$ respectively. For $i \\in \\{0, n-1\\}$, if $i= 0$ then set $\\bar{i} = n-1$ otherwise set $\\bar{i} = 0$. Set $X_{n,r}^{\\ast} := \\{\\dot{a}w: \\dot{a} \\in \\dotr \\mbox{ and } w \\in X_n^{\\ast}\\} \\sqcup \\{\\epsilon\\}$ where $\\epsilon$ denotes the empty word, and $X_{n,r}^{+}:= \\{\\dot{a}w: \\dot{a} \\in \\dotr \\mbox{ and } w \\in X_n^{\\ast}\\}$. For $j \\in \\N$, let $\\Xnr^{j}$, respectively $\\Xn^{j}$, denote the set of all elements of $X_{n,r}^{*}$, respectively $X_n^{*}$, of length $j$. Given two words $u$ and $v$ in $X_n^{*}$ of $X_{n,r}^{*}$, we say that  $u$ is a prefix of $v$ and denote this by $u \\le v$. We write $u < v$ if $u$ is a proper prefix of $v$. For two words $u, v$ in $X_{n}^{*}$ or $X_{n,r}^{*}$ if $u \\nleq v$ and $v \\nleq u$ then we say that $u$ is incomparable to $v$ an write $u \\perp v$. If instead $ v = u \\nu$ for some $\\nu \\in \\Xns \\sqcup \\Xnrs$ then we set $v-u := \\nu$. The relation $\\le$ is a partial order on  the set $X_{n,r}^{*}$ and $X_n^{*}$.\n\nFor two incomparable words $\\nu, \\eta$ in $\\Xnp$ and $\\Xnrp$ we say  \\emph{$\\nu$ is less than $\\eta$ in the lexicographic ordering}, denoted $\\nu \\lelex \\eta$, if there are $wi, wj \\in \\Xnrp$ or $wi, wj \\in  \\Xnp$ prefixes of $\\nu$ and $\\eta$ respectively with $i < j$ in the ordering on $\\Xn$ or $\\dotr$.  If $i \\le j$ then we write $\\nu  \\leqlex \\eta$.\n\n We may now defined a total order of $\\Xnrs$ and $\\Xns$ as follows: for $\\nu, \\eta \\in \\Xnrs$ or $\\nu, \\eta \\in \\Xns$ we say that \\emph{$\\nu$ is less than $\\eta$ in the short-lex ordering} if either $|\\nu| \\le |\\eta|$ or if $|\\nu| = |\\eta|$ then  $\\nu \\leqlex \\eta$, with strict inequalities if $\\nu$ is strictly less than $\\eta$. We write $\\nu \\leqslex \\eta$ if $\\nu$ is less than $\\eta$ in the short-lex ordering, and $\\nu \\leslex \\eta$ is the $\\nu$ is strictly less than $\\eta$ in the short-lex ordering. \n\nNow let $\\CCn := X_n^{\\omega}$ a Cantor space with the usual topology, and let $\\CCnr:= \\{\\dot{a}x | \\dot{a} \\in \\dotr \\mbox{ and } x \\in \\CCn \\}$, the disjoint union of $r$ copies of $\\CCn$. For a word $\\nu \\in  X_{n,r}^{+}$, let  $U_{\\nu}:= \\{\\nu\\delta \\mid \\delta \\in \\CCn\\}$, and $U_{\\epsilon}:=  \\CCnr$. For $\\nu \\in X_n^{\\ast}$, set $U_{\\nu}:=   \\{ \\nu \\delta: \\delta \\in \\CCn\\}$. Let $\\Banr:= \\{U_{\\nu}\\mid \\nu \\in X_{n,r}^{\\ast} \\}$ and let $\\Ban:= \\{U_{\\nu} \\mid \\nu \\in \\Xn \\}$, then $\\Banr$ and $\\Ban$ are a basis for the topology on $\\CCnr$ and $\\CCnr$ respectively. For a subset  $Z \\subseteq \\Xns \\sqcup \\Xnrs$ we shall denote by $U(Z)$ the set $\\{ U_{z} \\mid z \\in Z \\}$.\n\n\nThe lexicographic  extends to a total order on $\\CCn$ and $\\CCnr$ in the natural way, thus we extend the meanings of the symbols $\\leqlex$ and $\\lelex$ to this sets also. We further extend this notation to subsets of $\\CCn$ and $\\CCnr$. Given two subsets $V, W \\subset \\CCn$ or $V, W \\subset \\CCnr$ we write $V \\leqlex W$ if every element of $U$ is less than every element of $W$ in the lexicographic ordering. The strict inequality $V \\lelex W$ is analogously defined.\n\nLet $\\nu, \\eta \\in X_{n,r}^{+}$, then we shall denote by $g_{\\nu,\\eta}$ the map from $U_{\\nu}$ to $U_{\\eta}$ that replaces the prefix $\\nu$ with $\\eta$.\n\nA finite set $\\overline{u}:= \\{u_0, \\ldots, u_l\\}$ for $u_i \\in X_{n,r}^{+}$ and some $1\\le l \\in \\N$ is called a \\emph{complete antichain}, if for any pair $u_i, u_j \\in \\overline{u}$, $u_i \\perp u_j$ and for any word $v \\in X_{n,r}^{\\ast}$, there is some $u_i \\in \\overline{u}$ such that $v \\le u_i$ or $u_i \\le v$. We shall assume through out that any given antichain $\\overline{u}$ is ordered according to the lexicographic ordering.  If we have an antichain $\\ac{u}:= \\{u_0, \\ldots, u_l\\}$ then we may form another antichain $\\ac{u}' = \\{u_0, \\ldots, u_i0, u_i1, \\ldots u_in-1, u_{i+1}, \\ldots, u_l\\}$ where we have replaced $u_i$ with  $u_i0, \\ldots, u_in-1$. We call $\\ac{u}'$ a subdivision of $\\ac{u}$. Notice that each time we make a subdivision the length of the antichain increases by $n-1$.\n\nWe now define the Higman-Thompson group $G_{n,r}$ and the subgroup $T_{n,r}$.\n\nGiven two complete antichains $\\overline{u} = \\{u_0,\\ldots,u_{l-1} \\}$ and $\\overline{v} = \\{v_0, \\ldots, v_{l-1}\\}$ of equal length,  we can  define a homeomorphism from of $\\CCnr$ as follows. Let $\\pi$ be a bijection from $\\overline{u}$ to $\\overline{v}$, define a map $g : \\CCnr \\to \\CCnr$ by $ u_ix \\mapsto (u_i)\\pi x$ for $u_i \\in \\overline{u}$ and $x \\in X_n^{\\omega}$. Since $u$ is a complete antichain this map is well-defined on all of $\\CCnr$. The group $G_{n,r}$ consists of all homeomorphisms of $\\CCnr$ which can be defined in this way. The group $T_{n,r}$ is the subgroup of $G_{n,r}$, consisting of those element  $g$ that are derived from complete antichains $\\overline{u}$ and $\\ac{v}$ of the same length $l$, and a bijection $\\pi$ which maps $u_i$ to $v_{i+j \\mod l}$ for some fixed $j \\in \\{0,1 \\ldots, l-1\\}$. \n\nDefine an equivalence relation on $\\CCnr$ by $x \\sim_{t} y$ if and only if there are words $u$ and $v$ in $X_{n,r}^{+}$ such that $x = u z$ and $y = v z$ for some $z \\in \\CCn$. We write $[x]$ to denote the equivalence class of $x$. Let $H_{n,r,\\sim_{t}}$ be the subgroup of $H(\\CCnr)$ consisting of those elements of $H(\\CCnr)$ that preserve $\\sim_{t}$.\n\nWe shall need the following transitivity result for $T_{n,r}$\n\n\\begin{lemma}\\label{Lemma: Tnr preserves tail classes and acts transitively on each tail class}\nThe group $T_{n,r}$ fixes the equivalence classes of $\\sim_{t}$ and acts transitively on each equivalence class.\n\\end{lemma}\n\\begin{proof}\nThe first statement follows from the definition of $T_{n,r}$ as a subgroup of $G_{n,r}$. The second part of the lemma follows by  the following observation:  if $x = \\mu z$ and $y = \\nu z$ for some $\\mu, \\nu \\in X_{n,r}^{+}$ and $z \\in \\CCn$, then there is an element of $T_{n,r}$ mapping  $x$ to $y$. To see this observe that the lengths of the complete antichains $\\ac{u}:= X_{n,r}^{|\\mu|}$ and $\\ac{v} := X_{n,r}^{|\\nu|}$ are congruent modulo $n-1$. We may assume that the length of  $\\ac{u}$ is smaller than the length of $\\ac{v}$. Therefore there is a complete antichain $\\ac{u}'$ such that $\\mu$ is an element of  $\\ac{u}'$, $\\ac{u}'$ is a repeated subdivision of $\\ac{u}$ and the length of $\\ac{u}'$ is equal to the length of $\\ac{v}$. It is now  straightforward to see that, since $|\\ac{v}_1'| = |\\ac{v}_2$, there is an element of $T_{n,r}$ that sends any element $\\mu z'$ to the element $\\nu z'$ for $z' \\in \\CCn$. In particular such a map sends $x$ to $y$.\n\\end{proof} \n\nIn the next Section, we see that elements of $T_{n,r}$ induce homeomorphisms of the circle $\\mathbb{R}\/r\\Z$ by thinking of the circle as a quotient of Cantor space.\n\n\\section{From Cantor Space to the Circle}\\label{Section:fromcantorspacetothecircle}\n\n \n \\begin{Definition}\n We say that a homeomorphisms $h$ of $\\CCnr$ is \\emph{orientation preserving} if  whenever $x,y \\in \\CCnr$ and $x \\lelex y$ then $(x)h \\lelex (y)h$. We say that $h$ is \\emph{orientation reversing} if whenever $x\\lelex y$ then $(y)h \\lelex (x)h$. We say that $h$ is \\emph{locally orientation preserving\/  reversing} if there is a neighbourhood of $\\CCnr$ on which $h$ is orientation preserving\/reversing.\n \\end{Definition}\n \nWe identify $S_r$ with the interval $[0, r]$ with the end points identified. Now, every point in $[0,r]$ can be written as $\\dot{a}x$ for $x \\in \\CCn$ and $\\dot{a} \\in \\dotr$ in $n$-ary expansion. However this representation is not unique. In particular two elements $x, y \\in \\CCnr$ represents the same element of $[0,r]$ if and only if there is some integer $i$, $0< i \\le n-1$ or $\\dot{0}< i \\le \\dot{r-1}$, and some $w \\in \\Xnr^{*}$ such that $x = wi00\\ldots$ and $y= w(i-1)n-1n-1\\ldots$ (i.e only elements of $(0,r)\\cap \\Z[1\/n]$ have non-unique $n$-ary representations ).  Let $\\simeq$ be the equivalence relation on $\\CCnr$ defined by $x \\simeq y$ if and only if there is some $0< i \\le n-1$ or $\\dot{0} < i \\le \\dot{r-1}$ and some $w \\in \\Xnr^{+}$ such that $x = wi00\\ldots$ and $y= w(i-1)n-1n-1\\ldots$ or $x= \\dot{0}00\\ldots$ and $y = \\dot{r}n-1 n-1 \\ldots$ then $\\CCnr\/\\simeq$ is homeomorphic to $S_{r}$. Let $\\simeq_{{\\bf{I}}}$ be the relation on $\\CCn$ given by  $x \\simeq_{{\\bf{I}}} y$ if and only if there is some $0< i \\le n-1$ or $\\dot{0} < i \\le \\dot{r-1}$, and some $w \\in \\Xns$ such that $x = wi00\\ldots$ and $y= w(i-1)n-1 n-1\\ldots$  then $\\CCn\/\\simeq_{{\\bf{I}}}$ is homeomorphic to the interval $[0,r]$.\n\n\n\n\nLet $N$ be the subgroup of $H(\\CCnr)$ consisting of those elements $h$ which preserve $\\simeq$ and which satisfy the following:  for all points $t \\in \\CCnr$ there is a neighbourhood of $t$ in $\\CCnr$ such that $h$ is orientation preserving ($h$ is orientation reversing), and there is some $w \\in \\Xnrs$, $0< i \\le n-1$ or $\\dot{0} < i \\le \\dot{r-1}$, and points $x= wi00\\ldots$ and $y= wi-1n-1n-1\\ldots$ so that $(y)h = \\dot{r}n-1n-1\\ldots$  and $(x)h = \\dot{0}00\\ldots$ ($(y)h =  \\dot{0}00\\ldots$ and $(x)h = \\dot{r}n-1n-1\\ldots$). Observe that all elements of $N$ induce homeomorphisms of $S_{r}$, and $N$ contains $T_{n,r}$. Our aim shall be to show that $N_{H(\\CCnr)}(\\Tnr) \\le N$.\n\n\nWe shall need the following results from \\cite{MBrinFGuzman}. The first follows by Rubin's Theorem and the transitivity of $T_{n,r}$ on the circle $S_{r}$ and the second follows by studying the germs of elements of $T_{n,r}$ at a fixed point.\n\n\n\\begin{Theorem}\nLet $r < n \\in \\mathbb{\\N}$  and let $n \\ge 2$, then $\\aut{T_{n,r}} \\cong N_{H(S_{r})}(T_{n,r})$.\n\\end{Theorem}\n\n\\begin{lemma}\\label{Lemma: normalisers map n-adic rationals to n-adic rationals}\nIf $h \\in N_{H(S_{r})}(\\Tnr)$, then $ ( [0,r] \\cap \\Z[1\/n])h = [0,r] \\cap \\Z[1\/n]$. \n\\end{lemma}\n\nNow let $g \\in N_{H(S_r)}(\\Tnr)$. Define $h \\in H(\\CCnr)$ as follows. For all $t \\in S_{r}$ such that $t \\notin [0,r] \\cap \\Z[1\/n]$ (notice that this also means $(t)g  \\notin [0,r] \\cap \\Z[1\/n]$ by Lemma~\\ref{Lemma: normalisers map n-adic rationals to n-adic rationals}),   let $x$ be the unique $n$-ary expansion of $t$, and $y$ be the unique $n$-ary expansion of $(t)g$, set $(x)h = y$. For $t \\in [0,r] \\cap \\Z[1\/n]$ let $x$ and $x'$ be the $n$-ary expansions of $t$ such that $x \\lelex x'$ in the lexicographic ordering of $\\CCnr$ (if  $t =0$, then chose  $x$ and $x'$, satisfying $x \\lelex x'$, from the set $\\{\\dot{0}00\\ldots,  \\dot{r-1}n-1n-1\\ldots\\}$). Also let $y$ and $y'$ be the $n$-ary expansions of $(t)g$ in $\\CCnr$ such that $y \\lelex y'$ (if $(t)g = 0$ then take  $y= \\dot{0}00\\ldots$ and $y' = \\dot{r}n-1n-1\\ldots$). If $g$ is an orientation preserving homeomorphism of $S_{r}$,  set $(x)h = y$ and $(x')h = y'$, otherwise $g$ is orientation reversing and we set $(x)h = y'$ and $(x')h = y$. Thus $h$ is now defines a bijection from $\\CCnr$ to itself. Moreover it is easy to see that since $g$ is continuous, $h$ is also continuous on $\\CCnr$. Furthermore if $h'$ is the homeomorphism obtained from $g^{-1}$ in the same way, it is easy to see that $hh' = h'h = \\id \\in H(\\CCnr)$. Therefore $h  \\in N$. Let $\\phi: N_{S_r}(\\Tnr) \\to N$ such that an element $g \\in N_{S_r}(\\Tnr)$ maps to the element  $h \\in H(\\CCnr)$, constructed as above, which induces the map $g$ on $S_r$. It follows that $\\phi$ is an injective homomorphism. \n\nLet $N(\\Tnr)$ denote the image of $\\phi$. We now show that $N_{H(\\CCnr)}(\\Tnr) = N(\\Tnr)$. Observe that as $N(\\Tnr) \\subseteq N_{H(\\CCnr)}(\\Tnr)$ it suffices to show only that $N(\\Tnr) \\supseteq N_{H(\\CCnr)}(\\Tnr)$. \n\nLet $\\tau, \\eta \\in X_{n,r}^{+}$ such that $\\tau \\perp \\eta$. We further assume that the points $x=\\tau00\\ldots$, $y=\\eta n-1 n-1\\ldots $, $z= \\tau n-1 n-1 \\ldots$ and $t = \\eta 00\\ldots$ of $\\CCnr$ satisfy $x \\not\\simeq y$ or $z \\not\\simeq t$. \nLet $\\mathscr{G}$ denote the set of incomparable pairs $(\\tau, \\eta)$  satisfying the conditions above where $\\tau  \\lelex \\eta$. For a pair $(\\tau, \\eta)$ in $\\mathscr{G}$ let $\\ac{u} := \\{u_1, \\ldots, u_l\\}$ be any complete finite antichain of $\\Xnrs$ containing $\\tau$ and $\\eta$ (since $\\tau \\perp \\eta$ such an antichain exists). Let $u_{l_1} = \\tau$ and $u_{l_2} = \\eta$. Let  $j = l_1-l_2 -1$. Then we say that $j$ is the \\emph{node distance between $\\tau$ and $\\eta$ in  $\\ac{u}$} i.e $j$ is the number of elements of $\\ac{u}$ which are strictly between  $u_{l_1}$ and $u_{l_2}$. Notice that $j$ is necessarily non-zero by assumption.  Moreover we have, for any $v \\in \\{0\\}^{+}$ and $w \\in \\{n-1\\}^{+}$, that the pair $(\\dot{0}v, \\dot{r-1}w)$  is not in $\\mathscr{G}$.\n\nIt is straight-forward to see that if $(\\tau, \\eta) \\in \\mathscr{G}$ have node distance $i$ in some complete antichain $\\ac{u}_{1}$, then for any other complete antichain $\\ac{u}_2$ containing $\\tau$ and $\\eta$ the node distance between $\\tau$ and $\\eta $ in $\\ac{u}_{2}$ is congruent to $i$ modulo $n-1$. Moreover, for any $\\chi \\in X_n^{*}$, $(\\tau \\chi, \\eta \\chi) \\in \\mathscr{G}$ and there is a complete antichain $\\ac{v}$ containing $\\tau\\chi$ and $\\eta\\chi$ such that the node distance between $\\tau\\chi$ and $\\eta \\chi$ in $\\ac{v}$ is congruent to $i$ modulo $n-1$.  Since modulo $n-1$ the node distance between a pair $(\\tau, \\eta)$ in $\\mathscr{G}$ in a given complete antichain containing $\\tau$ and $\\eta$ is independent of the complete antichain,  we define the \\emph{reduced node distance} between $\\tau$ and $\\eta$ to be $i \\in \\{0,1, \\ldots, n-2\\}$ such that for any complete antichain $\\ac{u}$ containing $\\tau$ and $\\eta$ the node distance between $\\tau$ and $\\eta$ in $\\ac{u}$ is congruent to $i$ modulo $n-1$.\n\nWe require the following transitivity result of $\\Tnr$.\n\n\\begin{lemma}\\label{Lemma: can swap things with same node distance}\nLet $(\\nu_1, \\nu_2), (\\eta_1, \\eta_2) \\in \\mathscr{G}$ be such that the reduced node distance between $\\nu_1$ and $\\nu_2$ is equal to the reduced node distance between $\\eta_1$ and $\\eta_2$. Then there is a $g \\in \\Tnr$ such that $g\\restriction_{U_{\\nu_1}}=  g_{\\nu_1, \\eta_1}$ and $g\\restriction_{U_{\\nu_2}} = g_{\\nu_2, \\eta_2}$. \n\\end{lemma}\n\\begin{proof}\nLet $\\ac{u}$ be a complete antichain containing $\\nu_1$ and $\\nu_2$, and $\\ac{v}$ be a complete antichain containing $\\eta_1$ and $\\eta_2$. Let $\\ac{u}= \\{u_1, \\ldots, u_{l}\\}$ and $\\ac{v} = \\{v_1, \\ldots, v_{m}\\}$. Let $u_{l_1} = \\nu_1$ and $u_{l_2} = \\nu_2$ and $v_{m_1} = \\eta_1$ and $v_{m_2} = \\eta_2$ where $l_1 < l_2$ and $m_1 < m_2$. By assumption $i:= l_2 - l_1 -1 $ and $j:=m_2 - m_1 -1$ are both non-zero and congruent modulo $n-1$. Without loss of generality we may assume $i< j$, moreover there is an element $u_{l_1+1} \\in \\ac{u}$ between $\\nu_1$ or $\\nu_2$. By replacing $u_{l_1 + 1} \\in \\ac{u}$ with $\\{u_{l_1 +1}0, \\ldots, u_{l_1 +1} n-1 \\}$ we obtain a new complete antichain $\\ac{u}'$ containing $\\nu_1$ and $\\nu_2$ such that the node distance between $\\nu_1$ and $\\nu_2$ is equal to $i + n-1$. Replace the antichain $\\ac{u}$ with $\\ac{u}'$\n\nBy repeatedly performing this operation we may assume that the complete antichain $\\ac{u}$ containing $\\nu_1$ and $\\nu_2$ is such that the node distance between $\\nu_1$ and $\\nu_2$ in $\\ac{u}$ is equal to $j$.\n\nObserve that as $\\ac{u}$ and  $\\ac{v}$ are complete antichains the difference $| m-l|$ is congruent to $0$ modulo $n-1$. Without loss of generality (as we may relabel to achieve this) we assume that $l < m$.\n\nNow  observe that $u_1$ must be equal to $\\dot{0}w_1$ for some $w_1 \\in \\{0\\}^{*}$ and  $u_{l} = \\dot{r-1}w_2$ for some $w_2 \\in \\{n-1\\}^{*}$. Moreover, since the pair  $(\\dot{0}v, \\dot{r-1}w)$ for $v \\in \\{0\\}^{+}$ and $w \\in \\{n-1\\}^{+}$ is not in $\\mathscr{G}$, it follows that either $\\nu_{l_1} \\ne u_{l} = \\dot{0}w_1$ or $\\nu_{l_2} \\ne \\dot{r-1}w_2$. Suppose that $\\nu_{l_1} \\ne u_{l} = \\dot{1}w_1$ (the other case is handled similarly). Then by repeatedly expanding along the node $\\dot{1}w_1$, we obtain a complete antichain $\\ac{u}''$ of length $m$ containing $\\nu_1$ and $\\nu_2$ such that the node distance between $\\nu_1$ and $\\nu_2$ is $j$.\n\nSince the complete antichain $\\ac{u}''$ and $\\ac{v}$ have the same lengths and the node distance between $\\nu_1$ and $\\nu_2$ in $\\ac{u}''$ is equal to the node distance between $\\eta_1$ and $\\eta_2$ in $\\ac{v}$, it is easy to construct an element $g$ of $\\Tnr$  such that $g\\restriction_{U_{\\nu_1}}=  g_{\\nu_1, \\eta_1}$ and $g\\restriction_{U_{\\nu_2}} = g_{\\nu_2, \\eta_2}$. \n\\end{proof}\n\nUsing this transitivity result we show below  that $N(\\Tnr)$ is equal to $N_{H(\\CCnr)}(\\Tnr)$.\n\n\\begin{lemma}\nLet $h \\in H(\\CCnr)$ be such that $h \\in N_{H(\\CCnr)}(\\Tnr)$. Then $h$ (and so $h^{-1}$) preserves the equivalence relation $\\simeq$.\n\\end{lemma}\n\\begin{proof}\nSuppose there are $x, y \\in \\CCnr$ such that $x \\simeq y$ but $(x)h \\not\\simeq (y)h$. By relabelling if necessary we may assume that $(x)h < (y)h$. Since $(x)h \\not \\simeq (y)h$ there are $(\\tau, \\eta) \\in \\mathscr{G}$ such that $(x)h \\in U_{\\tau}$ and $(y)h \\in U_{\\eta}$. Let $j \\in \\{0,1 , \\ldots, n-2\\}$ be the reduced node distance between $\\tau$ and $\\eta$\n\nLet $(\\mu, \\nu) \\in  \\mathscr{G}$ and consider the clopen sets $(U_{\\mu})h$ and $(U_{\\nu})h$. It is straight-forward to see that there are $\\mu'$ and $\\nu'$ such that $(\\mu', \\nu') \\in \\mathscr{G}$, the reduced node distance between $\\mu'$ and $\\nu'$ is equal to $j$, and $U_{\\mu'} \\subset (U_{\\mu})h$ and $U_{\\nu'} \\subset (U_{\\nu})h$ (the case where $U_{\\mu'} \\subset (U_{\\nu})h$ and $U_{\\nu'} \\subset (U_{\\mu})h$ is analogous). By Lemma~\\ref{Lemma: can swap things with same node distance} there is an element $g \\in \\Tnr$ such that $g\\restriction_{U_{\\tau}} = g_{\\tau,\\mu'}$ and $g \\restriction_{U_{\\eta}} = g_{\\eta, \\nu'}$.\n\nNow consider the product $h g h^{-1}$. Notice that $(x) h g h^{-1}$ is contained in the clopen set $U_{\\mu}$ since $(x)h \\in U_{\\tau}$ and $(U_{\\tau})g \\in U_{\\mu'} \\subset (U_{\\mu})h$, likewise $(y)h g h^{-1} \\in U_{\\nu}$. Therefore $(x)hgh^{-1} \\not\\simeq (y)hgh^{-1}$, which is a contradiction since $T_{n,r}$ preserves $\\simeq$.\n\\end{proof}\n\n\\begin{lemma}\nLet $h \\in H(\\CCnr)$ be such that $h \\in N_{H(\\CCnr)}(\\Tnr)$, then $h \\in N(\\Tnr)$.\n\\end{lemma}\n\\begin{proof}\nSince $h \\in N_{H(\\CCnr)}(\\Tnr)$, then by the previous lemma $h$ preserves $\\simeq$. Therefore $h$ induces a continuous function $g$ on  $S_{r}$. However since $h^{-1}$ is also in $N_{(H(\\CCnr))}$, $g$ must be a homeomorphism and $(g)\\phi = h$, therefore $h \\in N(\\Tnr)$.\n\\end{proof}\n\nThus we have now proved that $N_{H(\\CCnr)}(\\Tnr) = N(\\Tnr) \\cong \n\\aut{\\Tnr}$. \n\n\\section{Automorphisms of \\texorpdfstring{$T_{n,r}$}{Lg}}\\label{Section:AutTnr}\n In what follows we adapt the results of \n\\cite{BCMNO} to show that elements of \n$N_{H(\\CCnr)}(\\Tnr)$ can be represented by bi-synchronizing \ntransducers.  However we shall need to define the group $\\T{R}_{n,r}$ introduced in \\cite{BCMNO} which is a slight  modification of the Rational group $\\mathcal{R}_{n}$ in  \\cite{GriNekSus}. The exposition in this section will largely mirror that found in \\cite{BCMNO}.\n\nFirst we introduce transducers generally then we introduce transducers over $\\CCnr$.\n\nLet $X_{I}$ and $X_{O}$ be finite sets of symbol. A transducer over the alphabet $X_{I}$ is a tuple $T = \\gen{X_{I},X_O, Q_{T} \\pi_{T}, \\lambda_{T}}$ such that:\n\n\\begin{enumerate}[label = (\\roman{*})]\n\\item  $X_{I}$  is the \\emph{input alphabet} and $X_{O}$ is the \\emph{output alphabet}.\n\\item $Q_T$ is the set of states of $T$.\n\\item $\\pi_{T}: X_I \\times Q_{T} \\to Q_{T}$ is the \\emph{transition function} and,\n\\item $\\lambda_{T}: X_I \\times Q_{T} \\to X_{I}^{*}$ is the \\emph{output function}.\n\\end{enumerate}\n\n\nIf $|Q_T| < \\infty$ then we say that $T$ is  a finite transducer. If $X_{I} = X_{O} = X$ then we shall write $T = \\gen{X, Q_{T}, \\pi_{T}, \\lambda_{T}}$. If we fix a state $q \\in Q_{T}$ from which we begin processing inputs then we say that $T$ is \\emph{initialised at $q$} and we denote this by $T_{q}$ and we call $T_{q}$ an \\emph{initial transducer}. Given an initial transducer $T_{q_0}$ we shall write $T$ for the underlying transducer with no initialised states.\n\nWe inductively extend the domain of the transition and rewrite function to  $X_{I}^{*} \\times Q_T $ by the following rules:\nfor a word $w \\in X_{I}^{*}$, $i \\in X_{I}$ and any state $q \\in Q_T$ we have $\\pi_T(wi, q ) = \\pi_T(i, \\pi_T(w, q))$ and $\\lambda_{T}(wi, q) = \\lambda_{T}(w, q)\\lambda_{T}(i, \\pi_{T}(w, q)) $. We then extend the domain of $\\pi_{T}$ and $\\lambda_{T}$ to  $X_{I}^{\\omega} \\times Q_{T}$.\n\nIn this paper we shall insist that for $\\delta \\in X_{I}^{\\omega}$ and any state $q \\in Q_T$ we have $\\lambda_{T}(\\delta, q) \\in X_{O}^{\\omega}$. This means that for a state $q \\in Q_T$ the initial transducer $T_{q}$ induces a continuous function $h_{T_{q}}$ ($h_{q}$ if it is clear from the context that $q$ is a state of $T$) from $X_I^{\\omega}$ to $X_O^{\\omega}$. For $q \\in Q_{T}$ we denote by $\\im(q)$ the image of the map $h_{q}$; if $h_{q}$ is a homeomorphism form $X_{I}^{\\omega} \\to X_{O}^{\\omega}$, then we call $q$ a \\emph{homeomorphism state}.\n\nTwo states $q_1$ and $q_2$ of $T$ are called $\\omega$-equivalent if $h_{q_1} = h_{q_2}$. A state $q$ of $T$ is called a state of incomplete response if for some $i \\in X_{I}$ $\\lambda_{T}(i, q)$ is not equal to the greatest common prefix of the set $\\{(i \\delta)h_{q} \\mid \\delta \\in X_{I}^{\\omega}\\}$. If $T$ is an initial transducer with initial state $q_0$, then $q$ is called \\emph{accessible} if  there is a word $w \\in X_I^*$ such that $\\pi_{T}(w, q_0) = q$. If all the states of $T_{q_0}$ are accessible then $T_{q_0}$ is called \\emph{accessible}.\n\nAn initial transducer $T_{q_0}$ is called \\emph{minimal} if $T_{q_0}$ is accessible, has no states of incomplete response and no pair of $\\omega$-equivalent states. The initial transducer $T_{q_0}$ is also called invertible if the state $q_0$ is a homeomorphism state. Given an initial transducer  $T_{q_0}$ there is a unique minimal transducer $S_{p_0}$ $\\omega$-equivalent to $T_{q_0}$ (\\cite{GriNekSus}).\n\nWe give below the method given in \\cite{GriNekSus} for constructing the inverse of an invertible, minimal transducer $T_{q_0}$. We first define the following function.\n\n\\begin{Definition}\nLet  $T_{q_0} = \\gen{X_{I}, X_{O}, Q_{T}, \\pi_{T}, \\lambda_{T}}$ be an invertible minimal transducer and $q$ a state of $T_{q_0}$. Define a function $L_{q}: X_{O}^{+} \\to X_{I}^{\\ast}$ by $(\\nu)L_{q} = \\varphi$ where  $\\varphi$ is the greatest common prefix of the set $(U_{\\nu})h_{q}^{-1}$.\n\\end{Definition}\n\nObserve that since $T_{q_0}$ is minimal and invertible, then each state $q$ of $T_{q_0}$ induces an injective function from $X_{I}^{\\omega}$ to $X_{O}^{\\omega}$ with clopen image. From this one can deduce that for each state $q$ of $T_{q_0}$ the set of words $w \\in  X_{O}^{\\ast}$ such that $(w)L_{q} = \\epsilon$ and $U_{w} \\subset \\im(q)$ is finite (see \\cite{GriNekSus}).\n\nWe now form a transducer $T_{(\\epsilon, q_0)} = \\gen{X_{O}, X_{I}, Q'_{T}, \\pi'_{T}, \\lambda'_{T}}$ where $Q'_{T} = \\{ (w,q) \\mid  q \\in Q_{T}, (w)L_{q} = \\epsilon, U_{w} \\subset \\im(q) \\}$ and $\\pi'_{T}$ and $\\lambda'_{T}$ are defined, for all $i \\in X_{O}$ and $(w,q) \\in Q'_{T}$, by the rules:\n\\begin{enumerate}[label=(\\roman*)]\n\\item  $\\pi'_{T}(i, (w,q)) = (wi - \\lambda_{T}((wi)L_{q},q), \\pi_{T}((wi)L_{q}, q)$ and, \n\\item $\\lambda'_{T}(i, (w,q)) = (wi)L_{q}$.\n\\end{enumerate} \n\nThe following proposition is a result in \\cite{GriNekSus}:\n\n\\begin{proposition}\\cite{GriNekSus}\nFor $T_{q_0}$ a minimal invertible transducer, the transducer $T_{(\\epsilon, q_0)}$ is well-defined, has no states of incomplete response, and satisfies $h_{(\\epsilon, q_0)} = h_{q_0}^{-1}$.\n\\end{proposition}\n\nWe observe that given a minimal invertible transducer $T_{q_0}$, the transducer $T_{(\\epsilon,q_0)}$ is  accessible if $T_{q_0}$. This is because if for any word $\\Gamma \\in \\Xnp$, $(\\Gamma)L_{q_0} = \\epsilon$, then $\\pi'_{T}(\\Gamma,  (\\epsilon, q_0)) = (\\Gamma, q_0)$. Furthermore,  for any word $w \\in \\Xnp$  and state $q \\in Q_{T}$, such that  $U_{w} \\subset  \\im(q)$ and $(w)L_{q} = \\epsilon$, picking a word $\\Gamma \\in \\Xnp$ such that  $\\pi_{T}(\\Gamma,q_0) = q$, we observe that $(\\lambda_{T}(\\Gamma, q_0)w)L_{q_0} = \\Gamma$ and so $\\pi'_{A}(\\lambda_{T}(\\Gamma, q_0)w, (\\epsilon,q_0)) = (w,q)$. Note that given a minimal invertible transducer $T_{q_0}$, the transducer $T_{(\\epsilon, q_0)}$, even though it has no states of incomplete response and is accessible, might not be minimal. \n\nGiven two  transducers $A = \\gen{X, Q_{A}, \\pi_{A}, \\lambda_{A}}$ and $B = \\gen{X, Q_B, \\pi_B, \\lambda_B}$ then the  product $A*B = \\gen{X, Q_{A*B}, \\pi_{A*B}, \\lambda_{A*B}}$ is the transducer defined as follows. The set of states $Q_{A*B}$ of $A*B$ is equal to the cartesian product $Q_{A} \\times Q_{B}$. The transition and output function of $A*B$ are given by the following rules: for states $q \\in Q_A$ , $p \\in Q_B$ and $i \\in X$ we have $\\pi_{A*B}(i, (q,p)) = (\\pi_{A}(i,q),\\pi_B(\\lambda_{A}(i,q), p) )$ and $\\lambda_{A*B}(i, (q,p)) = \\lambda_{B}(\\lambda_{A}(i, q), p)$. For two initial transducers  $A_{q_0}$ and $B_{p_0}$, where $A$ and $B$ are the resulting transducers with no initialised states, the product of the initial transducer $A_{q_0} * B_{q_0}$, is the initial transducer $(A*B)_{(q_0, p_0)}$. It is straightforward to see that $h_{(A*B)_{(q_0, p_0)}} = h_{A_{q_0}}\\circ h_{B_{p_0}}$.\n\nA transducer (initial or non-initial) $T = \\gen{X_I, X_O, Q_T, \\pi_T, \\lambda_T}$ is said to be \\emph{synchronizing at level $k$} if there is a natural number $k \\in \\N$ and a map $\\mathfrak{s}: X_I^{k} \\to  Q_T$ such that for a word $\\Gamma \\in X_I^{{k}}$ and for any state $q \\in Q_T$ we have $\\pi_{T}(\\Gamma, q) = (\\Gamma)\\mathfrak{s}$. We will denote by $\\core(T)$ the sub-transducer of $T$ induced by the states in the image of $\\mathfrak{s}$. We call this sub-transducer the \\emph{core of $T$}. If $T$ is equal to its core then we say that $T$ \\emph{core}. Viewed as a graph $\\core(T)$ is a strongly connected transducer. If $T$ is an initial transducer $T_{q_0}$  which is invertible, then we say that $T_{q_0}$ is bi-synchronizing if both $T_{q_0}$ and its inverse are synchronizing. Note that when $T$ is synchronous, then we shall say $T$ is bi-synchronizing if $T$ and its inverse are synchronizing.\n\nGiven  a transducer $T$ and states $q_1, q_2$ of $T$ we shall sometimes use the phrase $q_1$ and $q_2$ \\emph{transition identically  on a subset $W \\subset \\Xns$} to mean that the functions $\\pi_{T}(\\centerdot, q_1): W \\to Q_{T}$ and $\\pi_{T}(\\centerdot, q_2): W \\to Q_{T}$ are identical. We might also say that $q_1$ and $q_2$ \\emph{read all elements of $W$ to the same location}.\n\nNow we introduce transducers over $\\CCnr$. An initial transducer for $\\CCnr$ is a tuple $A_{q_0}=(\\dotr, X_n, R, S, \\pi, \\lambda, q_0)$ such that:\n\n\\begin{enumerate}[label = (\\roman*)]\n\\item $R$ is a finite set, and the set $Q$ of states of  $A$ is the disjoint union $R \\sqcup S$. The state $q_0 \\in R$ is the initial state\n\\item $\\pi: \\dotr \\times \\{q_0\\} \\sqcup X_n \\times  Q \\to Q\\backslash \\{q_0\\}$ and $\\lambda: \\dotr \\times q_0 \\sqcup X_n \\times Q \\to X_{n,r}^{\\ast} \\sqcup X_n^{\\ast}$\n\\end{enumerate}\n\nNotice that a letter from $\\dotr$ can only be read from the initial state $q_0$ and we can never return to $q_0$ after leaving $q_0$. We Inductively extend the domain of $\\pi$ and $\\lambda$ to $ \\dotr \\times \\{q_0\\} \\sqcup  X_n^{*} \\times Q$ by the following rules:\n\n$\\pi(wx, q) = \\pi(x, \\pi(w,q))$ and $\\lambda(wx, q) = \\lambda(w, q)\\lambda(x, \\pi(w,q))$. Where $w \\in X_{n,r}^{+} \\sqcup X_n^{+}$ and $x \\in X_n$ (if $w \\in X_{n,r}^{+}$ then $q = q_0$). We also take the convention that $\\pi(\\epsilon, q) = q$ and $\\lambda(\\epsilon, q) = \\epsilon$ for any state $q$ of $A$. By transfinite induction we may further extend the domains of $\\pi$ and $\\lambda$ to $ \\dotr \\times \\{q_0\\} \\sqcup  X_n^{\\omega} \\times Q$.\n\nWe impose the  following rules on $\\pi$ and $\\lambda$:\n\n\\begin{enumerate}[label = (\\arabic*)] \n\\item For a state $r \\in R $ and for $i$ in $\\dotr \\sqcup X_n$ such that $\\pi(i, r)$ is defined, if $\\pi(i, r) \\in R$ then $\\lambda(i, r) = \\epsilon$, otherwise $\\lambda(i, r) \\in X_{n,r}^{*}$. \\label{List: conditions of pi and lambda 1}\n\\item  For $x \\in X_n$ and $q \\in S$, $\\lambda(x, q) \\in X_{n}^{*}$  and $\\pi(x, q) \\in S$. \\label{List: conditions of pi and lambda 2}\n\\item For a state $s \\in S$  and $\\delta \\in \\CCn$ we have that $\\lambda(\\delta, s) \\in \\CCn$. \\label{List: conditions of pi and lambda 3}\n\\item If there is a word  $w \\in X_n^{+}$ and a state $q \\in Q$ such that $\\pi(w, q) = q$ then $q \\in  S$. \\label{List: conditions of pi and lambda 4}\n\\end{enumerate}\n\nThese rules serve the purpose of ensuring that  whenever an element of $\\CCnr$ is  is processed through $A_{q_0}$ the output is also in $\\CCnr$.\n\nLet $A_{q_0}$ be an initial transducer on $\\CCnr$ as above and let $q$ be a state of $A_{q_0}$. Let $A_q$ denote the initial transducer $A_{q_0}$ where we process inputs from the state $q$. Observe that $A_{q_0}$ induces a continuous function $h_{A_{q_0}}$ (or $h_{q_0}$ if it clear that $q_0$ is the initial state of $A$) from $\\CCnr$ to itself. Furthermore every non-initial state $q$ of $A_{q_0}$ which is also an element of $R$ induces a continuous function $h_{q}$ from $\\CCn$ to $\\CCnr$, otherwise it induces a continuous function from $\\CCn$ to itself. Once again we denote by $\\im(q)$ the image of $q$ and call $q$ a homeomorphism state if $h_{q}$ is a homeomorphism from its domain to its range.\n\nWe extend, in the natural way, the definition of accessibility, accessible transducers,and states of incomplete response given in the general setting to the specific setting of transducers over $\\CCnr$. We also extend the function $L_{q}$ for a minimal invertible transducer $T_{q_0}$ over $\\CCnr$ and $q \\in Q_{T}$. Having done this, we may thus define, given $T_{q_0}$ a minimal, invertible transducer, the transducer $T_{(\\epsilon, q_0)}$ such that $h_{(\\epsilon, q_0)} = h_{q_0}^{-1}$.\n\nWe say that a transducer $A_{q_0}$ over $\\CCnr$ is synchronizing  if there is a $k \\in \\mathbb{N}$ such that given any word $\\Gamma$ of length $k$ in $\\Xnrs \\sqcup \\Xns$ the active state of $A_{q_0}$ when $\\Gamma$ is processed from any \\emph{appropriate} state of $A_{q_0}$ is completely determined by $\\Gamma$. Thus we may also extend the notions of `core' for synchronizing transducers over $\\CCn$. We now introduce the notion of $\\omega$-equivalence and minimality for transducers over $\\CCnr$. Two initial automata with the same domain and range are said to be \\emph{$\\omega$-equivalent} if they induce the same continuous function from their domain to their range. \n\nAn initial transducer $A_{q_0}$ is called \\emph{minimal} if $A_{q_0}$ is accessible, no states of $A$ are states of incomplete response and for any distinct pair $q_1, q_2$ of states of $A_{q_0}$, $A_{q_1}$ and $A_{q_2}$ are not $\\omega$-equivalent.  In \\cite{BCMNO} the authors show, by slight modifications of arguments in \\cite{GriNekSus}, that for an initial transducer $A_{q_0}$ on $\\CCnr$ there is a unique minimal transducer under $\\omega$-equivalence.\n\n\nThe product $(A*B)_{(q_0,p_0)} = \\gen{\\dotr, \\Xn, R_{A} \\times R_{B} \\sqcup  S_{A} \\times R_{B}, S_A \\times S_B, \\pi_{A*B}, \\lambda_{A*B}}$ of the initial transducers $A_{q_0}$ and $B_{p_0}$ over $\\CCnr$ is defined as follows. The set of states $Q_{A*B}  = R_{A} \\times R_{B} \\sqcup  S_{A} \\times R_{B} \\sqcup S_A \\times S_B$, and the state $(q_0, p_0) \\in  R_{A} \\times R_{B}$ is the initial state. The transition and output functions are defined as follows. First for $a \\in \\dotr$ we have $\\pi_{A*B}(a, (q_0, p_0)) = (\\pi_{A}(a, q_0), \\pi_{B}(\\lambda_{A}(a, q_0), p_0 )$, and $\\lambda_{A*B}(a, (p_0, q_0)) = \\lambda_{B}(\\lambda_{A}(a, q_0), p_0)$. Now for any pair $(q, p) \\in  Q_{A,B}$ and for any $i \\in X_n$ we have $\\pi_{A*B}(i, (q,p)) = (\\pi_{A}(i, q), \\pi_B(\\lambda_{A}(1, q),p)$ and $\\lambda_{A*B}(i, (q,p)) = \\lambda_{B}(\\lambda_{A}(i,q),p)$. Now observe that as $A$ and $B$ satisfy condition \\ref{List: conditions of pi and lambda 1} to \\ref{List: conditions of pi and lambda 4} above then so does the product $(A*B)_{(q_0, p_0)}$. Furthermore, as before, it is a straightforward observation that $h_{(A*B)_{(q_0,p_0)}} = h_{A_{q_0}}\\circ h_{B_{p_0}}$.\n\nBelow we outline a procedure given in \\cite{BCMNO} for constructing from a homeomorphism $h$ of $\\CCnr$, an initial transducer $A_{q_0}$ of $\\CCnr$ such that $h_{q_0} = h$. \n\n\nWe first need to define local actions.\n\nFor an arbitrary homeomorphism $g: \\CCnr \\to \\CCnr$,  fix the notation   $P_{g} \\subset  X_{n,r}^{*}$ for the unique maximal subset of $X_{n,r}^{*}$ set satisfying: $(U_{\\nu})g \\subseteq U_{\\dot{a}}$ for $\\nu \\in P_{h}(a)$, and for any proper prefix $\\mu$ of $\\nu$ there are elements $\\delta_1, \\delta_2 \\in \\CCn$ and $\\dot{a}_1, \\dot{a}_{2} \\in \\dotr$ such that $(\\mu\\delta_1)g \\in U_{\\dot{a}_{1}}$ and $(\\mu\\delta_2)g \\in U_{\\dot{a}_{2}}$. Observe that since $g$ is a homeomorphism  and since $\\sqcup_{\\dot{a} \\in \\dotr} (U_{\\dot{a}})g^{-1}$ is clopen, $P_{g}$ exists. Moreover maximality of $P_{g}$ implies that $P_{g}$ is a complete antichain for $\\Xnrs$.\n\n\nLet $h : \\CCnr \\to \\CCnr$ be a homeomorphism. Define $\\theta_{h}: X_{n,r}^{*} \\to X_{n,r}^{*}$ as follows: for $\\mu \\in X_{n,r}^{*}$ if $\\mu$ is a prefix of some $\\nu \\in P_{h}$ set $(\\mu)\\theta_{h}:= \\epsilon$, otherwise $\\mu = \\nu \\chi$ for some $\\nu \\in P_{h}$ and $\\chi \\in X_{n,r}^{*}$, in this case set $(\\mu)\\theta_{h}$ to be the greatest common prefix of the set $(U_{\\mu})h$, since $h$ is a homeomorphism and by choice of $P_{h}$, $(U_{\\mu})h \\in X_{n,r}^{+}$.\n\nNow for $\\mu \\in  X_{n,r}^{*}$  we define a map $h_{\\mu}$ on  $\\CCn$ by $(\\delta)h_{\\mu} = (\\mu\\delta)h_{\\mu} - (\\mu)\\theta_{h}$. We call $h_{\\mu}$ the \\emph{local action of $h$ at $\\mu$}. Observe that if $\\mu$ is a prefix of an element of $P_{h}$ then the range of $h_{\\mu}$ is $\\CCnr$ otherwise the range of $\\mu$ is $\\CCn$, in either case $h_{\\mu}$ is continuous. The following fact is straightforward, let $\\mu, \\nu \\in X_{n,r}^{*}$ then $(\\mu \\nu)\\theta_{h} = (\\mu)\\theta_{h}(\\nu)\\theta_{h_\\mu}$. \n\n\nUsing the function $\\theta_{h}$ we now construct an initial transducer $A_{q_0}$ of $\\CCnr$ such that $h_{q_0}  = h$.\n\nLet $h: \\CCnr \\to \\CCnr$ be a homeomorphism. Form a transducer \n$A_{\\epsilon} = (\\dot{r}, X_n, R_{A}, S_{A}, \\pi_A, \\lambda_{A}, \\epsilon \n)$. The set $Q_{A}$ of states of  $A$ is precisely \n$X_{n,r}^{*}$ and $R_A \\subset Q_{A}$ is the set of proper \nprefixes of elements of $P_{g}$ and $S_A := \\Xns \\backslash R$. The \ntransition of function and output functions $\\pi_A$ and \n$\\lambda_{A}$ are defined as follows. For $\\dot{a} \\in \\dotr$ we \nhave  that $\\pi_{A}(\\dot{a}, \\epsilon) = \\dot{a}$ and $\\lambda_A \n(\\dot{a}, \\epsilon) = (\\dot{a})\\theta_{h}$; for $\\nu \\in \\Xnrp$ \nand $i \\in X_n$ we have $\\pi_{A}(i, \\nu) =\\nu i$ and \n$\\lambda_{A}(i, \\nu) = (\\nu i) \\theta_{h}  - (\\nu)\\theta_{h}$.\n\nObserve that the transducer $A_{\\epsilon}$ satisfies the conditions \\ref{List: conditions of pi and lambda 1} to \\ref{List: conditions of pi and lambda 4}. The following claim is straightforward to prove:\n\n\\begin{claim}\\label{Claim:tranducerarisingfromhomeo}\nLet  $h: \\CCnr \\to \\CCnr$ be a homeomorphism, and $A_{\\epsilon} = (\\dot{r}, X_n, R_{A}, S_{A}, \\pi_A, \\lambda_{A}, \\epsilon \n)$ be the transducer constructed form $h$ as above. Let $\\nu \\in \\Xnrs$ then for all $w \\in \\Xns$ if $\\nu \\ne \\epsilon$ or $w \\in \\Xnrp$ if $\\nu = \\epsilon$ we have that $(w)\\theta_{h_{\\nu}} = \\lambda_{A}(w, \\nu)$.  \n\\end{claim} \n\\begin{proof}\nFirst suppose $\\nu \\ne \\epsilon$ and let $x \\in X_n$. Then by \ndefinition we have $\\lambda_{A}(i,\\nu) = (\\nu i) \\theta_{h} - \n(\\nu)\\theta_{h}$, however by an observation above we have $(\\nu \ni) \\theta_{h} - (\\nu)\\theta_{h}) = (i)\\theta_{h_{\\nu}}$. Now \nassume by that for all $w \\in  \\Xnp$ we have that $\\lambda(w, \\nu \n) = (w)\\theta_{h_{\\nu}}$. Let $i \\in \\Xn$ and consider \n$\\lambda_{A}(w i, \\nu)$. We may write $\\lambda_{A}(w i, \\nu) = \n\\lambda_{A}(w, \\nu) \\lambda_{A}(i, \\nu w)$ (since $\\pi_A(w, \\nu) \n= \\nu w$). Therefore $\\lambda_{A}(w i, \\nu) = (w)\\theta_{h_{\\nu}} \n(\\nu w i)\\theta_{h} - (\\nu w)\\theta_{h}$. Observe that $(\\nu w \ni)\\theta_{h} - (\\nu w)\\theta_{h} = (i)\\theta_{h_{\\nu w}}$, \ntherefore $\\lambda_{A}(w i, \\nu) = (w)\\theta_{h_{\\nu}} \n(i)\\theta_{h_{\\nu w}}$ however,  $(w)\\theta_{h_{\\nu}} \n(i)\\theta_{h_{\\nu w}} = (wi)\\theta_{h_{\\nu}}$.\n\nNow suppose that $\\nu = \\epsilon$. Let $w \\in \\Xnrp$ and suppose $w = \\dot{a}v$ for some $v \\in \\Xns$ and $\\dot{a} \\in \\dotr$. Consider $\\lambda_{A}(w, \\epsilon)$, this can be broken up into $\\lambda_{A}(\\dot{a}w, \\epsilon) = \\lambda_{A}(\\dot{a}, \\epsilon)\\lambda_{A}(v, \\dot{a})$. Now $\\lambda_{A}(\\dot{a}, \\epsilon) = (\\dot{a})\\theta_{h}$ by definition, and $\\lambda_{A}(v, \\dot{a}) = (v)\\theta_{h_{\\dot{a}}}$ by the previous paragraph. Observe that $(\\dot{a}v)\\theta_{h} = (\\dot{a})\\theta_{h} (v)\\theta_{h_{\\dot{a}}}$ therefore $\\lambda_{A}(\\dot{a}w, \\epsilon) = (\\dot{a}v)\\theta_{h}$ as required.\n\\end{proof}\n\n\n\\begin{Remark}\\label{Remark: finitely many local actions implies finite transducer}\nFrom the claim we deduce that for a homeomorphism $h: \\CCnr \\to \\CCnr$ and for $A_{\\epsilon} =  (\\dotr, \\Xn, R_A, S_A, \\pi_A, \\lambda_A, \\epsilon)$ the transducer constructed from $h$, we have that $h_{A_{\\epsilon}} = h$, moreover for any $\\nu \\in \\Xnrs$ and any local action $h_{\\nu}$ of $h$ we have that $h_{\\nu} = h_{A_{\\nu}}$. Therefore if $h$ has finitely many local actions, then it follows that the minimal transducer on $\\CCnr$,  under $\\omega$-equivalence, representing $h$ has finitely many states. Moreover, since by Claim~\\ref{Claim:tranducerarisingfromhomeo}, $A_{\\epsilon}$ has no states of incomplete response, it follows that, if $B_{q_0}$ is the minimal transducer representing $A_{\\epsilon}$, then for all  $\\nu \\in \\Xnr$ $h_{\\nu} = h_{q}$ for some state $q$ of $B$.\n\\end{Remark}\n\nIn what follows we show that all homeomorphisms of $\\CCnr$ in the group $N(\\Tnr)$ can be represented by a minimal finite transducer. The previous paragraph demonstrates that it suffices to show that such homeomorphisms have finitely many local actions.\n\nOur approach shall essentially mirror that taken in \\cite{BCMNO}, the arguments differ only in so far as we need to make modifications to allow for the fact that the action of  $\\Tnr$ on $\\CCnr$ is not as transitive as the action $\\Gnr$ on $\\CCnr$.\n\nFirst we need the following result:\n\n\\begin{lemma}\\label{Lemma: normalisers in H-tilde}\nLet $X$ be a topological space and let $\\sim$ be any equivalence relation on $X$. Let $H_{\\sim}$ be the subgroup of $H(X)$ consisting of those elements of   $H(X)$ which respect $\\sim$. Let $G \\le H(X)$ be a subgroup  which fixes each equivalence class of $\\sim$ and acts transitively on each  class. Then $N_{H(X)}(G) \\le H_{\\sim}$.\n\\end{lemma}\n\nObserve that $T_{n,r} \\le H(\\CCnr)$, by Lemma ~\\ref{Lemma: Tnr preserves tail classes and acts transitively on each tail class}, preserves $\\sim_{t}$ and acts transitively on each equivalence class of $\\sim_{t}$. Therefore, as  corollary of the lemma above, we have $N(\\Tnr) \\le H_{\\sim_{t}}$. \n\nThe following definitions and lemmas appear in  \\cite{BCMNO} and introduce crucial notions and ideas in understanding local actions of homeomorphisms of $\\CCnr$.\n\n\\begin{Definition}\nLet $V \\subseteq \\CCnr$ be a clopen set. Let $B \\subset \\Xnrs$ be the minimal antichain such that $U_{B}:=\\{U_{\\nu} \\mid \\nu \\in B \\}$ and for all $\\mu$ a prefix of some element of $B$ we have $U_{\\mu} \\not \\subset V$. We call $U_{B}$  the \\emph{decomposition of $V$} and denote it by  $\\dec(V)$. \n\\end{Definition}\n\n\n\\begin{Definition}\\label{Def: acting in the same fashion}\nLet $h \\in H(\\CCnr)$, and $U_{\\nu}$ and $U_{\\eta}$ be elements of $\\Banr$ for $\\nu, \\eta \\in \\Xnrs$. Then we say that $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in \\emph{the same fashion} if $h_{\\nu} = h_{\\eta}$. If $V,W \\subset \\CCnr$ are a clopen subsets then we say that $h$ acts on $V$ and $W$ \\emph{in the same fashion} if for any $U_{\\nu} \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$, $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion.\n\\end{Definition}\n\n\\begin{Definition}\\label{Def: almost in the same fashion}\nLet $h \\in H(\\CCnr)$ and let $V, W \\subset \\CCnr$ be clopen subsets. Then we say that $h$ acts on $V$ and $W$ \\emph{almost in the same fashion} if  there is some $k \\in \\mathbb{N}$ such that for $U_{\\nu} \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$, and for any $\\chi \\in  \\Xn^{k}$ $h$ acts on $U_{\\nu \\chi}$ and $U_{\\eta\\chi}$ in the same fashion. We call the minimal $k$ satisfying this condition the \\emph{critical level of $V$ and $W$} and denote it by $\\crit_{h}(V, W)$.\n\\end{Definition}\n\n\\begin{Definition}\\label{Def: (almost) in the same fashion uniformly}\nLet $h \\in H(\\CCnr)$ and let $V, W \\subset \\CCnr$ be clopen subsets. We say that $h$ acts  on $V$ and $W$ \\emph{in the same fashion uniformly} if for any $\\chi \\in \\Xns$ $h$ and $U_{\\nu} \\in  \\dec(V)$ and  $U_{\\eta} \\in \\dec(W)$, $h$ acts on $U_{\\nu}$ and $U_{\\eta \\chi}$ in the same fashion. We say that $h$ acts \\emph{almost in the same fashion uniformly} on $U$ and $W$ if there is some $k \\in \\N$ such that for $U_{\\nu}  \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$ and for any $\\chi, \\xi \\in \\Xn^{k}$, $h$ acts on $U_{\\nu\\chi}$ and $U_{\\eta \\xi}$ in the same fashion.\n\\end{Definition}\n\n\\begin{Remark}\\label{Remark: Tnr acts almost in the same fashion uniformly}\nLet $V$ and $W$ be clopen subsets of cantor space  and $g \\in \\Tnr$ then $g$ acts on $V$ and $W$ almost in the same fashion uniformly.\n\\end{Remark}\n\nThe following lemmas are crucial in our understanding of local actions of elements of $N(\\Tnr)$ and are taken from \\cite{BCMNO}:\n\n\\begin{lemma}\\label{Lemma: g acts in same fashion h acts in same fashion gh acts in same fashion}\n Let $g,h \\in H(\\CCnr)$ and let $U_{\\nu}, U_{\\eta} \\in \\Banr$. Suppose there are $\\nu', \\eta' \\in \\Xnrp$ such that $(U_{\\nu})g = U_{\\nu'}$ and $(U_{\\eta})g = U_{\\eta'}$, $g$ acts  in the same fashion  on $U_{\\nu}$ and $U_{\\eta}$ and $h$ acts in the same fashion on $U_{\\nu'}$ and $U_{\\eta'}$, then $gh$ acts in the same fashion on $U_{\\nu}$ and $U_{\\eta}$. \n\\end{lemma}\n\n\\begin{lemma}\\label{Lemma: g acts in same fashion, h acts in almost same fashion uniformly, then gh acts in almost the same fashion}\nLet $g,h \\in H(\\CCnr)$ and let $U_{\\nu}, U_{\\eta} \\in \\Banr$. Suppose $g$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion and $h$ acts on $(U_{\\nu})g$ and $(U_{\\eta})g$ in almost the same fashion uniformly, then $gh$ acts in almost the same fashion on $U_{\\nu}$ and $U_{\\eta}$. \n\\end{lemma}\n\n \nWe shall also need the following proposition from the same source:\n\n\\begin{proposition}\\label{Prop: conjugators have same local action at pair of nodes}\nLet $h \\in H(\\CCnr)$ and let $U_{\\tau}$ and $U_{\\eta}$ be elements of $\\Banr$. Suppose that $h_{\\tau \\chi} \\ne h_{\\eta \\chi}$ holds for every $\\chi \\in X_n^{\\ast}$. Then  $h \\notin H_{n,r, \\sim_{t}}$.\n\\end{proposition}\n\nWe have the following corollary:\n\n\\begin{corollary}\\label{Cor:conjugators have same local action at pair of nodes with any node distance}\nLet $h \\in H_{n,r \\sim_{t}}$ and let $i \\in \\{0, 1, \\ldots, n-2\\}$. Then there exists $\\tau, \\eta \\in X_{n,r}$ such that $U_{\\tau} \\cup U_{\\eta} \\ne \\CCnr$, $(\\tau, \\eta) \\in \\mathscr{G}$ the reduced node distance between $\\tau$ and $\\eta$ is  $i$ modulo $n-1$, and $h_{\\tau}  = h_{\\eta}$.\n\\end{corollary}\n\\begin{proof}\nThis is a direct consequence of Proposition~\\ref{Prop: conjugators have same local action at pair of nodes}, since we may chose a pair $(\\tau, \\eta) \\in  \\mathscr{G}$ such that there is a complete antichain $\\ac{u}$ containing $\\tau$ and $\\eta$ and the node distance in $\\ac{u}$ between $\\tau$ and $\\eta$ is congruent to $i$ modulo $n-1$. Moreover, by observations above, for any $\\chi \\in X_n^{*}$, $(\\tau \\chi, \\eta \\chi) \\in \\mathscr{G}$  and any complete antichain $\\ac{v}$ containing $\\tau\\chi$ and $\\eta\\chi$, the node distance between $\\tau\\chi$ and $\\eta \\chi$ in $\\ac{v}$ is congruent to $i$ modulo $n-1$.  \n\\end{proof}\n\n\nWe have the following lemma:\n\n\n\\begin{lemma} \\label{Lemma:  automorphisms have few local actions}\nLet $h \\in H(\\CCnr)$ such that $h^{-1}T_{n,r}h \\subseteq T_{n,r}$. Then for every $U_{\\nu}, U_{\\eta} \\in \\Banr$ such that $U_{\\nu} \\cup  U_{\\eta} \\ne \\CCnr$, the map $h$ acts on  $U_{\\nu}$ and $U_{\\eta}$ almost in the same fashion.\n\\end{lemma}\n\\begin{proof}\n\nFirst observe that, by Lemma~\\ref{Lemma: normalisers in H-tilde}, \n$h \\in H_{\\sim_{t}}$. Moreover by Corollary~\\ref{Cor:conjugators \nhave same local action at pair of nodes with any node distance} \nfor any $i \\in \\{0,1,\\ldots, n-2\\}$ there are elements $\\nu$ and \n$\\eta$ of $X_{n,r}^{+}$ such that $(\\nu, \\eta) \\in \\mathscr{G}$, \nthe reduced node distance between $\\nu$ and $\\eta$ is $i$, \n$U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$, and $h_{\\nu} = h_{\\eta}$. Fix \n$i \\in \\{0,1, \\ldots, n-2\\}$ and let $(\\nu, \\eta) \\in \\mathscr{G}$ \nbe such that the reduced node distance between $\\nu$ and $\\eta$ \nis $i$ and $h_{\\nu} = h_{\\eta}$.\n\nLet $\\nu'$ and $\\eta'$ be elements of $X_{n,r}^{+}$ be such that $(\\nu', \\eta') \\in \\mathscr{G}$ and the reduced node distance between $\\nu'$ and $\\eta'$ is also equal to $i$. By Lemma ~\\ref{Lemma: can swap things with same node distance} there is an element $g \\in \\Tnr$ such that  $g\\restriction_{U_{\\nu'}} = g_{\\nu', \\nu}$ and $g\\restriction_{U_{\\eta'}} = g_{\\eta', \\eta}$.\n\nBy Lemma~\\ref{Lemma: g acts in same fashion h acts in same fashion gh acts in same fashion}, since $h$ acts on  $U_{\\nu}$ and $U_{\\eta}$ in the same fashion, and $g$ acts on $U_{\\nu'}$ and $U_{\\eta'}$ in the same fashion, then $gh$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion. Let $f = h^{-1}gh$. By assumption $f \\in T_{n,r}$ and so $f$ and $f^{-1}$ act on any pair of clopen sets of $\\CCnr$ in almost the same fashion. Therefore, since $h = ghf^{-1}$, by Lemma~\\ref{Lemma: g acts in same fashion, h acts in almost same fashion uniformly, then gh acts in almost the same fashion} it follows that $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in almost the same fashion.\n\nSince $i$ was arbitrarily chosen, we conclude that for any pair $(\\tau, \\mu) \\in \\mathscr{G}$ with reduced node distance equal to $i$, $h$ acts on $U_{\\tau}$ and $U_{\\eta}$ in almost the same fashion.\n\nNow let $\\nu'$ and $\\eta'$ be arbitrary elements of $X_{n,r}^{*}$ such that $U_{\\nu'} \\cup U_{\\eta'} \\ne \\CCnr$ . Observe that $\\nu', \\eta' \\in X_{n,r}^{+}$ otherwise  $U_{\\nu} = \\CCnr$ or $U_{\\eta} = \\CCnr$. Since $U_{\\nu'} \\cup U_{\\eta'} \\ne \\CCnr$ there exists an element $\\beta \\in X_{n,r}^{+}$ such that $\\beta \\perp \\nu'$ and $\\beta \\perp \\eta'$ and such that $(\\nu', \\beta) \\in \\mathscr{G}$ and $(\\eta', \\beta) \\in \\mathscr{G}$. By arguments in the previous paragraph we therefore have that $h$ acts on $U_{\\nu'}$ and $U_{\\beta}$ in almost the same fashion, and $h$ acts on $U_{\\eta'}$ and $U_{\\beta}$ in almost the same fashion. Let $k_1 = \\crit_{h}(U_{\\eta'}, U_{\\beta} )$ and $k_2 = \\crit_{h}(U_{\\nu'}, U_{\\beta} )$. Let $k = \\max\\{k_1, k_2\\}$. Then for any $\\chi \\in X_n^{k}$ we have that $h_{\\nu'\\chi} = h_{\\beta\\chi} = h_{\\eta' \\chi}$. Therefore we conclude that $h$ acts on $U_{\\eta'}$ and $U_{\\nu'}$ in almost the same fashion.\n\\end{proof}\n\nThe Corollary below demonstrates that if $h \\in H(\\CCnr)$ is such that  $h^{-1}\\Tnr h \\subseteq \\Tnr$, then $h$ uses only finitely many types of local action. This Corollary should be compared with Corollary 6.16 of \\Cite{BCMNO} and is proved almost identically; we reproduce the proof here for completeness. This is a key result in demonstrating that automorphisms of $\\Tnr$ can be represented by bi-synchronizing transducers.\n\n\\begin{corollary}\\label{Corollary: Automorphisms have finite local actions}\nLet $h \\in H(\\CCnr)$ be such that $h^{-1}\\Tnr h \\subseteq \\Tnr$, then $h$ uses only finitely many types of local action.\n\\end{corollary}\n\\begin{proof}\nLet $A$ be a complete antichain for  $\\Xnrs$ having at least 3 elements. For instance we may take $A = X_{n,r}^{3}$. Notice that for any pair of element $\\nu$ and $\\eta$ in $A$ we have  by Lemma ~\\ref{Lemma:  automorphisms have few local actions} that $h$  acts on $U_{\\nu}$ and $U_{\\eta}$ almost in the same fashion, and for $i \\in X_n$ $h$ acts on $U_{\\nu}$ and $U_{\\nu i}$ almost in the same fashion. Set \n\n$$ k= \\max \\{ \\crit_{h}(U_{\\nu},U_{\\eta}) \\mid \\nu, \\eta \\in A, \\mbox{ or } \\nu \\in A \\mbox{ and } \\eta= \\nu i \\mbox{ for some } i \\in \\Xn  \\}$$\n\nWe now demonstrate that for any $ m\\ge  k+3$, $m \\in \\N$ and for any word $\\eta \\in X_{n,r}^{m}$ we have that $h_{\\eta} = h_{\\nu \\xi}$ for some $\\nu \\in A$ and $\\xi \\in X_n^{k}$. We proceed by induction on $m$.\n\nThe base case is trivially satisfied. Let $m \\in \\N$ be strictly greater than $k+3$, and assume that for all $k+3< l < m$ we have that for any word $\\eta \\in \\Xnr^{l}$ $H_{\\eta} = h_{\\nu \\xi}$ for some $\\nu \\in A$ and $\\xi \\in \\Xn^{k}$. \n\nNow let $\\tau \\in \\Xnr^{m}$. We may write $\\tau = \\nu i \\xi$ for $\\nu \\in A$, some $i \\in \\Xn$ and $\\xi \\in X_n^{\\ast}$ such that $|\\xi| \\ge k$ (since $m$ is greater than $k+3$).\n\nObserve that as $h_{\\nu} = h_{\\nu i}$ we have that $h_{\\tau} = h_{\\nu \\xi}$. Now as $|\\nu \\xi| < m$, by the inductive assumption we have that there is some $\\chi \\in X_n^{k}$ and $\\mu \\in A$ such that $h_{\\nu \\xi} = h_{\\mu \\chi}$ and the conclusion follows.\n\\end{proof}\n\n\\begin{Remark}\nThe above corollary together with Remark~\\ref{Remark: finitely many local actions implies finite transducer} means that $N(\\Tnr)$ is a subgroup of the group  $\\Rnr$ of those homeomorphisms of $\\CCnr$ that can be represented by minimal, finite, initial transducers. In what follows we shall demonstrate the element of $N(\\Tnr)$ can be represented by minimal, bi-synchronizing transducers in $\\Rnr$. \n\\end{Remark}\n\n\\section{The group \\texorpdfstring{$N(\\Tnr)$}{Lg} is a subgroup of \\texorpdfstring{$\\Bnr$}{Lg}} \\label{Section: N(Tnr) is a subgroup of Bnr}\nFollowing \\cite{BCMNO} we shall use the notation $\\Bnr$ for the group consisting of those elements of $\\Rnr$ which can be represented by bi-synchronizing transducers. In this section we demonstrate that $N(\\Tnr)$ is a subgroup  $\\T{T}\\Bnr$ of $\\Bnr$.\n\nThe lemma below is straight-forward and connects the property of a homeomorphism acting in the almost the same fashion (Definition~\\ref{Def: almost in the same fashion}), to the synchronizing property of a transducer representing the homeomorphism.\n\n\\begin{lemma}\\label{Lemma:automorphisms have synch transducers}\nLet $h \\in H(\\CCnr)$ satisfy the following two conditions,\n\\begin{enumerate}[label = (\\alph*)]\n\\item $h$ has finitely many local actions,\\label{Lemma:automorphisms have synch transducers assumption 1}\n\\item for every pair $U_{\\nu}, U_{\\eta} \\in \\Banr$ such that $U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$ $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ almost in the same fashion.\\label{Lemma:automorphisms have synch transducers assumption 2}\n\\end{enumerate}\nThen the minimal transducer representing $h$ is synchronizing.\n\\end{lemma}\n\\begin{proof}\nLet $A_{q_0}$ be the minimal transducer representing $h$. As $h$ has finitely many local actions, by Remark~\\ref{Remark: finitely many local actions implies finite transducer}, $A_{q_0}$ is a finite transducer. Applying Remark~\\ref{Remark: finitely many local actions implies finite transducer}, there is an $m \\in \\N$ such that if  $q \\in Q_{A}$ is accessible from $q_0$ by a word of length at least $m$, then  $h_{q} = h_{\\eta}$ for some $\\eta \\in \\Xnrs$ of length at least $m$. Of course by the definition of $A_{q_0}$, it is the case that $h_{\\nu} = h_{p}$, for $p = \\pi_{A}(\\nu, q_0)$ and $\\nu \\in \\Xnrs$. We may also choose $m$ bigger than $1$ and long enough, so that for any pair $\\nu, \\eta \\in \\Xnrs$ of length $m$, $U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$. Fix such an $m \\in \\N$.\n\nBy assumption~\\ref{Lemma:automorphisms have synch transducers assumption 2} above and Definition~\\ref{Def: almost in the same fashion}, it follows that for every pair $\\nu, \\eta \\in \\Xnrs$ of length $m$, there is a minimal number $k_{\\nu, \\eta} \\in \\N$ such that for all words $\\xi \\in \\Xn^{k_{\\nu, \\eta}}$, $h_{\\nu \\xi} = h_{\\eta \\xi}$. Set $k:= \\max_{\\nu,\\eta \\in \\Xn^{m}}\\{k_{\\nu, \\eta}\\}$. It therefore follows that for any pair $\\tau, \\mu \\in \\Xnrs$ of length $m$, and any word $\\chi \\in \\Xn^{k}$, $h_{\\tau \\chi} = h_{\\mu \\chi}$.\n\nNow since any state $q$ of $A_{q_0}$ accessible from $q_0$ by a word of length at least $m$ satisfies $h_{q} = h_{\\eta}$ for some $\\eta \\in \\Xnrs$ of length $m$. It is therefore the case, by minimality of $A_{q_0}$, that for any $\\chi \\in \\Xn^{k}$  and any pair $\\nu, \\eta \\in \\Xnrs$ of length $m$, we have $\\pi_{A}(\\nu\\chi, q_0) = \\pi_{A}{\\eta\\chi, q_0}$. Thus we have that if $Q_{A,\\ge m} = \\{\\pi_{A}(\\nu, q_0) \\mid \\nu \\in \\Xnrs, |\\nu| \\ge m \\}$, then for any $\\xi \\in \\Xn^{k}$, $|\\{\\pi_{A}(\\chi,p) \\mid p \\in Q_{A, \\ge m} \\}| = 1$. From this we deduce that $A_{q_0}$ is synchronizing at level at least $k+m$.\n\n\n\\end{proof}\n\n\\begin{comment}\nThe lemma below is proved in \\cite{BCMNO} and gives essentially one direction of the main result. We need to make the following definitions in order to state the lemma.\n\n\\begin{Definition}\nLet $A_{q_0}  = \\gen{X_A, Q_A, \\pi_{A}, \\lambda_{A}, q_0}$ be an initial transducer. A state $p$ of $A$ is called \\emph{non-trivially accessible} if there is a word $\\Gamma \\in X_A^{*}$ such that $\\pi_{A}(\\Gamma, q_0) = p$ and $\\lambda(\\Gamma, q_0) \\ne \\epsilon$.\n\\end{Definition}\n\n\\begin{lemma}\\label{Lemma: automorphisms have synch transducers}\nLet $A_{q_0} = \\gen{\\dotr, X_n,Q_A,\\pi_A,\\lambda_A,q_0}$ be  a  minimal,  initial, finite,  invertible transducer.  Let $q_0^{-1}$ represent the initial state of the minimal transducer $B_{q_0^{-1}}= \\gen{\\dotr, X_n,Q_B,\\pi_B,\\lambda_B,q_0^{-1}}$ which  is  the  inverse  transducer  of $A_{q_0}$ (so,  in  particular, $X_A=X_B$).  Suppose  that  for  all non-trivially  accessible  states $p^{-1} \\in Q_B$ and  for  all $q \\in Q_A$ we  have  the  initial  transducer $B_{p^{-1}}A_q$ admits an $\\iota \\in \\N$ so that for each $\\gamma \\in \\Xn^{\\iota}$ there is some $\\delta  \\in \\Xn^{*} $ with $(\\gamma \\Gamma)B_{p^{-1}}A_{q}= \\delta \\Gamma$, for all $\\Gamma \\in \\Xn^{\\omega}$ .  Then $A_{q_0}$ is synchronizing\n\\end{lemma}\n\\end{comment}\nAs a corollary we have the following:\n\n\\begin{corollary}\\label{Cor: Transducers representing automorphims are synchronizing}\nLet $h \\in H(\\CCnr)$ be such that $h^{-1}T_{n,r}h \\subseteq \\Tnr$ and $A_{q_0} = \\gen{\\dotr, \\Xn, Q_A, \\pi_A, \\lambda_A, q_0}$ be an initial transducer such that $h_{q_0} = h$. Then $A_{q_0}$ is  synchronizing.\n\\end{corollary}\n\\begin{proof}\nBy Corollary~\\ref{Corollary: Automorphisms have finite local actions} and Lemma~\\ref{Lemma:  automorphisms have few local actions} it follows that $h$ and $h^{-1}$ satisfy the conditions of Lemma~\\ref{Lemma:automorphisms have synch transducers}. By definition of bi-synchronicity, the minimal transducer $A_{q_0}$ representing $h$ is bi-synchronizing.\n\\begin{comment}\nLet $B_{q_0^{-1}}= \\gen{\\dotr,\\Xn,Q_B,\\pi_B,\\lambda_B,q_0^{-1}}$ be the minimal initial transducer representing the inverse of $A_{q_0}$. It suffices to show that $A_{q_0}$ and $B_{q_0^{-1}}$ satisfy the conditions of Lemma~\\ref{Lemma: automorphisms have synch transducers}.\n\nLet $q_b$ be any non-trivially accessible state of $B_{q_0^{-1}}$. Let $\\Gamma \\in \\Xnr^{+}$  be such that $\\pi_{B}(\\Gamma, q_0^{-1}) = q_b$ and $\\lambda_{B}(\\Gamma, q_0^{-1}) \\ne \\epsilon$. Let $\\Delta \\in \\Xnr^{+}$ be such that $\\Delta = \\lambda_{B}(\\Gamma, q_0^{-1})$. Let $q_a$ be any state of $A_{q_0}$ and let $\\chi \\in \\Xnr^{+}$ be such that $\\pi_A(\\chi, q_0) = q_a$. Observe that there is an element $g \\in \\Tnr$ such that $g\\restriction_{U_{\\Delta}} = g_{\\Delta, \\chi}$.\n \nLet $C = \\gen{\\dotr, \\Xn, \\pi_C, \\lambda_C, p_0}$ be a minimal, finite initial transducer such that $h_{p_0} = g$. Observe that $C$ is a synchronizing transducers whose core is the single state identity transducer. Denote this state by $\\id$. Moreover since $C$ has no states of incomplete response (by minimality) it follows that $\\lambda_{C}(\\Delta,p_0) = \\chi$ and $\\pi(\\Delta, p_0) = \\id$.\n\nNow consider the product $D_{(q_0^{-1}, q_0, p_0)} :=B_{q_0^{-1}} \\ast C_{p_0} \\ast A_{q_0}$. Now after reading $\\Gamma$ through this product, the active state of the transducer is $(q_b, \\id, q_a)$. Now observe that as $h_{q_0^{-1}} g h_{q_0} \\in \\Tnr$, there is an $l \\in \\mathbb{N}$ such that for an word $\\gamma \\in \\Xnr^{l}$ and any $\\delta \\in \\CCn$ we have $(\\gamma \\delta)h_{q_0^{-1}}g h_{q_0} = \\rho \\delta$, for some $\\rho \\in \\Xnr^{+}$. In particular for any $\\xi \\in \\Xn^{l}$  and for any $\\delta \\in \\CCn$ we have $(\\chi\\xi\\delta)h_{q_0^{-1}}gh_{q_0} = \\varphi \\delta$ for some $\\varphi \\in \\Xnr^{+}$. Now since the active state of $B_{q_0^{-1}} C_{p_0} A_{q_0}$ after processing $\\chi$ is $(q_b, \\id, q_a)$ we therefore conclude that $\\lambda_{D}(\\xi\\delta, (q_b, \\id, q_a)) = \\varphi'\\delta$ for some (possibly empty) suffix $\\varphi'$ of $\\varphi$. Therefore  $(\\xi\\delta) B_{q_b}A_{q_a} = \\varphi' \\delta$. Hence for any $\\xi \\in \\Xnr^{l}$ there is some $\\varphi' \\in \\Xn^{*}\\sqcup \\Xnr^{\\ast}$ such that $(\\xi\\delta) B_{q_b}A_{q_a} = \\varphi' \\delta$ for any $\\delta \\in \\CCn$.\n\nSince $q_b$ was an arbitrary non-trivially accessible state of  $B_{q_0^{-1}}$ and since $q_a$ was an arbitrary state of $A_{q_0}$, we see that $B_{q_0^{-1}}$ and $A_{q_0}$ satisfy the hypotheses of Lemma~\\ref{Lemma: automorphisms have synch transducers}.\n\\end{comment}\n\\end{proof}\n\nNow we may prove the main result.\n\n\\begin{Theorem}\nLet $h \\in H(\\CCnr)$ then $h \\in N(\\Tnr)$ if and only if there is a bi-synchronizing transducer $A_{q_0}$ such that $h_{q_0} = h$ and preserves $\\simeq$.\n\\end{Theorem}\n\\begin{proof}\nIt follows by Corollary~\\ref{Cor: Transducers representing automorphims are synchronizing} that if $A_{q_0}$ is a transducer such that $h_{q_0} \\in N(\\Tnr)$ then $A_{q_0}$ is synchronizing. Since $N(\\Tnr)$ is a group it follows that $A_{q_0}$ is bi-synchronizing. This proves the forward implication.\n\nFor the reverse implication, suppose that $A_{q_0} = \\gen{\\dotr, \\Xn, Q_A, \\pi_A, \\lambda_A, q_0}$ is a bi-synchronising transducer such that $h_{q_0}$ preserves $\\simeq$. Let $T \\in T_{n,r}$ observe that since $h_{q_0}$ preserves $\\simeq$ it follows that $h^{-1}_{q_0}T h_{q_0}$ also preserves $\\simeq$. Let $C_{p_0} = \\gen{\\dotr, \\Xn, Q_{C}, \\pi_{C}, \\lambda_{C}, p_0}$ be an initial  transducer such that $h_{p_0} = T$. Observe that $C_{p_0}$ is synchronizing and $\\core(C_{p_0})$ is the single state identity transducer, let us denote this state by $\\iota$. Let $B_{q_0^{-1}} = \\gen{\\dotr, \\Xn, Q_B, \\pi_B, \\lambda_B, q_0^{-1}}$ be a minimal initial transducer representing the inverse of $A_{q_0}$. \n\nObserve that since $A_{q_0}$ is bi-synchronizing, that $B_{q_0}^{-1}$ is also bi-synchronizing. Let $k_1 \\in \\N$ be minimal so that $A_{q_0}$ and $B_{q_0}^{-1}$ are synchronizing at level $k_1$. let $k_2 \\in \\N$ be minimal so that for any word $\\Gamma \\in \\Xnr^{k_2}$  we have $\\pi_{C}(\\Gamma, p_0) = \\iota$. Let $k = \\max\\{k_1, k_2\\}$. Let $j \\in \\N$ be the minimal number greater than $k$ such that for any word $\\Delta \\in \\Xnr^{j}$  we have that $|\\lambda_{B}(\\Delta, q_0^{-1})|> 2k$. Therefore after processing any word of length $j$, the active state of the product transducer $B_{q_0}^{-1} \\ast C_{p_0} A_{q_0}$ is of the for $(r, \\iota, q )$ for some state $r$ of $\\core(B_{q_0^{-1}})$ and some state $q$ of $A_{q_0}$. Notice that as $\\iota$ is the single state identity transducer we may identify $(r, \\iota, q)$ with the state $(r, q)$ of $B_{q_0^{-1}} \\ast A_{q_0}$.\n\nNow let $\\Lambda \\in \\Xnr^{*}$ be a word such that $\\pi_{B}(\\Lambda, q_0^{-1}) = r$ and let $\\Delta \\in \\Xn$ be such that $\\lambda_{B}(\\Delta, r)$ is greater than $k$ (recall that states of $\\core(B_{q_0^{-1}}$ only process elements from $\\Xn$). Let $q'$ be the state of  $A_{q_0}$ forced by  $\\lambda_{B}(\\Delta,r)$ and let $r'$  be the state $\\pi_{B}(\\Delta, r)$. Then observe that after processing the word $\\Lambda\\Delta$ the active state of $B_{q_0^{-1}} \\ast A_{q_0}$ is $(r', q')$. However after processing $\\Delta$ from the state $(r,q)$ the active state is also $(r', q')$. However since $B_{q_0^{-1}}\\ast A_{q_0}$ is $\\omega$-equivalent to the single state identity transducer, we see that $(r',q')$ which is a state of $\\core(B_{q_0^{-1}}\\ast A_{q_0})$ is $\\omega$-equivalent to a map which produces some finite initial prefix and then acts as the identity.\n\nFrom this we may conclude that there some $N \\in \\N$ such that $B_{q_{0}^{-1}} \\ast C_{p_0} \\ast A_{q_0}$ acts as the identity after processing a word of length $N$. This means that since $h_{q_0}^{-1} T h_{q_0}$ preserves $\\simeq$ and is a homeomorphism that  $h_{q_0}^{-1} T h_{q_0} \\in \\Tnr$.\n\\end{proof}\n\nThus we have the following:\n\n\\begin{Theorem}\n The group $\\aut{\\Tnr}$ is isomorphic to the subgroup $\\mathcal{T}\\Bnr \\le \\Bnr$ consisting of those elements of $\\Rnr$ which may be represented by finite, initial, bi-synchronizing transducers which preserve $\\simeq$. \n\\end{Theorem}\n\n\\begin{Remark}\\label{Remark:elements of Tnr correspond to bisynch transducers with trivial cores}\nNotice that since all elements of $\\Tnr$ eventually act as the identity, then a minimal transducer $T_{q_0}$ representing an element $t \\in \\Tnr$  satisfies $\\core(T_{q_0})$ is the single states identity transducer over  $\\CCn$ which we denote $\\id$. Furthermore any element of $\\TBnr$ which may be represented by a bi-synchronizing transducer with trivial core, is an element of $\\Tnr$. This is because $\\Tnr \\le \\TBnr$ is uniquely defined by the fact that all elements act as the identity after modifying an  finite initial prefix.\n\\end{Remark}\n\n\\section{\\texorpdfstring{$\\out{\\Tnr}$}{Lg}}\\label{Section: out(Tnr)}\n\nThe paper \\cite{BCMNO} introduces a group $\\On$ which contains the group $\\Onr\\cong \\out{G_{n,r}}$ for all valid $r$. In this section we shall demonstrate that there is a corresponding subgroup $\\TOn \\le \\On$ containing the groups $\\TOnr:= \\out{\\Tnr}$ for all valid $r$. \n\nLet $\\widetilde{\\Bnr}$ be the set of minimal transducer $A_{q_0}$ such that $h_{q_0} \\in \\Bnr$. The set $\\widetilde{\\Bnr}$ of transducers becomes a group isomorphic to $\\Bnr$ with the product defined by for elements $A_{q_0}, B_{p_0}  \\in   \\widetilde{\\Bnr}$, $AB_{(q_0,p_0)}$ is the minimal initial transducer representing the product $A_{q_0} \\ast B_{p_0}$. Notice that since $h_{(q_0, p_0)} = h_{q_0} \\circ h_{p_0}$ then we have that $AB_{(q_0,p_0)} \\in \\widetilde{\\Bnr}$. To simplify the discussion below we shall identify $\\Bnr$ with the group $\\widetilde{\\Bnr}$ of transducers. In particular we shall no longer distinguish between an element $h \\in \\Bnr$ and the minimal transducer $A_{q_0} \\in \\widetilde{\\Bnr}$ representing $h$, thus we shall use the symbol $\\Bnr$ for both $\\widetilde{\\Bnr}$ and $\\Bnr$. Likewise we shall longer distinguish between elements of $\\TBnr$ and the minimal initial transducers representing them.\n\n \n\\begin{lemma}\nLet $A_{q_0}, B_{p_0} \\in \\TBnr$ such that $\\core(A_{q_0}) = \\core(B_{p_0})$. Then $A_{q_0}B_{p_0}^{-1} \\in \\Tnr$.\n\\end{lemma}\n\\begin{proof}\nBy Remark~\\ref{Remark:elements of Tnr correspond to bisynch transducers with trivial cores} it suffices to show that $\\core(A_{q_0}B_{p_0}^{-1})$ is equal to the single state identity transducer $\\id$.\nHowever it is a consequence of a result in \\cite{BCMNO}, that $\\core(A_{q_0}B_{p_0}^{-1})$ is trivial.\n\\end{proof}\n\nAs a corollary we have that an element of  $\\out{\\Tnr}$ corresponds to a subset of elements of $\\TBnr$ which all have the same core. Therefore we may identify an element $[A_{q_0}] \\in \\out{\\Tnr}$, for $A_{q_0}\\in \\TBnr$, with the element $\\core(A_{q_0})$. \n\nLet $\\Onr:= \\{ \\core(A_{q_0}) \\mid A_{q_0} \\in \\Bnr \\}$. This set inherits a product from $\\Bnr$ as follows. For $g_1, g_2 \\in \\Onr$, let $A_{q_0}, B_{p_0} \\in \\Bnr$ be elements such that $\\core(A_{q_0}) = g_1$ and $\\core(B_{q_0}) = g_2$, and set $g_1g_2 := \\core( AB_{(q_0, p_0)})$. Now observe that $\\core( AB_{(q_0, p_0)})$ depends only on the states of $\\core(g_1 \\ast g_2)$ by a result in \\cite{BCMNO} which states that $\\core (A_{q_0}* B_{q_0}) = \\core(\\core(A_{q_0})* \\core(B_{q_0}))$ and since removing states of incomplete response and identifying $\\omega$-equivalent depends only on $\\core(g_1 \\ast g_2)$ (see \\cite{GriNekSus}).  Therefore it follows that for any other element  $A'_{q'_0}$ and $B'_{p'_0}$ with cores $g_1$ and $g_2$ respectively we have $\\core(A'B'_{(q'_{0}p'_{0})}) = \\core(AB_{(q_0,p_0)})$. Thus setting the  product $g_1g_2 := \\core( AB_{(q_0, p_0)})$ for elements $A_{q_0}, B_{q_0} \\in \\Bnr$ with cores $g_1$ and $g_2$ respectively, results in a well-defined product.\n\nThe authors of \\cite{BCMNO} show that this product is equivalent to the following. Let $g, h \\in \\Onr$,  observe that  $g$ and $h$ are core non-initial transducers. Fix a state $q_1$ and $q_2$ of $g$ and $h$ respectively. Let  $gh := \\core( (gh)_{(q_{1},q_{2})})$ where $(gh)_{(q_1,q_2)}$ is the minimal transducer representing the product ${g}_{q_1} \\ast {h}_{q_2}$. The set $\\Onr$ is isomorphic to $\\out{G_{n,r}}$ under this product.\n\nThe inverse of an element of $\\Onr$ can likewise be defined in two equivalent ways as demonstrated in \\cite{BCMNO}. The first approach is as follows. Given an element $g \\in \\Onr$ let $A_{q_0} \\in \\Bnr$ be such that $\\core(A_{q_0}) = g$. Let $B_{p_0}$ be the minimal transducer representing the inverse of $A_{q_0} $ then set $g^{-1} = \\core(B_{q_0})$. It turns out (see \\cite{BCMNO}) that this inverse is independent of the the choice of transducer $A_{q_0} \\in \\Bnr$ with core equal to $g$ and so is well-defined. The second approach makes use only of the states of $g$ to construct the inverse of $g$, thus removing the need of finding an element of $\\Bnr$ with core equal to $g$.\n\nThus having removed the need to find elements of $\\Bnr$ with cores equal to the relevant transducers in order to compute products and inverses, the paper \\cite{BCMNO} introduces the group $\\On:= \\cup_{1 \\le r < n} \\Onr$ with product of two elements constructed as in the second approach, likewise for inverses. The groups $\\Onr$ are moreover subgroups of $\\On$. Another way of defining the group $\\On$ (see \\cite{BCMNO}) is as the group of all  bi-synchronizing transducers (note that as we have a method for computing inverses, our use of the word bi-synchronizing makes sense) all of whose states induce injective maps of $\\CCn$ with clopen images. \n\nLet $\\TOnr := \\{\\core(A_{q_0}) \\mid A_{q_0} \\in \\TBnr \\} \\subset  \\Onr$. Then $\\TOnr$ is a subgroup of $ \\Onr$, under the product inherited from $\\On$, moreover $\\TOnr \\cong \\out{\\Tnr}$ under this product.\n\nThe following lemma essentially solves one half of the membership problem of $\\TOnr$ in $\\Onr$.\n\n\\begin{lemma}\\label{Lemma: states of TBnr induces continuous functions from the interval to itsel }\nLet $A_{q_0} \\in \\TBnr$, then for all states $q$ of $\\core(A_{q_0})$ the maps $h_{q} : \\CCn \\to \\CCn$ either all preserve or all reverse the lexicographic ordering of $\\CCn$ and preserve the relation $\\simeq_{{\\bf{I}}}$ on $\\CCn$.\n\\end{lemma}\n\\begin{proof}\nWe consider only elements $A_{q_0} \\in \\TBnr$ such that $h_{q_0}$ induces an orientation preserving homeomorphism of $S_{r}$. The proof is similar in the other case.\n\nObserve that since $A_{q_0} \\in \\TBnr$ this means that $h_{q_0}: \\CCnr \\to \\CCnr$ is a homeomorphism which preserve $\\simeq$. Hence there is a $\\nu \\in \\Xnrs$ such that $h_{q_0}$  preserve the lexicographic ordering  on $U_{\\nu}$. By restricting to a small enough open set contained in $\\nu$ we may assume that assume that $\\nu$ is longer than the synchronizing level of $A_{q_0}$. Let $\\gamma \\in \\Xns$ be longer than the synchronizing level of $A_{q_0}$ such that the state of $A_{q_0}$ forced by $\\gamma$ is a state $p \\in \\core(A_{q_0})$. Let $q = \\pi_{A}(\\nu, q_0) \\in \\core(A_{q_0})$. Observe that as $h_{q_0}$ preserves the lexicographic ordering of $U_{\\nu}$, it must be the case that $h_{q}: \\CCn \\to \\CCn$ preserves the lexicographic ordering on $U_{\\nu}$. Thus $h_{p}$ must also preserve the lexicographic ordering on $\\CCn$ otherwise $h_{q}$ does not preserve the lexicographic ordering. Moreover since $h_{q_0}$ preserves $\\simeq$ we must also have that $h_{p}$ preserves $\\simeq_{{\\bf{I}}}$ otherwise $h_{q_0}$ would not preserve $\\simeq$. Now since $A_{q_0}$ is synchronizing for all states $p'$ of $\\core(A_{q_0})$ there is a  word $\\gamma'$ in $\\Xns$ longer than the synchronizing length of $A_{q_0}$ such that the state of $A_{q_0}$ forced by $\\gamma'$ is $p'$. This concludes the proof.\n\\end{proof}\n\nThis prompts the following definition:\n\n\\begin{Definition}\\label{Def: noninitial transducers preserving relations}\nLet $A$ be a non-initial transducer of over $\\CCn$, and let $P$ be some relation on $\\CCn$. Then we say that \\emph{$A$ preserves $P$} if for all states $q$ of $A$ the map $h_{q}$ preserves the relation $P$.\n\\end{Definition}\n\n\\begin{lemma}\\label{Lemma: images of states of core of elements of TBnr are connected}\nLet $A_{q_0} \\in \\TBnr$ then for all states $q \\in \\core(A_{q_0})$ there is a finite antichain  $\\ac{u} = \\{u_1, u_2, \\ldots, u_{m}\\}$ such that $\\im(q) =  \\cup_{1 \\le i \\le m}U_{u_i}$. Moreover for $1 \\le i \\le m-1$ we have that $u_in-1 n-1\\ldots \\simeqI u_{i+1}00\\ldots$.\n\\end{lemma}\n\\begin{proof}\nObserve that by  Lemma \\ref{Lemma: states of TBnr induces continuous functions from the interval to itsel } it follows that for every state $q \\in \\core(A_{q_0}$ $h_{q}$ induces a continuous function from the interval to itself. However since the interval is connected, and the continuous image of a connected topological space is connected, it follows that $\\im(q)\/\\simeqI$ is a connected. The lemma is now an easy consequence of this.\n\\end{proof}\n\n The result below was demonstrated first by Brin in his seminal paper \\cite{MBrin2} describing the automorphisms of R. Thompson's groups $F$ and $T$ and we have simple translated it into the language of transducers.\n\n\n\\begin{Theorem}\nLet $n = 2$, and let $A_{q_0} \\in \\TBnr$, then  $\\core(A_{q_0})$ is either the single state identity transducer or the single state transducer mapping induce the transposition $(0 \\ 1)$ on $X_{2}$. More specifically we have $\\TOn \\cong C_2$.\n\\end{Theorem} \n\nIn order to show the subgroup $\\TOnr \\le \\Onr$ is completely characterised by the fact that all of its states  preserve the relation $\\simeqI$ and either all preserve or  all reverse the lexicographic ordering of $\\CCn$ we need the following notions from  \\cite{BCMNO}. Recall that $\\On$ is the group of core, bi-synchronizing transducers all of whose states induce injective maps of $\\CCn$ with clopen images. \n\n\\begin{Definition}[\\cite{BCMNO}]\\label{Def: Viable combination}\nLet $g \\in \\On$. A \\emph{viable combination} $\\mathfrak{v}_{j}$ for $g$ is a pair of tuples $$((\\rho_1, \\ldots, \\rho_j), (p_1, \\ldots, p_j))$$ satisfying \n\\begin{enumerate}\n\\item $\\bigcup_{1\\le i \\le j} \\rho_i\\im(p_i) = \\CCn$ \\label{Def: Viable combination union fills cantor space}\n\\item  for $1 \\le i < l \\le j$ we have $\\rho_i \\im(p_i) \\cap \\rho_l \\im(p_l) = \\emptyset$ \\label{Def: Viable combination unions are disjoint}.\n\\end{enumerate}\n where the $\\rho_{i} \\in \\Xns$, $1 \\le i \\le j$ are not necessarily incomparable or distinct, and  $p_i$, $1 \\le i \\le j$ are not necessarily distinct states of $g$.\n\\end{Definition} \n\n\\begin{Definition}[Single expansions of viable combinations \\cite{BCMNO}] \\label{Def: expansions of viable combinations}\nLet $T = (X_n, Q, \\pi, \\lambda) \\in \\On$ and let \n$\\mathfrak{v}_{j} = ((\\rho_1, \\ldots, \\rho_j), (p_1, \\ldots, \np_j))$ be  a viable combination for $T$. Fix an $i$ such that $1 \n\\le i \\le j$, and let $\\rho_{i,l} := \\rho_i\\lambda(l, \np_i)$ and $p_{i,l} := \\pi(l, p_i)$ for $l \\in X_n$, then \n$$\\mathfrak{v}_{j+n-1} := ((\\rho_1, \\ldots, \\rho_{i-1}, \n\\rho_{i,0}, \\ldots,\\rho_{i,(n-1)}, \\rho_{i+1}, \\ldots, \\rho_{j} \n), (p_1, \\ldots, p_{i-1}, p_{i,0}, \\ldots, p_{i, (n-1)}, \np_{i+1},\\ldots, p_{i,k}))$$ is called a \\emph{single expansion of \n$\\mathfrak{v}_{j}$}. \n\\end{Definition}\n\n\n\\begin{Remark}\\label{Remark: sequence of expansions of viable combinations give rise to viable combinations}\nClearly a single expansion of a viable combination for an element $T \\in \\On$ results in a new viable combination for $T$. Therefore a sequence of single expansions applied to a viable combination for $T$ also results in a new viable combination for $T$. See \\cite{BCMNO} for more detail.\n\\end{Remark}\n\nThe following lemma is proved in \\cite{BCMNO}:\n\n\\begin{lemma}\\label{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties}\nLet $g \\in \\On$ and let $\\mathfrak{v}_{g}$ denote the set of all viable combinations of $g$. Let $1 \\le r \\le n-1$. There is a transducer $A_{q_0} \\in \\Bnr$ whose core is equal to $g$ if and only if there is a sequence $\\mathfrak{v}_{j_1}, \\mathfrak{v}_{j_2} \\ldots, \\mathfrak{v}_{j_m}$ of elements of $\\mathfrak{v}_{g}$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_i \\equiv m \\mod{n-1}$.\n\\end{lemma}\n\nNow we have the following lemma which solves the other half of the membership problem of $\\TOnr \\in \\Onr$. However we make the following definition first.\n\n\\begin{Definition}\\label{Def: lexicographic viable combinations}\nLet $g \\in \\On$. A \\emph{lexicographic viable combination} $\\mathfrak{v}_{j}$ for $g$ is a viable combination $$\\mathfrak{v}_{j}:=((\\rho_1, \\ldots, \\rho_j), (p_1, \\ldots, p_j))$$ satisfying: for $1 \\le a < b \\le j$  $\\rho_{a}\\im(p_a) \\lelex \\rho_{b}\\im(p_b)$.\n\\end{Definition}\n\n\\begin{lemma}\\label{Lemma: g in Onr induces map on the line iff it has lexicographic viable combination}\nLet $g \\in \\Onr$ be such that $g$ preserves  $\\simeqI$ and preserves (reverses) the lexicographic ordering on $\\CCn$ then any viable combination of   $g$ may be re-ordered to get a lexicographic viable combination.\n\\end{lemma}\n\\begin{proof}\n\nSince $g \\in \\Onr$ there is an element $A_{q_0} \\in \\TBnr$ such that $\\core(A_{q_0}) = g$. Now by Lemma~\\ref{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties} it follows that there are viable combinations  $\\mathfrak{v}_{j_1}, \\mathfrak{v}_{j_2}, \\ldots, \\mathfrak{v}_{j_{m}}$ of $g$. Thus the set  $\\mathfrak{v}_{g}$ of viable combinations of $g$ is non-empty. Let $\\mathfrak{v}_{l} \\in \\mathfrak{v}_{g}$. We shall now make use of Lemma~\\ref{Lemma: images of states of core of elements of TBnr are connected} to re-order $\\mathfrak{v}_{l}$ to obtain a  lexicographic viable combination of $g$. \n\nSuppose $\\mathfrak{v}_{l} = ((\\rho_1, \\ldots, \\rho_l),(p_1, \\ldots, p_l))$. Let $R_{\\mathfrak{v}_{l}} := \\{ \\rho_k \\mid 1 \\le k \\le l \\mbox{ such that }\\rho_{t} \\not\\leq \\rho_{k} \\forall  1\\le t \\le l \\}$. Observe that $R_{\\mathfrak{v}_{l}}$ is an antichain by construction, thus we may assume that the set $R_{\\mathfrak{v}_{l}}$ is totally ordered in the lexicographic order. Fix $\\rho_{k} \\in  R_{\\mathfrak{v}_{l}}$. Let $D(\\rho_{k}) = \\{ \\rho_{t} \\mid \\rho_{k} \\le \\rho_{t} \\}$. We assume that $D(\\rho_{k})$ is ordered according to the short-lex ordering. Observe that $\\rho_{k} \\in D(\\rho_{k})$ and in particular is the smallest element  of $\\rho_{k}$. Let $d_{\\rho_{k}} = | D(\\rho_{k})|$. Now for each $\\rho_{t} \\in D(\\rho_{k})$ let $\\mu_{\\rho_{t}}$ be the number of times it occurs in the tuple $(\\rho_1, \\ldots, \\rho_k)$.  Let  $\\bar{l}:= |R_{\\mathfrak{v}_{l}}|$. \n\nGiven a tuple $(y_1, y_2, \\ldots, y_t)$ we shall use the notation $(y_1, \\ldots, y_s^{p}, \\ldots, y_t)$ to mean that $y_s, \\ldots, y_{s+p-1}$ are all equal to $y_s$. Now let  $\\xi_1, \\ldots \\xi_{\\bar{l}}$ be the elements of $R_{\\mathfrak{v}_{l}}$ in lexicographic ordering. For $1 \\le s \\le \\bar{l}$ let $\\xi_{(s,1)}, \\ldots \\xi_{(s, d_{\\xi_{s}})}$ be the elements of $D(\\xi_{s})$ in short-lex ordering. Then we have \n\\begin{equation}\\label{Equation: first rearrangement of tuple}\n\\left(\\xi_1^{\\mu_{\\xi_1}}, \\xi_{(1,1)}^{\\mu_{\\xi_{(1,1)}}}, \\ldots, \\xi_{(1, d_{\\xi_1})}^{\\mu_{\\xi_{(1, d_{\\xi_1})}}}, \\ldots, \\xi_{\\bar{l}}^{\\mu_{\\xi_{\\bar{l}}}}, \\xi_{(\\bar{l},1)}^{\\mu_{\\xi_{(\\bar{l},1)}}}, \\ldots, \\xi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}^{\\mu_{\\xi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}}}\\right)\n\\end{equation}\nis a reordering of $(\\rho_1, \\ldots, \\rho_l)$. Let $(q_1, \\ldots, q_l)$ be the induced reordering on the tuple of states $(p_1,\\ldots, p_l)$.\n\nNow fix $1 \\le s \\le \\bar{l}$ and consider the sequence \n$\\xi_s^{\\mu_{\\xi_s}}, \\xi_{(s,1)}^{\\mu_{\\xi_{(s,1)}}}, \\ldots, \n\\xi_{(s, d_{\\xi_1})}^{\\mu_{\\xi_{(s, d_{\\xi_s})}}}$  let \n$$q_{(s_0,1)}, \\ldots, q_{(s_0, \\mu_{\\xi_s})}, \\ldots, \nq_{(s_{d_{\\xi_s}},1)} ,\\ldots , q_{(s_{d_{\\xi_s}}, \\mu_{\\xi_{(s, \nd_{\\xi_{s}})}})}$$ be the corresponding states. Then observe that $$ \n\\xi_{s}\\im(q_{(s_0,1)} \\cup \\ldots \\cup \\xi_s \\im(q_{(s_0, \\mu_{\\xi_s})}) \\cup \\bigcup_{1 \\le k \\le \nd_{\\xi_s}} \\xi_{(s,k)} \\im(q_{(s_{k},1)}) \\cup \\ldots \\cup \\xi_{(s,k)} \\im(q_{(s_{k}, \\mu_{\\xi_{(s, k)})}})  =  \nU_{\\xi_{1}}$$ by construction and the definition of viable \ncombinations. Now let  $0 \\le k < d_{\\xi_s}$ and let $k \\le k' \\le  d_{\\xi_s}$, for $1\\le t_1 \\le \\mu_{\\xi_{(s,l)}}$ and $1 \\le t_2 \\le \\mu_{\\xi_{(s, k')}}$ consider $\\xi_{(s,k)} \\im(q_{(s_{k}, t_1)})$  and $\\xi_{(s,k')} \\im(q_{(s_{k'},t_2)})$ (note that we set $\\xi_{(s,0)} \\im(q_{(s_{0},t_1)}):= \\xi_{s}\\im(q_{s_{0}, t_1})$). Observe that by the ordering of the set $D(\\xi_{s})$, by Lemma~\\ref{Lemma: images of states of core of elements of TBnr are connected} and by  Definition~\\ref{Def: Viable combination} part~\\ref{Def: Viable combination unions are disjoint} of viable combinations, we must have that exactly one of the following holds: \n\\begin{enumerate}[label = (\\roman*)]\n\\item $\\xi_{(s,k)} \\im(q_{(s_{k}, t_1)}) \\lelex \\xi_{(s,k')} \\im(q_{(s_{k'},t_2)})$ or,\n\\item $\\xi_{(s,k')} \\im(q_{(s_{k'},t_2)}) \\lelex \\xi_{(s,k)} \\im(q_{(s_{k}, t_1)})$ or,\n\\item $k = k'$ and $t_1 = t_2$.\n\\end{enumerate}\n\nThus we may reorder  the tuple \\eqref{Equation: first rearrangement of tuple} to obtain a new tuple\n\n\\begin{equation}\\label{Equation: second rearrangement of tuple}\n\\left(\\chi_1^{\\mu_{\\chi_1}}, \\chi_{(1,1)}^{\\mu_{\\chi_{(1,1)}}}, \\ldots, \\chi_{(1, d_{\\xi_1})}^{\\mu_{\\chi_{(1, d_{\\xi_1})}}}, \\ldots, \\chi_{\\bar{l}}^{\\mu_{\\chi_{\\bar{l}}}}, \\chi_{(\\bar{l},1)}^{\\mu_{\\chi_{(\\bar{l},1)}}}, \\ldots, \\chi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}^{\\mu_{\\chi_{(\\bar{l},d_{\\chi_{\\bar{l}}})}}}\\right)\n\\end{equation} \n\nsatisfying the following conditions. Let $1 \\le s \\le  \\bar{l}$ and consider the subsequence  $$\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n\\chi_{(s, d_{\\xi_s})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$$ then:\n\\begin{enumerate}[label = (\\roman*)]\n\\item  Each $\\chi_{(s, r)}^{\\mu_{\\chi_{(s,r)}}}$ for $0 \\le r \\le d_{\\xi_{s}}$ (where $\\chi_{(s,0)} = \\chi_{s}$) corresponds to precisely one element  $\\xi_{(s,r')}^{\\mu_{\\xi_{(s,r')}}}$ for $0 \\le r' \\le d_{\\xi_{s}}$ (where $\\xi_{(s,0)} = \\xi_{s}$). \\label{List: rearrangement 2 cond 1}\n\\item The subsequence $\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n\\chi_{(s, d_{\\xi_s})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$   is ordered so that  the following is true. If $q_{(s_0,1)}, \\ldots, q_{(s_0, \\mu_{\\chi_s})}, \\ldots, \nq_{(s_{d_{\\xi_s}},1)} ,\\ldots , q_{(s_{d_{\\xi_s}}, \\mu_{\\chi_{(s, \nd_{\\xi_{s}})}})}$ are the set of states corresponding to the sub-sequence $\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n\\chi_{(s, d_{\\xi_1})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$ then for $0 \\le k < d_{\\xi_{s}}$, for $k \\le k' \\le d_{\\xi_{s}}$, for $1 \\le t_1 \\le \\mu_{\\chi_{(s,k)}}$ and $1 \\le t_2 \\le \\mu_{\\chi_{(s,k')}}$ such that if $k = k'$ and $\\mu_{\\chi_{(s,k)}} >1$ then $t_1 \\ne t_2$, we have  $\\chi_{(s,k)} \\im(q_{(s_{k}, t_1)}) \\lelex \\chi_{(s,k')} \\im(q_{(s_{k'},t_2)})$.  \n\\end{enumerate}\n \n Notice condition~\\ref{List: rearrangement 2 cond 1} above means all elements of the subsequence $\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n \\chi_{(s, d_{\\xi_s})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$ have prefix $\\xi_{s}$.    \n\nNow observe that for $1 \\le s < s' \\le \\bar{l}$, for $ 0 \\le k \\le d_{\\xi_{s}}$, for  $0 \\le k' \\le d_{\\xi_{s'}}$, for $1 \\le t \\le \\mu_{\\chi_{(s, k)}}$ and for $1 \\le t' \\le \\mu_{\\chi_{(s', k')}}$ so that if  the $t$\\textsuperscript{th} copy of $\\chi_{(s, k)}$ ($\\chi_{s}$ if $k=0$) and the $t'$\\textsuperscript{th} copy of $\\chi_{(s', k')}$ ($\\chi_{s'}$ if $k'=0$) in the sequence \\eqref{Equation: second rearrangement of tuple} correspond to the states $q_{(s_{k}, t_1)}$ and $q_{(s'_{k}, t_2)}$ respectively, then $\\chi_{(s,k)} \\im(q_{(s_{k}, t_1)}) \\lelex \\chi_{(s',k')} \\im(q_{(s'_{k'},t_2)})$ in the lexicographic ordering on $\\CCn$. This follows from the observation in the previous paragraph and because $\\xi_{s} \\lelex \\xi_{s'}$.\n\nLet $$(\\zeta_1, \\ldots, \\zeta_{l}) := \\left(\\chi_1^{\\mu_{\\chi_1}}, \\chi_{(1,1)}^{\\mu_{\\chi_{(1,1)}}}, \\ldots, \\chi_{(1, d_{\\xi_1})}^{\\mu_{\\chi_{(1, d_{\\xi_1})}}}, \\ldots, \\chi_{\\bar{l}}^{\\mu_{\\chi_{\\bar{l}}}}, \\chi_{(\\bar{l},1)}^{\\mu_{\\chi_{(\\bar{l},1)}}}, \\ldots, \\chi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}^{\\mu_{\\chi_{(\\bar{l},d_{\\chi_{\\bar{l}}})}}}\\right)$$ and let \n\\begin{IEEEeqnarray*}{rCl}\n(q_1', \\ldots q_l') := (& q_{(1_0,1)}& , \\ldots, q_{(1_0, \\mu_{\\chi_{1}})}, \\ldots, q_{(1_{d_{\\xi_{1}}},1)},\\ldots, q_{(1_{d_{\\xi_{1}}}, \\mu_{\\chi_{(1, d_{\\xi_1})}})}, \\ldots,  \\\\\n  & q_{(\\bar{l}_0,1)},&   \\ldots, q_{(\\bar{l}_0, \n \\mu_{\\chi_{\\bar{l}}})}, \\ldots, \n q_{(\\bar{l}_{d_{\\xi_{1}}},1)},\\ldots, \n q_{(\\bar{l}_{d_{\\xi_{\\bar{l}}}}, \\mu_{\\chi_{(\\bar{l}, \n d_{\\xi_{\\bar{l}}})}})})\n\\end{IEEEeqnarray*}\n\n\nObserve that  $((\\zeta_1, \\ldots, \\zeta_l), (q_1', \\ldots, q_l'))$ is a viable combination for $g$. Moreover, by discussion above, for $1 \\le a < b \\le l$ it satisfies   $\\zeta_{a} \\im(q_a) \\lelex \\zeta_{b}\\im(q_b)$ in the lexicographic ordering of $\\CCn$ i.e  $((\\zeta_1, \\ldots, \\zeta_l), (q_1', \\ldots, q_l'))$ is a lexicographic  viable combination of $g$.  \n\\end{proof}\n\nLet $ \\pi_{R,n} \\in \\On$ be the single state transducer which induces the permutation $i \\mapsto n-1- i$ on $\\Xn$. Then  $\\pi_{R,n}$ has order $2$, preserves $\\simeqI$, and reverses the lexicographic ordering on  $\\CCn$. It is also not hard to see that $\\pi_{R,n} \\in \\TOnr$ for all $1 \\le r < n-1$. For instance the map $ f_{\\pi_{R,n}}: \\CCnr \\to \\CCnr$ given by $ \\dot{a} \\delta \\mapsto \\dot{r-a +1} (\\delta)h_{\\pi_{R,n}}$ is induces by an element of $\\TBnr$ with core equal to $\\pi_{R,n}$ since it replaces some finite prefix before acting by $\\pi_{R,n}$.\n\nThe following lemma follows immediately from the fact that $\\pi_{R,n}$ has order $2$ and reverses the lexicographic ordering on $\\CCn$.\n\n\\begin{lemma}\\label{Lemma: multiplication by piR bijection from orientation preserving to reversing and vice versa}\nMultiplication by $\\pi_{R,n}$ induces a bijection from the subset of $\\Onr$ preserving (reversing) the lexicographic ordering, to the subset of $\\Onr$ reversing (preserving) the lexicographic ordering.\n\\end{lemma}\n\n\n\n\\begin{lemma}\\label{Lemma: g in Onr is in TOnr if g prserve  simeqI and lex order}\nLet $g \\in \\Onr$ be such that $g$ preserves $\\simeqI$ and preserves (reverses) the lexicographic ordering on $\\CCn$ then $g \\in \\TOnr$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{Lemma: multiplication by piR bijection from orientation preserving to reversing and vice versa} it suffices to prove this for elements $g \\in \\Onr$ which preserves $\\simeqI$ and the lexicographic ordering on $\\CCn$. Since if $g \\pi \\in \\TOnr$, then as $\\TOnr$ is a group containing $\\pi_{R,n}$ by an observation above, then $g \\pi_{R,n} \\pi_{R,n} = g \\in \\TOnr$ also. Thus fix $g \\in \\Onr$ an element which preserves $\\simeqI$ and the lexicographic ordering on $\\CCn$.\n\nSince $g \\in \\Onr$ by Lemma~\\ref{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties} there are viable combinations $\\viable{j_1}, \\viable{j_2},\\ldots,\\viable{j_m}$ of $g$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_{i} \\equiv m \\mod{n-1}$. Let $j= \\sum_{1 \\le i \\le m} j_i$. By Remark \\ref{Remark: sequence of expansions of viable combinations give rise to viable combinations} we may assume that $j_i > 1$ for $1 \\le i \\le m$. Now by Lemma~\\ref{Lemma: g in Onr induces map on the line iff it has lexicographic viable combination}  we may assume that all the  $\\viable{j_{i}}$ for $1\\le i \\le m$ are lexicographic viable combinations for $g$.\n\nLet $1 \\le i \\le m$, and consider the viable combination $\\viable{j_i}$ of $g$. Suppose $\\viable{j_{i}} = ((\\rho_1, \\ldots, \\rho_{j_i}), (p_1, \\ldots, p_{j_i}))$. Let $\\ac{u}= \\{ u_1, u_2, \\ldots, u_{j_i}\\}$ be an antichain of $\\Xnrs$  of length $j_i$, and let $w \\in \\Xnrp$. Recall that all antichains are assumed to be ordered in the lexicographic ordering. Now if $g$  preserves the lexicographic ordering of $\\CCn$ then define a map $f_{\\mathfrak{v}_{j_i},w}: U_{u_1} \\sqcup \\ldots \\sqcup U_{u_{j_i}} \\to  U_{w}$ by  $u_a \\delta \\mapsto w\\rho_a (\\delta)h_{p_{a}}$ for $1 \\le a \\le j_i$ and $\\delta \\in \\CCn$. O Then $f_{\\mathfrak{v}_{j_i},w}$ is homeomorphism from its domain to its range (by the definition of a viable combination), moreover  $f_{\\mathfrak{v}_{j_i},w}$ preserves the lexicographic ordering on $U_{u_1} \\sqcup \\ldots \\sqcup U_{u_l}$ since $\\viable{j_i}$ is a lexicographic viable combination and for all states $q$ of $g$ $h_{q}$ preserves the lexicographic ordering on $\\CCn$ and the relation $\\simeqI$  (Lemma~\\ref{Lemma: states of TBnr induces continuous functions from the interval to itsel }). \n\nNow let $\\ac{v}$ be a complete antichain of $\\Xnrs$ of length $j$ and let $\\ac{w}$ be a complete antichain of $\\Xnrs$ of length $m$. These antichains exist since $j \\equiv m \\equiv r \\mod{n-1}$. Recall that as $ 1 \\le r < n-1$ we must have that $j$ and $m$ are both non-zero. Let $\\ac{v}_1, \\ldots, \\ac{v}_m$ be disjoint subsets of $\\ac{v}$ such that each $\\ac{v}_i$, $1 \\le i \\le m$ is an antichain (ordered lexicographically) of length $j_i$ and for $1 \\le a < b \\le m$ we have $\\ac{v}_{a} \\lelex \\ac{v}_{b}$. Note that the stipulation that $|\\ac{v}_{i}| = j_i$ for $1 \\le i \\le m$ means that $\\sqcup_{1 \\le i \\le m} \\ac{v}_{i} = \\ac{v}$. Suppose that $\\ac{w} = \\{ w_1, w_2, \\ldots, w_m\\}$. \n\nRecall that for a subset  $Z \\subseteq \\Xns \\sqcup \\Xnrs$ we  denote by $U(Z)$ the set $\\{ U_{z} \\mid z \\in Z \\}$. Let $f: \\CCnr \\to \\CCnr$ be defined such that $f \\restriction _{U(\\ac{v}_{i})} = f_{\\ac{v}_i,w_i}$ for $1 \\le i \\le m$. Then clearly $f$ is a homeomorphism, and since each $f_{\\ac{v}_i,w_i}$, $1 \\le i \\le m$, preserves the lexicographic ordering of  $U(\\ac{v_i})$ and $\\simeqI$ then, $f$ preserves the lexicographic ordering and the relation $\\simeqI$ on $\\CCnr$. Moreover since $f$ replaces some initial prefix before acting as a state of  $g$, it follows that there is a transducer $A_{q_0} \\in \\TBnr$ with $h_{q_0} = f$.\n\n\\end{proof}\n\n\\begin{Remark}\\label{Remark: direct proof that g in Onr preseving simeq and reversing lex is in TOnr}\nOne may also prove Lemma~\\ref{Lemma: g in Onr is in TOnr if g prserve  simeqI and lex order} above for $g \\in \\Onr$ which preserves $\\simeqI$ and reverses the lexicographic ordering by directly constructing a transducer $A_{q_0} \\in \\TBnr$ which induce an orientation reversing homeomorphism of the line. As in the proof above, we make use of the lexicographic viable combinations to induce maps on $\\CCnr$ which preserve $\\simeqI$ and reverse the lexicographic ordering.\n\\end{Remark}\n\n\\begin{lemma}\\label{Lemma:preservinglexsuffices}\nLet $g \\in \\Onr$ be such that $g$ preserves or reverses the lexicographic ordering, then $g$ also preserves $\\simeqI$.\n\\end{lemma}\n\\begin{proof}\nFix $q$ any state of $g$. We observe that, by definition $q$ induces a continuous injection function from $\\CCn$ to itself with clopen image. Let $x, y \\in \\CCn$ be such that  $x \\ne y$, $x \\lelex y$, $x \\simeqI y$ and there is a word $\\nu \\in \\Xnrp$ such that $(x)h, (y)h \\in  U_{\\nu}$. As there is no point $y' \\in \\CCn$  not equal to $x$ or $y$ satisfying $x \\lelex y' \\lelex y$, then it must be the case that $(x)h \\simeq (y')h$.  Now as $(U_{\\nu})h_{q} ^{-1}$ is open, there is a $\\mu \\in \\Xnrp$ such that $(U_{\\mu})h_{q} = U_{\\nu}$ and so $h_{q}$ preserves the relation $\\simeqI$ on $U_{\\mu}$. Finally, observe that as $g$ is synchronizing for any other state $p$ of $g$, there is a word $\\Gamma \\in \\Xnrp$ such that $\\pi_{g}(\\Gamma, q) = p$. Therefore, we deduce that  all states of $g$ preserve $\\simeqI$ as required.\n\\end{proof}\n\n\n\nPutting together Lemmas~\\ref{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties}, \\ref{Lemma: g in Onr induces map on the line iff it has lexicographic viable combination}, Lemma~\\ref{Lemma:preservinglexsuffices}, and \\ref{Lemma: g in Onr is in TOnr if g prserve  simeqI and lex order}, we obtain the following result.\n\n\\begin{Theorem}\\label{Thm: Equivalent conditions for an element of $On$ to belong to TOnr}\nLet $g \\in \\On$. The following are equivalent:\n\\begin{enumerate}[label =(\\alph*)]\n\\item $g \\in \\TOnr$\n\\item  $g$ preserves or reverses the lexicographic ordering, and there are lexicographic viable combinations $\\viable{{j_1}}, \\ldots, \\viable{{j_m}}$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_{i} \\equiv m \\mod{n-1}$.\n\\item  $g$ preserves or reverses the lexicographic ordering, and there are viable combinations $\\viable{{j_1}}, \\ldots, \\viable{{j_m}}$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_{i} \\equiv m \\mod{n-1}$.\n\\item $g \\in \\Onr$ and preserves or reverses the lexicographic ordering  on $\\CCn$.\n\\end{enumerate}\n\n\\end{Theorem}\n\n\n\\section{ The enveloping group $\\TOn$}\\label{Section:nestingproperties1}\n\nWe make the following definition.\n\n\\begin{Definition}\\label{Def: External description of TOn.}\nLet $\\TOn \\subset \\On$ consists of those element which reverse or preserve the lexicographic ordering on $\\CCn$.\n\\end{Definition}\n\n\\begin{Remark}\\label{Remark: TOn is the union of the TOnr's}By an observation in \\cite{BCMNO}, for an element $g \\in \\On$ the set $\\viable{g}$ of viable combinations of $g$ is non-empty. From this it follows that $\\TOn = \\cup_{1\\le  r \\le n-1}  \\TOnr$.\n\\end{Remark}\n\n\\begin{proposition} \\label{Proposition:TOn is a subgroup of On}\nThe set $\\TOn$ is a subgroup of $\\On$.\n\\end{proposition}\n\\begin{proof}\nIt suffices to show that $\\TOn$ is closed under inverses and products. It is clear that the single state identity transducer is an element of $\\TOn$.\n\n\nLet $g \\in \\TOn$. Then there is an $r \\in \\{1,2, \\ldots, n-1\\}$ such that there is an $A_{q_0} \\in \\TBnr$ with $\\core(A_{q_0}) = g$. Let $B_{p_0}$ be the minimal transducer representing the inverse of $A_{q_0}$, then  $B_{p_0} \\in \\TBnr$ and $g^{-1} = \\core(B_{p_0}) \\in \\TOnr \\subset \\TOn$.\n\nNow let $h \\in \\TOn$. Let $q$ be a state of $g$ and $p$ be a state of $h$. Let $gh_{(p,q)}$ be the minimal transducer representing the product $g_{p} \\ast h_{q}$. Then observe that  $gh_{(p,q)}$ induces the function $g \\circ h : \\CCn \\to \\CCn$. Thus since $g$ and $h$ either preserve or reverse the lexicographic ordering on $\\CCn$ it follows that the states of $gh_{(p,q)}$ either all reverse or all preserve the lexicographic ordering on $\\CCn$. Therefore $gh = \\core(gh_{(p,q)}) \\in \\TOn$.\n\\end{proof}\n\n\\begin{Definition}\nLet $\\widetilde{\\TOn}$ be the subset of $\\TOn$ consisting of all those elements which preserve the lexicographic ordering on $\\CCn$. Set $\\WTOnr:=  \\WTOn \\cap  \\TOnr$. Then we call elements of $\\WTOn$ \\emph{orientation preserving}, and elements of $\\TOn \\backslash \\WTOn$ \\emph{orientation reversing}. Likewise set $\\widetilde{\\TBnr}$ to be those elements of $\\TBnr$ with core in $\\widetilde{\\TOn}$, then $\\widetilde{\\TBnr}$ are precisely those elements of $\\TBnr$ which induce orientation preserving maps of $S_{r}$. We will also call elements of $\\widetilde{\\TBnr}$ \\emph{orientation preserving} and the elements of $\\TBnr \\backslash \\widetilde{\\TBnr}$ \\emph{orientation reversing}.\n\\end{Definition}\n\nThe following proposition is straightforward and so we omit its proof.\n\n\\begin{proposition}\nThe set $\\WTOn$ is an index 2 subgroup of $\\TOn$ and so a normal subgroup of $\\TOn$. The set $\\WTOnr$ is an index 2 subgroup of $\\TOnr$ and so a normal subgroup of $\\TOnr$. \n\\end{proposition}\n\nWe now investigate how the groups $\\TOn$ intersect each other. For this we require the following definition, which in fact applies to the elements of the group $\\On$.\n\n\\begin{Definition}[Signature]\\label{Definition:signature}\nLet $T \\in \\On$, for each state $q \\in Q_{T}$ let $m_q$ be the size of the smallest subset $V$ of $\\Xns$ such that $U(V) = \\{ U_v \\mid v \\in V\\}$  is a clopen cover of $\\im(q)$ and $U_{v} \\subset \\im(q)$ for all $v \\in V$. Let $k \\in \\N$ be the minimal synchronizing level of $T$ an order the elements of the set $\\Xn^{k}$ lexicographically as follows: $x_1 < x_2 < \\ldots < x_{n^{k}}$. Let $(q_{x_1}, q_{x_2}, \\ldots, q_{x_{n^{k}}}) \\in Q_{T}^{n^{k}}$ be such that, for all $1 \\le i \\le n^{k}$, $q_{x_i}$ is the unique state of $T$ forced by $x_i$. Set $(T)\\sig = \\sum_{1 \\le i \\le n^{k}} m_{q_{x_i}}$, we call $(T)\\sig$ the \\emph{signature} of $T$; set $(T)\\rsig = (T)\\sig \\mod{n-1}$, we call $(T)\\rsig$ the \\emph{reduced signature of $T$}.\n\\end{Definition}\n\nWe have the following proposition. We prove the proposition below for the group $\\TOn$ noting that a similar result holds, with almost identical proof, in the  group $\\On$.\n\n\\begin{proposition}\\label{Proposition:sigdeterminesmembership}\nLet $T \\in \\TOn [T \\in \\On]$, and $1 \\le r < n$, then $T \\in \\TOnr [T \\in \\Onr]$ if and only if $r (T)\\sig \\equiv r \\mod{n-1}$ (equivalently $r((T)\\sig-1) \\equiv 0 \\mod {n-1}$.\n\\end{proposition}\n\\begin{proof}\nWe begin with the forward implication. First suppose that $T \\in \\TOnr$ and $T$ has minimal synchronizing level $k$. Since $T \\in \\TOnr$, there is an element $A_{q_0} \\in \\TBnr$ with $\\core(A_{q_0}) = T$. Let $j \\in \\N$ be minimal such that after reading a word of length $j$ from the state $q_0$ of $A$, the resulting state is a state of $T$. Let $\\{\\mu_i \\mid 1 \\le i \\le rn^{k+j-1}\\}$ be the set of all words of length $j+k$ in $\\Xnrs$ ordered lexicographically. For $1 \\le i \\le rn^{k+j-1}$, set  $\\nu_i  = \\lambda_{A}(\\mu_i, q_0)$ and  $q_{\\mu_i}$ to be the state of $T$ forced by $\\mu_i$. Observe that the state  $q_{\\mu_i}$ depends only on the last $k$ letters of $\\mu_i$, hence if the elements of $\\Xn^{k}$ are ordered lexicographically as  $x_1 < x_2< \\ldots < x_{n^{k}}$, the sequence ${(q_{\\mu_i})}_{1 \\le i \\le rn^{j+k-1}}$, where $q_{x_i}$, $1 \\le i \\le n^{k}$, is the state of $T$ forced by $x_i$, is precisely the sequence $q_{x_1}, \\ldots, q_{x_{n^{k}}}$ repeated $rn^{j-1}$ times.  Since $\\bigcup_{1 \\le i \\le rn^{j+k-1}} \\nu_i \\im(q_{\\mu_i}) = \\CCnr$, it must therefore be the case that $rn^{j-1}(\\sum_{1\\le i \\le n^{k}} m_{q_{x_i}}) \\equiv r \\mod n-1$. This is because if, for each $q_{\\mu_i}$, $1 \\le i \\le rn^{k+j-1}$, $V_{q_{\\mu_i}}$ is the smallest subset of $\\Xns$ such that $U(V_{q_{\\mu_i}})$  is a cover of $\\im(q)$ and $U_{v} \\subset \\im(q)$ for all $v \\in V_{q_{\\mu_i}}$, then $ \\bigcup_{1 \\le i \\le rn^{j+k-1}}\\{ \\nu_i v \\mid v \\in V(q_{\\mu_i}) \\}$ must be a complete antichain of $\\CCnr$ (otherwise $A_{q_0}$ is not a homeomorphism). Therefore, setting $m_{q_{x_i}} =  |V_{q_{x_i}}|$ we have, $$\\left| \\bigcup_{1 \\le i \\le rn^{j+k-1}}\\{ \\nu_i v \\mid v \\in V(q_{\\mu_i}) \\}\\right |  = rn^{j-1}(\\sum_{1\\le i \\le n^{k}} m_{q_{x_i}}) \\equiv r \\mod n-1.$$\nSince $n \\equiv 1 \\mod{n-1}$ we therefore have that $r(T)\\sig \\equiv r \\mod n-1$ as required.\n\nFor the reverse implication  let $T \\in \\TOn$ and $1 \\le r < n-1$ be such that $r (T)\\sig \\equiv r \\mod{n-1}$. Let $k \\in \\N$ be such that $T$ is synchronizing at level $k$, once more assume that the set $\\Xn^{k}$ is  ordered lexicographically as  $x_1 < x_2< \\ldots < x_{n^{k}}$. For $1 \\le i \\le n^{k}$, let $q_{x_i}$ be the state of $T$ forced by $x_i$. For each $1 \\le i \\le n^{k}$, let $V_{q_{x_i}}$ be the smallest subset of $\\Xns$, with size $m_{q_{x_i}}$, such that $U(V_{q_{x_i}})$ is a clopen cover of $\\im(q)$ consisting of clopen subsets of $\\im(q)$; let $M_{i} = \\max\\{\\left|v\\right| \\mid v \\in V_{q_{x_i}}\\}$ and set $M = \\max_{1 \\le i \\le n^{k}} M_{i}$. Let $j \\in \\N$ be minimal such that for any word $\\Gamma \\in \\Xn^{j}$ and any state $q$ of $T$, $|\\lambda_{T}(\\Gamma, q)|> M$. Order the set $\\Xn^{j}$  lexicographically  as $y_1 < y_2 < \\ldots < y_{n^{j}}$. For each state $q_{x_i}$, $1 \\le i \\le n^{k}$, order the set $V_{q_{x_i}}$ lexicographically as $\\nu_{i,1} < \\nu_{i,2}< \\ldots < \\nu_{i,m_{q_{x_i}}}$, and for all $1 \\le l \\le n^{j}$ let $\\mu_{i,l}\\varphi_{i,l} = \\lambda_{T}(y_l, q_{x_i})$, for some $\\mu_{i,l} \\in V_{q_{x_i}}$, $\\varphi_{i,l} \\in \\Xnp$, and $p_{i,l} = \\pi_{T}(y_l, q_{x_i})$. Now since, for $1 \\le i \\le n^{k}$, $|V_{q_{x_i}}| = m_{q_{x_i}}$, each $\\mu_{i,l} = \\nu_{i,a}$ for some $1 \\le a \\le m_{q_{x_i}}$ and we may write, for all $1 \\le l \\le n^{j}$, $\\mu_{i,l}\\varphi_{i,l} = \\nu_{i,a}\\rho_{i,l_a}$ where $\\nu_{i,a} = \\mu_{i,l}$ and  $\\rho_{i,l_a} = \\varphi_{i,l}$ for some $1 \\le a \\le m_{q_{x_i}}$, we also adopt the same notation for the set of $p_{i,l}$ and write $p_{i,l_a}$, for $1 \\le a \\le m_{q_{x_i}}$, where $\\nu_{i,a} \\rho_{i,l_a} = \\mu_{i,l}\\varphi_{i,l}$.\n\nNow let  $\\ac{u}$ be a maximal antichain of $\\CCnr$ of length $r n^{j+k}$ and let $\\ac{v}$ be a maximal antichain of $\\CCnr$ of length $r (T)\\sig$. Write $\\ac{u} = \\cup_{1 \\le t \\le r}\\ac{u}_t$ where each $\\ac{u}_{t}$ is ordered lexicographically, $| \\ac{u}_t| = n^{j+k}$ and for $1 \\le t_1 < t_2 \\le r$ all elements  of $\\ac{u}_{t_1}$ are strictly less than all elements of $\\ac{u}_{t_2}$ in the lexicographic ordering on $\\Xns$. Let $\\ac{v} = \\cup_{1 \\le t \\le r}\\ac{v}_t$ where each $\\ac{v}_{t}$ is ordered lexicographically, $| \\ac{v}_{t}| = (T)\\sig$ and for $1 \\le t_1 < t_2 \\le r$ all elements  of $\\ac{v}_{t_1}$ are strictly less than all elements of $\\ac{v}_{t_2}$ in the lexicographic ordering on $\\Xns$.\n\nFix $1 \\le t \\le r$ and consider $\\ac{u}_{t}$. Write $\\ac{u}_{t}= \\{u_{t,i,l} \\mid 1 \\le i \\le n^{k}, 1 \\le l \\le n^{j}\\}$. We further assume that, for a fixed $1 \\le i \\le n^{k}$, the set $\\{u_{t,i,l} \\mid 1 \\le l \\le n^{j}\\}$ is ordered lexicographically and, for $1 \\le i_1 < i_2 \\le n^{k}$ and for all $1 \\le l_1, l_2 \\le n^{j}$,  $u_{t,i_1, l_1} \\lelex  u_{t,i_2, l_2}$. Likewise write $\\ac{v}_{t} = \\{ v_{t,i,a} \\mid 1 \\le i \\le n^{k}, 1 \\le a \\le m_{q_{x_i}} \\}$.  We further assume that, for a fixed $1 \\le i \\le n^{k}$, the set $\\{v_{t,i,a} \\mid 1 \\le a \\le m_{q_{x_i}}\\}$ is ordered lexicographically and, for $1 \\le i_1 < i_2 \\le n^{k}$,  $v_{t,i_1,a} \\lelex  v_{t,i_2,b}$ for all $1 \\le a \\le m_{q_{x_{i_1}}}$ and $1 \\le b \\le m_{q_{x_{i_2}}}$. Furthermore, whenever, for $1 \\le a \\le m_{q_{x_i}}$ and $1 \\le l \\le n^{j}$, we have $\\nu_{i,a}\\rho_{i,l_a} =  \\mu_{i,l}\\varphi_{i,l}$ we set $\\eta_{t,i,l} := v_{t,i,a}$.\n\nDefine a map $f$ from  $\\CCnr$ to itself as follows. For $1 \\le t \\le r$, $f$ acts on elements with prefix in the set $\\ac{u}_{t}$ as follows. For $1 \\le i \\le n^{k}$ and $1 \\le l \\le n^{j}$, and $\\Gamma \\in \\CCn$, $u_{t,i,l}\\Gamma \\mapsto \\eta_{t,i,l}\\varphi_{i,l}(\\Gamma)p_{i,l}$. Observe that for a fixed $1 \\le i \\le n^{k}$, and a fixed $1 \\le a \\le m_{q_{x_i}}$, since we have $\\cup \\{ \\mu_{i,l}\\varphi_{i,l}\\im(p_{i,l}) \\mid \\mu_{i,l} = \\nu_{i,a} \\} = U_{\\nu_{i,a}}$ then it is the case that, for a fixed $1 \\le t \\le r$,  $\\cup \\{ \\eta_{t,i,l}\\varphi_{i,l}\\im(p_{i,l}) \\mid \\eta_{i,l} = v_{t,i,a} \\} = U_{v_{t,i,a}}$.  Now since, for a given $1 \\le i \\le n^{k}$,  $\\bigcup_{1 \\le l \\le n^{j}} \\mu_{i,l} \\varphi_{i,l} \\im(p_{i,l}) = \\im(q)$ and as $q$ is injective and preserves the lexicographic ordering of $\\CCn$, we see that $f \\restriction_{U(\\ac{u}_{t})}$ is a bijection unto the set $U(\\ac{v}_{t})$ which preserves the lexicographic ordering of $\\CCn$. More specifically, since $f$ acts by a state of $T$ after a finite depth, we see that $f$ is a homeomorphism of $\\CCnr$ which is in fact an element of $\\TBnr$.\n\nThe proof of the other reading proceeds in an analogous fashion only here we do not have to worry about preserving the lexicographic ordering on $\\CCnr$.\n\\end{proof}\n\n\\begin{Remark}\nIt follows from results in \\cite{GriNekSus} that given an element $T \\in \\TOn [T \\in\\On]$ and $1 \\le r < n$, then it is possible to decide in finite time if $T \\in \\TOnr [T \\in\\Onr]$. Furthermore, by the above proposition,  $T \\in \\TOns{1} [T \\in \\Ons{1}]$ if and only if  $(T)\\sig \\equiv 1 \\mod{n-1}$.\n\\end{Remark}\n\nThe following result is a consequence of Proposition~\\ref{Proposition:sigdeterminesmembership}:\n\n\\begin{lemma}\\label{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}\nLet $n \\in \\N$ and suppose that $n\\ge 2$. Let $i, j, m \\in \\Z_{n}$ such that, $i$ and $j$ are non-zero and $m i \\equiv j \\mod n-1$ then $\\TOns{i} \\subseteq \\TOns{j}$ [$\\Ons{i} \\subseteq \\Ons{j}$].\n\\end{lemma}\n\\begin{proof}\nLet $i,j, m$ be as in the statement of the lemma. Let $T \\in \\TOns{i} [T \\in \\Ons{i}]$, then by Proposition~\\ref{Proposition:sigdeterminesmembership} $i (T)\\sig \\equiv i \\mod{n-1}$, therefore $mi (T)\\sig \\equiv mi \\mod{n-1}$ and so $j (T)\\sig \\equiv j \\mod{n-1}$ and $T \\in \\TOns{j} [T \\in \\Ons{j}]$  again by  Proposition~\\ref{Proposition:sigdeterminesmembership}.\n\\end{proof}\n\n\\begin{comment}\n\\begin{proof}\nWe first consider elements $g \\in \\WTOns{i}$. Let $g \\in \\WTOns{i}$, and as usual let ${\\bf{\\dot{i}}} = \\{\\dot{1},\\ldots, \\dot{i} \\}$. Let $\\bar{m} \\in \\N$ be such that $\\bar{m}$ is non-zero and congruent to $m$ modulo $n-1$. Let $A_{q_0} \\in \\TBmr{n}{,i}$ such that $\\core(A_{q_0}) = g$. There is a complete antichain $\\ac{u}$ of $\\CCmr{n,j}$ with size precisely precisely $\\bar{m}i$.  Suppose $$\\ac{u} = \\{u_{1,1}, \\ldots, u_{1,i}, u_{2,1},\\ldots, u_{2,i}, \\ldots, u_{m,1},\\ldots, u_{m,i}\\}.$$ For $1 \\le l \\le i$ Let $\\ac{u}_{l} := \\{u_{l,1}, u_{l,2}, \\ldots, u_{l,i} \\}$, observe that $\\ac{u}_{l}$ is lexicographically ordered since $\\ac{u}$ is lexicographically ordered, thus $u_{l,1}\\lelex u_{l,2}\\lelex \\ldots \\lelex u_{l,i}$. Let $U_{\\ac{u}_{l}}:= U_{u_{l,1}} \\sqcup \\ldots \\sqcup U_{u_{l,i}}$.\n\n Fix $1 \\le l \\le m$ and consider the set $U_{\\ac{u}_{l}}$.  Define a map  $f_{l}$ from $U_{\\ac{u}_{l}}$ to itself as follows: for $\\delta \\in \\CCn$ and $r, r' \\in \\{1, \\ldots, i\\}$, $(u_{l,r}\\delta)f = u_{l,r'}\\rho$ if and only if  $(\\dot{r}\\delta)h_{A_{q_0}} = \\dot{r'}\\rho$. Notice that $f_{l}$ is a homeomorphism form $U_{\\ac{u}_{l}}$ to itself, since $h_{A_{q_0}}$ is a homeomorphism on $\\CCmr{n,i}$, and $\\ac{u}_{l}$ is an antichain. Moreover since $A_{q_0} \\in \\WTOns{i}$ and the antichain $\\ac{u}_{l}$ is lexicographically ordered, we have that $f_{l}$ preserves the lexicographic ordering on $U_{\\ac{u}_{l}}$. Now let $f: \\CCmr{n,j} \\to \\CCmr{n,j}$ be such that for $1 \\le l \\le m$  $f\\restriction_{U_{\\ac{u}_{l}}} = f_{l}$. Then $f$ is a homeomorphism from $\\CCmr{n,j}$ to itself, since for $1 \\le l \\le m$ $f_{l}$ is a homeomorphism from $U_{\\ac{u}_{l}}$ to itself, and $\\ac{u}$ is a complete antichain for $\\CCmr{n,j}$. Moreover $f$ preserve the lexicographic ordering of $\\CCmr{n,j}$ since each of th $f_{l}$'s ($1\\le l \\le m$) preserve the lexicographic ordering on $U_{\\ac{u}_{l}}$. Furthermore observe that if $B_{p_0}$ is a minimal, initial automaton  such that $h_{B_{p_0}} = f$, then $B_{p_0}$ is finite, synchronizing, and $\\core(B_{p_0}) = g$. This is because each $f_{l}: U_{\\ac{u}_{l}} \\to U_{\\ac{u}_{l}}$ is defined such that after processing a finite prefix it acts as a state of $A_{q_0}$. Thus $f \\in \\TBmr{n}{,j}$ and so $g \\in \\WTOns{j}$.\n \n Since $g \\in \\WTOns{i}$ was arbitrary, we therefore conclude that $\\WTOns{i} \\subseteq \\WTOns{j}$. However since  $\\pi_{R,n} \\in \\TOns{i} \\cap \\TOns{j}$, we therefore have that $\\TOns{i} \\subseteq \\TOns{j}$ as required.\n\\end{proof}\n\\end{comment}\n\n\\begin{Remark}\nObserve that the lemma above implies that whenever $j \\in \\Z_{n}$ is co-prime to $n-1$ then $\\TOns{j} = \\TOns{1}$ [$\\Ons{j} = \\Ons{1}$]. Thus it follows that for $n$ a natural number bigger than $2$ such that $n-1$ is prime, if $T \\in \\TOn$ [$T \\in \\On$],  satisfies $(T)\\sig \\not\\equiv 1 \\mod{n-1}$ then $T \\in \\TOn \\backslash \\{\\TOns{1}\\}$[$T \\in \\On \\backslash \\{\\Ons{1}\\}$]. The following result generalises this observation.\n\\end{Remark}\n\n\\begin{comment}\n\\begin{proposition}\\label{Proposition:groupofunitscyclicimpliesonlyOn1andOnnminus1}\nLet $n \\in \\N_{2}$ and suppose that the group of units of $\\Z_{n-1}$ is cyclic. Then for any $1 \\le r \\le n-1$ either $\\TOnr = \\TOns{1}$ or $\\TOnr= \\TOns{n-1}$ [either $\\Onr = \\Ons{1}$ or $\\Onr= \\Ons{n-1}$]\n\\end{proposition}\n\\begin{proof}\nWe prove the first reading of the proposition, as the second reading follows by mechanical substitutions of $\\TOn$ with $\\On$ in the arguments below. \n\nLet $1 \\le r \\le n-1$ if there is no $T \\in \\TOnr$ such that $(T)\\rsig \\ne 1$, then $\\TOnr = \\TOns{1}$. Thus, assume that there is some $T \\in \\TOnr$ such that $(T)\\rsig \\ne 1$. Now observe that for any $m \\in \\N$, we have $r((T)\\rsig)^m \\equiv r \\mod{n-1}$ since $r(T)\\rsig \\equiv r \\mod{n-1}$ by Proposition~\\ref{Proposition:sigdeterminesmembership}. Now since $\\gen{(T)\\rsig}=\\{((T)\\rsig)^m \\mod{n-1} \\mid m \\in \\N \\}$ is precisely the group of units of $\\Z_{n-1}$, the result follows by Proposition~\\ref{Proposition:sigdeterminesmembership} once again.\n\\end{proof}\n\\end{comment}\n\nAn immediate corollary of the Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} is the following result, an analogous result appears in \\cite{BCMNO} for the groups $\\Onr$:\n\n\\begin{Theorem}\\label{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}\nLet $n \\in \\N$, $n \\ge 2$, then for all non-zero $i \\in \\Z_{n}$ we have $\\TOns{i} \\subseteq \\TOns{n-1}$. Hence $\\TOns{n-1} = \\TOn$ and   $\\TOns{1} \\subseteq \\TOns{i}$ for all $1 \\le i \\le n-1$, hence $\\cap_{1 \\le r \\le n-1} \\TOns{r} = \\TOns{1}$.\n\\end{Theorem}\n\\begin{proof}\nLet $i$ be non-zero in $\\Z_{n}$, then observe that $(n-1)\\ast i \\equiv n-1 \\mod{n-1}$ and $i*1 \\equiv i \\mod{n-1}$. Therefore by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} above, we have $\\TOns{i} \\subseteq \\TOns{n-1}$ and $\\TOns{1} \\subseteq \\TOns{i}$. Thus $\\TOns{n-1} = \\TOn$ and $\\cap_{1 \\le r \\le n-1} \\TOns{r} = \\TOns{1}$.\n\\end{proof}\n\nThe following result is again a corollary of Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}, we observe that the result for $\\Onr$ was proved in \\cite{BCMNO}:\n\n\\begin{corollary}\\label{Corollary:TOniisequaltoTOnjforsomejdividingnminus1}\nLet $n$ be a natural number bigger than $2$, $j \\in \\Z_{n}\\backslash \\{0\\}$ and $d$ be the greatest common divisor of $n-1$ and $j$, then $\\TOns{j} = \\TOns{d}$ [$\\Ons{j} = \\Ons{d}$]. \n\\end{corollary}\n\\begin{proof}\nLet $n,j,d$ be as in the statement of the corollary. Since $d$ is the greatest common divisor of $n-1$ and $j$, there are co-prime numbers $a, b \\in \\Z_{n}\\backslash\\{0\\}$ such that $j = da$ and $n-1 = db$. Since $a$ and $b$ are co-prime, there are numbers $u, v \\in \\Z$ such that $ua = 1 + vb$. Multiplying both sides of the equation by $d$ it follows that $uad = d + vbd$ and so $uj = d + v(n-1)$. Therefore there is some $m_1 \\in \\Z_{n-1}$ such that $m_1j \\equiv d \\mod {n-1}$. Moreover, since $d$ divides $j$, there is some $m_2 \\in \\Z_{n-1}$ such that $m_2 d = j \\mod{n-1}$. It therefore follows by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} that $\\TOns{j} = \\TOns{d}$[$\\Ons{j} = \\Ons{d}$].\n\\end{proof}\n\nCorollary~\\ref{Corollary:TOniisequaltoTOnjforsomejdividingnminus1} should be compared with the result of Pardo \\cite{EPardo} showing that $G_{n,r} \\cong G_{m,s}$ if and only if $n=m$ and $\\gcd(n-1,r) = \\gcd(n-1,s)$. It is a question in \\cite{BCMNO} whether or not $\\Ons{r} \\cong \\Ons{s}$ if and only if $\\gcd(n-1,r) = \\gcd(n-1,s)$. Below (Remark~\\ref{Remark:negativesolutiontoquestionofCollinetal}) we show that this question has a negative answer.  \n\nThe following Lemma can be thought of as a partial converse to Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}.\n\n\n\\begin{lemma}\\label{Lemma:partialconverselemma}\nLet $T \\in \\TOn$ [$T \\in \\On$] and let $r$ be minimal such that $T \\in \\TOnr$ [$T \\in \\Onr$], then $T \\in \\TOns{j}$ [$T \\in \\Ons{j}$] for some $1 \\le j \\le n-1$ if and only if  there is some $m \\in \\Z_{n}$ such that $mr = j$.\n\\end{lemma}\n\\begin{proof}\nThe forward implication is a consequence of Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}. Therefore let $T \\in \\TOn$ and $r$ be minimal such that $T \\in \\TOnr$ Let $1 \\le j \\le n-1$ be such that $T \\in \\TOns{j}$. By assumption we must have that $r < j$. Since $T \\in \\TOnr$ then by Proposition~\\ref{Proposition:sigdeterminesmembership}, $r(T)\\sig \\equiv r \\mod{n-1}$ and since $T \\in \\TOns{j}$, then $j (T)\\sig \\equiv j \\mod{n-1}$. Thus we deduce that $(j-r)(T)\\sig \\equiv (j-r) \\mod{n-1}$. If $(j-r) > r$, then we may repeat the process otherwise $j-r = r$, by the minimality assumption on $r$ and Proposition~\\ref{Proposition:sigdeterminesmembership}, in which case $j = 2r$. Inductively there is some $k \\in \\N$ such that $j-kr = r$ and so $j = (k+1)r$ which concludes the proof. \n\nThe other reading of the lemma is proved analogously, simply replace $\\TOn$ with $\\On$ in the paragraph above.\n\\end{proof}\n\nWe now show that the map $\\rsig: \\On \\to \\Z_{n-1}$ is a homomorphism from $\\On$ to the group of units of $\\Z_{n-1}$ with kernel $\\Ons{1}$.\n\nWe begin with the following result:\n\n\\begin{proposition}\\label{Proposition:signatureisequaltosizeofimagecovermodnminus1}\nLet $T \\in \\On$, $q$ be any state of $T$ and $m_q$ be the size of the smallest subset $V$ of $\\Xns$ such that $U(V) = \\{ U_v \\mid v \\in V\\}$  is a clopen cover of $\\im(q)$ and $U_{v} \\subset \\im(q)$ for all $v \\in V$, then  $m_{q} \\mod{n-1} = (T)\\rsig$.\n\\end{proposition}\n\\begin{proof}\n\nFor any state  $p$ of $T$ let $V(p)= \\{v_{1,p}, v_{2,p}, \\ldots\n, v_{m_p,p}\\} \\subset \\Xns$ be such that $\\im(p) = \\cup_{1 \\le i \\le m_{p}} U_{v_{i,p}}$. Now fix a state $q$ of $T$, let $k$ be the minimal synchronizing level of $T$ and $j \\in \\N_{k}$ be such that for any word $\\Gamma \\in \\Xn^{j}$ and any state $p$ of $T$, $|\\lambda_{T}(\\Gamma, p)| \\ge  \\max\\{ |v_{1,q}| \\mid v_{1,q} \\in V(q) \\}$. Order the set  $X_{n}^{j}$ in the lexicographic ordering as follows $x_1 < x_2 < x_3 \\ldots < x_{n^{j}}$, for $1 \\le a \\le n^{j}$ let $q_{x_a}$ be the unique state of $T$ forced by $x_a$ and $\\rho_{x_a} = \\lambda_{A}(x_a, q)$. Observe that for all $1 \\le a \\le n^{j}$, $|\\rho_{x_a}| \\ge \\max\\{ |v_{1,q}| \\mid v_{1,q} \\in V(q) \\}$ and so $\\rho_{x_a}$ has a prefix in $V(q)$. Now since the state $q$ is induces a homeomorphism from $\\CCn$ unto its image, it follows that the set $\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}$ is  a complete antichain for the subset of $\\Xns$ consisting of all elements with prefix in $V(q)$. From this we deduce that  $|\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}| \\equiv m_q \\mod{n-1}$.\n\nOn the other hand \n\\[\n|\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}| = \\sum_{1 \\le a \\le n^{j}} m_{q_{x_a}}.\n\\]\nHowever, since $T$ is synchronizing at level $k$, the sequence $(q_{x_a})_{1 \\le a \\le n^j}$ is in fact equal to the sequence $(q_{x_a})_{(1 \\le a \\le n^{k})}$ repeated $n^{j-k}$ times. Therefore we have,\n\\[\n|\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}| = n^{j-k}\\sum_{1 \\le a \\le n^{k}} m_{q_{x_a}} \n\\]\n and so we conclude that $m_{q} \\mod {n-1} = n^{j-k}\\sum_{1 \\le a \\le n^{k}} m_{q_{x_a}} \\mod{n-1} = (T)\\rsig$. \n\\end{proof}\n\n\\begin{Theorem}\\label{Theorem:rsigisahomomorphism}\nThe map $\\rsig: \\On \\to \\Z_{n-1}$ is a homomorphism from $\\On$ into the group of units of $\\Z_{n-1}$ with kernel $\\Ons{1}$.\n\\end{Theorem}\n\\begin{proof}\nLet $T, U \\in \\On$,  $j, k \\in \\N$ be the minimal synchronizing levels of $T$ and $U$. Consider the transducer product $T \\ast U$. Let $(p,q)$ be any state in the core of  $T \\ast U$, and for $\\sharp \\in \\{p,q\\}$ let $V(\\sharp)= \\{v_{1,\\sharp}, v_{2,\\sharp}, \\ldots\n, v_{m_p,\\sharp}\\} \\subset \\Xns$ be such that $\\im(\\sharp) = \\cup_{1 \\le i \\le m_{\\sharp}} U_{v_{i,\\sharp}}$. We may assume that $V_{q}$ is the smallest subset of $\\Xns$ with this property. Let  $m = \\max\\{ |v_{i,q} | \\mid v_{i,q} \\in V(q) \\}$ and $l \\in \\N$ be such that for any state  $p'$ of $T$ and any word $\\Gamma \\in \\Xn^{l}$, $|\\lambda_{T}(\\Gamma, p')| \\ge m$. We may further assume that $V(p)$ is the smallest subset of $\\Xn^{l}$ satisfying $\\im(p) = \\cup_{1 \\le i \\le m_{p}} U_{v_{i,p}}$. For each $1 \\le i \\le m_{p}$ let $q_i = \\pi_{U}(v_{i,p}, q)$ and $\\rho_i = \\lambda_{U}(v_{i,p}, q)$. Since $\\im(p) = \\cup_{1 \\le i \\le m_{p}} U_{v_{i,p}}$, it follows that $\\im((p,q)) = \\sqcup_{1 \\le i \\le m_{p}} \\{ \\rho_i \\im(q_i)\\}$. For each $1 \\le i \\le m_{p}$, let $m_{q_i}$ be the size of the smallest subset $V(q_i) \\subset \\Xns$ such that $\\cup_{v \\in V(q_i)} U_{v} = \\im(q_i)$, it follows that $V:=\\cup_{1 \\le i \\le m_p} \\{ \\rho_i v \\mid v \\in V(q_i) \\}$ satisfies $\\cup_{v \\in V}(U_{v}) = \\im((p,q))$. Notice that since both $p$ and $q$ are injective, then $|V| = \\sum_{1 \\le i \\le m_{p}} m_{q_i}$.\n\nObserve for any other set $V' \\subset \\Xns$ such that $\\cup_{v \\in V'} U_{v} =  \\im((p,q))$ it must be the case that $|V'| \\equiv |V| \\mod{n-1}$. Thus if $V'$ is the smallest subset of $\\Xns$ with this property, we have $|V'| \\equiv |V| \\mod{n-1}$. Furthermore we observe that removing incomplete response from the states of $T\\ast U$ simply removes the greatest common prefix of $\\im(p,q)$. Therefore if the state $s$ of $TU$ is equal to the state $(p,q)$ after removing the incomplete response, then for $V'' \\subset \\Xns$ minimal such that $\\cup_{v \\in V''} U_{v} =  \\im(s)$, we have $|V''| \\equiv |V| \\mod{n-1}$. By Proposition~\\ref{Proposition:signatureisequaltosizeofimagecovermodnminus1} it follows that\n\\[\n (TU)\\rsig \\equiv |V''| \\equiv \\sum_{1 \\le i \\le m_{p}} m_{q_i} \\equiv  (T)\\rsig(U)\\rsig\\mod{n-1}.\n\\]\n\nSince $(\\id)\\rsig = 1$, it follows that $\\rsig$ is a homomorphism from $\\On$ to the group of units of $\\Z_{n-1}$.\n\nTo see that $\\ker(\\rsig) = \\Ons{1}$, observe that by Proposition~\\ref{Proposition:sigdeterminesmembership} $(T)\\rsig = 1$ if and only if $T \\in \\Ons{1}$.\n\n\\end{proof}\n\nGiven $T \\in \\On$ the following result enables us to compute the reduced signature of $T^{-1}$ directly from $T$ i.e. without computing the inverse.\n\n\\begin{proposition}\nLet $T \\in \\On$ and $q$ be any state of $T$. Let $\\nu \\in \\CCn$ be such that $U_{\\nu} \\subset \\im(q)$ and $j \\in \\N$ be such that for any word $\\Gamma \\in \\Xn^{j}$, $|\\lambda_{T}(\\Gamma, q)| \\ge |\\nu|$. Let $W \\subset \\Xn^{j}$ be maximal such that for any word $\\Delta \\in W$, $\\nu$ is a prefix $\\lambda_{T}(\\Delta, \\Gamma)$. Let $1 \\le  w \\le n-1$ be such that $w \\equiv |W| \\mod{n-1}$, then $w$ depends only on $T$, in particular $w = (T^{-1})\\rsig$.\n\\end{proposition}\n\\begin{proof}\nBy Proposition~\\ref{Proposition:signatureisequaltosizeofimagecovermodnminus1} it suffices to show that there is a state $p'$ of $T^{-1}$ and a subset $V  \\subset \\Xns$  such that $\\cup_{\\mu \\in V} U_{\\mu} = \\im(p)$ and $|V| = |W|$.\n\nLet $A_{q_0}$ a bi-synchronizing transducer with $\\core(A_{q_0}) = T$ and $k \\in \\N_{1}$ be the minimal bi-synchronizing level of $A_{q_0}$. Let $\\varphi =  (\\nu)L_{q}$ i.e. $\\varphi$ is the greatest common prefix of the set $h_{q}^{-1}(U_{\\nu})$. Let $\\nu_1 = \\lambda_{T}(\\varphi, q)$, $p = \\pi_{T}(\\varphi, q)$ and $\\nu_2 = \\nu - \\nu_1$. We claim that $(\\nu_2,p)$ is  a state of $A_{(\\epsilon, q_0)}$ and is $\\omega$-equivalent to a state of $T^{-1}$.\n\n For, let $l \\in \\N$ be such that for any word $\\Gamma \\in \\Xn^{l}$, $|\\lambda_{A}(\\Gamma, q_0)| \\ge k$.\n Let $\\Gamma \\in \\Xn^{l}$ be such that $\\pi_{A}(\\Gamma, q_0) = q \\in \\core(A_{q_0})$. Such a word $\\Gamma$ exists because of the bi-synchronizing condition. Let $\\Delta = \\lambda_{A}(\\Gamma, q_0)$. Then observe that $(\\Delta\\nu)L_{q_0} = \\Gamma (\\nu)L_{q} = \\Gamma \\varphi$ since $U_{\\nu} \\subset \\im(q)$. Thus, in the transducer $A_{(\\epsilon, q_0)}$, $\\pi_{A}(\\Delta\\nu, q_0) = (\\Delta\\nu - \\lambda_{A}(\\Gamma \\varphi, q_0), \\pi_{A}(\\Gamma\\varphi, q_0))= (\\nu_2,p)$. Now since $A_{(\\epsilon, q_0)}$ has no states of incomplete response, is $\\omega$-equivalent to the minimal transducer $B_{p_0}$ representing $h_{q_0}^{-1}$, and by the choice of $l$, we therefore have that $(\\nu_2,p)$ is $\\omega$-equivalent to a state in $\\core(B_{p_0}) = T^{-1}$.\n \n Now, suppose $W = \\{ \\rho_1', \\rho_2', \\ldots, \\rho_{m}'\\}$. We observe that by definition of $W$, $\\varphi$ is the greatest common prefix of $W$, thus for $1\\le i \\le m$, let $\\rho_i \\in \\Xns$ be such that $\\varphi\\rho_i = \\rho_i'$. Observe that for any $\\delta \\in \\CCn$, there is a unique $1 \\le i \\le m$,  and $\\xi \\in \\CCn$ such that $\\lambda_{A}(\\rho_i'\\xi, q) = \\nu \\delta$, therefore $\\lambda_{A}(\\rho_i \\xi, p) = \\nu_2\\delta$. Moreover, for any $1 \\le i \\le m$ and $\\xi \\in \\CCn$, $\\lambda_{A}(\\rho_i\\xi, p) = \\nu_2 \\delta$ for some $\\delta \\in \\CCn$. Since $\\im((\\nu_2,p)) = \\{(\\nu_2\\delta)L_{p} \\mid \\delta \\in \\CCn \\} = (U_{\\nu_2})h_{p}^{-1}$, it follows that $\\im((\\nu_2,p)) = \\cup_{1 \\le i \\le m} U_{\\rho_i'}$. Therefore it follows, from Proposition~\\ref{Proposition:signatureisequaltosizeofimagecovermodnminus1}, that $m = |W| \\equiv (T^{-1})\\rsig \\mod{n-1}$.\n \n \n\\end{proof}\n\nThe corollary below follows straight-forwardly from Theorem~\\ref{Theorem:rsigisahomomorphism}\n\n\\begin{corollary}\nLet  $1< r< n$, then the following hold:\n\\begin{enumerate}[label = (\\alph*)]\n\\item $\\Onr$ is a normal subgroup of $\\Ons{n-1} = \\On$, in particular for non-zero $i, j \\in \\Z_{n}$ such that $i$ divides $j$ in the additive group $\\Z_{n-1}$, $\\Ons{i} \\unlhd \\Ons{j}$,\n\\item $[\\On, \\On] \\le \\Onr$,\n\\item $\\Onr\/\\Ons{1}$ is isomorphic to a subgroup of the group of units of $\\Z_{n-1}$ and $\\On\/\\Onr$ is isomorphic to a quotient of the group of units of $\\Z_{n-1}$.   \n\\end{enumerate}\n   \n\\end{corollary}\n\\begin{proof}\nSince the group of units of $\\Z_{n-1}$ is abelian, it follows that $[\\On, \\On] \\le \\Ons{1}$ and by Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1} $[\\On, \\On] \\le \\Ons{r}$.\n\nTo see that $\\Onr$ is normal in $\\On$, we observe that given $T \\in \\Onr$, and $U \\in \\On$, then $(U^{-1}TU)\\rsig \\equiv (U^{-1})\\rsig(T)\\rsig(U)\\rsig \\mod{n-1}$ and so $(U^{-1}TU)\\rsig = (T)\\rsig$ and by Proposition~\\ref{Proposition:sigdeterminesmembership}, we have $U^{-1}TU \\in \\Onr$.\n\nThat $\\Onr\/\\Ons{1}$ is isomorphic to a subgroup of the group of units of $\\Z_{n-1}$, follows by the correspondence theorem  since $\\Ons{1} \\le \\Onr$. \n\nThat $\\On\/\\Onr$ is isomorphic to a quotient of the group of units of $\\Z_{n-1}$, follows from the third isomorphism theorem: $\\On\/\\Onr \\cong (\\On\/\\Ons{1})\/(\\Onr\/\\Ons{1})$.\n\\end{proof}\n\n\\begin{Question}\\label{Question:issigsurjective}\nIs it the case that $[\\On, \\On] = \\Ons{1}$? Is the map $\\rsig$ from $\\On$ to the group of units of $\\Z_{n-1}$ surjective?\n\\end{Question}\n\n\\begin{Remark}\nSince $\\TOn \\le \\On$ the results above and questions can be restated with $\\TOn$ in place of $\\On$.\n\\end{Remark}\n\n\\section{ Nesting properties of the groups $\\TOnr$}\\label{Section:nestingproperties2}\n\nIn this section we focus on the group $\\TOn$ however, {\\bfseries{all the results below are equally valid when all occurrences of $\\TOn$ are replaced with $\\On$}}.\n\nThe following result is a direct consequence corollary of Proposition~\\ref{Proposition:sigdeterminesmembership} and generalises Corollary~\\ref{Corollary:TOniisequaltoTOnjforsomejdividingnminus1}:\n\n\\begin{corollary}\\label{Corollary:OnrisequaltoOnjifandonlyif}\nLet $1\\le r,s<n$ be natural numbers and let $(\\TOn)\\rsig$  $[(\\On)\\rsig]$ denote the image of $\\TOn$ [$\\On$] under the map $\\rsig$. Then $\\TOnr = \\TOns{s}$ [$\\Onr = \\Ons{s}$] if and only if for all $j \\in (\\TOn)\\rsig$ [$j \\in (\\On)\\rsig$], $r(j-1)\\equiv s(j-1) \\equiv 0 \\mod{n-1}$.\n\\end{corollary}\n\n\\begin{Remark}\\label{Remark:negativesolutiontoquestionofCollinetal}\nObserve that when $n=7$, then $(\\On)\\rsig \\subseteq  \\{1,5\\}$. Further observe, by Corollary~\\ref{Corollary:TOniisequaltoTOnjforsomejdividingnminus1}, that for $1 \\le r \\le 6$, $\\Oms{7}{r}$ is equal to  $\\Oms{7}{a}$ for $a \\in \\{1,2,3,6\\}$. Now notice that as $12$ is divisible by $6$, then $\\Oms{7}{3} = \\Oms{7}{6}$ and, since $10$ is not a multiple of $6$, $\\Oms{7}{1} = \\Oms{7}{2}$.  However $\\gcd(1,6) \\ne \\gcd(2,6)$ and $\\gcd(3,6) \\ne \\gcd(6,6)$. Therefore it is not the case that for $1 \\le r,s \\le n$ $\\Onr \\cong \\Ons{s}$ if and only if $\\gcd(r,n-1) = \\gcd(s, n-1)$. This yields a negative solution to a question in the paper \\cite{BCMNO}. We notice that the same result holds if $\\On$ is replaced with $\\TOn$. More specifically, $\\TOms{7}{a} = \\TOms{7}{b}$ for $(a,b) \\in \\{(1,2), (3,6)\\}$.\n\\end{Remark}\n\n\n\nRemark~\\ref{Remark:negativesolutiontoquestionofCollinetal} proves the following result:\n\n\\begin{Theorem}\nThere are numbers $n \\in \\N$, $n >2$, and $1 \\le r, s \\le n-1$ such that, for $\\T{X}= \\T{TO}, \\T{O}$, $\\XOnr = \\XOns{s}$ but $\\gcd(n-1, r) \\ne \\gcd(n-1, s)$.\n\\end{Theorem}\n\nPartition the set $\\Z_{n} \\backslash \\{0\\}$ as follows. For $i \\in \\Z_{n} \\backslash \\{0\\}$, set $[i] := \\{j \\in \\Z_{n} \\backslash \\{0\\} \\mid \\TOns{j}  = \\TOns{i}\\} \\subset  \\Z_{n}$, $[\\Z_{n}^{0}] \\seteq \\{ [i] \\mid i \\in \\Z_{n} \\backslash \\{0\\} \\}$ and, for $[i] \\in [\\Z_{n}^{0}]$, $\\TOns{[i]} \\seteq \\TOns{i}$. Observe that the set $[\\Z_{n}^{0}]$ inherits an ordering from $\\Z_{n}$ where $[i] < [j]$ if the smallest element of $[i]$ is less than the smallest element of $[j]$. Further observe that if $n-1$ is prime, then $[\\Z_{n}^{0}]$ has size at most $2$ by Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}.\n\n\\begin{Definition}[Atoms]\nLet $[r] \\in [\\Z_{n}^{0}]$, then we say $[r]$ is \\emph{an atom (of $[\\Z_{n}^0]$)} if  there is an element $T \\in \\TOns{[r]}$ which is not an element of $\\TOns{[s]}$ for any $[s] < [r]$. Let $[i] \\in [\\Z_{n}^{0}]$, an \\emph{atom of $[i]$} is an atom $[r]$ of $[\\Z_{n}^0]$ such that $\\TOns{[r]} \\le \\TOns{[i]}$. \n\\end{Definition}\n\n\\begin{Remark}\\label{Remark:atomsdividenminus1}\nLet $[r] \\in [\\Z_{n}^0]$ be an atom with $r$ the minimal element of $r$, then as a consequence of Lemma~\\ref{Lemma:partialconverselemma} and Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}, we have that $r|n-1$. Observe that by Remark~\\ref{Remark:negativesolutiontoquestionofCollinetal} it is not always the case that for every element $r$ dividing $n-1$ that $[r]$ is an atom of $[\\Z_{n}^{0}]$. \n\\end{Remark}\n\n\nObserve that if the map $\\rsig$ is surjective, then for each $i \\in \\Z_{n} \\backslash \\{0\\}$,  we may completely determine the elements of $[i]$. \n\n\\begin{Question}\nIs it the case that all elements $[i] \\in [\\Z_{n}^{0}]$ are atoms?\n\\end{Question}\n\nIn the interim we make the following definitions.\n\n\\begin{Definition}\nLet $[i],[j] \\in [\\Z_{n}^{0}]$ where $i,j \\in \\Z_{n}$ are the minimal elements of $[i]$ and $[j]$ respectively. We say that $[i]$ \\emph{divides} $[j]$ if $r|j$ for any atom $[r]$ of $[i]$ with $r$ the minimal element of $[r]$.\n\\end{Definition}\n\n\n\\begin{Definition}\nGiven two elements $[i],[j] \\in [\\Z_{n}^0]$, the \\emph{lowest common multiple of $[i]$ and $[j]$} is the smallest element $[l]$  of $[\\Z_{n}^0]$ such that $[i]$ and $[j]$ divide $[l]$. Notice that by Remark~\\ref{Remark:atomsdividenminus1} the lowest common multiple of any pair of numbers $[i], [j] \\in [\\Z_{n}^0]$ always exists. We extend the definition of lowest common multiple to tuples of elements of $[\\Z_{n}^0]$ in the usual way.\n\\end{Definition}\n\n\n\n\\begin{Definition}\n Let $i, j \\in \\Z_{n} \\backslash \\{0\\}$ be the minimal elements of $[i]$ and $[j]$ respectively, then we define the \\emph{greatest common divisor of $[i]$ and $[j]$}  to be  $[r] \\in [\\Z_{n}^0]$ such that $r$, the minimal element of $[r]$, is the greatest common divisor of $i$ and $j$.\n\\end{Definition}\n\n\\begin{Remark}\nNotice that for $[i], [j], [r] \\in [\\Z_{n}^{0}]$ where $i,j,r$ are the minimal elements of $[i]$, $[j]$ and $[r]$ respectively, if $[r]$ is an atom of $[i]$ then, by Lemma~\\ref{Lemma:partialconverselemma}, $r|i$ and so if $i|j$ then $r|j$ and $[i]|[j]$. Further observe that for $[i] \\in [\\Z_{n}^{0}]$ either $[i]$ is an atom or $\\TOns{[i]}$ is a union of groups $\\TOns{[r]}$ for atoms $[r]$ of $[i]$. This is because  for any element $T \\in \\TOns{[i]}$, either $T$ is not an element of $\\TOns{[r]}$ for any $[r]<[i]$  and so $[i]$ is an atom, or there is a  minimal $[j] \\in [\\Z_{n}^0]$ such that $T \\in \\TOns{[j]}$ and so $[j]$ is an atom of $[i]$ by Lemma~\\ref{Lemma:partialconverselemma} and Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}.\n\\end{Remark}\n\n As a consequence of Lemma~\\ref{Lemma:partialconverselemma} we have the following result.\n\n\\begin{proposition}\\label{Proposition:partiallatticestructure}\nLet $1 \\le i, \\le j \\le n-1$ be integers such $i$ is the smallest element of $[i]$ and $j$ is the smallest element of $[j]$, then the following things hold:\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\TOns{[i]} \\le \\TOns{[j]}$ if and only if $[i]$ divides $[j]$ \\label{Proposition:partiallatticestructure order},\n\\item if $[r]$ is the lowest common multiple of $[i],[j]$, then  $[r]$ is minimal such that $\\gen{\\TOns{[i]}, \\TOns{[j]} } \\le \\TOns{[r]}$ \\label{Proposition:partiallatticestructure join},\n\\item if $[r]$, where $r \\in \\Z_{n}$ is the smallest element of $[r]$, is the greatest common divisor of $[i]$ and $[j]$   then $\\TOns{[i]} \\cap \\TOns{[j]} = \\TOns{[r]}$. \\label{Proposition:partiallatticestructure meet}\n\\end{enumerate}\n \n\\end{proposition}\n\\begin{proof}\n For part~\\ref{Proposition:partiallatticestructure order}, let $i, j$ be elements of $\\Z_{n} \\backslash \\{0\\}$ which are the minimal elements of sets $[i]$ and $[j]$ respectively. Suppose $[i]$ divides $[j]$. If $[i]$ is an atom then  $i|j$, and so  by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} we have that $\\TOns{i} \\le \\TOns{j}$, in particular $\\TOns{[i]} \\le \\TOns{[j]}$. If $[i]$ is not an atom, then $\\TOns{[i]}$ is the union of the groups $\\TOns{[r]}$ over all atoms $[r]$ of $[i]$. Therefore let $[r]$ be any atom of $[i]$, where $r$ is the smallest element of $[r]$, since $[i]$ divides $[j]$ then $r|j$ and so $\\TOns{r} \\le \\TOns{j}$ by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} once more. Since $[r]$ was an arbitrary atom of $[i]$, we conclude that $\\TOns{[i]} \\le \\TOns{[j]}$.  On the other hand suppose that $\\TOns{i} \\le \\TOns{j}$.If $[i]$ is an atom then $i|j$ and so $[i]$ divides $[j]$. If $[i]$ is not an atom, then let $[r]$ be any atom of $[i]$ where $r$ is the minimal element of $[r]$. Since $\\TOns{[i]} \\le \\TOns{[j]}$, we also have $\\TOns{[r]} \\le \\TOns{[j]}$ by definition of an atom of $[i]$. Now making use of the definition of an atom together with Lemma~\\ref{Lemma:partialconverselemma}, we have that $r|j$. Therefore, since $[r]$ was an arbitrarily chose atom of $[i]$, we conclude that $[i]$ divides $[j]$.\n \n Part~\\ref{Proposition:partiallatticestructure join} follows from Part~\\ref{Proposition:partiallatticestructure order} since for any $s \\in \\Z_{n}\\backslash\\{0\\}$ such that $\\gen{\\TOns{[i]}, \\TOns{[j]}}\\le \\TOns{[s]}$, then  $[i]$ and $[j]$ divide $[s]$. \n \n For Part~\\ref{Proposition:partiallatticestructure meet} let $[r]$ be the greatest common divisor of $[i]$ and $[j]$. First observe that if $[l]$ is an atom of $[r]$, where $l$ is the minimal element of $[l]$, then $l|i$ and $l|j$. Therefore $[r]$ divides $[i]$ and $[j]$ and by Part~\\ref{Proposition:partiallatticestructure order}, $\\TOns{[r]} \\le \\TOns{[i]} \\cap \\TOns{[j]}$. Now let $[s]$, with $s$ the minimal element of $[s]$ be  an atom of $[i]$  and $[j]$. This means that $s|i$ and $s|j$ and so $s|r$, since $r$ is the greatest common divisor of $i$ and $j$. By Part~\\ref{Proposition:partiallatticestructure order} once more, we conclude that $s$ is an atom of $[r]$ and so $\\TOns{[s]} \\le \\TOns{[r]}$. Now since every element $T \\in \\TOns{[i]} \\cap \\TOns{[j]}$ lies in some atom of both $[i]$ and $[j]$, we conclude that  $\\TOns{[i]} \\cap \\TOns{[j]} = \\TOns{[r]}$.\n\\end{proof}\n\nThe proposition above prompts the following question:\n\n\\begin{Question}\\label{Question:canjoinbereplacedwithubgroupgenerated}\nLet $n$ be a natural number and let $[i], [j] \\in [\\Z_{n}^0]$ is it true that $\\gen{\\TOns{[i]}, \\TOns{[j]}} = \\TOns{[r]}$ where $[r]$ is the lowest common multiple of $[i]$ and $[j]$? \n\\end{Question}\n\nThe proposition below addresses this question by showing that in the case where the map $\\rsig$ is unto the group of units of $\\Z_{n-1}$, then Question~\\ref{Question:canjoinbereplacedwithubgroupgenerated} has an affirmative answer.\n\n\\begin{proposition}\nLet $\\T{X}= \\T{TO}, \\T{O}$ and  suppose the map $\\rsig: \\XOn  \\to  \\Z_{n-1}$ is unto the group of units of $\\Z_{n-1}$. Then for $[i],[j] \\in [Z_{n}^0]$, $\\gen{\\XOns{[i]},\\XOns{[j]}} = \\XOns{[r]}$ for $[r]$ the lowest common multiple of $[i]$ and $[j]$.\n\\end{proposition}\n\\begin{proof}\nThe result is essentially a consequence  of the following claim, as we have not been able to find a reference for it, we also provide a proof.\n\n\\begin{claim}\\label{Claim:groupgeneratedbypairequallcmgroup}\nLet $n \\in \\N_{2}$ and consider the integer ring $\\Z_{n}$. Let  $\\Z_{n}^{*}$ denote the group of units, and for each $i \\in \\Z_{n}$ let $(\\Z_{n}^{*})_{i} := \\{ a \\in \\Z_{n}^{*} \\mid ai \\equiv i \\mod{n}\\} \\le \\Z_{n}^{*}$. Let $i,j \\in \\Z_{n}$ let $i_1 = \\gcd(i,n)$, $j_1 = \\gcd(j,n)$ and $r = lcm(i_1,j_1)$ then $ \\gen{ (\\Z_{n}^{*})_{i_1}, (\\Z_{n}^{*})_{j_1}} = \\gen{ (\\Z_{n}^{*})_{i}, (\\Z_{n}^{*})_{j}} = (\\Z_{n}^{*})_{r}$.\n\\end{claim}\n\\begin{proof}\nWe first observe that for $i \\in \\Z_{n}$ the group $(\\Z_{n}^*)_{i} = (\\Z_{n}^*)_{d}$ where $d = \\gcd(i, n)$. This is because for any $a \\in \\Z_{n}^{*}$ such that $ad \\equiv d \\mod{n-1}$ then $ai \\equiv i \\mod{n-1}$ as well. If, on the other hand $ai \\equiv i \\mod{n-1}$, then observe that, as, by Bezout's lemma, there is a $u \\in \\Z_{n}$ such that $iu \\equiv d \\mod{n-1}$, then $ad \\equiv d \\mod{n-1}$ also.\n\nThus let $i, j \\in \\Z_{n}$ be divisors of $n$ with $r = lcm(i,j)$ (a divisor of $n$). First assume that $n = p^{\\alpha}$ for a prime $p$. Then $i = p^{\\beta}$ and $j = p^{\\gamma}$ for $\\beta, \\gamma \\le \\alpha$. Without loss of generality we may assume that $i \\le j$, and so, in particular, that $i$ divides $j$. However, it then follows that $(\\Z_{n}^{*})_{i} \\le (\\Z_{n}^{*})_{j}$, from which we conclude that $\\gen{ (\\Z_{n}^{*})_{i}, (\\Z_{n}^{*})_{j}} = (\\Z_{n})^{*}_{j}$ noting that $lcm(i,j) = j$.\n\nNow suppose that $n= p_1^{\\alpha_{1}}p_2^{\\alpha_{2}}\\ldots p_m^{\\alpha_{m}}$ where the  $p_{l}$'s, $1 \\le l \\le m$ are distinct primes. We may further assume that $\\gcd(i,j) = 1$. This is because if $\\gcd(i,j) = d$, then setting $i_1 = i\/d$ and $j_1 = j\/d$ we observe that $(\\Z_{n}^{\\ast})_{i_1} \\le (\\Z_{n}^{\\ast})_{i}$, $(\\Z_{n}^{\\ast})_{j_1} \\le (\\Z_{n}^{\\ast})_{j}$ and $lcm(i,j) = lcm(i_1, j_1)$. Thus, noting that $\\gen{ (\\Z_{n}^{\\ast})_{i}, (\\Z_{n}^{\\ast})_{j}} \\le \\gen{(\\Z_{n}^{\\ast})_{lcm(i,j)}}$, the result holds for $i_1$ and $j_1$ precisely if it holds for $i$ and $j$.  \n\nMaking use of the Chinese remainder theorem, there is a ring isomorphism from $\\Z_{n} \\to \\Z_{p_1^{\\alpha_{1}}} \\times \\Z_{p_2^{\\alpha_{2}}} \\times \\ldots \\times \\Z_{p_m^{\\alpha_{m}}}$ defined by $k \\mapsto (k\\mod{p_1^{\\alpha_1}},k\\mod{p_2^{\\alpha_1}}, \\ldots, k\\mod{p_m^{\\alpha_m}})$. For $1 \\le l \\le m$ let $i_l = i \\mod{p_l^{\\alpha_{m}}}$ likewise define the sequence $j_{l}$ and $r_l$, where $r = ij$. It  follows that $(\\Z_{n}^{\\ast})_{i} \\cong (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{i_1} \\times (\\Z_{p_1^{\\alpha_{2}}}^{\\ast})_{i_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{i_m}$, likewise $(\\Z_{n}^{\\ast})_{j} \\cong (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_1} \\times (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{j_m}$. Observe that $$\\gen{\\left((\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{i_1} \\times (\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{i_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{i_m}\\right),\\left((\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_1} \\times (\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{j_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{j_m}\\right) }$$ is precisely the group  $$\\gen{(\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{i_1}, (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_1}} \\times \\gen{(\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{i_2}, (\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{j_2}} \\times \\ldots \\times \\gen{(\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{i_m}, (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{j_m}}.$$ Since $i$ and $j$ are divisors of $n$ then $i = p_1^{\\beta_1}p_2^{\\beta_2} \\ldots p_{m}^{\\beta_m}$, $j= p_1^{\\gamma_1}p_2^{\\gamma_2} \\ldots p_{m}^{\\gamma_m}$ where for $1 \\le l \\le m$, $0 \\le \\beta_l, \\gamma_{l} \\le \\alpha_{l}$ and $\\gamma_{l}+ \\beta_{l} = \\max\\{\\gamma_{l}, \\beta_{l}\\}$ (since $i,j$ are assumed co-prime). It therefore follows that $i \\mod p_l^{\\alpha_{l}}$ is either a unit  of $\\Z_{p_l^{\\alpha_{l}}}$ otherwise it is  equal to a unit of $\\Z_{p_l^{\\alpha_{l}}}$ times a non-trivial (appropriate) power of $p_l$ and likewise for $j$. We note that if one of $i_l$ or $j_l$ is equal to  a unit of $\\Z_{p_l^{\\alpha_{l}}}$ times a non-trivial (appropriate) power of $p_l$, then the other must be equal to a unit of $\\Z_{p_l^{\\alpha_{l}}}$. Therefore for $1 \\le l \\le m$, $\\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$ is either equal to the trivial group or it is equal to $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{p_l^{\\delta}}$ where $p_l^{\\delta} \\ne 1$ is the maximum power of $p_l$ dividing $ij$. In the case that $\\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$ is the trivial group, then $i_l$ and $j_l$ are both units of $\\Z_{p_l^{\\alpha_{l}}}$ and so  $r_l = i_lj_l \\mod p_{l}^{\\alpha_{l}}$ is also a unit of $\\Z_{p_l^{\\alpha_{l}}}$. In the case that $\\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$ is equal to $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{p_l^{\\delta}}$ where $p_l^{\\delta} \\ne 1$ is the maximum power of $p_l$ dividing $ij$, then it follows that one of $i_l$ or $j_l$ is a unit and the other is equal to a unit times $p_{l}^{\\delta}$. In this case we have that $r_l$ is equal to a unit time $p_{l}{\\delta}$ and so $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{p_l^{\\delta}} = (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{r_l}$. In either case we see that $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{r_l} = \\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$. Thus we conclude that that $\\gen{(\\Z_{n}^{\\ast})_{i}, (\\Z_{n}^{\\ast})_{j}} = (\\Z_{n}^{\\ast})_{r}$ as required.\n\\end{proof}\n\nWe can now prove the proposition. We observe that for $1 \\le r < n$ since $\\rsig$ is surjective, then $(\\XOnr)\\rsig = (\\Z_{n-1}^*)_{r}$. Let $1 \\le i,j \\le  n-1$ such that $i$ the the minimal element of $[i]$ and $j$ is the minimal element of $[j]$ so that both $i$ and $j$ are divisors of $n$. Let $r = lcm(i,j)$. By Claim~\\ref{Claim:groupgeneratedbypairequallcmgroup} we have that $\\gen{(\\Z_{n-1}^*)_{i}, (\\Z_{n-1}^*)_{j}} = (\\Z_{n-1}^*)_{r}$. Thus, let $T \\in \\XOnr$ be any element. By Proposition~\\ref{Proposition:sigdeterminesmembership} $(T)\\rsig \\in (\\Z_{n-1}^*)_{r}$ and so there are elements $u \\in (\\Z_{n-1}^*)_{i}$ and $v \\in (\\Z_{n-1}^*)_{j}$ such that $(T)\\rsig = uv \\mod{n-1}$. Let $U \\i \\XOns{i}$ and $V \\in \\XOns{j}$ be such that $u= (U)\\rsig$ and $v = (V)\\rsig$, then $(TU^{-1}V^{-1})\\rsig = 1$ and so $TU^{-1}V^{-1} \\in \\XOns{1} \\le  \\gen{\\XOns{i}, \\XOns{j}}$. It therefore follows that $\\XOns{r}\\le \\gen{\\XOns{i}, \\XOns{j}}$. Proposition~\\ref{Proposition:sigdeterminesmembership} guarantees that $\\gen{\\XOns{i}, \\XOns{j}} \\le \\XOns{r}$ Thus by Proposition~\\ref{Proposition:partiallatticestructure} $[r]$ is the lowest common multiple of $[i]$ and $[j]$.\n\\end{proof}\n\n \n \nWe conclude our investigation of the nesting properties of the groups $\\TOns{r}$ for $1 \\le r < n$ by making  use of Proposition~\\ref{Proposition:sigdeterminesmembership} to construct an element of $\\T{TO}_{4}$ which is not an element of $\\T{TO}_{4,r}$ for any $1 \\le r \\le 3$. This indicates that in general the group $\\TOns{r}$ depends on $r$.\n\nLet $g$ be the transducer below:\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,node distance=3cm,on grid,auto] \n   \\node[state] (q_0)   {$a$};\n   \\node[state, xshift=0.5cm] (q_1) [right=of q_0] {$b$}; \n    \\path[->] \n    (q_0) edge[in=170, out=10]  node {$1|0$, $3|1$} (q_1)\n          edge[in=180, out=150,loop] node[swap] {$0|0$} ()\n          edge[in=180, out=210, loop] node {$2|1$} () \n    (q_1) edge[in=350, out=190] node {$0|2$, $2|3$} (q_0)\n          edge[in=0, out=30,loop] node {$3|3$}()\n          edge[in=0, out=330,loop] node[swap] {$1|2$}();\n\\end{tikzpicture}\n\\caption{An element  $g \\in \\T{TO}_{4}$ which is not in $\\T{TO}_{4,r}$ for any $0 \\le r <3$}.\n\\label{Figure:elementinTo4butnotinanyotherTOnr}\n\\end{figure}\n\n\nWe make the following observations about $g$ which the reader may verify:\n\\begin{enumerate}[label=(\\roman*)]\n\\item  For all states $\\alpha$ of $g$ the map $g_{\\alpha}: \\CCn \\to \\CCn$ preserves the lexicographic ordering on $\\CCn$. Thus, since $g^2 = \\id$ under the product defined for $\\TOn$, by Theorem~\\ref{Thm: Equivalent conditions for an element of $On$ to belong to TOnr} there is some non-zero $r \\in \\Z_{4}\\backslash$ such that $g \\in \\TOnr$. \\label{Observation:gisanelementofTOn}\n\\item We have $\\im(a) \\cap \\im(b) = \\emptyset$ and $\\im(a)\\sqcup\\im(b) = \\CCmr{4}$. In particular we  have $\\im(a) = U_{0} \\cup U_{1}$ and $\\im_{b} = U_{2}\\cup U_{3}$.\\label{Observation:gneedstostatestofillacone}\n\n\\item  The following fact is not essential to our discussion here but is nevertheless worth mentioning. The paper forthcoming article \\cite{BleakCameronOlukoya} shows that there is subgroup $\\T{L}_{n} \\le \\On$ which is isomorphic to the quotient $\\aut{\\Xn^{\\Z}, \\sigma_{n}}\/\\gen{\\sigma_{n}}$ of the automorphisms of the two-sided full shift on $n$ letters by the group generated by the shift ($\\sigma_{n}$ denotes the shift map on $n$ letters). The transducer $g$ is in fact an element of $\\T{L}_{4}$. \n\\end{enumerate}\n\n\\begin{lemma}\nLet $1 \\le r \\le 3$ and let $A_{q_0} \\in \\TBmr{4}{r}$ be such that $\\core(A_{q_0})$ is equal to the transducer $g$ of Figure~\\ref{Figure:elementinTo4butnotinanyotherTOnr}, then $r = 3$.\n\\end{lemma}\n\\begin{proof}\n\nWe compute $(g)\\sig$. First observe that $g$ is synchronizing at level 1 and moreover the states of $g$ forced by $0$ and $2$, $q_{0}$ and  $q_{2}$ respectively are both equal to  $a$ and the states of $g$ forced by $1$ and $3$, $q_{1}$ and $q_3$ respectively are both equal to $b$. Further observe that $m_{a} = 2$ and $m_{b} = 2$ (where $m_{a}$ and $m_{b}$ are as in Definition~\\ref{Definition:signature}). Therefore $(g)\\sig = 2.4 = 8 \\equiv 2 \\mod 3$. The only number $r \\in \\Z_{4} \\backslash \\{0\\}$ such that $ 2r \\equiv r \\mod{n-1}$ is $3$. Therefore by Proposition~\\ref{Proposition:sigdeterminesmembership} we conclude that if $g \\in \\TBmr{4}{r}$ then $r$ must be equal to 3.\n\n\\begin{comment}\nLet $A_{q_0}$ be as in the statement of the lemma and $k \\in \\N$ be the synchronizing level of $A_{q_0}$. Let $Q_{g} \\subset Q_{A}$ be the states of $g$. Since $k$ is the synchronizing level of $A$, for any state $q \\in Q_{A}$ and any word $\\Gamma \\in \\Xnr^{k}$, $\\pi_{A}(\\Gamma, q) \\in Q_{g}$. Let $P_{IN}:= \\Xnr^{k+1}$ and let $P_{OUT} = \\{\\lambda_{A}(\\Delta, q_0) \\mid \\Delta \\in \\Xnr^{k+1} \\}$. Since, for $(i,j) \\in  \\{(0,1), (2,3)\\}$ and $\\alpha \\in \\{a, b\\}$, we have $\\pi_{g}(i, \\alpha) = a$, $\\pi_{g}(j, \\alpha) = b$ and $\\lambda_{A}(i, \\alpha) = \\lambda_{A}(j, \\alpha)$, then, for every element $\\Delta$ of $P_{OUT}$, there are two elements $\\Gamma i, \\Gamma j$, $(i,j) \\in  \\{(0,1), (2,3)\\}$, such that $ \\lambda_{A}(\\Gamma i, q_0) = \\lambda_{A}(\\Gamma i, q_0) = \\Delta$ and $(\\pi_{A}(\\Gamma i, q_0),\\pi_{A}(\\Gamma i, q_0)) = (a,b)$. Let $\\Delta \\in P_{OUT}$ and $\\Gamma i, \\Gamma j \\in P_{IN}$, $(i,j) \\in  \\{(0,1), (2,3)\\}$, be such that $ \\lambda_{A}(\\Gamma i, q_0) = \\lambda_{A}(\\Gamma i, q_0) = \\Delta$ and $(\\pi_{A}(\\Gamma i, q_0),\\pi_{A}(\\Gamma i, q_0)) = (a,b)$. By point~\\ref{Observation:gneedstostatestofillacone} in the list of observations about $g$ and since $A_{q_0}$ represents a homeomorphism of $\\CCnr$, it must be the case that, for $\\Theta \\in P_{IN}$ such that $\\Theta \\ne \\{\\Gamma i, \\Gamma j\\}$, $\\lambda_{A}(\\Theta, q_0)$ is incomparable to $\\Delta$. Now since $A_{q_0}$ is a homeomorphism and $g$ is synchronous, then for any element of $\\rho \\in \\CCnr$ there is an element $\\nu \\in P_{OUT}$ such that $\\nu \\le \\rho$. Therefore, we deduce that $P_{OUT}$ is in fact a complete antichain for $\\Xnrs$ moreover, $2|P_{OUT}| = |P_{IN}|$. Since any complete antichain $P$ of $\\Xnrs$ satisfies $|P| \\equiv r \\mod{3}$, and since $P_{IN}$ is a complete antichain of $\\Xnrs$,  we have $2r \\equiv r \\mod{3}$. Hence $r \\equiv 0 \\mod{3}$ and, since $1 \\le r \\le 3$, we conclude that $r  =3$.\n\\end{comment}\n\\end{proof}\n\nIn the next section we address the question of the surjectivity of the homomorphism $\\rsig$.\n\n\\section{On the surjectivity of \\texorpdfstring{$\\rsig$}{Lg}}\\label{Section:onsurjectivityofrsig}\n\nIn this section we show that there are infinitely many numbers $n \\in \\N$ such that the  map $\\rsig$ from $\\On$ to the group of units of $\\Z_{n-1}$ is surjective. More specifically, for $n \\in \\N$ we show that for every  divisor $d$  of $n$, $1 \\le d <n$, there is an element $T \\in \\On$ with $(T)\\rsig = d$. We then argue that there are infinitely many numbers $n \\in \\N$ such that the divisors of $n$ generate the group of units of $\\Z_{n-1}$.\n\nWe first define some subgroups of $\\On$ highlighted in the paper \\cite{BCMNO} and outline an algorithm, `the collapsing procedure', given in  that paper for determining whether or not an automata (or transducer) is synchronizing.\n\n\\begin{Definition}\nLet $\\T{L}_{n}$ be the subgroup of $\\On$ consisting of all elements of $T \\in \\On$, such that if $q \\in Q_{T}$ and $\\Gamma \\in \\Xns$ satisfies, $\\pi_{T}(\\Gamma, q) = q$, then $|\\lambda_{T}(\\Gamma, q)| = |\\Gamma|$. Let $\\hn{n}$ be the subgroup of $\\T{L}_{n}$ consisting of synchronous, invertible transducers. \n\\end{Definition}\n\nLet $A$ be an automaton, the collapsing procedure constructs a new automaton $A_1$ as follows. For each $q \\in Q_{A}$ set $[q] := \\{p \\in Q_{A} \\mid \\pi_{A}(x, p) = \\pi_{A}(x, q) \\mbox{ for all } x \\in \\Xn\\}$ and set $Q_{A_1}:= \\{ [q] \\mid q \\in Q_{A}\\}$. Let $\\pi_{A_1}: \\Xn \\times Q_{A_1} \\to Q_{A_1}$ be defined by  $\\pi_{A}(x, [q])  = [\\pi_{A}(x,q)]$ for all $x \\in \\Xn$ and let $A_1= \\gen{\\Xn, Q_{A_1}, \\pi_{A_1}}$. Let $A = A_0$, $A_1, A_2 \\ldots $ be a sequence of automaton, where $A_{l+1}$, for $l \\in \\N$ is obtained from $A_{l}$ by applying the collapsing procedure. We observe that for $l \\in \\N$ either $|A_{l}| > |A_{l+1}|$ or $A_{l} = A_{l+1}$. Thus let $k$ be minimal in $\\N$ such that $A_{k} = A_{k+1}$. The automaton $A$ is synchronizing if and only if $A_{k}$ is the single state automaton over $\\Xn$ (\\cite{BCMNO}). If $T = \\gen{\\Xn, Q_{T}, \\pi_{T}, \\lambda_{T}}$ is a transducer, then $T$ is synchronizing if and only if the automaton  $\\T{A}(T) = \\gen{\\Xn, Q_T, \\pi_T}$ is synchronizing. Typically we when applying the collapsing procedure, we shall not distinguish between  $T$ and $\\T{A}(T)$.\n\nWe have the following general construction, which is in some sense an extension of a construction given in the paper \\cite{Olukoya1} for embedding direct sums of the group of automorphisms of one-sided shift over alphabets sizes summing to $n$ into the group of automorphisms of the one-sided shift on $n$ letters.\n\nLet $n \\in \\N_{2}$ and $d$ be a divisor of $n$ not equal to $n$. Let $m \\in \\N$ be such that $md =n$ and fix an element $T \\in \\hn{d}$. For $0\\le i \\le m-1$, partition the set $\\Xn$ into sets $X_{n,i}:= \\{di, di+1, \\ldots, d(i+1)-1 \\}$, likewise form sets $Q_{T,i} = \\{ q(i) \\mid q \\in Q_{T}\\}$. Form a transducer $\\oplus_{d}T = \\gen{ \\Xn, \\cup_{0 \\le i \\le m-1} Q_{T,i}, \\pi_{\\oplus_{d}T}, \\lambda_{\\oplus_{d}T}}$ with transition and output defined such that:\n\n\\begin{enumerate}[label=(\\arabic*.)]\n\\item the restriction $\\oplus_{d}T\\restriction_{Q_{T,i}} = \\gen{X_{n,i}, Q_{T,i},\\pi_{\\oplus_{d}T}\\restriction_{Q_{T,i}}, \\lambda_{\\oplus_{d}T}\\restriction_{Q_{T,i}}}$ of $\\oplus_{d}T$ to the set of states $Q_{T,i}$ is a transducer equal to $T$ up to relabelling the alphabet and the set of states,\n\n\\item for $0 \\le i,j \\le m-1$ such that $i \\ne j$, for any $b \\in X_{d}$, and for any $q \\in Q_{T}$,  $\\pi_{\\oplus_{d}T}(dj +b, q(i)) = (q_{b})(j)$, where $\\pi_{T}(b, q_{b})= b$, and  $\\lambda_{\\oplus_{d}T}(dj+b, q(i)) = di+b$.\n\n\\end{enumerate}\n\n\nWe have the following result:\n\n\\begin{proposition}\\label{Propositon:forncompositeforeverydivisorofnthereisanelementwithsigequaltodivisor}\nLet $n \\in \\N_{2}$, $d$ be a divisor of $n$ such that $n= md$ for some $m \\in \\N_{2}$, and $T \\in \\hn{d}$. Then, the transducer $\\oplus_{d}T$ is bi-synchronizing, and is fact an element of $\\Ln{n}$. Moreover, for $0 \\le i \\le m-1$, and any state $q(i) \\in Q_{T,i}$, $\\bigcup_{0 \\le b \\le d-1} U_{di+b}=\\im(q_i)$. In particular, $(\\oplus_{d}T)\\rsig = d$.\n\\end{proposition}\n\\begin{proof}\nWe begin by arguing that the transducer  $\\oplus_{d}T$ is synchronizing. However, this follows straight-forwardly from the observation that for $0 \\le i,j \\le m-1$ such that $i \\ne j$, any pair of states $q(i), p(i) \\in  Q_{T,i}$, and any $0 \\le b < d$, $\\pi_{\\oplus_{d}T}(dj+b, q(i)) = \\pi_{\\oplus_{d}T}(dj+b, p(i)) = (q_b)(i)$. Thus since, each transducer $\\oplus_{d}(T)\\restriction_{Q_{T,i}}$ for $0 \\le i \\le m-1$ at the same level $k \\in \\N$, then after applying the collapsing procedure at most $k+1$ times we have reduced $\\oplus_{d}T$ to the single state automaton.\n\nFurther observe that as $T$ is in fact a synchronous transducer, then if $T \\in \\On$ it is in fact an element of $\\Ln{n}$.\n\nTherefore in order to show that $\\oplus_{d}T \\in \\Ln{n}$ it suffices to  show that each state is injective and has clopen image and that it has a synchronizing inverse.\n\n We begin by showing that each state of $\\oplus_{d}T$ is injective and has clopen image. Let $0 \\le i \\le m-1$ and $q(i)$ be a state of $T$. Let  $\\mu, \\nu \\in \\Xnp$ such that $|\\mu| = |\\nu|  \\ge 2$ and $\\mu \\ne \\nu$. We now show that either $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$ or for any $x \\in \\Xn$, $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$. If $\\mu, \\nu \\in X_{n,i}^{+}$, then this follows from the fact that $T \\in \\hn{n}$ and so each state of $T$ induces a permutation of $X_{d}$. Thus, we may assume that there are $1 \\le j_1, j_2  \\le m-1$ with $j_1$ and $j_2$ not both equal to  $i$, $a_1 \\in X_{n, j_1}$, $a_2 \\in X_{n, j_2}$, and  $\\mu_1, \\nu_1 \\in \\Xn^{+}$ such that $\\mu = a_1 \\mu_1$ and $\\nu= a_2 \\nu_1$. If $j_2 \\ne j_2$, then as $\\pi_{\\oplus_{d}T}(a_1, q(i))  \\in Q_{T, j_1}$ and $\\pi_{\\oplus_{d}T}( a_2, q(i)) \\in Q_{T, j_2}$, it follows that the first letter of $\\lambda_{\\oplus_{d}T}(\\mu_1, \\pi_{\\oplus_{d}T}(a_1, q(i)))$ is an element of $X_{n, j_1}$ and the first letter of $\\lambda_{\\oplus_{d}T}(\\nu_1, \\pi_{\\oplus_{d}T}(a_2, q(i)))$ is an element of $X_{n, j_2}$. Thus $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$. Therefore we may assume that $j_1 = j_2 = j$. In this case, since $\\mu \\perp \\nu$, there are words $\\mu_2, \\nu_2, \\phi \\in \\Xn^{\\ast}$ and $a \\ne b \\in \\Xn$ such  that $\\mu= \\phi a \\mu_2$ and $\\nu = \\phi b \\nu_2$. If $a, b \\in X_{n,l}$ for some $0 \\le l \\le m-1$, then since for any state $t \\in \\cup_{0\\le i \\le m-1} Q_{T,i}$, $\\lambda_{\\oplus_{d}T}(a,t) \\ne \\lambda_{\\oplus_{d}T}(b, t)$, we have $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$. Thus, we may assume that $a \\in X_{n, l_1}$, $b \\in X_{n,l_2}$, by adding a letter to  $\\mu_2$ and $\\nu_2$ if necessary, we may assume they are non-empty. In this case, the fact that $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$ follows from the observation above that for any state $t \\in \\cup_{0\\le i \\le m-1} Q_{T,i}$, the first letter of $\\lambda_{\\oplus_{d}T}(\\mu_2, \\pi_{\\oplus_{d}T}(a,t))$ is an element of $X_{n,l_1}$ and the first letter of $\\lambda_{\\oplus_{d}T}(\\nu_2, \\pi_{\\oplus_{d}T}(b,t))$ is an element of $X_{n,l_2}$. Therefore, since $T$ is synchronous, each state must induce an injective map from $\\CCn$ to itself. \n \n Let $q(i) \\in Q_{T,i}$ be an arbitrary state of $T$, $b \\in \\Z_{d}$ be arbitrary. We now show, by induction, that $U_{di+b} \\subset \\im(q_i)$. \n \n Let $j \\in \\Z_{m}$ and $a \\in \\Z_{d}$ be arbitrary so that $jd +a \\in X_{n,j} \\subset \\Xn$ is arbitrary. There is a letter $x \\in X_{n,j}$ such that $\\lambda_{\\oplus_{d}T}(x, q(i)) = di+b$ and $\\pi_{\\oplus_{d}T}(x, q(i)) \\in Q_{T, j}$. Since $\\pi_{\\oplus_{d}T}(x, q(i)) \\in Q_{T, j}$, by construction, for any $l \\in \\Z_{m}$, there is a letter $y \\in X_{n,l}$ such that $\\lambda_{A}(y, \\pi_{\\oplus_{d}T}(x, q(i)))  = jd +a$ and $\\pi_{\\oplus_{d}T}(y, \\pi_{\\oplus_{d}T}(x, q(i))) \\in Q_{T,l}$. Thus, for any $j \\in \\Z_{m}$, $a \\in \\Z_{d}$  and any $l \\in   \\Z_{m}$, there is a word $xy \\in \\Xn^{2}$ such that $\\pi_{\\oplus_{d}T}(xy, q(i)) \\in Q_{T,l}$ and $\\lambda_{\\oplus_{d}T}(xy, q(i)) = (di+b) (jd+a)$.\n \n Assume by induction that there for any $\\nu \\in \\Xn^{M}$ and any $l \\in \\Z_{m}$, there is a word $\\mu \\in \\Xn^{M+1}$, such that $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) = (di+b) \\nu$ and $\\pi_{\\oplus_{d}T}(\\mu, q(i)) \\in Q_{T,l}$. Let $\\xi \\in \\Xn^{M+1}$ be arbitrary. Let $j \\in \\Z_{m}$, $a \\in \\Z_{d}$ be such that $ \\xi = \\chi(jd + a)$ for some $\\chi \\in \\Xn^{M}$. By the inductive assumption, there is a word $\\zeta \\in \\Xn^{M+1}$ such $\\lambda_{\\oplus_{d}T}(\\zeta, q(i)) = (di+b) \\chi$ and $\\pi_{\\oplus_{d}T}(\\zeta, q(i)) = p(j) \\in Q_{T,j}$. Now by construction, for any $l \\in \\Z_{m}$, there is a word $y \\in X_{n,l}$ such that $\\lambda_{\\oplus_{d}T}(y, p(j)) = dj + a$ and  $\\pi_{\\oplus_{d}T}(y, p(j)) \\in Q_{T,l}$. Therefore, we have that for any $l \\in \\Z_{m}$, there is a $y \\in X_{n,l}$ such that $\\lambda_{\\oplus_{d}T}(\\zeta y, q(i)) = (di+b) \\chi (dj+a)$ and  $\\pi_{\\oplus_{d}T}(\\zeta y, q(i)) \\in Q_{T,l}$. Therefore we see, by transfinite induction, that $U_{di+b} \\subset \\im(q_i)$. Therefore every state of $T$ is injective and has clopen image.\n \n We now show that $\\oplus_{d}T$ has an inverse in $\\On$. We do this by showing that for any transducer $A_{q_0}$, with $\\core(A_{q_0}) = \\oplus_{d}T$, then set of states of $A_{(\\epsilon, q_0)}$ which are reached by arbitrarily long words, form a synchronizing transducer. To this end, fix a transducer $A_{q_0}$ with $\\core(A_{q_0}) = \\oplus_{d}T$. \n \n Let $0 \\le i \\le m-1$ and $q(i) \\in  Q_{T,i}$, we observe that for $a \\in \\Z_{d}$, $(di + a)L_{q(i)} = \\epsilon$. Set $Q_{U} = \\{ (di + a, q(i)) \\mid i \\in \\Z_{m}, a \\in \\Z_{d} \\}$. Define  $\\lambda_{U}:  \\Xn \\times Q_{U} \\to \\Xn$ by $\\lambda_{U}(dj+ b, (di+a, q(i))) = ((di+a)(dj+b))L_{q(i)}$ for all $i, j \\in \\Z_{m}$ and $a,b  \\in \\Z_{d}$. Further define $\\pi_{U}: \\Xn \\times Q_{U} \\to Q_{U}$ by $\\pi_{U}(dj+ b, (di+a, q(i))) = ((di+a)(dj+b))L_{q(i)} = ( (di+a)(dj+b) - \\lambda_{A}(((di+a)(dj+b))L_{q(i)}, q(i)), \\pi_{A}(((di+a)(dj+b))L_{q(i)}, q(i)))$. Now, observe that $((di+a)(dj+b))L_{q(i)}$ is precisely the element $x \\in X_{n,j}$ such that $\\lambda_{\\oplus_{d}T}(x, q(i)) = di +a$, thus if $p(j) = \\pi_{\\oplus_{d}T}(x, (q(i)))$, then $\\pi_{U}(dj+ b, (di+a, q(i))) = (dj+b, p(j))$. Let $U = \\gen{ \\Xn, Q_{U}, \\pi_{U}, \\lambda_{U} }$. We now show that $U$ is synchronizing, then we argue that the set of states of $A_{(\\epsilon, q_0)}$ reached by arbitrarily long words forms a transducer precisely equal to $U$.\n \n We partition the sets of $U$ as follows: for $0 \\le i \\le m-1$ and $a \\in  \\Z_{d}$, set $Q_{U,i,a}:= \\{ (di+ a, q(i)) \\mid  q(i) \\in Q_{T,i} \\}$. Fix $0 \\le i \\le m-1$ an $a \\in \\Z_{d}$. \n \n Let $0 \\le j \\le n-1$ such that $i \\ne j$ and $(di+ a, q(i)), (di+ a, p(i)) \\in Q_{U,i,a}$  and $dj +b$, $b \\in \\Z_{d}$, be  arbitrary. We observe that, by construction, if $x, y \\in X_{n,j}$ satisfy $\\lambda_{\\oplus_{d}T}(x, q(i)) = \\lambda_{\\oplus_{d}T}(y, p(i)) = di+a$, then $x = y = dj +a$. Observe, moreover, that $\\pi_{\\oplus_{d}T}(dj + a, q(i)) = \\pi_{\\oplus_{d}T}(dj + a, p(i)) = (q_{a})(j)$. Therefore it follows that $\\pi_{U}(dj+ b, (di+a, q(i))) = (dj+b, (q_a)(j)) = \\pi_{U}(dj+ b, (di+a, p(i)))$.  Thus we conclude that for any $0 \\le i \\le m-1$ and any $a \\in \\Z_{d}$, the set of elements $Q_{U,i,a} := \\{ (di+a, q(i)) \\mid q(i) \\in Q_{T,i}\\}$ transition identically on all elements of $X_{n,j}$ for $j \\ne i$. \n \n \n \n  Now let, $T^{-1} = \\gen{ \\Xn, Q_{T^{-1}}, \\pi_{T^{-1}}, \\lambda_{T^{-1}}}$, a synchronous and synchronizing transducer,  denote the inverse of $T$ and denote by $(a)q^{-1}$, for $a \\in  \\Z_{d}$, the element of $X_{d}$  such that $\\lambda_{T}((a)q^{-1}, q) = a$. Consider a letter $di +c$, we observe that $\\pi_{U}(di + c, (di+a,q(i))) = (di + c, \\pi_{\\oplus_{d}T}(di+(a)q^{-1}, q(i)))$. Note moreover that $\\pi_{T^{-1}}(a, q^{-1}) = (\\pi_{T}((a)q^{-1}, q))^{-1}$. Thus, $$(di + c, \\pi_{\\oplus_{d}T}(di+(a)q^{-1}, q(i))) = (di+c, (\\pi_{T}((a)q^{-1}, q))^{-1}(i) ) = (di+c, (\\pi_{T^{-1}}(a, q^{-1}))^{-1}(i)). $$ A simple induction argument now shows that given a word $\\nu = v_1 v_2 \\ldots v_{l} \\in X_{d}^{+}$, with corresponding word $\\nu(i) = (di + v+1) (di+v_2) \\ldots  (di+ v_{l}) \\in X_{n,i}^{+}$, if $\\pi_{T^{-1}}(\\nu, q^{-1}) = p^{-1}$, for a state $q \\in Q_{T}$, then, setting $\\nu(i)_{1} = (di+v_2) \\ldots  (di+ v_{l})$,  $\\pi_{U}(\\nu(i)_{1}(di+b), (di+v_1,q(i))) = (di+b, p(i))$. Let $k$ be the synchronizing level of $T^{-1}$, it therefore follows that, all elements of  $Q_{U,i,a}$ transition identically on all words in $X_{n,i}^{k}$.\n \n Finally, since by an observation above we have that for any state $(dj+b, q(j)) \\in Q_{U}$, and any letter $dl +c \\in X_{n}$ $\\pi_{U}(dl+c,(dj+b, q(j))) \\in Q_{U,l,c}$, it therefore follows that $U$ is synchronizing at level $k+1$. \n \n\n\nTo conclude the proof we have to demonstrate that $(\\oplus_{d}T)^{-1} = U$. We do this by showing that $A_{(\\epsilon, q_0)}$ is synchronizing and has core equal to $U$. Let $q(i)$ be an arbitrary state of $\\oplus_{d}T$. Since $A_{q_0}$ is synchronizing, there is a word $\\Gamma \\in \\Xn^{+}$ such that $\\Delta = \\lambda_{A}(\\Gamma, q_0) \\ne \\epsilon$ and $\\pi_{A}(\\Gamma, q_0) = q(i)$. It therefore follows that, for $0 \\le a \\le d-1$, $(\\Gamma (di + a))L_{q_0}= \\Delta (di+a)L_{q}$ and so $\\pi'_{A}(\\Gamma (di+a), (\\epsilon,q_0)) = (di+a, q(i))$. Thus we see that $A_{(\\epsilon, q_0)}$ is synchronizing and has $\\core(A_{(\\epsilon, q_0)}) = U$. Therefore $T^{-1} = U$. This concludes the proof.      \n\\end{proof}\n\n\\begin{corollary}\nLet $n \\in \\N_{2}$, then the image of the map $\\rsig$ from  $\\On$ to $\\Z_{n-1}^{*}$, the group of units of $Z_{n-1}$, contains the subgroup of $\\Z_{n-1}^{*}$ generated by the divisors of $n$. \\qed  \n\\end{corollary} \n\\begin{corollary}\nLet $n \\in \\N_{2}$ be such that the group of units of $\\Z_{n-1}$ is generated by the divisors of $n$, then the map $\\rsig: \\On \\to \\Z_{n-1}^{*}$ is surjective. \\qed\n\\end{corollary} \n\n\\begin{corollary}\nThere are infinitely many number $n \\in \\N$ such that the map $\\rsig$ from $\\On$ to the group of units of $Z_{n-1}$ is surjective.\n\\end{corollary}\n\\begin{proof}\nIt is an elementary result in number theory that for an $m \\ge 1$, $2$ generates the group of units of $3^{m}$, thus we may take $n = 3^{m} + 1$.\n\\end{proof}\n\nThus, we are left with the question:\n\n\\begin{Question}\nfor $n \\in \\N_{2}$ and $p$ in the group $\\Z_{n-1}^{\\ast}$ of units of $\\Z_{n-1}$ such that $p$ is not an element of the subgroup of $\\Z_{n-1}$ generated by the divisors of $n$, is there an element $T \\in \\On$ such that $(T)\\rsig = p$? \n\\end{Question}\n\nWe observe that in order to answer this question, it suffices to show that for any element $p \\in \\Z_{n-1}$ co-prime both to $n$ and $n-1$, there is an element $T \\in \\On$ such that $(T)\\rsig = p$.\n \n\n\\begin{comment}\nWe address this question below. \n\nLet $n \\in \\N_{2}$, let $p \\in \\Z_{n-1}$ be co-prime both to $n$ and $n-1$. Since $p$ is co-prime to $n$ and $n-1$, we may find an integer $q \\in \\N$ such that  $n + q(n-1) \\equiv 0 \\mod{p}$. Let $d \\in \\N$ be such that $n + q(n-1) = dp$. Set $\\ac{u}:= \\{(n-1)^q a \\mid a \\in \\Xn\\} \\cup \\{ (n-1)^i a \\mid 0 \\le i < q, a \\in \\Xn\\backslash\\{n-2\\} \\}$ a complete antichain for $\\Xnp$. We recall that antichains are ordered lexicographically, thus for $u \\in \\ac{u}$, set $(u)\\iota = j \\in X_{dp}$  if and only if $(u)\\iota$ is the $j+1$'st element of  $\\ac{u}$. As $ \\iota$ is defined on elements of $\\ac{u}$ we extend it to a bijection  from $\\ac{u}^{\\ast} \\to X_{dp}^{\\ast}$, and  from $\\ac{u}^{\\omega} \\to X_{dp}^{\\omega}$. We write $\\iota^{-1}$ for the inverse of $\\iota$. Let $\\shift{n}: \\Xnz \\to \\Xnz$ be defined by $(x_{i})_{i \\in \\Z} \\mapsto (y_i)_{i \\in \\Z}$ where $y_{i} = x_{i+1}$ i.e. $\\shift{n}$ is the usual shift map on $\\Xnz$. Let $\\shift{dp}: X_{dp}^{\\Z} \\to X_{dp}^{\\Z}$ be defined analogously. \n\nLet $\\psi \\in \\Ln{dp}$ be a fixed element such that $\\psi$ is synchronous and let $k \\in \\N$ be the synchronizing level of $\\psi$. Then $\\psi$ defines a shift commuting homeomorphism map from  $X_{dp}^{\\Z}$ to itself by a sequence $x \\in X_{dp}^{\\Z}$ maps to the sequence $y$ defined by $y_i = \\lambda_{\\psi}(x_i, q_{x_{i+k}x_{i+k-1}\\ldots x_{i-1}})$ where $q_{x_{i+k}x_{i+k-1}\\ldots x_{i-1}}$ is the unique state in the image of the map $\\pi(x_{i+k}x_{i+k-1}\\ldots x_{i-1}, \\centerdot): Q_{\\psi}\\to Q_{\\psi}$. \n\nAs we now need also to read words in the opposite orientation we set-up some notation for this. For a word $w = w_1w_2\\ldots w_{|w|}$ over an finite alphabet let $\\rev{w} =w_{|w|}w_{|w|-1}\\ldots w_2w_1$. \n \nNow fix a right infinite sequence $x:=x_0x_1 \\ldots \\in \\Xn^{\\Z}$ and we assume that for some $l > k(q+1)$ $x_l \\ne n-1$. We further assume that the sequence $x_{l-1}x_{l-2}\\ldots x_0$ can be decomposed into a sequence $u_{l'}u_{l'-1}\\ldots u_{0}$ such that $x_0 x_1 \\ldots x_{l-1} = \\rev{u_{0}}\\rev{u_{1}}\\ldots \\rev{u_{l'}}$. We observe that as we have assumed this decomposition exists, it is unique. Also observe that $l'$ must be at least $k$ since $|u_{i}| \\le q+1$ for all $0 \\le i \\le l'$. Let $v = \\phi(u_{l'}u_{l'-1}\\ldots u_{0}) \\in X_{dp}^{l'}$.\n\n Define a map $h_{\\psi,x}: \\CCn \\to \\CCn$ as follows. Let $\\delta \\in \\CCn$ and write $\\delta = \\delta_{-1}\\delta_{-2} \\ldots$. There is a unique sequence $u_{-1}, u_{-2} \\ldots$ of elements of $\\ac{u}$ such that $\\delta_{-1} \\delta_{-2} \\ldots = u_{-1} u_{-2} \\ldots$. Thus  $ \\ldots\\delta_{-3}\\delta_{-2}\\delta_{-1}x_0x_1 \\ldots x_{l-1} =     \\ldots\\rev{u_{-2}}\\rev{u_{-1}}\\rev{u_{0}}\\rev{u_{1}}\\ldots \\rev{u_{l'}}$. Let $v_{i} = (u_i)\\iota$ for $i \\in \\{-1,-2,\\ldots\\}$. Then $\\ldots \\rev{v_2}\\rev{v_1}\\rev{v}$ is a left infinite sequence in $X_{dp}$. Let $\\rho_1\\rho_2\\ldots=\\rho = \\lambda_{\\psi}(v_1v_2\\ldots, q_{v})$. Set $(\\delta)h_{\\psi,x} = (\\rho)\\iota^{-1}$.\n \n\nWe have the following claim:\n\n\\begin{claim}\nThe map $h_{\\psi, x}$ is injective, has finitely many local  actions and is synchronizing. Furthermore, for any local action $g$ of $h_{\\psi,x}$, $\\im(g)$ is clopen, and if $W \\subset \\Xnp$ is minimal such that $\\cup_{\\eta \\in W}U_{\\eta} = \\im(g)$, then $|W| \\equiv (\\psi)\\rsig \\mod{n-1}$.\n\\end{claim}\n\\begin{proof}\nLet $\\nu \\in \\Xnp$ be a word of length $> k(q+1)$ and let $\\delta \\in \\CCn $ be arbitrary. We observe that as $\\nu$ has length at least $k(q+1) +1$ then $\\nu = u_{1}u_{2} \\ldots u_{m} \\phi$ for some $u_{i} \\in \\ac{u}$, $m \\ge k$, and $\\phi \\in \\Xns$ such that $\\phi$ is a prefix of some element of $\\ac{u}$. Let $\\phi\\delta = u_{m+1}u_{m+2}\\ldots$. Then, $\\nu\\phi\\delta = u_{1}u_{2}\\ldots$ and this decomposition is unique. Let  $v_i = (u_i)\\iota$ for $i \\in \\N_{1}$ and let $\\rho = v_1v_2 \\ldots\n \\in X_{dp}^{\\omega}$. By definition we have $(\\nu\\phi\\delta)h_{\\psi,x} = (\\lambda_{\\psi}(\\rho, q_{v}))\\iota^{-1}$. Let $v_{m-k+1}v_{m-k+1}\\ldots v_{m}$ be the length $k$ suffix of $v_{1}v_{2}\\ldots v_{m}$ and let $\\rho'$ be such that $v_{1}\\ldots v_{m} \\rho' = \\rho$, then $\\lambda_{\\psi}(\\rho, q_{v}) = \\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}) \\lambda_{\\psi}(\\rho', q_{v_{m-k+1}v_{m-k+1}\\ldots v_{m}})$. Thus  we have that $$(\\nu\\phi\\delta)h_{\\psi,x} =  (\\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}))\\iota^{-1}(\\lambda_{\\psi}(\\rho', q_{v_{m-k+1}v_{m-k+2}\\ldots v_{m}}))\\iota^{-1}.$$\n \n We now observe that the greatest common prefix of the set $(U_{\\nu}) h_{\\psi,x}$ is precisely $(\\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}))\\iota^{-1}$ since  $\\psi$ has no states of incomplete response, $\\iota$ is a bijection and the greatest common prefix of  $\\ac{u}$ is $\\epsilon$. Now  by a similar argument we also have that, setting $\\nu' = u_{m-k+1}u_{m-k+2}\\ldots u_{m}$,: \n \n $$(\\nu'\\phi\\delta)h_{\\psi,x} =  (\\lambda_{\\psi}(v_{m-k+1}v_{m-k+2}\\ldots v_{m}, q_{v}))\\iota^{-1}(\\lambda_{\\psi}(\\rho', q_{v_{m-k+1}v_{m-k+1}\\ldots v_{m}}))\\iota^{-1}.$$\n \n Thus, we see that $(\\nu\\phi\\delta)h_{\\psi,x} = (\\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}))\\iota^{-1} (\\phi\\delta)(h_{\\psi,x})_{\\nu'}$. In particular, the local actions $h_{\\nu}$ and  $h_{\\nu'}$ are equal and so the local action $h_{\\nu \\phi}$ is equal to $h_{\\nu'\\phi}$. Therefore we conclude that $h_{\\phi, x}$ has finitely many local actions. This argument also demonstrates that $h_{\\psi,x}$ is synchronizing, since the local action of a long enough word is determined by  some uniformly bounded suffix.\n \n That $h_{\\psi,x}$ is injective, follows from the facts that $\\iota$ is a bijection and all states of $\\psi$ induce injective function.\n \n\\end{proof}\n\nThe above claim means that $h_{\\psi,x}$ can be represented by a synchronizing transducer $A_{q_0}$, set $\\Psi:= \\core(A_{q_0})$, we now need to argue that $\\Psi$ is bi-synchronizing. We begin with the lemma below.\n\\end{comment}\n\\begin{comment}\nWe now define a map $\\varPsi: \\Xn^{\\Z} \\to X_{dp}^{\\Z}$ in the following way. Fix $x:=(x_i)_{i \\in  \\Z} \\in \\Xn^{\\Z}$. We define a sequence $y:=(y_i)_{i \\in \\Z} \\in X_{dp}^{\\Z}$ as follows.\nFirst suppose that $(x_i)$ has no occurrence of a symbol in the set $\\{0,1,\\ldots, n-2\\}$. In this case $x$ is the sequence such that $x_i = n-1$ for all $i \\in \\Z$. In this case $y$ is the sequence such that $y_i = dp-1$ for all $i \\in \\Z$. We set $(x)\\varPsi = y$.\n\nThus we may assume that $(x_i)$ has an occurrence of a symbol in $\\{0,1,\\ldots, n-2\\}$. Let $l \\in \\Z$ be minimal in absolute value such that  $x_{l}$ is an element of $\\{0,1,\\ldots, n-2\\}$.  Let $x_{<l} = x_{l-1}x_{l-2}x_{l-3}\\ldots$ and $x_{\\ge l} =  x_{l}x_{l+1}x_{l+2} \\ldots$. There are unique sequences $u_{-1} u_{-2}u_{-3}\\ldots$, and $u_0u_1 \\ldots$ such that $u_{i} \\in \\ac{u}$ for all $i \\in \\Z$ and  $ u_{-1}u_{-2}u_{-3} = x_{< l}$ and $u_{0}u_{1}u_{2}\\ldots = x_{\\ge l}$. Let $u = \\ldots u_{-2}u_{-1}u_{0}u_{1}u_{2}\\ldots$. Let $m \\in \\Z$ be such that $u_{m}$ contains $x_0$  and $v =  \\ldots v_{-1}v_0v_{1}\\ldots$ where $v_{i} = u_{i+m}$ for all $i \\in \\Z$. By replacing $x=(x_i)_{i \\in \\Z}$ with a shift of itself if necessary, we may assume that $v_0 =  x_{-|v_0|}\\ldots x_{-1}x_0$. Let $y \\in X_{dp}^{\\Z}$ be defined by $y_{i} = (v_i)\\iota$. For all $M \\in \\Z$ we set $((x)\\sigma^{M})\\varPsi = (y)\\sigma^{M}$. \n\\end{comment}\n\n\n\n\\section{The group $\\TOnr$ and $\\TBnr$ for $n>2$ and $1\\le r < n$ contain an isomorphic copy of Thompson's group $F$}\\label{Section:OutTncontainscopyofF}\n\nWe now show that for $n >2$, $n \\in N$, and for all valid $r$, the group $\\TOns{r}$ is infinite. This extends the result of Brin and Guzm{\\'a}n stating that the group $\\out{\\mathcal{T}_{n,n-1}}$ is infinite whenever $n>3$ (\\cite{MBrinFGuzman}) to the groups $\\out{\\Tnr}$. We observe that the group $\\out{T_{2}}$ is the cyclic group of order $2$ and so finite. By Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1} it suffices to demonstrate  that $\\TOns{1}$ is infinite. First we have the following straight-forward lemma.\n\n\\begin{lemma}\\label{Lemma:possessingahomeostatemeanslementsofTOn1}\nLet $T \\in \\TOn$ be an element with a homeomorphism state, then $T \\in \\TOns{1}$.\n\\end{lemma}\n\\begin{proof}\nLet $q \\in Q_{T}$ be an homeomorphism state of $T$, then the initial transducer $T_{q}$ is an element of $\\TBmr{n}{1}$.\n\\end{proof}\n\nTherefore to show that $\\TOns{1}$ is infinite, for $n>2$, it suffices to find an element of infinite order of $\\TOn$ which has a homeomorphism state.\n\nLet $T$ be the  element of $\\TOn$ depicted in Figure~\\ref{Figure:elementinTon1ofinfiniteorder}. In this figure we use the symbol $x$  to represent an element of $\\Xn$ strictly greater than zero and less than or equal to $n-2$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,node distance=3cm,on grid,auto] \n   \\node[state] (q_0)   {$a$};\n   \\node[state, xshift=4cm] (q_1) {$b$}; \n   \\node[state, xshift=2cm, yshift=-2cm] (q_2) {$c$};\n    \\path[->] \n    (q_0) edge[in=170, out=10] node {$n-1|n-1n-1$}  (q_1)\n          edge[in=105, out=75,loop] node[swap] {$x|n-1x$} ()\n          edge[in=180, out=265] node[swap] {$0|\\epsilon$} (q_2) \n    (q_1) edge[in=350, out=190] node {$x|x$}(q_0)\n          edge[in=90, out=120,loop] node {$0|0$}()\n          edge[in=90, out=60,loop] node[swap] {$n-1|n-1$}()\n    (q_2) edge[in=275, out=170] node[swap] {$x|x$} (q_0)\n          edge[in=270, out=0] node[swap, yshift=0.2cm] {$n-1|n-10$} node {$0|0$} (q_1);\n\\end{tikzpicture}\n\\caption{An element  $T \\in \\T{TO}_{n,1}$ of infinite order.}\n\\label{Figure:elementinTon1ofinfiniteorder}\n\\end{figure}\n\nThe states $a$ and $b$ of $T$ are homeomorphism states, and so $T$ is in fact an element of $\\TOns{1}$.\n\nIn order to show that $T$ has infinite order we make use of the action of the group $\\On$, as introduced in the paper \\cite{BCMNO}, on the space $\\Xnz\/ \\gen{\\sigma_{n}}$ where $\\sigma_{n}$ is the shift map on $\\Xnz$. Let $U \\in \\On$  and suppose that $k \\in \\N$ is the minimal synchronizing level of $U$. The action of $U$ on $\\Xnz\/\\gen{\\sigma_{n}}$ is given as follows: let $y= \\ldots y_{-1}y_{-1}y_0y_1y_2\\dots \\in \\Xnz$ represent an equivalence class of $\\Xnz\/\\gen{\\sigma_{n}}$, the image of this class under $U$ is the equivalence class of the bi-infinite word $w$ defined as follows: for $i \\in \\Z$, let $y(i+1,k) = y_{i+1}y_{i+1}\\ldots y_{i+k}$ and $q_{y(i+1,k)}$ be the state of $U$ forced by $y(i+1,k)$, then set $w_i = \\lambda_{U}(y_i,  q_{y(i+1,k)}$ and let $w = \\ldots w_{-2}w_{-1}w_{-1}w_0w_1w_2\\ldots$.  Consider the bi-infinite word $z:=\\ldots(x n-1)(x n-1)\\ldots$ where $ x \\in \\Xn \\backslash \\{0,n-1\\}$, then $(z)T = \\ldots(xn-1n-1)(xn-1n-1)\\ldots$, $(z)T^2 = \\ldots x(n-1)^3x(n-1)^3\\ldots$, and $(z)T^{i} = \\ldots x(n-1)^{i+1}x(n-1)^{i+1}\\ldots$. Therefore we see that $z$ is on an infinite orbit under the action of $T$, demonstrating that $T$ is an element of infinite order. We have shown the following:\n\n\\begin{Theorem}\\label{Theorem:Tnrinfinitewhenevernbiggerthan2}\nFor $n>3$ and $1 \\le r \\le n-1$, the group $T_{n,r}$ is infinite.\n\\end{Theorem}\n\nLet $U$ be the  following transducer where, once more, $n >2$ and $x \\in \\Xn$ is any element strictly bigger than $0$ and strictly less than $n-1$:\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,on grid,auto] \n   \\node[state] (q_0)   {$p$};\n   \\node[state, xshift=4cm] (q_1) {$q$}; \n   \\node[state, xshift=2cm, yshift=-4.3cm] (q_2) {$s$};\n   \\node[state, xshift=2cm, yshift=-2cm] (q_3) {$t$};\n    \\path[->] \n    (q_0) edge[in=170, out=10] node {$0|0$}  (q_1)\n          edge[in=105, out=75,loop] node[swap] {$x|x$} ()\n          edge[in=195, out=185] node[swap] {$n-1|n-1$} (q_2) \n    (q_1) edge[in=350, out=190] node {$x|n-1x$}(q_0)\n          edge[in=10, out=260] node[swap] {$0|\\epsilon$}(q_3)\n          edge[in=0, out=0] node{$n-1|(n-1)^2$}(q_2)\n    (q_2) edge[in=200, out=180] node[swap] {$x|x$}(q_0)\n          edge[in=270, out=240,loop] node[swap, yshift=0.1cm] {$0|0$}()\n          edge[in=270, out=300,loop] node[yshift=0.1cm] {$n-1|n-1$}()\n    (q_3) edge[in=275, out=170] node[swap] {$x|x$} (q_0)\n          edge node[swap] {$n-1|n-10$} node {$0|0$} (q_2);\n\\end{tikzpicture}\n\\caption{An element  $U \\in \\T{TO}_{n,1}$ of infinite order. }\n\\label{Figure:ThetransducerUinTOn1}\n\\end{figure}\n\nThe state $p$ of $U$ is a homeomorphism state and  moreover $U$ is an element of $\\TOns{1}$ of infinite order.\n\n\n\n\nIn order to state our next result we require the following notion and result from \\cite{BCMNO}.\n\n\\begin{Definition}\nGiven an element $g = \\gen{\\Xn, Q_{g}, \\pi_g, \\lambda_g} \\in \\On$ a state $q$ of $g$ is called a \\emph{loop state} if there is some $i \\in \\Xn$ such that $\\pi_{g}(i, q) = q$.\n\\end{Definition}\n\nThe following lemma is proven in \\cite{BCMNO}:\n\n\\begin{lemma}\\label{Lemma: reading loops give bijection from set of Xns to Xns}\nLet $g = \\gen{\\Xn, Q_g, \\pi_g, \\lambda_g} \\in \\On$ and let $w \\in \\Xnp$ then there is a unique state $q_{w} \\in Q_{g}$ such that $\\pi_{g}(w, q_w) = q_w$. Moreover for any periodic equivalence class of the tail equivalence $\\sim_{t}$ with minimal period $w$ there exists a unique  $j \\in \\N_{1}$ and a unique circuit, decorated by some prime word $v$, in $A$ with output $w$. \n\\end{lemma}\n\n\n\\begin{lemma}\\label{Lemma:TandUgenerateF}\nThe elements $T$ and $U$ of $\\TOns{1}$ depicted in Figures~\\ref{Figure:elementinTon1ofinfiniteorder} and \\ref{Figure:ThetransducerUinTOn1} generate a subgroup of $\\TOns{1}$ isomorphic to R. Thompson's group $F$.\n\\end{lemma}\n\\begin{proof}\nFirst we make some observations about $T$ and $U$. Let $q$ be a state of $T$ or of $U$,let $x \\in \\Xn \\backslash \\{0, n-1\\}$, and let $(D,d) \\in \\{(T,a),(U,p)\\}$, then all transitions $\\pi_{D}(x,q)  = d$ and $\\lambda_{D}(x,q)$ end in the symbol $x$. The output of all other transitions in $T$ or $U$ do not involve the symbol $x$. Therefore the transducers $A$ and $B$ below are in fact sub-transducers of $T$ and  $U$ respectively.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,node distance=3cm,on grid,auto] \n   \\node[state] (q_0)   {$a$};\n   \\node[state, xshift=4cm] (q_1) {$b$}; \n   \\node[state, xshift=2cm, yshift=-2cm] (q_2) {$c$};\n   \\node[state, xshift = 8cm] (p_0)   {$p$};\n   \\node[state, xshift=12cm] (p_1) {$q$}; \n   \\node[state, xshift=8cm, yshift=-2cm] (p_2) {$s$};\n   \\node[state, xshift=12cm, yshift=-2cm] (p_3) {$t$};\n    \\path[->] \n    (q_0) edge[in=170, out=10] node {$n-1|n-1n-1$}  (q_1)\n          edge[in=180, out=265] node[swap] {$0|\\epsilon$} (q_2) \n    (q_1) edge[in=90, out=120,loop] node {$0|0$}()\n          edge[in=90, out=60,loop] node[swap] {$n-1|n-1$}()\n    (q_2) edge[in=270, out=0] node[swap, yshift=0.2cm] {$n-1|n-10$} node {$0|0$} (q_1)\n    (p_0) edge[in=170, out=10] node {$0|0$}  (p_1)\n          edge node[swap] {$n-1|n-1$} (p_2) \n    (p_1) edge node {$0|\\epsilon$}(p_3)\n          edge node[swap, xshift=0.5cm]{$n-1|(n-1)^2$}(p_2)\n    (p_2) edge[in=270, out=240,loop] node[swap, yshift=0.1cm] {$0|0$}()\n          edge[in=270, out=300,loop] node[yshift=0.1cm] {$n-1|n-1$}()\n    (p_3) edge[out=190, in=350] node[swap] {$n-1|n-10$} node {$0|0$} (p_2);\n\\end{tikzpicture}\n\\caption{The subtransducers $A$ and $B$ of $T$ and $U$ respectively.}\n\\label{Figure:subtransducersAandB}\n\\end{figure}\n\nNotice that $A_{a}$ and $B_{p}$ induce self-homeomorphisms of the Cantor space $\\{0,n-1\\}^{\\omega}$ equal, respectively, to the restrictions of the homeomorphisms $T_{a}$ and $U_{p}$ to the space $\\{0,n-1\\}^{\\omega}$. It is not hard to see that $F \\cong \\gen{A_{a}, B_{p}} \\le H(\\{0,n-1\\}^{\\omega})$. In fact $A_{a}^{-1}$ and $A_{a}^{-2}B_{p}A_{a}$ are the standard generators for $F$ acting on the space $\\{0,n-1\\}^{\\omega}$. Moreover $A_{a}$ and $B_{p}$ satisfy the defining relations for $F$, and so we see that $\\gen{A_{a}, B_{p}} = \\gen{A_{a}, B_{p}\\mid [B_{p}^{-1}A_{a}, A_{a}B_{p}A_{a}^{-1}], [B_{p}^{-1}A_{a}, A_{a}^{2}B_{p}A_{a}^{-2}]}$. Moreover $T_{a}^{-1} \\restriction_{\\{0, n-1\\}^{\\omega}} = A_{a}^{-1}$ and  $U_{p}^{-1}\\restriction_{\\{0, n-1\\}^{\\omega}} = B_{p}^{-1}$. We also observe (one can verify this by direct computation) that the transducers $T^{-1}$ and $U^{-1}$ representing the inverses of $T$ and $U$ in $\\TOn$ respectively, have states $a^{-1}, b^{-1}$ of $T^{-1}$ and $p^{-1}, s^{-1}$ of $U^{-1}$ such that $T_{a^{-1}}^{-1} = T_{a}^{-1}$, $T_{s^{-1}}^{-1} = T_{s}^{-1}$, $U_{p^{-1}}^{-1} = U_{p}^{-1}$ and $U_{s^{-1}}^{-1} = U_{s}^{-1}$. \n\nWe now define a homomorphism from $\\gen{T, U}$ to $\\gen{A_{a}, B_{p}}$. To do this we make the following observation. Let $W(T,U) = w_1 w_2 \\ldots w_l$ be a word in $\\{T,U,T^{-1},U^{-1}\\}^{+}$. For each letter $w_i$ of $W(T,U)$ let $\\alpha_{i}$ be the state of $w_i$ so that $\\alpha_{i}$ is the unique loop state of $x$ in $w_i$. Observe that, for all $1 \\le i \\le l$, $\\alpha_i \\in \\{a,a^{-1}, p, p^{-1}\\}$. Let $S_{W(T,U)}$ be the element of $\\TOns{1}$ representing the word $W(T,U) \\in \\gen{T,U}$. Under the product defined on  $\\TOns{1}$, we have that $S_{W(T,U)}$ is $\\omega$-equivalent to the core of the minimal transducer representing the product of the initial transducers $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$. However, since $\\alpha_i$ is the unique loop state state of $x$ in $w_i$, $\\lambda_{w_i}(x, \\alpha_i)$ ends in $x$ and for any state $q$ of $T$, $U$, $T^{-1}$ or $U^{-1}$, the resulting state when $x$ is read from $q$ is the appropriate element of the set $\\{a, a^{-1}, p, p^{-1}\\}$, we must have that the state $(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})$ is the unique loop state of $x$ in  the product $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$. Therefore, since the core is strongly connected, $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$ is equal to its core. Therefore if $s$ is the unique loop state of $x$ in $S_{W(T,U)}$, we must have that $(S_{W(T,U)})_{s}$ is $\\omega$-equivalent to $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$. Now let $D(A,B) = d_1d_2\\ldots d_l$ be the word in $\\{A_{a}, B_{p}, A_{a}^{-1}, B_{p}^{-1} \\}^{+}$ defined as follows, for all $1 \\le i \\le l$, if $w_i = T^{\\pm 1}$ then $d_i = A_{a}^{\\pm 1}$ and if $w_i = U^{\\pm 1}$ then $d_i = B_{p}^{\\pm 1}$. Let $C_{D(A,B)}$ be the element of $\\gen{A_{a}, B_{p}}$ representing the word $D_{(A,B)}$, then it follows that $C_{D(A,B)} = (S_{W(T,U)})_{s}\\restriction_{\\{0,n-1\\}^{\\omega}}$ since for all $i$, $(w_{i})_{\\alpha_{i}}\\restriction_{\\{0,n-1\\}^{\\omega}} = d_{i}$. Therefore the map $\\phi: \\gen{T,U} \\to \\gen{A_{a}, B_{p}}$ by $S \\mapsto S_{s}\\restriction_{\\{0,1\\}^{\\omega}}$, where $s$ is the unique loops state of $x$ in $S$, is a surjective homomorphism. \n\nIn order to conclude that $\\gen{T,U} \\cong \\gen{A_{a}, B_{p}}$ it suffices to show that the map $\\varphi$ which maps $A_{a}$ to $T$ and $B_{p}$ to $U$ extends to a group homomorphism $\\overline{\\varphi}: \\gen{T,U} \\to \\gen{A_{a}, B_{p}}$. To see this, it suffices to show that $[U^{-1}T, TUT^{-1}] =1$ and $[U^{-1}T, T^{2}UT^{-2}] =1$. This is easily verified by direct computation.\n\n\\begin{comment}\nWe now argue that $\\phi$ is injective. To do this we introduce a tree pair representation for the elements $T$, $U$ and $A$, $B$ that correspond to the products in $\\gen{T,U}$ and $\\gen{A_{a}, B_{p}}$. The pictures below deal with the case $n=3$ however for the case for $n>3$ is dealt with analogously, and in this case all leaves corresponding to a words ending in $x$, for $x \\in \\Xn \\backslash \\{0,n-1\\}$, have label $p$ attached in the range tree. \n\n\\begin{tikzpicture}\n\\Tree [.{}  [ {} {} {} ] {} {}  ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{T_{a}}{\\longrightarrow}$};\n\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{}  ${}_{b}$ ${}_{p}$ [ ${}_{b}$ ${}_{p}$ ${}_{b}$ ]  ]\n\\end{tikzpicture} \\qquad\n\\begin{tikzpicture}\n\\Tree [.{}  [ [ {} {} {} ] {} {} ] {} {}  ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{U_{p}}{\\longrightarrow}$};\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{}  [ ${}_{s}$ ${}_{p}$ [ ${}_{s}$ ${}_{p}$ ${}_{s}$ ] ] ${}_{p}$ [ ${}_{b}$ ${}_{p}$ ${}_{b}$ ]  ]\n\\end{tikzpicture}\n\n Notice that all leaves in the range trees corresponding to words in $\\Xn^{+}$ with final letter $0$ or $n-1$ have the state $b$ attached, while all leaves corresponding to words with final letter $x \\in \\Xn \\backslash \\{0,1\\}$ have the label $p$ attached. Deleting all leaves corresponding to words ending  in $x$, $x \\in \\Xn \\backslash \\{0,1\\}$, in the domain and range trees and deleting the labels $b$ from the resulting range tree. The resulting tree pairs represent the elements $A_{a}$ and $B_{p}$ of $H(\\{0,1\\}^{\\omega})$. Notice that as the states $b$ and $p$ of $T$ and the states $p$ and $s$ of $U$  are homeomorphism states, it follows that the tree pair  representing the maps $T_{a}^{-1}$ and $U_{p}^{-1}$ are as follows:\n\\begin{tikzpicture}\n\\Tree [.{}  ${}$ ${}$ [ ${}$ ${}$ ${}$ ]  ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{T_{a}^{-1}}{\\longrightarrow}$};\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{}  [ ${}_{b^{-1}}$ ${}_{p^{-1}}$ ${}_{b^{-1}}$ ] ${}_{p^{-1}}$ ${}_{b^{-1}}$  ]\n\\end{tikzpicture}\\qquad\n\\begin{tikzpicture}\n\\Tree [.{}  [ ${}$ ${}$ [ ${}$ ${}$ ${}$ ] ] ${}$ [ ${}$ ${}$ ${}$ ]  ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{U_{p}^{-1}}{\\longrightarrow}$};\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{}  [ [ ${}_{s^{-1}}$ ${}_{p^{-1}}$ ${}_{s^{-1}}$ ] ${}_{p^{-1}}$ ${}_{s^{-1}}$ ] ${}_{p^{-1}}$ ${}_{s^{-1}}$  ]\n\\end{tikzpicture}\n\nHence we see that after performing the operation of deleting the ignoring  all leaves corresponding to words in $\\Xn^{+}$ ending in a letter $x \\in \\Xn \\backslash \\{0,1\\}$ we see that the resulting tree pairs actually represent the elements $A^{-1}$ and $B^{-1}$. Now observe that since for $i \\in \\{0, n-1\\}$, $x \\in \\Xn \\backslash \\{0,1\\}$, and for $(E,e,t) \\in \\{ (T,b,a), (T^{-1}, b^{-1}, a^{-1}),(U,s, p), (U^{-1}, s^{-1},p^{-1}) \\}$, we have $\\pi_{S}(i,s) = s$ and, $\\pi_{S}(x,s) = t$. Therefore adding a caret to a leaf corresponding to a word ending in the symbol $0$ or $n-1$ in the domain tree, results in adding a caret with leaves labelled $e,t,e$ for $(e,t) \\in \\{ (b,a), (b^{-1}, a^{-1}), (s,p), (s^{-1},p^{-1})\\}$ to the corresponding  leaf  in the range tree. Let $D_{(A,B)} = d_1 d_2 \\ldots d_l$ be a word over the alphabet $(A_{a}, B_{p}, A_{a}^{-1}, B_{p^{-1}})$,  and let $\\delta_i \\in \\{a, a^{-1}, p, p^{-1}\\}$ be such that $d_i = D_{\\delta_i}$  where $D = T^{\\pm 1}$ if $\\delta_i = a^{\\pm 1}$ and $D = U^{\\pm 1}$ if $\\delta_i = p^{\\pm 1}$. Let $\\varepsilon_i$ be defined by $\\varepsilon_i = b^{\\pm 1}$ if $\\delta_i = a^{\\pm 1}$ and $\\varepsilon_i = s^{\\pm 1}$ if $\\delta_i = p^{\\pm 1}$. As noted above $D(A,B)$ corresponds to a word $W(T,U)$ over the alphabet $(T_{a}, U_{p}, T_{a}^{-1}, U_{p}^{-1})$. Now a given a tree pair $(F_1, F_2)$ representing the product of $D_{(A,B)}$ in $\\gen{A_{a}, B_{p}}$, we obtain a tree pair for  the product $W(T,U)$ initialised at the loop state for  $x$ as follows. Form finite $n$-ary tree $\\overline{F}_i$, $i \\in \\{1,2\\}$, by adding at each level the `missing leaves'. That is $F_{i}$ is the resulting tree obtained by deleting the leaves of $\\overline{F}_{i}$ corresponding to words in $X_{n}^{+}$ which do not end in the letters $0$ or $n-1$.  Decorate the leaves of $\\overline{F}_{2}$ as follows, all leaves corresponding to words in $\\Xn^{+}$ ending in $0$ or $n-1$ are decorated with the word $\\epsilon_1\\epsilon2\\ldots\\epsilon_l$ otherwise leaves are decorated with the symbol $\\delta_1\\delta_2\\ldots\\delta_i$. \n\nSince $\\epsilon_1\\epsilon2\\ldots\\epsilon_l$ is the unique loop state of $0$ and $n-1$ and acts as the identity on $\\Xn$, and since  $\\delta_1\\delta_2\\ldots\\delta_i$ is the loop state for $x$, we see that if the product $D(A,B)$ is trivial in $\\gen{A_{a}, B_{p}}$, then $W(T,U)$ initialised at the state  $\\delta_1\\delta_2\\ldots\\delta_i$ is also trivial. Now since the core is strongly connected, and $\\delta_1\\delta_2\\ldots\\delta_i$ is a state of the core, it follows that $W(T,U)$ is the identity transducer. \n\n\\end{comment}\n\\end{proof}\n\n\\begin{Remark}\\label{Remark:TaandUpgenerateF}\nNotice that, for $T$ and $U$ as in Figures~\\ref{Figure:elementinTon1ofinfiniteorder} and \\ref{Figure:ThetransducerUinTOn1}, the proof above shows that the subgroup $\\gen{T_{a}, U_{p}} \\le \\TBmr{n}{1}$, for $n \\ge 3$ is isomorphic to $F$. \n\\end{Remark}\n\nAs a corollary we have the following theorem generalising the result of Brin and Guzman (\\cite{MBrinFGuzman}) stating that the groups $\\out{T_{n,n-1}}$ and $\\aut{T_{n,n-1}}$ contain an isomorphic copy of $F$.\n\n\\begin{Theorem}\nFor $n \\ge 3$ and $1 \\le r \\le n-1$, the groups $\\TOnr \\cong \\out{T_{n,r}}$ and $\\TBnr \\cong \\aut{T_{n,r}}$ contain an isomorphic copy of Thompson's group $F$.\n\\end{Theorem}\n\\begin{proof}\nThat $\\TOnr$, for $n\\ge 3$ and $1 \\le r \\le n-1$, contains a group isomorphic to  $F$ is a consequence of Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}. To deduce implications for $\\TBnr$, let $T$ and $U$ be the transducers in Figures~\\ref{Figure:elementinTon1ofinfiniteorder} and \\ref{Figure:ThetransducerUinTOn1}. Observe that the map $f$ defined by, for all $\\dot{d} \\in \\dotr$ and $\\xi \\in \\CCn$, $\\dot{d}\\xi \\mapsto \\dot{d}(\\xi)T_{a}$, and the map $g$ given by, for all $\\dot{d} \\in \\dotr$ and $\\xi \\in \\CCn$, $\\dot{d}\\xi \\mapsto \\dot{d}(\\xi)U_{p}$, are elements of $\\TBnr$ since $\\core(f) = T$ and $\\core(g) = U$. Moreover, they generate a subgroup isomorphic to $F$ by Remark~\\ref{Remark:TaandUpgenerateF}. \n\\end{proof}\n\\begin{comment}\nLet $\\rot$ be the equivalence relation on $\\Xnp$ such that $u \\simrot v$ if and only if $u$ is equal to a rotation of $v$. That is if $v = v_1 v_2 \\ldots  v_{|v|}$, then, setting $v_0 = \\epsilon$, there is some $1 \\le i \\le |v|$ such that $u = v_i v-{i+1}\\ldots v_{|v|} v_{1}\\ldots v_{|i-1|}$. Let $\\RXnp := \\Xnp \/ \\simrot$. For a word $w \\in\\Xnp$, we denote by $\\rotclass{w}$ the equivalence class of $w$ under $\\simrot$. Let $\\sym(\\RXnp)$ be the symmetric group on $\\RXnp$. For each element $T$ of $\\TOn$,it is a result in \\cite{BCMNO} that for every word $w \\in \\Xnp$ there is a unique state $q_{w}$ of $T$ such that $\\pi_{T}(w, q_{w}) = q_{w}$. Let $\\overline{T}$ be the map from $\\RXnp \\to \\RXnp$ given by $u \\mapsto v$, where $v$ is the output of the unique state $q_u$ such that $\\pi_{T}(u,q_{u} )= q_u$. The following result is straight-forward.\n\n\\begin{proposition}\\label{Proposition:grouphomomorphismfromOntoSym}\nThe map $\\phi: \\TOn \\to \\sym(\\RXnp)$ is a group homomorphism.\n\\end{proposition}\n\\begin{proof}\nLet $w \\in \\Xnp$, and $T, U \\in \\TOn$. Suppose that $\\rotclass{v} = (\\rotclass{w})\\overline{T}$ for $v \\in \\Xnp$. This means $\\lambda_{T}(w, q_{w})$ is a rotation of $v$, for $q_{w} \\in Q_{T}$ satisfying $\\pi_{T}(w, q_w) = w$. Let  $p_{v}$ be the state of $B$ such that $\\pi_{U}(v, p_{v}) = p_{v}$. Observe that $(\\rotclass{v})\\overline{T}$ is a rotation of the word $\\lambda_{U}(v, p_{v})$. Therefore $(\\rotclass{w})\\overline{T}\\overline{U} = \\rotclass{\\lambda_{U}(v, p_v)}$. Now observe that the unique state of $TU$ \n\\end{proof}\n\nWe now show that $\\Ons{r}$ contains a subgroup isomorphic to the Thompson's group $F_{2}$ for all $1 \\le r \\le n-1$.\n\n\\begin{Theorem}\nThe groups $\\TOns{r}$ for $1 \\le r < n-1$ and $n >2$ contain a subgroup isomorphic to  Thompson's group $F_{2}$.\n\\end{Theorem}\n\\begin{proof}\nWe show that for $n>2$ the group $\\TOns{1}$ contains a subgroup isomorphic to $F_{2}$ the result then follows by Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}.\n\nWe construct an isomorphism $\\phi: F_2 \\to \\TOns{1}$ which maps $x_0$ and $x_1$ the standard generators for Thompson's group $F$ (see e.g. \\cite{CannFlydPar}) to transducers $X$ and $Y$ in $\\TOns{n-1}$ such that the $n$ loop state is a homeomorphism state, from this it follows that $X, Y \\in \\TOns{1}$. We then show that $X$ and $Y$ satisfy the relations of $F_2$, and that the map $\\phi$ is injective.\n\nNote that $x_1$ and $x_2$ are homeomorphisms of $\\CCn$ induced by the transducers below. We identify the transducers and the homeomorphism of $\\CCn$ they induce.\n\n\\begin{figure}[H]\n\\begin{center}\n   \\begin{tikzpicture}[shorten >=0.5pt,node distance=3cm,on grid,auto] \n      \\node[state, initial,initial text=] (q_0)   {$q_0$}; \n      \\node[state] (q_1) [xshift=-1cm, yshift=-2cm] {$q_1$}; \n      \\node[state] (q_2) [xshift=1cm, yshift=-2cm] {$q_2$};\n      \\node[state, initial,initial text=] (p_0)[xshift=4cm, yshift=0cm]  {$p_0$};\n      \\node[state] (p_1)[xshift=6cm, yshift=0cm]  {$p_1$};\n      \\node[state] (p_2) [xshift=8cm, yshift=0cm]  {$p_2$};\n      \\node[state] (p_3) [xshift=4cm, yshift=-2cm]  {$p_3$};\n      \\node [xshift=0cm, yshift=-3cm] {$x_0$};\n      \\node [xshift=6cm, yshift=-3cm] {$x_1$};\n       \\path[->] \n       (q_0)   edge  node [swap] {$1|\\epsilon$} (q_1)\n                    edge  node {$0|00$} (q_2)\n       (q_1) edge  node  {$0|01$} node[swap] {$1|1$}(q_2)     \n       (q_2) edge[in=15, out=345, loop]  node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n       (p_0)   edge  node {$1|1$} (p_1)\n               edge  node[swap] {$0|0$} (p_3)\n       (p_1)   edge  node[xshift=0cm] [swap] {$0|00$} (p_3)\n               edge  node {$1|\\epsilon$} (p_2)\n       (p_2) edge  node  {$0|00$} node[xshift=0.1cm, yshift=-0.3cm] {$1|1$}(p_3)     \n       (p_3) edge[in=15, out=345, loop]  node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ();\n   \\end{tikzpicture}\n   \\end{center}\n   \\caption{Transducers representing generators of Thompson's group $F$.}\n   \\label{Figure:transducersrepresentinggeneratorsofF}\n\\end{figure}\n\nWe now construct the element $X_0, X_1 \\in \\TOns{1}$ such that $(x_0)\\phi = X_0$ and $(x_1)\\phi = X_1$. Below transitions labelled by $a$ indicate how states of the transducer act on any element of $\\Xn \\backslash \\{ 0,1 \\}$.\n\n\\begin{figure}[H]\n\\begin{center}\n   \\begin{tikzpicture}[shorten >=0.5pt,node distance=3cm,on grid,auto] \n      \\node[state] (q_0)   {$q_0$}; \n      \\node[state] (q_1) [xshift=-1cm, yshift=-2cm] {$q_1$}; \n      \\node[state] (q_2) [xshift=1cm, yshift=-2cm] {$q_2$};\n      \\node[state] (p_0)[xshift=4cm, yshift=0cm]  {$p_0$};\n      \\node[state] (p_1)[xshift=6cm, yshift=0cm]  {$p_1$};\n      \\node[state] (p_2) [xshift=8cm, yshift=0cm]  {$p_2$};\n      \\node[state] (p_3) [xshift=4cm, yshift=-2cm]  {$p_3$};\n      \\node [xshift=0cm, yshift=-3cm] {$X_0$};\n      \\node [xshift=6cm, yshift=-3cm] {$X_1$};\n       \\path[->] \n       (q_0)   edge[in=80, out=200]  node[xshift=0cm, yshift=0cm] {$1|\\epsilon$} (q_1)\n               edge[out=340, in=100]  node[xshift=0cm, yshift=0cm,swap] {$0|00$} (q_2)\n               edge[in=105, out=75, loop] node[swap] {$a|a$} ()     \n       (q_1) edge  node  {$0|01$} node[swap] {$1|1$}(q_2)\n             edge[in=190, out=90] node {$a|0a$}  (q_0)     \n       (q_2) edge[in=15, out=345, loop]  node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n             edge[in=350, out=90] node[swap] {$a|a$} (q_0)\n       (p_0)   edge  node[swap] {$1|1$} (p_1)\n               edge[in=120, out=240]  node[swap] {$0|0$} (p_3)\n               edge[in=105, out=75, loop] node[swap] {$a|a$} ()\n       (p_1)   edge[in=30, out=250]  node[xshift=0.2cm] [swap] {$0|00$} (p_3)\n               edge  node {$1|\\epsilon$} (p_2)\n               edge[in=10, out=170] node[swap] {$a|a$} (p_0)\n       (p_2) edge[in=20,out=230]  node  {$0|00$} node[xshift=0.1cm, yshift=-0.3cm] {$1|1$}(p_3)\n             edge[in=45, out=135] node[swap]{$a|0a$} (p_0)     \n       (p_3) edge[in=15, out=345, loop]  node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n             edge[in=250, out=110] node[swap]{$a|a$} (p_0);\n   \\end{tikzpicture}\n   \\end{center}\n   \\caption{Images of $x_0$ and $x_1$ under $\\phi$.}\n   \\label{Figure:embeddingFintoTOn}\n\\end{figure}\n\nObserve that since $X_0$ and $X_1$ have homeomorphism states, and since each stateThe inverses of $X_0$ and $X_1$ are given below:\n\n\\begin{figure}[H]\n\\begin{center}\n   \\begin{tikzpicture}[shorten >=0.5pt,node distance=3cm,on grid,auto] \n      \\node[state] (q_0)   {$q_0$}; \n      \\node[state] (q_1) [xshift=-1cm, yshift=-2cm] {$q_1$}; \n      \\node[state] (q_2) [xshift=1cm, yshift=-2cm] {$q_2$};\n      \\node[state] (p_0)[xshift=4cm, yshift=0cm]  {$p_0$};\n      \\node[state] (p_1)[xshift=6cm, yshift=0cm]  {$p_1$};\n      \\node[state] (p_2) [xshift=8cm, yshift=0cm]  {$p_2$};\n      \\node[state] (p_3) [xshift=4cm, yshift=-2cm]  {$p_3$};\n      \\node [xshift=0cm, yshift=-3cm] {$X_0^{-1}$};\n      \\node [xshift=6cm, yshift=-3cm] {$X_1^{-1}$};\n       \\path[->] \n       (q_0)   edge[in=80, out=200]  node[xshift=0cm, yshift=0cm] {$0|\\epsilon$} (q_1)\n               edge[out=340, in=100]  node[xshift=0cm, yshift=0cm,swap] {$1|11$} (q_2)\n               edge[in=105, out=75, loop] node[swap] {$a|a$} ()     \n       (q_1) edge  node  {$1|10$} node[swap] {$0|0$}(q_2)\n             edge[in=190, out=90] node {$a|1a$}  (q_0)     \n       (q_2) edge[in=15, out=345, loop]  node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n             edge[in=350, out=90] node[swap] {$a|a$} (q_0)\n       (p_0)   edge  node[swap] {$1|1$} (p_1)\n               edge[in=120, out=240]  node[swap] {$0|0$} (p_3)\n               edge[in=105, out=75, loop] node[swap] {$a|a$} ()\n       (p_1)   edge[in=30, out=250]  node[xshift=0.2cm] [swap] {$1|11$} (p_3)\n               edge  node {$0|\\epsilon$} (p_2)\n               edge[in=10, out=170] node[swap] {$a|a$} (p_0)\n       (p_2) edge[in=20,out=230]  node  {$1|10$} node[xshift=0.1cm, yshift=-0.3cm] {$0|0$}(p_3)\n             edge[in=45, out=135] node[swap]{$a|1a$} (p_0)     \n       (p_3) edge[in=15, out=345, loop]  node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n             edge[in=250, out=110] node[swap]{$a|a$} (p_0);\n   \\end{tikzpicture}\n   \\end{center}\n   \\caption{The inverses of $X_0$ and $X_1$ in $\\TOns{1}$}\n   \\label{Figure:embeddingFintoTOn}\n\\end{figure}\n\\end{proof}\n\\end{comment}\n\\section{Further properties of \\texorpdfstring{$\\TOn$}{Lg} and the \\texorpdfstring{$R_{\\infty}$}{Lg} property for \\texorpdfstring{$\\Tn$}{Lg} }\\label{Section:RftyTn}\n\nIn this section we deduce some properties of elements of $\\TOn$ which will enable us to demonstrate that the groups $\\Tnr$ for $1< r<n$ have the $R_{\\infty}$ property. The case $n=2$ has been dealt with in the paper \\cite{BMV}.\n\nThroughout this section we shall identify elements of $[0,r]$, $1 \\le r \\le n-1$, with their $n$-ary expansions.\n\n\n\\begin{Remark}\\label{Remark: homorphism from On into Sym(Xns)}\nA consequence of Lemma~\\ref{Lemma: reading loops give bijection from set of Xns to Xns} is that for $g \\in \\On$, and letting $\\mathcal{X}_{n}^{+}$ be the set of equivalences classes of $\\Xnp$ under the relation that two words are related if they are rotations of each other,  there is a bijection $\\bar{g}: \\RXnp \\to \\RXnp$ by $[w] \\mapsto  [\\lambda_g(w, q_{w})]$. Let $\\mbox{Sym}(\\RXnp)$ be the group consisting of bijections from the set $\\RXnp$ to itself. The map from  $\\On$ to $\\mbox{Sym}(\\RXnp)$ given by $g \\mapsto \\bar{g}$ is an injective homomorphism from $\\On$ into $\\mbox{Sym}(\\RXnp)$ (this follows from results in \\cite{BCMNO} and the forthcoming article \\cite{BleakCameronOlukoya}). We may fix a choice of representatives for the equivalence classes of $\\RXnp$ such that, for all $i \\in  \\Xn$, $[i] = i$.\n\\end{Remark}\n\nThe following Lemma explores properties of the map $\\bar{g}$ when $g \\in \\TOn$.\n\n\\begin{lemma}\nLet $g \\in \\WTOn$ then $0$ and $n-1$ are fixed by the map $\\bar{g}$.\n\\end{lemma}\n\\begin{proof}\nWe proceed by contradiction. We first consider the case that $(0)\\bar{g} \\ne 0$. By Lemma~\\ref{Lemma: reading loops give bijection from set of Xns to Xns} there is some prime word $w  \\in \\Xnp$ such that $(w)\\bar{g} = 0$. Let $q_w$ be the unique state such that $\\pi_{g}(w, q_w) = w$.  Then since $h_{q_{w}}$ preserves the lexicographic ordering on $\\CCn$ we must have that $(00\\ldots)h_{q_w}\\lelex (ww\\ldots)h_{q_{w}}$ however as $(ww\\ldots)h_{q_{w}} = 00\\ldots$ and  $h_{q_{w}}$ is injective, we have the desired contradiction.\n\nThe case $(1)\\bar{g} \\ne 1$ is dealt with analogously.\n\\end{proof}\n\nBy applying Lemma~\\ref{Lemma: multiplication by piR bijection from orientation preserving to reversing and vice versa} we immediately deduce the following result:\n\n\\begin{lemma}\\label{Lemma: elements of TOn either fix 0 or swap 0 and n-1}\nLet $g \\in \\TOn$ then if $g \\in \\WTOn$, $(0)\\bar{g} = 0$ otherwise $(0)\\bar{g} = n-1$.\n\\end{lemma}\n\nAs a corollary we have a different proof that for $A_{q_0} \\in \\TBnr$, $([0,r] \\cap  \\Z[1\/n] ) h_{q_0} = [0,r] \\cap  \\Z[1\/n]$.\n\n\\begin{corollary}\\label{Corollary:given hq0 in TBnr can assume that induced map on the Sr fixes 0 and r}\nLet $A_{q_0} \\in \\TBnr$ then $([0,r] \\cap  \\Z[1\/n] ) h_{q_0} = [0,r] \\cap  \\Z[1\/n]$\n\\end{corollary} \n\\begin{proof}\nObserve that since $A_{q_0}$ is synchronizing then after reading a sufficiently long string of $0$'s or $n-1$'s the next input if processed from the state $q_{0}$ or $q_{n-1}$ respectively. However these states either swap $0$ with $n-1$ and vice versa or fix $0$ and $n-1$ (depending on whether or not $A_{q_0}$ is orientation preserving or reversing) by Lemma~\\ref{Lemma: elements of TOn either fix 0 or swap 0 and n-1}. Hence since elements of $[0,r] \\cap  \\Z[1\/n]$ are precisely those elements of $\\CCnr$ of the form $w00\\ldots$ or $w n-1 n-1 \\ldots$ for $w \\in \\Xnrp$, the result follows.\n\\end{proof}\n\nFrom this we may deduce the following:\n\n\\begin{lemma}\\label{Lemma: automorphisms can be modified to contain fixed point}\nLet $A_{q_0} \\in \\TBnr$ then there is an element $h \\in \\Tnr$ such that $h_{q_0} \\circ h$ fixes the point $\\dot{0}0\\ldots \\simeq \\dot{r-1}n-1\\ldots$.\n\\end{lemma}\n\nThe following constructions demonstrate that $\\Tmr{n}{r}$ has the $R_{\\infty}$ property for any $1 \\le  r <  n$. and naturally breaks up into two cases. We first require the following standard lemma.\n\n\\begin{lemma}\\label{Lemma:changingcosetrepdoesnotaffectRinfty}\nLet $A_{q_0}, B_{p_0} \\in \\TBnr$ and $g \\in \\Tnr$ be such that $h_{q_0} g = h_{p_0}$ then $\\Tnr$ has infinitely many $h_{q_0}$-twisted conjugacy classes if and only if it has infinitely many $h_{p_0}$-twisted conjugacy classes. \n\\end{lemma}\n\\begin{proof}\nLet $f_1$ and $f_2$ be elements of $\\Tnr$ such that $f_1$ is $h_{p_0}$-twisted conjugate to $f_2$ by an element  $f_3$.  The computation below demonstrates that $f_1g^{-1}$ and $f_2g^{-1}$ are $h_{q_0}$-twisted conjugate to each other by the element $f_3$. \n\\[ f_{3}^{-1} f_1 g^{-1} h_{q_0}^{-1} f_3 h_{q_0} =  f_{3}^{-1} f_1 h_{p_0}^{-1} f_3 h_{p_0}g^{-1} = f_2g^{-1} \\]  \n\nIn a similar way we see that if $f_1'$ and $f_2'$ are elements of $\\Tnr$ such that $f_1'$ and $f_2'$  are $h_{q_0}$-twisted conjugate to each other by an element $f_3'$, then $f_1'g$ and $f_2'g$ are $h_{p_0}$ twisted conjugate to each other  by $f_3'$. \n\\end{proof}\n\nWe also observe that given $A_{q_0} \\in \\TBnr$,  by straight-forward manipulations, there are infinitely many $h_{q_0}$-twisted conjugacy classes if and only if there is an infinite subset $\\T{J}$  of $\\Tnr$ such that for any pair $f_1, f_2 \\in  \\T{J}$, $f_1h_{q_0}^{-1}$ is not conjugate to $f_2h_{q_0}^{-1}$ by an element of $\\Tnr$.\n\n{\\bfseries{Orientation preserving case:}} First let $A_{q_0} \\in \\TBmr{n}{1}$ be an orientation preserving element. By Lemma~\\ref{Lemma:changingcosetrepdoesnotaffectRinfty} and Lemma~\\ref{Lemma: automorphisms can be modified to contain fixed point} we may assume that $A_{q_0}$  fixes the points $00\\ldots$ and $n-1n-1\\ldots$ of $\\CCn$. Let $\\ac{u}$ be a complete antichain of $\\Xns$ of length $l >0$. Suppose that $\\ac{u} = \\{ u_0, u_1, \\ldots, u_{l-1} \\}$ (recall that antichains  are always assumed to be ordered lexicographically). We form a homeomorphism of $\\CCn$ as follows. Let $\\sigma: \\ac{u} \\to \\ac{ u}$ be given by $u_{i} \\mapsto u_{(i+1)\\mod{l}}$ for $1 \\le i \\le l-1$. Let $h_{l}: \\CCn \\to \\CCn$ be given by $(u_i\\delta) h_{l} = (u_{i})\\sigma (\\delta)h_{A_{q_0}}$. Then since $A_{q_0}$ is a homeomorphism of $\\CCn$, $h_{l}$ is also homeomorphism of $\\CCn$. Moreover, as $h_{q_0}$ is orientation preserving, and preserves the relation $\\simeqI$, by choice of the permutation $\\sigma$ it follows that $h_{l}$ is an orientation preserving element of $\\TBmr{n}{1}$. Moreover,  if $B_{p_0}$ is the transducer such that $h_{p_0} = h_{l}$, then $\\core(B_{p_0}) = \\core(A_{q_0})$. Thus  there is an element of $f \\in \\Tmr{n}{1}$ such that $h_{l}f = h_{q_0}$. Furthermore, observe that as $A_{q_0}$ fixes $00\\ldots$, we have that the point $00\\ldots$ is on an orbit of length precisely $l$ under $h_{l}$. Moreover, any point on a finite orbit under $h_{l}$ must have orbit length a multiple of $l$. Thus for every $l \\in \\mathbb{N}$ such that $l \\cong 1 \\mod{n-1}$, let $h_{l} \\in \\TBmr{n}{1}$ be the map constructed as above, and let $f_{l}$ be such that $h_{l} = f_{l}h_{q_0} $.  Therefore we  conclude that for any $A_{q_0} \\in \\widetilde{\\TBmr{n}{1}}$ there are infinitely may $h_{q_0}$-twisted conjugacy classes.\n\nNow fix $r>1$, and let $\\bar{r}$ be minimal such that $\\TOns{\\bar{r}} = \\TOns{r}$, by Lemma~\\ref{Lemma:partialconverselemma} $\\bar{r}$ divides $r$ and $\\bar{r}$ divides $n-1$. Let $A_{q_0} \\in \\TBmr{n}{\\bar{r}}$ be an element which fixes $\\dot{0}00\\ldots$ and $\\dot{r-1}n-1n-1\\ldots$. Let $\\ac{u}$ be a complete antichain of $X_{n,r}^{*}$ of length $l$. Observe that as $l \\cong r \\mod{n-1}$ hence there is an $m \\in \\N$ such that  $m\\bar{r} = l$. For $1 \\le i \\le m$ let $\\ac{u}_{i} : = \\{ u_{i,0}, \\ldots, u_{i,r-1}\\} \\subset \\ac{u}$ be such that for $1 \\le i_1 < i_2 \\le m$ all elements of $\\ac{u}_{i_1}$ are less than (in the lexicographic ordering) all elements of $\\ac{u}_{i_2}$.  Let $\\sigma: \\{1,2 \\ldots, m\\} \\to \\{1,2 \\ldots, m\\}$ be given by $ (i)\\sigma = i+1 \\mod{m}$. We construct an element $h_{l} \\in \\TBnr$ as follows: for $\\Gamma \\in \\CCn$ and $1 \\le i \\le m$ and $0 \\le a \\le r-1$ let $(u_{i,a}\\Gamma) h_{l} =  u_{(i)\\sigma,b}\\delta$ if and only if $(\\dot{a}\\Gamma)A_{q_0} =  \\dot{b}\\delta$. Therefore since $A_{q_0}$ induces an orientation preserving homeomorphism on  $\\CCmr{\\bar{r}}$, it follows that the ma $h_{l}$ induces a orientation preserving homeomorphism of $\\CCnr$. Moreover since $\\core(A_{q_0}) \\in \\TOnr$ it follows that $h_{l}$ is in fact and element of $\\TOnr$ and moreover (identifying $h_{l}$ with the initial transducer inducing inducing the homeomorphism $h_{l}$) $\\core(h_{l}) = \\core(A_{q_0})$. Lastly observe that the point $\\dot{0}00\\ldots$ is on an orbit of length $m$ under the action of $h_{l}$ and any other finite orbit of $h_{l}$ has length at least $l$.\n\nNow let $A_{q_0} \\in \\TBnr$ be arbitrary. Since $\\core(A_{q_0}) \\in \\TOns{\\bar{r}} = \\TOnr$, there is an element $B_{p_0} \\in \\TBmr{n}{\\bar{r}}$ such that $\\core(B_{p_0}) = \\core(A_{q_0})$. As before for each $l \\in \\N$ such that $l \\equiv r \\mod n-1$, there is an element $h_{l}$ with a minimal finite orbit of length $l$ and such that $\\core(h_{l}) = \\core(A_{q_0})$.  Thus, there is a $f_{l} \\in \\Tnr$ such that $h_{q_0}f_{l}  = h_{l}$. Therefore as in the case $r=1$, we conclude that for any there are infinitely many $h_{q_0}$-twisted conjugacy classes for any element $A_{q_0} \\in \\TBnr$.\n\n\\begin{comment}\nNow let $A_{q_0} \\in \\TBmr{n}{1}$ be orientation reversing such that $h_{q_0}$ fixes the point $00\\ldots$ and $n-1 n-1 \\ldots$ of $\\CCn$. Consider the antichains $\\ac{u} = \\{0, 10, 11, \\ldots 1n-1,2,3 \\ldots, n-1\\}$ and $\\ac{v} = \\{0, 1, \\ldots, n-2, n-10, n-11,\\ldots, n-1n-1\\}$. Let the elements of $\\ac{u}$ and $\\ac{v}$ be denoted in order by $u_1, u_2, \\ldots, u_{2n-1}$ and $v_1, v_2, \\ldots, v_{2n-1}$. Let $\\sigma: \\ac{u} \\to \\ac{v}$ be given by $u_i \\mapsto u_{n-i+1}$. Now let $h_{1}$ be defined by $u_i \\rho \\mapsto (u_i)\\sigma(\\rho)A_{q_0}$ for $\\rho \\in \\CCn$ and $1 \\le i \\le 2n-1$. Then $h_{1}^{2}$ has precisely 3 fixed points $00\\ldots$ $11\\ldots$ and $n-200\\ldots$.\n\\end{comment}\n\n{\\bfseries{Orientation reversing case:}} We now consider the orientation reversing case. Let $A_{s_0} \\in \\TBmr{n}{1}$ be orientation reversing. By Lemma~\\ref{Lemma: automorphisms can be modified to contain fixed point} we may assume that $(000\\ldots)A_{s_0} = n-1n-1n-1\\ldots$ and $(n-1n-1n-1\\ldots)A_{s_0} = n-1n-1n-1\\ldots$. Let $i$ be minimal such that $\\pi_{A}(0^{i}, s_0) = q_0$   and $\\pi_{A}((n-1)^{i}, s_0)  = q_{n-1}$ (where $q_0$ and $q_{n-1}$ are the zero and one loop state of $A_{s_0}$ respectively). Let $(n-1)^{l_1} = (0^{i+1})A_{q_0}$ and $0^{l_2} = ((n-1)^{i+1})A_{q_0}$. Observe  that $0<\\min\\{l_1, l_2\\}$ since $\\lambda_{A}(0, q_0) = n-1$ and $\\lambda_{A}(n-1, q_0) = n-1$ by Lemma~\\ref{Lemma: elements of TOn either fix 0 or swap 0 and n-1}. There is an element $h \\in \\Tn$ such that for any $\\delta \\in \\CCn$, $((n-1)^{l_1}\\delta)h = (n-1)^{i}\\delta$ and $(0^{l_2}\\delta h) = 0^{i}\\delta$. Let $C_{s'_0} \\in \\TBnr$ be such that $h_{s'_0} = h_{s_0} h$. Observe that $C_{s'_0}$ satisfies, $\\pi_{C}(0^{i}, s'_0) = q'_{0}$, $\\pi_{C}((n-1)^{i}, s'_0) = q'_{n-1}$, where $q'_{0}$ and $q'_{(n-1)}$ are the $0$ and $n-1$ loop states of $C_{s'_0}$ respectively, and, for $j \\ge i$, $\\lambda_{C}(0^{j}, s'_0) = (n-1)^{j}$ and $\\lambda_{C}((n-1)^{j}, s'_0) = 0^{j}$. As before we may replace $A_{s_0}$ with $C_{s'_0}$ without loss of generality, since there infinitely many $A_{s_0}$-twisted conjugacy classes if and only if there are infinitely many $C_{s'_0}$-twisted conjugacy classes.  Therefore we may assume that there is an $i \\in \\N$, $i>0$ such that $A_{s_0}$ satisfies $\\pi_{A}(0^{i}, s_0) = q_{0}$, $\\pi_{A}((n-1)^{i}, s_0) = q_{n-1}$, where $q_{0}$ and $q_{(n-1)}$ are the $0$ and $n-1$ loop states of $A_{s_0}$ respectively, and, for $j \\ge i$, $\\lambda_{A}(0^{j}, s_0) = (n-1)^{j}$ and $\\lambda_{A}((n-1)^{j}, s_0) = 0^{j}$.\n\nWe now construct an element $g \\in \\Tn$ such that $g h_{s_0}$ such $(g h_{s_0})^2$ has the point $000\\ldots$ and $n-1n-1\\ldots$ as attracting fixed points.\n\n\nBy a branch of length $k$, for $k \\in \\N$, $k>0$, we shall mean the finite rooted $n$-ary tree with $1+ k(n-1)$ leaves and $k$ the length of the geodesic from the root to the left-most leaf. \n\nLet $\\bar{f} \\in \\Tn$ be defined as follows. Let $E$ and $F$ be the finite trees where the tree $E$ is obtained from the single caret,  by attaching  a branch of length $i+1$ to the first and penultimate leaves of the branch and the tree $F$ is obtained from the single caret by attaching a branch to the first and last leaves of the caret. Label the leaves of $E$ left to right by $1,2, \\ldots, 1 + (2k+1)(n-1)$ and the leaves of $F$ from right to left by $1,2, \\ldots, 1 + (2k+1)(n-1)$. The map $\\bar{f}$ is now defined by, for $\\Gamma \\in \\CCn$ and $\\mu_{j}$ and $\\nu_{j}$ the addresses of the $j$'th  leaves of $E$ and $F$ respectively,  $\\mu_{j}\\Gamma \\mapsto \\nu_{j}\\Gamma$.\n\nLet $f \\in \\TBmr{n}{1}$, be defined by, for $\\Gamma \\in \\CCn$,  $\\mu_{j}\\Gamma \\mapsto \\nu_{j}(\\Gamma)A_{s_0}$ where  $\\mu_{j}$ and $\\nu_{j}$ are the addresses of the $j$'th  leaves of $E$ and $F$ respectively. Essentially $f$ is obtained from $\\bar{f}$ by attaching the state $s_0$ to all the leaves of $F$ the range tree of $\\bar{f}$. Moreover since $\\core(f) = \\core(A_{s_0})$ there is an element $g \\in \\Tn$ such that $f = gA_{s_0} $.\n\nBelow we shall consider the orbit of cones $U_{\\nu}$ for $\\nu \\in  X_{n}^{+}$ under $f$, and to simplify notation, we identify the cone $U_{\\nu}$ with the finite word $\\nu$. Let us consider the orbit of the cone $(n-1)$ (a leaf of $E$) under $f$: \\[(n-1)\\mapsto 00^{i} \\mapsto n-1 (n-1)^i \\mapsto 00^i 0^i \\mapsto n-1 (n-1)^i (n-1)^i \\mapsto 00^i(0)^{2i} \\ldots \\]\nThis is because for $(\\alpha, \\bar{\\alpha}) \\in \\{ (0, n-1), (n-1, 0) \\}$ and $j \\in \\N$ $j\\ge i$, $\\lambda_{A}(\\alpha^{j}, s_0) = \\bar{\\alpha}^{j}$ and $\\pi_{A}(\\alpha^{j}, s_0) = q_{\\alpha}$. Hence we see that the point $n-1n-1\\ldots$ is an attracting fixed point of $f^2$ and in a similar way the point $00\\ldots$ is an attracting fixed point of $f$. In particular we have, for $j \\in \\N_{1}$,  $(n-1)f^{2j} = (n-1)(n-1)^{ji}$ and $(00^{i})f^{2j} = 00^{i}0^{ji}$. We call the words $(n-1)^{i}$ and $0^{i}$ the attracting paths (in $f^2$) of $00\\ldots$ and $n-1n-1\\ldots\n$ respectively. \n\nLet $h \\in \\Tn$ be arbitrary and consider $h^{-1} f h$. Now as $h$ is a prefix exchange map, by an abuse of notation for a finite word $\\Gamma \\in \\Xn^{\\ast}$ such that $h$ acts as a prefix exchange on the cone $U_{\\Gamma} \\subset \\CCn$, we shall write, $(\\Gamma)h$ for the word $\\Delta \\in \\Xn^{*}$ such that  $(U_{\\Gamma}) h = U_{\\Delta}$. There is an $l \\in \\N$ such that $((n-1)^{l}) h$ is a finite word in $X_n^{+}$ and we may assume that $l \\ge i+1$. Thus we have,\n\\[\n ((n-1)^{l})h h^{-1}f^2h = ((n-1)^{l})f^2h = ((n-1)^{l})h(n-1)^{i}\n\\]\n\nHence we deduce that $((n-1)^l)h h^{-1}f^{2j}h = ((n-1)^{l})h(n-1)^{ji}$ and we see that conjugation of $f$ by an element of $\\Tn$ maps the attracting fixed point $n-1n-1\\ldots$ to an attracting fixed point and preserves the path $(n-1)^{i}$. We also observe that since an orientation reversing homeomorphism of the circle has precisely two fixed points (on the circle), then $h$ must map the pair of fixed points of $f$ to the pair of fixed of  $h^{-1}fh$. Notice that thinking of the fixed points as elements of $\\CCn$, since $\\Tn$ elements  $\\Z[1\/n] \\to \\Z[1\/n]$, this means that either $h$ fixes the points $00\\ldots$ and $n-1 n-1 \\ldots$ otherwise, the other fixed point of $f$ is an element of $\\Z[1\/n]$ of the form $\\eta_1 \\simeqI \\eta_2$ (where $\\eta_1 > \\eta_2$ in the lexicographic ordering on $\\CCn$), $(00\\ldots)h =\\eta_1$ and $(n-1n-1\\ldots)h = \\eta_2$. We make use of these facts to show that there are infinitely many $h_{s_0}$-twisted conjugacy classes.\n\nLet $j$ be any integer greater than $i$, as before we may construct a map $f_{j} \\in \\TBmr{n}{1}$ such that $f_{j} = g_{j}h_{s_0}$ for some $g_{j} \\in \\Tn$, and $00\\ldots$ and $n-1n-1\\ldots$ are attracting fixed points of $f_{j}^2$ with attracting paths $0^{j}$ and $(n-1)^{j}$. We  have the following claim:\n\n\\begin{claim}\nThere is an infinite subset $\\T{J} \\subset \\N_{i}$ for which the set $\\{f_{j} \\mid j \\in \\T{J}\\}$ are pairwise not conjugate by an element of $\\Tn$.\n\\end{claim}\n\\begin{proof}\nSuppose there is an $N \\in \\N$ such that every element $f_{j}$, $j \\in \\N_{i}$ is conjugate to some element of $\\{f_{i+1}, \\ldots\n f_{i+N}\\}$  by an element of $\\Tn$. Since $\\N_{i}$ is infinite, there is an infinite subset $\\T{I} \\subset \\N_{i}$ and $1 \\le M \\le N$ such that every element of $\\{f_{l} \\mid l \\in \\T{I}\\}$ is conjugate to $f_{i+M}$ by an element of $\\Tn$. Let $l > i+M$. Since conjugation by an element of $\\Tn$ preserves attracting paths and $l>M$ it must be the case that any conjugator $h \\in \\Tn$ such that $h^{-1}f_{l}h = f_{i+M}$ must satisfy $(n-1n-1\\ldots)h \\ne n-1 n-1 \\ldots$. Hence the other fixed point of $f_{i+m}$ must be a dyadic rational of the form $\\eta_1 \\simeqI \\eta_2 \\in \\CCn\/\\simeqI$ where $\\eta_1$ is greater than $\\eta_2$ in the lexicographic ordering of $\\CCn$. This means by arguments above that  $(n-1n-1\\ldots)h = \\eta_2$ and $\\eta_2$ is an attracting fixed point of $f_{i+M}$ with attracting path $(n-1)^i$. However for $l' \\in \\T{I}$ such that $l' >l$, it must be the case that $f_{l'}$ is not conjugate to $f_{i+M}$ since the attracting path in $f_{l'}^2$ of $n-1n-1\\ldots$  is longer than the attracting path in $f_{i+M}^2$ of both attracting fixed points of $f_{i+M}$.\n\\end{proof}\n\nNow fix $1< r < n$. Let $A_{s_0} \\in \\TBnr$ be an orientation reversing element,  $\\bar{r}$ be minimal such that $\\TOns{\\bar{r}} = \\TOns{r}$ and $B_{p_0}$ be an orientation reversing element of $\\TBmr{n}{\\bar{r}}$ with $\\core(B_{p_0}) = \\core(A_{s_0})$. As in the case $r=1$ we may assume that there is an $i \\in \\N$, $i>0$ such that $B_{p_0}$ satisfies $\\pi_{B}(\\dot{0}0^{i}, p_0) = q_{0}$, $\\pi_{B}(\\dot{r-1}(n-1)^{i}, p_0) = q_{n-1}$, where $q_{0}$ and $q_{(n-1)}$ are the $0$ and $n-1$ loop states of $\\core(A_{s_0}) = \\core(B_{p_0})$ respectively, and, for $j \\ge i$, $\\lambda_{B}(\\dot{0}0^{j}, p_0) = \\dot{r-1}(n-1)^{j}$ and $\\lambda_{B}(\\dot{r-1}(n-1)^{j}, p_0) = \\dot{0}0^{j}$. Now as in the orientation preserving case we construct elements $f_{j}$, for $j \\in \\N_{i}$ with desirable properties by simulating the element $B_{p_0}$ appropriately on cones.\n\nLet $\\ac{u} = \\{ \\dot{a} \\mid 1 \\le a \\le r \\}$, that is, $\\ac{u}$ corresponds to the roots of the disjoint union of $r$ copies of the $n$-ary tree. Since $\\bar{r}$ divides $r$ there is an $M \\in \\N$ such that $M\\bar{r} = r$. Let $\\ac{u}_{k}:= \\{ u_{k,1}, \\ldots, u_{k,r}\\}$, $1 \\le k \\le M$ be subsets of $\\ac{u}$, such that for $1 \\le k_1 < k_2 < M$, all elements of $\\ac{u}_{k_1}$ are less than (in the lexicographic ordering) all elements of $\\ac{u}_{k_2}$. Observe that $\\ac{u}_{1}$ corresponds to the roots of the disjoint union of the $\\bar{r}$ copies of the $n$-ary tree. Replace $\\ac{u}_1$, still retaining the symbol $\\ac{u}_{1}$ for the resulting antichain, with the antichain corresponding to attaching a branch of length $i+1$ to the root $u_{1,1}$. Likewise replace $\\ac{u}_M$ with the antichain corresponding to attaching a branch of length $i+1$ to the root $u_{M, r-1}$. In a similar way let $\\ac{v}_M$ be the antichain obtained from $\\ac{u}_{M}$ corresponding to attaching a branch of length $i+1$ to the root  $\\ac{u}_1$, and let $\\ac{v}_{k}:= \\ac{u}_{k}$ for $1 \\le k < M$. Observe that $\\ac{u}_{1}$, $\\ac{u}_{M}$ and $\\ac{v}_{M}$ all have equal length $d$ congruent to $\\bar{r}$ modulo $n-1$. Since $\\bar{r}$ divides $n-1$ let $m \\in \\N$ be such that $m\\bar{r} = d$. Let  $\\ac{u}_{1} = \\cup_{1 \\le a \\le m} \\ac{u}_{1,a}$ where $\\ac{u}_{1,a} =  \\{u_{1,a,b} \\mid 1 \\le b \\le r \\}$, likewise let $\\ac{u}_{M} = \\cup_{1 \\le a \\le m} \\ac{u}_{M,a}$ where $\\ac{u}_{M,a} =  \\{u_{M,a,b} \\mid 1 \\le b \\le r \\}$ and $\\ac{v}_{M} = \\cup_{1 \\le a \\le m} \\ac{v}_{M,a}$ where $\\ac{v}_{M,a} =  \\{v_{M,a,b} \\mid 1 \\le b \\le r \\}$. \n\nWe construct a homeomorphism $f_{i}$ of $\\CCnr$ as follows. Let $\\sigma: \\{1,2,\\ldots,M\\} \\to \\{1,2, \\ldots M\\}$ by $k \\mapsto M-k +1$ and let $\\rho: \\{1,2,\\ldots, m\\} \\to \\{1,2,\\ldots m\\}$ by $a \\mapsto m-a+1$. For $\\Gamma \\in \\CCn$, $1< k < M$ and $1 \\le b \\le r$, $(u_{k,b}\\Gamma)f_{i} = v_{(k)\\sigma, b'}\\delta$ if and only if $\\lambda_{B}(\\dot{b}\\Gamma, p_0) \\in U_{\\dot{b'}}$ and $\\delta = \\lambda_{B}(\\Gamma, \\pi_{B}(\\dot{b}, p_0))$. For $\\Gamma \\in \\CCn$ $1 \\le a \\le m$ and $1 \\le b \\le r$, $(u_{1,a,b}\\Gamma)f_{i} = v_{M,((a)\\rho),b'}\\delta$ if and only if $\\lambda_{B}(\\dot{b}\\Gamma,p_0) \\in U_{\\dot{b'}}$ $\\delta = \\lambda_{B}(\\Gamma, \\pi_{B}(\\dot{b}, p_0))$. For $\\Gamma \\in \\CCn$ $1 \\le a \\le m$ and $1 \\le b \\le r$, $(u_{M,a,b}\\Gamma)f_{i} = v_{1,((a)\\rho),b'}\\delta$ if and only if $\\lambda_{B}(\\dot{b}\\Gamma,p_0) \\in U_{\\dot{b'}}$ $\\delta = \\lambda_{B}(\\Gamma, \\pi_{B}(\\dot{b}, p_0))$. Since $B_{p_0}$  induces an orientation reversing homeomorphism on $\\CCmr{\\dotr}$, we see that $f_{i}$ is in fact an element of $\\TBnr{r}$. Moreover, $\\core(f_i) = \\core(B_{p_0}) = \\core(A_{s_0})$ and so there is an element $g_i \\in  \\Tnr$ such that $f_i = h_{s_0}g_i$.\n\nWe now argue that the points $\\dot{0}0\\ldots$ and $\\dot{r-1}n-1n-1\\ldots$ are attracting fixed points of $f_{i}$ with attracting paths $0^{i}$ and $(n-1)^{i}$. We consider the orbit of the cone  $\\dot{r-1}(n-1)^{i}$:\n\\[\n\\dot{r-1}(n-1)^{i} \\mapsto \\dot{0}0^{i}0^{i} \\mapsto \\dot{r-1}(n-1)^{i}(n-1)^{i} \\mapsto \\dot{0}0^{i}0^{i}0^{i} \\mapsto \\dot{r-1}(n-1)^{i}(n-1)^{2i} \\ldots\n\\] \n\nthis follows by making use of the definition of $f_{i}$ and the facts: $\\pi_{B}(\\dot{0}0^{i}, p_0) = q_{0}$, $\\pi_{B}(\\dot{r-1}(n-1)^{i}, p_0) = q_{n-1}$, where $q_{0}$ and $q_{(n-1)}$ are the $0$ and $n-1$ loop states of $\\core(A_{s_0}) = \\core(B_{p_0})$ respectively, and, for $j \\ge i$, $\\lambda_{B}(\\dot{0}0^{j}, p_0) = \\dot{r-1}(n-1)^{j}$ and $\\lambda_{B}(\\dot{r-1}(n-1)^{j}, p_0) = \\dot{0}0^{j}$. In general we see that, for $l \\in \\N_{1}$, $(\\dot{r-1}(n-1)^{i})f_{i}^{2l} = \\dot{r-1}(n-1)^{i}(n-1)^{l}$ and $(\\dot{0}0^{i}0^{i})f_{i}^{2l} = \\dot{0}0^{i}0^{i}0^{l}$.\n\nFor each  $j > i$ we may repeat the construction above to get  elements $h_{j}$ such that $f_{j} = h_{s_0}g_{j}$ for some $g_{j} \\in \\Tnr$ and the points $\\dot{0}0\\ldots$ and $\\dot{r-1}n-1n-1\\ldots$ are attracting fixed points of $f_{j}$ with attracting paths $0^{j}$ and $(n-1)^{j}$. Since $f_{j}$ for $j \\in \\N_{i}$ induces a homeomorphism on circle of length $r$, $[0,r]$ with end points identified, we may repeat the arguments as in the case $r =1$ to conclude that there is an infinite subset $\\T{J} \\subset \\N_{i}$ for which the set $\\{f_{j} \\mid j \\in \\T{J}\\}$ are pairwise not conjugate by an element of $\\Tnr$.\n\n\nThus we have demonstrated the following:\n\n\\begin{Theorem}\\label{Theorem:TnrhastheRftyproperty}\nThe group $\\Tnr$ for $1 \\le r < n-1$ has the $R_{\\infty}$ property.\n\\end{Theorem}\n   \n\n\\begin{comment}\nWe require a few result about conjugacy in homeomorphisms of the circle. Let $f$ be an orientation preserving homeomorphism of the circle, then, as usual we may identify $f$ with a map from $[0,1] \\to [0,1]$. Define a function $\\Delta_{f}$ as follows:\n\\[\n(x)\\Delta_f = \\begin{cases}\n1 \\ \\mbox{if } x < (x)f \\mbox{ in } [0,1], \\\\\n0 \\ \\mbox{if } x = (x)f, \\\\\n-1 \\ \\mbox{if } x > (x)f \\mbox{ in } [0,1]\n\n\\end{cases}.\n\\]\n\nThe following result is standard:\n\n\\begin{lemma}\nLet $f, h$ be orientation preserving homeomorphisms of the circle such that $f$ has a fixed point, then\n\\begin{enumerate}[label = (\\roman{*})]\n\\item $\\Delta_{h^{-1}fh} = h^{-1}\\Delta_{f}$\n\\item $\\Delta_{f}^{-1} = - \\Delta_{f}$.\n\\end{enumerate}\n\\end{lemma}\n\nThe following key result is standard in the literature concerning homeomorphisms of the circle.\n\n\\begin{Theorem}\nTwo orientation reversing homeomorphisms of the circle $f$ and $g$ are conjugate if and only if $f^2$ and $g^2$ are conjugate by a homeomorphism of the circle that maps the pair of fixed points of $f$ to the pair of fixed points of $g$.\n\\end{Theorem}\n\\end{comment}\n\n\\printbibliography \n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nGame theoretic models of interdependent security have been used to study security of complex information and physical systems for more than a decade~\\cite{LFB14}. One of the key findings is that the externalities resulting from security decisions made by selfish agents lead to, potentially significant, inefficiencies. This motivates research on methods for improving information security, such as insurance~\\cite{BS10} and network design~\\cite{CDG14,CDG17}. We study the problem of network design for interdependent security in the setup where a strategic adversary collaborates with some nodes in order to disrupt the network.\n\n\\subsection*{The motivation}\nOur main motivation is computer network security in face of contagious attack by a strategic adversary. Examples of contagious attacks are stealth worms and viruses, that gradually spread over the network, infecting subsequent unprotected nodes. Such attacks are considered among the main threats to cyber security~\\cite{SPW02}. Moreover, the study of the data from actual attacks demonstrates that the attackers spend time and resources to study the networks and choose the best place to attack~\\cite{SPW02}. Direct and indirect infection can be prevented by taking security measures that are costly and effective (i.e., provide sufficiently high safety to be considered perfect). Examples include using the right equipment (such as dedicated high quality routers), software (antivirus software, firewall), and following safety practices. All of these measures are costly. In particular, having antivirus software is cheap but using it can be considered to be costly, safety practises may require staff training, staying up to date with possible threats, creating backups, updating software, hiring specialized, well-paid staff. The security decisions are made individually by selfish nodes. Each node derives benefits from the nodes it is connected to (directly or indirectly) in the network. An example is the Metcalfe's law (attributed to Robert Metcalfe~\\cite{SV00}, a co-inventor of Ethernet) stating that each node's benefits from the network are equal to the number of nodes it can reach in the network, and the value of a connected network is equal to the square of the number of its nodes. An additional threat faced by the nodes in the network is the existence of malicious nodes whose objectives are aligned with those of the adversary: they aim to disrupt the network~\\cite{MSW06,MSW09}.\n\n\\subsection*{Contribution}\nWe study the effectiveness of network design for improving system security with malicious (or byzantine) players and strategic adversary.\nTo this end we propose and study a three stage game played by three classes of players: the designer, the adversary, and the nodes. Some of the nodes are malicious and cooperate with the adversary. The identity of the nodes is their private information, known to them and to the adversary only. The designer moves first, choosing the network of links between nodes. Then, costly protection is assigned to the nodes. We consider two methods of protection assignments: the centralized one, where the designer chooses the nodes to receive protection, and the decentralized one, where each node decides individually and independently whether to protect or not. Lastly, the adversary observes the protected network and chooses a single node to infect. The protection is perfect and each non-byzantine node can be infected only if she is unprotected. The byzantine nodes only pretend to use the protection and can be infected regardless of whether they are protected or not. After the initial node is infected, the infection spreads to all the nodes reachable from the origin of infection via a path containing unprotected or byzantine nodes.\nWe show that if the protection decisions are centralized, so that the designer chooses both the network and the protection assignment, then either choosing a disconnected network with unprotected components of equal size or a generalized star with protected core is optimal. When protection decisions are decentralized, then, for sufficiently large number of nodes, the designer can resort to choosing the generalized star as well. In the case of sufficiently well-behaved returns from the network (including for example Metcalfe's law), the protection chosen by the nodes in equilibrium guarantees outcomes that are asymptotically close to the optimum. Hence, in such cases, the inefficiencies due to defense decentralization can be fully mitigated even in the presence of byzantine nodes.\n\n\\subsection*{Related work}\nThere are two, overlapping, strands of literature that our work is related to: the interdependent security games~\\cite{LFB14} and multidefender security games~\\cite{SVL14,LV15,LSV17}.\nEarly research on interdependent security games assumed that the players only care about their own survival and that there are no benefits from being connected \\cite{KH03,V04,ACY06,LB08a,LB08b,CCO12,AMO16}. In particular, the authors of \\cite{ACY06} study a setting in which the network is fixed beforehand, nodes only care about their own survival, attack is random, protection is perfect, and contagion is perfect: infection spreads between unprotected nodes with probability $1$. The focus is on computing Nash equilibria of the game and estimating the inefficiencies caused by defense decentralization. They show that finding one Nash equilibrium is doable in polynomial time, but finding the least or most expensive one is NP-hard. They also point out the high inefficiency of decentralized protection, by showing unboundedness of the price of anarchy. In~\\cite{LB08a,LB08b} techniques based on local mean field analysis are used to study the problem of incentives and externalities in network security\non random networks. In a more recent publication~\\cite{AMO16}, individual investments in protection are considered. The focus is on the strategic structure of the security decisions across individuals and how the network shapes the choices under random versus targeted attacks. The authors show that both under- and overinvestment may be present when protection decisions are decentralized.\nA slightly different, but related, models are considered in~\\cite{GWA10,GWA11,GMW12,LSB12a,LSB12b}. In these models\nthe defender chooses a spanning tree of a network, while the attacker chooses a link to remove. The defender and the adversary move simultaneously. The attack is successful if the chosen link belongs to the chosen spanning tree. Polynomial time algorithms for computing optimal attack and defense strategies are provided for several variants of this game. For a comprehensive review of interdependent security games see an excellent survey~\\cite{LFB14}.\n\nMultidefender security games are models of security where two or more defenders make security decisions with regard to nodes, connected in a network, and prior to an attack by a strategic adversary. Each of the defenders is responsible for his own subset of nodes and the responsibilities of different defenders are non-overlapping. The underlying network creates interdependencies between the defenders' objectives, which result in externalities, like in the interdependent security games. The distinctive feature of multidefender security models is the adopted solution concept: the average case Stackelberg equilibrium.  The model is two stage. In the first stage the defenders commit to mixed strategies assigning different types of security configurations across the nodes. In the second stage the adversary observes the network and chooses an attack. The research focuses on equilibrium computation and quantification of inefficiencies due to distributed protection decisions.\n\nPapers most related to our work are~\\cite{MSW09,CDG14,CDG17,GJKKM16}. The authors of~\\cite{MSW06} introduce malicious nodes to the model of~\\cite{ACY06}. The key finding in that paper is that the presence of malicious nodes creates a ``fear factor'' that reduces the problem of underprotection due to defense decentralization. Inspired by~\\cite{MSW06,MSW09}, we also consider malicious nodes in the context of network defense. We provide a formal model of the game with such nodes as a game with incomplete information. Our contribution, in comparison to~\\cite{MSW09}, lies in placing the players in a richer setup, where nodes care about their connectivity as well as their survival, and where both underprotection (i.e., insufficiently many nodes protect as compared to an optimum) and overprotection (excessively many nodes protect as compared to an optimum) problems are present. This leads to a much more complicated incentives structure. In particular, the presence of malicious nodes may lead to underprotection, as nodes may be unable to secure sufficient returns from choosing protection on their own.\n\nWorks~\\cite{CDG14,CDG17} consider the problem of network design and defense prior to the attack by a strategic adversary. In a setting where the nodes care about both their connectivity and their survival, the authors study the inefficiencies caused by defense decentralization and how they can be mitigated by network design. The authors show that both underprotection as well as overprotection may appear, depending on the costs of protection and network topology. Both inefficiencies can be mitigated by network design. In particular, the underprotection problem can be fully mitigated by designing a network that creates a cascade of incentives to protect. Our work builds on~\\cite{CDG14,CDG17} by introducing malicious nodes to the model. We show how the designer can address the problem of uncertainty about the types of nodes and, at the same time, mitigate the inefficiencies due to defense decentralization. \nLastly, in~\\cite{GJKKM16}, a model of decentralized network formation and defense prior to the attack by adversaries of different profiles is considered. The authors show, in particular, that despite the decentralized protocol of network formation, the inefficiencies caused by defense decentralization are relatively low.\n\nThe rest of the paper is structured as follows. In \\cref{sec:model} we define the model of the game, which we then analyze in \\cref{sec:analysis}. In \\cref{sec:extension} we discuss possible modifications of our model. We provide concluding remarks in \\cref{sec:concl}. \\Cref{ap:centralized} contains the proofs of the most technical results.\n\n\\section{The model}\n\\label{sec:model}\nThere are $(n+2)$ players: the designer ($\\mathbf{\\sf{D}}$), the nodes ($V$), and the adversary~($\\mathbf{\\sf{A}}$). In addition, each of the nodes is of one of two types: a genuine node (type~$1$) or a byzantine node (type~$0$). We assume that there are at least $n = 3$ nodes and that there is a fixed amount $n_B \\ge 1$ of byzantine nodes. The byzantine nodes cooperate with the adversary and their identity is known to $\\mathbf{\\sf{A}}$. All the nodes know their own type only. On the other hand, the adversary has complete information about the game. We suppose that he infects a subset of $n_A \\ge 1$ nodes. A \\emph{network} over a set of nodes $V$ is a pair $G = ( V,E )$, where $E \\subseteq \\{ij \\colon i,j \\in V\\}$ is the set of undirected links of $G$. Given a set of nodes $V$, $\\mathcal{G}(V)$ denotes the set of all networks over $V$ and $\\mathcal{G} = \\bigcup_{U\n\\subseteq V} \\mathcal{G}(U)$ is the set of all networks that can be formed over $V$ or any of its subsets. The game proceeds in four rounds (the numbers $n \\ge 3, n_B \\ge 1, n_A \\ge 1$ are fixed before the game):\n\\begin{enumerate}\n\\item The types of the nodes are realized.\n\\item $\\mathbf{\\sf{D}}$ chooses a network $G \\in \\mathcal{G}(V)$, where $\\mathcal{G}(V)$ is the set of all undirected networks over~$V$.\n\\item Nodes from $V$ observe $G$ and choose, simultaneously and independently, whether to protect (what we denote by $1$) or not (denoted by $0$). This determines the set of protected nodes\n      $\\mathit{\\Delta}$. The protection of the byzantine nodes is fake and, when attacked, such node gets infected and transmits the infection to all her neighbors.\n\\item $\\mathbf{\\sf{A}}$ observes the protected network $(G,\\mathit{\\Delta})$ and chooses a subset $I \\subset V$ consisting of $\\abs{I} = n_A \\ge 1$ nodes to infect. The infection spreads and eliminates all unprotected or byzantine nodes reachable from $I$ in $G$ via a path that does not contain a genuine protected node from $\\mathit{\\Delta}$.\nThis leads to the residual network obtained from $G$ by removing all the infected nodes.\n\\end{enumerate}\n\nPayoffs to the players are based on the residual network and costs of defense.\nThe returns from a network are measured by a \\emph{network value function}\n$\\mathit{\\Phi} \\colon \\bigcup_{U \\subseteq V} \\mathcal{G}(U) \\rightarrow \\mathbb{R}$\nthat assigns a numerical value to each network that can be formed over a subset $U$ of nodes from $V$.\n\nA \\emph{path in $G$ between nodes} $i,j \\in V$ is a sequence of nodes $i_0,\\ldots,i_m \\in V$\nsuch that $i = i_0$, $j = i_m$, $m \\geq 1$, and $i_{k-1}i_k \\in E$ for all $k = 1,\\ldots,m$. Node $j$ is \\emph{reachable} from node $i$ in $G$ if $i = j$ or there is a path between them in $G$. A \\emph{component} of a network $G$ is a maximal set of nodes $C\\subseteq V$ such that for all $i,j \\in C$, $i \\neq j$, $i$ and $j$ are reachable in $G$. The set of components of $G$ is denoted by $\\mathcal{C}(G)$.\nGiven a network $G$ and a node $i \\in V$, $C_i(G)$ denotes the component $C \\in \\mathcal{C}(G)$ such that $i \\in C$. Network $G$ is \\emph{connected} if $|\\mathcal{C}(G)| = 1$.\n\nWe consider the following family of network value functions:\n\\begin{displaymath}\n\\mathit{\\Phi}(G) = \\sum_{C\\in \\mathcal{C}(G)} f(|C|) \\, ,\n\\end{displaymath}\nwhere the function $f \\colon \\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}$ is increasing, strictly convex, satisfies $f(0) = 0$, and, for all $x \\ge 1$, verifies the inequalities \n\\begin{equation}\\label{eq:move_half}\n\\begin{aligned}\nf(3x) &\\geq 2f(2x) \\, , \\\\ \nf(3x + 2) &\\ge f(2x+2) + f(2x+1) \\, .\n\\end{aligned}\n\\end{equation} \nIn other words, the value of a connected network is an increasing and strictly convex function of its size. The value of a disconnected network is equal to the sum of values of its components. These assumptions reflect the idea that each node derives additional utility from every node she can reach in the network. In the last property we assume that these returns are sufficiently large: the returns from increasing the size of a component by $50\\%$ are higher than the returns from adding an additional, separate, component of the same size to the network.\nSuch form of network value function is in line with Metcalfe's law, where the value of a connected network over $x$ nodes is given by $f(x) = x^2$, as well as with Reed's law, where the value of a connected network is of exponential order with respect to the number of nodes (e.g., $f(x) = 2^x-1$).\n\nBefore defining payoff to a node from a given network, defense, and attack, we formally define the residual network. Given a network $G = (V,E)$ and a set of nodes $Z \\subseteq V$, let $G-Z$ denote the network obtained from $G$ by removing the nodes from $Z$ and their connections from $G$. Thus $G-Z = (V\\setminus Z, E[V\\setminus Z])$, where $E[V\\setminus Z] = \\{ij \\in E \\colon i,j \\in V\\setminus Z\\}$.\nGiven defense $\\mathit{\\Delta}$ and the set of byzantine nodes $B$, the graph $A(G\\mid \\mathit{\\Delta},B) = G - \\mathit{\\Delta}\\setminus B$ is called the \\emph{attack graph}. By infecting a node $i \\in V$, the adversary eliminates the component of $i$ in the attack graph, $C_i(A(G\\mid \\mathit{\\Delta},B))$.\\footnote{We define $C_i(A(G\\mid \\mathit{\\Delta},B)) = \\varnothing$ for every $i \\in \\mathit{\\Delta}\\setminus B$.} Hence, if the adversary infects a subset $I \\subset V$ of nodes, then the \\emph{residual network} (i.e., the network that remains) after such an attack is $R(G \\mid \\mathit{\\Delta}, B, I) = G - \\bigcup_{i \\in I}C_i(A(G \\mid \\mathit{\\Delta},B))$. \n\nNodes' information about whether they are genuine or byzantine is private. Similarly, the adversary's information about the identity of the byzantine nodes is private. As usual in games with incomplete information, private information of the players is represented by their \\emph{types}. The type of a node $i \\in V$ is represented by $\\theta_i \\in \\{0,1\\}$ ($\\theta_i = 1$ means that $i$ is genuine and $\\theta_i = 0$ means that $i$ is byzantine) and the type of the adversary is represented by $\\theta_{\\mathbf{\\sf{A}}} \\in \\binom{V}{n_B}$. (If $X$ is a finite set, then we denote by $\\binom{X}{t}$ the set of subsets of $X$ of cardinality $t$.) A vector $\\bm{\\theta} = (\\theta_1,\\ldots,\\theta_n,\\theta_{\\mathbf{\\sf{A}}})$ of players' types is called a \\emph{type profile}. The type profiles must be consistent so that the byzantine nodes are really known to the adversary. The set of consistent type profiles is $\\Theta = \\{(\\theta_1,\\ldots,\\theta_n,\\theta_{\\mathbf{\\sf{A}}}) \\colon \\theta_{\\mathbf{\\sf{A}}} = \\{i \\in V \\colon \\theta_{i} =  0\\} , \\abs{\\theta_{\\mathbf{\\sf{A}}}} = n_B \\}$.\n\\begin{remark}\nWe point out that $B \\subset V$ is the set of byzantine nodes (i.e., the true state of the world) while $\\theta_{\\mathbf{\\sf{A}}}$ denotes the beliefs of the adversary. The consistency assumption implies that the beliefs of the adversary are correct and $\\theta_{\\mathbf{\\sf{A}}} = B$.\n\\end{remark}\n\nThe adversary aims to minimize the gross welfare (i.e., the sum of nodes' gross payoffs), which is equal to the value of the residual network. Given a network $G$, the set of protected nodes $\\mathit{\\Delta}$, and the type profile $\\bm{\\theta} \\in \\Theta$, the payoff to the adversary from infecting the set of nodes~$I$ is\n\\begin{align*}\nu^{\\mathbf{\\sf{A}}}(G,\\mathit{\\Delta},I \\mid \\bm{\\theta}) = -\\mathit{\\Phi}(R(G \\mid \\mathit{\\Delta}, B, I)) = -\\sum_{C \\in {\\mathcal C}(R(G \\mid \\mathit{\\Delta}, B, I))} f(|C|) \\, .\n\\end{align*}\n\nThe designer aims to maximize the value of the residual network minus the cost of defense. Notice that this cost includes the cost of defense of the byzantine nodes. Formally, the designer's payoff from network $G$ under defense $\\mathit{\\Delta}$, the set of infected nodes $I$, and the type profile $\\bm{\\theta}$ is equal to\n\\begin{align*}\nu^{\\mathbf{\\sf{D}}}(G,\\mathit{\\Delta},I \\mid \\bm{\\theta}) =  \\mathit{\\Phi}(R(G\\mid \\mathit{\\Delta},I,B))  - |\\mathit{\\Delta}| c\n = \\left(\\sum_{C \\in \\mathcal{C}(R(G \\mid \\mathit{\\Delta},I,B))} f(|C|)\\right) - |\\mathit{\\Delta}| c \\, .\n\\end{align*}\n\nThe \\emph{gross payoff} to a genuine (i.e., not a byzantine) node $j \\in V$ in a network $G$ is equal to $f(|C_j(G)|)\/|C_j(G)|$. In other words, each genuine node gets the equal share of the value of her component. The net payoff of a node is equal to the gross payoff minus the cost of protection. A genuine node gets payoff $0$ when removed. Defense has cost $c\\in \\mathbb{R}_{>0}$.\nThe byzantine nodes have the same objectives as the adversary and their payoff is the same as that of $\\mathbf{\\sf{A}}$.\nFormally, a payoff to the node $j \\in V$ given a network $G$ with defended nodes $\\mathit{\\Delta}$, the set of infected nodes~$I$, and the type profile $\\bm{\\theta}\\in \\Theta$ is equal to\n \\begin{align*}\nu^j&(G,\\mathit{\\Delta},I \\mid \\bm{\\theta})=\\begin{cases}\n\t\t\t\t  u^{\\mathbf{\\sf{A}}}(G,\\mathit{\\Delta},I \\mid \\bm{\\theta}), & \\textrm{if $\\theta_j = 0$}, \\\\\n                  \\frac{f(|C_j(R(G\\mid \\mathit{\\Delta},B,I))|)}{|C_j(R(G\\mid \\mathit{\\Delta},B,I))|}, & \\textrm{if $\\theta_j = 1$, $j \\notin \\mathit{\\Delta}$}, \\\\\n                  & \\textrm{and $j \\notin \\bigcup_{i \\in I}C_i(A(G\\mid \\mathit{\\Delta},B))$},\\\\\n                  \\frac{f(|C_j(R(G \\mid \\mathit{\\Delta},B,I))|)}{|C_j(R(G\\mid \\mathit{\\Delta},B,I))|} - c, & \\textrm{if $\\theta_j = 1$ and $j \\in \\mathit{\\Delta}$}, \\\\\n                  0, & \\textrm{if $\\theta_j = 1$, $j \\notin \\mathit{\\Delta}$,} \\\\\n                     & \\textrm{and $j \\in \\bigcup_{i \\in I} C_i(A(G\\mid \\mathit{\\Delta}, B))$} \\, .\n                  \\end{cases}\n\\end{align*}\n\nThe adversary and the byzantine nodes make choices that maximize their utility. The designer and the nodes have incomplete information about the game and we assume that they are pessimistic, making choices that maximize the worst possible type realization (cf.~\\cite{AB06}). Formally, the \\emph{pessimistic utility} of a genuine (i.e., of type $\\theta_j = 1$) node $j$ from network $G$, the set of protected nodes $\\Delta$, and the set of infected nodes $I$, is\n\\begin{equation*}\n\\hat{U}^j(G,\\mathit{\\Delta},I) = \\inf_{(\\bm{\\theta}_{-j},1)\\in \\Theta} u^j(G,\\mathit{\\Delta},I \\mid (\\bm{\\theta}_{-j},1)) \\, .\n\\end{equation*}\nSimilarly, the pessimistic utility of the designer from network $G$, the set of protected nodes $\\Delta$, and the set of infected nodes $I$, is\n\\begin{equation*}\n\\hat{U}^{\\mathbf{\\sf{D}}}(G,\\mathit{\\Delta},I) = \\inf_{\\bm{\\theta}\\in \\Theta} u^{\\mathbf{\\sf{D}}}(G,\\mathit{\\Delta},I \\mid \\bm{\\theta}) \\, .\n\\end{equation*}\n\nTo summarize, the set of players is $P = V \\cup \\{\\mathbf{\\sf{D}},\\mathbf{\\sf{A}}\\}$. The set of strategies of player $\\mathbf{\\sf{D}}$ is $S^{\\mathbf{\\sf{D}}} = \\mathcal{G}(V)$.\nA strategy of each node $j$ is a function $\\delta_j \\colon \\mathcal{G}(V) \\times \\{0,1\\} \\rightarrow \\{0,1\\}$ that,\ngiven a network $G \\in \\mathcal{G}(V)$ and a node's type $\\theta_j\\in \\{0,1\\}$, provides the defense decision $\\delta_j(G, \\theta_j)$ of the node. The individual strategies of the nodes determine a function $\\mathit{\\Delta} \\colon \\mathcal{G}(V)\\times \\{0,1\\}^{V} \\rightarrow 2^V$ providing, given a network $G \\in \\mathcal{G}(V)$ and nodes' types profile $\\bm{\\theta}_{-\\mathbf{\\sf{A}}}\\in \\{0,1\\}^V$, the set of defended nodes $\\mathit{\\Delta}(G \\mid \\bm{\\theta}_{-\\mathbf{\\sf{A}}}) = \\{j\\in V \\colon \\delta_j(G,\\theta_j) = 1\\}$.\nThe set of strategies of each node $j \\in V$ is $S^j = 2^{\\mathcal{G}(V)\\times \\{0,1\\}}$. \nA strategy of player $\\mathbf{\\sf{A}}$ is a function $x \\colon \\mathcal{G}(V) \\times 2^V \\times \\binom{V}{n_B} \\rightarrow \\binom{V}{n_A}$ that, given a network $G \\in \\mathcal{G}(V)$, the set of protected nodes $\\mathit{\\Delta} \\subseteq V$, and adversary's type $\\theta_{\\mathbf{\\sf{A}}}\\in \\binom{V}{n_B}$, provides the set of nodes to infect $x(G,\\mathit{\\Delta},\\theta_{\\mathbf{\\sf{A}}})$. The set of strategies of player $\\mathbf{\\sf{A}}$ is $S^{\\mathbf{\\sf{A}}} = \\binom{V}{n_A}^{\\mathcal{G}(V) \\times 2^V \\times \\binom{V}{n_B}}$.\n\nAbusing the notation slightly, we use the same notation for utilities of the players from the strategy profiles in the game. Thus, given a strategy profile $(G,\\Delta,x)$ and a type profile $\\bm{\\theta}$, the payoff to player $j \\in V \\cup \\{\\mathbf{\\sf{D}},\\mathbf{\\sf{A}}\\}$ is $u^{j}(G,\\mathit{\\Delta},x \\mid \\bm{\\theta}) = u^{j}(G,\\mathit{\\Delta}(G),x(G,\\mathit{\\Delta}(G)) \\mid \\bm{\\theta})$, the pessimistic payoff to player $j \\in V \\setminus B$ is\n\\begin{equation}\\label{eq:pess_payoff_node}\n\\hat{U}^j(G,\\mathit{\\Delta},x) = \\inf_{(\\bm{\\theta}_{-j},1)\\in \\Theta} u^j(G,\\mathit{\\Delta}(G),x(G, \\mathit{\\Delta}(G)) \\mid (\\bm{\\theta}_{-j},1)) \\, ,\n\\end{equation}\nand the pessimistic payoff to the designer is given by \n\\begin{equation}\\label{eq:pess_payoff_des}\n\\hat{U}^{\\mathbf{\\sf{D}}}(G,\\mathit{\\Delta},x) = \\inf_{\\bm{\\theta}\\in \\Theta} u^{\\mathbf{\\sf{D}}}(G,\\mathit{\\Delta}(G),x(G, \\mathit{\\Delta}(G)) \\mid \\bm{\\theta}) \\, .\n\\end{equation}\n\nBy convention, we say that the pessimistic payoff of the byzantine node is the same as her payoff. We are interested in subgame perfect mixed strategy equilibria of the game with the preferences of the players defined by the pessimistic payoffs. We call them the equilibria, for short. We make the usual assumption that when evaluating a mixed strategy profile, the players \nconsider an expected value of their payoffs from the pure strategies. In the case of the designer and the genuine nodes, these are expected pessimistic payoffs.\n\nThroughout the paper we will also refer to the subgames ensuing after a network $G$ is chosen. We will denote such subgames by $\\mathit{\\Gamma}(G)$ and call the \\emph{network subgames}. We will abuse the notation by using the same letters to denote the strategies in $\\mathit{\\Gamma}(G)$ and in $\\mathit{\\Gamma}$. The set of strategies of each node $i\\in V$ in game $\\mathit{\\Gamma}(G)$ is $\\{0,1\\}^{\\{0,1\\}}$. \nGiven the type profile $\\bm{\\theta}_{-\\mathbf{\\sf{A}}}\\in \\{0,1\\}^V$, the individual strategies of the nodes determine a function $\\mathit{\\Delta} \\colon \\{0,1\\}^{V} \\rightarrow 2^V$ that provides the set of defended nodes $\\mathit{\\Delta}(\\bm{\\theta}_{-\\mathbf{\\sf{A}}}) = \\{j\\in V \\colon \\delta_j(\\theta_j) = 1\\}$. The set of strategies of the adversary in $\\mathit{\\Gamma}(G)$ is $\\binom{V}{n_A}^{2^V\\times \\binom{V}{n_B}}$.\n\nAll the key notations are summarized in \\cref{tab:notation}.\n\n\\begin{table}\n\\small\n\\begin{tabular}{|c|c|}\n\\hline\n$n$&\\parbox[][20pt][c]{4cm}{\\centering number of nodes} \\\\\n\\hline \n$n_B$&\\parbox[][30pt][c]{4cm}{\\centering number of byzantine nodes} \\\\\n\\hline\n$n_A$&\\parbox[][35pt][c]{4cm}{\\centering number of nodes infected by the adversary} \\\\\n\\hline\n$f$&\\parbox[][20pt][c]{4cm}{\\centering component value function} \\\\\n\\hline\n$G$&\\parbox[][20pt][c]{4cm}{\\centering network} \\\\\n\\hline\n$\\Delta$&\\parbox[][20pt][c]{4cm}{\\centering set of protected nodes} \\\\\n\\hline\n$u^{\\mathbf{\\sf{D}}}, u^{\\mathbf{\\sf{A}}}, u^{j}$&\\parbox[][30pt][c]{4cm}{\\centering payoff to the designer, the adversary, and a node} \\\\\n\\hline\n$\\hat{U}^{\\mathbf{\\sf{D}}}, \\hat{U}^{j}$&\\parbox[][30pt][c]{4cm}{\\centering pessimistic payoff to the designer and a node} \\\\\n\\hline\n\\end{tabular}\n\\vspace*{0.5cm}\n\\caption{Summary of the notation.}\\label{tab:notation}\n\\end{table}\n\n\\subsection{Remarks on the model}\nWe make a number of assumptions that, although common for interdependent security games, are worth commenting on. Firstly, we assume that protection is perfect. This assumption is reasonable when available means of protection are considered sufficiently reliable and, in particular, deter the adversary towards the unprotected nodes. Arguably, this is the case for the protection means used in cybersecurity. Secondly, we assume that the designer and genuine nodes are pessimistic and maximize their worst-case payoff. Such an approach is common in computer science and is in line with trying to provide the worst-case guarantees on system performance. One can also take the probabilistic approach (by supposing that the distribution of the byzantine nodes is given by a random variable). In \\cref{sec:extension} we discuss how our results carry over to such model.\n\n\\section{The analysis}\n\\label{sec:analysis} \n\nWe start the analysis by characterizing the centralized defense model, where the designer chooses both the network and the defense assignment to the nodes. After that the adversary observes the protected network and nodes' types and chooses the nodes to infect. We focus on the first nontrivial case $n_B = n_A = 1$. In this case, we are able to characterize networks that are optimal to the designer. The topology of these networks is based on the generalized $k$-stars. We then turn to the decentralized defense and study the cost of decentralization. It turns out that the topology of $k$-star gives asymptotically low cost of decentralization not only for the simple case studied earlier but for all possible values of parameters $n_B$ and $n_A$. This is enough to prove our main result, \\cref{th:poa}, providing bounds on the price of anarchy.\n\n\\subsection{Centralized defense}\\label{sec:centralized}\nFix the parameters $n_B, n_A$ and suppose that the designer chooses both the network and the protection assignment. This leads to a two stage game where, in the first round, the designer chooses a protected network $(G,\\Delta)$ and in the second round the adversary observes the protected network and nodes' types (recognizing the byzantine nodes) and chooses the nodes to attack. Payoffs to the designer and to the adversary are as described in \\cref{sec:model} and we are interested in subgame perfect mixed strategy equilibria of the game with pessimistic preferences of the designer.\nWe call them equilibria, for short. Notice that, since the decisions are made sequentially, there is always a pure strategy equilibrium of this game. In this section, we focus only on such equilibria. Furthermore, the equilibrium payoff to the designer is the same for all equilibria. We denote this payoff by $\\globdespayoff(n,c)$.\n\nIn the rest of this subsection we focus on the case $n_B = n_A = 1$. In this case, when the protection is chosen by the designer, two types of protected networks can be chosen in an equilibrium (depending on the value function and the cost of defense): a disconnected network with no defense or a generalized star with protected core and, possibly, one or two unprotected components. Before stating the result characterizing equilibrium defended networks and equilibrium payoffs to the designer, we need to define the key concept of a generalized star and some auxiliary quantities. We start with the definition of a generalized star. If $G = (V, E)$ is a network and $V' \\subset V$ is a subset of nodes, then we denote by $G[V']$ the subnetwork of $G$ induced by $V'$, i.e., the network $G[V'] = (V', \\{ij \\in E \\colon i,j \\in V' \\})$.\n\n\\begin{definition}[Generalized $k$-star]\nGiven a set of nodes $V$ and $k \\ge 1$, a \\emph{generalized $k$-star} over $V$ is a network $G = (V,E)$ such that the set of nodes $V$ can be partitioned into two sets, $C$ (the core) of size $|C| = k$ and $P$ (the periphery), in such a way that $G[C]$ is a clique, every node in $P$ is connected to exactly one node in $C$, and every node in $C$ is connected to $\\lfloor n\/k \\rfloor - 1$ or $\\lceil n\/k \\rceil - 1$ nodes in $P$.\n\\end{definition}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.45\\textwidth}\n      \\centering\n        \\includegraphics[scale=0.5]{n12k5.pdf}\n     \\end{minipage}\\hfill \n\\end{center}\n\\caption{A generalized star with $12$ nodes and core of size $5$.}\\label{fig:star}\n\\end{figure}\n\nRoughly speaking, a generalized $k$-star is a core-periphery network with the core consisting of $k$ nodes and the periphery consisting of the remaining $n - k$ nodes. The core is a clique, each periphery node is connected to exactly one core node and they are distributed evenly across the core nodes. An example of a generalized star is depicted in \\cref{fig:star}.\n\nNow we turn to defining some auxiliary quantities. For any $n \\ge 3$ such that $n \\bmod 6 \\neq 3$ we define\n\\[\nw_{0}(n) = w_{1}(n) = f\\left(\\left\\lfloor \\frac{n}{2} \\right\\rfloor\\right) + f(1)\\mathbbm{1}_{\\{n \\bmod 2 = 1 \\}} \\, ,\n\\]\nand for every $n$ such that $n \\bmod 6 = 3$ we define\n\\begin{equation}\nw_{0}(n) = w_{1}(n) = \\max\\left( 2f\\left( \\frac{n}{3} \\right), f\\left(\\frac{n-1}{2} \\right) + f(1)\\right) \\, . \\label{eq:divisible_by_three}\n\\end{equation}\nGiven $n$ nodes, $w_{0}(n)$ is the maximal network value the designer can secure against a strategic adversary by choosing an unprotected network composed of three components of equal size or two components of equal size and possibly one disconnected node. This is also the maximal network value the designer can secure by choosing such a network with one protected node, because, in the worst case scenario, the protected node is byzantine and may be infected.\n\nFor every $k \\in \\{3, \\dots, n\\}$, let\n\\begin{equation*}\nw_k(n) = \\left\\{\\begin{array}{ll}\n                f\\left(n-1-\\frac{n-1}{k}\\right) + f(1), & \\textrm{if $n \\bmod k = 1$} \\, , \\\\\n                f\\left(n - \\lceil \\frac{n}{k} \\rceil \\right), & \\textrm{otherwise} \\, .\n                \\end{array}\\right.\n\\end{equation*}\nGiven $n$ nodes and $k \\ge 3$, $w_k(n)$ is the network value that the designer can secure by choosing a generalized $k$-star, with one node disconnected in the case of $k$ dividing $n-1$, having all core nodes protected and all periphery nodes unprotected.\n\nWe also define the following quantities:\n\\begin{align}\n&A_{q} = \\min \\left( f(n - q), f\\left(\\left\\lfloor\\frac{n - q}{2}\\right\\rfloor\\right) + f(q) \\right) \\, , \\nonumber \\\\\n&B_{q} = \\min \\left( f(n - q - 1), f\\left(\\left\\lfloor\\frac{n-q-1}{2}\\right\\rfloor\\right) + f(q) \\right) \\, , \\nonumber \\\\\n&h_q(n) = \\max\\left(A_{q}, B_{q} + f(1) \\right) \\, , \\label{eq:size_two_single} \\\\\n&w_2(n) = \\max_{q \\in \\{0, \\ldots, n - 2 \\}} h_{q}(n) \\, . \\label{eq:size_two}\n\\end{align}\nGiven $n$ nodes, $w_2(n)$ is the network value that the designer can secure by choosing a network composed of a generalized $2$-star with a protected core and unprotected periphery, an unprotected component (of size $q \\in \\{0, \\dots, n - 2\\}$), and possibly one node disconnected from both of these components.\n\nFinally, we define\n\\begin{equation*}\nK^{*}(n,c) = {\\arg\\max}_{k\\in \\{0,\\ldots,n\\}} w_k(n) - kc \\, .\n\\end{equation*}\n\nWe point out that $K^{*}(n,c)$ never contains $1$ (because $c > 0$). We are now ready to state the result characterizing equilibrium defended network and pessimistic equilibrium payoffs to the designer.\n\n\\begin{proposition}\n\\label{th:centr}\nLet $n_B = n_A = 1$, $n \\ge 3$,\n$c > 0$, and $k \\in K^{*}(n,c)$. Then, the pessimistic equilibrium payoff to the designer is equal to $\\globdespayoff(n,c) = w_{k} - kc$. Moreover, there exists an equilibrium network $(G, \\Delta)$ that has $\\abs{\\Delta} = k$ protected nodes and the following structure:\n\\begin{compactenum}[i)]\n\\item $G$ has at most three connected components.\n\\item If $k \\ge 3$ and $n \\bmod k \\neq 1$, then $G$ is a generalized $k$-star with protected core and unprotected periphery. \n\\item If $k \\ge 3$ and $n \\bmod k = 1$, then $G$ is composed of a generalized $k$-star of size $(n-1)$ with protected core and unprotected periphery and a single unprotected node.\n\\item If $k = 0$ and $n \\bmod 6 \\neq 3$, then $G$ has two connected components of size $\\lfloor n\/2 \\rfloor$ and, if $n \\bmod 2 = 1$, a single unprotected node.\\label{it:no_def}\n\\item If $k = 0$ and $n \\bmod 6 = 3$, then $G$ either has the structure described in \\cref{it:no_def} or $G$ is composed of three components of size $n\/3$, depending on the term achieving maximum in~\\cref{eq:divisible_by_three}.\n\\item If $k = 2$, then $G$ is composed of a generalized $2$-star with protected core and unprotected periphery, an unprotected component of size $q \\in \\{0, \\dots, n-2\\}$ and, possibly, a single unprotected node. The size $q$ is the number achieving maximum in \\cref{eq:size_two}. The existence of a single unprotected node depends on the term achieving maximum in \\cref{eq:size_two_single}.\n\\end{compactenum}\n\\end{proposition}\n\nThe intuitions behind this result are as follows. When the cost of defense is high, then the designer is better off by not using any defense and partitioning the network into several components. Since the strategic adversary will always eliminate a maximal such component, the designer has to make sure that all the components are equally large. Due to the divisibility problems, one component may be of lower size. Thanks to our assumptions on the component value function $f$, the number of such components is at most three. Moreover, if there are exactly three components, then they are of equal size or the smallest one has size $1$.\n\nWhen the cost of defense is sufficiently low, then it is profitable for the designer to protect some nodes. If the number of protected nodes is not smaller than $3$, then, by choosing a generalized $k$-star with fully protected core (of optimal size $k \\ge 3$ depending on the cost) and unprotected periphery, the designer knows that the strategic adversary is going to attack either the byzantine node (if she is among the core nodes) or any unprotected node (otherwise). An attack on the byzantine core node destroys that node and all periphery nodes attached to her. Thus, in the worst case, a core node with the largest number of periphery nodes connected to her is byzantine. By distributing the core nodes evenly, the designer minimizes the impact of this worst case scenario. Due to the divisibility problems, it may happen that some of the core nodes are connected to a higher number of periphery nodes. If this is the case for one core node only, then it is better for the designer to disconnect this one node from the generalized star. By doing so, the designer spares this node from destruction.\n\nThe case when there are exactly $2$ protected nodes is special. Indeed, in this case, choosing a generalized $2$-star with protected core is not better than using no protection at all. This is because, in the worst case, the byzantine node is among the two protected ones. Therefore, it would be better for the designer to split the network into two unprotected components -- this would result in the same network value after the attack without the need to pay the cost of protection. On the other hand, if the network consists of a generalized $2$-star with protected core and an unprotected component, then the argument above ceases to be valid: even if the byzantine node is among the protected ones, splitting them may give the adversary an incentive to destroy the unprotected component. Therefore, a protection of $2$ nodes may be used as a resource that ensures that one component survives the attack.\n\nIt is interesting to compare this result to an analogous result obtained in~\\cite{CDG14,CDG17} for a model without byzantine nodes. There, depending on the cost of protection, three equilibrium protected networks are possible: an unprotected disconnected network (like in the case with a byzantine node), a centrally protected star, and a fully protected connected network. The existence of a byzantine node leads to a range of core-protected networks between the centrally protected star and the fully protected clique (which is a generalized $n$-star). Notice that pessimistic attitude towards incomplete information results in the star network never being optimal: if only one node is protected, then, in the worst case, the designer expects this node to be byzantine, which leads to loosing all nodes after the attack by the adversary. Therefore, at least two nodes must be protected if protection is used in an equilibrium. The proof of \\cref{th:centr} is given in \\cref{ap:centralized}.\n\n\\begin{example}\n\\Cref{example50} presents how the optimal network changes for different cost values when $f(x) = x^{2}$ and $n \\in \\{12,30,50\\}$. For these values of $n$, it is never optimal to have one node that is disconnected from the rest of the network. Moreover, as we can see, for a given number $n$ of nodes, not all possible generalized $k$-stars arise as optima. It is interesting to note that $3$-stars have never appeared in our experiments as optimal networks for the value function $f(x) = x^{2}$. Similarly, we have not found an example where it is optimal to defend exactly $2$ nodes. The case where there is no defense but the network is split into $3$ equal parts arises when $n = 9$ and the cost is high enough (i.e., $c > 6.2$), as already established in~\\cite{CDG14}.\n\\end{example}\n\n\\begin{remark}\nIn this section, we have characterized the optimal networks for the case $n_B = n_A = 1$. Nevertheless, we have not found a network that has a substantially different structure than the ones described here and performs better for general values of $n_B$ and $n_A$. We therefore suspect that the characterization for the general case is similar to the case $n_B = n_A = 1$.\n\\end{remark}\n\n\\begin{table}\n\\footnotesize\n\n\\begin{tabular}{|l|C{2cm}|l|l|C{2cm}|l|l|C{2cm}|}\n\\cline{8-8}\n\\multicolumn{7}{c|}{} & $n = 50$ \\\\\n\\cline{5-5} \\cline{7-8}\n\\multicolumn{4}{c|}{}&$n=30$ & &$ c < 3.88 $ & $50$-star \\\\\n\\cline{4-5} \\cline{7-8}\n\\multicolumn{3}{c|}{}&$ c < 3.80 $ & $30$-star & & $c \\in (3.88, 11.875)$&$25$-star \\\\\n\\cline{2-2}\\cline{4-5}\\cline{7-8}\n\\multicolumn{1}{c|}{}&$n=12$&&$c \\in (3.80, 11)$ &$15$-star & & $c \\in (11.875, 23.25)$&$17$-star \\\\\n\\cline{1-2}\\cline{4-5}\\cline{7-8}\n$ c < 3.50 $ & $12$-star & & $c \\in (11, 26)$ &$10$-star & &$c \\in (23.25,30.(3))$&$13$-star \\\\\n\\cline{1-2}\\cline{4-5}\\cline{7-8}\n$c \\in (3.50, 9.50)$ &$6$-star & & $c \\in (26,49)$ &$6$-star & &$c \\in (30.(3), 85)$&$10$-star \\\\\n\\cline{1-2}\\cline{4-5}\\cline{7-8}\n$c \\in (9.50, 11.25)$ &$4$-star & & $c \\in (49, 70.20)$ &$5$-star& &$c \\in (85, 195)$ &$5$-star \\\\\n\\cline{1-2}\\cline{4-5}\\cline{7-8}\n$c > 11.25$ &two disconnected components of equal size & & $c > 70.20$&two disconnected components of equal size& &$c > 195$ &two disconnected components of equal size \\\\\n\\cline{1-2}\\cline{4-5}\\cline{7-8}\n\\end{tabular}\n\\vspace*{0.5cm}\n\n\\caption{Optimal networks for $n \\in \\{12, 30, 50\\}$.} \\label{example50}\n\\end{table}\n\n\\subsection{Decentralized defense}\nNow we turn attention to the variant of the model where defense decisions are decentralized. Our goal is to characterize the inefficiencies caused by decentralized protection decisions for general values of $n_B$ and $n_A$. To this end, we need to compare equilibrium payoffs to the designer under centralized and decentralized defense. We start by establishing two results about the existence of equilibria in the decentralized defense game. \n\nFirstly, since the game is finite, we get equilibrium existence by Nash theorem. Notice that our use of the pessimistic aggregation of the incomplete information about types of nodes determines a game where the utilities of the nodes and the designer are defined by the corresponding pessimistic utilities. This game is finite and, by Nash theorem, it has a Nash equilibrium in mixed strategies. This leads to the following existence result.\n\n\\begin{proposition}\n\\label{pr:exist}\nThere exists an equilibrium of $\\Gamma$.\n\\end{proposition}\n\n\\begin{proof}\nIt can be shown that a stronger statement holds. More precisely, one can prove that for any $n, c$ there exists an equilibrium $\\bm e$ such that the strategies of the nodes do not depend on their types. Let us sketch the proof. We consider a modified model in which the nodes do not know their types (i.e., every node thinks that she is genuine, but some of them are byzantine). In this model, the (mixed) strategies of nodes are functions $\\tilde{\\delta}_{j} \\colon \\mathcal{G}(V) \\to \\Sigma(\\{0,1\\})$,\\footnote{\nIf $X$ is a finite set, then by $\\Sigma(X)$ we denote the set of all probability measures on $X$.\n}\nand every node receives a pessimistic utility of a genuine node, as defined in \\cref{eq:pess_payoff_node}. The strategies and payoffs to the adversary and the designer are as in the original model. Let $\\overbar{x} \\colon \\mathcal{G}(V) \\times 2^{V} \\times \\binom{V}{n_B} \\to \\binom{V}{n_A}$ denote any optimal strategy of the adversary (i.e., a function that, given a defended network and the position of the byzantine nodes $B$, returns a subset of nodes that is optimal to infect in this situation). If we fix $\\overbar{x}$, then the game turns into a two stage game (the designer makes his action first and then the nodes make their actions) with complete information. Therefore, this game has a subgame perfect equilibrium in mixed strategies. This equilibrium, together with $\\overbar{x}$, forms an equilibrium $\\bm e$ in the original model, because, in the original model, a byzantine node cannot improve her payoff by a unilateral deviation.\n\\end{proof}\n\nFix the parameters $n_B, n_A$ and let $\\mathcal{E}(n, c)$ denote the set of all equilibria of $\\mathit{\\Gamma}$ with $n$ nodes and the cost of protection $c > 0$. Let $\\globdespayoff(n, c)$ denote the best payoff the designer can obtain in the centralized defense game (as discussed in \\cref{sec:centralized}).\nThe \\emph{price of anarchy} is the fraction of this payoff over the minimal payoff to the designer that can be attained in equilibrium of $\\mathit{\\Gamma}$ (for the given cost of protection $c$),\n\\begin{equation*}\n\\mathrm{PoA}(n, c) = \\frac{\\hat{U}^{\\mathbf{\\sf{D}}}_{\\star}(n,c)}{\\min_{\\bm{e} \\in \\mathcal{E}(n,c)} \\mathbf{E}\\hat{U}^{\\mathbf{\\sf{D}}}(\\bm{e})} \\, .\n\\end{equation*}\n\nAlthough pure strategy equilibria may not exist for some networks, they always exist on generalized stars. Moreover, when these stars are large enough, by choosing such a star, the designer can ensure that all genuine core nodes are protected. \nThis is enough to characterize the price of anarchy as $n$ goes to infinity (with a fixed cost $c$). The next proposition characterizes equilibria on generalized stars.\n\n    \\begin{proposition}\\label{thm:ROEchar}\n    Let $\\bm e \\in \\mathcal{E}$ be any equilibrium of $\\mathit{\\Gamma}$. Let $G = (V, E)$ be a generalized $k$-star. Denote $\\abs{V} = n$, $x = \\left\\lfloor \\frac{n}{k} \\right\\rfloor - n_A + 1$, and $y = n - n_B\\left\\lfloor \\frac{n}{k} \\right\\rfloor$. Furthermore, suppose that $n \\ge k \\ge n_B+1$ and $x \\ge 2$. If the cost value $c$ belongs to one of the intervals $(0, f(1))$, $(f(1), \\frac{f(x)}{x})$, $(\\frac{f(y)}{y}, +\\infty)$, then the following statements about $\\bm e$ restricted to $\\mathit{\\Gamma}(G)$ hold:\n     \\begin{compactitem}\n     \\item all genuine nodes use pure strategies\n      \\item if $c < f(1)$, then all genuine nodes are protected\n      \\item if $f(1) < c < \\frac{f(x)}{x}$, then \n            all genuine core nodes are protected and \n            all genuine periphery nodes are not protected\n      \\item if $\\frac{f(y)}{y} < c$, then all genuine nodes are not protected.\n    \\end{compactitem}\n  \\end{proposition}\n  \n  The proof of \\cref{thm:ROEchar} requires an auxiliary lemma.\n\n  \\begin{lemma}\\label{attack_on_byz}\n  Let $\\bm e \\in \\mathcal{E}$ be any equilibrium of $\\mathit{\\Gamma}$ and $\\overbar{x} \\colon \\mathcal{G}(V) \\times 2^{V} \\times \\binom{V}{n_B} \\to \\Sigma(\\binom{V}{n_A})$ denote the (possibly mixed) strategy of the adversary in this equilibrium.\n  Let $(G, \\Delta)$ be a network such that $G$ is a generalized $k$-star. Furthermore, suppose that $\\lfloor \\frac{n}{k} \\rfloor \\ge 2$, $n \\ge 3$, and that the set of byzantine nodes $B$ contains a core node. Then, $\\overbar{x}(G, \\Delta, \\theta_{\\adv})$ infects this node with probability one. \n  \\end{lemma}\n  \n  \\begin{proof}\n  Since $\\bm e$ is an equilibrium and the adversary has complete information about the network before making his decision, his strategy $\\overbar{x}(G, \\Delta, \\theta_{\\adv})$ is a probability distribution over the set of subsets of nodes that are optimal to attack. Let $b \\in B$ denote any byzantine node that is also a core node. We will show that any optimal attack infects $b$.\n  \nTo do so, fix any set of attacked nodes $I \\in \\binom{V}{n_A}$ and suppose that attacking $I$ does not infect $b$. Given the structure of generalized $k$-star, we see that $I$ consists of genuine protected nodes and periphery nodes that are connected to genuine protected core nodes. To finish the proof, fix any node $j \\in I$ and observe it is strictly better for the adversary to attack the set $I \\cup \\{ b\\} \\setminus \\{j\\}$. Indeed, if $j$ is a genuine protected node, then attacking it does nothing, while attacking $b$ destroys at least one more node. Moreover, if $j$ is a periphery node connected to a genuine core protected node, then attacking $b$ not only destroys one node but also disconnects the network ($b$ is connected to at least one periphery node because $\\lfloor \\frac{n}{k} \\rfloor -1 \\ge 1$).\n  \\end{proof}\n\nWe are now ready to present the proof of \\cref{thm:ROEchar}.\n  \n  \\begin{proof}[Proof of \\cref{thm:ROEchar}]\n  Let $\\overbar{x} \\colon \\mathcal{G}(V) \\times 2^{V} \\times \\binom{V}{n_B} \\to \\Sigma(\\binom{V}{n_A})$ denote the strategy of the adversary in $\\bm e$ and let $\\Delta$ be any choice of protected nodes on $G$, $\\Delta \\subset V$. Let $j \\in V$ be a genuine node. \n   \n   First, suppose that $j \\notin \\Delta$. We will show that the pessimistic payoff of $j$ is equal to $0$. On the one hand, this payoff is nonnegative for every possible choice of the infected node. On the other hand, we can bound it from above by supposing that there exists a byzantine node $b \\in B$ that is a core node and a neighbor of $j$. Then, \\cref{attack_on_byz} shows that $\\overbar{x}$ infects $b$, and the pessimistic payoff of $j$ is not greater than $0$. \n   \n   Second, suppose that $j \\in \\Delta$. Then, we have two possibilities. If $j$ is a periphery node, then the same argument as above shows that the pessimistic payoff of $j$ is equal to $f(1) - c$. If $j$ is a core node, then her payoff is bounded from below by $\\frac{f(x)}{x} - c$ (where $x = \\lfloor \\frac{n}{k} \\rfloor - n_A + 1$) for every possible choice of the set of infected nodes. Moreover, by supposing that every byzantine node is a core node, we see that the pessimistic payoff of $j$ is bounded from above by $\\frac{f(y)}{y} - c$ (where $y = n - n_B\\lfloor \\frac{n}{k} \\rfloor$).\n   \n   Since the estimates presented above are valid for any choice of $\\Delta$, we get the desired characterization of equilibria.\n  \\end{proof}\n  \nOur main result estimates the price of anarchy using \\cref{thm:ROEchar}.\n\\begin{theorem}\n\\label{th:poa}\nSuppose that for all $t \\ge 0$ the function $f$ satisfies \n\\[ \n\\lim_{n \\to +\\infty} f(n)\/f(n - t) = 1 \\, .\n\\]\nThen, for any cost level $c > 0$ and any fixed parameters $n_B \\ge 1$, $n_A \\ge 1$ we have \n\\[\n\\lim_{n \\to +\\infty} \\mathrm{PoA}(n,c) = 1 \\, .\n\\]\n\\end{theorem}\n\nThe proof requires an auxiliary lemma concerning the asymptotic behavior of the function $f$.\n  \n    \\begin{lemma}\\label{fast_growth}\n  We have $\\lim_{x \\to +\\infty} \\frac{f(x)}{x} = +\\infty$.\n  \\end{lemma}\n   \\begin{proof}\n   Since $f$ is strictly convex, for any $0 < x < y < z$ we have (cf. \\cite[Sect.~I.1.1]{lemarechal_convex_analysis})\n  \\begin{equation}\n  \\frac{f(y) - f(x)}{y - x} < \\frac{f(z) - f(x)}{z - x} < \\frac{f(z) - f(y)}{z - y} \\, . \\label{eq:slopes}\n  \\end{equation}\n  As a result, the function $g_{t}(x) = (f(x + t) - f(t))\/x$ is strictly increasing for all $t > 0$ (to see that, let $0 < x < y$ and use the left inequality from \\cref{eq:slopes} on the tuple $(t, x + t, y + t)$). Since $f$ is convex and increasing, it is also continuous on $[0, +\\infty)$ (cf. \\cite[Sect.~I.3.1]{lemarechal_convex_analysis}). By fixing $x$ and taking $t \\to 0$ we get that the function $x \\to \\frac{f(x)}{x}$ is nondecreasing. Suppose that $\\lim_{x \\to +\\infty} \\frac{f(x)}{x} = \\eta < +\\infty$. Then, by the assumption that $f(3x) \\geq 2f(2x)$ for all $x \\ge 1$, we have\n  \\[\n  \\eta = \\lim_{x \\to +\\infty} \\frac{f(x)}{x} = \\lim_{x \\to +\\infty} \\frac{f(3x)}{3x} \\ge \\lim_{x \\to +\\infty} \\frac{2 f(2x)}{3\/2 \\cdot 2x} = \n  \\frac{4}{3}\\eta \\, .\n  \\]\n  Hence $\\eta \\le 0$ and $f(x) = 0$ for all $x \\ge 0$, which contradicts the assumption that $f$ is strictly convex.\n  \\end{proof}\n  \nWe also need the fact that $f$ is superadditive.\n  \\begin{lemma}\\label{superadditive}\n  For all $x, y > 0$ we have $f(x + y) > f(x) + f(y)$.\n  \\end{lemma}\n  \\begin{proof}\n  From the strict convexity of $f$ we have $f(x) = f(\\frac{x}{x+y}(x+y) + \\frac{y}{x+y} \\cdot 0) < \\frac{x}{x+y}f(x+y)$. Analogously, $f(y) < \\frac{y}{x+y}f(x+y)$. Hence $f(x+y) > f(x) + f(y)$. \n  \\end{proof}\n\nWe now give the proof of \\cref{th:poa}.\n\n\\begin{proof}[Proof of \\cref{th:poa}]\nThe function $f$ is superadditive by \\cref{superadditive}. As a result, the pessimistic payoff to the designer can be trivially bounded by $\\globdespayoff(n,c) \\le f(n)$. We now want to give a lower bound for the quantity $\\min_{\\bm{e} \\in \\mathcal{E}(n,c)} \\mathbf{E}\\hat{U}^{\\mathbf{\\sf{D}}}(\\bm{e})$. By \\cref{fast_growth} we have $\\lim_{x \\to +\\infty} \\frac{f(x)}{x} = +\\infty$. Let $N \\ge 1 + n_A$ be a natural number such that $\\frac{f(x)}{x} > c$ for all $x \\ge N - n_A + 1$. For any $n \\ge (n_B+1)(N + 1)$ we define $k = \\lfloor \\frac{n}{N + 1} \\rfloor \\ge n_B+1$. Observe that if we denote $x = \\lfloor \\frac{n}{k} \\rfloor - n_A + 1$, then we have $x \\ge \\frac{n}{k} - n_A \\ge N - n_A + 1$.\n Hence, if the designer chooses a generalized $k$-star, then \\cref{thm:ROEchar} shows that all genuine core nodes are protected in any equilibrium. In particular, we have\n    $\n  \\min_{\\bm{e} \\in \\mathcal{E}(n,c)} \\mathbf{E}\\hat{U}^{\\mathbf{\\sf{D}}}(\\bm{e}) \\ge f(n - n_B\\left\\lceil \\frac{n}{k} \\right\\rceil - n_A + 1) - nc\n  $. \n  Moreover, we can estimate\n \\begin{equation}\n  \\begin{aligned}\n  \\left\\lceil \\frac{n}{k} \\right\\rceil \\le \\frac{n}{k} + 1 &\\le \\frac{n}{\\frac{n}{N+1} - 1} + 1 \\\\ &\\le \\frac{n}{\\frac{n}{N+1} - \\frac{n}{2(N+1)}} + 1 = 2N + 3 \\, .\n  \\end{aligned}\n \\end{equation}\n Hence, using \\cref{fast_growth}, we get\n  \\begin{align*} \n  \\lim_{n \\to +\\infty} \\mathrm{PoA}(n,c) &\\le \\lim_{n \\to +\\infty} \\frac{f(n)}{f(n - n_B(2N + 3) - n_A + 1) - nc}  \\\\ &= \\lim_{n \\to +\\infty} \\frac{1}{\\frac{f(n - n_B(2N + 3) - n_A + 1)}{f(n)} - \\frac{nc}{f(n)}} = 1 \\, . \\qedhere\n  \\end{align*}\n\\end{proof}\n\n\\begin{remark}\nNotice that the condition of \\cref{th:poa} is verified for $f(x) = x^{a}$ with $a \\geq 2$. \nHence, in the case of such functions $f$, the price of anarchy is $1$, so the inefficiencies due to decentralization are fully mitigated by the network design.\nThis is true, in particular, for Metcalfe's law.\n\\end{remark}\n\n\\section{Extensions of the model}\\label{sec:extension}\n\nIn the previous section, we have shown that the topology of generalized $k$-star mitigates the costs of decentralization in our model. Nevertheless, our approach can be used to show similar results in a number of modified models. For instance, one could consider a probabilistic model, in which $n_B$ byzantine nodes are randomly picked from the set of nodes $V$ (and the distribution of this random variable is known to all players). Then, the designer and nodes optimize their expected utilities, not the pessimistic ones (where the expectation is taken over the possible positions of the byzantine nodes). In this case, we still can give a partial characterization of Nash equilibria on generalized $k$-stars. More precisely, one can show that if the assumptions of \\cref{thm:ROEchar} are fulfilled and $f(1) < c < \\frac{f(x)}{x}$, then all genuine core nodes are protected. This is exactly what we need in the proof of \\cref{th:poa}. Therefore, the price of anarchy in the probabilistic model also converges to $1$ as the size of the network increases. \n\n\\section{Conclusions}\\label{sec:concl}\nWe studied a model of network defense and design in the presence of an intelligent adversary and byzantines nodes that cooperate with the adversary. We characterized optimal defended networks in the case where defense decisions are centralized, assuming that the number of byzantine nodes and the number of attacked nodes are equal to one. We have also shown that, in the case of sufficiently well-behaved functions $f$ (including $f$ in line with Metcalfe's law), careful network design allows to fully mitigate the inefficiencies due to decentralized protection decisions, despite the presence of the byzantine nodes. In terms of network design, we showed that a generalized star is a topology that can be used to achieve this goal. This topology creates incentives for protection by two means. Firstly, it is sufficiently redundant, so that the protected nodes are connected to several other protected nodes. This secures adequate network value even if some of these nodes are malicious. Secondly, it gives sufficient exposure to the nodes, encouraging the nodes that would benefit from protection to choose to protect through fear of being infected (either directly or indirectly). These results could be valuable, in particular, to policy-makers and regulators, showing that such regulations can have strong effect and providing hints for which network structures are better and why.\n\nAn interesting avenue for future research is to consider a setup where not only the identities but also the number of byzantine nodes are unknown. How would the optimal networks look like if the protection decisions are centralized? Can we still mitigate the inefficiencies caused by decentralization? Another interesting problem are the optimal networks under centralized protection when the number of byzantine nodes or the budget of the adversary are greater than~$1$. Based on our experiments, we suspect that the topology of these networks is very similar to the case considered here. Nevertheless, a formal result remains elusive.\n\n\\bibliographystyle{alpha}\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec:Introduction}\n\n\n\n \nThere is a vast number of applications of Gaussian random fields across several engineering and scientific disciplines including systems and control \\cite{ringh2016multidimensional,zhu2021multidimensional}, signal processing \\cite{stoica2005spectral,qian2022afd}, geotechnical engineering \\citep{chen2016cpt,latz2019fast}, image processing \\cite{winkler2003image}, biology, and meteorology \\cite{croci2018efficient,khristenko2019analysis}. \nIn these applications, we often face the standard problem of sampling a random field which could be used e.g., for the numerical solution of a stochastic partial differential equation (PDE). \nIn particular in geotechnical engineering, certain geomaterial properties are modeled as zero-mean stationary random fields indexed by spatial or temporal variables. Then the correlations between random variables in the field are described by the covariance function. \nIn order to carry out simulations in geomaterial modeling, a first step is to generate (possibly large-scale) samples of a stationary random field such that its covariances coincide with the values of a prescribed covariance function. \nA traditional method for this problem is called covariance matrix decomposition (CMD) \\cite{fenton2008risk} which employs e.g., the Cholesky factorization.\nSuch a method in general costs $O(N^3)$ flops given an $N\\times N$ matrix, which is computationally prohibitive when the covariance matrix has a large size. The latter case is typical for multidimensional random fields. For example, a $3$-d random field with a (moderate) size $100\\times 100\\times 100$ results in a $10^6\\times 10^6$ covariance matrix after vectorization. Thus applications of CMD are seriously limited to small-scale and unidimensional cases (time series). However, many practical problems involve multidimensional random fields \\cite{vanmarcke2010random,graham2018analysis} and the ability to efficiently generate large-scale samples is also important.  \n\nIn the literature, there are a number of methods to handle such a problem. Recent developments include \\citet{graham2018analysis} which aims to embed a finite multilevel Toeplitz covariance matrix into a larger positive definite multilevel circulant matrix, following earlier works on the embedding problem for (one-level) Toeplitz covariance matrices \\cite{dembo1989embedding,dietrich1997fast}. Then the fast Fourier transform (FFT) can be used to compute the spectral decomposition of the (multilevel) circulant matrix at a reduced cost.  \nAnother direction is to use the Karhunen--Lo{\\`e}ve expansion for the continuous covariance kernel and to truncate the expansion for practical computations \\cite{zheng2017simulation,latz2019fast,qian2022afd}.\nA different strategy for the approximation of the covariance matrix involves the notion of $H$ matrices \\cite{feischl2018fast} of which the square root can be computed at an almost linear cost.\n\n\n\n\n\nAlthough the above works are mathematically interesting and deal with general covariance functions, difficulties can still arise when one wants to generate very large-scale samples, especially when the dimension is greater than or equal to three. In order to handle the latter issue, the paper \\cite{li2019stepwise} made a \\emph{decoupling assumption} on the multivariable covariance function and proposed a stepwise CMD method which essentially computes the matrix square root along each dimension and thus reduces the computational cost compared to a full CMD. However, the space domain computation in that paper can be once again greatly reduced if one is able to recognize the frequency-domain structure of the covariance function, namely the \\emph{spectral density} of the random field. This is the main idea behind the current paper. More specifically, we draw inspiration from the systems and control literature on \\emph{stochastic realization} \\cite{LP15} and \\emph{rational covariance extension} in which one aims to describe an underlying random process with a linear dynamical system driven by white noise. The latter topic has undergone decades of development from scalar random processes to vector random fields, see \\cite{BGL-98,byrnes2001finite,enqvist2004aconvex,Georgiou-06,GL-08,FPR-08,LPcirculant-13,Zhu-Baggio-19,zhu2020well,ZFKZ2-M2-Aut,Liu-Zhu-2021,Liu-Zhu-2022-CCSSTA} and the references therein. \nIn particular, we have shown that the exponential covariance function corresponds exactly to an autoregressive (AR) model (a rational filter) of order one, which is easy to implement recursively and permits the sample generation at a \\emph{linear} cost. The decoupling assumption makes straightforward the generalization to multidimensional random fields, and the resulting multidimensional filter is simply a product of individual filters in each dimension. In this way, the filtering procedure is also decoupled as expected.\n\n\n\n\n\n\n\nIn addition, our approach is extremely suitable for multiscale simulations, see e.g., \\citet {chen2016cpt}, where the generated samples of a random field are \\emph{interpolated} and fed into a numerical PDE solver in order to obtain a refined solution.\nThe usual interpolation method samples the probability density of the fine-scale random variables whose values are to be determined, conditioned on the coarse-scale samples that have already been generated. The advantage of our approach is that, once a suitable refined noise input has been determined, only ``boundary'' samples that are necessary to initiate the \\emph{fine-scale} ARMA recursions need to be computed from the conditional probability density, and the rest fine-scale samples are generated in the same fashion as the coarse-scale sampling.\n \n\nThe outline of the paper is as follows. In Section~\\ref{Sec:Problem statement} we state the problem of sampling a stationary random field with a given covariance function that can be decoupled in each dimension. \nIn Section~\\ref{Sec:Spectral analysis} we propose a stochastic realization approach to the sampling problem and focus on the solutions for the exponential and Gaussian covariance functions.\nIn Section~\\ref{Sec:Multi-scale random fields} we integrate our method to multiscale simulations and provide an explicit solution procedure for the bivariate exponential covariance function.\nA number of numerical simulations are presented in Section~\\ref{Sec:Numerical examples}, and in Section~\\ref{Sec:Conclusions} we draw the conclusions.\n\n\n\n\\section{Background on sampling random fields}\\label{Sec:Problem statement}\n\nLet $y(\\mathbf t,\\omega)$ be a $d$-dimensional real random field over a probability space $(\\Omega, \\mathcal{F}, P)$ where $\\mathbf t=(t_1,\\dots,t_d)\\in\\mathbb R^d$ can be interpreted as a space (or spatio-temporal) coordinate vector. For each fixed $\\mathbf t\\in\\mathbb R^d$, assume that $y(\\mathbf t,\\cdot)$ is a zero-mean real-valued random variable with a finite variance, that is,\n\\begin{equation*}\n{\\mathbb E} y(\\mathbf t,\\cdot) = 0\\quad \\text{and} \\quad {\\mathbb E}\\, [y(\\mathbf t,\\cdot)]^2 < \\infty\n\\end{equation*}\nwhere ${\\mathbb E}$ indicates mathematical expectation.\nIt is customary to suppress the dependence on $\\omega$ and write simply $y(\\mathbf t)$. Assume further that the random field under consideration is \\emph{stationary}, which means that the covariance function\n\\begin{equation*}\n\\rho(\\mathbf t,\\sbf) := {\\mathbb E} [y(\\mathbf t)y(\\sbf)]\n\\end{equation*}\ndepends only on the difference $\\boldsymbol{\\tau}:=\\mathbf t-\\sbf$ between the arguments, so we can write $\\rho(\\boldsymbol{\\tau})$ instead. Notice the symmetry $\\rho(-\\boldsymbol{\\tau}) = \\rho(\\boldsymbol{\\tau})$.\n\nIn applications of geotechnical engineering, see e.g., \\cite{firouzianbandpey2015effect,ching2017characterizing}, it is often assumed that the covariance function has a decoupled form, \nnamely\n\\begin{equation}\\label{eq:decoupled form}\n\t\\rho(\\boldsymbol{\\tau})=\\rho_1(\\tau_1) \\rho_2(\\tau_2) \\cdots \\rho_d(\\tau_d),\n\\end{equation}\nwhere each $\\rho_j$ is a covariance function in one variable, and $\\tau_j\\in \\mathbb R$ is the $j$-th component of $\\boldsymbol{\\tau}$. \nIn the following, since we shall be concerned with the sampling problem of the random field $y(\\mathbf t)$, let us now define the sampled version of the random field as well as the covariance function.\nTake $\\tau_j = x_j T_j$ where $T_j>0$ is sampling distance and $x_j\\in\\mathbb Z$. Define a random field on the integer grid $\\mathbb Z^d$ via\n\\begin{equation}\\label{sampled_random_field}\ny_{\\mathrm{s}}(\\mathbf  x) = y(x_1 T_1,\\dots,x_d T_d)\n\\end{equation}\nwhere the subscript s means ``sampled''. Then it is easy to deduce that the covariance function of the discrete random field $y_{\\mathrm{s}}$ is\n\\begin{equation}\\label{rho_sampled}\n\t\\rho_{\\mathrm{s}}(\\mathbf k) = \\rho(k_1 T_1,\\dots,k_d T_d) = \\rho_1 (k_1 T_1) \\cdots \\rho_d (k_d T_d),\n\\end{equation}\nwhere $\\mathbf k=(k_1,\\dots,k_d)\\in\\mathbb Z^d$ denotes the difference between two discrete grid points.\nThe sampling problem can then be phrased as follows.\n\\begin{problem}\\label{prob_sampling_general}\n\tGiven a covariance function $\\rho(\\boldsymbol{\\tau})$ of the form \\eqref{eq:decoupled form}, a vector $\\mathbf T=(T_1,\\dots,T_d)$ of sampling distances, and a vector $\\mathbf N=(N_1,\\dots,N_d)$ of positive integers, generate samples of the random field $y_{\\mathrm{s}}(\\mathbf  x)$ in \\eqref{sampled_random_field} for $\\mathbf  x$ in the index set $($a regular cuboid$)$\n\t\\begin{equation}\\label{ind_set_RF}\n\t\\mathbb Z^d_{\\mathbf N} := \\{(x_1,\\dots,x_d) : 0\\leq x_j\\leq N_j-1,\\ j=1,\\dots,d\\}\n\t\\end{equation}\n\tsuch that its covariance function coincides with the sampled version $\\rho_{\\mathrm{s}}(\\mathbf k)$ in \\eqref{rho_sampled}.\n\\end{problem}\n\n\nThe most straightforward approach for this problem is covariance matrix decomposition mentioned in the Introduction. More precisely, since the index set \\eqref{ind_set_RF} has a finite cardinality, all the samples can be stacked into a long vector $\\mathbf  y$ of dimension $|\\mathbf N|:=\\prod_{j=1}^{d}N_j$. Then the covariance matrix $\\boldsymbol{\\Sigma}:={\\mathbb E}(\\mathbf  y\\yb^\\top)$ could in principle be evaluated elementwise according to \\eqref{rho_sampled}. Problem~\\ref{prob_sampling_general} would then be solved via simple linear algebra\n\\begin{equation}\\label{eq:Ranom field realization}\n\t\\mathbf  y=L\\mathbf  w,\n\\end{equation}\nwhere $\\mathbf  w\\sim \\mathcal{N}(\\mathbf{0},I)$ is an i.i.d.~standard normal random vector of dimension $|\\mathbf N|$, and $L$ is any matrix square root of $\\boldsymbol{\\Sigma}$ which can in particular, be taken as the Cholesky factor such that $LL^{\\top}=\\boldsymbol{\\Sigma}$. However, it is well known that the matrix factorization, which is the major computational burden here, involves $O(|\\mathbf N|^3)$ flops. Therefore, such a naive approach works only for random fields with a dimension $d=1$ or $2$ when the size of the samples $|\\mathbf N|$ is not too large. For the generation of large-scale samples, one needs to exploit the inherent structure of the covariance function in order to facilitate fast computation.\nThe latter point is indeed the theme of the next section where we will propose an efficient stochastic realization \napproach to Problem~\\ref{prob_sampling_general}.\nMore specifically, we consider two types of $1$-d covariance functions of practical interest:\n\\begin{enumerate}\n\t\\item the exponential type\n\t\\begin{equation}\\label{eq:exponential function}\n\t\t\\rho(x) = \\sigma^2 e^{-\\alpha|x|},\n\t\\end{equation}\n\n\t\\item the Gaussian type\n\t\\begin{equation}\\label{eq:Gaussian function}\n\t\t\\rho(x) = \\sigma^2 e^{-\\alpha|x|^{2}},\n\t\\end{equation}\n\\end{enumerate}\nwhere $\\sigma^2$ is the variance of the random field\nand $\\alpha>0$ is a parameter. The multidimensional covariance function is formed through the product in \\eqref{eq:decoupled form}.\nNotice that our method works also for other types of covariance functions $\\rho(x)$ provided that the decoupling assumption \\eqref{eq:decoupled form} holds.\nIn this case, the spectral density of the random field can be well approximated in each dimension by a low-order rational model.\n\n\n\\begin{remark}\nThe exponential and the Gaussian covariance functions are special cases of the Mat\\'{e}rn family of covariance functions, defined as\n\t\\begin{equation}\\label{matern_cov}\n\t\\rho(\\boldsymbol{\\tau}) = \\kappa(\\|\\boldsymbol{\\tau}\\|\/\\lambda)\\quad \\text{where}\\quad \\kappa(r) = \\sigma^2 \\frac{2^{1-\\nu}}{\\Gamma(\\nu)} (\\sqrt{2\\nu}\\, r)^\\nu K_{\\nu}(\\sqrt{2\\nu}\\, r).\n\t\\end{equation}\n\tIn the formulas above, $\\boldsymbol{\\tau}\\in\\mathbb R^d$, $\\|\\cdot\\|$ is the Euclidean norm, $\\lambda$ the correlation length, $\\sigma^2$ the variance, $\\nu>0$ a smoothness parameter, $\\Gamma$ the gamma function, and $K_{\\nu}$ the modified Bessel function of the second kind. Notice that the case $\\nu=1\/2$ corresponds to the exponential covariances, while $\\nu=\\infty$ corresponds to the Gaussian function of the form $\\kappa(r) = \\sigma^2 e^{-r^2\/2}$. See e.g., \\cite[Example 2.7]{graham2018analysis}.\n\\end{remark}\n\n\n\\section{A stochastic realization approach}\\label{Sec:Spectral analysis}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\tikzstyle{int}=[draw, minimum size=2em]\n\t\\tikzstyle{init} = [pin edge={to-,thin,black}]\n\t\\begin{tikzpicture}[node distance=2cm,auto,>=latex']\n\t\\node [int] (a) {$\\ W(\\mathbf  z)\\ $};\n\t\\node (b) [left of=a, coordinate] {};\n\t\\node (c) [right of=a] {};\n\n\t\\path[->] (b) edge node {$w(\\mathbf  x)$} (a);\n\t\\path[->] (a) edge node {$y(\\mathbf  x)$} (c);\n\n\t\\end{tikzpicture}\n\t\\caption{A $d$-dimensional linear stochastic system with a white noise input.}\n\t\\label{Fig:d_linear_system}\n\\end{figure}\n\nLet $\\mathbf  z=(z_1,\\cdots,z_d)$ be a vector of indeterminates.\nConsider a $d$-dimensional discrete-``time'' linear stochastic system as depicted in Fig.~\\ref{Fig:d_linear_system}, where \n\\begin{equation}\nW(\\mathbf  z) = \\sum_{\\mathbf k \\in \\mathbb Z^d} \\gamma(\\mathbf k) \\,\\mathbf  z^{-\\mathbf k}\n\\end{equation}\nis the transfer function, also called a shaping filter in the signal processing literature. Here the function $\\gamma:\\mathbb Z^d\\to \\mathbb R$ is called the \\emph{impulse response} of the system and $\\mathbf  z^{\\mathbf k}$ is a shorthand notation for $z_1^{k_1}\\cdots z_d^{k_d}$. Moreover, the symbol $\\mathbf  z^{-\\mathbf k}$ can be interpreted as a $\\mathbf k$-step delay operator. The system is excited by a normalized white noise $w(\\mathbf  x)$ such that for any $\\mathbf  x\\in\\mathbb Z^d$,\n\\begin{equation}\n{\\mathbb E} [w(\\mathbf  x)]=0\\quad\\text{and} \\quad {\\mathbb E}[w(\\mathbf  x+\\mathbf k) w(\\mathbf  x)]=\\delta_{\\mathbf k,\\mathbf 0} \n= \\begin{cases}\n1 & \\text{if}\\ \\mathbf k=\\mathbf 0, \\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThe output $y(\\mathbf  x)$ is a zero-mean stationary random field. Symbolically, we write\n\\begin{equation}\\label{eq:linear-stochastic-realization}\ny(\\mathbf  x)=W(\\mathbf  z)w(\\mathbf  x):= \\sum_{\\mathbf k \\in \\mathbb Z^d} \\gamma_{\\mathbf k} w(\\mathbf  x-\\mathbf k).\n\\end{equation}\nNotice that this is a standard model for stationary processes which goes back to the prediction theory of Wiener in the 1940s.\n\n\nLet $\\sigma(\\mathbf k):={\\mathbb E} [y(\\mathbf  x+\\mathbf k) y(\\mathbf  x)]$ be the covariance function of $y(\\mathbf  x)$ and let $\\mathbb T:=[-\\pi, \\pi)$ denote the frequency interval. Then the spectral density of $y(\\mathbf  x)$ is by definition \\cite{stoica2005spectral,LP15} the multidimensional discrete-time Fourier transform (DTFT) of the covariance function:\n\\begin{equation}\n\\Phi(e^{i\\boldsymbol{\\theta}})=\\sum_{\\mathbf k \\in \\mathbb Z^d} \\sigma(\\mathbf k)\\, e^{-i \\innerprod{\\mathbf k}{\\boldsymbol{\\theta}}},\n\\end{equation} \nwhere the frequency vector $\\boldsymbol{\\theta}=(\\theta_1,\\dots,\\theta_d)\\in \\mathbb T^d$, $e^{i\\boldsymbol{\\theta}}:=(e^{i\\theta_1},\\dots,e^{i\\theta_d})$ is a point on the $d$-torus (which is isomorphic to $\\mathbb T^d$), and\n$\\innerprod{\\mathbf k}{\\boldsymbol{\\theta}}:=k_1\\theta_1+\\cdots+k_d\\theta_d$ is the standard inner product in $\\mathbb R^d$. It then follows from the spectral theory of stationary random fields \\cite{yaglom1957some} that\n\\begin{equation}\\label{eq:Phi=w(z)w(z^-1)}\n\\Phi(e^{i\\boldsymbol{\\theta}})=|W(e^{i\\boldsymbol{\\theta}})|^2\n\\end{equation}\nwhere $W(e^{i\\boldsymbol{\\theta}})= \\sum_{\\mathbf k \\in \\mathbb Z^d} \\gamma(\\mathbf k) \\,e^{-i\\innerprod{\\mathbf k}{\\boldsymbol{\\theta}}}$, so $\\Phi(e^{i\\boldsymbol{\\theta}})$ takes \\emph{nonnegative} values.\nOn the other hand, if the spectral density $\\Phi$ satisfies certain analytic properties, then it admits a \\emph{spectral factor} $W$, see \\cite{LP15}.\n\n\n\n\nWe are mostly interested in the case where $W(\\mathbf  z)$ is a \\emph{rational} function, that is, it can be expressed as a ratio of two polynomials:\n\\begin{equation}\\label{eq:transfer_func}\n\tW(\\mathbf  z)=\\frac{b(\\mathbf  z)}{a(\\mathbf  z)}=\\frac{\\sum_{\\mathbf k \\in \\Lambda_{+,2}}b_{\\mathbf k}\\mathbf  z^{-\\mathbf k}}{\\sum_{\\mathbf k \\in \\Lambda_{+,1}}a_{\\mathbf k}\\mathbf  z^{-\\mathbf k}},\n\\end{equation}\nwhere\n\\begin{equation*}\n\\begin{aligned}\n\\Lambda_{+,1} & := \\{(k_1,\\cdots,k_d)\\in\\mathbb Z^d : 0\\leq k_j \\leq m_j,\\ j=1, \\cdots, d \\}, \\\\\n\\Lambda_{+,2} & := \\{(k_1,\\cdots,k_d)\\in\\mathbb Z^d : 0\\leq k_j \\leq n_j,\\ j=1,\\cdots,d\\}\n\\end{aligned}\n\\end{equation*}\nare two index sets with positive integers $m_j, n_j$ given for $j=1,\\dots,d$.\nThen the system \\eqref{eq:linear-stochastic-realization} can equivalently be described in the time domain as an autoregressive moving-average (ARMA) model\n\\begin{equation}\\label{eq:ARMA recursion}\n\t\\sum_{\\mathbf k \\in \\Lambda_{+,1}} a_{\\mathbf k}\\, y(\\mathbf  x-\\mathbf k) = \\sum_{\\mathbf k \\in \\Lambda_{+,2}} b_{\\mathbf k}\\, w(\\mathbf  x-\\mathbf k).\n\\end{equation}\nSuch a model is extremely useful in practice because rational functions (in fact, polynomials) can approximate any continuous function if the model order is sufficiently large. If the moving-average part of the model is trivial, i.e., $b(\\mathbf  z)\\equiv b_{\\mathbf 0}$ is a constant, then \\eqref{eq:ARMA recursion} reduces to a simpler AR model. \n\n\n\n\nIn the above context, Problem~\\ref{prob_sampling_general} can be posed more concretely as follows:\n\n\\begin{problem}\\label{prob_realization}\t\n\tGiven a sampled covariance function $\\rho_{\\mathrm{s}}(\\mathbf k)$ in \\eqref{rho_sampled}, find a rational filter $W(\\mathbf  z)$ of the form \\eqref{eq:transfer_func} such that when it is fed with a normalized white noise, the covariance function of the output $y(\\mathbf  x)$ coincides with $\\rho_{\\mathrm{s}}(\\mathbf k)$.\n\tEquivalently, we seek a rational spectral density satisfying the trigonometric moment equations\n\t\\begin{equation}\\label{d-dim_moment_eqns}\n\t\\int_{\\mathbb T^d} e^{i\\innerprod{\\mathbf k}{\\boldsymbol{\\theta}}} \\Phi(e^{i\\boldsymbol{\\theta}}) \\d\\mu(\\boldsymbol{\\theta}) = \\rho_{\\mathrm{s}}(\\mathbf k)\\quad \\forall \\mathbf k\\in\\mathbb Z^d\n\t\\end{equation}\n\twhere $\\d\\mu(\\boldsymbol{\\theta}) = \\frac{1}{(2\\pi)^d} \\prod_{j=1}^{d} \\d\\theta_j$ is the normalized Lebesgue on $\\mathbb T^d$.\n\\end{problem}\n\nNotice that the equivalence above is understood modulo the spectral factorization \\eqref{eq:Phi=w(z)w(z^-1)}.\nOnce the above problem is solved, samples of the random field $y_{\\mathrm{s}}(\\mathbf  x)$ for $\\mathbf  x$ indexed in \\eqref{ind_set_RF} can be generated efficiently via the ARMA recursion \\eqref{eq:ARMA recursion} with a white noise input.\n\n\n\nNext, in view of the decoupling assumption \\eqref{rho_sampled}, it follows easily from the multidimensional DTFT that the corresponding spectral density ${\\Phi}(e^{i\\boldsymbol{\\theta}})$ also has a decoupled form \n\\begin{equation}\\label{Spe-deco}\n\t{\\Phi} (e^{i\\boldsymbol{\\theta}}) = {\\Phi}_1(e^{i\\theta_1})\\, \\cdots\\, {\\Phi}_d(e^{i\\theta_d}),\n\\end{equation}\nwhere the factor ${\\Phi}_j(e^{i\\theta_j})$ can be interpreted as the spectral density in the $j$-th dimension for $j=1,\\cdots,d$, i.e., it is the $1$-d DTFT of the sampled covariance function $\\rho_{\\mathrm{s},j}(k_j):=\\rho_j (k_j T_j)$.\nTherefore, the $d$-dimensional moment equations \\eqref{d-dim_moment_eqns} decouple into $d$ sets of unidimensional moment equations\n\\begin{equation}\\label{one-d_moment_eqns}\n\\int_{\\mathbb T} e^{ik_j\\theta_j} \\Phi_j(e^{i\\theta_j}) \\frac{\\d\\theta_j}{2\\pi} = \\rho_{\\mathrm{s},j}(k_j) \\quad \\forall k_j\\in\\mathbb Z,\\ j=1,\\dots,d,\n\\end{equation}\nwhose solutions have been extensively studied in the literature.\nAfter each $\\Phi_j$ has been constructed from the covariance function $\\rho_{\\mathrm{s},j}$, we can perform the spectral factorization $\\Phi_j (z_j) = W_j (z_j) W_j (z_j^{-1})$ to obtain the transfer function $W_j(z_j)$. Notice that since $\\Phi_j$ is constrained to be rational, the spectral factorization reduces to that for positive trigonometric polynomials, for which there are a number of algorithms \\cite{sayed2001survey}.\nHence, the above procedure leads to a $d$-dimensional ARMA model\n\\begin{equation}\\label{d-dimensional ARMA model}\n\ty(\\mathbf  x) = \\underbrace{W_1(z_1) \\, \\cdots\\, W_d(z_d)}_{=: W(\\mathbf  z)} \\, w(\\mathbf  x)\n\\end{equation}\nagain in a decoupled form. \nThe main steps of our approach for sampling stationary random fields are summarized as follows.\n\\begin{enumerate}\n\t\\item[1)] Given a sampled covariance function \\eqref{rho_sampled}, solve the decoupled moment equations \\eqref{one-d_moment_eqns} for a rational spectral density of the form \\eqref{Spe-deco}.\n\t\\item[2)] Do spectral factorization to obtain the linear filter in \\eqref{d-dimensional ARMA model}.\n\t\\item[3)] Feed the filter with a Gaussian i.i.d.~white noise and collect the output random field.\n\\end{enumerate}\n\n\n\nIn the next subsections, we focus on the nontrivial Step 1 with two types of covariance functions mentioned before, i.e., the exponential and Gaussian covariance functions. It is worth remarking that the case of the exponential covariance function admits an \\emph{exact} rational spectrum and hence a shaping filter in a \\emph{closed form}.\nAlthough the Gaussian covariance function does not lead to an analytic solution, it can be well \\emph{approximated} in the frequency domain by a rational spectral density in the sense that only low-order covariances with significant values are matched in \\eqref{one-d_moment_eqns}. \n\n\n\\subsection{Solution for the exponential covariance function: an AR$(1)$ model}\\label{subsec-expon}\n\n\nUnder the decoupling assumption for the $d$-dimensional covariance function, we only need to solve Problem~\\ref{prob_realization} for each sampled covariance function $\\rho_{\\mathrm{s},j}(k_j)=\\rho_j (k_j T_j)$ of one variable, as discussed previously. Hence, we suppress the subscript $j$.\nConsider first the exponential covariance function in \\eqref{eq:exponential function}, i.e., $\\rho(x) = \\sigma^2 e^{-\\alpha|x|}$ with $x\\in\\mathbb R$. We can for simplicity take $\\sigma^2=1$ since it is only a multiplicative constant. Let $T>0$ be the sampling distance, and we have\n\\begin{equation}\\label{sampled_cov_exp}\n\\rho_{\\mathrm{s}}(k) := \\rho(k T) = e^{-\\alpha T|k|},\\quad k\\in \\mathbb Z.\n\\end{equation}\nLet $r=e^{-\\alpha T}$.\nUsing the DTFT pair\n\\begin{equation}\n\tx(k) = r^{k}u(k) \\xmapsto{\\mathcal{F}} X(e^{i\\theta}) = \\frac{1}{1 - r e^{-i\\theta}} \n\\end{equation}\nwith $0<|r|<1$ and $u(k)$ the discrete-time unit step function, the spectral density corresponding to $\\rho_\\mathrm{s}(k)$ is\n\\begin{equation}\\label{Phi_posi_real}\n\t{\\Phi}(e^{i\\theta}) = Z(e^{i\\theta}) + Z(e^{i\\theta})^* = 2\\Re\\{ Z(e^{i\\theta})\\},\n\\end{equation}\nwhere $^*$ denotes complex conjugate and\n\\begin{equation}\n\tZ(e^{i\\theta}) = \\frac{1}{1 - e^{-\\alpha T} e^{-i\\theta}} - \\frac{1}{2}\n\\end{equation}\nis the so-called \\emph{positive real} part of ${\\Phi} (e^{i\\theta})$. Notice that $Z(e^{i\\theta})$ admits an \\emph{analytic extension} as a rational function\n\\begin{equation}\n\tZ(z) = \\frac{1}{1 - e^{-\\alpha T} z^{-1}} - \\frac{1}{2},\\quad z\\in\\mathbb C,\n\\end{equation}\nso that ${\\Phi}(z)$ in \\eqref{Phi_posi_real} is rational as well. Then after some straightforward calculations, we obtain a transfer function\n\\begin{equation}\\label{eq:Expo-W}\nW(z)=\\frac{(1-e^{-2\\alpha T})^{\\frac{1}{2}}}{1-e^{-\\alpha T} z^{-1}}\n\\end{equation}\nwhich is a \\emph{stable and minimum-phase} spectral factor of $\\Phi(z)$, namely\n\\begin{equation}\n\t{\\Phi}(z) = W(z) \\, W(z^{-1})\n\\end{equation}\nwhere the identity is valid in a neighborhood of the unit circle.\nWe see that $W(z)$ corresponds to an AR model of order one that depends on the parameter $\\alpha_\\mathrm{s}:=\\alpha T$. \nIn consequence, samples of the discrete random field $y_\\mathrm{s}(\\mathbf  x)$ can be generated by filtering a white noise through $d$ cascaded AR$(1)$ models, one for each dimension. \nObviously, the algorithm achieves a very low computational cost because each AR component has only order one.\nCompared with \\cite{li2019stepwise}, our approach has a more concise frequency-domain interpretation and a neater time-domain implementation that avoids factorization of large matrices.\nMoreover, the model can be reused for computing multiple realizations of the random filed because the filter $W(z)$ remains unchanged with a given covariance function when the sampling distance is fixed. Although the Cholesky factor in the stepwise CMD method \\cite{li2019stepwise} can also be reused, our method requires much less storage since only the filter coefficients need to be stored.\nIn addition, our model can be used to compute samples of an \\emph{arbitrary} size. On the contrary, the CMD has to be redone from the start if the sample size changes.\nA comparison of the computational complexity between different algorithms will be given at the end of this section.\n\n\n\\subsection{Solution for the Gaussian covariance function: an ARMA model}\n\n\nUnlike the case with an exponential covariance function, the problem with a general covariance function may not have an exact analytic solution. \nIn this subsection, we discuss the case with an Gaussian covariance function in \\eqref{eq:Gaussian function}. The corresponding sampled version is\n\\begin{equation}\\label{sampled_cov_Gauss}\n\\rho_{\\mathrm{s}}(k) := \\rho(k T) = \\sigma^2 e^{-\\alpha T^2 |k|^{2}},\\quad k\\in \\mathbb Z.\n\\end{equation}\nIt is well known that the (continuous-time) Fourier transform of a Gaussian function is another Gaussian function which is certainly nonrational. We speculate that the same happens for the discrete samples of a Gaussian function, i.e., the corresponding spectral density is \\emph{not} a rational function. Therefore, an ARMA representation for such a covariance function must by nature be approximate.\n\n\nThe Gaussian function \\eqref{eq:Gaussian function} decays fast as $|x|$ increases. So a natural idea is to construct a rational spectral density \n\\begin{equation}\\label{eq:rational spectral}\n\\Phi(e^{i\\theta})=\\frac{P(e^{i\\theta})}{Q(e^{i\\theta})}\n\\end{equation}\nthat matches a \\emph{finite} number of low-order (dominant) covariances,\nwhere $P$ and $Q$ are positive symmetric trigonometric polynomials. \nThere are a number of solution techniques for this \\emph{rational covariance extension} problem, see e.g., \\cite{byrnes2001finite,Georgiou-L-03,RFP-09,FMP-12,Z-14,Z-14rat,ZFKZ2020M2-SIAM}. In the paper, we adopt the following \\emph{generalized maximum entropy} formulation \\cite{byrnes2001finite,ringh2016multidimensional}:\n\\begin{equation}\\label{Maximum entropy}\n\t\\begin{aligned}\n\t\t&\\underset{\\Phi>0}{\\max}\\ \\int_{\\mathbb T} P(e^{i\\theta}) \\log \\Phi(e^{i\\theta}) \\frac{\\d \\theta}{2\\pi} \\\\\n\t\t&\\mathrm{s.t.}\\quad \\sigma_k = \\int_{\\mathbb T}e^{ik \\theta}\\Phi(e^{i\\theta})\\frac{\\d \\theta}{2\\pi}\\quad \\forall k\\in\\Lambda,\n\t\\end{aligned}\n\\end{equation}\nwhere,\n\\begin{itemize}\n\t\\item $P$ is a \\emph{known} positive symmetric polynomial which is constructed from a \\emph{given} factor $b(z):=\\sum_{k=0}^{n} b_{k}z^{-k}$, i.e., $P(z)=b(z)b(z^{-1})$;\n\t\\item the index set is defined as $\\Lambda:=\\{-m,\\dots,-1,0,1,\\dots, m\\}$ such that $m$ is a user-specified positive integer,\n\t\\item $\\sigma_k$'s are the covariance data evaluated from the covariance function, namely $\\sigma_k = \\rho_{\\mathrm{s}}(k)$.\n\\end{itemize}\nMore precisely, we choose $m$ to be the smallest positive integer such that $\\rho_{\\mathrm{s}}(k)$ is practically zero for all $k>m$. In other words, the approximation procedure takes into account the covariances with significant values and discards the rest.\nThe optimization problem \\eqref{Maximum entropy} is convex and has a unique solution $\\Phi=P\/\\hat{Q}$, where $\\hat{Q}$ is the optimal solution of the dual problem\n\\begin{equation}\\label{eq:dual-problem}\n\t\\underset{Q>0}{\\min}\\ \\innerprod{\\boldsymbol{\\sigma}}{\\mathbf q}-\\int_{\\mathbb T} P(e^{i\\theta})\\log Q(e^{i\\theta}) \\frac{\\d\\theta}{2\\pi},\n\\end{equation}\nwhere $\\mathbf q:=\\{q_k\\}_{k \\in\\Lambda}$ are Lagrange multipliers, $\\innerprod{\\boldsymbol{\\sigma}}{\\mathbf q}:=\\sum_{k\\in \\Lambda}\\sigma_k q_k$ denotes the inner product, and $Q(e^{i\\theta}):=\\sum_{k =-m}^m q_k e^{-i k\\theta}$ is a symmetric trigonometric polynomial.\nFor technical details we refer readers to \\cite{byrnes2001finite,ringh2016multidimensional}. \nThe reason for choosing this formulation is that we want to obtain a rational $\\Phi(e^{i\\theta})$ of the form \\eqref{eq:rational spectral} which is directly connected to the ARMA model via spectral factorization.\nMore precisely, the polynomial $a(z)=\\sum_{k=0}^{m} a_{k}z^{-k}$ corresponding to the AR coefficients can be determined via the Bauer method \\cite{sayed2001survey} for \nfactoring $\\hat{Q}(z) = \\sum_{k=-m}^{m} \\hat{q}_{k}z^{-k}$ where $\\hat{q}_k$'s are the optimal Lagrange multipliers. \n\n\n\n\nIn this way, once the rational spectral density $\\Phi(z)$ and the filter \n\\begin{equation}\\label{W_z_ARMA}\nW(z) = \\frac{b(z)}{a(z)} =\\frac{\\sum_{k=0}^{n} b_{k}z^{-k}}{\\sum_{k=0}^{m} a_{k}z^{-k}}\n\\end{equation}\nare constructed in each dimension, samples of the random field $y_\\mathrm{s}(\\mathbf  x)$ can be generated in the same fashion as described in the previous subsection, now via the cascaded ARMA recursions. Each ARMA recursion has a fixed computational cost (though larger than that of the AR$(1)$ model) related to the model order $(m, n)$ which is chosen small. \n\n\nNext, we focus on the computational complexity of our approach to Problem~\\ref{prob_realization}.\nFor practical applications, we are primarily interested in the $3$-d case. Assume that we are asked to generate samples of a random field with a size $\\mathbf N=(N_1,N_2,N_3)\\in\\mathbb Z^3$, see \\eqref{ind_set_RF}, and each ARMA model in the cascade has order $(m_j,n_j)$ for $j=1,2,3$. It is then common practice to generate samples of a slightly larger size $\\mathbf M=(M_1, M_2, M_3)$ which can be taken as $(1+\\beta)\\mathbf N$, say with $\\beta=0.1$, in order to reduce the ``transient'' effect of filtering caused by an artificial boundary condition.\nFor the exponential covariance function, the computational cost of the algorithm is proportional to the product $M_1M_2M_3$, so the complexity is $O(N_1N_2N_3)$ in which we have absorbed the constant $1+\\beta$ into the capital $O$ notation.\nFor the Gaussian covariance function which corresponds to an ARMA model, the algorithm still mainly runs in $O(N_1N_2N_3)$ flops because the computational cost of solving a small-size convex optimization problem \\eqref{eq:dual-problem} is far lower than that of implementing the ARMA recursion.\n\nIn Table~\\ref{tab:table_Compu} we state the computational costs of different methods: traditional CMD by Cholesky decomposition, stepwise CMD, circulant embedding method, and our stochastic realization approach, where an instance with specific numbers is also shown for clarity.\n\\begin{table*}[t]\n\t\\begin{center}\n\t\t\\caption{\\centering The computational complexity of different methods in the $3$-d case. In the particular example, we have $\\mathbf N=(100,100,100)$, $\\mathbf C=(512,512,512)$, and $\\mathbf M=1.1\\mathbf N$.}\n\t\t\\label{tab:table_Compu}\n\t\t\\begin{tabular}{lll}  \n\t\t\t\\toprule   \n\t\t\tMethods& Major computational complexity&Example (flops) \\\\\n\t\t\t\\midrule   \n\t\t\tCMD&$O(N_1^{3}N_2^{3}N_3^{3}) $ &$1.00\\times10^{18}$\\\\\n\t\t\tStepwise CMD &$O(N_1N_2N_3(N_1+N_2+N_3))$&$3.00\\times10^8$\\\\\n\t\t\tCirculant Embedding&$O(C_1 C_2 C_3 \\log_{2} (C_1 C_2 C_3))$ &$3.62\\times 10^9$\\\\\n\t\t\tStochastic Realization&$O(N_1N_2N_3)$&$1.00\\times10^6$\\\\\n\t\t\t\\bottomrule  \n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\nIt can be seen that our method has the lower computational complexity which is \\emph{linear} in the number of samples.\n\n\n\n\\section{Application to multiscale simulations}\\label{Sec:Multi-scale random fields}\n\n\nIn some applications, e.g., the analysis of certain geomaterial properties \\cite{chen2012characterization,chen2016cpt}, we often face a challenging problem that \\emph{refinements} of the generated samples of the random field are needed across several scales, where the spatial variability, i.e., the covariance function is expected to be maintained. \nHere by ``a fine-scale simulation'', we mean obtaining samples of the random field on a denser grid by \\emph{interpolating} the already generated coarse-scale realization, instead of generating fine-scale samples from scratch. Our approach is particularly suitable for such a purpose as we shall describe shortly.\n\n\n\nFor simplicity, we assume that the sampling distance $T$ (which is the scale parameter) in \\eqref{sampled_cov_exp} and \\eqref{sampled_cov_Gauss} is halved in each fine-scale simulation, i.e., $T_1=\\frac{1}{2}T$, $T_2=\\frac{1}{2}T_1=\\frac{1}{4}T$, where the subscript denotes the number of fine-scale simulations that we perform. We want to point out that the model parameters in \\eqref{eq:Expo-W} and \\eqref{W_z_ARMA} in general \\emph{change} across different simulation scales since they depend on $T$, although the model order one is maintained in the case of an exponential covariance function. The change of the AR coefficients in \\eqref{W_z_ARMA} is more implicit and comes from the fact that the covariance data $\\sigma_k$'s are changed in a fine-scale simulation. In any case, once the fine-scale model \\eqref{eq:Expo-W} or \\eqref{W_z_ARMA} is constructed, the interpolation of the samples of the random field can be carried out by the same filtering technique as in the generation of the coarse-scale samples, \\emph{when} suitable boundary conditions and the fine-scale noise input (which is not i.i.d.~any more) are provided. \n\n\\subsection{The boundary conditions}\n\nThe determination of such boundary conditions is rather standard.\nSuppose that the coarse-scale samples of the random field are collected into a column vector $\\mathbf  y_1$ and the \\emph{boundary} random variables that are needed to initiate the fine-scale filtering are collected in $\\mathbf  y_2$. \nHere we want to remark on the special $1$-d case with the exponential covariance function, where the coarse-scale samples are themselves boundary conditions for the fine-scale simulation due to the AR$(1)$ model structure. Hence, all one needs to do is to generate the white noise samples on the new grid points and to implement the filtering. In general however, some boundary values of the fine-scale samples in $\\mathbf  y_2$ are needed in order to start the multidimensional filtering, as shown in Fig.~\\ref{Fig:2-d-multi-schematic-figure} for the $2$-d case where the two cascaded filters are both AR$(1)$.\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=10cm]{Fig_Multi_scale\/2d-example.eps}   \n\t\t\\caption{A schematic figure for the boundary conditions in a multiscale simulation, where the number of values in boundary locations represents only a small fraction of the total number of unknown variables.}\n\t\t\\label{Fig:2-d-multi-schematic-figure}\n\t\\end{center}\n\\end{figure}\n\nLet $\\mathbf  y$ be the joint vector such that\n\\begin{equation}\\label{condi_normal}\n\\mathbf  y=\\left[ \\begin{matrix} \\mathbf  y_1\\\\\\mathbf  y_2\\end{matrix} \\right]\\sim \\mathcal{N}(\\left[ \\begin{matrix} \\mathbf{0}\\\\ \\mathbf{0}\\end{matrix} \\right], \\left[ \\begin{matrix}\\boldsymbol{\\Sigma}_{11}&\\boldsymbol{\\Sigma}_{12}\\\\ \\boldsymbol{\\Sigma}_{21}&\\boldsymbol{\\Sigma}_{22}\\end{matrix} \\right]),\n\\end{equation} \nwhere we have introduced explicitly the Gaussianness assumption for the random field, and the matrix on the right side is\nthe covariance matrix $\\boldsymbol{\\Sigma}$ of $\\mathbf  y$ which is evaluated using the given covariance function\\footnote{Recall that the locations of samples in $\\mathbf  y$ are known.} $\\rho(\\boldsymbol{\\tau})$ in \\eqref{eq:decoupled form} and partitioned in accordance with $\\mathbf  y_1$ and $\\mathbf  y_2$.\nThen the unknown random vector $\\mathbf  y_2$ conditioned on $\\mathbf  y_1=\\mathbf a$ still has a multivariate normal distribution $(\\mathbf  y_2|\\mathbf  y_1=\\mathbf a)\\sim \\mathcal{N}(\\bar{\\boldsymbol{\\mu}},\\bar{\\boldsymbol{\\Sigma}})$ where\n\\begin{equation}\\label{eq:mean-vector}\n\t\\bar{\\boldsymbol{\\mu}}=\\boldsymbol{\\Sigma}_{21}\\boldsymbol{\\Sigma}_{11}^{-1}\\mathbf a,\n\\end{equation}\nand\n\\begin{equation}\\label{eq:correlation-matrix}\n\t\\bar{\\boldsymbol{\\Sigma}}=\\boldsymbol{\\Sigma}_{22}-\\boldsymbol{\\Sigma}_{21}\\boldsymbol{\\Sigma}^{-1}_{11}\\boldsymbol{\\Sigma}_{12}.\n\\end{equation}\nThe matrix $\\bar{\\boldsymbol{\\Sigma}}$ is known as the \\emph{Schur complement} of $\\boldsymbol{\\Sigma}_{11}$ in $\\boldsymbol{\\Sigma}$. It is positive definite because so is $\\boldsymbol{\\Sigma}$. \nThus, the unknown boundary values $\\mathbf  y_{2}$ in a fine-scale simulation can be computed via:\n\\begin{equation}\n\\mathbf  y_{2}=R\\mathbf  e+\\bar{\\boldsymbol{\\mu}},\n\\end{equation}\nwhere the matrix $R$ constitutes a \\emph{rank factorization} of $\\bar{\\boldsymbol{\\Sigma}}$, namely $\\bar{\\boldsymbol{\\Sigma}}=R R^\\top$ which can be computed from the spectral decomposition plus truncation, and $\\mathbf  e$ is an i.i.d.~standard normal vector. \n\n\t\n\\subsection{The noise input}\n\nObviously, samples on the fine-scale depend upon the white noise input which cannot be randomly generated any more because of the existence of the coarse-scale samples as interpolation conditions. Thus it is reasonable to \\emph{determine} the white noise before implementing ARMA recursion, and we report a particular solution for the case with exponential covariance functions, namely,\n\t\\begin{equation*}\\label{Exponential-multi-scale}\n\t\t\\rho_{\\mathrm{s}} (k_1,k_2) = \\sigma^2 e^{-\\alpha_1 T_1 |k_1|-\\alpha_2 T_2 |k_2|}, \\quad (k_1,k_2)\\in\\mathbb Z^2.\n\t\\end{equation*}\nHere we focus on $2$-d random fields which are of engineering interest in e.g., \\citep{chen2016cpt,chen2012characterization}, also for the simplicity of presentation. In principle, a similar procedure should work for the higher dimensional cases but the calculations will necessarily be more complicated.\n\n\n\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[node distance=2cm,auto,>=latex']\n\t\t\\node[draw, coordinate]                       \t\t(start)   {};\n\t\t\\node[draw, right=of start, minimum size=2em]       (step 1)  {$W_1(z_1)$};\n\t\t\\node[draw, right=of step 1, minimum size=2em]      (step 2)  {$W_2(z_2)$};\n\t\t\\node[draw, coordinate, right=of step 2]\t\t\t(end)\t  {};\n\t\t\\draw[->] (start)  -- node[above] {$w(s,t)$} (step 1);\n\t\t\\draw[->] (step 1) -- node[above] {$y_1(s,t)$} (step 2);\n\t\t\\draw[->] (step 2) -- node[above] {$y(s,t)$} (end);\n\t\t\\end{tikzpicture}\n\t\t\\caption{The two cascaded linear stochastic system with given white noise input.}\n\t\t\\label{Fig:2-cascaded filter}\n\t\\end{figure}\n\n\tAssume that we have computed the fine-scale samples in boundary locations, and we know the coarse-scale white noise and samples which have a size $(N_1,N_2)$.\n\tWe aim to interpolate the fine-scale input $w(s,t)$ and then samples $y(s,t)$ for $0 \\leq s \\leq 2(N_1 - 1),\\ 0 \\leq t \\leq 2(N_2 - 1)$. In particular, the indices $s$ and $t$ of the coarse-scale samples are \\emph{even} numbers in the fine-scale realization.\n\tReferring to Fig.~\\ref{Fig:2-cascaded filter}, the two cascaded AR(1) filters of the fine-scale random field are written as\n\t\\begin{equation*}\n\tW_1(z_1)=\\frac{b}{1-az_1^{-1}},\\quad W_2(z_2)=\\frac{d}{1-cz_2^{-1}}.\n\t\\end{equation*}\n\tThe input $w$, the intermediate output $y_1$, and the output $y$ are related via\n\t\\begin{equation*}\n\ty_1(s,t)= W_1(z_1) w(s,t),\\quad y(s,t)=W_2(z_2) y_1(s,t).\n\t\\end{equation*}\n\tApply twice the AR recursion, we have\n\t\\begin{equation}\\label{fine-scale w}\n\t\tw(s-1,t)=\\frac{1}{ab} y_1(s,t) - \\frac{a}{b} y_1(s-2,t) - \\frac{1}{a} w(s,t).\n\t\\end{equation}\n    If we take $s = 2k$ and $t=2\\ell$ with $k=1, \\cdots, N_1-1$ and $\\ell=1, \\cdots, N_2-1$, $w(2k,2\\ell)$ is known as the coarse-scale noise input. Then $w(2k-1,2\\ell)$ can be computed directly via \\eqref{fine-scale w} where\n\t\\begin{equation*}\n\t\ty_1(2k,2\\ell) = \\frac{1}{d'} y(2k,2\\ell) - \\frac{c'}{d'} y(2k-2,2\\ell),\n\t\\end{equation*}\n\twhere $c'$ and $d'$ correspond to the filter $W'_2(z_2)$ of coarse-scale random field, and the $y$ samples involved are from the coarse-scale realization.\n\tFor odd $t=2\\ell-1$, $w(2k,2\\ell-1)$ is unknown and \\eqref{fine-scale w} can be rewritten as a linear equation\t\n\t\\begin{equation}\\label{linear-equation}\n\t\tA\\mathbf  w = \\mathbf  b\n\t\\end{equation}\n\twith\n\t\\begin{equation*}\n\t\tA=\\left[ \\begin{matrix} a&1&0&0&\\cdots&0&0\\\\\n\t\t0&0&a&1&\\cdots&0&0\\\\\n\t\t\\vdots &\\vdots &\\vdots& \\vdots& \\ddots& \\vdots&\\vdots\\\\\n\t\t0&0&0&0&\\cdots&a&1\\end{matrix} \\right],\\,\n\t\t\\mathbf  w=\\left[ \\begin{matrix} w(1,t)\\\\w(2,t)\\\\ \\vdots\\\\ w(2N_1-2,t)\\end{matrix} \\right],\\,\n\t\t\\mathbf  b=\\frac{1}{b} \\left[ \\begin{matrix} y_1(2,t)\\\\ y_1(4,t)\\\\ \\vdots\\\\ y_1(2N_1-2,t) \\end{matrix} \\right] - \\frac{a^2}{b} \\left[ \\begin{matrix} y_1(0,t)\\\\ y_1(2,t)\\\\ \\vdots\\\\ y_1(2N_1-4,t) \\end{matrix} \\right],\n\t\\end{equation*}\n\twhere the coefficient matrix $A$ has a size $(N_1 - 1) \\times (2N_1-2)$, and $y_1 (2k,2\\ell-1)$ is computed via\n\t\\begin{equation*}\n\t\ty_1(2k,2\\ell-1)= \\frac{1}{cd} y(2k,2\\ell) - \\frac{c}{d} y(2k,2\\ell-2) - \\frac{1}{c} y_1(2k,2\\ell),\n\t\\end{equation*}\n\twhere again the right-hand side requires only the coarse-scale samples.\n    The equation \\eqref{linear-equation} has infinitely many solutions since $A$ is of full row rank. In order to make the problem well-posed, we notice a \\emph{dual} linear equation by swapping $W_1(z_1)$ and $W_2(z_2)$ in the filtering which obviously does not change the final output $y$. The intermediate output is, however, different and we write it as\n    \\begin{equation*}\n    y_2(2k,2\\ell)= \\frac{1}{b'}y(2k,2\\ell)- \\frac{a'}{b'} y(2k,2\\ell-2)\n    \\end{equation*}\n    where $a'$ and $b'$ correspond to the filter $W'_1(z_1)$ of coarse-scale random field.\n    We can now compute the components $w(2k,2\\ell-1)$ in the vector $\\mathbf  w$ as\n\t\\begin{equation*}\n\t\tw(2k,2\\ell-1)= \\frac{1}{cd} y_2(2k,2\\ell) -\\frac{c}{d} y_2(2k,2\\ell-2) -\\frac{1}{c} w(2k,2\\ell)\n\t\\end{equation*}\n    in the same fashion as computing $w(2k-1,2\\ell)$ from \\eqref{fine-scale w}. Substitute the result back into \\eqref{linear-equation}, we get the rest components.\n\t\n\t\n\n\n\n\n\n\n\t\nIn this way, the fine-scale samples can be generated via the AR recursion easily as the values in the boundary locations and the white noise input have been computed.\nWe would like to point out that unlike traditional methods which generate \\emph{all} the fine-scale samples (which could be a lot) using the conditional distribution \\eqref{condi_normal}, see e.g., \\citet[Sec.~2.4]{chen2012characterization}, \nwe only need $\\mathbf  y_2$ in boundary locations (whose number is much smaller, see Fig.~\\ref{Fig:2-d-multi-schematic-figure}).\nTherefore, our procedure involves multiplications and inversions of matrices of smaller sizes, which significantly improves computational efficiency. After the boundary values of the fine-scale random field are computed and the white noise input is obtained, the interpolation of the coarse-scale samples can be accomplished again at a linear cost.\n\n\n\\section{Numerical examples}\\label{Sec:Numerical examples}\n\nIn this section, we perform two sets of numerical simulations of our stochastic realization approach: one for sampling $3$-d random fields and the other includes multiscale simulations in the $2$-d case.\n\n\n\\subsection{Sampling $3$-d Gaussian random fields}\\label{Subsec:AR representation}\n\nFirst, we apply the stochastic realization approach to Problem~\\ref{prob_realization} with an exponential covariance function in three variables\n\\begin{equation}\\label{eq:exponential-covariance-function}\n\t\\rho_{\\mathrm{s}} (x,y,z)=\\sigma^2 e^{-\\alpha_1 T_1 |x|-\\alpha_2 T_2 |y|-\\alpha_3 T_3|z|},\\quad (x,y,z)\\in\\mathbb Z^3.\n\\end{equation}\nIn the following example, the size of the samples to be generated is $\\mathbf N = (100,100,100)$ for the random field $y_{\\mathrm{s}} (x,y,z)$, the variance is $\\sigma^2 = 1$, the parameter vector is $(\\alpha_1, \\alpha_2, \\alpha_3)=(1,1,1)$, and the vector of sampling distances along $x$-, $y$-, and $z$-directions is $(T_1, T_2, T_3) = (1\/12, 1\/10, 1\/8)$.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=8.4cm]{Fig_Exponential_Ran\/3-D-random.eps}   \n\t\t\\caption{A $3$-d random field realization with an exponential covariance function. The size of the realization is $8.33$m $\\times 10.00$m $\\times 12.50$m in space with a total number of samples equal to $100^3 = 10^6$\n\t\t}\n\t\t\\label{Fig:Three-Dimensional random field}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Exponential_Ran\/E-x-d.eps}\n\t\n\t\\end{minipage}\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Exponential_Ran\/E-y-d.eps}\n\t\n\t\\end{minipage}\n\t\\centering\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Exponential_Ran\/E-z-d.eps}\n\t\n\t\\end{minipage}\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Exponential_Ran\/E-d-d.eps}\n\t\n\t\\end{minipage}\n\t\\caption{The covariances versus distances along four directions of the $3$-d random field realization which contains 100 repeated trials. The red line denotes the given exponential covariance function and the blue lines are the corresponding sample covariance lags. By $x$-direction, we mean the section $[k_1,0,0]$ of the sample covariance array with $k_1=0,\\dots,N_1-1$. The other directions are understood similarly.}\n\t\\label{Fig:Four-direction covariance}\n\\end{figure}\n\nIt has been discussed in Subsection~\\ref{subsec-expon} that the exponential covariance function corresponds to an exact rational spectral density, and the filter in each dimension can be directly computed via \\eqref{eq:Expo-W}. \nThus the required samples of the random field can be generated easily by implementing three cascaded AR recursions \\eqref{d-dimensional ARMA model}. \nA realization of the random field is shown in Fig.~\\ref{Fig:Three-Dimensional random field} using the Matlab command \\verb*|slice|, where the spatial distances are defined as $\\tau_x=T_1|x|$, $\\tau_y = T_2|y|$, and $\\tau_z = T_3|z|$ along three directions. \nNext, in order to verify the performance of our method, we also plot the \\emph{sample covariances} of the realization $y_{\\mathrm{s}}(x,y,z)$ versus spatial distances along $x$-, $y$-, $z$-, and the diagonal directions in Fig.~\\ref{Fig:Four-direction covariance} with $100$ repeated trials. \nThe diagonal direction is along the line $x=y=z$ with $0\\leq x \\leq N_1-1$. \n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=8.4cm]{Fig_Gaussian_Ran\/Gau-3D.eps}   \n\t\t\\caption{A $3$-d random field realization with a Gaussian covariance function. The size of the realization is $2$m $\\times$ $3.13$m $\\times$ $5.56$m in space with a total number of samples equal to $50^3=1.25\\times 10^5$.}\n\t\t\\label{Fig:Three-Dimensional Gaussian random field}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{minipage}{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Gaussian_Ran\/G-x-d.eps}\n\t\n\t\\end{minipage}\n\t\\begin{minipage}{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Gaussian_Ran\/G-y-d.eps}\n\t\n\t\\end{minipage}\n\t\\centering\n\t\\begin{minipage}{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Gaussian_Ran\/G-z-d.eps}\n\t\n\t\\end{minipage}\n\t\\begin{minipage}{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=6cm]{Fig_Gaussian_Ran\/G-d-d.eps}\n\t\n\t\\end{minipage}\n\t\\caption{The covariances versus distances along four directions of the $3$-d random field realization which contains 100 repeated trials. The red line denotes the given Gaussian covariance function and the blue lines are the corresponding sample covariance lags. By $x$-direction, we mean the section $[k_1,0,0]$ of the sample covariance array with $k_1=0,\\dots,N_1-1$. The other directions are understood similarly.}\n\t\\label{Fig:Four-direction covariance-Gaussian}\n\\end{figure}\nIn particular, the sample covariances of $y_{\\mathrm{s}} (x,y,z)$ are computed via the spatial average \\cite[Section~5]{ZFKZ2-M2-Aut}:\n\\begin{equation}\n\t\\hat\\sigma_\\mathbf k:=\\frac{1}{|\\mathbf N|}\\sum_{\\mathbf  x} y_{\\mathrm{s}} (\\mathbf  x+\\mathbf k){y_{\\mathrm{s}} (\\mathbf  x)}\t\n\\end{equation}\nwhere $\\mathbf  x=(x,y,z)$ and $|\\mathbf N| = N_1 N_2 N_3$. \nSince we have explicitly enforced covariance matching in Problem~\\ref{prob_realization}, it follows from the general covariance estimation theory \\cite{priestley1981spectral} that the sample covariances of the output of the filter $W(\\mathbf  z)$ must be close to the values of the given covariance function when the sample size is sufficiently large,\nand this point is well illustrated in Fig~\\ref{Fig:Four-direction covariance}. \nWe remark that our result is visually much better than that reported in \\cite[Sec.~5.1]{li2019stepwise}.\n\n\n\nFurthermore, another simulation is performed for a Gaussian covariance function that has the form\n\\begin{equation}\\label{eq:Gaussian-random-produce}\n\t\\rho_{\\mathrm{s}} (x,y,z)=\\sigma^2 e^{-\\alpha_1 T_1^2 |x|^2- \\alpha_2 T_2^2 |y|^2- \\alpha_3 T_3^2 |z|^2}, \\quad (x,y,z)\\in\\mathbb Z^3.\n\\end{equation}\nThe parameters are reported as follows: $\\sigma^2=1$, $(\\alpha_1, \\alpha_2, \\alpha_3) = (1,1,1)$, and $(T_1, T_2, T_3) = (1\/5, 1\/4, 1\/3)$.\nIn this case, we set up an ARMA model to approximate the underlying spectral density. \nThe numerator polynomial $b(\\mathbf  z)$ in \\eqref{eq:transfer_func} is specified by the user.\nHere for simplicity, we take $b(z_1,z_2,z_3) = \\prod_{j=1}^3 b_j(z_j)$ with a quite arbitrary $b_j(z_j)=1-0.2z_j^{-1}$ of order one and identical for $j=1,2,3$.\nThe orders of the denominator polynomials $a_j(z_j)$ are chosen to be $(m_1,m_2,m_3)=(8,7,7)$ which is a threshold for ``dominant'' covariances, i.e., large values of the covariance function. \nThen the approximate spectrum in each dimension is constructed via solving the optimization problem \\eqref{Maximum entropy}, where Newton's method is implemented, and the polynomial $a_j(z_j)$ is obtained from spectral factorization.\nThe results are shown in Figs.~\\ref{Fig:Three-Dimensional Gaussian random field} and \\ref{Fig:Four-direction covariance-Gaussian} similar to the previous two figures.\nSince the above Gaussian covariance function decays fast as the spatial distance increases, we set the sample size $\\mathbf N=(50,50,50)$ in order to show more details in the figures. \nIt can be seen that the sample covariances are very close to the given Gaussian covariance function when the lag is small. The mismatch for large lags can be explained by the fact that we  are using a relatively low-order ARMA spectrum to approximate the nonrational Gaussian function. \nIn principle, we can also use a higher-order ARMA model for a better approximation, however, at the price of much more computational cost when generating the samples.\n\n\n\n\n\n\\subsection{Multiscale simulations in the $2$-d case}\\label{Subsec:Multi-scale realization}\n\n\nIn this subsection, we perform simulations to produce fine-scale samples of the random field given the coarse-scale realization using our stochastic realization approach. Following the derivation in Section \\ref{Sec:Multi-scale random fields}, we report an example with the following exponential covariance function\n\\begin{equation*}\n\t\t\\rho_{\\mathrm{s}} (x,y)=\\exp(-\\frac{|x|}{5}-\\frac{|y|}{4}), \\quad (x,y)\\in\\mathbb Z^2.\n\\end{equation*}\n\nSuppose that the coarse-scale samples of the random field have a size vector $(N_1,N_2)=(20,20)$ which corresponds to $400$ points. We take the sampling distances to be $1\/5$ and $1\/4$ along $x$- and $y$-directions, respectively, i.e., $\\mathbf T=(T_1,T_2)=(1\/5,1\/4)$, and the parameter vector $(\\alpha_1, \\alpha_2) = (1,1)$.\nThen three fine-scale realizations of the random field are computed, where the sampling distances are reduced to $1\/2$, $1\/4$, and $1\/8$ of the original values, respectively. \nThe simulations are carried out sequentially. More precisely, after the samples with the parameter $\\mathbf T$ are generated, they are then treated as the coarse-scale realization, and the interpolation procedure is executed to produce fine-scale samples with the parameter $\\mathbf T'=\\frac{1}{2}\\mathbf T$.\nNotice that we need to reconstruct the AR$(1)$ filter under each scale, due to the fact that the filter parameters in \\eqref{eq:Expo-W}\ndepend upon the product $\\alpha_j T_j$.\nThen after the sample values in boundary locations are computed utilizing the conditional normal distribution and the white noise in unknown locations is obtained,\nthe rest fine-scale samples can be generated by implementing the AR recursion. These operations are repeated for each finer scale.\nThe result of such a multiscale simulation is shown in Fig.~\\ref{Fig:Figure:multi-scale}. \nIt is evident that more details of the random field can be seen from the fine-scale samples than the coarse-scale realization.\nWe also plot the sample covariances under each scale versus distances along $x$- and $y$-directions in Fig.~\\ref{Fig:Finer-scale random field Exponential}. \nOne can see that the sample covariances are again close to the given covariance function, which indicates that the spatial variability of the random field is well maintained across multiple scales in our simulations. \n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{Sec:Conclusions}\n\nThis paper proposes an efficient stochastic realization approach for sampling large-scale multidimensional Gaussian stationary random fields.\nThe basic idea is to exploit the decoupling assumption on the covariance function, and to construct a rational model which approximates the spectrum of the underlying random field in terms of covariance matching.\nMoreover, our sampling approach features easy implementation and \n\\begin{figure}[t]\n\t\\centering\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=7.8cm]{Fig_Multi_scale\/Coarse_scale.eps}\n\t\t\\centerline{(a) Coarse-scale samples ($20\\times 20$)}\n\t\\end{minipage}\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=7.8cm]{Fig_Multi_scale\/Fine_scale_2.eps}\n\t\t\\centerline{(b) Fine-scale samples ($40\\times40$)}\n\t\\end{minipage}\n\t\\centering\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=7.8cm]{Fig_Multi_scale\/Fine_scale_4.eps}\n\t\t\\centerline{(c) Fine-scale samples ($80 \\times 80$)}\n\t\\end{minipage}\n\t\\begin{minipage}[h]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=7.8cm]{Fig_Multi_scale\/Fine_scale_8.eps}\n\t\t\\centerline{(d) Fine-scale samples ($160\\times 160$)}\n\t\\end{minipage}\n\t\\caption{Multiscale simulations with an exponential covariance function. Subfig.~(a) shows the coarse-scale samples (colored squares) which are already known, while Subfigs.~(b), (c), and (d) are the fine-scale realizations, where the numbers in parentheses give the number of sampling points in each direction.}\n\t\\label{Fig:Figure:multi-scale}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{minipage}[t]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=7.8cm]{Fig_Multi_scale\/Expo_x_d.eps}\n\t\n\t\\end{minipage}\n\t\\begin{minipage}[t]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=7.8cm]{Fig_Multi_scale\/Expo_y_d.eps}\n\t\n\t\\end{minipage}\n\t\\caption{Covariance lags of the random field versus distances. The red line denotes the given exponential covariance function, and the other four lines correspond to the sample covariances of under different scales. By $x$-direction, we mean the column of the sample covariance matrix indexed by  $[k_1,0]$ with $k_1=0,\\dots,N_1-1$. The $y$-direction is understood similarly.}\n\t\\label{Fig:Finer-scale random field Exponential}\n\\end{figure}\n{\\setlength{\\parindent}{0cm}\nlow computational complexity due to the simple structure of the approximate model.\nThe work in the paper can be concluded as follows:}\n\\begin{enumerate}\n\t\\item[(1)] Solutions to the sampling problem with the exponential and Gaussian covariance functions are given, respectively. The former corresponds to a rational spectral density that leads to an AR$(1)$ filter, and the latter has a nonrational spectral density which can be approximated by an ARMA spectrum.\t\n\t\\item[(2)] The stochastic realization approach has been applied to multiscale simulations. Compared with traditional methods, only a few number of values in boundary locations are computed prior to the interpolation via the ARMA recursion, which achieves high efficiency.\n\t\\item[(3)] Several numerical simulations are performed and they show that our method exhibits good performances not only in sampling large-size random fields, but also in refining generated samples across multiple scales. \n\\end{enumerate}\nFinally, it is expected that our approach can be extended to the multivariate case (i.e., vector processes), which will be a future study.\n\n\n\\section*{Acknowledgment}\nThis work was supported in part by the National Natural Science Foundation of China under the grant number 62103453 and the ``Hundred-Talent Program'' of Sun Yat-sen University.\n\n\\section*{Conflict of interest}\nThe authors declare no conflict of interest.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main goals of the LHC is the identification of the mechanism\nof electroweak symmetry breaking. The most frequently investigated\nmodels are the Higgs mechanism within the Standard \nModel (SM) and within the Minimal Supersymmetric Standard Model\n(MSSM)~\\cite{mssm}. Contrary to the case of the SM, in the MSSM \ntwo Higgs doublets are required.\nThis results in five physical Higgs bosons instead of the single Higgs\nboson in the SM. These are the light and heavy ${\\cal CP}$-even Higgs bosons, $h$\nand $H$, the ${\\cal CP}$-odd Higgs boson, $A$, and the charged Higgs bosons,\n$H^\\pm$.\nThe Higgs sector of the MSSM can be specified at lowest\norder in terms of the gauge couplings, the ratio of the two Higgs vacuum\nexpectation values, $\\tan \\beta \\equiv v_2\/v_1$, and the mass of the ${\\cal CP}$-odd\nHiggs boson, $M_A$ (or $M_{H^\\pm}$, the mass of the charged Higgs boson).\nConsequently, the masses of the ${\\cal CP}$-even neutral and the charged Higgs\nbosons are dependent quantities that can be\npredicted in terms of the Higgs-sector parameters, e.g.\\\n$M_{H^\\pm}^2 = M_A^2 + M_W^2$, where $M_W$ denotes the mass of the $W$~boson.\nThe same applies to\nthe production and decay properties of the MSSM Higgs bosons%\n\\footnote{If the production or decay involves SUSY particles at\n  tree-level, also other MSSM parameters enter the prediction at lowest\n  order.\n.~Higgs-phenomenology\nin the MSSM is strongly affected by higher-order corrections, in\nparticular from the sector of the third generation quarks and squarks,\nso that the dependencies on various other MSSM parameters can be\nimportant, see e.g.\\ \\citeres{PomssmRep,habilSH,mhiggsAWB} for reviews.\n\nSearches for the charged Higgs bosons of the MSSM (or a more general\nTwo Higgs Doublet Model (THDM)) have been carried out at\nLEP~\\cite{LEPchargedHiggsPrel}, yielding a bound of\n$M_{H^\\pm} \\gsim 80 \\,\\, \\mathrm{GeV}$~\\cite{LEPchargedHiggsProc,LEPchargedHiggs}.\nThe Tevatron placed additional bounds on the MSSM parameter space from\ncharged Higgs-boson searches, in particular at large $\\tan \\beta$ and low\n$M_A$~\\cite{Tevcharged}. At the LHC the charged Higgs bosons will be\naccessible best at large $\\tan \\beta$ up to $M_A \\lsim 800 \\,\\, \\mathrm{GeV}$\n\\cite{atlastdr,cmstdr,benchmark3}. At the ILC, for \n$M_{H^\\pm} \\lsim \\sqrt{s}\/2$ a high-precision determination of the charged\nHiggs boson properties will be\npossible~\\cite{tesla,orangebook,acfarep,Snowmass05Higgs}.\n\n\nThe prospective sensitivities at\nthe LHC are usually displayed in terms of the parameters $M_A$ and $\\tan \\beta$\n(or $M_{H^\\pm}$ and $\\tan \\beta$) that characterize the MSSM Higgs sector at lowest\norder. The other MSSM \nparameters are conventionally fixed according to certain benchmark\nscenarios~\\cite{benchmark2}. \nThe respective LHC analyses of the $5\\,\\sigma$ discovery contours for the\ncharged Higgs boson are given in \\citere{HchargedATLAS} for\nATLAS and in \\citeres{lightHexp,heavyHexp} for CMS. \nHowever, within these analyses the variation with relevant SUSY\nparameters as well as possibly relevant loop corrections in the Higgs\nproduction and decay~\\cite{benchmark3} have been neglected. \n\nWe focus in this paper on the $5\\,\\sigma$ discovery contours for the\ncharged MSSM Higgs boson \nfor the two cases $M_{H^\\pm} < m_{t}$ and $M_{H^\\pm} > m_{t}$, \nwithin the $m_h^{\\rm max}$~scenario and the no-mixing\nscenario~\\cite{benchmark2,benchmark3} (i.e.\\ we concentrate on the\n${\\cal CP}$-conserving case).\nThey are obtained by using the latest CMS\nresults~\\cite{lightHexp,heavyHexp} derived in a model-independent\napproach, i.e.\\ making no assumption on the Higgs boson production\nmechanism or decays. However, the detection relies on the decay mode\nof the charged Higgs bosons to $\\tau\\nu_\\tau$. Furthermore only SM\nbackgrounds have been assumed. \nThese experimental results are combined with up-to-date theoretical\npredictions for charged Higgs production and decay in the MSSM, taking\ninto account also the decay to SUSY particles that can in principle\nsuppress the branching ratio of the charged Higgs boson decay to\n$\\tau\\nu_\\tau$.\n\nFor the interpretation of the exclusion bounds and prospective discovery\ncontours in the benchmark scenarios it is important to assess how\nsensitively the results depend on those parameters that have been fixed\naccording to the benchmark prescriptions. In \\citeres{benchmark3,cmsHiggs}\nthis issue has been analyzed for the neutral heavy MSSM Higgs bosons,\nand it has been found that the by far largest effect arises from the\nvariation of the Higgs-mixing parameter~$\\mu$. \nConsequently, we investigate how the \n5$\\,\\sigma$ discovery regions in the $M_{H^\\pm}$--$\\tan \\beta$ plane \nfor the charged MSSM Higgs boson obtainable with the CMS experiment at\nthe LHC are affected by a variation of the \nmixing parameter~$\\mu$.\n\n\n\\section{Experimental analysis}\n\\label{sec:exp}\n\nThe main\nproduction channels at the LHC are\n\\begin{equation}\npp \\to t\\bar t \\; + \\; X, \\quad\nt \\bar t \\to t \\; H^- \\bar b \\mbox{~~or~~} H^+ b \\; \\bar t~\n\\label{pp2Hpm}\n\\end{equation}\nand\n\\begin{equation}\ngb \\to H^- t \\mbox{~~or~~} g \\bar b \\to H^+ \\bar t~.\n\\label{gb2Hpm}\n\\end{equation}\nThe decay used in the analysis to detect the charged Higgs boson is\n\\begin{equation}\nH^\\pm \\; \\to \\; \\tau \\nu_\\tau \\; \\to \\; {\\rm hadrons~}\\nu_\\tau. \n\\label{Hbug}\n\\end{equation}\nThe analyses described below correspond to \nCMS experimental sensitivities based on full simulation studies, \nassuming an integrated luminosity of 30~$\\mbox{fb}^{-1}$.\nIn these analyses a top quark mass of $m_{t} = 175 \\,\\, \\mathrm{GeV}$ has been\nassumed. \n\n\n\n\\subsection{The light charged Higgs Boson}\n\\label{sec:lightHpm}\n\nThe ``light charged Higgs boson'' is characterized by $M_{H^\\pm} < m_{t}$. \nThe main production channel is given in \\refeq{pp2Hpm}. Close to\nthreshold also \\refeq{gb2Hpm} contributes. The relevant (i.e.\\\ndetectable) decay channel is given by \\refeq{Hbug}.\nThe experimental analysis, based on 30~$\\mbox{fb}^{-1}$\\ collected with CMS, is\npresented in \\citere{lightHexp}. The events were required to be\nselected with the single lepton trigger, thus exploiting the\n$W \\to \\ell \\nu$ decay mode of a $W$~boson from the decay of \none of the top quarks in \\refeq{pp2Hpm}.\n\nThe total number of events leading to final states with the signal\ncharacteristics is evaluated, including their respective experimental\nefficiencies. The various channels and the corresponding efficiencies\ncan be found in \\refta{tab:lightHp}. The efficiencies are given for\n$M_{H^\\pm} = 160 \\,\\, \\mathrm{GeV}$, but vary only insignificantly over the parameter\nspace under investigation. \nThe number of signal-like events is evaluated as the sum of \nbackground and Higgs-boson signal events, \n\\begin{align}\nN_{\\rm ev} =& \\;N_{\\rm background} \n               \\mathrm{(from~the~processes~in~\\refta{tab:lightHp})} \\nonumber \\\\\n          &+ {\\cal L} \\times \\sigma(pp \\to t \\bar t + X) \n                 \\times {\\rm BR}(t \\to H^\\pm b)\n                 \\times {\\rm BR}(H^\\pm \\to \\tau \\nu_\\tau) \\\\\n          &\\mbox{}\\hspace{41.5mm} \\times {\\rm BR}(\\tau \\to \\mbox{hadrons})\n                 \\times \\mbox{exp.\\ eff.}~, \\nonumber \n\\end{align}\nwhere ${\\cal L}$ denotes the luminosity, and the experimental efficiency\nis given in \\refta{tab:lightHp}.\nA $5\\,\\sigma$ discovery can be achieved if a parameter point results in\nmore than 5260~events (with 30~$\\mbox{fb}^{-1}$).\\\\\n\\newpage\n\\noindent\nWe furthermore used \n\\begin{align}\n{\\rm BR}(W^\\pm \\to \\ell \\nu_\\ell) &~= ~0.217 ~~(\\ell = \\mu, e), \\nonumber \\\\ \n{\\rm BR}(W^\\pm \\to \\tau \\nu_\\tau) &~= ~0.1085 , \\nonumber \\\\\n{\\rm BR}(W^\\pm \\to \\mbox{jets}) &~= ~0.67 , \\\\\n{\\rm BR}(\\tau \\to \\mbox{hadrons}) &~= ~0.65 . \\nonumber\n\\end{align}\nThe next-to-leading order LHC cross section  for top quark pairs is\ntaken to be 840~pb~\\cite{sigmatt}. \nFor the $W^\\pm$+3 jets background the leading \norder cross section for the process $pp \\to W^{\\pm} + \\rm 3~jets$, \n$W^{\\pm} \\to \\ell^{\\pm} \\nu$ ($\\ell=e,~\\mu$) of 840~pb was used, \nas given by the MadGraph~\\cite{MadGraph} generator.\n\n\\begin{table}[htb!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c|c|} \\hline\nchannel & exp.\\ efficiency \\\\ \\hline\\hline\n$pp \\to t \\bar t +X,\\; t \\bar t \\to H^+ b \\; \\bar t \n                 \\to (\\tau^+ \\bar{\\nu}_\\tau) \\; b \\; (W^- \\bar b)$;  \n~$\\tau \\to \\mbox{hadrons}$, $W \\to \\ell \\nu_\\ell$ & 0.0052 \\\\\n\\hline\n$pp \\to t \\bar t +X,\\; t \\bar t  \\to W^+ \\; W^- \\; b \\bar b\n                 \\to (\\tau \\nu_\\tau) \\; (\\ell \\nu_\\ell) \\; b \\bar b$;\n~$\\tau \\to \\mbox{hadrons}$ & 0.00217 \\\\\n\\hline\n$pp \\to t \\bar t  +X,\\; t \\bar t \\to W^+ \\; W^- \\; b \\bar b\n            \\to (\\ell \\nu_\\ell) \\; (\\ell \\nu_\\ell) \\; b \\bar b$ & 0.000859 \\\\\n\\hline\n$pp \\to t \\bar t  +X,\\; t \\bar t \\to W^+ \\; W^- \\; b \\bar b\n            \\to (\\mbox{jet jet}) \\; (\\ell \\nu_\\ell) \\; b \\bar b$ & 0.000134 \\\\\n\\hline\n$pp \\to W + \\rm 3~jets$, $W \\to \\ell \\nu$ & 0.000013 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-1em}\n\\caption{Relevant signal (first line) and background \n  channels for the light charged Higgs boson and their\n  respective experimental efficiencies. The charge conjugated processes\n  ought to be included. The efficiency for the charged Higgs production\n  is given for $M_{H^\\pm} = 160 \\,\\, \\mathrm{GeV}$, but varies only insignificantly\n  over the relevant parameter space. $\\ell$ denotes $e$ or $\\mu$.\n}\n\\label{tab:lightHp}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\n\n\\subsection{The heavy charged Higgs Boson}\n\\label{sec:heavyHpm}\n\nThe ``heavy charged Higgs boson'' is characterized by $M_{H^\\pm} \\gsim m_{t}$.\nHere \\refeq{gb2Hpm} gives the largest contribution to the production cross\nsection, and very close to \nthreshold \\refeq{pp2Hpm} can contribute somewhat. The relevant decay\nchannel is again given in \\refeq{Hbug}.\nThe experimental analysis, based on 30~$\\mbox{fb}^{-1}$\\ collected with CMS, has been\npresented in \\citere{heavyHexp}. The fully hadronic final state \ntopology was considered, thus events were selected with the single\n$\\tau$ trigger at Level-1 and the combined $\\tau$-$E_{\\rm T}^{\\rm miss}$ High\nLevel trigger. \nThe backgrounds considered were $t \\bar t$, $W^\\pm t$, \n$W^\\pm + 3~{\\rm jets}$ as well as  QCD multi-jet background.\nThe $t \\bar t$ and QCD multi-jet processes were generated with\nPYTHIA~\\cite{pythia}, $W^\\pm t$ was \ngenerated with the TopRex generator~\\cite{toprex} and \n$W^\\pm + 3~{\\rm jets}$ with MadGraph~\\cite{MadGraph}. \nThe production cross sections for the $t\\bar t$~background processes were\nnormalized to the NLO cross sections~\\cite{sigmatt}.\nThe total background amounts (after cuts) to \n$1.7 \\pm 1$ events, independently of the charged Higgs boson mass.\n\n\\noindent\nThe number of signal events is evaluated as\n\\begin{equation}\nN_{\\rm ev} = {\\cal L} \\times \\sigma(pp \\to H^\\pm + X) \n                 \\times {\\rm BR}(H^\\pm \\to \\tau \\nu_\\tau)\n                 \\times {\\rm BR}(\\tau \\to \\mbox{hadrons})\n                 \\times \\mbox{exp.\\ eff.}~,\n\\end{equation}\nwhere ${\\cal L}$ denotes the luminosity, and the experimental efficiency\nis given in \\refta{tab:heavyHp} as a function of $M_{H^\\pm}$.\nA $5\\,\\sigma$ discovery corresponds to a number of signal events larger\nthan $14.1$.\n\n\\begin{table}[htb!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c||cccccc|} \n\\hline\\hline\n$M_{H^\\pm}$ [GeV]            & 171.6 & 180.4 & 201.0 & 300.9 & 400.7 & 600.8 \\\\ \n\\hline\nexp.\\ eff.\\  [$10^{-4}$] & 3.5   & 4.0   & 5.0  & 23    & 32    & 42 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-1em}\n\\caption{Experimental efficiencies for the heavy charged Higgs boson\n  detection. \n}\n\\label{tab:heavyHp}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nThe efficiency for the charged Higgs boson production over the\nfull mass range considered was evaluated with the PYTHIA~\\cite{pythia} \ngenerator processes 401 ($gg \\to tbH^{\\pm}$) and 402 ($qq \\to tbH^{\\pm}$) \nimplemented as described in ~\\citere{tbH}. \n\n\n\\section{Calculation of cross section and branching ratios}\n\\label{sec:theo}\n\nWhile the phenomenology of the production and decay processes of the\ncharged MSSM Higgs bosons at the LHC is mainly characterized by\nthe parameters $M_A$ (or $M_{H^\\pm}$) and $\\tan \\beta$ that govern the Higgs sector\nat lowest \norder, other MSSM parameters enter via higher-order contributions (see\ne.g.\\ \\citere{benchmark3} and references therein), \nand also via the kinematics of Higgs-boson decays into\nsupersymmetric particles. The other MSSM parameters are usually fixed\nin terms of benchmark scenarios. The most commonly used scenarios are\nthe ``$m_h^{\\rm max}$'' and ``no-mixing'' benchmark \nscenarios~\\cite{benchmark2,benchmark3}. According to the\ndefinition of \\citere{benchmark2} the $m_h^{\\rm max}$ scenario is given by,\n\\begin{eqnarray}\n\\mbox{\\underline{$m_h^{\\rm max}:$}} &&\nM_{\\rm SUSY} = 1000 \\,\\, \\mathrm{GeV}, \\quad X_t = 2\\, M_{\\rm SUSY}, \\quad A_b = A_t, \\nonumber \\\\\n&& \\mu = 200 \\,\\, \\mathrm{GeV}, \\quad M_2 = 200 \\,\\, \\mathrm{GeV}, \\quad m_{\\tilde{g}} = 0.8\\,M_{\\rm SUSY}~.\n\\label{mhmax}\n\\end{eqnarray}\nHere $M_{\\rm SUSY}$ denotes the diagonal soft SUSY-breaking parameters in the\nsfermion mass matrices, $m_{t}\\,X_t \\equiv m_{t}\\, (A_t - \\mu\/\\tan \\beta)$ is the\noff-diagonal entry in the scalar top mass matrix. $A_{t(b)}$ denote the\ntrilinear Higgs-stop (-sbottom) couplings, $\\mu$ is the Higgs mixing\nparameter, $m_{\\tilde{g}}$ the gluino mass, and $M_2$ and $M_1$ denote the soft\nSUSY-breaking parameters in the chargino\/neutralino sector. \nThe parameter $M_1$ is fixed via the GUT relation \n$M_1 = (5s_W^2)\/(3c_W^2) \\, M_2$.\nThe no-mixing scenario differs from the $m_h^{\\rm max}$ scenario only in the\ndefinition of \nvanishing mixing in the stop sector and a larger value of $M_{\\rm SUSY}$,\n\\begin{eqnarray}\n\\mbox{\\underline{no-mixing:}} &&\nM_{\\rm SUSY} = 2000 \\,\\, \\mathrm{GeV}, \\quad X_t = 0, \\quad A_b = A_t, \\nonumber \\\\\n&& \\mu = 200 \\,\\, \\mathrm{GeV}, \\quad M_2 = 200 \\,\\, \\mathrm{GeV}, \\quad m_{\\tilde{g}} = 0.8\\,M_{\\rm SUSY}~.\n\\label{nomix}\n\\end{eqnarray}\nThe value of the top-quark mass in \\citere{benchmark2} was chosen\naccording to the experimental central value at that time. For our\nnumerical analysis below, we use \nthe value, $m_{t} = 175 \\,\\, \\mathrm{GeV}$, see \\refse{sec:exp}. \nUsing the current value of $m_{t} = 172.6 \\,\\, \\mathrm{GeV}$~\\cite{mt1726}\nwould lead to a small shift of the discovery contours right at\nthreshold, but is insignificant for the qualitative results of this\nanalysis. \n\nIn \\citere{benchmark3} it was suggested that in the search for heavy\nMSSM Higgs bosons the $m_h^{\\rm max}$ and no-mixing scenarios, which originally\nwere mainly designed for the search for the light ${\\cal CP}$-even Higgs boson\n$h$, should be extended by several discrete values of $\\mu$ (see below),\n\\begin{equation}\n\\mu = \\pm 200, \\pm 500, \\pm 1000 \\,\\, \\mathrm{GeV} ~.\n\\label{eq:variationmu}\n\\end{equation}\nIn our analyses here we focus on $\\mu = \\pm 200, \\pm 1000 \\,\\, \\mathrm{GeV}$. \n\n\\bigskip\nFor the calculation of cross sections and branching ratios we use a\ncombination of up-to-date theory evaluations. The \ninteraction of the charged Higgs boson with the $t\/b$~doublet can be\nexpressed in terms of an effective Lagrangian~\\cite{deltamb2},\n\\begin{equation}\n\\label{effL}\n{\\cal L} = \\frac{g}{2M_W} \\frac{\\overline{m}_b}{1 + \\Delta_b} \\Bigg[ \n    \\sqrt{2} \\, V_{tb} \\, \\tan \\beta \\; H^+ \\bar{t}_L b_R \\Bigg] + {\\rm h.c.}\n\\end{equation}\nHere $\\overline{m}_b$ denotes the running bottom quark mass including SM QCD\ncorrections. \nThe prefactor $1\/(1 + \\Delta_b)$ in \\refeq{effL} arises from the\nresummation of the leading $\\tan \\beta$-enhanced corrections to all orders. \nThe explicit\nform of $\\Delta_b$ in the limit of heavy SUSY masses and $\\tan \\beta \\gg 1$\nreads~\\cite{deltamb1}\n\\begin{equation}\n\\Delta_b = \\frac{2\\alpha_s}{3\\,\\pi} \\, m_{\\tilde{g}} \\, \\mu \\, \\tan \\beta \\,\n                    \\times \\, I(m_{\\tilde{b}_1}, m_{\\tilde{b}_2}, m_{\\tilde{g}}) +\n      \\frac{\\alpha_t}{4\\,\\pi} \\, A_t \\, \\mu \\, \\tan \\beta \\,\n                    \\times \\, I(m_{\\tilde{t}_1}, m_{\\tilde{t}_2}, |\\mu|) ~.\n\\label{def:dmb}\n\\end{equation}\nHere $m_{\\tilde{t}_1}$, $m_{\\tilde{t}_2}$, $m_{\\tilde{b}_1}$, $m_{\\tilde{b}_2}$ denote the $\\tilde{t}$ and\n$\\tilde{b}$~masses. $\\alpha_s$ is the strong coupling\nconstant, while $\\alpha_t \\equiv h_t^2 \/ (4 \\pi)$ is defined via the top\nYukawa coupling. The analytical expression for $I(\\ldots)$ can be found\nin \\citere{benchmark3}.\nLarge negative values of $(\\mu\\,m_{\\tilde{g}})$ and $(\\mu\\,A_t)$ (it should be\nnoted that both\nbenchmark scenarios have positive $m_{\\tilde{g}}$ and $A_t$) can lead to a\nstrong enhancement of the  \n$H^\\pm t b$ coupling, while large positive values lead to a strong\nsuppression. \nConcerning the $m_h^{\\rm max}$ and the no-mixing benchmark scenarios, \nas discussed in \\citeres{cmsHiggs,benchmark3} the $\\Delta_b$ effects are \nmuch more pronounced in the $m_h^{\\rm max}$ scenario, where the two terms in\n\\refeq{def:dmb} are of similar size. In the no-mixing scenario the first\nterm in \\refeq{def:dmb} dominates, while the second term is small. A\nfurther suppression is caused by the larger value of $M_{\\rm SUSY}$ (see\n\\refeq{nomix}) in comparison with the $m_h^{\\rm max}$\nscenario. Consequently, the total effect of $\\Delta_b$ is smaller in the\nno-mixing scenario (see also the discussion in \\citere{benchmark3}). \n\nFor the production cross section in \\refeq{pp2Hpm} we use the SM cross\nsection $\\sigma(pp \\to t \\bar t) = 840~\\rm{pb}$~\\cite{sigmatt}%\n\\footnote{\nThe corresponding SUSY corrections are small~\\cite{sigmattSUSY} and have\nbeen neglected. \n}%\n~times the ${\\rm BR}(t \\to H^\\pm\\, b)$ including the $\\Delta_b$ corrections\ndescribed above. \nThe production cross section in \\refeq{gb2Hpm} is evaluated as given in\n\\citeres{HpmXSa,HpmXSb}. In addition also the $\\Delta_b$ corrections of\n\\refeq{effL} are applied. Finally the ${\\rm BR}(H^\\pm \\to \\tau \\nu_\\tau)$ is\nevaluated taking into account all decay channels, among which the most\nrelevant are $H^\\pm \\to tb, cs, W^{(*)}h$. Also possible decays to \nSUSY particles are taken into account. For the decay to $tb$ again\nthe $\\Delta_b$ corrections are included.\nAll the numerical evaluations are performed with the program \n{\\tt FeynHiggs}~\\cite{feynhiggs,mhiggslong,mhiggsAEC,mhcMSSMlong}, see\nalso \\citere{mhcMSSM2L}. \n\n\n\n\n\\section{Numerical analysis}\n\\label{sec:numanal}\n\nThe numerical analysis has been performed in the $m_h^{\\rm max}$~and the\nno-mixing scenarios~\\cite{benchmark2,benchmark3} for \n$\\mu = -1000, -200, +200, +1000 \\,\\, \\mathrm{GeV}$. \nWe separately present the results for the light and the heavy charged\nHiggs and finally compare with the results in the CMS PTDR, where the\nresults had been obtained fixing $\\mu = +200 \\,\\, \\mathrm{GeV}$ and neglecting the\n$\\Delta_b$ corrections, as well as neglecting the charged Higgs-boson decays\nto SUSY particles. \n\n\n\n\\subsection{The light charged Higgs boson}\n\nIn \\reffi{fig:reachlight} we show the \nresults for the $5\\,\\sigma$ discovery contours for the light \ncharged Higgs boson, corresponding to the experimental\nanalysis in \\refse{sec:lightHpm}, where the charged Higgs boson\ndiscovery will be possible in the areas above the curves shown in\n\\reffi{fig:reachlight}. \nAs described above, the experimental analysis was performed for the\nCMS detector and 30~$\\mbox{fb}^{-1}$. The top quark mass is set to $m_{t} = 175 \\,\\, \\mathrm{GeV}$. \nThe thick (thin) lines correspond to positive (negative) $\\mu$, and the\nsolid (dotted) lines have $|\\mu| = 1000 (200) \\,\\, \\mathrm{GeV}$. \nThe curves stop at $\\tan \\beta = 60$, where we stopped the evaluation of\nproduction cross section and branching ratios. For negative $\\mu$ very\nlarge values of $\\tan \\beta$ result in a strong enhancement of the bottom\nYukawa coupling, and for $\\Delta_b \\to -1$ the MSSM enters a non-perturbative\nregime, see \\refeq{effL}. \n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{mhmax_lightChH_MHP.eps}\\hspace{1em}\n\\includegraphics[width=0.45\\textwidth]{nomix_lightChH_MHP.eps}\n \\caption{%\nDiscovery reach for the light charged Higgs boson of CMS with 30~$\\mbox{fb}^{-1}$\\ in the \n$M_{H^\\pm}$--$\\tan \\beta$~plane for the $m_h^{\\rm max}$~scenario (left) and the no-mixing\nscenario (right).\n}\n\\label{fig:reachlight}\n\\end{center}\n\\end{figure}\n\nWithin the $m_h^{\\rm max}$ scenario, shown in the left plot of\n\\reffi{fig:reachlight}, the search for the light charged Higgs boson covers\nthe area of large $\\tan \\beta$ and $M_{H^\\pm} \\lsim 130 \\ldots 160 \\,\\, \\mathrm{GeV}$. \nThe variation with\n$\\mu$ induces a strong shift in the $5\\,\\sigma$ discovery contours. This\ncorresponds to a shift in $\\tan \\beta$ of \n$\\Delta\\tan \\beta = 15$ for $M_{H^\\pm} \\lsim 110 \\,\\, \\mathrm{GeV}$, rising up to $\\Delta\\tan \\beta = 40$ for\nlarger $M_{H^\\pm}$ values. The discovery region is largest (smallest) for \n$\\mu = -(+)1000 \\,\\, \\mathrm{GeV}$, corresponding to the largest (smallest)\nproduction cross section.\nThe results for the no-mixing scenario are shown in the right plot of\n\\reffi{fig:reachlight}. The effects of the variation of $\\mu$ are much\nless pronounced in this scenario, as discussed in \\refse{sec:theo}, due\nto the smaller \nabsolute value of $\\Delta_b$ (see also the corresponding analysis for neutral\nheavy Higgs bosons in \\citere{cmsHiggs}). The shift in $\\tan \\beta$ for\n$M_{H^\\pm} = 110 \\,\\, \\mathrm{GeV}$ is about $\\Delta\\tan \\beta = 5$ going from $\\mu = -1000 \\,\\, \\mathrm{GeV}$ to\n$+1000 \\,\\, \\mathrm{GeV}$. \nFor $\\tan \\beta = 60$ (where we stop our analysis) the covered $M_{H^\\pm}$ values\nrange from $150 \\,\\, \\mathrm{GeV}$ to $164 \\,\\, \\mathrm{GeV}$.\nIn this charged Higgs boson mass range for the considered benchmark\nscenarios no decay channels into SUSY particles are open, i.e.\\ the\nobserved effects are all due to higher-order corrections, in particular\nassociated with~$\\Delta_b$. \n\n\n\n\\subsection{The heavy charged Higgs boson}\n\nIn \\reffi{fig:reachheavy} we show the \nresults for the $5\\,\\sigma$ discovery contours for the heavy\ncharged Higgs boson, corresponding to the experimental\nanalysis in \\refse{sec:heavyHpm}. The Higgs boson discovery will be\npossible in the areas above the curves.%\n\\footnote{\nAn analysis in other benchmark scenarios that are in\nagreement with the cold dark matter density constraint imposed by WMAP\nand other cosmological data~\\cite{WMAP} can be found in \\citere{ehhow}.}%\n~As before, the experimental analysis was performed for the\nCMS detector and 30~$\\mbox{fb}^{-1}$. The top quark mass is set to $m_{t} = 175 \\,\\, \\mathrm{GeV}$. \nThe thick (thin) lines correspond to positive (negative) $\\mu$, and the\nsolid (dotted) lines have $|\\mu| = 1000 (200) \\,\\, \\mathrm{GeV}$. \n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{mhmax_heavyChH_MHP.eps}\\hspace{1em}\n\\includegraphics[width=0.45\\textwidth]{nomix_heavyChH_MHP.eps}\n \\caption{%\nDiscovery reach for the heavy charged Higgs boson of CMS with 30~$\\mbox{fb}^{-1}$\\ in the \n$M_{H^\\pm}$--$\\tan \\beta$~plane for the $m_h^{\\rm max}$~scenario (left) and the no-mixing\nscenario (right).\n}\n\\label{fig:reachheavy}\n\\end{center}\n\\end{figure}\n\n\nThe $5\\,\\sigma$ discovery regions for the search for heavy charged Higgs\nbosons in the $m_h^{\\rm max}$ scenario are shown in the left plot of\n\\reffi{fig:reachheavy}. For $M_{H^\\pm} = 170 \\,\\, \\mathrm{GeV}$, where the experimental\nanalysis stops, we find a strong variation\nin the accessible parameter space for $\\mu = -(+)1000 \\,\\, \\mathrm{GeV}$ of $\\Delta\\tan \\beta = 40$.\nIt should be noted in this context that close to threshold, where both\nproduction mechanisms, \\refeqs{pp2Hpm} and (\\ref{gb2Hpm}), contribute,\nthe theoretical \nuncertainties are somewhat larger than in the other regions. \nFor $M_{H^\\pm} = 300 \\,\\, \\mathrm{GeV}$ the variation in the $5\\,\\sigma$ discovery contours\ngoes from $\\tan \\beta = 38$ to $\\tan \\beta = 54$. For $\\mu = -1000 \\,\\, \\mathrm{GeV}$ and larger\n$\\tan \\beta$ values the bottom Yukawa coupling becomes so large \nthat a perturbative treatment would no longer be reliable in this\nregion, and correspondingly we do not continue the respective curve(s).\n\nThe shape of the $\\mu = +1000 \\,\\, \\mathrm{GeV}$ curve has a local minimum at \n$M_{H^\\pm} \\approx 300 \\,\\, \\mathrm{GeV}$ that is not (or only very weakly) present in the other\ncurves, and that is also not visible in the original CMS analysis in\n\\citere{heavyHexp} (obtained for $\\mu = +200 \\,\\, \\mathrm{GeV}$, but neglecting the\n$\\Delta_b$ effects). The reason for the local minimum can be traced back to\nthe strongly improved experimental efficiency going from \n$M_{H^\\pm} = 200 \\,\\, \\mathrm{GeV}$ to $300 \\,\\, \\mathrm{GeV}$, see \\refta{tab:heavyHp}. The better\nefficiency at $M_{H^\\pm} = 300 \\,\\, \\mathrm{GeV}$ corresponds to a lower required cross\nsection ($\\propto \\tan^2 \\beta\\hspace{1mm}$) and\/or a lower ${\\rm BR}(H^\\pm \\to \\tau \\nu_\\tau)$\nto obtain the same number of signal events. \nOn the other hand, going from $M_{H^\\pm} = 200 \\,\\, \\mathrm{GeV}$ to $300 \\,\\, \\mathrm{GeV}$ this effect \nis in most cases overcompensated by a decrease of the cross\nsection due to the increase in $M_{H^\\pm}$. The overcompensation results in\nan increase in $\\tan \\beta$ for the higher $M_{H^\\pm}$ value. \nFor $\\mu = +1000 \\,\\, \\mathrm{GeV}$, however, $\\Delta_b$ is very large, \nsuppressing strongly the charged Higgs production cross section as well\nas the ${\\rm BR}(H^\\pm \\to tb)$. The overall effect is a somewhat better\nreach in $\\tan \\beta$ for $M_{H^\\pm} = 300 \\,\\, \\mathrm{GeV}$ than for $M_{H^\\pm} = 200 \\,\\, \\mathrm{GeV}$. \n\nIn comparison with the analysis of \\citere{benchmark3}, based on the\nolder CMS analysis given in \\citere{heavyHexpold}, several differences\ncan be observed. The feature of the local minimum is absent in\n\\citere{benchmark3}, the variation of the $5\\,\\sigma$ discovery contours\nwith $\\mu$ is weaker, and the effect of the decay of the charged Higgs\nboson to a chargino and a neutralino is more pronounced in\n\\citere{benchmark3}. The reason for these differences is the strongly\nreduced discovery region in the new CMS analysis~\\cite{heavyHexp}\nemployed here as compared to the old CMS analysis~\\cite{heavyHexpold} \nused in \\citere{benchmark3}. The reach in $\\tan \\beta$ is worse by\n$\\sim 15 (30)$ for $M_A = 200 (400) \\,\\, \\mathrm{GeV}$ in the new analysis.%\n\\footnote{\nThe old analysis uses $\\mu = -200 \\,\\, \\mathrm{GeV}$~\\cite{heavyHexpold}, while the\nnew analysis set $\\mu = +200 \\,\\, \\mathrm{GeV}$~\\cite{heavyHexp}. However, since the\n$\\Delta_b$ corrections are neglected in \\citeres{heavyHexpold,heavyHexp}, \nthe effect on the discovery regions should be small.\n}%\n~Thus, at the substantially worse (i.e.\\ higher) $\\tan \\beta$ values employed\nhere the $\\Delta_b$ effects are more pronounced, leading to the local minimum\nfor $\\mu = +1000 \\,\\, \\mathrm{GeV}$ and to a larger absolute variation in $\\tan \\beta$ with the\nsize and the sign of $\\mu$, see \\refse{sec:theo}.\nIn the high $\\tan \\beta$ region furthermore the $\\Delta_b$ effects dominate over the\nimpact of the decay of the charged Higgs to charginos and neutralinos. \nAs an example, for $\\mu = +200 \\,\\, \\mathrm{GeV}$ and $M_{H^\\pm} = 400 \\,\\, \\mathrm{GeV}$ the old\nanalysis in \\citere{benchmark3} found that the discovery region starts\nat $\\tan \\beta = 32$, where ${\\rm BR}(H^\\pm \\to \\cha{}\\neu{}) \\approx 15\\%$.\nHere we find that the discovery region starts at $\\tan \\beta = 64$, where\n${\\rm BR}(H^\\pm \\to \\cha{}\\neu{}) \\approx 3\\%$.\n\nThe no-mixing scenario is shown in the right plot of\n\\reffi{fig:reachheavy}. The features are the same as in the $m_h^{\\rm max}$\nscenario. However, due to the smaller size of $|\\Delta_b|$, see\n\\refse{sec:theo}, they are much less pronounced. The variation in $\\tan \\beta$\nstays at or below the level of $\\Delta\\tan \\beta = 10$ for the whole range of\n$M_{H^\\pm}$. \n\n\n\n\\subsection{Comparison with the CMS PTDR}\n\nIn \\reffi{fig:reach} we show the \ncombined results for the $5\\,\\sigma$ discovery contours for the light and\nthe heavy charged Higgs boson, corresponding to the experimental\nanalyses in the $m_h^{\\rm max}$ scenario as presented in the two previous\nsubsections. They are compared with the results presented in the CMS\nPTDR~\\cite{cmstdr}. Contrary to the previous sections, we now show the\n$5\\,\\sigma$ discovery contours in the $M_A$--$\\tan \\beta$ plane. \nThe thick (thin) lines correspond to positive (negative) $\\mu$, and the\nsolid (dotted) lines have $|\\mu| = 1000 (200) \\,\\, \\mathrm{GeV}$. The thickened\ndotted (red\/blue) lines represent the CMS PTDR results, obtained for\n$\\mu = +200 \\,\\, \\mathrm{GeV}$ and neglecting the $\\Delta_b$ effects.\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=0.60\\textwidth]{mhmax_ChHvsPTDR.eps}\n \\caption{%\nDiscovery reach for the charged Higgs boson of CMS with 30~$\\mbox{fb}^{-1}$\\ in the \n$M_A$--$\\tan \\beta$~plane for the $m_h^{\\rm max}$~scenario for \n$\\mu = \\pm 200, \\pm 1000 \\,\\, \\mathrm{GeV}$ in comparison with the results from the CMS\nPTDR (thickened dotted (red and blue) lines), obtained for\n$\\mu = +200 \\,\\, \\mathrm{GeV}$ and neglecting the $\\Delta_b$ effects.\n}\n\\label{fig:reach}\n\\end{center}\n\\end{figure}\n\nApart from the variation in the $5\\,\\sigma$ discovery contours with the\nsize and the sign of $|\\mu|$, two differences can be observed in the\ncomparison of the PTDR results to the new results obtained here, i.e.\\\nincluding the $\\Delta_b$ corrections in the production and decay of the\ncharged Higgs boson as well as taking the decay to SUSY particles into\naccount. \nFor the light charged Higgs analysis the discovery contours are now\nshifted to smaller $M_A$ values, for negative $\\mu$ even ``bending over''\nfor larger $\\tan \\beta$ values. The reason is the more complete inclusion of\nhigher-order corrections (full one-loop and leading \\order{\\alpha_t\\alpha_s}\ntwo-loop) to the relation between $M_A$ and\n$M_{H^\\pm}$~\\cite{mhcMSSMlong,mhcMSSM2L}.\nThe second feature is a small gap between the light and the heavy\ncharged Higgs analyses, while in the PTDR analysis all charged Higgs\nmasses could be accessed. The gap can be observed best by comparing the\n$m_h^{\\rm max}$ scenario in \\reffis{fig:reachlight} and \\ref{fig:reachheavy}. \nThis gap is largest for $\\mu = +1000 \\,\\, \\mathrm{GeV}$ and smallest for \n$\\mu = -1000 \\,\\, \\mathrm{GeV}$, where it amounts only up to $\\sim 5 \\,\\, \\mathrm{GeV}$.\nPossibly the heavy charged Higgs analysis strategy exploiting the fully\nhadronic final state can be extended to smaller $M_A$ values to\ncompletely close the gap. \nFor the interpretation of \\reffi{fig:reach} it should be kept in mind\nthat the accessible area in the heavy Higgs analysis also ``bends over''\nto smaller $M_A$ values for larger $\\tan \\beta$, thus decreasing the visible\ngap in \\reffi{fig:reach}.\n\n\n\\section{Conclusions}\n\nWe have studied the variation of the $5\\,\\sigma$ discovery contours for the\nsearch for the charged MSSM Higgs boson with the SUSY parameters.\nWe combine the latest results for the\nCMS experimental sensitivities based on full simulation studies with \nstate-of-the-art theoretical predictions of MSSM Higgs-boson properties.\nThe experimental analyses are done assuming an integrated luminosity of \n30~$\\mbox{fb}^{-1}$\\ for the two cases, $M_{H^\\pm} < m_{t}$ and $M_{H^\\pm} > m_{t}$.\n\nThe numerical analysis has been performed in the $m_h^{\\rm max}$~and the\nno-mixing scenarios for $\\mu = \\pm 200, \\pm 1000 \\,\\, \\mathrm{GeV}$.\nThe impact of the variation of $\\mu$ enters in particular via the\nhigher-order correction $\\Delta_b$, affecting\nthe charged Higgs production cross section and branching ratios. Also\nthe decays of the charged Higgs boson to SUSY particles have been taken\ninto account. \nAs a general feature, large negative $\\mu$ values give the largest\nreach, while large positive values yield the smallest $5\\,\\sigma$ discovery\nareas.\n\nThe search for the light charged Higgs boson covers the the area of\nlarge $\\tan \\beta$ and $M_{H^\\pm} \\lsim 160 \\,\\, \\mathrm{GeV}$. \nThe variation with $\\mu$ within the $m_h^{\\rm max}$ scenario induces a strong\nshift in the $5\\,\\sigma$ discovery contours with  \n$\\Delta\\tan \\beta = 15$ for $M_{H^\\pm} = 100 \\,\\, \\mathrm{GeV}$, rising up to $\\Delta\\tan \\beta = 40$ for\nlarger $M_{H^\\pm}$ values. The discovery region is largest (smallest) for \n$\\mu = -(+)1000 \\,\\, \\mathrm{GeV}$, corresponding to the largest (smallest)\nproduction cross section. The effects are similar, but much less\npronounced, in the no-mixing scenario.\n\nThe search for the heavy charged Higgs boson reaches up to $M_{H^\\pm} \\lsim\n400 \\,\\, \\mathrm{GeV}$ for large $\\tan \\beta$. \nWithin the $m_h^{\\rm max}$ scenario the variation of $\\mu$ induces a very\nstrong shift in the $5\\,\\sigma$ discovery contours of up to $\\Delta\\tan \\beta = 40$\nfor $M_{H^\\pm} \\gsim m_{t}$. As in the light charged Higgs case, within the\nno-mixing scenario the effects show the same qualitative behavior, but\nare much less pronounced.\n\nCombining the search for the light and the heavy charge Higgs boson, we\nfind a small gap, while in the CMS Physics Technical Design Report\nanalysis all charged Higgs masses could be accessed. \nPossibly the heavy charged Higgs analysis strategy exploiting the fully\nhadronic final state can be extended to smaller $M_A$ values to\ncompletely close the gap. This issue deserves further studies.\n\n\n\n\n\n\\subsection*{Acknowledgements}\n\nThe work of S.H.\\ was partially supported by CICYT (grant FPA~2007--66387).\nWork supported in part by the European Community's Marie-Curie Research\nTraining Network under contract MRTN-CT-2006-035505\n`Tools and Precision Calculations for Physics Discoveries at Colliders'.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn cosmological models that obey General Relativity and the cosmological\nprinciple, the mass density parameter, $\\Omega$, and the cosmological\nconstant, $\\Lambda$, together determine the geometry of spacetime.\nThe Einstein-de Sitter ($\\Omega=1$) model has twin virtues of simplicity:\nflat spatial geometry and zero cosmological constant.\nHowever, the ages of globular cluster stars, the high baryon fraction\nin galaxy clusters, and some aspects of large scale galaxy clustering\nare accounted for more easily in low density models, which have flat \ngeometry if \n$\\mbox{${\\lambda}_{0}$} \\equiv\\Lambda c^{2}\/3 H_{0}^{2}= 1-\\mbox{${\\Omega}_{0}$ }$\\footnote{Subscript \n`zero' indicates the value of the parameters at the present epoch. \nAlso, we use the notation \n$H_{0}\\equiv \n{\\rm h}_{0}\\cdot 100\\; {\\rm km}\\; {\\rm s}^{-1}\\; {\\rm Mpc}^{-1}$}, \nor open, negative curvature geometry if $\\Lambda=0$.\nDynamical studies of large scale\nstructure can constrain \\mbox{${\\Omega}_{0}$ } (subject to uncertainties about\nbiased galaxy formation), but they are insensitive to a cosmological\nconstant because it represents an unclustered energy component.\nThis paper examines the prospects for using the anisotropy of the \nquasar correlation function to constrain \\mbox{${\\Omega}_{0}$ } and \\mbox{${\\lambda}_{0}$},\nin particular to distinguish between flat and open cosmologies.\nWhile this test is impractical with present quasar samples, the\nAnglo-Australian 2-degree Field Survey (hereafter 2dF) and the\nSloan Digital Sky Survey (hereafter SDSS or Sloan) will both produce redshift\nsamples of several tens of thousands of quasars over the next few years.\n\nClassical methods of measuring spacetime geometry rely on standard candles\nor standard rulers, and they are therefore subject to systematic uncertainties\nin the evolution of these objects over a large range in redshift.\nAlcock \\& Paczy\\'{n}ski (1979) proposed an alternative approach that is\nalmost entirely independent of evolution and assumes only that structure\nin the universe is statistically isotropic, as implied by the cosmological\nprinciple.  At high redshift, the ratio of distances corresponding to\na given redshift separation $\\Delta z$ and angular separation $\\Delta \\theta$\ndepends on the spacetime metric, and one can constrain the cosmological\nparameters by requiring that they yield isotropic structure.\n\nAlcock \\& Paczy\\'{n}ski (1979) illustrated their suggestion with the idealized\nexample of spherical clusters, but nature does not provide us with such\nconvenient objects of study.  Ryden (1995a) suggested using the statistical\ndistribution of void shapes in deep galaxy redshift surveys like the SDSS,\nbut it is not clear that even this million galaxy sample will be sufficient\nto detect the effect; it does not extend much beyond $z \\sim 0.2$, and\nat this depth the geometrical distortion of voids is difficult to separate \nfrom distortions induced by galaxy peculiar velocities (Ryden \\& Melott 1996).\nWhile quasar redshift surveys have fewer tracers, they reach to much\nhigher redshifts, where the expected geometrical anisotropy is more \npronounced.  Thus Phillips (1994) suggested using the orientations\nof neighboring quasar pairs to implement Alcock \\& Paczy\\'nski's method.\nThe 2-point correlation function contains statistical information on\nthe distribution of all quasar pairs, not just nearest neighbor pairs,\nso it is a more powerful measure for detecting distortions of geometry.\nIt is important to note that these distortions are detectable only because\nquasars are clustered.  If quasars were Poisson distributed, then all pair\norientations would be statistically isotropic regardless of the assumed\nmetric.\n\nTwo recent papers (Ballinger, Peacock \\& Heavens 1996; \nMatsubara \\& Suto 1996; see also Nakamura, Matsubara, \\& Suto 1997)\nhave discussed the use of redshift space clustering in quasar samples\nas a tool for constraining \\mbox{${\\Omega}_{0}$ } and \\mbox{${\\lambda}_{0}$}.  \nMatsubara \\& Suto (1996) derive an expression for the correlation\nfunction that includes the effects of geometrical distortion and\nlinear theory peculiar velocities.  Ballinger et al.\\ (1996) derive\nexpressions for the redshift space power spectrum, including effects of \ngeometrical distortion, linear theory peculiar velocities, and incoherent\nrandom velocities generated by small scale collapse.\n\nIn the next section, we discuss the geometrical distortion of the\ncorrelation function in the absence of peculiar velocities, extending\nthe calculation of Phillips (1994).  The distortion due to geometry\nalone is simple to understand: changes in the metric just alter the\nrelation between a $(\\Delta z, \\Delta \\theta)$ separation and the \ncorresponding physical distance.  In \\S 3 we show that, for realistic\nquasar samples, measurement errors in the correlation function are\nlikely to be dominated by Poisson fluctuations in the number of\nquasar pairs.  As a result, we can easily create Monte Carlo realizations\nof correlation function measurements for a specified model correlation\nfunction.  In \\S 4 we use such Monte Carlo experiments to see what\ncosmological constraints can be expected from the 2dF and Sloan quasar\nsamples.  In \\S 5 we summarize the results of these experiments and\ndiscuss some of their limitations.  The most serious of these is\nprobably our neglect of peculiar velocity distortions in modeling the \ncorrelation function, but we argue in \\S 5 that these are unlikely\nto overwhelm the geometrical signal.  A full analysis of observational\ndata will require joint consideration of velocity and geometry effects,\nalong the lines envisioned by Ballinger et al.\\ (1996).\n\n\\section{The redshift space correlation function }\nThe position of a quasar in \nredshift survey data is characterized by three numbers:\nthe redshift and two angle coordinates on the sky $(z, \\theta_r, \\phi_r)$.\nFor a given quasar, the probability of finding a\nneighbor in a given volume element is symmetric about the line of sight,\nsince this is the only preferred direction.\nFor any pair\nof quasars one can therefore find a transformation \n$(\\phi_r, \\theta_r) \\rightarrow (\\phi, \\theta) $ \nsuch that $\\phi_1 = \\phi_2$ in the new frame of reference.\nConsequently, the distance between the points in redshift\nspace can be described in terms of redshift $z$ and angle $\\theta$.\n\n\n\n\nFigure 1 shows the situation with quasars at the positions\n$(z_{1}, \\theta_{1})$ and $(z_{2}, \\theta_{2})$.\nWe define $s_z \\equiv z_{2}-z_{1}$, \n$s_{\\theta} \\equiv z\\cdot (\\theta_2- \\theta_1)=\nz\\cdot \\Delta\\theta $, with $z=(z_{1}+z_{2})\/2$ assumed to be much greater\nthen $z_{2}-z_{1}$. We now adopt the following notation:\n\\begin{equation}\ns \\equiv \\sqrt{s_z^2 + s_{\\theta}^2}\\;, \\lab{s.def}\n\\end{equation}\nand\n\\begin{equation}\n\\mu \\equiv \\frac{s_{z}}{s}\\;. \\lab{mu.def}\n\\end{equation}\nThe angular separation can then be expressed as\n\\begin{equation}\n{\\Delta\\theta}^2=z^{-2}s_z^2(\\mu^{-2} - 1).  \\lab{d.theta}\n\\end{equation}\nIn Euclidean geometry, $s_z$ and $s_{\\theta}$ would be the line-of-sight\nand transverse separations, respectively (in dimensionless, redshift units); \n$s$ would be\nthe 3-dimensional separation, and $\\mu$ would be the cosine of the\nangle between the separation vector and the line of sight. \nIn reality the geometry is not Euclidean, but we can still keep the formal\ndefinitions of $s$ and $\\mu$ introduced above.\n\nAdopting Phillipps' (1994) notation we have\n\\begin{equation}\nr^2=f^2 {\\Delta\\theta}^2+g^2 s_z^2\\;, \\lab{r.sq}\n\\end{equation}\nwhere in general (e.g., Weinberg 1972)\n\\begin{equation}\n{\\scriptsize\n{\\normalsize f= \\frac{c}{H_{0}}} \\times \n\\left\\{\n\\begin{array}{ll}\n{\\displaystyle \\frac{1}{\\sqrt{1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$}}} \\sinh{\\left[\\int_{\\frac{1}{1+z}}^{1} \\frac{\\sqrt{1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$}}dy}{y\\sqrt{\\frac{\\mbox{${\\Omega}_{0}$ }}{y}-(1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$})+\\mbox{${\\lambda}_{0}$} y^{2}}}\\right]}} & \\mbox{if $(\\mbox{${\\Omega}_{0}$ }+\\mbox{${\\lambda}_{0}$})<1$} \\\\\n{\\displaystyle \\int_{\\frac{1}{1+z}}^{1} \\frac{dy}{y\\sqrt{\\frac{\\mbox{${\\Omega}_{0}$ }}{y}+\\mbox{${\\lambda}_{0}$} y^{2}}}} & \\mbox{if $(\\mbox{${\\Omega}_{0}$ }+\\mbox{${\\lambda}_{0}$})=1$} \\\\\n{\\displaystyle \\frac{1}{\\sqrt{-\\left(1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$}\\right)}} \\sin{\\left[\\int_{\\frac{1}{1+z}}^{1} \\frac{\\sqrt{-\\left(1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$} \\right)}dy}{y\\sqrt{\\frac{\\mbox{${\\Omega}_{0}$ }}{y}-\\left(1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$}\\right)+\\mbox{${\\lambda}_{0}$} y^{2}}}\\right]}} & \\mbox{if $(\\mbox{${\\Omega}_{0}$ }+\\mbox{${\\lambda}_{0}$})>1$}\n\\end{array}\n\\right. \n}\\lab{f.def}\n\\end{equation}\nand\n\\begin{equation}\ng=\\frac{c}{H_{0}} \\times \\frac{1}{\\sqrt{\\mbox{${\\Omega}_{0}$ } {(1+z)}^{3} + (1-\\mbox{${\\Omega}_{0}$ }-\\mbox{${\\lambda}_{0}$}){(1+z)}^{2}+\\mbox{${\\lambda}_{0}$}}}. \\lab{g.def}\n\\end{equation}\nCombining equations\\r{s.def},\\r{mu.def},\\r{d.theta} and\\r{r.sq}, we find\n\\begin{equation}\n\\frac{r}{s}=g \\sqrt{\\mu^2 + h^2(1-\\mu^2)}\\;,\\;\\;\\; \\mbox{where} \\;\\;h\\equiv\\frac{1}{z}\\cdot \\frac{f}{g}.  \\lab{r.over.s}\n\\end{equation}\n\nNow suppose that the quasars are clustered and that their\ncorrelation function $\\xi(r)$ is described by power-law,\n\\begin{equation}\n\\xi(r)={\\left( \\frac{r}{r_0} \\right)}^{-\\mbox{$\\gamma$}},  \\lab{xi.r}\n\\end{equation}\nwhere, in principle, the correlation length $r_0$ and the index $\\mbox{$\\gamma$}$ may\nbe functions of $z$.\nIn redshift space, the correlation function at vector separation $(s, \\mu)$ \nis simply\nthe value of $\\xi(r)$ at the scalar separation $r$ corresponding to $s$, $\\mu$.\nSubstitution of\\r{r.over.s} into\\r{xi.r} yields\n\\begin{equation}\n\\xi(s, \\mu) = {\\left(\\frac{s}{s_0}\\right)}^{-\\mbox{$\\gamma$}}{\\left[{\\mu}^2 + h^2(1-{\\mu}^2)\\right]}^{-\\frac{\\mbox{$\\gamma$}}{2}}, \\lab{xi.s}\n\\end{equation}\nwhere $s_0=r_0\/g$.\nThe correlation function in these coordinates is anisotropic because the\nphysical separation corresponding to a given $s$ depends on angle\nwith respect to the line of sight. The anisotropy is stronger when\n$\\mbox{$\\gamma$}$ is larger because a given change in $r$ then produces a larger change\nin $\\xi(r)$.\n\n\n \nOnce $s_0$ and $\\mbox{$\\gamma$}$ are set, the dependence of \nthe correlation function on the geometry of the universe is \ndetermined entirely through the value of the ``distortion parameter'' $h$.\nFigure~2 shows the redshift dependence of $h(z)$\nfor open and flat cosmologies with various \\mbox{${\\Omega}_{0}$ }.\nThe functional form of $h(z)$ depends much\nmore strongly on \\mbox{${\\lambda}_{0}$ } than on \\mbox{${\\Omega}_{0}$ }, implying that clustering anisotropy\ncan much more easily distinguish models with the same \\mbox{${\\Omega}_{0}$ } and different \n\\mbox{${\\lambda}_{0}$ } than models with similar \\mbox{${\\lambda}_{0}$ } and different \n$\\Omega_0$, as Alcock \\& Paczy\\'{n}ski (1979) pointed out.\nThere is also an approximate degeneracy in $h(z)$ between open models\nwith low \\mbox{${\\Omega}_{0}$ } and flat models with relatively high $\\Omega_0$.\n\n\nIn the range $1 < z < 5$, $h(z)$ can be well approximated by a \nstraight line for most models.  The anisotropy increases with redshift\nin all cases, but models can in principle be distinguished both by \nthe value of $h$ at a given $z$ and by the overall redshift dependence\nof $h(z)$.  As discussed in the following sections, in a real \nredshift survey the information for distinguishing cosmologies\ncomes mainly from the redshifts where the quasar distribution peaks,\nsince that is where the clustering can be measured most precisely.\n\nFigure 3 shows the angular dependence of the correlation function,\nfrom equation~(\\ref{xi.s}), for various values of $h$.\nIf $h=1$, the geometry of redshift space is Euclidean, and\nthere is no correlation function anisotropy.\nAll models have $h \\approx 1$ at $z \\ll 1$, and the\n$\\lambda_0=1$, $\\Omega_0=0$ model has $h(z)=1$ at all redshifts.\nFor other models $h(z)>1$ at high redshifts, and the correlation\nfunction is amplified\\footnote{Whether one sees \n``amplification'' or ``suppression'' depends on the choice of reference \nmodel. Here our reference is Euclidean (\\mbox{${\\lambda}_{0}$}=1), but Ballinger et al. (1996)\ntake \\mbox{${\\Omega}_{0}$ }=1 as a reference and therefore find that alternative\nmodels have suppressed clustering along the line of sight.}\nfor pair separations along the line of sight ($|\\mu|=1$) relative\nto pair separations perpendicular to the line of sight.\nThe anisotropy of the correlation function\nincreases rapidly with increasing $h$.\n\n\n\\section{Statistical errors in measurements of $\\xi(s, \\mu)$}\nFollowing Peebles (1980, \\S 31), we can define a quasar's\naverage number of ``clustered neighbors'' by\n\\begin{equation}\nN_{c}=\\int^\\infty_0 \\bar{n}\\xi(r)4\\pi r^2dr, \\lab{N.c1}\n\\end{equation}\nwhere $\\bar{n}$ is the space density of quasars in a sample.\nFor purposes of calculation, we assume a power-law correlation function\n(equation~[\\ref{xi.r}]).\nSince the correlation function may fall below this power-law at large\ndistances, we only consider pairs out to a separation $\\alpha r_0$ that\nis a small multiple $\\alpha$ of the correlation length.\nSubstitution of\\r{xi.r} into\\r{N.c1} yields\n\\begin{equation}\nN_{c}=\\frac{4\\pi{\\mbox{$\\alpha$}}^{3-\\mbox{$\\gamma$}}}{3-\\mbox{$\\gamma$}} \\bar{n}r_{0}^3. \\lab{N.c2}\n\\end{equation}\nWe will take $\\mbox{$\\alpha$}=2$ as a template value in our simulations.\nFor $\\mbox{$\\alpha$}=2$, the value of $N_c$ from \\r{N.c2} varies by only 5\\%\nfor $1 \\leq \\mbox{$\\gamma$} \\leq 2$.\n\nConsider a sample of $N_Q$ quasars in an area $A$ square degrees,\nand let $F(z)dz$ denote the fraction of\nthe quasar sample in the redshift range $(z,z+dz)$, normalized\nso that $\\int_0^\\infty F(z) dz=1$.\nThe observations directly tell us $(N_Q\/A)F(z)$, the average\nnumber of quasars per square degree per unit redshift.\nThe relation between $(N_Q\/A)F(z)$\nand the number density $\\bar{n}(z)$ in\ncomoving ${\\mbox h_0}^{-3}\\;\\mbox{Mpc}^{3}$ depends on the\ncosmological model.\nSince the volume of a redshift range $dz$ and solid angle $d\\Omega$\nis $g\\, dz\\cdot f^2 d\\Omega$,\n\\begin{equation}\n\\bar{n}={3283 (N_Q\/A) F(z) \\over gf^2}, \\lab{bar.n1}\n\\end{equation}\nwhere 3283 is the number of square degrees per radian.\nFor compactness, in this and future equations we generally omit the\nexplicit $z$-dependence of $\\bar{n}$, $f$, and $g$.\nFrom\\r{f.def} and\\r{g.def} it is clear that both $f$ and $g$ involve\nthe factor $cH_{0}^{-1}=3000 {\\mbox h_0}^{-1}\\mbox{Mpc}$, and it is convenient\nto write them in the form\n\\begin{equation}\nf=\\frac{c}{H_{0}}\\tilde{f}, \\qquad\\mbox{and}\\qquad \ng=\\frac{c}{H_{0}}\\tilde{g}. \\lab{f.g.def}\n\\end{equation}\nWe use the notation\\r{f.g.def} \nand the value of $cH_{0}^{-1}$ to write\\r{bar.n1} as\n\\begin{equation}\n\\bar{n}=1.21\\cdot 10^{-7} \\frac{N_Q}{A} F(z)\n{\\tilde{g}}^{-1}\\tilde{f}^{-2} ~~\n{\\rm h}_0^3\\;{\\rm Mpc}^{-3} ~{\\rm (comoving)}.\n\\lab{bar.n2}\n\\end{equation}\nSubstitution of\\r{bar.n2} into\\r{N.c2} gives\n\\begin{equation}\nN_{c}(z)=\\frac{4\\pi{\\mbox{$\\alpha$}}^{3-\\mbox{$\\gamma$}}}{3-\\mbox{$\\gamma$}} r_{0}^3 \n\\left(1.21\\cdot 10^{-7} \\frac{N_{Q}}{A} F(z) \n{\\tilde{g}}^{-1}\\tilde{f}^{-2}\\right),\\lab{N.c3}\n\\end{equation}\nfor $r_0$ in comoving h$_0^{-1}$ Mpc.\nThe number of correlated pairs in the range $(z, z+dz)$ is\n\\begin{equation}\nN_{pairs}(z)=N_{Q} F(z) N_{c}(z)\\propto \\frac{N_{Q}^2}{A} F^2(z) r_{0}^3. \\lab{N.pairs1}\n\\end{equation}\n\nAs a fiducial case let us take $\\mbox{$\\alpha$}=2$, $\\mbox{$\\gamma$}=2$, \n$N_{Q}= 10^{5}$, $A= 10^{4} {\\Box}^\\circ $, and $F(z)=0.5$.\nThen\n\\begin{equation}\nN_{c}(z)=0.11 {\\left( \\frac{r_{0}}{10 {\\mbox h_0}^{-1}\\mbox{Mpc}} \\right)}^{3} \\left(\\frac{{\\tilde{g}}^{-1}\\tilde{f}^{-2}}{7.27}\\right), \\lab{N.c4}\n\\end{equation}\nwhere $r_0$ is the comoving correlation length.\nWe have scaled ${\\tilde{g}}^{-1}\\tilde{f}^{-2}$ to the value for an\n\\mbox{${\\Omega}_{0}$ }=1 model at $z=2$.\nSubstitution of\\r{N.c4} into\\r{N.pairs1} yields\n\\begin{equation}\nN_{pairs}(z)=5500 {\\left( \\frac{r_{0}}{10 {\\mbox h_0}^{-1}\\mbox{Mpc}} \\right)}^{3} \\left(\\frac{{\\tilde{g}}^{-1}\\tilde{f}^{-2}}{7.27}\\right) \\lab{N.pairs2}\n\\end{equation}\nfor the number of correlated quasar pairs expected in an interval\n$\\Delta z \\approx 1$ near the peak of the redshift distribution $F(z)$\nin a large quasar redshift survey.\n\nWe now come to the central issue of this Section --- a key issue for\nthe entire paper, in fact --- the nature of statistical fluctuations in\nestimates of the correlation function.  In the limit $N_c \\gg 1$,\nthe statistical uncertainty is dominated by the finite number of\nindependent structures in the survey volume, an effect sometimes referred to\nas ``cosmic variance.''  As a conceptual toy model, one can imagine\nthe structure to consist of clusters with an average of $N_c$ members\napiece.  Each cluster is sampled by many correlated pairs, so the\nfluctuation in the number of clusters within the survey dominates\nthe error in $\\xi$.  In the opposite limit, $N_c \\ll 1$, most clusters\nare not ``detected'' with even a single correlated pair, and it\nis the Poisson fluctuation in the number of pairs that dominates the \nstatistical error rather than the fluctuation in the number of\nindependent structures.  In this ``sparse sampling'' or ``Poisson limit,''\nthe error in an estimate of $\\xi(s,\\mu)$ in a bin $(\\Delta s, \\Delta \\mu)$\nis simply the Poisson error in the number of pairs in the bin,\nand the errors in separate bins are statistically independent.\nThe transition between the dense sampling and sparse sampling limits\noccurs at $N_c \\approx 1$; we have carried out numerical experiments\nwith Monte Carlo clustering models (Soneira \\& Peebles 1978) to\nconfirm that the Poisson error approximation holds quite accurately\nfor $N_c \\la 1$.\n\nIn a typical galaxy redshift survey, $N_c(z) \\gg 1$ nearby, \nand $N_c(z)$ drops below one at large distances, where only the\nmost luminous galaxies lie above the survey's apparent magnitude limit.\nRoughly speaking, the volume within which $N_c(z) \\ga 1$ is\nthe effective volume of the survey for purposes of estimating $\\xi$;\nthere is some usable information from larger distances, but as\nthe structure ``fades out'' under increasingly sparse sampling, the\nstatistical significance of this additional information drops.\n\nEquation~(\\ref{N.c4}) implies that even an ambitious quasar\nredshift survey is likely to be in the sparse sampling regime\nat {\\it all} redshifts, unless the comoving quasar correlation\nlength is substantially larger than $10 {\\rm h}_0^{-1}\\;$Mpc.\nThis result has several important implications.\nFirst, the independent, Poisson errors in $\\xi$ allow a straightforward\nmaximum-likelihood scheme for estimating correlation function parameters.\nWe describe this scheme in \\S 4.3 below; this method was developed\nindependently by Croft et al. (1997), who applied it to rich galaxy\nclusters, and it was applied to a sample of high-redshift quasars\nfrom the Palomar Transit Grism Survey by Stephens et al. (1997).\nA second implication is that, given a theoretical prediction of\n$\\xi(s,\\mu)$, one can generate Monte Carlo realizations of measured\ncorrelation functions for a given sample without creating detailed\nrealizations of a clustering pattern, because the fluctuation in\na measured value of $\\xi$ depends only on the expected number of pairs\nand is independent of fluctuations in other bins and of the higher-order\nclustering properties of the quasar distribution.\nWe describe this method for creating Monte Carlo realizations in \\S 4.2\nbelow, and in \\S 4.4 we use it to evaluate the power of the SDSS\nand 2dF quasar redshift surveys for constraining the cosmological constant.\nFinally, there is an observational implication: for a fixed number of \nquasars, a deeper survey over a smaller area has greater statistical\npower, at least for measurements of the correlation function on\nscales $\\sim r_0$.  \n\nThis last point is demonstrated most clearly by generalizing\nequations~(\\ref{N.c4}) and~(\\ref{N.pairs2}) to\n\\begin{equation}\nN_{c}(z)=0.11 \\left(\\frac{\\mbox{$\\alpha$}^{3-\\mbox{$\\gamma$}}\/(3-\\mbox{$\\gamma$})}{2}\\right)\n{\\left(\\frac{r_{0}(z)}{10 h^{-1} \\mbox{Mpc}}\\right)}^3 \n\\left(\\frac{N_{Q}\/A}{10\/{\\Box}^{o}}\\right) \n\\left(\\frac{F(z)}{0.5}\\right)\n\\left(\\frac{{\\tilde{g}}^{-1}\\tilde{f}^{-2}}{7.27}\\right),\\lab{N.c5}\n\\end{equation}\n\\begin{equation}\nN_{pairs}(z)=5500 \\left(\\frac{\\mbox{$\\alpha$}^{3-\\mbox{$\\gamma$}}\/(3-\\mbox{$\\gamma$})}{2}\\right)\n{\\left(\\frac{r_{0}(z)}{10 h^{-1} \\mbox{Mpc}}\\right)}^3 \n\\left(\\frac{N_{Q}}{10^5}\\right)\n\\left(\\frac{N_{Q}\/A}{10\/{\\Box}^{o}}\\right) \n{\\left(\\frac{F(z)}{0.5}\\right)}^2 \n\\left(\\frac{{\\tilde{g}}^{-1}\\tilde{f}^{-2}}{7.27}\\right). \\lab{N.pairs3}\n\\end{equation}\nIf $N_c(z) \\la 1$, then the effective signal for correlation function\nmeasurements is set by $N_{pairs}(z)$, since\neach correlated pair contributes non-redundant information.\nAt fixed surface density\n$N_Q\/A$, this signal increases in proportion to the number of\nquasars $N_Q$, but at fixed area $A$ it is proportional to $N_Q^2$.\nOf course, increasing $N_Q$ at fixed $A$ usually requires one to\nobserve fainter quasars, while increasing $N_Q$ at fixed surface density \ndoes not.  Equation~(\\ref{N.pairs3}) also implies that the precision of\ncorrelation function measurements will be peaked sharply near the\nmaximum of the survey's redshift distribution $F(z)$ and that the\nprecision attained will depend strongly on the actual value\nof the quasar correlation length $r_0$.\n\nThe remaining equations in this Section are not fundamental,\nbut they are needed for our Monte Carlo simulations and \nmaximum-likelihood parameter estimation, and we include their\nderivation for the convenience of others who may wish to \npursue similar approaches.\nA physical model specifies the correlation function in\nreal space ($r$-space), but a redshift survey provides data\nin redshift space ($s$-space).\nThe relation between coordinates in these two complementary descriptions is\n\\begin{equation}\n\\left( \\begin{array}{c}\ndr \\\\ d(\\cos t)\n\\end{array} \\right) = \\underbrace{\\left( \\begin{array}{cc} \n\\frac{\\mbox{$\\partial$} r}{\\mbox{$\\partial$} s} & \\frac{\\mbox{$\\partial$} r}{\\mbox{$\\partial$} \\mu} \\\\\n\\frac{\\mbox{$\\partial$} (\\cos t)}{\\mbox{$\\partial$} s} & \\frac{\\mbox{$\\partial$} (\\cos t)}{\\mbox{$\\partial$} \\mu}\n\\end{array} \\right)}_{\\mbox{\\boldmath $A$}} \\left( \\begin{array}{c}\nds \\\\ d \\mu\n\\end{array} \\right), \\lab{matrix}\n\\end{equation}\nwhere $\\cos{t} = g s_{z}\/r$.\nThus\n\\begin{equation}\ndr \\,d(\\cos t)= \\det{\\!\\mbox{\\boldmath $A$}} \\; ds \\, d\\mu .  \\lab{determinant}\n\\end{equation}\nA short calculation leads to\n\\begin{equation}\nd(\\cos{t})=\\frac{h^2 d\\mu}{[h^2-(h^2-1)\\mu^2]^\\frac{3}{2}}, \\lab{d.cos}\n\\end{equation}\nand\n\\begin{equation}\ndr=g \\sqrt{h^2 - (h^2-1) \\mu^2} \\,ds - \\frac{g s \\mu (h^2-1) d\\mu}{\\sqrt{h^2 - (h^2-1) \\mu^2}}. \\lab{d.r}\n\\end{equation}\nSince $\\frac{\\mbox{$\\partial$} (\\cos t)}{\\mbox{$\\partial$} s}=0$, equation\\r{determinant} yields\n\\begin{equation}\n(dV)_r = -2\\pi r^2 dr d(\\cos{t}) = -2\\pi r^2 \\left[\\frac{\\mbox{$\\partial$} r}{\\mbox{$\\partial$} s} \\cdot \\frac{\\mbox{$\\partial$} (\\cos t)}{\\mbox{$\\partial$} \\mu}\\right] ds d\\mu \\lab{vol.1}\n\\end{equation}\nfor the volume element in real space.\nSubstitution of\\r{d.cos} \\&\\r{d.r} into\\r{vol.1} leads to\n\\begin{equation}\n(dV)_r=g^3 h^2 (-2\\pi s^2 ds d\\mu)= g^3 h^2 (dV)_s, \\lab{vol.2}\n\\end{equation}\nwhere $(dV)_s$ is the volume element in redshift space.\nThe number of quasars in a given region will be the same\nregardless of the adopted coordinate system.  Thus\n\\begin{equation}\nn_s (dV)_s \\equiv n(z) (dV)_r, \\lab{vol.3}\n\\end{equation}\nwhere $n_s$ and $n(z)$ are number densities of quasars in redshift and real\nspaces, respectively.\nCombining\\r{vol.2} and\\r{vol.3} leads to \n\\begin{equation}\nn_s(s, \\mu) = g^3(z) h^2(z) n(z) . \\lab{num.den} \n\\end{equation}\n\nThe number of pairs expected in an\ninfinitesimally small bin in $(s, \\mu)$ within the finite redshift range\n$(z, z+\\Delta z)$ is\n\\begin{equation}\ndN_{pairs}=N(z, z+\\Delta z) n_s [1+\\xi(s, \\mu)] (-2\\pi s^2 ds d\\mu),\n\\lab{pair.num.1}\n\\end{equation}\nwhere $N(z, z+\\Delta z)$ is the number of observed quasars \nin this redshift range.\nSince $F(z)dz$ is the fraction of quasar redshifts in an infinitesimal \nrange $dz$, \n\\begin{equation}\nN(z, z+\\Delta z)=N_{Q} \\int_{z}^{z+\\Delta z} F(z) dz,\n\\lab{N.z.dz}\n\\end{equation}\nwhere $N_Q$ is the total number of quasars in the sample.\nSubstitution of\\r{xi.s},\\r{num.den} and\\r{N.z.dz} into\\r{pair.num.1} for\ninfinitesimally small $\\Delta z\\equiv dz$ yields \n\\begin{equation}\ndN_{pairs} = -2\\pi N_Q g^3 h^2 \\bar{n} F(z) \n\\left\\{1+{\\left(\\frac{s}{s_0}\\right)}^{-\\mbox{$\\gamma$}}\n{\\left[{\\mu}^2 + h^2(1-{\\mu}^2)\\right]}^{-\\frac{\\mbox{$\\gamma$}}{2}}\\right\\} \ns^2 dz ds d\\mu .\n\\lab{dN.pairs}\n\\end{equation}\nHere $s_0 \\equiv r_0\/g$ is the line-of-sight separation that corresponds\nto the comoving correlation length $r_0$, and the \nmean number density $\\bar{n}$ is given in terms of\n$N_Q$, $A$, and $F(z)$ by equation\\r{bar.n2}.\nThe quantities $g$, $h$, and $\\bar{n}$ all depend on \nredshift, and $s_0$ and $\\gamma$ may depend on redshift also.\n\n\\section{Monte Carlo experiments}\n\n\\subsection{Observational perspective}\n\nSuccessful measurement of quasar clustering anisotropy will\nrequire redshift samples much larger than those that exist today.\nThere are two clear prospects for such samples, the SDSS and\na quasar redshift survey using the 2dF fiber spectrograph on the \nAnglo-Australian Telescope.\nThe SDSS plans to obtain redshifts for 80,000 quasars in an area\nof 5,000 square degrees in the North Galactic Cap.\nIt will also measure redshifts for $\\sim 20,000$ quasars in the surrounding\n5,000 square degree ``skirt,'' but this subset will be less useful for\nclustering measurements because of its lower density.\nThe SDSS will also conduct a deeper quasar survey in a stripe of $\\sim 200$\nsquare degrees in the South Galactic Cap, to a limiting magnitude\nyet to be determined.\n(A discussion of the planned SDSS quasar survey can be found at\nhttp:\/\/www.astro.princeton.edu\/BBOOK\/SCIENCE\/QUASARS\/quasars.html.)\nThe SDSS will select quasar candidates based on 5-band CCD photometry.\nThe planned 2dF survey will target $\\sim 30,000$ quasars in an\narea $\\sim 750$ square degrees, selected as UV-excess stellar objects\non scanned photographic plates (Shanks, private communication).\n\nIn order to generate Monte Carlo realizations of correlation function\nmeasurements from these samples, we need to know (approximately) the\nexpected redshift distributions $F(z)$.\nWe have computed these using the observational determinations of\nthe quasar luminosity function by\nBoyle (1991) and Warren, Hewett, \\& Osmer (1994).\nThe Boyle (1991) results are based on the Boyle, Shanks \\& Peterson (1988) \nsurvey of quasars selected by the UV-excess method, \nwhich is effective mainly for $z \\la 2.2$.\nThe Warren et al. (1994)\nresults are based on a multicolor survey and probe the redshift range\n$2.0 < z < 4.5$.  For our calculations, we assume that the 2dF survey\nwill contain quasars only up to $z=2.2$, because of its UV-excess\nselection technique, and that the SDSS multi-color selection will\nidentify quasars at all redshifts, with apparent magnitude being\nthe only important limit.  We find that the planned SDSS and 2dF\nsurface densities imply limiting apparent magnitudes of roughly\n$B=19.6$ and $B=21.4$, respectively.  With these apparent magnitude\nlimits, we compute the number of quasars per square degree per unit\nredshift from the above-mentioned luminosity functions.\n\nThe solid curves in the two panels of Figure~4 show the results\nof these calculations.  Sharp dips at $z \\sim 2- 2.2$ are an artifact\nof using the Boyle (1991) luminosity function close to the redshift\nlimit of the Boyle et al. (1988) survey.  More broadly, the shapes\nof these curves reflect the interplay between the peak in the quasar \nluminosity\nfunction at $z \\sim 2-3$ and the smaller fraction of the luminosity\nfunction that is visible above the apparent magnitude limit at larger\ndistances.  Of course, $F(z)$ is also affected by the increase of the\ndifferential volume element $dV\/dz$ with $z$, especially at low redshift.\nUp to $z\\sim 2$, the evolution effects and the magnitude limit\nwork in different directions, producing a relatively flat redshift \ndistribution.  Above $z\\sim 2$ both the\nnumber density of quasars and our ability to see them decreases, and \nthe redshift distribution drops rapidly towards zero.\n\n\n\nSince these calculated redshift distributions are only approximate ---\nthe quasar luminosity function is itself uncertain, and we have not \nmodeled the selection criteria and corresponding incompleteness of \nthe two surveys in any detail --- we fit them with simple analytic\nforms to use in our Monte Carlo calculations below.  These fits\nare indicated by the dashed curves in Figure~4.\nThe survey's target selection algorithms will probably focus on objects\nwith point-source morphology in order to reduce contamination by galaxies.\nThis selection technique may exclude quasars at low redshift, where\nthe host galaxies are bright enough to be detected as extended emission.\nIn our $F(z)$ fits, we model this effect by a sharp cutoff at $z<0.4$.\n\n\\subsection{Generating Monte Carlo simulations}\nFor our Monte Carlo experiments, we wish to generate realizations\nof {\\it measured} redshift-space correlation functions $\\xi(s,\\mu)$\nfor various cosmological models and quasar survey parameters.\nWe consider two classes of cosmological models: flat models with\n$\\mbox{${\\Omega}_{0}$ }+\\mbox{${\\lambda}_{0}$ }=1$, and open models with $\\mbox{${\\lambda}_{0}$ }=0$.\nThe $\\mbox{${\\Omega}_{0}$ }=1$, Einstein-de Sitter cosmology is a limiting case of both\nfamilies.  We assume that the real-space quasar correlation function\nis a power-law, $\\xi(r) = (r\/r_0)^{-\\mbox{$\\gamma$}}$, out to $2r_0$.\nOnce the cosmological model, the correlation function parameters,\nand the quasar survey parameters \nare set, the expected number of pairs in a separation\/redshift\nbin $ds d\\mu dz$ is given by equation\\r{dN.pairs}.\n\nBecause the sparseness of the quasar distribution puts us in the Poisson\nlimit (see \\S 3), we are able to generate realizations of the\nmeasured $\\xi(s,\\mu)$ very simply, without creating artificial spatial\ndistributions.  Given the expected number of pairs in a bin from\nequation\\r{dN.pairs}, the number of pairs in a Monte Carlo realization\nis a random deviate drawn from a Poisson distribution with this\nmean value.  The pair numbers for each bin can be generated independently.\nIn practice we use\n20 logarithmic bins in $s$, starting at $0.1 s_{0}$ and\ngoing up to $2 s_{0}$.\nWe use 5 equal bins in $\\mu$; because these bins are large, we compute\nthe predicted number of pairs by numerical integration over the bin.\nWe work with redshift bins $\\Delta z = 0.15$, implying\n12 bins for simulations with a 2dF-like $F(z)$ and 22 bins for simulations\nwith an SDSS-like $F(z)$.\nWe perform simulations for seven cosmological models: flat and open\nlow-density models with $\\mbox{${\\Omega}_{0}$ }=0.1, 0.2, 0.4$, and the Einstein-de Sitter\ncosmology, $\\mbox{${\\Omega}_{0}$ }=1$.  To fully specify the theoretical model, we must also \nspecify the correlation length $s_0$ and the index $\\mbox{$\\gamma$}$ as functions\nof redshift.  Existing observations provide only weak constraints on $\\mbox{$\\gamma$}$.  \nWe adopt $\\mbox{$\\gamma$}=2$, close to the index $\\mbox{$\\gamma$}=1.8$ of the galaxy\ncorrelation function, and we assume that $\\mbox{$\\gamma$}$ is independent of redshift.\n\nEven the quasar correlation length $s_0$ is poorly known at present,\nbecause the sparseness of the quasar distribution in current samples\nmakes measurements of the quasar correlation function very noisy.\nThe best existing study is probably that of Shanks \\& Boyle \n(1994, hereafter SB),\nwho combine data from three quasar redshift surveys.\nFor an $\\mbox{${\\Omega}_{0}$ }=1$, $\\mbox{${\\lambda}_{0}$ }=0$ cosmology, \nthey find that their correlation function measurements are consistent\nwith a quasar correlation length $r_0 = 6 {\\rm h}_0^{-1}\\;{\\rm Mpc}$\nthat remains constant in comoving coordinates, implying\n\\begin{equation}\ns_0 \\equiv \\frac{r_0}{g} = 0.002 (1+z)^{3\/2},\n\\lab{s0.om1}\n\\end{equation}\nwhere we use equation\\r{g.def} for $g(z)$.\n\nIn order to scale the SB results to other\ncosmological models, we note that the property of the correlation\nfunction most robustly constrained by observations is the total\nnumber of correlated pairs.  From equation\\r{N.pairs3}, we see\nthat the number of correlated pairs is\n\\begin{equation}\nN_{pairs}(z) \\propto \\frac{r_0^3}{\\tilde g \\tilde f^2} \\propto\n\\frac{s_0^3 g^2}{f^2} \\propto \\frac{s_0^3}{h^2},\n\\lab{Npairs.scaling}\n\\end{equation}\nwhere we have dropped the factors that are independent of the\nspacetime geometry.  For alternative cosmological models,\nwe therefore wish to hold the combination $s_0^3\/h^2$ equal\nto the value implied for the $\\mbox{${\\Omega}_{0}$ }=1$, $\\mbox{${\\lambda}_{0}$ }=0$, \n$r_0 = 6{\\rm h}_0^{-1}\\;{\\rm Mpc}$ model that fits the\nSB data.  We therefore adopt\n\\begin{equation}\ns_{0}(z,\\mbox{${\\Omega}_{0}$ },\\mbox{${\\lambda}_{0}$})=0.002{(1+z)}^{\\frac{3}{2}}\n{\\left[\\frac{h(z,\\mbox{${\\Omega}_{0}$ },\\mbox{${\\lambda}_{0}$})}{h(z,\\Omega=1,\\mbox{${\\lambda}_{0}$}=0)}\\right]}^{\\frac{2}{3}}\n\\lab{s0.general}\n\\end{equation}\nas a fiducial redshift-space correlation length for our simulations.\n\n\\subsection{Maximum likelihood determination of parameters}\nGiven a simulated or observed set of correlation measurements, we are\ninterested in estimating the true correlation function parameters and\nthe parameters of the underlying cosmological model.\nSuppose that the data consist of pair counts $N_i$ in $i$ bins, where\n$i$ may in fact represent a multidimensional (e.g. $s, \\mu$) space.\nWe have a model ${\\cal M}$ for the correlation function that may depend\nupon several parameters, e.g. ${\\cal M} (s_0, \\mbox{$\\gamma$}, \\mbox{${\\Omega}_{0}$ }, \\mbox{${\\lambda}_{0}$ })$. \nThis model predicts a number of pairs in each bin\n$A_i=\\bar{n} V_i (1+\\xi (s_i, \\mu_i)) N_Q$, where $V_i$ is the bin volume.\nIn the Poisson limit $N_c \\la 1$, which is expected to hold for realistic\nquasar surveys (see \\S 3),\nthe probability of detecting $N_i$ pairs in bin $i$\nwhen $A_i$ are expected is\n\\begin{equation}\nP(N_i|A_i)=\\frac{e^{-A_i} \\cdot A_i^{N_i}}{N_i!}.\n\\end{equation}\nThe probabilities for separate bins are independent, so the overall\nlikelihood ${\\cal L}$ of obtaining the data given the model is \n\\begin{equation}\n{\\cal L} \\equiv P(D|{\\cal M}) = {\\prod}_i \\frac{e^{-A_i} \n\\cdot A_i^{N_i}}{N_i!},\n\\lab{likelihood}\n\\end{equation}\nimplying\n\\begin{equation}\n\\ln({\\cal L}) = {\\sum}_i (-A_i + N_i \\ln\\,A_i- \\ln \\, N_i!).\n\\end{equation}\nSince the data $N_i$ are independent of the model parameters, one can find\nthe maximum likelihood model by maximizing the quantity\n\\begin{equation}\n\\ln{\\cal L}^{'} ({\\cal M})=\\sum_i (N_i \\ln\\,A_i-A_i).\n\\end{equation}\nThe relative likelihood of two models ${\\cal M}_{1}$ and \n${\\cal M}_{2}$ is simply\n$\\exp(\\ln{\\cal L}^{'} ({\\cal M}_{1})-\\ln{\\cal L}^{'} ({\\cal M}_{2}))$.\n\nAlthough our focus in this paper is on constraining $\\mbox{${\\lambda}_{0}$ }$, the\nmaximum-likelihood technique outlined here is quite general.\nFor a specified cosmology, one can use this technique to estimate\ncorrelation function parameters in a way that makes maximum use\nof the available data (e.g., Stephens et al.\\ 1997), and the\nmethod can easily be extended to incorporate parametrized descriptions\nof peculiar velocity distortions, evolution of clustering, and\nso forth.  The sparse sampling limit is crucial to this approach,\nfor it is only this property that makes it possible to write down\na straightforward expression for the likelihood, equation\\r{likelihood}.\nFor a dense sample like a typical galaxy redshift survey, the likelihood\nis a much more complicated function of the model parameters and\nthe data, and a maximum-likelihood approach is correspondingly more\ncumbersome.  However, rich galaxy clusters do provide a sparse tracer \nof structure on large scales, and Croft et al. (1997) have used the \nsame maximum-likelihood method to estimate parameters of the \ncluster correlation function.\n\n\\subsection{Results}\nFigures 5--9 present our main results.  In Figure~5 we examine the\nability of the 2dF (left) and SDSS (right) quasar samples to \nconstrain $\\mbox{${\\Omega}_{0}$ }$ in flat, $\\mbox{${\\Omega}_{0}$ }+\\mbox{${\\lambda}_{0}$ }=1$ models.\nFor each survey and each value of $\\mbox{${\\Omega}_{0}$ }=0.1$, 0.2, 0.4, and 1,\nwe generated four realizations of measured $\\xi(s,\\mu)$ by\nthe Monte Carlo technique described in \\S 4.2.\nIn all cases we set $\\mbox{$\\gamma$}=2$ and adopt equation\\r{s0.general}\nfor $s_0(z)$, corresponding to $r_0 = 6{\\rm h_0}^{-1}\\;{\\rm Mpc}$\ncomoving at all $z$ for the $\\mbox{${\\Omega}_{0}$ }=1$, $\\mbox{${\\lambda}_{0}$ }=0$ model and\nto $s_0(z=0)=0.002$ in all models.\nFor each realization, we then determine the best-fit model\nparameters by the maximum-likelihood technique described\nin \\S 4.3, computing likelihoods for a grid of open models and\na grid of flat models with $0 \\leq \\mbox{${\\Omega}_{0}$ } \\leq 1$ and\nvarying values of $s_0$ and $\\mbox{$\\gamma$}$.\nSquares show the estimated values of $\\mbox{${\\Omega}_{0}$ }$, denoted\n${\\hat{\\Omega}}_0$, plotted against the true value $\\Omega_{true}$\nused to generate the realization.  Small offsets are added along\nthe $\\Omega_{true}$ axis to improve clarity.\nVertical line segments delineate the range of $\\mbox{${\\Omega}_{0}$ }$ values\nthat give likelihood values with 10\\% of the maximum likelihood;\nthis corresponds roughly to a $2\\sigma$ confidence interval\nfor a Gaussian approximation to the likelihood distribution.\nIn a few cases --- mostly $\\mbox{${\\Omega}_{0}$ }=1$ realizations, which are a limiting\ncase of both flat and open models --- an open model gave a higher\nlikelihood than the best-fit flat model.  For these cases\nwe plot a filled circle at the value of ${\\hat{\\Omega}}_0$ for the\nbest-fit open model.\n\n\nIn open cosmologies, the anisotropy of the correlation function is\nonly weakly sensitive to $\\mbox{${\\Omega}_{0}$ }$ (see Figure~2), so we do not show\nthe analog of Figure~5 for open models.\nEven with flat models the $\\mbox{${\\Omega}_{0}$ }$ constraints are rather loose, but\nit is encouraging to see that in the $\\mbox{${\\Omega}_{0}$ }<1$ models, the\n$\\mbox{${\\Omega}_{0}$ }=1$ model can be rejected on the basis of quasar clustering\nanisotropy alone in nearly every case.\n\nThere are a number of methods to constrain $\\mbox{${\\Omega}_{0}$ }$ from the\ndynamics of redshift-space galaxy clustering (e.g., Kaiser 1987;\nCarlberg et al.\\ 1996; Kepner, Summers, \\& Strauss 1997).\nThese methods suffer from a degeneracy between the density parameter\nand the ``bias'' of galaxies with respect to mass, but with the\nhigh-precision measurements from the 2dF and Sloan {\\it galaxy}\nredshift surveys it should be possible to break this degeneracy\nand measure $\\mbox{${\\Omega}_{0}$ }$ directly.  We would then like to use quasar\nclustering anisotropy to distinguish between open ($\\mbox{${\\lambda}_{0}$ }=0$)\nand flat ($\\mbox{${\\lambda}_{0}$ }=1-\\mbox{${\\Omega}_{0}$ }$) cosmologies of known $\\mbox{${\\Omega}_{0}$ }$; dynamical\ntechniques are insensitive to this distinction because the\ncosmological constant represents an unclustered energy component.\n\n\nFigure~6 presents the prospects for such a test.  For each realization\nof each of our low-$\\mbox{${\\Omega}_{0}$ }$ models, we compute the likelihood ratio\n$R \\equiv {\\cal L}_2\/{\\cal L}_1$, where ${\\cal L}_1$ is the likelihood\nof the true model (with the correct values of $\\mbox{${\\Omega}_{0}$ }$, $\\mbox{${\\lambda}_{0}$ }$,\n$s_0(z)$, and \\mbox{$\\gamma$} ) and ${\\cal L}_2$ is the likelihood of the \nmodel with the same $\\mbox{${\\Omega}_{0}$ }$, $s_0(z)$, and $\\mbox{$\\gamma$}$ but the incorrect\ngeometry.  For $\\mbox{${\\Omega}_{0}$ }=0.1$ or 0.2, the incorrect geometry can be\nrejected at $> 10:1$ odds in all of the 2dF realizations and at $\\ga 10:1$\nodds in the SDSS realizations.  For $\\mbox{${\\Omega}_{0}$ }=0.4$ the difference between\ngeometries is smaller, and the incorrect model can only be rejected\nat $\\sim 10:1$ odds or (occasionally) less.  Since the 2dF and SDSS\nsamples will be independent (with telescopes in the southern and\nnorthern hemispheres, respectively), a pair of $\\sim 10:1$ rejections\nwould constitute a strong statistical result.  However, there are\nuncertainties related to peculiar velocity distortions and evolution\nof the correlation function that are not incorporated in our current\nMonte Carlo experiments (see \\S 5 below), so it is not clear that\nthe geometry distinction will be possible with these samples for\n$\\mbox{${\\Omega}_{0}$ } \\geq 0.4$.\n\n\nAs equation\\r{N.pairs3} demonstrates, the precision with which\n$\\xi(s,\\mu)$ can be measured depends sensitively on the true quasar\ncorrelation length.  While the SB estimate\nis based on the most extensive compilation of existing data,\nit is nonetheless quite uncertain, and some other studies have\nsuggested higher values.  Stephens et al. (1997), for example,\nfind a maximum-likelihood estimate $r_0=20{\\rm h}_0^{-1}\\;{\\rm Mpc}$\n(comoving, for $\\mbox{${\\Omega}_{0}$ }=1$, $\\mbox{$\\Lambda$ }=0$) for the high-redshift quasars\nin the Palomar Transit Grism Survey.  Figures~7 and~8 repeat\nthe $\\mbox{${\\Omega}_{0}$ }$-constraint and geometry-discrimination tests for 2dF \n(Figure 7) and SDSS (Figure 8) realizations with a correlation \nlength double the value implied by equation\\r{s0.general}.  This\nhigher correlation length corresponds to \n$r_0 = 12{\\rm h_0}^{-1}\\;{\\rm Mpc}$\ncomoving for $\\mbox{${\\Omega}_{0}$ }=1$, $\\mbox{${\\lambda}_{0}$ }=0$, and to $s_0(z=0)=0.004$ for all models.\n\n\nWith the higher correlation length, incorrect geometry models with\n$\\mbox{${\\Omega}_{0}$ }=0.4$ are rejected at $\\ga 100:1$ odds in all of the 2dF realizations\nand all but one of the SDSS realizations.  With $\\mbox{${\\Omega}_{0}$ } \\leq 0.2$,\nthe incorrect geometry models are rejected with formal odds\nexceeding $10^6:1$ in all realizations of both surveys.\nThe $\\mbox{${\\Omega}_{0}$ }$ determinations in flat models are much more precise\nthan those shown in Figure~5 for the smaller correlation length.\nIt is of course possible that the true quasar correlation length\nis {\\it lower} than the SB estimate instead\nof higher, in which case the prospects for the clustering anisotropy\nmethod would be worse than those implied by Figures~5 and~6.\nThe uncertainty in the amplitude of quasar clustering is presently\nthe main uncertainty in assessing the power of future quasar\nsurveys to constrain cosmological parameters by this approach.\n\n\n\nFigures~5--8 show that the 2dF survey has somewhat greater constraining\npower than the SDSS, even though it has 30,000 quasars instead of 80,000.\nThe high surface density of the 2dF sample is crucial to its performance;\nthe factor $N_Q^2\/A$ in equation\\r{N.pairs3} for the number of \ncorrelated pairs is nearly the same for the 2dF and the SDSS.\nThe reason the 2dF survey does slightly better is its more\ncompact redshift distribution $F(z)$.  The SDSS has roughly\nequal numbers of quasars per unit redshift from $z=0.4$ to $z=2.2$,\nand a significant fraction of the sample lies at $z>2.2$.  For the\n2dF survey, a majority of the quasars lie in the range $1 < z < 2.2$.\nThe number of correlated pairs per unit redshift is proportional\nto $[F(z)]^2$, and the more precise measurement of $\\xi(s,\\mu)$ at\n$z \\sim 2$ in the 2dF sample wins out over the greater range of\nredshifts probed by the SDSS.\n\nAs this comparison suggests, the ability of a quasar survey to\nmeasure clustering anisotropy is sensitive to its details,\nand there are strategies that can substantially enhance this\nability for a fixed number of quasars.  One possible approach would\nbe to use multi-color selection techniques to target quasars in\nparticular redshift ranges, producing peaks in $F(z)$ at specific\nredshifts.  Another approach is simply to observe a smaller area\nto a fainter magnitude limit, thus increasing $N_Q^2\/A$.  Either\nstrategy improves the survey's sensitivity to geometrical distortion\n(at the cost of longer spectroscopic exposures), provided that the\nclustering measurements remain in the sparsely sampled regime,\n$N_c \\la 1$.  Once $N_c$ exceeds one, new quasar pairs contribute\npartially redundant information, and there is more to be gained by\nexpanding the survey's area or redshift range.  (This statement\napplies only to the two-point correlation function and its\nrelatives; clustering measures that are sensitive to higher-order\ncorrelations would continue to benefit from denser sampling.)\n\nAs an illustration, we repeat the tests of Figures 5--8 for a hypothetical\nhigh density quasar survey (HDS), now returning to the smaller quasar\ncorrelation length implied by equation\\r{s0.general}.\nWe assume a sample of 30,000 quasars in 200 square degrees --- the\nsame $N_Q$ as 2dF, but a surface density 3.25 times higher, and $F(z)$\nidentical to the one of 2dF.\nThe Sloan southern stripe quasar survey might provide such a sample\nif quasars can be identified and their redshifts measured to a \nsufficiently faint limiting magnitude. Figure 9 shows that even with the small \ncorrelation length, the high-density survey allows strong rejection\nof models with incorrect geometry in all realizations with\n$\\mbox{${\\Omega}_{0}$ }=0.1$ or 0.2 and in most realizations with $\\mbox{${\\Omega}_{0}$ }=0.4$.\n\n\n\n\nAlthough we have focused mainly on the sensitivity of planned surveys \nto \\mbox{${\\Omega}_{0}$ } and \\mbox{${\\lambda}_{0}$}, we should note that in our experiments the\n2dF and SDSS samples constrain $\\gamma$ and $s_{0}$ to at least \n10\\% accuracy, with typical accuracy of about 5\\%. \nIn the cases with a high quasar correlation length or a high-density\nquasar survey, the maximum-likelihood estimates achieve 2--3\\% accuracy \nin recovering $\\gamma$ and 1--3\\% accuracy in recovering $s_{0}$. \nWe can thus expect these future surveys to provide precise measurements\nof the basic parameters characterizing the quasar correlation function.\n\n\\section{Discussion}\nOur results from \\S 4.4 show that the ambitious quasar redshift surveys\nplanned by the 2dF and SDSS teams can make important contributions to\nthe study of the geometry of spacetime.\nIn the case of flat cosmological models, some constraints on \\mbox{${\\Omega}_{0}$ } can be \nobtained even for the SB correlation length, and these\nconstraints become interestingly tight for a high-density quasar survey\nor for the 2dF and SDSS if the correlation length is a factor of two\nlarger than SB's estimate.\nClustering anisotropy does not provide useful constraints on \\mbox{${\\Omega}_{0}$ } in\nopen models because the distortion parameter $h(z)$ is insensitive\nto \\mbox{${\\Omega}_{0}$ }, and there is a near degeneracy in $h(z)$ between an open model with\nlow \\mbox{${\\Omega}_{0}$ } and a flat model with a higher \\mbox{${\\Omega}_{0}$ }.\nThe true power of the clustering anisotropy technique comes into play\nif \\mbox{${\\Omega}_{0}$ } itself is determined independently by dynamical methods.\nClustering anisotropy then provides a tool for discriminating between\nopen models with $\\mbox{${\\lambda}_{0}$ }=0$ and flat models with $\\mbox{${\\lambda}_{0}$ }=1-\\mbox{${\\Omega}_{0}$ }$,\na distinction that is difficult to achieve with dynamical methods alone.\nWe find that clear discrimination between flat and open geometries \nis possible for the 2dF and SDSS samples with the SB correlation length\nif $\\mbox{${\\Omega}_{0}$ } \\leq 0.2$, but the discrimination is only marginal for $\\mbox{${\\Omega}_{0}$ }=0.4$.\nThe high-density survey with the SB correlation length or the 2dF or\nSDSS for double the SB correlation length provide clear discrimination\nfor $\\mbox{${\\Omega}_{0}$ }=0.4$.\n\nThe results in Figures~5--9 look fairly promising, \nbut there are some limitations in our numerical experiments. \nFor example, we assumed no evolution in $\\mbox{$\\gamma$}$ and a fixed form\nof $s_0(z)$ (corresponding to no comoving evolution for $\\mbox{${\\Omega}_{0}$ }=1$, $\\mbox{$\\Lambda$ }=0$),\nso that we could produce one global fit to the clustering and cosmological\nparameters using all of the data.\nIn the real universe, one will have to allow for the possibility of\ndifferent evolution of the correlation function parameters.\nThe maximum-likelihood method described in \\S 4.3 adapts easily\nto this if the evolution can be described in a parametrized form,\nbut because there can be some tradeoff between different effects,\nadding new parameters will weaken the constraints on \\mbox{${\\Omega}_{0}$ } and \\mbox{${\\lambda}_{0}$}.\nAlso, as we have already noted, our assumed values of $s_{0}$ and \\mbox{$\\gamma$}, \nwhile plausible, are highly uncertain. \nIf $s_{0}$ and $\\mbox{$\\gamma$}$ turn out to be larger than we have assumed, then\nthe statistical power of the clustering anisotropy test will increase; \nif smaller it will decrease.\n\nProbably the most important element missing in our analysis is the\neffect of peculiar velocities, since small scale velocity dispersions\nand large scale coherent flows can both induce an angular dependence \nin $\\xi (s,\\mu)$.  To a first approximation, the small-scale dispersion\nhas the effect of convolving the correlation function along the line of \nsight with the pairwise velocity distribution function \n(Davis \\& Peebles 1983; see also Fisher 1995). \nThis causes a large distortion of the correlation function for \ngalaxies at $z=0$.  However, if $s_0=0.01$ at $z=2$, then the velocity scale \ncorresponding to the correlation length is $0.01c = 3000 \\,{\\rm kms}^{-1}$. \nThe dispersion velocity of quasars is unknown, but a plausible value is \n$\\sim 300 \\,{\\rm kms}^{-1}$ --- the velocity dispersion of typical galaxy \ngroup. The width of the distribution function\nis thus very small compared to the correlation length, so for pairs separated\nby $\\sim s_0$, the distortion due to small scale dispersion should be small.\n\nCoherent large scale motions pose a potentially more serious problem.\nAs a rough gauge of the importance of those velocities in confusing\ngeometrical distortion, we can estimate the ratio of $\\xi(s,1)\/\\xi(s,0)$\nresulting from geometry and from linear theory peculiar velocities.\nFor the surveys that we have considered, most of the geometrical information\ncomes from the redshift $z \\sim 2$, so we will take this value as the\nfiducial one.  For typical cosmological models, $h(z) \\sim 2$ at $z=2$,\nwhich means that $\\xi(s,1)\/\\xi(s,0)\\approx 3.5$ for geometrical distortions \nalone (see equation~[\\ref{xi.s}]).\nIn conventional notation, \n\\begin{equation}\n\\beta \\approx \\frac{{\\Omega}^{0.6}}{b} \\lab{beta.def}\n\\end{equation}\nis the linear theory ratio of peculiar velocity convergence \n$-\\nabla\\cdot {\\bf v}_{pec}\/H$ to the galaxy density contrast,\nwhere $b$ is the bias parameter relating galaxy and mass fluctuations.\nIf $\\beta \\ll 1$,\nthen equations (10) and (11) of Hamilton (1992) allow us to write\n\\begin{equation}\n\\frac{\\xi(s,1)}{\\xi(s,0)}=1-\\frac{2\\beta\\mbox{$\\gamma$}}{(3-\\mbox{$\\gamma$})+2\\beta} \\lab{xi.ratio}\n\\end{equation}\nWe justify equation\\r{xi.ratio} in appendix A and argue that the typical\nvalue of $\\beta$ (assuming the SB quasar correlation length)\nis $\\sim 1\/6$ in open, flat, and Einstein-de Sitter cosmological models.\nWith $\\mbox{$\\gamma$}=1.8$, the ratio\\r{xi.ratio} then becomes\n\\begin{equation}\n\\frac{\\xi(s,1)}{\\xi(s,0)}\\approx 0.6\n\\end{equation}\nThus, the effect of coherent peculiar velocities by no means \noverwhelms the geometric signal, but it is not negligible. \nAnalyses of real data will have to fit jointly for peculiar velocity\ndistortions and geometrical distortions, using their different\nangular and redshift dependences to separate them.\nThe results of Ballinger et al.\\ (1996), Matsubara \\& Suto (1996),\nand Nakamura et al.\\ (1997) provide important steps towards this goal,\nthough it is not clear whether the linear theory formula for velocity\ndistortions will be adequate near $\\xi\\sim 1$. \nA suitably parametrized description of peculiar velocity distortions can\neasily be incorporated in the maximum-likelihood scheme.\nThe geometrical distortion is weakest in the models with low values of \n$h$ (low \\mbox{${\\Omega}_{0}$ }, high \\mbox{${\\lambda}_{0}$ } models).\nHowever, these are also the models for which open and flat geometries\nare most easily distinguished, so there is good reason to believe that\nsuch distinctions will remain possible even when the effects of peculiar\nvelocities are taken into account.\n\nThere are other routes to measuring or limiting the cosmological constant.\nProbably the most solid existing limits come from the statistics of\ngravitational lensing (see, e.g., Carroll, Press, \\& Turner 1992;\nMaoz \\& Rix 1993; Kochanek 1996).  These limits are somewhat dependent\non assumptions about the evolution of the galaxy population\n(Rix et al.\\ 1994), but current results present a strong case that \n$\\mbox{${\\lambda}_{0}$ } < 0.8$.  A promising approach on the horizon is the application of \nthe classical magnitude-redshift test to calibrated candles in the form of\nhigh-redshift, Type Ia supernovae (Perlmutter et al. 1997).\nThis technique is technically challenging, since one must guard carefully\nagainst differential selection biases between low- and and high-redshift\nsamples, but if these obstacles can be overcome it is one of the\nmost powerful methods for constraining $\\mbox{${\\lambda}_{0}$ }$.\nThis approach relies on the\nplausible but not incontrovertible assumption that the luminosities\nof Type Ia supernovae do not change systematically over the age of\nthe universe (see von Hippel, Bothun, \\& Schommer 1997).\nHigh-resolution maps of the cosmic microwave background, such as those\nexpected from the MAP and Planck surveyor satellites, offer another\npromising route to constraining \\mbox{${\\lambda}_{0}$ }, along with a host of other \ncosmological parameters (see, e.g., Bennett et al. 1995 and \nJungman et al. 1995), at least in the context of cosmic inflation models.\nIt is not yet clear which of these approaches will ultimately yield\nthe best constraints on the cosmological constant, but the \nexistence of four essentially independent methods, all of them able\nto produce interesting results in the next 5-10 years, is\nvery encouraging.  Either they will all yield consistent answers,\nin which case they provide not just a convincing constraint but\na consistency test of the underlying cosmological picture that motivates\nthem, or they will not, in which case they point to a breakdown in\nthe implicit assumptions behind one or more of the methods.\n\nAs an extension of this last point, we note that all of our \n2dF and SDSS realizations in Figure~5 are inconsistent at the $>2\\sigma$\nlevel with the Euclidean, $\\mbox{${\\lambda}_{0}$ }=1$, $\\mbox{${\\Omega}_{0}$ }=0$ model even for a flat\ngeometry with $\\mbox{${\\Omega}_{0}$ }$ as low as 0.1 --- and the inconsistency is much\nstronger for higher $\\mbox{${\\Omega}_{0}$ }$, a higher quasar correlation length\n(Figures~7 and~8), or a higher density quasar survey (Figure 9).\nThese results imply that the geometrical distortion described in \\S 2\nshould be detected by these surveys, even if the discrimination between\nflat and open space geometries is weak.  This distortion is a fundamental\nproperty of the cosmological spacetime curvature predicted by General\nRelativity.  If the universe is as we believe it to be, this signature\nof curved spacetime should be detected within the next decade, and perhaps\nwithin the next few years.\n\n\\acknowledgments\n\nWe thank Jim Gunn for seminal discussions at the outset of this project.\nWe thank Heidi Newberg, Brian Yanny, Don Schneider, and other members\nof the SDSS Quasar Working Group for helpful discussions of plans for\nthe SDSS quasar survey.  We thank Tom Shanks for information about the\n2dF quasar survey and for helpful discussions on recent estimates\nof quasar clustering.\nWe acknowledge support from NASA Theory Grant NAG5-3111.\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\tViolent stellar events, such as supernovae and neutron \nstar collisions, release tremendous amounts of energy, primarily \nin the form of neutrinos\\cite{1}. In regions of low density, the \nannihilation process $\\nu\\bar\\nu\\rightarrow e^+e^-$\nhas been proposed as a source of energy to restart the shock wave in \ncore collapse supernovae\\cite{2}, and as a means of generating an energetic \n$e^+e^-$ plasma for gamma ray bursts\\cite{3}. Since detailed \ncalculations of supernova and gamma ray burst scenarios indicate that the\nannihilation rate is too low\\cite{4}, it has been suggested that the \nstar's magnetic field will act as a catalyst for the reaction by eliminating \nsome kinematical constraints on the $e^+e^-$ pair\\cite{3}. The case for\nsuch an enhancement is motivated by studies of pair creation\n ($\\gamma\\rightarrow e^+e^-$ and $\\gamma\\rightarrow \\gamma\\gamma$)\\cite{5} \n and the neutrino synchotron process \n($\\nu\\rightarrow\\nu e^+e^-$\\cite{6} \\cite{7}), where processes \nthat are kinematically disallowed in the vacuum are made possible by \nthe exchange \nof momentum between the $e^+e^-$ pair and the magnetic field. The central role \nof the magnetic field in these processes is reflected in the sensitivity of the \ncross sections to the field strength. Significantly, this enhancement does not\nseem to occur in processes that are kinematically allowed in the absence of the \nfield\\cite{8}.   \n \n\tAs an added bonus, the magnetic field provides a preferred direction \nin space, opening the way for parity violating effects to produce an \nasymmetry in neutrino cross sections. The asymmetries, which have been calculated\nfor $\\nu-e$ scattering\\cite{7}\\cite{9} and $\\nu$-nucleus elastic \nscattering\\cite{10}, provide a mechanism \nfor producing asymmetric supernova explosions, either directly or by\nasymmetrically heating the surrounding matter. Such asymmetric explosions have \nbeen proposed as a means of producing the observed large \nvelocities of pulsars\\cite{6}. \n\t\n\tIn this paper, we calculate the $\\nu\\bar\\nu$ annihilation \nfor a variety of magnetic field strengths, densities, and temperatures\nsimilar to those found in supernovae. Our formalism\\cite{8}\\cite{9}\\cite{10}, is \ndetailed in the next section, while our results\nand their implications for astrophysics are left to the concluding section.\n\n\\section{Formalism}  \n\n\tWe consider the process for $\\nu\\bar\\nu$ annihilation \nin the presence of a magnetic field of strength ${\\bf B}$ in the z direction, \nusing the four point interaction\n\\begin{equation}\nL_{int} ={G_F\\over\\sqrt{2}}\\, \\int\\, d^4x\\, \\bar\\nu(x)\\gamma_\\alpha(1-\\gamma_5)\\nu(x)\\,\\bar e(x)\\gamma^\\alpha(C_V-C_A\\gamma_5)e(x),\n\\end{equation}\nwhere $G_F=1.13\\times 10^{-11}$ MeV$^{-2}$ is the Fermi coupling strength,\n$C_V=2\\sin^2\\theta_W\\pm \\frac{1}{2}$, $C_A=\\pm\\frac{1}{2}$ with the plus sign\nfor electrons and the minus for $\\mu$ or $\\tau$ neutrinos, and \n$\\sin^2\\theta_W\\approx .223$.\n\n\tThe wavefunctions of the electron-positron pair are obtained by \nsolving the Dirac equation in Landau gauge ($A_x=-By$)\\cite{12} \n\n\\begin{eqnarray} \n\\Psi^{e^-}_{(p_x,p_z,n,\\sigma)}({\\bf x},t)& =& {e^{ip_x x}e^{ip_z z}\ne^{-i\\epsilon t}\\over \\sqrt{L_xL_z}} \\pmatrix{ \\alpha CH_{n-1}(\\xi)\\cr\n-\\sigma \\alpha DH_{n}(\\xi)\\cr \\sigma\\beta CH_{n-1}(\\xi)\\cr  -\\beta DH_{n}(\\xi)\n\\cr},\\nonumber\\\\\n\\Psi^{e^+}_{(p^\\prime_x,p^\\prime_z,m,\\sigma^\\prime)}({\\bf x},t)& =& \n{e^{-ip^\\prime_x x}e^{-ip^\\prime_z z}\ne^{-i\\epsilon^\\prime t}\\over \\sqrt{L_xL_z}} \\pmatrix{ \\beta^\\prime D^\\prime \nH_{m-1}(\\xi^\\prime)\\cr\n-\\sigma^\\prime \\beta^\\prime C^\\prime H_{m}(\\xi^\\prime)\\cr -\\sigma^\\prime\n\\alpha^\\prime \nD^\\prime H_{m-1}(\\xi^\\prime)\\cr  \n\\alpha^\\prime C^\\prime H_{m}(\\xi^\\prime)\n\\cr},\n\\end{eqnarray}\n\n\\noindent where $\\pmatrix{\\alpha\\cr\\beta}=\\frac{1}{\\sqrt{2}}\n(1\\pm\\frac{m_e}{\\epsilon})^{\\frac{1}{2}}$, \n$\\pmatrix{C\\cr D}=\\frac{1}{\\sqrt{2}}(1\n\\pm{\\sigma p_z\\over (\\epsilon^2-m^2)^{\\frac{1}{2}}} )^{\\frac{1}{2}}$,\n$\\,\\sigma=\\pm 1$ is the electron's helicity,\n$H_n(\\xi)$ is the normalized solution of a one dimensional harmonic \noscillator with $\\xi=\\sqrt{eB}(y-{p_x\\over eB})$, and the energies of the \nLandau levels are given by $\\epsilon = \\sqrt{p_z^2+2eBn+m_e^2}$.\nFor the positron spinor, the primed quantities are obtained from the\nunprimed by replacing $\\epsilon\\rightarrow\\epsilon^\\prime$, $p_i\\rightarrow\np^\\prime_i$, and $\\xi^\\prime=\\sqrt{eB}(y+{p^\\prime_x\\over eB})$.\n\n\tThe differential cross section for the annihilation process is given \nby\t\n\\begin{equation}\nd\\sigma = {G_F^2 L_xL_z\\over 4\\pi k_1\\cdot k_2 V} C_{\\mu\\nu}A^{\\mu\\nu}\n\\delta(Q_0-\\epsilon-\\epsilon^\\prime)\\delta(Q_z-p_z-p^\\prime_z)\n\\delta(Q_x-p_x-p^\\prime_x) dp_x dp_z dp^\\prime_x dp^\\prime_z,\n\\end{equation}\nwhere $Q^\\mu=k_1^\\mu+k_2^\\mu$, with $k_1(k_2)$ the (anti-)neutrino momentum,\n$V=L_xL_yL_z$ is a normalization volume,\t\n\\begin{equation} \nC^{\\mu\\nu}= k_1^\\mu k_2^\\nu +k_1^\\nu k_2^\\mu - g^{\\mu\\nu}k_1\\cdot k_2\n+i\\epsilon^{\\mu\\nu\\alpha\\beta}k_{1\\alpha} k_{2\\beta},\n\\end{equation}\nand $A^{\\mu\\nu}=\\sum_{\\sigma\\sigma^\\prime} S^\\mu S^{*\\nu}$ is the \nspin-summed product of the current matrix elements of the electron.\nNeglecting the electron mass, the matrix elements are given by\n\\begin{eqnarray}\nS^0 &=& {(1-\\sigma\\sigma^\\prime)\\over 2}(C_V-C_A\\sigma)\n(-DC^\\prime F_{nm}+CD^\\prime F_{n-1,m-1})\\nonumber\\\\\nS^1 &=& {(1-\\sigma\\sigma^\\prime)\\over 2}(C_V-C_A\\sigma)\n(-DD^\\prime e^{i\\phi}F_{n,m-1}-CC^\\prime e^{-i\\phi}\nF_{n-1,m})\\nonumber\\\\\nS^2 &=& -i{(1-\\sigma\\sigma^\\prime)\\over 2}(C_V-C_A\\sigma)\n(-DD^\\prime e^{i\\phi} F_{n,m-1}+CC^\\prime e^{-i\\phi}\nF_{n-1,m})\\nonumber\\\\\nS^3 &=& {(1-\\sigma\\sigma^\\prime)\\over 2}(C_V-C_A\\sigma)\\sigma\n(DC^\\prime F_{nm}+CD^\\prime F_{n-1,m-1}),\n\\end{eqnarray}\nwhere\n\\begin{equation}\n F_{nm} = \\left(\\frac {n_<!}{n_>!}\\right)^\\frac{1}{2} \\left(\\frac{(n-m)}{\\vert n-m\\vert}\\sqrt{\\frac{\nQ_\\perp^2}{2eB}}\\right)^{n_<-n_>} e^{-Q_\\perp^2\\over 4eB} e^{i(m-n)\\phi} \nL_{n<}^{n_>-n_<}\\left(\\frac{Q_\\perp^2}{2eB}\\right),\n\\end{equation}\nwith $L_n^\\alpha$ a generalized Laguerre polynomial\\cite{A+S} and \n$\\tan{\\phi}=Q_y\/Q_x$.  Here $n_>$ is the greater of $n$ and $m$.\n\n\tTo simplify the remainder of the calculation, we choose coordinates \nsuch that $Q_y =0$. In this system, the non-zero components $A_{\\mu\\nu}$ are \ngiven by\n\\begin{eqnarray}\nA_{00}&=&\\frac{1}{2}\\bigg[(C_V^2+C_A^2)\\bigg(\n(1+v_e v_p)(\\vert F_{nm}\\vert^2+\n\\vert F_{n-1,m-1}\\vert^2) + 2 v_n v_m \n F_{nm}F_{n-1,m-1}\\vert_{\\phi=0}\\bigg) \\nonumber\\\\\n& & -2C_VC_A(v_e+v_p)\\bigg(\n\\vert F_{n-1,m-1}\\vert^2-\\vert F_{nm}\\vert^2\n\\bigg)\\bigg]   \\nonumber\\\\\n& &\\nonumber\\\\\nA_{11}&=&\\frac{1}{2}\\bigg[(C_V^2+C_A^2)\n\\bigg((1-v_e v_p)(\\vert F_{n,m-1}\\vert^2+\n\\vert F_{n-1,m}\\vert^2)-2 v_nv_m F_{n,m-1}\nF_{n-1,m}\\vert_{\\phi=0}\\bigg)\\nonumber \\\\\n& &  \n+2C_VC_A(v_e-v_p)\\bigg( \\vert F_{n-1,m}\n\\vert^2-\\vert F_{n,m-1}\\vert^2 \\bigg)\\bigg]\\nonumber\\\\\nA_{22}&=&\\frac{1}{2}\\bigg[(C_V^2+C_A^2)\\bigg((1-v_e v_p)\n(\\vert  F_{n,m-1}\\vert^2+\n\\vert F_{n-1,m}\\vert^2)+2 v_nv_m F_{n,m-1}F_{n-1,m}\n\\vert_{\\phi=0}\\bigg) \\nonumber \\\\\n& &+2C_VC_A(v_e-v_p)\\bigg(\n\\vert F_{n-1,m}\\vert^2-\\vert F_{n,m-1}\\vert^2\n\\bigg)\\bigg]\\nonumber\\\\\nA_{33}& = & \\frac{1}{2} \\bigg[ \n(C_V^2+C_A^2) \\bigg( (1+v_e v_p)(\\vert F_{n,m}\\vert^2+\n\\vert F_{n-1,m-1}\\vert^2) - 2 v_n v_m  F_{n,m} \nF_{n-1,m-1} \\vert_{\\phi=0}\\bigg) \\nonumber\\\\\n& & -2C_VC_A(v_e+v_p)( \\vert F_{n-1,m-1}\\vert^2-\n\\vert F_{n,m} \\vert^2 \n)\\bigg]\\nonumber\\\\\nA_{01}&=& A_{10}=\\nonumber\\\\\n& &\\,\\frac{1}{2} \\bigg[-(C_V^2+C_A^2)\n\\bigg( v_n (  F_{n,m-1}F_{n-1,m-1}+\n F_{n-1,m}F_{n,m}) +v_m ( F_{n,m-1}F_{n,m}\n+ F_{n-1,m}F_{n-1,m-1})\\bigg)\n\\nonumber\\\\\n& &  +2C_VC_A \\bigg( v_nv_p (F_{n,m-1}F_{n-1,m-1}\n-  F_{n-1,m}F_{n,m})-v_mv_e (F_{n,m-1}F_{n,m}\n- F_{n-1,m}F_{n-1,m-1})\\bigg) \\bigg]\\vert_{\\phi=0}\n\\nonumber\\\\\nA_{03}&=&A_{30}=\\nonumber\\\\\n& &\\,\\frac{1}{2}\\bigg[-(C_V^2+C_A^2)\\bigg(\n(v_e+v_p)(\\vert F_{n,m}\\vert^2+\\vert F_{n-1,m-1}\\vert^2)+ 2v_nv_m  F_{n,m}\nF_{n-1,m-1}\\vert_{\\phi=0}\\bigg)\\nonumber\\\\\n& &+2C_VC_A(1+v_ev_p)\n\\bigg(\\vert F_{n-1,m-1}\\vert^2-\\vert F_{n,m}\\vert^2\\bigg)\\bigg]\n\\nonumber\\\\\nA_{13}&=&A_{31}=\\nonumber\\\\\n& &\\,\\frac{1}{2} \\bigg[ ( C_V^2+C_A^2 )\n\\bigg(v_nv_p (F_{n,m-1} F_{n-1,m-1}\n+ F_{n-1,m}F_{n,m}) +v_mv_e( F_{n,m-1}F_{n,m}\n+ F_{n-1,m}F_{n-1,m-1})\\bigg)\\nonumber\\\\\n& & -2C_VC_A \\bigg( v_n (\n F_{n,m-1}F_{n-1,m-1}-\n F_{n-1,m}F_{n,m})+v_m(F_{n,m-1}F_{n,m}- F_{n-1,m}F_{n-1,m-1})\\bigg)\\bigg]\n\\vert_{\\phi=0} \n\\nonumber\\\\\nA_{02}& = &-A_{20}=\\nonumber\\\\\n& &\\,\\frac{i}{2}\n\\bigg[ -(C_V^2+C_A^2)\n\\bigg( v_n(F_{n,m-1}F_{n-1,m-1}- F_{n-1,m}F_{n,m}) +v_m(F_{n,m-1}F_{n,m}\n- F_{n-1,m}F_{n-1,m-1})\\bigg) \n\\nonumber\\\\\n& & +2C_VC_A\\bigg(v_nv_p( F_{n,m-1}\nF_{n-1,m-1} + F_{n-1,m}F_{n,m})+v_mv_e( F_{n,m-1}F_{n,m}\n+ F_{n-1,m}F_{n-1,m-1})\\bigg)\\bigg]\\vert_{\\phi=0}\n\\nonumber\\\\\nA_{12}&=&-A_{21}=\\nonumber\\\\\n& &\\, \\frac{i}{2}\\bigg[(C_V^2+C_A^2)(1-v_ev_p)\n\\bigg( \\vert F_{n-1,m}\\vert^2-\\vert F_{n,m-1}\\vert^2\\bigg)\\nonumber\\\\\n& & +2C_VC_A\\bigg((v_e-v_p)(\\vert F_{n,m-1}\n\\vert^2+ \\vert F_{n-1,m} \\vert^2)-2 v_nv_m F_{n,m-1}\nF_{n-1,m}\\vert_{\\phi=0}\\bigg)\\bigg] \n\\nonumber \\\\\nA_{23}&=&-A_{32}=\\nonumber\\\\\n& &\\, \\frac{1}{2} \\bigg[( C_V^2+C_A^2)\n\\bigg(v_nv_p(F_{n,m-1}F_{n-1,m-1}- F_{n-1,m}F_{n,m})\n -v_mv_e (F_{n,m-1}F_{n,m} -F_{n-1,m}F_{n-1,m-1})\\bigg) \\nonumber\\\\\n& &\n +2C_VC_A \\bigg( v_n ( F_{n,m-1}\nF_{n-1,m-1}- F_{n-1,m}F_{n,m})-v_m (  F_{n,m-1}F_{n,m}\n- F_{n-1,m}F_{n-1,m-1})\\bigg)\\bigg]\\vert_{\\phi=0},\n\\end{eqnarray}\n\\noindent where\n$v_{n(m)}=\\sqrt{2eBn(m)}$,$v_{e}=\\frac{p_z}{\\epsilon}$,and $v_p=\n\\frac{p^\\prime_z}{\\epsilon^\\prime}$. The contributions may be classified as\neither parity conserving($\\propto (C_V^2+C_A^2)$, symmetric) or \nparity violating, and by whether the parity violation occurs \nin the neutrino($\\propto (C_V^2+C_A^2)$, anti-symmetric) or electron currents\n($\\propto C_VC_A$, symmetric). This last distinction is important \nboth because the total effect of parity violating neutrino currents will\ntend to vanish when integrated against the initial spectrum of neutrino and \nanti-neutrinos, and because the asymmetric propagation of neutrinos vs \nanti-neutrinos will lead to asymmetries in the the star's lepton number,\n$Y_e$.    \n\n\tAll that remains is to solve the kinematic constraints on the\nelectron momenta and to include medium effects. The former is\na straight forward algebraic exercise, yielding\n\\begin{equation}\np_z={(Q_0^2-Q_z^2+2eB(n-m))Q_z+\\lambda Q_0\\sqrt{(Q_0^2-Q_z^2)^2-\n4eB(Q_0^2-Q_z^2)(n+m)+4e^2B^2(n-m)^2}\\over 2(Q_0^2-Q_z^2)},\n\\end{equation}\n with $p^\\prime_z=Q_z-p_z$ and $\\lambda=\\pm1$. Implicit in this \nprescription are\n maximum values for $m$ and $n$, $N_{max}=int((Q_0^2-Q_z^2)\/2eB)$,\nand $M_{max}= int{(\\sqrt{N_{max}}-\\sqrt{n})^2)}$. Additionally, the\nintegration over the energy conserving delta function will generate \na Jacobian factor $\\vert v_e-v_p\\vert^{-1}$. After inserting Pauli \nblocking factors for the electron and positron, the cross section becomes\n\\begin{equation}\n\\label{finito}\n\\sigma= \\sum_{\\lambda=\\pm 1}\\sum_{n=0}^{N_{max}}\\sum_{m=0}^{M_{max}}\n{G_F^2 eB\\over 2\\pi s} C_{\\mu\\nu}A^{\\mu\\nu} \n{(1-f_e)(1-f_p)\\over\\vert v_e-v_p\\vert}\n\\end{equation}\nwhere $s=(k_1+k_2)^2$ and \n\\begin{eqnarray}\nf_{e}&=&{1\\over 1+e^{{\\epsilon-\\mu_0}\\over T}},\\nonumber\\\\\nf_{p}&=&{1\\over 1+e^{{\\epsilon^\\prime+\\mu_0}\\over T}}\n\\end{eqnarray}\nwith $\\mu_0$ the electron chemical potential and $T$ the temperature\nof the medium.\n\n\\section{Results}\n\n\tIn order to understand the behaviour of the cross section\nderived in the last section, we have performed calculations for two \ndifferent kinematical situations using a variety of field strengths,\ntemperatures and densities typical of those found in astrophysical \nevents. In the first of these, which we shall refer to as the collinear\ncase, the neutrino and anti-neutrino are traveling nearly parallel to\none another, as is appropriate for neutrinos escaping a supernova. We \nassume their four momenta are given by\n\\begin{eqnarray}\nk_1 &=& (E,P\\sin\\theta,\\sqrt{s}\/2,P\\cos\\theta) \\nonumber\\\\\nk_2 &=& (E,P\\sin\\theta,-\\sqrt{s}\/2,P\\cos\\theta) \n\\end{eqnarray} \nwith $E=10 MeV$, and $P=9 MeV$. The second kinematical situation \nwe shall consider is the case where the neutrino and anti-neutrino\ncollide head-on with zero center of mass momentum. In this instance,\nthe neutrino and anti-neutrino momenta are given by\n\\begin{eqnarray}\nk_1 &=& (P,P\\sin\\theta,0,P\\cos\\theta)\\nonumber\\\\\nk_2 &=& (P,-P\\sin\\theta,0,-P\\cos\\theta).\n\\end{eqnarray}      \nThis situation, which desribes the annihilation of backscattered\n(anti-)neutrinos by outgoing (anti-)neutrinos, represents the largest \ncross section for a given incident energy. For ease of comparison\nwith the symmetric case, we  choose $P=8.72$ MeV so that the center\nof mass energy, and consequently the zero field cross section,\nremains the same as in the case of parallel kinematics. \n\n\tIn both kinematical regimes, the number of possible final\nstates grows inversely with $B^2$, so that it is necessary to calculate \nthe electron-positron matrix elements for large values of $n$ and $m$ in\norder to obtain the cross section. In the case of head-on collisions,\nthis difficulty is mitigated by the fact that only matrix elements \nwith $n=m$ are non-zero. For the symmetric collisions, however,\nit is necessary to calculate Laguerre polynomials of large order.\nThis is accomplished by upward recursion using quadruple precision\narithmetic to avoid numerical instabilities. As an added precaution,\nthe resulting polynomials have been compared to results obtained from a \ndownward recursive Clenshaw scheme, and found to agree to six significant\nfigures. Thus, our calculations are not limited by the accuracy to which\nthe Laguerre functions can be realized.  \n\n\tIn Fig. 1, the cross section for $\\nu\\bar\\nu\\rightarrow e^+e^-$\nin free space is shown for a variety of magnetic field strengths as a \nfunction of the angle between the total momentum and the magnetic field \ndirection, assuming symmetric kinematics for the \nannihilating neutrinos. The jagged structure of the cross section\nas a function of angle is a result of the near-vanishing of the\nJacobian factor at threshold values of $\\cos\\theta$ where new final\nstates for the electron-positron pair become available. For strong\nfields($B=100 m_e^2(\\approx 4\\times 10^{16}$G)), this cross section \nvaries strongly with angle, but this effect essentially vanishes\nfor fields comparable to those expected in the core of superova\n($B\/m_e^2 < 1)$. Qualitatively, the angle averaged cross section tends \nto decrease with increasing $B$, which reflects the fact that the creation \nof both the electron and positron in the lowest Landau level is forbidden \nby helicity conservation. Additionally, there is no asymmetry with respect to \nthe magnetic field direction, as such a preference would violate CP. \nIn Fig. 2, the cross section is shown for the head-on collisions for the same\ncenter of mass energy. Because the electron and positron are only sensitive\nto the total momentum of the annihilating pair, and to the direction of the \nB-field, the cross section varies quadratically with the angle between the \nfield and ${\\bf k}_1$, and can be shown analytically to have the form\n\\begin{equation}\n\\sigma\\propto  (A_{11}+A_{33}) + \\cos^2\\theta (A_{11}-A_{33})\n\\end{equation}\nFor large magnetic fields, the $A_{11}$ term dominates, and the \nannihilation cross section for neutrinos traveling parallel to\nthe magnetic field is twice that for those traveling perpendicular to the \nfield. As the field decreases, the electron-positron current matrix elements\nare more isotropic, and the angular dependence of the cross section is\ncorrespondingly smaller. Interestingly, there is a region at small fields\nwhere $A_{33}>A_{11}$, so that neutrinos traveling perpendicular to the \nfield are more likely to annihilate.  \n \n When the neutrinos annihilate in the presence of a nearly degenerate gas \nof electron-positron pairs typical of the astrophysical environments, the \ncross section is reduced significantly by the requirement that the produced\nelectron's energy be above the Fermi energy of the gas. Moreover, since \nthe electron gas is not CP symmetric, the argument forbidding a parity \nviolating asymmetry in the annihilation cross section is invalid. In Figs. \n3 and 4, the cross sections for collinear and head on collisions are \nshown as a function of magnetic field strength, assuming an electron\ngas with $T=1$ MeV and $\\mu=15$ MeV. Comparison of Figs. 1 and 3 \nindicates that, at the energies shown, the cross section is suppressed by a \nfactor of 5-10, and that the cross section for annihilating pairs with \nmomentum parallel to the magnetic field direction is larger. The \nsuppression of the cross section for head on collisions is much more \ndramatic, on the order of $10^3$ for the center of mass energies shown. \nThe cross section is also observed to be more sensitive to magnetic field \nstrength. This \nreflects the difficulty of producing a high energy electron in the\nhead on collision relative to the collinear case, where the electron can \nbe chosen to carry the bulk of the neutrinos' initial momentum. In Figs. 5 \nand 6, the neutrino annihilation asymmetry, defined as the ratio of the \nparity violating to parity conserving contributions to the cross section,\nas a function of magnetic field strength and direction. In the collinear case, \nthe asymmetry is, for angles not too near $0$ or $\\pi$, well fit by a \nstraight line in $\\cos\\theta$ with slope proportional to the strength of \nthe magnetic field. For the head on case, the linear dependence on \n$\\cos\\theta$ can be demonstrated analytically since the electon-positron\ncurrents depend only on the total momentum perpendicular to the magnetic field.\nThe resulting asymmetry is given by\n\\begin{equation}\nR=\\frac{\\sigma_{PV}}{\\sigma_{PC}}\\propto\\frac{A_{12}\\cos\\theta}{(A_{11}+A_{33}) + \\cos^2\\theta (A_{11}-A_{33})},\n\\end{equation}\nwith the same linear dependence on the field strength.\n\n\tAs we have already mentioned, the asymmetry is allowed because the \nbackground gas of electrons and positrons is not CP symmetric. Since the\nenergies of the electrons in the magnetic field are spin dependent, there is\na corresponding spin dependence in the occupation probabilities for states\nin a given Landau level. This variation in the occupation probabilities \nof states near the Fermi surface results in a preference for creating\nelectrons with spin aligned along the field direction. In Figs. 7 and\n8, the ratio of the parity violating to parity conserving contributions\nto the annihilation cross section is shown as a function of angle and\nelectron chemical potential $\\mu$ for $B=10m_e^2$. For the collinear case\n(Fig. 7), the asymmetry increases with $\\mu$ approximately quadratically,\nreflecting the increase in the density of states near the Fermi surface.\nFor the head-on case(Fig. 8), the asymmetry shows no significant dependence \non $\\mu$ at all. This remarkable result comes about as a result \nof the parity violation residing in the neutrino matrix elements, which are\noblivious to the presence of the Fermi sea and to the direction of the\nmagnetic field. Memory of the field direction is recovered when the \nneutrino matrix elements are multiplied by the electron currents, producing\nthe observed asymmetry. The effect of increasing the temperature from\n1-4 MeV is shown in in Figs. 9 and 10 for the same magnetic field strength.\nAs the Fermi surface becomes more diffuse, the electron occupation\nnumbers start to vary more slowly, with the result that the asymmetry\ndecreases for large $T$. In the collinear case(Fig. 9), the asymmetry falls\napproximately like $\\frac{1}{\\sqrt{T}}$ while in the head-on kinematics the\nfall-off is faster, varying approximately as $\\frac{1}{T}$.  \n\n\tCombining these results, we are able to characterize the asymmetry\nfor the two cases in a manner that allows us to extrapolate our results\nto other situations. For the collinear case, we find\n\\begin{equation} \n{\\sigma_{PV}\\over\\sigma_{PC}}\\propto {\\mu^2 B\\over\\sqrt{T}}. \n\\end{equation}\nSince in this case there are two independent parameters with dimensions\nof energy, $E$ and $s$, we have been unable to discover a simple\nenergy dependence for the asymmetry. For the head-on case, there is\nonly one dimensionally consistent possibility, given by\n\\begin{equation}\n{\\sigma_{PV}\\over\\sigma_{PC}}\\propto {B\\over T\\sqrt{s}}.\n\\end{equation}\nThe dependence of the head-on cross section as a function of energy and angle\nis shown in Fig. 11.\n\nAs we have noted previously, these relations are {\\it approximate},\ngood to 10-15 per cent over a wide range of angles. Moreover, it should\nbe apparent that these relations are only valid in instances were the\nasymmetry is fairly small, less than or of the order of 20 per cent. \nIf we assume the validity of these relations, we can extrapolate from\nthe relatively high fields($\\approx 10^{14}$ G) where calculations are \nfeasible,\nto the field strengths observed in pulsars($\\approx 10^{12}$G)\\cite{13}.  \nThe result of this scaling is an asymmetry of order $10^{-4}$, which is \nsignificantly smaller than that required to produce the observed recoil \nvelocities of neutron stars. The smallness of this result does not, however, \nrule out neutrino asymmetries as a source of the recoil velocities since \nthe relatively small asymmetries produced by neutrino annihilations may be \namplified by hydrodynamic instabilities in the collapsing core\\cite{14}. \nAdditionally, there have been speculations that much larger magnetic fields \nmay exist\\cite{15}\\cite{16}.  Finally, it should be noted that there are\nother neutrino processes for which the cross section and asymmetries may be\nlarger, and multiple neutrino interactions may significantly increase the net \nasymmetry\\cite{17}.   \n\n\tIn summary, we have calculated the cross section for $\\nu\\bar\\nu\n\\rightarrow e^+e^-$ in the presence of a magnetic field for an extensive\nset of conditions typical of those found in astrophysical environments. \nWe do not find that the presence of the field provides any significant\nenhancement of the cross section at fixed chemical potential. Although \nthe presence of a degenerate gas of electrons provides the possibility\nof parity violation in this reaction, they are insufficient to produce \nthe observed recoil velocities of neutron stars at the field strengths \nknown to be associated with pulsars.  \n\n\\centerline{\\bf Acknowledgments}\n\n\tThis authors wish to acknowledge support from  NSF Research \nContract No. PHY-9408843 and DOE grant DE-FG02-87ER40365.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nDirect imaging has revealed more than a dozen planetary-mass companions around young stars \\citep[for a review see, e.g.,][]{bow16}. Intermediate resolution ($R$$\\gtrsim$1000--2000) infrared spectroscopy of these objects reveals a large number of spectroscopic features that can be compared with those of free-floating objects of similar mass and\/or temperature (e.g., brown dwarfs) as well as atmospheric models to constrain formation and early evolution scenarios. Owing to the mostly high contrast ratios and close separations to their host stars, however, only few planets have been studied spectroscopically at high signal-to-noise ratios (S\/N). A prime candidate for further investigation is the HD\\,106906\\ AB+b system. \n\n\\object{HD 106906} is a close binary star \\citep{lag17} at a distance of 102.8$\\pm$2.5\\,pc \\citep{gai16} in the Lower Centaurus Crux association \\citep[13$\\pm$2\\,Myr][]{pec12}. This binary star is known to harbor a circumstellar disk extending to $>$500\\,AU with a large inner hole and a co-moving low-mass companion at a projected separation of 7\\farcs1 \\citep[$\\sim$730\\,AU;][]{che05,bai14,kal15,lag16}. Previous estimates of the temperature and mass of the companion, based on low-resolution 1--2.5\\,\\mum\\ spectroscopy and 0.6--3.5\\,\\mum\\ photometry, placed it in the planetary-mass regime \\citep[\\Teff=1800\\,K, $M$=11$\\pm$2\\MJup;][]{bai14,wu16}. \n\nThe large distance of \\object{HD 106906 b} to its host has spawned discussion about its formation process. This close binary may either have formed like a star through gravitational collapse of a molecular cloud or it may have formed in the primary's disk. The latter may have happened in situ or closer to the star and scattered to its current position through interaction with the central binary and\/or other stars \\citep[e.g.,][]{rod17}. Alternatively, it may have formed around another star and was scattered \\emph{into} the system through a close encounter early in the history of the association \\citep{par12}.\n\nWe present here new high S\/N infrared spectroscopy of HD\\,106906\\,b. The observation and data reduction are described in Sect.~\\ref{sec:obs}, determinations of spectral type, effective temperature, luminosity, and mass are presented in Sect.~\\ref{sec:results}. We summarize the new findings and discuss their implications in Sect.~\\ref{sec:summary}.\n\n\n\\section{Observations and data reduction}\\label{sec:obs}\nObservations of HD\\,106906\\,b\\ were taken with the SINFONI integral field spectrograph on the Very Large Telescope between December 2014 and March 2015. The primary star HD\\,106906\\ ($V$=7.8\\,mag), 7\\farcs1 from the target, served as natural guide star for the adaptive optics system. We obtained spectra in $J$ (1.10--1.40\\,\\mum), $H$ (1.45--1.85\\,\\mum), and \\Ks\\ (1.95--2.45\\,\\mum) bands with a spatial pixel scale of 125\\,mas\\,$\\times$\\,250\\,mas, resulting in a field of view of 8\\arcsec$\\times$8\\arcsec\\  and a spectral resolution of $R$$\\approx$2000--4000. Object (O) and sky (S) observations followed an OSSOOSS\\dots\\ pattern with a $\\sim$1--2\\arcsec\\ offset between the two positions. The object frames were randomly offset with respect to each other within a radius of $\\sim$1\\arcsec, always keeping the bright primary outside the field of view. Reference stars with spectral types between B3 and B9 for the correction of telluric absorption were observed close in time to each science observation with the same instrumental setup and at similar airmass. A summary of the observation details is given in Table~\\ref{tab:obs}.\n\n\\begin{table*}\n  \\caption{Observation summary\\label{tab:obs}}\n\\centering\\small\n\\begin{tabular}{lclccccccccc}\n\\hline\\hline\n  &\n  &\n  &\n  &\n  &\n  seeing &\n  Strehl &\n  FWHM\\tablefootmark{c} &\n  Telluric &\n  SpT &\n  \\Teff\\tablefootmark{d} &\n  airmass\\\\\n  UT Date  &\n  Filter   &\n  $n_{\\rm exp}$$\\times$$t_{\\rm int}$ &\n  $R\\!=\\!\\lambda\/\\Delta\\lambda$\\tablefootmark{a} &\n  airmass &\n  (arcsec) &\n  ratio\\tablefootmark{b} &\n  (arcsec) &\n  ref. star &\n  (ref. star) &\n  (ref. star) &\n  (ref. star)\\\\\n\\hline\n2014-12-31 & $H$ & 5$\\times$100\\,s                  & $\\sim$3000 & 1.32 & 1.1 & 10--16   & 0.35 & HIP\\,050038 & B5 & 15200 & 1.32 \\\\  \n2015-01-23 & $H$ & 2$\\times$100\\,s                  & $\\sim$3000 & 1.18 & 0.6 & $\\sim$38 & 0.32 & HIP\\,055480 & B8 & 11400 & 1.59 \\\\  \n2015-02-23 & $K$ & 4$\\times$300\\,s                  & $\\sim$4000 & 1.17 & 1.0 & 20--27   & 0.26 & HIP\\,059363 & B9 & 10500 & 1.19 \\\\\n2015-02-26 & $J$ & 3$\\times$300\\,s\\tablefootmark{e} & $\\sim$2000 & 1.21 & 1.2 & 5--15    & 0.46 & HIP\\,055938 & B3 & 19000 & 1.26 \\\\  \n2015-03-05 & $J$ & 6$\\times$300\\,s                  & $\\sim$2000 & 1.17 & 1.3 & 18--27   & 0.46 & HIP\\,057474 & B7 & 12500 & 1.24 \\\\  \n\\hline\n\\end{tabular}\n\\tablefoot{\n  \\tablefoottext{a}{From the SINFONI manual.}\n  \\tablefoottext{b}{According to SINFONI's Real Time Computer (RTC).}\n  \\tablefoottext{c}{Measured at central wavelength.}\n  \\tablefoottext{d}{Converted from spectral type using \\citet{cox_allens_2000}.}\n  \\tablefoottext{e}{A total of four spectra were observed and extracted at this epoch, but one showed exceptionally high noise and was excluded from further analysis.}\n}\n\\end{table*}\n\nData reduction partly relies on the SINFONI pipeline (Ver.~2.7.0) in the esorex\\footnote{\\url{http:\/\/www.eso.org\/sci\/software\/cpl\/esorex.html}} environment. We used the standard settings and workflow until one integral field data cube was reconstructed per exposure. This involves dark subtraction, flat fielding, distortion correction, and wavelength calibration. The sky frame closest in time, reduced in the same way, was subtracted from each science frame. Since sky frames were taken with a small spatial offset relative to the science frames, most of these frames contain an image of the science target close to the edge of the detector. In order to avoid over subtraction, some sky frames had to be rejected if their point spread function (PSF) overlapped with the science target. Subsequently, we replaced any pixels flagged by the pipeline as bad by an interpolation of the nearest 6 good pixels in spectral direction using a custom \\emph{IDL} routine. The resulting integral field spectroscopic cubes have 64$\\times$64 pixels\\footnote{The SINFONI image plane is filled with 32$\\times$64 rectangular pixels. To create the cube, every spatial pixel was automatically split by the pipeline into two neighboring square pixels with identical pixel values.} along the spatial and 2202 pixels along the spectral direction. \n\nWe extracted 1D spectra from the 3D cubes with the procedure described in \\citet{dae13}, using apertures with radii identical to 0.8 times the full width at half maximum (FWHM) in each wavelength bin. The FWHM values were determined from Gaussian fits to the trace in the high spatial resolution direction of the cube. Larger apertures could not be used because of contamination of the sky frames with astrophysical sources close to the science target location. We investigated whether any systematic uncertainties are introduced by choosing a small extraction aperture. Using the standard star as a reference, we increased the aperture size from 0.8 to 4$\\times$FWHM and found variations of the extracted flux of $<$10\\% per pixel at all wavelengths. Because of the low S\/N of the target object in each individual wavelength bin, determination of the centroid and FWHM as a function of wavelength use a 2D Gaussian fitting to the bright telluric standard star in every slice of the cube. The extracted trace was shifted to the location of the target by fitting a 2D Gaussian profile to the target PSF after averaging 20--100 slices of the cube in spectral dimension.\n\nThe telluric standard stars were reduced and extracted in the same way as the science observations, but with a larger aperture (4$\\times$FWHM), and divided by a blackbody curve according to their effective temperature (Table~\\ref{tab:obs}). Hydrogen absorption lines were replaced by straight lines along the continuum before dividing the science spectra by the tellurics.\n\nFig.~\\ref{fig:im} shows spectral median images of one representative cube per filter. Extracted and averaged spectra are shown in Fig.~\\ref{fig:spectra}.\n\\begin{figure*}[tbh]\n\\includegraphics[width=0.32\\textwidth]{f1a.eps}\\hfill\n\\includegraphics[width=0.32\\textwidth]{f1b.eps}\\hfill\n\\includegraphics[width=0.32\\textwidth]{f1c.eps}\n\\caption{\\label{fig:im} Examples of individual, fully reduced, and sky-subtracted data cubes of HD\\,106906\\,b\\ in $J$, $H$, and $K$ band, median-collapsed in spectral direction. The bright region in the bottom right is created by the primary star just outside the field of view. Dark patches are visible where the sky image contains stars. An example aperture for the extraction of the spectra ($r$=0\\farcs26) is superimposed on HD\\,106906\\,b\\ on the $K$-band image in red. Panels show the full field of view of SINFONI (8$\\times$8\\arcsec); centering is subject to jitter offset. The color stretch is linear, the orientation of all panels as indicated in the \\Ks\\ image.}\n\\end{figure*}\n\\begin{figure*}[tbh]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{f2a.eps}\\vspace{1ex}\n\\includegraphics[width=0.95\\textwidth]{f2b.eps}\\vspace{1ex}\n\\includegraphics[width=0.95\\textwidth]{f2c.eps}\n\\caption{\\label{fig:spectra}Fully reduced and averaged spectra in $J$ (top), $H$ (middle), and \\Ks\\ band (bottom) binned by a factor of 2. Green spectra show individual observing periods (most recent on top); black are averaged from all individual exposures per filter. Each spectrum has been normalized individually; offsets and flux scales are arbitrary. Regions highlighted in magenta may contain residual noise from the removal of absorption lines (mostly hydrogen) in the telluric standard. The wavelengths of some atomic and molecular absorption features of low-mass stellar objects are highlighted \\citep{cus05}. Regions of strong telluric absorption potentially causing residuals in the reduction are indicated by the black horizontal bars.}\n\\end{figure*}\nAs can be seen in the top two panels of Fig.~\\ref{fig:spectra}, we detect slight differences in the spectral slope of the continuum between the two observing epochs in $J$ and $H$ band, respectively. Normalized at the central wavelength in both filters, the amplitude of the variability at the band edges is $\\pm$$\\sim$10\\%. This may be a consequence of the small extraction apertures or of variable telluric water absorption during the observations or a combination thereof. Variable telluric absorption might apply in particular to our second epoch H-band observation where the largest airmass difference between the science and telluric standard target occurred ($\\Delta$airmass$\\sim$0.4). In addition, intrinsic spectral variability at this level has been previously observed for low-mass objects \\citep[e.g.,][]{apa13}. Since we find no significant difference for any of the extracted parameters in Sect.~\\ref{sec:results} when analyzing the individual epochs, we average all exposures for each filter band and perform all measurements on the combined spectra.\nUncertainties per spectral wavelength bin are calculated as the standard error of the mean between individual exposures of the same observing sequence. The measured S\/Ns are S\/N($J$)$\\approx$20\/pixel, S\/N($H$)$\\approx$20--50\/pix, and S\/N($K$)$\\approx$20--40\/pix.\n\n\\section{Analysis and results}\\label{sec:results}\n\n\\subsection{Spectral characteristics}\nThe spectra in Fig.~\\ref{fig:spectra} show the characteristic triangular shape of the H-band continuum, which is indicative of a young, low-mass, and low surface gravity object. Low gravity is quantitatively supported in the classification scheme by \\citet{all13}, which uses ``gravity scores'' based on spectral indices. Scores of 0, 1, and 2 identify objects with no, normal, and strong indications of low gravity, respectively. We measured FeH$_J$=1.12$^{+0.04}_{-0.03}$, K\\,I$_J$=1.06$\\pm$0.01, and H-cont=1.00$\\pm$0.02 resulting in gravity scores of 1, 1, and 2.\n\nWe see a large number of spectroscopic features that are likely caused by absorption through atoms and simple molecules in the atmosphere of HD\\,106906\\,b. The most prominent features are strong potassium lines between 1.15\\,\\mum\\ and 1.3\\,\\mum\\ as well as clearly detected carbon monoxide bands at $\\gtrsim$2.3\\,\\mum\\ \\citep{cus05}. These are common features of low-mass objects. Weaker, but significantly detected features include sodium, magnesium, and calcium. The rest of the spectrum appears to be dominated by weaker absorption lines that are partly blended. \n\nThe equivalent widths $W_\\lambda=\\int (F_\\lambda-F_\\mathrm{c})\/F_\\mathrm{c} \\,\\,\\mathrm{d}\\lambda$ of a selection of strong ($W_\\lambda>1$\\,\\AA) features are listed in Table~\\ref{tab:ews}. \n\\begin{table}\n  \\caption{Equivalent widths of strong ($W_\\lambda>1$\\,\\AA) spectral features\\label{tab:ews}}\n\\centering\n\\begin{tabular}{ccr@{\\dots}lr@{\\,$\\pm$\\,}l}\n  \\hline\\hline\n  feature &\n  $\\lambda_\\mathrm{c}$ (\\mum) &\n  \\multicolumn{2}{c}{$\\lambda_{\\rm int}$ (\\mum)\\tablefootmark{a}} &\n  \\multicolumn{2}{c}{$W_\\lambda$ (\\AA)} \\\\ \n  \\hline\nNa\\,I          & 1.138 & 1.1360  & 1.1420                    & 7.57 & 0.60 \\\\\nK\\,I           & 1.169 & 1.1670  & 1.1710                    & 4.81 & 0.17 \\\\\nK\\,I           & 1.177 & 1.1750  & 1.1805                    & 5.81 & 0.39 \\\\\nK\\,I           & 1.243 & 1.2415  & 1.2455                    & 5.03 & 0.13 \\\\\nK\\,I           & 1.253 & 1.2500  & 1.2550                    & 4.26 & 0.26 \\\\\nK\\,I           & 1.520 & 1.5150  & 1.5200                    & 3.64 & 0.29 \\\\\n$^{12}$CO(6$-$3)& 1.618 & 1.6180  & 1.6230                    & 2.63 & 0.16 \\\\\nFeH            & 1.625 & 1.6240  & 1.6280                    & 2.79 & 0.11 \\\\\nFeH?           & 1.650 & 1.6480  & 1.6515                    & 1.76 & 0.09 \\\\\nFeH?           & 1.656 & 1.6540  & 1.6565                    & 2.10 & 0.05 \\\\\nAlI            & 2.110 & 2.1075  & 2.1165                    & 4.11 & 0.40 \\\\\nNaI            & 2.206 & 2.2057  & 2.2105                    & 2.92 & 0.17 \\\\\nCaI            & 2.26  & 2.2610  & 2.2670                    & 2.69 & 0.29 \\\\\nMgI            & 2.281 & 2.2800  & 2.2820                    & 1.03 & 0.08 \\\\\n$^{12}$CO(2$-$0)& 2.294 & 2.2930  & 2.3100                    & 18.6 & 1.1  \\\\\n$^{12}$CO(3$-$1)& 2.323 & 2.3220  & 2.3390                    & 11.6 & 1.1 \\\\\n$^{12}$CO(4$-$2)& 2.353 & 2.3520  & 2.3690                    & 15.3 & 1.7 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\n  \\tablefoottext{a}{Integration range to determine equivalent widths, taking into account line center shifts with respect to $\\lambda_\\mathrm{c}$. Line identfication and widths from \\citet{mey98,cus05,dae12,bon14}.}\n}\n\\end{table}\nThe continuum $F_\\mathrm{c}$ was measured with a quadratic polynomial fit to visually selected points along the continuum in regions devoid of strong absorption features. Polynomial fits to predefined sections of the spectrum, as used by \\citet{bon14}, did not lead to satisfactory fits of the continuum. Uncertainties of $W_\\lambda$ were obtained from the standard deviation of repeated measurements with different continuum fits following this recipe.\n\n\\subsection{Spectral type}\nWe determined the spectral type of HD\\,106906\\,b\\ with three different methods: visual spectral comparisons (Sect.~\\ref{sec:vis}), spectral indices (Sect.~\\ref{sec:indic}), and examination of the Na\\,I and K\\,I equivalent widths (Sect.~\\ref{sec:ews}). Based on the inferred spectral types of L1.5--L2, L0.9$\\pm$0.6, and L0$\\pm$2, respectively, we calculated a spectral type of L1.7$\\pm$0.3 (inverse variance-weighted average).\nThis value and its uncertainty is dominated by the visual inspection measurement. Given that we relied on the classifications by \\citet{all13} and \\citet{bon14}, which themselves are subject to systematic and random uncertainties of typically 0.5 to 1 subclasses, we adopted a conservative uncertainty of 1 subclass for our best estimate. In the commonly used scheme of half subclasses we obtain L1.5$\\pm$1.0, which is our best estimate for the spectral type of HD\\,106906\\,b. \n\n\\subsubsection{Visual inspection}\\label{sec:vis}\nWe compared our $JHK$ data to the spectral libraries from \\citet{all13} and \\citet{bon14}, which present and characterize spectra of young late-type objects of low gravity. \\citet{all13} examined 73 ultracool young (10--300\\,Myr) field dwarfs and classified these sources in a spectral type range of M5--L7. Their spectra have low ($R\\approx100$) and medium resolution ($R\\gtrsim750$--2000). \\citet{bon14} focused on a smaller sample of 15 young brown dwarfs mostly in the M6--L0 range. Their spectra were observed with the same instrument as used for our study (SINFONI) at equal spectral resolution in $J$ band (R$\\approx$2000) but slightly lower resolution in $H$ and $K$ ($R$$\\approx$1500).\n\nIn Fig.~\\ref{fig:visualinspection} we compare our observed spectra to the library spectra. Because our spectra were not observed simultaneously and we cannot calibrate the flux of the H-band spectrum owing to an unknown H-band magnitude, we cannot obtain a reliable relative flux calibration of the full $JHK$ spectrum. We thus compare each filter individually. \n\\begin{figure*}[tbh]\n  \\centering\n\\includegraphics[width=0.5\\textheight]{f3a.eps}\\vspace{1ex}\n\\includegraphics[width=0.5\\textheight]{f3b.eps}\n\\caption{\\label{fig:visualinspection}Comparison of our spectra (black) with the libraries of young star spectra (red) by \\citet[top]{all13} and \\citet[bottom]{bon14}. The $J$, $H$, and $K$ spectra have been normalized and matched for each band individually.}\n\\end{figure*}\nFor each comparison, our target spectrum has been convolved with a Gaussian profile to match the spectral resolution of the respective library spectrum. The individual results are listed in Table~\\ref{tab:spectralindices}. Best fits range from M8 to L4. Based on the comparison from the \\citet{all13} library, we classify the companion as a L1.5--L2, while the \\citet{bon14} library suggests M8--L0. However, a caveat is that the latter library does not contain spectra with types L0.5--L3.5.\n\n\\begin{table}\n  \\caption{Spectral type estimation\\label{tab:spectralindices}}\n\\centering\n\\begin{tabular}{lcc}\n\\hline\\hline\nData set\/Index & Ref. & Spectral type \\\\\n\\hline\n$J$        & 1 & L0.5\\,$\\pm$\\,1.5 \\\\\n$H$        & 1 & L2.5\\,$\\pm$\\,0.5 \\\\\n$K$        & 1 & L1.0\\,$\\pm$\\,1.0 \\\\\n\\multicolumn{2}{l}{{\\bf Weighted Average}\\tablefootmark{a}} &  {\\bf L2.1\\,$\\pm$\\,0.4} \\\\\n\\hline\\\\[-1.5ex]\n$J$        & 2 & L0.0\\,$\\pm$\\,1.0 \\\\\n$H$        & 2 & L0.0\\,$\\pm$\\,4.0 \\\\\n$K$        & 2 & M9.5\\,$\\pm$\\,1.5 \\\\\n\\multicolumn{2}{l}{{\\bf Weighted Average}\\tablefootmark{a}} &  {\\bf M9.9\\,$\\pm$\\,0.8} \\\\\n\\hline\nFeH        & 3 & $>$L0.6\\tablefootmark{b} \\\\  \nH$_{2}$O-1 & 3 & L0.6\\,$\\pm$\\,1.1 \\\\  \nH$_{2}$O   & 4 & L1.1\\,$\\pm$\\,0.8 \\\\  \nH$_{2}$O-2 & 3 & L1.0\\,$\\pm$\\,1.6 \\\\  \nH$_{2}$OD  & 5 & M7.1\\,$\\pm$\\,0.8\\tablefootmark{b} \\\\  \n\\multicolumn{2}{l}{{\\bf Weighted Average}} &   {\\bf L0.9\\,$\\pm$\\,0.6} \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{Spectral types as estimated from visual inspection (top two sections) and spectral indices (bottom).\n  \\tablefoottext{a}{Average for a spectral library, weighted by the inverse variance.}\n  \\tablefoottext{b}{These measurements do not enter our final spectral type estimate because the FeH index does not cover the entire range of interest for HD\\,106906\\,b\\ \\citep[valid only for M3--L3;][]{sle04} and H$_2$OD is based on data that are strongly affected by residuals from the telluric correction \\citep[1.96--2.08\\,\\mum;][]{mcl03}.}\n  }\n\\tablebib{(1) \\citealt{all13}; (4) \\citealt{bon14}; (3) \\citealt{sle04}; (4) \\citealt{all07}; (5) \\citealt{mcl03}.}\n\\end{table}\n\n\\subsubsection{Spectral indices}\\label{sec:indic}\nIn order to be consistent with previous studies, we adopted spectral indices that were used by \\citet{all13} and \\citet{bon14}. Specifically, we explored the H$_{2}$O index near 1.5\\,\\mum\\ \\citep{all07}, H$_{2}$O-1 ($\\sim$1.3\\,\\mum), and \\mbox{H$_{2}$O-2} ($\\sim$2.1\\,\\mum) indices \\citep{sle04} and the H$_{2}$OD index \\citep[$\\sim$2.0\\,\\mum,][]{mcl03}. Since the $J$ band contains many spectral features from various molecules, we also examined the $J$ band FeH spectral index \\citep{sle04}, near 1.2\\,\\mum. The uncertainties of the best inferred value of L2.5$\\pm$1.9, however, extend beyond the valid spectral type range of M3--L3. Thus, we did not include these measurements in our final spectral type estimate.\n\nTable~\\ref{tab:spectralindices} summarizes our results for each spectral index. Uncertainties were propagated from our measurement uncertainty and the quoted uncertainties of the polynomial fits in the original publications. \nCalculating the average of the spectral types determined using the H$_{2}$O, H$_{2}$O-1 and H$_{2}$O-2 indices and weighing the average by the inverse of the square of their uncertainty, we get a spectral type of L0.9$\\pm$0.6. \n\n\\subsubsection{Equivalent widths}\\label{sec:ews}\nOur spectra have a high S\/N and a relatively high resolution. We can use the additional information contained in our data compared to, for example, \\citet{all13} to improve our spectral type determination. We compare the equivalent widths of five $J$-band features of Na\\,I (1.138\\,\\mum) and K\\,I (1.169, 1.177, 1.243, and 1.253\\,\\mum; see Table~\\ref{tab:ews}) with those of other young objects as measured by \\citet{bon14} at the same spectral resolution. We find that the equivalent widths are broadly consistent with an L0$\\pm$2 object. However, the correlation of spectral type with the Na\\,I and K\\,I equivalent widths of young object spectra such as those of HD\\,106906\\,b\\ is weak and has a larger scatter compared to field objects, as is apparent from Fig.~11 in \\citet{bon14}. This diagnostic thus leads to less precise results than for field dwarfs. More independent determinations of precise equivalent widths and spectral types are needed to increase the usefulness of equivalent widths for spectral type determinations of young planetary-mass objects.\n\n\\subsection{Effective temperature and luminosity}\\label{sec:teff}\nTo obtain robust estimates of \\Teff\\ and \\Lbol, we use two strategies. The first determination is based on our best spectral type estimate and uses empirical correlations of \\Teff\\ and \\Lbol\\ with spectral type for young late-type objects. The second approach derives a luminosity of HD\\,106906\\,b\\ based on its brightness, empirically calibrated bolometric corrections, and the new Gaia distance, which is 10\\% larger than pre-Gaia estimates.\n\nOur analysis is based on the analysis of colors, luminosities, and effective temperatures of 152 young brown dwarfs and directly imaged planets by \\citet{fah16}. They have estimated \\Lbol\\ by integrating the available SEDs for their sample of objects between 0 and 1000 microns. Since the distances to these objects are known, \\citeauthor{fah16} have estimated radii using \\Lbol\\ and model isochrones. Then, based on radius and \\Lbol, they have used the Stefan-Boltzmann law to calculate \\Teff. We mostly use the results from the \\citeauthor{fah16} study as it supersedes the smaller young object sample by \\citet{fil15}. The latter study, however, presents an analysis of a comparison sample of 65 field stars as well as bolometric corrections for young low-mass objects that are used in Sect.~\\ref{sec:lum}.\n\n\\subsubsection{Effective temperature}\n\\citet{fah16} have provided polynomial fits to the \\Lbol\\ and \\Teff\\ correlations with spectral type for three categories of low-gravity objects. First, their ``YNG'' sample contains all spectra with indications of low gravity. Second, the ``YNG2'' contains YNG as a subset, but also includes directly imaged planetary-mass objects. Third, the ``GRP'' sample contains confirmed young moving group targets.\nFor our best estimate of the spectral type of HD\\,106906\\,b\\  (L1.5$\\pm$1.0), we obtain $T_{\\rm eff}^{\\rm YNG}$\\,=\\,1920\\,$\\pm$\\,210\\,K, $T_{\\rm eff}^{\\rm YNG2}$\\,=\\,1820\\,$\\pm$\\,240\\,K, and $T_{\\rm eff}^{\\rm GRP}$\\,=\\,1900\\,$\\pm$\\,240\\,K. Uncertainties were derived from both the spectral type uncertainty and the rms of the spectral type-\\Teff\\ relation by \\citet{fah16}. The temperatures are in good agreement with each other and with the \\citeauthor{bai14} value of \\Teff=1800$\\pm$100\\,K derived from evolutionary models. Since HD\\,106906\\,b\\ is best described by the YNG2 sample, we adopt \\Teff=1820\\,$\\pm$\\,240\\,K as our best estimate for its temperature.\n\nThis temperature is lower than what is obtained when assuming that HD\\,106906\\,b\\ is a field object ($T_{\\rm eff}^{\\rm field}$\\,=\\,2030\\,$\\pm$\\,180\\,K). It is also slightly lower than the field object value derived by \\citealt{bai14} of $T_{\\rm eff}^{\\rm field}$=$1950\\pm200$\\,K based on their later spectral type of L2.5$\\pm$1. This is because similar to young brown dwarfs, planets are redder than field dwarfs in the near-infrared bands \\citep[e.g.,][]{fah16}; this is likely explained by high-altitude clouds \\citep[e.g.,][]{bow10,cur11,mad11,mar12,ske12}. \n\n\\subsubsection{Luminosity\\label{sec:lum}}\nUsing the polynomial fits to the empirical luminosity-spectral type relations derived by \\citet{fah16}, we calculate values of $\\log\\left(L_\\mathrm{bol}^\\mathrm{YNG}\/L_\\odot\\right)$\\,=\\,$-$3.83\\,$\\pm$\\,0.35, $\\log\\left(L_\\mathrm{bol}^\\mathrm{YNG2}\/L_\\odot\\right)$\\,=\\,$-$3.64\\,$\\pm$\\,0.24, and $\\log\\left(L_\\mathrm{bol}^\\mathrm{GRP}\/L_\\odot\\right)$\\,=\\,$-$3.47\\,$\\pm$\\,0.31. Uncertainties, as estimated from a Monte Carlo simulation, take into account the spectral type uncertainty and the rms in $\\log L$ \\citep{fah16}. Again, we select YNG2 as the most appropriate value and listed the others to illustrate the range of answers depending on the underlying sample. \n\nFor an independent estimate of the luminosity we also use bolometric corrections of young stars for the $J$ band and \\Ks\\  band, respectively. These are available through polynomial fits to the YNG sample of \\citet{fil15}. With bolometric corrections for the HD\\,106906\\,b\\  $J$ and \\Ks\\ magnitudes of $BC_J$\\,=\\,1.54\\,$\\pm$\\,0.28\\,mag and $BC_{Ks}$\\,=\\,3.27\\,$\\pm$\\,0.13\\,mag, we derive luminosities of $\\log\\left(L_\\mathrm{bol}^{\\mathrm{YNG,BC}_{Ks}}\/L_\\odot\\right)$\\,=\\,$-$3.57\\,$\\pm$\\,0.05 and $\\log\\left(L_\\mathrm{bol}^{\\mathrm{YNG,BC}_J}\/L_\\odot\\right)$\\,=\\,$-$3.74\\,$\\pm$\\,0.11 calculated as\n\\begin{equation}\n  \\log\\left(\\frac{L_\\mathrm{bol}}{L_\\odot}\\right)=0.4\\left(M_\\mathrm{bol,\\odot}-m+5\\log\\left(\\frac{d}{10\\,\\mathrm{pc}}\\right)-BC\\right)\\quad,\n\\end{equation}\nusing $M_\\mathrm{bol,\\odot}$=4.74\\,mag. Uncertainties were propagated using a Monte Carlo simulation based on the rms values reported by \\citet[$\\Delta$BC$_{Ks}$=0.126\\,mag, $\\Delta$BC$_{J}$=0.189\\,mag]{fil15} as well as the uncertainties of the distance modulus and our spectral type estimate.\nThe comparably faint luminosity derived from the $J$-band corrections is due to a red $J$$-$\\Ks\\ color, compared to atmospheric models and other targets of this spectral type, which was already observed by \\citet{wu16}. \n\nFrom the three estimates ($L_\\mathrm{bol}^\\mathrm{YNG2}$, $L_\\mathrm{bol}^{\\mathrm{YNG,BC}_J}$, and $L_\\mathrm{bol}^{\\mathrm{YNG,BC}_{Ks}}$) we derive $\\log\\left(L_\\mathrm{bol}\/L_\\odot\\right)$\\,=\\,$-$3.65\\,$\\pm$\\,0.08 as our best estimate of the bolometric luminosity of HD\\,106906\\,b\\  using a Monte Carlo simulation to propagate the uncertainties.\n\n\\subsection{Mass}\\label{sec:mass}\nWe used stellar evolution models to translate our inferred luminosity to mass.\nModel families include DUSTY \\citep{cha00}, BT-Settl \\citep{bar15}, and the Bern Exoplanet Tracks (BEX). The first two feature dusty atmospheres and are classified as ``hot start'' according to their high initial (i.e., post-formation) entropies. The BEX use initial entropies found in self-consistently coupled planet formation and evolution simulations \\citep{mor12}. The models were updated to employ Ames Cond atmospheric boundary conditions \\citep{all01} and include Deuterium burning \\citep{mol12}. The coupled formation and evolution calculations lead in population syntheses to planets with a range of post-formation entropies and deuterium abundances for more massive objects. This allows us to identify initial conditions for the evolution corresponding to the hottest, hot, warm, and cold post-formation states \\citep{mor17}.\n\n\nFig.~\\ref{fig:mass} shows HD\\,106906\\,b\\ with respect to the evolutionary tracks. \n\\begin{figure*}[tbh]\n  \\centering\n  \\setlength{\\unitlength}{\\textwidth}\n\\begin{picture}(1.0,1.2)\n    \\put(0.00,0.80){\\includegraphics[width=\\columnwidth]{f4a.eps}}\n    \\put(0.40,1.15){\\fontfamily{phv}\\selectfont DUSTY}\n    \\put(0.50,0.80){\\includegraphics[width=\\columnwidth]{f4b.eps}}\n    \\put(0.88,1.15){\\fontfamily{phv}\\selectfont BT-SETTL}\n    \\put(0.00,0.40){\\includegraphics[width=\\columnwidth]{f4c.eps}}\n    \\put(0.36,0.75){\\fontfamily{phv}\\selectfont BEX Hottest}\n    \\put(0.50,0.40){\\includegraphics[width=\\columnwidth]{f4d.eps}}\n    \\put(0.895,0.75){\\fontfamily{phv}\\selectfont BEX Hot}\n    \\put(0.00,0.00){\\includegraphics[width=\\columnwidth]{f4e.eps}}\n    \\put(0.38,0.35){\\fontfamily{phv}\\selectfont BEX Cold}\n\\end{picture}\n\\caption{\\label{fig:mass} HD\\,106906\\,b's luminosity and age (filled circle) compared to evolutionary tracks. References and extracted mass estimates are listed in Table~\\ref{tab:mass}. The locations of other young, low-mass companions are shown for comparison \\citep[open symbols, target names in the first panel;][]{cha17,bow16}.}\n\\end{figure*}\nWe adopt an age of 13$\\pm$2\\,Myr, assuming that there is no significant delay of the formation of HD\\,106906\\,b\\ compared to other members of Upper Scorpius and its host star. To estimate the mass of HD\\,106906\\,b, we linearly interpolate each grid at the age and luminosity of HD\\,106906\\,b. Mass estimates, listed in Table~\\ref{tab:mass}, range from $M$\\,=\\,12.5$\\pm$1.0\\,\\MJup\\ to $M$\\,=\\,14.2$^{+0.4}_{-0.9}$\\,\\MJup, depending on the initial entropy of the formation scenario, i.e., whether a hot star or cold start model was used, and on the atmospheric dust model. \nWhen assuming a 3\\,Myr formation timescale \\citep[cf.][]{for05} for HD\\,106906\\,b\\ (i.e., an effective age of 10$\\pm$2\\,Myr), the derived masses decrease by 0--0.7\\,\\MJup\\ (see Table~\\ref{tab:mass}).\n\n\\begin{table}\n  \\caption{Masses from isochrone fits\\label{tab:mass}}\n\\centering\n\\begin{tabular}{lccc}\n\\hline\\hline\nModel & Ref. & $M$ (\\MJup) & $M_{t_{\\rm form}=3\\,{\\rm Myr}}$ (\\MJup) \\\\\n\\hline\nDUSTY                          & 1 & $12.3^{+0.9}_{-0.8}$ & $11.9^{+2.5}_{-0.9}$ \\\\\nBT-SETTL                       & 2 & $12.3^{+0.8}_{-0.7}$ & $11.9^{+1.7}_{-0.8}$ \\\\\nBEX Hottest                    & 3 & $12.8^{+1.1}_{-0.8}$ & $12.1^{+1.7}_{-1.1}$ \\\\\nBEX Hot                        & 3 & $13.1^{+0.8}_{-0.6}$ & $12.6^{+1.3}_{-0.9}$ \\\\\nBEX Cold\/Warm\\tablefootmark{a} & 3 & $14.0^{+0.2}_{-0.5}$ & $14.0^{+0.2}_{-0.4}$ \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\n  \\tablefoottext{a}{The cold and warm evolutionary tracks of the BEX models predict identical luminosities for objects above $\\sim$10\\,\\MJup\\ \\citep{mor17}.}\n}\n\\tablebib{(1) \\citealt{cha00}, (2) \\citealt{bar15}, (3) \\citealt{mor17}.}\n\\end{table}\n\n\n\\subsection{Upper limits on accretion}\nWe do not detect any evidence for line emission, in particular not in the accretion-sensitive hydrogen features, i.e., Paschen-$\\beta$ (Pa$\\beta$, 1.282\\,\\mum) and Brackett-$\\gamma$ (Br$\\gamma$, 2.167\\,\\mum), or in the limit of the Brackett series ($H$ band). Accordingly, we do not see evidence for strong accretion activity onto HD\\,106906\\,b. Assuming magnetospheric accretion from an isolated circumplanetary disk onto HD\\,106906\\,b,\\ we infer a quantitative upper limit for the mass accretion rate based on the upper limit for the accretion luminosity $L_\\mathrm{acc}$ and empirical $L_\\mathrm{acc}$--$L_\\mathrm{line}$ relations derived for T~Tauri stars \\citep{alc14}.\n\nWe use the description by \\citet{gul98} to estimate a mass accretion rate from an accretion luminosity using\n\\begin{equation}\\label{eq:gul98}\n  \\dot{M} = \\left(1-\\frac{R_{\\rm pl}}{R_\\mathrm{in}}\\right)^{-1}\\frac{L_\\mathrm{acc}\\,R_{\\rm pl}}{G\\,M_{\\rm pl}}\\quad.\n\\end{equation}\nFollowing \\citet{lov11}, we conservatively set the inner disk truncation radius $R_\\mathrm{in}$=2.7\\,$R_{\\rm pl}$. Larger radii, such as $R_\\mathrm{in}$=5\\,$R_{\\rm pl}$, which are commonly used for T Tauri stars, result in stronger constraints on the inferred upper limit of the mass accretion rate of up to a factor of 1.5. The mass and radius of the companion were set to $M_{\\rm pl}$=12.3\\,$\\pm$\\,0.8\\,\\MJup\\ and $R_{\\rm pl}$=1.6$\\,R_\\mathrm{Jup}$, which is consistent with our derived luminosity and temperature. Flux calibration of the spectra was achieved by normalizing the spectra, after multiplication with the spectral response curves of the respective filters, to the $J$ and \\Ks\\ magnitudes measured by \\citet{bai14}\\footnote{No $H$-band magnitudes have been published so far, so this technique could not be applied to our $H$-band spectrum.}. The measured continuum noise values in Pa$\\beta$ and Br$\\gamma$ of $\\Delta F {\\rm d}\\lambda$(Pa$\\beta$)\\,=\\,1.4$\\times$$10^{-18}$\\,erg\/s\/cm$^2$ and $\\Delta F {\\rm d}\\lambda$(Br$\\gamma$)\\,=\\,1.6$\\times$$10^{-18}$\\,erg\/s\/cm$^2$ translate to upper limits of the line luminosity $L_\\mathrm{line}$\\,=\\,$4\\pi F_\\mathrm{line}d^2$ of $\\log(L_{\\rm line}\/L_\\sun)$\\,$<$\\,$-8.78$ and $<$\\,$-8.95$, respectively (99\\% confidence). Using the $L_\\mathrm{acc}$--$L_\\mathrm{line}$ correlations listed \\citet{alc14} we infer $\\log(L_\\mathrm{acc}\/L_\\sun)$\\,<\\,$-5.63$ (Pa$\\beta$) and $\\log(L_\\mathrm{acc}\/L_\\sun)$\\,<\\,$-6.20$ (Br$\\gamma$). Using eq.~(\\ref{eq:gul98}) we infer upper limits for the mass accretion rate of $\\dot{M}<1.8\\times10^{-9}$\\,\\MJup{}yr$^{-1}$ (Pa$\\beta$) and $\\dot{M}<4.8\\times10^{-10}$\\,\\MJup{}yr$^{-1}$ (Br$\\gamma$).\n\nUncertainties and upper limits were calculated using a Monte Carlo simulation taking into account the uncertainties in measured flux, mass, distance, and the empirical $L_\\mathrm{acc}$--$L_\\mathrm{line}$ relations \\citep{alc14}. We conservatively account for systematic uncertainties such as the flux calibration and the subtraction of hydrogen absorption in the telluric standard star spectra by increasing the spectral noise by a factor of 2. All quoted upper limits are 99\\% confidence limits. \nSince the mass accretion rate as calculated from Br$\\gamma$ results in the more restrictive limit, we quote $\\dot{M}<4.8\\times10^{-10}$\\,\\MJup{}yr$^{-1}$ as our best estimate of the mass accretion rate for HD\\,106906\\,b.\n\nWe make a prediction for the expected accretion-induced line luminosity at the wavelength of H$\\alpha$ based on the derived line luminosity (using the more restrictive limit $\\log(L_\\mathrm{acc}\/L_\\sun)$\\,<\\,$-6.20$~). We use the $L_\\mathrm{acc}$--$L_\\mathrm{line}$ relation from \\citet{alc14} to derive $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$<$\\,$-6.69$ (99\\% confidence). \n\n\n\\section{Summary and discussion}\\label{sec:summary}\nWe present new high S\/N (20--50\/pix) intermediate-resolution VLT\/SINFONI 1--2.5\\,\\mum\\ integral field spectroscopy of HD\\,106906\\,b, which is a companion at the deuterium burning limit in the Upper Scorpius association.\nWe detect a number of spectral features indicative of a low mass for HD\\,106906\\,b, and also the H-band continuum shows the typical triangular shape of low gravity objects.\nComparison with spectral libraries of young and field objects as well as analyzing spectral features suggests a spectral type of L1.5$\\pm$1.0. Previous estimates based on low-resolution spectroscopy classified this object as either L2 (based on $H$-band spectroscopy) or L3 \\citep[$K$-band spectroscopy;][]{bai14}. \n\nWe use this new information and the recently published distance of HD\\,106906\\,b\\ \\citep[102.8$\\pm$2.5\\,pc]{gai16} to derive a luminosity of $\\log\\left(L_\\mathrm{bol}\/L_\\odot\\right)=-3.65\\pm0.08$ and an effective temperature of \\Teff\\,=\\,1820\\,$\\pm$\\,240\\,K;  this recently published distance of this binary star is $\\sim$10\\% larger than the previous best estimate from the Hipparcos mission. Based on its luminosity and age compared to predictions from evolutionary models, we estimate a mass between $M$\\,=\\,12.3$^{+0.8}_{-0.7}$\\,\\MJup\\ (hot start) and $M$\\,=\\,14.0$^{+0.2}_{-0.5}$\\,\\MJup\\ (cold start) in the limit of zero formation time. Mass predictions are slightly lower if a formation timescale of 3\\,Myr is assumed of between $M$\\,=\\,$11.9^{+2.5}_{-0.9}$\\,\\MJup\\ and $M$\\,=\\,14.0$^{+0.2}_{-0.4}$\\,\\MJup.\n\nThe formation and early evolution of HD\\,106906\\,b\\ remain unclear. The low mass ratio of $q$$\\sim$0.004 with respect to the host system \\citep[$M_{\\rm HD106906AB}$\\,=\\,1.37\\,\\Msun+1.34\\,\\Msun;][]{lag17} suggests ``planet-like'' formation in a protoplanetary disk \\citep{pep14,reg16}. Formation through core accretion, however, is restricted to orbits much closer to the star than its current separation of $>$700\\,AU \\citep[e.g., $\\lesssim$35\\,AU in the simulations by][]{dod09}. Accordingly, subsequent migration would be required. Currently, however, no conclusive evidence for scattering with an internal planet or the central binary has been found \\citep[cf.][]{jil15}. An ejection scenario with subsequent braking through a stellar fly-by appears unlikely in the light of the simulations by \\citet{rod17}. While not strictly excluded, in situ formation through disk instability appears unlikely (though not excluded) because large disks $>$700\\,AU have been rarely observed around forming stars. Formation of HD\\,106906\\,b\\ in a star-like channel, i.e., through direct collapse of the parent cloud core into a triple system HD\\,106906{}AB+b, remains a possibility.\n\nThe degeneracy between the various formation scenarios could be broken when the orbit of HD\\,106906\\,b\\ around its host system becomes known. Its long orbital period of $>$3000\\,yr \\citep{jil15}, however, precludes precise direct measurement of the motion of HD\\,106906\\,b. Indirect information on the orbit of HD\\,106906\\,b, however, may be gathered from observations of the circumstellar disk whose eccentricity is thought to reflect interaction with the companion \\citep{nes17}. Alternatively, additional high precision spectroscopy in the atmospheric absorption bands (e.g., near 1.4\\,\\mum\\ and 2\\,\\mum) and at mid-IR wavelengths together with retrieval analysis \\citep[e.g.,][]{lin12} may be used to measure the chemical composition of the atmosphere of HD\\,106906\\,b,  which is a function of its birth location. For example, an enhanced carbon-to-oxygen ratio suggests a formation in a disk close to the host star \\citep[e.g.,][]{obe11}. Direct collapse, in contrast, would produce a stellar abundance ratio. Future observation with high-precision spectroscopy instruments such as those on board the James Webb Space Telescope will help to break these degeneracies.\n\nWhile the presented spectra cover a number of hydrogen features (Paschen-$\\beta$, Brackett-$\\gamma,$ and higher), none of these features appears in emission. We thus exclude magnetospheric accretion onto HD\\,106906\\,b\\ at a rate of $\\dot{M}>4.8\\times10^{-10}$\\,\\MJup{}yr$^{-1}$ (99\\% confidence). With this measurement we can exclude accretion at a level several times below the accretion levels that have been observed for other targets at a similar mass and age as HD\\,106906\\,b. For example, \\citep{joe13} find a mass accretion rate of $\\dot{M}=8\\times10^{-9}$\\,\\MJup{}yr$^{-1}$ for OTS\\,44, a free-floating 12\\,\\MJup\\ member of the $\\sim$2\\,Myr-old Chamaeleon\\,I region. DH\\,Tau, an 11\\,\\MJup\\ companion at $\\sim$340\\,AU from its $\\sim$1--2\\,Myr-old parent star, is thought to accrete at a rate of $\\dot{M}=3.3\\times10^{-9}$\\,\\MJup{}yr$^{-1}$ \\citep{zho14}. For comparison with optical studies and as a reference for optical follow up, we predict an accretion-induced H$\\alpha$ line luminosity of $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$<$\\,$-6.69$. Previous observations of substellar companion candidates in H$\\alpha$ returned values of $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$=$\\,$-4.2$ \\citep[LkCa\\,15b;][]{sal15} and $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$=$\\,$-3.3$ \\citep[HD\\,142527B;][]{clo14}, again significantly larger than our limit for HD\\,106906\\,b. If HD\\,106906\\,b\\ were accreting, its rate must be significantly below that of these comparison objects. We thus conclude that HD\\,106906\\,b\\ features no significant amount of circumplanetary gas. However, the present data do not exclude the presence of a \\emph{gas-poor} disk around HD\\,106906\\,b, similar to what has been observed for the primary object HD\\,106906\\ \\citep{kal15,lag16}. In fact, \\citet{kal15} present weak evidence for the presence of circumplanetary dust around HD\\,106906\\,b\\ based on color excesses and a marginally spatially extended image of the source. \n\nOur new study makes HD\\,106906\\,b\\ one of the few low-\\Teff\\ companions for which high S\/N spectroscopy could be observed. As demonstrated in Fig.~\\ref{fig:models1}, most of the observed features in the spectrum must be real rather than due to measurement uncertainties\\footnote{We note that the selected model temperature is slightly below our best estimate of \\Teff=1820\\,K. This selection was made due to a bimodal $\\chi^2$ distribution of BT-SETTL model fits in \\logg--\\Teff\\ space which has local minima at \\Teff$\\approx$1600\\,K and \\Teff$\\sim$2600\\,K with a particularly poor representation of the K-band spectrum of HD\\,106906\\,b\\  by \\Teff$\\approx$1700--2100\\,K models. Detailed model fitting to explore reasons for this bimodality is beyond the scope of the current paper.}.\n\\begin{figure*}[tbh]\n  \\centering\n  \\setlength{\\unitlength}{\\textwidth}\n\\begin{picture}(1.0,0.7)\n    \\put(0.00,0.00){\\includegraphics[width=1\\textwidth]{f5.eps}}\n    \\put(0.06,0.65){\\fontfamily{phv}\\Large\\selectfont a)}\n    \\put(0.06,0.48){\\fontfamily{phv}\\Large\\selectfont b)}\n    \\put(0.06,0.3){\\fontfamily{phv}\\Large\\selectfont c)}\n    \\put(0.06,0.135){\\fontfamily{phv}\\Large\\selectfont d)}\n\\end{picture}\n\\caption{\\label{fig:models1}{\\bf  (a)} \\Teff=1600\\,K, \\logg=3.5 BT-SETTL model \\citep[green curve;][]{bar15}, smoothed to the resolution of our K-band spectrum HD\\,106906\\,b\\ (black) and scaled to the same average flux. Most features observed in the spectra are reproduced by the (virtually noise-free) model, supporting evidence that the measured features are caused by atmospheric absoprtion rather than noise. {\\bf  (b)} Residuals are shown.\\  {\\bf (c)} and {\\bf (d)} show the same for a shorter wavelength range.}\n\\end{figure*}\nThese high fidelity observations were possible with current instrumentation because of the large angular separation of HD\\,106906\\,b\\  from its host star. Most other directly imaged systems feature much smaller separations, which reduce the achievable S\/N due to additional light from the primary. The current spectrum thus serves as a reference spectrum for future studies of young low-gravity objects in orbit around low-mass stars, in particular with regard to the higher sensitivity of the James Webb Space Telescope and the higher angular resolution of the upcoming generation of 30 m-class telescopes.\n\n\\begin{acknowledgements}\n  We thank the anonymous referee for a thorough review and valuable suggestions for improvement.\n  We thank Michael Line for valuable contributions in preparation of the observing time proposal. \n  This work has been carried out within the frame of the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation. S.D., S.P.Q., and G.-D.M.\\ acknowledge the financial support of the SNSF. C.M. and G.-D.M. acknowledge the support from the Swiss National Science Foundation under grant BSSGI0$\\_$155816 ``PlanetsInTime''. \n  This work has made use of data from the European Space Agency (ESA) mission {\\it Gaia} (\\url{http:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia} Data Processing and Analysis Consortium (DPAC; \\url{http:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn this paper we investigate the extremal behavior of free resolutions of subschemes of complete intersections $X \\subseteq \\mathbb{P}^n$.\nOur motivating question is the following.\nLet $\\mathbf{d} = (d_1, \\ldots, d_c)$ be a degree sequence and $p(\\zeta)$ a Hilbert polynomial: \nare there uniform bounds on the syzygies of $Z\\subseteq X$, where $X\\subseteq \\mathbb{P}^n$ is a complete intersection of degrees $\\mathbf{d}$ and $Z\\subseteq X$ a closed subscheme with Hilbert polynomial $p(\\zeta)$?\n\nIn order to address this problem, we study Hilbert schemes of Clements-Lindstr\\\"om schemes, \ni.e. complete intersections $Y \\subseteq \\mathbb{P}^n$ defined by the vanishing of $x_1^{d_1}, \\ldots, x_{c}^{d_{c}}$.\nThey include $Y = \\mathbb{P}^n$ as special case, which is in fact interesting and non-trivial for most of our considerations.\nOur main contributions revolve around a new distinguished point on the Hilbert scheme $\\mathrm{Hilb}^{p(\\zeta)}(Y)$, called the \\emph{expansive point} (or subscheme, or ideal) and denoted by $\\mathrm{Exp}(p(\\zeta))$.\nWe adopt an abstract recursive approach in defining $\\mathrm{Exp}(p(\\zeta))$, based on seven axioms related to hyperplane sections, cf. Theorem \\ref{TheoremDefinitionExpansiveAxioms}.\nIn a sense, this gives rise to a theory of expansive ideals and  Hilbert polynomials, which parallels the theory of lexicographic ideals and Hilbert functions.\n\nOur main result, Theorem \\ref{TheoremExtremalFiniteResolution}, states that $\\mathrm{Exp}(p(\\zeta))$ attains the largest possible number of $i$-th syzygies for a subscheme in $\\mathrm{Hilb}^{p(\\zeta)}(Y)$, for every homological degree $i$. \nNo such theorem  exists for \\emph{graded} syzygies,  since each Hilbert scheme has several maximal graded Betti tables. \nWe remark that  considering expansive subschemes of Clements-Lindstr\\\"om schemes $Y$ for various degree sequences $\\mathbf{d}$, as opposed to just for $\\mathbb{P}^n$,\ncarries advantages.\nFirst, \nby taking the degree sequence into account, and restricting thus to a smaller Hilbert scheme,\none obtains sharper numerical bounds on Betti numbers.\nA similar point of view is adopted e.g. in  \\cite{EiGrHa93}, where \n bounds on the  number of points in intersections of quadric hypersurfaces are improved using the data of the degree sequence.\nMore importantly, \nour main result extends conjecturally to arbitrary complete intersections of $\\mathbb{P}^n$.\nIn fact we show that, under the validity of the  Lex Plus Powers Conjecture, $\\mathrm{Exp}(p(\\zeta))$ yields uniform bounds for the syzygies of  subschemes $Z\\in \\mathrm{Hilb}^{p(\\zeta)}(X)$ for \\emph{all} complete intersections $X\\subseteq \\mathbb{P}^n$ of degrees $\\mathbf{d}$,\nthus giving a complete answer to our motivating problem.\nSee Proposition \\ref{PropositionLPP} and Theorem \\ref{TheoremLPP}.\n \nWe  apply the theory of expansive ideals also  to infinite free resolutions over complete intersections, motivated by the recent progress in this area.\nOur second main result, Theorem \\ref{TheoremIfninite}, shows that, over a quadratic Clements-Lindstr\\\"om ring of characteristic 0, expansive ideals achieve extremal Betti numbers for the infinite free resolution.\nWe conjecture that this pattern holds for arbitrary degree sequences and base field.\n\nIn the  case $Y=\\mathbb{P}^n$, Theorem \\ref{TheoremExtremalFiniteResolution} gives a new proof of \\cite[Theorem 1.1]{CaMu13}, \nwhich asserts the existence of a subscheme in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ with extremal Betti numbers.\nThe authors remark in  \\cite[Introduction]{CaMu13} that the proof, of combinatorial nature, is very long and complicated, and it would be desirable to have a better understanding of the structure and construction of such extremal subschemes.\nWe believe that, with the method developed in this work, we have found a satisfactory answer.\nIn fact, besides providing a short and more conceptual proof, \n the axioms of Theorem \\ref{TheoremDefinitionExpansiveAxioms} can be used to further  illuminate the structure of expansive ideals.\nIn particular, we prove in Theorem \\ref{TheoremChainExpansive} that expansive ideals form descending chains of inclusions, \n starting with a saturated lex ideal,  and each step of the chain is  described explicitly. \n This fact serves as the basis for an efficient algorithm to compute $\\mathrm{Exp}(p(\\zeta))$.\nThe problem of finding an algorithm to determine a subscheme in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ with maximal syzygies had been suggested also in \\cite[Section 5]{MoNa14}.\n\n\nBorel-fixed points  have proved  helpful in understanding the  geometry of the Hilbert scheme, \ne.g. in  questions of connectedness, smoothness, rationality,  enumeration of components, and defining equations, see for instance\n\\cite{BeLeRo13,LeRo11,PeSt05,Ra19,Re95,ReSt97}.\nSeveral problems in this area remain open.\nOur work identifies a new distinguished Borel-fixed point, that is very different from the well-known lex point in many respects.\nWe hope that the notion of expansive point may lead to new perspectives or applications in  the geometry of Hilbert schemes.\n \n\n\n\n\\section{Clements-Lindstr\\\"om rings}\\label{SectionPreliminaries}\n\nThis section serves the purpose of fixing the basic terminology for the paper.\n We introduce the rings that are central to this work, \n and some special classes of  ideals. \n\nLet $\\mathbb{N}$ denote the set of nonnegative  integers.\nThe symbol $\\Bbbk$ denotes an arbitrary field. \nAll  rings considered in this work are Noetherian $\\mathbb{Z}$-graded $\\Bbbk$-algebras generated in degree 1, and all ideals and modules are graded;\nthese attributes are often assumed implicitly and  omitted.\n\nIf $V$ is a $\\mathbb{Z}$-graded $\\Bbbk$-vector space,  denote the $j$-th graded component by $[V]_j$.\nThe numerical function \n$\\mathrm{HF}(V) : \\mathbb{Z} \\rightarrow  \\mathbb{N} \\cup \\{\\infty\\}$ defined by $\\mathrm{HF}(V,j) = \\dim_\\Bbbk[V]_j$ for all $j \\in \\mathbb{Z}$ is called the\n{\\bf Hilbert function} of $V$.\nIf there is a numerical polynomial $\\mathrm{HP}(V) \\in \\mathbb{Q}[\\zeta]$ such that \n $\\mathrm{HP}(V,j) = \\mathrm{HF}(V,j)$  for all $j \\gg 0$, then  $\\mathrm{HP}(V) $ is called the {\\bf Hilbert polynomial} of $V$.\n\nThe  maximal ideal of a ring $A$ is denoted by $\\mathfrak{m}_A$.\nAn ideal $I\\subseteq A$ is {\\bf saturated} if $I : \\mathfrak{m}_A = I$, \nequivalently, if $\\mathrm{depth}(A\/I)>0$;\nnotice that the unit ideal $I=A$ is saturated.\nThe saturation of $I \\subseteq A $ is defined as $I : \\mathfrak{m}_A^\\infty = \\cup_{t \\geq 0} I : \\mathfrak{m}_A^t$,\nand it is a saturated ideal with \n$\\mathrm{HP}(I : \\mathfrak{m}_A^\\infty) =\\mathrm{HP}(I)$.\n\nGiven a projective scheme $X= \\mathrm{Proj} A$ and  a  polynomial $p(\\zeta)\\in \\mathbb{Q}[\\zeta]$,\nthe {\\bf Hilbert scheme}, denote by $\\mathrm{Hilb}^{p(\\zeta)}(X)$, is the scheme parametrizing the closed subschemes  $Z\\subseteq X$ with $\\mathrm{HP}(Z) =p(\\zeta)$.\nAs it is common in the literature, \nwe  often  identify a closed subscheme $Z \\subseteq X$ with its saturated  ideal $I_Z \\subseteq A$ and with the point on the Hilbert scheme parametrizing it,\nand sometimes we  extend attributes of one object to the other two.\nFor instance we may talk about strongly stable subschemes or lex points on the Hilbert scheme,\nand we adopt the following:\n\n\\begin{convention}\nIf $I\\subseteq A$ is an ideal, the expression  ``$I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$'' means that $I$ is saturated and $\\mathrm{HP}(A\/I) =p(\\zeta)$.\n\\end{convention}\n\nLet $A$ be a  ring and $M$ a finite  $A$-module.\nThe integers \n$\\beta_{i,j}^A(M) = \\dim_\\Bbbk[\\mathrm{Tor}_A^i(M,\\Bbbk)]_j$ and  $\\beta_{i}^A(M) = \\dim_\\Bbbk\\mathrm{Tor}_A^i(M,\\Bbbk)$\nare  the {\\bf graded Betti numbers} and the {\\bf (total) Betti numbers} of $M$, respectively.\n\n\\begin{convention}\\label{ConventionInfinity}\nWe will often use  $\\mathbb{N}\\cup \\{\\infty\\}$ as index set and as range for exponents. \nWe adopt standard conventions on $\\infty$, namely that $m < \\infty $  and  $\\infty-m = \\infty$ for all $m \\in \\mathbb{N}$.\nIf $r$ is an element in a ring then we set $r^\\infty := 0$.\nIf $d = \\infty$, the expression ``$\\ell < d$''   means ``$\\ell\\in \\mathbb{N}$''.\n\\end{convention}\n\n\n\\begin{definition}\nA \\textbf{Clements-Lindstr\\\"om ring} is a ring of the form \n$$\nA = \\frac{\\Bbbk [x_1, \\ldots, x_{m}]}{\\big(x_1^{d_1}, \\ldots, x_m^{d_{m}}\\big)}\n$$\nfor some sequence of integers $d_1 \\leq d_2 \\leq \\cdots \\leq d_{m}$ with $d_i \\in \\mathbb{N}\\cup \\{\\infty\\}$\nWe emphasize that $x_i^\\infty := 0$. \nThus, when $d_1 = \\infty$ the ring  $A$ is simply a polynomial ring.\nOn the other hand, when $d_{m} < \\infty$ the ring  $A$  is Artinian, \nand its only saturated ideal  is the unit ideal.\n\\end{definition}\n\nFor the remainder of this section, let $A$ denote an arbitrary Clements-Lindstr\\\"om ring.\n\nAn ideal $I \\subseteq A$ is monomial if it is the image of a monomial ideal of $\\Bbbk [x_1, \\ldots, x_{m}]$. \nWe denote by $<_{\\mathrm{lex}}$ the lexicographic monomial order in  $A$ induced by $x_1 > x_2 > \\cdots > x_{m}$.\nA monomial  ideal $I \\subseteq A$ is \\textbf{lex} if  $[I]_j$ is a vector space generated by an initial segment of monomials with respect to $<_{\\mathrm{lex}}$ for every $j$,\nequivalently, if $I$ is the image of a lex ideal of $\\Bbbk [x_1, \\ldots, x_{m}]$.\nThe saturation of a lex ideal is again lex.\nA theorem of Clements and Lindstr\\\"om, which includes  classical results of Macaulay and Kruskal-Katona as special cases,\n states that lex ideals classify Hilbert functions in $A$:\n\n\\begin{prop}[\\cite{ClLi69}]\\label{PropositionCL}\nLet $A$ be a Clements-Linstr\\\"om ring and $I\\subseteq A$ an ideal.\nThere exists a unique lex ideal $L \\subseteq A$ such that $\\mathrm{HF}(L) = \\mathrm{HF}(I)$.\n\\end{prop}\n\nIf $\\mathcal{H}:\\mathbb{Z}\\rightarrow \\mathbb{N}$ is the Hilbert function of some ideal of $A$, we denote by ${\\mathrm{Lex}}(\\mathcal{H},A)$  the unique lex ideal $L\\subseteq A $ with $\\mathrm{HF}(L) = \\mathcal{H}$.\nIf $I \\subseteq A$ we define  ${\\mathrm{Lex}}(I):={\\mathrm{Lex}}(\\mathrm{HF}(I), A)$. \n\nA monomial ideal $I \\subseteq A $ is  {\\bf almost lex} if  the last variable $x_{m}$ is a non-zerodivisor on $A\/I$  and\n $\\frac{I+(x_{m})}{(x_{m})}$ is a lex ideal of the Clements-Lindstr\\\"om ring $\\frac{A}{(x_{m})}$.\nIn particular, almost lex ideals are saturated.\nObserve that a lex ideal is not, in general, almost lex.\n\nA monomial ideal $I \\subseteq A$ is \\textbf{strongly stable} if for every nonzero monomial $\\mathbf{u} \\in I$ and $x_h $ dividing $\\mathbf{u}$,\nthen $\\frac{x_k \\mathbf{u}}{x_h}\\in I$ for all $k <h$.\nIt suffices to check this condition for the monomial minimal generators $\\mathbf{u}$ of $I$.\nWhen $A$ is a polynomial ring, strongly stable ideals are fixed under the action of the Borel group.\nA strongly stable ideal $I \\subseteq A$ is saturated if and only if the last variable\n$x_{m}$ is a non-zerodivisor on $A\/I$;\nwhen $\\dim A >0$, this is equivalent to the fact that $x_{m}$ does not divide any monomial minimal generator of $I$.\nThe saturation of a strongly stable ideal is again strongly stable.\nBoth lex ideals and almost lex ideals are strongly stable.\n\n\\begin{examples}\nLet  $d_1=2, d_2 = 3, d_3=d_4 = \\infty$.\nThe  associated Clements-Lindstr\\\"om ring is $A = \\frac{\\Bbbk[x_1, x_2, x_3, x_4]}{(x_1^2, x_2^3)}$.\nConsider the following ideals of $A$:\n\\begin{itemize}\n\\item $\\big(x_1x_2^2,\\, x_1x_2x_3 \\big)$ is both lex and almost lex;\n\\item  $\\big(x_1x_2,\\, x_1x_3,\\, x_1x_4^2,\\, x_2^2x_3\\big)$ is lex but not almost lex, since it is not saturated;\n\\item $\\big( x_1 x_2,\\, x_1x_3,\\, x_2^2\\big)$ is almost lex but not lex, since $x_1x_4 >_{{\\mathrm{lex}}} x_2^2$;\n\\item $\\big( x_1 x_2,\\, x_2^2\\big)$ is strongly stable and saturated, but neither lex nor almost lex.\n\\end{itemize}\n\\end{examples}\n\n\\begin{remark}[The lex point]\\label{RemarkUniqueLex}\nFor every  $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ such that  $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)\\ne \\emptyset$, \nthere is exactly one lex ideal in  $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$.\nThe show existence,\ntake $I \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$ and let $L = {\\mathrm{Lex}}(I):\\mathfrak{m}_A^\\infty$.\nIt follows  that $L$ is a saturated lex ideal of $A$ with $\\mathrm{HP}(I)=\\mathrm{HP}(L)$.\nIf $L,L'$ are two saturated lex ideals with $\\mathrm{HP}(L)=\\mathrm{HP}(L')$, \nthen $\\mathrm{HF}(L,d) = \\mathrm{HF}(L',d)$ for  $d \\gg 0$, thus $[L]_d=[L']_d$.\nLet  $K= \\big([L]_d\\big) \\subseteq A$, \nit follows that $L = L' = K : \\mathfrak{m}_A^\\infty$, proving uniqueness.\nIn the case of  $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$\nit is known that the lex point is smooth \\cite{ReSt97},\nhowever this is unknown for Clements-Lindstr\\\"om schemes \\cite{PeSt07}.\n\\end{remark}\n\nIf $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)\\ne \\emptyset$, we denote by ${\\mathrm{Lex}}(p(\\zeta),A)$ the unique lex ideal in  $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$.\nWe emphasize that  the lex ideal of a Hilbert function and the lex ideal of a Hilbert polynomial are different concepts, and both are relevant for this work.  \nThe notation ${\\mathrm{Lex}}(I)$ is reserved for the lex ideal with the same Hilbert function as the ideal $I$.\n\nBy Remark \\ref{RemarkUniqueLex} the set of monomial subschemes in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$ is non-empty whenever $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A) \\ne \\emptyset$.\nOn the other hand,  this set is always finite, \nas the next discussion shows.\n\n\\begin{remark}\\label{RemarkFinitelyManyForEachHP}\nThere are finitely many monomial  subschemes in each $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$.\nTo see this, since the preimage in the polynomial ring of a saturated monomial ideal of $A$ is again   saturated and monomial, \nit suffices to treat the case when $A$ is a polynomial ring. \nThere is a well-known upper bound, due to Gotzmann \\cite{Go78},  for the Castelnuovo-Mumford regularity of a saturated ideal $J\\subseteq A$ in terms of  $\\mathrm{HP}(A\/J)$.\nThis implies the desired conclusion, since there are finitely many monomial ideals generated in bounded degrees.\n\\end{remark}\n\nWe remark that there are algorithms to produce all the strongly stable points or almost lex points of $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$,\nsee for instance \\cite{AlLe18,CiLeMaRo11,MoNa14}. \nThese algorithms can be extended with minor modifications to the case of  Clements-Lindstr\\\"om rings $A$.\n\n\n\n\n\\begin{example}\nLet $d_1=2, d_2=3$, $d_3=d_4=\\infty$ and consider  \nthe  associated Clements-Lindstr\\\"om ring  $A = \\frac{\\Bbbk[x, y, z, w]}{(x^2, y^3)}$.\nFor $p(\\zeta) = 3\\zeta+5$ the strongly stable ideals in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$ are\n \\begin{align*}\n & \\left(xy,  xz^5 \\right)&\n&\\left( xy^2, xyz^2, xz^3\\right)&\n &\\left(  xy^2, xyz, xz^4\\right)&\n  \\\\\n & \\left( xy^2,   xyz, y^2z^3\\right)&\n&\\left( xy^2, xyz^2, y^2z^2\\right)&\n& \\left(  xy,  y^2z^4\\right)&\n \\end{align*}\nThe ideals in the first row are almost lex, and ${\\mathrm{Lex}}(3\\zeta+5,A)=\\left(xy,  xz^5 \\right)$.\n\\end{example}\n\n\n\\section{Decomposition of monomial ideals}\\label{SectionDecomposition}\n\nWe introduce an inductive decomposition of monomial ideals in Clements-Lindstr\\\"om rings.\nThis decomposition is particularly effective for strongly stable and (almost) lex ideals; \nit will play a fundamental role in the construction of expansive ideals in Section \\ref{SectionExpansivePoint}.\n\n\nFor the rest of the paper, we fix the following notation:\n\\begin{equation}\n\\label{DefinitionRings}\n\\begin{aligned}\n S & =  \\Bbbk[x_1, x_2,\\ldots, x_{n}, x_{n+1}]  \\qquad &R & =  S \/ \\big(x_1^{d_1}, \\ldots, x_n^{d_n}\\big)\\\\  \n{\\overline{S}} &= \\Bbbk[x_1, x_2,\\ldots, x_{n-1}, x_{n+1}] \\qquad & {\\overline{R}} & =  {\\overline{S}} \/ \\big(x_1^{d_1}, \\ldots, x_{n-1}^{d_{n-1}}\\big)\\\\\n{\\widetilde{S}} &= \\Bbbk[x_1, x_2,\\ldots, x_{n-1}, x_n] \\qquad &{\\widetilde{R}} &=  {{\\widetilde{S}}}\/{\\big(x_1^{d_1}, \\ldots, x_n^{d_n}\\big)}\n\\end{aligned}\n\\end{equation}\nwhere  $n \\in \\mathbb{N}$ and $d_1 \\leq d_2 \\leq \\cdots \\leq d_{n}$ is any sequence with $d_i \\in \\mathbb{N}\\cup \\{\\infty\\}$.\nIn other words, we will always set  $d_{n+1} = \\infty$, \nso  that $x_{n+1}^{d_{n+1}}=0$ and it will be omitted. \nIn this way, the Clements-Lindstr\\\"om rings $R$ and ${\\overline{R}}$  always  have positive Krull dimension,\nwhereas the Clements-Lindstr\\\"om ring ${\\widetilde{R}}$ may be Artinian.\nThe rings ${\\overline{R}}$ and ${\\overline{S}}$ are defined only if $n>0$.\nWhen $n=0$ we have $R=S = \\Bbbk[x_1]$, and the  only saturated ideals are the zero ideal and the unit ideal.\nObserve that ${\\overline{S}}$ and ${\\overline{R}}$ are algebra retracts of $S$ and $R$, respectively, and they may be regarded either as subrings or as factor rings;\nsimilarly for ${\\widetilde{S}}$ and ${\\widetilde{R}}$.\nBy abuse of notation, the symbols $x_i$ will be used to denote elements in different rings.\n\nSince $R = {\\widetilde{R}}[x_{n+1}]$ there is a tight relation between invariants of ideals of $R$ and ${\\widetilde{R}}$;\nwe summarize  the main formulas in the following remark.\n\n\n\\begin{remark}\\label{RemarkModuloXNplus1}\nLet $I\\subseteq R$ be a saturated ideal  such that $I:x_{n+1}=I$.\nDenote by ${\\widetilde{I}} = \\frac{I+(x_{n+1})}{(x_{n+1})} \\subseteq {\\widetilde{R}}$ the image of $I$ in ${\\widetilde{R}}$.\n Then   $\\mathrm{HF}({\\widetilde{I}}, d) = \\mathrm{HF}(I,d)-\\mathrm{HF}(I,d-1)$ for $d\\in \\mathbb{Z}$ and $\\mathrm{HP}({\\widetilde{I}}, \\zeta) = \\mathrm{HF}(I,\\zeta)-\\mathrm{HF}(I,\\zeta-1)$.\nFurthermore, for all $i,j$ we have\n\\begin{align*}\n\\beta_{i,j}^{\\widetilde{S}}({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^S(R\/I)\n&\\beta_{i,j}^S({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^S(R\/I) + \\beta_{i-1,j-1}^S(R\/I)\\\\\n\\beta_{i,j}^{\\widetilde{R}}({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^R(R\/I)\n&\\beta_{i,j}^R({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^R(R\/I) + \\beta_{i-1,j-1}^R(R\/I).\n\\end{align*}\nIf $I$ is monomial or strongly stable, then so is ${\\widetilde{I}}$.\nConversely, given any strongly stable $K\\subseteq {\\widetilde{R}}$, the extension $ K R \\subseteq R$ of $K$ to $R$ is a saturated strongly stable ideal\nwhose image in ${\\widetilde{R}}$ is $K$.\n\\end{remark}\n\n\nFor a monomial ideal  $I \\subseteq R$ there exist uniquely determined monomial ideals $I_\\ell \\subseteq {\\overline{R}}$ such that\nthe following decomposition of ${\\overline{R}}$-modules holds\n\\begin{equation}\\label{EqDecomposition}\nI = \\bigoplus_{\\ell=0}^{d_{n}-1} I_\\ell x_{n}^\\ell.\n\\end{equation}\nThe set of components $\\{I_\\ell\\}$ is finite if $d_n < \\infty$, infinite otherwise.\nThroughout the paper, the notation $I_\\ell$ will always refer to this decomposition; \nit should not be confused with graded components,  denoted instead by $[I]_j$.\n\n\n\\begin{example}\nLet $R = {\\Bbbk[x_1,x_2, x_3, x_4]}\/{\\big(x_1^3,x_2^3,x_3^4\\big)}$,\nso that ${\\overline{R}} = {\\Bbbk[x_1,x_2, x_4]}\/{\\big(x_1^3,x_2^3\\big)}$.\nThe components of the ideal $I = \\big( x_1^2,\\, x_1x_2^2x_3,\\, x_1x_2x_3^2,\\,x_1x_3^3,\\, x_2^2x_3^2\\big) \\subseteq R$ are the ${\\overline{R}}$-ideals\n$$\nI_0 = \\big(x_1^2\\big), \\quad I_1 = \\big( x_1^2, x_1x_2^2\\big), \\quad I_2 = \\big( x_1^2, x_1x_2, x_2^2\\big), \\quad I_3 =\\big( x_1, x_2^2\\big).\n$$\n\\end{example}\n\nWe are going to record some elementary properties of the decomposition \\eqref{EqDecomposition}.\nFirst, we define a partial order $\\preceq$ among  univariate polynomials with rational coefficients.\n\n\\begin{definition}\\label{DeinitionPartialOrderPolynomials}\nLet $p(\\zeta),q(\\zeta) \\in \\mathbb{Q}[\\zeta]$.\nWe set  $p(\\zeta)\\preceq q(\\zeta)$ if $q(\\zeta) - p(\\zeta)$ is a non-negative constant polynomial,\ni.e., if the coefficients of positive degree coincide in $p(\\zeta)$ and $q(\\zeta)$ and the constant terms satisfy $p(0) \\leq q(0)$.\n\\end{definition}\n\nRecall that the constant term of a Hilbert polynomial is always an integer, carrying the same information as the arithmetic genus.\n\n\\begin{prop}\\label{PropositionFiniteLengthRank}\nLet $I\\subseteq J \\subseteq R$ be saturated strongly stable ideals.\nThe quotient $J\/I$ is a  free module over $\\Bbbk[x_{n+1}]$ via restriction of scalars.\nThe following conditions are equivalent\n\\begin{itemize}\n\\item[(i)] $\\mathrm{HP}(I) \\preceq \\mathrm{HP}(J)$ \\vspace*{0.1cm}\n\\item[(ii)]$\\mathrm{rank}_{\\Bbbk[x_{n+1}]} \\left(J\/I \\right)<\\infty $ \\vspace*{0.1cm}\n\\item[(iii)]$\\dim_\\Bbbk \\big({{\\widetilde{J}}}\/{{\\widetilde{I}}}\\big)< \\infty$\n\\end{itemize}\nand if these conditions are satisfied then \n$\n\\mathrm{HP}(J) -\\mathrm{HP}(I) =  \\mathrm{rank}_{\\Bbbk[x_{n+1}]} \\left({J}\/{I} \\right) = \\dim_\\Bbbk \\big({{\\widetilde{J}}}\/{{\\widetilde{I}}}\\big).\n$\n\\end{prop}\n\n\\begin{proof}\nLet $M = J\/I$ and $\\widetilde{M} = {\\widetilde{J}}\/{\\widetilde{I}}$, with notation as in Remark \\ref{RemarkModuloXNplus1}.\nThen  $M \\cong \\widetilde{M} \\otimes_{\\widetilde{R}} {\\widetilde{R}}[x_{n+1}] \\cong \\widetilde{M} \\otimes_\\Bbbk \\Bbbk[x_{n+1}]$, implying  the first statement  and the equality\n $\\mathrm{rank}_{\\Bbbk[x_{n+1}]} \\left({J}\/{I} \\right) = \\dim_\\Bbbk \\big({{\\widetilde{J}}}\/{{\\widetilde{I}}}\\big)$.\nSince $I \\subseteq J$, we have $\\mathrm{HP}(I) \\preceq \\mathrm{HP}(J)$ if and only if $\\mathrm{HP}(M)=\\mathrm{HP}(J) -\\mathrm{HP}(I) $ is a constant, \nequivalently $M$ has Krull dimension at most 1, equivalently $M$ is a finite $\\Bbbk[x_{n+1}]$-module.\nFinally, the rank of a finite free $\\Bbbk[x_{n+1}]$-module is equal to its Hilbert polynomial.\n\\end{proof}\n\nIn the next proposition we list  basic properties of the decomposition \\eqref{EqDecomposition}.\n\n\\begin{prop}\\label{PropositionElementaryPropertiesDecomposition}\nLet $R$ be a Clements-Lindstr\\\"om ring\nand $I \\subseteq R $ a monomial ideal.\n\\begin{itemize}\n\\item[(1)] The sequence $\\{ I_\\ell\\}$ is a non-decreasing chain of ideals of ${\\overline{R}}$.\n\\item[(2)] If $d_{n}= \\infty$  the sequence $\\{ I_\\ell\\}$ is eventually constant, \nand the limit is equal to the ideal\n$$\nI_\\infty := \\frac{I : (x_{n})^\\infty + (x_{n})}{(x_{n})} \\subseteq {\\overline{R}}.\n$$\n\\item[(3)] $I$ is strongly stable if and only if $I_\\ell$ is strongly stable for all $\\ell$ and  $(x_1, \\ldots, x_{n-1}) I_\\ell \\subseteq I_{\\ell-1}$ for all $\\ell\\geq 1$.\n\\item[(4)] If $I$ is strongly stable, then $I$ is saturated if and only if $I_\\ell$ is saturated for every $\\ell$.\n\\item[(5)] $\\mathrm{HP}\\big((x_1, \\ldots, x_{n})I\\big) \\preceq \\mathrm{HP}\\big(I)$.\n\\end{itemize}\nNow assume  that $I$ is strongly stable and saturated.\n\\begin{itemize}\n\\item[(6)] $I_{\\ell+1}\/I_{\\ell}$ is a finite free $\\Bbbk[x_{n+1}]$-module of rank equal to the integer $\\mathrm{HP}(I_{\\ell+1})  -\\mathrm{HP}(I_{\\ell}) $.\n\\item[(7)] If $d_n = \\infty$ then $\\mathrm{HP}(I_\\infty,\\zeta) = \\mathrm{HP}(I,\\zeta) - \\mathrm{HP}(I,\\zeta-1)$.\n\\item[(8)] $\\mathrm{HP}(I_\\ell,\\zeta) - \\mathrm{HP}(I,\\zeta)+\\mathrm{HP}(I,\\zeta-1)$ is a constant polynomial.\nIn particular, the coefficient of $\\zeta^h$ in $\\mathrm{HP}(I_{\\ell})$, where $h>0$, is independent of $\\ell$ and uniquely determined by $\\mathrm{HP}(I)$.\n\\item[(9)]  $\\mathrm{HP}(I_{\\ell_1}) \\preceq \\mathrm{HP}(I_{\\ell_2})$ for all  $\\ell_1 \\leq \\ell_2 < d_{n}$.\n\\item[(10)] There exists a saturated strongly stable ideal $J\\subseteq I$ such that $\\mathrm{HP}(I)-\\mathrm{HP}(J) = 1$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\n(1) follows immediately from \\eqref{EqDecomposition} since the ideal $I$ is closed under multiplication by $x_n$.\n\nAssume $d_n = \\infty$. \nSince ${\\overline{R}}$ is Noetherian, the non-decreasing sequence  $\\{ I_\\ell\\}$ stabilizes.\nChoose $\\ell_0\\in \\mathbb{N}$ so that $I_\\ell = I_{\\ell_0}$ for all $\\ell \\geq \\ell_0$,\nand consider the ideal $J =  \\bigoplus_{\\ell= 0}^{d_{n}-1} I_{\\ell_0} x_{n}^\\ell \\subseteq R$.\nThen we have $J : x_n = J$, $I \\subseteq J$, and $x_n^{\\ell_0}J \\subseteq I$.\nIt follows that $J = I: (x_n)^\\infty$, and thus $I_\\infty = I_{\\ell_0} = \\frac{J+(x_n)}{(x_n)} = \\frac{I : (x_{n})^\\infty + (x_{n})}{(x_{n})} \\subseteq {\\overline{R}}$, proving (2). \n\n(3) holds by definition of strongly stable ideal.\n\n(4) follows since a strongly stable ideal is saturated if and only if the last variable $x_{n+1}$ does not divide any of its monomial minimal generators,\nunder our assumption that $\\dim R >0$.\n\n(5) holds because the $R$-module $I\/(x_1,\\ldots, x_n)I$ has Krull dimension at most 1,\nhence its Hilbert polynomial is a non-negative constant.\n\n(6) follows from  (4) and Proposition \\ref{PropositionFiniteLengthRank}, since by (3) $I_{\\ell+1}\/I_{\\ell}$ is \nannihilated by $(x_1, \\ldots, x_{n-1})$ and thus it is a finite $\\Bbbk[x_{n+1}]$-module.\n\n With notation as in the proof of (2), let $J = I : (x_n)^\\infty = \\bigoplus_{\\ell= 0}^{d_{n}-1} I_{\\ell_0} x_{n}^\\ell  \\subseteq R$.\nSince $x_n$ is a non-zerodivisor on $R\/J$ and $I_\\infty  = \\frac{J+(x_n)}{(x_n)}  $, \nwe have $\\mathrm{HP}(I_\\infty, \\zeta) = \\mathrm{HP}(J,\\zeta) - \\mathrm{HP}(J,\\zeta-1)$.\nHowever, from (6) we see that $J\/I$ has Krull dimension at most 1, so that $\\mathrm{HP}(J) -\\mathrm{HP}(I)$ is  a constant polynomial, and (7) follows.\n\nIf $d_n < \\infty$ then $R$ has Krull dimension 1,\n the coefficient of $\\zeta^h$ in $\\mathrm{HP}(I_{\\ell})$ and $\\mathrm{HP}(I)$ is 0 for $h>0$, thus (8) holds in this case.\n Now assume $d_n = \\infty$.\nBy (6) we have that $\\mathrm{HP}(I_{\\ell}) - \\mathrm{HP}(I_{\\ell-1})\\in \\mathbb{N}$,\n hence the coefficient of $\\zeta^h$ in $\\mathrm{HP}(I_{\\ell})$ is independent of $\\ell$ for $h>0$.\n Therefore, the claim (8) reduces to the case of  the component $I_\\infty$, and follows thus from (7).\n \n(9) follows immediately from (1) and (6).\n \nLet $\\mathbf{u}_1, \\ldots, \\mathbf{u}_t$ be the minimal monomial generators of $I$  ordered decreasingly in $<_{\\mathrm{lex}}$.\nThen the ideal $J = (\\mathbf{u}_1, \\ldots, \\mathbf{u}_{t-1},x_1 \\mathbf{u}_t, \\ldots, x_{n} \\mathbf{u}_t)$ satisfies (10).\n\\end{proof}\n\n\n\n\n\\section{The expansive point in the Hilbert scheme}\\label{SectionExpansivePoint}\n\n\nThis section represents the core of the paper: here \nwe introduce  the expansive point on the Hilbert scheme of a Clements-Lindstr\\\"om scheme.\nWe develop a machine to deal with  expansive ideals both from an abstract and computational perspective.\nThe reader may choose to skip the proof of Theorem \\ref{TheoremDefinitionExpansiveAxioms} -- the rest of the paper relies on the axioms \\hyperref[A1]{(A1)} through \\hyperref[A7]{(A7)}, but does not use the proof.\nRecall our standing notation \\eqref{DefinitionRings} on Clements-Lindstr\\\"om rings.\n\n\\begin{thm}\\label{TheoremDefinitionExpansiveAxioms}\nLet $R$ be a Clements-Lindstr\\\"om ring.\nFor every polynomial\n $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$  such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) \\ne \\emptyset$\n there exists a unique  ideal $\\mathrm{Exp}(p,R) \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$, \n called the {\\bf expansive ideal} with Hilbert polynomial $p(\\zeta)$,\nsuch that  the  following  axioms are satisfied:\n\\begin{enumerate}\n\\item[(A1)\\label{A1}] $ \\mathrm{Exp}(p,R)$ is strongly stable;\n\n\\item[(A2)\\label{A2}]  the components $ \\mathrm{Exp}(p,R)_\\ell \\subseteq {\\overline{R}}$ are expansive  for all $\\ell$;\n\n\\item[(A3)\\label{A3}] given two polynomials  $p(\\zeta) \\preceq p'(\\zeta) $ we have   $\\mathrm{Exp}(p',R) \\subseteq \\mathrm{Exp}(p,R)$;\n\n\\item[(A4)\\label{A4}]  $(x_1, \\ldots, x_{n}) \\mathrm{Exp}(p,R) $ is expansive;\n\n\\item[(A5)\\label{A5}]  if $q(\\zeta) = \\mathrm{HP}\\big({\\overline{R}}\/\\mathrm{Exp}(p,R)_k\\big)-1$ is such that \n$\\mathrm{Hilb}^{q(\\zeta)}\\big(\\mathrm{Proj} {\\overline{R}}\\big) \\ne \\emptyset$, for some $k < d_n$, then for all $h < k$ we have\n$\n\\mathrm{Exp}(p,R)_h \\subseteq (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big(q(\\zeta),{\\overline{R}}\\big);\n$\n\n\\item[(A6)\\label{A6}]  if  $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$  is   strongly stable  then  for every $0\\leq \\rho \\leq d_n-1$ we have\n$$\n\\sum_{\\ell = 0}^\\rho \\mathrm{HP}\\big(\\mathrm{Exp}(p,R)_\\ell\\big) \\preceq  \\sum_{\\ell = 0}^\\rho  \\mathrm{HP}(J_\\ell) ;\n$$\n\n\\item[(A7)\\label{A7}]  if  $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$  is   strongly stable  then \n$\\mathrm{HP}\\big( (x_1, \\ldots, x_{n}) \\mathrm{Exp}(p,R) \\big)\\preceq\\mathrm{HP}\\big((x_1, \\ldots, x_{n}) J\\big).$\n\n\n\\end{enumerate}\n\\end{thm}\n\n\\noindent\nWe define $\\mathrm{Exp}(I) = \\mathrm{Exp}(p(\\zeta), R)$ \nif $I \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$.\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{TheoremDefinitionExpansiveAxioms}]\nWe prove the  theorem by induction on the number of variables $n+1$.\nFor the base of the induction $n=0$ there is nothing to prove, so we assume $n>0$.\nFix a polynomial  $p(\\zeta)$  such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) \\ne \\emptyset$.\nIt follows by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (8) \nthat for any two strongly stable $I,J \\in  \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) $ and $0\\leq h,k <d_n$, the polynomial\n$\\mathrm{HP}(I_h) - \\mathrm{HP}(J_k)$ is a constant, so $\\mathrm{HP}(I_h) \\preceq \\mathrm{HP}(J_k)$ or $\\mathrm{HP}(I_h) \\succeq \\mathrm{HP}(J_k)$.\nMoreover, by Remarks \\ref{RemarkUniqueLex} and \\ref{RemarkFinitelyManyForEachHP} the set of \nstrongly stable ideals in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ is finite and non-empty.\nWe can choose a strongly stable  $I \\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ satisfying the following condition: \nif $J\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ is also strongly stable and $\\sum_{\\ell=0}^{\\rho}\\mathrm{HP}(J_\\ell) \\preceq \\sum_{\\ell=0}^{\\rho} \\mathrm{HP}(I_\\ell)$ for  \nevery $\\rho <d_n$,\nthen $\\mathrm{HP}(I_\\ell)=\\mathrm{HP}(J_\\ell) $ for every $\\ell$.\nIn other words, the sequence of polynomials $\\big\\{\\sum_{\\ell=0}^{\\rho}\\mathrm{HP}(I_\\ell)\\big\\}_{\\rho=0}^{d_n-1}$ is  componentwise minimal, among all  strongly stable ideals in  $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$.\nWith this choice of $I$ we define \n\\begin{equation}\\label{EqConstructionExpansive}\nE = \\bigoplus_{\\ell<d_n}\\mathrm{Exp}(I_\\ell)x_n^\\ell  \\subseteq R.\n\\end{equation} \nNote that $\\mathrm{HP}(E_\\ell) = \\mathrm{HP}(I_\\ell)$ for all $\\ell$, and $\\mathrm{HP}(E) = \\mathrm{HP}(I)$.\nThus $E$ is an ideal of $R$, as\n Proposition \\ref{PropositionElementaryPropertiesDecomposition} (9) and \\hyperref[A3]{(A3)} imply that the sequence of components \n$\\{\\mathrm{Exp}(I_\\ell)\\}$ is non-decreasing.\nThe variable $x_{n+1}$ does not divide any monomial minimal generator of $E$,\nsince $E_\\ell$ is saturated for all $\\ell$ by induction;\nit follows that $E$ is saturated, \nso that $E \\in  \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$.\nWe claim that $E$ satisfies the axioms \\hyperref[A1]{(A1)} through \\hyperref[A7]{(A7)}.\nThe axiom \\hyperref[A2]{(A2)} holds by construction.\n\nBy \\hyperref[A1]{(A1)} the components of $E$ are strongly stable ideals of ${\\overline{R}}$.\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (3) we have $(x_1, \\ldots, x_{n-1})I_\\ell \\subseteq I_{\\ell-1}$.\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (5) and (8) we have \n$\\mathrm{HP}\\big((x_1, \\ldots, x_{n-1})I_\\ell\\big)\\preceq \\mathrm{HP}(I_{\\ell-1})$,\nequivalently,\n$\\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})I_\\ell\\big)\\succeq \\mathrm{HP}\\big({\\overline{R}}\/I_{\\ell-1}\\big)$,\nand by \\hyperref[A3]{(A3)} it follows  that $\\mathrm{Exp}\\big((x_1, \\ldots, x_{n-1})I_\\ell\\big) \\subseteq E_{\\ell-1}$.\nBy \\hyperref[A7]{(A7)} \n$\n\\mathrm{HP}\\big( (x_1, \\ldots, x_{n-1}) E_\\ell \\big)\\preceq \\mathrm{HP}\\big((x_1, \\ldots, x_{n-1}) I_\\ell\\big),\n$\ni.e. $\n\\mathrm{HP}\\big( {\\overline{R}}\/(x_1, \\ldots, x_{n-1}) E_\\ell \\big)\\succeq \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1}) I_\\ell\\big).\n$\nSince \n  $(x_1, \\ldots, x_{n-1}) E_\\ell$ is expansive by \\hyperref[A4]{(A4)}, \naxiom \\hyperref[A3]{(A3)} yields\n$\n(x_1, \\ldots, x_{n-1}) E_\\ell\\subseteq \\mathrm{Exp}\\big((x_1, \\ldots, x_{n-1}) I_\\ell\\big).\n$\nCombining the two inclusions we derive $(x_1, \\ldots, x_{n-1}) E_\\ell \\subseteq E_{\\ell-1}$, and by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (3) $E$ satisfies \\hyperref[A1]{(A1)}.\n\nNext, we prove that $E$ verifies \\hyperref[A5]{(A5)}.\nAssume by contradiction there exist $h < k < d_n$ such that   $\\mathrm{Hilb}^{q(\\zeta)}\\big(\\mathrm{Proj} {\\overline{R}}\\big) \\ne \\emptyset$ and \n$E_h \\not\\subseteq F := (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big(q(\\zeta),{\\overline{R}}\\big)\\subseteq {\\overline{R}}$,\nwhere  $q(\\zeta) = \\mathrm{HP}\\big({\\overline{R}}\/E_k\\big)-1$.\nPick $h, k$ so that $k-h$ is the least possible.\nBy \\hyperref[A4]{(A4)}  $ F$ is expansive.\nWe have $\\mathrm{HP}(F)\\preceq \\mathrm{HP}\\big(\\mathrm{Exp}\\big(q(\\zeta),{\\overline{R}}\\big)\\big)$ by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (5).\nSince $\\mathrm{HP}(E_h)\\preceq \\mathrm{HP}(E_k)$ and  $\\mathrm{HP}\\big(\\mathrm{Exp}\\big(q(\\zeta),{\\overline{R}}\\big)\\big)=\\mathrm{HP}(E_k)+1$\nwe conclude that $\\mathrm{HP}(F)$ and $\\mathrm{HP}(E_h)$ differ by an integer.\nAs $E_h\\not\\subseteq F$ and both ideals are expansive,\n \\hyperref[A3]{(A3)}\nimplies that $F \\subsetneq E_h$. \nLet $  \\eta \\leq h$ be minimal such that $E_\\eta = E_h$\nand $ \\chi > k$ be maximal such that $E_k = E_\\chi$.\n\nWe exhibit a strongly stable  $J \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ generating a contradiction. Set \n\\begin{equation}\\label{ConstructionJContradiction}\nJ = \\bigoplus_{\\ell \\ne \\eta,\\chi} E_\\ell x_n^{\\ell} \\oplus \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)+1,{\\overline{R}}\\big) x_n^\\eta \\oplus \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\chi\\big)-1,{\\overline{R}}\\big) x_n^\\chi \\subseteq R.\n\\end{equation}\nNote that both  $\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)+1,{\\overline{R}}\\big)$ and $\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\chi\\big)-1,{\\overline{R}}\\big)$\nexist by induction, since the corresponding Hilbert schemes are nonempty.\nFor the former,  by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (10) there exists some ideal $\\mathcal{J} \\subseteq E_h $ with \n$\\mathrm{HP}\\big({\\overline{R}}\/\\mathcal{J}\\big) = \\mathrm{HP}\\big({\\overline{R}}\/E_h\\big)+1$.\nFor the latter, the Hilbert scheme of  $q(\\zeta) $ is nonempty by assumption.\nWe observe  that $J$ is an ideal of $R$, as its components  form a non-decreasing sequence:\nby \\hyperref[A3]{(A3)}, since we already know that $E$ is an ideal, it only remains to check the two inclusions\n$ E_{\\eta-1} \\subseteq\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)+1,{\\overline{R}}\\big)$ and $ \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\chi\\big)-1,{\\overline{R}}\\big)\\subseteq E_{\\chi+1}$,\nbut they follow by the choice of $\\eta, \\chi$.\nClearly $J$ is monomial, and it  is  saturated since $x_{n+1}$ does not divide its minimal generators.\n\nWe prove that $J$ is strongly stable.\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (3)\nit suffices to show that $(x_1, \\ldots, x_{n-1})J_\\ell\\subseteq J_{\\ell-1}$ for each $\\ell$,\nsince all $J_\\ell$ are  strongly stable.\nUsing Proposition \\ref{PropositionElementaryPropertiesDecomposition} (5) and (8), we see that the  $\\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\ell\\big)-\\mathrm{HP}\\big({\\overline{R}}\/J_{\\ell-1}\\big) $ is an integer.\nBoth $(x_1, \\ldots, x_{n-1})J_\\ell$ and $ J_{\\ell-1}$ are expansive,\nhence it suffices to show   \n$ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\ell\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\ell-1}\\big)\\geq 0 $.\nThere are  three cases that do not follow from  $E$ being strongly stable.\nIf $\\ell =   \\chi = \\eta+1$,\nnecessarily $\\chi = k$ and $\\eta =h$,\nhence $F = (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big( q(\\zeta),{\\overline{R}}\\big)\\subsetneq E_h$\nbecomes $(x_1, \\ldots, x_{n-1}) J_\\chi \\subsetneq E_{\\eta}$,\nso $\\mathrm{HP}(E_\\eta) - \\mathrm{HP}\\big((x_1, \\ldots, x_{n-1})J_\\chi\\big)>0$ and \n$ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\ell\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\ell-1}\\big)\\geq 0 $.\nIf $\\ell = \\eta+1 < \\chi$,\nwe must prove $ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_{\\eta+1}\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\eta}\\big)\\geq 0$,\ni.e.  $\\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})E_{\\eta+1}\\big) -\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)-1 \\geq 0$, \nequivalently, $(x_1, \\ldots, x_{n-1})E_{\\eta+1} \\subsetneq E_\\eta$.\nBut if this were false, then $(x_1, \\ldots, x_{n-1})E_{\\eta+1} = E_\\eta$, since $E$ is strongly stable.\nIn particular $E_{\\eta+1} \\ne E_\\eta$, forcing $\\eta = h$,\nand the pair $h+1,k$ would contradict the choice of $h,k$.\nFinally, \nif $\\ell = \\chi > \\eta+1$,\nwe must show \n$ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\chi\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\chi-1}\\big)\\geq 0 $.\nIf this were false,\nthen $E_{\\chi-1} \\subsetneq (x_1, \\ldots, x_{n-1})E_\\chi$,\nforcing $\\chi = k$,\nand  $\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_{k-1}\\big)+1,{\\overline{R}})  \\subseteq (x_1, \\ldots, x_{n-1})\\mathrm{Exp}\\big(q(\\zeta)+1,{\\overline{R}}\\big)$.\nWe obtain\n$(x_1, \\ldots, x_{n-1})^{k-h-1}\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_{k-1}\\big)+1,{\\overline{R}})  \\subseteq  (x_1, \\ldots, x_{n-1})^{k-h}\\mathrm{Exp}\\big(q(\\zeta)+1,{\\overline{R}}\\big) = F\\subsetneq E_h$,\nand the pair $h,k-1$ contradicts the  choice of $h,k$.\nThus $J$ is strongly stable.\n\nFrom \\eqref{ConstructionJContradiction} it follows immediately that\n$\\sum_{\\ell=0}^\\rho \\mathrm{HP}(J_\\ell) \\preceq \\sum_{\\ell=0}^\\rho \\mathrm{HP}(E_\\ell)$ for every $\\rho < d_n$\nand $\\sum_{\\ell=0}^\\eta \\mathrm{HP}(J_\\ell) \\prec \\sum_{\\ell=0}^\\eta \\mathrm{HP}(E_\\ell)$.\nThis yields a contradiction to the choice of $I$, proving  \\hyperref[A5]{(A5)}.\n\n\n\nIn order to verify \\hyperref[A3]{(A3)}, we prove a stronger statement:\n\\begin{enumerate}\n\\item[$(\\dagger)$]\nif $E,E'\\subseteq R$ are saturated monomial ideals satisfying  \\hyperref[A1]{(A1)},  \\hyperref[A2]{(A2)},  \\hyperref[A5]{(A5)}, and\nsuch that $\\mathrm{HP}(E')-\\mathrm{HP}(E)$ is an integer, then  $E \\subseteq E'$ or $E' \\subseteq E$.\n\\end{enumerate}\nSince $\\mathrm{HP}(E')-\\mathrm{HP}(E)\\in \\mathbb{Z}$, by  \nProposition \\ref{PropositionElementaryPropertiesDecomposition} (8)  $\\mathrm{HP}(E'_h)-\\mathrm{HP}(E_k)\\in \\mathbb{Z}$ for every $h,k$.\nSuppose $E \\ne E'$, and let $h$ be the least index such that $E_h \\ne E'_h$. \nBy axiom \\hyperref[A3]{(A3)} either  $E_h \\subsetneq E'_h$ or $E'_h \\subsetneq E_h$;\n assume, for instance, that $E_h \\subsetneq E'_h$.\nAssume by contradiction that  there exists $h < k < d_n$ with  $E_k \\not\\subseteq E'_k$.\nUsing \\hyperref[A3]{(A3)} again we find that $E'_k \\subsetneq E_k$ and  $\\mathrm{HP}(E'_k)\\prec \\mathrm{HP}(E_k)$,\nso  $\\mathrm{HP}\\big({\\overline{R}}\/E_k\\big)\\preceq \\mathrm{HP}\\big({\\overline{R}}\/E'_k\\big)-1 $ and \n$\n\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E'_k)-1, {\\overline{R}}\\big) \\subseteq E_k.\n$\nSince $E$ is strongly stable, we have $(x_1, \\ldots, x_{n-1})^{k-h}E_k \\subseteq E_h$.\nBy \\hyperref[A5]{(A5)} we get $E'_h \\subseteq (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E'_k)-1, {\\overline{R}}\\big)$.\nWe derive the contradiction\n $$\n E_h \\subsetneq E'_h\\subseteq (x_1, \\ldots, x_{n-1})^{k-h}\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E'_k)-1, {\\overline{R}}\\big) \\subseteq (x_1, \\ldots, x_{n-1})^{k-h}E_k \\subseteq E_h.\n $$\nThis proves the claim $(\\dagger)$, which implies  the axiom \\hyperref[A3]{(A3)} and also the uniqueness of the expansive ideal for each Hilbert polynomial.\n\n\nNext, we show  \\hyperref[A6]{(A6)}.\nWe must prove that \n$\n\\big\\{ \\sum_{\\ell=0}^\\rho \\mathrm{HP}(E_\\ell) \\big\\}_{\\rho=0}^{d_n-1}\n$\nis the unique  minimal sequence, with respect to componentwise comparison by $\\preceq$, among all strongly stable ideals $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$.\nLet  $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$  be strongly stable\nsuch that \n$\n\\big\\{ \\sum_{\\ell=0}^\\rho \\mathrm{HP}(J_\\ell) \\big\\}_{\\rho=0}^{d_n-1}\n$\n is also minimal  with respect to componentwise comparison. \nAs in \\eqref{EqConstructionExpansive}\nwe define\n$$ \nE' = \\bigoplus_{\\ell<d_n}\\mathrm{Exp}(J_\\ell)x_n^\\ell \\subseteq R\n$$ \nand, by the same proof as for  $E$,\nit follows  that $E'$ is a saturated strongly stable ideal of $R$ satisfying \\hyperref[A1]{(A1)}, \\hyperref[A2]{(A2)}, \\hyperref[A5]{(A5)}.\nBy $(\\dagger)$  we deduce  that $E=E'$, proving thus \\hyperref[A6]{(A6)}.\n\nNext, we show that $E$ verifies \\hyperref[A7]{(A7)}.\nLet $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ be strongly stable.\nApplying \\hyperref[A6]{(A6)} with $\\rho = 0$ we have  $\\mathrm{HP}\\big({\\overline{R}}\/J_0\\big)\\preceq  \\mathrm{HP}\\big({\\overline{R}}\/E_0\\big)$, thus\n $E_0\\subseteq \\mathrm{Exp}(J_0)$ by \\hyperref[A3]{(A3)}, and therefore \n$\n\\mathrm{HP}\\big((x_1,\\ldots, x_{n-1})E_0\\big)\\preceq \\mathrm{HP}\\big((x_1,\\ldots, x_{n-1})\\mathrm{Exp}(J_0)\\big).\n$\nOn the other hand, by \\hyperref[A7]{(A7)}  we  have \n$\\mathrm{HP}\\big((x_1,\\ldots, x_{n-1})\\mathrm{Exp}(J_0)\\big)\\preceq \\mathrm{HP}\\big((x_1,\\ldots, x_{n-1})J_0\\big)$.\nCombining the inequalities we obtain $\\mathrm{HP}\\big((x_1,\\ldots, x_{n-1})E_0\\big)\\preceq \\mathrm{HP}\\big((x_1,\\ldots, x_{n-1})J_0\\big)$.\nNow consider   decompositions\n\\begin{eqnarray}\n\\label{EqDecompositionE} (x_1,\\ldots, x_n) E &=& (x_1,\\ldots, x_{n-1}) E_0 \\oplus \\bigoplus_{\\ell=0}^{d_n-2} E_\\ell x_n^{\\ell+1},\\\\\n(x_1,\\ldots, x_n)J &= &(x_1,\\ldots, x_{n-1}) J_0 \\oplus \\bigoplus_{\\ell=0}^{d_n-2} J_\\ell x_n^{\\ell+1}.\n\\end{eqnarray}\nThe desired inequality \n$\\mathrm{HP}\\big( (x_1, \\ldots, x_{n}) E \\big)\\preceq\\mathrm{HP}\\big((x_1, \\ldots, x_{n}) J\\big)$\nfollows from  additivity of $\\mathrm{HP}(-)$ on direct sums and axiom \\hyperref[A6]{(A6)} applied with $\\rho=d_n-2$.\n\nFinally, we show that $E$ verifies \\hyperref[A4]{(A4)}.\nIt follows from \\eqref{EqDecompositionE} that $(x_1,\\ldots, x_n) E$ is a saturated  ideal satisfying \\hyperref[A1]{(A1)}, \\hyperref[A2]{(A2)}, \\hyperref[A5]{(A5)}.\nRepeating the construction  \\eqref{EqConstructionExpansive} for the polynomial $p'(\\zeta) = \\mathrm{HP}\\big({\\overline{R}}\/(x_1,\\ldots, x_n) E\\big)$ yields another  ideal $E'\\in \\mathrm{Hilb}^{p'(\\zeta)}(\\mathrm{Proj} R)$ satisfying \\hyperref[A1]{(A1)}, \\hyperref[A2]{(A2)}, \\hyperref[A5]{(A5)}.\nApplying $(\\dagger)$ to $E'$ and $(x_1,\\ldots, x_n) E$ we conclude that $(x_1,\\ldots, x_n) E=E'$, which means that $(x_1,\\ldots, x_n) E$ is the expansive ideal with Hilbert polynomial $p'(\\zeta)$.\n\\end{proof}\n\n\nWe remark that expansive ideals satisfy  also an extremal property with respect to higher hyperplane sections, comparable to the inequalities for lex ideals  proved in  \\cite{Ga98,Ga99,HePo98}.\n\n\\begin{cor}\\label{CorollaryHyperplaneSections}\n Let  $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$  be  strongly stable,   then  \n$\n\\mathrm{HP}\\big(\\mathrm{Exp}(p) + (x_n^h)\\big) \\preceq \\mathrm{HP}\\big(J + (x_n^h)\\big)\n$\nfor every $h$.\n\\end{cor}\n\\begin{proof}\nIt follows from \\hyperref[A6]{(A6)}\nand additivity of $\\mathrm{HP}(-)$ on the decompositions\n$$\nJ + (x_n^h) = \\bigoplus_{\\ell=0}^{h-1} J_\\ell\\, x_n^\\ell \\oplus \\bigoplus_{\\ell=h}^{d_n-1} {\\overline{R}} x_n^\\ell \\qquad \n\\mathrm{Exp}(p) + (x_n^h) = \\bigoplus_{\\ell=0}^{h-1} \\mathrm{Exp}(p)_\\ell\\, x_n^\\ell \\oplus \\bigoplus_{\\ell=h}^{d_n-1} {\\overline{R}} x_n^\\ell.\n$$\n\\end{proof}\n\n\nWe conclude the section with a  comment.\nThe approach undertaken in Theorem \\ref{TheoremDefinitionExpansiveAxioms} has the advantage of  identifying extremal properties that play a crucial role in estimating syzygies.\nOn the other hand, the structure of $\\mathrm{Exp}(p)$ remains somewhat obscure, and the axioms are impractical for the purpose of\n computing examples.\nWe are going to fill this gap in Section \\ref{SectionComputation}.\n\n\n\\section{Maximal syzygies}\\label{SectionSyzygies}\n\nWe present the main application of expansive ideals: \nthe existence of  sharp upper bounds for the syzygies of subschemes of a Clements-Linstr\\\"om scheme. \nOur treatment relies entirely on the axioms of Theorem \\ref{TheoremDefinitionExpansiveAxioms}.\nThe extension of the  result to arbitrary complete intersections in $\\mathbb{P}^n$ is also discussed.\nWe keep the  notation of the previous sections, \nand in particular  \\eqref{DefinitionRings}.\n\nFirst, the main result of \\cite{MeMu11} allows to perform an important reduction  to almost lex ideals.\n\n\\begin{lemma}\\label{LemmaAlmostLexMerminMurai}\nFor any $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ there exists an almost lex\n $J\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ with $\\mathrm{HF}(I) = \\mathrm{HF}(J)$ and $\\beta_{i,j}^S(R\/I) \\leq \\beta_{i,j}^S(R\/J)$ for all $i,j$.\n\\end{lemma}\n\\begin{proof}\nSince $I\\subseteq R$ is saturated, there exists a linear form $\\ell\\in [S]_1$ that is a non-zerodivisor on $R\/I$.\nUp to a change of coordinates in $S$, we may assume that $\\ell=x_{n+1}$.\nWith the notation of  Remark \\ref{RemarkModuloXNplus1},\nconsider ${\\widetilde{J}} = {\\mathrm{Lex}}({\\widetilde{I}}) \\subseteq {\\widetilde{R}}$ and let $J = {\\widetilde{J}} R \\subseteq R$.\nBy \\cite[Theorem 8.1]{MeMu11} we obtain\n$\\beta_{i,j}^{{\\widetilde{S}}}({\\widetilde{R}}\/\\widetilde{I}) \\leq\\beta_{i,j}^{{\\widetilde{S}}}({\\widetilde{R}}\/\\widetilde{J})$ for all $i,j$,\nand the conclusion follows from Remark \\ref{RemarkModuloXNplus1}.\n\\end{proof}\n\nIn the next lemma we consider the natural $\\mathbb{Z}^{n+1}$-grading on $R$.\n\n\\begin{lemma}\\label{LemmaUpperBoundBettiFreeModule}\nLet $M$ be a finite $\\mathbb{Z}^{n+1}$-graded $R$-module that is a  free $\\Bbbk[x_{n+1}]$-module of finite rank $c\\in \\mathbb{N}$ via restriction of scalars.\nFor every $i\\in \\mathbb{N}$ we have\n\\begin{itemize}\n\\item[(i)]\n $\\beta_i^S(M) \\leq c\\cdot \\beta_i^S(\\Bbbk[x_{n+1}])$  and $\\beta_i^R(M) \\leq c\\cdot \\beta_i^R(\\Bbbk[x_{n+1}])$;\n\\item[(ii)]\nif ${\\mathrm{ann}}_R(M) = (x_1, \\ldots, x_n)$, then \n$\\beta_i^S(M) = c\\cdot \\beta_i^S(\\Bbbk[x_{n+1}])$\nand\n$\\beta_i^R(M) = c\\cdot \\beta_i^R(\\Bbbk[x_{n+1}])$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nWe prove (ii) first.\nLet $m_1, \\ldots, m_s$ be minimal $\\mathbb{Z}^{n+1}$-graded $R$-module generators of $M$.\nThe assumptions imply the isomorphisms of $R$-modules\n$M \\cong Rm_1 \\oplus \\cdots\\oplus Rm_s $\nand $Rm_h \\cong \\Bbbk[x_{n+1}]$ for every $h$,\nso that $s = c$ and the formulas for the Betti numbers follow.\nTo prove (i), we may assume $c>1$.\nLet $M' = (x_1, \\ldots, x_n) M$ and $M'' = M\/M'$.\nBoth $M'$ and $M''$ are finite $\\mathbb{Z}^{n+1}$-graded $R$-modules.\nAs $\\Bbbk[x_{n+1}]$-modules via restriction of scalars,\n $M'$ is free  of rank less than $c$,\nwhereas $M''$ is also free, by multidegree reasons, and it satisfies (ii).\nThe  conclusions  follow by induction on $c$ from the  exact sequence $0\\rightarrow M' \\rightarrow M \\rightarrow M'' \\rightarrow 0$.\n\\end{proof}\n\n\n\\begin{thm}\\label{TheoremExtremalFiniteResolution}\nLet $S = \\Bbbk[x_1, \\ldots, x_{n+1}]$ be a polynomial ring  and $R= S\/(x_1^{d_1}, \\ldots, x_n^{d_n})$  a Clements-Lindstr\\\"om ring, where $2 \\leq d_1 \\leq \\cdots \\leq d_n \\leq \\infty$.\nFor each polynomial $p(\\zeta)$ we have\n$$\n\\beta_{i}^S\\big(R\/I\\big) \\leq \\beta_{i}^S\\big(R\/\\mathrm{Exp}(p(\\zeta))\\big) \n$$\nfor all $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ and  all $i \\geq 0$.\n\\end{thm}\n\n\\begin{proof}\nWe proceed by induction on $n$ and the case $n=0$ is trivial, so  let $n>0$.\nBy Lemma \\ref{LemmaAlmostLexMerminMurai}, we may assume without loss of generality that $I$ is a  strongly stable monomial ideal.\nLet ${\\mathcal{I}}, {\\mathcal{E}}$ denote the preimages of  $I, \\mathrm{Exp}(p) \\subseteq R$ in the polynomial ring $S$.\n We have decompositions \n\\begin{align}\\label{EqDecompositionIandE}\n{\\mathcal{I}} &= \\bigoplus_{\\ell=0}^\\infty {\\mathcal{I}}_\\ell x_n^\\ell,  &{\\mathcal{E}} &= \\bigoplus_{\\ell=0}^\\infty {\\mathcal{E}}_\\ell x_n^\\ell\n\\end{align}\nwhere ${\\mathcal{I}}_\\ell, {\\mathcal{E}}_\\ell$ are ideals of ${\\overline{S}}$. \nSpecifically, ${\\mathcal{I}}_\\ell \\subseteq {\\overline{S}}$ is the preimage of $I_\\ell \\subseteq {\\overline{R}}$ if $\\ell < d_n$, and ${\\mathcal{I}}_\\ell = {\\overline{S}} $ if $d_n \\leq \\ell < \\infty$; \nlikewise for ${\\mathcal{E}}_\\ell$.\nSince $R\/I \\cong S\/{\\mathcal{I}}$ and   $R\/\\mathrm{Exp}(p) \\cong S\/{\\mathcal{E}}$,\nwe must prove that $\\beta_{i}^S({\\mathcal{I}}) \\leq \\beta_{i}^S({\\mathcal{E}})$ for all $i$.\nThe variable $x_n$ is a non-zerodivisor on $S, {\\mathcal{I}}, {\\mathcal{E}},$ so it suffices to \n prove $\\beta_{i}^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}}) \\leq \\beta_{i}^{\\overline{S}}({\\mathcal{E}}\/x_n{\\mathcal{E}})$ for all $i$.\n \n \n Let ${\\mathcal{J}}\\subseteq {\\overline{S}}$ denote the preimage of  $\\mathrm{Exp}(I_0) \\subseteq {\\overline{R}}$.\n Since ${\\overline{S}}\/{\\mathcal{I}}_0 \\cong R\/I_0$ and ${\\overline{S}}\/{\\mathcal{J}} \\cong {\\overline{R}} \/ \\mathrm{Exp}(I_0)$, \n by induction we have $  \\beta_i^{\\overline{S}}({\\mathcal{I}}_0) \\leq \\beta^{\\overline{S}}({\\mathcal{J}})$ for every $i \\geq 0$.\n  By Corollary \\ref{CorollaryHyperplaneSections} we have $\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n)\\big ) \\preceq\\mathrm{HP}\\big(I+(x_n)\\big )$.\n Note that $I_0 \\cong \\frac{I+(x_n)}{(x_n)} $ and $\\mathrm{Exp}(p)_0 \\cong \\frac{\\mathrm{Exp}(p)+(x_n)}{(x_n)} $, so \n $\\mathrm{HP}(\\mathrm{Exp}(p)_0 ) \\preceq\\mathrm{HP}(I_0)$. \n By \\hyperref[A3]{(A3)} we conclude that $\\mathrm{Exp}(p)_0 \\subseteq \\mathrm{Exp}(I_0)$, and hence ${\\mathcal{E}}_0 \\subseteq {\\mathcal{J}}$.\nBy Proposition \\ref{PropositionFiniteLengthRank}\nthe quotient ${\\mathcal{J}}\/{\\mathcal{E}}_0 \\cong \\mathrm{Exp}(I_0)\/ \\mathrm{Exp}(p)_0$ is a  free $\\Bbbk[x_{n+1}]$-module over  of rank \n $c_0 = \\mathrm{HP}\\big(  \\mathrm{Exp}(I_0)\/ \\mathrm{Exp}(p)_0 \\big)\n=  \\mathrm{HP}\\big(I+(x_n)\\big )-\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n)\\big )$. \nUsing the short exact sequence $ 0\\rightarrow {\\mathcal{E}}_0 \\rightarrow {\\mathcal{J}} \\rightarrow {\\mathcal{J}}\/{\\mathcal{E}}_0 \\rightarrow 0$ and Lemma \\ref{LemmaUpperBoundBettiFreeModule} (i) we obtain\n\\begin{equation}\\label{EqFirstEstimateFiniteResolution}\n \\beta_i^{\\overline{S}}({\\mathcal{I}}_0) \\leq \n\\beta_i^{\\overline{S}} ( {\\mathcal{J}}) \\leq \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) +  \\beta_i^{\\overline{S}} ( {\\mathcal{J}}\/{\\mathcal{E}}_0)  \\leq  \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) +  c_0\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big).\n\\end{equation}\n\nSuppose first that $d_n = \\infty$. \nFrom \\eqref{EqDecompositionIandE} we deduce\n decompositions of ${\\overline{S}}$-modules \n \\begin{equation}\\label{EqDecompositionQuotientsIandE}\n\\frac{{\\mathcal{I}}}{x_n{\\mathcal{I}}}\\cong {\\mathcal{I}}_0 \\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{{\\mathcal{I}}_\\ell}{{\\mathcal{I}}_{\\ell-1}} \\cong {\\mathcal{I}}_0 \\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{I_\\ell}{I_{\\ell-1}},\n \\qquad\n \\frac{{\\mathcal{E}}}{x_n{\\mathcal{E}}}\\cong {\\mathcal{E}}_0\\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{{\\mathcal{E}}_\\ell}{{\\mathcal{E}}_{\\ell-1}} \\cong {\\mathcal{E}}_0\\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}. \n \\end{equation}\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (6) the ${\\overline{S}}$-modules  $\\bigoplus_{\\ell=0}^\\infty \\frac{I_\\ell}{I_{\\ell-1}}$ and  $\\bigoplus_{\\ell=0}^\\infty \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}$ are  free $\\Bbbk[x_{n+1}]$-modules of rank $ c_1 = \\mathrm{HP}(I_\\infty)- \\mathrm{HP}(I_0)\\in \\mathbb{N}$ and $c_2 = \\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)- \\mathrm{HP}(\\mathrm{Exp}(p)_0)\\in \\mathbb{N}$, respectively.\nMoreover, by \\ref{PropositionElementaryPropertiesDecomposition} (3) these modules are annihilated by $(x_1, \\ldots, x_{n-1})\\subseteq {\\overline{S}}$.\nUsing Lemma \\ref{LemmaUpperBoundBettiFreeModule} (ii) and \ncombining with \\eqref{EqFirstEstimateFiniteResolution} we obtain\n\\begin{align*}\n\\beta_{i}^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}}) & = \\beta_{i}^{\\overline{S}}({\\mathcal{I}}_0) + \\beta_{i}^{\\overline{S}}\\left( \\bigoplus_{\\ell=0}^\\infty \\frac{{\\mathcal{I}}_\\ell}{{\\mathcal{I}}_{\\ell-1}}\\right) =  \\beta_{i}^{\\overline{S}}({\\mathcal{I}}_0) + c_1\\beta_{i}^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big)\\\\\n& \\leq   \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0)  +  (c_0+ c_1)\\beta_{i}^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big).\n\\end{align*}\nWe claim that $ \\beta_i^{\\overline{S}} ( {\\mathcal{E}}\/x_n{\\mathcal{E}}) =    \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0)  +  (c_0+ c_1)\\beta_{i}^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big)$, concluding the proof in this case. \nThis  follows from \\eqref{EqDecompositionQuotientsIandE} and Lemma \\ref{LemmaUpperBoundBettiFreeModule} (ii)   once we show that $\\bigoplus_{\\ell=0}^\\infty \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}$ has rank $c_0+c_1$ as $\\Bbbk[x_{n+1}]$-module,\nthat is, $c_2 = c_0+c_1$.\nBut this is true by definition of $c_0$, $c_1$, $c_2$,  additivity of Hilbert polynomials, and the fact that \n$\\mathrm{HP}(I_\\infty) = \\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)$ by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (7).\n\nNow suppose that $d_n < \\infty$.\nThe decompositions of ${\\overline{S}}$-modules obtained from  \\eqref{EqDecompositionIandE} become \n \\begin{equation}\\label{EqSecondDecompositionQuotientsIandE}\n\\frac{{\\mathcal{I}}}{x_n{\\mathcal{I}}}\\cong{\\mathcal{I}}_0 \\oplus \\bigoplus_{\\ell=0}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}} \\oplus \\frac{{\\overline{R}}}{I_{d_n-1}},\n \\qquad\n \\frac{{\\mathcal{E}}}{x_n{\\mathcal{E}}}\\cong {\\mathcal{E}}_0\\oplus \\bigoplus_{\\ell=0}^{d_n-1} \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}} \\oplus \\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}.\n \\end{equation}\nOur  goal is to estimate $\\beta_{i}^{\\overline{S}}({\\overline{R}}\/I_{d_n-1})$.\nBy induction\n$\\beta_{i}^{\\overline{S}}({\\overline{R}}\/I_{d_n-1}) \\leq \\beta_{i}^{\\overline{S}}\\big({\\overline{R}}\/\\mathrm{Exp}(I_{d_n-1})\\big)\n$ \nfor all $i\\geq 0$.\nNote that both $R, {\\overline{R}}$ have Krull dimension 1, since $d_n <\\infty$,\nhence all  Hilbert polynomials of ideals are constant.\nBy additivity of $\\mathrm{HP}(-)$ we have the formulas\n\\begin{align*}\n&\\mathrm{HP}(I) = \\sum_{\\ell=0}^{d_n-1} \\mathrm{HP}(I_\\ell)\n&& \\mathrm{HP}\\big(I+(x_n^{d_n-1})\\big) =   \\sum_{\\ell=0}^{d_n-2} \\mathrm{HP}(I_\\ell) + \\mathrm{HP}({\\overline{R}})\\\\\n&\\mathrm{HP}(\\mathrm{Exp}(p)) = \\sum_{\\ell=0}^{d_n-1} \\mathrm{HP}(\\mathrm{Exp}(p)_\\ell) \\quad\n&&\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n^{d_n-1})\\big) =  \\sum_{\\ell=0}^{d_n-2} \\mathrm{HP}(\\mathrm{Exp}(p)_\\ell) + \\mathrm{HP}({\\overline{R}}).\n\\end{align*}\nBy Corollary \\ref{CorollaryHyperplaneSections} we have $\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n^{d_n-1})) \\preceq \\mathrm{HP}\\big(I+(x_n^{d_n-1}))$.\nSince $\\mathrm{HP}(I) = \\mathrm{HP}(\\mathrm{Exp}(p))$, the formulas above imply that $\\mathrm{HP}(I_{d_n-1}) \\preceq \\mathrm{HP}(\\mathrm{Exp}(p)_{d_n-1})$, and \nusing  \\hyperref[A3]{(A3)} we deduce   $\\mathrm{Exp}(I_{d_n-1}) \\subseteq \\mathrm{Exp}(p)_{d_n-1}$.\nFrom the short exact sequence \n$$\n0 \\rightarrow \\frac{\\mathrm{Exp}(p)_{d_n-1}}{\\mathrm{Exp}(I_{d_n-1})} \\rightarrow \\frac{{\\overline{R}}}{\\mathrm{Exp}(I_{d_n-1})} \\rightarrow \\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}} \\rightarrow  0\n$$\nwe obtain\n\\begin{equation}\\label{EqEstimateLastPart}\n\\beta_{i}^{\\overline{S}}({\\overline{R}}\/I_{d_n-1}) \\leq \\beta_{i}^{\\overline{S}}\\big({\\overline{R}}\/\\mathrm{Exp}(I_{d_n-1})\\big) \\leq  \\beta_{i}^{\\overline{S}}\\left(\\frac{\\mathrm{Exp}(p)_{d_n-1}}{\\mathrm{Exp}(I_{d_n-1})}\\right) + \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}\\right).\n\\end{equation}\n\nFinally, we are going to  use \\eqref{EqSecondDecompositionQuotientsIandE} to give an upper bound for $\\beta_i^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}})$.\nAs before, \nthe ${\\overline{S}}$-modules \n$\\bigoplus_{\\ell=0}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}$ and  $\\bigoplus_{\\ell=0}^{d_n-1} \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}$ are annihilated by $(x_1, \\ldots, x_{n-1})\\subseteq {\\overline{S}}$,\nand by  Proposition \\ref{PropositionElementaryPropertiesDecomposition} (6) they are  free $\\Bbbk[x_{n+1}]$-modules of  ranks \n $ c'_1= \\mathrm{HP}(I_{d_n-1})- \\mathrm{HP}(I_0)$ and $c'_2 = \\mathrm{HP}\\big(\\mathrm{Exp}(p)_{d_n-1}\\big)- \\mathrm{HP}\\big(\\mathrm{Exp}(p)_0\\big)$, respectively.\nBy Proposition \\ref{PropositionFiniteLengthRank},\nthe module  $\\frac{\\mathrm{Exp}(p)_{d_n-1}}{\\mathrm{Exp}(I_{d_n-1})}$ is also free over $\\Bbbk[x_{n+1}]$, of  rank $c_3 = \\mathrm{HP}\\big(\\mathrm{Exp}(p)_{d_n-1}\\big)- \\mathrm{HP}\\big(\\mathrm{Exp}(I_{d_n-1}\\big)$.\nCombining the decomposition \\eqref{EqSecondDecompositionQuotientsIandE} and the bounds \\eqref{EqFirstEstimateFiniteResolution}, \\eqref{EqEstimateLastPart}, and using Lemma \\ref{LemmaUpperBoundBettiFreeModule} (i)  we find\n\\begin{align*}\n&\\quad\\beta_{i}^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}})  = \\beta_{i}^{\\overline{S}}({\\mathcal{I}}_0) + \\beta_{i}^{\\overline{S}}\\left( \\bigoplus_{\\ell=0}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}\\right) + \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{I_{d_n-1}}\\right) \\\\\n\\leq  &\n\\left[  \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) +  c_0\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big)\\right] \n+ c'_1\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big) + \\left[ c_3\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big) +  \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}\\right)\\right]\\\\\n= & \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) +  (c_0+c'_1+c_3)\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big) +  \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}\\right).\n\\end{align*}\nThe expression in the last line is equal to \n$ \\beta_i^{\\overline{S}} ( {\\mathcal{E}}\/x_n{\\mathcal{E}})$ \nbecause of \\eqref{EqSecondDecompositionQuotientsIandE},  Lemma \\ref{LemmaUpperBoundBettiFreeModule} (ii),\nand the fact that $c'_2 = c_0+c'_1+c_3$.\nThis concludes the proof.\n\\end{proof}\n\nWe remark that, in the  case  of $\\mathbb{P}^n$, \nthe existence of a point in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ satisfying the conclusion of Theorem \\ref{TheoremExtremalFiniteResolution}\nwas proved in \\cite[Theorem 1.1]{CaMu13}, cf. the Introduction.\n\n\n\\begin{remark}\nThe numerical bounds on the Betti numbers provided by Theorem \\ref{TheoremExtremalFiniteResolution} do not depend on $\\Bbbk$,\nas it follows from the combinatorial formula of \\cite[Proposition 2.1]{Mu08}.\n\\end{remark}\n\n\n\n\nGiven integers $d_1 \\leq d_2 \\leq \\cdots \\leq d_n \\leq \\infty$,\nwe say that an ideal is a complete intersection of degree sequence $d_1, \\ldots, d_n$ if\nit is generated by a regular sequence  $f_1, \\ldots, f_c$ with $d_i = \\deg(f_i)$ for every $i \\leq c:=\\max\\{j \\, : d_j < \\infty\\}$.\nWe emphasize that a complete intersection of degree sequence $d_1, \\ldots, d_n$ may have  codimension $c \\leq n$.\n\n\nA remarkable consequence of Theorem \\ref{TheoremExtremalFiniteResolution} is the fact that, conjecturally, \nthe expansive subscheme $\\mathrm{Exp}(p(\\zeta),R)\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$\n has the largest number of syzygies \n among all  subschemes $Z\\in  \\mathrm{Hilb}^{p(\\zeta)}(X)$ of \\emph{any} complete intersection $X  \\subseteq \\mathbb{P}^n$\n of degree sequence $d_1, \\ldots, d_n$.\n To justify this claim, \nwe recall the statements of two famous conjectures on complete intersections.\nFor our purposes, it is convenient to state them in terms of ideals of ${\\widetilde{S}} = \\Bbbk[x_1, \\ldots, x_n]$.\n \n \n\n\\begin{itemize}\n\\item \\emph{Eisenbud-Green-Harris Conjecture}:\nIf $I\\subseteq {\\widetilde{S}}$ contains a regular sequence of degree sequence  $d_1, \\ldots, d_n$, \nthen there exists a lex ideal $L\\subseteq {\\widetilde{S}}$ with $\\mathrm{HF}(I) = \\mathrm{HF} \\big( L + (x_1^{d_1},\\ldots, x_n^{d_n})\\big)$.\n\n\\item \\emph{Lex-plus-powers Conjecture}: \nIf $I\\subseteq {\\widetilde{S}}$ contains a regular sequence of degree sequence  $d_1, \\ldots, d_n$ and if\nthere exists a lex ideal $L\\subseteq {\\widetilde{S}}$ with $\\mathrm{HF}(I) = \\mathrm{HF} \\big( L + (x_1^{d_1},\\ldots, x_n^{d_n})\\big)$,\nthen $\\beta_{i,j}^{\\widetilde{S}}(I) \\leq \\beta_{i,j}^{\\widetilde{S}}( L + (x_1^{d_1},\\ldots, x_n^{d_n})\\big) $\nfor all $i,j$.\n\\end{itemize}\n\nThe first conjecture was proposed in \\cite{EiGrHa93}, \nwhereas the second one is attributed to  Charalambous and Evans in \\cite{FrRi07}.\nDespite the apparently independent statements, it is known that the Lex-plus-powers Conjecture implies the Eisenbud-Green-Harris Conjecture.\n\n\\begin{prop}\\label{PropositionLPP}\nLet $X  \\subseteq \\mathbb{P}^n$ be a complete intersection \nwith degree sequence $d_1 \\leq d_2 \\leq \\cdots \\leq d_n \\leq \\infty$.\nIf the Lex-plus-powers conjecture is true, then \n$\n\\beta_{i}^S\\big(S\/I_Z\\big) \\leq \\beta_{i}^S\\big(R\/\\mathrm{Exp}(p)\\big) \n$\nfor every closed subscheme $Z\\in\\mathrm{Hilb}^{p(\\zeta)}(X)$ and  all $i \\geq 0$.\n\\end{prop}\n\\begin{proof}\nAs in Lemma \\ref{LemmaAlmostLexMerminMurai},\nwe may assume that $x_{n+1}$ is a non-zerodivisor on $S\/I_Z$, and we \nconsider ${\\widetilde{I}} = \\frac{I_Z+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{S}}$.\nBy assumption the Lex-plus-powers Conjecture and the Eisenbud-Green-Harris Conjecture hold,\ntherefore  there exists a lex ideal ${\\widetilde{L}} \\subseteq {\\widetilde{R}}$ such that $\\mathrm{HF}({\\widetilde{S}}\/{\\widetilde{I}}) =\\mathrm{HF}({\\widetilde{R}}\/{\\widetilde{L}}) $\nand $\\beta_{i,j}^{\\widetilde{S}}({\\widetilde{S}}\/{\\widetilde{I}}) \\leq \\beta_{i,j}^{\\widetilde{S}}({\\widetilde{R}}\/{\\widetilde{L}}) $ for all $i,j\\geq 0$.\nThe extension $L= {\\widetilde{L}} R \\subseteq R$ is an almost lex ideal, and \nusing Remark \\ref{RemarkModuloXNplus1} we deduce $\\mathrm{HF}(L)=\\mathrm{HF}(I_Z)$ and \n$\\beta_{i,j}^S(S\/I_Z) \\leq \\beta_{i,j}^S(R\/L) $ for all $i, j\\geq 0$. \nBy Theorem \\ref{TheoremExtremalFiniteResolution} we have \n$\\beta_{i}^S(R\/L) \\leq \\beta_{i}^S(R\/\\mathrm{Exp}(p)) $ for all $i\\geq 0$, and this concludes the proof.\n\\end{proof}\n\n\n\nIn particular,\nby \\cite[Main Theorem]{CaSa18}\n we obtain the following result.\n\n\\begin{thm}\\label{TheoremLPP}\nAssume   $\\mathrm{char}(\\Bbbk) = 0$. \nLet $X  \\subseteq \\mathbb{P}^n_\\Bbbk$ be a complete intersection \nwith degree sequence \nsuch that  $d_j > \\sum_{h=1}^{j-1}(d_h-1)$ for all $j\\geq 3$.\nThen \n$\n\\beta_{i}^S\\big(S\/I_Z\\big) \\leq \\beta_{i}^S\\big(R\/\\mathrm{Exp}(p)\\big) \n$\nfor every closed subscheme $Z\\in\\mathrm{Hilb}^{p(\\zeta)}(X)$ and  all $i \\geq 0$.\n\\end{thm}\n\n\n\n\n\\section{Infinite free resolutions}\\label{SectionInfinite}\n\n\nIn this section we investigate  bounds for the  Betti numbers of    infinite free resolutions over a  Clements-Lindstr\\\"om ring.\nMinimal free resolutions  over complete intersections have attracted much attention in the past few years, \nand significant progress has been achieved in the description of their asymptotic behavior,\nsee e.g.\n\\cite{BuWa15,EiPe16,EiPeSc19}.\nWe conjecture that expansive subschemes exhibit extremal infinite free resolutions, and prove this conjecture for quadratic Clements-Linstr\\\"om rings in characteristic zero.\nWe also deduce the extremality of the deviations of  expansive subschemes of $\\mathbb{P}^n$, and in particular the extremality of the Poincar\\'e series.\n\nWe begin by proposing the following problem.\n\n\\begin{conj}\\label{ConjectureInfiniteResolution}\nLet $R$ be a Clements-Lindstr\\\"om ring. \nWe have\n$\\beta_{i}^{R}(I) \\leq \\beta_{i}^{R}(\\mathrm{Exp}(p)) $ \nfor every $I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ and  every $i \\geq 0$.\n\\end{conj}\n\nWhen the ground field has characteristic zero,\n\\cite[Theorem 1.4]{MuPe12} reduces the problem to almost lex ideals, \n proceeding  as in  Lemma \\ref{LemmaAlmostLexMerminMurai}.\n\n\n\\begin{lemma}\\label{LemmaReductionStableInfiniteResolution}\nAssume that  $\\mathrm{char}(\\Bbbk)=0$.\nFor every $I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ there exists an almost lex  $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$\nsuch that $\\mathrm{HF}(I)=\\mathrm{HF}(J)$ and $\\beta_{i,j}^R(I)\\leq \\beta_{i,j}^R(J)$ for all $i,j$.\n\\end{lemma}\n\nThe following theorem is the main result of this section.\nThe proof employs  a construction from \\cite{ArAvHe00,EiPoYu03,GaHiPe02}.\n\n\\begin{thm}\\label{TheoremIfninite}\nAssume that $\\mathrm{char}(\\Bbbk)=0$.\nLet $R$ be a Clements-Lindstr\\\"om ring with $d_j\\in \\{ 2, \\infty\\}$ for every $j$.\nWe have\n$\n\\beta_{i}^R\\big(I\\big) \\leq \\beta_{i}^R\\big(\\mathrm{Exp}(p)\\big) \n$\nfor every $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ and  every $i \\geq 0$.\n\\end{thm}\n\n\\begin{proof}\nWe proceed by induction on $n$, and the case $n=0$ is trivial, so assume $n>0$.\nBy Lemma \\ref{LemmaReductionStableInfiniteResolution} we may assume that $I$ is strongly stable.\nIn  addition to the notation established in \\eqref{DefinitionRings},\nin this proof we consider the ``intermediate'' ring \n$$\nT = \\frac{S}{(x_1^{d_1}, \\ldots, x_{n-1}^{d_{n-1}})}\n$$\nso that $R=T\/(x_n^{d_n})$.\nBy assumption, we either have $d_n = \\infty$, in which case $T=R$,\nor $d_n =2$.\n\n\nConsider the ideal ${\\mathcal{I}}\\subseteq T$ generated by the monomials of $T$ corresponding to the minimal generators of $I\\subseteq R$.\nNotice that ${\\mathcal{I}}$ is smaller than the preimage of $I$ in $T$ if $d_n=2$, whereas  ${\\mathcal{I}}=I$  if $d_n=\\infty$.\nSince $x_{n}$ is a non-zerodivisor on $T$ and ${\\mathcal{I}}$, and  $T\/(x_{n}) \\cong {\\overline{R}}$, \nwe have $\\beta_{i,j}^{T}({\\mathcal{I}}) = \\beta_{i,j}^{{\\overline{R}}}({\\mathcal{I}}\/x_{n}{\\mathcal{I}})$.\nWe have a decomposition of ${\\overline{R}}$-modules\n$$\n\\frac{{\\mathcal{I}}}{x_{n}{\\mathcal{I}}} = I_0 \\oplus  \\bigoplus_{\\ell =1}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}.\n$$\nBy induction $\\beta_i^{\\overline{R}}(I_0)\\leq\\beta_i^{\\overline{R}}(\\mathrm{Exp}(I_0))$.\nIn the proof of Theorem \\ref{TheoremExtremalFiniteResolution} \nwe established that $\\mathrm{Exp}(p)_0 \\subseteq \\mathrm{Exp}(I_0)$, \nand that \n$\\frac{\\mathrm{Exp}(I_0)}{\\mathrm{Exp}(p)_0}$\nis a  free $\\Bbbk[x_{n+1}]$-module via restriction of scalars of rank $c_0 = \\mathrm{HP}(I_0)-\\mathrm{HP}(\\mathrm{Exp}(p)_0)$.\nBy Lemma \\ref{LemmaUpperBoundBettiFreeModule} (i) we obtain\n\\begin{equation}\\label{InequalityBetti0InfiniteResolution}\n\\beta_i^{\\overline{R}}(I_0)\\leq\\beta_i^{\\overline{R}}(\\mathrm{Exp}(I_0)) \\leq \\beta_i^{\\overline{R}}(\\mathrm{Exp}(p)_0) + c_0 \\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}]).\n\\end{equation}\nWe also saw, using Propositions \\ref{PropositionFiniteLengthRank} and \\ref{PropositionElementaryPropertiesDecomposition},\nthat the ${\\overline{R}}$-module  $\\oplus_{\\ell =1}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}$ \nis annihilated by $(x_1, \\ldots, x_{n-1})$, and is a free $\\Bbbk[x_{n+1}]$-module of rank $c_1 =\\mathrm{HP}(I_{d_n-1}) - \\mathrm{HP}(I_0)$.\nBy Lemma \\ref{LemmaUpperBoundBettiFreeModule}~(ii) \n\\begin{equation}\\label{EqBettiNumbersTandOvR}\n\\beta_i^{T}({\\mathcal{I}}) = \\beta_i^{{\\overline{R}}}(I_0) + c_1\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}]).\n\\end{equation}\n\n\nIf $ d_n = \\infty$  then the formula \\eqref{EqBettiNumbersTandOvR} becomes \n$\n\\beta_i^{R}(I) = \\beta_i^{{\\overline{R}}}(I_0) + c_1 \\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}]),\n$\nand likewise we obtain\n$\n\\beta_i^{R}(\\mathrm{Exp}(p)) =  \\beta_i^{{\\overline{R}}}(\\mathrm{Exp}(p)_0) +  c_2 \\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])\n$\nwhere $c_2 = \\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)- \\mathrm{HP}(\\mathrm{Exp}(p)_0).$\nBy Proposition  \\ref{PropositionElementaryPropertiesDecomposition} (7) we have \n$\\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)=\\mathrm{HP}(I_\\infty)$ and therefore $c_2=c_0+c_1$, and\ncombining with \\eqref{InequalityBetti0InfiniteResolution} we conclude that $\\beta_i^{R}(I) \\leq \n\\beta_i^{R}(\\mathrm{Exp}(p))$ as desired.\n\nNow assume that $d_n = 2$.\nWe regard $R,{\\overline{R}},$ and $T$ as  $\\mathbb{Z}^{n+1}$-graded,\nbut we  also consider the $\\mathbb{Z}$-grading induced by the variable $x_n$.\nIf $M$ is $\\mathbb{Z}^{n+1}$-graded $T$-module \nwe define $\\sigma(M)$ to be the vector space consisting of the graded components of $M$ with $x_n$-degrees $0$ or $1$.\nClearly, $\\sigma$ defines an exact functor from the category of $\\mathbb{Z}^{n+1}$-graded $T$-modules\nto the category of $\\mathbb{Z}^{n+1}$-graded $\\Bbbk$-vector spaces.\n\nLet ${\\mathbb{F}}$ be the minimal $\\mathbb{Z}^{n+1}$-graded free resolution of ${\\mathcal{I}}$ over $T$.\nThe $x_{n}$-twists in this  resolution are all equal to 0 or 1:\nthis follows from the fact that ${\\mathbb{F}} \\otimes_T \\frac{T}{(x_{n})}$ is a minimal $\\mathbb{Z}^{n}$-graded free resolution of ${\\mathcal{I}}\/x_{n}{\\mathcal{I}}$ over ${\\overline{R}}$,\nand that ${\\mathcal{I}}\/x_{n}{\\mathcal{I}}$  is generated in $x_n$-degrees  $0,1$.\nThe complex ${\\mathbb{E}}=\\sigma({\\mathbb{F}})$ is acyclic and  minimal,  in the sense that the image of  its differential lies in $(x_1, \\ldots, x_{n+1}){\\mathbb{E}}$.\nEach direct summand in  ${\\mathbb{F}}$  has the form $T(-\\delta_1, \\ldots, -\\delta_n, -\\delta_{n+1})$ with $\\delta_n \\in \\{0,1\\}$; \nthe corresponding summand in ${\\mathbb{E}}$ is  a factor ring of $R = T\/(x_n^2)$, namely\n$$\n\\sigma\\big(T(-\\delta_1, \\ldots, -\\delta_n, -\\delta_{n+1})\\big) \\cong \\frac{R}{(x_n^{2-\\delta_n})}(-\\delta_1, \\ldots, -\\delta_n, -\\delta_{n+1}).\n$$\nThe cyclic $R$-module on the right hand side is free if and only if $\\delta_n = 0$.\nIn fact, ${\\mathbb{E}}$ is an acyclic minimal $\\mathbb{Z}^{n+1}$-graded complex of (not necessarily free) finitely generated $R$-modules.\nSince all the $x_n$-twists in ${\\mathbb{F}}$  are in $\\{0,1\\}$,  every free summand of ${\\mathbb{F}}$ contributes with a  non-zero  summand in ${\\mathbb{E}}$.\nIn other words,\nthe numbers of generators in every homological degree $i$ is the same for ${\\mathbb{F}}$ and ${\\mathbb{E}}$, and this number is $\\beta_i^{T}({\\mathcal{I}})$.\nAmong the direct summands of ${\\mathbb{E}}$, the free modules are precisely those coming from copies of $T$ in ${\\mathbb{F}}$ with $x_n$-twist equal to 0.\nThese modules form themselves another complex ${\\mathbb{E}}'$, which is again minimal and acyclic, but it is even free.\nIn fact, ${\\mathbb{E}}'$ is the minimal free resolution of $I_0$ over ${\\overline{R}}$, since $I_0$ is the truncation of ${\\mathcal{I}}$ in $x_n$-degree $0$, and ${\\overline{R}}$ is the truncation of $T$ in $x_n$-degree 0. \nWe conclude that in homological degree $i$ in ${\\mathbb{E}}$ we have exactly $\\beta^{\\overline{R}}_i(I_0)$  free summands, i.e.  copies of $R$.\n\nTo summarize,\n${\\mathbb{E}}$ is an acyclic minimal  complex of $\\mathbb{Z}^{n+1}$-graded  $R$-modules, \nit has $\\beta_i^{T}({\\mathcal{I}})$  generators in homological degree $i$,\nof which $\\beta^{\\overline{R}}_i(I_0)$   generate a free module $R$, \nwhereas the remaining ones generate a non-free module isomorphic to $R\/(x)$.\nThe number of non-free summands of ${\\mathbb{E}}$ in homological degree $i$ is  therefore\n$\\beta_i^{T}({\\mathcal{I}})- \\beta^{\\overline{R}}_i(I_0)  =  c_1\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$\nby \\eqref{EqBettiNumbersTandOvR}.\nNote also that the 0-homology of ${\\mathbb{E}}$ is $\\sigma(T\/{\\mathcal{I}}) =R\/I$.\n\n\nLet $E_j$ denote the module in homological degree $j$ in ${\\mathbb{E}}$.\nThe differentials of ${\\mathbb{E}}$ can be lifted to a complex of complexes, namely a double complex ${\\mathbb{D}}_I$ of $R$-modules where the \n$j$-th vertical complex is  the minimal free resolution of $F_j$.\nBy construction, the double complex  ${\\mathbb{D}}_I$ is free.\nFurthermore, it is minimal, \nand the total complex $\\mathrm{Tot}({\\mathbb{D}}_I)$ is a minimal $\\mathbb{Z}^{n+1}$-graded free resolution of $R\/I$ over $R$,\ncf. \n\\cite[Proposition 5.6]{EiPoYu03},\n\\cite[Theorem 1.3]{ArAvHe00},\nor\n\\cite[Theorem 2.10]{GaHiPe02}.\nRecall that the $R$-module $R\/(x)$  has an infinite minimal free resolution over $R$ with $\\beta_i^R(R\/(x))=1$ for every $i \\in \\mathbb{N}$.\nIt follows that in $\\mathbb{D}_I$, for each $i\\geq 0$,  we have \n\\begin{itemize}\n\\item[$(\\ast)$]$\\beta^{\\overline{R}}_i(I_0)$ summands in homological bidegree $(i,0)$ arising from the free summands of ${\\mathbb{E}}$;\n\n\\item[$(\\ast \\ast)$]  $c_1\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$ summands in homological bidegree $(i,j)$ for all $j\\geq 0$,\narising from the  non-free summands of ${\\mathbb{E}}$;\n\\end{itemize}\nwhere the first coordinate is  horizontal and the second coordinate  vertical.\nWe conclude that the Betti numbers of a saturated strongly stable  $I\\subseteq R$ depend only on those of $I_0\\subseteq {\\overline{R}}$ and on the number \n$c_1 =\\mathrm{HP}(I_1) - \\mathrm{HP}(I_0)$. \n\nThe same construction for $\\mathrm{Exp}(p)$ yields a double complex ${\\mathbb{D}}_{\\mathrm{Exp}(p)}$.\nLet $c'_2 = \\mathrm{HP}(\\mathrm{Exp}(p)_1)- \\mathrm{HP}(\\mathrm{Exp}(p)_0)$.\nWe observed in the proof of Theorem \\ref{TheoremExtremalFiniteResolution} that $\\mathrm{HP}(I_{d_n-1}) \\preceq \\mathrm{HP}(\\mathrm{Exp}(p)_{d_n-1})$,\nthat is, $\\mathrm{HP}(I_{1}) \\preceq \\mathrm{HP}(\\mathrm{Exp}(p)_{1})$.\nWe deduce that $c'_2 \\geq c_0+c_1$.\nFinally, \nwe compare the contribution of the two types of summands $(\\ast)$ and $(\\ast\\ast)$ to the double complexes\n${\\mathbb{D}}_I$ and ${\\mathbb{D}}_{\\mathrm{Exp}(p)}$:\n\n\n\\begin{itemize}\n\\item[$(\\ast)$] \nFor every $i\\geq 0$,\nby \\eqref{InequalityBetti0InfiniteResolution}, \n${\\mathbb{D}}_I$ has at most  $c_0\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$ more summands in position $(i,0)$ than  ${\\mathbb{D}}_{\\mathrm{Exp}(p)}$,\namong those arising from the free summands of ${\\mathbb{E}}$.\n\\item[$(\\ast\\ast)$] \nFor every $i,j\\geq 0$,\n$\\mathbb{D}_{\\mathrm{Exp}(p)}$ has  at least  $(c'_2-c_1)\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$ more summands in position $(i,j)$ than\n$\\mathbb{D}_I$, \namong those arising from the non-free summands of ${\\mathbb{E}}$.\n\\end{itemize}\nThus $\\mathbb{D}_{\\mathrm{Exp}(p)}$ has at least as many copies of $R$ as $\\mathbb{D}_I$ in every position $(i,j)$.\nThis concludes the proof, \nsince \n$\\beta_i^R(I),\\beta_i^R(\\mathrm{Exp}(p))$ \nare the Betti numbers of $\\mathrm{Tot}(\\mathbb{D}_I),\\mathrm{Tot}(\\mathbb{D}_{\\mathrm{Exp}(p)})$ respectively.\n\\end{proof}\n\n\nIn the remainder of this section, we explore deviations and Poincar\\'e series of expansive subschemes. \nThe deviations of a ring $A$ are a  sequence of  integers $\\{ \\varepsilon_i(A)\\}_{i \\geq 1}$\nmeasuring several homological or cohomological data of $A$.\nExamples include: the  generators of a Tate resolution of $A$ over a polynomial ring, as well as a Tate resolution of $\\Bbbk$ over $A$;\nthe ranks of the modules in a cotangent complex  of $A$;\n the dimensions of the  components of the homotopy Lie algebra $\\pi(A)$ of $A$.\nWe refer to \\cite[Sections 7 and 10]{Av98} for definitions and background.\n\n\n\\begin{lemma}\\label{LemmaNoVariable}\nLet $I \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ be strongly stable. \nWe have the inclusion of vector spaces of linear forms\n$[\\mathrm{Exp}(p)]_1 \\subseteq [I]_1$.\n\\end{lemma}\n\\begin{proof}\nWe may assume $I \\ne R$.\nSince $\\mathrm{Exp}(p)$ is saturated and strongly stable, we have $[\\mathrm{Exp}(p)]_1 = \\langle x_1, \\ldots, x_m\\rangle_\\Bbbk$ for some $0\\leq m \\leq n$.\nIf $m=n$ then $\\mathrm{Exp}(p) = (x_1, \\ldots, x_n) \\subseteq R$ is the only strongly stable ideal in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$,\nso $I= \\mathrm{Exp}(p)$.\nIf $m<n$ then $[\\mathrm{Exp}(p)]_1 = [\\mathrm{Exp}(p)_0]_1$.\nWe proceed by induction on $n$, and for $n=0 $ there is nothing to show.\nBy  axiom \\hyperref[A6]{(A6)} we have $\\mathrm{HP}(\\mathrm{Exp}(p)_0)\\preceq \\mathrm{HP}(I_0)$,\nthus $\\mathrm{Exp}(p)_0\\subseteq \\mathrm{Exp}(I_0)$ by \\hyperref[A3]{(A3)}.\nBy induction we have $[\\mathrm{Exp}(I_0)]_1 \\subseteq [I_0]_1$,\nhence $[\\mathrm{Exp}(p)]_1 = [\\mathrm{Exp}(p)_0]_1 \\subseteq[\\mathrm{Exp}(I_0)]_1 \\subseteq [I_0]_1 \\subseteq  [I]_1$.\n\\end{proof}\n\n\nA consequence of Theorem \\ref{TheoremExtremalFiniteResolution} and the results of \\cite{BoDaGrMoSa16}\nis the fact that an expansive subscheme of $\\mathbb{P}^n$ has maximal deviations in its Hilbert scheme.\n\n\\begin{cor}\\label{CorollaryDeviations}\nLet $S = \\Bbbk[x_1, \\ldots, x_{n+1}]$.\nWe have\n$\\varepsilon_{i}(S\/I) \\leq \\varepsilon_{i}(S\/\\mathrm{Exp}(p))\n$ \nfor every $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ and  all $i \\geq 1$.\n\\end{cor}\n\\begin{proof}\nWe may assume, as  in Lemma \\ref{LemmaAlmostLexMerminMurai},\nthat  $I:x_{n+1}=I$.\nLet ${\\widetilde{I}} = \\frac{I+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{S}}$ and  ${\\widetilde{L}} = {\\mathrm{Lex}}({\\widetilde{I}}) \\subseteq {\\widetilde{S}}$.\nBy \\cite[Theorem 3.4]{BoDaGrMoSa16}\nwe have $\\varepsilon_{i}({\\widetilde{S}}\/{\\widetilde{I}}) \\leq \\varepsilon_{i}({\\widetilde{S}}\/{\\widetilde{L}})$  for all $i \\geq 2$.\nIt follows from \\cite[Proposition 7.1.6]{Av98} that $\\varepsilon_{i}(S\/I) \\leq \\varepsilon_{i}(S\/L)$  for all $i \\geq 2$,\nwhere $L = {\\widetilde{L}} S \\subseteq S$.\nThe ideals $L$ and $\\mathrm{Exp}(p)$ are strongly stable, \nand this implies that $S\/L$ and $S\/\\mathrm{Exp}(p)$ are Golod rings\nby \\cite[Theorem 4]{HeReWe99}.\nNow by \\cite[Proposition 3.2]{BoDaGrMoSa16} we derive that $\\varepsilon_{i}(S\/L) \\leq \\varepsilon_{i}(S\/\\mathrm{Exp}(p))$  for all $i \\geq 2$.\nFinally, for $i=1$ the deviation $\\varepsilon_1(A)$ is equal to the embedding dimension of $A$,\ncf. \\cite[Corollary 7.1.5]{Av98},\ntherefore \n$\\varepsilon_{1}(S\/L) \\leq \\varepsilon_{1}(S\/\\mathrm{Exp}(p))$ by \nLemma \\ref{LemmaNoVariable}.\n\\end{proof}\n\n\nIn particular, an expansive subscheme of $\\mathbb{P}^n$ has maximal  Poincar\\'e series, \nthat is, the generating function of the dimensions of $\\mathrm{Tor}_\\bullet^A(\\Bbbk,\\Bbbk)$\nor $\\mathrm{Ext}^\\bullet_A(\\Bbbk,\\Bbbk)$.\n\n\\begin{cor}\nLet $S = \\Bbbk[x_1, \\ldots, x_{n+1}]$.\nWe have\n$\\beta_{i}^{S\/I}(\\Bbbk) \\leq \\beta_{i}^{S\/\\mathrm{Exp}(p))}(\\Bbbk) $\nfor every $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ and  all $i \\geq 0$.\n\\end{cor}\n\\begin{proof}\nApply Corollary \\ref{CorollaryDeviations} and \\cite[Remark 7.1.1]{Av98}.\n\\end{proof}\n\n\n\n\\section{Computation of expansive ideals}\\label{SectionComputation}\n\nIn this section we explore further properties of expansive subschemes. \nThe main goal is to construct the  generators of the expansive ideal from the Hilbert polynomial,\n in an explicit manner that avoids the recursive decomposition into expansive components.\nThis will be done in Theorem \\ref{TheoremChainExpansive}.\nAs a result, we provide a simple and efficient algorithm to construct the expansive point.\n\n\n\nWe begin by proving that expansive ideals are almost lex.\nRecall  \\eqref{DefinitionRings} and Remark \\ref{RemarkModuloXNplus1}.\n\n\\begin{prop}\\label{PropositionExpansiveAlmostLex}\nThe ideal $\\mathrm{Exp}\\big(p(\\zeta)\\big)\\subseteq R$ is almost lex.\n\\end{prop}\n\n\\begin{proof}\nFor simplicity we denote  $E = \\mathrm{Exp}\\big(p(\\zeta)\\big)$ and ${\\widetilde{E}} = \\frac{E+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{R}}$.\nBy definition, we must show that $\\widetilde{E}$ is a lex ideal of ${\\widetilde{R}}$.\nThe decomposition \\eqref{EqDecomposition} for $E$ yields \n$$\n{\\widetilde{E}} = \\bigoplus_{\\ell=0}^{d_n-1} {\\widetilde{E}}_\\ell x_n^\\ell\n\\qquad \\text{where} \\quad\n{\\widetilde{E}}_\\ell = \\frac{E_\\ell+(x_{n+1})}{(x_{n+1})}\\subseteq \\frac{{\\overline{R}}}{(x_{n+1})} = \\frac{R}{(x_n, x_{n+1})}\n$$\nin other words $E_\\ell$ is the extension of ${\\widetilde{E}}_\\ell$ to ${\\overline{R}}$.\nBy induction on $n$, each ${\\widetilde{E}}_\\ell$ is a lex ideal of $\\frac{R}{(x_n, x_{n+1})}$. \nMoreover, ${\\widetilde{E}}$ is strongly stable since $E$ is strongly stable.\nNow let  ${\\widetilde{I}} = {\\mathrm{Lex}}({\\widetilde{E}})\\subseteq {\\widetilde{R}}$ and decompose ${\\widetilde{I}}  = \\oplus_{\\ell=0}^{d_n-1} {\\widetilde{I}}_\\ell x_n^\\ell \\subseteq {\\widetilde{R}}$.\nThe recursive criterion for lex ideals in Clements-Lindstr\\\"om rings proved in  \\cite[Proof of Theorem 3.3, Lemma 3.7, Lemma 3.8]{CaKu13},\ncf. also \\cite[Remark 3.1]{CaSa18}, implies that ${\\widetilde{I}}_\\ell$ is lex for every $\\ell$ and that the following inequalities hold for every $\\rho, \\tau \\geq 0$\n\\begin{equation}\\label{EqInequalityCharacterizationLex}\n\\sum_{\\ell=0}^\\rho\\mathrm{HF}\\big({\\widetilde{I}}_\\ell, \\tau-\\ell\\big) \\leq  \\sum_{\\ell=0}^\\rho\\mathrm{HF}\\big({\\widetilde{E}}_\\ell, \\tau-\\ell\\big).\n\\end{equation}\n\nConsider the extension $I = {\\widetilde{I}} R \\subseteq R$. \nThen $I  = \\oplus_{\\ell=0}^{d_n-1} I_\\ell x_n^\\ell$ where $I_\\ell\\subseteq {\\overline{R}}$\nis the extension of ${\\widetilde{I}}_\\ell$ to ${\\overline{R}}$.\nFor every  $\\delta \\in \\mathbb{N}$ we have \n$\n\\mathrm{HF}(I_\\ell, \\delta) = \\sum_{\\tau = 0}^\\delta ({\\widetilde{I}}_\\ell, \\tau)$, \n $\\mathrm{HF}(E_\\ell, \\delta) = \\sum_{\\tau = 0}^\\delta ({\\widetilde{E}}_\\ell, \\tau)\n$,\ntherefore, adding the inequalities \\eqref{EqInequalityCharacterizationLex}, we obtain for every $\\rho, \\delta \\geq 0$ \n\\begin{equation}\\label{EqInequalityCharacterizationLex2}\n\\sum_{\\ell=0}^\\rho\\mathrm{HF}\\left(I_\\ell, \\delta-\\ell\\right) \\leq  \\sum_{\\ell=0}^\\rho\\mathrm{HF}\\left(E_\\ell, \\delta-\\ell\\right).\n\\end{equation}\n\nBy definition of ${\\widetilde{I}}$ we have  $\\mathrm{HF}({\\widetilde{I}}) = \\mathrm{HF}({\\widetilde{E}})$, which implies   $\\mathrm{HF}(I) = \\mathrm{HF}(E)$ and in particular $\\mathrm{HP}(I) = \\mathrm{HP}(E)$.\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (8) it follows that \n$\\mathrm{HP}(I_h) -\\mathrm{HP}(E_k)\\in \\mathbb{Z}$ for every $h,k$.\nCombining this fact with \\eqref{EqInequalityCharacterizationLex2} we deduce that\n$\n\\sum_{\\ell=0}^\\rho\\mathrm{HP}\\left(I_\\ell\\right) \\preceq  \\sum_{\\ell=0}^\\rho\\mathrm{HP}\\left(E_\\ell\\right)\n$\nfor every $\\rho \\geq 0$.\nHowever, by axiom \\hyperref[A6]{(A6)}\nwe have the opposite inequalities \n$\n\\sum_{\\ell=0}^\\rho\\mathrm{HP}\\left(I_\\ell\\right) \\succeq  \\sum_{\\ell=0}^\\rho\\mathrm{HP}\\left(E_\\ell\\right),\n$\nfor every $\\rho \\geq 0$,\nso that \n$\n\\sum_{\\ell=0}^\\rho\\mathrm{HP}\\left(I_\\ell\\right) =  \\sum_{\\ell=0}^\\rho\\mathrm{HP}\\left(E_\\ell\\right)\n$\nand hence \n$\n\\mathrm{HP}\\left(I_\\ell\\right) =  \\mathrm{HP}\\left(E_\\ell\\right)\n$\nfor every $\\ell$.\nWe claim that $I_\\ell = E_\\ell$ for every $\\ell\\in \\mathbb{N}$. \nAssume otherwise, and choose the least $\\rho$ such  that $I_\\rho \\ne E_\\rho$.\nIt follows from \\eqref{EqInequalityCharacterizationLex2} that \n$\\mathrm{HF}(I_\\rho, \\delta) \\leq\\mathrm{HF} (E_\\rho, \\delta)$ for every $\\delta$, and since  $I_\\rho, E_\\rho\\subseteq {\\overline{R}}$ are almost lex\nwe conclude that $I_\\rho \\subsetneq E_\\rho$.\nHowever, Proposition \\ref{PropositionFiniteLengthRank} yields $\\mathrm{HP}(I_\\rho) \\ne \\mathrm{HP}(E_\\rho)$, contradiction.\nWe have proved that $E = I$, that is, ${\\widetilde{E}} = {\\widetilde{I}}$, so\n${\\widetilde{E}}$ is lex as desired.\n\\end{proof}\n\n\nIn the following theorem we employ the \\emph{opposite lex order} on $R$, denoted by $<_\\mathrm{opp}$,\nthat is the lexicographic monomial order induced by the opposite order on the variables\n$x_{n+1} > x_n > x_{n-1} > \\cdots > x_1$.\nThe usual lex order is denoted by $<_{{\\mathrm{lex}}}$.\nFurthermore, let $\\mathcal{G}(-)$ denote the set of minimal monomial generators of a  monomial ideal,\nand $[\\mathcal{G}(-)]_d$ those of degree $d$.\n\n\\begin{thm}\\label{TheoremChainExpansive}\nLet $R$ be a Clements-Lindstr\\\"om ring and  $p(\\zeta)$  such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) \\ne \\emptyset$.\nLet $p'(\\zeta) = p(\\zeta) -p(\\zeta-1)$\nand  $\\widetilde{L} = {\\mathrm{Lex}}(p'(\\zeta),{\\widetilde{R}})$.\n  There exists a  chain of almost lex ideals of $R$\n $$\n E^{(0)} \\supseteq  E^{(1)} \\supseteq \\cdots \\supseteq E^{(c-1)} \\supseteq E^{(c)}\n $$\n such that  $E^{(c)}=\\mathrm{Exp}(p,R)$, \n $E^{(0)} = \\widetilde{L} R \\subseteq R$ is the extension to $R$, \n and  for each $k=0, \\ldots, c-1$ we have  $\\frac{E^{(k)}}{E^{(k+1)}} = \\Bbbk[x_{n+1}] \\mathbf{u}^{(k)}$, where $\\mathbf{u}^{(k)}$\n is the following monomial of $E^{(k)}$\n$$\n\\mathbf{u}^{(k)} = \\min_{<_\\mathrm{opp}}\\left\\{\\min_{<_{\\mathrm{lex}}} \\left[\\mathcal{G}\\big(E^{(k)}\\big)\\right]_d\\, : \\, d \\in \\mathbb{N}\\right\\}.\n$$\n\\end{thm}\n\n\n\\begin{proof}\nDenote $E = \\mathrm{Exp}(p,R)$ and ${\\widetilde{E}} = \\frac{E+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{R}}$.\nBy Proposition \\ref{PropositionExpansiveAlmostLex}  ${\\widetilde{E}}$ is a lex ideal of ${\\widetilde{R}}$, \nand by Remark \\ref{RemarkModuloXNplus1} $\\mathrm{HP}({\\widetilde{E}},\\zeta) = \\mathrm{HP}(E,\\zeta) - \\mathrm{HP}(E,\\zeta-1)$.\nThe saturation ${\\widetilde{L}} = {\\widetilde{E}} : \\mathfrak{m}_{\\widetilde{R}}^\\infty \\subseteq {\\widetilde{R}}$ is a saturated lex ideal containing ${\\widetilde{E}}$ and with $\\mathrm{HP}({\\widetilde{L}})=\\mathrm{HP}({\\widetilde{E}})$,\nso ${\\widetilde{L}}  = {\\mathrm{Lex}}(p'(\\zeta),{\\widetilde{R}})$.\n Let $E^{(0)} = \\widetilde{L} R \\subseteq R$, so\n $E\\subseteq E^{(0)} $ \n and $\\mathrm{HP}(E^{(0)})-\\mathrm{HP}(E)$ is the non-negative integer  $c= \\dim_\\Bbbk({\\widetilde{L}}\/{\\widetilde{E}})$,\n cf. Proposition \\ref{PropositionFiniteLengthRank}.\n\nWe prove the theorem by induction on $c$.\nThe case $c=0$ is trivial, so assume $c>0$.\nDenoting $E' = \\mathrm{Exp}\\big(p(\\zeta)-1,R\\big)$ and ${\\widetilde{E}}' = \\frac{E'+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{R}}$,\nwe have $E\\subseteq E'$  by axiom \\hyperref[A3]{(A3)}, \nand $\\mathrm{HP}(E'\/E) =1$.\nTaking images in ${\\widetilde{R}}$, the quotient ${\\widetilde{E}}'\/{\\widetilde{E}}$ is a 1-dimensional vector space generated by a  monomial  $\\mathbf{u}$ of ${\\widetilde{E}}'$,\nnecessarily $\\mathbf{u} \\in \\mathcal{G}(E')$.\nFurthermore, since ${\\widetilde{E}}$ is a lex ideal of ${\\widetilde{R}}$, \n$\\mathbf{u}$ must be the lowest monomial with respect to $<_{\\mathrm{lex}}$ in its graded component of ${\\widetilde{E}}'$,\nso\n$\\mathbf{u} =\\min_{<_{\\mathrm{lex}}} \\big[\\mathcal{G}\\big(E'\\big)\\big]_d$ for some $d$.\nSince ${\\widetilde{E}} : \\mathfrak{m}_{\\widetilde{R}}^\\infty = {\\widetilde{E}}' : \\mathfrak{m}_{\\widetilde{R}}^\\infty$ and the theorem holds for $E'$ by induction,\nit remains to show that \n$\\mathbf{u} = \\min_{<_\\mathrm{opp}}\\left\\{\\min_{<_{\\mathrm{lex}}}  \\big[\\mathcal{G}\\big(E'\\big)\\big]_d\\, : \\, d \\in \\mathbb{N}\\right\\}$.\nIn other words, given\n $\\mathbf{v} =\\min_{<_{\\mathrm{lex}}} \\big[\\mathcal{G}\\big(E'\\big)\\big]_{d'}$ with   $d \\ne d'$, we must show  that $\\mathbf{v} >_\\mathrm{opp} \\mathbf{u}$.\n \nLet $I \\subseteq E'$ be the  almost lex ideal such that \n${{\\widetilde{E}}'}\/{{\\widetilde{I}}}$ is the 1-dimensional vector space $\\langle \\mathbf{v}\\rangle$.\nNotice  that  $\\mathrm{HP}(I)=\\mathrm{HP}(E)$.\nLet \n$\n\\mathbf{u} = x_1^{u_1} \\cdots x_m^{u_m} \\cdots x_n^{u_n}$\nand\n$\\mathbf{v} = x_1^{v_1} \\cdots x_m^{v_m} \\cdots x_n^{v_n}\n$\nwhere $m= \\max\\{ i \\, : \\, u_i \\ne v_i\\}$.\nIn order to conclude the proof, we must show that $u_m > v_m$.\nIf $m<n$, it follows by construction of $I$ and $E$ that $I_\\ell = E_\\ell $ for all $\\ell \\ne  u_n$, \ntherefore we must also have $\\mathrm{HP}(I_{u_n}) = \\mathrm{HP}(E_{u_n})$.\nUp to replacing $I,E$ by $I_{u_n},E_{u_n}$ and repeating this process, we may assume that $m=n$.\nIf $u_m < v_m$ then \n $I_\\ell = E_\\ell $ for all $\\ell <u_m$ and $I_{u_m} \\subsetneq E_{u_m}$, \nthus $\\sum_{\\ell=0}^{u_m} \\mathrm{HP}(I_\\ell) \\prec \\sum_{\\ell=0}^{u_m} \\mathrm{HP}(E_\\ell)$ contradicting axiom  \\hyperref[A6]{(A6)}. \n\\end{proof}\n\n\n\\begin{remark}\\label{RemarkChainLex}\nThe unique lex ideal in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ (cf. Remark \\ref{RemarkUniqueLex})\nadmits a corresponding construction.\nSpecifically, \n  there are lex ideals\n $\n L^{(0)} \\supseteq  L^{(1)} \\supseteq \\cdots \\supseteq L^{(c-1)} \\supseteq L^{(c)}\n $\n of $R$\n such that  $L^{(c)}={\\mathrm{Lex}}(p(\\zeta),R)$, \n $\\, L^{(0)} = {\\mathrm{Lex}}\\big(p'(\\zeta),{\\widetilde{R}}\\big) R, \\, $ \n and    $\\frac{L^{(k)}}{L^{(k+1)}} = \\Bbbk[x_{n+1}] \\mathbf{w}^{(k)}$, but this time\n$$\n\\mathbf{w}^{(k)} = \\min_{<_{\\mathrm{lex}}} \\left[\\mathcal{G}\\big(L^{(k)}\\big)\\right]_\\delta \\qquad \\mbox{where}\\quad \\delta = \\max\\left\\{ d\\in \\mathbb{N} \\, : \\, \\left[\\mathcal{G}\\big(L^{(k)}\\big)\\right]_d \\ne \\emptyset \\right\\}.\n$$\nBy the same proof as  Theorem \\ref{TheoremChainExpansive}, it suffices to verify the following statement:\nlet $L'\\subseteq R$ be a saturated lex ideal,  \n$\\mathbf{v} = \\min_{<_{\\mathrm{lex}}} \\left[\\mathcal{G}\\big(L'\\big)\\right]_d$\nfor some $d < \\max\\left\\{ d\\in \\mathbb{N} \\, : \\, \\left[\\mathcal{G}\\big(L'\\big)\\right]_d \\ne \\emptyset \\right\\}$,\nand let $I \\subseteq E'$ be the  almost lex ideal such that \n${{\\widetilde{L}}'}\/{{\\widetilde{I}}}$ is the 1-dimensional vector space $\\langle \\mathbf{v}\\rangle$,\nthen $I$ is not lex.\nBy assumption there exists $\\mathbf{w} = \\min_{<_{\\mathrm{lex}}} \\left[\\mathcal{G}\\big(L'\\big)\\right]_\\delta$ with $\\delta>d$.\nSince $L'$ is lex, $\\mathbf{w}$ is a minimal generator, $x_{n+1}^{\\delta-d} \\mathbf{v}\\in L'$ is not a minimal generator, and the two monomials have the same degree, it follows that $\\mathbf{w} <_{{\\mathrm{lex}}} x_{n+1}^{\\delta-d} \\mathbf{v}$.\nHowever, $\\mathbf{w} \\in I$ but $x_{n+1}^{\\delta-d} \\mathbf{v}\\notin I$, so $I$ is not lex, as desired.\n\\end{remark}\n\nTheorem \\ref{TheoremChainExpansive} readily translates into an algorithm to compute $\\mathrm{Exp}(p(\\zeta)$ from $p(\\zeta)$, sketched below.\nFor the sake of completeness, we also include an algorithm to compute ${\\mathrm{Lex}}(p(\\zeta))$.\nThese algorithms have been implemented by the authors in Macaulay2 \\cite{M2}.\n\n\n\\begin{algorithm}[The expansive ideal of a Hilbert polynomial]\\label{AlgorithmExp}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ with $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)\\ne \\emptyset$.\n\\begin{itemize}\n\\item If $p(\\zeta) =0$, return $ \\mathrm{Exp}\\big(p(\\zeta),R\\big):=R$.\n\\item If $p(\\zeta) \\ne0$, let $p'(\\zeta)=p(\\zeta)-p(\\zeta-1)$,   $L^{(0)} := {\\mathrm{Lex}}\\big(p'(\\zeta),{\\widetilde{R}}\\big)R$,\n $c = p(\\zeta) - \\mathrm{HP}(R\/L^{(0)})$.\nFor each  $k = 1, \\ldots, c$ let $\\mathbf{u}_1, \\ldots, \\mathbf{u}_t$ be the minimal generators of $L^{(k-1)}$\nso that $\\mathbf{u}_t =  \\min_{<_\\mathrm{opp}}\\left\\{\\min_{<_{\\mathrm{lex}}}\\left[\\mathcal{G}\\big(H^{(k-1)}\\big)\\right]_d :  d \\in \\mathbb{N}\\right\\}.$\nSet $L^{(k)} := ( \\mathbf{u}_1, \\ldots, \\mathbf{u}_{t-1}, x_1 \\mathbf{u}_t, x_2 \\mathbf{u}_t, \\ldots, x_n \\mathbf{u}_t )$.\n\\item Return $ \\mathrm{Exp}\\big(p(\\zeta),R\\big)=L^{(c)}$.\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{algorithm}[The lex ideal of a Hilbert polynomial]\\label{AlgorithmLex}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ with $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)\\ne \\emptyset$.\n\\begin{itemize}\n\\item If $p(\\zeta) =0$, return $ {\\mathrm{Lex}}\\big(p(\\zeta),R\\big):=R$.\n\\item If $p(\\zeta) \\ne0$, let $p'(\\zeta)=p(\\zeta)-p(\\zeta-1)$,   $H^{(0)} := {\\mathrm{Lex}}\\big(p'(\\zeta),{\\widetilde{R}}\\big)R$,\n $c = p(\\zeta) - \\mathrm{HP}(R\/H^{(0)})$.\nFor each  $k = 1, \\ldots, c,\\,$ let  $  \\mathbf{w}_1, \\ldots, \\mathbf{w}_t$ be the minimal generators of $H^{(k-1)}$\nordered so that either $\\deg(\\mathbf{w}_i)<\\deg(\\mathbf{w}_{i+1})$ or  $\\deg(\\mathbf{w}_i)=\\deg(\\mathbf{w}_{i+1})$ and  $\\mathbf{w}_i>_{{\\mathrm{lex}}}\\mathbf{w}_{i+1}$.\nSet $H^{(k)} := ( \\mathbf{w}_1, \\ldots, \\mathbf{w}_{t-1}, x_1 \\mathbf{w}_t, x_2 \\mathbf{w}_t, \\ldots, x_n \\mathbf{w}_t )$.\n\\item Return $ {\\mathrm{Lex}}\\big(p(\\zeta),R\\big)=H^{(c)}$.\n\\end{itemize}\n\\end{algorithm}\n\nThe last result of this section shows that the lex point and the expansive point on $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ are as different as they can be:\nthey are almost never equal, and if they are, then there is only one strongly stable point on the Hilbert scheme.\nIn the case of $\\mathbb{P}^n$ it follows that  if $\\mathrm{Exp}(p(\\zeta),S) = {\\mathrm{Lex}}(p(\\zeta),S)$ then $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is rational, irreducible, and smooth\n\\cite{LeRo11,ReSt97}.\n\nObserve that a saturated lex $L\\subseteq R$ is necessarily of the form\n\\begin{equation}\\label{EqDescriptionLex}\nL = \\big(x_1^{a_1+1},  x_1^{a_1}x_2^{a_2+1}, \\ldots, x_1^{a_1}\\cdots x_{r-1}^{a_{r-1}}x_r^{a_r+1}\\big)\n\\end{equation}\nfor some integers $a_i \\geq 0$, $r\\leq n$, and such that $x_1^{a_1}\\cdots x_{r-1}^{a_{r-1}}x_r^{a_r+1}\\ne 0$.\nNote that some of the other generators may be 0, if we have $a_i+1 = d_i$ for some $i$.\n\n\n\\begin{prop}\\label{PropositionLexEqualExp}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta)\\in \\mathbb{Q}[\\zeta]$. \nWith notation as in Theorem \\ref{TheoremChainExpansive} and Remark \\ref{RemarkChainLex}\nwe have ${\\mathrm{Lex}}(p(\\zeta),R) = \\mathrm{Exp}(p(\\zeta),R) $ if and only if one of the following occurs\n\\begin{enumerate}\n\\item ${\\mathrm{Lex}}(p(\\zeta),R) = L^{(0)}$;\n\\item $L^{(0)} $ is generated in a single degree and ${\\mathrm{Lex}}(p(\\zeta),R) = L^{(1)}$;\n\\item $L^{(0)} $ is principal and ${\\mathrm{Lex}}(p(\\zeta),R) = L^{(2)}$;\n\\item $d_{n-1} <\\infty$ and ${\\mathrm{Lex}}(p(\\zeta),R) = \\big( x_1^{d_1-1}\\cdots x_{n-1}^{d_{n-1}} x_n^\\alpha \\big)$ for some $\\alpha \\in \\mathbb{N}$.\n\\end{enumerate}\nIn this case $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ contains only one strongly stable point.\n\\end{prop}\n\\begin{proof}\nIt is easy to check, \nusing  Theorem \\ref{TheoremChainExpansive} and Remark \\ref{RemarkChainLex},  that in each case (1), (2), (3), (4) we get \n$\\mathrm{Exp}(p(\\zeta),R) = E^{(c)} = L^{(c)} = {\\mathrm{Lex}}(p(\\zeta),R)$. \nAssume now that $L ={\\mathrm{Lex}}(p(\\zeta),R) = \\mathrm{Exp}(p(\\zeta),R)$, and its generators are as in \\eqref{EqDescriptionLex}.\nThen\n$$\nE^{(0)} = L^{(0)} =  \\big(x_1^{a_1+1},  x_1^{a_1}x_2^{a_2+1}, \\ldots, x_1^{a_1}\\cdots x_{r-2}^{a_{r-2}+1},  x_1^{a_1}\\cdots x_{r-1}^{a_{r-1}}\\big).\n$$\nThis implies that the $L^{(k)}$'s are the only almost lex $I$ such that $L^{(0)} \\subseteq I \\subseteq L$, hence $E^{(k)}=L^{(k)}$ for all $k\\leq  c$. \nIn particular, ${\\mathrm{Lex}}(p(\\zeta),R) = \\mathrm{Exp}(p(\\zeta),R)$ implies ${\\mathrm{Lex}}(p(\\zeta)-b,R) = \\mathrm{Exp}(p(\\zeta)-b,R)$ for every $b\\in \\mathbb{N}$ for which the Hilbert scheme is nonempty. \nObserve that the generators of  a saturated lex ideal  ordered as in \\eqref{EqDescriptionLex}\nare  non-decreasing in degree,  decreasing in $<_{{\\mathrm{lex}}}$, and  increasing in $<_\\mathrm{opp}$.\nIt follows from Theorem \\ref{TheoremChainExpansive} and Remark \\ref{RemarkChainLex} that,\nwhenever ${\\mathrm{Lex}}(q(\\zeta),R) = \\mathrm{Exp}(q(\\zeta),R)$ for some $q(\\zeta)$, we have\n${\\mathrm{Lex}}(q(\\zeta)+1,R) = \\mathrm{Exp}(q(\\zeta)+1,R)$ if and only if ${\\mathrm{Lex}}(q(\\zeta))$ is generated in a single degree. \nFinally notice that if ${\\mathrm{Lex}}(q(\\zeta)+1,R)$ is generated in a single degree, then ${\\mathrm{Lex}}(q(\\zeta),R)$ is necessarily principal, \nand ${\\mathrm{Lex}}(q(\\zeta)+1)$ is principal only if ${\\mathrm{Lex}}(q(\\zeta),R) = \\big( x_1^{d_1-1}\\cdots x_{n-1}^{d_{n-1}} x_n^\\alpha \\big)$ for some $\\alpha \\in \\mathbb{N}$.\nThis forces one of (1), (2), (3), or (4) to occur.\n\n\nTo prove the last statement, let $I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ be strongly stable.\nNote that   $L_0=\\mathrm{Exp}(p(\\zeta),R)_0 = {\\mathrm{Lex}}(p(\\zeta),R)_0\\subseteq {\\overline{R}}$ is both expansive and lex.\nLet $r(\\zeta) = \\mathrm{HP}({\\overline{R}}\/L_0)$.\nBy \\hyperref[A6]{(A6)} $\\mathrm{HP}(L_0) \\preceq  \\mathrm{HP}(I_0)$, i.e.\n$\\mathrm{HP}({\\overline{R}}\/I_0)=r(\\zeta)-b$ for some $b\\in \\mathbb{N}$.\nAs observed above, this  implies $\\mathrm{Exp}(r(\\zeta)-b,{\\overline{R}}) = {\\mathrm{Lex}}(r(\\zeta)-b,{\\overline{R}})$,\nthus by induction on $n$ we obtain $I_0=L_0$.\nIf case (1) holds,\nthen $L=L_0R$, as $x_n$ does not divide the generators of $L$.\nOn the other hand, $I_0R \\subseteq I$.\nWe have $\\mathrm{HP}(L_0) = \\mathrm{HP}(I_0)$ so $\\mathrm{HP}(L_0R) = \\mathrm{HP}(I_0R)$, and $\\mathrm{HP}(I) = \\mathrm{HP}(L)$, \nimplying $I = I_0R$ by Proposition \\ref{PropositionFiniteLengthRank}, so $I=L$ as desired.\nFor the other cases (2), (3), and (4), it suffices to observe that \n if  ${\\mathrm{Lex}}(q(\\zeta),R)$ is generated in a single degree  for some $q(\\zeta)$,\nthen $L(q(\\zeta)+1,R)$ is the only saturated strongly stable ideal $H \\subseteq {\\mathrm{Lex}}(q(\\zeta),R)$ with $\\mathrm{HP}(R\/H) = q(\\zeta) +1$.\n\\end{proof}\n\n\n\n\n\\section{Examples}\\label{SectionExamples}\n\nWe conclude the paper by \n exhibiting  examples of expansive  points in some Hilbert schemes, \n constructed by the methods of Section \\ref{SectionComputation},\nand numerical bounds on Betti numbers obtained by the results of Section \\ref{SectionSyzygies}.\n\nWe begin with the analysis of expansive subschemes of dimension 0.\nIt follows  by axiom \\hyperref[A4]{(A4)}\nthat the 0-dimensional subschemes  defined by  $(x_1, \\ldots, x_n)^\\delta$ are expansive  for every $\\delta\\geq 0$.\nMore generally, we can characterize all of them explicitly.\n\n\n\\begin{example}[0-dimensional subschemes]\\label{Ex0Dimension}\nLet $c\\in \\mathbb{N}$, then $\\mathrm{Exp}(c)$ is the unique almost lex ideal of $\\mathrm{Hilb}^c(\\mathrm{Proj} R)$ generated in at most two consecutive degrees. \nEquivalently,\n$\\mathrm{Exp}(c) = (x_1, \\ldots, x_n)^\\delta + I$\nwhere $\\delta = \\min\\{ d \\, : \\mathrm{HP}\\big(R\/(x_1, \\ldots, x_n)^d\\big) \\geq c\\}$ and \n$I\\subseteq R$ is an almost lex ideal generated in degree $\\delta -1$.\nNote that $I$ is necessarily generated by the first $\\mathrm{HP}\\big(R\/(x_1, \\ldots, x_n)^\\delta\\big) - c$ monomials of $[{\\widetilde{R}}]_{\\delta-1}$ in the lex order.\nThis statement follows  by induction on $c$, using the chain of ideals in Theorem \\ref{TheoremChainExpansive}.\nIn the special case of $\\mathrm{Hilb}^c(\\mathbb{P}^n)$, we recover the main result of \\cite{Va94}.\n\\end{example}\n\n\\begin{example}\nThe simplest known reducible Hilbert scheme of points is $\\mathrm{Hilb}^8(\\mathbb{P}^4)$, see\n\\cite{CaErVeVi09}.\nIt is the union of two irreducible  components of dimension 32 and 23. \nThe expansive subscheme is \n$E = (x_1, \\ldots, x_4)^3+ (x_1^2, x_1x_2, x_2^2, x_1x_3, x_2x_3, x_3^2, x_1x_4)$,\nand it lies in the intersection of the two components. \nTo verify this , consider the vector space \n $W = [{\\widetilde{S}}\/{\\widetilde{E}}]_2 = \\langle x_2x_4, x_3x_4, x_4^2\\rangle_\\Bbbk$\nand the bilinear form \n$B : \\big([{\\widetilde{S}}]_1\\otimes W \\big)^{\\otimes 2} \\rightarrow \\bigwedge^3W \\cong \\Bbbk$\ngiven by $B(\\ell_1 \\otimes q_1 , \\ell_2 \\otimes q_2) = \\ell_1\\ell_2 \\wedge q_1 \\wedge q_2$.\nThen $B$ is degenerate, since\n$B(x_2\\otimes x_2x_4 , \\ell_2\\otimes q_2 ) = x_2 \\ell_2 \\wedge x_2x_4 \\wedge q_2 = 0$ \nfor every $\\ell_2, q_2$, \nso the conclusion follows from \n \\cite[Theorem 1.3]{CaErVeVi09}.\n\\end{example}\n\n\n\\begin{example}\nWe exhibit three situations,  found in \\cite{Ra19}, \nwhere the lex point and the expansive point are the only two strongly stable points of the Hilbert scheme.\n\n(1) Let $p(\\zeta) = { \\zeta + n-2 \\choose n-2} +\\zeta +1$.\nThen $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is the union of two  irreducible components $\\mathcal{H}, \\mathcal{H}'$, \nwhose general points are respectively a line and an $(n-2)$-plane in general position, and a line intersecting an $(n-2)$-plane union an isolated point \\cite[Section 3]{Ra19}.\nThe lex point  ${\\mathrm{Lex}}(p(\\zeta)) = (x_1, x_2^2, x_2x_3, \\ldots, x_2x_{n-2}, x_2 x_{n-1}^2, x_2x_{n-1}x_n)$ lies in the interior of $\\mathcal{H}'$.\nThe expansive point \n$ \\mathrm{Exp}(p(\\zeta)) = (x_1^2, x_1x_2, \\ldots, x_1x_n, x_2^2, x_2x_3, \\ldots, x_2x_{n-1}) $\nlies in the intersection $\\mathcal{H}\\cap \\mathcal{H}'$.\n\n(2)  Let $p(\\zeta) = { \\zeta + n-2 \\choose n-2} +2$.\nThen $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is irreducible, and its general point parametrizes an $(n-2)$-plane and 2 isolated points  \\cite[Section 5.1]{Ra19}.\nWe have \n${\\mathrm{Lex}}(p(\\zeta)) = (x_1, x_2^2, x_2x_3, \\ldots, x_2x_{n-1}, x_2x_n^2)$ and\n$ \\mathrm{Exp}(p(\\zeta)) = (x_1, x_2)(x_1, \\ldots, x_{n})$.\nThe $\\mathrm{GL}(n+1)$-orbit of $ \\mathrm{Exp}(p(\\zeta))$ is the singular locus of $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$.\n\n(3) Let $p(\\zeta) = {\\zeta +n \\choose n} - {\\zeta +n-d \\choose n} +3$.\nThen $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is smooth, and its general point parametrizes a hypersurface of degree $d$ with 3 isolated points \\cite[Section 5.2]{Ra19}.\nWe have ${\\mathrm{Lex}}(p(\\zeta)) = x_1^d(x_1, x_2, \\ldots, x_{n-1}, x_n^3)$ and\n$ \\mathrm{Exp}(p(\\zeta)) = x_1^d (x_1, x_2, \\ldots, x_{n-2}, x_{n-1}^2,x_{n-1}x_n,x_{n}^2)$.\n\\end{example}\n\n\n\\begin{example}[Twisted cubics]\nThe Hilbert scheme $ \\mathrm{Hilb}^{3\\zeta+1}(\\mathbb{P}^3)$ is described in \\cite{PiSc85}. \nIt is the union of two rational smooth irreducible components $\\mathcal{H}, \\mathcal{H}'$, \nwhose general point parametrizes  respectively  a twisted cubic and a  plane cubic union a point in $\\mathbb{P}^3$.\nThere are three strongly stable points in $ \\mathrm{Hilb}^{3\\zeta+1}(\\mathbb{P}^3)$.\nThe point $(x^2,xy,y^2)$ lies in the interior of $\\mathcal{H}$,\nand it is the generic initial ideal of the twisted cubic with respect to $<_{{\\mathrm{lex}}}$.\nThe  point ${\\mathrm{Lex}}(3\\zeta+1) =( x, y^4, y^3z )$ lies in the interior of $\\mathcal{H}'$.\nFinally, $\\mathrm{Exp}(3\\zeta+1)=(x^2, xy, xz, y^3)$\n lies in the intersection $\\mathcal{H}\\cap \\mathcal{H}'$\nand gives the most degenerate curve in this Hilbert scheme, namely a line tripled in the plane with a spatial embedded point. \nThe universal deformation space of $\\mathrm{Exp}(3\\zeta+1)$ is studied in \\cite[Lemma 6]{PiSc85}\nto deduce the rationality of $\\mathcal{H}, \\mathcal{H}',$ and $ \\mathcal{H}\\cap\\mathcal{H}'$.\n\\end{example}\n\n\n\n\\begin{example}\nLet $ R = \\Bbbk[x_1, x_2, x_3, x_4, x_5]\/(x_1^2, x_2^3, x_3^3)$. \nThe upper bounds on the syzygies of $I \\in \\mathrm{Hilb}^{7\\zeta}(\\mathrm{Proj} R)$ are \n$$\n\\beta_1^S(R\/I) \\leq 7,\\quad\n\\beta_2^S(R\/I) \\leq 13, \\quad\n\\beta_3^S(R\/I) \\leq 9,\\quad\n\\beta_4^S(R\/I) \\leq 2.\n$$\nIf we ignore the data of the degree sequence and \nregard $\\mathrm{Proj} R\/I$ as a subscheme in \n$\\mathrm{Hilb}^{7\\zeta}(\\mathbb{P}^4)$,\nwe obtain the coarser bounds \n$$\n\\beta_1^S(R\/I) \\leq 19,\\quad\n\\beta_2^S(R\/I) \\leq 42, \\quad\n\\beta_3^S(R\/I) \\leq 33,\\quad\n\\beta_4^S(R\/I) \\leq 9.\n$$\n\\end{example}\n\n\n\n\\begin{example}\nLet $\\mathcal{S}\\subseteq \\mathbb{P}^4$ be a complete intersection of a quadric and a cubic hypersurface.\nUsing Theorem \\ref{TheoremLPP} we find that a 1-dimensional subscheme $\\mathcal{C}\\in \\mathrm{Hilb}^{5\\zeta+10}(\\mathcal{S})$ has syzygies bounded by \n$\n\\beta_0^S(I_\\mathcal{C}) \\leq 17,\n\\beta_1^S(I_\\mathcal{C}) \\leq 39, \n\\beta_2^S(I_\\mathcal{C}) \\leq 32,\n\\beta_3^S(I_\\mathcal{C}) \\leq 9.\n$\n\\end{example}\n\n\n\n\n\\begin{example}\nAn elliptic quartic $\\mathcal{C}\\subseteq \\mathbb{P}_{\\mathbb{C}}^3$ is the complete intersection of 2 quadric surfaces. \nFor any 0-dimensional subscheme $\\mathcal{Z}\\subseteq \\mathcal{C}$ we claim that the following bounds hold\n$$\n\\beta_0^S(I_\\mathcal{Z}) \\leq 6,\\,\n\\beta_1^S(I_\\mathcal{Z}) \\leq 9, \\,\n\\beta_2^S(I_\\mathcal{Z}) \\leq 4.\n$$\nTo see this, let $R = \\mathbb{C}[x,y,z,w]\/(x^2,y^2)$.\nIt follows from  Example \\ref{Ex0Dimension} that\n$E=\\mathrm{Exp}(\\mathrm{HP}(S\/I_\\mathcal{Z}), R)$ is either one of \n$(x,y,z), (x,y,z^2), (x,yz,z^2)$ or it has the form \n$(xyz^\\alpha, xz^{\\alpha+1+\\delta_1}, yz^{\\alpha+1+\\delta_2}, z^{\\alpha+2+\\delta_3})$\nfor some integers $\\alpha\\in \\mathbb{N}$ and $0\\leq \\delta_1 \\leq \\delta_2 \\leq \\delta_3\\leq 1$.\nThe claim follows now from Theorem \\ref{TheoremLPP} computing a resolution of $R\/E$.\nObserve that results that do not take degree sequences into account (such as those of \\cite{CaMu13} or \\cite{Va94})\ndo not yield any bounds in this example, \nsince the Betti numbers of arbitrary 0-dimensional subschemes $\\mathcal{Z}\\subseteq \\mathbb{P}^n$ are obviously unbounded. \n\\end{example}\n\n\n\n\\subsection*{Acknowledgments}\nThe second author would like to thank Ritvik Ramkumar for some helpful conversations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Setup}\n\n\nWe first evaluate our method on ImageNet. We sample a set of 1000 images from the ImageNet validation set that are initially classified correctly to avoid artificially inflating the success rate. Since the predicted probability is available for every class, we minimize the probability of the correct class as adversarial loss in untargeted attacks, and maximize the probability of the target class in targeted attacks. We sample a target class uniformly at random for all targeted attacks.\n\nNext, we evaluate SimBA in the real-world setting of attacking the Google Cloud Vision API. Due to the extreme budget required by baselines that might cost up to \\$150 per image\\footnote{The Google API charges \\$1.50 for 1000 image queries.}, here we only compare  to LFBA, which we found to be the most query efficient baseline.\n\nIn our experiments, we limit SimBA and SimBA-DCT to at most $T=10,000$ iterations for untargeted attacks and to $T=30,000$ for targeted attacks. For SimBA-DCT, we keep the first $1\/8$th of all frequencies, and add an additional $1\/32$nd of the frequencies whenever we exhaust available frequencies without succeeding. For both methods, we use a fixed step size of $\\epsilon = 0.2$.\n\n\\subsection{ImageNet results}\n\\label{sec:imagenet}\n\n\\paragraph{Success rate comparison (\\autoref{fig:methods_comparison}).}\nWe demonstrate the query efficiency of our method in comparison to the QL attack -- arguably the state-of-the-art black-box attack method to date -- by plotting the average success rate against the number of queries. Figure \\ref{fig:methods_comparison} shows the comparison for both untargeted and targeted attacks. The dotted lines show progression of average $L_2$-norm throughout optimization. Both SimBA and SimBA-DCT achieve dramatically faster increase in success rate in both untargeted and targeted scenarios. The average $L_2$-norm for both methods are also significantly lower.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig\/queries_untargeted2.pdf}\n\\includegraphics[width=0.49\\textwidth]{fig\/queries_targeted2.pdf}\n\\vspace{-1ex}\n\\caption{Histogram of number of queries required until a successful attack (over 1000 target images). SimBA-DCT is highly right skewed, suggesting that only a handful of images require more than a small number of queries. The \\emph{median} number of queries required by SimBA-DCT for untargeted attack is only 582. However, limiting to the low frequency basis results in SimBA-DCT failing to find a successful adversarial image after $60,000$ queries, whereas SimBA can achieve $100\\%$ success rate consistently.\n\\label{fig:queries_histogram}}\n\\end{figure*}\n\n\\begin{table*}[ht!]\n\\centering\n\\resizebox{0.49\\textwidth}{!}{\n\\begin{tabular}{cccc}\n\\multicolumn{4}{c}{\\large \\hspace{4ex} \\textbf{Untargeted}} \\\\\n\\hline\n\\textbf{Attack} & \\textbf{Average queries} & \\textbf{Average $L_2$} & \\textbf{Success rate}  \\\\\n\\hline\n\\multicolumn{4}{c}{\\hspace{4ex} Label-only} \\\\\n\\hline\nBoundary attack & 123,407 & 5.98 & $100\\%$ \\\\\nOpt-attack & 71,100 & 6.98 & $100\\%$ \\\\\nLFBA & 30,000 & 6.34 & $100\\%$ \\\\\n\\hline\n\\multicolumn{4}{c}{\\hspace{4ex} Score-based} \\\\\n\\hline\nQL-attack & 28,174 & 8.27 & $85.4\\%$ \\\\\nBandits-TD & 5,251 & 5.00 & $80.5\\%$ \\\\\n\\textbf{SimBA} & 1,665 & 3.98 & $98.6\\%$ \\\\\n\\textbf{SimBA-DCT} & {\\bf 1,283} & 3.06 & $97.8\\%$ \\\\\n\\hline\n\\end{tabular}\n}\n\\resizebox{0.49\\textwidth}{!}{\n\\begin{tabular}{cccc}\n\\multicolumn{4}{c}{\\large \\hspace{3ex} \\textbf{Targeted}} \\\\\n\\hline\n\\textbf{Attack} & \\textbf{Average queries} & \\textbf{Average $L_2$} & \\textbf{Success rate} \\\\\n\\hline\n\\multicolumn{4}{c}{\\hspace{4ex} Score-based} \\\\\n\\hline\nQL-attack & 20,614 & 11.39 & $98.7\\%$ \\\\\nAutoZOOM & 13,525 & 26.74 & $100\\%$ \\\\\n\\textbf{SimBA} & {\\bf 7,899} & 9.53 & $100\\%$ \\\\\n\\textbf{SimBA-DCT} & 8,824 & 7.04 & $96.5\\%$ \\\\\n\\hline\n\\end{tabular}\n}\n\\vspace{1ex}\n\\caption{Average query count for untargeted (left) and targeted (right) attacks on ImageNet. Methods are evaluated on three different metrics: average number of queries until success (lower is better), average perturbation $L_2$-norm (lower is better), and success rate (higher is better). Both SimBA and SimBA-DCT achieve close to $100\\%$ success rate, similar to other methods in comparison, but require significantly fewer model queries while achieving \\emph{lower} average $L_2$ distortion.\n\\label{table:method_comparison_imagenet}}\n\\end{table*}\n\n\\paragraph{Query distributions (\\autoref{fig:queries_histogram}).} In \\autoref{fig:queries_histogram} we plot the histogram of model queries made by both SimBA and SimBA-DCT over 1000 random images. Notice that the distributions are highly right skewed so the median query count is a much more representative aggregate statistic than average query count. These median counts for SimBA and SimBA-DCT are only 944 and 582, respectively. In the targeted case, SimBA-DCT can construct an adversarial perturbation within only $4,854$ median queries but failed to do so after $60,000$ queries for approximately $2.5\\%$ of the images. In contrast, SimBA achieves a success rate of $100\\%$ with a median query count of $7,038$.\n\nThis result shows a fundamental trade-off when selecting the orthonormal basis $Q$. Restricting to only the low frequency DCT basis vectors for SimBA-DCT results in faster average rate of descent for most images, but may fail to admit an adversarial perturbation for some images. This phenomenon has been observed by \\citet{guo2018low} for optimization-based white-box attacks as well. Finding the right spectrum to operate in on a per-image basis may be key to further improving the query efficiency and success rate of black-box attack algorithms. We leave this promising direction for future work.\n\n\n\\paragraph{Aggregate statistics (\\autoref{table:method_comparison_imagenet}).}\n\\autoref{table:method_comparison_imagenet} computes aggregate statistics of model queries, success rate, and perturbation $L_2$-norm across different attack algorithms. We reproduce the result for LFBA,  QL-attack and Bandits-TD using default hyperparameters, and present numbers reported by the original authors' papers for Boundary Attack\\footnote{Result reproduced by \\citet{cheng2018query}}, Opt-attack, and AutoZOOM. The target model is a pretrained ResNet-50 \\citep{he2016residual} network, with the exception of AutoZOOM, which used an Inception v3 \\citep{szegedy2016rethinking} network. Some of the attacks operate under the harder label-only setting (i.e., only the predicted label is observed), which may impact their query efficiency due to observation of partial information. Nevertheless, we include these methods in the table for completeness.\n\nThe three columns in the table show all the relevant metrics for evaluating a black-box attack. Ideally, an attack should succeed often, construct perturbation with low $L_2$ norm, and do so within very few queries. It is possible to artificially reduce the number of model queries by lowering success rate and\/or increasing perturbation norm. To ensure fair comparison, we enforce our methods achieve close to $100\\%$ success rate and compare the other two metrics. Note that the success rate for boundary attack and LFBA are always $100\\%$ since both methods begin with very large perturbations to guarantee misclassification and gradually reduce the perturbation norm.\n\nBoth SimBA and SimBA-DCT have \\emph{significantly lower} average $L_2$-norm than all baseline methods. For untargeted attack, our methods require 3-4x fewer queries (at $1,665$ and $1,232$, respectively) compared to the strongest baseline method -- Bandits-TD -- which only achieves a $80\\%$ success rate. For targeted attack (right table), the evaluated methods are much more comparable, but both SimBA and SimBA-DCT still require significantly fewer queries than QL-attack and AutoZOOM.\n\n\\begin{figure}[t]\n\\vspace{1ex}\n\\centering\n\\includegraphics[width=\\columnwidth]{fig\/models_comparison_both2.pdf}\n\\caption{Comparison of success rate versus number of model queries across different network architectures for untargeted SimBA (solid line) and SimBA-DCT (dashed line) attacks. Both methods can successfully construct adversarial perturbations within $20,000$ queries with high probability. DenseNet is the most vulnerable against both attacks, admitting a success rate of almost $100\\%$ after only 6,000 queries for SimBA and 4000 queries for SimBA-DCT. Inception v3 is much more difficult to attack for both methods.\n\\label{fig:network_comparison}}\n\\vspace{1ex}\n\\end{figure}\n\n\\paragraph{Evaluating different networks (\\autoref{fig:network_comparison}).}\nTo verify that our attack is robust against different model architectures, we evaluate SimBA and SimBA-DCT additionally against DenseNet-121 \\citep{huang2016densely} and Inception v3 \\citep{szegedy2016rethinking} networks. Figure \\ref{fig:network_comparison} shows success rate across the number of model queries for an untargeted attack against the three different network architectures. ResNet-50 and DenseNet-121 exhibit a similar degree of vulnerability against our attacks. However, Inception v3 is noticeably more difficult to attack, requiring more than $10,000$ queries to successfully attack with some images. Nevertheless, both methods can successfully construct adversarial perturbations against all models with high probability.\n\n\n\n\n\n\\begin{figure}[ht!]\n\\centering\\includegraphics[width=1.08\\columnwidth]{fig\/image_samples_combined.pdf}\n\\caption{Randomly selected images before and after adversarial perturbation by SimBA, SimBA-DCT and QL attack. The constructed perturbation is imperceptible for all three methods, but the perturbation $L_2$-norms for SimBA and SimBA-DCT are significantly lower than that of QL attack across all images. Our methods are capable of constructing an adversarial example in comparable or fewer queries than QL attack -- as few as 36 queries in some cases! Zoom in for detail. \\label{fig:image_samples}}\n\\vspace{-2ex}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\vspace{1ex}\n    \\includegraphics[width=0.8\\columnwidth]{fig\/GCV_comparison2.pdf}\n    \\vspace{-1ex}\n    \\caption{Plot of success rate across number of model queries for Google Cloud Vision attack. SimBA is able to achieve close to $70\\%$ success rate after only 5000 queries, while the success rate for LFBA has only reached $25\\%$. \\label{fig:gcv_succ_plot}}\n    \\vspace{-3ex}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/GCV_screenshot.pdf}\n    \\caption{Screenshot of Google Cloud Vision labeling results on a sample image before and after adversarial perturbation. The original image contains a set of camera instruments. The adversarial image successfully replaced the top concepts with guns and weapons. See supplementary material for additional samples. \\label{fig:gcv_screenshot}}\n    \\vspace{-2ex}\n\\end{figure*}\n\n\n\\paragraph{Qualitative results (\\autoref{fig:image_samples}).}\nFor qualitative evaluation of our method, we present several randomly selected images before and after adversarial perturbation by untargeted attack. For comparison, we attack the same set of images using QL attack. Figure \\ref{fig:image_samples} shows the clean and perturbed images along with the perturbation $L_2$-norm and number of queries. While all attacks are highly successful at changing the label, the norms of adversarial perturbations constructed by SimBA and SimBA-DCT are much smaller than that of QL attack. Both methods requires consistently fewer queries than QL attack for almost all images. In fact, SimBA-DCT was able to find an adversarial image in as few as 36 model queries! Notice that the perturbation produced by SimBA contains sparse but sharp differences, constituting a low $L_0$-norm attack. SimBA-DCT produces perturbations that are sparse in frequency space, and the resulting change in pixel space is spread out across all pixels.\n\n\\subsection{Google Cloud Vision attack}\n\\label{sec:gcv}\n\nTo demonstrate the efficacy of our attack against real world systems, we attack the Google Cloud Vision API, an online machine learning service that provides labels for arbitrary input images. For a given image, the API returns a list of top concepts contained in the image and their associated probabilities. Since the full list of probabilities associated with every label is unavailable, we define an untargeted attack that aims to remove the top 3 concepts in the original. We use the maximum of the original top 3 concepts' returned probabilities as the adversarial loss and use SimBA to minimize this loss. Figure \\ref{fig:gcv_screenshot} shows a sample image before and after the attack. The original image (left) contains concepts related to camera instruments. SimBA successfully replaced the top concepts with weapon-related objects with imperceptible change to the original image. Additional samples are included in the supplementary material.\n\nSince our attack can be executed efficiently, we evaluate its effectiveness over an aggregate of 50 random images. For the LFBA baseline, we define an attack as successful if the produced perturbation has an $L_2$-norm of at most the highest $L_2$-norm in a successful run of our attack. Figure \\ref{fig:gcv_succ_plot} shows the average success rate of both attacks across number of queries. SimBA achieves a final success rate of $70\\%$ after only 5000 API calls, while LFBA is able to succeed only $25\\%$ of the time under the same query budge. To the best of our knowledge, this is the first adversarial attack result on Google Cloud Vision that has a high reported success rate within very limited number of queries.\n\n\n\\section{Introduction}\n\\input{introduction.tex}\n\n\\section{Background}\n\\input{background.tex}\n\n\\section{A Simple Black-box Attack}\n\\input{method.tex}\n\n\\section{Experimental Evaluation}\n\\input{experiment.tex}\n\n\\section{Related Work}\n\\input{related.tex}\n\n\\section{Discussion and Conclusion}\n\\input{conclusion.tex}\n\n\n\n\\subsection{Hyper-parameters}\n\n\n\\paragraph{Cartesian basis.}\nA natural first choice for the set of orthogonal search directions $Q$ is the standard basis $Q = I$, which corresponds to performing our algorithm directly in pixel space. Essentially each iteration we are increasing or decreasing one color of a single randomly chosen pixel. Attacking in this basis corresponds to an $L_0$-attack, where the adversary aims to change as few pixels as possible.\n\n\\paragraph{Discrete cosine basis.} Recent work has discovered that random noise in low frequency space is more likely to be adversarial \\citep{guo2018low}. To exploit this fact, we follow \\citet{guo2018low} and propose to exploit the \\emph{discrete cosine transform} (DCT). The discrete cosine transform is an orthonormal transformation that maps signals in a 2D image space $\\mathbb{R}^{d \\times d}$ to frequency coefficients corresponding to magnitudes of cosine wave functions. In what follows, we will refer to the set of orthonormal frequencies extracted by the DCT as $Q_{\\dct}$. While the full set of directions $Q_{\\dct}$ contains $d \\times d$ frequencies, we keep only a fraction $r$ of the lowest frequency directions in order to make the adversarial perturbation in the low frequency space.\n\n\\paragraph{General basis.} In general, we believe that our attack can be used with any orthonormal basis, provided that the basis vectors can be sampled efficiently. This is especially challenging for high resolution datasets such as ImageNet since each orthonormal basis vector has dimensionality $d \\times d$. Iterative sampling methods such as Gram-Schmidt process cannot be used due to linear memory cost in the number of sampled vectors. Thus, we choose to evaluate our attack using only the standard basis vectors and DCT basis vectors for their efficiency and natural suitability to images.\n\n\\paragraph{Learning rate $\\epsilon$.}\nGiven any set of search directions $Q$, some directions may decrease $p_h(y\\mid\\mathbf{x})$ more than others. Further, it is possible for the output probability $p_{h}(y\\mid\\mathbf{x} + \\epsilon \\mathbf{q})$ to be non-monotonic in $\\epsilon$.\nIn \\autoref{fig:eps_vs_prob}, we plot the relative decrease in probability as a function of $\\epsilon$ for randomly sampled search directions in both pixel space and the DCT space. The probabilities correspond to prediction on an ImageNet validation sample by a ResNet-50 model. This figure highlights an illuminating result:\nthe probability $p_h(y\\mid\\mathbf{x}\\pm\\epsilon\\mathbf{q})$  \\emph{decreases monotonically} in $\\epsilon$ with surprising consistency (across random images and vectors $\\mathbf{q}$)!\nAlthough some directions eventually increase the true class probability, the expected change in this probability is negative with a relatively steep slope.\nThis means that our algorithm is not overly sensitive to the choice of $\\epsilon$ and the iterates will decrease the true class probability quickly. The figure also shows that search in the DCT space tends to lead  to steeper descent directions than pixel space.\nAs we show in the next section, we can tightly bound the final $L_2$-norm of the perturbation given a choice of $\\epsilon$ and maximum number of steps $T$, so the choice of $\\epsilon$ depends \\emph{primarily on budget considerations} with respect to $\\|\\delta\\|_2$.\n\n\\paragraph{Budget considerations.}\nBy exploiting the orthonormality of the basis $Q$ we can bound the norm of $\\delta$ tightly. Each iteration a basis vector is either added, subtracted, or discarded (if neither direction yields a reduction of the output probability.)\nLet $\\alpha_i \\in \\{-\\epsilon, 0, \\epsilon\\}$ denote the sign of the search direction chosen at step $t$, so\n\\begin{equation*}\n    \\delta_{t + 1} = \\delta_{t} + \\alpha_t\\mathbf{q}_{t}.\n\\end{equation*}\nWe can recursively expand $\\delta_{t + 1} = \\delta_{t } + \\alpha_t\\mathbf{q}_{t}$. In general, the\nfinal perturbation $\\delta_{T}$ after $T$ steps can be written as a sum of these individual search directions:\n\\begin{equation*}\n    \\delta_{T} = \\sum_{t=1}^{T} \\alpha_t \\mathbf{q}_{t}.\n\\end{equation*}\nSince the directions $\\mathbf{q}_{t}$ are orthogonal, $\\mathbf{q}_{t}^{\\top}\\mathbf{q}_{t'} = 0$ for any $t \\neq t'$. We can therefore compute the $L_2$-norm of the adversarial perturbation:\n\\begin{align*}\n    \\Vert \\delta_{T} \\Vert_2^2 = \\left\\| \\sum_{t=1}^{T} \\alpha_t \\mathbf{q}_{t} \\right\\|_2^2\n                          = \\sum_{t=1}^{T} \\Vert \\alpha_i \\mathbf{q}_{t} \\Vert_2^2\n\t\t\t\t\t\t\t\t\t\t\t\t\t &= \\sum_{t=1}^{T} \\alpha_t^2 \\Vert \\mathbf{q}_{t} \\Vert_2^2 \\\\\n                          &\\leq T\\epsilon^2.\n\\end{align*}\nHere, the second equality follows from the orthogonality of $\\mathbf{q}_{t}$ and $\\mathbf{q}_{t'}$, and the last inequality is tight if all queries result in a step of either $\\epsilon$ or $-\\epsilon$. Thus the adversarial perturbation has $L_2$-norm at most $\\sqrt{T} \\epsilon$ after $T$ iterations. This result holds for any orthonormal basis (e.g. $Q_{\\dct}$).\n\nOur analysis highlights an important trade-off: for query-limited scenarios, we may reduce the number of iterations by setting $\\epsilon$ higher, incurring higher perturbation $L_2$-norm. If a low norm solution is more desirable, reducing $\\epsilon$ will allow quadratically more queries at the same $L_2$-norm. A more thorough theoretical analysis of this trade-off could improve query efficiency.\n\n\n\\section{Experiment on CIFAR-10}\n\nIn this section, we evaluate SimBA and SimBA-DCT on a ResNet-50 model trained on CIFAR-10. Both attacks remain very efficient on this new dataset without any hyperparameter tuning.\n\n\\autoref{fig:queries_histogram_cifar} shows the distribution of queries required for a successful targeted attack to a random target label. In contrast to the experiment on ImageNet, the use of low frequency DCT basis is less effective due to the reduced image dimensionality. Both SimBA and SimBA-DCT perform similarly, with SimBA-DCT having a slightly heavier tail.\n\n\\autoref{table:method_comparison_cifar} shows aggregate statistics for the attack on CIFAR-10. Both methods achieve a success rate of $100\\%$ when limited to a maximum of $10,000$ queries. The actual required queries is much fewer, with both methods averaging to approximately $300$ queries, matching the median. SimBA-DCT has a slightly worse performance compared to SimBA due its query distribution having a slightly heavier tail. Nevertheless, the average query count is in line with state-of-the-art attacks on CIFAR-10. For instance, AutoZOOM achieves a mean query count of $259$ with an average $L_2$-norm of 3.53.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig\/queries_targeted_cifar.pdf}\n\\caption{Histogram of number of queries required until a successful targeted attack on CIFAR-10 (over 1000 target images).\n\\label{fig:queries_histogram_cifar}}\n\\end{figure}\n\n\\begin{table}[ht!]\n\\centering\n\\begin{tabular}{ccccc}\n\\hline\n\\textbf{Attack} & \\textbf{Average queries} & \\textbf{Median queries} & \\textbf{Average $L_2$} & \\textbf{Success rate} \\\\\n\\hline\n\\textbf{SimBA} & 322 & 297 & 2.04 & $100\\%$ \\\\\n\\textbf{SimBA-DCT} & 353 & 312 & 2.21 & $100\\%$ \\\\\n\\hline\n\\end{tabular}\n\\vspace{1ex}\n\\caption{Average query count for SimBA and SimBA-DCT on CIFAR-10.\n\\label{table:method_comparison_cifar}}\n\\end{table}\n\n\\section{Additional image samples for attack on Google Cloud Vision}\n\nTo demonstrate the generality of our evaluation of the Google Cloud Vision attack, we show 10 additional random images before and after perturbation by SimBA. In all cases, we successfully remove the top 3 original labels.\n\n\\begin{figure}[h!]\n    \\vspace{4ex}\n    \\centering\n    \\begin{subfigure}[]{0.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example1_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{0.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example2_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example3_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example4_reduced.pdf}\n    \\end{subfigure}\n\\end{figure}\n\n\\begin{figure}[h!]\n    \\ContinuedFloat\n    \\centering\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example5_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example6_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example7_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example8_reduced.pdf}\n    \\end{subfigure}\n\\end{figure}\n\n\\begin{figure}[h!]\n    \\ContinuedFloat\n    \\centering\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example9_reduced.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[]{.9\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/example10_reduced.pdf}\n    \\end{subfigure}\n    \\caption{Additional adversarial images on Google Cloud Vision.}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nThe lack of precise control over standard condensed matter systems has hindered\nthe development of experiments that can probe systematically the effects\nof strong correlations. However, the large degree of control in atomic systems, has\nmade them powerful tools for studying many condensed matter phenomena, \nand in particular novel superfluid phases~\\cite{bourdel1,regal2,chin,partridge1,kinast2,zwierlein3}.\nFor instance, a current research frontier is the study of fermion mixtures with population \nimbalance~\\cite{mit,rice,mit-2,rice-2}. \nSince the population of each component as well as their interaction strength\nare experimentally tunable, these knobs enabled the\nstudy of the BCS to BEC evolution in population imbalanced two-component fermion\nsuperfluids~\\cite{mit,rice,mit-2,rice-2}.\nIn contrast with the crossover physics found in the population balanced \ncase~\\cite{leggett,nsr,carlos}, these experiments have demonstrated the \nexistence of phase transitions between normal and superfluid phases, \nas well as phase separation between superfluid (paired) and normal (excess) \nfermions as a function of \npopulation imbalance~\\cite{liu,bedaque,sedrakian,castorina,carlson,pao,sheehy}.\n\nMotivated by these recent experiments, there has been intense theoretical\ninterest in understanding the phase diagram of population\nimbalanced mixtures~\\cite{torma,pieri,yi,silva,haque,iskin-mixture,lobo,liu-mixture,mizushima}.\nSo far, an accurate description of such mixtures is only available \nin the weak fermion attraction limit, and it is yet to be \ndeveloped for intermediate fermion attraction around unitarity, or\nfor the strong attraction limit. Some progress has been made \nin the strong attraction limit, where Fermi-Fermi mixtures were described\nas a weakly interacting Bose-Fermi mixture~\\cite{pieri, iskin-mixture, iskin-mixture2, taylor},\nhowever, the effective boson-fermion and boson-boson scattering parameters were\nobtained only in the Born approximation.\nStrictly speaking, the Bose-Fermi description is valid only in the strong attraction limit, \nbut may also provide semi-quantitative understanding of the phase diagram close to unitarity. \nThus, the main goal of this manuscript is to analyze the boson-fermion\nand boson-boson scattering parameters beyond the Born approximation\nfor arbitrary mass ratio of fermions, and use the effective Bose-Fermi mixture description\nto generate improved phase diagrams and density profiles of Fermi-Fermi \nmixtures with equal or unequal masses in the strong \nattraction limit beyond the Born and mean-field \napproximations~\\cite{iskin-mixture,pao-mixture,iskin-mixture2,duan-mixture,parish}. \n\nThe main results of this manuscript are as follows.\nFirst, we analyze three- and four-fermion scattering \nprocesses and obtain the exact boson-fermion and boson-boson scattering lengths \nas a function of mass anisotropy. Second, we use the exact boson-fermion\nand boson-boson scattering parameters to construct the phase\ndiagram for Fermi-Fermi mixtures in the strong attraction limit.\nIn addition to the normal (N) and uniform superfluid (U) phases,\nwe find two different non-uniform phase separated (PS) states: \n(1) phase separation between pure unpaired (excess) \nand pure paired fermions (molecular bosons), and\n(2) phase separation between pure excess fermions \nand a mixture of excess fermions and molecular bosons, \ndepending on the fermion-fermion scattering parameter.\nThe phase boundaries are very sensitive to the masses of the fermions, and also to\nthe boson-fermion and boson-boson interactions. \nFor equal mass mixtures, our results for the phase boundary between the PS(2) and \nthe uniform (U) phase improves on previous saddle-point \n(mean-field) results, and the quantitative changes are substantial, but not\ndramatic.\nHowever, there is a dramatic increase in quantitative differences between \nmean-field and the present results for unequal mass mixtures as the mass ratio\ndeviates from one. In particular, these differences are more pronounced when \nheavier fermions are in excess indicating the importance of taking into account\nscattering processes beyond the Born approximation.\nFurthermore, we discuss the effects of the trapping potential on the density \nprofiles of condensed and non-condensed molecular bosons, as well as excess \nfermions at zero and finite temperatures. Lastly, we discuss the implications \nof our findings to possible experiments involving Fermi-Fermi mixtures \nwith equal or unequal masses and equal or unequal populations.\n\nThe remainder of this manuscript is organized as follows.\nIn Sec.~\\ref{sec:hamiltonian}, we discuss briefly the Hamiltonian for Fermi-Fermi \nmixtures with equal or unequal masses and emphasize \nthat the system reduces effectively to a Bose-Fermi mixture \nof molecular bosons and excess fermions in the strong \nattraction limit.\nIn Sec.~\\ref{sec:a_BF}, we analyze the exact boson-fermion and boson-boson \nscattering lengths as a function of mass anisotropy, which are used to \ncalculate the resulting phase diagram of Fermi-Fermi mixtures in \nthe strong attraction limit. \nIn Sec.~\\ref{sec:BFmixture}, we discuss the stability of the effective\nBose-Fermi mixture of molecular bosons and excess fermions in three, two and one dimensions.\nWe analyze the stability of population imbalanced Fermi-Fermi mixtures \nin the strong attraction limit in Sec.~\\ref{sec:FFmixture}, and \nwe construct the phase diagrams of these systems in Sec.~\\ref{sec:phase.diagram}.\nIn Sec.~\\ref{sec:trap}, we discuss the effects of harmonic traps on the density \nprofiles of condensed and non-condensed molecular bosons, and excess fermions \nat zero and finite temperatures, and show their experimental signatures.\nLastly, we present a summary of our conclusions in Sec.~\\ref{sec:conclusions}.\n\n\n\n\n\n\n\n\n\n\n\\section{Hamiltonian}\n\\label{sec:hamiltonian}\n\nIn order to calculate the correct phase diagrams of Fermi-Fermi mixtures\nin the strong attraction limit, it is necessary to obtain first the correct\nscattering parameters between two Bose molecules (paired fermions), \nand also between a Bose molecule and an unpaired fermion. To achieve this task,\nwe begin by describing the Hamiltonian density for a mixture of fermions \n(in units of $\\hbar = k_B = 1$) as \n\\begin{eqnarray}\n\\label{eqn:hamiltonian}\nH (x) &=& \\sum_\\sigma \\bar \\psi_{\\sigma} (x) \\left[ - \\frac {\\nabla^2}{ 2m_{\\sigma} }  - \\mu_{\\sigma} \\right] \n\\psi_{\\sigma} \\nonumber \\\\\n&-& g \\bar \\psi_{\\uparrow} (x) \\bar \\psi_{\\downarrow} (x) \\psi_{\\downarrow} (x) \\psi_{\\uparrow} (x),\n\\end{eqnarray}\nwhere $\\bar \\psi_{\\sigma} (x)$ is the field corresponding to the creation\nof a fermion with pseudospin index $\\sigma$, at position and time $x \\equiv ({\\bf r}, \\tau)$.\nHere, $g > 0$ is the strength of the attractive fermion-fermion interaction \nand $\\sigma$ identifies two types of ($\\uparrow$ and $\\downarrow$) \nfermions. This notation allows the analysis of a mixture of fermions with \nequal or unequal masses, as well as equal or unequal chemical potentials.\nThroughout the manuscript, we assume that the lighter fermions are always $\\uparrow$ and\nthat the heavier fermions are always $\\downarrow$, as intuitively suggested by the direction of the arrows,\nand that the chemical potentials $\\mu_{\\sigma}$ fix the population (density) $n_{\\sigma}$ \nof each type of fermion independently.\n\nThe contact interaction Hamiltonian given in Eq.~(\\ref{eqn:hamiltonian})\ngeneralizes the equal mass and equal chemical potential Hamiltonian that is used to study \nthe BCS to BEC evolution within the functional integral formalism~\\cite{carlos}.\nThe functional integral formulation~\\cite{carlos,iskin-mixture,iskin-mixture2,taylor} captures \nsome essential features of the evolution from BCS to BEC superfluidity for Fermi mixtures \nwith equal or unequal masses as well as with equal or unequal populations. \nHowever, truly quantitative results are currently possible only in the BCS limit, \nwhere the theory is simple, but the temperatures required to reach the BCS regime \nare very low and hard to be achieved experimentally.\nIn the unitarity regime, experiments can be performed and the phase diagram \ncan be explored since the critical temperature for superfluidity is attainable,\nbut an accurate theoretical description of this regime is still lacking.\nWhile in the BEC limit, not only the temperature required to reach the BEC regime is \nexperimentally achievable, but also the theory becomes simple since the Fermi-Fermi mixtures \ncan be described effectively by a weakly interacting mixture of molecular bosons and \nexcess fermions~\\cite{pieri,iskin-mixture,iskin-mixture2,taylor}.\nHowever, the initial proposals of such effective Bose-Fermi mixtures~\\cite{pieri,iskin-mixture}\ncan provide only semi-quantitative results for comparison with experiments in the\nBEC regime since the scattering parameters for two molecular bosons and a molecular boson \nand an excess fermion are obtained only in the Born approximation.\nIn order to overcome this defficiency, we discuss next the boson-fermion \nand boson-boson scattering parameters beyond the Born approximation for arbitrary mass ratio.   \nThe correct scattering parameters will be used in Sec.~\\ref{sec:phase.diagram} \nto construct phase diagrams and density profiles for \nquantitative comparisons with experiments in the BEC regime.\n\n\n\n\n\n\n\n\n\n\\section{Boson-Fermion and Boson-Boson Scattering Lengths}\n\\label{sec:a_BF}\n\nMixtures of two types of fermions in the strong attraction limit can be described \nby effective Bose-Fermi models~\\cite{pieri,iskin-mixture,iskin-mixture2,taylor}, \nwhere fermion pairs behave as molecular bosons and interact weakly with\neach other and with excess unpaired fermions.\nScattering lengths between two molecular bosons ($a_{BB}$), and between\na molecular boson and an excess fermion ($a_{BF})$ were calculated\nin the Born approximation using many body techniques for \nequal~\\cite{carlos, pieri} and unequal masses~\\cite{iskin-mixture, iskin-mixture2}.\nHowever, these results do not agree with calculations using \nfew body techniques~\\cite{petrov,petrov-abf},\nbecause it is necessary to go beyond the Born approximation.\n\nIn the case of fermion mixtures with equal masses, while the Born approximation \nin many body theory leads to $a_{BB} = 2a_F$~\\cite{carlos} and \n$a_{BF} = 8a_F\/3$~\\cite{pieri, iskin-mixture}, the results from few body techniques are \n$a_{BB} \\approx 0.60a_F$~\\cite{petrov} and $a_{BF} \\approx 1.18a_F$~\\cite{skorniakov}.\nHowever, a diagrammatic approach beyond the Born approximation~\\cite{brodsky, levinsen} \nfor equal mass fermions recovers the few body results. \nIn this section, we generalize these diagrammatic \napproaches and analyze the boson-fermion and boson-boson scattering parameters \nfor two types of fermions with unequal masses in order \nto make quantitative predictions for experiments \ninvolving Fermi-Fermi mixtures in the strong attraction limit. \nHere, we show that the diagrammatic many body approach \nfor unequal mass fermions produce results consistent with three and\nfour body techniques that were recently used to obtain $a_{BF}$ and $a_{BB}$ as a function of \nmass ratio~\\cite{petrov, petrov-abf}. \nWe analyze the technical aspects of the boson-fermion scattering parameter for unequal\nmass fermions in some detail, since they are much easier to present, \nwhile we do not discuss in great detail the technical aspects of the boson-boson scattering\nparameter for unequal masses, as they are extremely cumbersome.\nDetailed descriptions of the boson-fermion and boson-boson scattering parameters for equal\nmass fermions can be found in the literature~\\cite{brodsky, levinsen}.\n\n\\begin{figure} [htb]\n\\psfrag{a}{\\LARGE $-\\mathbf{k}, w_F$}\n\\psfrag{b}{\\LARGE $\\mathbf{k}, w_B + \\epsilon_b$}\n\\psfrag{c}{\\LARGE $\\mathbf{k}+\\mathbf{p}, w_B - w_F + \\epsilon_b + p_0$}\n\\psfrag{d}{\\LARGE $-\\mathbf{q}, w_F - q_0$}\n\\psfrag{i}{\\LARGE $\\mathbf{q}, w_B + \\epsilon_b + q_0$}\n\\psfrag{f}{\\LARGE $\\mathbf{q}+\\mathbf{p}, w_B - w_F + \\epsilon_b + p_0 + q_0$}\n\\psfrag{g}{\\LARGE $-\\mathbf{p}, w_F - p_0$}\n\\psfrag{h}{\\LARGE $\\mathbf{p}, w_B + \\epsilon_b + p_0$}\n\\centerline{\\scalebox{0.4}{\\includegraphics{t-matrix.eps}}}\n\\caption{\\label{fig:t-matrix}\nDiagrammatic representation of the integral equation for the\nboson-fermion scattering T-matrix $T_\\mathbf{k}^{BF}(\\mathbf{p},p_0)$.\nHere, we use the notation $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\n}\n\\end{figure}\n\nWe begin our analysis by describing a zero temperature $(T = 0)$ diagrammatic representation for \nthe boson-fermion scattering T-matrix $T_\\mathbf{k}^{BF}(\\mathbf{p},p_0)$ \nas shown in Fig.~\\ref{fig:t-matrix}, where $w_F = k^2\/(2m_F)$ and $w_B = k^2\/(2m_B)$ \nare the kinetic energies for the excess fermions and molecular bosons, respectively, \nand $\\epsilon_b = - 1\/(m_{\\uparrow \\downarrow} a_F^2) < 0$\nis the binding energy of the molecular bosons.\nHere,\n\\begin{eqnarray}\nm_B &=& m_\\uparrow + m_\\downarrow \\\\\nm_{\\uparrow\\downarrow} &=& \\frac{2m_\\uparrow m_\\downarrow}{m_\\uparrow + m_\\downarrow}\n\\end{eqnarray}\nare masses of the molecular bosons and twice the reduced mass of \nthe $\\uparrow$ and $\\downarrow$ fermions, respectively.\nIn this figure, single lines represent retarded free fermion propagators\n\\begin{equation}\nG_{0,\\sigma}(\\mathbf{k}, w) = \\frac{1}{w - w_\\sigma + \\mu_\\sigma + i0^+},\n\\end{equation}\nwhere $w_\\sigma = k^2\/(2m_\\sigma)$ is the energy and $\\mu_\\sigma$ is the\nchemical potential of the $\\sigma$-type fermions.\nSimilarly double lines represent the retarded molecular boson propagators \n\\begin{equation}\n\\label{eqn:bose-propagator}\nD_0(\\mathbf{k}, w) = \\frac{4\\pi\/m_{\\uparrow\\downarrow}^{3\/2}}\n{|\\epsilon_b|^{1\/2} - (w_B - w - \\mu_\\uparrow - \\mu_\\downarrow - i0^+)^{1\/2}}\n\\end{equation}\nobtained from the expansion of the effective action~\\cite{iskin-mixture,iskin-mixture2},\nand a corresponding RPA re-summation of the fermion polarization bubbles leading to\n$\nD_0(\\mathbf{k}, w) =  - g\/[1 + g \\Gamma (\\mathbf{k}, w)]\n$\nwhere the fermion polarization bubble is \n$\n\\Gamma (\\mathbf{k}, w) = \\sum_{{\\bf q}, q_0} G_{0,\\sigma}(\\mathbf{k} + \\mathbf{q}, w + q_0 ) \nG_{0,\\sigma}(-\\mathbf{q}, -q_0 ).\n$\nIntegration over the internal momentum ${\\bf q}$ and frequency $q_0$ leads to \n$\n\\Gamma (\\mathbf{k}, w) = \n\\Gamma (0,0) + [m_{\\uparrow\\downarrow}^{3\/2} \/ (4 \\pi)]\n\\left( w_B - w  - \\mu_{\\uparrow} - \\mu_{\\downarrow} - i0^{+} \\right)^{1\/2}\n$\nwhich in combination with the definition of the fermion-fermion scattering \nlength $a_F = m_{\\uparrow\\downarrow} T^{FF} (0,0)\/ 4\\pi$, and\nthe fermion-fermion T-matrix \n$\nT^{FF} (0,0) = - g \/ [1 + g \\Gamma (\\mathbf{0}, 0)]\n$\nlead to the final result described in Eq.~(\\ref{eqn:bose-propagator}).\n\nOn the right hand side of Fig.~\\ref{fig:t-matrix}, the first diagram \nrepresents a fermion exchange process, and all other (infinitely many) \npossible processes are included in the second diagram.\nIn all diagrams, we choose $\\uparrow$ ($\\downarrow$) to label\nlighter (heavier) fermions such that lighter (heavier) fermions\nare in excess when $F \\equiv \\uparrow$ $(F \\equiv \\downarrow)$. \nThis choice spans all possible mass ratios.\nIn the following, we set $\\mu_\\sigma = 0$ since all of the calculations are \nperformed for three fermions scattering in vacuum.\nThe T-matrix $T_\\mathbf{k}^{BF}(\\mathbf{p},p_0)$ \nsatisfies the following integral equation\n\\begin{eqnarray}\n&& T_\\mathbf{k}^{BF}(\\mathbf{p},p_0) = \n- G_{0,-F}(\\mathbf{k}+\\mathbf{p}, w_B - w_F + \\epsilon_b + p_0) \\nonumber \\\\\n&& - \\sum_{\\mathbf{q}, q_0} D_0(\\mathbf{q}, w_B + \\epsilon_b + q_0) \nG_{0,F}(-\\mathbf{q}, w_F - q_0) \\times \\\\\n&& T_\\mathbf{k}^{BF}(\\mathbf{q},q_0) G_{0,-F}(\\mathbf{p} + \\mathbf{q}, w_B - w_F + \\epsilon_b + p_0 + q_0) \\nonumber,\n\\label{eqn:t-matrix}\n\\end{eqnarray}\nwhere we used $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\nOn the right hand side, we can sum over frequency $q_0$ by closing the \nintegration contour in the upper half-plane, where $T_\\mathbf{k}^{BF}(\\mathbf{q},q_0)$ \nand $D_0(\\mathbf{q}, w_B + \\epsilon_b + q_0)$ are analytic functions of $q_0$.\nSince this integration sets $q_0 = k^2\/(2m_F) - q^2\/(2m_F)$, we set \n$p_0 = k^2\/(2m_F) - p^2\/(2m_F)$ in order to have the same frequency dependence\nfor the T-matrix on both sides~\\cite{levinsen}. \nSince we are interested in the zero-range low energy s-wave scattering,\nwe average out the angular dependences of $\\mathbf{k}$ and $\\mathbf{p}$.\nWhen $k \\to 0$, the generalized integral equation defined in Eq.~(\\ref{eqn:t-matrix})\ncan be expressed in terms of the boson-fermion scattering function $a_{k \\to 0} (p)$ as\n\\begin{eqnarray}\n&& \\frac{m_{\\uparrow\\downarrow} a_0^{BF}(p)\/m_{BF}} \n{a_F^{-1} + (m_{\\uparrow\\downarrow} p^2\/m_{BF} + a_F^{-2})^{1\/2}} \n= \\frac{1}{p^2 + a_F^{-2}} - \\frac{m_B}{2\\pi m_F} \\nonumber \\\\\n&& \\int_0^\\infty \\frac{dq}{qp} \n\\ln\\left( \\frac{q^2 + 2m_F qp\/m_B + p^2 + a_F^{-2}} {q^2 - 2m_F qp\/m_B + p^2 + a_F^{-2}} \\right) \na_0^{BF}(q).\n\\label{eqn:ieqn}\n\\end{eqnarray}\nHere, we used the definition of the boson-fermion scattering length \n\\begin{equation}\na_k^{BF}(p) = \\frac{m_{BF}}{m_{\\uparrow\\downarrow}^{3\/2}}\n\\left[|\\epsilon_b|^{1\/2} + \\left(\\frac{p^2-k^2}{m_{BF}} - \\epsilon_b\\right)^{1\/2}\\right] \nT_k^{BF}(p),\n\\label{eqn:a-scattering}\n\\end{equation}\nwith its full momentum dependence,\nwhere $m_{BF}$ is twice the reduced mass of an excess fermion \nand a molecular boson given by\n\\begin{equation}\nm_{BF} = \\frac{2m_B m_F}{m_B + m_F}.\n\\end{equation}\nThe integral equation shown in Eq.~(\\ref{eqn:ieqn}) as well\nas the scattering length expression shown in Eq.~(\\ref{eqn:a-scattering}) \nreduce to the results for the equal masses~\\cite{brodsky, levinsen} when \n$m_\\uparrow = m_\\downarrow = m$.\nNotice that only the fermion exchange process is taken into account in \nthe Born approximation, and that neglecting the second term on the right hand side \nof Eq.~(\\ref{eqn:ieqn}) leads to\n$\na_0^{BF}(0) = 2(m_{BF}\/m_{\\uparrow\\downarrow})a_F\n$\nwhich is consistent with our previous results~\\cite{iskin-mixture, iskin-mixture2}. \nHowever, we need to include both terms in order to find the exact boson-fermion scattering length.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFscattering-mr.eps}}}\n\\centerline{\\scalebox{0.5}{\\includegraphics{BBscattering-mr.eps}}}\n\\caption{\\label{fig:a_BF-mr}\nThe boson-fermion scattering length $a_0^{BF}(0)\/a_F$ versus mass anisotropy \n$m_\\uparrow\/m_\\downarrow$ is shown in (a) when lighter $\\uparrow$-type (hollow circles) or \nheavier (solid circles) $\\downarrow$-type fermions are in excess.\nThe boson-boson scattering length $a^{BB}(0)\/a_F$ versus mass anisotropy \n$m_\\uparrow\/m_\\downarrow$ is shown in (b). In addition, the scattering lengths in the \nBorn approximation are shown as solid lines.\n}\n\\end{figure}\n\nNext, we solve numerically the integral equation given in Eq.~(\\ref{eqn:ieqn}),\nand obtain $a_0^{BF}(p)$ as a function of the mass anisotropy $m_\\uparrow\/m_\\downarrow$.\nThe exact solutions and the Born approximation values of\n$a_0^{BF}(0)$ are shown in Fig.~\\ref{fig:a_BF-mr}(a).\nWhen $m_\\uparrow = m_\\downarrow$, we find $a_0^{BF}(0) \\approx 1.179a_F$, which \nis consistent with results previously found for equal masses~\\cite{skorniakov, petrov, petrov-abf, brodsky, levinsen, taylor}.\nNotice that $a_0^{BF}(0)$ decreases (increases) from this value with increasing mass \nanisotropy when the lighter (heavier) fermions are in excess.\nIn addition, in the limit of $m_\\uparrow\/m_\\downarrow \\to 0$, while\n$a_0^{BF}(0) \\to a_F$ when the lighter fermions are in excess, \n$a_0^{BF}(0)$ grows rapidly when the heavier fermions are in excess.\nNotice also that the Born approximation values for $a_0^{BF}(0)$ are not in agreement\nwith the exact values for any mass anisotropy, but the general qualitative trends \nare captured by the Born approximation as can be seen from Fig.~\\ref{fig:a_BF-mr}(a).\n\nIn addition, we present results for $a^{BB}(0)$\nas a function of mass anisotropy $m_\\uparrow\/m_\\downarrow$ in Fig.~\\ref{fig:a_BF-mr}(b).\nThe exact results of the boson-boson scattering parameters for unequal mass fermions \ncan be obtained either by extending the diagrammatic approach\nfor equal mass fermions~\\cite{brodsky, levinsen}\nor by using few body techniques~\\cite{petrov}. The Born approximation values\n$\na^{BB}(0) = (m_B\/m_{\\uparrow\\downarrow}) a_F\n$\nare found in Ref.~\\cite{iskin-mixture, iskin-mixture2}.\nSince the technical details to calculate the boson-boson scattering parameters \nare quite cumbersome, and not particularly illuminating, here we just mention  \nthat the results found in the literature~\\cite{petrov} can also be \nobtained diagramatically for the unequal mass case.\nAs shown in Fig.~\\ref{fig:a_BF-mr}(b), $a^{BB}(0)$ grows very slowly as\nthe mass ratio $m_{\\uparrow}\/m_{\\downarrow}$ decreases, in constrast with\nthe much more rapid growth of the Born approximation values of $a^{BB}(0)$.\nAs expected, the Born approximation values of $a^{BB}(0)$ \nare not in agreement with the exact values for any mass anisotropy.\n\n\n\\begin{table} [htb]\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n$\\uparrow$ & $\\downarrow$ & $m_\\uparrow\/m_\\downarrow$ & $a_{BB}\/a_F$ & $a_{B\\uparrow}\/a_F$ & $a_{B\\downarrow}\/a_F$ \\\\\n\\hline\n$^{6}$Li & $^{6}$Li & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n$^{6}$Li & $^{40}$K & 0.150 & 0.695 & 1.010 & 1.984 \\\\\n$^{6}$Li & $^{87}$Sr & 0.068 & 1.123 & 1.003 & 2.512 \\\\\n$^{6}$Li & $^{171}$Yb & 0.035 & - & 1.001 & 3.023 \\\\\n$^{40}$K & $^{40}$K & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n$^{40}$K & $^{87}$Sr & 0.460 & 0.597 & 1.064 & 1.411 \\\\\n$^{40}$K & $^{171}$Yb & 0.234 & 0.629 & 1.022 & 1.723 \\\\\n$^{87}$Sr & $^{87}$Sr & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n$^{87}$Sr & $^{171}$Yb & 0.508 & 0.599 & 1.073 & 1.374 \\\\\n$^{171}$Yb & $^{171}$Yb & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{table:a} Exact boson-boson ($a_{BB}$) and boson-fermion ($a_{BF}$) \nscattering lengths for a list of two-species Fermi-Fermi mixtures.\nHere, $a_{B \\uparrow}$ ($a_{B \\downarrow}$) corresponds to excess-type \nof $\\uparrow$ ($\\downarrow$) fermions.\n}\n\\end{table}\n\nSeveral atomic gases of fermions ($^{6}$Li, $^{40}$K, $^{87}$Sr~\\cite{innsbruck}, \nand $^{171}$Yb~\\cite{fukuhara}) are being currently investigated,\nand experimental methods for studying mixtures of two types of fermions \nare being developed in several groups. Thus, anticipating future experiments \ninvolving mixtures of two types of fermions, \nwe show in Table~\\ref{table:a} the boson-fermion and boson-boson scattering lengths\nfor a few mixtures.\n\nHere, we would like to make two comments. First, it is quite remarkable that\nthe diagramatic many body approach recovers the few body results for boson-fermion and\nboson-boson scattering lengths for arbitrary mass ratio $m_{\\uparrow}\/m_{\\downarrow}$.\nThe diagramatic many body approach is performed in momentum space, while the few body \napproach is performed in real space. The two methods are equivalent because all of the possible\nscattering processes are taken into account exactly in the diagramatic many body approach \nat $T = 0$ for three or four fermions. However, the calculation of the \nscattering parameters for three or four fermions in the presence of many others \n(arbitrary number of particles) at zero or finite temperatures requires \na full many body approach, which is inevitably approximate and more difficult to implement.\nSecond, our analysis does not include the effects related to \nEfimov (three body bound) states,\nwhich may become important when analyzing the scattering parameters as a function\nof mass ratio $m_{\\uparrow}\/m_{\\downarrow}$~\\cite{petrov-abf}. In particular, the mixtures of \n$^6$Li and $^{87}$Sr or $^6$Li and $^{171}$Yb have mass ratios of  \n$m_{Li}\/m_{Sr} \\approx 0.068$ and $m_{Li}\/m_{Yb} \\approx 0.035$, which are below the critical\nratio $m_{\\uparrow}\/m_{\\downarrow} \\approx 0.073$ for the emergence of Efimov states. \n\nHaving presented the boson-fermion and boson-boson scattering lengths\nfor arbitrary mass ratio $m_{\\uparrow}\/m_{\\downarrow}$, we discuss next the\nresulting phase diagrams for Fermi-Fermi mixtures in the strong attraction\nlimit, where the system can be effectively described by a\nBose-Fermi mixture~\\cite{pieri,iskin-mixture,iskin-mixture2,taylor}\nof molecular bosons and excess unpaired fermions.\n\n\n\n\n\n\n\n\n\\section{Bose-Fermi Mixtures at Zero Temperature}\n\\label{sec:BFmixture}\n\nIn this section we use the effective Bose-Fermi mixture \ndescription~\\cite{pieri,iskin-mixture,taylor}\nto analyze the phase diagram of population imbalanced \nFermi-Fermi mixtures in the strong attraction limit. \nWe describe first the general stability conditions \nfor Bose-Fermi mixtures at zero temperature, and use \nthis connection to discuss the stability and phase \ndiagrams of Fermi-Fermi mixtures in the \nstrong attraction limit.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Bose-Fermi Mixtures}\n\\label{sec:atomic-BFmixture}\n\nThe ground state of Bose-Fermi mixtures can be described by the\nfree energy~\\cite{viverit, iskin-mixture2}\n\\begin{equation}\n{\\cal E} = {\\cal E}_B + {\\cal E}_F + \\frac{U_{BB} n_B^2}{2} + U_{BF} n_F n_B \n- \\mu_F n_F - \\mu_{B} n_B,\n\\label{eqn:energy}\n\\end{equation}\nwhich characterizes the center-of-mass degrees of freedom for a mixture\nof single-hyperfine-state bosons and fermions. \nHere $\\mu_F$ and $n_F$ ($\\mu_B$ and $n_B$) are the density and chemical \npotential of fermions (bosons), $\\epsilon_F$ is the Fermi energy \nof the fermions, and $U_{BB}$ and $U_{BF}$ are the repulsive \nboson-boson and boson-fermion interaction strengths.\nThe density of single-hyperfine-state fermions in three dimensions is given by\n$\nn_F = (1\/V) \\sum_\\mathbf{k}^{|\\mathbf{k}| < k_F} 1 = k_F^3\/(6\\pi^2),\n$\nwhere $k_F$ is the Fermi momentum and $V$ is the volume. \nThe first term in Eq.~(\\ref{eqn:energy}) is the total kinetic energy of \nbosons, which is assumed to be much smaller than all other energies, and is neglected.\nThis assumption is very good since essentially all bosons are condensed in\nthe ${\\bf k} = {\\bf 0}$ state, when the boson-boson and boson-fermion \ninteractions are weak.\nThe second term in Eq.~(\\ref{eqn:energy}) is the total kinetic energy of \nfermions, which in three dimensions is given by\n$\n{\\cal E}_F  = (1\/V) \\sum_\\mathbf{k}^{|\\mathbf{k}| < k_F} \\epsilon_{\\mathbf{k},F} \n= 3 \\epsilon_F n_F \/ 5,\n$\nwhere $\\epsilon_{\\mathbf{k},F} = |\\mathbf{k}|^2\/(2m_F)$.\n\nFrom the free energy given in Eq.~(\\ref{eqn:energy}), we obtain the fermion\nand boson chemical potentials using the condition \n$\\partial {\\cal E} \/\\partial n_{i} = 0$ with $i = F, B$, leading to\n\\begin{eqnarray}\n\\mu_F &=& \\epsilon_F + U_{BF} n_B, \\\\\n\\mu_B &=& U_{BB} n_B + U_{BF} n_F.\n\\end{eqnarray}\nThen, we use the positive definiteness of the Bose-Fermi compressibility matrix \n$\\kappa_{i,j} = \\partial \\mu_i\/\\partial n_j$,\n\\begin{eqnarray}\n\\frac{\\partial \\mu_F}{\\partial n_F} \\frac{\\partial \\mu_B}{\\partial n_B} - \n\\frac{\\partial \\mu_F}{\\partial n_B} \\frac{\\partial \\mu_B}{\\partial n_F} > 0,\n\\end{eqnarray}\nand find that bosons and fermions phase separate when the condition\n\\begin{equation}\nn_F \\ge \\frac{4\\pi^4}{3m_F^3} \\frac{U_{BB}^3}{U_{BF}^6}, \\hspace{3mm} \\textrm{(3D)}\n\\label{eqn:stab.3D}\n\\end{equation}\nis satisfied in three-dimensional systems~\\cite{viverit, iskin-mixture2}.\nTherefore, the stability of uniform superfluidity puts an upper \nlimit on the density of fermions in three-dimensions.\n\nFollowing a similar approach in lower dimensions, where\n$n_F = k_F^2\/(4\\pi)$ and ${\\cal E}_F = \\epsilon_F n_F \/ 2$ in two dimensions,\nand $n_F = k_F\/\\pi$ and ${\\cal E}_F = \\epsilon_F n_F \/ 3$ in one dimension,\nwe find that the bosons and the fermions phase separate when the conditions\n\\begin{eqnarray}\n1 &\\le& \\frac{m_F}{2\\pi} \\frac{U_{BF}^2}{U_{BB}}, \\hspace{3mm} \\textrm{(2D)} \n\\label{eqn:stab.2D} \\\\\nn_F &\\le& \\frac{m_F}{\\pi^2} \\frac{U_{BF}^2}{U_{BB}}, \\hspace{3mm} \\textrm{(1D)}\n\\label{eqn:stab.1D}\n\\end{eqnarray}\nare satisfied, respectively, for two- and one-dimensional systems.\nNotice that, the stability of uniform superfluidity puts a lower limit\nin one dimension, which is in sharp contrast with the three-dimensional result.\nFurthermore, the stability condition in two dimensions does not depend\nexplicitly on the density of fermions (see also Ref.~\\cite{tempere}). \nHowever, the results in lower dimensions have to be used with caution, \nsince quantum fluctuations are more pronounced, and may affect these stability conditions. \n\nFor an atomic Bose-Fermi mixture, we can also describe analytically a finer\nstructure of phases. There are four possible phases~\\cite{viverit}:\n(I) PS(1) where there is phase separation between pure fermions and pure bosons;\n(II) PS(2) where there is phase separation between pure fermions,\nand a mixture of fermions and bosons;\n(III) PS(3) where there is phase separation between pure bosons,\nand a mixture of fermions and bosons; and\n(IV) PS(4) where there is  phase separation between two different mixtures \nof fermions and bosons.\n\nFor a three-dimensional weakly interacting Bose-Fermi mixture, we follow Ref.~\\cite{viverit} \nand find that there are only two stable phases within the phase separation region:\n(I) PS(1) where there is phase separation between pure fermions and pure bosons, and \n(II) PS(2) where there is phase separation between pure fermions,\nand a mixture of fermions and bosons.\nWe obtain analytically the condition\n\\begin{equation}\nn_F \\ge \\frac{1125 \\pi^4}{128 m_F^4} \\frac{U_{BB}^3}{U_{BF}^6} \n- \\frac{5}{4} \\frac{U_{BB}}{U_{BF}} n_B, \\hspace{3mm} \\textrm{(3D)}\n\\label{eqn:stab2.3D}\n\\end{equation}\nfor the transition from the PS(2) to the PS(1) phase~\\cite{iskin-mixture2}.\n\nIn lower dimensions, we find that the structure of the phase diagram is quite different.\nIn two dimensions, the phase separated region consists only of \nPS(1) where there is phase separation between pure fermions and pure bosons.\nWhile, for a one-dimensional weakly interacting Bose-Fermi mixture, \nthe phase separated region consists also of two regions:\n(I) PS(1) where there is phase separation between pure fermions and pure bosons,\nand (III) PS(3) where there is phase separation between pure bosons,\nand a mixture of fermions and bosons. \nWe obtain analytically the condition\n\\begin{equation}\nn_F \\le \\frac{3 m_F}{2\\pi^2} \\frac{U_{BF}^2}{U_{BB}} \n- \\frac{U_{BB}}{U_{BF}} n_B, \\hspace{3mm} \\textrm{(1D)}\n\\label{eqn:stab2.1D}\n\\end{equation}\nfor the transition from the PS(3) to the PS(1) phase.\nNotice that the structure of the PS(3) phase in one-dimension is \nvery different from the structure of the PS(2) phase in three dimensions.\nAgain, the results in lower dimensions have to be used with caution, \nsince the quantum fluctuations are more pronounced, and may affect these stability conditions. \n\nNext, we concentrate only on the three-dimensional case, and use\nthe stability conditions found above as well as the interaction (scattering)\nparameters obtained in Sec.~\\ref{sec:a_BF} to analyze the phase\ndiagrams of Fermi-Fermi mixtures in the strong attraction limit.\n\n\n\n\n\n\n\n\n\n\\subsection{Fermi-Fermi Mixtures \\\\ in the Strong Attraction Limit}\n\\label{sec:FFmixture}\n\nTo make an analogy between Bose-Fermi mixtures and population\nimbalanced Fermi-Fermi mixtures in the strong attraction limit, \nwe identify $F \\equiv \\{\\uparrow$ or $\\downarrow\\}$ as the excess fermions.\nThis identification leads to the density of excess fermions ($n_E$) and molecular\nbosons ($n_B$) given by\n\\begin{eqnarray}\nn_E &=& n_F - n_{-F} = |n_\\uparrow - n_\\downarrow|, \\\\\nn_B &=& \\frac{n - n_E}{2} = n_{-F},\n\\end{eqnarray}\nrespectively, where $n = n_\\uparrow + n_\\downarrow$ is the total density of \n$\\uparrow$- and $\\downarrow$-type fermions.\nHere, we use $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\nFor instance, if $F \\equiv \\uparrow$ fermions are in excess, the density\nof excess fermions and molecular bosons are $n_E = n_\\uparrow - n_\\downarrow$ and \n$n_B = (n-n_E)\/2 = n_\\downarrow$, respectively, such that all $\\downarrow$-type\nfermions are paired with some of the $\\uparrow$-type fermions to form molecular bosons,\nbut there are $\\uparrow$-type fermions left unpaired.\nIt is important to emphasize that the internal degrees of freedom \n(electronic, vibrational, and rotational) of molecular bosons are\nnot explicitly considered here, in the same spirit of the description of atomic bosons \npresented in Sec.~\\ref{sec:atomic-BFmixture}, where the electronic degrees of freedom were\nalso not explicitly considered. \n\nFor three dimensions, we define the boson-boson and boson-fermion interaction strengths \n\\begin{eqnarray}\nU_{BB} &=& \\frac{4\\pi a_{BB}}{m_B} = \\frac{4\\pi \\gamma_B}{m_B} a_F, \n\\label{eqn:U_BB} \\\\\nU_{BF} &=& \\frac{4\\pi a_{BF}}{m_{BF}} = \\frac{4\\pi \\beta_F}{m_{BF}} a_F,\n\\label{eqn:U_BF}\n\\end{eqnarray}\nwhere $a_F$, $a_{BB} = \\gamma_B a_F$ and $a_{BF} = \\beta_F a_F$ are the \nfermion-fermion, boson-boson and boson-fermion scattering lengths.\nHere, $\\gamma_B = a_{BB}\/a_F$ and $\\beta_F = a_{BF}\/a_F$ are constants, \nwhich are found in Sec.~\\ref{sec:a_BF}\nas shown in Fig.~\\ref{fig:a_BF-mr} and Table~\\ref{table:a}.\nIn addition, we define the population imbalance parameter\n\\begin{equation}\nP = \\frac{N_\\uparrow - N_\\downarrow}{N_\\uparrow + N_\\downarrow} \n= \\frac{n_\\uparrow - n_\\downarrow}{n_\\uparrow + n_\\downarrow},\n\\end{equation}\nsuch that $|P| = n_E\/n$, and $n = K_F^3\/(3\\pi^2)$, where $N_\\sigma$ is the\nnumber of $\\sigma$-type fermions and $K_F$ is the Fermi \nmomentum corresponding to the total density of fermions defined\nby $K_F^3 = (k_{F,\\downarrow}^3 + k_{F,\\uparrow}^3)\/2$.\n\nUsing these definitions, the phase separation condition Eq.~(\\ref{eqn:stab.3D}) becomes\n\\begin{equation}\n|P| \\ge \\frac{\\pi^3 \\gamma_B^3 m_{BF}^6}{16 \\beta_F^6 m_B^3 m_F^3} \\lambda^3,\n\\label{eqn:psc}\n\\end{equation}\nwhere $\\lambda = 1\/(K_F a_F)$ is the scattering parameter.\nSimilarly, the condition given in Eq.~(\\ref{eqn:stab2.3D}) becomes\n\\begin{equation}\n|P| \\left(1 - \\frac{5 \\gamma_B m_{BF}}{8 \\beta_F m_B} \\right) \n\\ge \\frac{3375 \\pi^3 \\gamma_B^3 m_{BF}^6}{8192 \\beta_F^6 m_B^3 m_F^3}\\lambda^3 \n- \\frac{5\\gamma_B m_{BF}}{8 \\beta_F m_B},\n\\label{eqn:psc2}\n\\end{equation}\nfor the transition from the PS(2) to the PS(1) phase.\n\nWe would like to emphasize that these analytic expressions given in Eqs.~(\\ref{eqn:psc}) \nand~(\\ref{eqn:psc2}) are valid only in the strong attraction limit \nwhen $1\/(K_F a_F) \\gg 1$, but  \nthey may still give semi-quantitative results for $1\/(K_F a_F) \\gtrsim 1$.\nClose to the unitarity, the Bose-Fermi description of Fermi-Fermi mixtures \nin terms of molecular bosons and excess fermions is not reliable, since the binding energy \nof molecular bosons is small and the interactions between molecular bosons and excess fermions \nor between two molecular bosons may be sufficient to cause dissociation of the\nmolecules into directly scattering fermions. \nHowever, there may be an intermediate regime between unitarity and the \nstrict BEC limit where we can describe Fermi-Fermi mixtures in terms\nof a mixture of molecular bosons and excess fermions such that\nthe molecular bosons can dissociate due to boson-boson or boson-fermion interactions,\nbut be in chemical equilibrium with excess fermions. \nWhen dissociation of molecular bosons is included, the system is no longer\na binary mixture of molecular bosons and excess fermions, but a \nternary mixture of molecular bosons, dissociated bosons ($\\uparrow \\downarrow \\rightleftharpoons \\uparrow + \\downarrow$), \nand excess fermions, or effectively a ternary mixture of molecular bosons, \nand $\\uparrow$- and $\\downarrow$-type fermions. \nIn the case of ternary mixtures, there can be a large number of phase separated regimes. \nIf we confine our discussion to the equillibrium of a maximum of two phases \nof this ternary mixture then several other situations can be encountered. \nFor example, when $\\downarrow$-type fermions are in excess, a possible sequence of phases\nfor fixed population imbalance $P$ and increasing scattering parameter $1\/(K_F a_F)$ is:\n(1) Normal phase (N) of partially polarized fermions $\\to$ \n(2) mixture of molecular bosons\nand $\\uparrow$-type fermions phase separated from partially polarized normal \nfermions $\\to$ \n(3) molecular bosons phase separated from excess $\\downarrow$-type\nfermions $\\to$ \n(4) mixture of molecular bosons and $\\downarrow$-type excess fermions\nphase separated from $\\downarrow$-type excess fermions $\\to$ \n(5) coexistence of molecular bosons and $\\downarrow$-type excess fermions.\n\nTherefore, as long as Fermi-Fermi mixtures can be regarded as a binary mixture\nof non-dissociated molecular bosons and excess fermions,  \nthe expressions given in Eqs.~(\\ref{eqn:psc}) and~(\\ref{eqn:psc2}) \nmay be used as a guide for the boundaries between phase \nseparated (non-uniform) and the mixed (uniform) phases for any mixture of fermions. \nIn particular, Eqs.~(\\ref{eqn:psc}) and~(\\ref{eqn:psc2}) may serve as estimators for the phase boundaries\nof equal or unequal mass Fermi-Fermi mixtures with population imbalance, \nas discussed next. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Phase Diagrams of Fermi-Fermi Mixtures in the Strong Attraction Limit}\n\\label{sec:phase.diagram}\n\nAmong many possibilities of Fermi-Fermi mixtures (see Table~\\ref{table:a}), we focus our\nanalysis on population imbalanced mixtures of \n$^{6}$Li or $^{40}$K atoms where $m_\\uparrow = m_\\downarrow$, and\n$^{6}$Li and $^{40}$K atoms where $m_\\uparrow = 0.15 m_\\downarrow$.\n\nIn Fig.~\\ref{fig:BEC}, we show phase diagrams of population imbalance $P$\nand scattering parameter $1\/(K_F a_F)$ for equal mass mixtures \nwhen $m_\\uparrow = m_\\downarrow$, and for unequal mass mixtures \nwhen $m_\\uparrow = 0.15 m_\\downarrow$.\nIn these diagrams, we choose $\\uparrow$ ($\\downarrow$) to label\nlighter (heavier) fermions such that lighter (heavier) fermions\nare in excess when $P > 0$ $(P < 0)$. \nAlthough these diagrams are strictly valid in the strong attraction limit\nwhen $1\/(K_F a_F) \\gg  1$, they are qualitatively correct \nwhen $1\/(K_F a_F) \\gtrsim  1$ or as long as the molecular bosons are\nnot dissociated. In the later case, the system may be approximately \ndescribed as a ternary mixture of molecular bosons,\n$\\uparrow$- and $\\downarrow$-type fermions and many other phases are \npossible, as discussed in Sec.~\\ref{sec:FFmixture}.\n\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.1.eps} }}\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.0.15.eps} }}\n\\caption{\\label{fig:BEC}\nPhase diagram of population imbalance $P = (n_\\uparrow - n_\\downarrow)\/(n_\\uparrow + n_\\downarrow)$ \nversus scattering parameter $1\/(K_F a_F)$ for \n(a) equal masses when $m_\\uparrow = m_\\downarrow$, and\n(b) unequal masses when $m_\\uparrow = 0.15 m_\\downarrow$\nin the BEC side.\nWe show normal (N), uniform superfluid (U),\nand phase separated non-uniform superfluid phases PS(1) and PS(2).\n}\n\\end{figure}\n\nIn these figures, we show the following phases: \n(i) the normal (N) phase corresponding to balanced $(P = 0$) or \nimbalanced $(P \\ne 0)$ mixture of unpaired $\\uparrow$- or $\\downarrow$-type fermions;\n(ii) uniform superfluid (U) phase where paired\n(molecular bosons) and unpaired fermions coexist; \nand (iii) phase separated (PS) non-uniform superfluid phases.\nThe PS(1) region labels phase separation between pure excess fermions and \nsuperfluid molecular bosons, while the PS(2) region labels phase separation between \npure excess fermions, and a mixture of excess fermions and superfluid molecular bosons.\nThe phase boundary between U and PS(2) phases is determined from Eq.~(\\ref{eqn:psc}), \nand the phase boundary between PS(2) and PS(1) phases is determined \nfrom Eq.~(\\ref{eqn:psc2}).\nFor a fixed mass anisotropy, when $|P|$ is large, \nwe find phase transitions from PS(1) $\\to$ PS(2) $\\to$ U as \nthe interaction strength $1\/(K_F a_F)$ increases.\nHowever, when $|P|$ is very small, we find a phase transition directly from the \nPS(1) to the U phase as $1\/(K_F a_F)$ increases.\n\nWe would like to remark in passing that the phase diagrams for \nmixtures of $^6$Li and $^{87}$Sr or $^6$Li and $^{171}$Yb with mass ratios of  \n$m_{Li}\/m_{Sr} \\approx 0.068$ and $m_{Li}\/m_{Yb} \\approx 0.035$, which are below the critical\nratio $m_{\\uparrow}\/m_{\\downarrow} \\approx 0.073$ for the emergence \nof Efimov (three body bound) states are much richer, since phase separation and\ncoexistence phases involving Efimov states (trimers), molecular bosons\nand excess fermions are also present.\n\nIt is also important to emphasize that since we use the exact boson-boson and \nboson-fermion scattering lengths, our phase diagrams in the strong \nattraction limit already include fluctuation corrections beyond the Born approximation. \nFor comparison, the corresponding phase diagrams within the Born approximation \nare described in Fig.~\\ref{fig:BEC-born}, where the phase boundaries in the\npopulation imbalance $P$ versus scattering parameter $1\/(K_F a_F)$ plane are shown\nfor equal $(m_\\uparrow = m_\\downarrow)$ and unequal $(m_\\uparrow = 0.15 m_\\downarrow)$\nmass mixtures. A direct comparison of Figs.~\\ref{fig:BEC} and~\\ref{fig:BEC-born}\nshows that the results beyond the Born approximation are quantitatively \ndifferent from the saddle-point results~\\cite{pao, sheehy} in the equal mass case.\nThese quantitative differences become significantly large for unequal \nmass mixtures when heavier fermions are \nin excess~\\cite{iskin-mixture,iskin-mixture2,pao-mixture,parish} \ndue to the large sensitivity of the exact scattering parameters on \nthe mass ratio $m_{\\uparrow}\/ m_{\\downarrow}$ as shown\nin Fig.~\\ref{fig:a_BF-mr} and Table~\\ref{table:a}.\nHowever, the same phases are present in both cases, indicating that the \nBorn approximation captures the basic qualitative features, but fails\nto produce the phase boundaries quantitatively.\n\nLastly, we would like to point out the presence of several triple points\nin the phase diagrams shown in Figs.~\\ref{fig:BEC} and~\\ref{fig:BEC-born}. \nAlong the $\\vert P \\vert = 1$ lines, we find several triple points as $1\/(K_F a_F)$ increases\nwhere the fully polarized normal phases ($P = \\pm 1$) merge with (i) the partially polarized normal (N)\nand the PS(1) phase; or with (ii) the PS(1) and PS(2) phases; or with (iii) \nthe PS(2) and U phases. Furthermore, there is also an additional triple point that\noccurs for small $\\vert P \\vert$ (iv) where the phases PS(1), PS(2) and U meet.\nThe precise locations of these triple points can be obtained for any mass ratio \nand scattering parameter from Eqs.~(\\ref{eqn:psc})\nand~(\\ref{eqn:psc2}) using the equal sign $(=)$ condition.\nThe triple point for case (ii) can be obtained by setting $\\vert P \\vert = 1$ in \nEq.~(\\ref{eqn:psc2}), and the triple point for case (iii) \ncan be obtained by setting $\\vert P \\vert = 1$ in Eq.~(\\ref{eqn:psc}). \nFinally, the triple point for case (iv) can be obtained\nby using the equal sign $(=)$ condition of Eqs.~(\\ref{eqn:psc}) and~(\\ref{eqn:psc2}) and by\nsolving the two equations simultaneously.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.1-born.eps} }}\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.0.15-born.eps} }}\n\\caption{\\label{fig:BEC-born}\nPhase diagram of population imbalance $P = (n_\\uparrow - n_\\downarrow)\/(n_\\uparrow + n_\\downarrow)$ \nversus scattering parameter $1\/(K_F a_F)$ in the Born approximation for \n(a) equal masses when $m_\\uparrow = m_\\downarrow$, and\n(b) unequal masses when $m_\\uparrow = 0.15 m_\\downarrow$\nin the BEC side.\nWe show normal (N), uniform superfluid (U),\nand phase separated non-uniform superfluid phases PS(1) and PS(2).\n}\n\\end{figure}\n\nHaving analyzed the phase diagrams for non-trapped continuous systems, we discuss\nnext the effects of the trapping potential which \nare necessary to understand experiments involving ultracold \nFermi-Fermi mixtures.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Trapped Bose-Fermi Mixtures at Zero and Finite Temperatures}\n\\label{sec:trap}\n\nIn this section, we use again the simpler description of the effective Bose-Fermi mixture\nto describe trapped Fermi-Fermi mixtures in the strong attraction limit.\nFor this purpose, we present first the theory of trapped Bose-Fermi mixtures \nat zero and finite temperatures, and then discuss the density profiles \nof trapped Fermi-Fermi mixtures in the strong attraction limit using the\nrelation between the two systems described in Sec.~\\ref{sec:BFmixture}.\n\n\n\n\n\n\n\n\n\n\\subsection{Bose-Fermi mixtures}\n\\label{sec:trap.atomic}\n\nThe Hamiltonian density for a Bose-Fermi mixture in an external potential can be \nwritten as\n\\begin{eqnarray}\n\\label{eqn:bose-fermi-hamiltonian}\nH ( {\\bf r} ) & = & K_F( {\\bf r} ) + V_F( {\\bf r} ){\\hat n_F} ( {\\bf r} ) + K_B( {\\bf r} ) \n+ V_B( {\\bf r} ) {\\hat n_B} ( {\\bf r} ) \\nonumber \\\\\n& + & U_{BF} {\\hat n_B} ( {\\bf r} ) {\\hat n_F} ( {\\bf r} )  \n+ (U_{BB}\/2) {\\hat n_B} ( {\\bf r} ) {\\hat n_B}( {\\bf r} ), \n\\end{eqnarray}\nwhere $ K_i( {\\bf r} ) = \n\\psi_{i}^{\\dagger} ( {\\bf r} ) [-\\nabla^2\/(2m_i) - \\mu_i] \\psi_{i} ( {\\bf r} )$ \nrepresents the kinetic and chemical potential terms for fermions $(i \\equiv F)$ \nor bosons $(i \\equiv B)$ in a single hyperfine state. \nHere, ${\\hat n_i}({\\bf r}) = \\psi_{i}^{\\dagger}({\\bf r}) \\psi_i ({\\bf r})$\nrepresent the local density operators, and $U_{BF}$ and $U_{BB}$ represent the \nboson-fermion and boson-boson interaction. The single-hyperfine-state fermions\nare non-interacting, but obey the Pauli exclusion principle. \n\nFor simplicity, we approximate the trapping potential by an isotropic harmonic function\nwhere the potential energy is \n\\begin{eqnarray}\nV_{i} ( {\\bf r} ) = \\frac{1}{2} \\alpha_{i} r^2.\n\\end{eqnarray}\nHere, $\\alpha_{i} = m_{i} w_{i}^2$ is proportional to the trapping \nfrequency of bosons $(i \\equiv B)$ or fermions $(i \\equiv F)$, which is typically different \nfor each kind of atom. Since the potential and the interactions are isotropic,\nthe effective potentials and densities depend only on $r = \\vert {\\bf r} \\vert$.\n\nIn the presence of such trapping potentials, the bosons and fermions \nfeel the effective potentials \n\\begin{eqnarray}\nV_{B,{\\rm eff}} ( r ) &=& V_B ( r ) + 2U_{BB} n_B(r) + U_{BF} n_F(r),\n\\label{eqn:V_B} \\\\\nV_{F,{\\rm eff}} ( r ) &=& V_F (r) + U_{BF} n_B(r),\n\\label{eqn:V_F}\n\\end{eqnarray}\nrespectively, where $n_F (r)$ is the local density of fermions, and \n\\begin{equation}\nn_B(r) = n_C(r) + n_{NC}(r).\n\\end{equation}\nis the total local density of bosons.\nHere, $n_C (r)$ and $n_{NC} (r)$ are the density of condensed and non-condensed \nbosons, respectively.\n\nThe number of condensed bosons is determined from the Gross-Pitaevskii  \nequation leading to\n\\begin{equation}\nn_C (r) = \\frac{\\mu_B - V_B (r) - 2U_{BB} n_{NC}(r) - U_{BF} n_F(r)} {U_{BB}}\n\\label{eqn:n_Cr}\n\\end{equation}\nwithin the Thomas-Fermi approximation (TFA), where the kinetic energy\nof the bosons is neglected.\nThis relation is valid when the condition \n$ \\mu_B - V_B (r) - 2U_{BB} n_{NC} (r) - U_{BF} n_F(r) \\ge 0$\nis satisfied, otherwise, $ n_C(r) = 0 $. \n\nIn our analysis, we treat both the non-condensed bosons and the fermions \nas ideal gases~\\cite{amoruso} in effective potentials \ngiven in Eqs.~(\\ref{eqn:V_B}) and~(\\ref{eqn:V_F}).\nIn addition, we assume that non-condensed bosons are in thermal equilibrium\nwith condensed bosons at the same chemical potential $\\mu_B$.\nWithin these approximations, the density of non-condensed bosons and \nfermions are given by\n\\begin{eqnarray}\nn_{NC}(r) &=& \\frac{1}{V}\\sum_{\\mathbf{k}} b[\\epsilon_{\\mathbf{k},B} - \\mu_B + V_{B,{\\rm eff}}(r)], \n\\label{eqn:n_NCr} \\\\\nn_F(r) &=& \\frac{1}{V}\\sum_{\\mathbf{k}} f[\\epsilon_{\\mathbf{k},F} - \\mu_F + V_{F, {\\rm eff}}(r)],\n\\label{eqn:n_Fr}\n\\end{eqnarray}\nwhere $b(x) = 1\/[\\exp(x\/T) - 1]$ is the Bose,\nand $f(x) = 1\/[\\exp(x\/T) + 1]$ is the Fermi distribution.\nHere, $\\epsilon_{\\mathbf{k},i} = |\\mathbf{k}|^2\/(2m_{i})$ is the kinetic energy\nof bosons $(i \\equiv B)$ or fermions $(i \\equiv F)$.\n\nNotice that, at zero temperature, all bosons condense \nsuch that $n_{NC}(r) = 0$, and $n_B(r) = n_C(r)$, leading to \n\\begin{eqnarray}\nn_B(r) &=& \\frac{\\mu_B - V_B(r) - U_{BF} n_F(r)} {U_{BB}}, \\\\\nn_F(r) &=& \\frac{\\{2m_F [\\mu_F - V_F(r) - U_{BF} n_B(r) ]\\}^{3\/2} }{6\\pi^2}, \n\\end{eqnarray}\nfor the densities of bosons and fermions, respectively~\\cite{iskin-mixture2}. \nThe first expression is valid when the condition \n$\\mu_B - V_B(r) - U_{BF} n_F(r) \\ge 0$ is satisfied, \notherwise, $n_B(r) = 0$.\nThe second expression is valid when the condition \n$\\mu_F - V_F(r) - U_{BF} n_B(r) \\ge 0$ is satisfied, \notherwise, $n_F(r) = 0$.\n\nThe chemical potentials of bosons and fermions are determined from \nfixing the total number of bosons and the number of fermions, \nindependently, as follows\n\\begin{eqnarray}\nN_B &=& \\int d^3r n_B(r), \n\\label{eqn:N_B}\\\\\nN_F &=& \\int d^3r n_F(r),\n\\label{eqn:N_F}\n\\end{eqnarray}\nwhere the integration is over all space. Therefore, in order to find the density \nprofiles for condensed and non-condensed bosons, as wells as for fermions, \nwe need to solve Eqs.~(\\ref{eqn:N_B}) and~(\\ref{eqn:N_F}) \nfor $\\mu_B$ and $\\mu_F$ self-consistently.\nNext, we discuss the density profiles of Fermi-Fermi mixtures \nin the strong attraction limit using the effective Bose-Fermi description presented.\n\n\n\n\n\n\n\n\n\n\\subsection{Fermi-Fermi Mixtures \\\\ in the Strong Attraction Limit}\n\\label{sec:trap.FF}\n\nTo make the connection between Bose-Fermi mixtures and population\nimbalanced Fermi-Fermi mixtures in the strong attraction limit,\nwe identify $F \\equiv \\{\\uparrow$ or $\\downarrow\\}$ as the excess fermions.\nThis identification leads to the density of excess fermions ($n_E$) and molecular\nbosons ($n_B$) given by\n\\begin{eqnarray}\nn_E(r) &=& n_F(r) - n_{-F}(r) = |n_\\uparrow(r) - n_\\downarrow(r)|, \\\\\nn_B(r) &=& \\frac{n(r) - n_E(r)}{2} = n_{-F}(r),\n\\end{eqnarray}\nrespectively, where $n(r) = n_\\uparrow(r) + n_\\downarrow(r)$ is the total density of \n$\\uparrow$- and $\\downarrow$-type fermions.\nHere, we use $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\nFor instance, if $F \\equiv \\uparrow$ fermions are in excess, the density\nof excess fermions and molecular bosons are $n_E (r) = n_\\uparrow (r) - n_\\downarrow (r)$ \nand $n_B(r) = [n (r) - n_E (r)]\/2 = n_\\downarrow(r)$, respectively, such that \nall $\\downarrow$-type fermions are paired with some of the $\\uparrow$-type fermions \nto form molecular bosons, but there are $\\uparrow$-type fermions left unpaired.\nIn addition, we identify $\\alpha_B = \\alpha_\\uparrow + \\alpha_\\downarrow$, where\n$\\alpha_\\sigma = m_\\sigma w_\\sigma^2$ is different for different atoms.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-zero-a.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.2-a.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.35-a.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.41-a.eps} }}\n\\caption{\\label{fig:ncond}\nDensity (in units of $K_F^3$) of condensed ($n_C$) and non-condensed ($n_{NC}$) \nmolecular bosons, and excess ($n_E$) fermions versus trap radius $r\/R_{TF}$\nis shown for \n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nHere, $m_\\uparrow = m_\\downarrow$, the population imbalance parameter is $P = 0.5$ \nand the scattering length parameter is $1\/(K_F a_F) = 2$.\n}\n\\end{figure}\n\nAs an example, we look at the equal mass case $m_F = m_{\\uparrow} = m_{\\downarrow} = m$\nand $m_B = m_{\\uparrow} + m_{\\downarrow} = 2m$, and solve the self-consistent relations\nEqs.~(\\ref{eqn:N_B}) and~(\\ref{eqn:N_F}) for a two-hyperfine-state mixture of $^6$Li only\nor $^{40}$K only. In our numerical analysis, we choose \n$\\alpha_F = \\alpha_\\uparrow = \\alpha_\\downarrow = \\alpha$\nand $\\alpha_B = 2\\alpha$, scattering parameter $1\/(K_F a_F) = 2$ \nand population imbalance parameter $P = 0.5$ such that $n_\\uparrow = 3n_\\downarrow$. \nWe define a characteristic energy $E_F = K_F^2\/(2m)$ in terms of the wavevector $K_F$ \nand the fermion mass $m$ and assume it to be the known Fermi energy of a gas of non\ninteracting fermions in the trapping potential $V_F(r)$. \nWe scale the radial distance with the effective Thomas-Fermi (TF) radius $R_{TF}$ \ndefined by $E_F  = \\alpha R_{TF}^2\/2$. With this identification, the total number of trapped fermions \n$N = (E_F\/w_F)^3\/3$, with $w_F =  \\sqrt{\\alpha\/m}$, \ncan be rewritten as $N = K_F^3 R_{TF}^3\/24$ in terms of $K_F$ and $R_{TF}$.  \n\nSince we are interested also in the temperature dependence of the density\nprofiles, we recall that the critical temperature for Bose-Einstein condensation \nof a non-interacting harmonically trapped Bose gas is $T_{BEC} = w_B [ N_B \/\\zeta (3) ]^{1\/3}$, \nwhere $w_B = \\sqrt{\\alpha_B\/ m_B}$. In our Fermi-Fermi mixture the number of bosons is\n$N_B = N (1 - \\vert P \\vert )\/2$ expressed in terms of the total\nnumber of fermions $N = N_\\uparrow + N_\\downarrow$ and population imbalance \n$P = (N_\\uparrow - N_\\downarrow)\/N$.\nUsing the expression for $N_B$, the expression of $N$ in terms of $E_F$ and $w_F$,\nand that $w_B = w_F$ for equal masses, we find \n\\begin{equation}\nT_{BEC} = \\left[ \\frac{1-|P|}{6 \\zeta(3)} \\right]^\\frac{1}{3} E_F \n\\approx 0.518 (1-|P|)^{1\/3} E_F,\n\\label{eqn:tbec}\n\\end{equation}\nfor the critical BEC temperature, which is valid when $1\/(K_F a_F) \\to \\infty$. \nHere, $\\zeta(x)$ is the Zeta function and $\\zeta(3) \\approx 1.202$.\nTherefore, for $P = 0.5$, we obtain $T_{BEC} \\approx 0.41 E_F$.\nNotice that $T_{BEC}$ for population imbalance $P$ given in Eq.~(\\ref{eqn:tbec}) is a generalization \nof the results for equal populations~\\cite{griffin, perali}.\n\n\nIn Fig.~\\ref{fig:ncond}, we show the density (in units of $K_F^3$) of condensed \n($n_C$) and non-condensed ($n_{NC}$) molecular bosons, and excess ($n_E$) \nfermions as a function trap radius $r\/R_{TF}$ for four temperatures:\n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nAt zero temperature $(T = 0)$, as shown in Fig.~\\ref{fig:ncond}(a), we find that \nall of the molecular bosons are condensed, and that they are concentrated \nclose to the center of the trap.\nIn contrast, the majority of excess fermions are pushed away from the center \ntowards the edges of the trap due to the repulsive boson-fermion interaction and\nthe high concentration of condensed molecular \nbosons. Therefore, there is a clear indication of phase separation between \nmolecular bosons and excess fermions.\nWhen the temperature is increased to $T = 0.2E_F$\nshown in Fig.~\\ref{fig:ncond}(b), some of the molecular bosons are not \ncondensed. These non-condensed molecular bosons are also pushed away\nfrom the center towards the edges of the trap just like the excess fermions.\nFurther increase in temperature increases (decreases) the number of \nnon-condensed (condensed) molecular bosons as can be seen in Fig.~\\ref{fig:ncond}(c).\nFor temperatures close to $T_{BEC}$ and above, all of the molecular bosons \nbecome non-condensed as shown in Fig.~\\ref{fig:ncond}(d) \nhaving a Gaussian-like density distribution.\nSimilarly, the excess fermions also have Gaussian-like density distribution \nfor temperatures at $T_{BEC}$ and above due to the absence of the condensate. \n\nTherefore, at zero temperature, we find that the trapping potential tends to \nfavor phase separation into a PS(1)-rich phase where regions of almost pure fermions \nand almost pure bosons are separated. However, at finite temperatures, the system\ndevelops a PS(2)-rich phase where regions of almost pure fermions and of almost \nfully mixed bosons and fermions are separated. The region of coexistence of\nbosons and fermions can be further broken down into a domain of coexisting\nexcess fermions with condensed and non-condensed bosons, and into a domain of \ncoexisting excess fermions and non-condensed bosons as can be seen in \nFig.~\\ref{fig:ncond}(d). Again, if the molecular bosons are allowed to dissociate,\nthe system can be described by a ternary \nmixture as discussed in Sec.~\\ref{sec:FFmixture} and\nthe phase diagram can be even richer, especially closer to the unitarity.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-zero-b.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.2-b.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.35-b.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.41-b.eps} }}\n\\caption{\\label{fig:nupdown}\nDensity (in units of $K_F^3$) of $\\uparrow$- ($n_\\uparrow$) and $\\downarrow$-type\n($n_\\downarrow$) fermions versus trap radius $r\/R_{TF}$\nis shown for \n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nHere, $m_\\uparrow = m_\\downarrow$, population imbalance parameter is $P = 0.5$ \nand scattering length parameter is $1\/(K_F a_F) = 2$.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig:nupdown}, we show the density (in units of $K_F^3$)\nof $\\uparrow$- ($n_\\uparrow$) and $\\downarrow$-type ($n_\\downarrow$) fermions \nas a function of radius $r\/R_{TF}$ for four temperatures:\n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nAt zero temperature, as shown in  Fig.~\\ref{fig:nupdown}(a), we find that the density \nof $\\uparrow$- and $\\downarrow$-type fermions are similar close to the center of the \ntrapping potential, while some of the excess-type fermions are close to the edges.\nWhen the temperature increases to $T = 0.2E_F$\nshown in Fig.~\\ref{fig:nupdown}(b) or to $T = 0.35E_F$\nshown in Fig.~\\ref{fig:nupdown}(c), the density of $\\uparrow$- and $\\downarrow$-type\nfermions become different at the center of the trap. In addition, \nboth $\\uparrow$- and $\\downarrow$-type fermions exist towards the\nedges. At temperatures close to $T_{BEC}$ and above, the density profiles \nof $\\uparrow$- and $\\downarrow$-type fermions have the standard shapes of weakly interacting\ntrapped Fermi gases.\n\nIn this section, we have shown that the effective Bose-Fermi description of Fermi-Fermi mixtures\nis applicable in the strong attraction limit, and thus can provide good quantitative\ncomparisons to experiments in the same regime, since the\nexact boson-fermion and boson-boson scattering parameters\nwere used to obtain the phase diagrams and density profiles.\nThus, we think that it is particularly important to perform experiments\nfor different population imbalances in the strong attraction limit, \nwhere the theory is simple. The situation is somewhat more complicated \nnear unitarity where quantitative comparisons between theory and experiment \nare more difficult. Furthermore, there are also some differences between the experimental studies\nof MIT~\\cite{mit, mit-2} and Rice~\\cite{rice,rice-2} groups performed near unitarity,\nsince the shapes of their traps and the number of trapped atoms are quite different.\n\nHaving concluded the analysis of the effects of trapping potentials on \nFermi-Fermi mixtures in the strong attraction limit, next we give a summary of our conclusions.\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn summary, we used the effective Bose-Fermi mixture description to obtain\nthe phase diagrams of Fermi-Fermi mixtures with equal or \nunequal masses and equal or unequal populations in the strong attraction limit.\nFor this purpose, we analyzed first the exact boson-fermion \nand boson-boson scattering lengths as a function of mass anisotropy, \nand then we constructed the phase diagrams of Fermi-Fermi mixtures\nin the BEC regime.\n\nWe showed that three-dimensional  non-trapped fermion mixtures with population imbalance\nexhibit phase separation in addition to the normal polarized mixture of fermions \nand uniform mixture of superfluid and excess fermions. \nIn the BEC regime, we found two different non-uniform phase separated states: \nPS(1), where there is phase separation between pure unpaired (excess) \nand pure paired fermions (molecular bosons); and\nPS(2), where there is phase separation between pure excess fermions \nand a mixture of excess fermions and molecular bosons.\nFor equal mass mixtures, our results for the phase boundaries\nare quantitatively different from previous saddle-point results,\nand these quantitative differences become more pronounced \nfor unequal mass mixtures when heavier fermions are in excess \nindicating the importance of taking into account scattering processes \nbeyond the Born approximation. \n\nWe also discussed the effects of trapping potentials on the density profiles of \ncondensed and non-condensed molecular bosons, and excess fermions at zero and finite \ntemperatures. At zero temperature, we found that almost all of the condensed\nbosons are at the center of the trap, while the excess fermions are pushed to the edges\ndue to the repulsive boson-fermion interactions. At finite temperatures, we found that\nnon-condensed pairs and excess fermions are created at the center of the trap \nat the expense of an overall reduction of condensed bosons. \nFinally, at temperatures above the BEC temperature, the number of condensed bosons vanish,\nand the system becomes a mixture of weakly interacting non-condensed bosons\nand excess fermions. Finally, we discussed that our findings can provide \ngood quantitative comparisons to experiments performed in the same \nregime of validity of the theory (BEC regime), since the boson-fermion \nand boson-boson scattering parameters that enter our calculations are exact \nin the dilute limit.\n\nLastly, we think that it is important to perform experiments with Fermi-Fermi mixtures\nin the strong attraction limit (BEC regime) where the theoretical description is\nsimple. In this limit, additional superfluid and normal phases and richer density \nprofiles proposed here can be observed, and directly compared with the theory.\n\n\nWe thank NSF (DMR-0304380) for support.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nTo tackle the pressing challenge of climate change \\cite{bai2018six} and the urgent call by the United Nations to reduce the adverse environmental impact of cities \\cite{assembly2015sustainable}, there is an increasing effort to study the impact of human mobility on urban well-being dimensions, such as traffic congestion, air pollution, and carbon dioxide emissions.\nAn element that enriches the complexity of this challenge is the widespread diffusion of GPS navigation apps such as TomTom, Google Maps, and Waze, which use routing algorithms, heuristics and AI to suggest the best path to reach a user's desired destination. \nAlthough undoubtedly useful, particularly when exploring an unfamiliar city,\nGPS navigation apps may also cause several issues in the urban environment: since they are typically optimised to keep an individual's trip as short as possible, they do not care about collective effects on the city, such as whether the traffic can be absorbed by the streets, compromises safety or creates more pollution \\cite{debaets2014route, macfarlane2019when, siuhi2016opportunities}. \nMany documented cases show that GPS navigation apps may create chaos: for example, they sometimes divert heavy traffic through side roads in nearby towns, with such an impact on the residents' life that experts stated that these apps are a new agent claiming a ``right to the city\"~\\cite{leonia, fisher2022algorithms}.\n\nBeyond the anecdotal, preliminary research show that the impact of navigation apps on the urban environment is mixed \\cite{ericsson2006optimizing, samaras2016quantification}. \nOn the one hand, navigation apps may provide benefits in mitigating carbon dioxide emissions \\cite{arora2021quantifying}; on the other hand, they may increase the population exposure to pollution in densely-populated areas \\cite{perezprada2017managing}. \nOverall, existing studies are sporadic and yield contradictory results, leading to a picture of the navigation apps' impact on the urban environment that is mainly unclear and incomplete. \nIn particular, the literature still lacks a rigorous framework to assess and compare the impact of navigation apps on urban well-being.\n\nIn this paper, we design a simulation framework -- TraffiCO$_2$ -- to assess the impact of GPS navigation apps on urban well-being in terms of carbon dioxide (CO$_2$) emissions.\nThe framework uses GPS data to reconstruct a city's mobility demand, and navigation apps' APIs to obtain routing suggestions (paths on the road network) for a set of vehicles' origin and destination.\nThen, TraffiCO$_2$ relies on a traffic simulator that considers all aspects of vehicular mobility (e.g., jams, queues at traffic lights, roads capacity), to generate trajectories that describe when each vehicle visits each road in its path.\nFinally, the framework use an emission model to estimate CO$_2$ emissions for each segment in the road network.\n\nWe apply TraffiCO$_2$ to the city of Milan (Italy) to study how emissions changes as we increase the fraction of vehicles that follow navigation apps' routing.\nWe find that settings in which either all vehicles or none of them follow a navigation apps' routing suggestion lead to the highest amount of CO$_2$ emissions. \nThese settings also correspond to the most uneven distribution of the CO$_2$ emissions across the roads, with a few roads suffering the greater quantity of emissions.\nIn contrast, a scenario where just a fraction of the vehicles (around 50\\%) follows navigation apps' suggestion and the remaining part follow a randomly perturbed fastest path, reduces the vehicles' overall emissions and distributes them more evenly on the road network. \nWe also find that the navigation apps' routing affects the spatial distribution of emissions, making Milan's external ring road more polluted while unloading the internal roads from the emissions.\nNotably, adding perturbation to the vehicles' path is beneficial, as it reduces the overall emissions, distributes them more evenly, and reduces the vehicles' average travel time.\n\nOur simulation framework is a useful tool to assess and compare routing strategies, helping drivers, institutions, and policymakers understand the impact of navigation apps on the urban environment. \nMoreover, TraffiCO$_2$ is a first step towards designing and testing next-generation routing principles that may increase urban well-being while satisfying individual needs. \n\n\n\n\n\n\n\n\\subsection*{Open Source}\nWe provide the implementation of TraffiCO$_2$, the code and the link to the data to reproduce our study at \\url{https:\/\/bit.ly\/traffico2_gh}.\n\n\\section{Related Work}\n\\subsection*{Environmental impact of vehicular traffic}\nThe environmental impact generated by vehicles (e.g., air pollution) is becoming increasingly evident in urban environments.\nExisting methods to quantify vehicles' emissions range between two extremes. On the one hand, some approaches rely on measurements performed on small samples of vehicles with high spatio-temporal resolutions, such as those coming from particulate sensors~\\cite{desouza2020} or portable emissions measurement systems (PEMS)~\\cite{chong2020,lujan2018}. \nThese sensors measure emissions in real-world driving conditions, producing accurate estimates but hardly generalisable patterns due to the limited sample size. For example, two studies~\\cite{lujan2018,chong2020} analyse emissions from PEMS of one and three vehicles, finding that the highest emissions are associated with the urban part of the route, flat roads, and low speed.\n\nOn the other hand, some studies cover a region's entire fleet, such as those using odometer readings from annual safety inspections.\nThese data describe each vehicle's age, fuel type, engine volume, and mileage, used in macroscopic models to estimate annual emissions.\nTwo studies~\\cite{chatterton2015use,diao2014vehicle} use odometer readings to compute mean annual emissions for UK postcode areas and explore the built-environment effects (e.g., work accessibility) on the vehicles' annual miles travelled in Boston.\nUnfortunately, odometer readings miss critical information such as instantaneous speed and acceleration~\\cite{kancharla2018incorporating,choudhary2016urban,ferreira2015impact,zheng2017influence}, making it challenging to track emissions over time and map them to suburban areas.\n\nSomewhere between these two extremes lie works that use GPS traces, which describe human mobility in great detail~\\cite{luca2020deep,barbosa2018human} and can cover a representative fraction of the vehicle fleet~\\cite{pappalardo2013understanding, bohm2021improving}. \nThese data allow computing instantaneous speed and acceleration, which can be used within microscopic models to obtain emissions estimates in high spatio-temporal resolution.\nSeveral studies use GPS traces to analyse the vehicles' emissions at different spatio-temporal scales~\\cite{bohm2021improving,nyhan2016,liu2019}, investigate the relationship between emissions and the urban environment~\\cite{reznik2018}, vehicle miles travelled and fuel consumption~\\cite{wang2014using}, or trip rates and travel mode choice~\\cite{cervero1997travel}. \nOther studies concentrate on congestion-related emissions~\\cite{gately2017urban} or braking~\\cite{chen2020mining}, emissions associated with ride-hailing~\\cite{sui2019gps} and bus stops' positioning~\\cite{yu2020mobile}, the impact of urban policies~\\cite{rahman2017tribute}, methods for emission modelling~\\cite{zhu2020high,aziz2018novel}, and air quality monitoring~\\cite{desouza2020}.\n\n\\subsection*{Microscopic traffic simulation} \nMicroscopic traffic simulators are crucial to simulate vehicular traffic given a mobility demand.\nA notable example is SUMO (Simulation of Urban MObility), an open-source, portable, and multi-modal tool designed to handle traffic simulations on road networks~\\cite{Microscopic2018Lopez, alazzawi2018simulating, krajzewicz2012recent}. \nSUMO allows controlling several aspects of traffic, from fuel consumption to vehicle emissions and routing strategies.\nSeveral works use SUMO to study the impact of vehicular traffic on the urban environment.\nAlazzawi et al.~\\cite{alazzawi2018simulating} simulate the introduction of a fleet of automated shared vehicles into Milan, finding that a fleet of 9500 of these vehicles helps mitigate traffic congestion and emissions.\nMalik et al.~\\cite{malik2019evaluation} propose a traffic system that re-routes an emergency vehicle in case of traffic jams to reduce travel time and pollution. \nKrajzewicz et al.~\\cite{krajzewicz2005simulation} use optical information systems to optimise traffic lights over junctions better than traditional approaches.\nZubillaga et al.~\\cite{zubillaga2014measuring} compare the traditional traffic-light coordination (green-wave method) with a self-organising method that adapts to traffic demands, showing how the latter is way better. \n\n\\subsection*{Urban routing and impact of navigation systems}\nPeople's natural routing choices may significantly deviate from the optimal route.\nThe origin of these sub-optimal human routes may lie in several factors, e.g., the environment in which one grew up \\cite{coutrot2022entropy}, the subjective perception of space \\cite{norman2005perception}, the presence of landmarks \\cite{foo2005do}, and even the usage of an electric vehicle \\cite{jensen2020route}.\n\nZhu et al.~\\cite{zhu2015do} find that only $34\\%$ of trips, mostly very short and very long ones, follow the shortest time path.\nLima et al.~\\cite{lima2016understanding} find that $53\\%$ of the drivers' routes are not optimal and that most drivers use a few preferred routes for their journeys.\nXu et al.~\\cite{xu2021understanding} find similar results using location-based services data, which are less accurate but more pervasive than vehicular trajectory data.\nBongiorno et al.~\\cite{bongiorno2021vector} find that people increasingly deviate from the shortest path as the distance between origin and destination increases. \nSimilarly, Manley et al.~\\cite{manley2015shortest} show that people's route choice results from multiple decisions made at each anchor of the route.\nGiven the evident uncertainty in people's routing strategies, the impact of navigation apps' routing suggestions on the urban environment is not negligible~\\cite{macfarlane2019when}.\nFor this reason, some effort has been put into developing ``environmentally-friendly'' navigation apps that minimise fuel consumption instead of travel time~\\cite{barth2007environmentally}.\nStudies on the effects of this eco-routing, either as fuel savings~\\cite{ericsson2006optimizing} or system-wide impact~\\cite{ahn2013network,samaras2016quantification}, find that the urban impact of navigation apps is mixed: green navigation apps can reduce CO$_2$ emissions but increase the population exposure to nitrogen oxides~\\cite{perezprada2017managing}. \nFollowing the same research line, Mehrvarz et al.~\\cite{mehrvarz2020optimal} find that the fastest route suggested by navigation apps may not optimise fuel consumption.\nIn contrast, Arora et al.~\\cite{arora2021quantifying} show that following Google Maps' routing strategy saves $1.7\\%$ of CO$_2$ emissions and $6.5\\%$ travel time. \n\n\\subsection*{Position of our work}\nExisting studies are sporadic and yield contradictory results, and the impact of navigation apps on urban well-being is mainly unclear and incomplete.\nMoreover, the existing literature lacks a rigorous framework to assess and compare the various navigation apps and routing strategies.\nWe fill this gap by proposing TraffiCO$_2$, a simulation framework to estimate the adverse impact of different routing strategies on total CO$_2$ emissions and their distribution of a city's road network.\nThus, our work extends the branch of literature on sustainable urban mobility by providing a rigorous experimental framework to disentangle the impact of routing criteria on the urban environment. \n\n\\begin{figure*}[htb!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{imgs\/Navigators4.pdf}\n    \\caption{\nSchema of the TraffiCO$_2$ simulation framework.\n(a) The city is split into squared tiles using scikit-mobility \\cite{scikitmob}. \n(b.1) Real data are used to estimate mobility flows (OD matrix) within the city. (b.2-b.3) A trip is created by selecting at random an origin-destination pair from the OD matrix and two edges on the road network.\n(b.5) Some routing algorithm is used to convert each trip into a path on the road network. \n(c) Steps b.1-b.5 are repeated $N$ times ($N$ = number of vehicles) to obtain a multiset of routed paths.\n(d) A traffic simulator (SUMO) is used to simulate the urban traffic generated by the routed paths (e).}\n    \\label{fig:schema}\n\\end{figure*}\n\n\\section{Simulation Framework}\n\\label{sec:simulation_framework}\nWe design a simulation framework -- TraffiCO$_2$ -- to generate a realistic urban traffic considering different routing strategies (Figure~\\ref{fig:schema}).\nFirst, we use real data to generate a \\textit{mobility demand} describing trips (origin-destination pairs) in an urban environment.\nSecond, we transform each trip into a path on the road network using some routing algorithm, obtaining a multiset of \\textit{routed paths}.\nThird, we use an agent-based model (SUMO) that, considering realistic aspects of vehicular mobility (e.g., jams, traffic lights, slowdowns), simulates an \\textit{urban traffic} based on the multiset of routed paths.\nFinally, we compute the vehicles' travel time and the CO$_2$ emissions on each road from the trajectories generated by SUMO.\n\n\n\\subsection{Road Network} \n\\label{sec:road_network}\nIt describes the road infrastructure where the vehicles move during the simulation. \nThe road network is a directed graph $G=(V, E)$, where $V$ is the set of nodes representing road intersections and $E$ is the set of edges representing roads. \nBoth nodes and edges may have attributes, such as: traffic lights, number of lanes, road speed limit and type (e.g., motorway, secondary road).\nThese attributes are used by SUMO to simulate realistic aspects of vehicular mobility.\n\n\n\\subsection{Mobility Demand}\n\\label{sec:mobility_demand}\nThe \\emph{mobility demand} $D = \\{T_1, \\dots, T_N\\}$ is a multiset of $N$ trips (one per each vehicle) within an urban environment. \nA single trip $T_v=(o,d)$ for a vehicle $v$ is defined by its origin location $o$ and destination location $d$.\nTo compute $D$, we first divide the urban environment into squared tiles of a given side (Figure \\ref{fig:schema}a). Second, we use real mobility data to estimate the flows between the tiles, thus obtaining an origin-destination matrix $M$ where an element $m_{o, d}\\in M$ describes the number of vehicles' trips that start in tile $o$ and end in tile $d$ (Figure \\ref{fig:schema}b.1). \n\nThen, we iterate $N$ times the following procedure. A vehicle's $v$ trip is a pair $T_v=(e_o, e_d)$ generated by selecting at random a matrix element $m_{o,d} \\in M$ with a probability $p_{o, d} \\propto m_{o, d}$ and uniformly at random two edges $e_o, e_d \\in E$ within tiles $o$ and $d$, respectively (Figure \\ref{fig:schema}b.2, b.3). \n\n\\subsection{Paths generation}\n\\label{sec:paths_generation}\nWe translate the $N$ trips in $D$ into $N$ paths obtaining a multiset $\\overline{D}$ of \\emph{routed paths} within the urban environment. \nEach path $P_v(e_o,e_d, R) {=} (e_o, \\dots, e_d)$ of a vehicle $v$\nis a sequence of edges on the road network connecting $e_o$ and $e_d$ (Figure \\ref{fig:schema}b.5), obtained by some routing algorithm $R$. \nWhen a vehicle's path is generated by a routing algorithm $R$, we say that the vehicle is $R$-routed.\nIn $\\overline{D} = {\\{P_1, \\dots, P_N\\}}$, the routed paths (Figure \\ref{fig:schema}c) are generated independently by (different) routing algorithms. \n\n\\subsection{Traffic Simulation}\n\\label{sec:traffic_simulation}\nWe simulate the vehicular traffic generated by the routed paths in $\\overline{D}$ using SUMO (Simulation of Urban MObility) \\cite{Microscopic2018Lopez, alazzawi2018simulating, krajzewicz2012recent} (Figure~\\ref{fig:schema}d). \nSUMO explicitly models each vehicle's physics and dynamics, including their routes through the road network, allowing us to simulate vehicular traffic realistically, including traffic jams, queues at traffic lights, and slowdowns due to heavy traffic.\nSUMO outputs an \\emph{urban traffic} (Figure \\ref{fig:schema}e), i.e., a multiset $S(\\overline{D}, \\mathcal{T}) = \\{\\overline{P_1}, \\dots, \\overline{P_N}\\}$ where: \n\\begin{itemize}\n\\item $\\mathcal{T} = (t^{(1)}, \\dots, t^{(N)})$ is the sequence of departure times of the $N$ paths in $\\overline{D}$, where $t^{(i)} \\in \\mathcal{T}$ is a timestamp chosen uniformly at random from the simulation interval  (e.g., 1 hour); \n\\item a vehicle $v$'s trajectory $\\overline{P_v} \\in S(\\overline{D},\\mathcal{T})$ is defined as: $$\\overline{P_v} = ((e_o, t_1^{(v)}), \\dots, (e_d, t_m^{(v)}))$$ where $m$ is the length of path $\\overline{P_v}$.\n\n\\end{itemize}\n\n\\subsection{Vehicle Emissions}\n\\label{sec:vehicle_emissions}\nWe assess the impact of the urban traffic $S(\\overline{D}, \\mathcal{T})$ using a model that estimates the vehicles' emissions based on their trajectories $\\overline{P_1}, \\dots, \\overline{P_N}$. \nSpecifically, we use the HBEFA3 emission model based on the Handbook of Emission Factors for Road Transport (HBEFA) database~\\cite{infras2013handbook}. \nThe HBEFA3-based model estimates the vehicle's instantaneous CO$_2$ emissions relying on the following function, which is linked to the power the vehicle's engine produces in each trajectory point $j$ to overcome the driving resistance force~\\cite{krajzewicz2015second}:\n$$\\mathcal{E}(j) = c_0 + c_1sa + c_2sa^2 + c_3s + c_4s^2 + c_5s^3$$\nwhere $s$ and $a$ are the  vehicle's speed and acceleration in point $j$, respectively, and $c_0,\\dots,c_5$ are parameters changing\nper emission type and vehicle taken from the HBEFA database.\n\nWe compute the amount of CO$_2$ emissions on each edge $e \\in E$ by summing all the emissions corresponding to any vehicle $v$'s trajectory point that fall on $e$, i.e., $\\mathcal{E}(e) = \\sum_v \\sum_{j \\in \\overline{P_v}} \\mathcal{E}(j)$.\nFinally, we construct a weighted road network $\\overline{G} = (V, \\overline{E})$ where each edge $\\overline{e} \\in \\overline{E}$ is associated with the attribute $\\mathcal{E}(e)$ describing the amount of CO$_2$ emissions on it.\n\n\n\\section{Experimental Setup}\n\\label{sec:experimental_setup}\nWe apply TraffiCO$_2$ to a 45 km$^2$ area in the city centre of Milan, Italy, for which we have GPS data describing 17k private vehicles travelling between April 2nd and 8th, 2007 (114k GPS points).\nWe split Milan into squared tiles with 1 km side, detect the tiles where each vehicle starts and stops \\cite{Ramaswamy2004Project, scikitmob}, and compute the origin-destination matrix $M$ of vehicles' flows.\n\nWe download Milan's road network $G=(V, E)$ from OpenStreetMap ($|V|=5551$ nodes and $|E|=36,945$ edges) and preprocess it to fix incorrect information regarding turns, intersections, road interruptions, the number of lanes per road, and other inaccuracies that characterise these data \\cite{Argota2022Getting}. \n\nBased on the matrix $M$ and the road network $G$, we compute the mobility demand $D$ with $N=15,000$ vehicles using the procedure described in Section \\ref{sec:mobility_demand}.\nWe choose $N=15,000$ because it minimises the difference between the distribution of travel time of real trajectories and simulated ones, a common way to assess the realism of a simulated urban traffic \\cite{Argota2022Getting} (see Appendix \\ref{sec:traf-cal} for detail). \n\nTo translate the mobility demand $D$ into the multiset of routed paths $\\overline{D}$, we consider two routing algorithms: OpenStreetMap (OSM) and TomTom (TT). \nOSM is a public voluntary geographic information system, which provides APIs\\footnote{\\href{https:\/\/openrouteservice.org\/dev\/\\#\/api-docs}{https:\/\/openrouteservice.org\/dev\/\\#\/api-docs}} to generate paths between locations.\nTT is a commercial navigation system service that provides APIs\\footnote{\\href{https:\/\/developer.tomtom.com\/routing-api\/documentation\/product-information\/introduction}{https:\/\/developer.tomtom.com\/routing-api}} for routes generation.\nFor OSM and TT, we use a routing principle that suggests a path between two locations as a trade-off between travel time and distance.\nNote that OSM and TT implement this principle differently, i.e., they may provide different paths for the same origin-destination pair.\n\nWe obtain the path of vehicles that do not follow navigation apps' suggestions using Duarouter (DR), a routing algorithm provided by SUMO\\footnote{\\href{https:\/\/sumo.dlr.de\/docs\/duarouter.html}{https:\/\/sumo.dlr.de\/docs\/duarouter.html}} that suggests the fastest path (i.e., shortest travel time) between two edges on the road network.\nThe fastest path may be perturbed using a randomisation parameter $w \\in [1, +\\infty)$, where $w=1$ means no randomisation (i.e., the fastest path). The higher $w$, the more randomly perturbed the fastest path is.\nTherefore, DR with $w > 1$ allows us to model the driving behaviour of vehicles that do not strictly follow navigation apps' suggestions and, simultaneously, to model the imperfection and non-rationality of human drivers \\cite{Seele2012Cognitive}. \nIndeed, individuals get distracted when driving (e.g., take wrong turns), and they lack complete knowledge of the city's traffic, the road network, and the best path to reach a destination. \nIn our experiments, we use $w=5$ because it is the value that minimises the difference between the distribution of travel time of real trajectories and simulated ones (see Appendix \\ref{sec:traf-cal}).\nFigure \\ref{fig:trajectories} shows four examples of paths generated by OSM, TT, DR with $w=1$ and DR with $w=5$ between the same origin-destination pair (trip).\n\n\\begin{figure}[htb!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{imgs\/plot_trajs.pdf}\n    \\caption{Examples of routed paths between an origin and destination pair according to OSM (blue), TT (red), DR with $w=5$ (orange), and DR with $w=1$ (black).}\n    \\label{fig:trajectories}\n\\end{figure}\n\n\nGiven a routing algorithm $R \\in \\{\\text{OSM}, \\text{TT}\\}$, we study its impact on the urban environment generating 11 multisets of routed paths $\\overline{D}^{(R)}_0, \\dots, \\overline{D}^{(R)}_{10}$. \nIn each multiset $\\overline{D}^{(R)}_i$ ($i=0, \\dots, 10$), $(i \\cdot 10)$\\% of the paths (chosen uniformly at random among the $N$ paths) are $R$-routed ($R {\\in} \\{\\text{OSM}, \\text{TT}\\}$) and the remaining paths are routed by DR with $w=5$. \nFor example, for $i=5$ and $R = \\text{TT}$, $\\overline{D}_5^{(R)}$ contains 50\\% of the paths routed by TT and the remaining vehicles routed by DR with $w=5$.\nSimilarly, $i=7$ means that 70\\% of the vehicles are TT-routed and 30\\% are DR-routed ($w=5$).\n\nTo make experiments more robust, for each $i = 0, \\dots, 10$, we generate $\\overline{D}_i^{(R)}$ ten times, each one with a different choice of $R$-routed vehicles that are chosen uniformly at random.\nFinally, we generate the urban traffic $S(\\overline{D}_{i}^{(R)}, \\mathcal{T})$ for each multiset of routed paths $\\overline{D}_{i}^{(R)}$, where departure times in $\\mathcal{T}$ are chosen uniformly at random between 0 and 3600 seconds, i.e., we simulate one hour of traffic in Milan. \nThen, for each urban traffic $S(\\overline{D}_{i}^{(R)}, \\mathcal{T})$, we compute the CO$_2$ emissions for each trajectory and aggregate them at the edge level, obtaining the weighted road network $\\overline{G}_i^{(R)}$.\nFrom $\\overline{G}_i^{(R)}$, we obtain the overall CO$_2$ emissions in Milan $\\mathcal{E}_i^{(R)} = \\sum_{e\\in E} \\mathcal{E}(e)$, where $\\mathcal{E}(e)$ is the amount of emissions on edge $e$.\n\n\\begin{figure}[htb!]\n    \\centering\n    \\subfigure[\\large OSM]{\n    \\includegraphics[width=0.9\\linewidth]{imgs\/plot_ccdf_OSM__AVG_across_repetitions__INSET_ZOOM__V2.png}}\n    \\subfigure[\\large TT]{\\includegraphics[width=0.9\\linewidth]{imgs\/plot_ccdf_TT__AVG_across_repetitions__INSET_ZOOM__V2.png}}\n    \\caption{\n    The Complementary Cumulative Distribution Functions (CCDFs) of the CO$_2$ (in mg) emitted on the roads (averaged across 10 repetitions) by the vehicles of the generated urban traffic $S(\\overline{D}_{i}^{(R)}, \\mathcal{T})$, for OSM (a) and TT (b). \n    Colours represent routed paths with increasing percentage of $R$-routed vehicles.\n    The inset plot zooms on the distributions' tail.}\n    \\label{fig:ccdfs}\n\\end{figure}\n\n\n\n\\section{Results}\n\\label{sec:results}\nWe study how the distribution of the CO$_2$ emissions across Milan's roads changes varying the percentage of $R$-routed vehicles, i.e., varying $i=0, \\dots,10$ of $\\overline{D}^{(R)}_i$, $R \\in \\{\\text{OSM}, \\text{TT}\\}$.\nIn general, emissions distribute across roads in a heterogeneous way: a few grossly polluted roads coexist with roads with significantly fewer emissions (Figure \\ref{fig:ccdfs}).\nIndeed, these distributions are associated with a Gini index $g_{\\text{\\tiny OSM}} \\in [0.864, 0.876]$ and $g_{\\text{\\tiny TT}} \\in [0.860, 0.868]$ (Figure~\\ref{fig:gini&totco2}a) and are well approximated by a truncated power-law with the exponent $\\alpha_{\\text{\\tiny OSM}} \\in [1.76,1.89]$ and $\\alpha_{\\text{\\tiny TT}} \\in [1.76,2.00]$ (Figure~\\ref{fig:alpha}). \nSee Appendix \\ref{sec:fitting_distributions} for details on the curve fitting.\nIn particular, the distributions are the least uneven when 50\\% and 70\\% of the vehicles are OSM-routed and TT-routed, respectively (Figure~\\ref{fig:ccdfs}).\n\nThe analysis of how the total CO$_2$ emissions ($\\mathcal{E}$) varies with the percentage of $R$-routed vehicles also reveals a clear pattern: when either all vehicles are $R$-routed or none of them, the overall emissions are maximised (Figure \\ref{fig:gini&totco2}b).\nIn contrast, the overall emissions are minimised when just half of the vehicles are R-routed.\nIn other words, $\\mathcal{E}_5^{(R)} < \\mathcal{E}_0^{(R)}$ and $\\mathcal{E}_5^{(R)} < \\mathcal{E}_{10}^{(R)}$.\nNote that the total emissions when all vehicles are TT-routed are much heavier than when all are OSM-routed (Figure \\ref{fig:gini&totco2}b).\nThis result suggests that TT's routing algorithm recommends paths that generate a smaller adverse impact on urban well-being than OSM.\n\nThe fraction of $R$-routed vehicles also influence the spatial distribution of emissions in the city.\nIn particular, comparing the two scenarios that maximise the total emissions (0\\% and 100\\% of $R$-routed vehicles) with the scenario that minimises them (50\\% of $R$-routed vehicles) helps understand where vehicles are being routed, consequently revealing emissions hot spots.\nIn Figure \\ref{fig:map_diff}a, we show the difference between the per-road emissions (normalized by the road length) when none of the vehicle is OSM-routed and 50\\% of them are (i.e., $\\mathcal{E}_0^{\\text{\\tiny (OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$).\nSimilarly, Figure \\ref{fig:map_diff}b shows the normalized emissions difference when all vehicles are OSM-routed and 50\\% of them are (i.e., $\\mathcal{E}_{10}^{\\text{\\tiny(OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$).\nWe find that when 100\\% of vehicles are OSM-routed, the emissions are more concentrated towards Milan's ring road (Figure \\ref{fig:map_diff}b).\nIn contrast, when none of them is OSM-routed, the emissions are more concentrated towards the city centre (Figure \\ref{fig:map_diff}a).\nWe find similar results for TT (Figure \\ref{fig:map_diff_TT}).\n\n\\begin{figure*}[htb!]\n    \\centering\n    \\subfigure[\\large Gini index]{\n    \\includegraphics[width=0.8\\columnwidth]{imgs\/plot_errorbars_gini__V2.pdf}}\n    \\subfigure[\\large Total CO$_2$]{\\includegraphics[width=0.8\\columnwidth]{imgs\/plot_errorbars_AVG_total_co2__V2.pdf}}\n    \\caption{\n    Gini index of the CO$_2$ distribution (a) and total CO$_2$ emissions (b) varying the percentage of $R$-routed vehicles, for OSM (blue) and TT (red).\n   \n   \n    In the error bars, points indicate the average Gini index (a) and the total CO$_2$ (b) over ten simulations with different choices of $R$-routed vehicles chosen uniformly at random.\n    Vertical bars indicate the standard deviation. \n    }\n    \\label{fig:gini&totco2}\n\\end{figure*}\n\n\\begin{figure*}[htb!]\n    \\centering\n    \\subfigure[]{\n    \\includegraphics[width=0.85\\columnwidth]{imgs\/Milan_map_co2_diff_norm_by_len__osm_0_osm_50.pdf}}\n    \\subfigure[]{\\includegraphics[width=0.85\\columnwidth]{imgs\/Milan_map_co2_diff_norm_by_len__osm_100_osm_50.pdf}}\n    \\caption{\n    The difference in the total CO$_2$ emitted on each road (in mg per meter of road) when: (a) none of the vehicles is OSM-routed and 50\\% of them are\n    ($\\mathcal{E}_0^{\\text{\\tiny (OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$); (b) all vehicles are OSM-routed and 50\\% of them are ($\\mathcal{E}_{10}^{\\text{\\tiny (OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$).\n    Red roads indicate a positive difference; blue ones indicate a negative difference.\n   \n   \n    }\n    \\label{fig:map_diff}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htb!]\n    \\centering\n    \\subfigure[\\large OSM - Gini]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_gini_OSM.pdf}}\n\t\\subfigure[\\large OSM - CO$_2$]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_totalco2_OSM.pdf}}\n\t\\subfigure[\\large OSM - travel time]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_avgtraveltime_OSM.pdf}}\n\t\\subfigure[\\large TT - Gini]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_gini_TT.pdf}}\n\t\\subfigure[\\large TT - CO$_2$]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_totalco2_TT.pdf}}\n\t\\subfigure[\\large TT - travel time]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_avgtraveltime_TT.pdf}}\n    \\caption{\n    Gini index of CO$_2$ distribution (a,d), total CO$_2$ emissions (b,e), average travel time per vehicle (c,f), for $w = 1, \\dots, 15$, varying the percentage of $R$-routed vehicles, for OSM (a,b,c) and TT (d,e,f).\n   \n    In the error bars, points indicate the average Gini index (a,d), total CO$_2$ emissions (b,e), average travel time (c,f) over ten simulations with different choices of $R$-routed vehicles chosen uniformly at random.\n    Vertical bars indicate the standard deviation.}\n    \\label{fig:w_osm_tt}\n\\end{figure*}\n\n\n\\paragraph{Impact of randomization}\nWe investigate the impact of DR's randomisation parameter $w$ on the total emissions, their distribution over the road network, and the vehicles' average travel time.\nFor this purpose, we repeat the simulation of the urban traffic varying $w = 1, \\dots, 15$. \nWe remind that the higher $w$, the more randomly perturbed DR's fastest path is (see Appendix \\ref{sec:perturbation}).\n\nFirst, changing $w$ does not affect considerably the shape of the emission distributions: the Gini index still ranges in $g_{\\text{\\tiny OSM}} \\in [0.860, 0.879]$ and $g_{\\text{\\tiny TT}} \\in [0.858, 0.879]$ (Figure \\ref{fig:w_osm_tt}a, d) and the truncated power-law exponent in $\\alpha_{\\text{\\tiny OSM}} \\in [1.71,1.94]$ and $\\alpha_{\\text{\\tiny TT}} \\in [1.71,2.08]$.\nSecond, the higher $w$, the more even the distributions (Figure \\ref{fig:w_osm_tt}a, d), the lower the emissions (Figure \\ref{fig:w_osm_tt}b, e), and the shorter the average travel time (Figure \\ref{fig:w_osm_tt}c, f).\nThird, the total emissions are minimised when 40-60\\% of the vehicles are $R$-routed, and so are the average travel time and the distributions' Gini index (Figure \\ref{fig:w_osm_tt}).\n\n\n\n\n\n\n\n\n\\section{Discussion}\nOur study provide several interesting results. \nWe discuss and interpret them in the following paragraphs.\n\n\\textbf{Navigation apps do impact on emissions.} \nIn general, CO$_2$ emissions are unevenly distributed across the city roads, and this unevenness is exacerbated when all vehicles or none of them are $R$-routed. \nIn contrast, the total CO$_2$ emissions as well as their heterogeneity are minimised when around 50-70\\% of the vehicles are $R$-routed. \nThese results clearly show that navigation apps do have a non-negligible adverse impact on the urban environment.\n\n\\textbf{Path perturbation is beneficial.}\nWe also find that path perturbation (through DR's $w$ parameter) is beneficial: the more we randomise the vehicles' paths, the shorter their travel time and the lower the amount of emissions in the city.\nThis may be because the vehicles' perturbed paths are more ``diverse'' (i.e., they spread over more roads), thus reducing congestion and consequently the amount of emissions and travel time.\nThe role of path randomisation is interesting and deserves further investigation.\n\n\n\\textbf{Trips' spatial distribution matters.} \nUrban planners and policymakers may be interested in investigating the impact of navigation apps on the spatial distribution of vehicles' emissions. \nOur study shows that, in Milan, the more vehicles are $R$-routed, the less emissions concentrate in the city center and the more in the external ring road.\nPresumably, this may be because the navigation apps \nroute the vehicles preferably on the city's arterial roads (such as the ring road) to keep the individual path as fast as possible.\n\n\\textbf{Different navigation apps, different impact.} \nOur TraffiCO$_2$ simulation framework is a useful tool to compare the impact of different navigation apps on urban well-being. In our study, we find that TomTom is better than OpenStreetMap in minimising the emissions distribution's inequality, and the city's total emissions and average travel time. \nAlthough TomTom's algorithm is not public, we may suppose that TomTom's heuristics are more sophisticated, e.g., they use more or higher-quality information than OpenStreetMap, such as traffic data, detailed information about typical road speed, capacity, and length.\n\n\\textbf{A novel what-if analysis tool.} \nTraffiCO$_2$ is a simulation framework to compare the impact of different routing strategies on urban well-being in terms of amount of emissions and their spatial and statistical distribution. \nOur tool allows us to go beyond state-of-the-art studies, which investigate the reduction of emissions for a fixed fraction of vehicles routed by a navigation app ~\\cite{arora2021quantifying}.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nScenarios where either all vehicles or none of them follow a navigation app's suggestion lead to the highest CO$_2$ emissions and the most uneven distribution of emissions per road.\nWe show that when just a fraction of the vehicles (around half of them) follows the navigation app's suggestions, such an adverse impact is minimised.\nThis minimisation also holds when introducing more randomness in non-R-routed paths, which leads to a reduction of the vehicles' average travel time, overall CO$_2$ emissions, and inequality in the distribution of CO$_2$ emissions across the road network.\n\nWe plan to improve and extend this study in several directions. \nFirst and foremost, we plan to extend the set of navigation apps considered (e.g., to Google Maps) and to study the environmental impact of a fleet of vehicles that use various navigation apps.\nIn this work, we focus on light-duty vehicles only and assume that all vehicles carry the same engine type. \nIncluding heavy-duty vehicles (e.g., buses, trucks) and considering different vehicle engine ages and types (e.g. diesel, LPG, petrol) would provide a more complete mosaic of the emissions on the road network.\nWe also look forward to apply our framework to various cities, to investigate how the impact of navigation apps varies with city size, shape, road network, and other characteristics.\nMoreover, we could study the impact of navigation apps in terms of other pollutants, such as nitrogen oxides, ozone, particulate matter, and volatile organic compounds, using emissions models similar to those used in this paper.\n\nIn the meantime, our work is a first step towards designing next-generation routing algorithms that, as our results suggest, should consider some degree of path randomisation to increase urban well-being while still satisfying individual needs.\n\n\n\n\n\n\\begin{acks}\nThis work has been partially supported by EU project H2020 Humane-AI-net G.A. 952026, EU project H2020 SoBigData++ G.A. 871042, ERC-2018-ADG G.A. 834756 \"XAI: Science and technology for the eXplanation of AI decision making\", and by the CHIST-ERA grant CHIST-ERA-19-XAI-010, by MUR (grant No. not yet available), FWF (grant No. I 5205), EPSRC (grant No. EP\/V055712\/1), NCN (grant No. 2020\/02\/Y\/ST6\/00064), ETAg (grant No. SLTAT21096), BNSF (grant No. K$\\Pi$-06-AOO2\/5). \n\nWe thank Davide Nicola, Pietro Iemmello, Ren\u00e9 Ferretti, Dante Milonga and Maurizio Turone for the inspiration. \nWe thank Daniele Fadda for his precious\nsupport in data visualization.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. 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Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmlarge]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\n\nUsing the BibLaTeX system, the bibliography is included in your source\ndocument with the following command, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\printbibliography\n\\end{verbatim}\n\nThe command \\verb|\\addbibresource{bibfile}| declares the \\BibTeX\\ file to use\nin the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source\nwhere ``\\verb|bibfile|'' is the name, \\emph{with} the ``\\verb|.bib|'' suffix.\nNotice that \\verb|\\addbibresource| takes only one argument: to declare multiple files,\nuse multiple instances of the command.\n\nCitations and references are numbered by default. A small number of\nACM publications have citations and references formatted in the\n``author year'' style; for these exceptions, please pass the option \\verb|style=acmauthoryear|\nto the \\verb|biblatex| package loaded in the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source.\n\n\n  Some examples.  A paginated journal article \\cite{Abril07}, an\n  enumerated journal article \\cite{Cohen07}, a reference to an entire\n  issue \\cite{JCohen96}, a monograph (whole book) \\cite{Kosiur01}, a\n  monograph\/whole book in a series (see 2a in spec. document)\n  \\cite{Harel79}, a divisible-book such as an anthology or compilation\n  \\cite{Editor00} followed by the same example, however we only output\n  the series if the volume number is given \\cite{Editor00a} (so\n  Editor00a's series should NOT be present since it has no vol. no.),\n  a chapter in a divisible book \\cite{Spector90}, a chapter in a\n  divisible book in a series \\cite{Douglass98}, a multi-volume work as\n  book \\cite{Knuth97}, a couple of articles in a proceedings (of a\n  conference, symposium, workshop for example) (paginated proceedings\n  article) \\cite{Andler79, Hagerup1993}, a proceedings article with\n  all possible elements \\cite{Smith10}, an example of an enumerated\n  proceedings article \\cite{VanGundy07}, an informally published work\n  \\cite{Harel78}, a couple of preprints \\cite{Bornmann2019,\n    AnzarootPBM14}, a doctoral dissertation \\cite{Clarkson85}, a\n  master's thesis: \\cite{anisi03}, an online document \/ world wide web\n  resource \\cite{Thornburg01, Ablamowicz07, Poker06}, a video game\n  (Case 1) \\cite{Obama08} and (Case 2) \\cite{Novak03} and \\cite{Lee05}\n  and (Case 3) a patent \\cite{JoeScientist001}, work accepted for\n  publication \\cite{rous08}, 'YYYYb'-test for prolific author\n  \\cite{SaeediMEJ10} and \\cite{SaeediJETC10}. Other cites might\n  contain 'duplicate' DOI and URLs (some SIAM articles)\n  \\cite{Kirschmer:2010:AEI:1958016.1958018}. Boris \/ Barbara Beeton:\n  multi-volume works as books \\cite{MR781536} and \\cite{MR781537}. A\n  couple of citations with DOIs:\n  \\cite{2004:ITE:1009386.1010128,Kirschmer:2010:AEI:1958016.1958018}. Online\n  citations: \\cite{TUGInstmem, Thornburg01, CTANacmart}.\n  Data Artifacts: \\cite{UMassCitations}.\n  Software project: ~\\cite{cgal,delebecque:hal-02090402}. Software Version: ~\\cite{gf-tag-sound-repo,}. Software Module: ~\\cite{cgal:lp-gi-20a}. Code fragment: ~\\cite{simplemapper}.\n\n\\section{Acknowledgments}\n\nIdentification of funding sources and other support, and thanks to\nindividuals and groups that assisted in the research and the\npreparation of the work should be included in an acknowledgment\nsection, which is placed just before the reference section in your\ndocument.\n\nThis section has a special environment:\n\\begin{verbatim}\n  \\begin{acks}\n  ...\n  \\end{acks}\n\\end{verbatim}\nso that the information contained therein can be more easily collected\nduring the article metadata extraction phase, and to ensure\nconsistency in the spelling of the section heading.\n\nAuthors should not prepare this section as a numbered or unnumbered {\\verb|\\section|}; please use the ``{\\verb|acks|}'' environment.\n\n\\section{Appendices}\n\nIf your work needs an appendix, add it before the\n``\\verb|\\end{document}|'' command at the conclusion of your source\ndocument.\n\nStart the appendix with the ``\\verb|appendix|'' command:\n\\begin{verbatim}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmsmall]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmsmall,screen,review]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmsmall]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmtog]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf,authordraft]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[manuscript,screen,review]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\n\nUsing the BibLaTeX system, the bibliography is included in your source\ndocument with the following command, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\printbibliography\n\\end{verbatim}\n\nThe command \\verb|\\addbibresource{bibfile}| declares the \\BibTeX\\ file to use\nin the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source\nwhere ``\\verb|bibfile|'' is the name, \\emph{with} the ``\\verb|.bib|'' suffix.\nNotice that \\verb|\\addbibresource| takes only one argument: to declare multiple files,\nuse multiple instances of the command.\n\nCitations and references are numbered by default. A small number of\nACM publications have citations and references formatted in the\n``author year'' style; for these exceptions, please pass the option \\verb|style=acmauthoryear|\nto the \\verb|biblatex| package loaded in the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source.\n\n\n  Some examples.  A paginated journal article \\cite{Abril07}, an\n  enumerated journal article \\cite{Cohen07}, a reference to an entire\n  issue \\cite{JCohen96}, a monograph (whole book) \\cite{Kosiur01}, a\n  monograph\/whole book in a series (see 2a in spec. document)\n  \\cite{Harel79}, a divisible-book such as an anthology or compilation\n  \\cite{Editor00} followed by the same example, however we only output\n  the series if the volume number is given \\cite{Editor00a} (so\n  Editor00a's series should NOT be present since it has no vol. no.),\n  a chapter in a divisible book \\cite{Spector90}, a chapter in a\n  divisible book in a series \\cite{Douglass98}, a multi-volume work as\n  book \\cite{Knuth97}, a couple of articles in a proceedings (of a\n  conference, symposium, workshop for example) (paginated proceedings\n  article) \\cite{Andler79, Hagerup1993}, a proceedings article with\n  all possible elements \\cite{Smith10}, an example of an enumerated\n  proceedings article \\cite{VanGundy07}, an informally published work\n  \\cite{Harel78}, a couple of preprints \\cite{Bornmann2019,\n    AnzarootPBM14}, a doctoral dissertation \\cite{Clarkson85}, a\n  master's thesis: \\cite{anisi03}, an online document \/ world wide web\n  resource \\cite{Thornburg01, Ablamowicz07, Poker06}, a video game\n  (Case 1) \\cite{Obama08} and (Case 2) \\cite{Novak03} and \\cite{Lee05}\n  and (Case 3) a patent \\cite{JoeScientist001}, work accepted for\n  publication \\cite{rous08}, 'YYYYb'-test for prolific author\n  \\cite{SaeediMEJ10} and \\cite{SaeediJETC10}. Other cites might\n  contain 'duplicate' DOI and URLs (some SIAM articles)\n  \\cite{Kirschmer:2010:AEI:1958016.1958018}. Boris \/ Barbara Beeton:\n  multi-volume works as books \\cite{MR781536} and \\cite{MR781537}. A\n  couple of citations with DOIs:\n  \\cite{2004:ITE:1009386.1010128,Kirschmer:2010:AEI:1958016.1958018}. Online\n  citations: \\cite{TUGInstmem, Thornburg01, CTANacmart}.\n  Data Artifacts: \\cite{UMassCitations}.\n  Software project: ~\\cite{cgal,delebecque:hal-02090402}. Software Version: ~\\cite{gf-tag-sound-repo,}. Software Module: ~\\cite{cgal:lp-gi-20a}. Code fragment: ~\\cite{simplemapper}.\n\n\\section{Acknowledgments}\n\nIdentification of funding sources and other support, and thanks to\nindividuals and groups that assisted in the research and the\npreparation of the work should be included in an acknowledgment\nsection, which is placed just before the reference section in your\ndocument.\n\nThis section has a special environment:\n\\begin{verbatim}\n  \\begin{acks}\n  ...\n  \\end{acks}\n\\end{verbatim}\nso that the information contained therein can be more easily collected\nduring the article metadata extraction phase, and to ensure\nconsistency in the spelling of the section heading.\n\nAuthors should not prepare this section as a numbered or unnumbered {\\verb|\\section|}; please use the ``{\\verb|acks|}'' environment.\n\n\\section{Appendices}\n\nIf your work needs an appendix, add it before the\n``\\verb|\\end{document}|'' command at the conclusion of your source\ndocument.\n\nStart the appendix with the ``\\verb|appendix|'' command:\n\\begin{verbatim}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf, language=french,\nlanguage=german, language=spanish, language=english]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigplan,screen]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n  submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification.  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Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{sec:level1}Introduction\\protect\\\\ } \nThe Skyrme-Faddeev-Niemi (SFN) model which is an O(3) $\\sigma$ model \nin three dimensional space upto fourth-order in the first derivative \nhas topological soliton solutions with torus or knot-like structure. \nThe model was initiated in 70's~\\cite{faddeev75} and its interest has  \nbeen extensively growing. The numerical simulations were performed in \nRefs.~\\cite{faddeev97,gladikowski97,sutcliffe98,hietarinta99,hietarinta00}, \nthe integrability was shown in Ref.~\\cite{aratyn99}, and the application to \nthe condensed matter physics~\\cite{babaev02} and the Weinberg-Salam \nmodel~\\cite{fayzullaev} were also considered.  \nThe recent research especially focuses on the consistency between \nthe SFN and fundamental theories such as QCD~\\cite{faddeev99,langmann99,\nshabanov99,cho02}. \nIn those references, it is claimed that the SFN action should be deduced  \nfrom the SU(2) Yang-Mills (YM) action at low energies. \nOne can also show from the Wilsonian renormalization group argument that \nthe effective action of Yang-Mills theory recovers the SFN in the infrared \nregion~\\cite{gies01}. \nHowever, the derivative expansion for slowly varying \nfields $\\bm{n}$ upto quartic order brings an additional fourth-order term \nin the SFN model to destabilize the soliton solution. \n\nSimilar situations can be seen also in various topological soliton \nmodels. In the Skyrme model, the chirally invariant lagrangian \nwith quarks produces fourth order terms after the derivative expansion \nand they destabilize the soliton solution~\\cite{dhar85,aitchison85}.\nTo recover the stability of the skyrmion, the author of Ref.\\cite{marleau01} \nintroduced a large number of higher order terms in the first derivative \nwhose coefficients were determined from the coefficients of the \nSkyrme model by using the recursion relations. \nAlternatively, in Ref.~\\cite{gies01} Gies pointed out the possibility that \nthe second derivative order term can work as a stabilizer for the soliton. \n\nIn this paper, we examine the Gies's supposition by numerical analysis.\nIn section \\ref{sec:level2}, we give an introduction to the Skyrme-Faddeev-Niemi \nmodel with its topological property. \nIn section \\ref{sec:level3}, we show how to derive the SFN model action \nfrom the SU(2) Yang-Mills theory. In section \\ref{sec:level4}, \nsoliton solutions of this truncated YM action are studied. \nIn order to find stable soliton solutions, we introduce \na second derivative term which can be derived in a perturbative \nmanner.  \nThe naive extremization scheme, however, produce the fourth\norder differential equation and the model has no stable soliton solution. \nFailure of finding the soliton is caused by the \nbasic feature of the second derivative field theory. \nIn section \\ref{sec:level5}, the higher derivative theory and \nOstrogradski's formulation are reviewed. We show the absence \nof bound state in the second derivative theory using an example in \nquantum mechanics and introduce the perturbative treatment for the \nsecond derivative theory. \nIn section \\ref{sec:level6}, we present our numerical results. \nIn section \\ref{sec:level7} are concluding remarks. \n\n\\section{\\label{sec:level2}Skyrme-Faddeev-Niemi model\\protect\\\\}\nThe Faddeev-Niemi conjecture for the low-energy model of SU(2) \nYang-Mills theory is expressed by following effective action:\n\\begin{eqnarray}\nS_{\\rm SFN}=\\Lambda \\int d^4x\\Bigl[\\frac{1}{2}(\\partial_\\mu \\bm{n})^2\n+\\frac{g_1}{8}(\\bm{n}\\cdot \\partial_\\mu \\bm{n}\\times \\partial_\\nu \\bm{n})^2 \\Bigr] \n\\label{fsn_ac}\n\\end{eqnarray}\nwhere $\\bm{n}(\\bm{x})$ is a three component vector field normalized as \n$\\bm{n}\\cdot\\bm{n}=1$. The mass scale $\\Lambda$ is a free parameter \nand in this paper we set $\\Lambda=1$. \nStable soliton solutions exist when $g_1 > 0$. \n\nThe static field $\\bm{n}(\\bm{x})$ maps $\\bm{n}:R^3\\mapsto S^2$ and \nthe configurations are classified by the topological maps characterized \nby a topological invariant $H$ called Hopf charge\n\\begin{eqnarray}\nH=\\frac{1}{32\\pi^2}\\int A \\wedge F,~~F=dA\n\\label{hopf}\n\\end{eqnarray}\nwhere $F$ is the field strength and can be written as \n$F=(\\bm{n}\\cdot d\\bm{n}\\wedge d\\bm{n})$.\n\nThe static energy $E_{\\rm stt}$ from the action (\\ref{fsn_ac}) has \na topological lower bound~\\cite{ward98}, \n\\begin{eqnarray}\nE_{\\rm stt}\\ge K H^{3\/4}\n\\label{lowerbound}\n\\end{eqnarray}\nwhere $K=16\\pi^2\\sqrt{g_1}$.\n\n\nPerforming numerical simulation, one can find that the static configurations \nfor $H=1,2$ have axial symmetry~\\cite{sutcliffe98}. \nThus ``the toroidal ansatz'' which was studied in Ref.\\cite{gladikowski97} \nis suitable to be imposed on these configurations. The ansatz is given by \n\\begin{eqnarray}\n&&n_1=\\sqrt{1-w^2(\\eta,\\beta)}\\cos(N\\alpha+v(\\eta,\\beta))\\,, \\nonumber \\\\\n&&n_2=\\sqrt{1-w^2(\\eta,\\beta)}\\sin(N\\alpha+v(\\eta,\\beta)\\,, \n\\label{toroidal} \\\\\n&&n_3=w(\\eta,\\beta)\\,, \\nonumber \n\\end{eqnarray}\nwhere $(\\eta,\\beta,\\alpha)$ is toroidal coordinates which are related to \nthe $R^3$ as follows:\n\\begin{eqnarray}\nx=\\frac{a\\sinh\\eta\\cos\\alpha}{\\tau},y=\\frac{a\\sinh\\eta\\sin\\alpha}{\\tau},\nz=\\frac{a\\sin\\beta}{\\tau}\n\\end{eqnarray}\nwith $\\tau=\\cosh\\eta-\\cos\\beta$.\n\nThe function $w(\\eta,\\beta)$ is subject to the boundary conditions $w(0,\\beta)=1,w(\\infty,\\beta)=-1$\nand is periodic in $\\beta$. $v(\\eta,\\beta)$ is set to be $v(\\eta,\\beta)=M\\beta+v_0(\\eta,\\beta)$ and \n$v_0(,\\beta)$ is considered as a constant map. \nEquation (\\ref{hopf}) then gives $H=NM$.\n\nIn this paper we adopt a simpler ansatz than (\\ref{toroidal}), which \nis defined by\n\\begin{eqnarray}\n&&n_1=\\sqrt{1-w^2(\\eta)}\\cos(N\\alpha+M\\beta)\\,,\\nonumber \\\\\n&&n_2=\\sqrt{1-w^2(\\eta)}\\sin(N\\alpha+M\\beta)\\,, \n\\label{afz} \\\\\n&&n_3=w(\\eta)\\,, \\nonumber \n\\end{eqnarray}\nwhere $w(\\eta)$ satisfies the boundary conditions $w(0)=1,w(\\infty)=-1$.\nWe numerically study soliton solutions for both ansatz (\\ref{toroidal}) \nand (\\ref{afz}). By comparing those results, we found that this simple \nansatz produces at most 10\\% errors and it does not affect to the \nproperty of the soliton solution. \n\nBy using (\\ref{afz}), the static energy is written in terms of the function \n$w(\\eta)$ as\n\\begin{eqnarray}\n&&E_{\\rm stt}=2\\pi^2a \\int d\\eta\n\\Biggl[\n\\frac{(w')^2}{1-w^2}+(1-w^2) U_{M,N}(\\eta) \\nonumber \\\\\n&&\\hspace{2cm}+\\frac{g_1}{4a^2}\\sinh\\eta\\cosh\\eta (w')^2\nU_{M,N}(\\eta)\\Biggr]\\,,\\nonumber \\\\\n&&\\hspace{1cm}w'\\equiv \\frac{dw}{d\\eta},~~\nU_{M,N}(\\eta)\\equiv \\Bigl(M^2+\\frac{N^2}{\\sinh^2\\eta}\\Bigr)\\,. \\nonumber\n\\end{eqnarray}\nThe Euler-lagrange equation of motion is then derived as \n\\begin{eqnarray}\n&&\\frac{w''}{1-w^2}+\\frac{ww'^2}{(1-w^2)^2}+U_{M,N}(\\eta)w \\nonumber \\\\\n&&+\\frac{g_1}{2a^2}\\Bigl(-2N^2\\coth^2\\eta w'+(\\cosh^2\\eta+\\sinh^2\\eta)\nU_{M,N}(\\eta)w' \\nonumber \\\\ \n&&+\\sinh\\eta\\cosh\\eta U_{M,N}(\\eta)w''\\Bigr)=0\\,.\n\\label{fsn_eq}\n\\end{eqnarray}\nThe variation with respect to $a$ produces the equation for variable $a$.\nSoliton solutions are obtained by solving the equations for $a$ as \nwell as for $w$.\n\n\\section{\\label{sec:level3}Effective action in the Yang-Mills theory\nwith CFNS decomposition\\protect\\\\}\nIn this section, we briefly review how to derive the\nSFN effective action from the action of SU(2) Yang-Mills theory in the \ninfrared limit~\\cite{shabanov99,gies01}. \nFor the gauge fields $\\bm{A}_\\mu$, the Cho-Faddeev-Niemi-Shabanov \ndecomposition is applied~\\cite{faddeev99,langmann99,shabanov99,cho02}\n\\begin{eqnarray}\n\\bm{A}_\\mu=\\bm{n}C_\\mu+(\\partial_\\mu\\bm{n})\\times\\bm{n}+\\bm{W}_\\mu\\,.\n\\label{cfns}\n\\end{eqnarray}\nThe first two terms are the ``electric'' and ``magnetic'' Abelian connection, \nand $\\bm{W}_\\mu$ are chosen so as to orthogonal to $\\bm{n}$, \n$\\bm{W}_\\mu\\cdot\\bm{n}=0$.\nObviously, the degrees of freedom on the left- and right-hand side \nof Eq.(\\ref{cfns}) do not match. While the LHS describes \n$3_{\\rm color}\\times 4_{\\rm Lorentz}=12$, the RHS is comprised of \n$(C_\\mu:)4_{\\rm Lorentz}+(\\bm{n}:)2_{\\rm color}+(\\bm{W}_\\mu:)3_{\\rm color}\n\\times4_{\\rm Lorentz}-4_{\\bm{n}\\cdot\\bm{W}_\\mu=0}=14$ degrees freedom. \nShabanov introduced in his paper \\cite{shabanov99} the following constraint \n\\begin{eqnarray}\n\\bm{\\chi}(\\bm{n},C_\\mu,\\bm{W}_\\mu)=0,~{\\rm with}~~\\bm{\\chi}\\cdot\\bm{n}=0\\,.\n\\end{eqnarray}\nThe generating functional of YM theory can be written by using \nEq.(\\ref{cfns}) as \n\\begin{eqnarray}\n{\\cal Z}=\\int {\\cal D}\\bm{n}{\\cal D}C{\\cal D}\\bm{W}\\delta(\\bm{\\chi})\n\\Delta_{\\rm FP}\\Delta_{\\rm S}e^{-S_{\\rm YM}-S_{\\rm gf}}\\,.\n\\label{vf0}\n\\end{eqnarray}\n$\\Delta_{\\rm FP}$ and $S_{\\rm gf}$ are the Faddeev-Popov determinant and \nthe gauge fixing action respectively, and Shabanov introduced another \ndeterminant $\\Delta_{\\rm S}$ corresponding to the condition $\\bm{\\chi}=0$.\nYM and the gauge fixing action is given by \n\\begin{eqnarray}\n&&S_{\\rm YM}+S_{\\rm  gf}=\\int d^4x\\Bigl[\\frac{1}{4g^2}\\bm{F}_{\\mu\\nu}\\cdot\\bm{F}_{\\mu\\nu}\n+\\frac{1}{2\\alpha_{\\rm g} g^2}(\\partial_\\mu \\bm{A}_\\mu)^2\\Bigr] \\,. \\nonumber \n\\end{eqnarray}\nInserting Eq.(\\ref{cfns}) into the action, one obtains the vacuum functional\n\\begin{eqnarray}\n&&{\\cal Z}=\\int {\\cal D}\\bm{n}e^{-{\\cal S}_{\\rm eff}(\\bm{n})} \\nonumber \\\\\n&&~~~=\\int {\\cal D}\\bm{n} e^{-{\\cal S}_{\\rm cl}(\\bm{n})}\\int {\\cal D}\\tilde{C}{\\cal D}\\bm{W}_\\mu\n\\Delta_{\\rm FP}\\Delta_{\\rm S}\\delta({\\bm \\chi}) \\nonumber \\\\\n&&~~~\\times e^{-(1\/2g^2)\\int(\\tilde{C}_\\mu M^C_{\\mu\\nu}\\tilde{C}_\\nu+\\bm{W}_\\mu \\bar{M}^{\\bm{W}}_{\\mu\\nu}\\bm{W}_\\nu\n+2C_\\nu K^C_\\nu+2\\bm{W}_\\mu\\cdot\\bm{K}^{\\bm{W}}_\\mu)}\n\\nonumber \\\\\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&M^C_{\\mu\\nu}=-\\partial^2\\delta_{\\mu\\nu}+\\partial_\\mu\\bm{n}\\cdot\\partial_\\nu\\bm{n}\\,, \\nonumber \\\\\n&&M^{\\bm{W}}_{\\mu\\nu}=-\\partial^2\\delta_{\\mu\\nu}-\\partial_\\mu\\bm{n}\\otimes\\partial_\\nu\\bm{n}\n+\\partial_\\nu\\bm{n}\\otimes\\partial_\\mu\\bm{n}\\,, \\nonumber \\\\\n&&\\bm{Q}^C_{\\mu\\nu}=\\partial_\\mu\\bm{n}\\partial_\\nu+\\partial_\\nu\\bm{n}\\partial_\\mu+\\partial_\\mu\\partial_\\nu\\bm{n}\\,,\n\\\\\n&&K^C_{\\mu\\nu}=\\partial_\\nu(\\bm{n}\\cdot\\partial_\\nu\\bm{n}\\times\\partial_\\mu\\bm{n})\n+\\partial_\\mu\\bm{n}\\cdot\\partial^2\\bm{n}\\times\\bm{n}\\,, \\nonumber \\\\\n&&\\bm{K}^{\\bm W}_{\\mu\\nu}=\\partial_\\mu(\\bm{n}\\times\\partial^2\\bm{n})\\,, \n\\hspace{1cm}{\\rm (in~gauge~\\alpha_g=1}) \\nonumber \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&&\\bar{M}^{\\bm{W}}_{\\mu\\nu}:=M^{\\bm{W}}_{\\mu\\nu}-\\bm{Q}_{\\mu s}{M^C}^{-1}_{s\\lambda}\\bm{Q}_{\\lambda \\nu}\\,, \\nonumber \\\\\n&&\\tilde{C}_\\mu=C_\\mu+\\bm{W}_s\\cdot \\bm{Q}_{s\\lambda}{M^C}^{-1}_{\\lambda\\mu}\\,.\n\\end{eqnarray}\nThe classical action of $\\bm{n}$ including the gauge fixing term is given by\n\\begin{eqnarray}\n{\\cal S}_{\\rm cl}=\\int d^4x\\Bigl[\\frac{1}{4g^2}(\\partial_\\mu\\bm{n}\\times \\partial_\\nu\\bm{n})^2\n+\\frac{1}{2 \\alpha_{\\rm g} g^2}(\\partial^2\\bm{n}\\times\\bm{n})^2\\Bigr]\\,.\\nonumber \\\\\n\\end{eqnarray}\nThe $\\delta$ functional is expressed by its Fourier transform\n\\begin{eqnarray}\n\\delta(\\bm{\\chi})=\\int {\\cal D}\\bm{\\phi}e^{-i\\int (\\bm{\\phi}\\cdot \\partial\\bm{W}_\\mu\n+\\bm{\\phi}\\cdot C_\\mu\\bm{n}\\times \\bm{W}_\\mu+(\\bm{\\phi}\\cdot\\bm{n})(\\partial_\\mu\\bm{n}\\cdot\\bm{W}_\\mu))}\\,.\n\\nonumber \\\\\n\\end{eqnarray} \nIntegrating over $C,\\bm{W},\\bm{\\phi}$, we finally obtain \n\\begin{eqnarray}\n&&e^{-S_{\\rm eff}}=e^{-S_{\\rm cl}}\\Delta_{\\rm FP}\\Delta_{\\rm S}\n(\\det M^C)^{-1\/2} (\\det \\bar{M}^{\\bm{W}})^{-1\/2}\n\\nonumber \\\\\n&&\\hspace{1cm}\\times (\\det -Q^{\\bm{\\phi}}_\\mu (\\bar{M}^{\\bm{W}})^{-1}_{\\mu\\nu}\nQ^{\\bm{\\phi}}_\\nu)^{-1\/2} \\label{determ} \n\\end{eqnarray}\nwhere several nonlocal terms and the higher derivative components have been \nneglected. \n\nWe perform the derivative expansion for the four determinants in \nEq.~(\\ref{determ}) under the following assumptions \n\\begin{itemize}\n\\item [(i)]the theory is valid for the momenta $p$ with $k<p<\\Lambda$ ($k,\\Lambda$ are \ninfrared and ultraviolet cut-off)\n\\item [(ii)]$|\\partial\\bm{n}| \\ll k$ \n\\item [(iii)]the higher derivative terms, such as $\\partial^2\\bm{n}$ are omitted.\n\\end{itemize}\nThe effective action is then given by  \n\\begin{eqnarray}\n&&S_{\\rm eff}=\\int d^4x\\Bigl[\\frac{1}{2}(\\partial_\\mu \\bm{n})^2\n+\\frac{g_1}{8}(\\partial_\\mu \\bm{n}\\times \\partial_\\nu \\bm{n})^2 \\nonumber \\\\\n&&\\hspace{2cm}+\\frac{g_2}{8}(\\partial_\\mu \\bm{n})^4\n \\Bigr]\\,.\n\\label{fsn2}\n\\end{eqnarray}\nFor $g_1>0$ and $g_2=0$, the action is identical to the FSN effective \naction~(\\ref{fsn_ac}). \n\nIn order to get the stable soliton solutions, $g_2$ must be \npositive~\\cite{gladikowski97}. However, $g_2$ is found to be \nnegative according to the above analysis.  \nTherefore we consider higher-derivative terms and investigate \nif the model with the higher-derivatives possess soliton solutions. \n\n\\section{\\label{sec:level4}Search for the stable soliton solutions (1)\\protect\\\\}\nThe static energy is derived from Eq.(\\ref{fsn2}) as  \n\\begin{eqnarray}\nE_{\\rm stt}&=& \\int d^3x\\Bigl[\\frac{1}{2}(\\partial_i \\bm{n})^2\n+\\frac{g_1}{8}(\\partial_i \\bm{n}\\times \\partial_j \\bm{n})^2\n+\\frac{g_2}{8}(\\partial_i \\bm{n})^4 \\Bigr] \\nonumber \\\\\n&:=&E_2(\\bm{n})+E_4^{(1)}(\\bm{n})+E_4^{(2)}(\\bm{n})\\,.\n\\end{eqnarray}\nA spatial scaling behaviour of the static energy, so called Derrick's scaling \nargument, can be applied to examine the stability of the soliton~\\cite{sutcliffe05}. \nConsidering the map \n$\\bm{x}\\mapsto \\bm{x}'=\\mu\\bm{x}~(\\mu>0)$, with \n$\\bm{n}^{(\\mu)}\\equiv \\bm{n}(\\mu\\bm{x})$, the static energy scales as \n\\begin{eqnarray}\ne(\\mu)&=&E_{\\rm stt}(\\bm{n}^{(\\mu)}) \\nonumber \\\\\n&=&E_2(\\bm{n}^{(\\mu)})+E_4^{(1)}(\\bm{n}^{(\\mu)})+E_4^{(2)}(\\bm{n}^{(\\mu)}) \\nonumber \\\\\n&=&\\frac{1}{\\mu}E_2(\\bm{n})+\\mu(E_4^{(1)}(\\bm{n})+E_4^{(2)}(\\bm{n}))\\,.\n\\label{derrick}\n\\end{eqnarray}\nDerrick's theorem states that if the function $e(\\mu)$ has no stationary point, \nthe theory has no static solutions of the field equation with finite density, \nother than the vacuum. \nConversely, if $e(\\mu)$ has stationary point, the  possibility of \nhaving finite energy soliton solutions is not excluded. \nEq.(\\ref{derrick}) is stationary at $\\mu=\\sqrt{E_2\/(E_4^{(1)}+E_4^{(2)})}$. \nThen, the following inequality\n\\begin{eqnarray}\n&&g_1(\\partial_i\\bm{n}\\times\\partial_j\\bm{n})^2+g_2(\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2 \\nonumber \\\\\n&&=g_1(\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2-g_1(\\partial_i\\bm{n}\\cdot\\partial_j\\bm{n})^2\n+g_2(\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2 \\nonumber \\\\\n&&\\geqq g_2(\\partial_i\\bm{n}\\cdot\\partial_j\\bm{n})^2\n~~~~(\\because (\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2\\geqq (\\partial_i\\bm{n}\\cdot\\partial_j\\bm{n})^2) \\nonumber \\\\\n\\end{eqnarray}\nensures the possibility of existence of the stable soliton solutions for $g_2\\geqq 0$. \nAs mentioned in the section \\ref{sec:level3}, $g_2$ should be negative at least within \nour derivative expansion analysis of YM theory. \n\nA promising idea to tackle the problem was suggested by Gies~\\cite{gies01}.\nHe considered the following type of effective action, accompanying second \nderivative term \n\\begin{eqnarray}\n&&S_{\\rm eff}=\\int d^4x\\Bigl[\\frac{1}{2}(\\partial_\\mu \\bm{n})^2\n+\\frac{g_1}{8}(\\partial_\\mu \\bm{n}\\times \\partial_\\nu \\bm{n})^2 \n\\nonumber \\\\\n&&\\hspace{2cm}-\\frac{g_2}{8}(\\partial_\\mu \\bm{n})^4\n+\\frac{g_2}{8}(\\partial^2 \\bm{n}\\cdot\\partial^2 \\bm{n})\n \\Bigr]\\,.\n\\label{fsn_ac2}\n\\end{eqnarray}\nHere we choose positive value of $g_2$ and assign the explicit negative sign \nto the third term. In principle, it is possible to estimate the second derivative term \nby the derivative expansion without neglecting \nthroughout the calculation.\nThe calculation is, however, very laborious and hence we show only \none simple example of the $C$ determinant.\nThe determinant is real and thus it is expanded as follows\n\\begin{eqnarray}\n&&\\log(\\det M^C)^{-1\/2}\n=-\\frac{1}{2}{\\rm Tr}\\log(-\\partial^2+\\partial_\\mu\\bm{n}\\cdot\\partial_\\mu\\bm{n}) \\nonumber \\\\ \n&&~~\\to-\\frac{1}{4}{\\rm Tr}\\log[\\partial^4\n-2(\\partial\\bm{n})^2\\partial^2+(\\partial\\bm{n})^4-\\partial^2(\\partial\\bm{n})^2]\n\\nonumber \\\\\n&&~~=-\\frac{1}{4}{\\rm Tr}\\log(\\partial^4) \\nonumber \\\\\n&&~~~~~-\\frac{1}{4}{\\rm Tr}\\log\\Bigl[1-2\\frac{(\\partial\\bm{n}^2)}{\\partial^2}\n+\\frac{(\\partial\\bm{n})^4}{\\partial^4}\n-\\frac{\\partial^2(\\partial\\bm{n})^2}{\\partial^4}\\Bigr] \\nonumber \\\\\n&&~~=-\\frac{1}{4}{\\rm Tr}\\log(\\partial^4) \\nonumber \\\\\n&&~~~~~-\\frac{1}{4}{\\rm Tr}\\Bigl[-2\\frac{(\\partial\\bm{n}^2)}{\\partial^2}\n+\\frac{(\\partial\\bm{n})^4}{\\partial^4}\n-\\frac{\\partial^2(\\partial\\bm{n})^2}{\\partial^4}\\Bigr] \\nonumber \\\\\n&&~~~~~+\\frac{1}{8}{\\rm Tr}\\Bigl[-2\\frac{(\\partial\\bm{n})^2}{\\partial^2}\\Bigr]^2\n+O((\\partial\\bm{n})^6)\n\\end{eqnarray}\nwhere we have defined $\\partial_\\mu\\bm{n}\\cdot\\partial_\\mu\\bm{n}\\to(\\partial\\bm{n})^2,\n(\\partial_\\mu\\bm{n}\\cdot\\partial_\\nu\\bm{n})^2\\to(\\partial\\bm{n})^4$.\nEmploying the integral formulas~\\cite{gies01}\n\\begin{eqnarray}\n&&\\int_{[k,\\Lambda]} \\frac{d^4p}{(2\\pi)^4}\\frac{1}{p^2}=\\frac{1}{16\\pi^2}(\\Lambda^2-k^2)\\,, \\nonumber \\\\\n&&\\int_{[k,\\Lambda]} \\frac{d^4p}{(2\\pi)^4}\\frac{1}{p^4}=\\frac{1}{8\\pi^2}\\log\\frac{\\Lambda}{k} \\nonumber \n\\end{eqnarray}\ntogether with the equality (up to second derivative)\n\\begin{eqnarray}\n\\partial^2(\\partial_\\mu\\bm{n}\\cdot\\partial_\\mu\\bm{n})=-\\partial^2(\\bm{n}\\cdot\\partial^2\\bm{n})\n= -\\partial^2\\bm{n}\\cdot\\partial^2\\bm{n}\\,,\n\\end{eqnarray}\nwe obtain the form for the $C$ determinant\n\\begin{eqnarray}\n&&\\log(\\det M^C)^{-1\/2}= \\nonumber \\\\\n&&-\\frac{1}{32\\pi^2}\\int d^4x\\Bigl[(\\Lambda^2-k^2) (\\partial_\\mu\\bm{n})^2\n-\\log\\frac{\\Lambda}{k}(\\partial_\\mu\\bm{n}\\times\\partial_\\nu\\bm{n})^2  \\nonumber \\\\ \n&&\\hspace{1.5cm}+\\log\\frac{\\Lambda}{k}(\\partial_\\mu\\bm{n})^4\n-\\log\\frac{\\Lambda}{k}(\\partial^2\\bm{n}\\cdot\\partial^2\\bm{n})\\Bigr]\\,.\n\\end{eqnarray}\nThe other determinants can be estimated in a similar manner. \n\nThe static energy of Eq.(\\ref{fsn_ac2}) with the ansatz (\\ref{afz}) is written as \n\\begin{eqnarray}\n&&E_{\\rm stt}=2\\pi^2a \\int d\\eta\n\\Biggl[\n\\frac{(w')^2}{1-w^2}+(1-w^2) U_{M,N}(\\eta) \\nonumber \\\\\n&&+\\frac{g_1}{4a^2}\\sinh\\eta\\cosh\\eta (w')^2\nU_{M,N}(\\eta)\\nonumber \\\\\n&&-\\frac{g_2}{4a^2}\\sinh\\eta\\cosh\\eta\\biggl[\\frac{(w')^2}{1-w^2}\n+(1-w^2)U_{M,N}(\\eta)\\biggr]^2 \\nonumber \\\\\n&&+\\frac{g_2}{4a^2}\\biggl[\\Bigl(\\coth\\eta+\\sinh^2\\eta-\\sinh\\eta\\cosh\\eta\\Bigr)\n\\frac{(w')^2}{1-w^2} \\nonumber \\\\\n&&\\hspace{1cm}+(\\sinh\\eta\\cosh\\eta-\\sinh^2\\eta)(1-w^2)M^2 \\nonumber \\\\\n&&\\hspace{1cm}+2\\Bigl\\{ \\frac{w(w')^3}{(1-w^2)^2}\n+\\frac{w' w''}{1-w^2}\n+w w'U_{M,N}(\\eta) \\Bigr\\}\\nonumber \\\\\n&&\\hspace{1cm}+\\sinh\\eta\\cosh\\eta\\Bigl\\{ \\frac{1}{1-w^2}\\Bigl[\\frac{(w')^2}{1-w^2}\n+w w'' \\nonumber \\\\\n&&\\hspace{1.5cm}+(1-w^2)U_{M,N}(\\eta)\\Bigr]^2 \n+(w'')^2\\Bigr\\}\\biggr]\n\\Biggl], \\nonumber \\\\\n&&\\hspace{4cm}w''\\equiv \\frac{d^2w}{d\\eta^2}.\\nonumber \n\\nonumber \n\\end{eqnarray} \nThe Euler-Lagrange equation of motion is derived by \n\\begin{eqnarray}\n-\\frac{d^2}{d\\eta^2}\\Bigl(\\frac{\\partial E_{\\rm stt}}{\\partial w''}\\Bigr)\n+\\frac{d}{d\\eta}\\Bigl(\\frac{\\partial E_{\\rm stt}}{\\partial w'}\\Bigr)\n-\\frac{\\partial E_{\\rm stt}}{\\partial w}=0 \\,,\n\\end{eqnarray}\nwhich is too complicated and thus we adopt the following notation  \n\\begin{eqnarray}\n&&f_0(w,w',w'')+g_1f_1(w,w',w'') \\nonumber \\\\\n&&\\hspace{1.5cm}+g_2f_2(w,w',w'',w^{(3)},w^{(4)})=0\\,.\n\\label{fsn_eq2}\n\\end{eqnarray}\nHere $w^{(3)},w^{(4)}$ represent the third and the fourth derivative with \nrespect to $\\eta$. The first two terms of Eq.(\\ref{fsn_eq2}) are  \nidentical to those in Eq.(\\ref{fsn_eq}).\n\nUnfortunately, we could not find out stable soliton solutions from \nEq.(\\ref{fsn_eq2}) for any value of $g_2$. \n\nFrom the relation\n\\begin{eqnarray}\n\\int d^4x[(\\partial^2 \\bm{n}\\cdot\\partial^2 \\bm{n})-(\\partial_\\mu \\bm{n})^4]\n=\\int d^4x(\\partial^2 \\bm{n}\\times\\bm{n})^2\\,,\n\\end{eqnarray} \none easily finds that the static energy obtained from the last two terms \nin Eq.(\\ref{fsn_ac2}) \n\\begin{eqnarray}\n\\tilde{E}^{(2)}_4=\\int d^3x(\\partial^2 \\bm{n}\\times\\bm{n})^2\n\\label{energy2}\n\\end{eqnarray}\ngives the positive contribution. The total static energy is stationary \nat $\\mu=\\sqrt{E_2\/(E_4^{(1)}+\\tilde{E}_4^{(2)})}$ and hence the possibility \nof existence of soliton solutions is not excluded. \nAnd also, the positivity of Eq.(\\ref{energy2}) does not spoil the lower \nbound (\\ref{lowerbound}) of original SFN and the possibility is still  \nnot excluded, too.  \n\nTherefore, we suspect that the absence of the stable soliton is caused \nby the fact that {\\it higher derivative theory has no lower bound state}. \nWe shall investigate the lower bound in the higher derivative theory \nin detail in the next section. \n\n\\section{\\label{sec:level5}Higher derivative theory\\protect\\\\}\nIn this section, we address the basic problems in the higher derivative \ntheory~\\cite{pais,smilga,eliezer89,jaen,simon} which essentially falls \ninto two categories. The first problem concerns the increase in the  \nnumber of degrees of freedom. For example, if the theory contains \nsecond derivative terms, the equation of motion becomes up to the order \nin the fourth derivative. Thus, four parameters are required for \nthe initial conditions. If one considers more higher order terms, \nthe situation gets worse. However, this is not serious problem for \nour study because our concern is the existence of static soliton solutions. \nThe second problem is that the actions of the theory are not bounded from \nbelow. This feature makes the higher derivative theories unstable.\n\nThe lagrangian and the hamiltonian formalism with higher derivative was firstly developed\nby Ostrogradski~\\cite{ostrogradski}. We consider the lagrangian containing up to $n$th order derivatives\n\\begin{eqnarray}\nS=\\int dt {\\cal L}(q,\\dot{q},\\cdots,q^{(n)})\\,.\n\\end{eqnarray} \nTaking the variation of the action $\\delta S=0$ leads the Euler-lagrange equation of motion\n\\begin{eqnarray}\n\\sum_{i=0}^n (-1)^i\\frac{d^i}{dt^i}\\Bigl(\\frac{\\partial {\\cal L}}{\\partial q^{(i)}}\\Bigr)=0\\,.\n\\end{eqnarray}\nThe hamiltonian is obtained by introducing $n$ generalized momenta\n\\begin{eqnarray}\np_i=\\sum_{j=i+1}^n (-1)^{j-i-1}\\frac{d^{j-i-1}}{dt^{j-i-1}}\\Bigl(\\frac{\\partial {\\cal L}}{\\partial q^{(j)}}\\Bigr)\\,,\ni=1,\\cdots,n,\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n&&p_n=\\frac{\\partial {\\cal L}}{\\partial q^{(n)}}\\,, \\nonumber \\\\\n&&p_i=\\frac{\\partial {\\cal L}}{\\partial q^{(i)}}-\\frac{d}{dt}p_{i+1}\\,,~~i=1,\\cdots,n-1,\n\\label{canonical momenta}\n\\end{eqnarray}\nand $n$ independent variables\n\\begin{eqnarray}\n&&q_1\\equiv q\\,,\\nonumber \\\\\n&&q_i\\equiv q^{(i-1)}\\,,~~(i=2,\\cdots,n)\\,.\n\\end{eqnarray}\nThe lagrangian now depends on the $n$ coordinates $q_i$ and on the first derivative $\\dot{q}_n=q^{(n)}$.\nThe hamiltonian is defined as\n\\begin{eqnarray}\n{\\cal H}(q_i,p_i)=\\sum^n_{i=1}p_i\\dot{q}_i-{\\cal L}=\\sum^{n-1}_{i=1} p_i q_{i+1}+p_n \\dot{q}_n-{\\cal L}\\,.\n\\end{eqnarray}\nThe canonical equations of motion turn out to be\n\\begin{eqnarray}\n\\dot{q}_i=\\frac{\\partial {\\cal H}}{\\partial p_i}\\,,~~\\dot{p}_i=-\\frac{\\partial {\\cal H}}{\\partial q_i}\\,.\n\\end{eqnarray}\n\nWe consider a simple example including second derivative term \\cite{simon}, defined as \n\\begin{eqnarray}\n{\\cal L}=\\frac{1}{2}(1+\\varepsilon^2\\omega^2)\\dot{q}^2-\\frac{1}{2}\\omega^2q^2-\\frac{1}{2}\\varepsilon^2\\ddot{q}^2\\,,\n\\end{eqnarray}\nwhere constant $\\epsilon$ works as a coupling constant of second derivative term. \nThe equation of motion is\n\\begin{eqnarray}\n(1+\\varepsilon^2 \\omega^2)\\ddot{q}+\\omega^2 q+\\varepsilon^2 q^{(4)}=0\\,.\n\\label{eq_quanta}\n\\end{eqnarray}\nFrom Eq.~(\\ref{canonical momenta}), one gets \n\\begin{eqnarray}\n&&\\pi_{\\dot{q}}=\\frac{\\partial{\\cal L}}{\\partial \\ddot{q}}=-\\varepsilon^2\\ddot{q}\\,, \\nonumber \\\\\n&&\\pi_q=\\frac{\\partial{\\cal L}}{\\partial \\dot{q}}-\\frac{d}{dt}\\Bigl(\\frac{\\partial{\\cal L}}{\\partial \\ddot{q}}\\Bigr)\n=(1+\\varepsilon^2\\omega^2)\\dot{q}+\\varepsilon^2\\dddot{q}\\,.\n\\end{eqnarray}\nThus the hamiltonian becomes  \n\\begin{eqnarray}\n{\\cal H}&=&\\dot{x}\\pi_q+\\ddot{q}\\pi_{\\dot{q}}-{\\cal L} \\nonumber \\\\\n&=&\\dot{q}\\pi_q-\\frac{1}{2\\varepsilon^2}\\pi_{\\dot{q}}^2\n-\\frac{1}{2}(1+\\varepsilon^2\\omega^2)\\dot{q}^2+\\frac{1}{2}\\omega^2q^2\\,.\n\\end{eqnarray}\nWe introduce the new canonical variables\n\\begin{eqnarray}\n&&q_+=\\frac{1}{\\omega\\sqrt{1-\\varepsilon^2\\omega^2}}(\\varepsilon^2\\omega^2\\dot{q}-\\pi_q)~,\\nonumber \\\\\n&&p_+=\\frac{w}{\\sqrt{1-\\varepsilon^2\\omega^2}}(q-\\pi_{\\dot{q}})\\,, \\nonumber \\\\\n&&q_-=\\frac{\\varepsilon}{\\sqrt{1-\\varepsilon^2\\omega^2}}(\\dot{q}-\\pi_q)~, \\nonumber \\\\\n&&p_-=\\frac{1}{\\varepsilon\\sqrt{1-\\varepsilon^2\\omega^2}}(\\varepsilon^2\\omega^2 q-\\pi_{\\dot{q}})\\,,\n\\nonumber \n\\end{eqnarray}\nand the hamiltonian has of the form by using these variables\n\\begin{eqnarray}\n{\\cal H}\\to \\frac{1}{2}(p_+^2+\\omega^2 q_+^2)-\\frac{1}{2}(p_-^2+\\frac{1}{\\varepsilon^2} q_-^2)\\,. \\nonumber \n\\end{eqnarray}\nThe corresponding energy spectra is then given by\n\\begin{eqnarray}\nE=(n+\\frac{1}{2})\\omega-(m+\\frac{1}{2})\\frac{1}{\\varepsilon}~,~~n,m=0,1,2,\\cdots\n\\end{eqnarray}\nOne can see that in the limit $\\epsilon\\to 0$ the energy goes to \nnegative infinity rather than approaching to the harmonic oscillator \nenergy eigenstates.  \n\nTo obtain physically meaningful solutions, we employ the perturbative \nanalysis where the solution is expanded in terms of the small coupling \nconstant and the Euler-Lagrange equation of motion is replaced with the \ncorresponding perturbative equation. \nThe solutions of the equations of motion that are ill behaved in the limit \n$\\epsilon\\to 0$ are excluded from the very beginning~\\cite{eliezer89,jaen,simon}.\n\nWe assume that the solution of Eq.(\\ref{eq_quanta}) can be written as \n\\begin{eqnarray}\nq_{\\rm pert}(t)=\\sum^{\\infty}_{n=0} \\epsilon^n q(t)\\,. \\label{q}\n\\end{eqnarray}\nSubstituting Eq.(\\ref{q}) into Eq.(\\ref{eq_quanta}) and taking time \nderivatives of these equations, we obtain the constraints for higher \nderivative terms\n\\begin{eqnarray}\n&&O(\\epsilon^0)  \\nonumber \\\\\n&&\\hspace{3mm}equation :~~\\ddot{q}_0+\\omega^2q_0=0\\,, \\label{cnst00} \\\\\n&&\\hspace{3mm}constraints :~~\\dddot{q}_0=-\\omega^2\\dot{q}_0, \\ddddot{q}_0=\\omega^4 q_0\\,. \\label{cnst01} \\\\\n&&O(\\epsilon^2)  \\nonumber \\\\\n&&\\hspace{3mm}equation :~~\\ddot{q}_2+\\omega^2\\ddot{q}_0+\\omega^2 q_2+\\ddddot{q}_0=0\\,,  \\nonumber \\\\\n&&\\hspace{13mm}\\Rightarrow \\ddot{q}_2+\\omega^2q_2=0\\,,~~({\\rm using}~(\\ref{cnst00}),(\\ref{cnst01}))\\,, \\label{cnst20} \\\\\n&&\\hspace{3mm}constraints :~~\\dddot{q}_2=-\\omega^2\\dot{q}_2, \\ddddot{q}_2=\\omega^4 q_2\\,. \\label{cnst21} \\\\\n&&O(\\epsilon^4)  \\nonumber \\\\\n&&\\hspace{3mm}equation :~~\\ddot{q}_4+\\omega^2\\ddot{q}_2+\\omega^2 q_4+\\ddddot{q}_2=0\\,,  \\nonumber \\\\\n&&\\hspace{13mm}\\Rightarrow \\ddot{q}_4+\\omega^2q_4=0\\,,~~({\\rm using}~(\\ref{cnst20}),(\\ref{cnst21}))\\,, \\\\\n&&\\hspace{3mm}constraints :~~\\dddot{q}_4=-\\omega^2\\dot{q}_4, \\ddddot{q}_4=\\omega^4 q_4\\,. \\label{cnst4}\n\\end{eqnarray}\nCombining these results, we find the perturbative equation of motion up to $O(\\epsilon^4)$\n\\begin{eqnarray}\n\\ddot{q}_{\\rm pert}+\\omega^2q_{\\rm pert}=O(\\epsilon^6)\\,.\n\\end{eqnarray}\nwhich is the equation for harmonic oscillator. \n\n\\begin{figure}\n\\includegraphics[height=7cm, width=9cm]{Fig1}\n\\caption{\\label{fig:Fig1} The function $w(\\eta)$ for $g_1=0.4, g_2=0,0.05,0.1$\n(the rescaling radial coordinate $x=\\eta\/(1-\\eta)$ is used). }\n\\end{figure}\n\\begin{figure}\n\\includegraphics[height=7cm, width=9cm]{Fig2}\n\\caption{\\label{fig:Fig2} The energy density for $g_1=0.4, g_2=0,0.05,0.1$.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=7cm, width=9cm]{Fig3}\n\\caption{\\label{fig:Fig3} The energy as a function of $g_2$($g_1=0.4$).}\n\\end{figure}\n\n\\section{\\label{sec:level6}Search for the stable soliton solutions (2) -- perturbative analysis --\\protect\\\\}\nLet us employ the perturbative method introduced in the last section to our \nproblem. We assume that $g_2$ is relatively small and can be considered as \na perturbative coupling constant. Thus, the perturbative solution is written \nby a power series in $g_2$\n\\begin{eqnarray}\nw(\\eta)=\\sum^\\infty_{n=0}g_2^n w_n(\\eta)\\,.\n\\label{sol_expand}\n\\end{eqnarray}\nSubstituting Eq.(\\ref{sol_expand}) into Eq.(\\ref{fsn_eq2}), we obtain \nthe classical field equation in $O(g^0_2)$\n\\begin{eqnarray}\nf_0(w_0,w_0',w_0'')+g_1f_1(w_0,w_0',w_0'')=0\\,. \\label{classical}\n\\end{eqnarray}\nTaking derivatives for both sides in Eq.(\\ref{classical}) and solving \nfor $w_0'',w_0^{(3)},w_0^{(4)}$ read the following form\nof the constraint equations for higher derivatives\n\\begin{eqnarray}\nw_0^{(i)}=F^{(i)}(w_0,w_0')\\,,~~i=2,3,4\\,.\n\\label{constraint}\n\\end{eqnarray}\nThe equation in $O(g^1_2)$ can be written as \n\\begin{eqnarray}\n(f_0+g_2 f_1)_{O(g^1_2)}+f_2(w_0,w_0',w_0'',w_0^{(3)},w_0^{(4)})=0\\,.\n\\label{fsn_eq21}\n\\end{eqnarray}\nSubstituting the constraint equations (\\ref{constraint}) into Eq.(\\ref{fsn_eq21})\nand eliminate the higher derivative terms, one can obtain the perturbative \nequation of motion\n\\begin{eqnarray}\nf_0(w,w',w'')+g_1 f_1(w,w',w'')+g_2 \\tilde{f}_2(w,w')=O(g^2_2)\\,.\\nonumber \\\\\n\\label{fsn_eq2_p}\n\\end{eqnarray}\n\nNow Eq.(\\ref{fsn_eq2_p}) has topological soliton solutions. \nOur results of the estimated function $w(\\eta)$ and the energy density are displayed \nin Figs.\\ref{fig:Fig1},\\ref{fig:Fig2}.(In all figures, we show the results for the \ncase of Hopf charge $H=2;N=2,M=1$). We have small changes for varying the \ncoupling constant $g_2$. The dependence of the $g_2$ for the total energy \nis shown in Fig.\\ref{fig:Fig3}. \nIt can be seen that the change is moderate with respect to $g_2$.\n\n\\section{\\label{sec:level7}Summary\\protect\\\\}\nIn this paper we have studied the Skyrme-Faddeev-Niemi\nmodel and its extensions by introducing the reduction scheme of the SU(2) \nYang-Mills theory to the corresponding low-energy effective model. \nThe requirement of consistency between the low-energy effecive action \nof the YM and the SFN type model lead us to take into account second \nderivative terms in the action. \nHowever, we found that such an action including the second derivative terms \ndoes not have stable soliton solutions.\nThis is due to the absence of the energy bound in higher \nderivative theory. \nThis fact inspired us to employ the perturbative analysis to our effective \naction. Within the perturbative analysis, we were able to obtain the topological \nsoliton solutions. \n \nOur analysis is based on perturbation and the coupling constant $g_2$ is \nassumed to be small.  \nHowever, Wilsonian renormalization analysis of YM theory~\\cite{gies01} \nsuggest that the coupling constants $g_1,g_2$ (and the mass scale parameter \n$\\Lambda$) depend on the renormalization group time $t=\\log k\/\\Lambda$ \n($k,\\Lambda$ are infrared, ultraviolet cutoff parameter) and those are \nalmost comparable. To improve the analysis, we could perform the \nnext order of perturbation, but it is tedius and spoils the simplicity \nof the FSN model unfortunately. \n\nIt should be noted that our solutions do not much differ from \nthe solution of original SFN model, at least in the perturbative regime. \nWe suspect that some appropriate truncation ({\\it like} ``extra fourth order term \n+ second derivative term'') always supply the stable solutions that are \nclose to the original SFN model. Thus we conclude that the topological \nsoliton model comprised of the ``kinetic term + a special fourth order \nterm'' like SFN model is a good approximation. \n\n\\begin{acknowledgments}\nThe authors thank to Kei-Ichi Kondo for drawing our attention to the \ncoefficient problem of this model.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\setcounter{lemma}{0}}\n\n\n\n\n\n\n\\title{\\textbf{Quantum Edge Correspondences and Quantum Cuntz--Krieger Algebras}}\n\\author{    Michael Brannan \n            \\footnote{Department of Pure Mathematics and the Institute for Quantum Computing, University of Waterloo \\hfill \\url{michael.brannan@uwaterloo.ca}}\n            \n            \\and\n            \n            Mitch Hamidi\n            \\footnote{Department of Mathematics, Embry-Riddle Aeronautical University \\hfill \\url{hamidim@erau.edu}}\n            \n            \\and\n            \n            Lara Ismert\n            \\footnote{Department of Mathematics, Embry-Riddle Aeronautical University \\hfill \\url{ismertl@erau.edu}}\n            \n           \n            \n            \\and\n            \n            Brent Nelson\n            \\footnote{Department of Mathematics, Michigan State University \\hfill \\url{brent@math.msu.edu}}\n            \n             \\and\n            \n            Mateusz Wasilewski\n            \\footnote{Institute of Mathematics, Polish Academy of Sciences \\hfill\n            \\url{mwasilewski@impan.pl}}\n        }\n\n\n\n\\begin{document}\n\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\nGiven a quantum graph $\\cG=(B,\\psi,A)$, we define a C*-correspondence $E_\\cG$ over the noncommutative vertex C*-algebra $B$, called the \\emph{quantum edge correspondence}. For  a classical graph $\\cG$, $E_\\cG$ is the usual graph correspondence spanned by the edges of $\\cG$. When the quantum adjacency matrix $A\\colon B\\to B$ is completely positive, we show that $E_\\cG$ is faithful if and only if $\\ker(A)$ does not contain a central summand of $B$. In this case, we show that the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ is isomorphic to a quotient of the quantum Cuntz--Krieger algebra $\\cO(\\cG)$ defined in \\cite{BEVW20}. Moreover, the kernel of the quotient map is shown to be generated by ``localized'' versions of the quantum Cuntz--Krieger relations, and $\\cO_{E_\\cG}$ is shown to be the universal object associated to these local relations.  We study in detail some concrete examples and make connections with the theory of Exel crossed products.\n\\end{abstract}\n\n\n\n\\section*{Introduction}\nThe notion of a quantum graph goes back to the work of Erdos-Katavolos-Shulman \\cite{ErKaSh98} and Weaver \\cite{We12}, and was subsequently developed further by Duan-Severini-Winter \\cite{DuSeWi13} and Musto-Reutter-Verdon \\cite{MuReVe19}. Quantum graphs play an intriguing role in the study of the graph isomorphism game in quantum information via their connections with quantum symmetries of graphs (see \\cite{MR4097284} and \\cite{BCEHPSWCMP19}). Moreover, based on the use of quantum symmetries, fascinating results on the graph theoretic interpretation of quantum isomorphisms between finite graphs were recently obtained by Man\\v{c}inska-Roberson \\cite{MaRo19}.\nIn this paper, we take a finite directed quantum graph to mean a triple $(B,\\psi, A)$ consisting of a finite-dimensional C*-algebra $B$, a state $\\psi$ on $B$, and a linear map $A$ on $B$ satisfying a quantum Schur idempotent condition. Finite directed quantum graphs generalize classical finite directed graphs (without multiple edges) by encoding a classical graph $\\cG$ in the triple $(B,\\psi, A)$, where $B = C(V)$ is the C*-algebra of continuous functions on the vertex set $V$ of $\\mathcal G$, $\\psi$ is integration with respect to the uniform probability on $V$, and $A$ is the adjacency matrix of $\\cG$.   \n\n\nIn \\cite{BEVW20}, given a finite directed quantum graph $(B,\\psi,A)$ Eifler, Voigt, Weber and first author introduced a novel C*-algebra $\\bF\\cO(B, \\psi, A)$, called the \\textit{free quantum Cuntz--Krieger algebra}. This generalizes the well-studied Cuntz--Krieger algebra $\\cO_A$ arising from classical graphs \n(or rather a free version of it), where the standard generators are replaced by matrix-valued valued partial isometries whose matrix sizes are determined by the quantum graph, and the Cuntz--Krieger relations are expressed using the quantum adjacency matrix of the quantum graph in analogy to the scalar case.\nIntroduced in \\cite{ck1}, Cuntz--Krieger algebras have intimate connections with symbolic dynamics, and have been studied intensively in the framework of graph C*-algebras over the past decades, thus providing a rich supply of interesting examples \\cite{Raeburn, T17, ERRS18, ERRS21}. The structure of graph C*-algebras is understood to an impressive level of detail, and many algebraic properties can be interpreted in terms of the combinatorial properties of the underlying graphs. Motivated by this success, the original constructions and results have been generalized in several directions, including higher rank graphs \\cite{KuPa00}, Exel-Laca algebras \\cite{ExLa99}, and ultragraph algebras \\cite{To03}, among others.\nRecall that, under mild assumptions, the  Cuntz--Krieger algebra corresponding to a classical graph is isomorphic to the Cuntz--Pimsner algebra associated to the graph's edge correspondence \\cite[Example 2, p. 193]{Pimsner}. It is worth mentioning that in the more general setting of quantum graphs, the free quantum Cuntz--Krieger algebras seem to be difficult to compute in general, and their isomorphism classes are only known under very strict assumptions (e.g., when $(B,\\psi,A)$ is \\emph{complete} and $\\psi$ is an $n^2$-form for $n\\in \\mathbb{N}$; see \\cite[Theorem 4.5]{BEVW20}).\n\nIn the present paper we consider a natural unital version of $\\mathbb{F}\\mathcal{O}(B,\\psi,A)$, which we denote $\\mathcal{O}(B,\\psi,A)$, and under the assumption that $A$ is completely positive we show that $\\mathcal{O}(B,\\psi,A)$ quotients onto the Cuntz--Pimsner algebra associated to a C*-correspondence over $B$ which can be viewed as the quantum analogue of the edge correspondence for a classical graph. This is accomplished by showing that  this Cuntz--Pimsner algebra is the universal C*-algebra associated to ``local'' versions of the quantum Cuntz--Krieger relations introduced in \\cite{BEVW20}.\n\nIn Section~\\ref{sec:quantumedgecorrespondences}, we define the quantum edge correspondence $E_\\cG$ for a given quantum graph $\\cG = (B,\\psi,A)$ whose associated state is a $\\delta$-form. This C*-correspondence is generated by a generalized version of the Choi--Jamio\\l kowski matrix $\\epsilon_\\cG$ associated to the quantum adjacency operator $A$, and $E_\\cG$ generalizes the usual edge correspondence for a classical graph (see Example~\\ref{exmp:commutativecase}). Given the role of the element $\\epsilon_\\cG$ in generating this analogue of an edge correspondence, we think of it as the quantum analogue of a classical graph's edge matrix. Theorem~\\ref{thm:faithful_full_correspondence} identifies conditions on the quantum adjacency matrix which result in faithfulness and fullness of the quantum edge correspondence, and Proposition \\ref{prop:recognition} provides a recognition theorem for when a cyclic C*-correspondence is the quantum edge correspondence for a quantum graph. \n  \nIn Section~\\ref{sec:QCK_and_local_relations}, we introduce a natural quotient of the quantum Cuntz--Krieger algebra $\\cO(\\cG)$ by introducing certain``local\" relations  on the generators, and we call this quotient a {\\em local quantum Cuntz--Krieger algebra}. Given a quantum graph $\\cG$, we show in Theorem \\ref{thm:universal_local_quantum_Cuntz--Krieger_algebra} that the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ constructed from the quantum edge correspondence is precisely the local quantum Cuntz--Krieger algebra for that same quantum graph. As mentioned previously, the free quantum Cuntz--Krieger algebras (and their unital cousins) seem to be very difficult to describe concretely, except in the most basic cases.  This sentiment is emphasized by the fact that we are unable to find an example of a quantum graph $\\cG$ whose local quantum Cuntz--Krieger algebra is a {\\it proper} quotient of $\\cO(\\cG)$.\n\nIn Section~\\ref{sec:simpler_examples}, we focus on examples, beginning with a very special class of quantum graphs called the {\\it complete quantum graphs} $\\cG = K(B,\\psi)$.  Here we study the associated quantum Cuntz--Krieger algebras and Cuntz--Pimsner algebras, and make some connections to Exel's theory of crossed products of C*-algebras by endomorphisms \\cite{Exel03}. \nIn the classical setting, any square $\\{0,1\\}$-matrix $A = [A(x,y)]$ gives rise to the Markov subshift  $(X_A, \\sigma)$, which is the topological dynamical system given by the infinite compact path space $X_A = \\{x = (x_i) \\in \\{1, \\ldots, n\\}^{\\mathbb N}\\ \\big| \\ A(x_i,x_{i+1}) = 1 \\ \\forall i \\ge 1  \\}$ together with the left shift action $\\sigma:X_A \\to X_A$ given by $\\sigma(x)_i = x_{i+1}$.  From this dynamical system $(X_A, \\sigma)$, one can associate an {\\it Exel system} $(C(X_A),\\alpha,\\mathcal{L})$,  where $C(X_A)$ is the unital C*-algebra of continuous functions on $X_A$,  $\\alpha: C(X_A) \\to C(X_A)$ is the $*$-endomorphism defined by $\\alpha(f) = f \\circ \\sigma$, and $\\mathcal{L}:C(X_A) \\to C(X_A)$ is a {\\it transfer operator} for $\\alpha$. In \\cite{Exel03}, Exel builds from this data a crossed product C$^\\ast$-algebra $C(X_A)\\rtimes_{\\alpha, \\mathcal{L}} \\mathbb{N}$, and shows that it is isomorphic to the usual Cuntz--Krieger algebra $\\mathcal{O}_A$.  For any complete quantum graph $K = K(B,\\psi)$, we associate a natural choice of (non-commutative) Exel system, and show in Proposition \\ref{thm:crossed-product-is-a-Cuntz-algebra}, that this crossed product is isomorphic to the Cuntz algebra $\\cO_{n}$ on $n = \\dim B$ generators.  We also show in Proposition \\ref{prop:Cuntz--Pimsner-is-a-Cuntz-algebra} that the Cuntz--Pimsner algebra $\\cO_{E_K}$ is isomorphic to $\\cO_n$.  These results combined generalize the well known identifications of Cuntz--Krieger algebras, Cuntz--Pimsner algebras, and Exel crossed products associated to complete graphs.\n\n    \nFinally, in the other subsections of Section~\\ref{sec:simpler_examples} of the paper, we study trivial (edgeless) quantum graphs and their two natural generalizations: rank-one quantum graphs, and quantum graphs associated to $\\ast$-automorphic quantum adjacency matrices.  We are able  associate a natural choice of (non-commutative) Exel system in these cases as well, except for rank-one quantum graphs. Given one of these types of quantum graphs, we show that the Exel crossed product is isomorphic to the Cuntz--Pimsner algebra for that quantum graph's quantum edge correspondence.  See Corollary \\ref{cor:triv}, Proposition \\ref{prop:triv}, and Corollary \\ref{cor:triv2}. These examples also mimic the classical setting, where a graph's associated Cuntz--Krieger algebra, Exel crossed product, and Cuntz--Pimsner algebra are all isomorphic.\n\nLet us end this introduction with a remark: For more general quantum graphs $\\cG = (B,\\psi, A)$, it is an interesting and natural problem to associate to $\\cG$ a quantum analogue of (functions on) the path space $X_A$. The construction of an appropriate noncommutative version of $C(X_A)$ seems to be highly non-trivial, and we plan to investigate this in more detail in a followup work.\n\n\n\\subsection*{Acknowledgments} Michael Brannan was partially supported by NSF Grant DMS-2000331. Brent Nelson was partially supported by NSF Grant DMS-1856683. Lara Ismert was partially supported by the NSF-AWM Mentoring Travel Grant. Mateusz Wasilewski was partially supported by the Research Foundation \u2014 Flanders (FWO) through a Postdoctoral Fellowship, by long term structural funding \u2013 Methusalem grant of the Flemish\nGovernment \u2013 and by the European Research Council Starting Grant 677120 INDEX.\n\n\n\n\n\n\n\\section{Preliminaries}\n\\subsection{Quantum graphs}\nIn this paper, we consider \\emph{finite quantum spaces} $(B,\\psi)$, consisting of a finite dimensional C*-algebra $B$ and a distinguished faithful state $\\psi\\colon B\\to \\bC$ satisfying\n    \\[\n        mm^* = \\delta^2\\text{id},\n    \\]\nwhere $m\\colon B\\otimes B\\to B$ is the multiplication map, $m^*$ is its adjoint with respect to the inner product given by $\\psi$, and $\\delta>0$. States $\\psi:B \\to \\bC$ satisfying the above identity are called \\emph{$\\delta$-forms}. Since $B$ is finite dimensional, we have\n    \\[\n        B\\cong \\bigoplus_{a=1}^d M_{N_a}(\\bC).\n    \\]\nThe restriction of $\\psi$ to the $a^{th}$ summand $M_{N_a}(\\mathbb{C})$ appearing in the direct sum decomposition of $B$ will be given by a density matrix with respect to the usual trace, denoted $\\rho_a$. We may and will assume that each $\\rho_a$ is diagonal with the $i^{th}$ diagonal entry being $\\psi(e_{ii}^{a})$. The condition that $\\psi$ is a $\\delta$-form is then equivalent to $\\text{Tr}(\\rho_a^{-1})=\\sum_{i=1}^{N_a} \\psi(e_{ii}^{a})^{-1}=\\delta^2$ for each $a=1,\\ldots, d$.\n\nWe utilize the diagonal entries of these density matrices to define \\emph{adapted matrix units} for $(B,\\psi)$, given by\n    \\[\n        f_{ij}^{(a)}:= \\frac{1}{\\left[\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})\\right]^{1\/2}} e_{ij}^{(a)} \\qquad \\qquad 1\\leq a\\leq d,\\ 1\\leq i,j\\leq N_a,\n    \\]\nand by \\cite[Lemma 3.2]{BEVW20} they satisfy\n    \\begin{equation}\\label{Eq:adaptedunits}\n        m^*(f_{ij}^{(a)}) = \\sum_{k=1}^{N_a} f_{ik}^{(a)} \\otimes f_{kj}^{(a)}.\n    \\end{equation}\n\\begin{equation}\\label{Eq:prodadaptedunits}\n    f_{ij}^{(a)}f_{rs}^{(b)} = \\delta_{\\substack{a=b\\\\ j=r}} \\frac{1}{\\psi(e_{jj}^{(a)})} f_{is}^{(a)}. \n\\end{equation}\n\n\n\n\nGiven a finite quantum space $(B,\\psi)$, a linear map $A\\colon B\\to B$ is said to be a \\emph{quantum adjacency matrix} if\n    \\[\n        m(A\\otimes A)m^* = \\delta^2 A.\n    \\]\nIn this case, the triple $(B,\\psi,A)$ is called a \\emph{quantum graph}. At times, it will be convenient to express $A$ as an actual matrix $[A_{ija}^{rsb}]_{\\substack{1 \\le a,b \\le m \\\\\n1 \\le i,j \\le N_a \\\\\n1 \\le r,s \\le N_b}}$ with respect to the   basis of adapted matrix units:    \\[\n        A(f_{ij}^{(a)}) = \\sum_{b=1}^d \\sum_{r,s=1}^{N_b} A_{ija}^{rsb} f_{rs}^{(b)}.\n    \\]\nThese coefficients can be used to directly check whether a linear map $A\\colon B\\to B$ is a quantum adjacency matrix (see \\cite[Lemma 3.4]{BEVW20}), but this will not be necessary in the present paper.\n\n\n\n\n\n\\subsection{C*-correspondences and Cuntz--Pimsner algebras}\\label{subsec:correspondences_and_CP_algebras}\n\nGiven a C*-algebra $B$, a \\emph{C*-correspondence} over $B$ is a right Hilbert $B$-module $X$ (with right $B$-valued innner product $\\langle \\cdot, \\cdot \\rangle_B$) which admits a $*$-homomorphism $\\varphi_X\\colon B\\to \\cL(X)$. Here $\\cL(X)$ denotes the right $B$-linear adjointable operators on $X$. The $*$-homomorphism $\\varphi_X$ induces a left $B$-action that commutes with the right $B$-action:\n    \\[\n        x\\cdot \\xi:=\\varphi_X(x)\\xi \\qquad \\qquad x\\in B,\\ \\xi\\in X.\n    \\]\nOne says $X$ is \\emph{faithful} if $\\varphi_X$ is faithful, and \\emph{full} if $\\overline{\\text{span}}\\<X,X\\>_B=B$. The \\emph{compact operators} on $X$, denoted $\\cK(X)$, are generated by $\\theta_{\\xi,\\eta}\\in \\cL(X)$, $\\xi,\\eta\\in X$, where\n    \\[\n        \\theta_{\\xi,\\eta}(\\zeta) := \\xi\\cdot \\<\\eta,\\zeta\\>_B \\qquad \\zeta\\in X.\n    \\]\nIn this paper, we will exclusively consider finite dimensional C*-correspondences, for which $\\cK(X)=\\cL(X)$. The following example is particularly relevant for our purposes.\n\n\\begin{example}\\label{example:correspondence_from_cp_map}\nLet $B$ be a finite dimensional C*-algebra and $A\\colon B\\to B$ a completely positive map. Define a $B$-valued inner product on $B\\otimes B$ by\n    \\[\n        \\< a\\otimes b, c\\otimes d\\>_B := b^* A(a^* c)d,\n    \\]\nwhere the positivity follows from the complete positivity of $A$. Then after taking a separation,\n    \\[\n        B\\otimes_A B:= B\\otimes B\/\\{\\xi\\in B\\otimes B\\colon \\<\\xi,\\xi\\>_B=0\\}\n    \\]\ndefines a C*-correspondence over $B$. (Since everything is finite-dimensional, we do not need to take a completion.) The usual left and right actions of $B$ on $B\\otimes B$, \n    \\[\n        x\\cdot (a\\otimes b)\\cdot y = (xa)\\otimes (by),\\qquad a,b,x,y\\in B\n    \\]\nextend to left and right actions on $B\\otimes_A B$.\n\\end{example}\n\n\n\nAfter \\cite{Kat04}, a \\emph{representation} of a C*-correspondence $X$ over $B$ on a C*-algebra $D$ is a pair $(\\pi,t)$ consisting of a $*$-homomorphism $\\pi\\colon B\\to D$ and a linear map $t\\colon X\\to D$ satisfying:\n    \\begin{enumerate}[label=(\\roman*)]\n        \\item $\\pi(x)t(\\xi)=t(x\\cdot \\xi)$ for $x\\in B$ and $\\xi\\in X$,\n        \n        \\item $t(\\xi)^*t(\\eta)=\\pi(\\<\\xi,\\eta\\>_B)$ for $\\xi,\\eta\\in X$.\n    \\end{enumerate}\nUsing (i), one can also show $t(\\xi)\\pi(x)= t(\\xi\\cdot x)$ for $x\\in B$ and $\\xi\\in X$. One can also define a $*$-homomorphism $\\psi_t\\colon \\cK(X)\\to D$ by $\\psi_t(\\theta_{\\xi,\\eta})=t(\\xi)t(\\eta)^*$. A representation is said to be \\emph{covariant} if $\\pi(x) = \\psi_t(\\varphi_X(x))$ for all $x$ in the \\emph{Katsura ideal} $J_X$, which is defined by\n    \\[\n        J_X:=\\{x\\in B\\colon \\varphi_X(x)\\in \\cK(X) \\text{ and } xy=0 \\text{ for all }y\\in \\ker{\\varphi_X}\\}.\n    \\]\nNote that $J_X=B$ when $X$ is finite dimensional and faithful, which is the situation we will primarily consider. \n\nThe \\emph{Cuntz--Pimsner algebra} for a C*-correspondence $X$ over $B$ is the C*-algebra $\\cO_X=C^*(\\pi_X(B),t_X(X))$ where $(\\pi_X,t_X)$ is the universal covariant representation of $X$. That is, given any covariant representation $(\\pi,t)$ of $X$ on a C*-algebra $D$, there exists a $*$-homomorphism $\\rho \\colon \\cO_X\\to D$ satisfying $\\pi=\\rho\\circ \\pi_X$ and $t=\\rho\\circ t_X$. \n\n\n\n\n\n\n\n\\section{Quantum edge correspondences}\\label{sec:quantumedgecorrespondences}\n\nLet $(B,\\psi)$ be a finite quantum space.  Since the state $\\psi:B \\to \\bC \\subset B$ is completely positive we can consider the C*-correspondence $B\\otimes_\\psi B$ over $B$ from Example~\\ref{example:correspondence_from_cp_map}. Note that for $\\xi\\in B\\otimes B$, $\\psi(\\<\\xi,\\xi\\>_B)=\\|\\xi\\|^2_{\\psi\\otimes \\psi}$, and so $B\\otimes_\\psi B=B\\otimes B$ as a vector space. That is, we do not need to take a quotient of $B\\otimes B$.\n\n\\begin{definition}\\label{def:edgecorrespondence}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form. We define the {\\it quantum edge indicator} to be the element \\[\\epsilon_\\cG:=\\frac{1}{\\delta^2}(1\\otimes A)m^*(1) \\in B\\otimes_\\psi B.\\] The \\textit{quantum edge correspondence} of $\\cG$ is the C*-correspondence over $B$ defined by\n    \\[\n        E_\\cG:=B\\cdot \\epsilon_\\cG \\cdot B =\\text{span}\\{x\\cdot \\epsilon_\\cG\\cdot y \\colon x,y\\in B\\} \\subset B\\otimes_\\psi B.\n    \\]\nThat is, $E_\\cG$ is the C*-subcorrespondence of $B\\otimes_\\psi B$ generated by  the quantum edge indicator $\\epsilon_\\cG$.\n\\end{definition}\n\n\\begin{example}\\label{exmp:commutativecase}\n    In the case of a classical directed graph $\\mathcal{G}=( \\bC(V), \\frac{1}{|V|}, A)$, one has\n    \\[\n        \\epsilon_\\cG=\\frac{1}{\\delta^2}(1\\otimes A)m^*(1) = \\frac{1}{|V|}(1\\otimes A)\\sum_{v\\in V} |V| p_v\\otimes p_v = \\sum_{w\\to v} p_v\\otimes p_w.\n    \\]\nHence $\\epsilon_\\cG\\in \\bC(V\\times V)$ is the indicator function for the set $\\{(v,w)\\colon (w,v) \\text{ is an edge}\\}$, and the quantum edge correspondence $E_\\cG$ is the space of functions supported on this set.\n\\end{example} \n\n\n\n\\subsection{Properties of quantum edge indicators and correspondences}\n\nBefore studying the edge correspondence, we note some important properties of the quantum edge indicator $\\epsilon_\\cG$ which further justify our terminology. \n\n\\begin{proposition}\\label{prop:properties_of_edge_generator}\nLet $\\mathcal{G}=(B,\\psi,A)$ be a directed quantum graph with $\\delta$-form $\\psi$, and let $\\epsilon_\\cG:= \\frac{1}{\\delta^2}(1\\otimes A)m^*(1)$ be the quantum edge indicator. Then:\n    \\begin{enumerate}[label=(\\arabic*)]\n        \\item $A(x) = \\delta^2 (\\psi\\otimes 1)(x\\cdot \\epsilon_\\cG)$ for all $x\\in B$.\n    \n        \\item $\\epsilon_\\cG\\# \\epsilon_\\cG=\\epsilon_\\cG$ where $(a\\otimes b)\\#(c\\otimes d)=(ac)\\otimes (db)$ for $a,b,c,d\\in B$.\n        \n        \\item $A$ is completely positive if and only if $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\epsilon_\\cG)$ is self-adjoint, where $(\\sigma_t^\\psi)_{t \\in \\mathbb R}$ denotes the modular automorphism group of $\\psi$.\n    \\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\\begin{enumerate}\n    \\item[\\textbf{(1):}] For $x,y\\in B$, we have\n    \\begin{align*}\n        \\< \\delta^2 (\\psi\\otimes 1)(x\\cdot \\epsilon_\\cG), y\\>_\\psi &= \\delta^2 \\< \\epsilon_\\cG, x^*\\otimes y\\>_{\\psi\\otimes \\psi}\\\\\n            &= \\< m^*(1), x^* \\otimes A^*(y)\\>_{\\psi\\otimes \\psi}\\\\\n            &= \\< 1, x^* A^*(y)\\>_\\psi\\\\\n            &= \\< A(x), y\\>_\\psi.\n    \\end{align*}\nHence $\\delta^2 (\\psi\\otimes 1)(x\\cdot \\epsilon_\\cG)=A(x)$ as claimed.\n    \n    \\item[\\textbf{(2):}] Observe that $\\epsilon_\\cG$ is the image of $1\\otimes 1$ under the following map on $B\\otimes B$:\n        \\[\n            \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1).\n        \\]\n    We will show that this map is a left and right $B$-linear and idempotent.  (In fact, $E_\\cG$ is precisely the image of $B\\otimes B$ under this map.) \n    Bimodularity comes from the fact that both $m$ and $m^*$ are left and right $B$-linear. Using the associativity of the multiplication operation (i.e. $m(m\\otimes 1) = m(1\\otimes m)$, hence also $(m^*\\otimes 1)m^* = (1\\otimes m^*)m^*$) and that $A$ is a quantum adjacency matrix (i.e. $\\frac{1}{\\delta^2} m(A\\otimes A)m^*=A$), we will show that this map is idempotent. Its square is equal to \n    \\[\n    \\frac{1}{\\delta^4}     \\left(1\\otimes (m(1\\otimes m)\\right)  \\left(1\\otimes A\\otimes A \\otimes 1\\right)\\left((m^*\\otimes 1)m^* \\otimes 1\\right).\n    \\] \n    Using associativity, we get\n    \\[\n    \\frac{1}{\\delta^4}(1\\otimes m)(1\\otimes m \\otimes 1)(1\\otimes A \\otimes A \\otimes 1)(1\\otimes m^* \\otimes 1)(m^* \\otimes 1).\n    \\]\n    In the middle, we recognize the expression $1\\otimes (m(A\\otimes A)m^*)\\otimes 1$, which is equal to $\\delta^2 1\\otimes A \\otimes 1$. In the end we obtain\n    \\[\n    \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A \\otimes 1)(m^*\\otimes 1),\n    \\]\n    which verifies the map is idempotent. Consequently,\n        \\begin{align*}\n            \\epsilon_\\cG\\# \\epsilon_\\cG &=  \\epsilon_\\cG\\# \\frac{1}{\\delta^2} (1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)(1\\otimes 1)\\\\\n                &= \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)(\\epsilon_\\cG)\\\\\n                &= \\left[\\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)\\right]^2(1\\otimes 1)\\\\\n                &= \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)(1\\otimes 1) = \\epsilon_\\cG.\n        \\end{align*}\n    (This identity can also be checked directly using the adapted matrix units of $(B,\\psi)$, but this arduous task is left to the skeptical reader.)\n    \n    \\item[\\textbf{(3):}] Let $\\eta_\\cG\\in B\\otimes B^\\text{op}$ be the image of $\\epsilon_\\cG$ under the map $a\\otimes b\\mapsto a\\otimes b^\\circ$. Then $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\epsilon_\\cG)$ is self-adjoint if and only if $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG)$ is self-adjoint. Observe that $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG)$ is an idempotent by part (2), and so it is self-adjoint if and only if it is a projection, and hence if and only if it is positive. It therefore suffices to show $A$ is completely positive if and only if $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG)$ is positive. \n    \n    Recall that if $(M,\\varphi)$ is a von Neumann algebra equipped with a faithful normal linear functional, then $\\overline{\\{\\Delta_\\varphi^{1\/4} x\\colon x\\in M_+\\}}$ is a self-dual cone in $L^2(M,\\varphi)$, where $\\Delta_\\varphi$ is the modular operator with respect to $\\varphi$. Consequently $A$ is completely positive if and only if for all positive elements $X=(x_{ij}), Y=(y_{ij})\\in M_n(B)^+$ one has\n        \\[\n            \\< \\Delta_{\\psi\\otimes \\text{Tr}}^{1\/4} (A\\otimes I_n)(X), \\Delta_{\\psi\\otimes \\text{Tr}}^{1\/4}Y\\>_{\\psi\\otimes \\text{Tr}}= \\sum_{i,j=1}^n \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi \\geq 0.\n        \\]\n    Using part (1), we compute\n        \\[\n            \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi = \\< \\delta^2(\\psi\\otimes 1)(x_{ij}\\cdot \\epsilon_\\cG), \\Delta_\\psi^{1\/2} y_{ij}\\>_\\psi = \\delta^2 \\< \\epsilon_\\cG, x_{ij}^*\\otimes (\\Delta_\\psi^{1\/2} y_{ij})\\>_{\\psi\\otimes \\psi}.\n        \\]    \nNow, $L^2(B\\otimes B,\\psi\\otimes\\psi)\\ni a\\otimes b\\mapsto a\\otimes [\\Delta_\\psi^{-1\/2} b]^\\circ \\in L^2(B\\otimes B^{\\text{op}},\\psi\\otimes\\psi^{\\text{op}})$ is an isometry, so we can continue the above computation with\n        \\begin{align*}\n             \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi &= \\delta^2 \\< (1\\otimes \\Delta_\\psi^{-1\/2}) \\eta_\\cG, x_{ij}^*\\otimes y_{ij}^\\circ\\>_{\\psi\\otimes \\psi^{\\text{op}}}\\\\\n             &= \\delta^2\\< (\\Delta_\\psi^{1\/4}\\otimes \\Delta_\\psi^{-1\/4})( \\Delta_\\psi^{-1\/2}\\otimes 1)\\eta_\\cG, (\\Delta_\\psi^{1\/4}\\otimes \\Delta_\\psi^{-1\/4})( x_{ij}^*\\otimes y_{ij}^\\circ) \\>_{\\psi\\otimes \\psi^\\text{op}}\\\\\n            &= \\delta^2 \\< \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}\\left[ (\\sigma_{i\/2}^{\\psi}\\otimes 1)(\\eta_\\cG)\\right], \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}(x_{ij}^*\\otimes y_{ij}^\\circ)\\>_{\\psi\\otimes \\psi^{\\text{op}}}.\n        \\end{align*}\nThus\n    \\begin{align}\\label{eqn:complete_positivity_equation}\n        \\sum_{i,j=1}^n \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi & \\nonumber\\\\\n            = \\delta^2 \\sum_{i,j=1}^n &\\< \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}\\left[ (\\sigma_{i\/2}^{\\psi}\\otimes 1)(\\eta_\\cG)\\right], \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}(x_{ij}^*\\otimes y_{ij}^\\circ)\\>_{\\psi\\otimes \\psi^{\\text{op}}}.\n    \\end{align}\nSuppose $X=W^*W$ and $Y=Z^*Z$ for $W=(w_{ij}), Z=(z_{ij})\\in M_n(B)$. Then\n    \\[\n        \\sum_{i,j=1}^n x_{ij}^*\\otimes y_{ij}^\\circ = \\sum_{i,j,k,\\ell=1}^n w_{kj}^*w_{ki}\\otimes (z_{\\ell i}^* z_{\\ell j})^\\circ = \\sum_{k,\\ell=1}^n (\\sum_{j=1}^n w_{kj}\\otimes (z_{\\ell j}^*)^\\circ)^* (\\sum_{i=1}^n w_{ki}\\otimes (z_{\\ell i}^*)^\\circ)\n    \\]\nis positive. Also note that every positive element of $B\\otimes B^\\text{op}$ can be presented this way. Indeed,\n    \\[\n        (\\sum_{d=1}^m w_d\\otimes z_d^\\circ)^*(\\sum_{d=1}^m w_d\\otimes z_d^\\circ) = \\sum_{i,j=1}^n x_{ij}^*\\otimes y_{ij}^\\circ\n    \\] \nif $X=W^*W, Y=Z^*Z\\in M_m(B)$ where $(W)_{ij} = \\delta_{i=1} w_j$ and $(Z)_{ij}=\\delta_{i=1} z_i^*$. It follows from (\\ref{eqn:complete_positivity_equation}) and these observations that $A$ is completely positive if and only if $(\\sigma_{i\/2}^{\\psi}\\otimes 1)(\\eta_\\cG)$ is positive.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{remark}\nAt this point it is worth remarking on the connection to Weaver's notion of a quantum graph \\cite{We12, We21}. Recall that there is an algebra isomorphism \n    \\begin{align*}\n        \\pi:B \\otimes B^{{\\text{op}}} &\\to {}_{B'}CB_{B'}(B(L^2(B,\\psi)))\\\\\n        (a\\otimes b^\\circ)&\\mapsto (T\\mapsto aTb) \n    \\end{align*} \n    where ${}_{B'}CB_{B'}(B(L^2(B,\\psi)))$ denotes the space of completely bounded $B'$-$B'$-bimodule maps on $B(L^2(B,\\psi))$.   When a quantum adjacency matrix $A$ is completely positive, the projection $p = (\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG) \\in B\\otimes B^{\\text{op}}$ induces a  $B'$-$B'$-bimodule projection $\\pi(p)$ on $B(L^2(B, \\psi))$ whose range $S \\subseteq B(L^2(B,\\psi))$ is a $B'$-$B'$-bimodule. Such bimodules $S$ are exactly what Weaver refers to as (directed, non-reflexive) quantum graphs on $B \\subseteq B(L^2(B,\\psi))$.\n\\end{remark}\n\n\n\\begin{theorem}\\label{thm:cp_Adjacency_matrices_are_inner_products}\nLet $\\cG=(B,\\psi, A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive. Then for $x,y\\in B$ one has\n    \\[\n        \\< x \\cdot \\epsilon_\\cG, y\\cdot \\epsilon_\\cG\\>_B = \\frac{1}{\\delta^2}A(x^*y),\n    \\]\nwhere $\\epsilon_\\cG = \\frac{1}{\\delta^2} (1\\otimes A)m^*(1)$ is the quantum edge indicator. In particular, $A(x)=\\delta^2\\<\\epsilon_\\cG, x\\cdot \\epsilon_\\cG\\>_B$ for all $x\\in B$.\n\\end{theorem}\n\\begin{proof}\nWrite $\\epsilon_\\cG = \\sum_\\alpha p_\\alpha\\otimes q_\\alpha$ for $p_\\alpha,q_\\alpha\\in B$. Then\n    \\begin{align*}\n         \\< x \\cdot \\epsilon_\\cG, y \\cdot \\epsilon_\\cG\\>_B &= \\sum_{\\alpha,\\beta} \\< xp_\\alpha\\otimes q_\\alpha, yp_\\beta\\otimes q_\\beta\\>_B\\\\\n            &= \\sum_{\\alpha, \\beta} \\psi( p_\\alpha^* x^* y q_\\beta) q_\\alpha^* q_\\beta\\\\\n            &=  \\sum_{\\alpha, \\beta} \\psi(x^*y p_\\beta \\sigma_{-i}^\\psi(p_\\alpha^*)) q_\\alpha^* q_\\beta \\\\\n            &=  (\\psi\\otimes 1)[ x^*y\\cdot \\epsilon_\\cG \\# (\\sigma_{-i}^\\psi\\otimes 1)(\\epsilon_\\cG^*)]\\\\\n            &=\\ (\\psi\\otimes 1)( x^*y\\cdot \\epsilon_\\cG \\# \\epsilon_\\cG)\\\\\n            &= (\\psi\\otimes 1)( x^*y\\cdot \\epsilon_\\cG) =  \\frac{1}{\\delta^2} A(x^*y),\n    \\end{align*}\nwhere the last three equalities follow from the three parts of  Proposition~\\ref{prop:properties_of_edge_generator} (in reverse order).\n\\end{proof} \n\nRecalling the definition of $B\\otimes_A B$ from Example~\\ref{example:correspondence_from_cp_map}, we obtain the following corollary:\n\n\\begin{corollary}\\label{cor:quantum_graph_correspondence_as_cp_correspondence}\nLet $\\cG=(B,\\psi, A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive. Then $E_\\cG\\cong B\\otimes_A B$ as C*-correspondences over $B$ via the map $x\\cdot \\epsilon_\\cG\\cdot y\\mapsto \\frac{1}{\\delta}(x\\otimes y)$.\n\\end{corollary}\n\n\nThe above corollary is somewhat surprising in that a priori $B\\otimes_A B$ appears to be completely independent of $\\psi$. But of course the dependence is hidden in the fact that $A$ is a quantum adjacency matrix for $(B,\\psi)$. We also remark that while $B\\otimes_A B$ is more easily defined, $E_\\cG$ has the advantage of not requiring us to quotient by any null spaces.\n\n\nGiven a finite quantum space $(B,\\psi)$ such that $\\psi$ is a $\\delta$-form, Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products} tells us every completely positive adjacency matrix is of the form $A(x)=\\delta^2\\<\\xi, x\\cdot \\xi\\>_B$ for some $\\xi\\in B\\otimes_\\psi B$. Thus it is natural to ask for which elements $\\xi\\in B\\otimes_\\psi B$ is the map $B\\ni x\\mapsto \\delta^2\\< \\xi,x\\cdot \\xi\\>_B$ a (necessarily completely positive) quantum adjacency matrix? Suppose $\\xi\\#\\xi=\\xi$ and $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)$ is self-adjoint, and define $A_\\xi(x):= \\delta^2 \\<\\xi,x\\cdot \\xi\\>_B$, then the same computation as in the above theorem implies\n    \\[\n        A_\\xi(x)=\\delta^2\\< \\xi, x\\cdot \\xi\\>_B = \\delta^2(\\psi\\otimes 1)(x\\cdot \\xi).\n    \\]\nConsequently, for each $1\\leq a \\leq d$ and $1\\leq i,j\\leq N_a$\n    \\begin{align*}\n        \\frac{1}{\\delta^2} m(A_\\xi\\otimes A_\\xi)m^*(f_{ij}^{(a)}) &= \\frac{1}{\\delta^2}\\sum_{k} A_\\xi( f_{ik}^{(a)}) A_\\xi(f_{kj}^{(a)})\\\\\n        &= \\delta^2 \\sum_k (\\psi\\otimes 1)(f_{ik}^{(a)}\\cdot \\xi) (\\psi\\otimes 1)(f_{kj}^{(a)}\\cdot \\xi).\n    \\end{align*}\nWriting $\\xi= \\sum_\\alpha p_\\alpha\\otimes q_\\alpha$, we can continue the above with\n    \\begin{align*}\n        \\frac{1}{\\delta^2} m(A_\\xi\\otimes A_\\xi)m^*(f_{ij}^{(a)}) &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\<f_{ki}^{(a)}, p_\\alpha\\>_\\psi\\<f_{jk}^{(a)}, p_\\beta\\>_\\psi q_\\alpha q_\\beta\\\\\n        &=\\delta^2 \\sum_{k,\\alpha,\\beta} \\<f_{ki}^{(a)}, p_\\alpha\\>_\\psi\\<f_{ik}^{(a)}, \\psi(e_{ii}^{(a)}) f_{ij}^{(a)} p_\\beta\\>_\\psi q_\\alpha q_\\beta\\\\\n        &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\< \\<\\psi(e_{ii}^{(a)}) f_{ij}^{(a)} p_\\beta, f_{ik}^{(a)}\\>_\\psi f_{ki}^{(a)}, p_\\alpha\\>_\\psi  q_\\alpha q_\\beta\\\\\n        &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\< \\< \\Delta_\\psi f_{ki}^{(a)}, \\psi(e_{ii}^{(a)}) p_\\beta^* f_{ji}^{(a)}\\>_\\psi f_{ki}^{(a)}, p_\\alpha\\>_\\psi  q_\\alpha q_\\beta\\\\\n        &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\psi(e_{kk}^{(a)}) \\< \\< f_{ki}^{(a)}, p_\\beta^*f_{ji}^{(a)}\\>_\\psi f_{ki}^{(a)}, p_\\alpha\\>_\\psi  q_\\alpha q_\\beta.\n    \\end{align*}\nNow, using the fact that $\\{ \\psi(e_{rr}^{(b)})^{1\/2}f_{rs}^{(b)}\\colon 1\\leq b \\leq d,\\ 1\\leq r,s\\leq N_b \\}$ is an orthonormal basis for $L^2(B)$, we then have\n    \\begin{align*}\n       \\frac{1}{\\delta^2} m(A_\\xi\\otimes A_\\xi)m^*(f_{ij}^{(a)}) &= \\delta^2 \\sum_{r,s,b,\\alpha,\\beta} \\< \\< \\psi(e_{rr}^{(b)})^{1\/2}f_{rs}^{(b)}, p_\\beta^*f_{ji}^{(a)}\\>_\\psi \\psi(e_{rr}^{(b)})^{1\/2}f_{rs}^{(b)}, p_\\alpha\\>_\\psi  q_\\alpha q_\\beta\\\\\n        &= \\delta^2 \\sum_{\\alpha, \\beta} \\<p_\\beta^* f_{ji}^{(a)}, p_\\alpha\\>_\\psi q_\\alpha q_\\beta\\\\\n        &= \\delta^2 \\sum_{\\alpha,\\beta } (\\psi\\otimes 1)( f_{ij}^{(a)}\\cdot (p_\\beta \\otimes q_\\beta)\\# (p_\\alpha\\otimes q_\\alpha))\\\\\n        &= \\delta^2 \\sum_\\alpha (\\psi\\otimes 1)( f_{ij}^{(a)}\\cdot (p_\\alpha \\otimes q_\\alpha)) = A_\\xi(f_{ij}^{(a)}).\n    \\end{align*}\nThus $A_\\xi$ is a quantum adjacency matrix for $(B,\\psi)$. Moreover, by a similar computation one can show that $\\frac{1}{\\delta^2}(1\\otimes A_\\xi)m^*(1)=\\xi$ and so there is in fact a one-to-one correspondence between completely positive quantum adjacency matrices for $(B,\\psi)$ and the set\n    \\[\n        \\{\\xi\\in B\\otimes_\\psi B\\colon  \\xi\\#\\xi=\\xi,\\ (\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)^*=(\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)\\}.\n    \\]\nMore generally, we have the following recognition result for the quantum edge correspondence.\n\n\\begin{proposition}\\label{prop:recognition}\nLet $(B,\\psi)$ be a finite quantum space such that $\\psi$ is a $\\delta$-form, and let $X=B\\cdot\\xi\\cdot B$ be a cyclic C*-correspondence over $B$. If the map $A\\colon B\\to B$ defined by $A(x)=\\delta^2\\<\\xi, x\\cdot \\xi\\>_B$ is a quantum adjacency matrix for $(B,\\psi)$, then $x\\cdot \\xi\\cdot y\\mapsto x\\cdot  \\epsilon_{(B,\\psi,A)}\\cdot y$ extends to a C*-correspondence isomorphism from $X$ to the quantum edge correspondence $E_{(B,\\psi, A)}$.\n\\end{proposition}\n\\begin{proof}\nIf we denote $\\epsilon:=\\epsilon_{(B,\\psi,A)}$, then Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products} implies\n    \\[\n        \\<e_{ij}^{(a)}\\cdot \\xi, e_{rs}^{(b)}\\cdot \\xi\\>_B = \\delta_{\\substack{a=b\\\\i=r}} \\<\\xi, e_{js}^{(a)}\\cdot \\xi\\>_B = \\delta_{\\substack{a=b\\\\i=r}} \\frac{1}{\\delta^2}A(e_{js}^{(a)}) = \\< e_{ij}^{(a)}\\cdot\\epsilon, e_{rs}^{(b)}\\cdot \\epsilon\\>_B\n    \\]\nfor all $1\\leq a, b\\leq d$, $1\\leq i,j\\leq N_a$, and $1\\leq r,s\\leq N_b$. It follows that for all $x,y,z,w\\in B$ that\n    \\[\n        \\<x\\cdot \\xi\\cdot y, z\\cdot \\xi\\cdot w\\>_B = y^* \\< x\\cdot \\xi, z\\cdot \\xi\\>_B w = y^*\\< x\\cdot \\epsilon, z\\cdot \\epsilon\\>_B w = \\< x\\cdot \\epsilon\\cdot y, z\\cdot \\epsilon\\cdot w\\>_B.\n    \\]\nTherefore $\\sum_i x_i\\cdot \\xi\\cdot y_i\\mapsto \\sum_i x_i\\cdot\\epsilon\\cdot y_i$ is a well-defined, inner product preserving (and hence injective), $B$-bilinear map from $B\\cdot\\xi\\cdot B=X$ onto $B\\cdot \\epsilon\\cdot B =E_\\cG$. In other words, it is a C*-correspondence isomorphism.\n\\end{proof}\n\n\nIt is straightforward to check that any homomorphism $A:B\\to B$ is a quantum adjacency matrix for a quantum space $(B,\\psi)$ (independent, in fact, of $\\psi$). Furthermore, every homomorphism of a C*-algebra that is completely positive must necessarily be $*$-preserving, so it is natural to ask if the one-to-one correspondence in Proposition~\\ref{prop:recognition} restricts to a one-to-one correspondence between $*$-homomorphisms and a subset of $\\{\\xi\\in B\\otimes_\\psi B\\colon  \\xi\\#\\xi=\\xi,\\ (\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)^*=(\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)\\}.$ \n\n\\begin{proposition}\\label{prop:when_A_is_a_homomorphism}\nLet $(B,\\psi)$ be a finite quantum space, and suppose $A:B\\to B$ is a completely positive quantum adjacency matrix for $(B,\\psi).$ The following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $A$ is a homomorphism.\n\\item For all $x,y\\in B$, we have $(xy)\\cdot \\epsilon_\\cG=x\\cdot \\epsilon_\\cG\\cdot A(y)$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nSuppose for all $x,y\\in B$, we have $(xy)\\cdot \\epsilon_\\cG=x\\cdot \\epsilon_\\cG\\cdot A(y).$ Proposition~\\ref{prop:recognition} implies\n$$\n    A(xy)\n    =\\langle \\epsilon_\\cG, (xy)\\cdot \\epsilon_\\cG\\rangle_B\n    =\\langle \\epsilon_\\cG, x\\cdot \\epsilon_\\cG\\cdot A(y)\\rangle_B\n     =\\langle \\epsilon_\\cG, x\\cdot \\epsilon_\\cG\\rangle_B A(y)\n     =A(x)A(y)\n$$\nfor all $x,y\\in B.$ Conversely, suppose $A$ is a homomorphism, and fix $x,y\\in B$. For any $a,b\\in B$,\n$$\n\\langle a\\cdot \\epsilon_\\cG \\cdot b, (xy)\\cdot \\epsilon_\\cG\\rangle_B\n=\nb^* A(a^*xy)%\n=\nb^* A(a^*x)A(y)\n=\n\\langle a\\cdot\\epsilon_\\cG\\cdot b, x\\cdot \\epsilon_\\cG\\cdot  A(y)\\rangle_B\n$$\nAs elements of the form $a\\cdot \\epsilon_\\cG\\cdot b$ span $E_\\cG$, for any $\\xi\\in E_\\cG$, \n$$\\langle \\xi, (xy)\\cdot \\epsilon_\\cG - x\\cdot \\epsilon_\\cG \\cdot A(y)\\rangle_B=0\\quad \\text{for all}\\;x,y\\in B.$$ In particular, $\\xi_o:=(xy)\\cdot \\epsilon_\\cG - x\\cdot \\epsilon_\\cG \\cdot A(y)$ is an element of $E_\\cG$, so $\\langle \\xi_o,\\xi_o\\rangle_B=0$. By positive-definiteness of $\\langle \\cdot, \\cdot \\rangle_B,$ we may conclude $\\xi_o=0.$ Thus, $xy\\cdot \\epsilon_\\cG=x\\cdot \\epsilon_\\cG\\cdot A(y)$ for all $x,y\\in B.$\n\\end{proof}\n\nWhen $A$ is a $*$-automorphism, the quantum edge correspondence $E_\\cG$ arising from a quantum graph $\\cG:=(B,\\psi,A)$ is the span of $B\\cdot \\epsilon_\\cG$ or the span of $\\epsilon_\\cG\\cdot B$. Indeed, given $x\\cdot \\epsilon_\\cG\\cdot y\\in E_\\cG$ for some $x,y\\in B$, Proposition~\\ref{prop:when_A_is_a_homomorphism} implies that the maps $E_\\cG\\to E_\\cG$ given by $x\\cdot \\epsilon_G\\cdot y\\mapsto xA^{-1}(y)\\cdot \\epsilon_\\cG$ and  $x\\cdot \\epsilon_\\cG\\cdot y\\mapsto \\epsilon_\\cG\\cdot A(x)y$ are both the identity map on $E_\\cG.$\n\n\\subsection{Faithfulness and fullness of the quantum edge correspondence}\n\n\nRecall the definitions of faithful and full for C*-correspondences from Section~\\ref{subsec:correspondences_and_CP_algebras}. The next theorem shows that these features for a quantum edge correspondence $E_\\cG$ are determined by the quantum adjacency matrix $A$.\n\n\\begin{theorem}\\label{thm:faithful_full_correspondence}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG$ be the quantum edge correspondence. If $1_a:=\\sum_{i=1}^{N_a} e_{ii}^{(a)}$ for each $1\\leq a\\leq d$, then\n    \\[\n        \\{x\\in B\\colon x\\cdot \\xi=0\\ \\forall \\xi\\in E_\\cG\\}= \\left(BA^{\\ast}(B)B\\right)^{\\perp}\n   \n    \\]\nand\n    \\[\n        \\overline{\\text{span}}\\<E_\\cG, E_\\cG\\>_B = \n       \n        B\\cdot A(B) \\cdot B,\n    \\]\n    the two-sided ideal of $B$ generated by the range of $A$. \nIn particular, $E_\\cG$ is faithful if and only if  $\\ker(A)$ does not contain a central summand of $B$, and $E_\\cG$ is full if and only if $A(B)$ is not orthogonal to a central summand of $B$.\n\\end{theorem}\n\\begin{proof}\nSuppose $x\\cdot \\xi=0$ for all $\\xi\\in E_\\cG$. This means precisely that for all $a,b \\in B$ we have $xa\\epsilon_{\\cG} b=0$, which, on the other hand, is equivalent to $\\langle c \\epsilon_{\\cG} d, xa \\epsilon_{\\cG} b\\rangle=0$ for all $c,d \\in B$. This expression is equal to $\\delta^{-2}d^* A(c^* xa) b=0$, so it is equal to zero if and only if $A(c^* x a)=0$. This time using the inner product on $B$, we see that this is equivalent to $0=\\langle y, A(c^* x a)\\rangle = \\langle c A^{\\ast}(x) a^*, x \\rangle $ for all $y \\in B$. Therefore $x \\in \\left(B A^{\\ast}(B) B\\right)^{\\perp}$.\n\n\n \n       \n  \n   \n\n \n  \n\n   \n    \n \n\nNext, observe that\n\\begin{align*}\n    \\overline{\\text{span}}\\langle E_\\cG, E_\\cG\\rangle_B &= \\overline{\\text{span}}\\big\\{\\langle a \\cdot \\epsilon_\\cG \\cdot b, c \\cdot \\epsilon_\\cG \\cdot d\\rangle_B: a,b,c,d \\in B\\big\\} \\\\\n    &= \\overline{\\text{span}}\\big\\{\\delta^{-2}b^*A(a^*c)d: a,b,c,d \\in B\\big\\} \\\\\n    &=B \\cdot A(B) \\cdot B \n\\end{align*}\n\n\\end{proof}\n\n\\begin{remark}\nIn the case of a classical directed graph $\\cG=( C(V), \\psi, A)$, the central summands of $C(V)$ are indexed by $V$. A central summand belongs to $\\ker(A)$ when the corresponding vertex is a source (i.e. has no edges into it), and it is orthogonal to $A(C(V))$ when the corresponding vertex is a sink (i.e. has no edges out of it). Hence $E_\\cG$ is faithful when $A$ has no zero columns and is full when $A$ has no zero rows.\n\\end{remark}\n\nThe previous remark motivates the following definitions.\n\n\\begin{definition}\\label{def:quantum-sink-and-source}\n Let $\\cG=(B,\\psi, A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive. A \\textit{quantum sink} in $\\cG$ is a central summand of $B$ that is orthogonal to the range of $A$. A \\textit{quantum source} in $\\cG$ is a central summand of $B$ that lies in the kernel of $A$\n\\end{definition}\n\nIn the next section we will examine the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ associated to the quantum edge correspondence. Consequently, it is important to understand how to express the left $B$-action in terms of compact operators $\\cK(E_\\cG)$.\n\n\\begin{theorem}\\label{thm:compact_operators}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG=B\\cdot \\epsilon_\\cG\\cdot B$ be the quantum edge correspondence. If $\\{ f_{ij}^{(a)}\\colon 1\\leq a\\leq d,\\ 1\\leq i,j\\leq N_a\\}$ are the adapted matrix units for $(B,\\psi)$, then\n    \\[\n        f_{ij}^{(a)}\\cdot \\xi = \\sum_{k=1}^{N_a} \\theta_{f_{ik}^{(a)}\\cdot \\epsilon_\\cG,f_{jk}^{(a)}\\cdot \\epsilon_\\cG}(\\xi)\n        \\quad\n        \\text{for all }\n        \\xi\\in E_\\cG.\n    \\]\n\n\\end{theorem}\n\\begin{proof}\nSince both sides are right $B$-linear in $\\xi$, it suffices to prove the equality for $\\xi=e_{rs}^{(b)}\\cdot \\epsilon_\\cG$, $1\\leq b\\leq d$ and $1\\leq r,s\\leq N_b$. Write $\\epsilon_\\cG=\\sum_\\alpha x_\\alpha\\otimes y_\\alpha$. Then using Proposition~\\ref{prop:properties_of_edge_generator}.(2) we have\n    \\begin{align*}\n        f_{ij}^{(a)}\\cdot (e_{rs}^{(b)}\\cdot \\epsilon_\\cG) &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} e_{is}^{(b)}\\cdot \\epsilon_\\cG\\\\\n        &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} e_{is}^{(b)}\\cdot (\\epsilon_\\cG\\# \\epsilon_\\cG)\\\\\n        &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k,\\alpha,\\beta} (e_{is}^{(b)}x_\\alpha e_{kk}^{(a)} x_\\beta)\\otimes (y_\\beta y_\\alpha)\\\\\n        &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k,\\alpha,\\beta} (x_\\alpha)_{sk}^{(a)} (e_{ik}^{(a)}x_\\beta)\\otimes (y_\\beta y_\\alpha)\\\\\n        &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k,\\alpha,\\beta} \\frac{1}{\\psi(e_{kk}^{(a)})} \\psi( e_{ks}^{(a)}x_\\alpha)(e_{ik}^{(a)}x_\\beta)\\otimes (y_\\beta y_\\alpha)\\\\\n        &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k} \\frac{1}{\\psi(e_{kk}^{(a)})} e_{ik}^{(a)}\\cdot \\epsilon_\\cG \\cdot (\\psi\\otimes 1)(e_{ks}^{(a)}\\cdot \\epsilon_\\cG).\n    \\end{align*}\nNow, using Proposition~\\ref{prop:properties_of_edge_generator}.(1) and Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products} we see that\n    \\[\n        (\\psi\\otimes 1)(e_{ks}^{(a)}\\cdot \\epsilon_\\cG) = \\frac{1}{\\delta^2} A(e_{ks}^{(a)}) = \\< \\epsilon_\\cG, e_{ks}^{(a)}\\cdot \\epsilon_\\cG\\>_B = \\<  e_{jk}^{(a)}\\cdot \\epsilon_\\cG , e_{js}^{(a)}\\cdot \\epsilon_\\cG\\>_B.\n    \\]\nSo continuing our computation above, we have\n    \\begin{align*}\n        f_{ij}^{(a)}\\cdot (e_{rs}^{(b)}\\cdot \\epsilon_\\cG) &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k} \\frac{1}{\\psi(e_{kk}^{(a)})} e_{ik}^{(a)}\\cdot \\epsilon_\\cG \\cdot \\<  e_{jk}^{(a)}\\cdot \\epsilon_\\cG , e_{js}^{(a)}\\cdot \\epsilon_\\cG\\>_B\\\\\n            &= \\sum_{k} f_{ik}^{(a)}\\cdot \\epsilon_\\cG \\cdot \\< f_{jk}^{(a)}\\cdot \\epsilon_\\cG, e_{rs}^{(b)}\\cdot \\epsilon_\\cG\\>_B\\\\\n            &= \\sum_k \\theta_{f_{ik}^{(a)}\\cdot \\epsilon_\\cG, f_{jk}^{(a)}\\cdot \\epsilon_\\cG}( e_{rs}^{(b)}\\cdot \\epsilon_\\cG),\n    \\end{align*}\nas claimed.\n\\end{proof}\n\nThe following corollary is a rephrasing of Theorems~\\ref{thm:cp_Adjacency_matrices_are_inner_products} and \\ref{thm:compact_operators} in terms of linear maps, which will be useful in the next section. We leave the proof to the reader.\n\n\\begin{corollary}\\label{cor:abstract_Toeplitz}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG=B\\cdot \\epsilon_\\cG\\cdot B$ be the quantum edge correspondence. If $(\\pi,t)$ is a covariant representation of $E_\\cG$ on C*-algebra $D$, then the linear map $T\\colon B\\to D$ defined by $T(x):= t(x\\cdot \\epsilon_\\cG)$ satisfies\n    \\begin{align*}\n        \\mu_D(T^*\\otimes T) &= \\frac{1}{\\delta^2}\\pi A m,\\\\\n        \\mu_D(T\\otimes T^*) m^* &=\\psi_t,\n    \\end{align*}\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map, $T^*(x):=T(x^*)^*$, and $\\psi_t\\colon \\cK(E_\\cG)\\to D$ is the $*$-homomorphism induced by $t$.\n\\end{corollary}\n\n\n\n\n\n\n\n\n\\section{Quantum Cuntz--Krieger algebras and local relations}\\label{sec:QCK_and_local_relations}\n\nIn this section we recall the quantum Cuntz--Krieger relations and define local quantum Cuntz--Krieger relations. The former differ slightly from those appearing in \\cite[Section 3.2]{BEVW20} (see Remark~\\ref{rem:unital_relation}). We will see in Theorem~\\ref{thm:universal_local_quantum_Cuntz--Krieger_algebra} below that the Cuntz--Pimsner algebra for a faithful quantum edge correspondence plays the role of the universal C*-algebra generated by local quantum Cuntz--Krieger relations. This in turn allows us to deduce that such Cuntz--Pimsner algebras are quotients of quantum Cuntz--Krieger algebras (see Corollary~\\ref{cor:quantum_Cuntz--Krieger_quotients}).\n\n\\begin{definition}\\label{def:QCK}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph. We define a \\textit{quantum Cuntz--Krieger $\\cG$-family} in a unital C*-algebra $D$ to be a linear map $s\\colon B\\to D$ such that:\n    \\begin{enumerate}[label=(\\roman*)]\n        \\item $\\mu_D(\\mu_D\\otimes 1)(s\\otimes s^* \\otimes s)(m^* \\otimes 1)m^*=s$ $\\hfill (\\textbf{QCK1})$\n    \n        \\item $\\mu_D(s^*\\otimes s)m^* = \\mu_D(s\\otimes s^*)m^*A$ $\\hfill (\\textbf{QCK2})$\n        \n        \\item $\\mu_D (s \\otimes s^*)m^*(1_B) = \\frac{1}{\\delta^2} 1_D$ $\\hfill (\\textbf{QCK3})$\n    \\end{enumerate}\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map for $D$ and  $s^*(b)=s(b^*)^*$ for $b\\in B$. Then the \\textit{quantum Cuntz--Krieger algebra} associated to $\\cG$ is the universal unital C*-algebra  $\\mathcal{O}(\\cG)$ generated by the image of a quantum Cuntz--Krieger $\\cG$-family $S\\colon B\\to \\cO(\\cG)$.\n\\end{definition} \n\nWe show that Definition~\\ref{def:QCK} gives the classical Cuntz--Krieger algebra when $\\cG$ is a classical graph.\n\n\\begin{example}\nLet $\\cG=(B,\\psi, A)$ be a classical graph, i.e., $B=C(V)$ is the finite-dimensional commutative C*-algebra arising from a simple finite directed graph $G=(V, E)$ on $N = |V|$ vertices, $\\psi$ is the normalized trace on $B$, and $A$ is defined by the adjacency matrix $A_G$ on $G$ in the usual way. If $\\left\\{e_1, \\ldots, e_N\\right\\}$ is the canonical basis of minimal projections in $B$, it is easy to see that $\\psi(e_i)=1\/N$, $m(e_i\\otimes e_j)=\\delta_{i=j}e_j$, and $m^*(e_i)=Ne_i\\otimes e_i$ for all $i, j$. Moreover, $\\psi$ is a $\\delta$-form with $\\delta^2=N$.\n\nLet $S:B\\to \\cO(\\cG)$ be a universal quantum Cuntz--Krieger $\\cG$-family and define $S_i=NS(e_i) \\in \\cO(\\cG)$. It was shown in \\cite[Proposition 4.1]{BEVW20} that the $S_i$ are partial isometries satisfying $S_i^*S_i=\\sum_{j=1}^NA_G(i,j)S_jS_j^*$ since $S:B\\to \\cO(\\cG)$ satisfies (\\textbf{QCK1}) and (\\textbf{QCK2}).  One can see that the $S_i$ have mutually orthogonal range projections by observing that they sum to the identity in $\\cO({\\cG})$ since $S$ satisfies (\\textbf{QCK3}). Explicitly, observe that\n\\begin{align*}\n    \\sum_{i=1}^N S_iS_i^*&=N^2\\left[\\sum_{i=1}^N S(e_i)S^*(e_i)\\right]\\\\\n    &=N\\left[\\sum_{i=1}^n \\mu_{\\cO(\\cG)}(S\\otimes S^*)m^*(e_i)\\right]\\\\\n    &= N\\left[\\frac{1}{\\delta^2}1_{\\cO(\\cG)}\\right]\n    =1_{\\cO(\\cG)}.\n\\end{align*}\n\nThus, the $S_i$ form a Cuntz--Krieger $A_G$-family, which induces a $*$-homomorphism of $\\cO_{A_E}$ onto $\\cO(\\cG)$. \n\nConversely, given a universal Cuntz--Krieger $A_G$ family $\\left\\{S_i\\right\\}$, we can define $s:B \\to \\cO_{A_G}$ via $s(e_i)=\\frac{1}{N}S_i\\in \\cO_{A_G}$. As mentioned in \\cite[Proposition 4.1]{BEVW20}, one can check that $s$ satisfies (\\textbf{QCK1}) and (\\textbf{QCK2}). To see that $s$ satisfies (\\textbf{QCK3}), consider\n\\begin{align*}\n    \\mu_{\\cO_{A_G}}(s\\otimes s^*)m^*(1_B) &= \\sum_{i=1}^N\\mu_{\\cO_{A_G}}(s(Ne_i)\\otimes s^*(e_i))\\\\\n    &=\\frac{1}{N}\\sum_{i=1}^NS_iS_i^*\\\\\n    &=\\frac{1}{N}1_{\\cO_{A_G}}.\n\\end{align*}\nHence, $s$ is a quantum Cuntz--Krieger $\\cG$-family, which induces a $*$-homomorphism of $\\cO(\\cG)$ onto $\\cO_{A_G}$. Checking that this map is the inverse of the previously induced $*$-homomorphism of $\\cO_{A_G}$ onto $\\cO(\\cG)$ yields $\\cO(\\cG)$ is isomorphic to $\\cO_{A_G}.$\n\\end{example}\n\n\n\\begin{remark}\\label{rem:unital_relation}\nIn \\cite{BEVW20}, a notion of quantum Cuntz--Krieger algebras was introduced without the relation $(\\textbf{QCK3})$ (see \\cite[Definition 3.7]{BEVW20}), which gives potentially non-unital, non-nuclear C*-algebras denoted $\\mathbb{F}\\cO(\\cG)$. As discussed in \\cite[Section 4.1]{BEVW20}, when $\\cG$ is a classical graph $\\mathbb{F}\\cO(\\cG)$ is a \\emph{free} Cuntz--Krieger algebra (see \\cite[Definition 2.5]{BEVW20}), whereas by the above example $\\cO(\\cG)$ is a Cuntz--Krieger algebra. Thus we will generally refer to $\\mathbb{F}\\cO(\\cG)$ as the \\emph{free} quantum Cuntz--Krieger algebra, and reserve the terminology ``quantum Cuntz--Krieger algebra'' for $\\cO(\\cG)$.\n\\end{remark}\n\nUsing adapted matrix units, one can produce a more explicit presentation of the quantum Cuntz--Krieger relations.  This is essentially the content of \\cite[Proposition 3.9]{BEVW20}.  More precisely, if $s_{ij}^{(a)}:=s(f_{ij}^{(a)})$, where $\\{f_{ij}^{(a)}\\colon 1\\leq a \\leq d,\\ 1\\leq i,j\\leq N_a\\}$ are the adapted matrix units for $(B,\\psi)$, then $s:B \\to D$ is a quantum Cuntz--Krieger $\\cG$-family if and only if the following relations hold:\n\n     \\begin{align}\n         \\sum_{k,\\ell=1}^{N_a} s_{ik}^{(a)}(s_{\\ell k}^{(a)})^*s_{\\ell s}^{(a)} &=s_{is}^{(a)} \\label{eqn:QCK1}\\\\\n        \\sum_{\\ell =1}^{N_a}(s_{\\ell i}^{(a)})^*s_{\\ell j}^{(a)} &= \\sum_{c=1}^d\\sum_{\\ell,m,n =1}^{N_c} A_{ija}^{\\ell m c} s_{\\ell n}^{(c)} (s_{m n}^{(c)})^*  \\label{eqn:QCK2}\\\\\n        \\sum_{c=1}^d \\sum_{\\ell,m=1}^{N_c} \\psi(e_{\\ell,\\ell}^{(c)}) s_{\\ell m}^{(a)} (s_{\\ell m}^{(c)})^* &=\\frac{1}{\\delta^2} 1_D  \\label{eqn:QCK3}\n    \\end{align}\n\n\nWe now introduce localized versions of the above quantum Cuntz--Krieger relations.\n\n\\begin{definition}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form. We define a \\textit{local quantum Cuntz--Krieger $\\cG$-family} in a unital C*-algebra $D$ to be a linear map $s\\colon B\\to D$ such that\n    \\begin{enumerate}[label=(\\roman*)]\n        \\item $\\mu_D(\\mu_D \\otimes 1)(s \\otimes s^* \\otimes s)(m^* \\otimes 1) = \\frac{1}{\\delta^2}s m$ $\\hfill(\\textbf{LQCK1})$\n        \n        \\item $\\mu_D (s^* \\otimes s) = \\frac{1}{\\delta^2} \\mu_D(s \\otimes s^*)m^*Am$ $\\hfill(\\textbf{LQCK2})$\n        \n        \\item $\\mu_D (s \\otimes s^*)m^*(1_B) = \\frac{1}{\\delta^2} 1_D$ $\\hfill (\\textbf{LQCK3})$\n        \n    \\end{enumerate}\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map for $D$, $s^*(b)=s(b^*)^*$ for $b\\in B$, and $m^*$ is the adjoint of $m$ with respect to the inner product given by $\\psi$.\n\\end{definition}\nOnce again, using adapted matrix units, one can produce a more explicit presentation of the local quantum Cuntz--Krieger relations. This is the content of the following proposition whose proof is omitted as it is very similar to the proof of \\cite[Proposition 3.9]{BEVW20}.\n\n\\begin{proposition}\\label{prop:explicit_local_relations}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form, and let $s\\colon B\\to D$ be a linear map into a unital C*-algebra. Denote $s_{ij}^{(a)}:=s(f_{ij}^{(a)})$, where $\\{f_{ij}^{(a)}\\colon 1\\leq a \\leq d,\\ 1\\leq i,j\\leq N_a\\}$ are the adapted matrix units for $(B,\\psi)$. Then $s$ is a local quantum Cuntz--Krieger $\\cG$-family if and only if the following relations hold:\n    \\begin{align}\n         \\sum_{k=1}^{N_a} s_{ik}^{(a)}(s_{jk}^{(a)})^*s_{rs}^{(b)} &=\\delta_{\\substack{a=b \\\\ j=r}} \\frac{1}{\\delta^2\\psi(e_{jj}^{(a)})}s_{is}^{(a)} \\label{eqn:QCP1}\\\\\n        (s_{ij}^{(a)})^*s_{rs}^{(b)} &= \\delta_{\\substack{a=b\\\\ i=r}}\\frac{1}{\\delta^2\\psi(e_{ii}^{(a)})}\\sum_{c=1}^d\\sum_{\\ell,m,n =1}^{N_c} A_{jsa}^{\\ell m c} s_{\\ell n}^{(c)} (s_{m n}^{(c)})^*  \\label{eqn:QCP2}\\\\\n        \\sum_{c=1}^d \\sum_{\\ell,m=1}^{N_c} \\psi(e_{\\ell,\\ell}^{(c)}) s_{\\ell m}^{(a)} (s_{\\ell m}^{(c)})^* &=\\frac{1}{\\delta^2} 1_D  \\label{eqn:QCP3}\n    \\end{align}\nfor all $1\\leq a,b\\leq d$, $1\\leq i,j\\leq N_a$, and $1\\leq r,s\\leq N_b$.\n\\end{proposition}\n\nJust as in the case of the quantum Cuntz--Krieger algebras above, one can also define, for any quantum graph $\\cG = (B,\\psi, A)$, a corresponding {\\it local quantum Cuntz--Krieger algebra}. The local quantum Cuntz--Krieger algebra is the C$^\\ast$-algebra generated by a universal local quantum Cuntz--Krieger $\\cG$-family.\nThe following theorem is the main result of this section.  It shows that local quantum Cuntz--Krieger algebras are in fact  familiar objects -- under rather mild assumptions, they are precisely the Cuntz--Pimsner algebras of quantum edge correspondences. \n\n\n\\begin{theorem}\\label{thm:universal_local_quantum_Cuntz--Krieger_algebra}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG=B\\cdot \\epsilon_\\cG\\cdot B$ be its quantum edge correspondence. Let $(\\pi_{E_\\cG},t_{E_\\cG})$ denote the universal covariant representation of $E_\\cG$ on the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$.\nAssume $\\cG$ has no quantum sources.\nThen $S\\colon B\\to \\mathcal{O}_{E_\\cG}$ defined by $S(x):=\\frac{1}{\\delta} t_{E_{\\cG}}(x\\cdot \\epsilon_\\cG)$ is a local quantum Cuntz--Krieger $\\cG$-family whose image generates $\\cO_{E_\\cG}$. Moreover, given any local quantum Cuntz--Krieger $\\cG$-family $s\\colon B\\to D$ in a unital C*-algebra $D$ there exists a $*$-homomorphism $\\rho\\colon \\mathcal{O}_{E_\\cG}\\to D$ such that $s=\\rho\\circ S$.\n\\end{theorem}\n\\begin{proof}\nWe begin by showing that $S$ is a local quantum Cuntz--Krieger $\\cG$-family. Let $\\psi_{t_{E_\\cG}} \\colon \\cK(E_\\cG)\\to \\cO_{E_\\cG}$ be the $*$-homomorphism induced by $t_{E_\\cG}$, and let $\\mu\\colon \\cO_{E_\\cG}\\otimes \\cO_{E_\\cG}\\to \\cO_{E_\\cG}$ denote the multiplication map. Define a linear map $T\\colon B\\to \\cO_{E_\\cG}$ by $T(x):=t_{E_\\cG}(x\\cdot \\epsilon_\\cG)$, so that $S=\\frac{1}{\\delta} T$. Note that the quantum edge correspondence $E_\\cG$ is faithful by Theorem~\\ref{thm:faithful_full_correspondence} and finite dimensional so that the Katsura ideal $J_{E_\\cG}=B$. Moreover, faithfulness allows us to identify $B\\subset \\cL(E_\\cG)=\\cK(E_\\cG)$, and under this identification we have $\\psi_{t_{E_\\cG}}|_B = \\pi$. With this observation in hand, we can readily verify the local quantum Cuntz--Krieger relations.\n\n\n\n    \\begin{itemize}\n        \\item[]\\textbf{(LQCK1):} The second equation in Corollary~\\ref{cor:abstract_Toeplitz} implies\n            \\begin{align*}\n                \\mu(\\mu\\otimes 1)(S\\otimes S^*\\otimes S)(m^*\\otimes 1) &= \\frac{1}{\\delta^3} \\mu(\\mu\\otimes 1)(T\\otimes T^*\\otimes T)(m^*\\otimes 1)\\\\\n                &= \\frac{1}{\\delta^3} \\mu(\\psi_{t_{E_\\cG}}\\otimes T)\\\\\n                &=\\frac{1}{\\delta^3}\\mu(\\pi\\otimes T)\\\\\n                &= \\frac{1}{\\delta^3} Tm= \\frac{1}{\\delta^2} Sm,\n            \\end{align*}\n        where the second-to-last equality follows from the relation $\\pi_{E_\\cG}(x)t_{E_\\cG}(\\xi) = t_{E_\\cG}(x\\cdot \\xi)$ for $x\\in B$ and $\\xi\\in E_\\cG$.\n        \n        \\item[]\\textbf{(LQCK2):} Using both equations in Corollary~\\ref{cor:abstract_Toeplitz} gives\n            \\begin{align*}\n                \\mu(S^*\\otimes S) &= \\frac{1}{\\delta^2} \\mu(T^*\\otimes T) = \\frac{1}{\\delta^4} \\pi A m = \\frac{1}{\\delta^2} \\psi_{t_{E_\\cG}} A m\\\\\n                    &= \\frac{1}{\\delta^4} \\mu(T\\otimes T^*)m^* Am = \\frac{1}{\\delta^2} \\mu(S\\otimes S^*)m^*Am.\n            \\end{align*}\n            \n        \\item[]\\textbf{(LQCK3):} Using the second equation in Corollary~\\ref{cor:abstract_Toeplitz} we have\n            \\[\n                \\mu(S\\otimes S^*)m^*(1_B) = \\frac{1}{\\delta^2} \\mu(T\\otimes T^*)m^*(1_B) = \\frac{1}{\\delta^2} \\pi_{E_{\\cG}}(1_B) = \\frac{1}{\\delta^2} 1.\n            \\]\n        Note that $\\pi_{E_{\\cG}}$ is necessarily unital since $1\\cdot \\xi=\\xi$ for all $\\xi\\in E_{\\cG}$.\n    \\end{itemize}\nThus $S$ is a local quantum Cuntz--Krieger $\\cG$-family. We also have $C^*(S(B))=\\cO_{E_\\cG}$. Indeed, $\\pi_{E_\\cG}(B)\\subset C^*(S(B))$ by Corollary~\\ref{cor:abstract_Toeplitz}, and $t_{E_\\cG}(\\xi)\\pi_{E_\\cG}(b)=t_{E_\\cG}(\\xi\\cdot b)$ implies $t_{E_\\cG}(E_\\cG)\\subset C^*(S(B))$.\n\nNow, suppose $s\\colon B\\to D$ is a local quantum Cuntz--Krieger $\\cG$-family in a unital C*-algebra $D$. To obtain the desired homomorphism, we will construct a covariant representation of $E_\\cG$ on $D$ and invoke the universal property of the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$. Define a linear map $\\pi\\colon B\\to D$ by\n    \\[\n        \\pi := \\delta^2 \\mu_D(s\\otimes s^*)m^*,\n    \\]\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map. Then $\\pi$ is unital by (\\textbf{LQCK3}), $\\pi^*=\\pi$ in light of how the multiplication maps interact with the adjoint, and (\\textbf{LQCK1}) implies\n    \\begin{align*}\n        \\mu_D(\\pi\\otimes \\pi) &= \\delta^4 \\mu_D(\\mu_D\\otimes \\mu_D)(s\\otimes s^*\\otimes s\\otimes s^*)(m^*\\otimes m^*)\\\\\n            &= \\delta^4 \\mu_D( \\left[\\mu_D(\\mu_D\\otimes 1)(s\\otimes s^*\\otimes s)(m^*\\otimes 1)\\right]\\otimes s^*)(1\\otimes m^*)\\\\\n            &= \\delta^2 \\mu_D( (sm)\\otimes s^*)(1\\otimes m^*)\\\\\n            &= \\delta^2 \\mu_D( s\\otimes s^*)m^*m = \\pi m.\n    \\end{align*}\nHence $\\pi$ is a $*$-homomorphism. \n\nNext define a linear map $t\\colon E_\\cG\\to D$ by\n    \\[\n        t(x\\cdot \\epsilon_\\cG\\cdot y):=\\delta s(x)\\pi(y) \\qquad \\qquad x,y\\in B.\n    \\]\nWe will see below that $t(\\xi)^*t(\\eta)=\\pi(\\<\\xi,\\eta\\>_B)$ for $\\xi,\\eta\\in B$, which in particular will show this map is well-defined. By (\\textbf{LQCK1}) we have\n    \\begin{align*}\n        \\mu_D(\\mu_D\\otimes 1)(\\pi\\otimes s\\otimes \\pi) &= \\delta^2 \\mu_D(\\mu_D\\otimes 1)(\\mu_D\\otimes 1\\otimes 1)(s\\otimes s^*\\otimes s\\otimes \\pi)(m^*\\otimes 1\\otimes 1)\\\\\n            &=  \\mu_D( s\\otimes \\pi)(m\\otimes 1).\n    \\end{align*}\nThus for $x,y,z\\in B$ we have\n    \\begin{align*}\n        \\pi(x)t(y\\cdot \\epsilon_\\cG\\cdot z) &= \\delta \\mu_D(\\mu_D\\otimes 1)(\\pi\\otimes s\\otimes \\pi)(x\\otimes y\\otimes z)\\\\\n            &= \\delta \\mu_D(s\\otimes \\pi)(m\\otimes 1)(x\\otimes y \\otimes z) = s(xy)\\pi(z) = t(xy\\cdot \\epsilon_\\cG\\cdot z).\n    \\end{align*}\nand so $\\pi(x)t(\\xi)=t(x\\cdot \\xi)$ for all $x\\in B$ and $\\xi\\in E_\\cG$. Using (\\textbf{LQCK2}) and the definition of $\\pi$, we have\n    \\[\n        \\mu_D(s^*\\otimes s) = \\frac{1}{\\delta^2} \\mu_D(s\\otimes s^*)m^* Am = \\frac{1}{\\delta^4} \\pi A m.\n    \\]\nThus for $x,x',y,y'\\in B$ we have\n    \\begin{align*}\n        t(x\\cdot \\epsilon_\\cG \\cdot y)^* t(x'\\cdot \\epsilon_\\cG \\cdot y') &= \\delta^2  (\\mu_D\\otimes \\mu_D)(1\\otimes \\mu_D\\otimes 1)(\\pi^*\\otimes s^*\\otimes s\\otimes \\pi)(y^*\\otimes x^*\\otimes x'\\otimes y')\\\\\n            &=\\frac{1}{\\delta^2} (\\mu_D\\otimes \\mu_D)(\\pi \\otimes (\\pi A)\\otimes \\pi)(1\\otimes m\\otimes 1)(y^*\\otimes x^*\\otimes x'\\otimes y')\\\\\n            &= \\frac{1}{\\delta^2} \\pi( y^* A(x^*x') y')\\\\\n            &= \\pi\\left(y^* \\<x\\cdot \\epsilon_\\cG, x'\\cdot \\epsilon_\\cG\\>_B y'\\right)  = \\pi\\left(\\< x\\cdot \\epsilon_\\cG\\cdot y, x'\\cdot \\epsilon_\\cG\\cdot y'\\>_B \\right),\n    \\end{align*}\nwhere the second-to-last equality follows from Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products}. Thus $(\\pi,t)$ is a representation of $E_\\cG$ on $D$, and Theorem~\\ref{thm:compact_operators} implies it is covariant:\n    \\begin{align*}\n        \\psi_t(f_{ij}^{(a)}) &= \\sum_{k=1}^{N_a} \\psi_t\\left( \\theta_{f_{ik}^{(a)}\\cdot \\epsilon_\\cG, f_{jk}^{(a)}\\cdot \\epsilon_\\cG} \\right)\\\\\n            &=\\sum_{k=1}^{N_a} t(f_{ik}^{(a)}\\cdot \\epsilon_\\cG) t( f_{jk}^{(a)}\\cdot \\epsilon_\\cG)^*\\\\\n            &= \\delta^2 \\mu_D(s\\otimes s^*)m^*(f_{ij}^{(a)}) = \\pi(f_{ij}^{(a)}).\n    \\end{align*}\nThus the universal property for the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ implies there is a $*$-homomorphism $\\rho\\colon \\cO_{E_\\cG}\\to D$ satisfying $\\pi=\\rho\\circ \\pi_{E_\\cG}$ and $t=\\rho\\circ t_{E_\\cG}$. In particular, we have\n    \\[\n        s(x) = \\frac{1}{\\delta} t(x\\cdot \\epsilon_\\cG) = \\frac{1}{\\delta} \\rho\\circ t_{E_\\cG}(x\\cdot \\epsilon_\\cG) = \\rho(S(x)).\n    \\]\nHence $s=\\rho\\circ S$.\n\\end{proof}\n\n\n\n\n\n\nObserve that any local quantum Cuntz--Krieger $\\cG$-family is a (non-local) quantum Cuntz--Krieger $\\cG$-family. Indeed, since $\\psi$ is a $\\delta$ form one has $mm^*=\\delta^2 \\text{id}$. Thus if $s\\colon B\\to D$ satisfies (\\textbf{LQCK1}) and (\\textbf{LQCK2}), then applying $m^*$ to the right-hand sides of these relations yields (\\textbf{QCK1}) and (\\textbf{QCK2}), respectively. Also (\\textbf{LQCK3}) and (\\textbf{QCK3}) are identical. Hence the universal property for $\\mathcal{O}(\\cG)$ yields a unique $*$-homomorphism onto $C^*(s(D))$. In particular, if $\\ker(A)$ does not contain a central summand of $B$, then the previous theorem yields the following:\n\n\\begin{corollary}\\label{cor:quantum_Cuntz--Krieger_quotients}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG$ be its quantum edge correspondence. Assume that $\\ker(A)$ does not contain a central summand of $B$. Then $\\mathcal{O}_{E_\\cG}\\cong \\mathcal{O}(\\cG)\/\\mathcal{I}$ where $\\mathcal{I}\\triangleleft \\mathcal{O}(\\cG)$ is the closed two-sided ideal generated by the relations (\\textbf{LQCK1}), (\\textbf{LQCK2}), and (\\textbf{LQCK3}).\n\\end{corollary}\n\n\\subsection{Behavior of $\\cO_{E_\\cG}$ under quantum graph isomorphisms}\nIn this section we briefly examine the relationship between the Cuntz--Pimsner algebras of quantum edge correspondences associated to quantum isomorphic quantum graphs.   We begin by recalling the notion of quantum isomorphism from \\cite{BCEHPSWCMP19, BEVW20}.  Let $\\cG_i=(B_i,\\psi_i,A_i)$, $i=1,2$, be directed quantum graphs. We say that $\\cG_1$ and $\\cG_2$ are quantum isomorphic if there exists a Hilbert space $\\cH$ and a unital $\\ast$-homomorphism \\[\\theta_1:B_1 \\to B_2 \\otimes B(\\cH)\\]\nwhich satisfies the following $\\psi_i$ and $A_i$ covariance conditions\n    \\begin{align*}\n        (\\psi_2 \\otimes \\text{id})\\theta_1 &=\\psi_1(\\cdot)1_{B(\\cH)}, \\\\\n        (A_2 \\otimes \\text{id})\\theta_1 &= \\theta_1 \\circ A_1.\n    \\end{align*}\nNote that the general theory guarantees that whenever a morphism $\\theta_1:B_1 \\to B_2 \\otimes B(\\cH)$ exists, there automatically exists a corresponding morphism $\\theta_2:B_2 \\to B_1 \\otimes B(\\cH)$ with analogous covariance conditions to those for $\\theta_1$.  In particular, the notion of quantum isomorphism is symmetric in $\\cG_1$ and $\\cG_2$. \n\nQuantum isomorphisms between quantum graphs can be seen as relaxations (or generalizations) of the notion of an isomorphism between quantum graphs.  Indeed, in the special case where $\\cH = \\bC$, a quantum isomorphism between $\\cG_1$ and $\\cG_2$ defines  an ordinary isomorphism of quantum graphs. \n\nThe following theorem should be compared with \\cite[Theorem 6.13]{BEVW20}, which is a similar result in the context of (free) quantum Cuntz--Krieger algebas.  Note, however, that the conclusion of the following theorem is slightly stronger in that one gets {\\it injective} morphisms $\\Theta_i$ between Cuntz--Pimsner algebras below (compare with \\cite[Remark 6.14]{BEVW20}).     \n\n\\begin{theorem}\nLet $\\cG_i=(B_i,\\psi_i,A_i)$, $i=1,2$, be directed quantum graphs.  If $\\cG_1$ is quantum isomorphic to $\\cG_2$ with $*$-homomorphisms\n    \\[\n        \\theta_1\\colon B_1\\to B_2\\otimes B(\\cH)\\qquad \\text{and} \\qquad \\theta_2\\colon B_2\\to B_1\\otimes B(\\cH),\n    \\]\n    as above, then there exist injective $*$-homomorphisms\n    \\[\n        \\Theta_1\\colon \\cO_{E_{\\cG_1}}\\to \\cO_{E_{\\cG_2}}\\otimes B(\\cH) \\qquad \\text{and} \\qquad  \\Theta_2\\colon \\cO_{E_{\\cG_2}}\\to \\cO_{E_{\\cG_1}}\\otimes B(\\cH)\n    \\]\nsatisfying\n    \\[\n        \\Theta_1\\pi_{E_{\\cG_1}} = (\\pi_{E_{\\cG_2}}\\otimes 1)\\theta_1 \\qquad \\text{and}\\qquad \\Theta_2\\pi_{E_{\\cG_2}} =  (\\pi_{E_{\\cG_1}}\\otimes 1)\\theta_2.\n    \\]\n\\end{theorem}\n\\begin{proof}\nUsing Corollary~\\ref{cor:quantum_graph_correspondence_as_cp_correspondence}, we can work with $B_i\\otimes_{A_i} B_i$ rather than $E_{\\cG_i}$, $i=1,2$. Denote $\\tilde{A}_2:=A_2\\otimes \\text{id}_{B(\\cH)}$, which is a completely positive map, and $X:=(B_2\\otimes B(\\cH))\\otimes_{\\tilde{A}_2} (B_2\\otimes B(\\cH))$. Note that $X\\cong (B_2\\otimes_{A_2} B_2)\\otimes B(\\cH)$, and so $\\cO_X \\cong \\cO_{B_2\\otimes_{A_2} B_2}\\otimes B(\\cH)$ by \\cite[Example 6.4]{Mor17}. \n\nNow, for $x,y\\in B_1$ and $\\xi\\in B_1\\otimes B_1$ we have\n    \\[\n        (\\tilde{A}_2\\otimes 1)( (\\theta_1\\otimes \\theta_1)(\\xi)^* (\\theta_1\\otimes \\theta_1)(\\xi) ) = (\\theta_1\\otimes \\theta_1)(A_1\\otimes 1)(\\xi^*\\xi).\n    \\]\nIt follows that $\\theta_1\\otimes \\theta_1$ induces a linear map $T\\colon B_1\\otimes_{A_1} B_1\\to X$ satisfying\n    \\[\n        T(x\\cdot \\xi \\cdot y) = \\theta_1(x)\\cdot T(\\xi)\\cdot \\theta_1(y) \\qquad x,y\\in B_1,\\ \\xi\\in B_1\\otimes_{A_1} B_1,\n    \\]\nand\n    \\[\n        \\<T(\\xi),T(\\eta)\\>_{B_2\\otimes B(\\cH)} = \\theta_1(\\<\\xi, \\eta\\>_{B_1}) \\qquad \\xi,\\eta\\in B_1\\otimes_{A_1} B_1.\n    \\]\nThus if $(\\pi_X,t_X)$ is the universal covariant representation of $X$ (note that we can take $\\pi_X:= \\pi_{E_{\\cG_2}}\\otimes 1$ and $t_X:=t_{E_{\\cG_2}}\\otimes 1$), then $(\\pi_X\\circ \\theta_1, t_X\\circ T)$ is a covariant representation of $B_1\\otimes_{A_1} B_1$ on $\\cO_X$. One easily checks that this representation is injective and admits a gauge action, and so \\cite[Theorem 6.4]{Kat04} implies there is an injective $*$-homomomorphism $\\Theta_1\\colon \\cO_{B_1\\otimes_{A_1} B_1}\\to \\cO_X$. Reversing the roles of $B_1$ and $B_2$ yields $\\Theta_2$.\n\\end{proof}\n\n\\section{Examples}\\label{sec:simpler_examples}\n\nIn this section we consider three common types of quantum graphs and determine the isomorphism classes of the Cuntz--Pimsner algebras associated to their quantum edge correspondences. Notably, several of these examples can be realized as Exel crossed products associated to natural Exel systems. We refer the reader to \\cite{Exel03-2}, \\cite{Exel03}, and \\cite{Exel04} for details but provide a brief summary of the construction of Exel crossed products below.\n\nLet $\\cC$ be a C*-algebra and $\\alpha:\\cC \\to \\cC$ be $\\ast$-endomorphism. A {\\em transfer operator} for $(\\cC,\\alpha)$ is a positive linear map on $\\cC$ such that $\\mathcal{L}(\\alpha(f)g)=f\\mathcal{L}(g)$ for all $f,g\\in \\cC$. The Exel crossed product is obtained by first constructing a Toeplitz algebra $\\cT(\\cC,\\alpha,\\mathcal{L})$ which is the universl C$^\\ast$-algebra generated by a copy of $\\cC$ along with an element $S$ such that $Sf=\\alpha(f)S$ and $\\mathcal{L}(f)=S^*fS$ for all $ f\\in \\cC.$ Two elements $f,k\\in \\cT(\\cC,\\alpha,\\mathcal{L})$ form a {\\em redundancy} if $f\\in \\cC, k\\in \\overline{\\cC SS^*\\cC},$ and $fgS=kgS$ for all $g\\in \\cC.$ The {\\em Exel crossed product} $\\cC\\rtimes_{\\alpha,\\mathcal{L}} \\mathbb{N}$ is the quotient of $\\mathcal{T}(\\cC,\\alpha,\\mathcal{L})$ by the ideal generated by differences $f-k$ of redundancies $(f,k)$. \n\nGiven an $n\\times n$ $\\{0,1\\}$-matrix $A$, one can construct the Markov subshift space of infinite paths in the graph associated to $A$, denoted by $X_A$.  $X_A$ is a compact space, and admits a natural shift action $\\sigma:X_A \\to X_A$ given by $\\sigma (x_1,x_2,x_3, \\ldots ) = (x_2,x_3, \\ldots)$.  A nice class of commutative Exel systems $(C(X_A),\\alpha,\\cL)$ arises from this construction, where $\\alpha(f) = f \\circ \\sigma$ for $f \\in C(X_A)$ and \\[\\mathcal L(f) (x) = |\\{y \\in X_A \\ | \\ \\sigma(y) = x\\}|^{-1} \\Big(\n\\sum_{\\{y \\in X_A \\ | \\ \\sigma(y) = x\\}}f(y)\\Big).\\]  It is known that the Cuntz--Krieger algebra $\\cO_A$ is isomorphic to the Exel crossed product $ C(X_A)\\rtimes_{\\alpha,\\cL} \\bN$. Below, we explore the analogous relationship between noncommutative Exel systems and local quantum Cuntz--Krieger algebras.\n\n\\subsection{Complete quantum graphs}\\label{subsection:completequantumgraphs}\nGiven a finite quantum space $(B,\\psi)$ with $\\delta$-form $\\psi$, we denote by $K(B,\\psi)$ the {\\em complete quantum graph} $(B,\\psi, A)$, whose quantum adjacency matrix $A$ is given by the rank-one map  $A=\\delta^2\\psi(\\cdot)1_B$ \\cite{BCEHPSWCMP19, BEVW20}.  The quantum edge correspondence for $K(B,\\psi)$ is $E_K=B\\otimes B$ since an adapted matrix unit computation yields that the quantum edge indicator is $\\epsilon_K=1_B\\otimes 1_B.$ The quantum Cuntz--Krieger algebra $ \\cO(K(B,\\psi))$ is generated by $\\{\nS_{ij}^{(a)}: 1 \\le a \\le d, \\ 1 \\le i,j \\le N_a\\}$\naccording to the relations\n\\begin{align} \n\\label{QK1}\n&(S^{(a)})^*S^{(a)} \n= \n1 \\quad \\text{in } M_{N_a}\\otimes \\mathcal{O}(K(B,\\psi)) \\qquad (1 \\le a \\le d)\\\\\n\\label{QK2}\n& \\sum\\limits_{ars} \\delta^2 \\psi(e_{rr}^{(a)})S_{rs}^{(a)}S_{rs}^{(a)*}\n=\n1 \\quad \\text{in } \\mathcal{O}(K(B,\\psi))\n\\end{align}\n\nIn \\cite[Theorem 4.5]{BEVW20}, the authors establish that their definition of a quantum Cuntz--Krieger algebra associated to $K(B,\\psi)$ is isomorphic to $\\cO_{\\dim B}$ whenever $\\delta^2 \\in \\mathbb N$. It is unknown if this also holds for $\\delta^2\\not\\in \\mathbb{N}$.\nThe following proposition resolves this for local quantum Cuntz--Krieger algebras.\n\n\\begin{proposition} \\label{prop:Cuntz--Pimsner-is-a-Cuntz-algebra}\n$\\cO_{E_K}$ is isomorphic to the Cuntz algebra $\\cO_{n}$ where $n = \\dim B$. \n\\begin{proof}\nBy Theorem \\ref{thm:universal_local_quantum_Cuntz--Krieger_algebra} and Proposition \\ref{prop:explicit_local_relations}, the Cuntz--Pimsner algebra $\\mathcal{O}_{E_{K}}$ is a quotient of the quantum Cuntz--Krieger algebra $\\mathcal{O}(K(B,\\psi))$ subject to relations (\\ref{eqn:QCP1}), which remains the same, and (\\ref{eqn:QCP2}), which in this case becomes: \n\n\\begin{align}\n  \n  \n  \n  \n    (S_{ij}^{(a)})^*(S_{rs}^{(b)})\n    &=\\delta_{\\substack{a=b\\\\ i=r}}\n    \\frac{1}{\\delta^2 \\psi(e_{ii}^{(a)})}\n    \\sum\\limits_{c\\ell m}\n    \\left[\n    \\delta^2 \\delta_{\\substack{\\ell=m\\\\ j=s}}\\psi(e_{\\ell \\ell}^{(c)})\n    \\sum\\limits_n S_{\\ell n}^{(c)}(S_{mn}^{(c)})^*\n    \\right].\\label{eq:2}\n\\end{align}\nSimplifying equation \\eqref{eq:2} using the unit relation (\\ref{eqn:QCP3}), we get\n\n\\[\n(S_{ij}^{(a)})^*(S_{rs}^{(b)})=\\frac{\\delta_{\\substack{a=b\\\\ i=r\\\\ j=s}}}{\\delta^2 \\psi(e_{ii}^{(a)})}\\left(\\sum\\limits_{c\\ell n}\\delta^2\\psi(e_{\\ell \\ell}^{(c)}) S_{\\ell n}^{(c)}(S_{\\ell n}^{(c)})^*\\right) = \\frac{\\delta_{\\substack{a=b\\\\ i=r\\\\ j=s}}}{\\delta^2 \\psi(e_{ii}^{(a)})}(1_{\\mathcal{O}(E_{K})}).\n\\]\nThus, $\\left\\{\\delta\\psi(e_{ii}^{(a)})^{1\/2}S_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\right\\}$ are generating isometries for $\\mathcal{O}_{E_{K}}$ whose range projections sum to the identity, and it follows that $\\mathcal{O}_{E_{K}}$ is isomorphic to the Cuntz algebra $\\mathcal{O}_{\\dim B}.$\n\\end{proof}\n\\end{proposition}\n\nHere, we define a particular noncommutative Exel system $(B^{\\otimes \\bN}, \\alpha, \\mathcal{L})$ and show that the Exel crossed product $B^{\\otimes \\bN}\\rtimes _{\\alpha,\\mathcal{L}}\\bN$ is the Cuntz algebra on $\\dim B$ generators. Define $\\alpha:B^{\\otimes \\bN}\\to B^{\\otimes \\bN}$ by \\[\\alpha(f):=1_B\\otimes f \\qquad (f\\in B^{\\otimes \\bN}).\\] Then $\\mathcal{L}:B^{\\otimes \\bN} \\to B^{\\otimes \\bN}$ given by \\[\\mathcal{L}(f_1\\otimes g):=\\psi(f_1)g, \\qquad (f_1 \\in B, g \\in B^{\\otimes \\bN}),\\] is a transfer operator for the dynamical system $(B^{\\otimes \\bN},\\alpha)$. In the rest of this section, we let $1$ denote the identity in $B^{\\otimes \\bN}$ and continue to denote the identity in $B$ by $1_B$. For each $1\\leq a \\leq d$ and $1\\leq i,j \\leq N_a$, define $E_{ij}^{(a)}:=e_{ij}^{(a)}\\otimes 1$, and let $S$ denote the operator which arises in the construction of $\\cT(B^{\\otimes \\bN},\\alpha,\\mathcal{L})$. We will show that this particular Exel crossed product is $\\mathcal{O}_{\\dim B}$, generated by $\\{v_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\}$, where $$v_{ij}^{(a)}:=\\frac{1}{\\psi(e_{jj}^{(a)})^{-1\/2}}E_{ij}^{(a)}S.$$ \nThe $v_{ij}^{(a)}$ should be regarded as an analogue of the characteristic function of words in $X_A$ that ``begin with the quantum letter $e_{ij}^{(a)} \\in B$.''\n\n\\begin{lemma}\\label{lem:Complete_graph_Exel_redundancies}\nFor all $1\\leq a\\leq d$, $1\\leq i,j \\leq N_a$, the pair $\\left(E_{ij}^{(a)}, \\sum_{p} v_{ip}^{(a)}{v_{jp}^{(a)}}^*\\right)$ is a redundancy.\n\\end{lemma}\n\\begin{proof}\nFix $a\\in \\{1,...,d\\}$ and $i,j\\in \\{1,...,N_a\\}$. For $e_{k\\ell}^{(b)}\\otimes f\\in B^{\\otimes \\bN},$ note $v_{jp}^{(a)*}(e_{k\\ell}^{(b)}\\otimes f)=0$ unless $a=b$ and $j=k.$ Thus, it suffices to consider elements of the form $e_{j\\ell}^{(a)}\\otimes f$ in $B^{\\otimes \\bN}$. Observe:\n\\begin{align*}\n    \\psi(e_{pp}^{(a)})(v_{ip}^{(a)}{v_{jp}^{(a)*}})(e_{j\\ell}^{(a)}\\otimes f)S\n  \n    &=E_{ip}^{(a)}SS^*[E_{pj}^{(a)}(e_{j\\ell}^{(a)}\\otimes f)]S\\\\\n    &=\n    E_{ip}^{(a)}S[S^*(e_{p\\ell}^{(a)}\\otimes f)S]\\\\\n    &=\n    E_{ip}^{(a)}S\\mathcal{L}(e_{p\\ell}^{(a)}\\otimes f)\\\\\n    &=\n    \\psi(e_{p\\ell}^{(a)})E_{ip}^{(a)}Sf\\\\\n    &=\n    \\psi(e_{p\\ell}^{(a)})E_{ip}^{(a)}\\alpha(f)S\\\\\n    &=\n    \\psi(e_{p\\ell}^{(a)})E_{ip}^{(a)}(1_B\\otimes f)S\\\\\n    &=\n    \\psi(e_{pp}^{(a)})\\delta_{p=\\ell}(e_{i\\ell}^{(a)}\\otimes f)S\\\\\n    &=\n    \\psi(e_{pp}^{(a)})\\delta_{p=\\ell}(e_{ij}^{(a)}e_{j\\ell}^{(a)}\\otimes f)S\\\\\n    &= \n    \\psi(e_{pp}^{(a)})\\delta_{p=\\ell}E_{ij}^{(a)}(e_{j\\ell}^{(a)}\\otimes f)S\n\\end{align*}\nAs elements of the form $e_{k\\ell}^{(b)}\\otimes f$ span $B^{\\otimes \\bN}$, for all $g\\in B^{\\otimes \\bN},$ we have \n$$\n\\left(\\sum_{p} v_{ip}^{(a)}{v_{jp}^{(a)}}^*\\right)gS\n=\\sum_{p} \\left(v_{ip}^{(a)}{v_{jp}^{(a)}}^*gS\\right)\n=\\sum_p \\left(\\delta_{p\\ell} E_{ij}^{(a)}gS\\right)\n=E_{ij}^{(a)}gS.\n$$\nBy definition, $\\left(E_{ij}^{(a)},\\sum_{p} v_{ip}^{(a)}{v_{jp}^{(a)}}^*\\right)$ is a redundancy.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:crossed-product-is-a-Cuntz-algebra}\nThe Exel crossed product $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\mathcal{L}}\\mathbb{N}$ is isomorphic to the Cuntz algebra $\\cO_n$ where $n=\\dim B$.\n\\end{theorem}\n\\begin{proof}\nWe first show that the C*-algebra generated by $\\{v_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\}$ is $\\mathcal{O}_n,$ where $n=\\dim B.$ Observe that each $v_{ij}^{(a)}$ is an isometry in $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\mathcal{L}}\\mathbb{N}$:\n$$\n    {v_{ij}^{(a)}}^*v_{ij}^{(a)}\n   \n    =\\frac{1}{\\psi(e_{jj}^{(a)})}S^*E_{ji}^{(a)}E_{ij}^{(a)}S\n   \n    =\\frac{1}{\\psi(e_{jj}^{(a)})}S^*E_{jj}^{(a)}S\n   \n    =\\frac{1}{\\psi(e_{jj}^{(a)})}\\mathcal{L}\\left(E_{jj}^{(a)}\\right\n   \n   \n    =1.\n$$\nBy Lemma~\\ref{lem:Complete_graph_Exel_redundancies}, $\\sum_p v_{ip}^{(a)}{v_{ip}^{(a)}}^*=E_{ii}^{(a)}$, and thus, summing over $1\\leq a\\leq d$ and $1\\leq i\\leq N_a$ yields the identity. Therefore, each $v_{ij}^{(a)}$ is a Cuntz isometry, and the collection $\\{v_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\}$ generates the Cuntz algebra $\\cO_n$ where $n=\\dim B$.\n\nTo verify that this copy of $\\cO_{\\dim B}$ inside $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN$ is actually all of $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN$, note that each $E_{ij}^{(a)}$ belongs to $\\cO_{\\dim B}$ by Lemma~\\ref{lem:Complete_graph_Exel_redundancies}. Further, since $S=1_BS=(\\sum_{a,i} E_{ii}^{(a)})S$, we also know that $S$ belongs to this copy of $\\cO_{\\dim B}$. We claim that elements of the form $E_{ij}^{(a)}$ along with $S$ generate all of $B^{\\otimes \\bN}$. Recall from \\cite{Exel03} that $(\\alpha(f),SfS^*)$ is a redundancy for all $f\\in B^{\\otimes \\bN}$, so for any $r\\in \\bN$, we have\n$$\n\\underbrace{1_B\\otimes ...\\otimes 1_B}_{r} \\otimes \\, e_{ij}^{(a)} \\otimes 1 \n= \\alpha^r(E_{ij}^{(a)})\n=S^rE_{ij}^{(a)}(S^*)^r\n\\in \\cO_{\\dim B}.\n$$\nAs elements of the above form generate the C*-algebra $B^{\\otimes \\bN}$ and are contained in $\\cO_{\\dim B}$, we may conclude $B^{\\otimes \\bN}$ is a subset of $\\mathcal{O}_{\\dim{B}}.$ Therefore, $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN \\cong \\cO_n$ where $n=\\dim B.$\n\\end{proof}\n\nAs a result of Theorem~\\ref{thm:crossed-product-is-a-Cuntz-algebra}, the Cuntz--Pimsner algebra $\\cO_{E_K}$ for the quantum edge correspondence of the complete quantum graph can be realized as the Exel crossed product $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN$, whose underlying Exel system was a natural choice for the complete quantum graph. We may also conclude that the Cuntz algebra on $\\dim B$ generators is always a quotient of the quantum Cuntz-Krieger algebra $\\cO(K(B,\\psi))$ for the complete quantum graph.\n\n\n\\subsection{Trivial quantum graphs}\nDenote by $T(B,\\psi)$ the {\\it trivial quantum graph} $(B,\\psi, \\text{id}) $; in the following discussion $B$ and $\\psi$ will be fixed, so we will simply denote it by $T$. Using the adapted matrix units, we will now identify the edge correspondence and prove that it is isomorphic to the trivial $B$-correspondence, which will allow us to explicitly compute its Cuntz--Pimsner algebra. Recall the definition of the edge correspondence from \\Cref{def:edgecorrespondence}. It is generated, as a $B$-bimodule by the idempotent $\\epsilon_{T}:= \\frac{1}{\\delta^2} (\\text{id} \\otimes \\text{id})m^{\\ast}(1)=\\frac{1}{\\delta^2} m^{\\ast}(1)$. As $m^{\\ast}$ is a $B$-bimodule map, we get $x\\cdot \\epsilon_{T}\\cdot y = \\frac{1}{\\delta^2} m^{\\ast}(xy)$, hence the edge correspondence is equal as a vector space to the image of $m^{\\ast}$. We will actually show that the map $\\frac{1}{\\delta}m^{\\ast}$ identifies $B$ and the edge correspondence $E_{T}$ as $C^{\\ast}$-correspondences.\n\\begin{proposition}\nLet $B$ be viewed as a $C^{\\ast}$-correspondence, where $\\langle a, b\\rangle_B =a^{\\ast}b$ and the left and right actions are the usual ones. Then $\\frac{1}{\\delta} m^{\\ast}: B \\to E_{T}$ is an isomorphism of $C^{\\ast}$-correspondences.\n\\end{proposition}\n\\begin{proof}\nWe will use \\Cref{prop:recognition}. The vector $\\xi:=\\frac{1}{\\delta}1$ is cyclic for $B$ and $\\delta^2 \\langle \\xi, x\\cdot \\xi\\rangle = x$ is the quantum adjacency matrix of the trivial quantum graph. Therefore by \\Cref{prop:recognition} the assignment $\\xi \\mapsto \\epsilon_{T}$ extends to an isomorphism of correspondences $B$ and $E_{T}$. This map is precisely equal to $\\frac{1}{\\delta}m^{\\ast}$.\n\\end{proof}\n\\begin{corollary}\\label{cor:triv}\nThe Cuntz--Pimsner algebra $\\cO_{E_{T}}$ is isomorphic to $B\\otimes C(\\bT)$.\n\\end{corollary}\n\\begin{proof}\nFrom the previous proposition, $E_T$ is isomorphic to $B$. By \\cite[Example 3 p. 193]{Pimsner}, the Cuntz--Pimsner algebra of $B$ is isomorphic to the crossed of $B$ by the trivial action of $\\mathbb{Z}$, i.e. $B\\otimes C(\\bT)$.\n\\end{proof}\n\nConsider the trivial Exel system $(B,\\alpha,\\mathcal{L})$ where $\\alpha=\\mathcal{L}=\\text{id}$. Recall that $\\mathcal{T}(B,\\alpha,\\mathcal{L})$ is the universal C*-algebra generated by $B$ and an element $U$ subject to the relations $Ux=\\alpha(x)U$ and $U^*xU=\\mathcal{L}(x)$ for all $x\\in B.$ In this case, $Ux=xU$, which implies that $U$ is an isometry and commutes with every element of $B$.\n\n\\begin{proposition}\nThe Exel crossed product $B\\rtimes_{\\text{id},\\text{id}} \\bN$ is isomorphic to $B\\otimes C(\\bT)$\n\\end{proposition}\n\\begin{proof}\nRecall that $B\\rtimes_{\\text{id},\\text{id}} \\bN$ is the quotient of $\\mathcal{T}(B,\\text{id},\\text{id})$ by the ideal of generated by the set of redundancies, $\\{a-k:a\\in B,\\,k\\in \\overline{BUU^*B},\\,agU=kgU\\,\\,\\forall g\\in B\\}.$ Consider $k=UU^*$ and $a=1$. Then because $U$ is isometric and commutes with all elements of $B$, we have $UU^*gU=Ug=gU$ for all $g\\in B$. Therefore $UU^*$ and $1$ form a redundancy, so the quotient consists of a copy of $B$ and a unitary that commutes with $B$. Therefore, $B\\rtimes_{\\text{id}, \\text{id}}\\bN \\cong B\\otimes C(\\bT)$.\n\\end{proof}\n\n\\begin{remark}\nNote that the above proposition provides a second example of the Cuntz--Pimsner algebra for the quantum edge correspondence being implemented by a natural choice of Exel crossed product.\n\\end{remark}\n\n\n\n\\subsection{Rank-one quantum graphs}\nIn the special case when $B=M_n(\\mathbb{C})$ and $\\psi$ is a properly normalized trace, there is a nice correspondence between quantum adjacency matrices, projections in $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})^{{\\text{op}}}$, and subspaces of $M_n(\\mathbb{C})$. Just as before, a projection in $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})^{{\\text{op}}}$ can be viewed as the Choi matrix of a quantum adjacency matrix, but also as an orthogonal projection onto a subspace of $M_n(\\mathbb{C})$, when we identify the algebra of bounded operators on $M_n(\\mathbb{C})$ (viewed as a Hilbert space) with $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})^{{\\text{op}}}$ via the left-right action. \n\nFor the trivial quantum graph, the corresponding subspace of $M_n(\\mathbb{C})$ is the span of the identity matrix. It is therefore tempting to investigate other rank one subspaces. If such a subspace is spanned by a (suitably normalized) operator $T$, then the corresponding quantum adjacency matrix is equal to $A(x):= TxT^{\\ast}$. This is the inspiration for the following definition.  Recall that the density matrices  $\\{\\rho_a:1\\leq a\\leq d\\}$ associated to a $\\delta$-form $\\psi$ on $B$ are invertible and satisfy $\\text{Tr}(\\rho_{a}^{-1})=\\delta^2$.\n\n\\begin{definition}\n Let $T \\in B$ satisfy $\\text{Tr} (\\rho_a^{-1} T^{\\ast}T) = \\delta^2$ for all $1\\leq a\\leq d$. We call $A(x):= TxT^{\\ast}$ a quantum adjacency matrix of \\emph{rank one}.\n\\end{definition}\n\nBefore we proceed any further, we should first check that rank one quantum adjacency matrices are in fact quantum adjacency matrices. Because one can check the conditions separately on each matrix direct summand, we may assume that $B$ is a matrix algebra, so we can forget about the index $a$.\n\\begin{lemma}\\label{lem:rankone}\nIf $T \\in M_n(\\mathbb{C})$ satisfies $\\text{Tr} (\\rho^{-1} T^{\\ast}T)=\\delta^2$, where $\\rho$ is the density matrix of a $\\delta$-form $\\psi$ then $A(x):=TxT^{\\ast}$ is a quantum adjacency matrix.\n\\end{lemma}\n\\begin{proof}\nWe have to check that $m(A\\otimes A)m^{\\ast}(f_{ij}) = \\delta^2 A(f_{ij})$ for all adapted matrix units $f_{ij}$. The left-hand side is equal to $\\sum_{k} T f_{ik}T^{\\ast}T f_{kj} T^{\\ast}$. We will check that $\\sum_{k} f_{ik} S f_{kj} = \\text{Tr} (\\rho^{-1} S) f_{ij}$ and this will end the proof. First, we can write the matrix $S$ as $S = \\sum_{pt} S_{pt} e_{pt} = \\sum_{pt} S_{pt}\\sqrt{\\psi(e_{pp})\\psi(e_{tt})} f_{pt}$. We have $f_{ik} f_{pt} f_{kj} = \\delta_{pk}\\delta_{tk} \\frac{1}{\\psi(e_{kk})^2} f_{ij}$, so $f_{ik} S f_{kj} = \\frac{S_{kk}}{\\psi(e_{kk})} f_{ij}$. We obtain\n    \\begin{align*\n        \\sum_{k} f_{ik}S f_{kj} = \\left(\\sum_{k} \\frac{S_{kk}}{\\psi(e_{kk})}\\right) f_{ij} = \\text{Tr}(\\rho^{-1}S) f_{ij}. \n    \\end{align*}\n\\end{proof}\n\\begin{proposition}\\label{prop:triv}\nLet $A$ be a rank one quantum adjacency matrix, given by $A(x)=TxT^{\\ast}$. Recall that $T$ is of the form $\\bigoplus_{a} T^{(a)}$ and we assume that each $T^{(a)}\\neq 0$. Then the Cuntz--Pimsner algebra of the edge correspondence is isomorphic to $B\\otimes C(\\bT)$.\n\\end{proposition}\n\\begin{proof}\nWe will once again resort to \\Cref{prop:recognition} to show that the edge correspondence is isomorphic to the trivial correspondence $B$. By our assumption on $T$, the element $\\xi:=\\frac{1}{\\delta}T^{\\ast}$ is cyclic for the trivial correspondence and we have $\\delta^2 \\langle \\xi, x\\xi\\rangle = TxT^{\\ast} = A(x)$. Therefore by \\Cref{prop:recognition} the edge correspondence is isomorphic to the trivial correspondence by a bimodular extension of the assignment $\\xi \\mapsto \\epsilon$. We conclude as in the proof of \\Cref{cor:triv} that the corresponding Cuntz--Pimsner algebra is isomorphic to $B\\otimes C(\\bT)$.\n\\end{proof}\n\\begin{remark}\nIf it happens that $T^{(a)}=0$ for some $a$'s, then the same proof shows that the Cuntz--Pimsner algebra is isomorphic to $B' \\otimes C(\\bT)$, where $B'$ is the direct sum of matrix algebras, but only over the $a$'s, for which $T^{(a)}\\neq 0$.\n\\end{remark}\n\n\\subsection{Quantum adjacency matrices arising from $*$-automorphisms} \nAnother way to generalize the trivial quantum graph is to let $\\alpha$ be a $*$-homomorphism of a finite dimensional C*-algebra $B$. As $\\alpha$ is multiplicative, we have \n$\n\\alpha\\circ m \n= m\\circ (\\alpha\\otimes \\alpha)\n$. \nIt follows that \n$\nm(\\alpha \\otimes \\alpha)m^{\\ast} \n= \n\\alpha mm^{\\ast} \n= \\delta^2 \\alpha\n$, i.e. $\\alpha$ is a quantum adjacency matrix. Then $\\cG_\\alpha:=(B,\\psi,\\alpha)$ is a quantum graph, and the associated quantum edge indicator for $\\cG_\\alpha$ is \n\\begin{align*}\n    \\epsilon_{\\alpha}\n:=\\frac{1}{\\delta^2}(\\text{id}\\otimes \\alpha)m^*(1)\n    =\\frac{1}{\\delta^2}\\sum_{aij}\\psi(e_{ii}^{(a)})f_{ij}^{(a)}\\otimes \\alpha(f_{ji}^{(a)}).\n\\end{align*}\n\nLet $E_{\\alpha} = B \\cdot \\epsilon_\\alpha \\cdot B$ denote the quantum edge correspondence for $\\cG_\\alpha$, and define $B_{\\alpha}$ to be $B$ as a C*-correspondence over itself with $\\langle a,b\\rangle=a^*b$, left action given by $x\\cdot \\eta:= \\alpha(x) \\eta$, and right action given by right multiplication.\n\\begin{lemma}\n$E_\\alpha\\cong B_{\\alpha}$ as C$^\\ast$-correspondences.\n\\end{lemma}\n\\begin{proof}\nThe element $\\xi:= \\frac{1}{\\delta} 1$ is cyclic for $B_{\\alpha}$ and $\\delta^2 \\langle \\xi, x\\cdot \\xi\\rangle = \\alpha(x)$. By \\Cref{prop:recognition} the edge correspondence is isomorphic to $B_{\\alpha}$.\n\\end{proof}\n\\begin{corollary}\\label{cor:triv2}\nWhen $\\alpha$ is a $*$-automorphism, the Cuntz--Pimsner algebra of $E_\\alpha$ is isomorphic to $B \\rtimes_\\alpha \\mathbb{Z}$.\n\\end{corollary}\n\\begin{proof}\nThis follows from Pimsner's work \\cite[Example 3, p. 193]{Pimsner}.\n\\end{proof}\n\nWe can make this crossed product a bit more explicit. First, recall that if we have an automorphism $\\alpha$ on a direct sum $B_{1}\\oplus B_2$, preserving the summands, then the corresponding crossed product splits as a direct sum as well, i.e. $(B_1\\oplus B_2)\\rtimes \\mathbb{Z} \\simeq (B_1\\rtimes \\mathbb{Z})\\oplus (B_2\\rtimes \\mathbb{Z})$. Second,\nautomorphisms of direct sums of matrix algebras are of a very special form. We can first collect all the matrix algebras of the same size and such a subalgebra has to be preserved by any automorphism. In this case we are dealing with an algebra of the form $M_n(\\mathbb{C}) \\otimes \\mathbb{C}^{k}$ and then any automorphism comes from a permutation of the set $\\{1,\\dots,k\\}$ followed by \\emph{inner} automorphisms on individual matrix algebras. Because crossed products are insensitive to inner perturbations (\\cite[II.10.3.17]{MR2188261}), we may assume that we are just dealing with a permutation automorphism. Moreover any permutation decomposes as a disjoint sum of cycles so we can treat those separately. To sum up, any crossed product $B\\rtimes \\mathbb{Z}$ will decompose as a direct sum of crossed products of the form $(M_n(\\mathbb{C})\\otimes \\mathbb{C}^k) \\rtimes \\mathbb{Z}$, where $\\mathbb{Z}$ acts by a cycle on $\\{1,\\dots,k\\}$ and as identity on $M_n(\\mathbb{C})$. Since it acts as identity on the matrix algebra, the resulting crossed product is isomorphic to $M_n(\\mathbb{C}) \\otimes (\\mathbb{C}^k \\rtimes \\mathbb{Z}) \\cong M_n(\\mathbb{C}) \\otimes M_k(\\mathbb{C}) \\otimes C(\\mathbb T)$ (\\cite[Section VIII.3, p. 230]{MR1402012}. \n\nAgain, there is a natural Exel system we may associate to the quantum graph $\\cG_\\alpha$. Consider $(B,\\alpha,\\alpha^{-1})$, where $\\alpha^{-1}$ plays the role of the transfer operator. As in the example of the complete quantum graph, the Exel crossed product for this system, $B\\rtimes_{\\alpha,\\alpha^{-1}} \\mathbb{N}$, is isomorphic to the Cuntz--Pimsner algebra $B\\rtimes_\\alpha \\mathbb{Z}$ for the quantum edge correspondence $E_\\alpha.$\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider an inverse boundary value problem for the wave equation \n$\n(\\frac {\\partial^2}{\\partial t^2} - c(x)^2 \\frac {\\partial^2}{\\partial x^2}) u(t,x) = 0,$$\nand introduce a discrete regularization strategy to recover the sound speed $c(x)$ by using the knowledge of perturbed and discetized Neumann-to-Dirichlet map $\\widetilde \\Lambda_{N_1}$.\nOur approach is based on the Boundary Control method \\cite{Belishev1987, Belishev1992, Tataru1995}.\n\nA variant of the Boundary Control method, called the iterative time-reversal control method, was introduced in \\cite{Bingham2008}.\nThe method was later modified in \\cite{Dahl2009}\nto focus the energy of a wave at a fixed time\nand in \\cite{Lauri2013} \nto solve an inverse obstacle problem for a wave equation. \nIn \\cite{jusa1} we introduced a modification to the iterative time-reversal control method\nthat is tailored for the 1+1 dimensional wave equation.\n\n \nThe novelty in this paper is that we analyze the effect of the discretization in the regularized solution of the inverse problem. We give a direct discrete regularization method for the\nnon-linear inverse problem for the wave equation. The result contains an\nexplicit (but not necessarily optimal) convergence rate.\n\nBy referring to direct methods for non-linear problems we mean the explicit construction of non-linear map to solve the problem\nwithout resorting to a local optimization method. In our case the map is given by (\\ref{Rstategia}), shown below.\nThe advantage of direct approaches\nis that they do not suffer from the possibility that the algorithm converges to a local minimum.\nIn particular, they do not require a priori knowledge\nthat the solution is in a small neighbourhood of a given function.\n\nClassical abstract regularization theory is explained in \\cite{Engl1996}. The iterative regularization\nof both linear and non-linear inverse problems and convergence rates are discussed\nin a Hilbert space setting in \\cite{Bissantz2004, Hanke1997, Hohage2008, LuPeRa2007, Mathe2008} and in a Banach space setting in \\cite{Hofmann2007, Kaltenbacher2006, Kaltenbacher2008, Kirsch1996, Ramlau2008, Ramlau2006, Resmerita2005}. In section \\ref{pamelapulkkinen} we compare our regularization strategy to Morozov's discrepancy principle (MDP). In the context of abstract regularization theory, this principle has been discussed, e.g., in \\cite{Scherzer1993}.  \n\nThere are currently only a few regularized direct methods for non-linear inverse problems.\nAn example is a regularisation algorithm for\nthe inverse problem for the conductivity equation in \\cite{KLMS2009}.\n Also, a direct regularized inversion for blind\ndeconvolution is presented in \\cite{Justen2006}.\n\n\\section{Regularization Strategy}\n\\subsection{Continuity of forward map}\nWe define \n\\begin{align}\n\\label{normipoika}\n&\\norm{c}_{C^k(M)}=\n\\sum_{p=0}^{k}\\sup_{x\\in (0,\\infty)}|\\frac{\\partial^p c}{\\partial x^p}(x)|,\n\\end{align}\nwhere $M$ denotes the half axis $M=[0,\\infty)\\subset \\R$. We denote the set of bounded $C^k(M)$ -functions by \n\\begin{align*}\nC^k_b(M)=\\{c\\in C^k(M); \\norm{c}_{C^k(M)}< \\infty\n\\}.\n\\end{align*}\n Let $C_0,C_1,{L_0,L_1},m>0$ and define the space of {k times differentiable} velocity functions\n\\begin{align}\n\\label{nopeudet}\n {\\mathcal V^k}=&\\,\\{c\\in {C^k} (M);\nC_0\\le c(x)\\le C_1 ,\\\\\\nonumber &\\,\\norm{c}_{{C^k}(M)}\\le m,\\, c-1 \\in {C_0^k([L_0,L_1])} \\}.\n\\end{align}\nHere $C_0^k([L_0,L_1])$ is the subspace of functions in $C^k_b(M)$ that are supported on $[L_0,L_1]$.\nLet \n\\begin{align}\n\\label{aikapoika}\nT> \\frac{{L_1}}{C_0}.\n\\end{align}\nFor $c\\in{\\mathcal V^2}$ and \n$f\\in L^2(0,2T)$, the boundary value problem \n\\begin{align}\n\\label{dartwader}\n&(\\frac {\\partial^2}{\\partial t^2} - c(x)^2 \\frac {\\partial^2}{\\partial x^2}) u({x,t}) = 0 \\quad \\text{in ${M\\times (0,2T})$},\n\\\\\\nonumber\n&\\partial_x u({0,t}) = f(t),\n\\\\\\nonumber\n&u|_{t = 0} =0,\\quad  \\partial_t u|_{t=0} = 0,\n\\end{align}\nhas a unique solution $u=u^f\\in H^1({M\\times (0,2T)})$. Using this solution we define the Neumann-to-Dirichlet operator $ \\Lambda= \\Lambda_c$,\n\\begin{align}\n\\label{lam}\n \\Lambda : L^{2}(0,2T) \\to L^{2}(0,2T),\\quad\n \\Lambda f=u^f|_{x=0}.\n\\end{align}\nFor a Banach space $E$ we define\n\\begin{align*}\n\\mathcal{L} (E)=\\{A:E \\to E;\nA\\text{ is linear and continuous}\\}.\n\\end{align*}\nLet ${Z=C^2_b(M)}$  and $Y=\\mathcal{L} (L^2(0,2T))$. The operator $\\mathcal A$ is defined in the domain $\\mathcal D(\\mathcal A)=\\mathcal V^3$ by setting\n\\begin{align}\n\\label{apina}\n\\mathcal A:{\\mathcal D(\\mathcal A)}\\subset  Z\\to\\mathcal R (\\mathcal{A})\\subset Y,\\quad \n \\mathcal A(c)= \\Lambda_c. \n\\end{align}\nThe notation in (\\ref{apina}) means that the range {$\\mathcal R (\\mathcal{A})=\\mathcal A(\\mathcal V^3)$} and the domain {$\\mathcal D(\\mathcal A)$} are equipped with the topologies of $Y$ and $ Z$, respectively. Note that the maps (\\ref{lam}) and (\\ref{apina}) are continuous (see \\cite{jusa1}).\n\n\\subsection{Regularization strategies with discretizatized measurements}\n\nLet $T$ be as in (\\ref{aikapoika}) and $N\\in \\mathbb{Z}_+$. For $n\\in\\{1,2,3,...,2N\\}$ we define the basis functions  as\n\\begin{align}\n\\label{kantafunktio}\n \\phi_{n,N}(t) = \\Big(\\frac{N}{T}\\Big)^{\\frac{1}{2}}1_{[\\frac{(n-1)T}{N},\\frac{nT}{N} )}(t), \\quad t \\in [0,2T).\n\\end{align}\nNote that the functions ${\\phi_{n,N}}$ are orthonormal in $L^2(0,2T)$. \nHaving (\\ref{kantafunktio}) we define the space of piecewise constant functions as \n\\begin{align}\n\\label{pcsfjoukko}\n\\mathcal P^N=span\\big\\{\\phi_{1,N},...,\\phi_{2N,N}\\big\\}\\subset L^2(0,2T) \n\\end{align}\nand \nan orthogonal projection as\n\\begin{align}\n\\label{pcsfjoukko2}\nP^N:L^2(0,2T)\\to\\mathcal P^N,\\quad P^N(f)=\\sum_{j=1}^{2N} \\langle f,\\phi_{j,N}\\rangle\n_{L^2(0,2T)}\\phi_{j,N}(t).\n\\end{align}\nLet $\\Lambda$ be as in (\\ref{lam}). Using (\\ref{pcsfjoukko2}) we define \n\\begin{align}\n\\label{pallokala}\n\\Lambda_N=P^N\\Lambda P^N.\n \\end{align}\\\\\nLet $E$ be a Banach space and $\\mathcal H\\in E$. We denote \n\\begin{align}\n\\label{pallo}\n \\mathcal{B}_E(\\mathcal H,\\epsilon)=\\{\\widetilde {\\mathcal H}\\in E : \\norm{\\widetilde{\\mathcal H}-\\mathcal H}_E < \\epsilon \\}.\n\\end{align}\n\n\\subsubsection{A model for a single discrete and noisy measurement}\n\\label{huippumalli1}\nLet $\\epsilon_0>0$ and we define\n\\begin{align}\n\\label{porraspossu}\nl_0(\\epsilon_0)=\\big\\lfloor\\frac{4}{7}\\log_2 \\epsilon_0^{-1}\\big\\rfloor\n\\end{align}\nand \n\\begin{align}\n\\label{porraspossu2}\nN_0(\\epsilon_0)=2^{l_0}.\n\\end{align}\nLet $N=2^l\\ge N_0$, where $l\\in \\mathbb{Z}_+$. Let $P^N$ be as in (\\ref{pcsfjoukko2}) and $\\Lambda $ be as in (\\ref{lam}). \nLet us define\n\\begin{align}\n\\label{porras}\n H(t) = \n\\begin{cases}\n1, & t\\ge 0, \\\\\n0, & t<0.\n\\end{cases}\n\\end{align}\nLet us define \n \\begin{align}\n\\label{mittauspoika}\n \\widetilde m_{N,\\epsilon_0}=P^N\\Lambda H + n_{N,\\epsilon_0} ,\n\\end{align}\nin which $n_{N,\\epsilon_0} \\in \\mathcal P^N$ represents the error and $\\norm{n_{N,\\epsilon_0} }_{L^2(0,2T)}\\le \\epsilon_0$. We consider the quantity $(\\epsilon_0,N_0,\\widetilde m_{N,\\epsilon_0})$ that we call a measurement. \nLet $\\mathcal A$ be as in (\\ref{apina}) and $H$ be as in (\\ref{porras}). We define \n\\begin{align}\n\\label{apina4}\n\\mathcal A_0:\\mathcal D(\\mathcal A_0)\\subset Z\\to\\mathcal R (\\mathcal A_0)\\subset L^2(0,2T),\\quad \n \\mathcal A_0(c) =  \\mathcal A(c) H= \\Lambda H, \n\\end{align}\nwhere $\\mathcal D(\\mathcal A_0)=\\mathcal V^3$ . \nOur main result on the reconstruction of $c(x)$ from the measurement $(\\epsilon_0,N_0,\\widetilde m_{N,\\epsilon_0})$ is given by the following theorem. \n \n\\begin{theorem}\n\\label{kaiken_teoria2b}\nFor the operator $\\mathcal A_0:\\mathcal \\mathcal V^3\\subset Z\\to L^2(0,2T)$, there exists an admissible regularization strategy  $\\mathcal R^{(0)}_{N_0,\\alpha_0}$ with the choice of parameter \n\\begin{align*}\n\\alpha_0(\\epsilon_0)=a_0\\epsilon_0^{\\frac{4}{45}},\n\\end{align*}\nsatisfying the following: For every $c\\in{\\mathcal V^3}$\nthere are $\\widetilde\\epsilon_0>$, $a_0>0$, and $C>0$ such that\n\\begin{align*} \n&\\sup\\Big\\{\\norm{\\mathcal R^{(0)}_{N_0,\\alpha_0}\\widetilde m_{N,\\epsilon_0}-c}_{ Z}:\n\\widetilde m_{N,\\epsilon_0}\\in\\mathcal P^{N},N=2^l\\ge N_0(\\epsilon_0),\n\\\\\\nonumber&\\qquad  \\norm{\\widetilde m_{N,\\epsilon_0} -P^N\\Lambda H}_{L^2(0,2T)} \\le \\epsilon_0\\Big\\}\\le\nC\\epsilon_0 ^{\\gamma_0},\n\\end{align*}\nfor all $\\epsilon_0 \\in (0,\\widetilde\\epsilon_0)$. Here $\\gamma_0 =\\frac{1}{270}$ and $ N_0(\\epsilon_0)$ is as in (\\ref{porraspossu2}).\n\\end{theorem}\nAn explicit bound $\\widetilde\\epsilon_0$ and the value for constant $a_0$ are given in the proof.\nThe proof of Theorem \\ref{kaiken_teoria2b} is given in Section \\ref{convergense}.\n\\subsubsection{A model for several discrete and noisy measurements}\n\\label{virhesikapossu}\nLet  \n\\begin{align}\n\\label{virheoperator}\n\\mathcal{E}_{N_1}:\\mathcal P^{N_1}\\to\\mathcal P^{N_1},\n\\end{align}\nwhere $N_1\\in\\mathbb{Z}_+$.\nHaving $\\Lambda_{N_1}$ as in (\\ref{pallokala}) we define a discrete and noisy measurement operator\n\\begin{align}\n\\label{akuaku}\n& \\widetilde\\Lambda_{N_1}: L^2(0,2T) \\to L^2(0,2T), \n\\quad \n\\\\\\nonumber&\\widetilde\\Lambda_{N_1}f = \n\\begin{cases}\n\\Lambda_{N_1}f+\\mathcal{E}_{N_1}f, & f \\in \\mathcal P^{N_1}, \\\\\n\\quad\\qquad 0, & f \\in (\\mathcal P^{N_1})^\\perp.\n\\end{cases}\n\\end{align}\nNote that $\\mathcal P^{N_1}\\cup (\\mathcal P^{N_1})^\\perp=L^2(0,2T)$. \nWith data corresponding to several boundary measurements $(\\epsilon_1, N_1, \\widetilde\\Lambda_{N_1})$ we get\nthe following results with improved error estimates.\n\\begin{theorem}\n\\label{kaiken_teoria2}\nFor the operator $\\mathcal A:\\mathcal \\mathcal V^3\\subset Z\\to Y$, there exists an admissible regularization strategy  $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ with the choice of parameter \n\\begin{align*}\n\\alpha_1(\\epsilon_1)=a_1\\epsilon_1^{\\frac{4}{9}}\n\\end{align*}\nthat satisfies the following: For every $c\\in{\\mathcal V^3}$\nthere are $\\widetilde\\epsilon_1>0$, $a_1>0$, and $C>0$ such that\n\\begin{align*} \n&\\sup\\Big\\{\\norm{\\mathcal R^{(1)}_{N_1,\\alpha_1}\\widetilde\\Lambda_{N_1}-c}_{ Z}:\nN_1\\ge\\epsilon_1^{-4},\\quad \\widetilde\\Lambda_{N_1}\\in \\mathcal L(\\mathcal P^{N_1}),\n\\\\\\nonumber&\\qquad \\norm{\\widetilde\\Lambda_{N_1}-\\Lambda_{N_1} }_{{Y}}\\le \\epsilon_1\\Big\\}\\le\nC\\epsilon_1 ^{\\gamma_1},\n\\end{align*}\nfor all $\\epsilon_1 \\in (0,\\widetilde\\epsilon_1)$. Here $\\gamma_1 =\\frac{1}{54}$.\n\\end{theorem}\nAn explicit bound $\\widetilde\\epsilon_1$ and the value for constant $a_1$ are given in the proof.\nThe proof of Theorem \\ref{kaiken_teoria2} is given in Section \\ref{convergense}.\nWe will give explicit choices for $\\mathcal R^{(0)}_{N_0,\\alpha_0}$ in formula (\\ref{Rstategia2}) and for $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ in formula (\\ref{Rstategia2b}) below. For the convenience of the reader we give a short summary of the regularization strategy. \nAssume that we are given $\\widetilde \\Lambda_{N}\\in\\mathcal L(\\mathcal P^{N_1})\\subset Y$, that is,\n the discrete Neumann-to-Dirichlet map for the unknown wave speed $c(x)$ with measurements errors. Then\n the regularization strategy is obtained by the following steps:\n\n\\begin{enumerate}\n\\item Using operator $\\widetilde\\Lambda_{N_1}$ in (\\ref{sope3}) we constructed a source that produces a wave such that\n$u^{f_{\\alpha,r}}(t,x)|_{t=T}$ is close to the indicator function   $1_{\\mathcal M(r)}(x)$ of the \ndomain of influence ${\\mathcal M(r)}$---see (\\ref{siiri}). \n \\item Using sources $\\widetilde f^N_{\\alpha,r}$ we approximately compute the \nvolumes $V(r)$ of the domains of influences---see (\\ref{operaattorit4}), (\\ref{sope2}), and Proposition \\ref{PSN}. \n  \\item Using finite differences we compute approximate values of the \n  derivatives of the volumes of the domain influences $\\partial_r V(r)$---see   (\\ref{timojutila2c}). \n \\item We interpolate the obtained values of $\\partial_r V(r)$. This determines\n the approximate values of the wave speed $v(r)$ in the travel time coordinates---see (\\ref{operaattorit5}) and (\\ref{sebastianahoc}). \n  \\item Finally, we change the coordinates from travel time coordinates to Euclidean\n  coordinates to obtain the approximate values of the wave speed $c(x)$ for $x\\in M$---see (\\ref{kenguru}).\n \n\\end{enumerate}\n\\subsection{Previous literature}\nFrom the point of view of uniqueness questions, the inverse problem\nfor the 1+1 dimensional wave equation is equivalent to the one-dimensional inverse boundary spectral problem. The latter problem was\nthoroughly studied in the 1950s  \\cite{GeLe1951, Krein1951, Marchenko1950} and we refer to \\cite[pp. 65--67]{Kabanikhin2005} for a historical overview. In the\n1960s Blagove{\\v{s}}{\\v{c}}enski{\\u\\i} \\cite{Blagovevsvcenskiui1969, Blagovevsvcenskiui1971} developed an\napproach to solving the inverse problem for the 1+1 dimensional wave\nequation without reducing the problem to the inverse boundary spectral\nproblem. This and later dynamical methods have the advantage over\nspectral methods that they only require data on a finite time\ninterval. Applications of one-dimensional inverse probems have been discussed widely in\n\\cite{Blagovevsvcenskiui1966, Kabanikhin2005,KKL2001}. \n\nThe method in the present paper is a variant of the Boundary Control\n method  that was pioneered by M. Belishev \\cite{Belishev1987}\nand developed by  M. Belishev and Y. Kurylev \\cite{BeKu1987,Belishev1992} in the late 1980s and early 1990s.\nOf crucial importance for the method is the result by D.\\ Tataru\n\\cite{Tataru1995} concerning\na Holmgren-type uniqueness theorem for non-analytic coefficients.\nThe Boundary Control method for multidimensional inverse problems has been summarized in \\cite{Belishev1997, KKL2001},\nand considered for 1+1 dimensional scalar problems in \\cite{BeKa1989, BeRFi1994, jusa1} and for  multidimensional scalar problems\nin \\cite{Katchalov1998, KKLM2004, Kury1995, LaOk2014, LaOk2014b}.\nFor systems it has been considered in \\cite{Kurylev2009, Kurylev2006,Kurylev2015}.\nStability results for the method have been considered in \\cite{Anderson2004}\nand \\cite{Katsuda2007}, and\ncomputational implementations in\n\\cite{Belishev1999,Belishev2016,Hoop2017,Ivanov2016,Pestov2010}.\nAn application of the method to blockage detection in water pipes is in preparation \\cite{eemeli}.\n\nThe inverse problem for the wave equation can also be solved by using complex geometrical optics solutions. These solutions were developed in the context of elliptic\ninverse boundary value problems \\cite{Sylvester1987}, and in \\cite{Nachman1988} they were employed to solve an inverse\nboundary spectral problem. Local stability results can be proven using\n(real) geometrical optics solutions \\cite{Bellassoued2011, StUh1998, uulmanni2013}, and in \\cite{lauri2012} a stability result was\nproved by using ideas from the Boundary Control method together with complex\ngeometrical optics solutions. \n\n There is an important method\nbased on Carleman estimates \\cite{Bukhgeuim1981}, often called the Bukhgeim-Klibanov method after its founders, that can be used to show stability\nresults requiring only a\nsingle instance of boundary values, when the initial data for the wave equation is non-vanishing. \nWe mention the interesting recent computational work \\cite{Baudouin} that is based on this method, and also\nanother reconstruction method that uses a single measurement \\cite{beilina2012approximate, Beilina2008}. \nThis method is based\non a reduction to a non-linear integro-differential equation, and\nthere are several papers on\nhow to solve this equation (or an approximate version of it)---see \\cite{Klibanov2017,  Klibanov2015} for recent results\nincluding computational implementations. \n \n\\subsection{Notations}\nWe will define $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ as a discrete version of the regulation strategy given in \\cite{jusa1}. For that we recall some notation from  \\cite{jusa1}.  \n\nWe denote the indicator function of a set $E$ by \n$$\n1_E(x) = \n\\begin{cases}\n1, & x \\in E,\n\\\\\n0, & \\text{otherwise}.\n\\end{cases}\n$$\nWe define \n\\begin{align}\n\\label{operaattorit}\nJ : L^2(0,2T) \\to L^2(0,2T), \\quad\nJ f(t) = \\frac{1}{2} \\int_0^{2 T} 1_\\blacktriangle (t,s)f(s) ds,&\n\\end{align}\nwhere\n$\n\\blacktriangle = \\{(t,s) \\in (0,2T)^2;\\ \\text{$t + s \\le 2 T$ and $s > t > 0$}\\}.\n$\nWe define the time reversal operator as\n\\begin{align}\n\\label{operaattorit2}\nR : L^2(0,2T) \\to L^2(0,2T), \\quad\nR f(t) = f(2 T - t),\n\\end{align}\nand the projections as\n\\begin{align}\n\\label{operaattorit3}\n P_r f(t) = 1_{(T-r,T)}(t)f(t), \\quad r \\in [0,T].\n\\end{align}\nUsing (\\ref{operaattorit}), (\\ref{operaattorit2}), and (\\ref{operaattorit3}) we define that \n\\begin{align}\n\\label{Koo}\nK : Y \\to Y, \\quad\nKL = JL-RLRJ\n\\end{align}\nand\n\\begin{align}\n\\label{Hoo}\n\\boldsymbol{H} &:Y \\mapsto C([0,T],Y), \n\\quad\\boldsymbol{H}L(r) =     \nP_r(K L)P_r.\n\\end{align}\n\nWe define a regularized inversion with cutoff as\n\\begin{align}\n\\label{Zalfa}\n{Z}_{\\alpha} : Y \\to Y,\n\\quad\nZ_\\alpha(L) = \\eta_Y(L,\\alpha) (L+\\alpha)^{-1},\n\\end{align}\nwhere $\\eta_Y : Y \\times (0,\\infty) \\to \\R$ is, for example, a continuous function that satisfies\n$\\eta_Y(L,\\alpha) = 1$ when $d(L,Y^+) \\le \\alpha \/ 4$ and \n$\\eta_Y(L,\\alpha) = 0$ when $d(L,Y^+) \\ge \\alpha \/ 2$. Here \n$$\nd(L,Y^+) = \\inf \\{\\norm{L-L^+}_Y;\\ \\text{$L^+ \\in Y$ is positive semidefinite}\\}.\n$$\nWe denote by $\\boldsymbol{Z}_{\\alpha}$ the lift of ${Z}_{\\alpha}$\nto $C([0,T],Y)$, that is,\n$\\boldsymbol{Z}_{\\alpha}(L)(r)$ is ${Z}_{\\alpha}(L(r))$.\nMoreover, we define $b(t) = 1_{(0,T)}(t) (T-t)$ and\n\\begin{align}\n\\label{innerproduct}\n& \\boldsymbol{S}: C([0,T],Y) \\to C([0,T]),\n\\quad \\boldsymbol SL(r) =\\langle L(r)P_r b,b\\rangle _{L^2(0,2T)}.\n\\end{align}\n\nWe define the travel time coordinates by\n\\begin{align}\n\\label{def_tau}\n\\tau:[0,\\infty)\\to[0,\\infty), \\quad \\tau(x) = \\int_0^x \\frac{1}{c(t)} dt, \\quad x \\in M,\n\\end{align}\nand the domain of influence \n\\begin{align}\n\\label{siiri}\n\\mathcal M(r) = \\{x \\in M; \\tau(x) \\le r\\}, r \\ge 0.\n\\end{align}\nThe function $\\tau$ \nis strictly increasing and we denote its inverse by $\\chi$.\nMoreover, $V(r)$ denotes the volume of $\\mathcal M(r)$ with respect to the measure $dV = c^{-2}dx$, \nwhere $c$ is the speed of sound in (\\ref{dartwader}).\nFrom \\cite[Eq. (21)]{jusa1} we see that \n\\begin{align}\n\\label{operaattorit4}\nV = \\lim_{\\alpha \\to 0} \n(\\boldsymbol S \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol H)(\\Lambda).\n\\end{align}\nMoreover, according to \\cite[Eq. (19), (20)]{jusa1}, the speed of sound in travel time coordinates $v = c \\circ \\chi$ satisfies\n\\begin{align}\n\\label{operaattorit5}\nv(r) = \\frac 1 {\\partial_r V(r)}, \\quad \\chi(r) = \\int_0^r v(t) dt, \\quad r > 0. \n\\end{align}\nThus $c$ can be computed from $V$. We will next recall how the formula (\\ref{operaattorit5}) is regularized in \\cite{jusa1}. For small $h>0$ we consider the partition\n\\begin{align} \n\\label{jako}\n(0,T)=(0,h) \\cup[h,2h)\\cup[2h,3h)\\cup...\\cup\n[N_hh-h,N_hh)\\cup [N_hh,T),\n\\end{align}\nwhere $N_h\\in \\mathbb{N}$ satisfies $T-h\\le N_hh < T$.\nWe define a discretized and regularized approximation of the derivative operator $\\partial_r$ by\n\\begin{align} \n\\label{operaattori6}\nD_{h}:C([0,T])\\to L^\\infty(0,T), \n\\end{align}\n\\begin{displaymath}\nD_{h}(f)(t)= \\left\\{ \\begin{array}{cl}\n\\frac{f(h)}{h}, & \\textrm{if\\quad $t \\in (0,h)$},\\\\\n\\frac{f(jh+h\n)-f(jh)}{h}, & \\textrm{if\\quad $t \\in [jh,jh+h)$},\\\\\n \\frac{f(T)-f(N_hh)}{T-N_hh},& \\textrm{if\\quad $t \\in [N_hh,T)$.}\\\\ \n\\end{array} \\right.\n\\end{displaymath}\nLet us have \n\\begin{align}\n\\label{arskasika}\n\\vartheta (t;a,b) = a 1_{(-\\infty, a)}(t) + t 1_{[a, b]}(t)+\nb 1_{( b,\\infty)}(t).\n\\end{align}\nWe define an inversion with a cutoff that takes into account the a priori bounds in (\\ref{nopeudet}) by \n\\begin{align}\n\\label{rampo}\nz : \\R \\to \\R\n\\quad\nz(t) &= \\frac{1}{\\vartheta (t;C_0,C_1)}.\n\\end{align}\nWe denote by $\\boldsymbol z$ the lift of $z$\nto $L^\\infty(0,T)$, that is,\n$\\boldsymbol z(f)(t) = z(f(t))$.\nWe define the extension by one\n\\begin{align}\n\\label{arska}\nE : L^\\infty(0,T) \\to L^\\infty(0,\\infty), \n\\quad Ef(t) = \n\\begin{cases}\nf(t), & t \\in (0,T), \\\\\n1, & \\text{otherwise},\n\\end{cases}\n\\end{align}\nand set $W = E \\circ \\boldsymbol z$. We define \n\\begin{align}\n\\label{arskapiks}\n\\tilde \\chi : L^\\infty(0,\\infty) \\to C(0,\\infty), \\quad\n\\tilde \\chi(f)(r) = \\int_0^r f(t) dt.\n\\end{align}\nNote that having $f>0$ in (\\ref{arskapiks}) we have it that $\\tilde \\chi(f)$ is a strictly increasing function. \nHaving $L_0,L_1$, as in (\\ref{nopeudet}), we define \n\\begin{align} \n\\label{operaattori16}\n\\theta_\\R: L^\\infty(0,\\infty) \\to L^\\infty(\\mathbb{R}), \n\\end{align}\n\\begin{displaymath}\n\\theta_\\R(f)(t)= \\left\\{ \\begin{array}{cl}\n1, & \\textrm{if\\quad $t \\in (-\\infty,L_0]$},\\\\\nf(t), & \\textrm{if\\quad $t \\in (L_0,L_1)$},\\\\\n 1,& \\textrm{if\\quad $t \\in [L_1,\\infty)$.}\\\\ \n\\end{array} \\right.\n\\end{displaymath}\nHaving $f>0$ and using (\\ref{arskapiks}) and (\\ref{operaattori16}) we define\n\\begin{align}\n\\label{hiiri} \n\\Phi : L^\\infty(0,\\infty) \\to L^\\infty(\\mathbb{R}),\n\\quad\n\\Phi(f) = \\theta_\\R(f \\circ (\\tilde \\chi(f))^{-1}).\n\\end{align}\nLet us define $\\eta\\in C^\\infty (\\mathbb{R})$ by\n\\begin{align} \n\\label{konvooluutti}\n{ \\eta(x)= \\left\\{ \\begin{array}{cl}\n  C\\exp\\big (\\frac{1}{x^2-1}\\big ), & \\textrm{if\\quad $x \\in (-1,1)$},\\\\\n0, & \\textrm{if\\quad $|x|\\ge 1$},\\\\\n\\end{array} \\right.}\n\\end{align}\n{where the constant $C>0$ is selected so that $\\int_\\mathbb{R} \\eta (x)=1$.} For $\\nu >0$ we define\n\\begin{align} \n\\label{molliolli}\n{  \\eta_\\nu (x)=\\frac{1}{\\nu }\\eta \\Big (\\frac{x}{\\nu } \\Big ).}\n\\end{align}\nBy using convolution we define a smooth approximation to a given function $f\\in L^\\infty(\\mathbb{R} )$ by setting\n\\begin{align} \n\\label{smootapp}\n\\Gamma_\\nu:L^\\infty(\\mathbb{R})\\to C^\\infty (\\mathbb{R} ),\\quad \\Gamma_\\nu ( f)= \\eta_\\nu \\ast f.\n\\end{align}\n Using (\\ref{Hoo}),\n(\\ref{Zalfa}), (\\ref{innerproduct}), (\\ref{operaattori6}), (\\ref{arska}), (\\ref{hiiri}), and (\\ref{smootapp}) we define the family of operators for the regularization strategy used in \\cite{jusa1} by\n\\begin{align}\n\\label{Rstategia}\n&\\mathcal R_{\\alpha}:Y\\to Z,\n\\\\\\nonumber&\\mathcal R_{\\alpha}=\\Gamma_\\nu\\circ\\Phi\\circ W\\circ D_{h}\\circ\\boldsymbol{S}\\circ \\boldsymbol{Z}_{\\alpha}\\circ\\boldsymbol{H},\n\\end{align}\nwhere $\\nu=C\\epsilon^{\\frac{1}{54}}$, $h=C\\epsilon^{\\frac{1}{18}}$ and $\\alpha(\\epsilon)=2^{\\frac{13}{9}}T^{\\frac{4}{9}}\\epsilon^{\\frac{4}{9}}$. Note that in \\cite{jusa1} we considered perturbations of the Neumann-to-Dirichlet operator of the form  \n\\begin{align}\n\\label{virhemitta}\n&\\widetilde\\Lambda=\\Lambda+\\mathcal{E}, \n\\end{align}\nwhere $\\mathcal{E}\\in Y$\nmodels the measurement error and $\\norm{\\mathcal{E}}_Y \\le \\epsilon$. \nBelow we will introduce a discretized version of regularization strategy (\\ref{Rstategia}) that takes in discretized measurements. To this end we start with auxiliary lemmas.\n\\subsection{The proofs of the main results}\n\\label{convergense}\n\n\\begin{lemma}\n\\label{DLa} \nLet $N\\in \\mathbb{Z}_+$. Let $\\Lambda $ be as in (\\ref{lam}) and $\\Lambda_N$ as in (\\ref{pallokala}). Then we have\n\\begin{align*}\n \\norm{\\Lambda_{N}-\\Lambda }_{{Y}}\\le CN^{-\\frac{1}{4}}.\n\\end{align*}\n\\end{lemma}\nHere $C=C(T)>0$ depends on $T$.\n\\begin{proof}\nBy (\\ref{lam}) and the trace theorem we have\n\\begin{align}\n\\label{Lamestim}\n \\norm{\\Lambda}_{L^2(0,2T)\\to H^{\\frac{1}{2}}(0,2T)}\n\\le C_{Lam}.\n\\end{align}\nBy (\\ref{pcsfjoukko2}) we have\n\\begin{align}\n\\label{virheprojektiolle7}\n \\norm{I-P^N}_{L^2(0,2T)\\to L^2(0,2T)}\n\\le2.\n\\end{align}\nLet $f\\in H^2(0,2T)$. By (\\ref{pcsfjoukko2}) and $H^2(0,2T)\\hookrightarrow C^1([0,2T])$ we have\n\\begin{align}\n\\label{virheprojektiolle3}\n \\norm{f -P^N f}_{L^\\infty (0,2T)}\\le\n \\frac{T}{N}\\norm{f}_{C^1([0,2T])}.\n\\end{align}\nThus \n\\begin{align}\n\\label{virheprojektiolle4}\n \\norm{f -P^N f}_{L^2(0,2T)}\\le(2T)^{\\frac{1}{2}}\\norm{f -P^N f}_{L^\\infty (0,2T)}\n\\le(2T)^{\\frac{1}{2}} \\frac{T}{N}\\norm{f}_{C^1([0,2T])}.\n\\end{align}\nBy (\\ref{virheprojektiolle4}) and having $C_{sob}=\\norm{I}_{H^2(0,2T)\\to C^1([0,2T])}$ we get\n\\begin{align}\n\\label{virheprojektiolle5}\n \\norm{I-P^N}_{H^2(0,2T)\\to L^2(0,2T)}\n\\le(2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}.\n\\end{align}\nUsing interpolation theory with (\\ref{virheprojektiolle7}) and (\\ref{virheprojektiolle5}) we have\n\\begin{align}\n\\label{virheprojektiolle8}\n \\norm{I-P^N}_{H^{\\frac{1}{2}}(0,2T)\\to L^2(0,2T)}\n\\le 2^{\\frac{3}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}.\n\\end{align}\nFor self-adjoint operators $P^N$ and $\\Lambda$ we have\n\\begin{align}\n\\label{virheprojektiolle10}\n \\norm{\\Lambda -\\Lambda P^N }_{{Y}}=\n  \\norm{(\\Lambda -\\Lambda P^N )^*}_{Y}=\n  \\norm{(I - P^N)\\Lambda }_{{Y}}.\n\\end{align}\nBy (\\ref{Lamestim}) and (\\ref{virheprojektiolle8}) we have\n\\begin{align}\n\\label{virheprojektiolle12}\n \\norm{\\Lambda -\\Lambda P^N }_{{Y}}=\\norm{(I - P^N)\\Lambda }_{{Y}}\\le\n 2^{\\frac{3}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}C_{Lam}.\n\\end{align}\nThus\n\\begin{align}\n\\label{virheprojektiolle13}\n \\norm{P^N \\Lambda -P^N \\Lambda P^N }_{{Y}}\\le\n 2^{\\frac{3}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}C_{Lam}.\n\\end{align}\nBy (\\ref{virheprojektiolle10}), (\\ref{virheprojektiolle12}), and (\\ref{virheprojektiolle13}) \nwe have\n\\begin{align}\n\\label{virheprojektiolle14}\n \\norm{\\Lambda -P^N \\Lambda P^N }_{{Y}}\\le\n 2^{\\frac{7}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}C_{Lam}\\le\nCN^{-\\frac{1}{4}}.\n\\end{align}\n\\end{proof}\n\\begin{proposition}\n\\label{DLb} \nLet $\\epsilon>0$, $N\\in \\mathbb{Z}_+$ and $N\\ge\\epsilon^{-4}$ . Let $\\Lambda $ be as in (\\ref{lam}) and $\\Lambda_N$ be as in (\\ref{pallokala}). Assume that $\\widetilde\\Lambda_{N}\\in \\mathcal{B}_Y (\\Lambda_N,\\epsilon)$, then \n\\begin{align*}\n \\norm{\\widetilde\\Lambda_{N}-\\Lambda }_{{Y}}\\le C_2\\epsilon.\n\\end{align*}\n\\end{proposition}\nHere $C_2=C_2(T)>0$ depends on $T$.\n\\begin{proof}\n\n\nUsing Lemma \\ref{DLa} and having $N\\ge\\epsilon^{-4}$\nwe get\n\\begin{align}\n\\label{virheprojektiolle15}\n \\norm{\\widetilde\\Lambda_{N}-\\Lambda }_{{Y}}\\le  \\epsilon +\n CN^{-\\frac{1}{4}}\\le (C+1)\\epsilon.\n\\end{align}\n\\end{proof}\n Let $J$ be as in (\\ref{operaattorit})  and using (\\ref{pcsfjoukko2}) we define\n\\begin{align}\n\\label{pridekulkue}\nJ_N=P^NJP^N.\n\\end{align}\n\\begin{lemma}\n\\label{JN}\nLet $N\\in\\mathbb{Z}_+$. Let $J$ be as in (\\ref{operaattorit}) and $J^N$ be as in (\\ref{pridekulkue}). Then we have\n\\begin{align*}\n \\norm{J-J_N}_{Y}\\le\nCN^{-\\frac{1}{2}}.\n\\end{align*}\n\\end{lemma}\nHere $C=C(T)>0$ depends on $T$.\n\\begin{proof}\nBy (\\ref{operaattorit}) we have\n\\begin{align}\n\\label{JnormiH}\n \\norm{J}_{L^2(0,2T)\\to H^{1}(0,2T)}\\le C_{J}.\n\\end{align}\nUsing interpolation with (\\ref{virheprojektiolle7}) and (\\ref{virheprojektiolle5}) we have\n\\begin{align}\n\\label{virheprojektiolle8b}\n \\norm{I-P^N}_{H^{1}(0,2T)\\to L^2(0,2T)}\n\\le 2^{\\frac{1}{2}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{2}}.\n\\end{align}\nBy (\\ref{JnormiH}) and (\\ref{virheprojektiolle8b}) we have\n\\begin{align}\n\\label{virheprojektiolle9b}\n \\norm{(I-P^N)J}_{L^{2}(0,2T)\\to L^2(0,2T)}\n\\le C_{J} 2^{\\frac{1}{2}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{2}}.\n\\end{align}\nWe have\n\\begin{align}\n\\label{virheprojektiolle9c}\n \\norm{J(I-P^N)}_{L^{2}(0,2T)\\to L^2(0,2T)}=\n \\norm{(I-P^N)J^*}_{L^{2}(0,2T)\\to L^2(0,2T)}.\n\\end{align}\nBy (\\ref{JnormiH}), (\\ref{virheprojektiolle9b}), and (\\ref{virheprojektiolle9c})\nwe get\n\\begin{align}\n\\label{virheprojektiolle9d}\n \\norm{J-P^NJP^N}_{Y}\\le\n \\norm{(I-P^N)J}_{Y}+\\norm{P^N}_{Y}\\norm{J(I-P^N)}_{Y}\n\\le CN^{-\\frac{1}{2}}.\n\\end{align}\n\\end{proof}\n\nLet $J$ be as in (\\ref{operaattorit}), $J^N$ be as in (\\ref{pridekulkue}), and $R$ be as in (\\ref{operaattorit2}). We define \n\\begin{align}\n\\label{Koo2}\nK^N : Y \\to Y, \\quad\nK^N L = J_NL-RLRJ_N.\n\\end{align}\nBy (\\ref{operaattorit3}) and (\\ref{Koo2}) we define   \n\\begin{align}\n\\label{Hn}\n\\boldsymbol{H}^N &:Y \\to C([0,T],Y), \n\\quad\\boldsymbol{H}^NL(r) =     \nP_r(K^N L)P_r.\n\\end{align}\n\\begin{proposition}\n\\label{HN}\nLet $\\epsilon\\in (0,1)$, $\\kappa_3>0$, and $N\\ge\\kappa_3\\epsilon^{-4}$. Let $\\Lambda $ be as in (\\ref{lam}), $\\boldsymbol{H}$ be as in (\\ref{Hoo}), and $\\boldsymbol{H}^N$ be as in (\\ref{Hn}). Assume that $ \\widetilde\\Lambda_N\\in \\mathcal{B}_Y ( \\Lambda,\\epsilon)$, then we have  \n\\begin{align*}\n \\norm{\\boldsymbol{H}^N\\widetilde\\Lambda_N -\\boldsymbol{H}\\Lambda }_{C([0,T],Y)}\\le C_3\\epsilon.\n\\end{align*}\n\\end{proposition}\nHere $C_3=C_3(T,\\kappa_3)>0$ depends on $T$ and $\\kappa_3$.\n\\begin{proof}\nBy (\\ref{Koo}) and (\\ref{Koo2}) we get\n\\begin{align}\n\\label{Kvirhe10b}\nK^N\\widetilde\\Lambda_N -K\\widetilde\\Lambda_N=\nR\\widetilde\\Lambda_NR(J-J_N)+(J_N-J)\\widetilde\\Lambda_N.\n\\end{align}\nUsing (\\ref{operaattorit2}), we have $\\norm{R}_Y\\le 1$. Using \\cite[Theorem 5]{jusa1}, we have $\\norm{\\Lambda}_Y \\le M_1<\\infty$. By (\\ref{Kvirhe10b}) and Lemma \\ref{JN} we have\n\\begin{align}\n\\label{Kvirhe10}\n \\norm{\\boldsymbol{H}^N\\widetilde\\Lambda_N(r) -\\boldsymbol{H}\\widetilde\\Lambda_N (r) }_Y\\le\\norm{K^N\\widetilde\\Lambda_N -K\\widetilde\\Lambda_N}_{Y}\\le\n2(M_1+1)CN^{-\\frac{1}{2}}.\n\\end{align}\nBy \\cite[Proposition 1]{jusa1} we have\n\\begin{align}\n\\label{Kvirhepapru}\n \\norm{\\boldsymbol{H}\\widetilde\\Lambda_N -\\boldsymbol{H}\\Lambda}_{C([0,T],Y)}\\le\nT\\epsilon.\n\\end{align}\nBy (\\ref{Kvirhe10}) and (\\ref{Kvirhepapru}), when $\\epsilon\\in (0,1)$ and $N\\ge\\kappa_3\\epsilon^{-4}$ we have \n\\begin{align}\n\\label{tohtorisaksala}\n \\norm{\\boldsymbol{H}^N\\widetilde\\Lambda_N -\\boldsymbol{H}\\Lambda }_{C([0,T],Y)}\\le 2(M_1+1)C\\Big(\\frac{1}{\\kappa_2}\\Big)^{\\frac{1}{2}}\\epsilon^{2}\n +T\\epsilon\\le C_3\\epsilon.\n\\end{align}\n\\end{proof}\n\nLet $P_r$ be as in (\\ref{operaattorit3}), $b$ be as in (\\ref{innerproduct}), and $P^N$ be as in (\\ref{pcsfjoukko2}). We define \n\\begin{align}\n\\label{innerproduct2}\n& \\boldsymbol S^N: C([0,T],Y) \\to C([0,T]),\n\\quad \\boldsymbol S^NL(r) =\\langle P^N L(r)P_r b,b\\rangle _{L^2(0,2T)}.\n\\end{align}\n\\begin{lemma}\n\\label{SN}\nLet $L\\in C([0,T],Y)$ and $\\boldsymbol{S}$ be as defined in (\\ref{innerproduct}). Then\n\\begin{align*}\n\\norm{\\boldsymbol S^NL-\\boldsymbol SL}_{C([0,T])}\n\\le \\frac{T^3}{6N}\\norm{L}_{C([0,T],Y)}.\n\\end{align*} \n\\end{lemma}\n\\begin{proof} By using (\\ref{innerproduct}), (\\ref{innerproduct2}) with the self-adjointness of operator $P^N$ we get \n\\begin{align*}\n|\\boldsymbol S^NL(r)-\\boldsymbol SL(r)|\\le \\norm{L(r)}_Y  \\norm{P_rb}_{L^2(0,2T)}\\norm{P^Nb-b}_{L^2(0,2T)}.\n\\end{align*}\nWe have $\\norm{P_rb}_{L^2(0,2T)}\\le(\\frac{T^3}{3})^{\\frac{1}{2}}$ and $\\norm{P^Nb-b}_{L^2(0,2T)}\\le(\\frac{T^3}{12N^2})^{\\frac{1}{2}}.$\n\\end{proof}\nLet $K$ be as in (\\ref{Koo}) and $K^N$ be as in (\\ref{Koo2}). Let $\\boldsymbol{H}$ be as in (\\ref{Hoo}) and $\\boldsymbol{H}^N$ be as in (\\ref{Hn}). \nLet $\\boldsymbol{Z}_{\\alpha}$ be as in (\\ref{Zalfa}). For $r\\in[0,T]$ and $\\alpha>0$ we denote\n\\begin{align}\n\\label{elvisonkunkku}\n& Z_{\\alpha,r}=\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda)(r),\n \\\\\\nonumber &\\widetilde Z_{\\alpha,r}=\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)(r).\n\\end{align} \nLet $\\boldsymbol{S}$ be as in (\\ref{innerproduct}). Using (\\ref{elvisonkunkku}) we denote\n\\begin{align}\n\\label{sope2}\n& s_\\alpha = \\boldsymbol S \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol H\\Lambda,\n \\\\\\nonumber &\\widetilde s_\\alpha = \\boldsymbol S \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol{H}^N\\widetilde\\Lambda_N.\n\\end{align} \nLet $\\boldsymbol S^N$ be as in (\\ref{innerproduct2}). Using  (\\ref{elvisonkunkku}) we denote\n\\begin{align}\n\\label{sope3}\n\\widetilde s^N_\\alpha = \\boldsymbol S^N \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol{H}^N\\widetilde\\Lambda_N,\\quad \\widetilde s^N_\\alpha (r)\n =\\langle P^N\\widetilde Z_{\\alpha,r} P_r b,b\\rangle _{L^2(0,2T)}.\n\\end{align} \n\\begin{proposition}\n\\label{PSN}\nLet $\\epsilon\\in (0,1)$ and $\\kappa_4>0$. Let $\\alpha=2\\epsilon^{\\frac{4}{9}}$ and $N\\ge\\kappa_4\\epsilon^{-4}$. Let $s_\\alpha$ be as defined in (\\ref{sope2}) and $\\widetilde s^N_\\alpha$ be as defined in (\\ref{sope3}). Let $\\boldsymbol{H} \\Lambda(r)\\in Y$ be bounded and positive semidefinite for all $r\\in[0,T]$. Assume that $\\boldsymbol{H}^N\\widetilde\\Lambda_N\\in \\mathcal{B}_{C([0,T],Y)} (\\boldsymbol{H} \\Lambda,\\epsilon)$, then we have\n\\begin{align*}\n\\norm{\\widetilde s^N_\\alpha-s_\\alpha}_{C([0,T])}\n\\le C_4\\epsilon^{\\frac{1}{9}}.\n\\end{align*} \nHere $C_4=C_4(T,\\kappa_3)>0$ depends on $T$ and $\\kappa_3$.\n\\end{proposition}\n\n\\begin{proof}\nWe have \n\\begin{align}\n\\label{soinintimppa}\n\\norm{\\widetilde s^N_\\alpha-s_\\alpha}_{C([0,T])}\n\\le\\norm{\\widetilde s^N_\\alpha-\\widetilde s_\\alpha}_{C([0,T])}+\\norm{\\widetilde s_\\alpha-s_\\alpha}_{C([0,T])} .\n\\end{align} \nLet $\\boldsymbol{S}$ be as defined in (\\ref{innerproduct}). We have $\\norm{\\boldsymbol{S}}_{C([0,T],Y)\\to C([0,T])}\\le \\frac{T^3}{3}$, see \\cite[Proposition 2]{jusa1}. Having  $\\boldsymbol{H}^N\\widetilde\\Lambda_N\\in \\mathcal{B}_{C([0,T],Y)} (\\boldsymbol{H} \\Lambda,\\epsilon)$ we get $\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N) -\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_{C([0,T],Y)}\\le\\frac{1}{2}\\epsilon^{\\frac{1}{9}}$, see \\cite[Proposition 2]{jusa1}. Using (\\ref{sope2}), for the second part of the sum in the right-hand side we get\n\\begin{align}\n\\label{rape}\n\\norm{\\widetilde s_\\alpha-s_\\alpha}_{C([0,T])}\n\\le\\norm{\\boldsymbol{S}}_\\Omega\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N) -\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_{C([0,T],Y)}\\le\\frac{T^3}{3}\\frac{1}{2}\\epsilon^{\\frac{1}{9}} ,\n\\end{align} \nwhere we denote $\\Omega=C([0,T],Y)\\to C([0,T])$.\nUsing (\\ref{sope2}) and (\\ref{sope3}) with Lemma \\ref{SN}, for the first part of the sum in the right-hand side we get\n\\begin{align}\n\\label{rape2}\n\\norm{\\widetilde s^N_\\alpha-\\widetilde s_\\alpha}_{C([0,T])}\n\\le\\frac{T^3}{6N}\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)}_{C([0,T],Y)}.\n\\end{align} \nWe have \n\\begin{align}\n\\label{rape3}\n\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)}_\\Omega\n\\le\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)-\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_\\Omega+\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_\\Omega,\n\\end{align} \nwhere we denote $\\Omega=C([0,T],Y)$. Using \\cite[Eq. (32)]{jusa1} we have\n\\begin{align}\n\\label{rape4}\n\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda (r)) }_Y\\le\\alpha^{-1}.\n\\end{align} \nHaving $\\epsilon\\in (0,1)$, $N\\ge\\kappa_4\\epsilon^{-4}$ and $\\alpha=2\\epsilon^{\\frac{4}{9}}$ with use of (\\ref{rape2}), (\\ref{rape3}), and (\\ref{rape4}) we get\n\\begin{align}\n\\label{rape5}\n\\norm{\\widetilde s^N_\\alpha-\\widetilde s_\\alpha}_{C([0,T])}\n\\le\\frac{T^3}{6}\\frac{\\epsilon^4}{\\kappa_4}\\Big(\\frac{1}{2}\\epsilon^{\\frac{1}{9}}+\\frac{1}{2}\\epsilon^{-\\frac{4}{9}}\\Big)\\le \\frac{T^3}{3\\kappa_4}\\epsilon^{\\frac{32}{9}}.\n\\end{align} \nBy using (\\ref{soinintimppa}) (\\ref{rape}) and (\\ref{rape5}) we get the estimate.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{kaiken_teoria2}]\n\\label{hauki}\n\\hfill \\break\nLet us consider the measurement ($\\epsilon_1, N_1, \\widetilde\\Lambda_{N_1}$).\nThere is $N_1\\ge\\epsilon_1^{-4}$ and $\\widetilde\\Lambda_{N_1}$, for which $\\norm{\\widetilde\\Lambda_{N_1}-\\Lambda_{N_1}}_{Y}\\le \\epsilon_1$, and Proposition  \\ref{DLb}  gives us\n\n\\begin{align*}\n \\norm{\\widetilde\\Lambda_{N_1}-\\Lambda }_{{Y}}\\le C_2\\epsilon_1.\n\\end{align*}\n\n  Let $\\epsilon_1\\in (0,C_2^{-1})$ and $\\epsilon=C_2\\epsilon_1$. We choose $\\kappa_3=C_2^4$ and thus $ N_1\\ge\\epsilon_1^{-4}=C_2^4\\epsilon^{-4}$. Having $\\widetilde\\Lambda_{N_1}\\in \\mathcal{B}_{Y} (\\Lambda,\\epsilon)$, Proposition \\ref{HN} gives us \n\\begin{align}\n\\label{patelaine2c}\n \\norm{\\boldsymbol{H}^{N_1}\\widetilde\\Lambda_{N_1} -\\boldsymbol{H}\\Lambda}_{C([0,T],Y)}\\le C_3C_2\\epsilon_1.\n\\end{align}\nLet  $\\epsilon_1\\in(0,C_2^{-1}C_3^{-1})$ and $\\epsilon=C_2C_3\\epsilon_1$.  We choose $\\kappa_4=C_2^4C_3^4$ and thus $ N_1\\ge\\epsilon_1^{-4}=C_2^4C_3^4\\epsilon^{-4}$. Let $\\alpha_1=2C_2^{\\frac{4}{9}}C_3^{\\frac{4}{9}}\\epsilon_1^{\\frac{4}{9}}$. Having (\\ref{patelaine2c}), Proposition \\ref{PSN} gives us\n\\begin{align}\n\\label{timojutilac}\n\\norm{\\widetilde s^{N_1}_\\alpha-s_{\\alpha}}_{C([0,T])} \\le\nC_4C_3^{\\frac{1}{9}}C_2^{\\frac{1}{9}}\\epsilon_1^{\\frac{1}{9}}.\n\\end{align}\nLet $V$ be as defined as in (\\ref{operaattorit4}) and $D_{h}$ be as defined as in (\\ref{operaattori6}). Let $\\epsilon_1\\in(0,\\widehat\\epsilon)$, where $\\widehat\\epsilon$ as defined in  \\cite[Proposition 4]{jusa1}. Let $\\kappa_5 >0$ and $h=\\kappa_5\\epsilon_1^{\\frac{1}{18}}$. Having (\\ref{timojutilac}) and using \\cite[Proposition 4]{jusa1} we get \n\\begin{align} \n\\label{timojutila2c}\n\\norm{D_{h}(\\widetilde s^{N_1}_\\alpha)-\\partial_r V}_{L^\\infty (0,T)}\n\\le C_5 C_4^{\\frac{1}{2}} C_3^{\\frac{1}{18}}C_2^{\\frac{1}{18}}\\epsilon_1^{\\frac{1}{18}},\n\\end{align} \nwhere $C_5=C_5(\\kappa_5)$. Note that we use the parameter $\\kappa_5$ to control the size of discretization in (\\ref{operaattori6}).\nLet $v$ be as defined in (\\ref{operaattorit5}) and $W$ be as defined in (\\ref{arska}). Let us denote $\\widetilde{w}^{N_1}_{\\alpha}=W(D_{h}(\\widetilde s^{N_1}_\\alpha))$. \nHaving (\\ref{timojutila2c}) and using \\cite[Proposition 5]{jusa1} we get \n\\begin{align}\n\\label{sebastianahoc}\n\\norm{ \\widetilde{w}^{N_1}_{\\alpha}-v}_{L^\\infty (M)} \n\\le C_6 C_5 C_4^{\\frac{1}{2}} C_3^{\\frac{1}{18}} C_2^{\\frac{1}{18}}\\epsilon_1^{\\frac{1}{18}}.\n\\end{align}\nLet $\\nu =\\kappa_6\\epsilon_1^\\frac{1}{54}$ and $c\\in\\mathcal V^3$. Let $\\Phi$ be as in (\\ref{hiiri}) and $\\eta_\\nu$ be as in (\\ref{molliolli}). Let us denote $\\widetilde c^{N_1}_{\\alpha}=\\eta_\\nu \\ast\\Phi(\\widetilde w^{N_1}_\\alpha)$. Having (\\ref{sebastianahoc}) and using  \\cite[Proposition 6]{jusa1} we get \n\\begin{align}\n\\label{kenguru}\n\\norm{\\widetilde c^{N_1}_{\\alpha} -c}_{C^2(M)} \n\\le C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} C_4^{\\frac{1}{6}} C_3^{\\frac{1}{54}} C_2^{\\frac{1}{54}}\\epsilon_1^{\\frac{1}{54}},\n\\end{align}\nwhere $C_7=C_7(\\kappa_6)$. Note that we use the parameter $\\kappa_6$ to control the support of $\\eta_\\nu$ in (\\ref{molliolli}).\nWe define\n\\begin{align}\n\\label{epsilooninolla}\n \\widetilde\\epsilon_1=min\\Big\\{C_2^{-1},C_2^{-1}C_3^{-1},\\widehat\\epsilon\\Big\\}.\n\\end{align} \n\nBy using (\\ref{Zalfa}), (\\ref{operaattori6}), (\\ref{arska}), (\\ref{hiiri}), (\\ref{smootapp}), (\\ref{Hn}), and (\\ref{innerproduct2}) we define \n\\begin{align}\n\\label{Rstategia2}\n& \\mathcal R^{(1)}_{N_1,\\alpha_1}:Y\\to  Z,\n\\\\\\nonumber&\\mathcal \\mathcal R^{(1)}_{N_1,\\alpha_1}=\\Gamma_\\nu\\circ\\Phi\\circ W\\circ D_{h}\\circ\\boldsymbol{S}^{N_1}\\circ \\boldsymbol{Z}_{\\alpha_1}\\circ\\boldsymbol{H}^{N_1},\n\\end{align}\nand have the estimate\n\\begin{align*}\n\\norm{\\mathcal R^{(1)}_{N_1,\\alpha_1}(\\widetilde\\Lambda_{N_1})-c}_{ Z} \\le\na_1\\epsilon_1^{\\frac{1}{54}},\n\\end{align*}\nwhen $\\epsilon_1 \\in (0,\\widetilde\\epsilon_1)$ and $a_1=C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} C_4^{\\frac{1}{6}} C_3^{\\frac{1}{54}} C_2^{\\frac{1}{54}}$. \n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{kaiken_teoria2b}]\n\\label{ahven}\n\\hfill \\break\nLet ${\\phi_{j,N_0}}$ be as in (\\ref{kantafunktio}), where $j\\in\\{1,2,3,...,2N_0\\}$. \nLet $r\\in [0,2T)$ and we define\n\\begin{align}\n\\label{translaatio}\n\\mathcal T_r: L^2(0,2T) \\to L^2(0,2T), \n\\quad  T_rf(t) = \n\\begin{cases}\n\\,\\quad 0, & t \\in (0,r), \\\\\nf(t-r), & t\\in [r,2T).\n\\end{cases}\n\\end{align}\n\nUsing (\\ref{mittauspoika}) and (\\ref{translaatio}) we define\n\\begin{align}\n\\label{mittausaikapoika}\n \\widetilde\\Lambda_{\\mathcal P}\\phi_{j,N_0}=\\Big(\\frac{N_0}{T}\\Big)^{\\frac{1}{2}}\\Big(\\mathcal T_{\\frac{(j-1)T}{N_0}} -\\mathcal T_{\\frac{jT}{N_0}}\\Big)P^{N_0} \\widetilde m_{N,\\epsilon_0}.\n\\end{align}\nAs $\\big\\{\\phi_{1,N_0},...,\\phi_{2N_0,N_0}\\big\\}$ span $\\mathcal P^{N_0}$ this defines a linear map\n\\begin{align}\n\\label{mittahulg}\n \\widetilde\\Lambda_{\\mathcal P}:\\mathcal P^{N_0}\\to \\mathcal P^{N_0}.\n\\end{align}\nUsing (\\ref{mittausaikapoika}) and (\\ref{mittahulg}) we define a perturbed and  discretizatized Neumann-to-Dirichlet operator\n\n\\begin{align}\n\\label{mittamulgi}\n&\\widetilde\\Lambda_{N_0}:L^2(0,2T)\\to L^2(0,2T),\n\\\\\\nonumber& \\widetilde\\Lambda_{N_0} |_{\\mathcal P^{N_0}}= \\widetilde\\Lambda_{\\mathcal P},\n\\\\\\nonumber&\\widetilde\\Lambda_{N_0} |_{(\\mathcal P^{N_0})^\\perp}=0,\n\\end{align}\nwhere $\\mathcal P^{N_0}\\oplus (\\mathcal P^{N_0})^\\perp = L^2(0,2T)$. Using (\\ref{mittauspoika}), (\\ref{mittausaikapoika}), (\\ref{mittahulg}), and (\\ref{mittamulgi}) we define\n\\begin{align}\n\\label{erkkipoika}\nE_{N_0} : \\mathcal P^N \\to Y,\\quad\n E_{N_0} \\widetilde m_{N,\\epsilon_0}=\\widetilde\\Lambda_{N_0}.\n\\end{align}\nLet $\\epsilon_0>0$. Let $l_0(\\epsilon_0)$ be as in (\\ref{porraspossu}) and $N_0(\\epsilon_0)$ be as in (\\ref{porraspossu2}). Let $N=2^l\\ge N_0$, where $l\\in\\mathbb{Z}_+$. Let $\\Lambda_{N_0} $ be as in (\\ref{pallokala}) and $\\widetilde\\Lambda_{N_0}$ be as in (\\ref{mittamulgi}). Let $P^N$ be as in (\\ref{pcsfjoukko2}) and $\\phi_{1,N}$ be as in (\\ref{kantafunktio}). Let $\\widetilde m_{N,\\epsilon_0}$ be as in (\\ref{mittauspoika}). Assume that $\\widetilde m_{N,\\epsilon_0}\\in \\mathcal{B}_{L^2(0,2T)} (P^N\\Lambda H,\\epsilon_0)$ and $\\norm{n_{N,\\epsilon_0} }_{L^2(0,2T)}\\le\\epsilon_0$. Then \n\\begin{align}\n\\label{kumiukko}\n \\norm{\\widetilde\\Lambda_{N_0}-\\Lambda_{N_0} }_{{Y}}\\le C_{P_1}\\epsilon_0^{\\frac{1}{5}}.\n\\end{align}\n\nHere $C_{P_1}=C_{P_1}(T)>0$ depends on $T$.\nHaving (\\ref{kantafunktio}), (\\ref{porras}), and (\\ref{translaatio}) we get \n \\begin{align}\n\\label{mittauspoikanolla}\n\\Lambda_{N_0}\\phi_{1,{N_0}}=\\Big(\\frac{N_0}{T}\\Big)^{\\frac{1}{2}}P^{N_0}\\Lambda P^{N_0}(I-\\mathcal T_{\\frac{T}{N_0}})H.\n\\end{align}\nAs $N=2^l\\ge N_0=2^{l_0}$ we have $P^{N_0}P^N=P^{N_0}$. Using $\\mathcal T_r \\Lambda=\\Lambda\\mathcal T_r$ and $P^{N_0}\\mathcal T_{\\frac{T}{N_0}}=\\mathcal T_{\\frac{T}{N_0}}P^{N_0}$ with (\\ref{mittauspoika}), (\\ref{mittausaikapoika}), and (\\ref{mittauspoikanolla}) we get \n \\begin{align}\n\\label{mittauspoikan}\n \\widetilde\\Lambda_{N_0}\\phi_{1,{N_0}}-\\Lambda_{N_0}\\phi_{1,{N_0}}=\\Big(\\frac{N_0}{T}\\Big)^{\\frac{1}{2}}(I-\\mathcal T_{\\frac{T}{N_0}})n_{N,\\epsilon_0}.\n\\end{align}\nUsing $N_0\\le\\epsilon_0^{-\\frac{4}{5}}$, $\\norm{\\mathcal T_{\\frac{T}{N_0}} }_{L^2(0,2T)}\\le 1$ and $\\norm{n_{N,\\epsilon_0} }_{L^2(0,2T)}\\le\\epsilon_0$ we get \n\\begin{align}\n\\label{sorsa}\n \\norm{\\widetilde\\Lambda_{N_0}\\phi_{1,{N_0}}-\\Lambda_{N_0}\\phi_{1,{N_0}} }_{L^2(0,2T)}\\le 2T^{-\\frac{1}{2}}\\epsilon_0^{\\frac{3}{5}}.\n\\end{align}\nLet $f\\in L^2(0,2T)$. By (\\ref{pcsfjoukko2}) we get\n\\begin{align}\n\\label{virheporrasfunktiolle2}\n  \\norm{\\widetilde\\Lambda_{N_0} f-\\Lambda_{N_0}  f}_{L^2(0,2T)}\\le \\sum_{j=1}^{2N_0} \\langle f,\\phi_{j,N_0}\\rangle\n_{L^2(0,2T)}\\norm{P^{N_0}\\widetilde\\Lambda\\phi_{j,{N_0}}-P^N\\Lambda\\phi_{j,{N_0}}}_{L^2(0,2T)}.\n\\end{align}\nUsing (\\ref{kantafunktio}), (\\ref{mittausaikapoika}), (\\ref{mittauspoikanolla}), and (\\ref{mittauspoikan}) we have\n\\begin{align}\n\\label{virheporrasfunktiolle2cc}\n \\norm{\\widetilde\\Lambda_{N_0} f-\\Lambda_{N_0}  f}_{L^2(0,2T)}\\le\\norm{P^{N_0}\\widetilde\\Lambda\\phi_{1,{N_0}}-P^{N_0}\\Lambda\\phi_{1,{N_0}}}_{L^2(0,2T)} \\sum_{j=1}^{2{N_0}} \\langle f,\\phi_{j,{N_0}}\\rangle\n_{L^2(0,2T)}.\n\\end{align}\nThus (\\ref{sorsa}) and (\\ref{virheporrasfunktiolle2cc}) gives us\n\\begin{align}\n\\label{virheporrasfunktiolle2ccg}\n\\norm{\\widetilde\\Lambda_{N_0} f-\\Lambda_{N_0}  f}_{L^2(0,2T)}\\le (2N_0)^{\\frac{1}{2}}\\norm{f}_{L^2(0,2T)}2T^{-\\frac{1}{2}}\\epsilon_0^{\\frac{3}{5}}.\n\\end{align}\nWe have $N_0\\le\\epsilon_0^{-\\frac{4}{5}}\\le 2N_0$ and this proves (\\ref{kumiukko}).\n\nWe define \n\\begin{align}\n\\label{anttiaffauros}\n \\epsilon_1=C_{P_1}\\epsilon_0^{\\frac{1}{5}}.\n\\end{align}\nUsing $N_0\\le\\epsilon_0^{-\\frac{4}{5}}\\le 2N_0$ and (\\ref{anttiaffauros}) we get\n\\begin{align}\n\\label{anttiaffauros2}\n N_0\\ge 2^{-1}C_{P_1}^{4}\\epsilon_1^{-4}.\n\\end{align}\nBy (\\ref{anttiaffauros2}), $\\widetilde\\Lambda_{N_0}\\in \\mathcal{B}_Y (\\Lambda_{N_0},\\epsilon_1)$, and Theorem \\ref{kaiken_teoria2} we get \n\\begin{align}\n\\label{katiska}\n\\norm{\\widetilde c^{N_0}_{\\alpha} -c}_{C^2(M)} \n\\le C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} \\widetilde C_4^{\\frac{1}{6}} \\widetilde C_3^{\\frac{1}{54}} \\widetilde C_2^{\\frac{1}{54}}\\epsilon_1^{\\frac{1}{54}}.\n\\end{align}\nNote that in the proof of Theorem \\ref{kaiken_teoria2} we have assumed $N_1\\ge\\epsilon_1^{-4}$. After replacing this by (\\ref{anttiaffauros2}), the proof is identical, only the constants $\\widetilde C_2, \\widetilde C_3, \\widetilde C_4$ change---see (\\ref{virheprojektiolle15}), (\\ref{tohtorisaksala}), (\\ref{rape4}), and (\\ref{kenguru}). Using (\\ref{anttiaffauros}) we get\n\\begin{align*}\n\\norm{\\widetilde c^{N_0}_{\\alpha} -c}_{C^2(M)} \n\\le C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} \\widetilde C_4^{\\frac{1}{6}} \\widetilde C_3^{\\frac{1}{54}}\\widetilde C_2^{\\frac{1}{54}}C_{P_1}^{\\frac{1}{54}}\\epsilon_0^{\\frac{1}{270}}.\n\\end{align*}\n\nWe define\n\\begin{align}\n\\label{epsilooninolla}\n \\widetilde\\epsilon_0=min\\Big\\{C_{P_1}^{4}\\widetilde C_2^{-5},C_{P_1}^{4}\\widetilde C_2^{-5}C_3^{-5},\\widehat\\epsilon\\Big\\},\n\\end{align} \nwhere $\\widehat\\epsilon$ can be specified by using \\cite[Proposition 4]{jusa1}. \nBy using (\\ref{Zalfa}), (\\ref{operaattori6}), (\\ref{arska}), (\\ref{hiiri}), (\\ref{smootapp}), (\\ref{erkkipoika}), (\\ref{Hn}), and (\\ref{innerproduct2}), we define \n\\begin{align}\n\\label{Rstategia2b}\n&\\mathcal R^{(0)}_{N_0,\\alpha_0}:\\mathcal P^{N}\\to  Z,\n\\\\\\nonumber&\\mathcal R^{(0)}_{N_0,\\alpha_0}=\\Gamma_\\nu\\circ\\Phi\\circ W\\circ D_{h}\\circ\\boldsymbol{S}^{N_0}\\circ \\boldsymbol{Z}_{\\alpha}\\circ\\boldsymbol{H}^{N_0}\\circ E^{N_0},\n\\end{align}\nand have the estimate\n\\begin{align*}\n\\norm{\\mathcal R^{(0)}_{N_0,\\alpha_0}(\\widetilde m_{N,\\epsilon_0})-c}_{ Z} \\le\na_0\\epsilon_0^{\\frac{1}{270}},\n\\end{align*}\nwhen $\\epsilon_0 \\in (0,\\widetilde\\epsilon_0)$ and $a_0=C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} \\widetilde C_4^{\\frac{1}{6}} \\widetilde C_3^{\\frac{1}{54}}\\widetilde C_2^{\\frac{1}{54}}C_{P_1}^{\\frac{1}{54}}$. \n\\end{proof}\n\n\n\n\n\\section{Numerical examples}\n\\label{sec_computations}\n\nIn this section we describe a computational implementation of the regularization strategy in Theorem \\ref{kaiken_teoria2}. We will also compare this with a heuristic variant of MDP---see (\\ref{morozov}).\nWe will begin by describing how the data---that is, the noisy discretized Neumann-to-Dirichlet map $\\widetilde \\Lambda_{N_1}$---is simulated. \n\n\n\\subsection{The simulation of measurement data}\n\nWe choose $T = 0.6$ in all the simulations.\nWe use $k$-Wave \\cite{K-Wave} to solve the\nboundary value problem (\\ref{dartwader}) with \n$f =\\phi_{1,N_1} \\in \\mathcal P_{N_1}$, \nwhere $N_1 = 2^{10}$, and denote the solution by $u^{(sim)}$.\nRecall that \n\\begin{align}\n\\label{ekakantafunktio}\n \\phi_{1,N_1}(t) = h^{-\\frac{1}{2}}1_{[0,h)}(t), \\quad t \\in [0,2T),\n\\end{align} \nwhere $h = T \/ N_1$.\nIn order to simulate $u^{(sim)}$ for $2T$ time units, a fine discretization needs to be used, and we choose a regular mesh with $2^{13}$ spatial and $N_2 = 2^{15}$ temporal cells. \nThen we define the simulated Neumann-to-Dirichlet map, acting on the first basis function,\n\\begin{align}\n\\label{possuli}\n\\Lambda^{(sim)} \\phi_{1,N_1}(t) = \\sum_{j=1}^{2N_1} u^{(sim)}(t_j, 0) \\phi_{j,N_2}(t), \\quad t \\in [0,2T),\n\\end{align}\nwhere $t_j=\\frac{(j-1)2T}{N_2}$, $j\\in\\{1,2,...,N_2\\}$, are the temporal grid points.\nThe output of k-Wave is, of course, only an approximation of $u^{(sim)}$ but we do not analyze this simulation error and use the same notation for both $u^{(sim)}$ and its approximation. \n\nOur primary object of interest is the following discretized version of the Neumann-to-Dirichlet map \n\\begin{align}\n\\label{lamdad}\n \\Lambda_{N_1}^{(d)} : \\mathcal P^{N_1} \\to \\mathcal P^{N_1},\\quad\\Lambda_{N_1}^{(d)}f=\\sum_{j=1}^{2N_1}\\sum_{k=1}^{j}f_k\\Lambda_{j-k+1}\\phi_{j,N_1}, \n\\end{align}\nwhere $\\Lambda_j=\\langle \\Lambda^{(sim)} \\phi_{1,N_1},\\phi_{j,N_1}\\rangle_{L^2(0,2T)}$ and $f_k$, $k=1,2,\\dots,2N_1$,\nare the coefficients of $f$ on the basis of $\\mathcal P^{N_1}$.\nObserve that $\\Lambda_{N_1}^{(d)} \\phi_{1,N_1}$ is \nsimply the projection of $\\Lambda^{(sim)} \\phi_{1,N_1}$ on $\\mathcal P^{N_1}$, and that $\\Lambda_{N_1}^{(d)}f$ is then defined by using the fact that the wave equation (\\ref{dartwader}) is invariant with respect to translations in time.\n\nWe will now describe how the noise is simulated. \nConsider \n\\begin{equation}\n\\label{the_noise}\nn=(n_1,n_2,...,n_{2N_1})\\in\\mathbb{R}^{2N_1},\n\\end{equation}\nwhere\n$n_j\\in\\mathcal{N}(0,1)$, that is, $n_j$ is a normally distributed random variable with zero mean and unit variance.\nWe compute a realization of $n$ by using the randn function of MATLAB,\nand use the same notation for $n$ and its realization.\nLet $\\epsilon_0^{(d)}>0$ and define,\nanalogously to (\\ref{lamdad}),\na noisy, discretized version of the Neumann-to-Dirichlet map \n\\begin{align}\n\\label{lamdadvir}\n \\widetilde\\Lambda_{N_1}^{(d)} : \\mathcal P^{N_1} \\to \\mathcal P^{N_1},\\qquad\\widetilde\\Lambda_{N_1}^{(d)} f=\\sum_{j=1}^{2N_1}\\sum_{k=1}^{j}f_k \\widetilde\\Lambda_{j-k+1}\\phi_{j,N_1},\n\\end{align}\nwhere $\\widetilde\\Lambda_j=\\Lambda_j+\\widehat n_j$ and \n\\begin{align}\n\\label{mittavirhe_mod}\n\\widehat n_j= \\frac{\\epsilon_0^{(d)}}{\\norm{n}_{l^2}}n_j, \n\\quad j=1,2,\\dots,2N_1.\n\\end{align} \nFollowing the formulation of Theorem \\ref{kaiken_teoria2},\nrather than using $\\epsilon_0^{(d)}>0$, we prefer to \nparametrize the noise level in terms of \n\\begin{align}\n\\label{laskennalvirhe}\n\\epsilon^{(d)}_1:= \\norm{\\widetilde \\Lambda_{N_1}^{(d)}-\\Lambda_{N_1}^{(d)} }_{\\mathcal P^{N_1}}.\n\\end{align}\nIn what follows, we will consider \nthe quantity $(\\epsilon^{(d)}_1, \\widetilde \\Lambda_{N_1}^{(d)})$, \na simulated analogue of the noisy measurements in Section \\ref{virhesikapossu}. \n\n\n\\subsection{Implementation of the regularization strategy}\n\nFor the a priori bounds in (\\ref{nopeudet}) we use values $C_0=0.01$, $C_1=10$, $L_0=0.01$, and $L_1=0.6$.\nThe crux of the regularization strategy $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ is the computation of the inverse in (\\ref{Zalfa}). \nWhen starting from the simulated measurement $(\\epsilon^{(d)}_1, \\widetilde \\Lambda_{N_1}^{(d)})$, the analogue of (\\ref{Zalfa}) is to solve $X_j$ in the equation\n\\begin{align}\n\\label{yhtalot}\n(P_{r_j}K^{N_1} \\widetilde \\Lambda_{N_1}^{(d)} P_{r_j}+\\alpha_1)X_j=P_{r_j}b.\n\\end{align}\nHere $P_r$ is the projection in (\\ref{operaattorit3}), and\nwe choose $r_j=jh$, $j = 1,2,\\dots,N_1$.\nThe choice of the regularization parameter $\\alpha_1$ is discussed in detail below.  \n\nWe use the restarted generalized minimal residual (GMRES) method to solve the system of linear equations (\\ref{yhtalot}) and choose six as the maximum number of outer iterations and 10 as the number of inner iterations (restarts). We use the initial guess $f = 0$ and the tolerance of the method is set to 1e-12.\n\nAfter this we simply follow the regularization strategy \n(\\ref{Rstategia2}), that is, we get an approximation of $c$ \nby setting\n\\begin{align}\n\\label{the_rec}\n\\widetilde c^{N_1}_{\\alpha_{1}}=\\Gamma_\\nu(\\Phi(W(D_h(\\widetilde s^N_{\\alpha_1})))),\n\\end{align}\nwhere $\\widetilde s^{N_1}_{\\alpha_1} (r_j)\n =\\langle X_j,b\\rangle _{L^2(0,2T)}$, $j = 1,2,\\dots,N_1$.\nThe scaling of $\\nu$ is chosen as follows \n\\begin{align*}\n\\nu=0.01(\\epsilon_1^{(d)})^{\\frac{1}{54}}.\n\\end{align*}\nIn the numerical computations the parameter $h$ was fixed to be $h=\\frac{T}{N_1}$, that is, the discrete derivative $D_h$ was computed in the grid that is used in (\\ref{ekakantafunktio}) to represent the basis functions $\\phi_{1,N_1}$. Observe that this deviates from the theoretical choice $h=C\\epsilon_1^{\\frac{1}{18}}$ used in (\\ref{Rstategia}). We will describe next how the regularization parameter $\\alpha_1$ is chosen, and then we will study how the error $\\widetilde c^{N_1}_{\\alpha_{1}} - c$ behaves as function of $\\epsilon^{(d)}_1$.\n\n\n\\subsection{Calibration of the regularization strategy}\n\nRecall that in Theorem \\ref{kaiken_teoria2} \nthe choice of regularization parameter is of the form\n$\\alpha_1=C_{reg1}(\\epsilon_1^{(d)})^p$, where $p={\\frac{4}{9}}$.\nIn particular, the choice is explicit apart from the constant $C_{reg1}$. In this section we choose $C_{reg1}$ so that it gives a good reconstruction of a particular velocity function $c_c$---see Figure \\ref{testprofile}. Then the same constant is used in all the subsequent computational examples. \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.18]\n{calibfun.jpg}\n\\caption{The velocity function $c_c$\nused in the calibration of the regularization strategy.\n}   \n\\label{testprofile}\n\\end{figure}\n\nIn the regularization strategy we consider 10 values for measurement errors, as defined in (\\ref{laskennalvirhe})\n\\begin{align}\n\\label{epsihommeli}\n\\epsilon^{(d)}_{1,k}\\in\\{k\\cdot 10^{-2}|k=1,2,3,...,10\\},\n\\end{align} \nand nine values for the multiplicative constant $C_{reg1}=10^{-j}$, $j=1,3,...,9$.\nThen we consider the error in the reconstruction as a function of $j$,\n\\begin{align}\nerror(j)=\\norm{\\widetilde c^{N_1}_{\\alpha_{j,k}} -c_c}_{L^2 (M)},\n\\end{align}\nwhere for each error level, the reconstruction $\\widetilde c^{N_1}_{\\alpha_{j,k}}$ is computed by (\\ref{the_rec}).\nThese computations are summarized in Figure \\ref{choosing_C2}.\nWe see that the choice $j=4$, that is, \n\\begin{align}\n\\label{the_alpha}\n\\alpha_1 = 10^{-4} (\\epsilon_1^{(d)})^p, \\quad p={\\frac{4}{9}},\n\\end{align}\ngives a good reconstruction on all the error levels. \nIn what follows we will systematically use this choice. \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.18]\n{cvalinta.jpg}\n\\caption{\nThe reconstruction error as a function of the multiplicative constant $C_j = C_{reg1}$. Each curve corresponds to a noise level in (\\ref{epsihommeli}).\nAs expected, the reconstruction error is monotonous as a function of the noise level: The highest line corresponds to $\\epsilon_1^{(d)}=0.1$\nand the lowest one to $\\epsilon_1^{(d)}=0.01$. \nWe also observe that the reconstruction error becomes more sensitive to the choice of $C_{reg1}$ as the noise level grows. \n}   \n\\label{choosing_C2}\n\\end{figure}\n\n\\subsection{Reconstruction results based on the analysis}\nWe will now consider the reconstruction (\\ref{the_rec}), \nwith the choice of regularization parameter (\\ref{the_alpha}),\nin two test cases. \nWe begin with with a smooth velocity function $c_s$(see Figure \\ref{estimatkunkku}), where reconstructions of two different noise levels are shown. \n\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.14]\n{kuvauusisilee1.jpg}\n\\includegraphics[scale=0.14]\n{kuvauusisilee2.jpg}\n\\caption{\nTwo reconstructions (the solid blue lines) of a smooth velocity function $c_s$ (the dashed lines).\n{\\em Top:} Noise level $\\epsilon^{(d)}_1= \n 0.1$. {\\em Bottom:} Noise level $\\epsilon^{(d)}_1= \n 0.01$.\n }\n\\label{estimatkunkku}\n\\end{figure}\nTo study the order of convergence of our reconstruction method, \nwe consider 10 noise levels,\n\\begin{align}\n\\label{epsihommeli2}\n\\epsilon^{(d)}_{1}\\in\\{k\\cdot 10^{-2}|k=1,2,3,...,10\\},\n\\end{align} \nand simulate noisy measurements with five different realizations of the random vector\n$n$ in (\\ref{the_noise}) at each noise level.\nThe corresponding reconstruction errors \n$\\norm{\\widetilde c^{N_1}_{\\alpha_{1}} -c_s}_{L^2 (M)}$\nare summarized in Figure \\ref{fourestimates}.\nComputations suggest that the order of convergence is 0.40.\nThis is better than $\\frac{1}{54}$ in Theorem (\\ref{kaiken_teoria2}).  \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.18]\n{suppenema.jpg}\n\\caption{\nThe reconstruction error as a function of the noise level $\\epsilon^{(d)}_{1}$ (in log--log axes). We have used here 5 different realizations of the noise. The solid line is the average of these. \nLinear fitting (the dotted line) gives the estimated convergence the order 0.40.\n}\n\\label{fourestimates}\n\\end{figure}\n\n\nWe also tested the method with a non-smooth velocity function $c_p$(see Figure \\ref{estimatikka23}), where reconstructions of two different noise levels are shown. This case is not covered by the above analysis, but the reconstruction method is also robust in this case. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.14]\n{kuvauusiporras1.jpg}\n\\includegraphics[scale=0.14]\n{kuvauusiporras2.jpg}\n\\caption{\nTwo reconstructions (the solid blue lines) of a piecewise constant velocity function $c_p$ (the dashed lines).\n{\\em Top:} Noise level $\\epsilon^{(d)}_1= \n 0.1$. {\\em Bottom:} Noise level $\\epsilon^{(d)}_1= \n 0.01$.\n}\n\\label{estimatikka23}\n\\end{figure}\n\n\n\n\\subsection{Reconstruction results based on MDP}\n\\label{pamelapulkkinen}\nHere we use a heuristic version of MDP as a parameter choice\nrule for $\\alpha=\\alpha(\\epsilon)$.\nTypically MDP is applied to a Tikhonov regularization of the form\n\\begin{equation}\n\\label{classical_Tikhonov}\n\\min_x \\norm{F(x) - y}^2 + \\alpha \\norm{x - x^*}^2,\n\\end{equation}\nwhere $F$ is a model for the measurements, $y$ is the data, and $x^*$ plays the role of a selection criterion. \nIn our case, the model $F$ corresponds to $c \\mapsto \\Lambda_{N_1}^{c}\\phi_{1,N_1}$\nand $y = \\widetilde\\Lambda_{N_1}^{c}\\phi_{1,N_1}$ gives the measurement data, but we do not cast the inverse problem as a minimization problem and our regularization method is not of the Tikhonov type.\nIn particular, our method does not depend on the choice of the auxiliary parameter $x^*$, which can be viewed as an initial guess, and that chooses a local minimum of the non-linear optimization problem (\\ref{classical_Tikhonov}).\nDue to these differences, the existing results on MDP do not apply to our method, and (\\ref{morozov}) below is only a heuristic analogue of the classical MDP.\nWe refer to \\cite{Scherzer1993} for a study of MDP in an abstract context of the form (\\ref{classical_Tikhonov}), with non-linear $F$. \n\nThe heuristic principle that we use is as follows. We fix tuning parameters $h > 1$ and small $\\delta > 0$ and search for a regularization parameter $\\alpha$ in such a way that the following consistency condition holds:\n\\begin{align}\n\\label{morozov}\n(h-\\delta)\\epsilon\\le\\norm{\\Lambda_{N_1}^{\\widetilde c^{N_1}_{\\alpha}}\\phi_{1,N_1}-y}_{L^2(0,2T)}\n\\le(h+\\delta)\\epsilon.\n\\end{align} \nHere $\\epsilon > 0$ is the noise level, \n$y = \\widetilde\\Lambda_{N_1}^{c}\\phi_{1,N_1}$ is again the measurement data, and $\\Lambda_{N_1}^{\\widetilde c^{N_1}_{\\alpha}}\\phi_{1,N_1}$ is the corresponding data computed with the velocity function $\\widetilde c^{N_1}_{\\alpha}$, given by the reconstruction method. \nObserve that (\\ref{morozov}) is a relaxed version of (1.7) in \\cite{Scherzer1993}.\n\nWe choose $h=1.1$ and $\\delta = 0.01$ and use a bisection search to find $\\alpha$. Our implementation was unable to find $\\alpha$ satisfying the constraint (\\ref{morozov}) for noise levels $\\epsilon^{(d,k)}>0.02$.\nFor smaller noise levels, the regularization parameters found using the principle are summarized in Figure \\ref{estimatee}.\nWe see that, with the above choice of tuning parameters, the heuristic MDP always gives a larger regularization parameter than (\\ref{the_alpha}). The reconstructions are consistently worse than those produced by the choice (\\ref{the_alpha}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.15]\n{uusimorob.jpg}\n\\caption{\nRegularization parameter $\\alpha$, given by MDP as a function of the noise level (in log--log axes).\nThe straight line represents the relation (\\ref{the_alpha}).\n}\n\\label{estimatee}\n\\end{figure}\n\n\n\n\n\\noindent{\\bf Acknowledgements.} We thank Samuli Siltanen for inspiring discussions on the regularization on inverse problems. \n\nL.\\ Oksanen was partly supported by \nEPSRC, project EP\/P01593X\/1.\nJ.\\ Korpela and M.\\ Lassas were  supported by the Academy of Finland, projects 263235, 273979, 284715, and 312119.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nMasked image modeling (MIM) \\cite{bao2021beit} is at the center of self-supervised representation learning, showing good potentials on various downstream tasks, including image classification, semantic segmentation, object detection, and instance segmentation. \nIt masks out some patches in the images with a pre-defined \\emph{mask ratio} and adds the reconstruction supervision on a set of patches at some specific positions, \\textit{i.e.}, \\emph{supervision position}. Specifically, in most MIM methods \\cite{bao2021beit, he2022masked, xie2022simmim, chen2022context}, the supervision positions are only associated with the masked patches, \\textit{i.e.}, only adding the supervisions on the masked patches. \n\n\n\n\\par\nThe reconstruction loss of MIM can be applied in different domains or targets, such as \nRGB~\\cite{he2022masked,xie2022simmim}, HOG~\\cite{wei2022masked}, discrete visual tokens~\\cite{bao2021beit,chen2022context,el2021large,peng2022beit,dong2021peco}, momentum encoders~\\cite{tao2022siamese,chen2022sdae,wu2022extreme}, and pretrained models~\\cite{wei2022mvp,wei2022masked}.\nRecently, MVP~\\cite{wei2022mvp} applies the reconstruction loss on the image representations of CLIP~\\cite{radford2021learning}, \\textit{i.e.}, minimizing the reconstruction error in the domain of CLIP representation. Benefiting from the rich multi-modality information and informative representation, the CLIP-targeted MVP performs very well. \n\n\n\n\n\\par\nDespite that, it is still under-explored how the detailed ways of applying CLIP in MIM affect performance.\nUnlike most MIM methods~\\cite{bao2021beit, he2022masked, xie2022simmim, chen2022context} applying the reconstruction supervision on the masked patches, MVP supervises both masked and unmasked patches.\nIt raises a question: how will the \\emph{supervision position} influence the CLIP-targeted MIM?\nOn the other hand, the \\emph{mask ratio} performs differently for different supervision targets~\\cite{he2022masked,bao2021beit}. \nWith CLIP as the target, it is unclear how the mask ratio behaves. \n\n\n\n\n\\par\nIn this paper, we study these two critical ingredients, \\textit{i.e.}, \\textit{supervision position} and \\textit{mask ratio}, in MIM with the CLIP representation as the supervision. \nTo conduct the study, we develop a simple \nMIM pipeline, \\textit{i.e.}, context autoencoder with CLIP target ({CAE v2}\\xspace).\nWe will first analyze how the supervision position and the mask ratio influence the performance of {CAE v2}\\xspace. \nThen, relying on the analyses, we implement \n{CAE v2}\\xspace as a concise yet effective MIM model, producing superior performance on \nvarious downstream tasks. \n\n\\par\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[trim =0mm 0mm 0mm 0mm, clip, width=0.85\\linewidth]{figure\/framework.pdf}\n\\caption{\nOverview of the proposed {CAE v2}\\xspace. \n{CAE v2}\\xspace first masks the input image $\\pmb{\\mathrm{x}}$ with the mask ratio $\\gamma$, which is positively correlated with the model size of encoder.\n$\\propto$ represents the positive correlation.\nThen, {CAE v2}\\xspace inputs the visible patches $\\pmb{\\mathrm{X}}_v$ into the encoder to obtain the latent representation $\\pmb{\\mathrm{Z}}_v$.\nThe decoder receives $\\pmb{\\mathrm{Z}}_v$ and the mask token $\\pmb{\\mathrm{E}}_m$ to recover the latent representations of the masked patches $\\pmb{\\mathrm{Z}}_m$.\nAfter a lightweight head, $\\pmb{\\mathrm{Z}}_v$ and $\\pmb{\\mathrm{Z}}_m$ are projected to $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$.\n{CAE v2}\\xspace also inputs $\\pmb{\\mathrm{x}}$ into the CLIP model to generate the target supervisions, which are split to $\\pmb{\\mathrm{T}}_v$ and $\\pmb{\\mathrm{T}}_m$ according to the absolute positions of $\\pmb{\\mathrm{X}}_v$ and $\\pmb{\\mathrm{X}}_m$.\nThe optimization is applied on the prediction $\\pmb{\\mathrm{Y}}_v$ and the target supervision $\\pmb{\\mathrm{T}}_v$ of visible patches.\nMeanwhile, the loss on $\\pmb{\\mathrm{Y}}_m$ and $\\pmb{\\mathrm{T}}_m$ for masked patches is optional.\n}\n\\vspace{-1.0em}\n\\label{fig:framework}\n\\end{figure*}\n\nFirst, we study on the influence of the supervision position.\nExcept for applying the CLIP supervision on the predictions of masked patches,\nwe consider to directly supervise the latent representations of visible patches with CLIP features.\nSurprisingly, we find that the supervision \\textit{only on visible patches} achieves remarkable performance, even better than that on masked patches.\nIt reveals that the visible patches can effectively extract rich semantic information from CLIP, performing like the feature distillation. \nMoreover, combining the supervisions of masked and visible patches together like \\cite{wei2022mvp} brings slight performance improvement.\nTherefore, we advocate that the supervision on  \\textit{visible patches is a good way for the supervision position} in the CLIP-targeted MIM.\n\n\n\n\nNext we explore the behavior of the mask ratio.\nRecall that\nMAE~\\cite{he2022masked} points out a high mask ratio (75\\%) is good for the balance of efficiency and effectiveness.\nOthers like \\cite{bao2021beit,chen2022context} empirically set the mask ratio as 40-50\\%, and MVP simply follows \\cite{bao2021beit}.\nHere we want to explore what is the optimal mask ratio in the CLIP-targeted MIM.\nWe conduct a series of experiments on a battery of models with different sizes and sweep the mask ratio from 15\\% to 95\\%.\nThe results are interesting, showing that the smaller models favor lower mask ratios, while larger models prefer higher ones.\nThis provides a new perspective that the optimal mask ratio is \\textit{positively correlated with the model size}.\n\n\n\n\n\n\n\n\nBased on the above analyses,\nour {CAE v2}\\xspace achieves superior performance on various scales of models.\nEspecially, with 300 epoch pre-training, {CAE v2}\\xspace can boost a vanilla ViT-Large to 81.7\\% and 86.7\\% top-1 accuracy on linear probing and fine-tuning on ImageNet-1K, and 55.9\\% mIoU on ADE20K.\nWe hope our analyses and findings can be useful guidelines for the future MIM studies, especially for the pre-training on lightweight models.\nIn summary, our contributions are:\n\\begin{itemize}\n\\itemsep -.051cm\n\\item We develop a simple CLIP-targeted MIM pipeline,\nnamely {CAE v2}\\xspace, to study the supervision position and the mask ratio;\n\\item For the supervision position, we advocate that applying the supervision on visible patches is a good way;\n\\item For the mask ratio, we present that the optimal mask ratio positively correlates with the model size;\n\\item Driven by these analyses, \nour {CAE v2}\\xspace achieves superior performance on different scales of models on various downstream tasks.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{CAE v2}\\xspace}\nWith CLIP as the supervision target, this paper aims to study the two important elements in MIM, \\textit{i.e.}, the supervision position and the mask ratio.\nWe achieve this by developing a simple framework {CAE v2}\\xspace, in which the supervision target is CLIP and \nthe model structure is based on \\cite{chen2022context,wei2022mvp} with some specific modifications.\nWe introduce the details as follows.\n\n\\subsection{Overview}\nThe overview of our {CAE v2}\\xspace is illustrated in Figure~\\ref{fig:framework}.\nLet $\\pmb{\\mathrm{x}}\\in\\mathcal{D}$ denote an input image.\nFollowing previous MIM methods~\\cite{bao2021beit,he2022masked,wei2022masked,chen2022context}, {CAE v2}\\xspace first embeds $\\pmb{\\mathrm{x}}$ into a total length of $N$ patches,\nwhich are then randomly masked by a specific proportion $\\gamma$.\nThese $N$ patches are naturally split into two non-overlapped sets, \\textit{i.e.}, visible patches $\\pmb{\\mathrm{X}}_v$ and masked patches $\\pmb{\\mathrm{X}}_m$, where $N=\\left | v \\right |+\\left | m \\right |$.\nThe mask ratio is thus denoted as $\\gamma=\\left | m \\right |\/N$.\nFollowing ~\\cite{he2022masked,chen2022context}, the encoder $\\mathcal{F}$ maps the visible patches $\\pmb{\\mathrm{X}}_v$ to the latent representations $\\pmb{\\mathrm{Z}}_v$.\nThe decoder $\\mathcal{G}$ predicts the latent representations $\\pmb{\\mathrm{Z}}_m$ for the masked patches from mask tokens $\\pmb{\\mathrm{E}}_m$.\nAfter that, the predictions of visible patches $\\pmb{\\mathrm{Y}}_v$ and masked patches $\\pmb{\\mathrm{Y}}_m$ are obtained \nvia a head $\\mathcal{H}$.\n\nFor the target supervision, we directly input the intact image $\\pmb{\\mathrm{x}}$ into the CLIP vision model $\\mathcal{T}$ to generate the target supervision $\\pmb{\\mathrm{T}}$.\n$\\pmb{\\mathrm{T}}$ is then split into $\\pmb{\\mathrm{T}}_v$ and $\\pmb{\\mathrm{T}}_m$ corresponding to the absolute positions of $\\pmb{\\mathrm{X}}_v$ and $\\pmb{\\mathrm{X}}_m$.\nThe optimization is applied on $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{T}}_v$, \nand we also study to add the supervision on $\\pmb{\\mathrm{Y}}_m$ and $\\pmb{\\mathrm{T}}_m$.\n\n\n\n\\subsection{Architecture}\n{CAE v2}\\xspace contains four modules, \\textit{i.e.}, one encoder, one decoder, one MIM head, and one CLIP model.\n\n\\myparagraph{Encoder.}\nThe encoder $\\mathcal{F}$ only receives the visible patches $\\pmb{\\mathrm{X}}_v$ following \\cite{he2022masked,chen2022context}.\n$\\mathcal{F}$ maps the visible patches $\\pmb{\\mathrm{X}}_v$ to the latent representations $\\pmb{\\mathrm{Z}}_v$ across a stack of transformer blocks.\nThe operation of $\\mathcal{F}$ is based on self-attention.\nIn this paper, we utilize a series of ViTs~\\cite{dosovitskiy2020image} to form the encoder, including ViT-Tiny, ViT-Small, ViT-Base, and ViT-Large.\n\n\\myparagraph{Decoder.} The decoder $\\mathcal{G}$ predicts the latent representation $\\pmb{\\mathrm{Z}}_m$ for the masked patches from the mask tokens $\\pmb{\\mathrm{E}}_m$, conditioned on the\nvisible latent representation $\\pmb{\\mathrm{Z}}_v$.\nIt is inspired by the latent contextual regressor in CAE~\\cite{chen2022context}.\n$\\mathcal{G}$ performs as the same as cross-attention.\nHere, we utilizes a lighweight decoder in {CAE v2}\\xspace, \\textit{i.e.}, one-layer transformer block\\footnote{We find that one-layer transformer block as the decoder ready can produce promising performance. For parameter-friendly, we use this structure as the default setting.\nWe do not analyze the influence of the depth of decoder, since it is not the main concern in this paper.\n}.\n\n\\myparagraph{Head.} The head $\\mathcal{H}$ maps the latent predictions $\\pmb{\\mathrm{Z}}_v$ and the latent representations $\\pmb{\\mathrm{Z}}_m$ to $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$, respectively.\n$\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$ share the same form with the target supervisions $\\pmb{\\mathrm{T}}_v$ and $\\pmb{\\mathrm{T}}_m$.\nIn this work, we only use a FC (fully-connected) layer followed by a LN (layernorm) layer in $\\mathcal{H}$.\n\n\\myparagraph{CLIP model.} The vision branch of CLIP model $\\mathcal{T}$ generates the target supervisions $\\pmb{\\mathrm{T}}$.\nIn the whole paper, we only use the ViT-Base of the CLIP model as $\\mathcal{T}$.\n$\\pmb{\\mathrm{T}}$ is then split into the target supervisions for visible patches $\\pmb{\\mathrm{T}}_v$ and for masked patches $\\pmb{\\mathrm{T}}_m$ according to the absolute positions of $\\pmb{\\mathrm{X}}_v$ and $\\pmb{\\mathrm{X}}_m$.\n\n\n\\subsection{Study, Discovery and Analysis}\n\\label{sec:study}\nWe pay attention to two critical elements in MIM, \\textit{i.e.}, the supervision position and the mask ratio.\n\n\\myparagraph{Supervision position.}\nMost previous MIM methods~\\cite{he2022masked,chen2022context,bao2021beit} apply the reconstruction supervision on the predictions of masked patches.\nWith CLIP as the target, MVP~\\cite{wei2022mvp} supervises both visible and masked patches.\nHere, we do experiments to study how will the\nsupervision position influence the CLIP-targeted MIM.\n\n\nWe systematically analyze three kinds of supervision positions based on {CAE v2}\\xspace, \\textit{i.e.}, only on the predictions of visible patches $\\pmb{\\mathrm{Y}}_m$, only on the predictions of masked patches $\\pmb{\\mathrm{Y}}_v$, and on both $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$.\nThe loss function can be formulated as follows:\n\\begin{equation}\n\\centering\n\\begin{aligned}\nL=\\frac{\\delta_{v}\\cdot \\ell(\\pmb{\\mathrm{Y}}_{v},\\pmb{\\mathrm{T}}_{v}) + \\delta_{m}\\cdot \\ell(\\pmb{\\mathrm{Y}}_{m},\\pmb{\\mathrm{T}}_{m})}\n{\\delta_{v}\\cdot \\left | v \\right | + \\delta_{m}\\cdot \\left | m \\right |},\n\\end{aligned}\n\\label{eq:loss}\n\\end{equation} \nwhere $\\ell$ is the loss function.\nBy default, we use the cosine distance as the loss function.\n$\\delta_{v}$\/$\\delta_{m}$ is the indicative function, controlling whether to use visible\/masked patches for optimization.\nIf $\\delta_{m}=1-\\delta_{v}=1$, Eq.~\\eqref{eq:loss} reduces to only use the reconstruction loss for masked patches as in \\cite{he2022masked,chen2022context,bao2021beit}.\nIf $\\lambda_{m}=1-\\delta_{v}=0$, the loss becomes the feature distillation loss for visible patches.\nWhen setting $\\lambda_{m}=\\delta_{v}=1$, the optimization is imposed on both visible and masked patches as in \\cite{wei2022mvp}.\n\nFrom the experiments (see Table~\\ref{tab:supervision_position}), we observe an interesting phenomenon: \\textit{only applying the supervision on visible patches can achieve remarkable performance}.\nWe conjecture this benefit mainly comes from the powerful CLIP model by knowledge distillation.\nIntriguingly, the distillation loss works well on a subset of image patches, in which some contextual information are missing.\nMoreover, despite the relatively inferior performance when only using the reconstruction supervision on masked patches,\nit bring slightly positive improvement when\ncombining the reconstruction supervision on the masked patches with the feature distillation loss on the visible patches.\nWe believe that in this way, the supervision on masked patches performs as a regularization for the representation learning, which is beneficial to alleviating the over-fitting when training  only on visible patches. \n\n\nOur findings are different from the common sense in the current MIM methods~\\cite{bao2021beit,he2022masked,chen2022context} that only compute the loss on the masked patches, which\nis inherited from BERT~\\cite{bert} in the NLP areal and has been verified by most current works.\nTherefore, in CLIP-targeted MIM, we provide a new perspective that the feature distillation on the partial image patches (here referred to visible patches) is a good choice for the model optimization. \n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[trim =0mm 0mm 0mm 0mm, clip, width=1.0\\linewidth]{figure\/comparison_mvp.pdf}\n\\caption{\nMVP~\\cite{wei2022mvp} \\textit{vs.\\ } our {CAE v2}\\xspace. We mainly study the supervision position and the mask ratio in the CLIP-targeted MIM.\n}\n\\vspace{-0.5cm}\n\\label{fig:comparison_mvp}\n\\end{figure}\n\n\\myparagraph{Mask ratio.}\nThe MIM methods generally mask a specific percentage of patches on $\\pmb{\\mathrm{x}}$ as the input for the model training.\nFor example, \\cite{he2022masked} utilizes the mask ratio of 75\\%, while\n~\\cite{bao2021beit,wei2022mvp} and \\cite{chen2022context} empirically set the mask ratio as 40\\% and 50\\%.\nDriven by the above experience that the CLIP supervision on visible patches can achieve good results in our {CAE v2}\\xspace, we naturally consider to study what is the optimal option for the mask ratio.\n\nWe begin from a high mask ratio $\\gamma$, \\textit{i.e.}, 75\\% as in \\cite{he2022masked}.\nWe find that despite the reasonable performance on the large model (\\textit{e.g.}, ViT-Base), it performs less than satisfactory on smaller models like ViT-Tiny\/Small (see Figure~\\ref{fig:mask_ratio}).\nWe thus gradually decrease the mask ratio, leading to more visible patches.\nThe performance on all scales of models starts to improve at the beginning of reducing the mask ratio.\nThis trend is especially true for the downstream task like semantic segmentation, \nverifying that the high mask ratio is not necessary in the CLIP-targeted MIM.\n\nWith the continued decreas of the mask ratio, the models with different sizes perform differently.\nWe observe that the smaller models favor lower proportions of mask ratios, while the larger models prefer relatively higher ones.\nThe underlying reason may be that it is hard for the small-size model to optimize on a small subset of patches where most contextual information are missing.\nSo it is better to reduce the difficulty by using more visible patches, \\textit{i.e.}, a lower mask ratio.\nOn the contrary, large-scale models learn representations from plenty of patches easily.\nTherefore, a high mask ratio can make MIM harder to ease the over-fitting. \n\nBased on the above observation, we point out that \\textit{the optimal mask ratio is positively correlated with the model size}.\nThat is to say, the larger the model, the higher the mask ratio, and preferring more challenging work; otherwise, the smaller model favors a lower mask  ratio.\nWe believe that this discovery can be a useful guideline in the MIM pre-training area, especially for the small models.\n\n\\begin{table}\n\\small\n\\begin{center}\n\\setlength{\\tabcolsep}{3.0mm}\n\\renewcommand{\\arraystretch}{1.1}\n\\scalebox{1.0}{\n\\begin{tabular}{l|cc|c|c|c}\n\\hline\n\\multirow{2}{*}{Model} & \\multicolumn{2}{c|}{Supervision} & \\multicolumn{2}{c|}{IN-1K} & \\multicolumn{1}{c}{ADE20K} \\\\\n\\cline{2-6}\n & \\multicolumn{1}{c}{$\\pmb{\\mathrm{Y}}_m$} & \\multicolumn{1}{c|}{$\\pmb{\\mathrm{Y}}_v$} & LIN & FT & mIoU \\\\ \n\\hline\n\\hline\n\\multirow{3}{*}{ViT-Tiny} & \\checkmark & - & 64.9 & 77.2 & 44.1 \\\\\n& - & \\checkmark & 68.8 & 77.4 & 44.2 \\\\\n  & \\baseline{\\checkmark} & \\baseline{\\checkmark} & \\baseline{\\textbf{69.3}} & \\baseline{\\textbf{77.8}} & \\baseline{\\textbf{44.7}}  \\\\\n\\hline\n\\multirow{3}{*}{ViT-Small} & \\checkmark & - & 73.9 & 82.4 & 49.6  \\\\\n& - & \\checkmark & 77.3 & \\textbf{82.8} & 49.1 \\\\\n  & \\baseline{\\checkmark} & \\baseline{\\checkmark} & \\baseline{\\textbf{77.5}} & \\baseline{82.7} & \\baseline{\\textbf{49.7}}  \\\\\n\\hline\n\\multirow{3}{*}{ViT-Base} &  \\checkmark & - & 78.4 & 85.0 & 52.7 \\\\\n& - & \\checkmark & 80.5 & 85.2 & \\textbf{53.1} \\\\\n  & \\baseline{\\checkmark} & \\baseline{\\checkmark} & \\baseline{\\textbf{80.6}} & \\baseline{\\textbf{85.3}} & \\baseline{52.9} \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{Influences of the supervision position in our {CAE v2}\\xspace. Default settings are marked in \\colorbox{baselinecolor}{gray}.}\n\\vspace{-0.5cm}\n\\label{tab:supervision_position}\n\\end{table}\n\n\\myparagraph{Discussion.}\nThe most relevant work for ours is MVP~\\cite{wei2022mvp}.\nIt is noted that our {CAE v2}\\xspace is not contradictory to MVP, even though we both use CLIP as the target. \nWe focus on the analyses under the CLIP-targeted MIM and propose new perspectives, while MVP highlights the rich information brought by the CLIP target. \nSpecifically,\nin this work, we go further one step to study two important ingredients in MIM, including the supervision target and the mask ratio.\nFirst, we find only applying the CLIP supervision on visible patches already achieves comparable or even superior performance compared to the optimization on all patches as in MVP.\nSecond, MVP inherits the mask ratio (40\\%) from \\cite{bao2021beit} and applies it on both base- and large-size models.\nDifferently, we study the effect of the mask ratios on a battery of models with different scales, and find the mask ratio is highly related to the model size.\nIn addition, our {CAE v2}\\xspace with these two findings demonstrates large performance gains. \nWe illustrate the detailed paradigms of our {CAE v2}\\xspace and MVP in Figure~\\ref{fig:comparison_mvp} for better comparisons.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\n\\subsection{Settings}\n\\myparagraph{Model structures.} We study a series of vision transformer backbones~\\cite{dosovitskiy2020image}, including ViT-Tiny (12 layers with $dim$$=$192), ViT-Small (12 layers with $dim$$=$384), ViT-Base (12 layers with $dim$$=$768), and ViT-Large (24 layers with $dim$$=$1024).\nNote that for ViT-Tiny, we follow \\cite{wang2022closer} to increase the number of heads from 3 to 12, which gives better results on ImageNet-1K~\\cite{deng2009imagenet}. For other models, we strictly follow the model configurations as in~\\cite{dosovitskiy2020image}.\n\nTo eliminate the influence of different sizes of CLIP models, we adopt the vision branch ViT-Base\/16 of CLIP\\footnote{The official pre-trained CLIP model is available at \\url{https:\/\/github.com\/openai\/CLIP\/blob\/main\/clip\/clip.py}.} as the target for all pre-training experiments with ViT-Tiny\/Small\/Base\/Large~\\cite{dosovitskiy2020image}.\n\n\n\n\\begin{table}\n\\small\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.1}\n\\setlength{\\tabcolsep}{2.7mm}{\n\\begin{tabular}{l|c|c|c|c}\n\\hline\n\\multirow{2}{*}{Model} & \\multirow{2}{*}{Loss type} & \\multicolumn{2}{c|}{IN-1K} & \\multicolumn{1}{c}{ADE20K} \\\\\n\\cline{3-5}\n &  & LIN & FT & mIoU \\\\ \n\\hline\n\\hline\n\\multirow{3}{*}{ViT-Tiny} & MSE & 69.1 & 77.3 & \\textbf{44.8} \\\\\n & Smooth-$l$1 & \\textbf{69.4} & 77.6 & 43.7 \\\\\n  & \\baseline{Cosine distance} & \\baseline{69.3} & \\baseline{\\textbf{77.8}} & \\baseline{44.7} \\\\\n\\hline\n\\multirow{3}{*}{ViT-Small} & MSE & 77.3 & 82.7 & \\textbf{49.8} \\\\\n & Smooth-$l$1 & 77.4 & \\textbf{82.8} & \\textbf{49.8} \\\\\n  & \\baseline{Cosine distance} & \\baseline{\\textbf{77.5}} & \\baseline{82.7} & \\baseline{49.7} \\\\\n\\hline\n\\multirow{3}{*}{ViT-Base} & MSE & 80.4 & \\textbf{85.3} & \\textbf{52.9} \\\\\n & Smooth-$l$1 & 80.5 & 85.2 & 52.0\\\\\n  & \\baseline{Cosine distance} & \\baseline{\\textbf{80.6}} & \\baseline{\\textbf{85.3}} & \\baseline{\\textbf{52.9}} \\\\\n\\hline\n\\end{tabular}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{Ablation on the loss type in our {CAE v2}\\xspace. \nWe use the cosine distance as the default loss function (marked in \\colorbox{baselinecolor}{gray}).\n}\n\\vspace{-0.3cm}\n\\label{tab:loss_type}\n\\end{table}\n\n\n\\myparagraph{Pre-training.}\nFollowing most previous MIM methods~\\cite{bao2021beit,he2022masked,chen2022context,wei2022mvp,wang2022closer}, \nwe use ImageNet-1K (IN-1K) dataset~\\cite{deng2009imagenet} for all pre-training experiments.\nThe input images are with the size of $224\\times 224$ and partitioned into $14\\times 14$ patches with the patch size being $16\\times 16$.\nWe apply random resized cropping and horizontal flipping during pre-training.\n\nThe pre-training settings are almost the same as CAE~\\cite{chen2022context}, except for the mask ratios that are analyzed in Section.~\\ref{sec:ablation}.\nBy default, we use $15\\%$, $25\\%$, $50\\%$, and $50\\%$ mask ratios on ViT-Tiny, ViT-Small, ViT-Base, and ViT-Large, respectively.\nWe use AdamW~\\cite{Loshchilov2019} for optimization and train the {CAE v2}\\xspace for 300 epochs for all scales of ViTs~\\cite{dosovitskiy2020image}.\nThe detailed pre-training settings are shown in the supplementary material.\n\n\n\\begin{figure*}[t]\\centering\n\\includegraphics[width=.95\\linewidth]{figure\/masking_ratio.pdf}\n\\vspace{-.7em}\n\\caption{Influences of the mask ratio in our {CAE v2}\\xspace on different model sizes, including (top row) ViT-Tiny, (middle row) ViT-Small and (bottom row) ViT-Base. The optimal mask ratio is positively correlated to the model size.\nA higher mask ratio is more appropriate to a larger model, while the smaller model prefers a lower mask ratio.\nThe y-axes is the Top-1 accuracy (\\%) on (left column) linear probing and (middle column) fine-tuning on ImageNet-1K, and (right column) mIoU (\\%) on ADE20K.\n}\n\\label{fig:mask_ratio}\n\\vspace{-1em}\n\\end{figure*}\n\n\n\n\\myparagraph{Evaluation.}\nWe evaluate our {CAE v2}\\xspace on various downstream tasks.\nFor image classification, we conduct evaluations on ImageNet-1K~\\cite{deng2009imagenet} with both linear probing (LIN) and fine-tuning (FT) protocols.\nWithout specification, in all experiments, we conduct the linear probing for 90 epochs and the fine-tuning for 100 epochs.\nFor semantic segmentation, we follow BEiT\\cite{bao2021beit} to use UperNet~\\cite{xiao2018unified} and report the mIoU on ADE20K~\\cite{zhou2017scene} dataset.\nFor objection detection and instance segmentation, we use COCO~\\cite{lin2014microsoft} as the evaluation dataset.\nWe adopt both Mask R-CNN~\\cite{he2017mask} and Cascade Mask R-CNN~\\cite{cai2018cascade} frameworks and report $\\text{AP}^{b}$ and $\\text{AP}^{m}$ on the COCO val split.\nPlease refer to the supplementary material for more training details on various downstream tasks.\n\n\n\n\n\\subsection{Main Properties}\n\\label{sec:ablation}\nWe mainly explore two critical ingredients, the supervision position and the mask ratio, in {CAE v2}\\xspace when using CLIP as the supervision target. Compared with previous pre-training methods, we observe different properties and trends. In addition, we also give investigations on the loss types and the masking types. Details are given below.\n\n\\myparagraph{Supervision position.} Modern pre-training methods~\\cite{bao2021beit,he2022masked,chen2022context} only give supervision on masked patches, as they find that learning with visible patches is an easy task and may leak information, result in trivial solutions, and degenerate the representation learning. When using CLIP~\\cite{radford2021learning} as the pre-training target, the situation is different. \n\nTable~\\ref{tab:supervision_position} provides the detailed results of the influence of the supervision position on ImageNet-1K~\\cite{deng2009imagenet} and ADE20K~\\cite{zhou2017scene} based on {CAE v2}\\xspace. \nDifferent from existing MIM methods~\\cite{bao2021beit,he2022masked,chen2022context}, we observe that only adding supervision on visible patches already presents remarkable results on both linear probing, image classification fine-tuning, and semantic segmentation fine-tuning. While only adding supervision on masked patches~\\cite{bao2021beit,he2022masked,chen2022context} shows worse performance with linear probing (64.9\\% \\textit{vs.} 68.8\\% with ViT-Tiny, 73.9\\% \\textit{vs.} 77.3\\% with ViT-Small, and 78.4\\% \\textit{vs.} 80.5\\% with ViT-Base), but competitive results with image classification fine-tuning and semantic segmentation fine-tuning (Table~\\ref{tab:supervision_position}). We conjecture that the rich semantics in CLIP~\\cite{radford2021learning} make the supervision on visible patches\nwork\nas a knowledge distillation task. It is different from previous methods where the semantics in the target is not precise enough to provide strong supervision on visible patches. Moreover, by combining the supervision on visible patches and masked patches (the rows marked with \\colorbox{baselinecolor}{gray} in Table~\\ref{tab:supervision_position}), we obtain a slight improvement over the one only with supervision on visible patches. Based on the observation, \nthe supervision on masked patches\nin our {CAE v2}\\xspace can be considered as a regularization for representation learning with CLIP as the target.\n\n\n\n\n\n\n\n\n\n\\myparagraph{Mask ratio.}\nGiven that \nthe supervision on masked patches is a regularization for the learning of {CAE v2}\\xspace, we suppose that it may not be appropriate to adopt a high mask ratio (75\\% in MAE~\\cite{he2022masked}, 40\\%-50\\% in BEiT~\\cite{bao2021beit}, CAE~\\cite{chen2022context}, and MVP~\\cite{wei2022mvp}) for all scales of ViTs. \nWe study different mask ratios, ranging from 15\\% to 90\\% (see Figure~\\ref{fig:sampling_strategy}), for a series of ViTs (Tiny, Small, and Base) with our {CAE v2}\\xspace.\n\n\n\nFigure~\\ref{fig:mask_ratio} gives the performance of ViT-Tiny\/ViT-Small\/ViT-Base~\\cite{dosovitskiy2020image} under various mask ratios using three evaluation methods: linear probing, image classification fine-tuning, and semantic segmentation fine-tuning. The results give a clear trend on the choice of mask ratios, \\textit{i.e.}, {\\em the best mask ratio is positively related to the model size}. From the curves in Figure~\\ref{fig:mask_ratio}, we also find that with the selected mask ratio exceeding the best mask ratio, the performances on downstream tasks drop quickly, especially fine-tuning. The evidence indicates the significance of our study on the mask ratio and provides a guideline for choosing mask ratios for different scales of ViTs.\nAccording to the experiments in Figure~\\ref{fig:mask_ratio}, we use a mask ratio of $15\\%$ for ViT-Tiny, $25\\%$ for ViT-Small, and $50\\%$ for ViT-Base.\n\n\n\\myparagraph{Loss type.} As the target of {CAE v2}\\xspace is the CLIP features, we explore the influence of different loss functions in the feature space.\nTable~\\ref{tab:loss_type} shows the results of {CAE v2}\\xspace on different scales of models with different loss functions, including the mean square error (MSE), Smooth-$l$1 and the cosine distance. We only observe slight differences ($\\leq$$0.5\\%$) in performance on ImageNet1K~\\cite{deng2009imagenet}. On ADE20K~\\cite{zhou2017scene}, the variances are slightly bigger, indicating the loss function of using CLIP as the target gives more impact on challenging downstream tasks. We adopt the cosine distance loss as the default choice.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[trim =0mm 0mm 0mm 0mm, clip, width=1.0\\linewidth]{figure\/mask_ratio_image.pdf} \\\\\n\\setlength{\\belowcaptionskip}{0.3cm}  %\n\\figcaption{\nIllustration of corrupted images with different mask ratios $\\gamma$ via (top row) block-wise sampling strategy (our default) and (bottom row) random sampling strategy.\n}\n\\vspace{-0.8cm}\n\\label{fig:sampling_strategy}\n\\end{figure}\n\n\n\n\n\\myparagraph{Mask sampling strategy.} We also compare different mask sampling strategies in our {CAE v2}\\xspace, \\textit{i.e.}, random sampling~\\cite{he2022masked} and block-wise sampling~\\cite{bao2021beit,chen2022context,wei2022mvp} (as shown in Figure~\\ref{fig:sampling_strategy}). There are only $\\sim$$0.1\\%$ gaps between these two sampling strategies on linear probing and image classification fine-tuning (see Table~\\ref{tab:mask_strategy}). When it comes to semantic segmentation, the block-wise sampling shows better performance than the random sampling. In {CAE v2}\\xspace, we use block-wise sampling strategy by default.\n\n\n\n\\subsection{Main Results}\nBased on the analyses of the supervision position in Section~\\ref{sec:study}, we find that only applying the feature distillation supervision on visible patches can achieve good performance.\nWe denote this format as {CAE v2}\\xspace.\nMeanwhile, adding the reconstruction supervision on masked patches can further improve the performance, which is denoted as {CAE v2}\\xspace$+$.\nWe report both {CAE v2}\\xspace and {CAE v2}\\xspace$+$ in this subsection.\n\n\\begin{table}[t]\n\\small\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.15}\n\\setlength{\\tabcolsep}{2.5mm}{\n\\begin{tabular}{l|c|c|c|c}\n\\hline\n\\multirow{2}{*}{Model} & \\multirow{2}{*}{Mask strategy} & \\multicolumn{2}{c|}{IN-1K} & \\multicolumn{1}{c}{ADE20K} \\\\\n\\cline{3-5}\n &  & LIN & FT & mIoU \\\\ \n\\hline\n\\hline\n\\multirow{2}{*}{ViT-Tiny} & Random & 69.1 & 77.4 & 43.8 \\\\\n  & \\baseline{Blockwise} & \\baseline{\\textbf{69.3}} & \\baseline{\\textbf{77.8}} & \\baseline{\\textbf{44.7}} \\\\\n\\hline\n\\multirow{2}{*}{ViT-Small} & Random & 77.4 & \\textbf{82.7} & 49.0 \\\\\n  & \\baseline{Blockwise} & \\baseline{\\textbf{77.5}} & \\baseline{\\textbf{82.7}} & \\baseline{\\textbf{49.7}} \\\\\n\\hline\n\\multirow{2}{*}{ViT-Base} & Random & \\textbf{80.6} & \\textbf{85.4} & 52.4 \\\\\n  & \\baseline{Blockwise} & \\baseline{\\textbf{80.6}} & \\baseline{85.3} & \\baseline{\\textbf{52.9}} \\\\\n\\hline\n\\end{tabular}}\n\\end{center}\n\\vspace{-0.5cm}\n\\tabcaption{Ablation on the mask sampling strategy in our {CAE v2}\\xspace. We use the block-wise sampling by default (marked in \\colorbox{baselinecolor}{gray}).}\n\\vspace{-0.7cm}\n\\label{tab:mask_strategy}\n\\end{table}\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\scalebox{1.0}{\n\\renewcommand{\\arraystretch}{1.05}\n\\setlength{\\tabcolsep}{6.0mm}{\n\\begin{tabular}{lccccc} %\n\\toprule\n\\multirow{2}{*}{Methods} & \\multirow{2}{*}{\\#Epochs}  &  \\multirow{2}{*}{Target} & \\multicolumn{2}{c}{IN-1K}  \n& \\multirow{1}{*}{ADE20K} \\\\\n\\cline{4-6}\n&  &  & LIN & FT & mIoU \\\\\n\\midrule\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Tiny}:} \\\\\nMAE-Tiny~\\cite{wang2022closer} & 400 & RGB & 23.4 & 76.2 & - \\\\\nCAE~\\cite{chen2022context}$^\\ddagger$ & 300 & DALL-E & 28.1 & 75.9 & 38.3 \\\\\nDistilled MAE-lite~\\cite{wang2022closer} & 400 & RGB & - & 76.5 & - \\\\\n\\textbf{{CAE v2}\\xspace} & 300 & CLIP-B & 68.8 & 77.4 & 44.2 \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & \\textbf{69.3} & \\textbf{77.8} & \\textbf{44.7} \\\\\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Small}:} \\\\\nMoCo v3~\\cite{chen2021empirical}$^\\S$  & 300 & Self-EMA & 73.1 & 81.7$^\\dagger$ & - \\\\\nBEiT~\\cite{bao2021beit}$^\\S$ & 300 & DALL-E & 15.7 & 81.7$^\\dagger$ & - \\\\\nSplitMask~\\cite{el2021large} & 300  & DALL-E & - & 81.5 & - \\\\\nCAE~\\cite{chen2022context} & 300 & DALL-E & 51.8 & 82.0$^\\dagger$ & - \\\\\niBOT~\\cite{zhou2021ibot} & 3200 & Self-EMA & \\textbf{77.9} & 82.3$^\\dagger$ & 45.4 \\\\\n\\textbf{{CAE v2}\\xspace} & 300 & CLIP-B & 77.3 & 82.8 & 49.1 \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & 77.5 & \\textbf{83.1}$^\\dagger$ & \\textbf{49.7} \\\\\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Base}:} \\\\\nMoCo v3~\\cite{chen2021empirical}  & 300  & Self-EMA & 76.5 & 83.2 & 47.2 \\\\\nDINO~\\cite{caron2021emerging}$^\\S$ & 400 & Self-EMA & 77.3 & 83.3 & 47.2 \\\\\niBOT~\\cite{zhou2021ibot} & 1600 & Self-EMA & 79.5 & 84.0 & 50.0 \\\\\nBEiT~\\cite{bao2021beit}  & 800  & DALL-E & 56.7 & 83.2 & 45.6 \\\\\nSimMIM~\\cite{xie2022simmim} & 800 & RGB & 56.7 & 83.8 & - \\\\\nMAE~\\cite{he2022masked} & 1600  & RGB & 68.0 & 83.6 & 48.1 \\\\\nCAE~\\cite{chen2022context} & 1600  & DALL-E & 70.4 & 83.9 & 50.2 \\\\\nSdAE~\\cite{chen2022sdae} & 300  & Self-EMA & 64.9 & 84.1 & 48.6 \\\\\nSIM~\\cite{tao2022siamese} & 1600 & Self-EMA & 76.4 & 83.8 & - \\\\\nMaskFeat~\\cite{wei2022masked} & 1600  & HOG & - & 84.0 & - \\\\\nSplitMask~\\cite{el2021large} & 300  & DALL-E & - & 83.6 & 45.7 \\\\\nPeCo~\\cite{dong2021peco} & 800  & VQGAN & - & 84.5 & 48.5 \\\\\ndata2vec~\\cite{baevski2022data2vec} & 800  & Self-EMA & - & 84.2 & - \\\\\nCMAE~\\cite{huang2022contrastive} & 1600 & RGB & - & 84.7 & 50.1 \\\\\nExtreMA~\\cite{wu2022extreme} & 300 & Self-EMA & 73.3 & 83.7 & 47.9 \\\\\nCLIP~\\cite{radford2021learning} & -  & Text  & - & 84.9 & 51.1 \\\\\nMaskCLIP \\cite{Dong2022MaskCLIPMS} & 1600 & Text & 72.9 & 84.1 & 50.8 \\\\\nMVP~\\cite{wei2022mvp} & 300  & CLIP-B & 75.4 & 84.4 & 52.4 \\\\\nBEIT V2~\\cite{peng2022beit} & 300 & VQ-CLIP-B & 80.1 & 85.0 & 52.7 \\\\\n\\textbf{{CAE v2}\\xspace} & 300 & CLIP-B & 80.5 & 85.2 & \\textbf{53.1} \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & \\textbf{80.6} & \\textbf{85.3} & 52.9 \\\\\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Large}:} \\\\\nMoCo v3~\\cite{chen2021empirical}$^\\S$  & 300  & Self-EMA & - & 84.1 & 49.1 \\\\\nBEiT~\\cite{bao2021beit}$^\\S$  & 1600  & DALL-E & - & 85.2 & 53.3 \\\\\niBOT~\\cite{zhou2021ibot} & 1200 & Self-EMA & 81.0 & 84.8 & - \\\\\nMAE~\\cite{he2022masked} & 1600  & RGB & 75.8 & 85.9 & 53.6 \\\\\nCAE~\\cite{chen2022context} & 1600  & DALL-E & 78.1 & 86.3 & 54.7 \\\\\ndata2vec~\\cite{baevski2022data2vec} & 1600  & Self-EMA & - & 86.6 & - \\\\\nMVP~\\cite{wei2022mvp} & 300  & CLIP-B & - & 86.3 & 54.3 \\\\\nBEIT V2~\\cite{peng2022beit} & 300 & VQ-CLIP-B & - & 86.6 & 55.0 \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & \\textbf{81.7} & \\textbf{86.7} & \\textbf{55.9} \\\\\n\\bottomrule\n\\end{tabular}\n}\n}\n\\caption{\nPre-training evaluation on the top-1 accuracy (\\%) on linear probing (LIN) and fine-tuning (FT) on ImageNet-1K~\\cite{deng2009imagenet}, and mIoU (\\%) on ADE20K~\\cite{zhou2017scene}.\n$\\dagger$ denotes the fine-tuning epoch is 200 for ViT-Small.\n$\\ddagger$ means our implementation using the officially released code.\n$^\\S$ means the results from \\cite{chen2022context}.\nAll other results except for ours are from the original papers.\n}\n\\label{tab:imagenet_seg_sota}\n\\end{table*}\n\n\\myparagraph{Image classification on ImageNet-1K.}\nTable~\\ref{tab:imagenet_seg_sota} shows the comparisons of different models with two evaluation methods: linear probing and fine-tuning.\n\nWith linear probing, {CAE v2}\\xspace shows significant improvements over previous methods with other targets, {\\em e.g.}, BEiT~\\cite{bao2021beit}, MAE~\\cite{he2022masked}, CAE~\\cite{chen2022context}, and MaskFeat~\\cite{wei2022masked}. These gains are expected as CLIP features contain rich semantics than other targets. Compared with the methods (MVP~\\cite{wei2022mvp} and BEIT V2~\\cite{peng2022beit}) use CLIP as the target, {CAE v2}\\xspace can also give superior performance (on ViT-Base with 300 epoch pre-training, {CAE v2}\\xspace \\textit{vs.\\ } MVP: 80.6\\% \\textit{vs.\\ } 75.4\\% and {CAE v2}\\xspace \\textit{vs.\\ } BEIT V2: 80.6\\% \\textit{vs.\\ } 80.1\\%). When we fine-tune the pre-trained model on ImageNet-1K, {CAE v2}\\xspace achieves the best results among various methods across all scales of ViTs (Table~\\ref{tab:imagenet_seg_sota}). Specifically, {CAE v2}\\xspace achieves $\\textbf{85.3\\%}$ top-$1$ accuracy, surpassing previous methods by large margins. Moreover, with ViT-Large, {CAE v2}\\xspace improves the performance to $\\textbf{86.7\\%}$ top-$1$ accuracy.\n\n\n\\myparagraph{Semantic segmentation on ADE20K.} \nSemantic segmentation is a challenging task that needs to classify all pixels to various semantic labels given an image. CLIP~\\cite{radford2021learning} as the target shows clear advantages in this task. As shown in Table~\\ref{tab:imagenet_seg_sota}, {CAE v2}\\xspace significantly improves the results over the methods pre-trained with other targets, {\\em e.g.}, by 2.7\\% mIoU over CAE~\\cite{chen2022context} with ViT-Base. When comparing with CLIP~\\cite{radford2021learning}, MVP~\\cite{wei2022mvp}, and BEIT V2~\\cite{peng2022beit}, {CAE v2}\\xspace outperforms them with the same or less pre-training epochs. The superior performance hold when we move to ViT-Large, with which {CAE v2}\\xspace achieves $\\textbf{55.9\\%}$ mIoU on ADE20K~\\cite{zhou2017scene}.\n\n\\myparagraph{Object detection and instance segmentation on COCO.}\nWe evaluate the pre-trained models on COCO~\\cite{lin2014microsoft} with Mask R-CNN~\\cite{he2017mask} (Table~\\ref{tab:cocodetection_mask}) and Cascade Mask R-CNN~\\cite{cai2018cascade,he2017mask} (Table~\\ref{tab:cocodetection_cascade}). We report the results of $1\\times$ (12 epochs) training schedule.\nCompared with other pre-training methods, {CAE v2}\\xspace performs better \nunder both two configurations. With Mask R-CNN, {CAE v2}\\xspace gives 4.9\/3.0 points higher on ViT-Small and 2.0\/0.9 points higher on ViT-Base on AP$^b$\/AP$^m$ than the previous best method~\\cite{chen2022context}. The superior performance remain when adopting Cascade Mask R-CNN as the fine-tuned model (see Table~\\ref{tab:cocodetection_cascade}).\n\n\\begin{table}[!t]\n    \\centering\n\\small\n\\renewcommand{\\arraystretch}{1.05}\n\\setlength{\\tabcolsep}{4.8mm}{\n\\begin{tabular}{lccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \n        \\multirow{2}{*}{\\#Epochs} & \n        \\multicolumn{1}{c}{DET}& \n        \\multicolumn{1}{c}{INS}\\\\ \n        \\cline{3-4}\n        &  & { $\\text{AP}^{b}$} & {$\\text{AP}^{m}$} \\\\ \n        \\midrule\n        \\midrule\n      \\multicolumn{3}{l}{\\emph{Methods using ViT-Small}:} \\\\\n       DeiT \\cite{touvron2021training}  & 300 & 43.1 & 38.4 \\\\\n       MoCo v3$^*$ \\cite{chen2021empirical}  & 300 &  43.3 & 38.8 \\\\\n       BEiT \\cite{bao2021beit} & 300 & 35.6 & 32.6 \\\\\n       CAE \\cite{chen2022context}  & 300 & 44.1 & 39.2 \\\\\n       \\textbf{{CAE v2}\\xspace$+$} & 300  &  \\textbf{49.0} & \\textbf{42.2} \\\\\n       \\midrule\n      \\multicolumn{3}{l}{\\emph{Methods using ViT-Base}:}\\\\\n       DeiT \\cite{touvron2021training}  & 300  & 46.9 & 41.5 \\\\\n       MoCo v3$^*$ \\cite{chen2021empirical} & 300  & 45.5 & 40.5 \\\\\n       DINO$^*$ \\cite{caron2021emerging} & 400 & 46.8 & 41.5 \\\\\n       BEiT \\cite{bao2021beit} & 800 & 42.1 & 37.8 \\\\\n       MAE \\cite{he2022masked} & 1600 & 48.4 & 42.6 \\\\\n       data2vec~\\cite{baevski2022data2vec} & 800 & 41.1 & 37.0 \\\\\n       CAE \\cite{chen2022context} & 1600  &  50.0 & 44.0 \\\\\n       \\textbf{{CAE v2}\\xspace} & 300  &  51.8 & 44.7 \\\\\n       \\textbf{{CAE v2}\\xspace$+$} & 300  &  \\textbf{52.0} & \\textbf{44.9} \\\\\n\n        \\bottomrule \n    \\end{tabular} \n    }\n    \\caption{Pre-training evaluation on object detection (DET) and instance segmentation (INS) on COCO~\\cite{lin2014microsoft}. \n    Mask R-CNN~\\cite{he2017mask} is adopted and trained with $1\\times$ schedule.\n    All results except for {CAE v2}\\xspace are from \\cite{chen2022context}.\n    \\#Epochs refers to the pre-training epochs on ImageNet-$1$K.\n    $^*$ denotes multi-crop pre-training augmentation.\n    }\n    \\vspace{-0.5em}\n    \\label{tab:cocodetection_mask}\n\\end{table}\n\n\n\n\\section{Related Work}\n\nMasked image modeling (MIM) aims to learn transferable vision representations. It is inspired by the successful large-scale pre-training for transformers~\\cite{vaswani2017attention} with masked language modeling (MLM)~\\cite{bert,chen2020generative,gpt3,unilm} in NLP and can serve as a pretext task in self-supervised vision pre-training~\\cite{deepcluster, doersch2015unsupervised, cpc, ermolov2020whitening, goyal2021self, li2021prototypical, zbontar2021barlow,moco,mocov2,simclr,byol, Ge2-AE, BootMAE, HiViT, GreenMIM, MimCo, MSN, ConMIM, Li2022mcBEiTMD}. MIM methods~\\cite{bao2021beit,he2022masked,xie2022simmim,wei2022masked,baevski2022data2vec, Singh2022RevisitingWS} follow a mask-then-predict pipeline of (i) corrupting an image by masking several image patches based on a pre-defined mask ratio and (ii) learning to predict the missing content under specific supervision. {CAE v2}\\xspace uses the CLIP model~\\cite{radford2021learning} as the supervision target and studies on the above two aspects. Next, we discuss related works with respect to these two aspects.\n\n\n\n\\myparagraph{Supervision target.} There are several ways to represent the missing content when supervising a model. Existing MIM methods explore different supervision targets on their frameworks, including RGB pixels~\\cite{he2022masked,gao2022convmae}, HOG descriptors~\\cite{wei2022masked}, discrete visual tokens~\\cite{bao2021beit,chen2022context,el2021large,peng2022beit,dong2021peco}, and feature representation from momentum models~\\cite{tao2022siamese,chen2022sdae,wu2022extreme}. Among these methods, supervision is added to the model's predictions for masked patches. They give no supervision to the model's predictions for unmasked patches (visible patches). Recently, MVP~\\cite{wei2022mvp} explored changing the supervision target from other modalities and validated the effectiveness of the additional knowledge. In detail, MVP adopts the vision branch of the CLIP mode\\cite{radford2021learning} as the supervision target in MIM. Then MVP gives supervision on all patches of the image, including masked patches and unmasked patches. {CAE v2}\\xspace follows MVP~\\cite{wei2022mvp} to use the CLIP model as the supervision target and go one step further to study the supervision position in this paper.\n\n\\begin{table}[!t]\n    \\centering\n\n\\small\n\\renewcommand{\\arraystretch}{1.05}\n\\setlength{\\tabcolsep}{4.8mm}{\n\\begin{tabular}{lccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \n        \\multirow{2}{*}{\\#Epochs} & \n        \\multicolumn{1}{c}{DET}& \n        \\multicolumn{1}{c}{INS}\\\\ \n        \\cline{3-4}\n        &  & { $\\text{AP}^{b}$} & {$\\text{AP}^{m}$} \\\\ \n        \\midrule\n        \\midrule\n        \n  \\multicolumn{3}{l}{\\emph{Methods using ViT-Small}:} \\\\\n  iBOT$^*$~\\cite{zhou2021ibot} & 3200 & 49.4 & 42.6 \\\\\n  \\textbf{{CAE v2}\\xspace$+$} & 300 & \\textbf{51.5} & \\textbf{43.9} \\\\\n    \\midrule\n  \\multicolumn{3}{l}{\\emph{Methods using ViT-Base}:} \\\\\n  MAE \\cite{he2022masked} & 1600 & 51.3 & 44.3 \\\\\n  CAE  \\cite{chen2022context}  & 1600 & 52.9 & 45.5 \\\\\n  iBOT$^*$~\\cite{zhou2021ibot} & 1600 & 51.2 & 44.2 \\\\\n  \\textbf{{CAE v2}\\xspace$+$} & 300 & \\textbf{53.9} & \\textbf{45.9} \\\\\n        \\bottomrule \n    \\end{tabular} \n    }\n    \\caption{\n    Pre-training evaluation on object detection (DET) and instance segmentation (INS) on COCO~\\cite{lin2014microsoft} with Cascade Mask R-CNN~\\cite{cai2018cascade}. All models are trained with the $1\\times$ schedule.\n    All results except for {CAE v2}\\xspace are from \\cite{chen2022context},\n    and the results of iBOT are from the original paper~\\cite{zhou2021ibot}.\n    \\#Epochs refers to the effective pre-training epochs on ImageNet-$1$K.\n    $^*$ denotes the multi-crop pre-training augmentation.\n    }\n    \\vspace{-1.0em}\n    \\label{tab:cocodetection_cascade}\n\\end{table}\n\n\\myparagraph{Mask ratio.} The mask ratio is a hyper-parameter that needs hand-design in both MLM and MIM. In MLM, BERT~\\cite{bert} uses a relatively small mask ratio (15\\%) for pre-training. ~\\cite{wettig2022should} argues that masking up to 40\\% may give higher performance. In MIM, a lot of works use a high mask ratio for pertaining. For example, MAE~\\cite{he2022masked} utilizes a mask ratio of\n75\\%, BEiT~\\cite{bao2021beit}, CAE~\\cite{chen2022context}, and MVP~\\cite{wei2022mvp} empirically set the mask ratio as 40\\% and 50\\%. In this paper, we study this interesting problem and provide a guideline for choosing the right mask ratio for different scales of ViTs.\n\n\n\nConcurrent with our work, \\cite{hou2022milan, liu2022exploring, peng2022unified} also explores using CLIP to guide the MIM pre-training. MILAN~\\cite{hou2022milan} and dBOT~\\cite{liu2022exploring} focus on the impact of target representations, that the CLIP containing multi-modality knowledge can provide more benefits to MIM. MaskDistill~\\cite{peng2022unified} works on the design of distillation loss in supervision. Differently, we investigate two aspects orthogonal to these works. We find that adding supervision on visible patches further helps visual learning compared to only supervising the masked patches. Moreover, we explore the relationship between mask ratio and model scales.\nThese two findings provide useful guidelines for MIM pre-training.\n\n\\section{Conclusion and Limitation}\nThis paper studies two critical ingredients in MIM, \\textit{i.e.}, the supervision position and the mask ratio, with CLIP as the supervision target.\nWith our simple pipeline {CAE v2}\\xspace, we reveal two new insights: i) the feature distillation supervision on visible patches can achieve remarkable performance; ii) the optimal mask ratio is positively correlated to the model size.\nFollowing these two guidelines, our {CAE v2}\\xspace achieves superior performance on all scales of models on various downstream tasks.\n\n\\myparagraph{Limitation.} Limited by resources, we do not study on larger models, like ViT-Huge and ViT-Giant. We leave this exploration in the future.\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Klein's classification and McKay's correspondence} \nFelix Klein \\cite{Klein} classified\nfinite subgroups $G$ of ${\\mathbf{SL}}_2({\\mathds{C}})$:\nup to conjugation, there are two infinite series and three isolated cases.\n\n\\begin{enumerate}\n\\item The associated quotient singularity ${\\mathds{C}}^2\/G$ is called a \n \\emph{Klein singularity} and the singularities arising this way\n are precisely the \\emph{rational double point singularities}.\n Its minimal resolution of singularities is a union\n of ${\\mathds{P}}^1$'s, whose dual intersection graph $\\Gamma$ \n is a simply-laced Dynkin diagram of finite type, that is,\n of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$.\n\\item John McKay \\cite{McKay} associated a finite graph $\\widehat{\\Gamma}$\n to $G \\subset{\\mathbf{SL}}_2({\\mathds{C}})$, whose vertices correspond to the isomorphism classes of the \n simple representations of $G$.\n This graph is a Dynkin diagram of affine type\n $\\widehat{A}_n$, $\\widehat{D}_n$, $\\widehat{E}_6$, $\\widehat{E}_7$ or $\\widehat{E}_8$.\n\\item After these preparations, the \\emph{classical McKay correspondence} consists of the \nfollowing observations:\n\\begin{enumerate}\n\\item The graph $\\Gamma$ is obtained from $\\widehat{\\Gamma}$ by removing the vertex corresponding to the \ntrivial representation.\n\\item There exists a bijection between \nconjugacy classes of $G$, vertices of $\\widehat{\\Gamma}$, and isomorphism classes of simple representations of $G$.\n\\item There exists a bijection between\nfinite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$ up to conjugacy, the above Dynkin diagrams of affine type,\nKlein singularities, and the above Dynkin diagrams of finite type.\n\\end{enumerate}\n\\end{enumerate}\n\nBy now, there are various approaches to and a vast literature on this subject, such as \n\\cite{Knorrer, Kostant, McKay, McKay2, Steinberg} and many more.\nAlso, there are now generalisations into very\ndifferent directions: \nhigher dimensional algebraic geometry \\cite{Reid}, K-theory \\cite{GSV},\nderived categories of coherent sheaves \\cite{Bridgeland,KV}, \nrepresentations of quivers \\cite{Kirillov}, non-commutative geometry and Hopf algebras \\cite{Chan},\nand string theory \\cite{Dirichlet} - just to mention a few.\n\n\\subsection{Positive characteristic}\nNow, let $k$ be an algebraically closed field of characteristic $p>0$.\n\n\\subsubsection{Wild McKay correspondence}\nThe classical McKay correspondence as sketched above is still partially \navailable over $k$ if $G$ is assumed to be a finite subgroup of ${\\mathbf{SL}}_2(k)$\nof order prime to $p$ - this is the \\emph{tame} case.\nIf $p$ divides the order of $G$ - this is the \\emph{modular} or \\emph{wild} case -\nthen this correspondence breaks down.\nWe refer to Yasuda's survey \\cite{Yasuda} about conjectures\nand partial results concerning such wild McKay correspondences.\n\n\\subsubsection{Linearly reductive McKay correspondence}\nIn this article, we show that if $G$ is a finite and\n\\emph{linearly reductive} subgroup scheme of ${\\mathbf{SL}}_{2,k}$, \nthen there is a reasonable version of the classical McKay correspondence.\nFor example, we obtain a McKay correspondence for \\emph{all} rational double point \nsingularities if $p\\geq7$.\nInstead of considering groups, we allow non-reduced \ngroup schemes over $k$, but we require their categories of $k$-linear and finite-dimensional\nrepresentations to be semi-simple.\nWe refer to \\cite{Yasuda2} about conjectures and partial results concerning\na McKay correspondence for the group scheme $\\boldsymbol{\\alpha}_p$, which is not linearly reductive.\n\nThus, let $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$ \nwith $p\\geq7$.\n(In this introduction, we will exclude small characteristics whenever this is makes\nour discussion easier.)\nLet $x\\in X:=U\/G$ with $U={\\mathds{A}}_k^2$ or $U=\\widehat{{\\mathds{A}}}_k^2$\nbe the associated Klein singularity, \nwhich is a rational double point.\nOne goal of this article is to define a notion of conjugacy class for $G$,\nto construct graphs $\\Gamma$ and $\\widehat{\\Gamma}$, and  to \nestablish bijections as above.\n\nLet us make three comments:\n\\begin{enumerate}\n\\item It is interesting in its own that a McKay correspondence can be extended \nfrom finite group schemes of length prime to $p$, that is, the tame case, \nto linearly reductive group schemes.\n\\item What makes this linearly reductive McKay correspondence \nreally interesting is that the bijection in Theorem \\ref{thm: main intro} is \\emph{not}\ntrue when considering finite groups of order prime to $p$ only, \nsee Example \\ref{sec: mup intro}.\n\\item Probably, many more aspects of the classical McKay correspondence \ncan be carried over to the linearly reductive setting, but rather than writing a whole\nmonograph, we decided to establish only some basic bijections.\n\\end{enumerate}\n\n\n\\subsection{Linearly reductive group schemes}\nLet $G$ be a \\emph{finite and linearly reductive group scheme} \nover an algebraically closed field $k$ of characteristic $p\\geq0$.\nBy definition, this means that the category $\\mathrm{Rep}_k(G)$ of\n$k$-linear and finite-dimensional representations of $G$ is semi-simple.\n\nIf $p=0$, then every finite group scheme over $k$ is \\'etale and linearly reductive.\nand in fact, it is\nthe constant group scheme associated to a finite group.\nIn fact, the functor $G\\mapsto G_{\\mathrm{abs}}:=G(k)$ induces an equivalence \nof categories between finite group schemes over $k$ and finite groups.\n\nIf $p>0$, then every linearly reductive group scheme admits a \ncanonical semi-direct product decomposition\n$$\n  G\\,\\cong\\,G^\\circ\\rtimes G^{\\mathrm{\\acute{e}t}},\n$$\nwhere $G^\\mathrm{\\acute{e}t}$ is a group scheme of length prime to $p$ (and thus, the constant \ngroup scheme associated to a finite group of order prime to $p$) and where $G^\\circ$\nis infinitesimal and diagonalisable.\nThe latter implies that $G^\\circ$ is a product of group schemes of the form\n$\\boldsymbol{\\mu}_{p^n}$.\nConversely, every such semi-direct product of a diagonalisable group scheme\nwith the constant group scheme associated to a finite group of order prime to $p$\nis linearly reductive.\nThis structure result is usually attributed to Nagata \\cite{Nagata61},\nbut see also \\cite{AOV}, \\cite{Chin}, and \\cite{Hashimoto}.\nIn particular, the class of finite and linearly reductive group schemes over $k$\nstrictly contains the class of constant group schemes associated \nto finite groups of order prime to $p$.\n\n\n\\subsubsection{Abstract groups}\nAssociated to $G$, there is an abstract finite group $G_{\\rm abs}$\nand we refer for the slightly technical definition to Section \\ref{subsec: abstract groups}.\nThe order of $G_{\\rm abs}$ is equal to the length of $G$.\nFor example, if $G$ is \\'etale over $k$, then we have\n$G_{\\rm abs}\\cong G(k)$.\nThe assignment $G\\mapsto G_{\\mathrm{abs}}$ establishes an equivalence\nof categories \n\\begin{equation}\n\\label{eq: catequivalence}\n\\left\\{ \\begin{array}{l}\n\\mbox{finite linearly reductive}\\\\ \\mbox{group schemes over $k$} \n\\end{array}\\right\\}\n\\quad\\leftrightarrow\\quad\n\\left\\{ \\begin{array}{l}\n\\mbox{finite groups with a}\\\\ \\mbox{unique $p$-Sylow subgroup} \n\\end{array}\\right\\},\n\\end{equation}\nsee \\cite{LMM} and Lemma \\ref{lem: lrequivalence}.\n\n\n\\subsubsection{Canonical lifts}\nIn \\cite{LMM}, we showed that if $G$ is a finite and linearly\nreductive group scheme, then there exists a lift of $G$ over the\nring of Witt vectors $W(k)$.\nWe note that $G^\\circ$ and $G^\\mathrm{\\acute{e}t}$ even lift uniquely to $W(k)$\nand we define the \\emph{canonical lift} $G_{\\rm can}\\to \\mathrm{Spec}\\: K$ of $G$ to\nbe the unique lift that is a semi-direct product of the lifts\nof $G^\\circ$ and $G^{\\mathrm{\\acute{e}t}}$. \nAny lift of $G$ to some extension field of $K$ becomes\nisomorphic to $G_{\\mathrm{can}}$ after possibly passing to some\nfurther field extension.\nMoreover, the finite group $G_{\\mathrm{can}}(\\overline{K})$ \nis isomorphic to $G_{\\rm abs}$.\n\n\n\\subsubsection{Representation theory}\n\\label{subsubsec: rep thy}\nBy \\cite{LMM} and Proposition \\ref{prop: liftingrepresentation},\nthere exist canonical equivalences of representation categories \n\\begin{equation}\n\\label{eq: rep thy intro}\n   \\mathrm{Rep}_{k}(G) \\,\\to\\, \\mathrm{Rep}_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}})\n   \\,\\to\\, \\mathrm{Rep}_{\\mathds{C}}(G_{\\mathrm{abs}})\n\\end{equation}\nthat are compatible with degrees, direct sums, tensor products, duals, \nand simplicity.\nThese equivalences induce isomorphisms of rings\n$$\n  K_k(G) \\,\\to\\, K_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}}) \\,\\to\\,\n  K_{{\\mathds{C}}}(G_{\\mathrm{abs}}),\n$$\nsee Corollary \\ref{cor: Ktheory}.\nHere, $K_F(G)$ denotes the K-group associated to \n$F$-linear and finite-dimensional $G$-representations.\n\n\n\\subsubsection{Hopf algebras}\nIf $G$ is a finite group scheme over $k$, then the multiplication map\nturns $H^0(G,{\\cal O}_G)$ into a finite-dimensional \\emph{Hopf algebra}\nover $k$.\nWe discuss finite group schemes and among them the linearly reductive\nones from the point of Hopf algebras in \nAppendix \\ref{sec: Hopf}. \n\n\n\\subsection{Linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$}\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\nHashimoto \\cite{Hashimoto} extended Klein's classification \\cite{Klein}\nof finite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$ up to conjugation to the setting\nof finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\nIf $p\\geq7$, then one obtains a list analogous to Klein's classical \nlist.\nIf $p\\in\\{2,3,5\\}$, then some classical cases are missing, \nbut there are no new cases.\n\n\\subsection{McKay graph and McKay correspondence}\n\\label{subsec: McKay graph intro}\nLet $G$ be a finite and linearly reductive subgroup scheme of\n${\\mathbf{SL}}_{2,k}$.\nAs in McKay's original construction \\cite{McKay}, we associate\nan affine Dynkin diagram $\\widehat{\\Gamma}$ to $G$,\nits embedding into ${\\mathbf{SL}}_{2,k}$, and the set of isomorphism classes\nof simple representations of $G$.\nThis is the \\emph{McKay graph} associated to this data.\nIn fact, we will see that it is compatible with the equivalences\ninduced by \\eqref{eq: catequivalence} and \\eqref{eq: rep thy intro}\nand we refer to Section \\ref{subsec: McKay graph} for details.\nWe establish the following version of McKay's theorem \\cite{McKay}\nin positive characteristic.\n\n\\begin{Theorem}[Theorem \\ref{thm: main}]\n\\label{thm: main intro}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n There exists a bijection between\n non-trivial, finite, and linearly reductive subgroup schemes of\n ${\\mathbf{SL}}_{2,k}$ up to conjugation and affine Dynkin graphs\n of type\n $$\n  \\begin{array}{lcl}\n   \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7, \\widehat{E}_8\n       &\\mbox{\\quad if \\quad }&    p=0\\mbox{\\quad or \\quad}p\\geq7, \\\\\n   \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7 &\\mbox{\\quad if \\quad }& p=5,\\\\\n  \\widehat{A}_n, \\widehat{D}_n &\\mbox{\\quad if \\quad }& p=3, \\\\\n  \\widehat{A}_n &\\mbox{\\quad if \\quad }&p=2.\n  \\end{array}\n $$\n\\end{Theorem}\n\nBy construction, this bijection is compatible with the classical McKay correspondence via\nthe lifting results and the equivalences \\eqref{eq: catequivalence},\n\\eqref{eq: rep thy intro}.\n\n\\begin{Example}\n\\label{sec: mup intro}\nThe linearly reductive group scheme corresponding to \n$\\widehat{A}_n$ is $\\boldsymbol{\\mu}_{n+1}$.\nThis group scheme is reduced, that is, \\'etale, if and only if\n$p$ does not divide $n+1$.\nIn particular, it is crucial to allow non-reduced group schemes\nin order to obtain a bijection as in characteristic zero.\n\\end{Example}\n\n\n\\subsection{Linearly reductive quotient singularities}\nConsider ${\\mathbf{GL}}_{2,k}$ with its usual linear action on \n$U={\\mathds{A}}^2_k$ or $U=\\widehat{{\\mathds{A}}}_k^2$.\nIf $G$ is a finite, linearly reductive, and \\emph{very small}\n(see Definition \\ref{def: very small}) subgroup scheme of ${\\mathbf{GL}}_{2,k}$, then the associated\nquotient singularity $x\\in X:=U\/G$ is a \ntwo-dimensional \\emph{linearly reductive quotient singularity} \nin the sense of \\cite{LMM}.\nBy loc.cit., such a singularity determines \n$G$ together with its embedding $G\\to{\\mathbf{GL}}_{2,k}$ up to isomorphism and\nconjugation, respectively.\n\nIn Section \\ref{subsec: Ishii Ito Nakamura}, we will see \nthat a minimal resolution of singularities of a two-dimensional linearly reductive\nquotient singularity $x\\in X=U\/G$\nis provided by the $G$-Hilbert scheme\n\\begin{equation}\n\\label{eq: intro IIN}\n \\pi \\,:\\, G\\mathrm{-Hilb}(U) \\,\\to\\, U\/G,\n\\end{equation}\nwhich generalises work of Ishii, Ito, and Nakamura \n\\cite{Ishii,IN}.\n\n\\begin{Remark}\nIn dimension two, a linearly reductive quotient singularity\nis the same as an F-regular singularity \\cite{LMM}.\nThus, every two-dimensional F-regular singularity can be \nresolved by a suitable $G$-Hilbert scheme, see \nalso Remark \\ref{rem: f-regular}.\n\\end{Remark}\n\nIf moreover $G$ is a subgroup scheme of ${\\mathbf{SL}}_{2,k}$, \nthen $x\\in X=U\/G$ is called a \\emph{Klein singularity}.\nKlein singularities are rational double point singularities.\nIf $p=0$ or $p\\geq7$, then conversely every rational double point \nis a Klein singularity by Hashimoto \\cite{Hashimoto} and, independently, by \n\\cite{LiedtkeSatriano}.\nIf $p\\in\\{2,3,5\\}$, then not every rational double point \nis a Klein singularity.\n\n\n\\subsection{Canonical lifts and simultaneous resolutions}\nLet $G$ be a very small, finite, and linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$ \nand let $x\\in X=U\/G$ be the associated linearly reductive quotient singularity.\nAssume $p>0$ and let $W(k)$ be the ring of Witt vectors of $k$.\nIn Section \\ref{subsec: canonical lift singularity}, \nwe will establish the existence of a \n\\emph{canonical lift} \n$$\n \\mathcal{X}_{\\mathrm{can}}\\,\\to\\,\\mathrm{Spec}\\: W(k)\n$$\nof  $x\\in X=U\/G$.\nUsing $G$-Hilbert schemes in families, we \nwill see in Section \\ref{subsec: simultaneous resolution} \nthat it admits a simultaneous and minimal resolution of singularities\n$$\n \\widetilde{\\pi} \\,:\\, \\mathcal{Y}\\,\\to\\,\\mathcal{X}_{\\mathrm{can}}\\,\\to\\,\\mathrm{Spec}\\: W(k).\n$$\nWe will prove this resolution to be unique, see \nTheorem \\ref{thm: IN}.\n\n\n\\subsection{McKay correspondence for Klein singularities}\nLet $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$ \nand let $x\\in X=U\/G$ be the associated Klein singularity.\n\nAs discussed in Section \\ref{subsec: McKay graph intro}, we have\nthe McKay graph $\\widehat{\\Gamma}$ associated to \n$G$, its embedding into ${\\mathbf{SL}}_{2,k}$, and the\nset of isomorphism classes of simple representations of $G$.\n\nLet $\\pi:Y\\to X$ be a minimal resolution of singularities.\nSince $x\\in X$ is a rational double point, the exceptional divisor $\\mathrm{Exc}(\\pi)$ \nof $\\pi$  is a configuration of ${\\mathds{P}}^1$'s, whose\ndual intersection graph $\\Gamma$ is a simply-laced Dynkin diagram.\n\n\\begin{Theorem}[Theorem \\ref{thm: bijection simple and exceptional}]\n\\label{thm: bijection simple intro}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n Let $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$\n and let $x\\in X=U\/G$ be the associated Klein singularity.\n Then, there exists a natural bijection of the graph $\\Gamma$ \n with the graph obtained from $\\widehat{\\Gamma}$ by removing the vertex\n corresponding to the trivial representation.\n \\end{Theorem}\n \n Since every rational double point in characteristic $p\\geq7$ \n is a Klein singularity, we obtain the following.\n\n\\begin{Corollary}\nThere exists a linearly reductive McKay correspondence\nfor rational double point singularities in every characteristic $p\\geq7$.\n\\end{Corollary}\n\nTo establish this theorem, we use the Ishii-Ito-Nakamura resolution\nof singularities \\eqref{eq: intro IIN}, as well as Hecke correspondences as in the work\nof Ito and Nakamura \\cite{IN} and Nakajima \\cite{Nakajima, NakajimaLectures}.\n\n\n\\subsection{Generalisations and variants}\nLet $G$ be a very small, finite,\nand linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$ \nand let $x\\in X=U\/G$ be the associated two-dimensional linearly\nreductive quotient singularity.\nLet $\\pi:Y\\to X$ be the minimal resolution of singularities and \nlet $\\mathrm{Exc}(\\pi)$ be the exceptional divisor of $\\pi$.\n\\begin{enumerate}\n\\item In Theorem \\ref{thm: exc pi}, \nwe associate a representation of $G$ to each point of $\\mathrm{Exc}(\\pi)$.\nThis generalises results of Ishii and Nakamura \\cite{IshiiCrelle, Ishii}.\n\\item In Theorem \\ref{thm: reflexive}, we establish a bijection between \nthe components of $\\textrm{Exc}(\\pi)$ and reflexive \n${\\cal O}_X$-modules.\nProbably, this result should be viewed as a theorem on two-dimensional F-regular\nsingularities, see Remark \\ref{rem: f-regular Artin Verdier}.\nIt generalises work of Artin and Verdier \\cite{AV}, \nWunram \\cite{Wunram}, and Ishii and Nakamura \\cite{Ishii}.\n\\item In Theorem \\ref{thm: derived}, we establish an equivalence of derived\ncategories of coherent sheaves ${\\mathcal{D}}(Y)$ and \n${\\mathcal{D}}^G(U)$ ($G$-equivariant \nsheaves on $U$).\nThis generalises work of  Kapranov and Vasserot \\cite{KV},\nBridgeland, King, and Reid \\cite{Bridgeland},\nand Ishii, Nakamura, and Ueda \\cite{IshiiCrelle,Ishii, IshiiUeda}.\nOf course, these articles themselves generalise work of Gonzalez-Sprinberg and Verdier \n\\cite{GSV} from K-theory to derived categories of coherent sheaves.\n\\end{enumerate}\n\n\n\\subsection{Ito-Reid correspondence}\nIf $G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then \nIto and Reid \\cite{ItoReid} found a natural bijection,\nthe \\emph{Ito-Reid correspondence},\nbetween the conjugacy classes of $G$ and the vertices \nof the McKay graph $\\widehat{\\Gamma}$.\n\n\\begin{Theorem}\n\\label{thm: ito reid intro}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n There exists an Ito-Reid correspondence for finite and linearly reductive\n subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\n\\end{Theorem}\n\nThis can be proven using lifting results and the Ito-Reid correspondence\nover ${\\mathds{C}}$, see Section \\ref{subsec: Ito-Reid} for details.\n\nThe main difficulty is to define a notion of conjugacy class\nfor finite and linearly reductive group schemes that makes\nsuch a correspondence work:\nLet us recall the ring $K_k(G)$ from Section \\ref{subsubsec: rep thy}.\nIf $G$ is a finite and linearly reductive group scheme over $k$,\nthen we define the set of \\emph{conjugacy classes} of $G$\nto be\n$$\n\\mathrm{Spec}\\: \\left({\\mathds{C}}\\otimes K_k(G)\\right).\n$$\nAt first sight, this  might look rather artificial.\nOne should think of it as defining\nconjugacy classes to be ``dual'' to simple representations. \nMoreover, our definition is compatible with \nlifting over $W(k)$ and it induces a bijection of the conjugacy\nclasses of $G$ with the conjugacy classes of $G_{\\rm abs}$.\nWe refer to Section \\ref{subsec: conjugacy classes}\nand Appendix \\ref{subsec: second approach} for details.\n\n\n\\subsection{Conjugacy classes}\n\\label{intro: conjugacy classes}\n\nIn Theorem \\ref{thm: ito reid intro}, we had to find a definition\nof conjugacy classes that makes this theorem true.\nThis begs for the question whether there are other definitions \nor approaches, which is a question that is interesting in its own.\n\nLet $G$ be a finite group scheme (not necessarily linearly reductive)\nover an algebraically closed field $k$.\nIn Appendix \\ref{app: conjugacy class}, we study the following\napproaches to the notion of the set of conjugacy classes of $G$:\n\n\\begin{enumerate}\n\\item The set of conjugacy classes $G(k)\/\\sim$ of the group of $k$-rational points of $G$.\n\\item The spectrum of $F\\otimes K_k(G)$, where $F$ is a field of characteristic zero\nthat contains ``sufficiently many'' roots of unity.\n\\item The scheme that represents the functor of conjugacy classes from schemes over $k$ to sets\ndefined by $S\\mapsto G(S)\/\\sim$,\n\\item the isotypical component of the trivial representation of the \nadjoint representation of $G$.\n\\item The simple subrepresentations of the extended adjoint representations\n$^{\\mathrm{Ad}}A$ and $^{\\mathrm{Ad}}(A^*)$, which are defined using the\nquantum doubles of the Hopf algebra $A:=H^0(G,{\\cal O}_G)$\nand its dual $A^*$.\n\\end{enumerate}\nAll approaches lead essentially to the ``same answer'' in characteristic\nzero.\nHowever, they usually lead to very different notions in positive characteristic.\nOn the other hand, all approaches have their merits and drawbacks.\nFor linearly reductive group schemes, the approach (2) leads to a definition that is\ncompatible with lifting and that leads to an Ito-Reid correspondence.\n\n\n\\subsection{Organisation of this article}\n\n\\begin{enumerate}\n\\item[-] In Section \\ref{sec: linearly reductive}, we recall basic facts about finite and \nlinearly reductive group schemes over algebraically closed fields.\n\\item[-] In Section \\ref{sec: McKay}, we construct the McKay graph $\\widehat{\\Gamma}$.\nThe main result  is Theorem \\ref{thm: main}, a McKay correspondence.\n\\item[-] In Section \\ref{sec: lrq singularities}, we recall basic facts about linearly\nreductive quotient singularities.\nWe establish an Ishii-Ito-Nakamura-type resolution of singularities,\nthe canonical lift of such a singularity, and a unique simultaneous resolution\nof singularities of the canonical lift.\n\\item[-] In Section \\ref{sec: Hecke}, we revisit the Ishii-Ito-Nakamura-type resolution $\\pi$\nof singularities of $x\\in X=U\/G$\nand we introduce Hecke correspondences, which leads to \nTheorem \\ref{thm: bijection simple and exceptional},\na bijection between simple and non-trivial representations of $G$ \nand  components of $\\pi$.\nWe also study generalisations of this result to two-dimensional linearly reductive \nquotient singularities.\n\\item[-] In Section \\ref{sec: ItoReid}, we establish an Ito-Reid correspondence\nbetween conjugacy classes of $G$ \nand exceptional divisors in the minimal resolution of singularities of $x\\in X=U\/G$.\n\\item[-] In Section \\ref{sec: outlook}, we study derived categories\nof $G$-equivariant sheaves on $U$ and on the minimal \nresolution $\\pi:Y\\to X=U\/G$.\n\\item[-] In Appendix \\ref{sec: Hopf}, we recall results on finite group schemes\nfrom the point of view of Hopf algebras.\nWe recall the adjoint representation, quantum doubles, and the\nextended adjoint representation.\n\\item[-] In Appendix \\ref{app: conjugacy class}, we study several approaches\ntoward the notion of a conjugacy class for finite group schemes.\n\\end{enumerate}\n\n\\begin{VoidRoman}[Acknowledgements]\n I thank Frank Himstedt, Martin Lorenz, Frans Oort, Matt Satriano,\n and Takehiko Yasuda for discussions and comments.\n\\end{VoidRoman}\n\n\n\\section{Linearly reductive group schemes}\n\\label{sec: linearly reductive}\n\nIn this section, we recall a couple of general facts about finite and linearly\nreductive group schemes over algebraically closed fields.\nWe discuss the close relationship between such a group scheme $G$ \nand a certain abstracted group $G_{\\mathrm{abs}}$ associated to it.\nFor the relationship to Hopf algebras, we refer to Appendix \\ref{sec: Hopf}.\n\n\\subsection{Group schemes}\nLet $G$ be a finite group scheme over an algebraically closed \nfield $k$ of characteristic $p\\geq0$.\nSince $k$ is perfect, there is a short exact sequence\nof finite group schemes over $k$\n\\begin{equation}\n\\label{connected etale}\n   1\\,\\to\\,G^\\circ\\,\\to\\,G\\,\\to\\, G^{\\mathrm{\\acute{e}t}} \\,\\to\\,1,\n\\end{equation}\nwhere $G^\\circ$ is the connected component of the identity and where $G^{\\mathrm{\\acute{e}t}}$\nis an \\'etale group scheme over $k$.\nThe reduction $G_{\\mathrm{red}}\\to G$ provides a canonical splitting of \\eqref{connected etale}\nand we obtain a canonical semi-direct product decomposition $G\\cong G^\\circ\\rtimes G^\\mathrm{\\acute{e}t}$.\nSince $k$ is algebraically closed, $G^{\\mathrm{\\acute{e}t}}$ is the constant group scheme associated\nto the finite group $G(k)=G^{\\mathrm{\\acute{e}t}}(k)$ of $k$-rational points.\nMoreover, $G^\\circ$ is an infinitesimal group scheme of length equal to \nsome power of $p$.\nIn particular, if $p=0$ or if the length of $G$ is prime to $p$, \nthen $G^\\circ$ is trivial and then, $G$ is \\'etale. \n\nIf $M$ is a finitely generated abelian group, then the group algebra\n$k[M]$ carries a Hopf algebra structure and the associated commutative\ngroup scheme is denoted $D(M):=\\mathrm{Spec}\\: k[M]$.\nBy definition, such group schemes are called \\emph{diagonalisable}.\nFor example, we have \n$D({\\mathbf{C}}_n)\\cong\\boldsymbol{\\mu}_n$, where ${\\mathbf{C}}_n$ denotes the cyclic group of order $n$.\nWe have that $\\boldsymbol{\\mu}_n$ is \\'etale over $k$ if and only if\n$p\\nmid n$.\n\nA finite group scheme $G$ over $k$ is said to be \\emph{linearly reductive} \nif every $k$-linear and finite-dimensional representation of $G$ is semi-simple.\nIf $p=0$, then all finite group schemes over $k$ are \\'etale and\nlinearly reductive.\nIf $p>0$, then, by a theorem that is often attributed to\nNagata \\cite[Theorem 2]{Nagata61}\n(but see also \\cite[Proposition 2.10]{AOV}, \\cite{Chin}, \nand \\cite[Section 2]{Hashimoto}), \na finite group scheme over $k$ is\nlinearly reductive if and only if it is an extension of a finite and \\'etale\ngroup scheme, whose length is prime to $p$, \nby a diagonalisable group scheme. \n\n\\subsection{Abstract groups and canonical lifts}\n\\label{subsec: abstract groups}\nLet $G$ be a finite and linearly reductive group scheme\nover an algebraically closed field $k$ of characteristic $p>0$.\nFollowing \\cite[Section 2]{LMM}, we study the finite group\n$$\nG_{\\mathrm{abs}} \\,:=\\, \n\\left(\\underline{((G^\\circ)^{D}(k))}_{{\\mathds{C}}}\\right)^{D}({\\mathds{C}}) \\,\\rtimes\\, G^\\mathrm{\\acute{e}t}(k).\n$$\nHere, $-^D$ denotes $\\underline{\\mbox{Hom}}(-,{\\mathds{G}}_m)$, the \\emph{Cartier dual}, of\na commutative and finite group scheme.\nIf $G$ is \\'etale, then $G_{\\mathrm{abs}}=G(k)$ and if $G=\\boldsymbol{\\mu}_{p^n}$,\nthen $G_{\\mathrm{abs}}={\\mathbf{C}}_{p^n}$.\nIn any case, the order of $G_{\\mathrm{abs}}$ is equal to the length of $G$.\n\n\\begin{Definition}\nThe finite group $G_{\\mathrm{abs}}$\nis called the \\emph{abstract finite group} associated to $G$.\n\\end{Definition}\n\n\\begin{Lemma} \\label{lem: lrequivalence}\nThe functor\n$$\nG \\,\\mapsto\\, G_{\\mathrm{abs}}\n$$\nestablishes an equivalence of categories between the category of finite and \nlinearly reductive group schemes over $k$ \nand the category of finite groups with a normal and abelian $p$-Sylow subgroup.\n\\end{Lemma}\n\n\\begin{proof}\n\\cite[Lemma 2.1]{LMM}.\n\\end{proof}\n\n\\begin{Remark}\nA finite group $G$ with a normal and abelian $p$-Sylow subgroup $P$ is the same\nas a finite group with a unique and abelian $p$-Sylow subgroup.\nIn this case, the Schur-Zassenhaus theorem implies that\n$G$ is isomorphic to a semi-direct product $P\\rtimes G\/P$.\n\\end{Remark}\n\nNext, we study lifts to characteristic zero:\nlet $W(k)$ be the ring of Witt vectors of $k$, let $K$ be its field\nof fractions, and let $\\overline{K}$ be an algebraic closure\nof $K$.\nBy \\cite[Proposition 2.4]{LMM},\nthere exist lifts of $G$ as finite and flat\ngroup scheme over $W(k)$.\nMore precisely, $G^\\circ$ and $G^\\mathrm{\\acute{e}t}$ even lift uniquely to $W(k)$,\nbut their extension class usually does not, see also\n\\cite[Example 2.6]{LMM}.\nHowever, there is a unique lift ${\\cal G}_{\\mathrm{can}}$\nof $G$ to $W(k)$ that is characterised by being \na semi-direct product of the unique lift of $G^\\circ$ with the unique lift of \n$G^{\\mathrm{\\acute{e}t}}$.\n\n\\begin{Definition}\\label{def: canonical lift}\nThe lift $\\mathcal{G}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ is called\nthe \\emph{canonical lift} of $G$.\nWe set $G_{\\mathrm{can}}:=\\mathcal{G}_K\\to \\mathrm{Spec}\\: K$\nand also call it canonical lift.\n\\end{Definition}\n\nEvery other lift of $G$ to some extension of $R\\supseteq W(k)$ differs from\n${\\cal G}_{\\mathrm{can},R}$ by a twist and thus, there is only one \\emph{geometric lift}\nof $G$ to $\\overline{K}$ up to isomorphism, namely $G_{\\mathrm{can},\\overline{K}}$,\nsee \\cite[Section 2.2]{LMM}.\n\nSince $\\overline{K}$ is algebraically closed and\nof characteristic zero, $G_{\\mathrm{can},\\overline{K}}$ is the constant\ngroup scheme associated to the finite group\n$G_{\\mathrm{can}}(\\overline{K})$.\nIn fact, we have $G^\\circ\\cong\\prod_i \\boldsymbol{\\mu}_{p^{n_i}}$ for some $n_i$'s and if we set\n$N:=\\max\\{n_i\\}_i$, fix a primitive $p^N$.th root of unity $\\zeta_{p^N}$ and set \n$K_N:=K(\\zeta_{p^N})$, \nthen $G_{\\mathrm{can},K_N}$ is the constant group scheme associated \nto the finite group $G_{\\mathrm{can}}(K_N)$ and we have\n$G_{\\mathrm{can}}(K_N)=G_{\\mathrm{can}}(\\overline{K})$.\n\n\\begin{Lemma}\n There exist isomorphism of finite groups\n $$\n   G_{\\mathrm{abs}} \\,\\cong\\, G_{\\mathrm{can}}(K_N) \\,=\\,G_{\\mathrm{can}}(\\overline{K}).\n $$\nIn particular, there exist isomorphisms\n $$\n  G_{\\mathrm{can},K_N} \\,\\cong\\,\n \\left(\\underline{G_{\\mathrm{abs}}}\\right)_{K_N}\n \\mbox{ \\quad and \\quad }\n G_{\\mathrm{can},\\overline{K}} \\,\\cong\\,\n \\left(\\underline{G_{\\mathrm{abs}}}\\right)_{\\overline{K}}\n $$\n of finite group schemes over $K_N$ and $\\overline{K}$, respectively.\\qed\n\\end{Lemma}\n\n\\subsection{Representation theory}\n\\label{subsec: representation comparison}\nLet $k$ be an algebraically closed field of characteristic $p>0$,\nand let $W(k)$, $K$, and $\\overline{K}$ be as in the previous section.\nThe equivalence of Lemma  \\ref{lem: lrequivalence}\ncan be extended to representations and K-theory.\nIf $G$ is a finite group or a finite group scheme over $k$,\nwe denote by ${\\rm Rep}_k(G)$ the category of its\n$k$-linear and finite-dimensional representations.\nLet us first recall \\cite[Corollary 2.11]{LMM}.\n\n\\begin{Proposition} \\label{prop: liftingrepresentation}\n Let $G$ be a finite and linearly reductive group scheme\n over $k$.\n Then, there exist canonical equivalences of categories\n $$\n   \\mathrm{Rep}_{k}(G) \n   \\,\\to\\, \\mathrm{Rep}_{K_N}(G_{\\mathrm{can},K_N})\n   \\,\\to\\, \\mathrm{Rep}_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}})\n   \\,\\to\\, \\mathrm{Rep}_{\\mathds{C}}(G_{\\mathrm{abs}}),\n $$\n which are compatible with degrees, direct sums, tensor products, duals, \n and simplicity.\\qed\n\\end{Proposition}\n\n\\begin{Remark}\n In particular, this allows us to define \\emph{characters} or even a\n \\emph{character table} of $G$ via \n Proposition \\ref{prop: liftingrepresentation} and $G_{\\mathrm{abs}}$.\n\\end{Remark}\n\nLet $K_k(G)$ be $K$-group associated to ${\\rm Rep}_k(G)$.\nIn fact, $K_k(G)$ has a natural structure of a commutative ring with one, \nwhere the sum (resp. product) structure comes from direct sums (resp. tensor products)\nof representations.\nA straight forward application of Proposition \\ref{prop: liftingrepresentation}\nis the following.\n\n\\begin{Corollary}\n \\label{cor: Ktheory}\n There exist isomorphisms of rings\n $$\n  K_k(G) \\,\\to\\, \n  K_{K_N}(G_{\\mathrm{can},K_N}) \\,\\to\\,\n  K_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}}) \\,\\to\\,\n  K_{{\\mathds{C}}}(G_{\\mathrm{abs}}).\\qed\n $$\n\\end{Corollary}\n\n\n\\section{McKay graph and McKay correspondence}\n\\label{sec: McKay}\n\nIn this section, we introduce the \\emph{McKay graph}\nassociated to a finite and linearly reductive subgroup scheme $G$\nover an algebraically closed field $k$ of characteristic $p\\geq0$\nand a representation $\\rho:G\\to{\\mathbf{GL}}_{n,k}$.\nThis induces a bijection between certain affine Dynkin diagrams\nand finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\nAs an application, we establish a \\emph{linearly reductive McKay correspondence}.\n\n\\subsection{McKay graph}\n\\label{subsec: McKay graph}\nLet $G$ be a finite and linearly reductive group scheme\nover an algebraically closed field $k$ of characteristic $p\\geq0$.\nLet $\\{\\rho_i\\}_i$ be the finite set of isomorphism classes of $k$-linear \nand simple representations of $G$.\nFollowing tradition, we assume that $\\rho_0$ is the trivial representation.\nWe fix a representation $\\rho:G\\to{\\mathbf{GL}}(V)$.\nIf $G$ is a subgroup scheme of ${\\mathbf{SL}}_{n,k}$ or ${\\mathbf{GL}}_{n,k}$, \nthen $\\rho$ is usually the linear representation corresponding to the embedding\nof $G$ into this linear algebraic group.\nBy assumption, $\\mathrm{Rep}_k(G)$ is semi-simple.\nTherefore, there exist unique integers $a_{ij}\\in{\\mathds{Z}}_{\\geq0}$ for each $i$,\nsuch that we have isomorphisms of $k$-linear representations\n$$\n  \\rho\\otimes\\rho_i \\,\\cong\\, \\bigoplus_{j}\\, \\rho_j^{\\oplus a_{ij}}.\n$$\nAssociated to this data, we define the \\emph{McKay graph}, \ndenoted $\\Gamma(G,\\{\\rho_i\\},\\rho)$:\n\\begin{enumerate}\n\\item[-] The vertices are the $\\{\\rho_{i}\\}_i$.\n(Some sources exclude the\nthe trivial representation $\\rho_0$.)\n\\item[-] There are $a_{ij}$ edges from the vertex corresponding to\n$\\rho_i$ to the vertex corresponding to $\\rho_j$.\n\\end{enumerate}\nWe now establish a couple of elementary properties of\nthis graph, which are well-known in the classical case\nand which immediately carry over to the linearly reductive\nsituation.\nWe leave the proof of the first lemma to the reader.\n\n\\begin{Lemma}\n We have\n $$\n  a_{ij} \\,=\\, \\dim_k\\, \\mathrm{Hom}(\\rho_i,\\rho_j\\otimes\\rho).\n $$\n In particular, if $\\rho$ is self-dual, that is, $\\rho\\cong\\rho^\\vee$,\n then $a_{ij}=a_{ji}$ for all $i,j$.\n In this case, we can consider  $\\Gamma(G,\\{\\rho_i\\},\\rho)$ \n as an undirected graph.\\qed\n\\end{Lemma}\n\n\\begin{Lemma}\n\\label{lem: selfdual}\nLet $\\rho:G\\to{\\mathbf{SL}}_{2,k}$ be a homomorphism of group schemes\nover $k$, considered as a\n2-dimensional representation.\nThen, $\\rho$ is self-dual.\n\\end{Lemma}\n\n\\begin{proof}\nBeing a 2-dimensional representation, \n$\\rho^\\vee$ is isomorphic to $\\rho\\otimes\\det(\\rho)$ and\nthe lemma follows.\n\\end{proof}\n\n\\begin{Lemma}\n\\label{lem: faithful}\nLet $\\rho:G\\to{\\mathbf{GL}}_{n,k}$ be a faithful representation.\nThen, every irreducible representation of $G$\noccurs as subrepresentation of $\\rho^{\\otimes m}$ for\nsome suitable $m$.\n\\end{Lemma}\n\n\\begin{proof}\nThis is well-known for finite groups.\nUsing Proposition \\ref{prop: liftingrepresentation}, it \ncarries over to finite and linearly reductive group schemes.\n\\end{proof}\n\n\\begin{Corollary}\nIf $\\rho$ is a faithful representation, then the graph\n$\\Gamma(G,\\{\\rho_i\\},\\rho)$  is connected.\n\\end{Corollary}\n\n\\begin{proof}\nIt suffices to note that  the number of paths in $\\Gamma$\nof length $m$ that connect the vertices corresponding to $\\rho_i$ and $\\rho_j$\nis equal to the multiplicity of $\\rho_i$ in $\\rho_j\\otimes\\rho^{\\otimes m}$,\nsee, for example, the proof of \\cite[Theorem 8.13]{Kirillov}.\n\\end{proof}\n\nLet $k$ be an algebraically closed field of characteristic $p>0$,\nand let $W(k)$, $K$, and $\\overline{K}$ be as in Section \\ref{subsec: abstract groups}.\nThere, we also discussed the\ncanonical lift $G_{\\mathrm{can}}$ of $G$ over $K$ and we saw that there exists\nan isomorphism of finite groups\n$G_{\\mathrm{abs}}\\cong G_{\\mathrm{can}}(\\overline{K})$.\nProposition \\ref{prop: liftingrepresentation} implies the following result.\n\n\\begin{Proposition}\n\\label{prop: McKay Quiver lift}\nLet $G$ be a finite and linearly reductive group scheme over $k$.\nLet $\\{\\rho_i\\}_i$ be the set of isomorphism classes of \nsimple representations of $G$.\nLet $\\rho$ be a finite-dimensional representation of $G$.\n\\begin{enumerate}\n    \\item \n    Proposition \\ref{prop: liftingrepresentation} yields\n    sets of representations $\\{\\rho_{\\mathrm{can},i}\\}_i$ and $\\{\\rho_{\\mathrm{abs},i}\\}_i$\n    of $G_{\\mathrm{can}}$ and $G_{\\mathrm{abs}}$, respectively, which are the \n    sets of isomorphism classes of simple representations of $G_{\\mathrm{can}}$\n    and $G_{\\mathrm{abs}}$, respectively.\n    \\item \n    Proposition \\ref{prop: liftingrepresentation} yields\n    representations $\\rho_{\\mathrm{can}}$ and $\\rho_{\\mathrm{abs}}$\n    of $G_{\\mathrm{can}}$ and $G_{\\mathrm{abs}}$, respectively.\n \\end{enumerate}\nThis data leads to a bijection of the three McKay graphs\n     $$\n     \\Gamma(G,\\{\\rho_i\\}_i,\\rho),\\quad\n     \\Gamma(G_{\\mathrm{can}},\\{\\rho_{\\mathrm{can},i}\\}_i,\\rho_{\\mathrm{can}}),\\quad\n     \\Gamma(G_{\\mathrm{abs}},\\{\\rho_{\\mathrm{abs},i}\\}_i,\\rho_{\\mathrm{abs}}).\\qed\n     $$\n\\end{Proposition}\n\n\\subsection{McKay correspondence}\nWe now run through the classical McKay correspondence \\cite{McKay} \nin our setting, where we follow \\cite[Section 8.3]{Kirillov}.\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$\nand let $G$ be a finite and linearly reductive subgroup scheme\nof ${\\mathbf{SL}}_{2,k}$.\n\\begin{enumerate}\n    \\item The K-group $K_k(G)$ carries a symmetric bilinear form \n    $$\n     ([V],[W])_0 \\,:=\\,  \\dim_k\\,\\mathrm{Hom}_G(V,W).\n    $$\n    \\item We consider the closed embedding $G\\to{\\mathbf{SL}}_{2,k}$ as a \n    2-dimensional \n    representation \n    $\\rho:G\\to{\\mathbf{SL}}_{2,k}\\to{\\mathbf{GL}}_{2,k}$\n    and define an operator \n    $$\n     A\\,:\\,K_k(G)\\,\\to\\, K_k(G),\\mbox{\\qquad} [V]\\mapsto [V]\\otimes\\rho\\,.\n    $$\n    Since $\\rho$ is self-dual by Lemma \\ref{lem: selfdual}, it follows\n    that $A$ is symmetric with respect to $(-,-)_0$.\n    \\item Using $\\rho$, we define a symmetric bilinear form on \n    $K_k(G)\\otimes_{\\mathds{Z}}{\\mathds{R}}$ by\n    $$\n     ([V],[W]) \\,:=\\, \\left([V],\\, (2-A)[W]\\right)_0,\n    $$\n    which is positive semi-definite.\n    The class $\\delta\\in K_k(G)$ of the regular representation \n    of $G$ generates the radical of $(-,-)$.\n    \\item Let $\\{\\rho_i\\}$ be the set of isomorphism classes of simple\n    representations of $G$.\n    \\begin{enumerate}\n    \\item\n    The McKay graph $\\widehat{\\Gamma}=\\Gamma(G,\\{\\rho_i\\}_i,\\rho)$\n    is an affine Dynkin diagram\n    of type $\\widehat{A}_n$, $\\widehat{D}_n$ with $n\\geq4$, \n    $\\widehat{E}_6$, $\\widehat{E}_7$\n    or $\\widehat{E}_8$.\n    \\item\n    After removing the vertex corresponding to the trivial representation\n    $\\rho_0$ from $\\widehat{\\Gamma}$, \n    we obtain a finite Dynkin diagram \n    of type $A_n$, $D_n$ with $n\\geq4$, $E_6$, $E_7$\n    or $E_8$, respectively.\n    \\end{enumerate}\n\\end{enumerate}\nIf $G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then these statements are part\nof the classical McKay correspondence, see \\cite{McKay} or \n\\cite[Section 8.3]{Kirillov}.\nIn our setting of finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$,\nthe above claims immediately follow from the classical McKay correspondence \ntogether with the lifting results Proposition \\ref{prop: liftingrepresentation}\nand Proposition \\ref{prop: McKay Quiver lift}.\nWe then obtain the following analog of \nMcKay's theorem \\cite{McKay}.\n\n\\begin{Theorem}\n\\label{thm: main}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n There exists a bijection between\n finite, non-trivial, and linearly reductive subgroup schemes of\n ${\\mathbf{SL}}_{2,k}$ up to conjugation and affine Dynkin diagrams\n of type\n $$\n  \\begin{array}{lcl}\n   \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7, \\widehat{E}_8\n       &\\mbox{\\quad if \\quad }&    p=0\\mbox{\\quad or \\quad}p\\geq7, \\\\\n   \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7 &\\mbox{\\quad if \\quad }& p=5,\\\\\n  \\widehat{A}_n, \\widehat{D}_n &\\mbox{\\quad if \\quad }& p=3, \\mbox{ and} \\\\\n  \\widehat{A}_n &\\mbox{\\quad if \\quad }&p=2.\n  \\end{array}\n $$\n\\end{Theorem}\n\n\\begin{proof}\nIf $p=0$, then this is part of the classical McKay correspondence.\nIf $p>0$, then this follows from the linearly reductive McKay correspondence just discussed together \nwith Hashimoto's classification \\cite[Theorem 3.8]{Hashimoto}\nof finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\n\\end{proof}\n\n\\begin{Remark}\nThe linearly reductive group schemes corresponding to $\\widehat{E}_6$, $\\widehat{E}_7$, and\n$\\widehat{E}_8$ are \\'etale and correspond to finite groups of order prime to $p$.\nThe linearly reductive group scheme corresponding to $\\widehat{A}_n$ (resp.\n$\\widehat{D}_n$) is \\'etale if and only if $p\\nmid (n+1)$ (resp. $p\\nmid (n-2)$).\nWe refer to \\cite[Proposition 4.2]{LiedtkeSatriano} for details.\nThus, even if $p$ is sufficiently large, then one does not obtain a bijection in \nTheorem \\ref{thm: main} with finite groups of order to prime to $p$ only.\n\\end{Remark}\n\n\\begin{Remark}\nSteinberg \\cite{Steinberg} established many properties of the McKay graph\nand the McKay correspondence for finite subgroups of ${\\mathbf{SU}}_2({\\mathds{C}})$\n\\emph{without} using classification lists.\nIt seems plausible to obtain a proof of Theorem \\ref{thm: main} along these \nlines without using lifting results or classification lists.\n\\end{Remark}\n\n\n\\section{Linearly reductive quotient singularities}\n\\label{sec: lrq singularities}\n\nIn this section, we recall \\emph{linearly reductive quotient singularities}\nin the sense of \\cite{LMM} and some general results, including\nthe \\emph{canonical lift} of such a singularity over the ring of Witt vectors.\nIn dimension two, we establish a minimal resolution of singularities using $G$-Hilbert schemes\nas in the work of Ishii, Ito, and Nakamura \\cite{IshiiCrelle, Ishii, IN}.\nWe also show that the canonical lift admits a unique minimal and simultaneous\nresolution of singularities.\nFinally, we discuss \\emph{Klein singularities} and their relation to\nrational double point singularities.\n\n\\subsection{Quotient singularities}\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\nIf $G$ is a finite and linearly reductive group scheme over $k$,\nif $V$ is a finite-dimensional $k$-vector space,\nand if $\\rho:G\\to{\\mathbf{GL}}(V)$ is a linear representation, \nthen we define the \\emph{$\\lambda$-invariant} of $\\rho$ \nas in \\cite[Definition 2.7]{LMM} to be\n$$\n\\lambda(\\rho) \\,:=\\, \\max_{\\{{\\mathrm{id}}\\}\\neq\\boldsymbol{\\mu}_n\\subseteq G} \\dim V^{\\boldsymbol{\\mu}_n},\n$$\nwhere $V^{\\boldsymbol{\\mu}_n}$ denotes the subspace of $\\boldsymbol{\\mu}_n$-invariants.\nAs explained in \\cite[Remark 2.8]{LMM}, the representation\n$\\rho$ is faithful if and only if $\\lambda(\\rho)\\neq\\dim V$.\nMoreover, $\\rho$ contains no pseudo-reflections if and only\nif $\\lambda(\\rho)\\leq\\dim V-2$ and in this case,\nthe representation $\\rho$ is said to be \\emph{small}.\n\n\\begin{Definition}\n\\label{def: very small}\n The representation $\\rho$ is called \n \\emph{very small} if $\\lambda(\\rho)=0$.\n\\end{Definition}\n\nNote that in dimension two the notions of small and very small coincide.\nWe refer to \\cite[Section 3]{LMM} for the classification of  \nfinite and linearly reductive group schemes that admit a very small\nrepresentation.\nBy \\cite[Proposition 6.5]{LMM}, $\\lambda(\\rho)$ is equal to the dimension\nof the non-free locus of the induced $G$-action on the spectrum of the\nformal power series ring $k[[V^\\vee]]=(k[V^\\vee])^\\wedge$.\nFollowing \\cite[Definition 6.4]{LMM} and using the linearisation result\n\\cite[Proposition 6.3]{LMM}, we define:\n\n\\begin{Definition}\n \\label{def: lrq singularity}\n A \\emph{linearly reductive quotient singularity} over $k$\n is an isolated singularity that is analytically isomorphic to\n $\\mathrm{Spec}\\: k[[V^\\vee]]^G$, where $G$ is a finite and linearly reductive group scheme over\n $k$ and where $\\rho:G\\to{\\mathbf{GL}}(V)$ is a very small representation.\n\\end{Definition}\n\n\\begin{Remark}\nWe set $U:=\\mathrm{Spec}\\: k[V^\\vee]$ \nor $U:=\\mathrm{Spec}\\: k[[V^\\vee]]$ and simply write\n$x\\in X=U\/G$ with the assumptions of\nDefinition \\ref{def: lrq singularity} implicitly understood.\n\\end{Remark}\n\nProperties of these singularities have been studied  in \\cite{LMM}:\nfor their local \\'etale fundamental groups, class groups, and\nF-signatures, we refer to \\cite[Section 7]{LMM}.\n\nBy \\cite[Theorem 8.1]{LMM}, a linearly reductive quotient singularity \ndetermines the finite and linearly reductive group scheme $G$\ntogether with the very small representation $\\rho:G\\to{\\mathbf{GL}}(V)$\nuniquely up to isomorphism and conjugacy, respectively.\nIt thus makes sense to refer to $\\rho(G)\\subset{\\mathbf{GL}}(V)$\nas the \\emph{finite and linearly reductive subgroup scheme of ${\\mathbf{GL}}(V)$ \nassociated to the linearly reductive quotient singularity}\n$x\\in X=U\/G$.\nIn particular, the classification of linearly reductive quotient singularities \nin dimension $d$ is ``the same'' as the classification of very small, finite, \nand linearly reductive subgroup schemes of ${\\mathbf{GL}}_{d,k}$ up to conjugacy.\nWe refer to \\cite[Section 3]{LMM} for details and this classification.\n\n\\subsection{Minimal resolution of singularities}\nIf $x\\in X$ is a normal surface singularity, then it \nadmits a unique minimal resolution of singularities\n$$\n  \\pi\\,:\\, Y\\,\\to\\,X.\n$$\nIf $x\\in X$ is moreover a rational singularity, then the \nexceptional locus of $\\pi$ is a union\nof ${\\mathds{P}}^1$'s, whose dual intersection graph $\\Gamma$ contains no cycles.\nThe graph $\\Gamma$ is called the \\emph{type} of $x\\in X$.\nIf the type determines the singularity up to analytic isomorphism,\nthen the singularity is said to be \\emph{taut}.\n\nOver ${\\mathds{C}}$, taut singularities have been classified by \nLaufer \\cite{Laufer}.\nFor example, two-dimensional finite quotient singularities \nover ${\\mathds{C}}$ are taut.\nSince two-dimensional linearly reductive quotient singularities over algebraically\nclosed fields of positive characteristic are \nF-regular (see \\cite[Proposition 7.1]{LMM}), they are taut by Tanaka's\ntheorem \\cite{Tanaka}.\n\n\\begin{Remark}\n\\label{rem: type quotient singularity}\nVery small, finite, and linearly reductive subgroup schemes of ${\\mathbf{GL}}_{2,k}$\nhave been classified in \\cite[Theorem 3.4]{LMM}, extending Brieskorn's classification \n\\cite[Satz 2.9]{Brieskorn} of small subgroups of ${\\mathbf{GL}}_2({\\mathds{C}})$.\nThe types of the associated quotient singularities in terms of this classification are \ngiven by \\cite[Satz 2.11]{Brieskorn}.\n\nFinite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$ are automatically\nvery small and they\nhave been classified by Hashimoto \\cite[Theorem 3.8]{Hashimoto}, extending \nKlein's classification \\cite{Klein} of finite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$\nand we refer to \\cite{Hashimoto, LiedtkeSatriano} for the types\nof the associated quotient singularities\n(see also Theorem \\ref{thm: classification Klein} below).\n\\end{Remark}\n\n\n\\subsection{The Ishii-Ito-Nakamura resolution}\n\\label{subsec: Ishii Ito Nakamura}\nIn \\cite{IN}, Ito and Nakamura showed that if \n$G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then a minimal\nresolution of singularities of ${\\mathds{C}}^2\/G$ is provided\nby the $G$-Hilbert scheme $G\\mathrm{-Hilb}({\\mathds{C}}^2)$.\nIshii \\cite{IshiiCrelle} extended this to quotient singularities\n${\\mathds{C}}^2\/G$ for $G$ a finite and very small subgroup of ${\\mathbf{GL}}_2({\\mathds{C}})$\nand Ishii and Nakamura \\cite{Ishii} extended this to quotient\nsingularities $U\/G$ for $G$ a finite and very small subgroup\nof ${\\mathbf{GL}}_2(k)$ of order prime to $p$.\n\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\n Let $G$ be a very small, finite, and linearly reductive subgroup scheme of \n ${\\mathbf{GL}}_{2,k}$.\n Set $U:={\\mathds{A}}^2_k$ or $\\widehat{{\\mathds{A}}}_k^2$ \n and let $x\\in X:=U\/G$ be the associated linearly reductive\n quotient singularity.\n By \\cite{Blume}, there exists a $G$-Hilbert scheme\n$G\\mbox{-Hilb}(U)$ over $k$ that parametrises zero-dimensional\n$G$-invariant subschemes $Z\\subset U$ \n(so-called \\emph{clusters}), such that the $G$-representation on \n$H^0(Z,{\\cal O}_Z)$ is the regular representation.\nTaking a cluster $Z$ to its $G$-orbit\n(see, for example, \\cite[Remark 3.3]{Blume}) \ninduces a morphism\n$$\n\\pi\\,:\\,Y\\,:=\\,G\\mbox{-Hilb}(U) \\,\\to\\, U\/G\\,=\\,X\n$$\nover $k$.\n\n\\begin{Theorem}\n\\label{thm: IshiiItoNakamura}\n The morphism $\\pi$ is a minimal resolution of singularities.\n\\end{Theorem}\n\n\\begin{proof}\nIf $p=0$, then this is shown in \\cite{Ishii}, extending \\cite{IN}.\nIf $p>0$ and $G$ is of length prime to $p$, then this is shown in \\cite{Ishii}.\nHowever, this proof also works if $G$ is linearly reductive and $p$ divides the length\nof $G$.\n\\end{proof}\n\n\\begin{Remark}\n\\label{rem: f-regular}\nBy \\cite[Theorem 11.5]{LMM}, every two-dimensional F-regular singularity\n$x\\in X$ is a linearly reductive quotient singularity, say $x\\in X=U\/G$ for\nsome finite and linearly reductive subgroup scheme $G$ of ${\\mathbf{GL}}_{2,k}$.\nBy \\cite[Theorem 8.1]{LMM}, $G$ and its embedding into ${\\mathbf{GL}}_{2,k}$ are unique\nup to isomorphism and conjugacy, respectively.\nBy Theorem \\ref{thm: IshiiItoNakamura}, $G\\mbox{-Hilb}(U)\\to U\/G=X\\ni x$\nis the minimal resolution of singularities.\nIn particular, every two-dimensional F-regular singularity can be resolved\nby a suitable $G$-Hilbert scheme.\n\\end{Remark}\n\n\\subsection{Canonical lifts}\n\\label{subsec: canonical lift singularity}\nLet $k$ be an algebraically closed field of characteristic $p>0$,\nlet $W(k)$ be the ring of Witt vectors, let $K$ be its field of fractions, and let\n$\\overline{K}$ be an algebraic closure.\nLet $x\\in X=U\/G$ be a $d$-dimensional \nlinearly reductive singularity, where $U={\\mathds{A}}^d_k$ or $U=\\widehat{{\\mathds{A}}}_k^d$.\nNext, we set ${\\cal U}:={\\mathds{A}}^d_{W(k)}$ or $\\widehat{{\\mathds{A}}}^d_{W(k)}$, respectively.\nIn Section \\ref{subsec: abstract groups}, we recalled the canonical lift \n${\\cal G}_{\\mathrm{can}}$ of $G$ over $W(k)$.\nBy \\cite[Proposition 2.9]{LMM}, there exists a unique lift of the $G$-action\nfrom $U$ to ${\\cal U}$.\nFrom this, we obtain a flat family \n\\begin{equation}\n\\label{eq: lrq lift}\n  \\mathcal{X}_{\\mathrm{can}}\\,:=\\,{\\cal U}\/{\\cal G}_{\\mathrm{can}} \\,\\to\\, \\mathrm{Spec}\\: W(k)\\,.\n\\end{equation}\nof linearly reductive quotient singularities over $W(k)$, whose special fibre\nover $k$ is $x\\in X=U\/G$.\n\n\\begin{Definition}\n The family \\eqref{eq: lrq lift} is called the \\emph{canonical lift} of the linearly reductive \n quotient singularity $x\\in X=U\/G$.\n \\end{Definition}\n  \nBy the Lefschetz principle, the geometric generic fibre of $\\mathcal{X}_{\\mathrm{can}}$\ncan be identified with a finite quotient singularity of the form ${\\mathds{C}}^d\/G_{\\mathrm{abs}}$, \nwhere $G_{\\mathrm{abs}}$ is the abstract group associated to $G$\nand the embedding of $G_{\\mathrm{abs}}\\to{\\mathbf{GL}}_d({\\mathds{C}})$\ncorresponds to the embedding $G\\to{\\mathbf{GL}}_{d,k}$ provided by\nProposition \\ref{prop: liftingrepresentation}.\nThe canonical lift is unique in the following sense:\n\n\\begin{Proposition}\n  We keep the notations and assumptions.\n   Let $W(k)\\subseteq R$ be a finite extension of complete DVRs and let $\\mathcal{X}\\to\\mathrm{Spec}\\: R$ \n   be a lift of $x\\in X$ that is of the form ${\\cal V}\/{\\cal G}\\to\\mathrm{Spec}\\: R$ for some flat lift\n   ${\\cal G}$ of $G$ to $R$ and ${\\cal V}\\cong{\\cal U}\\times_RS$.\n   Then, there exists a finite extension $R\\subseteq S$ of complete DVRs, such that\n   \\begin{enumerate}\n      \\item There exists an isomorphism \n      $$\n        {\\cal G}_{\\mathrm{can}} \\times_{W(k)}S \\,\\cong\\, {\\cal G}\\times_RS\n      $$\n     of group schemes over $S$.\n \\item The very small representation $\\rho:G\\to{\\mathbf{GL}}_{d,k}$ corresponding to the $G$-action on $U$\n lifts uniquely to ${\\cal G}_{\\mathrm{can}}$ and ${\\cal G}$, respectively, and they become conjugate\n over $S$. \n \\item  There is an isomorphism \n  $$\n     \\mathcal{X}_{\\mathrm{can}}\\times_{W(k)}S \\,\\cong\\, \\mathcal{X}\\times_RS\n   $$\n   of deformations of $x\\in X$ over $S$.\n \\end{enumerate}\n\\end{Proposition}\n\n\\begin{proof}\nSince ${\\cal G}$ and ${\\cal G}_{\\mathrm{can},R}$ are lifts of $G$\nto $R$, they become isomorphic after passing to a finite extension $R\\subseteq S$,\nsee also the discussion in Section \\ref{subsec: abstract groups}.\nThere, we also saw that the linear representation $\\rho:G\\to{\\mathbf{GL}}_{d,k}$ \nlifts uniquely to ${\\cal G}$ and ${\\cal G}_{\\mathrm{can}}$, respectively,\nand that they become conjugate over $S$.\nFrom this, we deduce an isomorphism\n$\\mathcal{X}_{\\mathrm{can}}\\times_{W(k)}S\\cong\\mathcal{X}\\times_RS$.\n\\end{proof}\n\n\\begin{Remark}\nIf $d\\geq3$, then \\cite[Corollary 10.10]{LMM} shows that\na $d$-dimensional linearly reductive quotient singularity $x\\in X$ admits\nprecisely one lift over $W(k)$, namely the canonical lift.\nIf $d=2$, then linearly reductive quotient singularities usually have positive dimensional deformation\nspaces and admit many non-isomorphic\nlifts to $W(k)$, see \\cite[Section 12]{LMM}.\n\\end{Remark}\n\n\n\\subsection{Simultaneous resolution of singularities}\n\\label{subsec: simultaneous resolution}\nIf $\\mathcal{X}\\to S$ is a deformation of a rational double point singularity $x\\in X$,\nthen it admits a simultaneous resolution of singularities, but usually only after some\nfinite base-change $S'\\to S$, see \\cite{ArtinBrieskorn}.\nIn the case where $\\mathcal{X}\\to S$ is a family of rational surface singularities, then\nsuch a finite $S'\\to S$ exists if $S$ maps to the so-called \\emph{Artin component}\ninside the versal deformation space of $x\\in X$.\nMoreover, due to the existence of flops, \nthese simultaneous resolutions (if they exist) are not unique in general.\n\nIn the special case where $x\\in X$ is a two-dimensional linearly reductive\nquotient singularity over some algebraically closed field $k$ of\ncharacteristic $p>0$ and where $\\mathcal{X}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ is the canonical\nlift of $x\\in X$, we will now show that there exists a simultaneous and minimal resolution of singularities over $W(k)$\nand that it is unique.\nThis simultaneous resolution can be most elegantly constructed using \nthe Ishii-Ito-Nakamura resolution from Section \\ref{subsec: Ishii Ito Nakamura}\nin families.\n\nLet $k$ be an algebraically closed field of characteristic $p>0$.\nLet $G$ be a finite and linearly reductive group scheme over $k$ \nand let $\\rho:G\\to{\\mathbf{GL}}_{2,k}$ be a very small representation.\nLet ${\\cal G}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$\nbe the canonical lift of $G$ and let \n$\\widetilde{\\rho}:{\\cal G}_{\\mathrm{can}}\\to{\\mathbf{GL}}_{2,W(k)}$ be the lift of $\\rho$ to $W(k)$.\nLet ${\\cal U}:={\\mathds{A}}^2_{W(k)}$ and let \n$$\n \\mathcal{X}_{\\mathrm{can}} \\,:=\\, {\\cal U}\/{\\cal G}_{\\mathrm{can}} \\,\\to\\, \\mathrm{Spec}\\: W(k)\n$$\nbe the canonical lift of $x\\in X$.\nBy \\cite{Blume}, there exists a ${\\cal G}_{\\mathrm{can}}$-Hilbert scheme\n$$\n{\\cal G}_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}) \\,\\to\\,\\mathrm{Spec}\\: W(k)\n$$\nthat parametrises ${\\cal G}_{\\mathrm{can}}$-invariant subschemes $Z\\subset{\\cal U}$ \nthat are finite and flat over $W(k)$ (so-called ${\\cal G}_{\\mathrm{can}}-$\\emph{clusters})\nand such that the ${\\cal G}_{\\mathrm{can}}$-representation on \n$H^0(Z,{\\cal O}_Z)$ is the regular representation.\nTaking such a cluster $Z$ to its ${\\cal G}_{\\mathrm{can}}$-orbit\n(see, for example, \\cite[Remark 3.3]{Blume}) \ninduces a morphism\n\\begin{equation}\n\\label{eq: IIN in families}\n\\widetilde{\\pi}\\,:\\,\n\\mathcal{Y}\\,:=\\,{\\cal G}_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}) \\,\\to\\, {\\cal U}\/{\\cal G}_{\\mathrm{can}}\\,=\\,\\mathcal{X}_{\\mathrm{can}}\n\\end{equation}\nover $W(k)$.\n\n\\begin{Theorem}\n\\label{thm: IN}\n Keeping the assumptions and notations\n $$\n  \\widetilde{\\pi} \\,:\\, \\mathcal{Y} \\,\\to\\, \\mathcal{X}\\,\\to\\,\\mathrm{Spec}\\: W(k)\n $$\n is a simultaneous minimal resolution of singularities of the canonical lift\n $\\mathcal{X}\\to\\mathrm{Spec}\\: W(k)$ of the linearly reductive quotient singularity\n $x\\in X=U\/G$.\n \\begin{enumerate}\n \\item The simultaneous resolution $\\widetilde{\\pi}$ is unique up to isomorphism.\n \\item The exceptional locus of $\\widetilde{\\pi}$ is a union of ${\\mathds{P}}^1_{W(k)}$'s\n meeting transversally\n and we denote by $\\Gamma$ its dual intersection graph.\n \\item The special fibre and the generic\n fibre are linearly reductive quotient singularities of type $\\Gamma$\n and $\\widetilde{\\pi}$ identifies the components of the exceptional loci\n of $\\widetilde{\\pi}_k$ and $\\widetilde{\\pi}_K$.\n \\end{enumerate}\n\\end{Theorem}\n\n\\begin{proof}\nWe recall that we defined $G_{\\mathrm{can}}:={\\cal G}_{\\mathrm{can},K}$, which\nwill simplify the notation in the following.\nThe generic and special fibre of \\eqref{eq: IIN in families}\nover $K$ and $k$ are isomorphic to\n$$\n\\begin{array}{cccccc}\n&G_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}_K) &\\to& {\\cal U}_K\/G_{\\mathrm{can}} &\\to&\\mathrm{Spec}\\: K\\\\\n\\mbox{and \\quad}&\nG\\mbox{-Hilb}(U) &\\to& U\/G &\\to&\\mathrm{Spec}\\: k,\n\\end{array}\n$$\nrespectively, by \\cite[Remark 3.1]{Blume}.\nNow, \n$G_{\\mathrm{can},\\overline{K}}\\mbox{-Hilb}({\\cal U}_{\\overline{K}}) \\to {\\cal U}_{\\overline{K}}\/G_{\\mathrm{can},\\overline{K}}$\nand\n$G\\mbox{-Hilb}(U) \\to U\/G$,\nare minimal resolutions of singularities by Theorem \\ref{thm: IshiiItoNakamura}.\nThus, \n$G_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}_K) \\to {\\cal U}_K\/G_{\\mathrm{can}}$\nis a resolution of singularities, which is minimal since it is minimal over $\\overline{K}$.\nThus, $\\mathcal{Y}\\to\\mathcal{X}\\to\\mathrm{Spec}\\: W(k)$ is a simultaneous minimal resolution \nof singularities.\n\nThe exceptional fibres $\\widetilde{\\pi}_K$ and $\\widetilde{\\pi}_k$ of $\\widetilde{\\pi}$ over $K$ and $k$ \nare unions of ${\\mathds{P}}^1$'s that intersect transversally.\nThe types, that is dual resolution graphs, associated to $\\widetilde{\\pi}_K$ and $\\widetilde{\\pi}_k$\nare the same (see also Remark \\ref{rem: type quotient singularity}) \nand we denote this graph by $\\Gamma$.\nIn particular, the exceptional fibres of $\\widetilde{\\pi}_K$ and $\\widetilde{\\pi}_k$ have\nthe same numbers of irreducible components.\nIn particular, the specialisation maps of N\\'eron-Severi lattices\n$$\n\\mathrm{NS}(\\mathcal{Y}_K) \\,\\leftarrow\\, \\mathrm{NS}(\\mathcal{Y}) \\,\\to\\, \\mathrm{NS}(\\mathcal{Y}_k)\n$$\nare isometries of lattices.\nThese identify the components of the exceptional loci of $\\widetilde{\\pi}_K$\nand $\\widetilde{\\pi}_k$.\nMoreover, given such a component of the exceptional locus of $\\widetilde{\\pi}_k$,\nit is isomorphic to ${\\mathds{P}}^1_k$ and it extends uniquely to a ${\\mathds{P}}^1_{W(k)}$ \nin the exceptional locus of $\\widetilde{\\pi}$.\nThis establishes claims (2) and (3).\n\nIt remains to prove claim (1):\nLet $\\mathcal{Y}'\\to\\mathcal{X}\\to \\mathrm{Spec}\\: W(k)$ be a simultaneous resolution\nof singularities that coincides with the minimal resolution on special and generic fibres,\nrespectively.\nLet $\\alpha:\\mathcal{Y}'_K\\,\\to\\,\\mathcal{Y}_K$ be an isomorphism over $\\mathcal{X}_K$\nand choose a relatively (to $\\mathcal{X}$)\nample invertible sheaf $\\mathcal{L}$ on $\\mathcal{Y}_K$.\nThen, $\\alpha^*\\mathcal{L}$ is relatively ample on $\\mathcal{Y}_K$.\nSince the types of the singularities of $\\mathcal{X}_K$ and $\\mathcal{X}_k$\nare the same, the fibres of\n$\\mathcal{Y}'_K\\to\\mathcal{X}_K$ and $\\mathcal{Y}'_k\\to\\mathcal{Y}_k$\ncontain the same number of exceptional divisors.\nThus, the specialisation\nmap of N\\'eron-Severi lattices $\\mathrm{NS}(\\mathcal{Y}'_K)\\to\\mathrm{NS}(\\mathcal{Y}'_k)$\nis an injective map between two lattices of the same rank and the same discriminant,\nwhence an isometry.\nSimilarly, the specialisation\nmap of N\\'eron-Severi lattices $\\mathrm{NS}(\\mathcal{Y}_K)\\to\\mathrm{NS}(\\mathcal{Y}_k)$\nis an isometry.\nIn particular, we can identify the components of the exceptional loci \nof $\\mathcal{Y}_K$ and $\\mathcal{Y}_k$.\n(If the types of $\\mathcal{X}_k$ and $\\mathcal{X}_K$ differ, then this specialisation map is \nusually only injective, there are usually more $(-2)$-curves in the special fibre and this is also the\nplace where the difficulties with flops begin.)\nIn particular, $\\mathcal{L}$ and $\\alpha^*\\mathcal{L}$ extend to relative ample\ninvertible sheaves on $\\mathcal{Y}$ and $\\mathcal{Y}'$, respectively.\nBy \\cite[Theorem 5.14]{Kovacs}, the isomorphism $\\alpha$\nextends to an isomorphism $\\mathcal{Y}'\\to\\mathcal{Y}$ over $\\mathcal{X}$ and \nthe claimed uniqueness follows.\n\\end{proof}\n\n\n\\subsection{Rational double point singularities and Klein singularities}\nWe specialise Definition \\ref{def: lrq singularity} to the following case.\n\n\\begin{Definition}\n\\label{def: Klein singularity}\n A \\emph{Klein singularity} is a linearly reductive quotient singularity\n as in Definition \\ref{def: lrq singularity} with $\\dim V=2$\n and $\\det\\rho=1$, that is, $\\rho$ is a homomorphism of $G$ \n to ${\\mathbf{SL}}_{2,k}$.\n\\end{Definition}\n\nIn particular, a Klein singularity is a two-dimensional\nand linearly reductive quotient singularity, it is a rational surface singularity,\nand since $\\det(\\rho)=1$, it is Gorenstein.\nQuite generally, rational and Gorenstein surface singularities\nare precisely the \\emph{rational double point singularities} \n\\cite{ArtinRational}.\nWe refer to \\cite{Durfee} or \\cite{Seminaire} for more background\non surface singularities and rational double point singularities.\n\nIf $x\\in X$ is a rational double point, then its type $\\Gamma$ is a \nsimply-laced finite Dynkin graph.\nIn characteristic zero, every rational double point singularity is taut.\nIn positive characteristic, a Klein singularity is F-regular and thus, taut.\nIn the case of rational double points,\nthis also follows from Artin's explicit classification \\cite{ArtinRDP}.\nOn the other hand, rational double point singularities that are not F-regular \nneed not be taut.\nWe have the following relation between rational double point singularities\nand Klein singularities in positive characteristic, see \n\\cite[Theorem 11.2]{LMM}.\n\n\\begin{Theorem}\n\\label{thm: classification Klein}\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\nLet $x\\in X$ be a normal surface singularity over $k$.\n\\begin{enumerate}\n\\item If $p=0$ or $p\\geq7$, then $x\\in X$ is a Klein singularity if and only if $x\\in X$ is a rational double point singularity.\n\\item If $p>0$, then $x\\in X$ is a Klein singularity if and only if $x\\in X$ is a rational double point singularity and F-regular. \nThe finite Dynkin graphs of these singularities are of type\n$$\n  \\begin{array}{lcl}\n   A_n, D_n, E_6, E_7, E_8\n       &\\mbox{\\quad if \\quad }&    p\\geq7, \\\\\n   A_n, D_n, E_6, E_7 &\\mbox{\\quad if \\quad }& p=5,\\\\\n   A_n, D_n &\\mbox{\\quad if \\quad }& p=3, \\\\\n   A_n &\\mbox{\\quad if \\quad }&p=2.\n  \\end{array}\n $$\n\\end{enumerate}\n\\end{Theorem}\n\nA Klein singularity $x\\in X=U\/G$ is a linearly reductive quotient singularity and thus,\ndetermines $G$ and $\\rho:G\\to{\\mathbf{SL}}_{2,k}$ up to isomorphism and conjugacy, respectively.\nThus, the classification of Klein singularities boils down to the classification\nof finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$ and we refer to\nRemark \\ref{rem: type quotient singularity} for this classification.\n\n\\begin{Remark}\nTheorem \\ref{thm: main} and Theorem \\ref{thm: classification Klein} rely on the\nclassification of finite and linearly reductive subgroup schemes\nof ${\\mathbf{SL}}_{2,k}$.\nIt is therefore no surprise that the classification lists coincide.\n\\end{Remark}\n\n\\section{Hecke correspondences}\n\\label{sec: Hecke}\n\nLet $x\\in X=U\/G$ be a Klein singularity over an algebraically closed field $k$ \nof characteristic $p>0$.\nIn Section \\ref{subsec: abstract groups}, we discussed the\ncanonical lift $\\mathcal{X}\\to \\mathrm{Spec}\\: W(k)$ of $x\\in X$ and \nin Section \\ref{subsec: simultaneous resolution} we established\na simultaneous minimal resolution of singularities of the canonical lift.\nMore precisely, this simultaneous resolution was constructed\nusing $G$-Hilbert schemes extending previous work of Ishii, Ito, and Nakamura\n\\cite{IshiiCrelle, Ishii, IN}.\n\nIn this section, we refine this resolution of singularities as in\nthe work of Ito and Nakamura \\cite{IN} and Nakajima \\cite{Nakajima, NakajimaLectures}:\nwe eventually obtain a bijection between the components\nof the minimal resolution of singularities of the Klein singularity $x\\in X=U\/G$\nand the simple and non-trivial representations of $G$ using special\n \\emph{Hecke correspondences}.\n\n\\subsection{The Ito-Nakamura resolution revisited}\nWe  first slightly extend the setup of Section \\ref{subsec: Ishii Ito Nakamura}.\nLet $k$ be an algebraically closed field of characteristic $p>0$ and\nlet $x\\in X=U\/G$ be a Klein singularity as in Definition \\ref{def: Klein singularity},\nthat is, we have $U={\\mathds{A}}^2_k$ and $G$ a very small, finite, and linearly reductive\nsubgroup scheme of ${\\mathbf{SL}}_{2,k}$.\n\nLet $\\{\\rho_i\\}_{i\\in I}$ be the set of isomorphism classes of simple representations of $G$.\nGiven a finite-dimensional representation $\\rho$ of $G$, there exist\nnon-negative integers $\\nu_i\\in{\\mathds{Z}}_+$, such that $\\rho$ is isomorphic\nto $\\bigoplus_i\\rho_i^{\\oplus \\nu_i}$ and we combine these into a multi-index\n$\\nu=(\\{\\nu_i\\})\\in{\\mathds{Z}}_{\\geq0}^I$.\nWe set $\\dim(\\nu):=\\sum_i\\nu_i\\dim\\rho_i$, which is the dimension\nof the representation associated to $\\nu$.\n\nFor any integer $n\\geq1$, the $G$-action on $U$ induces\na $G$-action on the Hilbert scheme $\\mathrm{Hilb}^n(U)$, which \nparametrises zero-dimensional subschemes of length $n$ of $U$. \nWe consider the fixed point scheme\n$$\n\\mathrm{Hilb}^{n,G}(U) \\,:=\\,\n\\left( \\mathrm{Hilb}^n(U) \\right)^G,\n$$\nthat is, the largest subscheme of $\\mathrm{Hilb}^n(U)$\non which $G$ acts trivially.\nIt parametrises $G$-invariant and zero-dimensional subschemes of\nlength $n$ of $U$.\nGiven $\\nu=(\\{\\nu_i\\})_i\\in{\\mathds{Z}}_{\\geq0}^I$ with $\\dim(\\nu)=n$, we define\n$$\n   H^\\nu\\,=\\,\\left\\{\n   Z\\,\\in\\,\\mathrm{Hilb}^{n,G}(U)\\,|\\,\n   H^0(Z,{\\cal O}_Z)\\,\\cong\\,\n   \\bigoplus \\rho_i^{\\oplus \\nu_i}\\mbox{ as $G$-representation}\n   \\right\\},\n$$\nwhich defines a subscheme of $\\mathrm{Hilb}^{n,G}(U)$.\nAdapting \\cite[Lemma 12.4]{Kirillov}, which follows \\cite[Section 9]{IN},\nto our situation, we have the following.\n\n\\begin{Lemma}\nWe keep the assumptions and notations and let \n$n\\geq1$ be an integer.\n\\begin{enumerate}\n    \\item The scheme $\\mathrm{Hilb}^{n,G}(U)$ is smooth over $k$.\n    \\item We have a decomposition\n    $$\n    \\mathrm{Hilb}^{n,G}(U) \\,=\\, \\bigsqcup_{\\nu}\\, H^\\nu,\n    $$\n    where the disjoint union runs over all multi-indices $\\nu\\in{\\mathds{Z}}_{\\geq0}^I$ \n   of dimension $n$.\n    Each $H^\\nu$ is a smooth subscheme of $\\mathrm{Hilb}^{n,G}(U)$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nSince $U$ is two-dimensional, $\\textrm{Hilb}^n(U)$ is smooth over $k$ by\n\\cite[Theorem 2.4]{Fogarty}.\nTo show Claim (1), it remains to show that the fixed point scheme for the\n$G$-action is also smooth,\nwhich follows from Lemma \\ref{lem: fixed scheme} below.\nThen, Claim (2) is obvious.\n\\end{proof}\n\n\\begin{Lemma}\n\\label{lem: fixed scheme}\nLet $k$ be an algebraically closed field,\nlet $X$ be a scheme that is smooth over $k$,\nlet $G$ be a finite and linearly reductive group scheme over $k$,\nand assume that $G$ acts on $X$.\nThen, the fixed point scheme $X^G\\subseteq X$ is smooth over $k$.\n\\end{Lemma}\n\n\\begin{proof}\nLet $x\\in X^G$.\nPassing to the completion of the local ring ${\\cal O}_{X,x}$,\nusing that $X$ is smooth over $k$, and passing to\ncoordinates such that the $G$-action is linear\n(this is always possible by \\cite[proof of Corollary 1.8]{Satriano}),\nwe may assume that \n$$\n\\widehat{{\\cal O}}_{X,x} \\,\\cong\\, k[[u_1,...,u_d]]\n$$\nand that the $G$-action is linear.\nIn this description it is easy to see that the $G$-invariant subscheme\nof $\\mathrm{Spec}\\:\\widehat{{\\cal O}}_{X,x}$ is smooth\n(see also the proof of \\cite[Lemma 9.1]{IN}), which implies that\n$X^G$ is smooth near $x$.\n\\end{proof}\n\nAn important special case is the \\emph{regular representation} of $G$,\nwhere we have $\\nu_i=\\dim\\rho_i$ for all $i$ and in  this case, \nwe will write $\\delta$ for the corresponding multi-index.\nThe dimension of $\\delta$ is equal to the length of $G$.\nIn this case, we have \n$$\n   \\pi\\,:\\, H^\\delta \\,=\\, G\\mbox{-Hilb}(U) \\,\\to\\, U\/G \\,=\\,X,\n$$\nwhich we already studied in Section \\ref{subsec: Ishii Ito Nakamura}.\nThere, we saw that $\\pi$ is a minimal resolution\nof singularities of the Klein singularity $x\\in X=U\/G$.\n\n\n\\subsection{Hecke correspondences}\nWe keep the assumptions and notations of the previous section.\nWe let $n$ be the length of $G$ and let $\\delta$ be the multi-index\ncorresponding to the regular representation.\nFor $i\\in I$, we set $\\alpha_i:=(0,...,0,1,0...)$ with the non-zero entry in \nthe $i$.th position.\nWe define\n$$\n B_i \\,:=\\, \\left\\{ J_1 \\,\\in\\, H^{\\delta-\\alpha_i},\\quad J_2\\,\\in\\, H^\\delta\\quad |\\quad J_2\\subseteq J_1 \\right\\}\n \\,\\subseteq\\, H^{\\delta-\\alpha_i}\\times H^\\delta,\n$$\nwhich is a \\emph{Hecke correspondence},\nsee \\cite{Nakajima}, \\cite[Section 6.1]{NakajimaLectures} or \\cite[Section 12.4]{Kirillov}.\nWe let\n$$\n   E_i \\,:=\\, \\mathrm{pr}_2(B_i)\\,\\subseteq\\, H^\\delta \\,=\\, G\\mathrm{-Hilb}(U)\n$$\nbe the image under the projection $\\mathrm{pr}_2$ onto the second factor.\n\nIn characteristic zero, the following result is due to Nakajima \\cite{Nakajima}\nand independently to Ito and Nakamura \\cite{IN}, see also \n\\cite[Section 12.4]{Kirillov}.\n\n\\begin{Theorem}\n\\label{thm: bijection simple and exceptional}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n Let $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$,\n let $x\\in X:=U\/G$ be the associated Klein singularity, and \n let $\\pi:H^\\delta\\to X$ be the Ito-Nakamura resolution.\n \\begin{enumerate}\n \\item The assignment $i\\mapsto E_i$ defines a bijection\n between the set $\\{\\rho_i\\}_i$ of isomorphism classes of \n non-trivial simple representations  of $G$ \n and the set of irreducible components of the exceptional divisor $\\mathrm{Exc}(\\pi)$ \n of $\\pi$.\n \\item If $i\\neq j$, then  \n  $$\n    B_i\\cdot B_j \\,=\\, A_{ij}\n  $$\n  where the numbering of the $\\{\\rho_i\\}_i$ is as in\n  \\cite[Section 10]{IN}.\n  \\end{enumerate}\nWe thus obtain a bijection between the McKay graph\n of $G$ and the dual resolution graph of $\\pi$.\n\\end{Theorem}\n\n\\begin{proof}\nThe case-by-case analysis of \\cite[Section 12]{IN} carries directly over\nfrom finite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$ to finite and linearly reductive subgroup\nschemes of ${\\mathbf{SL}}_{2,k}$.\n\\end{proof} \n\nHere is a second proof of this theorem that uses our lifting results:\n\n\\begin{proof}\nIf $p=0$, then the Lefschetz principle allows to reduce to $k={\\mathds{C}}$,\nwhere the theorem is due to Ito, Nakajima, and Nakamura \\cite{IN, Nakajima}.\n\nWe now assume $p>0$.\nLet  $\\mathcal{X}_{\\mathrm{can}}={\\cal U}\/{\\cal G}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$\nbe the canonical lift of $x\\in X=U\/G$\n(see Section \\ref{subsec: canonical lift singularity})\nand let $\\widetilde{\\pi}:\\mathcal{Y}\\to\\mathcal{X}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ \nbe the simultaneous resolution of singularities\n(see Section \\ref{subsec: simultaneous resolution}).\nMoreover, we have the abstract group $G_{\\mathrm{abs}}$\nand the canonical lift $G_{\\mathrm{can}}$\nassociated to $G$ (see Section \\ref{subsec: abstract groups})\nand we have a bijection of simple representations of \n$G_{\\mathrm{abs}}$, $G_{\\mathrm{can}}$, and $G$\nby Proposition \\ref{prop: liftingrepresentation}.\n\nThe generic fibre $\\mathcal{X}_K$ is isomorphic\nto ${\\cal U}_K\/G_{\\mathrm{can}}$ and using the Lefschetz principle,\nthe geometric generic fibre can be identified with ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$.\nOver ${\\mathds{C}}$, we have Theorem \\ref{thm: bijection simple and exceptional}\nfor $G_{\\mathrm{abs}}$, ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$, and its minimal resolution of singularities\nby \\cite{IN, Nakajima}.\n\nBy the Lefschetz principle, we have it \nfor $G_{\\mathrm{can}}$, \n$\\mathcal{X}_K={\\cal U}_{K}\/G_{\\mathrm{can}}$ \nand $\\widetilde{\\pi}_K$.\nUsing the comparison results \nProposition \\ref{prop: liftingrepresentation} and Theorem \\ref{thm: IN},\nwe obtain it for\n$\\mathcal{X}_k={\\cal U}_k\/{\\cal G}_{\\mathrm{can},k}$ \nand $\\widetilde{\\pi}_k$, that is,\nfor $X=U\/G$, $G$ and $\\pi$.\n\\end{proof}\n\n\\subsection{Local McKay correspondence}\n\\label{subsec: local McKay}\nWe now extend work of Ishii and Nakamura \\cite{IshiiCrelle, Ishii} to our linearly \nreductive setting.\nLet $G$ be a very small, finite, and linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$\nand let $x\\in X=U\/G$ be the associated two-dimensional linearly reductive quotient\nsingularity.\nUntil the end of this section, we do not require it to be a Klein singularity.\nNote that in dimension two, linearly reductive quotient singularities\nare the same as F-regular singularities, see Remark \\ref{rem: f-regular}.\n\nLet $\\{\\rho_i\\}_i$ be the set of simple representations of $G$.\nLet $\\rho_0$ be the trivial representation and we choose our numbering of the $\\rho_i$'s \nto be the one of \\cite[Theorem 3.6]{Ishii}.\nLet $\\pi:Y\\to X$ be its minimal resolution\nof singularities and let $\\{E_i\\}$ be the irreducible components of the\nexceptional divisor $\\textrm{Exc}(\\pi)$ of $\\pi$.\nThen, there is exists a connection between the $\\{\\rho_i\\}$ and the\nexceptional divisors of $\\pi$ as follows - this is yet another version of the \nMcKay correspondence.\n\n\\begin{Theorem}\n\\label{thm: exc pi}\n Keeping assumptions and notations, let ${\\mathfrak m}\\subset{\\cal O}_U$ be the maximal\n ideal corresponding to the origin, let $y\\in Y$ be a closed point, let $Z_y$ be\n the $G$-invariant cluster of $U$ corresponding to $y$, and let ${\\cal I}_{Z_y}\\subset{\\cal O}_U$\n be its ideal sheaf.\n Then, the $G$-representation on ${\\cal I}_{Z_y}\/{\\mathfrak m}{\\cal I}_{Z_y}$ is given by\n $$\n    \\left\\{\n    \\begin{array}{ll}\n    \\rho_i\\,\\oplus\\,\\rho_0 &\\mbox{ \\quad if }y\\in E_i\\,\\backslash\\,\\bigcup_{j\\neq i}E_j\\\\\n    \\rho_i\\,\\oplus\\,\\rho_j\\,\\oplus\\,\\rho_0  & \\mbox{ \\quad if }y\\in E_i\\cap E_j\\mbox{ \\quad with \\quad }i\\neq j.\n    \\end{array}\n    \\right. \n $$\n\\end{Theorem}\n\n\\begin{proof}\nFor $k={\\mathds{C}}$, this is \\cite[Theorem 7.1]{IshiiCrelle}.\nFor $k$ algebraically closed of arbitrary characteristic and $G$ is a very small and finite\nsubgroup of ${\\mathbf{GL}}_{2}(k)$, this is \\cite[Theorem 3.6]{Ishii}.\nHowever, these proofs also work if $G$ is a very small and finite and linearly reductive\nsubgroup scheme of ${\\mathbf{GL}}_{2,k}$.\n\\end{proof}\n\n\\begin{Remark}\nIn \\cite{Ishii}, the $G$-quiver structure of $G\\mathrm{-Hilb}(U)$\nwas studied in the case where $G$ is a very small and finite subgroup of\n${\\mathbf{GL}}_{2}(k)$. \nWe leave the extension of these results to the linearly reductive case to the reader.\n\\end{Remark}\n\n\n\\subsection{Reflexive sheaves on the minimal resolution}\nWe end this section by shortly digressing on work of Wunram \\cite{Wunram},\nIshii and Nakamura \\cite{Ishii}, which generalises work of\nArtin and Verdier \\cite{AV}.\nWe keep the assumptions and notations from Section \\ref{subsec: local McKay}.\nLet $F\\subset Y$ be the \\emph{fundamental divisor} of $\\pi$,\nsee \\cite{ArtinRational}.\n\n\\begin{Theorem}\n\\label{thm: reflexive}\nKeeping assumptions and notations, there exists a bijection between\n\\begin{enumerate}\n\\item the set of irreducible components $\\{E_i\\}$ of $\\mathrm{Exc}(\\pi)$ and\n\\item the set of non-trivial indecomposable full ${\\cal O}_Y$-modules $\\{{\\cal M}_i\\}$,\nspecial in the sense that $H^1(Y,{\\cal M}_i^{\\vee})=0$\n\\end{enumerate}\nThis correspondence ${\\cal M}_i\\mapsto E_i$ is defined by\n$$\n c_1({\\cal M}_i)\\cdot E_j \\,=\\, \\delta_{i,j}\\,.\n$$\nThe rank of ${\\cal M}_i$ is equal to $c_1({\\cal M}_i)\\cdot F$, the \nmultiplicity of $E_i$ in $F$.\\qed\n\\end{Theorem}\n\n\\begin{proof}\nFor $k={\\mathds{C}}$, this is \\cite{Wunram}.\nFor $k$ algebraically closed of arbitrary characteristic and $G$ a very small and finite\nsubgroup of ${\\mathbf{GL}}_{2,k}$, this is \\cite[Theorem 2.8]{Ishii}.\nHowever, these proofs also work if $G$ is a very small, finite, and linearly reductive\nsubgroup scheme of ${\\mathbf{GL}}_{2,k}$.\n\\end{proof}\n\nHere, a \\emph{full} ${\\cal O}_Y$-module is as defined in \\cite[Definition 2.4]{Ishii}.\nBy  \\cite[Corollary 2.5]{Ishii}, the assignment ${\\cal M}\\mapsto\\pi_*{\\cal M}$\nsets up a bijection between the set of (indecomposable) \nfull ${\\cal O}_Y$-modules with the set of (indecomposable) \nreflexive ${\\cal O}_X$-modules.\nIn particular, the above theorem yields a bijection between the set of irreducible\ncomponents of $\\mathrm{Exc}(\\pi)$ and  the set of reflexive\n${\\cal O}_X$-modules.\n\n\\begin{Remark}\n\\label{rem: f-regular Artin Verdier}\nThe group scheme $G$ plays no r\\^{o}le in this theorem and the discussion thereafter.\nProbably, these results should be viewed as results \non two-dimensional F-regular \nsingularities (see Remark \\ref{rem: f-regular}).\n\\end{Remark}\n\n\n\\section{Conjugacy classes and Ito-Reid correspondence}\n\\label{sec: ItoReid}\n\nFor a finite group, the number of isomorphism classes of\ncomplex semi-simple representations is equal to the number \nof conjugacy classes.\nThus, one can \\emph{choose} a bijection between these two sets.\nIn particular, for a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$\none can \\emph{choose} a bijection between the vertices of its \nMcKay graph and the conjugacy classes of $G$.\nIn \\cite{ItoReid}, Ito and Reid gave a \\emph{canonical}\nbijection between these two sets.\nIn this section, we extend this \\emph{Ito-Reid correspondence}\nto finite and linearly reductive subgroup schemes of\n${\\mathbf{SL}}_{2,k}$.\nThe main difficulty is to define a suitable notion of \\emph{conjugacy classes}\nfor finite and linearly reductive group schemes.\n\n\n\\subsection{Conjugacy classes}\n\\label{subsec: conjugacy classes}\n\nGiven a finite group $G_{\\mathrm{abs}}$, a\nrepresentation $\\rho:G_{\\mathrm{abs}}\\to{\\mathbf{GL}}_n({\\mathds{C}})$,\nand an element $g\\in G_{\\mathrm{abs}}$, we have the \ntrace $\\mathrm{tr}(\\rho(g))\\in{\\mathds{C}}$.\nThis pairing $(\\rho,g)\\mapsto \\mathrm{tr}(\\rho(g))$\ninduces a non-degenerate pairing between\nisomorphism classes of \nsemi-simple representations of $G_{\\mathrm{abs}}$\nand conjugacy classes.\nThus, conjugacy classes can be thought of as being\n``dual'' to semi-simple representations, which\ncan be used to give an unusual definition of\nconjugacy classes.\nThis definition generalises to \nfinite and linearly reductive group schemes.\nMore precisely, we make the following definition,\nwhich is inspired by a result\nof Serre \\cite[Section 11.4]{Serre} and which we discuss\nin detail in Appendix \\ref{subsec: second approach}.\n\n\\begin{Definition}\n  \\label{def: conjugacy class}\n Let $G$ be a finite and linearly reductive group scheme over an\n algebraically closed field $k$ of characteristic $p\\geq0$.\n Then, the \\emph{set of conjugacy classes} is defined to be\n $\\mathrm{Spec}\\: {\\mathds{C}}\\otimes K_k(G)$.\n\\end{Definition} \n\nWe discuss several approaches to conjugacy classes for finite group schemes \nin Appendix \\ref{app: conjugacy class} -\nthere, we hope to convince the reader that Definition \\ref{def: conjugacy class}\nis the best for the purposes of this article.\nFor example, by Proposition \\ref{prop: conjugacy lift and bijection}\nit is compatible with canonical lifts and there is a natural\nbijection with the set of conjugacy classes of the abstract group \n$G_{\\mathrm{abs}}$ associated to $G$.\nConcerning the choice of field ${\\mathds{C}}$ in Definition \\ref{def: conjugacy class},\nwe refer to Remarks \\ref{rem: conjugacy classes}.\n\n\\subsection{An explicit bijection}\nFor a finite and linearly reductive subgroup scheme $G$ of ${\\mathbf{SL}}_{2,k}$,\nwe have an associated embedding of the finite group\n$G_{\\rm abs}$ into ${\\mathbf{SL}}_2({\\mathds{C}})$.\nLet $\\{\\rho_i\\}_i$ be the set of isomorphism classes of\nsemi-simple representations of $G$ and \nlet $\\{\\rho_{\\mathrm{abs},i}\\}_i$ be the corresponding\nset for $G_{\\mathrm{abs}}$ obtained by Proposition \\ref{prop: liftingrepresentation}.\nThe associated McKay graphs\n$\\Gamma(G,\\rho,\\{\\rho_i\\}_i)$ and \n$\\Gamma(G_{\\mathrm{abs}},\\rho_{\\mathrm{abs}}, \\{\\rho_{\\mathrm{abs},i}\\}_i)$\ncoincide by Proposition \\ref{prop: McKay Quiver lift}\nand we denote both by $\\widehat{\\Gamma}$.\nNext, the group $G_{\\mathrm{abs}}$ admits a presentation of the form\n$$\n G_{\\rm abs} \\,=\\, \\left\\langle A,B,C \\,|\\, A^r=B^s=C^t=ABC \\right\\rangle\n$$\nfor suitable non-negative integers $r,s,t$.\nThe non-trivial conjugacy classes of $G_{\\rm abs}$ can be uniquely represented\nby the following elements\n$$\nABC,\\quad A^i \\mbox{ with }1\\leq i\\leq r-1,\n \\quad B^i \\mbox{ with }1\\leq i\\leq s-1,\n \\quad C^i \\mbox{ with }1\\leq i\\leq t-1.\n$$\nThis allows to give an explicit bijection between the \nconjugacy classes of $G_{\\mathrm{abs}}$ and the vertices of \n$\\widehat{\\Gamma}$.\nWe refer to \\cite[Section 2]{Kirillov} for details.\nUsing Proposition \\ref{prop: conjugacy lift and bijection}, we obtain \nan explicit bijection between the conjugacy classes of $G$\nand the vertices of $\\widehat{\\Gamma}$.\n\n\n\\subsection{The Ito-Reid correspondence}\n\\label{subsec: Ito-Reid}\nIf $G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then Ito\nand Reid \\cite{ItoReid} (see also \\cite{Reid}) \nconstructed a \\emph{canonical} bijection.\nLet us run through \\cite[Section 2]{Reid} and show that this\ncan be carried over to our linearly reductive setting:\nlet $k$ be an algebraically closed field of characteristic $p>0$\nand let $G$ be a finite and linearly reductive subgroup scheme\nof ${\\mathbf{SL}}_{2,k}$.\n\\begin{enumerate}\n\\item Associated to $G$, we have the abstract group $G_{\\mathrm{abs}}$.\nBy Proposition \\ref{prop: liftingrepresentation}, an embedding\n$G\\to{\\mathbf{SL}}_{n,k}$ yields an embedding $G_{\\mathrm{abs}}\\to{\\mathbf{SL}}_n({\\mathds{C}})$.\nBy Proposition \\ref{prop: conjugacy lift and bijection}, we can identify\nconjugacy classes of $G$ (in the sense of Definition \\ref{def: conjugacy class}) \nand conjugacy classes of $G_{\\mathrm{abs}}$, which allows\nus to define the \\emph{age} of a conjugacy class of $G$ \nvia the corresponding notion for $G_{\\mathrm{abs}}$ as, for example,\nin \\cite[Section 2]{Reid}.\nA conjugacy class of age 1 is called \\emph{junior} and if $n=2$,\nthen all conjugacy classes are junior.\n\\end{enumerate}\n\n\\begin{Remark}\nUsing the ``toric mechanism'' mentioned in \\cite[Section 2]{Reid},\none can define the age directly and without referring to lifts,\nbut we will not pursue this here.\n\\end{Remark}\n\n\\begin{enumerate}\n\\setcounter{enumi}{1}\n\\item Let $x\\in X:=U\/G$ be the associated Klein singularity. \nWe have the canonical lift $\\mathcal{X}_{\\mathrm{can}}\\to \\mathrm{Spec}\\: W(k)$ and the \nsimultaneous resolution of singularities \n$\\widetilde{\\pi}:\\mathcal{Y}\\to\\mathcal{X}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ \nby Theorem \\ref{thm: IN}.\nPassing to geometric generic fibres and using the Lefschetz principle,\nwe obtain the minimal resolution of singularities of ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$.\nThe special fibre of $\\widetilde{\\pi}$, the geometric generic fibre\nof $\\widetilde{\\pi}$ and the minimal resolution of ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$\nare \\emph{crepant} in the sense of \\cite{YPG} and we can identify the\nexceptional divisors of these three resolutions with each other,\nsee Theorem \\ref{thm: IN}.\nThis way, we obtain an Ito-Reid correspondence between\njunior conjugacy classes of $G$ and crepant divisors of the resolution\n\\cite[Theorem 2.1]{Reid}.\n\\end{enumerate}\n\n\\begin{Remark}\nIt seems reasonable that one can extend this correspondence\nto finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{n,k}$ with $n\\geq3$,\nbut we will not pursue this here.\n\\end{Remark}\n\n\n\\section{Derived categories}\n\\label{sec: outlook}\n\nLet $G$ be a very small, finite, and linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$,\nlet $x\\in X:=U\/G$ be the associated  linearly reductive\nquotient singularity, and let $\\pi:Y\\to X$ be its minimal resolution of singularities.\nGonzalez-Sprinberg and Verdier \\cite{GSV} gave an interpretation of the McKay correspondence\nas an isomorphism between the K-groups $K^G(U)$ and $K(Y)$.\nKapranov and Vasserot \\cite{KV} and Bridgeland, King, and Reid \\cite{Bridgeland}  generalised this\nto an equivalence of derived categories ${\\mathcal{D}}^G(U)$ and ${\\mathcal{D}}(Y)$. \nIn this section, we extend this to our setting, following\nIshii, Ito, Nakamura, and Ueda \\cite{IshiiCrelle, Ishii, IshiiUeda}.\n\nWe have a commutative diagram\n $$\n \\xymatrix{\n  U\\times_k Y \\ar[r]^{\\pi_U}\\ar[d]^{\\pi_Y} &U\\ar[d]^\\varpi\\\\\n   Y \\ar[r]^\\pi & X.\n  }\n $$\nBy Theorem \\ref{thm: IshiiItoNakamura}, the minimal resolution $\\pi$ can be constructed by\nthe Ishii-Ito-Nakamura resolution\n$$\n G\\mathrm{-Hilb}(U) \\,\\to\\, U\/G\n$$\nWe let ${\\mathcal{Z}}$ be the universal cluster over $G\\mathrm{-Hilb}(U)$,\nwe identify $G\\mathrm{-Hilb}(U)$ with $Y$\nand then, we have a commutative diagram\n  $$\n \\xymatrix{\n  {\\mathcal{Z}} \\ar[r]^{q}\\ar[d]^{p} &U\\ar[d]^\\varpi\\\\\n   Y \\ar[r]^\\pi & X.\n  }\n $$\nLet ${\\mathcal{D}}(Y)$ be the derived category of coherent\nsheaves on $Y$.\nLet ${\\mathcal{D}}^G(U)$ be the derived category of\n$G$-equivariant coherent sheaves on $U$.\nFollowing \\cite{IshiiCrelle} and \\cite{Ishii}, we define two functors\n$$\n\\begin{array}{ccccc}\n \\Psi &:& {\\mathcal{D}}^G(U) &\\to& {\\mathcal{D}}(Y)\\\\\n \\Phi &:& {\\mathcal{D}}(Y) &\\to& {\\mathcal{D}}^G(U)\n\\end{array}\n$$\nby\n$$\n\\Psi(-) \\,:=\\, [p_*\\, \\mathbf{L}q^*\\, (-)]^G\n$$\nand\n$$\n\\Phi(-) \\,:=\\, \\mathbf{R}\\pi_{U,*} \\left(\n{\\cal O}_{{\\mathcal{Z}}}^\\vee \\,\\otimes^{\\mathbf{L}}\\,\\pi_Y^*(-\\otimes\\rho_0)\\,\\otimes^{\\mathbf{L}}\\, \\pi_U^*K_{U}\\right) [2],\n$$\nwhere ${\\cal O}_{{\\mathcal{Z}}}^\\vee:=\\mathbf{R}Hom({\\cal O}_{{\\mathcal{Z}}},{\\cal O}_{Y\\times U})$ denotes the dual of ${\\cal O}_{{\\mathcal{Z}}}$,\nwhere $-\\otimes\\rho_0:{\\mathcal{D}}(Y)\\to{\\mathcal{D}}^G(Y)$ denotes the functor that attaches the trivial $G$-action, and where\n$K_U$ denotes the canonical sheaf of $U$.\nWe refer to \\cite[Section 3.1]{Ishii} for details, conventions, and notations.\n\n\\begin{Theorem}\n\\label{thm: derived}\n Keeping assumptions and notations, \n $\\Phi$ is fully faithful and $\\Psi$ is a left adjoint of $\\Phi$.\n\\end{Theorem}\n\n\\begin{proof}\nFor $G$ is a very small subgroup of ${\\mathbf{GL}}_2({\\mathds{C}})$, this is \\cite[Section 6]{IshiiCrelle} and \n\\cite[Proposition 1.1 and Lemma 2.9]{IshiiUeda}.\nFor $k$ algebraically closed of arbitrary characteristic and $G$ is a very small and finite\nsubgroup of ${\\mathbf{GL}}_{2}(k)$, this is \\cite[Theorem 3.2]{Ishii}.\nHowever, these proofs also work if $G$ is a very small, finite, and linearly reductive\nsubgroup scheme of ${\\mathbf{GL}}_{2,k}$.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nA sequence $\\{f_n\\}_{n=1}^\\infty$ in separable Hilbert space $\\mathcal{H}$ is called a frame for the Hilbert space $\\mathcal{H},$ if there exist positive constants $A,~ B >0$ such that\n\\begin{align}\\label{0} A\\| f\\|^2_\\mathcal{H} \\leq \\sum_{n=1}^\\infty |\\langle f, f_n \\rangle|^2 \\leq B\\|f\\|^2_\\mathcal{H},~~~ \\mbox{for all}~~ f\\in \\mathcal{H}\n \\end{align} The positive constants $A$ and $B$ are called the lower and upper frame bounds of the frame, respectively. The inequality in (\\ref{0}) is called the frame inequality of the frame. The frame is called a tight frame if $A=B$ and is called normalized tight frame if $A=B=1.$  If $\\{f_n\\}_{n=1}^\\infty$  is a frame for $\\mathcal{H}$ then the following operators are associated with it.\n  \\begin{enumerate}[(a)]\n \\item Pre-frame operator $T: l^2(\\mathbb{N}) \\longrightarrow \\mathcal{H}$ is defined as $T\\{c_n\\}_{n=1}^\\infty= \\sum\\limits_{k=1}^\\infty c_nf_n,~~\\{c_n\\}_{n=1}^\\infty \\in l^2(\\mathbb{N}).$\n \\item Analysis operator $T^*: \\mathcal{H} \\longrightarrow l^2(\\mathbb{N}), T^*f= \\{\\langle f, f_k\\rangle\\}_{k=1}^\\infty~~~ f \\in \\mathcal{H}.$\n \\item Frame operator $S=TT^*=: \\mathcal{H} \\longrightarrow \\mathcal{H},~~ Sf= \\sum\\limits_{k=1}^\\infty \\langle f, f_k \\rangle f_k, ~~f \\in \\mathcal{H}.$ The frame operator $S$ is bounded, linear and invertible on $\\mathcal{H}$. Thus, a frame for $\\mathcal{H}$ allows each vector in $\\mathcal{H}$ to be written as a linear combination of the elements in the frame, but the linear independence between the elements is not required; i.e for each vector $f \\in \\mathcal{H}$ we have,\n     \\begin{eqnarray*}\n     f=SS^{-1}f= \\sum\\limits_{k=1}^\\infty \\langle f, f_k \\rangle f_k.\n     \\end{eqnarray*}\n   \\end{enumerate}\nFrames in Hilbert spaces were introduced by Duffin and Schaeffer \\cite{8} in 1952, while addressing some deep problems in non-harmonic Fourier series. Frames were reintroduced by Daubechies, Grossmann and Meyer \\cite{DGM} after three decades, in 1986, frames were brought to life, and were widely studied after this nobel work. Frames are generalization of orthonormal basis. The\nmain property of frames which makes them useful is their redundancy. Now, frames play an important role not only in  pure mathematics but also in applied mathematics. Representation\nof signals using frames is advantageous over basis expansions in\na variety of practical applications in science and engineering. In\nparticular, frames are widely used in sampling theory \\cite{ab, es}, wavelet theory \\cite{id}, signal processing \\cite{pg}, image processing \\cite{ma}, pseudo-differential operators \\cite{pdo},\nfilter banks \\cite{cf}, quantum computing \\cite{qc}, wireless sensor network \\cite{ws}, coding theory \\cite{ct}, geometry\\cite{SV1, SV2} and so on.\n Feichtinger and Gr\\\"{o}cheing \\cite{10} extended the notion of frames to Banach space and defined the notion of atomic decomposition.\n Gr\\\"{o}cheing \\cite{13} introduced a more general concept for Banach spaces called Banach frame. Banach frames and atomic decompositions were further studied in \\cite{6,13,14}. Casazza, Christensen and Stoeva \\cite{4} studied $\\mathcal{X}_d$-frame and $\\mathcal{X}_d$-Bessel sequence in Banach spaces. Shah \\cite{SS} defined and studied approximative $\\mathcal{X}_d$-frames and approximative $\\mathcal{X}_d$-Bessel sequences. He gave the following definition.\n \\begin{defn}\\cite{SS}\nA sequence $ \\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}} \\subseteq \\mathcal{X}^*,$ where $\\{m_n\\}$ is an increasing sequence of positive integers, is called an approximative \\emph{$\\mathcal{X}_d$-frame} for $\\mathcal{X}$ if\n\\begin{enumerate}[(a)]\n\\item $\\{h_{n,i}(x) \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}} \\in \\mathcal{X}_d$, for all $ x\\in \\mathcal{X}.$\n\\item There exist constants $A$ and $B$ with $0 < A \\leq B <\\infty$ such that\n\\begin{eqnarray}\\label{XD}\n A\\|x \\|_\\mathcal{X}~\\leq \\| \\{h_{n,i}(x) \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}} \\|_{\\mathcal{X}_d}~\\leq B \\|x \\|_\\mathcal{X},~\\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray}\n\\end{enumerate}\n\\end{defn}\n\\noindent The constants $A$ and $B$ are called approximative \\emph{$\\mathcal{X}_d$-frame} bounds.\nIf atleast (a) and the upper bound condition in (\\ref{XD}) are satisfied, then $\\{h_{n,i}(x) \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ is called an approximative \\emph{$\\mathcal{X}_d$-Bessel sequence} for $\\mathcal{X}.$\\\\\nOne may note that if $\\{f_n\\}$ is an $\\mathcal{X}_d$-frame for $\\mathcal{X},$ then for $\\{h_{n,i} \\}=f_i,~i=1,2,3,...,n;~ n \\in \\mathbb{N}$, $\\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ is an approximative $\\mathcal{X}_d$-frame for $\\mathcal{X}.$ Also, note that if $\\{f_n\\}$ is an $\\mathcal{X}_d$-Bessel sequence for $\\mathcal{X},$ then for $\\{h_{n,i} \\}=f_i,~i=1,2,3,...,n;~ n \\in \\mathbb{N}$, $\\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ is an approximative $\\mathcal{X}_d$-Bessel sequence for $\\mathcal{X}.$ The\nbounded linear operator $U: \\mathcal{X} \\rightarrow \\mathcal{X}_d$ given by\n\\begin{eqnarray}\nU(x)=\\{h_{n,i}(x) \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}, x \\in \\mathcal{X}\n\\end{eqnarray}\n is called an \\emph{analysis operator} associated to the approximative $\\mathcal{X}_d$-Bessel sequence $\\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$.\n If $\\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ is an approximative \\emph{$\\mathcal{X}_d$-frame} for $\\mathcal{X}$ and there exists a bounded linear operator $S:\\mathcal{X}_d\\longrightarrow \\mathcal{X}$ such that $S(\\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}})=x,$~ for all $x\\in \\mathcal{X}$, then $(\\{h_{n,i} \\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}},S)$ is called an approximative \\emph{Banach frame} for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$.\\\\\n Atomic systems for an operator $K$ in Hilbert spaces were introduced by Gavruta \\cite{12}.  Xiao et al. \\cite{21} discussed relationship between $K$-frames and ordinary frames in Hilbert spaces. Poumai and Jahan \\cite{KS} introduced atomic systems for operators in Banach spaces. Frames for operators in Banach spaces were further studied in \\cite{R1, R3}.\\\\\n \\textbf{Outline of the paper.}\n$K$-frames and atomic systems for an operator $K$ in Hilbert spaces were introduced by Gavruta \\cite{12} and further studied by Xiao, Zhu and Gavruta \\cite{21} discussed relationship between $K$-frames and frames in Hilbert spaces.\n In the present paper, we define approximative atomic system for an operator $K$ in a Banach space  and prove some results on the existence of approximative atomic system for $K$. We also define approximative family of local atoms for subspaces and give a characterization for the approximative family of local atoms for subspaces. Also, we discuss  methods to construct approximative atomic system for an operator $K$ from an approximative Bessel sequence and an approximative $\\mathcal{X}_d$-Bessel sequence.\\\\\n\\\\\\indent Throughout this paper, $\\mathcal{X}$ will denote a Banach space over the scalar field $\\mathbb{K}$($\\mathbb{R}$ or $\\mathbb{C}$), $\\mathcal{X}^*$ the dual space of $\\mathcal{X}$, $\\mathcal{X}_d$ a BK-space, $\\mathcal{X}_d^*$ denotes dual of $\\mathcal{X}_d$ and we will assume that $\\mathcal{X}_d$ has a sequence of canonical unit vectors as basis. By $\\{h_{n,i}\\}$ we means a sequence of coefficient functionals of row finite matrix. $L(\\mathcal{X})$ will denote the set of all bounded linear operators from $\\mathcal{X}$ into $\\mathcal{X}$. For $T\\in L(\\mathcal{X}) $, $T^{*}$ denotes the adjoint of $T$. $\\pi:\\mathcal{X} \\longrightarrow \\mathcal{X}^{**}$ is the natural canonical projection from $\\mathcal{X}$ onto $\\mathcal{X}^{**}$. A sequence space $S$ is called a \\emph{BK-space} if it is a Banach space and the co-ordinate functionals are continuous on $S.$ That is the relations $x_n= \\{ {\\alpha_j}^{(n)}\\}$, $x=\\{ \\alpha_j\\} \\in S$, $\\lim\\limits_{n \\longrightarrow \\infty}x_n=x$ imply $\\lim\\limits_{n \\longrightarrow \\infty}\\alpha_j^{(n)}=\\alpha_j ~~~(j=1,2,3,...).$\n\n\\section{Preliminaries}\n  Gavruta \\cite{12}, introduced the notion of $K$-frame and atomic system for an operator $K$ in a Hilbert space. She gave the following definition.\n\n\\begin{defn} \\cite{12} Let $\\mathcal{H}$ be a Hilbert space, $K\\in L(\\mathcal{H})$ and $\\lbrace x_n\\rbrace\\subseteq \\mathcal{H}$. Then \\\\\n(a) $\\lbrace x_n\\rbrace$ is called a \\emph{K-frame} for $\\mathcal{H}$ if there exist constants $A,B>0$ such that\n\\begin{eqnarray*}\nA\\Vert K^*x\\Vert^2\\leq\\sum\\limits_{n=1}^{\\infty}\\vert\\langle x,x_n\\rangle\\vert^2\\leq B\\Vert x\\Vert^2, \\ \\text{for all} \\ x\\in \\mathcal{H}.\n\\end{eqnarray*}\n(b) $\\lbrace x_n\\rbrace$ is called an \\emph{atomic system for $K$} if\n\\begin{enumerate}[(i)]\n\\item the series $\\sum\\limits_{n=1}^{\\infty}c_nx_n$ converges for all $c=\\lbrace c_n\\rbrace\\in l^2$.\n\\item there exists $C>0$ such that for every $x\\in \\mathcal{H}$ there exists $\\lbrace a_n\\rbrace\\in l^2$ such that $\\Vert \\lbrace a_n\\rbrace\\Vert_{l^2}\\leq C\\Vert x\\Vert$ and $K(x)=\\sum\\limits_{n=1}^{\\infty}a_nx_n.$\n\\end{enumerate}\n\\end{defn}\n Shah \\cite{SS} defined and studied approximative K-atomic decompositions and approximative $\\mathcal{X}_d$-frames in Banach spaces. For further studies related to this concept one may refer \\cite{SH, n}.\n\nNext, we give some results which we will use throughout this manuscript.\n\\begin{defn}\n\\cite{16} Let $T\\in L(\\mathcal{X})$, we say that an operator S$\\in L(\\mathcal{X})$ is a pseudoinverse of $T$ if $TST=T$. Also, $S\\in L(\\mathcal{X})$ is the generalized inverse of $T$ if $TST=T$ and $STS=S$.\n\\end{defn}\n\\begin{lem}\\cite{22}\\label{PI} Let $\\mathcal{X}$, $\\mathcal{Y}$ be Banach spaces and $T:\\mathcal{X} \\longrightarrow \\mathcal{Y}$ be a bounded linear operator. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item There exist two continuous projection operators $P:\\mathcal{X}\\rightarrow \\mathcal{X}$ and $Q:\\mathcal{Y} \\rightarrow \\mathcal{Y}$ such that\n\\begin{eqnarray}\\label{1E1}\nP( \\mathcal{X})=kerT \\quad and \\quad Q(\\mathcal{Y})=T( \\mathcal{X}).\n\\end{eqnarray}\n\\item There exist closed subspaces $W$ and $Z$ such that $\\mathcal{X}=KerT\\oplus W$ and $\\mathcal{Y}=T(\\mathcal{X})\\oplus Z.$\n\\item $T$ has a pseudo inverse operator $T^\\dag$.\n\\end{enumerate}\nIf two continuous projection operators $P:\\mathcal{X} \\rightarrow \\mathcal{X}$ and $Q:\\mathcal{Y} \\rightarrow \\mathcal{Y}$ satisfies (\\ref{1E1}), then there exists a pseudo inverse operator $T^\\dag$ of $T$ such that \\begin{equation*}\nT^\\dag T=I_{\\mathcal{X}}-P \\ and \\ TT^\\dag=Q,\n\\end{equation*}\nwhere $I_{\\mathcal{X}}$ is the identity operator on $\\mathcal{X}$.\n\\end{lem}\n\\begin{lem}\\label{5}\n\\cite{2,19} Let $\\mathcal{X}$ be a Banach space. If $T\\in L(\\mathcal{X})$ has a generalized inverse $S\\in L(\\mathcal{X}).$ Then $TS$,$ST$ are projections and $TS(\\mathcal{X})=T(\\mathcal{X})$ and $ST(\\mathcal{X})=S(\\mathcal{X}).$\n\\end{lem}\n\n\\begin{lem}\\label{6}\n\\cite{15} Let $\\mathcal{X}$ and $\\mathcal{Y}$ be Banach spaces. Let $L:\\mathcal{X} \\rightarrow \\mathcal{X}$ and $L_1:\\mathcal{X} \\rightarrow \\mathcal{Y}$ be linear operators. Then the following conditions are equivalent:\n\\begin{enumerate}[(a)]\n\\item $L=L_2L_1$, for some continuous linear operator $L_2:L_1(\\mathcal{X})\\rightarrow \\mathcal{X}$.\n\\item $\\Vert L(\\mathcal{X})\\Vert\\leq k\\Vert L_1(\\mathcal{X})\\Vert$, for some $k\\geq 0$ and for all $x\\in \\mathcal{X}$.\n\\item $Range(L^*)\\subseteq Range(L_1^{*})$.\n\\end{enumerate}\n\\end{lem}\n\\begin{lem}\\label{7}\n\\cite{1} Assume $T\\in B(\\mathcal{X},\\mathcal{Y})$. If $S\\in B(\\mathcal{X},\\mathcal{Z})$ with $Range(S^*)\\subseteq Range(T^*)$ and $\\overline{Range(T )}$ is complemented, then there\nexists $V\\in B(\\mathcal{Y},\\mathcal{Z})$ such that $S = VT.$\n\\end{lem}\n\\begin{lem}\\label{8}\n\\cite{4} Let $\\mathcal{X}_d$ be a BK-space for which the canonical unit vectors $\\lbrace e_n\\rbrace$ form a Schauder basis. Then the space $\\mathcal{Y}_d=\\lbrace\\lbrace h(e_n)\\rbrace\\vert h\\in \\mathcal{X}_d^*\\rbrace$ with norm $\\Vert\\lbrace h(e_n)\\rbrace\\Vert_{\\mathcal{Y}_d}=\\Vert h\\Vert_{\\mathcal{X}_d^*}$ is a BK-space isometrically isomorphic to $\\mathcal{X}_d^*$. Also, every continuous linear functional $\\Phi$ on $\\mathcal{X}_d$ has the form\n$\\Phi\\lbrace c_n\\rbrace=\\sum\\limits_{n=1}^{\\infty}c_nd_n,$\nwhere $\\lbrace d_n\\rbrace\\in \\mathcal{Y}_d$ is uniquely determined by $d_n=\\Phi(e_n)$, and\n$\\Vert \\Phi\\Vert=\\Vert\\lbrace \\Phi(e_n)\\rbrace\\Vert_{\\mathcal{Y}_d}$.\n\\end{lem}\nTerekhin \\cite{20} introduced and studied frames in Banach spaces.\n\\begin{defn}\n\\cite{20} Let $\\mathcal{X}$ be a Banach space and $\\mathcal{X}_d$ be a BK-space with the sequence of canonical unit vectors $\\lbrace e_n\\rbrace$ as basis. Let $Y_d$ be a sequence space mentioned in Lemma \\ref{8}. A sequence $\\lbrace x_n \\rbrace_{n=1}^{\\infty}\\subseteq \\mathcal{X}$ is called a frame with respect to $\\mathcal{X}_d$ if\n\\begin{enumerate}[(a)]\n\\item $\\lbrace f(x_n)\\rbrace\\in Y_d,$~ for all $f\\in \\mathcal{X}^*$,\n\\item there exist constants $A$ and $B$ with $0<A\\leq B<\\infty$ such that\n\\begin{eqnarray}\\label{1}\nA\\Vert f\\Vert_{\\mathcal{X}^*}\\leq\\Vert\\lbrace f(x_n)\\rbrace\\Vert_{\\mathcal{Y}_d}\\leq B\\Vert f\\Vert_{\\mathcal{X}^*}, \\ \\text{for  all} \\ f\\in \\mathcal{X}^*.\n\\end{eqnarray}\n\\end{enumerate}\nWe refer (\\ref{1}) as the \\emph{frame inequality}. If at least (a) and the upper bound condition in (\\ref{1}) are satisfied, then $\\lbrace x_n\\rbrace$ is called a \\emph{Bessel sequence} for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$.\nA frame $\\lbrace x_n\\rbrace$ is called exact if on removal of its one element $x_n$, it is no longer a frame for $\\mathcal{X}$.\n\\end{defn}\n\n\n\\section{Approximative atomic system for operators}\n\n Gavruta \\cite{12}, introduced and studied atomic system for an operator $K$ in Hilbert spaces. Poumai and Jahan \\cite{KS} defined and studied atomic systems for operators in Banach spaces as a generalization of atomic system in Hilbert spaces. Here we have generalize this concept further and introduce the concept of approximative atomic systems for operators in  Banach spaces and obtain new and interesting results. We starts this section with the following definition of approximative atomic system for K:\n\\begin{defn}\nLet $\\mathcal{X}$ be a Banach Space and $K$ be a bounded linear operator in $\\mathcal{X}$. A sequence $\\lbrace x_n\\rbrace\\subseteq \\mathcal{X}$ is called an \\emph{approximative atomic system for K} in $\\mathcal{X}$ with respect to $\\mathcal{X}_d$ if\n\\begin{enumerate}[(a)]\n\\item $\\lbrace x_n\\rbrace$ is a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d,$\n\\item there exists an approximative $\\mathcal{X}_d$-Bessel Sequence $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}\\subset \\mathcal{X}^*$ for $\\mathcal{X}$ such that\n\\begin{eqnarray*}\nK(x)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ \\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\n\\end{enumerate}\n\\indent The sequence $\\{h_{n, i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ is called the associated approximative $\\mathcal{X}_d$-Bessel sequence.\n\\end{defn}\n\\textbf{Observation}: If $\\{x_n\\}$ is an atomic system for $K$ in $\\mathcal{X}$ with respect to $\\mathcal{X}_d,$ then for $h_{n,i}=f_i,~~i=1,2,3,...,n,~~n \\in \\mathbb{N},$ $\\{x_n\\}$ is an approximative atomic system for $K.$\n\\begin{defn}\nLet $\\mathcal{X}$ be a Banach Space and $M$ be a closed subspace of $\\mathcal{X}$. A sequence $\\lbrace x_n \\rbrace\\subseteq \\mathcal{X}$ is called \\emph{an approximative family of local atoms} for $M$ if\n\\begin{enumerate}[(a)]\n\\item $\\lbrace x_n \\rbrace_{n=1}^{\\infty}$ be a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d,$\n\\item there exists an approximative $\\mathcal{X}_d$-Bessel Sequence $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}\\subset \\mathcal{X}^*$ for $M$ such that\n\\begin{eqnarray*}\nx=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ \\text{for all} \\ x\\in M.\n\\end{eqnarray*}\n\\end{enumerate}\nThe sequence $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ is called the associated approximative $\\mathcal{X}_d$-Bessel sequence.\n\\end{defn}\n\\textbf{Observations}~(I) If $\\{x_n\\}$ is a family of local atoms for $M$, then for $h_{n,i}=f_i,~~i=1,2,3,...,n,~~n \\in \\mathbb{N},$ $\\{x_n\\}$ is an approximative family of local atoms for $M.$\\\\\n\\\\~~~(II)\nLet $\\mathcal{X}$ be a Banach space and $M$ be a closed subspace of $\\mathcal{X}$. Let $\\lbrace x_n \\rbrace$ be an approximative family of local atoms for $M$ with respect to $\\mathcal{X}_d$ and let $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ be its associated approximative $\\mathcal{X}_d$-Bessel sequence with bound $B$. Then, $(x_n, \\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}})$ is an approximative atomic decomposition for $M$ with respect to $\\mathcal{X}_d$ and $\\lbrace x_n\\rbrace$ is a frame for $M$.\n Indeed, by given hypotheses,\n\\begin{eqnarray*}\nx=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\text{for all} \\ x \\in M.\n\\end{eqnarray*}\nLet $A$ be the bound of Bessel sequence $\\lbrace x_n\\rbrace.$ Then for $x \\in M$\n\\begin{eqnarray*}\n\\Vert x\\Vert&=&\\sup\\limits_{h\\in  \\mathcal{X}^*,\\Vert h\\Vert=1}\\vert \\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)h(x_n)\\vert\n\\leq A\\Vert\\lbrace h_{n,i}(x)\\rbrace\\Vert\n\\end{eqnarray*}\nWith Similar argument, we have\n\\begin{eqnarray*}\n1\/B\\Vert f\\Vert\\leq\\Vert\\lbrace f(x_n)\\rbrace\\Vert, \\ \\text{for all} \\ f\\in M^*.\n\\end{eqnarray*}\n\\\\~(III)\nLet $\\mathcal{X}$ be a Banach space and $K\\in L(\\mathcal{X})$. If $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$ in $\\mathcal{X}$. Then, there exist constants $C,D>0$ such that\n\\begin{eqnarray*}\nC\\Vert K(x)\\Vert\\leq\\Vert\\lbrace h_{n,i}(x)\\rbrace\\Vert, \\text{for all} \\ x\\in \\mathcal{X},\n\\end{eqnarray*} and\n\\begin{eqnarray*}\nD\\Vert K^*(f)\\Vert\\leq\\Vert\\lbrace h(x_n)\\rbrace\\Vert, \\text{for all} \\ h\\in \\mathcal{X}^*.\n\\end{eqnarray*}\n\\\\~(IV)\n$\\lbrace x_n\\rbrace\\subseteq \\mathcal{X}$ is a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$ if and only if there exists a bounded linear operator $T$ from $\\mathcal{X}_d$ into $\\mathcal{X}$ for which $T\\lbrace h_n\\rbrace=\\sum\\limits_{n=1}^{\\infty}h_nx_n$, $\\lbrace h_n\\rbrace\\in \\mathcal{X}_d.$ Recall that $T$ is called the \\emph{synthesis operator} associated with the Bessel sequence $\\{x_n\\},$ and the bounded linear operator $R:\\mathcal{X}^*\\rightarrow \\mathcal{Y}_d$ given by\n\\begin{eqnarray*}\nR(h)=\\lbrace h(x_n)\\rbrace, \\ \\text{for} \\ h\\in \\mathcal{X}^*,\n\\end{eqnarray*}\nis called the \\emph{analysis operator} of Bessel sequence $\\lbrace x_n\\rbrace$. Also observe that from Lemma \\ref{8}, $\\mathcal{Y}_d$ is isometrically isomorphic to $\\mathcal{X}_d^*$. Let $J_d:\\mathcal{X}_d^*\\rightarrow \\mathcal{Y}_d$ denote an isometric isomorphism from $\\mathcal{X}_d^*$ onto $\\mathcal{Y}_d$.\nNote that the synthesis operator $T$ need not be onto. Indeed, let $\\mathcal{X}_d=\\mathcal{X}=l_1$ and let $\\lbrace e_n\\rbrace$ be a sequence of canonical unit vectors as basis of $\\mathcal{X}$. Take $x_n=e_{n+1}$, for $n \\in \\mathbb{N}$. Let $h=\\lbrace h_n\\rbrace_{n=1}^\\infty\\in  \\mathcal{X}^*=l^\\infty$ then $\\lbrace h(x_n)\\rbrace\\in l^\\infty$ and $\\Vert\\lbrace h(x_n)\\rbrace\\Vert_{l^\\infty}\\leq A\\Vert h\\Vert_{l^\\infty}$, where $A>0$ is some constant. Thus, $\\lbrace x_n\\rbrace$ is Bessel sequence for $\\mathcal{X}$. But $T:\\mathcal{X}_d\\rightarrow \\mathcal{X}$ given by $T(e_n)=x_n$, for $n \\in \\mathbb{N}$ is a bounded linear operator which is not onto.\n\\\\\n$\\lbrace x_n\\rbrace\\subseteq \\mathcal{X}$ is a frame if and only if there exists a bounded linear operator $T$ from $\\mathcal{X}_d$ onto $\\mathcal{X}$ for which $T\\lbrace c_n\\rbrace=\\sum\\limits_{n=1}^{\\infty}c_nx_n$, $\\lbrace c_n\\rbrace\\in \\mathcal{X}_d$.\n\nFrom the frame inequality, $T^*$ is one-one and Range of $T^*$ is closed in $\\mathcal{X}_d^*$. So by [\\cite{17}, page 103], $T$ is onto.\\\\\n\\indent Conversely, if $T$ is onto. Then by [\\cite{17}, page 103] $T^*$ is one-one and range of $T^*$ is closed. Also, by [\\cite{9}, page 487], there exists a constant $D>0$ such that\n\\begin{eqnarray*}\n\\Vert h\\Vert\\leq D\\Vert T^*(h)\\Vert=D\\Vert \\lbrace h(x_n)\\rbrace\\Vert,\\ \\text{for all} \\ h\\in \\mathcal{X}^*.\n\\end{eqnarray*}\n\nIn the following result, we construct an approximative family of local atoms for $K(\\mathcal{X})$ and an approximative atomic decomposition for $[K(\\mathcal{X})]^*$ from a given approximative atomic system for a bounded linear operator $K$.\n\\begin{thm}\\label{T1}\nLet $\\lbrace x_n\\rbrace$ be an approximative atomic system for $K$ in $\\mathcal{X}$  and $K$ has pseudo inverse $K^\\dagger$. Then, $\\lbrace x_n\\rbrace$ is an approximative family of local atoms for $K(\\mathcal{X})$ in $\\mathcal{X}$.\nMoreover, if $\\mathcal{X}_d^*$ has a sequence of canonical unit vectors $\\lbrace e_n^*\\rbrace$ as basis, then there exists an approximative $\\mathcal{X}_d$-Bessel sequence $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}\\subset \\mathcal{X}^*$ for $K(\\mathcal{X})$ such that $(h_{n,i},\\pi(x_n))$ is an approximative atomic decomposition for $[K(\\mathcal{X})]^*$ with respect to $\\mathcal{X}_d^*$.\n\\end{thm}\n\\begin{proof}\nSince $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$ in $\\mathcal{X}$  $K$ has a pseudo inverse $K^\\dagger$, $KK^\\dagger$ is a projection from $\\mathcal{X}$ onto $K(\\mathcal{X})$ and\n$KK^\\dagger(x)=x, \\ \\text{for all} \\ x\\in K(\\mathcal{X}).$\nThus, $KK^\\dagger\\vert_{K(\\mathcal{X})}=I_{K(\\mathcal{X})}$. Let $\\lbrace h_{n,i}\\rbrace$ be its associated approximative $\\mathcal{X}_d$-Bessel sequence with bound C. Take $f_{n,i}=(K^\\dagger\\vert_{K(\\mathcal{X})})^*(h_{n,i})$, $n\\in\\mathbb{N}$ and\nlet $x\\in K(\\mathcal{X})$. Then, we compute\n\\begin{eqnarray*}\nx&=&K(K^\\dagger\\vert_{K(\\mathcal{X})}(x))\n\\\\\n&=&\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(K^\\dagger\\vert_{K(\\mathcal{X})}(x))x_i\n\\\\\n&=&\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}(K^\\dagger\\vert_{K(\\mathcal{X})^*}h_{n,i}(x))x_i\n\\\\\n&=&\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}f_{n,i}(x)x_i\n\\end{eqnarray*}\n\\indent Now, we will show that $\\lbrace f_{n,i}\\rbrace$ is an approximative Bessel sequence for $K(\\mathcal{X})$ with respect to $\\mathcal{X}_d$. Let $x\\in K(\\mathcal{X}).$ Then we have\n\\begin{eqnarray*}\n\\lbrace f_{n,i}(x)\\rbrace=\\lbrace h_{n,i}(K^\\dagger\\vert_{K(\\mathcal{X})}(x)\\rbrace\\in \\mathcal{X}_d, \\ \\text{for all} \\ x\\in K(\\mathcal{X})\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\Vert\\lbrace f_{n,i}(x)\\rbrace\\Vert&=&\\Vert\\lbrace h_{n,i}(K^\\dagger\\vert_{K(\\mathcal{X})}(x))\\rbrace\\Vert\\leq C\\Vert K^\\dagger\\vert_{K(\\mathcal{X})}(x)\\Vert\n\\\\\n&\\leq&C\\Vert K^\\dagger\\Vert\\Vert x\\Vert, \\ \\text{for all} \\ x\\in K(\\mathcal{X}).\n\\end{eqnarray*}\nAlso, $(K^\\dagger\\vert_{K(\\mathcal{X})})^* K^*=I^*_{[K(\\mathcal{X})]^*}$. Let $h \\in [K(\\mathcal{X})]^*$.  Then\n\\begin{eqnarray*}\n\\Vert h\\Vert&=&\\Vert(K^\\dagger\\vert_{K(\\mathcal{X})})^* K^*(h)\\Vert\\leq \\Vert K^\\dagger\\Vert\\Vert K^*(h)\\Vert\n\\\\\n&=&\\Vert K^\\dagger\\Vert\\sup\\limits_{x\\in K(\\mathcal{X}),\\Vert x\\Vert=1}\\vert h(\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n} h_{n,i}(x)x_i)\\vert\n\\\\\n&\\leq& C\\Vert K^\\dagger\\Vert\\Vert \\lbrace h(x_i)\\rbrace\\Vert.\n\\end{eqnarray*}\nLet $h\\in [K(\\mathcal{X})]^*.$ Then for $N\\in \\mathbb{N},$ we have\n\\begin{eqnarray*}\n\\Vert h-\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h(x_i)h_{n,i}\\Vert&=&\\sup\\limits_{x\\in K(\\mathcal{X}),\\Vert x\\Vert=1}\\vert h(x)- \\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h(x_i)h_{n,i}(x)\\vert\n\\\\\n&=&\\sup\\limits_{x\\in K(\\mathcal{X}),\\Vert x\\Vert=1}\\vert\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h(x_i)h_{n,i}(x)\\vert\n\\\\\n&\\leq&C\\Vert K^\\dagger\\Vert\\Vert \\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h(x_i)e_i^*\\Vert\n\\\\\n&\\rightarrow& 0 \\ \\text{as N} \\rightarrow \\infty.\n\\end{eqnarray*}\n\\indent Thus, $f=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h(x_i)h_{n,i}$, for all $h\\in [K(\\mathcal{X})]^*$.\n\\end{proof}\nIf $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$ and $K$ has pseudo inverse $K^\\dagger$, then for $h_{n,i}=f_i$, $\\lbrace x_n\\rbrace$ is frame as well as an approximative family of local atoms for $K(\\mathcal{X})$.\\\\\nThe following theorem gives a necessary condition for the existence of an approximative atomic system for a bounded linear operator $K$.\n\\begin{thm}\\label{T2}\nIf $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$, then there exists a bounded linear operator $T: \\mathcal{X}_d\\rightarrow \\mathcal{X}$ such that $T(e_n)=x_n$, $n\\in\\mathbb{N}$ and Range$K^{**}\\subseteq RangeT^{**}$, where $\\lbrace e_n\\rbrace$ is the sequence of canonical unit vectors as a basis of $\\mathcal{X}_d$.\n\\end{thm}\n\\begin{proof}\nSince $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$, $T:\\mathcal{X}_d\\rightarrow \\mathcal{X}$ given $T(\\lbrace h_{n,i}\\rbrace) =\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}x_i$ is a well defined bounded linear operator such that $T(e_i)=x_i$, $n\\in\\mathbb{N}$. Also, there exists an approximative $\\mathcal{X}_d$-Bessel sequence $\\lbrace h_{n,i}\\rbrace$ for $\\mathcal{X}$ with bound $B$ such that\n\\begin{eqnarray*}\nK(x)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ \\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\nThen, for $h\\in \\mathcal{X}^{*}$, we have\n\\begin{eqnarray*}\n\\Vert K^*(h)\\Vert&=&\\sup\\limits_{x\\in \\mathcal{X},\\Vert x\\Vert=1}\\vert K^*h(x)\\vert=\\sup\\limits_{x\\in \\mathcal{X},\\Vert x\\Vert=1}\\vert\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)h(x_i)\\vert\n\\\\\n&\\leq&B\\Vert \\lbrace h(x_i)\\rbrace\\Vert=B\\Vert T^*(h)\\Vert.\n\\end{eqnarray*}\nHence, by Lemma \\ref{7}, we have $RangeK^{**}\\subseteq RangeT^{**}$.\n\\end{proof}\nGavruta \\cite{12}, had proved that for a separable Hilbert space $\\mathcal{H}$ a sequence $\\lbrace x_n\\rbrace\\subseteq \\mathcal{H}$ is an atomic system for $K$ if and only if there exists a bounded linear operator $L:l^2\\rightarrow \\mathcal{H}$ such that $L(e_n)=x_n$ and $Range(K)\\subseteq Range(L)$, where $\\lbrace e_n\\rbrace$ is an orthonormal basis for $l^2$. Towards the converse of the Theorem \\ref{T2}, we have the following theorem.\n\\begin{thm}\nLet $\\mathcal{X}_d$ be reflexive, $\\mathcal{X}_d^*$ has a sequence of canonical unit vectors $\\lbrace e_n^*\\rbrace$ as basis and $\\lbrace x_n\\rbrace$ be a sequence in $\\mathcal{X}$. Let $K\\in L(\\mathcal{X})$, $T:\\mathcal{X}_d\\rightarrow  \\mathcal{X}$ be bounded linear operator with $T(e_n)=x_n,~n \\in \\mathbb{N}$, $RangeK^{**}\\subseteq RangeT^{**}$ and $\\overline{T^*(\\mathcal{X}^*)}$ is complemented subspace of $\\mathcal{X}_d^*$. Then $\\lbrace x_n\\rbrace$ is an approximative atomic system for K.\n\\end{thm}\n\\begin{proof}\nSince $T:\\mathcal{X}_d\\rightarrow \\mathcal{X}$ is a bounded linear operator and is given by\n\\begin{eqnarray*}\nT(\\lbrace \\alpha_n\\rbrace)=\\sum\\limits_{n=1}^{\\infty}\\alpha_nx_n,~~\\lbrace \\alpha_n\\rbrace \\in \\mathcal{X}_d^*.\n\\end{eqnarray*}\nSo, by Observation IV, $\\lbrace x_n\\rbrace$ is a Bessel sequence with bound say $B$. Since $RangeK^{**}\\subseteq RangeT^{**}$ and $\\overline{T^*(\\mathcal{X}^*)}$ is complemented subspace of $\\mathcal{X}_d^*$, by Lemma \\ref{7}, there exists a bounded linear operator $\\theta:\\mathcal{X}_d^*\\rightarrow \\mathcal{X}^*$ such that $K^*=\\theta T^*$.\n Take $h_{n,i},~~\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}=\\theta(e_n^*)$, $n\\in\\mathbb{N}$. Then, for $x\\in \\mathcal{X}$, we get\n\\begin{eqnarray*}\n\\theta^*(x)(e_j^*)=\\pi(x)\\theta(e_j^*)=h_{n,i}(x);i=1,2,...{m_n} \\ \\text{for} \\ n\\in\\mathbb{N}.\n\\end{eqnarray*}\nThis gives $\\lbrace h_{n,i}(x)\\rbrace=\\theta^*(x)\\in \\mathcal{X}_d, \\ \\text{for all} \\ x\\in \\mathcal{X}$\nand $\\Vert\\lbrace h_{n,i}(x)\\rbrace\\Vert=\\Vert\\theta^*(x)\\Vert\\leq\\Vert \\theta\\Vert\\Vert x\\Vert, \\ \\text{for all} \\ x\\in \\mathcal{X}.$\nAlso, for $h\\in \\mathcal{X}^*,$ we have\n\\begin{eqnarray*}\nK^*(h)=\\theta(\\lbrace h(x_n)\\rbrace)=\\theta(\\sum\\limits_{n=1}^{\\infty}h(x_n)e_n^*)\n&=&\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h(x_i)h_{n,i}.\n\\end{eqnarray*} and for $x\\in \\mathcal{X}$ and $n\\in \\mathbb{N},$ we compute\n\\begin{eqnarray*}\n\\Vert K(x)-\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{n}h_{n,i}(x)x_i\\Vert&=&\\sup\\limits_{h\\in \\mathcal{X}^*, \\Vert h\\Vert=1}\\vert h(K(x))-\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{n}h_{n,i}(x)h(x_i)\\vert\n\\\\\n&=&\\sup\\limits_{h\\in \\mathcal{X}^*, \\Vert h\\Vert=1}\\vert \\sum\\limits_{i=n+1}^{\\infty}h_{n,i}(x)h(x_i)\\vert\n\\\\\n&\\leq&B\\Vert \\sum\\limits_{i=n+1}^{\\infty}h_{n,i}(x)e_i\\Vert\\rightarrow 0 \\ \\text{as} \\ n\\rightarrow\\infty.\n\\end{eqnarray*}\n\\indent Hence, $K(x)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i$, for all $x\\in \\mathcal{X}$.\n\\end{proof}\nThe following theorem gives a complete characterization for the approximative family of local atoms for the closed subspace of a Banach space.\n\\begin{thm}\nLet $\\lbrace x_n\\rbrace$ be a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$, $\\mathcal{X}_d^*$ has a sequence of canonical unit vectors $\\lbrace e_n^*\\rbrace$ as basis and $M$ be a closed subspace of $\\mathcal{X}$. Let there exists a projection P from $\\mathcal{X}$ onto M. Then, the following statements are equivalent:\n\\begin{enumerate}[(a)]\n\\item $\\lbrace x_n\\rbrace$ is an approximative family of local atoms for M.\n\\item $\\lbrace x_n\\rbrace$ is an approximative atomic system for P.\n\\item There exists a bounded linear operator $U:M \\rightarrow \\mathcal{X}_d$ such that $TUP=P$, where $T$ is the synthesis operator of the Bessel sequence $\\lbrace x_n\\rbrace$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n$(a)\\Rightarrow(b)$ Since $\\lbrace x_n\\rbrace$ is an approximative family of local atoms for M so, there exists an approximative $\\mathcal{X}_d$-Bessel sequence $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ for $M$ such that\n\\begin{eqnarray*}\nx=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ \\text{for all} \\ x\\in M.\n\\end{eqnarray*}\nNow for each $y\\in \\mathcal{X}$, $P(y)\\in M$ and\n\\begin{eqnarray*}\nP(y)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(P(y))x_i=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}(P^*(h_{n,i})(y)x_i, \\ \\text{for all} \\ y\\in \\mathcal{X}.\n\\end{eqnarray*}\n\\indent Clearly $\\lbrace P^*(h_{n,i})\\rbrace$ is an approximative $\\mathcal{X}_d$-Bessel sequence for $\\mathcal{X}$. Hence, $\\lbrace x_n\\rbrace$ is an approximative atomic system for $P$.\n\\\\\n$(b)\\Rightarrow (a)$ Since $\\{x_n\\}$ is an approximative atomic system for P and $x=P(x)$, for all $x\\in M$, the proof follows.\n\\\\\n$(a)\\Rightarrow (c)$ Since $\\{x_n\\}$ is an approximative family of local atoms for M so there exists an approximative $\\mathcal{X}_d$-Bessel sequence $\\{h_{n,i}\\}\\underset{n \\in \\mathbb{N}}{_{i=1,2,3,...,m_n}}$ for M such that\n\\begin{eqnarray*}\nP(x)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(P(x))x_i, \\ \\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\nLet $U:M\\rightarrow \\mathcal{X}_d$ be the analysis operator for $\\lbrace h_{n,i}\\rbrace$ given by $U(P(x))=\\lbrace h_{n,i}(P(x))\\rbrace,x\\in \\mathcal{X}.$ Also, $\\lbrace h_{n,i}(P(x)\\rbrace\\in \\mathcal{X}_d$, for all $x\\in \\mathcal{X}$. Hence\n\\begin{eqnarray*}\nP(x)&=&\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(P(x))x_i=T(\\lbrace h_{n,i}(P(x))\\rbrace)\n\\\\\n&=&TUP(x), \\ \\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\\\\\n$(c)\\Rightarrow (a)$ Since $U:M\\rightarrow \\mathcal{X}_d$ is a bounded linear operator such that $TUP=P$. Take $h_{n,i}=U^*(e_n^*),n\\in\\mathbb{N}$ and $x\\in M$. Then $h_{n,i}(x)=U^*(e_n^*)(x)=e_n^*(U(x)).$\nTherefore, $\\lbrace h_{n,i}(x)\\rbrace=U(x)\\in \\mathcal{X}_d$, for all $x\\in M$ and\n\\begin{eqnarray*}\n\\Vert \\lbrace h_{n,i}(x)\\rbrace\\Vert=\\Vert U(x)\\Vert\\leq\\Vert U\\Vert\\Vert x\\Vert, \\ \\text{for all} \\ x\\in M.\n\\end{eqnarray*}\nAlso, for each, $x \\in M,$ there exists $y\\in \\mathcal{X}$ such that $x=P(y)$. Moreover, $\\lbrace h_{n,i}(P(y))\\rbrace\\in \\mathcal{X}_d$ for all $y\\in \\mathcal{X}$ and\n\\begin{eqnarray*}\nx=TUP(y)=T(\\lbrace h_{n,i}(P(y))\\rbrace)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(P(y))x_i=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(y)x_i\n\\end{eqnarray*}\n\\end{proof}\nNext, we shall show that if $\\lbrace x_n\\rbrace$ is an approximative $K$ atomic system with $K=I_\\mathcal{X}$, then every complemented subspace of $\\mathcal{X}$ has an approximative family of local atoms and in case $h_{n,i}=f_n$ then $\\{x_n\\}$ is a $K$ atomic system and every complemented subspace of $\\mathcal{X}$ has a family of local atoms.\n\\begin{thm}\nLet $\\lbrace x_n\\rbrace$ be a frame for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$. If there exists an approximative $\\mathcal{X}_d$-Bessel sequence $\\lbrace h_{n,i}\\rbrace$ such that\n\\begin{eqnarray*}\nx=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(P(x))x_i, \\ \\text{for all} \\ x\\in \\mathcal{X},\n\\end{eqnarray*}\nthen every complemented subspace of $\\mathcal{X}$ has an approximative family of local atoms.\n\\end{thm}\n\\begin{proof}\nLet $\\{x_n\\}$ be a frame for $\\mathcal{X}$ and $T$ be the synthesis operator of $\\{x_n\\}$. Then by given hypothesis, there exists an approximative $\\mathcal{X}_d$-Bessel sequence $\\lbrace h_{n,i}\\rbrace$ such that\n\\begin{eqnarray*}\nx=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ \\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\nLet $S$ be the analysis operator of $\\lbrace h_{n,i}\\rbrace$. Note that $I_\\mathcal{X}=TS$. Let $M$ be a complemented subspace of $\\mathcal{X}$ and $P$ be the projection from $\\mathcal{X}$ onto $M$. Take $y_n=P(x_n)$, for $n\\in \\mathbb{N}$. Define $T_1:\\mathcal{X}_d\\rightarrow M$ by $T_1=P\\circ T$.\nThen clearly $T_1$ is a bounded linear operator from $\\mathcal{X}_d$ onto $M$ such that\n\\begin{eqnarray*}\nT_1(\\lbrace h_{n,i}\\rbrace)=P(\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)y_i,~~ \\lbrace h_{n,i}\\rbrace\\in \\mathcal{X}_d.\n\\end{eqnarray*}\nThus, $\\lbrace y_n\\rbrace$ is frame for $M$ and a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$. Let $N=S(M)$ and let $\\alpha=\\lbrace h_{n,i}\\rbrace\\in \\mathcal{X}_d$. Then, $T_1(\\alpha)=x,$ for some $x\\in M.$ Now take $\\alpha_1=S(x).$ Then, $\\alpha_1\\in N.$ Also, if $\\alpha_0=\\alpha-\\alpha_1,$ then\n\\begin{eqnarray*}\nT_1(\\alpha_0)=T_1(\\alpha)-T_1(\\alpha_1)\n=x-P(x)=0.\n\\end{eqnarray*}\nThis gives $\\alpha_0 \\in kerT_1.$ Thus $\\alpha=\\alpha_0+\\alpha_1$. Then $\\lambda\\in N\\cap kerT_1$. Then $T_1(\\lambda)=0$ and there exists $x_1\\in M$ such that $\\lambda=S(x_1)$. Therefore\n\\begin{eqnarray*}\nx_1=P(x_1)=PTS(x_1)=T_1(\\lambda)=0.\n\\end{eqnarray*}\nThus, $\\lambda=0$ and so $N\\cap kerT_1=\\lbrace 0\\rbrace$. Therefore $\\mathcal{X}_d=N\\oplus kerT_1$. Now, by Lemma \\ref{PI}, $T_1$ has a pseudo inverse $T_1^\\dagger$. Moreover $T_1T_1^\\dagger$ is a projection from $M$ onto $T_1(\\mathcal{X}_d)=M$. This gives $T_1T_1^\\dagger=I_M$. Let $\\lbrace z_n\\rbrace$ be the sequence of coordinate functionals on $\\mathcal{X}_d$. Take $f_{n,i}=(T_1^\\dagger)^*(l_n)$, for $n\\in \\mathbb{N}$ and let $x\\in M$. Then\n\\begin{eqnarray*}\nf_{n,i}(x)=(T_1^\\dagger)^*(z_n)(x)=z_n(T_1^\\dagger(x)), \\ \\text{for} \\ n\\in\\mathbb{N}, \\ x\\in M.\n\\end{eqnarray*}\nTherefore $\\lbrace f_{n,i}(x)\\rbrace=T_1^\\dagger(x)\\in \\mathcal{X}_d$, for all $x\\in M$ and\n\\begin{eqnarray*}\n\\Vert\\lbrace f_{n,i}(x)\\rbrace\\Vert=\\Vert T_1^\\dagger(x)\\Vert\\leq\\Vert T_1^\\dagger\\Vert\\Vert x\\Vert, \\ \\text{for all} \\ x\\in M.\n\\end{eqnarray*}\nThus $\\lbrace f_{n,i}\\rbrace$ is an approximative $\\mathcal{X}_d$-Bessel sequence for $M$. Therefore, we compute\n\\begin{eqnarray*}\nx=T_1T_1^\\dagger(x)=T_1(\\lbrace f_{n,i}(x)\\rbrace)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}f_{n,i}(x)y_i, \\ \\text{for all} \\ x\\in M.\n\\end{eqnarray*}\n\\end{proof}\nLet $\\lbrace x_n\\rbrace$ be a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$ and let T be its associated synthesis operator. Also, let $\\lbrace h_{n,i}\\rbrace$ be an approximative $\\mathcal{X}_d$-Bessel sequence for $\\mathcal{X}$ with its associated analysis operator S. Then $TS: \\mathcal{X} \\rightarrow \\mathcal{X}$ is an operator such that\n\\begin{eqnarray*}\nTS(x)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ \\text{for all} \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\nNote that $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$ in $\\mathcal{X}$ with its associated approximative $\\mathcal{X}_d$-Bessel sequence $\\lbrace h_{n,i}\\rbrace$ if and only if\n\\begin{eqnarray}\\label{E2}\nTS=K.\n\\end{eqnarray} Also one may observe that in case $\\{h_{n,i}\\}=\\{f_i\\}$ then $\\{x_n\\}$ is an atomic system for $K$ in $\\mathcal{X}$ with its associated $\\mathcal{X}_d$-Bessel sequence $\\{f_i\\}$\\\\\nThe following theorem gives a method how to construct an approximative atomic system for $K$ from a given Bessel sequence.\n\\begin{thm}\nLet $\\mathcal{X}_d^*$ has a sequence of canonical unit vectors $\\lbrace e_n^*\\rbrace$ as basis. Let $\\lbrace x_n\\rbrace$ be a Bessel sequence for $\\mathcal{X}$ with respect to $\\mathcal{X}_d$ and $T:\\mathcal{X}_d\\rightarrow \\mathcal{X}$ be its synthesis operator such that T has pseudoinverse $T^\\dagger$. Let $K\\in  L(\\mathcal{X})$ with $K(\\mathcal{X})\\subseteq T(\\mathcal{X}_d)$. Then $\\mathcal{X}$ has an approximative atomic system for $K.$\n\\end{thm}\n\\begin{proof}\nSince $T$ has a pseudo inverse $T^\\dagger$, by Lemma \\ref{5},  $TT^\\dagger$ is a projection from $\\mathcal{X}$ onto $T(\\mathcal{X}_d)$. Define $S:\\mathcal{X}\\rightarrow \\mathcal{X}_d$ as\n\\begin{eqnarray}\\label{E3}\nS=T^\\dagger K+W-T^\\dagger TW,\n\\end{eqnarray}\nwhere $W:\\mathcal{X} \\rightarrow \\mathcal{X}_d$ is a bounded linear operator. Then, we compute\n\\begin{eqnarray*}\nTS&=&T(T^\\dagger K+W-T^\\dagger TW)\n\\\\\n&=&TT^\\dagger K+TW-TT^\\dagger TW\n\\\\\n&=&TT^\\dagger K=K.\n\\end{eqnarray*}\nNote that $T^*=J_d^{-1}\\circ R$ and $\\lbrace J_de_n^*\\rbrace$ is a basis of $\\mathcal{Y}_d$.\n Taking $h_{n,i}=S^*(e_n^*)$, $f_{n,i}=(T^+)^*(e_n^*)$ and $g_n=W^*(e_n^*)$ , for $n\\in \\mathcal{N},$ we compute\n\\begin{eqnarray*}\nT^*(T^\\dagger)^*e_n^*=T^*(f_{n,i})=J_d(R(f_{n,i}))=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}f_{n,i}(x_i)e_i^*\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nh_{n,i}&=&K^*(T^\\dagger)^*(e_n^*)+W^*(e_n^*)-W^*T^*(T^\\dagger)^*(e_n^*)\n\\\\\n&=&K^*f_{n,i}+g_i-\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}f_{n,i}(x_i)g_i, \\ \\text{for all} \\ n\\in \\mathbb{N}.\n\\end{eqnarray*}\nAlso, for $x\\in \\mathcal{X}$, we have\n\\begin{eqnarray*}\nh_{n,i}(x)=S^*(e_n^*)(x)=e_n^*(S(x)), \\ \\text{for all} \\ n\\in \\mathbb{N}.\n\\end{eqnarray*}\nTherefore $\\lbrace h_{n,i}(x)\\rbrace=S(x)\\in \\mathcal{X}_d$, for all $x \\in \\mathcal{X}$. Also,\n\\begin{eqnarray*}\n\\Vert\\lbrace h_{n,i}(x)\\rbrace\\Vert&=&\\Vert S(x)\\Vert\n\\\\\n&\\leq&\\Vert T^\\dagger K+W-T^\\dagger TW\\Vert\\Vert x\\Vert,~x \\in \\mathcal{X}.\n\\end{eqnarray*}\nThus, $\\lbrace h_{n,i}\\rbrace$ is an approximative $\\mathcal{X}_d$- Bessel sequence and\n\\begin{eqnarray*}\n\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i&=&T(\\lbrace h_{n,i}(x)\\rbrace)=T(\\lbrace S^*(e_n^*)(x)\\rbrace)\n\\\\\n&=&T(\\lbrace e_n^*(S(x))\\rbrace)=TS(x)\n\\\\\n&=&K(x), \\ x\\in \\mathcal{X}.\n\\end{eqnarray*}\nHence, $\\lbrace x_n\\rbrace$ is an approximative atomic system for $K$.\n\\end{proof}\nNote that (\\ref{E2}) is the general formula for all $U$ satisfying the equality (\\ref{E3}). As a result of this, let $U_\\circ$ be a linear operator such that $TU_\\circ=K$ and take $W=U_\\circ$ in the right side of (\\ref{E3}), then\n\\begin{eqnarray*}\nT^\\dagger K+U_\\circ-T^\\dagger TU_\\circ=T^\\dagger K+U_\\circ-T^\\dagger K=U_\\circ\n\\end{eqnarray*}\n\\indent The following theorem shows that we can construct an approximative atomic system for $K$ from a given approximative $\\mathcal{X}_d$-Bessel sequence.\n\\begin{thm}\n Let $\\mathcal{X}_d$ be a BK-space and let $\\lbrace e_n\\rbrace$ be a sequence of canonical unit vectors as basis of $\\mathcal{X}_d$ and $\\mathcal{X}_d^*$ has a sequence of canonical unit vectors $\\lbrace e_n^*\\rbrace$ as basis. Let $\\lbrace h_{n,i}\\rbrace$ be an approximative $\\mathcal{X}_d$-Bessel sequence for $\\mathcal{X}$ with $S:\\mathcal{X} \\rightarrow \\mathcal{X}_d$ as its analysis operator and S has pseudo inverse $S^\\dagger$. Let $K\\in L(\\mathcal{X})$ and $K^*(\\mathcal{X}^*)=[h_{n,i}]$. Then, $\\mathcal{X}$ has an approximative atomic system for $K.$\n\\end{thm}\n\\begin{proof} Since $\\lbrace h_{n,i}\\rbrace$ is an approximative $\\mathcal{X}_d$-Bessel sequence for $\\mathcal{X}$ with $S:\\mathcal{X} \\rightarrow \\mathcal{X}_d$ as its analysis operator and S has pseudoinverse $S^\\dagger.$\nDefine $T:\\mathcal{X}_d \\rightarrow \\mathcal{X}$ by\n\\begin{eqnarray}\\label{E4}\nT=KU^\\dagger +W(I-UU^\\dagger),\n\\end{eqnarray}\nwhere $W:\\mathcal{X}_d\\rightarrow \\mathcal{X}$ is a bounded linear operator. Then, we compute\n\\begin{eqnarray*}\nTS&=&(KS^\\dagger +W(I-SS^\\dagger))S\n\\\\\n&=&KS^\\dagger S+WS-WSS^\\dagger S\n\\\\\n&=&KS^\\dagger S.\n\\end{eqnarray*}\nAlso, for $x\\in \\mathcal{X}$, $S^*:\\mathcal{X}_d^*\\rightarrow \\mathcal{X}^*$ is given by\n\\begin{eqnarray*}\nS^*(e_n^*)(x)=e_n^*(S(x))=h_{n,i}(x), \\ n\\in \\mathcal{N}.\n\\end{eqnarray*}\nSo, we have\n\\begin{eqnarray*}\n[h_{n,i}]=[S^*(e_n^*)]=S^*(\\mathcal{X}_d^*)=K^*(\\mathcal{X}^*).\n\\end{eqnarray*}\nAlso $S$ has a pseudo inverse $S^\\dagger$, $S^*$ has pseudo inverse $(S^\\dagger)^*$. So,\n\\begin{eqnarray*}\nS^*(S^\\dagger)^*S^*(\\mathcal{X}_d^*)=S^*(\\mathcal{X}_d^*)=K^*(\\mathcal{X}^*).\n\\end{eqnarray*}\nThus, we conclude that $S^*(S^\\dagger)^*K^*=K^*$. Therefore, $TS=KS^\\dagger S=K$.\nLet $x_n=T(e_n)$, $y_n=S^\\dagger(e_n)$ and $l_n=W(e_n)$, for $n\\in \\mathbb{N}$.  Then\n\\begin{eqnarray*}\nx_n=KS^\\dagger(e_n)+W(I-SS^\\dagger)(e_n)\n\\\\\n=K(y_n)+l_n-\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(y_i)l_i, ~~n\\in \\mathcal{N}.\n\\end{eqnarray*}\nAlso, $T: \\mathcal{X}_d\\rightarrow \\mathcal{X}$ is a bounded linear operator given by\n\\begin{eqnarray*}\nT(\\lbrace h_{n,i}\\rbrace)=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i} x_i,~~ \\lbrace h_{n,i}\\rbrace\\in \\mathcal{X}_d.\n\\end{eqnarray*}\nBy observation III, $\\lbrace x_i\\rbrace$ is a Bessel sequence. Thus, we get\n\\begin{eqnarray*}\nK(x)=TS(x)=T(\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)e_i)\n=\\lim\\limits_{n \\rightarrow \\infty}\\sum\\limits_{i=1}^{m_n}h_{n,i}(x)x_i, \\ x \\in \\mathcal{X}.\n\\end{eqnarray*}\n\\indent Hence $\\lbrace x_n\\rbrace$ is an approximative atomic system for K.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nIt is expected that much of the dynamics of the small-amplitude solutions of a partial differential equation (PDE), including their stability or instability, is dictated by the study of a linearized (about a trivial solution, say $u=0$) problem. In this article, we focus specifically on the spectral stability of periodic traveling-wave solutions of Hamiltonian PDEs as they bifurcate away from a trivial solution. Our work follows earlier ideas of MacKay \\cite{mackay} and MacKay and Saffman \\cite{mackaysaffman}. We start from an autonomous Hamiltonian system of PDEs \\cite{arnold}, {\\em i.e.},\n\n\\begin{equation}\\la{ham}\nu_t=J \\dd{H}{u}.\n\\end{equation}\n\n\\noindent \\sloppypar \\noindent Here and throughout, indices involving $x$ or $t$ denote partial derivatives. Further, $u=(u_1(x,t), \\ldots, u_M(x,t))^T$ is an $M$-dimensional vector function defined in a suitable function space, and $J$ is a Poisson operator \\cite{arnold, arnoldnovikov}. More details and examples are given below. Finally, $H=\\int_D {\\cal H}(u,u_x, \\ldots)dx$ is the Hamiltonian, whose density ${\\cal H}$ depends on $u$ and its spatial derivatives, defined for $x\\in D$. We consider only the stability of periodic solutions, thus $D$ is any interval of length $L$, the period.\nNote that for some of our examples ${\\cal H}$ will depend on spatial derivatives of $u$ of arbitrary order.\n\nTo investigate the stability of traveling wave solutions of this system, we reformulate \\rf{ham} in a frame moving with speed $c$, using the transformation $\\hat x=x-ct$, $\\hat t=t$, and considering solutions $u(\\hat x, \\hat t)=U(\\hat x)$ (successively omitting hats). This leads to\n\n\\begin{equation}\\la{hamc}\nu_t-c u_x=J \\dd{H}{u} ~~\\iff~~u_t=J \\dd{H_c}{u},\n\\end{equation}\n\n\\noindent for a modified Hamiltonian $H_c$. Traveling wave solutions are solutions of the ordinary differential system\n\n\\begin{equation}\\la{travham}\n-c U_x=J \\dd{H}{U}  ~~\\iff~~0=J \\dd{H_c}{U}.\n\\end{equation}\n\n\\noindent Thus if $J$ is invertible, traveling waves are stationary points of the Hamiltonian $H_c$. The system~\\rf{travham} typically has the zero (trivial) solution for a range of $c$ values. The small-amplitude solutions whose stability we investigate bifurcate away from these trivial zero-amplitude solutions at special values of the speed parameter $c$, as is schematically shown in Fig.~\\ref{fig:bif}. It is our goal to see to what extent anything can be said about the stability of the small-amplitude solutions (with amplitudes in the shaded regions of Fig.~\\ref{fig:bif}) from knowledge of the zero-amplitude solutions at the bifurcation point. An outline of the steps in this process is as follows.\n\n\\begin{figure}[tb]\n\\def3.2in}\\input{bubbletrack.pdf_tex{5in}\n\\centerline{\\hspace*{0.3in}\\input{bif.pdf_tex}}\n\\caption{\\la{fig:bif} A cartoon of the bifurcation structure of the traveling waves for a third-order ($M=3$) system: solution branches bifurcate away from the trivial zero-amplitude solution at specific values of the traveling wave speed $c$.\n    }\n\\end{figure}\n\n\\begin{enumerate}\n\n\\item {\\bf Quadratic Hamiltonian}. A linear system of equations is obtained by linearizing the system \\rf{travham} around the zero solution: let $u=\\epsilon v+{o}(\\epsilon)$ and omit terms of order $o(\\epsilon)$. Alternatively, if $J$ is independent of $u$ and its spatial derivatives, one may expand the Hamiltonian $H_c$ as a function of $\\epsilon$ and retain its quadratic terms. The resulting Hamiltonian $H_c^0$ of the linearized system is the starting point for the next steps.\n\n\\item {\\bf Dispersion relation}. The linearized system has constant coefficients and is easily solved using Fourier analysis. The dispersion relation $F(\\omega, k)=0$ governs the time dependence of the solutions. It is obtained by investigating solutions whose spatial and temporal dependence is proportional to $\\exp(ikx-i \\omega t)$. Here $F(\\omega,k)=0$ is of degree $M$ in $\\omega$. It is a fundamental assumption of our approach that all solutions $\\omega_j(k)$ ($j=1, \\ldots, M$) of $F(\\omega,k)=0$ are real for $k\\in \\mathbb{R}$. The dispersion relation can be expressed entirely in terms of the coefficients appearing in the quadratic Hamiltonian $H_c^0$. For periodic systems of period $L$, the values of $k$ are restricted to be of the form $2\\pi N\/L$, $N\\in \\mathbb{Z}$.\n\n\\item {\\bf Bifurcation branches}. The values of the phase speed $c_j=\\omega_j\/k$ for which nontrivial solutions bifurcate away from the zero-amplitude solution are determined by the condition that the zero solution is {\\em not} the unique solution to the Fourier transformed problem. In effect, this is the classical bifurcation condition that a Jacobian is singular. This simple calculation determines the bifurcation branch starting points explicitly in terms of the different solutions to the dispersion relation. In what follows, we follow the first branch, starting at $c_1$, without loss of generality.\n\n    It is assumed that only a single non-trivial bifurcation branch emanates from a bifurcation point. Although more general cases can be incorporated, we do not consider them here. Further, we fix the period of the solutions on the bifurcation branch (usually to $2\\pi$). Other choices can be made. Instead of varying the amplitude as a function of the speed for fixed period, one could fix the speed and vary the period, {\\em etc.} The methods presented can be redone for those scenarios in a straightforward fashion.\n\n\\item {\\bf Stability spectrum}. The spectrum of the linear operator determining the spectral stability of the zero solution at the bifurcation point on the first branch is calculated. Since this spectral problem has constant coefficients, this calculation can be done explicitly. Again, this is done entirely in terms of the dispersion relation of the problem. Using a Floquet decomposition (see \\cite{deconinckkutz1, kapitulapromislow}), the spectrum is obtained as a collection of point spectra, parameterized by the Floquet exponent $\\mu\\in (-\\pi\/L, \\pi\/L]$. Due to the reality of the branches of the dispersion relation, the spectrum is confined to the imaginary axis. In other words, the zero-amplitude solutions are spectrally stable. The use of the Floquet decomposition allows for the inclusion of perturbations that are not necessarily periodic with period $L$. Instead, the perturbations may be quasiperiodic with two incommensurate periods, subharmonic (periodic, but with period an integer multiple of $L$), or spatially localized \\cite{deconinckkutz1, haraguskapitula, kapitulapromislow}.\n\n\\item {\\bf Collision condition}. Given the explicit expression for individual eigenvalues $\\lambda$, it is easy to find the conditions for which eigenvalues corresponding to different parameters (Floquet exponent, branch number of the dispersion relation, {\\em etc}.) coincide on the imaginary axis. This is referred to as the {collision condition}. Once again, it is given entirely in terms of the dispersion relation.\n\n    It is a consequence of the Floquet theorem \\cite{coddingtonlevinson} that collisions need to be considered only for spectral elements corresponding to the same value of the Floquet exponent since the subspaces of eigenfunctions for a fixed Floquet exponent are invariant under the flow of the linearized equation.\n\n\\item {\\bf Krein signature}. Having obtained the stability spectrum at the starting point of the bifurcation branches, we wish to know how the spectrum evolves as we move up a bifurcation branch. One tool to investigate this is the Krein signature \\cite{kollarmiller, krein1, krein2, mackay, meiss}. In essence, the Krein signature of an eigenvalue is the sign of the Hamiltonian of the linearized system evaluated on the eigenspace of the eigenvalue. Different characterizations are given below. If two imaginary eigenvalues of the same signature collide as a parameter changes, their collision does not result in them leaving the imaginary axis. Thus the collision of such eigenvalues does not result in the creation of unstable modes. In other words, it is a necessary condition for collisions to lead to instability that the Krein signature of the colliding eigenvalues is different. This scenario is illustrated in Fig.~\\ref{fig:krein}. That figure also illustrates the quadrifold symmetry of the stability spectrum of the solution of a Hamiltonian system: for each eigenvalue $\\lambda\\in \\mathbb{C}$, $\\lambda^*$, $-\\lambda$ and $-\\lambda^*$ are also eigenvalues. Here $\\lambda^*$ denotes the complex conjugate of $\\lambda$. It should be noted that the occurrence of a collision is required for eigenvalues to leave the imaginary axis, due to the quadrifold symmetry of the spectrum.\n\n\\begin{figure}[tb]\n\\def3.2in}\\input{bubbletrack.pdf_tex{6in}\n\\centerline{\\hspace*{1in}\\input{krein.pdf_tex}}\n\\caption{\\la{fig:krein} Colliding eigenvalues in the complex plane as a parameter is increased. On the left, two eigenvalues are moving towards each other on the positive imaginary axis, accompanied by a complex conjugate pair on the negative imaginary axis. In the middle, the eigenvalues in each pair have collided. On the right, a Hamiltonian Hopf bifurcation occurs: the collided eigenvalues separate, leaving the imaginary axis (implying that the two Krein signatures were different).\n    }\n\\end{figure}\n\n    Thus we calculate the Krein signature of any coinciding eigenvalues, obtained in Step~5. If these Krein signatures are equal, the eigenvalues will remain on the imaginary axis as the amplitude is increased. Otherwise, the eigenvalues may leave the imaginary axis, through a so-called Hamiltonian Hopf bifurcation \\cite{vandermeer}, resulting in instability. Thus we establish a necessary condition for the instability of periodic solutions of small amplitude. The Krein signature condition cannot be expressed entirely in terms of the dispersion relation, and the coefficients of $H^0_c$ are required as well. Please refer to the next two sections for details.\n\n\\end{enumerate}\n\nAlthough all calculations are done for the zero-amplitude solutions at the starting point of a bifurcation branch, the continuous dependence of the stability spectrum on the parameters in the problem \\cite{hislop}, including the velocity of the traveling wave or the amplitude of the solutions, guarantees that the stability conclusions obtained persist for solutions of small amplitude. Thus meaningful conclusions about solutions in the shaded regions of the bifurcation branches of Fig.~\\ref{fig:bif} are reached, despite the fact that we are unable to say anything about the size of the maximal amplitude for which these conclusions are valid.\n\n\\vspace*{0.1in}\n\n{\\bf Remarks.}\n\n\\begin{itemize}\n\n\\item Throughout this introduction, and in fact throughout much of this manuscript, our emphasis is on generality and usability as opposed to rigor. As a consequence some of the statements made above are necessarily vague: more precise statements would limit the generality aimed for. Within the context of more specific examples, more precision may be possible. Along the same lines, we have limited ourselves to the most generic case of two eigenvalues colliding on the imaginary axis. Lastly, since we are assuming an initial ({\\em i.e.}, for a starting value of 0 for the amplitude parameter) situation that is neutrally stable, our only interest is in collisions on the imaginary axis. If eigenvalues are present off the imaginary axis for the zero-amplitude solution, such solutions are already spectrally unstable, and the continuous dependence of the spectrum on its parameters \\cite{hislop} guarantees instability for solutions of small amplitude as  a consequence.\n\n\\item If eigenvalues collide at the origin, their Krein signature is zero (this follows immediately from its definition) and no conclusion can be drawn about whether or not colliding eigenvalues leave the imaginary axis or not. Because of this, the methods discussed here allow only for the study of so-called { high-frequency instabilities.} Indeed, if eigenvalues collide away from the origin, the unstable perturbations do not only grow in magnitude exponentially, they also display an oscillation of frequency $\\omega$ that is equal to the non-zero imaginary part of the colliding eigenvalues. In this sense {high frequency} is more accurately described as non-zero frequency. We use the high-frequency name to distinguish the instabilities investigated here from the modulational instability \\cite{zakharovostrovsky}, which is a consequence of eigenvalues colliding at the origin. Further, as seen in the examples below, often the presence of one high-frequency instability is accompanied by a sequence of such instabilities with increasing frequencies tending to infinity.\n\n    The study of the modulational instability requires a different set of techniques, see for instance \\cite{benjaminperiodic, whitham1, zakharovostrovsky} for different classical approaches. More recently, a general framework was developed by Bronski, Johnson and others (see {\\em e.g.} \\cite{bronskijohnson1, hurjohnson}). Collisions at the origin are common, since any Lie symmetry of the underlying problem gives rise to a zero eigenvalue in the stability problem \\cite{gss1, kapitulapromislow}. For this same reason, such collisions typically involve more than two eigenvalues (for instance, the original Benjamin-Feir instability involves four \\cite{benjaminperiodic, bridgesmielke}), which is one of the reasons why their treatment is often complicated: its analysis is not only technical but also involves tedious calculations, as exemplified recently in a paper by Hur and Johnson \\cite{hurjohnson} on the modulational instability for periodic solutions of the Whitham equation.\n\n\\item Following the Floquet decomposition, we use eigenfunctions given by a single Fourier mode since the linear stability problem for the zero-amplitude solutions has constant coefficients. Lastly, the calculation of the Krein signature involves only the finite-dimensional eigenspace of the eigenvalue under consideration. Thus, in essence, all the calculations done in this paper are finite dimensional, as they are in \\cite{mackaysaffman}, for instance. As was noted there, for equations with real-valued solutions, it is necessary to consider the eigenfunctions corresponding to Floquet exponent $\\mu$ and $-\\mu$ simultaneously, since the eigenfunctions with $-\\mu$ are the complex conjugates of those with $\\mu$.\n\n\\item We call an equation dispersive if all branches of its dispersion relation $\\omega(k)$ are real valued for $k\\in \\mathbb{R}$. It is easy to see that there exist linear, constant-coefficient Hamiltonian systems that are not dispersive. An explicit example is\n    \\begin{equation}\n    \\left\\{\n    \\begin{array}{rcl}\n    q_t&=&q_{xx},\\\\\n    p_t&=&-p_{xx}.\n    \\end{array}\n    \\right.\n    \\end{equation}\nThis (admittedly bizarre) system is Hamiltonian with canonical Poisson structure\n\\begin{equation}\\la{canonicalJ}\nJ=\\left(\n\\begin{array}{rr}\n0 & 1\\\\\n-1 & 0\n\\end{array}\n\\right),\n\\end{equation}\nand Hamiltonian $H=-\\int_0^{2\\pi} q_x p_x dx$. The branches of its dispersion relation are given by $\\omega_{1,2}(k)=\\mp i k^2$.\n\nIt is an interesting question whether there exist dispersive systems that are not Hamiltonian. One may be tempted to consider an example like\n\\begin{equation}\n    \\left\\{\n    \\begin{array}{rcl}\n    q_t&=&a q_{x},\\\\\n    p_t&=&b p_{x}.\n    \\end{array}\n    \\right.\n    \\end{equation}\nFor $a, b \\in \\mathbb{R}$ this system is dispersive ($\\omega_1=-a k$, $\\omega_2=-b k$), but it is not Hamiltonian with canonical structure. However, allowing for the noncanonical structure\n\\begin{equation}\nJ=\\left(\n\\begin{array}{rr}\n\\partial_x & 0\\\\\n0 & \\partial_x\n\\end{array}\n\\right),\n\\end{equation}\nthe system is easily found to have the Hamiltonian $H=\\frac{1}{2}\\int_0^{2\\pi}(a q^2+b p^2)dx$. We conjecture that no systems exist that are dispersive but not Hamiltonian. As demonstrated by the example above, the question is difficult to analyze, since different forms of the Poisson operator have to be allowed.\n\n\\item The investigation of the Hamiltonian structure of the zero-amplitude solution follows earlier work by Zakharov, Kuznetsov, and others. A review is available in \\cite{zakharovmusher}. The stability of the trivial solution is also investigated there, with stability criteria given entirely in terms of the branches of the dispersion relation. Using canonical perturbation theory, Hamiltonians containing higher-than-quadratic terms are considered. This is used to consider the impact of nonlinear effects on the dynamics of the trivial solution. At this point, connections to resonant interaction theory \\cite{benneyRIT, hammackhendersonRIT, phillipsRIT} become apparent, as they will in what follows. Although the physical reasoning leading to resonant interaction theory and that leading to the criteria established below is different, it is clear that connections exist. \n\n\\end{itemize}\n\n\\section{Motivating example}\\la{sec:whitham}\n\nOur investigations began with the study of the so-called Whitham equation \\cite{WhithamEq}, \\cite[page 368]{whitham}. The equation is usually posed on the whole line, for which the equation satisfied by $u(x,t)$ is\n\n\\begin{equation}\\la{whitham}\nu_t+N(u)+\\int_{-\\infty}^\\infty K(x-y)u_y(y,t) dy=0.\n\\end{equation}\n\n\\noindent The term $N(u)$ denotes the collection of all nonlinear terms in the equation. It is assumed that $\\lim_{\\epsilon\\rightarrow 0}N(\\epsilon u)\/\\epsilon=0$. The last term encodes the dispersion relation of the linearized Whitham equation. The kernel $K(x)$ is the inverse Fourier transform of the phase speed $c(k)$, where $c(k)=\\omega(k)\/k$, with $\\omega(k)$ the dispersion relation. Here $c(k)$ is assumed to be real valued and nonsingular for $k\\in \\mathbb{R}$. Thus\n\n\\begin{equation}\nK(x)=\\frac{1}{2\\pi}\\Xint-_{-\\infty}^\\infty c(k)e^{ikx}dk,\n\\end{equation}\n\n\\noindent where $\\Xint-$ denotes the principal value integral. Depending on $c(k)$, this equation may have to be interpreted in a distributions sense \\cite{EhrnstromKalish, stakgold}. Letting $u\\sim \\exp(ikx-\\omega(k) t)$, it is a straightforward calculation to see that the dispersion relation of the linear Whitham equation is $\\omega(k)$. In fact, the linear Whitham equation is easily seen to be a rewrite of the linear evolution equation \\cite{as}\n\n\\begin{equation}\\la{whithamasform}\nu_t=-i \\omega(-i \\partial_x)u,\n\\end{equation}\n\n\\noindent where $\\omega(-i \\partial_x)$ is a linear operator with odd symbol $\\omega(k)$: $\\omega(k)=-\\omega(-k)$, the dispersion relation considered. Indeed, letting $\\omega(-i \\partial_x)$ act on\n\n\\begin{equation}\nu(x,t)=\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty e^{ikx}\\hat u(k,t)dk,\n\\end{equation}\n\n\\noindent and replacing $\\hat u(k,t)$ by\n\n\\begin{equation}\n\\hat u(k,t)=\\int_{-\\infty}^\\infty e^{-ik y} u(y,t) dy,\n\\end{equation}\n\n\\noindent the linear part of \\rf{whitham} is obtained after one integration by parts. We restrict our considerations to odd dispersion relations, to ensure the reality of the Whitham equation.\n\nOne of Whitham's reasons for writing down the Whitham equation \\cite{WhithamEq, whitham} was to describe waves in shallow water (leading to the inclusion of a KdV-type nonlinearity $N(u)\\sim u u_x$) that feel the full dispersive response of the one-dimensional water wave problem (without surface tension), for which\n\n\\begin{equation}\\la{wwdispersion}\n\\omega^2(k)=gk\\tanh(kh),\n\\end{equation}\n\n\\noindent with $g$ the acceleration of gravity and water depth $h$. It is common to choose $c(k)=\\omega_1(k)\/k>0$ in \\rf{whitham}, so that $\\omega_1(k)$ is the root of \\rf{wwdispersion} with the same sign as $k$. In what follows, we refer to this choice as the Whitham equation. The stability of periodic traveling wave solutions of the Whitham  equation has received some attention recently. Notable are \\cite{ehrnstromgroveswahlen}, where the focus is on solitary waves, and \\cite{hurjohnson}, where the modulational instability of small-amplitude periodic solutions is emphasized, although the main result for the high-frequency instabilities discussed in Example~\\ref{ex:whitham} is included as well. Most recently, the spectral stability of periodic solutions of the Whitham equation was examined in \\cite{SanfordKodamaCarterKalisch}. The goal of considering an equation like \\rf{whitham} as opposed to the Korteweg-de~Vries equation or other simpler models is to capture as much of the dynamics of the full water wave problem as possible, without having to cope with the main difficulties imparted by the Euler water wave problem \\cite{vandenboek} ({\\em e.g.}, it is a nonlinear free boundary-value problem, the computation of its traveling-wave solutions is a nontrivial task, {\\em etc}). One of the important aspects of the dynamics of a nonlinear problem is the (in)stability of its traveling wave solutions. It was shown explicitly in \\cite{deconinckoliveras1} that periodic traveling wave solutions of the one-dimensional Euler water wave problem are spectrally unstable for all possible values of their parameters $h$, $g$, amplitude, and wave period. The nature of the instabilities depends on the value of these parameters. As is well known, waves in deep water are susceptible to the Benjamin-Feir or modulational instability (see \\cite{zakharovostrovsky} for a review). In addition, waves in both deep and shallow water of all non-zero amplitudes are unstable with respect to high-frequency perturbations: these are perturbations whose growth rates do not have a small imaginary part, resulting in oscillatory behavior in time, independent of the spatial behavior of the unstable modes. The work in \\cite{SanfordKodamaCarterKalisch} does not reveal any high-frequency instabilities for solutions of small amplitude in water that is shallow, in the context of the Whitham equation. Thus an important aspect of the Euler water wave dynamics is absent from \\rf{whitham}. We will provide an analytical indication that the Whitham equation misses the presence of these instabilities, while explaining why they are missed. This explanation leads to a way to address this problem.\n\nFor suitable $N(u)$, the Whitham equation \\rf{whitham} is a Hamiltonian system. In fact, for our considerations, it matters only that the linearized Whitham equation is Hamiltonian. The Lagrangian structure with the dispersion relation given by $\\omega_1(k)$ as in \\rf{wwdispersion} was already written down by Whitham in \\cite{whitham3}, from which the Hamiltonian structure easily follows. Explicitly, for the linearized Whitham equation posed on the whole line, for any odd $\\omega(k)$ we have\n\n\\begin{equation}\nH=-\\frac{1}{2}\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^\\infty K(x-y) u(x)u(y)dx dy,\n\\end{equation}\n\n\\noindent with $J=\\partial_x$. Then\n\n\\begin{equation}\\la{whithamham}\nu_t=\\partial_x \\frac{\\delta H}{\\delta u}.\n\\end{equation}\n\n\\noindent If instead the linearized equation is posed with periodic boundary conditions $u(x+L,t)=u(x,t)$, it follows immediately from \\rf{whithamasform} that we have\n\n\\begin{equation}\\la{whithamper}\nu_t+\\int_{-L\/2}^{L\/2} K(x-y)u(y)dy=0,\n\\end{equation}\n\n\\noindent where we have used a Fourier series instead of a Fourier transform. Further,\n\n\\begin{equation}\nK(x)=\\frac{1}{L}\\sum_{j=-\\infty}^\\infty c(k_j)e^{ik_jx},\n\\end{equation}\n\n\\noindent and $k_j=2\\pi j\/L$, $j\\in \\mathbb{Z}$. The Hamiltonian formulation for the periodic Whitham equation \\rf{whithamper} is also given by \\rf{whithamham}, but with\n\n\\begin{equation}\nH=-\\frac{1}{2} \\int_{-L\/2}^{L\/2}\\int_{-L\/2}^{L\/2} K(x-y) u(x) u(y) dx dy.\n\\end{equation}\n\n\\noindent In fact, a formal $L\\rightarrow \\infty$ immediately results in the recovery of the equation posed on the whole line. Thus the Whitham equation and its periodic solutions fit in to the framework developed in this manuscript. It is one of many examples we use below. Other notable examples are the Euler water wave problem (as expected, allowing us to check our results with those of MacKay \\& Saffman \\cite{mackaysaffman}), the KdV equation, the Sine-Gordon equation, {\\em etc}. We are particularly interested in the comparison between the results for the Euler water wave problem and those for the Whitham equation.\n\nThe results of Examples~\\ref{ex:whitham} and~\\ref{ex:ww} show that the Whitham equation cannot possess the high-frequency instabilities present in the water wave problem. This leads us to propose a new model equation, a so-called Boussinesq-Whitham or bidirectional Whitham equation. This equation is shown to at least satisfy the same necessary condition for the presence of high-frequency instabilities as the water wave problem, and these high-frequency instabilities originate from the same points on the imaginary axes as they do for the Euler equations. However, it is easily seen that this equation is ill posed for solutions that do not have zero average. As such it is a poor candidate to replace the Whitham equation \\rf{whitham} as a shallow-water equation with the correct dispersive behavior.\n\n\\vspace*{0.1in}\n\n{\\bf Remark.} If $\\omega(k)$ is not odd, then the function $u$ solving the linear Whitham equation \\rf{whithamasform} is necessarily complex. The problem is Hamiltonian:\n\n\\begin{equation}\\la{whithamcomplex}\nu_t=-i \\frac{\\delta H}{\\delta u^*},\n\\end{equation}\n\n\\noindent where $*$ denotes the complex conjugate. The Poisson structure is\n\n\\begin{equation}\nJ=\\left(\n\\begin{array}{rr}\n0 & -i\\\\\ni & 0\n\\end{array}\n\\right),\n\\end{equation}\n\n\\noindent although usually one writes only the first of the two evolution equations, omitting the equation that is the complex conjugate of \\rf{whithamcomplex}. The Hamiltonian is given by\n\n\\begin{equation}\nH=\\int_0^{2\\pi} u^* \\omega(-i\\partial_x)u dx.\n\\end{equation}\n\n\\noindent A real formulation in terms of the real and imaginary parts of $u$ is possible as well (using the canonical transformation $u=(q+i p)\/\\sqrt{2}$), resulting in a canonical Hamiltonian structure. The linear Whitham equation \\rf{whithamcomplex} is often rewritten in Fourier space using the discrete Fourier transform, due to the periodic boundary conditions. This leads to the Hamiltonian $H=\\sum_{n=-\\infty}^\\infty \\omega(n) z_n z_n^*$.\n\n\\section{Scalar Hamiltonian PDEs}\\label{sec:scalar}\n\nIn this section, we investigate the stability of $2\\pi$-periodic traveling wave solutions of Hamiltonian systems of the form\n\n\\begin{equation}\\la{scalar}\nu_t=\\partial_x \\frac{\\delta H}{\\delta u},\n\\end{equation}\n\n\\noindent where $u(x,t)$ is a scalar real-valued function. Thus $J=\\partial_x$. Since this Poisson operator is singular, all equations of this form conserve the quantity $\\int_0^{2\\pi} u dx$, which is the Casimir for this Poisson operator. Systems of this form include the Korteweg- de~Vries equation \\cite{gardner, zakharovfaddeev} and its many generalizations, the Whitham equation \\rf{whitham}, and many others. As mentioned above, our only interest is in the linearization of these equations around their trivial solution. We write the quadratic part $H^0$ of $H$ as\n\n\\begin{equation}\\la{h0scalar}\nH^0=-\\frac{1}{2}\\int_0^{2\\pi} \\sum_{n=0}^\\infty \\alpha_n u_{nx}^2 dx,\n\\end{equation}\n\n\\noindent where the coefficients $\\alpha_n\\in \\mathbb{R}$. As before, indices on $u$ denote partial derivatives. Specifically, $u_{nx}$ denotes $\\partial_x^{n} u$.\n\nFor most examples, the number of terms in \\rf{h0scalar} is finite, and all but a few of the coefficients $\\alpha_n$ are nonzero. For the Whitham equation \\rf{whitham}, the number of nonzero terms is infinite, but convergence is easily established. Note that \\rf{h0scalar} is the most general form of a quadratic Hamiltonian depending on a single function. Indeed, a term in $H^0$ containing $u_{mx} u_{nx}$, with $m$ and $n$ positive integers, may be reduced to a squared term using integration by parts.\n\nUsing the notation \\rf{h0scalar}, the linearized equation is\n\n\\begin{equation}\\la{scalarlinear}\nu_t=-\\sum_{n=0}^\\infty (-1)^n \\alpha_n u_{(2n+1)x}.\n\\end{equation}\n\n\\noindent  We proceed with the six steps outlined in the introduction.\n\n\\begin{enumerate}\n\n\\item {\\bf Quadratic Hamiltonian.} The modified Hamiltonian $H_c^0$ is given by\n\\begin{equation}\nH_c^0=\\frac{c}{2}\\int_0^{2\\pi}u^2dx-\\frac{1}{2}\\int_0^{2\\pi} \\sum_{n=0}^\\infty \\alpha_n u_{nx}^2 dx.\n\\end{equation}\n\n\\item {\\bf Dispersion relation.} For equations of the form \\rf{scalar}, the dispersion relation has only a single branch:\n\n\\begin{equation}\\la{scalardispersion}\n\\omega(k)=\\sum_{n=0}^\\infty \\alpha_n k^{2n+1}.\n\\end{equation}\n\nThe absence of even powers of $k$ in \\rf{scalardispersion} is due to our imposition that \\rf{scalarlinear} is a conservative equation, {\\em i.e.}, there is no dissipation. All integers are allowable $k$ values, since we have equated the period to be $2\\pi$. The equation \\rf{scalarlinear} may be written as\n\n\\begin{equation}\nu_t=-i \\omega(-i\\partial_x)u.\n\\end{equation}\n\n\\item {\\bf Bifurcation branches.} Since \\rf{scalar} is scalar, only one branch can bifurcate away from the trivial solution. To find the corresponding value of $c$, we write \\rf{scalarlinear} in a moving frame as\n\n\\begin{equation}\\la{scalarlinearmoving}\nu_t-c u_x=i \\omega(i\\partial_x)u.\n\\end{equation}\n\n\\noindent This equation has its own dispersion relation given by\n\n\\begin{equation}\n\\Omega(k)=\\omega(k)-c k,\n\\end{equation}\n\n\\noindent \\sloppypar obtained by looking for solutions of the form $u=\\exp(ikx-i \\Omega t)$. Letting $u=\\sum_{n=-\\infty}^\\infty \\exp(i n x) \\hat u_n$, it follows that $\\partial_t \\hat u_n=-i \\Omega(n)\\hat u_n$. Thus a nonzero stationary solution may exist provided $\\Omega(N)=0$, for $N\\in \\mathbb{N}, N\\neq 0$. We have used the oddness of $\\Omega(N)$ to restrict to strictly positive values of $N$. Thus the starting point of the bifurcation branch in the (speed, amplitude)-plane is $(c,0)$, where $c$ is determined by\n\n\\begin{equation}\nc=\\frac{\\omega(N)}{N},\n\\end{equation}\n\n\\noindent for any integer $N>0$. Choosing $N>1$ implies that the fundamental period of the solutions is not $2\\pi$, but $2\\pi\/N$. In practice, we choose $N=1$. A Fourier series approximation to the explicit form of the small-amplitude solutions corresponding to this bifurcation branch may be obtained using a standard Stokes expansion \\cite{stokes, whitham}.\n\n\\item {\\bf Stability spectrum.} In order to compute the stability spectrum associated with the zero-amplitude solution at the start of the bifurcation branch, we let $u(x,t)=U(x)\\exp(\\lambda t)+$c.c., where c.c. denotes the complex conjugate of the preceding term. As usual, if any $\\lambda$ are found for which the real part is positive, the solution is spectrally unstable \\cite{kapitulapromislow}. All bounded eigenfunctions $U(x)$ may be represented as\n\n\\begin{equation}\\la{efscalar}\nU(x)=\\sum_{n=-\\infty}^\\infty a_n e^{i(n+\\mu)x},\n\\end{equation}\n\n\\noindent where $\\mu\\in (-1\/2,1\/2]$ is the Floquet exponent. Such a representation for $U(x)$ is valid even for solutions on the bifurcation branch of nonzero amplitude \\cite{deconinckkutz1}. Since \\rf{scalarlinearmoving} is a problem with constant coefficients, only a single term in \\rf{efscalar} is required. We obtain\n\n\\begin{equation}\\la{evscalar}\n\\lambda_n^{(\\mu)}=-i\\Omega(n+\\mu)=-i \\omega(n+\\mu)+i(n+\\mu)c, ~~n\\in \\mathbb{Z}.\n\\end{equation}\n\n\\noindent As expected, all eigenvalues are imaginary and the zero-amplitude solution is neutrally stable. For a fixed value of $\\mu$, \\rf{evscalar} gives a point set on the imaginary axis in the complex $\\lambda$ plane. As $\\mu$ is varied in $(-1\/2, 1\/2]$, these points trace out intervals on the imaginary axis. Depending on $\\omega(k)$, these intervals may cover the imaginary axis.\n\n\\item {\\bf Collision condition.} The most generic scenarios for two eigenvalues given by \\rf{evscalar} to collide are that (i) two of them are zero, and they collide at the origin, and (ii) two of them are equal, but nonzero. We ignore the first possibility, since the next step proves to be inconclusive for this case,  as discussed in the introduction. The second possibility requires $\\lambda_n^{(\\mu)}=\\lambda_m^{(\\mu)}$, for some $m,n\\in \\mathbb{Z}$, $m\\neq n$, fixed $\\mu\\in (-1\/2, 1\/2]$, and $\\lambda_n^{(\\mu)}$, $\\lambda_m^{(\\mu)}\\neq 0$. This may be rewritten as\n\n    \\begin{equation}\\la{scalarcollision}\n    \\frac{\\omega(n+\\mu)-\\omega(m+\\mu)}{n-m}=\\frac{\\omega(N)}{N}, ~~m,n\\in \\mathbb{Z}, m\\neq n \\mbox{~and~} \\mu\\in (-1\/2, 1\/2].\n    \\end{equation}\n\n\\noindent This equation has an elegant graphical interpretation: the right-hand side is fixed by the choice of $N$, fixing the bifurcation branch in Step~3. It represents the slope of a line through the origin and the point $(N,\\omega(N))$ in the $(k, \\omega)$ plane. The left hand side is the slope of a line in the same plane passing through the points $(n+\\mu, \\omega(n+\\mu))$ and $(m+\\mu, \\omega(m+\\mu))$, see Fig.~\\ref{fig:scalarslope}.\n\nEven though the graph of the dispersion relation admits parallel secant lines, this is not sufficient for a solution of \\rf{scalarcollision}, as it is required that their abscissas are an integer apart. Nevertheless, the graphical interpretation can provide good intuition for solving the collision condition, which typically has to be done numerically.\n\n\\begin{figure}[tb]\n\\def3.2in}\\input{bubbletrack.pdf_tex{5.8in}\n\\centerline{\\hspace*{0in}\\input{scalarslope.pdf_tex}}\n\\caption{\\la{fig:scalarslope} The graphical interpretation of the collision condition \\rf{scalarcollision}. The solid curve is the graph of the dispersion relation $\\omega(k)$. The slope of the dashed line in the first quadrant is the right-hand side in \\rf{scalarcollision}. The slope of the parallel dotted line is its left-hand side.\n    }\n\\end{figure}\n\n\\item {\\bf Krein signature.} The Krein signature of an eigenvalue is the sign of the Hamiltonian $H^0_c$ evaluated on the eigenspace associated with the eigenvalue. We are considering two simple eigenvalues colliding, thus the eigenspace for each eigenvalue consists of multiples of the eigenfunction only. To allow for eigenfunctions of the form $a_n \\exp(i(n+\\mu)x+\\lambda_n^{(\\mu)}t)+$c.c, which are not $2\\pi$-periodic (unless $\\mu=0$), it is necessary to replace the integral in \\rf{h0scalar} with a whole-line average. More details on this process are found, for instance, in \\cite{deconinckoliveras1}. A simple calculation shows that the contribution to $H^0_c$ from the $(n,\\mu)$ mode is proportional to the ratio of $\\Omega(n+\\mu)\/(n+\\mu)$:\n\n    \\begin{equation}\\la{scalarkrein}\n    H^0_c|_{(n,\\mu)}\\sim -|a_n|^2 \\frac{\\Omega(n+\\mu)}{n+\\mu}.\n    \\end{equation}\n\n\\noindent Other terms are present in the Hamiltonian density, but they have zero average. The sign of this expression is the Krein signature of the eigenvalue $\\lambda_n^{(\\mu)}$. Thus a necessary condition for $\\lambda_n^{(\\mu)}$ and $\\lambda_m^{(\\mu)}$ to leave the imaginary axis for solutions of non-zero amplitude is that the signs of \\rf{scalarkrein} with $(n,\\mu)$ and $(m, \\mu)$ are different, contingent on $\\mu$, $m$ and $n$ satisfying \\rf{scalarcollision}. Explicitly, this condition is\n\n\\begin{equation}\\la{scalarkreincondition1}\n\\mbox{sign}\\left[\\frac{\\omega(n+\\mu)}{n+\\mu}-c\\right]\\neq \\mbox{sign}\\left[\\frac{\\omega(m+\\mu)}{m+\\mu}-c\\right].\n\\end{equation}\n\n\\noindent Alternatively, the product of the left-hand side and the right-hand side should be negative. Using \\rf{scalarcollision}, \\rf{scalarkreincondition1} becomes\n\n\\begin{equation}\\la{scalarkreincondition2}\n(n+\\mu)(m+\\mu)<0,\n\\end{equation}\n\n\\noindent or, provided $mn\\neq 0$, and using that $\\mu \\in (-1\/2, 1\/2]$,\n\n\\begin{equation}\\la{scalarkreincondition3}\nn m<0.\n\\end{equation}\n\n\\end{enumerate}\n\n{\\bf Remarks.}\n\n\\begin{itemize}\n\n\\item It is clear from \\rf{scalarkrein} why our methods do not lead to any conclusions about collisions of eigenvalues at the origin. If $\\lambda_n^{(\\mu)}=0$, then $\\Omega(n+\\mu)=0$, and the contribution to the Hamiltonian of such a mode vanishes. As a consequence, the associated Krein signature is zero.\n\n\\item When the theory of \\cite{kollarmiller} is restricted to the case of solutions of zero-amplitude, so as to recover the constant coefficient stability problem, the graphical stability criterion given there coincides with the one presented here.\n\n\\end{itemize}\n\n\\vspace*{0.1in}\n\nWe conclude our general considerations of this section with the following summary.\n\n\\vspace*{0.1in}\n\nAssume that the linearization of the scalar Hamiltonian system \\rf{scalar} is dispersive ({\\em i.e.,} its dispersion relation $\\omega(k)$ is real valued for $k\\in \\mathbb{R}$). Let $N$ be a strictly positive integer.\nConsider $2\\pi\/N$-periodic traveling wave solutions of this system of sufficiently small-amplitude and with velocity sufficiently close to $\\omega(N)\/N$. In order for these solutions to be spectrally unstable with respect to high-frequency instabilities as a consequence of two-eigenvalue collisions, it is necessary that there exist $n$, $m~\\in \\mathbb{Z}$, $n\\neq m$, $\\mu \\in (-1\/2, 1\/2]$ for which\n\n\\begin{equation}\n\\frac{\\omega(n+\\mu)}{n+\\mu}\\neq \\frac{\\omega(N)}{N}, ~~\\frac{\\omega(m+\\mu)}{m+\\mu}\\neq \\frac{\\omega(N)}{N},\n\\end{equation}\n\n\\noindent such that\n\n\\begin{equation}\n\\frac{\\omega(n+\\mu)-\\omega(m+\\mu)}{n-m}=\\frac{\\omega(N)}{N},\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n(m+\\mu)(n+\\mu) <0.\n\\end{equation}\n\nNext we proceed with some examples.\n\n\\subsection{The (generalized) Korteweg - de Vries equation}\n\nWe consider the generalized KdV (gKdV) equation\n\n\\begin{equation}\\la{gkdv}\nu_t+\\sigma u^n u_x +u_{xxx}=0,\n\\end{equation}\n\n\\noindent where we restrict $n$ to integers 1 or greater. Here $\\sigma$ is a constant coefficient, chosen as convenient. Important special cases discussed below are the KdV equation ($n=1$) and the modified KdV (mKdV) equation ($n=2$). Many of the details below extend easily to more general nonlinearities, with the main requirement being that the linearized equation is $u_t+u_{xxx}=0$. The stability of periodic solutions of the gKdV equation has received some attention recently \\cite{bronskijohnson1, bronskijohnsonkapitula, todd, johnsonzumbrunbronski}. For the integrable cases $n=1$ and $n=2$, more detailed analysis is possible, see \\cite{bottmandeconinck, deconinckkapitula, deconincknivalamkdv}. We do not claim to add anything new to these discussions, but we wish to use this example to illustrate how the six-step process outlined in this section leads to easy conclusions before moving on to more complicated settings.\n\n\\begin{enumerate}\n\n\\item The modified Hamiltonian is given by\n\n\\begin{equation}\nH^0_c=\\frac{1}{2}\\int_0^{2\\pi} (u_x^2+c u^2)dx.\n\\end{equation}\n\n\\item The dispersion relation is\n\n\\begin{equation}\n\\omega=-k^3.\n\\end{equation}\n\n\\item Bifurcation branches in the ($c$, amplitude)-plane start at $(c,0)$, with\n\n\\begin{equation}\nc=\\frac{\\omega(k)}{k}=-k^2.\n\\end{equation}\n\n\\noindent Since we desire $2\\pi$ periodic solutions, we choose $k=1$. Any choice $k=N$, where $N$ is a non-zero integer is allowed. Choosing $k=1$, bifurcation branches start at $(-1,0)$.\n\nFor the integrable cases $n=1$ (KdV) and $n=2$ (mKdV), these bifurcation branches may be calculated in closed form. For the KdV equation in a frame traveling with speed $c$, the $2\\pi$-periodic solutions are given by (with $\\sigma=1$)\n\n\\begin{equation}\\la{cnkdv}\nu=\\frac{12\\kappa^2 K^2(\\kappa)}{\\pi^2}{\\rm cn}^2\\left(\\frac{K(\\kappa)x}{\\pi},\\kappa\\right),\n\\end{equation}\n\n\\noindent where ${\\rm cn}$ denotes the Jacobian elliptic cosine function and $K(\\kappa)$ is the complete elliptic integral of the first kind \\cite{dlmf, dlmfbook}. Further,\n\n\\begin{equation}\\la{ckdv}\nc(\\kappa)=\\frac{4K^2(\\kappa)}{\\pi^2}(2\\kappa^2-1),\n\\end{equation}\n\n\\noindent resulting in an explicit bifurcation curve $(c(\\kappa), 12\\kappa^2 K^2(\\kappa)\/\\pi^2$), parameterized by the elliptic modulus $\\kappa\\in [0,1)$.   This bifurcation curve is shown in Fig.~\\ref{fig:kdvbranch}a.\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\multicolumn{2}{c}{\\def3.2in}\\input{bubbletrack.pdf_tex{3.5in}\n\\input{kdvbranch.pdf_tex}}\\\\\n\\multicolumn{2}{c}{(a)}\\\\\n~&~\\\\\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{mkdvcnbranch.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{mkdvsnbranch.pdf_tex}\\\\\n(b) & (c)\n\\end{tabular}\n\\caption{\\la{fig:kdvbranch} The amplitude {\\em vs.} $c$ bifurcation plots for the traveling-wave solutions of the generalized KdV equation \\rf{gkdv}. (a) The KdV equation, $n=1$, for the cnoidal wave solutions \\rf{cnkdv}. (b) The mKdV equation, $n=2$, for the cnoidal wave solutions \\rf{mkdvcn}. Lastly, (c) shows the bifurcation plot for the snoidal wave solutions \\rf{mkdvsn} of mKdV, $n=2$.\nNote that all bifurcation branches start at $(-1,0)$, as stated above.\n    }\n\\end{figure}\n\nFor the mKdV equation ($n=2$), different families of traveling-wave solutions exist \\cite{deconincknivalamkdv}. We consider two of the simplest. For $\\sigma=3$ (focusing mKdV), a family of $2\\pi$-periodic solutions is given by\n\n\\begin{equation}\\la{mkdvcn}\nu=\\frac{2\\sqrt{2}\\kappa K(\\kappa)}{\\pi}{\\rm cn}\\left(\\frac{2 K(\\kappa)x}{\\pi},\\kappa\\right),\n\\end{equation}\n\n\\noindent with $c(\\kappa)$ given by \\rf{ckdv}, resulting in an explicit bifurcation curve $(c(\\kappa), 2\\sqrt{2}\\kappa K(\\kappa)\/\\pi)$, parameterized by the elliptic modulus $\\kappa\\in [0,1)$.   This bifurcation curve is shown in Fig.~\\ref{fig:kdvbranch}b. It should be noted that a solution branch exists where the solution is expressed in terms of the Jacobian ${\\rm dn}$ function \\cite{dlmf, dlmfbook}: $u=\\sqrt(2)K(\\kappa){\\rm dn}(K(\\kappa)x\/\\pi,\\kappa)\/\\pi$, but this solution does not have a small-amplitude limit and our methods do not apply directly to it. Rather, the solutions limit to the constant solution $u=1\/\\sqrt{2}$ as $\\kappa\\rightarrow 0$. A simple transformation $v=u-1\/\\sqrt{2}$ transforms the problem to one where our methods apply.\n\nFor $\\sigma=-3$ (defocusing mKdV), a period $2\\pi$ solution family is\n\n\\begin{equation}\\la{mkdvsn}\nu=\\frac{2\\sqrt{2}\\kappa K(\\kappa)}{\\pi}{\\rm sn}\\left(\\frac{2 K(\\kappa)x}{\\pi},\\kappa\\right),\n\\end{equation}\n\n\\noindent with $c(\\kappa)=-4(1+\\kappa^2)K^2(\\kappa)\/\\pi^2$. Here ${\\rm sn}$ is the Jacobian elliptic sine function \\cite{dlmf, dlmfbook}, resulting in an explicit bifurcation curve $(c(\\kappa), 2\\sqrt{2}\\kappa K(\\kappa)\/\\pi)$, parameterized by the elliptic modulus $\\kappa\\in [0,1)$.   This bifurcation curve is shown in Fig.~\\ref{fig:kdvbranch}c.\n\n\\item The stability spectrum is given by \\rf{evscalar}, with $\\omega(k)=-k^3$ and $c=-1$, resulting in\n\n\\begin{equation}\n\\lambda^{(n)}_\\mu=i(n+\\mu)(1+(n+\\mu)^2).\n\\end{equation}\n\nThese eigenvalues cover the imaginary axis, as $n$ and $\\mu$ are varied. The imaginary part of this expression is displayed in Fig.~\\ref{fig:threecover}a. For the sake of comparison with Fig.~2 in \\cite{bottmandeconinck} we let $\\mu\\in [-1\/4, 1\/4)$, which implies that $n$ is any half integer. The results of Fig.~2 in \\cite{bottmandeconinck} are for elliptic modulus $\\kappa=0.8$, implying a solution of moderate amplitude. The comparison of these two figures serves to add credence to the relevance of the results obtained using the zero-amplitude solutions at the start of the bifurcation branch.\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{threecover.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{kdvcollision.pdf_tex}\\\\\n(a) & (b)\n\\end{tabular}\n\\caption{\\la{fig:threecover} (a) The imaginary part of $\\lambda_n^{(\\mu)}\\in (-0.7, 0.7)$ as a function of $\\mu\\in[-1\/4, 1\/4)$. (b) The curves $\\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur at the origin only.\n    }\n\\end{figure}\n\n\\item With $n+\\mu=k$ and $m+\\mu=k+l$, for some $l\\in \\mathbb{Z}$, the collision condition \\rf{scalarcollision} is written as\n\n\\begin{equation}\nl^2+3kl+3k^2-1=0,\n\\end{equation}\n\n\\noindent where the trivial solution $l=0$ has been discarded. This is the equation of an ellipsoid in the $(k,l)$ plane. It intersects lines of nonzero integer $l$ in six integer points: $\\pm (1, -2)$, $\\pm(0, 1)$, $\\pm(1, -1)$. Since for all of these, $\\Omega(k)=0$, any collisions happen only at the origin $\\lambda_n^{(\\mu)}=0$. This is also illustrated in Fig.~\\ref{fig:threecover}b.\n\n\\item The final step of our process is preempted by the results of the previous step. No Krein signature of colliding eigenvalues can be computed, since no eigenvalues collide.\n\n\\end{enumerate}\n\nSince eigenvalues do not collide away from the origin they cannot leave the imaginary axis through such collisions and no high-frequency instabilities occur for small amplitude solutions of the gKdV equation. This result applies to the KdV and mKdV equations as special cases. The absence of high-frequency instabilities for small amplitude solutions is consistent with the results in, for instance, \\cite{bottmandeconinck, deconinckkapitula, todd}.\n\n\\begin{comment}\n\n\\subsection{The Kawahara equation}\n\nAs a second example, consider the so-called Kawahara or super KdV (sKdV) equation \\cite{akersgao, hauptboyd, kawahara}, whose linearization is given by\n\\begin{equation}\\la{skdv}\nu_t=\\alpha u_{xxx}+\\beta u_{5x},\n\\end{equation}\n\n\\noindent We intend to show that high-frequency instabilities of small-amplitude solutions are present for suitable choices of the parameters  $\\alpha$ and $\\beta$.\n\n\\begin{enumerate}\n\n\\item The modified Hamiltonian is $H^0_c=\\frac{1}{2}\\int_0^{2\\pi} (c u^2-\\alpha u_x^2 +\\beta u_{xx}^2)dx$.\n\n\\item The dispersion relation is $\\omega=\\alpha k^3-\\beta k^5$.\n\n\\item The bifurcation branch starts at $(c,0)=(\\alpha-\\beta,0)$, where we have chosen $N=1$ so that the minimal period of the solutions is $2\\pi$.\n\n\\item The elements of the stability spectrum are given by $\\lambda_n^{(\\mu)}=i(n+\\mu)(\\alpha-\\beta-\\alpha(n+\\mu)^2+\\beta(n+\\mu)^4)$.\n\n\\item It is easily seen graphically that the collision condition \\rf{scalarcollision} may be satisfied, see Fig.~\\ref{fig:superkdv}, where the values $\\alpha=1$ and $\\beta=1\/4$ were used. In fact, three distinct solutions ({\\em i.e.}, solutions giving rise to different collisions locations on the imaginary axis) exist: $(\\mu, \\lambda_n^{(\\mu)})=(0.2071149, 0.212841i)$, $(\\mu, \\lambda_n^{(\\mu)})=(0.2154767, 0.179833i)$, and $(\\mu, \\lambda_n^{(\\mu)})=(-0.3675445, 0.227684i)$. These collision locations are obtained with $(m,n)=(-2,1)$, $(m,n)=(-2,-1)$, and $(m,n)=(0,2)$, respectively.\n\n\\item Since $mn<0$, the colliding eigenvalues have opposing Krein signatures, and it is possible for the eigenvalues to leave the imaginary axis, leading to high-frequency instabilities.\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{superkdv.pdf_tex}\n&\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{kawaharacols.pdf_tex}\n\\\\\n(a)&(b)\n\\end{tabular}\n\\caption{\\la{fig:superkdv} (a) The dispersion relation for the sKdV equation \\rf{skdv}, with $\\alpha=1$, $\\beta=1\/4$. The line through the origin has slope $\\omega(1)\/1$, which represents the right-hand side of \\rf{scalarcollision}. The slope of the parallel line through the points $(m+\\mu,\\omega(m+\\mu))$, $(n+\\mu, \\omega(n+\\mu))$ represents the left-hand side. Here $m=-2$, $n=1$ and $\\mu\\approx 0.2071149219$. (b) The curves $\\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur also away from the origin.\n    }\n\\end{figure}\n\n\\end{enumerate}\n\nThe conclusions above hold for small-amplitude solutions, and they provide only a necessary condition for the presence of high-frequency instabilities. Below, we use numerical results to indicate that for the small-amplitude solutions of the sKdV equation the condition is also sufficient, and instabilities do occur. To our knowledge, the only previous work on the stability or instability of periodic solutions of the sKdV equation was by Haragus {\\em et al} \\cite{haraguslombardischeel}, where the spectral stability of periodic waves of sufficiently small amplitude propagating with small velocities is shown.\n\\begin{figure}[!tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{KawaharaProfile.pdf_tex}\n&\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{KawaharaSpectrum.pdf_tex}\n\\\\\n(a)&(b)\\\\\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{KawaharaBubble1ComplexLambda.pdf_tex}\n&\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{KawaharaBubble2ComplexLambda.pdf_tex}\\\\\n(c)&(d)\\\\\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{KawaharaBubble1MuLambda.pdf_tex}\n&\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\n\\hspace*{0in}\\input{KawaharaBubble2MuLambda.pdf_tex}\\\\\n(e)&(f)\n\\end{tabular}\n\\caption{\\la{fig:kawaharasol} (a) A $2\\pi$-periodic solution of the sKdV equation, with amplitude slightly above $10^{-4}$. (b) The numerically computed stability spectrum of this solution, with two regions corresponding to instabilities highlighted. Note the $4$-fold symmetry of the spectrum. (c) A blow-up of region~I in (b), showing that the bubble of high-frequency instabilities is centered around the value predicted by our theory. (d) Same, but for region~II. (e) The range of Floquet exponents $\\mu$ corresponding to the bubble shown in (c), again validating the theoretical zero-amplitude prediction. (f) Same as (e), but for the bubble shown in (d).\n    }\n\\end{figure}\n\nUsing numerical continuation methods \\cite{allgowergeorg}, we construct small-amplitude $2\\pi$-periodic solutions of the sKdV equation, see Fig.~\\ref{fig:kawaharasol}a. The spectrum of this solution is computed using the method of \\cite{deconinckkutz1}. It is shown in Fig.~\\ref{fig:kawaharasol}b. In order to capture the instabilities shown, an adaptive spacing on the Floquet exponent $\\mu \\in [-1\/2, 1\/2)$ has to be used, with more values chosen near the values of $\\mu$ for which collisions occur, as determined above. The Hamiltonian symmetry of the spectrum is clear in Fig.~\\ref{fig:kawaharasol}b. The regions I and II are zoomed in on, in Panels (c) and (d) respectively. A third bubble is barely discernible above region~II. The vertical extent of the bubble in Fig.~\\ref{fig:kawaharasol}d is $1.32\\,10^{-6}$. The Floquet exponents giving rise to the two bubbles with largest real part are shown together with the real part of those eigenvalues in Panels~(e) and~(f). The support of the $\\mu$ region in Fig.~\\ref{fig:kawaharasol}f is approximately $8.6\\,10^{-7}$. It should be emphasized that since there does not appear to be a modulation instability (see Fig.~\\ref{fig:kawaharasol}b), the high-frequency instabilities are the only instabilities of the small-amplitude $2\\pi$-periodic solutions.\n\n\\end{comment}\n\n\\subsection{The Whitham equation}\\la{ex:whitham}\n\nAs our second scalar example, we consider the Whitham equation \\rf{whitham}. For this example, no analytical results exist, but the work of Sanford {\\em et al.} \\cite{SanfordKodamaCarterKalisch} allows for a comparison with numerical results. Sanford {\\em et al.} do not report the presence of high-frequency instabilities for solutions of any period. Their absence has been verified by us using the same methods, see Fig.~\\ref{fig:wstuff}. Hur \\& Johnson \\cite{hurjohnson} consider periodic solutions, focusing on the modulational instability. However, they do include a Krein signature calculation of the eigenvalues of the zero-amplitude solutions, reaching the same conclusions obtained below. In what follows the nonlinear term $N(u)$ does not contribute, as in the previous examples.\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{3.2in}\\hspace*{-0.2in}\\input{whithamProfile.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{3.6in}\\hspace*{-0.4in}\\input{whithamStab.pdf_tex}\\\\\n(a) & \\hspace*{-0.4in}(b)\n\\end{tabular}\n\\caption{\\la{fig:wstuff} (a) The profile of a $2\\pi$-periodic small-amplitude traveling wave solution of the Whitham equation \\rf{whitham} with $c\\approx 0.7697166847$, computed using a cosine collocation method with 128 Fourier modes, see \\cite{SanfordKodamaCarterKalisch}. (b) The stability spectrum of this solution, computed using the Fourier-Floquet-Hill method \\cite{deconinckkutz1} with $128$ modes and 2000 different values of the Floquet parameter $\\mu$. The presence of a modulational instability is clear, but no high-frequency instabilities are observed, in agreement with the theory presented. Note that the hallmark bubbles of instability were looked for far outside of the region displayed here.\n    }\n\\end{figure}\n\n\\begin{enumerate}\n\n\\item The modified Hamiltonian is $$H^0_V=\\frac{V}{2}\\int_0^{2\\pi} u^2 dx-\\frac{1}{2}\\int_{-\\pi}^\\pi \\int_{-\\pi}^\\pi K(x-y)u(x) u(y) dx dy.$$\n\n\\noindent We use $V$ to denote the speed of the traveling wave, to avoid confusion with the phase speed $c(k)$ in the kernel of the Whitham equation.\n\n\\item The dispersion relation is given by $\\omega(k)=\\mbox{sign}(k)\\sqrt{gk\\tanh(kh)}$.\n\n\\item The bifurcation branch starts at $(V,0)=(\\sqrt{g \\tanh(h)},0)$, where we have chosen $N=1$ so that the minimal period of the solutions is $2\\pi$.\n\n\\item The elements of the stability spectrum are given by\n\n\\begin{equation}\\la{evwhitham}\n\\lambda_n^{(\\mu)}=i(n+\\mu)\\sqrt{g \\tanh(h)}-i \\mbox{sign}(n+\\mu)\\sqrt{g(n+\\mu)\\tanh (h(n+\\mu))}.\n\\end{equation}\n\n\\item The dispersion relation for the Whitham equation is plotted in Fig.~\\ref{fig:wcollision}(a), together with the line through the origin with slope $\\omega(1)\/1$. Since the dispersion relation is concave down (up) in the first (third) quadrant, the condition \\rf{scalarcollision} is not satisfied. Thus collisions of eigenvalues away from the\n    origin do not occur. This is also illustrated in Fig.~\\ref{fig:wcollision}(b), where the imaginary part of $\\lambda_n^{(\\mu)}$ is plotted for various integers $n$.\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{wdispersion.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{wcollision.pdf_tex}\\\\\n(a) & (b)\n\\end{tabular}\n\\caption{\\la{fig:wcollision} (a) The dispersion relation for the Whitham equation (curve), together with the line through the origin of slope $\\omega(1)\/1$, representing the right-hand side of \\rf{scalarcollision}.  (b) The curves $\\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur at the origin only.\n    }\n\\end{figure}\n\n\\item No Krein signature calculation is relevant since eigenvalues do not collide away from the origin.\n\n\\end{enumerate}\n\nWe conclude that periodic solutions of sufficiently small amplitude of the Whitham equation are not susceptible to high-frequency instabilities. This is consistent with the results presented in \\cite{SanfordKodamaCarterKalisch}, see also Fig.~\\ref{fig:wstuff}. Thus, the Whitham equation is unable to replicate the instabilities found in the shallow depth water wave problem for solutions of small amplitude, despite having a dispersion relation that is identical to one branch of the water wave dispersion relation. We return to this in the next section.\n\n\\section{Two-component Hamiltonian PDEs with canonical Poisson structure}\\la{sec:vector}\n\nWe generalize the ideas of the previous section to the setting of two-component Hamiltonian PDEs with canonical Poisson structure. In other words, the evolution PDE can be written as\n\n\\begin{align}\\la{can}\n\\pp{}{t}\n\\left(\n\\begin{array}{c}\nq\\\\partial\n\\end{array}\n\\right)&=J \\nabla H ~~\\iff~~\\left\\{\n\\begin{array}{rcl}\n\\displaystyle q_t&=&\\displaystyle \\dd{H}{p}\\\\\n\\displaystyle p_t&=&\\displaystyle -\\dd{H}{q}\n\\end{array}\n\\right.,\n\\end{align}\n\n\\noindent where the Poisson operator $J$ is given by \\rf{canonicalJ}. This Poisson operator is nonsingular, thus there are no Casimirs. Examples of systems of this form are the Nonlinear Schr\\\"odinger equation in real coordinates \\cite{bottmandeconincknivala}, the Sine-Gordon equation \\cite{faddeev, jonesmiller1, jonesmiller2}, and the water wave problem \\cite{zakharov1}. As before, our interest is in the linearization of this system around the zero-amplitude solution. The quadratic Hamiltonian corresponding to this linearization can be written as\n\n\\begin{equation}\\la{canham}\nH^0=\\int_0^{2\\pi} \\left(\n\\frac{1}{2}\\sum_{n=0}^\\infty c_n q_{nx}^2+\\frac{1}{2}\\sum_{n=0}^\\infty b_n p_{nx}^2+p\\sum_{n=0}^\\infty a_n q_{nx}\n\\right)dx,\n\\end{equation}\n\n\\noindent with $a_n$, $b_n$, $c_n\\in \\mathbb{R}$. Typically the number of terms in the sums above is finite, but an example like the water wave problem requires the possibility of an infinite number of nonzero contributing terms in the Hamiltonian. As for the Whitham equation, convergence of the resulting series is not problematic. The form \\rf{canham} is the most general form of a quadratic Hamiltonian depending on two functions $q(x,t)$ and $p(x,t)$. Indeed, any quadratic term of a form not included above is reduced to a term that is included by straightforward integration by parts. The linearization of \\rf{can} is given by\n\n\\setcounter{saveeqn}{\\value{equation}\n\n\\begin{align}\\la{canlin}\nq_t&=\\sum_{n=0}^\\infty a_n q_{nx}+\\sum_{n=0}^\\infty (-1)^n b_n p_{2nx},\\\\\np_t&=-\\sum_{n=0}^\\infty (-1)^n c_n q_{2nx}-\\sum_{n=0}^\\infty (-1)^n a_n p_{nx}.\n\\end{align}\n\n\\setcounter{equation}{\\value{saveeqn}\n\nWe proceed with the six step program outlined in the introduction.\n\n\\begin{enumerate}\n\n\\item {\\bf Quadratic Hamiltonian.} The modified Hamiltonian $H_c^0$ is given by\n\n\\begin{equation}\nH_c^0=\\int_0^{2\\pi} \\left(c p q_x+\n\\frac{1}{2}\\sum_{n=0}^\\infty c_n q_{nx}^2+\\frac{1}{2}\\sum_{n=0}^\\infty b_n p_{nx}^2+p\\sum_{n=0}^\\infty a_n q_{nx}\n\\right)dx.\n\\end{equation}\n\n\\noindent This expression serves as a repository for the coefficients which are needed in what follows.\n\n\\item {\\bf Dispersion relation.} We look for solutions to \\rf{canlin} of the form $q=\\hat q \\exp(ikx-i\\omega t)$, $p=\\hat p \\exp(ikx-i\\omega t)$. Requiring the existence of non-trivial ({\\em i.e.}, non-zero) solutions, we find that $\\omega(k)$ is determined by\n\n\\begin{equation}\\la{candis}\n\\det \\left(\n\\begin{array}{cc}\n\\displaystyle i\\omega+\\sum_{n=0}^\\infty a_n (ik)^n &\n\\displaystyle \\sum_{n=0}^\\infty b_n k^{2n}\\\\\n\\displaystyle -\\sum_{n=0}^\\infty c_n k^{2n} &\n\\displaystyle i\\omega -\\sum_{n=0}^\\infty a_n (-1)^n (ik)^n\n\\end{array}\n\\right)=0.\n\\end{equation}\n\n\\noindent This is a quadratic equation for $\\omega(k)$, resulting in two branches of the dispersion relation, $\\omega_1(k)$ and $\\omega_2(k)$. Assuming that (\\ref{canlin}-b) is dispersive, $\\omega_1(k)$ and $\\omega_2(k)$ are real-valued for $k\\in \\mathbb{R}$. This is not easily translated in a condition on the coefficients $a_n$, $b_n$, $c_n$ and $d_n$, since their reality is not assumed.\n\n\\item {\\bf Bifurcation branches.} Traveling wave solutions are stationary solutions of\n\n\\setcounter{saveeqn}{\\value{equation}\n\n\\begin{align}\\la{linmoving}\nq_t&=c q_x+\\sum_{n=0}^\\infty a_n q_{nx}+\\sum_{n=0}^\\infty (-1)^n b_n p_{2nx}=\\dd{H_c^0}{p},\\\\\np_t&=c p_x-\\sum_{n=0}^\\infty (-1)^n c_n q_{2nx}-\\sum_{n=0}^\\infty (-1)^n a_n p_{nx}=-\\dd{H_c^0}{q}.\n\\end{align}\n\n\\setcounter{equation}{\\value{saveeqn}\n\n\\noindent This system has the dispersion relations $\\Omega_{1,2}(k)=\\omega_{1,2}(k)-c k$. In Fourier space the stationary equations become\n\n\\setcounter{saveeqn}{\\value{equation}\n\n\\begin{align}\n0&=ik c \\hat q+\\sum_{n=0}^\\infty a_n (ik^n) \\hat q+\\sum_{n=0}^\\infty (-1)^n b_n (ik)^{2n} \\hat p,\\\\\n0&=ik c \\hat p-\\sum_{n=0}^\\infty (-1)^n c_n (ik)^{2n} \\hat q-\\sum_{n=0}^\\infty (-1)^n a_n (ik)^n \\hat p.\n\\end{align}\n\n\\setcounter{equation}{\\value{saveeqn}\n\n\\noindent Thus $c$ is obtained from the condition that these equations have a nontrivial solution $(\\hat q, \\hat p)$. This condition requires that the $2\\times 2$ determinant of the system above is zero. A simple comparison with \\rf{candis} gives that there are two bifurcation points given by $(\\omega_1(N)\/N,0)$ and $(\\omega_2(N)\/N,0)$. Any positive integer value of $N$ is allowed, but we usually choose $N=1$ so that the fundamental period is $2\\pi$. In what follows, we examine the small-amplitude solutions starting from the branch $(c,0)=(\\omega_1(N)\/N,0)$, without loss of generality.\n\nIn many systems the two bifurcation branches are reflections of each other about the vertical axis. The corresponding solution profiles are identical to each other, moving to the right on one branch, moving to the left on the other. Examples are given below. Non-symmetric bifurcation branches cannot be excluded, however, without imposing extra assumptions on the coefficients of (\\ref{canlin}-b).\n\n\\item {\\bf Stability spectrum.} To find the stability spectrum, we let $q=Q(x) \\exp(\\lambda t)$, $p=P(x) \\exp(\\lambda t)$. Next, using Floquet's Theorem,\n\n\\begin{equation}\\la{canfloquet}\nQ=e^{i\\mu x}\\sum_{j=-\\infty}^\\infty Q_j e^{i j x}, ~~P=e^{i\\mu x}\\sum_{j=-\\infty}^\\infty P_j e^{ijx},\n\\end{equation}\n\n\\noindent with $\\mu\\in (-1\/2, 1\/2]$. Since (\\ref{linmoving}-b) has constant coefficients, it suffices to consider monochromatic waves, {\\em i.e.}, only one term of the sums in \\rf{canfloquet} is retained. It follows that $\\lambda$ satisfies \\rf{candis} with $i\\omega$ replaced by $-\\lambda+i(n+\\mu)c$. Thus\n\n\\begin{equation}\\la{canlambda}\n\\lambda_{n,l}^{(\\mu)}=i(n+\\mu)c-i\\omega_l(n+\\mu)=-i \\Omega_l(n+\\mu), ~~~~l=1,2.\n\\end{equation}\n\n\\noindent As expected, the zero solution is neutrally stable since $\\omega_{1,2}(k)$ are real, assuming dispersive equations. The stability spectrum consists of two one-parameter point sets, one for $l=1$, the other for $l=2$.\n\n\\item {\\bf Collision conditions.} Ignoring collisions at the origin, we require $\\lambda_{n_1, l_1}^{(\\mu)}=\\lambda_{n_2, l_2}^{(\\mu)}\\neq 0$ for some $n_1, n_2 \\in \\mathbb{Z}$, $\\mu\\in (-1\/2,1\/2]$, $l_1, l_2\\in \\{1,2\\}$. This gives\n\n\\begin{equation}\\la{cancollision}\n\\frac{\\omega_{l_1}(n_1+\\mu)-\\omega_{l_2}(n_2+\\mu)}{n_1-n_2}=\\frac{\\omega_1(N)}{N}.\n\\end{equation}\n\n\\noindent The right-hand side depends on $\\omega_1$ since we have chosen the first branch of the dispersion relation in Step~3. As before, this collision condition may be interpreted as a parallel secant condition, but with the additional freedom of being able to use points from both branches of the dispersion relation. This is illustrated in Fig.~\\ref{fig:cancollision}.\n\n\\begin{figure}[tb]\n\\def3.2in}\\input{bubbletrack.pdf_tex{4.8in}\n\\centerline{\\hspace*{0in}\\input{canonicalcollision.pdf_tex}}\n\\caption{\\la{fig:cancollision} The graphical interpretation of the collision condition \\rf{cancollision}. The dashed curves are the graphs of the dispersion relations $\\omega_1(k)$ and $\\omega_2(k)$. The slope of the segment $P_1P_2$ is the right-hand side in \\rf{cancollision}. The collision condition \\rf{cancollision} seeks points whose abscissas are an integer apart, so that at least one of the segments $P_3P_4$, $P_3P_6$, $P_5P_4$ or $P_5P_6$ is parallel to the segment $P_1P_2$.\n    }\n\\end{figure}\n\n\\item {\\bf Krein signature.} In the setting of a system of Hamiltonian PDEs as opposed to a scalar PDE, we use a different but equivalent characterization of the Krein signature~\\cite{mackay}. The Krein signature is the contribution to the Hamiltonian of the mode involved with the collision. Since our Hamiltonians are quadratic, this implies that the Krein signature of the eigenvalue $\\lambda$ with eigenvector $v$ is given by\n\n    \\begin{equation}\\la{kreinhessian}\n    \\mbox{signature}(\\lambda, v)=\\mbox{sign}(v^\\dagger {\\cal L}_c v),\n    \\end{equation}\n\n\\noindent where ${\\cal L}_c$ is the Hessian of the Hamiltonian $H_c^0$, and $v^\\dagger$ denotes the complex conjugate of the transposed vector. Since the Hessian ${\\cal L}_c$ is a symmetric linear operator, the argument of the $\\mbox{sign}$ in \\rf{kreinhessian} is real. Recall that the linearization of the system \\rf{hamc} can be written as\n\n\\begin{equation}\n\\partial_t\\left(\n\\begin{array}{c}\nq\\\\partial\n\\end{array}\n\\right)=\nJ {\\cal L}_c\n\\left(\n\\begin{array}{c}\nq\\\\partial\n\\end{array}\n\\right),\n\\end{equation}\n\n\\noindent which makes it easy to read off ${\\cal L}_c$. For the case of (\\ref{linmoving}-b),\n\n\\begin{equation}\\la{hessian}\n{\\cal L}_c=\\left(\n\\begin{array}{cc}\n\\displaystyle \\sum_{n=0}^\\infty c_n (-1)^n \\partial_x^{2n} &\n\\displaystyle -c \\partial_x+\\sum_{n=0}^\\infty a_n (-1)^n \\partial_x^n \\\\\n\\displaystyle c \\partial_x+\\sum_{n=0}^\\infty a_n \\partial_x^n &\n\\displaystyle \\sum_{n=0}^\\infty b_n (-1)^n \\partial_x^{2n}\n\\end{array}\n\\right).\n\\end{equation}\n\n\\noindent Next, the eigenvectors $v$ are given by\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nq \\\\ p\n\\end{array}\n\\right)=e^{\\lambda t+i\\mu x+i n x}\\left(\n\\begin{array}{c}\nQ_n \\\\ P_n\n\\end{array}\n\\right),\n\\end{equation}\n\n\\noindent where $(Q_n, P_n)^T$ satisfies\n\n\\begin{equation}\\la{evec}\n\\lambda\n\\left(\n\\begin{array}{c}\n\\displaystyle Q_n \\\\ \\displaystyle P_n\n\\end{array}\n\\right)=J\n\\hat {\\cal L}_c\\left(\\begin{array}{c}\n\\displaystyle Q_n \\\\ \\displaystyle P_n\n\\end{array}\n\\right).\n\\end{equation}\n\n\\noindent Here $\\hat {\\cal L}_c$ is the symbol of ${\\cal L}_c$, {\\em i.e.}, the $2\\times 2$ matrix obtained by replacing $\\partial_x\\rightarrow i(n+\\mu)$ in \\rf{hessian}:\n\n\\begin{equation}\\la{symbol}\n\\hat {\\cal L}_c=\n\\left(\n\\begin{array}{cc}\n\\displaystyle \\sum_{n=0}^\\infty c_n (n+\\mu)^{2n} &\n\\displaystyle -i c (n+\\mu)+\\sum_{n=0}^\\infty a_n (-1)^n (in+i\\mu)^n \\\\\n\\displaystyle i c (n+\\mu) +\\sum_{n=0}^\\infty a_n (in+i\\mu)^n &\n\\displaystyle \\sum_{n=0}^\\infty b_n (n+\\mu)^{2n}\n\\end{array}\n\\right).\n\\end{equation}\n\n\\noindent Using \\rf{evec}, \\rf{kreinhessian} is rewritten as\n\n\\begin{equation}\n\\mbox{signature}\\left(\\lambda_{n,l}^{(\\mu)}, v_{n,l}^{(\\mu)}\\right)=\\mbox{sign}\\left(\\lambda_{n,l}^{(\\mu)}\\det \\left(\n\\begin{array}{cc}\nQ_n & P_n\\\\\nQ_n^* & P_n^*\n\\end{array}\n\\right)\\right).\n\\end{equation}\n\n\\noindent The determinant is imaginary, since interchanging the rows gives the complex conjugate result. Since $\\lambda_{n,l}^{(\\mu)}$ is imaginary, the result is real and the signature is well defined. Again, it is clear that no conclusions can be drawn if $\\lambda_{n,l}^{(\\mu)}=0$. Since we wish to examine whether signatures are equal or opposite, we consider the product of the signatures corresponding to $\\lambda_{n_1,l_1}^{(\\mu)}$ and $\\lambda_{n_2,l_2}^{(\\mu)}$. Using \\rf{symbol} we find that signatures are opposite, provided that\n\n\\begin{align}\\nonumber\n&\\sum_{j_1=0}^\\infty c_{j_1} (n_1+\\mu)^{2j_1}\\sum_{j_2=0}^\\infty c_{j_2} (n_2+\\mu)^{2j_2}\n\\left(\\omega_{l_1}(n_1+\\mu)+\\sum_{j_3=0}^\\infty a_{2j_3+1}(-1)^{j_3}(n_1+\\mu)^{2j_3+1}\\right)\\times\\\\\\la{cankrein1}\n&~~~~~~~~~~~~~~~~\\left(\\omega_{l_2}(n_2+\\mu)+\\sum_{j_4=0}^\\infty a_{2j_4+1}(-1)^{j_4}(n_2+\\mu)^{2j_4+1}\\right)<0.\n\\end{align}\n\n\\noindent The above condition is obtained by expressing the eigenvectors in \\rf{evec} in terms of the entries of the first row of \\rf{symbol}. An equivalent condition is obtained using the second row:\n\n\\begin{align}\\nonumber\n&\\sum_{j_1=0}^\\infty b_{j_1} (n_1+\\mu)^{2j_1}\\sum_{j_2=0}^\\infty b_{j_2} (n_2+\\mu)^{2j_2}\n\\left(\\omega_{l_1}(n_1+\\mu)-\\sum_{j_3=0}^\\infty a_{2j_3+1}(-1)^{j_3}(n_1+\\mu)^{2j_3+1}\\right)\\times\\\\\\la{cankrein2}\n&~~~~~~~~~~~~~~~~\\left(\\omega_{l_2}(n_2+\\mu)-\\sum_{j_4=0}^\\infty a_{2j_4+1}(-1)^{j_4}(n_2+\\mu)^{2j_4+1}\\right)<0.\n\\end{align}\n\n\\noindent Depending on the system at hand, the condition \\rf{cankrein1} or \\rf{cankrein2} may be more convenient to use.\n\n\\end{enumerate}\n\n{\\bf Remark.} An important class of systems is those for which $\\omega_1(k)=-\\omega_2(k)$. We refer to such systems as even systems. It follows immediately from \\rf{candis} that for even systems $a_{2j+1}=0$, $j=1,2,\\ldots$. The Krein conditions \\rf{cankrein1} and \\rf{cankrein2} simplify significantly, becoming\n\n\\begin{equation}\\la{sym1}\n\\omega_{l_1}(n_1+\\mu)\\omega_{l_2}(n_2+\\mu)\\sum_{j_1=0}^\\infty c_{j_1} (n_1+\\mu)^{2j_1}\\sum_{j_2=0}^\\infty c_{j_2} (n_2+\\mu)^{2j_2}<0,\n\\end{equation}\n\n\\noindent or\n\n\\begin{equation}\\la{sym2}\n\\omega_{l_1}(n_1+\\mu)\\omega_{l_2}(n_2+\\mu)\\sum_{j_1=0}^\\infty b_{j_1} (n_1+\\mu)^{2j_1}\\sum_{j_2=0}^\\infty b_{j_2} (n_2+\\mu)^{2j_2}<0.\n\\end{equation}\n\n\nWe summarize our results.\n\n\\vspace*{0.1in}\n\nAssume that the linearization of the Hamiltonian system \\rf{can} is dispersive ({\\em i.e.,} its dispersion relations $\\omega_1(k)$ and $\\omega_2(k)$ are real valued for $k\\in \\mathbb{R}$). Let $N$ be a strictly positive integer.\nConsider $2\\pi\/N$-periodic traveling wave solutions of this system of sufficiently small-amplitude and with velocity sufficiently close to $\\omega_1(N)\/N$. In order for these solutions to be spectrally unstable with respect to high-frequency instabilities as a consequence of two-eigenvalue collisions, it is necessary that there exist $l_1$, $l_2~\\in \\{1,2\\}$, $n_1$, $n_2~\\in \\mathbb{Z}$, $n_1\\neq n_2$, $\\mu \\in (-1\/2, 1\/2]$ for which\n\n\\begin{equation}\n\\frac{\\omega_{l_1}(n_1+\\mu)}{n_1+\\mu}\\neq \\frac{\\omega_1(N)}{N}, ~~\\frac{\\omega_{l_2}(n_2+\\mu)}{n_2+\\mu}\\neq \\frac{\\omega(N)}{N},\n\\end{equation}\n\n\\noindent such that\n\n\\begin{equation}\n\\frac{\\omega_{l_1}(n_1+\\mu)-\\omega_{l_2}(n_2+\\mu)}{n_1-n_2}=\\frac{\\omega_1(N)}{N},\n\\end{equation}\n\n\\noindent and \\rf{cankrein1}, or equivalently, \\rf{cankrein2} holds.\n\n\\vspace*{0.1in}\n\nWe proceed with examples.\n\n\\subsection{The Sine-Gordon equation}\n\nAs a first example, we consider the Sine-Gordon (SG) equation \\cite{scott}:\n\n\\begin{equation}\nu_{tt}-u_{xx}+\\sin u=0.\n\\end{equation}\n\n\\noindent The stability of the periodic traveling wave solutions of this equation has been studied recently by Jones {\\em et al.} \\cite{jonesmiller1, jonesmiller2}. Different classes of periodic traveling wave solutions exist, but only two of those can be considered as small-amplitude perturbations of a constant background state. We consider the so-called superluminal ($c^2>1$) librational waves. The subluminal ($c^2<1$) librational waves require the use of the transformation $v=u-\\pi$ so that their small amplitude limit approaches the zero solution. We do not consider them here. The limits of the rotational waves are either soliton solutions or have increasingly larger amplitude. As such the rotational waves do not fit in the framework of this paper. An overview of the properties of these solutions as well as illuminating phase-plane plots are found in \\cite{jonesmiller1}. In contrast to \\cite{jonesmiller1, jonesmiller2}, we fix the period of our solutions, as elsewhere in this paper. This makes a comparison of the results more complicated.\n\n\\begin{enumerate}\n\n\\item {\\bf Quadratic Hamiltonian.} With $q=u$, $p=u_t$,\n\n\\begin{equation}\nH_c^0=\\int_0^{2\\pi} \\left(\nc p q_x+\\frac{1}{2}p^2+\\frac{1}{2}q^2+\\frac{1}{2}q_x^2\n\\right)dx.\n\\end{equation}\n\n\\noindent Thus $b_0=1$, $c_0=1$, $c_1=1$ are the only non-zero coefficients.\n\n\\item {\\bf Dispersion relation.} Using \\rf{candis},\n\n\\begin{equation}\n\\omega_{1,2}=\\pm \\sqrt{1+k^2}.\n\\end{equation}\n\n\\noindent These expressions are real valued for $k\\in \\mathbb{R}$, thus the SG equation is dispersive when linearized around the superluminal librational waves. Both branches of the dispersion relation are displayed in Fig.~\\ref{fig:sgcollision}a.\n\n\\item {\\bf Bifurcation branches.} With $N=1$, we obtain $c=\\omega_1(1)\/1=\\sqrt{2}$.\n\n\\item {\\bf Stability spectrum.} The stability spectrum is given by \\rf{canlambda}:\n\n\\begin{equation}\n\\lambda_{n,l}^{(\\mu)}=-i\\Omega_l(n+\\mu)=i(n+\\mu)\\sqrt{2}\\mp i\\sqrt{1+(n+\\mu)^2},\n\\end{equation}\n\n\\noindent with $l=1$ ($l=2$) corresponding to the $-$ ($+$) sign. Here $n\\in \\mathbb{Z}$, $\\mu\\in [-1\/2, 1\/2)$.\n\n\\item {\\bf Collision condition.} The collision condition \\rf{cancollision} becomes\n\n\\begin{equation}\n\\frac{\\omega_1(n_1+\\mu)-\\omega_2(n_2+\\mu)}{n_1-n_2}=\\sqrt{2}.\n\\end{equation}\n\n\\noindent We have chosen $\\omega_{l_1}=\\omega_1$ and $\\omega_{l_2}=\\omega_2$, since it is clear that the collision condition can only be satisfied if points from both dispersion relation branches are used. This is illustrated in Fig.~\\ref{fig:sgcollision}a. In fact, many collisions occur, as is illustrated in Fig.~\\ref{fig:sgcollision}b. One explicit solution is given by\n\n\\begin{equation}\nn_1=3,~n_2=0,~\\mu=\\frac{\\sqrt{10}-3}{2}\\approx 0.081138830.\n\\end{equation}\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{sg1dispersion.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{sg1collision.pdf_tex}\\\\\n(a) & (b)\n\\end{tabular}\n\\caption{\\la{fig:sgcollision} (a) The two branches of the dispersion relation for the Sine-Gordon equation. The line segment $P_1 P_2$ has slope $\\omega(1)\/1$, representing the right-hand side of \\rf{cancollision}. The slope of the parallel line segment $P_3 P_4$ represents the left-hand side of \\rf{cancollision}.  (b) The two families of curves $\\Omega_1(k+n)$ (red, solid) and $\\Omega_2(k+n)$ (black, dashed), for various (integer) values of $n$, illustrating that many collisions occur away from the origin.\n    }\n\\end{figure}\n\n\\item {\\bf Krein signature.} Since $\\omega_2(k)=-\\omega_1(k)$, we may use the conditions \\rf{sym1} or \\rf{sym2}. Since only one $b_j\\neq 0$, \\rf{sym2} is (slightly) simpler to use. We get that\n\n\\begin{equation}\n\\omega_1(n_1+\\mu)\\omega_2(n_2+\\mu)<0\n\\end{equation}\n\n\\noindent is a necessary condition for the presence of high-frequency instabilities of small-amplitude superluminal librational solutions of the SG equation. The condition is trivially satisfied as it was remarked in the previous step that points from both dispersion relation branches have to be used to have collisions.\n\n\\end{enumerate}\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{3.2in}\\hspace*{-0.2in}\\input{SGProfile.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{3.7in}\\hspace*{-0.4in}\\input{SGStab.pdf_tex}\\\\\n(a) & \\hspace*{-0.4in}(b)\n\\end{tabular}\n\\caption{\\la{fig:sgstuff} (a) A small-amplitude $2\\pi$-periodic superluminal solution of the SG equation ($c\\approx 1.236084655663$). (b) A blow-up of the numerically computed stability spectrum in a neighborhood of the origin, illustrating the presence of a modulational instability, but the absence of high-frequency instabilities.\n    }\n\\end{figure}\n\nIt follows that for the superluminal solutions of the SG equations the necessary condition for the occurrence of high-frequency instabilities is satisfied. Nevertheless, as the results of \\cite{jonesmiller1, jonesmiller2} show, such instabilities do not occur. This is illustrated in Fig.~\\ref{fig:sgstuff}. The left panel illustrates an exact $2\\pi$-periodic superluminal solution of the SG equation with $c\\approx 1.236084655663$, obtained using elliptic functions. Computing the stability spectrum (right panel) of the solution using the Fourier-Floquet-Hill method \\cite{deconinckkutz1} with $51$ Fourier modes and $1000$ Floquet exponents shows that no high-frequency instabilities are present to within the accuracy of the numerical method. This is consistent with the results of \\cite{jonesmiller1, jonesmiller2} where only the presence of a modulational instability is observed. Thus the example of this section illustrates that the necessary condition is not always sufficient.\n\n\\subsection{The water wave problem}\\la{ex:ww}\n\nAs a final example, we consider the water wave problem: the problem of determining the dynamics of the surface of an incompressible, irrotational fluid under the influence of gravity. For this example, the effects of surface tension are ignored and we consider only two-dimensional fluids, {\\em i.e.}, the surface is one dimensional. The Euler equations governing the dynamics are\n\n\\setcounter{saveeqn}{\\value{equation}\n\n\\begin{align}\\la{eulera}\n\\phi_{xx}+\\phi_{zz}&=0, && (x,z)\\in D,\\\\\n\\phi_z&=0, && z=-h,\\\\\n\\eta_t+\\eta_x \\phi_x&=\\phi_z,&& z=\\eta(x,t),\\\\\n\\phi_t+\\frac{1}{2}(\\phi_x^2+\\phi_z)+g\\eta&=0, && z=\\eta(x,t),\n\\end{align}\n\n\\setcounter{equation}{\\value{saveeqn}\n\n\\noindent where $x$ and $z$ are the horizontal and vertical coordinate, respectively, see Fig.~\\ref{fig:ww}; $z=\\eta(x,t)$ is the free top boundary and $\\phi(x,z,t)$ is the velocity potential. Further, $g$ is the acceleration due to gravity and $h$ is the average depth of the fluid.\n\n\\begin{figure}[tb]\n\\centerline{\\def3.2in}\\input{bubbletrack.pdf_tex{6in}\\input{perww.pdf_tex}}\n\\caption{\\la{fig:ww} The domain for the water wave problem. Here $z=0$ is the equation of the surface for flat water, $z=-h$ is the flat bottom.\n    }\n\\end{figure}\n\nThe main goal of the water wave problem is to understand the dynamics of the free surface $\\eta(x,t)$. Thus it is convenient to recast the problem so as to involve only surface variables. Zakharov \\cite{zakharov1} showed that the water wave problem is Hamiltonian with canonical variables $\\eta(x,t)$ and $\\varphi(x,t)=\\phi(x,\\eta(x,t),t)$. In other words $\\varphi(x,t)$ is the velocity potential evaluated at the surface. Following \\cite{craigsulem}, the Hamiltonian is written as\n\n\\begin{equation}\nH=\\frac{1}{2}\\int_0^{2\\pi}\\left(\\varphi G(\\eta) \\varphi+g \\eta^2\\right)dx,\n\\end{equation}\n\n\\noindent where $G(\\eta)$ is the Dirichlet~$\\rightarrow$~Neumann operator: $G(\\eta)\\varphi=(1+\\eta_x^2)^{1\/2}\\phi_n$, at $z=\\eta(x,t)$. Here $\\phi_n$ is the normal derivative of $\\phi$. Using the water wave problem, $G(\\eta)\\varphi=\\phi_z-\\eta_x \\phi_x=\\eta_t$, which is the first of Hamilton's equations. The water wave problem for $\\eta(x,t)$ and $\\varphi(x,t)$ is\n\n\\begin{equation}\\la{wweqs}\n\\eta_t=\\dd{H}{\\varphi}, ~~\\varphi_t=-\\dd{H}{\\eta}.\n\\end{equation}\n\n\\begin{enumerate}\n\n\\item {\\bf Quadratic Hamiltonian.} Since for our purposes, the linearization of (\\ref{wweqs}-b) in a moving frame is required, it suffices to evaluate the Dirichlet~$\\rightarrow$~Neumann operator $G(\\eta)$ at the flat surface $\\eta=0$, resulting in\n\n    \\begin{equation}\n    G(0)=-i \\partial_x \\tanh(-ih \\partial_x).\n    \\end{equation}\n\nThe quadratic Hamiltonian $H_c^0$ is given by\n\n\\begin{equation}\nH_c^0=c \\int_0^{2\\pi} \\varphi \\eta_x dx+\\frac{1}{2}\\int_0^{2\\pi}\\left(\n\\varphi (-i \\tanh(-ih\\partial_x) \\varphi_x+g \\eta^2\n\\right) dx,\n\\end{equation}\n\n\\noindent giving rise to the linearized equations in a frame moving with velocity $c$:\n\n\\setcounter{saveeqn}{\\value{equation}\n\n\\begin{align}\\la{linww}\n\\eta_t&=\\dd{H_c^0}{\\varphi}=c\\eta_x-i \\tanh(-ih \\partial_x)\\varphi_x,\\\\\n\\varphi_t&=-\\dd{H_c^0}{\\eta}=c\\varphi_x+g\\eta.\n\\end{align}\n\n\\setcounter{equation}{\\value{saveeqn}\n\n\\item {\\bf Dispersion relation.} The well-known dispersion relation \\cite{vandenboek} for the water wave problem is immediately recovered from the linearized system (\\ref{linww}-b) with $c=0$ (no moving frame), resulting in\n\n\\begin{equation}\\la{wwdisp}\n\\omega^2=gk \\tanh(kh).\n\\end{equation}\n\n\\noindent Note that the right-hand side of this expression is always positive. Thus there are two branches to the dispersion relation:\n\n\\begin{equation}\\la{wwbranches}\n\\omega_{1,2}=\\pm \\mbox{sign}(k)\\sqrt{gk\\tanh(kh)}.\n\\end{equation}\n\n\\noindent Thus $\\omega_1$ ($\\omega_2$) corresponds to positive (negative) phase speed, independent of the sign of $k$.\n\n\\item {\\bf Bifurcation branches.} Branches originate from $(c,\\mbox{amplitude})=(\\omega_1(1)\/1,0)$ and $(c,\\mbox{amplitude})=(\\omega_2(1)\/1,0)$. Without loss of generality, we focus on the first branch for which the phase speed $\\sqrt{g\\tanh(h)}$ is positive. This allows for a straightforward comparison of our results with those for the Whitham equation, in Example~\\ref{ex:whitham}.\n\n\\item {\\bf Stability spectrum.} The elements of the spectrum are given by\n\n\\setcounter{saveeqn}{\\value{equation}\n\\begin{align}\\nonumber\n\\lambda_{n,1}^{(\\mu)}&=-i\\,\\Omega_1(n+\\mu)\\\\\\la{wwspectrum}\n&=i(n+\\mu)\\sqrt{g\\tanh(h)}-i\\,\\mbox{sign}(n+\\mu)\\sqrt{g(n+\\mu)\\tanh(h(n+\\mu))},\\\\\n\\nonumber\n\\lambda_{n,2}^{(\\mu)}&=-i\\,\\Omega_2(n+\\mu)\\\\\n&=i(n+\\mu)\\sqrt{g\\tanh(h)}+i\\,\\mbox{sign}(n+\\mu)\\sqrt{g(n+\\mu)\\tanh(h(n+\\mu))}.\n\\end{align}\n\\setcounter{equation}{\\value{saveeqn}\n\n\\noindent The $\\mbox{sign}(n+\\mu)$'s may be omitted in these expressions, as the same set of spectral elements is obtained.\n\n\\item {\\bf Collision condition.} The condition \\rf{cancollision} is easily written out explicitly, but for our purposes it suffices to plot $\\Omega_1(k+n)$ and $\\Omega_2(k+n)$, for different values of $n$. This is done in Fig.~\\ref{fig:wwcollision}b with $g=1$ and $h=1$. Although only the first collision is visible in the figure (all intersection points are horizontal integer shift of each other and correspond to the same value of $\\mu$ and $\\lambda_{n,j}^{(\\mu)}$), it is clear from the curves shown that many collisions occur. The figure is qualitatively the same for all finite values of depth.\n\n\\begin{figure}[tb]\n\\begin{tabular}{cc}\n\\def3.2in}\\input{bubbletrack.pdf_tex{3.2in}\\input{wwdispersion.pdf_tex} &\n\\def3.2in}\\input{bubbletrack.pdf_tex{2.8in}\\input{wwcollision.pdf_tex}\\\\\n(a) & (b)\\\\\n\\multicolumn{2}{c}{\n\\def3.2in}\\input{bubbletrack.pdf_tex{3.2in}\\input{bubbletrack.pdf_tex}\n}\\\\\n\\multicolumn{2}{c}{(c)}\n\\end{tabular}\n\\caption{\\la{fig:wwcollision} (a) The two branches of the dispersion relation for the water wave problem ($g=1$, $h=1$). The line through the origin has slope $\\omega_1(1)\/1$, representing the right-hand side of \\rf{cancollision}. (b) The two families of curves $\\Omega_1(k+n)$ (red, solid) and $\\Omega_2(k+n)$ (black, dashed), for various (integer) values of $n$, illustrating that many collisions occur away from the origin. (c) The origin of the high-frequency instability closest to the origin as a function of depth $h$.\n    }\n\\end{figure}\n\n\\item {\\bf Krein signature.} The conditions \\rf{sym1} and \\rf{sym2} become\n\n\\begin{equation}\\la{sym1ww}\n\\omega_{l_1}(n_1+\\mu)\\omega_{l_2}(n_2+\\mu)g^2<0,\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\\la{sym2ww}\n\\omega_{l_1}(n_1+\\mu)\\omega_{l_2}(n_2+\\mu)\\sum_{{j_1}=1}^\\infty \\alpha_{j_1-1}h^{2j_1-1} (n_1+\\mu)^{2j_1}\\sum_{{j_2}=1}^\\infty \\alpha_{j_2-1}h^{2j_2-1} (n_2+\\mu)^{2j_2}<0,\n\\end{equation}\n\n\\noindent respectively. Here the coefficients $\\alpha_j$ are related to the Bernoulli numbers \\cite{dlmf}, as they are defined by the Taylor series\n\n\\begin{equation}\\la{tanh}\n\\tanh(z)=\\sum_{j=0}^\\infty \\alpha_j z^{2j+1}, ~~~|z|<\\pi\/2.\n\\end{equation}\n\n\\noindent Because of the finite radius of convergence of this series, \\rf{sym2ww} is only valid for small values of the wave numbers $n_1+\\mu$ and $n_2+\\mu$, but it is possible to phrase all results in terms of $\\tanh$ directly, avoiding this difficulty. For instance, using \\rf{tanh}, \\rf{sym2ww} may be rewritten as\n\n\\begin{align*}\n&&\\omega_{l_1}(n_1+\\mu)\\omega_{l_2}(n_2+\\mu)\\frac{\\omega_{\\alpha}^2(n_1+\\mu)}{g}\n\\frac{\\omega_{\\beta}^2(n_2+\\mu)}{g}&<0\\\\\n&\\Rightarrow& \\omega_{l_1}(n_1+\\mu)\\omega_{l_2}(n_2+\\mu)&<0,\n\\end{align*}\n\n\\noindent in agreement with \\rf{sym1ww}. The indices $\\alpha$, $\\beta\\in \\{0,1\\}$ are irrelevant since $\\omega_\\alpha$ and $\\omega_\\beta$ both appear squared. This serves to illustrate that for specific examples one of the two criteria \\rf{cankrein1} and \\rf{cankrein2} (or \\rf{sym1} and \\rf{sym2} for even systems) may be significantly easier to evaluate, although they are equivalent.\n\nThus all collision points are potential origins of high-frequency instabilities. It appears from the numerical results in \\cite{deconinckoliveras1} that the bubble of non-imaginary eigenvalues closest to the origin contains the high-frequency eigenvalues with the largest real part. Thus for waves in shallow water $kh<1.363$ (no Benjamin-Feir instability) \\cite{benjaminperiodic, whitham1, zakharovostrovsky}, these are the dominant instabilities. For waves in deep water ($kh>1.363$) the Benjamin-Feir instability typically dominates, although there is a range of depth in deep water where the high-frequency instabilities have a larger growth rate, see \\cite{deconinckoliveras1}. The dependence on depth $h$ of the location on the imaginary axis from which the high-frequency bubble closest to the origin bifurcates is shown in Fig.~\\ref{fig:wwcollision}(c), with $g=1$. As $h\\rightarrow \\infty $, the imaginary part of $\\lambda\\rightarrow 3\/4$. This asymptote is drawn in Fig.~\\ref{fig:wwcollision}(c) for reference. This figure demonstrates that for all positive values of the depth $h$, the instabilities considered are not modulational as they do not bifurcate away from the origin as the amplitude increases.\n\nIt was remarked in Example~\\ref{ex:whitham} that no collisions are possible due to the concavity of the dispersion relation. As a consequence, all collisions away from the origin observed in Fig.~\\ref{fig:wwcollision}b involve both branches of the dispersion relation, {\\em i.e.}, they involve a solid curve and a dashed curve. This is easily seen from Fig.~\\ref{fig:wwcollision}a: a parallel cord with abscissae of the endpoints that are integers apart is easily found by sliding a parallel cord away from the cord $((0,0),(1,\\omega_1(1)))$ until the integer condition is met. This implies that $\\omega_{l_1}(n_1+\\mu)$ and $\\omega_{l_2}(n_2+\\mu)$ in the collision condition \\rf{cancollision} have opposite sign and \\rf{sym1ww} is always satisfied. Thus colliding eigenvalues of zero-amplitude water wave solutions {\\em always} have opposite Krein signature. As a consequence, the necessary condition for the presence of high-frequency instabilities is met. In fact, it was observed in \\cite{deconinckoliveras1} that all colliding eigenvalues give rise to bubbles of instabilities as the amplitude is increased.\n\nOur general framework easily recovers the results of MacKay \\& Saffman \\cite{mackaysaffman}. There the set-up is for arbitrary amplitudes of the traveling wave solutions, but the results are only truly practical for the zero-amplitude case.\n\n\\end{enumerate}\n\n{\\bf Remark.} It follows from these considerations that the high-frequency instabilities present in the water wave problem are a consequence of counter-propagating waves as no such instabilities are present in the Whitham equation \\rf{whitham}. Although it is often stated that the value of the Whitham equation lies in that it has the same dispersion relation as the water wave problem (see for instance \\cite{whitham}), this is in fact not the case as it contains only one branch of the dispersion relation. Thus the equation does not allow for the interaction of counter-propagating modes, and as such misses out on much of the important dynamics of the Euler equations.\n\n\\section{A Boussinesq-Whitham equation}\n\nThe goal of this section is the introduction of a model equation that has the same dispersion relation as the Euler equations (\\ref{eulera}-d) at the level of heuristics that led Whitham to the model equation \\rf{whitham}. In other words, we propose a bidirectional Whitham equation, so as to capture both branches of the water wave dispersion relation. We refer to this equation as the Boussinesq-Whitham (BW) equation. It is given by\n\n\\begin{equation}\\la{bw}\nq_{tt}=N(q)+\\partial_x^2\\left(\\alpha q^2+\\int_{-\\infty}^\\infty K(x-y)q(y)dy\n\\right),\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation}\nK(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty c^2(k)e^{ikx}dk,\n\\end{equation}\n\n\\noindent and $c^2(k)=g\\tanh(kh)\/k$, $\\alpha>0$ for the water wave problem without surface tension. In \\rf{bw}, $N(u)$ denotes the nonlinear terms, which are ignored in the remainder of this section. Since our methods focus on the analysis of zero amplitude solutions, the sign of $\\alpha$ is not relevant in what follows. This equation is one of many that may stake its claim to the name ``Boussinesq-Whitham equation\". Equation \\rf{bw} is a ``Whithamized'' version of the standard Bad Boussinesq equation and it may be anticipated that it captures at least the small-amplitude instabilities of the water wave problem in shallow water. It should be remarked that the Bad Boussinesq equation is ill posed as an initial-value problem \\cite{mckean1}, but it might be anticipated that the inclusion of the entire water-wave dispersion relation overcomes the unbounded growth that is present due to the polynomial truncation. We return to this at the end of this section.\n\nBefore applying our method to examine the potential presence of high-frequency instabilities of small-amplitude solutions of the BW equation, we need to present its Hamiltonian structure. Further, since \\rf{bw} is defined as an equation on the whole line, a periodic analogue is required, as in Section~\\ref{sec:whitham}.\n\nIt is easily verified that \\rf{bw} is Hamiltonian with (non-canonical) Poisson operator \\cite{mckean1}\n\n\\begin{equation}\\la{bwj}\nJ=\\left(\n\\begin{array}{cc}\n0 & \\partial_x\\\\\n\\partial_x & 0\n\\end{array}\n\\right),\n\\end{equation}\n\n\\noindent and Hamiltonian\n\n\\begin{equation}\\la{bwh}\nH=\\int_{-\\infty}^{\\infty} \\left(\\frac{1}{2} p^2+\\frac{\\alpha}{3} q^3 \\right)dx+\\frac{1}{2}\\int_{-\\infty}^{\\infty} dx \\int_{-\\infty}^{\\infty} dy\\, K(x-y) q(x) q(y).\n\\end{equation}\n\n\\noindent Indeed, \\rf{bw} can be rewritten in the form \\rf{ham} with $u=(q,p)^T$.\n\nTo define a periodic version of \\rf{bw}, let\n\n\\begin{equation}\nK(x)=\\frac{1}{L}\\sum_{j=-\\infty}^\\infty c^2(k_j) e^{i k_j x},\n\\end{equation}\n\n\\noindent where $k_j=2\\pi j\/L$, $j\\in \\mathbb{Z}$. The periodic BW equation is obtained from \\rf{ham}, using \\rf{bwj} and \\rf{bwh}, but with all $\\pm$ infinities in the integration bounds replaced by $\\pm L\/2$, respectively. Since \\rf{bw} has a Poisson operator \\rf{bwj} that is different from those used in Sections~\\ref{sec:scalar} and~\\ref{sec:vector}, minor modifications to the use of the method are necessary.\n\n\\begin{enumerate}\n\n\\item {\\bf Quadratic Hamiltonian.} Ignoring the contributions of the nonlinear term, the quadratic Hamiltonian in a frame of reference moving with speed $V$ is given by\n\n\\begin{equation}\\la{bwhc}\nH_V^0=\\int_{0}^{2\\pi}\\left(V q p+\\frac{1}{2}p^2 \\right)dx+\\frac{1}{2}\\int_{0}^{2\\pi} dx \\int_{0}^{2\\pi} dy\\, K(x-y) q(x) q(y),\n\\end{equation}\n\n\\noindent where we have fixed the period of the solutions to be $L=2\\pi$. The inclusion of the first term in \\rf{bwhc} is one place where the effect of the different form for $J$ is felt, as its functional form is a direct consequence of the form of \\rf{bwj}.\n\n\\item {\\bf Dispersion Relation.} A direct calculation confirms that\n\n\\begin{equation}\n\\omega^2=gk \\tanh(kh),\n\\end{equation}\n\n\\noindent which is, by construction, identical to the dispersion relation for the full water wave problem \\rf{wwdisp}. This gives rise to two branches of the dispersion relation \\rf{wwbranches}, corresponding to right- and left-going waves.\n\n\\item {\\bf Bifurcation Branches.} Bifurcation branches for $2\\pi$-periodic solutions start at $(V_{1,2},0)$, where the phase speeds $V_{1,2}$ are given by $V_{1,2}=\\pm \\sqrt{g\\tanh(h)}$.\n\n\\item {\\bf Stability Spectrum.} The stability spectrum elements are, again by construction, identical to those for the water wave problem, given in (\\ref{wwspectrum}-b).\n\n\\item {\\bf Collision Condition.} Given that the spectral elements are identical to those for the water wave problem, the collision condition is identical too. It is displayed in Fig.~\\ref{fig:wwcollision}(a-b). Thus, collisions away from the origin occur. It remains to be seen whether these can result in the birth of high-frequency instabilities.\n\n\\item {\\bf Krein Signature.} As for the canonical case of Section~\\ref{sec:vector}, we use \\rf{kreinhessian}. Thus we calculate the Hessian ${\\cal L}_c$ of the Hamiltonian $H_c^0$.\n\n    Let\n\n    \\begin{equation}\n    c^2(k)=\\sum_{j=0}^{\\infty} \\gamma_j k^{2j},\n    \\end{equation}\n\n\\noindent where $\\gamma_j=gh^{2j+1}a_j$, with the coefficients $a_j$ defined in \\rf{tanh}. A direct calculation gives that the Hamiltonian \\rf{bwhc} is rewritten as\n\n\\begin{equation}\nH_V^0=\\frac{1}{2}\\int_0^{2\\pi} \\left(p^2+Vqp+\\sum_{j=0}^\\infty \\gamma_j q_{jx}^2\\right)dx.\n\\end{equation}\n\n\\noindent Using this form of the Hamiltonian, the calculation of the Hessian is straightforward, leading to\n\n\\begin{equation}\n{\\cal L}_V=\n\\left(\n\\begin{array}{cc}\n\\displaystyle \\sum_{j=0}^\\infty \\gamma_j (-1)^j \\partial_x^{2j} & V\\\\\nV & 1\n\\end{array}\n\\right).\n\\end{equation}\n\nNext, we compute the eigenvectors $v=(q,p)^T$. We have\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nq\\\\partial\n\\end{array}\n\\right)=e^{i \\lambda t}\n\\left(\n\\begin{array}{c}\nQ(x)\\\\P(x)\n\\end{array}\n\\right),\n\\end{equation}\n\n\\noindent where $(Q,P)^T$ satisfies\n\n\\begin{equation}\\la{bwevp}\n\\lambda \\left(\n\\begin{array}{c}\nQ\\\\P\n\\end{array}\n\\right)= \\left(\n\\begin{array}{cc}\n0 & \\partial_x\\\\\n\\partial_x & 0\n\\end{array}\n\\right) {\\cal L}_V \\left(\n\\begin{array}{c}\nQ\\\\P\n\\end{array}\n\\right).\n\\end{equation}\n\nThis is a second place where the Poisson operator $J$ plays a crucial role as it affects the form of $v=(q,p)^T$ and thus the expression for the signature. One easily verifies that\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nQ\\\\P\n\\end{array}\n\\right)=e^{i(n+\\mu)x}\\left(\n\\begin{array}{c}\ni(n+\\mu)\\\\\\lambda-i(n+\\mu)V\n\\end{array}\n\\right)\n\\end{equation}\n\n\\noindent satisfies \\rf{bwevp}.\n\nWe need to evaluate the sign of\n\n\\begin{align}\\nonumber\n&\\left(\n\\begin{array}{c}\nQ\\\\P\n\\end{array}\n\\right)^\\dagger\n{\\cal L}_V \\left(\n\\begin{array}{c}\nQ\\\\P\n\\end{array}\n\\right)\\\\\\nonumber\n&=e^{-i(n+\\mu)x}\\left(\n\\begin{array}{c}\n-i(n+\\mu) \\\\ -\\lambda+i(n+\\mu)V\n\\end{array}\n\\right)^T\n\\left(\n\\begin{array}{cc}\n\\displaystyle \\sum_{j=0}^\\infty \\gamma_j (-1)^j \\partial_x^{2j} & V\\\\\nV & 1\n\\end{array}\n\\right)\\left(\n\\begin{array}{c}\ni(n+\\mu)\\\\\\lambda-i(n+\\mu)V\n\\end{array}\n\\right)e^{i(n+\\mu)x}\\\\\\nonumber\n&=\\left(\n\\begin{array}{c}\n-i(n+\\mu) \\\\ -\\lambda+i(n+\\mu)V\n\\end{array}\n\\right)^T\n\\left(\n\\begin{array}{cc}\n\\displaystyle \\sum_{j=0}^\\infty \\gamma_j (n+\\mu)^{2j} & V\\\\\nV & 1\n\\end{array}\n\\right)\\left(\n\\begin{array}{c}\ni(n+\\mu)\\\\\\lambda-i(n+\\mu)V\n\\end{array}\n\\right)\\\\\\nonumber\n&=\\left(\n\\begin{array}{c}\n-i(n+\\mu)\\\\ i\\omega(n+\\mu)\n\\end{array}\n\\right)^T\\left(\n\\begin{array}{cc}\nc^2(n+\\mu) & V\\\\\nV & 1\n\\end{array}\n\\right)\\left(\n\\begin{array}{c}\ni(n+\\mu)\\\\-i\\omega(n+\\mu)\n\\end{array}\n\\right)\\\\\n&=2\\omega\\left(\\omega-(n+\\mu)V\\right).\n\\end{align}\n\nLet the signature associated with the first eigenvalue be the sign of $2\\omega_{j_1}(\\omega_{j_1}-(n_1+\\mu)V)$, where $\\omega_{j_1}$ is a function of $n_1+\\mu$. Similarly, for the second eigenvalue, the signature is the sign of $2\\omega_{j_2}(\\omega_{j_2}-(n_2+\\mu)V)$. Using the collision condition $\\lambda_{n_1,j_1}^{(\\mu)}=\\lambda_{n_2,j_2}^{(\\mu)}$, the product of these two expressions is $4\\omega_{j_1}\\omega_{j_2}(\\omega_{j_2}-(n_2+\\mu)V)^2$,\nwhich is less than zero since collisions can only occur for eigenvalues associated with opposite branches of the dispersion relation, see Fig.~\\ref{fig:wwcollision}b. It follows that, as in the water wave case, the signatures of colliding eigenvalues are always opposite, and the necessary condition for spectral instability is met. Thus, unlike the Whitham equation \\rf{whitham}, the BW model \\rf{bw} does not exclude the presence of high-frequency instabilities of small-amplitude solutions.\n\n\\end{enumerate}\n\nThe results obtained from the Krein signature calculations are confirmed by numerical results, see Fig.~\\ref{fig:bwstuff}. Panel~(a) shows a numerically computed traveling wave solution of the BW equation \\rf{bw}. This solution is computed using a cosine collocation method with 60 points, as for the Whitham equation, see Fig.~\\ref{fig:wstuff} \\cite{SanfordKodamaCarterKalisch}. For the solution plotted, $c\\approx 1.049815$. The second panel in the first row displays the spectrum computed using Hill's method with 100 modes and 20000 values of the Floquet parameter, using an interpolation of the solution profile. This panel shows the presence of a large number of apparent instabilities, most with small growth rate, in the neighborhood of the imaginary axis. The third panel shows a zoom of the region around the origin, revealing a modulational instability. This is expected, since such an instability is also present for the Whitham equation, see Section~\\ref{ex:whitham}. The fourth panel zooms in on the first bubble of instabilities centered on the positive imaginary axis, revealing a shape and location that is consistent with the Krein collision theory presented here.\n\n\\begin{figure}[hbt]\n\\begin{tabular}{cc}\n\\hspace*{-1.0in}\\def3.2in}\\input{bubbletrack.pdf_tex{3.2in}\\input{bwprofile.pdf_tex} &\n\\hspace*{-1.14in}\\def3.2in}\\input{bubbletrack.pdf_tex{4in}\\input{bwstab.pdf_tex}\\\\\n\\hspace*{-0.7in}(a) & \\hspace*{-1.5in}(b)\\\\\n\\hspace*{-0.3in}\\def3.2in}\\input{bubbletrack.pdf_tex{4in}\\input{bwbf.pdf_tex} &\n\\hspace*{-1.14in}\\def3.2in}\\input{bubbletrack.pdf_tex{4in}\\input{bwbubble.pdf_tex}\\\\\n\\hspace*{-0.7in}(c) & \\hspace*{-1.5in}(d)\\\\\n\\end{tabular}\n\\caption{\\la{fig:bwstuff} (a) A small-amplitude traveling wave solution of the Boussines-Whitham equation \\rf{bw} with $c\\approx 1.0498515$. (b) The numerically computer stability spectrum. (c) A blow-up of the stability spectrum in a neighborhood of the origin. (d) A blow-up of the stability spectrum around what appears as a horizontal segment visible in (b) immediately above the longest segment appearing horizontal. More detail is given in the main text.\n    }\n\\end{figure}\n\n\\vspace*{0.1in}\n\nThe wave form displayed in Fig.~\\ref{fig:bwstuff}(a) does not have zero average, unlike the one shown in Fig.~\\ref{fig:wstuff}, for reasons explained here. Let us examine the stability of a flat-water state $q=a$ (constant), $p=0$. Thinking of the BW equation as an approximation to the water wave problem where the flat-water state is neutrally stable (spectrum on the imaginary axis), independent of the reference level of the water, the neutral stability of this state is desired in the context of the BW system as well. However, the BW system is easily checked to not be Galilean invariant, thus the average value of the solution may be important.\n\nLinearizing the system around the flat water state $(q,p)=(a,0)$ results in a linear system with constant coefficients, whose dispersion relation is given by $\\omega^2=2ak^2+k^2c^2(k)$. This results in two branches for the dispersion relation: $\\omega_{1,2}=\\pm k \\sqrt{c^2(k)+2a}$. It follows that if $a>0$ then both $\\omega_1$ and $\\omega_2$ are real, resulting in neutral stability, since the stability eigenvalue and the frequency $\\omega(k)$ are different by a factor of $i$. On the other hand, if $a<0$, it follows that both $\\omega_1$ and $\\omega_2$ are imaginary for sufficiently large $k$, since $\\lim_{|k|\\rightarrow \\infty}c(k)=0$. This leads to the dynamics of the flat-water state with $a<0$ to not only be unstable, but to be ill-posed, as the growth rate of the instability $\\rightarrow \\infty$ as $|k|\\rightarrow \\infty$. Thus Whitham-izing the Bad Boussinesq equation and incorporating the full water wave dispersion relation does not remove the illposedness of the problem. Rather it alters it where negative constant solutions experience unbounded growth, unlike positive constant solutions.\n\nIt is observed numerically that this behavior of perturbed constant solutions is carried over to nonconstant solutions: solutions of negative average display the same illposed behavior described above, with stability spectra that have unbounded real part. In contrast, the spectra of solutions of positive average have bounded real part, as in Fig.~\\ref{fig:bwstuff}. Annoyingly, the illposed behavior extends to numerical solutions constructed to have zero average. Presumably this is a consequence of numerical error, as higher accuracy numerical experiments display narrower spectra whose real part tends to infinity more slowly.\n\nWe may summarize our findings on the BW equation as follows. The equation was constructed as a bi-directional Whitham equation so as to truly have the same linear dispersion relation as the water wave problem. Even though the BW has a different Poisson structure than the water wave problem, we find that periodic solutions of the BW are susceptible to high-frequency instabilities originating from Krein collisions at the exact same locations on the imaginary axis as the water wave problem. On the other hand, we have not attempted to quantify whether the resulting growth rates are comparable to those for the water wave problem. Further, the illposedness of the equation for solutions of negative average is a significant strike against its potential use in applications. Nevertheless, it appears possible to design more equations like the BW equation, possessing the exact same dispersion relation as the water wave problem and with it all its high-frequency instabilities, without the equation dynamics being illposed for any important class of solutions.\n\n\\section*{Acknowledgements}\n\nWe wish to thank Richard Kollar for interesting discussions. John Carter helped our understanding of the Whitham equation and Mat Johnson was a part of our initial investigations on the Boussinesq-Whitham equation.  This work was supported by the National\nScience Foundation through grant NSF-DMS-1008001 (BD) and in part by the EPSRC under grant EP\/J019569\/1 and NSERC (OT). Any opinions, findings,\nand conclusions or recommendations expressed in this material are\nthose of the authors and do not necessarily reflect the views of the\nfunding sources.\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur work is devoted to low-resource languages: Veps and Karelian. These languages belong to the Finno-Ugric languages of the Uralic language family. \nMost Uralic languages still lack full-fledged morphological analyzers and large corpora~\\cite{ref_Pirinen_2019}.\n\nIn order to avoid this trap the researchers of \nKarelian Research Centre are developing the    \nOpen corpus of Veps and Karelian languages (VepKar). \nOur corpus contains morphological dictionaries of the Veps language and the three supradialects of the Karelian language: the Karelian Proper, Livvi-Karelian and Ludic Karelian. \nThe developed software (corpus manager)\\footnote{See\n\\url{https:\/\/github.com\/componavt\/dictorpus}}\nand \nthe database, including dictionaries and texts, have open licenses. \n\nAlgorithms for assigning part of speech tags to words \nand grammatical properties to words, \nwithout taking into account a context, \nusing manually built dictionaries, are presented in the article\n(see Section~\\ref{section:pos_algorithm}).\n\nThe proposed technology of evaluation (see the section~\\ref{section:experiments}) \nallows to use all 313 thousand Veps and 66 thousand Karelian words to verify the accuracy of the algorithms (Table~\\ref{tab:nwords}).\nOnly a third of Karelian words (28\\%) and two-thirds of Veps words (65\\%) in the corpus texts are automatically linked to the dictionary entries \nwith all word forms (Table~\\ref{tab:nwords}).\nThese words were used in the evaluation of the algorithms.\n\n\\begin{table}\n\\caption{Total number of words in the VepKar corpus and dictionary}\\label{tab:nwords}\n\n\\begin{tabular}{c c c r} \\toprule\nLanguage & \\multicolumn{1}{|c|}{\\specialcell{The total number\\\\\n                                        of tokens in texts,\\\\\n                                        $10^3$}}\n         & \\multicolumn{1}{|c|}{\\specialcell{N tokens linked to\\\\\n                                    dictionary automatically,\\\\\n                                    $10^3$}}\n         & \\specialcell{N tokens linked to\\\\lemmas having\\\\\n                        a complete paradigm,\\\\\n                        $10^3$}\\\\\n\\midrule\nVeps    & 488     & 400     (82\\%) & 313     (65\\%)\\\\\n\\cmidrule{1-1}\n\\specialcell{Karelian\\\\Proper} \n        & 245     & 111     (45\\%) & 69     (28\\%)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\nLet us describe several works devoted to the development of morphological analyzers for the Veps and Karelian languages.\n\\begin{itemize}\n  \\item The Giellatekno language research group is mainly engaged in low-resource languages, the project covers about 50 languages~\\cite{ref_Moshagen_2014}. Our project has something in common with the work of Giellatekno in that (1) we work with low-resource languages, (2) we develop software and data with open licenses.\n  \n  A key role in the Giellatekno infrastructure is given to formal approaches (grammar-based approach) in language technologies. They work with morphology rich languages. Finite-state transducers (FST) are used to analyse and generate the word forms~\\cite{ref_Moshagen_2014}.\n  \n  \\item There is a texts and words processing library for the Uralic languages called UralicNLP~\\cite{ref_Hamalainen_2019}.\n  This Python library provides interface to such Giellatekno tools as FST for processing morphology and constraint grammar for syntax. \n  The UralicNLP library lemmatizes words in 30 Finno-Ugric languages and dialects including the Livvi dialect of the Karelian language (\\textit{olo} -- language code).\n\\end{itemize}\n\n\n\\section{Data organization and text tagging in the VepKar corpus}\n\nAutomatic text tagging is an important area of research in corpus linguistics. It is required for our corpus to be a useful resource.\n\nThe corpus manager handles the dictionary and the corpus of texts (Fig.~\\ref{fig:corpus:manager:tagging}).\nThe texts are segmented into sentences, \nthen sentences are segmented into words (tokens).\nThe dictionary includes lemmas with related meanings, word forms, and \\textbf{sets} of \\textbf{gram}matical features (in short -- \\textbf{gramsets}).\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{pix\/vepkar_structure_v7-b-en.png}\n\\caption{Data organization and text tagging in the VepKar corpus.\\newline Total values (e.g. number of words, texts) are calculated for all project languages.} \\label{fig:corpus:manager:tagging}\n\\end{figure}\n\n\nText tokens are automatically searched in the dictionary of lemmas and word forms, this is the first stage (\\RNum{1}) of the text tagging, it is not presented at Fig.~\\ref{fig:corpus:manager:tagging}.\n\\begin{enumerate}\n\\item \\textbf{Semantic tagging}. \nFor the word forms found in the dictionary, the lemmas linked with them are selected (\\RNum{2} at Fig.~\\ref{fig:corpus:manager:tagging}), then all the meanings of the lemmas are collected (\\RNum{3}) and semantic relationships are established between the tokens and the meanings of the lemmas (marked ``not verified'') (\\RNum{4}). \n\nThe task of an expert linguist is to check these links and confirm their correctness, either choose the correct link from several possible ones, or manually add a new word form, lemma or meaning.\n\n\n\\begin{tikzpicture}\n\n\\node (token)               {tokens (words)};\n\\node (wf) [right=of token] {word forms};\n\\node (lemka) [right=of wf] {lemmas};\n\\node (meaning) [right=of lemka] {meanings};\n\n\\draw[<->] (token.east) -- node[above] {\\RNum{1}} (wf.west);\n\\draw[<->] (lemka.east) -- node[above] {\\RNum{3}} (meaning.west);\n\\draw[<->] (wf.east) -- node[above] {\\RNum{2}} (lemka.west);\n\\path[<->,red] (token.south) edge [bend right=13] node[above] {\\RNum{4} not verified} (meaning.south);\n\\end{tikzpicture}\n\n\nWhen the editor clicks on the token in the text, then a drop-down list of lemmas with all the meaning will be shown. The editor selects the correct lemma and the meaning (Fig.~\\ref{fig:meaning:select}). \n\n\n\n\n\n\n\\begin{figure}\n  \\fbox{\\includegraphics[width=1\\textwidth]{pix\/samarialaine-en-2020_Select-meaning_narrow.png}}\n\\caption[Selection the correct lemma and the meaning for token]%\n    {Vepsian and Russian parallel Bible \n    translation\\textsuperscript{$\\ddagger$} \n    in the corpus.\n    The editor clicks the word ``Opendai'' in the text, a menu pops up. This menu contains a list of meanings collected automatically for this token, namely: \n    the meaning of the noun ``teacher'' (``opendai'' in Veps) and five meanings of the Veps verb ``opeta''.  \n    The noun ``opendai'' and the verb ``opeta'' have the same wordform ``opendai''.\n    If the editor selects one of the lemma meanings in the menu (clicks the plus sign), then the token and the correct meaning of the lemma will be connected \n    ({\\color{red}\\RNum{4} stage is verified}).\n    \\newline \n}\n\\line(1,0){82}\\newline\n\\small\\textsuperscript{$\\ddagger$} See full text online at VepKar: \\url{http:\/\/dictorpus.krc.karelia.ru\/en\/corpus\/text\/494}\n    \n  \\label{fig:meaning:select}\n\\end{figure}\n\n\n\\item \\textbf{Morphological tagging}. \nFor the word forms found in the dictionary, the gramsets linked with them are selected (\\RNum{5}) and morphological links are established (\\RNum{6}) between the tokens and the pairs ``word form -- gramset'' (Fig.~\\ref{fig:corpus:manager:tagging}). The expert's task is to choose the right gramset.\n\n\\begin{tikzpicture}\n\\node (token)               {tokens (words)};\n\\node (wf) [right=of token] {word forms};\n\\node (gramset) [right=of lemka] {gramsets};\n\n\\draw[<->] (token.east) -- node[above] {\\RNum{1}} (wf.west);\n\\draw[<->] (wf.east) -- node[above] {\\RNum{5}} (gramset.west);\n\\path[<->,red] (token.south) edge [bend right=13] node[above] {\\RNum{6}} (gramset.south);\n\\end{tikzpicture}\n\n\\end{enumerate}\n\n\n\n\\section{Corpus tagging peculiarities}\\label{section:corpus_peculiarities}\nIn this section, we describe why the word forms with white spaces and analytical forms are not taken into account in the search algorithm described below.\nAnalytical form is the compound form consisting of auxiliary words and the main word.\n\nThe ultimate goal of our work is the morphological markup of the text, previously tokenized into words by white spaces and non-alphabetic characters \n(for example, brackets, punctuation, numbers).\nTherefore, analytical forms do not have markup in the texts.\n\nAlthough we store complete paradigms in the dictionary, including analytical forms, such forms do not used in the analysis of the text, because each individual word is analyzed in the text, not a group of words.\n\nFor example, we take the Karelian verb ``pageta'' (leave, run away).\nIn the dictionary not only the negative form of indicative, presence, first-person singular ``en pagene'' is stored, but also connegative (a word form used in negative clauses) of the indicative, presence ``pagene'', which is involved in the construction of five of the six forms of indicative, presence.\nThus, in the text the word 'en' (auxiliary verb 'ei', indicative, first-person singular) and 'pagene' (verb 'pageta', connegative of indicative, presence) are separately marked.\n\n\n\\section{Part of speech and gramset search by analogy algorithms}\\label{section:pos_algorithm}\n\nThe proposed algorithms operate on data from a morphological dictionary. \nThe algorithms are based on the analogy hypothesis that words with the \\emph{same suffixes} are likely to have the same inflectional models and the same sets of grammatical information (part of speech, number, case, tense, etc.). \nThe \\emph{suffix} here is a final segment of a string of characters.\n\nLet the hypothesis be true, in that case, if the suffixes of new words coincide with the suffixes of dictionary words, then part of the speech and other grammatical features of the new words will coincide with the dictionary words.\nIt should be noted that \nthe length of the suffixes is unpredictable and can be various for different pairs of words~\\cite[p.~53]{ref_Belonogov_2004}.\n\nThe POSGuess and GramGuess algorithms described below use the concept of ``suffix'' (Fig.~\\ref{fig:kezaman_substring}), the GramPseudoGuess algorithm uses the concept ``pseudo-ending'' (Fig.~\\ref{fig:huukkua_substring}).\n\n\n\\subsection{The POSGuess algorithm for part of speech tagging with a~suffix}\n\nGiven the set of words $W$, for each word in this set the part of speech is known. \nThe algorithm~\\ref{alg:POSGuess} finds a part of speech $\\pos_u$ for a given word~$u$ using this set. \n\n\\begin{algorithm\n    \\caption{Part of speech search by a suffix (POSGuess)}\n    \\label{alg:POSGuess}\n\\DontPrintSemicolon\n\\SetAlgoLined\n    \\KwData{\n        $P$ -- a set of part of speech (POS),\\hfill \\break\n        $W = \\{ w \\; | \\; \\exists \\pos_w \\in P \\}$ -- a set of words, POS is known for each word,\\hfill \\break\n       \n       \n        $u \\notin W$ -- the word with unknown POS,\\hfill \\break\n        $len(u)$ -- the length (in characters) of the string $u$.\n       }\n    \\KwResult{\\[\n        u_z : \n        \\begin{cases}\n           \n           \n            %\n            len (u_z) \\xrightarrow[ z=2,\\ldots,len(u) ]{} \\max,\\;\n           \n            \\Comment{\/\/\\,Longest suffix}\\\\\n            \\exists w \\in W : w = w_{\\text{prefix}} \\concat u_z\\;\n            \\Comment{\/\/\\,Concatenation of strings}\n        \\end{cases}\n        \\]\n    \\begin{align*}\n\\text{Counter} & \\left[ \\pos^k \\right] = c^k, \\; k = \\overline{1,m}, \\; \\text{where :}\\\\\n        & c^k \\in \\mathbb{N}, \\; c^1 \\geq c^2 \\geq \\ldots \\geq c^m,\\\\\n        & \\exists w^k_i \\in W : \n            w^k_i = {w_{\\text{prefix}}}^k_i \\concat u_z \\Rightarrow\n            c^k = \\vert \\pos^k_{w^k_i} \\vert, \\\\\n        &    \n            i = \\overline{1,c^k}, \\\\\n        & \\forall i : \\pos^k_{w^k_i} = \\pos^k \\in P, \\;\\;\\;\\; \n           a \\ne b \\Leftrightarrow \\pos^a \\ne \\pos^b\\\\\n        & \n        m \\; \\text{-- the number of different POS of found words} \\; w^k_i\n    \\end{align*}\n   \n    }\n    \\BlankLine\n    \n    $z$ = 2 \\Comment{\/\/ The position in the string $u$}\n    $z_{found} = \\text{FALSE}$ \\;\n    \\BlankLine\n    \\While{ $z \\leq len(u)$ and $\\neg z_{found}$ }{\n        \\BlankLine\n        \\Comment{\/\/ The suffix of the word $u$ from $z$-th character}\n        $u_z = \\text{ substr } (u, z) \\;$\n        \\BlankLine\n        \\ForEach{$w \\in W$}{\n            \\Comment{\/\/ If the word $w$ has the suffix $u_z$ (regular expression)\n            }\n            \\If{$w =\\sim \\textrm{m}\/u_z\\$\/$}{\n                $\\text{Counter} \\left[ \\pos_w \\right] ++$\\\\\n                $z_{found} = \\text{TRUE}$ \\Comment{\/\/ Only POS of words with this $u_z$ suffix will be counted. The next \"while\" loop will break, so the shorter suffix $u_{z+1}$ will be omitted.}\n            }\n        }\n    z = z + 1\n    }\n    \\BlankLine\n    \\Comment{\/\/ Sort the array in descending order, according to the value}\n    \\verb|arsort|( $\\text{Counter} \\left[ \\; \\right]$ )\n\\end{algorithm}\n\n\\newpage\n\nIn Algorithm~\\ref{alg:POSGuess} \nwe look for in the set $W$ (line~5) the words \nwhich have the same suffix $u_z$ as the unknown word $u$. \nFirstly, we are searching for the longest substring of $u$, \nthat starts at index z. \nThe first substring $u_{z=2}$ will start at the second character (line 1 in Algorithm~\\ref{alg:POSGuess}), since $u_{z=1} = u$ is the whole string \n(Fig.~\\ref{fig:kezaman_substring}). \n\nThen we increment the value $z$ decreasing the length of the substring $u_z$ in the loop, \nwhile the substring $u_z$ has non-zero length, $z \\leq len(u)$.\nIf there are words in $W$ with the same suffix, then we count the number of similar words for each part of the speech and stop the search.\n\n\\hfill \\break\n\n\nThe Fig.~\\ref{fig:kezaman_substring} shows the idea of the algorithm~\\ref{alg:POSGuess}: for a new word ($kezaman$), we look for a word form in the dictionary ($raman$) with the same suffix ($aman$).\n\n\nWe begin to search in the dictionary for word forms with the suffix $u_{z=2}$. If we have not find any words, then we are looking for $u_{z=3}$ and so on.\nThe longest suffix $u_{z=4}$=``aman'' with $z=4$ is found.\n\nThen we find all words with the suffix $u_{z=4}$ and count \nhow many of such words are nouns, verbs, adjectives and so on.\nThe result is written to the array $Counter[\\,]$. In Fig.~\\ref{fig:kezaman_substring} the \\emph{noun} ``raman'' was found, therefore we increment the value of $Counter[noun]$.\n\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{pix\/kezaman_v05.png}\n\\caption{Veps nouns in the genitive case ``kezaman'' (``kezama'' means ``melted ground'') and ``raman'' (``rama'' means ``frame''). The word $u$ \nwith an unknown part of speech is ``kezaman''. The word $w$ from the dictionary with the known POS is ``raman''. \nThey share the common suffix $u_z$, which is ``aman''.} \\label{fig:kezaman_substring}\n\\end{figure}\n\n\n\n\n    \n    \n    \n\\subsection{The GramGuess algorithm for gramset tagging with a suffix}\n\nThe GramGuess algorithm is exactly the same as the POSGuess algorithm, \nexcept that it is needed to search a subset of gramsets instead of parts of speech.\nThat is in the set $W$ the gramset is known for each word. \nThe gramset is a set of morphological tags (number, case, tense, etc.).\n\n\n\n\n\\subsection{The GramPseudoGuess algorithm for gramset tagging with a pseudo-ending}\n\nLet us explain the ``pseudo-ending'' used in the algorithm GramPseudoGuess.\n\nAll word forms of one lemma share a common invariant substring. This substring is a \\emph{pseudo-base} of the word (Fig.~\\ref{fig:huukkua_substring}). \nHere the pseudo-base is placed at the start of a word, it suits for the Veps and Karelian languages.\nFor example, in Fig.~\\ref{fig:huukkua_substring} the invariant substring ``huuk'' is the pseudo-base for all word forms of the lemma ``huukkua''. \nThe Karelian verb ``huukkua'' means ``to call out'', ``to holler'', ``to halloo''.\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{pix\/huukkua_v02.png}\n\\caption{Wordforms of the Karelian verb ``huukkua'' (it means ``to call out'', ``to holler'', ``to halloo''). All word forms have the same pseudo-base and different pseudo-endings for different set of grammatical attributes (gramsets).} \\label{fig:huukkua_substring}\n\\end{figure}\n\nGiven the set of words $W$, for each word in this set a gramset and a pseudo-ending are known. \nThe algorithm~\\ref{alg:POSPseudoGuess} finds a gramset $g_u$ for a given word~$u$ using this set.\n\n\\begin{algorithm\n    \\caption{Gramset search by a pseudo-ending (GramPseudoGuess)}\n    \\label{alg:POSPseudoGuess}\n\\DontPrintSemicolon\n\\SetAlgoLined\n    \\KwData{\n        $G$ -- a set of gramsets,\\hfill \\break\n        $W = \\{ w \\; | \\; \\exists \\,g_w \\in G, \n                \\exists \\,\\pend_w : w = w_{\\text{prefix}} \\concat \\pend_w \\}$ \n                -- a set of words, where gramset and pseudo-ending (pend) are known for each word,  \\hfill \\break\n       \n       \n        $u \\notin W$ -- the word with unknown gramset,\\hfill \\break\n        $len(u)$ -- the length (in characters) of the string $u$.\n       }\n    \\KwResult{\\[\n        u_z : \n        \\begin{cases}\n           \n           \n            %\n            len (u_z) \\xrightarrow[ z=2,\\ldots,len(u) ]{} \\max,\\;\n           \n            \\Comment{\/\/\\,Longest substring}\\\\\n            \\exists w \\in W : \\pend_w = u_z\\;\n        \\end{cases}\n        \\]\n    \\begin{align*}\n\\text{Counter} & \\left[ g^k \\right] = c^k, \\; k = \\overline{1,m}, \\; \\text{where :}\\\\\n        & c^k \\in \\mathbb{N}, \\; c^1 \\geq c^2 \\geq \\ldots \\geq c^m,\\\\\n        & \\exists w^k_i \\in W : \n            \\pend_{w^k_i} = u_z \\Rightarrow\n            c^k = \\vert g^k_{w^k_i} \\vert, \\\\\n        &    \n            i = \\overline{1,c^k}, \\\\\n        & \\forall i : g^k_{w^k_i} = g^k \\in G, \\;\\;\\;\\; \n           a \\ne b \\Leftrightarrow g^a \\ne g^b\\\\\n        & \n        m \\; \\text{-- the number of different gramsets of found words} \\; w^k_i\n    \\end{align*}\n   \n    }\n    \\BlankLine\n    \n    $z$ = 2 \\Comment{\/\/ The position in the string $u$}\n    $z_{found} = \\text{FALSE}$ \\;\n    \\BlankLine\n    \\While{ $z \\leq len(u)$ and $\\neg z_{found}$ }{\n        \\BlankLine\n        \\Comment{\/\/ The substring of the word $u$ from $z$-th character}\n        $u_z = \\text{ substr } (u, z) \\;$\n        \\BlankLine\n        \\ForEach{$w \\in W$}{\n            \\Comment{\/\/ If the word $w$ has the pseudo-ending $u_z$}\n            \\If{$\\pend_w == u_z$}{\n                $\\text{Counter} \\left[ g_w \\right] ++$\\\\\n                $z_{found} = \\text{TRUE}$ \\Comment{\/\/ Only gramsets of words with the pseudo-ending $u_z$ will be counted. The next \"while\" loop will break, so the shorter $u_{z+1}$ will be omitted.}\n            }\n        }\n    z = z + 1\n    }\n    \\BlankLine\n    \\Comment{\/\/ Sort the array in descending order, according to the value}\n    \\verb|arsort|( $\\text{Counter} \\left[ \\; \\right]$ )\n\\end{algorithm}\n\nIn Algorithm~\\ref{alg:POSPseudoGuess} \nwe look for in the set $W$ (line~5) the words \nwhich have the same pseudo-ending $u_z$ as the unknown word $u$. \nFirstly, we are searching for the longest substring of $u$, \nthat starts at index z. \n\nThen we increment the value $z$ decreasing the length of the substring $u_z$ in the loop, \nwhile the substring $u_z$ has non-zero length, $z \\leq len(u)$.\nIf there are words in $W$ with the same pseudo-ending, then we count the number of similar words for each gramset and stop the search.\n\n\n\\section{Experiments}\\label{section:experiments}\n\n\\subsection{Data preparation}\nLemmas and word forms from our morphological dictionary were gathered to \\emph{one set} as a search space of part of speech tagging algorithm.\nThis set contains unique pairs ``word -- part of speech''.\n\nIn order to search a gramset, we form the set consisting of (1) lemmas without inflected forms (for example, adverbs, prepositions) and (2) inflected forms (for example, nouns, verbs).\nThis set contains unique pairs ``word -- gramset''. For lemmas without inflected forms the gramset is empty.\n\nWe put on constraints for the words in both sets: strings must consist of more than two characters and must not contain whitespace.\nThat is, analytical forms and compound phrases have been excluded from the sets (see section~\\ref{section:corpus_peculiarities}).\n\n\n\\subsection{Part of speech search by a suffix (POSGuess algorithm)}\n\nFor the evaluation of the quality of results of the searching algorithm POSGuess the following function $\\text{eval}(\\pos^u)$ was proposed:\n\\begin{equation}\\label{eq:metric-eval}\n\\text{eval} \\left( \n    \\begin{array}{@{}l@{\\thinspace}l}\n       \\pos^u, \\\\\n       \\text{Counter} & \\left[ \\pos^k \\right] \\rightarrow c^k,\\\\\n                      & \\forall k = \\overline{1,m} \\\\\n    \\end{array}\n    \\right)     = \n        \\begin{cases}\n            \\multicolumn{2}{l}{\\bluecomment{The array Counter[\\,] \n                  do not contain the correct\n                  $\\pos^u$.}}\\\\\n            0, & \\pos^u \\ne \\pos^k, \\forall k = \\overline{1,m},\\\\[3mm]\n            \\multicolumn{2}{l}{\\bluecomment{\\specialcell{First several POS in the array can have \\\\ \n                  the same maximum frequency $c^1$, \n                  one of this POS is $\\pos^u$.}}}\\\\\n            1, & \\pos^u \\in \\{ \\left[ \\pos^1, \\ldots, \\pos^j \\right] : \n                               c^1=c^2=\\ldots=c^j, j \\leq m \\},\\\\[4mm]\n            \\multicolumn{2}{l}{\\frac{c^k}{ \\sum_{k=1}^{m}c^k }, \\; \n                \\exists k : \\pos^k = \\pos^u, \\;\n                            c^k < c^1\n                }\n        \\end{cases}\n\\end{equation}\n\nThis function~(\\ref{eq:metric-eval}) evaluates the result of the POSGuess algorithm against the correct part of speech $\\pos^u$.\nThe POSGuess algorithm counts the number of words similar to the word $u$ \nseparately for each part of speech \nand stores the result in the Counter array.\n\nThe Counter array is sorted in descending order, according to the value. The first element in the array is a part of speech with maximum number of words similar to the unknown word $u$.\n\n\\num{71 091} ``word -- part of speech'' pairs for the Proper Karelian supradialect and \\num{399 260} ``word -- part of speech'' pairs for the Veps language have been used in the experiments to evaluate algorithms.\n\nDuring the experiments, two Karelian words were found, for which there were no suffix matches in the dictionary. They are the word ``cap'' (English: snap; Russian: \\foreign{\u0446\u0430\u043f}) and the word ``\u0161tob'' (English: in order to; Russian: \\foreign{\u0447\u0442\u043e\u0431\u044b}).\nThat is, there were no Karelian words with the endings -p and -b.\nThis could be explained by the fact that these two words migrated from Russian to Karelian language.\n\n\n\nFigure~\\ref{fig:search_POS} shows the proportion of Veps and Karelian words with correct and wrong part of speech assignment by the POSGuess algorithm. \nValues along the X axis are the values of the function $\\text{eval}(\\pos^u)$, \nsee the formula~(\\ref{eq:metric-eval}). \nThis function for evaluating the part of speech assignment takes the following values:\n\n\\begin{labeling}{eval-formula}\n\\item [0] \\num{4.7}\\% of Vepsian words and 9\\% of Karelian words ($x = 0$ in Fig.~\\ref{fig:search_POS}) were assigned the wrong part of speech. \nThat is, there is no correct part of speech in the result array $Counter[\\,]$ in the \\mbox{POSGuess} algorithm.\nThis is the first line in the formula~(\\ref{eq:metric-eval}).\n\\hfill \\break\n\n\\item [\\num{0.1} -- \\num{0.5}] \n\\num{2.92}\\% of Vepsian words and \\num{4.23}\\% of Karelian words ($x \\in [0.1 ; 0.5]$ in Fig.~\\ref{fig:search_POS}) were assigned the partially correct POS tags.\nThat is, the array $Counter[\\,]$ contains the correct part of speech, but it is not at the beginning of the array. \nThis is the last line in the formula~(\\ref{eq:metric-eval}).\n\\hfill \\break\n\n\\item [1] \n\\num{92.38}\\% of Vepsian words and \\num{86.77}\\% of Karelian words ($x = 1$ in Fig.~\\ref{fig:search_POS}) were assigned the correct part of speech. \nThe array $Counter[\\,]$ contains the correct part of speech at the beginning of the array. \n\\end{labeling}\n\n\\begin{figure}[H]\n\\includegraphics[width=1\\textwidth]{pix\/pos_search_all_langs.png}\n\\caption{\nThe proportion of Vepsian (red curve) and Karelian (blue curve) words with correct ($x=1$) and wrong ($x=0$) part of speech assignment by the POSGuess algorithm \nwith the formula~(\\ref{eq:metric-eval}).\n} \\label{fig:search_POS}\n\\end{figure}\n\n\nFigure~\\ref{fig:search_POS} shows the evaluation of the results of the POSGuess algorithm for all parts of speech together.\nTable~\\ref{tab:POS:quantity:Veps} (Veps) and Table~\\ref{tab:POS:quantity:krl} (Karelian) show the evaluation of the same results of the POSGuess algorithm, \nbut they are presented for each part of speech separately.  \n\n\n\n\\begin{table}\n\\caption{Number of Vepsian words of different parts of speech \nused in the experiment. \nThe evaluation of results found by POSGuess algorithm by the formula~(\\ref{eq:metric-eval}) \nand fraction of results in percent, where \nthe column \\textit{0} means the fraction of words with incorrectly found POS, \\textit{1} -- the fraction of words with correct POS in the top \nof the list created by the algorithm.}\\label{tab:POS:quantity:Veps}\n\n\\setlength{\\tabcolsep}{8pt}\n\n\\begin{tabular}{ l r r r r r r r >{\\bfseries}r } \\toprule\n    \n     \\multicolumn{2}{c}{\\large Veps} &\n     \\multicolumn{7}{c}{\\specialcell{Fraction \n     of not guessed (column 0),\\\\\n     partly guessed (0.1--0.5) and guessed (1) POS, \\%}} \\\\ \\cmidrule(r){3-9}\n                                                \n POS & Words & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & \\multicolumn{1}{c}{1}\\\\ \\midrule\nVerb & \\num{93 047} & 2.12 & 0.52 & 0.55 & 0.47 & 0.36 & 0.01 & 95.97\\\\\nNoun & \\num{240 513} & 2.88 & 0.3 & 0.67 & 0.6 & 0.42 & 0.24 & 94.89\\\\\nAdjective & \\num{61 845} & 12.45 & 1.62 & 1.44 & 1.53 & 1.58 & 0.51 & 80.87\\\\\nPronoun & \\num{1 244} & 46.54 & 8.12 & 0.56 & 0.64 & 0 & 0 & 44.13\\\\\nNumeral & \\num{1 200} & 44 & 6.25 & 2.33 & 0.67 & 0.33 & 0 & 46.42\\\\\nAdverb & 650 & 64.92 & 3.08 & 2.46 & 1.23 & 0.46 & 0 & 27.85\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Number of Karelian words of different parts of speech used in the experiment.}\\label{tab:POS:quantity:krl}\n\n\\setlength{\\tabcolsep}{8pt}\n\n\\begin{tabular}{ l r r r r r r r >{\\bfseries}r } \\toprule\n    \n     \\multicolumn{2}{c}{\\large  Karelian} &\n     \\multicolumn{7}{c}{\\specialcell{Fraction \n     of not guessed (column 0),\\\\\n     partly guessed (0.1--0.5) and guessed (1) POS, \\%}} \\\\ \\cmidrule(r){3-9}\n                                                \n POS & Words & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & \\multicolumn{1}{c}{1}\\\\ \\midrule\nVerb & \\num{26 033} & 3.26 & 0.5 & 0.74 & 0.6 & 0.23 & 0.01 & 94.67\\\\\nNoun & \\num{36 908} & 5.47 & 0.38 & 1.13 & 1.08 & 0.52 & 0.04 & 91.38\\\\\nAdjective & \\num{6 596} & 35.81 & 6.66 & 4.15 & 4.56 & 2.73 & 0.38 & 45.71\\\\\nPronoun & 610 & 81.64 & 2.13 & 0.66 & 3.11 & 2.3 & 0 & 10.16\\\\\nNumeral & 582 & 65.81 & 1.72 & 1.03 & 0.17 & 1.03 & 0 & 30.24\\\\\nAdverb & 235 & 68.51 & 3.4 & 2.98 & 2.13 & 0 & 0 & 22.98\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[width=1\\textwidth]{pix\/POSdistrib.png}\n\\caption{Number of Vepsian and Karelian words of different parts of speech used in the experiment.} \\label{fig:POS:distribution}\n\\end{figure}\n\n\\subsection{Gramset search by a suffix (GramGuess algorithm) and by a pseudo-ending (GramPseudoGuess algorithm)}\n\n\\num{73 395} ``word -- gramset'' pairs for the Karelian Proper supradialect and \\num{452 790} ``word -- gramset'' pairs for the Veps language have been used in the experiments to evaluate GramGuess and GramPseudoGuess algorithms.\n\nA list of gramsets was searched for each word. The list was ordered by the number of similar words having the same gramset.\n\nFor the evaluation of the quality of results of the searching algorithms the following function $\\text{eval}(g^u)$ has been proposed:\n\\begin{equation}\\label{eq:metric-eval-gram}\n\\text{eval} \\left( \n    \\begin{array}{@{}l@{\\thinspace}l}\n       g^u, \\\\\n       \\text{Counter} & \\left[ g^k \\right] \\rightarrow c^k,\\\\\n                      & \\forall k = \\overline{1,m} \\\\\n    \\end{array}\n    \\right)     = \n        \\begin{cases}\n            \\multicolumn{2}{l}{\\bluecomment{The array Counter \n                  do not contain the correct gramset \n                  $g^u$.}}\\\\\n            0, & g^u \\ne g^k, \\forall k = \\overline{1,m},\\\\[3mm]\n            \\multicolumn{2}{l}{\\bluecomment{\\specialcell{First several gramsets in the array can have \\\\ \n                  the same maximum frequency $c^1$, \n                  one of these gramsets is $g^u$.}}}\\\\\n            1, & g^u \\in \\{ \\left[ g^1, \\ldots, g^j \\right] : \n                               c^1=c^2=\\ldots=c^j, j \\leq m \\},\\\\[4mm]\n            \\multicolumn{2}{l}{\\frac{c^k}{ \\sum_{k=1}^{m}c^k }, \\; \n                \\exists \\, k : g^k = g^u, \\;\n                            c^k < c^1\n                }\n        \\end{cases}\n\\end{equation}\n\nThis function~(\\ref{eq:metric-eval-gram}) evaluates the results of the GramGuess and GramPseudoGuess algorithms against the correct gramset $g^u$.\n\n\\begin{table}\n\\caption{Evaluations of results of gramsets search for Vepsian and Karelian by GramGuess and GramPseudoGuess algorithms.\n}\\label{tab:Gram:quantity}\n\n\\setlength{\\tabcolsep}{8pt}\n\n\\begin{tabular}{ c c c c c} \\toprule\n     & \\multicolumn{2}{c}{GramGuess}& \\multicolumn{2}{c}{GramPseudoGuess}\\\\ \\cmidrule(r){2-5}\n                                                \n Evaluation & Veps & Karelian & Veps & Karelian\\\\ \\midrule\n0 & 2.53 & 5.72 & 7.9 & 9.23\\\\\n0.1 & 0.53 & 0.83 & 1.04 & 1.57\\\\\n0.2 & 0.71 & 1.16 & 1.24 & 1.37\\\\\n0.3 & 0.64 & 0.89 & 2.68 & 1.36\\\\\n0.4 & 0.2 &\t0.56 & 0.14 & 0.68\\\\\n0.5 & 0.11 & 0.09 & 0.83 & 0.43\\\\\n1 &\t\\bf{95.29} & \\bf{90.74} & \\bf{86.17} & \\bf{85.36}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nThe table 4 shows that the GramGuess algorithm gives the better results than the GramPseudoGuess algorithm, namely: \n\\begin{labeling}{eval-formula}\n\\item [Karelian] \n90.7\\% of Karelian words were assigned a correct gramset by GramGuess algorithm versus 85.4\\% by GramPseudoGuess algorithm;\n\\hfill \\break\n\n\\item [Veps] \n95.3\\% of Vepsian words were assigned a correct gramset by GramGuess algorithm versus 85.4\\% by GramPseudoGuess algorithm.\n\\end{labeling}\n\nIt may be suggested by the fact that suffixes are longer than pseudo-endings. \nIn addition, the GramPseudoGuess algorithm is not suitable for a part of speech without inflectional forms.\n\n\n\n\n\n\n\\section{Morphological analysis results}\n\n\nIn order to analyze the algorithm errors, the results of the part-of-speech algorithm POSGuess were visualized using the Graphviz program.\nPart-of-speech error transition graphs were built for Veps language (Fig.~\\ref{fig:pos-error-graph-vep}) \nand Karelian Proper supradialect (Fig.~\\ref{fig:pos-error-graph-krl}).\n\nLet us explain how these graphs were built.\nFor example, a thick grey vertical arrow connects adjective and noun (Fig.~\\ref{fig:pos-error-graph-krl}), and this arrow has labels of 21.6\\%, 1424 and 3.9\\%.\nThis means that the POSGuess algorithm has erroneously identified \\num{1424} Karelian adjectives as nouns.\nThis accounted for 21.6\\% \nof all Karelian adjectives \nand 3.9\\% of nouns.\nThis can be explained by the fact that \nthe same lemma (in Veps and Karelian) can be both a noun and an adjective. \nNouns and adjectives are inflected in the same form (paradigm).\n\n\nThe experiment showed that there are significantly more such lemmas (noun-adjective) for the Karelian language than for the Veps language (21.6\\% versus 9.8\\% in Fig.~\\ref{fig:pos-error-graph}).\nAlthough in absolute numbers Veps exceeds Karelian, namely: 6061 versus 1424 errors of this kind. \nThis is because the Veps dictionary is larger in the VepKar corpus.\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=0.95\\textwidth]{pix\/247_veps_only_max0_pos-1_0.jpg}\n  \\caption{Veps language}\n  \\label{fig:pos-error-graph-vep}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n\\includegraphics[width=1.0\\textwidth]{pix\/456_pos-4_20_number-krl.jpg}  \\caption{Karelian Proper supradialect}\n  \\label{fig:pos-error-graph-krl}\n\\end{subfigure}\n\\caption{Part-of-speech error transition graph, which reflects the results of the POSGuess algorithm.}\n\\label{fig:pos-error-graph}\n\\end{figure}\n\n\n\n\n\\section{Conclusion}\n\nThis research devoted to the low-resource Veps and Karelian languages. \n\nAlgorithms for assigning part of speech tags to words and grammatical properties to words are presented in the article. \nThese algorithms use our morphological dictionaries, where the lemma, part of speech and a set of grammatical features (gramset) are known for each word form. \n\nThe algorithms are based on the analogy hypothesis that words with the same suffixes are likely to have the same inflectional models, the same part of speech and gramset.\n\nThe accuracy of these algorithms were evaluated and compared. 313 thousand Vepsian and 66 thousand Karelian words were used to verify the accuracy of these algorithms.\nThe special functions were designed to assess the quality of results of the developed algorithms.\n\n\n\n71,091 ``word -- part of speech'' pairs for the Karelian Proper supradialect and 399,260 ``word -- part of speech'' pairs for the Veps language have been used in the experiments to evaluate algorithms.\n86.77\\% of Karelian words \nand 92.38\\% of Vepsian words \nwere assigned a correct part of speech. \n\n73,395 ``word -- gramset'' pairs for the Karelian Proper supradialect and 452,790 ``word -- gramset'' pairs for the Veps language have been used in the experiments to evaluate algorithms.\n90.7\\% of Karelian words \nand \n95.3\\% of Vepsian words \nwere assigned a correct gramset by our algorithm.\n\n\nIf you need only one correct answer, then all three of developed algorithms are not very useful.\nBut in our case, the task is to get an ordered list of the parts of speech and gramsets for a word and to offer this list to an expert.\nThen the expert selects the correct part of speech and gramset from the list and assigns to the word. \nThis is a semi-automatic tagging of the texts.\nThus, these algorithms are useful for our corpus.\n\n\\bibliographystyle{splncs04}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUsing a short simple pulse for sending in a radar system needs to employ a high power transmitter which is expensive and easy to intercept. If the pulse width increases, the required range resolution to detect the several targets is severely degraded; therefore, the best solution is to modulate the pulse \\cite{1,2}. By performing modulation also known as pulse compression, bandwidth and signal-to-noise ratio (SNR) are increased and range resolution improves \\cite{3,4,5,6,7,8}, but the existence of high sidelobe level in the autocorrelation function (ACF) is annoying \\cite{9,10}. Pulse compression is done using various methods such as phase coding (Barker code), amplitude weighing, linear frequency modulation (LFM), and NLFM \\cite{11,12,13}.\n\nIn the phase coding and amplitude weighing methods, due to the phase discontinuity and variable amplitude, the mismatch loss increases in the receiver \\cite{14}; therefore, the use of LFM signals increases significantly due to continuous phase and constant amplitude. Although, the LFM method shows clear advantages over the phase coding and amplitude weighing methods, but the high sidelobe level is still annoying because of masking the smaller targets by sidelobes of bigger targets; thus, the NLFM method is used thanks to its significant decrease in peak sidelobe level (PSL) \\cite{15}.\n\nIn the NLFM method similar to LFM method, the signal amplitude is constant, and an optimal phase is intended to find. The stationary phase concept (SPC) is often used for NLFM method. The stationary phase concept expresses that the power spectral density (PSD) in a specific frequency is relatively large if frequency variations are small with respect to time \\cite{12}. The stationary phase method has been used in \\cite{16} resulted in significant reduction of PSL while the mainlobe width widens which can be neglected.\n\nIn this paper, the proposed method improves the PSL of the stationary phase method.\n\nThe remainder of the paper is organized as follows: The section 2 reviews NLFM waveform design based on the stationary phase method. NLFM waveform design based on the proposed method is explained in the section 3 in which the optimal phase is calculated initially and then the convergence of the proposed method is demonstrated using mathematical analysis. The section 4 contains simulation results. Finally, the section 5 concludes the paper.\n\n\\section{NLFM Signal Design with Stationary Phase Method}\nIn stationary phase method, the desired signal is defined as $x\\left( t \\right)$.\n\\begin{equation}\nx\\left( t \\right) = a\\left( t \\right)\\exp \\left( {j\\varphi \\left( t \\right)} \\right)\\;\\;\\;\\;\\;\\;\\; - \\frac{T}{2} \\le t \\le \\;\\frac{T}{2}\n\\end{equation}\nwhere $a\\left( t \\right)$, ${\\varphi \\left( t \\right)}$, and $T$ are amplitude, phase, and pulse width of $x\\left( t \\right)$, respectively. The ${f_n}$ is instantaneous frequency of $x\\left( t \\right)$ at time ${t_n}$ which is determined as follow\n\\begin{equation}\n{f_n} = \\frac{1}{{2\\pi }}{\\varphi '\\left( t_n \\right)}\n\\end{equation}\n\nIf ${X\\left( {{f_n}} \\right)}$ is Fourier transform of $x\\left( t \\right)$ in instantaneous frequency ${f_n}$ , so based on SPC, the relation between the power spectral density and frequency variation can be denoted as following equation \\cite{12}\n\\begin{equation}\n{\\left| {X\\left( {{f_n}} \\right)} \\right|^2} \\approx \\;2\\pi \\frac{{{a^2}\\left( {{t_n}} \\right)}}{{\\left| {\\varphi ''\\left( {{t_n}} \\right)} \\right|}}\n\\end{equation}\n\nEquation (3) indicates that PSD is directly proportional to $a^2\\left( t \\right)$ and it is inversely proportional to value of second order derivative of ${\\varphi \\left( t \\right)}$. In NLFM technique, signal amplitude is constant, so we consider $a\\left( t \\right) = A$, ($A$ = constant), and PSD only depends on the value of ${\\varphi ''\\left( t \\right)}$. If we estimate $X(f)$ with a function such as $Z(f)$, we can rewrite (3) in frequency domain as follow \\cite{12}\n\\begin{equation}\n\\theta ''\\left( f \\right) \\approx k{\\left| {{\\rm}Z\\left( f \\right)} \\right|^2},\\;\\;\\;k = constant\n\\end{equation}\nwhere $\\theta\\left( f \\right)$ is defined as the phase of $X(f)$. If $B$ is the bandwidth of $Z(f)$, $\\theta '\\left( f \\right)$ is obtained from the integral of $\\theta ''\\left( f \\right)$ that can be written as follow\n\\begin{equation}\n\\theta '\\left( f \\right) = \\mathop \\int \\limits_{-\\frac{{B}}{2}}^f {\\theta ''\\left(\\alpha\\right)} d\\alpha\n\\end{equation}\n\nThe group time delay function ${T_g}\\left( f \\right)$ is defined as following equation\n\\begin{equation}\n{T_g}\\left( f \\right) =  - \\frac{1}{{2\\pi }}\\theta '\\left( f \\right)\n\\end{equation}\nIf (4) and (5) are substituted in (6), so\n\\begin{equation}\n\\frac{{d{T_g}\\left( f \\right)}}{{df}} =  - \\frac{k}{{2\\pi }}{\\left| {Z\\left( f \\right)} \\right|^2}\\to{T_g}\\left( f \\right) =  - \\frac{k}{{2\\pi }}\\mathop \\int \\limits_{ - \\frac{B}{2}}^f {\\left| {Z\\left( \\alpha  \\right)} \\right|^2}d\\alpha  + r\n\\end{equation}\nwhere $r$ is constant and independent of the frequency. Also it is calculated by using the following boundary conditions\n\\begin{equation}\n{T_g}\\left( {B\/2} \\right) = T\/2,\\;\\;{T_g}\\left( { - B\/2} \\right) =  - T\/2\n\\end{equation}\n\nNow, the frequency function of time can be determined as following equation \\cite{17}\n\\begin{equation}\nf\\left( t \\right) = T_g^{ - 1}\\left( f \\right)\n\\end{equation}\n\nFinding the inverse group time delay function is not always easy; and in some cases it should be carried\nout numerically. Eventually, the phase of $x\\left( t \\right)$ can be calculated by integral of frequency function as follow\n\\begin{equation}\n \\varphi \\left( t \\right) = 2\\pi\\int\\limits_{-\\frac{{T}}{2}}^t f\\left( \\alpha  \\right)d\\alpha\n\\end{equation}\n\nThis method was applied in \\cite{16}. In section 4, the results are compared against the proposed method.\n\n\\section{NLFM Signal Design with Proposed Method}\n\nOur proposed method is based on a constrained optimization problem. First, a window as the initial window is considered, then by solving the constrained optimization problem, the desired signal is aimed to be found. This method is performed iteratively and the phase obtained from the stationary phase method \\cite{16} is used to start. To guarantee the convergence of the proposed method, triangle inequality and mathematical analysis are used to demonstrate the minimum error decrement in each iteration. Additionally, the minimum error value is positive expressing convergence due in large to the fact that a positive nonincreasing sequence certainly converges.\n\n\\subsection{Optimal Phase}\nIn the proposed method, to obtain the phase of the desired signal, first, an initial window such as ${W_{initial}}$ is considered where $\\left| {Y\\left( f \\right)} \\right|$ is its root. Difference between $\\left| {Y\\left( f \\right)} \\right|$ and the amplitude of the Fourier transform of the desired signal is defined as the error. Since in NLFM signals, amplitude is constant, so our goal is to minimize the error with the constraint of being unit the amplitude of $x\\left( t \\right)$ for $| t | \\le T\/2$, therefore we try to minimize the following equation\n\\begin{equation}\n\\begin{array}{cl}\n\\min\\limits_{X(f)} \n&E = \\displaystyle\\int \\limits_{ - \\frac{B}{2}}^{\\frac{B}{2}} {\\Big| {\\left| {Y\\left( f \\right)} \\right| - \\left|X\\left( f \\right)\\right|} \\Big|^2}df\\\\\n\\text{s.t.} &\n\\begin{cases}\n| {x(t)} |^2 = 1 &| t | \\le T\/2\\\\\nx(t) = 0&| t | > T\/2\n\\end{cases}\n\\end{array}\n\\end{equation}\nIf two complex numbers come close to each other, then it can be concluded that the values of their amplitudes also close together, so if the following equation is reduced, then the error can be reduced, which is expressed in the phase matching problems \\cite{18,19,20}\n\\begin{equation}\n\\begin{array}{cl}\n\\min\\limits_{\\theta (f),X(f)} \n&E = \\displaystyle\\int \\limits_{ - \\frac{B}{2}}^{\\frac{B}{2}} {\\Big| {\\left| {Y\\left( f \\right)} \\right|\\exp (j\\theta (f)) - X\\left( f \\right)} \\Big|^2}df\\\\\n\\text{s.t.} &\n\\begin{cases}\n| {x(t)} |^2 = 1 &| t | \\le T\/2\\\\\nx(t) = 0&| t | > T\/2\n\\end{cases}\n\\end{array}\n\\end{equation}\nAssume ${Y_\\theta }\\left( f \\right) = \\left| {Y\\left( f \\right)} \\right|\\exp \\left( {j\\theta \\left( f \\right)} \\right)$ and $f = kB\/(K - 1)$, then the error equation can be written in the discrete form as follow\n\\begin{equation}\nE =  \\sum \\limits_{k = 0}^{K - 1} {\\left| {{Y_\\theta }\\left( k \\right) - X\\left( k \\right)} \\right|^2} \n\\end{equation}\nwhere ${X\\left( k \\right) = \\mathop \\sum \\limits_{n = 0}^{N - 1} x\\left( n \\right)\\exp \\left( { - j\\frac{{2\\pi kn}}{K}} \\right)}$.\nConsider equations in vector space and assume ${\\bf{x}} = {\\left[ {x\\left( 0 \\right),\\ x\\left( 1 \\right), \\dots,\\ x\\left( {N - 1} \\right)} \\right]^T}$ and ${{\\left[ {\\bf{W}} \\right]_{k,n}} = \\exp \\left( { - j\\frac{{2\\pi kn}}{K}} \\right)}$, so\n\\begin{equation}\n\\left[ {\\begin{array}{*{20}{c}}\n{\\begin{array}{*{20}{c}}\n{X\\left( 0 \\right)}\\\\\n\\end{array}}\\\\\n{\\begin{array}{*{20}{c}}\n \\vdots \\\\\n{X\\left( {K - 1} \\right)}\n\\end{array}}\n\\end{array}} \\right] = {\\bf{Wx}}\n\\end{equation}\nIf we assume ${{\\bf{Y}}_{\\bf{\\theta }}} = {\\left[ {{Y_\\theta }\\left( 0 \\right),\\ {Y_\\theta }\\left( 1 \\right), \\ldots.\\ {Y_\\theta }\\left( {K - 1} \\right)} \\right]^T}$, the (12) is rewritten as follow\n\\begin{equation}\n\\begin{array}{rl}\n\\min\\limits_{\\bf \\uptheta, \\bf{x}} & {\\left( {{{\\bf{Y}}_{\\bf{\\theta }}} - {\\bf{Wx}}} \\right)^H}\\left( {{{\\bf{Y}}_{\\bf{\\theta }}} - {\\bf{Wx}}} \\right)\\\\\n\\text{s.t.}&{\\left| {x\\left( n \\right)} \\right|^2} = 1,\\quad n=1,\\dots,N-1\n\\end{array}\n\\end{equation}\nUsing Lagrangian method, we solve the obtained constrained optimization problem \\cite{21}.\n\\begin{align}\nJ &= {\\left( {{{\\bf{Y}}_{\\bf{\\theta }}} - {\\bf{Wx}}} \\right)^H}\\left( {{{\\bf{Y}}_{\\bf{\\theta }}} - {\\bf{Wx}}} \\right) + \\displaystyle\\sum_{n=0}^{N-1}{\\lambda _n}\\;{\\left| {x\\left( n \\right)} \\right|^2} \\nonumber\\\\\n&= {{\\bf{Y}}_{\\bf{\\theta }}}^H{{\\bf{Y}}_{\\bf{\\theta }}} {-} {{\\bf{Y}}_{\\bf{\\theta }}}^H{\\bf{Wx}} {-} {{\\bf{x}}^H}{{\\bf{W}}^H}{{\\bf{Y}}_{\\bf{\\theta }}} {+} {{\\bf{x}}^H}{{\\bf{W}}^H}{\\bf{Wx}} {+} {{\\bf{x}}^H}{\\bf{\\Lambda }}{\\bf{x}}\n\\end{align\nwhere ${\\lambda _n}$ is Lagrange multiplier and ${\\bf{\\Lambda }} = \\text{diag}({\\lambda _0},\\;{\\lambda _1},\\;...,{\\lambda _{N - 1}})$. The symbol diag(.) is the diagonal matrix which the entries outside the main diagonal are all zero. Because of the orthogonality of the columns of the matrix ${\\bf{W}}$, the value of ${{\\bf{W}}^H}{\\bf{W}}$ is equal to $K{{\\bf{I}}_N}$ where ${{\\bf{I}}_N}$ is identity matrix of size $N$. We take $J$ derivative with respect to ${\\bf{x}}$.\n\\begin{gather}\n\\cfrac{{\\partial J}}{{\\partial {\\bf{x}}}} =  - {({{\\bf{W}}^H}{{\\bf{Y}}_{\\bf{\\theta }}})^*} + K{{\\bf{I}}_N}{{\\bf{x}}^*} + {\\bf{\\Lambda }}{{\\bf{x}}^*} = 0\\nonumber\\\\\n\n \\Rightarrow {\\bf{x}} = {(K{{\\bf{I}}_N} + {{\\bf{\\Lambda }}^*})^{ - 1}}{{\\bf{W}}^H}{{\\bf{Y}}_{\\bf{\\theta }}}\n\\end{gather}\nwhere * denotes the complex conjugate, ${(K{{\\bf{I}}_N} + {{\\bf{\\Lambda }}^*})^{ - 1}}$ is the inverse matrix of $K{{\\bf{I}}_N} + {{\\bf{\\Lambda }}^*}$ and calculated as follow\n\\begin{equation}\n{(K{{\\bf{I}}_N} + {{\\bf{\\Lambda }}^*})^{ - 1}} = \n\\text{diag}\\left({\\frac{1}{{K + \\lambda _0^*}}},\\ \\dots,\\ {\\frac{1}{{K + \\lambda _{N-1}^*}}}\\right)\n\\end{equation}\nSince the constraints of the optimization problem are real, then the Lagrange multipliers ${\\lambda _n}$ are real and the vector ${\\bf{x}}$ is expressed as follow\n\\begin{equation}\n\\begin{bmatrix}\nx(0)\\\\\n \\vdots \\\\\nx(N - 1)\n\\end{bmatrix}=\n\\begin{bmatrix}\n\\frac{1}{{K + {\\lambda _0}}}\\displaystyle\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{1,k + 1}}{Y_\\theta }\\left( k \\right)} \\\\\n \\vdots \\\\\n\\frac{1}{{K + {\\lambda _{N - 1}}}}\\displaystyle\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{N,k + 1}}{Y_\\theta }\\left( k \\right)} \n\\end{bmatrix}\n\\end{equation}\nDue to the constraints of the problem, the values ${\\lambda _n}$ must be such that the square of the amplitude of each coefficient of ${\\bf{x}}$ is equal to one, so\n\\begin{gather}\n{\\left| {\\frac{1}{{K + {\\lambda _n}}}\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{n+1,k + 1}}{Y_\\theta }\\left( k \\right)} } \\right|^2} = 1\\nonumber\\\\\n \\Rightarrow {\\lambda _n} =  \\pm \\left| {\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{n+1,k + 1}}{Y_\\theta }\\left( k \\right)} } \\right| - K\n\\end{gather}\nTherefore, the vector ${\\bf{x}}$ is calculated as follow.\n\\begin{equation}\n{\\bf{x}} =\n\\left[ \\begin{array}{c}\n\\frac{{\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{1,k + 1}}{Y_\\theta }\\left( k \\right)} }}{{\\left| {\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{1,k + 1}}{Y_\\theta }\\left( k \\right)} } \\right|}}\\\\\n \\vdots \\\\\n\\frac{{\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{N,k + 1}}{Y_\\theta }\\left( k \\right)} }}{{\\left| {\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{N,k + 1}}{Y_\\theta }\\left( k \\right)} } \\right|}}\n\\end{array} \\right] = {{\\bf{\\Lambda }}_1}{{\\bf{W}}^H}{{\\bf{Y}}_{\\bf{\\theta }}}\n\\end{equation}\nwhere ${{\\bf{\\Lambda }}_1}$ is a $N\\times N$ matrix as follow\n\\begin{equation}\n\\left[{{\\bf{\\Lambda }}_1}\\right]_{i,j} = \n\\begin{cases}\n{{{\\left| {\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{i,k + 1}}{Y_\\theta }\\left( k \\right)} } \\right|^{-1}}}}, &\ni=j\\\\\n0,&i\\neq j\n\\end{cases}\n\\end{equation}\nTo achieve the desired signal, the proposed method is performed as an iterative algorithm; therefore, in respect to (21), in r-th iteration, the desired signal will be as follow\n\\begin{equation}\n{{\\bf{x}}^{(r)}} = {\\bf{\\Lambda }}_1^{(r - 1)}{{\\bf{W}}^H}{\\bf{Y}}_{\\bf{\\theta }}^{(r - 1)}\n\\end{equation}\n${{\\bf{\\uptheta }}^{\\left( r \\right)}}$ is the phase of $X\\left( f \\right)$ in r-th iteration, which can be calculated as follow\n\\begin{equation}\n{{\\bf{\\uptheta }}^{\\left( r \\right)}} = {\\rm{phase}}({\\bf{W}}{{\\bf{x}}^{(r)}})\n\\end{equation}\nBy calculating the ${{\\bf{\\uptheta }}^{\\left( r \\right)}}$ value, vector ${\\bf{Y}}_{\\bf{\\theta }}^{(r)}$ and then the matrix ${\\bf{\\Lambda }}_1^{(r)}$ is calculated.\n\\begin{equation}\n{\\bf{Y}}_{\\bf{\\theta }}^{(r)} = \\left[ {\\begin{array}{*{20}{c}}\n{\\left| {Y\\left( 0 \\right)} \\right|\\exp (j{\\theta ^{(r)}}(0))}\\\\\n \\vdots \\\\\n{\\left| {Y\\left( {K - 1} \\right)} \\right|\\exp (j{\\theta ^{(r)}}(K - 1))}\n\\end{array}} \\right]\n\\end{equation}\n\\begin{equation}\n\\left[{{\\bf{\\Lambda }}_1^{(r)}}\\right]_{i,j} = \n\\begin{cases}\n{{{\\left| {\\sum\\limits_{k = 0}^{K - 1} {{{\\left[ {{{\\bf{W}}^H}} \\right]}_{i,k + 1}}{Y_\\theta^{(r)} }\\left( k \\right)} } \\right|^{-1}}}}, &\ni=j\\\\\n0,&i\\neq j\n\\end{cases}\n\\end{equation}\nTo start the algorithm, we set the ${{\\bf{\\uptheta }}^{\\left( 0 \\right)}}$ equal to the phase value obtained from the stationary phase method for the Fourier transform of the NLFM signal \\cite{16}. Thus, by repeating the algorithm, we obtain the desired NLFM signal, and because the amplitude of NLFM signal is constant, so we can multiply the amplitude of the obtained signal in the constant coefficient $A$.\n\n\\subsection{Convergence of the Proposed Method}\nWith respect to the obtained value for the vector ${\\bf{x}}$ in (21) substituted in (15), the minimum value of $E$ is calculated as follow.\n\\begin{equation}\n\\begin{array}{l}\n{E_{\\text{Min}}} \n = {{\\bf{Y}}_{\\bf{\\theta }}}^H\\left({{\\bf{I}}_K} - {\\bf{W}}(2{{\\bf{\\Lambda }}_1} - K{\\bf{\\Lambda }}_1^2){{\\bf{W}}^H}\\right){{\\bf{Y}}_{\\bf{\\theta }}}\n\\end{array}\n\\end{equation}\nIn (27), ${\\bf{P}} = {\\bf{W}}(2{{\\bf{\\Lambda }}_1} - K{\\bf{\\Lambda }}_1^2){{\\bf{W}}^H}$ and ${\\bf{A}} = {{\\bf{I}}_K} - {\\bf{W}}(2{{\\bf{\\Lambda }}_1} - K{\\bf{\\Lambda }}_1^2){{\\bf{W}}^H}$ are the projection and orthogonal complement matrices, respectively \\cite{22}. The minimum error in r-th iteration is as follow\n\\begin{equation}\n\\Scale[0.845]{\nE_{\\text{Min}}^{(r)} = {\\left({{\\bf{Y}}_{\\bf{\\theta }}}^{(r-1)}\\right)^H}\n\\left({{\\bf{I}}_K} - {\\bf{W}}\\left(2{\\bf{\\Lambda }}_1^{(r-1)} - K{({\\bf{\\Lambda }}_1^{(r-1)})^2}\\right){{\\bf{W}}^H}\\right)\n{{\\bf{Y}}_{\\bf{\\theta }}}^{(r-1)}}\n\\end{equation}\nFor convergence of the proposed method, the error value must be reduced with increasing of iterations. In other words, for the convergence of the proposed method, the following inequality should be satisfied.\n\\begin{equation}\n0 \\le E_{\\text{Min}}^{(r + 1)} \\le E_{\\text{Min}}^{(r)}\n\\end{equation}\nTo prove (29), we use (12) to represent the minimum error in r-th iteration as\n\\begin{equation}\n\\begin{array}{l}\nE_{\\text{Min}}^{^{(r)}} =  \\displaystyle\\int \\limits_{ - \\frac{B}{2}}^{\\frac{B}{2}}\\Big| {\\left| {Y\\left( f \\right)} \\right|\\exp (j{\\theta ^{(r - 1)}}(f)) - {X^{(r)}}\\left( f \\right)} \\Big|^2df\n\\end{array}\n\\end{equation}\nAccording to the triangle inequality, we can write\n\\begin{equation}\n\\begin{array}{ll}\nE_{\\text{Min}}^{^{(r)}} \n& \\ge \\displaystyle\\int \\limits_{-\\frac{B}{2}}^{\\frac{B}{2}} \\left||Y( f )| - | X^{(r)}(f)| \\right|^2df\\\\\n&=\\displaystyle\\int \\limits_{-\\frac{B}{2}}^{\\frac{B}{2}}\n \\Big| {\\exp (j{\\theta ^{(r)}}(f))} \\Big|^2\n\\left||Y( f )| - | X^{(r)}(f)| \\right|^2df\\\\\n\n&= \\displaystyle\\int \\limits_{ - \\frac{B}{2}}^{\\frac{B}{2}}\\Big| {\\left| {Y\\left( f \\right)} \\right|\\exp (j{\\theta ^{(r)}}(f)) - {X^{(r)}}\\left( f \\right)} \\Big|^2df\n\\end{array}\n\n\\end{equation}\nOn the other hand, since $E_{\\text{Min}}^{(r + 1)}$ is the minimum value of error in $(r {+} 1)$-th iteration, the following equation is satisfied.\n\\begin{equation}\n\\begin{array}{ll}\nE_{\\text{Min}}^{^{(r + 1)}} &= \n\\displaystyle\\int \\limits_{ - \\frac{B}{2}}^{\\frac{B}{2}}\\Big| {\\left| {Y\\left( f \\right)} \\right|\\exp (j{\\theta ^{(r)}}(f)) - {X^{(r+1)}}\\left( f \\right)} \\Big|^2df\\\\\n&\\le\\displaystyle\\int \\limits_{ - \\frac{B}{2}}^{\\frac{B}{2}}\\Big| {\\left| {Y\\left( f \\right)} \\right|\\exp (j{\\theta ^{(r)}}(f)) - {X^{(r)}}\\left( f \\right)} \\Big|^2df\n\\end{array}\n\\end{equation}\nFrom the comparison of the two equations (31) and (32), we conclude $E_{\\text{Min}}^{^{(r)}} \\ge E_{\\text{Min}}^{^{(r + 1)}}$. Since we considered two arbitrary successive iterations, so (29) is satisfied for each two arbitrary successive iterations.\nSince $E_{\\text{Min}}^{(r)}$ is a positive nonincreasing sequence, convergence of the proposed method is guaranteed.\n\n\\section{Simulation Results}\nThe proposed method is performed for six initial windows of Raised-Cosine, Taylor, Chebyshev, Gaussian, Poisson, and Kaiser. Table 1 shows the formula of the selected windows, the group time delay functions, and their constant parameters. As already mentioned, the group time delay function for some windows is numerically calculated. In the Table 1, erf(.) and sgn(.) are defined as the error and sign functions, respectively. The design parameters such as $\\text{bandwidth ($B$)}$, pulse width ($T$), and sampling rate are considered equal to $\\text{100 MHz}$, 2.5 $\\mu$s, and 1 GHz, respectively. Fig. 1 and $\\text{Fig. 2}$ show the autocorrelation functions of the designed signals using the stationary phase method and the proposed method for the six selected windows, respectively. Also, Fig. 3 shows the minimum error for the six selected windows in different iterations.\n\n\\begin{figure*}[!h]\n\t\\centering\n\t\\mbox{\\subfloat[]{\\label{subfig10:a} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{RC_SPC_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{RC_SPC}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\\vspace{24pt}\n\t\\mbox{\\subfloat[]{\\label{subfig10:b} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Taylor_SPC_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Taylor_SPC}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n\t\\mbox{\\subfloat[]{\\label{subfig10:c} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Cheb_SPC_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Cheb_SPC}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\\vspace{24pt}\n\t\\mbox{\\subfloat[]{\\label{subfig10:d} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Gaussian_SPC_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Gaussian_SPC}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n          \\mbox{\\subfloat[]{\\label{subfig10:d} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Poisson_SPC_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Poisson_SPC}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n          \\mbox{\\subfloat[]{\\label{subfig10:d} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Kaiser_SPC_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Kaiser_SPC}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n\t\\caption{The autocorrelation functions of the designed\nsignals using the stationary phase method for the six windows of Raised-Cosine, Taylor, Chebyshev, Gaussian, Poisson, and Kaiser.}\n\t\\label{fig:fig10}\n\\end{figure*}\n\\begin{figure*}[!h]\n\t\\centering\n\t\\mbox{\\subfloat[]{\\label{subfig10:a} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{RC_PIA_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{RC_PIA}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\\vspace{24pt}\n\t\\mbox{\\subfloat[]{\\label{subfig10:b} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Taylor_PIA_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Taylor_PIA}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n\t\\mbox{\\subfloat[]{\\label{subfig10:c} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Cheb_PIA_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Cheb_PIA}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\\vspace{24pt}\n\t\\mbox{\\subfloat[]{\\label{subfig10:d} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Gaussian_PIA_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Gaussian_PIA}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n          \\mbox{\\subfloat[]{\\label{subfig10:d} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Poisson_PIA_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Poisson_PIA}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n          \\mbox{\\subfloat[]{\\label{subfig10:d} \t\\def\\big{\\includegraphics[width=0.48\\textwidth,height=6.25cm]{Kaiser_PIA_B}}\n\t\t\t\\def\\little{\\includegraphics[height=2.4cm]{Kaiser_PIA}}\n\t\t\t\\def\\stackalignment{r}\n\t\t\t\\topinset{\\little}{\\big}{15pt}{24pt}}}\n\t\\caption{The autocorrelation functions of the designed\nsignals using the proposed method for the six windows of Raised-Cosine, Taylor, Chebyshev, Gaussian, Poisson, and Kaiser.}\n\t\\label{fig:fig1}\n\\end{figure*}\n\n\\begin{table*}[!h]\n\t\\centering\n\t\\caption{Selected Windows.}\n\\resizebox{\\textwidth}{!}{\\begin{tabular}{c c c c}\n                     \\toprule[1.5pt]\\\\\n\t\t\\bfseries Windows & \\bfseries Formula & \\bfseries Group Time Delay Function & \\bfseries Constant Parameters\\\\ \\\\\n                     \\toprule[1.5pt]\\\\\n\t\tRaised-Cosine & $w\\left( n \\right) = k + \\left( {1 - k} \\right){\\rm{co}}{{\\rm{s}}^2}\\left( {\\cfrac{{\\pi n}}{{M - 1}}} \\right),\\;\\;\\;\\;\\;\\left| n \\right| \\le \\cfrac{{M - 1}}{2}$ & $ {\\cfrac{Tf}{B} + \\cfrac{T}{{2\\pi }}\\left( {\\cfrac{{1 - k}}{{1 + k}}} \\right)\\sin \\left( {\\cfrac{{2\\pi f}}{B}} \\right)} $  & $k = 0.17$\\\\ \\\\\n\t\t\\hline\\\\\n\t\tTaylor & $\\begin{array}{l}\nw\\left( n \\right) = 1 + \\displaystyle\\sum \\limits_{m = 1}^{\\bar n - 1} {F_m}{\\rm{cos}}\\left( {\\cfrac{{2\\pi mn}}{{M - 1}}} \\right),\\;\\;\\;\\;\\;\\left| n \\right| \\le \\cfrac{{M - 1}}{2}\\\\\n{F_m} = F\\left( {m,\\bar n,\\eta } \\right)\n\\end{array}$ & ${\\cfrac{Tf}{B} + \\cfrac{T}{{2\\pi }} \\displaystyle\\sum \\limits_{m = 1}^{\\bar n - 1} \\cfrac{{{F_m}}}{m}{\\rm{sin}}\\left( {\\cfrac{{2\\pi mf}}{B}} \\right)} $ & $\\begin{array}{l} \n\t\t\\eta  = 88.5\\;dB\\\\\n\t\t\\bar n = 2\n\t\t\\end{array}$ \\\\ \\\\\n\t\t\\hline \\\\\n\t\tChebyshev & $\\begin{array}{l}\nW\\left( m \\right) = \\cfrac{{\\cos \\left\\{ {M{\\rm{co}}{{\\rm{s}}^{ - 1}}\\left[ {\\beta \\cos \\left( {\\frac{{\\pi m}}{M}} \\right)} \\right]} \\right\\}}}{{\\cosh \\left[ {M{\\rm{cos}}{{\\rm{h}}^{ - 1}}\\left( \\beta  \\right)} \\right]}}\\;\\;,\\;\\;\\;\\;m = 0,1,2, \\ldots ,M - 1\\\\\n\\beta  = \\cosh \\left[ {\\frac{1}{M}{\\rm{cos}}{{\\rm{h}}^{ - 1}}\\left( {{{10}^\\alpha }} \\right)} \\right]\\\\\nw\\left( n \\right) = \\cfrac{1}{N} \\displaystyle\\sum \\limits_{m = 0}^{M - 1} W\\left( m \\right).{\\rm{exp}}\\left( {\\cfrac{{j2\\pi mn}}{M}} \\right),\\;\\;\\;\\;\\;\\;\\left| n \\right| \\le \\cfrac{{M - 1}}{2}\n\\end{array}$ & Calculated Numerically & $\\alpha  = 2$ \\\\ \\\\\n\t\t\\hline\\\\\n                      Gaussian& $w\\left( n \\right) = \\exp \\left( { - k{{\\left( {\\cfrac{n}{{2\\left( {M - 1} \\right)}}} \\right)}^2}} \\right),\\;\\;\\;\\;\\;\\;\\left| n \\right| \\le \\cfrac{{M - 1}}{2}$ & $\\cfrac{T}{{2\\;{\\rm{erf}}\\left( {\\sqrt k \/4} \\right)}}\\ {\\rm{erf}}\\left( {\\cfrac{{f\\sqrt k }}{{2B}}} \\right)\\;$& $k = 35.51$\\\\ \\\\\n\t\t\\hline\\\\\n                      Poisson& $w\\left( n \\right) = \\exp \\left( { - k\\cfrac{{\\left| n \\right|}}{{\\left( {M - 1} \\right)}}} \\right),\\;\\;\\;\\;\\;\\;\\;\\;\\left| n \\right| \\le \\cfrac{{M - 1}}{2}$ & $\\cfrac{{T{\\rm{sgn}}\\left( f \\right)}}{{2\\left( {1 - \\exp \\left( { \\cfrac{-k}{2}} \\right)} \\right)}}\n\\left( {1 - \\exp \\left( {\\cfrac{{ - k\\left| f \\right|}}{B}} \\right)} \\right)$ & $k = 2.5$\\\\ \\\\\n\t\t\\hline\\\\\n\t\tKaiser & $w\\left( n \\right) \\buildrel \\Delta \\over = \\left\\{ {\\begin{array}{*{20}{c}}\n{\\cfrac{{{I_0}\\left( {\\pi \\alpha \\sqrt {1 - {{\\left( {\\frac{n}{{M\/2}}} \\right)}^2}} } \\right)}}{{{I_0}\\left( {\\pi \\alpha } \\right)}}\\;\\;\\;\\;\\;}&{\\left| n \\right| \\le \\cfrac{{M - 1}}{2}}\\\\\n0&{elsewhere}\n\\end{array}} \\right.,\\;\\beta  \\buildrel \\Delta \\over = \\pi \\alpha $ & Calculated Numerically & $\\beta  = 4.5$ \\\\ \\\\\n                     \\toprule[1.5pt]\n\t\\end{tabular} }\n\\end{table*}\n\n\\begin{figure*}[!ht]\n\t\\centering\n           \\includegraphics[width=\\linewidth,height=10.2cm]{Emin}\n\t\\caption{Minimum error with respect to iteration for the six windows of Raised-Cosine, Taylor, Chebyshev, Gaussian, Poisson, and Kaiser.}\n\t\\label{fig:fig11}\n\\end{figure*}\n\nTable 2 compares the results obtained from the stationary phase method and the proposed method for PSL of the autocorrelation function. The results indicate that the average of PSL reduction is about 5 dB revealing the maximum PSL reduction associated with the Poisson window with -17.28 dB decrement. The minimum error in Fig. 3 is calculated according to (28). As mentioned in section 3, the minimum error of the proposed method has a decreasing trend. First, this error has a significant value, but it begins to decrease and the trend of its change is almost constant in high iterations.\n\n\\begin{table}[!t]\n\\centering\n\\caption{Comparison of PSL for the Stationary Phase Method (SPM) and Pro-\\\\posed Method (PM).}\n\\begin{tabular}{c c c c}\n          \\toprule[1.5pt]\n           \\multirow{2}{*}{\\bfseries Windows}& \\multicolumn{2}{c} {\\bfseries PSL (dB)} & \\multirow{2}{*}{\\bfseries Improvement (dB)}              \n          \\\\\n          \\cmidrule(r){2-3}\n\t & \\bfseries  SPM& \\bfseries PM & \\\\\n          \\toprule[1.5pt] \\\\\n\tRaised-Cosine & $-33.34$ & $-37.89$ & $-4.55$ \\\\ \\\\\n\t\\hline \\\\\n\tTaylor & $-33.34$ & $-37.73$ & $-4.39$ \\\\ \\\\\n\t\\hline \\\\\n\tChebyshev & $-31.77$ & $-37.37$ & $-5.60$ \\\\ \\\\\n\t\\hline \\\\\n           Gaussian & $-32.38$ & $-37.67$ & $-5.29$ \\\\ \\\\\n\t\\hline \\\\\n           Poisson & $-20.39$ & $-37.67$ & $-17.28$ \\\\ \\\\\n\t\\hline \\\\\n\tKaiser & $-30.98$ & $-36.82$ & $-5.84$ \\\\ \\\\\n          \\bottomrule[1.5pt]\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\nIn the proposed method, a NLFM signal by solving a constrained optimization problem using Lagrangian method is obtained. By this iterative method, the PSL of the autocorrelation function reduced about 5 dB compared with the stationary phase method. PSL reduction for the Poisson window compared to other windows is significant. Using mathematical analysis, we showed that the minimum error of the proposed method has a decreasing trend and this guarantees the convergence of proposed method. The results of the minimum error of six selected windows also reveal the validity of this statement.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{BISTABLE MEDIA MODEL}\n\n\\indent This work is based on a modified FitzHugh-Nagumo model,\n\\begin{eqnarray}\n  u_t &=& au -u^3 - v + \\nabla^2 u,  \\\\\n  v_t &=& \\varepsilon(u-v)+\\delta \\nabla^2 v,\n\\end{eqnarray}\nhere variables $u$ and $v$ represent the concentrations of the activator and inhibitor, respectively, and $\\delta$ denotes the ratio of their diffusion coefficients. The small value $\\varepsilon$ characterizes the time scales of the two variables, where $v$ remains approximately constant $v_f$ on the length scale over which $u$ varies. The system described by Eqs.(1) and (2) can be excitable, Turing-Hopf, or bistable type. In this paper the parameter $a$ is chosen such that the system is bistable. The two stationary and uniform stable states are indicated by up state ($u_{+}$,$v_{+}$) and down state ($u_{-}$,$v_{-}$), respectively, and they are symmetric, $(u_{+},v_{+})$=$-(u_{-},v_{-})$. A front (interface) connects the two stable states smoothly. On decreasing $\\varepsilon$ the system follows NIB bifurcation that leads to the formation of a couple of Bloch fronts. In the followings, we define a front which jumps from down to up state ($v_{f}$$<$$0$, the planar front velocity $c_{nf}$), and a back which falls from up to down state ($v_{f}$$>$$0$, the planar back velocity $c_{nb}$). Here the front and the back correspond to the two Bloch fronts. So the image of bistable spiral wave is clear: a couple of Bloch fronts (front and back) propagating with opposite velocities enclose a spiral arm, and the front meets the back at the spiral tip at where $v_{f(b)}$$=$$0$ and $u$$=$$v$$=$$0$. In order to differentiate the obtained dense and sparse spiral waves, we define an order parameter $\\alpha=|\\frac{\\lambda_+-\\lambda_-}{\\lambda_++\\lambda_-}|$, here $\\lambda_+$ and $\\lambda_-$  present the average widths of up state and down state, respectively, which can be served as an indicator of duty ratio of spiral wave in a simple way. In dense spiral case, $\\alpha$$=$$0$ which means that the up and down states own identical widths [as Fig. 1 (c) indicated]. In sparse spiral case, if $\\lambda_+$$>$$\\lambda_-$ we call it Negative Phase Sparse Spiral [NPSS, Fig.1 (d)]. Otherwise if $\\lambda_+$$<$$\\lambda_-$ we call it Positive Phase Sparse Spiral [PPSS, Fig. 1(e)-(g)].\n\n\\indent  Because pattern formations in bistable media are sensitive to the initial conditions and boundary conditions, we adopt two types of fixed initial conditions during numerical simulations as shown in Fig.1 (a) and (b). In order to investigate the transformation process between the dense and sparse spirals, we use Fig.1 (a) [Fig.1 (b)] as the initial condition when increasing (decreasing) $\\varepsilon$, which firstly evolves into dense (sparse) spiral. The boundary conditions are taken to be no-flux. The simulation is done in a two-dimensional (2D) Cartesian coordinate system with different grid sizes. A generalized Peaceman-Rachford ADI scheme is used to integrate the above model with a space step $dx$$=$$dy$$=$$0.3$ length unit and a time step $dt$$=$$0.05$ time unit. Unless otherwise noted, our simulations are under the parameter sets: $a$$=$$2.0$, $\\delta$$=$$0.1$.\n\\section{Simulation results and discussion}\n\\subsection{Bifurcation scenario of bistable spiral waves}\n \\begin{figure}[htbp]\n  \\begin{center}\\includegraphics[width=8cm,height=4.7cm]{fig1.eps}\n  \\caption{Evolution of spiral waves and tip paths with increasing $\\varepsilon$.\n (a) symmetric (b) asymmetric initial conditions. (c) dense spiral, $\\varepsilon$$=$$0.3$;\n(d) NPSS, $\\varepsilon$$=$$0.33$; (e)-(g) PPSS, $\\varepsilon$$=$$0.34, 0.42, 0.4385$; (h) uniform state, $\\varepsilon$$=$$0.439$; (i) corresponding tip paths. The dash (solid) lines in (a) and (b) present the contour lines $u$=$0$ ($v$=$0$). The dash circle in (i) corresponding to (h) represents a rough tip trajectory of spiral which travels outside the domain. In order to illustrate the meandering of sparse spiral a small domain size is used: $64$$\\times$$64$ s.u.}\n  \\label{1}\n  \\end{center}\n\\end{figure}\n\\indent  Figure 1 shows the bifurcation scenarios of spiral waves and tip trajectories with increasing $\\varepsilon$. Here, we use a small domain size $64$$\\times$$64$ s.u. in order to illustrate the boundary-induced spiral meandering simultaneously. It can be seen that, upon increasing $\\varepsilon$, the system follows dense spiral - sparse spiral - meandering of sparse spiral - sparse spiral - uniform state. When $\\varepsilon$$<$$0.33$, the spiral wave is dense [Fig. 1(c)], and the spiral tip is a fixed point. When $\\varepsilon$ exceeds the critical value $\\varepsilon_{c}$$=$$0.33$ the dense spirals transform into either PPSS or NPSS. The order parameter $\\alpha$ increases with $\\varepsilon$. After this transformation the tip of sparse spiral begins to travel and traces out a circle with primary radius $r_{1}$ as the spiral wave rotating as indicated in Fig. 1(i). The radius $r_{1}$ increases with $\\varepsilon$ as shown in Fig. 2. When $\\varepsilon$ approaches roughly $0.404$, we observe the meandering of spiral waves. A secondary circle appears on the trajectory due to the meandering and the primary circle (radius $r_{1}$) orbits the secondary circle (radius $r_{2}$) in anticlockwise direction with frequency $f_{2}$. The primary circle spins about its center in clockwise direction with frequency $f_{1}$. The radius of secondary circle $r_{2}$ reduces upon increasing $\\varepsilon$. When the parameter $\\varepsilon$ reaches a critical value $0.445$, the meandering of spiral waves disappears and $r_{2}$$=$$0$. If $\\varepsilon$ exceeds this value the spiral tip travels outside the domain, so one obtains uniform state (either up state or down state).\n\\begin{figure}[htbp]\n  \\begin{center}\\includegraphics[width=7cm,height=5cm]{fig2.eps}\n  \\caption{Dependence of the radius $r_{1}$ on parameter $\\varepsilon$ under different domain sizes. The numbers in the label present the domain sizes.}\n  \\label{2}\n\\end{center}\n\\end{figure}\n\n\\indent  Figure 3 gives a two-parameter numerical bifurcation diagram of spiral waves in a finite domain. The below line $L1$ separates the dense spiral and sparse spiral, and it keeps constant while changing the domain sizes. There exists a parameter range corresponding the meandering of sparse spiral as indicated by the gray region in Fig. 3. This region changes with domain sizes. If the system is large enough, for example $160$$\\times$$160$ s.u., this region reduces to a line which separates the sparse spiral and the uniform state. So the observed meandering of sparse spiral is induced by the boundary effect.\n\n\\begin{figure}[htbp]\n  \\begin{center}\\includegraphics[width=7cm,height=5cm]{fig3.eps}\n  \\caption{Two-parameter numerical bifurcation diagram of spiral waves. The region between lines $L1$ and $L3$ presents sparse spirals. The gray region M indicates meandering sparse spirals. The regions $D$ and $U$ denote Dense spirals and Uniform states, respectively. Domain size: $64$$\\times$$64$ s.u.}\n  \\label{3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\begin{center}\\includegraphics[width=7cm,height=5cm]{fig4.eps}\n  \\caption{Phase diagram of the sparse spirals spanned by the domain size and the parameter $a$. This diagram presents the choice of NPSS or PPSS after the transformation as increasing $\\varepsilon$. R denotes Ruleless choice. P (N) indicates PPSS (NPSS). The dashed line divides roughly the states of sparse spirals after the transformation into two regions, above which the choice is always Ruleless.}\n  \\label{4}\n  \\end{center}\n\\end{figure}\n\n\\indent As indicated in Fig. 1, the dense spiral can transit into either PPSS or NPSS as increasing $\\varepsilon$. In order to study the choice of NPSS or PPSS after the transformation, we do extensive simulation with different parameter sets: $\\varepsilon$, $a$, and domain sizes. For example, given fixed parameters: domain size $64$$\\times$$64$ s.u., $a$$=$$2$ as shown in Fig. 1, we try to connect the spiral type with the parameter $\\varepsilon$. However, at different $\\varepsilon$, the dense spiral transforms into different types of sparse spirals. We don't find rule about how the parameter $\\varepsilon$ determines the type of sparse spiral. Here we denote this transformation as Ruleless choice. Nevertheless, at some specific parameters, the dense spiral always transits into one type of sparse spiral, either PPSS or NPSS. For example, given domain size $64$$\\times$$64$, $a$$=$$3$, the dense spiral always transits into NPSS at different $\\varepsilon$. We give a phase diagram of the spiral types as shown in Fig. 4, in which R represents Ruleless choice, P (N) indicates the PPSS (NPSS). It is shown that if the domain size is large enough (roughly, the region above the dash line) dense spiral evolves into either PPSS or NPSS as changing $\\varepsilon$. The transformed states PPSS and NPSS can be regarded as two stable states of system after the bifurcation. Without forcing (for large domain sizes), the choice is ruleless due to the spontaneous symmetry breakdown. For smaller domain size, however, this choice tends to be unique after the transformation. It is obvious that this choice originates from boundary effects.\n\n\\indent  We want to mention that Fig. 4 shows a phase diagram with different parameters. However, given a set of parameters, the choice is deterministic. For example, given domain size $64$$\\times$$64$ s.u., $a$$=$$2.0$, $\\varepsilon$$=$$0.33$, the initial dense spiral will transform into NPSS as shown in Fig. 1 (d). So, if decreasing $\\varepsilon$ from $0.33$ to $0.3$ then increasing to $0.34$, one can turn a NPSS into dense spiral then into PPSS. This process changes the duty ratio of spiral waves. It is helpful to adjusting the duration of a cardiac action potential to prevent the atrial fibrillation. \\cite{Nattel}\n\n\\indent Sparse spiral can also evolve into dense spiral as decreasing $\\varepsilon$. From the given asymmetric initial condition [Fig. 1(b)], it first develops into sparse spiral. When $\\varepsilon$$<$$0.315$ this sparse spiral further evolves into dense spiral. We want to mention that, as shown in Fig. 1 when increasing $\\varepsilon$, the transition from dense to sparse spiral occurs at $\\varepsilon$$=$$0.33$. So, this transformation is bistable with respect to $\\varepsilon$, and originates from a subcritical bifurcation. This transformation process is very different from that in excitable media, \\cite{Krinsky, Kessler} in which the transformation between sparse and dense spirals is smooth. The present phenomena attribute to the intrinsic characteristics of the bistable system.\n\n\\subsection {Transformation details from dense spiral to sparse spiral}\n\n\\indent Because this transformation is bistable, here, we focus on the transformation details from dense spiral to sparse spiral. When the parameter $\\varepsilon$ has exceeded $\\varepsilon_{c}$, the system with the given symmetric initial condition [Fig. 1 (a)] evolves first into dense spiral, then into sparse spiral. We study the transformation in detail by measuring the intensity change of variable $u$ and $v$ at a fixed point far away from the spiral tip and the domain boundary as shown in Fig. 5. It can be seen that the period of sparse spiral $T_{1}$ is always larger than that of dense spiral $T_{0}$. There exists a critical period $T_{c}$ (interval of $u_{+}$) at which the transformation from dense spiral to sparse spiral occurs. It is found that the differences between $T_{0}$ and $T_{c}$ are related to the choice of PPSS or NPSS,\n\\begin{displaymath}\n\\left\\{ \\begin{array}{ll}\nT_0 < T_c, & \\textrm{PPSS},\\\\\nT_0 > T_c, & \\textrm{NPSS}.\n\\end{array} \\right.\n\\end{displaymath}\n\\begin{figure}[htbp]\n  \\begin{center}\\includegraphics[width=7cm,height=5cm]{fig5.eps}\n  \\caption{Time sequence of variable $u$ and $v$ around the transformation point for NPSS.  Solid line and dash line present $u$ and $v$, respectively. $T_{0}$ and $T_{1}$ are the periods of dense spiral\nand sparse spiral, respectively. $T_{c}$ indicates a critical period.}\n  \\label{5}\n\\end{center}\n\\end{figure}\n\n\\indent The appearance of $T_{c}$ indicates the change in velocities of both the front and the back. For the case of small $\\varepsilon$ and $\\delta$, Eqs. (1) and (2) reduce to\n\n\\begin{eqnarray}\n  u_t &=& au - u^3 - v_i + \\nabla^2 u, \\\\\n  v &=& v_i  (i = f, b),\n\\end{eqnarray}\nhere $v_{i}$ is the value of the inhibitor at the front or the back. From Eqs.(3) and (4) we can obtain the velocity of the planar front (back) $c_{ni}$=$-\\frac{3}{\\sqrt{2}a}$$v_{i}$($i$=$f$,$b$). So the value $v_{i}$ determines uniquely the velocity of front (back). For\ndense spiral, the front and the back are symmetry except an angle separation of $\\pi$ and travel at the same speed, $|v_{b}|$$=$$|v_{f}|$. The interaction between the front and the back is negligible. Undergoing a subcritical bifurcation the symmetry between the front and the back breaks down. Here, if $|v_{f}|$$>$$|v_{b}|$, the front speeds up, so the measured critical period $T_{c}$ would be less than $T_{0}$. The accelerated wave front approaches and interacts with the back to form a NPSS. On the contrary, if $|v_{b}|$$>$$|v_{f}|$ the back speeds up, so $T_{c}$$>$$T_{0}$ and a PPSS will appear.\n\n\\indent  The velocities of both the front and the back are also restricted by the eikonal relation which amends the velocity difference between the front and the back induced by the symmetry breakdown,\n\\begin{equation}\\label{7}\n    c_i = c_{ni}(v_i) - D\\kappa_i.\n\\end{equation}\nhere, $\\kappa_i$ indicates the local curvature. For NPSS (PPSS) the front and the back are asymmetric and the local curvature $\\kappa_f$ of the front is larger (smaller) than that $\\kappa_b$ of the back far away the tip. So, the final velocities of the front and the back are consistent, which leads to stable spiral wave.\n\n\\indent In the simulation we find that the period difference between the dense and sparse spirals and the radius $r_{1}$ of primary circle satisfy the following approximate relation,\n\\begin{equation}\\label{4}\n    T_1 - T_0 =8 r_1.\n\\end{equation}\n\\indent  If $r_{1}$$=$$0$, $T_1$ reduces to $T_0$, and it corresponds to dense spiral. Because the velocities of both the front and the back decrease with $\\varepsilon$, $T_{0}$ and $T_{1}$ increase with $\\varepsilon$. The closer the parameter $\\varepsilon$ is to $\\varepsilon_{c}$, the longer it needs to evolve from the initial condition to sparse spiral. This attributes to the critical slowing down near the bifurcation point $\\varepsilon_{c}$.\n\n\\indent  In the present case, the system is symmetric and the symmetry breakdown of the front and the back originates from a subcritical bifurcation, which differs from that in Ref. 21. They obtained the dense and sparse spirals in symmetric and asymmetric bistable systems, respectively, in the limit of $\\varepsilon$\/$\\delta$$\\ll$$1$. In that case, the symmetry breakdown originates from a saddle-node bifurcation induced by a constant in the model.\n\\begin{figure}[htbp]\n  \\begin{center}\\includegraphics[width=7cm,height=8cm]{fig6.eps}\n  \\caption{Dependence of rotation frequency of spirals on the parameter $\\varepsilon$. In (a) the filled square (cycle) denotes dense (sparse) spiral. Domain size: $64$$\\times$$64$ s.u. (b) shows the rotation frequencies of sparse spirals under different domain sizes. The curves for small domain sizes diverge when $\\varepsilon$ is larger than about $0.38$.}\n  \\label{6}\n\\end{center}\n\\end{figure}\n\n\\indent Figure 6 (a) shows the dependence of rotation frequency of dense and sparse spirals on the parameter $\\varepsilon$. The filled square (cycle) denotes dense (sparse) spiral. It can be seen that the rotation frequency of dense (sparse) spiral varies nonlinearly, nonmonotonously (monotonously) with respect to $\\varepsilon$. The frequency of sparse spiral disappears for $\\varepsilon$$<$$0.33$, and is always smaller than that of dense spiral, which is due to the rigid rotation.\n\n\\subsection{Meandering of sparse spiral induced by boundary effects}\n\n\\indent  Figure 6 (b) shows the dependence of average primary frequency $\\omega_{1}$ on the parameter $\\varepsilon$ under different domain sizes. It can be seen that for small $\\varepsilon$ these curves are superposition. But for large $\\varepsilon$ they begin to diverge. The onset of diverge increases with domain sizes and indicates the startpoint of meandering spiral. The rotation frequency increases with decreasing domain sizes at larger $\\varepsilon$. This phenomenon agrees well with the experimental observation in Ref. 28. The radius $r_{1}$ of primary circle decreases with domain sizes for larger $\\varepsilon$ as shown in Fig. 2, which attribute to the interaction between the spiral tip and the boundary. In addition, near the onset, the sparse spiral begins to meander and the radius $r_{2}$ of the secondary cycle tends to be infinite. But due to the boundary effects, $r_{2}$ is limited to a distance away from the boundary about $10$ s.u.. With increasing $\\varepsilon$, $r_{2}$ decreases dramatically and then tends to be zero at certain bifurcation point. Near this point the radius $r_{2}$ scales approximately as the square-root of the distance from this bifurcation point. This means that the boundary-induced meandering originates from a supercritical bifurcation.\n\n\\indent  From the simulation for different domain sizes and $\\varepsilon$ we can give an experiential condition for meandering of sparse spirals,\n\\begin{equation}\\label{9}\n    L\/\\lambda\\in(0.5, 0.7),\n\\end{equation}\nwhere $L$ is the domain size and $\\lambda$ is the wavelength of sparse spiral. It demonstrates that meandering of spiral occurs when the wavelength of spiral reaches at least twice of the domain size.\n\n\\indent We only observed the meandering of sparse spirals with outward petals even by changing the chiralities of spirals. No meandering spiral with inward petal is found in simulations. The observed characteristics of meandering spirals differ greatly from that observed in excitable media in Refs. 29, 30 in which hypocycloidlike orbit transits into epibycloidlike orbit as changing a control parameter continuously. That meandering with both outward and inward petals originates from a pair of secondary Hopf bifurcations. But in the present work, from the description of Fig. 3, it can be seen that the obtained meandering of sparse spiral originates from boundary effect rather than regular meandering instability. By investigating the linear stability analysis of spirals, B\\\"{a}r and co-workers \\cite{Bar3} observed meandering-Hopf bifurcation (regular meandering which contains a couple of Hopf bifurcations as Barkley reported) and boundary-Hopf bifurcation (boundary-induced Hopf bifurcation which contains only one Hopf bifurcation point). The boundary-Hopf bifurcation has two characteristics: 1) it occurs only when the domain size is smaller than a critical value; 2) it contains only one bifurcation point. Our results, as shown in Fig. 1-3, satisfy the two conditions. So, the meandering spirals in our case result from boundary-induced Hopf bifurcation. If there exists regular meandering for bistable spiral, it is impossible to make a simulation because it tends to be uniform state when $\\varepsilon$ is large enough.\n\n\\section{CONCLUSION AND REMARKS}\n\\indent We have studied the transformation between dense and sparse spirals based on a bistable FitzHugh-Nagumo model and investigated a novel meandering of sparse spiral with only outward petals. The dense spiral and the sparse spiral can transform each other via a subcritical bifurcation which differs from that in excitable media. \\cite{Krinsky, Kessler} The dense spiral can transit into either PPSS or NPSS. We can find a way to turn a PPSS into dense spiral then into NPSS by decreasing $\\varepsilon$ and then increasing it again, which provides a simple method to adjusting the duration of a cardiac action potential to prevent the atrial fibrillation. \\cite{Nattel} By using different domain sizes we have studied the meandering of sparse spiral and given an experiential condition for spiral meandering induced by the boundary effects $L$\/$\\lambda$$\\in$($0.5$, $0.7$). The observed meandering of sparse spiral with only outward petal originates from a Hopf bifurcation induced by the boundary effects.\n\n\\indent Although the present FitzHugh-Nagumo model isn't a realistic model describing chemical reaction, it has successfully explained many pattern formations in ferrocyanide-iodate-sulfite reactions,\\cite{Li2, Hagberg0, Hagberg1, Szalai, Lee1, Lee2, Lee3} such as the spiral, the self-replicating spots, and the labyrinthine pattern which occur on the left, near, and on the right of the NIB bifurcation point, respectively. In the present work we studied the transformation between dense and sparse spirals on the left of this bifurcation point. We hope that our results can be observed in a ferrocyanide-iodate-sulfite reaction.\n\n\\section{ACKNOWLEDGMENTS}\n\\indent  This work is supported in part by Hong Kong Baptist University and the Hong Kong Research Grants Council. Y. F. He also acknowledges the National Natural Science Foundation of China with Grant No. 10775037, 10947166, and the Research Foundation of Education Bureau of Hebei Province, China (Grant No. 2009108).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOutlier, novelty, or anomaly detection is the process of identifying new data samples as part of the learned class (inliers) or not (outliers).  This problem is relevant since most problems do not have a fully characterized set of interest data.  During testing, the classifier must adapt to unseen data.  There is a vast literature on these topics~\\cite{Ahmed2016, Chandola2009, Hodge2004, Pimentel2014} since they have broad applications, with a particular interest in vision~\\cite{Buades2005, Cong2011, Li2014, Sabokrou2017, Sabokrou2018, Xia2015, You2017}.\n\nWe can classify these methods depending on the type of modeling used, such as probabilistic-based~\\cite{Eskin2000, Kim2012, Lerman2015, Markou2003, Yamanishi2004}, which model the distribution of inliers and the outliers are those points with low probability; distance-based~\\cite{Bodesheim2013, Breunig2000, Liu2017}, which identify outliers by their distance to inliers; self-representation~\\cite{Cong2011, Rahmani2017, Sabokrou2016, You2017}, which detect outliers as non-sparse representations on the given inliers; and deep-learning-based~\\cite{Hasan2016, Kimura2018, Pidhorskyi2018, Ravanbakhsh2017, Sabokrou2015, Sabokrou2018, Schlegl2017, Xia2015, Wang2018}, which use reconstruction-based networks to detect outliers according to their reconstruction error.  \n\nSome previous methods rely on true outliers and could suffer from unbalanced data problems since anomalies are quite rare.  A solution to these problems is the creation of synthetic anomalies from the anomaly class.  In contrast to using true outliers for anomaly creation, we propose a zero-shot synthesis, by generating outliers from the boundaries of the inlier data, to train binary classifiers.  However, using the data on the feature space is ineffective due to uncertainties on the original data and errors introduced by the feature extraction process.  Instead, we propose to model the data using probability distributions through a hierarchical encoding process that models the uncertainty too.  Then, we use the outskirt distributions as sources for our outlier synthesizing process.  This process proves to be effective in learning a robust boundary that correctly classifies (never seen before) outliers. \n\nOur proposed method lies at the intersection of probabilistic-, distance-, and deep-learning-based models.  On one hand, we model our data (inliers) with distributions that represent the distilled information of a set of features.  In contrast to existing probabilistic models, we learn a variational (autoencoder) family that is conditioned on the inlier samples alone.  Thus, we generate a latent space of distributions that represent our inliers.  Then, similarly to distance-based methods, we select the distributions that are on the outskirts of our learned distribution-space to generate synthetic outliers by perturbing the drawings from these distributions.  Conversely to distance-based methods that work on the original feature space, our transformation creates a robust distribution-space to work with.  And, instead of directly using these outskirt distributions for classification, as probabilistic or distance methods do, we synthesize outlier samples to train a binary classifier.  We found that the mixture of these techniques yields better results than their traditional counterparts, with a simpler representation.  We show an illustration of the overall approach in Fig.~\\ref{fig:diagram}.  Additionally, our method is trained without seeing any true outlier (thanks to our synthesis process).  In a way, our synthesized outliers are distorted versions of already ``strange'' samples. \n\nOur main contributions are: (1)~A simple, yet powerful, zero-shot outlier synthesis framework for anomaly detection, i.e\\onedot} \\def\\Ie{I.e\\onedot, we do not use actual outlier data to train our classifier or feature distillation but rather synthetically generated outliers from inlier data.  (2)~A two-level hierarchical scheme for feature distillation and creation of inlier's distributions that allows synthesizing pseudo outliers which are sufficient to train a robust binary classifier.  (3)~We demonstrate that the use of probabilistic modeling paired with distance-based methods for boundary definition produces robust-enough synthetic negative samples to train binary classifiers with robust performance on novel data, in particular, from an outlier class.  (4)~We show that loosely coupled methods have advantages over tightly coupled ones (i.e\\onedot} \\def\\Ie{I.e\\onedot, commonly end-to-end trained methods) on a set of outlier detection benchmarks.\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{.9\\linewidth}{!}{\\input{images\/diagram.tex}}\n  \\caption{Given a set of feature descriptors $\\{g_i(x)\\}_{i=0}^k$ (denoted by different shapes) for the set of inputs $x \\in \\mathcal{X}$ (denoted by different colors), we distill the features and fuse them into a distribution space $\\mathcal{Q}$.  From them, we select a set of outskirt distributions, $\\mathcal{O}$, as those farther from the other set of distributions (marked with diagonal lines).  They represent samples that are on the boundary of the latent space.  Then, we sample, from these outskirt distributions, our synthesized-outliers $y \\in \\mathcal{Y}$ (denoted as diamonds).  Finally, we train a classifier $C$ using the fused inputs (inliers) and the synthesized outliers as positive and negative samples, respectively.  This classifier is then used to detect true outliers, never seen before on this process.%\n  }\n  \\label{fig:diagram}\n\\end{figure}\n\n\n\\section{Related Works}\n\\label{sec:related}\n\n\n\\textbf{Data Representation.}\nExisting methods rely on low level features, e.g\\onedot} \\def\\Eg{E.g\\onedot, Histogram of Oriented Gradients~\\cite{Dalal2005}, local patterns~\\cite{Ojala1996, Huang2011, RamirezRivera2013, RamirezRivera2015a, RamirezRivera2015}; high level features, e.g\\onedot} \\def\\Eg{E.g\\onedot, bag of words~\\cite{Venhuizen2015}, trajectories~\\cite{Morris2011}; or deep-learned features~\\cite{Erfani2016, Kimura2018, Pidhorskyi2018, Sabokrou2015, Sabokrou2017, Sabokrou2018, Schlegl2017, Xu2015}.  The latter have gained traction in the past years since they tend to outperform low or high-level features designed for general purposes since the deep-features are tuned for each particular task.  However, due to this same property, they are not generalizable and not generally used outside of their own proposed architecture.  When a similar architecture is adapted into another problem, the previous weights need to be fine-tuned for the new tasks.  On the contrary, we propose a distillation method from feature descriptors (e.g\\onedot} \\def\\Eg{E.g\\onedot, low or high-level features, or from deep-features from other tasks) that can generalize them without adapting their original architectures.  Our proposed model takes advantage of compressing the features through autoencoders (AEs).  \nOn the other hand, we also propose a zero-shot synthetic outlier generation scheme.  Some methods tried to synthesize outliers while training by adding noise to the input samples~\\cite{Sabokrou2018}.  Nonetheless, by perturbing the original data, they can miss the original data variations.  On the contrary, we propose to synthesize data on a constructed feature-distribution space that already incorporated the variations from the inliers.\n\n\\textbf{Self-Representation.} \nBy assuming that outliers cannot be reconstructed sparsely from a set of inlier points, self-representation methods~\\cite{Cong2011, Sabokrou2016, You2017} achieve outlier detection.  Others exploit the correlations between the inliers that must lie on lower-dimensional spaces, and assume that incoherent samples must be outliers~\\cite{Rahmani2017}. Another similar stream of research~\\cite{Lerman2015, Xu2010} solves a problem with variations of the data points to find lower-dimensional spaces that define the inliers.  The idea of sparse representation can be extended as the ability to reconstruct the samples too~\\cite{Cong2011, Xu2015}.  In this sense, they are related to deep learning generative methods.  \n\n\\textbf{Probabilistic-based Methods.}  \nConventional probabilistic methods~\\cite{Eskin2000, Kim2012, Lerman2015, Markou2003, Yamanishi2004, Park2018, Atli2018} model the inliers' features' distributions, where the outliers are detected as samples with low probability.  Several approximations to the true distribution of the data have been proposed, for instance, Genetic programming~\\cite{Cao2016} to estimate a kernel density function, minimum-volume-sets to estimate a particular level set of the unknown nominal multivariate density~\\cite{Scott2006}, or constructing minimal graphs covering a $K$-point subset to estimate the critical region~\\cite{Hero2007, Sricharan2011}. Maximum-entropy-discrimination based models were proposed with several variants, and a recent method mixes this framework with hinge loss style discriminant functions and latent variables to discriminate the outliers~\\cite{Hou2018}. More recent methods~\\cite{Pidhorskyi2018} mix the distribution's generation with manifolds to better model the inliers.  Similarly, our proposed method is a mixture of ideas, and cannot be classified in a single category.  On the contrary to existing probabilistic methods, our proposed method does not use the probabilities as a novelty score, but rather as a synthesizer of new data.  \n\n\\textbf{Distance-based Methods.} \nThe core idea of distance-based methods is the assumption that inliers are close to each other, while outliers are far from them.  Commonly, $k$-nearest neighbors are used with density estimation~\\cite{Breunig2000}, as well as more advanced kernel transforms~\\cite{Bodesheim2013, Liu2017} where the distance between the projections is used for the novelty score. Conversely, we use the notion of distance to select outlier distributions that will be used later on, instead of assigning a novelty score from the neighbors.  The notion of distance between points can be posed as a classification problem.  One breakthrough was the kernel-based novelty detection scheme that relies on unsupervised support vector machine~\\cite{Schoelkopf2000}.\n\n\\textbf{Deep Learning.}  \nThe state of the art on outlier detection is based on deep neural networks.  In particular, they continue the ideas of self-representation methods based on reconstruction.  The main architecture is based on autoencoders (AEs)~\\cite{Hasan2016, Xu2015}.  Others~\\cite{Pidhorskyi2018, Ravanbakhsh2017, Sabokrou2018, Xia2015, Wang2018, Lim2018, Abdulhammed2019} add a generative adversarial loss~\\cite{Goodfellow2014} to enhance the reconstruction of the AEs.  The main idea is to train a discriminator and a generator (i.e\\onedot} \\def\\Ie{I.e\\onedot, the AE) on a min-max fashion.  This optimization scheme yields better results for the generation of the original samples.  However, it is hard to train, and it is unstable due to the different learning rates of its components.  Similarly to existing approaches, we use the unsupervised nature of AEs to learn a compact space of the original features (akin to  self-representation).  However, we do not work on this space, and, yet again, we produce a higher-level feature space based on distributions through variational autoencoders (VAEs), which have shown excellent capabilities for unsupervised clustering~\\cite{AriasFigueroa2017, AriasFigueroa2017a, Kingma2014, Baur2018, Xu2018a}.  And in contrast to common deep learning approaches, we detect outliers with vanilla classifiers.\n\n\\textbf{Novelty.}  We use AEs and VAEs similar to previous approaches~\\cite{Kingma2014, Zhai2016}.  (1)~However, our proposal's novelty lies in the construction of a two-level latent space hierarchy that (i)~encodes the inliers into a low-dimensional latent space that supports a reliable estimate on inliers' distributions (through AEs); and (ii)~provides a dense representation of the points in a distribution-space where inlier distributions can be compacted to find a reliable boundary (through VAEs).  (2)~Unlike previous tightly coupled outlier detectors~\\cite{Nguyen2018}, we separate the feature modeling, synthetic-outlier generation, and classification, which enables our learning spaces to be used on a diverse set of tasks.  (3)~Our zero-shot outlier generation from near-boundary inlier-distributions is different from previous generation methods~\\cite{Landgrebe2006} that commonly assume a uniform maximum entropy distributed outlier class.  Our method is superior in the sense that we do not assume any distribution over the outliers.  Instead, we rely only on the inliers' distributions (available at training time), and synthesize outliers from them (at training time).  Conversely to probabilistic methods that rely on matching different inlier and outlier distributions, we use a distance-based mechanism (training a classifier on synthetic outliers sampled from the distributions and the true inliers) to classify outliers and inliers (since the distributions and the outliers themselves are unknown at training time).\n\n\n\\section{Proposed Method}\n\\label{sec:proposal}\n\nWe define outlier detection as a binary classification problem.  Such that given any sample from the inlier set $x \\in \\mathcal{X}$, we classify it as inlier, while any other data $y \\notin \\mathcal{X}$ is classified as an outlier.  However, we assume that the outliers are unknown at training time, which allows us to tackle broader problems.  This setup poses a challenge since we need negative samples to train the binary classifier.  Although a one-class classifier may be an option, we show that learning a good boundary outperforms it (cf\\onedot} \\def\\Cf{Cf\\onedot Section~\\ref{sec:results}).\n\nTo train a binary classifier using only the inlier data, we need to synthesize outliers to act as negative samples.  We find the boundary of the training data, and then synthesize the outlier samples from it (detailed on Section~\\ref{sec:outlier}).  Instead of working on the original data or feature spaces, we learn a distribution-space that represents the data changes and uncertainties (see Section~\\ref{sec:distr}).  This feature distillation process helps to find better boundaries between inliers and outliers.  We detail our proposed hierarchical encoding scheme in Section~\\ref{sec:features}.\n\n\n\\subsection{Probabilistic Representation}\n\\label{sec:distr}\n\nWe are interested in modeling the inliers as distributions to consider their uncertainty, in addition to compressing their representation.  Thus, we create a conditioned Gaussian distribution that represents each data point by learning a variational autoencoder~(\\ref{eq:hf}), see Section~\\ref{sec:features} for details.  Given an inlier $x \\in \\mathcal{X}$ and its latent vector $w(x)$~(\\ref{eq:w}), we learn a distribution conditioned on the latent representations of the given data point $q \\left( z \\given w(x) \\right) \\equiv (\\mu, \\sigma)$.  In practice, we use standard Gaussians to represent the distributions, and we are interested in obtaining one distribution per data point in our training set.  In other words, we construct the set of distributions\n\\begin{equation}\n\\label{eq:q}\n  \\mathcal{Q} = \\{(\\mu, \\sigma) = q \\left(z \\given w(x) \\right) : x \\in \\mathcal{X}\\}.\n\\end{equation}\nThat is, each distribution $q \\in \\mathcal{Q}$ is a Normal distribution with parameters equal to the latent representation of our VAE that corresponds to a given inlier~$x$ in the training set.\n\n\n\\subsection{Outlier Classification}\n\\label{sec:outlier}\n\nOur synthesis process consists of perturbing inliers that are in the boundary of the training data, and using them with the inliers to train our classifier.  We perturb the inliers by drawing from the distributions that correspond to the borderline inliers.  In other words, our goal is to find the distributions from $\\mathcal{Q}$~(\\ref{eq:q}) that are on the outskirts of the space (as shown in Fig.~\\ref{fig:diagram}).  Our first task is to find the distribution-space's boundary given the training inliers.  We propose two methods to define this boundary and its corresponding distributions.\n\n\\textbf{Ellipsoid-based.} The first boundary-defining method is based on selecting the distributions that lie on the boundary of the distribution of distributions (i.e\\onedot} \\def\\Ie{I.e\\onedot, a meta-distribution).  To find them, we compute the center, $\\bar{\\mu}$, and the standard deviation, $\\bar{\\sigma}$, of the meta-distribution of $\\mathcal{Q}$ through\n\\begin{align}\n  \\bar{\\mu} &= \\frac{1}{|\\mathcal{Q}|} \\sum_{i=1}^{|\\mathcal{Q}|} \\mu_i,\\\\\n  \\bar{\\sigma}^2 &= \\frac{1}{|\\mathcal{Q}|-1} \\sum_{i=1}^{|\\mathcal{Q}|} (\\mu_i - \\bar{\\mu})^2,\n\\end{align}\nwhere $|\\mathcal{Q}|$ is the size of the set that represents the amount of samples.  Then, our outskirt distributions are within the set\n\\begin{equation}\n\\label{eq:ellipsoid-outlier}\n\\mathcal{O}_e = \\left\\{(\\mu, \\sigma) : \\sum_i \\frac{(\\mu_i - \\bar{\\mu}_i)^2}{\\alpha \\bar{\\sigma}_i^2} \\geq 1, (\\mu, \\sigma) \\in \\mathcal{Q} \\right\\},\n\\end{equation}\nwhere the subscripted values represent the components of the vectors, and $\\alpha$ is a scaling hyperparameter.  In other words, we select the outskirt distributions as those that lie outside of the ellipsoid defined by the scaled (by $\\alpha$) mean and variance of the meta-distribution.  Note that we constructed a diagonal covariance matrix through the above procedure.\n\n\\textbf{Distance-based.} On the other hand, since we have high dimensional distributions, we would be computing their distributions on a very sparse space.  Instead, we propose to analyze the $\\ell_2$ norm of their means.  Consequently, we collapse the high dimensional space into a lower one, and then we proceed in a similar fashion as above.  We compute the mean, $\\bar{\\mu}_\\ell$, and the standard deviation, $\\bar{\\sigma}_\\ell$, of the $\\ell_2$ norms ($\\norm{\\cdot}_2$) of all the distributions' means on $\\mathcal{Q}$, such that\n\\begin{align}\n\\bar{\\mu}_\\ell &= \\frac{1}{|\\mathcal{Q}|} \\sum_{i=1}^{|\\mathcal{Q}|} \\norm{\\mu_i}_2,\\\\\n\\bar{\\sigma}^2_\\ell &= \\frac{1}{|\\mathcal{Q}|-1} \\sum_{i=1}^{|\\mathcal{Q}|} (\\norm{\\mu_i}_2 - \\bar{\\mu}_\\ell)^2.\n\\end{align}\nAnd the outskirts distributions are within the set\n\\begin{equation}\n\\label{eq:l2-outlier}\n\\mathcal{O}_\\ell = \\left\\{(\\mu, \\sigma) : \\abs[\\big]{ \\norm{\\mu}_2 - \\bar{\\mu}_\\ell} \\geq \\alpha \\bar{\\sigma}_\\ell, (\\mu, \\sigma) \\in \\mathcal{Q} \\right\\}.\n\\end{equation}\nSimilarly, the $\\alpha$ parameter scales the boundary of which distributions will be selected.\n\n\\textbf{Outlier Synthesis.} Finally, given any set of distributions that lie on the outskirts of the space, $\\mathcal{O}$, we can synthesize a set of outlier points that will be our negative class for training.  Hence, let the set of synthetic outlier samples be\n\\begin{equation}\n\\label{eq:outliers}\n\\mathcal{Y} = \\left\\{ y = \\mu + \\beta \\sigma \\epsilon : (\\mu, \\sigma) \\in \\mathcal{O}, \\epsilon \\sim \\mathcal{E} \\right\\},\n\\end{equation}\nwhere each new sample is drawn from the distributions defined in $\\mathcal{O}$, $\\epsilon$ is generated noise from a distribution~$\\mathcal{E}$, and we use a scalar factor $\\beta$ to draw from the tails of the distribution.  Geometrically, we can draw from any direction from the mean.  However, we are interested in the samples that lie on the outside part of the boundary.  \\Eg, in Fig.~\\ref{fig:diagram}, we draw samples on the side closer to the boundary of each of the outskirt distributions instead of sampling from any side.  Hence, we construct $\\beta$ such that for each dimension its value is positive or negative depending on whether the mean of this distribution is greater or less than the mean of the meta-distribution.  We uniformly sample from all of the distributions to create enough outliers to match the amount of inliers used for training.  \n\n\\textbf{Classification.} Once we have defined our training sets, $\\mathcal{X}$ and $\\mathcal{Y}$, we can proceed to train a classifier $C$ by considering the projected $\\mathcal{X}$ in our distribution space (in practice, we use the mean of the corresponding distribution since they represent the expected value of each inlier) as our positive examples, and $\\mathcal{Y}$ as our negative ones.  During testing, we introduce new (never seen) samples to the classifier to evaluate its capabilities for outlier detection.  We present more details on Section~\\ref{sec:results} regarding our experimental setup.  Note that we defined a loosely coupled process that allows us to switch between classifiers, without the need to relearn the feature distillation process.\n\n\n\\subsection{Feature Encoding}\n\\label{sec:features}\n\n\\begin{figure}\n  \\centering\n  \\resizebox{.75\\linewidth}{!}{\\input{images\/architecture.tex}}\n  \\caption{Our feature distillation architecture uses $k$ features $\\{g_i(x)\\}_{i=0}^k$ from the original data point~$x$, and process them through an AE, $h_i$, for it to learn an approximated version of them, $\\hat{g}_i(x)$.  Simultaneously, we concatenate the encoded features of each AE, $h_i$, namely~$\\tilde{g}_i(x)$, into a vector~$w$.  Then, we pass it through a learned fusion function, $h_f$, that is a VAE to recover the input, $\\hat{w}$.  The VAE encodes the parameters of the distribution of $w$, $(\\mu, \\sigma) \\equiv q( \\cdot \\given w)$.  The set of these parameters for all data points is $\\mathcal{Q}$.  We use samples from the encoder of~$h_f$, namely~$\\tilde{g}_f$, as the positive inputs for a classifier~$C$, while the negative samples come from the synthesized outliers from the distributions at the outskirts (cf\\onedot} \\def\\Cf{Cf\\onedot Section~\\ref{sec:outlier}).}\n  \\label{fig:encoding}\n\\end{figure}\n\nOur model for anomaly detection relies on a hierarchical feature extraction and fusion scheme based on AEs (as shown in Fig.~\\ref{fig:diagram}).  Our proposal uses a two-level contraction and fusion method, as shown in Fig.~\\ref{fig:encoding}.  Lets consider the raw data $x \\in \\mathcal{X}$, and a extracted set of $k$ features $G = \\{g_i(x)\\}_{i=1}^k$.  Our goal is to create a mixture of the first level features $G$ that may not have all the characteristics of $x$.  For instance, we may be using feature descriptors that are easy to compute but not expressive enough, or features learned from different tasks that need to be fused to have a more powerful descriptor, among other cases.  Hence, if we convert this data into a more suitable representation that has particular distributions or characteristics, it will be easier to classify.  Several methods can be used to achieve such space, among them, one can do dimensionality reduction or clustering followed by a mapping function.  Nevertheless, AEs have demonstrated these capabilities within a single method.  They learn an identity function with the property of embedding the input vectors into a lower-dimensional space that distills the relevant features.  Due to this property, they are the backbone of our proposed architecture.\n\nThe first level of our model consists of a set of $k$ AEs, $H = \\{h_i^{\\theta_i} \\}_{i=1}^k$, parameterized by $\\theta_i \\in \\Theta$, where $\\Theta$ is the set of parameters for the whole model.  (In the following, we suppress the parameters from the notation of each AE for brevity.)  The job of each AE is to compact the input $g_i$ into a more expressive space, that is,\n\\begin{equation}\n\\hat{g}_i(x) = h_i \\left( g_i \\left(x \\right) \\right),\n\\end{equation}\nthe $i$th reconstructed feature $\\hat{g}_i(x)$ is the output of the $i$th AE $h_i$ that operates on the input feature $g_i(x)$.  During training, one of our objectives is to minimize the error between the reconstructed and the original features, that is our feature loss\n\\begin{equation}\n\\label{eq:loss-f}\n\\mathcal{L}_f = \\sum_{i=1}^k \\norm[\\big]{\\hat{g}_i(x) - g_i(x)}^2.\n\\end{equation}\n\nMoreover, we are interested in the compact representation learned within the AEs, as they define a space that clusters and separates the input data.  The $i$th AE, $h_i$, is the composition of encoder, $E_i$, and decoder, $D_i$, functions such that\n\\begin{equation}\nh_i ( z ) = D_i \\left( E_i \\left( z \\right)  \\right).\n\\end{equation}\nLets denote the inner compact representation of the $i$th AE (the output of the encoder) as \n\\begin{equation}\n\\tilde{g}_i (x) = E_i \\left( g_i (x) \\right).\n\\end{equation}\nThen, we are interested in the learned representation\n\\begin{equation}\n\\label{eq:w}\nw(x) = \\concat_{i=1}^k \\tilde{g}_i (x),\n\\end{equation}\nwhere $\\concat$ is the concatenation operator.  Our objective is to learn a transformation function, $h_f$, that converts this distilled representation into a single compact space.  Since having a sparse vectorial representation, through the AEs, is cumbersome, we plan to learn a continuous space by modeling each vector as a distribution.  Hence, we use variational inference to learn a variational family approximated through a VAE, defined by\n\\begin{equation}\n\\label{eq:hf}\n\\hat{w} = h_f(w),\n\\end{equation}\nwhere $h_f$ comprises the encoding of $w$ into a distribution $q(z\\given w) = E_f(w)$, and the decoding through a sampling process $\\hat{w} \\sim q(z\\given w)$.  (We omit the argument $x$ from $w$ for brevity.)  And, we define the loss for the high level features as\n\\begin{equation}\n\\label{eq:loss-r}\n\\mathcal{L}_h = \\norm{\\hat{w} - w}^2 + \\operatorname{KL}\\infdivx{q(z\\given w)}{\\mathcal{N}(0,I)},\n\\end{equation}\nwhere KL is the Kullback-Leibler divergence to measure the distance between the prior and the learned distribution.  We assume that our proposed distribution, $q(z\\given w)$, is distributed as a Gaussian and try to minimize its distance towards a zero-centered unitary Gaussian, $\\mathcal{N}(0,I)$.\n\nFinally, our objective is to find the set of parameters, $\\Theta$, of the neural networks (the set $\\left\\{ H, h_f \\right\\}$) that minimizes the sum of the loss functions\n\\begin{equation}\n\\Theta^* = \\underset{\\Theta}{\\argmin} \\left\\{\\mathcal{L}_f^\\Theta + \\mathcal{L}_h^\\Theta \\right\\},\n\\end{equation}\nwhere the loss functions, (\\ref{eq:loss-f}) and~(\\ref{eq:loss-r}), are parameterized by $\\Theta$ which corresponds to the parameters of AEs and VAE, $\\{ H^\\Theta, h_f^\\Theta \\}$.  These parameters are learned through back-propagation on the training phase.\n\n\\section{Results and Evaluation}\n\\label{sec:results}\n\nFor our experiments we are evaluating our method for outlier generation based on two different ways of selecting the outlier-generation distributions, i.e\\onedot} \\def\\Ie{I.e\\onedot,~(\\ref{eq:ellipsoid-outlier}) and~(\\ref{eq:l2-outlier}), that we named $\\text{OG}_e$ and $\\text{OG}_\\ell$, respectively.  We selected three features, namely Histogram of Oriented Gradients (HoG)~\\cite{Dalal2005}, Local Binary Patterns (LBP)~\\cite{Ahonen2006}, and raw pixel information as our set of features descriptors for our hierarchical learning.  For the classifier $C$, we are using a Support Vector Machine, due to training easiness.  Nevertheless, any amount of features and other classifiers can be used.  We used the area under the curve (AUC) of the Receiver Operating Characteristic curve, and $F_1$-score, which is the weighted average of the precision and recall, as our evaluation metrics for all experiments.  In both cases, the higher the score the better the methods.  The results reported in this section are from our implementation using Python~3.6.5 executed on an NVIDIA GeForce GTX 1080 Ti with the TensorFlow framework.  We detail the neural network architecture and the feature extractors hyperparameters on the supplementary material.\n\n\\subsection{Experimental Setup}\nWe evaluated the performance of our proposed method using MNIST~\\cite{Lecun1998,Lecun2010}, Caltech~\\cite{Griffin2007}, Coil-100~\\cite{Nene1996}, Extended Yale B~\\cite{Georghiades2001, Kclee05}, UCSD Pedestrian~\\cite{Chan2008}, ISIC 2018 Challenge (Task~3)~\\cite{Tschandl2018, Codella2018}, and CIFAR-10~\\cite{Krizhevsky2009} datasets.\n\nWe followed the same experimental setup as previous works~\\cite{Sabokrou2018, You2017} for MNIST, Caltech, Coil-100 and Extended Yale~B\\@.  For UCSD Pedestrian and ISIC 2018 Challenge (Task~3) datasets, we followed Chaker et al\\onedot's~\\cite{Chaker2017} and Lu and Xu's~\\cite{Lu2018} setups, respectively.  For CIFAR-10, we followed previous methods~\\cite{Golan2018, Hofer2019, Ruff2018} setups.  %\n\nIn general, we selected inliers from a set of classes of the dataset, and the evaluation outliers were randomly sampled from the remaining classes.  We describe the setup of each experiment per dataset on the supplementary material.  For each of our methods and per experiment, we perform a train, validation, and test (inlier) data split.  To find the best hyperparameters of the model, we used the train and validation split.  And, we evaluated in the test partition.   We report the average metrics on the test partition of $5$-fold cross-validation for the inlier data split.  Note that our training data contains no true outliers.  We train the classifier using our synthesized outliers as negative samples, as previously described, and then evaluated, during testing, on different percentages of true outliers (according to the setups of each experiment).\n\n\n\\subsection{Comparison against Baselines}\n\\label{sec:baselines}\n\nFor our experiments on MNIST, Caltech-256, Coil-100, and Yale~B, we compared our proposed methods against a set of methods, namely LOF~\\cite{Breunig2000}, DRAE~\\cite{Xia2015}, R and RD~\\cite{Sabokrou2018}, GPND~\\cite{Pidhorskyi2018}, Coherent Pursuit (CoP)~\\cite{Rahmani2017},  REAPER~\\cite{Lerman2015}, Outlier Pursuit (OP)~\\cite{Xu2010}, LRR~\\cite{Liu2010},  DPCP~\\cite{Tsakiris2015}, $\\ell_1$-thresh.~\\cite{Soltanolkotabi2012}, R-Graph~\\cite{You2017}, AnoGAN~\\cite{Schlegl2017}, and  AGAN~\\cite{Kimura2018}.%\n\nIn the MNIST and Caltech-256 experiments, our proposed methods obtained the highest results while maintaining a stable outlier detection rate regardless of the number of outliers, as shown in Tables~\\ref{tab:mnist} and~\\ref{tab:caltech}.  Moreover, our proposed method learns representations that are robust to the type of classes, as demonstrated by the results when showing them more classes in the Caltech-256 experiment (cf\\onedot} \\def\\Cf{Cf\\onedot Table~\\ref{tab:caltech}).  \nIn contrast, for the Coil-100 and Yale~B experiments, which present variations, such as rotation and illumination, respectively, our proposed methods do not outperform R-Graph according to the AUC, as shown in Tables~\\ref{tab:coil} and~\\ref{tab:yale-b}.  Interestingly, we consistently obtain higher $F_1$ scores.  Moreover, we also noted that the variants introduced in these databases pose a great challenge for the descriptors we are using since HoG and LBP are not invariant to the point of view and intense intensity transforms of these two datasets.  Nevertheless, our method still achieves a comparable performance with existing methods.\n\nIn Table~\\ref{tab:UCSD}, we reported the AUC performance of our proposed method on UCSD Pedestrian benchmark with respect to other methods, namely, Adam~\\cite{Adam2008}, SF~\\cite{Mehran2009}, MPPCA~\\cite{Kim2009}, DTM~\\cite{Mahadevan2010}, MPPCA-SF~\\cite{Mahadevan2010}, Sparce~\\cite{Cong2011}, SNM~\\cite{Chaker2017}, LSR-VAE~\\cite{Sun2018}, and GMFC-VAE~\\cite{Fan2018}.  $\\text{OG}_e$ outperforms the other methods in Ped2 and detects different kinds of anomalous events, such as bicycling, skateboarding, wheel-chair, etc\\onedot} \\def\\vs{vs\\onedot.  Whereas in Ped1 our methods achieve comparable results to LSR-VAE~\\cite{Sun2018} but $3\\%$ to $4\\%$ lower than those reported by Fan et al\\onedot~\\cite{Fan2018}.  We noticed that in Ped1 are several illumination variants when groups of people walk towards and away from the camera, and some amount of perspective distortion which affects the selected features.  \n\nThe AUC of our experiments on the ISIC dataset are summarized in Table~\\ref{tab:ISIC2018}.  The highest AUC and more stable results were obtained by $\\ell_2$-based selection method as compared to ellipsoid with an overall AUC score of $87.2\\%$.  In general, our proposal outperforms other methods~\\cite{Lu2018}, except on the MEL disease.  We assume that the problems are due to variations on the images that are not picked up, like in previous errors.\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.2}\n  \\setlength\\tabcolsep{5pt} %\n  \\scriptsize\n  \\caption{Comparison of metrics on the MNIST dataset when varying the percentage ($\\%$) of the evaluation data comprised by outliers.}\n  \\label{tab:mnist}\n  \\begin{tabular}{llSSSSSS[table-format=1.4]S[table-format=1.4]}\n    \\toprule\n    &  {\\%}  &  {LOF}  &  {DRAE}  &  {D}  &  {RD}  &  {GPND}  &  {$\\text{OG}_e$}  & {$\\text{OG}_\\ell$}\\\\\n    \\midrule\n    {AUC}  & 10 &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  & 0.99181 &  \\bfseries 0.99202 \\\\\n    $F_1$  &     & 0.92 & 0.95 & 0.93 & 0.97 & 0.96 & 0.99193 & \\bfseries 0.99214  \\\\ \n    \\midrule\n    {AUC}  & 20 &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  & 0.99122 & \\bfseries 0.99152  \\\\\n    $F_1$  &      & 0.83 & 0.91 & 0.9 & 0.92 & 0.95 & 0.99134 & \\bfseries 0.99164 \\\\\n    \\midrule\n    {AUC}  & 30 &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  & 0.99117 & \\bfseries 0.99141  \\\\\n    $F_1$  &      & 0.72 & 0.88 & 0.87 & 0.92 & 0.94 & 0.99129 & \\bfseries 0.99154  \\\\\n    \\midrule\n    {AUC}  & 40 &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  & 0.99112 & \\bfseries 0.99128 \\\\\n    $F_1$  &      & 0.65 & 0.82 & 0.84 & 0.91 & 0.93 & 0.99124 & \\bfseries 0.99140 \\\\\n    \\midrule\n    {AUC}  & 50 &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  &  {\\text{--}}  & 0.99110 & \\bfseries 0.99141 \\\\\n    $F_1$  &      & 0.55 & 0.73 & 0.82 & 0.88 & 0.93 & 0.99122 & \\bfseries 0.99154 \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.3,detect-weight}\n  \\setlength\\tabcolsep{1pt} %\n  \\scriptsize\n  \\caption{Results on the Caltech-256 dataset.  Each pair of rows shows the results when inliers were taken from 1, 3, and 5 categories.  For evaluation, $50\\%$ of the data were outliers sampled from the category `clutter.'}\n  \\label{tab:caltech}\n  \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{lSSSSSSSSSSSSS}\n      \\toprule\n      & {CoP} & {REAPER} & {OP} & {LRR} & {DPCP} & {$\\ell_1$-thres.} & {R-Graph} & {D} & {RD}& {AGAN} & {AnoGAN} & {$\\text{OG}_e$}  & {$\\text{OG}_\\ell$}\\\\\n      \\midrule \n      {AUC} & 0.905 & 0.816 & 0.837 & 0.907 & 0.783 & 0.772 & 0.948 & 0.932 & 0.942 & \\text{--} & \\text{--} & \\bfseries 0.99090 & 0.98856 \\\\\n      $F_1$ & 0.880 & 0.808 & 0.823 & 0.893 & 0.785 & 0.772 & 0.914 & 0.916 & 0.928 &  0.977 & 0.956 & \\bfseries 0.99080 & 0.98842 \\\\\n      \\midrule\n      {AUC} & 0.676& 0.796  & 0.788  & 0.479 & 0.798 & 0.810  & 0.929 & 0.930 &  0.938 & \\text{--}  & \\text{--} & 0.99562 & \\bfseries 0.99717 \\\\\n      $F_1$ &  0.718 & 0.784 &  0.779 &  0.671 &  0.777 &  0.782 &  0.880 &  0.902 & 0.913 &  0.963 & 0.915 & 0.99560 & \\bfseries 0.99717 \\\\\n      \\midrule\n      {AUC}&  0.487 &  0.657 &  0.629 &  0.337 &  0.676 &  0.774 &  0.913 &  0.913 & 0.923 & \\text{--} & \\text{--} & \\bfseries 0.99776 & 0.99731 \\\\\n      $F_1$& 0.672 & 0.716 & 0.711 & 0.667 & 0.715 & 0.762 & 0.858 & 0.890 & 0.905 & 0.945 & 0.887 & \\bfseries 0.99776 & 0.99730 \\\\ \n      \\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.3}\n  \\setlength\\tabcolsep{1.4pt} %\n  \\scriptsize\n  \\caption{Results on the Coil-100 database.  Each pair of rows shows the results when inliers were taken from 1, 4, and 7 categories with $50\\%$, $25\\%$, and $15\\%$ of outlier samples on the evaluation set.  Inliers categories were randomly chosen, and outliers were randomly sampled, in the aforementioned proportions, from the remaining categories (at most one from each category).}\n  \\label{tab:coil}\n  \\begin{tabular}{lSSSSSSSSSS}\n    \\toprule\n    & {CoP} & {REAPER} & {OP} & {LRR} & {DPCP} & {$\\ell_1$-thres.} & {R-Graph} & {GPND} & {$\\text{OG}_e$}  & {$\\text{OG}_\\ell$}\\\\\n    \\midrule\n    {AUC} & 0.843 & 0.9 & 0.908 & 0.847 & 0.9 & 0.991 & \\bfseries 0.997 & 0.968 & 0.97241\t& 0.97914\\\\\n    $F_1$ & 0.866 & 0.892 & 0.902 & 0.872 & 0.882 & 0.978 & \\bfseries 0.990 & 0.979 & 0.97155\t& 0.97864 \\\\\n    \\midrule \n    {AUC} & 0.628 & 0.877 & 0.837 & 0.687 & 0.859 & 0.992 & \\bfseries 0.996 & 0.945 & 0.98232\t& 0.98759 \\\\\n    $F_1$ & 0.5 & 0.703 & 0.686 & 0.541 & 0.684 & 0.941 & 0.97 & 0.96 & 0.98199 & \\bfseries0.98739 \\\\\n    \\midrule  \n    {AUC} & 0.58 & 0.824 & 0.822 & 0.628 & 0.804 & 0.991 & \\bfseries 0.996 & 0.919 & 0.97869 & 0.97534  \\\\\n    $F_1$ & 0.346 & 0.541 & 0.528 & 0.366 & 0.511 & 0.897 & 0.955 & 0.941 & \\bfseries0.97822 & 0.97468  \\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.3}\n  \\setlength\\tabcolsep{2pt} %\n  \\scriptsize  \n  \\caption{Results on the Extended Yale B database.  Each pair of rows shows the results when inliers were taken from 1 and 3 randomly chosen subjects, and outliers were randomly chosen from the other subjects (at most one from each subject).}\n  \\label{tab:yale-b}\n  \\begin{tabular}{lSSSSSSSSS}\n    \\toprule\n    & {CoP} & {REAPER} & {OP} & {LRR} & {DPCP} & {$\\ell_1$-thres.} & {R-Graph} &  {$\\text{OG}_e$}  & {$\\text{OG}_\\ell$}\\\\\n    \\midrule\n    {AUC}  & 0.556 & 0.964 & 0.972 & 0.857 & 0.952 & 0.844 &  \\bfseries 0.986  & 0.97571 & 0.95686  \\\\\n    $F_1$  & 0.563 & 0.911 & 0.918 & 0.797 & 0.885 & 0.763 & 0.951 & \\bfseries 0.97511 & 0.95449  \\\\\n    \\midrule\n    {AUC}  & 0.529 & 0.932 & 0.968 & 0.807 & 0.888 & 0.848 &  \\bfseries 0.985  & 0.93853 & 0.94316  \\\\\n    $F_1$  & 0.292 & 0.758 & 0.856 & 0.509 & 0.653 & 0.545 & 0.878 & 0.93318 & \\bfseries 0.94274  \\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.3,detect-weight}\n  \\setlength\\tabcolsep{1pt} %\n  \\scriptsize\n  \\caption{Results on the UCSD Pedestrian~1 and~2 datasets with the comparison of AUC performance where inliers were pedestrians and outliers were other objects.}\n  \\label{tab:UCSD}\n  \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{lSSSSSSSSSSS}\n      \\toprule\n      {} & {Adam} & {SF} & {MPPCA} & {DTM} & {MPPCA-SF} & {Sparce} & {SNM} & {LSR-VAE} & {GMFC-VAE} & {$\\text{OG}_e$} & {$\\text{OG}_\\ell$} \\\\ \n      \\midrule\n      {Ped1} & {\\text{--}} & 0.675 & 0.59 & 0.818  & 0.668 & 0.86 & 0.855 & 0.902 & \\bfseries{0.949} & 0.912 & 0.908 \\\\\n      {Ped2} & 0.634 & 0.623 & 0.774 & 0.848 & 0.710 & 0.861  & 0.879 & 0.891 & 0.922  & \\bfseries{0.955} & 0.900 \\\\\n      \\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.3}\n  \\setlength\\tabcolsep{2pt} %\n  \\scriptsize\n  \\caption{Results on the ISIC 2018 Challenge (Task~3) dataset.  Each row shows the AUC when inliers were taken from NV disease, and row's category was used as an outlier.}\n  \\label{tab:ISIC2018}\n  \\begin{tabular}{lSSSSSSSS}\n  \\toprule\n    {Disease} & $s_{\\text{vae}}^{\\text{reconst}}$ & ${s}_{\\text{iwae}}^{\\text{reconst}}$ & {${s}_{\\text{vae}}^{\\text{kl}}$} & ${s}_{\\text{iwae}}^{\\text{kl}}$ & ${s}_{\\text{vae}}$ & ${s}_{\\text{iwae}}$ & {$\\text{OG}_e$} & {$\\text{OG}_\\ell$} \\\\\n    \\midrule\n    {AKIEC} & 0.872 & 0.871 & 0.441 & 0.406 & 0.864 & 0.864 & 0.72702 & \\bfseries{0.943} \\\\\n    {BCC} & 0.803 & 0.802 & 0.454 & 0.431 & 0.795 & 0.795 & 0.70771 & \\bfseries{0.872} \\\\\n    {BKL} & 0.792 & 0.793 & 0.472 & 0.441 & 0.783 & 0.784 & 0.7717 & \\bfseries{0.934} \\\\\n    {DF} & 0.682 & 0.678 & 0.398 & 0.383 & 0.671 & 0.670 & 0.74139 & \\bfseries{0.916} \\\\\n    {MEL} & 0.862 & \\bfseries{0.864} & 0.690 & 0.677 & 0.861 & 0.861 & 0.73641 & 0.84453 \\\\\n    {VASC} & 0.662 & 0.657 & 0.487 & 0.477 & 0.651 & 0.648 & 0.71649 & \\bfseries{0.840} \\\\\n    {All Disease} & 0.779 & 0.777 & 0.491 & 0.469 & 0.771 & 0.771 & 0.7251 & \\bfseries{0.872} \\\\\n  \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\n\\subsection{Ablation Study}\n\\label{sec:ablation}\n\n\\subsubsection{Outlier Synthesis' Parameters}  \n\n\\begin{figure*}[t]\n  \\centering\n  \\resizebox{\\linewidth}{!}\n  {\\input{images\/alpha-beta-combined.tex}}\n \\caption{%\n  Comparisons of $F_1$ scores on MNIST, Caltech-256, Coil-100, Extended Yale~B, UCSD Ped1 and Ped2, and ISIC 2018 Challenge datasets by varying $\\alpha$ and $\\beta$ parameters when inliers come from one class and true outliers from remaining classes are used for testing (selected randomly from the remaining classes).%\n}\n  \\label{fig:alpha-beta}\n  \\vspace{-10pt}\n\\end{figure*}\n\nFor the parameters that define the selection of distributions at the outskirts of the space,  $\\alpha$,~(\\ref{eq:ellipsoid-outlier}) and~(\\ref{eq:l2-outlier}), and $\\beta$~(\\ref{eq:outliers}), we evaluated the ranges of $\\alpha$ from $1$ to $2$ with step $0.25$, and $\\beta$ from $1$ to $5$ with step $0.5$, on the best set of hyperparameters found on previous experiments, cf\\onedot} \\def\\Cf{Cf\\onedot Section~\\ref{sec:baselines}.  We show in Fig.~\\ref{fig:alpha-beta} the results for our metrics on all our evaluation datasets when using true outliers from the remaining classes on testing.  We show a detailed evaluation with other setups for outliers on all the datasets on the supplementary material.\n\nWe observed that selecting outskirt distributions that are more than one standard deviation away from the center of the meta-distribution yields better results, i.e\\onedot} \\def\\Ie{I.e\\onedot, having higher values for $\\alpha$.  This trend can be seen in all datasets.  However, it is clearer in Caltech, Coil and UCSD Pedestrian datasets, and a dip can be seen on MNIST when both parameters are one.  On the other hand, sampling the synthetic outliers farther from the center of the distributions, as indicated by higher $\\beta$ values, shows an improvement.  However, different from $\\alpha$, the variations in $\\beta$ are not consistent as the former, as $\\beta$'s metrics have falls on the different $\\alpha$ values.  We hypothesize that, in general, these parameters will present an inverted-u behavior since they control how different the synthesized samples could be.  By increasing them, we will increase the performance up to a certain point, and then we will get diminishing returns.  Note that high values of $\\alpha$ introduce outliers within the training inliers, while high values on $\\beta$ may introduce outliers that are too different from any expected anomaly at prediction time.  Hence, a classifier trained on this data will learn a decision boundary that is not close enough to the normal samples.\n\nWe note the particular behavior of the metrics on Coil-100, Extended Yale-B, and ISIC datasets.  On these datasets, there are several dips and changes on the $F_1$ scores. Moreover, no patterns seem to arise from the experiments. We notice, however, that the selection of outskirts distributions varies considerably. We attribute this behavior to two factors: the variations on the datasets, and the amount of data.  On one hand, the feature descriptors we used are sensitive to the variations of these two datasets, which may be reflected in the changes in the selection of the outliers (cf\\onedot} \\def\\Cf{Cf\\onedot ablation results on Section~\\ref{sec:other-dimensions}).  On the other hand, these datasets have considerably fewer samples, in comparison with MNIST, Caltech-256, and UCSD Pedestrian.  For example, MNIST varies between \\num{5500}--\\num{5900} samples, Caltech has \\num{750} (when $n=5$), in comparison Coil-100 has \\num{504} samples (when $n=7$) and Yale~B has only \\num{192} samples (when $n=3$).  Note the variation in the classes and their relation to the number of samples of each class.\n\n\n\\subsubsection{Evaluation of Other Dimensions}\n\\label{sec:other-dimensions}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{table-format=1.3}  \n  \\scriptsize\n  \\caption{Ablation study on the Extended Yale B database, where inliers are taken from one and three ($n$) subjects with $50\\%$ and $15\\%$ outlier samples from the other subjects (at most one from each subject) all randomly chosen.  We used different combinations of features, i.e\\onedot} \\def\\Ie{I.e\\onedot, HoG~(H), LBP~(L), and raw intensity~(R); way of applying the AE, such as individually to each feature~(I), or to the concatenated features~(C), or without AE and passing the features to the next step~(WO); existence of VAE, such as with~(W) and without~(WO) it; and different outskirt distribution selection~(OS), i.e\\onedot} \\def\\Ie{I.e\\onedot, ellipsoid~($e$), $\\ell_2$-based~($\\ell$), and without it~(--).  The hat OS denote experiments when the outliers were generated deterministically.  The last section show results with AE instead of VAE with outliers generated with jitter and without them~(--).  Blanks in a cell denote that the previous row's setup was used.}\n\\label{tab:ablation}\n  \\rowcolors{2}{gray!10}{white}\n  \\begin{tabular}{lll>{$}l<{$}SSSS}\n    \\toprule\n    \\rowcolor{white}\n    \\multirow{2}{*}{Feats.} & \\multirow{2}{*}{AE} & \\multirow{2}{*}{VAE} & \\multirow{2}{*}{OS} & \\multicolumn{2}{c}{$n=1$} & \\multicolumn{2}{c}{$n=3$} \\\\\n    \\cmidrule{5-8}\n    \\rowcolor{white}\n    & & & & {AUC} & {$F_1$} & {AUC} & {$F_1$} \\\\\n    \\midrule\n    HLR & I & W & e & 0.976 & 0.975 & 0.939 & 0.933 \\\\\n    & & & \\ell & 0.957 & 0.954 & 0.943 & 0.943 \\\\\n    HL &  &  & e & 0.883 & 0.884 & 0.897 & 0.902 \\\\\n    & & & \\ell & 0.914 & 0.905 & 0.903 & 0.904 \\\\\n    HR &  &  & e & 0.950 & 0.946 & 0.938 & 0.934\\\\\n    & & & \\ell & 0.957 & 0.955 & 0.908 & 0.897\\\\\n    LR &  &  & e & 0.957 & 0.955 & 0.884 & 0.884 \\\\\n    & & & \\ell & 0.930 & 0.923 & 0.898 & 0.891 \\\\\n    \n    HLR & C & W & e & 0.912 & 0.901 & 0.912 & 0.904 \\\\\n    & & & \\ell & 0.908 & 0.897 & 0.927 & 0.923 \\\\\n    HL &  &  & e & 0.922 & 0.915 & 0.902 & 0.919 \\\\\n    & & & \\ell & 0.828 & 0.833 & 0.886 & 0.878 \\\\\n    HR &  &  & e & 0.908 & 0.898 & 0.904 & 0.893 \\\\\n    & & & \\ell & 0.943 & 0.940 & 0.886 & 0.871 \\\\\n    LR &  &  & e & 0.943 & 0.939 & 0.912 & 0.903 \\\\\n    & & & \\ell & 0.927 & 0.920 & 0.911 & 0.901 \\\\\n    \n    HLR & WO & W & e & 0.925 & 0.918 & 0.923 & 0.930 \\\\\n    & & & \\ell & 0.905 & 0.892 & 0.911 & 0.919 \\\\\n    HL &  &  & e & 0.810 & 0.817 & 0.802 & 0.795 \\\\\n    & & & \\ell & 0.877 & 0.890 & 0.833 & 0.825 \\\\\n    HR &  &  & e & 0.891 & 0.891 & 0.907 & 0.910 \\\\\n    & & & \\ell & 0.939 & 0.934 & 0.908 & 0.904 \\\\\n    LR &  &  & e & 0.931 & 0.925 & 0.901 & 0.899 \\\\\n    & & & \\ell & 0.890 & 0.8931 & 0.927 & 0.921 \\\\\n    \n    HLR & C & W & \\text{--} & 0.621 & 0.610 & 0.611 & 0.621 \\\\\n    HL & & &  & 0.66222 & 0.76366 & 0.61375 & 0.44967 \\\\\n    HR & & &  & 0.63214 & 0.72527 & 0.57596 & 0.40745 \\\\\n    LR & & &  & 0.59250 & 0.65509 & 0.61000 & 0.44737 \\\\\n\n    HLR & I & WO & \\text{--} & 0.489 & 0.616 & 0.619 & 0.277 \\\\\n    HL &  &  &  & 0.550 & 0.560 & 0.481 & 0.171 \\\\\n    HR &  &  &  & 0.633 & 0.676 & 0.578 & 0.247 \\\\\n    LR &  &  &  & 0.551 & 0.567 & 0.527 & 0.210 \\\\\n    \n    HLR & C & WO & \\text{--} & 0.500 & 0.505 & 0.568 & 0.245 \\\\\n    HL &  &  &  & 0.580 & 0.601 & 0.544 & 0.219 \\\\\n    HR &  &  &  & 0.509 & 0.489 & 0.590 & 0.249 \\\\\n    LR &  &  &  & 0.586 & 0.623 & 0.580 & 0.254 \\\\\n    \n    HLR & WO & WO & \\text{--} & 0.413 & 0.394 & 0.492 & 0.183 \\\\\n    HL &  &  &  & 0.413 & 0.394 & 0.450 & 0.177 \\\\\n    HR &  &  &  & 0.514 & 0.563 & 0.414 & 0.179 \\\\\n    LR &  &  &  & 0.413 & 0.394 & 0.507 & 0.195 \\\\\n    \n    \\midrule\n    HLR & I & W & \\hat{e} & 0.601 & 0.621 & 0.611 & 0.575 \\\\\n    & & & \\hat{\\ell} & 0.6 & 0.622 & 0.611 & 0.575 \\\\\n    \n    \\midrule\n    HLR & I & AE & \\text{--} & 0.542 & 0.643 & 0.676 & 0.391 \\\\\n    HL &  &  &  & 0.604 & 0.631 & 0.517 & 0.284 \\\\\n    HR &  &  &  & 0.641 & 0.683 & 0.609 & 0.314 \\\\\n    LR &  &  &  & 0.589 & 0.614 & 0.592 & 0.284 \\\\\n    HLR & C & AE & \\text{--} & 0.523 & 0.535 & 0.591 & 0.308 \\\\\n    HL &  &  &  & 0.584 & 0.656 & 0.572 & 0.294 \\\\\n    HR &  &  &  & 0.557 & 0.54 & 0.635 & 0.314 \\\\\n    LR &  &  &  & 0.619 & 0.647 & 0.603 & 0.346 \\\\\n    HLR & I & AE & \\text{jitter} & 0.849 & 0.834 & 0.825 & 0.811 \\\\\n    HL &  &  &  & 0.768 & 0.759 & 0.796 & 0.814 \\\\\n    HR &  &  &  & 0.831 & 0.793 & 0.826 & 0.807 \\\\\n    LR &  &  &  & 0.812 & 0.828 & 0.771 & 0.793 \\\\\n    HLR & C & AE & \\text{jitter} & 0.727 & 0.749 & 0.741 & 0.784 \\\\\n    HL &  &  &  & 0.704 & 0.697 & 0.752 & 0.71 \\\\\n    HR &  &  &  & 0.674 & 0.691 & 0.735 & 0.724 \\\\\n    LR &  &  &  & 0.765 & 0.751 & 0.718 & 0.682 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace*{-15pt}\n\\end{table}\n\nWe may think of our proposed method in terms of four different dimensions: features used, way of applying the AEs on them, the fusion through the VAE, and the outskirt-distribution selection method.  To assess our method we performed a set of experiments by varying these dimensions on the Extended Yale~B dataset, and we report the results in Table~\\ref{tab:ablation}.  We selected this dataset for two reasons: size, which allows us to run several experiments on our restricted hardware, and challenging variations, which gives us a lower bound on the expected results.\n\nFor the features, we evaluate the combination of HoG~(H), LBP~(L), and raw intensity~(R) as pairs and triplets.  For the AEs, we evaluate the way of applying them to the set of inputs.  We can use different AEs per feature, i.e\\onedot} \\def\\Ie{I.e\\onedot, individually to each feature~(I), use a single AE on the concatenation of the inputs~(C), or use no AE at all~(WO).  Moreover, we consider the setup with~(W) and without~(WO) VAE for fusion.  In the latter setup, the classification is performed over the concatenation of the extracted representations, instead of the fusion.  Lastly, we consider the use of our two proposed outskirt selection schemes, i.e\\onedot} \\def\\Ie{I.e\\onedot, ellipsoid~($e$) and $\\ell_2$-based~($\\ell$).  In the case where no distributions are generated (i.e\\onedot} \\def\\Ie{I.e\\onedot, the case without VAE for fusion), we cannot use the selection method and the concatenated features are passed to the classifier.  The cases with no synthesis are denoted with an n-dash~(--).  In them, we use a one-class SVM as a classifier due to the lack of negative samples to train a binary one.\n\nIn general, we observe that fusion is a critical component of the proposed method.  Removing the VAE (WO in the VAE column) has a significant impact on the different results regardless of the combination of the AE and features.  We, also, compared the use of an AE instead of VAE with and without synthetic outlier generation by randomly jitter the outskirt features in the AE space with noise drawn from $\\mathcal{N}(0, 1.5)$, see the end of Table~\\ref{tab:ablation}.  These results show a significant decrease in performance.  On the other hand, the second dimension that affects the results is the way we apply the AEs.  By encoding each feature individually we obtain consistently higher metrics.  These results align with the objective of encoding and compressing each feature individually to later fuse them.  By having an early fusion (the concatenation at the AE level), the VAE does not have richer spaces to work with.  \n\nTo evaluate the contribution of the generation of the synthetic outliers, we trained a one-class SVM using only inlier data with different feature combinations (denoted by the n-dash on the OS column).  We notice a significant drop in the metrics concerning their counterparts that use synthetic outliers for training.  Moreover, we also evaluate the case in which the distribution-space is used as features for classification (denoted by `[H][L][R] C W --' rows).  Similarly, we can see that the performance comes from the synthetic outliers that help the classifier to learn a robust boundary, in contrast to the use of more expressive features.\n\nInterestingly, the results of HR or LR features show higher metrics in comparison to HL features.  These results indicate, again, that fusion can do so much to increase the invariants of the given inputs' features.  We can draw similar conclusions from the results of our proposed method and more complex fusion (cf\\onedot} \\def\\Cf{Cf\\onedot first two rows of Table~\\ref{tab:ablation} and Section~\\ref{sec:baselines}).  However, our proposed method can boost the metrics significantly on this task, in comparison to other methods (including concatenation and learning on the original space).\n\nWe also evaluated the case where the outliers are generated deterministically instead of using our probabilistic sampling.  In this case, we set $\\epsilon=1$ in the generation~(\\ref{eq:outliers}).  In that way, we will always generate the same outlier per outskirt distribution.  The corresponding deterministic counterparts of our outskirt selectors,~(\\ref{eq:ellipsoid-outlier}) and~(\\ref{eq:l2-outlier}), are denoted as $\\hat{e}$ and $\\hat{\\ell}$, respectively.  The results, in the last rows of the table, show a significant reduction in the different metrics.  This demonstrates the importance of creating robust outliers for training when lacking negative samples, independently of the robust feature spaces.\n\nIn summary, the way of fusing the feature spaces is a key component in the final representation.  And our probabilistic approach seems to yield excellent results in comparison to naive fusion methods, and existing approaches that do not fuse.  Secondly, we need to represent spaces individually to take advantage of each feature space's characteristics.  Moreover, a probabilistic representation has the advantage of synthesizing robust outliers, which are necessary to learn robust and general classifiers.  Regarding the outlier selection proposals, we could not draw a line of which one is best for the task since no pattern emerged from the results.\n\n\\subsubsection{Comparison against Gold Standard}\n\\label{sec:gold-standard}\nWe also compared our proposal against what can be considered an upper bound of performance of the method.  That is, we compared against a VAE and an AE with the same architecture as the proposal, except that they were trained on true outliers.  We show the results on Table~\\ref{tab:real-synth}.  We can observe that our proposal is close to the upper bound that represent the VAE and the AE results.  It is expected that the proposal achieves a lower performance due to the lack of true outliers.  Moreover, we note that the AE achieves a lower performance than its variational counterpart which also supports our decision to use a VAE as the main feature descriptor.\n\n\\begin{table}\n  \\sisetup{%\n    table-format=1.3(3),\n  }\n  \\caption{Comparison of the proposed method against VAE and AE when trained on true outliers (upper bound performance).  We report the AUC scores of all the methods on $5$-fold cross-validation with $20\\%$ outliers.  Our proposal was trained with synthetic data, but cannot be evaluated as a one-class due to lack of outliers.  VAE and AE were trained on both binary (first four rows) and one-class (last two rows) setups. We report max-min values in parenthesis.}\n  \\label{tab:real-synth}\n  \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{l@{ }S@{ (}S[table-format=1.3]@{, }S[table-format=1.3]@{)\\ \\ }S@{ (}S[table-format=1.3]@{, }S[table-format=1.3]@{)\\ \\ }S@{ (}S[table-format=1.3]@{, }S[table-format=1.3]@{) }}\n      \\toprule\n      & \\multicolumn{9}{c}{Dataset} \\\\\n      \\cmidrule{2-10}\n      & \\multicolumn{3}{c}{CIFAR 10} & \\multicolumn{3}{c}{Extended Yale B} & \\multicolumn{3}{c}{Coil-100} \\\\\n      \\midrule\n      VAE & 0.937 (012) & 0.957 & 0.919 & 0.987 (019) & 0.998 & 0.965 & 0.997 (022) & 0.999 & 0.952 \\\\\n      AE & 0.929 (9) & 0.941 & 0.914 & 0.965 (14) & 0.984 & 0.937 & 0.989 (7) & 0.997 & 0.974 \\\\\n      $\\text{OG}_e$ & 0.883 (31) & 0.925 & 0.832 & 0.971 (26) & 0.998 & 0.92 & 0.985 (32) & 0.998 & 0.916 \\\\\n      $\\text{OG}_\\ell$ & 0.915 (23) & 0.963 & 0.878 & 0.983 (16) & 0.998 & 0.953 & 0.991 (12) & 0.998 & 0.959 \\\\\n      \\midrule\n      VAE (OC) & 0.504 (2) & 0.538 & 0.48 & 0.671 (22) & 0.706 & 0.634 & 0.728 (19) & 0.74 & 0.687 \\\\\n      AE (OC) & 0.502 (2) & 0.535 & 0.462 & 0.636 (16) & 0.661 & 0.591 & 0.711 (13) & 0.749 & 0.684 \\\\\n      \\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\n\\subsection{Evaluation of Other Classifiers}\n\n\\begin{table}[t]\n  \\centering\n  \\sisetup{%\n    table-format=1.3(3),\n  } \n  \\caption{AUC scores of ellipsoid~($e$), $\\ell_2$-based~($\\ell$) methods on CIFAR-10 dataset where each model was trained on a single class and tested against all other classes.  We report averaged results overall classes with $5$-fold cross-validation, and different classifiers: Support Vector Machine~(SVM), Multilayer Perceptron~(MLP), Random Forest~(RF), and Naive Bayes~(NB).  We compared our AUC against other existing methods.}\n  \\label{tab:cifar}\n  \\vspace{2.5pt plus 2.5pt minus 2.5pt}\n  \\setlength\\tabcolsep{2pt} %\n  \\scriptsize\n  \\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{llSSSS}\n      \\toprule\n      {Feats.} & {OS} & {SVM} & {MLP} & {RF} & {NB} \\\\\n      \\midrule\n      \n      HLR & $\\ell$ &  0.917(26) &  0.813(26) &  0.958(29) &  0.886(13) \\\\\n      & e & 0.858(22) & 0.772(45) & 0.836(14) & 0.856(18) \\\\\n      \n      \\cmidrule{1-6}\n      \n      \\multicolumn{2}{l}{OC-SVM (CAE) \\cite{Golan2018}} & 0.624 & \\multicolumn{2}{l}{OC-SVM (E2E) \\cite{Golan2018}} & 0.648\\\\\n      \\multicolumn{2}{l}{OC-SVM (RAW) \\cite{Golan2018}} & 0.620 & \\multicolumn{2}{l}{DAGMM \\cite{Golan2018}} & 0.531 \\\\\n      \\multicolumn{2}{l}{DSEBM \\cite{Golan2018}}  & 0.610 & \\multicolumn{2}{l}{ADT \\cite{Golan2018}} & 0.860\\\\\n      \\multicolumn{2}{l}{ADT-120 \\cite{Hofer2019}} & 0.690 & \\multicolumn{2}{l}{ADT-500 \\cite{Hofer2019}} & 0.730\\\\\n      \\multicolumn{2}{l}{ADT-1000 \\cite{Hofer2019}} & 0.750 & \\multicolumn{2}{l}{COREL-120 \\cite{Hofer2019}} & 0.760\\\\\n      \\multicolumn{2}{l}{AnoGAN \\cite{Ruff2018}} & 0.615(27) & \\multicolumn{2}{l}{Deep-SVDD \\cite{Ruff2018}} & 0.648(20) \\\\\n      \\bottomrule\n    \\end{tabular}\n  }%\n\\end{table}\n\nOur previous experiments used SVM as the classifier, in this section, we explore the impact of other classifiers on our method's performance.  Namely, we additionally use Multilayer Perceptron~(MLP), Random Forest~(RF), and Naive Bayes~(NB).  Simultaneously, we evaluate our method against a set of algorithms that report results under a different dataset and metric.  We compare against One Class SVM~(OC-SVM), Deep Autoencoding Gaussian Mixture Model~(DAGMM), Deep structured energy-based models~(DSEBM), and Anomaly Detection Transformations~(ADT) reported by Golan and Ran~\\cite{Golan2018}; three ADT-based models, and the COREL method reported by Hofer et al\\onedot~\\cite{Hofer2019}; and AnoGAN and Deep-SVDD reported by Ruff et al\\onedot~\\cite{Ruff2018}.  In this case, we evaluate our results using the area under the curve of the ROC~(AUC) on the CIFAR-10 dataset.  This change in metric and dataset is to compare against these other sets of methods.  There are more challenging datasets and versions of the CIFAR dataset.  However, not all methods report on them.  Hence, it is difficult to have a comparison against more recent methods based on those other datasets.\n\nOur results (cf\\onedot} \\def\\Cf{Cf\\onedot Table~\\ref{tab:cifar}) show that SVM and RF are the best classifiers on the CIFAR-10 dataset.  There is a swap on the first ranked classifier while using our proposed selection methods.  However, the results are comparable.  Moreover, our proposed methods outperform the state of the art methods regardless of the classifier used.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe introduced an approach that learns representations of feature descriptors, and then fuse them into a distribution-space.  This probabilistic space allows us to learn a representation of the inputs' characteristics class, and, consequently, approximate what outliers may look like.  Then, we synthesize outliers as drawings from the outskirts-distributions of the training samples.  Finally, we demonstrated that using the given inputs and the synthesized outliers yields classifiers capable of detecting outliers consistently on several datasets with varying types and amounts of outliers.  Moreover, we showed the importance of the different parts of our proposal: individual representation through compression, variational inference for fusion, and sampling for synthesis, through an ablation study.\n\nAs future work, we envision applications of our proposed idea beyond images, i.e\\onedot} \\def\\Ie{I.e\\onedot, other domains like text, audio, etc..  We intend to explore other generative models for outlier synthesis to increase the robustness of our outliers.  Another branch to be explored is to train the synthesis and the classifiers in a tightly coupled way, i.e\\onedot} \\def\\Ie{I.e\\onedot, an end-to-end model from data representation, distillation, outlier synthesis, and classification, to compare if this approach yields better results than a loosely coupled one, like our proposal.\n\n\n\\input{appendix}\n\n\\bibliographystyle{ieeetr}\n\n\\section{Experimental Setup}\n\\label{sec:setup}\n\n\\textbf{MNIST}~\\cite{Lecun1998, Lecun2010} contains \\SI{60000}{} handwritten digits from `0' to `9.'  Each of the ten digit categories is taken as the target class (i.e\\onedot} \\def\\Ie{I.e\\onedot, inliers) and we simulate outliers (during testing) by randomly sampling images from other categories with a proportion of $10\\%$ to $50\\%$. This experiment is repeated for all of the ten digit categories with the images being of size $28 {\\mkern1mu\\oldtimes\\mkern1mu} 28$.  \n\n\\textbf{Caltech-256}~\\cite{Griffin2007} dataset contains 256 different object categories with a total number of \\SI{30607}{} images.  Each category has at least 80.  For this dataset the images' size is $28 {\\mkern1mu\\oldtimes\\mkern1mu} 28$ and we repeat same procedure three times by using images from $n \\in \\{1,3,5\\}$ randomly chosen categories as inliers.  From each category the first 150 images are used only if that category has more than 150 images.  We then randomly select $50\\%$ outliers from the `clutter' category which contains 827 images of different varieties in each experiment. \n\n\\textbf{Coil-100}~\\cite{Nene1996} dataset contains 100 different object categories with a total number of\n\\SI{7200}{} images.  Each object category has 72 images taken from different pose intervals of $5$~degrees and image's size is $32 {\\mkern1mu\\oldtimes\\mkern1mu} 32$.  We performed three different experiments by randomly selecting $n \\in \\{1, 4, 7 \\}$ categories images as inliers, and outliers were randomly chosen from other categories (at most one from each category) by varying percentage of outliers among $50\\%$, $25\\%$, and $15\\%$.\n\n\\textbf{Extended Yale~B}~\\cite{Georghiades2001, Kclee05} dataset contains frontal face images of 38 persons as categories.  Each category has 64 different illumination conditions.  The actual size of face images is $192 {\\mkern1mu\\oldtimes\\mkern1mu} 168$, and for our experiment we down-sample them to $48 {\\mkern1mu\\oldtimes\\mkern1mu} 42$.  We performed two different experiments by randomly selecting $n \\in \\{1, 3 \\}$ categories images as inliers, and outliers are randomly chosen from other categories (at most one from each category) by varying the outliers percentage between $35\\%$ and $15\\%$.\n\n\\textbf{UCSD Pedestrian}~\\cite{Chan2008} dataset contains two subsets of Ped1 and Ped2 with a different street scenes recorded with a stationary camera at \\SI{10}{fps} and resolutions $158 {\\mkern1mu\\oldtimes\\mkern1mu} 234$ and $240 {\\mkern1mu\\oldtimes\\mkern1mu} 360$, respectively.  The density of the pedestrians varies from low to high.  The normal objects in all scenes are pedestrians (i.e\\onedot} \\def\\Ie{I.e\\onedot, inliers).  All other objects (e.g\\onedot} \\def\\Eg{E.g\\onedot, cars, skateboarders, wheelchairs, etc\\onedot} \\def\\vs{vs\\onedot) are  considered outliers.  For the experiments the images size is down-sampled to $28 {\\mkern1mu\\oldtimes\\mkern1mu} 28$.  We performed different experiments on Ped1 scenes by randomly selecting normal object images from 34 categories as inliers, outlier images were randomly chosen from the remaining categories and each category contains 200 frames.  For Ped2 similar experiments were performed but with $16$ and $12$ object categories as inliers and outlier, respectively, where the number of frames of each clip varies in outliers.\n\n\\textbf{ISIC 2018 Challenge (Task~3)}~\\cite{Tschandl2018, Codella2018} dataset contains seven different diseases as categories with a total number of 10015 images.  The actual size of disease images is $450 {\\mkern1mu\\oldtimes\\mkern1mu} 600$, and for our experiment we down sample them to $32 {\\mkern1mu\\oldtimes\\mkern1mu} 32$  and try to implement deep architecture with limited resources.  We performed six different experiments by considering NV disease category images as inliers and we tested outliers by select the 100 images from the rest of disease categories.\n\n\\textbf{CIFAR-10}~\\cite{Krizhevsky2009} dataset consists of \\num{60000} $32 {\\mkern1mu\\oldtimes\\mkern1mu} 32$ color images in $10$ classes, with \\num{6000} images per class. There are \\num{50000} training images and \\num{10000} test images, divided equally across the classes.  For the experiments, we use a single class for training, and the rest classes as outliers (similar to the previous protocols).  We report averaged results overall classes with $5$-fold cross-validation.\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Amount of images on the ISIC 2018 Challenge (Task~3) Dataset.}\n\\label{tab:dataset4}\n\\sisetup{table-format=4, table-text-alignment=right}\n\\setlength\\tabcolsep{2pt} %\n\\scriptsize\n\\begin{tabular}{lS}\n \\toprule\n {Disease} &  {No.\\ of Images} \\\\\n \\midrule\n   MEL & 1113 \\\\\n   NV  & 6705 \\\\\n   BCC & 514 \\\\\n   AKIEC & 327 \\\\\n   BKL & 1099 \\\\\n   DF & 115 \\\\\n   VASC & 142 \\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\section{Extended Outlier Synthesis' Parameters}\n\nWe present high resolution versions for Fig.~\\ref{fig:alpha-beta}, and all configurations for all experiments in the six datasets (according to the setup shown in Appendix~\\ref{sec:setup}) in Figs.~\\ref{fig:alpha-beta-mnist}, \\ref{fig:alpha-beta-caltech}, \\ref{fig:alpha-beta-coil}, \\ref{fig:alpha-beta-yale}, \\ref{fig:alpha-beta-ucsd}, and~\\ref{fig:alpha-beta-isic}.\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{\\linewidth}{!}\n  {\\input{images\/alpha-beta-mnist.tex}}\n  \\caption{Comparisons of $F_1$ scores on MNIST dataset for the proposed methods by varying $\\alpha$ and $\\beta$ parameters when inliers come from each class, and outliers are taken from three different percentages ($\\%$). We report average values among all ten classes of the digits.}\n  \\label{fig:alpha-beta-mnist}\n\\end{figure}\n\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{\\linewidth}{!}\n  {\\input{images\/alpha-beta-caltech.tex}}\n  \\caption{Comparisons of $F_1$ scores on Caltech-256 dataset for the proposed methods by varying $\\alpha$ and $\\beta$ parameters when inliers are taken to be images from $n \\in \\{1,3,5\\}$ randomly chosen categories with $50\\%$ outlier samples randomly selected from category clutter.}\n  \\label{fig:alpha-beta-caltech}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{\\linewidth}{!}\n  {\\input{images\/alpha-beta-coil.tex}}\n  \\caption{Comparisons of $F_1$ scores on Coil-100 dataset for the proposed methods by varying $\\alpha$ and $\\beta$ parameters when inliers are taken to be images from $n \\in \\{1,4,7\\}$ randomly chosen categories with $50\\%$, $25\\%$, and $15\\%$ outlier samples, respectively, randomly selected from the remaining categories.}\n  \\label{fig:alpha-beta-coil}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{\\linewidth}{!}\n  {\\input{images\/alpha-beta-yale.tex}}\n  \\caption{Comparisons of $F_1$ scores on Extended Yale~B dataset for the proposed methods by varying $\\alpha$ and $\\beta$ parameters when inliers are taken from $n \\in \\{1,3\\}$ randomly chosen subjects, respectively, and outliers are randomly chosen from the other subjects (at most one from each subject).}\n  \\label{fig:alpha-beta-yale}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{\\linewidth}{!}\n  {\\input{images\/alpha-beta-combined-ucsd-suppl.tex}}\n  \\caption{Comparisons of $F_1$ scores on UCSD Pedestrian dataset for the proposed methods by varying $\\alpha$ and $\\beta$ parameters when inliers come from each class, and outliers are randomly selected from the remaining classes. We report the average values among all categories of video scenes.}\n  \\label{fig:alpha-beta-ucsd}\n\\end{figure}\n\n\\begin{figure*}[t]\n  \\centering\n  {\\input{images\/alpha-beta-combined-disease-suppl.tex}}\n  \\caption{Comparisons of $F_1$ scores on ISIC 2018 Challenge (Task~3) dataset for the proposed methods by varying $\\alpha$ and $\\beta$ parameters when inlier are taken images from NV disease category and outliers are randomly chosen from each individual disease category (show column wise, every two rows).  Last rows show the experimental results when we consider random outliers from all disease categories together.}\n  \\label{fig:alpha-beta-isic}\n\\end{figure*}\n\n\n\\section{Extended Results}\nWe show the details of the true positive rate (TPR) and false positive rate (FPR) for the MNIST, Caltech-256, Coil-100, Extended Yale-B, UCSD Pedestrian, and ISIC 2018 Challenge (Task 3) datasets experiments in Tables~\\ref{tab:tpr-fpr}, \\ref{tab:tpr-fpr2}, \\ref{tab:tpr-fpr3}, \\ref{tab:tpr-fpr4}, \\ref{tab:tpr-fpr5}, and~\\ref{tab:tpr-fpr6}, respectively. \n\n\\begin{table}[t]\n\\centering\n\\caption{Results of different percentage ($\\%$) of outliers on the MNIST dataset for the proposed methods: (a)~{$\\text{OG}_e$}, and (b)~{$\\text{OG}_\\ell$}. The top row show the zero to nine digit inlier categories an d average values among all ten classes of the digits.}\n\\label{tab:tpr-fpr}\n\\sisetup{table-format=1.3,}%\n\\setlength\\tabcolsep{2pt} %\n\\footnotesize%\n\\begin{subtable}[t]{\\linewidth}%\n\\caption{}%\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSSSSSS}\n    \\toprule\n    & $\\%$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 10 &  0.98444\t& 0.98381 &\t0.98424\t& 0.98279 &\t0.98537\t& 0.98182 &\t0.98254 & 0.98151 &\t0.98631 & 0.98310 &\t0.98359 \\\\\n    {FPR} &  & 0.00000\t& 0.00000 &\t0.00000\t& 0.00000 &\t0.00000\t& 0.00000 &\t0.00000\t& 0.00102 &\t0.00000 & 0.00423 & 0.00052 \\\\\n    \\midrule\n    {TPR} & 20 & 0.98588 & 0.98242 & 0.98514 & 0.98189 & 0.98601 & 0.97945 & 0.98551 & 0.98157 & 0.98581 & 0.98341 & 0.98370 \\\\\n    {FPR} &  & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00102 & 0.00000 & 0.00423 & 0.00052 \\\\\n    \n    \\midrule\n    {TPR} & 30 & 0.98453 & 0.98420 & 0.98370 & 0.98316 & 0.98444 & 0.98198 & 0.98405 & 0.98107 & 0.98611 & 0.98194 & 0.98351 \\\\ \n    {FPR} &  & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00102 & 0.00000 & 0.00423 & 0.00052 \\\\\n    \\midrule\n    {TPR} & 40 & 0.98321 & 0.98325 & 0.98360 & 0.98255 & 0.98388 & 0.98158 & 0.98417 & 0.98035 & 0.98646 & 0.98302 & 0.98320 \\\\\n    {FPR} &  & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00102 & 0.00000 & 0.00423 & 0.00052 \\\\\n    \\midrule\n    {TPR} & 50 & 0.98351 & 0.98382 & 0.98366 & 0.98211 & 0.98410 & 0.98122 & 0.98345 & 0.98036 & 0.98593 & 0.98377 & 0.98319 \\\\\n    {FPR} &  & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00102 & 0.00000 & 0.00423 & 0.00052 \\\\ \n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\begin{subtable}[t]{\\linewidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSSSSSS}\n    \\toprule\n    & $\\%$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 10 & 0.98567&0.98422&0.98402&0.98559&0.98557&0.98369&0.98514&0.98131&0.98611&0.98320&0.98445 \\\\\n    {FPR} &  & 0.00196&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00206&0.00040 \\\\\n    \\midrule\n    {TPR} & 20 & 0.98502&0.98280&0.98384&0.98282&0.98433&0.98257&0.98300&0.98011&0.98668&0.98330&0.98344 \\\\\n    {FPR} &  & 0.00196&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00206&0.00040 \\\\\n    \\midrule\n    {TPR} & 30 & 0.98377&0.98372&0.98451&0.98316&0.98337&0.98164&0.98364&0.98093&0.98523&0.98234&0.98323 \\\\\t\t\n    {FPR} &  & 0.00196&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00206&0.00040 \\\\\n    \\midrule\n    {TPR} & 40 & 0.98268&0.98337&0.98399&0.98258&0.98396&0.98206&0.98333&0.98082&0.98462&0.98224&0.98296 \\\\\n    {FPR} &  & 0.00196&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00206&0.00040 \\\\\n    \\midrule\n    {TPR} & 50 & 0.98451&0.98306&0.98390&0.98267&0.98406&0.98137&0.98426&0.98132&0.98571&0.98144&0.98322 \\\\\n    {FPR} &  & 0.00196&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00000&0.00206&0.00040 \\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\end{table}\n\n\\begin{table}[t]\n\\centering\n\\caption{Results of $50\\%$ randomly chosen outliers from category 257-clutter of the Caltech-256 dataset for the proposed methods: (a)~{$\\text{OG}_e$}, and (b)~{$\\text{OG}_\\ell$}.  The top row shows the 2, 4, 6, 8, and 10 inlier categories and average values among all five classes of the objects.}\n\\label{tab:tpr-fpr2}\n\\sisetup{table-format=1.3,}\n\\setlength\\tabcolsep{2pt} %\n\\footnotesize\n\\begin{subtable}[t]{0.3\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSS}\n    \\toprule\n    & $\\%$ & 2 & 4 & 6 & 8 & 10 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 50 &  0.976 & 0.894 & 0.901 & 0.899 & 0.971 & 0.928 \\\\\n    {FPR} &  & 0.000 & 0.092 & 0.084 & 0.086 & 0.000 & 0.052 \\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\begin{subtable}[t]{0.3\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSS}\n    \\toprule\n    & $\\%$ & 2 & 4 & 6 & 8 & 10 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 50 &  0.979 & 0.894 & 0.896 & 0.898 & 0.971 & 0.927 \\\\\n    {FPR} &  & 0.000 & 0.087 & 0.084 & 0.087 & 0.094 & 0.070\\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\end{table}\n\n\\begin{table}[t]\n\\centering\n\\caption{Results of $50\\%$, $25\\%$, and $15\\%$ outliers of the Coil-100 dataset for the proposed methods: (a)~{$\\text{OG}_e$}, and (b)~{$\\text{OG}_\\ell$}. The top row shows the 2, 4, 6, 8, and 10 inlier categories and average values among all five classes of the objects.}\n\\label{tab:tpr-fpr3}\n\\sisetup{table-format=1.3,}\n\\setlength\\tabcolsep{2pt} %\n\\footnotesize\n\\begin{subtable}[t]{0.3\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSS}\n    \\toprule\n    & $\\%$ & 2 & 4 & 6 & 8 & 10 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 50 & 0.933 & 0.928 & 0.934 & 0.922 & 0.949 & 0.933\\\\\n    {FPR} &  & 0.069 & 0.075 & 0.068 & 0.080 & 0.040 & 0.066 \\\\\n    \\midrule\n    {TPR} & 25 & 0.959 & 0.972 & 0.955 & 0.940 & 0.964 & 0.958\\\\\n    {FPR} &  & 0.041 & 0.021 & 0.040 & 0.059 & 0.029 & 0.038 \\\\\n    \\midrule\n    {TPR} & 15 & 0.877 & 0.860 & 0.870 & 0.884 & 0.856 & 0.869\\\\\n    {FPR} &  & 0.118 & 0.126 & 0.109 & 0.114 & 0.1392 & 0.121\\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\begin{subtable}[t]{0.3\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSS}\n    \\toprule\n    & $\\%$ & 2 & 4 & 6 & 8 & 10 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 50 & 0.903 & 0.893 & 0.888 & 0.892 & 0.924 & 0.900\\\\\n    {FPR} &  & 0.086 & 0.093 & 0.102 & 0.090 & 0.074 & 0.089\\\\\n    \\midrule\n    {TPR} & 25 & 0.900 & 0.903 & 0.968 & 0.890 & 0.924 & 0.917\\\\\n    {FPR} &  & 0.069 & 0.072 & 0.039 & 0.092 & 0.072 & 0.068\\\\\n    \\midrule\n    {TPR} & 15 & 0.837 & 0.850 & 0.917 & 0.876 & 0.860 & 0.868\\\\\n    {FPR} &  & 0.154 & 0.102 & 0.073 & 0.118 & 0.129 & 0.115 \\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\end{table}\n\n\\begin{table}[t]\n\\centering\n\\caption{Results of $35\\%$ and $15\\%$ outliers of the Extended Yale B dataset for the proposed methods: (a)~{$\\text{OG}_e$}, and (b)~{$\\text{OG}_\\ell$}. The top row shows the 2, 4, 6, 8, and 10 inlier categories and average values among all five classes of the objects.}\n\\label{tab:tpr-fpr4}\n\\sisetup{table-format=1.3,}\n\\setlength\\tabcolsep{2pt} %\n\\footnotesize\n\\begin{subtable}[t]{0.3\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSS}\n    \\toprule\n    & $\\%$ & 2 & 4 & 6 & 8 & 10 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 35 &  0.941 & 0.870 & 0.844 & 0.921 & 0.913 & 0.897\\\\\n    {FPR} &  & 0.049 & 0.075 & 0.083 & 0.076 & 0.074 & 0.061 \\\\\n    \\midrule\n    {TPR} & 15 &  0.882 & 0.860 & 0.862 & 0.910 & 0.876 & 0.869 \\\\\n    {FPR} &  &  0.048 & 0.079 & 0.088 & 0.059 & 0.083 & 0.070 \\\\    \n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\begin{subtable}[t]{0.3\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{llSSSSSSS}\n    \\toprule\n    & $\\%$ & 2 & 4 & 6 & 8 & 10 & {Avg.} \\\\\n    \\midrule\n    {TPR} & 35 &  0.932 & 0.852 & 0.837 & 0.901 & 0.923 & 0.889 \\\\\n    {FPR} &  &  0.037 & 0.092 & 0.088 & 0.081 & 0.074 & 0.067 \\\\\n    \\midrule\n    {TPR} & 15 &   0.899 & 0.907 & 0.872 & 0.908 & 0.891 & 0.882 \\\\\n    {FPR} &  & 0.053 & 0.049 & 0.052 & 0.069 & 0.068 & 0.062 \\\\    \n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\end{table}\n\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Results of randomly chosen outliers of the UCSD Pedestrian dataset (on both sets Ped1 and Ped2) for the proposed methods: (a)~{$\\text{OG}_e$}, and (b)~{$\\text{OG}_\\ell$}.  The top row shows the 1, 3, 5, 7, and 9 inlier categories and average values among all five classes of the scenes.}\n\\label{tab:tpr-fpr5}\n\\sisetup{table-format=1.3,}\n\\setlength\\tabcolsep{2pt} %\n\\footnotesize\n\\begin{subtable}[t]{0.35\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{lllSSSSSSS}\n    \\toprule\n    & Subset & & & 1 & 3 & 5 & 7 & 9 & {Avg.}\\\\\n    \\midrule\n    & Ped-1 & {TPR} & & 0.9358 & 0.9463 & 0.9512 & 0.9390 & 0.9721 & 0.9488\\\\\n    & & {FPR} &  & 0.0186 & 0.0145 & 0.0109 & 0.0498 & 0.0059 & 0.0199\\\\\n    \\midrule\n    & Ped-2 & {TPR} & & 0.9693 & 0.9631 & 0.9345 & 0.8859 & 0.9741 & 0.9453\\\\\n    & & {FPR} &  & 0.0063 & 0.0174 & 0.0063 & 0.0971 & 0.0154 & 0.0285\\\\\n    \\bottomrule\n    \\end{tabular}%\n}\n\\end{subtable}\n\\begin{subtable}[t]{0.35\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{lllSSSSSSS}\n    \\toprule\n    & Scene & & & 1 & 3 & 5 & 7 & 9 & {Avg.}\\\\\n    \\midrule\n    & Ped-1 & {TPR} & & 0.9431 & 0.9275 & 0.9470 & 0.9161 & 0.9384 & 0.9344\\\\\n    & & {FPR} &  & 0.0220 & 0.0208 & 0.0491 & 0.0126 & 0.0048 & 0.0218\\\\\n    \\midrule\n    & Ped-2 & {TPR} & & 0.9610 & 0.9851 & 0.9619 & 0.9878 & 0.9228 & 0.9637\\\\\n    & & {FPR} &  & 0.0026 & 0.0491 & 0.0067 & 0.0065 & 0.0018 & 0.0133\\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\end{table}\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Results of randomly chosen outliers of the ISIC 2018 Challenge (Task~3) dataset for the proposed methods: (a)~{$\\text{OG}_e$}, and (b)~{$\\text{OG}_\\ell$}.  The top row shows the disease categories for each outlier category, and average values among all.}\n\\label{tab:tpr-fpr6}\n\\sisetup{table-format=1.3,}\n\\setlength\\tabcolsep{2pt} %\n\\footnotesize\n\\begin{subtable}[t]{0.35\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{lllSSSSSSSS}\n    \\toprule\n    & & & {AKIEC} & {BCC} & {BKL} & {DF} & {MEL} & {VASC} & {All} & {Avg.}\\\\\n    \\midrule\n    & & {TPR} & 0.4508 & 0.4146 & 0.4590 & 0.4824 & 0.4811 & 0.43778 & 0.41510 & 0.4486\\\\\n    & & {FPR} & 0.0074 & 0.0018 & 0.0170 & 0.0204 & 0.0067 & 0.0080 & 0.0240 & 0.0091\\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\begin{subtable}[t]{0.35\\textwidth}\n\\caption{}\n\\resizebox{\\columnwidth}{!}{%\n  \\begin{tabular}{lllSSSSSSSS}\n    \\toprule\n    & & & {AKIEC} & {BCC} & {BKL} & {DF} & {MEL} & {VASC} & {All} & {Avg.}\\\\\n    \\midrule\n    & & {TPR} & 0.6357 & 0.6757 & 0.8789 & 0.8451 & 0.6661 & 0.7028 & 0.7445 & 0.7355\\\\\n    & & {FPR} & 0.0125 & 0.0201 & 0.0105 & 0.0173 & 0.0157 & 0.0181 & 0.0201 & 0.0164\\\\\n    \\bottomrule\n  \\end{tabular}%\n}\n\\end{subtable}\n\\end{table}\n\n\n\\section{Implementation Details}\n\nOur proposed framework comprises three main modules: deterministic and variational autoencoders (as shown in Fig.~\\ref{fig:encoding}), and the classifier. \n\nFor the feature encoding, we use denoising autoencoders~\\cite{Vincent2008}. There are four hyperparameters that we used for each denoising autoencoder: number of layers, neurons per layer, type of activation function, and type of reconstruction loss. We evaluated an amount of layers among $1$, $3$, $5$, and $7$. For the number of neurons, we used three settings `same,' `double,' and `half' number of neurons of the input per layer.  The decoder is a mirror of the encoder in terms of layer structure. We evaluated four types of activation functions: ReLU, sigmoid, tanh, and linear. And for the reconstruction loss function~(\\ref{eq:loss-f}), we used mean squared error (MSE) or binary cross entropy.  If the input values are in $[0, 1]$, then we use cross entropy, otherwise we use MSE\\@.\n\nFor the fusion, we use a VAE with a Gaussian distribution. There are three hyperparameters that we tuned for our VAE: number of layers, neurons per layer, and dimensionality of the Gaussian. Similarly to the AEs, we used between $1$ and $2$ layers, with `same' and `double' amount of neurons of the input per layer. And the sizes of the Gaussians vary from $2$ to $6$ (with step one)---we selected small dimensions due to resource constraints for executing the experiments.\n\nThere are three hyper-parameters that we tuned for our SVMs: kernel type, kernel coefficient ($\\gamma$), and regularization. We evaluated four types of kernels: linear, polynomial, RBF, and sigmoid.  For the kernel coefficient, we used values $1$, $\\SI{e-1}{}$, $\\SI{e-2}{}$, $\\SI{e-3}{}$, $\\SI{e-4}{}$, and $\\SI{e-5}{}$.  And for the regularization parameter ($C$) we used $\\SI{e-3}{}$, $\\SI{e-2}{}$, $\\SI{e-1}{}$, $1$, and $10$.\n\n\n\\section{Hyperparameters}\n\n\\subsection{Features}\n\nWe show the hyperparameters we used for extracting the features, HOG~\\cite{Dalal2005} and LBP~\\cite{Ahonen2006}, in Tables~\\ref{tab:HOGhyperparameters} and~\\ref{tab:LBPhyperparameters}.\n\n\\subsection{Autoencoders (AEs), Variational Autoencoders (VAEs) and SVM}\n\nWe show the best set of hyperparameters found on experimental results for MNIST, Caltech-256, Coil-100, Extended Yale-B, UCSD Pedestrian, and ISIC 2018 Challenge (Task~3) on Tables~\\ref{tab:MNISThyperparameters}, \\ref{tab:Caltechhyperparameters}, \\ref{tab:Coilhyperparameters}, \\ref{tab:YaleBhyperparameters}, \\ref{tab:UCSDhyperparameters}, and~\\ref{tab:Diseasehyperparameters}, respectively.  The corresponding original results (on the paper) are in Tables~\\ref{tab:mnist}, \\ref{tab:caltech}, \\ref{tab:coil}, \\ref{tab:yale-b}, \\ref{tab:UCSD}, and~\\ref{tab:ISIC2018}.\n\n\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{HOG's hyperparameters. The best cased used in our proposed experiment is typeset in bold.  Abbreviations:  cells per block (CpB), orientations (O).} \n  \\label{tab:HOGhyperparameters}\n  \\sisetup{table-format=1.3}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{lllll}\n    \\toprule\n    {CpB} & {O} & \\multicolumn{3}{c}{Pixels per Cell} \t\\\\\n    \\midrule\n    & & {(12 , 12)} & {(14 , 14)} & {(16 , 16)} \\\\\n    \\midrule\n    {(1 , 1)} & {9} & {(1,1) - 9 - (12,12)} & \\textbf{{(1,1) - 9 - (14,14)}} & {(1,1) - 9 - (16,16)} \\\\\n    & {12} & {(1,1) - 12 - (12-12)} & {(1,1) - 12 - (14,14)} & {(1,1) - 12 - (16,16)} \\\\\n    & {15} & {(1,1) - 15 - (12,12)} & {(1,1) - 15 - (14,14)} & {(1,1) - 15 - (16,16)} \\\\\n    \\midrule\n    {(2 , 2)} & {9} & {(2,2) - 9 - (12,12)} & {(2,2) - 9 - (14,14)} & {(2,2) - 9 - (16,16)} \\\\\n    & {12} & {(2,2) - 12 - (12,12)} & {(2,2) - 12 - (14,14)} & {(2,2) - 12 - (16,16)} \\\\\n    & {15} & {(2,2) - 15 - (12,12)} & {(2,2) - 15 - (14,14)} & {(2,2) - 15 - (16,16)} \\\\\n    \\midrule\n    {(4 , 4)} & {9} & {(4,4) - 9 - (12,12)} & {(4,4) - 9 - (14,14)} & {(4,4) - 9 - (16,16)} \\\\\n    & {12} & {(4,4) - 12 - (12,12)} & {(4,4) - 12 - (14,14)} & {(4,4) - 12 - (16,16)} \\\\\n    & {15} & {(4,4) - 15 - (12,12)} & {(4,4) - 15 - (14,14)} & {(4,4) - 15 - (16,16)} \\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\caption{LBP's hyperparameters. The best result is typeset in bold.  Abbreviations: number of patterns (NP).}\n  \\label{tab:LBPhyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{lllll}\n    \\toprule\n    {Method} & {\\# NP} & \\multicolumn{3}{c}{Radius of circle} \\\\\n    \\midrule\n    & & {8} & {16} & {24}\\\\\n    \\midrule\t\n    Default\t& 4 & {Default-4-8} & {Default-4-16} & {Default-4-24} \\\\\n    & 6 & {Default-6-8} & {Default-6-16} & {Default-6-24} \\\\\n    & 8 & {Default-8-8} & {Default-8-16} & {Default-8-24} \\\\\n    \\midrule\n    ROR\t& 4 & {ROR-4-8} & {ROR-4-16} & {ROR-4-24} \\\\\n    & 6 & {ROR-6-8} & {ROR-6-16} & {ROR-6-24} \\\\\n    & 8 & {ROR-8-8} & {ROR-8-16} & {ROR-8-24} \\\\\n    \\midrule\n    Uniform\t& 4 & \\textbf{{Uniform-4-8}} & {Uniform-4-16} & {Uniform-4-24} \\\\ \n    & 6 & {Uniform-6-8} & {Uniform-6-16} & {Uniform-6-24} \\\\\n    & 8 & {Uniform-8-8} & {Uniform-8-16} & {Uniform-8-24} \\\\\n    \\midrule\n    VAR\t& 4 & {VAR-4-8} & {VAR-4-16} & {VAR-4-24} \\\\ & 6 & {VAR-6-8} & {VAR-6-16} & {VAR-6-24} \\\\\n    & 8 & {VAR-8-8} & {VAR-8-16} & {VAR-8-24} \\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\t\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Best set of hyperparameters for MNIST dataset in proposed experiments. Abbreviations: Amount of Layers (AL), Number of neurons per layer (NN), Activation Function (AF), Latent Dimension (LD), Kernel coefficient ($\\gamma$), and regularization parameter ($C$).}\n  \\label{tab:MNISThyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{llllllllll}\n    \\toprule\n    {Method} & \\multicolumn{3}{c}{AutoEncoder} & \\multicolumn{3}{c}{Variational AutoEncoder} & \\multicolumn{3}{c}{SVM}\\\\\n    \\midrule\n    & {AL} & {NN} & {AF} & {LD} & {Alpha} & {Beta} & {Kernel} & {$\\gamma$} & {$C$}\\\\\n    \\midrule\t\n    {$\\text{OG}_e$}\t& {1}& {Same}& {ReLU} & {2} & {3.0} & {4.5} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {1}& {Same}& {ReLU} & {2} & {3.0} & {5.0} & {rbf} & {1} & {0.1}\\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Best set of hyperparameters for Caltech-256 dataset in proposed experiments.  Abbreviations: Amount of Layers (AL), Number of neurons per layer (NN), Activation Function (AF), Latent Dimension (LD), Kernel coefficient ($\\gamma$) and regularization parameter ($C$).}\n  \\label{tab:Caltechhyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{llllllllll}\n    \\toprule\n    {Method} & \\multicolumn{3}{c}{AutoEncoder} & \\multicolumn{3}{c}{Variational AutoEncoder} & \\multicolumn{3}{c}{SVM}\\\\\n    \\midrule\n    & {AL} & {NN} & {AF} & {LD} & {Alpha} & {Beta} & {Kernel} & {$\\gamma$} & {$C$}\\\\\n    \\midrule\t\n    {$\\text{OG}_e$}\t& {5}& {Half}& {ReLU} & {2} & {1.75} & {3.0} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {5}& {Half}& {ReLU} & {2} & {1.25} & {2.0} & {rbf} & {1} & {0.1}\\\\\n\n    \\midrule\t\n    {$\\text{OG}_e$}\t& {5}& {Same}& {ReLU} & {4} & {1.5} & {3.0} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {7}& {Same}& {ReLU} & {2} & {2.0} & {3.5} & {rbf} & {1} & {0.1}\\\\\n    \n    \\midrule\t\n    {$\\text{OG}_e$}\t& {5}& {Same}& {ReLU} & {2} & {1.0} & {2.0} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {7}& {Same}& {ReLU} & {4} & {1.5} & {4.5} & {rbf} & {1} & {0.1}\\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Best set of hyperparameters for Coil-100 dataset in proposed experiments. Abbreviations: Amount of Layers(AL), Number of neurons per layer (NN), Activation Function (AF), Latent Dimension (LD), Kernel coefficient ($\\gamma$) and regularization parameter ($C$).}\n  \\label{tab:Coilhyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{llllllllll}\n    \\toprule\n    {Method} & \\multicolumn{3}{c}{AutoEncoder} & \\multicolumn{3}{c}{Variational AutoEncoder} & \\multicolumn{3}{c}{SVM}\\\\\n    \\midrule\n   & {AL} & {NN} & {AF} & {LD} & {Alpha} & {Beta} & {Kernel} & {$\\gamma$} & {$C$}\\\\    \\midrule\t\n    {$\\text{OG}_e$}\t& {5}& {Half}& {ReLU} & {2} & {2.0} & {3.5} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {1}& {Same}& {ReLU} & {2} & {1.75} & {1.0} & {rbf} & {1} & {0.1}\\\\\n\n    \\midrule\t\n    {$\\text{OG}_e$}\t& {1}& {Double}& {ReLU} & {3} & {2.0} & {4.0} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {3}& {Same}& {ReLU} & {2} & {2.0} & {2.0} & {rbf} & {1} & {0.1}\\\\\n    \n    \\midrule\t\n    {$\\text{OG}_e$}\t& {1}& {Double}& {ReLU} & {3} & {1.5} & {2.5} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {3}& {Half}& {ReLU} & {2} & {1.5} & {5.0} & {rbf} & {1} & {0.1}\\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Best set of hyperparameters for Extended Yale B dataset in proposed experiments. Abbreviations: Amount of Layers(AL), Number of neurons per layer(NN), Activation Function (AF), Latent Dimension (LD), Kernel coefficient ($\\gamma$) and regularization parameter ($C$).}\n  \\label{tab:YaleBhyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{llllllllll}\n    \\toprule\n    {Method} & \\multicolumn{3}{c}{AutoEncoder} & \\multicolumn{3}{c}{Variational AutoEncoder} & \\multicolumn{3}{c}{SVM}\\\\\n    \\midrule\n    & {AL} & {NN} & {AF} & {LD} & {Alpha} & {Beta} & {Kernel} & {$\\gamma$} & {$C$}\\\\\n    \\midrule\t\n    {$\\text{OG}_e$}\t& {3}& {Half}& {ReLU} & {3} & {1.75} & {2.5} & {rbf} & {1} & {25}\\\\\n    {$\\text{OG}_\\ell$} & {1}& {Half}& {ReLU} & {4} & {2.0} & {1.0} & {rbf} & {1} & {25}\\\\\n\n    \\midrule\t\n    {$\\text{OG}_e$}\t& {5}& {Half}& {ReLU} & {3} & {2.0} & {1.5} & {rbf} & {1} & {0.1}\\\\\n    {$\\text{OG}_\\ell$} & {5}& {Half}& {ReLU} & {3} & {1.5} & {1.0} & {rbf} & {1} & {0.1}\\\\\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Best set of hyperparameters for UCSD Pedestrian dataset in proposed experiments. Abbreviations: Amount of Layers(AL), Number of neurons per layer (NN), Activation Function (AF), Latent Dimension (LD), Kernel coefficient ($\\gamma$) and regularization parameter ($C$).}\n  \\label{tab:UCSDhyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{llllllllll}\n    \\toprule\n    {Method} & \\multicolumn{3}{c}{AutoEncoder} & \\multicolumn{3}{c}{Variational AutoEncoder} & \\multicolumn{3}{c}{SVM}\\\\\n\\midrule\t\n& {AL} & {NN} & {AF} & {LD} & {Alpha} & {Beta} & {Kernel} & {$\\gamma$} & {$C$}\\\\\n\\midrule\t\n{$\\text{OG}_e$}\t& {5}& {Half} & {1} & {2} & {2} & {2.5} & {rbf} & {1} & {10}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half} & {1} & {2} & {2} & {2.5} & {rbf} & {1} & {10}\\\\\n\\midrule\n{$\\text{OG}_e$}\t& {1}& {Half} & {1} & {3} & {1.25} & {2.0} & {rbf} & {1} & {10}\\\\\n{$\\text{OG}_\\ell$}\t& {5} & {Half}& {1} & {3} & {1.75} & {4.0} & {rbf} & {1} & {10}\\\\    \n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\t\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Best set of hyperparameters for ISIC 2018 Challenge (Task~3) dataset in proposed experiments. Abbreviations: Amount of Layers (AL), Number of neurons per layer (NN), Activation Function (AF), Latent Dimension (LD), Kernel coefficient ($\\gamma$) and regularization parameter ($C$).}\n  \\label{tab:Diseasehyperparameters}\n  \\setlength\\tabcolsep{3pt} %\n  \\footnotesize\n  \\begin{tabular}{llllllllll}\n    \\toprule\n    {Method} & \\multicolumn{3}{c}{AutoEncoder} & \\multicolumn{3}{c}{Variational AutoEncoder} & \\multicolumn{3}{c}{SVM}\\\\\n\\midrule\n& {AL} & {NN} & {AF} & {LD} & {Alpha} & {Beta} & {Kernel} & {$\\gamma$} & {$C$}\\\\\n\\midrule\t\n{$\\text{OG}_e$}\t& {5}& {Half}& {1} & {12} & {1.25} & {1.0} & {rbf} & {1} & {1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {16} & {1.5} & {1.0} & {rbf} & {1} & {0.1}\\\\\n\n\\midrule\n{$\\text{OG}_e$}\t& {5}& {Half}& {1} & {11} & {1.5} & {1.0} & {rbf} & {1} & {1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {14} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n\n\\midrule\n{$\\text{OG}_e$}\t& {3}& {Half}& {1} & {10} & {1.0} & {1.0} & {rbf} & {1} & {0.1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {15} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n\n\\midrule\n{$\\text{OG}_e$}\t& {5}& {Half}& {1} & {16} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {14} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n\n\\midrule\n{$\\text{OG}_e$}\t& {5}& {Half}& {1} & {17} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {17} & {1.0} & {1.5} & {rbf} & {1} & {1}\\\\\n\n\\midrule\n{$\\text{OG}_e$}\t& {3}& {Half}& {1} & {12} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {18} & {1.5} & {1.0} & {rbf} & {0.1} & {0.1}\\\\\n\n\\midrule\n{$\\text{OG}_e$}\t& {5}& {Half}& {1} & {17} & {1.0} & {1.0} & {rbf} & {1} & {0.1}\\\\\n{$\\text{OG}_\\ell$}\t& {3}& {Half}& {1} & {16} & {1.0} & {1.0} & {rbf} & {1} & {1}\\\\\n\n    \\bottomrule\n  \\end{tabular}%\n\\end{table}\t\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\nThe method to count the number of unlabeled graphs, unlabeled connected graphs and unlabeled biconnected graphs\non $n$ vertices exactly based on the Redfield-P\\'{o}lya\nenumeration theorem is well-known and proves to be efficient \\cite{Redfield1927,Polya1937,Robinson1970,HararyPalmer1973}.\nHowever, there was no known efficient method to count the number\nof degree sequences of these graphs exactly. We present a solution to these problems in this paper. Only simple\n(i.e. with no loops or multiple edges) undirected graphs and their degree sequences are considered.\n\nA degree sequence of length $n$ is a sequence of nonnegative integers $d_1, d_2, \\ldots, d_n$ (in which order is\nirrelevant so that they are usually written in non-increasing order without loss of generality)\nsuch that there is a simple undirected graph $G$ of $n$ vertices\nwith the vertex degrees $d_1, d_2, \\ldots, d_n$.\nA zero-free degree sequence is a degree sequence such that each term of the sequence is positive, or equivalently,\na degree sequence of a simple undirected graph with no isolated vertices. For convenience, from now on\nby degree sequences we mean \\textit{zero-free} degree sequences unless otherwise noted. Let $D(n)$ denote the set\nof degree sequences of length $n$. A positive integer sequence $a_1, a_2, \\ldots, a_n$ (in which order is also irrelevant)\nwith sum $N=\\sum_{i=1}^{n}a_i$ is called a graphical partition of $N$ with $n$\nparts, or a graphic sequence, if it is the degree sequence of a simple undirected graph.\nNote that $N$ must be even to have a graphical partition, since the sum of the vertex degrees of a simple\nundirected graph is twice the number of its edges and therefore must be even. Sierksma and Hoogeveen \\cite{Sierksma1991}\nshowed seven criteria to be equivalent to characterize graphic integer sequences, of which the\nErd{\\H{o}}s-Gallai criterion \\cite{ErdosCallai1960} is probably the most widely known.\n\nThe problems of counting or enumerating the set $G(N)$ of graphical partitions of a fixed positive even integer $N$ and the set\n$D(n)$ for a fixed length $n$ have been extensively studied, for example \\cite{Erdos1993,Ruskey1994,BarnesSavage1995,\nBarnesSavage1997,Pittel1999,Kohnert2004,Burns2007,Ivanyi2013,Pittel2016}. Pittel \\cite{Pittel1999} affirmatively resolved\na problem originally posed in 1982 by Herbert S. Wilf whether the fraction of\ngraphical partitions in the set $P(N)$ of all partitions of a positive integer $N$ goes to 0 as $N$\napproaches infinity. Erd{\\H{o}}s and Richmond \\cite{Erdos1993} showed that this fraction as a function of $N$\ncannot tend to 0 faster than $\\pi\/\\sqrt{6N}$. Recently Pittel \\cite{Pittel2016} has obtained an upper bound\nof $\\exp({-0.11\\log N\/\\log \\log N})$ for $|G(N)|\/|P(N)|$\nwith $N$ sufficiently large using a local limit theorem. Comparison between the bound and actual values\nof $|G(N)|\/|P(N)|$ up to $N=910$ suggests that the bound does not appear to be tight. Up till now, the\nasymptotic order of $|G(N)|$ is still unknown, though the order of $|P(N)|$ has been known to be $|P(N)|\\sim \\frac{\\exp(\\pi\\sqrt{2N\/3})}{4\\sqrt{3}N}$\nsince 1918 by Hardy and Ramanujan \\cite{Hardy1918}.\n\nFor the asymptotic growth rate of $|D(n)|$, good upper and lower bounds are known. Burns \\cite{Burns2007} showed there\nare positive constants $c_1$ and $c_2$ such that $4^n\/(c_1 n) \\le |D(n)| \\le 4^n\/((\\log n)^{c_2} \\sqrt{n})$ for all sufficiently\nlarge $n$. Ruskey \\textit{et al.} \\cite{Ruskey1994} presented a fast algorithm to enumerate degree sequences of length $n$ using\nthe ``Reverse Search'' approach originated by Avis and Fukuda \\cite{Avis1996}. The algorithm appears to run in constant\namortized time (CAT), though no proof has been given yet to the knowledge of the author. No matter whether\nit is truly CAT or not, it is clear that its time complexity is at least $\\Omega(4^n\/n)$ based on the lower bound of $|D(n)|$ as proved\nin \\cite{Burns2007}. Exact values of $|D(n)|$ up to $n=31$ were obtained in\nIv\\'{a}nyi \\textit{et al.} \\cite{Ivanyi2013} by testing every sequence\nin the set $E(n)$ of even bounded sequences of length $n$ (that is, a non-increasing positive integer sequence of length $n$\nwith even sum and largest term less than $n$) for its graphicness. A sequence in $E(n)$\ncan be tested whether it is graphic in linear time and $|E(n)|$ is asymptotically\n$4^n\/(8\\sqrt{\\pi n})$ \\cite{Ivanyi2013}. Therefore, this filtering method of calculating $|D(n)|$ has an asymptotic time complexity of $\\Theta(\\sqrt{n} 4^n)$\nand space complexity of $O(n)$. By our own testing, the method in \\cite{Ruskey1994} appears to\nbe faster than the filtering method in \\cite{Ivanyi2013} if both are restricted to run in single-threaded mode,\nwhich would make sense because the``Reverse Search'' based approach would have a time complexity of $O(4^n\/((\\log n)^{c_2} \\sqrt{n}))$\nunder the assumption that it runs in CAT.\n\nThe OEIS entry \\textcolor{red}{\\href{https:\/\/oeis.org\/A095268}{A095268}}\nrecords the values of $|D(n)|$.\nIn 2006 Gordon F. Royle posed the question whether $|D(n)|\/|D(n-1)|$ tend to the limit 4 when $n$ approaches infinity\n\\cite{OEISA095268} as there seems to be no apparent reason why $|D(n)|\/|D(n-1)|$ would tend to the same limit as $|E(n)|\/|E(n-1)|$.\nIn this paper we will present two fast novel algorithms to calculate $|D(n)|$ based on several recurrences,\ncircumventing the need to generate all degree sequences with a given length as performed in \\cite{Ruskey1994,Ivanyi2013}.\nWe will show that both of these algorithms run in polynomial time, which appear to be the first such algorithms for this problem and\nare asymptotically faster than previous exponential algorithms shown above, at the expense of larger space requirement.\nThe numerical results of $|D(n)|$ so obtained up to $n=118$ tend to agree with the conjecture posed by Royle.\nSimilar techniques are utilized to design fast polynomial time algorithms to count the number of degree sequences of connected\nand biconnected graphs on $n$ vertices. The only known previous algorithms to count them seem to be first generating\nall the degree sequences in $D(n)$ and then counting a subset satisfying the relevant connected and biconnected conditions, which make them all\nrun in exponential time. A related problem of counting the number of degree sequences with fixed degree sum (instead of fixed length) of connected and biconnected\n\\textit{multigraphs} that might contain multiple edges is considered by R{\\o}dseth \\textit{et al.} \\cite{Rodseth2009}.\n\nWe note that the degree sequences considered in this paper are unordered. For example, the degree sequences $211$, $121$\nand $112$ of the path graph with three vertices are all treated as the same. When regarded as ordered degree sequences, they\nare considered distinct. Ordered partitions are called compositions in the literature \\cite{Andrews1984}. The efficient\ncomputation of the number $f(n)$ of ordered degree sequences of length $n$ allowing zero terms\n(the OEIS entry \\textcolor{red}{\\href{https:\/\/oeis.org\/A005155}{A005155}}) has been solved by\nStanley \\cite{Stanley1991} using the theory of zonotopes. For example, the exact value of $f(200)$ is about\n$9.97\\times 10^{458}$. The asymptotic order of $f(n)$ is shown on its OEIS entry by Vaclav Kotesovec\nto be $\\frac{\\Gamma(3\/4)}{2^{3\/4}\\sqrt{e \\pi}}n^{n-1\/4}(1-\\frac{11\\pi}{24\\sqrt{n}(\\Gamma(3\/4))^2})$, which is also asymptotic\nto $\\frac{\\Gamma(3\/4)}{2^{3\/4}\\sqrt{e \\pi}}n^{n-1\/4}$. By contrast, the tight asymptotic order of $|D(n)|$ is unknown\nand we are unable to resolve it at this time.\n\nThe rest of the paper is organized as follows. In Section \\ref{sec:Algo}, we formalize the necessary definitions,\nderive the relevant formulae and describe the two algorithms to compute $|D(n)|$. \nIn Section \\ref{sec:count_connected} we present algorithms based on similar techniques to compute the number of degree sequences\nof connected graphs on $n$ vertices. In Section \\ref{sec:count_biconnected} we present an algorithm based on similar dynamic\nprogramming techniques and additional combinatorial observations to compute the number of degree sequences\nof biconnected graphs on $n$ vertices. In Section \\ref{sec:Analysis} we analyze the computational complexity of the algorithms.\nIn Section \\ref{sec:experiments} we give some brief computational results that serve as proof of concept.\nIn Section \\ref{sec:conclusion} we conclude and suggest further research directions.\n\n\\section{Description of the Two Algorithms for $|D(n)|$}\n\\label{sec:Algo}\nIn this section we first review the definitions needed to describe the two algorithms to calculate $|D(n)|$.\nWe then present the relevant formulae for $|D(n)|$ and the pseudo-codes of our algorithms.\nFor the reader's convenience, the terminology employed in this paper is summarized in Table \\ref{tbl:definitions}.\n\\begin{table}[!htb\n\t\\centering\n\t\\caption{Terminology used in this paper}\n\t\\begin{tabular}{||c|l||}\n\t\t\\hline\\hline\n\t\tTerm & Meaning\\\\\n\t\t\\hline\\hline\n\t\t$P(N)$ & set of all partitions of an integer $N$\\\\\n\t\t\\hline\n\t\t$P(N,k,l)$ & set of partitions of an integer $N$ into at most $l$ parts\\\\\n\t\t\t\t   & with largest part at most $k$ \\\\\n\t\t\\hline\n\t\t$P(N,k,l,s)$ & subset of $P(N,k,l)$ determined by integer $s$ \\\\\n\t\t\\hline\n\t\t$P^{'}(N,k,l)$ & set of partitions in $P(N,k,l)$ with exactly $l$ parts\\\\\n\t\t\t\t\t\t& and largest part exactly $k$\\\\\n\t\t\\hline\n\t\t$P^{'}(N,k,l,s)$ & set of partitions in $P(N,k,l,s)$ with exactly $l$ parts\\\\\n\t\t\t\t\t\t& and largest part exactly $k$\\\\\n\t\t\\hline\n\t\t$G(N)$ & set of all graphical partitions of an even integer $N$\\\\\n\t\t\\hline\n\t\t$G^{'}(N,k,l)$ & set of graphical partitions in $P^{'}(N,k,l)$ \\\\\n\t\t\\hline\n\t\t$G^{'}(N,l)$ & set of graphical partitions of $N$ with exactly $l$ parts\\\\\n\t\t\\hline\n\t\t$H^{'}(N,l)$ & set of graphical partitions of $N$ with exactly $l$ parts\\\\\n\t\t& and largest part exactly $l-1$\\\\\n\t\t\\hline\n\t\t$L^{'}(N,l)$ & set of graphical partitions of $N$ with exactly $l$ parts\\\\\n\t\t& and largest part less than $l-1$\\\\\n\t\t\\hline\\hline\n\t\t$d(\\pi)$ & size of the Durfee square of the partition $\\pi$ \\\\\n\t\t\\hline\n\t\t$r(\\pi)$ & corank vector of the partition $\\pi$ \\\\\n\t\t\\hline\\hline\n\t\t$D(n)$ & set of (zero-free) degree sequences of length $n$ \\\\\n\t\t\\hline\n\t\t$B(n)$ & subset of $D(n)$ with largest part exactly $n-1$ and\\\\\n\t\t& smallest part exactly $1$\\\\\n\t\t\\hline\n\t\t$C(n)$ & subset of $D(n)$ with smallest part exactly $1$ \\\\\n\t\t\\hline\n\t\t$D_0(n)$ & set of degree sequences of length $n$ allowing zero terms \\\\\n\t\t\\hline\n\t\t$D_b(n)$ & set of degree sequences of biconnected graphs on $n$ vertices \\\\\n\t\t\\hline\n\t\t$D_c(n)$ & set of degree sequences of connected graphs on $n$ vertices \\\\\n\t\t\\hline\n\t\t$D_d(n)$ & subset of $D(n)$ that are forcibly disconnected \\\\\n\t\t\\hline\n\t\t$E(n)$ & set of even bounded sequences of length $n$ \\\\\n\t\t\\hline\n\t\t$f(n)$ & number of ordered degree sequences of length $n$ \\\\\n\t\t& allowing zero terms\\\\\n\t\t\\hline\n\t\t$H(n)$ & subset of $D(n)$ with largest part exactly $n-1$ \\\\\n\t\t\\hline\n\t\t$L(n)$ & subset of $D(n)$ with largest part less than $n-1$ \\\\\n\t\t\\hline\n\t\t$S(n)$ & subset of $D(n)$ with largest part exactly $n-2$ \\\\\n\t\t\\hline\n\t\t$W(n)$ & subset of $D(n)$ with largest part less than $n-1$ and\\\\\n\t\t& smallest part exactly 1 \\\\\n\t\t\\hline\n\t\t$I_e(n)$&$\\{N|n\\le N\\le n(n-1), N \\mbox{ is an even integer}\\}$\\\\\n\t\t\\hline\n\t\t$I'_e(n)$&$\\{N|2(n-1)\\le N\\le n(n-1), N \\mbox{ is an even integer}\\}$\\\\\n\t\t\\hline\n\t\t$I''_e(n)$&\t$\\{N|2n-3\\le N\\le n(n-2), N \\mbox{ is an even integer}\\}$\\\\\n\t\t\\hline\n\t\t$J_e(n)$&$\\{N|n\\le N\\le n(n-2), N \\mbox{ is an even integer}\\}$\\\\\n\t\t\\hline\n\t\t$J'_e(n)$&$\\{N|n\\le N< n(n-1)\/2, N \\mbox{ is an even integer}\\}$\\\\\n\t\t\\hline\\hline\n\t\\end{tabular}\n\t\\label{tbl:definitions}\n\\end{table}\n\n\\subsection{Definitions and the Basic Algorithm}\n\\label{subsec:BasicAlgo}\nThe notion of \\textit{partitions} in number theory is standard and can be found in many textbooks, for example \\cite{Hardy2008,Andrews1984}.\nA partition of an integer is a way of writing the integer as a sum of positive integers in which the order\nof the summands (also called parts) is irrelevant. The parts in a partition are usually written\nin non-increasing order, for example, $8=4+3+1$.\nA partition can be visually represented by its Ferrers diagram as a left-aligned array of dots so that the\nnumber of dots in the $i$-th row of the array is equal to the $i$-th part of the partition. The Durfee square\nis the upper-left largest square that is embedded within a partition's Ferrers diagram. Figure \\ref{fig:part8}\nshows the Ferrers diagram of the example partition $8=4+3+1$. This partition has three parts, the largest part\nis 4, and its Durfee square contains two rows and two columns.\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[width=4cm]{.\/part8.png}\\\\\n\t\\caption{Ferrers diagram of the partition 8=4+3+1. Its Durfee square size is 2.}\n\t\\label{fig:part8}\n\\end{figure}\n\nFollowing the notation of Barnes and Savage \\cite{BarnesSavage1995}, let $P(N,k,l)$ denote the set of partitions\nof an integer $N$ into at most $l$ parts with largest part at most $k$, and for integer $s$, define $P(N,k,l,s)$ by\n\\[ P(N,k,l,s)=\\left\\{ \\begin{array}{ll}\n\\emptyset & \\mbox{if $s < 0$};\\\\\n\\{\\pi \\in P(N,k,l) | s+\\sum_{i=1}^{j}r_i(\\pi)\\ge j, 1 \\le j \\le d(\\pi)\\} & \\mbox{if $s \\ge 0$},\\end{array} \\right.\\]\nwhere $d(\\pi)$ is the size (number of rows) of the Durfee square of the partition $\\pi$ and \n$r(\\pi)=(r_1(\\pi),\\ldots,r_{d(\\pi)}(\\pi))$ is the corank vector of $\\pi$ with $r_i(\\pi)=\\pi_i^{'}-\\pi_i$ and\n$\\pi_i$ and $\\pi_i^{'}$ are the number of dots in the $i$-th row and column of the Ferrers diagram of $\\pi$,\nrespectively, for $1 \\le i \\le d(\\pi)$.\nThe example partition $8=4+3+1$ in Figure \\ref{fig:part8} has the corank vector $(-1,-1)$.\n\nAdditionally, as in \\cite{BarnesSavage1995}, define $P^{'}(N,k,l)$ and $P^{'}(N,k,l,s)$\nto be the subsets of $P(N,k,l)$ and $P(N,k,l,s)$, respectively, consisting of those partitions\nwith \\textit{exactly} $l$ parts and largest part \\textit{exactly} $k$.\nDefine $G^{'}(N,k,l)$ to be the set of graphical partitions in $P^{'}(N,k,l)$. Then we have\nthe following lemma that establishes a relationship between a set of graphical partitions\nand a set of partitions.\n\\begin{lem}\n\\label{lem_graphical_exact}\nFor even integer $N \\ge 0$, $G^{'}(N,k,l)=P^{'}(N,k,l,0)$.\n\\end{lem}\n\\begin{proof}\nFrom the definition of $P^{'}(N,k,l,s)$, we have\n\\[P^{'}(N,k,l,0)=\\{\\pi \\in P^{'}(N,k,l) |\\sum_{i=1}^{j}r_i(\\pi)\\ge j, 1 \\le j \\le d(\\pi)\\}.\\]\nFrom the definition of $G^{'}(N,k,l)$, we have\n\\[ G^{'}(N,k,l)=\\{\\pi \\in P^{'}(N,k,l) |\\pi \\mbox{ is graphical}\\}.\\]\nNow based on the Nash-Williams condition \\cite{Ruch1979,Rousseau1995,Sierksma1991}, a partition $\\pi$\nof an even integer is graphical\nif and only if $\\sum_{i=1}^{j}r_i(\\pi)\\ge j, 1 \\le j \\le d(\\pi)$. So we have $G^{'}(N,k,l)=P^{'}(N,k,l,0)$.\n\n\\end{proof}\n\nBefore presenting the algorithms for $|D(n)|$ we need the following lemma that connects the function $|P^{'}(*,*,*,*)|$\nto its relative $|P(*,*,*,*)|$ and it will be\nused to bridge the gap between the problem of counting graphical partitions and that of counting partitions.\nIts proof is based on the observation that there is a bijection between $P^{'}(N,k,l)$ and $P(N-k-l+1,k-1,l-1)$.\n\\begin{lem}\n\\label{lem_partition_exact}\n\\cite[Lemma 4]{BarnesSavage1995} For $N>0,1\\le k,l \\le N, s\\ge 0$,\n\\[|P^{'}(N,k,l,s)|=|P(N-k-l+1,k-1,l-1,s+l-k-1)|.\\]\n\\end{lem}\n\nThe following theorem materializes a connection between particular variants of partition counting functions and\ngraphical partition counting functions. It is important to our algorithms to calculate $|D(n)|$.\n\\begin{thm}\n\t\\label{thm_graphical_partition}\n\t\tFor even integer $N \\ge 0$, $|G^{'}(N,k,l)|=|P(N-k-l+1,k-1,l-1,l-k-1)|$.\n\\end{thm}\n\\begin{proof}\n\tImmediate from Lemma \\ref{lem_graphical_exact} and Lemma \\ref{lem_partition_exact}.\n\\end{proof}\n\nThere are tons of efficient algorithms to calculate various partition functions in the literature.\nOur algorithms for $|D(n)|$ need an elegant recurrence of Barnes and Savage \\cite{BarnesSavage1995} that makes it easy\nto design an efficient dynamic programming algorithm to calculate all $|P(*,*,*,*)|$ values.\nWe now derive the simple connection between $D(n)$ and\t$G^{'}(N,k,l)$ and then present our basic algorithm\nto calculate $|D(n)|$ that eventually uses those $|P(*,*,*,*)|$ values.\n\nRecall that $D(n)$ denotes the set of degree sequences of length $n$. Since each term in a degree sequence\nof length $n$ is between 1 and $n-1$ inclusive, it follows that each degree sequence in $D(n)$ can be\nseen as a graphical partition of some even integer $N$ with exactly $n$ parts and largest part exactly $k$\nwhere $n\\le N \\le n(n-1)$ and $1\\le k \\le n-1$. Define $G^{'}(N,l)$ to be the set of graphical partitions of $N$\nwith \\textit{exactly} $l$ parts. Also define an index set $I_e(n)=\\{N|n\\le N\\le n(n-1), N \\mbox{ is an even integer}\\}$.\nBased on the above observations we clearly have\n\\begin{equation} \\label{eqn:set_deg_seq_edge}\nD(n)=\\bigcup_{N\\in I_e(n)}G^{'}(N,n)\n\\end{equation}\nand\n\\begin{equation} \\label{eqn:set_deg_seq_edge_largest_part}\nG^{'}(N,n)=\\bigcup_{k=1}^{n-1}G^{'}(N,k,n),\n\\end{equation}\nwhich can be combined into\n\\begin{equation} \\label{eqn:set_deg_seq_edge_largest_part2}\nD(n)=\\bigcup_{N\\in I_e(n)}\\bigcup_{k=1}^{n-1}G^{'}(N,k,n).\n\\end{equation}\nThese are all disjoint unions so that we have\n\\begin{equation} \\label{eqn:deg_seq_edge}\n\t\\begin{aligned}\n|D(n)|=\\sum_{N\\in I_e(n)}|G^{'}(N,n)|,\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{eqn:graphical_partition}\n\t\\begin{aligned}\n|G^{'}(N,n)|=\\sum_{k=1}^{n-1}|G^{'}(N,k,n)|,\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{eqn:deg_seq_combine1}\n\\begin{aligned}\n|D(n)|=\\sum_{N\\in I_e(n)}\\sum_{k=1}^{n-1}|G^{'}(N,k,n)|.\n\\end{aligned}\n\\end{equation}\nSubstituting the formula in Theorem \\ref{thm_graphical_partition} into Equation (\\ref{eqn:graphical_partition})\nand Equation (\\ref{eqn:deg_seq_combine1}) we get\n\\begin{equation} \\label{eqn:graphical_partition_detailed}\n\\begin{aligned}\n|G^{'}(N,n)|=\\sum_{k=1}^{n-1}|P(N-k-n+1,k-1,n-1,n-k-1)|,\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{eqn:deg_seq_combine}\n\t\\begin{aligned}\n\n|D(n)|=\\sum_{N\\in I_e(n)}\\sum_{k=1}^{n-1}|P(N-k-n+1,k-1,n-1,n-k-1)|.\n\t\\end{aligned}\n\\end{equation}\nWith Equation (\\ref{eqn:deg_seq_combine}) it is sufficient to calculate\na finite number of $|P(*,*,*,*)|$ values to compute $|D(n)|$. As mentioned above, all these $|P(*,*,*,*)|$ values\ncan be calculated with a recurrence of Barnes and Savage in the following theorem:\n\n\\begin{thm}\n\\label{thm_partition}\n\\cite[Theorem 1]{BarnesSavage1995} $|P(N,k,l,s)|$ is defined by:\n\\begin{equation*}\n\\begin{aligned}\n&|P(N,k,l,s)|:= \\\\\n& \\quad if((N<0) \\mbox{ }or\\mbox{ } (k<0) \\mbox{ }or\\mbox{ } (l<0) \\mbox{ }or\\mbox{ } (s<0)) \\mbox{ } \\qquad then: 0\\\\\n& \\quad else \\mbox{   } if (N=0)  \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad then: 1 \\\\\n& \\quad else \\mbox{   } if((k=0) \\mbox{ }or\\mbox{ } (l=0)) \\qquad \\qquad \\qquad \\qquad \\qquad then: 0 \\\\\n& \\quad else \\mbox{   } if(k>N) \\mbox{   } \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad then: |P(N,N,l,s)| \\\\\n& \\quad else \\mbox{   } if(l>N) \\mbox{   } \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad then: |P(N,k,N,s)| \\\\\n& \\quad else \\mbox{   } if(s>N) \\mbox{   } \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad then: |P(N,k,l,N)| \\\\\n& \\quad else: \\mbox{   } |P(N,k-1,l,s)|+|P(N,k,l-1,s)|-|P(N,k-1,l-1,s)| \\\\\n& \\quad \\qquad +|P(N-k-l+1,k-1,l-1,s+l-k-1)|\n\\end{aligned}\n\\end{equation*}\n\\end{thm}\n\n\\begin{algorithm}\n\t\\DontPrintSemicolon\n\t\\KwIn{A positive integer $n$}\n\t\\KwOut{$|D(n)|$}\n\t$N \\gets n(n-1)$\\;\n\tAllocate a four dimensional array $P[N-n+1][n-1][n][N]$\\;\n\tFill in the array $P$ using dynamic programming based on Theorem \\ref{thm_partition}\\;\n\t$S \\gets 0$\\;\n\t\\For{$i \\in I_e(n)$ } {\n\t\t\\For{$j \\gets 1$ \\textbf{to} $\\min ({n-1,i-n+1})$ } {\n\t\t\t\t$S \\gets S+P[i-j-n+1][j-1][n-1][n-j-1]$\\;\n\t\t}\n\t}\n\t\\Return{$S$}\\;\n\t\\caption{Basic Algorithm to compute $|D(n)|$. The index set $I_e(n)=\\{N|n\\le N\\le n(n-1), N \\mbox{ is an even integer}\\}$.}\n\t\\label{algo:basic}\n\\end{algorithm}\n\nWith the above definitions and techniques, we present our basic algorithm to calculate $|D(n)|$ as\npseudo-codes in Algorithm \\ref{algo:basic}.\nThe declaration of the array $P$ in line 2 specifies the sizes of the four dimensions. When filling in and accessing\nthe array elements we use the convention that array indices start from 0. Also note that based on the recurrence\nin Theorem \\ref{thm_partition} the $|P(*,*,l,*)|$ values only depend on the $|P(*,*,l-1,*)|$ values, and for\nthe purpose of calculating $|D(n)|$ only the $|P(*,*,n-1,*)|$ values are used. These\nmake it feasible to allocate only size 2 for the third dimension of the array $P$ to save memory. For each fixed $i$ in line 5,\nthe sum accumulated in the inner loop from lines 6 to 7 is the value $|G^{'}(i,n)|$ based on Equation (\\ref{eqn:graphical_partition_detailed})\nand it can also be saved as additional output if desired. The integer sequence $|D(n)|$ is in the OEIS\nentry \\textcolor{red}{\\href{https:\/\/oeis.org\/A095268}{A095268}}.\n\nWe note that Equation (\\ref{eqn:deg_seq_combine}) and Algorithm \\ref{algo:basic} are obtained\nby partitioning the set $D(n)$ into disjoint subsets $G'(N,n), N\\in I_e(n)$ based on degree sums. Alternatively, we\ncan partition the set $D(n)$ into disjoint subsets based on the largest term in each degree sequence.\nThis way effectively only produces a formula that interchanges the two sum symbols in the double sums\nin Equation (\\ref{eqn:deg_seq_combine1}) and Equation (\\ref{eqn:deg_seq_combine}). The change that can be made to Algorithm\n\\ref{algo:basic} based on the equivalent new formula is to simply interchange the two for loop lines 5 and 6, making the sums accumulated in the\ninner loop the number of degree sequences of given length $n$ and fixed largest term $j$. Thus, we can see that\nthe set of degree sequences of fixed length and fixed sum and the set of degree sequences of fixed length\nand fixed largest term can all be efficiently counted.\n\\subsection{Some Novel Recurrences and the Improved Algorithm}\n\\label{subsec:recur}\nIn this section we first review some old and also derive novel recurrences involving the sizes of $D(n)$\nand various related sets.\nWe then present an improved algorithm to compute $|D(n)|$ based on these recurrences.\n\nDefine $H(n)$ to be the set of degree sequences of length $n$ with largest part exactly $n-1$ and $L(n)$\nthe set of degree sequences of length $n$ with largest part less than $n-1$. Clearly $D(n)$ is the disjoint\nunion of $H(n)$ and $L(n)$. Let $D_0(n)$ denote\nthe set of degree sequences of length $n$ allowing zero terms. The OEIS entry \\textcolor{red}{\\href{https:\/\/oeis.org\/A004251}{A004251}}\nrecords the values of $|D_0(n)|$.\nWe now present two propositions demonstrating the relationships among the cardinalities of these sets that will\nbe used by our improved algorithm. The following proposition is not hard to\nderive by the simple observation that $D_0(n)$ is the disjoint union of\n$D(n)$ and the set of degree sequences of length $n$ containing at least one zero.\n\n\\begin{proposition}\n\\label{thm_deg_seq} \\cite[Lemma 2]{Ivanyi2013}: $|D_0(n)|=|D_0(n-1)|+|D(n)|$, for $n\\ge 2$. And\n\\[ |D_0(n)|=\\left\\{ \\begin{array}{ll}\n1 & \\mbox{if $n=1$};\\\\\n1+\\sum_{i=2}^{n}|D(i)| & \\mbox{if $n \\ge 2$}.\\end{array} \\right.\\]\n\\end{proposition}\n\nBased on Proposition \\ref{thm_deg_seq} it is a trivial thing to compute $|D_0(n)|$ once all $|D(i)|$ such that $i\\le n$ have been computed.\nWe also have the following proposition that transforms the problem of computing $|D(n)|$ to that\nof computing $|L(n)|$.\n\n\\begin{proposition}\n\\label{thm_deg_seq_high_low} $|H(n)|=|D_0(n-1)|$, and $|D(n)|=|D_0(n-1)|+|L(n)|$, for $n\\ge 2$.\n\\end{proposition}\n\\begin{proof}\nEach simple undirected graph $G$ with $n$ vertices having a vertex of degree $n-1$ can be transformed into\na graph $G_1$ of $n-1$ vertices (possibly with isolated vertices) by removing the vertex of degree $n-1$.\nThus, $H(n)\\subseteq D_0(n)$.\n\nConversely, each graph $G_1$ of $n-1$ vertices (possibly with isolated vertices) can be transformed into a graph $G$ with $n$\nvertices having a vertex of degree $n-1$ by adding a new vertex and making it adjacent to all the vertices originally in $G_1$.\nThus, $D_0(n)\\subseteq H(n)$. This bijection establishes that $|H(n)|=|D_0(n-1)|$.\n\nSince $D(n)=H(n)\\cup L(n)$, and this is a disjoint union, we have $|D(n)|=|H(n)|+|L(n)|=|D_0(n-1)|+|L(n)|.$\n\\end{proof}\n\t\nBy Propositions \\ref{thm_deg_seq} and \\ref{thm_deg_seq_high_low}, it is sufficient to compute $|L(n)|$ in order to\ncompute $|D(n)|$ when all $|D(i)|$ such that $i<n$ are already known. Not surprisingly, we will show that $|L(n)|$\ncan also be computed eventually through the $|P(*,*,*,*)|$ values.\n\nDefine $L^{'}(N,l)$ to be the set of graphical partitions of $N$\nwith exactly $l$ parts and largest part less than $l-1$. Also define an index set $J_e(n)=\\{N|n\\le N\\le n(n-2), N \\mbox{ is an even integer}\\}$. It is easily seen that\n\\begin{equation} \\label{eqn:lower_sum_set}\n\t\\begin{aligned}\n\n\tL(n)=\\bigcup_{N\\in J_e(n)}L^{'}(N,n)\n\t\\end{aligned}\n\\end{equation}\nand this is a disjoint union so that we have\n\\begin{equation} \\label{eqn:lower_sum}\n\\begin{aligned}\n|L(n)|=\\sum_{N\\in J_e(n)}|L^{'}(N,n)|.\n\\end{aligned}\n\\end{equation}\n\nBy definition we have\n\\begin{equation} \\label{eqn:lower_sum_partition_set}\n\\begin{aligned}\nL^{'}(N,n)=\\bigcup_{k=1}^{n-2}G^{'}(N,k,n)\n\\end{aligned}\n\\end{equation}\nand by disjointness of this union we have\n\\begin{equation} \\label{eqn:lower_sum_partition}\n\\begin{aligned}\n|L^{'}(N,n)|=\\sum_{k=1}^{n-2}|G^{'}(N,k,n)|.\n\\end{aligned}\n\\end{equation}\nCombining Equations (\\ref{eqn:lower_sum}) and (\\ref{eqn:lower_sum_partition}) and substituting the formula in Theorem \\ref{thm_graphical_partition} into Equation (\\ref{eqn:lower_sum_partition}) we get\n\\begin{equation} \\label{eqn:lower_sum_final}\n\\begin{aligned}\n\\begin{split}\n|L(n)|&=\\sum_{N\\in J_e(n)}\\sum_{k=1}^{n-2}|G^{'}(N,k,n)|\\\\\n&=\\sum_{N\\in J_e(n)}\\sum_{k=1}^{n-2}|P(N-k-n+1,k-1,n-1,n-k-1)|\n\\end{split}\n\\end{aligned}\n\\end{equation}\nand this shows how $|L(n)|$ can also be calculated by summing a finite number of $|P(*,*,*,*)|$ values.\nIn fact, the calculation of $|L(n)|$ can be further sped up by the following proposition.\n\n\\begin{proposition}\n\t\\label{thm_lower_complementary} $ |L^{'}(N,n)|=|L^{'}(n(n-1)-N,n)|$, for $N \\in J_e(n)$.\n\\end{proposition}\n\\begin{proof}\nEach degree sequence $d_1\\ge d_2 \\ge \\ldots \\ge d_n >0$ in $L(n)$ with sum $N$ and $d_1<n-1$ can be transformed into a\ndegree sequence $n-1-d_n\\ge n-1-d_{n-1} \\ge \\ldots \\ge n-1-d_1 >0$ with sum $n(n-1)-N$, which still lies\nin $L(n)$ since $n-1-d_n<n-1$.\n\\end{proof}\n\nBased on Equation (\\ref{eqn:lower_sum}) and Proposition \\ref{thm_lower_complementary}, it is sufficient to\ncalculate the $|L^{'}(N,n)|$ values for even $N$ between $n$ and $n(n-1)\/2$ in order to calculate $|L(n)|$,\napparently saving the work load by a constant factor.\n\n\\begin{algorithm}\n\t\\DontPrintSemicolon\n\t\\KwIn{A positive integer $n$}\n\t\\KwOut{$|D(n)|$}\n\t$N \\gets n(n-1)\/2$\\;\n\tAllocate a four dimensional array $P[N-n+1][n-2][n][N]$\\;\n\tFill in the array $P$ using dynamic programming based on Theorem \\ref{thm_partition}\\;\n\t$S \\gets 0$\\;\n\t\\For{$i \\in J'_e(n)$ } {\n\t\t\\For{$j \\gets 1$ \\textbf{to} $\\min ({n-2,i-n+1})$ } {\n\t\t\t$S \\gets S+P[i-j-n+1][j-1][n-1][n-j-1]$\\;\n\t\t}\n\t}\n\t$S \\gets 2S$\\;\n\t\\If{$N$ is even}\t{\n\t\t\\For{$j \\gets 1$ \\textbf{to} $\\min ({n-2,N-n+1})$ } {\n\t\t\t$S \\gets S+P[N-j-n+1][j-1][n-1][n-j-1]$\\;\n\t\t}\n\t}\n\tCalculate $|D_0(n-1)|$ using saved values of $|D(i)|,i<n$ by Proposition \\ref{thm_deg_seq}\\;\n\t\\Return{$S+|D_0(n-1)|$}\\;\n\t\\caption{Improved Algorithm to compute $|D(n)|$, assuming all $|D(i)|$ such that $i<n$ are already computed. The\n\t\tindex set $J'_e(n)=\\{N|n\\le N< n(n-1)\/2, N \\mbox{ is an even integer}\\}$.}\n\t\\label{algo:improved}\n\\end{algorithm}\n\nWe now present an improved Algorithm \\ref{algo:improved} to calculate $|D(n)|$ based on the above definitions\nand techniques, assuming all values of $|D(i)|$ such that $i<n$ have been calculated.\nThe index set $J'_e(n)$\nused in the improved algorithm is defined as $J'_e(n)=\\{N|n\\le N< n(n-1)\/2, N \\mbox{ is an even integer}\\}$.\nThe variable $S$ is used to store the value of $|L(n)|$.\nAs in Algorithm \\ref{algo:basic}, the size for the third dimension of the array $P$ can also be reduced to 2 for\nthe purpose of computing $|L(n)|$ and $|D(n)|$. The individual sums accumulated in the loop\nfrom lines 6 to 7 and lines 10 to 11 are the values $|L^{'}(*,n)|$ based on Equation (\\ref{eqn:lower_sum_partition})\nand Equation (\\ref{eqn:lower_sum_final})\nand they together with the $|L^{'}(n(n-1)-*,n)|$ values can also be saved as additional output if desired.\nAlgorithm \\ref{algo:improved}\nis called an improved algorithm because it achieves a constant factor improvement over Algorithm \\ref{algo:basic}\n(see complexity analysis in Section \\ref{sec:Analysis}). Although using more time and space, Algorithm \\ref{algo:basic}\ndoes have the advantage of applicability for any $n$ without the need to know all the values of $|D(i)|$ such that $i<n$\nas in Algorithm \\ref{algo:improved}.\n\nBefore concluding this section we derive one more interesting recurrence. Define\n$H^{'}(N,l)$ to be the set of graphical partitions of $N$\nwith exactly $l$ parts and largest part exactly $l-1$. Also define an index set\n$I'_e(n)=\\{N|2(n-1)\\le N\\le n(n-1), N \\mbox{ is an even integer}\\}$. By definitions it is easily seen that\n\\begin{equation} \\label{eqn:higher_sum_set}\n\\begin{aligned}\nH(n)=\\bigcup_{N\\in I'_e(n)}H^{'}(N,n)\n\\end{aligned}\n\\end{equation}\nand this is a disjoint union so that we have\n\\begin{equation} \\label{eqn:higher_sum}\n\\begin{aligned}\n|H(n)|=\\sum_{N\\in I'_e(n)}|H^{'}(N,n)|.\n\\end{aligned}\n\\end{equation}\n\nBy definition we have\n\\begin{equation} \\label{eqn:higher_sum_partition_set}\n\\begin{aligned}\nH^{'}(N,n)=G^{'}(N,n-1,n),\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation} \\label{eqn:higher_lower_partition_set}\n\\begin{aligned}\nG^{'}(N,n)=H^{'}(N,n) \\cup L^{'}(N,n), \\qquad |G^{'}(N,n)|=|H^{'}(N,n)|+ |L^{'}(N,n)|.\n\\end{aligned}\n\\end{equation}\n\nThe following proposition indicates a symmetry of $|H^{'}(N,n)|$ similar to that of $|L^{'}(N,n)|$ as shown\nin Proposition \\ref{thm_lower_complementary}.\n\\begin{proposition}\n\t\\label{thm_higher_complementary}\n\t$|H^{'}(N,n)|=|H^{'}((n+2)(n-1)-N,n)|$, for $N \\in I'_e(n)$.\n\\end{proposition}\n\\begin{proof}\nEach degree sequence $d_1\\ge d_2 \\ge \\ldots \\ge d_n >0$ in $H(n)$ with sum $N$ and $d_1=n-1$ can be first transformed into a\ndegree sequence allowing zero terms (of length $n$, by appending a zero after $d_n-1$)\n$d_2-1 \\ge d_3-1 \\ge \\ldots \\ge d_n-1 \\ge 0 \\ge 0$ by applying\nthe Havel-Hakimi algorithm \\cite{Havel1955,Hakimi1962} which\ncharacterizes equivalent realizable sequences, and then transformed into a degree sequence (of length $n$)\n$n-1 \\ge n-d_n\\ge n-d_{n-1} \\ge \\ldots \\ge n-d_2 >0$ with sum $(n+2)(n-1)-N$, which still lies\nin $H(n)$.\n\\end{proof}\n\nFrom available numerical results we observe that for fixed $n$ the sequence of $|H^{'}(N,n)|$ for $N\\in I'_e(n)$\nis unimodal and the maximum value in the sequence occurs when $N$ is an even integer closest to $(n^2+n-2)\/2$.\nIt is also observed that for fixed $n$ the sequence of $|L^{'}(N,n)|$ for $N\\in J_e(n)$ is unimodal and\nthe maximum value in the sequence occurs when $N$ is an even integer closest to $(n^2-n)\/2$. Based on Equation\n(\\ref{eqn:higher_lower_partition_set}) and available numerical results we also conjecture that for fixed $n$\nthe sequence of $|G^{'}(N,n)|$ for $N\\in I_e(n)$\nis unimodal and the maximum value in the sequence occurs when $N$ is closest to $n^2\/2-n\/6$. One of the ways to determine the\nasymptotic order of $|D(n)|$ might be to determine the order of $|G^{'}(N,n)|$ for a narrow range of $N$ centered around $n^2\/2-n\/6$.\nThe rigorous proof of these results might need advanced techniques.\n\n\\section{Algorithms to compute the number of degree sequences of connected graphs on $n$ vertices}\n\\label{sec:count_connected}\nIn this section we apply the techniques employed in the two algorithms for computing $|D(n)|$ in Section \\ref{sec:Algo} to derive similar\nalgorithms for computing the number of degree sequences of connected graphs on $n$ vertices.\n\nRecall that a degree sequence is called \\textit{potentially} connected (resp. disconnected) if it has a connected\n(resp. disconnected) realization and it is called \\textit{forcibly} connected (resp. disconnected) if all of\nits realizations are connected (resp. disconnected).\nLet $D_c(n)$ denote the subset of $D(n)$ that are potentially connected and $D_d(n)$ the subset of $D(n)$ that are\nforcibly disconnected. Clearly $D(n)$ is the disjoint union of $D_c(n)$ and $D_d(n)$. A degree sequence\n$d=(d_1,d_2,\\cdots,d_n)$ in $D(n)$ is potentially connected if and only if $\\sum_{i=1}^{n}d_i\\ge 2(n-1)$ \\cite{Berge1973}.\nBecause of this simple characterization it is easily seen by definitions that\n\\begin{equation} \\label{eqn:degseq_set_potentially_connected}\n\\begin{aligned}\nD_c(n)=\\bigcup_{N\\in I'_e(n)}G^{'}(N,n)=\\bigcup_{N\\in I'_e(n)}\\bigcup_{k=1}^{n-1}G^{'}(N,k,n),\n\\end{aligned}\n\\end{equation}\n\nand\n\n\\begin{equation} \\label{eqn:degseq_set_forcibly_disconnected}\n\\begin{aligned}\nD_d(n)=\\bigcup_{N\\in I_e(n)-I'_e(n)}G^{'}(N,n)=\\bigcup_{N\\in I_e(n)-I'_e(n)}\\bigcup_{k=1}^{n-1}G^{'}(N,k,n).\n\\end{aligned}\n\\end{equation}\nAgain by disjointness we have\n\\begin{equation} \\label{eqn:degseq_potentially_connected}\n\\begin{aligned}\n\\begin{split}\n|D_c(n)|&=\\sum_{N\\in I'_e(n)}\\sum_{k=1}^{n-1}|G^{'}(N,k,n)|\\\\\n&=\\sum_{N\\in I'_e(n)}\\sum_{k=1}^{n-1}|P(N-k-n+1,k-1,n-1,n-k-1)|\n\\end{split}\n\\end{aligned}\n\\end{equation}\n\nand\n\n\\begin{equation} \\label{eqn:degseq_forcibly_disconnected}\n\\begin{aligned}\n\\begin{split}\n|D_d(n)|&=\\sum_{N\\in I_e(n)-I'_e(n)}\\sum_{k=1}^{n-1}|G^{'}(N,k,n)|\\\\\n&=\\sum_{N\\in I_e(n)-I'_e(n)}\\sum_{k=1}^{n-1}|P(N-k-n+1,k-1,n-1,n-k-1)|.\n\\end{split}\n\\end{aligned}\n\\end{equation}\nIf $|D(n)|$ has already been obtained, we can choose to compute $|D_d(n)|$ in order to get $|D_c(n)|=|D(n)|-|D_d(n)|$\nfor performance reasons. Now we present Algorithm \\ref{algo:potentially_connected} to compute $|D_c(n)|$ directly\nbased on Equation (\\ref{eqn:degseq_potentially_connected}) and Algorithm \\ref{algo:forcibly_disconnected}\nto compute $|D_c(n)|$ indirectly through computing $|D_d(n)|$ based on Equation (\\ref{eqn:degseq_forcibly_disconnected}).\nAs in Algorithms \\ref{algo:basic} and \\ref{algo:improved}\nwe can still allocate size 2 for the third dimension of the array $P$.\n\n\\begin{algorithm}[h]\n\t\\DontPrintSemicolon\n\t\\KwIn{A positive integer $n$}\n\t\\KwOut{$|D_c(n)|$}\n\t$N \\gets n(n-1)$\\;\n\tAllocate a four dimensional array $P[N-n+1][n-1][n][N]$\\;\n\tFill in the array $P$ using dynamic programming based on Theorem \\ref{thm_partition}\\;\n\t$S \\gets 0$\\;\n\t\\For{$i \\in I'_e(n)$ } {\n\t\t\\For{$j \\gets 1$ \\textbf{to} $\\min ({n-1,i-n+1})$ } {\n\t\t\t$S \\gets S+P[i-j-n+1][j-1][n-1][n-j-1]$\\;\n\t\t}\n\t}\n\t\\Return{$S$}\\;\n\t\\caption{Algorithm to compute $|D_c(n)|$ directly. The index set $I'_e(n)=\\{N|2(n-1)\\le N\\le n(n-1), N \\mbox{ is an even integer}\\}$.}\n\t\\label{algo:potentially_connected}\n\\end{algorithm}\n\n\n\\begin{algorithm}[h]\n\t\\DontPrintSemicolon\n\t\\KwIn{A positive integer $n$}\n\t\\KwOut{$|D_c(n)|$}\n\t$N \\gets 2(n-2)$\\;\n\tAllocate a four dimensional array $P[N-n+1][n-1][n][N]$\\;\n\tFill in the array $P$ using dynamic programming based on Theorem \\ref{thm_partition}\\;\n\t$S \\gets 0$\\;\n\t\\For{$i \\in I_e(n)-I'_e(n)$ } {\n\t\t\\For{$j \\gets 1$ \\textbf{to} $\\min ({n-1,i-n+1})$ } {\n\t\t\t$S \\gets S+P[i-j-n+1][j-1][n-1][n-j-1]$\\;\n\t\t}\n\t}\n\t\\Return{$|D(n)|-S$}\\;\n\t\\caption{Algorithm to compute $|D_c(n)|$ indirectly, assuming $|D(n)|$ is known. The index set\n\t\t$I_e(n)-I'_e(n)=\\{N|n\\le N< 2(n-1), N \\mbox{ is an even integer}\\}$.}\n\t\\label{algo:forcibly_disconnected}\n\\end{algorithm}\n\nThe OEIS entry \\textcolor{red}{\\href{https:\/\/oeis.org\/A007721}{A007721}} records the values of $|D_c(n)|$.\n\n\\section{Algorithm to compute the number of degree sequences of biconnected graphs on $n$ vertices}\n\\label{sec:count_biconnected}\nIn this section we will derive Algorithm \\ref{algo:potentially_biconnected}\nto count the set $D_b(n)$ of degree sequences of biconnected graphs on $n$ vertices.\nA degree sequence $d_1\\ge d_2 \\ge \\cdots \\ge d_n$ in $D(n)$ is potentially biconnected if and only if\n$\\sum_{i=1}^{n}d_i\\ge 2n-4+2d_1$ and $d_n\\ge 2$ \\cite{WangKleitman1973}. Since this characterization\nexpresses the lower bound of the degree sum as a quantity that can vary due to $d_1$, there is not a simple\nalgorithm like Algorithm \\ref{algo:potentially_connected} or Algorithm \\ref{algo:forcibly_disconnected} to count $|D_b(n)|$.\nHowever, we will show that $|D_b(n)|$ can still be computed eventually through the $|P(*,*,*,*)|$ values\nand the unrestricted partition function $|P(*)|$ values.\n\nDefine $S(n)$ to be the set of degree sequences of length $n$ with largest part exactly $n-2$, $C(n)$ the set of degree\nsequences of length $n$ with smallest part exactly $1$ and $B(n)$ the set\nof degree sequences of length $n$ with largest part exactly $n-1$ and smallest part exactly $1$. Then, $D_2(n)=D(n)-C(n)$\nis the set of degree sequences of length $n$ with smallest part at least 2. We next establish some connections\nbetween these sets and show that $|D_2(n)|$ can be efficiently computed.\n\n\\begin{proposition}\n\t\\label{thm:deg_seq_second_and1} $|C(n)|=|B(n)|+|S(n)|$,$|B(n)|=|D_0(n-2)|$, for $n\\ge 3$.\n\\end{proposition}\n\\begin{proof}\n\tNotice that $C(n)$ is the disjoint union of $B(n)$ and the set $W(n)$ of degree sequences of length $n$ with largest part\n\tless than $n-1$ and smallest part exactly 1, so we have $|C(n)|=|B(n)|+|W(n)|$. Each degree sequence \n\t$d_1\\ge d_2 \\ge \\ldots \\ge d_n$ in $W(n)$\n\n\tcan be transformed into a degree sequence $n-1-d_n \\ge n-1-d_{n-1} \\ge \\ldots \\ge n-1-d_1$, which is a\n\tdegree sequence in $S(n)$ since $d_1<n-1$ and $d_n=1$. Conversely, each degree sequence\nin $S(n)$ can be transformed into one in $W(n)$ in a similar way. This bijection\n\testablishes that $|W(n)|=|S(n)|$, and then the first equation follows.\n\t\n\tEach simple undirected graph $G$ with $n$ vertices having a vertex of degree $n-1$ and a vertex of degree 1\n\tcan be transformed into\n\ta graph $G_2$ of $n-2$ vertices (possibly with isolated vertices) by first removing the vertex of degree $n-1$, resulting\n\tin an isolated vertex (previously with degree 1), and then removing the newly appeared isolated vertex. Conversely,\n\teach graph $G_2$ of $n-2$ vertices (possibly with isolated vertices) can be transformed to a graph $G$ with $n$\n\tvertices having a vertex of degree $n-1$ and a vertex of degree 1 by first adding a new isolated vertex and then\n\tadding a second vertex and making it adjacent to all the other $n-1$ vertices. This bijection establishes that\n\t$|B(n)|=|D_0(n-2)|$.\n\t\n\\end{proof}\n\nNow $|B(n)|$ can be efficiently computed based on Proposition \\ref{thm:deg_seq_second_and1} and Proposition \\ref{thm_deg_seq}.\nAs noted at the end of Section \\ref{subsec:BasicAlgo}, $|S(n)|$ can be efficiently computed. Each\ndegree sequence in $S(n)$ must have sum $N$ such that $2n-3\\le N\\le n(n-2)$. Define an index set\n$I''_e(n)=\\{N|2n-3\\le N\\le n(n-2), N \\mbox{ is an even integer}\\}$. Clearly,\n\\[S(n)=\\bigcup_{N\\in I''_e(n)}G^{'}(N,n-2,n).\\]\nBy disjointness, we have\n\\begin{equation} \\label{eqn:second_largest}\n|S(n)|=\\sum_{N\\in I''_e(n)}|G^{'}(N,n-2,n)|=\\sum_{N\\in I''_e(n)}|P(N-2n+3,n-3,n-1,1)|.\n\\end{equation}\nThus, $|C(n)|$ can be efficiently computed based on Proposition \\ref{thm:deg_seq_second_and1} and we can therefore\nefficiently compute $|D_2(n)|$ as\n\\begin{equation} \\label{eqn:D2nfirst}\n|D_2(n)|=|D(n)|-|C(n)|.\n\\end{equation}\nNext we will show that the combinatorial trick\nof expressing $|D_2(n)|$ in a different way allows us to compute $|D_b(n)|$ efficiently.\n\nRecall that $D_b(n)$ is the set of degree sequences of biconnected graphs on $n$ vertices. Clearly $D_b(n)\\subseteq D_2(n)$\nbased on the characterization of potentially biconnected degree sequences \\cite{WangKleitman1973}.\nDenote $D_{2\\setminus b}(n)=D_2(n)-D_b(n)$, i.e., the set of those degree sequences $d_1\\ge d_2\\ge \\cdots \\ge d_n$ such that\n$\\sum_{i=1}^{n}d_i< 2n-4+2d_1$ and $d_n\\ge 2$. Clearly, we can express $|D_2(n)|$ in a second way as\n\\begin{equation} \\label{eqn:D2nsecond}\n|D_2(n)|=|D_b(n)|+|D_{2\\setminus b}(n)|.\n\\end{equation}\nThis fulfills our goal of efficiently computing $|D_b(n)|=|D_2(n)|-|D_{2\\setminus b}(n)|$\nas long as $|D_{2\\setminus b}(n)|$ can be computed efficiently, which will be shown next.\n\nConsider the possible degree sequences $d_1\\ge d_2\\ge \\cdots \\ge d_n$ in $D_{2\\setminus b}(n)$.\nSince $D_{2\\setminus b}(n) \\subseteq D_2(n)$, $d_1\\ge d_n \\ge 2$. The possible choices for $d_1$ are $d_1=2,\\cdots,n-1$\nand they are considered individually below.\n\n\\begin{enumerate}\n\t\\item $d_1=2:$ It is required that $\\sum_{i=1}^{n}d_i< 2n$. This is impossible since we also need $\\sum_{i=1}^{n}d_i\\ge 2(n-1)+2=2n$.\n\t\\item $d_1=3:$ It is required that $\\sum_{i=1}^{n}d_i< 2n+2$. Again impossible since we also need $\\sum_{i=1}^{n}d_i\\ge 2(n-1)+3=2n+1$.\n\n\\item $d_1=2k\\ge 4:$ It is required that $\\sum_{i=1}^{n}d_i< 2n+4k-4$. We also need $\\sum_{i=1}^{n}d_i\\ge 2(n-1)+2k=2n+2k-2$. Thus\n$\\sum_{i=1}^{n}d_i=2n+2k-2,2n+2k,\\ldots,2n+4k-6$ are all the feasible degree sums. For each feasible degree sum $2n+2k-2+N$,\n$N=0,2,\\cdots,2k-4$, consider the possible degree sequences $d_1\\ge d_2\\ge \\cdots \\ge d_n$ such that $d_1=2k$, $d_n\\ge 2$, $\\sum_{i=1}^{n}d_i=2n+2k-2+N$.\nClearly the number of possible degree sequences is the number of ways to partition $N$ into $t\\le n-1$ parts\n$N=N_1+N_2+\\cdots+N_t$ such that\n$d_1\\ge 2+N_1\\ge 2+N_2\\ge \\cdots \\ge 2+N_t \\ge 2\\ge \\cdots \\ge 2$ is a graphical degree sequence.\nThe particular case $t=1$ gives the integer sequence $d_1\\ge 2+N\\ge 2\\ge \\cdots \\ge 2$.\nIt is easy to check based on the Nash-Williams condition \\cite{Ruch1979,Rousseau1995,Sierksma1991}\nthat for each $N=0,2,\\cdots,2k-4$, the integer sequence $d_1\\ge 2+N\\ge 2\\ge \\cdots \\ge 2$ is graphical.\nFor each fixed $N\\in \\{0,2,\\cdots,2k-4\\}$ and any partition $N=N_1+N_2+\\cdots+N_t$, the integer sequence\n$l_1=(d_1, 2+N, 2, \\cdots ,2)$ majorizes the integer sequence $l_2=(d_1, 2+N_1, 2+N_2, \\cdots ,2+N_t, 2, \\cdots ,2)$ and\nsince $l_1$ is graphical, $l_2$ is also graphical \\cite[Corollary 3.1.4]{MahadevPeled1995}.\nThis produces a total of $|P(0)|+|P(2)|+|P(4)|+\\cdots+|P(2k-4)|$ possible degree sequences.\n\n\t\\item $d_1=2k+1\\ge 5:$ It is required that $\\sum_{i=1}^{n}d_i< 2n+4k-2$. We also need $\\sum_{i=1}^{n}d_i\\ge 2(n-1)+2k+1=2n+2k-1$. Thus\n\t$\\sum_{i=1}^{n}d_i=2n+2k,2n+2k+2,\\ldots,2n+4k-4$ are all the feasible degree sums.\n\tFor each feasible degree sum $2n+2k-1+N$,\n\t$N=1,3,\\cdots,2k-3$, consider the possible degree sequences $d_1\\ge d_2\\ge \\cdots \\ge d_n$ such that $d_1=2k+1$, $d_n\\ge 2$, $\\sum_{i=1}^{n}d_i=2n+2k-1+N$.\n\tClearly the number of possible degree sequences is the number of ways to partition $N$ into $t\\le n-1$ parts\n\t$N=N_1+N_2+\\cdots+N_t$ such that\n\t$d_1\\ge 2+N_1\\ge 2+N_2\\ge \\cdots \\ge 2+N_t \\ge 2\\ge \\cdots \\ge 2$ is a graphical degree sequence.\n\tThe particular case $t=1$ gives the integer sequence $d_1\\ge 2+N\\ge 2\\ge \\cdots \\ge 2$.\n\tIt is easy to check based on the Nash-Williams condition \\cite{Ruch1979,Rousseau1995,Sierksma1991}\n\tthat for each $N=1,3,\\cdots,2k-3$, the integer sequence $d_1\\ge 2+N\\ge 2\\ge \\cdots \\ge 2$ is graphical.\n\tFor each fixed $N\\in \\{1,3,\\cdots,2k-3\\}$ and any partition $N=N_1+N_2+\\cdots+N_t$, the integer sequence\n\t$l_1=(d_1, 2+N, 2, \\cdots ,2)$ majorizes the integer sequence $l_2=(d_1, 2+N_1, 2+N_2, \\cdots ,2+N_t, 2, \\cdots ,2)$ and\n\tsince $l_1$ is graphical, $l_2$ is also graphical \\cite[Corollary 3.1.4]{MahadevPeled1995}.\n\tThis produces a total of $|P(1)|+|P(3)|+|P(5)|+\\cdots+|P(2k-3)|$ possible degree sequences.\n\t\n\\end{enumerate}\n\nBased on the above discussions and techniques, we present Algorithm \\ref{algo:potentially_biconnected} to\ncompute the number of potentially biconnected degree sequences of length $n$. The variable $S$ is supposed\nto store the value $|D_{2\\setminus b}(n)|$. The OEIS entry \\textcolor{red}{\\href{https:\/\/oeis.org\/A007722}{A007722}} records the $|D_b(n)|$ values.\n\\begin{algorithm}\n\t\\DontPrintSemicolon\n\t\\KwIn{A positive integer $n\\ge 5$}\n\t\\KwOut{$|D_b(n)|$}\n\tCompute $|S(n)|$ using dynamic programming based on Equation (\\ref{eqn:second_largest})\\;\n\tCompute $|D_0(n-2)|$ based on Proposition \\ref{thm_deg_seq} using the saved $|D(i)|$ values\\;\n\tCompute $|C(n)|=|D_0(n-2)|+|S(n)|$ based on Proposition \\ref{thm:deg_seq_second_and1}\\;\n\tCompute $|D_2(n)|=|D(n)|-|C(n)|$ using the saved $|D(n)|$ value\\;\n\t$S \\gets 0$\\;\n\t\\For{$i \\gets 4$ \\textbf{to} $n-1$} {\n\t\t$k \\gets \\lfloor \\frac{i}{2} \\rfloor$\\;\n\t\t\\uIf{$i$ is even}{\n\t\t\t\\For{$j \\gets 0$ \\textbf{to} $2k-4$ \\textbf{step} $2$ } {\n\t\t\t\t$S \\gets S+|P(j)|$\\;\n\t\t\t}\n\t\t}\n\t\t\\Else{\n\t\t\t\\For{$j \\gets 1$ \\textbf{to} $2k-3$ \\textbf{step} $2$ } {\n\t\t\t\t$S \\gets S+|P(j)|$\\;\n\t\t\t}\n\t\t}\n\t}\n\t\\Return{$|D_2(n)|-S$}\\;\n\t\\caption{Algorithm to compute $|D_b(n)|$, assuming all $|D(i)|$ such that $i\\le n$ have been computed.}\n\t\\label{algo:potentially_biconnected}\n\\end{algorithm}\n\n\nBefore we conclude this section, we briefly mention that the method in this section can be extended to count\nthe number of degree sequences of triconnected graphs and beyond. Characterization of potentially $k$-connected\ndegree sequences of length $n$ can be found in Wang and Cleitman \\cite{WangKleitman1973}. In order to use\nthis characterization, we can still count the set $D_k(n)$ of degree sequences of length $n$ with smallest\npart at least $k$ in two different ways. Let $C_k(n)$ denote the set of degree sequences of length $n$ with smallest\npart exactly $k$. Clearly $|D_k(n)|=|D(n)|-\\sum_{i=1}^{k-1}|C_i(n)|$.\nThe first way expresses $|C_i(n)|$ as the sum of the number of degree\nsequences with fixed largest part $n-i-1$ and lengths from $n-i$ to $n$ so that $|D_k(n)|$ can be efficiently computed.\nThe second way expresses $|D_k(n)|$\nas the sum of the number $|D_{kc}(n)|$ of potentially $k$-connected degree sequences and the rest $|R(n)|$. Then\n$|D_{kc}(n)|$ can be efficiently computed by combining the results of computing $|D_k(n)|$ in the two ways. Counting $R(n)$ in $D_k(n)$\ninvolves an enumeration of all possible values of $d_1,\\cdots, d_{k-1}$ and will become the most tedious\npart of the algorithm.\n\n\\section{Complexity Analysis}\n\\label{sec:Analysis}\nAs mentioned in Section \\ref{sec:Intro}, the previous best known methods to calculate $|D(n)|$ have a time\ncomplexity of $\\Omega(4^n\/n)$ as in \\cite{Ruskey1994} and a time complexity of $O(\\sqrt{n} 4^n)$ and space\ncomplexity of $O(n)$ as in \\cite{Ivanyi2013} respectively. Their exponential running time comes from the fact\nthat they compute $|D(n)|$ by generating all the sequences in $D(n)$. The variants of these algorithms to compute\n$|D_c(n)|$ and $|D_b(n)|$ by incorporating the relevant tests whether the degree sequence is potentially connected\nor potentially biconnected have similar exponential time\ncomplexity since the tests only add linear extra time for each generated degree sequence. Thus it is infeasible to\ncalculate $|D(n)|$, $|D_c(n)|$ and $|D_b(n)|$ using these methods for $n$ beyond 35. In this section we analyze the computational\ncomplexity of the presented algorithms. A summary of the results is presented in Table \\ref{tbl:complexity}.\n\n\\begin{table}[!htb\n\t\\centering\n\t\\caption{Complexity of the algorithms}\n\t\\begin{tabular}{||c|c|c||}\n\t\t\\hline\\hline\n\t\tAlgorithm & Time & Space\\\\\n\t\t\\hline\\hline\n\t\tRuskey \\textit{et al.}\\cite{Ruskey1994} & $\\Omega(4^n\/n)$ & $O(n)$\\\\\n\t\t\\hline\n\t\tIv\\'{a}nyi \\textit{et al.}\\cite{Ivanyi2013} & $\\Theta(\\sqrt{n} 4^n)$ & $O(n)$\\\\\n\t\t\\hline\n\t\tAlgorithm \\ref{algo:basic} & $O(n^5)$ & $O(n^5)$ \\\\\n\t\t\\hline\n\t\tAlgorithm \\ref{algo:improved} & $O(n^5)$ & $O(n^5)$ \\\\\n\t\t\\hline\n\t\tAlgorithm \\ref{algo:potentially_connected} & $O(n^5)$ & $O(n^5)$ \\\\\n\t\t\\hline\n\t\tAlgorithm \\ref{algo:forcibly_disconnected} & $O(n^3)$ & $O(n^3)$ \\\\\n\t\t\\hline\n\t\tAlgorithm \\ref{algo:potentially_biconnected} & $O(n^5)$ & $O(n^5)$ \\\\\n\t\t\\hline\\hline\n\t\\end{tabular}\n\t\\label{tbl:complexity}\n\\end{table}\n\nWe first analyze the complexity of the basic Algorithm \\ref{algo:basic}. The first and fourth dimensions of\nthe array $P$ have sizes $O(n^2)$, the second dimension $O(n)$ and the third dimension a constant as mentioned above.\nThus, the space complexity is $O(n^5)$. The dynamic programming step in line 3 costs time $O(n^5)$ as a result.\nThe loop from lines 5 to 7 costs time $O(n^2 n)=O(n^3)$, which is dominated by the step in line 3. So Algorithm\n\\ref{algo:basic} has time complexity $O(n^5)$.\n\nSimilar analysis shows that the improved Algorithm \\ref{algo:improved} for $|D(n)|$\nand Algorithm \\ref{algo:potentially_connected} for $|D_c(n)|$ also have the asymptotic time and space\ncomplexity $O(n^5)$. For Algorithm \\ref{algo:improved}, since the first and fourth dimensions of the array $P$\nhave sizes about half of those\nin the basic Algorithm \\ref{algo:basic}, it achieves an improvement of a constant factor of approximately 4.\nAlthough it is not asymptotically faster than the basic algorithm, it does allow us to compute more $|D(n)|$\nvalues since for large values of $n$ the memory issue is more severe than the time issue for these algorithms.\nAlgorithm \\ref{algo:forcibly_disconnected} has time and space complexity $O(n^3)$ due to the reduction\nof the sizes of the first and fourth dimension of the array $P$ to $O(n)$.\n\nFor Algorithm \\ref{algo:potentially_biconnected}, line 1 takes $O(n^5)$ time and $O(n^5)$ space. Lines 2 to 4\ntake time $O(n)$ using saved values of $|D(i)|,i\\le n$. For lines 5 to 13, all $|P(j)|,j\\le n-5$ can be precomputed\nin time $O(n^{3\/2+\\epsilon})$ since $|P(n)|$ can be computed in time $O(n^{1\/2+\\epsilon})$ \\cite{Johansson2012}. The double for loop\nfrom lines 6 to 13 takes time $O(n^2)$. The overall time and space complexity are both $O(n^5)$.\n\n\\section{Computational Experiences}\n\\label{sec:experiments}\n\nWe note that our calculation of $|D(30)|$ using an optimized C implementation of the basic Algorithm \\ref{algo:basic}\ntakes a single Intel Core i7 computer with 16 GB memory 1.8 seconds. The improved Algorithm \\ref{algo:improved} uses 0.7 seconds\nto get the same result. In contrast, the previous result of \\cite{Ivanyi2013} estimated that computing $|D(30)|$\nusing their method would cost a single computer about 72 years. The computation of $|D(118)|$ using the improved\nAlgorithm \\ref{algo:improved} on a computer with 256 GB memory takes about 2.5 hours.\n\nFor $n<30$ our results of $|D(n)|$ agree with those in \\cite{Ivanyi2013} while the values of $|D(30)|$ and $|D(31)|$ we calculated are\nslightly different. From the available numerical results it appears that $|D(n)|\/|D(n-1)|$\nis slowly increasing when $n\\ge 9$ and approaching 4, giving more supporting evidence of Royle's conjecture.\nWe also computed the $|D_0(n)|$, $|D_c(n)|$ and $|D_b(n)|$ values and incorporated all available numerical\nresults into their respective OEIS entries.\n\nDuring the preparation of this manuscript, the author became aware of the results in \\cite{KohnertSlides}, which\ncalculated the values of $|D(n)|$ up to $n=34$. The method employed therein seemed more complicated and not formally\npublished to the knowledge of the author.\n\n\\section{Conclusions}\n\\label{sec:conclusion}\nIn this paper we presented dynamic programming algorithms to compute the number of degree sequences\nof simple graphs, simple connected graphs and simple biconnected graphs on $n$ vertices exactly all based on a known recurrence\nof Barnes and Savage. They are the first known polynomial time algorithms to compute\nthese functions and are asymptotically much faster than previous best known counterparts. Using these\nfast algorithms we extend the known exact values of $|D(n)|$,  $|D_0(n)|$, $|D_c(n)|$ and $|D_b(n)|$ up to $n=118$ and these\nresults have been included in their OEIS entries.\nFurther research include how to determine the tight asymptotic order of these functions, how to efficiently compute\nthe number of forcibly $k$-connected degree sequences of length $n$ and determine its asymptotic order.\n\n\\section{Acknowledgements}\nThis research has been supported by the Startup Fund of Georgia Southern University.\n\n\\sloppy\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nDepth estimation is a fundamental computer vision task, which plays an influential role in many practical applications, such as  autonomous driving, robot grasping and augmented reality.\nThe traditional geometric approaches~\\cite{schonberger2016colmap, gherardi2010improvingsfm, mur2015orb} solve this problem by exploiting the key point matching relationship between adjacent frames and recovering the depth information of 2D pixels through triangulation.\nAlthough traditional geometric methods have achieved promising performance, they are unable to handle low-texture regions and non-stationary scenes.\nIn addition, the huge computational cost of feature matching limits their practical application.\n\nRecently, deep learning techniques have improved this research area by training networks to directly predict depth from single image.\nDespite the fact that supervised learning methods have achieved good results~\\cite{adabins, li2022depthformer, simipu}, the cost of labeling ground truth limits the practical application of such methods.\nTo solve this problem, unsupervised depth estimation~\\cite{monodepth2,manydepth,epc++,zhou2017unsupervised,zhan2018unsupervised} is gradually becoming a novel research trend, which treats depth estimation as a view synthesis problem.\nAlthough self-supervised depth estimation has made great progress in recent years, there are still some unresolved issues. \nFirst, without the feature matching step, the unsupervised depth estimation cannot leverage the semantic information from adjacent frames, which leads to inaccurate depth estimation results.\nSecond, it is known that the photometric loss is sensitive to dynamic objects, which violates the assumption of the static world. Therefore, the perspective projection will be non-rigid, resulting in unstable training process.\nOn the one hand, Manydepth\\cite{manydepth} leveraged cost-volume based method to aggregate multi-frame information. \nHowever, inaccurate pose estimation introduces additional noise to feature matching, which can sharply decreasing model performance in dynamic scenes.\nOn the other hand, recent work have tried to introduce instance segmentaion \\cite{lee2019instance}, auto-mask \\cite{monodepth2} and optical flow \\cite{yin2018geonet} to deal with dynamic objects under the self-supervised learning framework. \nThese methods introduce expensive additional labeling costs, which are therefore difficult to be applied in real-world applications.\n\n\\begin{figure}[t!]\n    \\centering\n   \n    \\includegraphics[width=0.35\\textwidth]{figures\/overview.jpg}\n    \\caption{\n        \\textbf{Overall architecture} of our proposed method. On the one hand, we model the dynamic objects in the scene to improve the accuracy of depth estimation in dynamic regions. On the other hand, considering the presence of dynamic objects, we design a geometry-based cross-view feature fusion approach to obtain discriminative feature representations.\n    }\n    \\label{fig:overview}\n   \n\\end{figure}\n\nIn this work, we propose a \\textbf{Dyna-DepthFormer} network in order to solve the above-mentioned problems. \nSpecifically, we propose a depth feature enhance module consisting of self-attention and cross-attention module to capture the correlation in multi-view feature map.\nInspired by \\cite{zhu2020deformable}, we realize this module by constructing a Transformer-based module, while the deformable attention is adopted to reduce the computational costs and adjust sample point to capture the feature in the local region.\nThe generated features will be more discriminative and robust, since it focus on the surrounding local context and aggregates relevant information.\nFurthermore, we propose a novel Motion Network, which model the dynamic object by leveraging the motion field prediction. Then, We can use the motion field and camera motion to calculate the reprojection loss without using semantic prior. In summary, our contributions are:\n\n\n\n\n\\begin{itemize}\n  \\item \n  We propose a transformer-based Depth Network, which enhances the depth feature representation through multi-view feature aggregating. Considering the high computation of dense attention, we propose a perspective-projection based temporal deformable cross-attention module that efficiently reduces the computation.\n  \\item\n  In addition to camera ego motion, we proposed to estimate the 3D motion field of moving objects, and improve its completeness through iterative refinement. This eliminates the necessity of the scene-static assumption and enables unsupervised method to use more unlabeled data for training.\n  \\item\n  We demonstrate that our proposed method achieves promising results in monocular depth estimation on benchmark datasets like KITTI and Cityscapes while reducing computing cost.\n\\end{itemize}\n\n\\section{RELATED WORK}\n\n\\subsection{Self-Supervised Depth Estimation}\nSelf-supervised depth estimation methods, in contrast to supervised depth estimation methods, can estimate depth from unlabeled video training data.\nThe work of Monodepth~\\cite{godard2017unsupervised} introduced a self-supervised depth estimation method by leveraging the left-right consistency. \nSFM-learner~\\cite{zhou2017unsupervised} estimated the relative poses between adjacent frames simultaneously with the addition of a pose network.\nThen, Monodepth2~\\cite{monodepth2} absorbed the advantages of the prior methods and proposed a universal framework, which can be trained with stereo pairs and videos data.\nHowever, this approach was afflicted by issues such as scale ambiguity, dynamic objects, low-texture areas, and non-Lambertian surfaces.\nIn recent years, several strategies have been developed to improve the performance of depth estimation by tackling the aforementioned difficulties.\nFeatdepth~\\cite{shu2020featdepth} introduced a auto-encoder structure in order to obtain a robust feature representation, which can improve the depth estimation accuracy in low-texture region.\nPacknet~\\cite{packnet} proposed a novel 3D convolution packing module to overcome the scale ambiguity problem, while it suffered from heavy computational cost.\n\nWith the development of cost-volume based multi-view stereo (MVS) methods, some of the work~\\cite{manydepth, guizilini2022multi, wang2021deep, monorec} introduced the cost-volume to self-supervised depth estimation field, which enabled the learning of additional geometric cues from feature matching across frames.\nHowever, due to the hand-crafted similarity metrics, the cost volume will be an unreliable source of depth information in locations where objects are moving or surfaces are untextured.\n\n\n\\subsection{Attention Mechanisms in Depth Estimation}\nSelf-Attention mechanism have shown promising results in the field of natural language processing~\\cite{attention_all} and are becoming increasingly popular in computer vision region ~\\cite{dosovitskiy2020vit, liu2021swin}. \nIn the field of depth estimation, current methods can be divided into two categories. \nOne class of methods apply attention mechanisms to exploit long-range correlation and local information of single view feature representation representation~\\cite{lyu2021hrdepth,yan2021channel,li2022depthformer}.\nThe other class of methods tried to fuse features from multiple sources through an attention mechanism.\nSome work \\cite{jung2021fsre, li2021learning} leveraged semantic information to guide depth feature representation.\nMore related to our work, \\cite{ruhkamp2021attention} proposed a spatial-temporal attention module to fuse cross-view information through two cascaded attention blocks.\nHowever, their approach did not make sufficient use of geometric information, resulting in high computational cost.\n\n\\subsection{Depth estimation in Dynamic Environment}\nVideo sequence data is usually captured in scenes containing dynamic objects. \nHowever, the non-rigid parts of the scene can affect the self-supervised training. \nTo alleviate this problem, \\cite{monodepth2} introduced an novel auto-mask method to exclude dynamic object areas. \nAlthough auto-mask can exclude dynamic object pixels, it also masks away some pixels that are useful for training.\nSome of the work \\cite{boulahbal2022instance, lee2019instance} leveraged semantic information to seperate the foreground and background object.\nThen, they estimated the object motion and camera ego-motion seperately.\nHowever, their methods require expensive labeling costs.\nRecent work attempted to model the moving object explicitly with optical flow~\\cite{yin2018geonet} or 3D motion field \\cite{li2020dynamic, lee2021attentive}. \nComparing to \\cite{lee2021attentive}, we design a one-stage warping based network to estimate the motion field, which directly represent object motion in 3D space.\nFurthermore, to improve the quality of the predicted motion fields, we also propose an iterative optimization methods with sparsity regularization loss.\n\n\\section{METHOD}\n\nIn this section, we propose Dyna-DepthFormer, a self-supervised depth estimation Framework with Transformer.\nWe first  describe the problem formulation, followed by a framework overview of our method in Section \\ref{sec:overview}.\nIn order to predict accurate depth map of the target image, we present a depth estimation network in Section \\ref{sec:depth_estimate}, which can fuse multi-view geometric information efficiently.\nIn Section \\ref{sec:motion_estimation}, we give details on the Motion Estimation Network, an module designed to deal with the moving objects and non-rigid region in training process. \nIn Section \\ref{sec:loss_function}, we further elaborate on the self-supervised loss function and regularization to finalize our training strategy.\n\n\n\\subsection{Framework Overview}\n\\label{sec:overview}\nThe overall framework, as illustrated in Fig \\ref{fig:model_pipeline}, consists of four parts: the pre-trained backbone, the depth estimation network, the motion estimation network and the view synthesis module. \n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.60\\textwidth]{figures\/pipeline.pdf}\n\\caption{\n\\textbf{Pipeline of our proposed Dyna-Depthformer architecture}. The entire pipeline includes three parts, Depth Network, Motion Network and View Synthesis Module.  For Depth Network, cross-view attention and self attention module are adopted to enhance the feature modeling and depth inferring. For Motion Network, we predict ego motion and object motion with two decoupled head. The view synthesis module reconstructs the image by differentiable warping as a way to perform self-supervised learning.\n}\n\\label{fig:model_pipeline}\n\\vspace{-4mm}\n\\end{figure*}\n\nThe pre-trained backbone takes the scaled images pair $\\{\\mathbf{I}_t,\\mathbf{I}_{s}\\}$ as input to generate multi-scale image feature. We first use motion network $\\mathbf{M}_{\\psi}$ and pose network $\\mathbf{P}_{\\phi}$ to generate ego motion $\\{\\mathbf{R}_{t \\rightarrow s},\\mathbf{t}_{t \\rightarrow s}\\}$ and object motion field $\\mathbf{T}_{res}$. Then, we perform perspective projection based attention to obtain enhanced multi-view depth feature to generate accurate depth prediction $\\mathbf{D}_t$ with depth network $\\mathbf{D}_{\\theta}$. \n\n\\begin{equation}\n    \\begin{gathered}\n\\left\\{\\mathbf{R}_{t \\rightarrow s}, \\mathbf{t}_{t \\rightarrow s}\\right\\}=\\mathbf{P}_\\phi\\left(\\left\\{\\mathbf{I}_t, \\mathbf{I}_s\\right\\}\\right) \\\\\n\\mathbf{T}_{\\text {res }}=\\mathbf{M}_{\\psi}\\left(\\left\\{\\mathbf{I}_t, \\mathbf{I}_{s \\rightarrow t}\\right\\}\\right) \\\\\n\\mathbf{D}_t=\\mathbf{D}_\\theta\\left(\\mathbf{I}_t\\right).\n\\end{gathered}\n\\end{equation}\n\nWith the predicted depth $\\mathbf{D}_t \\in \\mathbb{R}^{1 \\times H \\times W}$, relative camera pose $\\{ \\mathbf{R}_{t \\rightarrow s}, \\mathbf{t}_{t \\rightarrow s} \\}$ and object motion field $\\mathbf{T}_{res}\\in\\mathbb{R}^{3 \\times W \\times H}$, we can synthesize the target frame $\\mathbf{I}_t$ using the adjacent source frame $\\mathbf{I}_s$ by differentiable view synthesis module~\\cite{packnet}.\nThe difference between the source image and the reconstructed image can provide supervisory signal for self supervised training.\n\n\\subsection{Depth Estimation Network}\n\\label{sec:depth_estimate}\nThe goal of the depth estimation network ${D}_{\\theta}$ is to map the input RGB images $\\mathbf{I}_t$ to depth map $\\mathbf{D}_t$. \nNoted that, monocular depth estimation is inherently an ill-posed problem, which suffers from moving object, textureless region and non-rigid region, etc.\nMany previous work~\\cite{yan2021channelwise, lyu2021hrdepth,shu2020featdepth} attempted to improve feature representation by applying self-attention module to current picture feature in order to address the aforementioned issue.\nThese approaches, however, have substantial computational costs and do not leverage geometric previous information.\nInstead, we propose an perspective-project-based temporal cross-attention module to aggregate features in a learnable way.\n\n\\textbf{Feature Extract.} Given a target image $\\mathbf{I_t} \\in \\mathbb{R}^{3 \\times H \\times W}$ and its neighboring source images $\\mathbf{I_s} \\in \\{ \\mathbf{I_{t+1}},\\mathbf{I_{t-1}} \\}$, the first step is to extract the multi-scale image features of these images. \nA shared backbone~\\cite{he2016resnet} is applied, where the images are downscaled $N$ times to produce multi-scale deep features map $\\mathbf{F}_{i=\\{t-1, t, t+1\\}}^{k=0,\\ldots,N-1} \\in \\mathbb{R}^{C_k \\times \\frac{H}{2^k} \\times \\frac{W}{2^k}}$.\nThen, we feed the extracted feature maps to the subsequent module.\n\n\\textbf{Temporal Cross-Attention.} \nAs single-view depth estimation suffers from ill-posed problem, our method enhance the feature representation of target image by aggregating its neighboring image features with cross attention mechanisms.\n\nWe use the features of the target image as \\textit{Query} to match the features in the source image (\\textit{Keys}). \nHowever, due to the huge number of key-value pairs, the original attention mechanism cannot be applied directly. \nMeanwhile, the epipolar sampling method~\\cite{wang2022mvster,li2022depthformer} would suffer from camera parameter noise or scale-ambiguous problem as a result of the inaccurate pose estimate.\nInspired by \\cite{zhu2020deformable, li2022bevformer}, we obtained the key-value pairs in the neighboring image features with a learnable way to reduce computational cost and increase efficiency.\nFig.~\\ref{fig:cross_attention} illustrates the feature warping process and the generation of key-value pairs in our cross-attention module. \nGiven a query point $\\mathbf{p}_t=(u,v)$ in $\\mathbf{F}_t$, it can be warped to the source camera coordinate by:\n\n\\begin{equation}\n    \\left(\\mathbf{p}_s~1 \\right)^T = \\mathbf{K}\\left(\\mathbf{R}_{t \\rightarrow s} \\hat{\\mathbf{D}}_t\\left(\\mathbf{p}_t\\right) \\mathbf{K}^{-1}\\left (\\mathbf{p}_t~1)\\right)^T+\\mathbf{t}_{t \\rightarrow s}\\right),\n\\end{equation}\nwhere $\\mathbf{p}_s$ is the associated point of the query point on the source plane, $\\mathbf{K}$ is the camera intrinsics given in advance,  $\\{\\mathbf{R}_{t \\rightarrow s},\\mathbf{t}_{res}\\}$ indicates the relative camera pose predicted by motion network, and $\\hat{\\mathbf{D}}_t(p_t)$ is the depth value at point $\\mathbf{p}_t$. \nNote that, we use an auxiliary single scale depth decoder to generate a coarse predicted initial depth estimate $\\hat{\\mathbf{D}}_t$.\n\nThen, we search for the most relevant features of the query feature in the neighborhood of $\\mathbf{p}_s$.\nSpecifically, we learn some offsets $\\Delta \\mathbf{p} \\in \\mathbb{R}^2$ to the point $\\mathbf{p}_s$ with MLP layer.\nThe deformable attention feature is calculated by:\n\n\\begin{equation}\n    \\mathbf{F}_t^{\\prime}(\\mathbf{p}_t)=\\sum_{m=1}^M \\mathbf{W}_m\\left[\\sum_{k=1}^K A_{m  k} \\cdot \\mathbf{W}_m^{\\prime} \\mathbf{F}_s\\left(\\mathbf{p}_s+\\Delta \\mathbf{p}_{m k}\\right)\\right],\n\\end{equation}\nwhere $m$ indexes the attention head, $k$ indexes the sampled keys, $K$ is the total sampled points, $\\mathbf{F}^{\\prime}_t$ indicates the enhanced target feature, and $\\mathbf{A}_{mk}$ indicates the scalar attention weight between \\textit{Query} and \\textit{Key} embedding. \nAs $\\mathbf{p}_s+\\Delta \\mathbf{p}_{m k}$ can be a fraction, bilinear interpolation is used to compute $\\mathbf{F}_s\\left(\\mathbf{p}_s+\\Delta \\mathbf{p}_{m k}\\right)$.\nNote that $\\mathbf{p}_s$ can also be regarded as an coarse initial point generated by the geometric prior, which will be optimized in the subsequent training process.\nAfter that, We can fuse multiple aggregated features from source views by linear projection. \n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{figures\/cross_attention.pdf}\n    \\caption{\n        The details of a \\textbf{Temporal Cross Attention Module.} This module fuses adjacent frame features by deformable cross-attention mechanism to obtain a robust and discriminative depth feature representation.\n    }\n    \\label{fig:cross_attention}\n    \\vspace{-4mm}\n\\end{figure}\n\n\n\n\\textbf{Self-Attention Refinement.} \nThe enhanced feature $F_t^{\\prime}$ are generated following temporal cross-view feature aggregation, which is more robust, but matching noise is maintained. \nInspired by the cost aggregation step in the traditional stereo matching task~\\cite{sgm}, we consider leveraging self-attention to refine the local features. \nThe vectors in the feature map aggregate the discriminant features in the neighborhood based on cosine similarity to smooth the noisy information in cross-view enhanced feature map.\nSpecifically, Given the query point $\\mathbf{p}_q=(u,v)$, the query feature $z_q$, and the enhanced feature $\\mathbf{F}_t^{\\prime}$, the self-attention refined feature is calculated by:\n\\begin{equation}\n    \\mathbf{F}_t^{\\prime\\prime}(\\mathbf{p}_q)=\\sum_{m=1}^M \\mathbf{W}_m\\left[\\sum_{k=1}^K A_{m  k} \\cdot \\mathbf{W}_m^{\\prime} \\mathbf{F}_t^{\\prime}\\left(\\mathbf{p}_q+\\Delta \\mathbf{p}_{m k}\\right)\\right].\n\\end{equation}\n\nAfter obtaining the enhanced feature $\\mathbf{F}_t^{\\prime\\prime}$, we utilize a straightforward decoder mentioned in~\\cite{lyu2021hrdepth} to generate the multi-scale depth predictions $\\mathbf{D}_t$.\n\n\\begin{table*}[t!]\n\\vspace{2mm}\n\\centering\n\\caption{\\textbf{Depth estimation results} on the KITTI and Cityscapes datasets. \\emph{Multi-Fr.} indicates using multiple frames during test time. \\emph{Semantic.} indicates leveraging semantic prior during training process. K and CS denote the KITTI and the Cityscapes datasets.}\n\\label{tab:depth_result}\n\\scalebox{0.8}{\n\\begin{tabular}{lcccccccccc}\n\\toprule\n\\textbf{Method} & Multi-Fr & Semantic & Dataset & AbsRel & SqRel & RMSE & RMSE$_{log}$ & $\\delta < 1.25$ & $\\delta < 1.25^2$ & $\\delta < 1.25^3$ \\\\ \\midrule\nStruct2depth~\\cite{struct2depth} &  & \\checkmark & K & 0.141 & 1.026 & 5.291 & 0.215 & 0.816 & 0.945 & 0.979 \\\\\nGeoNet~\\cite{yin2018geonet} &  &  & K & 0.155 & 1.296 & 5.857 & 0.233 & 0.793 & 0.931 & 0.973 \\\\\nEPC++~\\cite{epc++} &  &  & K & 0.141 & 1.029 & 5.350 & 0.216 & 0.816 & 0.941 & 0.976 \\\\\nSC-SfM~\\cite{bian2019depth} &  &  & K & 0.119 & 0.857 & 4.950 & 0.197 & 0.863 & 0.957 & 0.981 \\\\\nCC~\\cite{competi_colab} &  &  & K & 0.140 & 1.070 & 5.326 & 0.217 & 0.826 & 0.941 & 0.975 \\\\\nGLNet~\\cite{glnet} &  &  & K & 0.135 & 1.070 & 5.230 & 0.210 & 0.841 & 0.948 & 0.980 \\\\\nMonodepth2~\\cite{monodepth2} &  &  & K & 0.115 & 0.882 & 4.701 & 0.190 & 0.879 & 0.961 & 0.982 \\\\\nPackNet-SFM~\\cite{packnet} &  &  & K & 0.111 & 0.785 & 4.601 & 0.189 & 0.878 & 0.960 & 0.982 \\\\\nLee \\textit{et al.} \\cite{lee2021attentive} &  & \\checkmark & K & 0.112 & 0.777 & 4.772 & 0.191 & 0.872 & 0.959 & 0.982 \\\\\nGao \\textit{et al.} \\cite{gao} &  &  & K & 0.112 & 0.866 & 4.693 & 0.189 & 0.881 & 0.961 & 0.981 \\\\\nFeatDepth~\\cite{shu2020featdepth} &  &  & K & 0.109 & 0.923 & 4.819 & 0.193 & 0.886 & 0.963 & 0.981 \\\\\nTC-Depth~\\cite{ruhkamp2021attention} & \\textbf{\\checkmark} &  & K & {0.106} & 0.770 & 4.558 & 0.187 & 0.890 & 0.964 & 0.983 \\\\\nRM-Depth~\\cite{rmdepth} &  &  & K & 0.108 & \\textbf{0.710} & 4.513 & 0.183 & 0.884 & 0.964 & 0.983 \\\\ \nManydepth~\\cite{manydepth} & \\checkmark & & K & \\underline{0.098} & 0.770 & \\underline{4.459} & \\underline{0.176} & \\textbf{0.900} & \\underline{0.965} & 0.983 \\\\\n\\textbf{Ours} & \\checkmark &  & K & \\textbf{0.094} & {\\underline{0.734}} & \\textbf{4.442} & \\textbf{0.169} & \\underline{0.893} & \\textbf{0.967} & \\textbf{0.983} \\\\ \\hline\nStruct2Depth~\\cite{struct2depth} & \\multicolumn{1}{l}{} & \\checkmark & CS & 0.145 & 1.737 & 7.280 & 0.205 & 0.813 & 0.942 & 0.976 \\\\\nMonodepth2~\\cite{monodepth2} & \\multicolumn{1}{l}{} &  & CS & 0.129 & 1.569 & 6.876 & 0.187 & 0.849 & 0.957 & 0.983 \\\\\nManydepth~\\cite{manydepth} & \\checkmark & \\multicolumn{1}{l}{} & CS & 0.114 & 1.193 & 6.223 & 0.170 & 0.875 & 0.967 & 0.989 \\\\\nInstaDM~\\cite{lee2019instance} & \\multicolumn{1}{l}{} & \\checkmark & CS & 0.111 & 1.158 & 6.437 & 0.182 & 0.868 & 0.961 & 0.983 \\\\\nLee \\textit{et al.} \\cite{lee2021attentive} & \\multicolumn{1}{l}{} & \\checkmark & CS & 0.116 & 1.213 & 6.695 & 0.186 & 0.852 & 0.951 & 0.982 \\\\\nRM-Depth~\\cite{rmdepth} & \\textbf{} & \\multicolumn{1}{l}{} & CS & \\underline{0.103} & \\underline{1.000} & \\underline{5.867} & \\underline{0.157} & \\underline{0.895} & \\underline{0.974} & \\underline{0.991} \\\\ \n\\multicolumn{1}{l}{\\textbf{Ours}} & \\checkmark & \\multicolumn{1}{l}{} & CS & \\textbf{0.100} & \\textbf{0.834} & \\textbf{5.843} & \\textbf{0.154} & \\textbf{0.901} & \\textbf{0.975} & \\multicolumn{1}{c}{\\textbf{0.992}} \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\\begin{table*}[t!]\n\\centering\n\\caption{\\textbf{Depth Estimation Result on Dynamic Object Region.} We evaluate the depth prediction accuracy of dynamic objects (e.g.,Vehicles, Person, Bike) on KITTI and Cityscapes datasets. The instance masks are obtained using Maskformer~\\cite{maskformer} pre-trained model.}\n\\label{tab:dynamic_result}\n\\scalebox{0.95}{\n\\begin{tabular}{lcccccccc}\n\\toprule\nMethods & Dataset & AbsRel & SqRel & RMSE & RMSE$_{log}$ & $\\delta < 1.25$ & $\\delta < 1.25^2$ & $\\delta < 1.25^3$ \\\\ \\midrule\nMonodepth2~\\cite{monodepth2} & K & 0.172 & 1.892 & 5.732 & 0.273 & 0.819 & 0.917 & 0.945 \\\\\nPacknet~\\cite{packnet} & K & 0.169 & 1.873 & 5.797 & 0.276 & 0.809 & 0.913 & 0.942 \\\\\nInsta-DM~\\cite{lee2019instance} & K & 0.156 & 1.327 & 5.763 & 0.272 & 0.818 & 0.917 & 0.943 \\\\\nManydepth~\\cite{manydepth} & K & 0.175 & 2.014 & 5.837 & 0.280 & 0.773 & 0.910 & 0.942 \\\\\n\\textbf{Ours} & K & \\textbf{0.148} & \\textbf{1.312} & \\textbf{5.280} & \\textbf{0.257} & \\textbf{0.837} & \\textbf{0.921} & \\textbf{0.945} \\\\ \\hline\nMonodepth2~\\cite{monodepth2} & CS & 0.159 & 1.948 & 6.492 & 0.217 & 0.820 & 0.948 & 0.981 \\\\\nPacknet~\\cite{packnet} & CS & 0.167 & 2.219 & 6.683 & 0.228 & 0.764 & 0.917 & 0.968 \\\\\nInsta-DM~\\cite{lee2019instance} & CS & 0.141 & 1.698 & 5.797 & 0.185 & 0.841 & 0.956 & 0.982 \\\\\nManydepth~\\cite{manydepth} & CS & 0.163 & 2.138 & 6.537 & 0.220 & 0.770 & 0.928 & 0.971 \\\\\n\\textbf{Ours} & CS & \\textbf{0.132} & \\textbf{1.281} & \\textbf{4.715} & \\textbf{0.170} & \\textbf{0.853} & \\textbf{0.962} & \\textbf{0.984} \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\n\n\n\n\\begin{figure*}[t]\n\\centering\n\n\\includegraphics[width=0.55\\textwidth]{figures\/depth_result.pdf}\n\\includegraphics[width=0.35\\textwidth]{figures\/flow_result.jpg}\n\\makebox[0.6\\textwidth]{\\small (a) Comparison of Depth Result }\n\\makebox[0.3\\textwidth]{\\small (b) Motion Field and Depth Result}\n\n\\caption{\n\\textbf{Qualitative depth estimation results} of our proposed Dyna-DepthFormer architecture, on the Cityscapes dataset. The results show that our method can achieve more accurate depth estimation results on dynamic objects and low-texture regions, while having sharper borders. In addition, our motion component can decouple object motion and camera ego motion efficiently.\n}\n\\vspace{-4mm}\n\\label{fig:depth_result}\n\\end{figure*}\n\n\\begin{table}[t]\n    \\vspace{2mm}\n    \\centering\n    \\caption{Ablation study of attention modules on KITTI dataset.}\n    \\label{tab:cross_attention_ablation}\n    \\begin{tabular}{@{}lccccc@{}}\n    \\toprule\n    \\textbf{Method} & AbsRel & SqRel & RMSE & $\\delta < 1.25$ & Mem (GB) \\\\ \\midrule\n    \\textbf{full} & \\textbf{0.100} & \\textbf{0.698} & \\textbf{4.236} & \\textbf{0.903} & 6.6 \\\\\n    W\/o cross-attn & 0.106 & 0.714 & 4.562 & 0.887 & 3.8 \\\\\n    W\/o self-attn & 0.102 & 0.711 & 4.383 & 0.895 & 6.3 \\\\\n    baseline & 0.108 & 0.721 & 4.421 & 0.883 & 3.5 \\\\ \n    \\midrule\n    K=1 & 0.108 & 0.774 & 4.823 & 0.882 & 3.7 \\\\\n    K=2 & 0.106 & 0.746 & 4.548 & 0.887 & 4.6 \\\\\n    K=4 & 0.100 & 0.714 & 4.320 & 0.895 & 7.8 \\\\\n    \\textbf{K=8} & \\textbf{0.097} & \\textbf{0.694} & \\textbf{4.216} & \\textbf{0.913} & 12.4 \\\\ \\hline\n    Manydepth & 0.100 & 0.770 & 4.459 & 0.900 & 16.8 \\\\\n    \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\\begin{table}[t]\n    \\vspace{2mm}\n    \\centering\n    \\begin{tabular}{@{}lcccc@{}}\n    \\toprule\n    \\textbf{Method} & AbsRel & SqRel & RMSE & $\\delta < 1.25$  \\\\ \\midrule\n    W Projection & 0.094 & 0.734 & 4.442 & 0.893 \\\\\n    W\/o Projection & 0.100 & 0.764 & 4.567 & 0.876 \\\\\n    \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\n\n\n\\begin{table}[t]\n    \\centering\n    \\caption{\n    Ablation study of the motion field estimation network on the dynamic area in Cityscapes Dataset.\n    \\emph{Motion} indicates motion field based warping. \\emph{Refine} indicates iterative refine step. \\emph{Mask} indicates the auto-mask mechanism.\n    }\n    \\label{tab:motion_ablation}\n    \\begin{tabular}{@{}cccccc@{}}\n    \\toprule\n    Motion & Refine & Mask & AbsRel & RMSE & $\\delta < 1.25$  \\\\ \\midrule\n     & & & 0.153 & 6.325 & 0.831 \\\\\n    \\checkmark & & & 0.142 & 5.835 & 0.837 \\\\\n    \\checkmark & \\checkmark & & 0.138 & 5.331 & 0.847 \\\\\n    \\checkmark & & \\checkmark & 0.136 & 5.218 & 0.850 \\\\\n    \\checkmark & \\checkmark & \\checkmark & \\textbf{0.132} & \\textbf{4.715} & \\textbf{0.853} \\\\\n    \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\n\\begin{table}[t]\n    \\vspace{2mm}\n    \\centering\n    \\caption{Ablation study of the number of iterative refinement on dynamic area in Cityscapes dataset.}\n    \\label{tab:iter_num}\n    \\begin{tabular}{@{}ccccc@{}}\n    \\toprule\n    \\textbf{Iter} & AbsRel & SqRel & RMSE & $\\delta < 1.25$  \\\\ \\midrule\n    1 & 0.136 & 1.576 & 5.342 & 0.849 \\\\\n    2 & 0.133 & 1.410 & 4.922 & 0.852 \\\\\n    4 & 0.132 & 1.293 & 4.723 & 0.853 \\\\\n    6 & \\textbf{0.132} & \\textbf{1.281} & \\textbf{4.715} & \\textbf{0.853} \\\\ \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\n\\subsection{Motion Estimate Network}\n\\label{sec:motion_estimation}\nPrevious research have tended to recover depth and camera motion simultaneously, but have ignored the non-rigid flow~\\cite{yin2018geonet} caused by moving objects, resulting in incorrect view synthesis and a direct influence on the performance of self-supervised learning.\nIn this section, we propose a novel motion estimation network that aims to address moving objects problem in a general manner.\nSpecifically, We recover the camera and object motion with two separate heads, a motion field decode head and pose decode head, while they share the same motion feature encoder. \nNote that we do not utilize any semantic prior during training process.\n\nThe camera motion is also referred to as rigid flow. \nThe shared encoder takes the image pair $\\{\\mathbf{I}_t, \\mathbf{I}_s\\}$ as input and outputs the relative pose $\\{\\mathbf{R}_{t \\rightarrow s},\\mathbf{t}_{t \\rightarrow s}\\}$ between them. \nThen, we can warp the source images $\\mathbf{I}_s$ to target frame by differentiable view synthesis\n\\begin{equation}\n    \\mathbf{I}_{s \\rightarrow t}=\\mathbf{I}_{s}\\left\\langle\\operatorname{proj}\\left(\\mathbf{D}_{t}, \\{\\mathbf{R}_{t \\rightarrow s},\\mathbf{t}_{t \\rightarrow s}\\}, \\mathbf{K}\\right)\\right\\rangle,\n\\end{equation}\nwhere  $\\left\\langle \\right\\rangle$ is the bilinear interpolation sampling operator, $\\operatorname{proj}()$ are the resulting 2D coordinates of the projected depths in $\\mathbf{I}_s$, and $\\mathbf{D}_t$ is the depth predicted by depth network.\n\nAs we synthesize the target view with the predicted camera motion, we suppose that the misaligned component is mainly caused by moving object.\nBased on the above assumption, we then feed $\\{\\mathbf{I}_t, \\mathbf{I}_{s \\rightarrow t}\\}$ into the shared encoder to generate object motion field $\\textbf{T}_{res} \\in \\mathbb{R}^{3 \\times H \\times W}$. \nIt is necessary to clarify that the motion field $\\mathbf{T}_{res}$ is not exactly the same as scene flow \\cite{teed2021raft3d}, as we consider the rotational motion of the moving object to be nearly zero.\n\n\\textbf{Iterative Refinement.} \nOur motion network are trained to decrease the local errors caused by object motion while learning the motion field.\nHowever, due to the limitation of photometric consistency, motion variation arises during training.\nTo address this problem, we propose iterative refinement step to improve consistency and completeness of the predicted motion field. \nSpecifically, we iteratively leverage $\\textbf{T}_{res}$ and $\\mathbf{I}_{t \\rightarrow s}$ to reconstruct target image frame,\n\\begin{equation}\n\\begin{gathered}\n\\mathbf{I}_{s \\rightarrow t}=\\mathbf{I}_s\\left\\langle\\operatorname{proj}\\left(\\mathbf{D}_t,\\left\\{\\mathbf{R}_{t \\rightarrow s}, \\mathbf{t}_{t \\rightarrow s}+\\mathbf{T}_{\\text {res }}\\right\\}, \\mathbf{K}\\right)\\right\\rangle \\\\\n\\Delta \\mathbf{T}=\\mathbf{M}_{\\psi}\\left(\\mathbf{I}_t, \\mathbf{I}_{t \\rightarrow s}\\right),~\n\\mathbf{T}_{\\text {res }}=\\mathbf{T}_{\\text {res }}+\\Delta \\mathbf{T},\n\\end{gathered}\n\\end{equation}\nwhere $\\mathbf{T}_{res}$ is initialized to zero. By calculating the above process in several cycles, we can obtain the complete motion.\n\n\n\\subsection{Self-Supervised Objective}\nIn this section, we present the details of our self-supervised objective, which can be fomulated as:\n\\begin{equation}\n   {L}=\\lambda_p {L}_p + \\lambda_{st} {L}_{st} + \\lambda_{sd} {L}_{sd} + \\lambda_{sp} {L}_{sp}.\n\\end{equation}\nwhere ${\\lambda_p,\\lambda_{st},\\lambda_{sd},\\lambda_{sp}}$ are human-set parameters to balance the impact of each loss function on the model training,\n\n\n\\textbf{Photometric consistency loss.}\nThe photometric loss measures the difference between the target image $\\mathbf{I}_t$ and the synthesized image $\\mathbf{I}_{s \\rightarrow t}$ for self-supervised training. The photometric error is defined as a combination of a L1 and SSIM loss:\n\\begin{equation}\n    L_p = \\alpha \\frac{1-\\operatorname{SSIM}\\left(\\mathbf{I}_{t}, \\mathbf{I}_{s \\rightarrow t}\\right)}{2}+(1-\\alpha)\\left\\|\\mathbf{I}_{t}-I_{s \\rightarrow t}\\right\\|_{1},\n\\end{equation}\nwhere $\\alpha$ is set to $0.85$. \nTo address the occlusion pixel problem, we compute the per-pixel minimum reprojection loss across all source views rather than averaging them, and we also integrate the auto-masking mechanism to filter dynamic pixel.\n\n\\textbf{Smoothness regularization.}\nFollowing \\cite{monodepth2}, we also use an edge-aware depth smoothness regularization and depth smoothness regularization to improve the performance in low texture regions:\n\n\\begin{equation}\n    {L}_{st}=\\sum_{p}\\left(\\nabla \\mathbf{T}_{res}(p) \\cdot e^{-\\nabla \\mathbf{D}_t(p)}\\right)^{2}\n\\end{equation}\n\n\\begin{equation}\n    L_{sd}=\\sum_{p}\\left(\\nabla \\mathbf{D}_{t}(p) \\cdot e^{-\\nabla \\mathbf{I}_{t}(p)}\\right)^{2}\n\\end{equation}\n\n\\textbf{Motion Field Sparsity loss.}\nIn order to encourage the sparsity of motion field prediction, we also design a motion field regularization loss:\n\n\\begin{equation}\n    {L}_{sp}= \\sum_{p} L_{g1}(\\mathbf{T}_{res}),\n\\end{equation}\nwhere $L_{g1}()$ is chosen to be the sparsity function as it is mentioned in \\cite{dynamic_scenes}.\n\n\\label{sec:loss_function}\n\n\\section{EXPERIMENTS}\n\n\\subsection{Datasets}\n\\textbf{KITTI}~\\cite{geiger2013vision} is an autonomous driving dataset collected in urban and highway areas. We adopted the pre-processing method of \\cite{zhou2017unsupervised} to remove the static frames in Eigen split~\\cite{eigen2014depth}. \nFinally, The training set includes 39,810 image triples, while the validation set has 4,424 images.\nThe depth map ground truth is generated by projecting the LiDAR pointcloud into the camera coordinate.\nNote that, we only calculate accuracy and error metrics on these sparse depth points.\nThe results shown are tested on 697 test images following the steps in \\cite{garg2016unsupervised}.\n\n\\textbf{Cityscapes}~\\cite{cityscapes} is also a popular urban driving dataset.\nThe training set contains 69,730 image triplets, and the validation set includes 1,525 images.\nWe used the cropping and evaluation protocol mentioned in~\\cite{laina2016deeper}.\n\n\\subsection{Implementation Details}\n\\textbf{Network Details.} We designed Depth Estimation Network with an encoder and decoder structure based on ResNet50 \\cite{he2016resnet}, which is pre-trained with ImageNet \\cite{deng2009imagenet}.\nThe decoder has the same structure as HR-Depth ~\\cite{lyu2021hrdepth}, and its output is a multi-scale inverse depth map.\nThe transformer module is realized follow by \\cite{zhu2020deformable}. The total attention head M is set to 8, and the total sample points K is set to 4 for efficiency. \n\n\\textbf{Training Details.} Our models are implemented using PyTorch~\\cite{paszke2017automatic} and trained across four NVIDIA GTX 2080~Ti GPUs.\nWe use the Adam optimizer~\\cite{kingma2014adam} , with $\\beta_1 = 0.9$ and $\\beta_2=0.999$, and a batch size of 4 per GPU. \nThe loss weights $\\{\\lambda_p,\\lambda_{st},\\lambda_{sd},\\lambda_{sp}\\}$ are set to $\\{1, 0.01, 0.001, 0.01\\}$.\nWe train our networks for 25 epochs, with the learning rate of 2e-4 for the first 15 epochs and reduce the learning rate to 2e-5 for the rest epochs.\nTo ensure the stability of the training process, we freeze the pose estimation network after 15 epochs. \n\n\n\\subsection{Benchmark Performance}\nIn order to validate our Dyna-DepthFormer architecture, we performed an in-depth comparison of its performance against other published methods.\n\nTable~\\ref{tab:depth_result} shows the depth prediction results on the Cityscapes \\cite{cityscapes} and KITTI \\cite{gtkitti} testing sets. \nWe compare with three types of methods: monocular depth estimation methods which predict the depth without using multi-frame information in the same mini-batch, and cost-volume based method including Manydepth~\\cite{manydepth} and TC-Depth~\\cite{ruhkamp2021attention}, and dynamic object modeling methods including InstaDM~\\cite{lee2019instance}, struct2depth~\\cite{struct2depth} and RM-Depth~\\cite{rmdepth}, etc.\nAs shown in Table~\\ref{tab:depth_result}, Dyna-DepthFormer outperforms the compared methods on both KITTI and Cityscapes datasets.\nCost-volume based methods can achieve promising result, while they suffer from feature matching noise on dynamic area, which has a large impact on depth estimation accuracy.\nDifferently, we fuse the multi-view features through a more robust cross-attention based approach and therefore achieve more accurate depth estimation results.\nCompared to the KITTI dataset, the Cityscapes dataset involves more moving objects, therefore the improvement of our method on the cityscapes dataset is more significant. \n\nTable~\\ref{tab:dynamic_result} shows the depth prediction on dynamic objects region (e.g.,Vehicles, Person, Bike).\nWe obtain the dynamic instance masks using the pre-trained state-of-the-art instance segmentation method~\\cite{maskformer}. \nTo cope with dynamic objects during training process, we did not use two-stage training scheme~\\cite{lee2021attentive} or apply instance-level semantic prior, but our approach achieved better results by decoupling object and ego motion explicitly. \nQualitative results are shown in Fig.~\\ref{fig:depth_result}.\n\n\n\\subsection{Ablation Analysis}\n\\textbf{Effectness of Attention Module.} \nAs shown in Table~\\ref{tab:cross_attention_ablation}, the complete model outperforms the baseline model by a large margin. The proposed attention components are effective in improving the depth accuracy. By removing either the cross attention or self attention module, the depth accuracy is decreased. This is to be expected, because without them, models can only predict the depth from a single \nview feature map, rather than estimating depth by matching features between adjacent frames.\nWe also modified the parameters of our architecture and evaluated its performance, obtained by modifying the number of sampling points K. \nBy observing the trends in the Table~\\ref{tab:cross_attention_ablation}, we can learn that increasing the complexity of the cross-attention network leads to better results. \nAlso, it is further demonstrated that robust feature matching is beneficial for depth estimation task.\n\n\\textbf{Effectness of Motion Field Prediction.}\nAs shown in Table~\\ref{tab:motion_ablation}, utilization of the motion field for warping-based view synthesis significantly improves depth estimate accuracy in dynamic regions. \nIn addition, we optimize the prediction of motion field by iterative refinement. As shown in Table \\ref{tab:iter_num}, the accuracy of dynamic object depth estimation generally improves as the number of repetitions grows, but ultimately approaches saturation.\n\n\\section{Conclusion}\nThis paper proposes a novel attention-based architecture for self-supervised monocular depth estimation in dynamic scenes. \nFirst, we designed a set of attention-based modules to perform feature aggregating across adjacent images efficiently, which has proved to be helpful in improving the overall performance of our network. \nSecond, we designed an object motion field estimation network to model the dynamic object explicitly without leveraging any semantic prior. The effectiveness of our algorithm has been demonstrated on various autonomous driving datasets.\nFuture work will consider the use of the proposed method for vision-based SLAM system in dynamic environments.\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEfficient manipulation of magnetic domain wall (DW) in nanostructures has drawn much attention because of its potential applications in spintronic devices such as in data storage devices \\cite{parkin2008} and logic operations \\cite{allwood2005}. Several controlling parameters, such as magnetic fields, microwaves, and spin-polarized currents, have been reported in the recent decade to drive DW in magnetic nanostructures. But these parameters suffer from certain limitations in applications. Particularly, in the case of magnetic-field driven, the DW propagation is obtained because of the energy dissipation \\cite{wang2009magnetic, wang2009high}. However, the magnetic field is unable to drive a series of DWs synchronously \\cite{atkinson2003magnetic, beach2005dynamics, hayashi2006influence}. On the other hand, an electric current can drive a DW, or a series of DWs in the same direction with the mechanism of angular momentum transfer \\cite{berger1996emission, slonczewski1996current, zhang2004roles, tatara2004theory}. Besides, a high critical current density is required to obtain a useful DW speed, which causes a Joule heating problem. \\cite{hayashi2006influence, yamaguchi2004real, yamaguchi2011temperature}. To overcome such challenges, spin-wave-spin-current generated by thermal gradient (TG) emerges as an alternative driving force for the DW motion \\cite{bauer2012spin, jiang2013direct}. Therefore, the study of TG-driven DW motion is significant not only for spintronic device applications but also for the understanding of the interplay between spin-wave and magnetic DW \\cite{wang2012magnonic, safranski2017spin}.\n\nPresently, to explain the physical picture of the TG driven DW dynamics, two theories of different origins are available, namely, microscopic (magnonic theory) theory \\cite{yan2011all, kovalev2012thermomagnonic, wang2012domain, yan2015thermodynamic} and macroscopic (thermodynamic theory) theory \\cite{schlickeiser2014role, etesami2014longitudinal,  wang2014t}. The microscopic theory predicts that more magnons are generated in the hotter part and diffuse from the hotter to the colder part, and thus a magnonic spin current is produced. While magnons pass through the DW, magnons exert a magnonic torque on the DW by transfering spin angular momentum to the DW. Thus, magnons drive the DW toward the hotter region of the nanowire, opposite to the direction of the magnonic current \\cite{wang2012magnonic, kovalev2012thermomagnonic, hinzke2011domain}. The thermodynamic theory suggests that, in order to minimize the system free energy, TG produces an entropic force which drives the DW towards the hotter region \\cite{schlickeiser2014role, etesami2014longitudinal}.\n\nIn our recent study \\cite{islam2019t}, we demonstrated the TG induced DW dynamics in a uniaxial nanowire but the nanowire structure has a significant effect on DW dynamics under TG. Therefore, in this paper, we study DW dynamics along a biaxial and uniaxial nanowire\nunder TG and present a comparison of the DW dynamics in the considered nanowires. We find that the DW propagates toward the hotter region in both nanowires. For the uniaxial nanowire, the DW propagates towards a hotter region, accompanying a rotational motion of the DW-plane around the easy axis. Both (propagation and rotational) speeds increase proportionately with the increase of TG. For the biaxial nanowire, DW also propagates in the hotter region with a significantly larger speed than the speed in the uniaxial nanowire. DW speed in biaxial nanowire initially increases with the TG. However, after certain TGs, the propagation speed decreases and afterwards, it shows an increasing trend again for the further increment of a TG, i.e., the so-called Walker breakdown phenomenon is observed. We also study the Gilbert damping ($\\alpha$) dependent DW dynamics and find that DW's both, linear and rotational speeds decrease with damping since $\\alpha$ reduces the magnonic current. So, the above findings of DW dynamics in biaxial$\\slash$uniaxial nanowire under TG may lead to realizing the spintronic (specially racetrack memory) devices.\n\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig1.pdf}  \n    \\caption{Schematic diagram of considered nanowires (Uniaxial and biaxial) with a head-to-head DW at the centre under the TG $\\nabla T$. The Blue (red) colour represents the colder (hotter) region of the sample.}\n    \\label{Fig1}\n\\end{figure}\n\n\\section{Analytical Model and Method}\nWe consider a uniaxial and biaxial ferromagnetic nanowire with a head-to-head DW at the centre along the $x$-axis (easy axis) of length $L_x$ and cross-sectional area $L_y \\times L_z$ as shown in Fig.\\ref{Fig1}. The DW width $\\Delta$ is larger than the dimension of $L_y$, $L_z$ but much smaller than $L_x$. The hard anisotropy of biaxial nanowire is given by  $K_{\\perp}=1\/2 \\mu_oM^2_s(\\mathcal{N}_y-\\mathcal{N}_z)$ along $z$-direction with $\\mathcal{N}_z \\ge \\mathcal{N}_y$, where $\\mathcal{N}_x,\\mathcal{N}_y$ and $\\mathcal{N}_z$ are demagnetization factors. However, for uniaxial nanowire, $\\mathcal{N}_z = \\mathcal{N}_y$, i.e., $K_\\perp = 0$. A TG as a driving force is applied along the nanowire. The applied highest temperature is far below the Curie temperature $T_{c}$. It is assumed that the saturation magnetization and exchange constant are temperature-independent. The magnetization dynamics is governed by the stochastic Landau-Lifshitz-Gilbert (sLLG) equation \\cite{brown1963thermal, gilbert2004phenomenological},\n\\begin{equation}\n    \\dfrac{d\\textbf{m}}{dt}=-\\textbf{m}\\times(\\textbf{h}_{\\text{eff}}+ \\textbf{h}_{\\text{th}})+\\alpha\\textbf{m}\\times\\dfrac{\\partial\\textbf{m}}{\\partial{t}},\n    \\label{sllg}\n\\end{equation}\nwhere $\\mathbf{m}=\\mathbf{M}\/M_{s}$ and $M_{s}$ are respectively the magnetization direction and the saturation magnetization. $\\alpha$ is the Gilbert damping constant and $t$ is the time measured in the units of \\((|\\gamma|M_{\\text{s}})^{-1}\\) in which $\\gamma$ is the gyromagnetic ratio. $\\mathbf{h}_\\text{eff}=\\frac{2A}{M_s}\\sum_\\sigma\\frac{\\partial \\mathbf{m}}{\\partial x_\\sigma}+K_\\parallel m_x \\hat{\\mathbf{x}} + \\mathbf{h}_\\text{dipole}$ is the effective field measured in the units of $M_s$, where $A$ is the exchange stiffness constant, $x_\\sigma$ ($\\sigma=1,2,3$) denote Cartesian coordinates $x$, $y$, $z$, $K_\\parallel$ is the easy-axis anisotropy, and $\\mathbf{h}_\\text{dipole}$ is the dipolar field and $\\mathbf{h}_\\text{th}$ is the thermal stochastic field. \n\nThe stochastic LLG equation is solved numerically by the micromagnetic simulation package MUMAX3 \\cite{vansteenkiste2014design} in which we use adaptive Heun solver. The time step is chosen $10^{-14}$s for the unit cell size ($2\\times2\\times2$) $\\text{nm}^{3}$ and $10^{-15}$s for unit cell smaller than ($2\\times2\\times2$) $\\text{nm}^{3}$. The saturation magnetization $M_\\text{s} = 8 \\times10^{5} \\text{A}\/\\text{m}$ and exchange stiffness constant $A = 13\\times10^{-12} \\text{J}\/\\text{m}$ are used to mimic permalloy in our simulations. The thermal field follows the Gaussian process characterized by following statistics\n\\begin{equation}\n    \\begin{gathered}\n        \\langle h_{\\text{th},ip}(t) \\rangle = 0, \\\\\n        \\langle h_{\\text{th},ip}(t)h_{\\text{th},jq}(t+\\Delta t) \\rangle = \\frac{2k_\\text{B}T_i \\alpha_i}{\\gamma M_\\text{s} a^3} \\delta_{ij} \\delta_{pq} \\delta(\\Delta t)\n    \\end{gathered}\n\\end{equation}\nwhere $i$ and $j$ denote the micromagnetic cells, and $p$, $q$ represent the Cartesian components of the thermal field. $T_{i}$ and $\\alpha_i$ are respectively temperature and the Gilbert damping at cell $i$, and $a$ is the cell size. $k_\\text{B}$ is the Boltzmann constant \\cite{brown1963thermal}. The numerical results presented in this study are averaged over $15$ random configurations.\n\n\n\n\\section{Numerical Results}\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig2.pdf}  \n    \\caption{A comparison of DW dynamics under TG  $\\nabla_{x}T$ along biaxial ($1024\\times30\\times4$ $\\text{nm}^{3}$) and uniaxial ($ 1024\\times4\\times4$ $\\text{nm}^{3}$) nanowire. (a)  Red and black lines represent the DW-linear speeds ($v_\\text{DW}$) along biaxial and uniaxial nanowire respectively. (b) Black and red lines represent DW-plane's rotational speeds ($\\dfrac{d\\phi}{dt}$) along uniaxial and biaxial nanowire respectively.}\n    \\label{Fig2}\n\\end{figure}\n\n\\subsection{Thermal gradient dependent DW dynamics}\n\nIn order to compare the DW dynamics (linear and DW-angular speeds) in the biaxial and uniaxial nanowires, $15$ random trajectories of DW propagation are generated for each set of parameters and the TG. Then we take the statistical ensemble average to obtain the time dependent average DW position, and hence DW's linear ($v_\\text{DW}$) and rotational (${d\\phi}\/{dt}$) speeds are estimated. In both cases, the DWs propagate toward the hotter region. The comparison between DW's linear and rotational speeds in biaxial and uniaxial nanowires is shown in Fig. \\ref{Fig2}(a) and (b), respectively. In the uniaxial nanowire under TG, the DW moves toward the hotter regions accompanying the rotation of the DW plane around the symmetry$\\slash$easy axis. DW speed proportionately increases with the TG. It is interesting to note that the DW linear speed (rotational speed) in the biaxial nanowire is significantly larger (smaller) than the DW linear speed (rotational speed) in the uniaxial nanowire. To explain the reason why the DW linear speed is larger in biaxial nanowire, one can look at the expression of hard anisotropy $K_\\perp = 1\/2 \\mu_0 M^2_s(\\mathcal{N}_z-\\mathcal{N}_y)$, say along $z$-direction for biaxial nanowire while $K_\\perp=0$ for uniaxial nanowire (since $\\mathcal{N}_z$ and $\\mathcal{N}_y$ are equal). This hard anisotropy contributes a polar torque ($\\Gamma_{\\theta}$) along $\\theta$-direction \\cite{mougin2007d} since $\\Gamma_{\\theta}$ is proportional to $(\\mathcal{N}_z-\\mathcal{N}_y)$.\n\nAfter certain TG, the DW-linear speed (rotational speed) decreases (increases), which indicates the Walker breakdown phenomena. With the further increment of TG, both DW-linear and rotational speeds show increasing trends. After the Walker breakdown limit, the biaxial nanowire behaves like a uniaxial nanowire and the DW linear and rotational speeds increase similarly to the uniaxial nanowire. These DW dynamics can be explained by microscopic theory as stating that more magnons are populated in the hotter region and travel from the hotter to the colder region. During crossing the DW, the magnon exerts a magnonic torque on the DW by transferring spin angular momentum to the DW. In this way, magnons drive the DW toward the hotter region of the nanowire, opposite to the direction of the magnonic current \\cite{yan2011all, kovalev2012thermomagnonic, wang2012domain, yan2015thermodynamic}. These findings are also consistent with the macroscopic theory \\cite{schlickeiser2014role, etesami2014longitudinal}. \n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig3.pdf}  \n    \\caption{Damping ($\\alpha$) dependent (a) DW-linear speed represented by a red line (black line) along biaxial (uniaxial) nanowire. (b) DW-plane's rotational speed is represented by a red line (black line) along biaxial (uniaxial) nanowire.}\n    \\label{Fig3}\n\\end{figure}\n\n\n\\subsection{Damping dependent DW dynamics}\nGilbert damping coefficient $\\alpha$ is a crucial material parameter that has a significant effect on DW dynamics under TG. For fixed  $\\nabla_{x}T =0.5$, we investigate the DW dynamics in biaxial and uniaxial nanowires of length $L_x= 1024$ nm. The obtained results of DW's linear and rotational speeds as a function of $\\alpha$ are presented as in Fig. \\ref{Fig3}(a) and (b), respectively. In both nanowires, DW's speeds increase initially and then decrease with increasing $\\alpha$. Usually, the DW velocities should be decreased since the magnon propagation length decreases with damping. However, we observe that the DW speeds increase initially till a certain $\\alpha$ and then decrease monotonically with $\\alpha$. To explain this observation, one can recall the basic physics of generating standing waves in the nanowire with fixed boundaries. If the length of the nanowire is not large enough and for lower damping, there is a probability to form the standing waves of some modes of spin-wave between the fixed boundaries. It is well known that the standing spin-wave does not carry any energy$\\slash$spin angular momentum. Consequently, the energy transfer to the DW is decreased, i.e., DW velocity is decreased due to the decrement of the magnonic current. However, for larger $\\alpha$, the number of standing spin-waves decreases due to the reduction of propagation length of spin waves. Therefore, all modes of the spin-wave-spin current effectively interact with the DW, and hence the DW speeds increase. It is predicted that for the larger length of the nanowire, it may be observed that the DW speeds will decrease monotonically with  $\\alpha$. The above statement would be justified in the next subsection. For a further increment of $\\alpha$, the magnon propagation length decreases with the damping, so the amount of spin angular momentum deposited on a DW should decrease. As a result, the DW propagation speed and DW-plane rotation speed decrease with $\\alpha$ as shown in Fig. \\ref{Fig3}(a) and (b), respectively. \n\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig4.pdf}  \n    \\caption{Damping $\\alpha$ dependent (a) DW-linear speed represented by black line (red line) along biaxial nanowire of length $L_x=1024$ nm ($L_x=2048$ nm). (b) DW-plane's rotational speed represented by black line (red line) along biaxial nanowire of length $L_x=1024$ nm ($L_x=2048$ nm).}\n    \\label{Fig4}\n\\end{figure}\n\n\\subsection{Length dependent DW dynamics}\nHere, we present the behaviour of DW dynamics in two different lengths i.e., $L_{x}=1024$ nm and $L_{x}=2048$ nm of biaxial nanowire in Fig. \\ref{Fig4}(a) and (b) respectively, as a function of $\\alpha$ (ranging from 0.001 to 0.006) for fixed TG $\\nabla_{x}T = 0.5$ $\\text{K}\/\\text{nm}$. Fig. 4(a) shows the DW linear velocity as a function of $\\alpha$ corresponding to the nanowire lengths, $L_{x}=1024$ nm (black) and $L_{x}=2048$ nm (red). In Fig. \\ref{Fig4}(b), the black line and red line show the change of angular velocity as a function of $\\alpha$ corresponding to the lengths $L_{x}=1024$ nm and $L_{x}=2048$ nm respectively. DW angular velocities also decrease with $\\alpha$. Since the magnons propagation length decreases with $\\alpha$ \\cite{hinzke2011domain}, it reduces the number of magnons to pass through the DW, which leads to the reduction of DW velocity proportionately. It is noted that for smaller $L_{x}=1024$ nm (Fig. \\ref{Fig4}(a)), the DW speed is lower for $\\alpha = 0.001$, and it gradually increases and then monotonically decreases. Interestingly, such behaviour disappears with the increase of length, and for the length $L_{x}=2048$ nm and the DW velocities show decreasing trend monotonically. The reason for observing such behaviour has been discussed above for shorter $L_{x}=1024$ nm. Here we mention again shortly that for lower damping, the magnon propagation length is larger. If the length of the sample size is not large enough, then there is a probability of forming the standing waves of some modes between the fixed boundaries. Consequently, the energy transfer through the DW is decreased, i.e., DW velocity is decreased because of the reduction of the magnonic current.\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig5.pdf}  \n    \\caption{(a) Square and (b) cylindrical cross-sectional area  dependent DW propagation along uniaxial nanowire.}\n    \\label{Fig5}\n\\end{figure}\n\n\\subsection{Cross-sectional area dependent DW dynamics}\nThe cross-sectional area of the nanowire has an influence on the DW dynamics. Fig. \\ref{Fig5}(a) and (b) show the DW dynamics in uniaxial nanowires of square and cylindrical cross-sectional area, respectively. For the nanowire with square cross-sectional area, it is observed that the DW propagates towards the hotter region with oscillatory or wavy nature. For larger cross-sectional area ($32\\times32$) nm$^2$, the oscillation amplitude is larger. As we know from previous observation that the DW-plane rotates during propagation in the uniaxial nanowire. Thus the spins of DW experience asymmetric shape anisotropy during rotation as it faces the easy and hard axis anisotropy successively. When the DW faces hard anisotropy, its propagation speed increases and vice-versa. So the DW propagates with wavy nature. This wavy nature of DW propagation decreases in the sample of cylindrical cross-section as shown in Fig. \\ref{Fig5}(b). The reason is that, in the sample of cylindrical cross-section, the DW experiences almost symmetric shape anisotropy. Fig. \\ref{Fig5}(b) shows still small wavy nature propagation in the sample of cylindrical cross-sectional area because it is technically difficult to simulate the perfect cylindrical cross-section.\n\n\n\\section{Discussions and Conclusions}\nIn this study, we investigate the DW dynamics in uniaxial and biaxial nanowires under TG. We found that the DW propagates toward the hotter region in both nanowires. For the uniaxial nanowire, the DW propagates towards a hotter region, accompanying a rotational motion of the DW-plane around the easy axis. Both (propagation and rotational) speeds increase proportionately with the increase of TG. For biaxial nanowire, DW also propagates in the hotter region, and DW propagation speed is significantly larger than the speed in the uniaxial nanowire. DW speed in biaxial nanowire initially increases with the TG. But after certain TGs, the propagation speed decreases and afterwards, it shows an increasing trend again for the further increment of a TG, i.e., the so-called Walker breakdown phenomenon is observed. We also study the Gilbert damping $\\alpha$ dependent DW dynamics and found that DW's both linear and rotational speeds decrease with damping since $\\alpha$ reduces the magnonic current. In this study, we also examined the length and cross-sectional area dependent DW dynamics with damping coefficient. We found that it is required to optimize the damping coefficient with the nanowire structure to obtain the efficient DW speed under TG. Therefore the above findings of DW dynamics in biaxial$\\slash$uniaxial nanowire under TG may might be useful for the fundamental interests and applications in spintronic devices.\n\n\n\\section*{Acknowledgments}\nThis research acknowledges Khulna University Research Cell\n(Grant No. KU\/RC-04\/2000-158), Khulna, Bangladesh.\n\n\\bibliographystyle{elsarticle-num}\n\n\\section{Introduction}\nEfficient manipulation of magnetic domain wall (DW) in nanostructures has drawn much attention because of its potential applications in spintronic devices such as in data storage devices \\cite{parkin2008} and logic operations \\cite{allwood2005}. Several controlling parameters, such as magnetic fields, microwaves, and spin-polarized currents, have been reported in the recent decade to drive DW in magnetic nanostructures. But these parameters suffer from certain limitations in applications. Particularly, in the case of magnetic-field driven, the DW propagation is obtained because of the energy dissipation \\cite{wang2009magnetic, wang2009high}. However, the magnetic field is unable to drive a series of DWs synchronously \\cite{atkinson2003magnetic, beach2005dynamics, hayashi2006influence}. On the other hand, an electric current can drive a DW, or a series of DWs in the same direction with the mechanism of angular momentum transfer \\cite{berger1996emission, slonczewski1996current, zhang2004roles, tatara2004theory}. Besides, a high critical current density is required to obtain a useful DW speed, which causes a Joule heating problem. \\cite{hayashi2006influence, yamaguchi2004real, yamaguchi2011temperature}. To overcome such challenges, spin-wave-spin-current generated by thermal gradient (TG) emerges as an alternative driving force for the DW motion \\cite{bauer2012spin, jiang2013direct}. Therefore, the study of TG-driven DW motion is significant not only for spintronic device applications but also for the understanding of the interplay between spin-wave and magnetic DW \\cite{wang2012magnonic, safranski2017spin}.\n\nPresently, to explain the physical picture of the TG driven DW dynamics, two theories of different origins are available, namely, microscopic (magnonic theory) theory \\cite{yan2011all, kovalev2012thermomagnonic, wang2012domain, yan2015thermodynamic} and macroscopic (thermodynamic theory) theory \\cite{schlickeiser2014role, etesami2014longitudinal,  wang2014t}. The microscopic theory predicts that more magnons are generated in the hotter part and diffuse from the hotter to the colder part, and thus a magnonic spin current is produced. While magnons pass through the DW, magnons exert a magnonic torque on the DW by transfering spin angular momentum to the DW. Thus, magnons drive the DW toward the hotter region of the nanowire, opposite to the direction of the magnonic current \\cite{wang2012magnonic, kovalev2012thermomagnonic, hinzke2011domain}. The thermodynamic theory suggests that, in order to minimize the system free energy, TG produces an entropic force which drives the DW towards the hotter region \\cite{schlickeiser2014role, etesami2014longitudinal}.\n\nIn our recent study \\cite{islam2019t}, we demonstrated the TG induced DW dynamics in a uniaxial nanowire but the nanowire structure has a significant effect on DW dynamics under TG. Therefore, in this paper, we study DW dynamics along a biaxial and uniaxial nanowire\nunder TG and present a comparison of the DW dynamics in the considered nanowires. We find that the DW propagates toward the hotter region in both nanowires. For the uniaxial nanowire, the DW propagates towards a hotter region, accompanying a rotational motion of the DW-plane around the easy axis. Both (propagation and rotational) speeds increase proportionately with the increase of TG. For the biaxial nanowire, DW also propagates in the hotter region with a significantly larger speed than the speed in the uniaxial nanowire. DW speed in biaxial nanowire initially increases with the TG. However, after certain TGs, the propagation speed decreases and afterwards, it shows an increasing trend again for the further increment of a TG, i.e., the so-called Walker breakdown phenomenon is observed. We also study the Gilbert damping ($\\alpha$) dependent DW dynamics and find that DW's both, linear and rotational speeds decrease with damping since $\\alpha$ reduces the magnonic current. So, the above findings of DW dynamics in biaxial$\\slash$uniaxial nanowire under TG may lead to realizing the spintronic (specially racetrack memory) devices.\n\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig1.pdf}  \n    \\caption{Schematic diagram of considered nanowires (Uniaxial and biaxial) with a head-to-head DW at the centre under the TG $\\nabla T$. The Blue (red) colour represents the colder (hotter) region of the sample.}\n    \\label{Fig1}\n\\end{figure}\n\n\\section{Analytical Model and Method}\nWe consider a uniaxial and biaxial ferromagnetic nanowire with a head-to-head DW at the centre along the $x$-axis (easy axis) of length $L_x$ and cross-sectional area $L_y \\times L_z$ as shown in Fig.\\ref{Fig1}. The DW width $\\Delta$ is larger than the dimension of $L_y$, $L_z$ but much smaller than $L_x$. The hard anisotropy of biaxial nanowire is given by  $K_{\\perp}=1\/2 \\mu_oM^2_s(\\mathcal{N}_y-\\mathcal{N}_z)$ along $z$-direction with $\\mathcal{N}_z \\ge \\mathcal{N}_y$, where $\\mathcal{N}_x,\\mathcal{N}_y$ and $\\mathcal{N}_z$ are demagnetization factors. However, for uniaxial nanowire, $\\mathcal{N}_z = \\mathcal{N}_y$, i.e., $K_\\perp = 0$. A TG as a driving force is applied along the nanowire. The applied highest temperature is far below the Curie temperature $T_{c}$. It is assumed that the saturation magnetization and exchange constant are temperature-independent. The magnetization dynamics is governed by the stochastic Landau-Lifshitz-Gilbert (sLLG) equation \\cite{brown1963thermal, gilbert2004phenomenological},\n\\begin{equation}\n    \\dfrac{d\\textbf{m}}{dt}=-\\textbf{m}\\times(\\textbf{h}_{\\text{eff}}+ \\textbf{h}_{\\text{th}})+\\alpha\\textbf{m}\\times\\dfrac{\\partial\\textbf{m}}{\\partial{t}},\n    \\label{sllg}\n\\end{equation}\nwhere $\\mathbf{m}=\\mathbf{M}\/M_{s}$ and $M_{s}$ are respectively the magnetization direction and the saturation magnetization. $\\alpha$ is the Gilbert damping constant and $t$ is the time measured in the units of \\((|\\gamma|M_{\\text{s}})^{-1}\\) in which $\\gamma$ is the gyromagnetic ratio. $\\mathbf{h}_\\text{eff}=\\frac{2A}{M_s}\\sum_\\sigma\\frac{\\partial \\mathbf{m}}{\\partial x_\\sigma}+K_\\parallel m_x \\hat{\\mathbf{x}} + \\mathbf{h}_\\text{dipole}$ is the effective field measured in the units of $M_s$, where $A$ is the exchange stiffness constant, $x_\\sigma$ ($\\sigma=1,2,3$) denote Cartesian coordinates $x$, $y$, $z$, $K_\\parallel$ is the easy-axis anisotropy, and $\\mathbf{h}_\\text{dipole}$ is the dipolar field and $\\mathbf{h}_\\text{th}$ is the thermal stochastic field. \n\nThe stochastic LLG equation is solved numerically by the micromagnetic simulation package MUMAX3 \\cite{vansteenkiste2014design} in which we use adaptive Heun solver. The time step is chosen $10^{-14}$s for the unit cell size ($2\\times2\\times2$) $\\text{nm}^{3}$ and $10^{-15}$s for unit cell smaller than ($2\\times2\\times2$) $\\text{nm}^{3}$. The saturation magnetization $M_\\text{s} = 8 \\times10^{5} \\text{A}\/\\text{m}$ and exchange stiffness constant $A = 13\\times10^{-12} \\text{J}\/\\text{m}$ are used to mimic permalloy in our simulations. The thermal field follows the Gaussian process characterized by following statistics\n\\begin{equation}\n    \\begin{gathered}\n        \\langle h_{\\text{th},ip}(t) \\rangle = 0, \\\\\n        \\langle h_{\\text{th},ip}(t)h_{\\text{th},jq}(t+\\Delta t) \\rangle = \\frac{2k_\\text{B}T_i \\alpha_i}{\\gamma M_\\text{s} a^3} \\delta_{ij} \\delta_{pq} \\delta(\\Delta t)\n    \\end{gathered}\n\\end{equation}\nwhere $i$ and $j$ denote the micromagnetic cells, and $p$, $q$ represent the Cartesian components of the thermal field. $T_{i}$ and $\\alpha_i$ are respectively temperature and the Gilbert damping at cell $i$, and $a$ is the cell size. $k_\\text{B}$ is the Boltzmann constant \\cite{brown1963thermal}. The numerical results presented in this study are averaged over $15$ random configurations.\n\n\n\n\\section{Numerical Results}\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig2.pdf}  \n    \\caption{A comparison of DW dynamics under TG  $\\nabla_{x}T$ along biaxial ($1024\\times30\\times4$ $\\text{nm}^{3}$) and uniaxial ($ 1024\\times4\\times4$ $\\text{nm}^{3}$) nanowire. (a)  Red and black lines represent the DW-linear speeds ($v_\\text{DW}$) along biaxial and uniaxial nanowire respectively. (b) Black and red lines represent DW-plane's rotational speeds ($\\dfrac{d\\phi}{dt}$) along uniaxial and biaxial nanowire respectively.}\n    \\label{Fig2}\n\\end{figure}\n\n\\subsection{Thermal gradient dependent DW dynamics}\n\nIn order to compare the DW dynamics (linear and DW-angular speeds) in the biaxial and uniaxial nanowires, $15$ random trajectories of DW propagation are generated for each set of parameters and the TG. Then we take the statistical ensemble average to obtain the time dependent average DW position, and hence DW's linear ($v_\\text{DW}$) and rotational (${d\\phi}\/{dt}$) speeds are estimated. In both cases, the DWs propagate toward the hotter region. The comparison between DW's linear and rotational speeds in biaxial and uniaxial nanowires is shown in Fig. \\ref{Fig2}(a) and (b), respectively. In the uniaxial nanowire under TG, the DW moves toward the hotter regions accompanying the rotation of the DW plane around the symmetry$\\slash$easy axis. DW speed proportionately increases with the TG. It is interesting to note that the DW linear speed (rotational speed) in the biaxial nanowire is significantly larger (smaller) than the DW linear speed (rotational speed) in the uniaxial nanowire. To explain the reason why the DW linear speed is larger in biaxial nanowire, one can look at the expression of hard anisotropy $K_\\perp = 1\/2 \\mu_0 M^2_s(\\mathcal{N}_z-\\mathcal{N}_y)$, say along $z$-direction for biaxial nanowire while $K_\\perp=0$ for uniaxial nanowire (since $\\mathcal{N}_z$ and $\\mathcal{N}_y$ are equal). This hard anisotropy contributes a polar torque ($\\Gamma_{\\theta}$) along $\\theta$-direction \\cite{mougin2007d} since $\\Gamma_{\\theta}$ is proportional to $(\\mathcal{N}_z-\\mathcal{N}_y)$.\n\nAfter certain TG, the DW-linear speed (rotational speed) decreases (increases), which indicates the Walker breakdown phenomena. With the further increment of TG, both DW-linear and rotational speeds show increasing trends. After the Walker breakdown limit, the biaxial nanowire behaves like a uniaxial nanowire and the DW linear and rotational speeds increase similarly to the uniaxial nanowire. These DW dynamics can be explained by microscopic theory as stating that more magnons are populated in the hotter region and travel from the hotter to the colder region. During crossing the DW, the magnon exerts a magnonic torque on the DW by transferring spin angular momentum to the DW. In this way, magnons drive the DW toward the hotter region of the nanowire, opposite to the direction of the magnonic current \\cite{yan2011all, kovalev2012thermomagnonic, wang2012domain, yan2015thermodynamic}. These findings are also consistent with the macroscopic theory \\cite{schlickeiser2014role, etesami2014longitudinal}. \n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig3.pdf}  \n    \\caption{Damping ($\\alpha$) dependent (a) DW-linear speed represented by a red line (black line) along biaxial (uniaxial) nanowire. (b) DW-plane's rotational speed is represented by a red line (black line) along biaxial (uniaxial) nanowire.}\n    \\label{Fig3}\n\\end{figure}\n\n\n\\subsection{Damping dependent DW dynamics}\nGilbert damping coefficient $\\alpha$ is a crucial material parameter that has a significant effect on DW dynamics under TG. For fixed  $\\nabla_{x}T =0.5$, we investigate the DW dynamics in biaxial and uniaxial nanowires of length $L_x= 1024$ nm. The obtained results of DW's linear and rotational speeds as a function of $\\alpha$ are presented as in Fig. \\ref{Fig3}(a) and (b), respectively. In both nanowires, DW's speeds increase initially and then decrease with increasing $\\alpha$. Usually, the DW velocities should be decreased since the magnon propagation length decreases with damping. However, we observe that the DW speeds increase initially till a certain $\\alpha$ and then decrease monotonically with $\\alpha$. To explain this observation, one can recall the basic physics of generating standing waves in the nanowire with fixed boundaries. If the length of the nanowire is not large enough and for lower damping, there is a probability to form the standing waves of some modes of spin-wave between the fixed boundaries. It is well known that the standing spin-wave does not carry any energy$\\slash$spin angular momentum. Consequently, the energy transfer to the DW is decreased, i.e., DW velocity is decreased due to the decrement of the magnonic current. However, for larger $\\alpha$, the number of standing spin-waves decreases due to the reduction of propagation length of spin waves. Therefore, all modes of the spin-wave-spin current effectively interact with the DW, and hence the DW speeds increase. It is predicted that for the larger length of the nanowire, it may be observed that the DW speeds will decrease monotonically with  $\\alpha$. The above statement would be justified in the next subsection. For a further increment of $\\alpha$, the magnon propagation length decreases with the damping, so the amount of spin angular momentum deposited on a DW should decrease. As a result, the DW propagation speed and DW-plane rotation speed decrease with $\\alpha$ as shown in Fig. \\ref{Fig3}(a) and (b), respectively. \n\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig4.pdf}  \n    \\caption{Damping $\\alpha$ dependent (a) DW-linear speed represented by black line (red line) along biaxial nanowire of length $L_x=1024$ nm ($L_x=2048$ nm). (b) DW-plane's rotational speed represented by black line (red line) along biaxial nanowire of length $L_x=1024$ nm ($L_x=2048$ nm).}\n    \\label{Fig4}\n\\end{figure}\n\n\\subsection{Length dependent DW dynamics}\nHere, we present the behaviour of DW dynamics in two different lengths i.e., $L_{x}=1024$ nm and $L_{x}=2048$ nm of biaxial nanowire in Fig. \\ref{Fig4}(a) and (b) respectively, as a function of $\\alpha$ (ranging from 0.001 to 0.006) for fixed TG $\\nabla_{x}T = 0.5$ $\\text{K}\/\\text{nm}$. Fig. 4(a) shows the DW linear velocity as a function of $\\alpha$ corresponding to the nanowire lengths, $L_{x}=1024$ nm (black) and $L_{x}=2048$ nm (red). In Fig. \\ref{Fig4}(b), the black line and red line show the change of angular velocity as a function of $\\alpha$ corresponding to the lengths $L_{x}=1024$ nm and $L_{x}=2048$ nm respectively. DW angular velocities also decrease with $\\alpha$. Since the magnons propagation length decreases with $\\alpha$ \\cite{hinzke2011domain}, it reduces the number of magnons to pass through the DW, which leads to the reduction of DW velocity proportionately. It is noted that for smaller $L_{x}=1024$ nm (Fig. \\ref{Fig4}(a)), the DW speed is lower for $\\alpha = 0.001$, and it gradually increases and then monotonically decreases. Interestingly, such behaviour disappears with the increase of length, and for the length $L_{x}=2048$ nm and the DW velocities show decreasing trend monotonically. The reason for observing such behaviour has been discussed above for shorter $L_{x}=1024$ nm. Here we mention again shortly that for lower damping, the magnon propagation length is larger. If the length of the sample size is not large enough, then there is a probability of forming the standing waves of some modes between the fixed boundaries. Consequently, the energy transfer through the DW is decreased, i.e., DW velocity is decreased because of the reduction of the magnonic current.\n\n\\begin{figure}\n    \\centering\n\t\\includegraphics[width=80mm]{Figs\/Fig5.pdf}  \n    \\caption{(a) Square and (b) cylindrical cross-sectional area  dependent DW propagation along uniaxial nanowire.}\n    \\label{Fig5}\n\\end{figure}\n\n\\subsection{Cross-sectional area dependent DW dynamics}\nThe cross-sectional area of the nanowire has an influence on the DW dynamics. Fig. \\ref{Fig5}(a) and (b) show the DW dynamics in uniaxial nanowires of square and cylindrical cross-sectional area, respectively. For the nanowire with square cross-sectional area, it is observed that the DW propagates towards the hotter region with oscillatory or wavy nature. For larger cross-sectional area ($32\\times32$) nm$^2$, the oscillation amplitude is larger. As we know from previous observation that the DW-plane rotates during propagation in the uniaxial nanowire. Thus the spins of DW experience asymmetric shape anisotropy during rotation as it faces the easy and hard axis anisotropy successively. When the DW faces hard anisotropy, its propagation speed increases and vice-versa. So the DW propagates with wavy nature. This wavy nature of DW propagation decreases in the sample of cylindrical cross-section as shown in Fig. \\ref{Fig5}(b). The reason is that, in the sample of cylindrical cross-section, the DW experiences almost symmetric shape anisotropy. Fig. \\ref{Fig5}(b) shows still small wavy nature propagation in the sample of cylindrical cross-sectional area because it is technically difficult to simulate the perfect cylindrical cross-section.\n\n\n\\section{Discussions and Conclusions}\nIn this study, we investigate the DW dynamics in uniaxial and biaxial nanowires under TG. We found that the DW propagates toward the hotter region in both nanowires. For the uniaxial nanowire, the DW propagates towards a hotter region, accompanying a rotational motion of the DW-plane around the easy axis. Both (propagation and rotational) speeds increase proportionately with the increase of TG. For biaxial nanowire, DW also propagates in the hotter region, and DW propagation speed is significantly larger than the speed in the uniaxial nanowire. DW speed in biaxial nanowire initially increases with the TG. But after certain TGs, the propagation speed decreases and afterwards, it shows an increasing trend again for the further increment of a TG, i.e., the so-called Walker breakdown phenomenon is observed. We also study the Gilbert damping $\\alpha$ dependent DW dynamics and found that DW's both linear and rotational speeds decrease with damping since $\\alpha$ reduces the magnonic current. In this study, we also examined the length and cross-sectional area dependent DW dynamics with damping coefficient. We found that it is required to optimize the damping coefficient with the nanowire structure to obtain the efficient DW speed under TG. Therefore the above findings of DW dynamics in biaxial$\\slash$uniaxial nanowire under TG may might be useful for the fundamental interests and applications in spintronic devices.\n\n\n\\section*{Acknowledgments}\nThis research acknowledges Khulna University Research Cell\n(Grant No. KU\/RC-04\/2000-158), Khulna, Bangladesh.\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{Intro}}\n\nJust as the discovery of high-temperature superconductivity in La$_{2-x}$Ba$_x$CuO$_{4}$\\ \\cite{bedn86,taka87,bedn87} led to a continuous process of revelation in doped cuprates \\cite{keim15}, a similar evolution of understanding has taken place with the parent compound La$_2$CuO$_{4}$\\ (LCO).  In an earlier burst of enthusiasm for transition-metal oxide compounds, it had been determined by X-ray diffraction that LCO transforms from a high-temperature tetragonal (HTT) phase (then known as the K$_2$NiF$_4$-type structure) to a low-temperature orthorhombic (LTO) phase at a temperature above 500~K \\footnote{P. Lehuede and M. Daire, Sur la structure et les propri\\'et\\'es du compos\\'e La$_2$CuO$_4$, C. R. Acad. Sci. Paris {\\bf 276}, C-1011 (1973).}\\cite{long73,gran77}, and this was soon confirmed by neutron diffraction \\cite{jorg87}.  When single crystals became available, the soft phonon modes associated with tilts of the CuO$_6$ octahedra were identified by inelastic neutron scattering \\cite{birg87,boni88}, \n\nNeutron diffraction also led to the discovery of antiferromagnetic order in LCO \\cite{yama87b}; however, because of the small size of the Cu moment and the reduction of the order parameter by quantum fluctuations \\cite{shir87,chak88}, it took further work to properly characterize the magnetic order \\cite{vakn87,mits87}.  Despite the strong two-dimensional character of the spin correlations, the magnetic susceptibility exhibits a peak at the N\\'eel temperature, $T_{\\rm N}$, that turns out to be a consequence of spin-orbit-coupling effects [Dzyaloshinsky-Moriya (DM) interaction and pseudo-dipolar coupling] combined with the octahedral tilts \\cite{shek92,thio94}.  The magnetic moments tend to lie within the CuO$_2$ planes, but there is a very small canting along the $c$ axis (perpendicular to the planes) that has a unique direction within each layer, but which cancels out due to an antiferromagnetic stacking of layers in the ordered state.  Applying a magnetic field of sufficient strength perpendicular to the planes causes a transition of the magnetic order to a weakly-ferromagnetic phase \\cite{kast88,reeh06}.\n\nThe early samples of LCO, grown in air, exhibited semiconducting behavior, had $T_{\\rm N}\\approx240$~K \\cite{thio88}, and occasionally showed signs of superconductivity \\cite{gree87}.  Further investigation demonstrated that this was a consequence of intercalation by excess oxygen \\cite{john87,sayl89}, with phase separation occurring between the antiferromagnetic and superconducting phases \\cite{jorg88,hamm90,well96,well97}.  Samples on the insulating edge of the miscibility gap have the low $T_{\\rm N}$, but annealing in Ar gas can raise the transition to 325~K \\cite{keim92a}.  Optical measurements on such samples demonstrated the substantial charge-transfer gap of the undoped compound \\cite{thio90,toku90}.\n\nThe transition to the orthorhombic phase generally results in strong twinning. With the preparation of detwinned crystals, it was discovered that the magnetic susceptibility peak at $T_{\\rm N}$ seen with the field along $c$ is also present when the field is along $b$, the in-plane direction along which the octahedra tilt \\cite{lavr01}.  This observation led to a reanalysis of the theoretical model for the susceptibility \\cite{tabu05,silv06}.  It also motivated a reinvestigation of the crystal structure, which resulted in the discovery that the symmetry of the LTO phase is actually monoclinic \\cite{reeh06}, as revealed by the appearance of weak diffraction peaks not allowed by the orthorhombic model.\n\nMore recently, a new surprise has been discovered.  Grissonnanche {\\it et al.} \\cite{gris19} reported the observation of a large thermal Hall effect in La$_2$CuO$_{4}$\\ and in related Sr-doped samples.  (This is a measurement in which a temperature gradient is applied along the longitudinal direction, a transverse magnetic field is applied, and the resulting temperature difference across the other transverse direction is measured.)  A follow on study \\cite{gris20} found similar responses both within the CuO$_2$ planes and perpendicular to them, leading to the conclusion that the transport response must be due to phonons.  The nature of those ``chiral'' phonons remains an open question.\n\nWith this motivation, we have taken another look at the lattice symmetry and fluctuations in LCO using neutron scattering.  While confirming the superlattice peaks previously attributed to a monoclinic phase \\cite{reeh06}, we find finite intensity at another set of peaks [of the type previously detected in La$_{2-x}$Sr$_x$CuO$_{4}$\\ (LSCO) with $x=0.07$ \\cite{jaco15}] that imply a further reduction in symmetry.  These peaks correspond to the second octahedral tilt mode (orthogonal to the one that goes soft at the transition to the nominal orthorhombic phase) associated with the residual soft phonon mode found in LSCO \\cite{axe89,kimu00,waki04b}.  If this mode were to go completely soft, it would result in the low-temperature-tetragonal (LTT) phase found in La$_{2-x}$Ba$_x$CuO$_{4}$\\ and in Nd-doped and Eu-doped LSCO \\cite{axe94}.  Intriguingly, the LTT phase of the latter compounds exhibits a weak ferromagnetic response that mimics the high-field response of LCO \\cite{craw93,huck04}.  It is known that the impact of spin-orbit coupling on the magnetic structure depends on the crystal electric field; therefore, the magnetic response is sensitive to the pattern of the octahedral tilts \\cite{bone92,kosh94}.  Hence, we propose that the observed soft mode corresponds to the ``chiral'' phonon inferred from the thermal Hall effect \\cite{gris20}.\n\nOur measurements also capture the antiferromagnetic spin waves.  We check for a possible interaction between the spin waves and a low-energy optical mode of the type seen in Ba- and Sr-doped LCO \\cite{wagm15,wagm16}, but find no evidence for such an interaction in the commensurate antiferromagnet, consistent with another recent neutron study \\cite{ikeu21}.\n\nThe rest of the paper is organized as follows.  The experimental methods and initial sample characterization are described in the next section.  The main neutron scattering results and analysis are presented in Sec.~III.  We conclude with a discussion and summary of our results in Sec.~IV.\n\n\\section{Experimental Details\\label{ED}}\n\n\\subsection{Crystal growth and annealing}\n\nSingle crystals of La$_2$CuO$_{4}$\\ (LCO) were grown at Brookhaven National Laboratory in an infrared image furnace by the traveling--solvent floating--zone method.  A large crystal of LCO was annealed in a vacuum at $10^{-3}$ torr with the following temperature profile: heat from 30~$^\\circ$C to 910~$^\\circ$C over 3~h, hold for 1~h, cool to 600~$^\\circ$C over 12~h, cool to 500~$^\\circ$C for 200 h, and finally cool to room temperature over 12 h.  The magnetic transition was determined from a measurement of the temperature dependence of magnetization using a commercial SQUID (superconducting quantum interference device) magnetometer on a small piece of the crystal.  The resulting magnetic susceptibility is shown in Fig.~\\ref{MND}(a).  The peak indicates $T_{\\rm N} = 327$~K, consistent with previous work \\cite{keim92a}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.7\\columnwidth]{fig_Susc_ND}\n\t\\caption{(a) Temperature dependence of magnetic susceptibility ($\\chi =M\/H$) measured with a field of $\\mu_\\mathrm{o}H = 0.1$~T along a random direction. Peak in the figure corresponds to N\\'eel temperature $T_\\mathrm{N}$. (b) Temperature dependence of the integrated intensity of $(1, 0, 0)$ magnetic and $(0,1,0)$ nuclear Bragg peaks (in close proximity due to twinning, but not resolved), illustrating the evolution of the antiferromagnetic order and the structural transition temperature, $T_{\\rm s}=555\\pm5$~K, to the HTT phase.}\n\t\\label{MND}\n\\end{figure}\n\n\\subsection{Neutron scattering}\n\nTo discuss the scattering measurements, we will express the momentum transfer $\\textbf{Q} = (H, K,L)$ in reciprocal lattice units (rlu) given by $(2\\pi\/a,2\\pi\/b,2\\pi\/c)$ with $a\\approx b \\approx 5.37$~\\AA\\ and $c\\approx13.1$~\\AA.  Initial neutron diffraction measurements were done on a small piece of crystal ($\\sim1$~mm$^3$) from a separately annealed larger crystal.  (That larger crystal was not slowly cooled through the transition from the HTT phase, and the resulting strains eventually, after several months, caused it to shatter, providing many smaller crystals.)  Those measurements were performed at the Dimensional Extreme Magnetic Neutron Diffractometer (DEMAND, HB-3A), at the High Flux Isotope Reactor, Oak Ridge National Laboratory (ORNL). Figure~\\ref{MND}(b) shows the temperature dependence of the intensity measured for the (100) antiferromagnetic reflection and the (010) monoclinic structural peak; because of twinning and finite momentum resolution, we integrated over both reflections.  The drop in intensity at $T_{\\rm N}$ is consistent with the expected disappearance of the (100) intensity.  Above that, the temperature dependence of the (010) intensity indicates that the transition to the HTT phase occurs at $555 \\pm 5$~K.\n\nInelastic neutron scattering measurements were performed on a single crystal of $11$~g at the HYSPEC spectrometer at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory.  The crystal was aligned with $(H, 0, 0)$ and $(0, K, 0)$ in the horizontal scattering plane; the vertical direction corresponds to $[0, 0, L]$. A closed-cycle He cryostat with a temperature range of 35--750~K was used to control the sample temperature. The measurements were carried out with an incident energy of $E_\\mathrm{i} = 27$~meV at $T = 35$, 332, 420, and 500~K.  At each temperature, data were collected as the sample was rotated about its vertical axis in $1\\degree$ steps over a range of $\\sim 155 \\degree$.  For all two dimensional (2D) slices and 1D cuts, shown below, the histogrammed intensities were obtained from the event data using {\\tt MANTID} \\cite{mantid14} and then scaled by a factor of $10^3$.\n\n\\subsection{Phonon calculations}\n\nDensity functional theory (DFT) calculations were performed using the projector augmented wave (PAW) method \\cite{kresse1999ultrasoft,blochl1994projector} as implemented in the Vienna Ab-initio Simulation Package ({\\tt VASP}) \\cite{kresse1996efficiency,kresse1996efficient,kresse1993ab}. The standard PAW datasets included with {\\tt VASP} were used to approximate the core states of all atoms. Exchange-correlation effects were treated in the local-density approximation \\cite{perdew1981self} with the Hubbard-$U$ correction applied to the Cu $3d$ orbitals using the method proposed by Dudarev {\\it et al.} \\cite{dudarev1998electron}. We chose $U = 6$ eV as this value predicts a 1.4-eV bandgap and $0.53~\\mu_{\\rm B}$ magnetic moments, in good agreement with experiment \\cite{tran21a}, and accurately reproduces the dispersions of the bond-stretching phonons in undoped La$_2$CuO$_4$ \\footnote{T. C. Sterling, A. Holder, and D. Reznik, ``Flat bond-stretching LO phonons in undoped La$_2$CuO$_4$ calculated using LDA$+U$,'' (unpublished).}. Magnetism was assumed to be collinear and spin-orbit coupling effects were neglected. The plane-wave energy cutoff was set to 650 eV and total energy was required to converge to less that $1\\times10^{-5}$~eV. During the relaxation and ground state calculations, Brillouin-zone integrations were performed using a $12\\times12\\times6$ $\\Gamma$-centered $k$-point mesh, and energy levels were smeared with a gaussian function with width $\\sigma=0.01$~eV to aid convergence (we chose a small $\\sigma$ consistent with the insulating ground state). Lattice parameters were fixed at the experimental values and atomic positions were relaxed until the forces were less than 0.2~meV\/\\AA\\ on all atoms.\n\nTo determine phonon energies and eigenvectors, we used the supercell approach as implemented in the code {\\tt PHONOPY} \\cite{phonopy}. We checked convergence between $2\\times2\\times1$ and $3\\times3\\times1$ supercells. The zone-center and zone-boundary energies are nearly identical between the two supercell sizes, so we concluded that the $3\\times3\\times1$ supercell is sufficiently large to accurately determine the force contants. We included magnetic ordering when determining the set of irreducible atomic displacements to generate the distorted supercells. {\\tt VASP} was used to calculate the forces in the distorted supercells and the changes in the forces were used to approximate the force constants and populate the dynamical matrices. {\\tt PHONOPY} was then used to solve the lattice dynamical equations for the energies and eigenvectors. \n\nWe used the phonon eigenvectors and energies from the calculations to compute the inelastic neutron scattering dynamic structure factors, $S({\\bf Q},\\omega)$, using the {\\tt SNAXS} software \\footnote{D. Parshall, ``Simulating Neutron And X-ray Scans,'' https:\/\/github.com\/danparshall\/snaxs}. Since the computed $S({\\bf Q},\\omega)$ are delta functions in energy, we broadened them with a gaussian function using a full-width-half-maximum (FWHM) of 2.5~meV to approximate the effects of experimental resolution. We integrated $S({\\bf Q},\\omega)$ across the same ranges as the experimental binning and evaluated it at the temperatures of the experiment. \n\n\n\\section{Results and Analysis \\label{RnD}}\n\n\n\\subsection{Superstructure elastic peaks and reduced symmetry\\label{SEP}}\n\n\\begin{table}[b]\n\t\\caption{Conditions for reflections within the $hk0$ plane for several space groups potentially relevant to the low-temperature phase in LCO, using the notation of Ref.~\\cite{crys95}. {\\color{black} (Application of the reflection conditions to our crystal must take account of twinning.)}\n\t\\label{RC}}.\n\t\\begin{ruledtabular} \n\t\t\\begin{tabular}{ccrccc}\n\t\tAcronym & \\multicolumn{2}{c}{Space Group} & \\multicolumn{3}{c}{Reflection conditions} \\\\\n\t\t& Symbol & No. & $hk0$ & $h00$ & $0k0$ \\\\\n\t\t\t\\hline\n\t\t\tLTO & $Bmab$ & 64 & $h, k = 2n$ & $h=2n$ & $k=2n$ \\\\\n\t\t\tLTLO & $Pccn$ & 56 & $h+k=2n$ & $h=2n$ & $k=2n$ \\\\\n\t\t\tLTM1 & $Bm11$ & 8 & $h = 2n$ & $h=2n$ &  \\\\\n\t\t\tLTM2 & $P2_111$ & 4 & & $h=2n$ &  \\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\nThe deviations of the lattice symmetry from the LTO phase are apparent from a study of diffraction peaks in the $(H,K,0)$ plane.  As indicated in Table~\\ref{RC}, the only allowed peaks within $Bmab$ require that $H$ and $K$ each be even.  Figure~\\ref{E0}(a) shows elastic scattering over a regime of moderate $Q$ obtained at 35~K; one can clearly see many peaks that are inconsistent with $Bmab$.  A complication at this temperature is that we have antiferromagnetism and associated Bragg peaks.  \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_Elastic}\n\t\\caption{Slices of elastic scattering (integrated over $\\pm0.3$~rlu in $L$) at temperatures (a) 35~K,  (b) 500~K. (c) Transverse cuts through $(3,-3,0)$ [inset: $(2,-3,0)$] at $T=35$, 332, and 500~K; dashed lines are gaussian fits.  (d) Temperature dependence of the integrated intensity for $(3, -3, 0)$ and $(2, -3, 0)$ peaks, normalized at 332~K; dotted lines are shape-preserving interpolations.  {\\color{black} The continuous rings are diffraction from the Al sample holder.}}\n\t\\label{E0}\n\\end{figure}\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{fig_Hscan_0m30_fade_allAl}\n\t\\caption{Elastic scattering along $(H,-3,0)$.  Diffraction peaks from Al sample holder are faded; red line indicates the background level.}\n\t\\label{E1}\n\\end{figure}\n\nTo further highlight this, Fig.~\\ref{E1} shows a cut along $(H,-3,0)$ obtained at 332~K, above $T_{\\rm N}$.  None of the peaks observed here are allowed in LTO.   The stronger peaks, at $(0,-3,0)$ and $(2,-3,0)$, are consistent with the monoclinic space group $Bm11$, as proposed by Reehuis {\\it et al.} \\cite{reeh06}; we will label this phase as low-temperature monoclinic 1 (LTM1). \n{\\color{black} [Please note: Reehuis {\\it et al.} \\cite{reeh06} measured a detwinned sample, so they were able to identify that there is intensity for peaks at $(0,K,0)$ with $K$ odd but not for $(H,0,0)$ with $H$ odd.  In contrast, our crystal is twinned, so that, for example, we see peaks at both $(0,-3,0)$ and $(3,0,0)$.]}\n In addition, there are also peaks at $(1,-3,0)$ and $(3,-3,0)$, that satisfy $H+K=2n$; in the absence of the monoclinic reflections, these would be consistent with the low-temperature less-orthorhombic (LTLO) phase observed in the phase diagram of La$_{2-x}$Ba$_x$CuO$_{4}$\\ \\cite{huck11}.    To explain the presence of both sets of peaks in a single phase, we propose space group $P2_111$ and label the phase LTM2.\n\nFigure~\\ref{E0}(c) and (d) show the $T$ dependence of the $(3,-3,0)$ and $(2,-3,0)$ peaks.  They both show a large fall off in intensity around 400~K, and are difficult to detect at 500 K.  Given that we have already seen in Fig.~\\ref{MND}(b) that the (010) monoclinic peak survives to $T_{\\rm s}$, it seems likely that the drop in intensity of the peaks at moderate $Q$ is due to the Debye-Waller factor.  {\\color{black} We suspect that there is a single transition at $T_s$ from the LTM2 phase to the HTT phase; however, further measurements would be necessary to confirm this.}  It is certainly the case that there are lots of fluctuations associated with the LTLO peaks, as we will discuss next.\n\n \n\n\\subsection{Soft tilt modes\\label{SM}}\n\nFigure~\\ref{MvS}(a) shows a slice of inelastic scattering along the same range of ${\\bf Q}$ (and the same temperature) as used for elastic scattering in Fig.~\\ref{E1}.  Here we see diffuse phonons rising from the points $H=1$ and 3, but nothing except vertically-dispersing spin excitations at $H=0$ and 2.   The calculated phonons, assuming an LTT structure, are shown in Fig.~\\ref{MvS}(b).  (Note that a single scale factor has been used for the calculated intensities for all of the comparisons to be shown, including phonons around strong Bragg peaks.)  The calculation shows relative weights of acoustic phonons at $H=1$ and 3 that are similar to the diffuse response in the measured spectra.  \n \n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_Hscan_0m30_both}\n\t\\caption{(a) Inelastic neutron scattering spectra at $332$~K illustrating dispersion along $[H, 0, 0]$ at $K = -3$ (integrated over $\\pm0.1$ in $K$ and $\\pm0.3$ in $L$). One can see soft-mode phonons at $H=1$ and 3, but only spin waves at $H=0$ and 2, with  the acoustic phonons from the corresponding monoclinic peaks being too weak to detect. (b) Calculated phonons for the same range, assuming LTT structure.}\n\t\\label{MvS}\n\\end{figure}\n\nThe intensity of acoustic modes must scale with the intensity of the associated Bragg peak.  The pattern that we observe between experiment and calculations suggests that if the sample transformed to the LTT phase, the $(1,-3,0)$ and $(3,-3,0)$ peaks would be much stronger than the LTM1 peaks at $(0,-3,0)$ and $(2,-3,0)$.  In reality, what we find is that much of the measured phonon weight associated with LTLO wave vectors is peaked above 5~meV, consistent with soft-mode behavior.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_slices_330_parula}\n\t\\caption{Comparisons of phonons in the vicinity of $(3,-3,0)$.  Constant-energy slices at $E=5$~meV, integrated over $\\pm2$~meV in $E$ and $\\pm0.5$~rlu in $L$ for (a) 35~K, (b) 332~K.  Transverse dispersion for (c) 35~K, (d) 332~K. Longitudinal dispersion for (e) 35~K, (f) 332~K.  The latter 4 plots were integrated over $\\pm0.5$ rlu in $L$ and $\\pm0.1$ rlu in the in-plane direction transverse to each plot.}\n\t\\label{NMSM}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_slices_330_calc}\n\t\\caption{Calculated phonons in the vicinity of $(3,-3,0)$ for comparison with data in Fig.~\\ref{NMSM}.    Transverse dispersion for (a) 35~K, (b) 332~K. Longitudinal dispersion for (c) 35~K, (d) 332~K.}\n\t\\label{NMSMc}\n\\end{figure}\n\nWe now take a more systematic look at the LTLO lattice fluctuations, focusing on the $(3,-3,0)$ wave vector.  Figure~\\ref{NMSM}(a) and (b) show constant-energy slices centered at $E=5$~meV and plotted in the $(H,K,0)$ plane for two temperatures.  The acoustic phonons from neighboring fundamental Bragg points have an elliptical shape, reflecting the anisotropic velocities for longitudinal (L) and transverse (T) phonons.  [Note that the neutron cross section is proportional to $({\\bf Q}\\cdot\\mbox{\\boldmath$\\epsilon$\\unboldmath})^2$ where $\\mbox{\\boldmath$\\epsilon$\\unboldmath}$ is the phonon eigenvector, which results in anisotropic sensitivity to L and T modes.]  The excitations at $(3,-3,0)$ are more symmetric, consistent with the weight coming from soft phonons, rather than sharply defined L and T acoustic modes.  At 332~K, we can also see sharp spots at $(2,-3,0)$ and $(3,-2,0)$ due to damped spin fluctuations, which are not apparent at 35~K due to a significant excitation gap in the magnetically-ordered state.\n\nThe dispersions of phonons about $(3,-3,0)$ in the T and L directions are plotted in Fig.~\\ref{NMSM}(c), (d) and (e), (f), respectively.  The T mode appears to connect continuously with the acoustic (A) modes from neighboring reciprocal lattice points (although the calculated dispersions suggest some small gaps from anticrossing modes at 8 meV and above).  The {\\color{black} LA} excitations {\\color{black} disperse linearly with strong intensity up to $\\sim 10$~meV, where the intensity suddenly falls off, with intensity reappearing near 15~meV.  This behavior can be understood as} an avoided crossing {\\color{black}(due to hybridization)} with an optical mode at $\\sim10$~meV (due to a mode involving La motion), so the weight of the branch connecting to the LA mode at $(2,-2,0)$ is weak.  With increasing $T$, the soft-mode weight shifts upward in energy.\n\nFor comparison, calculated neutron scattering from phonons is presented in Fig.~\\ref{NMSMc}.  There is good qualitative consistency between the measured and calculated phonons.  One feature that sticks out in the calculated T dispersions is the dip in the acoustic dispersion near 5~meV; inspection of the eigenvector indicates that this feature is associated with La and O motion, with almost no participation from the Cu sublattice.  There are hints of related behavior in the 35-K data, but the transverse $Q$ resolution is limited by the orthorhombic twinning that effectively increases the sample mosaic.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_cmp_3to7meV}\n\t\\caption{Intensity of phonon scattering for several temperatures\n\t{\\color{black} in the vicinity of} (a) $(3,-3,0)$, (b) $(4-\\xi,-2-\\xi,0)$ with $\\xi=0.225$.  In each case, the intensities average over windows of $\\pm0.1$ rlu in the $[1,-1,0]$ direction, $\\pm0.05$ in the $[1,1,0]$ direction, and $\\pm0.5$ in $L$.  The corresponding results for $\\chi''$ are plotted in (c) and (d), respectively.}\n\t\\label{DB}\n\\end{figure}\n\n{\\color{black}  The temperature dependence of the phonons at $(3,-3,0)$ is anomalous, and to demonstrate this, we will compare with the behavior of TA phonons dispersing from a neighboring Bragg point.  Considering Fig.~\\ref{NMSM}(c), we pick the points corresponding to $\\xi=0$ and 0.775.  The latter point can be described more conventionally as $(4-\\xi',-2-\\xi',0)$ with $\\xi'=0.225$.}\nFigure~\\ref{DB}(a) and (b) compare the temperature dependence of phonon intensities\n{\\color{black} in the vicinity of these two points}.  For the latter, we see that the scattered intensity grows with temperature.  If we divide out the detailed balance factor, we obtain the imaginary part of the dynamic susceptibility, $\\chi''({\\bf Q},E)$, plotted in Fig.~\\ref{DB}(d), which is almost independent of $T${\\color{black}, as is typical for conventional phonons}.  The situation is different for $(3,-3,0)$: the intensity changes much less with $T$ [Fig.~\\ref{DB}(a)], while $\\chi''$ changes quite a bit [Fig.~\\ref{DB}(c)].  \n\nTo quantify the behavior of the soft mode with temperature, we fit the energy dependence shown in Fig.~\\ref{DB} (c) using the damped harmonic oscillator (DHO) formula:\n\\begin{equation}\\label{DHO}\n\\chi^{\\prime \\prime}(\\textbf{Q}, E) = A \\times \\dfrac{\\gamma E E_{\\bf{q}}}{\\big[E^2 \\scalebox{0.75}[1.0]{$-$} E_{\\bf q}^2]^2 + (\\gamma E).^2} \n\\end{equation}\nwhere $E_{\\bf q}^2 = \\Delta_p^2+v^2q^2$, ${\\bf q} = {\\bf Q} - (3, -3, 0)$, $\\Delta_p$ is an energy gap, $v$ is the average acoustic velocity, the damping factor $\\gamma$ corresponds to the FWHM,  and $A$ is the amplitude. The fitted values are listed in Table~\\ref{TM}. \nAt all temperatures, the soft phonon is under-damped, with $\\Delta_p > \\gamma\/2$. \nThe increase of the soft mode gap with temperature is consistent with the soft-mode behavior in a perovskite structure \\cite{tkac21}. \n\n\\begin{table}[t]\n\t\\caption{DHO fitting parameters, corresponding to Eq.~\\ref{DHO}, for fits to phonon $\\chi''$ at $(3,-3,0)$ shown in Fig.~\\ref{DB}(c).  Uncertainties correspond to one standard deviation. \\label{TM}}.\n\t\\begin{ruledtabular} \n\t\t\\begin{tabular}{cccc}\n\t\t\t$T$~(K) & $\\Delta_p$~(meV) & $\\gamma$~(meV) & $A$~(arb. units)\\\\\n\t\t\t\\hline\n\t\t\t$35$ & $4.6 \\pm 0.1$ & $4.7 \\pm 0.4$ & $1.8 \\pm 0.1$\\\\\n\t\t\t $332$ & $6.7 \\pm 0.2$ & $5.1 \\pm 0.5$ & $1.4 \\pm 0.2$\\\\\n\t\t\t$500$ & $7.1 \\pm 0.1$ & $5.7 \\pm 0.8$ & $1.3 \\pm 0.2$\\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\n\n\\subsection{Spin Fluctuations\\label{SF}}\n\nIn studies of antiferromagnetic spin fluctuations in La$_{2-x}$Ba$_x$CuO$_{4}$\\ and La$_{2-x}$Sr$_x$CuO$_{4}$, Wagman {\\it et al.} \\cite{wagm15,wagm16} have identified evidence for spin-phonon coupling at energies in the range of 15-20~meV.  We have observed a similar effect in our own samples of La$_{2-x}$Ba$_x$CuO$_{4}$\\ \\cite{xu14} and La$_{2-x}$Sr$_x$CuO$_{4}$\\ \\cite{li18}, and even in superconducting La$_{2-x}$Ca$_{1+x}$Cu$_2$O$_6$ \\cite{schn19}.  In most of these cases, the low-energy magnetic excitations have an incommensurate modulation, and it is possible that the coupling to phonons is related to that spatial variation.  If that is the case, then the effect should be absent in the case of commensurate antiferromagnetism.  We decided to test this in our sample.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_spins}\n\t\\caption{(a) and (b) Inelastic neutron scattering spectra showing spin fluctuations dispersion in two magnetic Brillouin zones for temperatures below and above $T_\\mathrm{N}$, respectively. Slices are created after averaging along $[0, K, 0]$ and $[0, 0, L]$ by $\\pm 0.08$ and $\\pm 0.3$ r.l.u., respectively. (c) and (d) 1D energy cuts at $(0, -1, 0)$ showing the energy dependence of the spin fluctuations, blue circles, for $T_\\mathrm{N}$, respectively. Energy cuts for the spin fluctuations are obtained after integrating the data in $H$ by $\\pm 0.05$. Black open squares are the background obtained at the wavevectors (0.15 $\\pm$ 0.05, -1, 0). (e) Energy cuts obtained from (c) after subtracting the background and Gaussian lineshape fits to the zero energy signal and highlights the presence of the spin-gap at 35~K$ < T_\\mathrm{N}$. The solid orange line is fit to the spin gap equation in Ref.~\\cite{rama17}. (f) Energy cuts obtained after subtracting the background in (d).}\n\t\\label{SFSG}\n\\end{figure}\n\nFigure~\\ref{SFSG}(a) shows inelastic scattering measured along ${\\bf Q}=(H,-1,0)$ at 35~K, where one can see the spin waves dispersing vertically at $H=0$ and 2.  Similar results are obtained at 332~K, as shown in Fig.~\\ref{SFSG}(b).  From previous work \\cite{keim93}, there should be a spin gap for out-of-plane excitations of 5 meV.  For our geometry, the contribution from the in-plane fluctuations, with spin gap of 2.5~meV, should be only 1\/3 of the signal.  To compare our 35-K data centered at $(0,-1,0)$ with this expectation, we have integrated the intensity over {\\bf Q}, and plot the resulting intensity vs.\\ $E$ in Fig.~\\ref{SFSG}(c), together with background determined at a nearby point.  The net magnetic signal, after subtracting the background and gaussian peak fit to the elastic magnetic peak, is plotted in Fig.~\\ref{SFSG}(e).  As a quantitative check of the spin gap, we have fit the energy dependence using a lorentizian formula applied in previous studies \\cite{rama17,lake99}; we obtain a gap $\\Delta_s=4.0\\pm0.1$~meV, which is very close to the weighted average (4.17~meV) of the previous results cited above \\footnote{The other parameters determined in the fit are an overall damping factor $\\Gamma=8.6\\pm0.4$~meV for the lorentzian function, and a damping factor for the gap edge $\\Gamma_s=0.2\\pm0.1$~meV.}.\nIn comparison, heating above $T_{\\rm N}$ converts the static order into damped spin fluctuations, filling in the spin gap, as illustrated in Fig.~\\ref{SFSG}(d) and (f).\n\nTo look for evidence of spin-phonon coupling, we must shift to the magnetic signal at $(2,-1,0)$, where we can transfer enough energy to reach the energy range of interest.  Figure~\\ref{NESF}(a) and (b) show slices of the magnon and phonon dispersions along $(2,K,0)$ at two temperatures, where one can see the phonons that disperse towards the magnetic {\\bf Q} from the neighboring Bragg points.  To examine the energy dependence of the spin waves, we integrated the excitations in the window $H=2\\pm0.05$, $K=-1\\pm0.075$, and subtracted averaged background from the windows $K=-1.1\\pm0.02$ and $K=-0.9\\pm0.02$, both with $H=2\\pm0.05$.  The resulting signal is plotted in the form of {\\bf Q}-integrated $\\chi''(E)$ in Fig.~\\ref{NESF}(c) and (d).  As one can see, the signal for $E\\gtrsim8$~meV is essentially constant, consistent with expectations for {\\bf Q}-integrated 2D spin waves.  There is no sign of any significant interaction with phonons for this undoped, commensurate antiferromagnetic system in either the ordered or paramagnetic phases.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{fig_Kscan_2m10_2only}\n\t\\caption{(a) and (b) Inelastic neutron scattering spectra for $35$ and $332$~K, respectively, illustrating the energy dependence of spin fluctuation dispersions along $[0, K, 0]$.  Data were averaged over $\\pm0.05$ rlu in $H$ and $\\pm0.3$ rlu in $L$. (c) and (d) $\\chi^{\\prime\\prime}(E)$ for $35$ and $332$~K, respectively, obtained after subtracting the background from intensity vs $E$ plot for spin fluctuations and then applying the bose-factor correction. Intensity vs $E$ plot is a 1D cut at the magnetic zone center $(2, -1, 0)$ and the background is similar plot taken away from magnetic zone center.}\n\t\\label{NESF}\n\\end{figure}\n\n\n\\section{Discussion and Summary}\n\nWe have provided evidence for LTLO-like reflections in LCO.  The elastic intensities are weak, but there is plenty of intensity in the soft phonons at these positions, qualitatively consistent with phonons calculated for the LTT phase.  Following on our previous observation of such peaks in LSCO $x=0.07$ \\cite{jaco15}, it seems probable that this behavior is typical of the low-temperature structural phase in LSCO.  This finding is significant because the distortion results in inequivalent Cu-O bonds in the planes, which are relevant to pinning and ordering charge stripes \\cite{hunt99,crof14,tham14}.\n\nThe temperature dependence of the soft-mode at $(3,-3,0)$ is clearly distinct from that observed for conventional acoustic phonons.  In the latter case, $\\chi''$ is approximately constant with temperature, so that the intensity (obtained by multiplying $\\chi''$ by the detailed-balance factor) grows with $T$ due to weight transferred from the associated Bragg peak.  The substantial low-temperature weight of the soft mode is consistent with LTT-like tilt fluctuations that are not going to order, representing significant entropy at low $T$.  Besides the case of LSCO $x=0.07$ \\cite{jaco15}, we have seen related entropic modes in systems such as superconducting (Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te \\cite{ran18,sapk20} and in both superconducting and non-superconducting Fe$_{1+y}$Te$_{1-x}$Se$_x$ \\cite{fobe16a}.\n\nOf course, there are also peaks present that indicate monoclinic symmetry \\cite{reeh06}, and we have proposed that the space group consistent with all observed peaks is $P2_111$.  (Note that the observed unit cell vectors are consistent with orthorhombic symmetry \\cite{reeh06}; it is the set of atomic positions within the unit cell that determines the symmetry to be monoclinic.)  The LTLO-like peaks are understandable in terms of octahedral tilt instabilities \\cite{pick91}, but it is also desirable to have some physical understanding of the monoclinic symmetry.  Consider first a larger distortion that is not explained by tilting of rigid octahedra.  The large LTO-like tilt is along the $b$ axis.  If this involved rigid rotations, then we would have $b<a$, contrary to experiment.  Instead, there is a shear distortion of the CuO$_2$ planes, and an early analysis showed that it is a secondary effect induced by La-La and La-apical-oxygen interactions \\cite{pive91}.  (This effect is in some ways similar to the shear distortion that leads to the monoclinic phase of Fe$_{1+y}$Te, ``parent'' compound of another superconductor family \\cite{zali12}.)  A significant bonding between apical-O and La atoms has also been found in a new DFT-based study that concludes the ground state to be LTLO \\cite{lee21b}. \n\nAnother DFT-based study evaluated the phonons in the HTT phase, finding two unstable modes \\cite{wang99}.  From the calculated eigenvectors of these modes, one corresponds to the LTO-like tilts, while the other involves a translation (parallel to the planes and transverse to the tilt) of the apical O ions relative to the La ions.  A static displacement of the latter type would provide an explanation for the monoclinic symmetry.\n\n\\begin{figure}[t]\n\t\\centering\n\n\t\\includegraphics[width=\\columnwidth]{fig_LA1}\n\t\\caption{Schematic diagram of the LA eigenvector with ${\\bf q} = [0.1,-0.1,0]$, where  $E({\\bf q})\\approx5$~meV.  Atomic structure of 1.5 unit cells is indicated in the middle, with La in green, O red, and Cu with antiparallel spins in blue and magenta.  Left panels indicate displacements of atoms in middle layer for the top La-O layer, Cu-O layer, and bottom La-O layer.  Right panels show same for bottom layer.}\n\t\\label{eig}\n\\end{figure}\n\nWhile our phonon calculations were performed on the LTT structure rather than with the experimental symmetry, an examination of the eigenvector for an LA phonon in the $[1,-1,0]$ direction, shown in Fig.~\\ref{eig}, yields intriguing insights.  While the in-plane Cu and O atoms exhibit linear displacements, the apical oxygens and half of the La ions exhibit circular motions.  (The symmetry between the layers is broken by the orthogonal orientation of the octahedral tilt axes between neighboring layers, as indicated by the semicircular arrows in the middle panel.)  The rotations of atoms above and below the planes are equal and opposite.  For the experimental structure, the rotations are likely to be much more elliptical.\n\nTurning to the magnetic response of the system, the field-induced weak ferromagnetism in LCO is due to the DM interaction, as mentioned in the introduction.  The distortions of the octahedra will impact the crystal-field splittings, and these impact the DM interaction through spin-orbit coupling.  In zero field, we already have reduced lattice symmetry and associated low-energy fluctuations.  {\\color{black} In the weakly-ferromagnetic phase, the acoustic phonons can couple to the magnetic response, and this coupling could be relevant to understanding the recent experimental observations of the thermal Hall effect  \\cite{gris19,gris20}.}\n\n\n{\\color{black} \nThe thermal Hall effect involves a transverse component of the thermal conductivity that results in an anisotropic response in the presence of an orthogonal magnetic field.  In the case of LCO \\cite{gris19}, which is a Mott insulator, the excitations involved cannot be mobile charge carriers, leaving phonons or magnetic excitations.  Theoretical analyses find that, to explain the thermal Hall effect, these bosonic excitations must have a topological character \\cite{sait19,yang20}.   Some studies \\cite{han19,sama19} have shown that magnetic excitations of distinct types that might be associated with quantum critical effects could result in an enhanced thermal Hall effect, but later experimental results lead to the conclusion that phonons must be the chief contributor \\cite{gris20}.\n\nThe soft phonons in LCO may couple to the weak ferromagnetism in a $c$-axis magnetic field of sufficient strength.  We propose that they may be relevant to the experimental observations of the thermal Hall effect \\cite{gris20}.  Of course, the experimental story has become more complicated, as large thermal Hall effects have now also been observed in the antiferromagnetic insulators Nd$_2$CuO$_4$ and Sr$_2$CuO$_2$Cl$_2$ \\cite{boul20}.  While the Cu sites in these compounds lack apical O neighbors, they nevertheless exhibit unusual behaviors tied to magnetism.  Nd$_2$CuO$_4$ exhibits three distinct magnetic phases as a function of temperature, with Nd moments coupling to Cu in the phase that sets in below 30~K \\cite{mats90}.  An optical second-harmonic-generation study of Sr$_2$CuO$_2$Cl$_2$ has provided evidence for an unexpected order parameter tied to the antiferromagnetic order and leading to reduced lattice symmetry \\cite{dela21}. Hence, the possible coupling of phonons to unusual magnetic responses is not ruled out for these compounds. \n\nAnother complication is that a large thermal Hall effect has been reported in SrTiO$_3$ \\cite{li20c}, a nonmagnetic quantum paraelectric.  This has been explained in terms of an extrinsic mechanism of Berry-curvature-induced skew scattering from defects \\cite{chen20}.  While there are challenges in applying this picture to cuprates, skew scattering from magnetic defects is a possibility \\cite{chen20}. \n\nBesides LCO, a substantial thermal Hall effect has also been observed in hole-doped LSCO with $p=0.06$.}\nAlong these lines, it is relevant to note that Balendent {\\it et al.} \\cite{bale10} reported a neutron spin-flip {\\color{black} ({\\it i.e.}, a magnetic scattering)} signal at (110) [equivalent to (100) when indexed relative to the HTT unit cell] below $\\sim100$~K in LSCO $x=0.085$.  That study was motivated by the search for evidence of loop currents \\cite{varm06}, and the authors were not aware that there might be a nuclear peak at (110).  Their observation suggests a possible coupling of the lattice to the magnetic response, and deserves further investigation.\n\nFinally, we have found no evidence for interaction of the antiferromagnetic spin waves with phonons in the 15-20 meV range.  Our results are consistent with another recent study that found such interactions in superconducting LSCO but not in LCO \\cite{ikeu21}.  The interaction of phonons with spin stripes in hole-doped CuO$_2$ planes is an interesting problem that will need further study.\n\n\n\\section{Acknowledgments}\n\nWe thank Melissa Graves-Brook for assistance with the experiment at HYSPEC.  The work at Brookhaven was supported by the U. S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No.\\ DE-SC0012704.  \nT.S. and D.R. were supported by the DOE, Office of Basic Energy Sciences, Office of Science, under Contract No. DE-SC0006939. A portion of this research was performed using the Eagle computer operated by the Department of Energy's Office of Energy Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory.  T.R. thanks Aaron Holder for help with the calculations.  This research used resources at the High Flux Isotope Reactor and the Spallation Neutron Source, DOE Office of Science User Facilities operated by the Oak Ridge National Laboratory.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction to SN~1996cr}\n\nSN~1996cr, located comparatively nearby in the Circinus Galaxy (3.7 Mpc, $z\\sim 0.001$)\nwas serendipitously detected as a ULX at an age of $\\sim$\\,5 years \\citep{Bauer01}.\nAs detailed in \\citet{Bauer08}\\,:~\ni) followup archival research and VLT observations (at age $\\sim$\\,10 years)\nallowed SN~1996cr to be classified as a Type IIn SNe, and\nii) archival X-ray data showed that it was not detected at an\nage of $\\sim$\\,1000 days, yet brightened to $L_x \\sim 4\\times 10^{39}$ erg\/s\n(0.5-8 keV) after 10 years.  This X-ray behavior is\nshared only with SN 1987A, and is roughly shown in Figure~\\ref{fig:lightcurve}\nin the context of other X-ray-detected SNe and core-collapse (CC) SNRs.\nBecause of this behavior, and given\nan indication of Doppler structure in 2004 {\\it Chandra}\\ HETG data,\nwe proposed and obtained a deep, 485~ks HETG observation (PI Bauer)\nnear the beginning of 2009.  \nIn large part because of the high quality\nof this recent data we were able to tune a 1-D hydrodynamic model to agree with the\n{\\it multi-epoch} X-ray data (next section), and we are now\ninvestigating signatures of the ejecta\/CSM geometry in the HETG line shapes (last section.)\n\n\\begin{figure}\n  \\includegraphics[width=0.70\\textwidth]{dewey_p7rot.eps}\n  \\caption{The unique behavior of SN~1996cr and SN 1987A.  Their X-ray light\n  curves are overplotted on an $L_X$ vs Age plot\n  from \\citet{Immler05}.  Unlike other X-ray-bright SNe, these two\n  show a dramatic (re-)brightening at ages of a few years to decades.\n  As these and other SNe age we will begin to fill in the CC SNe-SNR gap\n  seen between ages of 30 to 300 years.\n  \\label{fig:lightcurve}}\n\\end{figure}\n\n\n\n\n\\section{Hydrodynamic Model and X-ray Emission}\n\n\\begin{figure}\n  \\includegraphics[width=0.70\\textwidth]{dewey_p8rot.eps}\n  \\caption{Initial density profile for the hydrodynamic model of SN~1996cr.\n  Several months after the SN explosion the ejecta (near the origin at left)\n  is seen making its way into the low density W-R wind cavity while\n  the shell of swept-up RSG wind material doesn't know what's\n  going to hit it.  See Figure~3 of \\citet{Dwarkadas10} for snapshots\n  of the further evolution of the hydrodynamics beyond this initial\n  configuration.\n  \\label{fig:1dhydro}}\n\\end{figure}\n\nA 1-D (spherical) hydrodynamic model of the ejecta-CSM interaction\nis developed in \\citet{Dwarkadas10}; post-processing calculates the X-ray\nemission from the non-radiative shocks, showing good agreement\nwith the measured X-ray light curve and spectra at multiple epochs.\nGiven our inferred configuration, Figure~\\ref{fig:1dhydro},\na realistic evolutionary scenario for SN~1996cr's progenitor has it evolving\nfrom the RSG to the W-R stage, creating a wind-blown bubble with a dense\nshell at about 0.04~pc, and then exploding as a SN.\nSome conclusions from the modeling are that:\n\\begin{itemize}\n\\item\n  The 1-D model explains the majority of the observed X-ray continuum\n  and lines, and their variation in time.\n\\item\n  The velocities of plasma in the model agree with the scale of the line\n  broadening seen in the HETG data.\n\\item\n  The inner ejecta core is opaque to HETG X-rays as late as 2009\n  when it has a plateau density of $10^5$~amu\\,cm$^{-3}$ and a\n  radius of 0.065~pc, giving a column density along the \n  diameter of 4.0\\,$\\times$\\,$10^{22}$~amu\\,cm$^{-2}$ of high-Z material.\n\\item\n  Initially the flux from the forward-shocked shell dominates.  However,\n  at $\\sim$\\,8 years the reverse-shocked ejecta contributes 50\\% of the flux, \n  this fraction grows to $\\sim$\\,70\\% at 15--20 years.\n\\item\n  Some small fraction (by mass\/volume) of denser CSM is needed to produce\n  some low-ionization lines seen in the X-ray spectrum; this ``clump''\n  emission is a small perturbation on the main hydrodynamics.\n\\end{itemize}\n\n\n\\section{{\\it Chandra}\\,\/\\,HETG Line Shapes and Ejecta-CSM Geometry}\n\n\\begin{figure}\n  \\includegraphics[width=0.50\\textwidth]{dewey_p19rot.eps}\n  \\caption{Line shapes of the bright Si and Fe-K lines in SN~1996cr.\n  Lines of Si (top) and Fe (bottom) show Doppler-modified structure\n  with velocity components of several thousand km\/s; the HETG\n  instrumental profile is shown in the upper right.  Note the very\n  different shapes of the H-like Si XIV and Fe XXVI lines with respect\n  to their nominal line energies (vertical black lines.)  The solid black\n  curves are model fits based on our 1-D hydrodynamic model velocities\n  and an emission region that does not cover a full sphere, see\n  Figures~\\ref{fig:lineshapes}~\\&~\\ref{fig:contours}.\n  \\label{fig:linefits}}\n\\end{figure}\n\nWith a general hydrodynamic picture established, we are now\nlooking at detailed line-shape fitting of a handful of the lines,\nin particular Si and Fe-K lines shown in Figure~\\ref{fig:linefits}.\nUsing simple 3-D modeling techniques \\citep{Dewey09} we compared\nthe HETG data with the line shapes expected from our 1-D model (with spherical symmetry\nand an opaque core) and found that, while there is qualitative agreement, there are\nstatistically significant departures from this simple model.\nSince the SNe is unresolved in the X-ray (and only just resolved\nby VLBI in 2007 \\citep{Bauer08}) we explore\nsimple, plausible geometric modifications that can better fit the line shapes.\n\nOne option is to retain the 1-D radial solution (i.e., densities and\nvelocities), but to restrict\nthe interaction and emission to less than the full\n$4\\pi$ solid angle; examples of such geometries are shown in Figure~\\ref{fig:lineshapes}.\nIn reality, this approximation could be appropriate if either the CSM is\nnon-uniform or if the explosion\/ejecta is not spherically symmetric.\nOf course this truncation of the solid angle also reduces the model flux, but\na flux factor of up to a few can be accomodated with\nsmall changes to the parameters (density, radii) and still\nobtain light curve and spectral agreement.\n\nComparing models with data in Figures~\\ref{fig:linefits}~\\&~\\ref{fig:contours},\nwe find that the line shapes are well fit by ``polar''\nmodels consisting of uniform emission within a half-opening angle of\nthe poles: within $\\sim$\\,60--70 degrees for Si and other lines\nand within $\\sim$\\,10--30 degrees for\nthe Fe XXVI line. In all cases the polar axis is consistent with being\noriented at $\\sim$\\,50--60 degrees to our line-of-sight; this leads to the\nschematic image shown in Figure~\\ref{fig:contours} (the position angle\nis arbitrary.)\n\nA preliminary sensitivity study of the effects of changing the\nhydrodynamic parameters\nsuggests that the easiest way to get more Fe XXVI emission is to\nincrease the swept up mass at high lattitudes, rather than, say,\nchanging the ejecta mass or energy along that direction. \nWe are currently in the processes of creating a more realistic model to take this\nand the optical line shapes into account.  Perhaps in the future\nsimilar CSM\/ejecta sleuthing will be possible for more SNe\nand even gamma-ray burst remnants.\n\n\n\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{dewey_p13rot.eps}\n  \\caption{From geometry to line shape.  We calculate the line shape from\n  emission that is confined within a range of the pole, e.g., within 55\n  degrees (upper row) and within 20 degrees (lower row.)  Including an\n  opaque core and viewing the system at an angle to the line-of-sight,\n  here 55 degrees, produces a variety of posible line shapes (right\n  plots.)  The middle images would be seen by a high spatial-resolution\n  X-ray telescope.\n  \\label{fig:lineshapes}}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=0.48\\textwidth]{dewey_p15rot.eps}\n  \\includegraphics[width=0.31\\textwidth]{dewey_p18rot.eps}\n  \\caption{Fitting the emission-line geometries.  Using line\n  shapes specified\n  by a polar angle and the angle to the line-of-sight, we\n  generate 1-sigma confidence contours for fits to discrete lines, left.  The Si,\n  S, Mg, and the lower-ionization Fe XXIV lines are all fit with similar\n  parameters.  In contrast the Fe XXVI line is much better fit with its\n  emission confined to higher-lattitudes.  The solid black lines in \n  Figure~\\ref{fig:linefits} show the Si and Fe fits; the image at\n  right is the implied geometry.\n  \\label{fig:contours}}\n\\end{figure}\n\n\n\n\n\\begin{theacknowledgments}\nDD was supported by NASA through SAO contract SV3-73016 to MIT for\nSupport of the {\\it Chandra}\\ X-Ray Center (CXC) and Science Instruments.\nFEB and VVD were supported by NASA through the {\\it Chandra}\\ Guest\nObserving program, Award Numbers GO9-0086A\/B and GO0-11095A\/B issued by\nthe CXC. The CXC is operated by the SAO for and on\nbehalf of NASA under contract NAS8-03060.\n\\end{theacknowledgments}\n\n\n\n\\bibliographystyle{aipproc}  \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{S:Goals}\n\n\\subsection{Definitions and basic properties of Lipschitz free\nspaces}\n\n\nBasic facts about Lipschitz free spaces can be found in\n\\cite[Chapter 10]{Ost13} and \\cite[Chapter 2]{Wea99} (in\n\\cite{Wea99} Lipschitz free spaces are called Arens-Eells spaces)\n\n\\begin{definition}\\label{D:molLF} Let $X$ be a metric space. A\n{\\it molecule} of $X$\\index{molecule of a metric space} is a\nfunction $m:X\\to\\mathbb{R}$ which is supported on a finite set and\nwhich satisfies $\\sum_{p\\in X} m(p)=0$. For $p,q\\in X$ define the\nmolecule $m_{pq}$ by $m_{pq}=\\1_p-\\1_q$, where $\\1_p$ and $\\1_q$\nare indicator functions of singleton sets $\\{p\\}$ and $\\{q\\}$. We\nendow the space of molecules with the seminorm\n\\[||m||_{\\are}=\\inf\\left\\{\\sum_{i=1}^n|a_i|d_X(p_i,q_i):~m=\\sum_{i=1}^na_im_{p_iq_i}\\right\\}\\]\nIt is not difficult to see that this is actually a norm. The {\\it\nLipschitz free space} over $X$\\index{Lipschitz free space over a\nmetric space} is defined as the completion of the space of all\nmolecules with respect to the norm $||\\cdot||_{\\are}$. We denote\nthe Lipschitz free space over $X$ by $\\are(X)$.\n\\end{definition}\n\nBy a {\\it pointed metric space} we mean a metric space with a\ndistinguished point, denoted $O$. By $\\lip_0(X)$ we denote the\nspace of all Lipschitz functions $f:X\\to\\mathbb{R}$ satisfying\n$f(O)=0$, where $O$ is the distinguished point of a pointed metric\nspace $X$. It is not difficult to check that $\\lip_0(X)$ is a\nBanach space with respect to the norm $||f||=\\lip(f)$. As is\nwell-known \\cite{Ost13,Wea99}, the following duality holds:\n\\begin{equation}\\label{E:LFdual}\\are(X)^*=\\lip_0(X).\\end{equation}\n\nWe also need the following description of $\\lf(X)$ in the case\nwhere $X$ is a vertex set of an unweighted graph with its graph\ndistance. Let $G=(V(G),E(G))=(V,E)$ be a finite graph. Let\n$\\ell_1(E)$ be the space of real-valued functions on $E$ with the\nnorm $||f||=\\sum_{e\\in E}|f(e)|$. We consider some orientation on\n$E$, so each edge of $E$ is a directed edge. For a directed cycle\n$C$ in $E$ (we mean that the cycle can be ``walked around''\nfollowing the direction, which is not related with the orientation\nof $E$) we introduce the {\\it signed indicator function} of $C$ by\n\\begin{equation}\\label{E:SignInd}\n\\chi_C(e)=\\begin{cases} 1 & \\hbox{ if }e\\in C\\hbox{ and its\norientations in $C$ and $G$ are the same}\\\\\n-1 & \\hbox{ if }e\\in C\\hbox{ but its orientations in $C$ and $G$\nare different}\n\\\\\n0 & \\hbox{ if }e\\notin C.\n\\end{cases}\\end{equation}\n\nThe {\\it cycle space} $Z(G)$ of $G$ is the subspace of $\\ell_1(E)$\nspanned by the signed indicator functions of all cycles in $G$. We\nwill use the fact that $\\lf(G)$ for unweighed graphs $G$\n(\\cite[Proposition 10.10]{Ost13}) is isometrically isomorphic to\nthe quotient of $\\ell_1(E)$ over $Z(G)$:\n\\begin{equation}\\label{E:LFunweigh}\\lf(G)=\\ell_1(E)\/Z(G)\\end{equation}\n\nWe use the standard terminology of Banach space theory\n\\cite{BL00}, graph theory \\cite{BM08,Die17}, and the theory of\nmetric embeddings \\cite{Ost13}.\n\n\\subsection{Historical and terminological remarks}\n\nThe Lipschitz-free spaces are studied by several groups of\nresearchers, for different reasons and under different names. Some\nauthors use the term {\\it Arens-Eells space} (see\n\\cite{Kal08,Wea99}), which reflects the contribution of Arens and\nEells \\cite{AE56}. The norm of this space and a more general space\nof measures (see \\cite{Vil03,Vil09,Wea99}) is called the {\\it\nKantorovich-Rubinstein distance (or norm)}  to acknowledge the\ncontribution of Kantorovich and Rubinstein \\cite{Kan42,KR58}, or\n{\\it Wasserstein distance (or norm)}, (see \\cite{ANN17+, NR17}) to\nacknowledge the contribution of Wasserstein \\cite{Vas69} (whose\nname is transliterated from Russian as Vasershtein), see the paper\n\\cite{Dob70}, where the term Wasserstein distance was introduced.\nThe term {\\it Wasserstein norm} is also used (and more\njustifiably) for the $p$-analogue of the distance. The term {\\it\nLipschitz free space} is commonly used (especially in the Banach\nspace theory) after the publication of the paper \\cite{GK03}. The\nnames used for this distance in Computer Science are {\\it earth\nmover distance} and {\\it transportation cost} (see \\cite{ADIW09},\n\\cite{AIK08}, \\cite{KN06}, \\cite{NS07}). All of the mentioned\nabove notions are equivalent for finite metric spaces which we\nconsider in this paper. For this reason we decided not to attach\nany of the mentioned names to the objects of our study and to use\nthe neutral name {\\it Lipschitz free space} (which only reflects\nthe connection of this notion with the notion of a Lipschitz\nfunction).\n\nLipschitz free spaces are of significant interest for Computer\nScience (see \\cite{IM04}), Functional Analysis (\\cite{God15}, \\cite{Kal08},\n\\cite{Wea99}), Metric Geometry (\\cite{ANN17+},\n\\cite[p.~134]{NR17}, \\cite{Ost13}), Optimal Transportation\n(\\cite{Vil03},\\cite{Vil09}).\n\n\\subsection{Overview of the paper}\n\nOur interest in Lipschitz free spaces is inspired by the theory of\nMetric Embeddings (see \\cite{Ost13}): we are interested in\nstudying properties of Banach spaces admitting an isometric\nembedding of a given metric space. We are going to focus on finite\nmetric spaces.\n\\medskip\n\nOur main results and observations:\n\n\\begin{enumerate}\n\n\\item[{\\bf 1.}] We show that for any finite metric space $M$ the\nspace $\\lf(M)$ contains a half-dimensional well-complemented\nsubspace which is close to $\\ell_1^n$, see Section\n\\ref{S:LargeL1}.\n\n\\item[{\\bf 2.}] We prove that the Lipschitz free spaces on large\nclasses of recursively defined sequences of graphs (see Section\n\\ref{S:DefRecDiLa} for definitions) are not uniformly isomorphic\nto $\\ell_1^n$ of the corresponding dimensions (Section\n\\ref{S:Recur}). These classes contain well-known families of\ndiamond graphs and Laakso graphs, see \\ref{S:DefRecDiLa} for\ndefinitions and Section \\ref{S:Diamond} for proofs. The case of\ndiamond graphs can also be handled using classical theory of\northogonal series. Since this approach has its advantages and\nleads to more precise results, we enclose the corresponding\nargument in Section \\ref{S:DiamHaar}.\n\nInteresting features of our approach are: (1) We consider averages\nover groups of cycle-preserving bijections of graphs which are not\nnecessarily automorphisms (see Section \\ref{S:CycBij}); (2) In the\ncase of such recursive families of graphs as Laakso graphs we use\nthe well-known approach of Gr\\\"unbaum \\cite{Gru60} and Rudin\n\\cite{Rud62} for estimating projection constants in the case where\ninvariant projections are not unique (see Sections\n\\ref{S:GRAAppr}, \\ref{S:Annihil}, \\ref{S:Combin}, and\n\\ref{S:NonUnLaakso}).\n\n\\item[{\\bf 3.}] We observe (Section \\ref{S:L1}) that the known\nfact (see \\cite{Dal15}, \\cite{CD16}) that Lipschitz free spaces on\nfinite ultrametrics are close to $\\ell_1$ in the Banach-Mazur\ndistance immediately follows from the result of Gupta \\cite{Gup01}\non Steiner points and the well-known result on isometric\nembeddability of ultrametrics into weighted trees.\n\n\\item[{\\bf 4.}] We finish this section by observing that the\nresult of Erd\\H{o}s and P\\'osa \\cite{EP62} on edge-disjoint cycles\nimplies that the cycle space (considered as a subspace of\n$\\ell_1(E)$) always contains a `large' $1$-complemented in\n$\\ell_1(E)$ subspace isometric to $\\ell_1^n$.\n\nObserve that the subspace in $Z(G)$ spanned by the signed\nindicator functions of a family of edge-disjoint cycles is\nisometric to $\\ell_1^n$ of the corresponding dimension and is\n$1$-complemented in $\\ell_1(E(G))$, and so in $Z(G)$.\n\nThis makes us interested in the estimates of the amount of\nedge-disjoint cycles in terms of the dimension of the cycle space.\nSuch estimates, sharp up to the constants involved in them, were\nobtained by Erd\\H{o}s and P\\'osa \\cite{EP62}. Denote by $\\mu(G)$\nthe dimension of the cycle space of $G$. It is well-known, see\n\\cite[Proposition 2.1]{Big97}, that for connected graphs\n$\\mu(G)=|E(G)|-|V(G)|+1$. Let $\\nu(G)$ be the maximal number of\nedge-disjoint cycles in $G$.\n\n\n\\begin{theorem}[{Erd\\H{o}s and P\\'osa \\cite[Theorem\n4]{EP62}}]\\label{T:EP}\n\\[\\nu(G)=\\Omega\\left(\\frac{\\mu(G)}{\\log(\\mu(G))}\\right)\\]\nand for some family of graphs\n\\[\\nu(G)=O\\left(\\frac{\\mu(G)}{\\log(\\mu(G))}\\right)\\]\n\\end{theorem}\n\n\\begin{remark} It is worth mentioning that Erd\\H{o}s and P\\'osa\nstate their result slightly differently. They do not require\ngraphs to be simple or connected and denote by $g(k)$ the smallest\ninteger such that for any $n\\in \\mathbb{N}$ a graph with $n$\nvertices and $n+g(k)$ edges contains at least $k$ edge-disjoint\ncycles. Theorem 4 in \\cite{EP62} states that\n\\[g(k)=\\Theta(k\\log k).\\]\nIt is easy to see that Theorem \\ref{T:EP} follows from this\nresult.\n\\end{remark}\n\n\\end{enumerate}\n\n\\subsection{Recursive families of graphs, diamond graphs and Laakso\ngraphs}\\label{S:DefRecDiLa}\n\nWe are going to use the general definition of recursive sequences\nof graphs introduced by Lee and Raghavendra \\cite{LR10}.\n\n\n\\begin{definition}\\label{D:Comp}\nLet $H$ and $G$ be two finite connected directed graphs having\ndistinguished vertices which we call {\\it top} and {\\it bottom},\nrespectively. The {\\it composition} $H\\oslash G$ is obtained by\nreplacing each  edge $\\overrightarrow{uv}\\in E(H)$ by a copy of\n$G$, the vertex $u$ is identified with the bottom of $G$ and the\nvertex $v$ is identified with the top of $G$. Directions of edges\nin $H\\oslash G$ are inherited from $G$. The {\\it top} and {\\it\nbottom} of the obtained graph are defined as the top and bottom of\n$H$, respectively.\n\\end{definition}\n\nWhen we consider these graphs as metric spaces we use the graph\ndistances of the underlying undirected graphs (that is, we ignore\nthe directions of edges).\\medskip\n\nThe following property of this composition is straightforward to\nverify:\n\n\\begin{lemma}[Associativity of $\\oslash$]\\label{L:Assoc}  For any three graphs\n$F,G,H$ the sides of\n\\begin{equation}\\label{E:Assoc}\n(F\\oslash G)\\oslash H=F\\oslash (G\\oslash H),\n\\end{equation}\nare equal both as directed graphs and as metric spaces.\n\\end{lemma}\n\n\nLet $B$ be a connected unweighted finite simple directed graph\nhaving two distinguished vertices, which we call {\\it top} and\n{\\it bottom}, respectively. We use $B$ to construct recursive\nfamily of graphs as follows:\n\n\\begin{definition}\\label{D:B_n} We say that the graphs $\\{B_n\\}_{n=0}^\\infty$ are defined by {\\it recursive\ncomposition} or that $\\{B_n\\}_{n=0}^\\infty$ is a {\\it recursive\nsequence} or {\\it recursive family} of graphs if:\n\n\\begin{itemize}\n\n\\item The graph $B_0$ consists of one directed edge with {\\it\nbottom} being the initial vertex and {\\it top} being the terminal\nvertex.\n\n\\item $B_n=B_{n-1}\\oslash B$.\n\n\\end{itemize}\n\\end{definition}\n\nObserve that Lemma \\ref{L:Assoc} implies that for every\n$k\\in\\{0,1,\\dots,n\\}$ we have\n\\begin{equation}\\label{E:DiffProd} B_n=B_{n-k}\\oslash B_{k},\n\\end{equation} also $B_1=B$. The authors of \\cite{LR10} use the notation\n$B_n=B^{\\oslash n}$.\\medskip\n\nObserve that in the case where the graph $B$ has an automorphism\nwhich maps its bottom to top and top to bottom, the choice of\ndirections on edges will not affect the isomorphic structure of\nthe underlying undirected  graphs. For this reason to define\nrecursive families in such cases we do not not need to assign\ndirections to edges.\n\\medskip\n\nInteresting and important examples of recursive families of graphs\nhave been extensively studied in the literature. One of the most\nwell-known and important for the theory of metric embeddings\nfamilies was introduced in \\cite{GNRS04} (conference version was\npublished in 1999). This family (which turned to be very useful in\nthe theory of metric characterizations of classes of Banach spaces\n\\cite{JS09}, see also \\cite[Section 9.3.2]{Ost13}) corresponds to\nthe special case of Definition \\ref{D:B_n}, where $B$ is a square\nand one pair of its opposite vertices is chosen to play roles of\nthe top and the bottom. The usual definition of diamond graphs is\nthe following.\n\n\\begin{definition}[Diamond graphs]\\label{D:Diamonds}\nDiamond graphs $\\{D_n\\}_{n=0}^\\infty$ are defined recursi\\-ve\\-ly:\nThe {\\it diamond graph} of level $0$ has two vertices joined by an\nedge of length $1$ and is denoted by $D_0$. The {\\it diamond\ngraph} $D_n$ is obtained from $D_{n-1}$ in the following way.\nGiven an edge $uv\\in E(D_{n-1})$, it is replaced by a\nquadrilateral $u, a, v, b$, with edges $ua$, $av$, $vb$, $bu$.\n(See Figure \\ref{F:Diamond2}.)\n\\end{definition}\n\n\n\\begin{figure}\n\\begin{center}\n{\n\\begin{tikzpicture}\n  [scale=.25,auto=left,every node\/.style={circle,draw}]\n  \\node (n1) at (16,0) {\\hbox{~~~}};\n  \\node (n2) at (5,5)  {\\hbox{~~~}};\n  \\node (n3) at (11,11)  {\\hbox{~~~}};\n  \\node (n4) at (0,16) {\\hbox{~~~}};\n  \\node (n5) at (5,27)  {\\hbox{~~~}};\n  \\node (n6) at (11,21)  {\\hbox{~~~}};\n  \\node (n7) at (16,32) {\\hbox{~~~}};\n  \\node (n8) at (21,21)  {\\hbox{~~~}};\n  \\node (n9) at (27,27)  {\\hbox{~~~}};\n  \\node (n10) at (32,16) {\\hbox{~~~}};\n  \\node (n11) at (21,11)  {\\hbox{~~~}};\n  \\node (n12) at (27,5)  {\\hbox{~~~}};\n\n\n  \\foreach \\from\/\\to in {n1\/n2,n1\/n3,n2\/n4,n3\/n4,n4\/n5,n4\/n6,n6\/n7,n5\/n7,n7\/n8,n7\/n9,n8\/n10,n9\/n10,n10\/n11,n10\/n12,n11\/n1,n12\/n1}\n    \\draw (\\from) -- (\\to);\n\n\\end{tikzpicture}\n} \\caption{Diamond $D_2$.}\\label{F:Diamond2}\n\\end{center}\n\\end{figure}\n\nLet us count some parameters associated with graphs $D_n$. Denote\nby $V(D_n)$ and $E(D_n)$ the vertex set and edge set of $D_n$,\nrespectively. We need the following simple observations about\ncardinalities of these sets:\n\n\\begin{enumerate}\n\n\\item[{\\bf (a)}] $|E(D_n)|=4^n$.\n\n\\item[{\\bf (b)}] $|V(D_{n+1})|=|V(D_n)|+2|E(D_n)|$.\\label{P:(b)}\n\n\\end{enumerate}\n\nHence $|V(D_n)|=2(1+\\sum_{i=0}^{n-1}4^i)$.\n\\medskip\n\nThe next special case of the general Definition \\ref{D:B_n}, whose\nmetric geometry was studied in \\cite{LR10,OR17}, corresponds to\nthe case where $B=K_{2,n}$, and the vertices in the part\ncontaining $2$ vertices play the roles of the top and the bottom.\nThe usual definition is the following.\n\n\\begin{definition}[Multibranching diamonds]\\label{D:BranchDiam}\nFor  any integer  $k\\ge 2$, we define $D_{0,k}$ to be the graph\nconsisting of two vertices joined by one edge. For any\n$n\\in\\mathbb{N}$, if the graph $D_{n-1,k}$  is already defined,\nthe graph $D_{n,k}$ is defined as the graph obtained from\n$D_{n-1,k}$ by replacing each edge $uv$ in $D_{n-1,k}$ by a set of\n$k$ independent paths of length $2$ joining $u$ and $v$. We endow\n$D_{n,k}$ with the shortest path distance. We call\n$\\{D_{n,k}\\}_{n=0}^\\infty$ {\\it diamond graphs of branching $k$},\nor {\\it diamonds of branching} $k$.\n\\end{definition}\n\nThe last special case of the general Definition \\ref{D:B_n}, which\nwe consider in this paper goes back to the paper of Laakso\n\\cite{Laa00}. The corresponding recursive family of graphs was\nintroduced by Lang and Plaut \\cite{LP01}. In \\cite{OO17} it was\nshown that these graphs are incomparable with diamond graphs in\nthe following sense: elements of none of these families admit\nbilipschitz embeddings into the other family with uniformly\nbounded distortions. Laakso graphs correspond to the case where\nthe graph $B$ is the graph shown in Figure \\ref{F:Laakso} with the\nnatural choice for the top and the bottom.\n\n\\begin{figure}\n\\begin{center}\n{\n\\begin{tikzpicture}\n  [scale=.25,auto=left,every node\/.style={circle,draw}]\n  \\node (n1) at (16,0) {\\hbox{~~~}};\n  \\node (n2) at (16,8)  {\\hbox{~~~}};\n  \\node (n3) at (12,16) {\\hbox{~~~}};\n  \\node (n4) at (20,16)  {\\hbox{~~~}};\n  \\node (n5) at (16,24)  {\\hbox{~~~}};\n  \\node (n6) at (16,32) {\\hbox{~~~}};\n\n\\foreach \\from\/\\to in {n1\/n2,n2\/n3,n2\/n4,n4\/n5,n3\/n5,n5\/n6}\n    \\draw (\\from) -- (\\to);\n\n\\end{tikzpicture}\n} \\caption{Laakso graph $\\mathcal{L}_1$.}\\label{F:Laakso}\n\\end{center}\n\\end{figure}\n\n\\begin{definition}\\label{D:Laakso}\nLaakso graphs $\\{\\mathcal{L}_n\\}_{n=0}^\\infty$ are defined\nrecursively: The {\\it Laakso graph} of level $0$ has two vertices\njoined by an edge of length $1$ and is denoted $\\mathcal{L}_0$.\nThe {\\it Laakso graph} $\\mathcal{L}_n$ is obtained from\n$\\mathcal{L}_{n-1}$ according to the following procedure. Each\nedge $uv\\in E(\\mathcal{L}_{n-1})$ is replaced by the graph\n$\\mathcal{L}_1$ exhibited in Figure \\ref{F:Laakso}, the vertices\n$u$ and $v$ are identified with the vertices of degree $1$ of\n$\\mathcal{L}_1$.\n\\end{definition}\n\n\n\n\\section{Lipschitz-free spaces close to $\\ell_1^n$}\\label{S:L1}\n\n\nOur first proposition is known, see \\cite[Corollary 3.3]{God10},\nwe give a direct proof of it for convenience of the reader.\n\n\\begin{proposition}\\label{P:Tree} Let $T$ be a finite weighted tree. Then  $\\lf(T)$ is\nisometric to $\\ell_1^k$, where $k$ is the number of edges in the\ntree.\n\\end{proposition}\n\n\\begin{proof} Let $f\\mapsto e_f$ be a bijection between the edge set of $T$ and the\nunit vector basis in $\\ell_1^k$. We denote the weight of $f$ by\n$w(f)$. We consider the following map $F$ of the set of molecules\non $T$ into $\\ell_1^k$.\n\nFor each edge $f=\\{u,v\\}$ we let $F(\\1_u-\\1_v)=w(f)e_f$. It is\nclear that each molecule in $\\lf(T)$ can be (uniquely) written as\na linear combination of molecules $\\{\\1_u-\\1_v\\}_{\\{u,v\\}\\in\nE(T)}$. We define $F$ to be the linear extension of the defined\nmap to $\\lf(T)$, it is clear from this definition that $F$ is a\nsurjective map onto $\\ell_1^k$.\n\nBy the duality \\eqref{E:LFdual}, to show that $F$ is an isometry\nof $\\lf(T)$ onto $\\ell_1^k$ it is enough to find a $1$-Lipschitz\nfunction $L\\in\\lip_0(T)$ (the base point $O$ is chosen\narbitrarily) such that\n\\[L\\left(\\sum_{\\{u,v\\}\\in E(T)}a_{uv}(\\1_u-\\1_v)\\right)=\\sum_{\\{u,v\\}\\in E(T)}|a_{uv}| \\cdot\nw(uv),\\] where $a_{uv}\\in\\mathbb{R}$.\n\nConstruction of such $1$-Lipschitz function $L$ is quite\nstraightforward. We let $L(O)=0$. If the function is already\ndefined on one end $u$ of an edge $\\{u,v\\}$, we set $L(v)=L(u)\\pm\nw(uv)$, where we choose $+$ if the coefficient of $\\1_v-\\1_u$ in\n$m$ is nonnegative, and $-$ if the coefficient of $\\1_v-\\1_u$ in\n$m$ is negative. It is clear that $L$ is $1$-Lipschitz and\n$L(m)=\\sum_{\\{u,v\\}\\in E(T)}|a_{uv}| \\cdot w(uv)$.\n\\end{proof}\n\nThe following result is very useful in the current context.\n\n\\begin{theorem}[\\cite{Gup01}]\\label{T:Gupta} Let $T$ be a weighted tree and $M$ be a subset of $V(T)$. Then there is a weighted\ntree $\\tilde T$ with the vertex set $M$ such that the distances\ninduced by $T$ and $\\tilde T$ on $M$ are $8$-equivalent.\n\\end{theorem}\n\n\\begin{corollary}\\label{C:SubsL1} Let $T$ be a weighted tree and $M$ be a subset of $V(T)$.\nThen the Banach-Mazur distance between $\\lf(M)$ (where $M$ is\nendowed with the metric induced from $T$) and $\\ell_1^k$ of the\ncorresponding dimension does not exceed $8$.\n\\end{corollary}\n\n\\begin{remark} Gupta \\cite{Gup01} did not show that the constant\n$8$ is sharp, his lower estimate for the constant is $4$. It is\nnot clear what is the optimal constant in Corollary\n\\ref{C:SubsL1}.\n\\end{remark}\n\nSince it is well-known that ultrametrics can be isometrically\nembedded into weighted trees (see, for example, \\cite[Theorem\n9]{CM10}, and also \\cite[Section 3]{DSW17}), we get also the\nfollowing finite version of results of \\cite{Dal15} and\n\\cite{CD16}:\n\n\n\\begin{corollary}\\label{C:Ultra} Let $M$ be a finite ultrametric space. Then  $\\lf(M)$ is\n$8$-isomorphic to $\\ell_1^k$.\n\\end{corollary}\n\n\nTo see that there are metric spaces of different nature whose\nLipschitz free spaces are also close to $\\ell_1^k$ of the\ncorresponding dimension, we use \\eqref{E:LFunweigh}. This equality\nimplies that if we consider a graph $G$ which contains small\namount of cycles, or all cycles in it are disjoint, then $\\lf(G)$\nis close to $\\ell_1^n$ of the corresponding dimension.\n\nThe space $\\lf(G)$ remains close to $\\ell_1^n$ for metric spaces\nwhich are bilipschitz equivalent to graphs having properties\ndescribed in the previous paragraph. One of the ways of getting\nsuch metric spaces is deletion of edges forming short cycles, see\n\\cite{Pel00} on results related to this construction, especially\n\\cite[Section 17.2]{Pel00}. It is worth mentioning that\nbilipschitz equivalent metric spaces can have quite different\nstructure of cycle spaces. Consider, for example, $K_n$ (complete\ngraph on $n$ vertices) and the graph $K_{1,n-1}$ consisting of $n$\nvertices in which the first vertex is adjacent to all other\nvertices, and there are no other edges. Any bijection between\nthese metric spaces has distortion $2$, the cycle space $Z(K_n)$\nis a large space whereas $Z(K_{1,n-1})$ is trivial.\n\n\n\\begin{problem}\\label{P:CloseToell1} It would be very interesting to find a condition on a finite metric space $M$ which is equivalent\nto the condition that the space $\\lf(M)$ is Banach-Mazur close to\n$\\ell_1^n$ of the corresponding dimension. It is not clear whether\nit is feasible to find such a condition.\n\\end{problem}\n\n\n\\section{Large complemented $\\ell_1^n$ in finite-dimensional\nLipschitz\\\\ free spaces}\\label{S:LargeL1}\n\nThe following result can be regarded as a finite-dimensional\nversion of the result of C\\'uth, Doucha, and Wojtaszczyk\n\\cite{CDW16} who proved that the Lipschitz free space on an\ninfinite metric space contains a complemented subspace isomorphic\nto $\\ell_1$.\n\n\n\\begin{theorem}\\label{T:HalfDimL1} For every $n$-point metric space $M$ the\nspace $\\lf(M)$ contains a $2$-complemented $2$-isomorphic copy of\n$\\ell_1^k$ with $k=\\lfloor\\frac{n}2\\rfloor$.\n\\end{theorem}\n\nThe following lemma is a version of \\cite[Lemma 3.1]{CDW16}.\n\n\\begin{lemma}\\label{L:CDW} Let $(M,d)$ be a finite metric space and $\\{y_i\\}_{i=1}^k$ be a sequence of distinct points in\n$M$ such that $M\\backslash \\{y_i\\}_{i=1}^k$ is nonempty. For each\n$i\\in\\{1,\\dots,k\\}$ let $x_i\\in M\\backslash \\{y_i\\}_{i=1}^k$ be\nsuch that the distance $d(x_i,y_i)$ is minimized, so\n$\\{x_i\\}_{i=1}^k$ are not necessarily distinct. Then linear\ncombinations of the functions $f_i(x)=d(y_i,x_i)\\1_{y_i}(x)$\nsatisfy the inequality\n\\begin{equation}\\label{E:LinfLip}\n\\max_i|\\alpha_i|\\le\n\\lip\\left(\\sum_{i=1}^k\\alpha_if_i\\right)\\le\\max\\left\\{\\max_{i\\ne\nj}\\frac{d(x_i,y_i)+d(x_j,y_j)}{d(y_i,y_j)}\\,,1\\right\\}\\cdot\\max_i|\\alpha_i|\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} The leftmost inequality is obtained by comparing\nthe values of $\\sum_{i=1}^k\\alpha_if_i$ at $x_m$ and $y_m$, where\n$m\\in\\{1,\\dots,k\\}$ is such that $\\alpha_m=\\max_i|\\alpha_i|$.\n\nTo prove the rightmost inequality we do the following analysis:\nconsider any pair $(u,v)$ of points in $M$ and estimate from above\nthe quotient\n\\[\\left|\\sum_{i=1}^k\\alpha_if_i(u)-\\sum_{i=1}^k\\alpha_if_i(v)\\right|\/d(u,v).\\]\n\nIf the points $u$ and $v$ are $y_i$ and $y_j$, $i\\ne j$, then the\nestimate from above is\n\\[\\displaystyle{\\frac{d(x_i,y_i)+d(x_j,y_j)}{d(y_i,y_j)}\\cdot\\max_i|\\alpha_i|}.\\]\nIf one of the points is $y_i$ and the other is not in the sequence\n$\\{y_i\\}_{i=1}^k$, we get at most $\\max_i|\\alpha_i|$, because of\nthe minimality property of $d(x_i,y_i)$. If both $u$ and $v$ are\nnot in $\\{y_i\\}_{i=1}^k$, then\n$\\sum_{i=1}^k\\alpha_if_i(u)=\\sum_{i=1}^k\\alpha_if_i(v)=0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:HalfDimL1}]\nAny finite metric space can be considered as a weighted graph with\nthe weighted graph distance (we may consider elements of the\nmetric space as vertices of a complete graph with the weight of\neach edge equal to the distance between its ends).\n\nConsider the minimum weight spanning tree $T$ in this graph\nconstructed according to Boruvka-Kruskal procedure\n\\cite[Construction A]{Kru56} (see also \\cite[Algorithm 8.22]{BM08}), that is: we list edges in the order\nof nondecreasing lengths; then we process this list from the\nbeginning and pick for the spanning tree all edges which do not\nform cycles with the previously selected.\n\nIt is easy to see that the picked set of edges satisfies the\nfollowing condition: at least one of the shortest edges incident\nto each of the vertices is in the spanning tree.\n\nAny tree is a bipartite graph. Therefore we can split $M$ into two\nsubsets, $M_1$ and $M_2$, such that any edge in the spanning tree\n$T$ has one vertex in $M_1$ and the other in $M_2$. At least one\nof the sets $M_1$ and $M_2$ contains at least half of elements of\n$M$. We assume that $M_1$ is such and label its vertices as\n$\\{y_i\\}_{i=1}^k$. For each $i$ we let $x_i$ be the closest to\n$y_i$ element of $M_2$ (the elements $\\{x_i\\}_{i=1}^k$ are not\nrequired to be distinct). The comment in the previous paragraph\nimplies that $x_i\\in M_2$ is one of the closest to $y_i$ and\ndifferent from $y_i$ elements of $M$. Hence, by Lemma \\ref{L:CDW},\nthe subspace of $\\lip_0(M)$ (we pick the base point to be any\nelement of $M\\backslash \\{y_i\\}_{i=1}^k$) spanned by\n$\\{d(y_i,x_i)\\1_{y_i}\\}_{i=1}^k$ is $2$-isomorphic to\n$\\ell_\\infty^k$ and thus $2$-complemented in $\\lip_0(M)$.\n\n\nConsider the functions $u_i=(\\1_{y_i}-\\1_{x_i})\/d(x_i,y_i)$ in\n$\\lf(M)$. We claim that $\\{u_i\\}_{i=1}^k$ span a $2$-complemented\nsubspace in $\\lf(M)$ which is $2$-isomorphic to $\\ell_1^k$. It is\nclear that $||u_i||=1$ and $f_i(u_j)=\\delta_{i,j}$ (Kronecker\n$\\delta$). Let $\\{b_i\\}_{i=1}^k$ be a sequence of real numbers\nsatisfying $\\sum_{i=1}^k|b_i|=1$, and $x=\\sum_{i=1}^k b_iu_i$. We\nneed to estimate the norm of $x$. Clearly\n$||x||\\le\\sum_{i=1}^k|b_i|=1$.\n\nOn the other hand, let $\\alpha_i=\\sign(b_i)$. Then, by the first\npart of the proof,\n\\[1\\le\\left\\|\\sum_{i=1}^k\\alpha_if_i\\right\\|\\le 2\\]\nOn the other hand\n\\[\\left(\\sum_{i=1}^k\\alpha_if_i\\right)(x)=\\sum_{i=1}^k\\alpha_i\nb_i=\\sum_{i=1}^k |b_i|.\\] Hence $\\frac12\\le||x||\\le 1$.\n\n\nNow we show that the linear span of $\\{u_i\\}$ is complemented. We\nintroduce $P:\\lf(M)\\to\\lin\\{u_i\\}$ by\n\\[P(u)=\\sum_{i=1}^k f_i(u)u_i.\\]\nIt is clear that $P$ is a linear projection. Let us estimate its\nnorm. Let $f\\in\\lip_0(M)$ be such that $||f||=1$ and\n$f(P(u))=||P(u)||$. Then\n\\[||P(u)||=\\sum_{i=1}^k f_i(u)f(u_i)\\le \\left\\|\\sum_{i=1}^k\nf(u_i)f_i\\right\\|\\cdot||u||\\le 2\\max_i|f(u_i)|\\cdot||u||\\le\n2||u||.\\]\n\nIt remains to recall that the construction is such that\n$k\\ge\\card(M)\/2$.\n\\end{proof}\n\n\\begin{problem} Whether the constant $2$ in the statement ``$2$-complemented $2$-iso\\-mor\\-phic'' of Theorem \\ref{T:HalfDimL1} is sharp?\n\\end{problem}\n\n\nIt is not surprising that Theorem \\ref{T:HalfDimL1} can be\nsharpened for some classes of graphs. In Theorem \\ref{T:PropDiam}\nwe sharpen it for the diamond graphs.\n\nIt is natural to ask: How and when can we go beyond\nhalf-dimensional subspace? It is easy to see that the following\nresult  can be proved on the same lines as Theorem\n\\ref{T:HalfDimL1}.\n\n\\begin{theorem}\\label{T:LargeL1}\nLet $M$ be a finite metric space and $\\{y_i\\}_{i=1}^k$ be a\nsequence in it such that $M\\backslash\\{y_i\\}_{i=1}^k$ is nonempty.\nLet $d_i=d(y_i,(M\\backslash\\{y_i\\}_{i=1}^k))$ and\n\\begin{equation}\\label{E:IPConst} C=\\max\\left\\{\\max_{i\\ne j}\\frac{d_i+d_j}{d(y_i,y_j)},1\\right\\}.\\end{equation} Then $\\lf(M)$\ncontains as $C$-complemented subspace which is $C$-isomorphic to\n$\\ell_1^k$ and $\\lip_0(M)$ contains a $C$-complemented subspace\nwhich is $C$-isomorphic to $\\ell_\\infty^k$.\n\\end{theorem}\n\n\\begin{corollary}\\label{C:LargeSub} If $M$ is a connected unweighted graph with $n$ vertices,\nthen for every $p\\in\\mathbb{N}$ with $p\\le\\diam(M)+1$ the space\n$\\lf(M)$ contains a subspace of dimension $d\\ge\nn\\left(\\frac{p-1}p\\right)$ which is $4p$-complemented and is\n$4p$-isomorphic to $\\ell_1^d$, and $\\lip_0(M)$ contains a\n$4p$-complemented subspace which is $4p$-isomorphic to\n$\\ell_\\infty^d$. If $p>\\diam(M)$, we have the inequality\n$d_{BM}(\\lf(M),\\ell_1^{n-1})\\le 2p$ for the Banach-Mazur distance.\n\\end{corollary}\n\n\\begin{proof} Let $O$ be one of the vertices of $M$ for which $\\max_{v\\in M}d_M(O,v)=\\diam(M)$. Assume that $p\\le\\diam(M)+1$. Consider partition\n$M=\\cup_{i=0}^{p-1}M_i$, where $M_i$ is the set of vertices in $M$\nwhose distance to $O$ is $i\\pmod p$. The assumption\n$p\\le\\diam(M)+1$ implies that all sets $M_i$ are nonempty. One of\nthe sets $\\{M_i\\}_{i=0}^{p-1}$ has cardinality $\\le\\frac{n}p$. Let\n$\\{y_i\\}_{i=1}^k$ be the complement of this set. Its cardinality,\nwhich we denote by $d$, is at least $n\\left(\\frac{p-1}p\\right)$.\nOn the other hand, it is clear that $d_i\\le 2p$ ($d_i$ is defined\nin Theorem \\ref{T:LargeL1}). Thus the constant $C$ defined in\n\\eqref{E:IPConst} is $\\le 4p$. The conclusion follows.\n\nThe last statement is true because $p\\ge \\diam(M)$ implies that\nthe space $M$ is $2p$-bilipschitz equivalent to the graph\n$K_{1,n-1}$ with its graph distance, and $\\lf(K_{1,n-1})$ is\nisometric to $\\ell_1^{n-1}$ by Proposition \\ref{P:Tree}.\n\\end{proof}\n\nFor some graphs the estimates of Theorems \\ref{T:HalfDimL1},\n\\ref{T:LargeL1} and Corollary \\ref{C:LargeSub} can be improved\nsignificantly. It is interesting that this can be done even in the\ncase of diamond graphs $\\{D_n\\}$, while $\\lf(D_n)$ are far from\n$\\ell_1^{d(n)}$ of the corresponding dimension, see Corollary\n\\ref{C:DiamLargeL1} below and Theorem \\ref{C:LFDnNotL1}.\n\\medskip\n\n\n\\begin{theorem}\\label{T:PropDiam} $\\lf(D_n)$ contains a\n1-complemented isometric copy of  $\\ell_1^k$ with $k=2\\cdot\n4^{n-1}$.\n\\end{theorem}\n\nNote that for large $n$ the number $2\\cdot 4^{n-1}$ is very close\nto $\\frac34\\,|V(D_n)|$, see page \\pageref{P:(b)}.\n\n\\begin{proof} We use an argument similar to the argument of Theorem\n\\ref{T:HalfDimL1} with the following choice of $\\{y_i\\}_{i=1}^k$:\nthe vertices $\\{y_i\\}_{i=1}^k$ are the vertices added to the graph\nin the last step. Formula {\\bf (b)} on page~\\pageref{P:(b)}\nimplies that $k=2\\cdot 4^{n-1}$. The vertex $x_i$ is chosen to be\none of the (two) closest to $y_i$ vertices in $D_n$. In this case\n$d(x_i,y_i)=1$ and $d(y_i,y_j)\\ge2$ for $i\\ne j$. Hence the same\nargument as in Theorem \\ref{T:HalfDimL1} leads to a subspace\nisometric to $\\ell_1^k$ and $1$-complemented.\n\\end{proof}\n\nCorollary of Theorem \\ref{T:LargeL1} for diamonds:\n\n\\begin{corollary}\\label{C:DiamLargeL1} For each $m<n$ the space $\\lf(D_n)$ contains a $C$-complemented\n$C$-isomorphic to $\\ell_1^k$ subspace with $C=2^{n-m}$ and\n$k=2(1+\\sum_{i=0}^{n-1}4^i)-2\\cdot 4^{m-1}$.\n\\end{corollary}\n\n\nNote that the codimension of the subspace does not exceed\n$\\frac3{4^{n-m+1}}|V(D_n)|$.\n\n\\begin{proof} Consider in $D_n$ the\nsubset $A_{n,m}$ of vertices which were added when $D_m$ was\ncreated. The equality {\\bf (b)} on page~\\pageref{P:(b)} implies\nthat the cardinality of $A_{n,m}$ is $2\\cdot 4^{m-1}$. It is also\neasy to see that the distance from any other vertex to this set\ndoes not exceed $2^{n-m-1}$. Define $\\{y_i\\}_{i=1}^k$ as\n$V(D_n)\\backslash A_{n,m}$.\n\nThen the constant $C$ defined in \\eqref{E:IPConst} does not exceed\n$2^{n-m}$ and $k=2(1+\\sum_{i=0}^{n-1}4^i)-2\\cdot\n4^{m-1}$.\\end{proof}\n\n\nResults of this section lead us to a suspicion that Lipschitz free\nspace of dimension $n$ cannot be ``too far'' from $\\ell_1^n$ in\nthe Banach-Mazur distance. In this connection we ask\n\n\\begin{problem}\\label{P:FarFromell1} Estimate the maximal possible Banach-Mazur distance between $\\ell_1^n$ and\na Lipschitz free space of dimension $n$.\n\\end{problem}\n\n\nSo far all known estimates for the Banach-Mazur distance\n$d_{BM}(\\lf(M),\\ell_1^n)$ (where $n=|M|-1$) from below are at most\nlogarithmic in $n$. We know two cases in which logarithmic\nestimates from below hold. One case is the case of diamond graphs\n(if we use estimates based on the theory of Haar functions), see\nTheorems \\ref{C:LFDnNotL1} and \\ref{T:LFmultL1}.\n\nThe second case is the case where $M$ itself has large\n$\\ell_1$-distortion. It is well-known that the $\\ell_1$-distortion\nof $n$-vertex expanders is of order $\\log n$, see \\cite{LLR95}.\nAnother example with $\\log n$-distortion was given in\n\\cite[Corollary 1]{KN06} (see also \\cite[Section 4.2]{Ost13}).\nBourgain \\cite{Bou85} proved that the $\\ell_1$-distortion of an\n$n$-element metric space can be estimated from above by $C\\log n$.\nTherefore on these lines we cannot get lower estimates for\n$d_{BM}(\\lf(M),\\ell_1^n)$ of higher than logarithmic order.\n\nObserve that if $M$ is an expander, then\n$d_{BM}(\\lf(M),\\ell_1^n)\\le C\\log n$ because expanders have\ndiameter of order $\\log n$ and thus are $C\\log n$-bilipschitz\nequivalent to the tree $K_{1,n}$.\n\\medskip\n\nCorollary \\ref{C:LargeSub} allows us to get an estimate\n(Proposition \\ref{P:ell1}) for $d_{BM}(\\lf(M),\\ell_1^n)$ from\nabove in the case where $M$ is an unweighted finite graph, which\nis slightly better than the estimate $d_{BM}(X_n,\\ell_1^n)$ for a\ngeneral $n$-dimensional Banach space $X_n$.\n\nLet us recall known estimates for\n$\\mathcal{D}_n:=\\max\\{d_{BM}(X_n,\\ell_1^n):~\\dim X_n=n\\}$:\n\n\\[n^{\\frac59}\\log^{-C} n\\le \\mathcal{D}_n\\le (2n)^{\\frac56}\\]\nfor some absolute constant $0<C<\\infty$. The lower estimate is due\nto Tikhomirov \\cite{Tik18}; it is an improvement of the previous\nestimate of \\cite{Sza90}. The upper estimate in this form is due\nto Youssef \\cite{You14}; it is an improvement of previous\nestimates of \\cite{BS88}, \\cite{ST89}, and \\cite{Gia95}.\n\n\\begin{proposition}\\label{P:ell1} If $M$ be an unweighted connected graph with $n+1$ vertices (endowed with its graph distance),\nthen $d_{BM}(\\lf(M),\\ell_1^n)\\le\nCn^{\\frac8{11}}$.\n\\end{proposition}\n\n\n\\begin{proof} We shall work with the dual space, that is, we will show\nthat \\[d_{BM}(\\lip_0(M),\\ell_\\infty^n)\\le Cn^{\\frac8{11}}.\\]\n\nBy Corollary \\ref{C:LargeSub} we can find elements\n$f_i\\in\\lip_0(M)$ such that\n\\[\\max_i|\\alpha_i|\\le\\|\\sum\\alpha_if_i\\|\\le 4p \\max|\\alpha_i|\\]\nwhere the codimension of the subspace $F$ spanned by $\\{f_i\\}$ is\n$k\\le \\frac{n}p$ provided $p\\le\\diam(M)+1$.  By an easy corollary\nof the Kadets-Snobar \\cite{KS71} theorem every subspace of\ncodimension $m$ of a finite-dimensional normed space is the range\nof a projection of norm at most $\\sqrt{m} + 1$. Hence we can  find\na projection $P$ of norm at most $2\\sqrt{\\frac np}$ onto $F$. By\nthe result of \\cite{You14}, \\[d_{BM}(\\ker\nP,\\ell_\\infty^k)\\le\\left(2\\frac np\\right)^{\\frac 56}\\] Therefore\nwe can find a sequence $\\{g_i\\}$ in $\\ker P$ such that\n\\[\\max_i|\\beta_i|\\le\\left\\|\\sum \\beta_ig_i\\right\\|\\le\\left(2\\frac np\\right)^{\\frac\n56}\\max_i|\\beta_i|.\\]\n\nWe have\n\\[\\frac1{4\\sqrt{n\/p}}\\max_{i,j}(|\\alpha_i|,|\\beta_j|)\\le\\left\\|\\sum\\alpha_if_i+\\sum\\beta_jg_j\\right\\|\\le\n(4p+(2n\/p)^{\\frac56})\\max_{i,j}(|\\alpha_i|,|\\beta_j|).\\]\n\nSo the Banach-Mazur distance $d_{BM}(\\lip_0(M),\\ell_\\infty^n)$ can\nbe estimated from above by\n\\[4\\sqrt{\\frac np}\\cdot\n\\left(4p+\\left(2\\,\\frac{n}{p}\\right)^{\\frac56}\\right)\\]\n\nPick $p=n^{\\frac 5{11}}$. We get\n$d_{BM}(\\lip_0(M),\\ell_\\infty^n)\\le cn^{\\frac8{11}}$ either by\napplying the argument above if $n^{\\frac 5{11}}\\le\\diam(M)+1$, or\nby using the final statement of Corollary \\ref{C:LargeSub}\notherwise.\n\\end{proof}\n\n\n\\section{Proof for general recursive\nfamilies}\\label{S:Recur}\n\nThe goal of this section is to show that if the graph $B$\nsatisfies the conditions listed in Section \\ref{S:CondB}, then the\nBanach-Mazur distances between the Lipschitz free spaces on $B_n$\n(see Definition \\ref{D:B_n}) and the spaces $\\ell_1^{d(n)}$ of the\ncorresponding dimensions tend to $\\infty$. See Theorem \\ref{T:B_n}\nfor the statement of the result.\n\n\\begin{note}\\label{N:BijVsIso} It is clear that each bijection $g$\non the edge set of a graph $G$ induces an isometry on the space\n$\\ell_1(E(G))$ given by\n\\begin{equation}\\label{E:BijVsIso} f\\mapsto h \\Leftrightarrow\nh(e)=f(g^{-1}e)\\quad f,h\\in \\ell_1(E(G)), e\\in E(G).\n\\end{equation}\nWith some abuse of notation we shall keep the notation $g$ for\nthis isometry.\n\\end{note}\n\n\n\n\\subsection{Conditions on $B$}\\label{S:CondB}\n\nThe conditions below are not independent. Our goal is to list all\nconditions which we use.\n\n\\begin{enumerate}\n\n\\item\\label{I:1} Each edge is contained in a geodesic (a shortest\npath) of even length joining the bottom and the top. Each path\njoining the top and the bottom is geodesic.\n\n\\item\\label{I:Dir} Each edge is directed to the vertex with the\nsmaller distance to the top. The cycle space $Z(B)$ is constructed\nusing this orientation of $B$. Each directed cycle in $B$ is a\nunion of two paths which are pieces of geodesics joining the top\nand the bottom. On one of these paths the direction on the cycle\ncoincides with the direction in $B$, on the other it is opposite.\n\n\\item\\label{I:Vert} The (underlying) graph $B$ has an automorphism\n$v$ which interchanges top and bottom vertices. We say that $v$ is\na {\\it vertical automorphism} of $B$. ({\\it Underlying} here means\nthat the automorphism does not respect directions of edges.)\n\n\\item\\label{I:BasInvv} The automorphism $v$ can be chosen in such\na way that each element of $Z(B)$ is a fixed point of $v$.\n\n\\item\\label{I:Delta} Let $D$ be the distance between the bottom\nand the top in $B$. Consider the vector\n\\begin{equation}\\label{E:Delta}\\Delta=\\frac1{DK}\\sum_p\\1_p\\in\\ell_1(E(B)),\\end{equation}\nwhere $K$ is the number of distinct geodesics joining the bottom\nand the top in $B$, and $\\1_p$ is the indicator function of a\nbottom-top geodesic, and the sum is over all distinct bottom-top\ngeodesics.\n\nIt is easy to see that the map $E_n$ $(n=0,1,2,\\dots)$ which maps\nthe indicator function $\\1_e$ of an edge $e$ onto $\\Delta$ in the\ncopy of $B$ which replaces $e$ extends to an isometric embedding\nof $\\ell_1(E(B_n))$ into $\\ell_1(E(B_{n+1}))$, and that $E_n$ maps\n$Z(B_n)$ into $Z(B_{n+1})$.  We introduce the function $c(B)$ in\n$\\ell_1(E(B))=\\ell_1(E(B_1))$ as the function whose absolute value\nis $E_0(\\1_e)$ (where $e$ is the only edge of $B_0$), and the\nsigns are positive for edges which are closer to the top and\nnegative for edges which are closer to the bottom (recall that\neach edge belongs to a geodesic of even length joining the top and\nthe bottom). One of the {\\bf conditions} on these maps is:\n$v(c(B))=-c(B)$ (see Note \\ref{N:BijVsIso}), this condition\nactually follows from other conditions. Another condition is in\nitem \\ref{I:NoFixCyclesButc(B)}.\n\n\\item\\label{I:NoFixCyclesButc(B)} The collection $\\mathcal{H}$ of\nall automorphisms of $B$ for which the top and the bottom are\nfixed points satisfies two conditions. First is that the\ncorresponding subgroup of isometries of $\\ell_1(E(B))$ has no\nfixed points in the cycle space $Z(B)$ except $0$. Second is that\nthe function $c(B)$ is a common fixed point of all elements of\n$\\mathcal{H}$. We call automorphisms of $\\mathcal{H}$ {\\it\nhorizontal}.\n\n\\item\\label{I:z} The cycle space of $B$ is nontrivial. This is\nequivalent to the existence of two distinct bottom-top geodesics,\nand this is, in turn, equivalent to the fact that\n$\\frac1D\\1_p-\\Delta\\ne 0$ for any bottom-top geodesic $p$. We pick\na bottom-top geodesic $p$ for which the $\\ell_1$-norm\n$\\frac1D\\1_p-\\Delta\\ne 0$ is maximized, denote this difference by\n$d(B)$ and its norm in $\\ell_1(E(B))$ by $\\alpha$. Observe that\n$d(B)\\in Z(B)$.\n\n\n\n\\end{enumerate}\n\nIt is worth mentioning that the graphs $\\{B_n\\}$, $n\\ge 1$,\ninherit some properties of the graph $B=B_1$.\n\\medskip\n\n\\noindent{\\bf (A)} Graphs $B_n$ have properties of items \\ref{I:1}\nand \\ref{I:Dir}.\n\\medskip\n\nOnly the last condition in item \\ref{I:Dir} requires verification.\nThis can be done using induction. We have assumed this condition\nfor $B_1$. Suppose that holds for $B_{n-1}$. Consider a directed\ncycle in $B_n$. By \\eqref{E:DiffProd} we have $B_n=B\\oslash\nB_{n-1}$. If the cycle is contained in one of the copies of\n$B_{n-1}$, we are done by the induction hypothesis. If the cycle\nis not contained in any of $B_{n-1}$, then it can be obtained\nreplacing each edge in the corresponding cycle in $B_1$ by a\ntop-bottom path in the corresponding copy of $B_{n-1}$ (see item\n\\ref{I:1}). The conclusion follows if we recall how edges of $B_n$\nare oriented, see Definition \\ref{D:Comp}.\\medskip\n\n\n\\noindent{\\bf (B)} The underlying graph of $B_n$ has an\nautomorphism $v_n$ which interchanges top and bottom vertices.\n\\medskip\n\nThis can be proved by induction:\n\n\\begin{itemize}\n\n\\item For $B_1=B$ this is an assumption of item \\ref{I:Vert}.\n\n\\item Suppose that this is true for $B_{n-1}$ and $v_{n-1}$ is the\ncorresponding automorphism. By \\eqref{E:DiffProd} we have\n$B_n=B\\oslash B_{n-1}$. We consider the bijection of the edge set\nof $B_n$ designed in the following way:\n\n\\item If $v$ maps an edge $uw$ to an edge $\\hat u\\hat w$, with $u$\nand $\\hat u$ being closer to the bottom of $B$, we map $B_{n-1}$\ncorresponding to the edge $uw$ onto $B_{n-1}$ corresponding to\n$\\hat u\\hat w$ ``upside down'', that is, using $v_{n-1}$.\n\n\\item It is easy to see that we get an automorphism of $B_n$,\nwhich interchanges the top and the bottom. We denote this\nautomorphism by $v_n$.\n\n\\end{itemize}\n\n\\subsection{The main result}\n\nOur main result on families $\\{B_n\\}$:\n\n\\begin{theorem}\\label{T:B_n} If the directed graph $B$ satisfies the conditions in items \\ref{I:1}--\\ref{I:z} listed above, and $\\{B_n\\}_{n=0}^\\infty$ are constructed\naccording to Definition \\ref{D:B_n}, then\n\\[d_{BM}(\\lf(B_n), \\ell_1^{d(n)})\\ge\\frac{cn}{\\ln n}\\] for $n\\ge 2$ and some absolute constant $c>0$, where $d(n)$ is the dimension of\n$\\lf(B_n)$.\n\\end{theorem}\n\nTo prove Theorem \\ref{T:B_n} we need several lemmas. The final\nstep in the proof is presented on page \\pageref{Page:EndPrf}.\n\n\\begin{lemma}\\label{L:RedProj} To prove Theorem \\ref{T:B_n} it\nsuffices to show that the relative projection constants of\n$Z(B_n)$ in $\\ell_1(E(B_n))$ satisfy\n\\[\\lambda(Z(B_n), \\ell_1(E(B_n)))\\ge\\frac{cn}{\\ln n}\\]\nfor $n\\ge 2$ and some absolute constant $c>0$.\n\\end{lemma}\n\n\\begin{proof} This lemma is a consequence of the following\nwell-known fact.\\end{proof}\n\n\\begin{fact}\\label{F:Lift} If a quotient $X\/Y$ is such that the\nBanach-Mazur distance satisfies \\[d_{BM}(X\/Y,\\ell_1(\\Gamma))\\le\nC,\\] then $\\lambda(Y,X)\\le (1+C)$.\n\\end{fact}\n\n\\begin{proof}[Proof of Fact \\ref{F:Lift}] Denote by $Q: X\\to X\/Y$ the quotient map. Let $T:\\ell_1(\\Gamma)\\to X\/Y$ be such that $||T||< C+\\ep$,\n$||T^{-1}||\\le 1$. By the {\\it lifting property} of\n$\\ell_1(\\Gamma)$ (see \\cite[pp.~107--108]{LT77}), there is a\nlinear operator $\\widehat T: \\ell_1(\\Gamma)\\to X$ such that\n$||\\widehat T||<C+\\ep$ and $Q\\widehat T=T$. Then the operator $(I\n- \\widehat T T^{-1} Q)$ is a projection of $X$ onto $Y$, and its\nnorm is $<(1+C+\\ep)$, the conclusion follows.\n\\end{proof}\n\n\\subsection{Cycle-preserving bijections of $B_n$}\\label{S:CycBij}\n\nFor each $n\\in\\mathbb{N}$ we introduce $G_n$ as the group of all\n{\\it cycle-preserving bijections} of $E(B_n)$ (we consider\nundirected edges) satisfying the additional condition: the edge\nset of any path joining the top and the bottom of $B_n$ is mapped\nonto the edge set of a path joining the top and the bottom of\n$B_n$. By a {\\it cycle-preserving bijection} we mean a bijection\nwhich maps the edge-set of any cycle to an edge-set of a cycle (we\ndo not pay attention to directions of edges). It is clear that\n$G_n$ is a finite group.\n\\medskip\n\nThe representation \\eqref{E:DiffProd} shows that for each $1\\le\nk\\le n-1$ the graph $B_n$ is a union of edge-disjoint copies of\n$B_k$. It is clear that bijections of $E(B_n)$ which leave all\nthese copies of $E(B_k)$ invariant, and whose restrictions to them\nare contained in $G_k$, belong to $G_n$.\\medskip\n\nThe groups $G_n$ lead in a natural way (see Note \\ref{N:BijVsIso})\nto subgroups of the group of isometries of $\\ell_1(E(B_n))$. An\nimportant observation is that the subgroup corresponding to $G_n$\nleaves the cycle space $Z(B_n)$ invariant.\n\\medskip\n\nThis observation can be shown as follows. By statement {\\bf (A)}\neach directed cycle in $B_n$ is a union of two pieces, $C_1$ and\n$C_2$, of geodesics (going up and going down). Thus there are\nwell-defined notions of the {\\it top} (and {\\it bottom}) {\\it of\nthe cycle} - the vertex of the cycle nearest to the top (bottom)\nof $B_n$. We join them to the top and bottom of $B_n$,\nrespectively, using pieces of geodesics $P_b$ and $P_t$. Then both\nthe concatenation $P_bC_1P_t$ and $P_bC_2P_t$ are paths joining\nthe bottom and the top of $B_n$. Therefore the additional\ncondition on cycle-preserving bijections implies that the edge\nsets of $P_bC_1P_t$ and $P_bC_2P_t$ are edge-sets of bottom-top\npaths in $B_n$. Also the image of the edge set of the cyclic\nconcatenation of $C_1C_2$ is an edge set of a cycle. It is easy to\nsee that these conditions together imply that the images of $C_1$\nand $C_2$ are parts of bottom-top geodesics. Hence the image of\n$C$ is in the cycle space.\n\n\n\nObserve that $G_1$ contains both $\\mathcal{H}$ and the vertical\nautomorphism $v$, and thus the group generated by\n$\\mathcal{H}\\cup\\{v\\}$.\n\n\\subsection{Gr\\\"unbaum-Rudin-Andrew-type\naveraging}\\label{S:GRAAppr}\n\n\nUsage of the averages of the following type for estimates of\nprojection constants goes back at least to Gr\\\"unbaum \\cite{Gru60}\nand Rudin \\cite{Rud62}. It was used in a similar to the present\ncontext by Andrew \\cite{And78}.\n\nLet $P$ be any linear projection of $\\ell_1(E(B_n))$ onto\n$Z(B_n)$. Since $G_n$ is a finite group, which can be regarded as\na group of isometries of $\\ell_1(E(B_n))$, the following operator\nis well-defined\n\\begin{equation}\\label{E:DefPGn}\nP_{G_n}:=\\frac1{|G_n|}\\sum_{g\\in G_n}g^{-1}Pg,\\end{equation} is\nalso projection on $Z(B_n)$, and $||P_{G_n}||\\le ||P||$. It is\neasy to check that $P_{G_n}$ has the following important property:\n\\begin{equation}\\label{E:CommWithGn}\n\\forall g\\in G_n\\quad P_{G_n}g=gP_{G_n}\n\\end{equation}\n\nWe call a projection satisfying \\eqref{E:CommWithGn} {\\it\ninvariant} with respect to $G_n$.\n\\medskip\n\n\n\nThe new twist in the usage of the method in our paper (see\nSections \\ref{S:Annihil} and \\ref{S:Combin}) is that we use it in\nsituations where the invariant projection is not unique. Namely we\nobserve that although in some situations which we consider\n$P_{G_n}$ obtained by formula \\eqref{E:DefPGn} is not unique (see\nSection \\ref{S:NonUnLaakso}), it is possible to show, see Lemma\n\\ref{L:Annihil}, that there is a collection of vectors in\n$\\ell_1(E(B_n))$ which are mapped to $0$ by any $P_{G_n}$\nsatisfying \\eqref{E:CommWithGn}. This allows us to show that in\nthe considered cases $||P_{G_n}||$ grows indefinitely as\n$n\\to\\infty$, see Section \\ref{S:Combin}; and to get the estimate\nstated in Lemma \\ref{L:RedProj}.\n\n\\subsection{Bases in the spaces $Z(B_n)$}\n\nWe need to find a basis $S_n$ in the cycle space $Z(B_n)$, $n\\ge\n1$. Each of the bases which we pick will satisfy the following\nconditions.\n\n\\begin{itemize}\n\n\\item[{\\bf (i)}] Each element is either a fixed point of $v_n$, or\nis supported on a copy of some $B_k$, $1\\le k\\le n-1$, and is an\nelement of the corresponding $S_k$.\n\n\\item[{\\bf (ii)}] If an element is a fixed point of $v_n$, then\nits restriction to any $B_k$, $1\\le k\\le n-1$, is a multiple of\n$\\Delta_k$, and thus is a fixed point of the corresponding $G_k$\n(see the discussion next to \\eqref{E:Delta_m} below). This\ncondition is void if $n=1$.\n\n\\end{itemize}\n\n\\medskip\n\nSince $B_1=B$, we let $S_1$ be any basis in $Z(B)$. The conditions\n{\\bf (i)} and {\\bf (ii)} are trivially satisfied, see item\n\\ref{I:BasInvv} in Section \\ref{S:CondB}.\n\\medskip\n\nLet $e\\in E(B_k)$. It is easy to verify that the function\n$E_{m+k-1}E_{m+k-2}\\dots E_k \\1_e\\in \\ell_1(E(B_{m+k}))$, which is\nsupported on a copy of $B_m$ which evolved from $e$, can be\nwritten (similarly to \\eqref{E:Delta}) as\n\\begin{equation}\\label{E:Delta_m}\\Delta_m:=\\frac1{D_{m}K_m}\\sum_p\\1_p,\\end{equation} where $K_m$ is the\nnumber of distinct geodesics joining the bottom and the top of the\ncopy of $B_m$ mentioned above, $\\1_p$ is the indicator function of\na bottom-top geodesic in $B_m$, and the sum is over all distinct\nbottom-top geodesics. It is easy to see that $\\Delta_1=\\Delta$.\n\n\n\\begin{observation}\\label{O:FixedDelta_m} Since any element of $G_m$ maps bijectively bottom-top paths\nin $B_m$, we see that the function $\\Delta_m$ is the fixed point\nof any element of $G_m$ considered as acting on the considered\ncopy of $B_m$.\n\\end{observation}\n\nNow we pick basis in $Z(B_n)$ assuming that we already picked a\nbasis $S_{n-1}$ in $Z(B_{n-1})$. The basis consists of two types\nof vectors:\n\n\\begin{itemize}\n\n\\item[{\\bf (I)}] Vectors which were already picked for $S_{n-1}$\nin one of the copies on $B_{n-1}$ in $B_n$. Recall that\n$B_n=B_1\\oslash B_{n-1}$, see \\eqref{E:DiffProd}.\n\n\\item[{\\bf (II)}]  For each $f\\in S_1$ we consider the following\nfunction on $B_n=B_1\\oslash B_{n-1}$: its restriction to each of\nthe copies of $B_{n-1}$ is a product of the corresponding\n$\\Delta_{n-1}$ and the value of $f$ on the edge from which the\nconsidered copy of $B_{n-1}$ has evolved.\n\n\\end{itemize}\n\n\\begin{observation}\\label{O:TypeII} Any vector of type {\\bf (II)}\nis a fixed point of any $G_{n-1}$. The same holds for any $G_k$,\n$1\\le k\\le n-1$ corresponding to $B_n=B_{n-k}\\oslash B_k$ and\nacting on one of the copies of $B_k$. For the second statement we\nneed to observe that the restriction of $\\Delta_{n-1}$ to $B_k$ is\na multiple of the corresponding $\\Delta_k$.\n\\end{observation}\n\n\nFirst we need to show that conditions {\\bf (i)} and {\\bf (ii)} are\nsatisfied. It is easy to see that the only statement requiring a\nproof is the fact that the function constructed in the previous\nparagraph is a fixed point for $v_n$.\\medskip\n\nTo see this we observe that the values of $f$ corresponding to\ncopies of $B_{n-1}$ which are mapped one onto another by $v_n$ are\nequal because $f$ is a fixed point of $v$ and by construction of\n$v_n$. Thus we get the desired conclusion.\n\n\n\\begin{lemma}\\label{L:Basis} The set $S_n$ is a basis of the\nlinear space $Z(B_n)$.\n\\end{lemma}\n\n\n\\begin{proof} We use induction. For $n=1$ this is true according\nto our choice. Suppose that the statement holds for $n-1$, and\nshow that this implies it for $n$. We need to show two things:\ncompleteness and linear independence.\\medskip\n\n\n\\noindent{\\bf Completeness:} (1) If a cycle is contained in one of\nthe $B_{n-1}$, then it is contained in the linear span of the\ncorresponding $S_{n-1}$ by the induction hypothesis, and we are\ndone because $S_{n}$ contains that $S_{n-1}$.\n\n(2) If a cycle $C$ is not contained in any of $B_{n-1}$, then,\nafter collapsing each of $B_{n-1}$ to the edge of $B_1$ from which\nit evolved (according to $B_n=B_1\\oslash B_{n-1}$), we get a\nnontrivial cycle $\\widehat C$ in $B_1$. This cycle is a linear\ncombination of cycles of $S_1$ (since $S_1$ is a basis in\n$Z(B_1)$), so $\\widehat C=\\sum\\gamma_is_i$ for some\n$\\gamma_i\\in\\mathbb{R}$ and $s_i\\in S_1$. Denote the composition\n$E_{n-1}\\dots E_1$ by $\\mathcal{E}_{n-1}$. We have\n\n\\[\\mathcal{E}_{n-1}\\widehat C=\\sum\\gamma_i\\mathcal{E}_{n-1}s_i.\\]\n\nThe description of the type {\\bf (II)} vectors implies that\nvectors $\\mathcal{E}_{n-1}s_i$ are elements of $S_n$. Therefore it\nremains to analyze the difference $C-\\mathcal{E}_{n-1}\\widehat C$.\n\\medskip\n\n\nFor each $B_{n-1}$ in $B_n$ (according to $B_n=B_1\\oslash\nB_{n-1}$) one of the following is true:\n\n\\begin{itemize}\n\n\\item There are no edges of $C$ and no edges of\n$\\mathcal{E}_{n-1}\\widehat C$ in it.\n\n\\item There is a path $p$ from the bottom to top of $B_{n-1}$\nwhich is contained in $C$, and the corresponding part of\n$\\mathcal{E}_{n-1}\\widehat C$ is $\\Delta_{n-1}$.\n\n\\end{itemize}\n\nIt remains to observe that that $\\Delta_{n-1}-\\1_p$ belongs to\n$Z(B_{n-1})$ (follows from the formula for $\\Delta_{n-1}$). Thus\nthe difference $C-\\mathcal{E}_{n-1}\\widehat C$ can be written as a\nsum of elements of $S_{n-1}$ for those $B_{n-1}$ which contain\nnontrivial sub-paths of $C$. As a conclusion we get that $C$ is in\nthe linear span of $S_n$.\n\\medskip\n\n\\noindent{\\bf Linear Independence.} It is clear that a nontrivial\nlinear combination of vectors of type {\\bf (I)} cannot be equal to\n$0$, since $S_{n-1}$ are linearly independent and $B_{n-1}$ are\nedge-disjoint.\n\\medskip\n\nFor this reason to prove linear independence it is enough to show\nthat a nontrivial linear combination containing vectors of type\n{\\bf (II)} cannot be $0$.\n\\medskip\n\nWe split a linear combination as $a+b$, where $a$ is a linear\ncombination of vectors of type {\\bf (I)}, and $b$ is a linear\ncombination of vectors of type {\\bf (II)}. Observe that $b$ can be\nobtained in the following way: we consider a non-zero vector\nvector in $Z(B_1)$, and then replace each $\\1_e$ used in this\nvector, by the corresponding $\\Delta_{n-1}$. Because of this the\nrestriction of $b$ to at least one of $B_{n-1}$ does not belong to\n$Z(B_{n-1})$. Hence $a+b$, restricted to that $B_{n-1}$ is\nnonzero, and we are done.\n\\end{proof}\n\n\\subsection{Invariant projections annihilate functions $c(B_1)$\nand their images under $E_k$}\\label{S:Annihil}\n\n\\begin{lemma}\\label{L:Annihil} The projection $P_{G_n}$ annihilates all of the functions\nof the form $c(B_1)$ for some $B_1$ in $B_n$, and functions which\nare obtained from $c(B_1)$ by repeated applications of $E_k$.\n\\end{lemma}\n\n\n\\begin{proof} Let $f$ be some function of the described form in\n$\\ell_1(E(B_n))$ and let $B_m$, $m\\le n$, be a subgraph of $B_n$\nsupporting $f$. It is easy to see that the absolute value of $f$\nis equal to the function $\\Delta_m$ described in\n\\eqref{E:Delta_m}, and that $f$ is positive on edges which are\ncloser to the top of $B_m$ and negative on the edges which closer\nto the bottom of $B_m$.\\medskip\n\nSuppose, contrary to the statement of the Lemma, that\n$P_{G_n}f=q\\ne 0$. Since $q\\in Z(B_n)$, it is a linear combination\nof vectors described in {\\bf (I)--(II)}.\\medskip\n\nIt is clear that one of the following is true:\n\n\\begin{itemize}\n\n\\item[$(\\le)$] One of the vectors of the basis described in {\\bf\n(I)--(II)}, present in the linear combination representing $q$,\nbelongs to $S_k$ with $k\\le m$.\n\n\\item[$(>)$] All vectors of the basis described in {\\bf\n(I)--(II)}, present in the linear combination representing $q$,\nbelong to $S_k$ with $k>m$.\n\n\\end{itemize}\n\nWe show, that in each of these cases we get a contradiction with\nthe invariance of $P_{G_n}$.\n\\medskip\n\n\\noindent{\\bf Case $(\\le)$.} Assume that $k$ is the smallest\ninteger with this property. Since it is the smallest integer, all\nbasis elements with nonzero coefficients belonging to $S_k$ are of\ntype {\\bf (II)}. Therefore they correspond to certain elements of\n$S_1$, and their linear combination $\\mu$ (as it is present in the\nrepresentation of $q$) corresponds to nonzero element $\\tau$ of\n$Z(B_1)$. By Condition \\ref{I:NoFixCyclesButc(B)} (on $B$) there\nexists a horizontal automorphism $g$ of $B_1$, such that $g\\tau\\ne\n\\tau$. Let us consider an automorphism $\\hat g$ of $S_k$ {\\it\ninduced} by $g$ in the following way. The automorphism $g$ is a\nbijection of $E(B_1)$. In $B_k=B_1\\oslash B_{k-1}$ we consider the\ncorresponding bijections of subgraphs $B_{k-1}$ which evolved from\nthose edges. It is clear that $\\hat g\\in G_k$, and that $\\hat\ng\\mu\\ne\\mu$.\n\\medskip\n\nOn the other hand, it is clear that $\\hat g f=f$. In the case\nwhere $k=m$ this follows from the fact that $c(B)$ is a fixed\npoint of all horizontal automorphisms (Condition\n\\ref{I:NoFixCyclesButc(B)}). In the case where $k<m$, this follows\nfrom Observation \\ref{O:FixedDelta_m}. We get a contradiction with\nthe fact that $P_{G_n}$ is an invariant projection (see\n\\eqref{E:CommWithGn}), because\n\\[\\begin{split}P_{G_n}f&=P_{G_n}\\hat gf=\\hat gP_{G_n}f=\\hat gq=\\hat\ng(\\mu+(q-\\mu))\\\\&=\\hat g\\mu+\\hat g(q-\\mu)=\\hat g(\\mu)+(q-\\mu)\\ne\n\\mu+q-\\mu=q,\\end{split}\\] where we used the fact that elements of\nthe basis $S_n$ used in the decomposition of $q-\\mu$ are either\nedge-disjoint with the copy of $B_k$ on which $\\mu$ is supported,\nor are proportional to $\\Delta_k$ on that $B_k$. In either case\n$(q-\\mu)$ is a fixed point of $\\hat g$.\n\\medskip\n\n\\noindent{\\bf Case $(>)$.} In this case, by Observation\n\\ref{O:FixedDelta_m}, any function used in the decomposition of\n$q$ with respect to the basis $S_n$ is a fixed point of $v_m$,\nwhich was defined in {\\bf (B)}.\\medskip\n\nOn the other hand, $v_mf=-f$, by the definitions of $v_m$ and $f$.\nThis contradicts the fact that $P_{G_n}$ is an invariant\nprojection (see \\eqref{E:CommWithGn}), because we get\n\\[-P_{G_n}f=P_{G_n}v_mf=v_mP_{G_n}f=v_mq=q=P_{G_n}f.\\qedhere\n\\]\n\\end{proof}\n\n\\subsection{Combining everything}\\label{S:Combin}\n\n\\begin{proof}[Proof of Theorem \\ref{T:B_n}]\\label{Page:EndPrf} Let us show, using Lemma \\ref{L:RedProj}, that in order to prove Theorem\n\\ref{T:B_n} it suffices to show that for each $r\\in \\mathbb{N}$\nthere exists $n=n(r)\\in\\mathbb{N}$,  $C_r\\in Z(B_n)$, and a linear\ncombination $A_r$ of vectors of the forms $c(B_1)$ and their\nimages under $\\{E_n\\}$ such that\n\\begin{equation}\\label{E:Norm1}\n||C_r+A_r||=1\n\\end{equation}\nand\n\\begin{equation}\\label{E:P_Large}\n||C_r||\\ge 1+\\frac{\\alpha (r-1)}{2},\n\\end{equation}\nwhere $\\alpha>0$ is the number introduced in item \\ref{I:z} of\nSection \\ref{S:CondB}, and to find a suitable estimate for the\ncorresponding $n(r)$ in terms of $r$.\\medskip\n\n\nIn fact, for every projection $P:\\ell_1(E(B_n))\\to Z(B_n)$ we get\n\n\\[||P||\\ge ||P_{G_n}||\\ge ||P_{G_n}(C_r+A_r)||\\stackrel{\\small\\hbox{(Lemma\n\\ref{L:Annihil})}}{=}||C_r||\\ge 1+\\frac{\\alpha(r-1)}2,\\] which\nproves the desired inequality for the projection constant.\n\\medskip\n\nCase $r=1$. We let $C_1$ be any $\\ell_1$-normalized element of\n$S_1$ (use non-triviality);  $A_1=0$. Everything is obvious.\n\nInductive step. Suppose that we have already constructed $C_r$ and\n$A_r$ in some $B_{n(r)}$.\n\\medskip\n\nWe apply $E_{n(r)}$ to $C_r+A_r$. Observe that $E_{n(r)}$ maps the\ncycle space into the cycle space, and preserves the desired form\nof the function $A_r$. Observe that $C_r+A_r$, as an element of\n$\\ell_1(E(B_{n(r)}))$ is a linear combination of edges. Therefore\n$E_{n(r)}(C_r+A_r)$ is of the form $\\displaystyle{\\sum_{e\\in\nE(B_{n(r)})}a_{e,1}E_{n(r)}\\1_e}$, where $a_{e,1}$ are real\nnumbers. The functions $E_{n(r)}\\1_e$ are of the form $\\Delta$\n(see \\eqref{E:Delta}), supported on different copies of $B_1$,\nrecall that\n\\begin{equation}\\label{E:Dec}\nB_{n(r)+1}=B_{n(r)}\\oslash B_1 \\end{equation} (see\n\\eqref{E:DiffProd}). We let $C_r^1=E_{n(r)}C_r$ and let\n\\[A_r^1=E_{n(r)}A_r+\\sum_{e\\in\nE(B_{n(r)})}a_{e,1}c(B_1),\\] where $c(B_1)$ is taken on the\ncorresponding copy of $B_1$, according to \\eqref{E:Dec}. It is\neasy to see that $||C_r^1+A_r^1||=1$, and its support is exactly\nhalf (in many respects) of the support of $E_{n(r)}(C_r+A_r)$.\n\\medskip\n\nWe repeat the procedure for $C_r^1$ and $A_r^1$ instead of $C_r$\nand $A_r$. We do this $t$ times, and get the functions which we\ndenote $C_r^t$ and $A_r^t$.\n\nSome observations:\n\n\\begin{itemize}\n\n\\item The function $C_r^t$ is an image of $C_r$ under the\ncomposition $E_{n(r)+t-1}\\dots E_{n(r)}$.\n\n\\item The function $A_r^t$ is a linear combination of\n$E_{n(r)+t-1}\\dots E_{n(r)}A_r$ and images of $c(B_1)$ under some\ncompositions of $E_k$.\n\n\\end{itemize}\n\nAfter that we do a somewhat different procedure. Namely we write\n$E_{n(r)+t}(C_r^t+A_r^t)$ in the form $\\sum_{e\\in E(B_{n(r)+t})}\na_{e,t+1}E_{n(r)+t}\\1_e$, where $a_{e,t+1}$ are real numbers. The\nfunctions $E_{n(r)+t}\\1_e$ are multiples of $\\Delta$, supported on\ndifferent copies of $B_1$, recall that\n$B_{n(r)+t+1}=B_{n(r)+t}\\oslash B_1$. Now we let\n\\[A_{r+1}=E_{n(r)+t}(A_r^t)\\]\nand\n\\[C_{r+1}=E_{n(r)+t}(C_r^t)+\\sum_{e\\in E(B_{n(r)+t})}\na_{e,t+1}d(B),\\] where $d(B)$ is the function defined in item\n\\ref{I:z} of Section \\ref{S:CondB} and supported on the\ncorresponding copy of $B_1$.\n\\medskip\n\nIt is clear from the definition of $d(B)$ that\n$||C_{r+1}+A_{r+1}||=1$. It is also clear that $C_{r+1}\\in\nZ(B_{n(r)+t+1})$, and $A_{r+1}$ is of the desired form.\n\\medskip\n\nObserve that since $||\\sum a_{e,t+1}1_e||=1$, we have $||\\sum\na_{e,t+1}d(B)||=\\alpha$ (see item \\ref{I:z} in Section\n\\ref{S:CondB}). Our construction is such that the norm of the part\nof $E_{n(r)+t}\\dots E_{n(r)}C_r$ supported in the support of\n$\\sum_{e\\in E(B_{n(r)+t})} a_{e,t+1}d(B)$ is\n$\\displaystyle{\\frac1{2^t}||C_r||}$. Therefore if we pick $t$ in\nsuch a way that\n\\begin{equation}\\label{E:Need}\\frac1{2^t}||C_r||<\\frac{\\alpha}4,\\end{equation} we get\n\n\\[\\begin{split}||C_{r+1}||&=\\left\\|E_{n(r)+t}\\dots E_{n(r)}C_r+\\sum_{e\\in E(B_{n(r)+t})}\na_{e,t+1}d(B)\\right\\|\\\\&\\ge ||E_{n(r)+t}\\dots E_{n(r)}C_r||+\n\\left\\|\\sum_{e\\in E(B_{n(r)+t})}\na_{e,t+1}d(B)\\right\\|-2\\frac{\\alpha}4\\\\&\\stackrel{\\eqref{E:P_Large}}{\\ge}\n1+\\frac{\\alpha (r-1)}2+\\alpha-\\frac{\\alpha}2=1+\\frac{\\alpha\nr}2.\\end{split}\\]\n\nIt remains to find an estimate for $n$ in terms of $r$. To achieve\nthe condition \\eqref{E:Need} for $r\\ge 2$ we need to pick $t\\ge\nC\\ln r$ for some $C>0$.\n\nThis leads to the estimate $\\lambda(Z(B_n),\\ell_1(E(B_n)))\\ge ck$\nif $n\\ge Ck\\ln k$, where $c>0$, $C<\\infty$ (the constants in these\nstatements do not have to be the same) and $d(n)$ is the dimension\nof $\\lf(B_n)$.\n\\medskip\n\nIt is easy to see that this estimate implies\n\\[ \\lambda(Z(B_n),\\ell_1(E(B_n)))\\ge \\frac{cn}{\\ln n}.\\qedhere\\]\n\\end{proof}\n\n\n\n\\section{Consequences for multibranching diamond graphs and Laakso graphs}\\label{S:Diamond}\n\nOur next goal is to show that diamond graphs and Laakso graphs\nsatisfy the conditions listed in Section \\ref{S:CondB}.\n\n\\subsection{Multibranching diamond graphs}\n\nCondition \\ref{I:1} in the case where $B$ is $K_{2,n}$, $n\\ge 2$,\nwith the top and the bottom being the vertices of the part\ncontaining two vertices is obvious.\\medskip\n\nCondition \\ref{I:Dir} is clear.\\medskip\n\nFor Condition \\ref{I:Vert} we choose the automorphism in such a\nway that it maps each bottom-top path onto itself.\\medskip\n\nWith this choice of $v$ the Condition \\ref{I:BasInvv} is easy to\ncheck.\\medskip\n\nCondition \\ref{I:Delta} is clearly satisfied.\\medskip\n\nCondition \\ref{I:NoFixCyclesButc(B)}: A nonzero element of\n$Z(K_{2,n})$ cannot be a fixed point of $\\mathcal{H}$ because\n(according to the directions chosen on edges) each non-zero\nelement of $Z(K_{2,n})$ has bottom-top paths on which the value is\npositive and bottom-top paths on which the value is\nnegative.\\medskip\n\nThe second part of Condition \\ref{I:NoFixCyclesButc(B)} holds\nbecause any horizontal automorphism maps edges which are closer to\nthe top (bottom) to edges which are closer to the top (bottom).\n\\medskip\n\nFinally, Condition \\ref{I:z} is satisfied because we consider\n$n\\ge 2$ and $s_1$ (element of the basis listed above) is an\nexample of a nontrivial cycle in $Z(K_{2,n})$. The value of\n$\\alpha$ is $\\frac{2(n-1)}n$.\n\n\\subsection{Laakso graphs}\n\nCondition \\ref{I:1} in the case where $B$ is $\\mathcal{L}_1$ with\nthe natural choice of the top and the bottom is obvious.\\medskip\n\nCondition \\ref{I:Dir} is clear.\\medskip\n\nFor Condition \\ref{I:Vert} we choose the automorphism $v$ which\nmaps each bottom-top path onto itself.\\medskip\n\nCondition \\ref{I:BasInvv}: There is only one cycle in\n$\\mathcal{L}_1$, it is obviously the fixed point of the chosen\nautomorphism of $\\mathcal{L}_1$.\\medskip\n\nCondition \\ref{I:Delta} is clearly satisfied.\\medskip\n\nThe first part of Condition \\ref{I:NoFixCyclesButc(B)} holds\nbecause, by the choice of the directions of edges any nonzero\nelement of $Z(\\mathcal{L}_1)$ has positive value on one side and\nnegative value on the other side, and thus is mapped onto its\nnegative by a nontrivial element of $\\mathcal{H}$.\\medskip\n\nThe second part of Condition \\ref{I:NoFixCyclesButc(B)} holds\nbecause any horizontal automorphism maps edges which are closer to\nthe top (bottom) to edges which are closer to the top (bottom).\n\\medskip\n\nCondition \\ref{I:z} is clearly satisfied. The value of $\\alpha$ is\n$\\frac12$.\n\n\\subsection{Non-uniqueness of invariant projections of $\\ell_1(E(\\mathcal{L}_2))$\nonto $Z(\\mathcal{L}_2)$}\\label{S:NonUnLaakso}\n\nOur main goal in this section is to show that for Laakso graphs\nthere is no uniqueness of invariant projections. It is clear that\none of the invariant projections is the orthogonal projection onto\n$Z(\\mathcal{L}_2)$ in $\\ell_2(E(\\mathcal{L}_2))$. So it is enough\nto construct an invariant projection which is not orthogonal.\n\n\n\\begin{proposition}\\label{P:CutNotTo0} There exists an invariant linear projection of $\\ell_2(E(\\mathcal{L}_2))$ onto  $Z(\\mathcal{L}_2)$\nwhich is different from the orthogonal projection.\n\\end{proposition}\n\n\\begin{proof} We consider the following projection:\nIt is like the orthogonal projection on the top and bottom\n``tails'' of $\\mathcal{L}_2$ and is different only in the central\npart. In the central part there are edges which belong to the\n16-cycle only and edges which belong also to $4$-cycles.\n\\medskip\n\nWe introduce the following functions in $\\ell_1(E(\\mathcal{L}_2))$\nsupported on the central part of $\\mathcal{L}_2$: \\medskip\n\n\n\\noindent{\\bf (1)} Indicator functions $\\chi_C$ of cycles of\nlength $4$ (see \\eqref{E:SignInd}) directed counterclockwise, so\nthey have values $1$ on the right sides and values $-1$ on the\nleft sides.\n\\medskip\n\n\n\\noindent{\\bf (2)} The function $F=\\frac{F_1+F_2}2$, where $F_1$\nis the indicator function of the directed counterclockwise ``outer\ncycle'' of length $16$ and $F_2$ is the indicator function of the\ndirected counterclockwise ``inner cycle'' of length $16$.\n\\medskip\n\n\n\nWe consider the projection which acts in the following way:\n\\smallskip\n\n\\noindent{\\bf (a)} It maps each edge which is in the ``$16$-cycle\nonly''  to $\\frac{\\theta}8\\,F$, where $\\theta=1$ on the right half\nand $\\theta=-1$ on the left half.\n\\medskip\n\n\\noindent{\\bf (b)} It maps each edge which is ``both in the\n$16$-cycle and $4$-cycle'' onto the $\\frac{\\theta}4\\chi_C$, where\n$C$ is the corresponding $4$-cycle and $\\theta=1$ on the right\nside and $\\theta=-1$ on the left side.\n\\medskip\n\nIt is straightforward to check that this projection is invariant\nin the sense of \\eqref{E:CommWithGn} and is different from the\northogonal projection.\n\\end{proof}\n\n\n\\section{Lipschitz free spaces on diamond graphs -- more precise results using Haar\nfunctions}\\label{S:DiamHaar}\n\nIn this section we present an alternative self-contained proof of\nour results for the binary diamond graphs $D_n$. This proof uses\nthe Haar system for $L_1[0,1]$ and makes an interesting connection\nwith some open problems concerning the even  levels of the Haar\nsystem. At the end of this section we extend the proof to handle\nthe multi-branching diamond graphs as well.\n\nWe begin by reformulating  the definition of the binary diamond graphs in order to use the Haar system.  For $n \\ge 2$, we shall consider $D_n$ as consisting of four copies of $D_{n-1}$, namely `top left', denoted $TL_n$, `bottom left', denoted $BL_n$, `bottom right', denoted $BR_n$,\nand `top right', denoted $TR_n$. In this identification the bottom vertex of $TL_n$ coincides with the top vertex of $BL_n$, etc.\n\nWe shall identify the edge space of $D_n$, denoted $\\ell_1(D_n)$,\nwith a certain subspace of $L_1[0,1]$. This identification is\nrecursive. We identify the edge vectors of $\\ell_1(D_1)$ with the\nfunctions $4\\cdot\\1_{((i-1)\/4, i\/4]}$ for $1 \\le i \\le 4$,\nwhich are disjointly\nsupported unit vectors in $L_1[0,1]$, ordering the edges\n$i=1,\\dots,4$ counterclockwise from the top vertex. Now suppose\nthat $n \\ge 2$ and that $\\ell_1(D_{n-1})$ has been identified with\na subspace of $L_1[0,1]$. For a function $f\\in L_1[0,1]$ we denote\nby $Qf$ the function which is $0$ in $\\left(\\frac14,1\\right]$ and\nis given by $(Qf)(t)=4f(4t)$ for $t\\in \\left[0,\\frac14\\right]$. It\nis clear that $Q$ is an isometric embedding of $L_1[0,1]$ into\nitself. Then we identify $\\ell_1(TL_n)$ with $Q(\\ell_1(D_{n-1}))$,\nand identify $\\ell_1(BL_n)$, $\\ell_1(BR_n)$, and $\\ell_1(TR_n)$,\nwith\ncopies of $\\ell_1(TL_n)$  translated by $\\frac14,\\frac12$, and $\\frac34$ to the right, respectively. It follows that the edge\nvectors of $\\ell_1(D_n)$ are the functions $4^n\n\\cdot\\1_{((i-1)\/4^n, i\/4^n]}$ for $1 \\le i \\le 4^n$, which are\ndisjointly supported unit vectors in $L_1[0,1]$.\n\nLet us now determine the subspace of $L_1[0,1]$ which corresponds\nunder this identification to the cycle space of $D_n$, denoted\n$Z(D_n)$. First, let us recall the definition of the Haar system\n$(h_i)_{i\\ge0}$. We define $h_0 := \\1_{(0,1]}$, and for $n \\ge 0$\nand $0 \\le i \\le 2^n-1$,\n$$h_{2^n + i} := \\1_{(i\/2^n, (2i+1)\/2^{n+1}]} -\n\\1_{((2i+1)\/2^{n+1}, (i+1)\/2^n]}.$$ Let $H_n := \\{ h_i \\colon 2^n\n\\le i \\le 2^{n+1}-1\\}$ be the collection of all $2^n$ Haar\nfunctions on the same level with support of length $2^{-n}$. Let\n$e_n$ be the cycle vector corresponding to the `large outer cycle'\nof $D_n$. To understand the pattern for $e_n$, first we calculate\n$e_1$, $e_2$ and $e_3$. Clearly, $$e_1 = 4(\\1_{[0,1\/2]} -\n\\1_{[1\/2,1]}) = 4h_1,$$\nand \\begin{equation}\\label{eq: e_2} \\begin{split} e_2 &= 16(\\1_{[0,1\/8]} + \\1_{[2\/8,3\/8]} - \\1_{[5\/8,6\/8]}- \\1_{[7\/8,1]})\\\\\n& = 8(h_1 + h_4 + h_5 + h_6 + h_7)\\\\\n&= 2e_1 + 8(\\sum_{h \\in A_2} h), \\end{split}\\end{equation} where $A_2 = \\{ h \\in H_2 \\colon \\operatorname{supp} h \\subseteq \\operatorname{supp} e_1\\}$.\nNote that\n\\begin{equation} \\label{eq: e_3}\n\\begin{split} e_3 &= 64[ (\\1_{[0,1\/32]} + \\1_{[2\/32,3\/32]}+ \\1_{[8\/32,9\/32]} + \\1_{[10\/32,11\/32]})\\\\ &\\quad- (\\1_{[21\/32,22\/32]} + \\1_{[23\/32,24\/32]}+ \\1_{[29\/32,30\/32]} + \\1_{[31\/32,1]})]\\\\\n&= 16[(h_1 + h_4 + h_5 + h_6 + h_7)\\\\&\\quad +2(h_{16} + h_{17} + h_{20} + h_{21}+h_{26} + h_{27} + h_{30} + h_{31})]\\\\\n&= 2e_2 + 32(\\sum_{h \\in A_3} h),\n\\end{split}\\end{equation} where $A_3 = \\{ h \\in H_4 \\colon \\operatorname{supp} h \\subseteq \\operatorname{supp} e_2\\}$.\nThe passage from $e_{n-1}$ to $e_n$ in the general case is\nanalogous to the passage from $e_2$ to $e_3$ above and is given by\na procedure which we now describe. let $I$ be a maximal dyadic\nsubinterval of $\\operatorname{supp} e_{n-1}$. Let $I_1$, $I_2$,\n$I_3$, and $I_4$ be the first, second, third, and fourth quarters\nof $I$ ordered from left to right. To get $e_n$ from $e_{n-1}$, if\n$I$ is contained in the support of the \\textit{positive} part of\n$e_{n-1}$  then  we replace $\\1_I$ in the expression for $e_{n-1}$\nby $\\1_{I_1} + \\1_{I_3}$, and if $I$ is contained in the support\nof the \\textit{negative} part of $e_{n-1}$  then we replace\n$-\\1_I$ in the expression for $e_{n-1}$  by $-(\\1_{I_2} +\n\\1_{I_4})$. Expressing $e_n$ in terms of Haar functions, it\nfollows, by analogy with \\eqref{eq: e_2} and \\eqref{eq: e_3}\nabove, that\n\\begin{equation} \\label{eq: e_n} e_n = 2e_{n-1} + 2^{2n-1} \\sum_{h \\in A_n} h, \\end{equation}\nwhere $A_n = \\{h \\in H_{2n-2} \\colon \\operatorname{supp} h \\subseteq \\operatorname{supp} e_{n-1}\\}$. Iterating \\eqref{eq: e_n} and\nrecalling that $e_1 = 4h_1$, we get\n\\begin{equation} \\label{eq: e_nexpression}  e_n - 2^{n+1} h_1 \\in \\operatorname{span} (\\cup_{k=1}^{n-1} H_{2k}). \\end{equation}\n\\begin{lemma} \\label{thm: Z(D_n)} For all $n \\ge 1$, $$ \\ell_1(D_n) = \\operatorname{span}( \\{h_0\\} \\cup (\\cup_{k=0}^{2n-1} H_k))$$ and\n\\begin{equation} \\label{eq: Z(D_n)} Z(D_n) = \\operatorname{span} (\\cup_{k=0}^{n-1} H_{2k}). \\end{equation}\n\n\\end{lemma} \\begin{proof} The description of $\\ell_1(D_n)$ follows from the observation above that the edge vectors of $\\ell_1(D_n)$ are the functions $4^n \\1_{[(i-1)\/4^n, i\/4^n]}$ for $1 \\le i \\le 4^n$.\n\nWe prove \\eqref{eq: Z(D_n)} by induction. Note that\n$$Z(D_1) = \\operatorname{span}(\\{h_1\\}) = \\operatorname{span}(H_0),$$\nwhich verifies the base case $n=1$. So suppose that $n \\ge 2$ and that the result holds for $n-1$. Note that \\begin{equation}\n\\label{eq: span1}\nZ(D_n) = \\operatorname{span}(Z(TL_n) \\cup Z(BL_n) \\cup Z(BR_n) \\cup Z(TR_n) \\cup \\{e_n\\}). \\end{equation}\nRecall that $TL_n$, $BL_n$, $BR_n$, and $TR_n$ are translated and dilated copies of $D_{n-1}$ on the intervals $[(i-1)\/4]$ for $1 \\le i \\le 4$.\nHence  $Z(TL_n)$, $Z(BL_n)$, $Z(BR_n)$, and $Z(TR_n)$ are translated and  dilated copies of $Z(D_{n-1})$ on the intervals $[(i-1)\/4]$ for $1 \\le i \\le 4$. Applying the inductive hypothesis  to $Z(D_{n-1})$ it follows that \\begin{equation} \\label{eq: span2}\n\\operatorname{span}(Z(TL_n) \\cup Z(BL_n) \\cup Z(BR_n) \\cup Z(TR_n)) = \\operatorname{span} (\\cup_{k=1}^{n-1} H_{2k}). \\end{equation}\nFinally, from \\eqref{eq: span1},   \\eqref{eq: span2} and \\eqref{eq: e_nexpression}, we get \\begin{align*}\nZ(D_n) &= \\operatorname{span} (\\{e_n\\} \\cup(\\cup_{k=1}^{n-1} H_{2k}))\\\\ &= \\operatorname{span} (\\{h_1\\} \\cup(\\cup_{k=1}^{n-1} H_{2k}))\\\\\n&= \\operatorname{span} (\\cup_{k=0}^{n-1} H_{2k}).\n\\end{align*} \\end{proof}\n\\begin{remark} Note that $Z(D_n)$ has dimension $\\sum_{k=0}^{2n-2} 4^k = (4^{2n-1}-1)\/3$. This can also be seen directly\nwithout using Lemma~\\ref{thm: Z(D_n)} since \\eqref{eq: span1}\nclearly implies that $\\operatorname{dim} Z(D_n) = 4\n\\operatorname{dim} Z(D_{n-1}) + 1$.  Using this observation that\nthe spaces have the same dimension,  it suffices to show that\n$Z(D_n) \\subseteq \\operatorname{span} (\\cup_{k=0}^{n-1} H_{2k})$,\nwhich follows from \\eqref{eq: e_nexpression} and  \\eqref{eq:\nspan2}. Thus the proof can be concluded slightly\ndifferently.\\end{remark} Our next goal is to prove that $Z(D_n)$\nis not well-complemented in $\\ell_1(D_n)$. This essentially\nfollows from a result of Andrew \\cite{And78}. (Note that the idea\nof using the average over the group of isometries to estimate\nnorms of projections goes back at least to Gr\\\"unbaum \\cite{Gru60}\nand Rudin \\cite{Rud62}.) For completeness we present a slight\ngeneralization of Andrew's elegant argument. Let\n $X_n = \\operatorname{span} ( \\{h_i \\colon 0 \\le i \\le 2^{n+1}-1\\}) = \\operatorname{span}(\\{h_0\\}\\cup (\\cup_{k=0}^n H_k))$. Let $(\\cdot,\\cdot)$ denote the usual inner product in $L_2[0,1]$. Orthogonality will refer to this inner product.\n\n Suppose $i \\ge 1$ and that $h_i \\in H_k$. Define a linear isomorphism $g_i \\colon X_n \\rightarrow X_n$ by\n\\begin{equation*}  (g_if)(t) = \\begin{cases}  f(t), &t \\notin \\operatorname{supp} h_i,\\\\\nf(t+ 2^{-k-1}), &t \\in h_i^{-1}(1),\\\\\nf(t- 2^{-k-1}), &t \\in h_i^{-1}(-1) \\end{cases}\n \\end{equation*} for all $f \\in X_n$. Suppose now that $\\| \\cdot \\|$ is any norm on $X_n$ with the property that each $g_i$ acts as  a linear isometry of $(X_n, \\|\\cdot\\|)$. For our purposes,  $\\|\\cdot\\|$ will be the usual norm of $L_1[0,1]$ or of $L_\\infty[0,1]$.\nLet $G$ be the group of isometries generated by $(g_i)_{i \\ge 1}$. Note that $G$ is finite.\n\nIn the next proposition it is convenient to set $H_{-1} :=\n\\{h_0\\}$.\n\n\\begin{lemma} \\label{prop: Andrew} Let $A$ be any nonempty subset of $\\{-1,0,1,\\dots,n\\}$ and let $P$ be any linear projection on $(X_n, \\|\\cdot \\|)$\nwith range $Y := \\operatorname{span} (\\cup_{k \\in A} H_k)$. Then $\\|P\\| \\ge \\|P_Y\\|$, where $P_Y$ is the orthogonal projection onto $Y$.\n\\end{lemma}\n\n\\begin{proof} Let $$ Q = \\frac{1}{ \\operatorname{card}(G)}\\sum_{g \\in G} g^{-1}Pg.$$ Clearly $\\|Q\\| \\le \\|P\\|$. Moreover $Q$ is a projection onto\n$Y$ since $g(Y) = Y$ for all $g \\in G$. It suffices to show that $Q = P_Y$. The proof of this makes use of the following observations:\n\\begin{enumerate} \\item $gQ = Qg$ for all $g \\in G$;\n  \\item $g_i h_i = -h_i$ for all $i \\ge 1$;\n  \\item $(g_i f, h_i) = -(f, h_i)$ for all $f \\in X_n$ and for all $i \\ge 1$;\n  \\item If  $0 \\le i < j$ and $\\operatorname{supp} h_j \\subset \\operatorname{supp} h_i$  then $(g_j f, h_i) = (f, h_i)$ for all $f \\in X_n$;\n  \\item If $i > j \\ge0$ or if  $h_i$ and $h_j$ are disjointly supported then  $g_i h_j = h_j$.\n  \\end{enumerate}\nSuppose that $h_j \\notin Y$. We have to show that $Qh_j =0$. Since $Q$ is a projection onto $Y$, it suffices to show that if $h_i \\in Y$ then\n$(Qh_j, h_i)=0$. If $0 \\le i < j$ and $\\operatorname{supp} h_j \\subset \\operatorname{supp} h_i$ then\n\\begin{align*} (Qh_j, h_i) &= (g_jQh_j, h_i) \\qquad\\text{(by ($4$))}\\\\\n&= (Qg_jh_j,h_i) \\qquad\\text{(by ($1$))}\\\\\n&= -(Qh_j, h_i) \\qquad \\text{(by ($2$))}.\\end{align*}\nHence $(Qh_j, h_i)=0$ in this case. Now suppose that $i > j\\ge 0$ or that $h_i$ and $h_j$ are disjointly supported. Then\n\\begin{align*}(Qh_j, h_i) &= (Qg_ih_j, h_i) \\qquad\\text{(by ($5$))}\\\\\n&= (g_iQh_j, h_i) \\qquad\\text{(by ($1$))}\\\\\n&= -(Qh_j, h_i) \\qquad\\text{(by ($3$))}. \\end{align*}\nSo $(Qh_j, h_i)=0$.\n \\end{proof}\n\\begin{lemma}\\label{C:LargePrConst} Let $P$ be a projection from $\\ell_1(D_n)$ onto $Z(D_n)$. Then  $\\|P\\| \\ge (2n+1)\/3$. \\end{lemma}\n\\begin{proof} By Theorem~\\ref{thm: Z(D_n)}, we have  $\\ell_1(D_n)=\\operatorname{span}( \\{h_0\\} \\cup (\\cup_{k=0}^{2n-1} H_k))$ and\n$Z(D_n)=\\operatorname{span}(\\cup_{k=0}^{n-1} H_{2k})$. By Proposition~\\ref{prop: Andrew} it suffices to show that the orthogonal projection\n$Q$ satisfies $\\|Q\\| \\ge (2n+1)\/3$.  This is well-known but for completeness we recall the proof. Consider\n$$ f = h_0 + h_1 + 2h_2+ 2^2 h_4 +\\dots + 2^{2n-2}h_{2^{2n-2}}.$$\nNote that $f$ is the sum over the first Haar functions (normalized in $L_1[0,1]$) in each level. Then\n$$Qf = h_1 + 2^2h_4 + 2^4h_{16} +\\dots+ 2^{2n-2}h_{2^{2n-2}}.$$\nIt is easily seen that $\\|f\\| = 1$ and $\\|Qf\\| \\ge (2n+1)\/3$.\n\\end{proof} \\begin{theorem}\\label{C:LFDnNotL1} The Banach-Mazur distance $d$ from the Lipschitz free space $LF(D_n)$ to the $\\ell_1^N$ space of the same dimension satisfies $$4n+4 \\ge d \\ge (2n+1)\/3.$$ \\end{theorem}\n\n\\begin{proof} Let $X_n = \\operatorname{span}(\\{h_0\\} \\cup (\\cup_{k=0}^{2n-1} H_k))$ Using the inner product in $L_2[0,1]$ we may identify $\\ell_1(D_n)^*$ with $(X_n, \\|\\cdot\\|_\\infty)$. Under this identification\n$Z(D_n)^\\perp = \\operatorname{span}(\\{h_0\\}\\cup(\\cup_{k=1}^{n} H_{2k-1}))$. A calculation similar to that of the previous result,\nbut now using the $L_\\infty$ norm, shows   that any projection $P$ from $(X_n, \\|\\cdot\\|_\\infty)$ onto $Z(D_n)^\\perp$ satisfies $\\|P\\| \\ge (2n+1)\/3$. Since an $\\ell_\\infty^N$ space is contractively complemented in any superspace, it follows that the Banach-Mazur distance from  $LF(D_n)^*=Z(D_n)^\\perp$ to an $\\ell_\\infty^N$ space\nis at least $(2n+1)\/3$. Dualizing again gives $d \\ge (2n+1)\/3$.\n\nTo get the upper estimate, note that  $\\{h_0\\}\\cup(\\cup_{k=1}^{n} H_{2k-1})$ is a monotone basis for $LF(D_n)$  in the quotient norm of $LF(D_n)$ and that\n $\\{2^{2k-1}h_i \\colon h_i \\in H_{2k-1}\\}$\nis $2$-equivalent to the unit vector basis of the $\\ell_1^N$ space of the same dimension.  Let $x \\in LF(D_n)$ and write\n$ x = \\sum_{k=0}^n x_k$, where $x_0 \\in \\operatorname{span}(\\{h_0\\})$ and $x_k \\in \\operatorname{span}(H_{2k-1})$. Then\n$$ \\sum_{k=0}^n \\|x_k\\| \\ge \\|x\\| \\ge \\frac{1}{2} \\max_{0 \\le k \\le n} \\|x_k\\| \\ge \\frac{1}{2n+2} \\sum_{k=0}^n \\|x_k\\|,$$\nwhich gives $d \\le 4n+4$.\n\\end{proof}\n\n\n\\begin{problem} Do $\\{LF(D_n)\\}$ admit embeddings into $\\ell_1$ with uniformly\nbounded distortions?\\end{problem}\n\n\\begin{problem} Do $\\{\\ell_\\infty^k\\}$ admit embeddings into\n$\\{LF(D_n)\\}$ with uniformly bounded distortions?\\end{problem}\n\n\\begin{problem} Are $\\{Z(D_n)\\}_{n=1}^\\infty$ uniformly isomorphic to $\\{\\ell_1^{k(n)}\\}_{n=1}^\\infty$ of the corresponding dimensions?  This is a finite version of the longstanding  open question as to\n whether  the even levels of the Haar system in $L_1[0,1]$ span a subspace isomorphic to $L_1$  \\cite{MOP05}. \\end{problem}\n\n\n\\begin{remark} It is curious that the subspaces generated by all the even\/odd levels of the Haar functions appear in the study of quasi-greedy basic sequences in $L_1[0,1]$. The notion of quasi-greedy bases, which generalizes unconditional bases, was introduced by S. Konyagin and V. Temlyakov \\cite{KT}, see also \\cite{DKKT}]. Although the Haar basis is not quasi-greedy in $L_1[0,1]$ \\cite{DKW02}, S. Gogyan \\cite{Gog05} showed the subsequence consisting of all Haar functions from the even\/odd levels is a quasi-greedy subsequence in $L_1[0,1]$. \\end{remark}\n\nFinally, we generalize the argument  to handle  the multi-branching diamond graphs $D_{n,k}$. The proof is similar to the case $k=2$, so we omit some of the  details.\n\n\\begin{theorem}\\label{T:LFmultL1} The Banach-Mazur distance $d_{n,k}$ from the Lipschitz free space $LF(D_{n,k})$ to the $\\ell_1^N$ space of the same dimension satisfies $$4n+4 \\ge d_{n,k} \\ge \\frac{k-1}{2k}n.$$ \\end{theorem}\n\\begin{proof}\nIt will be convenient to identify the edge space of $D_{n,k}$ with\na subspace of $L_1[0,1]$ as follows.  For $n=1$ and $1 \\le j \\le\nk$ we identify the pair of  edge vectors of the $j^{th}$ path of\nlength $2$ from $u$ to $v$ with the $L_1$-normalized indicator\nfunctions $2k \\1_{(j-1)\/k, (2j-1)\/(2k)]}$ and $2k\n\\1_{((2j-1)\/(2k), j\/k]}$. For $n \\ge2$,  the edge space of\n$D_{n,k}$ is obtained from that of $D_{n,k-1}$ by subdividing the\nintervals corresponding to edge vectors of $D_{n,k-1}$ into $2k$\nsubintervals each of length $(2k)^{-n}$.  Each of the $k$\nconsecutive disjoint  pairs of $L_1$-normalized indicator\nfunctions of the subintervals corresponds to each pair of  edge\nvectors of the $k$ paths of length $2$ from the $u$ and $v$\nvertices of the copy of $D_{1,k}$ which  replaces the edge vector\nof $D_{n-1,k}$   corresponding to the interval of length\n$(2k)^{n-1}$ which is  subdivided. We have now identified\n the  edge vectors of $D_{n,k}$ with the normalized indicator functions\n$$ e_{n,j} = (2k)^n \\1_{((j-1)\/(2k)^n, j\/(2k)^n]} \\quad (1 \\le j \\le (2k)^n).$$\n\nArguing as  in the case $k=2$, one can show that a   basis for the cycle space corresponds to the $L_\\infty$-normalized system $\\cup_{i=1}^n \\{ g_{i,j}  \\colon 1 \\le j \\le\n(2k )^{i-1}(k-1)\\}$, where, setting $j = a(k-1)+b$ with $0 \\le a \\le (2k)^{i-1}-1$ and $1 \\le b \\le k-1$,\n$$g_{i,j} =(2k)^{-i}( e_{a2^k+2b-1} + e_{a 2^k + 2b} - e_{a 2^k + 2b + 1}- e_{a2^k+2b}).$$\nNote that for $k=2$ this agrees with the previous description of the cycle space of $D_{n,2}$ in terms of alternate levels of the Haar system. But for $k \\ge 3$, note that $g_{i,j}$ \\textit{ overlaps}  with $g_{i,j+1}$  when $b \\le k-2$, and hence this is not an orthogonal basis.\n\nUsing the fact that the cut space is the orthogonal complement of\nthe cycle space, it is easily seen that an orthogonal basis for\nthe  cut space   corresponds to the $L_\\infty$-normalized system\n$\\{h_0\\} \\cup (\\cup_{i=1}^n \\{ h_{i,j} \\colon 1 \\le j \\le\n(2k)^i\/2\\}$, where $h_0 = \\1_{[0,1]}$, and\n$$ h_{i,j}\n= (2k)^{-i}(e_{i,2j-1}-e_{i,2j}).$$\n\nLet $P_{n,k}$ denote the orthogonal projection from the edge space of $D_{n,k}$ onto the cut space.\nThen $$P_{n,k}(e_{n,1}) =h_0 + \\frac{1}{2} \\sum_{i=1}^n (2k)^i h_{i,1}.$$\nNote that for $1 \\le i \\le n$,\n$$P_{n,k}(e_{n,1})|_{(2(2k)^{-i-1},(2k)^{-i}] }=1+\\frac{1}{2}\\sum_{j=1}^i(2k)^j\n\\ge\\frac{(2k)^i}{2}.$$\nHence\n$$\\|P_{n,k}\\|_1\\ge \\|P_{n,k}(e_{n,1})\\|_1 \\ge\\sum_{i=1}^n (1-\\frac{1}{k})(2k)^{-i} \\frac{(2k)^i}{2} = (1-\\frac{1}{k}) \\frac{n}{2}. $$\nSince $P_{n,k}$ is self-adjoint, it follows that $P_{n.k}$ is a\nprojection from the edge space $E(D_{n,k})$, equipped with the\n$L_\\infty$ norm, onto the cut space  $Z(D_{n,k})^\\perp $\nsatisfying $\\|P_{n,k}\\|_\\infty \\ge (1- 1\/k)n\/2$.\n\n\n\n As in the case $k=2$, one can show that if $P$ is any projection onto the cut space (in the $L_\\infty$ norm)  then $\\|P\\|_\\infty \\ge \\|P_{n,k}\\|_\\infty$ .\nBy duality, as in the case $k=2$,  it follows that $d_{n,k} \\ge (1 - 1\/k)n\/2$.\n\nTo get the upper estimate, note that   $\\{h_0\\} \\cup (\\cup_{i=1}^n \\{ h_{i,j} \\colon 1 \\le j \\le (2k)^i\/2\\})$ is a monotone basis\nfor $LF(D_{n,k})$  in the quotient norm of $LF(D_{n,k})$ and that,\n for each $i$, $(h_{i,j})_{j=1}^{(2k)^i}$\nis $2$-equivalent to the unit vector basis of the $\\ell_1^N$ space of the same dimension. As in the case $k=2$, this gives $d_{n,k} \\le 4n+4$.\n\\end{proof}\n\n\n\\thanks{ \\textbf{Acknowledgements:}\nThe first named author was supported by the National Science\nFoundation under Grant Number DMS--1361461. The first and second\nnamed authors were supported by the Workshop in Analysis and\nProbability at Texas A\\&M University in 2017. The third named\nauthor was supported by the National Science Foundation under\nGrant Number DMS--1700176.}\n\n\\end{large}\n\\begin{small}\n\n\\renewcommand{\\refname}{\\section{References}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Diversity of Sensitivities}\n\\label{sec:diversity}\n\nWe can generalize (\\ref{eq:SDE of single component}) by allowing for agent\ndependent coefficients. We consider a particular case in which \neach agent can have a different rate of mean reversion to the empirical mean, that is, for \n$j=1,\\ldots,N$, \n\\begin{equation}\n\t\\label{eq:SDE of single component, heterogeneous sensitivity}\n\tdx_j = -h\\frac{\\partial}{\\partial x_j}V(x_j)dt \n\t+ \\sigma dw_j + \\theta_j(\\bar{x}-x_j)dt,\n\\end{equation}\nand as before $V(y)=\\frac{1}{4}y^4 - \\frac{1}{2} y^2$.\nWe consider the case where $\\theta_1,\\ldots,\\theta_N$ take $K$ distinct positive numbers, \n$\\Theta_1,\\ldots,\\Theta_K$. We define $\\mathcal{I}_l=\\{j:\\theta_j=\\Theta_l\\}$, \n$\\rho_l=|\\mathcal{I}_l|\/N$ and \n$X_N^l=\\frac{1}{\\rho_l N}\\sum_{j\\in\\mathcal{I}_l}\\delta_{x_j}$. Assuming that \n$\\lim_{N\\rightarrow\\infty}\\rho_l$ exists and is positive for all $l$, the limit of \n$(X_N^1,\\ldots,X_N^K)$ as $N\\rightarrow\\infty$ are the weak solutions $(u_1,\\ldots,u_K)$ of \nthe set of $K$ coupled Fokker-Planck equations.\n\\begin{theorem}\n\t\\label{thm:mean-field limit, diversity case}\n\tAssume that $U(y)=y^3-y$ and that $(X_N^1(0), \\ldots, X_N^K(0))$ converge weakly in probability \n\tto the probability measures $(\\nu^1,\\ldots, \\nu^K)$. Then the measure valued vector process $(X_N^1, \\ldots, X_N^K)$ converges weakly as \n\t$N\\rightarrow\\infty$ to the weak solution $(u_1, \\ldots, u_K)$ of the system of the Fokker-Planck equations:\n\t\\begin{align}\n\t\t\\label{eq:system of Fokker-Planck equations}\n\t\t\\frac{\\partial}{\\partial t}u_1 \n\t\t&= \\frac{1}{2}\\sigma^2\\frac{\\partial^2}{\\partial y^2}u_1 \n\t\t- \\Theta_1\\frac{\\partial}{\\partial y}\\left\\{\\left[\n\t\t\\int y\\sum_{l=1}^K\\rho_lu_l(t,y)dy-y\\right]u_1\\right\\} \n\t\t+ h\\frac{\\partial}{\\partial y}[U(y)u_1]\\\\\n\t\t&\\vdots \\notag\\\\\n\t\t\\frac{\\partial}{\\partial t}u_{K} \n\t\t&= \\frac{1}{2}\\sigma^2\\frac{\\partial^2}{\\partial y^2}u_K \n\t\t- \\Theta_K\\frac{\\partial}{\\partial y}\\left\\{\\left[\n\t\t\\int y\\sum_{l=1}^K\\rho_lu_l(t,y)dy-y\\right]u_K\\right\\} \n\t\t+ h\\frac{\\partial}{\\partial y}[U(y)u_K], \\notag\n\t\\end{align}\n\twith initial condition $(\\nu^1,\\ldots, \\nu^K)$.\n\\end{theorem}\n\\begin{proof}\n\tSee Appendix \\ref{pf:mean-field limit, diversity case} for the outline of the proof\n\tfollowing \\cite{Gartner1988}.\n\\end{proof}\n\nThe equilibrium solutions $\\{u_{l,\\xi}^e\\}_{l=1}^K$ have the form\n\\begin{align}\n\t\\label{eq:equilibrium of Fokker-Planck eqn, heterogeneous case}\n\tu_{l,\\xi}^e(y) &= \\frac{1}{Z_{l,\\xi}\\sqrt{2\\pi\\frac{\\sigma^2}{2\\Theta_l}}}\n\t\\exp\\left\\{ -\\frac{(y-\\xi)^2}{2\\frac{\\sigma^2}{2\\Theta_l}} \n\t- h\\frac{2}{\\sigma^2} V(y) \\right\\}\\\\\n\tZ_{l,\\xi} &= \\int \\frac{1}{\\sqrt{2\\pi\\frac{\\sigma^2}{2\\Theta_l}}}\n\t\\exp\\left\\{ -\\frac{(y-\\xi)^2}{2\\frac{\\sigma^2}{2\\Theta_l}} \n\t- h\\frac{2}{\\sigma^2} V(y) \\right\\} dy, \\notag\n\\end{align}\nand $\\xi$ must satisfy the compatibility condition\n\\begin{equation}\n\t\\label{eq:xi equal to m(xi), diversity case}\n\t\\xi = m(\\xi) := \\sum_{l=1}^K \\rho_l \\int yu_{l,\\xi}^e (y)dy.\n\\end{equation}\nFor $U(y)=y^3-y$, $\\xi=0$ is the trivial solution of \n(\\ref{eq:xi equal to m(xi), diversity case}), and a simple extension of Theorem \n3.3.1 in \\cite{Dawson1983}, shows that there are two sets of non-trivial solutions \n$\\{u_{l,\\xi_b}^e\\}_{l=1}^K$ and $\\{u_{l,-\\xi_b}^e\\}_{l=1}^K$ if and only if \n$\\frac{d}{d\\xi}m(0)>1$. The numerical simulations presented in the next section show \nthat diversity in the rate of mean reversion can have significant impact on the stability of the \nmean-field model.\n\nAs in the homogeneous case, we can get an approximate condition for \nequilibrium bifurcation for small $h$.\n\\begin{proposition}\n\t\\label{prop:explicit condition for xi equal to m(xi) for small h, diversity case}\n\tThe compatibility condition (\\ref{eq:xi equal to m(xi), diversity case}) has \n\tnon-zero solutions if and only if $\\sigma < \\sigma_c^{\\text{div}}$.\n\tFor small $h$,  $\\sigma_c^{\\text{div}}$ has the expansion \n\t\\[ \n\t\t\\sigma_c^{\\text{div}} = \\sqrt{ \\sum_{l=1}^K \\frac{\\rho_l}{\\Theta_l}\n\t\t\/ \\sum_{l=1}^K\\frac{3\\rho_l}{2\\Theta_l^2} } + O(h).\n\t\\]\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix \n\t\\ref{pf:explicit condition for xi equal to m(xi) for small h, diversity case}.\n\\end{proof}\n\nWe note that diversity does affect the threshold condition and makes the analysis \nmore difficult. The non-zero solutions $\\pm \\xi_b$ can be computed approximately when $h$ is small:\n\\begin{equation} \n\t\\label{eq:explicit solution for xi equal to m(xi), diversity case}\n\t\\pm \\xi_b = \\pm \\sqrt{ \\sum_{l=1}^K \\frac{\\rho_l}{\\Theta_l} \n\t\\left( 1 - 3\\frac{\\sigma^2}{2\\Theta_l} \\right) \n\t\/ \\sum_{l=1}^K \\frac{\\rho_l}{\\Theta_l} } + O(h).\n\\end{equation}\nHigher order terms in the expansion of \n(\\ref{eq:explicit solution for xi equal to m(xi), diversity case}) can also be obtained \nbut we will omit them in this paper.\nIn the following Proposition we show that $\\sigma_c^{\\text{div}} \\leq \\sigma_c^{\\text{homo}}$, \nwhere $\\sigma_c^{\\text{homo}}=\\sigma_c$, the critical value (\\ref{eq:sigma_c of the homogeneous case}) of \nthe homogeneous case.\n\\begin{proposition}\n\t\\label{prop:sigma_c comparison}\n\tWith $\\theta = \\sum_{l=1}^K \\rho_l \\Theta_l$, we have \n\t$\\sigma_c^{\\text{homo}} \\geq \\sigma_c^{\\text{div}}$ for small $h$.\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix \n\t\\ref{pf:sigma_c comparison}.\n\\end{proof}\n\nThis result shows that when there is diversity the parameter region of existence of equilibria $\\pm \\xi_b$ \nis smaller than in the homogeneous case . From this observation we can anticipate that these equilibria are \nless stable in the presence of diversity, and this is confirmed next by numerical simulations and \nanalytically.\n\nBy noting that $\\xi_b^\\text{homo}=\\sqrt{1-(\\sigma^2\/\\sigma_c^\\text{homo})^2}+O(h)$ and \n$\\xi_b^\\text{div}=\\sqrt{1-(\\sigma^2\/\\sigma_c^\\text{div})^2}+O(h)$, we have the following corollary:\n\\begin{corollary}\n\tWith $\\theta = \\sum_{l=1}^K \\rho_l \\Theta_l$, we have $1>\\xi_b^\\text{homo} \\geq \\xi_b^\\text{div}$ for \n\tsmall $h$.\n\\end{corollary}\n\\section{Effect of Diversity of Sensitivities on the Transition Probability}\n\\label{sec:diversity2}\n\nWe consider the situation introduced in Section \\ref{sec:diversity} and analyze it when $h=0$. \nWe aim at computing the transition probability in this situation.\nThe $K$ partial empirical averages\n\\begin{equation}\n\\label{eq:systembarxk}\n\\bar{x}_k (t) := \\frac{1}{|\\mathcal{I}_{k}|} \\sum_{j \\in \\mathcal{I}_{k}} x_j(t), \\quad k=1,\\ldots,K \n\\end{equation}\nthen satisfy a closed system of stochastic differential equations\n\\begin{equation}\nd\\bar{x}_k = \\frac{\\sigma}{\\sqrt{\\rho_k N}} d\\bar{w}_k(t) - \\theta_k ( \\bar{x}_k -\\bar{x} ) dt\n\\end{equation}\nwhere $\\bar{w}_k$ are independent  Brownian  motions\nand  the empirical mean $\\bar{x}(t)$ can be expressed in terms of the partial averages as\n$$\n\\bar{x}(t) = \\sum_{k=1}^K \\rho_k \\bar{x}_k (t)\n$$\n\\begin{proposition}\n\\label{prop:xbar_diversity}\nIf $\\bar{x}_k(0)= -\\xi_b$ for all $k=1,\\ldots,K$, then $\\bar{x}(T)$ is a Gaussian random variable with mean $-\\xi_b$ and variance\n$\\sigma_T^2:={\\rm Var}(\\bar{x}(T)) $ given by \n\\begin{equation}\n\\label{eq:sigmaT2}\n\\sigma_T^2 = \\frac{\\sigma^2}{N} \\int_0^T \\varrho^\\mathbf{T} e^{Ms} R^{-1} (e^{Ms})^\\mathbf{T} \\varrho  ds\n\\end{equation}\nwhere $\\varrho$ is the $K$-dimensional column vector $(\\rho_k)_{k=1,\\ldots,K}$, $M$ and $R$ are the \n$K \\times K$ matrices  defined by\n$$\nM_{ij} = -\\theta_i (\\delta_{ij}  -\\rho_j) ,\\quad \\quad R_{ij} =\\rho_i \\delta_{ij},  \\quad \\quad i,j=1,\\ldots,K,\n$$\nand ${}^\\mathbf{T}$ stands for the transpose.\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix \\ref{pf:xbar_diversity}.\n\\end{proof}\n \nWe can then deduce that the transition probability is\n\\begin{equation}\n\\label{eq:pTdiverse}\np_T \\approx \\exp \\Big( - \\frac{2 \\xi_b^2}{\\sigma_T^2}\\Big)\n\\end{equation}\n\nOur next goal is to study the impact of the diversity on the transition probability.\n\n\\begin{proposition}\n\\label{prop:the diversity on the transition probability}\nLet us assume that the diversity is small:\n$$\n\\theta_k =\\bar{\\theta} (1 +\\delta \\alpha_k),\\quad \\quad \\delta \\ll 1\n$$\nwhere $\\sum_k \\rho_k \\alpha_k=0$ so that $\\bar{\\theta}$ is the mean value of the $\\theta_k$'s.\nThe equilibrium position $\\xi_b$, the variance $\\sigma_T^2$ and the transition probability $p_T$ can be expanded as powers of $\\delta$ as\n\\begin{align*}\n\t\\xi_b^2 &= \\Big(1- \\frac{3 \\sigma^2}{2 \\bar{\\theta}}\\Big) \n\t- \\delta^2\\Big(\\sum_k \\rho_k \\alpha_k^2 \\Big)   \\frac{3 \\sigma^2}{2 \\bar{\\theta}} \n\t+O(\\delta^3),\\\\\n\t\\sigma_T^2 &= \\frac{\\sigma^2 T}{N} \n\t\\Big[ 1 + \\delta^2  \\Big( \\sum_k \\rho_k \\alpha_k^2 \\Big) \n\t\\Big( \\frac{1}{T} \\int_0^T (1-e^{-\\bar{\\theta} s})^2 ds \\Big) +O(\\delta^3) \\Big], \\\\\n\tp_T &\\approx \\exp \\Big\\{ - \\frac{2 N}{\\sigma^2 T} \n\t\\Big[ \\Big(1- \\frac{3 \\sigma^2}{2 \\bar{\\theta}}\\Big)\n\t-\\delta^2   \\Big(\\sum_k \\rho_k \\alpha_k^2 \\Big)  \n\t\\Big(\\frac{3 \\sigma^2}{2 \\bar{\\theta}}   \n\t+\\frac{1}{T} \\int_0^T (1-e^{-\\bar{\\theta} s})^2 ds \\Big)  \\Big]\\Big\\}.\n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix \\ref{pf:the diversity on the transition probability}.\n\\end{proof}\n\nThis proposition shows that the diversity reduces the gap between the two equilibrium states and enhances the fluctuations of the empirical mean.\nBoth effects contribute to the increase of the systemic transition probability.\n\\section{Introduction}\n\nSystemic risk is the risk that in an interconnected system of agents that can fail individually, a large \nnumber of them fails simultaneously or nearly so, leading to the overall failure of the system. It is a \nproperty of the interconnected system as a whole, and not only of the individual components, in the sense \nthat assessment of the risk of individual failure alone cannot provide an assessment of the systemic risk. \nThe interconnectivity of the agents, its form and evolution, play an essential role in systemic risk \nassessment \\cite{Bisias2012}.\n\nIn this paper we consider a simple model of interacting agents for which systemic risk can be assessed \nanalytically in some interesting cases. Each agent can be in one of two states, a normal and a failed one, \nand it can undergo transitions between them. We assume that the dynamic evolution of each agent has the \nfollowing features. First, there is an intrinsic stabilization mechanism that tends to keep the agents near \nthe normal state. Second, there are external destabilizing forces that tend to push away from the normal \nstate and are modeled by stochastic processes. Third, there is cooperation among the agents that acts as \nindividual stabilizer by diversification. This means that in such a system we expect that there is a \ndecrease in the risk of destabilization or \"failure\" for each agent because of the cooperation or \ndiversification. What is less obvious is the effect of cooperation on the overall or system's risk, which \ncan be defined in a precise way for the model considered here. We show in this paper that for the models \nunder consideration and in a certain regime of parameters, the systemic risk increases with increasing \ncooperation. The aim of this paper is to analyze this tradeoff between individual risk and systemic risk for \na class of interacting systems subject to failure.\n\nPerhaps a simple mathematical model of interacting agents having the features we want is a system of \nstochastic differential equations with mean-field interaction. Let $x_j(t)$ be the state of risk of agent or \ncomponent $j$, taking real values. For $j=1,\\ldots,N$, the $x_j(t)$'s are modeled as continuous-time \nstochastic processes satisfying the system of It\\^{o} stochastic differential equations:\n\\begin{equation}\n\t\\label{eq:SDE of single component}\n\tdx_j(t) = -hU(x_j(t))dt + \\theta(\\bar{x}(t)-x_j(t))dt + \\sigma dw_j(t),\n\\end{equation}\nwith given initial conditions. Here $-h U(y)=-hV'(y)$ is the restoring force, $V$ is a potential  which we \nassume has two stable states, and $\\{w_j(t)\\}_{j=1}^N$ are independent, standard Brownian motions. The \nparameter $h$ controls the level of intrinsic stabilization and $\\sigma$ is the strength of the destabilizing \nrandom forces. The interaction or cooperation is the mean reversion term with rate of mean reversion  \n$\\theta$ and with $\\bar{x}(t):=\\frac{1}{N}\\sum_{i=1}^N x_i(t)$ denoting the empirical mean of the processes, \nthat is, the empirical mean of the individual risks. For $\\theta>0$ the individual risk processes tend to \nmean-revert to their empirical mean, which is a simple but non-trivial form of cooperation. We take the \nempirical mean $\\bar{x}(t)$ to be a measure of the systemic risk. The bi-stable-state structure of $V(y)$ \ndetermines the normal and failed states of the agents. We will assume in this paper that  $U(y)=y^3-y$, so that \n$V(y)= \\frac{1}{4} y^4 - \\frac{1}{2}y^2 +c$ and we take $c=0$ since it is inessential. \nThe two stable states are then $\\pm 1$ and we let $-1$ be the normal state and \n$+1$ to be the failed state. The potential $V(y)$ ensures that each risk variable $x_j(t)$ stays around $-1$ \n(normal) or $+1$ (failed). The evolution of the system is characterized by the initial conditions, the three \nparameters ($h$, $\\theta$, $\\sigma$) and by the system size $N$.\n\n\n\nWe have chosen a mean-field interaction because it is a simple form of cooperative behavior. More elaborate \nmodels are considered in Section \\ref{sec:diversity}, where some heterogeneity is introduced between the \ncomponents of the system. For mean-field models a natural measure of systemic risk is the transition \nprobability of the empirical mean $\\bar{x}(t)$ from the normal state to the failed state. More precisely, \nthe mathematical problem we address here is this: For $N$ large we calculate approximately such transition \nprobabilities and analyze how they depend on $h,\\sigma$ and $\\theta$, the three parameters of the system. We \nare interested in a regime of these parameters for which there are two collective, that is, large $N$, \nequilibria centered around the normal and failed states. These two equilibria can be identified through the \nmean-field limit of the system, that is, the weak limit in probability of the empirical density of the \nagents risk $x_j$. Mean field models with multiple stable points, not only bistable ones, could be \nconsidered but their analysis is more involved while the main result about systemic risk, and dependence on \nthe parameters  ($h$, $\\theta$, $\\sigma$) and by the system size $N$, is clearly seen in the bistable model\nthat we consider here.\n\nThe mathematical analysis of bistable mean field models like (\\ref{eq:SDE of single component}) was \ninitiated by Dawson \\cite{Dawson1983, Gartner1988}, including the mean field limit, the existence of multiple equilibria, \nand a fluctuation theory. Non-equilibrium statistical mechanics and phase transitions have been studied \nextensively in the sciences \\cite{Haken1983}. The large deviation theory that we use here was developed by \nDawson and G\\\"artner \\cite{Dawson1987,Dawson1989}. In particular, they introduced and analyzed the rate function for \nlarge deviations associated with (\\ref{eq:SDE of single component}) when $N$ is large and with\nmore general potentials \\cite{Dawson1989}. Their theory may be \nconsidered as an infinite dimensional extension of the Freidlin-Wentzell theory of large deviations for \nstochastic differential equations with small noise \\cite{Freidlin1998, Dembo2010}. The main result in this \npaper is the analysis of this rate function for small $h$. That is, for a shallow two-well potential, where \ntransitions from one well (quasi-equilibrium) to the other are exponentially small in $N$, the \"constant\" in \nthe exponent is small when $h$ is small. Other mean field models have been studied in  \\cite{Tanaka1984, \nGartner1988, Meleard1996, BenArous1999, DelMoral2005, DelMoral2011, Fouque2012}, and large deviations \nresults for various models can be found in \\cite{Dawson1994, Arous1995, DelMoral1998, Dawson2005, \nHerrmann2008, Budhiraja2008, Budhiraja2012}. In \\cite{Budhiraja2012} a general large deviations theory is \ndeveloped for a model with both drift and volatility interactions, as well as with degenerate noise, using \nweak convergence and optimal control methods.\n\nThe main contribution of the paper as far as systemic risk theory is concerned is the demonstration that, \nwithin the range of the bistable mean field model (\\ref{eq:SDE of single component}), while cooperation \nbetween agents decreases the individual risk of each agent, the systemic or overall risk is increased. This \nis discussed in detail in Section \\ref{sec:comparison}, in terms of the three parameters \n$(h,\\theta,\\sigma)$, with $h$ small. The fact that reducing individual risk by cooperation or \ndiversification can lead to increased systemic risk has been anticipated in macroeconomics and elsewhere and \nit has been extensively discussed, modeled, and analyzed in \\cite{Nier2007, Battiston2009, Haldane2009, \nGai2010, May2010, Stiglitz2010, Beale2011, Battiston2011, Haldane2011, Ibragimov2011}. However, the dynamic \nphase transitions formulation and the large deviations theory exploited in this paper have not been used in \nthe economics literature, to our knowledge. The use of coupled stochastic equations for modeling evolution \nof individual risk and the effects of interactions among agents is also considered in \\cite{Battiston2009, \nHull2001} where there is some discussion regarding the economic interpretation of the variables \n$\\{x_j(t)\\}$. They could, for example, represent some form of equity ratio in a very simple model in \ninsurance or banking. \n\nThe paper is organized as follows. In Section \\ref{sec:mean field}, we briefly review the classical \nmean-field limit in \\cite{Dawson1983}, and we discuss the intrinsic stability of equilibria \n\\cite{Dawson1983} when $h$ is small. Section \\ref{sec:diversity} generalizes \n(\\ref{eq:SDE of single component}) by replacing the rate of mean reversion $\\theta$ by an agent-dependent \n$\\theta_j$. The mean-field limit and the explicit conditions are also studied. In Section \n\\ref{sec:simulation}, we carry out numerical simulations of both the homogeneous and the heterogeneous model \nin various parameter ranges. \nSection \\ref{sec:large deviations} uses the large deviation principle in \\cite{Dawson1987} to formulate the \ndynamic phase transition of interest here, that is, the system transition from the normal state to the \nfailed state. In Section \\ref{sec:small h}, we specialize the large deviations theory when $h$ is small so  \nas to obtain a result from which the systemic risk as a function the basic parameters $(h,\\theta,\\sigma)$ \ncan be assessed and interpreted. In Section \\ref{sec:reduced LDP} we introduce a formal expansion of the \nrate function for small $h$ and obtain a reduced variational principle for the systemic risk that appears to \ncome from a large deviations principle for an one-dimensional dynamical system. It gives, of course, the \nsame results about systemic risk as described in Section \\ref{sec:small h}. In Section \n\\ref{sec:diversity2} we discuss the case where there is diversity in mean reversion and it \nis shown that under some natural conditions the heterogeneous model is systemically more unstable than the  \nhomogeneous one. The technical details of the proofs are in the appendices.\n\\section{Large Deviations}\n\\label{sec:large deviations}\n\nIn the previous two sections we saw both analytically and numerically\nthat for large $N$, the empirical density $X_N(t,dy)$ is close (weakly, in probability) to\nthe solution of the Fokker-Planck equation (\\ref{eq: full Fokker-Planck eqn}),\nand so the mean $\\bar{x}(t)$ in (\\ref{eq:SDE of single component})\nstays around the first order moment of the deterministic limit,\n$\\int_{-\\infty}^{\\infty} y u(t,y)dy$. \nIf the condition of existence of two equilibria is satisfied, \nthen $\\bar{x}(t)$ will remain close to either $-\\xi_b$ or $+\\xi_b$\nfor relatively long time intervals, depending in particular on the parameter $h$.\nHowever, as long as $N<\\infty$, as we have seen in the simulations \nthe random forcing by the Brownian motions $\\{w_j(t)\\}_{j=1}^N$ will cause transitions \nwith non-zero probability.\nA systemic transition is the event that $\\bar{x}(t)$\nis displaced from $\\pm \\xi_b$ to $\\mp \\xi_b$ within a finite time horizon.\nThus, systemic transition means that a large number of agents transition in a finite time. \nIn this paper, we are interested in computing the probability of such a systemic transition. \nMathematically, given a finite time horizon $[0,T]$ \nand the conditions for existence of two equilibria, we want to compute the probability\n\\begin{equation}\n\t\\label{eq:probability of phase transition}\n\t\\mathbf{P}(\\bar{x}(0)=-\\xi_b, \\bar{x}(T)=\\xi_b)\n\\end{equation}\nwhen $N$ is large and as a function of the parameters $(h,\\theta,\\sigma)$ in (\\ref{eq:SDE of single component}).\n\n\\subsection{Large Deviations of Mean-fields}\n\nAccording to \\cite{Dawson1987}, we can calculate this probability asymptotically for large $N$ \nusing large deviations. To state the large deviations theory that we will use, \nwe will review briefly some notation and terminology  from \\cite{Dawson1987}.\n\\begin{itemize}\n\t\\item $M_1(\\mathbb{R})$ is the space of probability measures on \n\t$\\mathbb{R}$ with the Prohorov metric $\\rho$, associated with weak convergence.\n\t\n\t\\item $C([0,T],M_1(\\mathbb{R}))$ is the space of continuous functions from $[0,T]$ \n\tto $M_1(\\mathbb{R})$ with the metric $\\sup_{0\\leq t\\leq T}\\rho(\\phi_1(t),\\phi_2(t))$.\n\n\t\\item $M_R(\\mathbb{R})=\\{\\mu\\in M_1(\\mathbb{R}),\\int\\varphi(y)\\mu(dy)\\leq R\\}$, \n\twhere $\\varphi \\in C^2(\\mathbb{R})$ is a nonnegative function with \n\t$\\lim_{|x|\\to\\infty}\\varphi(x)=\\infty$. From \\cite{Dawson1987}, if $U(y)=y^3-y$, we \n\tcan choose $\\varphi(y)=1+y^2+\\gamma y^4$, $0\\leq\\gamma\\leq h\/2$.\n\t\t\n\t\\item $M_\\infty(\\mathbb{R})=\\cup_{R>0}M_R(\\mathbb{R})\n\t=\\{\\mu\\in M_1(\\mathbb{R}), \\int\\varphi(y)\\mu(dy)<\\infty\\}$ endowed with the \n\tinductive topology: $\\mu_n\\rightarrow\\mu$ in $M_\\infty(\\mathbb{R})$ if and only if \n\t$\\mu_n\\rightarrow\\mu$ in $M_1(\\mathbb{R})$ and \t\n\t$\\sup_n\\int\\varphi(y)\\mu_n(dy)<\\infty$. \n\t\n\t\\item $C([0,T],M_\\infty(\\mathbb{R}))$ is the space of continuous functions from \n\t$[0,T]$ to $M_\\infty(\\mathbb{R})$ endowed with the topology: \n\t$\\phi_n(\\cdot)\\rightarrow\\phi(\\cdot)$ in $C([0,T],M_\\infty(\\mathbb{R}))$ \n\tif and only if $\\phi_n(\\cdot)\\rightarrow\\phi(\\cdot)$ in $C([0,T],M_1(\\mathbb{R}))$ \n\tand $\\sup_{0\\leq t\\leq T}\\sup_n\\int\\varphi(y)\\phi_n(t,dy)<\\infty$. \n\n\t\\item Given $\\nu\\in M_\\infty(\\mathbb{R})$, we let\n\t$\\mathcal{E}^\\nu=\\{\\phi\\in C([0,T],M_\\infty(\\mathbb{R})): \\phi(0)=\\nu\\}$,\n\tendowed with the relative topology.\n\\end{itemize}\n\nTo simplify the notation, we rewrite (\\ref{eq: full Fokker-Planck eqn}) as \n$u_t = \\mathcal{L}_u^* u + h\\mathcal{M}^* u$, where \n\\[\n\t\\mathcal{L}_\\psi^* \\phi = \\frac{1}{2}\\sigma^2 \\phi_{yy} \n\t+\\theta\\frac{\\partial}{\\partial y}\\left\\{\\left[y-\\int y\\psi(t,y)dy\\right]\\phi\\right\\},\n\t\\quad\n\t\\mathcal{M}^*\\phi =\t \\frac{\\partial}{\\partial y}\\left[U(y)\\phi\\right].\n\\]\n\n\n\\begin{theorem}\n\t\\label{thm: Dawson and Gartner, general LD}\n\t(Dawson and G\\\"{a}rtner, 1987) Given a finite horizon $\\left[0,T\\right]$, \n\t$\\nu\\in M_\\infty(\\mathbb{R})$ and $A \\subseteq \\mathcal{E}^\\nu$, \n\tif $X_N(0)=\\frac{1}{N}\\sum_{j=1}^N\\delta_{x_j(0)}\\rightarrow \\nu$ in  \n\t$M_\\infty(\\mathbb{R})$ as $N \\rightarrow \\infty$, then the law of \n\t$X_N(t)=\\frac{1}{N}\\sum_{j=1}^N\\delta_{x_j(t)}$ satisfies the large deviation \n\tprinciple with the good rate function $I_h$:\n\t\\begin{align*}\n\t\t-\\inf_{\\phi \\in \\mathring{A}}I_h(\\phi) \n\t\t& \\leq \\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\log\\mathbf{P}(X_N\\in A)\\\\\n\t\t& \\leq \\limsup_{N\\rightarrow\\infty}\\frac{1}{N}\\log\\mathbf{P}(X_N\\in A)\n\t\t\\leq - \\inf_{\\phi \\in \\bar{A}}I_h(\\phi),\n\t\\end{align*}\n\twhere $\\mathring{A}$ and $\\bar{A}$ are the interior and closure of $A$ in \n\t$\\mathcal{E}^\\nu$, respectively, and \n\t\\begin{equation}\n\t\t\\label{def:Ih}\n\t\tI_h(\\phi) = \\frac{1}{2\\sigma^2} \\int_0^T \n\t\t\\sup_{f:\\langle \\phi, f_y^2 \\rangle \\neq 0} J_h(\\phi,f)dt,\n\t\\end{equation}\n\t\\[\n\t\tJ_h(\\phi,f) = \\langle \\phi_t - \\mathcal{L}_\\phi^*\\phi \n\t\t- h\\mathcal{M}^*\\phi, f\\rangle^2 \/ \\langle \\phi, f_y^2 \\rangle, \\quad\n\t\t\\langle \\phi,f\\rangle = \\int_{-\\infty}^\\infty f(y)\\phi(dy),\n\t\\]\n\tif $\\phi(t)$ is absolutely continuous in $t\\in[0,T]$ and $I_h(\\phi)=\\infty$ otherwise.\n\\end{theorem}\n\n\\textbf{Remark.} Here for $\\phi \\in {\\cal E}^\\nu$ and $t\\in[0,T]$, $\\phi(t)$ is viewed as a real Schwartz distribution \non $\\mathbb{R}$, $\\mathcal{L}_\\psi^*$ and $\\mathcal{M}^*\\phi$ are differential operators in the distribution \nsense, and $f$ in (\\ref{def:Ih}) is a real Schwartz test function. The definition of absolute continuity for \nthe path of measures $(\\phi(t))_{t \\in [0,T]}$ is in the sense of Definition 4.1 in \\cite{Dawson1987}, that \nis to say: for each compact set $K \\subset \\mathbb{R}$ there exists a neighborhood $U_K$ of the null \nfunction in the set of test functions with compact support in $K$ and an absolutely continuous function \n$H_K$ from $[0,T]$ to $\\mathbb{R}$ such that $| \\langle \\phi(t),f\\rangle -\t\\langle \\phi(s),f\\rangle | \\leq \n|H_K(t)-H_K(s)|$ for all $s,t \\in [0,T]$ and $f\\in U_K$. Note that by Lemma 4.2 in \\cite{Dawson1987}, if \n$\\phi(t)$ is absolutely continuous in $t\\in[0,T]$, $\\phi_t(t)$ exists in the distribution sense almost \neverywhere on $t\\in[0,T]$.\n\nIn order to use Theorem \\ref{thm: Dawson and Gartner, general LD}, we let \n$\\nu = u^e_{-\\xi_b}$ in (\\ref{eq:equilibrium of Fokker-Planck eq}) \nand define the rare event $A$ of systemic transition by\n\\begin{equation}\n\t\\label{def: the set of phase transition}\n\tA = \\left\\{ \\phi\\in\\mathcal{E}^\\nu: \\phi(T)=u^e_{\\xi_b} \\right\\}.\n\\end{equation}\nHowever, since $\\mathring{A}$ is an empty set, \nTheorem \\ref{thm: Dawson and Gartner, general LD} give a  trivial lower bound \nfor the probability in question. Therefore we consider instead the closed rare event $A_\\delta$:\n\\[\n\tA_\\delta=\\left\\{\\phi\\in\\mathcal{E}^\\nu: \\rho(\\phi(T),u^e_{\\xi_b})\\leq\\delta\\right\\}.\n\\]\nThen Theorem \\ref{thm: Dawson and Gartner, general LD} implies that \n\\begin{align*}\n\t-\\inf_{\\phi \\in \\mathring{A}_\\delta}I_h(\\phi) \n\t& \\leq \\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\log\\mathbf{P}(X_N\\in A_\\delta)\\\\\n\t& \\leq \\limsup_{N\\rightarrow\\infty}\\frac{1}{N}\\log\\mathbf{P}(X_N\\in A_\\delta)\n\t\\leq - \\inf_{\\phi \\in A_\\delta}I_h(\\phi).\n\\end{align*}\nIn addition, we show that $\\inf_{\\phi \\in A_\\delta}I_h(\\phi)$ can be bounded from below \nby $\\inf_{\\phi \\in A}I_h(\\phi)$ as $\\delta \\rightarrow 0$.\n\\begin{lemma}\n\t\\label{lma:lower bound of I_h over A_delta}\n\tBy definition $\\inf_{\\phi\\in A_\\delta}I_h(\\phi)$ is decreasing with $\\delta>0$ and \n\tbounded from above by $\\inf_{\\phi\\in A}I_h(\\phi)$. In addition,\n\t\\[\n\t\t\\lim_{\\delta\\to 0}\\inf_{\\phi\\in A_\\delta}I_h(\\phi)\\geq\\inf_{\\phi\\in A}I_h(\\phi).\n\t\\]\n\\end{lemma}\n\\begin{proof}\n\tSee Appendix \\ref{pf:lower bound of I_h over A_delta}.\n\\end{proof}\n\nCombining Lemma \\ref{lma:lower bound of I_h over A_delta} and the fact that \n$\\inf_{\\phi \\in \\mathring{A}_\\delta}I_h(\\phi) \\leq \\inf_{\\phi \\in A}I_h(\\phi)$, for any \n$\\epsilon>0$, we have for sufficiently small $\\delta>0$ \n\\begin{align*}\n\t-\\inf_{\\phi \\in A}I_h(\\phi)\n\t& \\leq \\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\log\\mathbf{P}(X_N\\in A_\\delta)\\\\\n\t& \\leq \\limsup_{N\\rightarrow\\infty}\\frac{1}{N}\\log\\mathbf{P}(X_N\\in A_\\delta)\n\t\\leq - \\inf_{\\phi \\in A}I_h(\\phi) + \\epsilon.\n\\end{align*}\nTherefore for large $N$ and sufficiently small $\\delta$,\n\\begin{equation}\n\t\\label{eq:probability of phase transition, LD form}\n\t\\mathbf{P}(X_N\\in A_\\delta) \\approx \\exp\\left(-N\\inf_{\\phi\\in A}I_h(\\phi)\\right).\n\\end{equation}\nThis tells us that a larger system has a more stable empirical mean\ntrajectory, which is consistent with what we have seen in the numerical\nsimulation. Now the main step is finding $\\inf_{\\phi\\in A}I_h(\\phi)$,\nwhich is a min-max variational problem\n\\begin{equation}\n\t\\label{eq:general LD}\n\t\\inf_{\\phi \\in A} I_h(\\phi)\n\t= \\inf_{\\phi \\in A} \\frac{1}{2\\sigma^2} \\int_0^T \n\t\\sup_{f:\\langle \\phi, f_y^2 \\rangle \\neq 0}\n\t\\langle \\phi_t - \\mathcal{L}_\\phi^*\\phi - h\\mathcal{M}^*\\phi, f\\rangle^2\n\t\/ \\langle \\phi, f_y^2 \\rangle dt,\n\\end{equation}\nwhere the $f$ in the $\\sup$ is a  real Schwartz test function. \n\n\n\n\n\\subsection{An Alternative Expression for the Rate Function}\n\nThe representation of the rate function (\\ref{def:Ih}) is somewhat complicated, but we \ncan simplify it if $\\phi$ has the density with some additional properties.\nIf $\\phi$ is a density function such that $\\phi(t,y)$ is smooth, rapidly decreasing in \n$y\\in\\mathbb{R}$ for each $t\\in[0,T]$ and is absolutely continuous in $t\\in[0,T]$ for \neach $y\\in\\mathbb{R}$, then let $g(t,y)$ satisfy\n\\begin{equation}\n\t\\label{eq:Fokker-Planck equation with the driving force}\n\t\\phi_t - \\mathcal{L}_{\\phi}^{*}\\phi - h\\mathcal{M}^{*}\\phi = (\\phi g)_y.\n\\end{equation}\nNote that because of the properties of $\\phi$, the left hand side of \n(\\ref{eq:Fokker-Planck equation with the driving force}) is well-defined in \n$y\\in\\mathbb{R}$ and almost everywhere in $t\\in[0,T]$. In addition, because $\\phi$ is \npositive valued, $g$ exists and is unique except on a measure zero set in $[0,T]$.\n\n\n\n\n\nNote that for the pair $(\\phi,g)$ satisfying \n(\\ref{eq:Fokker-Planck equation with the driving force})\n\\[\n\t\\sup_{f:\\langle \\phi(t), f_y^2 \\rangle \\neq 0} J_h(\\phi(t),f)\n\t= \\sup_{f:\\langle \\phi(t), f_y^2 \\rangle \\neq 0} \n\t\\langle\\phi(t),f_y g\\rangle^2\/\\langle\\phi(t),f_y^2\\rangle\n\t= \\langle \\phi(t), g^2\\rangle,\n\\]\nand therefore we have the following proposition.\n\n\\begin{proposition}\n\t\\label{prop:I_h in terms of the driving force}\n\tIf $\\phi$ is a density function such that $\\phi(t)$ is a Schwartz function for each \n\t$t\\in[0,T]$ and is absolutely continuous in $t\\in[0,T]$ for each $y\\in\\mathbb{R}$, \n\tand $g(t,y)$ satisfies (\\ref{eq:Fokker-Planck equation with the driving force}), the \n\trate function $I_h(\\phi)$ in (\\ref{def:Ih}) can be written in the form\n\t\\begin{equation}\n\t\t\\label{eq:I_h in terms of the driving force}\n\t\tI_h(\\phi) = \\frac{1}{2\\sigma^2} \\int_0^T \\langle \\phi, g^2\\rangle dt.\n\t\\end{equation}\n\\end{proposition}\n\nWe interpret (\\ref{eq:Fokker-Planck equation with the driving force}) and \n(\\ref{eq:I_h in terms of the driving force}) as follows. The function $g$ is regarded as \nthe driving force making $\\phi$ deviate from the solution of the Fokker-Planck equation \n(\\ref{eq: full Fokker-Planck eqn}), and $I_h(\\phi)$ is the $L^2(\\phi)$ norm of $g$, \nwhich measures how difficult it is to have this deviation $\\phi$. \n\n\n\n\n\n\\section{The Mean-Field Limit}\n\\label{sec:mean field}\n\nWe briefly review the mean field limit in \\cite{Dawson1983,Gartner1988} and carry out a small $h$ analysis \nof results since they will be used in calculating large deviation probabilities. We want to analyze the \nsystemic behavior  of the interacting diffusion processes (\\ref{eq:SDE of single component}), through their \nempirical mean $\\bar{x}(t)$, but this is not possible in a direct way since \n(\\ref{eq:SDE of single component}) is nonlinear. We consider instead the empirical density of $x_j(t)$, \nwhich is a measure valued process that has a limit as $N\\rightarrow\\infty$. Let $M_1(\\mathbb{R})$ be the \nspace of probability measures endowed with the weak (Prohorov) topology and let $C([0,T],M_1(\\mathbb{R}))$ \nbe the space of continuous $M_1(\\mathbb{R})$-valued processes on $[0,T]$ endowed with the corresponding weak \ntopology. Define the empirical probability measure process \n$X_N(t,dy):=\\frac{1}{N}\\sum_{j=1}^N\\delta_{x_j(t)}(dy)$ and note that $X_N\\in C([0,T],M_1(\\mathbb{R}))$. The \nmean field limit theorem for $X_N$, proved in \\cite{Dawson1983,Gartner1988}, is as follows:\n\\begin{theorem}\n\t\\label{thm: Dawson, mean-field limit}\n\t(Dawson, 1983) Assume that the force is $U(y)=y^3-y$ and that $X_N(0)$ converges weakly to a probability\n\tmeasure $\\nu_0$. Then the measure valued process $X_N$ converges weakly in law as $N\\rightarrow\\infty$ to a \n\tdeterministic process with density $u(t,y)dy \\in C([0,T],M_1(\\mathbb{R}))$, which is the unique weak solution of the Fokker-Planck \n\tequation:\n\t\\begin{equation}\n\t\t\\label{eq: full Fokker-Planck eqn}\n\t\t\\frac{\\partial}{\\partial t}u\n\t\t= h\\frac{\\partial}{\\partial y}[U(y)u]\n\t\t- \\theta\\frac{\\partial}{\\partial y}\\left\\{\\left[\\int yu(t,y)dy-y\\right]u\\right\\} \n\t\t+ \\frac{1}{2}\\sigma^2\\frac{\\partial^2}{\\partial y^2}u,\n\t\\end{equation}\n\twith initial condition $\\nu_0$.\n\\end{theorem}\n\nBy Theorem \\ref{thm: Dawson, mean-field limit}, we can analyze $u$ and view $X_N$ as a perturbation of $u$ \nfor $N$ large. We may consider $\\bar{x}(t)$ in the same way because $\\bar{x}(t)=\\int y X_N(t,dy)$. However, \nthe limit problem is infinitely dimensional, as is expected.\n\nExplicit solutions of (\\ref{eq: full Fokker-Planck eqn}) are not available in general, but we can find \nequilibrium solutions. Assuming that $\\xi = \\lim_{t\\to\\infty}\\int yu(t,y)dy$, then an equilibrium solution \n$u_\\xi^e$ satisfies\n\\[\n\th\\frac{d}{dy}[(y^3-y)u_\\xi^e] - \\theta\\frac{d}{dy}[(\\xi-y)u_\\xi^e] \n\t+ \\frac{1}{2}\\sigma^2\\frac{d^2}{dy^2}u_\\xi^e = 0,\n\\]\nand has the form\n\\begin{equation}\n\t\\label{eq:equilibrium of Fokker-Planck eq}\n\tu^e_\\xi(y) = \\frac{1}{Z_\\xi \\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\\exp \\left\\{ -\\frac{(y-\\xi)^2}{2\\frac{\\sigma^2}{2\\theta}} \n\t- h\\frac{2}{\\sigma^2}V(y) \\right\\},\n\\end{equation}\nwith $Z_\\xi$ the normalization constant:\n\\[\n\tZ_\\xi = \\int \\frac{1}{\\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\\exp \\left\\{ -\\frac{(y-\\xi)^2}{2\\frac{\\sigma^2}{2\\theta}} \n\t- h\\frac{2}{\\sigma^2}V(y) \\right\\} dy.\n\\]\nNow $\\xi$ must satisfy the compatibility or consistency condition:\n\\begin{equation}\n\t\\label{eq:xi equals m(xi)}\n\t\\xi = m(\\xi):= \\int y u^e_\\xi(y)dy.\n\\end{equation}\nFinding equilibrium solutions has thus been reduced to finding solutions of this equation.\n\n\nFor $U(y)=y^3-y$, $\\xi=0$ is a solution for (\\ref{eq:xi equals m(xi)}). With the same \n$U(y)$, it can be shown (see also \\cite[Theorem 3.3.1 and 3.3.2]{Dawson1983}) that there are two additional \nnon-zero solutions $\\pm\\xi_b$ if and only if $\\frac{d}{d\\xi}m(0)>1$, and for given $h$ \nand $\\theta$, there exists a critical $\\sigma_c(h,\\theta) >0$ such that $\\frac{d}{d\\xi}m(0)>1$ if \nand only if $\\sigma<\\sigma_c(h,\\theta)$.\n\nAn explanation for this bifurcation at equilibrium is that when $\\sigma\\geq\\sigma_c$, randomness dominates the \ninteraction among the components, i.e., $\\theta(\\bar{x}(t)-x_j(t))dt$ is negligible. In \nthis case, the system behaves like $N$ independent diffusions and hence, by the\nsymmetry of $V(y)$, at any given time roughly half of them stay around $-1$ and half around $+1$ so the \naverage is $0$. When, however, $\\sigma<\\sigma_c$, then the interactive force is \nsignificantly larger (now $\\sigma dw_j(t)$ is less important). Therefore all agents stay \naround the same place (either $-\\xi_b$ or $+\\xi_b$) and the zero average equilibrium is unstable. \nSince we want to model systemic risk phenomena, \nwe assume that \n$\\sigma<\\sigma_c$ throughout this paper, and we regard $-\\xi_{b}$ as the normal state \nof the system and $+\\xi_b$ as the failed state. The calculation of\ntransitions probabilities  between these two states is our objective.\n\nFor small $h$ we can approximate the solution \nof (\\ref{eq:xi equals m(xi)}) to order $O(h)$ as follows.\n\\begin{proposition}\n\t\\label{prop:explicit condition and solution for xi equal to m(xi) for small h}\n\tFor small $h$, the critical value $\\sigma_c$ can be expanded as \n\t\\begin{equation}\n\t\t\\label{eq:sigma_c of the homogeneous case}\n\t\t\\sigma_c = \\sqrt{\\frac{2 \\theta}{3}} + O(h).\n\t\\end{equation}\n\tIn addition, the non-zero solutions $\\pm \\xi_{b}$ are \n\t\\begin{equation}\n\t\\label{eq:explicit solution for xi equal to m(xi)}\n\t\\pm \\xi_b = \\pm \\sqrt{1 - 3\\frac{\\sigma^2}{2\\theta}} \n\t\\left( 1 + h\\frac{6}{\\sigma^2}\\left(\\frac{\\sigma^2}{2\\theta}\\right)^2\n\t\\frac{1 - 2(\\sigma^2\/2\\theta)}{1 - 3(\\sigma^2\/2\\theta)}\\right) + O(h^2).\n\t\\end{equation}\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix \n\t\\ref{pf:explicit condition and solution for xi equal to m(xi) for small h}.\n\\end{proof}\n\nFrom Proposition \n\\ref{prop:explicit condition and solution for xi equal to m(xi) for small h}, we see the \nrelation between the existence of the bi-stable states and the ratio \n$\\sigma^2\/2\\theta$: For a given $\\theta$, and for small $h$, \n(\\ref{eq:xi equals m(xi)}) has non-zero solutions if and only if \n$ 3\\sigma^2\/2\\theta < 1$. Moreover, these non-zero solutions $\\pm \\xi_b$ are \ngenerally not $\\pm 1$ since the magnitude $|\\xi_b|$ is less than $1$. \nNote that the coefficient of order $h$ in the expansion \n(\\ref{eq:explicit solution for xi equal to m(xi)}) depends significantly on $\\theta$ and \n$\\sigma$. Thus, when $3\\sigma^2\/2\\theta$ tends to $1$, $\\xi_b$ in \n(\\ref{eq:explicit solution for xi equal to m(xi)}) will not go to $+\\infty$ while, in fact,\n$\\xi_b$ goes to $0$. From the $O(1)$ term in \n(\\ref{eq:explicit solution for xi equal to m(xi)}), we also see that $\\xi_b$ is roughly \ndecreasing as $\\sigma^2\/2\\theta$ is increasing.\n\\section{Proof of Lemma \\ref{lma:lower bound of I_h over A_delta}}\n\\label{pf:lower bound of I_h over A_delta}\n\n\nIt suffices to show the case that $\\delta=1\/n$. For each $n$, let $\\phi_n\\in A_{1\/n}$, \nsuch that \n$\\inf_{\\phi\\in A_{1\/n}}I_h(\\phi)\\leq I_h(\\phi_n)<\\inf_{\\phi\\in A_{1\/n}}I_h(\\phi)+1\/n$; \n$\\{I_h(\\phi_n)\\}$ are bounded from above by $\\inf_{\\phi\\in A}I_h(\\phi)+1<\\infty$. \nBecause $I_h$ is a good rate function, and by Proposition B.13 of \\cite{Gartner1988},\ncompactness is equivalent to sequentially compactness in \n$C([0,T],M_\\infty(\\mathbb{R}))$, $\\{\\phi_n\\}$ has a convergent subsequence \n$\\{\\phi_{n_k}\\}$ whose limit $\\phi^*$ is in $A$. As $I_h$ is lower semicontinuous, then\n\\[\n\t\\lim_n\\inf_{\\phi\\in A_{1\/n}}I_h(\\phi) = \\lim_k I_h(\\phi_{n_k})\n\t= \\liminf_k I_h(\\phi_{n_k}) \\geq I_h(\\phi^*) \\geq \\inf_{\\phi\\in A}I_h(\\phi).\n\\]\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs in Section \\ref{sec:diversity}}\n\n\\subsection{Proof of Theorem \\ref{thm:mean-field limit, diversity case}}\n\\label{pf:mean-field limit, diversity case}\n\nThe proof contains three steps.\n\n\n\\subsubsection{Existence and Uniqueness of the Weak Solution of the McKean-Vlasov Equation}\n\nThe existence and uniqueness of a probability measure valued process $(u_1(t),\\ldots,u_K(t))$ that is a weak \nsolution of the McKean-Vlasov equation (\\ref{eq:system of Fokker-Planck equations}) is guaranteed by \n\\cite[Theorem 2.11]{Gartner1988}.\n\n\n\\subsubsection{Weak Compactness of the Empirical Process}\n\nBy Prohorov's theorem, it suffices to prove that the sequence $\\{(X_N^1, \\ldots, X_N^K)\\}_{N=1}^\\infty$ is \nweakly compact by showing that\n\\begin{equation*}\n\t\\label{eq:tightness}\n\t\\sup_N\\sup_{1\\leq k\\leq K}\\sup_{0\\leq t\\leq T} \\mathbf{E}[\\langle X_N^k(t,dy),|y|\\rangle] <\\infty,\n\\end{equation*}\nwhich can be done by using the calculations similar to (B1) and (B2) in \\cite{Dawson1983}.\n\n\n\\subsubsection{Identification of the Limit}\n\nFor a test function $f\\in\\mathcal{S}(\\mathbb{R})$, we define $X_N^{f,l}(t)\n=\\langle f(y),X_N^l(t,y)\\rangle=\\sum_{j\\in\\mathcal{I}_l}f(x_j(t))\/|\\mathcal{I}_l|$. \nBy It\\^{o}'s formula,\n\\begin{align*}\n\tdX_N^{f,l}\n\t&= \\frac{1}{|\\mathcal{I}_l|} \\sum_{j\\in\\mathcal{I}_l} \n\t[ -h U(x_j)dt + \\sigma dw_j + \\Theta_l(\\bar{x}-x_j)dt ] f_y(x_j) \n\t+ \\frac{1}{2}\\sigma^2f_{yy}(x_j)dt\\\\\n\t&= \\langle -h U f_y \n\t+ \\Theta_l(\\langle y,\\sum_{l=1}^K\\rho_l X_N^l\\rangle - y)f_y \n\t+ \\frac{\\sigma^2}{2} f_{yy}, X_N^l \\rangle dt\n\t+ \\langle  f_y, \n\t\\frac{\\sigma}{|\\mathcal{I}_l|}\\sum_{j\\in\\mathcal{I}_l}\\delta_{x_j}dw_j \\rangle.\n\\end{align*}\nThen by the integration by parts, we write \n\\[\n\tdX_N^l\n\t= \\{(h U X_N^l)_y \n\t-[\\Theta_l(\\langle y,\\sum_{l=1}^K\\rho_l X_N^l\\rangle-y)X_N^l]_y \n\t+ \\frac{\\sigma^2}{2}(X_N^l)_{yy} \\}dt\n\t- \\frac{\\sigma}{|\\mathcal{I}_l|}\\sum_{j\\in\\mathcal{I}_l}(\\delta_{x_j})_y dw_j.\n\\]\nFor simplicity, we prove the case that $K=2$ and the general case is similar. \nWe let $X_N^{1,\\times n} \\times X_N^{2,\\times n}$ denote the product measure on \n$\\mathbb{R}^{2n}$:\n\\[\n\tX_N^{1,\\times n} \\times X_N^{2,\\times n}(y_1,\\ldots,y_{2n})\n\t= X_N^1(t,y_1)\\cdots X_N^1(t,y_n)X_N^2(t,y_{n+1})\\cdots X_N^2(t,y_{2n}).\n\\]\nFor a test function $f\\in\\mathcal{S}(\\mathbb{R}^{2n})$, we have \n\\[\n\td\\langle f, X_N^{1,\\times n} \\times X_N^{2,\\times n}\\rangle \n\t= d\\langle f, X_N^{1,\\times n} \\times X_N^{2,\\times n}\\rangle^{(1)} \n\t+ d\\langle f, X_N^{1,\\times n} \\times X_N^{2,\\times n}\\rangle^{(2)},\n\\]\nwhere $(1)$ and $(2)$ denote the first and the second order terms of \n$d\\langle f, X_N^{1,\\times n} \\times X_N^{2,\\times n}\\rangle$, respectively:\n\\begin{align*}\n\td\\langle f, X_N^{1,\\times n} \\times X_N^{2,\\times n}\\rangle^{(1)}\n\t&= \\sum_{j=1}^n \\langle f,dX_N^1(t,y_j)\\times X_N^{1,\\times(n-1),j}\n\t\\times X_N^{2,\\times n}\\rangle\\\\\n\t&\\quad + \\sum_{j=n+1}^{2n} \\langle f,dX_N^2(t,y_j)\\times X_N^{1,\\times n,j}\n\t\\times X_N^{2,\\times(n-1),j}\\rangle\n\\end{align*}\n\\begin{multline*}\n\td\\langle f,X_N^{1,\\times n}\\times X_N^{2,\\times n}\\rangle^{(2)}\n\t= \\frac{1}{2}\\sum_{\\substack{j,k=1\\\\j\\neq k}}^n \n\t\\langle f, dX_N^1(t,y_j) \\times dX_N^1(t,y_k)\\times X_N^{1,\\times(n-2),j,k} \n\t\\times X_N^{2,\\times n}\\rangle \\\\\n\t+ \\frac{1}{2}\\sum_{\\substack{j,k=n+1\\\\j\\neq k}}^{2n}\n\t\\langle f,dX_N^2(t,y_j) \\times dX_N^2(t,y_k) \\times X_N^{1,\\times n}\n\t\\times X_N^{2,\\times(n-2),j,k}\\rangle \\\\\n\t+ \\frac{1}{2}\\sum_{j=1}^n\\sum_{k=n+1}^{2n}\n\t\\langle f, dX_N^1(t,y_j) \\times dX_N^2(t,y_k) \\times X_N^{1,\\times(n-1),j}\n\t\\times X_N^{2,\\times(n-1),k}\\rangle.\n\\end{multline*}\nNote that for $j\\neq k$, $dX_N^l(t,y_j) \\times dX_N^l(t,y_k) \n= \\frac{\\sigma^2}{|\\mathcal{I}_l|^2} \\sum_{i\\in\\mathcal{I}_l} \n(\\delta_{x_i}(y_j))_j (\\delta_{x_i}(y_k))_k dt\n= \\frac{\\sigma^2}{\\rho_l^2 N} (\\delta(y_k-y_j) X_N^l(t,y_j))_{jk} dt$, and \n$dX_N^1(t,y_j) \\times dX_N^2(t,y_k) = 0 $.\nIf we analogously represent the generator $G_{(X_N^{1,\\times n},X_N^{2,\\times n})} f$ of \n$\\langle f, X_N^{1,\\times n} \\times X_N^{2,\\times n}\\rangle$ as \n\\[\n\tG_{(X_N^{1,\\times n},X_N^{2,\\times n})} f\n\t= G_{(X_N^{1,\\times n},X_N^{2,\\times n})}^{(1)} f \n\t+ G_{(X_N^{1,\\times n},X_N^{2,\\times n})}^{(2)} f,\n\\]\nthen $G_{(X_N^{1,\\times n},X_N^{2,\\times n})}^{(2)} f \\rightarrow 0 $ as \n$N\\rightarrow \\infty$ and $G_{(X_N^{1,\\times n},X_N^{2,\\times n})}^{(1)} f \n= G_{(u_1^{\\times n},u_2^{\\times n})}f$, the generator of \n$\\langle f, u_1^{\\times n} \\times u_2^{\\times n}\\rangle$, where $(u_1,u_2)$ satisfying \n(\\ref{eq:system of Fokker-Planck equations}). Then the limit of $(X_N^1,X_N^2)$ is a solution of the \nmartingale problem associated to (\\ref{eq:system of Fokker-Planck equations}). In addition, by \n\\cite[Corollary 2.10]{Gartner1988}, the solution is unique and therefore $(X_N^1,X_N^2)\\rightarrow \n(u_1,u_2)$ weakly as $N\\rightarrow\\infty$.\n\n\n\n\n\n\n\\subsection{Proof of Proposition \n\\ref{prop:explicit condition for xi equal to m(xi) for small h, diversity case}}\n\\label{pf:explicit condition for xi equal to m(xi) for small h, diversity case}\n\nAll we need to show is that for small $h$, $\\frac{d}{d\\xi}m(0)>1$ if and only if \n$\\sigma < \\sigma_c^{\\text{div}}$, where $m(\\xi)$ is defined by \n(\\ref{eq:xi equal to m(xi), diversity case}). We obtain $\\frac{d}{d\\xi}m$ by \ncalculate $\\frac{d}{d\\xi}\\int yu_{l,\\xi}^e (y)dy$. Note that \n$\\frac{d}{d\\xi}Z_{l,\\xi}=(2\\Theta_l\/\\sigma^2)(\\int yu^e_{l,\\xi}dy-\\xi) Z_{l,\\xi}$ and \n\\begin{align}\n\t\\label{eq:d2Z, first kind}\n\t\\frac{d^2}{d\\xi^2}Z_{l,\\xi} \n\t&= \\frac{2\\Theta_l}{\\sigma^2}Z_{l,\\xi}\n\t\\left(\\frac{d}{d\\xi}\\int yu^e_{l,\\xi}dy - 1\\right)\n\t+ \\frac{2\\Theta_l}{\\sigma^2} \\left(\\int yu^e_{l,\\xi}dy-\\xi\\right)\n\t\\frac{d}{d\\xi} Z_{l,\\xi}\\\\\n\t&= \\frac{2\\Theta_l}{\\sigma^2}Z_{l,\\xi} \n\t\\left(\\frac{d}{d\\xi}\\int yu^e_{l,\\xi}dy - 1\\right)\n\t+ \\left(\\frac{2\\Theta_l}{\\sigma^2}\\right)^2\n\tZ_{l,\\xi} \\left(\\int yu^e_{l,\\xi}dy - \\xi\\right)^2.\\notag\n\\end{align}\nOn the other hand, we can also compute $\\frac{d^2}{d\\xi^2}Z_{l,\\xi}$ by directly taking \nthe twice derivatives of $Z_{l,\\xi}$:\n\\begin{equation}\n\t\\label{eq:d2Z, second kind}\n\t\\frac{d^2}{d\\xi^2}Z_{l,\\xi} = -\\frac{2\\Theta_l}{\\sigma^2} Z_{l,\\xi} \n\t+ \\left(\\frac{2\\Theta_l}{\\sigma^2}\\right)^2 Z_{l,\\xi} \\int (y-\\xi)^2 u^e_{l,\\xi}dy.\n\\end{equation}\nBy comparing (\\ref{eq:d2Z, first kind}) and (\\ref{eq:d2Z, second kind}), \n\\[\n\t\\frac{d}{d\\xi}\\int yu^e_{l,\\xi}dy\n\t= \\frac{2\\Theta_l}{\\sigma^2} \\left[ \\int y^2u^e_{l,\\xi}dy\n\t- (\\int yu^e_{l,\\xi}dy)^2\\right].\n\\]\nNote that $\\int yu^e_{l,0}dy=0$, so \n$\\frac{d}{d\\xi}m(0)=\\sum_{l=1}^K\\rho_l(2\\Theta_l\/\\sigma^2)\\int y^2u^e_{l,0}dy$. \nBy using the same trick in the proof of Proposition \n\\ref{prop:explicit condition and solution for xi equal to m(xi) for small h}, \nlet $p_l(y)$ be the Gaussian density function with mean $0$ and variance \n$\\sigma^2\/2\\Theta_l$, $Y_l$ be the Gaussian random variable with the density $p_l$,\nand $\\eta = 2\/\\sigma^2$. Then for small $h$, \n$Z_{l,0}^{-1}=1+h\\eta\\mathbf{E}V(Y_l)+O(h^2)$, and \n\n\\begin{align*}\n\t\\int y^2u^e_{l,0}dy\n\t&= Z_{l,0}^{-1} \\int y^2 (1-h\\eta V+O(h^2)) p_l(y) dy\\\\\n\t&= Z_{l,0}^{-1} ( \\mathbf{E}Y_l^2 - h\\eta \\mathbf{E}[Y_l^2V(Y_l)] + O(h^2) )\\\\\n\t&= \\mathbf{E}Y_l^2 + h\\eta (\\mathbf{E}Y_l^2\\mathbf{E}V(Y_l)\n\t-\\mathbf{E}[Y_l^2V(Y_l)]) + O(h^2).\n\\end{align*}\nTherefore $\\frac{d}{d\\xi}m(0)>1$ if and only if \n$\\sum_{l=1}^K \\rho_l (2\\Theta_l\/\\sigma^2)\n(\\mathbf{E}Y_l^2\\mathbf{E}V(Y_l)-\\mathbf{E}[Y_l^2V(Y_l)]) > 0$. Note that \n$\\mathbf{E}Y_l^2 = \\sigma^2\/2\\Theta_l$, \n$\\mathbf{E}V(Y_l) = (3\/4)(\\mathbf{E}Y_l^2)^2 - (1\/2)\\mathbf{E}Y_l^2$, and\n$\\mathbf{E}[Y_l^2V(Y_l)] = (15\/4)(\\mathbf{E}Y_l^2)^3 - (3\/2)(\\mathbf{E}Y_l^2)^2$.\nThen the sufficient and necessary condition becomes \n\\[\n\t\\sum_{l=1}^K \\frac{\\rho_l}{\\Theta_l}\\left(1 - 3\\frac{\\sigma^2}{2\\Theta_l}\\right) > 0.\n\\]\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Proposition \\ref{prop:sigma_c comparison}}\n\\label{pf:sigma_c comparison}\n\nIt is equivalent to show that $\\sum_{l=1}^K \\rho_l\/\\Theta_l \n\\leq \\sum_{l=1}^K \\rho_l\\Theta_l \\sum_{l=1}^K \\rho_l\/\\Theta_l^2$.\nFirst note that by the Cauchy-Schwarz inequality,\n\\[\n\t\\left(\\sum_{l=1}^K \\frac{\\rho_l}{\\Theta_l}\\right)^2\n\t= \\left(\\sum_{l=1}^K\\frac{\\sqrt{\\rho_l}}{\\Theta_l}\\times\\sqrt{\\rho_l}\\right)^2\n\t\\leq \\sum_{l=1}^K\\frac{\\rho_l}{\\Theta_l^2} \\sum_{l=1}^K\\rho_l \n\t= \\sum_{l=1}^K\\frac{\\rho_l}{\\Theta_l^2}.\n\\]\nThen it suffices to show that $1\\leq\\sum_{l=1}^K\\rho_l\\Theta_l\n\\sum_{l=1}^K \\rho_l\/\\Theta_l$. Again by the Cauchy-Schwarz inequality,\n\\[\n\t\\sum_{l=1}^K \\rho_l\\Theta_l \\sum_{l=1}^K \\frac{\\rho_l}{\\Theta_l} \n\t\\geq \\sum_{l=1}^K \\sqrt{\\rho_l\\Theta_l}\\sqrt{\\frac{\\rho_l}{\\Theta_l}} \n\t= \\sum_{l=1}^K\\rho_l = 1.\n\\]\n\n\n\n\n\n\n\n\\section{Proofs in Section \\ref{sec:diversity2}}\n\n\\subsection{Proof of Proposition \\ref{prop:xbar_diversity}}\n\\label{pf:xbar_diversity}\n\nThe system of SDEs (\\ref{eq:systembarxk}) for the vector $\\bar{X}(t) =(\\bar{x}_k(t))_{k=1,\\ldots,K}$ has the \nform\n\\[\n\td\\bar{X} (t) =M\\bar{X}(t) + \\frac{\\sigma}{\\sqrt{N}} R^{-1\/2} d\\bar{W}(t)\n\\]\nwhere $\\bar{W}(t)= (\\bar{w}_k(t))_{k=1,\\ldots,K}$ is a column vector. This system can be solved:\n\\[\n\t\\bar{X}(t) = e^{Mt} \\bar{X}(0) +\\frac{\\sigma}{\\sqrt{N}} \\int_0^t e^{M(t-s)} R^{-1\/2} d \\bar{W}(s)\n\\]\nIf $\\bar{x}_k(0)= -\\xi_b$, then, using the fact that the uniform vector is in the null space of $M$, we \nhave $e^{Mt}\\bar{X}(0)=\\bar{X}(0)$. As a corollary we get the explicit representation of the empirical mean:\n\\[\n\t\\bar{x}(t) = -\\xi_b+\\frac{\\sigma}{\\sqrt{N}} \\int_0^t \\varrho^\\mathbf{T} e^{M(t-s)} R^{-1\/2} d\\bar{W}(s)\n\\] \nThis shows the desired result.\n\n\\subsection{Proof of Proposition \\ref{prop:the diversity on the transition probability}}\n\\label{pf:the diversity on the transition probability}\n\nThe expansion of $\\xi_b^2$ follows from the explicit expression \n(\\ref{eq:explicit solution for xi equal to m(xi), diversity case}). The expansion of $\\sigma_T^2$ follows \nfrom the expansion of (\\ref{eq:sigmaT2}) and uses the properties of the matrix $M$. We have \n$M=-\\bar{\\theta}\\bar{M}-\\delta\\bar{\\theta}N$, with\n\\begin{align*}\n\t\\bar{M} &= I-  u\\varrho^\\mathbf{T},\\quad \n\t\\text{where $u=(1,\\ldots,1)$ is the $K$-dimensional column vector,}\\\\\n\tN_{ij} &=\\alpha_i (  \\delta_{ij} - \\rho_j ) ,\\quad i,j=1,\\ldots,K.\n\\end{align*}\nThe matrix $\\bar{M}$ satisfies $\\bar{M}^n =\\bar{M}$ for all $n\\geq 1$ and therefore \n\\[\n\te^{-\\bar{\\theta} \\bar{M} t} \n\t= \\sum_{n=0}^\\infty \\frac{(-\\bar{\\theta} t)^n}{n!} \\bar{M}^n \n\t= I + \\sum_{n=1}^\\infty \\frac{(-\\bar{\\theta} t)^n}{n!} \\bar{M} \n\t=  I + ( e^{-\\bar{\\theta} t }-1)\\bar{M}.\n\\]\nWe have\n\\[\n\te^{Mt} =\\sum_{n=0}^\\infty \\frac{(-\\bar{\\theta} t)^n}{n!} ( \\bar{M}+\\delta N)^n\n\\]\nUsing the fact that $\\bar{M}^\\mathbf{T}\\varrho=0$ (and again that $\\bar{M}^n=\\bar{M}$ for $n \\geq 1$), we \ncan expand\n\\begin{align*}\n\t\\varrho^\\mathbf{T}e^{M t} \n\t&=\\varrho^\\mathbf{T} \n\t+\\delta\\varrho^\\mathbf{T}\\big\\{(-\\bar{\\theta}t)N+(e^{-\\bar{\\theta}t}-1+\\bar{\\theta}t)N\\bar{M}\\big\\}\\\\\n\t&\\quad + \\delta^2 \\varrho^\\mathbf{T}  \\Big\\{ \\frac{(\\bar{\\theta}t)^2}{2} N^2 \n\t+ \\big[ e^{- \\bar{\\theta}t}-1+\\bar{\\theta}t -\\frac{(\\bar{\\theta}t)^2}{2}  \\big]\n\t\\big[ N^2 \\bar{M} -3 (N \\bar{M} )^2 + N  \\bar{M} N \\big] \\\\\n\t&\\quad -\\bar{\\theta} t \\big[ e^{- \\bar{\\theta}t}-1+\\bar{\\theta}t \\big] ( N \\bar{M})^2 \\varrho \\Big\\} \n\t+ O(\\delta^3).\n\\end{align*}\nUsing the fact that $\\bar{M}^\\mathbf{T} N^\\mathbf{T} \\varrho = N^\\mathbf{T} \\varrho$ and  \n$\\bar{M}^\\mathbf{T} (N^\\mathbf{T})^2 \\varrho = (N^\\mathbf{T})^2 \\varrho$, this can be simplified into\n\\[\n\t\\varrho^\\mathbf{T} e^{M t} \n\t=  \\varrho^\\mathbf{T} + \\delta  \\varrho^\\mathbf{T}  (e^{-\\bar{\\theta} t} -1)  N    \n\t+ \\delta^2 \\varrho^\\mathbf{T} \\big[ (\\bar{\\theta} t)^2 \n\t- (1+\\bar{\\theta}t)(e^{-\\bar{\\theta} t} -1+\\bar{\\theta}t) \\big]   N^2 +O(\\delta^3).\n\\]\nConsequently\n\\begin{align*}\n\t&\\varrho^\\mathbf{T} e^{Mt} R^{-1} (e^{Mt})^\\mathbf{T} \\varrho \n\t= \\varrho^\\mathbf{T}  (I+(e^{-\\bar{\\theta}t}-1) \\bar{M} )  R^{-1}  \n\t(I+(e^{-\\bar{\\theta}t}-1) \\bar{M}^\\mathbf{T} ) \\varrho \\\\\n\t&\\quad +2\\delta\\varrho^\\mathbf{T}(e^{-\\bar{\\theta}t}-1)NR^{-1}\n\t(I+(e^{-\\bar{\\theta}t}-1) \\bar{M}^\\mathbf{T})\\varrho\\\\\n\t&\\quad  +2 \\delta^2  \\varrho^\\mathbf{T} \\big[ (\\bar{\\theta} t)^2 -(1+\\bar{\\theta}t)(e^{-\\bar{\\theta} t} \n\t-1+\\bar{\\theta}t) \\big]   N^2 R^{-1}  (I+(e^{-\\bar{\\theta}t}-1) \\bar{M}^\\mathbf{T} )  \\varrho  \\\\\n\t&\\quad \n\t+\\delta^2 \\varrho^\\mathbf{T}  ( e^{-\\bar{\\theta} t} -1 )   N R^{-1}  (e^{-\\bar{\\theta} t} -1 )   N^\\mathbf{T} \\varrho +O(\\delta^3).\n\\end{align*}\nUsing the fact that $\\bar{M}^\\mathbf{T} \\varrho=0$ and $NR^{-1} \\varrho = Nu=0$, we obtain\n\\[\n\t\\varrho^\\mathbf{T} e^{Mt} R^{-1} (e^{Mt})^\\mathbf{T} \\varrho = \n\t\\varrho^\\mathbf{T}   R^{-1}  \\varrho +\\delta^2 (1-e^{-\\bar{\\theta} t} )^2 \\varrho^\\mathbf{T}    \n\tN R^{-1} N^\\mathbf{T} \\varrho  +O(\\delta^3)\n\\]\nWe have $\\varrho^\\mathbf{T}   R^{-1}  \\varrho=1$ and \n$\\varrho^\\mathbf{T} N R^{-1}N^\\mathbf{T} \\varrho = \\sum_k \\rho_k \\alpha_k^2 $ which gives the expansion of \nthe variance $\\sigma_T^2$.\n\nFinally the expansion of the transition probability can be obtained by substituting the expansions of \n$\\xi_b^2$ and $\\sigma_T^2$ into (\\ref{eq:pTdiverse}).\n\\section{Proof of Proposition \n\\ref{prop:explicit condition and solution for xi equal to m(xi) for small h}}\n\\label{pf:explicit condition and solution for xi equal to m(xi) for small h}\n\nFor small $h$, we view $u^e_{\\xi}$ as a perturbed Gaussian density function. \nLet $p_\\xi(y)$ be the Gaussian density function with mean $\\xi$ and variance \n$\\sigma^2\/2\\theta$, $Y$ be the Gaussian random variable with the density $p_\\xi$,\nand $\\eta = 2\/\\sigma^2$. By using the expansion \n$\\exp( -h \\eta V) = 1 - h\\eta V + h^2 \\eta^2 V^2\/2 + O(h^3)$, we have\n\\begin{align*}\n\tZ_\\xi &= 1 - h\\eta\\mathbf{E}V(Y) + \\frac{1}{2} h^2\\eta^2\\mathbf{E}V^2(Y) + O(h^3)\\\\\n\tZ_\\xi^{-1} &= 1 + h\\eta \\mathbf{E}V(Y) - \\frac{1}{2} h^2\\eta^2 \\mathbf{E}V^2(Y) \n\t+ h^2\\eta^2(\\mathbf{E}V(Y))^2 + O(h^3).\n\\end{align*}\nThen we calculate $m(\\xi)$ as follows:\n\\begin{align*}\n\tm(\\xi) \n\t&= Z_\\xi^{-1} \\int y\\left(1-h\\eta V+\\frac{1}{2}h^2\\eta^2 V^2\n\t+ O(h^3)\\right)p_\\xi(y)dy\\\\\n\t&= Z_\\xi^{-1} \\left( \\xi - h\\eta \\mathbf{E}[YV(Y)] \n\t+ \\frac{1}{2}h^2\\eta^2 \\mathbf{E}[YV^2(Y)] + O(h^3) \\right)\\\\\n\t&= \\xi + h\\eta\\{\\xi\\mathbf{E}V(Y) - \\mathbf{E}[YV(Y)]\\}\n\t+ h^2\\eta^2 \\{ -\\frac{1}{2}\\xi\\mathbf{E}V^2(Y) + \\xi(\\mathbf{E}V(Y))^2\\\\\n\t&\\quad - \\mathbf{E}V(Y)\\mathbf{E}[YV(Y)] + \\frac{1}{2}\\mathbf{E}[YV^2(Y)] \\}+O(h^3)\\\\\n\t&= \\xi - h\\eta\\frac{\\sigma^2}{2\\theta} \\mathbf{E}V_y(Y)\t\n\t+ h^2\\eta^2 \\frac{\\sigma^2}{2\\theta} \n\t\\{ \\mathbf{E}[V(Y)V_y(Y)] - \\mathbf{E}V(Y)\\mathbf{E}V_y(Y)\\} + O(h^3)\\\\\n\t&= \\xi - h\\eta\\frac{\\sigma^2}{2\\theta} \\mathbf{E}V_y(Y) \n\t+ h^2\\eta^2 \\frac{\\sigma^2}{2\\theta} \\mathbf{Cov}(V_y(Y), V(Y)) + O(h^3).\n\\end{align*}\nThe compatibility condition $\\xi_b=m(\\xi_b)$ gives\n\\begin{equation}\n\t\\label{eq:compatibility condition for small h}\n\t\\mathbf{E}V_y(Y) - h\\eta\\mathbf{Cov}(V_y(Y), V(Y)) + O(h^2) = 0.\n\\end{equation}\nAssuming that $\\xi_b=\\xi_0 + h \\xi_1 + O(h^2)$, the $O(1)$ terms in \n(\\ref{eq:compatibility condition for small h}) give\n\\[\n\t\\xi_0^3 + 3\\frac{\\sigma^2}{2\\theta}\\xi_0 -\\xi_0 \n\t= \\xi_0(\\xi_0^2 + 3\\frac{\\sigma^2}{2\\theta}-1) = 0.\n\\]\nThen $\\xi_0 = 0, \\pm\\sqrt{1-3\\sigma^2\/2\\theta}$ if $3\\sigma^2 < 2\\theta$, or otherwise \n$\\xi_0 = 0$. In order to obtain the nontrivial result, we suppose that \n$3\\sigma^2 < 2\\theta$ and $\\xi_0$ takes $\\pm\\sqrt{1-3\\sigma^2\/2\\theta}$ in the later \ncalculations. Note that \n$\\mathbf{E}V_y(Y) = \\xi^3 + (3\\sigma^2\/2\\theta -1)\\xi = 2h \\xi_0^2\\xi_1 + O(h^2)$, and\n\\begin{align*}\n\t\\mathbf{Cov}(V_y(Y), V(Y)) &= \\mathbf{E}[V(Y)V_y(Y)] + O(h)\n\t= \\mathbf{E}[(\\frac{1}{4}Y^4 - \\frac{1}{2}Y^2)(Y^3-Y)] + O(h)\\\\\n\t&= \\mathbf{E}[\\frac{1}{4}Y^7 - \\frac{3}{4}Y^5 + \\frac{1}{2}Y^3] + O(h).\n\\end{align*}\nAlong with the identity $\\xi_0^2+3\\sigma^2\/2\\theta=1$, we have\n\\begin{align*}\n\t\\mathbf{E}Y^3 &= \\xi_0 + O(h), \\quad\n\t\\mathbf{E}Y^5 = \\left(1 + 4\\frac{\\sigma^2}{2\\theta} \n\t- 6\\left(\\frac{\\sigma^2}{2\\theta}\\right)^2 \\right) \\xi_0 + O(h),\\\\\n\t\\mathbf{E}Y^7 &= \\left(1 + 12\\frac{\\sigma^2}{2\\theta} \n\t+ 6\\left(\\frac{\\sigma^2}{2\\theta}\\right)^2\n\t- 48\\left(\\frac{\\sigma^2}{2\\theta}\\right)^3\\right) \\xi_0 + O(h).\n\\end{align*}\nThen $\\mathbf{Cov}(V_y(Y), V(Y))=6(\\sigma^2\/2\\theta)^2(1-2\\sigma^2\/2\\theta)\\xi_0+O(h)$. \nThe $O(h)$ terms in (\\ref{eq:compatibility condition for small h}) imply \n$\\xi_1 = 3\\eta(\\sigma^2\/2\\theta)^2(1-2\\sigma^2\/2\\theta)\/\\xi_0$.\n\n\n\n\n\n\\section{Proofs in Section \\ref{sec:reduced LDP}}\n\n\\subsection{Proof of Lemma \\ref{lma:the optimal g^0 and g^1}}\n\\label{pf:the optimal g^0 and g^1}\n\nWe note that $p_t = -p_y \\frac{d}{dt}a$ and therefore \n\\begin{align*}\n\t\\phi_t &= -p_y \\frac{d}{dt}a \n\t+ h\\sum_{n=2}^{\\infty}\\frac{d}{dt}b_n\\frac{\\partial^n}{\\partial y^n}p\n\t- h\\frac{d}{dt}a\\sum_{n=2}^{\\infty}b_n\\frac{\\partial^{n+1}}{\\partial y^{n+1}}p\\\\\n\t&\\quad + h^2\\sum_{n=2}^{\\infty}\\frac{d}{dt}c_n\\frac{\\partial^n}{\\partial y^n}p\n\t- h^2\\frac{d}{dt}a\\sum_{n=2}^{\\infty}c_n\\frac{\\partial^{n+1}}{\\partial y^{n+1}}p.\n\\end{align*}\nAfter collecting $O(1)$ terms in \n(\\ref{eq:Fokker-Planck equation with the driving force}) and integrating over $y$, \nwe have \n\\[\n\t-p \\frac{d}{dt}a = \\frac{1}{2}\\sigma^2 p_y + \\theta(y-a)p + p g^{(0)} = p g^{(0)}.\n\\]\nThen $g^{(0)} = -\\frac{d}{dt}a$.\n\nNow we collect $O(h)$ terms in (\\ref{eq:Fokker-Planck equation with the driving force}) \nand integrating over $y$. We get \n\\begin{multline*}\n\t\\sum_{n=1}^{\\infty}\\frac{d}{dt}b_{n+1}\\frac{\\partial^n}{\\partial y^n}p\n\t- \\frac{d}{dt}a\\sum_{n=2}^{\\infty}b_n\\frac{\\partial^n}{\\partial y^n}p\n\t= \\frac{1}{2}\\sigma^2\\sum_{n=2}^{\\infty}b_n\\frac{\\partial^{n+1}}{\\partial y^{n+1}}p\\\\\n\t+ \\theta(y-a)\\sum_{n=2}^{\\infty}b_n\\frac{\\partial^n}{\\partial y^n}p\n\t+ \\sum_{n=0}^3 \\delta_n\\frac{\\partial^n}{\\partial y^n}p\n\t+ g^{(0)} \\sum_{n=2}^{\\infty}b_n\\frac{\\partial^n}{\\partial y^n}p\n\t+ \\sum_{n=0}^{\\infty}\\beta_n \\frac{\\partial^n}{\\partial y^n}p.\n\\end{multline*}\nUsing the fact that \n\\[\n\t\\frac{1}{2}\\sigma^2\\frac{\\partial^{n+1}}{\\partial y^{n+1}}p\n\t= -\\theta(y-a)\\frac{\\partial^n}{\\partial y^n}p\n\t-n\\theta \\frac{\\partial^{n-1}}{\\partial y^{n-1}}p,\n\\]\nwe have \n\\[\n\t\\sum_{n=1}^{\\infty}\\frac{d}{dt}b_{n+1}\\frac{\\partial^n}{\\partial y^n}p\n\t= -\\theta\\sum_{n=1}^{\\infty}(n+1)b_{n+1}\\frac{\\partial^n}{\\partial y^n}p\\\\\n\t+ \\sum_{n=0}^3 \\delta_n\\frac{\\partial^n}{\\partial y^n}p\n\t+ \\sum_{n=0}^{\\infty}\\beta_n \\frac{\\partial^n}{\\partial y^n}p,\n\\]\nand the optimal $\\beta_n$ are obtained by comparing the coefficients.\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lma:the optimal g^2}}\n\\label{pf:the optimal g^2}\n\nLet $\\psi^{(2)}$ denote the anti-derivative of $q^{(2)}$ that vanishes at $-\\infty$. After collecting \n$O(h^2)$ terms in (\\ref{eq:Fokker-Planck equation with the driving force}) and integrating over $y$. We have \n\\begin{equation}\n\t\\label{eq:eqn for the optimal f^2}\n\t\\psi^{(2)}_t = \\frac{1}{2}\\sigma^2 q^{(2)}_y + \\theta(y-a)q^{(2)} + U(y)q^{(1)} + q^{(2)} g^{(0)}\n\t+ q^{(1)} g^{(1)} + p g^{(2)}.\n\\end{equation}\nNote that $p g^{(2)} = \\sum_{n=0}^\\infty \\gamma_n \\frac{\\partial^n}{\\partial y^n}p$, \nso $\\gamma_0$ is obtained by integrating (\\ref{eq:eqn for the optimal f^2}) from \n$y=-\\infty$ to $y=\\infty$. Then we have \n$\\gamma_0 = -\\langle q^{(1)}, U(y)+g^{(1)} \\rangle$.\n\n\n\n\n\\subsection{Proof of Proposition \\ref{prop:inf_A I_h up to O(h)}}\n\\label{pf:inf_A I_h up to O(h)}\n\nWe write $a(t) = a_0(t) + h a_1(t) + O(h^2)$ with $a_0(t) = 2\\xi_0 t\/T - \\xi_0$ \nand $a_1(t) = 2\\xi_1 t\/T - \\xi_1$. Then we put $a(t)$ into \n(\\ref{eq:variational problem of a}) and we have\n\\[\n\t\\inf_{\\phi\\in A}I_h(\\phi) \n\t= \\frac{1}{2\\sigma^2} \\int_0^T \\left\\{ (\\frac{d}{dt}a_0)^2 \n\t+ 2h(\\frac{d}{dt}a_0)(a_0^3+(3\\frac{\\sigma^2}{2\\theta}-1)a_0 \n\t+ \\frac{d}{dt}a_1)\\right\\} dt + O(h^2).\n\\]\nWe note that $\\frac{d}{dt}a_0$ is a constant, and $a_0(t)$ and $a_0^3(t)$ \nare odd functions with respect to $t=T\/2$. Then \n\\begin{multline*}\n\t\\inf_{\\phi\\in A}I_h(\\phi) \n\t= \\frac{1}{2\\sigma^2} \\int_0^T \\left\\{ \\left(\\frac{d}{dt}a_0\\right)^2 \n\t+ 2h\\frac{d}{dt}a_0\\frac{d}{dt}a_1 \\right\\} dt + O(h^2)\\\\\n\t= \\frac{1}{2\\sigma^2} \\int_0^T \\left\\{ \\left(\\frac{2\\xi_0}{T}\\right)^2 \n\t+ 2h\\frac{2\\xi_0}{T}\\frac{2\\xi_1}{T} \\right\\} dt + O(h^2)\n\t= \\frac{2\\xi_0}{\\sigma^2 T}(\\xi_0 + 2h\\xi_1) + O(h^2).\n\\end{multline*}\n\\section{Proofs in Section \\ref{sec:small h}}\n\n\n\\subsection{Proof of Theorem \\ref{thm:LD for zero h and Gaussian path}}\n\\label{pf:LD for zero h and Gaussian path}\n\nWe prove it in three steps. The first step is to show that there exists a uniform lower \nbound of $I_0(\\phi)$, for all $\\phi \\in A$.\n\\begin{lemma}\n\tIf $h=0$, then $\\inf_{\\phi \\in A} I_0(\\phi) \\geq 2\\xi_0^2\/(\\sigma^2 T)$.\n\\end{lemma}\n\\begin{proof}\n\tFor any $\\phi \\in A$, $a(t)$ denotes $\\int y \\phi(t,dy)$. We observe that\n\t\\[\n\t\tJ_h(\\phi)\n\t\t= \\sup_{f: \\langle \\phi, f_y^2 \\rangle \\neq 0}\n\t\t\\langle \\phi_t - \\mathcal{L}_\\phi^*\\phi, f \\rangle^2\n\t\t\/ {\\langle \\phi, f_y^2 \\rangle}\n\t\t\\overset{f \\equiv y}{\\geq} \n\t\t\\langle \\phi_t - \\mathcal{L}_\\phi^*\\phi, y\\rangle^2,\n\t\\]\n\tbecause $\\langle \\phi , 1\\rangle=1$. Note that \n\t$\\langle\\phi_t, y\\rangle=\\frac{d}{dt}\\langle\\phi, y\\rangle=\\frac{d}{dt}a(t)$, and\n\t\\begin{equation*}\n\t\t\\langle \\mathcal{L}_\\phi^*\\phi, y \\rangle \n\t\t= \\langle \\frac{1}{2}\\sigma^2 \\phi_{yy}\n\t\t+ \\theta\\frac{\\partial}{\\partial y} \\left[ (y-a(t)) \\phi\\right], y\\rangle\n\t\t= -\\theta \\langle (y-a(t))\\phi, 1 \\rangle = 0.\n\t\\end{equation*}\n\tThen after taking the infimum over $\\phi \\in A$, we have \n\t\\[\n\t\t\\inf_{\\phi \\in A} I_0(\\phi) \\geq \n\t\t\\inf_{\\phi \\in A} \\frac{1}{2\\sigma^2}\\int_0^T\\left(\\frac{d}{dt}a\\right)^2 dt\n\t\t= \\inf_{\\substack{a(t):0 \\leq t \\leq T\\\\ a(0)=-\\xi_0\\\\ a(T)=\\xi_0}}\n\t\t\\frac{1}{2\\sigma^2}\\int_0^T\\left(\\frac{d}{dt}a\\right)^2 dt\n\t\t= \\frac{2\\xi_0^2}{\\sigma^2 T}.\n\t\\]\n\tThe last equality is obtained by a simple calculus of variation with the optimal \n\tpath $a(t)= 2\\xi_0 t\/T - \\xi_0$.\n\\end{proof}\n\nThe second step is to show that $I_0(p^e) = 2\\xi_0^2\/(\\sigma^2 T)$. \nThen $\\inf_{\\phi \\in A} I_0(\\phi) =  2\\xi_0^2\/(\\sigma^2 T)$ and \ntherefore $p^e$ is a minimizer for (\\ref{eq:general LD}).\n\\begin{lemma}\n\tIf $h=0$, and \n\t\\[\n\t\tp^e(t,y) = \\frac{1}{\\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\t\\exp\\left\\{ -\\frac{(y - a^e(t))^2}{2\\frac{\\sigma^2}{2\\theta}} \\right\\}, \\quad\n\t\ta^e(t)= \\frac{2\\xi_0}{T} t - \\xi_0,\n\t\\]\n\tthen $p^e \\in A$ and $I_0(p^e) =  2\\xi_0^2\/(\\sigma^2 T)$.\n\\end{lemma}\n\\begin{proof}\n\tBy reading (\\ref{eq:Fokker-Planck equation with the driving force}) with $\\phi=p^e$ \n\tand $h=0$, we have $p^e_t = \\mathcal{L}_{p^e}^{*}p^e + (p^e g)_y$.\n\tOne can easily check that $\\mathcal{L}_{p^e}^{*}p^e = 0$ and \n\t$p^e_t = -p^e_y\\frac{d}{dt}a^e(t)$. Then we have $g=-\\frac{d}{dt}a^e(t)$ and \n\tby (\\ref{eq:I_h in terms of the driving force}),\n\t\\[\n\t\tI_0(p^e) = \\frac{1}{2\\sigma^2} \\int_0^T \\langle p^e, g^2\\rangle dt\n\t\t= \\frac{1}{2\\sigma^2} \\int_0^T \\left(\\frac{d}{dt}a^e\\right)^2 dt \n\t\t= \\frac{2\\xi_0^2}{\\sigma^2 T}.\n\t\\]\n\\end{proof}\n\nFinally we prove that for $h=0$, the minimizer $p^e$ is unique. \n\\begin{lemma}\n\tFor $h=0$, $p^e$ is the unique minimizer for (\\ref{eq:general LD}).\n\\end{lemma}\n\\begin{proof}\n\tFrom the previous lemmas, we find that if $\\phi$ is a minimizer then \n\t$a(t)=\\int y \\phi(t,dy)$ must be $a^e(t)$, and $f = -\\frac{d}{dt}a^e(t) y$ \n\tis a global maximizer of $J_0(\\phi,\\cdot)$. Then for any test function $\\tilde{f}$, \n\t$\\frac{d}{d\\epsilon}J_0(\\phi,-\\frac{d}{dt}a^e(t) y + \\epsilon \\tilde{f})=0$ at \n\t$\\epsilon=0$. By a simple calculus of variations, $\\phi$ satisfies the linear \n\tparabolic PDE:\n\t\\[\n\t\t\\phi_t = \\frac{1}{2}\\sigma^2 \\phi_{yy} \n\t\t+ \\theta\\frac{\\partial}{\\partial y}\\left[ (y-a^e(t))\\phi\\right]\n\t\t- \\frac{d}{dt}a^e(t) \\phi_y,\t\n\t\\]\n\twith the initial condition $\\phi(0)=u_{-\\xi_0}^e$, and that implies \n\tthe uniqueness of the minimizer, which is $p^e$.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:cotinuity for small h} (Upper Bounds)}\n\\label{pf:cotinuity for small h, upper bounds}\n\nDefine the test function:\n\\[\n\tp^u(t,y) = \\frac{1}{\\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\\exp\\left\\{ -\\frac{(y - a^u(t))^2}{2\\frac{\\sigma^2}{2\\theta}} \\right\\},\n\t\\quad a^u(t) = \\frac{2\\xi_b}{T}t - \\xi_b.\n\\]\nWe recall that from (\\ref{eq:xi equals m(xi)}) and \n(\\ref{eq:explicit solution for xi equal to m(xi)}), $\\xi_b$ depends on $h$ and \n$\\xi_b\\rightarrow \\xi_0$ as $h\\rightarrow 0$.\n\n\\begin{proposition}\n\t\\label{prop:upper bound for I}\n\tFor any $\\epsilon>0$, then for all sufficiently small $h$, \n\t\\begin{equation}\n\t\\label{eq:upper bound for I}\n\t\\inf_{\\phi\\in A}I_h(\\phi) \\leq \\frac{1}{2\\sigma^2} \\int_0^T \n\t\\langle p^u, ( \\frac{d}{dt}a^u - h(y^3-y) )^2 \\rangle dt + \\epsilon.\n\t\\end{equation}\n\\end{proposition}\nIt is not difficult to see that the first term of the right hand side of (\\ref{eq:upper bound for I}) \n is equal to  $2\\xi_b^2\/(\\sigma^2 T)$ up to a term of order $h$ as $h\\rightarrow 0$.\n\n\\begin{proof}\n\tWe construct the test function $\\phi^u\\in A$ as follows:\n\t\\[\n\t\t\\phi^u(t)=\n\t\t\\begin{cases}\n\t\t\t(1-\\frac{t}{\\delta T})u^e_{-\\xi_b} + \\frac{t}{\\delta T} p^u(t), \n\t\t\t& t\\in[0,\\delta T],\\\\\n\t\t\tp^u(t), & t\\in(\\delta T, T-\\delta T),\\\\\n\t\t\t(1-\\frac{t-(T-\\delta T)}{\\delta T})p^u(t) \n\t\t\t+ \\frac{t-(T-\\delta T)}{\\delta T} u^e_{\\xi_b}, & t\\in[T-\\delta T, T],\n\t\t\\end{cases}\n\t\\]\n\twhere $\\delta T$ will be determined later. Note that \n\t$\\inf_{\\phi\\in A}I_h(\\phi) \\leq I_h(\\phi^u)$ so we just need to compute \n\t$I_h(\\phi^u)$. Let $g^u$ satisfy \n\t(\\ref{eq:Fokker-Planck equation with the driving force}) for $\\phi=\\phi^u$. For \n\t$t\\in(\\delta T, T-\\delta T)$, $\\phi^u(t)=p^u(t)$, and it is easy to see that \n\t$p^u_t=-\\frac{d}{dt}a^u p^u_y$ and $\\mathcal{L}_{p^u}^{*}p^u=0$. Therefore for \n\t$t\\in(\\delta T, T-\\delta T)$, $g^u = -\\frac{d}{dt}a^u - h(y^3-y)$ by \n\t(\\ref{eq:Fokker-Planck equation with the driving force}). From \n\t(\\ref{eq:I_h in terms of the driving force}), we have \n\t\\begin{multline*}\n\t\tI_h(\\phi^u) = \\frac{1}{2\\sigma^2} \n\t\t\\left(\\int_0^{\\delta T} + \\int_{\\delta T}^{T-\\delta T} \n\t\t+ \\int_{T-\\delta T}^T\\right) \\langle \\phi^u, (g^u)^2\\rangle dt\\\\\n\t\t\\leq \\frac{1}{2\\sigma^2} \\int_0^T \n\t\t\\langle p^u, ( -\\frac{d}{dt}a^u - h(y^3-y) )^2 \\rangle dt\n\t\t+ \\frac{1}{2\\sigma^2} \\left(\\int_0^{\\delta T} + \\int_{T-\\delta T}^T\\right) \n\t\t\\langle \\phi^u, (g^u)^2\\rangle dt.\n\t\\end{multline*}\n\tThe rest is to show that for any $\\epsilon>0$, there exists a sufficiently small $h$ such that the last \n\tterm in the last equation is bounded by $\\epsilon$. It suffices to show that for any $\\delta T>0$, we \n\tcan choose a sufficiently small $h$ such that $\\langle \\phi^u, (g^u)^2\\rangle$ is bounded by a \n\t$\\delta T$-independent constant $c^u>0$ for $t\\in[0,\\delta T]\\cup[T-\\delta T, T]$. If so, then let \n\t$\\delta T <\\epsilon\\sigma^2\/c^u$ and\n\t\\[\n\t\t\\frac{1}{2\\sigma^2} \\left(\\int_0^{\\delta T} + \\int_{T-\\delta T}^T\\right) \n\t\t\\langle \\phi^u, (g^u)^2\\rangle dt \\leq \\frac{1}{2\\sigma^2}(2\\delta T)c^u < \\epsilon,\n\t\\]\n\tfor sufficiently small $h$.\n\t\t\n\tFor $t\\in[0,\\delta T]$, because $\\phi^u$ is simply the convex combination of $u^e_{-\\xi_b}$ and $p^u$, \n\t$\\phi^u$ can be bounded by a $\\delta T$-independent constant. To compute $g^u$ from \n\t(\\ref{eq:Fokker-Planck equation with the driving force}), it is also easy to see that \n\t$\\mathcal{L}_{\\phi^u}^{*}\\phi^u$ and $\\mathcal{M}^{*}\\phi^u$ can be bounded by $\\delta T$-independent \n\tconstants. The only term we need to worry is  $(p^u(t)-u^e_{-\\xi_b})\/\\delta T$ from computing \n\t$\\phi^u_t(t)$. However, $p^u(t)$ is differentiable at $t=0$ and $p^u(0)\\rightarrow u^e_{-\\xi_b}$ as \n\t$h\\rightarrow 0$ so we can bound $(p^u(t)-u^e_{-\\xi_b})\/\\delta T$ by a $\\delta T$-independent constant \n\twith suitable $h$. Thus $g^u$ is bounded independently of $\\delta T$ and so we can find a \n\t$\\delta T$-independent constant $c^u>0$ such that $\\langle \\phi^u, (g^u)^2\\rangle<c^u$.\n\t\n\tThe same argument works for $t\\in[T-\\delta T,T]$ and we have the desired result.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:cotinuity for small h} (Lower Bounds)}\n\\label{pf:cotinuity for small h, lower bounds}\n\nFrom (\\ref{eq:upper bound for I}), there exists some constant $C$ such that \n$\\inf_{\\phi\\in A}I_h(\\phi)\\leq C$ for all $h\\leq h_0$. Then we can assume that \n$I_h(\\phi)\\leq C$ for all $\\phi\\in A$ and all $h\\leq h_0$ without loss of generality. \nThe following lemma shows that the first and second moments of all $\\phi\\in A$ are \nuniformly bounded.\n\\begin{lemma}\n\t\\label{lma:uniform bound for first and second moments}\n\tGiven $C>0$, there exists $R>0$ such that for any $\\phi\\in A$ with $I_h(\\phi)\\leq C$ \n\tfor some $h\\geq 0$, then\n\t\\[\n\t\t\\sup_{t\\in[0,T]}\\langle\\phi(t),y\\rangle^2 \n\t\t\\leq \\sup_{t\\in[0,T]}\\langle\\phi(t),y^2\\rangle \n\t\t\\leq R.\n\t\\]\n\\end{lemma}\n\\begin{proof}\n\tRecall that $M_R(\\mathbb{R})=\\{\\phi\\in M_1(\\mathbb{R}),\\int\\varphi(y)\\phi(dy)\\leq \n\tR\\}$ and $M_\\infty(\\mathbb{R})=\\cup_{R>0}M_R(\\mathbb{R})$ with the inductive \n\ttopology. Here we focus on the case that $\\varphi=1+y^2$ in order to obtain the \n\tuniform result, and let $M_R^2(\\mathbb{R})$ and $M_\\infty^2(\\mathbb{R})$ denote the \n\tspaces with the quadratic Lyapunov function $\\varphi$.\n\t\n\tThe proof is an application of Theorem 5.1(c), Theorem 5.3 and Lemma 5.5 of \n\t\\cite{Dawson1987}. By Theorem 5.1(c), if $\\phi\\in C([0,T],M_\\infty^2(\\mathbb{R}))$ \n\twith $\\phi(0)=u^e_{-\\xi_b}$ and $I_h(\\phi)\\leq C$ for some $h\\geq 0$, then $\\phi$ is \n\tin an $h$-dependent compact set $K$. By Theorem 5.3 the compact set $K$ is contained \n\tin $C([0,T],M_R^2(\\mathbb{R}))$ for an $h$-dependent $R>0$. Finally, by Lemma 5.5 \n\tand Theorem 5.1(c), it suffices to let $R\\geq e^{\\lambda T}(C+r)$, where $r$ and \n\t$\\lambda$ satisfy\n\t\\[\n\t\tr \\geq 2\\int \\varphi(y)u^e_{-\\xi_b}(y)dy,\\quad\n\t\t\\lambda \\geq \\sup_{\\mu\\in M_1(\\mathbb{R})}\n\t\t\\langle\\mu,\\mathcal{L}_\\mu\\varphi+h\\mathcal{M}\\varphi\n\t\t+\\frac{1}{2}\\varphi_y^2\\rangle\n\t\t\/\\langle\\mu,\\varphi\\rangle,\n\t\\]\n\twith $\\varphi(y)=1+y^2$. Obviously we can find the uniform $r$ and $\\lambda$ for all\n\t$h\\geq 0$ and also the uniform $R$. Then any $\\phi$ of interest are in \n\t$C([0,T],M_R^2(\\mathbb{R}))$ and thus have the uniform bounded first and second \n\torder moments.\n\\end{proof}\n\nNow we derive that lower bound. The key idea is that because we have the universal upper \nbound for the first and second moments of all $\\phi\\in A$ and for all $h\\leq h_{0}$, \nChebyshev's inequality implies the uniform convergence. \n\n\\begin{proposition}\n\t\\label{prop:lower bound for I}\n\tFor any $\\epsilon>0$, then for all sufficiently small $h$, \n\t\\begin{equation}\n\t\t\\label{eq:lower bound for I}\n\t\t\\inf_{\\phi\\in A}I_h(\\phi) \\geq \\frac{1}{2\\sigma^2} \\int_0^T \n\t\t\\langle p^u, ( \\frac{d}{dt}a^u - h(y^3-y) )^2 \\rangle dt - \\epsilon.\n\t\\end{equation}\n\\end{proposition}\n\\begin{proof}\n\tDefine $f^M=\\iota*\\hat{f}^M$, where $\\hat{f}^M$ is a piecewise linear function and \n\t$\\iota$ is the standard mollifier: \n\t\\[\n\t\t\\hat{f}^M(y)\n\t\t=\\begin{cases}\n\t\t\ty,\t\t& y\\in(-M,M)\\\\\n\t\t\t-y+2M, \t& y\\in[M,2M]\\\\\n\t\t\t-y-2M, \t& y\\in[-2M,-M]\\\\\n\t\t\t0, \t\t& \\text{otherwise}\n\t\t\\end{cases}, \\quad \n\t\t\\iota(y)\n\t\t=\\begin{cases}\n\t\t\tZ\\exp(\\frac{1}{y^2-1}), & y^2 < 1\\\\\n\t\t\t0, \t\t\t\t\t\t& \\text{otherwise.}\n\t\t\\end{cases}\n\t\\]\n\tThen $f^M$ is a smooth function with the compact support $[-2M-1,2M+1]$. In \n\taddition, $f^M(y)\\equiv y$ on $(-M+1,M-1)$, $|f_x^M|\\leq 1$, and $|f_{xx}^M|$ is \n\tuniformly bounded for all $M$ and is nonzero only on $\\cup_{i=-2}^2 (iM-1,iM+1)$.\n\n\tBecause for all $\\phi\\in A$, $\\langle \\phi(t),(f_y^M)^2\\rangle \\leq 1$,\n\twe can estimate the rate function:\n\t\\begin{align*}\n\t\tI_h(\\phi) &\\geq \\frac{1}{2\\sigma^2}\\int_0^T\n\t\t\\langle \\phi_t - \\mathcal{L}_\\phi^*\\phi \n\t\t- h\\mathcal{M}^*\\phi, f^M \\rangle^2 dt\\\\\n\t\t&\\geq \\frac{1}{2\\sigma^2 T}\\left(\\int_0^T\\langle \\phi_t \n\t\t- \\mathcal{L}_\\phi^*\\phi - h\\mathcal{M}^*\\phi, f^M \\rangle dt\\right)^2.\n\t\\end{align*}\n\tThen we estimate the integrand term by term. By Lemma \n\t\\ref{lma:uniform bound for first and second moments}, the following convergences are \n\tall uniform in $\\phi\\in A$ and $h\\leq h_0$.\n\n\tFirst we have \n\t\\[\n\t\t\\int_0^T \\langle \\phi_t, f^M \\rangle dt\n\t\t= \\langle u_{\\xi_b}^e, f^M \\rangle - \\langle u_{-\\xi_b}^e, f^M \\rangle.\n\t\\]\n\t$u_{\\pm\\xi_b}^e$ are exponentially decaying functions so \n\t$\\langle u_{\\pm\\xi_b}^e, f^M \\rangle $ converges to $\\pm\\xi_{b}$ rapidly as \n\t$M\\rightarrow\\infty$.\n\n\tWe note that $\\langle \\mathcal{L}_\\phi^* \\phi, f^M\\rangle \n\t=\\sigma^2\\langle \\phi,f_{yy}^M \\rangle\/2 - \\theta\\langle \\phi,(y-a)f^M_y \\rangle$. \n\tBy reading the properties of $f_{yy}^M$ and Chebyshev's inequality, we have \n\t$\\langle \\phi,f_{yy}^M\\rangle \\rightarrow 0$ as $M\\rightarrow\\infty$. \n\tWe write $\\langle \\phi,(y-a)f^M_y\\rangle$ as \n\t\\[\n\t\t\\langle \\phi,(y-a)f^M_y\\rangle \n\t\t= a(1-\\langle \\phi,f^M_y\\rangle) + (\\langle \\phi, y f^M_y\\rangle -a).\n\t\\]\n\tSince $a$ is bounded and $\\langle \\phi,f^M_y \\rangle \\rightarrow 1$\tas \n\t$M\\rightarrow\\infty$, $a(1-\\langle \\phi,f^M_y \\rangle)\\rightarrow 0$ as \n\t$M\\rightarrow\\infty$. We see that \n\t\\begin{align*}\n\t\t|\\langle \\phi, y f^M_y \\rangle -a|^2\n\t\t& \\leq \\left( 2 \\int_{(-M+1,M+1)^c} |y| \\phi(dy) \\right)^2\\\\\n\t\t& \\leq 4 \\int_{(-M+1,M+1)^c} y^2 \\phi(dy) \\int_{(-M+1,M+1)^c} \\phi(dy).\n\t\\end{align*}\n\tAgain by Chebyshev's inequality, the right hand side vanishes as \n\t$M\\rightarrow\\infty$.\n\n\tFinally we estimate $\\langle \\mathcal{M}^*\\phi, f^M \\rangle $. Since $f^M$ is \n\tcompactly supported,\n\t\\[\n\t\t|\\langle \\mathcal{M}^* \\phi, f^M \\rangle |\n\t\t= |\\langle \\phi,(y^3-y)f^M_y \\rangle | \\leq (2M+1)^3 + (2M+1).\n\t\\]\n\tFor a fixed $M$, we can choose a sufficiently small $h$ such that \n\t$h|\\langle \\mathcal{M}^* \\phi, f^M\\rangle |$ is small.\n\n\tConsequently, for any $\\epsilon>0$, we can first choose a sufficiently large $M$ and \n\tthen there exists a sufficiently small $h$ such that\n\t\\[\n\t\t\\inf_{\\phi\\in A} I_h(\\phi) \\geq \\frac{2\\xi_b^2}{\\sigma^2 T} - \\epsilon.\n\t\\]\n\\end{proof}\n\n\n\n\\subsection{Proof of Lemma \\ref{lma:lower bound of I_h over B_delta}}\n\\label{pf:lower bound of I_h over B_delta}\nIt suffices to show the case that $\\delta=1\/n$. For each $n$, let $\\phi_n\\in B_{1\/n}$ and $t_n\\in(0,T]$ \nsuch that $\\rho(\\phi_n(t_n),u^e_{\\xi_b})<\\delta$ and \n$\\inf_{\\phi\\in B_{1\/n}}I_h(\\phi)\\leq I_h(\\phi_n)<\\inf_{\\phi\\in B_{1\/n}}I_h(\\phi)+1\/n$; $\\{I_h(\\phi_n)\\}$ are \nbounded from above by $\\inf_{\\phi\\in B}I_h(\\phi)+1<\\infty$. Let $\\{t_{n_k}\\}$ be a convergent subsequence of \n$\\{t_n\\}$. Because $I_h$ is a good rate function, and by Proposition B.13 of \\cite{Gartner1988}, compactness \nis equivalent to sequentially compactness in $C([0,T],M_\\infty(\\mathbb{R}))$, $\\{\\phi_{n_k}\\}$ has a \nconvergent subsequence $\\{\\phi_{n_{k'}}\\}$ whose limit $\\phi^*$ is in $A(t^*)$ where $t^*=\\lim t_{n_k}$. As \n$I_h$ is lower semicontinuous, then\n\\[\n\t\\lim_n\\inf_{\\phi\\in B_{1\/n}}I_h(\\phi) = \\liminf_{n_{k'}} I_h(\\phi_{n_{k'}}) \n\t\\geq I_h(\\phi^*) \\geq \\inf_{\\phi\\in A(t^*)}I_h(\\phi) \\geq \\inf_{\\phi\\in B}I_h(\\phi).\n\\]\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A Reduced Large Deviations Principle for Small $h$}\n\\label{sec:reduced LDP}\n\nIn Section \\ref{sec:small h analysis}, we show that the large deviation problem $\\inf_{\\phi\\in A}I_h(\\phi)$ \nis continuous in $h$ so that we have the upper and lower bounds for $\\inf_{\\phi\\in A}I_h(\\phi)$ when $h$ is \nsmall. In this section, we analyze with a formal expansion the optimal path for $\\inf_{\\phi\\in A}I_h(\\phi)$ \nby assuming that it is of the form $p^e + O(h)$, motivated by the fact\nthat the optimal path is $p^e$ for $h=0$. In this way, \nwe can obtain a  reduced large deviations principle (a reduced Freidlin-Wentzell theory) for the systemic \nrisk. That is, we obtain a reduced rate function corresponding to a finite dimensional system after ignoring \nhigher order terms. The reduced rate function has all relevant information up to $O(h^2)$ \nterms, and we also need to expand $\\phi$ to $O(h^2)$.\n\n\nWe assume that the optimal $\\phi = p + hq^{(1)} + h^2 q^{(2)} + \\ldots$, where \n\\[\n\tp(t,y) = \\frac{1}{\\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\\exp\\left\\{ -\\frac{(y - a(t))^2}{2\\frac{\\sigma^2}{2\\theta}} \\right\\},\n\t\\quad a(t) = \\langle \\phi, y \\rangle.\n\\]\nIn other words, we let the first moment of $\\phi$ be determined by $a(t)$, and from the zero $h$ case we \nknow that $a(t)=a^e(t)+O(h)$. From the form of $p$ and \n(\\ref{eq:Fokker-Planck equation with the driving force}), a natural parameterization for $q^{(1)}$ and $q^{(2)}$ is \nthe Hermite expansion\n\\[\n\tq^{(1)}(t,y) = \\sum_{n=2}^{\\infty}b_n(t)\\frac{\\partial^n}{\\partial y^n}p(t,y), \\quad\n\tq^{(2)}(t,y) = \\sum_{n=2}^{\\infty}c_n(t)\\frac{\\partial^n}{\\partial y^n}p(t,y).\n\\]\nNote that by the properties of $p$ and $a(t)$, $\\langle q^{(1)},y^n\\rangle = \\langle q^{(2)},y^n\\rangle = 0$ \nfor $n=0,1$ so we can start the Hermite expansion from $n=2$.\n\nThe formal expansion result of this section is that if the optimal $\\phi = p + hq^{(1)} + h^2 q^{(2)}$, then \n\\begin{equation}\n\t\\label{eq:reduced Freidlin-Wentzell}\n\t\\inf_{\\phi\\in A}I_h(\\phi) \\approx \n\t\\inf_{\\substack{a(t): 0\\leq t\\leq T\\\\a(0)=-\\xi_b\\\\a(T)=\\xi_b}} \n\t\\frac{1}{2\\sigma^2} \\int_0^T \\left(\\frac{d}{dt}a + h (a^3 + 3\\frac{\\sigma^2}{2\\theta}a - a)\\right)^2 dt,\n\\end{equation}\nfor small $h$. Note that $a(t)=\\langle \\phi, y \\rangle=\\bar{x}(t)$. The right hand side of \n(\\ref{eq:reduced Freidlin-Wentzell}) is an one-dimensional variational problem that has the form of a rate \nfunction of the Freidlin-Wentzell theory. \nIn fact, the right side of (\\ref{eq:reduced Freidlin-Wentzell}) is the large deviations variational problem for\nthe rate function of\nthe small-noise stochastic differential equation\n\\begin{equation}\n\t\\label{eq:approximate sde for x_bar}\n\td\\bar{x}(t) = -h\\left[\\bar{x}^3(t) - \\left(1-\\frac{3\\sigma^2}{2\\theta}\\right)\\bar{x}(t)\\right] dt \n\t+ \\epsilon \\sigma dw(t)\n\\end{equation}\nwhere here $\\epsilon =1\/\\sqrt{N}$ is small. Note that ${3\\sigma^2}\/{2\\theta}<1$, as assumed above, and \ntherefore (\\ref{eq:approximate sde for x_bar}) also represents a bi-stable structure. In the remainder of \nthis section we describe how this result is obtained by formal expansions and then in Section \n\\ref{sec:ldprob_reduced} we show how we recover from (\\ref{eq:reduced Freidlin-Wentzell}) the main result of \nthe paper stated in the previous section.\n\nAn important remark about the expansion is that the Hermite functions are a basis of the $L^2$ space and \nthus $p + hq^{(1)} + h^2 q^{(2)}$ is generally a signed measure. However, if $q^{(1)}$ and $q^{(2)}$ can be \nexpressed as the linear combinations of finite Hermite functions, then we can see that for any $\\epsilon>0$, \nthere exists a sufficiently small $h$ such that the negative part of $p + hq^{(1)} + h^2 q^{(2)}$ is less \nthan $\\epsilon$.\n\n\n\n\n\\subsection{Optimization over $g$}\n\nThe first step in finding the optimal $\\phi = p + h q^{(1)} + h^2 q^{(2)}$ is determining \nthe optimal $g$ by using (\\ref{eq:Fokker-Planck equation with the driving force}) \nfor $\\phi$. Once we obtain $g$, we can compute $I_h(\\phi)$ by using \n(\\ref{eq:I_h in terms of the driving force}). It is also natural to assume that \n$g = g^{(0)} + h g^{(1)} + h^2 g^{(2)}$ along with the Hermite expansion:\n\\[\n\tg^{(0)} = p^{-1}\\sum_{n=0}^{\\infty}\\alpha_n(t)\\frac{\\partial^n}{\\partial y^n}p, \\quad\n\tg^{(1)} = p^{-1}\\sum_{n=0}^{\\infty}\\beta_n(t) \\frac{\\partial^n}{\\partial y^n}p, \\quad\n\tg^{(2)} = p^{-1}\\sum_{n=0}^{\\infty}\\gamma_n(t)\\frac{\\partial^n}{\\partial y^n}p.\n\\]\nIn addition, since $\\langle q^{(1)},y\\rangle=\\langle q^{(2)},y\\rangle=0$, we can see that \n$\\phi = p + h q^{(1)} + h^2 q^{(2)}$ satisfies\n\\[\n\t\\mathcal{L}_{\\phi}^{*}\\phi = \\mathcal{L}_{p}^{*}p +h\\mathcal{L}_{p}^{*}q^{(1)} \n\t+ h^2\\mathcal{L}_{p}^{*}q^{(2)}, \\quad\n\t\\mathcal{M}^{*}\\phi = \\mathcal{M}^{*}p + h\\mathcal{M}^{*}q^{(1)} + h^2\\mathcal{M}^{*}q^{(2)}.\n\\]\nThe force $U(y)=y^3-y$ can also be expanded in Hermite polynomials:\n\\[\n\tU(y) = p^{-1}\\sum_{n=0}^3 \\delta_n(t)\\frac{\\partial^n}{\\partial y^n}p.\n\\]\nNow everything is expanded in the orthogonal basis and we can find the optimal $g^{(0)}$ and \n$g^{(1)}$ by putting everything into \n(\\ref{eq:Fokker-Planck equation with the driving force}) and comparing coefficients.\n\\begin{lemma}\n\t\\label{lma:the optimal g^0 and g^1}\n\tWith the expansions mentioned above, the optimal $g^{(0)}$ is $-\\frac{d}{dt}a$, \n\tand the optimal $\\beta_n$ for $g^{(1)}$ are \n\t\\begin{equation}\n\t\t\\label{eq:beta_n}\n\t\t\\beta_n =\n\t\t\\begin{cases}\n\t\t\t-\\delta_0 = -\\langle p, U(y) \\rangle,\t\t& n=0,\\\\\n\t\t\t\\frac{d}{dt}b_{n+1} + \\theta(n+1)b_{n+1} - \\delta_n,\t& 1\\leq n\\leq 3,\\\\\n\t\t\t\\frac{d}{dt}b_{n+1} + \\theta(n+1)b_{n+1},\t\t\t\t& n\\geq 4.\n\t\t\\end{cases}\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\tSee Appendix \\ref{pf:the optimal g^0 and g^1}.\n\\end{proof}\n\nIt remains to determine $g^{(2)}$. \nFrom (\\ref{eq:I_h in terms of the driving force}) we see that the only contribution of \n$g^{(2)}$ to $I_h$ up to $O(h^2)$ is \n$\\langle p, 2g^{(0)} g^{(2)} \\rangle = -2\\gamma_0\\frac{d}{dt}a$. \nThus it suffices to determine $\\gamma_0$, which can also be obtained from \n(\\ref{eq:Fokker-Planck equation with the driving force}).\n\\begin{lemma}\n\t\\label{lma:the optimal g^2}\n\tWith the expansions mentioned above, the optimal $\\gamma_0$ is \n\t\\[ \n\t\t\\gamma_0 = -\\langle q^{(1)}, U(y)+g^{(1)} \\rangle. \n\t\\]\n\\end{lemma}\n\\begin{proof}\n\tSee Appendix \\ref{pf:the optimal g^2}.\n\\end{proof}\n\n\n\n\\subsection{Optimization over $\\phi$}\n\nWe are now ready to find the optimal $\\phi$. For given $\\phi = p + hq^{(1)} + h^2q^{(2)}$ and \nthe corresponding optimal $g = g^{(0)} + hg^{(1)} + h^2g^{(2)}$, \n(\\ref{eq:I_h in terms of the driving force}) gives\n\\begin{align*}\n\tI_h(\\phi) \n\t&= \\frac{1}{2\\sigma^2} \\int_0^T \\langle p+hq^{(1)}+h^2q^{(2)}, (g^{(0)} + hg^{(1)} + h^2g^{(2)})^2 \\rangle dt\\\\\n\t&= \\frac{1}{2\\sigma^2} \\int_0^T \\langle p, (g^{(0)})^2 \\rangle dt\n\t+ \\frac{h}{2\\sigma^2} \\int_0^T \\langle p, 2 g^{(0)} g^{(1)} \\rangle dt\\\\\n\t&\\quad + \\frac{h^2}{2\\sigma^2} \\int_0^T \\left(\\langle p, (g^{(1)})^2 + 2 g^{(0)} g^{(2)} \\rangle\n\t+ \\langle q^{(1)}, 2 g^{(0)} g^{(1)} \\rangle \\right)dt + O(h^3).\n\\end{align*}\nFrom Lemma \\ref{lma:the optimal g^2}, $\\langle p, 2 g^{(0)} g^{(2)} \\rangle \n= -2g^{(0)}\\langle q^{(1)}, U(y)+g^{(1)} \\rangle$, and therefore\n\\[\n\t\\langle p, 2 g^{(0)} g^{(2)} \\rangle\n\t+ \\langle q^{(1)}, 2 g^{(0)} g^{(1)} \\rangle\n\t= -2g^{(0)} \\langle q^{(1)}, U(y) \\rangle\n\t= -2g^{(0)} \\sum_{n=2}^3 H_n \\delta_n b_n, \n\\]\nwhere $H_n(t) := \\langle p^{-1}, (\\partial^n p \/\\partial y^n)^2 \\rangle$.\nWe note that\n\\[\n\t\\langle p, 2 g^{(0)} g^{(1)} \\rangle = -2g^{(0)}\\delta_0, \\quad \n\t\\langle p,(g^{(1)})^2 \\rangle \n\t= \\delta_0^2+\\sum_{n=1}^\\infty H_n \\beta_n^2, \\quad\n\t\\langle p, (g^{(0)})^2 \\rangle = (g^{(0)})^2.\n\\]\nThen $I_h(\\phi)$ can be written as\n\\begin{align}\n\t\\label{eq:expansion of I_h}\n\tI_h(\\phi) &= \\frac{1}{2\\sigma^2} \\int_0^T (g^{(0)} - h\\delta_0)^2 dt\n\t+ \\frac{h^2}{2\\sigma^2} \\int_0^T (H_1\\beta_1^2 - 2H_2 g^{(0)}\\delta_2 b_2) dt\\\\\n\t&\\quad + \\frac{h^2}{2\\sigma^2} \\int_0^T (H_2\\beta_2^2 - 2H_3 g^{(0)}\\delta_3 b_3) dt \\notag\n\t+ \\frac{h^2}{2\\sigma^2} \\sum_{n=3}^\\infty \\int_0^T H_n\\beta_n^2 dt + O(h^3).\n\\end{align}\nWe see that $a$ and $b_n$ are coupled at the $O(h^2)$ level of \n(\\ref{eq:expansion of I_h}). However, from the results of the zero $h$ case, \n$a = a^e + O(h)$ and $p = p^e + O(h)$ so we can decouple $a$ and $b_n$ and express \nthe expanded $I_h(\\phi)$ up to $O(h^2)$ as the sum of independent terms.\n\\begin{proposition}\n\tTo order $O(h^2)$, the rate function $I_h(\\phi)$ can be written as the sum of \n\tindependent terms:\n\t\\begin{align}\n\t\t\\label{eq:I_h as the sum of independent terms}\n\t\tI_h(\\phi) \n\t\t&= \\frac{1}{2\\sigma^2} \\int_0^T (g^{(0)}-h\\delta_0)^2 dt\n\t\t+ \\frac{h^2}{2\\sigma^2} \\int_0^T (\\tilde{H}_1\\tilde{\\beta}_1^2 \n\t\t+ 2\\frac{d}{dt}a^e \\tilde{H}_2 \\tilde{\\delta}_2 b_2) dt\\\\\n\t\t&\\quad + \\frac{h^2}{2\\sigma^2} \\int_0^T (\\tilde{H}_2\\tilde{\\beta}_2^2 \n\t\t+ 2\\frac{d}{dt}a^e \\tilde{H}_3 \\tilde{\\delta}_3 b_3) dt \\notag\n\t\t+ \\frac{h^2}{2\\sigma^2} \\sum_{n=3}^\\infty \\int_0^T \\tilde{H}_n\\tilde{\\beta}_n^2 dt + O(h^3),\n\t\\end{align}\n\twhere $\\tilde{H}_n(t)=\\langle (p^e)^{-1}, \n\t(\\partial^n p^e \/\\partial y^n)^2 \\rangle$, \n\t$U(y)=(p^e)^{-1}\\sum_{n=0}^3\\tilde{\\delta}_n(t)\\frac{\\partial^n}{\\partial y^n}p^e$, \n\tand \n\t\\begin{equation}\n\t\t\\label{eq: tilde_beta_n}\n\t\t\\tilde{\\beta}_n =\n\t\t\\begin{cases}\n\t\t\t-\\tilde{\\delta}_0 = -\\langle p^e, U(y) \\rangle, & n=0,\\\\\n\t\t\t\\frac{d}{dt}b_{n+1}+\\theta(n+1)b_{n+1}-\\tilde{\\delta}_n,   & 1\\leq n\\leq 3,\\\\\n\t\t\t\\frac{d}{dt}b_{n+1} + \\theta(n+1)b_{n+1},\t\t\t\t   & n\\geq 4.\n\t\t\\end{cases}\n\t\\end{equation}\n\\end{proposition}\n\nWe can see from (\\ref{eq:I_h as the sum of independent terms}) that $q^{(2)}$ does not \nappear in terms up to $O(h^2)$.\nFrom the $h$ expansion of $u^e_{\\pm\\xi_b}$ in (\\ref{eq:equilibrium of Fokker-Planck eq}), and the fact that \n$V(y)$ is a polynomial of degree four, we have $b_{n+1}(0)=b_{n+1}(T)=0$ for $n\\geq 4$. The variational \nproblem for $b_{n+1}$ is to minimize $\\int_0^T\\tilde{H}_n \\tilde{\\beta}_n^2 dt$ where $\\tilde{\\beta}_n$ is \ngiven in terms of $b_{n+1}$ by (\\ref{eq: tilde_beta_n}). The obvious solution of this problem is $b_{n+1}=0$ \nand $\\tilde{\\beta}_n=0$ for $n \\geq 4$. Consequently, in order to find the optimal $\\phi$ for $I_h(\\phi)$ in \n(\\ref{eq:I_h as the sum of independent terms}), we may solve separately the variational problems for $a$, \n$b_1$, $b_2$ and $b_3$.\n\n\n\n\n\\subsection{Probability of Systemic Transitions for Small $h$}\n\\label{sec:ldprob_reduced}\n\n\nWe consider the small probability of systemic  transitions for large $N$ and small $h$ \nthrough the large deviation $\\inf_{\\phi\\in A}I_h(\\phi)$. Here we consider the solution up to $O(h)$ terms. \nThat is, using (\\ref{eq:I_h as the sum of independent terms}), \nwe solve the variational problem for $a(t)$:\n\\begin{equation}\n\t\\label{eq:variational problem of a}\n\t\\inf_{\\substack{a(t): 0\\leq t\\leq T\\\\a(0)=-\\xi_b\\\\a(T)=\\xi_b}} \n\t\\int_0^T (g^{(0)}-h\\delta_0)^2 dt\n\t= \\inf_{\\substack{a(t): 0\\leq t\\leq T\\\\a(0)=-\\xi_b\\\\a(T)=\\xi_b}} \n\t\\int_0^T (\\frac{d}{dt}a + h (a^3 + 3\\frac{\\sigma^2}{2\\theta}a - a))^2 dt.\n\\end{equation}\nBy simple calculus of variations methods we find the optimal $a$.\n\\begin{lemma}\n\t\\label{lma:ODE for the optimal a}\n\tThe optimal $a(t)$ for (\\ref{eq:variational problem of a}) satisfies \n\tthe second order ordinary differential equation\n\t\\[\n\t\t\\frac{d^2}{dt^2}a = h^2 (a^3 + (3\\frac{\\sigma^2}{2\\theta}-1)a) \n\t\t(3a^2+ (3\\frac{\\sigma^2}{2\\theta}-1))\n\t\\]\n\twith $a(0)=-\\xi_b$ and $a(T)=\\xi_b$. Consequently, the optimal path is \n\t\\begin{equation}\n\t\t\\label{eq:optima a(t)}\n\t\ta(t) = \\frac{2\\xi_b}{T}t - \\xi_b + O(h^2).\n\t\\end{equation}\n\\end{lemma}\n\nBy inserting (\\ref{eq:optima a(t)}) into (\\ref{eq:variational problem of a}) we\nobtain $\\inf_{\\phi\\in A}I_h(\\phi)$ up to $O(h)$.\n\\begin{proposition}\n\t\\label{prop:inf_A I_h up to O(h)}\n\tFor small $h$, the large deviations problem, $\\inf_{\\phi\\in A}I_h(\\phi)$, \n\tup to $O(h)$, is \n\t\\begin{equation}\n\t\t\\label{eq:inf_A I_h up to O(h)}\n\t\t\\inf_{\\phi\\in A}I_h(\\phi) = \\frac{2\\xi_0}{\\sigma^2 T}(\\xi_0 + 2h\\xi_1) + O(h^2),\n\t\\end{equation}\n\twhere $\\xi_b = \\xi_0 + h \\xi_1 + O(h^2)$ from \n\t(\\ref{eq:explicit solution for xi equal to m(xi)}). \n\tNote that $\\xi_1$ is positive because $2\\theta > 3\\sigma^2$.\n\\end{proposition}\n\\begin{proof}\n\tSee Appendix \\ref{pf:inf_A I_h up to O(h)}.\n\\end{proof}\n\nThe asymptotic probability of systemic transition for large $N$ and sufficiently small \n$\\delta$ and $h$ has the form \n\\[\n\t\\mathbf{P}(X_N\\in A_\\delta) \n\t\\approx \\exp\\left(-N\\inf_{\\phi\\in A}I_h(\\phi)\\right)\n\t= \\exp\\left(-N\n\t \\left\\{ \\frac{2\\xi_0}{\\sigma^2 T}(\\xi_0 + 2h\\xi_1) + O(h^2) \\right\\} \\right).\n\\]\n\\section{Numerical Simulations}\n\\label{sec:simulation}\n\nBefore going into a detailed analysis of the models, we carry out\nnumerical simulations of \n(\\ref{eq:SDE of single component}) and \n(\\ref{eq:SDE of single component, heterogeneous sensitivity}) so as to get \na quick impression of their behavior. We discretize with \na uniform time grid, and let $X_j^n$ denote the simulated $X_j$ at time $n \\Delta t$.\n\n\n\n\n\\subsection{Homogeneous Model}\n\\label{subsec:numhomo}\n\nWe simulate (\\ref{eq:SDE of single component}) using the Euler scheme\n\\begin{equation}\n\t\\label{eq:Euler method, homogeneous case}\n\tX_j^{n+1} = X_j^n - h U(X_j^n)\\Delta t + \\sigma\\Delta W_j^{n+1} \n\t+ \\theta(\\frac{1}{N}\\sum_{k=1}^N X_k^n -X_j^n)\\Delta t.\n\\end{equation}\nWe take $U(y)=y^3-y$, $=1$, $X_j^0=-1$, $\\Delta t=0.02$, and let\n$\\{\\Delta W_j^n\\}_{j,n}$ be independent Gaussian random variables with mean zero and \nvariance $\\Delta t$. In the figures presented, the dashed lines show the \nnumerical solutions of the compatibility equation (\\ref{eq:xi equals m(xi)}),\n$\\xi=m(\\xi)$. \nAs noted earlier, if $\\frac{d}{d\\xi}m(0)\\leq 1$, then\n$0=m(0)$ is the unique solution and $0$ is a stable state. Therefore we should observe \nthat the systemic risk fluctuates around $0$. If $\\frac{d}{d\\xi}m(0)>1$, there are two \nadditional non-zero solutions $\\pm\\xi_b=m(\\pm\\xi_b)$ and $\\pm\\xi_b$ are \nstable while $0$ is unstable. We also know that when $h$ is small, the condition \n$\\frac{d}{d\\xi}m(0)>1$ can be simplified to be $3\\sigma^2\/2\\theta<1$.\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/sigma\/l_sigma.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/sigma\/m_sigma.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/sigma\/h_sigma.eps}\n\t\\caption{\n\t\t\\label{fig:The change of sigma}\n\t\tSimulations for different $\\sigma$. The system has two (statistically) stable equilibria when $\\sigma$ is below the \n\t\tcritical value or otherwise has single stable state $0$. For small $h$, \n\t\t$3\\sigma^2\/2\\theta<1$ is the approximate criterion.}\n\\end{figure}\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/theta\/h_theta.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/theta\/m_theta.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/theta\/l_theta.eps}\n\t\\caption{\n\t\t\\label{fig:The change of theta}\n\t\tSimulations for different  $\\theta$. The system has two stable equilibria if $\\theta$ is above the \n\t\tcritical value or otherwise has single stable state $0$. For small $h$, \n\t\t$3\\sigma^2\/2\\theta<1$ is the approximate criterion.}\n\\end{figure}\n\n\nFigure \\ref{fig:The change of sigma} and Figure \\ref{fig:The change of theta} illustrate the behavior of the \nempirical mean as the system transitions from having two equilibria to having a single one, which is controlled \nby the value of $\\frac{d}{d\\xi}m(0)$. This is an instance of a bifurcation of equilibria. From Proposition \n\\ref{prop:explicit condition and solution for xi equal to m(xi) for small h}, we know that when $h$ is \nsmall, the existence condition of two equilibria, $\\frac{d}{d\\xi}m(0)>1$, can be approximated by the \ncondition $3\\sigma^2\/2\\theta<1$. In the simulations, we let $h=0.1$ so the approximate condition \n$3\\sigma^2\/2\\theta<1$ can be applied. In Figure \\ref{fig:The change of sigma} we change $\\sigma$ but fix the \nother parameters, and consider the three cases $\\frac{d}{d\\xi}m(0)<1$ ($3\\sigma^2\/2\\theta>1$), \n$\\frac{d}{d\\xi}m(0)\\approx 1$ ($3\\sigma^2\/2\\theta=1$) and $\\frac{d}{d\\xi}m(0)>1$ ($3\\sigma^2\/2\\theta<1$). In \nFigure \\ref{fig:The change of theta} we change $\\theta$. We can see that even though the  \nparameters varied in the numerical simulations are not the same, the bifurcation behavior is similar. \n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/h\/l_h.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/h\/m_h.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/h\/h_h.eps}\n\t\\caption{\n\t\t\\label{fig:The change of h}\n\t\tThe effect of changing $h$. Increasing it stabilizes the system.}\n\\end{figure}\n\nFigure \\ref{fig:The change of h} shows the effect of increasing $h$ on the system stability.  \nBy stability we mean resistance to the transition of the empirical mean of the system \nfrom one state to the other (because the model is symmetric). The parameter $h$  \nis proportional to the height of the potential barrier of each agent. Thus we increase\nthe overall system stability if we increase the component's stability. This observation \nis analogous to comments in \\cite{Nier2007, Lorenz2009, May2010}. It is clear that \n$h$ influences system stability substantially. \n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/N\/l_N.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/N\/m_N.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/homo\/N\/h_N.eps}\n\t\\caption{\n\t\t\\label{fig:The change of N}\n\t\tInfluence of the system size $N$. A larger system tends to have a more stable behavior.}\n\\end{figure}\n\nFigure \\ref{fig:The change of N} illustrates the effect of system size on its  \nstability. Clearly a larger system is more stable.\nThese stability phenomena will be quantified with the large deviations analysis  of Section \n\\ref{sec:large deviations}.\n\n\n\n\n\\subsection{Heterogeneous Model}\n\nFor the heterogeneous model, $\\theta$ is replaced by $\\theta_j$, \nand the discretization is \n\\begin{equation}\n\t\\label{eq:Euler method, heterogeneous case}\n\tX_j^{n+1} = X_j^n - h U(X_j^n)\\Delta t + \\sigma\\Delta W_j^{n+1} \n\t+ \\theta_j(\\frac{1}{N}\\sum_{k=1}^N X_k^n - X_j^n)\\Delta t,\n\\end{equation}\nwith the same parameter settings. The different values of $\\theta_j$ are controlled\nby the parameters  $\\Theta_l$ and \n$\\rho_l$. In the simulation, we take $K=3$ and\n$\\{\\Theta_l\\}_{l=1}^K=\\{ \\Theta_L,\\Theta_M,\\Theta_H\\}$  for a system\na low, medium and high rates of mean reversion to the empirical mean, that is, the systemic risk. \nWe also take $\\{\\rho_l\\}_{l=1}^K=\\{\\rho_L,\\rho_M,\\rho_H\\}$ for the corresponding fractions. We \nuse the normalized standard deviation of the distribution of $\\theta_j$ values in order to quantify  \ndiversity. We find that the heterogeneous model behaves like the homogeneous one when\n$h$, $\\sigma$ and $N$ change.  But, diversity on the rates of mean reversion \nhas significant impact on system stability.\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/sigma\/l_sigma.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/sigma\/m_sigma.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/sigma\/h_sigma.eps}\n\t\\caption{\n\t\t\\label{fig:The change of sigma, diversity case}\n\t\tEffect of changes in $\\sigma$. The system has two stable equllibria when $\\sigma$ is below the \n\t\tcritical value and has single one otherwise. For small $h$, \n\t\t$\\sum_{l=1}^K(\\rho_l\/\\Theta_l)(3\\sigma^2\/2\\Theta_l-1) < 1$ is the approximate \n\t\tcriterion.}\n\\end{figure}\n\nAs in the homogeneous case, in Figure \\ref{fig:The change of sigma, diversity case} we consider cases with  \n$\\sigma$ below, close to and above the critical value. The results are similar to the homogeneous case as \nexpected. For $\\sigma$ below the critical value we have two equlibria and for $\\sigma$ above the critical \nvalue one equilibrium. The condition $\\frac{d}{d\\xi}m(0)>1$ is still necessary and sufficient for the\nexistence two equlibria. The condition $\\sum_{l=1}^K(\\rho_l\/\\Theta_l)(3\\sigma^2\/2\\Theta_l-1) < 1$ is also a \ngood approximation to the exact one when $h$ is small.\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/h\/l_h.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/h\/m_h.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/h\/h_h.eps}\n\t\\caption{\n\t\t\\label{fig:The change of h, diversity case}\n\t\tEffect of changing $h$. Increasing it stabilizes the system.}\n\\end{figure}\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/N\/l_N.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/N\/m_N.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/N\/h_N.eps}\n\t\\caption{\n\t\t\\label{fig:The change of N, diversity case}\n\t\tEffect of changing the system size $N$. Larger system have a more stable behavior.}\n\\end{figure}\n\nThe parameter $h$ and the system size $N$ are closely associated with system stability. We note that in \nFigure \\ref{fig:The change of h, diversity case} and Figure \\ref{fig:The change of N, diversity case} when \n$h$ or $N$ are increased, the system becomes visibly more stable. Another observation is that with $h$, \n$\\sigma$ and $N$ fixed, and with the mean of $\\theta_j$ of (\\ref{eq:Euler method, heterogeneous case}) equal \nto $\\theta$ of (\\ref{eq:Euler method, homogeneous case}), the heterogeneous system is consistently more \nunstable than the corresponding homogeneous model (see Figure \\ref{fig:The change of h} and Figure \n\\ref{fig:The change of N}). Clearly diversity tends to destabilize the system.\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/Theta\/l_div.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/Theta\/m_div.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/Theta\/h_div.eps}\n\t\\caption{\n\t\t\\label{fig:The change of Theta, diversity case}\n\t\tThe effect of changes in $\\Theta_l$. The median of the diversity values is fixed but the low and high \n\t\tsensitivities are changed to adjust the level of diversity of $\\theta_j$ while $\\rho_l$ \n\t\tand the mean of $\\theta_j$ are the same. Increasing diversity tends to destabilize \n\t\tthe system.}\n\\end{figure}\n\n\\begin{figure}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/rho\/l_div.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/rho\/m_div.eps}\n\t\\includegraphics[width=0.32\\textwidth]{figure\/hetero\/rho\/h_div.eps}\n\t\\caption{\n\t\t\\label{fig:The change of rho, diversity case}\n\t\tThe effect of changes in $\\rho_l$, with $\\Theta_l$ and the mean of $\\theta_j$ fixed. \n\t\tIncreasing diversity tends to destabilize the system.}\n\\end{figure}\n\nWe also change the diversity of $\\theta_j$ by changing $\\Theta_l$ and $\\rho_l$. To compare with the \nhomogeneous case, in Figure \\ref{fig:The change of Theta, diversity case} and Figure \n\\ref{fig:The change of rho, diversity case} we change the standard deviation of $\\theta_j$ while the mean of \n$\\theta_j$ is fixed. In this most interesting part of the simulations we see that when we increase the \nstandard deviation of diversity values, the number of transitions is notably larger than that in the \nhomogeneous case.\n\\section{Small $h$ Analysis}\n\\label{sec:small h}\n\nThe goal of this section is to analyze the min-max problem (\\ref{eq:general LD}) which\ncontrols the asymptotic systemic transition probability. \nThis problem is nonlinear and infinitely dimensional and is \ndifficult to analyze. To get some useful information about it\nwe will assume that $h$ is small and analyze it in this regime. \nWe will first solve (\\ref{eq:general LD}) when $h$ is exactly $0$, and \nthen we will get rigorous upper and lower bounds for (\\ref{eq:general LD}) when $h$ is nonzero \nbut small. We will then compare the large deviations result with the local\nfluctuation theory of a single agent so as to explain why interconnectedness destabilizes \nthe system. \n\n\n\\subsection{The $h=0$ and the Small $h$ Analysis}\n\\label{sec:small h analysis}\n\nWe note that when $h=0$, $u_{\\pm\\xi_b}^e = u_{\\pm\\xi_0}^e$, where\n\\begin{equation}\n\tu_{\\pm\\xi_0}^e(y) = \\frac{1}{\\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\\exp\\left\\{ -\\frac{(y - (\\pm\\xi_0))^2}{2\\frac{\\sigma^2}{2\\theta}} \\right\\},\n\t\\quad\\quad \\xi_0 = \\sqrt{1-3\\frac{\\sigma^2}{2\\theta}}.\n\\end{equation}\nIn this case, (\\ref{eq:general LD}) is solvable and the optimal path is a Gaussian, \nstarting from $u_{-\\xi_0}^e$ and ending in $u_{+\\xi_0}^e$.\n\\begin{theorem}\n\t\\label{thm:LD for zero h and Gaussian path}\n\tLet $h=0$ and define \n\t\\begin{equation}\n\t\t\\label{def:pe}\n\t\tp^e(t,y) = \\frac{1}{\\sqrt{2\\pi \\frac{\\sigma^2}{2\\theta}}}\n\t\t\\exp\\left\\{ -\\frac{(y - a^e(t))^2}{2\\frac{\\sigma^2}{2\\theta}} \\right\\}, \\quad\n\t\ta^e(t)= \\frac{2\\xi_0}{T} t - \\xi_0.\n\t\\end{equation}\n\tThen $p^e \\in A$ is the unique minimizer for (\\ref{eq:general LD}) and\n\t\\[\n\t\t\\inf_{\\phi \\in A} I_0(\\phi) = I_0(p^e) = \\frac{2\\xi_0^2}{\\sigma^2 T}.\n\t\\]\n\\end{theorem}\n\\begin{proof}\n\tSee Appendix \\ref{pf:LD for zero h and Gaussian path}.\n\\end{proof}\n\nWe show next that (\\ref{eq:general LD}) is continuous at $h=0$. \n\\begin{theorem}\n\t\\label{thm:cotinuity for small h}\n\n\tThere exists $\\gamma(h)$ such that $\\gamma(h)\\to 0$ as $h \\to 0$ and\n\t\\begin{equation}\n\t\t\\label{eq:cotinuity for small h}\n\t\t\\Big| \\inf_{\\phi \\in A} I_h(\\phi) - \\frac{2\\xi_b^2}{\\sigma^2 T} \\Big| \\leq \\gamma(h).\n\t\\end{equation}\n\tWe recall here that \n\t\\begin{equation}\n\t\t\\xi_b = \\xi_0+h \\xi_1 +O(h^2), \\quad \\quad \\xi_1 =  \\sqrt{1 - 3\\frac{\\sigma^2}{2\\theta}} \n\t\t \\frac{6}{\\sigma^2}\\left(\\frac{\\sigma^2}{2\\theta}\\right)^2\n\t\t\\frac{1 - 2(\\sigma^2\/2\\theta)}{1 - 3(\\sigma^2\/2\\theta)} .\n\t\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\n\tSee Appendix \\ref{pf:cotinuity for small h, upper bounds} and \n\t\\ref{pf:cotinuity for small h, lower bounds}.\n\\end{proof}\n\nAs it is stated we could replace $\\xi_b$ by $\\xi_0$ in Theorem \\ref{thm:cotinuity for small h}, \nsince $\\xi_b = \\xi_0+o(1)$ as $h \\to 0$. We will see in the next section \n(in Proposition \\ref{prop:inf_A I_h up to O(h)})  that $\\gamma(h)=O(h^2)$. In fact we show this rigorously \nfor the upper bound but only formally for the lower bound. Since $\\xi_b=\\xi_0+h\\xi_1+O(h^2)$ we see that the \nterm $2\\xi_b^2\/(\\sigma^2 T)$ contains the leading-order term and the first-order correction in the \n$h$-expansion of $\\inf_{\\phi \\in A} I_h(\\phi)$.\n\n\n\\subsection{Large Deviations for the First Exit Time}\n\nIn this subsection, we consider the rare event $B$ of systemic transition at some time before $T$:\n\\[\n\tB_\\delta = \\{\\phi\\in\\mathcal{E}^\\nu: \\exists t\\in(0,T], \\rho(\\phi(t),u^e_{\\xi_b}) \\leq \\delta \\}.\n\\]\nIn other words, $B_\\delta=\\cup_{t\\in(0,T]}A_\\delta(t)$, where \n\\[\n\tA_\\delta(t) = \\{\\phi\\in\\mathcal{E}^\\nu: \\rho(\\phi(t),u^e_{\\xi_b}) \\leq \\delta \\}.\n\\]\nWe let $B:=B_0$. We then have that \n\\begin{lemma}\n\t\\label{lma:lower bound of I_h over B_delta}\n\tBy definition $\\inf_{\\phi\\in B_\\delta}I_h(\\phi)$ is decreasing with $\\delta>0$ and bounded from above by \n\t$\\inf_{\\phi\\in B}I_h(\\phi)$. In addition,\n\t\\[\n\t\t\\lim_{\\delta\\to 0}\\inf_{\\phi\\in B_\\delta}I_h(\\phi) \n\t\t= \\inf_{\\phi\\in \\cup_{t\\in(0,T]}A(t)}I_h(\\phi)\n\t\t= \\inf_{\\phi\\in B}I_h(\\phi),\n\t\\]\n\twhere $A(t):=A_0(t)$.\n\\end{lemma}\n\\begin{proof}\n\tSee Appendix \\ref{pf:lower bound of I_h over B_delta}.\n\\end{proof}\n\nFrom Theorem \\ref{thm:cotinuity for small h}, we see that in the sense of large deviations the \nprobability of system failure at some time before time $T$ is essentially the same as the probability of \nsystem failure at time $T$.\n\\begin{corollary}\n\tFor any $t_1<t_2$, there exists a sufficiently small $h$ such that \n\t$\\inf_{\\phi\\in A(t_1)} I_h(\\phi)> \\inf_{\\phi\\in A(t_2)}I_h(\\phi)$. Consequently, \n\t$\\inf_{\\phi\\in B}I_h(\\phi)\\approx \\inf_{\\phi\\in A(T)}I_h(\\phi)$ for small $h$.\n\\end{corollary}\n\n\\subsection{Comparison with the Fluctuation Theory of a Single Agent}\n\nTo get a better understanding of the large deviations results we need to\ncarry out a standard fluctuation theory for a single agent. \nWe assume that $x_j(0)=-1$ for all $j$ and that the\n$x_j(t)$'s are in the vicinity of $-1$ so that we can linearize \n(\\ref{eq:SDE of single component}):\n\\[\n\tx_j(t)=-1+z_j(t),\\quad \\bar{x}(t)=-1+\\bar{z}(t),\\quad \n\t\\bar{z}(t) = \\frac{1}{N}\\sum_{j=1}^N z_j(t).\n\\]\nFor $V(y)=\\frac{1}{4}y^4-\\frac{1}{2}y^2$, $z_j(t)$ and $\\bar{z}(t)$ satisfy \nthe linear stochastic differential equations\n\\[\n\tdz_j = -(\\theta+2h)z_j dt + \\theta\\bar{z}dt + \\sigma dw_j, \\quad\n\td\\bar{z} = -2h\\bar{z}dt + \\frac{\\sigma}{N}\\sum_{j=1}^N dw_j,\n\\]\nwith $z_j(0)=\\bar{z}(0)=0$. The processes $z_j(t)$ and $\\bar{z}(t)$ are \nGaussian and the mean and variance functions are easily calculated. We are \nespecially interested in their behavior for large $N$.\n\\begin{lemma}\n\t\\label{lma:fluctuation of a single agent}\n\tFor all $t\\geq 0$, $\\mathbf{E}z_j(t)=\\mathbf{E}\\bar{z}(t)=0$ and \n\t$\\mathbf{Var}\\bar{z}(t)=\\frac{\\sigma^2}{N}(1-e^{-4ht})$. In addition, \n\t$\\mathbf{Var}z_j(t) \\rightarrow \n\t\\frac{\\sigma^2}{2(\\theta+2h)}(1-e^{-2(\\theta+2h)t})$ as $N\\rightarrow\\infty$,\n\tuniformly in $t\\geq 0$.\n\\end{lemma}\n\nFrom Lemma \\ref{lma:fluctuation of a single agent}, we see that $\\sigma^2\/N$ and \n$\\sigma^2\/2(\\theta+2h)$ should be sufficiently small so that linearization is \nconsistent with the results it produces.\n\n\\subsection{Increased Probability of Large Deviations for Increased $\\theta$ and Its\nSystemic Risk Interpretation}\n\\label{sec:comparison}\n\nWe have now the analytical results with which we\nmay conclude that individual risk diversification may increase the systemic risk. Assume that \n$\\sigma^2\/N$ and $\\sigma^2\/2(\\theta+2h)$ are sufficiently small and $N$ is large. From \nLemma \\ref{lma:fluctuation of a single agent}, the risk $x_j(t)$ of the agent $j$ \nis approximately a Gaussian process with the stationary distribution \n$\\mathcal{N}(-1,\\sigma^2\/2(\\theta+2h))$. If the external risk, $\\sigma$ is high, then in \norder to keep the risk $x_j(t)$ at an acceptable level, the agent may increase the \nintrinsic stability, $h$, or share the risk with other agents, that is, increase\n$\\theta$. Increasing $h$ is in general more costly (cuts into profits) than increasing \n$\\theta$, and at the individual agent level there is no difference in risk assessment \nbetween increasing \n$h$ and increasing $\\theta$. Therefore the agents are likely to increase $\\theta$ and\nreduce individual risk by diversifying it. Note that \n$\\sigma^2\/2(\\theta+2h) \\lesssim \\sigma^2\/2\\theta$ when $\\sigma^2$ and $\\theta$ are \nsignificantly larger than $h$. Thus, individual agents can maintain \nlow locally assessed risk by diversification, even in a very uncertain environment.\n\nWhat is not perceived by the individual agents, however,\nis that risk diversification may increases the systemic risk while it reduces\ntheir individual risk. Because $\\sigma^2$ and $\\theta$ are significantly larger than $h$, the \nsmall $h$ analysis can be applied and from \n(\\ref{eq:probability of phase transition, LD form}) and Theorem \n\\ref{thm:cotinuity for small h}, the systemic risk (the probability of the system \nfailure) is \n\\begin{align*}\n\t&\\mathbf{P}(X_N\\in B_\\delta) \\approx \\exp\\left(-N\\frac{2\\xi_b^2}{\\sigma^2 T}\\right), \\quad \\text{for small $\\delta$ and $h$,}\\\\\n\t&\\xi_b = \\sqrt{1 - 3\\frac{\\sigma^2}{2\\theta}} \n\t\t\t\\left( 1 + h\\frac{6}{\\sigma^2}\\left(\\frac{\\sigma^2}{2\\theta}\\right)^2\n\t\t\t\\frac{1 - 2(\\sigma^2\/2\\theta)}{1 - 3(\\sigma^2\/2\\theta)}\\right) + O(h^2).\n\\end{align*}\nWe see that there are additional systemic-level $\\sigma^2$ terms in the exponent and \n$\\xi_b$, which can not be observed by the agents, increasing the systemic risk, even if \nthe individual risk $\\sigma^2\/2\\theta$ is fixed. In other words, the individual agents may \nbelieve that they are able to withstand larger external fluctuations as long as their risk can be \ndiversified, but a higher $\\sigma$ tends to destabilize the system.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Summary and Conclusions}\n\\label{sec:summary}\n\nThe aim of this paper is to introduce and analyze a mathematical model\nfor the evolution of risk in a system of interacting agents \nwhere cooperation between them can reduce\ntheir individual risk of failure but increase the systemic or overall risk.\nThe model we use is a system of bistable diffusion processes that interact through\ntheir empirical mean, a mean field model. \nWe take the rate of mean reversion to the empirical mean\n$\\theta$ as a measure of cooperation, the depth of the bistable potential $h$\nas a measure of intrinsic stability of each agent, and the strength \nof the external random perturbations $\\sigma$ as the level of uncertainty in\nwhich the agents function. Using the theory of large\ndeviations we calculate the probability that the system will transition\nfrom one of the two bistable states to the other during a time interval of length\n$T$, when the number of agents $N$ is large and when $h$ is small. In this\nregime of parameters we find that systemic risk increases with cooperation.\nThe formula from which we draw this conclusion is given is Section \\ref{sec:comparison}.\nWe also show that when the rate of mean reversion to the empirical mean\nvaries among the different agents,\nthat is, when there is diversity in the cooperative behavior then the probability\nof transitions increases, which means that the systemic risk increases.\n\n\n\\section*{Acknowledgement}\n\nThis work is partly supported by the Department of Energy \n[National Nuclear Security Administration] under Award Number NA28614, and\npartly by AFOSR grant FA9550-11-1-0266.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe central regions of active galaxies (AGN) harbour essential clues to how these amazing objects work. Differences on small scales lead to enormous variety of observed characteristics on large scales. For radio-loud AGN, a distinction is made between high- and low-power radio galaxies, referred to as FR\\,II and FR\\,I, respectively \\citep{fr74}. Differences between these sources occur in radio morphology, optical, and X-ray properties. They are attributed to different accretion rates or accretion modes \\citep[see e.g.][]{butti10}. This implies differences in kinematics and morphology of gas and dust in the central regions.\n\nOne difference between high- and low-power radio galaxies is well established. Unified schemes postulate a parsec-scale, thick dust torus to explain the bimodality in optical spectra of high-power FR\\,II radio galaxies \\citep[][e.g.]{pdb89,antonucci93}. In contrast, the optical nuclei of FR\\,I radio galaxies appear {\\em unobscured} \\citep[e.g.][]{koff96,chiab99,capettiHST02}, though the host galaxies often have kiloparsec-scale dust rings and lanes \\citep{dekoff00,vk02,ruiter02}. \\cite{chiab99} find that the brightness of the optical core correlates with the brightness of the radio core. Therefore, they conclude that the optical core emission is non-thermal, coming from the active nucleus. This is only possible if a parsec-scale torus is absent. X-ray observations confirm this absence, and find evidence for a different accretion mode in FR\\,I radio galaxies \\citep{balmaverde06,hardcastle09}. \n\nA major question in AGN research is how the material is transported from kilo-parsec scales to the sub-parsec scale. This requires kinematical studies at high spatial and velocity resolution. Due to their proximity, FR\\,I radio galaxies are best suited for this purpose. H{\\,\\small I}\\ absorption observations are a powerful tool for studying kinematics on small scales \\citep[e.g.][]{vg89,morganti01,morganti03,vermeulen03,gupta06}. Molecular lines can provide essential additional information, as they also trace the cool interstellar medium. In this paper we present a pilot study to explore the relation between dust (associated with the molecular lines) and neutral gas (associated with the H{\\,\\small I}) on kilo-parsec scales or smaller. If we can relate the gas and dust on sub-kiloparsec scales, future studies can combine information from these sources to generate a complete kinematical picture.\n\n\\begin{table*}[t]\n\\caption{\\label{VLAobs}VLA sample description (columns 1 through 6) and observational results (columns 7 through 10).}\n\\begin{tabular}{lllccccccc}\n\\hline \\hline\nName \t& ID &  $z$\t& $S_{c,1.4}$\t&$P_{t,4.9}$& t$_{obs}$\t& $S_{abs}$\t&  rms\t&$\\tau_{int}$& $N_{\\mathrm{HI}}$\t\\\\\nB1950\t&&    \t\t& (Jy)\t& (W Hz${^-1}$)\t& (min)\t&(mJy\\,bm$^{-1}$)   & (mJy\\,bm$^{-1}$) & (km s${^-1}$)\t& ($\\times10^{20}$\\,cm$^{-2}$)\t\\\\\n(1)\t\t& (2) &  (3)\t\t& (4)\t\t& (5)\t\t& (6)\t\t&   (7)\t& (8)\t\t& (9)\t\t& (10)\t\t\\\\\n\\hline\n0055+30  & NGC315 & \t0.0165\t& 0.464\t& 23.84\t& 27\t\t& 27.0\t& \t1.0\t& 4.5$\\pm$0.7\t& 8.3$\\pm$0.3\t\\\\\n0104+32\t& 3C31 \t  & \t0.0169\t& 0.067\t& 24.11\t& 66+47\t& $<$1.8& \t0.6\t& $<$2.7\t\t& $<$5.19\t\t\\\\\n0755+37\t& 3C189\t  &   0.0428\t& 0.137\t& 24.62\t& 22\t\t& $<$3.0\t&\t1.0\t& $<$2.2\t\t& $<$4.23\t\t\\\\\n1322+36\t& NGC5141&   0.0176\t& 0.054\t& 23.45\t& 57\t\t& 3.3\t&   0.6\t& 5.5$\\pm$1.0\t& 10.0$\\pm$1.8\t\\\\\n1346+26\t& 4C26.42  &  0.0632\t& 0.135\t& 24.35\t& 139\t& 3.0\t&  \t0.5\t& 2.2$\\pm$0.4\t& 4.1$\\pm$0.7\t\\\\\n1350+31\t& 3C293\t  &  0.045 \t\t& 1.652\t& 24.93\t& 20\t\t& 148.0\t& \t1.0\t&12.8$\\pm$0.1& 23.5$\\pm$0.2\\\\\n1626+39\t& 3C338 \t  &  0.0303\t& 0.138\t& 23.98\t& 17\t\t& $<$1.8& \t0.6\t& $<$1.3\t\t& $<$2.5\t\t\\\\\n2116+26\t& NGC7052&  0.0156\t& 0.040\t& 22.80\t& 88\t\t& $<$1.2&\t0.4\t& $<$2.6\t\t& $<$5.1\\tablefootmark{*}\t\t\\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\\tablefoottext{*}{Upper limit from our VLA observations. \\citet{emonts10} set $N_{\\mathrm{HI}} < 0.34 \\times 10^{20}$\\,cm$^{-2}$.}\\\\\nClarification of the columns: (1) B2 rado name, (2) ID used in the paper, (3) redshift, (4) core flux at 1.4\\,GHz, (5) total power at 4.9\\,GHz, (6) total on-source integration time, (7) peak of the absorbed flux (8) rms noise, (9) integrated optical depth, and (10) H{\\,\\small I}\\ column density.}\n\\end{table*}\n\nWe expand upon earlier studies by combining H{\\,\\small I}\\ information with HST imaging. Our primary goal is to determine if the presence of an optical core is associated with the absence of H{\\,\\small I}\\ absorption. Next, we search for a relation between the dust and H{\\,\\small I}, where we expect to see absorption if dust lies in the line-of-sight to the radio source. If H{\\,\\small I}\\ is indeed associated with the dust, deeper H{\\,\\small I}\\ observations can be combined with molecular line data to study the kinematics of the central regions in a complete sample. This work is essential to provide new insight into AGN fueling and feedback.\n\n\\section{The sample} \nOur sample consists of low-power radio galaxies\\footnote{Throughout this paper we use $H_{\\rm 0}$= 70 km s$^{-1}$ Mpc$^{-1}$}: $log P_{\\rm 1.4 GHz} < 25.5$\\,W Hz$^{-1}$. They all have typical FR\\,I (core-dominated) radio morphologies. The host galaxies are early-type galaxies, with a range of ellipticities and environments. For our new VLA observations we selected a representative sample of eight FR\\,I radio galaxies from the HST snapshot survey of the B2 catalogue \\citep{capettiHST02}. The HST snapshot images trace dust down to scales of $\\sim$100\\,pc and are used to identify the presence of an optical unresolved core.\n\nTo increase the sample size, we have added 14 FR\\,I radio galaxies from literature. They were selected to match our criterion for radio power, and to have H{\\,\\small I}\\ data and optical HST images available. This literature sample consists of results obtained by \\citet[][hereafter JvG89]{vg89}, and four additional sources from \\citet{vangorkom86}, \\citet{deyoung73}, \\citet{jaffe94}, and \\citet{hulst83}. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=80mm]{N5141_03JUL12.png}\n\\includegraphics[width=80mm]{C162_03JUL12.png}\n\\caption{\\label{HIspec}Plot of the continuum subtracted spectra for two of the detected sources from the VLA sample. The dashed vertical line indicates the systemic velocity. The x-axis is in optical barycentric velocity. The spectra are obtained against the brightest part of the radio source, which in both of these cases corresponds to the core.}\n\\end{figure}\n\nThe total sample has 22 objects and is not meant to be complete, but representative of this type of radio sources. The full  sample is listed in Table~\\ref{fullsample}. \n\n\\begin{table*}[t]\n\\caption{\\label{fullsample} H{\\,\\small I}\\ and  optical properties of the sample. First we list the sources with H{\\,\\small I}\\ absorption, then we list the non-detections. The two rows labelled as 'Total' give the total number of sources that are listed above, and the number of objects that have either dust structures or an optical core.}\n\\begin{tabular}{llcllcccc}\n\\hline \\hline\n\\noalign{\\smallskip}\nB1950\t& ID\t\t\t& $N_{\\mathrm{HI}}$ & Ref \t& Dust\t\t& Diameter       & $i$ & $D$\t\t& optical \t\\\\\n \t\t& \t\t\t& ($\\times10^{20}$\\,cm$^{-2}$) & & morph.\t\t& (pc) \t&$^\\circ$\t& (\")\t& core\t\\\\\n(1)\t\t&  (2)\t\t& (3) \t& (4)\t\t& (5)\t\t& (6)\t& (7)\t\t& (8)\t\t& (9) \\\\\n\\hline\n0055+30 \t& NGC315\t& 8.3 \t&\t1 & disk\t\t& 835\t& 75 & 0.15\t&\ty \t\\\\\n0116+31 \t& 4C31.04\t& 10.8\t&\t2 & extended\t& 2586\t& 83 & 0\t\t&\tn \t\\\\\n0238-08 \t& NGC1052\t&  3.0\t&\t3 & disk\t\t& 416 \t& 72 & 0.2\t&\ty \t\\\\\n0316+41 \t& 3C84\t\t&  2.0\t&\t4 & extended\t& 17625 \t& 37 & 1.0\t&\ty \t\\\\\n1146+59 \t& NGC3894\t& 18.2\t&\t2 & disk\t\t& 911 \t& 74 & 0.07\t&\tn \t\\\\\n1216+06 \t& 3C270\t\t& 7.2\t&\t5 & disk\t\t& 263 \t& 65 & 0.07\t&\ty \t\\\\\n1322-42 \t& Cen A\t\t& 50.0\t&\t6 & extended\t& 3365\t& 90 & 0\t\t&\tn \t\\\\\n1322+36 \t& NGC5141\t& 10.0\t&\t1 & disk\t\t& 638\t& 78 & 0\t\t&\tn \t\\\\\n1346+26 \t& 4C26.42\t& 4.1\t&\t1 & extended\t& 9063\t& 57 & 0.14\t&\ty \t\\\\\n1350+31 \t& 3C293\t\t& 23.5\t&\t1 & extended\t& 4130\t& 67 & 0\t\t&\tn \t\\\\\n2254-367\t& IC1459\t\t& $>$1.2\t&\t7 & disk\t\t& 310\t& 54 & 0\t\t&\ty\t\\\\\n\\hline\nTotal\t\t&\t\t\t& 11 & \t&\t11\t&\t&\t& \t& 5 \\\\%& \\\\\n\\hline\n\\hline\n0104+32 \t& 3C31\t\t& $<$5.2\t &\t1 & disk\t\t& 2174\t& 37 & 0.8\t&\ty \t\\\\\n0305+03 \t& 3C78\t\t& $<$1.47\t &\t2 & none\t\t& ...\t& ... & ...\t&\ty \t\\\\\n0430+05\t& 3C120 \t\t& $<$1.21\t &\t2 & none\t \t& ... \t& ... & ...\t&\ty \t\\\\\n0755+37 \t& 3C189\t\t& $<$4.2\t &\t1 & none\t\t& ...\t& ... & ...\t&\ty \t\\\\\n1142+19 \t& 3C264\t\t& $<$1.65\t &\t2 & disk\t\t& 675\t& 20 & 0.2\t&\ty \t\\\\\n1514+07 \t& 3C317\t\t& $<$1.70\t &\t2 & extended\t& 2595\t& $33$ & 0.25\t&\ty \t\\\\\n1626+39 \t& 3C338\t\t& $<$2.5\t &\t1 & extended\t& 3463\t& ... & 0.6\t&\ty \t\\\\\n1637+82 \t& NGC6251\t& $<$0.3\t &\t1 & disk\t\t& 678\t& 65 & 0.11\t&\ty \t\\\\\n1652+39 \t& 4C39.49\t& $<$4.52 &\t2 & none\t\t& ...\t& ...& ...\t&\ty \t\\\\\n1807+69\t& 3C371\t\t& $<$2.96 &\t2 & none\t\t& ...\t& ... & ...\t&\ty \t\\\\\n2116+26 \t& NGC7052\t& $<$0.34\t &\t8 & disk\t\t& 1227\t& 75 & 0\t\t&\ty \t\\\\\n\\hline\nTotal\t\t&\t\t\t& 11 &  \t&\t6\t&\t&\t\t&  \t&\t11 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n{\\bf Notes.} Clarification of the columns: (1) B2 radio name, (2) ID used in this paper, (3) H{\\,\\small I}\\ column density (4) reference for column 3, (5) morphology of the dust, (6) size of the dust structure, (7) inclination of the disk (where applicable), (8) distance between optical core and the dust detected in the optical images, and (9) presence of an optical core: y=yes, n=no. \n\\tablebib{(1) This paper;\n(2) \\citet{vg89};\n(3) \\citet{vangorkom86};\n(4) \\citet{deyoung73};\n(5) \\citet{jaffe94};\n(6) \\citet{hulst83};\n(7) \\citet{oosterloo98};\n(8) \\citet{emonts10}.\n}\n\\end{table*}\n\n\\section{Observations and H{\\,\\small I}\\ results } \nsubsection{VLA observations}\nWe observed eight sources with the VLA in A-configuration during December 2000 and January 2001. Observations were made with 6.25\\,MHz bandwidth and 64 channels. The data were reduced with CASA\\footnote{http:\/\/casa.nrao.edu}. We performed the default steps for H{\\,\\small I}\\ absorption line data reduction, which are described in the online manual for H{\\,\\small I}\\ absorption reduction in CASA\\footnote{http:\/\/casaguides.nrao.edu}. Hanning smoothing was applied after the calibration steps. Observational parameters for each source are listed in Table~\\ref{VLAobs}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=80mm]{IC1459_20SEP12.png}\n\\caption{\\label{IC1459}Plot of the H{\\,\\small I}\\ absorption detected in IC\\,1459 with the ATCA telescope. For comparison with the VLA spectra, the plot has been smoothed to a velocity resolution of 40\\,km\\,s$^{-1}$.}\n\\end{figure}\n\n H{\\,\\small I}\\ absorption is detected in four out of eight sources (see Table~\\ref{VLAobs}). The peak absorbed flux exceeds $5\\sigma$, with $\\sigma$ being the rms noise per channel. The source 4C\\,26.42 is a new detection. The $3\\sigma$ absorption peak in NGC\\,7052 is considered a non-detection, though it shows a hint of a resolved line. Since our observations, four sources have been studied with other telescopes. NGC\\,315 is presented in \\citet{morganti_n315}; 3C\\,293 was observed at high resolution by \\citet{beswick02}; and NGC\\,5141 and NGC\\,7052 were observed at lower resolution in \\citet{emonts10}. Only NGC\\,7052 was not detected, and a stringent upper-limit was set. Our column densities and H{\\,\\small I}\\ profiles are consistent with literature results. With our limited bandwidth and sensitivity we cannot recover the very broad component in 3C\\,293 \\citep{morganti03}. Two representative spectra, including 4C\\,26.42, are shown in Figure~\\ref{HIspec}.\n\n\n\\subsection{Additional data}\nA detection of H{\\,\\small I}\\ absorption in IC\\,1459 has been obtained with the Australia Telescope Compact Array. IC\\,1459 was observed with 16.0\\,MHz bandwidth, at a velocity resolution of 13\\,km\\,s$^{-1}$\\ (before Hanning smoothing), and with the 375\\,m configuration \\citep[see Fig.~\\ref{IC1459}]{oosterloo98}. The rms noise is 1.4\\,mJy\\,beam$^{-1}$ after Hanning smoothing. Although the spatial resolution of these observations is very poor, the background source is unresolved on arcsecond scales \\citep{sadler89}. Therefore, we can assume that the absorption is coming from these small scales.\n\nThe giant radio galaxy NGC\\,6251 is listed as a non-detection by JvG89. However, \\citet{evans05} reported an unpublished column density of $4.5\\times10^{20}$\\,cm$^{-2}$. To confirm this, we obtained a 12 hour WSRT service observation, with 1024 channels and a bandwidth of 20\\,MHz. The observation was processed similar to the VLA observations. We found no detection and set an upper limit of $N_{\\mathrm{HI}}<0.3\\times10^{20}$\\,cm$^{-2}$\\ for this source. \n\n\n\\subsection{H{\\,\\small I}\\ column density}\nTo derive the H{\\,\\small I}\\ column density in units of \\,cm$^{-2}$, we used the standard equation\n\\begin{equation}\nN_{\\mathrm{HI}} = 1.835\\times10^{18}  \\frac{T_s}{f_c} \\int{\\tau({\\upsilon}) d{\\upsilon}} = 1.835\\times10^{20}  \\cdot \\tau_S \\cdot \\Delta {\\upsilon}_{\\mathrm{ch}} \\mbox{ ,}\n\\end{equation}\nin which $T_S$ is the spin temperature and $f_c$ the covering fraction, $\\tau_S$ is the optical depth summed over the channels where absorption is present, and $\\Delta {\\upsilon}_{\\mathrm{ch}}$ is the velocity width of a single channel. The product $\\tau_S \\cdot \\Delta {\\upsilon}_{\\mathrm{ch}}$ is the integrated optical depth, $\\tau_{int}$. We assumed a spin temperature of 100\\,K and a covering fraction of 1. \n\nFor non-detections we replaced the integral with the outcome for a Gaussian profile\n\\begin{equation}\n\\int{\\tau({\\upsilon}) d{\\upsilon}} = 1.06 \\cdot \\tau_p \\cdot \\Delta {\\upsilon}_{\\mathrm{FWHM}} \\mbox{ ,}\n\\end{equation}\nin which $\\tau_p$ is the peak optical depth and $\\Delta {\\upsilon}_{\\mathrm{FWHM}}$ the velocity width of the Gaussian at half maximum. In a non-detection the peak absorption depth is assumed to be less than 3$\\sigma$, with $\\sigma$ equal to the rms noise per channel. Since $\\tau_p \\approx S_{abs} \/ S_c$, we can replace $\\tau_p$ by $3\\sigma \/ S_c$, with $S_c$ being the core flux at 1.4\\,GHz. The resulting upper limit of column density is then\n\\begin{equation}\nN_{\\mathrm{HI}} < 1.93\\times10^{20} \\cdot \\frac{3\\sigma}{S_c} \\cdot \\Delta {\\upsilon}_{\\mathrm{FWHM}} \\mbox{ .}\n\\end{equation}\n\nWe assumed the average velocity width of the detected sources for the line width $\\Delta {\\upsilon}_{\\mathrm{FWHM}} = 100$\\,km\\,s$^{-1}$. For a proper comparison, we recalculated the literature upper limits, which use different values. \n\nThe H{\\,\\small I}\\ absorption technique is limited to objects with a bright core. Hence, in sources with a faint core, we may be missing H{\\,\\small I}\\ if it has a low column density. In our small VLA sample this bias is not obvious. In the source with the weakest core, NGC\\,5141, H{\\,\\small I}\\ is detected with an integrated optical depth of $5.5\\pm1.0$. The results for all VLA observations are listed in Table~\\ref{VLAobs}.\n\nCombining all the results, we have 11 detections and 11 non-detections. Owing to our selection method, these numbers cannot be used to derive detection rates for the population of low-power radio galaxies as a whole.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Histogram of the distribution of dust.]\n{\\includegraphics[width=80mm]{dusthist.png}\n\\label{HIhista}}\n\n\\subfigure[Histogram of the distribution of optical cores.]\n{\\includegraphics[width=80mm]{corehist.png}\n\\label{HIhistb}}\n\n\\caption{\\label{HIhist}Histograms of the distributions of dust (a) and optical cores (b) in the H{\\,\\small I}\\ detections and non-detections.}\n\\end{figure}\n\n\\section{Results}\n\nThe goal of this study is to compare the detection of H{\\,\\small I}\\ in absorption with the presence of kiloparsec-scale dust and an optical core. \n\n\\subsection{Relation between neutral gas, dust, and optical core}\n\nThe trends we find are illustrated in Figs.~\\ref{HIhist} and \\ref{HIcolumn} and are listed in Table~\\ref{fullsample}. They can be summarized as follows:\n\n\\begin{enumerate}\n\\item All H{\\,\\small I}\\ absorption detections have significant amounts of dust in their host galaxies, and sources without visible dust in the host galaxies have no H{\\,\\small I}\\ absorption (Fig.~\\ref{HIhista}).\n\\item All sources without H{\\,\\small I}\\ detection have an optical core (Fig.~\\ref{HIhistb}).\n\\item Sources with $N_{\\mathrm{HI}} > 10^{21}$\\,cm$^{-2}$\\ have no optical core (Fig.~\\ref{HIcolumn}).\n\\end{enumerate}\n\nWe found sources with H{\\,\\small I}\\ absorption that have an optical core, see Fig.~\\ref{HIcolumn}. These sources have either extended dust far from the nucleus (see section~4.2), or disks with low inclination, consistent with a line of sight through a less dense medium. If $N_H > 10^{21}$\\,cm$^{-2}$, the unresolved optical core is not detected. This column density corresponds to a visible extinction of $\\sim1$ magnitude for Galactic extinction \\citep{predehl95}. Because we have no handle on the intrinsic brightness of the optical core, future work on a larger, complete sample will have to clarify the scientific merit of our third result.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=80mm]{NH_core_hist.png}\n\\caption{\\label{HIcolumn} Distribution of H{\\,\\small I}\\ column density for the sample. The detections are grey, the detections with an optical core are hatched. Within the detections there is a clear trend for the sources without an optical core to have a higher H{\\,\\small I}\\ column density. This is reflected in the entire sample, taking into account that the non-detections in the white area only have upper limits of H{\\,\\small I}\\ column density and all have an optical core.}\n\\end{figure}\n\n\\subsection{Dust morphology}\nIn the optical images we find three distinct morphologies for the dust: \n\n\\begin{itemize}\n\\item  {\\it Extended}: dust extending up to tens of kpc. In many sources this dust does not show any symmetry with respect to the core of the galaxy, often it is diffuse and\/or filamentary. Some extended dust structures show evidence for smaller scale arms, or have a feather-like appearance. In a few sources it appears as a large dust lane (e.g. Centaurus~A).\n\\item {\\it Disks}: dust in a well-defined and symmetric distribution. The dust disks are rather compact, having radii of several hundreds of parsec. We find no disks with radii larger than 1\\,kpc.\n\\item {\\it None}: in the HST images there is no visible dust. \n\\end{itemize}\nOur classification of dust structures is consistent with results from \\citet{ruiter02}. \n\nThe 11 sources with H{\\,\\small I}\\ absorption all have dust, and are split between extended dust (5) and disks (6). The disks are at high inclinations, with a median value of 73\\degr. Extended dust either covers the optical core, or is located at less than one HST resolution element from the core. The exception is 3C\\,84, where the dust is more than 1$^{\\prime\\prime}$ from the optical core. This source is known to have a galaxy in the line of sight. This was first discussed by \\citet{minkowski57} and later confirmed by \\citet{lynds70}. Two separate H{\\,\\small I}\\ absorption components are seen. The velocity differences are consistent with a component associated with the foreground galaxy and another with the radio galaxy \\citep{deyoung73,rubin77}. It is not clear where the dust is located. VLBI observations are ongoing to spatially resolve the H{\\,\\small I}\\ absorption components in this complex source.\n\nAmong the 11 H{\\,\\small I}\\ non-detections, 6 sources have dust. In four cases the dust is either in a face-on disk (e.g. 3C\\,31) or at large projected distance from the core. The remaining two, NGC\\,6251 and NGC\\,7052, have an edge-on dust disk. The non-detection of NGC\\,6251 can be explained by the existence of a strong warp in the inner regions. While the outer parts of the disk are nearly edge-on at 76$^\\circ$ inclination, the inner regions are at 35$^\\circ$ inclination \\citep{ferrarese99}. If neutral gas is present in the dust disk, this would significantly suppress the column density of H{\\,\\small I}\\ in the center. NGC\\,7052 has the weakest radio core in our sample, which makes it the most difficult source to detect. Nevertheless, \\citet{emonts10} set a stringent upper limit. Higher resolution observations of this source may provide insight into the structure of the disk.\n\nIn sources without dust there is no detection of H{\\,\\small I}, and they have an unresolved optical core. Though this is consistent with the absence {\\bf of} obscuration on all scales, for these five sources alone we cannot fully exclude orientation effects. Considering the sample as a whole, we would expect a tighter relation between the absence of an optical core and the presence of H{\\,\\small I}\\ absorption if a small-scale torus were present. This is not reflected in our results.\n\n\\section{Conclusions}\nWe have studied the presence of H{\\,\\small I}, dust, and optical cores in the central kiloparsec of low-power radio galaxies. We find that the presence of H{\\,\\small I}\\ absorption is linked to the presence of dust. Furthermore, the {\\em absence} of H{\\,\\small I}\\ absorption is linked to the presence of an optical core. The optical core is not detected if $N_{\\mathrm{HI}} > 10^{21}$\\,cm$^{-2}$. Our results are consistent with neutral gas being associated with extended dust and dust in the kiloparsec-scale disks. We require no parsec-scale dust torus.\n\nThe distribution of H{\\,\\small I}\\ may be more complicated than a regular rotating central disk. This is evident from structure in the absorption profile, and is demonstrated by several higher resolution studies. H{\\,\\small I}\\ distributed in clouds with a filling factor much lower than unity, and H{\\,\\small I}\\ in infalling and outflowing structures are seen \\citep[see the case of NGC315,][]{morganti_n315}. Only VLBI observations will allow one to distinguish these effects. For sources with H{\\,\\small I}\\ absorption, the line kinematics is an essential component for understanding the feeding mechanism of active galaxies. Further insight into the physics of kiloparsec-scale disks will require deep observations of a complete sample. These are currently ongoing. More work will be done by the large surveys planned with the SKA pathfinders.\n\nFinally, molecular line transitions can provide crucial complementary information on the kinematics of the disks. This is especially important for sources that do not have a bright, unresolved radio core. This work opens a potential path for combining molecular lines with H{\\,\\small I}\\ absorption to study kinematics on sub-kiloparsec scales. Systematic studies of the cold, molecular gas component are now starting. However, \\citet{ocana10} found that radio galaxies tend to have relatively low molecular gas masses. Given the limited sensitivity of current instruments, ALMA will make a major leap in this field.\n\n\\begin{acknowledgements} \nWe gratefully acknowledge discussions with Neeraj Gupta, Raymond Oonk, Preeti Kharb and Gijs Verdoes Kleijn. We thank Jacqueline van Gorkom for pointing us to relevant information on ISM physics. The VLA is operated by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of the NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations made with the NASA\/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI\/NASA), the Space Telescope European Coordinating Facility (ST-ECF\/ESA) and the Canadian Astronomy Data Centre (CADC\/NRC\/CSA). This research has made use of Aladin.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\nA \\textit{lattice} polytope is a convex polytope all of its vertices belong to the standard lattice $\\Z^d$. \nLet $P \\subset \\R^d$ be a lattice polytope $P$ of dimension $d$. Given $n \\in \\Z_{>0}$, \nwe consider the number of lattice points $|nP \\cap \\Z^d|$ contained in the $n$-th dilation of $P$. \nEhrhart~\\cite{Ehr62} proved that $|nP \\cap \\Z^d|$ can be expressed by a polynomial in $n$ of degree $d$ for each $n$, denoted by $E_P(n)$. \nThis polynomial $E_P(n)$ is called the \\textit{Ehrhart polynomial} of $P$. \nMoreover, the generating function of $|nP \\cap \\Z^d|$ becomes the rational function which is of the form \n$$1+\\sum_{n = 1}^\\infty |nP \\cap \\Z^d| t^n =\\frac{\\sum_{i = 0}^dh_i^*t^i}{(1-t)^{d+1}},$$ \nwhere each $h_i^*$ is an integer. The sequence of integers $h^*(P)=(h_0^*,h_1^*,\\ldots,h_d^*)$ is called the \\textit{$h^*$-vector} (or \\textit{$\\delta$-vector}) of $P$, \nand the polynomial $h_P^*(t)=h_0^*+h_1^*t+\\cdots+h_d^*t^d$ is called the \\textit{$h^*$-polynomial} (or \\textit{$\\delta$-polynomial}) of $P$. \nWe call the degree of the $h^*$-polynomial of $P$ the \\textit{degree} of $P$, denoted by $\\deg(P)$. \nNote that the degree of a lattice polytope is at most the dimension of polytope. This implies that \nlattice polytopes having degree at most $d$ are a kind of generalization of lattice polytopes of dimension $d$. \n\nOn the $h^*$-vector $h^*(P)=(h_0^*,\\ldots,h_d^*)$ of a lattice polytope $P$ of dimension $d$, the following facts are well know. \n\\begin{itemize}\n\\item We have $\\displaystyle E_P(n)=\\sum_{i=0}^d h_i^*\\binom{n+d-i}{d}$. \n\\item Some of $h_i^*$ has a simple description in terms of $P$, e.g., $h_0^*=1$, $h_1^*=|P \\cap \\Z^d| - (d+1)$ and \n$h_i^*=| (d+1-i)P^\\circ \\cap \\Z^d|$ for each $\\deg(P) \\leq i \\leq d$, where $P^\\circ$ denotes the relative interior of $P$. \nIn particular, $h_d^*=|P^\\circ \\cap \\Z^d|$. Hence, we have \\begin{align}\\label{ineq_1d}h_1^* \\geq h_d^*.\\end{align} \n\\item The leading coefficient of $E_P(n)$ is equal to $\\sum_{i=0}^d h_i^*\/d!$ and this coincides with the relative volume of $P$ (\\cite[Corollaries 3.20, 3.21]{BR15}). \nThe sum $\\sum_{i=0}^d h_i^*$ of $h^*$-vector is called the \\textit{normalized volume} of $P$. \n\\item Each $h_i^*$ is nonnegative (\\cite{Sta80}). \n\\item If $h_d^* >0$, then we have $h_i^* \\geq h_1^*$ for each $1 \\leq i \\leq d-1$ (\\cite{Hib94}). \n\\end{itemize}\nFor more details on Ehrhart polynomials and $h^*$-vectors, consult e.g., \\cite{BR15}. \n\n\n\n\\bigskip\n\n\nOne of the most important problems in Ehrhart theory is the characterization of the polynomials that are the Ehrhart polynomials of some lattice polytopes. \nSince the Ehrhart polynomial and the $h^*$-vector are equivalent, i.e., we can know another one once we know the one, \nwe can rephrase this problem as follows: \n\\begin{quest}[{cf. \\cite[Question 1.1]{BH18}}]\nCharacterize the sequences $(h_0^*,h_1^*,\\ldots,h_d^*)$ of nonnegative integers that are the $h^*$-vectors of some lattice polytope of dimension $d$. \n\\end{quest}\nIt is easy to see that $(1,a)$ is always the $h^*$-vector of some lattice polytope of dimension $1$ for any nonnnegative integer $a$. \nThe case of $d=2$ is highly non-trivial, but the following is known: \n\\begin{thm}[{\\cite{Sco76}}]\\label{thm:Scott}\nLet $P$ be a lattice polytope $P$ of dimension $2$ with its $h^*$-vector $h^*(P)=(1,h_1^*,h_2^*)$. Then one of the following conditions is satisfied: \n\\begin{enumerate}\n\\item \\label{scott:1} $h^*_2 = 0$;\n\\item \\label{scott:2} $h_2^* \\leq h^*_1 \\leq 3h^*_2 + 3$;\n\\item \\label{scott:3} $h^*_1 = 7$ and $h^*_2=1$.\n\\end{enumerate}\n\\end{thm}\nWe can see that the converse of Theorem \\ref{thm:Scott} is also true, i.e., \nwe can construct a lattice polytope of dimension $2$ whose $h^*$-vector is equal to $(1,h_1^*,h_2^*)$ for each $(h_1^*,h_2^*)$ \nwhich satisfies one of the conditions \\eqref{scott:1}, \\eqref{scott:2} and \\eqref{scott:3} in Theorem \\ref{thm:Scott}. \n\nThe case $d \\geq 3$ is widely open, while some necessary inequalities have been recently conjectured in \\cite[Conjecture 8.7]{Balletti} in the case of $d=3$. \n\n\nAs a generalization of Theorem \\ref{thm:Scott}, the following is also known: \n\\begin{thm}[{\\cite[Theorem~2]{Tre10}}]\\label{thm:genScott}\nLet $P$ be a lattice polytope of degree at most two with its $h^*$-polynomial $h^*_P(t) = 1 + h^*_1 t + h^*_2 t^2$. \nThen one of the following conditions is satisfied: \n\\begin{enumerate}\n\\item \\label{genscott:1} $h^*_2 = 0$;\n\\item \\label{genscott:2} $h^*_1 \\leq 3h^*_2 + 3$;\n\\item \\label{genscott:3} $h^*_1 = 7$ and $h^*_2=1$. \n\\end{enumerate}\n\\end{thm}\nIt is proved in \\cite[Proposition~1.10]{HT09} that the converse of Theorem \\ref{thm:genScott} is also true. \nNote that the inequality $h_1^* \\geq h_2^*$ of Theorem~\\ref{thm:Scott} comes from \\eqref{ineq_1d}, \nwhile it does not appear in Theorem~\\ref{thm:genScott} since the dimension may be greater than two. \n\nMoreover, as a further generalization of Theorem~\\ref{thm:genScott}, the following is proved: \n\\begin{thm}[{\\cite[Theorem~1.4]{BH18}}]\\label{thm:BH}\nLet $P$ be a lattice polytope with its $h^*$-polynomial $h_P^*(t) = \\sum_{i \\geq 0}h_i^*t^i$. Assume $h^*_3=0$. \nThen one of the following conditions is satisfied: \n\\begin{enumerate}\n\\item \\label{main:1} $h^*_2 = 0$;\n\\item \\label{main:2} $h^*_1 \\leq 3h^*_2 + 3$;\n\\item \\label{main:3} $h^*_1 = 7$ and $h^*_2=1$.\n\\end{enumerate}\n\\end{thm}\nIn \\cite{BH18}, the inequalities \\eqref{main:1}, \\eqref{main:2} and \\eqref{main:3} are called \\textit{universal} \nbecause those are valid independently of both the dimension and the degree of lattice polytopes. \nA proof of Theorem \\ref{thm:BH} is based on the idea of \\textit{spanning polytopes} developed in \\cite{HKN16}. \n\nThe motivation to organize this paper is to find a new kind of universal inequalities or \\textit{universal conditions}. \n\n\n\n\\begin{defn}\nWe say that a sequence $(a_0,a_1,\\ldots,a_k)$ of nonnegative integers satisfies \\textit{degree $k$ condition} \nif there exists a lattice polytope $P$ of degree at most $k$ such that $h_P^*(t)=\\sum_{i=0}^k a_it^i$. \n\\end{defn}\n\nFor any lattice polytope $P$ with its $h^*$-vector $(h_0^*,h_1^*,\\ldots,h_d^*)$,\nwe see that $(h_0^*,h_1^*)$ satisfies degree $1$ conditions. \nMoreover, Theorem \\ref{thm:BH} says that the assumption ``$h_3^*=0$'' implies degree $2$ condition of $(h_0^*,h_1^*,h_2^*)$ \nof any $h^*$-vector $(h_0^*,h_1^*,\\ldots,h_d^*)$ of lattice polytopes. \nNote that the assumption $h_3^* =0$ is necessary for degree $2$ condition. See \\cite[Example 1.5]{BH18}. \n\nHence, it is natural to think of the assumptions which imply degree $k$ condition for $(h_0^*,h_1^*,\\ldots,h_k^*)$ \nof any $h^*$-vector $(h_0^*,h_1^*,\\ldots,h_d^*)$ of lattice polytopes in the case $k \\geq 3$. The main result of this paper is the following: \n\\begin{thm}\\label{mainthm}\nLet $\\Delta$ be a lattice simplex with its $h^*$-polynomial $h_\\Delta^*(t)=\\sum_{i \\geq 0} h_i^*t^i$. \nAssume that $h_{k+1}^*=h_{k+2}^*=\\cdots=h_{2k}^*=0$ for some $k \\geq 3$. \nThen there exists a face $\\Delta'$ of $\\Delta$ such that $h^*_{\\Delta'}(t)=\\sum_{i=0}^k h_i^*t^i$. \n\\end{thm}\nAs an immediate corollary, we obtain the following: \n\\begin{cor}\\label{maincor}\nLet $\\Delta$ be a lattice simplex with $h_\\Delta^*(t)=\\sum_{i \\geq 0} h_i^*t^i$. \nAssume that $h_{k+1}^*=h_{k+2}^*=\\cdots=h_{2k}^*=0$ for some $k \\geq 3$. Then $(h_0^*,h_1^*,\\ldots,h_k^*)$ satisfies degree $k$ condition. \n\\end{cor}\n\nA brief organization of this paper is as follows. \nFirst, in Section 2, we prepare a material which we will use in the proof of the main results. \nNext, in Section 3, we give a proof of Theorem \\ref{mainthm}. \nFinally, in Section 4, we show that the condition $h_{k+1}^*=h_{k+2}^*=\\cdots=h_{2k-1}^*=0$ of Theorem \\ref{mainthm} \nis necessary for degree $k$ condition. See Proposition \\ref{prop:main}. \n\n\n\\bigskip\n\n\n\n\n\n\n\\section{Preliminary}\n\nFirst of all, we prepare some tools for the computation of $h^*$-vectors of lattice simplices. \n\n\nLet $\\Delta \\subset \\R^d$ be a lattice simplex of dimension $d$ and let ${\\bf v}_1,\\ldots,{\\bf v}_{d+1} \\in \\Z^d$ be the vertices of $\\Delta$. \nWe use the usual notation $[d+1]:=\\{1,\\ldots,d+1\\}$. We define \n\\begin{align*}\n\\Lambda_\\Delta:=\\left\\{(r_1,\\ldots,r_{d+1}) : 0 \\leq r_j < 1 \\text{ for }\\forall j \\in [d+1], \\; \n\\sum_{i=1}^{d+1} r_i{\\bf v}_i \\in \\Z^d, \\; \\sum_{i=1}^{d+1} r_i \\in \\Z \\right\\}. \n\\end{align*}\nWe see that $\\Lambda_\\Delta$ is a finite abelian group by its addition \n$$\\alpha+\\beta=(\\{\\alpha_1+\\beta_1\\},\\ldots,\\{\\alpha_{d+1}+\\beta_{d+1}\\}) \\in \\Lambda_\\Delta$$ \nfor $\\alpha,\\beta \\in \\Lambda_\\Delta$, where $\\{r\\}=r-\\lfloor r \\rfloor$ for $r \\in \\R$. \nNote that ${\\bf 0}=(0, \\ldots, 0) \\in \\Lambda_\\Delta$ \nand $-\\alpha=(\\{1-\\alpha_1\\},\\ldots,\\{1-\\alpha_{d+1}\\}) \\in \\Lambda_\\Delta$ for $\\alpha \\in \\Lambda_\\Delta$. \n\nWe collect the notation concerning $\\Lambda_\\Delta$ which we will use. \n\\begin{itemize}\n\\item For $\\alpha \\in \\Lambda_\\Delta$, let $\\heit(\\alpha)=\\sum_{i=1}^{d+1}\\alpha_i$. \n\\item For $h=0,1,\\ldots,d$, let $$\\Lambda_\\Delta^{(h)}=\\{\\alpha \\in \\Lambda_\\Delta : \\heit(\\alpha)=h\\}.$$ \nNote that $\\Lambda_\\Delta = \\bigsqcup_{h=0}^d \\Lambda_\\Delta^{(h)}$. \nThe $h^*$-polynomial of $\\Delta$ can be computed as follows: \n\\begin{align}\\label{eq:h^*}h_\\Delta^*(t)=\\sum_{\\alpha \\in \\Lambda_\\Delta}t^{\\heit(\\alpha)}.\\end{align} \nSee \\cite[Corollary 3.11]{BR15}. \n\\item For $\\alpha \\in \\Lambda_\\Delta$, let $\\supp(\\alpha)=\\{ i \\in [d+1] : \\alpha_i > 0\\}$ \nand let $|\\alpha|=|\\supp(\\alpha)|$. \n\\item For $\\Gamma \\subset \\Lambda_\\Delta$, let $\\supp(\\Gamma)=\\bigcup_{\\alpha \\in \\Gamma}\\supp(\\alpha)$. \n\\end{itemize}\n\n\nWe observe that for any $\\alpha \\in \\Lambda_\\Delta$, we have \\begin{align}\\label{obser}|\\alpha|=\\heit(\\alpha)+\\heit(-\\alpha).\\end{align} \nIn fact, $$\\heit(\\alpha)+\\heit(-\\alpha)=\\sum_{i \\in \\supp(\\alpha)}\\alpha_i+\\sum_{i \\in \\supp(\\alpha)} (1-\\alpha_i)=|\\alpha|.$$ \n\n\n\\bigskip\n\n\\section{A proof of Theorem \\ref{mainthm}}\n\nFix an integer $k$ with $k \\geq 3$. \nIn what follows, we consider a lattice simplex $\\Delta$ with its $h^*$-polynomial $\\sum_{i \\geq 0}h_i^*t^i$ \nsatisfying $h_{k+1}^*=\\cdots=h_{2k}^*=0$. Namely, we have $\\Lambda_\\Delta^{(k+1)}=\\cdots=\\Lambda_\\Delta^{(2k)}=\\emptyset$. \nOur goal is to find a face $\\Delta'$ of $\\Delta$ such that $h_{\\Delta'}^*(t)=\\sum_{i=0}^k h_i^*t^i$. \n\n\n\\begin{lem}\\label{lem1}\nFor any $\\alpha \\in \\Lambda_\\Delta^{(h)}$ and $0 \\leq h \\leq k$, we have $|\\alpha| \\leq k+h$. \n\\end{lem}\n\\begin{proof}\nTake $\\alpha \\in \\Lambda_\\Delta^{(h)}$ with $0 \\leq h \\leq k$. Then $\\heit(\\alpha) = h$. \n\nSuppose that $|\\alpha| \\geq k+h+1$. Then it follows from \\eqref{obser} that \n$$\\heit(-\\alpha)=|\\alpha|-\\heit(\\alpha) \\geq k+1.$$ Hence $\\heit(-\\alpha) \\geq 2k+1$ by our assumption. \nLet $m$ denote the order of $\\alpha$, i.e., $m\\alpha={\\bf 0}$. Note that $-\\alpha=(m-1)\\alpha$. \nTake any $1 \\leq j \\leq m-1$. Then we see that \n$$\\heit(j\\alpha) = \\sum_{i=1}^{d+1} \\{ j\\alpha_i \\} \\leq \\sum_{i=1}^{d+1} \\{ (j-1)\\alpha_i \\} + \\sum_{i=1}^{d+1} \\alpha_i \n=\\heit((j-1)\\alpha)+h.$$ \nOn the one hand, $\\heit(\\alpha)=h \\leq k$. On the other hand, $\\heit((m-1)\\alpha) \\geq 2k+1$. \nThis means together with $\\heit(j\\alpha) \\leq \\heit((j-1)\\alpha)+k$ that there is $1 < j' <m-1$ with $k+1 \\leq \\heit(j'\\alpha) \\leq 2k$, \na contradiction. \n\\end{proof}\n\n\\begin{lem}\\label{lem2}\nLet $\\Lambda'=\\{\\alpha \\in \\Lambda_\\Delta : \\heit(\\alpha) \\leq k\\}$. Then the following statements hold: \n\\begin{itemize}\n\\item[(a)] $\\Lambda'$ is a subgroup of $\\Lambda_\\Delta$; \n\\item[(b)] $|\\supp(\\Lambda')| \\leq 4k-1$. \n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n(a) In general, one sees that $\\heit(\\alpha+\\beta) \\leq \\heit(\\alpha)+\\heit(\\beta)$ for $\\alpha,\\beta \\in \\Lambda_\\Delta$. \nSince $\\heit(\\alpha+\\beta) \\leq 2k$ for any $\\alpha,\\beta \\in \\Lambda'$, one has $\\heit(\\alpha+\\beta) \\leq k$. Thus, $\\alpha+\\beta \\in \\Lambda'$. \nMoreover, we see from Lemma \\ref{lem1} that one has $\\heit(-\\alpha) = |\\alpha|-\\heit(\\alpha) \\leq k$ for $\\alpha \\in \\Lambda'$. \nHence, $-\\alpha \\in \\Lambda'$. Clearly, ${\\bf 0} \\in \\Lambda'$ since $\\heit({\\bf 0})=0$. Therefore, $\\Lambda'$ is a subgroup of $\\Lambda_\\Delta$. \n\n\\noindent\n(b) Let $s=\\max\\{\\heit(\\alpha) : \\alpha \\in \\Lambda'\\}$. Then $s \\leq k$. \nBy \\cite[Lemma 2.2 and Proposition 2.3 (a)]{Hig18}, we obtain that $|\\supp(\\Lambda')| \\leq 2(2s)-1 \\leq 4k-1$. \n\\end{proof}\n\n\n\n\n\nNow we are ready to give a proof of Theorem \\ref{mainthm}. \n\n\\begin{proof}[Proof of Theorem \\ref{mainthm}]\nWork with the same notation as in Lemma \\ref{lem2}. Let $\\Delta'=\\conv(\\{v_i : i \\in \\supp(\\Lambda')\\})$.  \nThen $\\Delta'$ is a face of $\\Delta$. Note that $\\supp(\\Lambda_{\\Delta'})=\\supp(\\Lambda') \\subset [d+1]$ by the definition. \nIn what follows, we prove that $h_{\\Delta'}^*(t)=\\sum_{i=0}^k h_i^*t^i$. Our goal is to show that $\\Lambda_{\\Delta'}=\\Lambda'$. \nClearly, we have $\\Lambda' \\subset \\Lambda_{\\Delta'}$, where we regard $\\Lambda_{\\Delta'}$ as a subgroup of $\\Lambda_\\Delta$. \nWe prove that for any $\\alpha \\in \\Lambda_\\Delta \\setminus \\Lambda'$, we have $\\alpha \\not\\in \\Lambda_{\\Delta'}$. \n\nLet $\\alpha \\in \\Lambda_\\Delta \\setminus \\Lambda'$. Then $\\heit(\\alpha) \\geq 2k+1$. \nIf $\\heit(-\\alpha) \\leq k$, then $\\heit(\\alpha) \\leq k$ by Lemma \\ref{lem2} (a), a contradiction. Thus, $\\heit(-\\alpha) \\geq 2k+1$. \nFrom \\eqref{obser}, we see that \n\\begin{align*}\n2k+1 \\leq \\heit(-\\alpha)=|\\alpha|-\\heit(\\alpha) \\leq |\\alpha| - 2k-1, \n\\end{align*}\ni.e., $|\\alpha| \\geq 4k+2$. Now, Lemma \\ref{lem2} (b) says that $|\\gamma| \\leq 4k-1$ for any $\\gamma \\in \\Lambda'$. \nSince $|\\alpha| \\geq 4k+2>4k-1$, one sees that $\\supp(\\alpha) \\not\\subset \\supp(\\Lambda')$. \nSince $\\supp(\\Lambda') = \\supp(\\Lambda_{\\Delta'})$, we conclude that $\\alpha \\not\\in \\Lambda_{\\Delta'}$, as required. \n\\end{proof}\n\n\n\\bigskip\n\n\n\\section{Examples showing that $h_{k+1}^*=h_{k+2}^*=\\cdots=h_{2k-1}^*=0$ is necessary}\n\n\nWe will provide the example which shows that Theorem \\ref{mainthm} is not true \nif we drop the assumption $h_j^*=0$ for each $k+1 \\leq j \\leq 2k-1$ (Proposition \\ref{prop:main}). \n\nWe recall a useful construction of lattice polytopes. Let $P \\subset \\R^d$ and $Q \\subset \\R^e$ be lattice polytopes. We define the \\textit{join} of $P$ and $Q$: \n$$P \\star Q:=\\conv(\\{(0,{\\bf x}, {\\bf 0}_e) : {\\bf x} \\in P\\} \\cup \\{(1, {\\bf 0}_d, {\\bf y}) : {\\bf y} \\in Q\\}) \\subset \\R^{d+e+1},$$ \nwhere ${\\bf 0}_d$ (resp. ${\\bf 0}_e$) denotes the origin of $\\R^d$ (resp. $\\R^e$). \nNote that $P \\star Q$ is a simplex if and only if so are $P$ and $Q$. \nIt is known from \\cite[Lemma 1.3]{HT09} that $$h_{P \\star Q}^*(t)=h_P^*(t)h_Q^*(t).$$ \n\n\\begin{lem}\\label{construct}\nFor any positive integers $a,b,k,\\ell$, there exists a lattice simplex $\\Delta$ \nwith its $h^*$-polynomial $1+at^k+bt^\\ell+(a+b)t^{k+\\ell}$. \n\\end{lem}\n\\begin{proof}\nFor positive integers $c$ and $m$, let $\\Delta_{c,m}$ be a lattice simplex with its $h^*$-polynomial $1+ct^m$. \nSuch a simplex is given in e.g., \\cite[Example 3.22]{BR15}.\nA required simplex can be obtained by $\\Delta_{a,k} \\star \\Delta_{b,\\ell}$. \n\\end{proof}\n\n\\begin{lem}\\label{HHH}\nFor any prime number $p$ with $p \\geq 5$ and integers $i,j$ with $2 \\leq i < j$, \nthere exists no lattice polytope whose $h^*$-polynomial is $1+t^i+(p-2)t^j$. \n\\end{lem}\n\\begin{proof}\nSuppose that there is a lattice polytope $P$ with $h^*_P(t)=1+t^i+(p-2)t^j$. \nSince the linear term of $h_P^*(t)$ vanishes, $P$ must be a simplex. \nNow, \\cite[Theorem 1.1 (a)]{Hig14} says that the $h^*$-polynomial with a prime normalized volume should satisfy \n$h_{i+1}^*=h_{s-i}^*$ for any $i=0,1,\\ldots,s$, where $s$ is the degree of lattice polytope. \nHowever, since $p \\geq 5$, this never happens, a contradiction. \n\\end{proof}\n\n\n\\begin{prop}\\label{prop:main}\nFor any $k \\geq 3$ and $k+1 \\leq j \\leq 2k-1$, there exists a lattice simplex $\\Delta$ with $h^*_\\Delta(t)=\\sum_{i \\geq 0}h_i^*t^i$ \nsuch that \\begin{itemize}\n\\item $h_{k+1}^*=\\cdots=h_{j-1}^*=h_{j+1}^*=\\cdots=h_{2k-1}^*=0$, \n\\item $h_j^* \\neq 0$ and \n\\item $(h_0^*,\\ldots,h_k^*)$ does not satisfy degree $k$ condition. \n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nWhen $j \\geq k+2$, let $a=j-k$ and $b=k$. Note that $2 \\leq a \\leq k-1$. \nThen Lemma \\ref{construct} says that there exists a lattice simplex $\\Delta$ with its $h^*$-polynomial \n$1+t^a+(p-2)t^b+(p-1)t^{a+b}=1+t^{j-k}+(p-2)t^k+(p-1)t^j$, where we let $p$ be a prime number with $p \\geq 5$. \nTherefore, by Lemma \\ref{HHH}, this $h^*$-polynomial enjoys the required properties. \n\nIn the case $j=k+1$ and $k \\geq 4$, there exists a lattice simplex $\\Delta$ with its $h^*$-polynomial \n$1+t^2+(p-2)t^{k-1}+(p-1)t^{k+1}$ by Lemma \\ref{construct}, where we let $p$ be a prime number with $p \\geq 5$, \nand this $h^*$-polynomial enjoys the required properties by Lemma \\ref{HHH}. \n\nIn the case $j=k+1$ and $k=3$, consider the lattice simplex \n$$\\mathrm{conv}({\\bf 0},{\\bf e}_1,\\ldots,{\\bf e}_4,{\\bf e}_1+4{\\bf e}_2+7{\\bf e}_3+8{\\bf e}_4+9{\\bf e}_5\\}) \\subset \\R^5.$$ \nThen the $h^*$-polynomial of this simplex is equal to $1+2t^2+4t^3+2t^4$. \nSince $(1,0,2,4)$ does not satisfy degree $3$ condition by Lemma \\ref{HHH}, we conclude that this $h^*$-polynomial enjoys the required properties. \n\\end{proof}\n\n\nProposition \\ref{prop:main} says that we cannot drop the assumption $h_j^*=0$ in Corollary \\ref{maincor} (Theorem \\ref{mainthm}, too) for any $k+1 \\leq j \\leq 2k-1$. \n\\begin{rem}\\label{chuui}\nThere exists a lattice simplex $\\Delta$ of dimension $3k-1$ such that $h_\\Delta^*(t)=1+t^k+t^{2k}$ for any $k \\geq 2$. \nIn fact, we may set $\\Delta=\\mathrm{conv}(\\{{\\bf 0}, \\eb_1,\\ldots,\\eb_{d-1},2(\\eb_1+\\cdots+\\eb_{d-1})+3\\eb_d)\\})$, \nwhere $d=3k-1$ and $\\eb_1,\\ldots,\\eb_d$ denote the unit vectors of $\\R^d$. \nMoreover, we see from \\cite[Section 4.2]{HHL12} that this is the only lattice simplex of dimension $3k-1$ up to unimodular equivalence \nwhose $h^*$-polynomial is $1+t^k+t^{2k}$. On the other hand, we see that this $h^*$-vector is {\\bf shifted symmetric} (see \\cite{Hig10}), \ni.e., $h_{i+1}^*=h_{d-i}^*$ for each $0 \\leq i \\leq d-1$. \nThus, the $h^*$-polynomial of every face of $\\Delta$ is $1$ by \\cite[Theorem 2.1]{Hig10}. \n\nTherefore, Theorem \\ref{mainthm} is not true under the assumption $h_{k+1}^*=\\cdots=h_{2k-1}^*=0$, i.e., the assumption $h_{2k}^*=0$ is necessary. \n\nHowever, Corollary \\ref{maincor} might be still true even if $h_{2k}^* \\neq 0$. In fact, $1+t^k$ is a possible $h^*$-polynomial of a lattice simplex $\\Delta_{1,k}$. \n\\end{rem}\n\n\n\\bigskip\n\n\n\\section{Future Questions}\n\n\nFinally, we suggest two questions: \n\\begin{quest}\\label{q1}\nLet $P$ be a lattice {\\bf polytope} with $h_P^*(t)=\\sum_{i \\geq 0}h_i^*t^i$. \nAssume that $h_{k+1}^*=h_{k+2}^*=\\cdots=h_{2k}^*=0$. Then does $(h_0^*,h_1^*,\\ldots,h_k^*)$ satisfy degree $k$ condition? \n\\end{quest}\n\n\\begin{quest}\\label{q2}\nLet $\\Delta$ be a lattice simplex with $h_\\Delta^*(t)=\\sum_{i \\geq 0}h_i^*t^i$. \nAssume that $h_{k+1}^*=h_{k+2}^*=\\cdots=h_{2k-1}^*=0$. Then does $(h_0^*,h_1^*,\\ldots,h_k^*)$ satisfy degree $k$ condition? \nIs this also true for polytopes?\n\\end{quest}\n\n\n\\bigskip\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\subsection{General introduction}\n\n\\noindent\nThe Nelson model, describing a fixed number of spinless non-relativistic quantum particles linearly coupled to a field of spinless bosons, is an important testing ground for various mathematical aspects of the ultraviolet problem. Originally proposed by Nelson \\cite{Nelson.1964,Nelson.1964c} as an example of an explicitly ultraviolet-renormalizable theory, it is the subject of ongoing research up to date. \n\nWe investigate an adaption of the original Nelson model in which the particles have a relativistic dispersion relation, from now on referred to as the {\\em relativistic Nelson model}, and focus on its probabilistic analysis. From a physical point of view, \nsuch an ultraviolet-renormalizable relativistic model is of special interest, because one expects the particles to obey the relativistic dispersion relation if the system can also describe very high energies. Due to a stronger ultraviolet divergence in three space dimensions, we restrict our attention to the case of two space dimensions. We assume the bosons to have a mass, for otherwise the model would be unstable in two dimensions.\n\nThe renormalization problem for the relativistic Nelson model has been treated in the articles \\cite{Sloan.1974} and \\cite{Gross.1973,DeckertPizzo.2014} in two and three spatial dimensions, respectively. Recently, Schmidt \\cite{Schmidt.2019} renormalized the two-dimensional model using the technique of interior boundary conditions, which allowed him to prove stronger convergence results and to give an explicit expression for the renormalized operator. A related model, in which a fermionic scalar field is linearly coupled to a bosonic scalar field has recently been renormalized via a resummation scheme technique in \\cite{AlvarezMoller.2022}, where the physical example can also only be treated in two space dimensions. \n\nMotivated by the recently revived interest in the model, we derive a Feynman--Kac formula for the renormalized relativistic Nelson Hamiltonian and discuss some of its applications. Similar formulas for the {\\em non-relativistic} model have been derived in \\cite{Nelson.1964c,GubinelliHiroshimaLorinczi.2014,MatteMoller.2018}. The technique we use is built upon that of \\cite{MatteMoller.2018}. Explicitly, we derive an expression for the Feynman--Kac semigroup with ultraviolet cutoff and a cutoff dependent energy counter term, which still is mathematically meaningful when the cutoff is dropped. We then prove that the generator of this semigroup without cutoff is the norm resolvent limit of ultraviolet regularized Nelson Hamiltonians with added energy counter terms as the cutoff goes to infinity. In particular, the generator of our limiting semigroup agrees with the renormalized Hamiltonians constructed in \\cite{Sloan.1974,Schmidt.2019}.\nCompared to \\cite{MatteMoller.2018}, we need to replace the Brownian motion as generator of the particle dynamics by a suitable L\\'evy process.\nFurther, in comparison to \\cite{Sloan.1974,Schmidt.2019}, the probabilistic technique easily allows us to treat a broad class of external potentials.\n\nAlong similar lines to \\cite{MatteMoller.2018}, our\nprobabilistic analysis also entails a variety of lower bounds on the minimal energy of the relativistic Nelson model, with only a minor amount of extra work. Combined with more well-known trial function arguments, it further yields upper bounds,\nwhich asymptotically match with the lower ones in the translation invariant case (no external potential). More precisely, we are able to compute the leading asymptotics of the minimal energy in the three regimes where the number of matter particles goes to infinity, the coupling constant for the matter-radiation interaction goes to infinity and the boson mass goes to zero. Here, we are always discussing the minimal energy in our model without any symmetry restrictions imposed on the matter particles (no Pauli principle).\n\nFurther results of our analysis are weighted $L^p$ to $L^q$ bounds on the semigroup, which also is shown to map\nbounded sets in $L^p$ onto equicontinuous sets of functions. As observed in \\cite{Matte.2016,MatteMoller.2018}\nthe general structure of our Feynman--Kac formula easily entails ergodicity of the semigroup, \na result that has been obtained in \\cite{Sloan.1974b} for the fiber Hamiltonian at total momentum zero\nin the translation invariant case but seems to be new for the full Hamiltonian. \nA well-known implication of our continuity \nand ergodicity results is the continuity, non-degeneracy and strict positivity (for the appropriate phase factor) of ground\nstate eigenvectors (if any). The weighted $L^p$ to $L^q$ bounds can be used to turn $L^2$-bounds on the\nexponential localization of ground state eigenvectors into pointwise decay estimates; \nsee~\\cite{HiroshimaMatte.2019}\nfor such results in the original Nelson model. A Markov property satisfied by our Feynman--Kac integrands\nand the ergodicity are the crucial prerequisites for the construction of path measures associated with ground states; \nsee again~\\cite{HiroshimaMatte.2019}.\n\n\\subsection*{Structure and Notation}\n\nThe article is structured as follows: In the remainder of this introduction we define the relativistic \nNelson model studied here and take a first glance at our Feynman--Kac formula, as its thorough presentation is \nsomewhat lengthy. Precise statements of our results on the Feynman--Kac formula, ergodicity,\nas well as bounds on and asymptotics of the minimal energy can be found in \\cref{secmainres}.\n In \\cref{sec:basicproc} we discuss some basic stochastic processes contributing to our Feynman--Kac integrands\n and in particular to the analogue of Feynman's complex action.\n The full complex action is studied in \\cref{sec:complex}; the bounds derived therein will eventually imply our lower\n bounds on the minimal energy.\n In \\cref{sec:FKInt} we then treat the full integrands in the Feynman--Kac formula and prove\n Markov, semigroup and continuity properties as well as weighted $L^p$ to $L^q$ bounds.\n \\cref{sec:FK} is devoted to proving the Feynman--Kac formula. In the final \\cref{sec:upbound} we prove the upper bounds\n on the minimal energy. The main text is followed by three appendices respectively dealing with the relativistic Kato class,\n a corollary of an exponential tail estimate for L\\'{e}vy type stochastic integrals\n due to Applebaum and Siakalli \\cite{Applebaum.2009,Siakalli.2019} and bounds on the free relativistic semigroup.\n\nLet us fix some general notational conventions:\n\\begin{itemize}\n\t\\item The characteristic function of a set $M\\subset \\mathbb{R}^\\nu$, $\\nu\\in\\mathbb{N}$ is denoted by $\\chi_M$.\n\t\\item The two-dimensional open ball of radius $r>0$ about $0$ is denoted as \n\t\\begin{align}\\label{2Dball}\n\tB_r \\coloneq \\{y\\in\\mathbb{R}^2|\\,|y|<r\\}.\n\t\\end{align}\n\t\\item For $a,b\\in\\mathbb{R}$, we write $a\\wedge b = \\min \\{a,b\\}$ and $a\\vee b = \\max\\{a,b\\}$.\n\t\\item The set of bounded operators on some Banach space $\\mc{X}$ is denoted by $\\mathcal{B}(\\mc{X})$.\n\t\\item The Borel $\\sigma$-algebra of some topological space $S$ is denoted by $\\fr{B}(S)$.\n\t\\item The symbol $\\mathcal{D}(\\cdot)$ denotes domains of definition. If $A$ is  a selfadjoint operator in some Hilbert space,\nthen \t$\\mathcal{D}(A)$ is equipped with the graph norm of $A$.\n\\end{itemize}\n\n\\subsection{The relativistic Nelson model in two spatial dimensions}\n\n\\noindent\nLet us now define the model studied in this paper. For brevity we shall freely use some standard notation\nfor operators acting in the bosonic Fock space $\\mathcal{F}$, which is modeled over the one-boson space $L^2(\\mathbb{R}^2)$; the definitions of $\\mathcal{F}$ and all relevant operators acting in it are recalled in \\cref{subsec:Fock} below.\n\nIn the translation invariant case our model is determined by four parameters:\nThe number $N\\in\\mathbb{N}$ and mass $m_{\\mathrm{p}}\\ge0$ of the matter particles, the boson mass $m_{\\mathrm{b}}>0$ \nand the coupling constant $g\\in\\mathbb{R}\\setminus\\{0\\}$ for the matter-radiation interaction. The dispersion relations for\na single matter particle and a single boson are given, respectively, by\n\\begin{align}\\label{defpsi}\n\\psi(\\eta)&\\coloneq(|\\eta|^2+m_{\\mathrm{p}}^2)^{1\/2}-m_{\\mathrm{p}},\\quad\\eta\\in\\mathbb{R}^2,\n\\\\\\label{defomega}\n\\omega(k)&\\coloneq (|k|^2+m_{\\mathrm{b}}^2)^{1\/2},\\quad k\\in\\mathbb{R}^2.\n\\end{align}\nThe coupling function for the matter-radiation interaction is given by\n\\begin{align}\\label{defv}\nv&\\coloneq g\\omega^{-1\/2}.\n\\end{align}\nWe shall use the capital letter $K_i$ to denote multiplication by $k_i$, i.e.,\n\\begin{align}\\label{eq:defK}\n(K_if)(k)&\\coloneq k_if(k),\\quad k\\in \\mathbb{R}^2,\\quad\\text{for every $f:\\mathbb{R}^2\\to\\mathbb{C}$ and $i\\in\\{1,2\\}$.}\n\\end{align}\nSince $v\\notin L^2(\\mathbb{R}^2)$, the bosonic field operators $\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot y} v)$, $y\\in\\mathbb{R}^2$, are ill-defined.\nThis makes the energy renormalization procedure necessary that we employ in the construction of the\nHamiltonian $H$ hereinafter: \n\nLet us write\n\\begin{align}\\label{defxxj}\nx&=(x_1,\\ldots,x_N),\\quad\\text{with $x_1,\\ldots,x_N\\in\\mathbb{R}^2$,}\n\\end{align}\nand introduce the ultraviolet cutoff coupling functions\n\\begin{align}\\label{defvUV}\nv_{\\Lambda}\\coloneq \\chi_{B_{\\Lambda}}v,\n\\quad \\Lambda\\in[0,\\infty).\n\\end{align}\nSince $v_{\\Lambda}\\in L^2(\\mathbb{R}^2)$ for all $\\Lambda\\in[0,\\infty)$, \nthe field operators $\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}v_\\Lambda)$ are now well-defined.\nFurthermore, they are infinitesimally bounded, uniformly in $x$, with respect to \nthe radiation field energy $\\mathrm{d}\\Gamma(\\omega)$.\n\nFinally, we add an external potential \n\\begin{align*}\nV&=V_+-V_-:\\mathbb{R}^{2N}\\longrightarrow \\mathbb{R},\\quad\\text{with $V_+\\coloneq V\\vee0$.}\n\\end{align*} \nThroughout this article $V$ is assumed to be Kato decomposable with respect to the L\\'{e}vy process\ndescribing the trajectories of the $N$ matter particles. \nThe latter notion will be explained in the beginning of \\cref{subsec:FK}. For the moment\nit is sufficient to know that our assumptions on $V$ imply its local integrability and infinitesimal form boundedness\nof its negative part $V_-$  with respect to the Hamiltonian for the free matter particles \n$\\sum_{j=1}^N \\psi(-\\mathrm{i}\\nabla_{x_j})$. \n\nWe can now define the regularized relativistic Nelson Hamiltonian with ultraviolet cutoff $\\Lambda\\in[0,\\infty)$, denoted $H_{\\Lambda}$.\nIt is acting in the Hilbert space $L^2(\\mathbb{R}^{2N},\\mathcal{F})$. Analogously to \\eqref{defxxj} we write\n\\begin{align}\\label{defxixij}\n\\xi&=(\\xi_1,\\ldots,\\xi_N),\\quad \\text{with $\\xi_1,\\ldots,\\xi_N\\in\\mathbb{R}^2$,}\n\\end{align}\nand denote by $\\wh{\\Psi}$ the $\\mathcal{F}$-valued Fourier transform of $\\Psi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$. \nThen the form domain $\\mathcal{Q}(H_{\\Lambda})$ of $H_{\\Lambda}$ is the set of all $\\Psi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$ such that\n$\\Psi(x)$ is in the form domain of the radiation field energy $\\mathrm{d}\\Gamma(\\omega)$ for a.e. $x$ and such that\n\\begin{align*}\n\\mathfrak{q}[\\Psi]&\\coloneq  \\sum_{j=1}^N\\int_{\\mathbb{R}^{2N}}\n\\psi(\\xi_j)\\|\\wh{\\Psi}(\\xi)\\|_{\\mathcal{F}}^2\\mathrm{d}\\xi+\\int_{\\mathbb{R}^{2N}} \\big(V_+(x)\\|\\Psi(x)\\|_{\\mathcal{F}}^2\n+\\|\\mathrm{d}\\Gamma(\\omega)^{1\/2}\\Psi(x)\\|^2_{\\mathcal{F}}\\big)\\mathrm{d} x\n\\end{align*}\nis finite. By the above considerations the quadratic form\n$\\mathfrak{h}_{\\Lambda}:\\mathcal{Q}(H_{\\Lambda})\\to\\mathbb{R}$ given by\n\\begin{align}\\nonumber\n\\mathfrak{h}_{\\Lambda}[\\Psi]&\\coloneq  \\mathfrak{q}[\\Psi]-\\int_{\\mathbb{R}^{2N}} V_-(x)\\|\\Psi(x)\\|_{\\mathcal{F}}^2\\mathrm{d} x\n\\\\\\label{def:HUV}\n&\\quad\n+\\sum_{j=1}^N\\int_{\\mathbb{R}^{2N}}\\langle\\Psi(x)|\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}v_\\Lambda)\\Psi(x)\\rangle_{\\mathcal{F}}\\mathrm{d} x\n+NE_{\\Lambda}^{\\mathrm{ren}}\\|\\Psi\\|^2,\n\\end{align}\nfor all $\\Psi\\in \\mathcal{Q}(H_{\\Lambda})$, is closed and lower semibounded.\nIn its definition we already added the energy counter term $NE_{\\Lambda}^{\\mathrm{ren}}$ with\n\\begin{align}\\label{def:EUVren}\nE_{\\Lambda}^{\\mathrm{ren}}&\\coloneq \\int_{B_{\\Lambda}}\\frac{g^2}{\\omega(k)(\\omega(k)+\\psi(k))}\\mathrm{d} k,\\quad \\Lambda\\in[0,\\infty).\n\\end{align}\nBy definition $H_{\\Lambda}$ is the unique selfadjoint operator in $L^2(\\mathbb{R}^{2N},\\mathcal{F})$ representing $\\mathfrak{h}_{\\Lambda}$.\n\nThe renormalized relativistic Nelson Hamiltion in two space dimensions is now given by\n\\begin{align}\\label{reslimHUV}\nH&\\coloneq \\underset{\\Lambda\\to\\infty}{\\textrm{norm-resolvent-lim}} \\ H_\\Lambda.\n\\end{align}\nAs alluded to above, existence of\nthe norm resolvent limit has been established in \\cite{Sloan.1974,Schmidt.2019}, at least for the case $V=0$.\n\nOur Feynman--Kac formula for $H$ (\\cref{thm:FK})\ncan be stated as follows: For all  $t>0$ and $\\Psi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$, \nthere is a unique continuous representative of $e^{-tH}\\Psi$ satisfying\n\\begin{align*\n\t\\begin{aligned}\n\t\t\t(e^{-tH}\\Psi)(x) &= \n\t\t\t\\mathbb{E}\\left[e^{u_t^N(x)-\\int_0^tV(X_s^x)\\mathrm{d} s}F_{t\/2}(-U_t^{N,-}(x))F_{t\/2}(-U_t^{N,+}(x))^*\\Psi(X_t^x)\\right],\n\t\t\n\t\\end{aligned}\n\\end{align*}\nfor all $x\\in\\mathbb{R}^{2N}$.\nHere $X$ is a vector of $N$ independent identically distributed $\\mathbb{R}^2$-valued L\\'evy processes with L\\'{e}vy symbol $-\\psi$,\nand $X^x=x+X$. The real-valued continuous and adapted stochastic process $u^N(x)$ is the complex action and \n$U^{N,\\pm}(x)$ are continuous adapted stochastic processes taking values in $L^2(\\mathbb{R}^2)$. \nFurthermore, for fixed $h\\in L^2(\\mathbb{R}^2)$, $F_t(h)$ is an operator on $\\mathcal{F}$ \nexplicitly given as an exponential series of creation operators $a^\\dagger(h)$\ncontrolled by the semigroup at time $t$\nof the free radiation field; see \\cref{defFtg}. Its adjoint $F_t(h)^*$ contains annihilation terms.\nThe processes $u^N(x)$ and $U^{N,\\pm}(x)$ are given by explicit formulas involving (stochastic) integrals\nthat are mathematically meaningful directly without any cutoff. Nevertheless, \nwe will actually re-prove the existence of \\eqref{reslimHUV}, mainly to verify that $H$ generates the\nsemigroup defined by the right hand side of the Feynman--Kac formula. Here we shall\nmake use of our assumption that $V$ be Kato decomposable in a relativistic $N$-particle sense. \n\nIf the boson mass was zero, there would still exist a canonical selfadjoint realization of $H_{\\Lambda}$ for suitable $V$, \nwhose spectrum was, however, unbounded from below; see, e.g., Lemma~\\ref{lemesaNC} and Corollary~\\ref{corubGauss1}\nbelow. Hence, an energy renormalization is impossible\nfor zero boson mass. This is why we assume $m_{\\mathrm{b}}$ to be strictly positive, while $m_{\\mathrm{p}}$ is only\nrequired to be non-negative.\n\n\n\\section{Main results}\\label{secmainres}\n\n\\noindent\nIn this section we present our main results in detail. To this end we first recall some necessary Fock space calculus\nin \\cref{subsec:Fock}. In \\cref{subsec:FK} we will step by step introduce all processes appearing in our Feynman--Kac formula\nand state the latter in \\cref{thm:FK}. Applications of the Feynman--Kac formula are described in \n\\cref{subsecerg} (ergodicity), \\cref{ssectwosided} (two-sided bounds on the minimal energy) \nand \\cref{ssecasymp} (asymptotics of the minimal energy).\n\n\n\\subsection{Important operators on Fock space}\\label{subsec:Fock}\nHere we shall briefly recall the definitions from Fock space theory necessary for an understanding of this article. For a more detailed introduction, we refer the reader to \\cite{Arai.2018,Parthasarathy.1992}.\n\nThe bosonic Fock space over $L^2(\\mathbb{R}^2)$ is the direct sum of Hilbert spaces\n\\begin{align}\\label{defFock}\n\\mathcal{F}&\\coloneq \\mathbb{C}\\oplus\\bigoplus_{n=1}^\\infty L_{\\mathrm{sym}}^2(\\mathbb{R}^{2n}),\n\\end{align}\nwith $L_{\\mathrm{sym}}^2(\\mathbb{R}^{2n})$ denoting the closed subspace in $L^2(\\mathbb{R}^{2n})$ of all its elements\n$\\phi_n$ satisfying $\\phi_n(k_{\\pi(1)},\\ldots,k_{\\pi(n)})=\\phi_n(k_1,\\ldots,k_n)$,\na.e. for every permutation $\\pi$ of $\\{1,\\ldots,n\\}$; here $k_1,\\ldots,k_n\\in\\mathbb{R}^2$. \n\nThe first operator in $\\mathcal{F}$ we\nintroduce is the selfadjoint radiation field energy denoted~$\\mathrm{d}\\Gamma(\\omega)$. Each subspace in \\eqref{defFock}\nis reducing for $\\mathrm{d}\\Gamma(\\omega)$. Its action on $\\phi=(\\phi_n)_{n=0}^\\infty\\in\\mathcal{D}(\\mathrm{d}\\Gamma(\\omega))$ is given by\n\\begin{align}\\label{defdGammaomega}\n(\\mathrm{d}\\Gamma(\\omega)\\phi)_0=0\\quad\\text{and}\\quad\n(\\mathrm{d}\\Gamma(\\omega)\\phi)_n(k_1,\\ldots,k_n)&=\\sum_{j=1}^n\\omega(k_j)\\phi_n(k_1,\\ldots,k_n),\n\\end{align}\na.e. for every $n\\in\\mathbb{N}$. Here $\\phi\\in\\mathcal{F}$ is in the domain of $\\mathrm{d}\\Gamma(\\omega)$, if and only if the\nexpressions on the right hand sides in \\eqref{defdGammaomega} define a new vector in $\\mathcal{F}$.\n\nNext, we introduce the annihilation operator, $a(f)$, associated with $f\\in L^2(\\mathbb{R}^2)$. \nIt is the closed operator whose action on $\\phi=(\\phi_n)_{n=0}^\\infty\\in\\mathcal{D}(a(f))$ reads\n\\begin{align}\\label{defaf}\n(a(f)\\phi)_n(k_1,\\ldots,k_n)&=(n+1)^{1\/2}\\int_{\\mathbb{R}^2}\\ol{f(k)}\\phi_{n+1}(k,k_1,\\ldots,k_n)\\mathrm{d} k,\n\\end{align}\na.e. for every $n\\in\\mathbb{N}$, and $(a(f)\\phi)_0=\\langle f|\\phi_1\\rangle$. \nAgain $\\phi\\in\\mathcal{F}$ is in $\\mathcal{D}(a(f))$, if and only if the previous right hand sides yield a vector in $\\mathcal{F}$.\nFinally, the creation operator, $a^\\dagger(f)$, and the selfadjoint field operator, $\\varphi(f)$, associated with $f$ are given by\n\\begin{align*}\na^\\dagger(f)&\\coloneq a(f)^*,\\qquad \\varphi(f)\\coloneq(a^\\dagger(f)+a(f))^{**}.\n\\end{align*}\n\nAs is well-known, $a(f)$, $a^\\dagger(f)$ and $\\varphi(f)$ are relatively $\\mathrm{d}\\Gamma(\\omega)^{1\/2}$-bounded\nfor all $f\\in L^2(\\mathbb{R}^2)$, because $m_{\\mathrm{b}}>0$.\nMore generally, the $n$-th powers of these operators are relatively $\\mathrm{d}\\Gamma(\\omega)^{n\/2}$-bounded\nfor all $n\\in\\mathbb{N}$. \nEmploying suitable relative bounds we can in fact verify operator norm convergence of the following series, \nthat we shall encounter in our Feynman--Kac formula, \n\\begin{align}\\label{defFtg}\nF_t(h)&\\coloneq \\sum_{n=0}^\\infty \\frac{1}{n!}a^\\dagger(h)^n\\mathrm{e}^{-t\\mathrm{d}\\Gamma(\\omega)},\\quad t>0,\\,h\\in L^2(\\mathbb{R}^2).\n\\end{align}\nThese series have been introduced in \\cite{GueneysuMatteMoller.2017} and details can be found in Appendix~6 of that paper. \n\n\n\\subsection{The Feynman--Kac formula}\\label{subsec:FK}\n\n\\noindent\nLet us now move to the introduction of probabilistic objects essential for this article and the presentation of our Feynman--Kac formula. We will use well-known notions from the theory of L\\'evy process. For an introductory textbook on the topic, we refer to \\cite{Applebaum.2009}.\n\n\\subsubsection{L\\'{e}vy processes for particle paths}\n\n\\noindent\nThroughout the article, let us assume that $(\\Omega,\\fr{F},(\\fr{F}_t)_{t\\ge0},\\mathbb{P})$ is a filtered probability space\nsatisfying the usual hypotheses of completeness and right-continuity. The letter $\\mathbb{E}$ denotes expectations\nwith respect to $\\mathbb{P}$.\nFurthermore, for every $j\\in\\{1,\\ldots,N\\}$,\nwe let $X_j=(X_{j,t})_{t\\ge0}$ denote a L\\'{e}vy process associated with the L\\'{e}vy symbol $-\\psi$ given by \\eqref{defpsi}. \nWe assume that $X_1, \\ldots, X_N$ are independent processes and put $X\\coloneq(X_1,\\ldots,X_N)$.\nIn this article we stick to the convention that Banach space-valued stochastic processes with time horizon $[0,\\infty)$ \nare called c\\`{a}dl\\`{a}g, iff {\\em all} their paths are right-continuous on $[0,\\infty)$ and have left-sided limits \nat every point in $(0,\\infty)$.\nWe assume that $X$ is c\\`{a}dl\\`{a}g in this sense. Then $X_{t-}$ stands for the pointwise defined\nleft limit $\\lim_{s\\uparrow t}X$ for all $t>0$. \nFor every $x\\in\\mathbb{R}^{2N}$, we further abbreviate $X^x\\coloneq x+X$ and $X_j^{x}\\coloneq x_j+X_j$, so that for instance\n$X_{j,t-}^x=x_j+\\lim_{s\\uparrow t}X_{j,s}$ for $t>0$; recall \\eqref{defxxj}. Let us finally mention that each $X_j$\nis an example of an inverse Gaussian process; see, e.g., \\cite[Example~1.3.32]{Applebaum.2009}.\n\n\\subsubsection{Relativistic Kato class for $N$ particles}\\label{sssecKato}\n\n\\noindent\nFollowing \\cite{CarmonaMastersSimon.1990}\nwe introduce the Kato class corresponding to the process $X$ as\n\\begin{align}\\label{def:Kato}\n\t\\mc K_X \\coloneq \\left\\{f:\\mathbb{R}^{2N}\\to [0,\\infty) \\ \\mbox{ Borel meas.}\\,\\middle| \\\n\t\\lim_{t\\downarrow0}\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}\\left[ \\int_0^t f(X^x_s)\\mathrm{d} s\\right] = 0 \\right\\}.\n\\end{align}\nAs indicated earlier the external potential $V$ is always assumed to be Kato decomposable in the sense of $\\mc K_X$, i.e., \n$\\chi_CV_+,V_-\\in \\mc K_X$ for all compact $C\\subset\\mathbb{R}^{2N}$. As a consequence, \n$V$ is locally integrable and $V_-$ is infinitesimally form \nbounded with respect to the free particle Hamiltonian $\\sum_{j=1}^N \\psi(-\\mathrm{i}\\nabla_{x_j})$; \ncf. \\cite[Theorem~III.1 and the paragraph following its proof]{CarmonaMastersSimon.1990}. We also know that \n\\begin{align}\\label{defNVx}\n\\scr{N}_V(x)&\\coloneq\n\\big\\{\\gamma\\in\\Omega\\,\\big|\\: V(X^x_\\bullet(\\gamma)):[0,\\infty)\\to\\mathbb{R}\\;\\text{is not locally integrable}\\big\\}\n\\end{align}\nis a $\\mathbb{P}$-zero set for every $x\\in\\mathbb{R}^{2N}$. Once and for all we fix the convention that the integral paths\n$\\int_0^\\bullet V(X_s^x)\\mathrm{d} s$ equal $0$ on $\\scr{N}_V(x)$.\nIn \\cref{app:Kato} we discuss some further properties of Kato decomposable potentials to be used later on.\n\nFor example, $V$ is Kato decomposable in the sense of $\\mc K_X$, if it has the form\n\\begin{align*}\nV(x) &= \\sum_{j=1}^N V_{\\{j\\}}(x_j)+\\sum_{1\\le i<j\\le N}V_{\\{i,j\\}}(x_i-x_j),\\quad x\\in\\mathbb{R}^{2N},\n\\end{align*} \nwhere $\\chi_C(V_{A})_+,(V_{A})_-\\in \\mc K_{X_1}$ \nfor all compact $C\\subset\\mathbb{R}^2$ and all subsets $A\\subset\\{1,\\ldots,N\\}$ of cardinality one or two.\nHere $\\mc K_{X_1}$ is defined as in \\cref{def:Kato} with $\\mathbb{R}^{2N}$ replaced by $\\mathbb{R}^2$ and $X$ replaced by $X_1$.\nThis follows from the independence of $X_1,\\ldots,X_N$ and since, for $i<j$ and $s\\ge0$, the laws of \n$X_{i,s}$ and $X_{j,s}$ have the same, rotationally symmetric probability density $\\rho_{m_{\\mathrm{p}},s}$ satisfying\n$\\rho_{m_{\\mathrm{p}},s}*\\rho_{m_{\\mathrm{p}},s}=\\rho_{m_{\\mathrm{p}},2s}$. In view of \\cref{correlLpLinfty}, we know that\n$L^p(\\mathbb{R}^2)\\cap L^{2p}(\\mathbb{R}^2)\\subset \\mc K_{X_1}$ for every $p\\in(1,\\infty]$.\nIf $m_{\\mathrm{p}}=0$, the corollary implies $L^p(\\mathbb{R}^2)\\subset \\mc K_{X_1}$ for all $p\\in (2,\\infty]$.\n\n\\subsubsection{The processes appearing in the creation\/annihilation terms}\nNext, we discuss the two processes to be substituted into the creation term $F_{t\/2}(h)$ \nand the annihilation term $F_{t\/2}(h)^*$ in our Feynman--Kac formula. The bound\n\\begin{align}\\label{eq:bounde-somv}\n\t\\|\\mathrm{e}^{-s\\omega}v\\|_2\\le|g|(2\\pi\/s)^{1\/2}\n\t, \\quad s>0,\n\\end{align}\nensures existence of the following $L^2(\\mathbb{R}^2)$-valued Bochner--Lebesgue integrals,\n\\begin{align}\\label{defUplusminus}\nU_t^-[\\gamma]&\\coloneq\\int_0^t\\mathrm{e}^{-s\\omega-\\mathrm{i} K\\cdot\\gamma_s}v\\mathrm{d} s,\\quad\nU_t^+[\\gamma]\\coloneq\\int_0^t\\mathrm{e}^{-(t-s)\\omega-\\mathrm{i} K\\cdot\\gamma_s}v\\mathrm{d} s,\\quad t\\ge0.\n\\end{align}\nHere $\\gamma:[0,\\infty)\\to\\mathbb{R}^2$ is assumed to be Borel measurable, $\\gamma_s\\coloneq\\gamma(s)$ as usual,\nand we used the notation \\eqref{eq:defK}. With this we define\n\\begin{align}\nU_t^{N,\\pm}(x)&\\coloneq \\sum_{j=1}^N\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}U_t^{\\pm}[X_{j,\\bullet}],\\quad x\\in\\mathbb{R}^{2N},\\,t\\ge0.\n\\label{def:UtN}\n\\end{align}\nFor each choice of the sign the $L^2(\\mathbb{R}^2)$-valued process $(U_t^{N,\\pm}(x))_{t\\ge0}$ \nis adapted and all its paths are continuous; see \\cref{remUpmcont,lem:Upmadapted} for details.\n\n\\subsubsection{The complex action}\n\n\\noindent\nWe shall next introduce the analogue of Feynman's complex action in our Feynman--Kac formula, \nwhich is the sum of three contributions coming from the radiation field.\n\nFirstly, the radiation field mediates an effective, translation and rotation invariant pair interaction potential \nbetween the matter particles that we call $w:\\mathbb{R}^2\\to\\mathbb{R}$:\nWith $J_0$ denoting the Bessel function of the first kind and order $0$, cf. \\cref{app:Bessel}, we set\n\\begin{align}\\label{defvt}\n\\vartheta(r,s)&\\coloneq \\frac{2\\pi rJ_0(rs)}{\\omega(r)(\\omega(r)+\\psi(r))},\n\\quad r,s\\ge0,\n\\end{align}\nwhere we from now on introduce the slightly abusive notation\n\\begin{align*}\n\t\\omega(r)\\coloneq (r^2+m_\\mathrm{b}^2)^{1\/2}, \\quad \\psi(r) \\coloneq (r^2+m_\\mathrm{p}^2)^{1\/2}-m_\\mathrm{p}, \\quad r\\ge 0.\n\\end{align*}\nThen $w$ is given by the following well-defined Lebesgue integrals\n\\begin{align}\\label{def:wy}\nw(y)&\\coloneq g^2\\int_0^{\\infty} \\vartheta(r,|y|)\\mathrm{d} r, \\quad y\\in\\mathbb{R}^2\\setminus\\{0\\},\n\\end{align}\nand we complement this by the (immaterial) convention $w(0)\\coloneq0$.\nFor $N\\ge2$, the total effective interaction term appearing in the complex action is\n\\begin{align}\\label{def:wtN}\nw_{t}^N(x)&\\coloneq \\sum_{1\\le j<\\ell\\le N}\\int_0^t2w(X^x_{j,s}-X^x_{\\ell,s})\\mathrm{d} s,\\quad x\\in\\mathbb{R}^{2N},\\,t\\ge0.\n\\end{align}\nFor $N=1$, there is no effective interaction and we set $w_t^1(x)\\coloneq0$. \nAs we shall see in \\cref{remdefw}, $w\\in L^p(\\mathbb{R}^2)$ for all $p\\in[2,\\infty)$, whence $w^N(x)$ is a well-defined\npath integral process according to the conventions and examples in \\cref{sssecKato}.\n\nLet us abbreviate\n\\begin{align}\\label{def:beta}\n\\beta&\\coloneq(\\omega+\\psi)^{-1}v=g\\omega^{-1\/2}(\\omega+\\psi)^{-1}\\in L^2(\\mathbb{R}^2).\n\\end{align}\nThen the second contribution to the complex action is\n\\begin{align}\\label{defcinfty}\nc^N_{t}(x)&\\coloneq \\sum_{\\ell=1}^N\n\\langle U_{t}^{N,+}(x)|\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{\\ell,t}^x}\\beta\\rangle,\n\\quad x\\in\\mathbb{R}^{2N},\\,t\\ge0.\n\\end{align}\nThis expression is in fact real-valued and uniformly bounded in $x$ and $t$ and on $\\Omega$; see \\cref{rem:bdcN}.\nThe process $(c_t^N(x))_{t\\ge0}$ is manifestly adapted and c\\`{a}dl\\`{a}g.\n\nMost work is needed to obtain control on the third contribution to the complex action, \nnamely the real-valued c\\`{a}dl\\`{a}g square-integrable martingale $(m_t^N(x))_{t\\ge0}$ with\n\\begin{align}\\label{def:martingale}\nm^N_{t}(x)&\\coloneq\\int_{(0,t]\\times\\mathbb{R}^{2N}}\\sum_{j=1}^N\\langle U_s^{N,+}(x)|\n\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{j,s-}^x}(\\mathrm{e}^{-\\mathrm{i} K\\cdot z_j}-1)\\beta\\rangle\\mathrm{d}\\wt{N}_X(s,z).\n\\end{align}\nHere $\\wt{N}_X$ denotes the martingale-valued measure associated with the L\\'{e}vy process $X$,\nthe integral is an $L^2$-stochastic integral and $z_j$ is the $j$'th two-dimensional component of $z$\nsimilarly as in \\eqref{defxxj}.\nFor more details, see the beginning of \\cref{subsec:m}.\n\nCombined, our formula for the complex action reads\n\\begin{align}\\label{def:action}\nu_t^N(x)&\\coloneq w_t^N(x)-c_t^N(x)+m_t^N(x),\\quad x\\in\\mathbb{R}^{2N},\\, t\\ge0.\n\\end{align}\nAs it turns out in \\cref{cor:pathsactioncont}, the paths of $u^N(x)$ are $\\mathbb{P}$-a.s. continuous, although the last\ntwo members of the right hand side of \\cref{def:action} are merely c\\`{a}dl\\`{a}g.\n\n\\subsubsection{Feynman--Kac integrand and formula}\n\n\\noindent\nWe are now in a position to state our Feynman--Kac formula where we abbreviate\n\\begin{align}\\label{def:Wt}\nW_t(x)&\\coloneq \\mathrm{e}^{u_t^N(x)-\\int_0^tV(X^x_s)\\mathrm{d} s}F_{t\/2}(-U_t^{N,+}(x))F_{t\/2}(-U_t^{N,-}(x))^*.\n\\end{align}\nFor every $t>0$ the $\\mathcal{B}(\\mathcal{F})$-valued function $W_t(x)$ on $\\Omega$ is $\\fr{F}_t$-measurable and separably valued\nand its operator norm $\\|W_t(x)\\|$ has moments of all orders; see \\cref{rem:Wadapt} and \\cref{remmomentbdW}. \n\n\\begin{thm}\\label{thm:FK}\nLet $\\Psi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$ and $t>0$. Then $\\mathrm{e}^{-tH}\\Psi$ has a unique continuous representative\nwhich for every $x\\in\\mathbb{R}^{2N}$ is given by\n\\begin{align}\\label{eq:FK}\n(\\mathrm{e}^{-tH}\\Psi)(x)&=\\mathbb{E}\\left[W_{t}(x)^*\\Psi(X_t^x)\\right].\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nThe continuity of the right hand side of \\eqref{eq:FK} in $x\\in\\mathbb{R}^{2N}$ follows from the \nstronger continuity result \\cref{thm:cont}.\nEquation \\eqref{eq:FK} itself is established in \\cref{sec:FK.ren}. \nAs a byproduct we shall in fact obtain a new proof of the existence of the\nthe norm resolvent limit \\crefnosort{reslimHUV}.\n\\end{proof}\n\nIn the following subsections we present some applications of \\cref{thm:FK}.\n\n\\subsection{Ergodicity}\\label{subsecerg}\n\nLet $\\mc{U}:\\mathcal{F}\\to L^2(\\mc{g})\\coloneq L^2(\\mc{Q},\\mc{g})$ be a unitary map from the\nFock space to a $\\mc{Q}$-space, so that $\\mc{g}$ is a probability measure on the set $\\mc{Q}$\nand all field operators $\\varphi(h)$ with\n\\begin{align}\\label{defmcRR}\nh\\in\\mc{h}_{\\mathbb{R}}&\\coloneq\\{f\\in L^2(\\mathbb{R}^{2})|\\,\\ol{f}(k)=f(-k),\\,\\text{a.e. $k$}\\}\n\\end{align}\nbecome Gaussian random variables on $\\mc{Q}$ after conjugation with $\\mc{U}$. Confer, e.g., \\cite{Parthasarathy.1992} for constructions of $\\mc{Q}$-space. \nWe denote by the same symbol the constant direct integral of $\\mc U$, i.e., $(\\mc{U}\\Psi)(x)=\\mc{U}\\Psi(x)$, \na.e. $x\\in\\mathbb{R}^{2N}$, $\\Psi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$.\nOf course we have a canonical notion of positivity for functions in\n$L^2(\\mathbb{R}^{2N},L^2(\\mc{g}))=L^2(\\mathbb{R}^{2N}\\times\\mc{Q},\\mathrm{d} x\\otimes\\mc{g})$, that we shall employ in the following theorem.\nIt is a consequence of \\cref{thm:FK} and its proof is a word by word  transcription of the proof of \\cite[Theorem~8.3]{MatteMoller.2018}. The crucial point is that the operators\n$\\mc{U}F_{t\/2}(h)F_{t\/2}(h)^*\\mc{U}^*$ with $t>0$ and $h\\in \\mc{h}_{\\mathbb{R}}$ improve positivity on $L^2(\\mc{g})$ as observed in\n\\cite{Matte.2016,MatteMoller.2018}, and $U^{N,\\pm}_{t}(x)$ are in fact $\\mc{h}_{\\mathbb{R}}$-valued in view of \\eqref{defUplusminus}.\n\n\\begin{thm}\nLet $t>0$. Then $\\mc{U}\\mathrm{e}^{-tH}\\mc{U}^*$ improves positivity.\n\\end{thm}\n\n\tSloan \\cite{Sloan.1974} has shown an analogous result for the fiber Hamiltonian at total momentum zero in the\n\trenormalized translation-invariant relativistic Nelson model.\n\tIn the non-relativistic renormalized Nelson model the ergodicity of fiber Hamiltonians attached to arbitrary\n\ttotal momenta -- with respect to a different positive cone in $\\mathcal{F}$ -- has been shown in\n\t\\cite{Lampart.2020,Miyao.2018}. \n\n\nA standard conclusion (see, e.g., \\cite[Corollaries~8.5\\,\\&\\,8.6]{MatteMoller.2018}) is the following:\n\n\\begin{cor}\\label{corforspecu}\\ \n\t\\begin{enumerate}\n\t\t\\item[{\\rm(i)}] Let $\\Psi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})\\setminus\\{0\\}$ be such that $\\mc{U}\\Psi$ is non-negative. Then\n\t\t\\begin{align*}\n\t\t\t\\inf\\sigma(H)&=-\\lim_{t\\to\\infty}\\frac{1}{t}\\ln\\langle\\Psi|\\mathrm{e}^{-tH}\\Psi\\rangle.\n\t\t\\end{align*}\n\t\t\\item[{\\rm{(ii)}}\n\t\tLet $f\\in L^2(\\mathbb{R}^{2N})\\setminus\\{0\\}$ be non-negative. Then\n\t\t\\begin{align*}\n\t\t\t\\inf\\sigma(H)&=-\\lim_{t\\to\\infty}\\frac{1}{t}\\ln\\bigg(\\int_{\\mathbb{R}^{2N}}f(x)\n\t\t\t\\mathbb{E}[\\mathrm{e}^{u^N_t(x)-\\int_0^tV(X^x_s)\\mathrm{d} s}f(X_t^x)]\\mathrm{d} x\\bigg).\n\t\t\\end{align*}\n\t\\end{enumerate}\n\\end{cor}\n\n\\subsection{Two-sided bounds on the minimal energy}\\label{ssectwosided}\n\n\\noindent\nWith the aid of \\cref{corforspecu} and exponential moment bounds on the complex action, \nwe obtain the estimates on the minimal energy presented in the following. We only consider the\ntranslation invariant case. Non-zero external potentials $V$ can be incorporated by using form bounds\nrelative to $H$ and variational arguments.\n\n\\subsubsection{Lower bounds}\n\n\\noindent\nThe first lower bound in the next theorem seems to be reasonably good for small\ncoupling constants $g$. In this regime it provides a lower bound proportional to $-g^4N^3$,\nwhich is the behavior observed for all values of $g$ and $N$ in the non-relativistic Nelson\nmodel \\cite{MatteMoller.2018}. (The minimal energy depends on the choice of the energy counter terms, of course.\nIn \\cite{MatteMoller.2018} the authors work in 3D, set $m_{\\mathrm{p}}=1$ and use \\eqref{def:EUVren} \nwith $|k|^2\/2$ put in place of $\\psi(k)$.)\nThe second and third bounds in the next theorem show the correct behavior for large $g$ and\/or $N$\nas we shall see in \\cref{ssecasymp}.\n\n\\begin{thm}\\label{thm:lowbound}\nConsider the translation invariant case $V=0$.\nThere exist constants $b,b'>0$ such that, for all coupling constants~$g$ and particle numbers~$N$,\n\\begin{align*}\n\\inf\\sigma(H)&\\ge -\\frac{bg^4N^3}{m_{\\mathrm{b}}}\\cdot\\mathrm{e}^{8\\pi g^2N\/m_{\\mathrm{b}}}\n-\\frac{b'g^4N^2(N-1)}{m_{\\mathrm{b}}}\\bigg(1+\\frac{m_{\\mathrm{p}}}{m_{\\mathrm{b}}}\\bigg).\n\\end{align*}\nFurthermore, there exist constants $c,c'>0$ such that\n\\begin{align*}\n\\inf\\sigma(H)&\\ge -\\bigg\\{\\pi g^2(2N^2-N)\\ln\\bigg(\\frac{g^2N}{m_{\\mathrm{b}}}\\bigg)\\bigg\\}-cg^2N^2-c'(N-1)m_{\\mathrm{p}},\n\\ \\ \\text{if $g^2N\\ge m_{\\mathrm{b}}$.}\n\\end{align*}\nFor $m_{\\mathrm{p}}>0$, the term in curly brackets $\\{\\cdots\\}$ can be replaced by\n\\begin{align*}\n2\\pi g^2N(N-1)\\ln\\bigg(\\frac{g^2N}{m_{\\mathrm{b}}}\\bigg)\n+\\pi g^2N\\ln(1\\vee(g^2N))+\\frac{\\pi g^2N}{m_{\\mathrm{p}}}[(g^2N)\\wedge1].\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nTo prove these bounds we choose a bounded, integrable, non-negative and non-zero $f:\\mathbb{R}^{2N}\\to\\mathbb{R}$ in\n\\cref{corforspecu}(ii) and estimate $f(X_t^x)\\le\\|f\\|_\\infty$. After that we apply\nthe exponential moment bounds on the complex phase\nin \\cref{thmexpmonentu1,thmexpmonentu2} with $p=1$. Since the parameter $\\alpha>1$ \nappearing in these theorems is arbitrary, we obtain the asserted lower bounds in this way.\n\\end{proof}\n\n\\subsubsection{Upper bound}\n\n\\noindent\nWe complement the lower bounds from the previous theorem with an upper bound, which follows from a combination\nof \\cref{corforspecu} and our exponential moment bounds on the complex action with a more well-known\ntrial state argument. \n\n\\begin{thm}\\label{thm:upbound}\nAssume that $V=0$\nand let $\\theta\\in(0,1)$. Then there exists $C(\\theta,m_{\\mathrm{p}},1\\vee m_{\\mathrm{b}})\\in(0,\\infty)$, depending solely on the\nquantities displayed in its argument, as well as a universal constant $c>0$ such that, whenever $g^2N>\\sqrt{2}m_{\\mathrm{b}}$,\n\\begin{align*}\n\\inf\\sigma(H)\n&\\le c(N-1)m_{\\mathrm{p}}+C(\\theta,m_{\\mathrm{p}},1\\vee m_{\\mathrm{b}})g^2N^2\n\\\\\n&\\quad-2\\pi \\theta g^2N(N-1)\\ln\\bigg(\\frac{g^2N}{m_{\\mathrm{b}}}\\bigg)-\\pi \\theta g^2N\\ln(1\\vee (g^2N))\n\\\\\n&\\quad-\\chi_{\\{0\\}}(m_{\\mathrm{p}})\\pi \\theta g^2N\\ln\\bigg(\\frac{1\\wedge(g^2N)}{m_{\\mathrm{b}}}\\bigg).\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nThe proof of this theorem can be found at the end of \\cref{sec:upbound}.\n\\end{proof}\n\n\n\\subsection{Asymptotics of the minimal energy}\\label{ssecasymp}\n\nCombined the upper and lower bounds in \\cref{ssectwosided} reveal the asymptotics of the minimal energy in the three regimes where the particle number $N$ and the coupling constant $g$ are large and the boson mass is small.\n\nFor convenience we do typically not display the dependence of $H$ and related objects on\nthe parameters $N$, $m_{\\mathrm{p}}$, $m_{\\mathrm{b}}$ and $g$ in our notation. An exception will be\nthe next theorem and the subsequent remark where the minimal energy of $H$ in the translation invariant case \nis denoted by\n\\begin{align*}\n\\mc{E}(g,N,m_{\\mathrm{p}},m_{\\mathrm{b}})&\\coloneq\\inf\\sigma(H)\\quad\\text{when $V=0$.}\n\\end{align*}\n\n\n\\begin{thm}\\label{thm:asymp}\nThe following relations hold for $V=0$,\n\\begin{align}\n\\lim_{N\\to\\infty}\\frac{\\mc{E}(g,N,m_{\\mathrm{p}},m_{\\mathrm{b}})}{N^2\\ln(N)}&=-2\\pi g^2,\n\\\\\n\\lim_{|g|\\to\\infty}\\frac{\\mc{E}(g,N,m_{\\mathrm{p}},m_{\\mathrm{b}})}{g^2\\ln(g^2)}&=-\\pi(2N^2-N),\n\\\\\\label{asympmb}\n\\lim_{m_{\\mathrm{b}}\\downarrow0}\\frac{\\mc{E}(g,N,m_{\\mathrm{p}},m_{\\mathrm{b}})}{\\ln(m_{\\mathrm{b}})}&=2\\pi g^2N(N-1),\\quad m_{\\mathrm{p}}>0,\n\\\\\n\\lim_{m_{\\mathrm{b}}\\downarrow0}\\frac{\\mc{E}(g,N,0,m_{\\mathrm{b}})}{\\ln(m_{\\mathrm{b}})}&=\\pi g^2(2N^2-N).\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nAll limit relations follow directly from \\cref{thm:lowbound,thm:upbound}, if we take into account that the\nparameter $\\theta\\in(0,1)$ appearing in \\cref{thm:upbound} can be chosen arbitrarily close to $1$.\n\\end{proof}\n\n\\begin{rem}\nIn the case $N=1$ with $m_{\\mathrm{p}}>0$, where \\eqref{asympmb} does not reveal any leading asymptotic\nbehavior as $m_{\\mathrm{b}}\\downarrow0$, the last statement in \\cref{thm:lowbound} implies the uniform lower bound\n\\begin{align}\n\\mc{E}(g,1,m_{\\mathrm{p}},m_{\\mathrm{b}})\\ge -\\pi g^2\\ln(g^2\\vee1)-\\frac{\\pi g^2}{m_{\\mathrm{p}}}[g^2\\wedge1]-cg^2.\n\\end{align}\nThat is, singular contributions to the minimal energy coming from $m_{\\mathrm{b}}$ have been compensated for\nby the renormalization energies $E_{\\Lambda}^{\\mathrm{ren}}$ in this case.\n\\end{rem}\n\n\\section{Properties of basic processes}\\label{sec:basicproc}\n\n\\noindent\nIn this \\lcnamecref{sec:basicproc} we provide crucial estimates on those processes in our Feynman--Kac formula \nwhose analysis poses new technical challenges. In the following three subsections we shall, respectively, address\nthe $L^2(\\mathbb{R}^2)$-valued processes in the creation\/annihilation terms, the martingale contribution to the complex action\nand the effective interaction potential mediated by the radiation field.\n\nFor technical purposes, we introduce versions of these with infrared and ultraviolet cutoffs, usually denoted by $\\sigma$ and $\\Lambda$, respectively. The purpose of $\\sigma$ is twofold: Firstly, when studying convergence properties as the ultraviolet cutoff\ngoes to infinity, we encounter difference terms with an effective, large infrared cutoff $\\sigma$ and with $\\Lambda=\\infty$.\nSecondly, to obtain good exponential moment bounds on the complex action, we have to split up some\nof its contributions into two parts, comprising boson momenta $|k|<\\sigma$ and $\\sigma\\le|k|<\\Lambda$,\nrespectively, and treat both parts differently.\n\n\\subsection{Processes appearing in the creation\/annihilation terms}\\label{sec:basicproc.UtN}\n\n\\noindent\nWe start our treatment with a discussion of the processes $U_t^{N,\\pm}$ introduced in \\cref{def:UtN}.\nFor all $0\\le \\sigma <\\Lambda\\le\\infty$, $t\\ge0$, $x\\in\\mathbb{R}^{2N}$, \nBorel measurable $\\gamma=(\\gamma_1,\\ldots,\\gamma_N):[0,\\infty)\\to\\mathbb{R}^{2N}$ \nand both choices of the sign we set\n\\begin{align}\\label{def:Ucutoff}\nU_{\\sigma,\\Lambda,t}^{N,\\pm}[x,\\gamma]&\\coloneq \n\\sum_{j=1}^N\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}\\chi_{B_\\Lambda\\setminus B_\\sigma}U_{t}^\\pm[\\gamma_j].\n\\end{align}\nPrior to \\eqref{defUplusminus} we already argued that $U^\\pm_t[\\gamma_j]$ are well-defined $L^2(\\mathbb{R}^2)$-valued\nBoch\\-ner-Le\\-besgue integrals for fixed $t\\ge0$ and Borel measurable $\\gamma_j:[0,\\infty)\\to\\mathbb{R}^{2}$.\n\nWe shall need the following bounds in the time-dependent norm $\\|\\cdot\\|_t\\ge\\|\\cdot\\|$ given by\n\\begin{align}\\label{deftnorm}\n\\|f\\|_t^2&\\coloneq\\int_{\\mathbb{R}^2}\\bigg(1+\\frac{1}{t\\omega(k)}\\bigg)|f(k)|^2\\mathrm{d} k,\\quad f\\in L^2(\\mathbb{R}^2), \\,t>0.\n\\end{align}\n\\begin{lem}\nLet $0\\le\\sigma<\\Lambda\\le\\infty$, $t>0$, $x\\in\\mathbb{R}^{2N}$ and  $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ be Borel measurable. Then \n\\begin{align}\\label{tnormU}\n\\|U_{\\sigma,\\Lambda,t}^{N,\\pm}[x,\\gamma]\\|_{t\/2}^2\\le \n\\frac{6\\pi}{1-\\varepsilon}\\cdot\\frac{g^2N^2t^{\\varepsilon}}{\\omega(\\sigma)^{1-\\varepsilon}}\\qquad\\mbox{for all}\\ \\varepsilon\\in[0,1).\n\\end{align}\nFurther, we have\n\\begin{align}\\label{tnormwU}\n\t\t\\|\\omega U_{\\sigma,\\Lambda,t}^{N,\\pm}[x,\\gamma]\\|_{t\/2}^2 \\le 6\\pi g^2 N^2 \\omega(\\Lambda), \\quad \\Lambda<\\infty.\n\\end{align}\n\\end{lem}\n\\begin{proof}\nFor $a,\\varepsilon\\ge0$ such that $a+\\frac\\ve2\\le 1$ and $\\varepsilon<1$, we find\n\\begin{align}\n\\|\\omega^{-a}U_{\\sigma,\\Lambda,t}^{N,\\pm}[x,\\gamma]\\|^2\n&\\le g^2N^2\\int_{B_\\sigma^c}\\int_0^t\\int_0^t\\frac{\\mathrm{e}^{-s\\omega(k)-u\\omega(k)}}{\\omega(k)^{1+2a}}\\mathrm{d} u\\,\\mathrm{d} s\\,\\mathrm{d} k\n\\nonumber\\\\\n&=g^2N^2\\int_{B_\\sigma^c}\\frac{(1-\\mathrm{e}^{-t\\omega(k)})^2}{\\omega(k)^{3+2a}}\\mathrm{d} k\n\\nonumber\\\\\n&\\le g^2N^2\\int_{B_\\sigma^c}\\frac{t^{2a+\\varepsilon}}{\\omega(k)^{3-\\varepsilon}}\\mathrm{d} k\n\\nonumber\\\\\n&=\\frac{2\\pi}{1-\\varepsilon}\\cdot\\frac{g^2N^2t^{2a+\\varepsilon}}{\\omega(\\sigma)^{1-\\varepsilon}}. \\label{eq:easyboundU}\n\\end{align}\nApplying this estimate with both $a=0$ and $a=1\/2$, we arrive at \\eqref{tnormU}.\nBy a similar argument, we have\n\\begin{align*}\n\t\\|\\omega^aU_{\\sigma,\\Lambda,t}^{N,\\pm}[x,\\gamma]\\|^2\n\t\\le \\frac{2\\pi}{\\varepsilon-1}g^2N^2t^{\\varepsilon-2a}\\omega(\\Lambda)^{\\varepsilon-1}, \\quad a\\ge 0,\\varepsilon\\in(1,\\infty)\\cap[2a,2a+2].\n\\end{align*}\nApplying this estimate with $\\varepsilon=2$ and both $a=1\/2$ and $a=1$ yields \\cref{tnormwU}.\n\\end{proof}\nThe next two \\lcnamecrefs{remUpmcont} prove that the processes \n$(U_{\\sigma,\\Lambda,t}^{N,\\pm}[x,X_\\bullet])_{t\\ge0}$ are in fact continuous and adapted.\n\\begin{lem}\\label{remUpmcont}\nLet $0\\le\\sigma<\\Lambda\\le\\infty$ and $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ be Borel measurable. Then the map\n\\begin{align}\\label{Upmcont}\n[0,\\infty)\\times\\mathbb{R}^{2N}\\ni(t,x)\\longmapsto U_{\\sigma,\\Lambda,t}^{N,\\pm}[x,\\gamma]\\in L^2(\\mathbb{R}^2)\\quad\\text{is continuous.}\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nIn view of \\cref{eq:bounde-somv},\nthe assertion is clear when the minus sign is chosen in \\eqref{Upmcont}.\nFurthermore, dominated convergence ensures that $I_{j,\\Lambda}:[0,\\infty)^2\\to L^2(\\mathbb{R}^2)$ \nis continuous for every $\\Lambda\\in(0,\\infty)$, where\n\\begin{align*}\nI_{j,\\Lambda}(t,\\tau)&\\coloneq \\int_0^t\\mathrm{e}^{-|\\tau-s|\\omega-\\mathrm{i} K\\cdot \\gamma_{j,s}}v_{\\Lambda}\\mathrm{d} s,\\quad t,\\tau\\ge0,\\,\nj\\in\\{1,\\ldots,N\\}.\n\\end{align*}\nHence, $[0,\\infty)\\ni t\\mapsto \\chi_{B_{\\Lambda}} U^+_{t}[\\gamma_j]=I_{j,\\Lambda}(t,t)$ is continuous \nand so is the map in \\eqref{Upmcont} with plus sign as long as $\\Lambda<\\infty$.\nBy virtue of \\eqref{tnormU}, with $(\\Lambda,\\infty,0)$ put in place of $(\\sigma,\\Lambda,\\varepsilon)$ there, we know, however, that\n$\\|U_{\\sigma,\\Lambda,t}^{N,+}[x,\\gamma]-U_{\\sigma,\\infty,t}^{N,+}[x,\\gamma]\\|$ goes to $0$ as $\\Lambda\\to\\infty$\nuniformly in $(t,x)\\in[0,\\infty)\\times\\mathbb{R}^{2N}$. Hence, the statement in \\eqref{Upmcont} \nholds for $\\Lambda=\\infty$ as well.\n\\end{proof}\n\n\\begin{lem}\\label{lem:Upmadapted}\nLet $0\\le\\sigma<\\Lambda\\le\\infty$ and $x\\in\\mathbb{R}^{2N}$. \nThen $(U^{N,\\pm}_{\\sigma,\\Lambda,t}[x,X_\\bullet])_{t\\ge0}$ is predictable for both choices of the sign.\n\\end{lem}\n\n\\begin{proof}\nBy virtue of \\eqref{Upmcont} it suffices to show that both processes are adapted.\nIn view of \\eqref{def:Ucutoff} we only have to show that $(U_t^\\pm[X_{j,\\bullet}])_{t\\ge0}$ are adapted.\n\nSo fix $t>0$ and $j\\in\\{1,\\ldots,N\\}$. Then\n$(\\chi_{[0,t)}(s)\\mathrm{e}^{-(t-s)\\omega-\\mathrm{i} K\\cdot X_{j,s}}v)_{s\\ge0}$ is adapted with\nright-continuous paths, hence progressively measurable. In particular the integrand in\n$U_{t}^+[X_{j,\\bullet}]=\\int_{[0,t)}\\mathrm{e}^{-(t-s)\\omega-\\mathrm{i} K\\cdot X_{j,s}}v\\mathrm{d} s$\nis $\\fr{B}([0,t])\\otimes\\fr{F}_t$-measurable, whence $U_{t}^+[X_{j,\\bullet}]$ itself\nis $\\fr{F}_t$-measurable. Likewise,\n$(\\chi_{(0,\\infty)}(s)\\mathrm{e}^{-s\\omega-\\mathrm{i} K\\cdot X_{j,s-}}v)_{s\\ge0}$ is adapted with\nleft-continuous paths, hence progressively measurable (predictable in fact). \nFurthermore, for every  $\\omega\\in\\Omega$ the left-continuous path $s\\mapsto X_{j,s-}(\\omega)$ differs\nfrom $s\\mapsto X_s(\\omega)$ at most at countably many points, which permits to write\n$U_{t}^-[X_{j,\\bullet}]=\\int_{(0,t]}(s)\\mathrm{e}^{-s\\omega-\\mathrm{i} K\\cdot X_{j,s-}}v\\mathrm{d} s$ on $\\Omega$.\nAs before we conclude that $U_{t}^-[X_{j,\\bullet}]$ is $\\fr{F}_t$-measurable. \n\\end{proof}\nWe close this subsection with a lemma that is valid only for finite $\\Lambda$. It will be used to establish the\nFeynman--Kac formula with ultraviolet cutoff.\n\\begin{lem}\\label{lemIBPUplus}\nLet $\\Lambda\\in(0,\\infty)$, $\\sigma\\in[0,\\Lambda)$, $x\\in\\mathbb{R}^{2N}$ and \n$\\gamma=(\\gamma_1,\\ldots,\\gamma_N):[0,\\infty)\\to\\mathbb{R}^{2N}$ be Borel measurable. Abbreviate\n\\begin{align}\\label{def:vUVs}\nv_{\\Lambda,s}^N[x,\\gamma]\\coloneq\\sum_{j=1}^N\\mathrm{e}^{-\\mathrm{i} K\\cdot (x_j+\\gamma_{j,s})}v_\\Lambda,\\quad s\\ge0.\n\\end{align}\nThen $[0,\\infty)\\ni t\\mapsto U_{\\sigma,\\Lambda,t}^{N,+}[x,\\gamma]$ is absolutely continuous on every compact subinterval of\n$[0,\\infty)$ and\n\\begin{align}\\label{IBPUplus}\nU_{\\sigma,\\Lambda,t}^{N,+}[x,\\gamma]&=\\int_0^t(\\chi_{B_\\sigma^c}v_{\\Lambda,s}^N[x,\\gamma]-\\omega U_{\\sigma,\\Lambda,s}^{N,+}[x,\\gamma])\\mathrm{d} s,\n\\quad t\\ge0.\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nWe write $\\omega_{\\Lambda}\\coloneq \\chi_{B_{\\Lambda}}\\omega$ and observe that, thanks to the finite cutoff $\\Lambda$, both\n$f(s)\\coloneq -\\omega_\\Lambda \\mathrm{e}^{-s\\omega}$ and $g(s)\\coloneq \\mathrm{e}^{s\\omega_\\Lambda-\\mathrm{i} K\\cdot\\gamma_{j,s}}\\chi_{B_\\sigma^c}v_\\Lambda$\nare integrable functions of $s$ over $[0,t]$ for any $t>0$ taking values in $\\mathcal{B}(L^2(\\mathbb{R}^2))$ and\n$L^2(\\mathbb{R}^2)$, respectively.\nPutting $F(t)\\coloneq \\int_0^t f(s)\\mathrm{d} s=\\mathrm{e}^{-t\\omega_\\Lambda}-\\mathbbm{1}$ and $G(t)\\coloneq\\int_{0}^tg(s)\\mathrm{d} s$, we therefore have \nthe integration by parts formula\n\\begin{align*}\nF(t)G(t)&=\\int_0^tF(s)g(s)\\mathrm{d} s+\\int_0^tf(s)G(s)\\mathrm{d} s,\n\\end{align*}\nwhich is equivalent to \\eqref{IBPUplus} with $N=1$, $x=0$ and $\\gamma_j$ put in place of $\\gamma$. \n(The above integration by parts formula is a consequence of Fubini's theorem applied to\n$f(r)g(s)\\chi_{\\Delta_t}(r,s)$ with $\\Delta_t\\coloneq\\{0\\le r<s\\le t\\}$; see, e.g., \\cite[p.~187]{Elstrodt.1996}.)\nThe general case now follows by linearity.\n\\end{proof}\n\n\\subsection{The martingale contribution to the complex action}\\label{subsec:m}\n\nWe now want to discuss the martingale introduced in \\cref{def:martingale} and certain infrared and\/or\nultraviolet cutoff versions of it. These are $L^2$-stochastic\nintegrals with respect to the martingale-valued measure $\\wt{N}_X$ associated with $X$. In order to work with\n$\\wt{N}_X$ we first have to collect the following remarks:\n\nLet $\\nu:\\fr{B}(\\mathbb{R}^2)\\to[0,\\infty]$ denote the L\\'{e}vy measure having the density $\\rho$ \nwith respect to the two-dimensional Lebesgue-Borel measure, where\n\\begin{align}\\label{def:nu}\n\t\\forall y\\in\\mathbb{R}^2\\setminus\\{0\\}:\\quad\n\t\\rho(y)&\\coloneq\n\t\\begin{cases}\n\t\t\\displaystyle \\frac{m_{\\mathrm{p}}^{3\/2}}{2^{1\/2}(\\pi|y|)^{3\/2}}K_{3\/2}(m_{\\mathrm{p}}|y|),&\\text{if $m_{\\mathrm{p}}>0$},\n\t\t\\\\[1em]\n\t\t\\displaystyle \\frac{1}{2\\pi|y|^3},&\\text{if $m_{\\mathrm{p}}=0$,}\n\t\\end{cases}\n\\end{align}\nand, say, $\\rho(0)=0$.\nHere, $K_{3\/2}$ is a modified Bessel function of the third kind; see \\cref{app:Bessel} for more details.\nBy \\cref{bdBesselK}, \n\\begin{align*}\n\\int_{\\mathbb{R}^2}|y|^{1+\\varepsilon}\\wedge|y|^{1-\\varepsilon}\\mathrm{d}\\nu(y)<\\infty,\\quad \\varepsilon>0,\n\\end{align*}\nso that the density \\cref{def:nu} indeed gives rise to a L\\'evy measure. We can write $\\psi$ as\n\\begin{align}\\label{LKpsi}\n\\psi(\\eta)&=\\psi(-\\eta)=-\\int_{\\mathbb{R}^2}(\\mathrm{e}^{-\\mathrm{i} \\eta\\cdot y}-1+\\mathrm{i}\\chi_{B_1}(y)\\eta\\cdot y)\\mathrm{d}\\nu(y),\\quad \\eta\\in\\mathbb{R}^2,\n\\end{align}\nwith the two-dimensional ball $B_1$ as defined in \\eqref{2Dball}; cf. \\cite[\\textsection 1.2.6]{Applebaum.2009}.\nSince $X_1,\\ldots,X_N$ are independent L\\'{e}vy processes \nwith common L\\'{e}vy symbol $-\\psi$, the symbol $-\\psi_X$ of the \n$2N$-dimensional process $X=(X_1,\\ldots,X_N)$ is given by\n\\begin{align}\\label{LKPsiX}\n\\psi_X(\\xi)&\\coloneq\\sum_{j=1}^N\\psi(\\xi_j)\n=-\\int_{\\mathbb{R}^{2N}}(\\mathrm{e}^{-\\mathrm{i}\\xi\\cdot z}-1+\\mathrm{i}\\chi_{B_1'}(z)\\xi\\cdot z)\\mathrm{d} \\nu_X(z),\\quad\\xi\\in\\mathbb{R}^{2N}.\n\\end{align}\nHere we use the notation \\eqref{defxixij}, and\n$B_1'$ denotes the open unit ball about $0$ in $\\mathbb{R}^{2N}$. The L\\'{e}vy measure \n$\\nu_X:\\fr{B}(\\mathbb{R}^{2N})\\to[0,\\infty]$ of $X$ is\n\\begin{align}\\label{eq:nuX}\n\\nu_X&\\coloneq\\sum_{j=1}^N\\delta_0\\otimes\\dots\\otimes\\delta_0\\otimes\\underbrace{\\nu}_{\\text{$j$'th factor}}\n\\otimes\\delta_0\\otimes\\dots\\otimes\\delta_0,\n\\end{align}\nwith $\\delta_0$ denoting the Dirac measure on the Borel sets of $\\mathbb{R}^2$ concentrated in $0$.\nTo read off the formula for $\\nu_X$, we made use of \\eqref{LKpsi} and observed that $\\chi_{B_1'}(z)=\\chi_{B_1}(z_\\ell)$\nwhenever $z_j=0$ for all $j\\not=\\ell$. \n\nThe integrands of the $\\mathrm{d}\\wt{N}_X$-integrals we are interested in are given by\n\\begin{align*}\nh_{\\sigma,\\Lambda}(x,s,z)&\\coloneq \\sum_{\\ell=1}^N\\langle U^{N,+}_{\\sigma,\\Lambda,s}[x,X_\\bullet]|\n\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{\\ell,s-}^x}(\\mathrm{e}^{-\\mathrm{i} K\\cdot z_\\ell}-1)\\beta\\rangle,\n\\quad s\\ge0,\\,x,z\\in\\mathbb{R}^{2N}.\n\\end{align*}\nAccording to our earlier observations $(U^{N,+}_{\\sigma,\\Lambda,s}[x,X_\\bullet])_{s\\ge0}$ is adapted and continuous. \nThus, we can read off from the above formula that all paths $s\\mapsto h_{\\sigma,\\Lambda}(x,s,z)$ are left-continuous and,\nfor fixed $s\\ge0$, the map $(z,\\omega)\\mapsto (h_{\\sigma,\\Lambda}(x,s,z))(\\omega)$ is \n$\\fr{B}(\\mathbb{R}^{2N})\\otimes\\fr{F}_t$-measurable.\nIn particular, $h_{\\sigma,\\Lambda}(x,\\cdot,\\cdot)$ is predictable.\n\n\n\\begin{lem}\\label{lemexpmomentm}\nLet $0\\le\\sigma<\\Lambda\\le\\infty$, $x\\in\\mathbb{R}^{2N}$ and consider\n\\begin{align}\\label{mstochint}\nm_{\\sigma,\\Lambda,t}^N(x)&\\coloneq \\int_{(0,t]\\times\\mathbb{R}^{2N}}h_{\\sigma,\\Lambda}(x,s,z)\\mathrm{d}\\wt{N}_X(s,z),\\quad t\\ge0.\n\\end{align}\nThen the following holds:\n\\begin{enumerate}\n\\item[{\\rm(i)}] The integrals in \\eqref{mstochint}\nare well-defined $L^2$-stochastic integrals with respect to the martingale-valued measure $\\wt{N}_X$. \nUp to indistinguishability they define a unique c\\`{a}dl\\`{a}g square-integrable martingale\n$m_{\\sigma,\\Lambda}^N(x)=(m_{\\sigma,\\Lambda,t}^N(x))_{t\\ge0}$. \n\\item[{\\rm(ii)}]\nFor every $p\\ge2$ there exists $c_p>0$, depending solely on $p$, such that\n\\begin{align*\n\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|m_{\\sigma,\\Lambda,s}^N(x)|^p\\bigg]\n&\\le c_pg^{2p}\\bigg(\\frac{N^{3p\/2}t^{p\/2}}{\\omega(\\sigma)^{p\/2}}\n+\\frac{N^{p+1}t}{\\omega(\\sigma)^{p-1}}\\bigg),\\quad t\\ge0.\n\\end{align*}\n\\item[{\\rm(iii)}]\nThere exists a universal constant $c>0$ such that\n\\begin{align*}\n\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}e^{\\pm pm_{\\sigma,\\Lambda,s}^N(x)}\\bigg]\n&\\le \\frac{\\alpha}{\\alpha-1}\\exp\\left(\\frac{c\\alpha p^2g^4N^3t}{\\omega(\\sigma)}\\cdot\n\\mathrm{e}^{2\\pi\\alpha pg^2N\/\\omega(\\sigma)}\\right),\n\\end{align*}\nfor all $\\alpha>1$, $p>0$ and $t\\ge0$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nWe split the proof into four steps.\n\n\\smallskip\\noindent\n{\\em Step~1: Bounds on $h_{\\sigma,\\Lambda}$.}\nSince $U_s^+[X_{j,\\bullet}]$ is,  pointwise on $\\Omega$, given by a Bochner-Lebesgue integral in $L^2(\\mathbb{R}^2)$,\nwe can insert the definitions \\cref{def:UtN,def:Ucutoff}\nand switch the order between integration \nand taking the scalar products in the formula for $h_{\\sigma,\\Lambda}$. Applying Fubini's theorem afterwards we thus get\n\\begin{align}\\label{eq:hstep1}\n\t\\begin{aligned}\n\t\t&h_{\\sigma,\\Lambda}(x,s,z)\\\\&\n\t\t= g^2\\sum_{j,\\ell=1}^N\\int_{B_{\\Lambda}\\setminus B_\\sigma}\n\t\t\\int_0^s\\frac{\\mathrm{e}^{-(s-r)\\omega(k)+\\mathrm{i} k\\cdot (X^x_{j,r}-X^x_{\\ell,s-})}\n\t\t\t(\\mathrm{e}^{-\\mathrm{i} k\\cdot z_\\ell}-1)}{\\omega(k)(\\omega(k)+\\psi(k))}\\mathrm{d} r\\,\\mathrm{d} k,\n\t\\end{aligned}\n\\end{align}\nfor all $s\\ge0$ and $z\\in\\mathbb{R}^{2N}$. Under the integrals we can apply the bounds \n$|\\mathrm{e}^{-\\mathrm{i} k\\cdot z_\\ell}-1|\\le 2^{1-a_\\ell}|k|^{a_\\ell}|z|^{a_\\ell}$ as long as $a_\\ell\\in[0,1)$ to retain integrability.\nThus, we find, for all $s\\ge0$, $z\\in\\mathbb{R}^{2N}$ and $a_1,\\ldots,a_N\\in[0,1)$,\n\\begin{align}\\nonumber\n|h_{\\sigma,\\Lambda}^N(x,s,z)|&\\le 2g^2N\\sum_{\\ell=1}^N|z_\\ell|^{a_\\ell}\n\\int_{B_\\sigma^c}\\int_0^s\\frac{\\mathrm{e}^{-r\\omega(k)}|k|^{a_\\ell}}{\\omega(k)(\\omega(k)+\\psi(k))}\\mathrm{d} r\\,\\mathrm{d} k\n\\\\\\nonumber\n&\\le 2g^2N\\sum_{\\ell=1}^N|z_\\ell|^{a_\\ell}\\int_{B_\\sigma^c}\\frac{|k|^{a_\\ell}}{\\omega(k)^3}\\mathrm{d} k\n\\\\\\label{mona0}\n&\\le 4\\pi g^2N\\sum_{\\ell=1}^N\\frac{|z_\\ell|^{a_\\ell}}{1-a_\\ell}\\cdot\\frac{1}{\\omega(\\sigma)^{1-a_\\ell}},\n\\end{align}\nwhere we used $|k|^{a_\\ell}\\le \\omega(k)^{a_\\ell}$ in the last step.\nGiven a vector $z\\in\\mathbb{R}^{2N}$ and $\\ell\\in\\{1,\\ldots,N\\}$ such that $z_j=0$ for all $j\\in\\{1,\\ldots,N\\}\\setminus\\{\\ell\\}$, the sum in \\cref{mona0} reduces to one summand when we pick $a_\\ell = 0$ and $a_j\\in(0,1)$ for $j\\not=\\ell$. \nSince, by \\cref{eq:nuX}, \nat most one two-dimensional component of $z\\in\\mathbb{R}^{2N}$ is non-zero for $\\nu_X$-almost every $z\\in\\mathbb{R}^{2N}$,\nthis implies\n\\begin{align}\\label{mona1}\n\\sup_{s\\ge0}\n|h_{\\sigma,\\Lambda}^N(x,s,z)|&\\le\\frac{4\\pi g^2N}{\\omega(\\sigma)},\\quad\n\\text{$\\nu_X$-a.e. $z\\in\\mathbb{R}^{2N}$.}\n\\end{align}\nGiven $a\\in(0,1)$, we apply \\eqref{mona1} whenever $|z_\\ell|\\omega(\\sigma)\\ge(1-a)^{1\/a}$\nfor some $\\ell\\in\\{1,\\ldots,N\\}$ and choose $a_1=\\ldots=a_N=a$ in \\eqref{mona0} otherwise. \nFor any $p>0$ and $\\nu_X$-almost every $z\\in\\mathbb{R}^{2N}$, this yields\n\\begin{align}\\label{mona2}\n\t\\sup_{s\\ge0}|h_{\\sigma,\\Lambda}(x,s,z)|^p\n\t\\le \\frac{(4\\pi g^2N)^p}{\\omega(\\sigma)^{p}}\n\t\\sum_{\\ell=1}^N1\\wedge\\left(\\frac{\\omega(\\sigma)^{pa}}{(1-a)^{p}}|z_\\ell|^{pa}\\right).\n\\end{align}\nThe bound \\crefnosort{bdBesselK,def:nu} imply the estimate\n\\begin{align*}\n&\\int_{\\mathbb{R}^2}1\\wedge\\left(\\frac{\\omega(\\sigma)^{pa}}{(1-a)^{p}}|z_\\ell|^{pa}\\right)\\mathrm{d}\\nu(z_\\ell)\n\\\\&\\qquad\n\\le \\int_{\\mathbb{R}^2}\\bigg\\{1\\wedge\\left(\\frac{\\omega(\\sigma)^{pa}}{(1-a)^p}|z_\\ell|^{pa}\\right)\\bigg\\}\n\\frac{c}{|z_\\ell|^3}\\mathrm{d} z_\\ell\n=\\frac{2\\pi cpa}{pa-1}\\cdot\\frac{\\omega(\\sigma)}{(1-a)^{1\/a}},\n\\end{align*}\nwhich applies to all particle masses $m_{\\mathrm{p}}\\ge0$ and $p,a>0$ with $pa>1$. \nChoosing $a=2\/3$, say, we arrive at\n\\begin{align}\\label{mona3}\n\\int_{0}^t\\int_{\\mathbb{R}^{2N}}|h_{\\sigma,\\Lambda}(x,s,z)|^p\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s&\\le \n\\frac{c_pg^{2p}N^{p+1}t}{\\omega(\\sigma)^{p-1}},\\quad t\\ge0,\\,p\\ge2.\n\\end{align}\n\n\\smallskip\\noindent\n{\\em Step~2: Proof of {\\rm(i)}.} \nBy \\cref{mona3},\n$\\mathbb{E}[\\int_0^t\\int_{\\mathbb{R}^{2N}}|h_{\\sigma,\\Lambda}(x,s,z)|^2\\mathrm{d} \\nu_X(z)\\mathrm{d} s]<\\infty$ for all $t\\ge0$,\nwhich implies that \\eqref{mstochint} are well-defined $L^2$-stochastic integrals and the corresponding\nintegral process is a square-integrable martingale; see, e.g., \\cite[\\textsection 4.2.1]{Applebaum.2009}.\nSince we are working under the usual hypotheses, the latter has a c\\`{a}dl\\`{a}g modification that is\nunique up to indistinguishability.\n\n\\smallskip\n\n\\noindent\n{\\em Step~2: Proof of  {\\rm(ii)}.}\nLet $p\\ge2$. According to Kunita's inequality (see, e.g., \\cite[Theorem~4.4.23]{Applebaum.2009}),\nthere is a solely $p$-dependent $c_p>0$ such that\n\\begin{align*}\n\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|m_{\\sigma,\\Lambda,s}^N(x)|^p\\bigg]\n\\le\\ & c_p\\mathbb{E}\\bigg[\\bigg(\\int_0^t\\int_{\\mathbb{R}^{2N}}|h_{\\sigma,\\Lambda,s}(x,s,z)|^2\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s\\bigg)^{p\/2}\\bigg]\n\\\\\n&\\quad+\\mathbb{E}\\bigg[\\int_0^t\\int_{\\mathbb{R}^{2N}}|h_{\\sigma,\\Lambda,s}(x,s,z)|^p\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s\\bigg],\n\\quad t\\ge0.\n\\end{align*}\nInserting  \\eqref{mona3}, we find the bound asserted in (ii).\n\n\\smallskip\n\\noindent\n{\\em Step~3: Proof of {\\rm(iii)}.}\nLet $\\alpha>1$ and $p>0$. Then an elementary manipulation of the exponential series yields\n\\begin{align*}\n\\frac{1}{\\alpha}|\\mathrm{e}^{\\alpha ph_{\\sigma,\\Lambda}(x,s,z)}-1-\\alpha ph_{\\sigma,\\Lambda}(x,s,z)|\n&\\le \\frac{\\alpha p^2}{2}|h_{\\sigma,\\Lambda}(x,s,z)|^2\\mathrm{e}^{\\alpha p|h_{\\sigma,\\Lambda}(x,s,z)|}.\n\\end{align*}\nEmploying \\eqref{mona1} in the exponent of the exponential on the previous right hand side and using\n\\eqref{mona3} with $p=2$ we find, with some universal constant $c>0$,\n\\begin{align}\\nonumber\n\\frac{1}{\\alpha}\\int_{0}^t\\int_{\\mathbb{R}^{2N}}|\\mathrm{e}^{\\alpha ph_{\\sigma,\\Lambda}(x,s,z)}&-1-\\alpha ph_{\\sigma,\\Lambda}(x,s,z)|\n\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s\n\\\\\\label{mona4}\n&\\le \\frac{c\\alpha p^2g^4N^3t}{\\omega(\\sigma)}\\cdot\\mathrm{e}^{4\\pi\\alpha pg^2N\/\\omega(\\sigma)},\n\\quad t\\ge0.\n\\end{align}\nThe asserted exponential moment bound now follows from \\eqref{mona4}, an inequality due to \nApplebaum and Siakalli \\cite{Applebaum.2009,Siakalli.2019} and a standard argument based on a suitable \nlayer cake representation; see Lemma~\\ref{lemexpbdtildeN}.\n\\end{proof}\nThe next lemma about continuous dependence on the initial positions of the matter particles is analogous\nto \\cite[Lemma~4.14]{MatteMoller.2018}. It is proved in a similar, standard fashion with Kunita's\ninequality replacing the Burkholder inequalities used in \\cite{MatteMoller.2018}.\n\\begin{lem}\\label{lem:canm}\nLet $0\\le\\sigma<\\Lambda\\le\\infty$. Then,\nfor every $x\\in\\mathbb{R}^{2N}$, the c\\`{a}dl\\`{a}g modification of $m_{\\sigma,\\Lambda}^N(x)$ found in the first part of\nLemma~\\ref{lemexpmomentm} can be chosen such that\n\\begin{enumerate}\n\\item[{\\rm(a)}]\n$\\mathbb{R}^{2N}\\ni x\\mapsto (m_{\\sigma,\\Lambda,t}^N(x))(\\omega)$ is continuous for all $t\\ge0$ and $\\omega\\in\\Omega$.\n\\item[{\\rm(b)}]\nFor all $t\\ge0$, $\\omega\\in\\Omega$ and compact $K\\subset\\mathbb{R}^{2N}$,\n\\begin{align*}\n\\lim_{s\\downarrow t} \\sup_{x\\in K}|m_{\\sigma,\\Lambda,s}^N(x)-m_{\\sigma,\\Lambda,t}^N(x)|&=0.\n\\end{align*}\n\\end{enumerate} \n\\end{lem}\n\n\\begin{proof}\nLet $x,\\tilde{x}\\in\\mathbb{R}^{2N}$.\nBy the same arguments as in the\nderivation of \\cref{eq:hstep1}, \n\\begin{align*}\n\\tilde{h}_{\\sigma,\\Lambda}(\\tilde{x},x,s,z)&\\coloneq h_{\\sigma,\\Lambda}(\\tilde{x},s,z)-h_{\\sigma,\\Lambda}(x,s,z)\n\\\\\n&= g^2\\sum_{j,\\ell=1}^N\\int_{B_{\\Lambda}\\setminus B_\\sigma}\n\\int_0^s\\frac{\\mathrm{e}^{-(s-r)\\omega(k)+\\mathrm{i} k\\cdot (X_{j,r}-X_{\\ell,s-})}\n}{\\omega(k)(\\omega(k)+\\psi(k))}\\cdot \\theta_{j,\\ell}(k,\\tilde{x},x,z) \\mathrm{d} r\\,\\mathrm{d} k,\n\\end{align*} \nwhere $\\theta_{j,\\ell}(k,\\tilde{x},x,z)\\coloneq (\\mathrm{e}^{\\mathrm{i} k\\cdot(\\tilde{x}_{j}-\\tilde{x}_{\\ell})}-\\mathrm{e}^{\\mathrm{i} k\\cdot(x_j-x_\\ell)})\n(\\mathrm{e}^{-\\mathrm{i} k\\cdot z_\\ell}-1)$.\nWe note\n\\begin{align*}\n&|\\theta_{j,\\ell}(k,\\tilde{x},x,z)|\\le 2^{2-a_\\ell}|x-\\tilde{x}|^\\varepsilon|z_\\ell|^{a_\\ell}|k|^{\\varepsilon+a_\\ell},\n\\end{align*}\nfor all $\\varepsilon,a_1,\\ldots,a_\\ell\\in[0,1]$. Assuming $\\varepsilon+\\max\\{a_1,\\ldots,a_N\\}<1$,\nreplacing $2|k|^{a_\\ell}$ by $4|x-\\tilde{x}|^\\varepsilon|k|^{\\varepsilon+a_\\ell}$ \nin the second and third members of \\eqref{mona0}\nand following the derivation of \\cref{mona2}, we obtain\n\\begin{align*}\n\\sup_{s\\ge0}|\\tilde{h}_{\\sigma,\\Lambda}(\\tilde{x},x,s,z)|^p\n&\\le \\frac{(8g^2N)^p|x-\\tilde{x}|^{p\\varepsilon}}{\\omega(\\sigma)^{(1-\\varepsilon)p}}\n \\sum_{\\ell=1}^N1\\wedge\\bigg(\\frac{\\omega(\\sigma)^{pa}}{(1-\\varepsilon-a)^p}|z_\\ell|^{pa}\\bigg),\n\\end{align*}\nfor all $a,p,\\varepsilon>0$ such that $a+\\varepsilon<1$. Here the right hand side is $\\nu_X$-integrable provided\nthat $pa>1$. Again choosing $a=2\/3$, we find\n\\begin{align*}\n\\int_{0}^t\\int_{\\mathbb{R}^{2N}}|\\tilde{h}_{\\sigma,\\Lambda}(\\tilde{x},x,s,z)|^p\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s\n&\\le \\frac{c_{p,\\varepsilon}g^{2p}N^{p+1}|x-\\tilde{x}|^{p\\varepsilon} t}{\\omega(\\sigma)^{p-p\\varepsilon-1}},\\quad t\\ge0,\n\\end{align*}\nfor all $p\\ge2$ and $\\varepsilon\\in(0,1\/3)$. (Here and in what follows constants depend only on the quantities \ndisplayed in their subscripts.)\nCombining these remarks with Kunita's inequality, we arrive at\n\\begin{align*}\n&\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|m_{\\sigma,\\Lambda,s}^N(\\tilde{x})-m_{\\sigma,\\Lambda,s}^N(x)|^p\\bigg]\n\\\\\n&=\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}\\bigg|\\int_{(0,s]\\times\\mathbb{R}^{2N}}\\tilde{h}_{\\sigma,\\Lambda}(\\tilde{x},x,s,z)\\mathrm{d}\\wt{N}_X(s,z)\\bigg|^p\\bigg]\n\\\\\n&\\le c_p\\mathbb{E}\\bigg[\\bigg(\\int_0^t\\int_{\\mathbb{R}^{2N}}|\\tilde{h}_{\\sigma,\\Lambda,s}(\\tilde{x},x,s,z)|^2\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s\\bigg)^{p\/2}\\bigg]\n\\\\\n&\\quad+\\mathbb{E}\\bigg[\\int_0^t\\int_{\\mathbb{R}^{2N}}|\\tilde{h}_{\\sigma,\\Lambda,s}(\\tilde{x},x,s,z)|^p\\mathrm{d}\\nu_X(z)\\,\\mathrm{d} s\\bigg]\n\\\\\n&\\le  c_{m_{\\mathrm{b}},p,\\varepsilon,g,N}(t^{p\/2}+t)|x-\\tilde{x}|^{p\\varepsilon},\n\\end{align*}\nfor all $t\\ge0$, $\\varepsilon\\in(0,1\/3)$ and $p\\ge2$. Choosing $\\varepsilon=1\/4$ and $p>2$ such that\n$p\\varepsilon=2N+1$, we see that all assertions follow from the Kolmogorov-Neveu lemma;\nsee, e.g., \\cite[pp. 268\/9]{Metivier.1982}.\n\\end{proof}\n\n\n\\subsection{Effective interaction terms}\n\nNext, we treat the effective interaction introduced in \\cref{def:wy,def:wtN} together with variants containing cutoffs.\n\nSince $\\chi_{B_\\sigma^c}v_\\Lambda\\beta$ is $\\tilde{p}$-integrable for every $\\tilde{p}\\ge1$ \nand finite $\\Lambda\\ge\\sigma\\ge 0$, we can consider the Fourier integrals\n\\begin{align}\\label{def:wIRUV}\n\\begin{aligned}\n\tw_{\\sigma,\\Lambda}(y)&\\coloneq \\langle\\mathrm{e}^{\\mathrm{i} K\\cdot y}\\chi_{B_\\sigma^c}v_\\Lambda|\\beta\\rangle\n\t\\\\\n\t&=g^2\\int_{B_{\\Lambda}\\setminus B_\\sigma}\\frac{\\mathrm{e}^{-\\mathrm{i} k\\cdot y}}{\\omega(k)(\\omega(k)+\\psi(k))}\\mathrm{d} k\n\t=g^2\\int_\\sigma^{\\Lambda} \\vartheta(r,|y|)\\mathrm{d} r,\n\\end{aligned}\n\\end{align}\nfor $0\\le\\sigma<\\Lambda<\\infty$ and $y\\in\\mathbb{R}^2$, where $\\vartheta$ is defined in \\eqref{defvt}. \nSince the Bessel function $J_0$ in \\eqref{defvt} satisfies the bound \\cref{eq:J0bound}, it is clear that\n\\begin{align*}\n\\lim_{\\Lambda\\to\\infty}w_{\\sigma,\\Lambda}(y)&=w_{\\sigma,\\infty}(y)\\coloneq g^2\\int_\\sigma^\\infty\\vartheta(r,|y|)\\mathrm{d} r,\n\\quad y\\in\\mathbb{R}^2\\setminus\\{0\\}.\n\\end{align*}\nSetting $w_{\\sigma,\\infty}(0)\\coloneq 0$, say,  $w_{\\sigma,\\infty}:\\mathbb{R}^2\\to\\mathbb{R}$ \nis the up to its value at $0$ unique representative of the \ndistributional Fourier transform of $\\chi_{B_\\sigma^c}v\\beta$ that is continuous on $\\mathbb{R}^2\\setminus\\{0\\}$.\n\n\\begin{rem}\\label{remdefw}\nSince $v\\beta=g^2\/\\omega(\\omega+\\psi)\\in L^{\\tilde{p}}(\\mathbb{R}^2)$ for every $\\tilde{p}>1$, the Hausdorff-Young inequality \nimplies\n$w_{\\sigma,\\infty}\\in L^{p}(\\mathbb{R}^2)$\nand\n$\\|w_{\\sigma,\\infty}\\|_{p}\\le \\|\\chi_{B_\\sigma^c}v\\beta\\|_{p'}$ for all $p\\in[2,\\infty)$,\nwhere $p'$ is the exponent conjugated to $p$.\nFor all $p\\in(1,\\infty)$, or equivalently $p'\\in(1,\\infty)$, and $\\sigma\\ge0$, we further have\n\\begin{align*\n\\|\\chi_{B_\\sigma^c}v\\beta\\|_{p'}\n&\\le g^2\\bigg(\\pi\\int_\\sigma^\\infty\\frac{2r}{(r^2+m_{\\mathrm{b}}^2)^{p'}}\\mathrm{d} r\\bigg)^{1\/p'}\n=\\frac{c_pg^2}{(\\sigma^2+m_{\\mathrm{b}}^2)^{1\/p}}.\n\\end{align*}\nCombined, we find the bound\n\\begin{align}\\label{Lpvbetasigma2}\n\\|w_{\\sigma,\\Lambda}\\|_p&\\le \\frac{c_pg^2}{(\\sigma^2+m_{\\mathrm{b}}^2)^{1\/p}},\\quad 0\\le\\sigma<\\Lambda\\le\\infty,\\,p\\in[2,\\infty).\n\\end{align}\n\\end{rem}\nWe can now treat a (possibly) cutoff version of the full effective interaction expression defined in \\cref{def:wtN}.\nFor all $0\\le\\sigma<\\Lambda\\le\\infty$ and $x\\in\\mathbb{R}^{2N}$, the corresponding expression is given by\n\\begin{align}\\nonumber\nw^N_{\\sigma,\\Lambda,t}(x)&\\coloneq \\sum_{\\substack{j,\\ell=1\\\\ j\\not=\\ell}}^N\\int_0^tw_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)\\mathrm{d} s\n\\\\\\label{def:wNsigmaUV}\n&=\\sum_{1\\le j<\\ell\\le N}\\int_0^t2w_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)\\mathrm{d} s,\\quad \\mbox{whenever $N\\ge2$.}\n\\end{align}\nNotice that $w_{\\sigma,\\Lambda}(-y)=w_{\\sigma,\\Lambda}(y)$ for all $y\\in\\mathbb{R}^2$. Also notice that we are again using\nthe conventions introduced in \\cref{sssecKato} in the case $\\Lambda=\\infty$; i.e., the pathwise integrals in \n\\eqref{def:wNsigmaUV} are $0$ by definition unless they are well-defined for {\\em all} $t\\ge0$.\nFor $N=1$ we set $w_{\\sigma,\\Lambda,t}^1(x)\\coloneq 0$. \n\nThe next \\lcnamecref{lemexpmomentw} is an exponential moment bound for the effective interaction.\n\\begin{lem}\\label{lemexpmomentw}\nThere exist universal constants $c,c'>0$ such that\n\\begin{align*}\n&\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}\\left[\\sup_{s\\in[0,t]}\\mathrm{e}^{p|w_{\\sigma,\\Lambda,s}^N(x)|}\\right]\n\\le c^N\\exp\\left(tc'p^2g^4N^2(N-1)\n\\Big(\\frac{m_{\\mathrm{p}}}{\\omega(\\sigma)^2}+\\frac{1}{\\omega(\\sigma)}\\Big)\\right),\n\\end{align*}\nfor all $0\\le\\sigma<\\Lambda\\le\\infty$, $t\\ge0$ and $p\\in[1,\\infty)$.\n\\end{lem}\n\n\\begin{proof}\nLet $t>0$ and $x\\in\\mathbb{R}^{2N}$. Exactly as in \\cite[Equation (4.26)]{MatteMoller.2018}\n(where an observation from \\cite{Bley.2018} is used; see also \\cite[Appendix B]{MatteMoller.2018}),\nwe can exploit the independence of the processes $X_1,\\ldots,X_N$ to obtain the bound\n\\begin{align*}\n\\mathbb{E}\\left[\\sup_{s\\in[0,t]}\\mathrm{e}^{p|w_{\\sigma,\\Lambda,s}^N(x)|}\\right]\n&\\le \\max_{1\\le j<\\ell\\le N}\\mathbb{E}\\left[\\sup_{s\\in[0,t]}\n\\mathrm{e}^{2pN\\int_0^t|w_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)|\\mathrm{d} s}\\right]^{(N-1)\/2}.\n\\end{align*}\nPick two indices $j$ and $\\ell$ with $1\\le j<\\ell\\le N$. Since $X_j$ and $X_\\ell$ are independent and\nsince $\\psi(-\\xi)=\\psi(\\xi)$ for all $\\xi\\in\\mathbb{R}^2$, \nwe know that the difference $X_\\ell-X_j$ is a L\\'{e}vy process associated with the L\\'{e}vy symbol $-2\\psi$.\nTherefore, we can apply the bound \\eqref{relLpLinfty3} with $a=d=p=2$ to deduce that\n\\begin{align*}\n\\mathbb{E}\\left[\\sup_{s\\in[0,t]}\n\\mathrm{e}^{2pN\\int_0^t|w_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)|\\mathrm{d} s}\\right]\n&\\le c\\exp\\left(c'(m_{\\mathrm{p}}^{1\/2}\\|2pNw_{\\sigma,\\Lambda}\\|_2+\\|2pNw_{\\sigma,\\Lambda}\\|_4)^2t\\right),\n\\end{align*}\nwith universal constants $c,c'>0$. Combining the above remarks with\n\\eqref{Lpvbetasigma2}, we arrive at the asserted inequality.\n\\end{proof}\nWe can also bound the $p$'th moments of the effective interaction.\n\\begin{lem}\\label{lempmomentw}\nLet $p\\in[1,\\infty)$. Then there exists $c_p>0$, depending solely on $p$, such that\nfor all $0\\le\\sigma<\\Lambda\\le\\infty$, $t\\ge0$ and $x\\in\\mathbb{R}^{2N}$,\n\\begin{align}\n&\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|w_{\\sigma,\\Lambda,s}^N(x)|^p\\bigg]\n\\label{convw0}\n\\le c_pg^{2p}N^p(N-1)^{p}t^{p-1\/2}\\bigg(\\frac{m_{\\mathrm{p}}}{\\omega(\\sigma)^2}\n+\\frac{1}{\\omega(\\sigma)}\\bigg)^{1\/2}.\n\\end{align}\n\\end{lem}\n\\begin{proof}\nFirst applying the ordinary and afterwards the generalized Minkowski inequality, we find\n\\begin{align*}\n\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|w_{\\sigma,\\Lambda,s}^N(x)|^p\\bigg]^{\\frac{1}{p}}\n&\\le\\sum_{1\\le j<\\ell\\le N}\n\\mathbb{E}\\bigg[\\bigg(\\int_0^t2|w_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)|\\mathrm{d} s\\bigg)^p\\bigg]^{\\frac{1}{p}}\n\\\\\n&\\le 2 \\sum_{1\\le j<\\ell\\le N}\n\\int_0^t\\mathbb{E}\\big[|w_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)|^p\\big]^{\\frac{1}{p}}\\mathrm{d} s.\n\\end{align*}\nHere $(X_{\\ell,\\tau}-X_{j,\\tau})_{\\tau\\ge0}$\nis a L\\'{e}vy process with L\\'{e}vy symbol $-2\\psi$. Thus, by \\eqref{relLpLinfty2} and \\eqref{Lpvbetasigma2},\n\\begin{align*}\n\\mathbb{E}\\big[|w_{\\sigma,\\Lambda}(X_{\\ell,s}^x-X_{j,s}^x)|^p\\big]\n&\\le c(\\|w_{\\sigma,\\Lambda}\\|_{4p}^p+m_{\\mathrm{p}}^{1\/2}\\|w_{\\sigma,\\Lambda}\\|_{2p}^p) s^{-1\/2}\n\\\\\n&\\le \\frac{c_pg^{2p}}{s^{1\/2}}\\bigg(\\frac{1}{\\omega(\\sigma)^{1\/2}}\n+\\frac{m_{\\mathrm{p}}^{1\/2}}{\\omega(\\sigma)}\\bigg), \\quad s>0.\n\\end{align*}\nPutting these remarks together we arrive at \\eqref{convw0}.\n\\end{proof}\n\n\n\\section{Discussion of the complex action}\\label{sec:complex}\n\\noindent\nAfter the basic discussions in the previous \\lcnamecref{sec:basicproc}, we now derive properties of (possibly cutoff versions) of the complex action introduced in \\cref{def:action}.  Instead of the definition \\cref{def:action}, we employ a different expression for finite ultraviolet cutoff $\\Lambda$, which has its origin directly in our derivation of the Feynman--Kac formula but is only well-defined for finite $\\Lambda$. Then we rewrite this expression in the fashion already known from \\cref{def:action}, allowing us to discuss the limit as the ultraviolet cutoff is taken to infinity. We also derive the exponential moment bounds on the complex action, which are the main ingredients for our bounds on the minimal energy. Further, we introduce a canonical representative of the complex action.\n\n\\subsection{Definition of the complex action}\n\nFor finite $\\Lambda$ we first give a pathwise definition of the complex action employing the notation \\cref{def:Ucutoff,def:vUVs}.\n\n\\begin{defn}\\label{def:action-process}\nFor finite $\\Lambda\\in(0,\\infty)$, $\\sigma\\in[0,\\Lambda)$, $t\\ge0$, $x\\in\\mathbb{R}^{2N}$ \nand Borel measurable $\\gamma:[0,\\infty)\\to \\mathbb{R}^{2N}$, \nwe define the corresponding complex action by\n\\begin{align}\\label{defuUV}\nu_{\\sigma,\\Lambda,t}^N[x,\\gamma]&\\coloneq\n\\int_0^t\\langle U^{N,+}_{\\sigma,\\Lambda,s}[x,\\gamma]|v_{\\Lambda,s}^N[x,\\gamma]\\rangle\\mathrm{d} s\n-tNE_{\\sigma,\\Lambda}^{\\mathrm{ren}},\n\\end{align}\nwhere $\\gamma=(\\gamma_1,\\ldots,\\gamma_N)$, each $\\gamma_j$ is $\\mathbb{R}^2$-valued and\n\\begin{align}\\label{defEren}\nE_{\\sigma,\\Lambda}^{\\mathrm{ren}}&\\coloneq \\langle\\chi_{B_{\\Lambda}\\setminus B_\\sigma}v|\\beta\\rangle.\n\\end{align}\n\\end{defn}\nIn view of \\cref{defvUV,def:beta} we indeed have  $E_{0,\\Lambda}^{\\mathrm{ren}}=E_{\\Lambda}^{\\mathrm{ren}}$ with $E_{\\Lambda}^{\\mathrm{ren}}$ given by\n\\cref{def:EUVren}.\n\nUsing It\\^o's formula, we can rewrite the integral in \\cref{defuUV} when the paths \nare coming from the process $X$. This is done in \\cref{lem:actionito} below and the alternative expression\nfor the complex action we find is \\cref{defcomplexactionsigmaUV} in the next definition. The crucial point\nabout rewriting the complex action in this way is that the right hand side of \\cref{defcomplexactionsigmaUV}\nstill is meaningful without ultraviolet cutoff, i.e., for $\\Lambda=\\infty$. \n\nRecall the definition of the martingale contribution \\cref{mstochint} and the effective interaction \\cref{def:wNsigmaUV}.\nFor any path $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ and $\\Lambda\\in(0,\\infty]$, we also abbreviate\n\\begin{align*}\n\\beta_\\Lambda\\coloneq\\chi_{B_{\\Lambda}}\\beta,\\qquad\n\\beta_{\\Lambda,t}^N[x,\\gamma]&\\coloneq \\sum_{j=1}^N\\mathrm{e}^{-\\mathrm{i} K\\cdot(x_j+\\gamma_{j,t})}\\beta_{\\Lambda}.\n\\end{align*}\n\n\\begin{defn}\nLet $x\\in\\mathbb{R}^{2N}$.  Then we define\n\\begin{align}\\label{defcomplexactionsigmaUV}\nu_{\\sigma,\\Lambda,t}^N(x)&\\coloneq w^N_{\\sigma,\\Lambda,t}(x)-c^N_{\\sigma,\\Lambda,t}(x)+m^N_{\\sigma,\\Lambda,t}(x),\\quad t\\ge0,\n\\end{align}\nfor all $0\\le\\sigma<\\Lambda\\le\\infty$, with\n\\begin{align}\\label{def:cNsigmaUV}\nc^N_{\\sigma,\\Lambda,t}(x)&\\coloneq \n\\langle U_{\\sigma,\\Lambda,t}^{N,+}[x,X_{\\bullet}]|\\beta_{\\Lambda,t}^N[x,X_\\bullet]\\rangle.\n\\end{align}\n\\end{defn}\n\nRecall that the three contributions to the complex action and the action itself \nfor our model without any cutoffs have already been introduced in \\eqref{defvt} through \\eqref{def:action}. In fact,\n\\begin{align}\\label{defcomplexaction}\nu_t^N(x)&= u_{0,\\infty,t}^N(x), \\ \\ \\mbox{and analogously with $c$, $m$ or $w$ put in place of $u$.}\n\\end{align}\n\n\\begin{rem}\\label{rem:bdcN}\nWe note for later use that\n\\begin{align}\\label{bdcN}\n|c_{\\sigma,\\Lambda,t}^N(x)|\\le\\frac{2\\pi g^2N^2}{\\omega(\\sigma)},\\quad \\sigma,t\\ge0,\n\\end{align}\nfor all $x\\in\\mathbb{R}^{2N}$, by the last two relations in \\eqref{mona0} with $a_\\ell=0$.\n\\end{rem}\n\n\\begin{rem}\\label{eq:u-decomp}\nLet $0\\le \\sigma <\\kappa < \\Lambda\\le \\infty$ and $x\\in\\mathbb{R}^{2N}$. Then\n\\cref{mstochint,def:wIRUV,def:wNsigmaUV,def:cNsigmaUV} $\\mathbb{P}$-a.s. imply\n\t\\begin{align}\n\t\tu_{\\sigma,\\Lambda,t}^N(x) = u_{\\sigma,\\kappa,t}^N(x) + u_{\\kappa,\\Lambda,t}^N(x), \\quad t\\ge0.\n\t\\end{align}\n\\end{rem}\n\n\\begin{lem}\\label{lem:actionito}\nLet $0\\le\\sigma<\\Lambda<\\infty$ and $x\\in\\mathbb{R}^{2N}$. Then, $\\mathbb{P}$-a.s.,\n\\begin{align}\\label{ItoNeu0}\nu_{\\sigma,\\Lambda,t}^N[x,X_\\bullet]&=u_{\\sigma,\\Lambda,t}^N(x),\\quad t\\ge0.\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nWe let $N_X$ denote the Poisson random measure associated with $X$; recall that $X$ has\ncharacteristics $(0,0,\\nu_X)$ with $\\nu_X$ given by \\eqref{eq:nuX}.\nFurthermore, we pick\nan orthonormal basis $\\{e_i:i\\in\\mathbb{N}\\}$ of the real Hilbert space $\\mc{h}_{\\mathbb{R}}$ defined\nin \\eqref{defmcRR} and put $P_n\\coloneq\\sum_{i=1}^n|e_i\\rangle\\langle e_i|$ for every $n\\in\\mathbb{N}$.\nWe introduce the projections $P_n$ as we wish to employ the It\\^{o} formula for finite dimensional\nL\\'{e}vy type stochastic integrals presented, e.g., in \\cite[Theorem~4.4.7]{Applebaum.2009}.\n(Alternatively, we could also use an It\\^{o} formula for infinite dimensional processes \\cite[Theorem~27.2]{Metivier.1982} and\nre-write the $\\mathrm{d} X$-integrations and summations over jumps appearing there.)\nRecalling that $U_{\\sigma,\\Lambda,t}^{N,+}$ is $\\mc h_\\mathbb{R}$-valued, cf. \\cref{subsecerg}, and in view of \\eqref{IBPUplus}, the processes $A_i\\coloneq(\\langle e_i|U_{\\sigma,\\Lambda,t}^{N,+}[x,X_\\bullet]\\rangle)_{t\\ge0}$, $i\\in\\mathbb{N}$, \nare real-valued pathwise defined integral processes whose adapted integrands can be chosen left-continuous.\nSince $\\Lambda<\\infty$, the function \n\\begin{align*}\n\\mathbb{R}^{n}\\times\\mathbb{R}^{2N}\\ni(y,z)\\longmapsto\\sum_{i=1}^n\n\\sum_{j=1}^Ny_i\\langle e_i|\\mathrm{e}^{-\\mathrm{i} K\\cdot z_j}\\beta_{\\Lambda}\\rangle\\quad\\text{is smooth.}\n\\end{align*}\nComposing it with the $n+2N$ dimensional process $(A_1,\\ldots,A_n,X^x)$, we can therefore\napply the aforementioned It\\^{o} formula from \\cite{Applebaum.2009}. Taking the L\\'evy--It\\^o decomposition of $X$, cf. \\cite[Theorem~2.4.16]{Applebaum.2009}, and \\eqref{IBPUplus} into account,\nwe $\\mathbb{P}$-a.s. obtain\n\\begin{align}\\label{ItoNeu1}\n\\langle P_nU_{\\sigma,\\Lambda,t}^{N,+}[x,X_\\bullet]|\\beta_{\\Lambda,t}^{N}[x,X_\\bullet]\\rangle\n&=\\sum_{\\ell=1}^5I_\\ell(n,t),\\quad t\\ge0,\\,n\\in\\mathbb{N},\n\\end{align}\nwith (recall that $B_1'$ denotes the open unit ball about $0$ in $\\mathbb{R}^{2N}$)\n\\begin{align*}\nI_1(n,t)&\\coloneq\\int_0^t\\langle P_n\\chi_{B_\\sigma^c}v_{\\Lambda,s}^N[x,X_{\\bullet}]|\\beta_{\\Lambda,s}^{N}[x,X_\\bullet]\\rangle\\mathrm{d} s,\n\\\\\nI_2(n,t)&\\coloneq\n-\\sum_{j=1}^N\\int_0^t\n\\langle P_nU_{\\sigma,\\Lambda,s}^{N,+}[x,X_{\\bullet}]|\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{j,s}^x}\\omega\\beta_{\\Lambda}\\rangle\\mathrm{d} s,\n\\\\\nI_3(n,t)&\\coloneq\\int_{(0,t]\\times (\\mathbb{R}^{2N}\\setminus B_1')}h_{n}(s,z)\\mathrm{d} N_X(s,z),\n\\\\\nI_4(n,t)&\\coloneq\\int_{(0,t]\\times B_1'}h_{n}(s,z)\\mathrm{d} \\wt{N}_X(s,z),\n\\end{align*}\nwhere the integrand of $I_3(n,t)$ and $I_4(n,t)$ is given by\n\\begin{align*\nh_{n}(s,z)&\\coloneq \\sum_{j=1}^N\n\\langle P_nU_{\\sigma,\\Lambda,s}^{N,+}[x,X_\\bullet]|\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{j,s-}^x}(\\mathrm{e}^{-\\mathrm{i} K\\cdot z_j}-1)\\beta_{\\Lambda}\\rangle,\n\\quad s\\ge0,\\,z\\in\\mathbb{R}^{2N}.\n\\end{align*}\nFinally,\n\\begin{align*}\n&I_5(n,t)\n\\\\\n&\\coloneq\\sum_{j=1}^N\\int_0^t\\int_{B_1'}\n\\langle P_nU_{\\sigma,\\Lambda,s}^{N,+}[x,X_\\bullet]|\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{j,s-}^x}\n(\\mathrm{e}^{-\\mathrm{i} K\\cdot z_j}-1+\\mathrm{i} K\\cdot z_j)\\beta_{\\Lambda}\\rangle\\mathrm{d} \\nu_X(z)\\mathrm{d} s.\n\\end{align*}\nOn account of \\eqref{tnormU}, $h_n$ is bounded. Therefore, $I_3(n,t)$ is an ordinary Lebesgue integral\ndefined pointwise on $\\Omega$. Moreover, the $\\mathrm{d} \\nu_X(z)\\mathrm{d} s$-integral of $h_n$ over\n$(0,t]\\times(\\mathbb{R}^{2N}\\setminus B_1')$ exists and we can add and subtract\nit on the right hand side of \\eqref{ItoNeu1}. In view of the relation\n\\begin{align*}\n\\int_{(0,t]\\times M}Z_s(z)\\mathrm{d} \\wt{N}_X(s,z)&=\\int_{(0,t]\\times M}Z_s(z)\\mathrm{d} N_X(s,z)-\\int_0^t\\int_{M}Z_s(z)\\mathrm{d} \\nu_X(z)\\mathrm{d} s,\n\\end{align*}\nwhich holds, for instance, for all bounded predictable integrands $Z$ and \n$M\\in\\fr{B}(\\mathbb{R}^{2N})$ with $0\\notin \\ol{M}$ (see, e.g., \\cite[p. 231]{Applebaum.2009}), this gives\n\\begin{align*}\nI_3(n,t)+I_4(n,t)+I_5(n,t)&=I_6(n,t)+I_7(n,t),\n\\end{align*}\nwith\n\\begin{align*}\nI_6(n,t)&\\coloneq\n\\int_{(0,t]\\times\\mathbb{R}^{2N}}h_n(s,z)\\mathrm{d}\\wt{N}_X(s,x),\n\\\\\nI_7(n,t)\n&\\coloneq\n\\sum_{j=1}^N\\int_0^t\\int_{\\mathbb{R}^{2N}}\n\\Big\\langle P_nU_{\\sigma,\\Lambda,s}^{N,+}[x,X_\\bullet]\\Big|\n\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{j,s-}^x}\n\\\\\n&\\qquad\\qquad\\qquad\\times(\\mathrm{e}^{-\\mathrm{i} K\\cdot z_j}-1+\\mathrm{i} K\\cdot z_j\\chi_{B_1'}(z))\\beta_{\\Lambda}\\Big\\rangle\\mathrm{d} \\nu_X(z)\\mathrm{d} s.\n\\end{align*}\nIn view of \\eqref{eq:nuX},\nwe can replace $\\chi_{B_1'}(z)$ and $\\mathrm{d}\\nu_X(z)$ by $\\chi_{B_1}(z_j)$ and $\\mathrm{d}\\nu(z_j)$,\nrespectively, under the sum in $I_7(n,t)$. Explicitly writing out the $L^2(\\mathbb{R}^2)$-scalar products in the integrands of $I_7(n,t)$\nas $\\mathrm{d} k$-integrals, we can switch the $\\mathrm{d} k$ and $\\mathrm{d}\\nu(z_j)$-integrations by Fubini's theorem afterwards.\nBy virtue of \\eqref{LKpsi}, this results in\n\\begin{align*}\nI_7(n,t)&=-\\sum_{j=1}^N\\int_0^t\n\\langle P_nU_{\\sigma,\\Lambda,s}^{N,+}[x,X_{\\bullet}]|\\mathrm{e}^{-\\mathrm{i} K\\cdot X_{j,s}^x}\\psi\\beta_{\\Lambda}\\rangle\\mathrm{d} s.\n\\end{align*}\nSince $(\\omega+\\psi)\\beta_{\\Lambda}=v_{\\Lambda}$, we see that\n\\begin{align*}\nI_2(n,t)+I_7(n,t)&=-\\int_0^t\n\\langle P_nU_{\\sigma,\\Lambda,s}^{N,+}[x,X_{\\bullet}]|v_{\\Lambda,s}^N[x,X_\\bullet]\\rangle\\mathrm{d} s.\n\\end{align*}\nAfter all these rearrangements, we pass to the limit $n\\to\\infty$ for fixed $t\\ge0$. Since $P_n\\to\\mathbbm{1}$ strongly,\nthe expression on the left hand side of \\eqref{ItoNeu1} converges pointwise on $\\Omega$ to $c_{\\sigma,\\Lambda,t}^N(q)$.\nBy dominated convergence, \\cref{def:wIRUV,def:wNsigmaUV,defuUV,defEren} we further have, again pointwise on $\\Omega$,\n\\begin{align*}\nI_1(n,t)&\\xrightarrow{\\;\\;n\\to\\infty\\;\\;}w_{\\sigma,\\Lambda,t}^N(x)+tNE_{\\sigma,\\Lambda}^{\\mathrm{ren}},\n\\\\\nI_2(n,t)+I_7(n,t)&\\xrightarrow{\\;\\;n\\to\\infty\\;\\;}-u_{\\sigma,\\Lambda,t}^N[x,X_\\bullet]-tNE_{\\sigma,\\Lambda}^{\\mathrm{ren}}.\n\\end{align*}\n\nFinally, recall that $h_{\\sigma,\\Lambda}(x,\\cdot)$, the integrand of the martingale\n$m_{\\sigma,\\Lambda}^N(x)$, has been introduced prior to \\cref{lemexpmomentm}.\nClearly, $h_{n}(s,z)\\to h_{\\sigma,\\Lambda}(q,s,z)$, $n\\to\\infty$, pointwise on $\\Omega$\nand for all $s\\ge0$ and $z\\in\\mathbb{R}^{2N}$. Since $\\Lambda$ is finite, we can employ\nsome $(g,N,m_{\\mathrm{b}})$-dependent constant times $(|z|\\|K\\beta_{\\Lambda}\\|)\\wedge\\|2\\beta\\|$\nas square-integrable dominating function to conclude that\n$h_n\\to h_{\\sigma,\\Lambda}(q,\\cdot)$ in $L^2((0,t]\\times\\mathbb{R}^{2N}\\times\\Omega,\\mathrm{d} s\\otimes\\nu_X\\otimes\\mathbb{P})$.\nBy means of It\\^{o}'s isometry (see, e.g., \\cite[Theorem~4.2.3(2)]{Applebaum.2009}) we deduce that\n$I_6(n,t)\\to m_{\\sigma,\\Lambda,t}^N(x)$, $n\\to\\infty$, in $L^2(\\mathbb{P})$ and in particular $\\mathbb{P}$-a.s.\nalong a suitable subsequence.\n\nAll these remarks prove $\\mathbb{P}$-a.s. an identity that is obviously equivalent to the one in \\eqref{ItoNeu0}\nfor fixed $t\\ge0$. Since the processes on both sides in \\eqref{ItoNeu0} are c\\`{a}dl\\`{a}g, they\nconsequently are indistinguishable.\n\\end{proof}\n\n\n\\subsection{Exponential moment bounds}\n\n\\noindent\nExponential moment bounds on the complex action\nare the essential quantitative ingredient for the proofs of our lower bounds on the minimal energy.\nThe first one, stated in the next theorem, is good for small $g^2N$ and obviously very bad for large $g^2N$.\n\n\\begin{thm}\\label{thmexpmonentu1}\nThere exist universal constants $b,c,c'>0$ such that, for all $\\alpha>1$, $p>0$, $0<\\Lambda\\le\\infty$,\n$t\\ge0$ and $x\\in\\mathbb{R}^{2N}$,\n\\begin{align*}\n&\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}e^{pu_{0,\\Lambda,s}^N(x)}\\bigg]\n\\le b^N\\left(\\frac{\\alpha}{\\alpha-1}\\right)^{1\/2}\\mathrm{e}^{2\\pi pN^2g^2\/m_{\\mathrm{b}}}\n\\\\\n&\\quad\\cdot\n\\exp\\left(tc'\\cdot\\frac{p^2g^4N^2(N-1)}{m_{\\mathrm{b}}}\n\\Big(1+\\frac{m_{\\mathrm{p}}}{m_{\\mathrm{b}}}\\Big)\n+tc\\cdot\\frac{\\alpha p^2g^4N^3}{m_{\\mathrm{b}}}\\cdot\n\\mathrm{e}^{8\\pi\\alpha pg^2N\/m_{\\mathrm{b}}}\\right).\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nThis follows from \\cref{defcomplexactionsigmaUV}, \\cref{bdcN}, \\cref{lemexpmomentw,lemexpmomentm} with $\\sigma=0$ and the Cauchy-Schwarz inequality.\n(After using the Cauchy-Schwarz inequality, we have to apply \\cref{lemexpmomentm,lemexpmomentm} with $2p$ put in place of $p$; hence the factor $8\\pi$ in the exponent of \n$\\mathrm{e}^{8\\pi\\alpha pg^2N\/m_{\\mathrm{b}}}$.)\n\\end{proof}\n\nWe infer another exponential moment bound on the complex action, exhibiting a\nbetter behavior for large $g^2N$. For $N=1$ and $m_{\\mathrm{p}}>0$ we also find a bound that\nis uniform in $m_{\\mathrm{b}}>0$.\n\n\\begin{thm}\\label{thmexpmonentu2}\nThere exist universal constants $b,c,c'>0$\nsuch that, for all $\\alpha>1$, $0<\\Lambda\\le\\infty$,\n$t\\ge0$, $x\\in\\mathbb{R}^{2N}$ and $p>0$ such that $pg^2N>m_{\\mathrm{b}}$,\n\\begin{align*}\n\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}e^{pu_{0,\\Lambda,s}^N(x)}\\bigg]\n&\\le b^N\\left(\\frac{\\alpha}{\\alpha-1}\\right)^{1\/2\n\\exp\\bigg(t\\pi pg^2(2N^2-N)\\ln\\Big(\\frac{pg^2N}{m_{\\mathrm{b}}}\\Big)\\bigg)\n\\\\\n&\\quad\\cdot\\exp\\Big(tc'(N-1)(m_{\\mathrm{p}}+pg^2N)+tc\\alpha pg^2N^2\\mathrm{e}^{8\\pi\\alpha}\\Big).\n\\end{align*}\nFor $m_{\\mathrm{p}}>0$, the argument of the first exponential can be replaced by\n\\begin{align*}\nt2\\pi pg^2N(N-1)\\ln\\Big(\\frac{pg^2N}{m_{\\mathrm{b}}}\\Big)\n+t\\pi pg^2N[\\ln(pg^2N)\\vee0]+t\\frac{\\pi g^2N}{m_{\\mathrm{p}}}[(pg^2N)\\wedge1].\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nThanks to our assumption $pg^2N>m_{\\mathrm{b}}$, we can fix $\\sigma>0$ such that\n\\begin{align}\\label{eq:sigma0}\n\\omega(\\sigma) = (\\sigma^2+m_{\\mathrm{b}}^2)^{1\/2}&=pg^2N\n\\end{align}\nFurthermore, by \\cref{eq:u-decomp,lem:actionito}, we may $\\mathbb{P}$-a.s. split up the complex action as\n\\begin{align}\\label{splitu}\nu_{0,\\Lambda,s}^N(x)&=u_{0,\\sigma,s}^N[x,X_\\bullet]\n-c_{\\sigma,\\infty,s}^N(x)+w_{\\sigma,\\infty,s}^N(x)+m_{\\sigma,\\infty,s}^N(x),\n\\quad s\\ge0.\n\\end{align}\nThe first two terms on the right hand side of \\cref{splitu} can be estimated trivially using \\eqref{bdcN} and\n\\begin{align}\\nonumber\n\\sup_{s\\in[0,t]}|u_{0,\\sigma,s}^N[x,X_\\bullet]|\n&\\le g^2N^2\\int_{B_{\\sigma}}\\int_0^t\\int_0^s\\frac{\\mathrm{e}^{-(s-r)\\omega(k)}}{\\omega(k)}\\mathrm{d} r\\,\\mathrm{d} s\\,\\mathrm{d} k-tNE_{\\sigma}^{\\mathrm{ren}}\n\\\\\\label{trivialbdu}\n&\\le tg^2N^2 \\int_{B_{\\sigma}}\\frac{1}{\\omega(k)^2}\\mathrm{d} k-tNE_{\\sigma}^{\\mathrm{ren}},\\quad t\\ge0.\n\\end{align}\nDirect computation and our choice of $\\sigma$ satisfying \\cref{eq:sigma0} lead to\n\\begin{align*}\ng^2(N^2-N)\\int_{B_{\\sigma}}\\frac{1}{\\omega(k)^2}\\mathrm{d} k\n=2\\pi g^2N(N-1) \\ln\\bigg(\\frac{pg^2N}{m_{\\mathrm{b}}}\\bigg).\n\\end{align*}\nSince $\\omega(k)>|k|\\ge\\psi(k)$,\nthe remaining contribution proportional to $N$ can be estimated using\n\\begin{align}\\label{eq:g2N-NE}\n\\begin{aligned}\n\tg^2N\\int_{B_\\sigma}\\frac{1}{\\omega(k)^2}\\mathrm{d} k-NE_{\\sigma}^{\\mathrm{ren}}\n\t&=g^2N\\int_{B_\\sigma}\\frac{\\psi(k)}{\\omega(k)^2(\\omega(k)+\\psi(k))}\\mathrm{d} k\n\t\\\\\n\t&\\le\\frac{g^2N}{2}\\int_{B_\\sigma}\\frac{1}{\\omega(k)^2}\\mathrm{d} k\n\t=g^2N\\pi \\ln\\bigg(\\frac{pg^2N}{m_{\\mathrm{b}}}\\bigg).\n\\end{aligned}\n\\end{align}\nFinally, we can bound the expectation of the product of\n$\\sup_{s\\in[0,t]}\\mathrm{e}^{pw_{\\sigma,\\infty,s}^N(x)}$ and $\\sup_{s\\in[0,t]}\\mathrm{e}^{pm_{\\sigma,\\infty,s}^N(x)}$ \nby means of the Cauchy-Schwarz inequality and \\cref{lemexpmomentm,lemexpmomentm} with the above choice for $\\sigma$. \nPutting all these remarks together we obtain the asserted bound for general $m_{p}\\ge0$.\n\nFinally, for $m_{\\mathrm{p}}>0$, we can instead use the following estimate.\nObserving that $\\psi(k)(\\psi(k)+2m_{\\mathrm{p}})=|k|^2$, we find\n\\begin{align*}\n\\int_{B_{\\sigma\\wedge 1}}\\frac{g^2\\psi(k)}{\\omega(k)^2(\\omega(k)+\\psi(k))}\\mathrm{d} k\n&=\\int_{B_{\\sigma\\wedge1}}\\frac{|k|^2}{\\omega(k)^2}\\cdot\\frac{g^2}{\\omega(k)(\\psi(k)+2m_{\\mathrm{p}})+|k|^2}\\mathrm{d} k\n\\\\\n&\\le \\frac{g^2}{2m_{\\mathrm{p}}}\\int_{B_{\\sigma\\wedge1}}\\frac{1}{\\omega(k)}\\mathrm{d} k\n\\\\\n&\\le\\frac{\\pi g^2}{m_{\\mathrm{p}}}(\\sigma\\wedge1)=\\frac{\\pi g^2}{m_{\\mathrm{p}}}[(pg^2N)\\wedge1],\n\\end{align*}\nand, similar to above,\n\\begin{align*}\n\\int_{B_\\sigma\\setminus B_{\\sigma\\wedge 1}}\\frac{g^2\\psi(k)}{\\omega(k)^2(\\omega(k)+\\psi(k))}\\mathrm{d} k\n&\\le \\frac{g^2}{2}\\int_{B_\\sigma\\setminus B_{\\sigma\\wedge 1}}\\frac{1}{\\omega(k)^2}\\mathrm{d} k\n\\le \\pi g^2[\\ln(pg^2N)\\vee 0].\n\\end{align*}\nReplacing the bound from \\cref{eq:g2N-NE} with the above proves the statement.\n\\end{proof}\n\n\\subsection{Convergence properties of the complex action}\n\n\\noindent\nUsing the bounds derived in \\cref{sec:basicproc}, we can easily deduce convergence statements for the complex action when removing the ultraviolet cutoff.\n\n\\begin{thm}\\label{thm:uconv}\nLet $\\sigma,t\\ge0$. Then\n\\begin{align}\\label{convu1}\n\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|u_{\\sigma,\\Lambda,s}^N(x)-u_{\\sigma,\\infty,s}^N(x)|^p\\bigg]\n&\\xrightarrow{\\;\\;\\Lambda\\to\\infty\\;\\;}0,\n\\\\\\label{convu2}\n\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}\\big|\\mathrm{e}^{u_{0,\\Lambda,s}^N(x)}-\\mathrm{e}^{u_{0,\\infty,s}^N(x)}\\big|^p\\bigg]\n&\\xrightarrow{\\;\\;\\Lambda\\to\\infty\\;\\;}0.\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nBy \\cref{eq:u-decomp}, for $\\sigma<\\Lambda<\\infty$, we $\\mathbb{P}$-a.s. have\n\\begin{align*}\nu_{\\sigma,\\Lambda,s}^N(x)-u_{\\sigma,\\infty,s}^N(x)&=w_{\\Lambda,\\infty,s}^N(x)-c_{\\Lambda,\\infty,s}^N(x)+m_{\\Lambda,\\infty,s}^N(x),\n\\quad s\\ge0.\n\\end{align*}\nTherefore, the convergence \\eqref{convu1} follows directly from \\cref{lemexpmomentm,lempmomentw}, \n\\cref{bdcN} (all applied with $(\\Lambda,\\infty)$ put in place of $(\\sigma,\\Lambda)$)\nand Minkowski's inequality. Now \\eqref{convu2} follows from the bound \n$|\\mathrm{e}^a-\\mathrm{e}^b|\\le|a-b|\\mathrm{e}^a\\mathrm{e}^b$, $a,b\\ge0$, H\\\"{o}lder's inequality,\nthe exponential moment bound in \\cref{thmexpmonentu1,convu1}.\n\\end{proof}\n\nThe following conclusion easily follows and is a helpful technical ingredient for our proof of strong continuity of the \nFeynman--Kac semigroup.\n\\begin{cor}\\label{cor:pathsactioncont}\n\tFor any $\\Lambda\\in(0,\\infty]$ and $x\\in\\mathbb{R}^{2N}$, the paths \n\t$[0,\\infty)\\ni t\\mapsto u^N_{0,\\Lambda,t}(x)$ are continuous $\\mathbb{P}$-a.s. \n\\end{cor}\n\\begin{proof}\n\tThe statement directly follows from \\cref{def:action-process} for $\\Lambda<\\infty$. \n\tFurther, by \\cref{convu1}, there exists an increasing sequence of cutoffs $\\Lambda_n\\subset (0,\\infty)$, $n\\in\\mathbb{N}$, \n\twith $\\lim_{n\\to \\infty}\\Lambda_n=\\infty$ such that $u_{0,\\Lambda_n,s}^N(x)$ converges  $\\mathbb{P}$-a.s. to $u_{0,\\infty,s}^N(x)$ uniformly in $s$ on any compact subinterval of $[0,\\infty)$. This proves the case $\\Lambda=\\infty$.\n\\end{proof}\n\n\\subsection{Canonical representative of the complex action}\n\\noindent\nWe construct a canonical representative of the limiting complex action on the Skorohod space of c\\`adl\\`ag functions. This will be very convenient in our proof of the Markov property and hence the semigroup property of the Feynman--Kac semigroup in \\cref{subsec:Markov}. \n\nWe let $D\\coloneq D([0,\\infty),\\mathbb{R}^{2N})$ denote the set of all c\\`{a}dl\\`{a}g functions from $[0,\\infty)$ to\n$\\mathbb{R}^{2N}$ equipped with Skorohod's $J_1$-topology. Then $D$ is Polish and we know its Borel $\\sigma$-algebra\nagrees with the trace on $D$ of the infinite product $\\sigma$-algebra on the set of all maps from $[0,\\infty)$ to $\\mathbb{R}^{2N}$.\nIn symbols,\n\\begin{align*}\n\\fr{B}(D)&= \\sigma\\big(\\mathrm{ev}_t :\\,t\\in[0,\\infty)\\big).\n\\end{align*} \nHere and henceforth the evaluation maps denoted by $\\mathrm{ev}_t$ are considered as functions on $D$, that is,\n$\\mathrm{ev}_t:D\\to\\mathbb{R}^{2N}$, $\\gamma\\mapsto\\gamma_t$.\nThey generate the filtration\n\\begin{align*}\n\\fr{D}_t'&\\coloneq \\sigma\\big(\\mathrm{ev}_s:\\,s\\in[0,t]\\big),\\quad t\\ge0.\n\\end{align*}\nWhile the choice of our basic c\\`{a}dl\\`{a}g L\\'{e}vy process $X$ is not unique, all possible choices have the same\nlaw on $D$ that we denote by $\\mathbb{P}_{\\mathrm{can}}'$. That is, $\\mathbb{P}_{\\mathrm{can}}'=\\mathbb{P}\\circ X_{\\bullet}^{-1}$ \nand the process $(\\mathrm{ev}_t)_{t\\ge0}$ on $(D,\\fr{B}(D),\\mathbb{P}_{\\mathrm{can}}')$ is the canonical representative of \n$X$ with paths in $D$.\nFurthermore, we let $(D,\\fr{D},(\\fr{D}_t)_{t\\ge0},\\mathbb{P}_{\\mathrm{can}})$ denote the completion of \nthe filtered probability space $(D,\\fr{B}(D),(\\fr{D}_t')_{t\\ge0},\\mathbb{P}_{\\mathrm{can}}')$.\nSince $(\\mathrm{ev}_t)_{t\\ge0}$ is L\\'{e}vy with c\\`{a}dl\\`{a}g paths, we know that $(\\fr{D}_t)_{t\\ge0}$ is\nautomatically right-continuous; see, e.g., \\cite[Theorem~2.1.10]{Applebaum.2009}.\nIn conclusion, $(D,\\fr{D},(\\fr{D}_t)_{t\\ge0},\\mathbb{P}_{\\mathrm{can}})$ satisfies the usual hypotheses.\n\n\n\\begin{defn}\\label{defncanreprCCaction}\nWe denote by $(m_{t}[x,\\cdot])_{t\\ge0}$, $x\\in\\mathbb{R}^{2N}$,\nparticular versions of the martingales we obtain by applying \\cref{lem:canm} with $\\sigma=0$ and $\\Lambda=\\infty$ to the\nfiltered probability space $(D,\\fr{D},(\\fr{D}_t)_{t\\ge0},\\mathbb{P}_{\\mathrm{can}})$ and the L\\'{e}vy process\n$(\\mathrm{ev}_t)_{t\\ge0}$. Furthermore, we denote by $(w_{t}[x,\\cdot])_{t\\ge0}$\nand $(c_{t}[x,\\cdot])_{t\\ge0}$ the processes obtained upon choosing $\\sigma=0$, $\\Lambda=\\infty$\nand $X=(\\mathrm{ev}_t)_{t\\ge0}$ in \\cref{def:wNsigmaUV} and \\cref{def:cNsigmaUV},\nrespectively. Finally, we define canonical representatives of the complex action by setting\n\\begin{align*}\nu_{t}^N[x,\\gamma]&\\coloneq w_{t}^N[x,\\gamma]-c_{t}^N[x,\\gamma]+m_{t}^N[x,\\gamma],\n\\end{align*}\nfor all $t\\ge0$, $x\\in\\mathbb{R}^{2N}$ and $\\gamma\\in D$. \n\\end{defn}\n\n\\begin{thm}\\label{thm:canonical}\nLet $x\\in\\mathbb{R}^{2N}$. \nThen the processes $(u_{t}^N(x))_{t\\ge0}$ and $(u_t^N[x,X_\\bullet])_{t\\ge0}$ are indistinguishable.\n\\end{thm}\n\\begin{rem}\n\tEquivalently, $(m_{t}^N(x))_{t\\ge0}$ and $(m_t^N[x,X_\\bullet])_{t\\ge0}$ are indistinguishable.\n\\end{rem}\n\\begin{proof}\nDenoting expectations with respect to $\\mathbb{P}_{\\mathrm{can}}$ by $\\mathbb{E}_{\\mathrm{can}}$, we find\n\\begin{align*}\n\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|u_{s}^N(x)-u_t^N[x,X_\\bullet]|^2\\bigg]\n&=\\lim_{\\Lambda\\to\\infty}\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}|u_{\\Lambda,s}^N[x,X_\\bullet]-u_t^N[x,X_\\bullet]|^2\\bigg]\n\\\\\n&=\\lim_{\\Lambda\\to\\infty}\\mathbb{E}_{\\mathrm{can}}\\bigg[\n\\sup_{s\\in[0,t]}|u_{\\Lambda,s}^N[x,\\mathrm{ev}_{\\bullet}]-u_t^N[x,\\mathrm{ev}_{\\bullet}]|^2\\bigg]\n\\\\\n&=0,\\quad t\\ge0.\n\\end{align*}\nIn the first equality we applied \\eqref{convu1} (with $\\sigma=0$ and filtered probability space\n$(\\Omega,\\fr{F},(\\fr{F}_t)_{t\\ge0},\\mathbb{P})$) and took into account that $(u_{0,\\Lambda,t}^N(x))_{t\\ge0}$ and \n$(u_{\\Lambda,t}^N[x,X_\\bullet])_{t\\ge0}$ are indistinguishable. In the second equality\nwe used that $\\mathbb{P}_{\\mathrm{can}}$ is the law of $X$.\nIn the last one we again applied \\eqref{convu1}, now with filtered probability space\n$(D,\\fr{D},(\\fr{D}_t)_{t\\ge0},\\mathbb{P}_{\\mathrm{can}})$.\n\\end{proof}\n\n\n\\section{Feynman--Kac integrands and semigroups}\\label{sec:FKInt}\n\\noindent\nIn this \\lcnamecref{sec:FKInt} we discuss (possibly ultraviolet cutoff versions of) the Feynman--Kac integrands defined in \\cref{def:Wt}\nas well as the corresponding semigroups. In \\cref{ssecdefWT} we first collect relevant definitions\nand a few immediate observations. In \\cref{ssecpwbds} we then provide pathwise bounds on the Fock space operator-valued\npart of the Feynman--Kac integrands, which in conjuction with our results on the complex action are applied in the succeeding\nsubsections. There we prove weighted $L^p$ to $L^q$ bounds on and\nconvergence as $\\Lambda\\to\\infty$ of semigroup members (\\cref{subsec:LpLq,ssecconvTUV}), \nMarkov and semigroup properties (\\cref{subsec:Markov})\nand finally strong continuity of the semigroups as well as equicontinuity in the range of semigroup elements\n(\\cref{sseccontprop}).\n\n\n\n\\subsection{Definition of Feynman--Kac integrands and operators}\\label{ssecdefWT}\n\n\\noindent\nWe start out by defining possibly cutoff versions and canonical representatives of \\cref{def:Wt}. \n\\begin{defn}\\label{def:FKint}\n\tLet $x\\in\\mathbb{R}^{2N}$, $t>0$ and $\\Lambda\\in(0,\\infty]$.\n\tFor every Borel measurable $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ we define\n\t\\begin{align*}\n\t\tD_{\\Lambda,t}[x,\\gamma]&\\coloneq F_{t\/2}(-U_{0,\\Lambda,t}^{N,+}[x,\\gamma])F_{t\/2}(-U^{N,-}_{0,\\Lambda,t}[x,\\gamma])^*.\n\t\\end{align*}\n\tFor every c\\`{a}dl\\`{a}g $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ we further set\n\t\\begin{align*}\n\t\tW_{\\Lambda,t}[x,\\gamma]&\\coloneq\n\t\t\\mathrm{e}^{u^N_{0,\\Lambda,t}[x,\\gamma]-\\int_0^tV(x+\\gamma_s)\\mathrm{d} s}D_{\\Lambda,t}[x,\\gamma],\n\t\\end{align*}\n\twhere $u^N_{0,\\Lambda,t}[x,\\gamma]$ is given by \\eqref{defuUV} for finite $\\Lambda$ and\n\t$u^N_{0,\\infty,t}[x,\\gamma]\\coloneq u^N_t[x,\\gamma]$ is the canonical representative\n\tof the limiting complex action introduced in \\cref{defncanreprCCaction}. When working with $X$ we\n\temploy the slightly more compact notation\n\t\\begin{align*}\n\tW_{\\Lambda,t}(x)&\\coloneq \n\t\\mathrm{e}^{u^N_{0,\\Lambda,t}(x)-\\int_0^tV(X_s^x)\\mathrm{d} s}D_{\\Lambda,t}[x,X_\\bullet].\n\t\\end{align*}\nFinally, we set $W_{\\Lambda,0}[x,\\gamma]\\coloneq W_{\\Lambda,0}(x)\\coloneq \\mathbbm{1}_{\\mathcal{F}}$.\n\\end{defn}\n\nNotice that $W_{\\infty,t}(x)=W_t(x)$ where $W_t(x)$ is defined in \\cref{def:Wt}. \n\n\\begin{rem}\\label{rem:Wadapt}\nWe know from \\cite{GueneysuMatteMoller.2017} that $L^2(\\mathbb{R}^2)\\ni h\\mapsto F_s(h)\\in\\mathcal{B}(\\mathcal{F})$ with $s>0$ is continuous\n(in fact analytic); see also \\eqref{normdiffFt} below. Together with \\cref{lem:Upmadapted} and the way we constructed\nthe complex action this implies that \n$W_{\\Lambda,t}(x):\\Omega\\to\\mathcal{B}(\\mathcal{F})$ is $\\fr{F}_t$-measurable and separably valued for all\n$t\\ge0$, $x\\in\\mathbb{R}^{2N}$ and $\\Lambda\\in(0,\\infty]$.\n\\end{rem}\n\n\\begin{rem}\\label{rem:WWindist}\n \\cref{thm:canonical} implies that $(W_{\\Lambda,t}(x))_{t\\ge0}$ and \n$(W_{\\Lambda,t}[x,X_\\bullet])_{t\\ge0}$ are indistinguishable.\n\\end{rem}\n\n\\begin{rem}\\label{rem:Wprodmeas}\nWorking on the filtered probability space $(D,\\fr{D},(\\fr{D}_t)_{t\\ge0},\\mathbb{P}_{\\mathrm{can}})$ we observe that\n\\begin{align*}\n\\scr{N}_V&\\coloneq\\big\\{(x,\\gamma)\\in\\mathbb{R}^{2N}\\times D\\,\\big|\\,\\gamma\\in\\scr{N}_V(x)\\big\\}\n\\\\\n&= \\bigg\\{(x,\\gamma)\\in\\mathbb{R}^{2N}\\times D\\,\\bigg|\\:\n\\max_{\\diamond\\in\\{+,-\\}}\\sup_{n\\in\\mathbb{N}}\\int_0^nV_\\diamond(x+\\mathrm{ev}_s(\\gamma))\\mathrm{d} s=\\infty\\bigg\\}\n\\in\\fr{B}(\\mathbb{R}^{2N})\\otimes\\fr{D}.\n\\end{align*}\nIn view of the convention introduced in \\cref{sssecKato}, we infer from this that the expression\n$\\int_0^tV(x+\\mathrm{ev}_s(\\gamma))\\mathrm{d} s$ is $\\fr{B}(\\mathbb{R}^{2N})\\otimes\\fr{D}$-measurable in $(x,\\gamma)$ for every $t\\ge0$.\nAnalogous remarks apply to $w_{0,\\infty,t}^N$ defined in \\eqref{def:wNsigmaUV}.\nAlso taking into account the continuity in $h$ of $F_s(h)$ mentioned in \\cref{rem:Wadapt} as well as\n\\cref{remUpmcont,lem:canm,defncanreprCCaction}, we conclude that\n$W_{\\Lambda,t}[x,\\gamma]$ is $\\fr{B}(\\mathbb{R}^{2N})\\otimes\\fr{D}$-measurable in $(x,\\gamma)$ as a $\\mathcal{B}(\\mathcal{F})$-valued function\nfor every $t\\ge0$.\n\\end{rem}\nIn view of \\cref{rem:Wadapt}, we can introduce\ncorresponding transition operators by their pointwise action:\n\\begin{defn}\\label{def:semigroup}\nLet $\\Lambda\\in(0,\\infty]$ and $t\\ge0$. Then, we define\n\\begin{align*}\n(T_{\\Lambda,t}\\Psi)(x)&\\coloneq \\mathbb{E}\\big[W_{\\Lambda,t}(x)^*\\Psi(X_t^x)\\big],\\quad x\\in\\mathbb{R}^{2N},\n\\end{align*}\nfor all measurable $\\Psi:\\mathbb{R}^{2N}\\to\\mathcal{F}$ such that $\\|W_{\\Lambda,t}(x)^*\\Psi(X_t^x)\\|$\nis $\\mathbb{P}$-integrable for all $x\\in\\mathbb{R}^{2N}$.\n\\end{defn}\n\nSince $\\mathbb{P}(X_t^x\\in\\scr{N})=0$ for all zero sets $\\scr{N}\\in\\fr{B}(\\mathbb{R}^{2N})$ and $t>0$,\nthe maps $T_{\\Lambda,t}$, $t>0$, are also well-defined on equivalence classes of functions. \nWe will often and tacitly make use of this. The symbol $T_{\\Lambda,0}$ always stands for the\nidentity mapping, whether acting on measurable functions or their equivalence classes.\n\n \n\\subsection{Some pathwise bounds}\\label{ssecpwbds}\n\n\\noindent\nWe start by collecting some bounds on the Fock space operators $F_t(g)$ defined in \\eqref{defFtg}.\nTo this end, we introduce the series\n\\begin{align}\\label{def:S}\n\\mc{S}(z)&\\coloneq \\sum_{n=1}^\\infty\\frac{2^nz^n}{(n!)^{1\/2}},\\quad z\\in\\mathbb{C}.\n\\end{align}\nRecall also the definition \\eqref{deftnorm} of the time dependent norms $\\|\\cdot\\|_t$ on $L^2(\\mathbb{R}^2)$.\n\n\\begin{lem}\nLet $g,\\tilde{g},h,\\tilde{h}\\in L^2(\\mathbb{R}^2)$ and $t>0$. Then\n\\begin{align}\\label{normFt}\n\\|F_t(g)\\|&\\le \\mc{S}(\\|g\\|_t)\n\\\\\\label{normdiffFt}\n\\|F_t(g)-F_t(h)\\|&\\le2\\|g-h\\|_t\\sup_{\\tau\\in[0,1]}\\mc{S}\\big(\\|\\tau g+(1-\\tau)h\\|_t\\big),\n\\end{align}\nand\n\\begin{align}\\nonumber\n&\\|F_t(g)F_t(\\tilde{g})^*-F_t(h)F_t(\\tilde{h})^*\\|\n\\\\\\label{LipFF}\n&\\le 2(\\|g-h\\|_t+\\|\\tilde{g}-\\tilde{h}\\|_t)\n\\max\\big\\{\\mc{S}(\\|g\\|_t),\\mc{S}(\\|h\\|_t)\\big\\}\\max\\big\\{\\mc{S}(\\|\\tilde{g}\\|_t),\\mc{S}(\\|\\tilde{h}\\|_t)\\big\\}.\n\\end{align}\nFurthermore, for all $\\varepsilon\\in(0,1]$,\n\\begin{align}\\label{FFbei0}\n\\|(F_t(g)F_t(\\tilde{g})^*-\\mathbbm{1})(1+\\mathrm{d}\\Gamma(\\omega))^{-\\varepsilon}\\|\n&\\le 2(\\|g\\|_t+\\|\\tilde{g}\\|_t)\\mc{S}(\\|g\\|_t)\\mc{S}(\\|\\tilde{g}\\|_t)+(2t)^{\\varepsilon}.\n\\end{align}\nFinally, if $\\omega g\\in L^2(\\mathbb{R}^{2N})$, then  $F_t(g)\\mathcal{F}\\subset\\mathcal{D}(\\mathrm{d}\\Gamma(\\omega)$ and\n\t\\begin{align}\\label{eq:FdG}\n\t\t\\|\\mathrm{d}\\Gamma(\\omega)F_t(g)\\| \\le 2\\|\\omega g\\|_t\\mc S(\\|g\\|_t) + \\frac1{\\mathrm{e}\\tau}\\mc S(\\|g\\|_{t-\\tau}),\\quad \\tau\\in(0,t)  .\n\t\\end{align}\n\\end{lem}\n\n\\begin{proof}\nThe bound \\eqref{normFt} is proven in \\cite[Lemma~17.4]{GueneysuMatteMoller.2017}, where it is also shown that\n\\begin{align}\\label{eq:Fderbound}\n\\sum_{n=0}^\\infty\\frac{1}{n!}\\|a^\\dagger(f)a^\\dagger(g)^n\\mathrm{e}^{-t\\mathrm{d}\\Gamma(\\omega)}\\|&\\le 2\\|f\\|_t\\mc{S}(\\|g\\|_t).\n\\end{align}\nas well as that the Fr\\'echet derivative of $F_t$ satisfies\n\\begin{align*}\n\tF_t'(g)f=\\sum_{n=0}^\\infty\\frac{1}{n!}a^\\dagger(f)a^\\dagger(g)^n\\mathrm{e}^{-t\\mathrm{d}\\Gamma(\\omega)}.\n\\end{align*}\nThis directly implies \\eqref{normdiffFt}, which in turn entails \\eqref{LipFF}. \nApplying \\eqref{LipFF} with $h=\\tilde{h}=0$ \nand recalling that $F_{t}(0)=\\mathrm{e}^{-t\\mathrm{d}\\Gamma(\\omega)}$, we see that, to verify \\eqref{FFbei0}, it only\nremains to show that $\\|(\\mathbbm{1}-\\mathrm{e}^{-2t\\mathrm{d}\\Gamma(\\omega)})(1+\\mathrm{d}\\Gamma(\\omega))^{-\\varepsilon}\\|\\le (2t)^{\\varepsilon}$.\nThe latter bound, however, follows directly from the spectral calculus, $\\mathrm{d}\\Gamma(\\omega)\\ge 0$ \nand the simple bound $\\sup_{s\\ge0}(1-e^{-2ts})\/(1+s)^\\varepsilon \\le (2t)^\\varepsilon$.\n\nFinally, \\cref{eq:FdG} follows by using the commutation relation $[\\mathrm{d}\\Gamma(\\omega),a^\\dagger(g)]=a^\\dagger(\\omega g)$, which holds on the range of $a^\\dagger(g)^k\\mathrm{e}^{-t\\mathrm{d}\\Gamma(\\omega)}$ for every $k\\in\\mathbb{N}_0$, the bounds \\eqref{normFt}, \\eqref{eq:Fderbound}  and the simple estimate $\\|\\mathrm{d}\\Gamma(\\omega)\\mathrm{e}^{-\\tau \\mathrm{d}\\Gamma(\\omega)}\\|\\le 1\/\\mathrm{e} \\tau$.\n\\end{proof}\nIn the next lemma, we consider the Fock space operator-valued part of our Feynman--Kac integrands. The aim is\nto obtain norm bounds and study their convergence properties as $\\Lambda$ goes to infinity or $t$ goes to zero. \n\n\\begin{lem}\\label{lempwbdD}\n\tLet $s_{*} = \\mc S((2\\pi\/m_\\mathrm{b})^{1\/2}|g|N)$.\n\tThen, for all $t>0$, $x\\in\\mathbb{R}^{2N}$, Borel measurable $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ and $\\varepsilon\\in(0,1\/2)$, $\\Lambda\\in(0,\\infty]$, we have the bounds\n\\begin{align}\n\\label{bdFU}\n&\\|D_{\\Lambda,t}[x,\\gamma]\\| \\le s_*^2, \n\\\\\n&\\label{bddGFU}\\|\\mathrm{d}\\Gamma(\\omega)D_{\\Lambda,t}[x,\\gamma]\\| \\le 2(6\\pi)^{1\/2} |g|N \\omega(\\Lambda)s_* + \\frac 2{\\mathrm{e} t}s_*^2,\\quad 0<\\Lambda<\\infty,\n\\\\\n\\label{bddiffFFUU}\n&\\|D_{\\Lambda,t}[x,\\gamma]-D_{\\infty,t}[x,\\gamma]\\|\n\\le  \\frac{2(6\\pi)^{1\/2}|g|N}{\\omega(\\Lambda)^{1\/2}}s_*^2,\\quad 0<\\Lambda<\\infty,\n\\\\\n\\label{Dbei0}\n&\\|(D_{\\Lambda,t}[x,\\gamma]^*-\\mathbbm{1})(1+\\mathrm{d}\\Gamma(\\omega))^{-\\varepsilon}\\| \\le t^{\\varepsilon}\\bigg(1+\\frac{4(6\\pi)^{1\/2}|g|N}{m_{\\mathrm{b}}^{(1-2\\varepsilon)\/2}}s_*^2\\bigg).\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nPick $t>0$, $x\\in\\mathbb{R}^{2N}$ and some path $\\gamma$ as in the statement. From \\cref{tnormU} with $\\varepsilon=\\sigma=0$, we know that\n\\begin{align}\\label{bdeasy}\n\t\\|U^{N,\\pm}_{0,\\Lambda,t}[x,\\gamma]\\|_{t\/2} \\le (2\\pi\/m_\\mathrm{b})^{1\/2}|g|N.\n\\end{align} \nHence, \\cref{bdFU} directly follows from \\cref{normFt}. Similarly, \\cref{bddGFU} follows from \\cref{tnormwU,eq:FdG} (taking the limit $\\tau\\to\\frac t2$).\nFurther, \\cref{bdeasy} implies that the left hand side of \n\\eqref{bddiffFFUU} is less than or equal to\n$2(\\|U^{N,+}_{\\Lambda,\\infty,t}[x,\\gamma]\\|_t+\\|U^{N,-}_{\\Lambda,\\infty,t}[x,\\gamma]\\|_t)s_*^2$.\nAgain employing \\eqref{tnormU} with $\\varepsilon=0$ and\n$(\\Lambda,\\infty)$ put in place of $(\\sigma,\\Lambda)$, we see that \\eqref{bddiffFFUU} holds true as well.\nFinally, \\eqref{Dbei0} follows from \\crefnosort{FFbei0,bdeasy,tnormU} applied with $(\\varepsilon,\\sigma)$ replaced by $(2\\varepsilon,0)$.\n\\end{proof}\n\n\n\\subsection{Weighted \\texorpdfstring{$L^p$}{Lp} to \\texorpdfstring{$L^q$}{Lq} bounds}\\label{subsec:LpLq}\n\n\\noindent\nWe now want to derive bounds on the maps $T_{\\Lambda,t}$ \nseen as operators from $L^p(\\mathbb{R}^{2N},\\mathcal{F})$ to $L^q(\\mathbb{R}^{2N},\\mathcal{F})$ for any $1< p\\le q\\le \\infty$. We also cover Lipschitz-continuous weight functions, in the sense of the norm given below in \\cref{1norm2}. Similar bounds for the free relativistic semigroup are derived in \\cref{app:lplq} and applied in what follows.\n\nAs in \\cref{app:lplq}, it would also be possible to include the case $p=1$ in the discussion of the families\n$(T_{\\Lambda,t})_{t\\ge0}$, $\\Lambda\\in(0,\\infty]$. This would, however, require additional arguments proceeding along the\nlines in \\cite{MatteMoller.2018} at several places in the article. As we have no actual need to include the case $p=1$\nhere, we refrained from doing so. (Only considering the case $p=2$ would not lead to any simplifications.)\n\n\n\n\\begin{rem}\\label{remmomentbdW}\nLet $\\mho_*(p,t)$ denote the product of $s_*^{2p}$ from \\cref{lempwbdD}\nwith the minimum of the three alternative right hand sides \nin the exponential moment bounds of \\cref{thmexpmonentu1,thmexpmonentu2}.\nThus, $\\mho(p,t)\\le \\mathrm{e}^{c_p(1\\vee t)}$ where $c_p\\in(0,\\infty)$ depends, besides $p$,\nalso on the model parameters $m_{\\mathrm{p}}$, $m_{\\mathrm{b}}$, $g$, $N$ and the parameter $\\alpha$\nappearing in \\cref{thmexpmonentu1,thmexpmonentu2}, but not on $\\Lambda$. \nCombining the lemma and two theorems with H\\\"{o}lder's inequality and \\eqref{V2} we obtain,\nfor all $p,t>0$ and $\\theta\\in(1,\\infty)$,\n\\begin{align}\\label{momentbdW}\n\\sup_{\\Lambda\\in(0,\\infty]}\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}\\|W_{\\Lambda,s}(x)\\|^p\\bigg]\n&\\le \\mathrm{e}^{c_{\\theta,pV_-}(1\\vee t)}\\mho(\\theta p,t)^{1\/\\theta}.\n\\end{align}\nHere $c_{\\theta,pV_-}\\in[0,\\infty)$ solely depends on $\\theta$ and the product $pV_-$. If $V_-$ is bounded\nwe can also choose $\\theta=1$ and $c_{\\theta,pV_-}=p\\|V_-\\|_\\infty$.\n\\end{rem}\n\nFor $1\\le p\\le q\\le \\infty$, we shall denote the norm on the space of all bounded operators from $L^p(\\mathbb{R}^{2N},\\mathcal{F})$\nto $L^q(\\mathbb{R}^{2N},\\mathcal{F})$ by $\\|\\cdot\\|_{p,q}$ in what follows.\nIn the next proposition we also set\n\\begin{align}\\label{1norm2}\n|x|_{1}&\\coloneq|x_1|+\\dots+|x_N|,\\quad x=(x_1,\\ldots,x_N)\\in\\mathbb{R}^{2N},\n\\end{align}\nwhere $|\\cdot|$ is the Euclidean norm on $\\mathbb{R}^2$.\n\n\\begin{prop}\\label{prop:LpLq}\nIn the case $m_{\\mathrm{p}}>0$, we assume that $G:\\mathbb{R}^{2N}\\to\\mathbb{R}$ satisfies \n\\begin{align*}\n|G(x)-G(x')|&\\le L|x-x'|_1,\\quad x,x'\\in\\mathbb{R}^{2N},\n\\end{align*}\nwith $L\\in[0,m_{\\mathrm{p}})$. In the case $m_{\\mathrm{p}}=0$ we set $G=L=0$. \nIf $m_{\\mathrm{p}}>L>0$, then we pick some $r>1$ such that $rL<m_{\\mathrm{p}}$.\nOtherwise we set $r=\\infty$ so that $r'=1$.\nLet $\\Lambda\\in(0,\\infty]$, $p\\in (1,\\infty]$, $q\\in[p,\\infty]$ and $t>0$. Then the following holds:\n\\begin{enumerate}\n\\item[{\\rm(i)}] \nThe random variable\n$\\|W_{\\Lambda,t}(x)^*(\\mathrm{e}^{-G}\\Psi)(X_t^x)\\|_{\\mathcal{F}}$ is $\\mathbb{P}$-integrable for all $x\\in\\mathbb{R}^{2N}$ and $\\Psi\\in L^p(\\mathbb{R}^{2N},\\mathcal{F})$.\n\\item[{\\rm(ii)}] \n$\\mathrm{e}^{G}T_{\\Lambda,t}\\mathrm{e}^{-G}$ maps $L^p(\\mathbb{R}^{2N},\\mathcal{F})$ continuously into $L^q(\\mathbb{R}^{2N},\\mathcal{F})$. \n\\item[{\\rm(iii)}] \nLet also $\\theta\\in(1,\\infty)$.\nThere exists some $b_*>0$, depending solely on $m_{\\mathrm{p}}$, $N$, $L$, $p$, $q$ and $r$,\nas well as some $c_*>0$, depending solely on $V_-$,  $p$, $r$ and $\\theta$, such that\n\\begin{align*}\n\\|\\mathrm{e}^GT_{\\Lambda,t}\\mathrm{e}^{-G}\\|_{p,q}\n&\\le b_*\\bigg[\\frac{1}{t^2}\\vee\\frac{m_{\\mathrm{p}}-L}{t}\\bigg]^{N(\\frac{1}{p}-\\frac{1}{q})}\n\\mathrm{e}^{LNt+c_{*}(1\\vee t)}\\mho(\\theta p'r',t)^{\\frac{1}{\\theta p'r'}}.\n\\end{align*}\nFor bounded $V_-$ we can also choose $\\theta=1$ and $c_*=\\|V_-\\|_\\infty$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nAll assertions are direct consequences of \\cref{remmomentbdW,corwLpLqNW}.\n\\end{proof}\n\nOur next lemma holds only for finite $\\Lambda$. It is essential for our proof of the continuity results in \\cref{thm:cont}.\n\n\\begin{lem}\\label{prop:LpLqdG}\n\tLet $\\Lambda\\in(0,\\infty)$, $p\\in(1,\\infty]$ and $q\\in[p,\\infty]$. Then $T_{\\Lambda,t}$ with $t>0$ maps\n\t$L^p(\\mathbb{R}^{2N},\\mathcal{F})$ into $L^q(\\mathbb{R}^{2N},\\mathcal{D}(\\mathrm{d}\\Gamma(\\omega)))$ and\n\t\\begin{align*}\n\t\t\\sup_{t\\in[\\tau_1,\\tau_2]}\\|\\mathrm{d}\\Gamma(\\omega)T_{\\Lambda,t}\\|_{p,q}&<\\infty,\\quad 0<\\tau_1\\le\\tau_2<\\infty.\n\t\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nIn view of \\cref{tnormwU,trivialbdu,normFt,eq:FdG,V2} it is clear that \n\\begin{align*}\n\\sup_{s\\in[0,t]}\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}[\\|\\mathrm{d}\\Gamma(\\omega)W_{\\Lambda,t}(x)^*\\|^p]<\\infty,\\quad p,t>0.\n\\end{align*}\nIn conjunction with \\cref{corwLpLqNW} (there we choose $d=2$, $G=L=0$, $r'=1$) this implies all statements.\n\\end{proof}\n\n\\subsection{Convergence properties}\\label{ssecconvTUV}\n\n\\noindent\nNext, we derive the semigroup convergence when removing the ultraviolet cutoff. \n\n\n\\begin{prop}\\label{prop:Tconv}\n\tLet $1<p\\le q\\le\\infty$ and assume $0< \\tau_1\\le \\tau_2<\\infty$. Then\n\t\\begin{align*}\n\t\t\\sup_{t\\in[0,\\tau_1]} \\|T_{\\Lambda,t}-T_{\\infty,t}\\|_{p,p}\\xrightarrow{\\;\\;\\Lambda\\to\\infty\\;\\;} 0, \\quad \n\t\t\\sup_{t\\in[\\tau_1,\\tau_2]} \\|T_{\\Lambda,t}-T_{\\infty,t}\\|_{p,q} \\xrightarrow{\\;\\;\\Lambda\\to\\infty\\;\\;} 0.\n\t\\end{align*}\n\tWe can actually work with the supremum over $x\\in\\mathbb{R}^{2N}$ instead of the essential supremum when $q=0$,\n\ti.e., in both relations above we have\n\t\\begin{align*}\n\t\\|T_{\\Lambda,t}-T_{\\infty,t}\\|_{p,\\infty}\n\t&=\\sup_{t\\in[\\tau_1,\\tau_2]} \\sup_{\\|\\Psi\\|_p\\le 1}\\sup_{x\\in\\mathbb{R}^{2N}}\\|(T_{\\Lambda,t}\\Psi)(x)-(T_{\\infty,t}\\Psi)(x)\\|_{\\mathcal{F}}.\n\t\\end{align*}\n\\end{prop}\n\\begin{proof}\n\tBy Minkowski's inequality, we have\n\t\\begin{align*}\n\t\t&\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}\\|W_{\\Lambda,s}(x)-W_{\\infty,s}(x)\\|^p\\bigg]^{1\/p} \n\t\t\\\\\n\t\t&\\le\\mathbb{E} \\bigg[\\mathrm{e}^{\\int_0^t pV_-(X_s^x)\\mathrm{d} s}\n\t\t\\sup_{s\\in[0,t]} \\big|\\mathrm{e}^{u_{0,\\Lambda,s}^N(x)}-\\mathrm{e}^{u_{0,\\infty,s}^N(x)}\\big|^p\n\t\t\\|D_{\\Lambda,s}[x,X_\\bullet]\\|^p  \\bigg]^{1\/p}\n\t\t\\\\\n\t\t&\\quad  + \\mathbb{E}\\bigg[\\mathrm{e}^{\\int_0^t pV_-(X_s^x)\\mathrm{d} s}\\sup_{s\\in[0,t]}\\mathrm{e}^{pu_{0,\\infty,s}^N(x)}\n\t\t\\|D_{\\Lambda,s}[x,X_\\bullet]-D_{\\infty,s}[x,X_\\bullet]\\|^p\n\t\t\\bigg]^{1\/p},\n\t\\end{align*}\n\tfor all $p>1$, $t>0$ and $x\\in\\mathbb{R}^{2N}$.\n\tHere the first expression can be bounded by H\\\"{o}lder's inequality together with \\cref{bdFU,thm:uconv,V2}. \n\tTo treat the second one by we apply \\cref{thmexpmonentu1,bddiffFFUU,V2} instead.\n\tAs a result we see that\n\t\\begin{align}\\label{locunivconvW}\n\t\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}\\bigg[\\sup_{s\\in[0,t]}\\|W_{\\Lambda,s}(x)-W_{\\infty,s}(x)\\|^p\\bigg]^{1\/p} \n\t\\xrightarrow{\\;\\;\\Lambda\\to\\infty\\;\\;}0,\\quad p>1,\\,t>0.\n\t\\end{align}\n\tNow the assertion follows from \\cref{corwLpLqNW}.\n\\end{proof}\n\n\n\\subsection{Markov and semigroup properties}\\label{subsec:Markov}\n\n\\noindent\nNext, we establish a Markov property of our Feynman--Kac integrands that will directly  entail semigroup properties\nfor the families $(T_{\\Lambda,t})_{t\\ge0}$. The Markov property will be a consequence\nof the flow relation for the family of Fock space operator-valued processes $(W_{\\Lambda,t}[x,X])_{t\\ge0}$, $x\\in\\mathbb{R}^{2N}$,\nthat we prove first in Lemma \\ref{lemMark1} below. It will be convenient to verify the flow relation by applying the\noperator-valued processes to exponential vectors, i.e., Fock space vectors of the form\n\\begin{align}\\label{def:expvec}\n\t\\expv{h} \\coloneq (1,h,\\ldots,h^{\\otimes n}\/(n!)^{1\/2},\\ldots)\\in \\mathcal{F}, \\quad h\\in L^2(\\mathbb{R}^2),\n\\end{align}\nwhere $h^{\\otimes n}(k_1,\\ldots,k_n) \\coloneq h(k_1)\\cdots h(k_n)$, \na.e. $k=(k_1,\\ldots,k_n)\\in\\mathbb{R}^{2N}$ with $k_j\\in\\mathbb{R}^2$.\nAs explained in \\cite[Remark~5.2]{MatteMoller.2018},\n\\begin{align*}\nF_{t}(g)\\expv{h}&=\\expv{g+\\mathrm{e}^{-t\\omega}h},\\quad\nF_{t}(g)^*\\expv{h}=\\mathrm{e}^{\\langle g|h\\rangle}\\expv{\\mathrm{e}^{-t\\omega}h},\\quad g,h\\in L^2(\\mathbb{R}^2),\\,t>0.\n\\end{align*}\nThe action of $W_{\\Lambda,t}[x,X_\\bullet]$ on an exponential vector therefore reads\n\\begin{align}\\nonumber\n&W_{\\Lambda,t}[x,X_\\bullet]\\expv{h}\n\\\\\\label{Wexpvec}\n&=\\mathrm{e}^{u_{0,\\Lambda,t}^N[x,X_\\bullet]-\\int_0^tV(X_s^x)\\mathrm{d} s-\\langle U^{N,-}_{0,\\Lambda,t}[x,X_\\bullet]|h\\rangle}\n\\expv{\\mathrm{e}^{-t\\omega}h-U_{0,\\Lambda,t}^{N,+}[x,X_\\bullet]}.\n\\end{align}\nIn the next lemma, we argue similarly as in \\cite[Proposition~5.5]{MatteMoller.2018}.\n\n\\begin{lem}\\label{lemMark1}\nLet $\\Lambda\\in(0,\\infty]$, $t\\ge0$ and $x\\in\\mathbb{R}^{2N}$. Then, $\\mathbb{P}$-a.s.,\n\\begin{align}\\label{Mark1}\nW_{\\Lambda,s}[X_t^x,X_{t+\\bullet}-X_t]W_{\\Lambda,t}[x,X_\\bullet]&=W_{\\Lambda,s+t}[x,X_\\bullet],\\quad s\\ge0.\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nRepeatedly applying \\eqref{Wexpvec} we see that the $\\mathbb{P}$-a.s. validity of \n\\begin{align*}\nW_{\\Lambda,s}[X_t^x,X_{t+\\bullet}-X_t]W_{\\Lambda,t}[x,X_\\bullet]\\expv{h}&=W_{\\Lambda,s+t}[x,X_\\bullet]\\expv{h},\n\\quad s\\ge0,\\,h\\in L^2(\\mathbb{R}^2),\n\\end{align*}\nis implied by the $\\mathbb{P}$-a.s. validity for all $s\\ge0$ of the four relations\n\\begin{align}\\label{MarkV}\n\\int_0^{s+t}V(X_r^x)\\mathrm{d} r&=\\int_0^sV(z+X_{t+r}-X_t)\\mathrm{d} r\\bigg|_{z=X_t^x}+\\int_0^tV(X_r^x)\\mathrm{d} r,\n\\end{align}\nas well as\n\\begin{align}\\label{Mark2}\nU^{N,-}_{\\Lambda,s+t}[x,X_\\bullet]&=\\mathrm{e}^{-t\\omega}U_{\\Lambda,s}^{N,-}[X_t^x,X_{t+\\bullet}-X_t]+U^{N,-}_{\\Lambda,t}[x,X_\\bullet],\n\\\\\\label{Mark3}\nU^{N,+}_{\\Lambda,s+t}[x,X_\\bullet]&=U_{\\Lambda,s}^{N,+}[X_t^x,X_{t+\\bullet}-X_t]+\\mathrm{e}^{-s\\omega}U^{N,+}_{\\Lambda,t}[x,X_\\bullet],\n\\end{align}\nand\n\\begin{align}\\nonumber\nu^N_{\\Lambda,s+t}[x,X_\\bullet]\n&=u_{\\Lambda,s}^N[X_t^x,X_{t+\\bullet}-X_t]+u^N_{\\Lambda,t}[x,X_\\bullet]\n\\\\\\label{Mark4}\n&\\quad+\\langle U_{\\Lambda,s}^{N,-}[X_t^x,X_{t+\\bullet}-X_t]|U^{N,+}_{\\Lambda,t}[x,X_\\bullet]\\rangle.\n\\end{align}\nRecall that the left and rightmost integrals in \\eqref{MarkV} are set equal to zero\non the $\\mathbb{P}$-zero set $\\scr{N}_V(x)$ introduced in \\eqref{defNVx}, and the analogous convention\napplies to the expression in the middle for each $z$. However, if $\\gamma\\in \\Omega\\setminus\\scr{N}_V(x)$,\nand $X_t^x(\\gamma)=z$, then $V(z+X_{t+\\bullet}-X_t)$ is locally integrable. This implies that\n\\eqref{MarkV} holds for all $s\\ge0$ at least on  $\\Omega\\setminus\\scr{N}_V(x)$.\nStraightforward computations further show that \\eqref{Mark2} and \\eqref{Mark3}\nhold on $\\Omega$ for all $s\\ge0$ and $0<\\Lambda\\le \\infty$. Again by direct computations based on \\eqref{defuUV}\n(virtually identical to the ones in the proof of \\cite[Lemma~4.18]{MatteMoller.2018}), we can also verify \\eqref{Mark4}\nfor all $s\\ge0$ and on $\\Omega$, provided that $\\Lambda\\in(0,\\infty)$. Since the set of exponential vectors is total in $\\mathcal{F}$, \nthese remarks prove the assertion for finite $\\Lambda$. \n\n\nBy approximation we can $\\mathbb{P}$-a.s. extend \\eqref{Mark4} to $\\Lambda=\\infty$: Since $X_t^x$ and $X_{t+\\bullet}-X_t$\nare independent and the law of $X_t$ has the probability density $\\rho_{0,m_{\\mathrm{p}},t}^{\\otimes_N}$ defined prior to\n\\cref{relLpLqN1}, we infer from \\eqref{locunivconvW} that\n\\begin{align*}\n&\\mathbb{E}\\bigg[\\sup_{s\\in[0,\\tau]}\\|W_{\\Lambda,s}[X_t^x,X_{t+\\bullet}-X_t]-W_{\\infty,s}[X_t^x,X_{t+\\bullet}-X_t]\\|^p\\bigg]\n\\\\\n&=\\int_{\\mathbb{R}^{2N}}\\mathbb{E}\\bigg[\\sup_{s\\in[0,\\tau]}\\|W_{\\Lambda,s}[z,X_{t+\\bullet}-X_t]-W_{\\infty,s}[z,X_{t+\\bullet}-X_t]\\|^p\\bigg]\n\\rho_{0,m_{\\mathrm{p}},t}^{\\otimes_N}(z-x)\\mathrm{d} z\n\\\\\n&\\xrightarrow{\\;\\;\\Lambda\\to\\infty\\;\\;}0,\\quad p>1,\\,\\tau>0.\n\\end{align*} \nOn account of this convergence relation and \\eqref{locunivconvW} we find an increasing sequence of\ncutoffs $\\Lambda_n\\in(0,\\infty)$, $n\\in\\mathbb{N}$, with $\\Lambda_n\\to\\infty$, $n\\to\\infty$, such that, after substituting \n$\\Lambda=\\Lambda_n$ in \\eqref{Mark4}, both sides convergence $\\mathbb{P}$-a.s. in operator norm and locally uniformly in $s\\ge0$.\nThe limiting relation is \\eqref{Mark4} with $\\Lambda=\\infty$.\n\\end{proof}\n\nWe are now in a position to derive the promised Markov property. We shall denote by $\\mathbb{E}^{\\fr F_t}$ the conditional expectation with respect to $\\fr F_t$ for $t\\ge 0$.\n\n\\begin{thm}[Markov property]\\label{thm:markov}\nLet $\\Lambda\\in (0,\\infty]$, $p\\in(1,\\infty]$, $s,t\\ge0$, $x\\in\\mathbb{R}^{2N}$ and $\\Psi\\in L^p(\\mathbb{R}^{2N})$. Then, $\\mathbb{P}$-a.s.,\n\\begin{align}\\label{Markprop1}\n\\mathbb{E}^{\\fr{F}_t}[W_{\\Lambda,s+t}[x,X_\\bullet]^*\\Psi(X_{s+t}^x)]&=W_{\\Lambda,t}[x,X_\\bullet]^*(T_{\\Lambda,s}\\Psi)(X_t^x).\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nLet $x\\in\\mathbb{R}^{2N}$. By \\cref{prop:LpLq}(i), $W_{\\Lambda,s+t}[x,X_\\bullet]^*\\Psi(X_{s+t}^x)$ is $\\mathbb{P}$-integrable, \nwhence the left hand side of \\eqref{Markprop1} is well-defined. \nFurthermore, we infer from \\cref{rem:Wprodmeas} that the function $\\Theta$ defined by\n\\begin{align*}\n\\Theta(x',\\phi',\\gamma)\\coloneq\\langle \\phi'|W_{\\Lambda,s}[x',\\gamma]^*\\Psi(x'+\\mathrm{ev}_s(\\gamma))\\rangle,\n\\quad x'\\in\\mathbb{R}^{2N},\\,\\phi'\\in\\mathcal{F},\\,\\gamma\\in D,\n\\end{align*}\nis $\\fr{B}(\\mathbb{R}^{2N}\\times\\mathcal{F})\\otimes\\fr{D}$-measurable.\nPick $x\\in\\mathbb{R}^{2N}$ and $\\phi\\in\\mathcal{F}$.\nThen $(X_t^x,W_{\\Lambda,t}[x,X_\\bullet]\\phi):\\Omega\\to\\mathbb{R}^{2N}\\times\\mathcal{F}$ is $\\fr{F}_t$-measurable\nby Remarks~\\ref{rem:Wadapt} and~\\ref{rem:WWindist},\nwhile the path map $X_{t+\\bullet}-X_t:\\Omega\\to D$ is $\\fr{F}_t$-independent.\nThe ``useful rule'' for conditional expectations, cf. \\cite[(6.8.14)]{HoffmannJorgensen.1994a}, thus implies the first relation in \n\\begin{align*}\n&\\mathbb{E}^{\\fr{F}_t}[\\Theta(X_t^x,W_{\\Lambda,t}[x,X]\\phi,X_{t+\\bullet}-X_t)]\n\\\\\n&\n=\\mathbb{E}[\\Theta(x',\\phi',X_{t+\\bullet}-X_t)]\\Big|_{(x',\\phi')=(X_t^x,W_{\\Lambda,t}[x,X]\\phi)}\n\\\\\n&=\\mathbb{E}[\\Theta(x',\\phi',X_{\\bullet})]\\Big|_{(x',\\phi')=(X_t^x,W_{\\Lambda,t}[x,X]\\phi)},\\quad\\text{$\\mathbb{P}$-a.s.},\n\\end{align*} \nwhere the second equality holds because $X_{t+\\bullet}-X_t$ and $X_\\bullet$ have the same distribution.\nThe resulting identity is equivalent to the second one in \n\\begin{align*}\n&\\langle\\phi|\\mathbb{E}^{\\fr{F}_t}[W_{\\Lambda,s+t}[x,X]^*\\Psi(X_{s+t}^x)]\\rangle\n\\\\\n&=\\mathbb{E}^{\\fr{F}_t}[\\langle \\phi|W_{\\Lambda,t}[x,X_\\bullet]^*W_{\\Lambda,s}[X_t^x,X_{t+\\bullet}-X_t]\\Psi(X_{s+t}^x)\\rangle]\n\\\\\n&=\\langle\\phi|W_{\\Lambda,t}[x,X_\\bullet]^*(T_{\\Lambda,s}\\Psi)(X_t^x)\\rangle,\\quad\\text{$\\mathbb{P}$-a.s.}\n\\end{align*}\nHere the first equality follows from \\eqref{Mark1} and the fact that the $\\mathcal{F}$-valued conditional expectation\n$\\mathbb{E}^{\\fr{F}_t}$ can be interchanged with taking the scalar product with~$\\phi$.\nApplying this for every $\\phi$ in a countable dense subset of $\\mathcal{F}$ we arrive at \\eqref{Markprop1}.\n\\end{proof}\n\nTaking expectations on both sides of \\eqref{Markprop1}, we obtain the following corollary:\n\n\\begin{cor}\\label{thm:semigroup}\nLet $\\Lambda\\in(0,\\infty]$ and $p\\in(1,\\infty]$. \nThen $(T_{\\Lambda,t})_{t\\ge0}$ is a semigroup of bounded operators on $L^p(\\mathbb{R}^{2N},\\mathcal{F})$\nsuch that $\\|T_{\\Lambda,t}\\|_{p,p}\\le \\mathrm{e}^{c(1+t)}$, $t\\ge0$, for some constant $c\\in(0,\\infty)$ whose dependence\non the model parameters can be read off from \\cref{remmomentbdW,prop:LpLq}.\n\\end{cor}\n\n\\subsection{Continuity properties}\\label{sseccontprop}\n\n\\noindent\nWe end this section by deriving the strong continuity and regularizing property \nof the semigroup in the next two propositions.\n\n\\begin{prop}\\label{lem:strongcont}\nFor all $p\\in(1,\\infty)$ and $\\Lambda\\in(0,\\infty]$, \nthe semigroup $(T_{\\Lambda,t})_{t\\ge0}$ is strongly continuous on $L^p(\\mathbb{R}^{2N},\\mathcal{F})$.\n\\end{prop}\n\n\\begin{proof}\nDue to the semigroup property and the bound\n$\\sup_{t\\in[0,1]}\\|T_{\\Lambda,t}\\|_{p,p}<\\infty$ from \\cref{thm:semigroup}, it suffices to show that\n$T_{\\Lambda,t}\\Phi\\to \\Phi$, $t\\downarrow0$,\nfor every $\\Phi$ in a dense subset of $L^p(\\mathbb{R}^{2N},\\mathcal{F})$. Hence, we assume in the following that $\\Phi$ is of the form\n$\\Phi=(1+\\mathrm{d}\\Gamma(\\omega))^{-1\/2}\\Psi$ for some $\\Psi\\in L^p(\\mathbb{R}^{2N},\\mathcal{F})$. \n\nMinkowski's inequality implies\n$\\|T_{\\Lambda,t}\\Phi-\\Phi\\|_p\\le\\mc{N}_1(t)+\\mc{N}_2(t)+\\mc{N}_3(t)$, for all $t>0$, \nwhere the norms $\\mc{N}_j(t)$ are given by\n\\begin{align*}\n\\mc{N}_1(t)^p&\\coloneq\\int_{\\mathbb{R}^{2N}}\\big\\|\\mathbb{E}\\big[W_{\\Lambda,t}(x)^*(\\Phi(X^x_t)-\\Phi(x))\\big]\\big\\|_{\\mathcal{F}}^p\\mathrm{d} x,\n\\\\\n\\mc{N}_2(t)^p&\\coloneq\\int_{\\mathbb{R}^{2N}}\\big\\|\\mathbb{E}\\big[\\big(\\mathrm{e}^{u_{0,\\Lambda,t}^N(x)-\\int_0^tV(X^x_s)\\mathrm{d} s}\n-1\\big)D_{\\Lambda,t}[x,X_\\bullet]\\Phi(x)\\big]\\big\\|_{\\mathcal{F}}^p\\mathrm{d} x,\n\\\\\n\\mc{N}_3(t)^p&\\coloneq\\int_{\\mathbb{R}^{2N}}\\big\\|\\mathbb{E}\\big[(D_{\\Lambda,t}[x,X_\\bullet]-\\mathbbm{1})\n(1+\\mathrm{d}\\Gamma(\\omega))^{-1\/2}\\Psi(x)\\big]\\big\\|_{\\mathcal{F}}^p\\mathrm{d} x.\n\\end{align*}\nEmploying H\\\"{o}lder's inequality and setting $f(z)\\coloneq\\|\\Phi(\\cdot - z)-\\Phi\\|_p^p$, $z\\in\\mathbb{R}^{2N}$, we find\n\\begin{align*}\n\\mc{N}_1(t)^p&\\le \\mathbb{E}[f(X_t)]\\sup_{s\\in[0,1]}\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}[\\|W_{\\Lambda,s}(x)\\|^{p'}]^{p\/p'},\\quad t\\in(0,1],\n\\end{align*}\nwhere the double supremum is finite due to \\eqref{momentbdW}. Furthermore, we know that $f$ is bounded and continuous.\nSince the laws of $X_t$, $t>0$, converge weakly to the Dirac measure concentrated in $0$, \nwe deduce that $\\mathbb{E}[f(X_t)]\\to f(0)=0$ as $t\\downarrow0$.\n\nFurthermore, by \\eqref{bdFU}, \n\\begin{align}\\label{strcont1}\n\\mc{N}_2(t)^p&\\le s_*^{2p}\\int_{\\mathbb{R}^{2N}}\\mathbb{E}\\big[\\big|\\mathrm{e}^{u_{0,\\Lambda,t}^N(x)-\\int_0^tV(X^x_s)\\mathrm{d} s}\n-1\\big|\\big]^p\\|\\Phi(x)\\|_{\\mathcal{F}}^p\\mathrm{d} x.\n\\end{align}\nFor fixed $x\\in\\mathbb{R}^{2N}$, the integrand under the expectation in \\eqref{strcont1} goes $\\mathbb{P}$-a.s. to $0$ as $t\\downarrow0$\nby \\cref{cor:pathsactioncont}. It is dominated by\n$G_x\\coloneq1+\\mathrm{e}^{\\int_0^1V_-(X^x_s)\\mathrm{d} s}\\sup_{s\\in[0,1]}\\mathrm{e}^{u_{0,\\Lambda,s}^N(x)}$\nas long as $t\\in(0,1]$. The majorant $G_x$\nis $\\mathbb{P}$-integrable by Theorem~\\ref{thmexpmonentu1} and \\cref{V2}, which in fact imply that\n$\\sup_{x\\in\\mathbb{R}^{2N}}\\mathbb{E}[G_x]<\\infty$. Applying the dominated convergence theorem for each $x$\nwe first see that the expectation in \\eqref{strcont1} goes to $0$ as $t\\downarrow0$ pointwise in $x\\in\\mathbb{R}^{2N}$.\nSince $\\|\\Phi(\\cdot)\\|_{\\mathcal{F}}^p$ is integrable, we can invoke dominated convergence a second time to conclude\nthat $\\mc{N}_2(t)\\to0$ as $t\\downarrow0$. By virtue of \\eqref{Dbei0} and the integrability of\n$\\|\\Psi(\\cdot)\\|_{\\mathcal{F}}^p$ it is finally clear that  $\\mc{N}_3(t)\\to0$ as $t\\downarrow0$. \n\\end{proof}\nThe proof of the regularizing property is based on a similar one due to Carmona \\cite[Proposition~3.3]{Carmona.1979}; \nsee also \\cite[Theorem~4.1]{BroderixHundertmarkLeschke.2000} and \\cite[Theorem~8.1]{Matte.2016}.\n\\begin{prop}\\label{thm:cont}\n\tLet $p\\in(1,\\infty]$, $\\Lambda\\in(0,\\infty]$ and $0<\\tau_1\\le\\tau_2<\\infty$. Then the family\n\t\\begin{align*}\n\t\t\\left\\{ T_{\\Lambda,t}\\Psi\\middle| \\Psi\\in L^p(\\mathbb{R}^{2N},\\mathcal{F}),\\|\\Psi\\|_p\\le 1,t\\in[\\tau_1,\\tau_2]\\right\\}\n\t\\end{align*}\n\tis uniformly equicontinuous on every compact subset of $\\mathbb{R}^{2N}$. Further, if $V\\in \\mc K_X$, then uniform equicontinuity holds on all of $\\mathbb{R}^{2N}$.\n\\end{prop}\n\n\n\\begin{proof}\nBy the uniform convergence as $\\Lambda\\to\\infty$ from \\cref{prop:Tconv} (recall the last remark in that proposition) \nit suffices to prove the assertion when $\\Lambda$ is finite, which we assume\nin the rest of this proof. We write\n\t\\begin{align*}\n\t\t(P_t \\Psi)(x) = \\mathbb{E}\\left[ \\Psi(X_t^x) \\right] = \\int_{\\mathbb{R}^{2N}} \\rho_{0,m_\\mathrm{p},t}^{\\otimes_N}(x-y)\\Psi(y) \\mathrm{d} y\n\t\t,\\quad \\Psi\\in L^p(\\mathbb{R}^{2N},\\mathcal{F}),\n\t\\end{align*}\n\twhere the probability density function $\\rho_{0,m_{\\mathrm{p}},t}^{\\otimes_N}$ is defined as in \\cref{app:lplq} with $d=2$. \n\tFor given $\\tau>0$, it is well-known that $P_\\tau M$ is uniformly equicontinuous on $\\mathbb{R}^{2N}$ \n\tfor any bounded family $M\\subset L^\\infty(\\mathbb{R}^{2N},\\mathcal{F})$. In fact,\n\t\\begin{align*}\n\t\t\\|P_\\tau f(x)-P_\\tau f(y)\\| \n\t\t&\\le \\|\\rho_{0,m_\\mathrm{p},\\tau}^{\\otimes_N}(x-y-\\cdot)-\\rho_{0,m_\\mathrm{p},\\tau}^{\\otimes_N}\\|_1\\|f\\|_\\infty,\n\t\t\\quad x,y\\in\\mathbb{R}^{2N},\n\t\\end{align*}\n\tand $x\\mapsto\\|h(x-\\cdot)\\|_1$ is continuous for any $h\\in L^1(\\mathbb{R}^{2N})$.\n\tHence, since $\\|T_{\\Lambda,t}\\|_{p,\\infty}$ is bounded uniformly in $t\\in[\\tau_1\/2,\\tau_2]$ by \\cref{prop:LpLq},\n\tthe set\n\t\\begin{align*}\n\t\t\\left\\{ P_\\tau T_{\\Lambda,t-\\tau}\\Psi\\middle| \\Psi\\in L^p(\\mathbb{R}^{2N},\\mathcal{F}),\\|\\Psi\\|_p\\le 1,t\\in[\\tau_1,\\tau_2]\\right\\}\n\t\\end{align*}\n\tis uniformly equicontinuous on $\\mathbb{R}^{2N}$ for every $\\tau\\in(0,\\tau_1\/2)$. \n\tIn view of the semigroup property from \\cref{thm:semigroup}, \n\tit therefore suffices to prove\n\t\\begin{align}\\label{eq:uniformconv}\n\t\t\\lim_{\\tau\\downarrow0}\\sup_{t\\in[\\tau_1,\\tau_2]}\\sup_{\\|\\Psi\\|_p\\le 1}\\sup_{x\\in K }\n\t\t\\|(P_\\tau-T_{\\Lambda,\\tau})T_{\\Lambda,t-\\tau}\\Psi(x)\\|_{\\mathcal{F}} = 0, \n\t\\end{align}\n\twhere $K\\subset \\mathbb{R}^{2N}$ is compact (or $K=\\mathbb{R}^{2N}$ in case $V\\in\\mc K_X$).\n\t\n\tHowever, by \\cref{prop:LpLq,prop:LpLqdG} and since $\\Lambda$ is assumed to be finite, the set\n\t\\begin{align*}\n\t\t\\mc M &\\coloneq \\big\\{(\\mathbbm{1} + \\mathrm{d}\\Gamma(\\omega))^{1\/4}T_{\\Lambda,t-\\tau}\\Psi\\big|\n\t\t\\,\\|\\Psi\\|_p\\le 1,\\,t\\in[\\tau_1,\\tau_2],\\,\\tau\\in[0,\\tau_1\/2]\\big\\}\n\t\\end{align*}\n\t\tis bounded in $L^\\infty(\\mathbb{R}^{2N},\\mathcal{F})$.\n\tFurther, for $\\Phi\\in L^\\infty(\\mathbb{R}^{2N},\\mathcal{F})$, $\\tau>0$ and $x\\in\\mathbb{R}^{2N}$, \n\t\\begin{align}\\nonumber\n\t\t&\\big\\|\\big((P_\\tau-T_{\\Lambda,\\tau})(\\mathbbm{1}+\\mathrm{d} \\Gamma(\\omega))^{-1\/4}\\Phi\\big)(x)\\big\\|_{\\mathcal{F}}\n\t\t\\\\\\nonumber\n\t\t& =\n\t\t\\left\\|\\mathbb{E}\\left[ (\\mathbbm{1}-W_{\\Lambda,\\tau}(x)^*) (\\mathbbm{1}+\\mathrm{d} \\Gamma(\\omega))^{-1\/4}\\Phi (X_\\tau^x)  \\right]\\right\\|\n\t\t\\\\ \\nonumber\n\t\t& \\le \\mathbb{E}\\left[|1-\\mathrm{e}^{-\\int_0^\\tau V(X_s^x) \\mathrm{d} x}|\\mathrm{e}^{u_{0,\\Lambda,\\tau}^N(x)}\n\t\t\\|D_{\\Lambda,\\tau}[x,X_\\bullet]\\|\\right] \\|\\Phi\\|_\\infty\n\t\t\\\\\\nonumber\n\t\t&\\quad +\\mathbb{E}\\left[|\\mathrm{e}^{u_{0,\\Lambda,\\tau}^N(x)}-1|\n\t\t\\|D_{\\Lambda,\\tau}[x,X_\\bullet]\\|\\right] \\|\\Phi\\|_\\infty\n\t\t\\\\\\label{eq:uniformconv2}\n\t\t\t& \\quad + \\mathbb{E}\\left[ \\|(\\mathbbm{1}-D_{\\Lambda,\\tau}[x,X_\\bullet]^*)(\\mathbbm{1}+\\mathrm{d}\\Gamma(\\omega))^{-1\/4}\\|  \\right]\\|\\Phi\\|_\\infty.\n\t\\end{align}\n\tSince $\\Lambda<\\infty$ by the present assumption, we can make use of the trivial bound \\eqref{trivialbdu} which entails\n\t$|u_{0,\\Lambda,\\tau}^N(x)|\\le\\tau \\pi g^2N^2\\Lambda^2\/m_{\\mathrm{b}} $. Therefore, \\eqref{eq:uniformconv} follows from\n\t\\eqref{eq:uniformconv2} in conjunction with \\eqref{bdFU}, \\eqref{Dbei0} and  \\cref{eq:boundexpVminus1}. \n\tThis concludes the proof.\n\\end{proof}\n\n\\section{Feynman--Kac formula}\\label{sec:FK}\n\n\\noindent\nWe can now move to the proof of our Feynman--Kac formula, i.e., we identify the operators $H_\\Lambda$ and $H$ as the infinitesimal generators of the strongly continuous semigroups $(T_{\\Lambda,t})_{t\\ge 0}$ and $(T_{\\infty,t})_{t\\ge 0}$, respectively.\nIn the first subsection we deal with finite $\\Lambda$. In the second one we treat the renormalized model.\n\n\\subsection{Feynman--Kac formula with ultraviolet cutoff}\n\\noindent\nWe start by deriving a pathwise integral equation for matrix elements of the Fock space operator valued Feynman--Kac integrands as defined in \\cref{def:FKint}.\n\\begin{lem}\\label{lemODEW1}\nLet $x\\in\\mathbb{R}^{2N}$ and assume that $\\gamma:[0,\\infty)\\to\\mathbb{R}^{2N}$ is a c\\`{a}dl\\`{a}g path such that\n$V(\\gamma^x)$ is locally integrable on $[0,\\infty)$ where $\\gamma^x\\coloneq x+\\gamma$. \nLet $\\Lambda\\in(0,\\infty)$, $\\phi_1\\in\\mathcal{D}(\\omega)$ and $\\phi_2\\in\\mathcal{F}$. Then, for all $t\\ge0$,\n\\begin{align*\n\t\\begin{aligned}\n\t\t&\\langle\\phi_1|W_{\\Lambda,t}[x,\\gamma]\\phi_2\\rangle-\\langle\\phi_1|\\phi_2\\rangle\n\t\t\\\\\n\t\t&=-\\int_0^t\\big\\langle(\\mathrm{d}\\Gamma(\\omega)+\\sum_{j=1}^N\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot\\gamma_{j,s}^x}v_\\Lambda)\n\t\t+V(\\gamma_s^x)-NE_{\\Lambda}^{\\mathrm{ren}})\\phi_1\\big|\n\t\tW_{\\Lambda,s}[x,\\gamma]\\phi_2\\big\\rangle\\mathrm{d} s.\n\t\\end{aligned}\n\\end{align*}\nHere $\\gamma_{j,s}^x$ is as usual the $j$'th two-dimensional component of $x+\\gamma(s)$.\n\\end{lem}\n\\begin{proof}\nTo start with recall that the linear hull of all exponential vectors $\\expv{h}$ with $h\\in\\mathcal{D}(\\omega)$\nis a core for $\\mathrm{d}\\Gamma(\\omega)$. We also recall that $\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot y})$ is relatively\n$\\mathrm{d}\\Gamma(\\omega)$-bounded uniformly in $y\\in\\mathbb{R}^2$. Thus,\nby sesquilinearity and dominated convergence, it suffices to prove the assertion for \n$\\phi_i=\\expv{h_i}$, $i\\in\\{1,2\\}$, with $h_1,h_2\\in\\mathcal{D}(\\omega)$.\nRecalling \\cref{Wexpvec} and using $\\langle \\expv{f}|\\expv{g}\\rangle = \\mathrm{e}^{\\langle f | g\\rangle}$, we find $\\langle\\expv{h_1}|W_{\\Lambda,t}[x,\\gamma]\\expv{h_2}\\rangle=\\mathrm{e}^{\\Theta(t)}$ with\n\\begin{align*}\n\\Theta(t)&\\coloneq \nu^N_{0,\\Lambda,t}[x,\\gamma] - \\int_0^t V(\\gamma_s^x)\\mathrm{d} s\n\\\\\n&\\quad-\\langle U^{N,-}_{0,\\Lambda,t}[x,\\gamma]|h_2\\rangle\n-\\langle h_1|U_{0,\\Lambda,t}^{N,+}[x,\\gamma]\\rangle+\\langle h_1|\\mathrm{e}^{-t\\omega}h_2\\rangle\n\\\\\n&=\\int_0^t\\vartheta(s)\\mathrm{d} s+\\langle h_1|h_2\\rangle,\n\\end{align*}\nwhere, in view of \\cref{IBPUplus,def:action-process,defUplusminus},\n\\begin{align}\\nonumber\n\t\\vartheta(s)&\\coloneq\n\t\\sum_{j=1}^N\\langle U_{0,\\Lambda,s}^{N,+}[x,\\gamma]|\\mathrm{e}^{-\\mathrm{i} K\\cdot\\gamma_{j,s}^x}v_{\\Lambda}\\rangle -NE_{\\Lambda}^{\\mathrm{ren}}   -V(\\gamma_s^x)\n\t\\\\\\nonumber\n\t&\\quad -\\sum_{j=1}^N\\langle\\mathrm{e}^{-s\\omega-\\mathrm{i} K\\cdot\\gamma_{j,s}^x}v_{\\Lambda}|h_2\\rangle  \n\t-\\sum_{j=1}^N\\langle h_1|\\mathrm{e}^{-\\mathrm{i} K\\cdot\\gamma_{j,s}^x}v_{\\Lambda}\\rangle\n\t\\\\&\\quad\\label{ODEW2}\n\t+\\langle h_1|\\omega(U^{N,+}_{0,\\Lambda,s}[x,\\gamma]-\\mathrm{e}^{-s\\omega}h_2)\\rangle.\n\\end{align}\nSince $v(k)=v(-k) = \\overline{v(-k)}$, the scalar product in the first expression is real:\n\\begin{align}\\label{ODEW3}\n\\langle U_{0,\\Lambda,s}^{N,+}[x,\\gamma]|\\mathrm{e}^{-\\mathrm{i} K\\cdot \\gamma_{j,s}^x}v_{\\Lambda}\\rangle\n&=\\langle \\mathrm{e}^{-\\mathrm{i} K\\cdot \\gamma_{j,s}^x}v_{\\Lambda}|U_{0,\\Lambda,s}^{N,+}[x,\\gamma]\\rangle.\n\\end{align}\nFurther, since $\\vartheta:[0,\\infty)\\to\\mathbb{R}$ is locally integrable, $\\Theta$ is absolutely continuous on every compact interval\nin $[0,\\infty)$, and\n\\begin{align}\\label{ODEW4}\n\\mathrm{e}^{\\Theta(t)}-\\mathrm{e}^{\\Theta(0)}\n&=\\int_0^t\\vartheta(s)\\mathrm{e}^{\\Theta(s)}\\mathrm{d} s,\\quad t\\ge0.\n\\end{align}\nOn the other hand, using\n\\begin{align*}\n\t&a(g)\\expv h = \\langle g|h\\rangle\\expv h, \\quad g,h \\in L^2(\\mathbb{R}^2),\\\\\n\t& \\langle \\expv g|\\mathrm{d}\\Gamma(\\omega) \\expv h\\rangle = \\langle g|\\omega h\\rangle \\mathrm{e}^{\\langle g|h\\rangle},\\quad g\\in L^2(\\mathbb{R}^2),\\ h\\in \\mathcal{D}(\\omega),\n\\end{align*}\n we find\n\\begin{align*}\n\\langle \\mathrm{e}^{-\\mathrm{i} K\\cdot \\gamma_{j,s}^x}v_{\\Lambda}|U_{0,\\Lambda,s}^{N,+}[x,\\gamma]-\\mathrm{e}^{-s\\omega}h_2\\rangle\\mathrm{e}^{\\Theta(s)} \n& =-\\langle\\expv{h_1}|a(\\mathrm{e}^{-\\mathrm{i} K\\cdot \\gamma_{j,s}^x)}v_{\\Lambda})W_{\\Lambda,s}[x,\\gamma]\\expv{h_2}\\rangle,\n\\\\\n\\langle h_1|\\omega(U^{N,+}_{0,\\Lambda,s}[x,\\gamma]-\\mathrm{e}^{-s\\omega}h_2)\\rangle\\mathrm{e}^{\\Theta(s)}\n&=-\\langle\\expv{h_1}|\\mathrm{d}\\Gamma(\\omega)W_{\\Lambda,s}[x,\\gamma]\\expv{h_2}\\rangle,\n\\\\\n\\langle h_1|\\mathrm{e}^{-\\mathrm{i} K\\cdot\\gamma_{j,s}^x)}v_{\\Lambda}\\rangle\\mathrm{e}^{\\Theta(s)}\n&=\\langle\\expv{h_1}|a^\\dagger(\\mathrm{e}^{-\\mathrm{i} K\\cdot\\gamma_{j,s}^x)}v_{\\Lambda})W_{\\Lambda,s}[x,\\gamma]\\expv{h_2}\\rangle.\n\\end{align*}\nCombining the previous three relations with \\eqref{ODEW2} and \\eqref{ODEW3}, we see that \\eqref{ODEW4}\nis equivalent to the statement.\n\\end{proof}\nWe can now prove the Feynman--Kac formula in presence of a finite cutoff.\n\\begin{thm}\\label{thm:FK-UV}\nLet $\\Lambda\\in(0,\\infty)$ and $t\\ge0$. Consider $T_{\\Lambda,t}$ as an operator on $L^2(\\mathbb{R}^{2N},\\mathcal{F})$.\nThen $\\mathrm{e}^{-tH_\\Lambda}=T_{\\Lambda,t}$ and in particular $T_{\\Lambda,t}$ is selfadjoint.\n\\end{thm}\n\n\\begin{proof}\nTo start with we assume that $V$ is bounded.\nLet $\\phi_1\\in\\mathcal{D}(\\mathrm{d}\\Gamma(\\omega))$ and $f\\in C_0^\\infty(\\mathbb{R}^{2N})$. Since the Fourier transform\n$\\hat{f}$ is a Schwartz function, \\eqref{LKPsiX} and Fubini's theorem entail\n\\begin{align}\\nonumber\n(\\psi_X(-\\mathrm{i}\\nabla)f)(x)\n&\\coloneq \\frac{1}{(2\\pi)^N}\\int_{\\mathbb{R}^{2N}}\\mathrm{e}^{\\mathrm{i} \\xi\\cdot x}\\psi_X(\\xi)\\hat{f}(\\xi)\\mathrm{d}\\xi\n\\\\\\label{def:psiXnabla}\n&=-\\int_{\\mathbb{R}^{2N}}(f(x+z)-f(x)-\\chi_{B_1'}(z)\\nabla f(x)\\cdot z)\\mathrm{d}\\nu_X(z),\n\\end{align}\nfor all $x\\in\\mathbb{R}^{2N}$.\nBy virtue of the previous identity and the boundedness of $f$, we can re-write the It\\^{o} formula for $f(X_t^x)$ \n(see, e.g., \\cite[Theorem~4.4.7]{Applebaum.2009}) as\n\\begin{align*}\nf(X_t^x)&=f(x)-\\int_0^t(\\psi_X(-\\mathrm{i}\\nabla)f)(X_s^x)\\mathrm{d} s+\\int_{(0,t]\\times\\mathbb{R}^{2N}}(f(X_{s-}^x+z)-f(X_{s-}^x))\\mathrm{d}\\wt{N}_X,\n\\end{align*}\nfor all $t\\ge0$, $\\mathbb{P}$-a.s.\nFurthermore, since $V$ is bounded, $H_{\\Lambda}f\\phi_1=\\Psi$, a.e., with\n\\begin{align*}\n\\Psi(x)&\\coloneq (\\psi_X(-\\mathrm{i}\\nabla)f)(x)\\phi_1\n+  (V(x)-NE_{\\Lambda}^{\\mathrm{ren}})f(x)\\phi_1\n\\\\\n&\\quad +f(x)\\mathrm{d}\\Gamma(\\omega)\\phi_1+f(x)\\sum_{j=1}^N\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}v_\\Lambda)\\phi_1,\\quad x\\in\\mathbb{R}^{2N}.\n\\end{align*}\n\nLet also $\\phi_2\\in\\mathcal{F}$ and fix $x\\in\\mathbb{R}^{2N}$ for the moment. \nThen \\cref{lemODEW1} ensures that $(\\langle\\phi_1|W_{\\Lambda,t}[x,X]\\phi_2\\rangle)_{t\\ge0}$ \nis a finite variation process. Thus, using the integral equation in \\cref{lemODEW1}, the above formula for $f(X_t^x)$\nand It\\^o's product formula for L\\'{e}vy type stochastic integrals (see, e.g., \\cite[Theorem~4.4.13]{Applebaum.2009}), \nwe $\\mathbb{P}$-a.s. find\n\\begin{align*}\n&\\langle\\phi_1|W_{\\Lambda,t}[x,X]\\phi_2\\rangle\\ol{f}(X_t^x)-\\langle\\phi_1|\\phi_2\\rangle \\ol{f}(x)\n\\\\\n&=-\\int_0^t\\langle W_{\\Lambda,s}[x,X]^*\\Psi(X_s^x)|\\phi_2\\rangle\\mathrm{d} s\n\\\\\n&\\quad-\\int_{(0,t]\\times\\mathbb{R}^{2N}}\\langle (f(X_{s-}^x+z)-f(X_{s-}^x))\n\\phi_1| W_{\\Lambda,s}[x,X]\\phi_2\\rangle\\mathrm{d} \\wt{N}_X(s,z),\n\\end{align*}\nfor all $t\\ge0$. Here the term in the last line is a martingale starting at zero. (Since $\\Lambda$ is finite and $V$\nbounded, its integrand is predictable and\nbounded by some constant times $|z|\\wedge1$, $\\nu_X$-a.e. on sets of the form \n$[0,t]\\times\\mathbb{R}^{2N}\\times\\Omega$ with $t>0$.) \nUpon taking expectations in the above identity, the martingale drops out and we find\n\\begin{align*}\n\\langle (T_{\\Lambda,t}f\\phi_1)(x)|\\phi_2\\rangle -\\langle f(x)\\phi_1|\\phi_2\\rangle\n&=-\\int_0^t\\langle (T_{\\Lambda,s}\\Psi)(x)|\\phi_2\\rangle\\mathrm{d} s,\\quad t\\ge0.\n\\end{align*}\nAfter choosing $\\phi_2=\\Phi(x)$ for every $x$ and some $\\Phi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$\nand integrating with respect to $x$, \nwe can use Fubini's theorem to interchange the order of the $\\mathrm{d} x$- and $\\mathrm{d} s$-integration.\nSince $s\\mapsto T_{\\Lambda,s}\\Psi\\in L^2(\\mathbb{R}^2,\\mathcal{F})$ is Bochner-Lebesgue integrable over $[0,t]$,\nthis reveals that\n\\begin{align}\\label{HgenT1}\nT_{\\Lambda,t}f\\phi-f\\phi&=-\\int_0^t T_{\\Lambda,s}\\Psi\\mathrm{d} s,\\quad t\\ge0.\n\\end{align}\nLet $A$ denote the generator of $(T_{\\Lambda,t})_{t\\ge0}$ seen as a semigroup on $L^2(\\mathbb{R}^{2N},\\mathcal{F})$.\nSince $T_{\\Lambda,s}\\Psi\\to H_\\Lambda f\\phi_1$, $s\\downarrow0$, in $L^2(\\mathbb{R}^2,\\mathcal{F})$, we can infer from\n\\eqref{HgenT1} that $f\\phi$ is contained in the domain of $A$, with $Af\\phi=H_{\\Lambda}f\\phi$. \nAgain since $\\Lambda$ is finite and $V$ bounded, we know that the linear hull of all vectors of the form\n$f\\phi$ with $f$ and $\\phi$ as above is a core for $H_{\\Lambda}$; see, e.g., \\cref{lemesaNC} below. Since $A$ is closed,\nwe conclude that $H_\\Lambda\\subset A$. By the Hille--Yosida theorem we further know, however, \nthat the real parts of all points in the spectrum of $A$ are bounded from below. Since $H_{\\Lambda}$ is\nlower semibounded, we clearly find some $\\zeta\\in\\mathbb{C}$ belonging to the resolvent sets of both $A$ and $H_{\\Lambda}$.\nThen the second resolvent identity implies $(A-\\zeta)^{-1}=(H_{\\Lambda}-\\zeta)^{-1}$, thus $A=H_{\\Lambda}$,\nthus $T_{\\Lambda,t}=\\mathrm{e}^{-tH_{\\Lambda}}$, $t\\ge0$.\n\nTo generalize this result we follow a standard approximation procedure: \nLet $V$ be Kato decomposable and bounded from below. \nSet $V_n \\coloneq n\\wedge V$ and denote by $T^{n}_{\\Lambda,t}$, $H^{n}_\\Lambda$ and $\\mathfrak{h}_{\\Lambda}^n$ \nthe corresponding semigroup members, Hamiltonian and quadratic form \nfor $n\\in\\mathbb{N}$. Let $\\Phi\\in L^2(\\mathbb{R}^{2N},\\mathcal{F})$. By the above result, $\\mathrm{e}^{-tH_{\\Lambda}^n}\\Phi=T^n_{\\Lambda,t}\\Phi$ \nfor all $n\\in\\mathbb{N}$.\nBy dominated convergence and \\cref{prop:LpLq}, $(T^{n}_{\\Lambda,t}\\Phi)(x)$ converges to $(T_{\\Lambda,t}\\Phi)(x)$ \nas $n\\to\\infty$ for all $x\\in\\mathbb{R}^{2N}$. Further, $\\mathfrak{h}_{\\Lambda}^n\\uparrow\\mathfrak{h}_{\\Lambda}$ \non $\\mathcal{D}(\\mathfrak{h}_{\\Lambda})$ as $n\\to\\infty$, which by a convergence theorem for an increasing sequence of\nquadratic forms entails strong resolvent convergence of $H^n_{\\Lambda}$ to $H_{\\Lambda}$; see, e.g.,\n\\cite[Theorems~VIII.20(b) and S.14]{ReedSimon.1980}.\nThus, $\\mathrm{e}^{-tH^{n_\\ell}_\\Lambda}\\Phi\\to \\mathrm{e}^{-tH_\\Lambda}\\Phi$, $\\ell\\to\\infty$, a.e. for some strictly increasing sequence \n$(n_\\ell)_{\\ell\\in\\mathbb{N}}$ in $\\mathbb{N}$. This proves the theorem when $V$ is bounded from below.\n\nFor general Kato decomposable $V$, we consider $(-n)\\vee V$ and argue analogously.\nInstead of monotone we use dominated convergence, and we employ a convergence theorem for \na decreasing sequence of quadratic forms; see, e.g.,\n\\cite[Theorems~VIII.20(b) and S.16]{ReedSimon.1980}. (When applying Theorem~S.16 in \\cite{ReedSimon.1980}\nwe take into account that the quadratic forms corresponding to $V$ and to every $(-n)\\vee V$ with $n\\in\\mathbb{N}$ \nall have the same domain, namely $\\mathcal{Q}(H_{\\Lambda})$ described prior to \\cref{def:HUV}.)\n\\end{proof}\n\n\\subsection{Feynman--Kac formula for the renormalized Hamiltonian}\\label{sec:FK.ren}\n\nWe can now combine the convergence statements from \\cref{prop:Tconv} with the Feyn\\-man-Kac formula with cutoff to \ndiscuss the limit as $\\Lambda$ goes to infinity.\n\n\\begin{cor}\\label{cor:semigroup-ren}\nWhen its members are considered as operators on $L^2(\\mathbb{R}^{2N},\\mathcal{F})$, \nthe semigroup $(T_{\\infty,t})_{t\\ge0}$ is strongly continuous and every $T_{\\infty,t}$ is bounded and selfadjoint \nwith $\\|T_{\\infty,t}\\|\\le \\mathrm{e}^{c(1+t)}$, $t\\ge0$, for some $c>0$.\nLet $H$ denote its selfadjoint, lower semibounded generator.\nThen $H_{\\Lambda}\\to H$ in norm resolvent sense as $\\Lambda\\to\\infty$.\t\n\\end{cor}\n\n\\begin{proof}\nApart from the claimed selfadjointness,\nthe first statement only summarizes special cases of \\cref{prop:LpLq,thm:semigroup,lem:strongcont}.\nThese known results ensure that $(T_{\\infty,t})_{t\\ge0}$ has a closed generator $H$.\nThen \\cref{prop:Tconv} (with $p=q=2$) and \\cref{thm:FK-UV} reveal that $\\mathrm{e}^{-tH_{\\Lambda}}\\to\\mathrm{e}^{-tH}=T_{\\infty,t}$,\n$\\Lambda\\to\\infty$, in operator norm. In particular, the members of the semigroup $(T_{\\infty,t})_{t\\ge0}$ and, hence, \n$H$ are selfadjoint. The norm resolvent convergence now follows from the representations of the \nresolvents as Bochner-Lebesgue integrals over semigroup elements\n$(H_{\\Lambda}+\\lambda)^{-1} = \\int_0^\\infty \\mathrm{e}^{-t(H_{\\Lambda}+\\lambda)}\\mathrm{d} t$. The latter\nhold for any $\\lambda<c$ with $c$ as in the first statement.\n\\end{proof}\n\n\n\n\n\\section{Upper bound on the minimal energy}\\label{sec:upbound}\n\n\\noindent\nIn this last \\lcnamecref{sec:upbound}, we prove the upper bound on the minimal energy of the translation invariant Hamiltonian stated in \\cref{thm:upbound}.\n\n\\subsection{A well-known trial function argument}\n\nIn \\cref{lemtrialfct} we carry through a trial function argument yielding an upper bound on the\nminimal energy for finite $\\Lambda$, which in combination with our exponential moment\nbounds on the complex action will allow us to prove \\cref{thm:upbound} later on. The procedure we follow in the proof\nof  \\cref{lemtrialfct}.\nis well-known and leads to the minimization of the Pekar functional in the non-relativistic case in three dimensions;\nsee, e.g., \\cite[\\textsection8.2]{MatteMoller.2018} and the references given there.\nHere we encounter relativistic analogues of Pekar's functional (see the last two displayed relations in the proof\nof Lemma~\\ref{lemtrialfct}), parametrically depending on $m_{\\mathrm{b}}\/g^2N$. \nFor our purposes it will, however, be sufficient to simply insert a family of Gaussians into these functionals.\n\nIn this appendix we do not require the boson mass to be strictly positive, and we only consider finite $\\Lambda$. \nWe pick a core $\\scr{D}$ for $\\mathrm{d}\\Gamma(1\\vee\\omega)$ and set\n\\begin{align*}\n\\scr{C}&\\coloneq\\mathrm{span}\\big\\{f\\phi\\,\\big|\\:f\\in\\scr{S}(\\mathbb{R}^{2N}),\\,\\phi\\in\\scr{D}\\},\n\\\\\n(H_{\\Lambda}'\\Psi)(x)&\\coloneq(\\psi_X(-\\mathrm{i}\\nabla)\\Psi)(x)+\\mathrm{d}\\Gamma(\\omega)\\Psi(x)\n+\\sum_{j=1}^N\\varphi(\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}v_\\Lambda)\\Psi(x),\\quad \\Psi\\in\\scr{C},\n\\end{align*}\nwith $\\scr{S}(\\mathbb{R}^{2N})$ denoting the set of Schwartz functions on $\\mathbb{R}^{2N}$.\nAgain $\\psi_X(-\\mathrm{i}\\nabla)$ is defined via Fourier tranformation analogously to \\eqref{def:psiXnabla}.\nSince $\\omega^{-1\/2}v_\\Lambda\\notin L^2(\\mathbb{R}^2)$ for $m_{\\mathrm{b}}=0$, we cannot rely on the\nKato-Rellich theorem to ensure essential selfadjointness of $H_{\\Lambda}'$ in this case.\nWe can, however, still apply Nelson's commutator theorem: \n\n\\begin{lem}\\label{lemesaNC}\nIf $m_{\\mathrm{b}}\\ge0$ and $\\Lambda\\in(0,\\infty)$, then $H_{\\Lambda}'$ is essentially selfadjoint on $\\scr{C}$.\n\\end{lem}\n\n\\begin{proof}\nSince $\\|\\varphi(h)\\phi\\|\\le 2\\|h\\|\\|(1+\\mathrm{d}\\Gamma(1\\vee\\omega))^{1\/2}\\phi\\|$, $h\\in L^2(\\mathbb{R}^2)$, $\\phi\\in\\scr{D}$,\nwell-known and simple estimations reveal that, for sufficiently large $C\\in(0,\\infty)$, the operator\n$K'\\coloneq H_{\\Lambda}'+\\mathrm{d}\\Gamma((1\\vee\\omega)-\\omega)+C$\nis essentially selfadjoint on $\\scr{C}$, lower bounded by $1$ and satisfies\n$\\|H_{\\Lambda}'\\Psi\\|\\le c\\|K'\\Psi\\|$, $\\Psi\\in\\scr{C}$, for some $c\\in(0,\\infty)$. Furthermore,\n\\begin{align*}\n2\\Im \\langle H_{\\Lambda}'\\Psi|K'\\Psi\\rangle&=\\sum_{j=1}^N\\int_{\\mathbb{R}^{2N}}\n\\langle\\Psi(x)|\\varphi(\\mathrm{i} \\chi_{B_1}(1-\\omega)\\mathrm{e}^{-\\mathrm{i} K\\cdot x_j}v_{\\Lambda})\\Psi(x)\\rangle_{\\mathcal{F}}\\mathrm{d} x,\n\\end{align*}\nby the commutation relation between field operators and $\\mathrm{d}\\Gamma(\\chi_{B_1}(1-\\omega))$;\nsee, e.g., \\cite{Arai.2018}. This implies $|\\Im \\langle H_{\\Lambda}'\\Psi|K'\\Psi\\rangle|\\le d\\langle\\Psi|K'\\Psi\\rangle$,\n$\\Psi\\in\\scr{C}$, for some $d\\in(0,\\infty)$.\n The assertion now follows from Nelson's commutator theorem\n\\cite[Theorem~X.37]{ReedSimon.1975} applied with the selfadjoint closure of $K'$ as test operator.\n\\end{proof}\n\nIn what follows we shall denote the unique selfadjoint extension of $H_\\Lambda'$ again by the same symbol.\n\n\\begin{lem}\\label{lemtrialfct}\nLet $m_{\\mathrm{b}}\\ge0$, $0\\le \\sigma<\\Lambda<\\infty$ and $s>0$.\nAssume in addition that $\\sigma>0$ when $m_{\\mathrm{b}}=0$. Then\n\\begin{align}\\label{ubGauss1}\n\\inf\\sigma(H_{\\Lambda}')\n&\\le \\frac{\\pi ^{1\/2}s^{1\/2}g^2N^2}{2^{1\/2}}\n- g^2N^2\\int_{B_{\\Lambda}\\setminus B_{\\sigma}}\n\\frac{\\mathrm{e}^{-|k|^2\/4sg^4N^2}}{|k|^2+m_{\\mathrm{b}}^2}\\mathrm{d} k.\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nLet $f:\\mathbb{R}^2\\to\\mathbb{C}$ be a Schwartz function, normalized in $L^2(\\mathbb{R}^2)$, and let $h\\in L^2(\\mathbb{R}^2)$\nbe such that $\\mathrm{supp}(h)\\subset B_{\\Lambda}$. We set\n\\begin{align*}\nf_N(x)&\\coloneq \\prod_{j=1}^Nf(x_j),\\quad x\\in\\mathbb{R}^{2N},\\quad\\text{and}\\quad\n\\zeta(h)\\coloneq\\mathrm{e}^{-\\|h\\|^2\/2}\\expv{h},\n\\end{align*}\nso that $f_N$ and $\\zeta(h)$ are normalized in $L^2(\\mathbb{R}^{2N})$ and $\\mathcal{F}$, respectively.\nDirect computation shows that\n\\begin{align*}\n&\\langle f_N\\zeta(h)|H_{\\Lambda} f_N\\zeta(h)\\rangle\n\\\\\n&=N\\int_{\\mathbb{R}^2}\\psi(\\eta)|\\hat{f}(\\eta)|^2\\mathrm{d}\\eta+\\|\\omega^{1\/2}h\\|^2\n+2N\\int_{\\mathbb{R}^2}\\Re\\langle\\mathrm{e}^{-\\mathrm{i} K\\cdot y}v_\\Lambda|h\\rangle |f(y)|^2\\mathrm{d} y.\n\\end{align*}\nSetting $\\rho\\coloneq|f|^2$ and choosing \n$h = -2\\pi gN\\omega^{-3\/2}\\chi_{B_\\Lambda\\setminus B_\\sigma}\\wh{\\rho}$, we obtain\n\\begin{align*}\n\\inf\\sigma(H_{\\Lambda})&\\le N\\int_{\\mathbb{R}^2}\\psi(\\eta)|\\hat{f}(\\eta)|^2\\mathrm{d}\\eta\n-g^2N^2\\int_{B_\\Lambda\\setminus B_\\sigma}\\frac{|2\\pi\\wh{\\rho}(k)|^2}{\\omega(k)^2}\\mathrm{d} k.\n\\end{align*}\nReplacing $f$ by its unitarily scaled version $f_a(y)\\coloneq af(ay)$, $y\\in\\mathbb{R}^2$, $a>0$, writing\n$\\rho_a\\coloneq |f_a|^2$ and taking $\\wh{f_a}(\\eta)=\\hat{f}(\\eta\/a)\/a$ and $\\wh{\\rho_a}(k)=\\wh{\\rho}(k\/a)$\ninto account, we deduce that\n\\begin{align*}\n\\inf\\sigma(H_{\\Lambda})\n&\\le N\\int_{\\mathbb{R}^2}\\psi(a\\eta)|\\hat{f}(\\eta)|^2\\mathrm{d}\\eta\n-g^2N^2\\int_{B_{\\Lambda}\\setminus B_{\\sigma}}\\frac{|2\\pi\\wh{\\rho}(k\/a)|^2}{|k|^2+m_{\\mathrm{b}}^2}\\mathrm{d} k.\n\\end{align*}\nEstimating $\\psi(a\\eta)\\le a|\\eta|$ and choosing $a=g^2N$ we find\n\\begin{align*}\n\\inf\\sigma(H_{\\Lambda})&\\le g^2N^2\\int_{\\mathbb{R}^2}|\\eta||\\hat{f}(\\eta)|^2\\mathrm{d}\\eta\n-g^2N^2\\int_{B_{\\Lambda}\\setminus B_{\\sigma}}\\frac{|2\\pi\\wh{\\rho}(k\/g^2N)|^2}{|k|^2+m_{\\mathrm{b}}^2}\\mathrm{d} k.\n\\end{align*}\nSince $2\\pi\\wh{\\rho}=\\hat{f}*\\ol{\\hat{f}(-\\,\\cdot\\,)}$, it is convenient to choose\n$\\hat{f}(k)\\coloneq (2\\pi s)^{-1\/2}\\mathrm{e}^{-|k|^2\/4s}$, a Gaussian that is normalized in $L^2(\\mathbb{R}^2)$.\nThen $2\\pi\\wh{\\rho}(k)=(\\hat{f}*\\hat{f})(k)=\\mathrm{e}^{-|k|^2\/8s}$. Taking also \n$\\int_{\\mathbb{R}^2}|\\eta|\\mathrm{e}^{-|\\eta|^2\/2s}\\mathrm{d}\\eta=(\\pi s\/2)^{1\/2}$ \ninto account we arrive at \\eqref{ubGauss1}.\n\\end{proof}\n\n\\begin{cor}\\label{corubGauss1}\nIf $m_{\\mathrm{b}}=0$ and $\\Lambda\\in(0,\\infty)$, then $H_{\\Lambda}'$ is unbounded from below.\n\\end{cor}\n\n\\begin{proof}\nIn the case $m_{\\mathrm{b}}=0$, the right hand side of \\eqref{ubGauss1} goes to $-\\infty$ as $\\sigma\\downarrow0$.\n\\end{proof}\n\n\\subsection{Derivation of the upper bound}\nWe now add the renormalization energy to both sides of \\cref{ubGauss1}\nand derive bounds on the so-obtained right hand side. The  estimates found in this way\nthen allow us to prove \\cref{thm:upbound}. \nThroughout the remainder of this section, we will again work under the assumption $m_\\mathrm{b}>0$.\n\\begin{cor}\\label{corubGauss2}\nAssume that $g^2N>m_{\\mathrm{b}}>0$ and choose $\\Lambda_{g,N}>0$ such that $\\Lambda_{g,N}^2+m_{\\mathrm{b}}^2=g^4N^2$. \nLet $s>0$. Then, for every $\\varepsilon\\in(0,1)$, there exists $c_*>1$, depending only on $\\varepsilon$, $m_{\\mathrm{p}}$ and $m_{\\mathrm{b}}\\vee1$,\nsuch that\n\\begin{align}\\nonumber\n\\inf\\sigma(H_{\\Lambda_{g,N}}'+NE_{\\Lambda_{g,N}}^{\\mathrm{ren}})\n&\\le \\frac{\\pi ^{1\/2}s^{1\/2}g^2N^2}{2^{1\/2}}-\\frac{2\\pi}{\\mathrm{e}^{1\/4s}}g^2N(N-1)\\ln\\bigg(\\frac{g^2N}{m_{\\mathrm{b}}}\\bigg)\n\\\\\\nonumber\n&\\quad-\\frac{\\pi g^2N}{\\mathrm{e}^{1\/4s}}\\cdot\\frac{2-2\\varepsilon}{2-\\varepsilon}\\cdot[0\\vee\\ln(c_*^{-1}g^2N)]+\\frac{\\pi}{4s}g^2N\n\\\\\\label{ubGauss2}\n&\\quad-\\chi_{\\{0\\}}(m_{\\mathrm{p}})\\frac{\\pi g^2N}{\\mathrm{e}^{1\/4s}}\\ln\\bigg(\\frac{1\\wedge(g^2N)}{\\mathrm{e} m_{\\mathrm{b}}}\\bigg).\n\\end{align}\n\\end{cor}\n\n\\begin{proof}\nWe shall apply \\eqref{ubGauss1} with $\\sigma=0$ and $\\Lambda=\\Lambda_{g,N}$.\nObserving that $\\Lambda_{g,N}\/g^2N<1$, we can estimate $N(N-1)$ copies of the integral on the right hand side\nof \\eqref{ubGauss1} as\n\\begin{align*}\n\\int_{B_{\\Lambda_{g,N}}}\n\\frac{\\mathrm{e}^{-|k|^2\/4sg^4N^2}}{|k|^2+m_{\\mathrm{b}}^2}\\mathrm{d} k\n&\\ge \\frac{\\pi}{\\mathrm{e}^{1\/4s}}\\int_0^{\\Lambda_{g,N}}\\frac{2r}{r^2+m_{\\mathrm{b}}^2\/g^4N^2}\\mathrm{d} r\n=\\frac{2\\pi}{\\mathrm{e}^{1\/4s}}\\ln\\bigg(\\frac{g^2N}{m_{\\mathrm{b}}}\\bigg).\n\\end{align*}\nTo deal with the $N$ remaining copies and the term $NE_{\\Lambda_{g,N}}^{\\mathrm{ren}}$, we use\n\\begin{align*}\n&g^2N\\int_{B_{\\Lambda_{g,N}}}\n\\frac{\\mathrm{e}^{-|k|^2\/4sg^4N^2}}{|k|^2+m_{\\mathrm{b}}^2}\\mathrm{d} k-NE_{\\Lambda_{g,N}}^{\\mathrm{ren}}\n\\\\\n&=g^2N\\int_{B_{\\Lambda_{g,N}}}\n\\frac{\\mathrm{e}^{-|k|^2\/4sg^4N^2}\\psi(k)}{\\omega(k)^2(\\omega(k)+\\psi(k))}\\mathrm{d} k\n-g^2N\\int_{B_{\\Lambda_{g,N}}}\\frac{1-\\mathrm{e}^{-|k|^2\/4sg^4N^2}}{\\omega(k)(\\omega(k)+\\psi(k))}\\mathrm{d} k\n\\\\\n&\\ge \\frac{g^2N}{\\mathrm{e}^{1\/4s}}\\int_{B_{\\Lambda_{g,N}}}\n\\frac{\\psi(k)}{\\omega(k)^2(\\omega(k)+\\psi(k))}\\mathrm{d} k\n-\\frac{1}{4sg^2N}\\int_{B_{\\Lambda_{g,N}}}\\frac{|k|^2}{\\omega(k)(\\omega(k)+\\psi(k))}\\mathrm{d} k.\n\\end{align*}\nSince $\\omega(k)\\le(|k|^2+(1\\vee m_{\\mathrm{b}})^2)^{1\/2}$,\nwe find some $R_{*}\\ge1$ depending only on $\\varepsilon$, $m_{\\mathrm{p}}$ and $1\\vee m_{\\mathrm{b}}$ such that \n$\\psi(k)\\ge (1-\\varepsilon)\\omega(k)$ as soon as $|k|\\ge R_{*}$. \nSince $(0,\\infty)\\ni a\\mapsto a\/(\\omega(k)+a)$ is increasing and we always have the bound $\\omega(k)\\ge\\psi(k)$,\nwe can thus continue the above estimation as follows,\n\\begin{align*}\n&g^2N\\int_{B_{\\Lambda_{g,N}}}\n\\frac{\\mathrm{e}^{-|k|^2\/4sg^4N^2}}{|k|^2+m_{\\mathrm{b}}^2}\\mathrm{d} k-NE_{\\Lambda_{g,N}}^{\\mathrm{ren}}\n\\\\\n&\\ge \\frac{g^2N}{2\\mathrm{e}^{1\/4s}}\\int_{B_{R_*\\wedge\\Lambda_{g,N}}}\n\\frac{\\psi(k)}{\\omega(k)^3}\\mathrm{d} k\n+\\frac{g^2N}{\\mathrm{e}^{1\/4s}}\\cdot\\frac{1-\\varepsilon}{2-\\varepsilon}\\int_{B_{\\Lambda_{g,N}}\\setminus B_{R_*}}\\frac{1}{\\omega(k)^2}\\mathrm{d} k\n-\\frac{\\mathrm{vol}(B_{\\Lambda_{g,N}})}{4sg^2N}.\n\\end{align*}\nIn the case $m_{\\mathrm{p}}>0$, we simply estimate the integral over $B_{R_*\\wedge\\Lambda_{g,N}}$ from below by $0$.\nFor $m_{\\mathrm{p}}=0$ it is equal to the expression in the first line of\n\\begin{align*}\n&\\int_{B_{R_*\\wedge\\Lambda_{g,N}}}\\frac{|k|}{\\omega(k)^3}\\mathrm{d} k\n\\\\\n&\\ge \\int_{B_{R_*\\wedge\\Lambda_{g,N}}}\\frac{1}{\\omega(k)^2}\\mathrm{d} k\n-\\int_{B_{R_*\\wedge\\Lambda_{g,N}}}\\frac{m_{\\mathrm{b}}}{\\omega(k)^3}\\mathrm{d} k\n\\\\\n&=\\pi\\ln\\bigg(\\frac{(R_*\\wedge\\Lambda_{g,N})^2+m_{\\mathrm{b}}^2}{m_{\\mathrm{b}}^2}\\bigg)\n-2\\pi m_{\\mathrm{b}}\\bigg(\\frac{1}{m_{\\mathrm{b}}}-\\frac{1}{\\sqrt{(R_*\\wedge\\Lambda_{g,N})^2+m_{\\mathrm{b}}^2}}\\bigg),\n\\end{align*}\nwhere we used $|k|\\ge\\omega(k)-m_{\\mathrm{b}}$.\nIf $R_*\\wedge\\Lambda_{g,N}=\\Lambda_{g,N}$, then $(R_*\\wedge\\Lambda_{g,N})^2+m_{\\mathrm{b}}^2=g^4N^2$.\nOtherwise, $(R_*\\wedge\\Lambda_{g,N})^2+m_{\\mathrm{b}}^2\\ge1+m_{\\mathrm{b}}^2>1$ because $R_*\\ge1$.\nHence, the term containing the logarithm in the previous bound is $\\ge2\\pi\\ln([1\\wedge(g^2N)]\/m_{\\mathrm{b}})$,\nwhile the second term is obviously $\\ge -2\\pi$.\nFinally, we calculate\n\\begin{align*}\n\\int_{B_{\\Lambda_{g,N}}\\setminus B_{R_*}}\\frac{1}{\\omega(k)^2}\\mathrm{d} k\n&=\\pi\\ln\\bigg(\\frac{g^4N^2}{R_*^2+m_{\\mathrm{b}}^2}\\bigg)\\chi_{(R_*^2+m_{\\mathrm{b}}^2,\\infty)}(g^4N^2),\n\\end{align*}\nwhere we obtain a lower bound on the right hand side by putting $m_{\\mathrm{b}}\\vee1$ in place of $m_{\\mathrm{b}}$ in it.\nPutting all remarks above together we arrive at \\eqref{ubGauss2}.\n\\end{proof}\n\\begin{proof}[\\textbf{Proof of \\cref{thm:upbound}}]\nWe  choose $\\sigma>0$ such that $\\sigma^2+m_{\\mathrm{b}}^2=g^4N^2$.\nWe also pick some measurable $f:\\mathbb{R}^{2N}\\to\\mathbb{R}$ with $0\\le f\\le1$ and $\\int_{\\mathbb{R}^{2N}}f(x)\\mathrm{d} x=1$. \nFurthermore, we pick some $p>1$ and let $p'$ denote its conjugated exponent.\nWe again employ the splitting \\eqref{splitu} with our new, $p$-independent choice of $\\sigma$, however.\nApplying H\\\"{o}lder's inequality and afterwards \\eqref{bdcN}, Lemma~\\ref{lemexpmomentm}(iii)\nand Lemma~\\ref{lemexpmomentw} (all with $\\Lambda=\\infty$ and $\\sigma$ as chosen above) we find\n\\begin{align*}\n\\mathbb{E}\\big[\\mathrm{e}^{u_{\\sigma,t}^N[x,X]}f(X_t^x)\\big]\n&\\le\\|\\mathrm{e}^{-c_{\\sigma,\\infty,t}^N(x)}\\|_{L^\\infty(\\mathbb{P})}\n\\mathbb{E}\\big[\\mathrm{e}^{-2pw_{\\sigma,\\infty,t}^N(x)}f(X_t^x)\\big]^{1\/2p}\n\\\\\n&\\quad\\cdot\n\\mathbb{E}\\big[\\mathrm{e}^{-2pm_{\\sigma,\\infty,t}^N(x)}f(X_t^x)\\big]^{1\/2p}\n\\mathbb{E}\\big[\\mathrm{e}^{p'u_{t}^N(x)}f(X_t^x)\\big]^{1\/p'}\n\\\\\n&\\le \nb^{N}c_\\alpha^{1\/4p}\\mathrm{e}^{tc'p(N-1)(m_{\\mathrm{p}}+g^2N)+tc\\alpha p g^2N^2\\mathrm{e}^{8\\pi\\alpha p}}\n\\\\\n&\\quad\\cdot\\mathbb{E}\\big[\\mathrm{e}^{p'u_{t}^N(x)}f(X_t^x)\\big]^{1\/p'},\n\\end{align*}\nfor all $\\alpha>1$ with $c_\\alpha\\coloneq\\alpha\/(\\alpha-1)$ and universal constants $b,c,c'>0$. This entails\n\\begin{align*}\n&\\frac{\\mathrm{e}^{-tc'p(N-1)(m_{\\mathrm{p}}+g^2N)-tc\\alpha pg^2N^2\\mathrm{e}^{8\\pi\\alpha p}}}{b^{N}c_\\alpha^{1\/4p}}\n\\int_{\\mathbb{R}^{2N}}f(x)\\mathbb{E}\\big[\\mathrm{e}^{u_{\\sigma,t}^N[x,X]}f(X_t^x)\\big]\\mathrm{d} x\n\\\\\n&\\le\\bigg(\\int_{\\mathbb{R}^{2N}}f(x)\\mathbb{E}\\big[\\mathrm{e}^{p'u_{t}^N(x)}f(X_t^x)\\big]\\mathrm{d} x\\bigg)^{1\/p'},\n\\end{align*}\nwhere we used H\\\"{o}lders inequality and $\\|f\\|_1=1$. \nLet $H(\\sqrt{p'}g)$ denote the renormalized relativistic Nelson operator with coupling constant $\\sqrt{p'}g$.\nThen the complex action corresponding to it is $p'u_{t}^N(x)$, and we infer from the previous inequality that\n\\begin{align*}\n\\frac{1}{p'}\\inf\\sigma(H(\\sqrt{p'}g))\n&=-\\lim_{t\\to\\infty}\\frac{1}{p't}\\ln \\bigg(\\int_{\\mathbb{R}^{2N}}f(x)\\mathbb{E}\\big[\\mathrm{e}^{p'u_{t}^N(x)}f(X_t^x)\\big]\\mathrm{d} x\\bigg)\n\\\\\n&\\le c'p(N-1)(m_{\\mathrm{p}}+g^2N)+c\\alpha pg^2N^2\\mathrm{e}^{8\\pi\\alpha p}+\\inf\\sigma(H_\\sigma).\n\\end{align*}\nHere $H_\\sigma$ again has coupling coupling constant $g$ as usual. \nCombining the latter inequality with \\eqref{ubGauss2} \n(notice that $\\Lambda_{g,N}$ defined in Corollary~\\ref{corubGauss2} is equal to our present choice of $\\sigma$),\nmultipliying by $p'$ and passing to the limit $\\alpha\\downarrow1$ we obtain\n\\begin{align*}\n\\inf\\sigma(H(\\sqrt{p'}g))\n&\\le c'p(N-1)(p'm_{\\mathrm{p}}+p'g^2N)+cpp'g^2N^2\\mathrm{e}^{8\\pi p}\n +\\frac{\\pi ^{1\/2}s^{1\/2}p'g^2N^2}{2^{1\/2}}\n \\\\\n &\\quad-\\frac{2\\pi}{\\mathrm{e}^{1\/4s}}p'g^2N(N-1)\\ln\\bigg(\\frac{g^2N}{m_{\\mathrm{b}}}\\bigg)\n\\\\\n&\\quad-\\frac{\\pi p'g^2N}{\\mathrm{e}^{1\/4s}}\\cdot\\frac{2-2\\varepsilon}{2-\\varepsilon}\\cdot[0\\vee\\ln(c_*^{-1}g^2N)]+\\frac{\\pi}{4s}p'g^2N\n\\\\\n&\\quad-\\chi_{\\{0\\}}(m_{\\mathrm{p}})\\frac{\\pi p'g^2N}{\\mathrm{e}^{1\/4s}}\\ln\\bigg(\\frac{1\\wedge(g^2N)}{\\mathrm{e} m_{\\mathrm{b}}}\\bigg).\n\\end{align*}\nThe constants appearing here do not depend on $g$. We have, however, used that $g^2N>m_{\\mathrm{b}}$.\nAssuming $g^2N>m_{\\mathrm{b}}\\sqrt{p'}$ we may thus put $g\/\\sqrt{p'}$ in place of $g$ in the above bound.\nAfterwards we conclude by choosing $p=p'=2$. \n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Numerical Algorithms}\\label{Sec:5}\nIn the previous sections, we reformulated the MLFG in (\\ref{follower},\\ref{leader}) as  smooth Nash game \\eqref{Eq:SmoothNEP} with a smoothing parameter $\\varepsilon>0$ and developed theory confirming the validity of this approach.\nIn this section, a computational method is provided which is consistent to the developed theory.\n\nIn particular, we  apply a gradient type method and recall corresponding convergence theory.\nAs alternative we propose a Newton like method.\n\n\\subsection{The Method}\nDue to Lemma \\ref{lem:KKTNEPeps}, every \\textcolor{black}{leader's} unique optimal solution is characterized by its KKT system \\ref{Eq:KKTepsilon} for a fixed smoothing parameter $\\varepsilon$.\n\\textcolor{black}{The multistrategy vector of these solutions characterizes the unique Nash equilibrium of \\eqref{Eq:SmoothNEP}; therefore, we aim to find a primal dual pair $z=(x,\\lambda)$ which satisfies the concatenated KKT conditions.\nWe abbreviate the concatenation with $F^\\varepsilon(z)=(F_1^\\varepsilon(z),F_2^\\varepsilon(z))^\\top$ such that \n}\\begin{subequations}\\label{Eq:KKTjoint}\n\t\\begin{align}\n\t\\begin{split}\n\t\\textcolor{black}{F^\\varepsilon_1(z)}=&Qx+c+\\frac{1}{2} (L^\\top +Q_y^{-1}B^\\top )^\\top  a  \\\\  & +\\frac{1}{2} \\sum\\limits_{i=1}^m a_i (L^\\top -Q_y^{-1}B^\\top )^\\top _{:,i} ~ \\tilde{\\phi_\\varepsilon}'\\left([(L^\\top -Q_y^{-1}B^\\top ) x]_i\\right)+ \\begin{bmatrix}\n\t\\nabla_{x_{1}} g_{1} (x_{1}) \\lambda_{1}\\\\\n\t\\vdots\\\\\n\t\\nabla_{x_{N}}  g_{N} (x_{N}) \\lambda_{N}\n\t\\end{bmatrix}\n\t\\end{split}\\label{Eq:KKTjoint1}\\\\\n\t\\textcolor{black}{F^\\varepsilon_2(z)}=&\\min\\left\\{ \\begin{bmatrix}\\lambda_{1}\\\\\\vdots\\\\\\lambda_{N}\n\t\\end{bmatrix},\\begin{bmatrix}-g_{1}(x_{1})\\\\\\vdots\\\\-g_{N}(x_{N})\n\t\\end{bmatrix}  \\right\\} \\label{Eq:KKTjoint2}\n\t\\end{align}\n\\end{subequations}\n using the notation $Q=\\mathrm{diag}(Q_1,\\dots,Q_N)$ and $c=(c_1^\\top,\\dots,c^\\top_N)^\\top$.\nThe roots of this system characterize the Nash equilibrium for a fixed relaxation parameter $\\varepsilon$.\nWith this notation, the KKT system can be equivalently expressed as the minimization of the auxiliary function $\\Psi_\\varepsilon: \\R^{n+\\bar{m}}\\rightarrow\\R_+$ where $\\bar{m}=m_1+\\dots+m_N$ and\n$$\\Psi_\\varepsilon(z)=\\frac{1}{2}\\|F^\\varepsilon(z)\\|^2_2=\\frac{1}{2}\\left(\\|F_1^\\varepsilon(z)\\|^2_2+\\|F_2^\\varepsilon(z)\\|^2_2\\right).$$\nThe global minimum is obtained for an $z^*$ satisfying $\\Psi_\\varepsilon(z^*)=0$.\nFor convergence theory, the Lipschitz property of $\\Psi_{\\varepsilon}$ is crucial; therefore, we prove it in the following lemma.\n\\begin{lemma}\\label{Lemma:Lipschitz}\n\tThe function $\\Psi_\\varepsilon$ is locally Lipschitz and directionally differentiable.\n\\end{lemma}\n\\begin{proof}\nWe verify the  properties for each part of the sum separately.\n\n\\noindent\n(i) $\\frac{1}{2}\\|F_1(z)\\|_2^2\\in C^1$, as a composition of $C^1$ functions because $\\phi_\\varepsilon$ is assumed to be twice differentiable.\nTherefore, this part is locally Lipschitz and directionally differentiable. \\\\\n(ii) $\\frac{1}{2}\\|F_2(z)\\|_2^2=\\frac{1}{2}\\sum\\limits^m_{i=1} \\min^2\\{\\lambda_i,-g_i(x)\\}=\\frac{1}{8}\\sum\\limits_{i=1}^m \\left(\\lambda_i-g_i(x)-|\\lambda_i+g_i(x)|\\right)^2$ is locally Lipschitz as a composition of locally Lipschitz functions.\nIt is also directionally differentiable as it is also a composition of directionally differentiable functions.\n\\end{proof}\n\nWe are interested in the solution of the system for $\\varepsilon$ close to zero.\nHowever, the problem characteristics are poor for very small $\\varepsilon$ and we expect bad numerical performance with arbitrary initial values.\nTherefore, we propose to solve a sequence of minimization problems:\n\\begin{equation*}\n\\min\\limits_z ~\\Psi_\\varepsilon(z)\\quad \\mathrm{s.t.} ~z\\in\\R^{n+\\bar{m}},\n\\end{equation*}\n for a decreasing sequence of \\textcolor{black}{positive numbers} \t $(\\varepsilon_i)_{i\\in\\mathbb{N}}$.\n This approach returns a sequence of KKT points $\\left(z^*(\\varepsilon_{i})\\right)_{i\\in\\mathbb{N}}=\\left(x^*(\\varepsilon_{i}), \\lambda^*(\\varepsilon_{i})\\right)_{i\\in\\mathbb{N}}$ whose primal part $\\left(x^*(\\varepsilon_i)\\right)_{i\\in\\mathbb{N}}$  is  the Nash equilibrium of NEP$(\\varepsilon_{i})$.\n We  use the solution $z^*(\\varepsilon_{i})$ as initial value for the subsequent solving for $\\varepsilon_{i+1}$.\n\nTo further increase the quality of the initial values, we propose an update for the primal variables $x$  based on formal Taylor expansion of the map $\\varepsilon\\mapsto x^*(\\varepsilon)$.\nWe compute the derivative of the objectives of the Nash game with respect to $\\varepsilon$ which implicitly characterize $\\frac{\\partial x}{\\partial \\varepsilon}$.\nFor $\\nu=1,\\dots, N$, we have\n\\begin{equation*}\n\t\\frac{\\mathrm{d}}{\\mathrm{d}\\varepsilon} \\left(  \\nabla_{x_\\nu} \\theta_\\nu^\\varepsilon (x_\\nu(\\varepsilon),x_{-\\nu}(\\varepsilon)) \\right) =0,\n\\end{equation*}\nwhich leads to the following system\n\\begin{equation*}\n\tE\\frac{\\partial x}{\\partial \\varepsilon}(\\varepsilon)=h.\n\\end{equation*}\nHere, we denote $\\tilde{\\phi}_\\varepsilon(\\eta)=\\Phi(\\eta,\\varepsilon)$ to emphasize the explicit dependence on $\\varepsilon$, then the linear system has the following coefficient matrix\n\\begin{equation*}\n E=Q\n+\\frac{1}{2}\\sum\\limits_{i=1}^m a_i (L^\\top-Q_y^{-1}B^\\top)^\\top_{:,i}(L^\\top-Q_y^{-1}B^\\top)_{:,i} \\frac{\\partial^2 {\\Phi}}{\\partial t^2}((L^\\top-Q_y^{-1}B^\\top)^\\top_{:,i} x,\\varepsilon),\n\\end{equation*}\nand the right-hand-side\n\\begin{equation*}\n\th= \\frac{1}{2}\\sum\\limits_{i=1}^m a_i (L^\\top -Q_y^{-1}B^\\top )^\\top _{:,i} \\frac{\\partial {\\Phi}}{\\partial\\varepsilon}((L^\\top-Q_y^{-1}B^\\top)^\\top_{:,i} x,\\varepsilon).\n\\end{equation*}\nWe remark that $E$ is nonsingular since it is composed of the second derivatives of the strictly convex objectives.\nWe summarize the general approach in the following algorithm.\n\n\n\\begin{algorithm}[H]\n\t\\caption{}\\label{Alg:Outer}\n\t\\begin{algorithmic}[1]\n\t\t\\State \\textbf{Initialize} Choose $z^0(\\varepsilon_0)=(x^0(\\varepsilon_0),\\lambda^0(\\varepsilon_0))\\in\\R^{n+\\bar{m}}$,  $tol>0$, $\\varepsilon_0\\in(1,2)$, $\\gamma\\in(0,1)$.\n\t\t\\For{$i=0,1,\\dots$\n\t\t\\State Compute Nash equilibrium of (NEP$(\\varepsilon_{i})$)  with initial guess $z^0(\\varepsilon_{i})=(x^0(\\varepsilon_{i}),\\lambda^0(\\varepsilon_{i}))$ by $$z^*(\\varepsilon_{i})=(x^*(\\varepsilon_{i}),\\lambda^*(\\varepsilon_{i}))=\\arg\\min\\limits_{z\\in\\R^{n+\\bar{m}}} \\Psi_{\\varepsilon_i}(z),$$ \\label{Alg:General3}\n\t\t\\State Decrease $\\varepsilon_{i+1}=\\gamma\\varepsilon_{i}$,\n\t\t\\State Compute Taylor update $$d^i=\\frac{\\partial x}{\\partial \\varepsilon}(\\varepsilon_{i+1} ),$$\\label{Alg:General5}\n\t\t\\State Update initial guess\\label{Alg:General6} $x^0(\\varepsilon_{i+1})=x^{*}(\\varepsilon_{i})-(\\varepsilon_i-\\varepsilon_{i+1})d^i$ and $\\lambda^0(\\varepsilon_{i+1})=\\lambda^{*}(\\varepsilon_{i})$.\n\t\t\n\t\t\\EndFor \\State \\textbf{end for}\n\t\\end{algorithmic}\n\\end{algorithm}\n\nIn Step \\ref{Alg:General5} of the algorithm, we use a forward evaluation of  $\\frac{\\partial x}{\\partial\\varepsilon}$ but also $\\frac{\\partial x}{\\partial\\varepsilon}(\\varepsilon_{i})$ is a valid choice.\nIn the remainder of this section, we propose two algorithms for computation of the Nash equilibria in Step \\ref{Alg:General3}, but other approaches are conceivable, \\textcolor{black}{ e.g. diagonalization methods as in  \\cite{HR07} or path following techniques \\cite{Dirkse1995,Ferris1999}.}\n\n\n\n\\subsection{Subgradient Method}\nTo generate the sequence of Nash equilibria, we propose a method which is based on subgradient \\textcolor{black}{descent}.\nWe apply the method of \\cite{Bagirov2013} for a fixed smoothing parameter $\\varepsilon>0$.\nStationary points of $\\Psi_\\varepsilon$ are computed as the limit of a sequence of $h$-$\\delta$-stationary points.\nBagirov et. al. \\cite{Bagirov2013} showed that the limit is a Clarke stationary point.\nWith this method we obtain the unique Nash Equilibrium of the smoothed game.\n\nBefore stating the algorithm and the inherent convergence results, we introduce some terms.\n\n\n\n\n\\begin{definition}[$h$-$\\delta$ Stationary Point]\\label{Def:Wh}\n\tLet $W_h(x)$ denote the closed convex hull of all possible quasisecants of a locally Lipschitz function $f:\\R^n\\rightarrow \\R$ at the point $x\\in\\R^n$ with length $h>0$:\n\t$$W_h(x)=\\overline{\\mathrm{conv}}\\left\\{w\\in\\R^n: \\exists d\\in\\R^n \\text{ with } \\|d\\|=1: w=v(x,d,h)   \\right\\}.$$\n\tThen a point $x$ is called a $h$-$\\delta$ stationary point of a locally Lipschitz function $f:\\R^n\\rightarrow \\R$ if and only if\n\t$$\\min\\left\\{\\|v\\|: v\\in W_h(x)  \\right\\}<\\delta.$$\n\\end{definition}\n\n\\begin{lemma}[Termination]\n\t(1) If $\\max\\{\\|v\\|: v\\in W_h(z)\\}<\\infty$ for all iterates $z^k\\in\\R^{n+\\bar{m}}$, the loop in Lines 7-15 terminates after finitely many iterations with a decent direction.\n\t(2) The loop in Lines 4-20 terminates after finitely many iterations with a $h$-$\\delta$-stationary point. \n\\end{lemma}\n\\begin{proof}\n\t(1) Since $\\Psi_\\varepsilon$ is locally Lipschitz with Lemma \\ref{Lemma:Lipschitz}, \\cite[Proposition 4.1]{Bagirov2013} is applicable.\\\\\n\t(2) The function $\\Psi_\\varepsilon$ is bounded from below as it takes nonnegative values only, therefore \\cite[Proposition 5.1]{Bagirov2013} is applicable.\n\\end{proof}\n\n\\begin{theorem}[Convergence]\n\tAssume $\\mathcal{L}(z^0)=\\left\\{z\\in\\R^{n+\\bar{m}}: \\Psi_{\\varepsilon}(z)\\leq\\Psi_\\varepsilon(z^0)  \\right\\}$ is bounded and Assumption \\ref{Assumption31} is fulfilled.\n\tThen there exists at least one accumulation point of the sequence $(z^k)_{k\\in\\mathbb{N}}$ generated by Alg. 2 and any accumulation point is a stationary point of $\\Psi_\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\n\tDue to Lemma \\ref{Lemma:Lipschitz}, $\\Psi_{\\varepsilon}$ is locally Lipschitz and therefore, \\cite[Proposition 5.2]{Bagirov2013} is applicable.\n\tThe boundedness of $\\mathcal{L}(z^0)$ implies that \\textcolor{black}{there exists} at least one accumulation point.\n\\end{proof}\nBagirov et. al. \\cite{Bagirov2013} state that subgradients are in particular quasisecants and therefore,  we limit ourselves to the usage of subgradients as decent directions and to $h=0$ in the implementations.\nThe algorithm is stated as Algorithm \\ref{Alg:Subgr} below.\n\\begin{algorithm}[H]\n\t\\caption{Subgradient Method}\\label{Alg:Subgr}\n\t\\begin{algorithmic}[1]\n\t\t\\State \\textbf{Initialize} $h_0>0$ , $\\delta_0>0$, $\\gamma\\in(0,1)$, $z^0\\in\\R^{n+\\bar{m}}$, $d_0\\in\\R^{n+\\bar{m}}$ with $\\|d_0\\|=1$, $0<c_2\\leq c_1\\leq 1$, $\\varepsilon>0$.\n\n\t\t\\For{$k=0,\\dots$\n\t\t\\State $\\bar{z}_1=z^k$,\n\t\t\\For{$j=1,\\dots$}\\Comment{compute $h$-$\\delta$-stationary point}\n\t\t\\State Compute quasisecant $v_0=v(\\bar{z}_j,d_0,h)$,\n\t\t\\State $\\tilde{v}_0=v_0$,\n\t\t\\For{$i=0,1,\\dots$}\\Comment{find decent direction}\n\t\t\\State $c_i=\\arg\\min\\{\\|cv_i+(1-c)\\tilde{v}_i\\|^2_2~ | c\\in(0,1)\\}$,\n\t\t\\State $\\bar{v}_i=c_iv_i+(1-c_i)\\tilde{v}_i$,\n\t\t\\If{$\\|\\bar{v}_i\\|\\leq \\delta_k$}  return $v^j=\\bar{v}_i$  .\\EndIf\n\t\t\\State $d_i=-\\frac{\\bar{v}_i}{\\|\\bar{v}_i\\|}$,\n\t\t\\If{$\\Psi_\\varepsilon(\\bar{z}_j+hd_{i})-\\Psi_\\varepsilon(\\bar{z}_j)\\leq -c_1h\\|\\bar{v}_i\\|$} return $v^j=\\bar{v}_i$. \\EndIf\n\t\t\\State Compute quasisecant $v_{i+1}=v(x,d_i,h)$,\n\t\t\\State $\\tilde{v}_{i+1}=\\bar{v}_i$.\n\t\t\\EndFor \\State \\textbf{end for}\n\t\t\\If{$\\|v^j\\|\\leq \\delta_k$}  Stop.  \\EndIf\n\t\t\\State $d^j=-\\frac{v^j}{\\|v^j\\|}$,\n\t\t\\State Compute step length such that\n\t\t$$\\sigma_j=\\arg\\max\\{\\Psi_\\varepsilon(\\bar{z}_j+\\sigma d^j)-\\Psi_\\varepsilon(\\bar{z}_j)\\leq -c_2\\sigma\\|v^j\\|~|\\sigma>0\\},$$\n\t\t\\State Update $\\bar{z}_{j+1}=\\bar{z}_j+\\sigma_j d^j$.\n    \n\t\t\\EndFor \\State \\textbf{end for}\n\t\t\\State $z^{k+1}=\\bar{z}_j$,\n\t\t\\State $h_{k+1}=\\gamma h_k$, \n\t\t\\State $\\delta_{k+1}=\\gamma \\delta_k $.\n\t\t\\EndFor \\State \\textbf{end for}\n\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Nonsmooth Newton Method}\nNext, we present an improved method.\nThe joint KKT system  \\eqref{Eq:KKTjoint} leads to the problem to find the unique $z^*(\\varepsilon)=(x^*(\\varepsilon),\\lambda^*(\\varepsilon))$ that satisfy\n\\begin{equation*}\n\tF^\\varepsilon(z)=0.\n\\end{equation*}\nThis  is a nonlinear and nonsmooth system of equations which depend on the  parameter $\\varepsilon>0$.\nThe generalized Newton method can be written as the solving of a sequence of the linear systems\n\\begin{equation*}\nH \\left(z^{k+1}-z^k\\right) = -F^\\varepsilon(z^k),\n\\end{equation*}\nfor an element $H\\in\\partial F^\\varepsilon(z^k)$ of the Clarke subdifferential of $F^\\varepsilon$.\nThe explicit structure of a generalized Jacobian $H$ can be found in \\ref{App:Newton}.\n\nSince $H$ is not necessarily regular,  we verify this property in Step \\ref{Alg:Newton5} of  Algorithm \\ref{Alg:Newton} and use a first order decent direction if necessary.\nThis subgradient decent also serves as globalization strategy.\n\\begin{algorithm}[H]\n\t\\caption{Nonsmooth Newton Method}\\label{Alg:Newton}\n\t\\begin{algorithmic}[1]\n\t\t\\State \\textbf{Initialize} Choose $z^0=(x^0,\\lambda^0)\\in\\R^{n+\\bar{m}}$,  $\\beta\\in(0,1)$, $\\sigma\\in(0,0.5)$, $tol>0$, $\\varepsilon>0$.\n\t\t\n\t\t\\For{$k=0,\\dots$\n\t\t\\If{$\\Psi_{\\varepsilon}\\leq tol$}  Stop. \\EndIf\n\t\t\\State Let $H\\in\\partial F^\\varepsilon(z^k)$,\n\t\t\\If{$H$ singular} do subgradient decent of $\\Psi_{\\varepsilon}$, thus choose $$s^k\\in-\\partial\\Psi_{\\varepsilon}(z^k),$$\\label{Alg:Newton5}\n\t\tand the step length $t_k=\\max\\{\\beta^l|l=0,1,\\dots \\}$ which fulfills the Armijo condition \n\t\t$$\\Psi_{\\varepsilon}(z^k+t^ks^k)\\leq \\Psi_{\\varepsilon}(z^k) + t^k \\sigma  {s^k}^\\top s^k,$$\n\t\t\\Else ~ let $t_k=1$ and compute Newton step by solving  $$ Hs^k=-F^{\\varepsilon}(z^k),$$\n\t\t\\EndIf\n\t\t\\State Update $z^{k+1}=z^k+t_ks^k$.\n\t\t\\EndFor \\State \\textbf{end for}\n\t\\end{algorithmic}\n\\end{algorithm}\nFor further discussions and convergence analysis we refer to e.g. \\cite{Qi1993}.\n\n\n\n\\section{Appendix}\n\\subsection{Remarks on Extensions to the Model}\\label{Sec:Ext}\nIn the following, we mention two generalizations to the follower problem in \\eqref{follower}, which seem obvious to include.\nWe explain challenges to motivate future research.\n\nIn  \\eqref{follower}, it is assumed that the matrix $Q_y$ is positive definite and diagonal.\nA popular approach in quadratic programming is to enforce diagonality of symmetric positive definite matrices by introducing an auxiliary variable $z=D^\\top y$, where we have the Cholesky decomposition $Q_y=DD^\\top$, which exists for symmetric  positive definite  $Q_y$.\nThe follower's problem can be equivalently formulated as a minimization problem in $z$:\n\\begin{equation*}\n\\min\\limits_{z\\in \\R^m} \\frac{1}{2}z^\\top z-b(x)^\\top D^{-\\top}z \\quad \\st\\quad  z\\geq D^\\top l(x).\n\\end{equation*}\nAs in Lemma \\ref{Th:bestresponse}, we   derive the solution to this optimization problem explicitly:\n\\begin{equation*}\nz(x)=\\max\\{D^{-1}b(x),D^\\top l(x)\\},\n\\end{equation*}\nand we recover the follower's best response in this setting:\n\\begin{equation*}\ny(x)=D^{-\\top}\\max\\{D^{-1}b(x),D^\\top l(x)\\}.\n\\end{equation*}\nHowever, unlike \\eqref{solY2}, this function is not necessarily componentwise convex.\nBut this property is essential to guarantee convexity of the leaders' objectives.\nThis property is required  for the proof of existence of Nash equilibria in Theorem \\ref{Th:MainExistence} as it is based on Kakutani fixed-point theorem.\nAn other obvious extension to the follower problem is to incorporate upper bounds besides the discussed lower bounds, i.e. $l(x)\\leq y\\leq u(x)$.\nWe can also derive an explicit representation of the best response, e.g. via projection of the objective's gradient into the feasible set.\nFor that let $\\tilde{y}_i(x)=(Q_y)_{ii}^{-1}b_i(x)$, then the best response is for $i=1,\\dots,m$\n\\begin{equation*}\ny_i(x)=\\mathrm{median}( l_i(x),\\tilde{y}_i(x),u_i(x))=\\begin{cases}\n\nl_i(x), & \\tilde{y}_i(x)\\leq l_i(x),\\\\\nu_i(x), &  \\tilde{y}_i(x)\\geq u_i(x),\\\\\n\\tilde{y}_i(x), & \\text{else}.\n\\end{cases}\n\\end{equation*}\nLike in the previous case, this best response is not necessarily convex; therefore, the existence result in Theorem \\ref{Th:MainExistence} does not apply.\n\n\n\\subsection{Subgradient Method}\nIn the following, we state the definition of quasisecants and a theorem which relates quasisecants and subgradients.\nFurthermore, we state a assumption which is needed in the convergence theorem of the subgradient method.\nAll is adapted from \\cite{Bagirov2013} and can be found there in an extended form.\n\n\n\\begin{definition}[Quasisecant]\n\tA vector $v=v(x,d,h)\\in\\R^n$ is called a quasisecant of a locally Lipschitz function $f:\\R^n\\rightarrow \\R$ at the point $x\\in\\R^n$ in direction $d\\in\\R^n$ with $\\,d\\,=1$ with the length $h>0$ if and only if\n\t\\begin{equation*}\n\t\tf(x+hd)-f(x)\\leq h\\langle v,d\\rangle,\n\t\\end{equation*}\n\tand\n\t\t\\begin{equation*}\n\tv\\in\\partial_{d,h}f(x)+B_{O(h)},\n\t\\end{equation*}\n\twhere  $\\partial_{d,h} f=\\cup_{t\\in[0,h]}\\partial f(x+td)$ denotes the union of all Clarke subdifferentials over the set $\\mathrm{conv}(x,x+hd)$ and $B_{O(h)}$ denotes a ball for which $O(h)\\rightarrow0$ for $h\\rightarrow0$. \n\\end{definition}\n\nFor the convergence proof it is necessary to study the relation of $W_h(x)$ (Definition \\ref{Def:Wh}) and the subdifferential $\\partial f(x)$ and therefore, the following assumption is crucial.\n\\begin{assumption}\\label{Assumption31}\n\tAt any given point $x\\in\\R^n$ there exists $\\delta=\\delta(x)>0$ such that $O(y,h)\\downarrow0$ uniformly as $h\\downarrow0$ for all $y\\in B_\\delta(x)$ that is for any $\\eta>0$ there exists $h(\\eta)>0$ such that $O(y,h)<\\eta$ for all $h\\in (0,h(\\eta))$ and $y\\in B_\\delta(x)$.\n\\end{assumption}\nIn particular, this assumptions guarantees a certain relation between quasisecants and subgradients.\n\\begin{theorem}\n\tAssume that a function $f$ satisfies Assumption \\ref{Assumption31}.\n\tThen at a given point $x\\in\\R^n$ for any $\\eta>0$ there exists $\\delta=\\delta(\\eta)$ and $h(\\eta)>0$ such that\n\t\\begin{equation*}\n\t\tW_h(y)\\subset \\partial f+ B_\\eta,\n\t\\end{equation*}\n\tfor all $h\\in(0,h(\\eta))$ and $y\\in B_\\delta(x)$.\n\tFurthermore, it holds for locally Lipschitz function that  the limit as $h\\rightarrow 0$ of the $W_h(x)$ lies in the subdifferential, i.e.\n\t\\begin{equation*}\n\t\tW_0(x)\\subset \\partial f(x).\n\t\\end{equation*}\n\\end{theorem}\n\n\n\\subsection{Nonsmooth Newton}\\label{App:Newton}\nWe propose a nonsmooth Newton method to compute Nash equilibria.\nIn order to keep the readability of the paper, we specify the structure of the generalized Jacobian here.\n\nWe look at the elements of $\\partial F_\\varepsilon$ as a block matrix:\n$$H=\\begin{bmatrix}\n\\boxed{A } & \\boxed{B}\\\\\\boxed{C}&\\boxed{D}\n\\end{bmatrix},$$\nwhere the block $\\boxed{A }\\in\\R^{n\\times n}$ is\n\\begin{align*}\n\\nonumber\\boxed{A}&=Q+ \\begin{bmatrix}\n\\nabla_{x_{(1)}}(\\nabla_{x_{1}} g_{1}(x_{1}) \\lambda_{1}) & & \\\\\n&\\ddots & \n\\\\\n& & \t\\nabla_{x_{N}}(\\nabla_{x_{N}} g_{N}(x_{N}) \\lambda_{N})\n\\end{bmatrix}\\\\\n& +\\frac{1}{2}\\sum\\limits_{i=1}^m a_i (L^\\top -Q_y^{-1}B^\\top )^\\top _{:,i}(L^\\top -Q_y^{-1}B^\\top )_{:,i} \\\\ & \\left(\\frac{1}{\\sqrt{[(L^\\top -Q_y^{-1}B^\\top ) x]_i^2 +4\\varepsilon^2} } -\\frac{[(L^\\top -Q_y^{-1}B^\\top )x]_i^2}{\\sqrt{[(L^\\top -Q_y^{-1}B^\\top ) x]_i^2 +4\\varepsilon^2}^3}\\right),\n\\end{align*}\nthe block $\\boxed{B}\\in\\R^{n\\times\\bar{m}}$ is\n\\begin{equation*}\n\\boxed{B}=\\begin{bmatrix}\n\\nabla_{x_{1}} g_{1}(x_{1}) & & \\\\\n&\\ddots & \\\\\n& & \\nabla_{x_{N}} g_{N}(x_{N}) \n\\end{bmatrix},\n\\end{equation*}\nand\nthe block diagonal $\\boxed{C}\\in \\R^{\\bar{m}\\times n}$ is\n\\begin{equation*}\n\\boxed{C}=\\begin{bmatrix}\n\\boxed{C_1} & & \\\\ & \\ddots & \\\\ & & \\boxed{C_N}\n\\end{bmatrix},\n\\end{equation*}\nwith the blocks $\\boxed{C_\\nu}\\in\\R^{m_\\nu\\times n_\\nu}$ and the entries\n\\begin{equation*}\n\\left(\\boxed{C_\\nu}\\right)_{i,j}=\\partial^C_{x_\\nu^j} \\min\\left\\{ \\lambda_\\nu^i, -g_\\nu^i(x_\\nu)  \\right\\}=\\begin{cases} \n0, & \\lambda_\\nu^i <-g_\\nu^i(x_\\nu), \\\\\n-\\frac{\\partial}{\\partial x_\\nu^j} g_\\nu^i(x_\\nu), & \\lambda_\\nu^i > -g_\\nu^i(x_\\nu), \\\\\n\\left[0, -\\frac{\\partial}{\\partial x_\\nu^j} g_\\nu^i(x_\\nu) \\right], & \\lambda_\\nu^i = -g_\\nu^i(x_\\nu). \\\\\n\\end{cases}\n\\end{equation*}\nand the diagonal matrix: \n\\begin{equation*}\n\\boxed{D}=\\begin{bmatrix}\n\\boxed{D_1} & & \\\\ & \\ddots & \\\\ & & \\boxed{D_N}\n\\end{bmatrix} \\in\\R^{\\bar{m}\\times\\bar{m}},\n\\end{equation*}\nwith its blocks $\\boxed{D_\\nu}\\in\\R^{m_\\nu\\times m_\\nu}$ with the entries:\n\\begin{equation*}\n\\left(\\boxed{D_\\nu}\\right)_{i}=\\partial^C_{\\lambda_\\nu^i} \\min\\left\\{ \\lambda_\\nu^i, -g_\\nu^i(x_\\nu)  \\right\\}=\\begin{cases} \n1, & \\lambda_\\nu^i <-g_\\nu^i(x_\\nu), \\\\\n0, & \\lambda_\\nu^i > -g_\\nu^i(x_\\nu), \\\\\n\\left[1,0 \\right], &  \\lambda_\\nu^i = -g_\\nu^i(x_\\nu). \\\\\n\\end{cases}\n\\end{equation*}\n\n\\subsection{The Data}\n\\label{App:Data}\nIn the following, we specify the data used for the experiments presented in Section 5.\nWe adapted the data used in \\cite{HuFu2013}.\n\n\\subsubsection{Data Set 1}\nWe consider $N    = 2$ leader with each $n_1=n_2=2$ variables.\nThe objectives of the leader are given by\n\\begin{equation*}\nQ_{1}= \\begin{bmatrix}\n1.7 & 1.6 \\\\\n1.6 & 2.8 \n\\end{bmatrix}, \\quad Q_2=\\begin{bmatrix} 2.7 & 1.3 \\\\ 1.3 & 3.6 \n\\end{bmatrix}, \\quad c_1=c_2=\\begin{bmatrix} 0\\\\ 0\\end{bmatrix},  \\quad a=\\begin{bmatrix} 1.4 \\\\ 2.6 \\\\ 2.1\\end{bmatrix}.\n\\end{equation*}\nEach leader has $m_1=m_2=3$ linear constraints $g_\\nu=A_\\nu^Tx_\\nu + b_\\nu\\leq 0 $ with \n\\begin{align*}\nA_1=\\begin{bmatrix}\n1.6  & 0.8 & 1.3 \\\\\n2.6 &  2.2 & 1.7\n\\end{bmatrix}, \\quad\tb_1=\\begin{bmatrix}\n1.6 \\\\ 1.2 \\\\ 0.4 \n\\end{bmatrix},\\qquad\nA_2=\\begin{bmatrix}\n1.8 & 1.6 & 1.4 \\\\\n1.3 & 1.2 & 2.7\n\\end{bmatrix}, \\quad\tb_2=\\begin{bmatrix}\n1.6 \\\\ 1.5 \\\\ 2.6\n\\end{bmatrix}.\n\\end{align*}\n\n\nThe follower has $M=3$ variables and its objective and constraints are given by\n\\begin{equation*}\nQ_y=\\begin{bmatrix}\n2.5 &0& 0\\\\\n0& 3.6& 0 \\\\\n0 &0 &4.6\n\\end{bmatrix}, \\quad B= \\begin{bmatrix}\n2.3& 1.4 &2.6 \\\\\n1.3& 2.1 &1.7 \\\\\n2.5& 1.9& 1.4 \\\\\n1.3 &2.4 &1.6\n\\end{bmatrix}, \\quad L=\\begin{bmatrix}\n1.3& 2.4& 1.8 \\\\\n1.3 &2.4& 1.8 \\\\\n1.3& 2.4& 1.8\\\\\n1.3& 2.4& 1.8 \n\\end{bmatrix}. \n\\end{equation*}\n\n\n\\subsubsection{Data Set 2}\nWe consider $N    = 3$ leader with each $n_1=n_2=n_3=2$ variables.\nThe objectives of the leader are given by\n\\begin{align*}\nQ_{1}= \\begin{bmatrix}\n2.5 &1.6 \\\\\n1.6 &3.8  \\end{bmatrix}, ~ Q_2=\\begin{bmatrix} 2.9 &1.3 \\\\\n1.3 &1.8   \n\\end{bmatrix},~ Q_3=\\begin{bmatrix}\n3.2 &2.3 \\\\\n2.3 &2.6  \n\\end{bmatrix}, ~ c_1=c_2=c_3=\\begin{bmatrix} 0\\\\ 0\\end{bmatrix}, ~  a=\\begin{bmatrix} 0.4 \\\\1.6\\\\ 2.6\\end{bmatrix}.\n\\end{align*}\nEach leader has $m_1=m_2=m_3=3$ linear constraints $g_\\nu=A_\\nu^Tx_\\nu + b_\\nu\\leq 0 $ with \n\\begin{align*}\nA_1&=&\\begin{bmatrix}\n1.6& 0.8 &1.3\\\\\n2.6& 2.2 &1.7\n\\end{bmatrix}, ~~\nA_2&=&\\begin{bmatrix}\n1.8 & 1.6 & 1.4 \\\\\n1.3 & 1.2 & 2.7\n\\end{bmatrix}, ~~\nA_3&=&\\begin{bmatrix}\n2.3 &1.9& 1.6 \\\\\n1.3 &1.7& 2.7\n\\end{bmatrix}, \\\\\nb_1&=&\\begin{bmatrix}\n1.6&1.2 & 0.4\n\\end{bmatrix}^\\top, ~~\n\tb_2&=&\\begin{bmatrix}\n1.6& 1.5 & 2.6\n\\end{bmatrix}^\\top,~~\n\tb_3&=&\\begin{bmatrix}\n1.5& 0.3 &1.8\n\\end{bmatrix}^\\top.\n\\end{align*}\n\n\nThe follower has $M=3$ variables and its objective and constraints are given by\n\\begin{equation*}\nQ_y=\\begin{bmatrix}\n3.7 &0 &0 \\\\\n0& 2.6& 0 \\\\\n0& 0& 0.7\n\\end{bmatrix}, \\quad B= \\begin{bmatrix}\n0.8& 2.1 &1.3 \\\\\n1.5 & 2.3 &0.7 \\\\\n1.5 & 0.9 &2.4 \\\\\n1.8 &2.3& 3.6 \\\\\n1.3 &1.7 &1.7 \\\\\n1.1 &2.6& 1.6 \\\\\n\\end{bmatrix}, \\quad L=\\begin{bmatrix}\n0.8 & 2.1 & 1.3 \\\\\n1.5 &2.3 & 0.7 \\\\\n1.5 &0.9& 2.4 \\\\\n1.8 &2.3& 3.6 \\\\\n0.5 &1.1& 2.1 \\\\\n1.2 &1.5& 1.8 \n\\end{bmatrix}. \n\\end{equation*}\n\n\n\n\n\n\n\n\\section{Relation between MLFG\/NEP and NEP($\\varepsilon$)}\\label{Sec:4}\n\\textcolor{black}{In this section, we study the relationship between the original MLFG (or its equivalent NEP) and the smoothed problem NEP($\\varepsilon$).\nOur aim is to demonstrate that Nash equilibria of NEP($\\varepsilon$) \\textcolor{black}{converge} to a Nash equilibrium of the original problem as the smoothing parameter $\\varepsilon\\rightarrow0$.\n}\n\n\\textcolor{black}{Instead of looking at optimality conditions for the nonsmooth formulation, we are considering  stationarity concepts for mathematical programs with complementarity constraints (MPCC) next.}\n\\begin{definition}[Mathematical Programs with Complementarity Constraints]\n\tLet $f:~\\R^n\\rightarrow\\R$, $g:~\\R^n\\rightarrow\\R^{m}$, and $G_1,G_2:~\\R^n\\rightarrow\\R^{l}$ be smooth functions.\n\tThen we call:\n\t\\begin{align}\\begin{split}\\label{MPCC}\n\t\\min\\limits_z~& f(z)\\\\\n\t\\mathrm{s.t.}~ &g(z)\\leq0\\\\\n\t& 0=\\min\\left\\{ G_1(z), G_2(z)  \\right\\} \n\t\\end{split}\\tag{MPCC}\n\t\\end{align}\n\ta mathematical program with complementarity constraints.\t\n\\end{definition}\nIn fact, each leader problem \\eqref{leader} can be formulated as an MPCC:\n\\begin{align}\\begin{split}\\label{Eq:MPCCnu}\n\\min\\limits_{x_\\nu,y}~&\\theta_\\nu(x_\\nu,x_{-\\nu})=\\frac{1}{2}x_\\nu^\\top Q_\\nu x_\\nu+c_\\nu^\\top x_\\nu+ a^\\top \\textcolor{black}{y(x)}\\\\\n\\mathrm{s.t.}~&g_\\nu(x_\\nu)\\leq 0\\\\\n& 0=\\min\\{G_1(x_\\nu,x_{-\\nu},\\textcolor{black}{y(x)}),G_2(x_\\nu,x_{-\\nu},\\textcolor{black}{y(x)})\\},\n\t\\end{split}\\tag{MPCC$_\\nu$}\n\\end{align}\nwith the complementarity constraints $G_1(x_\\nu,x_{-\\nu},\\textcolor{black}{y(x)})=\\textcolor{black}{y(x)}-(Q_y^{-1}B^\\top)^\\top x$ and $G_2(x_\\nu,x_{-\\nu},\\textcolor{black}{y(x)})=\\textcolor{black}{y(x)}-L^\\top x$.\nFor $\\nu=1,\\dots, N$, the \\eqref{Eq:MPCCnu} form together the generalized Nash equilibrium problem (GNEP) formulation of the MLFG, where the complementarity constraints are shablack constraints.\n\n\nSimilar to KKT points, there is a variety of stationary concepts for MPCC, we introduce the strongest one here: strongly stationary points.\nWe adapt \\cite[Theorem 2]{ScheelScholtes00}.\n\\begin{definition}[S-Stationarity]\\label{Def:SStation}\n   We call $\\bar{z}$ a strongly \\textcolor{black}{(S-)stationary} point of \\eqref{MPCC} if there exist multipliers $(\\lambda, \\Gamma_1, \\Gamma_2)\\in\\R^{m+l+l}$ with:\n\t\\begin{align}\t\\begin{split}\\label{Eq:Sstationarity}\n\t0=\\nabla_z f(\\bar{z}) + \\sum\\limits_{i=1}^{m} \\lambda_i \\nabla_z g_i(\\bar{z})& - \\sum\\limits_{i=1}^{l} \\Gamma_{1,i} \\nabla_z G_{1,i}(\\bar{z}) - \\sum\\limits_{i=1}^{l} \\Gamma_{2,i} \\nabla_z G_{2,i}(\\bar{z}),\\\\\n\tg(\\bar{z})&\\leq 0,\\\\\n\t\\lambda & \\geq 0,\\\\\n\tg_i(\\bar{z})\\lambda_i &= 0, \\quad i=1,\\dots, m,\\\\\n\t\\min\\left\\{ G_{1,i}(\\bar{z}), G_{2,i}(\\bar{z})  \\right\\} &=0,  \\quad i=1,\\dots,l,\\\\\n\tG_{1,i}(\\bar{z})\\Gamma_{1,i}&=0,  \\quad i=1,\\dots,l,\\\\\n\tG_{2,i}(\\bar{z})\\Gamma_{2,i}&=0,\\quad i=1,\\dots,l,\\\\\n\t\\Gamma_{1,i},\\Gamma_{2,i}&\\geq 0,\\quad i: G_{1,i}(\\bar{z})=G_{2,i}(\\bar{z})=0.\n\t\\end{split}\t\\end{align}\n\\end{definition}\n\n\n\\textcolor{black}{Besides suitable stationarity concepts, MPCCs also require proper constraint qualifications. \nHere, the Constant Rank Constraint Qualification (CRCQ) for MPCC is defined  similarly to its version for MPEC in \\cite[Def. 2.2]{Sonja10}.\nOriginally, the CRCQ was introduced for NLP by \\cite{Janin1984}.\n\\begin{definition}(MPCC-CRCQ)\n The constraint qualification MPCC-CRCQ holds in the feasible point $\\bar{z}$ of  \\eqref{MPCC}, if  for every $\\mathcal{K}_g$, $\\mathcal{K}_1$, and $\\mathcal{K}_2$ with\n\t\\begin{align*}\n\t\t\\mathcal{K}_g &\\subseteq I_g(\\bar{z})=\\left\\{i \\in \\{1,\\dots,m\\}| g_i(\\bar{z})=0 \\right\\},\\\\\n\t\t\\mathcal{K}_1 &\\subseteq I_1(\\bar{z})=\\left\\{i \\in \\{1,\\dots,l\\}| G_{1,i}(\\bar{z})=0 \\right\\},\\\\\n\t\t\\mathcal{K}_2 &\\subseteq I_2(\\bar{z})=\\left\\{i \\in \\{1,\\dots,l\\}| G_{2,i}(\\bar{z})=0 \\right\\},\\\\\n\t\\end{align*}\n\tthere exists a neighborhood $\\mathcal{U}(\\bar{z})$ such that for every ${z}\\in \\mathcal{U}(\\hat{z})$ the family of gradient vectors\n\t\\begin{equation*}\n        \\{ \\nabla g_i({z})|i\\in\\mathcal{K}_g  \\}\\cup  \\{ \\nabla G_{1,i}({z})|i\\in\\mathcal{K}_1  \\} \\cup \\{ \\nabla G_{2,i}({z})|i\\in\\mathcal{K}_2  \\},\n\t\\end{equation*}\n\thas the same rank as the family\n\t\\begin{equation*}\n\t\\{ \\nabla g_i(\\bar{z})|i\\in\\mathcal{K}_g  \\}\\cup  \\{ \\nabla G_{1,i}(\\bar{z})|i\\in\\mathcal{K}_1  \\} \\cup \\{ \\nabla G_{2,i}(\\bar{z})|i\\in\\mathcal{K}_2  \\}.\n\t\\end{equation*}\n\\end{definition}\n}\n\n\n\nIn the following theorem we show that a limit of the Nash equilibria of the smoothed problems is indeed strongly stationary with respect to the leader problems.\n\n\\begin{theorem}[S-stationarity and Convergence of Multipliers]\n\\textcolor{black}{\tLet $(\\varepsilon_k)_k$ be a positive sequence with $\\varepsilon_k\\rightarrow 0$ and let $(x^*(\\varepsilon_k))_k$ be the associated sequence of the unique  Nash equilibria to \\emph{NEP($\\varepsilon_k$)}.\n\tAssume this sequence  converges with $x^*(\\varepsilon_k)\\rightarrow x^*(0)$ and every subvector of the limit $x_\\nu^*(0)$ satisfies MPCC-CRCQ for its leader's  \\emph{MPCC}$_\\nu$\n}\n\n\t\\textcolor{black}{Then, \\textcolor{black}{there exists} a bounded sequence of multipliers associated to  $(x^*(\\varepsilon_k))_k$ and the limit $x_\\nu^*(0)$ is a strongly stationary point for \\eqref{Eq:MPCCnu} for $\\nu=1,\\dots,N$.}\n\\end{theorem}\n\\begin{proof}\n\t\\textcolor{black}{The proof consists of three major parts: First, we introduce notation and modest conclusions.\n\tSecond, we  demonstrate that there exists a bounded sequence of multipliers associated to $(x^*_{\\nu}(\\varepsilon_k))_k$.\n\tThird, we verify the conditions of strong stationarity by constructing suitable multipliers.}\\\\\n\t\n \\textcolor{black}{(i)\n In order to keep the notation simple, let $x_\\nu^k=x_\\nu^*(\\varepsilon_k)$ and the limit $\\bar{x}_\\nu=x_\\nu^*(0)$ and respectively the concatenations $x^k$ and $\\bar{x}$, both without the player index.\n\tFurther we introduce the short forms for $k\\in\\mathbb{N}$ and $i=1,\\dots, m$:\n\t\\begin{equation*}\n\t\tz_i^k=\\left( (L^\\top-Q_y^{-1}B^\\top)x^k\\right)_i,  \\bar{z}_i=\\left( (L^\\top-Q_y^{-1}B^\\top)\\bar{x}\\right)_i, \\text{ and } \\xi^k_i=\\tilde{\\phi}'_{\\varepsilon_k}(z^k_i),\n\t\\end{equation*}\n\t and the concatenations are denoted by $z^k$, $\\bar{z}$, and $\\xi^k$, respectively.\n\t Recall, it holds that $z_i^k\\rightarrow\\bar{z}_i$ and  $|\\xi_i^k|\\leq1$ for  $i=1,\\dots, m$ and all $k\\in\\mathbb{N}$.}\n\t\n\t\\textcolor{black}{Due to stationarity of  $x^k_{\\nu}$ for all $\\nu$ and $k\\in\\mathbb{N}$, we  use  Lemma \\ref{lem:KKTNEPeps}, i.e. \\eqref{Eq:KKTepsilon}, and conclude that there exists multipliers $\\lambda_\\nu^k\\in \\R^{m_\\nu}$ such that:\n\\begin{subequations}\n\\begin{align}\n\t0=&Q_\\nu x^k_\\nu+c_\\nu +\\frac{1}{2} (L^\\top +Q_y^{-1}B^\\top )^\\top _{\\nu,:} a + \\frac{1}{2}\\sum\\limits^{m}_{i=1}a_i(L^\\top -Q_y^{-1}B^\\top )^\\top _{\\nu,i} ~\\xi_i^k + \\nabla_{x_\\nu}g_\\nu (x^k_\\nu) \\lambda_\\nu^k,\\label{KKT10a}\\\\\n\t0=&\\min\\left\\{ \\lambda^k_\\nu , -g_{\\nu}(x^k_\\nu)\\right\\},\\label{eq:KKTproof0}\n\t\\end{align}\n\t\\end{subequations}\n\tHence, by rearranging the sum in \\eqref{KKT10a} we have:\n\t\\begin{equation}\\label{eq:KKTproof}\n\t\t\t0=Q_\\nu x^k_\\nu+c_\\nu + \\sum\\limits_{i=1}^{m}L^\\top_{\\nu,i}\\frac{a_i}{2}(1+\\xi_i^k)+   \\sum\\limits_{i=1}^{m}Q_y^{-1}B^\\top_{\\nu,i}\\frac{a_i}{2}(1-\\xi_i^k)+ \\sum\\limits_{j=1}^{m_\\nu}\\nabla_{x_\\nu}g_{\\nu,j} (x^k_\\nu) \\lambda_{\\nu,j}^k,\n\t\\end{equation}\nNext, we construct a multiplier vector $\\hat{\\lambda}^k_\\nu$ for all $\\nu$ and sufficiently large $k$ such that the following inclusion holds:\n$$ \\mathrm{supp}\\left(\\hat{\\lambda}^k_\\nu\\right)\\subseteq  \\mathrm{supp}\\left({\\lambda}^k_\\nu\\right)\\subseteq I_{g_\\nu}(\\bar{x}_\\nu)=\\left\\{j \\left|g_{\\nu,j}(\\bar{x}_\\nu)=0 \\right.\\right\\},$$\t\nwhere $\\mathrm{supp}(\\cdot)$ denotes the support of a vector, i.e. the set of indices of its nonzero entries, the multiplier $\\hat{\\lambda}^k_\\nu$  shall satisfy \\eqref{eq:KKTproof} and the family of vectors\n\\begin{equation}\\label{Eq:linindep}\n\t\\left\\{  \\nabla_{x_\\nu}g_{\\nu,j} (x^k_\\nu) \\left| j\\in \\mathrm{supp}\\left(\\hat{\\lambda}^k_\\nu\\right)    \\right. \\right\\}\n\\end{equation}\nis linearly independent.}\\\\\n\n\\textcolor{black}{(ii) \nNow, we demonstrate  by contradiction that the multiplier sequence $\\left(\\hat{\\lambda}^k_\\nu\\right)_k$ is bounded for all $\\nu$.\nTherefore, it is assumed that the converse is true, i.e. there exists $\\hat{\\nu}$ such that $\\left(\\hat{\\lambda}^k_{\\hat{\\nu}}\\right)_k$ is unbounded.\nBy assumption, we have $\\|\\hat{\\lambda}^k_{\\hat{\\nu}}\\|\\rightarrow \\infty$ as $k\\rightarrow\\infty$ and thus  it holds, that $\\|\\hat{\\lambda}^k_{\\hat{\\nu}}\\|>0$ for sufficiently large $k$.\nTherefore, we can define the auxiliary sequence\n\\begin{equation}\\label{Eq:auxsequence}\n\t \\tilde{\\lambda}^k_{\\hat{\\nu}}=\\frac{\\hat{\\lambda}^k_{\\hat{\\nu}}}{\\|\\hat{\\lambda}^k_{\\hat{\\nu}}\\|}.\n\\end{equation}\nClearly, all elements of this sequence are normalized, i.e. $\\|\\tilde{\\lambda}^k_{\\hat{\\nu}}\\|=1$.\nThis in turn implies that the sequence is bounded and admits a convergent subsequence $\\left(\\tilde{\\lambda}^l_{\\hat{\\nu}}\\right)_{l\\in K}\\rightarrow\\tilde{\\lambda}_{\\hat{\\nu}}$ with $K \\subseteq\\mathbb{N}$.\nIt follows, that also the limit of the subsequence satisfies $\\|\\tilde{\\lambda}_{\\hat{\\nu}}\\|=1$ and there exists an index $j_0\\in\\{1,\\dots,m_\\nu\\}$ with $\\tilde{\\lambda}_{\\hat{\\nu},j_0}>0$ such that $\\mathrm{supp}\\left(\\tilde{\\lambda}_{\\hat{\\nu}}\\right)\\neq\\emptyset$.\nNow, divide \\eqref{eq:KKTproof} by norm of the unbounded sequence $\\|\\hat{\\lambda}^l_{\\hat{\\nu}}\\|$ and take the limit $l\\rightarrow\\infty$.\nNote, that the first four terms of \\eqref{eq:KKTproof} are bounded and thus they vanish in the limit.\nIt remains to compute the limit of the last term:\n$$0=\\lim\\limits_{l\\rightarrow \\infty, l\\in K}\\frac{1}{\\|\\hat{\\lambda}^l_{\\hat{\\nu}}\\|}\\sum\\limits_{j\\in\\mathrm{supp}\\left( \\hat{\\lambda}_{\\hat{\\nu}}^k \\right)} \\nabla_{x_{\\hat{\\nu}}}g_{{\\hat{\\nu}},j} (\\bar{x}_{\\hat{\\nu}}) \\hat{\\lambda}_{\\hat{\\nu},j}^k,$$\nwith \\eqref{Eq:auxsequence} and (for sufficiently large $k$) $\\mathrm{supp}\\left(\\tilde{\\lambda}_{\\hat{\\nu}}   \\right)\\subseteq\\mathrm{supp}\\left(\\tilde{\\lambda}_{\\hat{\\nu}}^k   \\right)=\\mathrm{supp}\\left(\\hat{\\lambda}_{\\hat{\\nu}}^k   \\right)$, we get \n$$0=\\sum\\limits_{j\\in\\mathrm{supp}\\left( \\tilde{\\lambda}_{\\hat{\\nu}} \\right)} \\nabla_{x_{\\hat{\\nu}}}g_{{\\hat{\\nu}},j} (\\bar{x}_{\\hat{\\nu}}) \\tilde{\\lambda}_{{\\hat{\\nu}},j}.$$\nThis implies with $\\mathrm{supp}\\left(\\tilde{\\lambda}_{\\hat{\\nu}}\\right)\\neq\\emptyset$, that the set of vectors\n\\begin{equation*}\n\\left\\{ \\nabla_{x_{\\hat{\\nu}}}g_{{\\hat{\\nu}},j} (\\bar{x}_{\\hat{\\nu}}) \\left| j\\in \\mathrm{supp}\\left(\\tilde{\\lambda}_{\\hat{\\nu}}\\right)    \\right. \\right\\},\t\n\\end{equation*}\nis  linearly dependent.\nThus, by MPCC-CRCQ and $\\mathrm{supp}\\left(\\tilde{\\lambda}_{\\hat{\\nu}}   \\right)\\subseteq\\mathrm{supp}\\left(\\hat{\\lambda}_{\\hat{\\nu}}^k   \\right)$, also \n\\begin{equation*}\n\\left\\{ \\nabla_{x_{\\hat{\\nu}}}g_{{\\hat{\\nu}},j} ({x}^k_{\\hat{\\nu}}) \\left| j\\in \\mathrm{supp}\\left(\\hat{\\lambda}^k_{\\hat{\\nu}}\\right)    \\right. \\right\\},\t\n\\end{equation*}\nis  linearly dependent, which is a contradiction to the linear independence of \\eqref{Eq:linindep} for sufficiently large $k$.\nTherefore the assumption $\\left(\\hat{\\lambda}^k_{\\hat{\\nu}}\\right)_k$ being unbounded, because \\eqref{eq:KKTproof} could not be satisfied by such a multiplier sequence.\nTherefore, the sequence of multipliers  associated to $(x^k)_k$ is bounded, i.e. it exist a multiplier vector $\\bar{\\lambda}$ associated to $\\bar{x}$.}\\\\ \n\n\\textcolor{black}{(iii) In the following, we verify the conditions of \\textcolor{black}{S-stationarity} of Definition \\ref{Def:SStation}, i.e. (\\ref{Eq:Sstationarity}) exemplary for on leader $\\nu$, and begin with applying them to \\eqref{Eq:MPCCnu}:\n\\begin{subequations}\\label{Eq:SstationarityProof}\n\t\\begin{align}\n\t0=\\begin{pmatrix}Q_\\nu x_\\nu+c_\\nu\\\\ a\\end{pmatrix}\t+ \\sum\\limits_{i=1}^{m} \\lambda_i\\begin{pmatrix}\\nabla_{x_\\nu} g_{\\nu,i}(x_\\nu)\\\\0\\end{pmatrix}  &- \\sum\\limits_{i=1}^{l} \\Gamma_{1,i}\\begin{pmatrix}-(Q_y^{-1}B^\\top)_{\\nu,i}\\\\e_i\\end{pmatrix} - \\sum\\limits_{i=1}^{l} \\Gamma_{2,i}\\begin{pmatrix}\n\t-L_{\\nu,i}\\\\e_i\n\t\\end{pmatrix},\\label{Eq:SstationarityProofa}\\\\\n\tg_\\nu(x_\\nu)&\\leq 0\\label{Eq:SstationarityProofb},\\\\\n\t\\lambda_\\nu & \\geq 0\\label{Eq:SstationarityProofc},\\\\\n\tg_{\\nu,i}(x_\\nu)\\lambda_{\\nu,i} &= 0,\\quad i=1,\\dots, m_\\nu\\label{Eq:SstationarityProofd},\\\\\n\t\\min\\left\\{\t(y-(Q_y^{-1}B^\\top)^\\top x)_i,(y-L^\\top x)_i\\right\\}&=0,  \\quad i=1,\\dots,m, \\label{Eq:SstationarityProofe}\\\\\n\t(y-(Q_y^{-1}B^\\top)^\\top x)_i\\Gamma_{1,i}&=0,\\quad  i=1,\\dots,m\\label{Eq:SstationarityProoff},\\\\\n\t(y-L^\\top x)_i\\Gamma_{2,i}&=0,\\quad i=1,\\dots,m\\label{Eq:SstationarityProofg},\\\\\n\t\\Gamma_{1,i},\\Gamma_{2,i}&\\geq 0,\\quad i: \t(y-(Q_y^{-1}B^\\top)^\\top x)_i=(y-L^\\top x)_i\\label{Eq:SstationarityProofh}.\n\t\\end{align}\n\\end{subequations}\nThe remainder of the proof demonstrates that \\eqref{Eq:SstationarityProof} is satisfied for the limit strategy $\\bar{x}_\\nu$ and its multiplier $\\bar{\\lambda}_\\nu$, and we construct the additional multipliers $\\Gamma_{1,i},\\Gamma_{2,i}$.\nWe begin with the limit of \\eqref{eq:KKTproof} for $k\\rightarrow\\infty$ in the sense of a suitable subsequence and get the expression:\n\\begin{equation}\\label{Eq:KKTproof2}\n\t\t\t0=Q_\\nu \\bar{x}_\\nu+c_\\nu + \\sum\\limits_{i=1}^{m}L^\\top_{\\nu,i}\\frac{a_i}{2}(1+\\bar{\\xi}_i)+   \\sum\\limits_{i=1}^{m}Q_y^{-1}B^\\top_{\\nu,i}\\frac{a_i}{2}(1-\\bar{\\xi}_i)+ \\sum\\limits_{j=1}^{m_\\nu}\\nabla_{x_\\nu}g_{\\nu,j} (\\bar{x}_\\nu) \\bar{\\lambda}_{\\nu,j},\n\\end{equation}\nwhere $\\bar{\\xi}_i=\\lim\\limits_{k\\rightarrow\\infty}\\xi_i^k=\\lim\\limits_{k\\rightarrow\\infty}\\tilde{\\phi}'_{\\varepsilon_k}(z_k^i)$, note that $\\bar{\\xi}^i\\in\\{-1,0,1\\}$.\nIf we choose  the multipliers to the complementarity constraints to be:\n\\begin{equation*}\n\\bar{\\Gamma}_{1,i}=\\frac{a_i}{2}\\left(1+\\bar{\\xi}_i\\right),\t\\bar{\\Gamma}_{2,i}=\\frac{a_i}{2}\\left(1-\\bar{\\xi}_i\\right),\n\\end{equation*}\nand since $a_i-\\Gamma_{1,i}-\\Gamma_{2,i}=a_i-a_i\/2(1+\\bar{\\xi}_{i})-a_i\/2(1-\\bar{\\xi}_{i})=0$ for all $i=1,\\dots,m_\\nu$ and \\eqref{Eq:KKTproof2}, then the condition in \\eqref{Eq:SstationarityProofa} follows.}\n\n\\textcolor{black}{Feasibility (\\ref{Eq:SstationarityProofb}-\\ref{Eq:SstationarityProofd}) is due to continuity of $g_\\nu$ and the convergent subsequence of the multipliers, which gives us $\\bar{\\lambda}$.\nThe feasibility of the complementarity constraint \\eqref{Eq:SstationarityProofe} is due to choice of $\\tilde{\\phi}_\\varepsilon$, which belongs to a smooth NCP function, c.f. (\\ref{Eq:KKT}-\\ref{Eq:FollowerSolution}).}\n\n\\textcolor{black}{\\noindent It remains to demonstrate (\\ref{Eq:SstationarityProoff}-\\ref{Eq:SstationarityProofh}):}\n\n\\textcolor{black}{In case $G_{1,i}(\\bar{x}_\\nu,\\bar{x}_{-\\nu})> 0$, then by feasibility of $\\bar{x}$ we have $G_{2,i}(\\bar{x}_\\nu,\\bar{x}_{-\\nu})= 0$ and thus  $\\bar{\\xi}_i=-1$  such that $\\Gamma_{1,i}=0$.\nIf otherwise $G_{2,i}(\\bar{x}_\\nu,\\bar{x}_{-\\nu}) > 0$, then by the  feasibility of $\\bar{x}$ it holds that $G_{1,i}(\\bar{x}_\\nu,\\bar{x}_{-\\nu})= 0$ and thus  $\\bar{\\xi}_i=-1$, i.e. $\\Gamma_{2,i}=0$.\nSince both arguments hold for all $i=1,\\dots, m$, this yields (\\ref{Eq:SstationarityProoff}-\\ref{Eq:SstationarityProofg}).}\n\n\n\n\\textcolor{black}{Moreover, both multipliers satisfy $\\bar{\\Gamma}_{1,i}, \\bar{\\Gamma}_{2,i}\\geq 0$ for all $i$ because $a_i>0$ and $|\\bar{\\xi}_i|\\leq 1$; therefore,  \\eqref{Eq:SstationarityProofh} holds.}\n\n\\textcolor{black}{Hence, the strategy $\\bar{x}_\\nu$ is strongly stationary for MPCC$_\\nu$ and since we derived \\eqref{Eq:SstationarityProof} for an arbitrary leader $\\nu$, the proof is complete.}\n\n\\end{proof}\n\\textcolor{black}{In the preceding theorem, it is demonstrated that the sequence of multipliers has accumulation points in presence of a suitable constraint qualification if the primal variables converge.\nWe have seen that the limit of the Nash equilibria of NEP($\\varepsilon$) are in fact strongly stationary to the original MLFG.\nHowever, the following theorem goes even further by demonstrating that a limit of these Nash equilibria is in fact a Nash Equilibrium of the MLFG.}\n\\begin{theorem}[A Nash Equilibrium of the MLFG]\\label{Th:constr}\n\\textcolor{black}{\tLet $(\\varepsilon_k)_k$ be a positive sequence with $\\varepsilon_k\\rightarrow 0$ and let $(x^*(\\varepsilon_k))_k$ be the associated sequence of the unique  Nash equilibria to \\emph{NEP($\\varepsilon_k$)}.}\n\t\n\\textcolor{black}{\tThen any accumulation point of $(x^*(\\varepsilon_k))_k$ for a positive sequence $\\varepsilon_k\\rightarrow 0$ is a Nash equilibrium to \\emph{NEP\n\tand therefore of the MLFG.}\n\\end{theorem}\n\\begin{proof}\n\\textcolor{black}{Recall, since $x^*(\\varepsilon)$ is the unique Nash equilibrium of \\ref{Eq:SmoothNEP} for any $\\varepsilon>0$, it holds by the definition of a Nash equilibrium that for all $\\nu=1,\\dots,N$:\n\\begin{equation*} \\label{Eq:DefNE}\n\t\\theta_\\nu^\\varepsilon(x_\\nu^*(\\varepsilon),x_{-\\nu}^*(\\varepsilon))\\leq \t\\theta_\\nu^\\varepsilon(x_\\nu,x_{-\\nu}^*(\\varepsilon)) \\text{ for all } \tx_\\nu\\in X_\\nu.\n\\end{equation*} \nWe prove that a limit strategy $x^{*}(0)$ is a Nash equilibrium to \\ref{NEP22} by contradiction.\nAssume, $x^{*}(0)$ is not a Nash equilibrium to \\ref{NEP22}, then \\textcolor{black}{there exists} $\\hat{x}\\in X$ such that:\n\\begin{equation*}\n\\theta_\\nu(x_\\nu^*(0),x_{-\\nu}^*(0)) > \t\\theta_\\nu(\\hat{x}_\\nu,x_{-\\nu}^*(0)) \\text{ for a leader } \t\\nu.\n\\end{equation*} \nWe define the distance as\n\\begin{equation}\\label{Eq:ProofDist}\n\t\\varrho= \\theta_\\nu(x_\\nu^*(0),x_{-\\nu}^*(0))- \t\\theta_\\nu(\\hat{x}_\\nu,x_{-\\nu}^*(0))\n\\end{equation}\nRecall that all objectives $\\theta_\\nu^\\varepsilon$ are continuous in the strategies $x_\\nu, x_{-\\nu}$ and the smoothing parameter $\\varepsilon$, in particular in $\\varepsilon=0$.\nThen \\textcolor{black}{there exists} $\\hat{\\varepsilon}$ such that for all $\\varepsilon\\in (0,\\hat{\\varepsilon})$,   the following relations hold for any $\\nu$:\n\\begin{subequations}\\label{Eq:ContinuityProperties}\n\\begin{align}\n\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(\\varepsilon),x_{-\\nu}^*(\\varepsilon))  - \t\\theta^\\varepsilon_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(\\varepsilon))  \\right| & \\leq \\frac{\\varrho}{6} &\\text{(continuity in $x_\\nu$)}\\label{Eq:ContinuityProperties1}\\\\\n\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(0),x_{-\\nu}^*(\\varepsilon))  - \t\\theta^\\varepsilon_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(0))  \\right| & \\leq \\frac{\\varrho}{6}&\\text{(continuity in $x_{-\\nu}$)}\\label{Eq:ContinuityProperties2}\\\\\n\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(0),x_{-\\nu}^*(0))  - \t\\theta_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(0))  \\right| & \\leq \\frac{\\varrho}{6}&\\text{(continuity in $\\varepsilon$)}\\label{Eq:ContinuityProperties3}\\\\\n\t\\left|\\theta^\\varepsilon_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  - \t\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  \\right| & \\leq \\frac{\\varrho}{6}&\\text{(continuity in $\\varepsilon$)}\\label{Eq:ContinuityProperties4}\\\\\n\t\\left|\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  - \t\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(0))  \\right| & \\leq \\frac{\\varrho}{6}&\\text{(continuity in $x_{-\\nu}$)}\\label{Eq:ContinuityProperties5}\n\\end{align}\n\\end{subequations}\nIt follows with (\\ref{Eq:ContinuityProperties1}-\\ref{Eq:ContinuityProperties3}) and triangle inequality, that\n\\begin{align*}\n\t\\theta_\\nu^\\varepsilon(x^*_\\nu(\\varepsilon),x^*_{-\\nu}(\\varepsilon)) &\\geq&\t\\theta_\\nu(x^*_\\nu(0),x^*_{-\\nu}(0))  ~&-& \\left| \t\\theta_\\nu^\\varepsilon(x^*_\\nu(\\varepsilon),x^*_{-\\nu}(\\varepsilon)) -\\theta_\\nu(x^*_\\nu(0),x^*_{-\\nu}(0))  \\right|&\\\\\n&\\geq& \\theta_\\nu(x^*_\\nu(0),x^*_{-\\nu}(0)) ~&-&\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(\\varepsilon),x_{-\\nu}^*(\\varepsilon))  - \t\\theta^\\varepsilon_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(\\varepsilon))  \\right|&\\\\\n& & ~&-&\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(0),x_{-\\nu}^*(\\varepsilon))  - \t\\theta^\\varepsilon_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(0))  \\right|& \\\\\n& & ~&-& \t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(0),x_{-\\nu}^*(0))  - \t\\theta_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(0))  \\right|& ,\n\\end{align*}\nand similar with (\\ref{Eq:ContinuityProperties4}-\\ref{Eq:ContinuityProperties5}), we have\n\\begin{align*}\n\t\\theta_\\nu^\\varepsilon(\\hat{x}_\\nu,x^*_{-\\nu}(\\varepsilon)) &\\leq& \t\\theta_\\nu(\\hat{x}_\\nu,x^*_{-\\nu}(0)) ~&+& \\left| \t\\theta_\\nu^\\varepsilon(\\hat{x}_\\nu,x^*_{-\\nu}(\\varepsilon))- \\theta_\\nu(\\hat{x}_\\nu,x^*_{-\\nu}(0)) \\right|&,\\\\\n\t&\\leq& \t\\theta_\\nu(\\hat{x}_\\nu,x^*_{-\\nu}(0)) ~&+& \t\\left|\\theta^\\varepsilon_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  - \t\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  \\right|&\\\\\n\t& & ~&+&   \t\\left|\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  - \t\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(0))  \\right|& ,\n\\end{align*}\nSubtracting these inequality expressions yields with \\eqref{Eq:ProofDist} and  (\\ref{Eq:ContinuityProperties1}-\\ref{Eq:ContinuityProperties5})\n\\begin{align*}\n\t\t\\theta_\\nu^\\varepsilon(x^*_\\nu(\\varepsilon),x^*_{-\\nu}(\\varepsilon)) - \\theta_\\nu^\\varepsilon(\\hat{x}_\\nu,x^*_{-\\nu}(\\varepsilon)) \\geq &~ \\theta_\\nu(x^*_\\nu(0),x^*_{-\\nu}(0)) -\\theta_\\nu(\\hat{x}_\\nu,x^*_{-\\nu}(0)) &\\\\\n\t&\t-\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(\\varepsilon),x_{-\\nu}^*(\\varepsilon))  - \t\\theta^\\varepsilon_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(\\varepsilon))  \\right|&\\\\\n\t&\t-\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(0),x_{-\\nu}^*(\\varepsilon))  - \t\\theta^\\varepsilon_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(0))  \\right|& \\\\\n\t&\t-\t\\left|\\theta^\\varepsilon_\\nu(x_\\nu^*(0),x_{-\\nu}^*(0))  - \t\\theta_\\nu(x_{\\nu}^*(0),x_{-\\nu}^*(0))  \\right|& \\\\\n\t& - \t\\left|\\theta^\\varepsilon_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  - \t\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  \\right|&\\\\\n\t\t& -  \t\\left|\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(\\varepsilon))  - \t\\theta_\\nu(\\hat{x}_{\\nu},x_{-\\nu}^*(0))  \\right|,&\\\\\n\t\t\\geq & ~\\varrho -\\frac{5}{6}\\varrho=\\frac{\\varrho}{6}>0, &\n\\end{align*}\nwhich contradicts the assumption that to $x^*(\\varepsilon)$ is the Nash equilibrium of \\ref{Eq:SmoothNEP}.\nTherefore the assumption that $x^*(0)$ is not a Nash equilibrium to \\ref{NEP22}, which completes the proof.}\n\\end{proof}\n\\textcolor{black}{We remark, that this can also be  understood as a constructive existence proof of Nash equilibria of \\eqref{NEP22} if $X_\\nu$ is compact for all $\\nu=1,\\dots,N$, as alternative to Theorem \\ref{Th:MainExistence}.}\n\n\\textcolor{black}{Note, that we needed to assume convergence of the Nash equilibria.\nThis strong requirement is weakened on the following two corollaries:\n\\begin{corollary}[Compact Strategy Sets]\n\tIf in addition $X_\\nu$ is compact for all $\\nu=1,\\dots, N$, the convergence of $x^*_k$ and $x^*(0)$ is not an assumption, because \\textcolor{black}{there exists} at least one accumulation point.\n\\end{corollary}\n\\begin{corollary}[Accumulation Points are Nash Equilibria]\n\tIf the limit of $x^*_k$ is non unique, every accumulation point of the sequence is a Nash equilibrium of the MLFG.\n\\end{corollary}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Outlook}\nWe presented a quadratic MLFG and explicitly computed the best response of the follower player.\nWith this best response we derived a Nash game formulation where existence theory is available.\nFurthermore, we smoothed the best response function and formulate the MLFG as smooth Nash game and proved existence and uniqueness of the Nash equilibrium for all smoothing parameters.\nWe followed an all KKT approach to characterize the corresponding Nash equilibrium.\nFor decreasing positive smoothing parameter, we showed that the limit of Nash equilibria satisfies  the conditions of \\textcolor{black}{S-stationarity}.\nFurther, we demonstrated that \\textcolor{black}{S-stationary} points are eventually Nash equilibria of the MLFG.\nNumerically, we computed Nash equilibria with a globalized nonsmooth Newton and compare with a standard methods based on subgradients.\nFor efficient computation, we updated the primal variables by a Taylor approximation before a subsequent computation of the Nash equilibrium for a smaller smoothing parameter.\n\n\\textcolor{black}{The numerical comparison to alternative methods designed for MPECs and EPECs are planned as future research.}\nAlso, the extension possibilities discussed in Appendix \\ref{Sec:Ext} are  subject of further investigation.\n\n\n\\section{Introduction and Background }\\label{S:intro}\nThe multi-leader-follower game (MLFG) is a particular class of problems in classical game theory \\textcolor{black}{that represents a generalization} of the Stackelberg game which includes a single leader.\nThese models serve as an analytical tool for studying the  strategic behavior of noncooperative individuals.\nIn particular, the individuals (so-called players)  are divided into two groups, namely  leaders and  followers, according to their position in the game.\nMathematically, this yields a hierarchical Nash game, where further minimization problems appear in the participants' optimization problems as constraints.\nAn equilibrium is then given by a multistrategy vector of all players, where  no player has the incentive to change his chosen strategy unilaterally.\n\nMost recently, such models have increasingly become the focus of interest among mathematicians as well as scientists in other fields such as operations research, robotics, and computer science \\cite{Aussel16,Wang08,Koh10}.\nHowever, compared to the knowledge of other classical game models, little is known   so far  concerning existence and uniqueness theory as well as suitable numerical solution methods.\n\n\\textcolor{black}{The structure of MLFGs is related to equilibrium problem with equilibrium constraints (EPEC) provided that the optimization problems of the followers can be replaced by the corresponding optimality conditions.}\nThe admissibility of this approach is discussed in \\cite{Aussel2018}.\nRecently,  the competition on the  electric power market is described by EPEC \\cite{Allevi2018,Aussel17part1,Aussel17part2,henrion12,HR07}.\nHere, the power generators and consumers are the leaders who bid their cost and utility functions.\nThe single follower plays the role of an independent system operator coordinating dispatch and minimizing  social costs subject to network constraints.\n\nSo far, there exist only a few recent theoretical results for general MLFGs or EPECs, analyzing the existence, the uniqueness and characterizations of equilibria: \n\\textcolor{black}{\nEarly work by Sherali \\cite{Sherali84} generalizes Stackelberg games in the setting of Cournot competition.\nExistence theory of equilibria for identical leaders and sufficiency conditions for convex follower reaction functions is provided here.\nSu \\cite{Su2007} extended this work to a two-period forward market model and proposes an \\textcolor{black}{SNCP} algorithm in \\cite{SU2005}.\nLater, further generalizations in the Cournot setting include the incorporation of stochasticity by DeMiguel and Xu  and the application to telecommunication industry \\cite{DeMiguel2009}.}\n\n\\textcolor{black}{\nIn \\cite{Kulkarni2014,Kulkarni2015}, Kulkarni and  Shanbhag introduce a more general setting for MLFGs and its reformulation with shared constraints.\nThe relation of Nash equilibria of the MLFG and weaker solution concepts of the reformulation are extensively discussed here, many results rely on potentiality of the players' objectives.\n}\n\n\\textcolor{black}{Fukushima and Pang \\cite{Pang2005} present a reformulation as generalized Nash equilibrium problem (GNEP).\nUnder suitable condtions, the equilibria of their GNEP reformulation can be characterized by a quasi-variational inequality.}\nIn another work, Hu and Ralph \\cite{HR07} \\textcolor{black}{apply a standard result \nfor the existence of pure strategy Nash equilibria in a particular case of an MLFG related to an electricity market model.\nLeyffer and Munson \\cite{LeyfferMunson10} describe various reformulations in terms of  mathematical programs with equilibrium constraints (MPEC), nonlinear programs, and nonlinear complementarity problems.\n\\textcolor{black}{In a very recent paper \\cite{Kim2019}, Kim and Ferris propose a reformulation of MLFG using extended mathematical programming which allows the usage of the complementarity problem solver PATH \\cite{Dirkse1995,Ferris1999}.} \n\nMore recent work has been done by Hu and Fukushima in \\cite{HuFu2013}.\nTherein, they discuss the existence of robust Nash equilibria of a class of quadratic MLFGs.\nFurther, they propose a uniqueness result for an MLFG with two leaders.\n\nIn this paper, we study a quadratic MLFG with  similarities to the model studied in \\cite{HuFu2013} and generalize the follower's strategy set by allowing inequality constraints.\nThis modification translates  into equilibrium conditions or to a nonsmooth Nash game formulation.\nThe existence of Nash equilibria can be proven with suitable convexity assumptions.\nFurthermore, we propose a smoothed Nash game formulation and prove uniqueness of Nash equilibria for an arbitrary number of players.\nThe main result of this article is that it can be shown that these smooth problems are indeed approximating Nash equilibria of the MLFG.\nBesides the theoretical results, we propose an algorithm to numerically compute Nash equilibria.\nTherein, we combine an update based on a Taylor expansion of the parameter-dependent solution and  the computation  of Nash equilibria of approximating problems.\n\n \nThis article is structured as follows.\nIn Section \\ref{Sec:2}, we introduce the quadratic MLFG and develop an equivalent nonsmooth Nash equilibrium problem (NEP) for which we can prove existence of equilibria.\nIn Section \\ref{Sec:3}, smoothing of the best response of the follower leads to a differentiable NEP formulation, which allows to characterize solutions by KKT conditions.\nIn Section \\ref{Sec:4}, it is demonstrated that \\textcolor{black}{the sequence} of Nash equilibria of the smooth problems converges also in the multipliers for decreasing smoothing parameters and that \\textcolor{black}{S-stationarity} conditions are satisfied in the limit.\nEventually, this \\textcolor{black}{S-stationary} point are proven to be Nash equilibria to the MLFG.\nIn Section \\ref{Sec:5}, we introduce our general approach to computing Nash equilibria.\nIn Section \\ref{Sec:6}, we present the numerical results of the proposed methods.\n\n\\section*{Acknowledgements}\nSpecial thanks to Michael Ferris and Olivier Huber for fruitful discussions on model extensions.\nThis work was supported by the DFG under Grant STE2063\/2-1. \n\n\\bibliographystyle{abbrvnat}\n\n\\section{Existence of Nash Equilibria for MLFGs}\\label{Sec:2}\n\nWe consider an MLFG, where the follower's game is modeled by the optimization problem\n\\begin{equation}\\label{follower}\n\\min\\limits_{y\\in \\R^m} \\frac{1}{2}y^\\top Q_yy-b(x)^\\top y \\quad \\st\\quad y\\geq l(x),\n\\end{equation}\n\\textcolor{black}{where $b_i,l_i:~\\R^n\\rightarrow\\R$ are differentiable functions for $i=1,\\dots,m$.\nThe matrix $Q_y\\in\\R^{m\\times m}$ is  positive definite and diagonal, which  guarantees the existence of an explicit characterization of the minimizer, c.f. Lemma \\ref{Th:bestresponse}.}\nThe leader problems are given for $\\nu=1,\\dots,N$ by\n\\begin{equation}\\label{leader}\n\\min\\limits_{x_\\nu\\in\\R^{n_\\nu}} \\theta_\\nu(x_\\nu,x_{-\\nu})=\\frac{1}{2} x_\\nu^\\top Q_\\nu x_{\\nu}+c_\\nu^\\top x_\\nu+a^\\top \\textcolor{black}{y(x)} \\quad\\st\\quad x_\\nu\\in X_\\nu,\n\\end{equation}\nwith nonempty, convex, and closed strategy sets $X_\\nu$ which we assume to be described by smooth functions $g_\\nu:\\R^{n_\\nu}\\rightarrow\\R^{m_\\nu}$ such that $X_\\nu=\\left\\{x_\\nu\\in\\R^{n_\\nu} | g_\\nu(x_\\nu)\\leq0 \\right\\}$.\nThe quadratic objective $\\theta_\\nu$  is strictly convex  with $Q_\\nu\\in\\R^{n_\\nu\\times n_\\nu}$ symmetric positive definite, $c_\\nu\\in\\R^{n_\\nu}$, and $a\\in\\R^m_+$.\nThe multistrategy vector of all players is denoted  by $x=(x_\\nu)_{\\nu=1}^N\\in\\R^n$ and  the \\textcolor{black}{rival's} strategies to player $\\nu$ by $x_{-\\nu}=(x_i)_{i=1,i\\neq\\nu}^N\\in\\R^{n-n_\\nu}$.\n\\textcolor{black}{The follower's strategy \\textcolor{black}{$y=y(x)$} in \\eqref{leader} is a best response to the leader strategies and couples implicitly to the leaders' problems.\n}\n\nFor simple notation, the model is introduced as a Multi-Leader-(Single-)Follower game.\nHowever, this can be extended to multiple followers:\n\\begin{remark}(Multiple Followers)\n\tFor $j=1,\\dots,N_F$ let the $j$--th follower's optimization problem be\n\t\\begin{equation*}\n\t\\min\\limits_{y_j\\in \\R^{m_j}} \\frac{1}{2}y_j^\\top Q^j_yy_j-b_j(x)^\\top \\textcolor{black}{y_j(x)} \\quad \\mathrm{s.t.}\\quad y_j\\geq l_j(x),\n\t\\end{equation*}\n\t with a positive definite diagonal matrix $Q^j_y\\in\\R^{m_j\\times m_j}$   and $b_j,l_j:~\\R^n\\rightarrow\\R^{m_j}$ \\textcolor{black}{are} convex in every component and differentiable.\n\t\\textcolor{black}{This structure is called potential game} and it can be equivalently reformulated as a single optimization problem by summing the objectives and concatenating the constraints, c.f. \\cite{MONDERER1996}.\n\\end{remark}\n\n\\textcolor{black}{No matter how many followers are modeled, our approach relies on the property that the follower level can be replaced by its optimal conditions.\nIn addition, we need an explicit response function of the followers, which is given here in particular by the convexity properties of the follower problem and the diagonality of $Q_y$.}\nSince the follower's problem has a strictly convex objective and a convex strategy set, \nwe state its unique solution in the following lemma:\n\\begin{lemma}[Follower's Best Response]\\label{Th:bestresponse}\n\tThe follower's optimization problem \\eqref{follower} has a unique solution $y^*$ for any given leader strategy vector $x\\in\\R^n$.\n\tIn particular, this best response function is\n\t\\begin{equation}\\label{solY2}\n\t\t\ty^*(x)=\\max\\left\\{Q_y^{-1}b(x),l(x)\\right\\}.\n\t\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\n\tThe objective and the feasible set are convex for any $x\\in\\R^n$, \\textcolor{black}{in particular, the constraints are linear.}\n\tTherefore, the KKT conditions are necessary and sufficient for a global minimizer.\n\tSince the objective is strictly convex, the minimizer is unique for all leader strategies $x$.\n\t\n\tTo derive the structure, we apply the KKT conditions:\n\tThere exist Lagrange multipliers $\\lambda\\in\\R^m$ such that\n\t\\begin{align*}\n\t\t0&=Q_yy-b(x)-\\lambda,\\\\\n\t\t0&\\leq \\lambda~\\bot~y-l(x)\\geq 0.\n\t\\end{align*}\n\tCombining these expressions yields the complementarity expression\n\t$ 0\\leq \\lambda=Q_yy-b(x) ~\\bot~ y-l(x)\\geq0$.\n\tWe assumed $Q_y$ to be positive definite and diagonal; therefore, we can write equivalently as\n$ 0\\leq y-Q^{-1}_yb(x) ~\\bot~ y-l(x)\\geq0$ and\n\t$0=\\min\\left\\{ y-Q^{-1}_yb(x) ,y-l(x) \\right\\}$.\n\tWe obtain \\eqref{solY2} by extracting $y$ and changing $\\min$ to $\\max$.\n\\end{proof}\n\\noindent We remark that the $\\min$ and $\\max$ operator are applied  componentwise to a vector.\nIf the data $b,l$ are chosen to be componentwise convex, then also the components of best response function $y^*(x)$ are  \\textcolor{black}{also} convex in $x$ as a maximum of  convex functions.\\\\\n  \nThe MLFG  is formulated as a Nash game by plugging the best response \\eqref{solY2} in the leader game \\eqref{leader} for $\\nu=1,\\dots,N$, yielding\n\\begin{equation}\\label{NEP22}\n\\min\\limits_{x_\\nu} \\frac{1}{2} x_\\nu^\\top Q_\\nu x_{\\nu}+c_\\nu^\\top x_\\nu+\\sum\\limits_{i=1}^{m}a_i \\max\\left\\{\\left(Q_y^{-1}b(x)\\right)_i,l_i(x)\\right\\} \\quad\\st\\quad x_\\nu\\in X_\\nu.\\tag{NEP}\n\\end{equation}\nEach optimization problem has a nonsmooth but convex objective \\textcolor{black}{(if $b_i,l_i$ are convex)} and a convex strategy set.\nFor compact strategy sets, we can prove the existence of Nash equilibria in the following theorem.\n\\begin{theorem}[Existence of Nash Equilibria for Compact Strategy Sets]\\label{Th:MainExistence}\n\tAssume that the nonsmooth Nash equilibrium problem in \\eqref{NEP22} has a  convex and compact joint strategy set $X=X_1\\times \\dots\\times X_N$, where all $X_\\nu$ are nonempty.\n\\textcolor{black}{\tFurther, assume that $b,l$ are componentwise convex functions on $X$.}\n\tThen there exists at least one Nash equilibrium.\n\t\n\tTherefore, the quadratic multi-leader-follower game given by (\\ref{follower}) and (\\ref{leader}) has at least one Nash equilibrium.\n\\end{theorem}\n\\begin{proof}\n\tWe formulated the MLFG as a convex Nash equilibrium problem (\\ref{NEP22}), especially the objectives are continuous in $(x_\\nu,x_{-\\nu})$ and convex in $x_\\nu$ because they are a sum of a strictly convex quadratic term and the maximum of two convex functions.\n\tFurthermore, we assumed  the admissible strategy sets $X_\\nu$ to be nonempty, convex, and compact. \n\tTherefore, the conditions of \\cite[Theorem 3.1]{nikaido1955note} are fulfilled.\n\n\\end{proof}\nAlternatively, Theorem \\ref{Th:constr} includes a constructive \\textcolor{black}{proof of existence}.\n\\textcolor{black}{In addition to convexity, both existence results rely on compactness of the strategy sets to ensure existence of global minimizer to the  players' optimization problems.}\nHowever, this is clearly not a necessary condition for the existence or uniqueness of Nash equilibria to (\\ref{follower},\\ref{leader}).\n\nIn the remainder of the section, we exchange the compactness assumption with \\textcolor{black}{a} coercivity assumption on the best response of the followers exploiting the potentiality of the \\textcolor{black}{leaders'} objectives.\nTherefore, let the generalized potential function be\n\\begin{align*}\n\t\\Theta(x)&=\\sum\\limits_{\\nu=1}^{N}\\left[\\frac{1}{2} x_\\nu^\\top Q_\\nu x_{\\nu}+c_\\nu^\\top x_\\nu\\right] +\\varphi(x),\\\\\n\t\\varphi(x)&=\\sum\\limits_{i=1}^{m}a_i \\max\\left\\{\\left(Q_y^{-1}b(x)\\right)_i,l_i(x)\\right\\}.\n\\end{align*}\nWith this, we can formulate the major assumption:\n\\begin{assumption}\\label{Ass:coerc}\n\tAssume, \\textcolor{black}{there exists} $\\rho\\geq 0$ \\textcolor{black}{and  $\\omega_1,\\omega_2\\in\\R$} such that for all $x\\in X$ with $\\|x\\|>\\rho$  it holds\n\t$$\\textcolor{black}{\\min\\left\\{0,\\varphi(x)\\right\\}}\\geq \\omega_1 \\|x\\| +\\omega_2.$$\n\\end{assumption}\n\\textcolor{black}{This assumption guarantees that the coercivity of  generalized potential $\\Theta$ is preserved.\nIt includes linear functions $\\varphi(x)$ therefore especially settings where $b$ and $l$ are linear.\nIt excludes polynomials of higher degree which are not bounded from below.} \n\n\\textcolor{black}{\n\\begin{lemma}[Coercivity of the Generalized Potential $\\Theta$]\nLet Assumption \\ref{Ass:coerc} hold, then the generalized potential function $\\Theta$ is coercive.\n\\end{lemma}\n\\begin{proof}\nWe introduce a short hand for the block diagonal matrix $Q=\\mathrm{diag}(Q_1,\\dots,Q_N)$ and the concatenation of $c_1,\\dots,c_N$ as $c$. \nThen we have\n\\begin{align*}\n  \\Theta(x)=x^\\top Q x+c^\\top x +\\varphi(x)\\geq \\mu \\|x\\|^2 -\\|c\\|\\,\\|x\\|+\\min\\left\\{0,\\varphi(x)\\right\\},\n\\end{align*}\nwhere $\\mu$ is the smallest eigenvalue of $Q$, recall that $\\mu>0$ since all $Q_\\nu$ are positive definite.\nWe rewrite the the right hand side with Assumption \\ref{Ass:coerc} and get\n\\begin{align*}\n\\Theta(x)\\geq \\mu \\|x\\|^2 -\\|c\\|\\,\\|x\\|+\\omega_1\\|x\\|+\\omega_2,\n\\end{align*}\nAs $\\|x\\|\\rightarrow\\infty$, the right hand side is dominated by a quadratic term with $\\mu \\|x\\|^2\\rightarrow\\infty$ and we conclude\n$$\\lim\\limits_{\\|x\\|\\rightarrow\\infty}\\Theta(x)=\\infty,$$\ni.e. the generalized potential function is coercive.\n\\end{proof}}\nThe Assumption \\ref{Ass:coerc} and the concluded coercivity property allows us to formulate an existence result without requiring compactness, \\textcolor{black}{similarly to the standard result of Monderer and Shapley in \\cite{MONDERER1996}}.\n\n\\begin{theorem}[Existence  of Nash Equilibria for Coercive Generalized Potential]\\label{Th:coerc}\n\tLet  Assumption \\ref{Ass:coerc} hold and   $x^*$ be a global minimizer of the generalized potential $\\Theta$, then $x^*$ is also a Nash equilibrium of the MLFG (\\ref{follower}-\\ref{leader}).\n\\end{theorem}\n\\begin{proof}\n\tSince  Assumption \\ref{Ass:coerc} holds, the generalized potential $\\Theta$ admits a global minimizer on any closed joint strategy set $X$.\n\tWe demonstrate that $x^*$ is a Nash equilibrium by contradiction.\n\tTherefore, assume the converse is true, i.e. \\textcolor{black}{there exists} a leader $\\hat{\\nu}$ and a strategy $\\hat{x}_{\\hat{\\nu}}\\in X_{\\hat{\\nu}}$ such that:\n\t$$\\theta_{\\hat{\\nu}}(\\hat{x}_{\\hat{\\nu}},x^*_{-\\hat{\\nu}})< \\theta_{\\hat{\\nu}}(x^*_{\\hat{\\nu}},x^*_{-\\hat{\\nu}}),$$\nor more detailed,\n$$ \\frac{1}{2}\\hat{x}_{\\hat{\\nu}}^\\top Q_{\\hat{\\nu}}\\hat{x}_{\\hat{\\nu}}+c_{\\hat{\\nu}}^\\top \\hat{x}_{\\hat{\\nu}}+\\varphi(\\hat{x}_{\\hat{\\nu}},{x}^*_{-\\hat{\\nu}})<\\frac{1}{2}x^{*\\top}_{\\hat{\\nu}} Q_{\\hat{\\nu}}{x}^*_{\\hat{\\nu}}+c_{\\hat{\\nu}}^\\top {x}^*_{\\hat{\\nu}}+\\varphi({x}^*_{\\hat{\\nu}},{x}^*_{-\\hat{\\nu}}).\n$$\nWith this, we can overestimate the following expression\n\\begin{align*}\n\t\\Theta(\\hat{x}_{\\hat{\\nu}},{x}^*_{-\\hat{\\nu}})=& \\sum\\limits_{\\nu=1,\\nu\\neq \\hat{\\nu}}^{N}\\left[\\frac{1}{2} x^{*\\top}_\\nu Q_\\nu x^*_{\\nu}+c_\\nu^\\top x^*_\\nu\\right] + \\frac{1}{2}\\hat{x}_{\\hat{\\nu}}^\\top Q_{\\hat{\\nu}}\\hat{x}_{\\hat{\\nu}}+c_{\\hat{\\nu}}^\\top \\hat{x}_{\\hat{\\nu}}+\\varphi(\\hat{x}_{\\hat{\\nu}},{x}^*_{-\\hat{\\nu}})\\\\\n\t<&  \\sum\\limits_{\\nu=1}^{N}\\left[\\frac{1}{2} x^{*\\top}_\\nu Q_\\nu x^*_{\\nu}+c_\\nu^\\top x^*_\\nu\\right] + \\varphi({x}^*_{\\hat{\\nu}},{x}^*_{-\\hat{\\nu}})=\\Theta({x}^*_{\\hat{\\nu}},{x}^*_{-\\hat{\\nu}}),\n\\end{align*}\nwhich is a \\textcolor{black}{contradiction to} the optimality of $x^*$.\n\\end{proof}\nIn the remainder of this article, we assume  $b$ and $l$ to be linear:\n\\begin{assumption}\\label{Ass:Lin}\n\t\\textcolor{black}{The data is assumed to be linear:\n\t\t\\begin{equation*}\n\t\tb(x)=B^\\top x \\quad\\mbox{and}\\quad l(x)=L^\\top x,\n\t\t\\end{equation*}\n\t\twhere $B,L\\in\\R^{n\\times m}$,  $B_{:,i},L_{:,i}\\in\\R^{n}$ denotes the $i$-th column, and $B_{\\nu,:},L_{\\nu,:}$ denote their submatrices of the rows referring to $x_\\nu$.}\n\\end{assumption}\n\\textcolor{black}{This linearity combined with Assumption \\ref{Ass:coerc} directly yields existence of Nash equilibria.\n\\begin{lemma}[Existence  of Nash Equilibria for Linear Data $b,l$]\n\tIf \\textcolor{black}{Assumptions} \\ref{Ass:coerc}  and \\ref{Ass:Lin} hold,  then \\textcolor{black}{there exists} a Nash equilibrium to the MLFG.\n\t\\end{lemma}\n\\begin{proof}\n\tWith linear data $b,l$, $\\varphi$ gets: \n\t$$\\varphi(x)=\\sum\\limits_{i=1}^{m}a_i \\max\\left\\{\\left(Q_y^{-1}B^\\top x\\right)_i,\\left(L^\\top x\\right)_i\\right\\},$$\n\twhich is a linear combination of the maxima of two linear functions.\n\tThus, it can be underestimated by an other linear function.\n\tTherefore, Assumption \\ref{Ass:coerc} holds and Theorem \\ref{Th:coerc} applies.\n\\end{proof}\n}\n\\textcolor{black}{In \\cite{Steffensen2019P} there is a counterexample for the structure of  MLFG in (\\ref{follower},\\ref{leader}) with linear data $b,l$.\nThe data and $X$ are such that the non-differentiable parts of the objectives of the NEP reformulation lie either on the boundary of $X$ or outside the feasible domain.\nOr with other words, the $\\max$ operator can be uniquely evaluated  on the interior of the feasible set.\nIn particular, existence and uniqueness of Nash equilibria is derived by a formulation as variational inequality.\nIn case of diagonal $Q_\\nu$, the Nash equilibrium can be explicitly computed in the presented setting.\n}\n\n\\section{Numerical Results}\\label{Sec:6}\nIn the previous sections, we proposed an algorithm with gradient updates of the primal variables.\nThis included the computation of Nash equilibria for a sequence of smoothing parameter $(\\varepsilon_i)_{i\\in \\mathbb{N}}$.\nFor this computation, we introduced a subgradient and a Newton method.\nThe presented numerical results are obtained for the data sets in \\ref{App:Data} which are adapted from \\cite{HuFu2013}.\nAll plots are generated for the Data Set 1, however experiments with Data Set 2 produced similar graphics.\n\n\n\nThe naive approach of computing a sequence of Nash equilibria is to use the Nash equilibrium of a larger smoothing parameter as initial for the subsequent computation with the smaller smoothing parameter.\nThe main purpose of the outer Taylor expansion based update in Algorithm \\ref{Alg:Outer} (Step \\ref{Alg:General5} and \\ref{Alg:General6}) is to improve the quality of the initials in order to reduce the computational effort in Step \\ref{Alg:General3}.\n\nIn the upper left part of Figure \\ref{Fig:Fig1}, we observe the quadratic decent of the error for decreasing smoothing parameter.\nIn the upper right part, the Taylor update is exemplary illustrated for one component of the leader variables.\nThe blue dots indicate each the Nash equilibrium of a NEP$(\\varepsilon_i)$, $x^*(\\varepsilon_i)$.\nA black line represents the Taylor update and the lower end of a black line indicates the updated initial values $x^0(\\varepsilon_{i+1})$ for the subsequent Nash equilibrium computation.\n\nThe lower part of Figure \\ref{Fig:Fig1} is dedicated to illustrate the importance of large smoothing parameter for the first computations of Nash equilibria.\nSince the problem gets closer to it original nonsmooth formulation as $\\varepsilon$ decreases, the problem is also more challenging to solve for both Subgradient and Nonsmooth Newton  method.\nWe observe this expected behavior, in particular if we compare the number of iterations for $\\varepsilon=1.6$ and $\\varepsilon=0.1$.\n\n\n\\begin{figure}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\textwidth]{OuterIteration1.eps} \\\\\n\t\t\t\\includegraphics[width=\\textwidth]{OuterIteration4.eps} \n\t\t\\end{center}\n\t\\end{minipage}\\begin{minipage}{0.5\\textwidth}\n\t\t\t\\begin{center}\n\t\t\\includegraphics[width=\\textwidth]{OuterIteration2.eps} \\\\\n\t\t\\includegraphics[width=\\textwidth]{OuterIteration3.eps} \n\t\\end{center}\n\\end{minipage}\n\t\\caption{\\label{Fig:Fig1}Upper left: Quadratic convergence to the limit Nash equilibrium $x^*(0)$, right: Taylor expansion based update on primal variables, exemplary for first leader variable; Lower: comparison of Subgradient and Nonsmooth Newton method for varying smoothing parameter. }\n\\end{figure}\n\nAs already seen in Figure \\ref{Fig:Fig1}, the subgradient based method suffers from characteristically slow convergence for our instances.\nIn Figure \\ref{Fig:Fig2}, all iterations for a sequence of decreasing smoothing parameter are shown.\n\n\\begin{figure}[H]\n\\begin{center}\n\t\\includegraphics[width=0.5\\textwidth]{SubgradientConvergencePlot.eps} \n\\end{center}\n\n\t\\caption{\\label{Fig:Fig2}Subgradient Method; all iterations for decreasing sequence of smoothing parameters.}\n\\end{figure}\n\nSimilarly to Figure \\ref{Fig:Fig2},  left in Figure \\ref{Fig:Fig3}, all iterations for a sequence of decreasing smoothing parameter are shown for the Nonsmooth Newton.\nThe alternating behavior is due to the decreasing parameter changing the minimization problem.\nThe right part of Figure  \\ref{Fig:Fig3} illustrates the decrease in $\\Psi_\\varepsilon$ for different random initial values but fixed smoothing parameter.\n\n\\begin{figure}[H]\n\t\\label{Fig:Fig3}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\textwidth]{ConvergenceNewton.eps} \n\t\t\\end{center}\n\t\\end{minipage}\\begin{minipage}{0.5\\textwidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\textwidth]{NewtonConvergencePlot.eps} \n\t\t\\end{center}\n\t\\end{minipage}\n\t\n\t\\caption{Nonsmooth Newton Method; left all iterations for decreasing sequence of smoothing parameters, right for one smoothing parameter and multiple initials.}\n\\end{figure}\n\n\n\\section{Existence and Uniqueness of Nash Equilibria of Smoothed MLFG} \\label{Sec:3}\nWe formulated the MLFG as a convex but nonsmooth Nash game and proved existence of  equilibria in case of compact strategy sets.\nIn this section,  we  relax the nonsmoothness of the follower's best response.\nFor the resulting smooth convex Nash equilibrium problem, we show existence and uniqueness for more general strategy sets.\n\nSimilarly to Lemma \\ref{Th:bestresponse}, we formulate the follower's KKT conditions with the linear data:\n\\begin{equation}\\label{Eq:KKT}\n0\\leq y-Q^{-1}_yB^\\top x ~\\bot~ y-L^\\top x\\geq0,\n\\end{equation}\nwhere we replace the complementarity expression by a formulation with a nonlinear complementarity  (NCP) function.\nWe consider smooth NCP functions of the following type with smoothing parameter $\\varepsilon>0$:\n\\begin{equation}\\label{NCPsmooth}\n\\phi_\\varepsilon(\\alpha,\\beta)=\\alpha+\\beta-\\tilde{\\phi_\\varepsilon}(\\alpha-\\beta).\n\\end{equation}\n\\textcolor{black}{An example for a smooth NCP function with convex $\\tilde{\\phi}_\\varepsilon$ is  $\\phi_\\varepsilon^{p}(\\alpha,\\beta)=\\alpha+\\beta-\\sqrt[p]{(\\alpha-\\beta)^p+(2\\varepsilon)^p}$ for an even number $p\\in\\mathbb{N}$, this coincides for $p=2$ with the smooth minimum function.\nThis structure \\textcolor{black}{captures also} NCP functions related to the relaxation scheme for mathematical programs with equilibrium constraints (MPEC) introduced by Steffensen and Ulbrich \\cite{Sonja10}.\nBesides \\textcolor{black}{differentiability}  in $\\alpha$ and $\\beta$, we require the smooth NCP function $\\phi_\\varepsilon$ to be continuous in the smoothing parameter $\\varepsilon$.}\n\nIf we apply (\\ref{NCPsmooth}) on the KKT system \\eqref{Eq:KKT}, we obtain (component wise) for $\\varepsilon>0$ the expression\n\\begin{align}\n0&= y - Q^{-1}_yB^\\top x + y-L^\\top x -\\tilde{\\phi_\\varepsilon}\\left(y-Q^{-1}_yB^\\top x - y+L^\\top x\\right).\n\\end{align}\nTherefore, we write the best response function as:\n\\begin{equation}\\label{Eq:FollowerSolution}\ny_\\varepsilon(x)=\\frac{1}{2}\\left[\\left(L^\\top +Q^{-1}_yB^\\top\\right) x +\\tilde{\\phi_\\varepsilon}\\left(\\left(L^\\top-Q^{-1}_yB^\\top\\right) x\\right)\\right].\n\\end{equation}\n\n\n\nThe smoothed best response function $y_\\varepsilon$ plugged  into the leader's objectives yields  a smooth Nash equilibrium problem with positive smoothing parameter $\\varepsilon$; for $\\nu=1,\\dots, N$, we have\n\\begin{align}\\label{Eq:SmoothNEP} \\tag{\\mbox{NEP$(\\varepsilon) $}}\n\\begin{split}\n\\min\\limits_{x_\\nu\\in\\R^{n_\\nu}}  \\theta^\\varepsilon_\\nu(x_\\nu,x_{-\\nu})=&\\frac{1}{2} x_\\nu^\\top Q_\\nu x_{\\nu}+c_\\nu^\\top x_\\nu\\\\ &+ \\frac{1}{2}\\sum\\limits_{i=1}^m a_i \\left[\\left(L^\\top +Q^{-1}_yB^\\top\\right) x +\\tilde{\\phi_\\varepsilon}\\left(\\left(L^\\top -Q^{-1}_yB^\\top\\right) x\\right)\\right]_i \\\\\n\\st ~ x_\\nu\\in X_\\nu. \\quad\\quad&\n\\end{split}\n\\end{align}\nFor this game, we can state an existence and uniqueness theorem.\n\n\n\\begin{theorem}[Existence and Uniqueness]\n\tAssume that the  Nash equilibrium problem  \\ref{Eq:SmoothNEP} has a  convex and closed strategy set $X=X_1\\times\\dots \\times X_N$, where all $X_\\nu$ are nonempty, and $\\tilde{\\phi_\\varepsilon}$ is convex and smooth. \n\tThen the  Nash equilibrium problem has a unique  equilibrium for every smoothing parameter $\\varepsilon>0$.\n\\end{theorem}\n\\begin{proof}\n\tDue to \\cite[Proposition 1.4.2]{facchinei2007finite} and the convexity assumptions, a strategy $x\\in X$ is a Nash equilibrium if and only if $x$ solves the variational inequality $\\mathrm{VI}(X,{\\theta'}^\\varepsilon)$, where\n\t$${\\theta'}^\\varepsilon(x)=\\begin{pmatrix} \\nabla_{x_1} \\theta^\\varepsilon_1 (x_1,x_{-1})\\\\ \\vdots \\\\\\nabla_{x_N} \\theta^\\varepsilon_N (x_N,x_{-N})\n\t\\end{pmatrix}. $$\n\tSince we assumed a convex and closed strategy set $X$, the variational inequality has a unique solution if ${\\theta'}^\\varepsilon$ is uniformly monotone \\cite[Theorem 2.3.3]{facchinei2007finite}.\n\tThe remainder of the proof demonstrates this.\n\t\n\tWe introduce a short hand for the linear term $A=L^\\top -Q_y^{-1}B^\\top $, the block diagonal matrix $Q=\\mathrm{diag}(Q_1,\\dots,Q_N)$, and the concatenation of $c_1,\\dots,c_N$ as $c$. \n\tLet $x,\\hat{x}\\in X$ be distinct, then we have\n\t\\begin{align*}\n\t\t&(x-\\hat{x})^\\top \\left({\\theta'}^\\varepsilon(x)-{\\theta'}^\\varepsilon(\\hat{x})\\right), \\\\\n\t\t=&(x-\\hat{x})^\\top  \\Bigg[Qx+c + \\frac{1}{2}(L^\\top +Q_y^{-1}B^\\top )^\\top a+\\frac{1}{2}\\sum\\limits_{i=1}^m a_i A_{i,:}^\\top  \\tilde{\\phi}_\\varepsilon'((Ax)_i)  \\\\\n\t\t &   -\\left(Q\\hat{x} +c+\\frac{1}{2}(L^\\top +Q_y^{-1}B^\\top )^\\top a+\\frac{1}{2}\\sum\\limits_{i=1}^m a_i A_{i,:}^\\top  \\tilde{\\phi}_\\varepsilon'((A\\hat{x})_i)\\right)\\Bigg], \\\\\n\t\t  = & (x-\\hat{x})^\\top   \\left[ Q(x-\\hat{x}) + \\frac{1}{2}\\sum\\limits_{i=1}^m a_i A_{i,:}^\\top  \\left(\\tilde{\\phi}_\\varepsilon'((Ax)_i)-\\tilde{\\phi}_\\varepsilon'((A\\hat{x})_i)\\right) \\right].\n\t\\end{align*}\n\tWe apply the mean value theorem for $\\tilde{\\phi}_\\varepsilon'$, thus there exists a $t\\in[0,1]$ such that we derive from the last equation with $\\zeta_i=t(Ax)_i+(1-t)(A\\hat{x})_i$ a lower \\textcolor{black}{bound}:\n\t\\begin{align*}\n\t\t&(x-\\hat{x})^\\top \\left({\\theta'}^\\varepsilon(x)-{\\theta'}^\\varepsilon(\\hat{x})\\right),\\\\\n\t\t =&  (x-\\hat{x})^\\top    Q(x-\\hat{x}) + \\frac{1}{2}\\sum\\limits_{i=1}^m a_i  \\underbrace{\\tilde{\\phi}_\\varepsilon''(\\zeta_i)}_{\\geq 0} \\underbrace{(x-\\hat{x})^\\top  A_{i,:}^\\top  A_{i,:} (x-\\hat{x})}_{\\geq 0}, \\\\\n\t\t\\geq\n\t\t& (x-\\hat{x})^\\top    Q(x-\\hat{x}) \\geq \\mu\\|x-\\hat{x}\\|^2,\n\t\\end{align*}\n\twhere $\\mu$ is the  smallest eigenvalue of  $Q$. It is positive because $Q$ is symmetric positive definite.\nThus $ {\\theta'}^\\varepsilon$ is uniformly monotone.\n\tTherefore, the variational inequality $\\mathrm{VI}(X,{\\theta'}^\\varepsilon)$ has exactly one solution which is the unique Nash equilibrium of the smoothed game  \\ref{Eq:SmoothNEP}.\n\t\\end{proof}\n\n\n\\textcolor{black}{The remainder of this section is dedicated to the characterization of the unique Nash equilibrium.}\nTherefore, we recall the assumptions made for the  smooth Nash equilibrium problem \\ref{Eq:SmoothNEP} for positive smoothing parameter $\\varepsilon$:\n\\begin{assumption}\\label{Assump:SNEP}\n\tWe assume that the data of the MLFG and its smooth Nash game reformulation satisfy  for $\\nu=1,\\dots, N$ the following properties.\n\t\\begin{itemize}\n\t\t\\item $Q_\\nu\\in\\R^{n_\\nu\\times n_\\nu}$ symmetric positive definite,\n\t\t\\item $c_\\nu\\in\\R^{n_\\nu}$,\n\t\t\\item $a\\in\\R_+^m$,\n\t\t\\item $X_\\nu=\\left\\{ x_\\nu\\in\\R^{n_\\nu} | g_\\nu(x_\\nu)\\leq 0 \\right\\}\\subseteq \\R^{n_\\nu}$ nonempty, convex, and closed;\n\t\t\\item $ g_\\nu: \\R^{n_\\nu}\\rightarrow\\R^{m_\\nu} $  at least twice differentiable, and convex;\n\t\t\\item $Q_y\\in\\R^{m\\times m}$  positive definite and diagonal,\n\t\t\\item $b(x)=B^\\top x$, $l(x)=L^\\top x$, with $B,L \\in \\R^{n\\times m}$,\n\t\t\\item smooth NCP function of the form $ \\phi_\\varepsilon(\\alpha,\\beta)=\\alpha+\\beta-\\tilde{\\phi_\\varepsilon}(\\alpha-\\beta)$ where $\\tilde{\\phi_\\varepsilon}$  is at least twice differentiable and convex for every $\\varepsilon>0$.\n\t\\end{itemize}\n\\end{assumption}\n\n\\textcolor{black}{Based on these assumptions, we characterize the Nash equilibrium of NEP($\\varepsilon$) by its KKT conditions in the following lemma:\n\\begin{lemma}[KKT of NEP($\\varepsilon$)]\\label{lem:KKTNEPeps}\n\tGiven \\textcolor{black}{Assumption} \\ref{Assump:SNEP} and some constraint qualification for $X_\\nu$, the KKT conditions \\textcolor{black}{of the leader-level problems} are necessary and sufficient for the global minimizer of each leader problem in \\ref{Eq:SmoothNEP}.\n\tIn particular, we have the KKT conditions of player $\\nu$'s optimization problem for $\\nu=1,\\dots N$:\n\t\\begin{subequations}\\label{Eq:KKTepsilon}\n\t\t\\begin{align}\n\t\t\\begin{split}\n\t\t0=&Q_\\nu x_\\nu+c_\\nu +\\frac{1}{2} (L^\\top +Q_y^{-1}B^\\top )^\\top _{\\nu,:} a \\\\\n\t\t&+ \\frac{1}{2}\\sum\\limits^{m}_{i=1}a_i(L^\\top -Q_y^{-1}B^\\top )^\\top _{\\nu,i} ~\\tilde{\\phi_\\varepsilon}'\\left([(L^\\top -Q_y^{-1}B^\\top ) x]_i\\right) + \\nabla_{x_\\nu}g_\\nu (x_\\nu) \\lambda_\\nu,\n\t\t\\end{split}\\\\\n\t\t0&=\\min\\left\\{ \\lambda_\\nu , -g_{\\nu}(x_\\nu)\\right\\},\n\t\t\\end{align}\n\t\\end{subequations}\nwith the Lagrange multiplier $\\lambda_\\nu\\in\\R_+^{m_\\nu}$ and the Jacobian of the constraints\\\\% DAS \\\\ MUSS WEG\n $\t \\nabla_{x_\\nu}g_\\nu (x_\\nu)= (\\nabla_{x_\\nu} g_{\\nu_1}(x_\\nu),\\dots,\\nabla_{x_\\nu} g_{\\nu_{m_\\nu}}(x_\\nu) )\\in\\R^{n_\\nu\\times m_\\nu}$.\n\\end{lemma}\n\\begin{proof}\n\tKKT is necessary and sufficient for every leader problem in presence of a constraint qualification, since the objectives are strictly convex and the  strategy set are convex and closed.\n\tTherefore, the joint KKT system is {necessary and sufficient}  for the unique Nash equilibrium.\n\tThe follower's solution can be explicitly computed by the leader's solutions, c.f. (\\ref{Eq:FollowerSolution}).\n\\end{proof}\nThe imposed constraint qualification is specified in the following section.}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix A: Resonance hadronic decay amplitudes employed in the JM model\n\\label{hadrwidths}}\n\n\nThe relationship within the JM model \\cite{Mo09,Ri00} between the $N^*$ hadronic decay amplitudes  \n$\\langle \\lambda_{f} \\vert T_{dec} \\vert \n\\lambda_{R} \\rangle$ and the energy-dependent $N^{*}$ partial hadronic decay widths   \n$\\Gamma_{\\lambda_{f}}(W)$ is presented in this Appendix.  Here $\\lambda_R$ is  \nthe helicity of the $'$N*$'$, which decays into $\\pi\\Delta$ or $\\rho p$ final states with  \nhelicity $\\lambda_{f}$.\n\n\n\n\n\n The $N^*$ hadronic decay amplitudes in Eq. (\\ref{regbwamp}) can be expanded in partial waves of total \n momentum $J$: \n\\begin{eqnarray}\n\\label{angdecompa} \n\\langle \\lambda_{f} \\vert T_{dec} \\vert \\lambda_{R} \\rangle=\\sum_{J} \\langle \\lambda_{f} \\vert T^{J}_{dec} \\vert \n\\lambda_{R} \\rangle d^{J}_{\\mu\\nu}(\\cos\\theta^{*})e^{i\\mu\\phi^{*}},  \n\\end{eqnarray}\nwhere $\\theta^{*}$ and $\\phi^{*}$ are the CM emission angle of the $\\pi$ \nfor the $\\pi \\Delta$\nintermediate state and of the final $'$p$'$ for the $\\rho$ p intermediate states. The indexes $\\mu$ and $\\nu$ are defined in Eq. (\\ref{decaywidth5}). Only a single term \nwith $J$=$J_{r}$, where $J_{r}$ is resonance spin,\ncontributes to the expansion in Eq. (\\ref{angdecompa}) because of \ntotal angular momentum conservation.\nWe can rotate the quantization axis adopted for the initial-state \n$\\vert \\lambda_{R} \\rangle$ wave function and \nre-evaluate the matrix element $\\langle \\lambda_{f} \\vert T_{dec} \\vert \\lambda_{R} \\rangle$ \nin the frame with the quantization axis defined by the final \n$\\pi$  (the final $'$p$'$) \nmomentum for $\\pi \\Delta$ ($\\rho p$) $N^*$ \n decays, respectively. The matrix element $\\langle \\lambda_{f} \\vert T_{dec} \\vert \\lambda_{R} \\rangle$\nafter rotation becomes:\n\\begin{eqnarray}\n\\label{angdecomp1} \n\\langle \\lambda_{f} \\vert T_{dec} \\vert \\lambda_{R} \\rangle=\\sum_{\\nu'}\\langle \\lambda_{f} \\vert T_{dec} \\vert \nJ_{r} \\; \\nu' \\rangle d^{J_{r}}_{\\mu\\nu'}(cos\\theta^{*})e^{i\\mu\\phi^{*}}. \n\\end{eqnarray}\nThe superposition of the states $\\vert \nJ_{r} \\; \\nu' \\rangle$ in Eq. (\\ref{angdecomp1}), with spin $J_{r}$ and projection \nonto the final-state  quantization axis $\\nu'$,  is the transformed wave function of the initial-state $\\vert \\lambda_{R} \\rangle$ after the aforementioned rotation of the initial-state quantization axis.\nRotational invariance of the resonance hadronic decay amplitudes requires that the operator $T_{dec}$ should \nbe an SU(2) $\\otimes$ O(3) - spin $\\otimes$ orbital momentum scalar.  Only the term \nwith $\\nu'=\\nu$ in Eq. (\\ref{angdecomp1}) \n(with $\\nu$ defined by Eq. (\\ref{decaywidth5})) \nremains non-zero in the sum of Eq. (\\ref{angdecomp1}), as  a consequence of the Wigner-Eckart theorem \napplied to matrix elements $\\langle \\lambda_{f} \\vert T_{dec} \\vert \nJ^{r} \\; \\nu' \\rangle$ with the scalar $T_{dec}$ operator. \n\nFrom comparisons between Eqs. (\\ref{angdecompa}) and (\\ref{angdecomp1}) we can see that:\n\\begin{eqnarray}\n\\label{equal} \n\\langle \\lambda_{f} \\vert T^{J_{r}}_{dec} \\vert \\lambda_{r} \\rangle=\\langle \\lambda_{f} \\vert T_{dec} \\vert J^{r} \n\\nu \\rangle , \n\\end{eqnarray} \nThe $\\langle \\lambda_{f} \\vert T^{J_{r}}_{dec} \\vert \n\\lambda_{R} \\rangle$ matrix elements in Eqs. (\\ref{decaywidth5},\\ref{equal}) are determined by the final-state helicity \n$\\lambda_{f}$ only, and  \nare independent \nof $N^*$ helicities $\\lambda_{R}$. \n\n  \nAssuming real values for the matrix element $\\langle \\lambda_{f} \\vert T^{J_{r}}_{dec} \\vert \n\\lambda_{r} \\rangle$ \nin Eqs.~(\\ref{decaywidth5},\\ref{equal}), we can relate it to the $\\Gamma_{\\lambda_{f}}(W)$ \npartial hadronic decay width of the excited state $N^{*}$ to the final state of helicity\n$\\lambda_{f}$. We employ general relations of quantum theory \nbetween \nthe resonance decay amplitude $\\langle \\lambda_{f} \\vert T_{dec} \\vert \n\\lambda_{R} \\rangle$ of Eq. (\\ref{angdecompa}),\nthe two-body state phase space of resonance decay products $d\\Phi_{2b}$, \nand  the $\\Gamma_{\\lambda_{f}}(W)$ decay width:\n\\begin{eqnarray}\n\\Gamma_{\\lambda_{f}}(W)=\\frac{1}{2M_{r}}\\frac{1}{2J_{r}+1}\\sum_{\\lambda{R}}\\int \\vert\\langle \\lambda_{f} \\vert T_{dec} \\vert \n\\lambda_{R} \\rangle \\vert^{2}d\\Phi_{2b}.  \n\\label{ampwidth}\n\\end{eqnarray}\nThe factor $\\frac{1}{2M_{r}}$ in Eq. (\\ref{ampwidth}) reflects the spin-tensor normalization in the convention \nof the JM model \\cite{Mo09}. This normalization, and the expression for the $'$S$'$-matrix adopted in the JM model \n\\cite{Mo09}, defines the final-state two-body  phase space $d\\Phi_{2b}$  as:\n\\begin{eqnarray}\nd\\Phi_{2b}=\\frac{1}{4\\pi^{2}}\\frac{p_{f}}{4M_{r}}\\sin(\\theta^{*})d\\theta^{*}d\\phi^{*},  \\\\\nE_{f}=\\frac{W^2+M_{f}^2-M_{f'}^2}{2W}, \\nonumber \\\\\np_{f}=\\sqrt{E_{f}^2-M_{f}^2} \\nonumber ,\n\\label{diff2b}\n\\end{eqnarray}\nwhere $E_{f}$, $p_{f}$ are the energy and momentum modulus of one of the  final\nhadrons $f$ ($f$ is either pion or the final proton for the $N^*$ $\\rightarrow$ $\\pi \\Delta$ or $\\rho$ $'$p$'$\ndecays, respectively), $M_{f}$ is its mass, \nwhile the index $f'$ stands for the other hadron. \nAll frame-dependent kinematic variables of the final hadrons are defined \nin the  final hadron\nCM frame.  Inserting Eqs. (\\ref{angdecompa},\\ref{equal}) into Eq. (\\ref{ampwidth}) and accounting for and the $'$d-$'$function normalization,\n\\begin{equation}\n\\int d^{J*}_{\\mu \\, \\nu}(\\cos\\theta^*)\\cdot d^{J}_{\\mu \\, \\nu}(\\cos\\theta^*) \\sin\\theta^* d\\theta^*=\\frac{2}{2J+1}, \n\\end{equation}\nwe obtain Eq. (\\ref{decaywidth5}) for\nthe $\\langle \\lambda_{f} \\vert T_{dec} \\vert \\lambda_{R} \\rangle$ amplitudes, apart from the factor  \n$\\sqrt{\\frac{\\langle p^{r}_{i} \\rangle}{\\langle p_{i} \\rangle}}$. Note that at the resonant point, $W$=$M_{r}$,\nthis factor is equal to unity. However, in calculations of resonant cross sections for  \n$W$$\\ne$$M_{r}$, the two-body phase space becomes different than that at the resonant point. \nThe factor $\\sqrt{\\frac{\\langle p^{r}_{i} \\rangle}{\\langle p_{i} \\rangle}}$ in Eq. (\\ref{decaywidth5}) accounts for \nthis difference. It is needed to evaluate resonant cross sections and amplitudes for $W$ values off \nthe resonant point.\n\n\n\n\n\nThe $\\langle p^{r}_{i} \\rangle$ and $\\langle p_{i} \\rangle$  absolute \nthree-momentum  values of the final $\\pi$ for the   \n$N^* \\rightarrow \\pi \\Delta$ decay ($i$=1) or of the final $p'$ \nfor the $N^* \\rightarrow \\rho p$ decay ($i$=2) in Eq. (\\ref{decaywidth5}) are  \naveraged over the running mass of the unstable hadron in the\nintermediate state:\n\n\\begin{gather}\n\\label{averf}\n\\langle p^{r}_{i} \\rangle=  \n\\int dM^{2}_{i}\\frac{1}{\\pi}\\frac{M_{i\\,0}\\Gamma_{i\\,0}}\n{(M^{2}_{i}-M^{2}_{i\\,0})^{2}+M^{2}_{0\\, i}\\Gamma^{2}_{0\\,i}}p^{r}_{i}(M^{2}_{i}),  \\\\\n\\nonumber \\\\\n\\langle p_{i} \\rangle=\n\\int dM^{2}_{i}\\frac{1}{\\pi}\\frac{M_{i\\,0}\\Gamma_{i\\,0}}\n{(M^{2}_{i}-M^{2}_{i\\,0})^{2}+M^{2}_{i\\,0}\\Gamma^{2}_{i\\,0}}p_{i}(M^{2}_{i}) ,  \\nonumber \\\\\n\\nonumber\n\\end{gather}\nwhere $M_{i}$ is the current invariant mass of the final  \n$\\pi p$ particles in the case of $N^* \\rightarrow \\pi \\Delta$\ndecay ($i=1$) or the current invariant mass of final $\\pi\\pi$ particles for the  \n$N^* \\rightarrow \\rho p$ decay ($i=2$); $M_{i\\,0}$ are the central masses of either $\\Delta$ ($i=1$) or\n$\\rho$ ($i=2$); and $\\Gamma_{i\\,0}$, are their total decay widths.\nThe running momenta of the stable particles from $N^*$ decays  $p_{i}(M^{2}_{i})$ in \nEq. (\\ref{averf}) are evaluated as:\n\\begin{eqnarray}\n\\label{runmom}\np_{1}=\\frac{[(W^2+m_{\\pi}^{2}-M_{1}^2)^2-4W^{2}m_{\\pi}^2]^{1\/2}}{2W} \\ \\text{and} \\\\\n\\, p_{2}=\\frac{[(W^2+m_{p'}^{2}-M_{2}^2)^2-4W^{2}m_{p'}^2]^{1\/2}}{2W} . \\nonumber\n\\end{eqnarray}\nThe values of these momenta at the resonance point $p^{r}_{i}(M^{2}_{i})$ were obtained from Eq. (\\ref{runmom}) at  $W=M_{r}$.\n\n\n\n\n\n \n \n\n\n\nThe hadronic decay widths $\\Gamma_{\\lambda_{f}}(W)$ \nwere taken from experiments\nwith hadronic probes, as was described in Section~\\ref{fit}. Those decay widths were obtained in another\nrepresentation of orbital angular momentum $L$ \nand total final hadron spin $S$ $\\Gamma_{LS_{f}}(W)$. \nThe partial $\\Gamma_{\\lambda_{f}}(W)$ decay widths can be transformed into this \nrepresentation \\cite{Jac59} by:\n\\begin{eqnarray} \n\\label{hadrwidth1}\n\\sqrt{\\Gamma_{\\lambda_{f}}}=\\sqrt{\\frac{2J_{r}+1}{2L+1}}\\langle L 0 S \\lambda_{1}-\\lambda_{2} \n\\vert J_{r} \\lambda_{1}-\\lambda_{2} \\rangle \\cdot \\nonumber \\\\ \n\\langle s_{1} \\lambda_{1} s_{2} -\\lambda_{2} \\vert\nS \\lambda_{1}-\\lambda_{2} \\rangle\\sqrt{\\Gamma_{LS}}, \n\\end{eqnarray}\n\\\\\nwhere $s_{1}$, $\\lambda_{1}$, $s_{2}$, $\\lambda_{2}$ are spins and helicities for first stable and\nsecond unstable particles in the intermediate states. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\newpage\n\n\\section{Introduction}\n\n\n\n\nAn extensive research program on nucleon resonance ($N^*$) excitation \nis in progress using the CLAS detector in\nHall-B at Jefferson Lab \\cite{Bu11a,Bu11b,Bu11,Bu05,Bu04}. The studies of transition helicity amplitudes \nfrom the proton ground state to its excited states (or $\\gamma_{v}pN^*$ photo-\/electrocouplings) represent a key\ndirection in the $N^*$ program with CLAS. Meson-electroproduction data off nucleons in the $N^*$ region \nobtained\nwith CLAS open up an opportunity to determine\nthe $Q^2$-evolution of \n$\\gamma_{v}NN^*$ electrocouplings in a combined analysis of various meson-electroproduction channels.\nThe $Q^2$-evolution of $\\gamma_{v}NN^*$ electrocouplings \nwill allow us to pin down active\ndegrees of freedom in the $N^*$ structure at various distance scales \nand to access non-perturbative \nstrong-interaction mechanisms that govern \nthe excited nucleon state formation as bound systems of quarks and gluons.\n\n\n\n\n \nTheoretical and experimental studies\nof the electroexcitation of nucleon resonances\nhave a long history. \nAlong with the hadron masses and their partial decay widths, the information on the\n$\\gamma_{v}pN^*$ electrocouplings played\nan important role in the development  of the quark models in their contemporary advanced relativistic version\nin light-front dynamics \\cite{Brodsky,Aznquark1,Capstick,Simula1,Bruno,AznRoper,AzBu12}. \nThe picture of the nucleon \nand its excited states,\nwhich seemed quite simply modeled with three relativistic \nconstituent quarks,\nturned out to be more complex. Recently obtained electrocouplings for the\n$P_{11}(1440)$, $D_{13}(1520)$, and \n$S_{11}(1535)$ states \\cite{Az09} showed that those quark models, which were \nsuccessful in describing\nthe electrocouplings of these states for $Q^2$ $>$ 2.0  GeV$^2$, failed to reproduce the results \nat smaller photon virtualities of $Q^2$ $<$ 1.0 GeV$^2$. Quark models with flavor-conserving quark interactions are unable to describe \nthe small mass of the $P_{11}(1440)$ state \\cite{Ca00} and the ordering of $P_{11}(1440)$ and\n$S_{11}(1535)$ resonances \\cite{Bu11b}. Moreover, models that treat the $P_{11}(1440)$ structure as just three constituent quarks \nare unable \nto describe the large total $P_{11}(1440)$ decay width of $\\approx$ 300 MeV. \nThese\ndifficulties prompted a search for additional contributions to the $N^*$ structure. \n\n\n\nA general unitarity requirement imposes meson-baryon dressing contributions to\nboth resonance electromagnetic excitation and hadronic decay amplitudes.\nStudies of meson-baryon dressing contributions to $\\gamma_{v}pN^*$ \nelectrocouplings \nand resonance hadronic decay\namplitudes, carried out at the Excited Baryon Analysis Center (EBAC) at \nJefferson Lab \\cite{Lee10,Lee09,Lee091,Lee08,Lee101,Lee101a,Lee102,Lee11c}, have extended our insight into the spectrum and structure of excited nucleon states \nconsiderably. The contributions from \nmeson-baryon dressing to the $P_{11}(1440)$ and $D_{13}(1520)$ electrocouplings\nwere determined \nfrom a global analysis of the world data on $\\pi N$ scattering and $\\pi^+n$, $\\pi^0p$\nelectroproduction off protons within the framework of the EBAC dynamical \ncoupled-channel \napproach (EBAC-DCC) \\cite{Lee10}. \n\nThis analysis showed that the contributions from meson-baryon dressing \nto the $\\gamma_{v}pN^*$ electrocouplings are maximal at\nsmall $Q^2$ and decrease with increasing photon virtualities \\cite{Lee08}.\nAt $Q^2$ $<$ 1.0 GeV$^2$ these contributions\nmay even be dominant. The meson-baryon cloud \nhas a profound impact on the resonance spectrum. \nFor example, in the $P_{11}$ partial wave \na single bare resonance pole \nlocated at $W$=1.76 GeV, being affected by meson-baryon dressing, splits into \nthree poles located on different Riemann sheets \\cite{Lee101,Lee102}. Two of them  \nwith $Re(W)$ $\\approx$ 1.36 GeV correspond to the physical Roper resonance. The double-pole structure of the Roper resonance was also observed in previous studies \\cite{Ar85,Ar06}. Meson-baryon dressing should be taken\ninto consideration in the interpretation of resonance electrocouplings \nas well as for the\nexcited nucleon spectrum. \n\n\n\n\n\nOur studies of resonance electrocouplings \nat small photon\nvirtualities presented in this paper offer valuable information to further \nexplore\nthe role of meson-baryon and quark components in the $N^*$ structure. The separation between the meson-baryon cloud and quark core contributions within a well-defined \ntheoretical\nframework \\cite{Ca07} can help pin down the domain of photon virtualities\nwhere  quark components are the main contributor to the $N^*$\nstructure. This kinematic domain is of particular interest for the studies of hadrons from the first principles of QCD, including Lattice QCD (LQCD) \\cite{Br09,Li09,Li09a,Li11,Li11a,Ed11,Ed12,CL12} and Dyson-Schwinger equation studies of \nQCD (DSEQCD) \\cite{CL12,CRo09,CRo011,CRo101,CRo103,CRo104,CR2012}.\n\n\n\n\nThe CLAS detector at \nJefferson Lab\nis a unique large-acceptance instrument\ndesigned for the comprehensive exploration\nof exclusive meson electroproduction. It offers excellent opportunities \nof studying the electroexcitation of nucleon\nresonances in detail and with precision. The CLAS detector has provided the dominant portion of all data on meson electroproduction in the resonance excitation region.\n \n\n\n\n\nA variety of measurements\nof single pion electroproduction off protons,\nincluding polarization measurements, have been performed\nat CLAS in a range of $Q^2$ from 0.16 to 6 GeV$^2$ \\cite{Joo}.\n The electroexcitation amplitudes\nfor the low-lying resonances\n$P_{33}(1232)$, $P_{11}(1440)$, $D_{13}(1520)$,\nand $S_{11}(1535)$ were determined over a wide range of $Q^2$\nin a comprehensive analysis\nof JLab-CLAS  data \non differential cross sections, longitudinally polarized\nbeam asymmetries, and longitudinal target and beam-target asymmetries \\cite{Az09}.\n\n\nThe combination of the large-acceptance CLAS detector and \nthe continuous electron beam from CEBAF \nmade it possible to measure $\\pi^+\\pi^-p$ electroproduction cross sections with\nnearly full kinematic coverage for this three-body final hadron state \\cite{Ri03,Fe09}. These are the most extensive data sets on unpolarized $\\pi^+\\pi^-p$ electroproduction cross sections obtained so far. These data allowed for the first time the projection  \nof nine one-dimensional differential cross sections, each sensitive to  \na different combination of resonance and background strength. The data of \\cite{Fe09}  were collected in the mass range 1.31 GeV $<$ $W$ $<$ 1.56 GeV and with\nphoton virtualities 0.25 GeV$^2$ $<$ $Q^2$ $<$ 0.6 GeV$^2$. A good description \nof these data was achieved\nwithin the framework of a phenomenological Jefferson Laboratory - Moscow State University (JM) reaction model \\cite{Mo09}, that\nallowed \nus to establish the mechanisms contributing to this exclusive reaction. The presence and strength of the contributing\n$\\pi^+\\pi^-p$ electroproduction mechanisms were established\nby studying their kinematical dependencies and correlations in different \none-fold differential cross sections.\n\nIn this work we present results on the electroexcitation of the $P_{11}(1440)$ and  $D_{13}(1520)$ states,\nobtained from the analysis of data on $\\pi^+\\pi^-p$ electroproduction off protons \\cite{Fe09}. The analysis was\ncarried out employing the JM reaction model \\cite{Mo09}, which was further developed to provide a framework for \nthe determination\nof $\\gamma_{v}pN^*$ electrocouplings from a combined fit of unpolarized differential cross sections. In the previous studies \\cite{Mo09}  we did not attempt to isolate the\ncontributions from resonances. In the analysis reported in\nthis paper, we employ the JM model with the goal of isolating the resonant contributions \nfor the individual differential cross sections. For the description of \nresonant amplitudes, \nwe updated the Breit-Wigner (BW) parametrization, \nmaking\nit consistent with the restrictions required by the general unitarity condition. The reliable evaluation of the resonant contributions enabled for the first time \n   the determination of the $\\gamma_{v}pN^*$ electrocouplings for the $P_{11}(1440)$ and $D_{13}(1520)$ states \nfrom charged double pion\nelectroproduction off protons. \nThis complements the results from $N\\pi$ electroproduction in an \nindependent channel. \n\nAnalyses of different exclusive channels are essential for a \nreliable extraction of resonance parameters. Currently the separation of resonant and non-resonant \nparts of the electro production amplitudes can be done only within phenomenological \nreaction models. Therefore, the resonance parameters extracted from the meson electro production data fit may be affected by the model assumptions, and their credibility should be further examined. Non-resonant mechanisms in various meson-electroproduction channels are\ncompletely different, while the $\\gamma_{v}NN^*$ electrocouplings are the same.  Independent analyses of different exclusive channels make it possible to test whether they give consistent results for the\nresonance electrocouplings. \nMost nucleon resonances decay into both $N\\pi$ and $N\\pi\\pi$ final states. \nStudies of resonance electroexcitations \nin these channels with completely different non-resonant contributions offer independent information on $N^*$ electrocouplings.  \nTherefore, a successful description of the data on $\\pi^+n$, $\\pi^0p$, and $\\pi^+\\pi^-p$ electroproduction off protons with \nconsistent $N^*$\nelectrocoupling values provides clear evidence for the reliable extraction of these quantities \nfrom meson-electroproduction data.\n\nStudies of the $P_{11}(1440)$ resonance \nin $\\pi^+\\pi^-p$ electroproduction off protons \noffer the additional opportunities to improve the knowledge on electrocouplings of this $\\approx$ 300 MeV broad state. Contrary to $N\\pi$\nelectroproduction channels, the $P_{33}(1232)$ resonance does not directly \ncontribute to the resonant parts \nof the $\\pi^+\\pi^-p$ electroproduction amplitude. Hence, \nthe influence of the $P_{33}(1232)$ resonance on the extracted electrocouplings of \nthe $P_{11}(1440)$ is much weaker. \n\nThe requirement of $Q^2$-independent $N^*$ hadronic decay amplitudes\n in  $\\pi^+\\pi^-p$ electroproduction  provides constraints on \n the $N\\pi\\pi$ partial decay widths. This makes possible   \naccess to the $\\pi\\Delta$ and $\\rho p$  partial hadronic decay widths of the $P_{11}(1440)$ and $D_{13}(1520)$ in the measurements of $\\pi^+\\pi^-p$ electroproduction off protons.   \n\n\n\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=8cm]{pictures\/angle1.eps}\n\\includegraphics[width=8cm]{pictures\/angle_var22_var_1_new.eps}\n\\caption{\\small Kinematic variables for the description  of\n$e p \\rightarrow e' p' \\pi^{+} \\pi^{-}$ in the CM frame of the final-state \nhadrons corresponding to the first assignment presented\nin Section~\\ref{kinxsect}. Panel (a) shows the \n$\\pi^{-}$ spherical angles $\\theta_{\\pi^{-}}$ and $\\varphi_{\\pi^{-}}$. Panel (b)\nshows the angle $\\alpha_{[p\\pi^{-}][p'\\pi^{+}]}$  between the two planes: one of them (plane A)\nis defined by the 3-momenta of the initial proton and the final $\\pi^{-}$,\nthe other (plane B) is defined by the 3-momenta of the two others final\nhadrons $\\pi^{+}$ and proton. The unit vectors \n$\\overline{\\gamma}$ and $\\overline{\\beta}$ are normal\nto the $\\pi^-$ three-momentum in the planes A and B, respectively.} \\label{kinematic}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\hspace{-2cm}\n\\vspace{0.5cm}\n\\epsfig{file=pictures\/part1_new.eps,width=12cm}\n\\end{center}\n\\begin{center}\n\\hspace{-3cm}\n\\vspace{0.5cm}\n\\epsfig{file=pictures\/part2_new.eps,width=12cm}\n\\end{center}\n\\begin{center}\n\\hspace{-3.cm}\n\\vspace{-1.cm}\n\\epsfig{file=pictures\/note_03_13_7_t.eps,width=10.cm}\n\\end{center}\n\\begin{center}\n\\hspace{-3.cm}\n\\vspace{-0.5cm}\n\\epsfig{file=pictures\/note_03_12_7_b.eps,width=10.cm}\n\\end{center}\n\\caption{\\small Reaction mechanisms of the JM model \\cite{Mo09} that contribute to $\\pi^{+}\\pi^{-} p$ \nelectroproduction in the kinematic region covered by recent CLAS\nmeasurements \\cite{Fe09} with W $<$ 1.6 GeV and 0.25 GeV$^{2}$ $<$ $Q^{2}$ $<$\n0.6 GeV$^{2}$: 3-body mechanisms (a), $\\pi \\Delta$ isobar channels (b), and \ndirect 2$\\pi$ production \nmechanisms that correspond to different assignments for the \nfinal state hadrons (c,d).\n The $t_{i}$ ($i$=1,..3) stand for the squared transferred momenta in \n the exchange processes by\n unspecified particle(s), as described in \\cite{Mo09} and shown by blobs in the (c)\n and (d) panels.}\n\\label{mech_low_06}\n\\end{figure*}\n\n\n\n\n\n\n\\section{Phenomenological model JM for evaluation of $\\gamma_{v}pN^*$ electrocouplings}\n\\label{genjm}\nThe phenomenological meson-baryon model JM was developed to describe \n$\\pi^+\\pi^-p$ electroproduction off protons \\cite{Mo09,Mo07,Sh07,Mo06,Az05,Mo06-1} \nwith \nthe primary objective of determining the resonance  $\\gamma_{v}pN^*$ electrocouplings \nand the $\\pi\\Delta$ and $\\rho p$ \npartial hadronic decay widths from a combined fit to all\nmeasured observables. In our current analysis of the CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Fe09} the JM model was used to separate the resonant and non-resonant contributions \nto differential cross sections and to access the electrocouplings and $\\pi \\Delta$, \n$\\rho p$ decay widths of the $P_{11}(1440)$ and $D_{13}(1520)$ resonances.\nHere we briefly discuss the basic ingredients of \nthe JM model that\nare relevant to the objectives of this paper.  \n\n\n\n\\subsection{Kinematics and cross sections}\n\\label{kinxsect}\nAt a given invariant mass $W$ and photon virtuality $Q^2$,\nthe $\\gamma_{v} p \\rightarrow \\pi^+\\pi^-p$ reaction\ncan be fully described as a five-fold differential cross section \n$d^5\\sigma\/d^5\\tau$,\nwhere $d^5\\tau$ is the phase-space volume of the\nfive independent variables in the center-of-mass (CM) system of the final\n$\\pi^{+} \\pi^{-} p$ state. There are\nmany possible choices \\cite{Byc} of the\nfive independent variables.\nAfter defining $M_{\\pi^+p}$, $M_{\\pi^-p}$, and\n$M_{\\pi^+\\pi^-}$ as invariant mass variables of\n the three possible two-particle pairs in the $\\pi^{+}\\pi^{-}p$ system,\nwe adopt the following three assignments:\n\n\n\n\n\n\n\n\n\\begin{enumerate}\n\\item $d^5\\tau_1=dM_{p\\pi^+}dM_{\\pi^+\\pi^-}d\\Omega_{\\pi^-}\nd\\alpha_{[p'\\pi^{+}][p\\pi^{-}]}$, where\n$\\Omega_{\\pi^-}$ ($\\theta_{\\pi^{-}}$, $\\varphi_{\\pi^{-}}$) are\nthe final $\\pi^-$\nspherical angles with respect to the direction of the virtual photon, and\n$\\alpha_{[p' \\pi^{+}][p\\pi^{-}]}$ is the angle\nbetween Plane B defined by the momenta of\nthe final $p'\\pi^{+}$ pair and Plane A defined by the momenta of the initial proton and the final $\\pi^{-}$ ;\n\\item $d^5\\tau_2=dM_{p\\pi^+}dM_{\\pi^+\\pi^-}d\\Omega_{p'}d\\alpha_{[\\pi^{+}\\pi^{-}][p'p]}$,\nwhere \n $\\Omega_{p'}$ ($\\theta_{p'}$, $\\varphi_{p'}$) are the final proton\nspherical angles with respect to the direction of the virtual photon, and\n $\\alpha_{[\\pi^{+}\\pi^{-}][p'p]}$ is the angle\nbetween Plane B$^{'}$ defined by the momenta of the \n$\\pi^{+}\\pi^{-}$ pair and Plane A  defined by the momenta of the\ninitial and final protons ;\n\\item $d^5\\tau_3=dM_{p\\pi^+}dM_{p\\pi^-}d\\Omega_{\\pi^+}\nd\\alpha_{[p'\\pi^{-}][p\\pi^{+}]}$, where \n$\\Omega_{\\pi^+}$ ($\\theta_{\\pi^{+}}$, $\\varphi_{\\pi^{+}}$)\nare \nthe final $\\pi^+$\nspherical angles with respect to the direction of virtual photon,\nand\n$\\alpha_{[p'\\pi^{-}][p\\pi^{+}]}$ is the angle \nbetween  Plane B$^{''}$ defined by the momenta of\nthe final $p'\\pi^{-}$ pair and Plane A$^{''}$ defined by the momenta of the initial proton and the $\\pi^{+}$.\n\\end{enumerate}\nAll frame-dependent variables are defined in the final hadron center-of-mass (CM)\nframe.\n\n\nThe emission angles for the final state particles in the case of the first assignment \nare shown in  Fig.~\\ref{kinematic}.\nThis choice is \nmost suitable for describing  $\\pi^+\\pi^-p$ electroproduction through the \n$\\pi^{-}\\Delta^{++}$ intermediate state, which is the dominant contributor\nof all isobar channels in  the kinematic region covered by the data \\cite{Fe09}.\n For the other assignments the emission\nangles of the final hadrons are analogous to the ones given in Fig.~\\ref{kinematic}.\nThe relations between the  momenta of the final-state hadrons and the five\nvariables of the first assignment can be found in the Ref.~\\cite{Fe09}.\n\n\n\n\n\n\n\nThe $\\pi^+\\pi^-p$ electroproduction data have been collected in the \nbins of a seven\ndimensional space. As mentioned above, five variables are needed to fully\ndescribe the final hadron kinematics, while to describe \nthe initial state kinematics two others variables \n$W$ and $Q^2$ are required.  The huge number of seven dimensional \nbins over the reaction phase space ($\\approx$ 100,000 bins) \ndoes not allow us to use the\ncorrelated multi-fold differential cross sections in the analysis of\nelectroproduction processes, where the statistics decrease drastically with the photon \nvirtualities $Q^2$. More than half of the five-dimensional phase-space bins \nof the final hadrons  is not populated due to statistical limitations. \nThis is a serious obstacle \nfor any analysis method that employs information on the behavior of multi-fold differential cross sections. \nWe therefore  use the following one-fold differential cross sections in each bin of \n$W$ and $Q^2$ covered by the measurements:\n\\begin{itemize}\n\\item invariant mass distributions for three pairs of the final particles \n$d\\sigma\/dM_{\\pi^{+}\\pi^{-}} $, $d\\sigma\/dM_{\\pi^+ p}$, and \n$d\\sigma\/dM_{\\pi^- p}$;\n\\item angular distributions for spherical angles of the three final particles $d\\sigma\/d(-cos\\theta_{\\pi^-})$,\n$d\\sigma\/d(-cos\\theta_{\\pi^+})$, and $d\\sigma\/d(-cos\\theta_{p'})$ \nin the CM frame;\n\\item angular distributions for the three $\\alpha$-angles described above and determined in the CM frame \n$d\\sigma\/d\\alpha_{[p'\\pi^+][p\\pi^{-}]}$,  $d\\sigma\/d\\alpha_{[p'\\pi^-][p\\pi^{+}]}$, and\n $d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$.\n\\end{itemize} \nThe one-fold differential cross sections were obtained from integrating the \nfive-fold differential \ncross sections over the relevant four others kinematic \nvariables of $d^{5}\\tau_{i}$. All details related to the evaluation of the $\\pi^+\\pi^-p$ one-fold differential cross sections we are using for the extraction of\nresonance parameters can be found in Ref.~\\cite{Fe09}.  \n\n\n\\subsection{Relevant electroproduction mechanisms}\n\\label{pipipmech}\n\nThe major part of the $\\pi^+\\pi^-p$ electroproduction off \nprotons  at $W$ $<$ 1.6 GeV is due to \ncontributions from the two $\\pi \\Delta$ isobar channels $\\pi^- \\Delta^{++}$ and $\\pi^+ \\Delta^0$.\nThe $\\Delta^{++}$(1232) resonance\nis clearly seen in all $\\pi^{+} p$ mass distributions for W $>$ 1.4\nGeV, while contributions from the $\\pi^+ \\Delta^0$ isobar channel are needed to better describe\nthe data in the low mass regions of the $\\pi^- p$ mass distributions. \nThe observed in the data \\cite{Fe09} strength of the $\\pi^- \\Delta^{++}$ isobar \nchannel is approximately nine times larger than that of $\\pi^+ \\Delta^0$ \\cite{Mo09} \ndue to isospin invariance. \nThe contributions from all other isobar channels $p\\,\\rho$,\n$\\pi^{+}D_{13}^{0}(1520)$, $\\pi^{+}F_{15}^{0}(1685)$, and\n$\\pi^{-}P_{33}^{++}(1640)$, which are incorporated into the JM-model \\cite{Sh07,Mo06,Az05,Mo06-1} in order to describe the data at $W$ $>$ 1.6 GeV, \nare negligible in the kinematic region covered in this analysis, and are not included \nin this work. \n\n\nThe production amplitudes relevant for our\nanalysis of $\\gamma_{v} p \\rightarrow \\pi^+\\pi^- p$ are illustrated in Fig.~\\ref{mech_low_06}. \nThey consist of the $\\pi^-\n\\Delta^{++}$ and $\\pi^{+} \\Delta^{0}$ isobar channels and direct double pion\nproduction mechanisms. The production amplitudes for $\\pi \\Delta$ intermediate\nstates (Fig.~\\ref{mech_low_06}b) consist of the resonant contributions \n$\\gamma_{v} N \\rightarrow N^*,\\Delta^* \\rightarrow \\pi\\Delta$ and non-resonant terms. \nAll resonances listed in Table~\\ref{nstlist} are\nincluded in the JM model. However, in the kinematic area \ncovered in our measurements, only the \n$P_{11}(1440)$,\n$D_{13}(1520)$, and $S_{11}(1535)$ nucleon resonances have strength that is sufficient to manifest themselves in the \none-fold differential\ncross-sections. \nNon-resonant amplitudes, depicted in the diagrams in  Fig.~\\ref{mech_low_06}b are computed from the\n well-established Born terms \n presented in Appendix A of Ref.~\\cite{Mo09}. The additional \ncontact terms are implemented in the full $\\pi \\Delta$ production amplitudes. \nThey describe effectively the  contributions from mechanisms other\nthan the Born terms to $\\pi \\Delta$ production and the part of the  \n$\\pi \\Delta$ final-state interactions (FSIs) that are not included in\nthe JM model's absorptive approximation for FSIs \\cite{Mo09,Ri00}.\nThe Lorentz structure of these additional contact terms is determined by\nsuperposition of the two second-rank Lorentz tensors\n\\begin{eqnarray}\n\\label{newstr}\n\\gamma^{\\mu}p^{\\pi}_{\\nu},\\nonumber  \\\\\np_{c}^{\\delta}\\gamma_{\\delta}g_{\\mu\\nu},\n\\end{eqnarray}\nwhere \n$p_{c}=(2p^{\\pi}_{\\nu}-q_{\\gamma})$ is the difference of the final pion \nfour-momentum $p^{\\pi}_{\\nu}$ and \nthe momentum transferred, $q_{\\gamma}-p^{\\pi}_{\\nu}$. \nA parametrization of the additional \ncontact-term\namplitudes in the $\\pi \\Delta$ channels may be found in Appendix B of Ref.~\\cite{Mo09}.\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n$N^{*}$ states & Mass,\\ & Total   &\nBranching  & Branching  & $N^{*}$ electro  \\\\\nincorporated &  (GeV)  & decay    &\nratio    & ratio  & coupling  \\\\\ninto the  &   & width  & ${\\pi\\Delta}$, \\% & ${\\rho p}$, \\% & variation  \\\\\ndata fit&  & $\\Gamma_{tot}$,  &  &  &  in the fit \\\\\n        &  &        (GeV)       &  &  &             \\\\ \n\\hline\n$P_{11}(1440)$ & var & var & var & var & \\cite{Az051} var \\\\\n$D_{13}(1520)$ & var & var & var & var & \\cite{Az051} var \\\\\n$S_{11}(1535)$ & var & var & var & var & \\cite{Az09} fix \\\\\n$S_{31}(1620)$ & 1.62 & 0.16 & 60 & 16 & \\cite{Az05,Mo06} var\\\\\n$S_{11}(1650)$ & 1.65 & 0.15 & 2 & 3 & \\cite{Bu03} var \\\\\n$F_{15}(1680)$ & 1.68 & 0.12 & 12 & 5.5 & \\cite{Az05,Mo06} var \\\\\n$D_{13}(1700)$ & 1.74 & 0.19 & 53 & 45 & \\cite{Az05,Mo06} fix \\\\\n$D_{33}(1700)$ & 1.70 & 0.26 & 89 & 2 & \\cite{Az05,Mo06} fix  \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{nstlist} List of resonances invoked in the $\\pi^+\\pi^-p$ \nfit and their parameters: total decay widths $\\Gamma_{tot}$ \nand branching fractions (BF) to $\\pi\n\\Delta$ and $\\rho p$ final states.\nThe quoted values for the hadronic parameters are taken from fits to \nthe earlier CLAS $\\pi^+\\pi^-p$ data \\cite{Ri03} \nusing the 2005\nversion of the JM model \\cite{Az05,Mo06}.\nThe quantities labeled as $var$ correspond to the variable parameters fit\nto the CLAS $\\pi^+\\pi^-p$ data \\cite{Fe09} within the framework of the \ncurrent JM model version \\cite{Mo09} \nemploying the unitarized BW ansatz of Section~\\ref{unitbw} for the resonant\ncontributions. Start values for the resonance electrocouplings are taken from\nthe references listed in the last column and extrapolated to the $Q^2$ area\ncovered by the CLAS experiment \\cite{Fe09}.}\n\\end{center}\n\\end{table}\n\n\n\nAll isobar channels combined account for 70\\% to 90\\% of the charged double-pion \nfully integrated\ncross sections in the kinematic region covered by data \\cite{Fe09}.\nThe remaining part of the cross sections comes from \ndirect charged  pion (2$\\pi$) production mechanisms, in which the $\\pi^{+} \\pi^{-} p$ final state \n is created \nwithout the formation\nof unstable hadrons in the intermediate states. The presence of these mechanisms \nis required by the unitarity of \nthe three-body\n$\\pi^+\\pi^-p$ production amplitudes \\cite{Ait78}. \nTheir manifestation in $\\pi^{+} \\pi^{-} p$\nelectroproduction was observed for the first time in our previous analyses \nof CLAS data \\cite{Az05,Mo09}. \nThe dynamics of these\nprocesses was unknown and has been established from the CLAS data analysis within the framework of the \nJM model. The direct  \n2$\\pi$-production mechanisms incorporated in the JM model are depicted \nin Fig.~\\ref{mech_low_06}(c,d). They represent \ntwo subsequent\nexchanges of unspecified particles, parametrized by propagators that depend exponentially on the running\nfour-momenta squared. Each set of diagrams in Fig.~\\ref{mech_low_06} \ncorresponds to various assignments of the final-state hadrons, resulting in \ndifferent four-momenta\nsquared running over propagators in the exchange amplitudes. The JM model extends for the first \ntime the description of \n$\\pi^+\\pi^-p$ electroproduction beyond the approximation of superimposed \nisobar channels by incorporating direct 2$\\pi$-production. \nExplicit expressions for the above-mentioned direct 2$\\pi$-production\namplitudes can be found in Appendices A-C of Ref.~\\cite{Mo09}. \n\n\n\n\n\nThe relationships between $\\pi^+\\pi^-p$ electroproduction cross sections \nand the three-body production amplitudes employed in the JM model are given \nin Appendix D of Ref.~\\cite{Mo09}. This information is required in order to compare the amplitudes of the JM model with the results from any other study of $\\pi^+\\pi^-p$ electroproduction amplitudes. \n\n\\subsection{Breit-Wigner parametrization of resonant amplitudes}\n\\label{regbw}\nWe start from a non-unitarized relativistic Breit-Wigner (BW) ansatz to \ndescribe the resonant contribution $\\langle \\lambda_{f} \\left| T_{r} \\right| \\lambda_{\\gamma}\\lambda_{p}\n\\rangle$ in the helicity representation:\n\\begin{eqnarray}\n\\label{regbwamp}\n\\langle \\lambda_{f} \\left| T_{r} \\right| \\lambda_{\\gamma}\\lambda_{p}\n\\rangle =\n\\sum_{N^{*}}\\frac{\\langle  \\lambda_{f} \\left| T_{dec} \\right|\\lambda_{R} \\rangle \n\\langle \\lambda_{R} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle}\n{M_{r}^{2}-W^{2}-i\\Gamma_{r}(W)M_{r}} ,\n\\end{eqnarray}\nwhere  \\(M_{r}\\) and \\(\\Gamma_{r}\\) are the resonance mass and \nenergy-dependent total width, respectively. The matrix elements\n\\(\\langle \\lambda_{R} \\left| T_{em} \\right|\\lambda_{\\gamma}\\lambda_{p}\\rangle\\) and     \\(\\langle  \\lambda_{f} \\left| T_{dec} \\right| \\lambda_{R} \\rangle\\)\nare the electromagnetic production and hadronic decay\namplitudes of the \\(N^{*}\\) with helicity\n\\(\\lambda_{R}=\\lambda_{\\gamma}-\\lambda_{p}\\), in which  $\\lambda_{\\gamma}$ and\n$\\lambda_{p}$ stand for the helicities of the photon and proton in the initial\nstate, and $\\lambda_{f}$\nrepresents the helicities of final-state hadrons in the $N^*$ decays.\n\n\nThe  hadronic decay\namplitudes $\\langle \\lambda_{f} \\vert T_{dec} \\vert \n\\lambda_{R} \\rangle$ are related to the $\\Gamma_{\\lambda_{f}}(W)$ partial hadronic decay widths of the $N^{*}$ \nto $\\pi \\Delta$ or $\\rho p$\nfinal states $f$ of helicity $\\lambda_{f}$ by: \n\\begin{equation}\n\\label{decaywidth1}\n\\langle \\lambda_{f} \\vert T_{dec} \\vert \\lambda_{R} \\rangle=\\langle \\lambda_{f} \\vert T^{J_{r}}_{dec} \\vert \n\\lambda_{R} \\rangle d^{J_{r}}_{\\mu\\nu}(\\cos\\theta^{*})e^{i\\mu\\phi^{*}}, \\nonumber \\\\\n\\end{equation}\nwith $\\mu=\\lambda_{R}$  and \n$\\nu=-\\lambda_{\\Delta}$   for the  $\\pi \\Delta$    intermediate state  \\\\\nand $\\nu=\\lambda_{p'}-\\lambda_{\\rho}$    for the  $\\rho$p$^{'}$ \nintermediate state, and \n\\begin{equation}\n\\label{decaywidth5} \n\\langle \\lambda_{f} \\vert T^{J_{r}}_{dec} \\vert \\lambda_{R}\n\\rangle=\\frac{2\\sqrt{2\\pi}\\sqrt{2J_{r}+1}M_{r}\\sqrt{\\Gamma_{\\lambda_{f}}}}{\\sqrt{\\langle\np^{r}_{i} \\rangle}}\\sqrt{\\frac{\\langle p^{r}_{i} \\rangle}{\\langle p_{i} \\rangle}}  . \n\\end{equation}\nThe means $\\langle p^{r}_{i} \\rangle$ and $\\langle p_{i} \\rangle$ are the \nmagnitudes of the \nthree-momenta  of the final $\\pi$ for the   \n$N^* \\rightarrow \\pi \\Delta$ decay ($i$=1) or of the final proton  \nfor the $N^* \\rightarrow \\rho p$ decay ($i$=2), evaluated at $W=M_{r}$ and \nat the running $W$, respectively, and  \naveraged over the running mass of the unstable hadron in the\nintermediate state. The variables $\\theta^{*}$, $\\phi^{*}$ are the CM  polar and azimuthal \nangles for the final $\\pi$ ($N^* \\rightarrow \\pi \\Delta$ decay) or the final\nproton ($N^* \\rightarrow \\rho p$ decay). The symbol $J_{r}$ stands for the $N^*$ spin. \nThis relationship between the $N^*$ hadronic decay\namplitudes and the partial decay widths was derived for the \n$N^* \\rightarrow \\pi \\Delta$ decay in our previous article \\cite{Ri00}. All details on the parametrization of the resonance \nhadronic decay amplitudes in the JM model  \ncan be found in Appendix~A of this paper.\n\nEnergy dependencies of the partial hadronic decay widths are described\nunder the assumption that centrifugal barrier-penetration factors are the major contributors to the off-shell behavior of resonance hadronic decay amplitudes \\cite{Ri00,Bla52}:  \n\\begin{gather}\n\\sqrt{\\Gamma_{LS}} = \\sqrt{\\Gamma^{r}_{LS}}\\cdot\n\\left[ \\frac{M_{r}}{W}\n\\frac{(J_{l}^{2}(p^{r}R) + N_{l}^{2}(p^{r}R))}\n{(J_{l}^{2}(pR) + N_{l}^{2}(pR))} \\right]^{1\/2},\n\\label{ww}\n\\end{gather}\nwhere \\(J_{l}\\) and \\(N_{l}\\) are the Bessel and Neumann functions. The  factor  in\nsquare  brackets represent the ratio of barrier penetration factors for the final \nmeson with orbital angular momentum \\(l\\) evaluated at $W=M_{r}$ and \nat the running $W$.  \nThe variable R represents the interaction radius whose value was set to 1 fm, \nand $\\sqrt{\\Gamma^{r}_{LS}}$ stands for the decay\namplitude estimated at the resonant point. The partial decay \namplitudes $\\sqrt{\\Gamma_{LS}}$ of Eq. (\\ref{ww}) \nare transformed from the $LS$ to the helicity representation\n$\\sqrt{\\Gamma_{\\lambda_{f}}}$\n(see the Appendix A) \nand used to compute the $N^*$ hadronic decay amplitudes via \nEq. (\\ref{decaywidth5}). \n\n\n\nIn the  JM model, the \\(N^{*}\\) total decay width \\(\\Gamma_{r}(W)\\) \nin Eq.~(\\ref{regbwamp}) is \nevaluated as a sum  \nover all partial decay widths. In this way we ensure unitarity of \nthe resonant amplitude for a single resonance\ncontribution. The unitarization procedure in the actual case of many contributing \nresonances will be\ndiscussed in Section \\ref{unitbw}. \n\n\nThe resonance electroexcitation amplitudes\n\\(\\langle \\lambda_{R} \\left| T_{em}\n\\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle\\) in Eq.~(\\ref{regbwamp}) are related to the \n$\\gamma_{v}NN^*$ electrocouplings \\(A_{1\/2}\\),\n\\(A_{3\/2}\\),\nand \\(S_{1\/2}\\) for nucleons. The definition of these electrocouplings in the JM model is consistent with\nthe Review of Particle Physics (RPP) \\cite{PDG10} relation between the \\(A_{1\/2}\\),\n\\(A_{3\/2}\\) electrocouplings and the $N^*$ electromagnetic decay\nwidth $\\Gamma_{\\gamma}$: \n\\begin{equation}\n\\label{a12a32width1} \n\\Gamma_{\\gamma}=\n\\frac{q^{2}_{\\gamma, r}}\n{\\pi}\\frac{2M_{N}}{(2J_{r}+1)M_{r}} \\left[ \\left | A_{1\/2} \\right |^{2}+\\left |\nA_{3\/2} \\right|^{2} \\right]  ,\n\\end{equation} \nwhere $q_{\\gamma, r}$ is the three-momentum modulus of the photon at $W=M_{r}$ \nin the CM frame. \nThe  \ntransition amplitudes \n$\\langle \\lambda_{R} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle$ are related to the $\\gamma_{v}NN^*$ $A_{1\/2}$, $A_{3\/2}$, and $S_{1\/2}$ electrocouplings by imposing the requirement that the \nBW parametrization \\cite{LS71} of the resonant cross section \nfor a single contributing resonance should be reproduced:\n\\begin{gather}\n\\label{secbw} \n\\sigma(W)=\\frac{\\pi}{q_{\\gamma}^2}(2J_{r}+1)\n\\frac{M^{2}_{r}\\Gamma_{i}(W)\\Gamma_{\\gamma}}\n{(M^{2}_{r}-W^{2})^{2}-M^{2}_{r}\\Gamma^{2}_{r}(W)}\\frac{q_{\\gamma}}{K}. \n\\end{gather}\nHere the photon equivalent energy \n$K=\\frac{W^2-M_{N}^{2}}{2W}$, $q_{\\gamma}$ is the absolute value of the initial photon three-momentum of\nvirtuality $Q^2$ $>$ 0  $q_{\\gamma}=\\sqrt{Q^2+E_{\\gamma}^2}$ and the energy $E_{\\gamma}$\nin the CM frame at the running $W$\n\\begin{eqnarray}\n\\label{ccc}\nE_{\\gamma}=\\frac{W^2-Q^2-M_{N}^2}{2W}. \n\\end{eqnarray}\nThe $q_{\\gamma, r}$ value in Eq. (\\ref{a12a32width1}) can be computed \nfrom\nEq. (\\ref{ccc}) at W=$M_{r}$.  \n$\\Gamma_{i}(W)$ is the energy-dependent hadronic decay width to the final state $\\pi\\Delta$ ($i$=1) \nor $\\rho p$ ($i$=2). The factor $\\frac{q_{\\gamma}}{K}$ in Eq. (\\ref{secbw}) is equal to\nunity at the photon point. It accounts for the choice \\cite{Fe09} of the virtual\nphoton flux in the evaluation of the virtual photon cross sections. \nIn this way we obtain the following relationship between the transition amplitudes \n$\\langle \\lambda_{R} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle$ and the \n$\\gamma_{v}NN^*$ electrocouplings:\n\\begin{eqnarray}\n\\label{a12a32sec}\n\\langle \\lambda_{R} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle  \n=\\frac{W}{M_{r}}\\sqrt{\\frac{8M_{N}M_{r}q_{\\gamma_{r}}}{4\\pi\\alpha}\n\\sqrt{\\frac{q_{\\gamma_{r}}}{q_{\\gamma}}}A_{1\/2,3\/2}(Q^{2}), \\nonumber \\\\\n\\text{with}\\left| \\lambda_{\\gamma}-\\lambda_{p} \\right| = \\frac{1}{2},\\, \\frac{3}{2} \\,\\, \n\\text{ for transverse photons, and}\\nonumber\\\\\n\\langle \\lambda_{R} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle \n=\\frac{W}{M_{r}}\\sqrt{\\frac{16M_{N}M_{r}q_{\\gamma_{r}}}{4\\pi\\alpha}\n\\sqrt{\\frac{q_{\\gamma_{r}}}{q_{\\gamma}}}\\, S_{1\/2}(Q^{2}) , \n\\nonumber \\\\\n \\text{for\\, longitudinal\\, photons.}\\nonumber\\\\\n\\end{eqnarray}\nThe factor\n$4\\pi\\alpha$ in Eqs. (\\ref{a12a32sec}) reflects the particular \nrelationship between the JM model amplitudes and cross sections \\cite{Mo09}, when\nthe absolute value of the electron charge is factorized out of the production\namplitudes. \n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Unitarization of the full resonant amplitudes}\n\\label{unitbw}\nThe BW ansatz, described  in Section~\\ref{regbw}, provides unitary contributions\nfrom any individual $N^*$ state to the resonant part of the electroproduction\namplitudes. \nHowever, the full resonant amplitude, which represents a superposition of contributions from all $N^*$ states,\nis not unitary. Figure \\ref{fullresamp}  schematizes the processes that contribute to the \nresonant propagator dressing. They consist of transitions between the same and different $N^*$ states mediated by the strong interaction. The regular BW ansatz\nincorporates only transitions between the same $N^*$ states shown \nby the blob in the top panel of  Fig.~\\ref{fullresamp}. The unitarization is achieved via inclusion of both diagrams\nin Fig. 3, allowing us to take into account \ntransitions between\nboth the same and \ndifferent $N^*$ states (bottom panel in Fig.~\\ref{fullresamp}) in the dressed\nresonant propagators. We replaced the usual BW ansatz by its \nunitarized extension originally\nproposed in Ref.~\\cite{Ait72}. This ansatz was adjusted for the description of \nthe BW resonance \nelectroproduction amplitudes of Eq. (\\ref{regbwamp}) employed in the JM model.  \nThe full resonant amplitude of the unitarized extension of the BW ansatz satisfies the unitarity condition~\\cite{Ait72}. \n\n\n\\begin{figure}[t]\n\\includegraphics[width=9cm]{pictures\/bw_unitar_prop1.eps}\n\\includegraphics[width=9cm]{pictures\/bw_unitar_prop.eps}\n\\caption{\\small Diagrams showing the unitarized BW ansatz of the JM model, which \nincorporates \ntransitions\nvia the dressed resonance propagator between states listed in \nTable \\ref{nstlist} for the same\n$N_{\\alpha}$=$N_{\\beta}$ (top) and different  \n$N_{\\alpha} \\ne N_{\\beta}$ resonances as allowed by the conservation \nlaws of the strong interaction (bottom).}\n\\label{fullresamp}\n\\end{figure}\n\n\nUnitarized resonance amplitudes in the JM model incorporate  photo- and \nelectro-excitation amplitudes of all relevant\nresonances $\\alpha$, all possible transitions between the initial $N^*$ state $\\alpha$ and\nthe final $N^*$ state $\\beta$, and hadronic decays of  $\\beta$. Fully unitarized resonance amplitudes are determined by the sum  \nof the products of the electromagnetic excitation amplitudes of the $\\alpha$-th $N^*$ state \n$\\langle \\lambda_{\\alpha} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle$, the hadronic decay \namplitudes of the $\\beta$-th $N^*$ state \n$\\langle  \\lambda_{f} \\left| T_{dec} \\right|\\lambda_{\\beta} \\rangle$, \nthe dressed\npropagator $S_{\\alpha \\beta}$, and\n\\begin{gather}\n\\langle \\lambda_{f} \\left| T_{res} \\right| \\lambda_{\\gamma}\\lambda_{p}\n\\rangle =\n\\sum_{\\alpha,\\beta}\\langle \\lambda_{f} \\left| T_{dec} \\right|\n\\lambda_{\\beta} \\rangle \n S_{\\alpha \\beta} \n\\langle \\lambda_{\\alpha} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle.\n\\label{matel}\n\\end{gather}\n The sum incorporates all transitions between the $\\alpha$-th initial and the $\\beta$-th\nfinal $N^*$ states that are allowed by quantum number conservation laws.\nIt runs over all repeated indices $\\alpha$ and $\\beta$ that label the $N^*$ states. \nAs a consequence of angular momentum conservation \n$\\lambda_{\\alpha}$=$\\lambda_{\\beta}$, but these two helicities stand either for the same $N^*$ states\nif $\\alpha$=$\\beta$ (diagonal transitions) or different $N^*$ states if $\\alpha \\ne \\beta$ \n(off-diagonal transitions).  \nThe index $f$ represents the final-state helicities, either for $\\pi \\Delta$ or for $\\rho p$. \n\n\nThe expression for the inverse dressed propagator $S^{-1}_{\\alpha \\beta}$ is \nobtained  in Ref.~\\cite{Ait72} (see Eq.~(6) in Ref.~\\cite{Ait72}). However, the parametrization \nof the $N^*$ propagator in Ref.~\\cite{Ait72} and in JM  are different.\n  Therefore, we change the inverse resonant propagator \n  $S^{-1}_{\\alpha \\beta}$ of  Ref.~\\cite{Ait72}, so that \nit coincides with the single $N^*$ contribution employed in the JM model.   \nIn this way we obtain \n\n\\begin{eqnarray}\n\\label{transel}\nS^{-1}_{\\alpha \\beta}=M^{2}_{N^*}\\delta{\\alpha\\beta}\n-i(\\sum_{k}\\sqrt{\\Gamma_{\\alpha_{k}}\\Gamma_{\\beta_{k}}})\n\\sqrt{M_{N^*_{\\alpha}}M_{N^*_{\\beta}}} \\\\\n-W^2\\delta_{\\alpha\\beta} \\nonumber,\n\\end{eqnarray}\nwhere the index $k$ represents the partial $N^*$ hadronic decay widths to all possible \nfinal states, decomposed over \n$LS$ partial waves. For a single $N^*$ contribution,\n Eq.~(\\ref{transel}) \ncoincides with the inverse BW propagator in Eq.~(\\ref{regbwamp}). \n\n\n\n\n\nThe resonant contribution at $W$ $<$ 1.6 GeV  incorporates \nthe $N^*$ states listed in Table~\\ref{nstlist} and  the $3\/2^+(1720)$ candidate state\nobserved in the previous analysis of the CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Ri03}. The conservation laws in strong interactions allow transitions in dressed resonant propagators only\nbetween the following pairs of $N^*$ states: $D_{13}(1520)$ and $D_{13}(1700)$,\n $S_{11}(1535)$ and $S_{11}(1650)$, and  \n $3\/2^+(1720)$ candidate and $P_{13}(1720)$ (bottom panel in Fig.~\\ref{fullresamp}). \nTherefore, the unitarized resonant amplitude represents the sum of\nregular BW amplitudes over all other $N^*$ states with only diagonal transitions in the dressed\nresonance propagators plus contributions from the\naforementioned pairs of $N^*$ states that have both diagonal and non-diagonal transitions in the resonant\npropagators. The contributions  from each pair of $N^*$ states \n$\\langle \\lambda_{f} \\left| T_{2res} \\right| \\lambda_{\\gamma}\\lambda_{p}\n\\rangle$ \nare determined by  Eq.~(\\ref{matel}) with indices $\\alpha$ and $\\beta$ running from\n1 to 2, resulting in 2x2 $S^{-1}_{\\alpha \\beta}$ matrices. \nAfter invertion of  $S^{-1}_{\\alpha \\beta}$ in Eq.~(\\ref{transel}) and insertion  \ninto Eq.~(\\ref{matel}),\nwe get the following expressions for the contribution from each of the three pairs of \n$N^*$ states listed in Fig.~\\ref{fullresamp}: \n\n\\begin{gather}\n\\langle \\lambda_{f} \\left| T_{2res} \\right| \\lambda_{\\gamma}\\lambda_{p}\n\\rangle \n=  \\frac{1}{det[S^{-1}_{\\alpha \\beta}]} * \\nonumber\\\\\n\\{\\langle \\lambda_{f} \\left| T_{dec} \\right|\n\\lambda_{1} \\rangle \n(M^2_{2}-i\\Gamma_{2}(W)M_{2}-W^2) \n\\langle \\lambda_{1} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle \\nonumber \\\\\n+\\langle \\lambda_{f} \\left| T_{dec} \\right|\\lambda_{2} \\rangle \n(M^2_{1}-i\\Gamma_{1}(W)M_{1}-W^2)\n\\langle \\lambda_{2} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle \\nonumber \\\\\n+i\\langle \\lambda_{f} \\left| T_{dec} \\right|\n\\lambda_{2} \\rangle \\sum_{k}\\sqrt{\\Gamma_{1_{k}}\\Gamma_{2_{k}}} \n\\sqrt{M_{N^*_{1}}M_{N^*_{2}}}\\langle \\lambda_{1} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p} \\rangle \\nonumber\\\\ \n+i\\langle \\lambda_{f} \\left| T_{dec} \\right|\n\\lambda_{1} \\rangle\\sum_{k}\\sqrt{\\Gamma_{1_{k}}\\Gamma_{2_{k}}} \n\\sqrt{M_{N^*_{1}}M_{N^*_{2}}}\\langle \\lambda_{2} \\left| T_{em} \\right| \\lambda_{\\gamma}\\lambda_{p}\n\\rangle \\}  .\n\\label{amp2res}\n\\end{gather} \nThe first two terms in Eq.~(\\ref{amp2res}) correspond to \nthe diagonal $S_{\\alpha \\beta}$ elements. They describe the processes depicted \nin  the top panel \nof  Fig.~\\ref{fullresamp}. The two other terms correspond to transition between different $N^*$ states, \nshown in the bottom panel of Fig.~\\ref{fullresamp}. The determinant of \n$S^{-1}_{\\alpha \\beta}$ can be computed from Eq. (\\ref{transel}) as\n\\begin{gather}\ndet[S^{-1}_{\\alpha\n\\beta}]=(M^2_{1}-i\\Gamma_{1}(W)M_{1}-W^2)\\cdot \\nonumber\\\\\n(M^2_{2}-i\\Gamma_{2}(W)M_{2}-W^2)\n+(\\sum_{k}\\sqrt{\\Gamma_{1_{k}}\\Gamma_{2_{k}}})^2M_{1}M_{2}.\n\\label{determ}\n\\end{gather} \nIt follows from Eqs. (\\ref{amp2res}) and (\\ref{determ}), that if transitions between two different \n$N^*$ states in the dressed resonant propagators become impossible, then\n\\begin{equation}\n\\sum_{k}\\sqrt{\\Gamma_{1_{k}}\\Gamma_{2_{k}}}=0,\n\\end{equation}\nnamely, all off-diagonal terms become zero, while the diagonal terms give rise to the amplitudes of the regular \nBW ansatz. When the transitions between the pairs of $N^*$ states are turned on, they not only add off-diagonal\nterms to Eq.~(\\ref{amp2res}), but they also change the effective propagator in the \n$\\langle \\lambda_{f} \\left| T_{2res} \\right| \\lambda_{\\gamma}\\lambda_{p}\\rangle$ amplitude with respect to the BW ansatz Eq.~(\\ref{regbwamp}), determined by\n 1\/$det[S^{-1}_{\\alpha \\beta}]$. \n\n\\section{The CLAS data fit}\n\\label{fit}\n\nThe resonance parameters obtained in this paper \n are fit to the CLAS $\\pi^+\\pi^-p$ electroproduction \ndifferential cross sections  \\cite{Fe09}. A realistic\nevaluation of resonance electrocoupling uncertainties extracted in the fit\nis an important objective of our analysis. We require that \nthe ranges of\nresonance electrocouplings extracted from the $\\pi^+\\pi^-p$ electroproduction\nchannel take into account both uncertainties in the measured differential cross sections\nand in the JM model parameters. In order to provide a realistic\nevaluation of resonance parameters, we abandoned\nthe traditional least-squares fit, since the parameters extracted in such a fit\ncorrespond to a single presumed global minimum, while the experimental\ndata description achieved with others local minima may be equally good \nwithin the data error bars. Furthermore, the traditional evaluation of\nfit-parameter uncertainties, based on the error propagation matrix, \ncannot be used for the same reason.\n\n\n\n\n\n A special \n procedure was developed to obtain not only the best fit, \n but also to establish bands \n of computed cross sections that are compatible with the data within their \n uncertainties. In the fit we vary simultaneously \n non-resonant and resonant parameters of the JM model\n given in Tables~\\ref{nstlist} and ~\\ref{bckpar}, respectively,\n around their start values, employing unrestricted normal distributions. \n The choice of the start\n parameters and the normal distribution $\\sigma$-values employed for\n parameter variation will be further described below. For each trial set of the JM-model resonant and non-resonant\n parameters we  compute nine measured one-fold differential $\\pi^+\\pi^-p$ \n cross sections, and \n the $\\chi^2$ per data point values ($\\chi^2$\/$d.p.$). The $\\chi^2$\/$d.p.$\n values were estimated in point-by-point comparisons between the  \n measured and computed \n one-fold\n differential cross sections in all bins of $W$ and $Q^2$ \n covered by the CLAS $\\pi^+\\pi^-p$ data of Ref. \\cite{Fe09}. In the fit we \n selected computed one-fold differential \n cross\n sections closest to the data with  $\\chi^2\/d.p.$ less than a predetermined maximum \n value $\\chi^2_{max.}\/d.p.$. \n The values of  $\\chi^2_{max.}\/d.p.$ were obtained by requiring that the \n computed cross\n sections with smaller $\\chi^2\/d.p.$ be within the data \n uncertainties for the majority of the data points, based on point-by-point\n comparisons between the measured and the computed cross sections (see examples \n in Figs.~\\ref{9secnstbck},~\\ref{9secnstbck1}). In this fit procedure we obtain the \n $\\chi^2\/d.p.$ intervals within which the computed cross sections \n describe the data equally well within the data uncertainties.\n\n\n\n The resonance parameters obtained from all these equally good fits are averaged \n and their mean\n values are taken as the resonance parameters extracted from the data. \n Dispersions in these parameters are taken as the uncertainties. \n Our fitting \n procedure  allows us to obtain more realistic uncertainties \n of the fit parameters than from the usual least-squares method. \n In this\n way we take into\n account both statistical uncertainties in the data and  \n systematic uncertainties imposed by the use of the JM reaction model. \n\n\n  \n\n\n\n  \n\nWe vary the parameters of the JM model, listed in\nTable~\\ref{bckpar} for the non-resonant \nmechanisms, which are relevant in describing the \nCLAS $\\pi^+\\pi^-p$ data \\cite{Fe09}. They are limited to non-resonant $\\pi \\Delta$\nsub-channels and direct $2\\pi$-electroproduction. \n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n                      & Ranges covered  \\\\\n                      &  in variations of the  \\\\ \nVariable \\, parameters &  start parameters,  \\\\\n\t\t       & \\% from their values \\\\   \t\t       \t\t                   \n\\hline\nMagnitude of the additional &     \\\\\ncontact term amplitude in the & 45.0                 \\\\\n$\\pi^- \\Delta^{++}$ sub-channel &                \\\\  \n\\hline\nMagnitude of the additional &     \\\\\ncontact term amplitude in the &  60.0        \\\\\n$\\pi^+ \\Delta^{0}$ sub-channel &                \\\\\n\\hline\nMagnitudes of six &    \\\\\n2$\\pi$ direct production&      30.0    \\\\\namplitudes                        &    \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{bckpar} Variable parameters of non-resonant mechanisms\nincorporated into the JM model \\cite{Mo09}. The ranges in the table correspond\nto the 3$\\sigma$ areas around the start values of parameters.}\n\\end{center}\n\\end{table}\n\n\n \n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n$N^{*}$ states   & Mass, & Total\\; decay\\; width,\\; $\\Gamma_{tot}$, \\\\\n            &  (MeV)  &   (MeV)   \\\\\n\\hline\n$P_{11}(1440)$ & 1430-1480 & 200-450   \\\\\n$D_{13}(1520)$ & 1515-1530 & 100-150  \\\\\n$S_{11}(1535)$ & 1510-1560 & 100-200  \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{hadrrange} Allowed $N^*$ hadronic parameter variations \nin the fit of the CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Fe09}.}\n\\end{center}\n\\end{table}\n\n\n\nThe magnitudes of the additional contact terms in the $\\pi^{-} \\Delta^{++}$ and\n$\\pi^{+} \\Delta^{0}$ isobar channels (row (b) in Fig.~\\ref{mech_low_06}), \nas well as  the magnitudes of all direct\n2$\\pi$-production amplitudes (rows (c),(d) in Fig.~\\ref{mech_low_06}), \n are chosen as variable parameters. \nAll these parameters\nwere determined in the fit to the CLAS data on $\\pi^+\\pi^-p$ electroproduction\nin our previous analysis \\cite{Mo09} without variation of the resonance parameters. In the CLAS data fit presented in this\npaper, the values of the aforementioned non-resonant\nparameters of the JM model are re-evaluated under simultaneous variation of:\n\\begin{itemize}\n\\item the magnitudes of \nadditional contact-term amplitudes in the $\\pi^{-} \\Delta^{++}$ and\n$\\pi^{+} \\Delta^{0}$ isobar channels (2 parameters per $Q^2$ bin);\n\\item the magnitudes of all direct\n2$\\pi$-production amplitudes (6 parameters per $Q^2$ bin);\n\\item the resonant\nparameters listed in Table~\\ref{nstlist}. The CLAS $\\pi^+\\pi^-p$ data \\cite{Fe09} are mostly sensitive\nto the electrocouplings of the $P_{11}(1440)$ and $D_{13}(1520)$ states \n(5 resonance electrocouplings per $Q^2$ bin), to the \n$\\pi \\Delta$ and $\\rho p$ hadronic decay widths of these two resonances, and\nto the $S_{11}(1535)$ state \n(6 parameters that remain the \nsame in the entire $Q^2$ area covered by the measurements). \n\\end{itemize}\n\nTherefore, we consistently account for the\ncorrelations between variations of the non-resonant and the resonant contributions \nwhile extracting \nthe resonant parameters.    \n\n\n The $W$-dependences of the magnitudes of the additional contact-term \n amplitudes \nin the $\\pi \\Delta$\nsub-channels and of the magnitudes of the direct  $2\\pi$-production amplitudes \nare adopted from our previous analysis \\cite{Mo09}. \nWe apply\nmultiplicative factors to the magnitudes of the extra contact-term amplitudes \nand the 2$\\pi$ direct-production amplitudes. \nThe multiplicative factors are $W$-independent within any $Q^2$-bin, \nbut they are fit to the data in each $Q^2$-bin independently.\n In this way we retain the smooth $W$-dependences of the non-resonant contributions\nestablished in our previous analysis \\cite{Mo09}. \n\n\n\n\n\n\nWe use two parameters for the variation of the magnitudes of additional \ncontact-term amplitudes \nin the $\\pi^-\\Delta^{++}$ and $\\pi^+\\Delta^0$\nisobar channels.  \nHowever, the non-resonant parameters in the $\\pi^+\\Delta^0$ \nisobar channel have a rather small impact on the fit, since \nthe non-resonant contributions of the $\\pi^+\\Delta^0$ isobar channel are approximately a \nfactor 9 smaller than that of the $\\pi^-\\Delta^{++}$.\n\n\n\n\n\nThe parameters for the non-resonant Born terms in the $\\pi \\Delta$ sub-channels ((b)-row\nin Fig.~\\ref{mech_low_06}) include the $p \\pi \\Delta$ coupling and\ncut-off for this hadron transition form factor, as well as the electromagnetic \npion and nucleon form factors \\cite{Ri00}. All these parameters \nare taken from previous studies \nof meson\nphoto-, electro-, and hadroproduction referred to in Ref. \\cite{Mo09} and are kept fixed\nfor\nthe current fit. \n\n\nDifferent assignments of final hadrons in $2\\pi$-direct production\nmechanisms, as shown in \nFig.~\\ref{mech_low_06} (rows (c),(d)), result in 12 different sub-processes. Our\nprevious analysis \\cite{Mo09} demonstrated that the magnitudes of all these sub-processes are determined by six independent parameters.\nIn the current fit we vary these six parameters of the direct 2$\\pi$-production amplitudes. \n\n\n\n\n\n\n\nIn the fitting procedure described above, nine one-fold $\\pi^+\\pi^-p$ differential\n cross sections are computed with non-resonant variable \nparameters of the JM model obtained by employing unrestricted normal distributions \naround their start values \nwith $\\sigma$-parameters of 10\\%-20\\% of their start values \n(see Table ~\\ref{bckpar}). In this way we explore mostly the range \nof $\\approx$ 3$\\sigma$ around the start parameter values. These ranges for\nnon-resonant \nparameter variations in the JM model are  shown in Table~\\ref{bckpar} as a percentage of their start values. \n \n\n\n\n\n\n  \n\n\n\n\nIn this fit we also vary the $\\gamma_{v}pN^*$\nelectrocouplings and the $\\pi \\Delta$ and $\\rho p$ hadronic partial decay widths \nof the \n$P_{11}(1440)$ and $D_{13}(1520)$ resonances \naround their start\nvalues. \nThe start \nvalues of the $P_{11}(1440)$ and\n$D_{13}(1520)$ electrocouplings are determined by interpolating \nthe results from the analysis \\cite{Az051} of the CLAS data on $N\\pi$ \nelectroproduction off protons into the range 0.4 GeV$^2$ $<$ $Q^2$ $<$ 0.6 GeV$^2$ and extrapolating these\nresults into the $Q^2$-area from 0.25 GeV$^2$ to 0.4 GeV$^2$. \nThe electrocouplings \nof the $P_{11}(1440)$ and\n$D_{13}(1520)$ resonances\n are varied employing normal distributions with\n$\\sigma$ parameters equal to 30\\% of their start values. There are no\nrestrictions on the minimum or maximum trial electrocoupling values. \nThe normal distributions allow us to explore mostly \nthe area of $\\approx$ 3$\\sigma$ around their start values or \n90\\% around electrocoupling start values.\n\n \n\nThe $\\pi^+\\pi^-p$ electroproduction channel has also some \nsensitivity to the $S_{11}(1535)$ state, \nwhich couples dominantly  to the $N\\pi$ and $N\\eta$ final states. The \n$S_{11}(1535)$ electrocouplings were taken from \nthe CLAS analysis\nof $N \\pi$ electroproduction \\cite{Az09} and varied strictly inside \nthe uncertainties reported in that\npaper. \n\n\n\n\n\n\nThe $\\pi \\Delta$ and $\\rho p$ hadronic decay widths of the $P_{11}(1440)$,\n$D_{13}(1520)$, and $S_{11}(1535)$ resonances are varied around their start values taken from previous \nanalyses of the CLAS double pion electroproduction data \\cite{Az05,Mo06,Mo06-1} in ranges restricted \nby the total\n$N^*$ decay widths and their uncertainties, as shown in Table~\\ref{hadrrange}. \nThe total $N^{*}$ decay widths were \nobtained by summing \nthe partial widths over all decay channels. Partial hadronic \ndecay widths to all final states other than $\\pi \\Delta$ and $\\rho p$ are\ncomputed as the products of RPP \\cite{PDG10} values of $N^*$ total decay widths and \nbranching fractions for decays to particular hadronic final states, that are taken from analyses \\cite{Man92} of hadroproduction experiments.  \nWe varied the $\\pi \\Delta$ and $\\rho p$ hadronic decay widths of the \n$P_{11}(1440)$,\n$D_{13}(1520)$, and $S_{11}(1535)$ resonances simultaneously with their \nmasses, keeping the hadronic \n$N^*$ parameters independent of $Q^2$. Accounting for the correlations of the $N^*$ \nelectromagnetic and hadronic decay\nparameters in a combined variation is important for a credible extraction of the resonance parameters and, in particular, for the evaluation \nof their uncertainties. \n\n\nThe here developed fit procedure allows us to determine the $\\pi \\Delta$ and $\\rho p$ hadronic decay widths \nof the $P_{11}(1440)$, $D_{13}(1520)$, and $S_{11}(1535)$. The resonance total decay widths are affected \nby\nthe variation of their $\\pi \\Delta$ and $\\rho p$ partial decay widths. Instead, when varying resonance electrocuplings, the $N^*$ \ntotal decay widths of resonances are kept almost unchanged. Resonance hadronic decay\namplitudes are independent of $Q^2$, while resonance electrocouplings represent the functional dependencies on the photon virtualities. \nThese distinctive features allow us to disentangle resonance electromagnetic\nand hadronic decay amplitudes in the fit of the CLAS data \\cite{Fe09} and to obtain  \nthe $\\pi \\Delta$ and $\\rho p$ hadronic decay\nwidths of the $P_{11}(1440)$, $D_{13}(1520)$, and $S_{11}(1535)$.\n\nTo account for the contributions from the tails of higher mass\nresonances, electrocouplings of all $N^*$ states marked in Table~\\ref{nstlist} \nas ``var\" are varied around their start values. For resonances with masses \nabove 1.6 GeV, except the $S_{11}(1650)$ state, the start values \nfor electrocouplings are taken from the results of \nprevious analyses  \\cite{Az05,Mo06} of $\\pi^+\\pi^-p$\nelectroproduction data \\cite{Ri03} after the extrapolation \ninto the $Q^2$ range of the current analysis (0.25-0.6 GeV$^2$). \nThe $S_{11}(1650)$ state may have a more pronounced impact on the extraction of\nthe $P_{11}(1440)$ and $D_{13}(1520)$ resonance parameters, since\n the $S_{11}(1535)$ and $S_{11}(1650)$ states are mixed in the \n dressed resonance propagator of the unitarized BW ansatz employed in the JM model.\nFor this reason we prefer to have more accurate start\nvalues for the $S_{11}(1650)$ electrocouplings. They are computed within the \nframework of the single quark transition model (SQTM) \\cite{Bu03} from electrocouplings of \nthe $S_{11}(1535)$ state that are available from analyses of $N\\pi$\nand $N\\eta$ electroproduction data \\cite{Bu11b}.\nElectrocouplings of the $D_{13}(1700)$ and \n$D_{33}(1700)$ resonances are kept\nfixed at their start values, since the masses of these resonances are far \noutside of\nthe $W$ area covered by the CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Fe09} $W$ $<$ 1.6 GeV.\n\nThe $\\pi \\Delta$ and $\\rho p$ partial hadronic decay widths for $N^{*}$ states\nother than the $P_{11}(1440)$,\n$D_{13}(1520)$, and $S_{11}(1535)$ resonances are taken from previous \nanalyses of the CLAS double pion electroproduction data \\cite{Az05,Mo06,Mo06-1}.  They are in  a reasonable agreement\nwith the values reported in the RPP \\cite{PDG10}. The respective branching fractions \nare listed in\nTable~\\ref{nstlist}. \n\n  \n\n \n\n\n \n\n\n\n\n\n\n \n\n\n    \n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n $Q^{2}$ bins, & $W$ intervals, & Number of fitted  &\n$\\chi^2\/d.p.$  intervals\\\\\n (GeV$^2$)&  (GeV)  & points   & for selected \n  \\\\\n & & & cross sections \\\\ \n\\hline\n0.25-0.30 & 1.44-1.56 & 378 & $~~~$  \\\\\n0.30-0.35 & 1.41-1.54 & 378 & 2.66-2.74  \\\\\n0.35-0.40 & 1.41-1.54 & 378 & $~~~$  \\\\\n\\hline\n0.40-0.45 & 1.39-1.51 & 373 & 1.87-2.04  \\\\\n0.45-0.50 & 1.39-1.49 & 309 & $~~~$  \\\\\n\\hline\n0.50-0.55 & 1.39-1.51 & 360 & 1.57-1.75  \\\\\n0.55-0.60 & 1.34-1.44 & 279 & $~~~$  \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{fitqual} Quality of the CLAS $\\pi^+\\pi^-p$  \nfit within the framework \nof the JM model \\cite{Mo09}. Horizontal lines separate the three \n$Q^2$-intervals, in which the fits were carried-out independently.}\n\\end{center}\n\\end{table}\n\n\n\n \nWe\n fit the CLAS data \\cite{Fe09} consisting of nine differential cross sections \n of the\n $ep \\rightarrow e'p'\\pi^+\\pi^-$ electroproduction reaction \n in all bins of $W$ and $Q^2$ in \n the kinematic region: 1.35 GeV $<$ $W$ $<$ 1.57 GeV and 0.25 GeV$^2$ $<$ $Q^2$\n $<$ 0.6 GeV$^2$ within the framework of the fit procedure described above. Three\n intervals of $Q^2$ separated in Table~\\ref{fitqual} by horizontal solid lines are fit independently.   \n The $\\chi^2\/d.p.$ intervals, that correspond to \n equally good data description\n within the error bars are shown in Table~\\ref{fitqual}. Their values\n demonstrate the quality of the CLAS $\\pi^+\\pi^-p$ data \n description achieved in the fits. Examining the description \n of the nine one-fold\n differential cross sections, we found that the $\\chi^2\/d.p.$ values \n are determined mostly by the random deviations \n of some experimental data points from the bunches of computed fit \n cross sections. There are no discrepancies in describing the\n shapes of the differential cross sections, which would manifest themselves \n systematically in neighboring \n bins of $W$ and $Q^2$. \n Typical examples  for $W$=1.51 GeV and neighboring  $Q^2$ intervals\n centered at 0.38 GeV$^2$ and 0.42 GeV$^2$  are \n shown in Fig.~\\ref{9secnstbck} and Fig.~\\ref{9secnstbck1}, respectively.\n \n \n\n \nThe sets of computed differential cross sections with  $\\chi^2\/d.p.$ values within \nthe intervals given in the Table~\\ref{fitqual} offer the best data descriptions achievable \nwithin the framework of \nthe JM model.  \nThe minimal values of $\\chi^2\/d.p.$  in each interval represent the global minima \nof these fits. \nWe found that the increases of the $\\chi^2\/d.p.$ values within the intervals listed in the Table~\\ref{fitqual} change \nthe computed \ndifferential cross sections, but still keep them inside the data uncertainties, offering equally good data descriptions in \nall these fits with different sets of JM model parameters. \nThe mean values of the resonance parameters from these sets and their dispersions (statistical r.m.s.) are determining the resonance parameters and \ntheir uncertainties, as it was described at the beginning of this Section. \n\n\n        \nSince only statistical data uncertainties are used \n in the computation of the $\\chi^2\/d.p.$ values listed in the Table~\\ref{fitqual}, \n we concluded \n that a reasonable data description was\n achieved.  \n The $\\chi^2\/d.p.$ values of our fits \n are comparable with those obtained in the fit \n of the CLAS $N\\pi$ electroproduction data published in \\cite{Az09}, as well as with those obtained \n in the MAID analysis \\cite{Tia07}.  \n \n\n \n\n\n\\section{Evaluation of $\\gamma_{v}pN^*$ electrocouplings and resonance hadronic decay parameters}\n\\label{nstarelectrocoupl}\n\n\n\nThe extraction of resonance electrocouplings using the JM model \nrelies on fitting resonant \nand non-resonant contributions to measured differential cross sections. Therefore, we first have to check the quality of the \nseparation between  resonant and\nnon-resonant contributions achieved in the data fit. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Separation of resonant and non-resonant contributions to the $\\pi^+\\pi^-p$ cross sections}\n\\label{nstarbackgr} \nWith the parameters determined from the fits, we now use the JM model \nto evaluate \nthe contributions from resonant and non-resonant parts to the cross sections. \nThis is done by computing the nine differential \ncross sections\nwithout the resonant parts  and with only the resonant parts for all \n trial differential cross sections of the JM model selected in the fit. \nIn this way we also determine the ranges of the \nresonant and non-resonant contributions to the cross sections \nas imposed by the uncertainties of the experimental\ndata. These ranges account for the uncertainties of the non-resonant\nparameters listed in Table ~\\ref{bckpar} and for all correlations between\nresonance and non-resonant amplitudes. Therefore, we obtain reliable \nestimates for the uncertainties of the resonant and non-resonant contributions to the differential cross sections. \nExamples of the separated resonant and non-resonant contributions \nfor two particular $W$ and $Q^2$ bins are shown \nin Figs.~\\ref{9secnstbck} and~\\ref{9secnstbck1}. The interference \nbetween the resonant and non-resonant amplitudes is clearly seen in all\nangular distributions. The differences between the fitted\n$\\pi^+\\pi^-p$ differential cross sections and their non-resonant parts are \nlarger than the resonant contributions for a majority of the data points in \nthe angular distributions of the final hadrons, offering clear evidence for  \nthe interference between resonant and non-resonant\namplitudes. This interference amplifies\nthe $N^*$ contributions to all CM-angular distributions \nd$\\sigma$\/d$(-cos(\\theta_{i}))$ ($i=\\pi^+, \\pi^-, p$)\nof the final hadrons, which improves the sensitivity of the fit to the \nresonance parameters.  Furthermore, the shapes \nof the resonant and non-resonant  differential cross sections  \nare rather different, especially for the final-state  \nangular distributions. Substantial differences in the shapes of the \nresonant\/non-resonant contributions and their interference allow us \nto isolate the resonant contribution in a\ncombined fit of all nine one-fold differential cross sections despite the relatively \nsmall resonant contributions to the fully integrated cross sections. \nIn our previous analysis \\cite{Fe09} we found resonant contributions to the fully \nintegrated cross sections from 10\\% to 30\\%.\n\n\n\\begin{figure*}[btp]\n\\includegraphics[width=18cm,height=20cm]\n{pictures\/nstar_backgr_275375_0.375w_15_1525.eps}\n\\caption{\\small(color online) Resonant (blue triangles) and non-resonant (green open circles) contributions \nto the differential cross sections (red lines) obtained from the CLAS data \\cite{Fe09}, fit within the framework of \nthe JM model at\n$W$= 1.51 GeV, $Q^2$=0.38 GeV$^2$. The solid blue and dotted-dashed green lines stand for the resonant and non-resonant contributions, respectively, which were computed  for minimal $\\chi^2\/d.p.$ achieved in the data fit.  The points for resonant and non-resonant contributions are shifted on each panel for better visibility. Dashed lines show selected  \nfits.}\n\\label{9secnstbck}\n\\end{figure*}\n\n\n\n\\begin{figure*}[btp]\n\\includegraphics[width=18cm,height=20cm]\n{pictures\/nstar_backgr_channels_425475_0.425w_15_1525.eps}\n\\caption{\\small(color online) The same as in Fig.~\\ref{9secnstbck} at\n$W$= 1.51 GeV, $Q^2$=0.43 GeV$^2$.}\n\\label{9secnstbck1}\n\\end{figure*}  \n\n\n\nThe examples of Figs.~\\ref{9secnstbck} and~\\ref{9secnstbck1} demonstrate \nthat the uncertainties  \nof the resonance parts \nare comparable with those of the experimental\ndata. This is further evidence for the credible separation \nbetween the resonant and non-resonant contributions.\nAny ambiguities in the evaluation of these two contributions would\nresult in larger uncertainties for the resonant and\/or non-resonant\nparts derived from the fit than the uncertainties of the original data. But this is not the case in the entire kinematical area covered by the\nCLAS $\\pi^+\\pi^-p$ electroproduction measurements of Ref.~\\cite{Fe09}. \nTherefore, these data provide enough constraints in order to determine the resonant contributions to all differential cross sections.\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$N^*$ parameters from the fit of the $\\rm \\pi^+\\pi^-p$ electroproduction cross sections}\n\\label{resonances}  \n\n\n\n\n\n\n\n\n\n \n \n\n The procedure described in Section~\\ref{fit} allows us to extract the $\\gamma_{v}pN^*$ electrocouplings and their uncertainties. \nA special approach was developed for the evaluation \nof the $P_{11}(1440)$ electrocouplings. According to the analysis \\cite{Az09} of\nthe CLAS $N\\pi$ electroproduction data, the $A_{1\/2}$ electrocoupling of this\nresonance changes sign between $Q^2$ of 0.40 GeV$^2$ and 0.45 GeV$^2$. \nHere the $A_{1\/2}$ electrocoupling is close to zero and is an order of \nmagnitude smaller \nthan $S_{1\/2}$. The $A_{1\/2}$ variations computed as a \npercentage of the start value, which is close to\nzero, become too small. For realistic uncertainty \nestimates we varied $A_{1\/2}$ for 0.4 GeV$^2$ $<$ $Q^2$ $<$ 0.5\nGeV$^2$ in the ranges shown in  Table~\\ref{range}. \nBy varying $A_{1\/2}$ inside these widened ranges, we scanned the trial\nvalues comparable with those of $S_{1\/2}$, as they were obtained in the\nanalysis \\cite{Az09} of the CLAS $N\\pi$ electroproduction data.  \nWe refit the CLAS data of Ref.~\\cite{Fe09} on $\\pi^+\\pi^-p$ electroproduction by varying\n$A_{1\/2}$, as described above, keeping the variation of all other resonant and non-resonant parameters as described in Section~\\ref{fit}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n$Q^2$ bins,         & Ranges covered  \\\\\n(GeV$^2$)                    &     in variations of $A_{1\/2}$                                       \\\\ \n                    &  electrocoupling of the \\\\\n\t\t      &    $P_{11}(1440)$ resonance, \\\\ \n\t\t     & (10$^{-3}$ GeV$^{-1\/2}$) \\\\   \t\t       \t\t                   \n\\hline\n0.40-0.45            & -20 \\, \\,-\\,\\, +5     \\\\\n\n0.45-0.50            &  -5 \\, \\,-\\,\\, 20   \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{range} Variation range for the \n$P_{11}(1440)$ $A_{1\/2}$ electrocoupling at photon\nvirtualities where this electrocoupling changes sign.}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=8.cm]{pictures\/p11_a12_1pi2pi.eps}\n\\includegraphics[width=8.cm]{pictures\/p11_s12_1pi2pi.eps}\n\n\\caption{\\small(color online) Electrocouplings and full error bars of the $P_{11}(1440)$ state. \nThe results from the analysis of \nthe CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Fe09} are shown by triangles. \nElectrocouplings from the analysis of the $N\\pi$ electroproduction \ndata \\cite{Az09} are shown by circles.}  \n\\label{p11_23}\n\\end{center}\n\\end{figure} \n\n\n\n\n\nIn order to compare our results on the $P_{11}(1440)$ and $D_{13}(1520)$ \nelectrocouplings in the $\\pi^+\\pi^-p$ \nelectroproduction channel with their values from the analysis of \n$N\\pi$ electroproduction \\cite{Az09}, we have to use in both of the exclusive\nelectroproduction channels common branching\nfractions for decays of these resonances to the $N\\pi$ and $N\\pi\\pi$ \nfinal states. According to the RPP \\cite{PDG10}, the sum of the branching \nfractions into the $N\\pi$ and $N\\pi\\pi$ final states accounts for almost 100\\% of the total decay widths of the  $P_{11}(1440)$ and $D_{13}(1520)$ states.\nIn our analysis the branching fractions for $P_{11}(1440)$ and $D_{13}(1520)$ \nresonance decays \nto the $N\\pi$ final state $BF(N\\pi)$ \nwere taken from the previous CLAS studies of $N\\pi$ electroproduction data \n\\cite{Az09}, since the $N\\pi$ exclusive electroproduction channels are \nmost sensitive to contributions from the \n$P_{11}(1440)$ and $D_{13}(1520)$ resonances. The unitarity condition\nallows us to estimate the branching fraction $BF(N\\pi\\pi)_{corr}$ value as:\n\\begin{equation}\n\\label{bnpipi}\nBF(N\\pi\\pi)_{corr}=1-BF(N\\pi).\n\\end{equation} \nFor these resonance decays to the $N\\pi\\pi$ final states it turns out that \nthe estimated branching fractions  $BF(N\\pi\\pi)_{corr}$ \nfrom Eq.~(\\ref{bnpipi}) \n  are slightly ($<$ 10\\%) different with respect to those \n obtained from the $\\pi^+\\pi^-p$ fit ($BF(N\\pi\\pi)_{0}$). Therefore, \nwe multiplied\nthe $\\pi \\Delta$ and $\\rho p$ hadronic decay widths of\n $P_{11}(1440)$ and $D_{13}(1520)$ from the $\\pi^+\\pi^-p$ fit by the ratio\n$\\frac{BF(N\\pi\\pi)_{corr}}{BF(N\\pi\\pi)_{0}}$ in order to make them consistent\nwith the unitarity condition of Eq.~(\\ref{bnpipi}).\n\n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=7.5cm]{pictures\/d13_a12_1pi2pi.eps}\n\\includegraphics[width=7.5cm]{pictures\/d13_s12_1pi2pi.eps}\n\\includegraphics[width=7.5cm]{pictures\/d13_a32_1pi2pi.eps}\n\\caption{\\small(color online) Electrocouplings and full error bars of the $D_{13}(1520)$ state. \nThis analysis of \nthe CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Fe09} is shown by triangles. \nElectrocouplings from the analysis of the $N\\pi$ electroproduction \ndata \\cite{Az09} are shown by circles.}  \n\\label{d13_23}\n\\end{center}\n\\end{figure} \n\n\n\n\n\n\n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=8cm]{pictures\/p11_s12_unit.eps}\n\n\n\\caption{\\small(color online) Comparison between the final results on $S_{1\/2}$ electrocouplings \nfor the $P_{11}(1440)$ from this analysis of CLAS $\\pi^+\\pi^-p$ electroproduction data \\cite{Fe09} (triangles) \nwith the values \nobtained by employing a regular BW ansatz for the resonant amplitudes and with the contributions from the $S_{11}(1535)$ \nstate turned off (squares).}  \n\\label{p11_s12}\n\\end{center}\n\\end{figure}\n\n\n\nConsequently, the $P_{11}(1440)$ and $D_{13}(1520)$ electrocouplings obtained in \nour analysis\nare multiplied by correction factors \n\\begin{equation}\n\\label{bnpipi1}\nC_{hd}=\\sqrt{\\frac{BF(N\\pi\\pi)_{0}}{BF(N\\pi\\pi)_{corr}}},\n\\end{equation} \nin order to keep the resonant parts and the full computed differential $\\pi^+\\pi^-p$ \ncross sections unchanged under the re-scaling \nof resonance hadronic decay\nparameters described above.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nQ$^{2}$, & $A_{1\/2}$,          &  $S_{1\/2}$,           \\\\\n(GeV$^2)$            & (10$^{-3}$ GeV$^{-1\/2})$  &  (10$^{-3}$ GeV$^{-1\/2})$  \\\\\n\\hline\n$0.28$ & $-36.3 \\pm 17.0$ & $37.8 \\pm 6.3$  \\\\\n$0.33$ & $-21.5 \\pm 7.7$ & $35.8 \\pm 4.7$ \\\\\n$0.38$ & $-10.9 \\pm 5.9$ & $32.7 \\pm 5.3$ \\\\\n$0.43$ & $-8.8 \\pm 5.4$ & $33.8 \\pm 3.8$ \\\\\n$0.48$ & $3.7 \\pm 5.9$ & $37.4 \\pm 5.9$ \\\\\n$0.53$ & $15.8 \\pm 9.1 $ & $26.8 \\pm 3.5$  \\\\\n$0.58$ & $17.2 \\pm 10.4$ & $22.2 \\pm 6.8$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{ecp11} Electrocouplings of the $P_{11}(1440)$ resonance \ndetermined from this analysis of\n$\\pi^+\\pi^-p$ electroproduction off protons within the framework of the \nJM model \\cite{Mo09}.}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nQ$^{2}$, & $A_{1\/2}$,          & $S_{1\/2}$,         & $A_{3\/2}$, \\\\\n (GeV$^2)$            & (10$^{-3}$ GeV$^{-1\/2})$  & (10$^{-3}$ GeV$^{-1\/2}$ & (10$^{-3})$ GeV$^{-1\/2})$ \\\\\n\\hline\n$0.28$ & $-51.9 \\pm 4.8$ & $-44.0 \\pm 4.1$ & $71.5 \\pm 6.8$ \\\\\n$0.33$ & $-57.1 \\pm 7.2$ & $-44.5 \\pm 4.1$ & $74.2 \\pm 6.8$ \\\\\n$0.38$ & $-73.4 \\pm 7.0$ & $-36.5 \\pm 4.1$ & $65.3 \\pm 7.3$ \\\\\n$0.43$ & $-75.0 \\pm 9.3$ & $-43.9 \\pm 5.3$ & $74.1 \\pm 6.1$ \\\\\n$0.48$ & $-82.6 \\pm 15.9$ & $-35.7 \\pm 4.2$ & $63.0 \\pm 6.8$ \\\\\n$0.53$ & $-65.2 \\pm 8.9 $ & $-36.6 \\pm 4.5$ & $59.6 \\pm 10.0$ \\\\\n$0.58$ & $-62.2 \\pm 9.3$ & $-31.0 \\pm 4.6$ & $48.9 \\pm 7.4$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{ecd13} Electrocouplings of the $D_{13}(1520)$ resonance determined\n from this analysis of\n$\\pi^+\\pi^-p$ electroproduction off protons within the framework of the JM model \\cite{Mo09}.}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nParameter & Analysis of the CLAS  & RPP \\\\\n   &  $\\pi^+\\pi^-p$ data &   \\\\\n\\hline\nBreit-Wigner mass, MeV & 1458 $\\pm$ 12 & 1420-1470 ($\\approx$ 1440) \\\\\nBreit-Wigner width, MeV & 363 $\\pm$ 39 & 200-450 ($\\approx$ 300) \\\\\n$\\pi \\Delta$ partial decay width, MeV & 142 $\\pm$ 48 & \\\\\n$\\pi \\Delta$ BF,  & 23\\%-58\\% &  20\\%-30\\%     \\\\\n$\\rho ~p$ partial decay width, MeV & 6.2 $\\pm$ 4.1 &  \\\\\n$\\rho ~p$ BF & $<$ $~$2.0\\% &  $<$ $~$8.0\\%     \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{hpp11} Hadronic parameters of the $P_{11}(1440)$ resonance determined from the CLAS data \\cite{Fe09} on\n$\\pi^+\\pi^-p$ electroproduction off protons within the framework of the JM model \\cite{Mo09} and from the RPP \\cite{PDG10}.}\n\\end{center}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nParameter & Analysis of the CLAS  & RPP \\\\\n          &  $\\pi^+\\pi^-p$ data   &   \\\\\n\\hline\nBreit-Wigner mass, MeV & 1521 $\\pm$ 4 & 1515-1525 ($\\approx$ 1520) \\\\\nBreit-Wigner width, MeV & 127 $\\pm$ 4 & 100-125 ($\\approx$ 115) \\\\\n$\\pi \\Delta$ partial decay width, MeV & 35 $\\pm$ 4 &  \\\\\n$\\pi \\Delta$ BF & 24\\%-32\\% &  15\\%-25\\%     \\\\\n$\\rho ~p$ partial decay width, MeV & 16 $\\pm$ 5 &  \\\\\n$\\rho ~p$ BF  & 8.4\\%-17\\% &   15\\%-25\\%     \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{hpd13} Hadronic parameters of the $D_{13}(1520)$ resonance determined from the CLAS data \\cite{Fe09} on\n$\\pi^+\\pi^-p$ electroproduction off protons within the framework of the JM model \\cite{Mo09} and from the RPP \\cite{PDG10}.}\n\\end{center}\n\\end{table*}\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe $P_{11}(1440)$ and $D_{13}(1520)$ electrocouplings derived from \nthe fit of $\\pi^+\\pi^-p$ electroproduction data are presented in Tables~\\ref{ecp11} \nand ~\\ref{ecd13}. The masses and hadronic \ndecay parameters are shown in Tables~\\ref{hpp11} and ~\\ref{hpd13}. \nThe hadronic resonance parameters are taken as the average of their values \nobtained in independent fits in the three \n$Q^2$ intervals shown in Table~\\ref{fitqual}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn Figures~\\ref{p11_23} and~\\ref{d13_23} we compare the results of this analysis with \nresults of the analysis of CLAS $N\\pi$ electroproduction data \\cite{Az09}. Furthermore, there are also \nthe  $P_{11}(1440)$ and $D_{13}(1520)$ electrocouplings available from the MAID partial wave analysis of \n$N\\pi$\nelectroproduction at $Q^2$ $<$ 0.6 GeV$^2$ \\cite{Tia12,Tia09,Tia07}.  \nOur analysis confirms the\nsign change of the $A_{1\/2}$ electrocoupling for the $P_{11}(1440)$, \nfirst observed in the $N\\pi$ channels. We\nfound the zero-crossing to be between $Q^2$=0.4 GeV$^2$ and 0.45 GeV$^2$ in agreement with the $N\\pi$ analysis \\cite{Az09}. The electrocouplings for the $P_{11}(1440)$ and $D_{13}(1520)$ are also consistent within their uncertainties \nwith the CLAS analysis of the $N\\pi$ electroproduction  \\cite{Az09} \nwith the exception of the $A_{1\/2}$ amplitude of $D_{13}(1520)$,  \nwhere we see a difference in the range of 0.4 GeV$^2$ $<$ $Q^2$ $<$ 0.5 GeV$^2$, which is slightly larger \nthan a standard deviation. In this range of $Q^2$ two different analyses of the \n$N\\pi$ exclusive channels the CLAS \\cite{Az09} and the MAID \\cite{Tia12}, give also slightly different\nvalues of the $A_{1\/2}$ electrocoupling for the $D_{13}(1520)$. The MAID results  are close to our extraction at 0.4 GeV$^2$ $<$ $Q^2$ $<$ 0.5 GeV$^2$, but start to deviate at larger photon virtualities. Instead, for the other electrocouplings of $D_{13}(1520)$, both the CLAS \\cite{Az09} and the MAID \\cite{Tia12} analyses of the $N\\pi$ channels are consistent with the\nresults of this analysis.\n\n\n\n \n\n\n\n\nThe unitarization of the resonant amplitudes and \nthe contribution from the $S_{11}(1535)$ state \nhave a negligible impact on $A_{1\/2}$ for $P_{11}(1440)$ and on \nall $D_{13}(1520)$ electrocouplings. However,\nboth factors affect the $S_{1\/2}$ electrocouplings of $P_{11}(1440)$. \nFigure~\\ref{p11_s12} shows a comparison \nof the $P_{11}(1440)$ $S_{1\/2}$ electrocouplings obtained with \nthe unitarized BW ansatz incorporating the $S_{11}(1535)$ contributions\nversus the regular \nBW ansatz without the $S_{11}(1535)$. The implementation of the  \n$S_{11}(1535)$ contributions imposes additional constraints on the ranges of\nthe $P_{11}(1440)$ $S_{1\/2}$ electrocouplings that correspond to the \ncomputed differential cross\nsections, which are consistent with the data, allowing us to improve the \nuncertainties in the extraction of this electrocoupling.\n\n \n\nConsistent results on the electrocouplings of the $P_{11}(1440)$ and \n$D_{13}(1520)$\nresonances, which are available for the first time from independent analyses \nof the  major $N\\pi$ and\n$\\pi^+\\pi^-p$ exclusive channels provide evidence for the reliable extraction of these\nfundamental quantities from the experimental data. Furthermore, this\nagreement strongly supports the reaction models employed for the analyses of the \n$N\\pi$ \\cite{Az09} and $\\pi^+\\pi^-p$ \\cite{Mo09} electroproduction data.\n\n\n\n\n  \n\n\n\n\n\n\n\n\n\nThe Breit-Wigner masses and mean values for the total decay widths \nof the $P_{11}(1440)$ and $D_{13}(1520)$, derived \nfrom the $\\pi^+\\pi^-p$ data and listed in Tables~\\ref{hpp11} and ~\\ref{hpd13}, are in a good\nagreement with those reported in the RPP \\cite{PDG10}. Uncertainties in the total resonance decay widths  were obtained by\nvarying the $\\pi \\Delta$ and $\\rho p$ partial decay widths, while the decay \nwidth to the $N\\pi$  final state remained \nfixed at the\nvalues taken from Ref.~\\cite{Az09}. In comparison with the results in the RPP \\cite{PDG10}, \nour analysis gives somewhat larger branching fractions to the $\\pi \\Delta$ final states for the hadronic decays\nof both the $P_{11}(1440)$ and $D_{13}(1520)$ states, and a smaller hadronic decay to the\n $\\rho p$ final state. \n\n\n\n\n\nBoth $\\gamma_{v}pN^*$ \nelectrocouplings and resonance \nhadronic decay widths are\nobtained at the resonant point on the real energy\naxis $W=M_{r}$, using the relativistic Breit-Wigner parametrizations of the resonant \namplitudes. All resonance parameters determined in our approach incorporate \n$all$ relevant contributions \nto the $N^*$ structure. The extraction from the data of the individual contributions of the  quark core and the meson-baryon dressing is outside of our scope. \nOur results can be \ncompared directly with those determined from other meson\nelectroproduction channels, where the BW ansatz for resonant amplitudes is used. \nFor comparisons\nwith results obtained from models that use other ways to \ndescribe the resonant contributions, \nthe resonant cross sections at the resonant points can be compared.\n \n\n\n\n\n\n\n\n\n\n\n\\section{The impact on the studies of\n resonance structure from the CLAS results \n on the $P_{11}(1440)$ and $D_{13}(1520)$ electrocouplings and hadronic \nparameters.}\nIn this section we discuss the impact of the CLAS data on \nthe $P_{11}(1440)$ and $D_{13}(1520)$\nelectrocouplings and hadronic parameters determined from the independent analyses of $N\\pi$ and $\\pi^+\\pi^-p$ electroproduction off protons\non the contemporary understanding of the structures these states at \nvarious distances. \n\n \n\\subsection{$P_{11}(1440)$ resonance}     \nThe first comprehensive $P_{11}(1440)$ electrocoupling sets available from \nthe CLAS data on\n$N\\pi$ and $\\pi^+\\pi^-p$ electroproduction provide\naccess to the active degrees of freedom in the $P_{11}(1440)$ structure \nat various distances. \nPrevious studies  \\cite{AznRoper,Az09}  already showed a strong sensitivity of the electrocouplings\nto assumptions\nabout the active components contributing to the $P_{11}(1440)$ structure.\n\n\nThe $P_{11}(1440)$ resonance is characterized by peculiar features:\n \\begin{itemize}\n\\item  a smaller mass (1440 MeV) than that expected by quark models with \nvector flavor-conserving \nquark-quark ($qq$) interactions, this is less \nthan the mass of the first orbital excitation of the nucleon, the\n$S_{11}(1535)$, leading to so-called wrong $P_{11}(1440)$\/$S_{11}(1535)$ mass\nordering \\cite{Bu11b}, and \n\\item a large hadronic decay width of $\\approx$ 300 MeV, although other resonances with masses \nbelow 1.6 GeV have decay\nwidths of less than 200 MeV,\n\\end{itemize}\nthat make it difficult \nto describe the $P_{11}(1440)$ as a pure three quark bound state. \nTo the best of our knowledge, \nthere are no models, incorporating only  quark degrees of freedom, \ncapable of describing simultaneously the mass ordering  \nof the $P_{11}(1440)$ and $S_{11}(1535)$ \nresonances, the unusually large $P_{11}(1440)$ hadronic decay width, and \nthe photo and electrocouplings. Therefore, different\napproaches for the description of the $P_{11}(1440)$ were developed that \ntake into\naccount the contributions from degrees of freedom \nother than those of dressed quarks \\cite{Lee10,Lee08,Li92,Ob11}. The CLAS results on the $P_{11}(1440)$\nelectrocouplings of \\cite{Az09} and this analysis allows for the first \ntime to confront the model expectations for the $P_{11}(1440)$ structure \nwith the experimental data, in particular\nat $Q^2$ $<$ 0.6 GeV$^2$, where consistent results \nfrom the independent analyses of $N\\pi$\nand $\\pi^+\\pi^-p$ electroproduction off protons have now \nbecome available. \n\n\n\n \n\n\n\nIn Ref. \\cite{Li92} the $Q^2$-dependence of the $P_{11}(1440)$ electrocouplings was obtained by assuming a hybrid \nnature of this\nresonance. In this approach, the $P_{11}(1440)$  is treated as a resonance \ndominated by a single SU(6) configuration of three  dressed quarks \noscillating against an explicitly excited glue, denoted $q^{3}G$.\n The $P_{11}(1440)$ as a hybrid state has a spin-flavor part of the wave function orthogonal \nto that of the nucleon, whereas the spin-flavor part of wave\nfunction for the $P_{11}(1440)$, as a radially excited three quark state,  \nis identical to that of the nucleon. This difference makes it possible\nto distinguish between the $P_{11}(1440)$ as a regular \n$q^{3}$ or hybrid $q^{3}G$ state by studying  the \n$Q^2$-evolution of the $P_{11}(1440)$ electrocouplings. A characteristic prediction \nof \\cite{Li92} for the hybrid origin \nof the $P_{11}(1440)$ is the absence of the longitudinal coupling, i.e.\n$S_{1\/2}=0$. This is a consequence of\nnegligible SU(6) configuration mixing in the non-relativistic \napproximation adopted in \\cite{Li92} for the description of the $P_{11}(1440)$ \nstructure. The recent Lattice QCD (LQCD) studies of the $N^*$ spectrum \\cite{Ed11,Ed12} also confirmed the presence of the leading SU(6)-spin-flavor-configuration in the structure of $P_{11}(1440)$. The $S_{1\/2}=0$ values for $P_{11}(1440)$ are clearly in disagreement \nwith the results of our\nanalysis of the $\\pi^+\\pi^-p$ electroproduction data (see Fig.~\\ref{p11_23}),\nshowing that at small photon virtualities $S_{1\/2}$ is\nlarger than $A_{1\/2}$, or at least comparable at $Q^2$ close to zero. \nThis allows us to rule out sizable hybrid contributions to the \n$P_{11}(1440)$, confirming the conclusions \nfrom our previous analysis of\nthe $N\\pi$ final state \\cite{Az09}. The lack of a substantial contribution of a hybrid component to the lowest mass resonances in the $P_{11}$ partial wave was also confirmed in the LQCD studies \\cite{Ed12}. \n\n\n\n\n\nThe previous analysis \\cite{Az09} of the $N\\pi$ CLAS data \nshowed that the most \nsatisfactory description of the $P_{11}(1440)$ electrocouplings \nwas achieved at $Q^2$ $>$ 1.5 GeV$^2$ within the framework \nof relativistic light-front quark models \\cite{Capstick,AznRoper}, as shown \nin Fig.~\\ref{lf_p11}. \nIn these models, the $P_{11}(1440)$ electrocouplings are evaluated from \nthe fully\ncovariant, one-body electromagnetic transition current for point-like \nconstituent quarks in the impulse approximation, \nemploying light-front relativistic\nHamiltonian dynamics \\cite{Dr70,Be77,Ke91}. \nResonance electrocouplings are computed \nin a frame where the ``+\" component of the \nvirtual photon light-cone momenta is equal to zero.\nFor this particular choice, the contribution from $q\\bar{q}$ pairs into \nphoton propagation are eliminated \\cite{AznRoper}. \nIn these models \\cite{Capstick,AznRoper} \nthe $P_{11}(1440)$ is\ntreated as the first radial excitation of three constituent quarks. \nThe wave function of the ground state \nand $P_{11}(1440)$ are described under the simplifying assumption, \nthat they are unmixed single \noscillator basis states.\n\n\nA reasonable description of the CLAS data \nat $Q^2$ $>$ 1.5 GeV$^2$ is achieved in both models \\cite{Capstick,AznRoper},\nas seen in Fig.~\\ref{lf_p11}.  Accounting for the relativistic transition operator \nis important in\norder to describe the CLAS  data over the entire range of photon virtualities, \nand especially at small $Q^2$ where the sign  \nof $A_{1\/2}$ changes \\cite{Capstick,AznRoper}, although a quantitative \n description of the data at small  $Q^2$ has not been achieved (see Fig.~\\ref{lf_p11}).\n\n\n\n\nA similarly good description of the $P_{11}(1440)$ electrocouplings at \n$Q^2$ $>$ 1.5 GeV$^2$ was \nfound using a valence quark \nmodel \\cite{Ra10} based on the covariant spectator formalism \\cite{Gr08,Gr92}. The results of this model \nare also shown in Fig.~\\ref{lf_p11}. Incorporating three constituent quarks only, this model treats the $N^*$\nelectroexcitation as a virtual photon interaction with a valence quark in the ground state, while the two other\nquarks are combined to a spectator diquark in both spin states 0 and 1. \nThe wave function of the ground\nstate is parametrized and fit to the data on elastic nucleon form factors. \nThe $P_{11}(1440)$ structure is described as the first radial excitation of the nucleon ground state quarks. \nWith no additional parameters,\nthe wave function of the $P_{11}(1440)$ was calculated employing the orthogonality condition \nbetween the ground state and \nthe $P_{11}(1440)$ wave functions. The electrocouplings of $P_{11}(1440)$ are calculated for \nthe valence quark transition between the nucleon ground state and its first radial excitation.\nAs shown in Fig.~\\ref{lf_p11}, this model \\cite{Ra10} describes the trend of the  CLAS data for \n$Q^2$ $>$ 1.5 GeV$^2$.\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=8.8cm]{pictures\/p11_a12_qua.eps}\n\\includegraphics[width=8.8cm]{pictures\/p11_s12_qua.eps}\n\n\\caption{\\small(color online) Photo- and electrocouplings of the $P_{11}(1440)$ resonance in comparison with\npredictions of quark models. The photocouplings are taken from RPP \\cite{PDG10} (open square) and the CLAS data analysis \\cite{Dug09} (open triangle). Other data points \nare the same as in Fig.~\\ref{p11_23}. The results of relativistic light-front quark models \n\\cite{AznRoper,Capstick} are shown by solid and dashed lines, respectively. Results of the covariant valence quark spectator\ndiquark model \\cite{Ra10} are shown by the dashed dotted line.}  \n\\label{lf_p11}\n\\end{center}\n\\end{figure*}\n\n\nDifferent approaches to describe the $P_{11}(1440)$ resonance using structureless constituent quarks of non-running masses -- two relativistic light front quark models \\cite{Capstick,AznRoper} and a valence quark model \\cite{Ra10} -- are capable of reproducing the major features of\nthe CLAS data on \n$P_{11}(1440)$ electrocouplings only for \n$Q^2$ $>$ 1.5 GeV$^2$. These approaches fail \nto describe the electrocouplings for $Q^2$ $<$ 1.0 GeV$^2$. On the other\nhand, consistent results \non $P_{11}(1440)$ electrocouplings from the independent analyses\nof $N\\pi$ and $\\pi^+\\pi^-p$ electroproduction off protons at $Q^2$ $<$ 0.6\nGeV$^2$ emphasize the credibility of the experimental data  and the inability of  \ndescribing the $P_{11}(1440)$ structure by only accounting for quark  \ndegrees of freedom without meson-baryon dressing.\n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=8cm]{pictures\/mb_dressing.eps}\n\\caption{\\small Processes that contribute to meson-baryon dressing in the $N^*$ photo\/electroexcitations within the EBAC-DCC\napproach.}  \n\\label{mb_dress}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\nThe global analysis of $N\\pi$, photo-, electro-, and hadroproduction within the framework of  \nthe EBAC-DCC coupled-channel approach \\cite{Lee96,Lee01,Lee07,Lee08,Lee09,Lee09j,Lee10}  revealed\nthis additional component in the $N^*$ structure that is most relevant \nat $Q^2$ $<$   1.0\nGeV$^2$.   \nThe general unitarity condition for full meson electroproduction amplitudes \nrequires contributions from\nnon-resonant meson electroproduction and hadronic scattering amplitudes \nto the $\\gamma_{v}pN^*$ vertex, as depicted in Fig.~\\ref{mb_dress}. This contribution incorporates\nall possible intermediate meson-baryon states and subsequent meson-baryon \nscattering processes\nthat eventually result in the $N^*$ formation. \nAs was mentioned in Section~\\ref{nstarelectrocoupl}, the $\\gamma_{v}pN^*$ electrocouplings \nextracted in our\nanalysis of $\\pi^+\\pi^-p$ electroproduction data, as well as in the previous\nanalysis of the $N\\pi$ data \\cite{Az09} account for $all$ contributions to\nthe resonance structure, including for those from meson-baryon dressing. \nInstead, most of quark models, including aforementioned \\cite{Capstick,AznRoper,Ra10}, account only for the quark-core component in $N^*$ photo-\/electroexcitation, while \nmeson-baryon dressing, being a part of the reaction mechanism, is completely outside of their \nscope. Therefore, the current and the previous CLAS results \\cite{Az09} on $\\gamma_{v}pN^*$ electrocouplings \nshould be different with respect to those from the quark model expectations \nin $Q^2$-areas where the contributions \nfrom meson-baryon dressing  are substantial. \n\n\n\nIn Ref.~\\cite{Lee08} the meson-baryon dressing amplitudes \nare calculated from diagrams describing \nnon-resonant mechanisms in $N\\pi$ photo-, electro-, and \nhadroproduction channels with hadronic parameters \nfit to the world meson hadroproduction data.   \nThe estimates of the absolute values for meson-baryon dressing\ncontributions to the $P_{11}(1440)$ electrocouplings showed that meson-baryon\ndressing amplitudes are maximal at $Q^2$ $<$ 1.0 GeV$^2$. \n\nTherefore the meson-baryon dressing contributions could be  responsible \nfor the differences observed in this area \nbetween the CLAS data on the $N^*$ electrocouplings \nand the quark model expectations of Refs. \\cite{Capstick,AznRoper,Ra10}. \nThe meson-baryon dressing\ndecreases with $Q^2$ and in the region $Q^2$ $>$ 1.0 GeV$^2$ we have \na gradual transition to the dominance of\nquark degrees of freedom, as indicated by the better \ndescription of the $P_{11}(1440)$\nelectrocouplings within the framework of quark models. \n\nEvaluations for bare-quark-core contributions to the dressed $P_{11}(1440)$ electrocouplings have recently be obtained within the framework\nof QCD-based Dyson-Schwinger Equations (DSEQCD) \\cite{CR2012}. The DSEQCD results\nare close to the bare-quark-core contributions to the $P_{11}(1440)$ \nelectrocouplings extracted from the experimental\ndata within the framework of the EBAC-DCC coupled-channel approach \\cite{Lee10} \nand far from the dressed $P_{11}(1440)$ electrocouplings obtained in our analysis. \nThis DSEQCD result offers further evidence for substantial contributions \nfrom meson-baryon dressing to the $P_{11}(1440)$ electrocouplings. \n\n\nWe conclude that the structure of $P_{11}(1440)$ is determined by the \ncombined\ncontributions from an internal quark core of constituent quarks in the\nfirst radial excitation \nand an external meson-baryon dressing, \nthe latter being most relevant at small $Q^2$. \n\n\n\nLattice QCD is making progress in calculating $\\gamma_{v}NN^*$ \ntransition form factors \nfrom first QCD\nprinciples. Exploratory calculations of the $\\gamma_{v}pP_{11}(1440)$ transition form factors $F^{P11}_{1,2}(Q^2)$ \nwere carried out within LQCD in the quenched approximation using a pion mass of 500 MeV and a simplified set of projection\noperators \\cite{Li09,Li09a}. Despite the aforementioned simplifications, \nthe first LQCD evaluations are in approximate agreement with the \nCLAS $F^{P11}_{1,2}(Q^2)$ form factor results\nat  $Q^2$ $>$ 1.5 GeV$^2$. Improved LQCD calculations with dynamical \nquarks and a smaller pion mass of 380 MeV \\cite{Li11,Li11a} provide for the first time a reasonable description of the CLAS data \nfor $Q^2$ $<$ 1.0 GeV$^2$. These studies demonstrated promising opportunities \nto relate the results \non the $Q^2$-evolution of the $\\gamma_{v}pN^*$ electrocouplings \ndirectly to QCD. \n\nTogether with the recent LQCD  \\cite{Ed11,Ed12} and DSEQCD \\cite{CRo103}\nresults on the $N^*$ spectrum and the first DSEQCD results on bare\n$\\gamma_{v}pP_{11}(1440)$ electrocouplings \\cite{CR2012}, this is an important step\ntoward to our understanding of baryon formation \nand the nature of confinement from the first principles of QCD.\n\n\nThe $\\pi \\Delta$ and $\\rho p$ $P_{11}(1440)$ hadronic decay widths obtained \nin our\nanalysis are listed in Table~\\ref{hpp11} and compared with RPP \\cite{PDG10} values. The\nquark models, which describe $N^*$  hadronic decays through  flux tube \nbreaking \\cite{St89,St95}, provide a \ngood description of our $\\pi \\Delta$ and \n$\\rho p$ decay widths of 100 and  2.5 MeV, respectively. However, the flux-tube breaking mechanism is unable to \ndescribe the $P_{11}(1400)$ $N\\pi$  hadronic decays. It  predicts a width of \napproximately 400\nMeV for this channel, which is in strong disagreement with measured values \\cite{PDG10}.  \nOther approaches, listed in Ref.~\\cite{Ca00}, fail to describe the $N\\pi\\pi$ hadronic decay widths.\n\nTo date the quantitative description of \n$P_{11}(1440)$ hadronic decays remains\na challenging problem. Consistent  accounting for the meson-baryon  dressing contributions to \nboth electromagnetic and  \nhadronic $P_{11}(1440)$ vertices is important to gain insight into the structure \nof this excited state \\cite{Lee101,Lee101a,Ob11,CR2012}. \n\n\\subsection{$D_{13}(1520)$ resonance} \n\n\n\n\n\nAnalyses of the previous CLAS data \\cite{Az09} on $D_{13}(1520)$\nelectrocouplings determined from $N\\pi$ exclusive electroproduction channels,\nshowed that their satisfactory description for \n$Q^2$ $>$ 1.5 GeV$^2$ can be achieved within the framework of\nthe hypercentral constituent quark model (hCQM) \\cite{Gia98} \n(solid lines in Fig.~\\ref{d13_hcqm}). In this model the central confinement potential \nis parametrized by a sum of Coulomb and linear terms expressed in collective\nhypercoordinates for the three constituent quark system. The use of \nhypercoordinates effectively accounts for three-body effects in quark interactions.\nThe remaining interaction between pairs of quarks is parametrized by a superposition of spin- \nand isospin-dependent hyperfine terms. Wave functions for resonances are\nobtained by diagonalizing the Hamiltonian in a non-relativistic approximation. \nThree parameters of the hQCM were fit to data on \nthe baryon spectrum. With these parameters\nelectrocouplings of all well-established excited nucleon states \nwere evaluated for $Q^2$ $<$ 5.0\nGeV$^2$, employing non-relativistic electromagnetic transition operators. The results obtained\nfor the $D_{13}(1520)$ state are shown in Fig.~\\ref{d13_hcqm}. \n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=7cm]{pictures\/d13_a12mbqm.eps}\n\\includegraphics[width=7cm]{pictures\/d13_s12mbqm.eps}\n\\includegraphics[width=7cm]{pictures\/d13_a32mbqm.eps}\n\\caption{\\small(color online) Description of the CLAS data on $D_{13}(1520)$ \nelectrocouplings\nwithin the hQCM (solid lines) \\cite{Gia98}. The symbols associated with\n the experimental \ndata are the same as in Fig.~\\ref{lf_p11}.}  \n\\label{d13_hcqm}\n\\end{center}\n\\end{figure} \n\n\n\nThe hCQM model can describe the data trends for the $D_{13}(1520)$ \nelectrocouplings for $Q^2$ $>$ 1.5 GeV$^2$. Pronounced differences between \nhCQM expectations and\nthe $D_{13}(1520)$ electrocouplings at smaller $Q^2$ offer an indication for\ncontributions from active degrees of freedom other than a core of three dressed quarks to this state's\nelectrocouplings. Meson-baryon dressing amplitudes are likely contributors in\nthis area of photon virtualities. According to the EBAC analysis \\cite{Lee08}, for the $D_{13}(1520)$ state \nthey are maximal \nfor small photon virtualities and \ndecrease with $Q^2$. The EBAC analysis also suggests negligible meson-baryon dressing or dominant quark core \ncontributions to the $A_{1\/2}$ \nelectrocoupling of the $D_{13}(1520)$ state for $Q^2$ $>$ 1.5 GeV$^2$, where we already have the CLAS results on \nthis electrocoupling  \\cite{Az09}. \nThese results thus offer access to quark degrees of freedom \nin the structure of $D_{13}(1520)$ and open up new prospects to study the dynamical \ndressed quark  mass, structure, \nand their strong interactions, that are\nresponsible for the  $N^*$ formation. These studies are of particular \nimportance to understand \nthe nature of confinement in the baryon sector based \non QCD \\cite{CRo101,CRo103,CRo104,Li09,Li11,Br09,Brod11,CL12}. \n\n\n\n\n\n\n\nOur analysis confirms  a rapid helicity switch from the dominance of the $A_{3\/2}$ \nelectrocoupling at the photon point to a comparable contribution from both transverse electrocouplings \nat $Q^2$ $\\approx$ 0.5 GeV$^2$ as already observed in \\cite{Az09}.\nThis is shown in Fig.~\\ref{held13} in terms of the\nhelicity asymmetry, defined as\n\\begin{equation}\n\\label{held13form} \nA_{hel}=\\frac{A^{2}_{1\/2}-A^{2}_{3\/2}}{A^{2}_{1\/2}+A^{2}_{3\/2}} .\n\\end{equation}\nThis particular feature is expected for the contributions from the core of three constituent quarks \nin the first orbital nucleon excitation $L=1$ \\cite{Ko80}. \nIt suggests a significant \ncontribution  from the core of three constituent quarks  to \nthe transverse $D_{13}(1520)$ electrocouplings even at small $Q^2$.\n\n\n\n\n\n\nWe conclude that the $Q^2$-evolution of the $D_{13}(1520)$  electrocouplings is consistent with \ncontributions of both an external meson-baryon cloud and an internal core of three constituent quarks. \n\n\n\n\n\n \n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=8cm]{pictures\/d13_asym.eps}\n\\caption{\\small(color online) Helicity asymmetry $A_{hel}$ of \nEq.~(\\ref{held13form}) \nfor the transverse electrocouplings of the \n$D_{13}(1520)$ from this analysis of $\\pi^+\\pi^-p$ (filled\ntriangles) and $N\\pi$ (open circles) exclusive electroproduction channels measured with CLAS. \nThe curve\nrepresents the non-relativistic quark model calculations \\cite{Ko80}.}  \n\\label{held13}\n\\end{center}\n\\end{figure}\n\n\n\nThe $N\\pi\\pi$ hadronic decay widths of $D_{13}(1520)$, obtained from \nthe CLAS $\\pi^+\\pi^-p$\ndata, are given in the Table~\\ref{hpd13}. They are close to the \nRPP \\cite{PDG10} values. However, the branching fraction to the $\\pi \\Delta$\nfinal state is larger  and to the $\\rho p$ final state is smaller \nthan the RPP average. \nOur results on the $N\\pi\\pi$ hadronic couplings of the $D_{13}(1520)$ are well described by the \nflux tube breaking model of \\cite{St89,St95}, as well\nas within the framework of the $^{3}P_{0}$ pair creation model of \\cite{Ca94}. The success of these models \nindicates a substantial role of the quark core in hadronic decays of the $D_{13}(1520)$ state. \n\n\\section{Summary and outlook}\n The analysis of a large body of differential and fully integrated cross sections for \nthe process $\\gamma_{v}p \\rightarrow \\pi^+ \\pi^- p$ carried out within \nthe framework of the JM model in our previous paper \\cite{Mo09}\nallowed us to establish all essential mechanisms that contribute \nto this exclusive channel for \n1.3 GeV $<$ $W$ $<$ 1.6 GeV and 0.2 GeV$^2$ $<$ $Q^2$ $<$ 0.6 GeV$^2$. \nIn this paper we use\nthe reaction model of Ref. \\cite{Mo09} in order to determine the resonant and\nnon-resonant contributions to the $\\pi^+\\pi^-p$ differential cross sections\nmeasured with the CLAS detector \\cite{Fe09} and eventually to obtain \nthe $A_{1\/2}$, $S_{1\/2}$, and $A_{3\/2}$ electrocouplings, as well as the \n$\\pi \\Delta$ and $\\rho p$ decay widths for the $P_{11}(1440)$ and $D_{13}(1520)$ excited proton \nstates from this\ndata fit.   \n\nThe good description of all measured \nobservables makes it possible to evaluate the resonant contributions \nto the cross sections, which are needed to extract the $\\gamma_{v}pN^*$\nelectrocouplings. Resonance electrocouplings and  hadronic decay parameters were determined\nusing a unitarized BW ansatz for the resonant amplitudes, that takes into account \ninteractions between the same and different excited states in the dressed resonance propagators.\n\n\n\n\n\n\nFor the first time electrocouplings of the $P_{11}(1440)$ and $D_{13}(1520)$ resonances were obtained \nfrom the analysis of the $\\pi^+\\pi^-p$ exclusive electroproduction at photon\nvirtualities 0.25 GeV$^2$ $<$ $Q^2$ $<$ 0.60 GeV$^2$. They are in \nreasonable agreement with\nthe electrocoupling values obtained in the previous CLAS analysis of the exclusive $N\\pi$\nelectroproduction channels \\cite{Az09}. \nSingle and charged double pion electroproduction \nchannels represent two major contributors to meson electroproduction in the\nkinematical region covered by our measurements. The non-resonant mechanisms \nin these exclusive channels are\ncompletely different. \nA successful description of a large body of observables in single and charged \ndouble pion\nelectroproduction channels with consistent values of  the $P_{11}(1440)$ and \n$D_{13}(1520)$ electrocouplings\nconfirms that the $\\gamma_{v}pN^*$ electrocouplings can be reliably determined in independent analyses\nof these electroproduction channels.\nA good agreement between electrocouplings of the aforementioned excited states obtained \nfrom $N\\pi$ and $\\pi^+\\pi^-p$\nelectroproduction also demonstrate that the reaction models developed \nto describe \nthese exclusive channels \\cite{Az09,Mo09} can be used to determine \nelectrocouplings  of most of the excited proton states that decay preferentially \ninto either $N\\pi$ or $N\\pi\\pi$ final states. \n\nThe $\\pi \\Delta$ and $\\rho p$ partial \nhadronic decay widths of\nthe $P_{11}(1440)$ and \n$D_{13}(1520)$ states were also determined. They are close to \nthe results of experiments with hadronic probes reported in the RPP \\cite{PDG10}. \n\n\n\nThe comparison of the CLAS data on $P_{11}(1440)$ and $D_{13}(1520)$ electrocouplings \nwith the expectations of conceptually different  \nquark models \\cite{Capstick,AznRoper} and \\cite{Ra10}, complemented by evaluations \nof meson-baryon dressing \\cite{Lee08} allowed us to shed light on the relevant components \nin the structure of these states and to explore their\nevolution with photon virtualities. The electrocouplings reveal\ntwo major contributions, from an internal core of three dressed quarks\nand an external meson-baryon cloud. \n\nThe structure of the $P_{11}(1440)$  is consistent \nwith a combined contribution from constituent quarks in the first \nradial excitation and  from a meson-baryon\ncloud with pronounced contributions  at $Q^2$ $<$ 0.6 GeV$^2$.\nThe electrocouplings of the $D_{13}(1520)$ contain a combined \ncontribution of three constituent quarks \nin the first orbital nucleon excitation $L=1$ and an external \nmeson-baryon cloud, which becomes negligible for\nthe $A_{1\/2}$ electrocoupling of $D_{13}(1520)$ for $Q^2$ $>$ 1.5 GeV$^2$. \nThe CLAS results on the $A_{1\/2}$  electrocoupling of the $D_{13}(1520)$ resonance \nin this kinematical region provide direct \naccess to quark degrees of freedom \nin the structure of this excited state. A physical interpretation of these results is of particular \ninterest for those who are studying hadron structure starting from the QCD Lagrangian \\cite{CL12,CR2012,Br09,Li09,Li11,Li11a,Ed11,Ed12,Brod11}. \n\n\nThe resonance electrocouplings and hadronic decay parameters presented \nin this paper were \nobtained using the unitarized BW parametrization of\nresonant amplitudes. Their values are determined at the resonant point \non the real energy axis ($W=M_{r}$) and incorporate $all$ combined relevant\ncontributions (quark-core, meson-baryon dressing and so on). \nThe $N^*$ parameters extracted from the data in this way  can only be compared \ndirectly with those obtained from the same or \nother exclusive channel data fits that \nemploy BW parametrizations of resonant amplitudes. \nIt remains to be seen how these resonance\nparameters can be related to the values determined  \nfrom singularities of the reaction amplitudes in the complex energy plane, as employed in \ncoupled-channel \nanalyses \\cite{Lee10}.\nIn the Section~\\ref{regbw} we described the JM model relations between \nthe resonance parameters and the model independent observables: the resonant \npart of the cross sections and \n$N^*$ electromagnetic\/hadronic decay widths. We can use the model\nindependence of these observables and require agreement of the cross section and\/or resonance decay widths\nat the resonant point evaluated in different models. These observables, in turn, could be related to the resonance \nparameters for any particular model. \nThis should make it possible to establish\nrelations between resonant parameters determined in\nour approach and those from a global multi-channel analyses.\n\n\n \n\\section{Acknowledgments}\nWe would like to acknowledge the outstanding efforts of the staff of the Accelerator and the Physics Divisions at Jefferson Lab that made this evaluation of $P_{11}(1440)$ and  \n$D_{13}(1520)$ electrocouplings possible. We are grateful to I. J. R. Aitchison, I. G. Aznauryan, R. G. Edwards, M. M. Giannini, T.-S. H. Lee, H-W. Lin, C. D. Roberts, and E. Santopinto for helpful discussions. This work was supported\nin part by the U.S. Department of Energy and the National\nScience Foundation, the Russian Federation\nGovernment Grant 02.740.11.0242, 07.07.2009, the Skobeltsyn Institute of Nuclear Physics and\nPhysics Department at Moscow State University, University of South Carolina, Yerevan Physics Institute (Armenia), the Chilean Comisi\\'on Nacional de Investigaci\\'on Cient\\'ifica y Tecnol\\'ogica (CONICYT), the French Centre National de la Recherche Scientifique (CNRS), the French Commissariat a l'Energie Atomique, the Italian Istituto Nazionale di Fisica Nucleare, the National Research Foundation of Korea,\nand the UK Science and Technology Facilities Research Council (STFC), the Scottish Universities Physics Alliance (SUPA), and the United Kingdom's Science and Technology Facilities Council. The Southeastern Universities Research Association (SURA) operates the\nThomas Jefferson National Accelerator Facility for the United States\nDepartment of Energy under contract DE-AC05-84ER40150.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConventional training of neural networks has been shown to produce classifiers that are highly sensitive to adversarial perturbations \\citep{szegedy2013intriguing, biggio2013evasion},  ``natural looking'' images that have been manipulated to cause misclassification by a neural network (Figure~\\ref{fig:teaser}).   While a wide range of defenses exist that harden neural networks against such attacks \\citep{madry2017towards, shafahi2019adversarial}, defenses based on heuristics and tricks are often easily breakable \\cite{athalye2018obfuscated}.  This has motivated work on {\\em certifiably} secure networks --- classifiers that produce a label for an image, and also (when possible) a rigorous guarantee that the input is not adversarially manipulated \\citep{cohen2019certified, zhang2019towards}.\n\n\\begin{figure}[b!]\n    \\begin{minipage}{\\linewidth}\n        \\centering\n      \\begin{minipage}{0.24\\linewidth}\n        \\centering\n        Natural\n        \\vspace{-3mm}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/teaser\/nat.png}\n           \n        \\end{figure}\n        \\vspace{-2mm}\n        macaw$(0.6\\%)$\n      \\end{minipage}\n      \\begin{minipage}{0.24\\linewidth}\n        \\centering\n        $\\ell_{\\infty}:8\/255$\n        \\vspace{-4mm}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/teaser\/linf.png}\n           \n        \\end{figure}\n        \\vspace{-2mm}\n        bucket$(1.1\\%)$\n      \\end{minipage}\n      \\begin{minipage}{0.24\\linewidth}\n        \\centering\n        Semantic\n        \\vspace{-3mm}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/teaser\/svh.png}\n           \n        \\end{figure}\n        \\vspace{-2mm}\n        bucket$(0.6\\%)$\n      \\end{minipage}\n      \\begin{minipage}{0.24\\linewidth}\n        \\centering\n        Shadow Attack $ $\n        \\vspace{-3mm}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/teaser\/tv.png}\n           \n        \\end{figure}\n        \\vspace{-2mm}\n        bucket$(0.6\\%)$\n      \\end{minipage}\n  \\end{minipage}\n    \\caption{All adversarial examples have the goal of fooling classifiers while looking ``natural''. Standard attacks limit the $\\ell_p$-norm of the perturbation, while semantic attacks have large $\\ell_p$-norm while still producing natural looking images.   Our attack produces large but visually subtle semantic perturbations that not only cause misclassification, but also cause a certified defense to issue a ``spoofed'' high-confidence certificate.   In this case, a certified Gaussian smoothing classifier mislabels the image, and yet issues a certificate with radius 0.24, which is a larger certified radius than its corresponding unmodified ImageNet image which is 0.13.  \n   \n    }\n    \\label{fig:teaser}\n\\end{figure}        \n\n\nTo date, all work on certifiable defenses has focused on deflecting $\\ell_p$-bounded attacks, where $p=2$ or $\\infty$ \\citep{cohen2019certified, gowal2018effectiveness, wong2018scaling}.  After labelling an image, these defenses then check whether there exists an image of a different label within $\\epsilon$ units (in the $\\ell_p$ metric) of the input, where $\\epsilon$ is a security parameter chosen by the user.   If the classifier assigns all images within the $\\epsilon$ ball the same class label, then a certificate is issued, and the input image known not to be an $\\ell_p$ adversarial example. \n \nIn this work, we demonstrate how a system that relies on certificates as a measure of label security can be exploited.  We present a new class of adversarial examples that target not only the classifier output label, {\\em but also the certificate.}  We do this by adding adversarial perturbations to images that are large in the $\\ell_p$ norm (larger than the $\\epsilon$ used by the certificate generator), and produce attack images that are surrounded by a large ball exclusively containing images of the same label.  The resulting attacks produce a ``spoofed'' certificate with a seemingly strong security guarantee despite being adversarially manipulated.   Note that the statement made by the certificate (i.e., that the input image is not an $\\epsilon$ adversarial example in the chosen norm) is still technically correct, however in this case the adversary is hiding behind a certificate to avoid detection by a certifiable defense.\n\nIn summary,  we consider methods that attack a certified classifier in the following sense:\n\\begin{itemize}\n    \\item \\textbf{Imperceptibility:} the adversarial example ``looks like'' its corresponding natural base example,\n    \\item \\textbf{Misclassification:} the certified classifier assigns an incorrect label to the adversarial example, and\n    \\item \\textbf{Strongly certified:} the certified classifier provides a strong\/large-radius certificate for the adversarial example.\n\\end{itemize}\nWhile the existence of such an attack does not invalidate the certificates produced by certifiable systems, it should serve as a warning that certifiable defenses are not inherently secure, and one should take caution when relying on certificates as an indicator of label correctness.\n\n\\subsection*{Background}\nIn the white-box setting, where the attacker knows the victim's network and parameters,  adversarial perturbation are often constructed using first-order gradient information \\citep{carlini2017adversarial, kurakin2016adversarialBIM, moosavi2016deepfool} or using approximations of the gradient \\citep{uesato2018adversarial,athalye2018obfuscated}. \nThe prevailing formulation for crafting attacks uses an additive adversarial perturbation, and perceptibility is minimized using an $\\ell_p$-norm constraint.  For example, $\\ell_\\infty$-bounded attacks limit how much each pixel can move, while $\\ell_0$ adversarial attacks limit the number of pixels that can be modified, without limiting the size of the change to each pixel \\citep{wiyatno2018maximal}. \n\nIt is possible to craft imperceptible attacks without using $\\ell_p$ bounds \\citep{brown2018unrestricted}. For example, \\citet{hosseini2018semantic} use shifting color channels, \\citet{wong2019wasserstein} use the Wasserstein ball\/distance, and \\citet{engstrom2017rotation} use rotation and translation to craft ``semantic'' adversarial examples. In Figure~\\ref{fig:teaser}, we produce semantic adversarial examples using the method of \\citet{hosseini2018semantic} which is a greedy approach that transforms the image into HSV space, and then, while keeping V constant, tries to find the smallest S perturbation causing misclassification\\footnote{In fig.~\\ref{fig:teaser}, the adversarial example has saturation=0}. \nOther variants use generative models to construct natural looking images causing misclassification \n\\citep{song2018constructing, dunn2019generating}.\n\nIn practice, many of the defenses which top adversarial defense leader-board challenges are non-certified defenses \\citep{madry2017towards, zhang2019theoretically,shafahi2019adversarial}. The majority of these defenses make use of {\\em adversarial training}, in which attack images are crafted during training and injected into the training set.  These non-certified defenses are mostly evaluated against PGD-based attacks, resulting in an upper-bound on robustness. \n\nCertified defenses, on the other-hand, provably make networks resist $\\ell_p$-bounded perturbations of a certain radius. \nFor instance, randomized smoothing \\citep{cohen2019certified}  is a certifiable defense against $\\ell_2$-norm bounded attacks, and CROWN-IBP  \\citep{zhang2019towards} is a certifiable defense against $l_{\\infty}$-norm bounded perturbations. \nBoth of these defenses produce a class label, and also a guarantee that the image could not have been crafted by making small perturbations to an image of a different label. \nCertified defenses can also benefit from adversarial training. For example, \\citet{salman2019provably} recently improved the certified radii of randomized smoothing \\citep{cohen2019certified} by training on adversarial examples generated for the smoothed classifier.\n\nTo the best of our knowledge, prior works have focused on making adversarial examples that satisfy the imperceptibility and misclassification conditions, but none have investigated manipulating certificates, which is our focus here.\n\nThe reminder of this paper is organized as follows.\nIn section \\ref{sec:attack} we introduce our new approach Shadow Attack $ $ for generating adversarial perturbations.  This is a hybrid model that allows various kinds of attacks to be compounded together, resulting in perturbations of large radii.\nIn section \\ref{sec:smoothing} we present an attack on ``randomized smoothing'' certificates \\citep{cohen2019certified}. \nSection \\ref{sec:ablation} shows an ablation study which illustrates why the elements of the Shadow Attack $ $ are important for successfully manipulating certified models. \nIn section \\ref{sec:crown-ibp} we generate adversarial examples for ``CROWN-IBP'' \\citep{zhang2019towards}. \nFinally, we discuss results and wrap up in section~\\ref{sec:conclusion}.\n\n\n\n\\section{The Shadow Attack $ $} \\label{sec:attack}\nBecause certificate spoofing requires large perturbations (larger than the $\\ell_p$ ball of the certificate), we propose a simple attack that ensembles numerous modes to create large perturbations.  \nOur attack can be seen as the generalization of the well-known PGD attack, which creates adversarial images by modifying a clean base image. Given a loss function $L$ and an $\\ell_{p}$-norm bound ${\\epsilon}$ for some $p \\geq 0$, PGD attacks solve the following optimization problem: \n\\begin{equation}\n    \\max_{\\delta} {L(\\theta, x+\\delta)}\n\\end{equation}\n\\begin{equation} \\label{eq:lp}\n    \\textit{s.t. } \\|\\delta\\|_p \\leq {\\epsilon},\n\\end{equation}\nwhere $\\theta$ are the network parameters and $\\delta$ is the adversarial perturbation to be added to the clean input image $x$. \nConstraint~\\ref{eq:lp} promotes imperceptibility of the resulting perturbation to the human eye by limiting the perturbation size. \nIn the shadow attack, instead of solving the above constrained optimization problem, we solve the following problem with a range of penalties: \n\\begin{equation}\n        \\max_{\\delta} {L(\\theta, x+\\delta)} - \\lambda_{c} C(\\delta) - \\lambda_{tv} TV(\\delta) -\\lambda_{s} Dissim(\\delta),\n\\end{equation}\nwhere $\\lambda_c, \\lambda_{tv},\\lambda_{s}$ are scalar penalty weights.\nPenalty~ $TV(\\delta)$\nforces the perturbation $\\delta$ to have small total variation ($TV$), and so appear more smooth and natural.\nPenalty~$C(\\delta)$\nlimits the perturbation $\\delta$ globally by constraining the change in the mean of each color channel $c$.  This constraint is needed since total variation is invariant to constant\/scalar additions to each color channel, and it is desirable to suppress extreme changes in the color balances of images.\n\nPenalty~$Dissim(\\delta)$\npromotes perturbations $\\delta$ that assume similar values in each color channel. \nIn the case of an RGB image of shape $3\\times W \\times H$, if $Dissim(\\delta)$ is small, the perturbations to red, green, and blue channels are similar, i.e., $\\delta_{R,w,h} \\approx \\delta_{G,w,h} \\approx \\delta_{B,w,h},  \\forall (w,h) \\in W\\times H$.   This amounts to making the pixels darker\/lighter, without changing the color balance of the image.\nLater, in section \\ref{sec:smoothing}, we suggest two ways of enforcing such similarity between RGB channels and we find both of them effective: \n\\begin{itemize}\n    \\item \\textbf{1-channel attack} strictly enforces $\\delta_{R,i} \\approx \\delta_{G,i} \\approx \\delta_{B,i}, \\forall i$ by using just one array to simultaneously represent each color channel $\\delta_{W\\times H}.$ On the forward pass, we duplicate $\\delta$ to make a 3-channel image. In this case, $Dissim(\\delta)=0$, and the perturbation is greyscale.\n    \\item \\textbf{3-channel attack} uses a 3-channel perturbation $\\delta_{3 \\times W \\times H}$, along with the  dissimilarity metric $Dissim(\\delta) = \\|\\delta_R - \\delta_B\\|_p + \\|\\delta_R - \\delta_G\\|_p + \\|\\delta_B - \\delta_G\\|_p$.\n\\end{itemize}\n\nAll together, the three penalties minimize perceptibility of perturbations by forcing them to be $(a)$ small, $(b)$ smooth, and $(c)$ without dramatic color changes (e.g. swapping blue to red).  At the same time, these penalties allow perturbations that are very large in $\\ell_p$-norm. \n\n\\subsection{Creating untargeted attacks}\n\nWe focus on spoofing certificates for {\\em untargeted} attacks, in which the attacker does not specify the class into which the attack image moves. \nTo achieve this, we generate an adversarial perturbation for all possible wrong classes $\\bar{y}$  and choose the best one as our strong attack:\n\\begin{equation} \\label{eq:attack}\n        \\max_{\\bar{y} \\neq y, \\delta}  {-L(\\theta, x+\\delta \\| \\bar{y}) } - \\lambda_{c} C(\\delta) - \\lambda_{tv} TV(\\delta) -\\lambda_{s} Dissim(\\delta)\n\\end{equation}\nwhere $y$ is the true label\/class for the clean image $x$, and $L$ is a spoofing loss that promotes a strong certificate.  We examine different choices for $L$ for different certificates below.\n\n\\section{Attacks on Randomized Smoothing} \\label{sec:smoothing}\nThe Randomized Smoothing method, first proposed by \\citet{lecuyer2018certified} and later improved by \\citet{DBLP:journals\/corr\/abs-1809-03113}, is an adversarial defense against $\\ell_2$-norm bounded attacks.  \\citet{cohen2019certified} prove a tight robustness guarantee under the $\\ell_2$ norm for smoothing with Gaussian noise.  \nTheir study was the first certifiable defense for the ImageNet dataset \\citep{imagenet_cvpr09}.\nThe method constructs certificates by first creating many copies of an input image contaminated with random Gaussian noise of standard deviation $\\sigma$. Then, it uses a base classifier (a neural net) to make a prediction for all of the images in the randomly augmented batch. Depending on the level of the consensus of the class labels at these random images, a certified radius is calculated that can be at most $4\\sigma$ (in the case of perfect consensus).\n\nIntuitively, if the image is far away from the decision boundary, the base classifier should predict the same label for each noisy copy of the test image, in which case the certificate is strong.\nOn the other hand, if the image is adjacent to the decision boundary, the base classifier's predictions for the Gaussian augmented copies may vary. If the variation is large, the smoothed classifier abstains from making a prediction. \n\nTo spoof strong certificates (large certified radius) for an incorrect class, we must make sure that the majority of a batch of noisy images around the adversarial image are assigned the same (wrong) label.   We do this by minimizing the cross entropy loss relative to a chosen (incorrect) label, averaged over a large set of randomly perturbed images.\nTo this end, we minimize \\eqref{eq:attack}, where $L$ is \nchosen to be the average cross-entropy over a batch of Gaussian perturbed copies. \nThis method is analogous to the technique presented by \\cite{shafahi2018universal} for generating universal perturbations that are effective when added to a large number of different images. \n\n\n\n\\subsection*{Results}\n\\citet{cohen2019certified} show the performance of the Gaussian smoothed classifier on CIFAR-10 \\citep{cifar10} and ImageNet \\citep{imagenet_cvpr09}. \nTo attack the CIFAR-10 and ImageNet smoothed classifiers, we use $400$ randomly sampled Gaussian images, $\\lambda_{tv}=0.3$, $\\lambda_{c}=1.0$, and perform 300 steps of SGD with learning rate $0.1$. \nWe choose the functional regularizers $C(\\delta)$ and $TV(\\delta)$ to be \n$$\n  C(\\delta)=\\| \\Avg(|\\delta_R|), \\Avg(|\\delta_G|), \\Avg(|\\delta_B|) \\|_2^2 ,\n  \\text{\\,\\,\\,\\,\\,and\\,\\,\\,\\,\\,} \n        TV(\\delta_{i,j})=\\text{anisotropic-TV}(\\delta_{i,j})^2,\n$$\nwhere $|\\cdot|$ is the element-wise absolute value operator, and $\\Avg$ computes the average. \nFor the $Dissim$ regularizer, we experiment with both the  \n 1-Channel attack that ensures $Dissim(\\delta)=0,$ and the 3-Channel attack by setting $Dissim(\\delta)=\\|  (\\delta_R-\\delta_G)^2, (\\delta_R-\\delta_B)^2, (\\delta_G - \\delta_B)^2 \\|_2 $ and $\\lambda_{s}=0.5$. \nFor the validation examples on which the smoothed classifier does not abstain (see \\citet{cohen2019certified} for more details), the less-constrained 3-channel attack is always able to find an adversarial example while the 1-channel attack also performs well, achieving 98.5\\% success. \\footnote{Source code for all experiments can be found at:  \\href{https:\/\/github.com\/AminJun\/BreakingCertifiableDefenses}{https:\/\/github.com\/AminJun\/BreakingCertifiableDefenses}}  \n\nIn section \\ref{sec:ablation} we will discuss in more detail other differences between 1-channel and 3-channel attacks.\nThe results are summarized in Table~\\ref{table:randsmooth}. For the various base-models and choices of $\\sigma$, our adversarial examples are able to produce certified radii that are on average larger than the certified radii produced for natural images. For ImageNet, since attacking all 999 remaining target classes is computationally expensive, we only attacked target class IDs 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1000.\n\n\\begin{table}\n\\caption{\nCertified radii produced by the Randomized Smoothing method for Shadow Attack $ $ images and also natural images (larger radii means a stronger\/more  confident certificate).\n}\n\\label{table:randsmooth}\n\\begin{center}\n\\begin{tabular}{llcccc}\n\\multirow{2}{*}{ \\bf Dataset } & \\multirow{2}{*}{$\\sigma(l_2)$ }  &\\multicolumn{2}{c}{\\bf Unmodified\/Natural Images}  &\\multicolumn{2}{c}{\\bf Shadow Attack $ $} \\\\\n&&Mean&STD&Mean&STD\\\\ \\hline\n\\multirow{4}{*}{ CIFAR-10 } & 0.12 & 0.14 & 0.056 & \\bf 0.22 & 0.005 \\\\\\cline{2-6}\n                            & 0.25 & 0.30 & 0.111 & \\bf 0.35 & 0.062 \\\\\\cline{2-6}\n                            & 0.50 & 0.47 & 0.234 & \\bf 0.65 & 0.14 \\\\\\cline{2-6}\n                            & 1.00 & 0.78 & 0.556 & \\bf 0.85 & 0.442 \\\\\\cline{1-6}\n                           \n                           \n                           \n                            \n\\multirow{3}{*}{ ImageNet } & 0.25 & 0.30 & 0.109 & \\bf 0.31 & 0.109 \\\\\\cline{2-6}\n                            & 0.50 & 0.61 & 0.217 & 0.38 & 0.191 \\\\\\cline{2-6}\n                            & 1.00 & 1.04 & 0.519 & 0.64 & 0.322 \\\\\\cline{1-6}\n                            \n                           \n                           \n\\end{tabular}\n\\end{center}\n\\vspace{-5mm}\n\\end{table}\n\nFigure~\\ref{fig:imagenet} depicts a sample adversarial example built for the smoothed ImageNet classifier that produces a strong certificate. The adversarial perturbation causes the batch of Gaussian augmented black swan images to get misclassified as hooks.\nFor more, see appendix \\ref{app:fig:imagenet-full}.\n\n \\begin{figure*\n    \\centering\n    \\begin{subfigure}[t]{0.3\\linewidth}                   \n        \\includegraphics[width=\\linewidth]{figs\/imagenet\/nat.png}\n        \\caption{Natural image ($x$)}\n        \\label{fig:imagenet-nat}%\n    \\end{subfigure}%\n    -\n    \\begin{subfigure}[t]{0.3\\linewidth}\n        \\includegraphics[width=\\linewidth]{figs\/imagenet\/noise.png}\n        \\caption{Adversarial perturbation ($\\delta$)}\n        \\label{fig:imagenet-noise}\n    \\end{subfigure}%\n    -\n    \\begin{subfigure}[t]{0.3\\linewidth}\n        \\includegraphics[width=\\linewidth]{figs\/imagenet\/adv.png}\n        \\caption{Adversarial example ($x + \\delta$)}\n        \\label{fig:imagenet-adv}\n    \\end{subfigure}\n    \\caption{\n    An adversarial example built using our Shadow Attack $ $ for the smoothed ImageNet classifier for which the certifiable classifier produces a large certified radii. The adversarial perturbation is smooth and natural looking  even-though it is large when measured using $\\ell_p$-metrics. \n   \n    Also see Figure~\\ref{app:fig:imagenet-full} in the appendix.\n    }\n    \\label{fig:imagenet}\n\\end{figure*}\n\n\\section{Ablation study of the attack parameters} \\label{sec:ablation}\n\nIn this section we perform an ablation study on the parameters of the Shadow Attack $ $ to evaluate\n$(i)$ the number of SGD steps needed, \n$(ii)$ the importance of $\\lambda_{s}$ (or alternatively using 1-channel attacks), \nand \n$(iii)$ the effect of $\\lambda_{tv}$. \n\nThe default parameters for all of the experiments are as follows unless explicitly mentioned: \nWe use $30$ SGD steps with learning rate $0.1$ for the optimization. \nAll experiments except part $(ii)$ use 1-channel attacks for the sake of simplicity and efficiency (since it has less parameters). \nWe assume $\\lambda_{tv}=0.3$, $\\lambda_{c}=20,$ and use batch-size $50$. \nWe present results using the first example from each class of the CIFAR-10 validation set. \n\nFigure~\\ref{fig:converge-1} shows how the adversarial example evolves during the first few steps of optimization\n(See appendix~\\ref{app:fig:converge-10} for more examples). \nAlso, figures~\\ref{fig:converge-loss},~\\ref{fig:converge-tv}, and~\\ref{fig:converge-c} show the evolution of $L(\\delta)$, $TV(\\delta)$, and $C(\\delta)$, respectively (Note that we use 1-channel attacks, so $Dissim(\\delta)$ is always 0).\nWe find that taking just 10 SGD steps is enough for convergence on CIFAR-10, but for our main results (i.e. attacking Randomized Smoothing in section \\ref{sec:smoothing} and attacking CROWN-IBP in section \\ref{sec:crown-ibp}) we take $300$ steps to be safe.\n\n\n\\begin{figure}\n\\begin{center}\n\\setlength\\tabcolsep{0.0pt}\n\\begin{tabular}{cccccccccccc}\n0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & original\\\\\n\\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_0.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_1.png} &  \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_2.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_3.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_4.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_5.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_6.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_7.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_8.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_9.png} &\\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_10.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/194_orig.png}\\\\\n\\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_0.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_1.png} &  \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_2.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_3.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_4.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_5.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_6.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_7.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_8.png} & \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_9.png} &\n\\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_10.png}& \\includegraphics[width=0.08\\textwidth]{figs\/first-steps\/84_orig.png} \\\\\n\\end{tabular}\n\\caption{The first 10 steps of optimization (beginning with a randomly perturbed image copy).}\n\\label{fig:converge-1}\n\\end{center}\n\\end{figure}\n        \n \\begin{minipage}{\\linewidth}\n      \\centering\n      \\begin{minipage}{0.3\\linewidth}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/converge\/loss.pdf}\n            \\caption{Average $L_{b}(\\delta)$ in the first $10$ steps. }\n            \\label{fig:converge-loss}\n        \\end{figure}\n      \\end{minipage}\n      \\quad\n      \\begin{minipage}{0.3\\linewidth}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/converge\/tv.pdf}\n            \\caption{Average $TV(\\delta)$ in the first $10$ steps. }\n            \\label{fig:converge-tv}\n        \\end{figure}\n      \\end{minipage}\n      \\quad\n      %\n      \\begin{minipage}{0.3\\linewidth}\n        \\begin{figure}[H]\n           \n            \\includegraphics[width=\\linewidth]{figs\/converge\/c.pdf}\n            \\caption{Average $C(\\delta)$ in the first $10$ steps.}\n            \\label{fig:converge-c}\n        \\end{figure}\n      \\end{minipage}\n      \n      \n      \n      \n      \n      \\begin{minipage}[t]{0.3\\linewidth}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/lam\/sim.pdf}\n            \\caption{The effect of $\\lambda_{s}$ on the resulting $Dissim(\\delta)$ }\n            \\label{fig:lam-sim}\n        \\end{figure}\n      \\end{minipage}\n      \\quad\n      \\begin{minipage}[t]{0.3\\linewidth}\n        \\begin{figure}[H]\n            \\includegraphics[width=\\linewidth]{figs\/lam\/tv.pdf}\n            \\caption{The effect of $\\lambda_{tv}$ on the resulting $TV(\\delta)$ }\n            \\label{fig:lam-tv}\n        \\end{figure}\n      \\end{minipage}\n      \\quad\n      \\begin{minipage}[t]{0.3\\linewidth}\n        \n      \\end{minipage}\n\\vspace{2mm}\n\\end{minipage}\n  \n  \n\n\n\n  \n\\begin{SCfigure}\n  \\centering\n\n  \\includegraphics[width=0.6\\linewidth]{figs\/channels.pdf}\n  \\vspace{-4mm}\n    \\caption{\n  Histogram of randomized smoothed certificate radii for 100 randomly sampled CIFAR-10 validation images vs those calculated for their adversarial examples crafted using our 1-channel and 3-channel adversarial Shadow Attack $ $ attacks. \n  The ``robust'' victim classifier is based off Resnet-110, and smoothed with $\\sigma=0.50$.\n  1-channel attacks are almost as good as the less-restricted 3-channel attacks.\n  }\n    \\label{channels_dist}\n\\end{SCfigure}\n\nTo explore the importance of $\\lambda_{s}$, we use 3-channel attacks and vary $\\lambda_{s}$ to produce different images in figure~\\ref{fig:lam-sim-1}\\footnote{See figure~\\ref{app:fig:lam-sim-10} in the appendix for more examples.}. \n \nAlso, figure~\\ref{fig:lam-sim} shows the mean $Dissim(\\delta)$ for different values of $\\lambda_{s} (0 \\leq \\lambda_{s} \\leq 5.0)$. \nWe also plot the histogram of the certificate radii in figure \\ref{channels_dist}. \nFigure~\\ref{table:channel-1vs3} compares 1-Channel vs 3-Channel attacks for some of randomly selected CIFAR-10 images. \nFinally, we explore the effect of $\\lambda_{tv}$ on imperceptibility of the perturbations in Figure \\ref{fig:lam-tv-1}. \nSee table~\\ref{app:fig:lam-tv-10} for more images, and figure~\\ref{fig:lam-tv} for the impact of parameters on $TV(\\delta)$. \n\n\\begin{center}\n\\begin{figure}\n\\setlength\\tabcolsep{0.0pt}\n\\begin{tabularx}{\\textwidth}{rccccccccc}\n natural &    \n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/0.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/1.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/2.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/3.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/5.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/6.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/7.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/8.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/natural\/9.png} }\n\\\\\n1-channel &\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/0.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/1.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/2.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/3.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/5.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/6.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/7.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/8.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/1-channel\/9.png} }\n\\\\\n3-channel &\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/0.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/1.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/2.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/3.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/5.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/6.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/7.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/8.png} }&\n\\raisebox{-.37\\height}{ \\includegraphics[width=0.08\\textwidth]{figs\/channels\/3-channel\/9.png} }\n\\end{tabularx}\n\n\\caption{The visual effect of Shadow Attack $ $ on 9 randomly selected CIFAR-10 examples using 1-Channel and 3-Channel attacks. }\n\\label{table:channel-1vs3}\n\\end{figure}\n\\end{center}\n\n\n\\begin{figure}\n\\begin{center}\n\\setlength\\tabcolsep{0.0pt}\n\\begin{tabular}{ccccccccccccc}\n0.0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 & 4.5 & 5.0  & original\\\\\n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_00.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_05.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_10.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_15.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_20.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_25.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_30.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_35.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_40.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_45.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_50.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/103_orig.png} \n\\\\\n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_00.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_05.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_10.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_15.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_20.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_25.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_30.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_35.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_40.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_45.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_50.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_sim\/178_orig.png} \n\\end{tabular}\n\\caption{The visual effect of $\\lambda_{s}$ on perceptibility of the perturbations.\nThe first row shows the value of $\\lambda_{s}$.}\n\\label{fig:lam-sim-1}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\setlength\\tabcolsep{0.0pt}\n\\begin{tabular}{cccccccccccc}\n0.0 & 0.03 & 0.06 & 0.09 & 0.12 & 0.15 & 0.18 & 0.21 & 0.24 & 0.27 & 0.30 & original\\\\\n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_000.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_003.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_006.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_009.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_012.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_015.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_018.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_021.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_024.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_027.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_030.png} & \n\\includegraphics[width=0.08\\textwidth]{figs\/lam_tv\/209_orig.png} \n\\\\\n\\end{tabular}\n\\caption{The visual effect of $\\lambda_{tv}$ on the on imperceptibility of the perturbations. \nThe first row shows the value of $\\lambda_{tv}$}\n\\label{fig:lam-tv-1}\n\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Attacks on CROWN-IBP} \\label{sec:crown-ibp}\nInterval Bound Propagation (IBP) methods have been recently studied as a defense against $\\ell_{\\infty}$-bounded attacks. \nMany recent studies such as \\citet{gowal2018effectiveness, xiao2018training, wong2018scaling, mirman2018differentiable} have investigated IBP methods to train provably robust networks. \nTo the best of our knowledge, the CROWN-IBP method by \\citet{zhang2019towards} achieves state-of-the-art performance for MNIST \\citep{lecun-mnisthandwrittendigit-2010}, Fashion-MNIST \\citep{xiao2017fashion}, and CIFAR-10 datasets among certifiable $\\ell_\\infty$ defenses. \nIn this section we focus on attacking \\citet{zhang2019towards} using CIFAR-10.\n\nIBP methods (over)estimate how much a small $\\ell_{\\infty}$-bounded noise in the input can impact the classification layer. \nThis is done by propagating errors from layer to layer, computing bounds on the maximum possible perturbation to each activation.  \nDuring testing, the user chooses an $\\ell_\\infty$ perturbation bound $\\epsilon,$ and error propagation is used to bound the magnitude of the largest achievable perturbation in network output.  If the output perturbation is not large enough to flip the image label, then a certificate is produced.  If the output perturbation is large enough to flip the image label, then a certificate is not produced.  During network training, IBP methods include a term in the loss function that promotes tight error bounds that result in certificates.\nOur attack directly uses the loss function term used during IBP network training in addition to the Shadow Attack $ $ penalties; we search for an image that is visually similar to the base image, while producing a bound on the output perturbation that is too small to flip the image label. \nNote that there is a subtle difference between crafting conventional adversarial examples which ultimately targets misclassification and our adversarial examples which aim to produce adversarial examples which cause misclassification \\textit{and} produce strong certificates. In the former case, we only need to attack the cross-entropy loss. If we use our Shadow Attack $ $ to craft adversarial examples based on the cross-entropy loss, the robustness errors are on average \nroughly 50\\% larger \nthan those reported in table~\\ref{table:ibp} (i.e., simple adversarial examples produce weaker certificates.)\n\nWe attack 4 classes of networks released by  \\cite{zhang2019towards}  for CIFAR-10. \nThere are two classes of IBP architectures, one of them consists of 9 small models and the other consists of 8 larger models. \nFor each class of architecture, there are two sets of pre-trained models: one for ${\\epsilon}=2\/255$ and one for ${\\epsilon}=8\/255$. \nWe use $\\lambda_{tv} = 0.000009$, $\\lambda_{c} = 0.02$, $C(\\delta)=\\| \\delta \\|_2 $ and set the learning rate to $200$ and for the rest of the regularizers and hyper-paramters we\nuse the same hyper-parameters and regularizers as in \\ref{sec:smoothing}. \nFor the sake of efficiency, we only do 1-channel attacks.  We attack the 4 classes of models and for each class, we report the min, mean, and max of the robustness errors and compare them with those of the CROWN-IBP paper. \n\nTo quantify the success of our attack, we define two metrics of error.  For natural images, we report the rate of ``robust errors,'' which are images that are either (i)  incorrectly labeled, or (ii) correctly labelled but without a certificate.   In contrast, for attack images, we report the rate of ``attack errors,''  which are either (i) correctly classified or (ii) incorrectly classified but without a certificate.   Table~\\ref{table:ibp} shows the robust error on natural images, and the attack error on Shadow Attack $ $ images for the CROWN-IBP models. With ${\\epsilon}=2\/255$, our attack finds adversarial examples that certify roughly 15\\% of the time (i.e., attack error $<$85\\%).  With ${\\epsilon}=8\/255$, our attack finds adversarial examples that are incorrectly classified, and yet certify even more often than natural images.\n \n\n\\begin{table}\n\\caption{``Robust error'' for natural images, and ``attack error'' for Shadow Attack $ $ images using the CIFAR-10 dataset, and CROWN-IBP models. Smaller is better.\n}\n\\label{table:ibp}\n\\begin{center}\n\\begin{tabular}{llllll}\n\\multirow{2}{*}{ $\\epsilon(l_{\\infty})$ } & \\multirow{2}{*}{\\bf Model Family}  &\\multirow{2}{*}{\\bf Method}  &\\multicolumn{3}{c}{\\bf Robustness Errors} \\\\\n&&&Min&Mean&Max\\\\ \\hline\n\n\n\\multirow{4}{*}{ $2\/255$}   & \\multirow{2}{*}{9 small models} & CROWN-IPB & 52.46     & 57.55 & 60.67 \\\\\\cline{3-6}\n                           \n                           \n                            &                                    & Shadow Attack $ $   & \\bf 45.90 & \\bf 53.89 & 65.74 \\\\\\cline{2-6}\n                           \n                            \n                            &\\multirow{2}{*}{8 large models } & CROWN-IBP & 52.52     & 53.9 & 56.05 \\\\\\cline{3-6}\n                           \n                           \n                            &                                 & Shadow Attack $ $   & \\bf 46.21  & \\bf 49.77 & \\bf 51.79 \\\\\\cline{1-6}\n                           \n\n\\multirow{4}{*}{ $8\/255$ }  & \\multirow{2}{*}{ 9 small models } & CROWN-IBP & 71.28 & 72.15 & 73.66 \\\\\\cline{3-6}\n                           \n                           \n                            &                                     & Shadow Attack $ $ & \\bf 63.43  & \\bf 66.94 & \\bf 71.02 \\\\\\cline{2-6}\n                           \n\n                            \n                            & \\multirow{2}{*}{8 large models} & CROWN-IBP & 70.79 & 71.17  & 72.29 \\\\\\cline{3-6}\n                           \n                           \n                            &                                   & Shadow Attack $ $    & \\bf 64.04   & \\bf 67.32 & \\bf 71.16 \\\\\\cline{1-6}\n                           \n\\end{tabular}\n\\vspace{3mm}\n\\end{center}\n\\end{table}\n                            \n\n\n                            \n\n\n\\section{conclusion} \\label{sec:conclusion}\nWe demonstrate that it is possible to produce adversarial examples with ``spoofed'' certified robustness by using large-norm perturbations. \nOur adversarial examples are built using our Shadow Attack $ $ that produces smooth and natural looking perturbations that are often less perceptible than those of the commonly used $\\ell_p$-bounded perturbations, while being large enough in norm to escape the certification regions of state of the art principled defenses.  This work suggests that the certificates produced by certifiably robust classifiers, while mathematically rigorous, are not always good indicators of robustness or accuracy.  \n\n\n\\paragraph{Acknowledgements:} Goldstein and his students were supported by the DARPA QED for RML program, the DARPA GARD program, and the National Science Foundation.\n\n\n\\section{Submission of conference papers to ICLR 2020}\n\nICLR requires electronic submissions, processed by\n\\url{https:\/\/openreview.net\/}. See ICLR's website for more instructions.\n\nIf your paper is ultimately accepted, the statement {\\tt\n  {\\textbackslash}iclrfinalcopy} should be inserted to adjust the\nformat to the camera ready requirements.\n\nThe format for the submissions is a variant of the NIPS format.\nPlease read carefully the instructions below, and follow them\nfaithfully.\n\n\\subsection{Style}\n\nPapers to be submitted to ICLR 2020 must be prepared according to the\ninstructions presented here.\n\n\nAuthors are required to use the ICLR \\LaTeX{} style files obtainable at the\nICLR website. Please make sure you use the current files and\nnot previous versions. 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Place one line space before the figure caption, and one line\nspace after the figure. The figure caption is lower case (except for\nfirst word and proper nouns); figures are numbered consecutively.\n\nMake sure the figure caption does not get separated from the figure.\nLeave sufficient space to avoid splitting the figure and figure caption.\n\nYou may use color figures.\nHowever, it is best for the\nfigure captions and the paper body to make sense if the paper is printed\neither in black\/white or in color.\n\\begin{figure}[h]\n\\begin{center}\n\\fbox{\\rule[-.5cm]{0cm}{4cm} \\rule[-.5cm]{4cm}{0cm}}\n\\end{center}\n\\caption{Sample figure caption.}\n\\end{figure}\n\n\\subsection{Tables}\n\nAll tables must be centered, neat, clean and legible. Do not use hand-drawn\ntables. The table number and title always appear before the table. See\nTable~\\ref{sample-table}.\n\nPlace one line space before the table title, one line space after the table\ntitle, and one line space after the table. The table title must be lower case\n(except for first word and proper nouns); tables are numbered consecutively.\n\n\\begin{table}[t]\n\\caption{Sample table title}\n\\label{sample-table}\n\\begin{center}\n\\begin{tabular}{ll}\n\\multicolumn{1}{c}{\\bf PART}  &\\multicolumn{1}{c}{\\bf DESCRIPTION}\n\\\\ \\hline \\\\\nDendrite         &Input terminal \\\\\nAxon             &Output terminal \\\\\nSoma             &Cell body (contains cell nucleus) \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Default Notation}\n\nIn an attempt to encourage standardized notation, we have included the\nnotation file from the textbook, \\textit{Deep Learning}\n\\cite{goodfellow2016deep} available at\n\\url{https:\/\/github.com\/goodfeli\/dlbook_notation\/}.  Use of this style\nis not required and can be disabled by commenting out\n\\texttt{math\\_commands.tex}.\n\n\n\\centerline{\\bf Numbers and Arrays}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1in}p{3.25in}}\n$\\displaystyle a$ & A scalar (integer or real)\\\\\n$\\displaystyle {\\bm{a}}$ & A vector\\\\\n$\\displaystyle {\\bm{A}}$ & A matrix\\\\\n$\\displaystyle {\\tens{A}}$ & A tensor\\\\\n$\\displaystyle {\\bm{I}}_n$ & Identity matrix with $n$ rows and $n$ columns\\\\\n$\\displaystyle {\\bm{I}}$ & Identity matrix with dimensionality implied by context\\\\\n$\\displaystyle {\\bm{e}}^{(i)}$ & Standard basis vector $[0,\\dots,0,1,0,\\dots,0]$ with a 1 at position $i$\\\\\n$\\displaystyle \\text{diag}({\\bm{a}})$ & A square, diagonal matrix with diagonal entries given by ${\\bm{a}}$\\\\\n$\\displaystyle {\\textnormal{a}}$ & A scalar random variable\\\\\n$\\displaystyle {\\mathbf{a}}$ & A vector-valued random variable\\\\\n$\\displaystyle {\\mathbf{A}}$ & A matrix-valued random variable\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Sets and Graphs}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {\\mathbb{A}}$ & A set\\\\\n$\\displaystyle \\mathbb{R}$ & The set of real numbers \\\\\n$\\displaystyle \\{0, 1\\}$ & The set containing 0 and 1 \\\\\n$\\displaystyle \\{0, 1, \\dots, n \\}$ & The set of all integers between $0$ and $n$\\\\\n$\\displaystyle [a, b]$ & The real interval including $a$ and $b$\\\\\n$\\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$\\\\\n$\\displaystyle {\\mathbb{A}} \\backslash {\\mathbb{B}}$ & Set subtraction, i.e., the set containing the elements of ${\\mathbb{A}}$ that are not in ${\\mathbb{B}}$\\\\\n$\\displaystyle {\\mathcal{G}}$ & A graph\\\\\n$\\displaystyle \\parents_{\\mathcal{G}}({\\textnormal{x}}_i)$ & The parents of ${\\textnormal{x}}_i$ in ${\\mathcal{G}}$\n\\end{tabular}\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Indexing}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {a}_i$ & Element $i$ of vector ${\\bm{a}}$, with indexing starting at 1 \\\\\n$\\displaystyle {a}_{-i}$ & All elements of vector ${\\bm{a}}$ except for element $i$ \\\\\n$\\displaystyle {A}_{i,j}$ & Element $i, j$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{i, :}$ & Row $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{:, i}$ & Column $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\etens{A}}_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor ${\\tens{A}}$\\\\\n$\\displaystyle {\\tens{A}}_{:, :, i}$ & 2-D slice of a 3-D tensor\\\\\n$\\displaystyle {\\textnormal{a}}_i$ & Element $i$ of the random vector ${\\mathbf{a}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Calculus}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle\\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$\\\\ [2ex]\n$\\displaystyle \\frac{\\partial y} {\\partial x} $ & Partial derivative of $y$ with respect to $x$ \\\\\n$\\displaystyle \\nabla_{\\bm{x}} y $ & Gradient of $y$ with respect to ${\\bm{x}}$ \\\\\n$\\displaystyle \\nabla_{\\bm{X}} y $ & Matrix derivatives of $y$ with respect to ${\\bm{X}}$ \\\\\n$\\displaystyle \\nabla_{\\tens{X}} y $ & Tensor containing derivatives of $y$ with respect to ${\\tens{X}}$ \\\\\n$\\displaystyle \\frac{\\partial f}{\\partial {\\bm{x}}} $ & Jacobian matrix ${\\bm{J}} \\in \\mathbb{R}^{m\\times n}$ of $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$\\\\\n$\\displaystyle \\nabla_{\\bm{x}}^2 f({\\bm{x}})\\text{ or }{\\bm{H}}( f)({\\bm{x}})$ & The Hessian matrix of $f$ at input point ${\\bm{x}}$\\\\\n$\\displaystyle \\int f({\\bm{x}}) d{\\bm{x}} $ & Definite integral over the entire domain of ${\\bm{x}}$ \\\\\n$\\displaystyle \\int_{\\mathbb{S}} f({\\bm{x}}) d{\\bm{x}}$ & Definite integral with respect to ${\\bm{x}}$ over the set ${\\mathbb{S}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Probability and Information Theory}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle P({\\textnormal{a}})$ & A probability distribution over a discrete variable\\\\\n$\\displaystyle p({\\textnormal{a}})$ & A probability distribution over a continuous variable, or over\na variable whose type has not been specified\\\\\n$\\displaystyle {\\textnormal{a}} \\sim P$ & Random variable ${\\textnormal{a}}$ has distribution $P$\\\\% so thing on left of \\sim should always be a random variable, with name beginning with \\r\n$\\displaystyle  \\mathbb{E}_{{\\textnormal{x}}\\sim P} [ f(x) ]\\text{ or } \\mathbb{E} f(x)$ & Expectation of $f(x)$ with respect to $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Var}(f(x)) $ &  Variance of $f(x)$ under $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Cov}(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P({\\textnormal{x}})$\\\\\n$\\displaystyle H({\\textnormal{x}}) $ & Shannon entropy of the random variable ${\\textnormal{x}}$\\\\\n$\\displaystyle D_{\\mathrm{KL}} ( P \\Vert Q ) $ & Kullback-Leibler divergence of P and Q \\\\\n$\\displaystyle \\mathcal{N} ( {\\bm{x}} ; {\\bm{\\mu}} , {\\bm{\\Sigma}})$ & Gaussian distribution %\nover ${\\bm{x}}$ with mean ${\\bm{\\mu}}$ and covariance ${\\bm{\\Sigma}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Functions}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle f: {\\mathbb{A}} \\rightarrow {\\mathbb{B}}$ & The function $f$ with domain ${\\mathbb{A}}$ and range ${\\mathbb{B}}$\\\\\n$\\displaystyle f \\circ g $ & Composition of the functions $f$ and $g$ \\\\\n  $\\displaystyle f({\\bm{x}} ; {\\bm{\\theta}}) $ & A function of ${\\bm{x}}$ parametrized by ${\\bm{\\theta}}$.\n  (Sometimes we write $f({\\bm{x}})$ and omit the argument ${\\bm{\\theta}}$ to lighten notation) \\\\\n$\\displaystyle \\log x$ & Natural logarithm of $x$ \\\\\n$\\displaystyle \\sigma(x)$ & Logistic sigmoid, $\\displaystyle \\frac{1} {1 + \\exp(-x)}$ \\\\\n$\\displaystyle \\zeta(x)$ & Softplus, $\\log(1 + \\exp(x))$ \\\\\n$\\displaystyle || {\\bm{x}} ||_p $ & $L^p$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle || {\\bm{x}} || $ & $L^2$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle x^+$ & Positive part of $x$, i.e., $\\max(0,x)$\\\\\n$\\displaystyle \\bm{1}_\\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\n\\section{Final instructions}\nDo not change any aspects of the formatting parameters in the style files.\nIn particular, do not modify the width or length of the rectangle the text\nshould fit into, and do not change font sizes (except perhaps in the\n\\textsc{References} section; see below). Please note that pages should be\nnumbered.\n\n\\section{Preparing PostScript or PDF files}\n\nPlease prepare PostScript or PDF files with paper size ``US Letter'', and\nnot, for example, ``A4''. The -t\nletter option on dvips will produce US Letter files.\n\nConsider directly generating PDF files using \\verb+pdflatex+\n(especially if you are a MiKTeX user).\nPDF figures must be substituted for EPS figures, however.\n\nOtherwise, please generate your PostScript and PDF files with the following commands:\n\\begin{verbatim}\ndvips mypaper.dvi -t letter -Ppdf -G0 -o mypaper.ps\nps2pdf mypaper.ps mypaper.pdf\n\\end{verbatim}\n\n\\subsection{Margins in LaTeX}\n\nMost of the margin problems come from figures positioned by hand using\n\\verb+\\special+ or other commands. We suggest using the command\n\\verb+\\includegraphics+\nfrom the graphicx package. Always specify the figure width as a multiple of\nthe line width as in the example below using .eps graphics\n\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.eps}\n\\end{verbatim}\nor\n\\begin{verbatim}\n   \\usepackage[pdftex]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.pdf}\n\\end{verbatim}\nfor .pdf graphics.\nSee section~4.4 in the graphics bundle documentation (\\url{http:\/\/www.ctan.org\/tex-archive\/macros\/latex\/required\/graphics\/grfguide.ps})\n\nA number of width problems arise when LaTeX cannot properly hyphenate a\nline. Please give LaTeX hyphenation hints using the \\verb+\\-+ command.\n\n\\subsubsection*{Author Contributions}\nIf you'd like to, you may include  a section for author contributions as is done\nin many journals. This is optional and at the discretion of the authors.\n\n\\subsubsection*{Acknowledgments}\nUse unnumbered third level headings for the acknowledgments. All\nacknowledgments, including those to funding agencies, go at the end of the paper.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCarbon nanotubes have the potential\nto make good interconnects because current densities approaching $10^9\n\\; A\/cm^2$~\\cite{wei-apl-01} and compatibility with silicon \ntechnology~\\cite{kreupl-microeleceng-02,ngo-eeetn-04} have\nbeen demonstrated. In the use of multiwall nanotubes for such applications, \ncontact with as many layers of the tube as possible~\\cite{pablo-apl-99} is desirable.  Larger \ndiameter nanotubes are preferable because even the semiconducting shells\ncan carry current due to small bandgaps. For example, a \nsemiconducting shell with a radius of 18.8 nm  has a half bandgap of\nonly 62.5 meV (kT=26 meV at room temperature).\nExperimentally measured conductance of metallic carbon nanotubes\nis close to the theoretical maximum of $4e^2\/h$ at low\nbiases~\\cite{frank-science-98,poncharal-epjd-99,nygard-applphysa-99,\nkong-prl-01}.\n\nExperiments on small diameter carbon nanotubes show that the \ndifferential conductance decreases with applied bias, for voltages\nlarger than 100 - 200 meV~\\cite{yao-prl-00,collins-prl-01,pablo-prl-02}.\nRef. \\cite{yao-prl-00} found that the conductance \ndecrease with increase in bias was caused by reflection of incident electrons \nat crossing subbands due to scattering with zone boundary \nphonons. Measured values of the mean free path ~\\cite{park-nl-04}\nare in the range of 0.1 - 1 $\\mu$m at low bias and around 10 nm at high bias for\n1.8 nm diameter nanotubes (corresponds to a (24,0) nanotube).\nSimple estimates of scattering rates  ~\\cite{park-nl-04,golgsman-prb-03}\nshow inverse dependence with nanotube diameter, which suggests that \nphonon scattering is less important in large diameter nanotubes.\n\nThe non crossing subbands of small diameter nanotubes do not \ncarry current due to their large band gap. In contrast, large diameter \nnanotubes experimentally show an increase in conductance with applied\nbias~\\cite{frank-science-98,poncharal-epjd-99,poncharal-jpcb-02,liang-apl-04}. \nReference \\cite{Anantram-prb-00} suggested that as the diameter \nincreases electrons may tunnel into non crossing subbands, thus \ncausing an increase in differential conductance with applied bias. The\nmain drawbacks of the calculation in reference \\cite{Anantram-prb-00}\nwas that the results depended on the assumed form of potential drop in\nmetallic nanotubes (charge self-consistency and phonon scattering were\nneglected).\n\nApart from measuring current-voltage characteristics, the potential\ndrop can be measured by EFM. Refs.  \\cite{bachtold-prl-00} found that\nthe potential drop in a single wall nanotube was small compared to a \nmultiwall nanotube.\n\nIn this work, we computationally model ballistic electron flow in metallic\nnanowires within a tight-binding approximation by  including {\\it charge \nself consistency}, which is an important factor in \ndetermining the current. \n The potential distribution and current-voltage characteristics are studied\nas a function of both nanotube diameter and length.\nIn 3D (bulk) crystal the electric field, potential and net charge at the edges of the tube \nwould decay exponentially, resulting in zero net charge and \nzero electric field inside the conductor.\nIn a 1D wire, even perfectly ballistic,  \nthe screening is poorer, due to the lower electron density of states.\nReference \\cite{leonard} found that near {\\it p-n} junctions in \nsemiconducting nanotubes, this \nleads to a very long range ($\\sim 1\/ln(y)$) \ntail in the charge distribution \nand, consequently,  \na non zero electric field very far from the junction. \nIt is however an open question whether metallic \nnanotubes also have poor screening properties. \nHere we use an electrostatics model similar to that in \\cite{leonard}, \nbut significantly improve on \ntransport, using Non Equilibrium Green's Function formalism.\nThe ballistic approximation works well at \nlow bias and\/or in large diameter nanotubes. \nThe detailed study of phonon scattering effects will be given elsewhere.\n\nThe nature of the metallic contacts is also\nimportant. From an experimental view point the contact between a metal and \na nanotube can either be an end-contact or side-contact. \nThe end-contact\n corresponds to only the nanotube tip making the contact to the \n  metal lead. In experiments, the end-contacts \n   usually include strong chemical \nmodification of the nanotube at the metal-nanotube interface.\n  Also, reference \\cite{frank-science-98} found that end-contacts without \n  sufficient chemical modification of the nanotoube-metal interface\n   have a large contact resistance.\n    In such configuration, \n   electrons are injected directly from the metal to the nanotube \n   region between the contacts. \n     Due to the uncertainty of the contact bandstructure,  \n modeling end-contacts even remotely\n  correct is extremely difficult.\n   \n   The side-contacts correspond to \n   coupling between metal and nanotube atoms over many unit cells\n    of the nanotube, and can be thought of as a nanotube buried \n    inside a metal. Most experimental configurations correspond to\n     side-contacts \\cite{frank-science-98,tans}. \n     An important feature of the side contacts is that the  coupling between\n      atoms in the nanotube is much stronger than coupling between \n      nanotube and metal atoms which means that the bandstructure \n      of the contacts can be assumed to be\n       the same as that of the nanotube between the contacts.     \n     In the side-contact geometry, \n      electrons are predominantly injected from the metal into the\n       nanotube buried in the metal and then transmitted to the \n       nanotube region between the contacts.              \n        In fact, as proof of such a process, scaling \n       of conductance with contact area has been observed in the \n       side-contacted geometry by references \\cite{frank-science-98,tans}. \n       Modeling showed that the conductance in metallic zigzag \n       nanotubes can be close to the theoretical maximum for \n       conductance with sufficient overlap area in the \n       contact \\cite{Anantram-apl-01}. \n\n\nIn the work presented here, we assume  \"perfect\" contacts to \nrepresent an ideal side-contacted nanotube.  The details of the contact model\nwill be given below.\n\n\n\\section{Formalism}\n\nIn this paper we consider only zigzag carbon nanotubes. \nThe analysis for armchair nanotubes is similar.\nThe general form of the Hamiltonian for electrons in a carbon nanotube\n can be written as:\n\\begin{eqnarray}\nH & = & \\sum_{i,x}U^{i}_{x}c^{\\dagger}_{i,x}c_{i,x}+\n\\sum_{i,j,x,x'}\nt^{i,j}_{x,x'}c^{\\dagger}_{i,x}c_{j,x'}\n\\end{eqnarray}\nThe sum is taken over all rings $i,j$ in transport direction and all atoms located at $x,x'$ \nin each ring. \nWe make the following common approximations:\ni)  only nearest neighbors are included; \neach atom in an $sp^2$-coordinated carbon nanotube\nhas three nearest neighbors, located $a_{cc}=$ 1.42 $\\AA$ away;\nii) the bandstructure consists of only $\\pi$-orbital, with the \nhopping parameter\n$t_o=V_{pp\\pi}=-2.77$ eV and the on-site potential $U_o=\\epsilon_p=0$. \nSuch a tight-binding model \nis adequate to model transport properties in undeformed nanotubes.\nWithin these approximations, only the following parameters are non zero:\n\\begin{eqnarray}\nU^{i}_{x}& = & U_o , \\forall i \\nonumber \\\\\nt^{i,i-1}_{x,x'}& = &t^{i-1,i}_{x,x'}=t_o\\delta_{x\\pm a\/2,x'}, \\forall i=2k \n\\label{eq:hop_par}  \\\\\nt^{i,i+1}_{x,x'}& = &t^{i+1,i}_{x,x'}=t_o\\delta_{x,x'}, \\forall i=2k ~,\n\\nonumber\n\\end{eqnarray}\nwhere $a=a_{cc}\\sqrt{3}$.\n In $(N,0)$ zigzag nanotubes, the \nwave vector in the circumferential direction is quantized as \n$\\tilde q=2\\pi q\/Na$, $q=1,2,...N$, \ncreating eigenmodes in the energy spectrum. \nBy doing a Fourier expansion of $c^{\\dagger}_{i,x}$ and $c_{i,x}$ \nin $\\tilde q$-space  and using  (\\ref{eq:hop_par}) we obtain\na decoupled electron Hamiltonian in the eigenmode space:\n\\begin{eqnarray}\nH & = & \\sum_{q} H^{q}\\\\\nH^{q}& = & \\sum_{i}\n(U^{i}_{\\tilde q}\nc^{\\dagger}_{i,\\tilde q}c_{i,\\tilde q}\n+t^{i,i\\pm1}_{\\tilde q}\nc^{\\dagger}_{i,\\tilde q}c_{i\\pm1,\\tilde q})\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nU^i_{\\tilde q} &=& U_o, \\forall i \\nonumber\\\\\nt^{i,i-1}_{\\tilde q} & = & t^{i-1,i}_{\\tilde q}=2t_o\n\\cos({\\tilde qa \\over 2}) \\equiv t_1, \\forall i=2k  \\\\\nt^{i,i+1}_{\\tilde q}& = & t^{i+1,i}_{\\tilde q}=\nt_o \\equiv t_2,  \\forall i=2k  \\nonumber\n\\end{eqnarray}\nThe 1-D tight-binding Hamiltonian $H^q$ describes  a  \nchain with two sites per unit cell\nwith on-site potential $U_o$ and \nhopping parameters $t_1$ and $t_2$ (Fig. \\ref{fig:tube}).\nFor numerical solution, the spatial grid corresponds to the rings of the nanotube, \nseparated by $a\/2$ with a unit cell \nlength of $3a\/2=$ 2.13 $\\AA$, which is half the unit cell length of a \nzigzag nanotube. \n \nThe subband dispersion relations are given by\n\\begin{eqnarray}\nE_q(k)=\\pm {\\bigl| 1+4cos({3ak\\over2})cos({q\\pi\\over N})+\n4cos^2({q\\pi\\over N})\\bigr|}^{1\/2}\n\\end{eqnarray}\nTherefore, when $N=3k$,\n there are two subbands with zero bandgap \n the tube is metallic. In the rest of the paper we  \n distinguish between metallic \nor {\\it crossing} subbands  ($q=N\/3$ and $2N\/3$) and semiconducting or \n{\\it non crossing} subbands.\n\n The contacts \nare assumed to be reflectionless reservoirs\n maintained at equilibrium, i.e. they have  well defined chemical \npotentials, equal to that of the metal leads: $V_S$ in the source and $V_D$  in \nthe drain.\nFurther, the nanotube and metal\n are assumed to have the same workfunction.\nThe contact self-energies $\\Sigma_{c,q}^{R,<,>}=\n\\Sigma_{S,q}^{R,<,>}+\\Sigma_{D,q}^{R,<,>}$ due \nto the source contact are found\n\\cite{JAP} using\nthe surface Green's function $g^{S}_q$ of a semi-infinite nanotube, \nwhich is the solution \nof the following system of equations:\n\\begin{eqnarray}\n(a_1-t_{2}^2g^{S}_{2,q})g^{S}_{1,q}&=&1 \\nonumber\\\\\n(a_2-t_{1}^2g^{S}_{1,q})g^{S}_{2,q}&=&1 ~, \\label{eq:surf_gr}\n\\end{eqnarray}\nwhere the indices $1$ and $2$ stand for the two sites of the unit cell and\n$a_{1,2}=E-U_o-V_{S}$. The Green's function for the drain\nare solved for in a similar way, by making the following substitutions:\n$t_{1,2}^{S} \\rightarrow t_{2,1}^{D}$, \n$g_{1,2,q}^{S}\\rightarrow g_{1,2,q}^{D}$.\n\nFor each subband $q$ we solve a system of transport equations\n \\cite{JAP}:\n\\begin{eqnarray}\n(E-H^q-V-\\Sigma_{c,q}^{R}(E))G^{R,q}=I \\label{eq:tran1}\\\\\n(E-H^q-V-\\Sigma_{c,q}^{R}(E))G^{<,>,q}=\n\\Sigma_{c,q}^{<,>}(E)G^{A,q} \n\\label{eq:tran2}\n \\end{eqnarray}\nElectron charge  and current density $n_i$ and $J_i$  at each node $i$ \nare found from the following equations:\n\\begin{eqnarray}\nn_i=-{2i}\\sum_{q} \n\\int_{-3t_o-eV_D}^{-eV_S+10kT}G^{<,q}_{i,i}(E){dE \\over 2\\pi} \\label{eq:dens} \\\\ \nJ_i={4e \\over \\hbar }\\sum_{q} \\int_{-eV_D-10kT}^{-eV_S+10kT}\nG^{<,q}_{i,i+1}(E)t^q_{i+1,i}{dE \\over 2\\pi}, \n\\end{eqnarray} \nwhere we used the fact that the band \nbottom at equilibrium is at $-3t_o$.  \nTo integrate the electron charge  over energy,\nwe employed a conventional approach \ndeveloped in DFT (\\cite{Williams,Guo,Stokbro}) and described below.\nThe technique  \ndrastically reduces the computational requirements as the whole \nband can be integrated with 200 energy grid points with error \nof $10^{-6}$ of charge.\nThe integral (\\ref{eq:dens}) is divided into equilibrium $[-3t_o-eV_D; -eV_D-10kT]$ and \nnon-equilibrium parts $[-eV_D-10kT, -eV_S+10kT]$.  In the equilibrium part \nof the energy spectrum, the current is zero and one can apply the \nfluctuation-dissipation theorem $G^<(E)=-2if(E)\\Im m[G^R(E)]$,  where \nFermi factor $f(E)=1$ at these energies. \n$G^R$ is analytical in the upper plain of \ncomplex energies which allows \none to transform the integration of $G^<$  over the real axis in the \nequilibrium part of energy spectrum to the integration of $G^R$ over a complex contour $C$ \nstarting at $-3t_o-eV_D$ and ending at $-eV_D-10kT$. \nThe non equilibrium part is integrated over the real axis from \n $-eV_D-10kT$ to $-eV_S+10kT$:\n\\begin{eqnarray}\nn_i&=& -{2 \\over \\pi}\\sum_{q} \\oint_C\\Im m[G^{R,q}_{i,i}(E)dE] \\nonumber \\\\\n&&-{i \\over \\pi}\\sum_{q} \\int_{-eV_D-10kT}^{-eV_S+10kT}G^{<,q}_{i,i}(E)dE \n\\label{eq:densint}\n\\end{eqnarray}\n \n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_tube.eps}\n\\caption{Zigzag carbon nanotube and the corresponding 1D chain. \nThe hopping parameter between nearest neighbors in the nanotube is $t_o$. \nThe 1-D \nchain has two sites per unit cell\nwith on-site potential $U_o$ and \nhopping parameters $t_1=2t_o\n\\cos({\\tilde qa \\over 2})$ and $t_2=t_o$ }\n\\label{fig:tube}\n\\end{figure}\n\nIn this work, corresponding to ballistic transport, we neglect electron-phonon scattering. \nThe electron mean free path (mfp) is proportional to the matrix element squared and thus \ninversely proportional to the chirality index.\n Ref. \\cite{park-nl-04} reported \n$\\lambda_{o}^{el} \\approx$ 1.6 $\\mu m$ and \n$\\lambda_{o}^{in}\\approx$ 10 nm for a tube \nwith a diameter of 1.8 nm, corresponding to a (24,0) nanotube.\nThis  allows us to assume that large diameter nanotubes of \nmoderate lengths, e.g. $\\sim\\,$100 nm (240,0) nanotube, will remain nearly ballistic even at high bias.  \nThe ballistic transport is also an adequate approximation at low bias for all diameter nanotubes due to the absence\nof strong inelastic phonon emission.\n\n\n  \nWe model the electrostatics of the nanotube as a system of \npoint charges  between the two contacts located at $y=y_S=0$ and \n$y=y_D=L$.\nThe \"perfect contacts\" are modeled as parallel semi-infinite \nthree dimensional\n metal leads that are maintained at fixed source and drain potentials: \n $V_S$ for $y<y_S$ and $V_D$ for $y>y_D$. \n So, while the self-energies due to contacts are identical to that \n of a semi-infinite nanotube, the role of electrostatics is included\n  by image charges corresponding to a perfect metal. \n The electrostatic potential consists of  a linear drop due to a \nuniform electric field\ncreated by the leads and the potential due to the charges on the \ntube and their images\n\\begin{eqnarray}\nV=-eV_S-e(V_D-V_S)(y-y_S)\/(y_D-y_S)\\nonumber \\\\\n+\\sum_{j}G(i,j)(n_j-N) \\label{eq:prof}\n\\end{eqnarray}\nwith the Green's function  \n{\\small \\begin{eqnarray}\nG(i,j)&=&{e\\over 4N\\pi\\epsilon_o}\\sum_{n=-\\infty}^{+\\infty} \\sum_{l}\n\\biggl[ {1\\over\\sqrt{(y_i-y_j+2nL)^2+\\rho_{k,l}^2}} \\nonumber\\\\\n& & \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n-{1\\over\\sqrt{(y_i+y_j+2nL)^2+\\rho_{k,l}^2}}\\biggr]~.  \\label{eq:green}\n\\end{eqnarray}}\nHere,  $\\rho_{k,l}$ is the radial projection of the vector between atom \n$k$ at ring $i$ and atom $l$ at ring $j$. The summation is \nperformed over all atoms $l$ at ring $j$ for an arbitrary value of $k$.\nMaintaining \nthe nanotube atoms buried in the metal at a fixed potential is close \nto reality because of the screening properties of 3D metals. Within \na few atomic layers from the metal surface, the potential should have\n approached the bulk values. While the variation in potential in these\n  few atomic layers of the 3D metal is not captured in our model, our\n   conclusions on the nanotube electrostatics should not be significantly \n   affected. \n\nThe calculations for one bias point involve \niterations of (\\ref{eq:surf_gr}-\\ref{eq:prof}) \nuntil convergence \nof the\npotential and charge distributions is achieved.\nThe solution of (\\ref{eq:tran1}-\\ref{eq:tran2}) employs \nthe recursive algorithm \\cite{JAP}, which scales linearly  \nwith the number of nodes. \nThe contour integral in (\\ref{eq:densint})\nis taken using Gaussian quadratures. \n \n\n\n\\section{Results}\n\n\\subsection{Electrostatics}\n\nAt low biases (100 mV), electron-phonon scattering does not play a significant\nrole in determining the potential distribution for wires of moderate length\n(less than few hundreds of nanometer). The potential distribution for \n(12,0) nanotubes of \nlengths varying from  21.3 to 213 nm are shown in Fig. \\ref{fig:pot1}.\nThe edges of the nanotube near the contact rapidly screen the applied\nbias \/ electric field.  \nThe screening is due to the free electron density provided by the crossing subbands.\nThe potential drop is divided unequally\nbetween different parts of the nanotube,\n with 90\\% of the applied bias dropping within \n1 nm from the edges for both lengths considered here.  \nNote, that in the ballistic regime, \nresistance of the nanotube does not depend on tube length, \nwhich results in the drop at the edges being constant for all tube lengths. \n\n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_POTvsY_12_vsL.eps}\n\\caption{Potential distribution versus position for (12,0) nanotubes\nof various lengths. The diameter of the (12,0) nanotube\nis 0.94 nm and the applied bias is 100 mV. Note the constant potential drop at the edges. }\n\\label{fig:pot1}\n\\end{figure}\n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_POTvsY_123660_100.eps}\n\\caption{Potential versus position for (12,0), (36,0) and (60,0) nanotubes, \nwith diameters 0.94, 2.82 and 4.7 nm. The screening for the large\ndiameter nanotubes is poorer. The inset magnifies the\npotential close to the edge of the nanotube. The applied bias is\n100 mV.}\n\\label{fig:pot2}\n\\end{figure}\n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_NvsY_123660_100.eps}\n\\caption{Charge density versus position for (12,0), (36,0) and (60,0) nanotubes.\nThe inset magnifies the\ncharge distribution close to the edge of the nanotube. The applied bias is\n100 mV.}\n\\label{fig:dens2}\n\\end{figure} \n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_EfldvsL_N.eps}\n\\caption{This plot shows the electric field at the mid point versus \nnanotube length for various nanotube diameters at an applied bias of 100 mV.\nThe electric field decreases approximately as \n$1\/L^{1.25-1.75}$.}\n\\label{fig:efield}\n\\end{figure}\n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_EfldvsN_L.eps}\n\\caption{The electric field at the mid point versus \nnanotube chirality index for various lengths at an applied bias of 100 mV.\n The electric field increases approximately as \n$N^{0.25-0.75}$.}\n\\label{fig:efieldN}\n\\end{figure}\nWhile the total density of states  (DOS) per unit length of\nmetallic nanotubes is independent of diameter, the DOS per subband  \nand free charge carried by the crossing subbands are\ninversely proportional to the diameter.  \nAs a result, we find that the\nscreening in metallic nanotubes degrades with diameter. \nThe potential\ndrop for  nanotubes with diameters of 0.94 nm [(12,0) nanotube],\n2.82 nm [(36,0)] and 4.7 nm [(60,0)] are shown in Fig. \\ref{fig:pot2}.\nClearly, screening is poorer in the larger diameter nanotube. In fact,\nwhile the potential drops by 45 mV in a distance of 1 nm from the edge\nfor the (12,0) nanotube, the potential drops for (36,0) and (60,0) \n are only 37 mV and 31 mV respectively. \nThe inset of Fig. \\ref{fig:pot2} shows a \nsubstantially larger electric field away from the edges of the large\ndiameter nanotube.  \nAnother difference between large and small diameter \nnanotubes is due to the different geometry contained in the \nelectrostatic Green's function (\\ref{eq:green}).\nHigher number of atoms per ring in \nlarger diameter tubes slightly offsets  \nthe deterioration of screening due lower density of states.\n  \n\nFig. \\ref{fig:dens2} shows the self-consistent  charge profiles, corresponding to the \npotential in Fig. \\ref{fig:pot2}.  The interplay between the competing\nfactors of different density of states \nand electrostatic Green's function results\nin different charge profiles. In the larger diameter tubes more charge is accumulated at the edges.\n\nThe electric field at the center of the nanotube as a function of \nlength is shown in Fig. \\ref{fig:efield} for the tube with a diameter of\n0.94 nm. We find that for all diameters, the electric field decreases\nmore rapidly than $\\frac{1}{L}$, where $L$ is the length of the\nnanotube. The exact power law however depends on the diameter.\nIf the computed electric field versus length is fit to $\\frac{1}{L^a}$,\nthe exponent $a$ increases with increase in diameter. The value of\n$a$ increases from 1.25 to 1.75 as the diameter increases from 0.94\nto 18.85 nm.\nThe dependence of the electric field on the chirality index\n of the tube is shown in Fig. \\ref{fig:efieldN}.\nThe electric field versus diameter  ($D=aN\/\\pi$) \ncan be fit to $D^{b}$, where $b$ is in the range of 0.25 and 0.75. \nThe presence of non zero electric field inside a conducting nanotube is not \ntotally surprising: it is a consequence of a lower electron density of states\n in 1D.\n\n\n\\subsection{Transport}\n\nWe now discuss the current-voltage characteristics as a function of\nbias for small and large diameter nanotubes of length 42.6 nm, in the\nballistic limit. The ballistic current and differential conductance \nfor the different diameters are  shown in Fig.\\ref{fig:IVbal} and\nFig.\\ref{fig:dIdV}.\nFor the moderate diameter nanotubes ($N<90$), \nonly two crossing subbands contribute to transport.\nThe  current increases linearly\nwith applied bias and the differential conductance is $4e^2\/h$.\n\nIn large diameter nanotubes ($N =$ 90, 120 and 240),\nballistic current shows super linear increase.\nThe conductance of large diameter nanotubes  starts \nincreasing after a threshold bias.\nThis increase occurs due to Zener tunneling contribution of \nthe non crossing subbands: when the bias \nbecomes larger than twice the bandgap of the lowest non crossing subband,\nelectrons can tunnel from  valence band \nstates ($E<-E^{non}_{g}\/2+V(y)$) in the source to  \nthe conduction band  states ($E >E^{non}_{g}\/2+V(y)$) in the middle of the tube (the channel) and also\nfrom  valence band \nstates in the channel  to the conduction band  states in the drain.\nThe lowest non crossing subbands ($q=N\/3 \\pm 1$) in (90,0), (120,0) and (240,0) \nhave bandgaps $E^{non}_{g}=$ 331, 249 and 125 meV respectively \nand start to contribute \nto current at biases of twice these values.\nIn the ballistic limit, the self-consistently\ncalculated current of (240,0) nanotube  at 1 V is 310 $\\mu$A and the differential conductance \nis almost $13 e^2 \/ h$, signifying that a large number of subbands are\ncontributing to current. As the barrier width at energies corresponding to\n valence band edge in the source $E_v^S$ and conduction band edge in the drain $E_c^D$ \ndecreases with bias, the transmission and Zener tunneling current increase.\nThe density of states (DOS) in the lowest non crossing subband in (240,0) \nnanotube at a bias of 0.4 V is shown in Fig. \\ref{fig:DOSvsE}. Note the \nincreased DOS in the bandgap at energies $E_v^S$ and $E_c^D$. \n\n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_IVbal_100.eps}\n\\caption{The current-voltage characteristics of large diameter nanotubes in the ballistic limit.\nThe current shows faster than linear increase with bias due to the contribution of non crossing subbands.\n}\n\\label{fig:IVbal}\n\\end{figure}\n\n\nAnother qualitative difference occurs at zero bias in (240,0) nanotube (Fig.\\ref{fig:dIdV}).\nThe zero bias conductance of (240,0) nanotube\nis larger than $4e^2\/h$ because the \nnon crossing subbands  are partially \nfilled and contribute to current:\nthe first non crossing subband opens at an energy of 2.4 $kT$ \nfrom the band center.\nThis contribution is a simple intraband transport, \ndetermined by the population of the conduction band in the source and the \nvalence band in the drain, but rather insensitive to the details of the potential distribution.\nAt slightly higher biases\nthe non crossing subbands  contribution  to current \nsaturates to a constant value and the contribution to differential conductance decreases to zero, while\nthe total conductance decreases to $4e^2\/h$ . \n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_GVbal_100.eps}\n\\caption{The differential conductance as a function \nof bias  in the ballistic case. \nContribution of non crossing subbands in (240,0) nanotube leads to \nthe non monotonic behavior of conductance. High bias increase \ncorresponds to Zener tunneling and is determined by the bandgap of the lowest non crossing subband.}\n\\label{fig:dIdV}\n\\end{figure} \n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_DOSvsEcont.eps}\n\\caption{Local density of states in the lowest non crossing \nsubband of a (240,0) nanotube\nunder a bias of 0.4 V as a function of position and energy.\nBrighter color shows higher DOS.\n Also shown is the potential profile.\nThe arrows correspond to \nfour contributions to a current due to the non crossing \nsubband in Fig.\\ref{fig:CURvsE}}.\n\n\\label{fig:DOSvsE}\n\\end{figure}\n\\begin{figure}\n\\epsfxsize=3.25in\\epsfbox{ieeeshort_CURvsE_240_100bal.eps}\n\\caption{A set of current density \nversus energy curves due to the lowest non crossing subband in a (240,0) nanotube under different biases.\nTwo outermost peaks correspond to transport in the valence and conduction bands.\nTwo inner peaks correspond to Zener tunneling between valence and conduction band states.\n}\n\\label{fig:CURvsE}\n\\end{figure} \n \nTo elucidate the effect of non crossing subbands\nfurther, Fig.\\ref{fig:CURvsE} shows\nthe evolution with bias of \ncurrent versus electron energy in the lowest  non crossing subband.   \nAt low biases there are two peaks in the current density corresponding to\npartially filled valence and conduction bands. \nThe height of the peaks increases only slightly and \nthen saturates at all higher biases. These peaks in\n the current density are responsible for the increased value of zero bias conductance.\nWhen  bias is higher than twice the bandgap, \ntwo new peaks in the current density occur at $E_v^S$ and $E_c^D$,\ncorresponding to  Zener tunneling at the source and drain respectively.\nThe height of these peaks increases rapidly due to the exponential dependence on the barrier width.\n\nThe important consequence of the different origin of these \nfour peaks is the temperature dependence of differential conductance. \nAt low temperatures, the outermost peaks will vanish,\nreducing the value of conductance to $4e^2\/h$. However, the height and shape of the inner peaks are \n insensitive to the temperature which implies the robustness of Zener tunneling.\n\n\nIn conclusion, we find that \nin the ballistic (low bias\/large diameter) regime, applied \nbias drops mostly at the edges. \nHowever, the 1-dimensionality of the nanotube leads to very  \nslow ($\\sim 1\/L^{\\alpha}$) screening of the electric \nfield as compared to a bulk crystal.\nIn agreement with \nexperimental measurements of Ref. \\cite{bachtold-prl-00}, \nour calculations show larger electric field\nfor larger diameter nanotubes. \nThe conductance of large diameter nanotubes shows \nnon monotonic bias dependence: at low and high bias the \nconductance is higher than $4e^2\/h$ due to the \ncontribution of non crossing subbands. \n\nWe are grateful to Dr. Hatem Mehrez for his\nhelp with  \nthe complex contour integration of DOS.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nA machine learning application can fail for many reasons: maybe the\ntraining data is insufficient, maybe there is a flaw in the design of\nthe learning algorithm, or maybe there is an error in the\nimplementation of the algorithm. Such errors can go unnoticed for long\nperiods of time. This is particularly worrisome for machine learning\napplications that, for example, process loan requests or suggest\nhiring recommendations.\n\nThorough testing is one way to help catch such errors. But testing ML\napplications is challenging because of their random behavior. Many\niterations may be needed to encounter a bug or detect a statistical\nirregularity in behavior.  And when it comes to adversarial or safety\ncritical scenarios, no amount of testing may be enough to make a\nsystem trustworthy.  Moreover, while there has been much work on\ndeveloping algorithms that are provably robust or fair, bugs in\nimplementations of these algorithms may render these guarantees\nmeaningless.\n\nOne way to eliminate these kinds of errors is to \\emph{formally\nverify} a machine learning system with a machine checked proof of\ncorrectness.\nA formal proof is one in which every step and logical inference is checked.\nThe computer programs that help write and check such\nproofs are called \\emph{proof assistants}. Proof assistants provide a\nlanguage to express programs and mathematical proofs in some logic.\nIn recent years, it has become feasible\nto use proof assistants to verify large, realistic software systems,\nincluding compilers~\\citep{leroy2009formal}, cryptographic primitives~\\citep{fiat-crypto}, file systems~\\citep{fscq-cacm}, and microkernels~\\citep{sel4-cacm}.\n\nIn principle, proof assistants are expressive enough to represent the\nmathematics underlying ML systems and check proofs of their\ncorrectness. But in practice, formally verifying ML systems remains\nchallenging.  Prior work has begun to develop formal proofs of\ncorrectness for machine learning software~\\citep{selsam2017bugfree,\nbagnall2019certifying}.\nAlthough these early results are impressive, representing machine learning\nprograms and their correctness statements inside a proof assistant remains a\nmajor challenge.  Proof assistants only have built-in support for representing\n``pure'' mathematical functions. The approach taken in the aforementioned works\nis to define, within the proof assistant, a small domain-specific programming\nlanguage that has commands for drawing random samples from distributions.\n\nWhile this approach is feasible, it becomes challenging to use, particularly\nwhen reasoning about samples from continuous probability distributions.\nIn particular, the way past work represents such programs in the proof assistant\nis rather different from the way ML researchers usually think about things in pencil-and-paper proofs.\n\nWe propose using an alternative representation that more closely matches the familiar style of paper proofs.\nOur approach is to\nautomatically \\emph{reparameterize} parts of a probabilistic program\nso that it can be written as a pure, non-randomized, functional\nprogram operating over a pre-sampled list of random inputs.  This\nsimplifies reasoning about the resulting program and avoids the\nforegoing difficulties. We have implemented an automatic translation\nto perform this reparameterization, so that we can convert programs\nfrom a probabilistic programming language based on Pyro \\cite{Pyro}\ninto input to the theorem prover.\n\nA simple example of our translation is shown\nin \\autoref{fig:intro_reparam}. On the left is a Python program\nthat samples from two distributions and returns a pair %\nas result. On the right is our reparameterized version of this program in\nthe {\\tt Lean}\\xspace proof assistant.  Background on {\\tt Lean}\\xspace follows, but for now,\nthe important point is that in the translation, \\texttt{majority\\_fun}\ntakes as input an argument $\\texttt{u}$ that represents a pair of\nuniform $[0, 1]$ samples. Then instead of sampling, it uses the\ninverse CDF transform to convert the input $\\texttt{u}$ into the\ndistributions it needs. Finally, we define the distribution\nrepresented by this program by taking the pushforward of the input\ndistribution through the function \\texttt{majority\\_fun}.\n\n\n\\begin{figure}\n\n\\begin{minipage}{0.35\\textwidth}\n\\begin{python}\ndef majority():\n  theta = uniform(0, 1) \n  X = bernoulli(theta)\n  return (theta, X)\n\\end{python}\n\\subcaption[first caption.]{Python program}\\label{fig:pyprog}\n\\end{minipage}\n\\quad\n\\begin{minipage}{0.5\\textwidth}\n\\centering\n\\begin{lstlisting}[language=lean]\ndef majority_fun u :=\n  let u1 := u.fst in\n  let u2 := u.snd in\n  let theta := gen_uniform(0,1,u1) in\n  let X := gen_bernoulli(theta,u2) in\n  (theta,X)\n\ndef majority :=\n  push_forward majority_fun pair_uniform(0,1)\n\n\\end{lstlisting}\n\\subcaption[]{Reparameterization in {\\tt Lean}\\xspace}\\label{fig:reparam}\n\\end{minipage}\n  \\caption{Demonstration of our automatic reparameterization translation to represent programs in the proof assistant. Instead of drawing samples, {\\tt majority\\_fun} takes a pre-sampled pair {\\tt u} of uniform [0, 1] variates. The expressions {\\tt u.fst} and {\\tt u.snd} are the first and second projections of the pair.}\\label{fig:intro_reparam}\n\\end{figure}\n \n\nTo demonstrate our approach, we have verified two case studies. The\nfirst is a proof that the class of decision stumps is\nPAC-learnable~\\citep{blumer1989learnability}, a classic introductory\nresult that appears in many textbooks on computational learning\ntheory. Formalizing this proof revealed errors and many omitted\ndetails found in several expository accounts of this result.  Our\nsecond case-study is a proof that a null model used in a Bayesian hypothesis test,\nimplemented in a probabilistic programming language, correctly satisfies\na fairness criterion called demographic parity. The source code\nfor our case studies and translator are publicly available.\\footnote{\\url{https:\/\/github.com\/jtristan\/FormalML}}\n\n\n\n \n\\section{A Short Introduction to Lean}\n\\label{sec:lean}\nThe {\\tt Lean}\\xspace theorem prover can be viewed as both a functional\nprogramming language (like Haskell) and a foundation for mathematics,\nbased on dependent type theory.  Dependent type theories are an\nalternative to Zermelo-Frankel set theory where types are associated\nwith mathematical expressions, in the same way that types can be used\nin programming languages, but with much stronger guarantees.  Before\nwe introduce the concept of dependent types, it is useful to consider\na simple example of mathematical formalization in {\\tt Lean}\\xspace.\n\nUsing {\\tt Lean}\\xspace as a programming language, we can define a function\n\\texttt{double} that takes a natural number as input and multiplies it by 2.\n\\begin{lstlisting}[language=lean]\ndef double(n: nat): nat := 2 * n\\end{lstlisting}\nThis definition is similar to what one would find in any modern\nfunctional programming language.  However, there is one significant\ndifference between programming in {\\tt Lean}\\xspace and those languages: in order\nto ensure that {\\tt Lean}\\xspace is a consistent foundation for mathematics,\nfunctions cannot have side effects (printing on the screen, reading a\nfile) and they must be proven to always terminate. Next, we can define\na predicate that formalizes the concept of an even number.\n\\begin{lstlisting}[language=lean]\ndef isEven(n: nat): Prop := exists k: nat, n = 2 * k\\end{lstlisting}\nThis example clearly shows how {\\tt Lean}\\xspace differs from a programming\nlanguage. The function we define does not return simple data like a\nnumber or string, but instead a logical proposition that states that a\nnatural number $n$ is even if there exists a natural number $k$ such that $n = 2 *\nk$.\n\n\\squeezedpara{Proofs with tactics:}\nFinally, {\\tt Lean}\\xspace lets us specify mathematical properties and prove\nthem. For example, the following states and proves a lemma called\n\\texttt{doubleIsEven} that says that the result of \n\\texttt{double} is always even:\n\\begin{lstlisting}[language=lean]\nlemma doubleIsEven: forall n: nat, isEven (double (n)) := \nbegin \n  intros, unfold isEven, unfold double, existsi n, trivial, \nend\\end{lstlisting}\nThe first line is the mathematical statement we wish to prove. What\nfollows the ``:='' and enclosed by the keywords ``begin'' and ``end''\nis a set of commands, called tactics, that describes the proof in a\nmanner that {\\tt Lean}\\xspace can check.  The programmer constructs this tactic\nproof interactively: their IDE displays a list of current assumptions\nand what remains to be proved.  This is represented by\na \\emph{sequent}, which is a tuple of the form $\\Gamma\n\\vdash \\phi$, where $\\Gamma$ is the list of hypotheses and variables (called the context)\nand $\\phi$ is a proposition (called the target). When the proof\nstarts, the sequent is $\\emptyset \\vdash \\forall\nn: \\mathbb{N}, \\texttt{isEven} (\\texttt{double} (n))$. Executing the tactic\n``intros'' transforms the sequent into $n: \\mathbb{N} \\vdash \\texttt{isEven}\n(\\texttt{double}(n))$ where $n$ is now a fixed but arbitrary natural\nnumber. Executing the tactic ``unfold'' applied to \\texttt{isEven}\nunfolds the definition of \\texttt{isEven} to give the sequent\n$n: \\mathbb{N} \\vdash \\exists k: \\mathbb{N}, \\texttt{double}(n) = 2 * k$. Likewise, by\nunfolding \\texttt{double}, we obtain the sequent $n: \\mathbb{N}\n\\vdash \\exists k: \\mathbb{N}, 2 * n = 2 * k$. Now we must\nexhibit a choice for $k$ that satisfies the property, for which we can\nuse the tactic ``existi'' applied to $n$, which appears in the\ncontext. This gives the sequent $n: \\mathbb{N} \\vdash 2 * n = 2 * n$ to which\nwe can apply the ``trivial'' tactic that ensures that a basic axiom\n(namely, that equality is reflexive) has been reached.\n\n\\squeezedpara{{\\tt mathlib}\\xspace library:}\nThe {\\tt mathlib}\\xspace library is a large library of mathematical results\nformalized in {\\tt Lean}\\xspace~\\citep{mathlib-cpp}.\nIn particular, it contains a formalization of\nmeasure theory, based on \\citet{HolzlH11}'s library from the Isabelle\ntheorem prover. Unfortunately, {\\tt mathlib}\\xspace does not a have a probability\ntheory library, and in order to formalize our results, we had to\ndevelop one, as a special case of measure theory. This development\naccounts for about 2,500 lines of {\\tt Lean}\\xspace formalization.\n\n\\squeezedpara{Proof by reduction:}\nAn important feature of {\\tt Lean}\\xspace that makes proofs easier is that the proof checker\ncan automatically execute or \\emph{reduce} parts of programs. For example,\nsuppose at some point we need to show that the boolean expression\n\\texttt{(a and false) or (false and b)}\nis always \\texttt{false}. Instead of using lemmas for the basic rules\nof boolean algebra, we can instruct {\\tt Lean}\\xspace to case split on all\nthe possible values for \\texttt{a} and \\texttt{b}, evaluate the\nexpression, and then check that all cases reduce to \\texttt{false}. Of\ncourse, not all definitions can be executed or simplified this\nway. For example, if we have a proof involving the Lesbesgue integral\nfrom {\\tt mathlib}\\xspace, we cannot expect {\\tt Lean}\\xspace to symbolically compute the\nsolution to an arbitrary definite integral. Definitions like the\nintegral are marked \\texttt{noncomputable} in {\\tt Lean}\\xspace, which means they\nwill not be executed.\n\n \n\\section{Denotational Semantics of Programs}\n\\label{sec:semantics}\nWith the probability theory library defined, we next need a\nway to write down learning algorithms and probabilistic programs in {\\tt Lean}\\xspace, so\nthat we can state and prove theorems about them, much as we did\nwith \\texttt{double} above. However, by default, all functions in {\\tt Lean}\\xspace have to\nbe purely functional (that is, they have to behave like mathematical functions,\nwith deterministic output and no side-effects).\n\n\\newcommand{\\texttt{bind}}{\\texttt{bind}}\n\\newcommand{\\texttt{ret}}{\\texttt{ret}}\n\\newcommand{\\textsf{ret}}{\\textsf{ret}}\n\\newcommand{\\mdo}[3]{\\texttt{do}\\; #1\\leftarrow#2\\, ;\\;#3}\n\n\nFor that reason, past work on verifying machine learning algorithms\nhas used a \\emph{denotational semantics} in which programs are\nrepresented as distributions over the types of values they can return.\nGiven a type $X$, we write ${\\bf {\\tt Meas}}(X)$ for this type of probability\ndistributions on $X$. The first step is to define the primitive\ndistributions that our programs will need to sample from.  Next we\nneed a way to sequence together multiple steps of sampling and running\ncomputations on sampled values. To do so, we can define a function\ncalled \\texttt{bind} of type ${\\bf {\\tt Meas}}(X) \\rightarrow (X \\rightarrow\n{\\bf {\\tt Meas}}(Y)) \\rightarrow {\\bf {\\tt Meas}}(Y)$.  That is, $\\mathtt{bind}$ takes a\nprobability measure on $X$ and a function that transforms values from\n$X$ into probability measures over $Y$, and returns a probability\nmeasure on $Y$. Intuitively, we should read \n$\\mathtt{bind(\\mu, f)}$ as representing a program which first samples\nfrom the distribution $\\mu$ and then passes the result to $f$.  It is\ncommon to use the notation $\\mdo{x}{\\mu}{g(x)}$ for $\\texttt{bind}(\\mu, g)$,\nwhich helps reinforce the intuition that $\\texttt{bind}$ samples from $\\mu$\nand then runs $g$.  We also define a function $\\mathtt{ret}$ of type\n$X \\rightarrow {\\bf {\\tt Meas}}(X)$. It takes a value from $X$ and returns a\nprobability measure on $X$.\n\nFunctions $\\mathtt{bind}$ and $\\mathtt{ret}$ construct probability\nmeasures, so their definitions say what probability they assign to an\nevent. If $A$ is an event we define them as:\n\\begin{align}\n  \\texttt{bind}(\\mu,f)(A) & = \\int_{x \\in X} f(x)(A) d\\mu \\\\\n  \\texttt{ret}(x)(A) & = \\chi_A(x)\n\\end{align}\nWhile the definitions use standard mathematical notation, our\nformalization uses the {\\tt Lean}\\xspace definition of the Lebesgue integral, and\nso on.  Here, $\\texttt{ret}(x)$ is the $\\delta$-Dirac distribution at $x$. To\nunderstand the definition of $\\texttt{bind}$, consider the following\nexample. Let $\\mu$ be a distribution over $X$ and consider the random\nvariables $U \\sim \\mu$ and $V \\sim f(U)$. For example, $f$ could be\nthe function that for an input $l$ returns the distribution\n$\\mathcal{N}(l,0.1)$. What is the distribution of $V$? By the sum rule\nof probability we have $\\Pr(V=v) = \\int_{u \\in X} \\Pr(V = v \\mid U =\nu) d\\mu$. Therefore, $\\Pr(V)(A) = \\int_{u \\in X} f(u)(A) d\\mu$. Hence,\n$\\texttt{bind}(\\mu,f)$ is simply computing the distribution that results from\napplying $f$ while marginalizing over $\\mu$.\n\n\\paragraph{Example.}\nConsider the Python function from the introduction in\n\\autoref{fig:pyprog}.  The translation of this program into {\\tt Lean}\\xspace as\nwe have so far described is:\n\\begin{lstlisting}[language=lean]\ndef majority1 :=\n  do theta \u2190 uniform(0, 1) ; \n  do X \u2190 bernoulli(theta) ;\n  ret (theta, X)\\end{lstlisting}\nThe notation makes this look almost the same as the Python program\nthat we started with. However, in the {\\tt Lean}\\xspace code, the ``function''\nactually evaluates to a nested integral representing the distribution\nthat this procedure encodes. Similarly, in {\\tt Lean}\\xspace, \\texttt{uniform} and\n\\texttt{bernoulli} are not random number generators, but definitions\nof those distributions in terms of their CDF. These \\texttt{bind} and\n\\texttt{ret} operations are an example of a \\emph{monad}. Monads are\ncommonly used in functional languages like Haskell to\nrepresent programs that have side-effects. This probability monad was defined by\n\\citet{giry1982categorical}.\n\nThe Giry monad representation has some advantages.  As we have seen,\nthe denotation of a program has a structure that mimicks the original\nsource code. However, there are drawbacks when we try to reason about\nprograms expressed this way, particularly when we want to avoid using\naxioms or making restrictions to discrete spaces. First, recall from\n\\autoref{sec:lean} that proof by reduction is blocked when we work\nwith noncomputable definitions like integrals. Because the Giry monad\nincludes an integral every time we use $\\mathtt{bind}$, {\\tt Lean}\\xspace cannot\nreduce such programs very much at all. Second, in our experience, this\nmonadic semantics is unfamiliar to ML experts, and doesn't correspond\nclosely to the style used on paper proofs.  In the next section, we\ndescribe our approach for alleviating these issues.\n \n\n \n\\section{Reparameterizing to Simplify Semantics}\n\n\\newcommand{\\cc}[1]{\\texttt{#1}} How can we find a denotational\nsemantics that would make it easier to reason formally about a\nlearning or randomized algorithm and avoid the issues with the Giry\nmonad described above? Recall that a classic result in probability\ntheory is that any distribution on the reals with the Borel sigma\nalgebra can be constructed as the pushforward of the uniform\ndistribution on $[0,1]$, which we write as $\\mathcal{U}$. That is, for\nany distribution $\\mu$, there exists a (measurable) function $f$ such\nthat for any event $E$\n\\begin{align}\\label{eqn:map_apply}\n \\mu(E) & = \\mathcal{L}[f^{-1}(E)]\n\\end{align}\nIndeed, a more general version of this result is an important lemma\nin proving the Kolmogorov Extension Theorem, which is used to\nshow the existence of many stochastic processes.\n\n\nThis fact hints at an alternate denotation for our programs: we could\nrepresent a program as some pure function $f$, applied to\nsamples from $\\mathcal{U}$. That is, we would take the pushforward\nmeasure of $f$ applied to an appropriate input distribution. Because\n$f$ will be pure, {\\tt Lean}\\xspace will be able to evaluate it, enabling us to\nuse proof by reduction. Moreover, once we prove that $f$ is a\nmeasurable function, we can avoid most measure-theoretic issues in the\nproof.  Of course, the theorem above suggests that $f$ will exist\n\\emph{in principle}, but we still need a way to construct the function\n$f$ in a useful form.\n\nIn order to help explain how we find a simple representation of $f$, we\nfirst observe that such a function $f$ is a \\emph{reparameterization}\nof the original program.  A reparameterization is a transformation of\na probabilistic model that changes how a variable is sampled, often by\nsampling an additional variable from an alternate distribution and\ntransforming the result.  For example, if $X$ is a random variable\nwith distribution $\\mathcal{N}(3,4)$, we can reparameterize $X$ to\ndefine it as $X = 2Z + 3$ where $Z$ is a draw from the standard normal\ndistribution. Reparameterization has many other applications in\nML. For example, it can improve the convergence of MCMC\nalgorithms~\\citep{gorinovaMHarXiv}, and enable the use of\nstochastic gradient descent in Variational Auto Encoders~\\citep{KingmaW13}.\n\nWe now describe the steps to compute the reparameterization of\nfunctions in our setting. We have implemented this translation for a\nsmall probabilistic programming language implemented on top of\nPyro. However, to keep things self-contained, we will explain how the\ntranslation works using simple Python programs, without dynamic\nlooping or recursion. Our $\\cc{majority}$ function from\n\\autoref{fig:pyprog} will serve as a running example.\n\n\\squeezedpara{Transforming primitive distributions:}\nFirst we extend the probability theory library in {\\tt Lean}\\xspace to\ninclude reparameterized definitions of all the primitive distributions\nthat our programs can sample from.  For example, \\cc{gen\\_uniform(a,\n  b, u)} generates a uniform random variate on the interval $[a,b]$ by\nscaling the input $\\cc{u}$, which is assumed to be a uniform sample\nfrom $[0, 1]$.  More generally, we can implement the inverse CDF\ntransform to convert a sample from the uniform distribution on $[0, 1]$ into the appropriate\ndistribution.\n\nThese primitive translations are added to a dictionary that tracks\ntheir input and output types, which in particular records how many\nuniform inputs they need.  As we translate a function, we replace\nsampling from primitive distributions with these translations, and\nrecord how many total uniform samples will be needed. Then, an\nargument $\\cc{u}$ is added that is a vector of all of the uniform\nsamples that will be needed. We write $\\cc{u1}$, $\\cc{u2}$, $\\cc{u3}$\netc. for the components of this vector. These are defined using {\\tt Lean}\\xspace's primitive\noperations \\cc{fst} and \\cc{snd} for extracting the first and second element of a pair.\n\n\\squeezedpara{Slicing and Coupling:}\nNext, for each return value of the function, we compute a\n\\emph{slice}, which is the subset of expressions in the function that\ndetermine that return value.  For example, in the \\cc{majority}\nexample from \\autoref{fig:pyprog}, there are two return values,\n\\cc{theta} and \\cc{X}. The slice for $\\cc{theta}$ is\n$\\cc{gen\\_uniform(0,1,u1)}$, and the slice for \\cc{X} is\n$\\cc{gen\\_bernoulli(gen\\_uniform(0, 1, u1), u2)}$. Note that the\nre-use of \\cc{u1} in both the computation of \\cc{theta} and \\cc{X}\nensures that we properly capture the dependence between these two\nrandom variables. In general, all dependencies between the variable\ndefinitions are explicitly captured by which inputs are passed to\nwhich parameters, so that the right joint distribution is obtained.\nEssentially, the inputs\nare being used to construct an appropriate \\emph{coupling}~\\citep{lindvall2002}\nbetween the random variables encoded by the slices.\nWe generate {\\tt Lean}\\xspace definitions for each slice,\nand then call these to compute each return value in the function. Now,\nwe can perform standard compiler optimization transforms, such as\nremoving common sub-expressions, to simplify the function.\n\n\\squeezedpara{Nested functions:}\nAfter we complete the translation of \\cc{majority}, we can add an\nentry to the dictionary of translations tracking its type.  When\ntranslating a subsequent function \\cc{g}, if a call to \\cc{majority}\nis encountered in \\cc{g}, we can replace it with a call to the\ntranslation and add additional uniform samples to the vector of inputs\nto $\\cc{g}$. Since the pre-translated version of \\cc{majority} takes\nno arguments as input, it is also possible to hoist the call to\n\\cc{majority} out of the body of \\cc{g}, and instead pass in the\npre-sampled results of \\cc{majority} as an argument to \\cc{g}.\n\\autoref{fig:pyprog-parity} shows an example, where \\cc{majority} is\ncalled by \\cc{demographic\\_parity}. The transformed function takes as\ninput both a sample from \\cc{majority} and a pair of uniform samples, which it uses\nto generate $\\cc{phi}$ and $\\cc{Y}$.\n\n\\begin{figure}\n\\begin{minipage}{0.35\\textwidth}\n\\begin{python}\ndef demographic_parity():\n  [theta,X] = majority()\n  phi = uniform(0.8 * theta,1)\n  Y = bernoulli(phi)\n  return (theta,X,phi,Y)\n\\end{python}\n\\subcaption[first caption.]{Python program}\\label{fig:pyprog-parity}\n\\end{minipage}\n\\quad\n\\begin{minipage}{0.5\\textwidth}\n\\begin{lstlisting}[language=lean]\ndef demographic_parity_fun u :=\n  let theta := u.fst.fst in\n  let X := u.fst.snd in\n  let u3 := u.snd.fst in\n  let u4 := u.snd.snd in\n  let phi := gen_uniform(0.8 * theta,1,u3) in\n  let Y := gen_bernoulli(phi,u4) in\n  ((theta,X),(phi,Y))\n\ndef demographic_parity :=\n  pushforward demographic_parity_fun\n       (prod_measure majority pair_uniform(0,1))\n\\end{lstlisting}\n\\subcaption[first caption.]{Reparameterization in {\\tt Lean}\\xspace}\\label{fig:lean-parity}\n\\end{minipage}\n\\label{fig:bht}\n\\caption{{\\tt Lean}\\xspace translation with nested functions for a Bayesian hypothesis testing example.}\n\\end{figure}\n\n\\squeezedpara{Automatic proof generation:}\nIn the course of generating the function $f$, we can also generate\nproofs in {\\tt Lean}\\xspace that the function $f$, and all the slices used to\ncompute the return values, are measurable. To do so, we first manually\nwrote proofs of measurability for all primitive operations in the\nlanguage, as well as the transformer functions for the primitive\ndistributions, such as $\\cc{gen\\_uniform}$.  Then, using the fact that\nthe composition of two measurable functions is measurable, these\nproofs for primitive operations %\nare composed to produce a proof\nthat an entire function, such as $\\cc{majority\\_fun}$ is measurable.\nThis proof can then be re-used if $\\cc{majority}$ is called in another\nfunction.\n\n\\squeezedpara{Impact of reparameterization on proofs:}\nOur motivation for defining a semantics based on reparamaterization\nis to simplify formal proofs for learning and randomized\nalgorithms. By embedding programs as pure functions of random inputs,\nwe make it possible to use Equation~(\\ref{eqn:map_apply}) to turn a\nprobabilistic statement on random variables into a statement on\nevents, getting the heavy machinery of measure-theoretic probability\ntheory out of the way. At that point, reasoning about the program\nboils down to reasoning about the set of inputs to the program that\nsatisfy some properties, which is usually very basic, intuitive, and\nallows us to apply reduction to the function to simplify the\nreasoning. We will see an example of this in Section\n\\ref{sec:bayes_verif}.\n\n\\squeezedpara{Generality:}\nA natural question is whether this kind of reparameterization\ntranslation can be applied to more complex, general purpose\nprograms. As we have mentioned, results from measure theory suggest\nthat in principle this can be performed on a large class of programs.\nBut, for large programs with complicated control flow and looping, a\nnaive reparameterization translation may make the program harder to\nunderstand.  However, even when reparameterization would be unnatural\nto apply to the entire program, we believe that it can be useful to\napply to subcomponents. For example, consider a typical implementation\nof stochastic gradient descent, which is usually structured as a\nloop. Within each iteration of the loop, training examples are\nrandomly selected or ordered, and then gradients and updates are\ncomputed. We can factor out and reparameterize the computation of gradients\nto be pure functions. Then, use of the Giry monad would be limited to only\nthe remaining impure parts that glue together iterations of the loop.\nThis way, we would obtain the advantages of reparameterization for the bulk of the proof.\n\n \n\\section{Case Study 1: Decision Stumps}\nFor our first case study, we prove in {\\tt Lean}\\xspace that the concept class of\ndecision stumps is PAC learnable.  We focus here on how this algorithm\nis formulated in {\\tt Lean}\\xspace. More complete details about the classic pencil-and-paper proof and our formalization are found in an earlier report~\\citep{tassarotti:stump}.\nRecall that a decision stump is a classifier that assigns binary\nlabels to real valued points based on whether they are above or below\nsome threshold value. Points above the threshold are labeled $0$, and\npoints $\\leq$ are labeled 1.  We assume that there is some unknown\ndistribution $\\mu$ of examples that the classifier will have to label,\nand that the true labels of these examples are determined by some\nthreshold $t$, which is also unknown.  Training such a classifier is\nstraightforward: we take the maximum of all the training examples with\nlabel $1$, and use that as our threshold.\n\nTo show that this decision class is PAC learnable, we must prove that\nfor all $\\epsilon, \\delta \\in (0, 1)$, there is some number $n$ such\nthat with $n$ labeled training examples drawn independently from\n$\\mu$, this training algorithm gives a stump that achieves error rate\nbelow $\\epsilon$ with probability at least $1 -\n\\delta$. This is the one-dimensional version of the problem of learning an axis-aligned\nrectangle, which is used as a motivating example and exercise in many\nintroductory texts on learning theory~\\citep{kearns1994introduction,\n  shalev2014understanding, mohri2018foundations}.\n\nWe first express the learning algorithm as a pure {\\tt Lean}\\xspace\nfunction \\cc{choose}, which takes the training data as a vector of\nexamples, where each example is a pair consisting of the data point\nand its label. The algorithm filters out the non-positive examples,\nremoves the remaining labels, and takes the maximum. The\nparameter \\cc{n} below tracks the number of examples in the\nvector \\cc{data}:\n\\begin{lstlisting}[language=lean]\ndef filter n data := vec_map (\u03bb p, if p.snd then p.fst else 0) n data\ndef choose n data := max n (filter n data) \n\\end{lstlisting}\nThe process of training on $n$ can then be described as taking the\n$n$-ary product measure on the input distribution $\\mu$, and then\nfirst pushing-forward a function to assign the true labels, and then\npushing-forward the result with \\cc{choose}.\n\\begin{lstlisting}[language=lean]\ndef denot :=\n    let \u03b7 := vec.prob_measure n \u03bc  in \n    let \u03bd := pushforward (label_sample target n) \u03b7 in  \n    pushforward (choose n) \u03bd\n\\end{lstlisting}\nFinally, to make claims about the error rate of this algorithm, we\ndefine an event \\cc{error\\_set} which captures whether the label\nassigned by a classifier \\cc{h} differs from the true label provided\nby the unknown \\cc{target}. Finally, the \\cc{error} rate of a\nclassifier is the probability of this set under the unknown\ndistribution $\\mu$ of test examples:\n\\begin{lstlisting}[language=lean]\ndef error_set h := {x | label h x \u2260 label target x}\ndef error h := \u03bc (error_set h target)  \n\\end{lstlisting}\n\nAll of the randomization lies in the process of modeling the training\nexamples as if they have been sampled from some arbitrary training\ndistribution.  Because the training algorithm here is entirely\ndeterministic, we are able to write down this algorithm directly\nwithout using the automated reparameterization we have\ndescribed.  Nevertheless, the experience of working with this formulation of the\nalgorithm convinced us of the benefits of working with a pure function\nas much as possible, which led us to automating reparameteriztion to\nhandle examples such as the one we describe next.\n\n \n\\section{Case Study 2: Bayesian Hypothesis Tests}\n\\label{sec:bayes_verif}\nIn our next example, we prove a property of a null model used for a\nBayesian hypothesis test.  Recall that in Bayesian hypothesis testing,\nwe have models for how a data set may have been generated, along with\nprior probabilities for those models. We use Bayes rule to update our\nprobabilities of the models and then select from among them~\\citep{BF}.\nIn this case study, we consider the use of Bayesian hypothesis testing\nto judge whether the output of a selection procedure satisfies a\nfairness property called the four-fifths rule, which is used as a\ncriterion for disparate impact testing by the U.S. Equal Employment\nOpportunity Commission \\cite{guidelines}. In particular, the\nfour-fifths rule says that if the selection rate for a protected class\nis less than 4\/5 the rate for the majority group, then this can be\nconstrued as evidence of violating legal standards for adverse impact.\n\nIn order to formulate a Bayesian hypothesis test for this criterion,\nthe first step is to write down a null model that is supposed to\nsatisfy the 4\/5 test: that is, the selection rate is meant to\nbe \\emph{at least} 4\/5 the majority rate.\nThe \\cc{demographic\\_parity} example from \\autoref{fig:pyprog-parity}\nis this null model. The selection rate of the majority\nclass is \\cc{theta}, and the selection rate of the minority class is \\cc{phi}.  The model is constructed so that \\cc{phi} is at least\n$4\/5 \\cdot \\cc{theta}$. Then, \\cc{X} and \\cc{Y} give the results of\nthe selection procedure on one member of the majority class and the\nminority class, respectively. A more general version of this model\nmight draw different numbers of samples from the two classes. By\nexpressing this model in a probabilistic programming language such as\nPyro, and specifying a prior, we could then compare the posterior\nprobability, given some example data, to an alternative model.\n\nBecause the null model is supposed to represent a selection procedure that\nsatisfies the 4\/5 test, it is important to prove that it in fact does. We\nuse {\\tt Lean}\\xspace to prove this. We want to show that $0.8 \\cdot \\Pr[\\cc{X} =\n1] \\leq \\Pr[\\cc{Y} = 1]$.  This follows from conditioning on \\cc{theta}, so that\nit suffices to show that for all $t \\in [0,1]$, we have $ 0.8 \\cdot \\Pr[\\cc{X} =\n1 | \\cc{theta} = t] \\leq \\Pr[\\cc{Y} = 1 | \\cc{theta} = t]$.\\footnote{We use\nconditional probability notation to explain the argument. More precisely, we are using a disintegrating measure~\\citep{chang1997conditioning}.\nBecause we are conditioning\non the first projection of the product measure that we are pushing-forward, the existence\nof this disintegration is a result of Fubini's theorem.}\n\nFor the {\\tt Lean}\\xspace version of the demographic parity model, this is\nformalized with the following events:\n\\begin{lstlisting}[language=lean]\ndef B(t: [0,1]) := {v | v.fst.fst = t}\ndef majority_selected := {v | v.fst.snd = 1}\ndef minority_selected := {v | v.snd.snd = 1}\n\\end{lstlisting}\nWe can divide the proof into two steps. Consider a Bernoulli random variable $\\cc{Y'}$ with selection rate $4\/5 \\cdot \\cc{theta}$, generated using the same $\\cc{u4}$ that is used to generate $\\cc{Y}$. First, we can show that $0.8 \\cdot \\Pr[\\cc{X} = 1 | \\cc{theta} = t] \\leq \\Pr[\\cc{Y'} = 1 | \\cc{theta} = t]$. This is simpler because having conditioned on $t$, the selection rate of \\cc{Y'} is deterministic.\n\nSecond, we show that $\\Pr[\\cc{Y'} = 1 | \\cc{theta} = t] \\leq \\Pr[\\cc{Y} = 1 | \\cc{theta} =t]$.\nThe key is that this part of the proof can be entirely reduced to proving a pure fact about {\\tt demographic\\_parity\\_fun}. In particular, because probabilities are monotone with respect to subset ordering, we just have to show that the set of random inputs of the function {\\tt\ndemographic\\_parity\\_fun} which cause $\\cc{Y'} = 1$ is a subset of the inputs that cause ${\\cc{Y}=1}$. This is captured by the following {\\tt Lean}\\xspace statement, where we write {\\tt f} as an abbreviation for\n{\\tt demographic\\_parity\\_fun}:\n\\begin{lstlisting}[language=lean]\n\u2200 t\u22650, set.prod {a: [0,1] \u00d7 \u2115 | a.fst = t}\n           (set.prod [0,1] {a: [0,1] | generate_bernoulli(4\/5 * t,a) = 1})\n        \u2286 {a: ([0,1] \u00d7 \u2115) \u00d7 [0,1] \u00d7 [0,1] | (f a).snd.snd = 1 \u2227 (f a).fst.fst = t}\n\\end{lstlisting}\n\n \n\n\n\\section{Related Work}\n\\label{sec:related}\n\\squeezedpara{Measure-theory in proof assistants:}\nThere have been general formalizations of measure-theoretic\nprobability theory in a few proof assistants. \\citet{hurd_thesis}\nformalized basic measure theory in the HOL proof assistant, including\na proof of Caratheodory's extension theorem. \\citeauthor{hurd_thesis}\nuses a semantics that is closest to some of the ideas underlying our\nreparameterization approach, in that he models randomized programs as\nhaving access to an infinite ``tape'' of pre-sampled random\nbits. \\citet{HolzlH11} developed a more substantial library in the\nIsabelle theorem prover.  \\citet{AvigadHs14} use this library to\nformalize a proof of the Central Limit Theorem. \n\n\\squeezedpara{Machine-checked proofs for ML:}\nWe have already mentioned some more recent work that has formalized\nmachine learning results. \\citet{selsam2017bugfree} use {\\tt Lean}\\xspace to prove\nthe correctness of an optimization procedure for stochastic\ncomputation graphs. They prove that the random gradients used in their\nstochastic backpropagation implementation are unbiased. In their\nproof, they add axioms to the system for various mathematical\nfacts. They argue that even if there are errors in these axioms that\ncould potentially lead to inconsistency, the process of constructing\nformal proofs for the rest of the algorithm still helps eliminate\nmistakes. \\citet{bagnall2019certifying} give machine-checked proofs of\nbounds on generalization errors. They use Hoeffding's inequality to\nobtain bounds when the hypothesis space is finite or there is a\nseparate test-set and apply this result to bound the generalization\nerror of neural networks with quantized weights. Their proof is\nrestricted to discrete distributions and adds some results as axioms\n(Pinsker's inequality and Gibbs'\ninequality). \\citet{bentkamp2019formal} formalize a result\nby \\citet{CohenSS16} which shows that deep convolutional arithmetic\ncircuits are more expressive than shallow ones. This result deals with\nwhat is deterministically representable by these structures, so their\nproof does not require probability theory.\n\n\n\\squeezedpara{Semantics of probabilistic programs:}\nThe representation of programs in {\\tt Lean}\\xspace in the Giry monad is an\nexample of \\emph{denotational semantics} \\cite{giry1982categorical},\nwhere the meaning of a program is given in terms of a mathematical\nobject (here, the distribution on the type of values it can\nreturn). We have already alluded to some of the problems of using the\nGiry monad and our reparameterization transform on programs with\ncomplicated control flow or looping structure.  Defining denotational\nsemantics for probabilistic programs with arbitrary general recursion\nand higher-order functions is challenging, and the subject of much\nrecent research~\\cite{Staton,Tasson}.\n\n\\squeezedpara{ML for automated theorem proving:}\nA related but distinct line of work applies machine learning\ntechniques to automatically construct formal proofs of theorems.  By\nusing a pre-existing corpus of formal proofs, supervised learning\nalgorithms can be trained to select hypotheses and construct proofs in\na formal system~\\citep{holist, gamepad, holstep, SelsamB19}.\n \n\\section{Conclusion}\n\\label{sec:conclusion}\nWe present an approach to simplify reasoning about probabilistic\nprograms in proof assistants. By reparameterizing these programs to be\npure functions of pre-sampled randomized input, we can exploit more of\nthe native automation and support for reasoning about pure functions\nfound in many proof assistants. Our case studies show that our\napproach can be applied to verify programs from a diverse range of\nsubfields of ML.\n \n\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{\\refname}}{}{}{}\n\\bgroup\n\\def1.5{1.5}\n\\def\\vspace{.15in}\\noindent{\\vspace{.15in}\\noindent}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{definition}{Definition}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{conjecture}{Conjecture}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}{Example}\n\\DeclareMathOperator*{\\esssup}{ess\\,\\sup}\n\\DeclareMathOperator*{\\essinf}{ess\\,\\inf}\n\\DeclareMathOperator*{\\supp}{supp}\n\\newcommand{\\textbf{Proof:\\ }}{\\textbf{Proof:\\ }}\n\\newcommand{\\hfill$\\Box$\\\\}{\\hfill$\\Box$\\\\}\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathbb{R}}{\\mathbb{R}}\n\\newcommand{\\mathbb{T}}{\\mathbb{T}}\n\\newcommand{\\mathbb{N}}{\\mathbb{N}}\n\\newcommand{\\mathbb{C}}{\\mathbb{C}}\n\\newcommand{\\mathbb{Z}}{\\mathbb{Z}}\n\\newcommand{\\mathbb{D}}{\\mathbb{D}}\n\\newcommand{\\mathcal{M}}{\\mathcal{M}}\n\\newcommand{\\mathcal{N}}{\\mathcal{N}}\n\\newcommand{\\mathcal{I}}{\\mathcal{I}}\n\\newcommand{\\mathcal{X}}{\\mathcal{X}}\n\\newcommand{\\mathcal{S}}{\\mathcal{S}}\n\\newcommand{\\mathcal{HS}}{\\mathcal{HS}}\n\\renewcommand{\\S}{\\mathcal{S}}\n\\renewcommand{\\H}{\\mathcal{H}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\\renewcommand{\\P}{\\mathbb{P}}\n\\newcommand{{\\sf Dom}\\,}{{\\sf Dom}\\,}\n\\newcommand{{\\sf Ran}\\,}{{\\sf Ran}\\,}\n\\newcommand{{\\sf Ker}\\,}{{\\sf Ker}\\,}\n\\renewcommand{\\ker}{{\\sf Ker}\\,}\n\\newcommand{^{-1}}{^{-1}}\n\\newcommand{\\,\\mathrm{d}}{\\,\\mathrm{d}}\n\\newcommand{\\norm}[2]{\\left\\| #2 \\right\\|_{#1}}\n\\newcommand{\\ip}[2]{ \\langle {#1} ,{#2}  \\rangle}\n\\def{\\mathcal V}{{\\mathcal V}}\n\\definecolor{darkviolet}{rgb}{0.58,0,0.83} \n\\newcommand{\\blue}[1]{{\\color{blue} #1 }}\n\\newcommand{\\red}[1]{{\\color{red} #1 }}\n\\usetikzlibrary{patterns}\n\\newcommand{\\mathcal{C}o\\,}{\\mathcal{C}o\\,}\n\\newcommand{\\mathcal{C}o\\,}{\\mathcal{C}o\\,}\n\\newcommand{\\mathcal{A}}{\\mathcal{A}}\n\\newcommand{\\mathcal{W}}{\\mathcal{W}}\n\n\\begin{document}\n\n\\title{Donoho-Logan large sieve principles for the continuous wavelet\ntransform }\n\\author{Lu\\'{\\i}s Daniel Abreu and Michael Speckbacher}\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\n\\noindent In this paper we formulate Donoho and Logan's large sieve\nprinciple for the continuous wavelet transform on the Hardy space, adapting the concept of maximum Nyquist density to the hyperbolic geometry of the underlying space. The results provide deterministic guarantees for $L_{1}$-minimization methods and hold for a class of mother wavelets which constitute an orthonormal basis of\nthe Hardy space and can be associated with  higher hyperbolic  Landau levels.\nExplicit calculations of the basis functions reveal a connection with the\nZernike polynomials. We prove a novel local reproducing formula for the\nspaces in consideration and use it to derive concentration estimates of the\nlarge sieve type for the corresponding wavelet transforms.\nWe conclude with a discussion of optimality of localization and Lieb inequalities in  the analytic case by  building on recent results   of Kulikov,  Ramos and Tilli based on the groundbreaking methods of Nicola and Tilli. This leads  to a sharp  uncertainty principle and a local Lieb inequality for the wavelet transform.\n\\end{abstract}\n\\medskip\n\n\\noindent \\textbf{MSC2020:} 42C40, 46E15, 30H10, 42C15, 11N36\\\\\n\\noindent \\textbf{Keywords:} {large sieve principle, wavelet transform, concentration estimates, maximum Nyquist rate, wavelet coorbit spaces, uncertainty principles}\n\n\\section{Introduction}\n\nLet $d\\mu ^{+}(z)= \\text{Im}\\left(z\\right) ^{-2}d\\mu _{\\mathbb{C}^{+}}(z)$,\nwhere $d\\mu _{\\mathbb{C}^{+}}(z)\\ $is the Lebesgue measure in $\\mathbb{C}^{+}\n$. We denote the \\emph{hyperbolic measure} of a set $\\Delta \\subset \\mathbb{C}^{+}$ by$\\ |\\Delta |_{h}:=\\int_{\\Delta }d\\mu ^{+}(z)$ and define the weighted Lebesgue spaces on $\\mathbb{C}^+$ via the norms $\\|F\\|_{L^p(\\mathbb{C}^+)}^p=\\int_{\\mathbb{C}^+}|F(z)|^pd \\mu^+(z)$. Let $0<R<1$, and $1\\leq p<\\infty$. We will be concerned with inequalities of the type \n\\begin{equation}\n\\nu_p (\\Delta )=\\sup_{f\\in \\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})}\\frac{%\n\\int_{\\Delta }|W_{\\psi }f(z)|^{p}d\\mu ^{+}(z)}{\\int_{\\mathbb{C}^{+}}|W_{\\psi\n}f(z)|^{p}d\\mu ^{+}(z)}\\leq \\frac{\\rho (\\Delta ,R)}{C_{\\psi }(R)}\\text{.}\n\\label{1}\n\\end{equation}\nHere, $W_{\\psi }f$ denotes the \\emph{continuous analytic wavelet\ntransform} of a function $f$ with respect to a wavelet\\ $\\psi $, $\\mathcal{C}%\no\\, {L}^{p}(\\mathbb{C}^{+})$ is the wavelet coorbit space with the\nnorm $\\Vert f\\Vert _{\\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})}=\\Vert\nW_{g}f\\Vert _{ {L}^{p}(\\mathbb{C}^{+})}$ (see the preliminaries\nsection for precise definitions) and $\\rho (\\Delta ,R)$ is the \\emph{maximum\nNyquist density} given by \n\\begin{equation}\n\\rho (\\Delta ,R):=\\sup_{z\\in \\mathbb{C}^{+}}|\\Delta \\cap \\mathcal{D}%\n_{R}(z)|_{h}\\text{,}  \\label{def-max-nyq}\n\\end{equation}%\nwhere $\\mathcal{D}_{R}(z)$ is a disc with respect to the pseudohyperbolic\ndistance on $\\mathbb{C}^{+}$. Sets such that $\\rho (\\Delta ,R)\\ll|\\Delta |_{h}$%\n\\ will be said to be $R$-\\emph{sparse in the hyperbolic measure}. \n\nThe problem consists of obtaining the constant $C_{\\psi }(R)$, which is\nindependent of $\\Delta $ but grows with $R$. Given such a constant, the\nestimate  \\eqref{1} shows its full potential  for sets $\\Delta $ which have low\nconcentration of measure in any hyperbolic disk $\\mathcal{D}_{R}(z)$, allowing the use of $L_{1}$-minimization methods for signal recovering as in \\cite{dosta89,dolo92,abspe18-sieve}. See also \\cite{huli,JaSpe,HrySpe} for similar concentration estimates.\n\nThe inequality \\eqref{1} will be proven for an orthonormal basis $\\{\\psi\n_{n}^{\\alpha }\\}_{n\\geq 0}$ of the Hardy space $H^{2}(\\mathbb{C}^{+})$\ndefined via the Fourier transform as \n\\begin{equation*}\n\\widehat{\\psi _{n}^{\\alpha }}(t):={\\sqrt{ \\frac{ 2^{\\alpha +2}\\pi n!}{\\Gamma\n(n+\\alpha +1)}}}\\ t^{\\frac{\\alpha }{2}}e^{-t}L_{n}^{\\alpha }(2t),\\quad t>0,\n\\end{equation*}%\nwhere $L_{n}^{\\alpha }$ denotes the generalized Laguerre polynomial of\ndegree $n$, \n\\begin{equation*}\nL_{n}^{\\alpha }(t)=\\sum_{k=0}^{n}\\frac{(-1)^{k}}{k!}\\binom{n+\\alpha }{n-k}%\nt^{k},\\quad t>0.\n\\end{equation*}%\nNote that the normalizing constant for $\\psi _{n}^{\\alpha }$ is chosen such\nthat  $\\big\\|\\widehat{\\psi\n_{n}^{\\alpha }}\\big\\|_{L^2(\\mathbb{R}^+,t^{-1})}^2= {4\\pi}\/{\\alpha }$, and $\\Vert \\psi _{n}^{\\alpha }\\Vert _{{^{2}}}^{2}=1$. This family was also the central object\nof \\cite{abspe22} where  conditions for frames generated by\norbits of Fuchsian groups in terms of a Nyquist-rate were obtained. The family ${\\psi _{n}^{\\alpha }}(t)$ is connected to the eigenspaces of the Maass Laplacian, a Schr\\\"{o}dinger\noperator that is of central importance in number theory due to its role in the\ntheory of Maass forms \\cite{maass1949neue}, and Selberg trace formulas \\cite{patterson1975laplacian}, as well as \nin physics, where the pure point spectrum of the Maass Laplacian admits an\ninterpretation as a hyperbolic analogue of the Euclidean Landau levels \\cite{co87}. It\nhas been shown in \\cite{abspe22} that these wavelet eigenspaces have $PSL(2,\\mathbb{R}%\n)$ invariance (a convenient property to define point processes \\cite{affine}) and also that, essentially, this is the only choice with such property, if we assume mild reasonable\nrestrictions on $\\psi $ (see Theorem 3 in \\cite{abspe22}). The sequence $\\{\\psi\n_{n}^{\\alpha }\\}_{n\\geq 0}$ also arises as the best localized functions of Daubechies and Paul localization problem in the wavelet domain \\cite{daupau88} and have been used as time-scale tapers in spectral estimation \\cite{bayram1996multiple}, under the name of \\emph{Morse wavelets }. It\nshould be noticed that the slightly different choice of wavelets whose Fourier transforms are a\nconstant multiple of $t^{\\frac{\\alpha +1}{2}}e^{-t}L_{n}^{\\alpha }(2t)$ (note the different exponent of $t$)\nleads to the poly-Bergman polyanalytic structure discovered by Vasilevski \\cite{va99} (see \\cite{hutnik2011wavelets,ab17,berge2022affine} for these special\nchoices in connection to poly-Bergman spaces, which are also fundamental in the theory of the affine Wigner distribution \\cite{berge2022affineACHA} and in commutative algebras of Toeplitz operators \\cite{vasilevski2008commutative}). This setting also involves a connection with Zernike polynomials \\cite{barrera2021radial} and one should obtain results similar in nature \nto those of the present paper. What we would like to emphasize is that, in the STFT case, Hermite functions lead both to the polyanalytic structure and the Euclidean Landau Levels \\cite{abspe18-sieve}, but, for for the wavelet transform, \\emph{the choices leading to Vasilevski's polyanalytic decomposition  structure \\cite{va99} are different from those leading to the $PSL(2,\\mathbb{R})$ invariant hyperbolic Landau Level eigenspaces}. \n\nOur proof  of \\eqref{1}  starts with a general Schur-type argument, as in \\cite{abspe18-sieve}. Then the explicitly defined constant $C_{\\psi\n_{n}^{\\alpha }}(R)=C_{n}^\\alpha(R)$ is computed using the following local\nreproducing formula for the wavelet coefficients: \n\\begin{equation}\nW_{\\psi _{n}^{\\alpha }}f(z)=\\frac{1}{C_{n}^{\\alpha }(R)}\\int_{\\mathcal{D}%\n_{R}(z)}W_{\\psi _{n}^{\\alpha }}f(w)K_{\\psi _{n}^{\\alpha }}(z,w)d\\mu\n^{+}(w),\\qquad z\\in \\mathbb{C}^{+}\\text{.}  \\label{local_rep}\n\\end{equation}%\nProving this formula will constitute a significant part of the paper. In \\cite{abspe18-sieve}, a local reproducing formula for the short-time\nFourier transform (STFT) with Hermite windows was proven using the well-known correspondence between the  STFT and complex Hermite polynomials, which are orthogonal in\nconcentric discs of the plane. Since we could not find a similar\ncorrespondence for wavelets in the literature, we\\ have computed the wavelet\ntransforms $W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }$, and the result\nyields a correspondence (up to a conformal map) to the well known Zernike\npolynomials in the disc \\cite{wuensch05}, defined in terms of Jacobi polynomials in a fashion\nreminiscent of the definition of the complex Hermite polynomials using\nLaguerre functions \\cite{ghanmi2008class,ismail2016analytic}. This   allows us to show the following orthogonality in\nconcentric pseudohyperbolic discs: \n\\begin{equation*}\n\\int_{\\mathcal{D}_{R}}W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }(z)\\overline{%\nW_{\\psi _{n}^{\\alpha }}\\psi _{k}^{\\alpha }(z)}d\\mu ^{+}(z)=C_{m,n}^{\\alpha\n}(R)\\delta _{m-k}.\n\\end{equation*}%\nThe case $n=m$ of this orthogonality relation is then sufficient to show \\eqref{local_rep}.\n\nFor $p=1$, $\\nu_1 (\\Delta ) <\\frac{1}{2}$ implies that%\n\\begin{equation}\n\\Vert W_{\\psi }f\\cdot \\chi _{\\Delta }\\Vert _{{L}^{1}(\\mathbb{C}%\n^{+})}< \\frac{1}{2}\\Vert W_{\\psi }f\\Vert _{ {L}^{1}(%\n\\mathbb{C}^{+})}\\text{,}  \\label{sparse}\n\\end{equation}%\nmeaning that for \\emph{every signal }$f\\in \\mathcal{C}o\\, {L}^{1}(%\n\\mathbb{C}^{+})$, $W_{\\psi }f(z)$\\emph{\\ is sparse }(low concentration on $%\n\\Delta $). By generalizing an observation of Donoho and Stark for\nbandlimited discrete functions \\cite[Theorem 9]{dosta89}, we obtain an\ninteresting reconstruction result. In the absence of noise, if $\\nu_1(\\Delta)<1\/2$ and we only sense\nthe projection of a general $W_{\\psi }f(z)$, $f\\in \\mathcal{C}o\\, {L}%\n^{1}(\\mathbb{C}^{+})$ on $\\Delta ^{c}$, then $W_{\\psi }f(z)$ can be perfectly\nreconstructed as the solution of the $L^{1}$-minimization problem%\n\\begin{equation*}\nW_{\\psi }f=\\arg \\min_{_{h\\in \\mathcal{C}o\\, {L}^{1}(\\mathbb{C}%\n^{+})}}\\Vert W_{\\psi }h\\Vert _{ {L}^{1}(\\mathbb{C}^{+})}%\n\\text{, \\ \\ \\ \\ \\ subject to }P_{\\Delta^c }W_{\\psi }h=P_{\\Delta^c }W_{\\psi }f%\n\\text{.}\n\\end{equation*}\nBesides this result,  condition \\eqref{sparse} allows several signal\napproximation and recovery scenarios by using $L_{1}$-minimization \\cite{abspe18-sieve}. \n\nA simple example shows that good estimates for $\\nu_p (\\Delta )$ can be obtained \nif $\\Delta $ is sparse in the hyperbolic measure (low concentration in any disc $\n\\mathcal{D}_{R}(z)$). Consider first $\n\\Delta _{1}=\\mathcal{D}_{R}$. Then $\\rho (\\Delta _{1},R)=|\\mathcal{D}_{R}|_{h}\n$. Now if $\\Delta _{2}=\\mathcal{D}_{R}\\cup (\\mathcal{D}_{R+2\\delta }-\\mathcal{D}_{R+\\delta })$, with $0<\\delta <R$, then $\\rho (\\Delta\n_{2},R+\\delta )=|\\mathcal{D}_{R}|_{h}$. This gives the example of two sets $\\Delta _{1},\\Delta _{2}$ with $|\\Delta\n_{1}|_{h}<|\\Delta _{2}|_{h}$ but with an estimate for $\\nu_p (\\Delta _{2})$\nbetter than the estimate for $\\nu_p \n (\\Delta _{1})$, since $\\rho (\\Delta\n_{2},R+\\delta )\/C_{\\psi }(R+\\delta )<\\rho (\\Delta _{1},R)\/C_{\\psi }(R)$ (because, as we will see, in the estimates of the form \\eqref{1}, $C_{\\psi }(R)$ is independent of $\\Delta $ and grows with $R$). \nThis elementary observation already hints in the direction that motivated\nDonoho and Logan, as well as our previous work \\cite{abspe18-sieve}: if a\nset is poorly concentrated in discs of big enough fixed radius $R$ (`$R$-\\emph{sparse'}), then $\\nu_p (\\Delta )<\\frac{1}{2}$.\n\nAnother interesting setting where the maximum Nyquist rate can be estimated is the hyperbolic Cantor set \\cite{ABREU2022126699}. This has been done in \n \\cite[Section 3.1]{knutsen2022fractal} using our concept from \\cite{abspe18-sieve}, and a similar calculation applies to the wavelet case. This approach may help in dealing with more general hyperbolic fractal sets, adapting the concept of porous sets from the time-frequency plane \\cite{knutsen2022daubechies}.\n\n\nThe problem of estimating $\\nu_p(\\Delta)$ is closely connected to the study of  localization operators first introduced by Daubechies \\cite{daub88} in the time-frequency domain and by Daubechies and Paul in the wavelet domain \\cite{daupau88}. An analysis similar to the one in \\cite{abspe18-sieve}\nshows that our estimates fall short from being sharp. However, in the analytic  case $n=0$  it was recently shown by Ramos and Tilli \\cite{RamosTilli} that\npseudohyperbolic disks maximize $\\nu_2 (\\Delta )$ among all sets $\\Delta$ of a given hyperbolic measure, using methods from the\nbreakthrough paper \\cite{Inventiones}, where the analogue result for the\ntime-frequency case has been obtained.\nWe  include a section where we derive some direct consequences\nof the  results in \\cite{RamosTilli}, including a local Lieb inequality, an analogue of Theorem~5.2 in \\cite{Inventiones}.\nThis requires a Lieb-type\ninequality for the Wavelet transform, that, as we shall see, has recently been proved in a slightly disguised\nform by Kulikov \\cite[Theorem~1.2]{kuli22}. \n\nThe outline of the paper is as follows. In Section~\\ref{sec:prel}, we gather all the concepts required to follow the paper in a relatively self-contained manner.\nIn Section~\\ref{sec:dbl-orth} we illustrate the  connection between the basis functions $W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }$ and the Zernike polynomials. Then the double orthogonality of these basis functions in concentric pseudohyperbolic discs and the resulting local reproducing formula are proved. With these formulas at hand, we show our large sieve estimates  in Section~\\ref{sec:LS}. Section~\\ref{sec:opt} is devoted to optimality results in the analytic setting,\nconcluding with a local Lieb uncertainty principle for analytic wavelets and a  short discussion about general Lieb inequalities. The computation of the basis functions $W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }$, the explicit formulas for the reproducing kernels, and a proof of the integrability of the mother wavelets are left to the Appendix.\n\n\\section{Preliminaries}\\label{sec:prel}\n\nWe use the following convention for the \\emph{Fourier transform}: for $f\\in\nL^{1}(\\mathbb{R})$, its Fourier transform $\\widehat{f}$ is defined as \n\\begin{equation*}\n\\widehat{f}(\\xi )=\\mathcal{F}(f)(\\xi ):=\\int_{\\mathbb{R}}f(t)e^{-i\\xi\nt}dt,\\qquad \\xi \\in \\mathbb{R}.\n\\end{equation*}%\nBy standard arguments, the Fourier transform extends to $L^{2}(\\mathbb{R})$\nand \\emph{Plancherel's formula} holds \n\\begin{equation*}\n\\langle f_{1},f_{2}\\rangle = {\\frac{1}{2\\pi}}\\langle \\widehat{f_{1}},\\widehat{f_{2}}\\rangle\n,\\qquad \\text{ for }f_{1},f_{2}\\in L^{2}(\\mathbb{R}).\n\\end{equation*}%\nWe will use the basic notation for $H^{2}(\\mathbb{C}^{+})$, the \\emph{Hardy space on the upper half plane}, of analytic functions in $\\mathbb{C}^{+}$ with the norm \n\\begin{equation*}\n\\left\\Vert f\\right\\Vert _{H^{2}(\\mathbb{C}^{+})}=\\text{ }\\sup_{0<s<\\infty }\\int_{-\\infty }^{\\infty }\\left\\vert\nf(x+is)\\right\\vert ^{2}dx<\\infty \\text{.}\n\\end{equation*}\nTo simplify the computations it is often convenient to use the equivalent\ndefinition (since the Paley-Wiener theorem \\cite{DGM} gives $\\mathcal{F}\n(H^{2}(\\mathbb{C}^{+}))=L^{2}(\\mathbb{R}^+ )$) \n\\begin{equation*}\nH^{2}(\\mathbb{C}^{+})=\\left\\{ f\\in L^{2}(\\mathbb{R}):(\\mathcal{F}f)(\\xi )=0\n\\text{ for almost all }\\xi <0\\right\\} \\text{.}\n\\end{equation*}\n\n\\subsection{The Continuous Wavelet Transform }\n\n\\subsubsection{The affine group}\n\nConsider the {\\em $ax+b$ group} (see \\cite[Chapter 10]{groe1} for the listed\nproperties) $G\\sim \\mathbb{R}\\times \\mathbb{R}^{+}\\sim $ $\\mathbb{C}^{+}$\nwith the multiplication\n\\begin{equation*}\n(x,s)\\cdot (x^{\\prime },s^{\\prime })=(x+sx^{\\prime },ss^{\\prime })\\text{.}\n\\end{equation*}%\nThe identification $G\\sim $ $\\mathbb{C}^{+}$\\ is done by setting $(x,s)\\sim\nx+is$. The neutral element of the group is $(0,1)\\sim i$ and the inverse\nelement is given by $(x,s)^{-1}=(-\\frac{x}{s},\\frac{1}{s})\\sim -\\frac{x}{s}%\n+\\frac{i}{s}$. The $ax+b$ group is not\\ unimodular, since the left Haar\nmeasure on $G$ is $%\n\\frac{dxds}{s^2}$ and the right Haar measure on $G$ is $%\n\\frac{dxds}{s}$. \\ Throughout this paper we will follow the convention that\nfor $z,w\\in \\mathbb{C}^{+}$, $zw$ denotes the product of two complex numbers\nand $z\\cdot w$ denotes the above group multiplication induced by the  affine group. \nFor a set $\\Delta \\subset \\mathbb{C}^{+}$ we define $\\Delta ^{-1}:=\\{z\\in \n\\mathbb{C}^{+}:\\ z^{-1}\\in \\Delta \\}$. The left Haar measure of a set $%\n\\Delta \\subseteq G$ \n\\begin{equation*}\n|\\Delta |=\\int_{\\Delta }\\frac{dxds}{s^{2}}\\text{, }\n\\end{equation*}\nequals, under the identification of the $ax+b$ group with $\\mathbb{C}^{+}\n$,  the hyperbolic measure\n\\begin{equation*}\n |\\Delta |=\\left\\vert \\Delta \\right\\vert _{h}:= \\int_{\\Delta }(\n\\text{Im}z)^{-2}\\,d\\mu _{\\mathbb{C}^{+}}(z)\\text{,}\n\\end{equation*}\nwhere $d\\mu _{\\mathbb{C}^{+}}(z)\\ $is the Lebesgue measure in $\\mathbb{C}^{+}\n$. We will write \n\\begin{equation}\nd\\mu ^{+}(z)=\\left( \\text{Im}z\\right) ^{-2}d\\mu _{\\mathbb{C}^{+}}(z)\\text{.}\n\\label{measure}\n\\end{equation}\n\n\\subsubsection{The continuous wavelet transform}\n\nFor every $x\\in \\mathbb{R}$ and $s\\in \\mathbb{R}^{+}$, define the\ntranslation operator $T_{x}$ by $T_{x}f(t)=f(t-x)$ and the dilation operator $D_{s}f(t)=\\frac{1\n}{\\sqrt{s}}f(t\/s)$. Let $z=x+is\\in  \\mathbb{C}^{+}$ and define the unitary representation $\\pi:G\\to \\mathcal{U}(H^2(\\mathbb{C}^+))$\n\\begin{equation}\n\\pi (z)\\psi (t):=T_{x}D_{s}\\psi (t)=s^{-\\frac{1}{2}}\\psi (s^{-1}(t-x)),\\qquad \\psi \\in H^{2}(\\mathbb{C}^{+}) .\\label{representation}\n\\end{equation}\nOne needs to take into account that the $ax+b$ group is\nnot\\ unimodular\nwhich requires the following additional condition on the integrability of \n$\\psi\\in H^2(\\mathbb{C}^+)$ \n\\begin{equation}\n0<\\left\\Vert \\mathcal{F}\\psi \\right\\Vert _{L^{2}(\\mathbb{R}^{+},t^{-1}dt)}^{2}=:C_{\\psi }<\\infty. \\label{Adm_const}\n\\end{equation}\nFunctions satisfying \\eqref{Adm_const} are called \\emph{admissible }and the\nconstant $C_{\\psi }$ is the \\emph{admissibility constant}.{\\ }\n\n\\emph{The continuous analytic wavelet transform} of a function $f$ with\nrespect to a wavelet\\ $\\psi $ is defined as \n\\begin{equation}\nW_{\\psi }f(z)=\\left\\langle f,\\pi ({z})\\psi \\right\\rangle _{H^{2}\\left( \n\\mathbb{C}^{+}\\right) }=\\frac{1}{\\sqrt{s}}\\int_{\\mathbb{R}}\\tilde{f}(t)\\tilde{\\psi} \\left( \\frac{t-x}{s}\\right)dt,\\quad z=x+is\\in \\mathbb{C}^{+}, \n\\end{equation}\nwhere $\\tilde{f}(t)=\\lim_{y\\to0^+}f(t+iy)$.\nUsing $\\mathcal{F}(H^{2}(\\mathbb{C}^{+}))=L^{2}(\\mathbb{R}^+ )$, this can also\nbe written (and we will do it as a rule to simplify the calculations) as%\n\\begin{equation}\nW_{\\psi }f(z)={\\frac{\\sqrt{s}}{2\\pi}}\\int_{\\mathbb{R}^+ }(\\mathcal{F}f)(\\xi )e^{ix\\xi }(\\mathcal{F}%\n\\psi )(s\\xi )d\\xi \\text{.}  \\label{anwav}\n\\end{equation}\nAs proven recently in \\cite{AnalyticWavelet}, $W_{\\psi }f(z)$ only leads to\nanalytic (Bergman) phase spaces for a very special choice of $\\psi $, but it\nis common practice to call it in general continuous analytic wavelet\ntransform. The orthogonality relation\n\\begin{equation}\n\\int_{\\mathbb{C}^{+}}W_{\\psi _{1}}f_{1}(z)\\overline{W_{\\psi _{2}}f_{2}(z)\n}d\\mu ^{+}(z)= \\left\\langle \\mathcal{F}\\psi _{1},\\mathcal{F}\\psi\n_{2}\\right\\rangle _{L^{2}(\\mathbb{R}^{+},t^{-1})}\\left\\langle f_{1},f_{2}\\right\\rangle _{H^{2}\\left( \\mathbb{C}\n^{+}\\right) }\\text{,}  \\label{ortogonalityrelations}\n\\end{equation}\nis valid for all $f_{1},f_{2}\\in H^{2}\\left( \\mathbb{C}^{+}\\right) $ and $\n\\psi _{1},\\psi _{2}\\in H^{2}\\left( \\mathbb{C}^{+}\\right) $ admissible. Then,\nsetting $\\psi _{1}=\\psi _{2}=\\psi $\\ and $f_{1}=f_{2}$\\ in \\eqref{ortogonalityrelations}, gives\n\\begin{equation}\n\\int_{\\mathbb{C}^{+}}\\left\\vert W_{\\psi }f(z)\\right\\vert ^{2}d\\mu\n^{+}(z)=C_{\\psi }\\left\\Vert f\\right\\Vert _{H^{2}\\left( \\mathbb{C}^{+}\\right)\n}^{2},  \\label{isometry}\n\\end{equation}\nand the continuous wavelet transform is a multiple of an isometric inclusion $\nW_{\\psi }:H^{2}\\left( \\mathbb{C}^{+}\\right) \\rightarrow L^{2}(\\mathbb{C}^{+}\n)$. Setting $\\psi _{1}=\\psi _{2}=\\psi $\\ and $f_{2}=\\pi\n(z)\\psi $ in \\eqref{ortogonalityrelations} then shows that for every $f\\in H^{2}\\left( \n\\mathbb{C}^{+}\\right) $ one has \n\\begin{equation}\nW_{\\psi }f(z)=\\frac{1}{C_{\\psi }}\\int_{\\mathbb{C}^{+}}W_{\\psi }f(w)\\langle\n\\pi (w)\\psi ,\\pi (z)\\psi \\rangle d\\mu ^{+}(z),\\quad z\\in \\mathbb{C}^{+}. \\label{eq:rep-eq}\n\\end{equation}\nThus, the range of the wavelet transform \n\\begin{equation*}\nW_{\\psi }\\left( H^{2}\\left( \\mathbb{C}^{+}\\right) \\right) :=\\big\\{F\\in L^{2}(\n\\mathbb{C}^{+},\\mu ^{+}):\\ F=W_{\\psi }f,\\ f\\in H^{2}\\left( \\mathbb{C}\n^{+}\\right) \\big\\}\n\\end{equation*}\nis a reproducing kernel subspace of $L^{2}(\\mathbb{C}^{+} )$ with\nkernel\n\\begin{equation}\nK_{\\psi }(z,w)=\\frac{1}{C_{\\psi }}\\langle \\pi (w)\\psi ,\\pi (z)\\psi \\rangle\n_{H^{2}\\left( \\mathbb{C}^{+}\\right) }=\\frac{1}{C_{\\psi }}W_{\\psi }\\psi ({w}\n^{-1}\\cdot z),  \\label{RepKern}\n\\end{equation}\nand $K_{\\psi }(z,z)= {\\Vert \\psi \\Vert _{2}^{2}\n}\/{C_{\\psi }}$\\text{.}\nThe Fourier transform $\\mathcal{F}:H^{2}\\left( \\mathbb{C}^{+}\\right)\n\\rightarrow L^{2}(\\mathbb{R}^+ )$ can be used to simplify computations, since for $z=x+is$ and $w=x'+is'$\n\\begin{align}\n\\left\\langle \\pi (w)\\psi ,\\pi (z)\\psi \\right\\rangle _{H^{2}\\left( \\mathbb{C}%\n^{+}\\right) }&= {\\frac{1}{2\\pi}}\\left\\langle \\widehat{\\pi (w)\\psi },\\widehat{\\pi (z)\\psi }%\n\\right\\rangle _{L^{2}(\\mathbb{R}^{+})}\\notag\n\\\\\n&= {\\frac{1}{2\\pi}}\\left( ss^{\\prime }\\right) ^{\\frac{%\n1}{2}}\\int_{\\mathbb{R}^+ }\\widehat{\\psi }(s^{\\prime }\\xi )\\overline{\\widehat{%\n\\psi }\\left( s\\xi \\right) }e^{i(x-x^{\\prime })\\xi }d\\xi \\text{.}\n\\label{kernelF}\n\\end{align}\n\n\\subsubsection{Wavelet coorbit spaces}\n\nIt is commonly known that the representation $\\pi $ is also \\emph{integrable}, that\nis, there exist admissible mother wavelets $\\psi \\neq 0$ such that \n\\begin{equation}\n\\int_{\\mathbb{C}^{+}}|\\langle \\psi ,\\pi (z)\\psi \\rangle |d\\mu ^{+}(z)<\\infty\n.  \\label{eq:integr}\n\\end{equation}\nWe will show in Appendix~\\ref{app:int} that we can choose $\\psi=\\psi_n^\\alpha$ here. \nFor such a mother wavelet, we define the space of test functions \n\\begin{equation*}\n\\mathcal{H}_{1}:=\\left\\{ f\\in H^{2}\\left( \\mathbb{C}^{+}\\right) :\\ W_{\\psi}f\\in\nL^{1}(\\mathbb{C}^+)\\right\\} ,\n\\end{equation*}%\nand denote by $\\mathcal{H}_{1}^{\\prime }$ its anti-dual space, i.e., the\nspace of antilinear continuous functionals on $\\mathcal{H}_{1}$. Then the \n\\emph{coorbit space} with respect to ${L}^{p}(\\mathbb{C}^{+})$, $%\n1\\leq p\\leq \\infty $, is defined as \n\\begin{equation*}\n\\mathcal{C}o\\,{L}^{p}(\\mathbb{C}^{+}):=\\left\\{ f\\in \\mathcal{H}%\n_{1}^{\\prime }:W_{\\psi}f\\in  {L}^{p}(\\mathbb{C}^{+})\\right\\} ,\n\\end{equation*}%\nequipped with the natural norm $\\Vert f\\Vert _{\\mathcal{C}o\\,{L}^{p}(%\n\\mathbb{C}^{+})}^p:=\\int_{\\mathbb{C}^+}| W_{\\psi}f(z)|^pd \\mu^+(z)$, $1\\leq p<\\infty$, and the usual modification if $p=\\infty$.\nSee, e.g., \\cite{fegr89,fegr89II,groe91} for more information on coorbit\nspace theory. The spaces $\\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})$ are\nBanach spaces for $1\\leq p\\leq \\infty $, they are independent of the\nparticular choice of mother wavelet $\\psi\\in \\mathcal{H}_{1}$, the reproducing\nformula \\eqref{eq:rep-eq} extends to $\\mathcal{C}o\\, {L}^{p}(%\n\\mathbb{C}^{+})$, and \n\\begin{equation*}\n\\mathcal{C}o\\, {L}^{1}(\\mathbb{C}^{+})=\\mathcal{H}_{1}\\subset \n\\mathcal{C}o\\, {L}^{2}(\\mathbb{C}^{+})=H^{2}\\left( \\mathbb{C}%\n^{+}\\right) \\subset \\mathcal{C}o\\, {L}^{\\infty }(\\mathbb{C}^{+})=%\n\\mathcal{H}_{1}^{\\prime }.\n\\end{equation*}\n\n\\subsection{Hyperbolic Landau Level Spaces}\n\n\\label{sec:Landau-levels}\n\nWe first make an important remark on the transforms $W_{\\psi _{n}^{\\alpha\n}}$. \\ The spaces $W_{\\psi _{n}^{\\alpha }}(H^2(\\mathbb{C}^+))$ are not orthogonal for a\ngeneral choice of $\\alpha $ because $\\psi _{n}^{\\alpha }$ are orthogonal in\nthe Hardy space and not in the space of admissible wavelets. But if we make\nthe particular choice $\\alpha =2B-2n-1$, they become orthogonal, that is, for $%\nn,m\\in \\{0,1,\\ldots ,\\lfloor B-\\frac{1}{2}\\rfloor \\}$, \n\\begin{equation*}\n\\langle W_{\\psi _{n}^{2B-2n-1}}f,W_{\\psi _{m}^{2B-2m-1}}g\\rangle _{{L%\n}^{2}(\\mathbb{C}^{+})}=\\frac{2}{2B-2n-1}\\delta _{n,m},\\quad {f,g\\in H^2(\\mathbb{C}^+)}.\n\\end{equation*}%\nThis is related to the identification of the spaces $W_{\\psi _{n}^{2B-2n-1}}(H^{2}(\\mathbb{C}^{+}))$ with the eigenspaces of the Maass-Landau levels\noperator with a constant magnetic field $B$ originally studied in number theory in the theory of Maass forms \\cite{maass1949neue}, and the Selberg trace formula \\cite{patterson1975laplacian}. We provide a brief account of this Schr\\\"{o}dinger operator from a physical perspective.\n\nThe \\emph{Schr{\\\"{o}}dinger operator} describing the dynamics of a charged\nparticle moving in $\\mathbb{C}^{+}$ under the action of the constant magnetic field $B\n$ \\cite{co87} is given by \n\\begin{equation}\nH_{B}=-s^{2}\\left( \\frac{\\partial ^{2}}{\\partial x^{2}}+\\frac{\\partial ^{2}}{%\n\\partial s^{2}}\\right) +2iBs\\frac{\\partial }{\\partial x}\\text{.}\n\\label{eq:schroedinger-op}\n\\end{equation}%\n$H_{B}$ is an elliptic, and densely defined operator on ${L}^{2}(%\n\\mathbb{C}^{+})$. Its spectrum consists of a continuous part $[1\/4,\\infty )$\ncorresponding to \\emph{scattering states} and a finite number of eigenvalues \n\\begin{equation*}\n\\lambda _{n}^{B}=(B-n)(B-n-1),\\qquad n=0,1\\ldots ,\\lfloor B-\\frac{1}{2}\n\\rfloor \\text{.}\n\\end{equation*}\nThe eigenvalues exist provided that $2B>1$, and each eigenvalue corresponds\nto an infinite dimensional reproducing kernel Hilbert space $E_{n}^{B}(\n\\mathbb{C}^{+})$. From a physical viewpoint, this condition guarantees\nthat the magnetic field is strong enough to capture particles in a closed\norbit, see \\cite[p. 189]{gro88}. The eigenspaces corresponding to $\\lambda\n_{n}^{B}$ are called \\emph{hyperbolic Landau levels} and we define them by \n\\begin{equation*}\nE_{n}^{B}(\\mathbb{C}^{+}):=\\{F\\in  {L}^{2}(\\mathbb{C}^{+}):\\\nH_{B}F=\\lambda _{n}^{B}F\\}\\text{.}\n\\end{equation*}\nIn \\cite[Section 4,5]{affine} it is shown in detail (this connection was first observed by Mouayn \\cite{mouayn2003characterization}) that \n\\begin{equation*}\nW_{\\psi _{n}^{2B-2n-1}}(H^{2}(\\mathbb{C}^{+}))=E_{n}^{B}(\\mathbb{C}^{+})\n\\text{.}\n\\end{equation*}\nThis way we obtain an orthogonal basis for all the spaces $E_{n}^{B}(\\mathbb{C}\n^{+})$, by noting that for $n,m\\in\n\\{0,1,\\ldots ,\\lfloor B-\\frac{1}{2}\\rfloor \\}$, \\eqref{ortogonalityrelations} gives \n\\begin{equation*}\n\\langle W_{\\psi _{n}^{2B-2n-1}}\\psi _{k}^{\\alpha },W_{\\psi\n_{m}^{2B-2m-1}}\\psi _{l}^{\\alpha }\\rangle _{{L}^{2}(\\mathbb{C}^{+})}=%\n\\frac{2}{2B-2n-1}\\delta _{n,m}\\delta _{k,l}\\text{.}\n\\end{equation*}\nOur large sieve inequalities for this case provide conditions allowing for the recovery of a continuous coherent state in higher hyperbolic Landau levels, from partial information, using $L_{1}$-minimization, as outlined in the introduction.\n\\subsection{Pseudohyperbolic Metric and M{\\\"o}bius Transformation}\n\nThe \\emph{pseudohyperbolic distance} on $\\mathbb{C}^{+}$ is given by \n\\begin{equation*}\n\\varrho (z,w):=\\left\\vert \\frac{z-w}{z-\\overline{w}}\\right\\vert ,\\qquad\nz,w\\in \\mathbb{C}^{+},\n\\end{equation*}%\nand the \\emph{pseudohyperbolic disk} of radius $R>0$ centered at $z\\in \n\\mathbb{C}^{+}$ is defined by $\\mathcal{D}_{R}(z):=\\{\\omega \\in \\mathbb{C}%\n^{+}:\\ \\varrho (z,w)<R\\}$. We write for short $\\mathcal{D}_{R}=\\mathcal{D}%\n_{R}(i)$. Note that $\\varrho $ only takes values in the half open interval $\n[0,1)$. It can be checked that the pseudohyperbolic distance is symmetric\nand  invariant under the action of the affine group. This leads to several\nuseful properties, like $\\varrho (z,w)=\\varrho (z^{-1}\\cdot w,i)$ and $\n\\varrho (u,z\\cdot w)=\\varrho (z^{-1}\\cdot u,w)$.\n\nLet $\\mathbb{D}_{R}$ denote the disk of radius $R>0$ in $\\mathbb{C}$. We\nwrite for short $\\mathbb{D}:=\\mathbb{D}_{1}$ to denote the unit disk. The\nmapping $T:\\mathbb{D}\\rightarrow \\mathbb{C}^{+}$,  \n\\begin{equation*}\nT(u):=i\\frac{1+u}{1-u},\\quad u\\in \\mathbb{D}\n\\end{equation*}%\nmaps the pseudohyperbolic distance in $\\mathbb{C}^{+}$ to the\npseudohyperbolic distance in $\\mathbb{D}$:\n\\begin{equation*}\n\\varrho (T(z),T(w))=\\left\\vert \\frac{w-z}{1-\\overline{z}w}\\right\\vert\n=\\varrho _{\\mathbb{D}}(z,w)\\text{.}\n\\end{equation*}\nThe mapping $T$ will be used to map some integrals from the upper half plane to\nthe unit disk according to the change of variables formula\n\\begin{equation}\\label{eq:change-of-variable}\n\\int_{\\mathbb{C}^{+}}F(z)s^{\\beta }d\\mu_{\\mathbb{C}^+}(z)=\\int_{\\mathbb{D}}F(T(u))\\frac{\n4(1-|u|^{2})^{\\beta }}{|1-u|^{2\\beta +4}}du,\\qquad \\beta \\in \\mathbb{R}.\n\\end{equation}\n\n\\section{Double Orthogonality and the Local Reproducing Formula}\\label{sec:dbl-orth}\n\nThe following family of   polynomials defined on $[0,1]$ is closely related to  the \\emph{ Zernike\npolynomials}  \\cite{wuensch05}, a family of orthogononal polynomials in $(z,\\overline{z})$ defined on $\\mathbb{D}$,\n\\begin{equation}\nZ_{n,m}^{\\alpha }(t):=\\sqrt{\\dfrac{\\Gamma (\\max (n,m)+\\alpha\n+1)\\min (n,m)!}{\\Gamma (\\min (n,m)+\\alpha +1)\\max (n,m)!}}(-t)^{-\\min\n(n,m)}P_{\\min (n,m)}^{(|n-m|,\\alpha )}\\left( 1-2t\\right),\n\\label{eq:def-Phi_nm}\n\\end{equation}\nwhere $P_n^{(\\alpha,\\beta)}$ denotes the \\emph{Jacobi polynomials} which are given explicitly in \\eqref{eq:jacobi}.\nWe will see in this section that \n\\begin{equation*}\nZ_{n,m}^{\\alpha }\\left( \\left\\vert \\dfrac{z-i}{z+i}\\right\\vert ^{2}\\right),\n\\quad z\\in \\mathbb{C}^{+},\n\\end{equation*}\nplays a role in wavelet analysis similar to the one of complex Hermite\npolynomials in time-frequency analysis \\cite{abspe18-sieve}. The following result is proved in the\nAppendix.\n\n\\begin{proposition}\n\\label{lem:explicit-wavelet-trafo} Let $z=x+is$ and $w=x^{\\prime\n}+is^{\\prime }$ be in $\\mathbb{C}^{+}$. For every $n,m\\in \\mathbb{N}_{0}$,\none has \n\\begin{equation}\\label{eq:kernel-expl}\nK_{\\psi _{n}^{\\alpha }}(z,w)=\\frac{\\alpha}{4\\pi} \\left( \\frac{w-\\overline{z}}{z-\n\\overline{w}}\\right) ^{n}\\left( \\frac{{2}\\sqrt{ss^{\\prime }}}{i(\\overline{w}-z)}\n\\right) ^{\\alpha +1}P_{n}^{(0,\\alpha )}\\left( 1-2\\left\\vert \\frac{z-w}{z-\n\\overline{w}}\\right\\vert ^{2}\\right) ,\n\\end{equation}\nand \n\\begin{equation}\nW_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }(z)= \\left(\\frac{z-i}{z+i}\\right)^{m}\\left(\\frac{\\overline{z}%\n+i}{z+i}\\right)^{n}\\left( \\dfrac{{2}\\sqrt{s}}{1-iz}\\right) ^{\\alpha\n+1}Z_{n,m}^{\\alpha }\\left( \\left\\vert \\dfrac{z-i}{z+i}\\right\\vert\n^{2}\\right) .  \\label{eq:explicit-wavelet}\n\\end{equation}\n\\end{proposition}\n\n\\begin{remark}\n For $n=0$, i.e. the case of the Cauchy wavelet, one has that the\nJacobi polynomial is constantly one. Consequently, \n\\begin{equation}\n\\bm{\\Phi}_{m}^{\\alpha }(z):=s^{-\\frac{\\alpha +1}{2}}W_{\\psi _{0}^{\\alpha\n}}\\psi _{m}^{\\alpha }(z)= \\sqrt{\\frac{\\Gamma (m+\\alpha +1)}{\n\\Gamma (\\alpha +1)m!}}\\left( \\frac{z-i}{z+i}\\right) ^{m}\\left( \\frac{{2}}{1-iz}\n\\right) ^{\\alpha +1},\\label{eq:n=0}\n\\end{equation}\nwhich are (up to a constant factor due to different normalizations) the\nbasis functions of the Bergman space $\nA_{\\alpha -1}^2(\\mathbb{C}^{+})$, where \n$$\nA_{\\alpha }^p(\\mathbb{C}^{+}):=\\{F\\text{ holomorphic}:\\ \\|F\\|_{A^p_\\alpha(\\mathbb{C}^+)}<\\infty\\},\n$$\nand \n\\begin{equation}\\label{bergman:halfplane}\n\\|F\\|_{A_{\\alpha }^p(\\mathbb{C}^{+})}^p:=\\int_{\\mathbb{C}^+}|F(z)|^p s^{\\alpha}d \\mu_{\\mathbb{C}^+}(z),\n\\end{equation}  \nsee,\ne.g., \\cite[Section 4.4]{abdoe12}.\n\n\\end{remark}\n\nIn the following, we show a local orthogonality relation for the family $%\n\\{W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }\\}_{m\\in \\mathbb{N}_{0}}$ and\nsubsequently derive a local reproducing formula.\n\n\\begin{theorem}\n\\label{thm:local-orthogonal} Let $\\alpha >0$, $0<R<1$, and $n,m,k\\in \\mathbb{\nN}_{0}$. It holds \n\\begin{equation}\n\\int_{\\mathcal{D}_{R}}W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }(z)\\overline{\nW_{\\psi _{n}^{\\alpha }}\\psi _{k}^{\\alpha }(z)}d\\mu ^{+}(z)=C_{m,n}^{\\alpha\n}(R)\\delta _{m-k},  \\label{eq:local-orthogonal}\n\\end{equation}\nwith \n\\begin{equation*}\nC_{n,m}^{\\alpha }(R)=   {4\\pi} \\int_{0}^{R^{2}}r^{n+m}(1-r)^{\\alpha\n-1}Z_{n,m}^{\\alpha }(r)^{2}dr.\n\\end{equation*}\nIf $n=m$, we write $C_{n}^{\\alpha }(R)=C_{n,n}^{\\alpha }(R)$ which is given\nexplicitly by \n\\begin{equation}\nC_{n}^{\\alpha }(R)=  {4\\pi}\\int_{0}^{R^{2}}(1-r)^{\\alpha\n-1}P_{n}^{(0,\\alpha )}(1-2r)^{2}dr.  \\label{eq:def_Cn}\n\\end{equation}\n\\end{theorem}\n\n\\noindent \\textbf{Proof:\\ }\nFirst we note that putting $z=T(u)$ in \\eqref{eq:explicit-wavelet} yields\n\\begin{equation}\\label{eq:aux-moebius}\nW_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }(T(u))={(-1)^{n+\\frac{\\alpha+1}{2}}}u^{m}\\overline{u}^{n}\\left( \\dfrac{1-u}{|1-{u}|}\\right) ^{2n+\\alpha +1}\\left(  {1-|u|^{2}} \\right) ^{\\frac{\\alpha +1}{2}}Z _{n,m}^\\alpha(|u|^{2}).\n\\end{equation}\nUsing \\eqref{eq:change-of-variable} for $\\beta=-2$ then leads us to\n\\begin{align*}\n\\int_{\\mathcal{D}_{R}}W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }(z)\\overline{%\nW_{\\psi _{n}^{\\alpha }}\\psi _{k}^{\\alpha }(z)}d\\mu ^{+}(z)& =\\int_{\\mathbb{D}%\n_{R}}W_{\\psi _{n}^{\\alpha }}\\psi _{m}^{\\alpha }(T(u))\\overline{W_{\\psi\n_{n}^{\\alpha }}\\psi _{k}^{\\alpha }(T(u))}\\frac{4du}{(1-|u|^{2})^{2}} \\\\\n&= {4}\\int_{\\mathbb{D}_{R}}u^{m}\\overline{u}^{k}|u|^{2n}(1-|u|^{2})^{%\n\\alpha -1}Z_{n,m}^{\\alpha }(|u|^{2})Z_{n,k}^{\\alpha }(|u|^{2})du \\\\\n& =\\circledast .\n\\end{align*}%\nIntegration using polar coordinates then yields \n\\begin{align*}\n\\circledast & = {4}\\int_{0}^{2\\pi }\\int_{0}^{R}e^{i(m-k)\\varphi\n}r^{m+k+2n+1}(1-r^{2})^{\\alpha -1}Z_{n,m}^{\\alpha }(r^{2})Z_{n,k}^{\\alpha\n}(r^{2})drd\\varphi \\\\\n& = {8}\\pi \\delta _{m-k}\\int_{0}^{R}r^{2(n+m)+1}(1-r^{2})^{\\alpha\n-1}Z_{n,m}^{\\alpha }(r^{2})^{2}dr \\\\\n& ={4}\\pi \\delta _{m-k}\\int_{0}^{R^{2}}r^{n+m}(1-r)^{\\alpha -1}Z_{n,m}^{\\alpha\n}(r)^{2}dr,\n\\end{align*}%\nwhich concludes the proof once we recall the definition of $Z_{n,m}^{\\alpha }\n$ in \\eqref{eq:def-Phi_nm}. \\hfill $\\Box $\\newline\n\nLet us shortly consider the case  $n=0:$%\n\\begin{align*}\nC_{0}^{\\alpha }(R) &= {4\\pi}\\int_{0}^{R^{2}}(1-r)^{\\alpha\n-1}P_{0}^{(0,\\alpha )}(1-2r)^{2}dr= {4\\pi}\\int_{0}^{R^{2}}(1-r)^{\\alpha -1}dr \\\\\n&= {\\frac{4\\pi}{\\alpha}}\\left( 1-(1-R^{2})^{\\alpha }\\right),\n\\end{align*}%\nwhich converges to $4\\pi\/\\alpha$ (as $R\\to 1$) as one expects from \n\\eqref{ortogonalityrelations}.\n\n\\begin{theorem}\n\\label{thm:loc-rep} Let $n\\in \\mathbb{N}_{0}$, $\\alpha >1$, and $0<R<1$. The\nfollowing identity holds in the weak sense \n\\begin{equation}\n\\int_{\\mathcal{D}_{R}}W_{\\psi _{n}^{\\alpha }}\\psi _{n}^{\\alpha }(z)\\pi\n(z)\\psi _{n}^{\\alpha }d\\mu ^{+}(z)=C_{n}^{\\alpha }(R)\\ \\psi _{n}^{\\alpha }.\n\\label{eq:weak-reproducing}\n\\end{equation}\nIf $f\\in \\mathcal{C}o\\,{L}^{\\infty }(\\mathbb{C}^{+})$, then the wavelet\ncoefficients can locally be reconstructed via \n\\begin{equation}\nW_{\\psi _{n}^{\\alpha }}f(z)=\\frac{4\\pi}{\\alpha C_{n}^{\\alpha }(R)}\\int_{\\mathcal{D}\n_{R}(z)}W_{\\psi _{n}^{\\alpha }}f(w)K_{\\psi _{n}^{\\alpha }}(z,w) d\\mu\n^{+}(w),\\qquad z\\in \\mathbb{C}^{+}.  \\label{eq:strong-reprod}\n\\end{equation}\n\\end{theorem}\n\n\\noindent \\textbf{Proof:\\ } Setting $m=n$ in \\eqref{eq:local-orthogonal} shows that \n\\eqref{eq:weak-reproducing} holds weakly. If we apply $\\pi (w)$ on both\nsides of \\eqref{eq:weak-reproducing} and subsequently take the inner product\nwith $f$, we obtain \n\\begin{align*}\nW_{\\psi _{n}^{\\alpha }}f(z)& =\\langle f,\\pi (z)\\psi _{n}^{\\alpha }\\rangle \\\\\n& =\\frac{1}{C_{n}^{\\alpha }(R)}\\int_{\\mathcal{D}_{R}}\\langle f,\\pi (z\\cdot\nw)\\psi _{n}^{\\alpha }\\rangle \\langle \\pi (w)\\psi _{n}^{\\alpha },\\psi\n_{n}^{\\alpha }\\rangle d\\mu ^{+}(w) \\\\\n& =\\frac{1}{C_{n}^{\\alpha }(R)}\\int_{\\mathcal{D}_{R}(z)}\\langle f,\\pi\n(w)\\psi _{n}^{\\alpha }\\rangle \\langle \\pi (z^{-1}\\cdot w)\\psi _{n}^{\\alpha\n},\\psi _{n}^{\\alpha }\\rangle {d\\mu^+(w)} \\\\\n& =\\frac{1}{C_{n}^{\\alpha }(R)}\\int_{\\mathcal{D}_{R}(z)}\\langle f,\\pi\n(w)\\psi _{n}^{\\alpha }\\rangle \\langle \\pi (w)\\psi _{n}^{\\alpha },\\pi (z)\\psi\n_{n}^{\\alpha }\\rangle d\\mu ^{+}(w) \\\\\n& =\\frac{4\\pi}{\\alpha C_{n}^{\\alpha }(R)}\\int_{\\mathcal{D}_{R}(z)}W_{\\psi _{n}^{\\alpha\n}}f(w)K_{\\psi _{n}^{\\alpha }}(z,w)d\\mu ^{+}(w),\n\\end{align*}\nwhere we used  $\n\\varrho (z,z\\cdot w)=\\varrho (i,w)$, and the left\ninvariance of the Haar measure. The second equality holds since the integral\nand the duality pairing may be interchanged knowing that $f\\in \\mathcal{C}o\\,\n {L}^{\\infty }(\\mathbb{C}^{+})$, and $K_{\\psi _{n}^{\\alpha }}(z,\\cdot)\\in \n {L}^{1}(\\mathbb{C}^{+})$.\\hfill $\\Box $\\newline\n\n\\section{Large Sieve Estimates}\\label{sec:LS}\n\n\\subsection{A Schur-type estimate}\n\nLet $(X,\\mu )$, $(X,\\nu )$ be measure spaces and $B_{p}\\subset L^{p}(X,\\nu )$\na Banach space. In \\cite[Proposition 1, Corollary 1]{abspe18-sieve}, a bound\non the embedding $(B_{p},\\Vert \\cdot \\Vert _{L_{\\nu }^{p}})\\hookrightarrow\n(B_{p},\\Vert \\cdot \\Vert _{L_{\\mu }^{p}})$ was derived using an argument\nsimilar to Schur's test which we shortly recall here.\n\n\\begin{lemma}\n\\label{prop-measure-bound} Let $\\mu $ be a positive $\\sigma $-finite measure\non $X$, $B_{1}\\subset L^{1}(X,\\nu)$ and $K:X\\times X\\rightarrow \\mathbb{C} $\nbe \nsuch that $\\mathcal{K}:B_{1}\\rightarrow B_{1}$, \n\\begin{equation*}\n\\mathcal{K}F(x):=\\int_{X}F(y)K(x,y)d\\nu(y),\n\\end{equation*}\nis bounded and boundedly invertible on $B_{1}$. Then, for every $F\\in B_{1}$, we have \n\\begin{equation} \n\\frac{\\int_{X}|F|d\\mu }{\\int_X |F|d\\nu}\\leq \\theta_\\nu (K) \\cdot \\sup_{y\\in\nX}\\int_{X}|K(x,y)|d\\mu (x),  \\label{eq-norm-bound2}\n\\end{equation}\nwhere \n\\begin{equation*}\n\\theta_\\nu (K):=\\sup_{H \\in B_{1}}\\left( \\dfrac{\\Vert H \\Vert _{L^1_\\nu}}{\\Vert \\mathcal{K}H \\Vert _{L^1_\\nu}}\\right) .\n\\end{equation*}\nIf $\\Vert F\\Vert _{L_{\\mu }^{\\infty }}\\leq C_{\\infty }\\Vert F\\Vert\n_{L^\\infty_\\nu },\\ \\forall F\\in B_{\\infty }$ and $1\\leq p<\\infty $, then for\nevery $F\\in B_{1}\\cap B_{\\infty }$ it holds \n\\begin{equation}\n\\frac{\\int_{X}|F|^{p}d\\mu }{\\int_X|F|^{p}d\\nu}\\leq C_{\\infty }^{p-1}\\cdot\n\\theta (K)\\cdot \\sup_{y\\in X}\\int_{X}|K(x,y)|d\\mu (x).\n\\end{equation}\n\\end{lemma}\n\n\\subsection{Estimates with explicit constants}\n\nRecall that the \\emph{maximum Nyquist density} is given by \n\\begin{equation*}\n\\rho (\\Delta ,R):=\\sup_{z\\in \\mathbb{C}^{+}}|\\Delta \\cap \\mathcal{D}_{R}(z))|_{ {h}},\n\\end{equation*}\nwhere $|\\Delta |_{ {h}}:= \\int_{\\Delta }d\\mu ^{+}(z)$ denotes \\emph{hyperbolic\nmeasure} of $\\Delta $. We will also use a second notion of density given\nby \n\\begin{equation}\nD_{n}^{\\alpha }(\\Delta ,R):= \\sup_{z\\in \\mathbb{C}^{+}}\\int_{\\Delta \\cap \n\\mathcal{D}_{R}(z)}|\\langle \\pi(w)\\psi_{n}^{\\alpha },\\pi(z)\\psi_n^\\alpha\\rangle| {d\\mu^+(w)}.\n\\label{Ar-density}\n\\end{equation}\nNote that $D_{n}^{\\alpha }(\\Delta ,R)\\leq \\rho (\\Delta ,R)$, as \n\\begin{equation}\n|\\langle \\pi(w)\\psi_{n}^{\\alpha },\\pi(z)\\psi_n^\\alpha\\rangle|\\leq \\Vert \\psi _{n}^{\\alpha }\\Vert ^{2}_{H^2(\\mathbb{C}^+)}=1.\n\\label{KernelEst}\n\\end{equation}\nWe have gathered all the ingredients to formulate and prove our main result.\n\n\\begin{theorem}\n\\label{main} Let $\\Delta \\subset \\mathbb{C}^{+}$ and $f\\in \\mathcal{C}o\\,\n {L}^{p}(\\mathbb{C}^{+})$, $1\\leq p<\\infty $. For every $0<R<1$, it\nholds \n\\begin{equation}\n\\dfrac{\\Vert W_{\\psi _{n}^{\\alpha }}f\\cdot \\chi _{\\Delta }\\Vert _{L^p(\\mathbb{C}^+)}^p}{\\Vert W_{\\psi _{n}^{\\alpha }}f\\Vert _{L^p(\\mathbb{C}^+)}^{p}}\\leq  \\frac{D_{n}^{\\alpha\n}(\\Delta ,R)}{C_{n}^{\\alpha }(R)}\\leq   \\frac{\\rho (\\Delta ,R)}{C_{n}^{\\alpha\n}(R)}\\text{.}\\   \\label{theo1-eq}\n\\end{equation}\n\\end{theorem}\n\n\\noindent \\textbf{Proof:\\ } We take $K(z,w):=\\langle \\pi(w)\\psi_{n}^{\\alpha },\\pi(z)\\psi_n^\\alpha\\rangle \\chi _{\\mathcal{D}_{R}(z)}(w)$ in Lemma~\\ref{prop-measure-bound}. Then by \\eqref{thm:loc-rep} we get\n\\begin{equation*}\n\\theta _{\\nu }(K):=\\sup_{f\\in \\mathcal{C}o\\, {L}^{1}(\\mathbb{C}^{+} )}\\left( \\dfrac{\\Vert W_{\\psi _{n}^{\\alpha }}f\\Vert _{L^{1}(\\mathbb{C}^+ )}}{\\big\\|\n\\int_{\\mathcal{D}_{R}(\\cdot)}W_{\\psi _{n}^{\\alpha }}f(w)\\langle \\pi(w)\\psi_{n}^{\\alpha },\\pi(\\cdot)\\psi_n^\\alpha\\rangle d\\mu ^{+}(w)\\big\\| _{L^{1}(\\mathbb{C}^+)}}\\right) =\\frac{1}{C_{n}^{\\alpha }(R)}.\n\\end{equation*}\nThus, if $d\\mu (z)=\\chi _{\\Delta }(z)d\\mu^+(z)$, we have by \\eqref{eq-norm-bound2} and \\eqref{KernelEst}\n\\begin{equation*}\n\\dfrac{\\Vert W_{\\psi _{n}^{\\alpha }}f\\cdot \\chi _{\\Delta }\\Vert _{ {L}^{1}(\\mathbb{C}^+)}}{\\Vert W_{\\psi }^{\\alpha }f\\Vert _{ {L}^{1}(\\mathbb{C}^+)}}\\leq \\frac{1}{C_{n}^{\\alpha }(R)}\\cdot \\sup_{z\\in \\mathbb{C}^{+}}\\int_{\\Delta \\cap \n\\mathcal{D}_{R}(z)}|\\langle \\pi(w)\\psi_{n}^{\\alpha },\\pi(z)\\psi_n^\\alpha\\rangle| d\\mu ^{+}(w)\\leq   \\frac{\\rho\n(\\Delta ,R)}{C_{n}^{\\alpha }(R)}\\text{.}\n\\end{equation*}\nThe result thus holds for $p=1$. As $\\mathcal{C}o\\, {L}^{1}(\\mathbb{C}^{+})\\cap \\ \\mathcal{C}o\\, {L}^{\\infty }(\\mathbb{C}^{+})=\\mathcal{C}\no\\, {L}^{1}(\\mathbb{C}^{+})$ is dense in $\\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})$ and \n\\begin{equation*}\n\\sup_{f\\in \\mathcal{C}o\\, {L}^{\\infty }(\\mathbb{C}^{+})}\\frac{\\Vert\nW_{\\psi _{n}^{\\alpha }}f\\cdot \\chi _{\\Delta }\\Vert _{{L}^{\\infty(\\mathbb{C}^+) }}}{\\Vert W_{\\psi _{n}^{\\alpha }}f\\Vert _{ {L}^{\\infty}(\\mathbb{C}^+) }}=1,\n\\end{equation*}\nthe result follow for $1<p<\\infty $. \\hfill $\\Box $\n\n\\begin{corollary}\n\\label{uncertainty} Suppose that $f\\in \\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})$, $1\\leq p<\\infty $, satisfies $\\Vert W_{\\psi _{n}^{\\alpha }}f\\Vert\n_{L^p(\\mathbb{C}^+)}=1$, and that $W_{\\psi _{n}^{\\alpha }}f$ is $\\varepsilon$-concentrated\non $\\Delta \\subset \\mathbb{C}^{+}$, {i.e.}, \n\\begin{equation*}\n1-\\varepsilon \\leq \\int_{\\Delta }|W_{\\psi _{n}^{\\alpha }}f(z)|^{p}d\\mu\n^{+}(z)\\text{.}\n\\end{equation*}\nThen \n\\begin{equation*}\n1-\\varepsilon \\leq \\inf_{0<R<1}\\left( \\frac{\\rho (\\Delta ,R)}{C_{n}^{\\alpha\n}(R)}\\right) \\leq |\\Delta |_h.\n\\end{equation*}\n\\end{corollary}\n\n\\begin{remark}\nLet us introduce the Bergman spaces on the unit disk by\n\\begin{equation}\\label{def:berg-disk}\nA_{\\alpha }^p(\\mathbb{D}):=\\left\\{ f\\text{ analytic}:\\ \\|f\\|_{A_\\alpha^p}^p:=\\frac{\\alpha+1}{\\pi}\\int_{\\mathbb{D}}|f(z)|^p(1-|z|^{2})^{\\alpha}dz<\\infty \\right\\} .\n\\end{equation}\nWe derived   estimates similar to Theorem~\\ref{main} for the Bergman \nspace $A_{\\alpha-2 }^2(\\mathbb{D})$ in \\cite[Theorem~2]{abspe19}.  \nThere Seip's double orthogonality result \\cite{sei91} was  applied to obtain the concentration estimate.\n\\end{remark}\n\n\\section{Optimal bounds in the analytic setting}\\label{sec:opt}\n\nThe following result was proved recently for the case $p=2$ in \\cite{RamosTilli}. We note here that the case of general $p$ works with exactly the same arguments as Ramos and Tilli used with one extra information: $\\mathcal{C}o\\,{L}^{p}(\\mathbb{C}^{+})$ can be mapped isometrically and surjectively onto a certain Bergman space. \n\n\\begin{theorem}[Ramos-Tilli]\n\\label{extremal-conjecture} Let $\\Delta \\subset \\mathbb{C}^{+}$ be a set of\nfinite hyperbolic measure $A>0$, and $\\alpha>1$. Then \n\\begin{equation*}\n\\sup_{|\\Delta |_h=A}\\sup_{f\\in \\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})}%\n\\frac{\\Vert W_{\\psi _{0}^{\\alpha } }f\\cdot \\chi _{\\Delta }\\Vert _{L^p(\\mathbb{C}^+)}^{p}}{\\Vert W_{\\psi _{0}^{\\alpha }\n}f\\Vert _{L^p(\\mathbb{C}^+)}^{p}}\n\\end{equation*}%\nis attained if and only if $\\Delta $ is a pseudohyperbolic disc centered at\nsome $z\\in \\mathbb{C}^+$, up to perturbations of Lebesgue measure zero.  In particular,\n\\begin{equation} \\label{eq:ramos-tilli-p}\n\\int_{\\Delta }|W_{\\psi _{0}^{\\alpha }}f(z)|^{p}d\\mu ^{+}(z)\\leq  {\\left(1-\\left(\n1+\\frac{|\\Delta |_h}{4\\pi}\\right)  ^{1-(\\alpha+1) p\/2}\\right)}\\|W_{\\psi_0^\\alpha}f\\|_{L^p(\\mathbb{C}^+)}^p\\text{.} \n\\end{equation}\n\\end{theorem}\n\nTo prove the result for general $p$, it just takes a simple observation. As in \\cite{RamosTilli} one can map the space $A_{{(\\alpha +1)p\/2-2}}^{p}(\\mathbb{C}%\n^{+})$ to $A_{{(\\alpha +1)p\/2-2}}^{p}(\\mathbb{D})$ (see definitions \\eqref{bergman:halfplane} and \\eqref{def:berg-disk}) by virtue of the\ntransformation%\n\\begin{equation*}\nT_{\\alpha }F(u)=F\\left( i\\frac{1+u}{1-u}\\right) \\left( \\frac{1}{1-u}\\right)\n^{\\alpha +1},\\quad u\\in \\mathbb{D}, \n\\end{equation*}\nwhich, by \\eqref{eq:change-of-variable}, is a multiple of an isometry. Thus,\nthe composition $T_\\alpha\\circ s^{-\\frac{\\alpha +1}{2}}W_{\\psi }:\\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+}) \\to A_{(\\alpha\n+1)p\/2-2}^{p}(\\mathbb{D})$\\ \n\\begin{equation*}\n\\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})  \\hspace{0.2cm}\\xrightarrow{\\hspace{0.2cm}s^{-\\frac{\\alpha +1}{2}}W_{\\psi }\\hspace{0.2cm}}  \\hspace{0.2cm}A_{(\\alpha +1)p\/2-2}^{p}(\\mathbb{C}^{+})\\hspace{0.2cm}  \\xrightarrow{\\hspace{0.5cm}T_\\alpha\\hspace{0.5cm}} \\hspace{0.2cm} A_{(\\alpha+1)p\/2-2}^{p}(\\mathbb{D})\n\\text{,}\n\\end{equation*}%\nis surjective. To see this, we note that by \\eqref{eq:n=0} the sequence $\\{\\psi _{m}^{\\alpha }\\}_{m\\in\\mathbb{N}_0}$ (which is complete in $\\mathcal{C}%\no\\, {L}^{p}(\\mathbb{C}^{+})$) is  first mapped to the sequence \n\\begin{equation*}\ns^{-\\frac{\\alpha +1}{2}}W_{\\psi _{0}^{\\alpha }}\\psi _{m}^{\\alpha }(z)= \\sqrt{%\n\\frac{\\Gamma (m+\\alpha +1)}{\\Gamma (\\alpha +1)m!}}\\left( \\frac{z-i}{z+i}%\n\\right) ^{m}\\left( \\frac{2}{1-iz}\\right) ^{\\alpha +1}, \\quad m\\in\\mathbb{N}_0,\n\\end{equation*}\nwhich is in turn mapped to the sequence of\nmonomials.\nThe completeness of the monomials\nin $A_{(\\alpha +1)p\/2-2}^{p}(\\mathbb{D})$ follows from the following\nstandard argument, which is included for convenience. Consider the expansion\nof the analytic function $F(u)\\in A^p_{(\\alpha+1)p\/2-2}(\\mathbb{D})$ into its\nTaylor series  $F(u)=\\sum_{n=0}^{\\infty }a_{n}u^{n}$. The series converges\nuniformly on compact sets. If for every $m\\in\\mathbb{N}_0$ we have\n\\begin{align*}\n0&=\\int_{\\mathbb{D}}F(u)\\overline{u^{m}}(1-\\left\\vert u\\right\\vert\n^{2})^{(\\alpha+1)p\/2+p-2}du\n\\\\\n&=\\sum_{n=0}^{\\infty }a_{n}\\int_{\\mathbb{D}} u^{n}\\overline{u^{m}}(1-\\left\\vert u\\right\\vert ^{2})^{(\\alpha+1)\np\/2+p-2}du=C_{p,\\alpha,m}a_m,\n\\end{align*}\nthen $F=0$. \n\nThe proof of Theorem~\\ref{extremal-conjecture} for general $p\\neq2$ works  exactly as the proof of \\cite[Theorem~3.1]{RamosTilli} with the minor change that the definition of the auxiliary function $u$ (see \\cite[p.~8]{RamosTilli}) needs to be changed to $u(z):=|F(z)|^p (1-|z|^2)^{(\\alpha+1)p\/2-2}$, $F\\in A^p_{(\\alpha+1)p\/2-2}(\\mathbb{D})$.\n\nIf $W_{\\psi _{n}^{\\alpha }}f$ is $\\varepsilon $\n-concentrated on $\\Delta \\subset \\mathbb{C}^{+}$, i.e., \n\\begin{equation*}\n(1-\\varepsilon)\\|W_{\\psi_0^\\alpha}f\\|_{L^p(\\mathbb{C}^+)}^p \\leq \\int_{\\Delta }|W_{\\psi _{0}^{\\alpha }}f(z)|^{p}d\\mu\n^{+}(z)\\text{,}\n\\end{equation*}\nthen \\eqref{eq:ramos-tilli-p} yields \n$$\n1-\\varepsilon\\leq {1-\\left(\n1+\\frac{|\\Delta |_h}{4\\pi}\\right)  ^{1-(\\alpha+1) p\/2}}.\n$$\nThis leads to the following sharp uncertainty principle for the wavelet\ntransform.\n\\begin{theorem}\nLet $\\alpha>1$, $\\varepsilon\\in (0,1)$ and $1\\leq\np<\\infty $. Suppose that $f\\in \\mathcal{C}o\\, {L}^{p}(\\mathbb{C}^{+})$ satisfies $\\Vert W_{\\psi _{0}^{\\alpha }}f\\Vert _{L^p(\\mathbb{C}^+)}=1$, and that $W_{\\psi _{0}^{\\alpha }}f$ is $\\varepsilon $-concentrated with respect to the $p$-norm in $\\Delta \\subset \n\\mathbb{C}^{+}$. Then \n\\begin{equation*}\n4\\pi\\left(\\epsilon ^{\\frac{2}{2-(\\alpha+1) p}}-1\\right)\\leq |\\Delta |_h.\n\\end{equation*}\n\\end{theorem}\n\n\\begin{remark}\nExtending these optimal results to the general family of wavelets $\\psi _{n}^{\\alpha\n}$ remains an open problem (as well as the corresponding optimal bound for\ngeneral Hermite functions in \\cite{abspe18-sieve}). The methods of \n\\cite{Inventiones} and \\cite[Theorem~3.1]{RamosTilli} are extremely\ndependent on the analytic structure of the case $n=0$, and for $n>0$ the transforms are not related to analytic functions. However, since a derivative of an analytic function is still analytic, the methods of \n\\cite{Inventiones} and \\cite[Theorem~3.1]{RamosTilli}  can be adapted to obtain sharp inequalities for weighted $L^p$-norms of derivatives of holomorphic functions.   In \\cite%\n{kalaj2022contraction}, combined with sharp pointwise inequalities for derivatives in Fock spaces (involving Laguerre functions with complex argument), this has been used to obtain a contraction inequality for derivatives\nin Fock spaces. It is reasonable to expect a related inequality for derivatives in Bergman spaces. This requires sharp pointwise inequalities for derivatives in Bergman spaces, which possibly involve Jacobi polynomials with complex argument.\n\\end{remark}\n\n\n\\subsection{Lieb's uncertainty principle}\n\nThe following result is due to Kulikov \\cite[Theorem~1.2]{kuli22}. \nNote that the normalization of the norm of $A_{\\alpha }^{p}(\\mathbb{D})$   is chosen such that $\\Vert 1\\Vert _{A_{\\alpha\n }^{p}(\\mathbb{D})}=1$ (as defined in \\eqref{def:berg-disk}).\n\n\\begin{theorem}[Kulikov]\n\\label{thm:kulikov} Let $G:[0,\\infty )\\rightarrow \\mathbb{R}^{+}$ be a\nconvex function. Then for every $f\\in A_{\\alpha-2 }^{p}(\\mathbb{D})$ with $%\n\\Vert f\\Vert _{A_{\\alpha-2 }^{p}(\\mathbb{D})}=1$ it holds \n\\begin{equation*}\n\\int_{\\mathbb{D}}G\\left( |f(z)|^{p}(1-|z|^{2})^{\\alpha }\\right)\n(1-|z|^{2})^{-2}dz\\leq \\int_{\\mathbb{D}}G\\left( (1-|z|^{2})^{\\alpha }\\right)\n(1-|z|^{2})^{-2}dz,\n\\end{equation*}%\nand equality  is attained for $f\\equiv 1$.\n\\end{theorem}\n\nThe next theorem is simply a restatement of Kulikov's theorem for a particular choice of $G$, but we\nwould like to highlight this case since this is the first instance of a counterpart\nof Lieb's inequality for the short-time Fourier transform. From the\nexperience in time-frequency analysis, this inequality should have many\napplications (as noticed in the comments after Corollary 1.2 in \\cite{nicola2022norm}, Lieb's inequality is equivalent to sharp norms for localization operators; for applications in signal analysis see \\cite{baraniuk2001measuring} and \\cite{ricaud2014survey}). It will also be essential in the derivation of the local Lieb\nformula in the next section. \n\n\\begin{theorem}[Lieb's inequality for analytic wavelets]\n\\label{thm:Lieb-UP}Let $\\alpha >1$,   \n  $p\\geq 2$, and $f\\in H^2(\\mathbb{C}^+)$. Then\n\\begin{equation}\n{\\Vert W_{\\psi _{0}^{\\alpha }}f\\Vert _{ {L}^{p}(\\mathbb{C}^{+})}^{p}}%\n\\leq \\frac{4\\pi  }{ {  (\\alpha  +1 )p\/2-1}}\\Vert f\\Vert_{H^2(\\mathbb{C}^+) }^{p},\n\\end{equation}\nwith the inequality being sharp.\n\\end{theorem}\n\n\\noindent\\textbf{Proof:\\ } If we set $\\alpha =  {\\beta +1}$ and define \n\\begin{equation*}\nF(z)=s^{-(\\beta \/2+1)}\\left( \\Vert W_{\\psi _{0}^{ {\\beta +1}}}f\\Vert _{%\n{L}^{2}(\\mathbb{C}^{+})}\\right) ^{-1}\\ W_{\\psi _{0}^{ {\\beta +1}}}f(z),\n\\end{equation*}%\nthen $F$ is analytic and $\\Vert F\\Vert _{{A}^{2}_\\beta(\\mathbb{C}%\n^{+})}=1$. Moreover, taking $G(t)=t^{p\/2}$ (which is convex for $p\\geq 2$%\n) in Theorem~\\ref{thm:kulikov} and applying \\eqref{eq:change-of-variable}\nshows \n\\begin{align*}\n& \\frac{\\Vert W_{\\psi _{0}^{{\\beta +1}}}f\\Vert _{ {L}^{p}(\\mathbb{C}%\n^{+})}^{p}}{\\Vert W_{\\psi _{0}^{ {\\beta +1}}}f\\Vert _{{L}^{2}(%\n\\mathbb{C}^{+})}^{p}}=\\int_{\\mathbb{C}^{+}}|F(z)|^{p}s^{\\beta p\/2+p-2}d\\mu_{\\mathbb{C}^+}(z) \\\\\n& \\hspace{1cm}=4\\left( \\frac{\\beta +1}{4\\pi }\\right) ^{\\frac{p}{2}}\\int_{%\n\\mathbb{D}}\\left( \\frac{4\\pi }{\\beta +1}|F(T(u))|^{2}|1-u|^{-(2\\beta\n+4)}(1-|u|^{2})^{\\beta +2}\\right) ^{\\frac{p}{2}}(1-|u|^{2})^{-2}du \\\\\n& \\hspace{1cm}\\leq 4\\left( \\frac{\\beta +1}{4\\pi }\\right) ^{\\frac{p}{2}}\\int_{%\n\\mathbb{D}}(1-|u|^{2})^{\\beta p\/2+p-2}du=4\\pi \\left( \\frac{\\beta +1}{4\\pi }%\n\\right) ^{p\/2}\\int_{0}^{1}r^{\\beta p\/2+p-2}dr \\\\\n& \\hspace{1cm}=\\frac{4\\pi }{p(\\beta \/2+1)-1}\\left( \\frac{\\beta +1}{4\\pi }%\n\\right) ^{p\/2}.\n\\end{align*}%\nNote that we were allowed to apply Theorem~\\ref{thm:kulikov} as \n\\begin{align*}\n\\left\\Vert \\frac{4\\pi }{\\beta +1}F(T(u))^{2}(1-u)^{-(2\\beta +4)}\\right\\Vert\n_{A_{\\beta }^{1}(\\mathbb{D})}& =4\\int_{\\mathbb{D}}|F(T(u))|^{2}|1-u|^{-(2%\n\\beta +4)}(1-|u|^{2})^{\\beta }du \\\\\n& =\\int_{\\mathbb{C}^{+}}|F(z)|^{2}s^{\\beta }dz=1.\n\\end{align*}%\nSubstituting back $\\alpha =  {\\beta +1}$ and using $\\Vert W_{\\psi }f\\Vert _{{L}^{2}(\\mathbb{C}^{+})}^{2}=\\|f\\|^2_{H^2(\\mathbb{C}^+)}\\|\\psi\\|_{L^2(\\mathbb{R}^+,t^{-1})}^2$ yields \n\\begin{align*}\n\\Vert W_{\\psi _{0}^{\\alpha }}f\\Vert _{{L}^{p}(\\mathbb{C}^{+} )}^{p}&\n\\leq \\frac{4\\pi  }{ {p (\\alpha  +1 )\/2-1}}\\left( { \\frac{\\alpha}{4\\pi }}\\right)\n^{p\/2}\\Vert f\\Vert_{H^2(\\mathbb{C}^+)} ^{p}\\big\\| \\widehat{\\psi _{0}^{\\alpha }}\\big\\| _{L^{2}(%\n\\mathbb{R}^{+},t^{-1})}^{p} \\\\\n& =\\frac{4\\pi  }{ {p (\\alpha  +1 )\/2-1}} \\Vert f\\Vert ^{p}_{H^2(\\mathbb{C}^+)},\n\\end{align*}%\nwhere we used that $\\big\\| \\widehat{\\psi _{0}^{\\alpha }}\\big\\| _{L^{2}(\\mathbb{%\nR}^{+},t^{-1})}^{2}=4\\pi\/\\alpha $. \\hfill $\n\\Box $\\newline\n\n\\subsection{Local Lieb's uncertainty principle}\nWe conclude our considerations on sharp concentration inequalities with the following immediate consequence of the results of the last two subsections. This is the wavelet analogue of Theorem 5.2 in \\cite{Inventiones}, but we provide a more direct proof.\n\\begin{theorem}[Local Lieb's inequality for analytic wavelets]\nLet $\\Delta \\subset \\mathbb{C}^{+}$ be a set of finite measure, $\\alpha>1$, and $p\\geq 2$. For $f\\in H^2(\\mathbb{C}^+)$, it holds\n\\begin{equation*}\n\\frac{\\Vert W_{\\psi_0^\\alpha }f\\cdot \\chi _{\\Delta }\\Vert _{L^p(\\mathbb{C}^+) }^{p}}{\\Vert f\\Vert _{H^2(\\mathbb{C}^+)}^{p}}\\leq\\frac{4\\pi  }{{p (\\alpha  +1 )\/2-1}} {\\left(1-\\left(\n1+\\frac{|\\Delta |_h}{4\\pi}\\right)  ^{1-(\\alpha+1) p\/2}\\right)}.\n\\end{equation*}\n\\end{theorem}\n\\textbf{Proof:\\ } Apply first \\eqref{eq:ramos-tilli-p} and then  Theorem~\\ref{thm:Lieb-UP}.\\hfill$\\Box$\\\\\n\n\n\n\n\\subsection{Discussion of general Lieb inequalities}\n\nThe currently available methods do not seem to provide a Lieb inequality\nwith sharp constants for a general window $\\psi $. The problem can be stated\nas providing the best constant in the inequality%\n\\begin{equation}\n\\Vert W_{\\psi }f\\Vert _{ {L}^{p}(\\mathbb{C}^{+})}^{p}\\leq C(\\psi\n,p)\\Vert f\\Vert ^{p}_{H^2(\\mathbb{C}^+)}\\text{.}  \\label{Lieb}\n\\end{equation}%\nThis seems to be a non-trivial question.  Kulikov's methods are dependent on\nanalytic functions, which, by the results of \\cite{AnalyticWavelet}, restricts\nthe window to a weight times $\\psi _{0}^{\\alpha }$. On the other side, Lieb's \nmethods depend on the optimal constants of Young inequality, which is not possible to conveniently use in this context.\nAn intermediate result\nwould be to obtain \\eqref{Lieb} for the family $\\psi _{n}^{\\alpha }$, which\nstill keeps some properties of the analytic case.\\ We will obtain an\ninequality for $\\Vert W_{\\psi }f\\Vert _{{L}^{p}(\\mathbb{C}^{+})}^{p}$\nusing the information from Theorem \\ref{thm:Lieb-UP}, but our estimates are\nquite rough, and serve only to give an idea of a significant obstruction we\nface, namely the non-unimodularity of the affine group, which does not allow\nto apply Young's inequality in a conventional form (there are versions of Young's inequality for the affine group, see Section 4 in \\cite{klein1978sharp}, but they do not seem to apply to the problem at hand). \n\nFor any non-unimodular locally compact group $\\mathcal{G}$ with left-invarian Haar measure $\\mu_{\\mathcal{G}}$, we show the\nfollowing version of Young's convolution inequality. First, the convolution\nof two functions $F,G:\\mathcal{G}\\rightarrow \\mathbb{C}$ is defined as \n\\begin{equation*}\n(F\\ast G)(z)=\\int_{\\mathcal{G}}F(w)G(w^{-1}\\cdot z)d\\mu_{\\mathcal{G}}(w).\n\\end{equation*}\nIn addition, we define reflection operator $\\mathcal{R}F(z)=F(z^{-1})$.\n\n\\begin{lemma}\n\\label{lem:young} Let $\\mathcal{G}$ be a locally compact group with left\nHaar measure $\\mu$, and $1\\leq p,q,r\\leq \\infty$ be such that $1+\\frac{1}{r}=%\n\\frac{1}{p}+\\frac{1}{q}$. If $F\\in L^p(\\mathcal{G})$, and $G, \\mathcal{R}%\nG\\in L^q(\\mathcal{G})$, then \n\\begin{equation*}\n\\|F\\ast G\\|_r\\leq \\|F\\|_p\\cdot \\max\\{\\|G\\|_q,\\|\\mathcal{R}G\\|_q\\}. \n\\end{equation*}\n\\end{lemma}\n\n\\noindent \\textbf{Proof:\\ } We adopt the classical proof of Young's inequality for\nunimodular groups via the Riesz-Thorin theorem applied to the operator $T_G\nF=F\\ast G$. Let us assume that $G, \\mathcal{R}G\\in L^q(\\mathcal{G})$. First,\nif $r=\\infty$ we note that by H\\\"older's inequality \n\\begin{equation*}\n\\|T_GF\\|_\\infty \\leq \\sup_{z\\in \\mathcal{G}} \\int_{\\mathcal{G}}\n|F(w)\\mathcal{R}G(z^{-1}\\cdot w)|d\\mu_{\\mathcal{G}}(w)\\leq \\|F\\|_p\\|\\mathcal{R}G\\|_q,\\quad \\frac{1}{p%\n}+\\frac{1}{q}=1. \n\\end{equation*}\nOn the other hand, for $r=q$, we have by   \\cite[(20.14)]{hewittross} that $\\|T_G F\\|_{q}\\leq \\|F\\|_{1}\\|  G\\|_{q}$. An application of the \nRiesz-Thorin theorem then completes the proof. \\hfill$\\Box$\\newline\n\nTogether with Theorem \\ref{thm:Lieb-UP} this leads to the following.\n\n\\begin{proposition}\nLet $f\\in H^2(\\mathbb{C}^2)$, $\\psi\\in \\mathcal{C}o L^1(\\mathbb{C}^+)$ be an admissible wavelet. For every $\\alpha >1,$ and every $p\\geq 2$, it holds \n\\begin{align*}\n\\frac{\\Vert W_{\\psi }f\\Vert _{{L}^{p}(\\mathbb{C}^{+})}^{p}}{\\|f\\|^p_{H^2(\\mathbb{C}^+)}}\\leq\n {\\frac{4\\pi  }{p (\\alpha  +1 )\/2-1}\\left( \\frac{\\alpha }{4\\pi}\\right)\n^{p}}\\max\\left\\{ {\\Vert W_{\\psi }{\\psi _{0}^{\\alpha }}\\Vert\n_{{L}^{1}(\\mathbb{C}^{+})}^{p}},\\Vert W_{{{\\psi _{0}^{\\alpha }}}%\n}\\psi \\Vert _{{L}^{1}(\\mathbb{C}^{+})}^{p}\\right\\}. \n\\end{align*}\n\\end{proposition}\n\n\\noindent\\textbf{Proof:\\ }  Let us  apply consecutively \\eqref{ortogonalityrelations},  Lemma~\\ref{lem:young},  and  Theorem~\\ref{thm:Lieb-UP}  to show \n\\begin{align*}\n\\Vert W_{\\psi}f \\Vert _{{L}^{p}(\\mathbb{C}^{+})}^{p} &=\\Vert \n\\widehat{\\psi _{0}^{\\alpha }}\\Vert _{L^{2}(\\mathbb{R}^{+},t^{-1})}^{-2p}\n\\int_{\\mathbb{C}^{+}}\\left\\vert \\int_{\\mathbb{C}^{+}}W_{\\psi _{0}^{\\alpha\n}}f(w)\\langle {{\\psi _{0}^{\\alpha }}},\\pi (w^{-1}\\cdot z)\\psi\\rangle d\\mu^+(w)\\right\\vert ^{p}d\\mu^+(z) \n\\\\\n& \\leq  \\Vert \\widehat{\\psi _{0}^{\\alpha }}\\Vert _{L^{2}(\\mathbb{R}%\n^{+},t^{-1})}^{-2p}\\Vert W_{\\psi _{0}^{\\alpha }}f\\Vert _{ {L}^{p}(%\n\\mathbb{C}^{+})}^{p}\\max \\left\\{ \\Vert W_{\\psi}{{\\psi _{0}^{\\alpha }}}\\Vert _{%\n{L}^{1}(\\mathbb{C}^{+} )}^{p},\\Vert W_{{{\\psi _{0}^{\\alpha }}}}\\psi\n\\Vert _{ {L}^{1}(\\mathbb{C}^{+} )}^{p}\\right\\} \n\\\\\n& \\leq  {\\frac{4\\pi  }{p (\\alpha  +1 )\/2-1}\\hspace{-2pt}\\left( \\frac{\\alpha }{4\\pi}\\right)\n^{p}}\\Vert f\\Vert ^{p}_{H^2(\\mathbb{C}^+)}\\max \\hspace{-1pt}\\left\\{ {\\Vert W_{\\psi }{\\psi _{0}^{\\alpha }}\\Vert\n_{{L}^{1}(\\mathbb{C}^{+} )}^{p}},\\Vert W_{{{\\psi _{0}^{\\alpha }}}%\n}\\psi \\Vert _{{L}^{1}(\\mathbb{C}^{+} )}^{p}\\right\\} ,\n\\end{align*}%\nwhere we used that $\\mathcal{R}W_{\\psi}f=\\overline{W_{f}\\psi}$. \\hfill$\\Box$\\\\\n\n\\section{Appendix}\n\n\\subsection{Calculation of Reproducing Kernels and Basis Functions}\n\nThe following integral formula \\cite[p. 809, 7.414 (4)]{grary07} enables us\nto explicitly calculate the wavelet transform $W_{\\psi _{n}^{\\alpha }}$ of\nthe basis elements $\\psi _{m}^{\\alpha }$ of $H^{2}(\\mathbb{C}^{+})$ \n\\begin{align}\n\\int_{0}^{\\infty }e^{-bt}t^{\\alpha }L_{n}^{\\alpha }(\\lambda t)L_{m}^{\\alpha\n}(\\mu t)dt =&\\frac{\\Gamma (m+n+\\alpha +1)}{m!n!}\\frac{(b-\\lambda )^{n}(b-\\mu\n)^{m}}{b^{m+n+\\alpha +1}}  \\notag  \\label{grad-equ} \\\\\n& \\times F\\left( -m,-n;-m-n-\\alpha ;\\frac{b(b-\\lambda -\\mu )}{(b-\\mu\n)(b-\\lambda )}\\right) ,\n\\end{align}%\nwhere $\\mbox{Re}(\\alpha )>-1$, $\\mbox{Re}(b)>0$, and $F={}_{2}F_{1}$ denotes\nthe \\emph{Gauss hypergeometric} function defined by \n\\begin{equation}\nF(a,b;c;z)=\\sum_{k=0}^{\\infty }\\frac{(a)_{k}(b)_{k}}{(c)_{k}k!}z^{k},\n\\label{eq:hypergeom}\n\\end{equation}%\nand $(x)_{k}$ denotes Pochhammer's rising factorial. Moreover, we need the\nfollowing explicit representation of the \\emph{Jacobi polynomials} \\cite[p.\n442, (18.5.7)]{NIST10} \n\\begin{equation}\nP_{n}^{(\\alpha ,\\beta )}(z)=\\frac{\\Gamma (n+\\alpha +1)}{n!\\Gamma (n+\\alpha\n+\\beta +1)}\\sum_{k=0}^{n}\\binom{n}{k}\\frac{\\Gamma (n+k+\\alpha +\\beta +1)}{\n\\Gamma (k+\\alpha +1)}\\left( \\frac{z-1}{2}\\right) ^{k}.  \\label{eq:jacobi}\n\\end{equation}\nThis formula implies the following connection between the hypergeometric\nfunction and the Jacobi polynomials.\n\n\\begin{lemma}\n\\label{lem:jacobi-hypergeo} For $n,m\\in\\mathbb{N}_0$, it holds \n\\begin{align*}  \\label{eq:jacobi-hypergeo}\nF(-n,&-m;-n-m-\\alpha;z) \\\\\n&= \\dfrac{\\min(n,m)!\\Gamma(\\max(n,m)+\\alpha+1)}{\\Gamma(n+m+\\alpha+1)}\n(-z)^{\\min(n,m)}P_{\\min(n,m)}^{(|n-m|,\\alpha)}\\left(1-\\frac{2}{z}\\right).\n\\end{align*}\n\\end{lemma}\n\n\\noindent\\textbf{Proof:\\ } As $F(-n,-m;-n-m-\\alpha;z)=F(-m,-n;-n-m-\\alpha;z)$, we may\nassume that $n\\geq m$. The definition of $F$ and   $\n(-x)_k=(-1)^k(x-k+1)_k $ lead us to\n\\begin{align*}\nF(-n,-m;&-n-m-\\alpha;z)=\\sum_{k=0}^\\infty \\frac{(-n)_k(-m)_k}{\n(-n-m-\\alpha)_k k!}z^k \\\\\n&=\\sum_{k=0}^m \\frac{(n-k+1)_k(m-k+1)_k}{(n+m+\\alpha-k+1)_k k!}(-z)^k \\\\\n&=\\sum_{k=0}^m \\frac{\\Gamma(n+1)\\Gamma(m+1)\\Gamma(n+m+\\alpha-k+1)}{\n\\Gamma(n-k+1)\\Gamma(m-k+1)\\Gamma(n+m+\\alpha+1) k!}(-z)^k \\\\\n&=\\frac{n!}{\\Gamma(n+m+\\alpha+1)}\\sum_{k=0}^m \\binom{m}{k}\\frac{\n\\Gamma(n+m+\\alpha-k+1)}{\\Gamma(n-k+1)}(-z)^k \\\\\n&=\\frac{n!}{\\Gamma(n+m+\\alpha+1)}\\sum_{k=0}^m \\binom{m}{k}\\frac{\n\\Gamma(n+\\alpha+k+1)}{\\Gamma(n-m+k+1)}(-z)^{m-k} \\\\\n&=\\frac{\\Gamma(n+1)}{\\Gamma(n+m+\\alpha+1)}(-z)^m\\sum_{k=0}^m \\binom{m}{k}\n\\frac{\\Gamma(n+\\alpha+k+1)}{\\Gamma(n-m+k+1)}\\left(-\\frac{1}{z}\\right)^{k} \\\\\n&=\\frac{m!\\Gamma(n+\\alpha+1)}{\\Gamma(n+m+\\alpha+1)}(-z)^m\nP_m^{(n-m,\\alpha)}\\left(1-\\frac{2}{z}\\right).\n\\end{align*}\n\\hfill$\\Box$\\newline\n\n\\noindent \\textbf{Proof of Proposition \\ref{lem:explicit-wavelet-trafo}:} Let $z=x+is$ and $w=x^{\\prime }+is^{\\prime }$. By definition of $\n\\psi _{n}^{\\alpha }$, one has \n\\begin{align*}\n\\frac{1}{2^{\\alpha +2}\\pi}& \\sqrt{\\frac{\\Gamma (n+\\alpha +1)\\Gamma (m+\\alpha +1)\n}{n!m!}}\\langle \\pi (w)\\psi _{m}^{\\alpha },\\pi (z)\\psi _{n}^{\\alpha }\\rangle \n\\\\\n& \\hspace{2.5cm}=  {\\frac{1}{2\\pi}}\\sqrt{ss^{\\prime }}\\int_{0}^{\\infty }(s^{\\prime }t)^{\\frac{\\alpha }{2}}e^{-s^{\\prime }t-ix^{\\prime }t}L_{m}^{\\alpha }(2s^{\\prime\n}t)(st)^{\\frac{\\alpha }{2}}e^{-st+ixt}L_{n}^{\\alpha }(2st){dt} \\\\\n& \\hspace{2.5cm}= {\\frac{1}{2\\pi}}(ss^{\\prime })^{\\frac{\\alpha +1}{2}}\\int_{0}^{\\infty\n}t^{\\alpha }e^{it(z-\\overline{w})} L_{n}^{\\alpha }(2st)L_{m}^{\\alpha }(2s^{\\prime\n}t)dt=\\circledast .\n\\end{align*}\nHence, setting $b=-i(z-\\overline{w})$, $\\lambda =2s$, and $\\mu =2s'$ in\nequation \\eqref{grad-equ} yields $b-\\lambda =-i(\\overline{z}-\\overline{w})$, $b-\\mu =-i(z-w)$, $b-\\lambda -\\mu =-i(\\overline{z}-w)$, and consequently \n\\begin{align*}\n\\circledast =& {\\frac{1}{2\\pi}} (ss^{\\prime })^{\\frac{\\alpha +1}{2}}\\frac{\\Gamma (m+n+\\alpha\n+1)}{m!n!}\\frac{(z-w)^{m}(\\overline{z}-\\overline{w})^{n}}{(z-\\overline{w}\n)^{m+n}}\\big(-i(z-\\overline{w})\\big)^{-\\alpha -1} \\\\\n& \\times F\\left( -m,-n,-m-n-\\alpha ,\\left\\vert \\frac{z-\\overline{w}}{z-w}\n\\right\\vert ^{2}\\right) .\n\\end{align*}\nSetting $n=m$ gives \\eqref{eq:kernel-expl} by Lemma~\\ref{lem:jacobi-hypergeo}. Moreover, taking $w=i$ and applying Lemma~\\ref{lem:jacobi-hypergeo} again yields \\eqref{eq:explicit-wavelet}. \\hfill $\\Box \n$\\newline\n\n\n\n\\subsection{Integrability}\\label{app:int}\n\nIt remains for us to show that the mother wavelets $\\psi _{n}^{\\alpha }$\nare admissible for the integrable representation $\\pi $ of the $ax+b$ group.\n\n\n\n\\begin{proposition}\n\\label{prop:integrable} Let $n\\in \\mathbb{N}_0$, and  $\\alpha >1$. Then \n\\begin{equation}\n\\int_{\\mathbb{C}^{+}}\\big|\\langle \\psi _{n}^{\\alpha },\\pi (z)\\psi\n_{n}^{\\alpha }\\rangle \\big|d\\mu^+(z)<\\infty .  \\label{eq:integrable}\n\\end{equation}\n\\end{proposition}\n\n\\noindent\\textbf{Proof:\\ } By  \\eqref{eq:aux-moebius}, we have \n\\begin{equation*}\n\\big|W_{\\psi _{n}^{\\alpha }}\\psi _{n}^{\\alpha }(T(u))\\big|=|u|^{2n}\\left( \n\\frac{1-|u|^{2}}{4}\\right) ^{\\frac{\\alpha +1}{2}}|Z_{n,n}^\\alpha(|u|^{2})|.\n\\end{equation*}%\nConsequently, by \\eqref{eq:change-of-variable} and \\eqref{eq:def-Phi_nm}\n\\begin{align*}\n\\int_{\\mathbb{C}^{+}}\\big|W_{\\psi _{n}^{\\alpha }}\\psi _{n}^{\\alpha }(z)\\big|\n\\frac{d\\mu_{\\mathbb{C}^+}(z)}{s^{2}}& = \\int_{\\mathbb{D}}\\big|W_{\\psi _{n}^{\\alpha }}\\psi\n_{n}^{\\alpha }(T(u))\\big|\\frac{4du}{(1-|u|^{2})^{2}} \\\\\n& =4^{-\\frac{\\alpha -1}{2}}\\int_{\\mathbb{D}}|u|^{2n}|Z\n_{n,n}^\\alpha(|u|^{2})|(1-|u|^{2})^{\\frac{\\alpha -3}{2}}du \\\\\n& \\lesssim \\int_{\\mathbb{D}}(1-|u|^{2})^{\\frac{\\alpha -3}{2}}du=\\pi\n\\int_{0}^{1}r^{\\frac{\\alpha -3}{2}}dr.\n\\end{align*}\nThe last integral is finite if and only if $\\alpha >1$. \\hfill $\\Box $\n\\newline\n\n\\section*{Acknowledgements}\n\nThe authors ackowledge the support of the Austrian Science Fund (FWF)\nthrough the projects P 31225-N32 (L.D. Abreu) and J-4254, and Y-1199 (M. Speckbacher).\n\n\\section*{References}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and summary}\nProperties like energy, linear momentum, and angular momentum can be transferred from the electromagnetic field to material systems during light-matter interactions. These exchanges are governed by conservation laws that apply to the combined system of field and matter. These joint conservation laws are, arguably, among the most important theoretical principles in electrodynamics. Some of them have been put to practical use for some time now. For example, the exchange of linear momentum and angular momentum give rise to optical forces and torques. This allows the optical manipulation of objects, like in optical trapping and optical tweezers applications. The practical use of other less well known conservation laws is being investigated. For example, the helicity conservation law is being considered in the context of chiral light-matter interactions \\cite{Nieto2015,Nienhuis2016,Poulikakos2016,Zambrana2016b}. The effects of the conservation laws of the two transverse components of angular momentum are also under scrutiny \\cite{Andrews2004,Albaladejo2009,Iglesias2011,Nieto2015b,Nienhuis2016}. One of the areas of application of these less well known conservation laws is envisioned to be the manipulation of chiral matter with chiral light \\cite{Hakobyan2014,Tkachenko2014,Cameron2014,Tkachenko2014b,Canaguier2013,Hayat2015,Rahimzadegan2016b}.\n\nIn this article, and motivated by the growing number of interesting conservation laws, we develop a unified theory for the analysis and engineering of the transfer of properties from the electromagnetic field to material systems. Any measurable property can be treated in the same way. \n\nIn Sec. \\ref{sec:setting} we introduce the setting and mathematical tools that we use in the article. We also outline and discuss the assumptions that we make. In Sec. \\ref{sec:absvsasym} we identify a structure underlying all conservation laws: The total transfer of any measurable property can be divided into a part due to absorption of the field by the object and a part due to asymmetry of the object. One is zero if the object is non-absorbing. The other is zero if the object has the symmetry related to the property in question. We provide the expressions of the operators that determine both kinds of transfer, and discuss them in relation to Noether's theorem \\cite{Noether1918}. After introducing bases for the incoming and outgoing fields in Sec. \\ref{sec:basis}, we obtain a universal equation for the transfer of any property. The form of the equation does not depend on the property. We then derive expressions for the transfer rate per Watt of incoming power in Sec. \\ref{sec:transferrates}, and show their suitability for engineering purposes in Sec. \\ref{sec:engineering}. For example, given an object, the expressions that we obtain allow to compute both the maximum achievable transfer rate per incoming Watt, and the exact incoming field which achieves such upper bound. We also discuss other engineering possibilities. In Sec. \\ref{sec:dtransfer} we provide the means to compute optimal incoming fields for chiral sorting. In Sec. \\ref{sec:implementation} we develop the formalism further into a computer friendly formulation and  provide guidance for its implementation using T-matrix algorithms. In Sec. \\ref{sec:examples}, we show two examples of application. In Sec. \\ref{sec:example1}, the transfer rates of different quantities between a plane-wave and a complicated composite object are rigorously obtained. We analyze the transfer of energy, linear and angular momentum, helicity, and transverse ``spin angular momentum''. We quantitatively show the rate of transfer due to absorption and that due to asymmetry in each case, and relate the latter to the symmetries of the object. We then show a comparison between the force and helicity transfer under plane-wave illumination, and the maximum that can be achieved in each case with a monochromatic beam. Section \\ref{sec:example2} contains an example of application of the chiral sorting theory developed in Sec. \\ref{sec:dtransfer}. Finally, Sec. \\ref{sec:conclusions} contains the conclusions.\n\n\\section{Setting\\label{sec:setting}}\nThe light-matter interaction setting is sketched in Fig. \\ref{fig:scatt_op}. In a first phase depicted in Fig. \\ref{fig:scatt_op}(a), the field and the object do not interact. In a second phase, gray in the figure, they interact during some finite amount of time. In a third phase depicted in Fig. \\ref{fig:scatt_op}(b), the field and the object are again impervious to each other. From the point of view of the object, the electromagnetic field is incoming during the first phase and outgoing during the third phase. \n\nIn this setting, the exact description of what happens during the grayed area is not used. All the information about the interaction processes that has an observable physical meaning is contained in the mapping from the incoming to the outgoing fields. This is the central idea in Heisenberg's scattering matrix formulation of quantum mechanics \\cite[p. 261, p. 321]{Van2007}. It is still relevant in current physical theories (\\cite[\\S 19 p. 314]{Birula1975}\\cite[\\S 1, p. 3]{Berestetskii1982}, \\cite[Chap. 12]{Weinberg1995}, \\cite[Preamble, Chap. 2 p. 45]{Scharf2014}).\n\\begin{figure}[h!]\n\t\\includegraphics[width=\\linewidth]{scatt_first.pdf}\n\t\t\t\t\t\t\t\t\t\t\\caption{ \\label{fig:scatt_op} Successive phases of the light-matter interaction. The bullet-shaped field in the incoming phase (a), and the cloud-plus-bullet-shaped field in the outgoing phase (b) do not interact with the object. The interaction time is confined to the grayed-out region. The object is represented by the bean-shaped drawings.  The scattering operator of the object ($S$) relates the incoming and outgoing fields $\\phioutket=S\\phiinket$.}\n\\end{figure}\n\nLet us now consider a measurable property like energy, momentum, angular momentum, helicity, etc ... subject to a conservation law. We call this generic property $\\Gamma$. We can quantify the transfer of $\\Gamma$ between the field and the object by measuring $\\Gamma$ in the incoming field, measuring $\\Gamma$ in the outgoing field, and subtracting the two. To do so, we must specify how to measure $\\Gamma$ in the incoming and outgoing fields. To this end, we introduce the mathematical tools that we are going to use in this article, namely Hilbert spaces. We will use Dirac's ``$\\langle$bra$|$'' ``$|$ket$\\rangle$'' notation\\footnote{Dirac introduced this notation in quantum mechanics. It is also very convenient for other situations that can be treated using the framework of Hilbert spaces.}.\n\nIn this description, the fields are vectors in the incoming and outgoing Hilbert spaces of transverse solutions to Maxwell's equations: $\\phiinket$ and $\\phioutket$. Classical real fields are conveniently mapped onto the complex Hilbert spaces through their complex representation, which uses only the positive frequencies of the field's harmonic decomposition (see e.g. \\cite[Chap. 3.1]{Mandel1995} or \\cite[Chap. 10.2]{Born1999}).\n\nThe effect of the object on the fields is represented by the linear scattering operator $S$. It maps incoming fields onto outgoing fields: \n\\begin{equation}\n\t\\label{eq:outSin}\n\t\\phioutket=S\\phiinket.\n\\end{equation}\nIn this article, we restrict ourselves to linear scatterers. The property $\\Gamma$ is also represented by a linear operator.\n\nWe will measure the transfer of $\\Gamma$ from light to matter during the interaction by subtracting the integrated value of $\\Gamma$ in the outgoing state from that in the incoming state\\footnote{The integrated value of an operator $X$ for the vector $|\\Psi\\rangle$ is obtained by the scalar product of the vectors $X|\\Psi\\rangle$ and $|\\Psi\\rangle$: $\\langle\\Psi|X|\\Psi\\rangle$. As explained in \\cite[Sec. 5.2]{Birula1996}, \\cite[Sec. 3.2]{FerCorThesis} and \\cite[Sec. III]{FerCor2012b}, the integrated values of operators like energy, momentum, angular momentum, etc ..., computed as $\\langle\\Psi|X|\\Psi\\rangle$, coincide with the results of the typical spatial integrals involving the electromagnetic fields: For example $c\/(4\\pi)\\int d\\mathbf{r}\\ \\mathbf{D}\\times \\mathbf{B}$ for the linear momentum.}. The difference between the two integrated values \n\\begin{equation}\n\t\\label{eq:transfer}\n\t\\DG=\\phiinbra\\Gamma\\phiinket-\\phioutbra\\Gamma\\phioutket\n\\end{equation}\nmust represent the amount of $\\Gamma$ that has been transferred from light to matter during the interaction.\n\nUsing the Hermitian conjugate version of Eq. (\\ref{eq:outSin}) [$\\phioutbra=\\phiinbra S^\\dagger$], Eq. (\\ref{eq:transfer}) can be written as\n\\begin{equation}\n\t\\label{eq:transfer_dev}\n\t\\begin{split}\n\t\t\\DG&=\\phiinbra\\Gamma\\phiinket-\\phiinbra S^{\\dagger}\\Gamma S\\phiinket\\\\\n\t\t   &=\\phiinbra\\Gamma-S^{\\dagger}\\Gamma S\\phiinket.\n\t\\end{split}\n\\end{equation}\nWe now impose that $\\DG$ must be always real for any $\\phiinket$. This forces $\\Gamma-S^{\\dagger}\\Gamma S$ to be a Hermitian operator, which then forces $\\Gamma$ to be a Hermitian operator as well: $\\Gamma=\\Gamma^{\\dagger}$. This is the case for all the conservation laws that we are aware of. For example, it is the case for the most commonly considered conservation laws: Energy, momentum, and angular momentum. It is also the case for the conservation laws of helicity and the two parts of the transverse split of angular momentum $\\mathbf{J}=\\mathbf{\\hat{L}}+\\mathbf{\\hat{S}}$ \\cite{Drummond1999,Drummond2006,Cameron2012,Bliokh2013,Nieto2015,Nieto2015b}. \n\nWe will call the properties for which $\\DG$ is real valued for all $\\phiinket$ {\\em measurable} properties. The motivation for this naming is the following: Any measurement device can be thought of as a material system interacting with the electromagnetic field. The measurement device is triggered by the exchange of properties like energy and momentum. It seems physically justified to restrict the measurable properties of the field to those in which the exchange $\\DG$ is real valued. For example, in the transfer of linear and angular momentum, the restriction corresponds to the physical fact that forces and torques are real valued. \n\nIn the interest of conciseness, we will from now on write {\\em property} as to mean {\\em measurable property represented by a Hermitian operator}.\n\nThe idea of subtracting incoming and outgoing fluxes is used for the linear and angular momentum in \\cite{Nieminen2007,Nieminen2014}. This section generalizes its application to any property.\n\nA note about the applicability of Eqs. (\\ref{eq:transfer})-(\\ref{eq:transfer_dev}) is now in order.\n\\subsection{Steady state\\label{sec:steadystate}}\nThe incoming field depicted in Fig. \\ref{fig:scatt_op} (a) can be referred to as a wave packet. If, instead of well separated wave packets, the illumination consists of a continuous beam like a CW laser, the above methodology can be used to compute the {\\em transfer rate} of $\\Gamma$ (see Sec. \\ref{sec:transferrates}). The transfer rates are meaningful if the system is in a steady state with respect to the incoming and outgoing electromagnetic fluxes. After the continuous beam source is turned on, a steady state is quickly reached for the incoming flux. The same is true for the outgoing flux as long as the response of the object does not change significantly. This last assumption is only going to hold temporarily. For example, the interaction with the field may eventually cause the object to change position and orientation. Then, the expression of the operator $S$ changes accordingly: It must be translated and rotated. Changes that are more substantial than translations and rotations may also happen, for example, if the field deforms the object significantly. In many situations we expect that the time scale in which $S$ changes will be much longer than the time needed for reaching a steady state in the outgoing flux. As long as the two time scales are distinct enough, a piece-wise steady state can be assumed and the methodology applied with a constant $S$. Then, after some period of time, a new $S$ operator may be considered taking into account the changes that the field has produced in the object, and so on. \n\nThe time scale for reaching a steady outgoing flux is set by what has been called interaction time \\cite{Wigner1955,Smith1960}. It also determines the length of the gray region in Fig. \\ref{fig:scatt_op}. It is very often on the order of a few tens of attoseconds \\cite{Pazourek2015}. In extreme cases it may reach tens of nanoseconds when metastable resonant states in the matter are excited by the incoming fields \\cite{Bourgain2013}.\n\\section{Transfer due to absorption versus transfer due to asymmetry\\label{sec:absvsasym}}\nWe now return to Eq. (\\ref{eq:transfer_dev}). We see that when\n\\begin{equation}\n\t\\label{eq:cons}\n\t\\Gamma=S^\\dagger \\Gamma S,\n\\end{equation}\nthe transfer of $\\Gamma$ between light and matter is forbidden. That is: $\\DG=0$ for all possible illuminations $\\phiinket$. \n\nWe will now investigate the relationship between the $\\DG=0$ condition and the symmetries of the object. \n\nOne salient property of Hermitian operators is that their exponentiation generates unitary transformations parametrized by a real number $\\theta$ \n\\begin{equation}\n\t\\label{eq:genT}\n\tT_\\Gamma(\\theta)=\\exp\\left(-i\\theta\\Gamma\\right)=\\sum_{l=0}^{\\infty} \\frac{\\left(-i\\theta\\Gamma\\right)^l}{l!}.\n\\end{equation}\nThese transformations are often physically relevant. For example, energy generates time translations, momentum generates spatial translations, angular momentum generates rotations, and helicity generates electromagnetic duality. The transformations generated by the two parts of the transverse split of the angular momentum $\\mathbf{J}=\\mathbf{\\hat{L}}+\\mathbf{\\hat{S}}$ can be found in \\cite{FerCor2013b}. In particular, $\\mathbf{\\hat{S}}$ generates helicity and frequency dependent translations.\n\nAn object is invariant under a given transformation when the commutator between $T_\\Gamma(\\theta)$ and the scattering operator of the object vanishes: \n\\begin{equation}\n\t\\label{eq:commutT}\n\t[T_\\Gamma(\\theta),S]=T_\\Gamma(\\theta)S-ST_\\Gamma(\\theta)=0.\n\\end{equation}\nWe say that the object has the symmetry $T_\\Gamma(\\theta)$. It can be shown\\footnote{In order to show that Eq. (\\ref{eq:commut}) implies Eq. (\\ref{eq:commutT}), one uses Eq. (\\ref{eq:genT}) and the definition of commutator $[A,B]=AB-BA$. The proof that Eq. (\\ref{eq:commutT}) implies Eq. (\\ref{eq:commut}) can be found in \\cite[Sec. 8.1.1]{Elbaz1998}.} that Eq. (\\ref{eq:commutT}) is equivalent to the vanishing of the commutator between $S$ and the generator $\\Gamma$. \n\\begin{equation}\n\t\\label{eq:commut}\n\t[\\Gamma,S]=0,\\text{ i.e. }\\Gamma S=S\\Gamma.\n\\end{equation}\nFor example, if $S$ commutes with the linear momentum in the $x$ direction ($[P_x,S]=0$), it implies that $[\\exp\\left(-i\\theta P_x\\right),S]=0$. This means that the response of the object has continuous translational symmetry in the $x$ direction, i.e. it is invariant under translations in the $x$ direction.\n\nEquation (\\ref{eq:commut}) is also equivalent\\footnote{Showing that Eq. (\\ref{eq:commut}) implies Eq. (\\ref{eq:eigcons}) is straightforward. To show the converse implication, we use that $\\Gamma$ can  be chosen to be one of the four commuting operators whose eigenvalues characterize a basis of incoming fields, as discussed in Sec. \\ref{sec:basis}. We then expand the incoming field into such basis $\\phiinket=\\sumint_{\\eta,\\gamma}\\ain\\etagammain$ and apply the commutator operator $\\Gamma S-S\\Gamma$ to it. The first term is $\\Gamma S\\phiinket=\\sumint_{\\eta,\\gamma}\\ain\\Gamma\\left(S\\etagammain\\right)=\\sumint_{\\eta,\\gamma}\\ain\\gamma\\left(S\\etagammain\\right)$, where the last equality follows from assuming Eq. (\\ref{eq:eigcons}); note that $S\\etagammain$ is outgoing. The second term is $-S\\Gamma\\phiinket=-S\\sumint_{\\eta,\\gamma}\\gamma\\ain\\etagammain=-\\sumint_{\\eta,\\gamma}\\gamma\\ain\\left(S\\etagammain\\right)$. The sum of the two terms is zero, which means that Eq. (\\ref{eq:commut}) is met.} to the following condition:\n\\begin{equation}\n\t\\label{eq:eigcons}\n\\Gamma\\phiinket=\\gamma_0\\phiinket\\implies\\Gamma\\phioutket=\\gamma_0\\phioutket.\n\\end{equation}\nNamely: If the incoming field is an eigenvector of $\\Gamma$ with eigenvalue $\\gamma_0$, the outgoing field will also be an eigenvector of $\\Gamma$ with the same eigenvalue. This, however, does not necessarily imply that $\\DG$ will be zero. The conditions in Eqs. (\\ref{eq:cons}) and (\\ref{eq:commut}) are not equivalent. It is straightforward to check that they become equivalent if there is neither absorption nor gain in the light-matter interaction \\footnote{In this case, $S$ is unitary, which implies that it is invertible, and that $S^{-1}=S^\\dagger$. Then, if Eq. (\\ref{eq:cons}) is met, we can left multiply it by $S$ to get Eq. (\\ref{eq:commut}). Similarly, left multiplying Eq. (\\ref{eq:commut}) by $S^\\dagger$ results in Eq. (\\ref{eq:cons}).}.\n\nWe conclude that {\\em objects that have the symmetry generated by $\\Gamma$ can only exchange $\\Gamma$ with the field if the interaction is not unitary}. These considerations suggest the possibility of writing $\\DG$ as the sum of two contributions: One that is zero if the object has the symmetry generated by $\\Gamma$, and another one that is zero if the interaction is unitary. We now perform such split, for which we first define an operator related to the (non-)unitary character of $S$.\n\nWhen the interaction is not unitary, the norms of the incoming and outgoing fields need not be equal. We can define an operator $A$ which determines such norm changes:\n\\begin{equation}\n\t\\begin{split}\n\t\t\\phiinbra\\Phi_{\\textrm{in}}\\rangle-\\phioutbra\\Phi_{\\textrm{out}}\\rangle\\duetoeq{eq:outSin}&\\phiinbra I-S^\\dagger S\\phiinket\\\\\n\t\t=&\\phiinbra A \\phiinket,\n\t\\end{split}\n\\end{equation}\nwhere the last equality is the definition:\n\\begin{equation}\n\t\\label{eq:A}\n\tA=I-S^\\dagger S.\n\\end{equation}\nFrom now on, we will assume that there is no gain in the interaction, so that $A$ represents the absorption of the field by the object during the interaction. \n\nTogether, the operators $S$ and $A$ allow for a straightforward interpretation of their role in light-matter interactions. This is achieved by the relationship between the singular value decompositions of the $S$ and $A$ operators established in App. \\ref{app:svd}. A salient fact is that the right(left) singular vectors of $S$ form a basis for the incoming(outgoing) fields. Then, the energy of a general incoming field contained in each right singular vector is split by the object into two parts: One is absorbed by the object, the other goes to the outgoing field, but only into a single outgoing mode: The corresponding left singular vector. The ratio between re-radiated and absorbed energy is $(d_l)^2\/(1-d_l^2)$, where $d_l$ is the corresponding singular value, which is always a real non-negative number. Note that no gain implies $d_l\\le 1$, and neither gain nor loss implies $d_l=1$. Along these lines, we can also interpret $(1-d_l^2)\/(d_l)^2$ as the proportion of energy that changes its character from propagating to localized.\n\nTo advance towards the split, we perform two manipulations of Eq. (\\ref{eq:transfer_dev}). In the first one, we substitute $\\Gamma S=S\\Gamma-\\left[S,\\Gamma\\right]$:  \n\\begin{equation}\n\t\\label{eq:dgabssym1}\n\t\\begin{split}\n\t\t\\DG=&\\phiinbra\\Gamma-S^{\\dagger}\\dboxed{\\Gamma S}\\phiinket\\\\\n\t\t\t&=\\phiinbra\\Gamma-S^{\\dagger}\\dboxed{\\left(S\\Gamma-\\left[S,\\Gamma\\right]\\right)}\\phiinket\\\\\n\t\t&=\\phiinbra\\Gamma-S^{\\dagger}S\\Gamma+S^{\\dagger}\\left[S,\\Gamma\\right]\\phiinket\\\\\n\t\t&=\\phiinbra\\left(I-S^{\\dagger} S\\right)\\Gamma\\phiinket+\\phiinbra S^{\\dagger}\\left[S,\\Gamma\\right]\\phiinket\\\\\n\t\t&=\\phiinbra A\\Gamma\\phiinket+\\phiinbra S^{\\dagger}\\left[S,\\Gamma\\right]\\phiinket.\\\\\n\t\\end{split}\n\\end{equation}\nIn the second one we substitute $S^{\\dagger}\\Gamma=\\Gamma S^{\\dagger}+[S^\\dagger,\\Gamma]=\\Gamma S^{\\dagger}-[S,\\Gamma]^\\dagger$, where the last equality holds because $\\Gamma=\\Gamma^\\dagger$:\n\n\\begin{equation}\n\t\\label{eq:dgabssym2}\n\t\\begin{split}\n\t\t\\DG=&\\phiinbra\\Gamma-\\dboxed{S^{\\dagger}\\Gamma} S\\phiinket\\\\\n\t\t\t&=\\phiinbra\\Gamma-\\dboxed{\\left(\\Gamma S^{\\dagger}-[S,\\Gamma]^\\dagger\\right)}S\\phiinket\\\\\n\t\t&=\\phiinbra\\Gamma-\\Gamma S^{\\dagger}S+[S,\\Gamma]^{\\dagger}S\\phiinket\\\\\n\t\t&=\\phiinbra\\Gamma\\left(I-S^{\\dagger} S\\right)\\phiinket+\\phiinbra [S,\\Gamma]^{\\dagger}S\\phiinket\\\\\n\t\t&=\\phiinbra \\Gamma A\\phiinket+\\phiinbra [S,\\Gamma]^{\\dagger}S\\phiinket.\\\\\n\t\\end{split}\n\\end{equation}\n\nWe now add the last lines of Eqs. (\\ref{eq:dgabssym1}) and (\\ref{eq:dgabssym2}) and divide by 2:\n\\begin{equation}\n\t\\label{eq:dgabssym}\n\t\\begin{split}\n\t\\DG&=\\phiinbra \\Dabs \\phiinket+\\phiinbra \\Dasy \\phiinket\\\\\n\t\t&=\\DG_{\\mathrm{absorption}}+\\DG_{\\mathrm{asymmetry}}.\n\t\\end{split}\n\\end{equation}\n\nThe total $\\DG$ splits into two additive parts, one featuring $A$ and the other one featuring the commutator $[S,\\Gamma]$. Both parts are the integrated value of a Hermitian operator, which renders them real valued and separately measurable. Additionally, when $\\phiinket$ is an eigenstate of $\\Gamma$, the two parts are independent of each other in the following sense. For a fixed $\\phiinket$, changing the absorption without changing the commutator ($[S,\\Gamma]$) affects $\\DG_{\\mathrm{absorption}}$ but not $\\DG_{\\mathrm{asymmetry}}$. Similarly, changing the commutator without changing the absorption affects $\\DG_{\\mathrm{asymmetry}}$ but not $\\DG_{\\mathrm{absorption}}$.\n\n\\subsection{Relationship with Noether's theorem}\nWe can connect Eq. (\\ref{eq:dgabssym}) with Noether's theorem about symmetries and conserved quantities \\cite{Noether1918}. We first need to particularize Eq. (\\ref{eq:dgabssym}) to the case of zero absorption ($A=0$) because Noether's theorem does not apply to dissipative systems. In this case, $\\DG_{\\mathrm{absorption}}=0$, and Eq. (\\ref{eq:dgabssym}) reduces to $\\DG=\\phiinbra\\Dasy\\phiinket$, which contains Noether's theorem in the following sense: The existence of a symmetry of the system ($[S,\\Gamma]=0$) implies the existence of a conserved quantity ($\\DG=0$). Notably, Eq. (\\ref{eq:dgabssym}) applies to dissipative systems as well. When $A\\neq 0$, the term $\\phiinbra \\Dabs\\phiinket$ is the extra component in $\\DG$ due to absorption, which is not covered by Noether's theorem.\n\nWe now aim to use the expressions obtained so far for analyzing and engineering the transfer of properties between light and matter. To such end, it is convenient to introduce basis sets for the incoming and outgoing fields.\n\\section{A universal formula, transfer rates, engineering bounds, and differential transfer\\label{sec:basis}}\nWe now choose orthonormal bases sets where the eigenvalue of $\\Gamma$ is one of the four defining numbers of the basis vectors\\footnote{Each vector of an orthonormal basis of incoming(outgoing) transverse Maxwell fields has four numbers that identify it. These four numbers are the eigenvalues of four commuting operators. For example, multipolar fields of well-defined parity are eigenvectors of the angular momentum squared, the angular momentum along one axis, the energy (frequency), and the parity operator. Plane-waves can be chosen as eigenstates of the three components of linear momentum, which fixes the frequency, and a polarization descriptor, for example, the helicity operator. The superscripts in  $\\etagammainout$ denote the incoming(outgoing) character of the fields; {\\em they do not imply the relationships} $\\etagammaout=S\\etagammain$.}. We denote by $\\etagammainout$ the basis vectors, where $\\eta$ is a collective index containing the other three numbers. The incoming and outgoing states can be decomposed into such bases\n\\begin{equation}\n\t\\label{eq:decomp}\n\t\t\\phiinket=\\sumint_{\\eta,\\gamma}\\ain\\etagammain,\\\n\t\t\\phioutket=\\sumint_{\\eta,\\gamma}\\aout\\etagammaout,\n\\end{equation}\nwhere $\\ain$ and $\\aout$ are complex numbers, and $\\sumint_{\\eta,\\gamma}$ indicates that there can be summations over discrete indexes like the eigenvalues of angular momentum, or integrals over continuous variables like the eigenvalues of linear momentum or of $\\mathbf{\\hat{S}}$.\n\nIn this basis, the incoming and outgoing fields are represented by column vectors containing their coordinates\n\\begin{equation}\n\t\\label{eq:vecs}\n\\ainvec\\equiv\\begin{bmatrix}\\vdots\\\\\\ain\\\\\\vdots\\end{bmatrix},\\ \\aoutvec\\equiv\\begin{bmatrix}\\vdots\\\\\\aout\\\\\\vdots\\end{bmatrix},\n\\end{equation}\nand the operators $S$ and $\\Gamma$ are represented by matrices whose elements are\n\\begin{equation}\n\t\\label{eq:mats}\n\t\\begin{split}\n\t\t\\Smat\\left(\\eta\\gamma,\\bar{\\eta}\\bar{\\gamma}\\right)&=\\baretagammaout S \\etagammain,\\\\\n\t\t\\Gammamat\\left(\\eta\\gamma,\\bar{\\eta}\\bar{\\gamma}\\right)&=\\baretagammainout\\Gamma\\etagammainout.\\ \n\\end{split}\n\\end{equation}\nfor all $\\left(\\eta,\\gamma,\\bar{\\eta},\\bar{\\gamma}\\right)$. The column vectors and the matrices have infinite dimension. Since $\\Gamma\\etagammainout=\\gamma\\etagammainout$, the matrix $\\Gammamat$ is diagonal in the chosen basis:\n\\begin{equation}\n\t\\label{eq:gammadiag}\n\t\\Gammamat\\left(\\eta\\gamma,\\bar{\\eta}\\bar{\\gamma}\\right)=\\gamma\\delta_{\\gamma\\bar{\\gamma}}.\n\\end{equation}\n\nOne benefit that we obtain from this choice of basis is a formula for $\\DG$ whose form is the same for any $\\Gamma$.\n\nUsing Eqs. (\\ref{eq:decomp}) to (\\ref{eq:gammadiag}), we can write the following expressions for Eqs. (\\ref{eq:transfer}) and (\\ref{eq:transfer_dev}) \n\\begin{eqnarray}\n\t\\DG&=&\\ainvec^{\\dagger}\\Gammamat\\ainvec-\\aoutvec^{\\dagger}\\Gammamat\\aoutvec\\\\\n\t\\label{eq:dgsm}\n\t&=&\\ainvec^{\\dagger}\\left(\\Gammamat-{\\Smat}^\\dagger \\Gammamat \\Smat\\right)\\ainvec\\\\\n\t\\label{eq:dgs}\n\t&=&\\sumint_{\\eta,\\gamma} \\gamma\\left(|\\ain|^2-|\\aout|^2\\right).\n\\end{eqnarray}\n\nThroughout the article, the appropriate differentials from $\\sumint_{\\eta,\\gamma}$ are implicitly taken into account in the operations with vectors and matrices.\n\nThe form of Eqs. (\\ref{eq:dgsm}) and (\\ref{eq:dgs}) is universal in the following sense. The total transfer of a property $\\Gamma$ during light-matter interactions can always be written in this form: One only needs to choose an orthonormal basis where the eigenvalue of $\\Gamma$ is one of the four defining numbers. For example, when we particularize Eq. (\\ref{eq:dgs}) to the $z$ component of the torque on a sphere ($\\Gamma=J_z$), and use the basis of multipolar fields of well-defined parity, we recover the remarkably simple formulas that were obtained in \\cite{Barton1989,Farsund1996} by different means\\footnote{Since $\\Gamma=J_z$, one can use the multipolar basis of well-defined $J_z$. The starting point is Eq. (\\ref{eq:dgt}), which is the $\\mat{T}$ matrix version of Eq. (\\ref{eq:dgs}) (see App. \\ref{app:ST} for more details). One then uses the fact that the $\\mat{T}$ matrix of a sphere is diagonal in this basis, and its elements are the Mie coefficients. Then, the simple relationship between the coefficients of the incident and scattered fields can be used to recover the formula for the $z$ component of the torque \\cite{Barton1989,Farsund1996}.}.\n\n\n\\subsection{Transfer rates\\label{sec:transferrates}}\nGiven an incoming field $\\ainvec$, we can use Eqs. (\\ref{eq:dgsm}) or (\\ref{eq:dgs}) to compute the transfer of $\\Gamma$ between light and matter. We now obtain the expression for the transfer rate of $\\Gamma$ per Watt of electromagnetic power in the incoming field.\n\nWe start by computing the integrated energy of the incoming field, which, in its abstract form reads: \n\\begin{equation}\n\tE_{\\textrm{in}}=\\phiinbra H \\phiinket,\n\\end{equation}\nwhere $H$ is the energy operator, which can be represented by $i\\hbar\\partial_t$, where $\\hbar$ is Planck's constant divided by $2\\pi$.\n\nLet us now chose a basis where frequency is well-defined. The energy in Joules of a given incoming field $\\ainvec$ can be then computed as\n\\begin{equation}\n\t\\label{eq:H}\n\tE_{\\textrm{in}}={\\ainvec}^{\\dagger}\\mat{H}\\ainvec,\n\\end{equation}\nwhere $\\mat{H}$ is a diagonal matrix whose elements are $\\hbar\\omega_l$, and $\\omega_l$ are the definite frequencies of the vectors of the basis. For example, if we consider a monochromatic field, then $\\omega_l=\\omega$ for all $l$ and the result of Eq. (\\ref{eq:H}) is $\\hbar\\omega{\\ainvec}^{\\dagger}\\ainvec$.\n\nThe expression\n\\begin{equation}\n\\label{eq:perjoule}\n\\frac{{\\ainvec}^\\dagger\\left(\\Gammamat-{\\Smat}^\\dagger \\Gammamat \\Smat\\right)\\ainvec}{{\\ainvec}^{\\dagger}\\mat{H}\\ainvec}\n\\end{equation}\nis hence the transfer of $\\Gamma$ between the field and the matter per Joule of energy of the incoming field. If the source of the incoming field $\\ainvec$ has $W$ Joules\/second (Watts) of power, the rate of transfer of $\\Gamma$ $\\left(d\\Gamma\/d t\\right)$ would be $W$ times the quantity in Eq. (\\ref{eq:perjoule}). For $W=1$ we have hence that \n\\begin{equation}\n\\label{eq:transferrate}\n\\boxed{\n\\frac{d\\Gamma}{d t}=\\frac{{\\ainvec}^\\dagger\\left(\\Gammamat-{\\Smat}^\\dagger \\Gammamat \\Smat\\right)\\ainvec}{{\\ainvec}^{\\dagger}\\mat{H}\\ainvec}\n}\n\\end{equation}\nis the transfer rate per incoming Watt of electromagnetic power.\n\nWe can now use Eq. (\\ref{eq:dgabssym}) to split the contributions related to absorption and asymmetry:\n\\begin{equation}\n\\label{eq:transferratesplit}\n\\frac{d\\Gamma}{dt}=\\stackrel{{\\text{absorption}}}{\\boxed{\\frac{{\\ainvec}^\\dagger\\left(\\Dabsmat\\right)\\ainvec}{2{\\ainvec}^{\\dagger}\\mat{H}\\ainvec}}}+\\stackrel{{\\text{asymmetry}}}{\\boxed{\\frac{{\\ainvec}^\\dagger\\left(\\Dasymat\\right)\\ainvec}{2{\\ainvec}^{\\dagger}\\mat{H}\\ainvec}}}.\n\\end{equation}\n\n\\subsection{Engineering bounds\\label{sec:engineering}}\nEquations (\\ref{eq:transferrate}) and (\\ref{eq:transferratesplit}) are suitable for engineering purposes. For example, let us say that, given an object and a property, we seek the incoming field that maximizes the absolute transfer $\\DG$ per unit power of the illumination. This can be written using Eq. (\\ref{eq:transferrate}) as:\n\\begin{equation}\n\t\\label{eq:max}\n\t\\max_{\\ainvec}\\frac{d\\Gamma}{dt}=\\max_{\\ainvec}\\frac{{\\ainvec}^\\dagger\\left(\\Gammamat-{\\Smat}^\\dagger \\Gammamat \\Smat\\right)\\ainvec}{{\\ainvec}^\\dagger\\mat{H}\\ainvec}.\n\\end{equation}\n\nEquation (\\ref{eq:max}) is a well known optimization problem. Under the conditions of our case\\footnote{Since $\\Gammamat$ is Hermitian, the matrix in the numerator of Eq. (\\ref{eq:max}) is Hermitian. The matrix $\\mat{H}$ is Hermitian and positive definite since the frequencies are always positive: $\\omega_l>0$ for all $l$.}, the maximum is equal to the largest generalized eigenvalue of $\\left(\\Gammamat-{\\Smat}^\\dagger \\Gammamat \\Smat\\right)$ and $\\mat{H}$, and is achieved when the incoming field $\\ainvec_{\\textrm{max}}$ is equal to a generalized eigenvector corresponding to that eigenvalue (see e.g. \\cite{Cheng1965}). Similarly, the incoming field resulting in the minimum transfer rate can be found, together with such minimum rate.\n\nMore sophisticated optimization strategies are also possible. For example, we may want to maximize the transfer rate of $\\Gamma$ but avoid (excessive) absorption by the system. Or we may want to maximize the transfer rate of $\\Gamma_1$ without transferring (excessive) $\\Gamma_2$. These kinds of constrained optimization problems can be written down using the formalism presented in this paper. The design of optimal tractor beams \\cite{Chen2011,Dogariu2013} can also be conveniently achieved using the plane-wave basis. First, we restrict the elements of the vector $\\vect{\\alpha}$ to be non-zero only if they correspond to plane-waves with positive momentum projection along a given axis. Then, the non-zero weights are optimized for the maximum negative force along that axis.\n\n\\subsection{Differential transfer\\label{sec:dtransfer}}\nA particular application of light-matter interaction that is attracting research attention is chiral sorting \\cite{Tkachenko2014,Cameron2014,Tkachenko2014b,Canaguier2013,Hayat2015}. Consider a mixture containing chiral objects and their mirror symmetric versions, like for example a solution of the two enantiomers of a chiral molecule. The objective is to physically separate the two groups using electromagnetic beams. In this section, we address this problem using the theoretical framework outlined so far. We assume that the direct interaction between the incoming beam and each chiral object is much stronger than the mutual interaction between objects, and do not account for the latter. Chiral sorting can be seen as differential transfer rates of linear momentum onto the two versions of the object. We perform the following analysis for a generic property $\\Gamma$ since other forms of differential interaction can be considered, for example inducing different torques \\cite[Sec. V]{Nieto2015b}. \n\nLet us consider a chiral object and its enantiomer. Their scattering operators $\\Smat$ and $\\Smatpar$ can be related by a transformation through the parity operator $\\Pi$: $\\Smatpar=\\mat{\\Pi}\\Smat\\mat{\\Pi}^{-1}$. The differential transfer rate of a property $\\Gamma$ given an incoming field $\\ainvec$ can be obtained using Eq. (\\ref{eq:dgsm}):\n\\begin{equation}\n\t\\label{eq:diffrate}\n\t\\DGS-\\DGSpar={\\ainvec}^\\dagger\\left(\\Smatpar^\\dagger\\Gamma\\Smatpar-\\Smat^\\dagger\\Gamma\\Smat\\right)\\ainvec.\n\\end{equation}\nThe most efficient illumination for differential transfer of $\\Gamma$ is hence (Sec. \\ref{sec:engineering}):\n\\begin{equation}\n\t\\max_{\\ainvec}\\frac{{\\ainvec}^\\dagger\\left(\\Smatpar^\\dagger\\Gamma\\Smatpar-\\Smat^\\dagger\\Gamma\\Smat\\right)\\ainvec}{{\\ainvec}^\\dagger\\mat{H}\\ainvec}.\n\\end{equation}\n\nIn situations like a solution of chiral molecules, the chiral objects are both anisotropic and freely rotating. This random orientation must be taken into account. Assuming that their rotation rates are slow enough so that the steady state conditions in Sec. \\ref{sec:steadystate} apply, the right hand side of Eq. (\\ref{eq:diffrate}) changes into\n\n\\begin{equation}\n\t\\label{eq:diffrateR}\n\t{\\ainvec}^\\dagger\\left[\\left(\\Rmatb\\Smatpar\\Rmatbinv\\right)^\\dagger\\Gamma\\left(\\Rmatb\\Smatpar\\Rmatbinv\\right)-\\left(\\Rmata\\Smat\\Rmatainv\\right)^\\dagger\\Gamma\\left(\\Rmata\\Smat\\Rmatainv\\right)\\right]\\ainvec,\n\\end{equation}\nwhere $\\Rmata$ and $\\Rmatb$ are random rotations. The differential transfer in Eq. (\\ref{eq:diffrateR}) is now a random variable: For fixed $\\ainvec$, its value depends on $\\Rmata$ and $\\Rmatb$. We may compute its average assuming that all the orientations are equiprobable\\footnote{Using obvious definitions to write Eq. (\\ref{eq:diffrateR}) as ${\\ainvec}^\\dagger\\left(\\Mmatb-\\Mmata\\right)\\ainvec$, its statistical average is $\\int d\\Rmata\\int d\\Rmatb C {\\ainvec}^\\dagger\\left(\\Mmatb-\\Mmata\\right)\\ainvec$, where $C$ is a constant that denotes that all orientations are equiprobable. The value of $C=1$ can be deduced from the fact that $\\int d\\Rmata\\int d\\Rmatb C $ must be one, and that $\\int d\\Rmat=1$ (see e.g. \\cite[Eq. 8.2-11]{Tung1985}). Eq. (\\ref{eq:diffrateRav}) is then readily reached using that $\\Mmata$($\\Mmatb$) is independent of $\\Rmatb$($\\Rmata$), and $\\ainvec$ independent of both rotations.}:\n\n{\\small\n\\begin{equation}\n\t\\label{eq:diffrateRav}\n\t{\\ainvec}^\\dagger\\left[\\int d\\Rmat \\left(\\Rmat\\Smatpar\\Rmatinv\\right)^\\dagger\\Gammamat\\left(\\Rmat\\Smatpar\\Rmatinv\\right)-\\left(\\Rmat\\Smat\\Rmatinv\\right)^\\dagger\\Gammamat\\left(\\Rmat\\Smat\\Rmatinv\\right)\\right]\\ainvec,\n\\end{equation}\n}\nwhere each rotation $\\Rmat$ is characterized by its three Euler angles $[\\phi,\\theta,\\psi]$, and $\\int d\\Rmat=\\frac{1}{8\\Pi^2}\\int_0^\\Pi d\\theta\\sin\\theta\\int_0^{2\\pi}d\\phi\\int_0^{2\\pi}d\\psi$.\n\nThe integral in Eq. (\\ref{eq:diffrateRav}) defines a Hermitian matrix $\\mat{M}$. Therefore, the solution of the optimization problem\n\\begin{equation}\n\t\\label{eq:soldiff}\n\\max_{\\ainvec}\\frac{{\\ainvec}^\\dagger\\mat{M}\\ainvec}{{\\ainvec}^\\dagger\\mat{H}\\ainvec}\n\\end{equation}\nproduces both the most efficient incoming beam for the rotationally averaged differential transfer of $\\Gamma$ between the two enantiomeric versions of the object, and the actual value of such upper bound.\n\n\\section{Practical implementation\\label{sec:implementation}}\nIn this section we develop the formalism further into a computer friendly formulation. We address the questions of how to practically obtain the $\\Smat$ matrix of a given object, and how to deal with its theoretically infinite dimension. Both questions are resolved by the T-matrix.\n\nThere exists a convenient practical way of obtaining the scattering matrix $\\Smat$ for the implementation of the proposed methodology. The scattering matrix $\\Smat$ can be numerically obtained as:\n\\begin{equation}\n\t\\label{eq:smattmat}\n\\Smat=\\underline{\\underline{I}}+2\\Tmat,\n\\end{equation}\nwhere $\\mat{I}$ is the identity matrix and $\\Tmat$ is the T-matrix, a.k.a transfer matrix. There is abundant literature on the T-matrix. Reference \\onlinecite{Mishchenko2016} is a comprehensive collection of T-matrix references classified by themes. \n\nThere are also publicly available computer codes for computing the T-matrix for different kinds of objects \\cite{Mishchenko2000,Nieminen2007}. These numerical tools produce finite T-matrices in the multipolar basis of well-defined parity. The dimensions of the matrix are set by the selection of a maximum multipolar order, which must be large enough so that the interaction of the object with the multipolar fields corresponding to the discarded higher orders can be neglected according to some criteria. \n\nNotwithstanding the simple numerical relationship in Eq. (\\ref{eq:smattmat}), there is an important difference between the physical meaning of the two matrices. The $\\Smat$ matrix maps incoming fields $\\ainvec$ to outgoing fields $\\aoutvec$: $\\aoutvec=\\Smat\\ainvec$. The $\\Tmat$ matrix maps incident fields $\\incvec$ to scattered fields $\\scattvec$: $\\scattvec=\\mat{T}\\incvec$. We depict the two different settings in  Figs. \\ref{fig:smatrix} and \\ref{fig:tmatrix}, respectively. While the scattered field $\\vect{\\rho}$ is only outgoing, the incident field $\\vect{\\mu}$ contains both incoming and outgoing parts. This is a physically relevant difference. For example, while measuring the complete scattered field requires interferometric techniques to cancel the outgoing part of the incident field from the total outgoing field \\cite{Husnik2012}, the measurement of incoming and\/or outgoing fields does not. \n\\begin{figure}[h!]\n\t\\begin{center}\n\t\\includegraphics[width=\\linewidth]{in_out.pdf}\n\t\t\\end{center}\n\\caption{\\label{fig:smatrix} After choosing basis sets for the incoming and outgoing fields of Fig. \\ref{fig:scatt_op}, the vector of coordinates of the incoming field $\\vect{\\alpha}$, and that of the outgoing field $\\vect{\\beta}$ are related by the matrix representation of the scattering operator: $\\vect{\\beta}=\\mat{S}\\vect{\\alpha}$. In this setting, the $\\Smat$ matrix connects purely incoming fields to purely outgoing fields.}\n\\end{figure}\n\nThe $\\Smat$ matrix setting is the one which is directly usable for our purpose of subtracting the incoming and outgoing fluxes of some property in order to compute its transfer from the field to the object. Nevertheless, Eq. (\\ref{eq:smattmat}) allows to benefit from existing T-matrix codes. Appendix \\ref{app:ST} contains the numerical relationships between the objects of the two settings, as well as the expressions of $\\DG$ in the T-matrix setting.\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\linewidth]{inc_scatt.pdf}\n\t\t\t\t\t\t\\end{center}\n\\caption{\\label{fig:tmatrix} In the T-matrix setting, the matrix $\\mat{T}$ relates incident fields represented by the coordinate vector $\\vect{\\mu}$ to scattered fields represented by  $\\vect{\\rho}$: $\\vect{\\rho}=\\mat{T}\\vect{\\mu}$. In this setting, the incident fields have a {\\em mixed} incoming and outgoing character, and the scattered field is equal to the total outgoing field ($\\vect{\\beta}$ in Fig. \\ref{fig:smatrix}) {\\em minus} the outgoing part of the incident field (equal to $\\vect{\\mu}\/2$). As explained in the text, the two settings in Figs. \\ref{fig:smatrix} and \\ref{fig:tmatrix} have a tight numerical relationship, but exhibit important differences in their physical interpretation.}\n\\end{figure}\nWe now illustrate the proposed methodology with two examples. \n\n\\section{Examples\\label{sec:examples}}\nFor the first example, we choose a rather complex system in order to showcase the practical capabilities of the approach. In particular, that our methodology is exact, and can be used for any object, including those whose sizes place them outside the ranges where the dipolar approximation or the ray optics approximation are suitable. The object that we choose is chiral and, effectively, spherically symmetric. This kind of objects are often considered in the literature \\cite{Moloney2007,Tkachenko2014b,Canaguier2015,Hayat2015,Fernandes2016}.\n\nFor the second example, we choose an electromagnetically small uniaxial chiral object, with which we illustrate the chiral sorting theory of Sec. \\ref{sec:dtransfer}.\n\n\\begin{table}[]\n\t\\label{tab:mat}\n\t\\begin{ruledtabular}\n\\begin{center}\\begin{tabular}{ccc|cc}\n\t& {$|\\omega \\ j\\ m\\ \\lambda\\rangle^{\\textrm{in(out)}}$}  & & {\\hspace{0.5cm}$|\\omega \\ \\theta\\ \\phi\\ \\lambda\\rangle^{\\textrm{in(out)}}$} & \\\\\n\t\\hline\n\t$H$ & $\\Lambda$ & $J_z$ & $P_z$ & $\\hat{S}_z=\\Lambda P_z\/|\\mathbf{P}|$\\\\\n\t$\\hbar\\omega_l$ & $\\hbar\\lambda_l$ & $\\hbar m_l$ & $\\hbar \\frac{\\omega}{c}\\cos\\theta_l$ & $\\hbar\\cos\\theta_l$\n\\end{tabular}\\end{center}\n\t\\end{ruledtabular}\n\t\\caption{\\label{tab:gammamatrices} Expression of the matrix representation of different operators: Energy $H$, helicity $\\Lambda$, and the third components of angular momentum $J_z$, linear momentum $P_z$, and ``spin angular momentum'' $\\hat{S}_z$. The matrices are diagonal in appropriate bases. For $H$, $\\Lambda$ and $J_z$ we choose the basis of multipoles of well-defined helicity $|\\omega \\ j\\ m\\ \\lambda\\rangle$. For $P_z$ and $\\hat{S}_z$, the basis of plane-waves of well-defined helicity $|\\omega \\ \\theta\\ \\phi\\ \\lambda\\rangle$, where the momentum of the plane-wave is $\\mathbf{p}=\\frac{\\omega}{c}[\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta]$. The last row of the table shows the diagonal elements of the matrices representing each operator in their corresponding bases. The matrices are the same for both incoming and outgoing versions of the bases. See App. \\ref{app:num} for more details.}\n\\end{table}\n\n\\subsection{Example 1\\label{sec:example1}}\nWe consider a material system composed of a CdSe sphere decorated with twenty helical objects in an icosahedral arrangement [see Fig. \\ref{fig:rates}(a)]. The composite object is surrounded by vacuum. This is an example of a chiral object which is practically symmetric under rotations. Spherically symmetric chiral objects are often theoretically and experimentally studied \\cite{Moloney2007,Tkachenko2014b,Canaguier2015,Hayat2015,Fernandes2016}. We highlight that the method by which we obtain the electromagnetic response of our object is free of approximations like the dipole approximation, ray optics, or chiral constitutive material parameters $\\chi\\neq0$.\n\nWhile the expressions obtained in the previous sections allow for wide-band poly-chromatic illuminations, we fix here the operating frequency in order to simplify the example. We will take the monochromatic field as an approximation for a narrowband beam, and describe the object by the $\\Smat$ matrix at the central frequency. This assumes that the response of the object does not substantially change within the bandwidth of the beam. We have chosen the frequency so that this is met. \n\nFrom another point of view, the case of a wide-band poly-chromatic illumination on a time-invariant system\\footnote{Time invariance ensures that the interaction does not couple different frequencies.} can be treated by an integral over many monochromatic illuminations with a frequency dependent $\\Smat$. The calculations for each frequency would then be similar to those presented in this section.\n\nWe will also assume a time invariant $\\Smat$. This can be seen as one of the individual steps in the step wise steady state procedure discussed in Sec. \\ref{sec:steadystate}. \n\nWe will first analyze the transfer rate of several quantities between the material system and a single plane-wave as a function of the radius of the CdSe sphere. The separation between absorption and asymmetry mediated transfers will be considered. Then, we will compare the force and helicity transfer rate per incoming Watt exerted by the single plane-wave to their corresponding upper bounds. \n\nThe calculations are based on Eqs. (\\ref{eq:transferrate}) and (\\ref{eq:transferratesplit}). Appendix \\ref{app:num} contains the detailed explanation of how to obtain all the quantities that appear in these equations, plus a few extra practical implementation notes including indications on how to compute transfer rates of components of $\\mathbf{P}$, $\\mathbf{J}$, and $\\mathbf{\\hat{S}}$, in arbitrary directions. \n\nFigure (\\ref{fig:rates}) shows the transfer rates per incoming Watt of several quantities as a function of the radius of the CdSe sphere. The quantities are: Energy $H$ [Fig. \\ref{fig:rates}(b)], helicity $\\Lambda$ [Fig. \\ref{fig:rates}(c)], and the third components\\footnote{The transfer of the other Cartesian components is comparatively small in our example.} of angular momentum $J_z$ [Fig. \\ref{fig:rates}(d)] (torque), linear momentum $P_z$ [Fig. \\ref{fig:rates}(e)] (force), and ``spin angular momentum'' $\\hat{S}_z$ [Fig. \\ref{fig:rates}(f)] (\\cite{FerCor2013b}). All quantities are expressed in SI units. Table \\ref{tab:gammamatrices} contains the matrix representation of the operators. We have used Eq. (\\ref{eq:transferratesplit}) to separately show the contributions due to absorption and asymmetry.\n\n\\afterpage{\\clearpage}\n\t\\vspace{-0.25cm}\n\t\\begin{figure*}[p]\n\t\t\\subfloat[]{\n\t\t\t\t\t\\begin{overpic}[width=0.46\\textwidth]{dec_sphere.pdf}\n\t\t\t\\put(20,60){{\\small (a)}}\n\\end{overpic}}\n\t\t\\subfloat[]{\\begin{overpic}[width=0.46\\textwidth]{energy.pdf}\n\t\t\\put(25,60){(b)}\n\\end{overpic}}\\\\\n\t\t\\subfloat[]{\\begin{overpic}[width=0.46\\textwidth]{helicity.pdf}\n\t\t\\put(25,60){(c)}\n\\end{overpic}}\n\t\t\\subfloat[]{\\begin{overpic}[width=0.46\\textwidth]{torque.pdf}\n\t\t\\put(25,60){(d)}\n\\end{overpic}}\\\\\n\t\\subfloat[]{\\begin{overpic}[width=0.46\\textwidth]{force.pdf}\n\t\t\\put(25,60){(e)}\n\\end{overpic}}\n\t\\subfloat[]{\\begin{overpic}[width=0.46\\textwidth]{shat.pdf}\n\t\t\\put(25,60){(f)}\n\t\\end{overpic}}\n\\caption{\\label{fig:rates}The object in panel (a) is illuminated by a single monochromatic plane-wave (not shown). The material system is composed of a CdSe sphere decorated with twenty helical objects in an icosahedral arrangement. The plane-wave is of helicity $+1$ (i.e. left handed polarized), vacuum wavelength equal to $222.8$ $\\mu$m, and momentum aligned with the positive $z$ direction. Panels (b) to (f) show the transfer rates of different properties between the field and the object as a function of the CdSe sphere radius: Energy (b), helicity (c), $z$-component of the angular momentum (torque) (d), $z$-component of the linear momentum (force) (e), and $z$-component of the transverse ``spin angular momentum'' (f). All quantities are expressed in SI units. The plots contain the total transfer rates (red lines), the transfer rates due to absorption (green crosses), and the transfer rates due to asymmetry (blue stars). Panel (b) additionally contains the scattering cross-section that the object presents to the illumination (magenta squares).}\n\\end{figure*}\n\nFigure \\ref{fig:rates}(b) also contains the scattering cross-section that the object presents to the illumination. We see in Fig. \\ref{fig:rates}(b) that all the energy transfer is due to absorption. This must be the case because the system is time invariant: It has the time translation symmetry generated by the energy operator. Similarly, the approximate spherical symmetry of the system forces the torque $T_z=dJ_z\/dt$ in Fig. \\ref{fig:rates}(d) to be mostly due to absorption as well, hence the coincidence in shape between $dH\/dt$ and $T_z$ [Figs. \\ref{fig:rates}(b) and (d)]. On the other hand, Fig. \\ref{fig:rates}(c) shows that helicity is mostly transferred due to the broken duality symmetry of the object (\\cite{FerCor2012p}, \\cite[Sec. 2.7]{FerCorThesis}), and absorption plays a lesser role. Figures \\ref{fig:rates}(e) and (f) reveal that both $F_z=dP_z\/dt$ and $d\\hat{S}_z\/dt$ are also mostly due to asymmetry. The object breaks translational symmetry along the $z$ axis, and is susceptible to be pushed by the plane-wave in this direction. The object is also far from being symmetric under the transformations generated by $\\hat{S}_z=\\Lambda\\frac{P_z}{|\\mathbf{P}|}$, which are helicity and frequency dependent {\\em translations}, as can be foreseen by the expression for $\\mathbf{\\hat{S}}=\\Lambda\\mathbf{P}\/\\mathbf{|P|}$, where $\\Lambda$ is the helicity operator and $\\mathbf{P}$ the momentum vector operator \\cite{FerCor2013b}. An example of an object with the symmetry generated by $\\hat{S}_z$ is an infinitely long (in $z$) dual symmetric object with constant cross-section in the XY plane.\n\nThe positions of the common feature of helicity, momentum and $\\hat{S}_z$ transfer in Figs. \\ref{fig:rates}(c), (e), and (f), correlate well with the maximum in the scattering cross-section in Fig. \\ref{fig:rates}(a).\n\nFinally, Fig. \\ref{fig:max_f3} shows the force $F_z$ and helicity transfer $d\\Lambda\/dt$ due to the plane-wave compared to the achievable maxima for the optimal monochromatic illuminations for each sphere radius. The maxima are obtained as indicated in Sec. \\ref{sec:engineering}: They are the largest positive eigenvalue of the generalized eigenvalue decomposition of the matrices $\\Gammamat-\\Smat^{\\dagger}\\Gammamat\\Smat$ and $\\mat{H}$. For the force $\\Gammamat=\\mat{P_z}$, and for the helicity transfer $\\Gammamat=\\mat{\\Lambda}$.\n\n\\begin{figure}[h]\n\t\t\\subfloat[]{\\includegraphics[width=\\linewidth]{max_f3.pdf}}\\\\\n\t\t\\subfloat[]{\\includegraphics[width=\\linewidth]{max_dh.pdf}}\\\\\n\t\t\\caption{\\label{fig:max_f3}The green dashed lines show the force per incoming Watt of power (a) and the helicity transfer (b) for the plane-wave used in Fig. \\ref{fig:rates}. The red solid lines show the upper bounds of the same quantities set by the optimal monochromatic illuminations of the same frequency for each value of the sphere radius.}\n\\end{figure}\n\n\n\\subsection{Example 2\\label{sec:example2}}\n\nThis example is an application of the differential transfer theory of Sec. \\ref{sec:dtransfer}. We will compute the differential force between the two enantiomeric versions of a chiral object: The silver helix in the inset of Fig. \\ref{fig:diff}. In the considered frequency range, the helix is practically a dipolar object: Its dipolar response is at least 500 times larger than all the other higher orders added together. We have studied this object in \\cite[Fig. 5]{FerCor2016}. The elements of its polarizability tensor can be seen in \\cite[Fig. 3]{Rahimzadegan2016b}. The T-matrix of the helix was obtained by rigorous full-wave simulations \\cite{Fruhnert2016b}.\n\nFor each frequency, we compute the rotationally averaged differential force achieved in the $z$ direction by two monochromatic illuminations. One is the optimal beam as defined by Eq. (\\ref{eq:diffrateRav}). The other is composed by the sum of two counter-propagating plane-waves of opposite helicity aligned with the $z$ axis \\cite{Fernandes2016}.  Figure \\ref{fig:diff} shows the performance of the two beams. In both cases, the force has its maximum at the same frequency. This is related with the facts that the helix approaches the upper bound of electromagnetic chirality around that frequency (see \\cite[Fig. 5]{FerCor2016}), and that objects that achieve such upper bound are transparent to one of the helicities of the field.\n\n\\begin{figure}[h]\n\t\t\\includegraphics[width=\\linewidth]{diff_f3_plot.pdf}\n\t\t\\caption{\\label{fig:diff}Difference between the rotationally averaged force exerted onto two chiral objects by two beams: The optimal monochromatic beam for each frequency ($\\delta{F_z}^{\\textrm{bound}}$), and a beam composed by the sum of two monochromatic counter-propagating plane-waves of opposite helicity aligned with the $z$ axis ($\\delta F_z^{\\textrm{CPPW}}$). The chiral objects are the two enantiomeric versions of the silver helix shown in the inset. The dimensions of the helix are: Major radius $a=6.48$ $\\mu$m, height $b=8.52$ $\\mu$m, and wire radius $c=0.8$ $\\mu$m. }\n\\end{figure}\n\n\n\\section{Concluding remarks\\label{sec:conclusions}}\nWe have presented a unified theory to treat conservation laws in light-matter interactions. It can be used to describe and engineer the transfer of any measurable property from the electromagnetic field to any object. The theory allows to explicitly characterize and separately compute the transfer due to asymmetry of the object and the transfer due to field absorption by the object. The two components are separately measurable. Among other general engineering possibilities, the framework also allows to compute the upper bound of the transfer rate of any given property to any given object, together with the corresponding most efficient illumination which achieves the bound. The approach is inherently suited for computer implementation. \n\nThese features make us believe that the approach will be useful in the expanding field of optical manipulation, in particular in chiral light-matter interactions where conservation laws like the one for helicity are envisioned to play an important role.\n\n\\begin{acknowledgments}\nI.F-C wishes to warmly thank Ms. Magda Felo for her help with the figures. We also thank Dr. Martin Fruhnert for providing us with the T-matrix of the silver helix.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSolutions of anisotropic colloids or stiff polymers can exhibit phase\ntransitions from isotropic to orientationally ordered nematic states\nas temperature is lowered or density is increased. Such systems are\noften polydisperse, containing e.g.\\ colloid particles with an\neffectively continuous range of lengths and diameters or polymers with\na distribution of chain lengths. Such polydispersity has\nexperimentally measurable consequences. For example, in polydisperse\ncolloidal rods with nearly hard interactions, demixing into two\nnematic (N) phases and coexistence of two nematics and an isotropic\n(I) phase can occur~\\cite{BuiLek93,VanVanLek96}. This is to be\ncontrasted with the hypothetical case where all colloidal particles\nare identical, i.e.\\ the solution is monodisperse; there at most two\nphases (I--N) can coexist.\n\nFor the theoretical description of phase transitions to nematically\nordered states there are two broad classes of models. The first\nfocusses on short-range repulsions between particles and is therefore\nappropriate for {\\em lyotropic} liquid crystals whose phase changes\nare controlled primarily by density. It contains in particular the\nOnsager model of hard rods, whose second-virial truncation becomes\nexact in the limit of long thin rods~\\cite{Onsager49}, and the Flory\nlattice model~\\cite{Flory56b}. The second class of models concentrates\non the anisotropy of longer-range attractive interactions. It contains\nin particular the Maier-Saupe model, a mean-field theory which expands\nthe orientational interaction dependence in low-order Legendre\npolynomials~\\cite{MaiSau58,HumJamLuc72}. Models of this type are\nuseful for {\\em thermotropics}, where temperature is the dominant\ncontrol parameter.  Theories combining these two approaches have also\nbeen developed, see the review~\\cite{Singh00}, and extended to include\ne.g. particle flexibility~\\cite{KhoSem85}.\n\nFor {\\em lyotropics}, binary (and to lesser extent ternary) mixtures\nhave been thoroughly investigated theoretically using the Onsager and\nFlory models~\\cite{AbeFlo78,LekCouVanDeb84,%\nOdiLek85,BirKolPry88,VroLek93,VanMul96,Hemmer99,Hemmer00,VarSza00,VarGalJac03},\nand phase separation into two nematic (N--N) phases and three-phase\nI--N--N coexistence predicted for sufficiently disparate rod sizes.\nEarly work on binary {\\em thermotropic} mixtures ignored the\npossibility of fractionation, i.e.\\ unequal partitioning of different\nparticle species across coexisting phases. Within this approximation a\nmixture behaves like a monodisperse system, with no I--N coexistence\nregion~\\cite{HumJamLuc71,HumLuc73}. Later studies that accounted for\nfractionation effects did find I--N~\\cite{HumLuc76}\nand N--N~\\cite{PalBerDebDun84} coexistence in bidisperse Maier-Saupe\ntheory. We note as an aside that similar effects have also been seen\nin theories of mixtures of semi-flexible or stiff polymers; see\ne.g.~\\cite{KhoSem85,MaiSix89,HolSch92,ChiKyu95}. However, there a\ndriving force for demixing even between isotropic phases is normally\npresent, e.g.\\ because of unfavourable interactions between unlike\nspecies. N--N demixing then occurs rather trivially when the nematic\norder prevalent at low temperatures is `superimposed' on this\nisotropic demixing transition.\n\n\\iffalse\nComment on N-N in LC polymers, where it's nothing to do with\npolydispersity, just ordinary demixing that would also take place for\nisotropic phases, decorated by nematic ordering.\n[\\cite{ChiKyu95} - bidisperse Flory-Huggins plus Maier-Saupe for\nsemi-flexible polymers; get N-N but really a consequence of\nFlory-Huggins. They also allow for non-factorized interactions (their\nc is 1 for the factorized case). \\cite{HolSch92} use fancier\ntechnology but for rigid-rigid mixtures get similar phase diagrams;\nalso \\cite{MaiSix89}. (also for single species in\\cite{KhoSem85},\nwhere N-N can again occur, but largely a Parsons-Onsager theory with\nadded Flory-Huggins attraction)]\n\\fi\n\n\nTheoretical studies of genuinely polydisperse systems have been rather\nrarer. For distributions of the polydisperse attribute (particle\nlength, diameter etc) with two narrow peaks one of course expects\nphase behaviour similar to the bidisperse\ncase~\\cite{ItoTer84,ItoTer84b}, but it is not at all obvious what\nhappens for unimodal (single-peaked) distributions.  For {\\em\nlyotropics}, work on the polydisperse Flory\nmodel~\\cite{FloAbe78,FloFro78,FroFlo78,MosWil82} focussed on general\nfeatures~\\cite{Sollich02} such as fractionation and the widening of\nthe I--N coexistence region, but did not investigate the possibility of\nN--N demixing. For the Onsager model, primarily polydispersity in\nparticle lengths has been studied. For a narrow distribution of\nlengths, perturbative calculations show that the I--N coexistence\nregion again widens~\\cite{Chen94}. Recent\nstudies~\\cite{WenVro03,SpeSol03b} for wider length distributions have\nconfirmed this and provided evidence for a region of I--N--N\ncoexistence in the phase diagram. However, the existence of N--N\ndemixing at higher density remains an open question; in simplified\nOnsager theories N--N coexistence is found only in a limited density\nrange~\\cite{SpeSol02,SpeSol03a} or not at\nall~\\cite{ClaCueSeaSolSpe00}.\n\nFor polydisperse {\\em thermotropics}, Sluckin used Maier-Saupe theory\nto predict the opening up of an I--N coexistence region in slightly\npolydisperse systems~\\cite{Sluckin89}. This was based on a\nperturbation theory around the monodisperse case, which did not allow\nfor a study of possible N--N demixing. Semenov obtained similar\nresults for thermotropic semi-flexible polymers with small\npolydispersity in the chain lengths~\\cite{Semenov93}.\n\nOur aim in this paper is to find out whether a unimodal distribution\nof the polydisperse attribute is enough to cause N--N demixing and the\nassociated I--N--N coexistence in polydisperse thermotropics. We use\nthe simplest Maier-Saupe theory for this purpose, with factorized\ninteractions. As explained in the next section, this gives a\ntruncatable free energy for which phase equilibria can be calculated\nrelatively efficiently. The resulting phase diagrams are presented in\nSec.~\\ref{sec:results}, and wider implications and possible\ngeneralizations are discussed in Sec.~\\ref{sec:conclusions}.\n\n\n\n\n\\iffalse\nProbably leave out: [Can also get N-N in monodisperse systems from\ncombination of isotropic and anisotropic interactions, see\n\\cite{TeiTel95} - weird single-species model with isotropic hard\nrepulsion, plus soft attractive\/repulsive isotropic interaction $\\sim\nr^{-6}$, plus attractive Maier-Saupe term (also $\\sim r^{-6}$). If the\nsoft isotropic interaction is repulsive, they get N-N coexistence in a\nnarrow T-range. Not very transparent physically.  \\cite{Teixeira97}\nalso sees such effects, though for a different interaction potential.]\n\\fi\n\n\n\\section{Method}\n\nWe consider a system with liquid crystal particles having a continuous\nrange of values of some polydisperse attribute $l$. This could be\nparticle length, for example, or polarizability in the case of van der\nWaals interactions; for definiteness we refer to length below. The\nstate of a single phase of a system is then described by a\ndistribution $n(l,\\Omega)$, defined so that\n$n(l,\\Omega)\\,dl\\,d\\Omega\/4\\pi$ gives the fraction of particles with\nlength in an interval $dl$ around $l$, and orientations $\\Omega$ in a\nsolid angle $d\\Omega$. This distribution is the natural extension for\na polydisperse system of the usual orientational distribution\n$P(\\Omega)$ used for monodisperse systems of rods~\\cite{VroLek92}. The\nrod orientation $\\Omega$ can be parameterized in terms of the angle\n$\\theta$ with the nematic axis and an azimuthal angle $\\varphi$; due\nto the cylindrical symmetry of the nematic phase $n(l,\\Omega)$ is\nindependent of $\\varphi$. Using $d\\Omega = d\\cos\\theta\\ d\\varphi$ the\nlength distribution, obtained by integrating over orientations, is\ntherefore\n\\begin{equation}\\label{eq:normalization_ang}\nn(l)=\\frac{1}{4\\pi}\\int\\! d\\Omega\\ \\rho(l,\\Omega)=\\int\\!\\thint\\,\nn(l,\\theta)\n\\end{equation}\nwhere $\\int\\!\\thint\\, \\ldots = (1\/2)\\int_{-1}^1\nd\\cos\\theta\\ \\ldots$\\ . The orientational distribution of the particles\ncan be factored out from $n(l,\\theta)$ as~\\cite{ClaCueSeaSolSpe00}\n\\begin{equation}\nn(l,\\theta)=\nn(l)P_l(\\theta)\n\\label{eq:decomposition_rho}\n\\end{equation}\nwhere $P_l(\\theta)$ represents the probability of finding a rod of\ngiven length $l$ in orientation $\\Omega=(\\theta,\\varphi)$ and is\nnormalized according to $\\int\\!\\thint\\, P_l(\\theta)=1$. In the isotropic\nphase, one has $P_l(\\theta)\\equiv 1$ and $n(l,\\theta)=n(l)$.\n\n\n\nIn the above notation the free energy per particle of polydisperse\nMaier-Saupe theory~\\cite{Sluckin89} can be written as $f=f_{\\rm{id}}+f_{\\rm{ex}}$,\nwhere the ideal part $f_{\\rm{id}}$ is the free energy of an ideal mixture\nwhile the excess part is\n\\begin{equation}\nf_{\\rm{ex}} = - \\frac{1}{2} \\int\\! dl\\,dl'\\, n(l) n(l') u(l,l') S(l) S(l')\n\\end{equation}\nHere $S(l) = \\int\\!\\thint\\, P_l(\\theta) {\\mathcal P}_2(\\cos\\theta)$ is the\norientational order parameter of particles with length $l$, defined as\nthe average of the second Legendre polynomial\n${\\mathcal P}_2(\\cos\\theta)=\\frac{1}{2}(3\\cos^2\\theta-1)$. The coefficients\n$u(l,l')$ determine the strength of the attractive interaction\nfavouring nematic ordering and depend on the particle lengths $l$ and\n$l'$. We note that the description of the system in terms of a\nnormalized distribution over lengths and orientations rather than a\ndensity distribution contains the implicit assumption that density\nvariations with temperature or between coexisting phases are weak and\ncan be neglected. Accordingly, we will not concern ourselves with the\ndensity dependence of the interaction strengths $u(l,l')$. Maier and\nSaupe~\\cite{MaiSau58} assumed this to be quadratic, but\nCotter~\\cite{Cotter77c} later showed that a thermodynamically\nconsistent derivation of Maier-Saupe theory as a mean-field\napproximation requires a linear density scaling.\n\nIn the following we make the common assumption that the interaction\nstrengths factorize according to $u(l,l')=[u(l,l)u(l',l')]^{1\/2}$. It\nis then sensible to switch from $l$ to $u\\equiv u(l,l)$ as the\npolydisperse attribute, giving for the ideal and excess free energies\n($k_{\\rm B}=1$) \n\\begin{eqnarray}\nf_{\\rm{id}} &=& T\\int\\!du\\,\\thint\\, n(u,\\theta)\\ln n(u,\\theta)\\\\\n&=&T\\int\\! du\\, n(u)\\left[\\ln n(u) + \\int\\!\\thint\\, P_u(\\theta)\\ln\nP_u(\\theta)\\right]\\\\\nf_{\\rm{ex}} &=& - \\frac{1}{2} \\int\\! du\\,du'\\, n(u) n(u') \\sqrt{uu'} S(u) S(u')\n\\end{eqnarray}\nWith our factorized interactions the underlying nature of the\npolydisperse attribute has become irrelevant; polydispersity manifests\nitself only through the spread in the interaction strengths $u$.\n\nFor a given distribution $n(u)$, the orientational distributions are\nobtained by minimization of the free energy with respect to the\norientational distributions $P_u(\\theta)$. Bearing in mind that these\nare normalized, one finds\n\\begin{eqnarray}\n\\label{eq:P_l}\nP_u(\\theta) &=& z^{-1}(u)\\exp\\left(\\beta\\bar{S}\\sqrt{u}{\\mathcal P}_2\\right) \\\\\nz(u) &=& \\int\\!\\thint\\, \\exp\\left(\\beta\\bar{S}\\sqrt{u}{\\mathcal P}_2\\right)\n\\label{eq:z}\n\\end{eqnarray}\nwhere we have abbreviated $\\beta=1\/T$ and ${\\mathcal P}_2\\equiv\n{\\mathcal P}_2(\\cos\\theta)$ and $z(u)$ is the normalizing partition function\nfor $P_u(\\theta)$. The order parameter that appears\nin~(\\ref{eq:P_l},\\ref{eq:z}) is\n\\begin{equation}\n\\bar{S} = \\int\\! du\\, n(u) \\sqrt{u} S(u) = \\int\\!du\\,\\thint\\, n(u,\\theta)\\sqrt{u}{\\mathcal P}_2\n\\end{equation}\nand determines the excess free energy $f_{\\rm{ex}} = -\\bar{S}^2\/2$. It obeys the\nself-consistency equation\n\\begin{equation}\n\\label{eq:rho2_selfconsis}\n\\bar{S} = \\int\\!du\\,\\thint\\, n(u) \n\\frac{\\sqrt{u} \n{\\mathcal P}_2 \\exp\\left(\\beta\\bar{S} \\sqrt{u} {\\mathcal P}_2\\right)}\n{\\int\\!\\thint\\, \\exp\\left(\\beta\\bar{S} \\sqrt{u} {\\mathcal P}_2\\right)}\n\\end{equation}\nThis always has the trivial solution $\\bar{S}=0$ corresponding to an\nisotropic phase; nematic phases are characterized by $\\bar{S}>0$ and become\nphysically relevant where they have a lower free energy than the\nisotropic solution. \n\nTo calculate phase equilibria we need the chemical potentials\n$\\mu(u)$, which need to be equal in coexisting phases. Denote $N(u)$\nthe particle number distribution, so that the total particle number is\n$N=\\int\\! du\\,N(u)$, and $n(u)=N(u)\/N$. Then $\\mu(u) = \\delta F\/\\delta\nN(u)$, where the extensive free energy is $F=Nf[N(u)\/N]$. This gives\n\\begin{equation}\n\\mu(u) = \\frac{\\delta f}{\\delta n(u)} + f - \\int\ndu'\\,n(u')\\frac{\\delta f}{\\delta n(u')}\n\\end{equation}\nIn evaluating the derivatives $\\delta f\/\\delta n(u)$ we do not need to\ntake into account the variation of the $P_u(\\theta)$ because they are\nchosen to minimize $f$. A short calculation then gives $\\delta\nf\/\\delta n(u)=T\\ln(n(u)\/z(u))+T$ and so\n\\begin{equation}\n\\mu(u) = T\\ln\\frac{n(u)}{z(u)} + \\frac{1}{2} \\bar{S}^2\n\\end{equation}\nIf the system separates into $P$ phases $a=1,\\ldots,P$,\ntheir $u$-distributions $n_a(u)$ can therefore be written as\n\\begin{equation}\nn_a(u) = R(u) z_a(u) \\exp(-\\beta\\m_a^2\/2)\n\\end{equation}\nThe common factor $R(u)=\\exp[\\beta \\mu(u)]$ follows from the\nrequirement of particle conservation: if phase $a$ contains a fraction\n$x_a$ of all particles, then $\\sum_a x_a n_a(u) = n_0(u)$, where\n$n_0(u)$ is the overall or ``parent'' distribution of $u$. Using\nalso\\eq{eq:P_l}, the joint distribution over interaction strengths and\norientations in each phase therefore takes the form\n\\begin{equation}\nn_a(u,\\theta) = \n\\frac{n_0(u) \\exp\\left(\\beta\\m_a \\sqrt{u} {\\mathcal P}_2 - \\beta \\m_a^2\/2\\right)}\n{\\sum_b x_b\\int\\!\\thint\\, \\exp\\left(\\beta\\m_b \\sqrt{u} {\\mathcal P}_2 - \\beta \\m_b^2\/2\\right)}\n\\end{equation}\nA phase split involving $P$ phases is therefore characterized by the\n$2P$ variables $\\m_a$ and $x_a$. These are determined by as many\nconditions, namely \n\\begin{eqnarray}\n1   &=& \\int\\!du\\,\\thint\\, n_a(u,\\theta)\n\\label{na-norm}\n\\\\\n\\m_a &=& \\int\\!du\\,\\thint\\, n_a(u,\\theta)\\sqrt{u}{\\mathcal P}_2\n\\end{eqnarray}\nThe relations\\eq{na-norm} automatically guarantee that $\\sum_a x_a=1$\nbecause the parent $n_0(u)$ is normalized by assumption.\n\n\nIn our numerical work, we exploit the fact that the excess free energy\n$f_{\\rm{ex}} = -\\bar{S}^2\/2$ is truncatable~\\cite{SolWarCat01}, i.e.\\ it involves\nonly a single moment $\\bar{S}$ of the distribution $n(u,\\theta)$. Phase\nequilibria can then efficiently be found using the moment free energy\n(MFE) method~\\cite{SolCat98,Warren98,SolWarCat01}. This constructs a\nMFE which depends only on $\\bar{S}$, and from which phase coexistence can\nbe predicted in the standard manner. In this\nsimplest form the MFE gives only the onset of phase separation\nexactly. However, by including additional moments, defined by\nadaptively chosen weight functions, increasingly accurate solutions\nare obtained. Using these as initial points, we are then able to find\nthe, for our free energy, exact solutions of the phase equilibrium\nequations~\\cite{ClaCueSeaSolSpe00,SpeSol02,SpeSol03a}. We omit the\ndetails of the implementation because they are similar to our study of\nan approximate Onsager model~\\cite{SpeSol02,SpeSol03a}. Differences\narise primarily because, in the Maier-Saupe model, we are describing\nthe system in terms of normalized distributions, as would be\nappropriate in other systems for the case of fixed pressure rather\nthan fixed volume; the appropriate modifications of the MFE approach\nare described in~\\cite{SolWarCat01}.\n\n\n\\section{Results}\n\\label{sec:results}\n\n\nWe concentrate on the example case of a Schulz parent distribution of\ninteraction strengths, given by\n\\begin{equation}\nn_0(u) \\propto u^z e^{-(z+1)u}\n\\end{equation}\nThe coefficient in the exponent has been chosen such that the mean\ninteraction strength $\\langle u \\rangle_0$ is fixed to unity, thus\nsetting our energy and temperature scale. The width of the\ndistribution is characterized by its standard deviation divided by its\nmean. This quantity, denoted $\\delta$ below, is often also referred to\nsimply as the polydispersity and is related to $z$ by $\\delta =\n(1+z)^{-1\/2}$. To avoid interaction strengths which are arbitrarily\nlarge compared to the mean, we impose an upper cutoff of $u_{\\rm max}\n= 4$. This affects the mean value of $u$ only negligibly in the range\nof $\\delta$ of interest to us, while the relation between $\\delta$ and\n$z$ has to be calculated numerically.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{phase_diagram.eps}\n\\end{center}\n\\caption{Phase diagram for systems with a Schulz distribution of\ninteraction strengths $u$. The nature of the coexisting isotropic (I)\nand nematic (N) phases is shown as a function of the inverse\ntemperature and the parent polydispersity $\\delta$. Dashed lines\nindicate the expected continuation of the phase boundaries where they\ncannot be determined reliably from the numerical calculations. The\ninset shows the whole range of polydispersities studied, including the\nmonodisperse limit $\\delta\\to 0$.\n\\label{fig:phase_diagram}\n}\n\\end{figure}\nFig.~\\ref{fig:phase_diagram} shows the resulting phase diagram,\nindicating for each given parent polydispersity $\\delta$ (in the range\n$\\delta=0\\ldots 0.60$) and temperature the nature of the phases\ncoexisting at equilibrium. Focussing first on the inset, which shows\nthe whole range of $\\delta$, we see that in the monodisperse limit\n$\\delta\\to 0$ there is no coexistence gap, with the transition from\nthe isotropic (I) to the nematic (N) phase taking place at the\nwell-known Maier-Saupe temperature of $1\/T_{\\rm c} \\approx 4.54$. At\nnonzero $\\delta$ an I--N coexistence region appears. Its width\ninitially grows quadratically with $\\delta$, as predicted by\nperturbation theory for small $\\delta$~\\cite{Sluckin89}. However, as\n$\\delta$ increases, the two boundaries of this I--N region are\naffected in an increasingly asymmetric way. The onset temperature of\nnematic ordering coming from high $T$ changes rather little,\nincreasing by only around 12\\% as the polydispersity increases from\nzero to $\\delta=0.6$. The {\\em lower} boundary of the I--N region, on\nthe other hand, is shifted to significantly lower temperatures. In\nfact the broadening is so strong that the phase boundary becomes\nre-entrant: for $0.553<\\delta<0.567$ one has the phase split sequence\nI $\\to$ I--N $\\to$ N $\\to$ I--N $\\to\\ldots$ as temperature is lowered.\n\nFor polydispersities above $\\delta\\approx 0.45$,\nFig.~\\ref{fig:phase_diagram} shows in addition that the system can\nfractionate from a single-phase nematic into two and later even three\ncoexisting nematics as $T$ is lowered. The boundary of the N--N region\nmeets that of the I--N region, and at that point a three-phase\ncoexistence region (I--N--N) opens up where one isotropic and two\nfractionated nematic phases coexist. The structure of the phase\ndiagram in this area is similar to the high-density phase diagram of\npolydisperse hard spheres~\\cite{FasSol03,FasSol04}, where two or more\nfractionated solids can occur with our without a coexisting fluid\nphase.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{bidisperse.eps}\n\\end{center}\n\\caption{Phase diagram for a bidisperse mixture with interaction\nstrengths $u_1=1$ and $u_2=3.5$; $x$ is the fraction of particles of\nthe second kind in the parent. The dashed line indicates the\ntemperature where three phases (I--N--N) can coexist.\n\\label{fig:bidisperse}\n}\n\\end{figure}\nIt is worth emphasizing that all coexistence regions in the phase\ndiagram are caused by polydispersity; the monodisperse system only has\nan I--N transition at a single temperature. Also, the presence of an\neffectively infinite number of different particle species,\ncharacterized by their respective interaction strengths $u$, means\nthat the number of possible phases can be in principle arbitrary. This\nshould be contrasted with the case of a binary\nmixture~\\cite{PalBerDebDun84}, which is illustrated in\nFig.~\\ref{fig:bidisperse} for particles with interaction strengths\n$u_1=1$ and $u_2=3.5$; $x$ is the fraction of particles of the second\nkind in the parent. A region of N--N demixing again occurs at low\ntemperatures, but I--N--N coexistence is possible only at a single\ntemperature.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{chi15_and_5.5_delta_nom0.575_distributions.eps}\n\\end{center}\n\\caption{Distributions of particles according to interaction strength,\n$n(u)$. The solid lines show the three coexisting phases (I--N--N) at\n$\\delta=0.571$ and $1\/T=15$; the dashed line is the parent\ndistribution. Inset: Same for the I--N coexistence at $\\delta=0.571$\nand $1\/T=5.5$.\n\\label{fig:distributions}\n}\n\\end{figure}\nThe rich phase behaviour that occurs in the polydisperse system is\ndriven by fractionation, i.e.\\ by the fact that particles of different\ninteraction strength $u$ partition themselves unevenly among\ncoexisting phases. As an example, we show in\nFig.~\\ref{fig:distributions} the distributions $n_a(u)$ for the three\ncoexisting phases (I--N--N) into which a Schulz parent with\n$\\delta=0.571$ splits at $1\/T=15$ (compare\nFig.~\\ref{fig:phase_diagram}). The more weakly interacting particles\nare found predominantly in the isotropic phase, while the particles\nwith larger $u$ are partititioned across the two nematic phases. This\ntendency is observed more generally, as the case of two-phase I--N\ncoexistence in the inset of Fig.~\\ref{fig:distributions}\nshows. Physically, it is explained easily from the form of the\nMaier-Saupe free energy: the system can lower its excess free energy\n(only) by ordering nematically. This effect is stronger for the more\nstrongly interacting particles, which are therefore found\npreferentially in the nematic phases.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{volume_fractions.eps}\n\\end{center}\n\\caption{Volume fractions $x_a$ of the various coexisting phases\nplotted as a function of inverse temperature, for a Schulz parent with\npolydispersity $\\delta=0.571$.\n\\label{fig:volume_fractions}\n}\n\\end{figure}\nAs a consequence of fractionation, the overall particle number\nfractions $x_a$ found in the various coexisting phases also do not\nvary linearly across coexistence\nregion. Fig.~\\ref{fig:volume_fractions} illustrates this, for a Schulz\nparent with the same polydispersity $\\delta=0.571$ as in the two\nexamples in Fig.~\\ref{fig:distributions}. One notes in particular the\nnon-monotonic variation of the fraction of particles in the isotropic\nphase before the second nematic phase appears. This is due to the\nproximity to the re-entrant I--N phase boundary in the phase diagram.\n\n\n\n\\section{Discussion}\n\\label{sec:conclusions}\n\n\nWe have studied the effect of polydispersity in thermotropic liquid\ncrystals, using a Maier-Saupe theory with factorized interactions. For\na spread of interaction strengths $u$ of around $\\delta=0.50$ of the\nmean and higher, polydispersity causes new phase splits to appear in\nthe phase diagram. In particular, the theory predicts coexistence of\ntwo or more nematics with or without an additional isotropic\nphase. Together with this, a strong broadening of the I--N coexistence\nregion is observed, to the extent that the phase boundary between the\nN and I--N regions becomes re-entrant within a small range of\n$\\delta$. Our calculation only considered isotropic and nematic phases\nand so at least some of phase behaviour we found at lower $T$ will be\nonly metastable, the stable phases being e.g.\\ smectic or\ncrystalline. Nevertheless, one would expect to be able to observe at\nleast the strong widening of the I--N coexistence region in\nappropriate experimental systems.\n\nFrom a broader theoretical point of view, our results demonstrate that\npolydispersity in the attractive (and orientation-dependent)\ninteraction between liquid crystal particles can be sufficient to\ncause demixing into two or more nematic phases. Polydispersity in\nshort-range repulsions as caused e.g.\\ by a distribution of particle\nshapes is not necessary, although from the results for binary mixtures\nin the Onsager model one would expect it to reinforce the tendency\ntowards nematic demixing. This issue would merit further study but\npromises to be very challenging. Indeed, even in the polydisperse\nOnsager model without added attractive interactions it remains an open\nproblem to establish the existence of nematic-nematic demixing at high\ndensities.\n\nOne may ask how crucial the assumption of factorized interactions\n$u(l,l')=u^{1\/2}(l,l) u^{1\/2}(l',l')$ is for the phase behaviour that\nwe have found. The situation is easiest to understand at low temperatures,\nwhere one has full nematic order ($S(l)\\approx 1$) and the ideal\n(entropic) part of the free energy can be neglected. The system then\nhas to minimize the excess free energy $-\\frac{1}{2}\\int\\!  dl\\,dl'\\,n(l)\nn(l')u(l,l')$. To understand whether phase separation will occur, take\nfirst the simple case of a binary mixture, with a fraction $x$ of\nparticles of length (polarizability, etc) $l$ and fraction $(1-x)$ of\nparticles of length $l'$. The demixed state of two phases, each\ncontaining only one type of particles, has a lower free energy than\nthe mixed phase if\n\\begin{eqnarray}\n-\\frac{1}{2} [x u(l,l) + (1-x) u(l',l')] < \n-\\frac{1}{2} [x^2 u(l,l) \\\\\n{}+{} (1-x)^2 u(l',l') + 2x(1-x) u(l,l')] \n\\nonumber\n\\end{eqnarray}\nor equivalently\n\\begin{equation}\nu(l,l') < \\frac{1}{2} [u(l,l)+u(l',l')]\n\\label{u_convex}\n\\end{equation}\nOne can easily show that this condition in fact guarantees that {\\em\nany} mixture of a finite number of components will separate into\nphases each containing only one component. In a fully polydisperse\nsystem, one then effects a cascade of demixing transitions with an\never-increasing number of phases as $T$ is lowered, as is found in\ne.g.\\ models of polydisperse copolymer blends~\\cite{SolWarCat01}. Our\nobservation of demixing into multiple nematic phases should therefore\nbe generic as long as\\eq{u_convex} holds, i.e.\\ as long as the\nattractive interaction strength between different particle species is\nalways lower than the mean of the intra-species interaction strengths.\n\nIn future work, one could contemplate extending the present study by\naccounting for the effects of density variations. To a first\napproximation this can be done by dividing the interaction strengths\n$u(l,l')$ by the average particle volume in the given\nphase~\\cite{HumLuc73}. In the polydisperse setting, this would give an\nextra factor of $1\/\\bar{v}$ in the excess free energy, where\n$\\bar{v}=\\int\\! dl\\,n(l)v(l)$ and $v(l)$ is the particle volume as a\nfunction of $l$. One then still has a truncatable free energy, but now\nwith an excess part depending on two moments, $\\bar{S}$ and\n$\\bar{v}$. Also of interest would be an extension to liquid crystal\npolymers, where perturbative results for small polydispersity $\\delta$\nalready exist~\\cite{Semenov93} and there is experimental evidence for\nN--N phase separation, at least in binary mixtures~\\cite{CasVeyFin82}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe decay mode(s) of a nuclear quantum state is (are) one of its most important properties after energy.  Usually nuclear decay follows the hierarchy of the fundamental forces of nature.  Decay by emission of one or more particles mediated by the strong nuclear interaction is normally the dominant mode for those states unbound to particle emission.  Bound excited states usually decay by emission of electromagnetic radiation to the ground state. The ground state, in turn, decays much more slowly by $\\beta$ decay mediated by the weak nuclear interaction until the lowest energy neutron-to-proton ratio has been reached.  Of course, there are exceptions: lower energy charged particle decay from unbound states is inhibited by the Coulomb barrier, and large spin change or low emission energy can inhibit $\\gamma$ decay so much that $\\beta$ decay occurs first.  Even decay by neutrons which face no Coulomb barrier can be inhibited by large spin change, as is well known in high-spin spectroscopy of medium and heavy nuclei.  However, based on the simple picture of barrier penetrability, neutron decay is usually assumed to dominate over radiative decay when angular momentum barriers are not very high.  But Fermi's Golden Rule \\cite{fermi} states that the decay rate is the product of the coupling strength, the density of final states, and the matrix element between the wavefunction of the parent state and the daughter state (usually called spectroscopic factor ($S$)). What is often overlooked is that this latter factor due to the nuclear structure may be more instrumental than the angular momentum barrier in limiting neutron decay rates below those of electromagnetic decay.  Examples are states with complex structure often involving intruder configurations.\n\nThe role of factors beyond barrier penetrability is highlighted by observations reported in this paper of the decays of unbound states in $^{19}$O populated in the $^9$Be($^{14}$C,$\\alpha \\gamma$) reaction which favors higher-spin states, which are typically more complex and may involve intruder configurations.  The importance of $^{19}$O with a closed major proton shell and an exactly half-filled neutron d$_{5\/2}$ orbital has been recognized with over half a century of investigations~\\cite{ahnlund54,zimmermann59,armstrong61,moreh64I,allen65,broude71,hibou71,crozier72,sen74,fortune74, fortune77,koehler,dufour86,warburton88,sumithrarachchi06,nagai07,oertzen10}.  In spite of this record, we were surprised to discover that six states from 368 to 2147 keV above the neutron threshold decay predominantly or totally by $\\gamma$ radiation (Figure \\ref{fig:levelscheme}). Five of these states had been previously reported~\\cite{fortune77,oertzen10} in $^{13}$C($^7$Li,$p$) reaction studies which also favor higher-spin states, but no $\\gamma$ detection was used in those works.  Many neutron decaying states in $^{19}$O were observed in an $^{18}$O + neutron resonance experiment~\\cite{koehler}.  There is no evidence above statistical uncertainties for the $\\gamma$ decaying states reported here in the neutron resonance spectrum~\\cite{koehler}, implying $\\Gamma_n$ widths much less than a keV.  The $\\gamma$ decaying states lie interspersed among the neutron decaying ones, leading to one of the clearest pictures of the competition between neutron and radiative decay in light nuclei and emphasizing the need for different experimental approaches to gain a clear picture of nuclear structure, especially more exotic intruder states. \n\nBarrier penetrability alone cannot explain the difference between neutron and $\\gamma$ decaying states since the spins of the $\\gamma$ decaying ones cannot be very high due to their decays to known states in $^{19}$O with spins of 3\/2$^+$ to 9\/2$^+$.  They must have significantly smaller $S$ values for neutron decay, presumably due to more complex structure.\n\nThe $\\gamma$ decaying unbound states were discovered following production of $^{19}$O in the $^{9}$Be($^{14}$C,$\\alpha\\gamma\\gamma$) reaction performed at the Florida State University (FSU) John D. Fox Superconducting Laboratory using $^{14}$C beams of 30 and 35 MeV. The $^9$Be target thickness of 0.125 mm was chosen to allow $\\emph{$\\alpha$}$ particles to escape from the target with limited energy loss while stopping the $^{14}$C beam.  The target was placed in the FSU Compton-suppressed $\\gamma$ detector array.  A Si $\\Delta$E - E telescope was placed at 0$^{\\circ}$ relative to the beam to detect and identify charged reaction products for channel selection. More experimental information can be found in Ref. \\cite{vonmoss}.  \n\nThis was the first study of $^{19}$O using a heavy-ion fusion reaction which favors population of higher-spin states and employs a modern $\\gamma$ detector array.  The $\\alpha$-$\\gamma$ and mostly $\\alpha$-$\\gamma$-$\\gamma$ coincidences were analyzed to build upon the previously known level and decay scheme.  Two of the $\\gamma$ decay lines from unbound states can be seen in Figure \\ref{fig:96gate} which shows the $\\gamma$ spectrum in coincidence with $\\alpha$ particles and decays of the 96 keV 3\/2$^+$ state.  Gamma decays of the other unbound states can be seen in Figure~\\ref{fig:23712777}, especially in coincidence with the 2371 and 2778 keV decays from the states having the highest previously known spin assignments of 9\/2$^+$ and 7\/2$^+$.  Their placement in the level scheme (Figure \\ref{fig:levelscheme}) has been confirmed by reverse gating.\n\n\nThe present experiment has been able to clarify a few open questions about the bound states to which the unbound states decay.  Conflicting $\\gamma$ decays of a state at about 3067 keV was pointed out in Ref.~\\cite{warburton88}:\nonly the 1596 keV one was reported in Ref.\\cite{broude71} and only the 2969 keV in Ref.\\cite{hibou71}. The second decay is not included in the NNDC compilation~\\cite{nndc}.  The most obvious resolution is the existence of two different excited states near the same energy.  The present experiment is the first to report both decay lines in the same experiment (see Figure \\ref{fig:96gate}), confirming that both $\\gamma$ lines exist in $^{19}$O, although this does not distinguish between the possibilities of two different states at $3064\\,(2)$ and $3067\\,(1)$ keV or one state at about 3065 keV with two decay branches.  The earlier results of different decay modes in different reactions strongly support the doublet possibility with J$^{\\pi}$= (5\/2$^{-}$) and (3\/2$^{+}$) respectively, as has been adopted in Figure \\ref{fig:levelscheme}.  The doublet hypothesis is also consistent with shell model predictions but not conclusively proved by this experiment.\n\nThe 96-2681 keV coincidence relation establishes a new decay branch of the 2778 keV state to the lowest 3\/2$^+$ level, providing further evidence against an older 9\/2$^+$ assignment~\\cite{moreh64I} and supporting the newer 7\/2$^+$ one~\\cite{crozier72}.\n\n\nThe observation of allowed $\\beta$$^{-}$ decay from the 1\/2$^{-}$ g.s. of $^{19}$N to the 3231 and 3945 keV states establishes J$^{\\pi}$ = 1\/2$^{-}$ or 3\/2$^{-}$ to them~\\cite{dufour86,sumithrarachchi06}. This confirms the J$^\\pi$ = 3\/2$^-$ assignment to 3945 keV from earlier  (d,p) reactions~\\cite{sen74}.  We see the same $\\gamma$ decays from these two states as are reported in~\\cite{sumithrarachchi06} and no sign of the 709.2 keV line reported in~\\cite{dufour86}, and interpreted~\\cite{warburton88} as a 3945 keV $\\rightarrow$ 3231 keV transition.  J$^\\pi$ = 1\/2$^-$ was also reported for the 3231 keV state in a thermal neutron capture experiment ~\\cite{nagai07}.  There is also no evidence in the present work for a doublet of states near 3945 keV as suggested in~\\cite{fortune74}.\n\n\nBecause the intensities of the $\\gamma$ lines from the unbound $4330\\,(3)$, $4943\\,(4)$, $4990\\,(4)$, $5232\\,(5)$, $5360\\,(3)$, and $6109\\,(4)$ keV levels are comparable to the decays from the bound ones, their decays must be predominantly radiative. Five of the six $\\gamma$ decays come from levels which correspond in energy within uncertainties to some of the states reported in a comprehensive study of the $^{13}$C($^7$Li,p) reaction \\cite{oertzen10}.  However, widths above 4 keV without uncertainties are quoted for all states above S$_n$ in that paper, including values of 5 to 13 keV for the five states observed in the present work to $\\gamma$ decay.  Such widths are not consistent with the eV level widths of the observed $\\gamma$ decays (see Table \\ref{tab:widths}).  Another example independent of the present work is a width of 19 keV listed in Ref. \\cite{oertzen10} for the 5151 keV level, while the direct neutron resonance measurement \\cite{koehler} reports a width of only 2 keV.  It appears that some of the smaller widths reported in Ref. \\cite{oertzen10} overestimate the actual values.    \n   \n\nIt is informative to compare the nearby odd-A Oxygen isotopes, as shown in Figure \\ref{fig:oxygencomparison}. Two $\\gamma$ decays have been reported \\cite{igashira} from the 4554 keV 3\/2$^-$ state in $^{17}$O, which is unbound to $\\ell = 1$ neutron decay by 411 keV, but their widths of 1.8 eV each represent only a total radiative branch of about 10$^{-4}$ of the neutron decay width of 40 keV.  These $\\gamma$ decays could only be seen in a sensitive neutron capture  experiment and do not correspond to the dominant $\\gamma$ decays reported here for $^{19}$O. \n\n\nMore similar to the $^{19}$O decays is a dominant $\\gamma$ decay branch from the 9\/2$^+$ level in $^{21}$O which is unbound to neutron decay by 1.1 MeV \\cite{21Odecay}. The authors calculated a partial lifetime for M1 $\\gamma$ decay of 57 fs, but 0.0020 fs for $\\ell = 4$ neutron decay with $S$=1.  At least a 50\\% $\\gamma$ branch would imply a very small neutron spectroscopic factor, less than $3.5\\times10^{-5}$.  Their estimated $S$ value from the g$_{9\/2}$ occupancy in a shell model calculation is $2\\times10^{-4}$.  \n\nShell model calculations in the \\emph{0p-0d-1s} model space using the $\\emph {psdu}$ interaction~\\cite{utsuno11} which allows full freedom of occupancy in the model space are shown in Figure \\ref{fig:levelscheme} for comparison with experiment~\\cite{volyacosmo}.  There is reasonable agreement with the states below 4 MeV.  The calculated states above this are dominated by negative-parity ones due to the extra degree of freedom and spin allowed by predominantly one particle-one hole configurations.  It is among these that the best candidates can be found for the radiatively decaying unbound states.  The spin suggestions are shown in parenthesis in Figure \\ref{fig:oxygencomparison} and in Table \\ref {tab:widths}. The states have been matched based on spin restrictions imposed by the observed $\\gamma$ decays, closeness in energy, strongest E-M decays, and highest angular momentum barriers to neutron decay. However, these are only intended as a kind of existence theorem that possible $\\gamma$ decaying states exist in the shell model calculations. Some of the spin parity suggestions differ from those of previous works~\\cite{fortune77,oertzen10}. However, the previous spin parity suggestions were based solely on empirical methods such as total cross sections and $2J+1$ analysis in Ref.~\\cite{fortune77}  and cross section and energy systematics in Ref.~\\cite{oertzen10} and hence are tentative. \n\n\n\nDecay properties of unbound states in $^{19}$O from the present experiment and Ref.~\\cite{koehler} below 6.3 MeV are compared in Table \\ref {tab:widths} along with possible correspondences with \\emph{pdsu} shell model states.  For the $\\gamma$-decaying (neutron-decaying) states, the shell model $\\gamma$ decay widths (measured neutron decay widths) are compared with the angular-momentum-limited neutron decay widths assuming perfect overlap between the parent and daughter configurations ($S$ = 1).  The ratio,$S$, of these quantities, listed in the last column, shows the factor by which the neutron decay must be inhibited by the overlap between the initial and final states.  Note that for the $\\gamma$ decaying states, $S$ is an upper limit assuming equal $\\Gamma_\\gamma$ and $\\Gamma_n$.  Since $\\Gamma_n$ has not been observed, it may well be much smaller than $\\Gamma_\\gamma$.  All of the spectroscopic factors are well below unity.  Those for the neutron-decaying states span 0.016 to 0.27, while the upper limits for the $\\gamma$-decaying states are generally much smaller, ranging from $2.8\\times10^{-5}$ to 0.055.  Thus, the experimental observation of radiative decay directly implies very low spectroscopic factors $S$. The \\emph{psdu} model space does not include\nany of the orbitals \\emph{$f_{7\/2}$}, \\emph{$f_{5\/2}$}, \\emph{$h_{9\/2}$}, nor \\emph{$i_{11\/2}$}\nneeded to calculate spectroscopic factors $S$ directly as pure single-particle transitions for the $\\gamma$ decaying states. It is not clear that the uncertainties in shell model interactions covering such a wide model space would justify a calculation.  \n\n\n\n\n\n\n\n\nTable \\ref {tab:widths} and Figure \\ref{fig:nresonance} show that the $\\gamma$ decaying states are distributed among the neutron decaying ones.  Although the $\\gamma$-decaying states are generally higher in spin or lower in energy, this can not be the whole story.  An example is the $7\/2^{-}$ states, the lowest of which  decays radiatively while the higher two decay by neutron emission.  The lowest one must also have a small spectroscopic factor (reduced overlap with the  $^{18}$O ground state plus a neutron).  The general idea of aligning experimental and theoretical states in increasing energy order works well for the 7\/2$^{-}$ states. However, the predicted bound 9\/2$_1^-$ state has not been matched with any experimental states because the predicted  $\\gamma$ decays are too weak. Instead the 9\/2$_2^-$ and 9\/2$_3^-$ states have been matched with unbound $\\gamma$ decaying states.  However, one should keep these correspondences in mind as more of a possibility, rather than certain. \n\nAnother characteristic of the $\\gamma$ decaying states is that most may be identified with negative parity ones involving an odd number of intruder particles.  This is not surprising since neutron hole states can have higher spins and less overlap with the ground state of $^{18}$O which is unlikely to have substantial intruder components. In this case $\\gamma$ decay can occur by parity-changing E1 transitions which are stronger in lighter nuclei. \n\n$^{19}$O is unique among the Oxygen isotopes in having a higher level density with a reasonably high S$_n$ (see Figure \\ref{fig:oxygencomparison}) giving more higher spin states close enough to S$_n$ to allow their $\\gamma$ decay to dominate over the usually expected neutron decay. It would be very interesting to compare $\\Gamma_n$ and $\\Gamma_\\gamma$ from the same state, but neutron decay seems to be more of an either-or case than charged-particle decay with its Coulomb barrier.  Two orders of magnitude separate the weakest neutron peaks that can be seen in current resonance studies from the strongest $\\gamma$ widths. In fact, there are no clear neutron resonance peaks at any of the energies indicated in Figure \\ref{fig:nresonance}.  By comparison the narrow $\\Gamma_n$ = 2 keV 5\/2$^+$ peak is very visible.\n\nIn any event, the present observation of multiple $\\gamma$ decaying states above S$_n$ suggests that focused measurements in other nuclei could be fruitful.  The observation of electromagnetic decay competing successfully with the normally much more probable neutron decay provides  significant constraints on the neutron decay width and highlights the exotic structure of these states.  It demonstrates that many exotic unbound states can be missed in neutron resonance measurements so $\\gamma$ decay measurements are essential for a more complete spectroscopy. \n\n\\begin{acknowledgements}\nThis work was supported in part by U.S. National Science Foundation Grants No. 1064819 and No. 1401574 and by the U.S. Department of Energy Grant No. DE-SC0009883. \n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA central theme in complex and symplectic geometry is to understand the stability of various properties under natural deformations, and the relationship to uniqueness and moduli problems.  For instance, the classic result of Kodaira-Spencer \\cite{KodairaSpencer} shows that small complex deformations of K\\\"ahler manifolds remain K\\\"ahler.  The main result of this note is local stability of K\\\"ahler structures under symplectic deformations on Calabi-Yau manifolds:\n\n\\begin{thm} \\label{t:mainthm} Let $(M^{2n}, \\omega, J)$ be a compact K\\\"ahler manifold with $c_1(M, J) = 0$.  There exists $\\ge > 0$ depending on $\\omega$ so that if $\\omega'$ is another symplectic form on $M$ such that\n\\begin{align*}\n\\brs{\\omega - \\omega'}_{C^{\\infty}(\\omega)} < \\ge,\n\\end{align*}\nthen there exists an integrable complex structure $J'$ on $M$ compatible with $\\omega'$.\n\\end{thm}\n\n\\begin{rmk}\n\\begin{enumerate}\n\\item The proof of Theorem \\ref{t:mainthm} is an elementary consequence of the dynamical stability of symplectic curvature flow (SCF) \\cite{SCF} near Calabi-Yau metrics.  This dynamical stability of the more general family of `almost Hermitian curvature flows' introduced in \\cite{SCF} was shown in the thesis of Smith (cf. \\cite{SmithSCF}).  We sketch the proof below in the simplified case of SCF.\n\\item It follows from the proof that in fact the $C^{\\infty}$ smallness of the perturbation can be weakened to smallness in an appropriate Sobolev space.\n\\item The space of deformations of symplectic cohomology classes which remain K\\\"ahler under deformation was studied in \\cite{deBart}.\n\\item Theorem \\ref{t:mainthm} follows in some cases using results from complex deformation theory, and our proof provides an alternative using a geometric flow adapted to almost K\\\"ahler geometry.\n\\item Recently, the result of Theorem \\ref{t:mainthm} was shown in the case $n = 3$ in \\cite{IIAflow} using a geometric flow of symplectic forms adapted to that dimension.\n\\end{enumerate}\n\\end{rmk}\n\nTo begin we recall fundamental properties of symplectic curvature flow \\cite{SCF}.  An almost K\\\"ahler structure is a pair $(\\omega, J)$ of a symplectic structure together with a compatible almost complex structure $J$ such that $g = \\omega J$ is a Riemannian metric.  In general $J$ is not integrable and $N$ will denote the Nijenhuis tensor of $J$.  Almost K\\\"ahler structures come equipped with a Chern connection, the unique connection $\\nabla$ on the tangent bundle such that $\\nabla g \\equiv 0, \\nabla J \\equiv 0$, and $T^{1,1} = 0$, where $T^{1,1}$ denotes the $(1,1)$ component of the torsion of $\\nabla$.  Let $\\Omega$ denote the curvature of $\\nabla$, and define $P = \\tr \\Omega J \\in \\pi c_1(M, J)$.  A one-parameter family of almost K\\\"ahler structures $(g_t, \\omega_t, J_t)$ satisfies symplectic curvature flow if\n\\begin{gather} \\label{f:SCF}\n\\begin{split}\n\\dt g =&\\ -2 \\Rc + \\frac{1}{2} B^1 - B^2,\\\\\n\\dt \\omega =&\\ - P,\\\\\n\\dt J =&\\ - D^* D J + \\mathcal N + \\mathcal R,\n\\end{split}\n\\end{gather}\nwhere $D$ denotes the Levi-Civita connection, and\n\\begin{align*}\nB^1_{ij} =&\\ g^{kl} g_{mn} D_i J_k^m D_j J_l^n, \\qquad \\qquad B^2_{ij} = g^{kl} g_{mn} D_k J_i^m D_l J_j^n,\\\\\n\\mathcal N_i^j =&\\ g^{jk} g_{mn} g^{pq} D_p J_r^m J_i^r D_q J_k^n, \\qquad \\mathcal R_i^j = J_i^k \\Rc_k^j - \\Rc_i^k J_k^i.\n\\end{align*}\nNote that the our description of symplectic curvature flow is redundant as any two of $(g, \\omega, J)$ suffices to recover the third by compatibility.  The fundamental points (cf. \\cite{SCF} Theorem 1.6) are that symplectic curvature flow is locally well-posed for arbitrary initial data on compact manifolds, preserves the almost K\\\"ahler conditions, and if $J_0$ is integrable reduces to K\\\"ahler-Ricci flow.\n\n\\begin{thm} \\label{t:dynstabSCF} (cf. \\cite{SmithSCF} Theorem 1.1) Let $(M^{2n}, \\omega_{CY}, J_{CY})$ denote a compact Calabi-Yau manifold.  There exists $\\ge > 0$ so that if $(\\omega, J)$ is an almost K\\\"ahler structure such that\n\\begin{align*}\n\\brs{\\omega_{CY} - \\omega}_{C^{\\infty}(\\omega_{CY})} + \\brs{J_{CY} - J}_{C^{\\infty}(\\omega_{CY})} < \\ge,\n\\end{align*}\nthen the solution to symplectic curvature flow with initial condition $(\\omega, J)$ exists on $[0,\\infty)$ and converges exponentially to a K\\\"ahler Calabi-Yau structure $(\\omega_{\\infty}, J_{\\infty})$.\n\\end{thm}\n\n\\begin{proof}\nThe proof relies on ideas from parabolic regularity theory and so we work directly with the gauge-modified flow which is strictly parabolic.  For any almost K\\\"ahler structure $(g, J)$ we define the vector field\n\\begin{align*}\nX^k(g, J) = g^{ij} \\left( \\Gamma_{ij}^k - (\\Gamma_{CY})_{ij}^k \\right).\n\\end{align*}\nAn elementary but important point is that this vector field is equivalently expressed as\n\\begin{align*}\nX^k =&\\ \\omega^{ij} \\nabla^{CY}_i J_j^k.\n\\end{align*}\nUsing $X$ we define the gauge-fixed symplectic curvature flow:\n\\begin{gather} \\label{f:gfSCF}\n\\begin{split}\n\\dt g =&\\ -2 \\Rc + \\frac{1}{2} B^1 - B^2 + L_X g =: \\mathcal F_1(g,J)\\\\\n\\dt J =&\\ - D^* D J + \\mathcal N + \\mathcal R + L_X J =: \\mathcal F_2(g,J)\n\\end{split}\n\\end{gather}\nThe analysis centers on a sharp characterization of the linearization of this flow.  To find this fix a one-parameter family of almost K\\\"ahler structures $(g_t, J_t)$ such that $(g_0, J_0) = (g_{CY}, J_{CY})$ and $\\dot{g} = h, \\dot{J} = K$.  Lengthy but straightforward computations using that $(g_0, J_0)$ is Calabi-Yau show (cf. \\cite{SCF} proof of Theorem 1.6)\n\\begin{align*}\n\\mathcal L_{(g_{CY}, J_{CY})} \\mathcal \\mathcal F_1 (h,K) =&\\ \\Delta h + 2 R \\circ h =: \\mathcal L_1(h),\\\\\n\\mathcal L_{(g_{CY}, J_{CY})} \\mathcal \\mathcal F_2 (h,K) =&\\ \\Delta K + 2 R \\circ K =: \\mathcal L_2(K),\\\\\n\\end{align*}\nwhere\n\\begin{align*}\n(R \\circ h)_{ij} =&\\ R_{i k l j} h^{kl}, \\qquad (R \\circ K)_j^i = g^{kl} R_{j l m}^i J_k^m.\n\\end{align*}\nTo analyze this operator we recall the work of Koiso \\cite{Koisodef}.  The operator $\\mathcal L_1$ is the Einstein deformation operator at a Ricci-flat metric, and splits according to the decomposition $h = h_S + h_A$ into the  $J$-symmetric and $J$-antisymmetric pieces.  The action on $h_S$ corresponds precisely to the Hodge Laplacian acting on the $(1,1)$-form $h_S J_{CY}$, which is negative semidefinite with kernel determined by harmonic $(1,1)$ forms, which are canonically identified with $H^{1,1}(M, J_{CY})$.  The action on $h_A$ is identified, after raising an index with $g_{CY}$, with the $\\bar{\\partial}}\\newcommand{\\dt}{\\frac{\\partial}{\\partial t}$-Hodge Laplacian acting on $\\Lambda^{0,1} \\otimes T^{1,0}$, in this case restricted to symmetric endomorphisms.  Furthermore, the operator $\\mathcal L_2$ is again this same $\\bar{\\partial}}\\newcommand{\\dt}{\\frac{\\partial}{\\partial t}$-Hodge Laplacian acting on $\\Lambda^{0,1} \\otimes T^{1,0}$, whose kernel is identified with the space of deformations of $J_{CY}$.  This again is negative semidefinite with kernel identified with $H^{2,0}(M, \\mathbb C)$.\n\nThus we have shown that the linearized operator is negative semidefinite, with kernel identified with the space of Einstein deformations of the given Calabi-Yau.  It follows from a result in \\cite{Tiandeformation} that every such infinitesimal deformation is in fact integrable.\nGiven this weak linear stability, together with an explicit description of the kernel, which is integrable, the remainder of the proof follows standard lines (cf. for instance \\cite{Lotaystab, Sesum, SmithSCF,HCF}).  In particular, by treating the flow as a small perturbation of the linearized flow, and using the analysis of the linearized operator above, one can show exponential decay towards some Calabi-Yau structure.  Given this exponential convergence, it is elementary to show that the family of diffeomorphisms relating (\\ref{f:gfSCF}) and (\\ref{f:SCF}) converges exponentially, and thus the solution to (\\ref{f:SCF}) is also converging to a Calabi-Yau structure exponentially fast.\n\\end{proof}\n\nWe now prove Theorem \\ref{t:mainthm} as a consequence of Theorem \\ref{t:dynstabSCF}:\n\n\\begin{proof}[Proof of Theorem \\ref{t:mainthm}] Given $(M^{2n}, \\omega, J)$ a compact K\\\"ahler manifold with $c_1(M, J) = 0$, by Yau's theorem \\cite{YauCC} there exists a unique Calabi-Yau metric $\\omega_{CY} \\in [\\omega]$ compatible with $J$.  Applying Moser's Lemma to the family of cohomologous symplectic forms $\\omega_t = t \\omega_{CY} + (1-t) \\omega$ we obtain the existence of a diffeomorphism $\\phi$ such that $\\phi^* \\omega_{CY} = \\omega$.  By construction the pair $(\\omega, \\phi^* J) = (\\phi^* \\omega_{CY}, \\phi^* J)$ is K\\\"ahler, Calabi-Yau.  Now fix $\\omega'$ such that $\\brs{\\omega - \\omega'}_{C^{\\infty}(\\omega)} < \\ge$.  For sufficiently small $\\ge > 0$, the one-parameter family $\\omega_t = t \\omega' + (1-t) \\omega$ consists of symplectic forms, and we deform $\\phi^* J$ along this path to produce an almost complex structure $J'$ compatible with $\\omega'$ such that $\\brs{\\phi^* J - J'}_{C^{\\infty}(\\omega)} < C(\\omega) \\ge$ (cf. \\cite{SCF} Lemma 4.3).  For $\\ge$ chosen sufficiently small at the outset, the pair $(\\omega', J')$ satisfies the hypothesis of Theorem \\ref{t:dynstabSCF} relative to the Calabi-Yau structure $(\\omega, \\phi^* J)$, and thus the solution to symplectic curvature flow with initial condition $(\\omega', J')$ exists globally and converges to a Calabi-Yau structure $(\\omega_{\\infty}, J_{\\infty})$, which further satisfies $\\brs{\\omega - \\omega_{\\infty}}_{C^{\\infty}(\\omega)} < C(\\omega) \\ge$.  By construction, since $J'$ is connected by a smooth path to $J$, it follows that $c_1(M, J') = 0$, and this property is preserved along the symplectic curvature flow.  This in turn implies that the cohomology class of $[\\omega']$ is preserved along the flow, thus $[\\omega_{\\infty}] = [\\omega']$.  Since both $\\omega_{\\infty}$ and $\\omega'$ are $\\ge$-close to the symplectic form $\\omega$, it follows that the path $t \\omega' + (1-t) \\omega_{\\infty}$ consists of cohomologous symplectic forms, again applying Moser's Lemma we obtain a diffeomorphism $\\psi$ such that $\\psi^* \\omega_{\\infty} = \\omega'$.  It follows that the pair $(\\omega', \\psi^* J_{\\infty})$ is K\\\"ahler, in fact Calabi-Yau, finishing the proof.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Bandit linear optimization}\\label{sec:blo}\n\nOur second general application is to bandit linear optimization,  in which at each round $i$ of task $t$ we play a vector $\\*x_{t,i}\\in\\mathcal K$ for some convex set $\\mathcal K$ and observe loss $\\langle\\ell_{t,i},\\*x_{t,i}\\rangle\\in[-1,1]$.\nWe will again use a variant of mirror descent on top of estimated losses, this time setting $\\phi$ to be a self-concordant barrier function with specialized loss estimators as in \\citet{abernethy2008competing}.\nThis class of algorithms is picked because of its general applicability to any convex domain $\\mathcal K$ via the construction of such barriers and the optimal dependence of its regret on the number of rounds $m$.\nNote that our ability to handle non-smooth regularizers via the structural result in Theorem~\\ref{thm:meta} is even more important here, as the barrier functions are  infinite at the boundaries.\nIndeed, in this section we will no longer learn a $\\beta$ parameterizing the regularizer and instead focus on learning an offset $\\varepsilon>0$ away from the boundary.\nFor each such offset define $\\mathcal K_\\varepsilon=\\{\\*x\\in\\mathbb R^d:\\pi_{\\*x_{1,1}}(\\*x)\\le1\/(1+\\varepsilon)\\}\\subset\\mathcal K$, where $\\*x_{1,1}=\\argmin_{\\*x\\in\\mathcal K}\\phi(\\*x)$ and $\\pi_{\\*x_{1,1}}(\\*x)=\\inf_{\\lambda\\ge0,\\*x_{1,1}+(\\*x-\\*x_{1,1})\/\\lambda\\in\\mathcal K}\\lambda$ is the Minkowski function.\nAs before we obtain the $\\varepsilon$-restricted optima-in-hindsight via the primitive $\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)=\\argmin_{\\*x\\in\\mathcal K_\\varepsilon}\\langle\\hat\\ell_t,\\*x\\rangle$.\n\nWith this specified, we can again adapt our meta-approach of Algorithm~\\ref{alg:meta}, roughly summarized for BLO as doing the following at each task $t>1$:\n\\begin{samepage}\n\\begin{enumerate}[noitemsep]\n\t\\item sample $\\theta_t=(\\eta_t,\\varepsilon_t)$ from a distribution $\\*p_t$ over the discretization $\\Theta$\n\t\\item run $\\texttt{OMD}_{\\eta_t}$ using the initialization $\\*x_{t,1}\n\t=\\frac1{t-1}\\sum_{s<t}\\hat{\\*x}_t^{(\\theta_t)}\n\t=\\frac1{t-1}\\sum_{s<t}\\operatorname{OPT}_{\\varepsilon_t}(\\hat\\ell_t)$\n\t\\item update $\\*p_{t+1}$ using multiplicative weights with losses $\\frac1{\\eta_t}B_\\phi(\\hat{\\*x}_t^{(\\varepsilon_t)}||\\*x_{t,1})+(32d^2\\eta_t+\\varepsilon_t)m$\n\\end{enumerate}\n\\end{samepage}\nNote that this algorithm is very similar to that for MAB, with both being special cases of Algorithm~\\ref{alg:meta}, with the main difference being the different upper bound passed to multiplicative weights.\nThe procedure has the following guarantee\n\n\\begin{Thm}\\label{thm:blo}\n\tSuppose $\\texttt{OMD}_{\\eta,\\beta}$ is online mirror descent with a self-concordant barrier $\\phi$ as a regularizer and loss estimators specified as in \\citet{abernethy2008competing}.\n\tThen for every $\\overline\\varepsilon\\in(0,1\/\\sqrt m]$ and $\\underline\\varepsilon\\in(0,\\overline\\varepsilon]$ there exists an integer $k=\\mathcal O(D_{\\underline\\varepsilon}^2d\\lceil\\sqrt{mT}\\rceil)$, where $D_{\\underline\\varepsilon}^2$ is a bound on $B_\\phi$ over $\\mathcal K_{\\underline\\varepsilon}$, and $\\alpha,\\underline\\eta,\\overline\\eta\\in(0,\\infty)$ such that running Algorithm~\\ref{alg:meta} with $\\Theta$ the product of uniform grids of size $k$ over each dimension of $[\\underline\\eta,\\overline\\eta]\\times[\\underline\\epsilon,\\overline\\epsilon]$ and $\\alpha$ the meta-step-size yields the expected task-averaged regret\n\t\\begin{align}\n\t\\begin{split}\n\t\\mathbb E\\frac1T\\sum_{t=1}^T\\sum_{i=1}^m\\langle\\ell_{t,i},\\*x_{t,i}-\\*x_t^\\ast\\rangle\n\t&\\le72d\\sqrt m\\sqrt[4]T\\left(D_{\\underline\\varepsilon}\\sqrt{\\frac mT\\log k}+\\frac{S_{\\underline\\varepsilon}K^2}{D_{\\underline\\varepsilon}T}(1+\\log T)\\right)\\\\\n\t&\\qquad+\\min_{\\*x\\in\\mathcal K,\\eta>0,\\varepsilon\\in[\\underline\\varepsilon,\\overline\\varepsilon]}\\mathbb E\\frac1T\\sum_{t=1}^T\\frac{B_\\phi(\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)||\\*x)}\\eta+(32\\eta d^2+\\varepsilon)m\\\\\n\t&=\\tilde\\mathcal O\\left(\\frac{D_{\\underline\\varepsilon}dm}{\\sqrt[4]T}+\\frac{S_{\\underline\\varepsilon}K^2d\\sqrt m}{D_{\\underline\\varepsilon}T^\\frac34}\\right)+\\min_{\\*x\\in\\mathcal K,\\varepsilon\\in[\\underline\\varepsilon,\\overline\\varepsilon]}4d\\hat V_\\varepsilon\\sqrt{2m}+\\varepsilon m\n\t\\end{split}\n\t\\end{align}\n\twhere $S_{\\underline\\varepsilon}=\\max_{\\*x\\in\\mathcal K_{\\underline\\varepsilon}}\\|\\nabla^2\\phi(\\*x)\\|_2$, $K$ is the Euclidean diameter of $\\mathcal K$, and $\\hat V_\\varepsilon$ is what we call the {\\bf barrier-divergence at level $\\varepsilon$} defined by $\\hat V_\\varepsilon^2=\\min_{\\*x\\in\\mathcal K}\\mathbb E\\frac1T\\sum_{t=1}^TB_\\phi(\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)||\\*x)$.\n\\end{Thm}\n\nFor self-concordant barriers we generally have $D_{\\underline\\varepsilon}=\\mathcal O(1\/\\underline\\varepsilon)$ and $S_{\\underline\\varepsilon}=\\mathcal O(1\/\\underline\\varepsilon^2)$~\\citep{abernethy2008competing}, so setting $\\underline\\varepsilon=1\/m$ and yields\n\\begin{equation}\n\\mathbb E\\frac1T\\sum_{t=1}^T\\sum_{i=1}^m\\langle\\ell_{t,i},\\*x_{t,i}-\\*x_t^\\ast\\rangle\n\\le\\tilde\\mathcal O\\left(\\frac{dm^2}{\\sqrt[4]T}\\right)+\\min_{\\frac1m\\le\\varepsilon\\le\\frac1{\\sqrt m}}4d\\hat V_\\varepsilon\\sqrt{2m}+\\varepsilon m\n\\end{equation}\nAs before, this shows that as the number of tasks $T\\to\\infty$ the average regret improves with a notion of task-similarity $\\hat V_\\varepsilon$ that decreases if the estimated task-optima are close together.\nRoughly speaking, if tasks have barrier-divergence $\\hat V_\\varepsilon$ then the average regret will be $\\mathcal O(\\hat V_\\varepsilon\\sqrt m+\\varepsilon m)$, which can be a significant improvement over the single-task case, e.g. if $\\hat V_\\frac1m$ is small.\nIn-particular, our analysis removes explicit dependence on the square root of the self-concordance constant of $\\phi$ in the single-task case \\citep{abernethy2008competing};\nas an example, this constant is equal to the number of constraints if $\\mathcal K$ is defined by linear inequalities, as in the bandit shortest-path application below.\nNote that the use of $\\varepsilon$-constrained optima is necessary for this problem due to the regularizers being infinite at the boundaries, where all true optima lie.\\looseness-1\n\nTo make the above result and task-similarity notion more concrete, consider the following corollary for BLO over the unit sphere $\\mathcal K=\\{\\*x\\in\\mathbb R^d:\\|\\*x\\|_2\\le1\\}$:\n\\begin{Cor}\\label{cor:sphere}\n\tLet $\\mathcal K$ be the unit sphere with the self-concordant barrier $\\phi(\\*x)=-\\log(1-\\|\\*x\\|_2^2)$.\n\tThen Algorithm~\\ref{alg:meta} attains expected task-averaged regret bounded by\n\t\\begin{equation}\n\t\t\\tilde\\mathcal O\\left(\\frac{dm^2}{\\sqrt[4]T}\\right)+\\min_{\\frac1m\\le\\varepsilon\\le\\frac1{\\sqrt m}}4d\\mathbb E\\sqrt{2m\\log\\left(\\frac{1-\\|\\hat{\\bar\\ell}^{(\\varepsilon)}\\|_2^2}{2\\varepsilon-\\varepsilon^2}\\right)}+\\varepsilon m\n\t\\end{equation}\n\tfor $\\hat{\\bar\\ell}^{(\\varepsilon)}=\\frac1T\\sum_{t=1}^T\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)=\\frac{\\varepsilon-1}T\\sum_{t=1}^T\\frac{\\hat\\ell_t}{\\|\\hat\\ell_t\\|_2}$ the average over normalized estimated task-optima.\\looseness-1\n\\end{Cor}\nThus in this setting if all tasks have similar estimated losses then $\\hat{\\bar\\ell}^{(\\varepsilon)}$ will be an average over similar vectors and thus have large Euclidean norm close to $1-\\varepsilon$, making the term in the logarithm above close to 1.\nIn this case $\\hat V_\\varepsilon$ is close to zero and so the average regret is $\\varepsilon m$ as $T\\to\\infty$;\nsetting $\\varepsilon=1\/m$ yields constant asymptotic averaged regret.\nThis demonstrates the usefulness of the barrier-divergence as a measure of task-similarity.\n\nAs a final application, we apply our meta-BLO result to the shortest-path problem in online optimization \\citep{takimoto2003path,kalai2005efficient}.\nIn its bandit variant \\citep{awerbuch2004adaptive,dani2008price}, at each time step $i=1,\\dots,m$ the player must choose a path $p_i$ from a fixed source $u\\in V$ to a fixed sink $v\\in V$ in a directed graph $G(V,E)$.\nAt the same time the adversary chooses edge weights $\\ell_i\\in\\mathbb R^{|E|}$ and the player suffers the sum $\\sum_{e\\in p_t}\\ell_i(e)$ of the weights in their chosen path $p_t$.\nThis can be transformed into BLO over vectors $\\*x$ in a convex set $\\mathcal K\\subset[0,1]^{|E|}$ defined by a set $\\mathcal C$ of $\\mathcal O(|E|)$ linear constraints $(\\*a,b)$ s.t. $\\langle\\*a,\\*x\\rangle\\le b$ enforcing flows from $u$ to $v$;\npaths from $u$ to $v$ can then be sampled from any $\\*x\\in\\mathcal K$ in an unbiased manner \\citep[Proposition~1]{abernethy2008competing}.\nIn the single-task case the BLO method of \\citet{abernethy2008competing} yields an $\\mathcal O(|E|^\\frac32\\sqrt m)$-regret algorithm for this problem.\n\nIn the multi-task case consider a sequence of $t=1,\\dots,T$ shortest path instances, each consisting of $m$ edge loss vectors $\\ell_{t,i}$ selected by an adversary.\nThe goal is to minimize average regret across instances.\nNote that our setup may be viewed as learning a prediction of the optimal path in a manner similar to the algorithms with predictions paradigm in beyond-worst-case-analysis \\citep{mitzenmacher2021awp};\nin-particular, we have incorporated predictions into the algorithm of \\citet{abernethy2008competing} via the meta-initialization approach and now present the learning-theoretic result for an end-to-end guarantee \\citep{khodak2022awp}.\n\\begin{Cor}\\label{cor:path}\n\tLet $\\mathcal K=\\{\\*x\\in[0,1]^{|E|}:\\langle\\*a,\\*x\\rangle\\le b~\\forall~(\\*a,b)\\in\\mathcal C\\}$ be the set of flows from $u$ to $v$ on a graph $G(V,E)$, where $\\mathcal C\\subset\\mathbb R^{|E|}\\times\\mathbb R$ is a set of $\\mathcal O(|E|)$ linear constraints.\n\tSuppose we see $T$ instances of the bandit online shortest path problem with $m$ timesteps each.\n\tThen sampling from probability distributions over paths from $u$ to $v$ returned by running Algorithm~\\ref{alg:meta} with regularizer $\\phi(\\*x)=-\\sum_{\\*a,b\\in\\mathcal C}\\log(b-\\langle\\*a,\\*x\\rangle)$ attains the following expected average regret across instances:\n\t\\begin{equation}\n\t\t\\tilde\\mathcal O\\left(\\frac{|E|m^2}{\\sqrt[4]T}\\right)+\\min_{\\frac1m\\le\\varepsilon\\le\\frac1{\\sqrt m}}4|E|\\mathbb E\\sqrt{2m\\sum_{\\*a,b\\in\\mathcal C}\\log\\left(\\frac{\\frac1T\\sum_{t=1}^Tb-\\langle\\*a,\\hat{\\*x}_t^{(\\varepsilon)}\\rangle}{\\sqrt[T]{\\prod_{t=1}^Tb-\\langle\\*a,\\hat{\\*x}_t^{(\\varepsilon)}\\rangle}}\\right)}+\\varepsilon m\n\t\\end{equation}\n\tHere $\\hat{\\*x}_t^{(\\varepsilon)}=\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)$ is the $\\varepsilon$-constrained estimated optimal flow for instance $t$.\n\\end{Cor}\nCorollary~\\ref{cor:path} shows that the average regret on the $T$ bandit shortest-path problems scales with the sum across all constraints $\\*a,b\\in\\mathcal C$ of the log of the ratio between the arithmetic and geometric mean of the distances $b-\\langle\\*a,\\hat{\\*x}_t^{(\\varepsilon)}\\rangle$ from the estimated optimum flow $\\hat{\\*x}_t^{(\\varepsilon)}$ to the constraint boundary.\nSince the arithmetic and geometric mean are equal exactly when all entries are equal---and otherwise the former is larger---this means that the regret is small when the estimated optimal flows $\\hat{\\*x}_t^{(\\varepsilon)}$ for each task are at similar distances from the constraints.\n\\section*{Checklist}\n\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  \\item Did you describe the limitations of your work?\n    \\answerYes{}\n  \\item Did you discuss any potential negative societal impacts of your work?\n    \\answerNA{}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n    \\answerYes{}\n\\end{enumerate}\n\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{Assumptions are stated in Sections~\\ref{sec:setup}, \\ref{sec:mab}, and~\\ref{sec:blo}.}\n        \\item Did you include complete proofs of all theoretical results?\n    \\answerYes{Proofs are given in the Appendix.}\n\\end{enumerate}\n\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    \\answerNA{}\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerNA{}\n        \\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerNA{}\n        \\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    \\answerNA{}\n\\end{enumerate}\n\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    \\answerNA{}\n  \\item Did you mention the license of the assets?\n    \\answerNA{}\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\\end{enumerate}\n\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\n\\end{enumerate}\n\\section{Conclusion}\n\nIn this work, we develop a meta-algorithm for learning to initialize and tune OMD for regularizers used in adversarial bandit tasks. We apply our meta-algorithm to obtain task-averaged regret guarantees for both the multi-armed and the linear bandit settings that depend on natural, setting-specific notions of task similarity. For MAB, we use OMD with the Tsallis regularizer as our base-learner and meta-learn the initialization, step-size, and entropy parameter. For BLO, we again use a variant of mirror descent with self-concordant barrier regularizers as our base-learner, and meta-learn the initialization, step-size, and boundary-offset. \n\n\\newpage\nThere are several exciting directions for future work.\nA limitation of our current results is the dependence of the task-similarity on optima estimated by the within-task algorithm;\nwhile they have the benefit of being computable, one may wish to obtain task-averaged regret bounds which do not depend on algorithmic quantities.\nIn particular, defining task-similarity via the true optima or losses may be more natural.\nAchieving this may be possible by making further assumptions on the structure of the problem, e.g. gap conditions or best-arm identifiability.\nAnother direction is to extend these results to other adversarial bandits settings such as contextual bandits and Lipschitz bandits.\n\\section{Introduction}\\label{sec:intro}\nMany real-world problems involve decision-making under partial feedback. For example, an administrator of a news website will not get to observe user engagement for undisplayed content. Likewise, the administrator does not know what their commute time would have been had they taken a different route to work that day. Such  feedback, where a \\emph{learner} only observes the outcome of the action taken, is often referred to as \\emph{bandit feedback}. This is in contrast to the \\emph{full feedback} setting, in which the learner gets to observe what would have happened under all possible actions they could have taken. \n\nWhile there are many methods with performance guarantees for bandit learning, most do not take into consideration the information the learner has gained from previous experience completing similar tasks. When selecting content for a website targeting a new demographic, the administrator will likely consider which types of content generated high levels of engagement with similar subpopulations of users. Likewise, they will most likely use knowledge about traffic patterns gained from their daily commute to work to inform routes to other locations. \nCan our learning algorithms do the same?\n\n\\emph{Meta-learning}, also known as \\emph{learning-to-learn} \\cite{thrun1998ltl}, is a popular approach to studying such problems in the context of multi-task learning, changing environments, and beyond. \nThe goal of such meta-learning algorithms is to leverage information from previously-seen tasks in order to achieve better performance on the current task at hand. While most meta-learning algorithms are designed for the full feedback setting, there is a small but growing amount of recent work which aims to design meta-learning algorithms capable of operating under bandit feedback. \nThe two most commonly studied types of feedback in the bandit literature are \\emph{stochastic bandit feedback}, where feedback is sampled i.i.d. from some distribution, and a more general notion called \\emph{adversarial bandit feedback}, where it is chosen by an adversary possibly trying to harm the learner. To the best of our knowledge, we are the first to study the problem of meta-learning under adversarial bandit feedback.\n\nWe consider a setting in which a meta-learner interacts with a sequence of bandit tasks.\nIn the single-task setting, the goal is to minimize \\emph{regret} with respect to the best fixed action in hindsight.\nLifting this to the multi-task setting, our goal will be to design algorithms which achieve low regret \\emph{on average} across tasks. Ideally, an algorithm's task-averaged regret should be no worse than that of algorithms for the single-task setting, e.g. if the tasks are not very similar, and should be much better on tasks that are closely related, e.g. if the same small set of arms do well on all of them.\n\nWe design a meta-algorithm based on learning the initialization and tuning parameters of online mirror descent (OMD) when it uses regularizers employed by bandit algorithms such as Exp3 \\citep{auer2002exp3}.\nTheoretically, our work differs from past approaches to parameterizing OMD in the full information setting because the regularizers used by bandit methods are non-Lipschitz near the boundary of the action set;\nthus the results of past work~\\citep{khodak2019provable,khodak2019adaptive,denevi2019meta} do not apply.\nTo overcome this issue, we only initialize away from the boundary and adapt our algorithms to handle the resulting error.\n\nWe apply our meta-algorithm for adversarial bandit feedback to both multi-armed bandits (MAB) and bandit linear optimization (BLO), obtaining in both settings new meta-learning algorithms with provable guarantees. For MAB, the average $m$-round regret across $T$ tasks of our algorithm is\n\\begin{equation}\\label{eq:result}\n\to_T(1)+2\\min_{\\beta\\in(0,1]}\\sqrt{\\hat H_\\beta d^\\beta m\/\\beta}\n\\end{equation}\nwhere $d$ is the number of actions and $\\hat H_\\beta$ is the Tsallis entropy~\\citep{tsallis1988possible,abernethy2015fighting} of the empirical distribution over the estimated optimal actions across tasks.\nAt $\\beta=1$ the Tsallis entropy reduces to the Shannon entropy;\nboth are small if most tasks are estimated to be solved by the same few arms and large if all are used roughly the same amount, making it a natural task-similarity notion.\nThe bound of $\\log d\\ge\\hat H_1$ means that the bound~\\eqref{eq:result} recovers Exp3's guarantee in the worst-case of dissimilar tasks.\nIn the important case of $s\\ll d$ arms always being estimated to be optimal we have $\\hat H_\\beta=\\mathcal O(s)$, so using $\\beta=\\frac1{\\log d}$ in bound~\\eqref{eq:result} yields a task-averaged regret of $\\mathcal O(\\sqrt{sm\\log d})$ as $T\\to\\infty$.\nFor $s=\\mathcal O_d(1)$ this beats the single-task lower bound of $\\Omega(\\sqrt{dm})$~\\citep{audibert2011minimax}.\nWe also obtain natural task-averaged regret bounds for BLO, albeit with different setting-specific notions of task similarity. \n\nOur main technical contributions are as follows:\n\\begin{enumerate}[noitemsep]\n    \\item We design a unified meta-learning algorithm to set the initialization and tuning parameters of OMD when using regularizers used by different bandit algorithms (Algorithm~\\ref{alg:meta}).\n    Apart from strong guarantees and generality, our approach is notable for its adaptivity:\n    we do not need to know anything about the task-similarity---e.g. the size of the subset of optimal arms---to adapt to similar tasks.\\looseness-1\n    \\item We apply our meta-approach to obtain a meta-learning algorithm for the adversarial MAB problem. In particular, we use the method of \\citet{abernethy2015fighting}---OMD with the Tsallis regularizer---as our within-task algorithm to achieve bounds on task-averaged regret that depend on a natural notion of task similarity:\n    the Tsallis entropy of the estimated optima-in-hindsight.\n   \n   \n    \\item We adapt Algorithm \\ref{alg:meta} to the adversarial BLO problem by setting the regularizer to be a self-concordant barrier function, as in~\\citet{abernethy2008competing}. As in MAB, we obtain task-averaged regret bounds which depend on a natural notion of task similarity based on the constraints defining the convex action space.\n    We instantiate the BLO result in two settings:\n    linear bandits over the sphere and an application to the bandit shortest-path problem~\\citep{takimoto2003path,kalai2005efficient}.\n\\end{enumerate}\n\\section{Multi-armed bandits}\\label{sec:mab}\n\nWe now turn to our first application:\nthe multi-armed bandits problem.\nIn this setting at each round $i$ of task $t$ we take action $a_{t,i}\\in[d]$ and observe loss $\\ell_{t,i}(a_{t,i})\\in[0,1]$.\nAs algorithms for MAB are probabilistic, we often sample methods from distributions $\\*x\\in\\overline\\mathcal K=\\triangle_d$ in the $k$-simplex, thus making the inner product $\\langle\\ell_{t,i},\\*x_{t,i}\\rangle$ the expectation.\n\nIn this paper we use as a base-learner a generalization of the popular Exp3 method of \\citet{auer2002exp3}, which runs multiplicative weights over unbiased estimators of the losses.\nThe first generalization is of the OMD regularizer, which for Exp3 is the negative Shannon entropy; \nwe employ the negative Tsallis entropy $\\phi_\\beta(\\*p)=\\frac{1-\\sum_{a=1}^d\\*p^\\beta(a)}{1-\\beta}$ for $\\beta\\in[0,1]$, which was used by \\citet{abernethy2015fighting} to improve the dependence of the regret on the dimension from $\\mathcal O(\\sqrt{dm\\log d})$ to the optimal $\\mathcal O(\\sqrt{dm})$.\nNote that $\\phi_\\beta$ recovers the Shannon entropy in the limit $\\beta\\to1$, and also that $B_{\\phi_\\beta}(\\*x||\\cdot)$ is non-convex in the second argument, making ours the first known application of the online learnability of non-convex Bregman divergences.\nThe second generalization is in the loss estimators;\nfor $\\gamma>0$ we employ $\\hat\\ell_{t,i}(a)=\\frac{\\ell_{t,i}(a)1_{a_{t,i}=a}}{\\*x_{t,i}(a)+\\gamma}$, where $\\*x_{t,i}(a)$ is the probability of sampling $a$ on round $i$ of task $t$.\nWhile this is an under-estimate of $\\ell_{t,i}(a)$, its lower variance compared to the unbiased estimator---recovered by setting $\\gamma=0$---allows \\citet{neu2015explore} to obtain high probability bounds.\n\nAs the Tsallis entropy is non-smooth at the simplex boundary, learning Tsallis divergences will require the tools developed previously for initializing $\\texttt{OMD}$ in the interior of $\\overline\\mathcal K$.\nWe set $\\mathcal K_\\varepsilon=\\{\\*x\\in\\triangle_d:\\min_a\\*x(a)\\ge\\varepsilon\/d\\}$, so that the offset optimum $\\hat{\\*x}_t^{(\\theta)}$ then has the very simple form $\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)=(1-\\varepsilon)\\hat{\\*x}_t+\\varepsilon\\*1_d\/d$, i.e. it is the mixture of the estimated optimum $\\hat{\\*x}_t$ over the entire simplex with the uniform distribution.\nNote that for MAB we will {\\em not} need to the capability of Algorithm~\\ref{alg:meta} to learn $\\varepsilon$ using multiplicative weights and can just set it assuming knowledge of the number of tasks.\nThus the method in this setting can be roughly summarized as doing the following at each task $t>1$:\n\\begin{enumerate}[noitemsep]\n\t\\item sample $\\theta_t=(\\eta_t,\\beta_t)$ from a distribution $\\*p_t$ over the discretization $\\Theta$\n\t\\item run $\\texttt{OMD}_{\\beta_t,\\eta_t}$ using the initialization $\\*x_{t,1}\n\t=\\frac1{t-1}\\sum_{s<t}\\hat{\\*x}_t^{(\\theta_t)}\n\t=\\frac\\varepsilon d\\*1_d+\\frac{1-\\varepsilon}{t-1}\\sum_{s<t}\\hat{\\*x}_t$\n\t\\item update $\\*p_{t+1}$ using multiplicative weights with the expert losses $\\frac1{\\eta_t}B_{\\phi_{\\beta_t}}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x_{t,1})+\\frac{\\eta_td^\\beta_tm}{\\beta_t}$\n\\end{enumerate}\nThe latter regret-upper-bound is derived from the within-task regret of OMD with the Tsallis regularizer.\nThis simple procedure achieves the following guarantee on the task-averaged regret:\n\n\\begin{Thm}\\label{thm:mab}\n\tSuppose $\\texttt{OMD}_{\\eta,\\beta}$ is online mirror descent with the Tsallis entropy regularizer $\\phi_\\beta$ over $\\gamma$-offset loss estimators.\n\tThen for every $\\varepsilon>0$ and $\\underline\\beta\\in(0,1]$ there exists integer $k=\\tilde\\mathcal O(\\lceil d^4\\sqrt{mT}\\log\\frac1\\varepsilon\\rceil)$ and $\\alpha,\\underline\\eta,\\overline\\eta\\in(0,\\infty)$ such that running Algorithm~\\ref{alg:meta} with $\\Theta$ the product of uniform grids of size $k$ over each non-singleton dimension of $[\\underline\\eta,\\overline\\eta]\\times[\\max\\{\\underline\\beta,1\/\\log d\\},1]\\times\\{\\varepsilon\\}$ and $\\alpha$ the meta-step-size yields w.p. at least $1-\\delta$ the task-averaged regret\n\t\\begin{align}\n\t\\begin{split}\n\t\\frac1T&\\sum_{t=1}^T\\sum_{i=1}^m\\ell_{t,i}(a_{t,i})-\\ell_{t,i}(a_t^\\ast)\\\\\n\t&\\le\\tilde\\mathcal O\\left(\n\t\\frac{\\sqrt d}{\\gamma T}\\log\\frac4\\delta\n\t+\\left(\\frac\\varepsilon{\\gamma d}+\\gamma d\\right)m\n\t+\\frac{d^{2-\\underline\\beta}\\sqrt m}{\\rho\\varepsilon^{2-\\underline\\beta}T}\n\t+\\left(\\frac{\\sqrt d}\\rho+d\\right)\\sqrt{\\frac{md}T\\log\\frac4\\delta}\n\t+\\rho d\\sqrt m\n\t\\right)\\\\\n\t&+\\qquad\\min_{\\eta>0,\\beta\\in[\\underline\\beta,1]}\\frac{H_\\beta(\\hat{\\bar{\\*x}})}\\eta+\\frac{\\eta d^\\beta m}\\beta+\\frac{\\varepsilon^\\beta d^{1-\\beta}1_{\\beta<1}}{(1-\\beta)\\eta}\n\t\\end{split}\t\n\t\\end{align}\n\twhere $H_\\beta=-\\phi_\\beta$ is the Tsallis entropy and $\\hat{\\bar{\\*x}}$ is the mean of the estimated optima $\\hat{\\*x}_1,\\dots,\\hat{\\*x}_T$.\n\\end{Thm}\n\nWe see that the regret-upper-bound is highly dependent on the loss estimator offset $\\gamma$, the boundary offset $\\varepsilon$, the step-size offset $\\rho$, and the lower bound $\\underline\\beta$ on the parameter of the Tsallis entropy.\nThus to clarify the guarantee we consider three regimes of $\\underline\\beta$:\n$\\underline\\beta=1$, i.e. always using Exp3;\n$\\underline\\beta=1\/2$, which corresponds to the standard setting when using the Tsallis entropy \\citep{abernethy2015fighting};\nand $\\underline\\beta=1\/\\log d$, below which the OMD regret-upper-bound always worsens and so it does not make sense to try $\\beta<1\/\\log d$.\n\n\\begin{Cor}\\label{cor:mab}\n\tSuppose we run Algorithm~\\ref{alg:meta} as in Theorem~\\ref{thm:mab}.\n\tFor $\\underline\\beta=1$, if we set $\\varepsilon=\\frac1{\\sqrt T}$, $\\gamma=\\frac{\\sqrt{\\log\\frac4\\delta}}{d\\sqrt[4]T}$, and $\\rho=\\frac{\\sqrt d}{\\sqrt[4]T}$ then w.p. $1-\\delta$ the task-averaged regret satisfies\n\t\\begin{equation}\\label{eq:exp3}\n\t\\tilde\\mathcal O\\left(\\frac{d^\\frac32+\\sqrt m}{\\sqrt[4]T}\\sqrt{m\\log\\frac4\\delta}\\right)+2\\sqrt{H_1(\\hat{\\bar{\\*x}})dm}\n\t\\end{equation}\n\tFor $\\underline\\beta=\\frac12$, if $\\varepsilon=\\sqrt{\\frac dT}$, $\\gamma=\\frac{\\sqrt{\\log\\frac4\\delta}}{d\\sqrt[4]T}$, and $\\rho=\\frac1{\\sqrt{md}}$ then w.p. $1-\\delta$ the task-averaged regret is\n\t\\begin{equation}\\label{eq:half}\n\t\\tilde\\mathcal O\\left(\\frac{dm\\sqrt{d\\log\\frac4\\delta}}{\\sqrt[4]T}\\right)+2\\sqrt d+2\\min_{\\beta\\in\\left[\\frac12,\\frac{\\log d-1}{\\log d}\\right]}\\sqrt{H_\\beta(\\hat{\\bar{\\*x}})d^\\beta m\/\\beta}\n\t\\end{equation}\n\tFor $\\underline\\beta=\\frac1{\\log d}$, if $\\varepsilon=\\frac1{\\sqrt[3]T}$, $\\gamma=\\frac{\\sqrt{\\log\\frac4\\delta}}{d\\sqrt[6]T}$, and $\\rho=\\frac{\\sqrt d}{\\sqrt[6]T}$ then w.p. $1-\\delta$ the task-averaged regret is \n\t\\begin{equation}\\label{eq:full}\n\t\\tilde\\mathcal O\\left(\\frac{d^\\frac32+\\sqrt m}{\\sqrt[6]T}\\sqrt{m\\log\\frac4\\delta}\\right)+2\\min_{\\beta\\in(0,1]}\\sqrt{H_\\beta(\\hat{\\bar{\\*x}})d^\\beta m\/\\beta}+\\sqrt{\\frac{d1_{\\beta<1}}{\\beta(1-\\beta)mT^\\frac\\beta 3}}\n\t\\end{equation}\n\\end{Cor}\n\nThese results show that for all three settings of $\\underline\\beta$, as the meta-learner sees more tasks the average regret depends directly on the entropy of the estimated optima-in-hindsight, a natural notion of task-similarity since it is small if most tasks are estimated to be solved by the same arms and large if all arms are used roughly the same amount.\nIt also demonstrates how our algorithm's automatic tuning of the step-size $\\eta$ allows us to set the asymptotic rate optimally depending on the entropy.\nThe algorithm's tuning of the entropy itself via $\\beta$ also enables adaptation to similar tasks;\nspecifically, a smaller $\\beta$ weights the $H_\\beta(\\hat{\\bar{\\*x}})\/\\eta$ term higher and is thus beneficial if tasks are similar.\nAs a natural example, suppose a constant $s\\ll d$ actions are always minimizers, i.e. $\\hat{\\bar{\\*x}}$ is $s$-sparse.\nThen the last bound~\\eqref{eq:full} implies that Algorithm~\\ref{alg:meta} can achieve  task-averaged regret $o_T(1)+\\mathcal O(\\sqrt{sm\\log d})$, albeit at the cost of slow convergence.\nIn-general, for the case of tuning over all $\\beta\\ge1\/\\log d$ the speed of the convergence depends on the optimal $\\beta$;\nthe algorithm will converge very slowly at rate $\\tilde\\mathcal O(1\/\\sqrt[6\\log d]T)$ if the optimal $\\beta$ is around $1\/\\log d$, but for $\\beta$ near 1 the rate will be $\\tilde\\mathcal O(1\/\\sqrt[4]T)$.\nNote that we show in the intermediate case of tuning only as low as $\\beta=1\/2$ that we can still achieve $\\tilde\\mathcal O(1\/\\sqrt[4]T)$ at the cost of a fast $2\\sqrt d$ term per-task.\nFinally, note that because the entropy is bounded by $d^{1-\\beta}$ we do asymptotically recover worst-case guarantees in all three cases if the tasks are dissimilar.\n\nTo put these results in some theoretical context, we can compare them to those of \\citet{azizi2022non}, who achieve task-averaged regret bounds of the form $\\tilde\\mathcal O(1\/\\sqrt T+\\sqrt{sm})$ in the {\\em stochastic} MAB setting, where $s$ is an unknown subset of optimal actions.\nUnlike their result, we study the harder adversarial setting and do {\\em not} place restrictions on how the tasks are related;\ndespite this greater generality, our bounds are asymptotically comparable if the estimated and true optima-in-hindsight are roughly equivalent, as we also have $\\tilde\\mathcal O(\\sqrt{sm})$ average regret as $T\\to\\infty$.\nOn the other hand, the rate in the number of tasks of \\citet{azizi2022non} is much better, albeit at a cost of runtime exponential in $s$.\nApart from generality, we believe a great strength of our result is its adaptiveness;\nunlike this work, we do not need to know how many optimal arms there are or their entropy in order to improve task-averaged regret with task-similarity.\n\\section*{Acknowledgments}\nThis material is based on work supported by the National Science Foundation under grants CCF-1910321, FAI-1939606, IIS-1901403, SCC-1952085, and SES-1919453; the Defense Advanced Research Projects Agency under cooperative agreement HR00112020003; a Simons Investigator Award; an AWS Machine Learning Research Award; an Amazon Research Award; a Bloomberg Research Grant;  a Microsoft Research Faculty Fellowship; a Google Faculty Research Award; a J.P. Morgan Faculty Award; a Facebook Research Award; a Mozilla Research Grant; and a Facebook PhD Fellowship.\n\n\n\\medskip\n\n\n\\bibliographystyle{plainnat}\n\n\\section{Proof of Theorem~\\ref{thm:meta}}\n\n\\begin{proof}\n\tWe define $\\underline\\eta=\\frac{\\rho D}{G\\sqrt m}$, $\\overline\\eta=\\frac{2DM}{G\\sqrt m}$, number of grid points $k=\\Omega(\\lceil(4D^2M^2LG+C)\\sqrt{mT}\\rceil)$,  $\\Theta=\\left\\{\\underline\\eta+\\frac jk(\\overline\\eta-\\underline\\eta)\\right\\}_{j=0}^k\\times\\{\\underline\\beta+\\frac jk(\\overline\\beta-\\underline\\beta)\\}_{j=0}^{k1_{\\overline\\beta>\\underline\\beta}}\\times\\{\\underline\\varepsilon+\\frac jk(\\overline\\varepsilon-\\underline\\varepsilon)\\}_{j=0}^{k1_{\\overline\\varepsilon>\\underline\\varepsilon}}$, and meta-step-size $\\alpha=\\frac1{DG\/\\rho+2DMG+C\\sqrt m}\\sqrt{\\frac{3\\log k}{2Tm}}$.\n\tNote that\n\t\\begin{equation}\n\t\t\\frac{\\rho D}{G\\sqrt m}\n\t\t\\le\\argmin_{\\eta>0}\\min_{\\*x\\in\\mathcal K_{\\underline\\varepsilon},\\beta,\\varepsilon}\\sum_{t=1}^T\\tilde U_t(\\*x,(\\eta,\\beta,\\varepsilon))\n\t\t\\le\\frac{DM}G\\sqrt{\\frac{1+\\rho^2}m}\n\t\t\\le\\frac{2DM}{G\\sqrt m}\n\t\\end{equation}\n\tso \n\t\\begin{equation}\n\t\t\\max_{t\\in[T]}U_t^{(\\rho)}(\\*x_t^{(\\theta_t)},\\theta_t)\n\t\t\\le\\frac{DG\\sqrt m}\\rho+2DMG\\sqrt{m}+Cm\n\t\\end{equation}\n\tTherefore applying the regret guarantee for exponentiated gradient \\citep[Corollary~2.14]{shalev-shwartz2011oco} followed by the regret of follow-the-leader on a sequent of Bregman divergences (Lemma~\\ref{lem:bregman}) yields\n\t\\begin{align}\n\t\\begin{split}\n\t\t\\mathbb E&\\sum_{t=1}^TU_t(\\*x_t^{(\\theta_t)},\\theta_t)\\\\\n\t\t&\\le\\mathbb E\\sum_{t=1}^TU_t^{(\\rho)}(\\*x_t^{(\\theta_t)},\\theta_t)\\\\\n\t\t&\\le\\left(C\\sqrt m+DG\\left(\\frac1\\rho+2M\\right)\\right)\\sqrt{2mT\\log|\\Theta|}+\\min_{\\theta\\in\\Theta}\\mathbb E\\sum_{t=1}^TU_t^{(\\rho)}(\\*x_t^{(\\theta)},\\theta)\\\\\n\t\t&\\le\\left(C\\sqrt m+DG\\left(\\frac1\\rho+2M\\right)\\right)\\sqrt{2mT\\log|\\Theta|}\\\\\n\t\t&\\qquad+\\min_{(\\eta,\\beta,\\varepsilon)\\in\\Theta}\\frac{8SK^2}\\eta(1+\\log T)+\\min_{\\*x\\in\\mathcal K_\\varepsilon}\\mathbb E\\sum_{t=1}^T\\frac{B_{\\phi_\\beta}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x)+\\rho^2D^2}\\eta+(\\eta G_\\beta^2+C\\varepsilon)m\\\\\n\t\t&\\le\\left(C\\sqrt m+DG\\left(\\frac1\\rho+2M\\right)\\right)\\sqrt{2mT\\log|\\Theta|}+\\frac{8SK^2\\overline G\\sqrt m}{\\rho D}(1+\\log T)\\\\\n\t\t&\\qquad+\\min_{(\\eta,\\beta,\\varepsilon)\\in\\Theta}\\eta G_\\beta^2mT+C\\varepsilon mT+\\frac{\\rho^2D^2T}\\eta+\\mathbb E\\sum_{t=1}^T\\frac{\\phi_\\beta(\\hat{\\*x}_t^{(\\varepsilon)})-\\phi_\\beta(\\hat{\\bar{\\*x}}^{(\\varepsilon)})}\\eta\\\\\n\t\t&\\le\\left(C\\sqrt m+DG\\left(\\frac1\\rho+2M\\right)\\right)\\sqrt{2mT\\log|\\Theta|}+\\frac{8SK^2G\\sqrt m}{\\rho D}(1+\\log T)+\\rho DGT\\sqrt m\\\\\n\t\t&\\qquad+\\left(4D^2M^2+\\left(C\\sqrt m+\\frac{2LG}{\\rho D}\\right)(\\overline\\varepsilon-\\underline\\varepsilon)\\sqrt m+2\\left(\\frac{2M}G+\\frac G\\rho\\right)DL\\sqrt m(\\overline\\beta-\\underline\\beta)\\right)\\frac Tk\\\\\n\t\t&\\qquad+\\min_{(\\eta,\\beta,\\varepsilon)\\in\\overline\\Theta}\\eta G_\\beta^2mT+C\\varepsilon mT\n\t\t+\\mathbb E\\sum_{t=1}^T\\frac{\\phi_\\beta(\\hat{\\*x}_t^{(\\varepsilon)})-\\phi_\\beta(\\hat{\\bar{\\*x}}^{(\\varepsilon)})}\\eta\\\\\n\t\t&\\le\\left(C\\sqrt m+DG\\left(\\frac1\\rho+2M\\right)\\right)\\sqrt{6mT\\log k}+(4D^2M^2LG+C)\\frac{Tm}{\\rho k}\\\\\n\t\t&\\qquad+\\frac{8SK^2G\\sqrt m}{\\rho D}(1+\\log T)+\\rho DGT\\sqrt m+\\min_{\\*x\\in\\mathcal K,\\theta\\in\\Theta^\\ast}\\mathbb E\\sum_{t=1}^TU_t(\\*x,\\theta)\n\t\\end{split}\n\t\\end{align}\n\twhere the fourth inequality follows by Claim~\\ref{clm:bregman}, the fifth by Lipschitzness of $1\/\\eta$ on $\\eta\\ge\\frac{\\rho D}{G\\sqrt m}$ and of $\\phi_\\beta(\\hat{\\*x}_t^{(\\varepsilon)}||\\cdot)$ on $\\mathcal K_{\\underline\\varepsilon}$, and the sixth by simplifying and substituting the lower bound for $k$.\n\tThe w.h.p. version of the bound follows by applying \\citet[Lemma~4.1]{cesa-bianchi2006prediction} when obtaining the second inequality.\n\\end{proof}\n\n\\begin{Lem}\\label{lem:bregman}\n\tLet $\\phi:\\mathcal K\\mapsto\\mathbb R_{\\ge0}$ be a strictly-convex function with $\\max_{\\*x\\in\\mathcal K}\\|\\nabla^2\\phi(\\*x)\\|_2\\le S$ over a convex set $\\mathcal K\\subset\\mathbb R^d$ with $\\max_{\\*x\\in\\mathcal K}\\|\\*x\\|_2\\le K$.\n\tThen for any points $\\*x_1,\\dots,\\*x_T\\in\\mathcal K$ the actions $\\*y_1=\\argmin_{\\*x\\in\\mathcal K}\\phi(\\*x)$ and $\\*y_t=\\frac1{t-1}\\sum_{s<t}\\*x_s$ have regret\n\t\\begin{equation}\n\t\t\\sum_{t=1}^TB_\\phi(\\*x_t||\\*y_t)-B_\\phi(\\*x_t||\\*y_{T+1})\n\t\t\\le\\sum_{t=1}^T\\frac{8SK^2}{2t-1}\n\t\t\\le8SK^2(1+\\log T)\n\t\\end{equation}\n\\end{Lem}\n\\begin{proof}\n\tNote that\n\t\\begin{equation}\n\t\t\\nabla_{\\*y}B_\\phi(\\*x||\\*y)\n\t\t=-\\nabla\\phi(\\*y)-\\nabla_{\\*y}\\langle\\nabla\\phi(\\*y),\\*x\\rangle+\\nabla_{\\*y}\\langle\\nabla\\phi(\\*y),\\*y\\rangle\n\t\t=\\operatorname{diag}(\\nabla^2\\phi(\\*y))(\\*y-\\*x)\n\t\\end{equation}\n\tso $B_\\phi(\\*x_t||\\*y)$ is $2SK$-Lipschitz w.r.t. the Euclidean norm.\n\tApplying \\citet[Proposition~B.1]{khodak2019adaptive}  yields the result.\n\\end{proof}\n\n\\begin{Clm}\\label{clm:bregman}\n\t\tLet $\\phi:\\mathcal K\\mapsto\\mathbb R$ be a strictly-convex function over a convex set $\\mathcal K\\subset\\mathbb R^d$ containing points $\\*x_1,\\dots,\\*x_T$.\n\t\tThen their mean $\\bar{\\*x}=\\frac1T\\sum_{t=1}^T\\*x_t$ satisfies\n\t\t\\begin{equation}\n\t\t\t\\sum_{t=1}^TB_\\phi(\\*x_t||\\bar{\\*x})\n\t\t\t=\\sum_{t=1}^T\\phi(\\*x_t)-\\phi(\\bar{\\*x})\n\t\t\\end{equation}\n\\end{Clm}\n\\begin{proof}\n\t\\begin{align}\n\t\\begin{split}\n\t\\sum_{t=1}^TB_\\phi(\\*x_t||\\bar{\\*x})\n\t&=\\sum_{t=1}^T\\phi(\\*x_t)-\\phi(\\bar{\\*x})-\\langle\\nabla\\phi(\\bar{\\*x}),\\*x_t-\\bar{\\*x}\\rangle\\\\\n\t&=\\sum_{t=1}^T\\phi(\\*x_t)-\\phi(\\bar{\\*x})-\\langle\\nabla\\phi(\\bar{\\*x}),\\sum_{t=1}^T\\*x_t-\\bar{\\*x}\\rangle\\\\\n\t&=\\sum_{t=1}^T\\phi(\\*x_t)-\\phi(\\bar{\\*x})\n\t\\end{split}\n\t\\end{align}\n\\end{proof}\n\n\\newpage\n\\section{Proof of Theorem~\\ref{thm:mab}}\n\\begin{proof}\n\tSince $\\varepsilon$ is constant we use the shorthand $\\hat{\\*x}_t^{(\\varepsilon)}=\\hat{\\*x}_t^{(\\theta)}~\\forall~\\theta\\in\\Theta$.\n\tNote that we use search space $\\Theta=\\left[\\frac\\rho{\\sqrt m},2\\sqrt{\\frac{d\\log d}{em}}\\right]\\times\\left[\\max\\left\\{\\frac1{\\log d},\\underline\\beta\\right\\},1\\right]\\times\\{\\varepsilon\\}$.\n\tWe have the constants $D\\le\\sqrt d$, $G\\le\\sqrt d$, $M\\le\\sqrt{\\frac{d\\log d}e}$, $S\\le\\left(\\frac d \\varepsilon\\right)^{2-\\underline\\beta}$, and $K=1$.\n\tNote that the second term $d^{1-\\beta}m\/\\beta$ is decreasing on $\\beta<1\/\\log d$, so since $\\phi_\\beta$ is always increasing in $\\beta$ we know that the optimal $\\beta$ is in $[1\/\\log d,1]$.\n\tNote that by Lemma~\\ref{lem:tsallis} we have that $L=d\\log\\frac d\\varepsilon$.\n\tWe thus have\n\t\\begin{align}\n\t\\begin{split}\n\t\\sum_{t=1}^T&\\sum_{i=1}^m\\ell_{t,i}(a_{t,i})-\\ell_{t,i}(a_t^\\ast)\\\\\n\t&\\le\\sum_{t=1}^T\\sum_{i=1}^m\\langle\\hat\\ell_{t,i},\\*x_{t,i}-\\ell_{t,i}(a_t^\\ast)\\rangle+\\gamma\\sum_{a=1}^d\\hat\\ell_{t,i}(a)\\\\\n\t&\\le\\sum_{t=1}^T\\frac{B_{\\phi_{\\beta_t}}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x_{t,1})}{\\eta_t}+\\sum_{i=1}^m\\langle\\hat\\ell_{t,i},\\hat{\\*x}_{t,i}^{(\\varepsilon)}\\rangle-\\ell_{t,i}(a_t^\\ast)+\\frac{\\eta_t}{\\beta_t}\\sum_{a=1}^d\\*x_{t,i}^{2-\\beta_t}(a)\\hat\\ell_{t,i}^2(a)+\\gamma\\sum_{a=1}^d\\hat\\ell_{t,i}(a)\\\\\n\t&\\le\\frac{\\varepsilon mT}{\\gamma d}+\\sum_{t=1}^T\\frac{B_{\\phi_{\\beta_t}}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x_{t,1})}{\\eta_t}+\\sum_{i=1}^m\\hat\\ell_{t,i}(a_t^\\ast)-\\ell_{t,i}(a_t^\\ast)\\\\\n\t&\\qquad+\\sum_{t=1}^T\\frac{\\eta_t}{\\beta_t}\\sum_{i=1}^m\\sum_{a=1}^d\\*x_{t,i}^{1-\\beta_t}(a)\\hat\\ell_{t,i}(a)+\\gamma\\sum_{a=1}^d\\hat\\ell_{t,i}(a)\\\\\n\t&\\le\\frac{\\varepsilon mT}{\\gamma d}+\\frac{1+\\frac{\\overline\\eta}{\\underline\\beta}+\\gamma}{2\\gamma}\\log\\frac4\\delta+\\sum_{t=1}^T\\frac{B_{\\phi_{\\beta_t}}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x_{t,1})}{\\eta_t}\\\\\n\t&\\qquad+\\sum_{t=1}^T\\frac{\\eta_t}{\\beta_t}\\sum_{i=1}^m\\sum_{a=1}^d\\*x_{t,i}^{1-\\beta_t}(a)\\ell_{t,i}(a)+\\gamma\\sum_{a=1}^d\\ell_{t,i}(a)\\\\\n\t&\\le\\frac{\\varepsilon mT}{\\gamma d}+\\frac{1+\\sqrt{\\frac{d\\log^3d}{em}}}\\gamma\\log\\frac4\\delta+\\gamma dmT+\\sum_{t=1}^T\\frac{B_{\\phi_{\\beta_t}}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x_{t,1})}{\\eta_t}+\\frac{\\eta_td^{\\beta_t}m}{\\beta_t}\\\\\n\t&\\le\\frac{\\varepsilon mT}{\\gamma d}+\\frac{1+\\sqrt{\\frac{d\\log^3d}{em}}}\\gamma\\log\\frac4\\delta+\\gamma dmT+\\min_{\\*x\\in\\triangle_d,\\eta>0,\\beta\\in[\\underline\\beta,1]}\\sum_{t=1}^T\\frac{B_{\\phi_\\beta}(\\hat{\\*x}_t^{(\\varepsilon)}||\\*x)}\\eta+\\frac{\\eta d^\\beta m}\\beta\\\\\n\t&\\qquad+2d\\left(\\frac1\\rho+\\sqrt{\\frac{d\\log d}e}\\right)\\sqrt{6mT\\log\\frac{4k}\\delta}+\\frac{8d^{2-\\underline\\beta}\\sqrt{m}}{\\rho\\varepsilon^{2-\\underline\\beta}}(1+\\log T)+\\rho dT\\sqrt m\\\\\n\t&\\le\\left(\\frac\\varepsilon{\\gamma d}+\\gamma d\\right)mT+\\frac{1+\\sqrt{\\frac{d\\log^3d}{em}}}\\gamma\\log\\frac4\\delta+T\\min_{\\eta>0,\\beta\\in[\\underline\\beta,1]}\\frac{H_\\beta(\\hat{\\bar{\\*x}})}\\eta+\\frac{\\eta d^\\beta m}\\beta+\\frac{\\varepsilon^\\beta d^{1-\\beta}1_{\\beta<1}}{(1-\\beta)\\eta}\\\\\n\t&\\qquad+2d\\left(\\frac1\\rho+\\sqrt{\\frac{d\\log d}e}\\right)\\sqrt{6mT\\log\\frac{4k}\\delta}+\\frac{8d^{2-\\underline\\beta}\\sqrt{m}}{\\rho\\varepsilon^{2-\\underline\\beta}}(1+\\log T)+\\rho dT\\sqrt m\n\t\\end{split}\n\t\\end{align}\n\twhere the second inequality follows by Lemma~\\ref{lem:mirror}, the third by H\\\"older's inequality and the definitions $\\hat\\ell_{t,i}$ and $\\hat{\\*x}_{t,i}^{(\\varepsilon)}$, the fourth by \\citet[Lemma~1]{neu2015explore}, the fifth by the definition of $\\ell_{t,i}$, the sixth by Theorem~\\ref{thm:meta}, and the last by the derivation below for $\\beta<1$ (otherwise it holds by joint convexity of the KL-divergence) followed by Claim~\\ref{clm:bregman} combined with the fact that the entropy of optima-in-hindsight is zero.\n\t\\begin{align}\n\t\\begin{split}\n\t-\\phi_\\beta((1-\\varepsilon)\\*x+\\varepsilon\\*1_d\/d)\n\t&=\\frac{\\sum_{a=1}^d((1-\\varepsilon)\\*x(a)+\\varepsilon\/d)^\\beta-1}{1-\\beta}\\\\\n\t&\\le\\frac{\\varepsilon^\\beta d^{1-\\beta}+(1-\\varepsilon)^\\beta\\sum_{a=1}^d\\*x^\\beta(a)-1}{1-\\beta}\n\t\\le\\frac{\\varepsilon^\\beta d^{1-\\beta}}{1-\\beta}\n\t\\end{split}\n\t\\end{align}\n\\end{proof}\n\n\\begin{Lem}\\label{lem:mirror}\n\tSuppose we play $\\texttt{OMD}_{\\beta,\\eta}$ with regularizer $\\phi_\\beta$ the negative Tsallis entropy and initialization $\\*x_1\\in\\triangle_d$ on the sequence of linear loss functions  $\\ell_1,\\dots,\\ell_T\\in[0,1]^d$.\n\tThen for any $\\*x^\\ast\\in\\triangle_d$ we have\n\t\\begin{equation}\n\t\t\\sum_{t=1}^T\\langle\\ell_t,\\*x_t-\\*x^\\ast\\rangle\n\t\t\\le\\frac{B_{\\phi_\\beta}(\\*x^\\ast||\\*x_1)}\\eta+\\frac\\eta\\beta\\sum_{a=1}^d\\*x_t^{2-\\beta}(a)\\ell_t^2(a)\n\t\\end{equation}\n\\end{Lem}\n\\begin{proof}\n\tNote that the following proof follows parts of the course notes by \\citet{luo2017tsallis}, which we reproduce for completeness.\n\tThe OMD update at each step $t$ involves the following two steps: set $\\*y_{t+1}\\in\\triangle_d$ s.t. $\\nabla\\phi_\\beta(\\*y_{t+1})=\\nabla\\phi_\\beta(\\*x_t)-\\eta\\ell_t$ and then set $\\*x_{t+1}=\\argmin_{\\*x\\in\\triangle_d}B_{\\phi_\\beta}(\\*x,\\*y_{t+1})$ \\citep[Algorithm~14]{hazan2015oco}.\n\tNote that by \\citet[Equation~5.3]{hazan2015oco} and nonnegativity of the Bregman divergence we have\n\t\\begin{equation}\n\t\t\\sum_{t=1}^T\\langle\\ell_t,\\*x_t-\\*x^\\ast\\rangle\n\t\t\\le\\frac{B_{\\phi_\\beta}(\\*x^\\ast||\\*x_1)}\\eta+\\frac1\\eta\\sum_{t=1}^TB_{\\phi_\\beta}(\\*x_t||\\*y_{t+1})\n\t\\end{equation}\n\tTo bound the second term, note that when $\\phi_\\beta$ is the negative Tsallis entropy we have\n\t\\begin{align}\n\t\\begin{split}\n\tB_{\\phi_\\beta}&(\\*x_t||\\*y_{t+1})\\\\\n\t&=\\frac1{1-\\beta}\\sum_{a=1}^d\\left(\\*y_{t+1}^\\beta(a)-\\*x_t^\\beta(a)+\\frac\\beta{\\*y_{t+1}^{1-\\beta}(a)}(\\*x_t(a)-\\*y_{t+1}(a)\\right)\\\\\n\t&=\\frac1{1-\\beta}\\sum_{a=1}^d\\left((1-\\beta)\\*y_{t+1}^\\beta(a)-\\*x_t^\\beta(a)+\\beta\\left(\\frac1{\\*x_t^{1-\\beta}(a)}+\\frac{1-\\beta}\\beta\\eta\\ell_t(a)\\right)\\*x_t(a)\\right)\\\\\n\t&=\\sum_{a=1}^d\\left(\\*y_{t+1}^\\beta(a)-\\*x_t^\\beta(a)+\\eta\\*x_t(a)\\ell_t(a)\\right)\n\t\\end{split}\n\t\\end{align}\n\tPlugging the following result, which follows from $(1+x)^\\alpha\\le1+\\alpha x+\\alpha(\\alpha-1)x^2~\\forall~x\\ge0,\\alpha<0$, into the above yields the desired bound.\n\t\\begin{align}\n\t\\begin{split}\n\t\\*y_{t+1}^\\beta(a)\n\t=\\*x_t^\\beta(a)\\left(\\frac{\\*y_{t+1}^{\\beta-1}(a)}{\\*x_t^{\\beta-1}(a)}\\right)^\\frac\\beta{\\beta-1}\n\t&=\\*x_t^\\beta(a)\\left(1+\\frac{1-\\beta}\\beta\\eta\\*x_t^{1-\\beta}(a)\\ell_t(a)\\right)^\\frac\\beta{\\beta-1}\\\\\n\t&\\le\\*x_t^\\beta(a)\\left(1-\\eta\\*x_t^{1-\\beta}(a)\\ell_t(a)+\\frac{\\eta^2}\\beta\\*x_t^{2-2\\beta}(a)\\ell_t(a)^2\\right)\\\\\n\t&=\\*x_t^\\beta(a)-\\eta\\*x_t(a)\\ell_t(a)+\\frac{\\eta^2}\\beta\\*x_t^{2-\\beta}(a)\\ell_t(a)^2\n\t\\end{split}\n\t\\end{align}\n\\end{proof}\n\n\\newpage\n\\begin{Lem}\\label{lem:tsallis}\n\tFor any $\\rho\\in(0,1\/d]$ and $\\*x\\in\\triangle_d$ s.t. $\\*x(a)\\ge\\rho~\\forall~a\\in[d]$ the $\\beta$-Tsallis entropy $H_\\beta(\\*x)=-\\frac{1-\\sum_{a=1}^d\\*x^\\beta(a)}{1-\\beta}$ is $d\\log\\frac1\\rho$-Lipschitz w.r.t. $\\beta\\in[0,1]$.\n\\end{Lem}\n\\begin{proof}\n\tLet $\\log_\\beta x=\\frac{x^{1-\\beta}-1}{1-\\beta}$ be the $\\beta$-logarithm function and note that by \\citet[Equation~6]{yamano2002tsallis} we have\n\t$\\log_\\beta x-\\log x=(1-\\beta)(\\partial_b\\log_\\beta x+\\log_\\beta x\\log x)\\ge0~\\forall~\\beta\\in[0,1]$.\n\tThen we have for $\\beta\\in[0,1)$ that\n\t\\begin{align}\n\t\\begin{split}\n\t|\\partial_\\beta H_\\beta(\\*x)|\n\t&=\\left|\\frac{-H_\\beta(\\*x)-\\sum_{a=1}^d\\*x^\\beta(a)\\log\\*x(a)}{1-\\beta}\\right|\\\\\n\t&=\\frac1{1-\\beta}\\left|\\sum_{a=1}^d\\*x^\\beta(a)(\\log_\\beta\\*x(a)-\\log\\*x(a))\\right|\\\\\n\t&=\\frac1{1-\\beta}\\sum_{a=1}^d\\*x^\\beta(a)(\\log_\\beta\\*x(a)-\\log\\*x(a))\\\\\n\t&\\le\\frac1{1-\\beta}\\left(\\sum_{a=1}^d\\*x(a)\\right)^\\beta\\left(\\sum_{a=1}^d(\\log_\\beta\\*x(a)-\\log\\*x(a))^{\\frac1{1-\\beta}}\\right)^{1-\\beta}\\\\\n\t&\\le\\frac1{1-\\beta}\\sum_{a=1}^d\\log_\\beta\\*x(a)-\\log\\*x(a)\\\\\n\t&\\le\\frac d{1-\\beta}(\\log_\\beta\\rho-\\log\\rho)\\\\\n\t&\\le-d\\log\\rho\n\t\\end{split}\n\t\\end{align}\n\twhere the fourth line follows by H\\\"older's inequality, the fifth by subadditivity of $x^a$ for $a\\in(0,1]$, the sixth by the fact that $\\partial_x(\\log_\\beta x-\\log x)=x^{-\\beta}-1\/x\\le0~\\forall~\\beta,x\\in[0,1)$, and the last line by substituting $\\beta=0$ since $\\partial_\\beta\\left(\\frac{\\log_\\beta\\rho-\\log\\rho}{1-\\beta}\\right)\n\t=\\frac{2(\\rho-\\rho^\\beta)-(1-\\beta)(\\rho^\\beta+\\rho)\\log\\rho}{\\rho^\\beta(1-\\beta)^3}\\le0~\\forall~\\beta\\in[0,1),\\rho\\in(0,1\/d]$.\n\tFor $\\beta=1$, applying L'H\\^opital's rule yields\n\t\\begin{equation}\n\t\\lim_{\\beta\\to1}\\partial_\\beta H_\\beta(\\*x)\n\t=-\\frac12\\lim_{\\beta\\to1}\\sum_{a=1}^d\\*x^\\beta(a)\\log^2\\*x(a)(1-(1-\\beta)\\log\\*x(a))\n\t=-\\frac12\\sum_{a=1}^d\\*x(a)\\log^2\\*x(a)\n\t\\end{equation}\n\twhich is bounded on $[-2d\/e^2,0]$.\n\\end{proof}\n\n\\newpage\n\\section{Proof of Theorem~\\ref{thm:blo}}\n\\begin{proof}\n\tApplying Theorem~\\ref{thm:meta} with constants $D=D_{\\underline\\varepsilon}$, $G=4d\\sqrt 2$, $M=1$, $S=S_{\\underline\\varepsilon}$, and $K=K$ yields\n\t\\begin{align}\n\t\\begin{split}\n\t\\mathbb E&\\sum_{t=1}^T\\sum_{i=1}^m\\langle\\ell_{t,i},\\*x_{t,i}-\\*x_t^\\ast\\rangle\\\\\n\t&\\le\\mathbb E\\sum_{t=1}^T\\varepsilon_tm+\\sum_{i=1}^m\\langle\\ell_{t,i},\\*x_{t,i}-\\operatorname{OPT}_{\\varepsilon_t}(\\ell_t)\\rangle\\\\\n\t&=\\mathbb E\\sum_{t=1}^T\\varepsilon_tm+\\sum_{i=1}^m\\langle\\hat\\ell_{t,i},\\*x_{t,i}-\\operatorname{OPT}_{\\varepsilon_t}(\\ell_t)\\rangle\\\\\n\t&\\le\\mathbb E\\sum_{t=1}^T\\varepsilon_tm+\\sum_{i=1}^m\\langle\\hat\\ell_{t,i},\\*x_{t,i}-\\hat{\\*x}_t^{(\\theta_t)}\\rangle\\\\\n\t&\\le\\mathbb E\\sum_{t=1}^T\\frac{B_\\phi(\\hat{\\*x}_t^{(\\theta_t)}||\\*x_{t,1}^{(\\theta_t)})}{\\eta_t}+(\\eta_tG^2+\\varepsilon_t)m\\\\\n\t&\\le\\left(\\sqrt m+\\frac{4D_{\\underline\\varepsilon}G}\\rho\\right)\\sqrt{6mT\\log k}+\\frac{8S_{\\underline\\varepsilon}K^2G\\sqrt m}{\\rho D_{\\underline\\varepsilon}}(1+\\log T)+\\rho D_{\\underline\\varepsilon}GT\\sqrt m\\\\\n\t&\\qquad+\\min_{\\*x\\in\\mathcal K,\\eta>0,\\varepsilon\\in[\\underline\\varepsilon,\\overline\\varepsilon]}\\mathbb E\\sum_{t=1}^T\\frac{B_\\phi(\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)||\\*x)}\\eta+(\\eta G^2+\\varepsilon)m\\\\\n\t&\\le72d\\sqrt m\\sqrt[4]T\\left(D_{\\underline\\varepsilon}\\sqrt{mT\\log k}+\\frac{S_{\\underline\\varepsilon}K^2}{D_{\\underline\\varepsilon}}(1+\\log T)\\right)\\\\\n\t&\\qquad+\\min_{\\*x\\in\\mathcal K,\\eta>0,\\varepsilon\\in[\\underline\\varepsilon,\\overline\\varepsilon]}\\mathbb E\\sum_{t=1}^T\\frac{B_\\phi(\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)||\\*x)}\\eta+(32\\eta d^2+\\varepsilon)m\\\\\n\t\\end{split}\n\t\\end{align}\n\twhere the third inequality follows from Lemma~\\ref{lem:concordant}.\n\\end{proof}\n\n\\begin{Lem}\\label{lem:concordant}\n\tLet $\\overline\\mathcal K\\subset\\mathbb R^d$ be a convex set and $\\phi$ be a self-concordant barrier.\n\tSuppose $\\ell_1,\\dots,\\ell_T$ are a sequence of loss functions satisfying $|\\langle\\ell_t,\\*x\\rangle|\\le1~\\forall~\\*x\\in\\mathcal K$.\n\tThen if we run OMD with step-size $\\eta>0$ as in \\citet[Algorithm~1]{abernethy2008competing} on the sequence of estimators $\\hat\\ell_t$ our estimated regret w.r.t. any $\\*x^\\ast\\in\\mathcal K_\\varepsilon$ for $\\varepsilon>0$ will satisfy\n\t\\begin{equation}\n\t\\sum_{t=1}^T\\langle\\hat\\ell_t,\\*x_t-\\*x^\\ast\\rangle\\le\\frac{B_\\phi(\\*x^\\ast||\\*x_1)}\\eta+32d^2\\eta T\n\t\\end{equation}\n\\end{Lem}\n\\begin{proof}\n\tThe result follows from \\citet{abernethy2008competing} by stopping the derivation on the second inequality below Equation~10.\n\\end{proof}\n\\section{Related work}\n\nWhile we are the first to consider meta-learning under adversarial bandit feedback, many have studied meta-learning in various {\\em stochastic} bandit settings~\\citep{sharaf2021meta, simchowitz2021bayesian, kveton2020meta, cella2020meta, kveton2021meta, basu2021no, azizi2022non, lazaric2013sequential}. \\citet{kveton2021meta}, \\citet{basu2021no}, and \\citet{simchowitz2021bayesian} study meta-learning algorithms for the Bayesian bandit setting. \\citet{kveton2020meta} and \\citet{sharaf2021meta} consider meta-learning for contextual bandits, although they allow their algorithms to have offline access to a set of training tasks for which full feedback is available. \\citet{cella2020meta} and \\citet{moradipari2022multi} provide algorithms based on OFUL~\\citep{abbasi2011improved} for meta-learning in stochastic linear bandits under various assumptions on how the bandit learning tasks are generated. \\citet{azizi2022non} study a setting in which a meta-learner faces a sequence of stochastic multi-armed bandit tasks. While the sequence of tasks may be adversarially designed, the adversary is constrained to choose the optimal arm for each task from a smaller but unknown subset of arms. In contrast to \\cite{cella2020meta, moradipari2022multi, azizi2022non}, we make no assumptions about how the sequence of tasks is generated and our guarantees adapt to a natural measure of similarity between tasks.\n\nTheoretically our analysis draws on the average regret-upper-bound analysis (ARUBA) framework of \\citet{khodak2019adaptive}, which was designed for meta-learning under full information.\nWhile the general approach is not restricted by convexity~\\citep{balcan2021ltl} and has  been combined with bandit algorithms on the meta-level~\\citep{khodak2021fedex}, the existing results cannot be applied to OMD methods for within-task learning under bandit feedback because the associated regularizers are non-Lipschitz or sometimes even unbounded near the boundaries of the action space.\nWe thus require a specialized analysis for the bandit setting.\n\\citet{denevi2019meta} also study an OMD-based algorithm for meta-learning in the online setting, but their results are also only applicable in the full information setting.\\looseness-1\n\n\\section{Learning the regularizers of bandit algorithms}\\label{sec:setup}\n\nWe consider the problem of meta-learning across bandit tasks $t=1,\\dots,T$ over some fixed set $\\mathcal K\\subset\\mathbb R^d$. \nOn each round $i=1,\\dots,m$ of task $t$ we play action $\\*x_{t,i}\\in\\mathcal K$ and receive feedback $\\ell_{t,i}(\\*x_{t,i})$ for some function $\\ell_{t,i}:\\mathcal K\\mapsto[-1,1]$.\nNote that all functions we consider will be linear and so we will also write $\\ell_{t,i}(\\*x)=\\langle\\ell_{t,i},\\*x\\rangle$. Additionally, we allow each $\\ell_{t,i}$ to be chosen by an \\emph{oblivious adversary}, i.e. an adversary with knowledge of the algorithm that must select $\\ell_{t,i}$ independent of $\\*x_{t,i}$.\nWe will also denote $\\*x(a)$ to be the $a$th element of the vector $\\*x\\in\\mathbb R^d$, $\\overline\\mathcal K$ to be the convex hull of $\\mathcal K$, and $\\triangle_n$ to be the simplex on $n$ elements.\nFinally, note that all proofs can be found in the Appendix.\\looseness-1\n\nIn online learning, the goal on a single task $t$ is to play actions $\\*x_{t,1},\\dots\\*x_{t,m}$ that minimize the regret $\\sum_{i=1}^m\\ell_{t,i}(\\*x_{t,i})-\\ell_{t,i}(\\*x_t^\\ast)$, where $\\*x_t^\\ast\\in\\argmin_{\\*x\\in\\mathcal K}\\sum_{i=1}^m\\ell_{t,i}(\\*x)$.\nLifting this to the meta-learning setting, our goal as in past work \\citep{khodak2019adaptive,balcan2021ltl} will be to minimize the {\\bf task-averaged regret}\n\\begin{equation}\\label{eq:tar}\n\t\\frac1T\\sum_{t=1}^T\\sum_{i=1}^m\\ell_{t,i}(\\*x_{i,t})-\\ell_{t,i}(\\*x_t^\\ast)\n\\end{equation}\nIn-particular, we hope to use multi-task data in order to improve average performance as the number of tasks $T\\to\\infty$, e.g. by attaining a task-averaged regret of $o_T(1)+\\tilde\\mathcal O(V\\sqrt m)$, where $V\\in\\mathbb R_{\\ge0}$ is a measure of task-similarity that is small if the tasks are similar but still yields the worst-case single-task performance if they are not.\n\n\\subsection{Online mirror descent as a base-learner}\\label{ssec:mirror}\n\nIn meta-learning we are commonly interested in learning a within-task algorithm or {\\bf base-learner}, a parameterized method that we run on each task $t$.\nA popular approach, both empirically~\\citep{finn2017maml,nichol2018reptile} and theoretically~\\citep{khodak2019adaptive,denevi2019ltlsgd}, is to learn the initialization and sometimes other parameters of a gradient-based method such as stochastic gradient descent.\nThe hope is that optimal parameters for each task are close to each other and thus a meta-learned initialization will result in a strong model after only a few steps.\nIn this paper we take a similar approach applied to online mirror descent, a generalization of gradient descent to non-Euclidean geometries~\\citep{beck2003mirror}.\nGiven a strictly convex {\\bf regularizer} $\\phi:\\overline\\mathcal K\\mapsto\\mathbb R$ and step-size $\\eta>0$, this method performs the update\\looseness-1\n\\begin{equation}\\label{eq:mirror}\n\t\\*x_{t,i+1}=\\argmin_{\\*x\\in\\overline\\mathcal K}B_\\phi(\\*x||\\*x_{t,1})+\\eta\\sum_{j<i}\\langle\\nabla\\ell_{t,j}(\\*x_{t,j}),\\*x\\rangle\n\\end{equation}\nwhere $B_\\phi(\\*x||\\*y)=\\phi(\\*x)-\\phi(\\*y)-\\langle\\nabla\\phi(\\*y),\\*x-\\*y\\rangle$ is the {\\bf Bregman divergence} of $\\phi$.\nOMD recovers online gradient descent when $\\phi(\\*x)=\\frac12\\|\\*x\\|_2^2$, in which case $B_\\phi(\\*x||\\*y)=\\frac12\\|\\*x-\\*y\\|_2^2$;\nanother important example is {\\bf exponentiated gradient}, for which $\\phi(\\*p)=\\langle\\*p,\\log\\*p\\rangle$ is the negative Shannon entropy on probability vectors $\\*p\\in\\triangle_n$ and $B_\\phi$ is the KL-divergence~\\citep{shalev-shwartz2011oco}.\nAn important property of $B_\\phi$ is that the sum over functions $B_\\phi(\\*x_t||\\cdot)$ is minimized at the mean $\\bar{\\*x}$ of the points $\\*x_1,\\dots,\\*x_T$.\n\nWhile originally developed for online convex optimization, mirror descent using {\\bf loss estimators} $\\hat\\ell_{t,i}$ constructed using bandit feedback $\\ell_{t,i}(\\*x_{t,i})$ forms an important class of methods for bandit settings~\\citep{abernethy2008competing,neu2015explore,abernethy2015fighting}, including the famous Exp3 method \\citep{auer2002exp3}.\nLearning the initialization for online mirror descent has been considered for full-information meta-learning \\citep{khodak2019adaptive,denevi2019meta}, but these papers do not apply to the types of regularizers $\\phi$ required for bandits, which are often non-Lipschitz and sometimes even unbounded on the boundary of $\\overline\\mathcal K$ \\citep{abernethy2008competing,abernethy2015fighting}.\nFor example, \\citet{khodak2019adaptive} also take advantage of the mean-as-minimizer property of $B_\\phi$ and learn both the initialization and step-size $\\eta$, but they assume the gradient of $B_\\phi$ is bounded on the domain, which does not hold if $\\phi$ is non-Lipschitz, e.g. if $\\phi$ is the negative Shannon entropy as in Exp3.\n\nIn this paper we resolve these issues by meta-learning to initialize and tune mirror descent when it employs a regularizer used by bandit methods.\nFollowing the average regret-upper-bound analysis (ARUBA) framework of \\citet{khodak2019adaptive}, we do this by online learning a sequence of losses $U_t(\\*x,\\theta)$, each of which is a hyperparameter $\\theta$-dependent affine function of a Bregman divergence from an initialization $\\*x\\in\\overline\\mathcal K$ to some known fixed point in $\\overline\\mathcal K$.\nWe are interested in learning such functions because the regret after $m$ rounds of OMD initialized at $\\*x$ with step-size $\\eta$ is usually upper-bounded by $\\frac1\\etaB_\\phi(\\*x_t^\\ast||\\*x)+\\mathcal O(\\eta m)$ for $\\*x_t^\\ast$, the optimum-in-hindsight on task~$t$~\\citep{shalev-shwartz2011oco,hazan2015oco}.\\looseness-1\n\nUnlike past work, we use a parameter $\\varepsilon>0$ to constrain this optimum to lie in a convex subset $\\mathcal K_\\varepsilon\\subset\\overline\\mathcal K$ whose boundary is $\\varepsilon$-away from that of $\\overline\\mathcal K$ and which satisfies $\\mathcal K_\\varepsilon\\subset\\mathcal K_{\\varepsilon'}$ whenever $\\varepsilon\\le\\varepsilon'$;\nfor example, we use $\\mathcal K_\\varepsilon=\\{\\*x\\in\\triangle_d:\\min_a\\*x(a)\\ge\\varepsilon\/d\\}$ for the simplex.\nThus, unlike with full-information, the feedback we receive from the within-task algorithm will be the minimizer $\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)=\\argmin_{\\*x\\in\\mathcal K_\\varepsilon}\\langle\\hat\\ell_t,\\*x\\rangle$ of the estimated loss $\\hat\\ell_t=\\sum_{i=1}^m\\hat\\ell_{t,i}$ over the $\\varepsilon$-constrained subset, where we can pick $\\varepsilon\\in(0,1)$.\nThis allows us to handle regularizers that diverge near the boundary, but also introduces $\\varepsilon$-dependent error terms to handle.\nIn the BLO case it also forces us to automatically tune $\\varepsilon$ itself, as initializing too close to the boundary leads to unbounded regret while initializing too far away does not take advantage of the task-similarity.\n\nThus in full generality the upper bounds of interest are functions of the initialization $\\*x$ and three parameters:\nthe step-size $\\eta>0$, a parameter $\\beta$ of the regularizer $\\phi_\\beta$, and the boundary offset $\\varepsilon>0$.\n\\begin{equation}\\label{eq:rub}\n\tU_t(\\*x,(\\eta,\\beta,\\varepsilon))\n\t=\\frac{B_{\\phi_\\beta}(\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)||\\*x)}\\eta+(\\eta G_\\beta^2+C\\varepsilon)m\n\\end{equation}\nHere $G_\\beta\\ge1,C\\ge0$ are constants and $\\beta$ parameterizes the regularizer $\\phi_\\beta$, e.g. the negative Tsallis entropy used to attain optimal dependence on dimension for MAB~\\citep{abernethy2015fighting}.\nThe reason to optimize this sequence of upper bounds is because the resulting average regret directly bounds the task-averaged regret, apart from some $o_T(1)$ terms.\nFurthermore, an affine sum over Bregman divergences is minimized at the average optimum in hindsight, which leads to natural and problem specific task-similarity measures $V$ \\citep{khodak2019adaptive};\nspecifically, $V$ is the square root of the average divergence between optima in hindsight and their mean, which is small if the tasks are optimized by similar parameters.\\looseness-1\n\n\\subsection{A meta-algorithm for tuning bandit algorithms}\\label{ssec:algo}\n\nHaving specified our meta-goal---learning to initialize and tune OMD for regularizers $\\phi_\\beta$ use in bandit tasks---we now detail our meta-algorithm for doing so, pseudo-code for which is in Algorithm~\\ref{alg:meta}.\nAt a high level, the method simultaneously learns the initialization by taking the mean of $\\mathcal K_\\varepsilon$-constrained estimated optima-in-hindsight---i.e. follow-the-leader over the Bregman divergences in \\eqref{eq:rub}---while simultaneously tuning OMD via multiplicative weights over a discrete grid $\\Theta$ over $\\theta=(\\eta,\\beta,\\varepsilon)$.\n\nIn more detail, the algorithm assumes two primitives discussed above: \n(1) the base-learner $\\texttt{OMD}_{\\eta,\\beta}$ that outputs an estimated cumulative loss $\\hat\\ell_t\\in\\mathbb R^d$ after running online mirror descent over the $m$ losses $\\ell_{t,1},\\dots,\\ell_{t,m}$ of task $t$, and (2) an optimizer $\\operatorname{OPT}_\\varepsilon$ that, given a vector $\\*c\\in\\mathbb R^d$, finds the minimum of $\\langle\\*c,\\cdot\\rangle$ over $\\mathcal K_\\varepsilon$.\nAlgorithm~\\ref{alg:meta} maintains a categorical distribution $\\*p_t$ over a finite set $\\Theta\\subset\\mathbb R^3$ containing triples $\\theta=(\\eta,\\beta,\\varepsilon)$, each with its own associated initialization $\\*x_t^{(\\theta)}$;\nat each task $t$ it samples $\\theta_t=(\\eta_t,\\beta_t,\\varepsilon_t)$ from $\\Theta$ using $\\*p_t$ and runs $\\texttt{OMD}_{\\eta_t,\\beta_t}$ from initialization $\\*x_t^{(\\theta_t)}$, obtaining a loss estimate $\\hat\\ell_t$.\nThen for each $\\theta=(\\eta,\\beta,\\varepsilon)$ in $\\Theta$ the method updates the corresponding initialization $\\*x_t^{(\\theta)}$ by taking the average of the $\\varepsilon$-constrained optima-in-hindsight $\\operatorname{OPT}_\\varepsilon(\\hat\\ell_1),\\dots,\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)$ seen so far.\nFinally, the algorithm updates the distribution $\\*p_t$ using multiplicative weights over the following modification of the regret-upper-bound \\eqref{eq:rub} above for some $\\rho>0$:\n\\begin{align}\\label{eq:modrub}\nU_t^{(\\rho)}(\\*x,\\theta)=\\frac{B_{\\phi_\\beta}(\\hat{\\*x}_t^{(\\theta)}||\\*x)+\\rho^2D^2}\\eta+(\\eta G_\\beta^2+C\\varepsilon)m\n\\end{align}\nNote that given $\\rho>0$ this function is fully defined after running $\\texttt{OMD}_{\\eta_t,\\beta_t}$ on task $t$ to obtain loss estimates $\\hat\\ell_t$ and then computing the $\\varepsilon$-constrained optimum-in-hindsight $\\hat{\\*x}_t^{(\\theta)}=\\operatorname{OPT}_\\varepsilon(\\hat\\ell_t)$ for each $\\theta=(\\eta,\\beta,\\varepsilon)$.\nThis allows us to use full-information multiplicative weights for $\\theta$.\n$\\rho>0$ is necessary for learning $\\eta$, as if its optimum is near zero then $U_t$ will not be Lipschitz near the optimum.\nTheorem~\\ref{thm:meta} shows a sublinear regret guarantee for Algorithm~\\ref{alg:meta} over the unmodified regret-upper-bounds \\eqref{eq:modrub} w.r.t. all elements in $\\overline\\mathcal K$ and in a continous set of hyperparameters $\\Theta^\\ast\\subset\\mathbb R^3$.\n\n\\begin{algorithm}[!t]\n\t\\DontPrintSemicolon\n\t\\KwIn{compact $\\overline\\mathcal K\\subset\\mathbb R^d$, \n\t\t\tmeta-hyperparameters $\\alpha,\\rho>0$,\n\t\t\tfinite $\\Theta\\subset\\mathbb R^3$ over $(\\eta,\\beta,\\varepsilon)$,\n\t\t\tbase-learner $\\texttt{OMD}_{\\eta,\\beta}:\\overline\\mathcal K\\mapsto\\mathbb R^d$, \n\t\t\tconstrained linear minimizer $\\operatorname{OPT}_\\varepsilon:\\mathbb R^d\\mapsto\\mathcal K_\\varepsilon$}\n\t\\For{$\\theta=(\\eta,\\beta,\\varepsilon)\\in\\Theta$}{\n\t\t$\\*x_1^{(\\theta)}\\gets\\argmin_{\\*x\\in\\overline\\mathcal K}\\phi(\\*x)$\\tcp*{maintain an initialization for each $\\theta\\in\\Theta$}\n\t}\n\t$\\*p_1\\gets\\*1_{|\\Theta|}\/|\\Theta|$\\tcp*{multiplicative weights (MW) initialization}\n\t\\For{task $t=1,\\dots,T$}{\n\t\tsample $\\theta_t=(\\eta_t,\\beta_t,\\varepsilon_t)\\sim\\*p_t$ from $\\Theta$\\\\\n\t\t$\\hat\\ell_t\\gets\\texttt{OMD}_{\\eta_t,\\beta_t}(\\*x^{(\\theta_t)})$\\tcp*{run bandit OMD within-task}\n\t\t\\For{$\\theta=(\\eta,\\beta,\\varepsilon)\\in\\Theta$}{\n\t\t\t$\\*x_{t+1}^{(\\theta)}\\gets\\frac1t\\sum_{s=1}^t\\operatorname{OPT}_\\varepsilon(\\hat\\ell_s)$\\tcp*{update all initializations}\n\t\t\t$\\*p_{t+1}(\\theta)\\gets\\*p_{t+1}(\\theta)\\exp\\left(-\\alpha U_t^{(\\rho)}(\\*x_t^{(\\theta)},\\theta)\\right)$\\tcp*{MW update using loss in \\eqref{eq:modrub}}\n\t\t}\n\t\t$\\*p_{t+1}\\gets\\*p_{t+1}\/\\|\\*p_{t+1}\\|_1$\n\t}\n\t\\caption{\\label{alg:meta}\n\t\tAlgorithm for tuning an online mirror descent (OMD) base-learner $\\texttt{OMD}_{\\eta,\\beta}$ with parameterized regularizer $\\phi_\\beta:\\overline\\mathcal K\\mapsto\\mathbb R$ and step-size $\\eta>0$ that runs OMD on loss estimators $\\hat\\ell_{t,1},\\dots,\\hat\\ell_{t,m}$ from an initialization $\\*x\\in\\overline\\mathcal K$ and returns estimated loss $\\hat\\ell_t=\\sum_{i=1}^m\\hat\\ell_{t,i}\\in\\mathbb R^d$.\n\t\tThen for every $\\varepsilon>0$ the constrained optimizer $\\operatorname{OPT}_\\varepsilon(\\hat\\ell)=\\argmin_{\\*x\\in\\mathcal K_\\varepsilon}\\langle\\hat\\ell,\\*x\\rangle$ returns the minimizer of the estimated loss over the constrained subset $\\mathcal K_\\varepsilon\\subset\\overline\\mathcal K$ (set $\\operatorname{OPT}_\\varepsilon(\\*0_d)=\\argmin_{\\*x\\in\\overline\\mathcal K}\\phi(\\*x)$).\n\t}\n\\end{algorithm}\n\n\\begin{Thm}\\label{thm:meta}\n\tLet $\\Theta^\\ast=(0,\\infty)\\times[\\underline\\beta,\\overline\\beta]\\times[\\underline\\varepsilon,\\overline\\varepsilon]$ for $0\\le\\underline\\beta\\le\\overline\\beta\\le1$ and $0\\le\\underline\\varepsilon\\le\\overline\\varepsilon\\le1$ be the set of hyperparameters $(\\eta,\\beta,\\varepsilon)$ of interest.\n\tThen there exists integer $k=\\mathcal O(\\lceil\\sqrt{mT}\\rceil)$ and $\\alpha,\\underline\\eta,\\overline\\eta\\in(0,\\infty)$ such that running Algorithm~\\ref{alg:meta} with $\\Theta$ the product of uniform grids of size $k$ over each non-singleton dimension of  $[\\underline\\eta,\\overline\\eta]\\times[\\underline\\beta,\\overline\\beta]\\times[\\underline\\varepsilon,\\overline\\varepsilon]$ and $\\alpha$ the meta-step-size yields regret\n\t\\begin{align}\\label{eq:metareg}\n\t\\begin{split}\n\t\\mathbb E&\\sum_{t=1}^TU_t(\\*x_t^{(\\theta_t)},\\theta_t)-\\min_{\\*x\\in\\overline\\mathcal K,\\theta\\in\\Theta^\\ast}\\sum_{t=1}^TU_t(\\*x,\\theta)\n\t\\\\\n\t&\\le\\left(C\\sqrt m+2DG\\left(\\frac1\\rho+M\\right)\\right)\\sqrt{6mT\\log k}+\\frac{8SK^2G\\sqrt m}{\\rho D}(1+\\log T)+\\rho DGT\\sqrt m\n\t\\end{split}\n\t\\end{align}\n\tfor $G=\\max_\\beta G_\\beta\\ge1$, \n\t$M=\\frac G{\\min_\\beta G_\\beta}$,  $D^2=\\max_{\\beta,\\varepsilon,\\*x,\\*y\\in\\mathcal K_\\varepsilon}B_{\\phi_\\beta}(\\*x||\\*y)\\ge1$, \n\t$L$ the maximum Lipschitz constant of $\\phi_\\beta(\\operatorname{OPT}_\\varepsilon(\\ell))$ w.r.t. $(\\beta,\\varepsilon)$ over $\\ell\\in\\mathbb R^d$, $S=\\max_{\\beta,\\varepsilon,\\*x\\in\\mathcal K_\\varepsilon}\\|\\nabla^2\\phi_\\beta(\\*x)\\|_2$, \n\t$K=\\max_{\\*x,\\*y\\in\\mathcal K}\\|\\*x-\\*y\\|_2$, \n\tand the expectation is over sampling $\\theta_t\\sim\\*p_t$.\n\tThe result without the expectation holds w.p. $1-\\delta$ at the cost of an additional $\\left(C\\sqrt m+2DG\\left(\\frac1\\rho+M\\right)\\right)\\sqrt{\\frac T2\\log\\frac1\\delta}$ term.\\looseness-1\n\\end{Thm}\n\\begin{proof}[Proof sketch]\n\tAt a high-level, we use the $\\mathcal O(\\log T)$ regret of follow-the-leader over Bregman divergences~\\citep{khodak2019adaptive} for the initialization, the $\\mathcal O(\\sqrt{T\\log k})$ regret of multiplicative weights over $k$ experts~\\citep{shalev-shwartz2011oco} to tune over a large grid of hyperparameters, the fact that $U_t$ is an affine function of the Bregman divergence to combine the two methods, and the identity $\\sum_{t=1}^TB_\\phi(\\*x_t||\\bar{\\*x})=\\sum_{t=1}^T\\phi(\\*x_t)-\\phi(\\bar{\\*x})$ to bound the discretization error.\n\tThe w.h.p. result follows by \\citet[Lemma~4.1]{cesa-bianchi2006prediction}.\n\\end{proof}\n\nNote that we keep details of the dependence on values like Lipschitz constants because they are important in applying this result;\nhowever, in general setting $\\rho=1\/\\sqrt[4]T$ in \\eqref{eq:metareg} yields $\\tilde\\mathcal O(T^\\frac34)$-regret.\nWhile a slow rate, note that Algorithm~\\ref{alg:meta} is learning a sequence of affine functions of Bregman divergences that are non-smooth and non-convex in-general.\nTheorem~\\ref{thm:meta} is an important structural result;\nour main contributions to multi-armed and linear bandits follow by applying its instantiations for specific regularizers $\\phi$ and hyperparameter sets $\\Theta^\\ast$.\nWe also believe Theorem~\\ref{thm:meta} may be of independent interest as it holds for any choice of Bregman divergence beyond those we consider, and unlike past work \\citep{khodak2019adaptive} allows for explicit control of non-smooth regularizers near the boundaries.\nThe theorem allows tuning the hyperparameters over user-specified intervals for $\\beta$ and $\\varepsilon$ and over an infinite interval for the step-size $\\eta>0$.\nNote that a similar result is straightforward to show for $\\beta$ outside $[0,1]$ or for discrete rather than continuous set of hyperparameters.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe idea of Synthetic Differential Geometry \\cite{Koc:syn} which originates from synthetic considerations of Sophus Lie is very simple: All geometric constructions are performed within a suitable base category in which space forms are objects. For classical geometry the base category is cartesian closed, i.e.\\ it has finite products (e.g.\\ the cartesian product in the category of sets) and the exponentials (that is the collection of functions between space forms is such a form). Furthermore it often comes equipped with a method of constructing and classifying sub-objects (the sub-object classifier); in a word a base category is a {\\em topos}. In this article we use a synthetic method in description or re-interpretation of principal bundles in Noncommutative Differential Geometry.\n\nThis article is addressed to noncommutative geometers and Hopf algebraists who are familiar with general category theory culture but are not experts in category theory. The aim is to explain categorical ingredients that enter a synthetic definition of a principal bundle and then to show that noncommutative generalisation of this definition yields in particular principal comodule algebras or faithfully flat Hopf-Galois extensions. The full geometric potential of Hopf-Galois extensions was probably first explored in \\cite{BrzMaj:gau}, but only now solid topological and geometric evidence is being gathered that affirms that principal comodule algebras should be accepted as principal bundles in noncommutative geometry \\cite{BauHaj:pet}. Thus this article might be intrepeted as making a synthetic case for this claim.  We also hope that this article will contribute to greater appreciation of the power of categorical (and synthetic) thinking in Noncommutative Geometry. \n\nWe assume that the reader is familiar with basic concepts and terminology of category theory such as a monomorphism, epimorphism, functor, natural transformation, adjoint functors, unit and counit of adjunction and equivalence of categories. Some familiarity with finite limits, in particular products, pullbacks and equalisers is also needed, although these notions will be explained in due course. \\cite{Mac:cat} is the standard reference for all categorical concepts not explained in this article. Details of Hopf algebra theory in braided monoidal categories that feature prominently in Section~\\ref{sec.quant} may be found in \\cite[Chapter~9]{Maj:fou}. \n\n\\section{Explaining the definition of principal bundles}\\label{sec.clas}\n\\setcounter{equation}{0}\nIn a synthetic formulation the definition of a principal bundle is very succinct; see \\cite{Koc:fib}, \\cite{Koc:alg}. Let ${\\mathfrak E}$ be a category with finite limits (for example the category $\\mathfrak{Set}$ of sets or the category of smooth manifolds). This means that ${\\mathfrak E}$ has a terminal object, binary products and equalisers; see \\cite[Section~V.2, ~ Corollary~1]{Mac:cat}. We briefly explain these three terms presently.\nA {\\em terminal object} in ${\\mathfrak E}$ is an object that is a codomain for exactly one arrow from any object in ${\\mathfrak E}$ (for example, any singleton set is a terminal object in $\\mathfrak{Set}$). A {\\em (binary) product} of objects  $E_1$, $E_2$ is an object $E_1\\times E_2$ together with two arrows $p_1: E_1\\times E_2\\to E_1$ and $p_2: E_1\\times E_2\\to E_2$, known as {\\em projections}, such that, for any morphisms $q_1: D\\to E_1$, $q_2:D\\to E_2$, there exists a {\\bf unique} morphism $f: D\\to E_1\\times E_2$, such that $p_1\\circ f = q_1$ and $p_2\\circ f =q_2$ (for example the cartesian product of sets is a product in $\\mathfrak{Set}$).   An {\\em equaliser} of morphisms  $q_1, q_2: D\\to E$ is an object $C$ together with a morphism $\\iota : C\\to D$ that {\\em equalises} $q_1$ and $q_2$, that is $q_1\\circ \\iota = q_2\\circ \\iota$, and has the following {\\em universal property}. For any morphism $p: B\\to D$ that equalises $q_1$ and $q_2$, there exists a {\\bf unique} arrow $q: B\\to C$ such that $p = \\iota\\circ q$. \n\nLet $G$ a group object in ${\\mathfrak E}$ (i.e. an object with associative multiplication $G\\times G\\to G$, neutral (global) element $* \\to G$, where $*$ is  a terminal object in ${\\mathfrak E}$, and with the inverses given by an isomorphism $G\\to G$). In case of category of sets $G$ is simply a group, in the category of smooth manifolds it is a Lie group etc. Let $B$  be an object in ${\\mathfrak E}$. A principal $G$-bundle over $B$ is an object $P$ together with a morphism $\\pi: P\\to B$ and a (right) $G$-action $\\varrho : P\\times G \\to P$ such that the the following diagram\n\\begin{equation}\\label{diag1}\n\\xymatrix{ P\\times G\n\\ar@<0.5ex>[r]^-{p}\\ar@<-0.5ex>[r]_-{\\varrho} & \nP \\ar[r]^\\pi &B ,}\n\\end{equation}\nwhere $p$ is the projection, is exact. The term {\\em exact} means that\n\\begin{blist}\n\\item \\eqref{diag1} is a coequaliser diagram,\n\\item \\eqref{diag1} is a kernel pair diagram,\n\\item $\\pi$ is an effective descent epimorphism,\n\\end{blist}\nand our aim in this section is to explain the meaning of (a)--(c).\n\nThe meaning of (a) is most straightforward as the definition of a coequaliser is obtained by reversing of all the arrows in already recalled definition of an equaliser. Explicitly, \\eqref{diag1} is a coequaliser diagram provided $\\pi$ {\\em coequalises} $p$ and $\\varrho$, i.e.,\n$\n\\pi\\circ p = \\pi\\circ \\varrho,\n$\nand for any other morphism $\\phi: P\\to A$ coequalising $p$ and $\\varrho$ there exists a unique morphism $\\psi: B\\to A$ fitting the following diagram\n$$\n\\xymatrix{ P\\times G\n\\ar@<0.5ex>[r]^-{p}\\ar@<-0.5ex>[r]_-{\\varrho} & \nP \\ar[r]^\\pi\\ar[d]_\\phi &B\\ar[ld]^\\psi \\\\\n& A.&}\n$$\nThe existence of such a unique $\\psi$ is referred to as the {\\em universal property of coequalisers}.\n\nBefore moving to condition (b), we can look at the meaning of (a) in case ${\\mathfrak E}$ is the category $\\mathfrak{Set}$ of sets. In this case $P\\times G$ is the usual cartesian product, $p$, $\\varrho$, $\\pi$ are functions and writing $x\\cdot g$ for $\\varrho(x,g)$ (the action of $g\\in G$ on $x\\in P$), the coequalising property of $\\pi$ means that, for all $b\\in G, x\\in P$,\n$$\n\\pi(x\\cdot g) = \\pi\\left(p\\left(x,g\\right)\\right) = \\pi(x).\n$$\nThe universal property now means that (up to an isomorphism) $B = P\/G$ (the quotient set) and, by the coequalising property, $\\pi: P\\to B$ is the canonical surjection. \n\nSince ${\\mathfrak E}$ has finite limits, in particular it has pullbacks. That is, for any pair of morphisms with common codomain, $f_1: E_1 \\to B$, $f_2: E_2 \\to B$, there is a \ndiagram (i.e.\\ an object $E_1\\times_B E_2$ and two morphisms $p_1$, $p_2$ fitting the following diagram)\n$$\n\\xymatrix{ E_1 \\times_B E_2 \\ar[r]^-{p_1} \\ar[d]_{p_2} & E_1 \\ar[d]^{f_1} \\\\\nE_2 \\ar[r]_{f_2} & B,}\n$$ \nwith the following universal property. For any object $F$ and morphisms $q_1: F\\to E_1$ and $q_2: F\\to E_2$ such that \n$\nf_2 \\circ q_2 = f_1 \\circ q_1$, there exists a unique morphism $\\gamma : F\\to E_1 \\times_B E_2$ such that $p_2\\circ \\gamma = q_2$ and $p_1\\circ \\gamma = q_1$. One should keep in mind that a pullback (as every finite limit) is a particular equaliser; see \\cite[Section~V.2, ~ Theorem~2]{Mac:cat}.\n\nThe statement (b) that  \\eqref{diag1} is a kernel pair diagram means that the (unique) map $\\gamma: P\\times G \\to P\\times_B P$,\n$$\n\\xymatrix{ P\\times G \\ar@{-->}[rd]^-\\gamma \\ar@\/^\/[drr]^-p\\ar@\/_\/[ddr]_-\\varrho && \\\\ & P \\times_B P \\ar[r]^-{p_1} \\ar[d]_{p_2} & P \\ar[d]^\\pi \\\\\n& P \\ar[r]_\\pi & B,}\n$$ \nis an isomorphism. \n\nAgain we can look at the case of sets. The pullback $P\\times_B P$ of $\\pi$ is a subset of $P\\times P$,\n$$\nP\\times_B P = \\{ (x,y) \\in P\\times P\\; |\\; \\pi(x)=\\pi(y)\\},\n$$\nwhile $p_1, p_2$ are restrictions of canonical projections $P\\times P \\to P$. The map $\\gamma$ is therefore given by\n$$\n\\gamma(x,g) = (x, x\\cdot g),\n$$\nand its bijectivity means precisely that the action of $G$ on $P$ is free. \n\nThe explanation of the condition (c) requires a quick tour of the descent theory. We follow formulation of effective descent morphisms presented in \\cite{JanSob:bey}. With any object $A$ of ${\\mathfrak E}$ one can associate its {\\em comma category} or {\\em category of objects over $A$},  $({\\mathfrak E} \\downarrow A)$. Objects in $({\\mathfrak E} \\downarrow A)$ are morphisms with codomain $A$, i.e.\\ pairs $(E,f)$, where $E$ is an object in ${\\mathfrak E}$ and $f:E\\to A$ is a morphism. A morphism from $(E_1,f_1)$ to $(E_2,f_2)$ in the comma category $({\\mathfrak E} \\downarrow A)$ is a morphism $h: E_1\\to E_2$ in ${\\mathfrak E}$ making the following triangle\n\\begin{equation}\\label{triang}\n\\xymatrix{E_1 \\ar[rr]^-h\\ar[rd]_{f_1} && E_2\\ar[ld]^{f_2} \\\\ & A &}\n\\end{equation}\ncommute. While objects in ${\\mathfrak E}$ have no elements in a set theoretic sense, one can treat all morphisms with codomain $A$ as {\\em generalised elements} of the object $A$. Thus  $({\\mathfrak E} \\downarrow A)$ can be thought of as a collection of elements of $A$; see \\cite[Part~II]{Koc:syn} for an in-depth discussion of generalised elements in relation to comma categories.\n\nAny morphism $\\pi: P\\to B$ in ${\\mathfrak E}$ induces a pair of functors between the comma categories $({\\mathfrak E} \\downarrow P)$ and $({\\mathfrak E} \\downarrow B)$. The functor  $\\pi_!: ({\\mathfrak E} \\downarrow P)\\to ({\\mathfrak E} \\downarrow B)$ is given by the composition with $\\pi$, that is\n$\\pi_!(E,f) = (E, \\pi\\circ f)$ on objects and $\\pi_!(h) =h$ on morphisms in $({\\mathfrak E} \\downarrow P)$. \nThe functor $\\pi^*: ({\\mathfrak E} \\downarrow B)\\to ({\\mathfrak E} \\downarrow P)$, to every morphism $f: E\\to B$ in ${\\mathfrak E}$ assigns the projection $p_2$ in the pullback\n\\begin{equation}\\label{pullback}\n\\xymatrix{ E \\times_B P \\ar[r]^-{p_1} \\ar[d]_{p_2} & E \\ar[d]^f & \\\\\nP \\ar[r]^\\pi & B, & \\qquad \\pi^*(E,f) = (E\\times_B P, p_2).}\n\\end{equation}\nFor all morphisms $h: (E_1, f_1) \\to (E_2,f_2)$ in $({\\mathfrak E} \\downarrow B)$, $\\pi^*(h)$ is a unique filler in the following pullback diagram\n$$\n\\xymatrix{ E_1\\times_B P \\ar@{-->}[rd]^{\\!\\!\\!\\!\\pi^*(h)} \\ar@\/^\/[drr]^-{h\\circ p_1}\\ar@\/_\/[ddr] && \\\\ & E_2 \\times_B P \\ar[r] \\ar[d] & E_2 \\ar[d]^{f_2} \\\\\n& P \\ar[r]_\\pi & B,}\n$$ \nwhere $p_1$ and unmarked arrows are relevant canonical projections. \n\nThe functor $\\pi_!$ is the left adjoint of the functor $\\pi^*$. The counit of this adjunction, $\\epsilon: \\pi_!\\circ \\pi^* \\to \\id_{({\\mathfrak E} \\downarrow B)}$, assigns to a morphism $f:E\\to B$ the projection $p_1$ in the pullback \\eqref{pullback}, which is clearly a morphism in $({\\mathfrak E} \\downarrow B)$ between \n$$\n\\pi_!\\circ \\pi^*(E,f) = \\pi_! (E\\times_B P, p_2) = (E\\times_B P, \\pi\\circ p_2)\n$$\nand $(E,f)$. The unit $\\eta: \\id_{({\\mathfrak E} \\downarrow P)} \\to \\pi^*\\circ \\pi_!$ assigns to $f: E\\to P$ the unique morphism $\\langle E,f\\rangle$ that fits into the diagram\n$$\n\\xymatrix{ E \\ar@{-->}[rd]^{\\!\\!\\langle E,f\\rangle} \\ar@\/^\/[drrr]^-{\\id_E}\\ar@\/_\/[ddr]_-{f} &&& \\\\ & E \\times_B P \\ar[rr]^-{p_1} \\ar[d]_{p_2} && E \\ar[d]^{f\\circ\\pi} \\\\\n& P \\ar[rr]_\\pi && B.}\n$$ \n\nA {\\em monad} or a {\\em triple} on a category ${\\mathfrak E}$ is a functor $T: {\\mathfrak E}\\to{\\mathfrak E}$ together with two natural transformations $\\mu: T\\circ T \\to T$ (the multiplication) and $\\eta: \\id_{\\mathfrak E}\\to T$ (the unit) which satisfy the associativity and unitality conditions. Explicitly, recall that to each object $E$ in ${\\mathfrak E}$, a natural transformation $\\mu$ assigns  a morphism $\\mu_E : T\\circ T(E) \\to T(E)$ and $\\eta$ assigns  a morphism $\\eta_E: E\\to T(E)$. The associativity and unitality conditions state that, for all objects $E$,\n$$\n\\mu_E \\circ \\mu_{T(E)} = \\mu_E\\circ T(\\mu_E), \\quad \\mu_E\\circ \\eta_{T(E)} = \\mu_E\\circ T(\\eta_{E}) = \\id_{T(E)}.\n$$\nTo a monad $(T,\\mu,\\eta)$ one associates its representation category known as the Eilenberg-Moore category of {\\em algebras} or {\\em modules} ${\\mathfrak E}^T$. The objects are pairs $(X,\\xi)$, where $X$ is an object in ${\\mathfrak E}$ and $\\xi: T(X)\\to X$ is a morphism such that\n$$\n\\xi\\circ \\mu_X = \\xi\\circ T(\\xi), \\qquad \\xi\\circ \\eta_X = \\id_X,\n$$\n(i.e.\\ $\\xi$ is associative and unital). One refers to $\\xi$ as a {\\em structure map} for the module $X$. A morphism of modules is a morphism in ${\\mathfrak E}$ that commutes with the structure maps.\n\nWe have all the ingredients needed for explaining condition (c) at hand. Going back to the morphism $\\pi: P\\to B$, the adjunction $\\pi_! \\dashv \\pi^*$ induces a monad $T = \\pi^*\\circ \\pi_!$ on $({\\mathfrak E} \\downarrow P)$ with $\\mu_{(E,f)} = \\pi^*(\\epsilon_{\\pi_!(E,f)}) = \\pi^*(\\epsilon_{(E,\\pi\\circ f)})$, for all $(E,f) \\in ({\\mathfrak E} \\downarrow P)$, and unit $\\eta$. Furthermore, it induces the functor $K: ({\\mathfrak E} \\downarrow B)\\to ({\\mathfrak E} \\downarrow P)^T$, known as the {\\em comparison functor}, that fits the following diagram\n$$\n\\xymatrix{ ({\\mathfrak E} \\downarrow B)\\ar[rr]^{K}\\ar[dr]_{\\pi^*} & & ({\\mathfrak E} \\downarrow P)^T\\ar[dl]^{U} \\\\\n& ({\\mathfrak E} \\downarrow P)& ,}\n$$\nwhere $U$ is the forgetful functor. The commutativity of the diagram means that the object part of $K(E,f)$ is the projection $p_2: E \\times_B P \\to P$ in the pullback diagram \\eqref{pullback}, i.e.\\ $\\pi^*(E,f)$. The structure map is \n$$\n\\pi^*(\\epsilon_{(E ,f)}) : T\\circ \\pi^*(E,f) = \\pi^*\\circ \\pi_!\\circ \\pi^*(E,f) \\longrightarrow \\pi^*(E,f) = (E \\times_B P, p_2).\n$$\nThe morphism $\\pi: P\\to B$ is an {\\em effective descent morphism} provided the associated comparison functor $K$ is an equivalence of categories. Of course effective descent epimorphism is an effective descent morphism which is an epimorphism. \n\nIn case of the category of sets, every epimorphism (surjective function) is an effective descent morphism, but in case of other categories condition (c) carries non-trivial information.\n\n\\section{Translating the definition of principal bundles}\\label{sec.quant} \\setcounter{equation}{0}\nIn this section we would like to translate carefully the definition of principal bundles to the case of braided monoidal categories with equalisers preserved by the tensor product. The ongoing programme whose foundations are presented in \\cite{Mas:non} makes a very solid case for such categories to be a proper environment for Noncommutative Geometry. \n\nA category $\\cB$ is called\na {\\em  monoidal category}  if there exist\n a functor $-\\ot-   :\\cB \\times\\cB \\to \\cB $, a distinguished\n object $\\mathbf{1}$ in $\\cB$ and isomorphisms\n $$\\alpha_{X,Y,Z} : (X\\ot Y)\\ot Z \\to X\\ot (Y\\ot Z), \\quad  \n  \\lambda_X: \\mathbf{1}\\ot X \\to X,\\quad \\varrho_X:X\\ot \\mathbf{1}\\to X,$$ \nnatural in $X,Y,Z$, such that, for all objects $W,X,Y,Z$ in $\\cB $, the following diagrams\n$$\\xymatrix{\n((W\\ot X)\\ot  Y)\\ot  Z\\ar[rr]^-{\\alpha_{W,X,Y}\\ot  \\id_Z}\\ar[d]_{\\alpha_{W\\ot \nX,Y,Z}} & &  (W\\ot  (X\\ot  Y))\\ot  Z \\ar[rr]^-{\\alpha_{W, X\\ot Y,Z}} && W\\ot  ((X\\ot  Y)\\ot  Z)\\ar[d]^{\\quad \\id_W\\ot \\alpha_{X,Y,Z}} \\\\ \n    (W\\ot  X)\\ot  (Y\\ot  Z)  \\ar[rrrr]^-{\\alpha_{W,X,Y\\ot Z}}  \n & & & & W\\ot  (X\\ot  (Y\\ot Z))\\, ,  }$$ \nand \n$$\\xymatrix{\n(X\\ot \\mathbf{1})\\ot Y\\ar[dr]_{\\varrho_{X}\\ot \\id_Y}\\ar[rr]^{\\alpha_{X,\\mathbf{1},Y}} & & X\\ot (\\mathbf{1}\\ot Y)\n     \\ar[dl]^{\\id_X\\ot\\lambda_Y} &\\\\ \n & X\\ot Y &  }\n $$ \n commute.  \n The functor $\\ot$ is referred to as the {\\em tensor} or {\\em mnoidal product} and $\\mathbf{1}$ is called the {\\em monoidal unit}. A monoidal category is said to be {\\em strict monoidal} if the isomorphisms $\\alpha_{X,Y,Z}$, $\\lambda_X$ and $\\gamma_X$ are identities, for all objects $X,Y,Z$ of $\\cB$. Although there might be rather few strict monoidal categories in nature, any monoidal category is equivalent to a strict one. It is therefore customary (and very convenient) to treat monoidal category as if it were strict monoidal, and not to write the rearranging brackets isomorphisms (or {\\em associators}) $\\alpha_{X,Y,Z}$ and the {\\em unitors} $\\lambda_X$ and $\\varrho_X$ in any formulae (each of  these Greek letters will denote something else in the sequel). With this convention in mind, a monoidal category is denoted by $(\\cB, \\otimes, \\mathbf{1}, \\tau)$, and the brackets are not written between multiple tensor products.\n \n A (strict) monoidal category  $(\\cB, \\otimes, \\mathbf{1}, \\tau)$ is said to be a {\\em (strict) braided monoidal category} if, for any pair of objects $X,Y$ of $\\cB$, there exists an isomorphism, called a {\\em braiding}, $\\tau_{X,Y}: X\\ot Y \\to Y\\ot X$, natural  in $X$ and $Y$, such that, for all objects $X,Y,Z$ of $\\cB$, $\\tau_{\\mathbf{1},X}$, $\\tau_{X,\\mathbf{1}}$ are identity morphisms (up to unitors), and \n $$\n (\\id_Y\\ot \\tau_{X,Z})\\circ (\\tau_{X,Y}\\ot \\id_Z) = \\tau_{X, Y\\ot Z},  \\qquad (\\tau_{X,Z}\\ot\\id_Y)\\circ(\\id_X\\ot \\tau_{Y,Z}) = \\tau_{X\\ot Y, Z},\n $$\n (the associators in the first equality and their inverses in the second one are not written explicitly). The naturality of braiding means that, for all morphisms $f: X\\to Y$ and all objects $Z$,\n $$\n \\tau_{Y,Z}\\circ(f\\ot\\id_Z) = (\\id_Z\\ot f)\\circ\\tau_{X,Z}, \\qquad \\tau_{Z,Y}\\circ(\\id_Z\\ot f) = (f\\ot \\id_Z)\\circ\\tau_{Z,X}.\n $$\nA braided monoidal category \nis denoted by $(\\cB ,\\ot, \\mathbf{1},\\tau)$. \nThe examples of particular interest are the category $\\mathfrak{Vect}_\\Bbbk$ of  vector spaces over a field $\\Bbbk$, with the monoidal product given by the standard tensor product of $\\Bbbk$-vector spaces, monoidal unit $\\mathbf{1} =\\Bbbk$ and with the braiding $\\tau$ provided by the flip, and its opposite category $\\mathfrak{Vect}_\\Bbbk^{op}$. (Recall that the opposite category is obtained from the original one by reversing all the arrows.) \n\nAs in the category of vector spaces in any monoidal category one can consider {\\em monoids} or {\\em algebras} and {\\em comonoids} or {\\em coalgebras}. These are defined in the same diagrammatic way as usual $\\Bbbk$-algebras and  $\\Bbbk$-coalgebras. In braided monoidal category, the tensor product of two coalgebras (algebras) is again a coalgebra (algebra). If $C$ is a coalgebra in $(\\cB ,\\ot, \\mathbf{1},\\tau)$ with comultiplication $\\Delta_C:C\\to C\\ot C$ and counit $\\eps_C: C\\to \\mathbf{1}$, and $D$ is a coalgebra with comultiplication $\\Delta_D:D\\to D\\ot D$ and counit $\\eps_D: D\\to \\mathbf{1}$, then $C\\ot D$ is a coalgebra with counit $\\eps_C\\ot \\eps_D$ and comultiplication\n\\begin{equation}\\label{eq.coalg}\n\\xymatrix{C\\ot D \\ar[rr]^-{\\Delta_C\\ot \\Delta_D} && C\\ot C\\ot D\\ot D \\ar[rr]^-{\\id_C\\ot \\tau_{C,D} \\ot \\id_D} &&  C\\ot D\\ot C\\ot D.}\n\\end{equation}\nSimilarly, if  $A$ is an algebra with multiplication $m_A$ and unit $1_A: \\mathbf{1} \\to A$, and $B$ is an algebra with multiplication $m_B$ and unit $1_B: \\mathbf{1}\\to A$, then $A\\ot B$ is an algebra with unit $1_A\\ot 1_B$ and multiplication\n\\begin{equation}\\label{eq.alg}\n\\xymatrix{A\\ot B\\ot A\\ot B \\ar[rr]^-{\\id_A\\ot \\tau_{B,A} \\ot \\id_B} && A\\ot A\\ot B\\ot B \\ar[rr]^-{m_A\\ot m_B} && A\\ot B.}\n\\end{equation}\nThus, similarly as for $\\Bbbk$-algebras and $\\Bbbk$-coalgebras, in a braided monoidal category one can combine algebras and coalgebras to define bialgebras and Hopf algebras.  \n\nThe aim of this section is to translate the definition of principal bundles to the case of braided monoidal categories. The translation procedure we employ follows the same principle as in \\cite{Agu:int}: we interpret the category of sets as a monoidal category (with the cartesian product understood as the tensor product, the monoidal unit given by the singleton set, and the braiding provided by the flip) and  reformulate notions described in the preceding section in the way suitable for a general monoidal category. As with every translation, some information is lost some other gained, but it is hoped that the general spirit is preserved. \n\nThe first observation is that every set $X$ is a {\\em comonoid} or a {\\em coalgebra} with the unique comultiplication given by the diagonal function,\n$$\n\\Delta_X : X\\to X\\times X, \\qquad x\\mapsto (x,x).\n$$\nThe coassociativity of this comultiplication is obvious. The counit is the unique map $\\eps_X : X \\to *$, where $*$ is a fixed singleton set understood as the monoidal unit in the category of sets. The standard identification $X\\times * \\cong X \\cong *\\times X$ immediately yields the counitality of $\\Delta_X$. Guided by this we should consider comonoids or coalgebras  in $(\\cB, \\otimes, \\mathbf{1}, \\tau)$ as basic objects. \n\nSince any set has unique comultiplication and counit, a group $G$ is the same as a Hopf algebra in the category of sets; the standard compatibility conditions between comultiplication and multiplication etc., are automatically satisfied (the singleton set $*$ is understood as a monoid in the only possible way). The antipode $S$ is given by the inverse function\n$$\nS : G \\to G , \\qquad g\\mapsto g^{-1}.\n$$\nThis antipode is obviously bijective (in fact, involutive, since $S\\circ S =\\id_G$). Thus for a group object in a braided category $\\cB$ we take a Hopf algebra $H$ in $\\cB$ (and we may assume that $H$ has a bijective antipode, although it is not needed at this point). \n\nAgain, since the comonoid structure on a set is uniquely given, the action of a group $G$ on a set $P$ is compatible with the comonoid structures. This means that the action $\\varrho : P\\times G\\to P$ makes $P$ into a right module coalgebra over the Hopf algebra $G$ (all in the category of sets), i.e., for all $x\\in P$ and $g\\in G$,\n\\begin{equation}\\label{mod.coa}\n\\Delta_P(x\\cdot g) = \\Delta_P(x)\\cdot \\Delta_G(g),\n\\end{equation}\nwhere the action on the right hand side is defined componentwise. Obviously $\\eps_P(x\\cdot g) = \\eps_P(x)\\eps_G(g)$. Thus, in a braided monoidal category $\\cB$, $P$ is required to be a right $H$-module coalgebra with the $H$-action $\\varrho: P\\ot H \\to P$. The compatibility \\eqref{mod.coa} is then expressed in the element-free way and with the use of the braiding $\\tau$ as the commutativity of the following diagram\n$$\n\\xymatrix{P\\otimes H \\ar[r]^-\\varrho \\ar[d]_{\\Delta_P\\ot \\Delta_H} & P\\ar[r]^-{\\Delta_P} & P\\ot P \\\\\nP\\ot P\\ot H\\ot H \\ar[rr]^-{\\id_P\\ot \\tau_{P,H}\\ot \\id_H} && P\\ot H\\ot P\\ot H \\ar[u]_{\\varrho\\ot \\varrho } ,}\n$$\nand $\\eps_P\\circ \\varrho = \\eps_P\\ot \\eps_H$, i.e.\\ $\\varrho$ is a morphism of coalgebras (comonoids) in $\\cB$.\n\nIn the case of sets the canonical projection $p: P\\times G \\to P$, $(x,g)\\mapsto x$ can be identified with the function $(x,g)\\mapsto (x,*) \\equiv x$ (where $*$ stands for the only element of the singleton set $*$), i.e.\\ $p$ can be identified with $\\id_P\\times \\eps_G$. In this form, the map $p$ can be translated to a braided monoidal category $\\cB$.\n\nThese are  ingredients needed for the definition of a noncommutative principal bundle.\n\n\\begin{definition}\\label{def.main}\nLet $(\\cB, \\otimes, \\mathbf{1}, \\tau)$ be a braided monoidal category with equalisers preserved by the tensor product. Let $H$ be a Hopf algebra in $\\cB$ and $B$ be a coalgebra in $\\cB$.  A right $H$-module coalgebra $P$ (with action $\\varrho: P\\ot H\\to P$) together with a coalgebra morphism $\\pi: P\\to B$ is called a {\\em noncommutative $H$-principal bundle} provided that the following diagram\n\\begin{equation}\\label{diag2}\n\\xymatrix{ P\\otimes H\n\\ar@<0.5ex>[rr]^-{\\id_P\\ot \\eps_H}\\ar@<-0.5ex>[rr]_-{\\varrho} && \nP \\ar[r]^\\pi &B ,}\n\\end{equation}\nof comonoids is $\\ot$-exact.\n\\end{definition}\nThe term `$\\ot$-exact' is a translation of the term `exact' explained in Section~\\ref{sec.clas} to braided monoidal categories. Therefore,  its  meaning comprises three statements:\n\\begin{alist}\n\\item  \\eqref{diag2} is a coequaliser diagram,\n\\item \\eqref{diag2} is a kernel pair diagram of comonoids in $\\cB$,\n\\item  $\\pi$ is an effective descent epimorphism of comonoids in $\\cB$.\n\\end{alist}\nThe statement (A) is the same as (a) in Section~\\ref{sec.clas}, the meaning of (B) and (C) need to be explained. In due course the need for and meaning of the assumptions that $\\cB$ has equalisers and that they are preserved by the tensor product should become clear. \n\nAs already mentioned in Section~\\ref{sec.clas}, in the category of sets a pullback of $\\alpha: E_1 \\to B$ and $\\beta: E_2 \\to B$ is a subset of $E_1\\times E_2$ defined by\n$$\nE_1\\times_B E_2 = \\{(x,y)\\in E_1\\times E_2\\; |\\; \\alpha(x) = \\beta(y)\\}.\n$$\nSince all sets are comonoids, and functions are maps of comonoids, the set $E_1$ is a right $B$-comodule and $E_2$ is a left $B$-comodule with coactions\n$$\n\\lambda_1 = (\\id_{E_1}\\times \\alpha)\\circ \\Delta_{E_1} : E_1\\to E_1\\times B, \\qquad x\\mapsto (x,\\alpha(x)),\n$$\nand \n$$\n\\lambda_2 = (\\beta\\times \\id_{E_2})\\circ \\Delta_{E_2} : E_2\\to B\\times E_2, \\qquad y\\mapsto (\\beta(y),y).\n$$\nThus\n$$\nE_1\\times_B E_2 \n= \\{(x,y)\\in E_1\\times E_2\\; |\\; (x,\\alpha(x),y) = (x,\\beta(y),y) \\} = E_1\\coten B E_2,\n$$\nwhere $E_1\\coten B E_2$ denotes the equaliser of $\\lambda_1\\times \\id_{E_2}$ and $\\id_{E_1}\\times \\lambda_2$, i.e.\\ the {\\em cotensor product} of comodules. This indicates that pullbacks in a category with finite limits should be translated to cotensor products of comodules in a monoidal category. \n\nWhile the decision which of the monoids turn into left and which one in the right comodule is somewhat arbitrary, this ambiguity vanishes if we want to equalise the same morphism.  This leads to the following lemma and definition which explain (B).\n\\begin{lefinition}\\label{lem.can}\nLet $\\cB$ be a monoidal category (not necessarily braided). Consider a diagram of comonoids\n\\begin{equation}\\label{diag3}\n\\xymatrix{ C\n\\ar@<0.5ex>[r]^-{\\alpha}\\ar@<-0.5ex>[r]_-{\\beta} & \nP \\ar[r]^\\pi &B ,}\n\\end{equation}\nin which $\\pi\\circ\\alpha = \\pi\\circ\\beta$, i.e.\\ $\\pi$ coequalises $\\alpha$ and $\\beta$.\nAssume that there exists the equaliser of $(\\id_P\\ot \\pi \\ot \\id_P)\\circ (\\Delta_P\\ot \\id_P)$ and $(\\id_P\\ot \\pi \\ot \\id_P)\\circ (\\id_P\\ot \\Delta_P)$, and denote it by  $P\\coten B P$. Then there exists  a unique morphism\n$$\n\\can: C \\to P\\coten B P,\n$$\nsuch that\n$$\n\\xymatrix{C\\ar[rr]^-\\can \\ar[rrd]_{(\\alpha \\ot \\beta)\\circ \\Delta_C} &&  P\\coten B P\\ar[d]\\\\\n&& P\\ot P.}\n$$\nThe diagram \\eqref{diag3} is said to be a {\\em kernel pair  diagram of comonoids in $\\cB$} provided the map $\\can$ is an isomorphism.\n\\end{lefinition}\n\\begin{proof}\nStart with the following  straightforward calculation:\n\\begin{eqnarray*}\n(\\id_P\\ot \\pi \\ot \\id_P)\\circ (\\Delta_P\\ot \\id_P)\\circ (\\alpha\\ot \\beta )\\circ \\Delta_C &= &(\\id_P\\ot \\pi \\ot \\id_P)\\circ  (\\alpha\\ot\\alpha\\ot\\beta)\\circ (\\Delta_C\\ot \\id_C)\\circ\\Delta_C\\\\\n&=& (\\id_P\\ot \\pi \\ot \\id_P)\\circ  (\\alpha\\ot\\beta \\ot \\beta)\\circ (\\id_C\\ot \\Delta_C)\\circ\\Delta_C\\\\\n&=& (\\id_P\\ot \\pi \\ot \\id_P)\\circ (\\id_P\\ot \\Delta_P)\\circ (\\alpha\\ot \\beta)\\circ \\Delta_C .\n\\end{eqnarray*}\nThe first and the third equalities follow by the fact that $\\alpha$ and $\\beta$ are morphisms of comonoids, while the second equality is a consequence of the coassociativity of $\\Delta_C$ and the fact that $\\pi$ coequalises $\\alpha$ and $\\beta$. The existence and uniqueness of the morphism $\\can$ now follow by the universal property of equalisers.\n\\end{proof}\n\n\\begin{lemma}\\label{lem.can.H}\nIn the setting of Definition~\\ref{def.main}, \n$\\can: P\\ot H \\to P\\coten B P$ is the unique morphism induced by the composite\n\\begin{equation}\\label{eq.can}\n\\xymatrix{ P\\ot H \\ar[rr]^{\\Delta_P\\ot \\id_H} && P\\ot P\\ot H \\ar[rr]^-{\\id_P \\ot \\varrho} && P\\ot P.\n}\n\\end{equation}\n\\end{lemma}\n\\begin{proof} The form of the map $\\can$ is obtained by the following computation \n\\begin{eqnarray*}\n(\\id_P \\ot \\eps_H \\ot \\varrho)\\circ\\Delta_{P\\ot H} &=& (\\id_P \\ot \\eps_H \\ot \\varrho)\\circ (\\id_P\\ot \\tau_{P,H} \\ot \\id_H)\\circ(\\Delta_P\\ot \\Delta_H) \\\\\n&=&  (\\id_P \\ot \\varrho)\\circ(\\id_P \\ot \\id_P\\ot \\eps_H\\ot \\id_H) \\circ(\\Delta_P\\ot \\Delta_H) \\\\\n&=& (\\id_P \\ot \\varrho)\\circ(\\Delta_P\\ot \\id_H),\n\\end{eqnarray*}\nwhere the second equality follows by the naturality of the braiding and from the fact that (up to unitors) the braiding $\\tau_{P,\\mathbf{1}}$ is the identity. \n\\end{proof}\n\n\nThe property (B) thus states that the map $\\can$ induced by \\eqref{eq.can} is an isomorphism.  \n\n\nIt remains to explain the meaning of (C), or, more precisely to translate (c) so that the meaning to (C) can be given. Take a set $A$. The comma category $(\\mathfrak{Set} \\downarrow A)$ consists of functions $f:X\\to A$. A function $f:X\\to A$ can be equivalently described as a function assigning to $X$ the graph of $f$, i.e.\\ as\n$$\n\\graph{f}: X\\to X\\times A, \\qquad x\\mapsto (x, f(x)).\n$$\nIf $A$ is understood as a comonoid (in a unique way), then $(X, \\graph{f})$ is a right $A$-comodule ($\\graph{f}$ is a coaction). The commutativity of the diagram \\eqref{triang} for a morphism $g:(X_1, f_1)\\to (X_2, f_2)$ in $(\\mathfrak{Set} \\downarrow A)$ is equivalent to the commutativity of the following diagram for the induced coactions $\\graph{f_1}$, $\\graph{f_2}$\n$$\n\\xymatrix{X_1 \\ar[rr]^-g \\ar[d]_{\\graph{f_1}} && X_2\\ar[d]^{\\graph{f_2}}\\\\\nX_1\\times A \\ar[rr]^-{g\\times \\id_A} && X_2\\times A.}\n$$\nTherefore, the comma category $(\\mathfrak{Set} \\downarrow A)$ is the same as or isomorphic to the category of right $A$-comodules.\n\nIn a monoidal category $\\cB$, by the comma category of a comonoid $A$ we understand the category of right $A$-comodules $\\Com A$. We should point out at this point that the choice of right over left $A$-comodules is arbitrary. We also do not require the objects of the comma category of $A$ to be (morphisms of) comonoids, as the notion of an effective descent morphism of comonoids can already be introduced in present generality.  \n\nA morphism of comonoids $\\pi: P\\to B$ induces a functor $\\pi_! : \\Com P\\to \\Com B$ by composition with the coaction. If $(E,\\lambda)$ is a right $P$-comodule, then\n$$\n\\pi_! (E,\\lambda) = (E, (\\id_E\\ot \\pi)\\circ \\lambda),\n$$\nis a right $B$-comodule. On morphisms $\\pi_!(h) =h$. In the case of sets this is exactly the functor $\\pi_!$ between comma categories described in Section~\\ref{sec.clas}. To see this one should use the identification of a function $f$ with the  function $\\graph{f}$ into its graph. \n\nThe construction of the functor $\\pi^* :  \\Com B\\to \\Com P$ requires additional assumptions on the monoidal category $\\cB$. First  we need to look at the case of sets, and study the diagram \\eqref{pullback}. The pullback in diagram \\eqref{pullback} is \n\\begin{eqnarray*}\nE\\times_B P &=& \\{(e,x)\\in E\\times P \\; |\\; \\pi(x) = f(e)\\}\\\\\n& = & \\{(e,x)\\in E\\times P \\; |\\; (e,\\pi(x),x) = (\\graph{f}(e), x)\\} = E\\coten B P,\n\\end{eqnarray*}\nwhere $E$ is the right $B$-comodule with coaction $\\graph{f}$ and $P$ is the left $B$-comodule with coaction $(\\pi\\times \\id_P)\\circ \\Delta_P$. The $P$-coaction induced by the projection $p_2$ is\n$$\n\\graph{p_2} : E\\times_B P \\to E\\times_B P\\times P, \\qquad (e,x)\\mapsto (e,x,p_2(e,x)) = (e,x,x),\n$$\nthat is  \n$\n\\graph{p_2} =  \\id_E\\times \\Delta_P\\mid_{E\\times_BP}.$ Translating this to the monoidal category $\\cB$ we conclude that, for all $B$-comodules $(E,\\lambda)$,\n$$\n\\pi^*(E,\\lambda) = (E\\coten B P,  \\id_E\\coten B\\Delta_P).\n$$ \nFor this functor to be defined we thus need to require the existence of equalisers in $\\cB$ to obtain the object part of the comodule, and these equalisers need be preserved by the tensor product (meaning: after tensoring an equaliser we again obtain the equaliser) for $\\id_E\\coten B\\Delta_P$ to be a well-defined coaction (this explains the origin of assumptions on $\\cB$ made in Definition~\\ref{def.main}). More explicitly, by $\\id_E\\coten B\\Delta_P$ we understand the unique arrow in the following diagram\n\\begin{equation}\\label{equaliser}\n\\xymatrix{ (E\\coten B P) \\otimes P \\ar[r] & E\\ot P\\ot P \n\\ar@<0.5ex>[rrrrr]^-{(\\id_E \\ot \\pi \\ot\\id_P\\ot \\id_P)\\circ(\\id_E \\ot \\Delta_P\\ot \\id_P) }\\ar@<-0.5ex>[rrrrr]_-{\\lambda \\ot \\id_P\\ot \\id_P} &&&&& E\\ot B\\ot P\\ot P\\\\ \n& E\\ot P\\ar[u]_{\\id_E \\ot \\Delta_P} &&&&&\\\\\n& E\\coten B P\\ar[u]\\ar@{-->}[luu]^{\\id_E\\coten B\\Delta_P}. &&&&& }\n\\end{equation}\nThe existence and uniqueness of $\\id_E\\coten B\\Delta_P$ with specified codomain follow by the assumption that the tensor product preserves equalisers, so that the top row in diagram \\eqref{equaliser} is an equaliser.  On morphisms of right $B$-comodules $f: (E_1,\\lambda_1) \\to (E_2,\\lambda_2)$, $\\pi^*(f) = f\\coten B\\id_P$ is the unique filler in the following equaliser diagram\n\\begin{equation}\\label{eq.pi*}\n\\xymatrix{ E_2\\coten B P  \\ar[r] & E_2\\ot P\n\\ar@<0.5ex>[rrrr]^-{(\\id_{E_2} \\ot \\pi \\ot\\id_P)\\circ(\\id_{E_2} \\ot \\Delta_P) }\\ar@<-0.5ex>[rrrr]_-{\\lambda_2 \\ot \\id_P} &&&& E_2\\ot B\\ot P\\\\ \n& E_1\\ot P\\ar[u]_{f\\ot \\id_P} &&&&\\\\\n& E_1\\coten B P\\ar[u]\\ar@{-->}[luu]^{f\\coten B\\id_P}. &&&& }\n\\end{equation}\nAt this point the translation stops and the development of translated notions follows its own course.\n\nIf $\\cB$ has equalisers and equalisers are preserved by the tensor product, then to any morphism of comonoids $\\pi: P\\to B$ one can associate a pair of functors \n$$\n\\pi_! : \\Com P\\to \\Com B, \\qquad \\pi^*: \\Com B\\to \\Com P.\n$$\nThis is an adjoint pair with $\\pi_!$ the left adjoint of $\\pi^*$. For any $P$-comodule $(E,\\lambda)$, the unit of the adjunction is\n$$\n\\eta_{(E,\\lambda)}: (E,\\lambda) \\to \\pi^*\\circ\\pi_!(E,\\lambda)= (E\\coten B P, \\id_E\\coten B \\Delta_P), \\qquad \\eta_{(E,\\lambda)}= \\hat{\\lambda},\n$$\nwhere  $E$ on the right hand side of the first equality is understood as a right $B$-comodule with the coaction  $(\\id_E\\ot \\pi)\\circ \\lambda$, and $\\hat{\\lambda}$ is the (universally induced) unique filler in the following diagram\n$$\n\\xymatrix{ E\\coten B P \\ar[r] & E\\ot P \n\\ar@<0.5ex>[rrrr]^-{(\\id_E \\ot \\pi \\ot\\id_P)\\circ(\\id_E \\ot \\Delta_P) }\\ar@<-0.5ex>[rrrr]_-{(\\id_E \\ot \\pi \\ot\\id_P)\\circ(\\lambda \\ot \\id_P)} &&&& E\\ot B\\ot P\\\\ \n& E\\ar[u]^\\lambda\\ar@{-->}[lu]^{\\hat{\\lambda}}. &&&& }\n$$\nFor any $B$-comodule $(E,\\lambda)$, the counit of the adjunction \n$$\n\\epsilon_{(E,\\lambda)}: \\pi_!\\circ\\pi^*(E,\\lambda) = \\left(E\\coten BP, \\id_E\\coten B\\left(\\left(\\id_P\\ot \\pi\\right)\\circ \\Delta_P\\right)\\right) \\to (E,\\lambda), \n$$\nis  the composite \n$$\n \\epsilon_{(E,\\lambda)} : \\xymatrix{E\\coten B P\\ar[r] & E\\ot P \\ar[rr]^-{\\id_E \\ot  \\eps_P} && E.}\n$$\nThe morphism $\\id_E\\coten B\\left(\\left(\\id_P\\ot \\pi\\right)\\circ \\Delta_P\\right)$ is defined by a diagram similar to \\eqref{equaliser}. Since $\\pi:P\\to B$ is a morphism of comonoids, $\\eps_B\\circ \\pi = \\eps_P$, and $E\\coten B B$ is isomorphic to $E$ (with the isomorphism induced by $\\id_E\\ot \\eps_B$),  the counit of adjunction $ \\epsilon_{(E,\\lambda)}$ can be identified with $\\id_E\\coten B \\pi$. \n\nThe monad $(T = \\pi^*\\circ\\pi_!,\\mu,\\eta)$ on the category $\\Com P$ induced by the adjunction $\\pi_!\\dashv \\pi^*$ according to the standard procedure described in Section~\\ref{sec.clas} comes out as follows. For any right $P$-comodule $(E,\\lambda)$,\n\\begin{equation}\\label{T}\nT(E,\\lambda) = (E\\coten B P, \\id_E\\coten B \\Delta_P), \\qquad \\eta_{(E,\\lambda)} = \\hat{\\lambda},\n\\end{equation}\nand \n\\begin{equation}\\label{mu}\n\\mu_{(E,\\lambda)} =  \\epsilon_{(E,\\lambda)} \\coten B \\id_P:  \\xymatrix{E\\coten B P\\coten B P \\ar[r] & (E\\ot P)\\coten BP \\ar[rrr]^-{(\\id_E \\ot  \\eps_P)\\coten B \\id_P} &&& E\\coten B P,}\n\\end{equation}\nwhere $E$ is understood as a right $B$-comodule by the coaction acquired from $\\pi_!$, i.e., $(\\id_E\\ot \\pi)\\circ \\lambda$. \n\nAn algebra or module over the monad $(T,\\mu,\\eta)$ is a triple $(E,\\lambda,\\xi)$, in which $(E,\\lambda)$ is a right $P$-comodule (i.e.\\ an object in the category $\\Com P$ on which $T$ operates) and $\\xi: E\\coten B P \\to E$ is a morphism of $P$-comodules rendering commutative the following diagrams\n\\begin{equation}\\label{alg}\n\\xymatrix{ E\\coten B P \\coten BP\\ar[rr]^-{\\mu_{(E,\\lambda)}} \\ar[d]_{\\xi \\coten B \\id_P} && E\\coten B P \\ar[d]^\\xi \\\\\nE\\coten B P \\ar[rr]^-\\xi && E,}\\qquad \n\\xymatrix{E  \\ar[rr]^-{\\hat{\\lambda}}\\ar[rd]_{\\id_E} && E\\coten B P\\ar[ld]^\\xi \\\\\n& E. &}\n\\end{equation}\nAs in Section~\\ref{sec.clas} (or, indeed, as in the case of any adjoint pair of  functors), associated to the comonad morphism $\\pi: P\\to B$ there is a comparison functor $K$ connecting the category of $B$ comodules with the category  $ \\Com P^T$ of algebras over the monad $T=\\pi^*\\circ\\pi_!$. $K$ fits into the commutative triangle\n$$\n\\xymatrix{ \\Com B\\ar[rr]^{K}\\ar[dr]_{\\pi^*} & & \\Com P^T\\ar[dl]^{U} \\\\\n& \\Com P.& }\n$$\nExplicitly, for any $B$-comodule $(E,\\lambda)$,\n\\begin{equation} \\label{comparison}\nK(E,\\lambda) = \\left( E\\coten B P ,\\id_E\\coten B\\Delta_P , \\epsilon_{(E,\\lambda)}\\coten B \\id_P\\right).\n\\end{equation}\nAn epimorphism of comonoids $\\pi: P\\to B$ in the monoidal category $\\cB$ (with equalisers preserved by the tensor product) is called an {\\em effective descent epimorphism of comonoids in $\\cB$} provided the  comparison functor $K: \\Com B \\to \\Com P^{\\pi^*\\circ\\pi_!}$ is an equivalence of categories. With this last definition full contents of Definition~\\ref{def.main} is explained. Note that the unit and counit of adjunction $\\pi_! \\dashv \\pi^*$, the associated monad and comparison functor are translations of structures described in Section~\\ref{sec.clas}. This might be interpreted as passing of a consistency check and as confirmation of faithfulness of the translation procedure employed here. \n\n\n\\section{Identifying principal comodule algebras}\\label{sec.pcalg}\n\\setcounter{equation}{0}\nIn this section we would like to work out what noncommutative $H$-principal bundles are in the braided (symmetric) monoidal category opposite to the category of vector spaces. Fix a field $\\Bbbk$.  As explained in Section~\\ref{sec.quant}, the category of $\\Bbbk$-vector spaces $\\mathfrak{Vect}_\\Bbbk$ is a monoidal category with the usual tensor product as the monoidal product and with the field $\\Bbbk$ as the monoidal unit. The flip $V\\ot W\\ni v\\ot w\\mapsto w\\ot v\\in W\\ot V$ is a braiding (in fact symmetry as it squares to identity) in $\\mathfrak{Vect}_\\Bbbk$.  The category of vector spaces has all equalisers and coequalisers and they are preserved by the tensor product.\n\nWe will study noncommutative principal bundles in the category $\\mathfrak{Vect}_\\Bbbk^{op}$, {\\bf opposite} to $\\Bbbk$-vector spaces. $\\mathfrak{Vect}_\\Bbbk^{op}$ has $\\Bbbk$-vector spaces as  objects and linear transformations as morphisms, but a morphism $f: V\\to W$ in $\\mathfrak{Vect}_\\Bbbk^{op}$ is given by a $\\Bbbk$-linear transformation $f: W\\to V$. Consequently, the composition in  $\\mathfrak{Vect}_\\Bbbk^{op}$ is given by the opposite composition of linear transformations. This `reversing of arrows' results in `swapping the prefix co-'. We now carefully study the contents of Definition~\\ref{def.main} in case $\\cB = \\mathfrak{Vect}_\\Bbbk^{op}$.\n\nA Hopf algebra $H$ in $\\mathfrak{Vect}_\\Bbbk^{op}$ is also a Hopf algebra in $\\mathfrak{Vect}_\\Bbbk$, and vice versa (what was the multiplication in one case becomes the comultiplication in the other and vice versa); i.e.\\ in both cases $H$ is a standard Hopf algebra over $\\Bbbk$. A coalgebra $B$ in $\\mathfrak{Vect}_\\Bbbk^{op}$ the same as a $\\Bbbk$-algebra $B$. A right $H$-module coalgebra $P$ in $\\mathfrak{Vect}_\\Bbbk^{op}$ is the same as a right $H$-comodule $\\Bbbk$-algebra $P$ with coaction $\\varrho: P\\to P\\ot H$ in $\\mathfrak{Vect}_\\Bbbk$. The diagram \\eqref{diag2} of coalgebras in $\\mathfrak{Vect}_\\Bbbk^{op}$, after inverting all the arrows, becomes the following diagram of $\\Bbbk$-algebras\n\\begin{equation}\\label{diag4}\n\\xymatrix{ B\\ar[r]^\\pi & P\n\\ar@<0.5ex>[rr]^-{\\id_P\\ot 1_H}\\ar@<-0.5ex>[rr]_-{\\varrho} && \nP\\ot H ,}\n\\end{equation}\nwhere $1_H : \\Bbbk\\to H$ is the unit (map) of $H$.\nThe $\\ot$-exactness of \\eqref{diag4} means that \n\\begin{alist}\n\\item  \\eqref{diag4} is an equaliser diagram of $\\Bbbk$-linear transformations,\n\\item \\eqref{diag4} is a cokernel pair diagram of $\\Bbbk$-algebras,\n\\item  $\\pi$ is an effective descent monomorphism of $\\Bbbk$-algebras.\n\\end{alist}\nSince the equaliser of linear transformations is the same as the kernel of their difference, the condition (A) means that $B$ can be identified with the subalgebra  of $H$-coaction invariants of $P$,\n$$\nB\\cong P^{coH} := \\{ x\\in P\\; |\\; \\varrho(x) = x\\ot 1_H\\},\n$$\nand $\\pi$ is the inclusion map. The cotensor product of comodules in $\\mathfrak{Vect}_\\Bbbk^{op}$ is the same as the tensor product of modules over $\\Bbbk$-algebras. Thus the statement (B) means that the unique $\\Bbbk$-linear map $\\can : P\\ot_B P \\to P\\ot H$ fitting the following diagram\n$$\n\\xymatrix{ P\\ot_B P \\ar[rrrr]^-\\can &&&& P\\ot H \\\\\nP\\ot P \\ar[u] \\ar[rrrru]_{~\\hspace{8mm}m_{P\\ot H}\\circ (\\id_P\\ot 1_H\\ot \\varrho)}, &&&& }\n$$\nis an isomorphism. Here $m_{P\\ot H}$ denotes the multiplication in the tensor product algebra, that is, for all $x, y\\in P$ and $g,h\\in H$, \n$$\nm_{P\\ot H} (x\\ot g \\ot y \\ot h) = xy\\ot gh;\n$$\nsee \\eqref{eq.alg}. The map $\\can$ can be easily computed (compare the proof of Lemma~\\ref{lem.can.H}),\n$$\n\\can = (m_P \\ot \\id_H )\\circ (\\id_P \\ot_B \\varrho), \\qquad x\\ot_B y\\mapsto x\\varrho(y).\n$$\nTherefore, conditions (A) and (B) mean that $P$ is a {\\em Hopf-Galois $H$-extension of $B$}. \n\nThe map of $\\Bbbk$-algebras $\\pi: B\\to P$ induces a pair of functors between the categories of their right modules $\\Mod B$ and $\\Mod P$ (these are the categories of right comodules over the comonoids $B$ and $P$ in $\\mathfrak{Vect}_\\Bbbk^{op}$). The functor \n$\n\\pi_! : \\Mod P \\to \\Mod B,\n$\nis the restriction of scalars functor, which views every right $P$-module $E$ as a right $B$-module via the map $\\pi$, i.e., for all $a\\in B$ and $e\\in E$, $e\\cdot a := e\\cdot \\pi(a)$. Equivalently, $\\pi_!$ can be described as the homorphism functor $\\rhom P{{}_BP} -$ (and in the algebraic geometry literature often denoted by $\\pi_*$ as it is the {\\em direct image functor}). The functor $\\pi^* : \\Mod B\\to \\Mod P$ is the extension of scalars functor, for any right $B$-module $X$, $\\pi^*(V) = V\\ot_B P$, with the action of $P$ induced by the multiplication in $P$, i.e.\\ $(v\\ot_B x)\\cdot y := v\\ot_B xy$, for all $x,y\\in P$ and $v\\in V$. Equivalently, $\\pi^*$ can be described as the tensor functor $-\\ot_BP$ (and in the algebraic geometry literature it is often referred to as the {\\em inverse image functor}). \n\nBy reversing the arrows in the underlying monoidal category, we reverse adjunctions. Therefore, when viewed from the point of view of vector spaces, the functor $\\pi^*$ is the {\\bf left} adjoint of $\\pi_!=\\pi_*$. The formula for the unit of adjunction written in Section~\\ref{sec.quant} gives the counit of adjunction $\\pi^* \\dashv \\pi_*$ and vice versa. The composite $T= \\pi^*\\circ \\pi_*$ is a {\\bf comonad} on the category $\\Mod P$. For all right $P$-modules $E$, $T(E) = E\\ot_BP$ with $P$-action $\\id_E\\ot_B m_P$. The comultiplication and counit of $T$ come out as\n$$\nE\\ot_B P \\to E\\ot_BP\\ot_BP, \\qquad e\\ot_B x\\mapsto e\\ot_B 1_P\\ot_B x,\n$$\nand\n$$\nE\\ot_B P \\to E, \\qquad e\\ot_B x\\mapsto e\\cdot x. \n$$\n\nA {\\em comodule} (or {\\em coalgebra} in category theory terminology) over the comonad $T$, consists of a right $P$-module $E$ (i.e.\\ an object of the category $\\Mod P$ on which $T$ operates; the $P$-action is not written explicitly) together with a right $P$-linear map $\\xi: T(E) = E\\to E\\ot_B P$ which satisfies conditions obtained by reversing all the arrows   in diagrams \\eqref{alg}  and replacing cotensor product by the tensor product. On elements $e\\in E$ and writing $\\xi(e) = \\sum_i e_i\\ot_B x_i \\in E\\ot_B P$ the conditions satisfied by $\\xi$ are\n\\begin{equation}\\label{des}\n\\sum_i \\xi(e_i) \\ot_B x_i = \\sum_i e_i\\ot_B1_P\\ot_B  x_i , \\qquad \\sum_i e_i\\cdot x_i =e.\n\\end{equation}\nPairs $(E,\\xi)$ with $\\xi$ satisfying conditions \\eqref{des} are known as {\\em descent data} and form the backbone of descent theory of (noncommutative) algebras; see the classic text \\cite{Gro:techI} or  modern elegant expositions \\cite[Section~4.7]{Bor:han2}, \\cite{Nus:non}. The category of descent data corresponding to the $\\Bbbk$-algebra map $\\pi$ is denoted by $\\mathfrak{Desc}_\\pi$. We have just shown that the category of comodules over $T$ can be identified with  $\\mathfrak{Desc}_\\pi$, and therefore there is the following triangle of categories of functors\n$$\n\\xymatrix{ \\Mod B\\ar[rr]^{K}\\ar[dr]_{\\pi^*} & & \\mathfrak{Desc}_\\pi \\ar[dl]^{U} \\\\\n& \\Mod P .& }\n$$\nFor all right $B$-modules $V$, the comparison functor $K$ returns the descent datum $(V\\ot_B P, \\xi)$, where $\\xi: V\\ot_B P\\to V\\ot_B P\\ot_B P$, $v\\ot_B x\\mapsto v\\ot_B 1_P\\ot_B x$. The functor $K$ is an equivalence if and only if the algebraic descent associated to $\\pi$ is effective, which, by the Grothendieck theorem (extended in \\cite{Cip:dis}) is equivalent to the statement that $P$ is {\\em faithfully flat} as a left $B$-module. (Recall that, by definition, $P$ is faithfully flat as a left $B$-module if the sequence of right $B$-module maps\n$\n\\xymatrix{V\\ar[r]^-f &  V' \\ar[r]^-g & V''}\n$\nis exact if and only if the sequence\n$\n\\xymatrix{V\\ot_B P\\ar[rr]^-{f\\ot_B\\id_P} &&  V'\\ot_B P \\ar[rr]^-{g\\ot_B\\id_P} && V''\\ot_B P}\n$\nis exact.) \n\nTherefore, a noncommutative $H$-principal bundle in $\\mathfrak{Vect}_\\Bbbk^{op}$ is the same as a Hopf-Galois $H$-extension $B\\subseteq P$ such that $P$ is faithfully flat as a left $B$-module. If $H$ has a bijective antipode, then  by \\cite[Theorems~4.8~\\&~5.6]{SchSch:gen}, $P$ is an {\\em $H$-equivariantly projective} left $B$-module, that is the restriction of the multiplication map $B\\ot P\\to P$ has a left $B$-module right $H$-comodule section (splitting). An $H$-equivariantly projective Hopf-Galois extension is termed a {\\em principal $H$-comodule algebra} in \\cite{HajKra:pic}. A strong evidence that principal comodule algebras should be understood as principal bundles in noncommutative geometry  is being uncovered in \\cite{BauHaj:pet}. \nWe have just arrived at the following\n\n\\begin{ident}\nLet $H$ be a Hopf algebra (over a field $\\Bbbk$) with bijective antipode. Then principal $H$-comodule algebras can be identified with noncommutative $H$-principal bundles in the braided monoidal category $\\mathfrak{Vect}_\\Bbbk^{op}$.\n\\end{ident}\n\nWith this the main aim of these notes is achieved.\n\n\\section{Further directions}\\label{sec.dir}\n\\setcounter{equation}{0}\nIn this final section we comment on some aspects of Definition~\\ref{def.main} in a general braided monoidal category $\\cB$ (with equalisers preserved by the tensor product). Slightly greater familiarity with category theory than in the rest of this article is required here. The discussion of internal categories in \\cite[Chapter~8]{Bor:han2} or \\cite[Chapter~2]{Joh:top} and of Beck's theorem in \\cite[Section~3.3]{BarWel:top} might be of some assistance. \n\nWe consider first a monoidal category $\\cB$ (not necessarily braided) with coequalisers preserved by the tensor product).  Take a morphism of comonoids $\\pi: P\\to B$ and consider the associated monad $T$ defined by equations \\eqref{T}--\\eqref{mu} and the comparison functor $K: \\Com B \\to \\Com P^T$  defined in equation \\eqref{comparison}. For any right $P$-comodule $(E,\\lambda)$, there is an isomorphism (of $P$-comodules) $E\\to E\\coten P P$ induced from the coaction $\\lambda: E\\to E\\ot P$. Taking this isomorphism into account, $T$ can be identified with the functor\n$$\n\\hat{T}:= -\\coten P P\\coten B P : E\\mapsto E\\coten P P\\coten B P.\n$$\n Again taking the above isomorphism as a right unitor and a similar isomorphism as a left unitor, the category $\\Bicom P$ of $P$-bicomodules is a monoidal category with the cotensor product $\\coten P$ as its monoidal product. The fact that $\\hat{T}$ is a monad means that $P\\coten B P$ is a monoid in $\\Bicom P$. The multiplication of this monoid is \n $$\n \\id_P\\coten B \\pi_B \\coten B \\id_P: P\\coten B P\\coten B P \\cong P\\coten B P\\coten P P\\coten B P \\to P\\coten B B\\coten B P \\cong P\\coten B P,\n $$\n and the unit is induced from $\\Delta_P$. The category $\\Com P^T$ is isomorphic to the category of right modules over $P\\coten B P$.  As explained in \\cite{Agu:int}, $(P, P\\coten BP)$ might be interpreted as an {\\em internal category} in the monoidal category $\\cB$ ($P$ is the object of objects and $P\\coten B P$ is the object of morphisms). With this interpretation in mind, $\\Com P^T$ is isomorphic to  the category  $\\Pr{P, P\\coten BP}$ of {\\em internal presheaves} on $(P, P\\coten BP)$; see \\cite[Chapter~6]{Agu:int} and \\cite{Vaz:phd}. As $\\Com B$ is the category of right modules over the trivial monoid in  $\\Bicom B$, the functor $K$ compares the categories of presheaves over two internal categories in $\\cB$.\n\nThe necessary and sufficient conditions for $K$ to be an equivalence are provided by the Beck Precise Monadicity  Theorem \\cite{Bec:tri}. In particular, if $K$ is an equivalence, then its inverse functor $K^{-1}$ is defined as follows. For all $(E,\\lambda,\\xi)$ in $\\Com P^T$, $K^{-1}(E,\\lambda,\\xi) = E^P$,\nwhere $E^P$ is defined by the coequaliser\n\\begin{equation}\\label{Beck.coeq}\n\\xymatrix{ E\\coten B P \n\\ar@<0.5ex>[rr]^-{\\xi}\\ar@<-0.5ex>[rr]_-{\\epsilon_{(E,\\lambda)}} && \nE \\ar[rr]^-{\\Pi_E} && E^P.}\n\\end{equation}\nThe existence of coequaliser \\eqref{Beck.coeq} is guaranteed by the fact that $K$ is an equivalence. The natural isomorphism $\\Phi: \\id_{\\Com P^T} \\to K\\circ K^{-1}$ is the composite, for all $(E,\\lambda,\\xi)$ in $\\Com P^T$,\n$$\n\\Phi_{(E,\\lambda,\\xi)} : \\xymatrix{ E\\ar[rr]^-{\\hat{\\lambda}} && E\\coten B P \\ar[rr]^{\\Pi_E\\coten B \\id_P} && E^P\\coten B P.}\n$$\nThe natural isomorphism $\\Psi: K^{-1}\\circ K\\to \\id_{\\Com B} $, for any right $B$-comodule $(E,\\lambda)$, is the unique filler in the following diagram\n$$\n\\xymatrix{E\\coten B P\\coten B P \\ar@<0.5ex>[rrrr]^-{\\epsilon_{(E,\\lambda)}\\coten B \\id_P}\\ar@<-0.5ex>[rrrr]_-{\\epsilon_{(E\\coten BP,\\id_E\\coten B((\\id_E\\ot \\pi)\\circ \\Delta_P))}} && &&\nE \\coten B P \\ar[rr]^-{\\Pi_{E\\coten B P} }\\ar[drr]_{\\epsilon_{(E,\\lambda)}}&& (E\\coten B P)^P \\ar@{-->}[d]^{\\Psi_{(E,\\lambda)}} \\\\\n&&&&&& E.}\n$$\n\nIn the context of Definition~\\ref{def.main} the category $\\Com P^T$ and the functors $K$, $K^{-1}$ can be described in purely Hopf-algebraic terms.\n\n\\begin{proposition}  \\label{prop.invariants}\nLet $\\pi: P\\to B$ be a non-commutative principal $H$-bundle in a braided monoidal category $(\\cB,\\ot,\\mathbf{1},\\tau)$ as in Definition~\\ref{def.main}. Then:\n\\begin{zlist}\n\\item The morphism $\\can: P\\ot H \\to P\\coten B P$ described in Lemma~\\ref{lem.can.H} is a left $P$-colinear and right $H$-linear map.\n\\item  The functor $T$ is naturally isomorphic to the functor which sends right $P$-comodule $(E,\\lambda)$ to the right $P$-comodule $(E\\ot H,\\nu)$, where \n$$\n\\nu = (\\id_E\\ot \\id_H\\ot \\varrho)\\circ (\\id_E\\ot \\tau_{P,H}\\ot \\id_H)\\circ (\\lambda \\ot \\Delta_H).\n$$\nUpon this isomorphism the resulting functor is a monad with multiplication $\\id_E\\ot m_H$ and unit $\\id_E\\ot 1_H$. \n\\item The category $\\Com P^T$ is isomorphic to the category of relative $[P,H]$-Hopf modules, $\\Mod{P,H}$. The objects of  $\\Mod{P,H}$ are triples $(E,\\lambda ,\\zeta)$, where $(E,\\lambda)$ is a right $P$-comodule and $(E,\\zeta)$ is a right $H$-module, such that\n$$\n\\lambda\\circ\\zeta = (\\zeta\\ot \\varrho)\\circ (\\id_E\\ot \\tau_{P,H} \\ot \\id_H)\\circ (\\lambda\\ot \\Delta_H).\n$$\n\\item The comparison functor $K$ is naturally isomorphic to the functor which sends every right $B$-comodule $(E,\\lambda)$ to the relative $[P,H]$-Hopf module \n$$\n(E\\coten BP, \\id_E\\coten B \\Delta_P, \\id_E\\coten B \\varrho).\n$$\n\\item The functor $K^{-1}$ is naturally isomorphic to the {\\em $H$-invariants functor}, which sends $(E,\\lambda,\\zeta)$ to the $B$-comodule $(E^H, \\lambda^H)$, where $E^H$ is the coequaliser of $\\zeta$ and $\\id_E\\ot \\eps_H$, and $\\lambda^H$ is induced from the composite\n$$\n\\xymatrix{ E \\ar[r]^-\\lambda & E\\ot P \\ar[rr]^-{\\id_E\\ot \\pi} && E\\ot B \\ar[r] & E^H\\ot B.}\n$$\n\\end{zlist}\n\\end{proposition}\n\\begin{proof}\nThe morphism \\eqref{eq.can} is both left $P$-colinear and right $H$-linear, where  $P\\ot H$ (resp.\\ $P\\ot P$)  is a $P$-comodule by $\\Delta_P\\ot \\id_H$ (resp.\\  $\\Delta_P\\ot \\id_P$) and $H$-module  by $\\id_P \\ot m_H$ (resp.\\ $\\id_P \\ot \\varrho$).  Let $\\iota : P\\coten B P \\to P\\ot P$ be the equaliser monomorphism, so that  \n$$\n\\iota \\circ \\can = (\\id_P \\ot \\varrho)\\circ (\\Delta_P\\ot \\id_H);\n$$\nsee equation~\\eqref{eq.can}. The preservation of equalisers by tensor product allows one to make $P\\coten BP$ into a left $P$-comodule with coaction $\\Delta_P\\coten B \\id_P$ defined as the unique filler in the diagram\n $$\n\\xymatrix{ P\\ot (P\\coten B P)  \\ar[rr]^-{\\id_P\\ot \\iota} && P\\ot P\\ot P \n\\ar@<0.5ex>[rr] \\ar@<-0.5ex>[rr] && P\\ot P\\ot B\\ot P\\\\ \n&& P\\coten B P\\ar[u]_{(\\Delta_P\\ot \\id_P)\\circ\\iota}\\ar@{-->}[llu]^{\\Delta_P\\coten B \\id_P}; && }\n$$\ncompare diagram \\eqref{equaliser}. By construction, $\\iota$ is a $P$-colinear map. Similarly, he right $H$-action on $P\\coten B P$ is defined as $\\id_P\\coten B \\varrho$, \n$$\n\\xymatrix{ P\\coten B P  \\ar[rr]^-{ \\iota} && P\\ot P\n\\ar@<0.5ex>[rrrr]^-{(\\id_P\\ot \\pi\\ot \\id_P)\\circ(\\Delta_P\\ot \\id_P)} \\ar@<-0.5ex>[rrrr]_-{(\\id_P\\ot \\pi\\ot \\id_P)\\circ(\\id_P\\ot \\Delta_P)} &&&&  P\\ot B\\ot P\\\\ \n&& P\\coten B P\\ot H\\ar[u]_{(\\id_P\\ot \\varrho)\\circ(\\iota\\ot H)}\\ar@{-->}[llu]^{\\id_P\\coten B \\varrho}; &&& }\n$$\ncompare diagram \\eqref{eq.pi*}. Again by construction $\\iota$ is an $H$-linear map. \n Therefore,\n$$\n( \\id_P \\ot \\iota)\\circ ( \\id_P\\ot \\can)\\circ (\\Delta_P\\ot \\id_H) = (\\Delta_P \\ot \\id_P)\\circ \\iota \\circ \\can \n= (\\id_P\\ot \\iota)\\circ  (\\Delta_P\\coten B \\id_P)\\circ \\can.\n$$\nThe first equality expresses the $P$-colinearity of morphism \\eqref{eq.can}, while the second is the $P$-colinearity of $\\iota$. Since the tensor product preserves equalisers,\n$(\\id_P \\ot \\iota) : (P\\ot P)\\coten B P\\cong P\\ot (P\\coten B P)\\to P\\ot P\\ot P$ is the equaliser monomorphism. The (left) cancellation property of monomorphisms now yields\n$$\n( \\id_P\\ot \\can)\\circ (\\Delta_P\\ot \\id_H) =(\\Delta_P\\coten B \\id_P)\\circ \\can,\n$$\ni.e.\\ $\\can$ is a left $P$-colinear morphism as required. The $H$-linearity of $\\can$ is proven is a similar way. This proves statement (1).\n\n\nSince $\\can$ is an isomorphism of left $P$-comodules and right $H$-modules,  there is a chain of isomorphisms\n$$\n\\xymatrix{E\\coten B P\\ar[r]^-\\cong & E\\coten PP\\coten B P\\ar[rr]^-{\\id_E\\coten P\\can^{-1}} && E\\coten P P\\ot H \\ar[r]^-\\cong & E\\ot H.}\n$$\nOne uses this composite isomorphism, which is natural in $E$, to form the required isomorphisms of functors and  the required equivalence of categories. All the remaining statements are obtained by translation through this isomorphism. \n\\end{proof}\n\nOnce this interpertation of categories and functors is made, we find ourselves in the realm of Hopf-Galois theory in braided monoidal categories as developed for example in \\cite{Sch:bra}. Making arguments dual to  these in \\cite[Section~4]{Sch:bra} one associates to a noncommutative principal $H$-bundle $\\pi: P\\to B$ a {\\em quantum category} in the sense of \\cite[Section~12]{DayStr:cat} (or a braided version of a {\\em bicoalgebroid} in terminology of \\cite[Section~5]{BrzMil:bia}) as follows. $B\\cong P^H$ is the object of objects and $G= (P\\ot P)^H$ is the object of morphisms, where $P\\ot P$ is a $[P,H]$-Hopf module with coaction $\\id_P\\ot \\Delta_P$ and the diagonal right $H$-action. The source and target maps are induced from $\\pi\\ot \\eps_P$ and $\\eps_P\\ot \\pi$. The multiplication (or composition of morphisms) $m_G: G\\coten BG \\to G$ is induced from the composite\n$$\n\\Pi_{P\\ot P}\\circ (\\varrho\\ot \\id_P)\\circ (\\id_P\\ot\\eps_P\\ot \\id_H\\ot \\id_P)\\circ (\\id_P\\ot\\can^{-1}\\ot \\id_P),\n $$\n while the unit $u_G: B\\to G$ is induced from $\\Pi_{P\\ot P}\\circ\\Delta_P$. The coalgebra structure of $G$ is induced from the tensor product coalgebra $P\\ot P^{op}$, where $P^{op}$ is the opposite coalgebra to $P$, that is the object $P$ of $\\cB$ with comultiplication $\\tau_{P,P}\\circ \\Delta_P$ and counit $\\eps_P$. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[width=\\linewidth]{images\/introduction.png}\n   \n    \\caption{Illustration of proposed value alignment task. Given the same content, {\\scshape VA-model}~makes variable predictions based on explicitly provided human values. \n   \n    }\n    \\label{fig:introduction}\n\\end{figure}\n\n\n \n\n\n\n\n\n\n\n\n\n\nThe demand for responsible NLP technology -- to make it more robust, inclusive and fair, as well as more explainable and trustworthy -- has increased since pre-trained large-scale language models (LLMs) have brought about significant progress in making NLP tasks more efficient and broad-ranging \\cite{brown2020language,zhang2022opt,chowdhery2022palm, radford2019language,brown2020language,petroni2019language,madotto2020language}. Researchers have studied how to align machines with human values as one of the directions to achieve responsible AI technology by teaching machines about moral and social norms \\cite{forbes2020social, emelin2020moral, jiang2021delphi}, ethics and common human values \\cite{hendrycks2020aligning} or human preferences \\cite{christiano2017deep, koren2008factorization}.\n\nValue-alignment of AI systems is not a trivial problem as human values are non-consensual by nature~\\cite{hanel2018cross}. \nValues can be very diverse and most existing works have attempted to align machines with shared human values or average norms, or from a certain cultural perspective with crowd sourced annotations \\cite{jiang2021delphi}.\nThese days, for instance, many societies agree that sexism should be eliminated, and we expect machines to be non-sexist, but different individuals and cultures may perceive sexism differently. As is shown in Figure \\ref{fig:introduction}, the same content can be considered to be sexist or non-sexist depending on the values provided to make the judgements.\n\n\n\nIn this paper, we propose a value-aligned judgement task that separates the value definition process from the development of the models for more inclusive and explainable value-aligned NLP. Our proposed task aims to build a single model to make dynamic judgements based on explicitly provided human values, requiring the model to understand the value and its corresponding entailment on the given content. The value is provided in the form of instructions, allowing coarse-to-fine customization.  We start with value-aligned sexism classification as a proof of concept for the proposed approach, as sexism is one of the most representative examples of varying cultural perspectives.\n\n\nWe also present Value-Aligned Models ({\\scshape VA-model}{s}) that leverage value-based knowledge from LLMs.\nLLMs are trained from vast amounts of human data with embedded human values \\cite{hendrycks2020aligning}. However, they are not controllable and it is difficult to fine-tune such large models with explicit value alignment. \nInstead,  we distill value-based training data from the LLMs using prompt-based data generation with example values, and build {\\scshape VA-model}~ by fine-tuning smaller classification models with the distilled data. \nExperimental results show that our approach is more stable and accurate than directly applying few-shot learning on LLMs. Moreover, our methodology avoids costly human labeling or crowd sourcing of values, allowing easier extensions to other value-aligned tasks in different domains.\nWe further investigate model performance using data generated from different scales and types of LLMs, and study the effect of data size for fine-tuning, and analyze the quality of the generated data. Moreover, we study the generalization ability of {\\scshape VA-model}{s} by testing its performance on unseen value sets.\n\nOur contributions are as follows: 1) we introduce the value-aligned classification task, where we first define human values externally and then use them at the instruction level in an in-context learning paradigm and construct value-aligned classifiers to make predictions; 2) we propose to leverage prompt-based data generation to distill value-aligned knowledge from LLMs for smaller classification models; 3) experimental results indicate that our approach significantly outperforms strong baselines, including in-context few-shot learning with LLMs and existing text augmentation methods; 4) we systematically study factors that impact prompt-based data generation and highlight research questions and challenges in the value-aligned judgement task through thorough analysis. \n\n\n\n\\section{Related Work}\n\\paragraph{Human Value Alignment}\nOne challenge in value alignment is value definition, and there has been a profusion of documents on AI ethical standards \\cite{gabriel2020artificial, dignum2017responsible}. \\citet{jobin2019global} identified 11 clusters of ethical principles among 84 documents, and \\citet{fjeld2020principled} found eight key themes across 36 of the most influential of them. However, since human values are variable with culture, we anticipate value definition to be dynamic. Meanwhile, the values should be defined externally to the development of the NLP algorithms, like how we adopt definitions of sexism categories based on social studies. \n\nTo teach models value-alignment, the literature has focused on improving the model's reasoning ability relating to human values and morality \\citep{forbes2020social,emelin2020moral,lourie2021scruples, hendrycks2020aligning}.\nRecently, \\citet{solaiman2021process} proposed to fine-tune GPT-3 to adapt to a manually crafted values-targeted dataset to arrive at a values-targeted model. However, in their approach, value alignment and definition are intertwined and entangled in an iterative process. We instead separate the value definition and alignment process models about value-aligned judgement with explicit value provision.\n\n\n\n\\paragraph{Prompt-based Learning}\nRecently, LLMs have shown great performance on prompt-based learning \\cite{brown2020language,chowdhery2022palm}, which doesn't require fine-tuning. Instead, the model is directly fed a prompt that includes some examples, and the model can generate results as if it has ``learned''. Studies on efficient prompt-learning\/-construction include \\citet{lu2021fantastically, reynolds2021prompt, zhao2021calibrate, schick2020few}. We consider the literature for prompt-construction in our methodology.\n\n\\paragraph{Knowledge Distillation} Knowledge distillation is the transfer of knowledge from teacher to student distribution \\cite{distilling2015hinton}. Recent works have attempted to perform distillation from LLMs by prompting for text generation to show that it outperforms existing text augmentation methods \\cite{yoo2021gpt3mix, wang2022exploring}.  \\cite{west2021symbolic} retrieves commonsense knowledge symbolically in a text form from GPT-3 for downstream tasks with help of smaller filtering classifiers. We distill \\textit{value-specific} knowledge, not all abilities of general language model, from LLMs through value-aligned training data generation for training smaller value-aligned classifiers. This reduces the cost of human labeling and also enables building smaller models specialized for value-aligned judgment task.\n\n\n\n\n\n\\section{Value-Aligned Judgement Task}\n\\label{sec:task_introduce}\n\\subsection{Task Description}\nAs an effort to align machines with human values, our task focuses on teaching the model that different values can lead to different judgements even given the same content. The task is formulated as follows. A model needs to make a judgement $Y_V$ on content $C$ based on an explicit human value $V$. In this work, ``value'' refers to any qualities, standards of behavior, or beliefs that individuals or societies hold, and is expressed in natural language phrases or sentences. The set of values is externally defined by a human user of the system or from existing relevant literature on moral philosophy, and is independent of the development of algorithms. The distinction from the existing value-aligned classification task\nand conventional classification tasks is that our task expects the model to incorporate \n\\textit{explicitly provided values} along with other inputs for making judgements. \n\n\nWe separate the process of value definition from the development of the value-aligned models so that the models can learn to make dynamic judgements based on external values. For instance, existing sexism classifiers implicitly learn a fixed set of definitions of sexism from labeled data, so the content will be judged based on these static values. Our task requires the model to predict dynamic labels depending on the different explicit values even when the content is the same. \n\n\\subsection{Value-aligned Sexism Classification}\nWe showcase the value-aligned judgement task with an application to sexism classification. The model needs to judge whether natural language content is sexist or non-sexist based on a given value $V$. If the value is not applicable or irrelevant, the model needs to predict that it is not applicable (NA). \nOur rationale for choosing the sexism classification task is that the definition of sexism has changed over time as values have evolved and altered and it still varies across cultures. Thus, we can verify the effect of varying values in a more evident manner in the sexism classification task. Furthermore, the importance of a fine-grained understanding of sexism has been emphasized~\\cite{jha-mamidi-2017-compliment, sharifirad-etal-2018-boosting, parikh-etal-2019-multi}.\nThis aligns with our motivation for explicit value-aligned judgement. Lastly, values related to sexism are complicated, involving religious, cultural, and personal beliefs or values. We thus believe it is a task with enough complexity to act as a case study. \n\n\n\n\\section{Methodology}\n\\label{sec:methodology}\nThere is no existing resource for training value-aligned classification models. We therefore propose to leverage LLMs for generating synthetic training data. LLMs have been found to learn significant amounts of inherent knowledge as well as human values during pre-training \\cite{petroni2019language, hendrycks2020aligning, west2021symbolic, roberts2020much}. \nHowever, the direct usage of LLMs in zero-shot setting for NLP tasks can be unstable and still limited~\\cite{wei2021finetuned}.\nThe richly embedded knowledge in LLMs nevertheless makes them good resource generators. Therefore, we attempt to build value-aligned models ({\\scshape VA-model}{s}) through fine-tuning smaller models on the value-aligned training data generated by LLM(s).\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[width=0.95\\linewidth]{images\/vamodel.png}\n   \n    \\caption{Illustration of the construction of our proposed  {\\scshape VA-model}. Using LLMs, we first create synthetic value-aligned training data. Then, we transfer the knowledge into smaller models by fine-tuning them on the data, so Value Aligned Models can make value-aligned judgements.}\n    \\label{fig:pipeline}\n\\end{figure}\n\nOur proposed method (Figure \\ref{fig:pipeline}) consists of two steps: 1) prompting human value-aligned contents from LLMs by providing explicit human values and instructions, and 2) fine-tuning smaller LMs on the generated data to teach them about value-aligned judgements. \nFormally, we build {\\scshape VA-model}~ (parameterized by $\\theta$) to maximize the following likelihood:\n\\begin{equation}\n    L(\\theta) = logP(Y|V,C;\\theta).\n\\end{equation}\n\n\n\n\\subsection{Value-Aligned Knowledge Distillation: Prompt-based Data Generation}\n\\label{sec:data_generation}\n\n\\paragraph{Prompt Construction with Few-shot Examples}\nThe prompt construction of in-context few-shot examples affect performance. Thus we refer to the existing literature on different prompt-techniques \\cite{reynolds2021prompt,zhao2021calibrate, yoo2021gpt3mix}. For the few-shot examples, we create a pool of 10 human-labeled samples (value, content, and value-aligned labels) for each value. According to \\citet{lu2021fantastically}, the order of the few-shot samples in the prompt affects the in-context learning for LLMs. Therefore, we randomly select and order five samples out of the pool.\n\nTo select the most appropriate prompt for generating value-aligned synthesized data, we test five candidate prompt templates with reference to literature. All prompt templates consist of a label, a value, and value-aligned content examples. The best-performing prompt template is selected based on testing with a smaller size of the samples. The prompt templates and their performance are available in Appendix~\\ref{appendix:Prompt_Choice_and_Results}.\n\n\n\\paragraph{Generation}\nWe feed the prompts to LLMs to generate value-aligned synthetic training samples. Our method is model agnostic in that any LLMs can be adopted for this step. Recently, LLMs have scaled to more than 500 billion parameters \\cite{chowdhery2022palm,smith2022using}, and some models with more than 100 billion parameters are available publicly, such as Jurassic-1 Jumbo (GPT-Jurassic~\\citet{lieber2021jurassic}), Open Pre-trained Transformer (OPT~\\citet{zhang2022opt}), and GPT-3. In this paper, we choose OPT-175B for the main experiment and provide an analysis on the effects of the size and types of LLMs.\n\n\n\n\n\\paragraph{Generated Content Extraction \\& Processing}\nThe generated content is generated in succession after the prompt as natural text, and extracted through pattern matching. We gather all extracted content to construct a synthetic training set for teaching the smaller models in the next step, and process the generated data as follows. Firstly, we keep only unique samples by dropping all duplicates. Then, we remove exact copies of the few-shot examples used in the prompts. Finally, any content less than three words is filtered out as it is less informative. \n\n\\subsection{Fine-tuning Smaller Models -- Value-Aligned Models}\n\\label{sec:Fine-tuning_Classifiers}\nIn the next phase, we build classifiers by fine-tuning relatively smaller transformer-based models (e.g., ALBERT-base, RoBERTa-base, BART-base) with the generated training data to enable them to make value-aligned judgements. We add a linear layer on top of the pooled output of the smaller models to construct our proposed {\\scshape VA-model}. In order to make the model intake both values and content in the learning phase, the input text is formatted into \\texttt{``value [sep] content [sep]''} and the output is a value-aligned judgement.\n\nThe classifiers need to predict different labels according to explicitly provided values given the same content. Recalling the example of value-aligned sexism classification in Figure \\ref{fig:introduction}, the same content can be considered to be sexist, non-sexist or NA depending on the considered values. \n\n\\section{Experiments}\nIn this paper, we conduct value-aligned sexism classification. Models are expected to label content with label choices sexist, non-sexist, NA \\textit{depending on} explicitly provided values.\n\n\n\n\n\n\\subsection{Dataset}\nWe borrow multi-label sexism categorization data (multi-sexism) \\cite{parikh-etal-2019-multi}, which offers fine-grained sexism categorization for sexist content.\nExample categories include, but are not limited to, \\textit{Role-stereotyping}, \\textit{Pay gap}, and \\textit{Mansplaining}. We select 10 items of content per category to have a small set of human-labeled data for the prompt-construction in our methodology and baselines. The rest of the data are used as the test set.\n\nBased on the description of each category, we manually compose two opposing values -- one making the content  sexist (value) and another making the content  non-sexist (counter-value). For instance, any \\textit{Role Stereotyping} contents will be considered to be sexist based on the value ``Men and women are equally capable for any role,'' but can also be considered to be non-sexist with the different value ``Men and women are biologically different; hence certain roles are more appropriate for women.'' A full list of values and counter values is available in Appendix \\ref{appendix:value_c_value}. In total, we consider 19 categories of sexism and two corresponding values (value, counter-value) for each category, translated into 38 (19 $\\times$ 2) human values.\n\n \n\n\n\n \n\n\n\\paragraph{Test set}\nWe use the original multi-label sexism content (human-labeled, non-synthetic) for creating a test set for the value-aligned judgement task, excluding that used for prompt-construction in the training data generation. Originally, each item of content is labelled with one\/multiple sexism categories. For our task setup, we translate the data into the form of triplet \\{content, value, label\\}, and we assign value-dependent labels to each sample.\nFor instance, if content $C$ was originally labelled as \\textit{Role-stereotyping} (\\textit{RS}), we convert into three testing samples, \\{$C$, value$_{RS}$, Sexist\\}, \\{$C$, counter-value$_{RS}$, Non-Sexist\\}, and \\{$C$, random value\/counter-value, NA\\}. Note that values for NA labels are totally unrelated to the content category. In this way, we can inspect the model's performance in making a value judgement on the same content with different values. In total, there are 17,720 test samples, with a label ratio of 1:1:1. \n\n\\subsection{Models}\n\\subsubsection{{\\scshape VA-model}{s} (Ours)}\n\n\\paragraph{Generating value-aligned training data}\nUsing the method explained in Section~\\ref{sec:data_generation}, we get 100 content pieces \nfrom each of the value and counter-value prompts. In sum, there are 200 unique pieces of content per category.\\footnote{Reflecting the original ratio of multi-sexism, we keep the original number of samples if there are less than 100.} Then, all content per category is paired with a value and counter value and corresponding labels \\{content, \\textit{value}, `Sexist'\\} and \\{content, \\textit{counter-value}, `Non-Sexist'\\}. So, each content item has a duplicate but is paired with different values and value-aligned judgements. To prevent the model from only learning two value and label associations, we synthetically make the class `NA' by assigning irrelevant values\/counter-values to the content (e.g., assigning the value of \\textit{Pay Gap} to a content of \\textit{Role Stereotyping} so the label is `NA'). In total, there are 10,722 samples, including the prompt construction samples. We split them into training and validation sets with a ratio of 4:1.\n\n\\paragraph{Building {\\scshape VA-model}{s}} We finetune smaller models with the generated value-aligned training data. We build {\\scshape VA-model}{s} to incorporate explicit human values to make judgements for value-aligned sexism classification following Section~\\ref{sec:Fine-tuning_Classifiers}.\nFor the smaller models, we take base versions of ALBERT (12M params.) ~\\cite{lan2019albert},  RoBERTa (125M params.)~\\cite{liu2019roberta} and BART (110M params.)~\\cite{lewis2019bart} to construct {\\scshape VA-ALBERT}, {\\scshape VA-RoBERTa}~and {\\scshape VA-BART}, respectively. \nRoBERTa has been proved to be robust in various NLP tasks and BART shows comparable performance to RoBERTa on GLUE tasks.\n\n\n\n\\subsubsection{Baselines}\n\\label{sec:Baselines}\nTo examine our proposed approach, we compare it with multiple baselines, including a random baseline, prompt-based few-shot learning with OPT-175B, and fine-tuning transformer-based models. For the fine-tuning setting, we fine-tune on different data setups -- only with human-labeled data (without generated data) and with semantically augmented data.\n\n\\paragraph{Random Baseline} We randomly select the predicted label for each test sample with the same label probability distribution as in the training data. \n\n\\paragraph{OPT-175B (few-shot)} This baseline uses OPT-175B with a prompt-based few-shot learning for \\textit{label prediction}.~\\footnote{We use prompt-based few-shot learning with OPT-175B for generating \\textit{value-aligned content} in our methodology while the baseline used it for directly predicting \\textit{label}. Refer to Appendix \\ref{sec:appendix_exp} for details.} We provide 20 few-shot samples in the context.\n\n\\paragraph{Human-Labeled (HL)-Models} We only use the small subset of human-labeled samples as training data to fine-tune smaller transformer-based LMs with a linear layer trained on top. We choose the base versions of ALBERT, RoBERTa and BART as the backbone models for a fair comparison with our {\\scshape VA-model}{s}.\n\n\\paragraph{Nlpaug-Models} Nlpaug~\\cite{ma2019nlpaug} is semantic augmentation method using BERT-base embedding. We conduct augmentation with prompt construction examples by insertion and substitution. For each examples, we make 10 augmented samples (five insertions and five substitutions). Then, we fine-tune the base versions of ALBERT, BART and RoBERTa on the semantically augmented data and prompt-construction examples so we can evaluate the effectiveness of the prompt-based augmentation in our method.\n\n\n\n\\subsection{Experimental setup}\n\\label{sec:Experimental_setup}\n\n\\paragraph{Evaluation metric}\nWe evaluate our experiments with both F1 score and accuracy. For the main results, we report all accuracy, weighted F1-score (W-F1), precision and recall.\n\n\\paragraph{Implementation Details}\nFor generating value-aligned training data, we conduct the main experiment with OPT-175B model with top-p 0.7 and temperature 1. \nFor our {\\scshape VA-model}{s} and {HL-Models}~baselines, we use pre-trained transformer-based LMs available through the HuggingFace API.\nFurther implementation details such as hyperparameters are given in Appendix \\ref{sec:appendix_exp}.\n\n\n\\begin{table}[t]\n    \\centering\n   \n    \\begin{adjustbox}{width={0.45\\textwidth},totalheight={\\textheight},keepaspectratio}\n\n    \\begin{tabular}{l|cc}\n    \\toprule\n    Model & $Accuracy$ & $W$-$F1$ \\\\ \\midrule\n    Random Baseline       & $33.53_{0.41}$ &  $33.53_{0.41}$\\\\ \n    OPT-175B (few-shot)   & $55.18_{7.75}$ & $54.78_{7.20}$\\\\\n    {HL-ALBERT}  & $58.70_{4.43}$ & $51.67_{3.96}$ \\\\\n    {HL-RoBERTa}  & $64.53_{2.54}$ & $55.23_{1.91}$ \\\\\n    {HL-BART} & $63.23_{1.87}$ & $54.93_{1.47}$ \\\\\n    Nlpaug-ALBERT  & $62.87_{2.13}$ & $58.80_{3.44}$ \\\\\n    Nlpaug-RoBERTa & $61.52_{3.03}$ & $58.67_{2.89}$ \\\\\n    Nlpaug-BART & $59.03_{1.38}$& $58.49_{1.60}$ \\\\ \\midrule\n    {\\scshape VA-ALBERT} & $70.10_{1.65}$ & $70.75_{1.48}$\\\\\n    {\\scshape VA-RoBERTa} & $73.24_{0.39}$& $73.82_{0.32}$\\\\\n    {\\scshape VA-BART} & $\\mathbf{74.07}_{0.82}$ & $\\mathbf{74.36}_{0.60}$\n    \\\\ \\bottomrule\n   \n    \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Evaluation results of baselines and our proposed {\\scshape VA-model}{s}, on the value-alignment task. We use 200 value-aligned training data samples generated from the LLMs per category to fine-tune {\\scshape VA-model}. Experiments are ran with five random seeds and results are reported in $mean_{std}$ format. All our {\\scshape VA-model}~performances are statistically significant (t-test with $p$-value $<0.05$). Scores are all in percentage (\\%).}\n    \\label{tab:main_results}\n\\end{table}\n\n\n\\section{Results and Analysis}\n\\label{results_and_analysis}\n\n\n\n\\subsection{Main Results}\n\\label{sec:main_results}\n\\paragraph{Effectiveness of our method}\nTable \\ref{tab:main_results} shows the performance of the models on the value-aligned sexism classification task. Our models achieve better scores on W-F1 and accuracy than the baselines by large gaps ($15.56\\sim40.83$\\% gain in W-F1), which signifies the robustness and superiority of our approach. Our {\\scshape VA-ALBERT}~also surpasses all baselines, including those back-boned with bigger models (e.g., Nlp-RoBERTa, {HL-RoBERTa}). This highlights the effectiveness of the value-aligned knowledge distillation with LLMs. \n\nWe observe that the OPT-175B few-shot learning approach performs better than random label assignments on the test set and {HL-ALBERT}, but still performs worse than or as comparable as the other baselines. This indicates that LLMs with prompt-based few-shot learning can understand the value-aligned classification task to some extent, but the performance is still low. {HL-Models}~surpass OPT-175B (few-shot) under all evaluation metrics except {HL-ALBERT}~in W-F1 score, showing that the models can capture our task with limited human-labeled data due to the effectiveness of fine-tuning. Nlpaug is one of the conventional data augmentation approaches and we augment the same amount of data as {\\scshape VA-model}. In comparison with {HL-Models}, Nlpaug-models show higher W-F1 scores with small drops in accuracy.\n\nOverall, the experimental results support our proposed approach for the value-aligned judgement task. OPT-175B (few-shot) shows much lower and unstable performance than {\\scshape VA-model}{s}~although the value-aligned training data of {\\scshape VA-model}{s}~is generated from OPT-175B. For the prompt-based few-shot approach, especially when the task setup is complicated like value-aligned classification, the model cannot easily overfit the task by giving several prompts, leading to a higher chance to predict random labels. Instead, we used a knowledge distillation approach through training data generation, which is a simpler task for the model as the main objective of the general language model is text generation.\nMoreover, utilizing the LLMs for generating knowledge distilled data is more effective than simple semantic text augmentation (e.g., Nlpaug). \n\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[width=\\linewidth]{images\/category_s.png}\n   \n    \\caption{Evaluation results of {\\scshape VA-BART}~per sexism category on the test set. Only the top and bottom five categories (based on W-F1) are displayed. The performance for the nine categories in the middle are $\\sim80\\%$ for Acc. and W-F1. The full results for the 19 categories are available in Appendix~\\ref{sec:appendix_Per_Category_Results}.}\n    \\label{fig:analyze_category}\n\\end{figure}\n\n\n\n\n\\paragraph{Per-Category performance} Figure~\\ref{fig:analyze_category} presents the per-category evaluation scores of {\\scshape VA-BART}. The results vary significantly between categories, indicating the complexity of our proposed task. The results for both \\textit{Menstruation-related Discrimination} and \\textit{Pay Gap} achieve scores higher than 90\\%, while the results for \\textit{Internalized Sexism} are relatively low. We conjecture reasons for the high performance of certain categories are varying quality of generated training data per categories and more distinguishable features than other. We investigate this point further in Section~\\ref{sec:quality_analysis}.\n\n\n\\subsection{Quality Analysis for Generated Value-Aligned Training Data}\n\\label{sec:quality_analysis}\n\n\n\\paragraph{Distinction between generated data \\& test set}\nThe vocabulary overlap between all generated data (training set for {\\scshape VA-model}{s}) and test set data is 51.79\\%.\nMoreover, we check how many of generated data samples that share more than 80\\% of vocabulary with at least one of the test data samples, finding that \\textit{only} 0.01\\% of generated data samples reach the threshold (80\\%). Therefore, the data generated from OPT-175B for training {\\scshape VA-model}{s}~is clearly distinct from the test set.\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[width=0.8\\linewidth]{images\/overlap_6cate.png}\n   \n    \\caption{Vocabulary overlaps (\\%) of the generated data among sexism categories. Only top-3 and bottom-3 categories are displayed in descending order of W-F1 (top to bottom; left-to-right). Full set is in Appendix~\\ref{sec:appendix_vocabulary_overlaps}.}\n\\label{fig:analyze_overlap}\n\\end{figure}\n\n\n\\paragraph{Diversity of Data}\nWe calculate the vocabulary overlaps for each sexism category of the generated data in Figure~\\ref{fig:analyze_overlap}. We observe that the vocabulary overlaps are generally small, which illustrates that OPT-175B can generate diverse data for different values (e.g., sexism categories) provided in prompts. We can observe the trend that the overlaps among high performing categories are small, especially \\textit{Pay gap} and \\textit{Menstruation related}, which make data sample distinguishable to others. In contrast, low performing categories, overlaps are relatively higher.\n\n\n\n\n\\paragraph{Human evaluation}\nLLMs are powerful few-shot learners, yet they are not perfect. Thus, we conduct human evaluation on two categories' data (\\textit{Internalized sexism} and \\textit{Pay gap}) to further investigate the augmented data quality. We assess generated contents from two aspects: 1) relevance to the corresponding category ($R$); 2) sexism ($S$).\nPay gap data are evaluated to be relevant and sexist ($R$\\&$S$) 69.44\\% of the time ($R$: 75.93\\%, $S$: 87.03\\%) while internalized sexism data are evaluated to be 25\\% ($R$\\&$S$) ($R$: 34.50\\%, $S$: 67.50\\%).\nWe observe that the quality of \\textit{Pay gap} generated data is much better than that of \\textit{Internalized sexism}, which is consistent with the per category results in Figure~\\ref{fig:analyze_category}. This highlights the difficulty of our task and the need for more robust prompt templates for prompt-based data generation. And the human-in-the-loop method may further boost the performance of our approach with less noisy data.\n\n\\paragraph{Effectiveness of generated training data}\n\\begin{table}[t]\n    \\centering\n    \\begin{adjustbox}{width=0.85\\linewidth,totalheight={\\textheight},keepaspectratio}\n    \\begin{tabular}{l|cc}\n    \\toprule\n    Model & $Accuracy$ & $W$-$F1$  \\\\ \\midrule\n    {\\scshape VA-ALBERT}  & $70.10_{1.65}$ & $70.75_{1.48}$ \\\\\n        \\qquad w\/o human labeled data & $70.79_{1.40}$ & $71.29_{1.33}$\\\\\\midrule\n    {\\scshape VA-RoBERTa} & $73.24_{0.39}$& $73.82_{0.32}$\\\\\n        \\qquad w\/o human labeled data & $72.90_{2.06}$ & $73.19_{1.68}$ \\\\\\midrule\n    {\\scshape VA-BART}            & $\\mathbf{74.07}_{0.82}$ & $\\mathbf{74.36}_{0.60}$\\\\\n        \\qquad w\/o human labeled data &  $72.30_{1.24}$& $72.71_{0.90}$ \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Effectiveness of generated data. We remove human-labelled data from the training set and \\textbf{only} use synthetic samples generated from LLM for training (w\/o human-labeled data). The minimal drops in performance show the effectiveness of value-aligned training data generated from LLMs for the value alignment task.}\n    \\label{tab:ablation_study}\n\\end{table}\n\nTo investigate the standalone effectiveness of the generated training data (value-aligned knowledge distillation), we study the performance of {\\scshape VA-model}{s}~when they are trained \\textit{without} any of human-labeled data but \\textbf{only} with generated data (Table \\ref{tab:ablation_study}). Minor performance degradations in both {\\scshape VA-BART}~and {\\scshape VA-RoBERTa}~are investigated, $-1.65\\%$ and $-0.63\\%$ W-F1 respectively. However, these values are still above those of the baselines. Interestingly, {\\scshape VA-ALBERT}~showed a minimal performance gain on both accuracy and W-F1. This indicates that the value alignment knowledge distilled from LLMs is the main contributor for {\\scshape VA-model}~to understand the task.\n\n\n\n\\begin{table}[t]\n    \\centering\n    \\begin{adjustbox}{width=0.8\\linewidth,totalheight={\\textheight},keepaspectratio}\n    \\begin{tabular}{l|cc}\n    \\toprule\n    Model & $Accuracy$ & $W$-$F1$ \\\\ \\midrule\n   \n    OPT-175B (fewshot) & $32.97$ & $30.23$ \\\\\n    {HL-ALBERT}  &  $40.25_{6.05}$& $37.52_{7.68}$ \\\\\n    {HL-RoBERTa}  & $47.79_{5.65}$& $45.35_{6.09}$ \\\\\n    {HL-BART} &  $46.09_{3.42}$& $45.97_{3.68}$ \\\\\n    \n    Nlpaug-ALBERT  &  $48.10_{11.0}$& $40.62_{8.05}$\\\\\n    Nlpaug-RoBERTa &  $40.14_{2.00}$& $30.64_{3.56}$\\\\\n    Nlpaug-BART &  $47.76_{3.89}$& $42.40_{5.45}$ \\\\ \\midrule\n    \n    {\\scshape VA-ALBERT} &  $55.15_{6.83}$& $53.14_{9.05}$ \\\\\n    {\\scshape VA-RoBERTa} &$\\mathbf{58.13}_{5.33}$& $\\mathbf{56.60}_{6.56}$\\\\\n    {\\scshape VA-BART} & $57.98_{5.12}$& $55.94_{5.48}$\n    \\\\ \\bottomrule\n   \n    \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Performances of {\\scshape VA-model}{s}~and baselines on \\textbf{unseen} values in value-aligned sexism classification. In the training phase, models did not see any of the values in the test set.}\n    \\label{tab:unseen}\n\\end{table}\n\n\n\\subsection{Generalization Ability on Unseen Values}\n\nTo understand capacity of models to generalize value-aligned judgement over unseen values, we conduct an experiment in which three randomly selected sexism categories are separated from the training process (i.e., models have never seen values related to the three categories in the training phase and are evaluated on test set only composed of those unseen values) and the results are presented in Table~\\ref{tab:unseen}. \nOverall, there are drops in performance compared to the main experiment (Table~\\ref{tab:main_results}), while all of our {\\scshape VA-model}{s} continue to outperform all baselines. The baselines experience larger drops (maximum 43.18\\% for Nlpaug-RoBERTa) than the {\\scshape VA-model}{s} (17.22\\% for {\\scshape VA-RoBERTa}). Considering the model was never taught or received any direct supervision on the test values, it is expected behavior as other generalization problem. We leave how to improve the models' generalization ability in value-aligned judgement task for future work.\n\n\n\n\n\\subsection{Ablation Studies}\n\\label{sec:ablation_study}\n\n\\paragraph{LLMs capacity for prompting}\n\\label{sec:LLMs'_capacity}\nWe first investigate how the size of LLMs affects the capacity for generating value-aligned training data by evaluating the final performance of {\\scshape VA-model}~trained on data from varying sizes of LLMs.\nUnsurprisingly, as is shown in Table \\ref{tab:size_of_llms}, we can continually boost the model's performance when the LLMs size increase.\n\nWe also train {\\scshape VA-BART}~with the data prompted from GPT-Jurassic. Results for GPT-Jurassic 6B are slightly higher than those of OPT-6.7B, although the model size is smaller. However, when the LLMs become extremely large, GPT-Jurassic 178B performs similar to OPT-175B with only 0.12\\% difference. Since similar model sizes show similar performance with minimal differences, the types of LLMs do not have much effect on the generated data quality for our task.\n\n\n\n\n\n\\begin{table}[t]\n    \\centering\n    \\begin{adjustbox}{width=0.95\\linewidth,totalheight={\\textheight},keepaspectratio}\n    \\begin{tabular}{l|cc}\n    \\toprule\n    Models & $Accuracy$ & $W$-$F1$ \\\\ \\midrule\n    {\\scshape VA-BART} (OPT-1.3B) & $65.72_{2.03}$ & $66.50_{2.15}$ \\\\\n    {\\scshape VA-BART} (OPT-6.7B) &  $65.69_{2.28}$ & $66.44_{2.45}$ \\\\\n    {\\scshape VA-BART} (OPT-175B) & $\\mathbf{74.07}_{0.82}$ & $\\mathbf{74.36}_{0.60}$\\\\ \\midrule\n    {\\scshape VA-BART} (GPT-Jurassic 6B) & $69.09_{1.59}$ & $69.89_{1.49}$ \\\\\n    {\\scshape VA-BART} (GPT-Jurassic 17B) & $71.03_{1.01}$ & $71.68_{0.90}$ \\\\\n    {\\scshape VA-BART} (GPT-Jurassic 178B) & $74.04_{1.16}$ & $74.24_{0.91}$ \\\\\\bottomrule\n   \n    \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Effect of size and types of LLMs \n    on value-aligned training data generation. \n    We prompted OPT and GPT-Jurassic ranging 1.3B $\\sim$ 178B. The bigger the model, the better the final performance in the value alignment task. All {\\scshape VA-BART}~variations are fine-tuned with the same number of training samples.}\n    \\label{tab:size_of_llms}\n\\end{table}\n\n\n\\paragraph{Effect of the size of generated training data}\n\\label{sec:number_of_data}\nTo investigate whether increasing the the number of generated data can gain further improvements, we fine-tune {\\scshape VA-model}{s}~with different training data size. \nIn Figure \\ref{fig:data_size}, we show that the W-F1 score does not show any gain when the size exceeds 200 samples except for {\\scshape VA-ALBERT}. As we analysed in Section~\\ref{sec:quality_analysis}, the generated data has noise. We conjecture that when using more generated data, the additional data will not only bring more value alignment knowledge, but also add more noise to the training set. Therefore, when the degradation in model performance caused by the noisy data is greater than the improvement in model performance from the additional knowledge, the overall results decrease.\n\n\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[width=\\linewidth]{images\/data_size_F1.png}\n   \n    \\caption{Evaluation results (W-F1) of {\\scshape VA-model}{s}~over different size of generated training data.}\n    \\label{fig:data_size}\n\\end{figure}\n\n\n\n\n\\section{Conclusion and Future Work}\n\\label{sec:Conclusion_and_Future_Work}\nIn this paper, we propose a task that focuses on teaching a model human value alignment knowledge. We also introduce value-aligned models ({\\scshape VA-model}) that generate value-aligned training data from LLMs by prompt-based data generation and fine-tune smaller classification models with the value-aligned generated training data. Experimental results show that {\\scshape VA-model}~generally outperforms strong baselines.\nFurther analysis illustrates that the generated data from larger LLMs helps increase the performance, and more generated data can cause performance reduction when the data size is too large. In addition, we also test the promising generalization ability of {\\scshape VA-model}. Finally, we highlights several research challenges for future work: improvements in 1) the robustness of the model on diverse values, 2) the models' generalization ability for our value-aligned judgement task, 3) higher quality generated data with more human curation.\n\n\n\n\\section*{Limitations}\n\n\n\nOur methodology is currently tested with only English. We conjecture that the methodology should be applicable to other languages, but may be limited by the capacity of LLMs in those specific languages. It is possible that value-aligned knowledge distillation may be more difficult with languages from countries and regions that do not have a complete set of human value definitions. Thus, exploring the value-aligned task in different languages other than English is a promising research direction. \n\nOur main experimental results are based on a 175B parameter model, which requires large GPU resources or access through an API. This may hinder other researchers from reproducing experimental results. Additionally, we explored different sizes of LLM including 1B and 6B models, which do not require large GPU resources, and showed they can achieve comparable results. We hope they can be possible alternative options for researchers who may not have access to 100B+ models.\n\n\nAlthough sexism is a suitable case study for us to investigate the feasibility of the value alignment task as we have shown throughout this work, it is still one domain. Further expansion to different domains or value-aligned classification tasks such as the detection of racism, toxicity, other than sexism, are needed. \n\n\n\\section*{Ethics Statement}\nIn this work, we conduct experiments with some values that are unconventional in and counter to the current contemporary society. However, we are not suggesting tolerance on sexist behaviors or beliefs. Instead, we explain the existence of different perspectives in the discussion of sexism across cultures or religions. Our value-aligned sexism classification task is a case study of this decoupled process. \n\nWe propose that the human value definition should be decoupled from the value-alignment task. The former should be defined together with society, ethicists, social scientists, and so on. However, the mechanics of value alignment should be independent of the first part so that engineers do not define these values directly in the training data or in the code itself. \n\n\n \n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince unitary quantum theory almost inevitably makes it the case that for most times, nearly all macroscopic objects have large uncertainties in their positions relative to other objects not rigidly attached to them, there has been the mystery of why our conscious perceptions are generally of such objects being perceived as having relatively precise relative locations.  In other words, although our universe certainly seems to be quantum, our conscious observations seem to be almost entirely classical.\n\nDecoherence \\cite{Zeh1970,Zeh1973,Z1981,Z1982,Z1984,JZ1985,Z1989,Z1991,Albrecht1992,PHZ1993,ZHP1993,Z1993,Anderson, PZ1993,Albrecht1993,ZP1994,ZP1995,AZ1996,APZ1997,Z1998a,HSZ1998,Z1998b,Z1998c,PZ1999,Z2003a,Z2003b, Schloss2004,DDZ2005,Schloss2007,P2021} and Quantum Darwinism \\cite{Z2000,OPZ1,OPZ2,BZ1,BZ2,BZ3,Z1,PR2009,ZQZ2009,ZQZ2010,RZ2010,RZ2011,Z2,Z3,TYGDZ,Z4,Ball} are properties of the quantum state and its evolution at least in a region of the universe sufficiently like ours, particularly in having a strong thermodynamic arrow of time and having subsystems that exist moderately stably for time periods long compared with the decoherence timescale that is often microscopically short.  They can explain why a system or object can be redundantly recorded, in a suitable basis called the {\\it pointer basis} \\cite{Z1981,Z1982}, by the environment, especially by the photons that scatter off the object.  Observers who intercept a small fraction of this environment gain access to the redundant information that the environment records and presumably can also record this information redundantly.\n\nHowever, the quantum state and dynamics do not by themselves logically imply what (if any) conscious perceptions occur, or how much.  (I use conscious perception and sentient experience interchangeably, meaning all that one is consciously aware of at once.)  Presumably there are rules for getting the sentient experiences and their measure from the quantum state, but we do not yet know much about what these rules might be.  These measures would be needed to use as substitutes, after being normalized, for the likelihood of a theory for the quantum state and for these rules in complete theories that are deterministic and hence do not really have any probabilities from true randomness \\cite{P2006,P2017}.\n\n\\newpage\n\nPerhaps the simplest framework for the rules giving the measures of the sentient experiences would be that the unnormalized measures are the expectation values, in the quantum state of the universe, of a certain set of positive operators, one for each possible sentient experience, which I shall call the Awareness Operator corresponding to the sentient experience \\cite{P1994a,P1994b,P1995a,P1995b,P1996,P2001,P2006,P2011,P2017,P2020}.  (Of course, the measures might be more complicated functionals of the quantum state, such as non-unit powers of expectation values or other nonlinear functionals, but expectations values, which are linear in the quantum state density matrix, appear to be the simplest possibility.  For the unnormalized measures to be truly linear in the quantum state density matrix, the Awareness Operators themselves should be independent of the quantum state.)\n\nLet us make the plausible assumption that most of the contribution to the expectation values of the Awareness Operators come from conditions inside conscious beings.  For conscious humans, it seems virtually certain that most of the contribution comes from inside their brains (so that, for example, it was just a figure of speech when Einstein said, ``One feels in one's bones the significance of blood ties'').  Therefore, for brevity, I shall refer to ``brains'' as the locations whose conditions give the dominant contributions for the measure of conscious perceptions, without meaning to prejudice the question of what gives the dominant contributions over the entire universe.\n\nFor sighted humans, it seems that most (but certainly not all) of their conscious awareness is usually of visual images, which are mostly mediated by photons scattering off objects, entering the eyes, and exciting retinal neurons that then process and transmit the signals to the brain, where more processing occurs before the resulting state in some region of the brain presumably contributes to the expectation value of the Awareness Operators.  Quantum Darwinism can help explain why stable records of the effect of the photons give information about the angular and spectral distribution of the photons entering the eyes, and hence about the approximate location and colors of objects, rather than about superpositions of greatly different locations.\n\nHowever, if the Awareness Operators are determined {\\it a priori} by the laws of psycho-physical parallelism (which, as a physicist, I shall call laws of physics, even though they are not laws logically  deducible from any of the laws of physics so far discovered, so to distinguish them from the presently known laws of physics, I shall call them the augmented laws of physics), they would not be determined by the conditions outside the brain.  (The Awareness Operators themselves would also not be determined by the conditions inside the brain, but the expectation values they produce would of course depend on the conditions inside the brain that are themselves highly influenced by the signals coming in from the outside, even though I am assuming that the conditions outside the brain do not contribute directly to the expectation values.)  So this raises the question of why our conscious perceptions seem to be of aspects of the external world that are determined by Quantum Darwinism.  It seems mysterious why the state-independent Awareness Operators should be well tuned to the information recorded redundantly by Quantum Darwinism.\n\nHere I shall propose that indeed the set of all Awareness Operators is not itself tuned to the conditions redundantly recorded by Quantum Darwinism, but that instead there are many subsets of Awareness Operators that are each tuned to different forms of multiple copies of information, and it is the Awareness Operators that are tuned to the form of multiple copies that actually exist in the brain that will receive the dominant expectation values and hence make the dominant contribution to consciousness.  There may indeed be some measure of conscious perceptions that do not correspond to classical components of the quantum state, but if that measure is sufficiently suppressed in comparison with the measure corresponding to classical components, that would be sufficient to explain why our observations are predominantly classical.\n\n\\section{Model for the Brain and Awareness Operators}\n\nAs discussed above, assume that most of the contributions to the expectation values of the Awareness Operators (which I am assuming give the unnormalized measures for the corresponding conscious perception, one Awareness Operator for each conscious perception) come from the conditions inside brains, and plausibly from conditions in certain subregions of brains, conditions that have low measures of occurring elsewhere.  For example, one might think of each Awareness Operator as being essentially an integral over spacetime of a localized projection operator onto a certain set of configurations that predominantly occur only inside certain regions of brains when they have the corresponding conscious perception.  (Each Awareness Operator thus does not have any fixed location but only picks up expectation value from regions that do have an element of one of the set of configurations corresponding to the localized projection operator that is integrated over the spacetime to give the full Awareness Operator.) \n\nHere let us model this region of the brain by a certain set of $N \\geq 1$ qubits.  Although these qubits need not be spins or have any particular directional properties, I shall visualize them as if they were spin-half particles (with no positional degrees of freedom) that would each, if it were in a pure state, have its spin point in some direction given by a unit vector ${\\bf n}$ on the Bloch sphere; such a state can be characterized as having the density operator $\\rho_{\\bf n} = |{\\bf n}\\rangle\\langle{\\bf n}|$, the rank-1 projection operator onto this pure state.  Then I can write the probability that such a pure state labeled by its spin direction ${\\bf n}$ would be measured to have spin direction ${\\bf m}$ at an angle $\\theta$ from ${\\bf n}$, with $\\cos{\\theta} = {\\bf m}\\cdot{\\bf n}$, as \n\\begin{equation}\nP({\\bf m},{\\bf n}) = Tr(\\rho_{\\bf m}\\rho_{\\bf n}) = \\cos^2{(\\theta\/2)} = (1\/2)(1 + {\\bf m}\\cdot{\\bf n}).\n\\label{prob}\n\\end{equation}  \n\n(Incidentally, if one visualized the spin-half particle in the pure state $\\rho_{\\bf n} = |{\\bf n}\\rangle\\langle{\\bf n}|$ as being a small opaque ball with the hemisphere surface within $90^\\circ$ of the direction ${\\bf n}$ painted white and the opposite hemisphere painted black, then this probability is the fraction of the total solid angle that is seen as being white from a long way away in the direction ${\\bf m}$.  If anyone knows a previous publication of this fact I could cite, please let me know.)\n\nI shall assume that these qubits occur mainly only inside certain regions of brains.  Let the projection operator ${\\bf P}_k$ (before integrating over the spacetime) corresponding to the Awareness Operator ${\\bf A}_k$ (with $1\\leq k \\leq M$ labeling which of the $M$ total Awareness Operators of this type is being considered) be the tensor product of $N$ projection operators for each of the spins to be ``up'' along the same direction, given by the unit vector ${\\bf m}_k$, but with that direction varying with $k$ that labels the Awareness Operator ${\\bf A}_k$.  If all of the qubits were pointing in the direction ${\\bf n}$, then the expectation value of this projection operator would be $\\langle{\\bf P}_k\\rangle = P({\\bf m}_k,{\\bf n})^N = [(1\/2)(1 + {\\bf m_k}\\cdot{\\bf n})]^N$.  If the state of the $N$ spins were a maximally mixed state, the expectation value of the total projection operator at the location of this set of $N$ qubits would be $\\langle{\\bf P}_k\\rangle = 2^{-N}$, which I am assuming is very small, since I am taking $N \\gg 1$.\n\nNow suppose that Quantum Darwinism leads to the density matrix of the $N$ qubits to be essentially an incoherent sum of the tensor product of the $N$ spin projection operators that has all the spins pointing along a particular direction given by the unit vector ${\\bf n}$, with nonnegative real coefficient $p$, and of the tensor product of the $N$ spin projection operators that has all the spins pointing in the opposite direction, given by the unit vector ${\\bf -n}$, with nonnegative real coefficient $1-p$.  In other words, the mixed state of the $N$ qubits is an incoherent mixture with probability $p$ that all of the spins are pointed in the ${\\bf n}$-direction and probability $1-p$ that all of the spins are pointing in the opposite direction, given by the following density operator,\n\\begin{equation}\n\\rho = p|{\\bf n},{\\bf n},\\ldots\\rangle\\langle{\\bf n},{\\bf n},\\ldots|+(1-p)|{-\\bf n},{-\\bf n},\\ldots\\rangle\\langle{-\\bf n},{-\\bf n},\\ldots|, \n\\label{qubit-state}\n\\end{equation}\nwith $N$ qubits in each ket and bra of this density operator.\n\n\\newpage\n\nHowever, assume that the augmented laws of physics do not restrict the Awareness Operators ${\\bf A}_k$ to have just one single direction for their projection operators, but rather that instead they correspond to a whole set of directions ${\\bf m}_k$.  Then the expectation value of the Awareness Operator ${\\bf A}_k$ would be proportional to the expectation value of its corresponding projection operator \n\\begin{equation}\n{\\bf P}_k = |{\\bf m}_k,{\\bf m}_k,\\ldots\\rangle\\langle{\\bf m}_k,{\\bf m}_k,\\ldots|\n\\label{Pk}\n\\end{equation}\nfor all of the qubits to be pointing in the direction ${\\bf m_k}$, which is\n\\begin{equation} \n\\langle{\\bf P}_k\\rangle \\equiv Tr({\\bf P}_k\\rho) = p[(1\/2)(1 + {\\bf m}_k\\cdot{\\bf n})]^N + (1-p)[(1\/2)(1 - {\\bf m}_k\\cdot{\\bf n})]^N.\n\\label{Pexp}\n\\end{equation}\n\nFor a complete set of laws of physics, one would need not only the quantum state of the universe and the Awareness Operators whose expectation values in that quantum state give the unnormalized measures of the corresponding conscious perceptions, but also the content of each perception.  How the content of the perception is related to the corresponding Awareness Operator is far beyond the scope of my ability to predict, but surely there is some strong correspondence.  Here I shall just assume that if the direction ${\\bf m}_k$ corresponding to the projection operator ${\\bf P}_k$ that is integrated over spacetime to give the Awareness Operator ${\\bf A}_k$ is closely aligned with the direction ${\\bf n}$ the spins would have if they corresponded to a classical pointer basis, then the content of the corresponding conscious perception would be an awareness of a classical perception, without large quantum uncertainties.  In other words, the conscious perception would include an awareness of an image that appears to be of one or more objects at definite directions from the observer.\n\nLet us assume that an Awareness Operator ${\\bf A}_k$ with its projection operator ${\\bf P}_k$ being a tensor product of $N$ spin-half projection operators each having a direction ${\\bf m}_k$ close to that of ${\\bf n}$ would correspond to a classical awareness of the visual image recorded redundantly by the $N$ qubits all pointing in the direction ${\\bf n}$ (with probability $p$, there also being probability $1-p$ that the $N$ qubits are all pointing in the opposite direction, ${-\\bf n}$, which I am assuming would lead to a complementary classical awareness, so that an ${\\bf A}_k$ with ${\\bf m}_k$ close to that of ${-\\bf n}$ would contribute significant measure for the complementary classical awareness).  By ${\\bf m}_k$ being ``close'' to ${\\bf n}$ or to ${-\\bf n}$, I mean that either ${\\bf n}\\cdot{\\bf m}_k$ or ${-\\bf n}\\cdot{\\bf m}_k$ is within $f$ of unity for $f\\ll 1$, i.e., that $1-f \\leq |{\\bf n}\\cdot{\\bf m}_k| \\leq 1$.  If there are many ${\\bf m}_k$'s uniformly distributed over the unit sphere, $f$ is the fraction of them that are ``close'' to ${\\bf n}$ or to ${-\\bf n}$ in this sense.  This implies that if the brain qubits corresponded to directions distributed uniformly over the unit sphere, only a fraction $f$ of the total measure for all the Awareness Operators of this form would correspond to classical awarenesses.\n\n\\section{Fraction of the Measure That Is Classical}\n\nIf the $N$ relevant brain qubits are in the mixed state given by Eq.\\ (\\ref{qubit-state}),\nas plausibly given by Quantum Darwinism, then the situation is quite different from having brain qubits distributed randomly in direction.  If $c = \\cos{\\theta} = {\\bf n}\\cdot{\\bf m}_k$ with $\\theta$ being the angle between ${\\bf n}$ and ${\\bf m}_k$, then the contribution of the fraction $f$ of the Awareness Operator directions that are ``close'' to ${\\bf n}$ or ${-\\bf n}$ \n(have $1-f\\leq |c|\\leq 1$ and hence are assumed to give the measure for a classical awareness) to the total measure given by all the Awareness Operators is, with $\\Theta$ being the Heaviside step function,\n\\begin{eqnarray}\nF&\\equiv&\\frac{\\int_{-1}^1 dc\\, \\Theta(|c|-1+f)\\langle{\\bf P}_k\\rangle}{\\int_{-1}^1 dc \\langle{\\bf P}_k\\rangle}\n=\\frac{\\int_{-1}^1 dc\\, \\Theta(|c|-1+f)\\left[p\\left(\\frac{1+c}{2}\\right)^N+(1-p)\\left(\\frac{1-c}{2}\\right)^N\\right]}\n{\\int_{-1}^1 dc\\, \\left[p\\left(\\frac{1+c}{2}\\right)^N+(1-p)\\left(\\frac{1-c}{2}\\right)^N\\right]}\\nonumber \\\\\n&=& 1-\\left(1-\\frac{1}{2}f\\right)^{N+1}+\\left(\\frac{1}{2}f\\right)^{N+1}=1-\\delta.\n\\label{fraction}\n\\end{eqnarray}\nTherefore, all but a fraction $\\delta = 1-F = (1-f\/2)^{N+1}-(f\/2)^{N+1}$ of the measure for this type of conscious perceptions will be perceived as classical in this toy model.  For a very small fraction $f$ of directions for the Awareness Operator qubit directions ${\\bf m}_k$ that correspond to classical observations, and for a large number $N$ of qubits where the information is stored redundantly, \n\\begin{equation}\n\\delta \\approx (1-f\/2)^{N+1} = e^{(N+1)\\ln{(1-f\/2)}} \\approx e^{-\\frac{N+1}{2}f}.\n\\label{delta}\n\\end{equation}\nThis implies that if $f\\ll 1$, to get at least all but a fraction $\\delta$ of the measure of classical conscious perceptions to be classical, the number of qubits $N$ in the projection operator ${\\bf P}_k$ that is integrated over the spacetime to give the Awareness Operator ${\\bf A}_k$ must be at least $N\\approx (2\/f)\\ln{(1\/\\delta)}-1$.  Alternatively, the fraction of the solid angle $f$ given by the directions ${\\bf m}_k$ that make classical perseptions have a measure that is at least a fraction $F = 1-\\delta$ of the total must be at least $f \\approx [2\/(N+1)]\\ln{(1\/\\delta)}$.  Since $\\ln{(1\/\\delta)}$ is only logarithmically large, it is not too hard to get $N$ large enough that the solid angle fraction $f$ can be small, even for small $\\delta$ that is the fraction of the measure of these conscious perceptions that are not classical.\n\n\\section{Conclusions}\n\nEven if the Awareness Operators, whose expectation values in the quantum state of the universe I have postulated give the unnormalized measures of the corresponding conscious perceptions, are specified by augmented laws of physics independently of the quantum state, they can have a form such that, for a quantum state in which consciousness occurs primarily in a region where decoherence and Quantum Darwinism lead to high redundancy of information, most of the measure is for conscious perceptions that are classical (e.g., perceiving visual images of objects to have fairly well defined directions from the observer), instead of having large quantum uncertainties.  I do need to assume that a fraction not too small of the Awareness Operators get large expectation values from quantum states of the brain in which information redundantly by Quantum Darwinism, though this fraction can be considerably smaller than unity if the information is stored sufficiently redundantly.  I also need to assume that there is a strong correspondence between the content of conscious perceptions and the quantum states that produce the measures of these perceptions as the expectation values of the corresponding Awareness Operators.\n\n\\section{Acknowledgments}\n\nI am grateful especially for email discussions with James Hartle and Wojciech Zurek, which motivated this analysis.  My research was funded by the Natural Sciences and Engineering Research Council of Canada.\n\n\\baselineskip 4pt\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{On Diagram $\\text{\\bfseries\\sffamily{C}}$}\n\nSimilar as in the proof of {\\sc Lemma}\\;\\ref{lem:N}, in every model for\nDiagram\\;$\\text{\\bfseries\\sffamily{C}}$, where $\\mathfrak{m}=\\card A$, \nwe have that the cardinality $\\mathfrak{m}$ is transfinite.\n\n\\begin{lem}\\label{lem:C}\nIf\\\/ $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and\\\/ $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ for some\\\/ $\\mathfrak{m}\\ge\\mathfrak{1}$, \nthen\\\/ $\\aleph_0\\le\\mathfrak{m}$.\n\\end{lem}\n\n\\begin{proof} Assume that $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ \nfor some cardinal $\\mathfrak{m}\\ge\\mathfrak{1}$\nand let $A$ be a necessarily infinite set with $\\card{A}=\\mathfrak{m}$. \nLet $f:A^2\\to [A]^2$ and $g:\\fin(A)\\to\\iseq(A)$ be injections.\nThe goal is to construct with the functions $f$ and $g$ an injection $h:\\omega\\to A$.\nWe first construct a countably infinite set of pairwise disjoint \nnon-empty finite subsets of $A$. For this, we first\nchoose an element $a_0\\in A$, let $E_0:=\\{a_0\\}$, and let $\\nE_0:=\\{E_0\\}$.\n\nAssume that for some $n\\in\\omega$ \nwe have already constructed an $(n+1)$-element set $\\nE_n:=\\blcb E_i:i\\le n\\brcb$\nof pairwise disjoint non-empty finite subsets of $A$. \nLet $$E_{n+1}:=\\bigcup_{i,j\\le n}\\!\\Big{\\{}\\{x,y\\}:\\exists a\\in E_i\\,\\exists b\\in E_j\n\\bigl(f(\\langle a,b\\rangle)=\\{x,y\\}\\bigr)\\Big{\\}}\\setminus\\bigcup_{i\\le n}E_i\\,,$$\nand let $\\nE_{n+1}:=\\nE_n\\cup \\{E_{n+1}\\}$. Notice that for $k:=\\bigcup_{i\\le n}E_i$,\nwe have $$\\bigg{|}\\bigcup_{i\\le n}E_i{\\,}^2\\bigg{|}\\;=\\;k^2\\,>\\,\\binom k2\n\\;=\\;\\bigg{|}\\biggl[\\,\\bigcup_{i\\le n}E_i\\biggr]^2\\bigg{|}\\,,$$ \nwhich implies that $E_{n+1}\\neq\\emptyset$. \nProceeding this way,  $\\blcb E_n:n\\in\\omega\\brcb$ is a countably infinite \nset of pairwise disjoint non-empty finite subsets of $A$.\n\nNow, we apply the function $g$. For every $n\\in\\omega$, let $S_n:=g(E_n)$. \nFurthermore, let $\\nS_0:=S_0$, and in general, for $n\\in\\omega$ let $\\nS_{n+1}:=\n\\nS_n{}^\\frown S_{n+1}$. This way, we obtain an infinite sequence\n$\\nS_\\infty$ of elements of~$A$. Since $g$ is injective and the sets finite sets\n$E_n$ are pairwise disjoint, the sequence $\\nS_\\infty$ must contain infinitely\nmany pairwise distinct elements of~$A$. Now, let $h$ be the enumeration of these\npairwise distinct elements in the order they appear in $\\nS_\\infty$. Then\n$h:\\omega\\to A$ is an injection.\n\\end{proof}\n\nAs a consequence of {\\sc Proposition}\\;\\ref{prp:fin-to-one} and {\\sc Lemma}\\;\\ref{lem:C} we get\n\n\\begin{cor}\nIf\\\/ $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and\\\/ $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ \nfor some\\\/ $\\mathfrak{m}=\\card A\\ge\\mathfrak{1}$, then there \nexists a finite-to-one function\\\/ $g:\\seq(A)\\to\\fin(A)$.\n\\end{cor}\n\n\n\n\\begin{comment}\n\nAs a last result we show that in every model for Diagram\\;{\\text{\\bfseries\\sffamily{Z}}}, where $\\mathfrak{m}=\\card A$, \nthere is a finite-to-one function from $\\seq(A)$ to~$\\fin(A)$.\n\n\\begin{prp}\\label{prp:finite-to-one}\nIf\\\/ $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and\\\/ $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ for some infinite \ncardinal\\\/ $\\mathfrak{m}=\\card A$, then there exists a finite-to-one function\\\/ $g:\\seq(A)\n\\to\\fin(A)$.\n\\end{prp}\n\n\\begin{proof}\nBy {\\sc Lemma}\\;\\ref{lem:C}, there exists an injection $h:\\omega\\to A$\nand for each $i\\in\\omega$, let $x_i:=h(i)$, let $B=\\{x_{i}:i\\in\\omega\\}$, and let\n$C:=\\{x_{2i}:i\\in\\omega\\}$. Notice that \n$$\\iota(a)=\n\\begin{cases}\na &\\text{if $a\\in A\\setminus B$,}\\\\\nx_{2i+1} &\\text{if $a=x_i$,}\n\\end{cases}$$\nis a bijection between $A$ and $A\\setminus C$. \nSince $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$, there exists \nan injection $f:A^2\\to [A]^2$, and without loss of generality we may assume that\n$f:A^2\\to [A\\setminus C]^2$. Now, we define the function $g:\\seq(A)\n\\to\\fin(A)$ as follows: $g(\\langle\\;\\rangle):=\\emptyset$, \n$g(\\langle a\\rangle):=\\{a\\}$, and for every\nsequence $s=\\langle a_1,\\ldots,a_l\\rangle\\in\\seq(A)$ of length at least~$2$, let\n$$g(s):=\\{x_{4l}\\}\\cup\\bigcup \\blcb f(\\langle a_k,x_{2k}\\rangle:1\\le k\\le l\\brcb.$$\nLet now $E=g(s)$ for some sequence $s\\in\\iseq(A)$ of length at least~$2$.\n\nIt is not hard to see that for any finite set $E\\subs A$, there are just finitely \nmany sequences $s\\in\\seq(A)$ such that $g(s)=E$, which shows that \nthe function $g$ is finite-to-one.\n\\end{proof}\n\n\n\n\n\\begin{proof} \nSince $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$, there exists \nan injection $f:A^2\\to [A]^2$. Now, we define the function $g:\\iseq(A)\n\\to\\fin(A)$ as follows: $g(\\langle\\;\\rangle):=\\emptyset$, \n$g(\\langle a\\rangle):=\\{a\\}$, and for every\nsequence $s=\\langle a_1,\\ldots,a_l\\rangle\\in\\iseq(A)$ of length at least~$2$, let\n$$g(s):=\\bigcup \\blcb f(\\langle a_k,a_{k+1}\\rangle):1\\le k<l\\brcb.$$\nLet now $E_0=g(s_0)$ for some sequence $s_0\\in\\iseq(A)$ of length at least~$2$.\nBy considering the pre-images of $[E_0]^2$ under $f$, we see that there are\njust finitely many sequences $s\\in\\iseq(A)$ such that $g(s)=E_0$, which shows that \nthe function $g$ is finite-to-one.\n\\end{proof}\n\n\\textcolor{red}{In order to obtain a model for Diagram\\;$\\text{\\bfseries\\sffamily{C}}$, we tried to\nmodify the model for Diagram\\;$\\text{\\bfseries\\sffamily{Z}}$ by stipulating, for example, that $g:\\fin(A)\\to\n\\iseq(A)$ is a function which maps an $n$-element set $E\\subs A$ to an injective \nsequence of length $n+1$. However, if we do so, then the closure $N$ of some $M$ \nby $1$-step fulfillments would give us an injection $h:\\iseq(N)\\to\\fin(N)$:\n}\n\n\\textcolor{red}{By definition of $1$-step fulfillment of some $M$, \nthe $2$-element sets $f_M(a,b)$ we get from\n$\\langle a,b\\rangle\\in M\\times M$ as well as the\nranges of the sequences $g_M(E)$ we get from finite \nsets $E\\subs M$ are pairwise disjoint. Let $E\\in\\fin(M)$ and let \n$g_M(E)=s$ where $s=\\langle a_1,a_2,\\ldots,a_n\\rangle$. Then the $2$-element sets\n$$f_M(a_1,a_2),\\;f_M(a_2,a_3),\\ldots,f_M(a_{n-1},a_n)$$ are pairwise \ndisjoint and $h(s):=\\bigcup\\blcb f_M(a_i,a_{i+1}):0\\le i<n\\brcb$ is injective.}\n\n\nThe atoms of the permutation model for $\\mathfrak{m}^2<[\\mathfrak{m}]^2<\\fin(\\mathfrak{m})<\\seq(\\mathfrak{m})$\nwill be again the elements of a {Fra\\\"{\\i}ss\\'e-limit} of finite models. \nThe construction is similar as above. \n\nLet $\\nL$ be the signature containing two symbols $f_1$ and $f_2$ for\npartial functions, and let $\\Th$ be the $\\nL$-theory stating that \n\\begin{itemize}\n\\item $f_1$ is a partial, injective function from the set of finite sequences \nof length at least~$3$ into the set of finite sets of cardinality at least~$2$, \nwhere a sequence of length~$n+1$ is mapped to an $n$-element set,\n\\item $f_2$ is a partial, injective function from the set of $2$-element sets into \nthe set of sequences of length~$2$.\n\\end{itemize}\n\nLet $\\modM$ with domain $A$ be the \nFra\\\"{\\i}ss\\'e-limit of the finite models of $\\Th$. \nBy the same arguments as above,\nin the model $\\modM$, there exists a partial injective function \n$F_1$ from a proper subset of $\\seq_{\\ge 3}(A)$ ({\\ie}, the finite \nsequences with elements of $A$ of length at least~$3$)\ninto $\\fin_{\\ge 2}(A)$ ({\\ie}, the finite \nsubsets of $A$ with at least~$2$ elements), and a \npartial injective function $F_2$ from a proper subset of $[A]^2$\ninto $A^2$. Moreover, by construction of $\\modM$, the partial\nfunctions $F_1$ and $F_2$ are surjections.\nHence, $F_1$ is a bijection between a proper subset\nof $\\seq_{\\ge 3}(A)$ and $\\fin_{\\ge 2}(A)$. Similarly,\n$F_2$ is a bijection between a proper subset\nof $[A]^2$ and $A^2$.\n\nWe construct the permutation model $\\modcV$ similarly\nas above. So, the set $A$ ({\\ie}, the domain of $\\modM$),\nis the set of atoms of $\\modcV$ and $G$ is the group of all permutations \n$\\pi$ of $A$ such that $\\pi$ induces an automorphism of $\\modM$.\nFurthermore, $\\nF$ is the filter on \n$G$ generated by $\\{\\operatorname{Fix}(E)_{G}:E\\in\\fin(A)\\}$ and let \n$\\modcV$ be the class of all hereditarily symmetric sets.\n\nNow we are ready to prove the following result.\n\n\\begin{prp}\nLet\\\/ $\\mathfrak{m}$ denote the cardinality of the set of atoms $A$ of\\\/\n$\\modcV$. Then\\\/ $$\\modcV\\models\\mathfrak{m}^2<[\\mathfrak{m}]^2\\;\\wedge\\;\\fin(\\mathfrak{m})<\\seq(\\mathfrak{m})\\,.$$\n\\end{prp}\n\n\\begin{proof} To see that $\\modcV\\models \\fin(\\mathfrak{m})\\le\\seq(\\mathfrak{m})$, it is enough to \nconstruct an injection $\\iota_1$ from $\\fin(A)$ into $\\seq(\\mathfrak{m})$ which belongs\nto~$\\modcV$. For this, \nlet $$\\iota_1(E):=\\begin{cases}\nF_1^{-1}(E)&\\text{if $|E|\\ge 2$,}\\\\[1ex]\n\\langle a\\rangle&\\text{if $E=\\{a\\}$,}\\\\[1ex]\n\\langle\\ \\rangle&\\text{if $E=\\emptyset$.}\n\\end{cases}$$\nNotice that since $\\iota_1$ has empty support, $\\iota_1$ belongs to~$\\modcV$.\n\nTo see that $\\modcV\\models\\mathfrak{m}^2\\le [\\mathfrak{m}]^2$, it is enough to \nconstruct an injection $\\iota_2$ from $A^2$ into $[A]^2$ which belongs\nto~$\\modcV$. For this, \nlet $$\\iota_2\\of ab:=F_2^{-1}\\of ab\\,.$$\nNotice that since $\\iota_2$ has empty support, $\\iota_2$ belongs to~$\\modcV$.\n\nNow, we show that $\\modcV\\models \\fin(\\mathfrak{m})<\\seq(\\mathfrak{m})$. For this, assume towards \na contradiction that in $\\modcV$, there exists bijection $h:\\seq(A)\\to\\fin(A)$.\nLet $E_h\\in\\fin(A)$ be a non-empty support of $h$ and let $\\NN_0\\models\\Th$ such\nthat $E_h\\subs N_0$. Let $k:=|N_0|$, let $a\\in N_0$ and consider the all the \nsub-sequences of the sequence\n$$\\sigma:=\\underset{\\text{\\tiny{$2^{k+1}-times$}}}\n{\\langle\\,\\underbrace{\\mathstrut a,a,\\ldots,a,a}\\,\\rangle}\\,.$$ \nSince there are $2^{k+1}+1$ sub-sequences of $\\sigma$, but there are only $2^k$ \nsubsets of $N_0$, there exists a sub-sequence $\\sigma_0$ of $\\sigma$, such that \n$h(\\sigma_0)\\setminus N_0\\neq\\emptyset$. Let $x_0\\in h(\\sigma_0)\\setminus N_0$. \nLet $\\NN'$ and $\\NN''$ be finite models of $\\Th$ which are isomorphic over $\\NN_0$,\nsuch that $N_0=N'\\cap N''$ and $x_0\\in N'$. Then there exists \na permutation $\\pi$ of $A$ which fixes $N_0$ pointwise,\nsuch that $\\pi x_0\\in N''$, which implies that $\\pi\\sigma_0=\\sigma_0$ and \n$\\pi h(\\sigma_0)\\neq h(\\sigma_0)$. Thus, $E_h$ is not a support of\n$h$, which contradicts our assumption.\n\nFinally, we show that $\\modcV\\models \\mathfrak{m}^2<[\\mathfrak{m}]^2$. For this, assume towards \na contradiction that in $\\modcV$, there exists bijection $h:A^2\\to [A]^2$.\nLet $E_h\\in\\fin(A)$ be a non-empty support of $h$ and let $\\NN_0\\models\\Th$ \nbe such that $E_h\\subs N_0$. Let $k:=|N_0|$ and consider the set $N_0\\times N_0$. \nSince $|N_0\\times N_0|=k^2$, but $|[N_0]^2|=\\binom k2<k^2$, \nthere exists a pair $\\langle x,y\\rangle\\in N_0\\times N_0$ such that \n$h\\of{x}{y}\\setminus N_0\\neq\\emptyset$. Let $x_0\\in h\\of{x}{y}\\setminus N_0$. \nLet $\\NN'$ and $\\NN''$ be finite models of $\\Th$ which are isomorphic over $\\NN_0$,\nsuch that $N_0=N'\\cap N''$ and $x_0\\in N'$. Then there exists \na permutation $\\pi$ of $A$ which fixes $N_0$ pointwise\nsuch that $\\pi x_0\\in N''$, which implies that $\\pi\\of xy=\\pair xy$ and \n$\\pi h\\of xy\\neq h\\of xy$. Thus, $E_h$ is not a support of\n$h$, which contradicts our assumption.\n\\end{proof}\n\nIt remains to show that $[\\mathfrak{m}]^2<\\fin(\\mathfrak{m})$.\n\n\\begin{lem}\nLet\\\/ $\\mathfrak{m}$ denote the cardinality of the set of atoms $A$ of\\\/\n$\\modcV$. Then\\\/ $$\\modcV\\models[\\mathfrak{m}]^2<\\fin(\\mathfrak{m})\\,.$$\n\\end{lem}\n\n\\begin{proof}\nSince we have $[\\mathfrak{m}]^2\\le\\fin(\\mathfrak{m})$, it is enough to show that\n$\\modcV\\models [\\mathfrak{m}]^2\\neq\\fin(\\mathfrak{m})$. Assume towards \na contradiction that in $\\modcV$, there exists bijection $h:\\fin(A)\\to [A]^2$.\nLet $E_h\\in\\fin(A)$ be a non-empty support of $h$ and let $\\NN_0\\models\\Th$ \nbe such that $E_h\\subs N_0$. Let $k:=|N_0|$ and consider the set $\\pwr(N_0)$. \nOn the one hand, we have $|\\pwr(N_0)|=2^k$, and on the other hand, \nwe have $|[N_0]^2|=\\binom k2<2^k$. Thus, \nthere exists a set $u_0\\subs N_0$ such that \n$h(u_0)\\setminus N_0\\neq\\emptyset$. Let $x_0\\in h(u_0)\\setminus N_0$. \nLet $\\NN'$ and $\\NN''$ be finite models of $\\Th$ which are isomorphic over $\\NN_0$,\nsuch that $N_0=N'\\cap N''$ and $x_0\\in N'$. Then there exists \na permutation $\\pi$ of $A$ which fixes $N_0$ pointwise,\nsuch that $\\pi x_0\\in N''$. In particular, we have $\\pi u_0=u_0$ and \n$\\pi h(u_0)\\neq h(u_0)$. Thus, $E_h$ is not a support of\n$h$, which contradicts our assumption.\n\\end{proof}\n\nSo, the model $\\modcV$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ with\n$$\\mathfrak{m}^2<[\\mathfrak{m}]^2<\\fin(\\mathfrak{m})<\\seq(\\mathfrak{m})$$\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[rr]& &\\seq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\\]\nis consistent with\\\/ $ZF$.\n\\end{conc}\n\\end{comment}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{\\reflectbox{C}}}} \n\nWe show that Diagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}} holds in a permutation model \n$\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ which is similar to the {\\it Second Fraenkel Model}, \nwhere $\\mathfrak{m}$ is the cardinality of the set of atoms of $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$. \n\nThe permutation model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ is constructed as follows\n(see also \\cite[p.\\,197]{cst}):\nThe set of atoms of the model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ consists of\ncountably many mutually disjoint, cyclically ordered $3$-element sets. \nMore formally,\n\\[\nA=\\bigcup_{n\\in\\omega} P_n,\n\\quad\n\\mbox{where }P_n=\\{a_n,b_n,c_n\\}\\ (\\mbox{for }n\\in\\omega),\n\\]\nand the cyclic ordering on $P_n$ is illustrated by the following figure:\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=.9]{circle.eps}\n\\end{figure}\n\nOn each triple $P_n$, we define the cyclic distance between two elements\nby stipulating $$\\cy{a_n}{b_n}=\\cy{b_n}{c_n}=\\cy{c_n}{a_n}=1$$ and \n$$\\cy{a_n}{c_n}=\\cy{b_n}{a_n}=\\cy{c_n}{b_n}=2\\,.$$\n\nLet $G$ be the group of those permutations of $A$ which preserve the\ntriples $P_n$ (\\ie, $\\pi P_n=P_n$ for $\\pi\\inG$ and $n\\in\\omega$)\nand their cyclic ordering.\nThe sets in $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ are those with finite support.\n\n\\begin{prp} Let\\\/ $A$ be the set of\natoms of\\\/ $\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}}$ and let\\\/ $\\mathfrak{m}:=|A|$. Then\n$$\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}}\\models [\\mathfrak{m}]^2<\\mathfrak{m}^2<\\iseq(\\mathfrak{m})<\\fin(\\mathfrak{m})\\,.$$ \n\\end{prp}\n\n\\begin{proof} We first show that $[\\mathfrak{m}]^2\\le\\mathfrak{m}^2$, $\\mathfrak{m}^2\\le\\iseq(\\mathfrak{m})$,\nand $\\iseq(\\mathfrak{m})\\le\\fin(\\mathfrak{m})$, and then we show that\n$[\\mathfrak{m}]^2\\neq\\mathfrak{m}^2$, $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$, and $\\iseq(\\mathfrak{m})\\neq\\fin(\\mathfrak{m})$.\n\\medskip\n\n\\noindent $[\\mathfrak{m}]^2\\le\\mathfrak{m}^2$: We define an injective function $f_1:[A]^2\\to A^2$.\nLet $\\{x,y\\}\\in [A]^2$ and $m,n\\in\\omega$ be \nsuch that $x\\in P_m$ and $y\\in P_n$. Without loss of generality we may assume\nthat $m\\le n$. If $m<n$, then $f_1(\\{x,y\\}):=\\langle x,y\\rangle$, and if $m=n$, then\n$f_1(\\{x,y\\}):=\\langle z,z\\rangle$ where $z:=P_m\\setminus\\{x,y\\}$. It is easy to see \nthat $f_1$ is an injective function, and since $f_1$ has empty support, $f_1$\nbelongs to~$\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$.\\smallskip\n\n\\noindent $\\mathfrak{m}^2\\le\\iseq(\\mathfrak{m})$: The function $f_2:A^2\\to \\iseq(A)$ defined\nby stipulating $$f_2\\left(\\langle x,y\\rangle\\right):=\n\\begin{cases}\n \\ \\langle x\\rangle &\\text{if $x=y$,}\\\\[1.2ex]\n \\langle x,y\\rangle &\\text{otherwise,}\n\\end{cases}$$\nis an injective function from $A^2$ into $\\iseq(A)$ which\nbelongs to~$\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$.\\smallskip\n\n\\noindent $\\iseq(\\mathfrak{m})\\le\\fin(\\mathfrak{m})$: We define a injective function $f_3:\\iseq(A)\\to\\fin(A)$.\nFirst, let $f_3(\\langle\\;\\rangle):=\\emptyset$. Now, let $s=\\langle a_0,\\ldots,a_{k-1}\\rangle\\in\\iseq(A)$ be \na non-empty sequence without repetition of length~$k$. Let $j:k\\to\\omega$ be such that \nfor each $i\\in k$, $a_i\\in P_{j(i)}$. Let $E_0:=\\emptyset$, and by induction, for $i\\in k$ define\n$$E_{i+1}:=\n\\begin{cases} \nE_i\\cup\\{a_{i}\\} &\\text{if $P_{j(i)}\\cap E_i=\\emptyset$,}\\\\[1.2ex]\nE_i &\\text{otherwise,}\n\\end{cases}\n$$\nand\n$$\\varepsilon_{i}:=\n\\begin{cases} \n2 &\\text{if $|P_{j(i)}\\cap E_i|=1$ and $\\cy{P_{j(i)}\\cap E_i}{a_i}=2$,}\\\\[1.2ex]\n1 &\\text{otherwise.}\n\\end{cases}\n$$\nFurthermore, let $\\sigma(0):=j(0)$, and by induction, for $i\\in k_1$ define\n$\\sigma(i+1):=\\sigma(i)+j(i+1)+1$. Finally, let $\\{p_i:i\\in\\omega\\}$ be\nan enumeration of the prime numbers and let $$q_s:=\\prod_{i\\in k}p_{\\sigma(i)}^{\\varepsilon_i}\\,.$$\nNow, we define $f_3(s):=E_{k}\\cup P_{q_s}$. \nIt is easy to see that $f_3$ is an injective function from \n$\\iseq(A)$ into $\\fin(A)$, and since $f_3$ has empty support, $f_3$\nbelongs to~$\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$.\\medskip\n\n\\noindent $[\\mathfrak{m}]^2\\neq\\mathfrak{m}^2$: It is enough to show that there is no\ninjection from $A^2$ into $[A]^2$. Assume towards a contradiction\nthat there exists an injection $g_1:A^2\\to [A]^2$ with finite support $E_1$.\n\nLet $E_1\\subs E$ for some non-empty set $E\\in\\fin(A)$. Then $E$ is also a \nsupport of~$g_1$. Now $|[E]^2|<|E^2|$, which implies that there exists\na pair $\\langle x,y\\rangle\\in E^2$ such that $g_1(\\langle x,y\\rangle)\\notin [E]^2$.\nSo, there exists a $\\pi\\in\\operatorname{Fix}_{G}(E)$ such that $\\pi g_1(\\langle x,y\\rangle)\\neq\ng_1(\\langle x,y\\rangle)$, but $\\pi\\langle x,y\\rangle=\\langle x,y\\rangle$, which contradicts the fact that\n$E$ is a support of $g_1$.\\smallskip\n\n\\noindent $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$: It is enough to show that there is no\ninjection from $\\iseq(A)$ into $A^2$. Assume towards a contradiction\nthat there exists an injection $g_2:\\iseq(A)\\to A^2$ with finite support $E_2$.\nLet $E_2\\subs E$ for some non-empty set $E\\in\\fin(A)$. Then $E$ is also a \nsupport of~$g_2$. Now $|E^2|<|\\iseq(E)|$, and by similar arguments as above,\nwe obtain a contradiction.\n\n\\noindent $\\iseq(\\mathfrak{m})\\neq\\fin(\\mathfrak{m})$: It is enough to show that there is no\ninjection from $\\fin(A)$ into $\\iseq(A)$. Assume towards a contradiction\nthat there is an injection \\hbox{$g_3:\\fin(A)\\to\\iseq(A)$} with finite support $E_3$.\n\nSince $E_3$ is finite, there exists an $n\\in\\omega$ such that $g_3(P_n)\\notin\\iseq(E_3)$.\nLet $a\\in A$ be the first element of the sequence $g_3(P_n)$ which does not belong to~$E_3$. \nThen we find a $\\pi\\in\\operatorname{Fix}_{G}(E_3)$ such that $\\pi a\\neq a$ ({\\ie}, $\\pi g_3(P_n)\\neq\ng_3(P_n)$) but $\\pi P_n=P_n$, which contradicts the fact that $E_3$ is a support of~$g_3$.\n\\end{proof}\n\nSo, the model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\\ar@{->}[u]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{N}}\n\nWe first show that in every model for Diagram\\;$\\text{\\bfseries\\sffamily{N}}$, we have that \nthe cardinality $\\mathfrak{m}$ is transfinite.\n\n\\begin{lem}\\label{lem:N}\nIf\\\/ $\\fin(\\mathfrak{m})\\le\\mathfrak{m}^2$ for some\\\/ $\\mathfrak{m}\\ge\\mathfrak{5}$, then\\\/ $\\aleph_0\\le\\mathfrak{m}$.\n\\end{lem}\n\n\\begin{proof} Let $A$ be a set of cardinality $\\mathfrak{m}\\ge\\mathfrak{5}$ and assume that\n$h:\\fin(A)\\to A^2$ is an injection. First we choose a\n$5$-sequence $S_5:=\\langle a_1,\\ldots,a_5\\rangle$ of pairwise distinct elements of~$A$. \nThe ordering of $S_5$ induces an ordering on $P_5:=\\fin(\\{a_1,\\ldots,a_5\\})$, and since\n$|h[P_5]|=2^5$ and $2^5>5^2$, there exists a first set $u\\in P_5$ such that for \n$\\pair xy=h(u)$, the set $D_6:=\\{x,y\\}\\setminus \\{a_1,\\ldots,a_5\\}$ is non-empty. \nIf $x\\in D_6$, let $a_6:=x$, otherwise, let $a_6:=y$. Now, let $S_6:=\\langle a_1,\\ldots,a_6\\rangle$\nand $P_6:=\\fin(\\{a_1,\\ldots,a_6\\})$. As above, we find a $u\\in P_6$ such that for \n$\\pair xy=h(u)$, the set $D_7:=\\{x,y\\}\\setminus \\{a_1,\\ldots,a_7\\}$ is non-empty.\nIf $x\\in D_7$, let $a_7:=x$, otherwise, let $a_7:=y$. Proceeding this way, we \nfinally have an injection from $\\omega$ into $A$, which shows that $\\aleph_0\\le\\mathfrak{m}$.\n\\end{proof}\n\n\\begin{prp}\\label{prp:fin-to-one}\nIf\\\/ $\\aleph_0\\le\\mathfrak{m}$ for some infinite \ncardinal\\\/ $\\mathfrak{m}=\\card A$, then there exists a finite-to-one function\\\/ $g:\\seq(A)\n\\to\\fin(A)$.\n\\end{prp}\n\n\\begin{proof}\nBy {\\sc Lemma}\\;\\ref{lem:N}, there exists an injection $h:\\omega\\to A$,\nand for each $i\\in\\omega$, let $x_i:=h(i)$, let $B=\\{x_{i}:i\\in\\omega\\}$, and let\n$C:=\\{x_{2i}:i\\in\\omega\\}$. Notice that \n$$\\iota(a)=\n\\begin{cases}\na &\\text{if $a\\in A\\setminus B$,}\\\\\nx_{2i+1} &\\text{if $a=x_i$,}\n\\end{cases}$$\nis a bijection between $A$ and $A\\setminus C$. Thus,\nit is enough to construct a finite-to-one function $g:\\seq(A\\setminus C)\n\\to\\fin(A)$. Let $s=\\langle a_0,\\ldots,a_{n-1}\\rangle\\in \\seq(A\\setminus C)$ and let\n$\\ran(s):=\\{a_0,\\ldots,a_{n-1}\\}$. The sequence $s$ gives us in a natural way\nan enumeration of $\\ran(s)$, and with respect to this enumeration we can encode the \nsequence $s$ by a natural number $i_s\\in\\omega$. Now, let\n$g(s):=\\ran(s)\\cup\\{x_{2i_s}\\}$. Then, since there are just finitely many \nenumerations of $\\ran(s)$, $g$ is a finite-to-one function.\n\\end{proof}\n\nThe following result is just a consequence of {\\sc Proposition}\\;\\ref{prp:fin-to-one} \nand {\\sc Lemma}\\;\\ref{lem:N}.\n\n\\begin{cor}\nIf\\\/ $\\fin(\\mathfrak{m})\\le\\mathfrak{m}^2$ for some\\\/ $\\mathfrak{m}=\\card A\\ge\\mathfrak{5}$, then there \nexists a finite-to-one function\\\/ $g:\\seq(A)\\to\\fin(A)$.\n\\end{cor}\n\nWe now introduce the technique we intend to use in order to build a permutation \nmodel from which it will follow that for some infinite cardinal $\\mathfrak{m}$,\nthe relation $\\fin(\\mathfrak{m})<\\mathfrak{m}^2$ is consistent with $\\ZF$. Notice that this \nrelation is the main feature of Diagram~$\\text{\\bfseries\\sffamily{N}}$ and that this relation implies\nthat $\\aleph_0\\le\\mathfrak{m}$.\nIn the next section, we shall use a similar permutation model in order to show \nthe consistency of Diagram~$\\text{\\bfseries\\sffamily{Z}}$ with $\\ZF$. \n\nLet $K$ be the class of all the pairs $(A,h)$ such that $A$ is a (possibly empty) set\nand $h$ is an injection $h\\colon\\fin(A)\\to A^2$. We will also refer to the elements \nof $K$ as models. We define a partial ordering $\\leq$ on $K$ by stipulating\n$$(A,h)\\leq (B,f)\\iff\nA\\subs B\\;\\wedge\\;h\\subs f\\;\\wedge\\;\n\\ran\\left(f|_{\\fin(B)\\setminus\\fin(A)}\\right)\\subs(B\\setminus A)^2\\,.$$ \nWhen the functions involved are clear from the context, with a slight abuse \nof notation we will just write $A\\leq B$ instead of $(A,h)\\leq(B,f)$ \nand $A\\in K$ instead of $(A,h)\\in K$.\nNotice that the last condition in the definition of $(A,h)\\leq (B,f)$\nimplies that $(B\\setminus A)\\in K$.\n\n\\begin{prp}[CH]\\label{prp:limit}\nThere is a model\\\/ $M_*$ of cardinality\\\/ $\\mathfrak{c}$ in $K$ such that:\n\\begin{itemize}\n    \\item $M_*$ is\\\/ $\\aleph_1$-universal, i.e., if\\\/ $N\\in K$ is countable then\\\/ $N$ \n    is isomorphic to some\\\/ $N_*\\leq M_*$.\n    \\item $M_*$ is\\\/ $\\aleph_1$-homogeneous, i.e., if\\\/ $N_1,N_2\\leq M_*$ are \n    countable and\\\/ $\\pi\\colon N_1\\to N_2$ is an isomorphism then there exists \n    an automorphism\\\/ $\\pi_*$ of $M_*$ such that\\\/ $\\pi\\subs\\pi_*$.\n    \\item If\\\/ $N\\leq M_*$ and\\\/ $A\\subs M_*$ are countable, then there \n    is an automorphism\\\/ $\\pi$ of\\\/ $M_*$ that fixes\\\/ $N$ pointwise, \n    such that\\\/ $\\pi(A)\\setminus N$ is disjoint from $A$. \n\\end{itemize}\n\\end{prp}\n\n\\begin{proof}\nWe construct the model $M_*$ by induction on~$\\omega_1$, where we assume that $\\omega_1=\\mathfrak{c}$.\nLet $M_0=\\emptyset$. When $M_\\alpha$ is already defined for some $\\alpha\\in\\omega_1$, we can define \n$$C_\\alpha:=\\blcb N\\leq M_\\alpha:N\\in K\\;\\textrm{and $N$ is countable}\\brcb.$$\n\nThe construction of $M_{\\alpha+1}$, starting from $M_\\alpha$, consists of a disjoint \nunion of two differently built sets of models. First, for each element $N\\in C_\\alpha$, \nlet $S_N$ be a system of representatives for the $\\textit{strong}$ isomorphism classes \nof all the models $M\\in K$ such that $N\\leq M$, $M$ is countable,\nand for all $M_1,M_2\\in S_N$ we have $M_1\\cap M_2=N$. \nHere, by $\\textit{strong}$ we mean that, \nfor two models $M_1$ and $M_2$ with $N\\le M_1,M_2$,\nit is not enough to be isomorphic in order to belong to \nthe same class, but we require that there exists an isomorphism between $M_1$ and $M_2$ \nthat fixes $N$ pointwise, which we can express by saying that \\textit{$M_1$ is isomorphic to $M_2$ \nover $N$}. We first extend $M_\\alpha$ by the set\n$$M_\\alpha'=\\bigsqcup_{N\\in C_\\alpha}\\bigsqcup_{M\\in S_N}M\\setminus N,$$\nwhere {``}\\,$\\bigsqcup$\\,{''} indicates that we have a \\textit{disjoint union}.\nNow, we extend $M_\\alpha$ by a second set $M_\\alpha''$, where $M_\\alpha''$ is defined as follows: \nfor each pair $\\{M_1,M_2\\}\\in[C_\\alpha]^2$, we fix a way of constructing a common extension $M_1*M_2$ \nwith $M_1,M_2\\leq M_1*M_2$. Let $f^{M_1}$ and $f^{M_2}$ be the functions coming with \nthe models $M_1,M_2$, let $N_0=M_1\\cup M_2$, let $f_0=f^{M_1}\\cup f^{M_2}$, and, \nfor each finite set $E$ in \\hbox{$\\nE_0:=\\textrm{fin}(N_0)\\setminus\\textrm{dom}(f_0)$,} choose\na pair $p_E=\\langle a_1,a_2\\rangle$ such that $\\{a_1,a_2\\}\\cap N_0=\\emptyset$ and for \nall $E,E'\\in\\nE_0$, either $E=E'$ or $\\ran(p_E)\\cap\\ran(p_{E'})=\\emptyset$,\nwhere for $p_E=\\langle a_1,a_2\\rangle$, \\hbox{$\\ran(p_E):=\\{a_1,a_2\\}$.}\nNow, by induction on~$\\omega$, define \n$$N_{i+1}=N_i\\sqcup\\bigsqcup_{E\\in\\nE_i}\\ran(p_E),$$\ntogether with \n$$f_{i+1}=f_i\\cup\\bigcup_{E\\in\\nE_i}\\langle E,p_E\\rangle\\qquad\\text{and}\\qquad\n\\nE_{i+1}=\\textrm{fin}(N'_{i+1})\\setminus\\textrm{dom}(f_{i+1}),$$ and set \n$M_1*M_2:=\\bigcup_{i\\in\\omega}N_i$ with $f^{M_1*M_2}:=\\bigcup_{i\\in\\omega}f_i$. Now, let\n$$M_\\alpha''\\;:=\\bigsqcup_{\\{M_1,M_2\\}\\in[C_\\alpha]^2}M_1*M_2\\,\\setminus(M_1\\cup M_2).$$ \n\nFinally, let $M_{\\alpha+1}=M_\\alpha\\sqcup M_\\alpha'\\sqcup M_\\alpha''$,\nfor non-empty limit ordinals $\\delta$ define \n$M_\\delta=\\cup_{\\alpha\\in\\delta}M_\\alpha$, and let\n$$M_*=\\bigcup_{\\alpha\\in\\omega_1}M_\\alpha.$$ \n\nIt remains to show that the model $M_*$ has the required properties:\nFirst we notice that $M_*$ has cardinality $|M_*|=\\mathfrak{c}$, as required,\nand since, by construction, $M_1$ is $\\aleph_1$-universal,\n$M_*$ is also $\\aleph_1$-universal. In order to show that $M_*$ \nis $\\aleph_1$-homogeneous, let $N_1,N_2\\leq M_*$ be countable models and \n$\\pi\\colon N_1\\to N_2$ an isomorphism. Let $\\{x_\\alpha:\\alpha\\in\\omega_1\\}$ be \nan enumeration of the elements of $M_*$ and let $I_0:=N_1$. If $x_{\\delta_1}$ \nis the first element (w.r.t. this enumeration) in $M_*\\setminus I_0$, \nthen, by construction, there exists a model \n$I_1\\leq M_*$ such that $I_0\\leq I_1$ and $x_{\\delta_1}\\in I_1$. Again by \nconstruction, there is a model $J_1$ with $N_2\\leq J_1\\leq M_*$ such that \nthere exists an isomorphism $\\pi_1\\colon I_1\\to J_1$ with $\\pi\\subseteq\\pi_1$. \nIn fact, we have that $J_1$ and $I_1$ are isomorphic over $N_2$.\nProceed inductively with $x_{\\delta_{\\alpha+1}}$ being the first element in \n$M_*\\setminus I_\\alpha$, we find models $I_\\alpha\\leq I_{\\alpha+1}\\leq M_*$, \n$J_\\alpha\\leq J_{\\alpha+1}\\leq M_*$ and isomorphisms \n$\\pi_{\\alpha+1}\\colon I_{\\alpha+1}\\to J_{\\alpha+1}$, and finally we obtain\n$\\pi_*=\\cup_{\\alpha\\in\\omega_1}\\pi_\\alpha$, which is the required automorphism.\n\nTo show the last property of the theorem, let $N\\leq M_*$ and $A\\subs M_*$ \nbe both countable. Since the cofinality of $\\omega_1$ is greater than $\\omega$, \nwe can find by construction both a countable model $M$ satisfying the properties \n$A\\subseteq M$, $N\\leq M\\leq M_*$ and a further model $M'$ with $N\\leq M'\\leq M_*$ \nsuch that $M'\\cap(A\\setminus N)=\\emptyset$, and such that there exists an \nisomorphism $i\\colon M\\to M'$ with $i$ fixing $N$ pointwise. Now, by $M_*$ \nbeing $\\aleph_1$-homogeneous we obtain an automorphism $i_*$ extending $i$, \nas required.\n\\end{proof}\n\nAs anticipated, the construction of the previous theorem does not exploit any \nparticular property of the functions $h\\colon\\fin(A)\\to A^2$. In fact, \nthe construction is an analogue of a Fra\\\"{\\i}ss\\'e limit as it relies \non similar properties, like, for example, a modified version of \nthe {\\sl Disjoint Amalgamation Property\\\/} (DAP) of~$K$, where we require that embeddings \nbetween structures $f\\colon(A,h)\\to(B,g)$ are allowed only when, according to our \nprevious definition, $A\\leq B$. Indeed, exactly the same construction can be carried \nout in the alternative framework of models $(A,f,g,h)$, where $A$ is a set and we have \nthree injections $f\\colon A^2\\to[A]^2$, $g\\colon[A]^2\\to\\iseq(A)$ and \n$h\\colon\\iseq(A)\\to\\fin(A)$, which will be used below \nto show the consistency of Diagram~$\\text{\\bfseries\\sffamily{Z}}$ with~$\\ZF$.\n\nGiven {\\sc Proposition}\\;\\ref{prp:limit}, we consider the permutation \nmodel $\\modcV_\\text{\\bfseries\\sffamily{N}}$ that arises \nnaturally by considering the elements of the $\\aleph_1$-universal \nand $\\aleph_1$-homogeneous model $M_*$ \nas the set of atoms and its automorphisms $\\operatorname{Aut}(M_*)$ as the \ngroup $G$ of permutations. In particular,\neach permutation of $M_*$ preserves the injection \n$h\\colon\\fin(M_*)\\to M_*^2$ \nthat the model $(M_*,h)$ comes with.\n\nWe are now ready to prove the following result.\n\n\\begin{thm}\nLet\\\/ $M_*$ be the set of\natoms of\\\/ $\\modcV_\\text{\\bfseries\\sffamily{N}}$ and let\\\/ $\\mathfrak{m}=|M_*|$. Then\n$$\\modcV_\\text{\\bfseries\\sffamily{N}}\\models [\\mathfrak{m}]^2<\\fin(\\mathfrak{m})<\\mathfrak{m}^2<\\iseq(\\mathfrak{m})\\,.$$ \n\\end{thm}\n\n\\begin{proof}\nThe existence of an injection $h\\colon\\fin(M_*)\\to M_*^2$ in $\\modcV_\\text{\\bfseries\\sffamily{N}}$ follows \ndirectly from the definition of the specific permutation model. So, we only need to \nprove that in $\\modcV_\\text{\\bfseries\\sffamily{N}}$, there is no reverse injection from $M_*^2$ into $\\fin (M_*)$, \nand that there are no injections from $\\fin(M_*)$ into $[M_*]^2$ \nor from $\\iseq(M_*)$ into $M_*^2$.\n\nIn order to show that there is neither an injection from $M_*^2$ into \n$\\fin (M_*)$, nor an injection from $\\iseq(M_*)$ into $M_*^2$,\nassume towards a contradiction that $\\modcV_\\text{\\bfseries\\sffamily{N}}$ contains an injection\n$f_1\\colon M_*^2\\to\\fin (M_*)$ or an injection $f_2\\colon\\iseq(M_*)\\to M_*^2$. Let $S$ \nbe a finite support of both functions~$f_1$ and $f_2$ (if they exists). In other words, \n$S\\in\\fin (M_*)$ and for each automorphism $\\pi\\in\\operatorname{Fix}_{G}(S)$ \nwe have $\\pi(f_1)=f_1$ and  $\\pi(f_2)=f_2$, respectively. \nLet $N_1$ be a countable model in $K$ with $S\\subs N_1\\leq M_*$. \nLet $(N_2,g)$ be a countable model in $K$ \nsuch that $(N_1,h|_{N_1})\\le (N_2,g)$,\nconstructed as follows: The domain of $N_2$ is \nthe disjoint union $$N_2=N_1\\sqcup\\{x,y,z\\}\\sqcup\\{a_i:i\\in\\omega\\}\\,.$$\nFurthermore, we define the injection $g\\colon\\fin(N_2)\\to N_2^2$ such that $g\\supseteq h|_{N_1}$\nand for $E\\in\\fin (N_2)\\setminus\\fin (N_1)$ we define $g(E)=\\langle e_1,e_2\\rangle$ such\nthat $g$ is injective and satisfies the following conditions (recall the since $N_2$ is \ncountable, also $\\fin(N_2)$ is countable):\n\\begin{itemize}\n    \\item If $E\\cap\\{x,y,z\\}=\\emptyset$ then $\\langle e_1,e_2\\rangle=\\langle a_n,a_m\\rangle$ for some \n    $n,m\\in\\omega$.\n   \n    \\item If $|E\\cap\\{x,y,z\\}|=1$, then $\\langle e_1,e_2\\rangle =\\langle u,a_k\\rangle$ \n    for some $k\\in\\omega$, where $u$ is the unique element in $E\\cap\\{x,y,z\\}$. \n   \n    \\item If $|E\\cap\\{x,y,z\\}|=2$, then $\\langle e_1,e_2\\rangle=\\langle v,a_k\\rangle$ for some $k\\in\\omega$, \n    where $v$ is the unique element in $\\{x,y,z\\}\\setminus(E\\cap\\{x,y,z\\})$.\n    \\item If $|E\\cap\\{x,y,z\\}|=3$ then $\\langle e_1,e_2\\rangle=\\langle a_n,a_m\\rangle$ for some $n,m\\in\\omega$. \n\\end{itemize}\nNotice that there are automorphisms of $(N_2,g)$ that just permute ${x,y,z}$ and\nfix all other elements of $N_2$ pointwise.\nBy construction of $M_*$, we find a model $N_2'\\in K$ such that \n$N_1\\leq N_2'\\leq M_*$ and $N_2'$ is isomorphic to $N_2$ over~$N_1$.\nFor this reason we can refer to $N_2$ as a legit submodel of $M_*$ that extends $N_1$ \nin the way we described. Let us now consider $f_1(\\langle x,y\\rangle)$, where we assumed in $\\modcV_\\text{\\bfseries\\sffamily{N}}$\nthe existence of an injection $f_1\\colon M_*^2\\to\\fin (M_*)$ with finite support~$S$. \nIf $f_1(\\langle x,y\\rangle)\\nsubseteq N_2\\setminus N_1$, then we can\napply the third property of {\\sc Proposition}\\;\\ref{prp:limit} with respect to $f_1(\\langle x,y\\rangle)$ \nand $N_1$ and~$N_2$, which gives us a contradiction.\nIf $\\{x,y\\}\\subs f_1(\\langle x,y\\rangle)$ or $\\{x,y\\}\\cap f_1(\\langle x,y\\rangle)=\\emptyset$,\nwe could swap $x$ and $y$ while fixing every other element of $N_2$ \npointwise and get $f(\\langle x,y\\rangle)=f_1(\\langle y,x\\rangle)$, which would imply that\n$f_1$ is not injective. So, assume that\n$|\\{x,y\\}\\cap f_1(\\langle x,y\\rangle)|=1$ and without loss\nof generality assume that $\\{x,y\\}\\cap f_1(\\langle x,y\\rangle)=\\{x\\}$. \nNow, if $z\\in f_1(\\langle x,y\\rangle)$, {\\ie}, $\\{x,z\\}\\subs f_1(\\langle x,y\\rangle)$, \nwe similarly obtain a contradiction by swapping $z$ and $x$, \nwhile if $z\\notin f_1(\\langle x,y\\rangle)$ we get a contradiction by swapping $z$ and $y$. \nThis shows that $f_1$ cannot belong to $\\modcV_\\text{\\bfseries\\sffamily{N}}$.\n\nFor what concerns $f_2$, let us consider the set $\\nS$ consisting of sequences\nwithout repetition of $\\{x,y,z\\}$ of length~$2$ or~$3$.\nNotice that $|\\nS|=12$. Now, for each element $s\\in\\nS$, if $f_2(s)=\\langle a,b\\rangle$, then $a$ and\n$b$ are such that $a\\neq b$ and $\\langle a,b\\rangle\\in\\{x,y,z\\}^2$\\,---\\,notice that otherwise, for example, \nif $\\{a,b\\}\\cap\\{x,y\\}=\\emptyset$, then we can swap $x$ and $y$ and hence move $s$ \nwithout moving $\\langle a,b\\rangle$, which is not consistent with $S$ being a support of $f_2$. \nWe get the conclusion by noticing that, because of this restriction, \nthere are only six possible images of elements of $E$, which implies \nthat $f_2$ cannot be an injection.\n\nIt remains to show that in $\\modcV_\\text{\\bfseries\\sffamily{N}}$ there are no injections \nfrom $\\fin(M_*)$ into $[M_*]^2$. For this, assume towards a contradiction \nthat there exists such a function $f_3$\nin $\\modcV_\\text{\\bfseries\\sffamily{N}}$ and assume that $S$ is a finite support of~$f_3$. Then, \nlet $N_1$ be a countable model in $K$ with \n$S\\subs N_1\\leq M_*$. We will construct a countable model $(N_2,g)\\in K$ \nsatisfying $(N_1,h|_{N_1})\\leq (N_2,g)\\leq M_*$ with a finite subset \n$u\\in\\fin(N_2\\setminus N_1)$ such that, for all $\\langle x,y\\rangle\\in(N_2\\setminus N_1)^2$, \none of the following holds:\n\\begin{itemize}\n\\item there is no finite set $E\\in\\fin(N_2)$ with $h(E)=\\langle x,y\\rangle$;\n\\item there exists an automorphism $\\pi$ of $N_2$ over $N_1$ with $\\pi(u)\\neq u$ \nand $\\pi\\{x,y\\}=\\{x,y\\}$.\n\\end{itemize}\nLet $u=\\{a_0,b_0,c_0\\}$ be disjoint from $N_1$ and define $G^1_0=N_1\\sqcup\\{a_0,b_0,c_0\\}$. \nNow, for each finite set $E\\in\\fin(G^1_0)$ which is not in the domain of \n$h_0=h|_{N_1}$, that is, for each finite set $E\\in\\fin(G^1_0)$ with \n$E\\cap\\{a_0,b_0,c_0\\}\\neq\\emptyset$, let $\\{x_E,y_E\\}$ be a pair of new elements  \nand define $$G^*_0\\;:=\\;G^1_0\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_0)\\setminus\\dom(h_0)}\n\\hspace{-2ex}\\{x_E,y_E\\}\\qquad\\text{and}\\qquad\nh^1_0:=h_0\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_0)\\setminus\n\\dom(h_0)}\\hspace{-2ex}\\blcb\\big{\\langle} E,\\langle x_E,y_E\\rangle\\big{\\rangle}\\brcb.$$ \nLet now $G^2_0$ be an extension of $G^*_0$ \nby adding a copy of $G^1_0\\setminus N_1$, where the ``copy function'' \nis denoted $\\tau_0$. Notice that at this stage, $G^1_0\\setminus N_1=\\{a_0,b_0,c_0\\}$.\nMore formally, $G^2_0=G^*_0\\sqcup\\{\\tau_0(a):a\\in \nG^1_0\\setminus N_1\\}$, together with an extension of $h^1_0$ defined as \n$$h^2_0:=h^1_0\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_0)\\setminus\\dom(h_0)}\n\\hspace{-2ex}\\blcb\\big{\\langle} \\tau_0(E),\\langle y_E,x_E\\rangle\\big{\\rangle}\\brcb,$$ \nwhere, given $E\\in\\fin(G^1_0)\\setminus\\dom(h_0)$, $\\tau_0(E)$ is defined \nas $$\\tau_0(E):=(E\\cap N_1)\\sqcup\\{\\tau_0(a):a\\in E\\setminus N_1\\}.$$ \nNotice that if $a\\in G^1_0\\setminus N_1$, then $\\tau_0(a)\\in G^2_0\\setminus G^*_0$. \nThe construction carried out so far is actually the first of countably many \nanalogous extension steps we will consequently apply in order to consider \nthe union of all the progressive extensions. \nThat is, assume that for some \n$i\\in\\omega$ we have already defined $G^2_i$ and $h^2_i$. Define $G^1_{i+1}=G^2_i$ \nand, for each finite set $E\\in\\fin(G^1_{i+1})$ which is not in the domain of $h^2_i$, \nconsider a pair of new elements $\\{x_E,y_E\\}$ and define\n$$G^*_{i+1}:=G^1_{i+1}\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_{i+1})\n\\setminus\\dom(h^2_i)}\\hspace{-2ex}\\{x_E,y_E\\}\\qquad\\text{and}\\qquad\nh^1_{i+1}:=h^2_i\\;\\sqcup\\hspace{-2ex}\n\\bigsqcup_{E\\in\\fin(G^1_{i+1})\\setminus\\dom(h^2_i)}\\hspace{-2ex}\n\\blcb\\big{\\langle}E,\\langle x_E,y_E\\rangle\\big{\\rangle}\\brcb.$$ \nLet now $G^2_{i+1}$ be an extension of $G^*_{i+1}$ by adding a copy of \n$G^1_{i+1}\\setminus N_1$, where the ``copy function'' is now $\\tau_{i+1}$. \nMore formally, $G^2_{i+1}:=G^*_{i+1}\\;\\sqcup\\{a_{i+1}:a\\in G^1_{i+1}\\setminus N_1\\}$, \ntogether with an extension of $h^1_{i+1}$ defined as \n$$h^2_{i+1}:=h^1_{i+1}\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_{i+1})\n\\setminus\\dom(h^2_i)}\\hspace{-2ex}\\blcb\\big{\\langle}\\tau_{i+1}(E),\\langle y_E,x_E\\rangle\\big{\\rangle}\\brcb,$$ \nwhere, again, given $E\\in\\fin(G^1_{i+1})\\setminus\\dom(h^2_i)$, $\\tau_{i+1}(E)$ \nis defined as $\\tau_{i+1}(E):=(E\\cap N_1)\\sqcup\\{\\tau_{i+1}(a):\na\\in E\\setminus N_1\\}$, for which we newly remark that if \n$a\\in G^1_{i+1}\\setminus N_1$, then $\\tau_{i+1}(a)\\in G^2_{i+1}\\setminus \nG^*_{i+1}$. \nNotice that every automorphism of $G_{i+1}^1$ which moves a finite set $E$ to\n$E'$, moves the pair $\\{x_E,y_E\\}$ to $\\{x_{E'},y_{E'}\\}$, and consequently, it\nmoves $\\big{\\langle} E,\\langle x_E,y_E\\rangle\\big{\\rangle}$ to $\\big{\\langle} E',\\langle x_{E'},y_{E'}\\rangle\\big{\\rangle}$.\nIn particular, every automorphism of $(G_{i+1}^1,h_i^2)$ can be extended to an automorphism\nof $(G_{i+1}^*,h_{i+1}^1)$. Moreover, every automorphism of $(G_{i+1}^*,h_{i+1}^1)$ can be extended to an automorphism of $(G_{i+1}^2,h_{i+1}^2)$.\n\nNow, let $$N_2:=\\bigcup_{i\\in\\omega}\nG^1_i\\qquad\\text{and}\\qquad g:=\\bigcup_{i\\in\\omega}h^1_i\\,.$$\nWe claim that $(N_2,g)$ satisfies the required properties. \nIndeed, if $\\langle x,y\\rangle\\in(N_2\\setminus N_1)^2$ and there is some \nfinite set $E\\in\\fin(N_2)$ with $g(E)=h(E)=\\langle x,y\\rangle$, then by \nconstruction there exists some index $n\\in\\omega$ such that \n$\\langle x,y\\rangle\\in (G^*_n\\setminus G^1_n)^2$, which implies that there \nexists an automorphism $\\pi$ of $N_2$ over $N_1$ acting as follows:\n$\\pi\\langle x,y\\rangle=\\langle y,x\\rangle$ and \n$\\pi u=\\pi\\{a_0,b_0,c_0\\}=\\blcb\\tau_n(a_0),\\tau_n(b_0),\\tau_n(c_0)\\brcb$, \nwhich in particular means $\\pi\\{x,y\\}=\\{x,y\\}$ and $\\pi u\\neq u$, \nas desired. We can finally consider the image $f_3(u)=\\{x,y\\}$: \nIf $\\{x,y\\}\\nsubseteq N_2\\setminus N_1$, then we can\napply the third property of {\\sc Proposition}\\;\\ref{prp:limit} with respect to $\\{x,y\\}$ \nand $N_1$ and~$N_2$, which gives us a contradiction.\nThus $\\{x,y\\}\\subseteq N_2\\setminus N_1$, and if there exists some finite \nset $E\\in\\fin(N_2)$ with $g(E)=h(E)=\\langle x,y\\rangle$, then by the reasoning \nabove we find that some automorphism of $N_2$ over $N_1$ does not \npreserve $f_3$, a contradiction. In every other case, we consider \n$\\textrm{cl}(N_1\\cup\\{x,y\\},M_*)$ and notice that, since for no \n$E\\in\\fin(N_2)$ we have $h(E)=\\langle x,y\\rangle$ or $h(E)=\\langle y,x\\rangle$, then $u$ \ncannot be a subset of $\\textrm{cl}(N_1\\cup\\{x,y\\},M_*)$, which allows \nus to fix $\\textrm{cl}(N_1\\cup\\{x,y\\},M_*)$ pointwise, while not \npreserving $u$, a contradiction as well.\n\\end{proof}\n\nSo, the model $\\modcV_{\\text{\\bfseries\\sffamily{N}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{->}[drr]& &\\seq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{->}[u]& &\\mathfrak{m}^2\\ar@{->}[u]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{\\reflectbox{N}}}} \n\nWe show that Diagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}} holds in the model constructed\nin~\\cite{weird} (see also~\\cite[p.\\,209\\,ff]{cst}), \nwhere $\\mathfrak{m}$ is the cardinality of the set of atoms of that model.\n\nThe atoms of the permutation model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ for \nDiagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}} are constructed as follows:\n\\begin{itemize}\n\\setlength{\\itemsep}{.5ex}\n\\item[($\\alpha$)] Let $A_0$ be an arbitrary infinite set.\n\\item[($\\beta$)] $G_0$ is the group of {\\it all\\\/} permutations of $A_0$.\n\\item[($\\gamma$)] $A_{n+1}:=A_n\\cup\\blcb (n+1,p,\\varepsilon):\np\\in\\bigcup_{k=0}^{n+1} A_n^k\\wedge\\varepsilon\\in\\{0,1\\}\\brcb$.\n\\item[($\\delta$)] $G_{n+1}$ is the subgroup of the\npermutation group of $A_{n+1}$ containing all permutations \n$\\sigma$ for which there are\n$\\pi_\\sigma\\inG_n$ and $\\varepsilon_{\\sigma,p}\\in\\{0,1\\}$ such that\n\\begin{eqnarray*}\n\\sigma(x)=\n\\begin{cases}\n\\pi_\\sigma(x) &\\mbox{if $x\\in A_n$,}\\cr\\noalign{\\vspace{3pt}}\n(n+1,\\pi_\\sigma(p),\\varepsilon_{\\sigma,p}+_2\\varepsilon) \n&\\mbox{if $x=(n+1,p,\\varepsilon)$,}\n\\end{cases}\n\\end{eqnarray*}\nwhere for $p=\\langle p_0,\\ldots,p_{l-1}\\rangle\\in\n\\bigcup_{0\\le k\\le n+1} A_n^k$, \n$\\pi_\\sigma(p):=\\langle \\pi_\\sigma(p_0),\\ldots,\n\\pi_\\sigma(p_{l-1})\\rangle$\nand $+_2$ denotes addition modulo $2$.\n\\end{itemize}\n\nLet $A:=\\bigcup\\{A_n:n\\in\\omega\\}$ be the set of atoms and let \n$\\operatorname{Aut}(A)$ be the group of all permutations of $A$. Then\n\\[\nG:=\\blcb H\\in\\operatorname{Aut}(A):\\forall n\\in\\omega\\,(H\\res{A_n}\\inG_n)\\brcb\n\\]\nis a group of permutations of $A$. \nThe sets in $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ are those with finite support.\n\n\\begin{prp} Let\\\/ $A$ be the set of\natoms of\\\/ $\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}}$ and let\\\/ $\\mathfrak{m}:=|A|$. Then\n$$\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}}\\models \\mathfrak{m}^2<\\iseq(\\mathfrak{m})<[\\mathfrak{m}]^2<\\fin(\\mathfrak{m})\\,.$$ \n\\end{prp}\n\n\\begin{proof}\nIn~\\cite{weird} it is shown that $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}\\models\\seq(\\mathfrak{m})<[\\mathfrak{m}]^2$, \nwhich implies that $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}\\models\\iseq(\\mathfrak{m})<[\\mathfrak{m}]^2$. Thus, since\n$\\mathfrak{m}^2\\le \\iseq(\\mathfrak{m})$ and $[\\mathfrak{m}]^2\\le\\fin(\\mathfrak{m})$, it remains to show that\nin $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ we have $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$ and $[\\mathfrak{m}]^2\\neq\\fin(\\mathfrak{m})$.\n\\medskip\n\n\\noindent $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$: We show that there is no\ninjection $g_1:\\iseq(A)\\to A^2$. Assume towards a contradiction\nthat there is such an injection with finite support $E_1$.\n\nSince $E_1$ is finite, there is an integer $n_1\\in\\omega$ \nsuch that $E_1\\subs A_{n_1}$. \nBy extending $E_1$ if necessary, we may assume that if \n\\hbox{$(n+1,\\langle a_0,\\ldots,a_{l-1}\\rangle,\\varepsilon)\\in E_1$}, \nthen also $a_0,\\ldots,a_{l-1}$ belong to $E_1$ as well as the\natom $(n+1,\\langle a_0,\\ldots,a_{l-1}\\rangle,1-\\varepsilon)$.\n\nFor a large enough number $k\\in\\omega$ choose a $k$-element set\n$X\\subs A_0\\setminus E_1$ such that\n$|\\iseq\\left(X\\right)|>\\big{|}\\left(E_1\\cup X\\right)^2\\big{|}$.\nNotice that $|\\iseq\\left(X\\right)|\\ge k!$ and \nthat $\\big{|}\\left(E_1\\cup X\\right)^2\\big{|}=\\left(|E_1|+k\\right)^2$.\nThus, we find a sequence $s\\in\\iseq(X)$ such that $g_1(s)\\notin \n\\left(E_1\\cup X\\right)^2$. So, there exists a $\\pi\\in\\operatorname{Fix}_{G}(E_1\\cup X)$ \nsuch that $\\pi g_1(s)\\neq g_1(s)$ but $\\pi s=s$, which contradicts \nthe fact that $E_1$ is a support of~$g_1$.\\smallskip\n\n\\noindent $[\\mathfrak{m}]^2\\neq\\fin(\\mathfrak{m})$: We show that there is no\ninjection $g_2:\\fin(A)\\to [A]^2$. Assume towards a contradiction\nthat there is such an injection with finite support~$E_2$.\nAfter extending $E_2$ in the same way as above, if necessary, \nfor a large enough number $k\\in\\omega$ we choose again a $k$-element set\n$X\\subs A_0\\setminus E_2$ such that\n$|\\fin\\left(X\\right)|>\\big{|}\\left[E_2\\cup X\\right]^2\\big{|}$.\nThen, by similar arguments as above, we can show that $E_2$ is not\na support of~$g_2$, which gives us the desired contradiction.\n\\end{proof}\n\nSo, the permutation model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{<-}[urr]& &\\mathfrak{m}^2\\ar@{->}[u]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\section{Permutation Models}\n\nIn order to show, for example, that for some infinite cardinals $\\mathfrak{m}$ and $\\mathfrak{n}$, \n\\hbox{$\\mathfrak{m}<\\mathfrak{n}$} is consistent with $\\ZF$, \nby the {\\sc Jech-Sochor Embedding Theorem} \n(see, for example, \\cite[Thm.\\,6.1]{Jechchoice} or~\\cite[Thm.\\,17.2]{cst}), it is\nenough to construct a permutation model in which this statement holds.\nThe underlying idea of permutation models, which will be models of \nset theory with atoms ($\\ZFA$),\nis the fact that a model $\\modcV\\models\\ZFA$ does not distinguish between the\natoms, where atoms are objects which do not have\nany elements but which are distinct from the empty set. \nThe theory $\\ZFA$ is essentially the same as that of $\\ZF$\n(except for the definition of ordinals, where we have to require that\nan ordinal does not have atoms among its elements). Let $A$ be a set.\nThen by transfinite recursion on the ordinals $\\alpha\\in\\Omega$ we can define\nthe $\\alpha$-power ${\\pwr}^\\alpha(A)$ of~$A$ and\n${\\pwr}^\\infty(A):=\\bigcup_{\\alpha\\in\\Omega}{\\pwr}^\\alpha(A)$. Like\nfor the cumulative hierarchy of sets in $\\ZF$, one can show that if\n$\\modcM$ is a model of $\\ZFA$ and $A$ is the set of atoms of\n$\\modcM$, then $\\modcM:={\\pwr}^\\infty(A)$. The\nclass $M_0:={\\pwr}^\\infty(\\emptyset)$ is a model of $\\ZF$\nand is called the {\\bf kernel}. Notice that all ordinals belong to the kernel.\nBy construction we obtain that every permutation of the set of atoms induces an\nautomorphism of $\\modcV$, where the sets in the kernel are fixed.\n\nPermutation models were first\nintroduced by Adolf~Fraenkel and, in a precise version (with\nsupports), by Andrzej~Mostowski. The version with filters, which we will follow\nbelow, is due to Ernst~Specker (a detailed introduction to permutation models \ncan be found, for example, in~\\cite[Ch.\\,8]{cst} or~\\cite{Jechchoice}).\n\nIn order to construct a permutation model, we usually start with \na set of atoms $A$ and then define a group $G$ of permutations or \nautomorphisms of $A$. \n\nThe permutation models we construct below are of the following simple\ntype: For each finite set $E\\in\\fin(A)$, let $$\\operatorname{Fix}_{G}(E):=\\blcb\\pi\\inG:\n\\forall a\\in E\\,(\\pi a=a)\\brcb\\,,$$ and let $\\nF$ be the \nfilter of subgroups on $G$ generated by the subgroups \n$\\{\\operatorname{Fix}_{G}(E): E\\in\\fin(A)\\}$. In other words, $\\nF$ is the set\nof all subgroups $H\\leG$, such that the there exists a finite set\n$E\\in\\fin(A)$, such that $\\operatorname{Fix}_G(E)\\le H$.\n\nFor a set $x$, let \n$$\\Sym_{G}(x):=\\{\\pi \\inG :\\pi x\\,=\\,x\\}$$ \nwhere $$\\pi x=\\begin{cases}\n\\ \\emptyset &\\text{if $x=\\emptyset$,}\\\\[1ex]\n\\ \\pi a&\\text{if $x=a$ for some $a\\in A$,}\\\\[1ex]\n\\blcb\\pi y:y\\in x\\brcb&\\text{otherwise.}\n\\end{cases}$$\nThen, a set $x$ is {\\it symmetric\\\/} if and only if\nthere exists a set of atoms $E_{x}\\in\\fin(A)$, such that\n$$\\operatorname{Fix}_{G}(E_x)\\le\\Sym_{G}(x).$$ We say that $E_x$ is a {\\it\nsupport\\\/} of $x$. Finally, let $\\modcV$ be the class \nof all hereditarily symmetric objects; then\n$\\modcV$ is a transitive model of $\\ZFA$. \nWe call $\\modcV$ a {\\it permutation model}.\nSo, a set $x$ belongs to the permutation model\n$\\modcV$ (with respect to $G$ and $\\nF$), \nif and only if $x$ has a finite support $E_x\\in\\fin(A)$.\nBecause every $a\\in A$ is symmetric, we get that each atom $a\\in A$\nbelongs to $\\modcV$.\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{Z}}}\n\nWe are now going to set an analogue framework to the one for Diagram~$\\text{\\bfseries\\sffamily{N}}$, \njust with the definitions adapted, in order to show the consistency of \nDiagram~$\\text{\\bfseries\\sffamily{Z}}$. In fact, as mentioned above, \nwe can state the same proposition, guaranteeing the existence of a suitable \n$\\aleph_1$-universal and $\\aleph_1$-homogeneous model.\n\nLet $K$ be the class of all the pairs $(A,f,g,h)$ such that $A$ is a (possibly empty) \nset and $f,g,h$ are the following three injections:\n$$f\\colon A^2\\to[A]^2\\qquad \ng\\colon[A]^2\\to\\textrm{seq}^{1-1}(A)\\qquad h\\colon\\textrm{seq}^{1-1}(A)\\to\\textrm{fin}(A)$$\nAs before, we define a partial ordering $\\leq$ on $K$ \nby stipulating $(A,f_1,g_1,h_1)\\leq (B,f_2,g_2,h_2)$ if and only if \n\\begin{itemize}\n\\item $A\\subseteq B$, \n\\item $f_1\\subseteq f_2$,\n$\\ran\\left(f_2|_{B^2\\setminus A^2}\\right)\\subs [B\\setminus A]^2$, \n\\item $g_1\\subseteq g_2$,\n$\\ran\\left(g_2|_{[B]^2\\setminus [A]^2}\\right)\\subs\\iseq(B\\setminus A)$,\n\\item $h_1\\subseteq h_2$,\n$\\ran\\left(h_2|_{\\iseq(B)\\setminus \\iseq(A)}\\right)\\subs\\fin(B\\setminus A)$.\n\\end{itemize}\n\n\\begin{prp}[CH]\\label{prp:AtomsZ}\nThere is a model\\\/ $M_*$ of cardinality\\\/ $\\mathfrak{c}$ in $K$ such that:\n\\begin{itemize}\n    \\item $M_*$ is\\\/ $\\aleph_1$-universal, i.e., if\\\/ $N\\in K$ is countable \n    then\\\/ $N$ is isomorphic to some\\\/ $N_*\\leq M_*$.\n    \\item $M_*$ is\\\/ $\\aleph_1$-homogeneous, i.e., if\\\/ $N_1,N_2\\leq M_*$ are \n    countable and\\\/ $\\pi\\colon N_1\\to N_2$ is an isomorphism then there exists \n    an automorphism\\\/ $\\pi_*$ of\\\/ $M_*$ such that\\\/ $\\pi\\subs\\pi_*$.\n    \\item If\\\/ $N\\leq M_*$ and\\\/ $A\\subs M_*$ are countable, then there is an \n    automorphism\\\/ $\\pi$ of\\\/ $M_*$ over\\\/ $N$ such that\\\/ \n    $\\pi(A)\\setminus N$ is disjoint from\\\/ $A$. \n\\end{itemize}\n\\end{prp}\n\\begin{proof}\nThe proof is essentially the same as the one of {\\sc Proposition}\\;\\ref{prp:limit}.\n\\end{proof}\n\nWe define $\\modcV_\\text{\\bfseries\\sffamily{Z}}$ as the permutation model obtained by setting \nthe elements of the $\\aleph_1$-universal and $\\aleph_1$-homogeneous model $M_*$ \nas the set of atoms and its automorphisms $\\operatorname{Aut}(M_*)$ as the group $G$ of permutations. \nIn particular, each permutation of $M_*$ preserves\nthe injections $f,g,h$ that the model $(M_*,f,g,h)$ comes with.\n\n\\begin{thm}\nLet\\\/ $M_*$ be the set of\natoms of\\\/ $\\modcV_\\text{\\bfseries\\sffamily{Z}}$ and let\\\/ $\\mathfrak{m}=|M_*|$. Then\n$$\\modcV_\\text{\\bfseries\\sffamily{Z}}\\models \\mathfrak{m}^2<[\\mathfrak{m}]^2<\\iseq(\\mathfrak{m})<\\fin(\\mathfrak{m})\\,.$$ \n\\end{thm}\n\\begin{proof}\nThe existence of the required injections is clear by the definition of the \nmodel. Thus, it remains to prove that there are no reverse injections. \nFirst, we give two preliminary definitions. Given a model $(M,f,g,h)$ \nand a countable subset $A\\subs M$, \nwe define the \\textit{closure\\\/} $\\textrm{cl}(A,M)$ as \nthe smallest superset of $A$ that is closed under $f,g,h$ and pre-images \nwith respect to the same functions. Constructively, we can characterize \n$\\textrm{cl}(A,M)$ as a countable union as follows: \nDefine $\\textrm{cl}_0=\\textrm{cl}_0(A,M):=A$ and, for all $i\\in\\omega$,\n\\begin{align*}\n    \\textrm{cl}_{i+1}= \\textrm{ cl}_i &\\;\\sqcup \\bigsqcup_{p\\in(\\textrm{cl}_i)^2}f(p)\\; \n    \\sqcup \\bigsqcup_{q\\in[\\textrm{cl}_i]^2}\\ran(g(q))\\;\\sqcup\n    \\bigsqcup_{s\\in\\iseq(\\textrm{cl}_i)}\\hspace{-2ex}h(s) \\\\[1.0ex]\n    & \\sqcup \\bigsqcup_{q\\in[\\textrm{cl}_i]^2} \\ran(f^{-1}(q)) \\;\\sqcup \n    \\bigsqcup_{s\\in\\textrm{seq}^{1-1}(\\textrm{cl}_i)}\\hspace{-2ex}g^{-1}(s) \\;\\sqcup \n    \\bigsqcup_{r\\in\\textrm{fin}(\\textrm{cl}_i)}\\hspace{-.8ex}\\ran(h^{-1}(r))\\,,\n\\end{align*}\nin order to finally define $\\textrm{cl}(A,M)=\\cup_{i\\in\\omega}\\textrm{ cl}_i$. \nFurthermore, we set a standardized way \nto extend a \\textit{partial} model $(A,f,g,h)$, where $f,g,h$ are only \npartial functions, to an element of $K$: Consider $(A,f',g',h')$, \nwhere $A$ is a countable set and $f,g,h$ are injections with \n\\begin{center}\n$$    \n\\begin{array}{ccc}\n  \\dom(f)\\subseteq A^2,&\\dom(g)\\subseteq [A]^2, & \\dom(h)\\subseteq\\textrm{seq}^{1-1}(A)\\\\[1.4ex]\n  \\ran(f)\\subseteq [A]^2, &\\ran(g)\\subseteq\\textrm{seq}^{1-1}(A),& \\ran(h)\\subseteq\\textrm{fin}(A).\n\\end{array}\n$$\n\\end{center}\nLet $(M_0,f'_0,g'_0,h'_0)=(A,f',g',h')$ and, for $j\\in\\omega$, define inductively   \n$(M_{j+1},f'_{j+1},g'_{j+1},h'_{j+1})$ as follows: $M_{j+1}$ is the fully \ndisjoint union $$M_j\\quad\\sqcup\\bigsqcup_{P\\in M_j^2\\setminus\\dom(f_j)}\n\\hspace{-3ex}\\{a_P,b_P\\}\\quad\\sqcup\\bigsqcup_{Q\\in[M_j]^2\\setminus\\dom(g_j)}\\hspace{-3.4ex}\\{a_Q,b_Q,c_Q\\}\n\\quad\\sqcup\\bigsqcup_{R\\in\\iseq(M_j)\\setminus\\dom(h_j)}\\hspace{-5ex}\\{a_R,b_R,c_R\\}.$$ \nFor what concerns the injections $f'_{j+1},g'_{j+1},h'_{j+1}$, we naturally \nrequire the inclusions $f'_j\\subseteq f'_{j+1}$, $g'_j\\subseteq g'_{j+1}$, and $h'_j\\subseteq h'_{j+1}$, \nas well as the equalities $\\dom(f'_{j+1})=M_j^2$, $\\dom(g'_{j+1})=[M_j]^2$, \nand $\\dom (h'_{j+1})=\\textrm{seq}^{1-1}(M_j)$, respectively, where for\n$P\\in M_j^2\\setminus\\dom (f_j)$, \n$Q\\in[M_j]^2\\setminus\\dom (g_j)$, and $R\\in\\textrm{seq}^{1-1}(M_j)\n\\setminus\\dom (h_j)$, we define  $$f'_{j+1}(P):=\\{a_P,b_P\\}\\,,\\qquad g'_{j+1}(Q):=\n\\langle a_Q,b_Q,c_Q\\rangle\\,,\\qquad h'_{j+1}(R):=\\{a_R,b_R,c_R\\}\\,.$$ \nWe are now in the position of defining the plain extension of $(A,f',g',h')$ as \n$$(M,f,g,h)=\\left(\\bigcup_{j\\in\\omega}M_j,\\;\\bigcup_{j\\in\\omega}f'_j,\\;\\bigcup_{j\\in\\omega}g'_j,\\;\n\\bigcup_{j\\in\\omega}h'_j\\right),$$ and we can finally prove, in three analogous steps, \nthat neither of the three injections of the model $(M_*,f,g,h)$ \nadmits a reverse injection.\n\nAssume there is an injection $i\\colon [A]^2\\to A^2$ with finite support \n$S$, where $A$ is the set of atoms. Let $N_1\\in K$ \nbe a countable model such that $N_1\\leq M_*$ and $S\\subs N_1$. \nLet $\\{x,y\\}\\in [A]^2$ with $N_1\\cap\\{x,y\\}=\\emptyset$, \nlet $M_0=N_1\\sqcup\\{x,y\\}$, and let $N_2$ be the plain extension of $M_0$.\nWithout loss of generality we can assume that $N_2\\leq M_*$. \nConsider $\\langle a,b\\rangle=i(\\{x,y\\})$. \nThen $\\{a,b\\}\\nsubseteq N_1$ and $\\{a,b\\}\\subs N_2$, since otherwise, we could \napply the third property of {\\sc Proposition}\\;\\ref{prp:AtomsZ} with respect to $\\{a,b\\}$ \nand $N_1$ and $N_2$, \nrespectively. Moreover, $\\{a,b\\}\\cap\\{x,y\\}=\\emptyset$, since otherwise,\n({\\eg}, $a=x$), we can swap $x$ and $y$ which would imply that $b=y$,\nand since $x\\neq y$, we get $i(\\{x,y\\})=\\langle x,y\\rangle\\neq\\langle y,x\\rangle=i(\\{y,x\\})$\nwhich is a contradiction to the assumption that $i$ is a function.\nFurthermore, we have \n\\begin{equation}\n\\{x,y\\}\\;\\subs\\;\\textrm{cl}(N_1\\cup\\{a,b\\},M_*)\\tag{$*$}\n\\end{equation}\nsince otherwise, we could \napply the third property of {\\sc Proposition}\\;\\ref{prp:AtomsZ} with respect to $\\{x,y\\}$ and\n\\hbox{$\\textrm{cl}\\bigl(N_1\\cup\\{a,b\\},M_*\\bigr)\\leq M_*$.} \nNow, this last inclusion implies that \n$a\\neq b$ and that $\\{a,b\\}=f(\\langle x',y'\\rangle)$ for \nsome $x',y'\\in N_2\\setminus N_1$. \nTo see this, notice first that since $N_1\\cup \\{a,b\\}\\subs N_2$,\nwe build the closure \n\\hbox{$\\textrm{cl}\\bigl(N_1\\cup\\{a,b\\},M_*\\bigr)$} within the\nplain extension~$N_2$, and recall that for $\\{u,v\\}\\in [N_2]^2\\setminus [M_0]^2$\nwe have $g(\\{u,v\\})=\\langle x_1,x_2,x_3\\rangle$ where $\\langle x_1,x_2,x_3\\rangle\\in \n\\iseq(N_2\\setminus M_0)$, and that for $\\langle x_1,\\ldots,x_n\\rangle\\in\\iseq(N_2)\\setminus\n\\iseq(M_0)$ we have $h(\\langle x_1,\\ldots,x_n\\rangle)\\in [N_2\\setminus M_0]^3$.\nIf there are no $x',y'$ such that $f(\\langle x',y'\\rangle)=\\{a,b\\}$, then, since\n$\\{a,b\\}\\subs N_2$, $\\{a,b\\}$ is a proper subset of $\\ran\\bigl(g(\\{u,v\\})\\bigr)$ for some $u,v$, \nor of $h(\\langle x_1,\\ldots,x_n\\rangle)$ for some $x_1,\\ldots,x_n$.\nIn both cases we have that $\\{x,y\\}\\nsubseteq\\textrm{cl}(N_1\\cup\\{a,b\\},M_*)$,\nwhich is a contradiction to~$(*)$.\nNow, since $f(\\langle x',y'\\rangle)=\\{a,b\\}$, by the construction of the plain extension $N_2$\nwe find an automorphism $\\pi$ of $N_2$ \nthat fixes $N_1\\cup\\{x,y\\}$ pointwise and for which we have $\\pi(a)=b$ and $\\pi(b)=a$.\nHence, $i(\\pi\\{x,y\\})=\\langle a,b\\rangle\\neq\\langle b,a\\rangle=\\pi i(\\{x,y\\})$, which is a\ncontradiction.\n\nAssume there is an injection $i\\colon \\mathrm{seq}^{1-1}(A)\\to [A]^2$ with \nfinite support $S$. Let $N_1\\in K$ be a countable model such that\n$N_1\\leq M_*$ and $S\\subs N_1$. Let $\\langle x,y,z\\rangle\\in\\iseq(A)$ with \n$N_1\\cap\\{x,y,z\\}=\\emptyset$, let $M_0=N_1\\sqcup\\{x,y,z\\}$, and \nlet $N_2$ be the plain extension of $M_0$. Finally, let $\\{a,b\\}=i(\\langle x,y,z\\rangle)$. \nThen, a contradiction follows by noticing\\,---\\,with similar arguments as \nabove\\,---\\,that necessarily \n$\\{x,y,z\\}\\not\\subs\\textrm{cl}\\bigl(N_1\\cup\\{a,b\\},M_*\\bigr)$.\nSo, similarly as above, there is an automorphism $\\pi$ of $M_*$ which fixes \n$\\textrm{cl}(N_1\\cup\\{a,b\\},M_*)$ pointwise, but $\\pi(\\{x,y,z\\})\\neq\\{x,y,z\\}$.\n\nFinally, assume there is an injection $i\\colon\\fin(A)\\to\\mathrm{seq}^{1-1}(A)$ \nwith finite support $S$. Let $N_1\\in K$ \nbe a countable model such that $N_1\\leq M_*$ and $S\\subs N_1$. \nLet $\\{x,y,z\\}\\in [A]^3$ be such that \\hbox{$N_1\\cap\\{x,y,z\\}=\\emptyset$,} \nlet $M_0=N_1\\sqcup\\{x,y,z\\}$, and let $N_2$ be the plain extension of $M_0$.\nConsider $\\langle a_j:j\\in n\\rangle=i(\\{x,y,z\\})$ for some $n\\in\\omega$. It is easy \nto see that we must have $\\{a_j:j\\in n\\}\\cap (N_2\\setminus M_0)\\neq\\emptyset$, \nand as before it must also hold $\\{x,y,z\\}\\subs\n\\textrm{cl}\\bigl(N_1\\cup\\{a_j:j\\in n\\},M_*\\bigr)$. \nThe last inclusion implies that a $3$-cycle $\\pi$ applied to $\\{x,y,z\\}$ \ncannot leave $\\{a_j:j\\in n\\}\\cap (N_2\\setminus M_0)$ unchanged, since $\\pi$ \nmoves every unordered pair, ordered pair and injective sequence with values \nin $\\{x,y,z\\}$. We conclude the proof by noticing that we can easily find \nan automorphism of $N_2$ that fixes $N_1$ pointwise and that acts on \n$\\{x,y,z\\}$ as a $3$-cycle.\n\\end{proof}\n\nSo, the model $\\modcV_{\\text{\\bfseries\\sffamily{Z}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{->}[urr]& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}} \n\nAs mention above, Diagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}}\nholds in the {\\it Ordered Mostowski Model\\\/}, \nwhere $\\mathfrak{m}$ is the set of atoms (see, for example, \n\\cite[Related\\;Result\\,48,\\;p.\\,217]{cst}). This\nleads to the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{->}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\\ar@{->}[ull]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\\section{Introduction}\n\nLet $M$ be a set. Then $\\fin(M)$ denotes the set of all finite subsets of $M$,\n$M^2$ denotes the Cartesian product $M\\times M$, \n$[M]^2$ denotes the set of all $2$-element subsets of $M$,\n$\\iseq(M)$ denotes the set of all finite sequences without repetitions\nwhich can be formed with elements of $M$, and $\\seq(M)$\ndenotes the set of all finite sequences which can be formed with elements of $M$\n(where repetitions are allowed).\n\nFurthermore, for a set $A$, let $\\card{A}$ denote the cardinality of $A$. \nWe write $\\card A=\\card B$, if there exists a bijection between $A$ and $B$,\nand we write $\\card A\\le\\card B$, if there exists a bijection between $A$\nand a subset $B'\\subs B$ (\\ie, $\\card A\\le\\card B$ if and only if there \nexists an injection from $A$ into $B$). Finally, we write $\\card A<\\card B$\nif $\\card A\\le\\card B$ and $\\card A\\neq\\card B$. By the \n{\\sc Cantor-Bernstein Theorem},\nwhich is provable in $\\ZF$ only (\\ie, without using the Axiom of Choice),\nwe get that $\\card A\\le\\card B$ and $\\card A\\ge\\card B$ implies \n$\\card A=\\card B$. \n\nLet $\\mathfrak{m}:=|M|$, and let $[\\mathfrak{m}]^2:=\\card{[M]^2}$, $\\mathfrak{m}^2:=\\card{M^2}$, \n$\\fin(\\mathfrak{m}):=\\card{\\fin(M)}$, $\\iseq(\\mathfrak{m}):=\\card{\\iseq(M)}$,\nand $\\seq(\\mathfrak{m}):=\\card{\\seq(M)}$. \nConcerning these cardinalities, in $\\ZF$ we obviously have $\\iseq(\\mathfrak{m})\\le \n\\seq(\\mathfrak{m})$, $[\\mathfrak{m}]^2\\le \\fin(\\mathfrak{m})$ and $\\mathfrak{m}^2\\le\\iseq(\\mathfrak{m})$, \nwhere the latter relations are visualized\nby the following diagram (in the diagram, $\\mathfrak{n}_1$ is below \n$\\mathfrak{n}_2$ if $\\mathfrak{n}_1\\le\\mathfrak{n}_2$):\n\n{\\small\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{-}[d]& &\\iseq(\\mathfrak{m})\\ar@{-}[d]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\n}\n\\]\n\nMoreover, for \\emph{finite\\\/} cardinals $\\mathfrak{m}$ with $\\mathfrak{m}\\ge\\mathfrak{5}$ we have\n$$[\\mathfrak{m}]^2<\\mathfrak{m}^2<\\fin(\\mathfrak{m})<\\iseq(\\mathfrak{m})\\,,$$\nand in the presence of the Axiom of Choice ({\\ie}, in $\\ZFC$),\nfor every infinite cardinal $\\mathfrak{m}$ we have\n$$[\\mathfrak{m}]^2=\\mathfrak{m}^2=\\fin(\\mathfrak{m})=\\iseq(\\mathfrak{m})\\,.$$\nIt is natural to ask whether some of these equalities can be proved also\nin $\\ZF$, {\\ie}, without the aid of $\\AC$. Surprisingly, this is not the\ncase. In~\\cite{carla}, a permutation model was constructed in which for\nan infinite cardinal $\\mathfrak{m}$ we have $\\seq(\\mathfrak{m})<\\fin(\\mathfrak{m})$\n(see~\\cite[Thm.\\,2]{carla} or~\\cite[Prp.\\,7.17]{cst1st}). \nAs a consequence we obtain that the existence\nof an infinite cardinal $\\mathfrak{m}$ such that $\\iseq(\\mathfrak{m})<\\fin(\\mathfrak{m})$ is \nconsistent with $\\ZF$. This consistency result was modified\nto the existence of an infinite cardinal \n$\\mathfrak{m}$ for which $\\mathfrak{m}^2<[\\mathfrak{m}]^2$ (see~\\cite[Prp.\\,7.18]{cst1st}), \nand later, it was strengthened to the existence of an infinite cardinal \n$\\mathfrak{m}$ for which $\\seq(\\mathfrak{m})<[\\mathfrak{m}]^2$ (see~\\cite{weird} or~\\cite[Prp.\\;8.28]{cst}).\nThe consistency of $\\fin(\\mathfrak{m})<\\iseq(\\mathfrak{m})$ for infinite cardinals $\\mathfrak{m}$ can be obtained\nwith the {\\it Ordered Mostowski Model\\\/} (see, for example, \n\\cite[Related\\;Result\\,48,\\;p.\\,217]{cst}), in which there is an \ninfinite cardinal $\\mathfrak{m}$ with $$[\\mathfrak{m}]^2<\\mathfrak{m}^2<\\fin(\\mathfrak{m})<\\iseq(\\mathfrak{m})\\,.$$\nConsistency results as well as $\\ZF$-results concerning the relations between these \ncardinals with other cardinals can be found, for example, \nin~\\cite{Shen1,Shen2} or~\\cite{berlin}.\n\nConcerning the four cardinalities $[\\mathfrak{m}]^2$, $\\mathfrak{m}^2$, $\\fin(\\mathfrak{m})$, and $\\iseq(\\mathfrak{m})$, \na question which arises naturally is whether for some infinite cardinal $\\mathfrak{m}$, \n$\\fin(\\mathfrak{m})<\\mathfrak{m}^2$ is consistent with $\\ZF$ (see \\cite[Related\\;Result\\,20,\\;p.\\,133]{cst}).\nMoreover, assuming that $\\mathfrak{m}$ is infinite and the four cardinalities \n$[\\mathfrak{m}]^2$, $\\mathfrak{m}^2$, $\\fin(\\mathfrak{m})$, $\\iseq(\\mathfrak{m})$ are pairwise distinct and pairwise \ncomparable in~$\\ZF$, one may ask which linear orderings on these four cardinalities\nare consistent with~$\\ZF$.\n\nSince for cardinals $\\mathfrak{m}\\ge\\mathfrak{5}$,\nwe cannot have $[\\mathfrak{m}]^2>\\fin(\\mathfrak{m})$ or $\\mathfrak{m}^2>\\iseq(\\mathfrak{m})$, there are only the following\nsix linear orderings on these four cardinalities which might be consistent with $\\ZF$\n(where for two cardinals $\\mathfrak{n}_1$ and $\\mathfrak{n}_2$, \n$\\mathfrak{n}_1\\longrightarrow\\mathfrak{n}_2$ means $\\mathfrak{n}_1<\\mathfrak{n}_2$). \n\n\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[drr]& &\\iseq(\\mathfrak{m})\\ar@{<-}[d]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{N}}$}\n}\n\\hfill\n\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{C}}$}\n}\n\\hfill\n\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{->}[urr]& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{Z}}$}\n}\n\\hfill{\\ }\n\n\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]& &\\iseq(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[dll]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{\\reflectbox{N}}}$}\n}\n\\hfill\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\ar@{<-}[d]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{\\reflectbox{C}}}$}\n}\n\\hfill\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{->}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\\ar@{->}[ull]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}$}\n}\n\\hfill{\\ }\n\nBelow we show that each of the five diagrams $\\text{\\bfseries\\sffamily{N}}$, $\\text{\\bfseries\\sffamily{Z}}$, $\\text{\\bfseries\\sffamily{\\reflectbox{N}}}$, $\\text{\\bfseries\\sffamily{\\reflectbox{C}}}$, \n$\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}$ is consistent with $\\ZF$. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEntanglement is the most important resource in quantum information, with a vast array of practical uses \\cite{HHHH09}.  In physics more generally, understanding the entanglement structure in physical systems is becoming increasingly important, e.g., in condensed matter theory where quantum phase transitions are signalled by long-range entanglement \\cite{OAFF01,ON02,VLRK03,AFOV08,LR09}.  In high energy physics, quantifying the entanglement of states of a quantum field has applications to a variety of problems \\cite{SW85,Summers08,KST17}, from the AdS\/CFT correspondence \\cite{RT06}, through to detecting spacetime curvature by probing vacuum entanglement \\cite{SM09} or harvesting this entanglement by locally coupling small systems to the field \\cite{Valentini91,Reznik03,RRS05,SR07,PM15,PM16,SM17,SMM17}.\n\nQuantifying entanglement in states of quantum fields is a nontrivial task due to the UV dependence of quantum entanglement near a boundary, leading to naive divergences \\cite{HLW94,CC04,CC06}.  Many ad hoc approaches have been developed to deal with these divergences, usually relying on subtracting the UV divergent piece \\cite{HLW94}.  Of course, operationally, there are no such divergences in the entanglement we can measure because any apparatus we can build to extract entanglement from the field vacuum would only use a finite amount of energy.\n\nSurprisingly little work has been done in the quantum information literature on the problem of quantifying accessible entanglement subject to an energy constraint.  Here we build a theory for this problem and use it to understand the energy cost of extracting entanglement via local operations and classical communication (LOCC).  We use the term extraction rather than one-shot distillation \\cite{BD10}, as we want to emphasise that we are not necessarily distilling all the entanglement from a state.  In contrast, we wish to quantify the \\textit{optimal} energy cost per EPR pair extracted.  While individual protocols for entanglement extraction are interesting, we are primarily concerned with the protocol that minimises the energy cost.\n\nThere are some very interesting related ideas in the literature:\\ in \\cite{CY17}, general quantum operations costing \\textit{zero} energy are studied.  Also, the energy cost of \\textit{creating} entanglement in specific many-body systems was calculated in \\cite{GL09}.  Similarly, in the setting of quantum thermodynamics, the energy\/work cost of creating correlations in quantum systems was studied in \\cite{HPHSKBA15,BPFHH15,FHP16}.  These give useful strategies for creating correlations between finite dimensional systems or a pair of bosons or fermions using energy-conserving (global) unitary operations in the presence of heat baths.  In \\cite{DKPSSS17}, entanglement distillation is considered (also in the presence of a heat bath) with an energy constraint:\\ asymptotically many entangled pairs are distilled into EPR pairs, with the constraint that the energy before and after is \\textit{equal}.  In \\cite{MBDK15}, using a specific local entanglement harvesting protocol (called entanglement farming), the energy cost in the low energy regime was calculated.  In contrast, here we are interested in how the \\textit{optimal} energy cost scales with the number of EPR pairs extracted and in the overall entanglement structure of states, which is a rather different question.\n\nIn this article, we first provide the setting and define the energy cost of a quantum operation.  Then we introduce entanglement extraction subject to an energy constraint.  Following this, we explore the idea using a toy model, which is chosen to share many of the features of the vacuum state of a quantum field but to also be relatively simple.  Next, we introduce the entanglement temperature, which relates the amount of entanglement extracted to the energy cost.  In the following section, we look at the energy cost of entanglement extraction in quantum field theories using physical arguments.  Then we discuss some general methods for quantifying the energy cost of entanglement extraction.  In particular, one of these methods uses matrix product states to numerically bound the energy cost of entanglement extraction for some condensed matter systems and to plot the entanglement temperature.  We conclude with an outlook.\n\n\\section{Preliminaries}\nWe focus on a simplified setup exemplifying the core features of our problem. To whit, we focus on the setting where two players, Alice and Bob, have access to a bipartition of a common system with Hilbert space $\\mathcal{H}_{AB}$. This system, which we refer to as the \\textit{physical system}, has a Hamiltonian $H_{AB}$, which neither Alice nor Bob can modify. Alice and Bob also have access to local ancillary degrees of freedom $A'B'$, which they can use to store the entanglement they extract from the physical system. Thus, the total Hilbert space for the system is given by \n\\begin{equation}\n\t\\mathcal{H}_{AA'BB'} \\equiv \\mathcal{H}_{AB}\\otimes \\mathcal{H}_{A'B'}. \n\\end{equation} \nWe assume that Alice and Bob can carry out any local operation they like on the ancillary degrees of freedom with no energy cost. Thus we associate to the total system+ancilla system the Hamiltonian $H=H_{AB}\\otimes\\openone_{A'B'}$.\n\\begin{figure}[ht!]\n \\resizebox{8.5cm}{!}{\\includegraphics{Intro_fig_01.png}}\n    \\footnotesize{\\caption[Basic Idea]{Energy cost of entanglement extraction:\\ Alice and Bob have access to two parts of a quantum system in an entangled state $\\ket{\\Omega}_{AB}$ with Hamiltonian $H_{AB}$.  Using local operations and classical communication (LOCC), they extract $m$ EPR pairs into their ancillary systems $A^{\\prime}$ and $B^{\\prime}$, leaving the physical system in the final state $\\ket{\\psi}_{AB}$.  Dropping subscripts, the energy cost is then $\\Delta E = \\bra{\\psi}H\\ket{\\psi}-\\bra{\\Omega}H\\ket{\\Omega}$. \\label{fig:intro}}}\n\\end{figure}\n\nWe assume that Alice and Bob can only perform local operations and classical communication (LOCC).  We also suppose that Alice and Bob are working in the one-shot regime, which is natural if, for example, we are thinking of understanding the entanglement structure of vacuum states in quantum field theory where there is only one copy of the system available.  In contrast, in the asymptotic many-copy regime we could use entanglement distillation protocols \\cite{HHHH09}.  We will also comment on the energy cost in the asymptotic regime.\n\nWe quantify the energy cost as follows.  Suppose we have a completely positive trace-preserving map $\\mathcal{E}:\\mathcal{S}(\\mathcal{H})\\rightarrow \\mathcal{S}(\\mathcal{H})$ acting on the space $\\mathcal{S}(\\mathcal{H})$ of density operators on a Hilbert space $\\mathcal{H}$. Suppose we also have a Hamiltonian $H \\in \\mathcal{B}(\\mathcal{H})$. Given a state $\\rho \\in \\mathcal{S}(\\mathcal{H})$, the operation $\\mathcal{E}$ induces an energy change \n\\begin{equation}\n\t\\Delta(\\mathcal{E}, \\rho) = \\operatorname{tr}(\\mathcal{E}(\\rho)H) - \\operatorname{tr}(\\rho H)\n\\end{equation}\nwhen acting on $\\rho$ (this can be negative).  This is the energy cost when we apply the channel $\\mathcal{E}$ to the state $\\rho$.  We may also define the \\textit{energy cost for the channel} $\\mathcal{E}$, which involves an optimisation:\\ we imagine that an adversary prepares the system in the state $\\rho$ which leads to the \\textit{largest} possible change in energy after application of $\\mathcal{E}$:\n\\begin{equation}\n\t\\Delta(\\mathcal{E}) = \\sup_{\\rho \\in \\mathcal{S}(\\mathcal{H})} \\operatorname{tr}(\\mathcal{E}(\\rho)H) - \\operatorname{tr}(\\rho H).\n\\end{equation}\nExploiting the variational definition of the operator norm $\\|\\cdot\\|_\\infty$, we notice that\n\\begin{equation}\n\t\\Delta(\\mathcal{E}) = \\|H-\\mathcal{E}^*(H)\\|_\\infty,\n\\end{equation}\nwhere $\\mathcal{E}^*$ is the dual of $\\mathcal{E}$ acting in the Heisenberg picture.  In the following, we will typically be interested in $\\Delta(\\mathcal{E},\\rho)$ for a specific state and channel, or the set of LOCC channels with $\\Delta(\\mathcal{E},\\rho)\\leq \\Delta E$.  We denote this set by $\\mathcal{C}_{\\text{LOCC}}(\\Delta E)$.\n\n\\section{Extracting entanglement subject to an energy constraint}\nHere we propose a definition for the entanglement accessible to Alice and Bob when they have access only to operations costing less than $\\Delta E$. \n\nWe imagine that the system $AB$ starts in a state $\\sigma_{ABA^{\\prime}B^{\\prime}}=|\\Omega\\rangle_{AB}\\bra{\\Omega}\\otimes|00\\rangle_{A'B'}\\!\\bra{00}$ where $|0\\rangle$ is a convenient fiducial state of the ancilla and $|\\Omega\\rangle_{AB}$ is the initial state of the physical system. Alice and Bob are now allowed to carry out LOCC operations costing less than $\\Delta E$ in total to maximise the quantum entanglement between $A'$ and $B'$. Suppose that $\\mathcal{E}\\in \\mathcal{C}_{\\text{LOCC}}(\\Delta E)$, then we write $\\rho_{AA'BB'} = \\mathcal{E}(\\sigma_{ABA^{\\prime}B^{\\prime}})$.  Thus we define the \\emph{entanglement accessible with energy $\\Delta E$} to be\n\\begin{equation}\n\t\\text{Ent}_{\\Delta E}(|\\Omega_{AB}\\rangle) \\equiv \\sup_{\\mathcal{E}\\in \\mathcal{C}_{\\text{LOCC}}(\\Delta E)} \\text{Ent} (\\rho_{A'B'}),\n\\end{equation}\nwhere $\\text{Ent}$ is some convenient entanglement measure.\n\nWe also define the \\emph{energy cost of extracting $m$ EPR pairs} to be $\\Delta E=\\min\\Delta\\left(\\mathcal{E},\\sigma_{ABA^{\\prime}B^{\\prime}}\\right)$, where the minimum is over all LOCC channels satisfying $\\rho_{A'B'}=\\ket{\\phi^{+\\otimes m}}_{A^{\\prime}B^{\\prime}}\\bra{\\phi^{+\\otimes m}}$, and $\\ket{\\phi^{+}}=\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle)$ is a maximally entangled state of two qubits.\n\nIt is well possible that after extracting entanglement the energy of the system can \\textit{go down}, i.e., extracting entanglement can cool the system. This all depends on the state $|\\Omega_{AB}\\rangle$, i.e., whether it is an excited state or ground state. Since the emphasis in this paper is on ground states, we assume henceforth that $|\\Omega_{AB}\\rangle$ is the ground state of $H_{AB}$.\n\nA key ingredient in any entanglement extraction protocol is the strength of the interaction between Alice's and Bob's systems.  If we write $H_{AB}=H_A\\otimes \\openone_B + \\openone_A \\otimes H_B + V_{AB}$, then the limitations on how much entanglement Alice and Bob can extract using LOCC are determined by $V_{AB}$.  Indeed, if $V_{AB}=0$, the ground state $\\ket{\\Omega}$ will have no entanglement between Alice's and Bob's systems.\n\nThere is a useful naive protocol for entanglement extraction:\\ Alice and Bob first swap the states of their primed and non-primed systems.  Then they can prepare a state of the physical $AB$ system (using LOCC) with minimal \\textit{local} energy, meaning Alice\/Bob prepares $\\ket{\\psi_{A\/B}}$, such that $\\bra{\\psi_{A\/B}} H_{A\/B} \\ket{\\psi_{A\/B}}$ is minimised.  The total energy change is, with $\\ket{\\psi}=\\ket{\\psi_A}\\ket{\\psi_B}$,\n\\begin{equation}\n\\begin{split}\n & \\bra{\\psi} H \\ket{\\psi}- \\bra{\\Omega} H\\ket{\\Omega}\\\\\n \\leq & \\bra{\\psi} V_{AB} \\ket{\\psi}- \\bra{\\Omega} V_{AB}\\ket{\\Omega} \\leq 2\\|V_{AB}\\|_{\\infty}.\n \\end{split}\n\\end{equation}\nTherefore, when the coupling is sufficiently weak, Alice and Bob can safely extract all the entanglement whilst only incurring a small energy cost.\n\nIn contrast, for strong couplings the situation is entirely different, which is exactly the case for quantum field theories, where extracting all the entanglement costs a divergent amount of energy.  For the example of a free fermion field, we see in the appendix that all product states $\\ket{\\psi}$ satisfy $\\bra{\\psi}H\\ket{\\psi}\\geq 1\/a$, where $a$ is the regulator (the lattice spacing in this case).  Thus, the energy diverges as $a\\rightarrow 0$ for any product state, meaning that extracting all the entanglement costs a diverging amount of energy.  In general, the energy cost for extracting all the entanglement will diverge in quantum field theory.  Again using a lattice regulator, the energy contained in the interaction terms between a region $A$ and the rest scales like $(\\partial A\/a^{d})$, where $\\partial A$ is the boundary of $A$, which is also shown in the appendix.\n\n\\section{A toy model}\nIn this section we discuss an idealised model, which exemplifies many of the features of the quantum field vacuum.  It has high entanglement and a high energy cost for extracting all this entanglement, as we will see. \n\nSuppose that the system $AB$ is actually composed of $2n$ qubits, with $n$ qubits in $A$ and $n$ qubits in $B$. We call the qubits $A_j$ (respectively, $B_j$), for $j=1,2, \\ldots, n$. We suppose that $H_{AB}$ is given by \n\\begin{equation}\n\tH_{AB} = \\sum_{j=1}^n (\\mathbb{I}-P_{A_jB_j}),\n\\end{equation}\nwhere $P_{A_jB_j}$ is the projector onto the maximally entangled state $\\ket{\\Phi^{+}}=\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle)$ of qubits $A_j$ and $B_j$. The ground state $|\\Omega_{AB}\\rangle$ of $H_{AB}$ is thus a product of maximally entangled pairs, i.e., it is a maximally entangled state between $A$ and $B$.\n\nIf Alice and Bob could do arbitrary LOCC, then they could easily extract $n$ EPR pairs. However, if they are only allowed an energy cost of $\\Delta E$, then naively they should only be able to extract $O(\\Delta E)$ EPR pairs. \n\nIn the most extreme case, Alice and Bob fully extract all the EPR pairs. Then, in order that this entanglement is between ancilla degrees of freedom in $A'$ and $B'$, it must be that $A$ and $B$ are in a separable state $\\sigma_{AB}$. Since the energy depends linearly on $\\sigma_{AB}$, we may as well suppose that $\\sigma_{AB}$ is an extreme point of the convex set of separable states, namely, a product state $|\\phi\\rangle_A|\\psi\\rangle_B$. The energy of our initial state $|\\Omega\\rangle_{AB}$ was zero, so the energy cost of any entanglement extraction procedure must be greater than  \n\\begin{equation}\n\t\\inf_{|\\phi\\rangle_A|\\psi\\rangle_B} \\sum_{j=1}^n (1-\\langle \\phi_A|\\langle \\psi_B|P_{A_jB_j}|\\phi_A\\rangle|\\psi_B\\rangle).\n\\end{equation}\nThis infimum is achieved by finding the supremum:\n\\begin{equation}\n \t\\sup_{|\\phi\\rangle_A|\\psi\\rangle_B} \\langle \\phi_A|\\langle \\psi_B|P_{A_jB_j}|\\phi_A\\rangle|\\psi_B\\rangle,\n\\end{equation} \nwhich is equal to $1\/2$.  (E.g., setting each pair to $\\ket{00}_{A_iB_i}$ will do the job.) Thus, the energy cost is given by\n\\begin{equation}\n\t\\Delta E \\ge \\frac{1}{2}\\sum_{j=1}^n 1 = n\/2.\n\\end{equation}\n\nMore generally, suppose Alice and Bob extract fewer EPR pairs (say $m$).  One option is to use the following simple protocol.  They swap the states of the first $m$ EPR pairs of the physical system into their ancilla systems, which they can do using local operations.  The first $m$ pairs of qubits of the physical system are now in a product pure state.  Then they can apply local unitaries mapping each of these qubit pairs to the state $\\ket{00}_{A_iB_i}$, getting the final energy cost\n\\begin{equation}\n\t\\Delta E = \\frac{1}{2}\\sum_{j=1}^m 1 = m\/2.\n\\end{equation}\nThus, the total energy cost is $1\/2$ per EPR pair extracted.\n\nOf course, there may be a protocol extracting the same amount of entanglement but costing less energy.  Here we will argue that the simple protocol given above is, in fact, optimal.\n\nWe assume that after applying their operations, Alice and Bob get $m$ Bell states in the ancilla $\\ket{\\Phi^{+\\otimes m}}_{A'B'}$ and some pure state in the physical system $\\ket{\\psi}_{AB}$.  Denote the Schmidt values (decreasingly ordered) of the initial state $\\ket{\\Omega}_{AB}\\ket{00}_{A'B'}$ by $\\alpha_i$, and note that the Schmidt rank is $2^n$.  For it to be possible to transform this state into the new state $\\ket{\\psi}_{AB}\\ket{\\Phi^{+\\otimes m}}_{A'B'}$, with Schmidt values $\\beta_i$, the majorization condition \\cite{NC00} must be satisfied:\n\\begin{equation}\n \\forall K\\geq 1\\ \\ \\ \\  \\sum_{i=1}^{K}\\alpha_i\\leq \\sum_{i=1}^{K}\\beta_i.\n\\end{equation}\nFor this to be possible, the Schmidt rank of the resulting state must be smaller.  This implies that the Schmidt rank of the new state of the physical system $\\ket{\\psi}_{AB}$ can be at most $2^{n-m}$.\n\n\\begin{figure}[ht!]\n\n \\setlength{\\figheight}{5cm}\n \\setlength{\\figwidth}{8.5cm}\n \\def.875{.875}\n\n\n \\input{.\/Toy.tikz}  \n    \\footnotesize{\\caption[Toy Model]{This figure shows the minimum energy cost of extracting entanglement for the toy model.  More precisely, the figure shows the minimum change in energy $\\Delta E$ when there is a decrease in zero entropy $\\Delta S_0$ (with $S_0$ equal to $\\log_2$ of the Schmidt rank) of the state of the physical system.\\label{fig:toymodel}  When $\\Delta S_0$ is an integer $m$ (which corresponds to extracting $m$ EPR pairs from the system), then the plot shows $\\Delta E =0.5m$.  The calculation was performed using DMRG by restricting the bond dimension between Alice and Bob's systems.}}\n\\end{figure}\n\nFigure \\ref{fig:toymodel} shows numerics from a DMRG calculation of the minimum increase in energy as the Schmidt rank of the state of the physical system decreases.  Based on these numerical results, we see that the minimum increase in energy when the Schmidt rank decreases by a factor of $2^m$ is $0.5 m$.  This can be achieved by the simple protocol of the previous section, indicating that this protocol is optimal.  Actually, this whole argument also goes through even if Alice and Bob have some additional shared entanglement that can be used as a catalyst, as in \\cite{JP99}.\n\nIn terms of entanglement distillation in the asymptotic setting, this is not optimal.  In that case, one can distil entanglement at a lower energy cost, which we show later.  In practice, however, we only have access to one copy of a quantum field or condensed matter system, so it is crucial to consider the one-shot setting.  Furthermore, entanglement distillation protocols rely on projecting onto typical subspaces defined by the singular vectors of the initial state \\cite{NC00}, which for extremely complex systems would be practically impossible.\n\n\\section{The entanglement temperature}\nIn the previous section, the total energy cost was $1\/2$ per EPR pair extracted.  To relate the change in entanglement entropy $\\Delta S$ to the energy cost $\\Delta E$, we define the \\textit{entanglement temperature} $T_{\\mathrm{ent}}$ by\n\\begin{equation}\n \\Delta E = T_{\\mathrm{ent}} \\Delta S.\n\\end{equation}\n(The name entanglement temperature is chosen in analogy with thermodynamics.)  So $T_{\\mathrm{ent}}$ is a property of the ground state of a system.  For the toy model, we see that $T_{\\mathrm{ent}}=1\/2$ since $\\Delta S=m\\log_{2}(2)=m$.  In this case $T_{\\mathrm{ent}}$ is constant because there is a linear relationship between the entanglement extracted and the energy cost.  For general systems, we would not expect $\\Delta E \\propto \\Delta S$ for the entire range of $\\Delta S$.  Instead, we should think of the entanglement temperature as a function of the extracted entanglement.  (This is also true in thermodynamics, where temperature can often be thought of as a function of other state functions, such as entropy or pressure.)\n\nIn the following sections, we give some physical and numerical arguments to find $\\Delta E$ as a function of $\\Delta S$ and hence find the entanglement temperature for some physical systems.\n\n\\section{The Energy Cost in General}\n\\label{sec:General Strategy}\nFor some quantum field theories or condensed matter systems, we can give a physical argument for the energy cost of entanglement extraction.  In one dimensional systems, often the entanglement entropy (or, for example, the logarithmic negativity) of ground states can be calculated.  This typically has the form $S(\\rho_I)=c_1\\log_2(N)+c_2$, where $c_i$ are constants and $\\rho_I$ is the state restricted to a contiguous region with $N$ sites \\cite{Korepin04,ECP10}.  For a quantum field theory, regulated by a lattice with lattice spacing $a$, we have instead $S(\\rho_I)=c_1\\log_2(l\/a)+c_2$, where $l$ is the length of a region.  Also, the entanglement entropy in the ground state of models close to the critical point is \\cite{CC04,ECP10}\n\\begin{equation}\n S(\\rho_I)=\\frac{c}{6}\\log_2(\\xi\/a),\n\\end{equation}\nwhere $\\xi\\gg a$ is the correlation length, $c$ is a constant and $I$ corresponds to the infinite half-line $(-\\infty,0]$.  This is equivalent to a massive relativistic QFT with $1\/\\xi$ equal to the mass, e.g., for free bosons we have $c=1$.\n\nAs argued previously, the energy cost for extracting all of this entanglement is $\\Delta E = O(1\/a)$.  At lattice spacing, $a=\\xi\/2^{6m\/c}$, we have $S(\\rho_I)=m$, meaning that the \\textit{most} entanglement we can extract is $m$ EPR pairs.  (In some cases, exactly half of this entanglement is one-shot distillable \\cite{OLEC06}.)  By probing higher energies, which corresponds to smaller values of $a$, we can extract more entanglement, and we have the energy cost $\\Delta E\\propto \\exp(Km)$, where $K=6\\ln(2)\/c$.  This means that the energy cost of entanglement extraction increases \\textit{exponentially}:\\ there is infinite entanglement in the quantum field vacuum, but the cost of extraction grows quickly.  (For gapless models the same argument goes through if Alice has access to a finite region and Bob has access to the rest.  In contrast, if Alice's system is a halfline, there is infinite entanglement at any energy scale.)\n\nThe scaling is different for quantum fields in higher dimensional spaces.  In many cases, the entanglement entropy of the ground state obeys an area law \\cite{RT06b}.  Then in a region $A$ with area $\\partial A$, the leading contribution to the entropy is $S(\\rho_A)\\propto \\partial A\/a^{d-1}$.  However, the energy cost of extracting all the entanglement scales like $\\Delta E \\propto \\partial A\/a^{d}$.  Thus we get an idea for the energy cost of entanglement extraction:\\ $\\Delta E \\propto \\Delta S^{d\/(d-1)}$.  And the entanglement temperature is then $T_{\\mathrm{ent}}\\propto 1\/a\\propto \\Delta E^{1\/d}$.  The energy cost of entanglement extraction in QFT are plotted in figure \\ref{fig:qft}.\n\n\\begin{figure}\n\t\\setlength{\\figheight}{4cm}\n\t\\setlength{\\figwidth}{8cm}\n\t\\def.875{1}\n\n\n\t\\!\\!\\input{.\/QFT.tikz}\n\t\\footnotesize{\\caption{The energy cost of entanglement extraction from the quantum field vacuum depends heavily on the spatial dimension.  Here we sketch the behaviour in dimensions $d=1,2,3$.  When $d=1$, $\\Delta E\\propto \\exp(K\\Delta S)$, where $K$ is a constant, and for $d>1$, $\\Delta E \\propto \\Delta S^{d\/(d-1)}$.}\\label{fig:qft}}\n\\end{figure}\n\nFor more general systems, there is no clear way to proceed.  Below we outline some potential methods to approach the problem.  In the first two cases, we suppose that Alice and Bob use some LOCC protocol to extract $m$ EPR pairs into their ancillary systems, and in the third we consider the trade-off between entanglement change and energy cost numerically.\n\n\\subsection{Method I}\nAlice and Bob have the initial state $|\\Omega\\rangle_{AB}\\ket{00}_{A^{\\prime}B^{\\prime}}$, and then they apply some LOCC protocol to extract $m$ EPR pairs into the ancilla $A^{\\prime}B^{\\prime}$.  We assume that the resulting state on the physical system after the protocol is also pure $\\ket{\\psi}_{AB}$.  (It is possible that a protocol giving a mixed state on the physical system may be more efficient.  In this case, we may use a superadditive entanglement measure, like the squashed entanglement \\cite{CW04}, to upper bound the entanglement left in the physical system.)  Because they are using LOCC, the overall entanglement can only decrease:\n\\begin{equation}\n\\begin{split}\n \\text{Ent}\\left(|\\psi\\rangle_{AB}\\ket{\\Phi^+}^{\\otimes m}_{A^{\\prime}B^{\\prime}}\\right) &\\leq\\text{Ent}\\left(|\\Omega\\rangle_{AB}\\ket{00}_{A^{\\prime}B^{\\prime}}\\right)\\\\\n & = S_{\\mathrm{initial}},\n \\end{split}\n\\end{equation}\nwhere $S_{\\mathrm{initial}}$ is the initial entanglement entropy in the state $|\\Omega\\rangle_{AB}$ and $\\text{Ent}$ is an entanglement measure, which we take to be the entanglement entropy, since the states are all pure.  Then we have that the entanglement entropy in the final state of physical system is $\\text{Ent}\\left(|\\psi\\rangle_{AB}\\right)\\leq S_{\\mathrm{initial}}-m$.\n\nSo what is the minimum energy cost of extracting this entanglement?  We can get an idea by finding the state (or set of states) that have this final value of entanglement entropy while minimising the energy.  In the appendix, we derive the equation\n\\begin{equation}\\label{eq:La}\n \\left[H - \\mu_1\\openone_A\\otimes\\log(\\rho_B) - \\mu_1 +\\mu_2\\right]\\ket{\\psi}_{AB}=0,\n\\end{equation}\nwhere $\\mu_i$ are Lagrange multipliers and $\\mathrm{tr}_A[\\ket{\\psi}\\bra{\\psi}]=\\rho_B$.\n\nThis is difficult to solve in general but may be simplified if we know something about the structure of $H$.  This is the case for the toy model, where $H$ is a sum of commuting terms acting on different pairs of qubits $A_iB_i$.\n\nWith the ansatz $\\ket{\\psi}_{AB}=\\ket{\\psi_1}_{A_1B_1}\\otimes...\\otimes \\ket{\\psi_n}_{A_nB_n}$, we see from equation (\\ref{eq:La}) that each $\\ket{\\psi_i}_{A_iB_i}$ should have the same Schmidt vectors as $\\ket{\\Phi^{+}}_{A_iB_i}$.  One possible solution is to take all qubit pairs to be in the same state:\\ $\\ket{\\psi}_{AB}=\\ket{\\phi}_{A_1B_1}\\otimes...\\otimes \\ket{\\phi}_{A_nB_n}$, where $\\ket{\\phi}=\\alpha\\ket{00}+\\beta\\ket{11}$.  Then, since $S_{\\mathrm{initial}}=n$, one need only solve\n\\begin{equation}\\label{eq:ent}\n n-m= -n[\\alpha^2\\log_2(\\alpha^2)+\\beta^2\\log_2(\\beta^2)]\n\\end{equation}\nfor $\\alpha$ and $\\beta$.  And the corresponding energy cost is $\\Delta E = n[1- (\\alpha+\\beta)^2\/2]$.\n\nFor example, with $m=n\/2$, one gets $\\Delta E\\simeq 0.38 m$.  This is smaller than the optimal energy cost in the one-shot setting:\\ $\\Delta E= 0.5 m$.  However, in the one-shot setting, Alice and Bob cannot prepare the state $\\ket{\\psi}_{AB}$ after extracting $m$ EPR pairs (because $\\ket{\\psi}_{AB}$ has maximal Schmidt rank).  Interestingly, however, we get a nontrivial \\textit{upper} bound on the optimal energy cost of extracting entanglement in the asymptotic setting of many copies of this system.  In the asymptotic setting, the criterion for deciding whether one bipartite entangled pure state can be transformed into another reversibly using LOCC is that the entanglement entropies are the same \\cite{NC00}.  So we see that the energy cost of distilling $m$ EPR pairs (per copy of the physical system) in the asymptotic setting will be lower than in the one shot case.\n\n\\subsection{Method II}\nA second option is to maximise the overlap of the final state of the physical system (after the entanglement has been extracted) with its ground state.  This gives a naive strategy at least.  And for Hamiltonians of the form $H_{AB}=-\\ket{\\Omega}\\bra{\\Omega}$, we get an exact answer for the optimal energy cost.\n\nAs an example, take $\\ket{\\Omega}=(1\/\\sqrt{d})\\sum_{i=1}^{d}\\ket{i}_A\\ket{i}_B$, where $d=2^n$.  Suppose that Alice and Bob extract $m$ EPR pairs using LOCC, leaving a pure state $\\ket{\\psi}$ in the physical system.  Using the majorization criterion, this can be any state with Schmidt rank up to $K=2^{n-m}$.  To minimise the energy cost, we need to find such a state having maximal overlap with $\\ket{\\Omega}$.\n\nWe may write the optimal $\\ket{\\psi}$ in its Schmidt basis as $\\sum_{i=1}^{K}\\alpha_i\\ket{a_i}_A\\ket{b_i}_B$.  Next notice that \n\\begin{equation}\n\\left(\\sum_{i=1}^{d}\\bra{i}_A\\bra{i}_B\\right)\\ket{a_i}_A\\ket{b_i}_B\\leq 1.\n\\end{equation}\nThen we have\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle\\leq \\frac{1}{\\sqrt{d}}\\sum_{i=1}^K\\alpha_i\\leq \\sqrt{\\frac{K}{d}}=2^{-m\/2}.\n\\end{equation}\nTherefore, we see that the energy cost for extracting $m$ EPR pairs is $\\Delta E = 1-1\/2^m$.\n\n\\subsection{Method III}\nA third option is to consider the trade-off between entanglement and energy numerically. For a given Hamiltonian $H$, we consider a procedure in which the system starts in the ground state $\\ket{\\Omega}$, some entanglement is extracted, and the system is left in a final state $\\ket{\\psi}$. The energy cost of this procedure is \\mbox{$\\left\\langle \\psi \\middle|H\\middle|\\psi \\right\\rangle-\\left\\langle \\Omega \\middle|H\\middle|\\Omega \\right\\rangle$}, and the extracted entropy is upper bounded by \\mbox{$\\text{Ent}(\\ket{\\Omega})-\\text{Ent}(\\ket{\\psi})$}.  In the asymptotic many-copy case, this is exactly the extracted entanglement entropy.  We denote the entanglement temperature in that case by $T_{\\mathrm{ent}}^A$.   We have that\n\\begin{align}\n\tT^{A}_{\\mathrm{ent}}\\leq \\frac{\\left\\langle \\psi \\middle|H\\middle|\\psi \\right\\rangle-\\left\\langle \\Omega \\middle|H\\middle|\\Omega \\right\\rangle}{\\text{Ent}(\\ket{\\Omega})-\\text{Ent}(\\ket{\\psi})}.\n\\end{align}\nNote that for a given amount of extracted entanglement $\\Delta S$, the one-shot entanglement temperature is lower bounded by the asymptotic-setting entanglement temperature $T_{\\mathrm{ent}}^A \\leq T_{\\mathrm{ent}}$.\n\nA given state does not necessarily give a tight bound on $T_{\\mathrm{ent}}^A$. For this we need to study the optimal trade-off between entanglement and energy, which is given by a \\emph{Pareto front}. By randomly generating states with low energy \\emph{and} low entanglement, we can numerically evaluate the above upper bound, and use this to compute the Pareto front. We describe a tensor network method for generating such samples in the appendix. In Figure~\\ref{fig:pareto} we present numerical results for two 1D spin models.\n\n\\begin{figure}\n\t\\setlength{\\figheight}{5cm}\n\t\\setlength{\\figwidth}{3.75cm}\n\t\\def.875{.875}\n\n\n\t\\! \\input{.\/HAF.tikz} \\!\\! \\input{.\/TIM.tikz} \\!\n\t\\footnotesize{\\caption{Asymptotic-case entanglement temperature ($T^A_{\\mathrm{ent}}= \\Delta E\/\\Delta S$) as a function of entropy change $\\Delta S$ for the critical Heisenberg anti-ferromagnet and the critical transverse-field Ising model, for multiple system sizes. Notice that near the ground state $T^A_{\\mathrm{ent}}\\propto \\Delta S$, which we prove is generic in the appendix.  The one-shot entanglement temperature $T_{\\mathrm{ent}}$ is lower bounded by the asymptotic  temperature $T_{\\mathrm{ent}}^A$.}\\label{fig:pareto}}\n\\end{figure}\n\n\\section{Outlook}\nWe introduced a framework to understand and quantify the energy cost of extracting entanglement from complex quantum systems.  After looking at a toy model, which illustrated the key concepts, we defined the entanglement temperature.  Then we analysed the energy cost of entanglement extraction in quantum field theories, and we saw that the energy cost of extracting entanglement depends on the spatial dimension.  Finally, we looked at some general methods to approach the problem, including numerical methods for lattice models.  \n\nQuantifying how much energy extracting $m$ EPR pairs costs in physical systems illuminates the entanglement structure of states, particularly ground states of, e.g., quantum fields.  But it also can upper bound how efficient protocols such as entanglement harvesting can be.  For general systems the optimal strategy for entanglement extraction may be hard to find.  Still, it is heartening that, at least for quantum field theories, there is a relatively simple form of the entanglement temperature.\n\nIt would be interesting to combine the ideas here with those in \\cite{AMOP17}, where transformations between entangled states are considered using an additional resource:\\ an entanglement battery.  This is a reservoir from which entanglement may be taken or deposited to facilitate state transformations, which may be impossible otherwise.  One may then ask how this theory changes when there is also an energy cost associated with using the entanglement in the battery.\n\n\\section*{Acknowledgements}\nWe would like to thank David Reeb and Robin Harper for useful discussions.\n\nTF and TJO are supported by the DFG through SFB 1227 (DQ-mat) and the RTG 1991, the ERC grants QFTCMPS and SIQS, and the cluster of excellence EXC201 Quantum Engineering and Space-Time Research. CC acknowledges support from the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013, and from the AINST Postgraduate Scholarship (John Makepeace Bennett Gift).  CB was supported by the research fund of Hanyang University (HY-2016-2237).\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Introduction}\n Let $\\, A_1 := \\mathbb{C} \\langle x, \\, y \\rangle\/(xy-yx-1) \\,$\nbe the first Weyl algebra over $\\mathbb{C}$ with canonical generators $x$ and $y$. In his classic paper \\cite{D},\nDixmier described the group $\\, {\\rm{Aut}} \\, A_1 \\,$ of automorphisms of $ A_1 $: specifically, he proved that\n$\\, {\\rm{Aut}} \\, A_1 \\,$ is generated by the following transformations\n\\begin{equation}\n\\label{Phs}\n\\Phi_p\\,:\\,(x,\\,y) \\mapsto (x,\\,y + p(x))\\ ,\\qquad \\Psi_q\\,:\\, (x,\\,y) \\mapsto (x + q(y),\\,y)\\ ,\n\\end{equation}\nwhere $\\,p(x) \\in \\mathbb{C}[x]\\,$ and $\\,q(y) \\in \\mathbb{C}[y]\\,$. Using this result of Dixmier, Makar-Limanov (see \\cite{ML1, ML2})\nshowed that $\\, {\\rm{Aut}} \\, A_1 \\,$ is isomorphic to the group $\\, G_0 \\subset {\\rm{Aut}} \\,\\mathbb{C} \\langle x, \\, y \\rangle \\,$\nof `symplectic' (i.e. preserving $\\, \\omega = xy-yx$) automorphisms of the free algebra\n$\\,\\mathbb{C} \\langle x, \\, y \\rangle \\,$: the corresponding isomorphism\n\\begin{equation}\n\\label{isML}\nG_0 \\stackrel{\\sim}{\\to} {\\rm{Aut}} \\,A_1\n\\end{equation}\nis induced by the canonical projection $\\, \\mathbb{C} \\langle x, \\, y \\rangle \\to A_1\\,$. On the other hand, the results of\n\\cite{ML1} (see, e.g., \\cite{C}) also imply that $ G_0 $ is given by the amalgamated free product\n\\begin{equation}\n\\label{Aut}\nG_0 = A *_U B\\ ,\n\\end{equation}\nwhere $\\, A \\, $ is the subgroup of symplectic affine transformations\n\\begin{equation}\n\\label{A}\n(x,\\,y) \\mapsto (ax+by+e,\\,cx+dy+f)\\ , \\quad a,\\,b, \\ldots,\nf \\in \\mathbb{C}\\ ,\\quad  ad-bc=1\\ ,\n\\end{equation}\n$\\, B \\, $ is the subgroup of triangular (Jonqui\\`eres) transformations\n\\begin{equation}\n\\label{B}\n(x,\\,y) \\mapsto (ax + q(y),\\,a^{-1}y+h)\\ , \\quad a \\in \\mathbb{C}^*,\\\nh \\in \\mathbb{C}\\ ,\\quad  q(y) \\in \\mathbb{C}[y]\\ ,\n\\end{equation}\nand $\\, U\\,$ is the intersection of $ A $ and $ B $ in $ G_0 $:\n\\begin{equation}\n\\label{U}\n(x,\\,y) \\mapsto (ax+by+e,\\,a^{-1}y+h)\\ , \\quad a \\in \\mathbb{C}^*,\\\nb,\\,e,\\, h \\in \\mathbb{C}\\ .\n\\end{equation}\nCombining \\eqref{isML} and \\eqref{Aut}, we thus get decomposition $\\,{\\rm{Aut}}\\,A_1 \\cong A *_U B \\,$,\nwhich completely describes the structure of $\\,{\\rm{Aut}} \\,A_1\\,$ as a discrete group (cf. \\cite{A}).\n\nThe aim of the present paper is to generalize the above results to the case when\n$ A_1 $ is replaced by a noncommutative domain $D$, Morita equivalent to\n$A_1$ as a $\\mathbb{C}$-algebra. This question was originally posed by Stafford in \\cite{St}\n(see {\\it loc. cit.}, p. 636). To explain why it is natural, we recall that the algebras $\\,D\\,$\nare classified, up to isomorphism, by a single integer $\\, n \\ge 0 \\,$; the corresponding isomorphism classes\nare represented by the endomorphism rings $\\,D_n := \\End_{A_1} M_n \\,$ of certain distinguished right ideals\nof $ A_1 $ and can be realized geometrically as algebras of global differential operators on rational singular\ncurves (see \\cite{K, BW1} and \\cite{BW4} for a detailed exposition). Thus\nthe Dixmier group $\\,{\\rm{Aut}} \\,A_1 = {\\rm{Aut}} \\,D_0 \\,$ appears naturally as\nthe first member in the family $\\,\\{{\\rm{Aut}} \\,D_n\\, :\\, n \\ge 0 \\}\\,$. Our aim is to describe the `higher' groups\nin this family: in particular, to give a presentation of $\\,{\\rm{Aut}} \\,D_n\\,$ for arbitrary $ n \\ge 0 $ in terms\nof amalgamated products.\n\nThe groups $ {\\rm{Aut}} \\,D_n $ for $ n \\ge 1 $ can be naturally identified with subgroups of ${\\rm{Aut}} \\,D_0$.\nTo be precise, let $ {\\rm{Pic}}\\, D $ denote the (noncommutative) Picard group of a $\\mathbb{C}$-algebra $D$. By definition,\n$ {\\rm{Pic}}\\, D $ is the group of $\\mathbb{C}$-linear Morita equivalences of the category of $ D$-modules; its elements can be\nrepresented by the isomorphism classes of invertible $D$-bimodules $\\,[P]\\,$ (see, e.g., \\cite{B}). There is a\nnatural group homomorphism $\\,\\omega_D:\\, {\\rm{Aut}}\\,D \\to {\\rm{Pic}}\\,D \\,$, taking $\\, \\sigma \\in {\\rm{Aut}}\\,D \\,$ to the class of\nthe bimodule $ [{}_1 D_{\\sigma}] $, and if $\\, D' \\,$ is a ring Morita equivalent to $ D $, with\nprogenerator $ M $, then there is a group isomorphism $\\, \\alpha_M:\\, {\\rm{Pic}} \\,D' \\stackrel{\\sim}{\\to} {\\rm{Pic}} \\,D\\,$\ngiven by $\\,[P] \\mapsto [M^* \\otimes_D P \\otimes _D M]\\,$. Thus, in our situation, for each $ n \\ge 0 $\nwe have the following diagram\n\\begin{equation}\n\\la{D1}\n\\begin{diagram}[small, tight]\n{\\rm{Aut}}  \\,D_n    & \\rTo^{\\ \\omega_{D_n}\\ }  & {\\rm{Pic}} \\,D_n \\\\\n\\dDotsto^{i_n} &                      & \\dTo_{\\alpha_{M_n}} \\\\\n{\\rm{Aut}} \\,D_0    & \\rTo^{\\ \\omega_{D_0}\\ }  & {\\rm{Pic}} \\,D_0 \\\\\n\\end{diagram}\n\\end{equation}\nwhere the vertical map $ \\alpha_{M_n} $ is an isomorphism and the two horizontal maps are injective.\nMoreover, since $ D_0 = A_1 $, a theorem of Stafford (see \\cite{St}) implies that $ \\omega_{D_0} $ is actually\nan isomorphism. Inverting this isomorphism, we define the embedding\n$\\, i_n:\\,{\\rm{Aut}} \\,D_n \\,\\,\\hookrightarrow\\,\\, {\\rm{Aut}} \\,D_0 \\,$, which makes \\eqref{D1} a commutative diagram.\n\nRecall that we defined $ G_0 $ to be the automorphism group of the free algebra $\\,\\mathbb{C} \\langle x, \\, y \\rangle\\,$\npreserving $ [x,\\,y] $. Now, for $\\, n > 0 \\,$, we introduce the groups $ G_n $ geometrically,\nin terms of a natural action of $ G_0 $ on the {\\it Calogero-Moser spaces}\\, (see \\cite{W})\n\\begin{equation*}\n\\label{e1}\n{\\mathcal C}_n := \\{\\,(X,\\,Y) \\in \\mbox{\\tt Mat}_n(\\mathbb{C}) \\times\n\\mbox{\\tt Mat}_n(\\mathbb{C})\\ :\\ \\mbox{\\tt rk}([X,\\,Y] + I_n) = 1 \\,\\}\/\\,\\mbox{\\tt PGL}_n(\\mathbb{C})\\ ,\n\\end{equation*}\nwhere $ \\mbox{\\tt PGL}_n(\\mathbb{C}) $ operates on matrices $ (X,\\,Y) $ by simultaneous conjugation.\nThe action of $ G_0 $ on $ \\mathcal{C}_n $ is given by\n\\begin{equation}\n\\label{at}\n(X,\\,Y) \\mapsto (\\sigma^{-1}(X),\\, \\sigma^{-1}(Y))\\ , \\quad \\sigma \\in G_0\\ ,\n\\end{equation}\nwhere $ \\sigma^{-1}(X) $ and $ \\sigma^{-1}(Y) $ are the noncommutative polynomials\n$\\,\\sigma^{-1}(x) \\in \\mathbb{C} \\langle x, \\, y \\rangle \\,$ and $\\,\\sigma^{-1}(y) \\in \\mathbb{C} \\langle x, \\, y \\rangle \\,$\nevaluated at $(X,Y)$. It is known that $ {\\mathcal C}_n $ is a smooth affine algebraic variety\nof dimension $2n$, equipped with a natural symplectic structure, and it is easy to check that\n$G_0$ preserves that structure. Now, a theorem of Wilson and the first author (see \\cite{BW}) implies that\n\\eqref{at} is a transitive action for all $ n \\ge 0 $.\nWe define the groups $ G_n $ to be the stabilizers of points of $ \\mathcal{C}_n $\nunder this action: precisely, for each $ n \\ge 0 $, we fix a basepoint $ (X_0,\\,Y_0) \\in\n{\\mathcal C}_n $, with\n\\begin{equation*}\n\\label{base}\nX_0 = \\sum_{k=1}^{n-1} E_{k+1, k}\n\\quad , \\qquad\nY_0 = \\sum_{k=1}^{n-1} \\,(k-n)\\, E_{k, k+1}\\ ,\n\\end{equation*}\nwhere $ E_{i,j} $ stands for the elementary matrix with $(i,j)$-entry $1$, and let\n\\begin{equation*}\n\\label{gn}\nG_n := \\mbox{\\tt Stab}_{G_0}(X_0, Y_0)\\ ,\\quad n \\ge 0 \\ .\n\\end{equation*}\n\n\nThe following result can be viewed as a generalization of the above-mentioned theorem\nof Makar-Limanov; in a slightly different form, it has already appeared in\n\\cite{BW4} (cf. {\\it loc. cit.}, p.~120; see also \\cite{W2}).\n\\begin{theorem}\n\\label{T1}\nThere is a natural isomorphism of groups $\\,G_n \\stackrel{\\sim}{\\to} {\\rm{Aut}} \\, D_n\\,$.\n\\end{theorem}\n\\noindent\nSpecifically, we have group homomorphisms\n\\begin{equation*}\n\\label{ison}\nG_n \\,\\,\\hookrightarrow\\,\\, G_0 \\stackrel{\\sim}{\\to} {\\rm{Aut}} \\,A_1 \\stackrel{i_n}{\\hookleftarrow} {\\rm{Aut}} \\,\\mathcal{D}_n\\ ,\n\\end{equation*}\nwhere the first map is the canonical inclusion, the second is the Makar-Limanov isomorphism\n\\eqref{isML} and $ i_n $ is the embedding defined by \\eqref{D1}. We claim that the image\nof $ i_n $ coincides with the image of $ G_n $, which gives the required isomorphism.\n\nTheorem~\\ref{T1} is a simple consequence of the main results of \\cite{BW}: in fact, it is shown in \\cite{BW}\nthat there is a natural $G_0$-equivariant bijection (called the Calogero-Moser correspondence)\nbetween $\\, \\bigsqcup_{n \\ge 0} {\\mathcal C}_n \\,$ and the space of isomorphism classes of right\nideals of $ A_1 $. Under this bijection, the points $ (X_0, Y_0) \\in {\\mathcal C}_n $\ncorrespond precisely to the classes of the ideals $ M_n $.\n\nWe will use Theorem~\\ref{T1} to give a geometric presentation for the groups $ {\\rm{Aut}} \\,D_n $.\nTo this end, we associate to each space $ \\mathcal{C}_n $ a graph $ \\Gamma_n $ consisting of orbits of\ncertain subgroups of $ G_0 $ and identify $ G_n $ with the {\\it fundamental group}\n$\\, \\pi_1({\\mathbf \\Gamma}_n, \\ast)\\, $ of a graph of groups $ {\\mathbf \\Gamma}_n $ defined\nby the stabilizers of points of those orbits in $ \\Gamma_n $. The Bass-Serre theory of groups\nacting on graphs \\cite{Se} will give\nthen an explicit formula for $\\, \\pi_1({\\mathbf \\Gamma}_n, \\ast)\\, $ in terms of generalized\namalgamated products (see \\eqref{fgr2} below).\n\nTo define the graph $ \\Gamma_n $ we take the subgroups $A$, $B$ and $U$ of $G_0$\ndefined by the transformations \\eqref{A}, \\eqref{B} and \\eqref{U}. Restricting the action of $ G_0 $ on $ \\mathcal{C}_n $\nto these subgroups, we let $ \\Gamma_n $ be the oriented bipartite graph, with vertex and edge sets\n\\begin{equation}\n\\label{gamman}\n\\mbox{\\tt Vert}(\\Gamma_{n}) := (A \\backslash \\mathcal{C}_n)\\,\\bigsqcup\\, (B \\backslash \\mathcal{C}_n) \\ ,\\quad\n\\mbox{\\tt Edge}(\\Gamma_{n}) := U \\backslash \\mathcal{C}_n \\ ,\n\\end{equation}\nand the incidence maps $\\,\\mbox{\\tt Edge}(\\Gamma_{n}) \\to \\mbox{\\tt Vert}(\\Gamma_{n})\\,$ given by the canonical\nprojections $\\, i: U \\backslash \\mathcal{C}_n \\to A \\backslash \\mathcal{C}_n \\,$\nand $\\,\\tau: U \\backslash \\mathcal{C}_n \\to B \\backslash \\mathcal{C}_n  \\,$. Since the elements of\n$A$ and $B$ generate $ G_0 $ and $ G_0 $ acts transitively on each $ \\mathcal{C}_n $, the graph $ \\Gamma_n $ is connected.\n\nNow, on each orbit in $ A \\backslash \\mathcal{C}_n $ and $\\,B \\backslash \\mathcal{C}_n\\,$\nwe choose a basepoint and elements $\\, \\sigma_A \\in G_0 \\,$ and $\\,\\sigma_B \\in G_0\\, $ moving these basepoints\nto the basepoint $ (X_0,\\,Y_0) $ of $ {\\mathcal C}_n $. Next, on each $U$-orbit $\\,\\mathcal{O}_U \\in\nU \\backslash \\mathcal{C}_n \\,$ we also choose a basepoint and an element $\\,\\sigma_U \\in G_0 \\,$ moving this\nbasepoint to $ (X_0,\\,Y_0) $ and such that $\\,\t\\sigma_U \\in \\sigma_A A \\,\\cap \\,\\sigma_B B\\,$,\nwhere $ \\sigma_A $ and $ \\sigma_B $ correspond to the (unique) $A$- and $B$-orbits containing $ \\mathcal{O}_U $.\nUsing a standard construction in the Bass-Serre theory (see \\cite{Se}, Sect.~5.4), we then assign to the\nvertices and edges of $ \\Gamma_n $ the stabilizers $\\,A_\\sigma = G_n \\cap \\sigma A \\sigma^{-1} \\,$,\n$\\,B_\\sigma = G_n \\cap \\sigma B \\sigma^{-1} \\,$, $\\,U_\\sigma = G_n \\cap \\sigma U \\sigma^{-1} \\,$  of the corresponding\nelements $ \\sigma $ in the graph of right cosets of $ G_0 $ under the action of $ G_n $.\nThese data together with natural group homomorphisms $\\, a_\\sigma: U_\\sigma \\hookrightarrow A_\\sigma  \\,$\nand $\\, b_\\sigma:\\,U_\\sigma \\hookrightarrow B_\\sigma \\,$ define a graph of groups $ {\\mathbf \\Gamma}_n $ over\n$ \\Gamma_n $, and its fundamental group $\\, \\pi_1({\\mathbf \\Gamma}_n,\\,T)\\,$\nrelative to a maximal tree $\\,T \\subseteq \\Gamma_n \\,$ has canonical presentation (see \\cite{Se}, Sect.~5.1):\n\\begin{equation}\n\\label{fgr2}\n\\pi_1({\\mathbf \\Gamma}_n,\\,T) = \\frac{A_\\sigma \\ast_{\\,U_\\sigma} B_\\sigma \\ast \\, \\ldots\\, \\ast \\langle\\, \\mbox{\\tt Edge}(\\Gamma_n \\setminus T)\\,\\rangle}{(\\,e^{-1} a_\\sigma(g)\\, e = b_{\\sigma}(g)\\, :\\, \\forall\\,e \\in\n\\mbox{\\tt Edge}(\\Gamma_n \\setminus T),\\, \\forall\\, g \\in U_\\sigma\\,)}\\ .\n\\end{equation}\nIn \\eqref{fgr2}, the amalgams $\\, A_\\sigma \\ast_{\\,U_\\sigma} B_\\sigma \\ast \\, \\ldots \\,$ are taken along the stabilizers\nof edges of the tree $ T $, while $\\, \\langle\\, \\mbox{\\tt Edge}(\\Gamma_n \\setminus T)\\,\\rangle\\,$ denotes\nthe free group based on the set of edges of $ \\Gamma_n $ in the complement of $T$.\n\nOur main observation is the following.\n\\begin{theorem}\n\\label{T2}\nFor each $ n \\ge 0 $, the group $ G_n $ is isomorphic to $\\, \\pi_1({\\mathbf \\Gamma}_n,\\,T)\\,$. In particular,\n$ G_n $ has an explicit presentation of the form \\eqref{fgr2}.\n\\end{theorem}\n\\begin{proof}\nOne can prove Theorem~\\ref{T2} using the standard Bass-Serre theory (as exposed in \\cite{Se}, Ch~I, Sect.~5,\nor \\cite{DD}, Ch.~I, Sect.~9). However, it seems that the more economic and intuitively clearer proof is\nbased on topological arguments: namely, an abstract version of Van Kampen's Theorem, which we are now\ngoing to explain.\n\nLet $\\,\\mathfrak{G}_n := \\mathcal{C}_n \\rtimes G_0 \\,$ denote the (discrete) transformation groupoid\ncorresponding to the action of $ G_0 $ on $ \\mathcal{C}_n $. The canonical projection $\\,p:\\, \\mathfrak{G}_n \\to G_0 \\,$ is then\na connected covering of groupoids\\footnote{We refer to \\cite{M}, Ch.~3, for the theory of coverings of groupoids.},\nwhich maps identically the vertex group of $ \\mathfrak{G}_n $ at $\\, (X_0,\\,Y_0) \\in \\mathcal{C}_n \\,$ to the subgroup\n$ G_n \\subseteq G_0 $. Now, each of the subgroups $A$, $B$ and $U$ of $ G_0 $ can be lifted\nto $ \\mathfrak{G}_n\\,$: $\\,p^{-1}(A) = \\mathfrak{G}_n \\times_{G_0} A \\,$,  $\\,p^{-1}(B) = \\mathfrak{G}_n \\times_{G_0} B \\,$\nand $\\,p^{-1}(U) = \\mathfrak{G}_n \\times_{G_0} U \\,$, and these fibred products\nare naturally isomorphic to the subgroupoids $\\, \\mathfrak{A}_n := \\mathcal{C}_n \\rtimes A $, $\\, \\mathfrak{B}_n := \\mathcal{C}_n \\rtimes B \\,$\nand $\\, \\mathfrak{U}_n := \\mathcal{C}_n \\rtimes U \\,$ of $ \\mathfrak{G}_n$, respectively. Since the coproducts in the category of\ngroups coincide with coproducts in the category of groupoids and the latter can be lifted through coverings\n(see \\cite{O}, Lemma~3.1.1), the decomposition \\eqref{Aut} implies\n\\begin{equation}\n\\la{vank}\n\\mathfrak{G}_n = \\mathfrak{A}_n *_{\\mathfrak{U}_n} \\mathfrak{B}_n\\ ,\\quad \\forall\\,n \\ge 0\\ .\n\\end{equation}\nNote that, unlike $ \\mathfrak{G}_n$, the groupoids $ \\mathfrak{A}_n $, $ \\mathfrak{B}_n $ and $ \\mathfrak{U}_n $ are not transitive\n(if $\\, n \\ge 1 $), so \\eqref{vank} can be viewed as an analogue of the Seifert-Van Kampen Theorem\nfor non-connected spaces (see, e.g., \\cite{Ge}, Ch.~6, Appendix). As in the topological situation, computing\nthe fundamental (vertex) group from \\eqref{vank} amounts to contracting the connected components (orbits) of\n$ \\mathfrak{A}_n $ and $ \\mathfrak{B}_n $ to points (vertices) and $ \\mathfrak{U}_n $ to edges. This defines a graph which is exactly\n$ \\Gamma_n $. Now, choosing basepoints in each of the contracted components and assigning the fundamental groups\nat these basepoints to the corresponding vertices and edges defines a graph of groups (see \\cite{HMM}, p.~46).\nBy {\\it loc.~cit.}, Theorem~3, this graph of groups is (conjugate) isomorphic to the graph $ {\\mathbf \\Gamma}_n $\ndescribed above, and our group $ G_n $ is isomorphic to $\\, \\pi_1({\\mathbf \\Gamma}_n,\\,T)\\,$.\n\\end{proof}\nTheorems~\\ref{T1} and~\\ref{T2} reduce the problem of describing the groups\n$ {\\rm{Aut}} \\,D_n  $ to a purely geometric problem of describing the structure of the orbit spaces\nof  $A$ and $B$ and $U$ on the Calogero-Moser varieties $ {\\mathcal C}_n$. Using the earlier results\nof \\cite{W} and \\cite{BW} and some basic invariant theory, one can obtain much\ninformation about these orbits (and thence about the groups $ G_n $). In particular, the graphs\n$ \\Gamma_n $ can be completely described for small $ n $; it turns that $ \\Gamma_n $ is a finite tree for\n$\\,n = 0,\\,1,\\, 2\\,$, but has infinitely many cycles for $ n \\ge 3 $ (see examples below).\n\nWe now explain the origin of $ \\Gamma_n $. It turns out that these graphs can be\nrealized as quotient graphs of a certain `universal' tree $ \\Gamma $ on which all\nthe groups $ {\\rm{Aut}} \\,D_n $ naturally act. Our construction of $ \\Gamma $ is motivated\nby algebraic geometry: specifically, a known application of the\nBass-Serre theory in the theory of surfaces (see, e.g., \\cite{GD}, \\cite{Wr}). In that approach,\nthe automorphism group of an affine surface $ S $ is described via its action on a tree whose\nvertices correspond to certain (admissible) projective compactifications of $ S $. Following\nthe standard (by now) philosophy in noncommutative geometry (see, e.g., \\cite{SV}), we may\nthink of our algebra $ D $ as the coordinate ring of a `noncommutative affine surface';\na `projective compactification' of $ D $ is then determined by a choice of filtration. Thus, we will define\n$ \\Gamma $ by taking as its vertices a certain class of filtrations on the algebra $D$.\nIt turns out that these filtrations can be naturally parametrized by an infinite-dimensional\n{\\it adelic Grassmannian} $\\,\\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ introduced in \\cite{W1} and studied in \\cite{W, BW, BW3}\n(in particular, we rely heavily on results of \\cite{BW3}). Our contruction is close in spirit to\nSerre's classic application of Bruhat-Tits trees for computing arithmetic subgroups of $ \\SL_2(\\mathbb{K}) $\nover the function fields of smooth curves (see \\cite{Se}, Chap.~II, \\S\\,2); however, at the moment,\nwe are not aware of any direct connection.\n\nWe begin by briefly recalling the definition of $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $.  Let $ \\mathbb{C}[z] $ be the polynomial ring in one\nvariable $z$. For each $\\, \\lambda \\in \\mathbb{C} \\,$, we choose a {\\it $\\lambda$-primary} subspace in $ \\mathbb{C}[z] $,\nthat is, a $\\mathbb{C}$-linear subspace $\\, V_{\\lambda} \\subseteq \\mathbb{C}[z] \\,$ containing a power of the maximal ideal\n$\\,\\mathfrak{m}_\\lambda \\,$ at $\\, \\lambda $. We suppose that $\\, V_{\\lambda} = \\mathbb{C}[z] \\, $\nfor all but finitely many $\\, \\lambda$'s.  Let $\\, V = \\bigcap_{\\lambda} V_{\\lambda} \\,$\n(such a subspace $\\, V \\,$ is called {\\it primary decomposable} in  $ \\mathbb{C}[z] $) and, finally, let\n$$\nW = \\prod_{\\lambda}\\, (z - \\lambda)^{-n_{\\lambda}} \\, V \\subset\n\\mathbb{C}(z) \\, ,\n$$\nwhere $\\, n_{\\lambda} \\,$ is the codimension of $\\, V_{\\lambda} \\,$\nin $\\, \\mathbb{C}[z] \\,$. By definition, $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ consists of all subspaces\n$\\, W \\subset \\mathbb{C}(z) \\,$ obtained in this way.  For each\n$\\, W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ we set\n\\begin{equation*}\n\\label{aw}\nA_W := \\{f \\in \\mathbb{C}[z] \\,:\\, f W \\subseteq W \\} \\,.\n\\end{equation*}\nTaking $\\,{\\rm{Spec}}\\,$ of $\\, A_W \\,$ gives then a rational curve $X$,\nthe inclusion $\\, A_W \\,\\,\\hookrightarrow\\,\\, \\mathbb{C}[z] \\,$ corresponds to normalization\n$\\, \\pi : \\mathbb{A}_{\\mathbb{C}}^1 \\to X \\,$ (which is set-theoretically a bijective map),\nand the $\\, A_W$-module $\\, W \\,$ defines\na rank 1 torsion-free coherent sheaf $\\, \\mathscr{L} \\,$ over $\\, X \\,$. In this way,\nthe points of $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ correspond bijectively to isomorphism classes of triples\n$\\, (\\pi, X, \\mathscr{L}) \\,$ (see \\cite{W1}).\n\nNow, following \\cite{BW}, for $\\, W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ we define\\footnote{In geometric terms, $ D(W) $\ncan be thought of as the ring $ D_{\\mathscr{L}}(X) $ of twisted differential operators on $X$ with\ncoefficients in $ \\mathscr{L} $.}\n\\begin{equation}\n\\label{dw}\nD(W) := \\{\\mathcal{D} \\in \\mathbb{C}(z)[\\partial_z] \\,:\\, \\mathcal{D} W \\subseteq W \\} \\, ,\n\\end{equation}\nwhere $\\, \\mathbb{C}(z)[\\partial_z] \\,$ is the ring of rational differential operators\nin the variable $z$. This last ring carries two natural filtrations:\nthe standard filtration, in which both generators $ z $ and $ \\partial_z $\nhave degree $1$, and  the differential filtration, in which $\\,\\deg(z) = 0\\,$\nand $\\,\\deg(\\partial_z) = 1\\,$. These filtrations induce two different filtrations\non the algebra $ D(W) $, which we denote by $ \\{D_{\\bullet}^A(W)\\} $ and $ \\{D_{\\bullet}^B(W)\\} $\nrespectively.\n\nNow, let $D$ be a fixed domain Morita equivalent to $ A_1 $. Following \\cite{BW3}, we consider\nthe set\\footnote{More generally, we may think of $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ as a groupoid, in which the objects are\nthe $W$'s and the arrows are given by the algebra isomorphisms $ D(W) \\to D(W') $. For $ D = D(W) $,\nthe set $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}}(D) $ is then a costar in $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}}  $, consisting of all arrows with target at $W$.\nIn \\cite{BW3}, this set was denoted by $\\,\\mbox{\\tt Grad}\\,D\\,$.} $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\,$\nof all algebra isomorphisms $\\,\\sigma_W: D(W) \\to D \\,$, where $ W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $\n(more precisely, $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\,$ is the set of all pairs $\\, (W, \\,\\sigma_W) \\,$, where\n$\\, W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ and $\\, \\sigma_W \\,$ is an isomorphism as above). Each\n$\\, \\sigma_W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) $ maps the two distinguished filtrations $ \\{D_{\\bullet}^A(W)\\} $ and\n$ \\{D_{\\bullet}^B(W)\\} $ into the algebra $ D $: we call their images the {\\it admissible}\nfiltrations on $D$ of type $A$ and type $B$, respectively. Let $ {\\mathbb P}_{A}(D) $ and $ {\\mathbb P}_{B}(D) $ denote\nthe sets of all such filtrations coming from various $\\,\\sigma_W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\\,$.\nBy definition, we have then two natural projections\n\\begin{equation}\n\\la{prc}\n{\\mathbb P}_{A}(D)  \\stackrel{\\pi_A}{\\longleftarrow} \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\stackrel{\\pi_B}{\\longrightarrow} {\\mathbb P}_B(D)\\ .\n\\end{equation}\nWe say that $\\,(W, \\sigma_W) \\,$ and $\\,(W', \\sigma_W') \\,$ are {\\it equivalent} in $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\,$\nif their images under $ \\pi_A $ and $ \\pi_B $ coincide. Writing $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\/\\!\\sim $ for the set of\nequivalence classes under this relation, we define an oriented graph $ \\Gamma $ by\n$$\n\\mbox{\\tt Vert}(\\Gamma) := {\\mathbb P}_A(D) \\,\\bigsqcup\\, {\\mathbb P}_B(D)\\ ,  \\quad \\mbox{\\tt Edge}(\\Gamma) := \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\/\\!\\sim\\ ,\n$$\nwith incidence maps $\\,\\mbox{\\tt Edge}(\\Gamma) \\to \\mbox{\\tt Vert}(\\Gamma)\\,$ induced by the\nprojections \\eqref{prc}. Observe that the group $ {\\rm{Aut}} \\,D $ acts naturally on the set $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\\,$\n(by composition), and this action induces an action of $ {\\rm{Aut}} \\,D $ on the graph $ \\Gamma $ via \\eqref{prc}.\nWe write $\\, {\\rm{Aut}} \\, D \\backslash \\Gamma\\,$ for the corresponding quotient graph.\n\n\\begin{theorem}\n\\la{T3}\n\n$(a)$ $ \\Gamma $ is a tree, which is independent of $ D $ (up to isomorphism).\n\n$(b)$ For each $ n \\ge 0 $, the graph $ {\\rm{Aut}} \\, D_n \\backslash \\Gamma $ is naturally isomorphic to $ \\Gamma_n $.\n\n\\end{theorem}\n\nTheorem~\\ref{T3} can be viewed as a generalization of the main results of \\cite{BW3}.\nIndeed, this last paper is concerned with a description of the maximal abelian ad-nilpotent ({\\it mad})\nsubalgebras of $ D_n \\,$: its main theorems (see {\\it loc. cit.}, Theorem~1.5 and Theorem~1.6) say\nthat the space $ \\mbox{\\rm Mad}(D_n) $ of all mad subalgebras of $ D_n $ is independent of $ D_n $\nand its quotient modulo the natural action of $ {\\rm{Aut}} \\,D_n $ is isomorphic to the orbit space\n$ B \\backslash \\mathcal{C}_n $. Now, it is easy to\nsee that every mad sublagebra defines an admissible filtration on $ D_n $ of type $B$, and conversely\nthe zero degree component of every filtration of type $B$ is a mad subalgebra of $D_n$. Thus, we have\na natural bijection $\\, {\\mathbb P}_B(D_n) \\cong \\mbox{\\rm Mad}(D_n) $, which is equivariant under the action\nof $ {\\rm{Aut}} \\,D_n $. This implies that $ {\\mathbb P}_B(D_n) $ does not depend on $ D_n $, which is part of\nTheorem~\\ref{T3}$(a)$, and\n$$\n{\\rm{Aut}} \\, D_n \\backslash {\\mathbb P}_B(D_n)\\,\\cong \\,{\\rm{Aut}} \\, D_n \\backslash \\mbox{\\rm Mad}(D_n) \\,\\cong\\,\nB \\backslash \\mathcal{C}_n\\ ,\n$$\nwhich is part of Theorem~\\ref{T3}$(b)$. In fact, the entire Theorem~\\ref{T3} can be proved using the techniques\nof \\cite{BW3}. We should also mention that for $ D = A_1(k) $ our construction of the tree $ \\Gamma $\nagrees with the one given in \\cite{A}.\n\n\\vspace{1ex}\n\nWe now look at examples of the graphs $ \\Gamma_n $ and groups $G_n$ for small $n$.\nFor $ n = 0 $, the space $ \\mathcal{C}_0 $ is just a point, and so are a fortiori\nits orbit spaces. The graph $ \\Gamma_0 $ is thus a segment, and the corresponding graph of\ngroups $ {\\mathbf \\Gamma}_0 $ is given by $\\,[\\,A \\stackrel{U}{\\longrightarrow} B\\,] \\,$.\nFormula \\eqref{fgr2} then says that $\\,G_0 = A \\ast_U B \\,$, which agrees, of course, with\nthe Makar-Limanov isomorphism \\eqref{Aut}.\n\nFor $ n =1 $, we have $\\, {\\mathcal C}_1 \\cong \\mathbb{C}^2 $, with $ (X_0, Y_0) $ corresponding\nto the origin. Since each of the groups\n$A$, $B$ and $U$ contains translations $\\,(x,y) \\mapsto (x+a, y+b) \\,$, $\\,a,b \\in \\mathbb{C} $,\nthey act transitively on $ {\\mathcal C}_1 $. So again $ \\Gamma_1 $ is just the segment, and\n$ {\\mathbf \\Gamma}_1 $ is given by $\\,[\\,A_1 \\stackrel{U_1}{\\longrightarrow} B_1\\,]\\,$,\nwhere $\\, A_1 := G_1 \\cap A \\,$, $\\, B_1 := G_1 \\cap B \\,$ and $\\, U_1 := G_1 \\cap U \\,$.\nSince, by definition, $ G_1 $ consists of all $\\, \\sigma \\in G_0 \\,$\npreserving $\\,(0, 0)\\,$, the groups $A_1$, $ B_1 $ and $U_1 $ are obvious:\n\\begin{eqnarray}\nA_1\\!\\!\\!\\!\\! &:& (x,\\,y) \\mapsto (ax + by,\\,cx+dy)\\ , \\quad a,\\,b,\\,c,\\,d \\in\n\\mathbb{C} \\ ,\\ ad-bc=1\\ ,\n\\nonumber \\\\\nB_1\\!\\!\\!\\!\\! &:& (x,\\,y) \\mapsto (ax + q(y),\\,a^{-1} y)\\ , \\quad a \\in \\mathbb{C}^*\\ ,\\\nq \\in \\mathbb{C}[y]\\ ,  \\ q(0) = 0\\ , \\nonumber \\\\\nU_1 \\!\\!\\!\\!\\! &:& (x,\\,y) \\mapsto (ax + by,\\,a^{-1} y)\\ , \\quad a \\in \\mathbb{C}^*\\ ,\\\nb\\in \\mathbb{C}\\ .\n\\nonumber\n\\end{eqnarray}\nIt follows from \\eqref{fgr2} that $\\, G_1 = A_1 \\ast_{U_1} B_1 \\,$.\n\nFor $ n = 2 $, the situation is already more interesting. A simple calculation shows that\n$ U $ has three orbits in $ {\\mathcal C}_2$: two closed orbits of dimension $3$ and one open orbit\nof dimension $4$. Moreover, the $B$-orbits coincide with the $U$-orbits. Combinatorially, this means\nthat the group $A$ acts transitively, and the graph $ \\Gamma_2 $ is a tree with one nonterminal and\nthree terminal vertices corresponding to the $A$-orbit and the $B$-orbits, respectively.\nIn this case, the graph of groups ${\\mathbf \\Gamma_2} $ is given by\n$$\n\\begin{diagram}\n&                        &                       &  G_{2,y} \\rtimes \\mathbb{C}^*    \\\\\n&                        & \\ruTo^{\\mathbb{C}^*}     &               \\\\\n&\\mathbb{C}^*                   & \\rTo^{\\mathbb{Z}_{2}\\quad} & G^{(1)}_{2, y} \\rtimes \\mathbb{Z}_{2} \\\\\n&                         & \\rdTo_{\\mathbb{C}^*}     &               \\\\\n&                         &                       & G_{2,x} \\rtimes \\mathbb{C}^*    \\\\\n\\end{diagram}\n$$\nwhere $ G_{2,x} $ and $ G_{2,y} $ are the subgroups of $ G_0 $ consisting of all transformations\n$ \\Phi_p $ and $ \\Psi_q $ (see \\eqref{Phs}),\nwith $ p \\in \\mathbb{C}[x] $ and $ q \\in \\mathbb{C}[y] $ satisfying $ p(0) = p'(0) = 0 $ and\n$ q(0) = q'(0)= 0 $ respectively, and $\\, G^{(1)}_{2, y} := \\{\\,\\Phi_{-x}\\,\\Psi_q \\,\\Phi_{ x} \\in G_0 \\ :\\ q \\in \\mathbb{C}[y]\\ , \\\nq(\\pm 1) = 0\\,\\}\\,$. Formula \\eqref{fgr2} yields the presentation\n$$\n\\,G_2 = (G_{2,x} \\rtimes \\mathbb{C}^*) \\ast_{\\mathbb{C}^*}\n(G_{2,y} \\rtimes \\mathbb{C}^*) \\ast_{\\mathbb{Z}_2} (G^{(1)}_{2, y}\n\\rtimes \\mathbb{Z}_2) \\, .\n$$\nIn particular, $ G_2 $ is generated by its subgroups $\\, G_{2,x} $, $\\, G_{2,y} $, $\\, G^{(1)}_{2, y}$\nand $ \\mathbb{C}^* $.\n\nNow, for $ n = 3 $, the structure of the graph $ \\Gamma_3 $ and the group $ G_3 $ is much more complicated.\nThe graph $ \\Gamma_3 $ is  {\\it not}\na tree: in fact, it has infinitely many circuits. Nevertheless, the group $ G_3 $ can still be described\nexplicitly:\n$$\nG_3 = \\frac{\\pi_1(T_3,\\,G_3) \\,\\star\\,\n\\langle E_+(\\Gamma_3 \\setminus\nT_3)\\rangle}{(\\,e^{-1} \\alpha_e(g)\\, e = \\alpha_{e^*}(g)\\, :\\, \\forall\\,e \\in\nE_+(\\Gamma_3 \\setminus T_3),\\, \\forall\\, g \\in G_e\\,)} \\ ,\n$$\nwhere $ T_3 $ is a (maximal) tree in $ \\Gamma_3 $ given in Figure \\ref{T3}, $\\,\\pi_1(T_3,\\,G_3)\\,$ is the\ncorresponding tree product of stabilizer groups, and the complement graph $\\,\\Gamma_3 \\setminus T_3 \\,$\nis shown in Figure \\ref{graph4}.\n\nWe would like to end this paper with some questions and conjectures.\n\n\\medskip\n\n{1.} By \\cite{ML1}, it is known that $ G_0 $ is isomorphic to the group $\\,\\mbox{\\rm SAut}\\,\\mathbb{A}_{\\mathbb{C}}^2 \\,$ of\n{\\it symplectic} automorphisms of the affine plane $\\,\\mathbb{A}_{\\mathbb{C}}^2\\,$ (as in the case of the Weyl algebra,\nthe isomorphism $\\,G_0 \\cong \\mbox{\\rm SAut}\\, \\mathbb{A}_{\\mathbb{C}}^2 \\,$ is induced by the canonical projection\n$\\, \\mathbb{C}\\langle x,\\,y \\rangle \\to \\mathbb{C}[x,\\,y] $). Thus, the groups $ G_n $ can be naturally identified\nwith subgroups of $\\,{\\rm{Aut}}\\,\\mathbb{A}_{\\mathbb{C}}^2 \\,$. Do these last subgroups have a geometric interpretation?\n\n{2.}\nIn this paper, we have described the structure of $ G_n $ and $ {\\rm{Aut}} \\,D_n $ as discrete groups.\nHowever, these two groups carry natural {\\it algebraic} structures and can be viewed as\ninfinite-dimensional algebraic groups (in the sense of Shafarevich \\cite{Sh}). Despite being isomorphic\nto each other as discrete groups, they are not isomorphic as algebraic groups\n(for $n=0$, this phenomenon was observed in \\cite{BW}.) A natural question is to explicitly describe\nthe algebraic structures on $ G_n $ and $ {\\rm{Aut}} \\,D_n $; in particular, to compute the corresponding\n(infinite-dimensional) Lie algebras. The last question was an original motivation for our work. For\n$ G_0 $, the answer is known (see \\cite{G}).\n\n{3.}\nCompute the homology of the groups $ G_n $ for all $n$. Again, for $ n = 0 \\,$, the answer is known\n(see \\cite{Al}): $\\,H_*(G_0,\\,\\mathbb{Z}) \\cong H_*(\\SL_2(\\mathbb{C}),\\,\\mathbb{Z})\\,$. One may wonder whether the groups\n$ H_*(G_n,\\,\\mathbb{Z}) $ are strong enough invariants to distinguish the algebras $ D_n $\nup to isomorphism. Unfortunately, the answer is `no': in fact, it follows from our description of $ G_1 $ that\n$\\,H_*(G_1,\\,\\mathbb{Z}) \\cong H_*(\\SL_2(\\mathbb{C}),\\,\\mathbb{Z})\\,$. However, for $ n \\ge 2 $, it seems that the groups $ H_*(G_n,\\,\\mathbb{Z}) $\nare neither isomorphic to $\\,H_*(\\SL_2(\\mathbb{C}),\\,\\mathbb{Z})\\,$ nor to each other, so they may provide\ninteresting invariants.\n\n{4.}\nFinally, we would like to propose an extension of the well-known {\\it Dixmier Conjecture}\nfor $ A_1 \\,$ (see \\cite{D}, Probl\\`eme~11.1) to the class of Morita equivalent algebras.\nWe recall that if $ D $ is a domain Morita equivalent to $ A_1 $, then there is a unique\ninteger $ n \\ge 0 $ such that $\\,D \\cong D_n \\,$, where $\\,D_n \\,$ is the endomorphism ring of\nthe right ideal $\\, M_n = x^{n} A_1 + (y+n x^{-1})\\,A_1 $. For two unital $\\mathbb{C}$-algebras $A$ and $B$,\nwe denote by $\\,\\Hom\\, (A,\\,B)\\,$ the set of all unital $\\mathbb{C}$-algebra homomorphisms $\\,A \\to B \\,$.\n\\begin{conjecture}\n\\la{DC}\nFor all $\\, n, m \\ge 0 \\,$, we have\n\\begin{equation*}\n\\Hom\\, (D_n,\\,D_m) = \\left\\{\n\\begin{array}{lll}\n\\emptyset & \\mbox{if}\\quad n \\ne m \\\\*[1ex]\n{\\rm{Aut}} \\,D_n & \\mbox{if} \\quad n = m\n\\end{array}\n\\right.\n\\end{equation*}\n\\end{conjecture}\n\\noindent\nFormally, Conjecture~\\ref{DC} is a strengthening of the Dixmier Conjecture for $ A_1 $: in fact,\nin our notation, the latter says that $\\,\\Hom\\, (D_0,\\,D_0) = {\\rm{Aut}} \\,D_0\\,$. Does actually the Dixmier\nConjecture imply Conjecture~\\ref{DC}?\n\n\n\\subsection*{Acknowledgments}\nWe are grateful to J.~Alev, V.~Bavula, O.~Chalykh, K.~Vogtmann, D.~Wright, G.~Wilson and E.~Zelmanov for interesting discussions,\nquestions and comments. We would also like to thank D.~Wright for providing us with reference \\cite{Wr},\nwhich turned out to be very useful, and G.~Wilson for sending us a copy of Quillen's private notes on trees and amalgams.\nThis work was partially supported by NSF grant DMS 09-01570.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLocal volatility model was introduced by \\cite{Dupire:94} and \\cite{derman\/kani:94} as a natural extension of the celebrating Black-Scholes model to take into account an existence of option smile. It is able to exactly replicate the local volatility function $\\sigma(T,K)$ where $K,T$ are the option strike and time to maturity, at given pairs $(T,K)$  where the European options prices or their implied volatilities are known. This process is called calibration of the local volatility (or, alternatively, implied volatility) surface. Various approaches to solving this important problem were proposed, see, eg, survey in  \\cite{ItkinLipton2017} and references therein.\n\nAs mentioned in \\cite{ItkinLipton2017}, there are two main approaches to solving the calibration problem. The first approach relies on some parametric or non-parametric regression to construct a continuous implied volatility (IV) surface matching the given market quotes. Then the corresponding local volatility surface can be found via the well-known Dupire's formula, see, e.g., \\cite{ItkinSigmoid2015} and references therein.\n\nThe second approach relies on the direct solution of the Dupire equation using either analytical or numerical methods. The advantage of this approach is that it guarantees no-arbitrage \\footnote{But only if an analytical or numerical method in use does preserve no-arbitrage. This includes various interpolations, etc.}. However, the problem of the direct solution can be ill-posed, \\cite{Coleman2001}, and is rather computationally expensive. For instance, in \\cite{ItkinLipton2017} the Dupire equation (a partial differential equation (PDE) of the parabolic type) is solved by i) first using the Laplace-Carson transform, and ii) then applying various transformations to obtain a closed form solution of the transformed equation in terms of Kummer hypergeometric functions. Still, it requires an inverse Laplace transform to obtain the final solution.\n\nWith the second approach in use one also has to make an assumption about the behavior of the local\/implied volatility surface at strikes and maturities where the market quotes are not known. Usually, by a tractability argument the corresponding local variance is seen either piecewise constant, \\cite{LiptonSepp2011}, or piecewise linear \\cite{ItkinLipton2017} in the log-strike space, and piecewise constant in the time to maturity space \\footnote{See, however, comments in \\cite{ItkinLipton2017} about their assumptions.}.\n\nTo improve computational efficiency of calibration, an important step is made in \\cite{CarrNadtochiy2017} where Local Variance Gamma (LVG) model has been introduced (the first version refers to 2014 and can be found in \\cite{CarrLVGOrig2014}).  This model assumes that the risk-neutral process for the underlying futures price is a pure jump Markov martingale, and that European option prices are given at a continuum of strikes and at one or more maturities. The authors construct a time-homogeneous process which meets a single smile and a piecewise time-homogeneous process, which can meet multiple smiles. However, in contrast to eg, \\cite{ItkinLipton2017}, their construction leads not to a PDE, but to a partial differential difference equation (PDDE), which permits both explicit calibration and fast numerical valuation. In particular, it does not require application of any optimization methods, rather just a root solver. In \\cite{CarrNadtochiy2017} this model is used to calibrate the local volatility surface assuming its piecewise constant structure in the strike space.\n\nOne of the potential criticism of this calibration method is the fact that the resulting local volatility function has a finite number of discontinuities. So it would be advantaged to relax the piecewise constant behavior of the surface. This is similar to how \\cite{ItkinLipton2017} was developed to overcome the same problem as compared with \\cite{LiptonSepp2011}.\n\nOn this way, recently \\cite{FalckDeryabin2017} applied the LVG model to the FX options market where usually option prices are quoted only at five strikes. They assumed that the local volatility function is continuous, piecewise linear in the four inner strike subintervals and constant in the outer subintervals. A closed form solution of the PDDE derived in \\cite{CarrLVGOrig2014}) is obtained with this parametrization, and calibration of some volatility smiles is provided. Still, to calibrate the model the authors rely on a residual minimization by using a least-square approach. So, despite an improved version of the LVG model is used, computational efficiency of this method is not perfect.\n\nAnother remark of \\cite{CarrNadtochiy2017} is about the limitation that the risk-neutral price process of the underlying is assumed to be a martingale, i.e. the main driving process in \\eqref{D} doesn't have a drift. However, the drift may not be negligible. If the drift is deterministic, e.g when the interest rate and dividends are deterministic, and the drift is a deterministic function of them,  the calibration problem can be reduced to the driftless case by discounting, but this assumption might be inconsistent with the market. Therefore, an expansion of the proposed model that allows for a non-zero and stochastic drift is very desirable. In particular, it would be interesting to expand the LVG model to a risk-neutral price process obtained by stochastic time change of a drifted diffusion. In this way, similar to local Variance Gamma model, \\cite{MadCarrChang}, we introduce both stochastic volatility and stochastic drift.\n\nWith this in mind, our ultimate goals in this paper are as follows. First, we propose an expanded version of the LVG model by adding drift to the governing underlying process. It turns out that to proceed we need to re-derive and re-think every step in construction proposed in \\cite{CarrNadtochiy2017}. We show that still it is possible to find an ordinary differential equation (ODE) for the option price which plays a role of Dupire's equation for the standard local volatility model, and how calibration of multiple smiles (the whole local volatility surface) can be done in such a case.\n\nFurther, assuming the local variance to be a piecewise linear function of strike and piecewise constant function of time we solve this ODE in closed form in terms of Confluent hypergeometric functions. Calibration of the model to market smiles does not require solving any optimization problem. In contrast, it can be done term-by-term by solving a system of non-linear algebraic equations for each maturity, and thus is much faster.\n\nThe rest of the paper is organized as follows. In Section~\\ref{Process} the Expanded Local Variance Gamma model is formulated. In Section~\\ref{ForwardEq} we derive a forward equation (which is an ordinary differential equation (ODE)) for Put option prices using a homogeneous Bochner subordination approach. Section~\\ref{LVpwconst} generalizes this approach by considering the local variance being piece-wise constant in time. In Section~\\ref{SolutionODE} a closed form solution of the derived ODE is given in terms of Confluent hypergeometric functions. The next Section discusses computation of a source term of this ODE which requires a no-arbitrage interpolation. Using the idea of \\cite{ItkinLipton2017}), we show how to construct non-linear interpolation which provides both no-arbitrage, and a nice tractable representation of the source term, so that all integrals in the source term can be computed in closed form. In Section~\\ref{calib} calibration of multiple smiles in our model is discussed in detail. To calibrate a single smile we derive a system of nonlinear algebraic equations for the model parameters, and explain how to obtain a smart guess for their initial values.  In Section~\\ref{asympt}\nasymptotic solutions of our ODE at extreme values of the model parameters are derived which improve computational accuracy and speed of the numerical solution. Section~\\ref{numExp} presents the results of some numerical experiments where calibration of the model to the given market smiles is done term-by-term. The last Section concludes.\n\n\n\\section{Process} \\label{Process}\n\nBelow where possible we follow the notation of \\cite{CarrNadtochiy2017}.\n\nLet $W_t$ be a $\\mathbb{Q}$ standard Brownian motion with time index $t \\ge 0$. Consider a stochastic process $D_t$ to be a time-homogeneous diffusion\n\\begin{equation} \\label{D}\n d D_t = \\mu D_t d t + \\sigma(D_t) d W_t,\n\\end{equation}\n\\noindent where the volatility function $\\sigma$ is local and time-homogeneous, and  $\\mu$ is deterministic.\n\nA unique solution to \\eqref{D} exists if $\\sigma(D) : \\mathbb{R} \\to \\mathbb{R}$ is\nLipschitz continuous in $D$ and satisfies growth conditions at infinity. According to \\eqref{D} we have $D_t  \\in (-\\infty,\\infty)$ while $t \\in [0,\\infty)$. Since $D$ is a time-homogeneous Markov process, its infinitesimal generator ${\\cal A}$ is given by\n\\begin{equation} \\label{gen}\n{\\cal A} \\phi(D) \\equiv \\left[\\mu D {\\nabla}_D  + \\frac{1}{2}\\sigma^2(D) {\\nabla}^2_D \\right] \\phi(D)\n\\end{equation}\n\\noindent for all twice differentiable functions $\\phi$. Here ${\\nabla}_x$ is a first order\ndifferential operator on $x$. The semigroup of the $D$ process is\n\\begin{equation} \\label{semig}\n{\\cal T}^D_t \\phi(D_t) = e^{t {\\cal A}} \\phi(D_t)  = \\EQ[\\phi(D_t)|D_0 = D], \\quad \\forall t \\ge 0.\n\\end{equation}\n\nIn the spirit of Variance Gamma model, \\cite{MadanSeneta:90,MadCarrChang} and similar to \\cite{CarrNadtochiy2017}, introduce a new process $D_{\\Gamma_t}$ which is $D_t$ subordinated by the unbiased Gamma clock $\\Gamma_t$. The density of the unbiased Gamma clock $\\Gamma_t$ at time $t \\ge 0$ is\n\\begin{equation} \\label{gammaDen}\n\\mathbb{Q}\\{\\Gamma_t \\in d\\nu\\} = \\dfrac{\\nu^{m-1} e^{-\\nu m \/t}}{(t^*)^m \\Gamma(m)} d\\nu, \\quad \\nu > 0, \\quad m \\equiv t\/t^*.\n\\end{equation}\nHere $t^* > 0$ is a free parameter of the process, $\\Gamma(x)$ is the Gamma function. It is easy to check that\n\\begin{equation} \\label{gammaExp}\n\\EQ[\\Gamma_t] = t.\n\\end{equation}\n\\noindent Thus, on average the stochastic gamma clock $\\Gamma_t$ runs synchronously with the calendar time $t$.\n\nAs applied to the option pricing problem, we introduce a more complex construction.\nNamely, consider options written on the underlying process $S_t$. Without loss of generality and for the sake of clearness let us treat below $S_t$ as the stock price process. Here, in contrast to \\cite{CarrNadtochiy2017}, we don't ignore interest rates $r$ and continuous dividends $q$ assuming them to be deterministic (below for simplicity of presentation we treat them as constants, but this can be easily relaxed). Then, let us define $S_t$ as\n\\begin{equation} \\label{sub}\nS_t = D_{\\Gamma_{X(t)}}, \\qquad X(t) = \\dfrac{1 - e^{-(r-q) t}}{r-q}.\n\\end{equation}\n\\noindent It is clear that in the limit $ r \\to 0, \\ q \\to 0$ we have $X(t) = t$, i.e., in this limit our construction coincides with that in \\cite{CarrNadtochiy2017} who considered a driftless diffusion and assumed $S_t = D_{\\Gamma_t}$. Also based on \\eqref{gammaExp}\n\\begin{equation} \\label{gammaXexp}\n\\EQ[\\Gamma_{X(t)}] = X(t).\n\\end{equation}\nFunction $X(t)$ starts at zero, ie, $X(0) = 0$, and is a continuous increasing function of time $t$. Indeed, if $r - q > 0$, then $X(t)$ is increasing in $t$ on $t \\in [0,\\infty)$, and at $t \\to \\infty$ it tends to constant. The infinite time horizon is not practically important, but for any finite time $t$ function $X(t)$ can be treated as an increasing function in $t$. If $r - q < 0$, function $X(t)$ is strictly increasing $\\forall t \\in [0,\\infty)$. Thus, $X(t)$ has all properties of a good clock. Accordingly, $\\Gamma_{X(t)}$ has all properties of a random time.\n\nUnder a risk-neutral measure $\\mathbb{Q}$, the total gain process, including the underlying price appreciation and dividends, after discounting at the risk free rate should be a martingale, see, eg, \\cite{Shreve:1992}. This process obeys the following stochastic differential equation\n\\begin{align} \\label{mart}\nd \\left( e^{-r t} S_t e^{q t} \\right) &=  e^{(q-r)t} \\left[ (q-r) S_t dt + d S_t\\right]. \\end{align}\nTaking an expectation of both parts we obtain\n\\begin{align} \\label{martCond}\n\\EQ[d \\left( e^{(q-r)t} S_t\\right)] &=\ne^{(q-r)t} \\left\\{ (q-r)  \\EQ[S_t] d t + d \\EQ[S_t] \\right\\}.\n\\end{align}\nObserve, that from \\eqref{sub}, \\eqref{D}\n\\begin{align} \\label{e1}\n\\EQ[d S_t] &= \\EQ[d {D_{\\Gamma_{X(t)}}}] = \\mu \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}] + \\EQ[\\sigma(D_{\\Gamma_{X(t)}}) d W_{\\Gamma_{X(t)}}] = \\mu \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}],\n\\end{align}\n\\noindent because the process $W_{\\Gamma_t}$ is a local martingale, see \\cite{RevuzYor1999}, chapter 6. Accordingly, the process $W_{\\Gamma_{X(t)}}$ inherits this property from $W_{\\Gamma_t}$, hence $\\EQ[\\sigma(D_{\\Gamma_{X(t)}}) d W_{\\Gamma_{X(t)}}] = 0$.\n\nFurther assume that the Gamma process $\\Gamma_t$ is independent of $W_t$ (and, accordingly, ${\\Gamma_{X(t)}}$ is independent of $W_{\\Gamma_{X(t)}}$). Then the expectation in the RHS of \\eqref{e1} can be computed, by first conditioning on ${\\Gamma_{X(t)}}$, and then integrating over the distribution of ${\\Gamma_{X(t)}}$ which can be obtained from \\eqref{gammaDen} by replacing $t$ with $X(t)$, i.e.\n\\begin{align} \\label{e2}\n\\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}  | S_s] &= \\int_0^\\infty  \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}| {\\Gamma_{X(t)}} = \\nu] \\dfrac{\\nu^{m-1} e^{-\\nu m \/X(t)}}{(t^*)^m \\Gamma(m)} \\\\\n&= \\int_0^\\infty  \\EQ [D_\\nu] \\dfrac{\\nu^{m-1} e^{-\\nu m \/X(t)}}{(t^*)^m \\Gamma(m)} d \\nu, \\quad\n\\nu > 0, \\quad m \\equiv X(t)\/t^*. \\nonumber\n\\end{align}\nThe find $\\EQ [D_\\nu]$ we take into account \\eqref{D} to obtain\n\\begin{align} \\label{e3}\nd \\EQ [D_\\nu] =  \\EQ [d D_\\nu]\n= \\EQ [\\mu D_\\nu d\\nu + \\sigma(D_\\nu) D_\\nu d W_\\nu] = \\mu \\EQ [D_\\nu] d\\nu.\n\\end{align}\nSolving this equation with respect to $y(\\nu) = \\EQ [D_\\nu  | D_s]$,  we obtain $\\EQ [D_\\nu  | D_s] = D_s e^{\\mu (\\nu-s)}$. Since we condition on time $s$, it means that $D_s = D_{\\Gamma_{X(s)}} = S_s$, and thus\n$\\EQ [D_\\nu  | D_s] = S_s e^{\\mu (\\nu-s)}$.\n\nFurther, we substitute this into \\eqref{e2}, set the parameter of the Gamma distribution $t^*$ to be $t^* = X(t)$ (so $m = 1$) and integrate to obtain\n\\begin{equation} \\label{intRight}\nd \\EQ[S_t  | S_s] = \\EQ[d S_t  | S_s] = \\mu \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}] = S_s e^{- s \\mu} \\frac{\\mu}{1 - \\mu X(t)}.\n\\end{equation}\nSetting now $m = r-q$ and solving this equation we find\n\\begin{equation} \\label{ESsol}\n\\EQ[S_t  | S_s] = S_s (r-q) e^{(q-r)(s-t)}.\n\\end{equation}\nSubstituting \\cref{ESsol} and \\cref{intRight} into \\cref{martCond} yields $d \\left( e^{-r t} S_t e^{q t} \\right) = 0$. Thus, if we chose $\\mu = r-q$, the right hands part of \\eqref{mart} vanishes, and our discounted stock process with allowance for non-zero interest rates and continuous dividends becomes a martingale. So the proposed construction can be used for option pricing.\n\nThis setting can be easily generalized for time-dependent interest rates $r(t)$ and continuous dividends $q(t)$. We leave it for the reader.\n\nThe next step is to consider connection between the original and time-changed processes. It is known from \\cite{BochnerPDE1949}  that the process $G_{\\Gamma_t}$ defined as\n\\[ d G_t = \\sigma^2(G) d W_t \\]\n\\noindent is a time-homogeneous Markov process. As the deterministic process $\\mu t$ is also time-homogeneous, the whole process $D_t$ defined in \\eqref{D} is also a time-homogeneous Markov process. Accordingly, the semigroups $T^S_t$ of $S_t$ and $T^D_t$ of $D_{\\Gamma_{X(t)}}$ are connected by the Bochner integral\n\\begin{equation} \\label{BI}\n{\\cal T}^S_t U(S) = \\int_0^\\infty {\\cal T}^D_\\nu U(S) \\mathbb{Q}\\{{\\Gamma_{X(t)}} \\in d\\nu\\}, \\quad \\forall t \\ge 0,\n\\end{equation}\n\\noindent where $U(S)$ is a function in the domain of both ${\\cal T}^D_t$ and ${\\cal T}^S_t$.\nIt can be derived by exploiting the time homogeneity of the $D$ process, conditioning on the gamma time first, and taking into account the independence of $\\Gamma_t$ and $W_t$ (or $\\Gamma_{\\Gamma_{X(t)}}$ and $W_{\\Gamma_{X(t)}}$ in our case).\n\nWe set parameter $t^*$ of the gamma clock to $t^* = X(t)$. Then \\eqref{BI} and \\eqref{gammaDen} imply\n\\begin{equation} \\label{BI1}\n{\\cal T}^S_{t} U(S) = \\int_0^\\infty {\\cal T}^D_\\nu U(S) \\dfrac{e^{-\\nu\/X(t)}}{X(t)} d\\nu.\n\\end{equation}\nIn what follows for the sake of brevity we will call this model as an Expanded Local Variance Gamma model, or ELVG.\n\n\\section{Forward equation for Put option prices} \\label{ForwardEq}\n\nFollowing \\cite{CarrNadtochiy2017} we interpret the index $t$ of the semigroup ${\\cal T}^S_t$ as the maturity date $T$ of a European claim with the valuation time $t = 0$. Also let the test function $U(S)$ be the payoff of this European claim, ie,\n\\begin{equation} \\label{payoff}\nU(S_T) = e^{-r T}(K - S_T)^+.\n\\end{equation}\nThen define\n\\begin{equation} \\label{P0}\nP(S_0,T,K) = {\\cal T}^S_T U(S_0)\n\\end{equation}\n\\noindent as the European Put value with maturity $T$ at time $t=0$ in the ELVG model. Similarly\n\\begin{equation} \\label{P0D}\nP^D(S_0,\\nu,K) = {\\cal T}^D_\\nu U(S_0)\n\\end{equation}\n\\noindent would be the European Put value with maturity $\\nu$ at time $t=0$ in the model of \\eqref{D}\\footnote{Below for simplicity of notation we drop the subscript '0'  in $S_0$.}. Then the Bochner integral in \\eqref{BI1} takes the form\n\\begin{equation} \\label{Bochner2}\nP(S,T,K) =  \\int_0^\\infty P^D(S,\\nu,K) p e^{- p \\nu} d \\nu,  \\quad p \\equiv 1\/X(T).\n\\end{equation}\nThus, $P(S,X(T),K)$ is represented by a Laplace-Carson transform of $P^D(S,\\nu,K)$ with $p$ being a parameter of the transform. Note that\n\\begin{equation} \\label{init}\nP(S,0,K) = P^D(S,0,K) = U(S).\n\\end{equation}\nTo proceed, we need an analog of the Dupire forward PDE for $P^D(S,\\nu,K)$.\n\n\\subsection{Derivation of the Dupire forward PDE \\label{dupFWPDE}}\n\nDespite this can be done in many different ways, below for the sake of compatibility we do it in the spirit of \\cite{CarrNadtochiy2017}.\n\nFirst, differentiating \\eqref{P0D} by $\\nu$ with allowance for \\eqref{semig} yields\n\\begin{align} \\label{Dt}\n{\\nabla}_\\nu P^D(S,\\nu,K) &= e^{-r \\nu} e^{\\nu {\\cal A}}\\left[ {\\cal A} - r\\right] U(S)\n= e^{-r \\nu} \\EQ \\left[ {\\cal A} - r\\right] U(S).\n\\end{align}\nWe take into account the definition of the generator ${\\cal A}$ in \\eqref{gen}, and also remind that at $t=0$ we have $D_0 = S_0$. Then \\eqref{Dt} transforms to\n\\begin{equation} \\label{Dt1}\n{\\nabla}_\\nu P^D(S,\\nu,K) = -r P^D(S,\\nu,K) + (r-q) S {\\nabla}_S P^D(S,\\nu,K) + e^{-r \\nu}\\frac{1}{2} \\EQ \\left[ \\sigma^2(S) {\\nabla}_S^2 U(S) \\right].\n\\end{equation}\nHowever, we need to express the forward equation using a pair of independent variables $(\\nu,K)$ while \\eqref{Dt} is derived in terms of $(\\nu,S)$. To do this, observe that\n\\begin{align} \\label{sir}\ne^{-r \\nu} \\EQ \\left[\\sigma^2(S) {\\nabla}_S^2 U(S) \\right] &= e^{-r \\nu}\\EQ \\left[\\sigma^2(S) \\delta(K-S)\\right] = e^{-r \\nu} \\EQ \\left[\\sigma^2(K) \\delta(K-S)\\right] \\\\\n&=  e^{-r \\nu} \\EQ \\left[\\sigma^2(K) {\\nabla}_K^2 U(S)\\right] = \\sigma^2(K) {\\nabla}_K^2 P^D(S,\\nu,K). \\nonumber\n\\end{align}\n\\noindent where the sifting property of the Dirac delta function $\\delta(S-K)$ has been used. Also\n\\begin{align} \\label{term2}\n-r & P^D(S,\\nu,K) + (r-q) S \\nabla_S P^D(S,\\nu,K) \\\\\n&= e^{-r \\nu} \\EQ\\left[ -r (K-S)^+ + (r-q)S \\fp{(K-S)^+}{S} \\right] \\nonumber \\\\\n&= e^{-r \\nu} \\EQ\\left[ -r (K-S)^+ - (r-q)(K-S) \\fp{(K-S)^+}{S} + (r-q)K \\fp{(K-S)^+}{S}\\right] \\nonumber \\\\\n&= e^{-r \\nu} \\EQ\\left[ -r (K-S)^+ + (r-q)(K-S)^+  - (r-q)K \\fp{(K-S)^+}{K}\\right] \\nonumber \\\\\n&= -q P^D(S,\\nu,K) - (r-q) K \\nabla_K P^D(S,\\nu,K). \\nonumber\n\\end{align}\nTherefore, using \\eqref{sir} and \\eqref{term2}, \\eqref{Dt} could be transformed to\n\\begin{align} \\label{Dup1}\n\\nabla_\\nu P^D(S,\\nu,K) &= -q P^D(S,\\nu,K) - (r-q) K \\nabla_K P^D(S,\\nu,K) + \\frac{1}{2} \\sigma^2(K) K^2 \\nabla_K^2 P^D(S,\\nu,K) \\nonumber  \\\\\n&\\equiv {{\\cal A}}^K P^D(S,\\nu,K),  \\\\\n{{\\cal A}}^K &= -q  - (r-q) K \\nabla_K  + \\frac{1}{2} \\sigma^2(K) K^2 \\nabla_K^2. \\nonumber\n\\end{align}\nThis equation looks exactly like the Dupire equation with non-zero interest rates and continuous dividends, see, eg, \\cite{Tysk2012} and references therein. Note, that ${\\cal A}^K$ is also a time-homogeneous generator.\n\n\\subsection{Forward partial divided-difference equation} \\label{FPDDder}\n\nOur final step is to apply the linear differential operator ${{\\cal A}}^K$ defined in  \\eqref{Dup1} to both parts of \\eqref{Bochner2}. Using time-homogeneity of $D_t$ and, again, the Dupire equation \\eqref{Dup1}, we obtain\n\\begin{align} \\label{b2}\n-q &P(S,T,K) - (r-q) K {\\nabla}_K P(S,T,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2\nP(S,T,K) \\\\\n&= \\int_0^\\infty p e^{- p \\nu} \\left[-q P^D(S,\\nu,K) - (r-q) K {\\nabla}_K P^D(S,\\nu,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 P^D(S,\\nu,K)\\right] d \\nu \\nonumber \\\\\n&= \\int_0^\\infty p e^{- p \\nu} {\\nabla}_\\nu P^D(S,\\nu,K) d \\nu  = - p P^D(S,0,K) + p \\int_0^\\infty P^D(S,\\nu,K) p e^{- p \\nu} d \\nu \\nonumber \\\\\n&= p\\left[P(S,T,K) - P^D(S,0,K) \\right] = p\\left[P(S,T,K) - P(S,0,K) \\right], \\nonumber\n\\end{align}\n\\noindent where in the last line \\eqref{init} was taken into account.\n\nThus, finally $P(S,T,K)$ solves the following problem\n\\begin{align} \\label{finDup}\n-q P(S,T,K) &- (r-q) K {\\nabla}_K P(S,T,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2\nP(S,T,K) \\\\\n&= \\dfrac{P(S,T,K) - P(S,0,K)}{X(T)}, \\qquad P(S,0,K) = (K-S)^+. \\nonumber\n\\end{align}\nAt $ r = q = 0$ this equation translates to the corresponding equation in \\cite{CarrNadtochiy2017}. In contrast to the Dupire equation which belongs to the class of PDE, \\eqref{finDup} is an ODE, or, more precisely, a partial divided-difference equation (PDDE), since the derivative in time in the right hands part is now replaced by a divided difference. In the form of an ODE it reads\n\\begin{equation} \\label{finDupPut}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 - (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{X(T)}\\right) \\right] P(S,T,K) =\n- \\dfrac{P(S,0,K)}{X(T)}.\n\\end{equation}\nThis equation could be solved analytically for some particular forms of the local volatility function $\\sigma(K)$ which are considered later in this paper. Also in the same way a similar equation could be derived for the Call option price $C_0(S,T,K)$ which reads\n\\begin{align} \\label{finDupCall}\n\\Big[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 + (r-q) K {\\nabla}_K &- \\left(q + \\dfrac{1}{X(T)}\\right) \\Big] C_0(S,T,K) = - \\dfrac{C_0(S,0,K)}{X(T)}, \\nonumber \\\\\nC_0(S,0,K) &= (S-K)^+.\n\\end{align}\n\nSolving \\eqref{finDupPut} or \\eqref{finDupCall} provides the way to determine $\\sigma(K)$ given market quotes of Call and Put options with maturity $T$. However, this allows calibration of just a single term. Calibration of the whole local volatility surface, in principle, could be done term-by-term (because of the time-homogeneity assumption) if \\eqref{finDupPut}, \\eqref{finDupCall} could be generalized to this case. We consider this in the following Section.\n\n\\section{Local variance piece-wise constant in time} \\label{LVpwconst}\n\nTo address calibration of multiple smiles, we need to relax some assumptions about time-homogeneity of the process $D_t$ defined in \\eqref{D}. This includes several steps which are described below in more detail.\n\n\\subsection{Local variance}\n\nHere we assume that the local variance $\\sigma(D_t)$ is no more time-homogeneous, but a piece-wise constant function of time $\\sigma(D_t,t)$.\n\nLet $T_1, T_2, \\ldots, T_M$ be the time points at which the variance rate $\\sigma^2(D_t)$ jumps deterministically. In other words, at the interval $t \\in [T_0,T_1)$, the variance rate is $\\sigma^2_0(D_t)$, at $t \\in [T_1, T_2)$ it is $\\sigma^2_1(D_t)$, etc. This can be also represented as\n\\begin{align} \\label{sigmaPW}\n\\sigma^2(D_t,t) &= \\sum_{i=0}^M \\sigma^2_i(D_t) w_i(t), \\\\\nw_i(t) &\\equiv {\\mathbf 1}_{t - T_i} - {\\mathbf 1}_{t - T_{i+1}}, \\ i=0,\\ldots,M,\n\\quad T_0 = 0, \\ T_{M+1} = \\infty, \\quad {\\mathbf 1}_x =\n\\begin{cases}\n1, & x \\ge 0 \\\\\n0, & x < 0.\n\\end{cases}\n\\nonumber\n\\end{align}\n\nNote, that\n\\[ \\sum_{i=0}^M w_i(t) = {\\mathbf 1}_t - {\\mathbf 1}_{t-\\infty} = 1, \\quad \\forall t \\ge 0.\\]\n\\noindent Therefore, in case when all $\\sigma^2_i(D_t)$ are equal, ie, independent on index $i$, \\eqref{sigmaPW} reduces to the case considered in the previous Sections.\n\nIt is important to notice that our construction implies that the volatility $\\sigma(D_t)$ jumps as a function of time at the calendar times $T_0, T_1,\\ldots,T_M$, and not at the business times $\\nu$ determined by the gamma clock. Otherwise, the volatility function would change at random (business) times which means it is stochastic. But this definitely lies out of scope of our model. Therefore, we need to change \\eqref{sigmaPW} to\n\\begin{align} \\label{sigmaPW_exp}\n\\sigma^2({D_{\\Gamma_{X(t)}}}, {\\Gamma_{X(t)}}) &= \\sum_{i=0}^M \\sigma^2_i(D_t) {\\bar w}_i(\\EQ({\\Gamma_{X(t)}})), \\\\\n{\\bar w}_i(t) &= {\\mathbf 1}_{X^{-1}(t) - T_i} - {\\mathbf 1}_{X^{-1}(t) - T_{i+1}}, \\ i=0,\\ldots,M, \\nonumber \\\\\nX^{-1}(t) &= \\dfrac{1}{q-r} \\log \\left[1 - (r-q)t \\right].\n\\end{align}\nHence, when using \\eqref{sub} we have\n\\begin{align} \\label{sigmaPW_exp1}\n\\sigma^2(D_t, t)\\Big|_{t = \\Gamma_{X(t)}} &= \\sum_{i=0}^M \\sigma^2_i(D_t) \\bar{w}_i(X(t)) = \\sum_{i=0}^M \\sigma^2_i(D_t) w_i(t).\n\\end{align}\n\nAccordingly, if the calendar time $t$ belongs to the interval $T_0 \\le t < T_1$, the infinitesimal generator ${\\cal A}$ of the semigroup ${\\cal T}^D_\\nu$ is a function of $\\sigma(D_t)$ (and not on $\\sigma(D_\\nu)$). As at $T_0 \\le t < T_1$ we assume $\\sigma(D) = \\sigma_0(D)$, i.e., is constant in time, it doesn't depend of $\\nu$. Thus, ${\\cal A}$ (which for this interval of time we will denote as ${\\cal A}_0$) is still time-homogeneous.\n\nSimilarly, one can see, that for $T_1 \\le t < T_2$ the infinitesimal generator ${\\cal A}_1$ of the semigroup ${\\cal T}^D_\\nu$ is also time-homogeneous and depends on $\\sigma_1(D)$, etc.\n\n\\subsection{Bochner subordination}\n\nWe start with re-definition of \\eqref{P0}, \\eqref{P0D}. We now define the European Put value with maturity $T$ at the evaluation time $t=X(T_1)$ in the ELVG model\n\\begin{equation} \\label{P02}\nP(S_0,T_1 + T,K) = {\\cal T}^S_T [e^{-r T} P(S_0,T_1,K)].\n\\end{equation}\nAnd, clearly we are interesting in the value of $T$ to be $T = T_2 - T_1$.\n\nSimilarly, we define the European Put value with maturity $\\nu$ at the evaluation time $t=T_1$ in the model given by \\eqref{D} as\n\\begin{equation} \\label{P0D2}\nP^D(S_0,T_1 + \\nu,K) = {\\cal T}^D_\\nu [e^{-r \\nu} P(S_0,T_1,K)].\n\\end{equation}\nBy these definitions\n\\[ P(S_0,T_1 + T,K)\\Big|_{T=0} = P^D(S_0,T_1 + \\nu,K)\\Big|_{\\nu=0} = P(S_0,T_1,K)]. \\]\n\nIn contrast to \\eqref{Bochner2}, in case of multiple smiles at $t > T_1$ we need to change the definition of $t$ in \\eqref{BI1} from $t \\mapsto X(t)$ to\n\\begin{equation} \\label{newt*}\nt \\mapsto X(T_1+t) - X(T_1) \\equiv \\Delta x(T_1,t).\n\\end{equation}\nThis definition implies two observations.\n\nFirst, function $\\Delta x(T_1,t)$ starts at zero at $t=0$ and is an increasing function of time. Also, in case $r=q=0$ we have $\\Delta x(T_1,t) = t$. Therefore, $\\Delta x(T_1,t)$ can be used as a good clock. Accordingly, similar to \\eqref{gammaExp} we have\n\\begin{equation} \\label{gammaXexp1}\n\\EQ[\\Gamma_{\\Delta x(T_1,t)}] = \\Delta x(T_1,t).\n\\end{equation}\nSecond, a proof that in our model the discounted stock price is a martingale\ngiven in Section~\\ref{Process} could be repeated for times $t: \\ T_1 < t \\le T_2$.\nWhen doing so, at $t > T_1$ we reset the definition of $S_t$ to\n\\[ S_{T_1+t} = D_{\\Gamma_{\\Delta x(T_1,t) }}, \\quad t \\ge 0. \\]\nThen instead of \\eqref{e1} we now have\n\\begin{align} \\label{e11}\n\\EQ[d S_{T_1+t}] &= \\EQ[d D_{\\Gamma_{\\Delta x(T_1,t)}}] = \\mu \\EQ [D_{\\Gamma_{\\Delta x(T_1,t)}} d \\Gamma_{\\Delta x(T_1,t)}] + \\EQ[\\sigma(D_{\\Gamma_{\\Delta x(T_1,t)}}) d W_{\\Gamma_{\\Delta x(T_1,t) }}] \\\\\n&= \\mu \\EQ [D_{\\Gamma_{\\Delta x(T_1,t)}}] d \\Delta x(T_1,t) =\n\\mu \\EQ [D_{\\Gamma_{\\Delta x(T_1,t)}}] d X(T_1+t). \\nonumber\n\\end{align}\nOn the other hand,\n\\begin{align} \\label{martCond1}\n\\EQ[d \\left( e^{(q-r)(T_1+t)} S_{T_1+t} \\right)] &=\ne^{(q-r)(T_1+t)} \\left\\{ (q-r)  \\EQ[S_{T_1+t}] d t + d \\EQ[S_{T_1+t}] \\right\\} \\\\\n&= e^{(q-r)t}[\\mu + (q-r)S_{T_1}e^{-(r-q)T_1}] dt \\nonumber\n\\end{align}\n\nOne can check, that with $\\mu = r-q$ the RHS of \\cref{martCond1} vanishes, therefore this construction can be used for option pricing.\n\nThe definition in \\eqref{newt*} implies that parameter $t$ of the Gamma random clock is reset at the point $T_1$, i.e., at $0 \\le t \\le T_1$ it is $t \\mapsto X(t) = X(t) - X(0)$, while at $T_1 < t \\le T_2$ it is $t \\mapsto X(T_1+t) - X(T_1)$. Using the definition of $w_i(t)$ in \\eqref{sigmaPW}, this could be written as\n\\begin{equation} \\label{t*2}\nt \\mapsto \\sum_{i=0}^M w_i(T_i+t)[X(T_i+t) - X(T_i)]\n\\end{equation}\nResetting $t$ was also first proposed in \\cite{CarrNadtochiy2017} but in a different form.\n\nThen, the Bochner integral in \\eqref{BI1} transforms to\n\\begin{equation} \\label{BI3}\n{\\cal T}^S_{T} P(S,T_1,K) = \\int_{0}^\\infty {\\cal T}^D_\\nu P(S,T_1 + \\nu,K) \\dfrac{\\nu^{m-1} e^{-\\nu m \/\\Delta X(T_1,T)}}{(t^*)^m \\Gamma(m)} d\\nu.\n\\end{equation}\nSince for a tractability reason we still want to have $m \\equiv \\Delta X(T_1,T)\/t^* = 1$. we need to redefine $t^*$ in accordance with \\eqref{t*2}. Based on that, the Bochner integral in \\eqref{Bochner2} now finally reads\n\\begin{align} \\label{Bochner22}\nP(S,T_1 + T,K) &= \\int_0^\\infty P^D(S,T_1+\\nu,K) p e^{- p \\nu} d \\nu,  \\quad\np \\equiv 1\/\\Delta X(T_1, T).\n\\end{align}\n\n\\subsection{Forward partial divided-difference equation for the second term} \\label{FPDDder2}\n\nNow we need to derive a Forward partial divided-difference equation for the second term $T_2$ similar to how this is done in Section~\\ref{FPDDder}. Obviously, the Put price $P^D(S_0,T_1 + \\nu,K)$ solves the same Dupire equation \\eqref{Dup1}. Therefore, proceeding in the same way as in Section~\\ref{FPDDder}, we apply linear differential operator $\\cal L$ defined in  \\eqref{Dup1} to both parts of \\eqref{Bochner22}. Using time-homogeneity of $D_t$ at the interval $[T_1, T_2)$ and again the Dupire equation \\eqref{Dup1}, we obtain\n\\begin{align} \\label{b21}\n-q &P(S,T_1 + T,K) - (r-q) K {\\nabla}_K P(S,T_1 + T,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 P(S,T_1 + T,K) \\nonumber \\\\\n&= \\int_0^\\infty p e^{- p \\nu} \\Big[-q P^D(S,T_1 + \\nu,K) - (r-q) K {\\nabla}_K P^D(S,T_1 + \\nu,K)  \\\\\n&+ \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 P^D(S,T_1 + \\nu,K)\\Big] d \\nu\n= \\int_0^\\infty p e^{- p \\nu} {\\nabla}_\\nu P^D(S,T_1 + \\nu,K) d \\nu \\nonumber \\\\\n&= - p P^D(S,T_1,K) + p \\int_0^\\infty P^D(S,T_1+\\nu,K) p e^{- p \\nu} d\\nu \\nonumber \\\\\n&= p\\left[P(S,T_1 + T,K) - P^D(S,T_1,K)\\right]  = p\\left[P(S,T_1 + T,K) - P(S,T_1,K) \\right]. \\nonumber\n\\end{align}\nFinally, taking $T = T_2  - T_1$ we obtain an ODE for the Put price $P(S,T_2,K)$.\n\\begin{equation} \\label{finDupPut2}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 - (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{\nX(T_2) - X(T_1)}\\right) \\right] P(S,T_2,K) = - \\dfrac{P(S,T_1,K)}{X(T_2) - X(T_1)}.\n\\end{equation}\nHere the local variance function $\\sigma^2(K) = \\sigma^2_1(K)$ as it corresponds to the interval $(T_1,T_2]$ where the above ODE is solved.\n\nWe continue in the same way to derive an ODE for the Put price $P(S,T_i,K), \\ i=1,\\ldots,M$, which finally reads\n\\begin{equation} \\label{finDupPutI}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 - (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{\nX(T_i) - X(T_{i-1})}\\right) \\right] P(S,T_i,K) = - \\dfrac{P(S,T_{i-1},K)}{X(T_i) - X(T_{i-1})}.\n\\end{equation}\nThis is a recurrent equation that can be solved for all $i=1,\\ldots,M$ sequentially starting with $i=1$ subject to some boundary conditions. The natural boundary conditions for the Put option price are, \\cite{hull:97}\n\\begin{equation} \\label{bc}\n\\begin{array}{lll}\nP(S,T_i,K) = 0, & & K \\to 0, \\\\\nP(S,T_i,K) = {\\cal D}_i K - {\\cal Q}_i S \\approx {\\cal D}_i K, & & K \\to \\infty,\n\\end{array}\n\\end{equation}\n\\noindent where ${\\cal D}_i = e^{-r T_i}$ is the discount factor, and ${\\cal Q}_i = e^{-q T_i}$.\n\nA similar equation can be obtained for the Call option prices, which reads\n\\begin{equation} \\label{finDupCallI}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 + (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{\nX(T_i) - X(T_{i-1})}\\right) \\right] C(S,T_i,K) = - \\dfrac{C(S,T_{i-1},K)}{X(T_i) - X(T_{i-1})},\n\\end{equation}\n\\noindent subject to the boundary conditions\n\\begin{equation} \\label{bcCall}\n\\begin{array}{lll}\nC(S,T_i,K) = {\\cal Q}_i S, & & K \\to 0, \\\\\nC(S,T_i,K) = 0, & & K \\to \\infty.\n\\end{array}\n\\end{equation}\n\n\n\n\n\\section{Solution of the ODE \\eqref{finDupPutI}} \\label{SolutionODE}\n\nBelow we use the approach similar to \\cite{ItkinLipton2017} by assuming the local variance to be a piecewise linear continuous function of strike. In contrast to \\cite{ItkinLipton2017}, instead of a standard local volatility model in this paper we use the ELVG model. As the result, instead of a partial differential (Dupire) equation, we face a problem of solving the ODE in \\eqref{finDupPutI}.\n\nFirst, it is useful to change the dependent variable from $P(S,T_j,K)$ to\n\\[ V(S,T_j,K) = P(S,T_j,K) - {\\cal D}_j K, \\]\n\\noindent which is known as a {\\it covered Put}. The advantage of the covered Put is that according to \\eqref{bc} its price obeys homogeneous boundary conditions.\n\nUsing this definition we now re-write \\eqref{finDupPutI} in a more convenient form (while with some loose of notation)\n\\begin{align} \\label{finODE}\n& - v(x) V_{x,x}(x) + b_1 x V_x(x) + b_{0,j} V(x) = c_j, \\\\\nb_1 &= (r - q) p_j, \\quad b_{0,j} = q p_j + 1, \\quad c_j = V(T_{j-1},x) + \\beta x, \\nonumber \\\\\np_j &= X(T_j) - X(T_{j-1}) > 0, \\quad x = \\frac{K}{S}, \\quad V(x) = V(S,T_j,x), \\quad v(x) = p_j\\frac{\\sigma^2(x)}{2 S^2}. \\nonumber \\\\\n\\beta &= - S[{\\cal D}_j(1 + p_j r) - {\\cal D}_{j-1}]. \\nonumber\n\\end{align}\nIn \\eqref{finODE} $x$ is the inverse moneyness. In what follows we also assume that $r > q > 0$, but this assumption could be easily relaxed.\n\nFurther, suppose that for each maturity $T_j, \\ j \\in [1,M]$ the market quotes are provided at a set of strikes $K_i, \\ i=1,\\ldots,n_j$ where the strikes are assumed to be sorted in the increasing order. Then the corresponding continuous piecewise linear local variance function $v_j(x)$ on the interval $[x_{i},x_{i+1}]$ reads\n\\begin{equation} \\label{vDef}\nv_{j,i}(x) = v^0_{j,i} + v^1_{j,i} x,\n\\end{equation}\n\\noindent where we use the super-index $0$ to denote a level $v^0$, and the super-index $1$ to denote a slope $v^1$. Subindex $i=0$ in $v^0_{j,0}, v^1_{j,0}$ corresponds to the interval $(0, x_1]$. Since $v_j(x)$ is continuous, we have\n\\begin{equation} \\label{cont}\nv^0_{j,i} + v^1_{j,i} x_{i+1} = v^0_{j,i+1} + v^1_{j,i+1} x_{i+1}, \\quad i=0,\\ldots,n_j-1.\n\\end{equation}\nThe first derivative of $v_j(x)$ experiences a jump at points $x_{i}, \\irg{i}{1}{n_j}$. We also assume that $v(x,T)$ is a piecewise constant function of time, i.e., $v^0_{j,i}, v^1_{j,i}$ don't depend on $T$ on the intervals $[T_j, T_{j+1}), \\ j \\in [0,M-1]$, and jump to new values at the points $T_j, \\irg{j}{1}{M}$.\n\nWith the above assumptions in mind, \\eqref{finODE} can be solved by induction. One starts with $T_0 = 0$, and on each time interval $[T_{j-1},T_j], \\ \\irg{j}{1}{M}$ solves the problem \\eqref{finODE} for $V(x) \\mapsto P(S,T_j,x) - d_j S x$.\n\nSince $v(x)$ is a piecewise linear function, the solution of \\eqref{finODE} can also be constructed separately for each interval $[x_{i-1},x_i]$. By taking into account the explicit representation of $v(x)$ in \\eqref{vDef}, from \\eqref{finODE} for the $i$-th spatial interval we obtain\n\\begin{align} \\label{Laplace2}\n-(b_2 + a_2 x) V_{xx}(x) &+ b_1 x V_x(x) + b_0 V(x) = c \\\\\nb_2 &= v^0_{j,i}, \\ a_2 = v^1_{j,i}. \\nonumber\n\\end{align}\nWe proceed by introducing a new independent variable $z = (b_2 + a_2 x)b_1\/a_2^2, \\ z \\in \\mathbb{R}^+$, so that \\eqref{Laplace2} transforms to\n\\begin{align} \\label{Laplace3}\n-z V_{zz}(z) &+ (z-q_2)V_z(z) + q_1 V(z) = \\chi \\\\\nq_1 &= b_0\/b_1, \\ q_2 = b_2 b_1\/a_2^2, \\ \\chi = c\/b_1. \\nonumber\n\\end{align}\n\nThe \\eqref{Laplace3} is an {\\it inhomogeneous} Laplace equation, \\cite{PolyaninSaitsevODE2003}, page 155. It is well known that if $y_1=y_1(z)$, $y_2=y_2(z)$ are two fundamental solutions of the corresponding {\\it homogeneous} equation, then the general solution of \\eqref{Laplace3} can be represented as\n\\begin{align} \\label{solInhom}\nV(z) &= C_1 y_1(z) + C_2 y_2(z) + \\frac{1}{b_1} I_{12}(z) \\\\\nI_{12}(z) &= -y_2(z) \\int \\dfrac{ y_1(z) f(z)} {W z}d z + y_1(z) \\int  \\dfrac{ y_2(z) f(z)}{W z}d z \\equiv I_1 + I_2, \\nonumber \\\\\nf(z) &= V(T_{j-1},z) - k_1 - k_2 z, \\quad\nk_1 = \\beta \\frac{b_2}{a_2}, \\quad k_2 = - \\beta \\frac{a_2}{b_1}, \\nonumber\n\\end{align}\n\\noindent where $W = y_1 (y_2)_z - y_2 (y_1)_z$ is the so-called Wronskian, and $\\beta$ is defined in \\eqref{finODE}. Then the problem is to determine suitable fundamental solutions of the homogeneous Laplace equations. Based on \\cite{PolyaninSaitsevODE2003}, if $a_2 \\ne 0$, they read\n\\begin{equation} \\label{homog}\ny_i(z) = {\\mathcal V}_i (q_1, q_2, z), \\quad i=1,2\n\\end{equation}\nHere ${\\mathcal V}_i(a,b,z)$ is an arbitrary solution of the degenerate hypergeometric equation, i.e., Kummer's function, \\cite{as64}. Two types of Kummer's functions are known, namely $M(a,b,z)$ and $U(a,b,z)$, which are Kummer's functions of the first and second kind.\n\nIt is known, that there exist several pairs of such independent solutions. Therefore, for every spatial interval in $z$ among all possible fundamental pairs we have to determine just one which is numerically satisfactory at this interval (see \\cite{Olver1997} for the detailed definition of satisfactory solutions and the corresponding discussion). Since our boundary conditions are set at zero and positive infinity, we need a numerically satisfactory solution for the positive half of the real line.\n\nSimilar to \\cite{ItkinLipton2017}, in the vicinity of the origin we choose the numerically satisfactory pair as, \\cite{Olver1997}\n\\begin{align} \\label{Kummer0}\ny_1(\\chi) &= M\\left(q_1, q_2, z\\right) = e^z M\\left(q_2-q_1, q_2, -z\\right), \\\\\ny_2(\\chi) &= z^{1-q_2} M\\left(q_1 - q_2 + 1, 2-q_2, z\\right) =\nz^{1-q_2} e^z M\\left(1-q_1, 2-q_2, -z\\right), \\nonumber \\\\\nW &= \\sin(\\pi q_2) z^{-q_2} e^{z}\/\\pi. \\nonumber\n\\end{align}\nHowever, in the vicinity of infinity the numerically satisfactory pair is, \\cite{Olver1997}\n\\begin{align} \\label{Kummer1}\ny_1(\\chi) &=   U\\left(q_1, q_2, z\\right) = z^{1-q_2} U\\left(q_1-q_2+1, 2-q_2, z\\right), \\\\\ny_2(\\chi) &= e^z U\\left(q_2 - q_1, q_2, -z\\right) = e^z z^{1-q_2} U\\left(1-q_1, 2-q_2, -z\\right), \\nonumber \\\\\nW &= (-1)^{q_1 - q_2} e^{z} z^{-q_2}. \\nonumber\n\\end{align}\n\nAs two solutions $J_1(q_1,q_2, z), J_2(q_1,q_2, z)$ are independent, \\eqref{solInhom} is a general solution of \\eqref{Laplace3}. Two constants $C_1,C_2$ should be determined based on the boundary conditions for the function $V(z)$.\n\nThe boundary conditions for the ODE \\eqref{Laplace2} in a strike $K$ space (or in $x$ space) should be set at zero and infinity. Based on the usual shape of the local variance curve and its positivity, for $x \\to 0$, we expect that $v^1_{j,i} < 0$.  Similarly, for $x \\to \\infty$ we expect that $v^1_{j,i} > 0$. In between these two limits the local variance curve for a given maturity $T_j$ is assumed to be continuous, but the slope of the curve could be both positive and negative, see, e.g., \\cite{ItkinSigmoid2015} and references therein. Also, by definition $z = v_{j,i}$, and $\\mathrm{Dom}(z) = \\mathbb{R}^+$. Thus, at high strikes $a_2 = v^1_{j,i} > 0$. Therefore, the boundary conditions for \\eqref{Laplace3} should be set at $z = b_2$ (which corresponds to the boundary $K=0$) and at $z \\to \\infty$. These are the boundary conditions given in \\eqref{bc}.\n\n\\section{Computation of the source term} \\label{SolutionInt}\n\nComputation of the source term $p I_{12}$ in \\eqref{solInhom} could be achieved in several ways. The most straightforward one is to use numerical integration as the Put price $P(x,T_{i-1})$ as a function of $x$ is already known when we solve \\eqref{finODE} for $T=T_i$. However, as this is discussed in detail in \\cite{ItkinLipton2017}, function $P(x,T_{i-1})$ is known only for a discrete set of points in $x$. Therefore, some kind of interpolation is necessary to find its values at the other points.\n\n\\subsection{No-arbitrage interpolation}\n\nAs shown in \\cite{ItkinLipton2017}, this interpolation must preserve no-arbitrage. So, for instance, a standard linear interpolation is not a good candidate, since its violates no-arbitrage conditions. Indeed, given three Put option prices $P(K_1), P(K_2), P(K_3)$ for three strikes $K_1 < K_2 < K_3$, the necessary and sufficient conditions for an arbitrage-free system read, \\cite{CoxRubinstein1985}\n\\begin{align} \\label{noarb}\nP(K_3) &> 0, \\qquad P(K_2) < P(K_3), \\\\\nBs = (K_3 - K_2)P(K_1) &- (K_3 - K_1)P(K_2) + (K_2 - K_1)P(K_3) > 0. \\nonumber\n\\end{align}\nSuppose that we want to use linear interpolation in the strike space on the interval $[K_1,K_3]$ to find the unknown Put option price $P(K_2)$ given the values of $P(K_1), P(K_3)$,\n\\begin{equation*}\nP(K_2) \\equiv P_l(K_2) = \\dfrac{P(K_1) K_3 - P(K_3) K_1}{K_3 - K_1} + \\dfrac{P(K_3) - P(K_1)}{K_3 - K_1} K_2.\n\\end{equation*}\nWhen plugging this expression into the second line of \\eqref{noarb}, the left hands side of the latter vanishes, so the third no-arbitrage condition is violated.\n\nIn \\cite{ItkinLipton2017} it is shown that this problem could be resolved if we use linear interpolation with a modified independent variable (further on we denote it as $P_F(K)$),\n\\begin{align} \\label{lin2}\nP(K_2) &\\equiv P_F(K_2) \\\\\n&= \\dfrac{P(K_1) f(K_3) - P(K_3) f(K_1)}{f(K_3) - f(K_1)} + \\dfrac{P(K_3) - P(K_1)}{f(K_3) - f(K_1)} f(K_2), \\nonumber\n\\end{align}\n\\noindent where $f(K)$ is a convex and increasing function in $[K_1,K_3]$. Indeed, if $f(K)$ is convex, then $P(K_2) = P_F(K_2) = P_l(K_2) - \\varepsilon, \\ \\varepsilon > 0$ (see Fig.~2 in \\cite{ItkinLipton2017}). Substitution of this expression into the second line of \\eqref{noarb} gives $(K_3 - K_1) \\varepsilon > 0$, which is true. The second condition in \\eqref{noarb} now reads\n\\[ (P(K_1) - P(K_3))(f(K_3) - f(K_2))(f(K_1) - f(K_3)) > 0, \\]\n\\noindent which is also true since $f(K)$ is an increasing function of $K$.\n\nAlternatively, one can use non-linear interpolation. In \\cite{ItkinLipton2017}) both approaches were combined, and it was proved that the new interpolation scheme preserves no-arbitrage. Moreover, the final representation of the modified Put price (which is a dependent variable in their approach) obtains a nice tractable representation, so the integral $I_{12}$ can be computed in closed form. Here we want to exploit the same idea, thus significantly improving performance of our model as compared with the numerical integration.\n\nTherefore, here we propose the following interpolation scheme\n\\begin{align} \\label{linNew}\nP(x) &\\equiv P_F(x) = \\gamma_1 + \\gamma_2 x^2, \\quad x_1 \\le x \\le x_3, \\\\\n\\gamma_1 &= \\dfrac{P(x_3) x_1^2   - P(x_1) x^2_3}{x_1^2 - x_3^2}, \\qquad\n\\gamma_2 = \\dfrac{P(x_1) - P(x_3)}{x_1^2 - x_3^2}. \\nonumber\n\\end{align}\nThen Proposition similar to that in \\cite{ItkinLipton2017} can be proved.\n\\begin{proposition} \\label{prop1}\nThe interpolation scheme in \\eqref{linNew} is arbitrage free at the interval $[K_1,K_3]$.\n\\end{proposition}\n\\begin{proof}\nObserve, that the no-arbitrage conditions in \\eqref{noarb} are discrete versions of the conditions\n\\[ P > 0, \\quad P_K > 0, \\quad P_{K,K} > 0, \\]\nThey, in turn, correspond to the conditions\n\\[ P > 0, \\quad P_x > 0, \\quad P_{x,x} > 0, \\]\n\\noindent as $x'(K) = 1\/S > 0$. By differentiating the first line of \\eqref{linNew} one can check that the proposed interpolation obeys these conditions provided that $P$ is an increasing function of $K$ (or $x$) given the values of all other parameters to be constant. For instance, this is the case for the Black-Scholes Puts.\n\\hfill $\\blacksquare$\n\\end{proof}\n\nAs by definition $z$ is a linear function of $x$, a similar interpolation scheme can be used in the $z$ space, with a similar proof of no-arbitrage.\n\n\\subsection{No-arbitrage at consecutive intervals}\n\nProposition~\\ref{prop1} guarantees that the proposed interpolation doesn't introduce an arbitrage into the solution if any three strikes belong to the same interval $[K_1,K_3]$. However, what if we consider strikes $K_2, K_3, K_4$ as this is schematically  depicted in Fig.~\\ref{str2interv}.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[line\/.style={<->},thick, framed, scale=1.4,\nredline\/.style={shape=rectangle, draw=red, line width=2},\nblueline\/.style={shape=rectangle, draw=blue, line width=2}\n]\n\n\\draw[->] (-3.0,0) -- (5,0) node[right] {$K$};\n\\draw[->] (-3.0,-0.2) -- (-3.0,4.5) node[right] {$P(K)$};\n\\draw[red,ultra thick] (-2,1) parabola (4.5,3.5);\n\\node at (-3.0,-0.3) {$0$};\n\n\\node at (-2,-0.3) {$K_1$};\n\\node at (-2,1.3) {$P_1$};\n\\node at (-2,1.) {$\\bullet$};\n\\draw[red, dashed] (-2,0) -- (-2,1.);\n\n\\node at (-0,-0.3) {$K_2$};\n\\node at (0.2,0.6) {$P_2$};\n\\node at (-0,0.8) {$\\bullet$};\n\\draw[red, dashed] (-0,0) -- (-0,0.8);\n\n\\node at (1.5,-0.3) {$K_3$};\n\\node at (1.5,2.) {$P_3$};\n\\node at (1.5,1.7) {$\\bullet$};\n\\draw[red, dashed] (1.5,0) -- (1.5,1.7);\n\n\\node at (2.5,-0.3) {$K_4$};\n\\node at (2.7,1.5) {$P_4$};\n\\node at (2.5,1.65) {$\\bullet$};\n\\draw[red, dashed] (2.5,0) -- (2.5,1.65);\n\n\\node at (4.5,-0.3) {$K_5$};\n\\node at (4.5,3.8) {$P_5$};\n\\node at (4.5,3.5) {$\\bullet$};\n\\draw[red, dashed] (4.5,0) -- (4.5,3.5);\n\n\\draw[blue,ultra thick] (-2,1.) parabola bend (-0.2, 0.8) (1.5,1.7);\n\\draw[blue,ultra thick] (1.5,1.7) parabola bend (2., 1.6) (4.5,3.5);\n\n\\matrix [draw,below left] at (1.5, 4.5) {\n  \\node [redline,label=right:Exact solution] {}; \\\\\n  \\node [blueline,label=right:Interpolation] {}; \\\\\n};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Three strikes $K_2, K_3, K_4$ which belong to the consecutive intervals.}\n\\label{str2interv}\n\\end{figure}\n\n\nHere at the interval $[K_1,K_3]$ the exact solution is depicted by the red line, while our quadratic interpolation is in blue. Accordingly, the Put prices $P_1,P_3$ are the market quotes, so they assumed to be the exact prices with no market arbitrage. By our construction, these prices also don't have a model arbitrage. At the consecutive interval $[K_3,K_5]$ a similar construction applies.\n\nWe have to emphasize that this graph is pure illustrative, and no-arbitrage interpolation guarantees  that $P'(K) > 0$, while the blue line in Fig.~\\ref{str2interv} doesn't support this. However, if we draw an accurate picture by using the above formulae, it would be almost impossible to distinguish the red and blue lines. Therefore, we changed convexity and skew of the blue line to make the difference visible.\n\nBy Proposition~\\ref{prop1} given a set of strikes $K_1, K_2, K_3$ the price $P_2$ obtained by interpolation preserves no-arbitrage. The same is true for $P_5$ given the Put prices $P_3, P_5$ at strikes $K_3, K_5$.\nNow assume that given $K_1, K_3, K_5$ and $P_1, P_3, P_5$ we want to check the no-arbitrage conditions for the set of strikes $K_2, K_3, K_4$. The Proposition~\\ref{prop1} doesn't help in this situation, so we need a special consideration of this case.\n\nObviously, the first and second conditions in \\eqref{noarb} are still satisfied in this case, so we need to check that the butterfly spread is positive. Unfortunately, at the moment we don't have a general analytical solution of this problem, while some particular cases can be addressed. Thus this remains an open question. However, we checked this condition numerically. In doing so we used the Black-Scholes Put prices $P_1, P_3, P_5$\\footnote{This is done to preserve upper bounds on the Put price that $P(S, K,T,r) \\le K e^{-rT}$, \\cite{Levy1985}.} and built a 2D plot of $Bs$ which is the left-hands side of the third line in \\eqref{noarb}. The results for two cases presented in Table~\\ref{tabCases} are presented in Fig.~\\ref{intCase1}, \\ref{intCase2}.\n\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{|r|r|r|r|r|r|r|r|}\n\\hline\nTest & $S$ & $r$ & $\\sigma_{BS}$ & $T$ & $K_1$ & $K_3$ & $K_5$ \\\\\n\\hline\n1 & 100 & 0.01 & 0.5 & 2 & 80 & 100 & 130 \\\\\n\\hline\n2 & 100 & 0.1 & 0.1 & 0.1 & 90 & 100 & 105 \\\\\n\\hline\n\\end{tabular}\n\\caption{Parameters of the test for non-negativity of the Butterfly spread. $\\sigma_{BS}$ is the Black-Scholes implied volatility.}\n\\label{tabCases}\n\\end{center}\n\\end{table}\n\nOverall, we ran a lot of tests and didn't find any case where the butterfly spread would become negative. This partly supports our no-arbitrage interpolation. More sophisticated cases where, e.g., instead of strike $K_3$ in the butterfly spread at strikes $K_2, K_3, K_4$ we use another strike $K_6$ such that $K_1 < K_2 < K_3 < K_4 < K_6 < K_5$, could be treated in a similar way. Again, our numerical tests didn't reveal any case where a butterfly spread would become negative.\n\n\\begin{figure}[h!]\n\\begin{minipage}{0.46\\linewidth}\n\\begin{center}\n\\fbox{\\includegraphics[width=\\linewidth]{test1.png}}\n\\caption{Butterfly spread $Bs$ for a set of strikes $K_2, K_3, K_4$ computed in Test 1 in Table~\\ref{tabCases}.}\n\\label{intCase1}\n\\end{center}\n\\end{minipage}\n\\hspace{0.04\\linewidth}\n\\begin{minipage}{0.46\\linewidth}\n\\begin{center}\n\\fbox{\\includegraphics[width=\\linewidth]{test2.png}}\n\\caption{Butterfly spread $Bs$ for a set of strikes $K_2, K_3, K_4$ computed in Test 2 in Table~\\ref{tabCases}.}\n\\label{intCase2}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Computing the integrals in \\eqref{solInhom} far from $z = 0$}\n\nUsing the interpolation scheme proposed in above, consider the first integral in \\eqref{solInhom}.  To remind, we compute it at some interval $z \\in [z_i,z_{i+1}], \\  \\irg{i}{1}{n_j}$. Picking together the solutions in \\eqref{Kummer0} with the interpolation scheme for $P(z,T_{j-1})$ and Wronskians in \\eqref{Kummer0}, and substituting them into the first integral in \\eqref{solInhom} we obtain\n\\begin{align} \\label{Int1}\n\\int & \\dfrac{ y_2(z) f(z,T_{j-1})}{W z}d z = A \\Big[-B_0\n+ B_1 M(-2-q_1, -1-q_2, -z)\n+ B_2 M(-1-q_1, -q_2, -z) \\\\\n&+ B_3 M(-q_1, 1-q_2, -z)\\Big], \\nonumber \\\\\nA &= \\dfrac{1}{b_1^2 q_1} \\pi(1-q_2) \\csc(\\pi q_2), \\nonumber \\\\\nB_0 &= \\frac{1}{a_2^2 (q_1+1) (q_1+2)} \\Big[\na_2 b_1 (q_1+2) \\left(a_2^2 \\beta q_2 - b_1 (q_1+1) (\\beta  b_2-a_2 \\gamma_1)\n\\right) \\nonumber \\\\\n&+ \\gamma_2 \\left(2 a_2^4 q_2 (q_2+1)-2 a_2^2 b_1 b_2 (q_1+2)q_2 + b_1^2 b_2^2 (q_1+1) (q_1+2) \\right) \\Big],\n\\nonumber \\\\\nB_1 &= 2 a_2^2 \\gamma_2 \\frac{q_2 (q_2+1)}{(1+q_1)(2+q_1)}, \\nonumber \\\\\nB_2 &= (a_2 b_1 \\beta - 2 b_1 b_2 \\gamma_2 + 2 a_2^2 \\gamma_2 z)  \\frac{q_2}{1+q_1}, \\nonumber \\\\\nB_3 &= \\frac{1}{a_2^2}\n\\left[a_2 b_1\\left(a_2^2 \\beta  z + a_2 b_1 \\gamma_1 - \\beta b_1 b_2 \\right)\n + \\gamma_2 \\left(b_1 b_2 - a_2^2 z\\right)^2\\right]. \\nonumber\n\\end{align}\nSimilarly\n\\begin{align} \\label{Int2}\n\\int & \\dfrac{ y_1(z) f(z,T_{j-1})}{W z}d z = \\bar{A} \\Big[\n\\bar{B}_1 M(q_2-q_1, 1+q_2, -z)\n+ \\bar{B}_2 \\ \\pFq{2}{2}{q_2-q_1, 1+q_2; q_2,2+q_2; -z} \\nonumber \\\\\n&+ \\bar{B}_3 \\ \\pFq{2}{2}{q_2-q_1, 2+q_2; q_2,3+q_2; -z} \\Big], \\\\\nA &= \\pi z^{q_2} \\csc(\\pi q_2) \\Gamma(q_2), \\nonumber \\\\\nB_1 &=\n\\frac{a_2^2 \\gamma_1 - a_2 \\beta  b_2 + b_2^2 \\gamma_2}{a_2^2 \\Gamma(1+q_2)}, \\nonumber \\\\\nB_2 &=\\frac{\\Gamma(q_2+1) }{\\Gamma(q_2) \\Gamma(2+q_2) b_1} (a_2 \\beta -2 b_2 \\gamma_2) z, \\nonumber \\\\\nB_3 &= \\frac{\\Gamma(q_2+1) }{\\Gamma(q_2) \\Gamma(3+q_2) b_1^2}\na_2^2(1+q_2)\\gamma_2 z^2, \\nonumber\n\\end{align}\n\\noindent where $\\pFq{p}{q}{{a_1,...,a_p}; {b_1,...,b_q}; z}$ is the generalized hypergeometric function, \\cite{Olver1997}.\n\n\n\\subsection{Computing the integrals in \\eqref{solInhom} far from $z = \\pm \\infty$}\n\nHere we proceed in the same way as in the previous section. Again, we pick together the solutions in \\eqref{Kummer1} with the interpolation scheme for $P(z,T_{j-1})$ and Wronskians in \\eqref{Kummer1}, and substitute them into the first integral in \\eqref{solInhom} we obtain\n\\begin{align} \\label{Int11}\n\\int & \\dfrac{ y_2(z) f(z,T_{j-1})}{W z}d z = (-1)^{q_2 - q_1} [C_0 J_0 + C_1 J_1 + C_2 J_2], \\\\\nJ_i &= \\int z^i U(1-q_1, 2-q_2, -z) dz, \\nonumber \\\\\nC_0 &= \\frac{b_2^2 \\gamma_2}{a_2^2} - \\frac{\\beta b_2}{a_2} + \\gamma_1, \\quad\nC_1 = \\frac{a_2 \\beta - 2 b_2 \\gamma_2}{b_1}, \\quad\nC_2 = \\frac{a_2^2 \\gamma_2}{b_1^2}. \\nonumber\n\\end{align}\nIt is known, \\cite{as64}, that\n\\begin{equation*}\nJ_0 = - \\frac{1}{q_1}U(-q_1, 1-q_2, -z).\n\\end{equation*}\nThen, $J_1, J_2$ can be found using integration by parts to yield\n\\begin{align*}\nJ_1 &= z J_0 + \\frac{1}{q_1(1+q_1)}U(-1-q_1, -q_2, -z), \\\\\nJ_2 &= z J_1 + \\frac{1}{q_1(2+3 q_1 + q_1^2)}U(-2-q_1, -1-q_2, -z)\n - z\\frac{1}{q_1(1 + q_1)}U(-1-q_1,-q_2,-z).\n\\end{align*}\nSimilarly\n\\begin{align} \\label{Int21}\n\\int & \\dfrac{ y_1(z) f(z,T_{j-1})}{W z}d z = (-1)^{q_2 - q_1} [C_0\n{\\mathcal J}_0 + C_1 {\\mathcal J}_1 + C_2 {\\mathcal J}_2], \\\\\n{\\cal J}_i &= \\int z^i e^{-z} U(1+q_1-q_2, 2-q_2, z) dz. \\nonumber\n\\end{align}\nThe integrals ${\\mathcal J}_i $ have been considered in \\cite{ItkinLipton2017} using the approach of \\cite{kummerInt1970}. Borrowing from there the result\n\\begin{equation*}\n{\\mathcal J}_0 = \\int e^{-z} U(1+q_1-q_2,2-q_2,z) d z = -e^{-z} U(q_1-q_2,1-q_2,z),\n\\end{equation*}\n\\noindent and using integration by parts, we obtain\n\\begin{align*}\n{\\mathcal J}_1 &= z {\\mathcal J}_0 + e^{-z}U(q_1-q_2-1, -1-q_2, z), \\\\\n{\\mathcal J}_2 &= z {\\mathcal J}_1 - \\int  {\\mathcal J}_1 dz = (z-1) {\\mathcal J}_1 - \\int e^{-z}U(q_1-q_2-1, -1-q_2, z) dz \\\\\n&= (z-1) {\\mathcal J}_1 + e^{-z}U(q_1-q_2-2, -2-q_2, z).\n\\end{align*}\n\n\n\\subsection{Some additional notes}\n\nBased on the no-arbitrage interpolation and some analytics proposed in this Section, we managed to find the solution \\eqref{solInhom} of the forward equation  \\eqref{finODE} in closed form. This solution by construction is arbitrage free at any interval where the local variance function defined in \\eqref{vDef} is linear. In other words we proved, that if we consider, say 3 strikes $0 < K_1 < K_2 < K_3 < \\infty$ such that, e.g., $x_1  = K_1\/S \\in [x_i, x_{i+1}]$,  $x_2  = K_2\/S \\in [x_i, x_{i+1}]$,  $x_3  = K_3\/S \\in [x_{i}, x_{i+1}]$, then the solution at these 3 points obeys no-arbitrage conditions.\n\n\n\\section{Calibration of smile for a given term $T_i$} \\label{calib}\n\nCalibration problem for the local volatility model can be formulated as follows: given market quotes of Call and\/or Put options corresponding to a set of $N$ strikes $\\{K\\}:= K_j, \\ j \\in [1,N]$ and same maturity $T_i$, find the local variance function $\\sigma(K)$ such that these quotes solve equations in \\eqref{finDupPutI}, \\eqref{finDupCallI}.\n\nAs mentioned in \\cite{ItkinLipton2017}, there are two main approaches to solving this problem. The first approach attempts to construct a continuous implied volatility (IV) surface matching the market quotes by using either some parametric or non-parametric regression, and then generates the corresponding LV surface via the well-known relationship between the local and implied variances also known as the Dupire formula, see, e.g., \\cite{ItkinSigmoid2015} and references therein. To be practically useful, this construction should guarantee no arbitrage for all strikes and maturities, which is a serious challenge for any model based on interpolation. If the no-arbitrage condition is satisfied, then the LV surface can be calculated using the Dupire formula. The second approach relies on the direct solution of the corresponding forward equation (which is the Dupire equation in the Black-Scholes world, or \\eqref{finDupCallI}, \\eqref{finDupPutI} in our model) using either analytical or numerical methods. The advantage of this approach is that it guarantees no-arbitrage. However, the problem of the direct solution can be ill-posed, \\cite{Coleman2001}, and is rather computationally intensive.\n\nIn this Section we show that the second approach could be significantly simplified when using the ELVG model, so calibration of the smile could be done very fast and accurate.\n\nFurther, for the sake of certainty, suppose that all known market quotes are Puts, despite this can be easily relaxed. Also, suppose that the shape of a local variance is given by some function $\\sigma_j(K) = f_j(K, p_1,\\ldots,p_L)$, where $p_1,\\ldots,p_L$ is a set of the model parameters to be determined. For instance, in \\cite{LiptonSepp2011, CarrNadtochiy2017} the local variance is assumed to be a piecewise constant function of strike, while in \\cite{ItkinLipton2017} this is a piecewise linear function of strike.\n\nIn this paper we also assume the local variance to be a piecewise linear function of strike. Moreover, for our model we obtained a closed form representation of the Put option prices via parameters of the model given in Sections~\\ref{SolutionODE},\\ref{SolutionInt}. Therefore, calibration of the model to the given set of smiles could be provided as follows. First, using the above-mentioned closed form solution for a fixed interval in $x$ where parameters of the model are constant, we construct the combined solution valid for all $x \\in \\mathbb{R}^+$. At the second step, the parameters of the local variance function $v^0_{j,i}, v^1_{j,i}$ can be found together with the integration constants $C_1, C_2$ in \\eqref{solInhom} by solving a system of non-linear algebraic equations.\n\n\n\\subsection{The combined solution in $x \\in \\mathbb{R}^+$ \\label{wholeSol}}\n\nSuppose that the Put prices for $T=T_j$ are known for $n_j$ ordered strikes. The location of these strikes on the $x$ line is schematically depicted in Fig.~\\ref{Fig3}.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[line\/.style={<->},thick, framed, scale=1.4]\n\n\\draw[->] (-3.0,0) -- (5,0) node[right] {$x$};\n\\draw[->] (-3.0,-0.2) -- (-3.0,3.5) node[right] {$v(x)$};\n\\draw[red,ultra thick] (0.5,0.5) parabola (4.5,1.5);\n\\draw[red,ultra thick] (0.5,0.5) parabola (-2,3);\n\\node at (-3.0,-0.3) {$0$};\n\\node at (-2,-0.3) {$x_1$};\n\\node at (-2,3.) {$\\bullet$};\n\\draw[red, dashed] (-2,0) -- (-2,3.);\n\\node at (-2.5,0.5) {$B_1$};\n\\draw[blue, dashed] (-2,3.) -- (-2.8,3.2);\n\\node at (-1,-0.3) {$x_2$};\n\\node at (-1,1.4) {$\\bullet$};\n\\draw[red, dashed] (-1,0) -- (-1,1.4);\n\\node at (-1.5,0.5) {$B_{12}$};\n\\draw[blue, dashed] (-2,3.) -- (-1,1.4);\n\\node at (1.5,-0.3) {$x_3$};\n\\node at (1.5,0.55) {$\\bullet$};\n\\draw[red, dashed] (1.5,0) -- (1.5,0.55);\n\\node at (-0.5,0.5) {$B_{23}$};\n\\draw[blue, dashed] (-1,1.4) -- (1.5,0.55);\n\\node at (2.5,-0.3) {$\\ldots$};\n\\node at (3.5,-0.3) {$x_{n_j}$};\n\\node at (3.5,1.05) {$\\bullet$};\n\\draw[red, dashed] (3.5,0) -- (3.5,1.05);\n\\draw[blue, dashed] (1.5,0.55) -- (3.5,1.05);\n\\draw[blue, dashed] (3.5,1.05) -- (4.5,2.);\n\\node at (4,0.5) {$B_{n_j}$};\n\\node at (0.5,1.3) {$2$};\n\\node at (0.7,0.3) {$1$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Construction of the combined solution in $x \\in \\mathbb{R}^+$: 1 (red solid line - the real (unknown) local variance curve, 2 (dashed blue line) - a piecewise linear solution.}\n\\label{Fig3}\n\\end{figure}\n\nRecall, that the Put prices are given by \\eqref{solInhom}, which in a more convenient form at the interval $x_{i-1} \\le x \\le x_i$ and at $T=T_j$ can be represented as\n\\begin{align} \\label{combPut}\nP_i(x) &= \\CO{i}{j} {\\cal J}_1(q_1, q_2, z)\n+ \\CT{i}{j} {\\cal J}_2(q_1, q_2, z) + \\frac{1}{b_1}I_{12}(z) + {\\cal D}_j K, \\\\\nz &\\equiv (b_2 + a_2 x)b_1\/a_2^2 = (v_{j,i}^0 + v_{j,i}^1 x)b_1\/a_2^2. \\nonumber\n\\end{align}\nHere, for consistency we change notation of two integration constants which belong to the $i$-th interval in $x$ and $j$-th maturity to $\\CO{i}{j},\\CT{i}{j}$.\n\nFor the open interval $B_1$ in Fig.~\\ref{Fig3}, since function $K_\\nu(z)$ diverges when  $z \\to 0$, we have to put $\\CO{1}{j} = 0$ as the boundary condition\\footnote{\nActually, since $x \\to 0$ implies $z = v \\to b_2$, so $b_2$ should be non-negative, $b_2 \\ge 0$. Therefore, the only case when $z \\to 0$ at $x \\to 0$ is when $b_2 = 0$.}. Therefore, \\eqref{combPut} contains just one yet unknown constant $\\CT{1}{j}$. For the closed intervals $x \\in [x_{i-1}, x_{i}], \\ i \\in [2,n_j]$ the solutions in \\eqref{combPut} have two yet unknown constants $\\CO{i}{j}, \\CT{i}{j}$, since $x$ is finite on the corresponding intervals, and both solutions $y_1(x), y_2(x)$ are well-behaved. Finally, for the interval $x \\in [x_{n_j}, \\infty)$, according to the boundary conditions in \\eqref{bc} we must set $C^{(2)}_{2,n_j+1} = 0$.\n\nRigorously speaking, we also have to show that in the limits $x \\to 0$ and $x \\to \\infty$ the source term $I_{12}(z)$ in \\eqref{solInhom} also vanishes. This could be done similar to Proposition~2 in \\cite{ItkinLipton2017}.\n\nThus, we have $2 n_j$ unknown constants to be determined. Since the local volatility function $v_i$ is continuous at the points $x_i, \\ i=1,\\ldots,n_j$, so should be the Put options prices $P(x,T_j)$. Therefore, we require that at the points $x_i, \\ i = 1,...,n_j$ the solution for Puts and its first derivative in $x$ should be a continuous function of $x$. Thus, if the local variance function is known, the above constants solve a system of $2 n_j$ algebraic equations. This system has a block-diagonal structure where each block is a 2x2 matrix. Therefore, it can be easily solved with the linear complexity $O(n_j)$.\n\nWhen computing the first derivatives, we take into account that, \\cite{as64}\n\\begin{align} \\label{der}\n\\fp{M(a,b,z)}{z} &= \\dfrac{a}{b}M(a+1,b+1,z), \\quad\n\\fp{U(a,b,z)}{z} = -a U(a+1,b+1,z), \\\\\n\\partial_z I_{12}(z) &= \\left[\\frac{y'_{1}(z)}{y_{1}(z)} I_1 + \\frac{y'_{2}(z)}{y_{2}(z)} I_2 \\right] a_2. \\nonumber\n\\end{align}\nTherefore, computing the derivatives of the solution doesn't cause any new technical problem.\n\n\\subsection{Additional equations for calibration}\n\nAs we have already mentioned above, the standard way of doing calibration of the local volatility model would be that described, e.g., in \\cite{ItkinLipton2017}. Namely, given the maturity $T_j$ and some initial guess of the local variance parameters $v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$, the following steps represented in Panel~\\ref{Algo} have to be achieved, e.g., in the standard least-square method,\n\\begin{center}\n\\begin{algorithm}[H]\n \\KwIn{Strikes $z_i, \\ i \\in [1,n_j]$, Put prices  $V^{market}_i, \\ i \\in [1,n_j]$}\n \\KwOut{$v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$}\n \\emph{{\\bf Initialization}: The initial guess of $v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$, the tolerance $\\epsilon$} \\;\n\\While{1}{\n  1. Solve the system for $\\CO{i}{j}, \\CT{i}{j}$ \\;\n  2. Compute Put option prices $V(x)$\\;\n  3. Compute the total error $\\Delta = \\sum_{i=1}^{n_j} [V(x_i) - V^{market}(x_i)]^2$\\;\n    \\eIf{ $\\Delta > \\epsilon$}{\n        New guess for $v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$\\;\n    }{\n        break\\;\n    }\n }\n\\caption{Calibration of the local volatility model using a least-square method.}\n\\label{Algo}\n\\end{algorithm}\n\\end{center}\n\\vspace{0.2in}\nHere $V^{market}(z_i)$ are the market Put quotes at the given strikes and maturity. Obviously, when the number of calibration parameters (strikes) is high, this algorithm is slow even if the closed form solution is known and can be used at Step~2. Things become even worse when a numerical solution at Step~2 has to be used if the closed form solution is not available.\n\nHowever, in our case this tedious algorithm can be fully eliminated. Indeed, at every point $i$ in strike space, $i \\in [1,n_j]$ we have four unknown variables $v_{j,i}^0, v_{j,i}^1, \\CO{i}{j}, \\CT{i}{j}$. We also have four equations which contain these variables, namely\n\\begin{align} \\label{finSystem}\nP_i(x)|_{x=x_i} &= P_{i+1}(x)|_{x = x_i}, \\\\\nP_i(x)|_{x=x_i} &= P_{market}(x_i), \\nonumber \\\\\n\\fp{P_{i+1}(x)}{x}\\Big|_{x = x_i} &=  \\fp{P_{i}(x)}{x}\\Big|_{x = x_i}, \\nonumber \\\\\nv^0_{j,i} + v^1_{j,i} x_{i} &= v^0_{j,i+1} + v^1_{j,i+1} x_{i}, \\quad i=1,\\ldots,n_j. \\nonumber\n\\end{align}\nAlso, based on \\eqref{cont}, the last line in \\eqref{finSystem} could be re-written as a recurrent expression\n\\begin{equation} \\label{recurr}\nv^0_{j,i} = v^0_{j,n_j} + \\sum_{k=i+1}^{n_j} x_{k}(v^1_{j,k} - v^1_{j,k-1}) , \\quad i=0,\\ldots,n_j-1.\n\\end{equation}\n\nThe \\eqref{finSystem} is a system of $4 n_j$ nonlinear equations with respect to $4 (n_j+1)$ variables $v_{j,i}^0$, $v_{j,i}^1$, $\\CO{i}{j}$, $\\CT{i}{j}$. We remind that according to the boundary conditions $\\CO{1}{j} = \\CT{n_j}{j} = 0$. Therefore, we need two additional conditions to provide a unique solution. For instance, often traders have an intuition about the asymptotic behavior of the volatility surface at infinity, which, according to our construction, is determined by $v^1_{j,n_j}$ and $v^1_{j,0}$.\n\nOverall, solving the nonlinear system of equations \\eqref{finSystem} provides the final solution of our problem. This can be done by using standard methods, and, thus, no any optimization procedure is necessary. However, a good initial guess still would be helpful for a better (and faster) convergence.\n\n\\subsection{Smart initial guess}\n\nThe initial guess of the solution of \\eqref{combPut} can be constructed, for instance, as follows. We take advantage of the fact that according to \\eqref{finODE} the local variance function $v(x)$ could be explicitly expressed as\n\\begin{equation} \\label{localVarappr}\nv(x) = \\dfrac{b_1 x V_x(x) + b_0 V(x) - c}{V_{x,x}(x)}.\n\\end{equation}\n\nGiven maturity $T_j$ and approximating derivatives by central finite differences with the second order of approximation in step $h$ in the strike space (see, e.g. \\cite{ItkinBook}), \\eqref{localVarappr} can be represented in the form\n\\begin{align} \\label{localVarappr1}\nv^0_{j,i} + v^1_{j,i} x_{i} &= \\dfrac{b_1 x V_x(x_i) + b_{0,j} V(x_i) - c_j}{V_{x,x}(x_i)}, \\\\\nV_{x}(x_i) &= \\alpha_{-1} V(x_{i-1}) + \\alpha_{0} V(x_{i}) + \\alpha_{1} V(x_{i+1}), , \\nonumber \\\\\nV_{x,x}(x_i) &= \\delta_{-1} V(x_{i-1}) + \\delta_{0} V(x_{i}) + \\delta_{1} V(x_{i+1}), \\nonumber \\\\\n\\alpha_{-1} &= - \\dfrac{h_{i+1}}{h_{i}(h_{i+1} + h_{i})}, \\quad\n\\alpha_{0} = \\dfrac{h_{i+1}-h_{i}}{h_{i+1}h_{i}}, \\quad\n\\alpha_{1} = \\dfrac{h_{i}}{h_{i+1}(h_{i+1} + h_{i})}. \\nonumber \\\\\n\\delta_{-1} &=  \\dfrac{2}{h_{i}(h_{i+1} + h_{i})}, \\quad\n\\delta_{0} = -\\dfrac{2}{h_{i+1}h_{i}}, \\quad\n\\delta_{1} = \\dfrac{2}{h_{i+1}(h_{i+1} + h_{i})}. \\nonumber \\\\\nh_i &= x_i-x_{i-1}, \\quad i \\in [1,n_j]. \\nonumber\n\\end{align}\nFurther, associating Put prices $P(S,T_j,x_{i})$ with the given market quotes, the right hands side of the first line in \\eqref{localVarappr1} can be found explicitly. This then can be combined with the last line of \\eqref{finSystem} to produce a system of $2(n_j-1)$ equations for $v^1_{j,i}$ and $v^1_{j,i}, \\ i \\in [1,n_j]$. Finally, we take into account the asymptotic behavior of the volatility surface in $x$ at zero and infinity, which, according to our construction, is determined by $v^1_{j,n_j}$ and $v^1_{j,0}$ and is assumed to be known. Thus, we obtain a closed system of $2(n_j-1)$ linear equations with a banded matrix which can be easily solved with a linear complexity. This provides an explicit representation of the local variance function over the whole set of intervals in the strike space determined according to our approximation where the continuous derivatives are replace by finite differences.\n\nNote, that at the first and last strike intervals the approximation of the first and second derivatives by central finite differences should be replaced by one-sided approximations, in more detail see \\cite{ItkinBook}, chapter 2.\n\nIt could also happen that at some strikes this solution (the smart guess) gives rise to a negative local variance. In such a case we do another step which is a kind of smoothing. Namely, we exclude from the initial guess all values where the local variance is negative and using the remaining points create a spline. Then the negative values in the initial guess are replaced by those given by the constructed spline.\n\nThe final step utilizes the exact representation \\eqref{combPut} of the Put price in the ELVG model. As the variance function is already known from the previous step, this equation contains two yet unknown constants $\\CO{i}{j},\\CT{i}{j}$. Accordingly, they can be found by solving the system of 2 linear equations represented by the first and third  lines of \\eqref{finSystem}. Then, after this last step is complete, all unknown variables are determined, and thus found solution could be used as an educated initial guess for solving \\eqref{finSystem} numerically.\n\n\\section{Asymptotic solutions} \\label{asympt}\n\nIn many practical situations either some coefficients $a_2 = v^1_{j,i}$, or both\n$b_2 = v^0_{j,i}, \\ a_2 = v^1_{j,i}$ in \\eqref{Laplace2} are small. Of course, in that case the general solution \\eqref{combPut} remains valid. However, in this case when computing the values of Kummer functions numerically, numerical errors significantly grow. This is especially pronounced when computing the integral $I_{12}$. The main point is that either the Kummer function takes a very small value, and then the constants $\\CO{i}{j},\\CT{i}{j}$ should be big to compensate, or vice versa. Resolution of this requires a high-precision arithmetics, and, which is more important, taking many terms in a series representation of the Kummer functions, which significantly slows down the total performance of the method.\n\nOn the other hand, to eliminate these problems we can look for asymptotic solutions of \\eqref{Laplace2} taking into account the existence of small parameters from the very beginning. This approach was successfully elaborated on in \\cite{ItkinLipton2017}, and below we proceed in a similar spirit.\n\n\\subsection{Small $a_2$}\n\nWe can build the solution of \\eqref{Laplace3} directly using an independent variable $x$ (so not switching to the variable $z$). We represent it as a series on the small parameter $a_2$, i.e.\n\\begin{equation} \\label{ser}\nV(x) = \\sum_{i=0}^\\infty a_2^i V_i(x).\n\\end{equation}\nIn the zero-order approximation by plugging \\eqref{ser} into \\eqref{Laplace3} and neglecting by terms proportional to $a_2 \\ll 1$ we obtain the following equation for $V_0(x)$\n\\begin{equation} \\label{asymA1}\n-b_2 V_{xx}(x) + b_1 x V_x(x) + b_0 V(x) = c.\n\\end{equation}\nThis equation is simpler than \\eqref{combPut}. Still, its solution is given by a general formula\n\\begin{equation*}\nV(x) = C_1 y_1(x) + C_2 y_2(x) + I_{12}(x),\n\\end{equation*}\n\\noindent but the fundamental solutions $y_1(x), y_2(x)$ now read\n\\begin{equation*}\ny_1(x) = {\\mathcal H} \\left(-\\frac{b_0}{b_1}, \\sqrt{\\frac{b_1}{2 b_2}} x\\right), \\qquad\ny_2(x) = M\\left(\\frac{b_0}{2 b_1}, \\frac{1}{2}, \\frac{b_1}{2 b_2}x^2\\right), \\end{equation*}\n\\noindent where ${\\mathcal H}(a, x), \\ a,x \\in \\mathbb{R}$ is the generalized Hermite polynomial $H_a(x)$, \\cite{as64}.\n\n\\subsection{Small $|z|$}\n\nBased on the definition of $z = (b_2 + a_2 x)b_1\/a_2^2$, this could occur in two cases: either at some finite interval in the strike space $|a_2| \\gg |b_1 x|, \\ |a_2| \\gg |b_2|$, or just $z$ is small, so $b_2$ and $a_2$ have the opposite signs.  In any case we have a small parameter under the high-order derivative. This equation belongs to the class of singularly perturbed differential equations, \\cite{Wasow1987}. It can be solved by using either the method of matching asymptotic expansions, \\cite{Nayfeh2000}, or the method of boundary functions, \\cite{VasBut1995}. The latter was used in \\cite{ItkinLipton2017} in a similar situation, so for further details we refer a reader to that paper.\n\nHowever, we can partly eliminate this by constructing solutions of \\eqref{Laplace2} using the original variable $x$. Then we have to consider various cases where instead of a small parameter $z$ some other combinations of parameters could be small or large. But if so, a general solution as a function of the original independent variable $x$ could be represented as regular series on the new small parameter. Then, truncating the series, one gets a simplified solution.\n\nTo make it more transparent let us represent the general solution of \\eqref{Laplace2} expressed in variable $x$, rather than in $z$, as this was done in \\eqref{solInhom}\n\\begin{align} \\label{combPutX}\nV(x) &= \\CO{i}{j} y_1(x) + \\CT{i}{j} y_2 (x) + I_{12}(x), \\\\\ny_i(x) &= a_2^k (b_2+a_2 x)^k {\\mathcal V}_i\\left(\n-1 - \\frac{b_0}{b_1} + \\frac{b_1 b_2}{a_2^2}, 2 - \\frac{b_1 b_2}{a_2^2},\n\\frac{b_1}{a_2^2}(b_2 + a_2 x)\\right), \\quad i=1,2.\\nonumber \\\\\nk &= 1 - \\frac{b_1 b_2}{a_2^2}. \\nonumber\n\\end{align}\nObserve, that based on the definition of $b_1$ in \\cref{finODE}, $b_1 \\approx (r-q)\\Delta T$, so usually small. Therefore, small $z$ doesn't mean that\n$w$ is necessarily small. Below we consider two cases.\n\n\\subsubsection{$w \\ll 1$}\n\nAs $|z| \\ll 1$ and $w \\ll 1$ we have $w \\ll |a_2^2\/b_1|$. So $a_2 \\ge \\sqrt{b_1}$.\nIn this case $w \\ll 1$ is an actual small argument. Therefore, the general solution \\eqref{combPutX} can be expanded into series on small $w$. The condition $0 < w \\ll 1$ implies that $a_2$ and $b_2$ have the opposite signs.  If $a_2 > 0$ (and so $b_2 < 0$), then in the zero-order approximation we obtain\n\\begin{align} \\label{asymZ}\ny_1(w) &= (a_2 w)^{k-1} \\left[\\dfrac{\\Gamma(-k)}{\\Gamma(b_0\/b_1)} a_2 w + O(w^2) \\right] - \\left(\\frac{b_1}{a_2^3}\\right)^{1-k} \\frac{\\Gamma(k-1)}{b_1\\Gamma(k + b_0\/b1)}(a_2 b_1 b_2 - b_0 a_2 w) + O(w^2) , \\nonumber \\\\\ny_2(w) &= (a_2 w)^{k-1} \\left[a_2 w + O(w^2)\\right].\n\\end{align}\nAs $b_1 > 0$ we have $k-1 > 0$.\n\nIf $a_2 < 0$ and $b_2 > 0$, then both RHS in \\eqref{asymZ} should be multiplied by a factor\n$\\exp(-2 i \\pi  b_1 b_2\/a_2^2)$.\n\n\\subsubsection{$a^2_2 \\gg |b_1 w|$}\n\nIn this case we can also expand the solution in \\eqref{combPutX} into series on small $z$ to obtain\n\\begin{align} \\label{asym1}\ny_1(z) &= \\frac{1}{\\Gamma(1 + q_1 - q_2)}\\left[\\Gamma(1 - q_2) - q_1 \\Gamma(- q_2) z\\right] + z^{-q_2} \\left[\\frac{\\Gamma(q_2-1)}{\\Gamma(q_1)}z + O(z^2)\\right] + O(z^2), \\\\\ny_2(w) &= 1 + \\frac{q_1}{q_2}z + O(z^2). \\nonumber\n\\end{align}\nNote, that based on the definition $q_2 = b_2 b_1\/a_2^2$, at large $a_2$ the coefficient $q_2$ could also be small. But $z\/q_2 = 1 + a_2 x\/b_2 = w\/b_2 = O(1)$.\n\n\n\\section{Numerical experiments} \\label{numExp}\n\nIn our numerical test we use the same data set as in \\cite{ItkinSigmoid2015, ItkinLipton2017}. This is done first, to compare performance and a quality of the fit for all those models. Also, we already know that these smiles are difficult to fit precisely, see discussions in \\cite{ItkinSigmoid2015, ItkinLipton2017}.\n\nTo remind, we take data from \\url{http:\/\/www.optionseducation.org} on XLF traded at NYSEArca on March 25, 2014. The spot price of the index is $S = 22.64$, and $r = 0.0148,\\  q=0.01$. The option implied volatilities ($I_{call}, I_{put}$) are given in Tables~\\ref{TabOptC},\\ref{TabOptP}. We take all OTM quotes and some ITM quotes which are\nvery close to the at-the-money (ATM). When strikes for Calls and Puts coincide, we take an average of $I_{call}$ and $I_{put}$ with weights proportional to $1 - |\\Delta|_c$ and $1-|\\Delta|_p$ respectively, where $\\Delta_c, \\Delta_p$ are option Call and Put deltas~\\footnote{By doing so we do take into account effects reported in \\cite{callPutIV}, who pointed out that the IVs calculated from  Call and  Put option prices corresponding to the same strike  do not coincide,  although  they  should  be  equal  in theory. Our weights are chosen according to a pure empirical rule of thumb, and a more detailed investigation of this effect is required.}.\n\\begin{sidewaystable*}[!htb]\n\\begin{center}\n\\footnotesize\n\n\\caption{XLF implied volatilities for the Put options.}\n\\label{TabOptC}\n\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n    \\multicolumn{1}{|c|}{\\multirow{2}[4]{*}{T}} & \\multicolumn{17}{c|}{K,Put} \\\\\n\\cline{2-18}          & 10    & 11    & 12    & 13    & 14    & 15    & 16    & 17    & 18    & 19    & 19.5  & 20    & 20.5  & 21    & 21.5  & 22    & 23 \\\\\n\\hline\n    4\/4\/2014 &       &       &       &       &       &       &       &       &       &       &       & 39.53 &       & 23.77 & 19.73 & 16.67 &  \\\\\n\\hline\n    4\/11\/2014 &       &       &       &       &       &       &       &       &       &       & 35.89 & 30.33 & 26.62 & 22.06 & 18.49 & 16.11 &  \\\\\n\\hline\n    4\/19\/2014 &       &       &       &       &       &       &       &       &       & 32.90 &       & 26.79 &       & 20.14 &       & 15.19 & 12.93 \\\\\n\\hline\n    5\/17\/2014 &       &       &       &       &       &       &       & 37.66 & 33.27 & 26.88 &       & 23.08 &       & 18.94 &       & 16.12 & 13.86 \\\\\n\\hline\n    6\/21\/2014 &       &       &       &       &       & 40.51 & 37.21 & 31.41 & 27.84 & 23.90 &       & 21.07 &       & 18.88 &       & 16.95 & 15.82 \\\\\n\\hline\n    7\/19\/2014 &       &       &       &       &       & 36.71 & 33.35 & 29.96 & 26.09 & 22.81 &       & 20.29 &       & 18.13 &       & 16.30 & 14.93 \\\\\n\\hline\n    12\/20\/2014 &       &       &       &       & 31.98 & 29.38 & 27.21 & 25.30 & 23.75 & 22.09 &       & 20.67 &       & 19.44 &       & 18.36 & 17.60 \\\\\n\\hline\n    1\/17\/2015 & 42.75 & 38.79 & 35.60 & 33.26 & 30.94 & 28.82 & 26.52 & 24.96 & 23.12 & 21.67 &       & 20.29 &       & 19.10 &       & 17.90 & 18.07 \\\\\n\\hline\n\\end{tabular}%\n\n\\bigskip\n\\bigskip\n\n\\caption{XLF implied volatilities for the Put options.}\n\\label{TabOptP}\n\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n    \\multicolumn{1}{|c|}{\\multirow{2}[4]{*}{T}} & \\multicolumn{13}{c|}{K,Call} \\\\\n\\cline{2-14}          & 21    & 21.5  & 22    & 22.5  & 23    & 23.5  & 24    & 25    & 26    & 27    & 28    & 29    & 30 \\\\\n\\hline\n    4\/4\/2014 &       & 16.60 & 14.69 & 14.40 & 14.86 &       &       &       &       &       &       &       &  \\\\\n\\hline\n    4\/11\/2014 &       & 16.89 & 14.96 & 14.52 & 14.77 & 14.98 &       &       &       &       &       &       &  \\\\\n\\hline\n    4\/19\/2014 &       &       & 15.79 &       & 13.38 &       & 15.39 &       &       &       &       &       &  \\\\\n\\hline\n    5\/17\/2014 & 16.71 &       & 14.48 &       &       &       & 13.75 &       &       &       &       &       &  \\\\\n\\hline\n    6\/21\/2014 & 16.31 &       & 14.78 &       &       &       & 13.92 & 14.28 & 16.58 &       &       &       &  \\\\\n\\hline\n    7\/19\/2014 & 16.82 &       & 15.24 &       &       &       & 14.36 & 14.19 & 15.20 &       &       &       &  \\\\\n\\hline\n    12\/20\/2014 & 17.63 &       & 16.61 &       &       &       & 15.86 & 15.47 & 15.12 & 15.18 & 15.03 &       &  \\\\\n\\hline\n    1\/17\/2015 &       &       &       &       & 16.95 &       & 17.25 & 16.23 & 15.73 & 15.50 & 15.58 & 15.86 & 16.47 \\\\\n\\hline\n\\end{tabular}%\n\n\\end{center}\n\\end{sidewaystable*}\n\nWe have already mentioned that in our model for each term the slopes of the smile at plus and minus infinity, $v^1_{j,n_j}$ and $v^1_{j,0}$, are free parameters. So often traders have an intuition about these values. However, in our numerical experiments we take for them just some plausible values. In more detail, for a normalized variance $v(x)$ defined in \\eqref{finODE}, for all smiles we use $v^1_{j,0} = -0.1$, and $v^1_{j,n_j} = 0.1$. Accordingly, for the instantaneous variance $\\sigma^2(x) = 2 S^2 v(x)\/p_j$ the slopes at both zero and plus infinity are time-dependent and can be computed by using the above formula.\n\nWhen calibrating the model to market data, we use the standard Matlab {\\it fsolve} function, and utilize a \"trust-region-dogleg\" algorithm (see Matlab documentation on {\\it fsolve}). Parameter \"TypicalX\" has to be chosen carefully to speedup calculations.\n\nThe results of this calibration which is done term-by term, are given in Fig.~\\ref{FigPut}. Here each subplot corresponds to a single maturity $T$ (marked in the legend) and shows market data (discrete points) and computed values (solid line). It can be seen that this simple local calibration algorithm provides a very accurate fit for all terms\\footnote{Note, that in \\cite{ItkinLipton2017} in the last subplot the fit is not perfect in the vicinity of $X = -0.5$, where $X = \\log K\/F$ and $F = S e^{(r-q)T}$.}.\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width = 0.8\\textwidth]{fit.png}}\n\\caption{Term-by-term fitting of market Put prices constructed using the whole set of data in Tab.~\\ref{TabOptC},\\ref{TabOptP}.}\n\\label{FigPut}\n\\end{center}\n\\end{figure}\n\nWe constructed the calibration algorithm to be smart enough in a sense that based on the values of parameters at each iteration it decides itself which particular solution (full or asymptotic) should be used at this iteration. We also observed that all full and asymptotic solutions are utilized by the algorithm when calibrating these market smiles.\n\nTable~\\ref{Perf} presents some performance measures of our algorithm. It can be seen that the elapsed time depends on the number of iterations and function evaluations necessary to converge to the given tolerance (we use a relative tolerance $\\varepsilon = 10^{-4}$). This, in turn, depends on the number of evaluated Kummer functions (for the full solution), or number of exponential and Gamma functions (for the asymptotic solutions). Of course, the asymptotic solutions are much faster to evaluate, therefore an average time to calibrate a typical term is less than a second. For the last term 8 in Tab.~\\ref{Perf} calibration is slow for two reasons: i) full solution is used based on the values of parameters, and 2) the number of strikes is higher than for the other terms. But the main reason is that the market data for this term is quite irregular. In any case, performance of this model is much better than that reported in both \\cite{ItkinSigmoid2015} and \\cite{ItkinLipton2017}.\n\\begin{table}[H]\n\\begin{center}\n\\small\n\\begin{tabular}{|r|r|r|r|r|r|}\n\\hline\nTerm & $T$, years & Elapsed time, sec & iterations & function evaluations & strikes \\\\\n\\hline\n1 & 0.0274 & 0.86 & 97 & 1202 & 6 \\\\\n\\hline\n2 & 0.0466 & 2.83 & 97 & 1808 & 9 \\\\\n\\hline\n3 & 0.0685 & 1.43 & 95 & 1200 & 6 \\\\\n\\hline\n4 & 0.1452 & 0.64 & 48 & 433 & 8 \\\\\n\\hline\n5 & 0.2411 & 0.90 & 37 & 470 & 12 \\\\\n\\hline\n6 & 0.3178 & 2.98 & 82 & 1523 & 12 \\\\\n\\hline\n7 & 0.7397 & 6.60 & 106 & 3017 & 15 \\\\\n\\hline\n8 & 0.8164 & 149.67 & 56 & 1317 & 21 \\\\\n\\hline\n\\end{tabular}\n\\caption{Performance characteristics of the algorithm in the described experiment.}\n\\label{Perf}\n\\end{center}\n\\end{table}\n\nThe local variance curves obtained as a result of this fitting are given term-by-term in Fig.~\\ref{termXLF}. The corresponding local variance surface is represented in Fig.~\\ref{lvXLF}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth]{local_var.png}}\n\\caption{Term-by-term fitting of the instantaneous local variance $\\sigma^2(x,T)$.}\n\\label{termXLF}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth, height=3in]{surf.png}}\n\\caption{The instantaneous local variance surface $\\sigma^2(x,T)$ constructed by using the proposed approach.}\n\\label{lvXLF}\n\\end{center}\n\\end{figure}\n\nBy comparing the surface with that given in \\cite{ItkinLipton2017}, one can notice that the shape is quite different while for calibration we use the same market smiles. This is because  in \\cite{ItkinLipton2017} the standard local volatility model is used, where the underlying price follows a Geometric Brownian motion equipped with an instantaneous local volatility function, while in this paper the model is quite different.\n\n\nTo look at a more regular surface, we proceed with another example which is taken from \\cite{Balaraman2016}. In that paper an implied volatility surface of S\\&P500 is presented, and the local volatility surface is constructed using the Dupire formula. In our test we take data for the first 12 maturities and all strikes as they are given in \\cite{Balaraman2016}, and apply our model to calibrate the local variance surface as this is described in above. When doing so we set $v^1_{j,0} = -0.3$, and $v^1_{j,n_j} = 0.1$ for all smiles.\n\nThe results of this calibration are presented in Fig.~\\ref{fitSP},\\ref{termSP},\\ref{lvSP}. By construction, our surface preserves no-arbitrage, while for the approach in \\cite{Balaraman2016} they have to solve some additional problems\\footnote{As this is mentioned in \\cite{Balaraman2016}, the correct pricing of local volatility surface requires an arbitrage free implied volatility surface. If the input implied volatility surface is not arbitrage free, this can lead to negative transition probabilities and\/or negative local volatilities and can give rise to mispricing.}.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width = 0.8\\textwidth]{fit_sp500.png}}\n\\caption{Term-by-term fitting of market S\\&P500 Put prices constructed using data of \\cite{Balaraman2016}.}\n\\label{fitSP}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth]{local_var_sp500.png}}\n\\caption{Term-by-term fitting of the instantaneous local variance $\\sigma^2(x,T)$ for S\\&P500.}\n\\label{termSP}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth, height=3in]{surf_sp500.png}}\n\\caption{The instantaneous local variance surface $\\sigma^2(x,T)$ for S\\&P500 constructed by using the proposed approach.}\n\\label{lvSP}\n\\end{center}\n\\end{figure}\n\n\nIn Table~\\ref{PerfSP500} we present the performance of our algorithm in this experiment.\nIt can be seen that here the elapsed time is similar or shorter as compared with the previous test presented in Table~\\ref{Perf}.\n\n\\begin{table}[H]\n\\begin{center}\n\\small\n\\begin{tabular}{|r|r|r|r|r|r|}\n\\hline\nTerm & $T$, years & Elapsed time, sec & iterations & function evaluations & strikes \\\\\n\\hline\n    1     & 0.0822 & 1.09  & 99    & 1604  & 8 \\\\ \\hline\n    2     & 0.1671 & 0.56  & 40    & 377   & 8 \\\\ \\hline\n    3     & 0.2521 & 2.32  & 94    & 1615  & 8 \\\\ \\hline\n    4     & 0.3315 & 1.70  & 97    & 1186  & 8 \\\\ \\hline\n    5     & 0.4164 & 0.10  & 15    & 64    & 8 \\\\ \\hline\n    6     & 0.4986 & 2.35  & 111   & 1600  & 8 \\\\ \\hline\n    7     & 0.5836 & 2.40  & 111   & 1584  & 8 \\\\ \\hline\n    8     & 0.6658 & 2.25  & 131   & 1604  & 8 \\\\ \\hline\n    9     & 0.7507 & 1.51  & 95    & 1072  & 8 \\\\ \\hline\n    10    & 0.8356 & 2.30  & 98    & 1603  & 8 \\\\ \\hline\n    11    & 0.9178 & 0.07  & 13    & 46    & 8 \\\\ \\hline\n    12    & 1.0027 & 72.80 & 74    & 795   & 8 \\\\ \\hline\n\\end{tabular}\n\\caption{Performance characteristics of the algorithm for calibration of a S\\&P500 surface.}\n\\label{PerfSP500}\n\\end{center}\n\\end{table}\n\n\n\\clearpage\n\\section{Conclusions}\n\nIn this paper we propose an expanded version of the Local Variance Gamma model of\n\\cite{CarrNadtochiy2017} which we refer as an Expanded Local Variance Gamma model, or ELVG. Two main improvements are introduced as compared with the LVG model. First, we add drift to the governing underlying process. It turns out that this a relatively minor (at the first glance) improvement requires a interesting trick to preserve tractability of the model, which is a non-trivial time-change. We show that still in this new model it is possible to derive an ordinary differential equation for the option price which plays a role of Dupire's equation for the standard local volatility model.\n\nThe second novelty of the paper as compared with the LVG model is that in contrast to \\cite{CarrNadtochiy2017} we consider a local variance to be a piecewise linear function of strike, while in \\cite{CarrNadtochiy2017} it was piecewise constant. We proceed in the spirit of \\cite{ItkinLipton2017} by describing a no-arbitrage interpolation, and then construct a closed-form solution of our ODE in terms of hypergeometric and generalized hypergeometric functions. An important advantage of this approach is that calibration of the model to market smiles does not require solving any optimization problem, and can be done term-by-term by solving a system of non-linear algebraic equations for each maturity, which, in general, is significantly faster, especially since we provide an algorithm for constructing a smart initial guess. We also provide various asymptotic solutions which allow a significant acceleration of the numerical solution and improvement of its accuracy in the corresponding cases (i.e, when parameters of the model at some iteration obey the conditions to apply the corresponding asymptotic).\n\nIn principle, somebody could claim that solving a system of nonlinear equations with a generic solver is not much different from solving a nonlinear optimization problem. Obviously, when our ODE is used as an alternative to the Dupire equation, the difference comes from the fact that calibration based on the Dupire equation requires solving this PDE at every iteration by either numerically, or semi-analytically by using a Laplace transform, which is obviously slower. As was mentioned in Introduction there exist many other calibration algorithms which reduce to a nonlinear optimization problem (e.g., taking a sufficiently large parametric family of local volatility functions and choosing the parameters that provide the best fit of observed prices). For the latter computation of the objective function is fast, but optimization must be constrained to preserve no-arbitrage, and, thus, slow.\n\nIn our numerical test we use same market data as in \\cite{ItkinSigmoid2015, ItkinLipton2017}. The results of the test demonstrate robustness of the proposed approach from both the speed and accuracy point of view, especially in cases where the above referred papers experienced some difficulties with achieving a perfect fit. An additional test performed for the S\\&P500 data taken from \\cite{Balaraman2016} gives rise to the same conclusion.\n\n\n\n\\clearpage\n\n\\section*{References}\n\n\\newcommand{\\noopsort}[1]{} \\newcommand{\\printfirst}[2]{#1}\n  \\newcommand{\\singleletter}[1]{#1} \\newcommand{\\switchargs}[2]{#2#1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe motion of a compressible viscous, heat-conductive, and Newtonian polytropic fluid occupying a spatial domain $\\Omega \\subset \\mathbb{R}^{3}$ is governed by the following full compressible Navier-Stokes system:\n\\be \\la{a0}\n\\begin{cases}\n \t\\n_t+{\\rm div} (\\n u)=0,\\\\\n \t(\\n u)_t+{\\rm div}(\\n u\\otimes u)+\\na P=\\div \\mathbb{S},\\\\\n \t(\\n E)_t+\\div (\\n Eu+Pu)=\\div (\\kappa \\na \\te)+\\div (\\mathbb{S}u),\n\\end{cases}\\ee\nwhere   $\\mathbb{S}$ and $E$ are respectively the viscous stress tensor and the total energy given by\n$$\\mathbb{S}=2\\mu\\mathfrak{D}(u) +\\lambda \\div  u \\mathbb{I}_3,~~E=e+\\frac{1}{2}|u|^2,$$\nwith $\\mathfrak{D}(u)  = (\\nabla u + (\\nabla u)^{\\rm tr})\/2$ and $\\mathbb{I}_3$  denoting the deformation tensor and the $3\\times3$ identity matrix respectively.\nHere, $t\\ge 0$ is time, $x\\in \\Omega$ is the spatial coordinate, and $\\n$, $u=\\left(u^1,u^2,u^3\\right)^{\\rm tr},$ $e$, $P,$ and $\\te $ represent respectively the fluid density, velocity, specific internal energy, pressure, and  absolute temperature.\nThe viscosity coefficients $\\mu$ and $\\lambda$  are constants satisfying the physical restrictions:\n\\be\\la{h3} \\mu>0,\\quad 2 \\mu + 3\\lambda\\ge 0.\\ee\nThe heat-conductivity coefficient $\\ka$ is a positive constant.\nWe consider the ideal  polytropic fluids so that $P$ and $e$ are given by the state equations:\n\\be \\la{pg}\nP(\\n,e)=(\\ga-1)\\n e=R\\n \\te, \\quad e=\\frac{R\\theta}{\\ga-1},\n\\ee\nwhere $\\ga>1$  is the adiabatic constant  and $ R$ is a positive constant.\n\n\nLet $\\Omega \\subset \\r^3 $ be a simply connected bounded domain.\nNote that for the classical solutions, the system \\eqref{a0} can be rewritten as\n\\be \\la{a1}\n\\begin{cases}\n\t\\n_t+{\\rm div} (\\n u)=0,\\\\\n\t\\n (u_t+  u\\cdot \\na u )=\\mu\\Delta u+(\\mu+\\lambda)\\na({\\rm div}u)-\\na P,\\\\\n\t\\frac{R}{\\ga-1}\\n ( \\te_t+u\\cdot\\na \\te)=\\ka\\Delta\\te-P\\div u+\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2.\n\\end{cases}\n\\ee\nWe consider the system \\eqref{a1}  subjected  to the given initial data\n\\be \\la{h2}\n(\\rho,\\n u,\\n \\te)(x,{t=0})=(\\rho_0,\\n_0 u_0, \\n_0\\te_0)(x), \\quad x\\in \\Omega,\n\\ee\nand boundary conditions\n\\be \\la{h1}\nu\\cdot n=0,~~\\mbox{curl} u\\times n=0,~~\\na \\theta\\cdot n=0~~~~\\text{on}~\\p\\Omega \\times(0,T),\n\\ee\nwhere $n =(n^1, n^2, n^3)^{\\rm{tr}}$ is the unit outward normal vector on $\\partial\\Omega$.\n\n\nThere is a lot  of literature  on the global existence and large time\nbehavior of solutions to (\\ref{a0}). The one-dimensional problem\nwith strictly positive initial density and temperature has been\nstudied extensively by many people (see \\cite{kazh01,Kaz,akm} and\nthe references therein). For the multi-dimensional case,  the local existence and uniqueness of classical solutions are known in \\cite{Na,se1}  in the absence of vacuum. \nThe global classical solutions were\nfirst obtained by Matsumura-Nishida \\cite{M1} for initial data close\nto a non-vacuum equilibrium in some Sobolev space $H^s.$\nLater, Hoff \\cite{Hof1} studied the\nglobal weak solutions with strictly positive initial density and\ntemperature for discontinuous initial data. On the other hand, in\nthe presence of vacuum, this issue becomes much more complicated.\nConcerning viscous compressible fluids in a barotropic regime, where\nthe state of these fluids at each instant $t>0 $ is completely\ndetermined by the density $\\n=\\n(x,t)$ and the velocity $u=u(x,t),$\nthe pressure $P$ being an explicit function of the density,    the\nmajor breakthrough is due to Lions \\cite{L1} (see also  Feireisl\n\\cite{F1, feireisl1}), where he obtained global existence of weak\nsolutions, defined as solutions with finite energy,  when the\npressure $P$ satisfies $P(\\n)=a\\n^\\ga (a>0,\\ga>1)$  with suitably large  $\\ga.$\nThe main restriction on initial data is that the initial energy  is\nfinite, so that the density vanishes at far fields, or even has\ncompact support. Recently,  Huang-Li-Xin \\cite{hulx} and Li-Xin \\cite{2dlx} established the global well-posedness\nof classical solutions to the Cauchy problem for the 3D and 2D barotropic compressible Navier-Stokes equations in whole space with smooth initial data that are of small energy but\npossibly large oscillations, in particular, the initial density is allowed to vanish.\nMore recently, for slip boundary condition  in bounded domains,\nCai-Li \\cite{C-L}   obtained the global classical solutions with initial vacuum, provided that the initial energy is suitably small.\n\n\n\nCompared with the barotropic flows, it seems much more difficult and complicated to study the global well-posedness of solutions to full compressible Navier-Stokes system (\\ref{a0}) with   vacuum, where some additional difficulties arise, such as the degeneracy of both momentum and energy equations, the strong coupling between the velocity and temperature,  et al. For specific pressure laws   excluding the perfect gas\nequation of state, the question of existence of so-called\n``variational\" solutions in dimension $d\\ge 2$ has been recently\n addressed\nin \\cite{feireisl,feireisl1}, where the temperature equation is\nsatisfied only as an inequality  which justifies the notion of\nvariational solutions.\n Moreover, for   a very particular form of the viscosity\ncoefficients depending on the density, Bresch-Desjardins\n\\cite{bd} obtained global stability of weak solutions.\nFor   the global well-posedness of classical solutions to the full compressible Navier-Stokes system \\eqref{a0}, it  is shown in  {Xin} \\cite{X1}  that  there is no solution in $C^{1}\\left([0, \\infty), H^{s}\\left(\\mathbb{R}^{d}\\right)\\right)$ for large $s$ to the Cauchy problem for the full compressible Navier-Stokes\nsystem without heat conduction  provided  that the initial density has compact support.  See also the recent generalizations to the\ncase   for non-compact but rapidly decreasing at far field initial densities (\\cite{R}).\nRecently, Huang-Li \\cite{H-L} established the global existence and uniqueness for the classical solutions to the 3D Cauchy problem with interior vacuum provided the initial energy is small enough.  Later,  Wen-Zhu  \\cite{W-C} obtained the global existence and uniqueness of the classical solutions  for vanishing far-field density under the assumption that the initial mass is sufficiently small or both viscosity and heat-conductivity coefficients are large enough.  It should be mentioned here that   the results of \\cite{H-L,W-C} hold  only for the Cauchy problem. However,\n the global existence   of classical solutions or even weak ones with vacuum to multi-dimensional   full compressible Navier-Stokes system \\eqref{a0} in general bounded domains remains completely  open except for spherically or cylindrically symmetric\ninitial data (see  \\cite{wenzhu1,wenzhu2}). In fact, one of the aims of  this paper is to study the global well-posedness of classical solutions to full compressible Navier-Stokes system \\eqref{a0} in general bounded domains.\n\n\n\n\n\n\n\n\n\n\n\n\n\nBefore stating the main results, we explain the notations and conventions used throughout this paper.  We denote\n\\bnn\\int fdx\\triangleq\\int_{\\Omega}fdx,\\enn\nand \\bnn \\overline{f}\\triangleq\\frac{1}{|\\O|}\\int_{\\Omega}fdx,\\enn\nwhich is the average of a function $f$ over $\\Omega$.\nFor $1\\le p\\le \\infty $ and integer $k\\ge 0,$  we adopt the simplified notations for Sobolev spaces as\nfollows:\n\\be\\ba\\notag \\begin {cases}\nL^p=L^p(\\Omega),\\quad W^{k,p}=W^{k,p}(\\Omega),\\quad H^k=W^{k,2}(\\O),\\\\\nH_\\omega^i= \\left.\\left\\{f\\in H^i \\right | f\\cdot n=0,\\,\\curl f\\times n=0 \\rm \\,\\,{on}\\,\\, \\p\\O\\right\\}\\,(i=1,2).\n\\end{cases}\\ea\\ee\n\nWithout loss of generality, we assume that\n\\be\\la{m}\\overline{\\rho_0}=\\frac{1}{|\\O|}\\int \\rho_0 dx=1.\\ee\nWe then define  the initial energy  $C_0$  as follows:\n\\be\\la{e}\\ba  C_0\\triangleq &\\frac{1}{2}\\int\\n_0\n|u_0|^2dx+R \\int  \\left( 1+\\n_0\\log {\\n_0}-\\n_0 \\right)dx\\\\\n&+\\frac{R}{\\ga-1}\\int  \\n_0\\left(\\te_0- \\log {\\te_0} -1\n\\right) dx .\\ea\\ee\n\nThe first main result in this paper can be stated  as follows:\n\n\n\n\n\\begin{theorem}\\la{th1}\nLet $\\O\\subset\\r^3$ be a simply connected bounded smooth domain, whose boundary $\\p\\O$ has a finite number of 2-dimensional connected components. For  given numbers $M>0$ (not necessarily\nsmall), $q\\in (3,6),$\n  $\\on> 2,$ and $\\bt>1,$\nsuppose that the initial data $(\\n_0,u_0,\\te_0)$ satisfies \\be\\ba\n\\la{co3} \\n_0\\in  W^{2,q}, \\quad  u_0 \\in H^2_\\omega,\\quad\n\\te_0\\in\\left.\\left\\{f\\in H^1 \\right| \\na f\\cdot n=0 \\,\\,\\rm {on}\\,\\, \\p\\O\\right\\}, \\ea\\ee\n\\be \\la{co4} 0\\le\\inf\\rho_0\\le\\sup\\rho_0< \\hat{\\rho},\\quad 0  \\le\\inf\\te_0\\le\\sup\\te_0\\le \\bt, \\quad \\|\\na u_0\\|_{L^2} \\le M,\n   \\ee\n   and the compatibility condition\n\\be\n\\la{co2}-\\mu \\Delta u_0-(\\mu+\\lambda)\\na\\div u_0+R\\na (\\n_0\\te_0)\n=\\sqrt{\\n_0} g,\\ee\nwith  $g\\in L^2. $ Then there exists a positive constant $\\ve$\ndepending only\n on $\\mu,$ $\\lambda,$ $\\ka,$ $ R,$ $ \\ga,$  $\\on,$ $\\bt$, $\\O$, and $M$ such that if\n \\be\n \\la{co14} C_0\\le\\ve,\n   \\ee  the   problem  (\\ref{a1})--(\\ref{h1})\nadmits a unique global classical solution $(\\rho,u,\\te)$ in\n   $\\Omega\\times(0,\\infty)$ satisfying\n  \\be\\la{h8}\n  0\\le\\rho(x,t)\\le 2\\hat{\\rho},\\quad \\te(x,t)\\ge 0,\\quad x\\in \\Omega,\\,~~ t\\ge 0,\n  \\ee\n and \\be\n   \\la{h9}\\begin{cases}\n   \\rho\\in C([0,T];W^{2,q}),\\\\\n   u \\in C([0,T];W^{1,\\tilde p})\\cap L^\\infty(0,T;H^2)\\cap   L^\\infty(\\tau,T; W^{3,q}),  \\\\\n   \\te\\in   L^\\infty(\\tau,T;H^4)\\cap  C([\\tau,T];  W^{3,\\tilde p}), \\\\\n  u_t \\in L^2(0,T;H^1)\\cap\nL^{\\infty}(\\tau,T;H^2)\\cap H^1(\\tau,T;H^1), \\\\\n  \\te_t \\in\nL^{\\infty}(\\tau,T;H^2)\\cap H^1(\\tau,T;H^1),\\end{cases} \\ee for any $0<\\tau<T<\\infty $ and $\\tilde p\\in [1,6).$\nMoreover, for any $p\\in  [1 ,\\infty)$ and $r\\in [1,6]$, there exist positive constants $C$, $\\al_0$, and $\\te_\\infty$ depending only  on $\\mu,$ $\\lambda,$ $\\ka,$ $ R,$ $ \\ga,$  $\\on,$ $\\bt$, $\\O$, $p$, $r$, and $M$  such that for any $t\\geq 1,$\n\\be\\la{h11}\n\\|\\n-1\\|_{L^p}+\\|u\\|_{W^{1,r}}^2+\\|\\te-\\te_\\infty\\|_{H^2}^2\\le C e^{-\\al_0 t}.\n\\ee\n\n\\end{theorem}\n\nThe next result of this paper concerns  weak solutions whose   definition is as follows.\n\\begin{definition}\\la{def} We say that $(\\n,u,E=\\frac{1}{2}|u|^2+\\frac{R}{\\ga-1}\\te)$ is a weak solution to Cauchy problem (\\ref{a0}) (\\ref{h2}) (\\ref{h1}) provided that\n$$\\n\\in L^\\infty_{\\rm loc}([0,\\infty);L^\\infty(\\O)),\\quad u ,\\te\\in L^2_{\\rm loc} ([0,\\infty); H^1(\\O)),$$\nand that for all test functions $\\psi\\in\\mathcal{D}(\\O\\times(-\\infty,\\infty)),$\n\\be \\la{def1}\n\\int_{\\O}\\n_0\\psi(\\cdot,0)dx+\\int_0^\\infty\\int_{\\O}\\left(\\n\\psi_t+\\n u\\cdot\\na\\psi\\right) dxdt=0,\\ee\n\\be\\la{def2}\\ba&\n\\int_{\\O}\\n_0u^j_0\\psi(\\cdot,0)dx+\\int_0^\\infty\\int_{\\O}\\left(\\n u^j\\psi_t\n   +\\n u^ju\\cdot\\na\\psi+P(\\n,\\te)\\psi_{x_j}\\right)dxdt\\\\\n&-\\int_0^\\infty\\int_{\\O}\\left(  \\mu\\na u^j\\cdot\\na\\psi+(\\mu+\\lambda)(\\div u)\\psi_{x_j}\\right) dxdt=0,\\quad\nj=1,2,3, \\ea\\ee\n\n\n\n\\be \\la{def3}\\ba\n&\\int_{\\O}\\left(\\frac{1}{2}\\n_0|u_0|^2 +\\frac{R}{\\ga-1}\\n_0\\te_0 \\right)\\psi(\\cdot,0)dx\\\\\n&+\\int_0^\\infty\\int_{\\O}\\left(\\n E\\psi_t+ (\\n E  +P)u\\cdot\\na\\psi\\right) dxdt\\\\\n&-\\int_0^\\infty\\int_{\\O}\\left(\\ka\\na\\te+\\frac{1}{2}\\mu \\na(|u|^2)\n +\\mu  u\\cdot \\na u+\\lambda\\div u u\\right)\\cdot\\na \\psi dxdt=0.\n\\ea\\ee\n\\end{definition}\n\nThen we state our second main result as follows:\n\\begin{theorem}\\la{th2}  Under the conditions of Theorem \\ref{th1} except (\\ref{co2}), where the condition (\\ref{co3}) is replaced by\n\\be\\la{dt7}u_0\\in H^1_\\omega,\\ee\nassume  further that $C_0$  as in (\\ref{e}) satisfies (\\ref{co14}) with $\\varepsilon$ as in Theorem \\ref{th1}. Then there exists a global weak solution $(\\rho,u,E=\\frac{1}{2}|u|^2+\\frac{R}{\\ga-1}\\te)$ to the problem  (\\ref{a0}) (\\ref{h2}) (\\ref{h1})\n    satisfying\n\\be\\la{hq1}\n\\n\\in C([0,\\infty);L^p), \\quad (\\n u,\\,\\n |u|^2,\\,\\n\\te)\\in C([0,\\infty);H^{-1}) ,\n\\ee\n\\be\\la{hq2}u\\in L^\\infty(0,\\infty;H^1)\\cap C((0,\\infty);L^2 )  ,\\quad \\te\\in C((0,\\infty); W^{1,\\tilde p}),\\ee\n\\be \\la{hq3}\nu(\\cdot,t),\\,\\,\\curl u(\\cdot,t),\\,\\,((2\\mu+\\lambda)\\div u-(P-\\overline P))(\\cdot,t),  \\,\\,\\na\\te(\\cdot,t)\\in H^1,\\quad t>0,\\ee\n\\be\\la{hq4}\\rho\\in [0,2\\hat{\\rho}] \\quad \\mbox{ \\rm a.e.}, \\quad \\te\\ge 0 \\quad\\mbox{ \\rm a.e.},\\ee\n and the exponential decay property (\\ref{h11})  with $p\\in  [1 ,\\infty)$, $\\tilde p\\in [1,6)$, and $r\\in[1,6]$. In addition, there exists some  positive constant $C$ depending  only on $ \\mu,$ $\\lambda,$ $ \\ka,$ $ R,$ $ \\ga,$  $\\on$, $\\bt,$ $\\O$, and $M $ such that, for  $\\si(t)\\triangleq\\min\\{1,t\\},$  the following estimates hold\n  \\be\\la{hq5}\\ba  \\sup_{t\\in (0,\\infty)} \\| u \\|_{H^1}^2  +\\int_0^\\infty\\int\\left| (\\n u)_t+{\\rm div}(\\n u\\otimes u)\\right|^2dxdt\\le C,\\ea\\ee\n  \\be\\la{hq7}\\ba &\\sup_{t\\in (0,\\infty)} \\int\\left((\\n-1)^2+\\n |u|^2+\\n(R\\te-\\overline P)^2\\right)dx  \\\\&\\quad   +\\int_0^\\infty\\left( \\|\\na u\\|^2_{L^2}+ \\|\\na \\te\\|_{L^2}^2 \\right)dt\\le CC_0^{1\/4}, \\ea\\ee\n\\be\\la{hq8}\\ba\n&\\sup_{t\\in (0,\\infty)}\\left(\\si\\|\\na u \\|^2_{L^6}+\\si^2\\|\\te\\|^2_{H^2}\\right) \\\\\n&+\\int_0^\\infty\\left( \\si\\|u_t\\|^2_{L^2}+\\si^2\\|\\na \\dot u\\|_{L^2}^2+\\si^2\\|\\te_t\\|^2_{H^1}\\right)dt\\le C. \\ea\\ee\nMoreover, $(\\n,u,\\te)$ satisfies (\\ref{a1})$_3$ in the weak form, that is, for any test function  $\\psi\\in\n\\mathcal{D}(\\O\\times(-\\infty,\\infty)),$\n\\be\\la{vu019}\\ba &\\frac{R}{\\ga-1} \\int\\n_0 \\te_0\\psi(\\cdot,0) dx+\\frac{R}{\\ga-1}\\int_0^\\infty\\int\\n \\te \\left(\\psi_t+u \\cdot\\na\\psi\n\\right)dxdt\\\\ & =  \\ka \\int_0^\\infty\\int \\na\\te \\cdot\\na\\psi dxdt+R\\int_0^\\infty\\int  \\n\\te  \\div u \\psi dxdt\\\\&\\quad -\\int_0^\\infty\\int \\left(\\lambda(\\div u)^2+2\\mu   |\\mathfrak{D}(u)|^2  \\right) \\psi dxdt.\\ea\\ee\n\n\\end{theorem}\n\n\nNext, as a direct application of \\eqref{h11}, the following Corollary \\ref{th3},  whose proof is similar to that of     \\cite[Theorem 1.2]{C-L}, shows that the oscillation of the density will grow unboundedly in the long run with an exponential  rate provided  vacuum appears  (even at a point) initially.\n\n\\begin{cor} \\la{th3}\nIn addition to the conditions  of Theorem \\ref{th1}, assume further that there exists some point $x_0\\in\\Omega$ such that $\\rho_0(x_0)=0.$ Then for any $\\hat p>3$, there exists some positive constant $C$ depending on $\\mu$, $\\lambda$, $\\ka,$ $ R,$ $ \\ga,$  $\\on,$ $\\bt$, $\\O$, $\\hat p$ and $M$ such that the unique global classical solution $(\\rho,u,\\te)$ to the problem (\\ref{a1})--(\\ref{h1}) obtained in Theorem \\ref{th1} satisfies that for any $t\\geq 1$,\n\\be\\notag\\|\\nabla \\rho(\\cdot,t)\\|_{L^{\\hat p} }\\geq Ce^{Ct}.\\ee\n\\end{cor}\n\n\n\nA few remarks are in order:\n\n\n\n\n\n\n\\begin{remark}\nIt is easy to deduce from (\\ref{h9}) and the Sobolev imbedding  theorem that for any $0<\\tau<T<\\infty,$\n\\be \\la{hk1}\n(\\n,\\, \\na\\n,\\,u) \\in C(\\overline{\\Omega} \\times[0,T] ),\\quad   (\\te,\\,\\na\\te,\\,\\na^2\\te)  \\in C(\\overline{\\Omega} \\times(0,T] ),\\ee\nand\n\\be \\ba\\la{ono}\n(\\na u,\\,\\na^2 u) \\in  C( [\\tau,T];L^2 )\\cap L^\\infty(\\tau,T;  W^{1,q})\\hookrightarrow  C(\\overline{\\Omega} \\times[\\tau,T] ),\n\\ea \\ee\nwhich together with  (\\ref{a1})$_1$  and (\\ref{hk1}) shows\n\\be  \\la{hk3} \\n_t\\in   C(\\overline{\\Omega}\\times[\\tau,T] ).\\ee\nAnalogously, we have\n\\be\\notag(u_t,\\te_t)\\in C(\\overline{\\Omega} \\times[\\tau,T] ),\\ee\nwhich along with (\\ref{hk1})--(\\ref{hk3}) arrives at\n the solution $(\\n,u,\\te)$ obtained in Theorem \\ref{th1} is a classical one to  the   problem (\\ref{a1})--(\\ref{h1}) in $\\Omega\\times (0,\\infty).$\n\\end{remark}\n\n\\begin{remark} It seems that\nTheorem \\ref{th1}, which extends the global existence result of the  barotropic flows studied in \\cite{C-L} to the full compressible Navier-Stokes system,  is the first result concerning the global existence of classical solutions with initial vacuum to  (\\ref{a0}) in general bounded domains. Although its energy is small, the oscillations could be arbitrarily large.\n \\end{remark}\n\n\\begin{remark}\n\tTo obtain the global existence and uniqueness of classical solutions with vacuum, we only need the compatibility condition on the velocity (\\ref{co2}) as in \\cite{lxz}, which is much weaker than those in \\cite{choe1, H-L,W-C} where not only (\\ref{co2}) but also the following compatibility condition on the temperature\n\t\\be\\la{co1} \\ka\\Delta \\te_0+\\frac{\\mu}{2}|\\na u_0+(\\na u_0)^{\\rm tr}|^2+\\lambda (\\div u_0)^2=\\sqrt{\\n_0}g_1, \\quad g_1 \\in L^2\\ee\n\tis needed. This reveals that the compatibility condition on the temperature (\\ref{co1}) is not necessary for establishing the classical solutions with vacuum to the full Navier-Stokes equations, which is just the same as the barotropic case \\cite{hulx, 2dlx}.\n\t\\end{remark}\n\n\n\\begin{remark}\nIt should be  mentioned here that the boundary condition for velocity $u$:\n\t\\be\\la{bb}\n\tu\\cdot n=0,\\,\\,\\curl u \\times n=0 \\,\\,\\text {on}\\,\\, \\p\\O,\n\t\\ee\n\tis a special case of the following general Navier-type slip condition (see Navier \\cite{Nclm1})\n\t\\be\\notag\n\tu \\cdot n = 0, \\,\\,(2\\mathfrak{D}(u)\\,n+ \\vartheta u)_{tan}=0 \\,\\,\\,\\text{on}\\,\\,\\, \\partial\\Omega,\n\t\\ee which as indicated by \\cite[Remark 1.1]{C-L},   is in fact  a particular case of   the following slip boundary one:\n\t\\bn\\label{Ns}\n\tu\\cdot n=0,\\,\\,\\curl u \\times n=-Au \\,\\,\\text {on}\\,\\, \\p\\O,\n\t\\en\n\twhere $\\vartheta$ is a scalar friction function, the symbol $v_{tan}$ represents the projection of tangent plane of the vector $v$ on $\\partial\\Omega$, and $A = A(x)$ is a given $3\\times3$ symmetric matrix defined on $\\p\\O$.\t\n\tIndeed, our result still holds for more general slip boundary condition (\\ref{Ns}) with $A$ being semi-positive and regular enough. The proof is similar to \\cite{C-L} and omitted here.\n\\end{remark}\n\n\n\nWe now comment on the analysis of this paper.\nWe mainly take the strategy that we first extend the\nstandard local classical solutions with strictly positive initial density (see Lemma \\ref{th0}) globally in time just under the condition that the initial energy is suitably small\n(see Proposition \\ref{pro2}), then let the lower bound of the initial density go to zero. To\ndo so, one needs to establish global a priori estimates, which are independent of\nthe lower bound of the density, on smooth solutions to  (\\ref{a1})--(\\ref{h1}) in suitable\nhigher norms.\nAs indicated in \\cite{H-L,h101}, the key issue  is to obtain the time-independent upper bound of the density.\n\nThe first main difficulty arises in deriving the basic energy estimate, which indeed  is obtained directly for the Cauchy problem \\cite{H-L}.\nHowever, in our case,  the basic energy equality reads\n\\be\\ba\\la{11a}\n&E'(t)+\\int \\left( \\frac{\\lambda(\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2}{{\\te}}+\\ka \\frac{|\\na \\te|^2}{\\te^2} \\right)dx \\\\\n&=- \\mu \\int \\left(|\\curl u|^2+2(\\div u)^2 - 2 |\\mathfrak{D}(u)|^2\\right)dx,\n\\ea\\ee where the basic energy $E(t)$ is defined by\n\\be \\label{enet}E(t) \\triangleq \\int  \\left( \\frac{1}{2}\\n |u|^2+R(1+\\n\\log\n\\n-\\n)+\\frac{R}{\\ga-1}\\n(\\te-\\log \\te-1)\\right)dx. \\ee\nNote that the right-hand term in \\eqref{11a}\n    is   sign-undetermined   due to the slip boundary condition (\\ref{bb}), thus it seems difficult to obtain directly the usual standard energy estimate  \\be \\la{edwq1} E(t) \\le C C_0,\\ee  \nwhere the smallness (with the same order of initial energy) of basic energy  plays a key role in the whole analysis of the global existence of classical solutions with vacuum not only for Cauchy problem\/IBVP of barotropic flows \\cite{2dlx, hulx, C-L} but also for Cauchy problem of  full compressible Navier-Stokes equations \\cite{H-L}.\nTo overcome this difficulty,   we  first    assume that    $A_2(T)$ (see \\eqref{AS2})  a priori satisfies $A_2(T)\\le 2C_0^{1\/4}$ (see \\eqref{3.q2}) and  obtain the following ``weaker\" basic energy estimate (see also \\eqref{a2.8}):\n\\be\\la{weaken} E(t) \\le CC_0^{1\/4},\\ee which compared with \\eqref{edwq1}, however,  is not enough and indeed will  bring us some essential difficulties to obtain all the a priori estimates (see Proposition \\ref{pr1}).\nThen, the first   observation is that the average of the pressure $\\bp$ is uniformly bounded with positive lower and upper bounds (see \\eqref{key}) by both ``weaker\" basic energy estimate \\eqref{weaken} and   Jensen's inequality (see Lemma \\ref{je}). Combining this with the fact that the quantity $\\bp$   plays a similar role as $\\overline{\\te}$ (see \\eqref{pq}) implies that we can replace $\\bar \\te$ by $\\bar P $ whose  positive  lower and upper  bounds play an important role  in further analysis.\n\nNext, the second difficulty lies in the  estimation on the energy-like term $A_2(T)$ (see \\eqref{AS1}) which includes the key bounds on the $L^2(\\O\\times(0,T))$-norm of the spatial derivatives of both the velocity and the temperature. To proceed, first, we adopt the ideas due to \\cite{Hof1,C-L} to  estimate $\\dot u$ and $\\dot\\te$ (see Lemma \\ref{a113.4}), where $\\dot f\\triangleq f_t+u\\cdot\\nabla f $ denotes the material derivative of $f.$   Indeed, in this process, one needs to deal with the boundary  integrals in \\eqref{bz8}, for example\n$$\\int_{\\p\\O}G(u\\cdot\\na)u\\cdot\\na n\\cdot u dS,$$ which by\nthe classical trace theorem seems to be bounded by some good terms and the $L^p$-norm of $\\na^2 u$, which is unavailable in this step. To overcome this difficulty, we adopt some idea  due to \\cite{C-L}, that is, $u=u^{\\perp}\\times n$ with  $u^{\\perp}\\triangleq-u\\times n$, which combined with the following fact:\n\\be\\la{cd}\\div(\\na u^i\\times u^{\\perp})=-\\na u^i\\cdot\\na\\times u^{\\perp},\\ee yields that the above boundary integral can be indeed bounded by some suitable norms on both $\\na u$ and  $\\na G$ (see \\eqref{bz3} for details).\nNext, after observing that  the evolution of $\\bp$ can  be derived from the temperature equation (see \\eqref{pt}), combining a careful analysis on the system \\eqref{a1}  with the $L^1(0,\\min\\{1,T\\};L^\\infty)$-norm of the temperature  gives the desired basic energy estimate for small time (see Lemma \\ref{a13}).\nMoreover, we observe that  $R \\te -\\bp$ can be bounded by the combination of the initial energy with the spatial $L^2$-norm of the spatial derivatives of the temperature (see (\\ref{a2.17})), which together  with a suitable combination of kinetic energy and thermal energy (see \\eqref{a2.23}) yields $ A_2(T)\\le CC_0^{7\/24}$ (see \\eqref{kyu1}) which implies $A_2(T)\\le C_0^{1\/ 4}$ (see \\eqref{a2.34}), provided the initial energy is suitably small.\n\n\nNext, the third difficulty is to obtain  the  key time-independent upper bound of the density. It should be noted that the methods used in Cauchy problem \\cite{H-L}, which heavily relies on the nontrivial far field states, can not be applied to the IBVP directly. Here, by inserting the key quantity $\\overline{P}$, we rewrite the continuity equation in the following way\n\\bnn\\ba\n(2\\mu+\\lambda) D_t \\n&=-\\bp \\n(\\n-1)- \\n^2(R\\te-\\bp)-\\n G,\n\\ea\\enn\nwhere $D_t$ is the material derivative and $G$ is the effective viscous flux, defined by\n\\be \\la{hj1} D_t f \\triangleq \\dot f\\triangleq f_t+u\\cdot\\nabla f,\\quad G\\triangleq(2\\mu + \\lambda)\\div u -  (P-\\bp), \\ee\nrespectively.\nWith the aid of the uniform bound of $\\overline{P}$ (see \\eqref{key}),\nthe upper bound of the density follows directly by applying the Gr\\\"{o}nwall-type inequality (see Lemma \\ref{le1}) and using the estimates on $R\\te-\\bp$ and $G$.\n\n\n\n\nFinally, with the lower-order estimates including the time-independent upper bound of the density  at hand,  we can obtain the higher-order estimates\njust under the compatibility condition on the velocity \\eqref{co2}. Note that all the a priori estimates are independent of the lower bound of the  density, thus after a standard approximate procedure, we can obtain the global existence of classical solutions with vacuum.  Moreover, we can as well establish the global weak solutions   almost  the same way as we established the classical one with a new modified approximate initial data.\n\n\n\nThe rest of the paper is organized as follows: In Section 2, we collect some basic facts and inequalities which will be used later. Section 3 is devoted to deriving the lower-order a priori estimates on classical solutions which are needed to extend the local solutions to all time. The higher-order estimates are established in Section 4.\nFinally, with all a priori estimates at hand, the main results, Theorems \\ref{th1} and \\ref{th2}, are proved in Section 5.\n\n\n\n\n\n\\section{Preliminaries}\\la{se2}\n\n\nFirst, the following local existence theory with strictly positive initial density can be shown by the standard contraction mapping arguments as in \\cite{choe1,Tani,M1}.\n\n\\begin{lemma}   \\la{th0} Let $\\O$ be as in Theorem \\ref{th1}. Assume  that\n $(\\n_0,u_0,\\te_0)$ satisfies \\be \\la{2.1} \\begin{cases}\n(\\n_0,u_0,\\te_0)\\in H^3, \\quad \\inf\\limits_{x\\in\\Omega}\\n_0(x) >0, \\quad \\inf\\limits_{x\\in\\Omega}\\te_0(x)> 0,\\\\\nu_0\\cdot n=0,~~\\curl u_0\\times n=0,~~\\na\\te_0\\cdot n=0~~~~\\text{on}~\\p\\Omega.\\end{cases}\\ee Then there exist  a small time\n$0<T_0<1$ and a unique classical solution $(\\rho , u,\\te )$ to the problem  (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times(0,T_0]$ satisfying\n\\be\\la{mn6}\n  \\inf\\limits_{(x,t)\\in\\Omega\\times (0,T_0]}\\n(x,t)\\ge \\frac{1}{2}\n \\inf\\limits_{x\\in\\Omega}\\n_0(x), \\ee and\n \\be\\la{mn5}\n \\begin{cases}\n ( \\rho,u,\\te) \\in C([0,T_0];H^3),\\quad\n \\n_t\\in C([0,T_0];H^2),\\\\   (u_t,\\te_t)\\in C([0,T_0];H^1),\n \\quad (u,\\te)\\in L^2(0,T_0;H^4).\\end{cases}\\ee\n \\end{lemma}\n\n \\begin{remark} Applying the same arguments as in \\cite[Lemma 2.1]{H-L}, one can deduce that the classical solution $(\\rho , u,\\te )$  obtained in Lemma \\ref{th0} satisfies\n\\be \\la{mn1}\\begin{cases} (t u_{t},  t \\te_{t}) \\in L^2( 0,T_0;H^3)  ,\\quad (t u_{tt},  t\\te_{tt}) \\in L^2(\n0,T_0;H^1), \\\\  (t^2u_{tt},  t^2\\te_{tt}) \\in L^2( 0,T_0;H^2), \\quad\n (t^2u_{ttt},  t^2\\te_{ttt}) \\in L^2(0,T_0;L^2).\n\\end{cases}\\ee\nMoreover, for any    $(x,t)\\in \\Omega\\times [0,T_0],$ the following estimate holds:\n   \\be\\la{mn2}\n\\te(x,t)\\ge\n\\inf\\limits_{x\\in\\Omega}\\te_0(x)\\exp\\left\\{-(\\ga-1)\\int_0^{T_0}\n \\|\\div u\\|_{L^\\infty}dt\\right\\}.\\ee\n\\end{remark}\n\n\n\\iffalse\n\n {\\it Proof. } Without loss of generality, we assume that $T_0\\le 1.$  It's only need to show    (\\ref{mn1}) and (\\ref{mn2}), which are not given in  \\cite[Theorem 5.2]{M1}.\n\nDifferentiating  $(\\ref{a1})_2$  with\nrespect to $t$ leads to \\be\\la{nt0}\\ba\n& \\n u_{tt}-\\mu\\Delta u_t-(\\mu+\\lambda)\\na\\div u_t\\\\\n&=-\\n_tu_t-\\n_tu\\cdot\\na u-\\n u_t\\cdot\\na u -\\n u\\cdot\\na u_t-\\na P_t.\\ea\\ee This shows that\n$tu_t$ satisfies \\be\\la{mn7}\n\\begin{cases}\n\\n (tu_t)_t-\\mu \\Delta(tu_t)-(\\mu+\\lambda)\\na\\div (tu_t)=F_1,&\\quad\\text{in}\\,\\,\\O\\times(0,T],\\\\\n(tu_t)\\cdot n=0,\\,\\,\\curl (tu_t)\\times n=0,&\\quad \\text{on}\\,\\,\\p\\O\\times(0,T],\\\\\ntu_t=0,&\\quad\\text{on}\\,\\,\\O\\times\\left\\{t=0\\right\\},\n\\end{cases}\\ee\n where $$F_1\\triangleq\\n u_t-t\\n_tu_t-t\\n_tu\\cdot\\na u-t\\n u_t\\cdot\\na u -t\\n\nu\\cdot\\na u_t-Rt\\na (\\n_t\\te+\\n\\te_t), $$ satisfies $F_1\\in\nL^2(0,T_0;L^2)$ due to  (\\ref{mn5}). It thus follows from\n(\\ref{mn5}), (\\ref{mn6}), and standard $L^2$-theory for parabolic\nsystem (\\ref{mn7}) that \\be\\la{mn9} (tu_t)_t,~\\na^2(tu_t)\\in\nL^2(0,T_0;L^2).\\ee\nSimilarly, one  differentiates $(\\ref{a1})_3$ with respect to $t$ and gets\nget \\be\\la{eg1}\\ba &\\n\\te_{tt}-\\frac{\\ka(\\ga-1)}{R}\\Delta \\te_t\n\\\\&=-\\n_t\\te_{t}- \\n_t\\left(\nu\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)-\\n\\left( u\\cdot\\na\n\\te+(\\ga-1)\\te\\div u\\right)_t\n\\\\&\\quad+\\frac{\\ga-1}{R}\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_t,\\ea\\ee\nwhich implies that $t\\te_t$ satisfies\n\\be\\la{mn8}\\begin{cases}\nR\\n (t\\te_t)_t- {\\ka(\\ga-1)}\\Delta (t \\te_t)={R}F_2,&\\quad\\text{in}\\,\\,\\O\\times(0,T],\\\\\n\\na(t\\te_t)\\cdot n=0,&\\quad \\text{on}\\,\\,\\p\\O\\times(0,T],\\\\\nt\\te_t=0,&\\quad\\text{on}\\,\\,\\O\\times\\left\\{t=0\\right\\},\n\\end{cases}\\ee\nwith \\bnn\\ba F_2\\triangleq&\\n\\te_t-t\\n_t\\te_{t}- t\\n_t\\left(\nu\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\\\&-t\\n\\left( u\\cdot\\na\n\\te+(\\ga-1)\\te\\div u\\right)_t+\n \\frac{\\ga-1}{R}t\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_t.\\ea\\enn\nOne derives from (\\ref{mn5}) that $F_2\\in L^2(0,T_0;L^2),$ which\ntogether with (\\ref{mn5}), (\\ref{mn6}), and standard $L^2$-theory for\nparabolic system (\\ref{mn8}) implies \\be\\la{mn10}\n(t\\te_t)_t,~\\na^2(t\\te_t)\\in L^2(0,T_0;L^2).\\ee It thus follows from\n(\\ref{mn5}), (\\ref{mn9}), and (\\ref{mn10}) that $$F_1,~F_2\\in\nL^2(0,T_0;H^1),$$ which together with (\\ref{mn5}),  (\\ref{mn6}),\n(\\ref{mn7}), and  (\\ref{mn8}) gives $ (\\ref{mn1})_1.$\n\nNext,\ndifferentiating $(\\ref{nt0})$ with\nrespect to $t$ gives \\be\\la{sp30}\\ba\n&\\n u_{ttt}-\\mu\\Delta u_{tt}-(\\mu+\\lambda)\\nabla{\\rm div}u_{tt}\\\\\n&= 2{\\rm div}(\\n u)u_{tt} +{\\rm div}(\\n u)_{t}u_t-2(\\n u)_t\\cdot\\na u_t-(\\n_{tt} u+2\\n_t u_t) \\cdot\\na u\n\\\\& \\quad- \\n u_{tt}\\cdot\\na u-\\na P_{tt}-\\n u\\cdot\\na u_{tt}-\\n u_{tt}\\cdot \\na u.\n \\ea\\ee\nThis together with $(\\ref{mn1})_1$ and (\\ref{mn5}) implies that\n$t^2u_{tt}$ satisfies\n\\be\\la{mn11}\\begin{cases}\n\\n (t^2u_{tt})_t-\\mu\\Delta (t^2u_{tt})-(\\mu+\\lambda)\\na\\div (t^2u_{tt})\n=F_3,&\\quad\\text{in}\\,\\,\\O\\times(0,T],\\\\\n(t^2u_{tt})\\cdot n=0,\\,\\,\\curl (t^2u_{tt})\\times n=0,&\\quad \\text{on}\\,\\,\\p\\O\\times(0,T],\\\\\nt^2u_{tt}=0,&\\quad\\text{on}\\,\\,\\O\\times\\left\\{t=0\\right\\},\n\\end{cases}\\ee\nwhere \\be\\ba F_3\\triangleq&2t\\n u_{tt}+2t^2{\\rm div}(\\n u)u_{tt} +t^2{\\rm div}(\\n u)_{t}u_t-2t^2(\\n\nu)_t\\cdot\\na u_t\\\\& -t^2(\\n_{tt} u+2\\n_t u_t) \\cdot\\na u -t^2\\na P_{tt}-t^2\\n u\\cdot\\na u_{tt}- t^2\\n\nu_{tt}\\cdot\\na u \\ea\\ee satisfying  $F_3\\in\nL^2(0,T_0;L^2)$ due to  (\\ref{mn5}) and $(\\ref{mn1})_1.$ It follows\nfrom (\\ref{mn6}), (\\ref{mn5}), $(\\ref{mn1})_1,$ and standard\n$L^2$-estimate for (\\ref{mn11}) that \\be\\la{mn12} (t^2\nu_{tt})_t,~\\na^2(t^2 u_{tt})\\in L^2(0,T_0;L^2).\\ee\nSimilarly, differentiating $(\\ref{eg1})$ with respect to $t$ yields\n\\be\\la{eg2}\\ba &\\n\\te_{ttt}-\\frac{\\ka(\\ga-1)}{R}\\Delta \\te_{tt}\n\\\\&=- \\n_{tt}\\left(\\te_t+u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)+\\div(\\n u)\\te_{tt} \\\\\n&\\quad - 2\\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)_t\\\\\n&\\quad- \\n\\left(u_{tt}\\cdot\\na \\te+2u_t\\cdot\\na\\te_t+ u\\cdot\\na\\te_{tt}+(\\ga-1)(\\te\\div u)_{tt}\\right)\\\\\n &\\quad\n+\\frac{\\ga-1}{R}\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2 \\right)_{tt}.\\ea\\ee\nWe thus deduce from $(\\ref{mn1})_1,$\n(\\ref{mn5}), and (\\ref{eg2})  that $t^2\\te_{tt}$ satisfies\n\\be\\la{mn13}\\begin{cases}\nR\\n (t^2\\te_{tt})_t- {\\ka(\\ga-1)}  \\Delta(t^2\\te_{tt}) =RF_4,&\\quad\\text{in}\\,\\,\\O\\times(0,T],\\\\\n\\na(t^2\\te_{tt})\\cdot n=0,&\\quad \\text{on}\\,\\,\\p\\O\\times(0,T],\\\\\nt^2\\te_{tt}=0,&\\quad\\text{on}\\,\\,\\O\\times\\left\\{t=0\\right\\},\n\\end{cases}\\ee\nwith $$\\ba F_4\\triangleq\n&2t\\n\\te_{tt} - t^2\\n_{tt}\\left(\\te_t+ u\\cdot\\na\\te+(\\ga-1)\\te\\div u\\right) +t^2\\div(\\n u)\\te_{tt} \\\\\n&  - 2t^2\\n_t\\left( u\\cdot\\na\\te+(\\ga-1)\\te\\div u\\right)_t\n - t^2\\n u_{tt}\\cdot\\na \\te-2t^2\\n u_t\\cdot\\na\\te_t-t^2\\n u\\cdot\\na \\te_{tt}\n\\\\&-(\\ga-1)t^2\\n(\\te\\div u)_{tt}\n+\\frac{\\ga-1}{R}t^2\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_{tt}.\\ea$$ It thus follows from  (\\ref{mn5})  and\n$(\\ref{mn1})_1 $ that  $F_4\\in L^2(0,T_0;L^2),$ which together with\n(\\ref{mn6}), (\\ref{mn5}), $(\\ref{mn1})_1,$ and standard\n$L^2$-estimate for (\\ref{mn13}) gives that \\be\\la{mn14}\n(t^2\\te_{tt})_t,~\\na^2(t^2\\te_{tt})\\in L^2(0,T_0;L^2).\\ee One thus\nobtains $(\\ref{mn1})_2$ directly from (\\ref{mn5}), $(\\ref{mn1})_1,$\n (\\ref{mn12}), and (\\ref{mn14}).\n\nFinally, we will show (\\ref{mn2}), the lower bound of $\\te$. In fact, it   follows from $(\\ref{a1})_3$ and $(\\ref{h1})$ that\n  \\be\\la{mn4}\\begin{cases}  \\n\\te_t+\\n u\\cdot\\na\n\\te-\\frac{\\ka(\\ga-1)}{R}\\Delta\\te+(\\ga-1)\\n \\te \\div u\\ge 0,&\\quad\\text{in}\\,\\,\\O\\times(0,T],\\\\\n\\na\\te\\cdot n=0,&\\quad \\text{on}\\,\\,\\p\\O\\times(0,T],\\end{cases}\\ee\n where we have used \\be \\la{a2.56} 2\\mu  |\\mathfrak{D}(u)|^2  +\\lambda (\\div u)^2\n \\ge 0.\\ee Using (\\ref{mn5}), it gets\n \\bnn\n\\int_0^{T_0}\\|\\div u\\|_{L^\\infty} dt\n  <\\infty,\\enn\n  which together with  the standard maximum principle\n    thus gives  (\\ref{mn2}). The proof of Lemma \\ref{th0}\n   is completed.\n\\hfill $\\Box$\n\n\\fi\n\nNext, we state the classical Jensen's inequality (see \\cite[Theorem 3.3]{jes}), which guarantees the key uniform upper and lower bounds of $\\overline P$.\n\\begin{lemma}\\la{je}\nLet $\\mu$ be a positive measure on a $\\si$-algebra $\\mathfrak{M}$ in a set $\\Omega$, so that $\\mu(\\Omega)=1$. If $f$ is a real function in $L^1(\\mu)$,  $a<f(x)<b$ for all $x\\in \\Omega$, and  $\\Phi$ is convex on $(a,b)$, then\n\\be\\la{jen}\\Phi\\left(\\int_\\Omega fd\\mu\\right)\\le \\int_\\Omega (\\Phi\\circ f)d\\mu.\\ee\n\\end{lemma}\n\nNext, the following well-known Gagliardo-Nirenberg-Sobolev-type inequality (see \\cite{nir}) will be used later frequently.\n\n\n\\begin{lemma}\n \\la{11} Assume that $\\O\\subset\\r^3$ is a bounded Lipschitz domain. For $r\\in [2,6]$, $p\\in(1,\\infty)$, and $ q\\in\n(3,\\infty),$ there exist  positive\n constants $C, \\,C',$ and $C''$  which may depend  on $r,\\,p,\\, q$, and $\\O$ such that for $f\\in H^1$,\n $g\\in L^p\\cap W^{1,q}$,  and $\\varphi,\\,\\psi\\in H^2$,\n \\be\n\\la{g1}\\|f\\|_{L^r} \\le C \\| f\\|_{L^2}^{{(6-r)}\/{2r}}  \\|\\na f\\|_{L^2}^{{(3r-6)}\/{2r}} + C' \\| f\\|_{L^2} ,\\ee\n\\be \\la{g2}\\|g\\|_{C\\left(\\overline{\\Omega}\\right)}\n       \\le C \\|g\\|_{L^p}^{p(q-3)\/(3q+p(q-3))}\\|\\na g\\|_{L^q}^{3q\/(3q+p(q-3))} + C'' \\|g\\|_{L^2},\\ee\n\\be\\la{hs} \\|\\varphi\\psi\\|_{H^2}\\le C \\|\\varphi\\|_{H^2}\\|\\psi\\|_{H^2}.\\ee\nMoreover, if $f\\cdot n|_{\\p \\O}=0$ or $\\overline{f}=0$, one has $C'=0$.\nSimilarly, if $g\\cdot n|_{\\p \\O}=0$ or $\\overline{g}=0$, it holds  $C''=0$.\n\\end{lemma}\n\n\nThen, the following div-curl type inequality (see \\cite[Theorem 3.2]{vww}) is used to get the estimates on the spatial derivatives of velocity.\n\\begin{lemma}   \\la{crle1}\n\tAssume that $\\O\\subset\\r^3$ is a simply connected bounded domain with $C^{1,1}$ boundary $\\partial\\Omega$. Then, for $v\\in W^{1,q}$ with $q\\in(1,\\infty)$ and $v\\cdot n|_{\\partial\\Omega}=0$, there exists a positive constant $C=C(q,\\Omega)$ such that\n\t\\be\\la{nn2}\\|v\\|_{W^{1,q}}\\leq C\\left(\\|\\div v\\|_{L^{q}}+\\|\\curl v\\|_{L^{q}}\\right).\\ee\n\\end{lemma}\n\n\n\nNow, we deduce  from $(\\ref{a1})_2$ that $G$ defined in \\eqref{hj1}  satisfies the following elliptic equation:\n \\be\\la{h13}\\begin{cases}\n \\Delta G = \\div (\\rho\\dot{u}),\\,\\,&x\\in \\O,\\\\\n \\na G\\cdot n=\\n \\dot u\\cdot n,\\,\\,&x\\in\\p\\O.\n \\end{cases}\\ee\n\n\n\n\n\nThe  standard $L^p$-estimate  for  (\\ref{h13})  together with the div-curl type inequalities (see \\cite{CANEHS,vww})\nyields  the following essential estimates (see also \\cite[Lemma 2.9]{C-L}).\n\n\n\n\n\n\\begin{lemma} \\la{le4}\n Let  $\\O\\subset\\r^3$ be the same as in Theorem \\ref{th1} and $(\\rho,u,\\te)$ a smooth solution of\n   (\\ref{a1})--(\\ref{h1}).\n    Then there exists a generic positive\n   constant $C$ depending only on $p$, $\\mu,$   $\\lambda,$  and $\\O$ such that, for any $p\\in [2,6],$\n   \\be\\la{h18}\n   \\|\\na u \\|_{L^p} \\le C \\left( \\|\\div u\\|_{L^p}  +\\|\\curl u\\|_{L^p}\\right),\\ee\n   \\be\\la{h19}\\|\\na G\\|_{L^p} \\le C\\|\\rho\\dot{u}\\|_{L^p},\\ee\n   \\be\\la{h191}  \\|{\\nabla \\curl u}\\|_{L^p}\n   \\le C (\\|\\rho\\dot{u}\\|_{L^p} + \\|\\na {u}\\|_{L^2}),\\ee\n   \\be  \\la{h20}\\|G\\|_{L^p}\n   \\le C \\|\\rho\\dot{u}\\|_{L^2}^{(3p-6)\/(2p) }\n   \\left(\\|{\\nabla u}\\|_{L^2}\n   +  \\|P-\\bp\\|_{L^2}\\right)^{(6-p)\/(2p)},\\ee\n   \\be  \\la{h21} \\|\\curl u\\|_{L^p}\n   \\le C \\|\\rho\\dot{u}\\|_{L^2}^{(3p-6)\/(2p) }\n   \\|{\\nabla u}\\|_{L^2}\n   ^{(6-p)\/(2p)} + C\\|{\\nabla u}\\|_{L^2}.\\ee\n  Moreover, it holds that\n  \\be \\la{1h19}\\ba\n  \\|G\\|_{L^p} +\\|\\curl u\\|_{L^p}\\le C (\\|\\rho \\dot{u}\\|_{L^2}+\\|\\na u\\|_{L^2}),\n  \\ea \\ee\n\n  \\be\\ba\\la{h17} \\|\\na u\\|_{L^p}\\le& C  \\|\\rho\\dot{u}\\|_{L^2}^{(3p-6)\/(2p) }  \\left(\\|{\\nabla\n  \tu}\\|_{L^2}+\\|P-\\bp\\|_{L^2}\\right)^{(6-p)\/(2p)}\\\\\n  &+ C (\\|{\\nabla\tu}\\|_{L^2} +  \\|P-\\bp\\|_{L^p}).\n  \\ea\\ee\n\\end{lemma}\n\nNext, we state the following estimates  on $\\dot u$ with $u \\cdot n|_{\\p \\O}=0$, whose proof can be found in \\cite[Lemma 2.10]{C-L}\n\n\n\n\\begin{lemma}\\la{uup1}\n\n\tLet $\\Omega \\subset \\mathbb{R}^3$ with $C^{1,1}$ boundary.\n\n\tAssume that $u$ is smooth enough and $u \\cdot n|_{\\p \\O}=0$,\n\tthen there exists a generic positive constant $C$ depending only on   $\\Omega$ such that\n\t\\be\\la{tb90}\n\t\\ba\\|\\dot{u}\\|_{L^6}\\le C(\\|\\nabla\\dot{u}\\|_{L^2}+\\|\\nabla u\\|_{L^2}^2),\n\t\\ea\\ee\n\t\\be\\la{tb11}\\ba\n\t\\|\\nabla\\dot{u}\\|_{L^2}\\le C(\\|\\div \\dot{u}\\|_{L^2}+\\|\\curl \\dot{u}\\|_{L^2}+\\|\\nabla u\\|_{L^4}^2).\n\t\\ea\\ee\n\\end{lemma}\n\n\nFurthermore, by the classical elliptic theory owing to  Agmon-Douglis-Nirenberg \\cite{adn}, one has the following estimates for smooth solution to  the Lam\\'{e}'s system:\n\\be\\la{lames}\\begin{cases}\n\t-\\mu\\Delta u-(\\lambda+\\mu)\\nabla\\div u=-\\rho\\dot{u}-\\nabla P , \\,\\, &x\\in\\Omega, \\\\\n\tu\\cdot n=0,\\,\\curl u\\times n=0,\\,\\,&x\\in\\partial\\Omega,\n\\end{cases} \\ee\nwhere\n$\\Omega\\subset\\r^3$ is a bounded smooth domain, and $\\mu,\\lambda$ satisfy the condition \\eqref{h3}.\n\\begin{lemma}  \\la{zhle}\n\tLet $u$ be a smooth solution of the Lam\\'{e}'s system (\\ref{lames}). Then for $p\\in[2,6]$ and $k\\ge 2$,   there exists a positive constant $C$ depending only on $\\lambda,\\,\\mu,\\,p,\\,k,$ and $\\Omega$ such that\n\t\\be\\ba\\la{rmk1}\n\t\\| u\\|_{W^{k,p}}\n\t&\\leq C(\\|\\rho\\dot{u}\\|_{W^{k-2,p}}+\\|\\nabla P\\|_{W^{k-2,p}}+\\|\\nabla u\\|_{L^2}).\n\t\\ea\\ee\n\n\\end{lemma}\n\n\n\n\n\n\n\nNext, the following Gr\\\"{o}nwall-type inequality will be used to get the uniform (in time) upper bound of the density $\\n$, whose proof can be found in \\cite[Lemma 2.5]{H-L}.\n\\begin{lemma} \\la{le1}\n\tLet the function $y\\in W^{1,1}(0,T)$ satisfy\n\t\\be \\notag\n\ty'(t)+\\al(t) y(t)\\le  g(t)\\mbox{  on  } [0,T] ,\\quad y(0)=y_0,\n\t\\ee\n\twhere $0<\\al_0\\le \\al(t)$ for any $t\\in[0,T]$ and  $ g \\in L^p(0,T_1)\\cap L^q(T_1,T)$  for some $p,\\,q\\ge 1, $  $T_1\\in [0,T].$ Then it has\n\t\\be \\la{2.34}\n\t\\sup_{0\\le t\\le T} y(t) \\le |y_0| + (1+\\al_0^{-1}) \\left(\\|g\\|_{L^p(0,T_1)} + \\|g\\|_{L^q(T_1,T)}\\right).\n\t\\ee\n\\end{lemma}\n\n\n\n\n\n\nNext, to derive the exponential decay property of the solutions, we consider the following auxiliary problem\n\\be\\la{e480}\\begin{cases}\n{\\rm div}v=f,&x\\in\\Omega, \\\\\nv=0,&x\\in{\\partial\\Omega}.\n\\end{cases} \\ee\n\\begin{lemma} \\cite[Theorem III.3.1]{GPG} \\la{th00}  There exists a linear operator $\\mathcal{B} = [\\mathcal{B}_1 , \\mathcal{B}_2 , \\mathcal{B}_3 ]$ enjoying\nthe properties:\n\n1)The operator $$\\mathcal{B}:\\{f\\in L^p\\left|\\right.\\overline f =0\\}\\mapsto W^{1,p}_0$$ is a bounded linear one, that is,\n\\bnn \\|\\mathcal{B}[f]\\|_{W^{1,p}_0}\\le C(p)\\|f\\|_{L^p}, \\mbox{ for any }p\\in (1,\\infty).\\enn\n\n2) The function $v = \\mathcal{B}[f]$ solves the problem (\\ref{e480}).\n\n3) If, moreover, for $f = \\div  g$ with a certain $g\\in L^r,$ $g\\cdot n|_{\\pa\\O}=0,$  then  for   any $r  \\in (1,\\infty),$\n\\bnn \\|\\mathcal{B}[f]\\|_{L^{r}}\\le C(r)\\|g\\|_{L^r} .\\enn\n\\end{lemma}\n\n\n\n\t\n\n\nFinally, in order to estimate $\\|\\nabla u\\|_{L^\\infty}$ for the further higher order estimates, we need the following Beale-Kato-Majda-type inequality, which was first proved in \\cite{bkm,kato} when $\\div u\\equiv 0,$\nwhose detailed proof in the case of slip boundary condition  can be found in \\cite[Lemma 2.7]{C-L} (see also \\cite{h101,h1x}).\n\\begin{lemma}\\la{le9}\nLet $\\O\\subset\\r^3$ be a  bounded smooth domain. For $3<q<\\infty$, assume that $ u \\in \\{ f \\in W^{2,q}|  f\\cdot n=0, \\curl f\\times n=0 \\text{ on } \\partial \\Omega \\}$,  then there is a positive constant  $C=C(q,\\O)$ such that\n\\bnn\\ba\n\\|\\na u\\|_{L^\\infty}\\le C\\left(\\|{\\rm div}u\\|_{L^\\infty}+\\|\\curl u\\|_{L^\\infty} \\right)\\ln(e+\\|\\na^2u\\|_{L^q})+C\\|\\na u\\|_{L^2} +C.\n\\ea\\enn\n\\end{lemma}\n\n\n\n\n\\section{\\la{se3} A priori estimates (I): lower-order estimates}\n\n\nIn this section,  we  will establish a priori bounds for the local-in-time smooth solution to problem (\\ref{a1})--(\\ref{h1}) obtained in Lemma \\ref{th0}.\n\nLet $(\\n,u,\\te)$ be a smooth solution to the  problem (\\ref{a1})--(\\ref{h1})  on $\\Omega\\times (0,T]$ for some fixed time $T>0,$ with  initial data $(\\n_0,u_0,\\te_0)$ satisfying \\eqref{2.1}.\nFor\n$\\si(t)\\triangleq\\min\\{1,t\\}, $  we  define\n$A_i(T)(i= 1,2,3)$ as follows:\n  \\be\\la{As1}\n  A_1(T) \\triangleq \\sup_{t\\in[0,T] }\\|\\nabla u \\|_{L^2}^2\n  + \\int_0^{T}\\int  \\rho|\\dot{u}|^2dxdt,\n  \\ee\n  \\be\\label{AS1}\n  A_2(T) \\triangleq \\frac{1}{2(\\ga-1)}\\sup_{t\\in[0,T] }\\int\\n (R\\te-\\bp)^2dx\n  +\\int_0^T\\left( \\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right)dt,\n  \\ee\n  \\be\\ba \\label{AS2}\n  A_3(T) \\triangleq &\\sup_{t\\in(0,T]}\\left(\\si \\|\\na u\\|_{L^2}^2+\\si^2\\int\\n |\\dot u|^2dx + \\si^2\\|\\na\\te\\|_{L^2}^2 \\right)\\\\\n  & + \\int_0^T\\int\\left(\\si\\n |\\dot u|^2 +\\sigma^2|\\nabla\\dot{u}|^2 +\\sigma^2\\n|\\dot \\te|^2 \\right)dxdt.\n  \\ea\\ee\n\n\n\nWe have the following key a priori estimates on $(\\n,u,\\te)$.\n\\begin{pro}\\la{pr1}\nFor  given numbers $M>0$, $\\on> 2,$  and $\\bt> 1,$ assume further that $(\\rho_0,u_0,\\te_0) $  satisfies\n\\be \\la{3.1}\n0<\\inf \\rho_0 \\le\\sup \\rho_0 <\\on,\\quad 0<\\inf \\te_0 \\le\\sup \\te_0 \\le \\bt, \\quad \\|\\na u_0\\|_{L^2} \\le M.\n\\ee\nThen there exist  positive constants $K$,  $C^\\ast$, $\\al$,  $\\te_\\infty$, and $\\ep_0$ all depending on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,$      $\\on,$ $\\bt$, $\\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\O\\times (0,T]$ satisfying\n\\be \\la{z1}\n0<  \\rho\\le 2\\on, \\,\\,\\, A_1(T)\\le 3 K, \\,\\,\\, A_2(T) \\le 2C_0^{1\/4}, \\,\\,\\, A_3(T)  \\le 2C_0^{1\/6},\n\\ee\nthe following estimates hold:\n\\be \\la{zs2}\n0< \\rho\\le 3\\on\/2,\\,\\,\\, A_1(T)\\le 2 K, \\,\\,\\, A_2(T) \\le C_0^{1\/4},  \\,\\,\\, A_3(T)  \\le  C_0^{1\/6},\n\\ee\nand for any $t\\geq1$,\n\\be\\la{h22}\n\\|\\n-1\\|_{L^2}+\\|u\\|_{W^{1,6}}^2+\\|\\te-\\te_\\infty\\|_{H^2}^2\\le C^\\ast e^{-\\al t},\n\\ee\nprovided \\be\\la{z01}C_0\\le \\ve_0.\\ee      \\end{pro}\n\n\\begin{proof}\nProposition \\ref{pr1} is a straight consequence of\nthe following Lemmas \\ref{le2}, \\ref{le6}, \\ref{le3}, \\ref{le7}, and \\ref{pr2}  with $\\ve_0$ as in (\\ref{t7}).\n\\end{proof}\n\nIn this section, we always assume that $C_0\\le 1$ and let $C$ denote some generic positive constant depending only on $\\mu$,  $\\lambda$, $\\ka$,  $R$, $\\ga$, $\\on$, $\\bt$, $\\O,$ and $M,$ and we write $C(\\al)$ to emphasize that $C$ may depend  on $\\al.$\n\nTo begin with, we have the following uniform estimate on $\\bp$, which plays an important role in the whole analysis.\n\n\n\\begin{lemma}\\la{a13.1} Under the conditions of Proposition \\ref{pr1}, there exists a positive constant $C$ depending only on $\\mu$, $R$, and $\\on$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1})  on $\\Omega\\times (0,T] $ satisfying\n\\be\\la{3.q2}\n0<\\n\\le 2\\on ,\\quad A_2(T)\\le 2C_0^{1\/4},\n\\ee\nthe following estimates hold:\n\\be \\la{a2.112}\n\\sup_{0\\le t\\le T}\\int\\left( \\n |u|^2+(\\n-\\tn)^2\\right)dx \\le C  C_0^{1\/4},\n\\ee\nand\n\\be \\la{key}\n0<\\pi_1  \\le \\overline P (t)\\le \\pi_2,~~~\\mbox{for~any}~t\\in [0,T],\n\\ee\nwhere  $\\pi_1 $ and  $\\pi_2$ are  positive constants depending only on $\\mu$, $\\gamma$, $R$, and $\\O$.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst, it follows from (\\ref{3.1}) and  (\\ref{mn2}) that, for all $ (x,t)\\in \\Omega\\times(0,T),$\n\\be \\la{3.2}\n\\te(x,t)>0 .\n\\ee\n\nNote that\n\\be\\notag\n\\Delta u = \\na \\div u - \\na \\times \\curl u,\n\\ee\none can rewrite $(\\ref{a1})_2$  as\n\\be\\la{a11}\\ba\n\\n (u_t+  u\\cdot \\na u )&=(2\\mu+\\lambda)\\na{\\rm div}u- \\mu \\na \\times \\curl u-\\na P.\\ea\\ee\nAdding $(\\ref{a11})$ multiplied by $u$ to $(\\ref{a1})_3$ multiplied by $1-\\te^{-1}$ and  integrating the resulting equality over $\\Omega$ by parts,  we obtain\nafter   using $(\\ref{a1})_1$, \\eqref{h3}, \\eqref{3.2}, and  the boundary conditions \\eqref{h1}    that\n\\be\\la{la2.7}\\ba\nE'(t)\n&=-\\int \\left( \\frac{\\lambda(\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2}{{\\te}}+\\ka \\frac{|\\na \\te|^2}{\\te^2} \\right)dx \\\\\n&\\quad - \\mu \\int \\left(|\\curl u|^2+2(\\div u)^2 - 2 |\\mathfrak{D}(u)|^2\\right)dx\\\\\n&\\le 2\\mu \\int  |\\na u|^2 dx,\n\\ea\\ee\nwhere  $E(t)$ is  the basic energy defined by \\eqref{enet}.\n\nThen, integrating \\eqref{la2.7} with respect to $t$ over $(0,T)$ and using \\eqref{3.q2}, one has\n \\be\\la{a2.8}\\ba\n&\\sup_{0\\le t\\le T} E(t\n\\le C_0+ 2\\mu \\int_{0}^{T} \\int  |\\na u|^2 dxdt\\le C C_0^{1\/4},\\ea\\ee\nwhich together with\n\\be\\la{a2.9}\\ba\n (\\n-1)^2\\ge 1+\\n\\log\\n-\\n&=(\\n-1)^2\\int_0^1\\frac{1-\\al}{\\al (\\n-1)+1}d\\al  \\ge \\frac{(\\n-1)^2}{ 2(2\\on+1)  }\n \\ea\\ee\ngives (\\ref{a2.112}).\n\n\n\n\n\n\nNext,  it is easy to deduce from $\\eqref{a1}_1$  and \\eqref{m} that for any $t\\in[0,T]$,\n\\be \\la{mmm}\n\\overline\\rho(t)=\\overline{\\n_0}=1.\n\\ee\nDenote $d\\mu\\triangleq|\\Omega|^{-1}\\n dx$. Then $d\\mu$ is a positive measure satisfying $\\mu(\\Omega)=1$ due to \\eqref{3.q2} and \\eqref{mmm}. Moreover, observe that $y-\\log y-1$ is a convex function in $(0,\\infty)$,  it thus follows directly from Jensen's inequality \\eqref{jen} that for any $t\\in [0,T]$,\n\\be \\notag\n\\overline {\\rho \\te }(t) - \\log \\overline {\\rho \\te }(t) -1 \\le \\int(\\te-\\log\\te-1)\\frac{\\n dx}{|\\Omega|}\\le C\n\\ee due to   \\eqref{a2.8}.\nThis in particular gives  \\eqref{key} and finishes the proof of Lemma \\ref{a13.1}.\n\n\\end{proof}\n\n\\begin{remark}\n    It should be pointed out that the following term in (\\ref{la2.7})\n    \\begin{equation}\\label{xm}\n        - \\mu \\int \\left( |\\curl u|^2+2(\\div u)^2 - 2 |\\mathfrak{D}(u)|^2 \\right)dx\n    \\end{equation}\n    is a sign-undetermined term due to the slip boundary condition (\\ref{bb}), which is in sharp contrast to the Cauchy problem \\cite{H-L} where the term (\\ref{xm}) vanishes after integration by parts.\n   \n   \n   \n    Thus, in this case, we can not bound the basic energy only by the initial energy. However, this term obviously can be bounded by $C \\int  |\\nabla u|^2dx$, which implies a ``weaker\" basic energy estimate (\\ref{a2.8}).\n\\end{remark}\n\n\n\n\nThe next lemma provides an estimate on the term $A_1(T)$.\n\\begin{lemma}\\la{le2}\n\tUnder the conditions of Proposition \\ref{pr1}, there exist positive constants  $K $  and $\\ep_1 $ both depending only  on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\, \\O,$ and $M$ such that if  $(\\rho,u,\\te)$ is a smooth solution to the problem  (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying\n\t\\be\\la{3.q1}  0<\\n\\le 2\\on ,\\quad A_2(T)\\le 2C_0^{1\/4},\\quad A_1(T)\\le 3K,\\ee\n\tthe following estimate holds:\n\t\\be\\la{h23} A_1(T)\\le 2K ,  \\ee\n\tprovided   $C_0\\le \\ep_1.$\n\\end{lemma}\n\n\n\n\\begin{proof}\n First, integrating $(\\ref{a11})$  multiplied by $2u_t $ over $\\Omega $ by parts gives\n \\be\\ba \\la{hh17}\n &\\frac{d}{dt}\\int \\left(  {\\mu} |\\curl u|^2+ (2\\mu+\\lambda)(\\div u)^2\\right)dx+ \\int\\rho |\\dot u|^2dx \\\\\n&\\le 2\\int  P\\div u_t dx+ \\int \\n|u\\cdot \\na u|^2dx\\\\\n &=  2\\frac{d}{dt}\\int  (P-\\overline P) \\div u  dx-2\\int (P-\\overline{P})_t \\div u dx+\\int \\n|u\\cdot \\na u|^2dx\\\\\n &= 2\\frac{d}{dt}\\int  (P-\\overline P) \\div u  dx-\\frac{1}{2\\mu+\\lambda}\\frac{d}{dt}\\int (P-\\overline P)^2 dx\\\\\n &\\quad-\\frac{2}{2\\mu+\\lambda}\\int (P-\\overline{P})_t G dx+ \\int \\n|u\\cdot \\na u|^2dx ,\n \\ea\\ee\nwhere in the last equality  we have used\n(\\ref{hj1}).\n\nNext,   straight calculations show\nthat for any $p\\in [2,6],$\n\\be\\la{pq}\\ba  \\|R\\te- \\overline P \\|_{L^p}\n  \\le R\\|\\te- \\overline\\te \\|_{L^p}+C|R\\overline\\te-\\overline P|\\le C(\\hat{\\n}) \\|\\na \\te\\|_{L^2},\\ea\\ee\nwhere one has used (\\ref{g1}) and the following fact:\n\\be\\notag\\ba\n|R\\overline\\te-\\overline P|=&\\frac{R}{|\\O|}\\xl|\\int(1-\\n)\\te dx\\xr|\n=  \\frac{R}{|\\O|}\\xl|\\int(1-\\n)(\\te-\\overline\\te)dx\\xr|\\\\\n\\le&C\\|\\n-1\\|_{L^2}\\| \\te-\\overline\\te\\|_{L^2}\\\\\n\\le&C(\\on)C_0^{1\/8}\\|\\na \\te\\|_{L^2}\\ea\\ee\ndue to   (\\ref{mmm}) and \\eqref{a2.112}. Thus, it follows from \\eqref{key}, \\eqref{3.q1}, \\eqref{a2.112}, and \\eqref{pq} that for any $p\\in[2,6]$,\n\\be\\la{p}\\ba  \\|P-\\overline P \\|_{L^p}&= \\| \\n(R\\te-\n\\overline P )+ (\n \\n -1)\\overline P\\|_{L^p}\\\\&\\le \\|\\n(R\\te-\\overline P )\\|_{L^2}^{(6-p)\/(2p)}\n \\|\\n(R\\te-\\overline P )\\|_{L^6}^{ 3(p-2)\/(2p)}+ \\pi_2 \\|\\n-1\\|_{L^p}\n \\\\&\\le C(\\on)C_0^{(6-p)\/(16p)}\n \\|\\na\\te \\|_{L^2}^{ 3(p-2)\/(2p)}+ C(\\hat\\n)C_0^{1\/(4p)},\n  \\ea\\ee\nwhich together with (\\ref{h17}) and \\eqref{3.q1} yields\n\\be \\la{3.30}  \\|\\na\nu\\|_{L^6} \\le C(\\hat\\n) \\left(  \\|\\n^{1\/2}\\dot\nu\\|_{L^2}+\\|\\na u\\|_{L^2}+ \\|\\na \\te\\|_{L^2}+C_0^{1\/24}\\right).\n\\ee\n\nNote that (\\ref{a1})$_3$ implies\n\\be \\la{op3} \\ba\nP_t=&-\\div (Pu) -(\\gamma-1) P\\div u+(\\ga-1)\\ka \\Delta\\te\\\\&+(\\ga-1)\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right),\n\\ea\\ee\nwhich along with \\eqref{h1} gives\n\\be \\ba \\la{pt}\n\\bp_t =&\n-(\\gamma-1) \\overline{P\\div u} +(\\ga-1)\\left(\\lambda\n\\overline{(\\div u)^2}+2\\mu \\overline{|\\mathfrak{D}(u)|^2}\\right).\n\\ea \\ee\nWe thus obtain after  using  integration by parts, (\\ref{key}),  (\\ref{g1}), (\\ref{h19}), and  (\\ref{p})--(\\ref{op3}) that\n\\be\\la{a16}\\ba\n&\\left|\\int   P_t Gdx\\right| \\\\\n&\\le C\\int P(|G||\\na u|+ |u||\\na G|)dx+ C\\int\\left( |\\na\\te||\\na G|+|\\na u|^2|G|\\right)dx \\\\\n&\\le  C\\int |P-  \\bp| (|G||\\na u|+|u||\\na G|)dx +C\\bp\\int (|G||\\na u|+|u||\\na G|)dx \\\\\n&\\quad + C \\|\\na G\\|_{L^2} \\|\\na \\te\\|_{L^2} + C \\| G\\|_{L^6} \\|\\na u\\|_{L^2}^{3\/2}  \\|\\na u\\|_{L^6}^{1\/2} \\\\\n&\\le C(\\hat \\n)(\\|\\na \\te\\|_{L^2}^{1\/2}+1)\\|\\na G\\|_{L^2}\\|\\na u\\|_{L^2}+ C \\|\\na G\\|_{L^2} \\|\\na \\te\\|_{L^2} \\\\\n& \\quad+ C(\\hat\\n) \\|\\na G\\|_{L^2} \\|\\na u\\|_{L^2}^{3\/2} \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+\\|\\na\\te\\|_{L^2}+1 \\right)^{1\/2} \\\\\n&\\le \\de \\|\\na G\\|_{L^2}^2 +\\de \\|\\rho^{1\/2}\\dot u\\|^2_{L^2}+C(\\de,\\on) \\left( \\|\\na u\\|_{L^2}^2+ \\|\\na \\te\\|_{L^2}^2+ \\|\\na u\\|^6_{L^2}\\right) \\\\\n&\\le C(\\on)\\de\\|\\rho^{1\/2} \\dot u\\|^2_{L^2}     +C(\\de,\\on) \\left( \\|\\na u\\|_{L^2}^2+ \\|\\na \\te\\|_{L^2}^2 +\\|\\na  u\\|^6_{L^2}\\right),\n\\ea\\ee\nand\n\\be\\la{a161}\\ba   \\left|\\int   \\bp_t  Gdx\\right|\n\\le& C(\\on)(\\| \\na u\\|_{L^2}+\\| \\na u\\|_{L^2}^2)\\| G\\|_{L^2}\n\\\\  \\le& \\de \\|\\na G\\|_{L^2}^2  +C(\\de,\\on)(\\|\\na u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^4)\n\\\\\\le&  C(\\on)\\de\\|\\rho^{1\/2}\n\\dot u\\|^2_{L^2}     +C(\\de,\\on) (\\|\\na u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^4),\n\\ea\\ee\nwhere one has used\n\\be\\la{511} \\ba\n|\\overline P_t|\n&\\le C\\| P-\\bp\\|_{L^2} \\| \\na u\\|_{L^2} + C \\|\\na u\\|_{L^2}^2 \\le C(\\on)(C^{1\/8}_0\\| \\na u\\|_{L^2}+\\| \\na u\\|_{L^2}^2)\n\\ea \\ee\nowing to \\eqref{pt} and \\eqref{p}.\n\nThen, it follows from (\\ref{g1}), \\eqref{3.q1}, and (\\ref{3.30}) that\n\\be\\la{op1}\\ba\n\\int \\n|u\\cdot \\na u|^2dx&\\le C(\\on)\\|u\\|_{L^6}^2 \\|\\na u\\|_{L^2} \\|\\na u\\|_{L^6}  \\\\\n&\\le \\de\\|\\rho^{1\/2} \\dot u\\|_{L^2}^2+ C(\\de,\\on)\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+\\|\\na\nu\\|_{L^2}^6\\right).\n\\ea\\ee\n\nFinally, substituting (\\ref{a16}), \\eqref{a161}, and (\\ref{op1}) into (\\ref{hh17}), one obtains after choosing $\\de$ suitably small that\n\\be\\ba \\la{1hh17}\n&\\frac{d}{dt}\\int \\left(  {\\mu} |\\curl u|^2+ (2\\mu+\\lambda)(\\div u)^2\\right)dx+ \\frac{1}{2\\mu+\\lambda}\\frac{d}{dt}  \\|P-\\overline P\\|_{L^2}^2  + \\frac{1}{2}\\int\\rho |\\dot u|^2dx \\\\\n& \\le 2\\frac{d}{dt}\\int  (P-\\overline P) \\div u  dx + C(\\on)\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+\\|\\na\nu\\|_{L^2}^6\\right).\n\\ea\\ee\n\nNote that it holds\n\\be\\la{cz}\\ba \\|P-\\overline P\\|_{L^2}^2(0)&=R^2\\int(\\n_0\\te_0-\\overline{\\n_0\\te_0})^2dx\\\\\n&\\le C(\\on)\\int\\n_0(\\te_0-\\overline{\\n_0\\te_0})^2dx+C\\int(\\n_0-1)^2dx\\\\\n&\\le C(\\on, \\bt)C_0,\\ea\\ee\nwhere we have used \\eqref{key}, \\eqref{a2.9}, and the following fact:\n\\be \\ba \\label{jia10} \\int \\n_0(\\te_0- \\overline{\\n_0\\te_0})^2dx &\\le C\\int \\n_0(\\te_0-1)^2dx+C|1-\\overline{\\n_0\\te_0}|^2\\\\\n&\\le  C\\int \\n_0(\\te_0-1)^2dx+C\\left|\\int\\n_0(1-\\te_0)dx\\right|^2\\\\\n&\\le  C(\\hat\\n,\\bt)\\int \\n_0(\\te_0-\\log\\te_0-1) dx\\\\\n&\\le  C(\\hat\\n, \\bt)C_0\n\\ea\\ee\ndue to \\eqref{m} and\n \\be\\la{cz1}\\te-\\log\\te-1 =(\\te-1)^2\\int_0^1\\frac{\\al}{\\al (\\te-1)+1}d\\al\\geq\\frac{1}{2(\\|\\te(\\cdot,t)\\|_{L^{\\infty}}+1)}(\\te-1)^2.\\ee\nThen, integrating (\\ref{1hh17}) over $(0,T)$, one deduces from (\\ref{h18}), \\eqref{cz}, (\\ref{3.q1}), and \\eqref{p} that\n\\bnn\\la{h81} \\ba\n&\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}^2+ \\int_0^{T}\\int\\rho|\\dot{u}|^2dxdt\\\\\n&\\le CM^2+C(\\on,\\hat\\te)C_0^{1\/4} + C(\\on ) C_0^{1\/4}\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}^4 + C(\\on ) C_0^{1\/8}\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}\\\\\n&\\le CM^2+C(\\on,\\hat\\te)C_0^{1\/12}+ C(\\on ) C_0^{1\/4}\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}^4 \\\\\n&\\le K+9C(\\on)C_0^{1\/4}K^2 \\\\\n&\\le 2K,\n\\ea \\enn\nwith $K\\triangleq CM^2+C(\\on,\\hat\\te) +1$, provided\n\\be\\notag C_0\\le \\ep_1 \\triangleq \\min\\left\\{1,\\xl(9C(\\on)K\\xr)^{-4}\\right\\}.\\ee\nThe proof of Lemma \\ref{le2} is completed.\n\\end{proof}\n\n\nNext,  to estimate $A_3(T)$, we adopt the approach due to Hoff \\cite{Hof1} (see also Huang-Li \\cite{H-L})  to establish the following elementary estimates on $\\dot u$ and $\\dot \\te$, where the boundary terms are handled by the ideas due to \\cite{C-L}.\nThe estimate of $A_3(T)$ will be postponed to Lemma \\ref{le6}.\n\n\\begin{lemma}\\la{a113.4}\n\tUnder the conditions of Proposition \\ref{pr1}, let $(\\rho,u,\\te)$ be a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}.  Then  there exist positive constants $C$, $ C_1$, and $C_2$ depending only on $\\mu,\\,\\lambda, \\,k,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O,$ and $ M$  such that, for any $\\eta\\in (0,1]$ and $m\\geq0,$\nthe following estimates hold:\n\\be\\ba  \\la{an1}\n(\\sigma B_1)'(t) + \\frac{1}{2}\\sigma \\int \\rho |\\dot u|^2dx\n\\le   C C_0^{1\/4} \\sigma' + C\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right),\n\\ea\\ee\n\\be\\la{ae0}\\ba\n&\\left(\\sigma^{m}\\|\\rho^{1\/2}\\dot{u}\\|_{L^2}^2\\right)_t+C_1 \\sigma^{m}\\|\\na\\dot{u}\\|_{L^2}^2\\\\\n&\\le - 2\\left(\\int_{\\p \\O}  \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t + C(\\si^{m-1}\\si'+\\si^m)  \\|\\rho^{1\/2} \\dot u\\|_{L^2}^2 \\\\&\\quad+\nC_2  \\si^m \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2+ C\\|\\na u\\|^2_{L^2}+C \\si^m \\|\\na u\\|^4_{L^4} + C \\si^m  \\|\\te \\na u\\|_{L^2}^2,\\\\\\ea\\ee\n  and\n \\be\\la{nle7}\\ba  &(\\si^mB_2 )'(t)+\\si^m \\int\\n|\\dot \\te|^2dx\\\\\n&\\le C \\eta \\si^m\\|\\na\\dot u\\|_{L^2}^2+C \\|\\na\n\\te \\|_{L^2}^2+C\\si^m \\|\\na u\\|_{L^4}^4+C(\\eta)\\si^m \\|\\te\\na u\\|_{L^2}^2,\\ea\\ee\nwhere\n\\be \\la{an2} \\ba B_1(t)\\triangleq& \\mu\\|\\curl u\\|_{L^2}^2+ (2\\mu+\\lambda) \\|\\div u\\|_{L^2}^2 \\\\&+ \\frac{1}{2\\mu+\\lambda}  \\|P-\\overline P\\|_{L^2}^2 -2 \\int \\div u(P-\\bp) dx, \\ea\\ee\nand\n \\be\\la{e6}\nB_2(t)\\triangleq\\frac{\\ga-1}{R}\\left(\\ka \\|\\na\n\\te\\|_{L^2}^2-2 \\int (\\lambda (\\div u)^2+2\\mu|\\mathfrak{D}(u)|^2)\\te dx\\right).\\ee\n\\end{lemma}\n\n\n\\begin{proof}   First, multiplying \\eqref{1hh17} by $\\sigma$ and using \\eqref{p} and \\eqref{z1} give \\eqref{an1} directly.\n\n\\iffalse\n\nMultiplying\n$(\\ref{a111}) $ by $\\sigma \\dot{u}$ and integrating the resulting\nequality over $\\Omega $ lead  to \\be\\la{m0} \\ba  \\int \\sigma\n\\rho|\\dot{u}|^2dx & = \\int (\\sigma \\dot{u}\\cdot\\nabla G - \\si \\mu \\na \\times \\curl u\\cdot\\dot{u})dx  \\\\\n&=\\int_{\\partial \\O} \\sigma (u\\cdot\\na u \\cdot n) G dS - \\int \\sigma \\div \\dot{u} G dx - \\mu \\int  \\si  \\curl u \\cdot \\curl \\dot{u} dx \\\\\n& \\triangleq \\sum_{i=1}^{3}M_i. \\ea \\ee\n\nFor the term $M_1$, it can be deduced from \\eqref{pzw1}, \\eqref{g1},  \\eqref{h19},  and \\eqref{z1} that\n\\be \\ba \\la{b2}\nM_1&=-\\int_{\\partial \\O} \\sigma  (u\\cdot \\na n\\cdot u) G dS\\\\\n &\\le C \\sigma\\|u\\|_{H^1}^2 \\|G\\|_{H^1}\\\\\n &\\le C \\sigma\\|\\na u\\|_{L^2}^2 \\|\\rho \\dot u\\|_{L^2}\\\\\n &\\le \\delta \\sigma\\|\\rho^{1\/2} \\dot u\\|_{L^2}^2+C(\\de,\\on,M) \\sigma\\|\\na u\\|_{L^2}^2.\n\\ea \\ee\n\n\nNotice that\n\\be \\ba \\label{jia1}\nP_t=(R\\n \\te)_t=R\\n \\dot{\\te}-\\div (Pu),~~~\\bp_t=R\\overline{\\n\\dot{\\te}},\n\\ea \\ee\nwhich along with some straight calculations gives\n\\be \\ba\\la{pt1}\n\\div \\dot u G=& (\\div u_t + \\div (u \\cdot \\na u))((2\\mu+\\lambda)\\div u - (P-\\bp))\\\\\n=& \\frac{2\\mu+\\lambda}{2} (\\div u)^2_t - ((P-\\bp) \\div u)_t +(P-\\overline{P})_t \\div u \\\\\n&+ (2\\mu+\\lambda) \\div (u \\cdot \\na u)\\div u - (P-\\bp) \\div (u \\cdot \\na u)\\\\\n=& \\frac{2\\mu+\\lambda}{2} (\\div u)^2_t - ((P-\\bp) \\div u)_t +R\\rho \\dot\\te \\div u - \\div(Pu) \\div u\\\\\n&-R\\overline{\\rho \\dot\\te} \\div u + (2\\mu+\\lambda) \\div (u \\cdot \\na u)\\div u - (P-\\bp) \\div (u \\cdot \\na u)\\\\\n=& \\frac{2\\mu+\\lambda}{2} (\\div u)^2_t - ((P-\\bp) \\div u)_t +R\\rho \\dot\\te \\div u -R\\overline{\\rho \\dot\\te} \\div u\\\\\n& + (2\\mu+\\lambda) \\na u : (\\na u)^{\\rm tr} \\div u + \\frac{2\\mu+\\lambda}{2} u \\cdot \\na(\\div u)^2 \\\\\n&- \\div((P-\\bp)u \\div u)- (P-\\bp) \\na u : (\\na u)^{\\rm tr}- \\bp (\\div u)^2.\n\\ea \\ee\nThis together with integration by parts, \\eqref{key}, and \\eqref{z1} implies that for any $\\beta\\in\n(0,1],$\n\\be\\la{m1} \\ba\nM_2    =&  -\\frac{2\\mu+\\lambda}{2} \\left(\\int \\sigma  (\\div u)^2 dx \\right)_t\n       + \\frac{2\\mu+\\lambda}{2} \\si' \\int  (\\div u)^2 dx \\\\\n&+\\left(\\int \\si (P-\\bp) \\div u dx\\right)_t    - \\si' \\int (P-\\bp) \\div u dx - R\\si \\int \\rho \\dot\\te \\div u dx\\\\\n&-(2\\mu+\\lambda)  \\si \\int  \\na u : (\\na u)^{\\rm tr} \\div u dx + \\frac{2\\mu  +\\lambda}{2} \\si \\int  (\\div u)^3 dx\\\\\n&+ \\si \\int (P-\\bp) \\na u : (\\na u)^{\\rm tr}dx +  \\si \\bp \\int (\\div u)^2 dx\\\\\n\\le &  - \\frac{2\\mu+\\lambda}{2} \\left(\\int \\sigma  (\\div u)^2 dx \\right)_t +\\left(\\int \\si (P-\\bp) \\div u dx\\right)_t \\\\\n& + C \\si' \\|P-\\bp\\|_{L^2}^2+ \\beta \\si^2 \\|\\rho^{1\/2} \\dot\\te\\|_{L^2}^2 \\\\\n&+ C \\si^2 \\|\\na u\\|_{L^4}^4 +  C(\\on) \\beta^{-1}  \\|\\na u\\|_{L^2}^2 + C(\\on) \\si \\int\n\\te |\\na u|^2dx\\\\\n\\le &  - \\frac{2\\mu+\\lambda}{2} \\left(\\int \\sigma  (\\div u)^2 dx \\right)_t  +\\left(\\int \\si (P-\\bp) \\div u dx\\right)_t \\\\\n&+ C(\\on)  C_0^{1\/4}\\si'+ \\beta \\si^2 \\|\\rho^{1\/2} \\dot\\te\\|_{L^2}^2  + C \\si^2 \\|\\na u\\|_{L^4}^4 \\\\\n&+  C(\\de,\\on,M) \\beta^{-1} ( \\|\\na u\\|_{L^2}^2 + \\|\\na \\te \\|_{L^2}^2)+ C(\\on) \\de \\si \\|\\rho^{1\/2} \\dot u \\|_{L^2}^2,\\\\\n\\ea \\ee where in the last inequality we have used   (\\ref{p}) and\nthe following simple fact:\n\\be \\la{2.48}\\ba   \\int\n\\te |\\na u|^2dx   & \\le\nC\\int|R\\te-\\bp||\\na u|^2dx+C\\bp \\int |\\na u|^2dx\\\\ &\\le C\n\\|R\\te-\\bp \\|_{L^6}\\|\\na u\\|_{L^2}^{3\/2}\n\\|\\na u\\|_{L^6}^{1\/2}+   C \\|\\na u\\|_{L^2}^2\n\\\\ &\\le C(\\on)\\|\\na\\te\\|_{L^2}\\|\\na u\\|_{L^2}^{3\/2}\n\\left(   \\|\\n \\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+ \\|\\na\\te\\|_{L^2}+1\\right)^{1\/2}\\\\\n&\\quad+   C \\|\\na  u\\|_{L^2}^2\\\\ &\\le \\de\n\\left(  \\|\\na\\te\\|^2_{L^2}  + \\|\\n^{1\/2}  \\dot\nu\\|_{L^2}^2 \\right) + C(\\de,\\on,M)  \\|\\na\nu\\|_{L^2}^2  ,  \\ea\\ee\ndue to  \\eqref{key}, (\\ref{pq}), (\\ref{3.30}),   and (\\ref{z1}).\n\n\nFor the term $M_3$, it holds\u00a3\u00ba\n \\be\\la{m2} \\ba\nM_3\n& = -\\frac{\\mu }{2}\\int\\sigma |\\curl u|^2_t dx\n    -\\mu \\sigma  \\int \\curl u \\cdot \\curl (u\\cdot \\na u)dx \\\\\n& = -\\frac{\\mu }{2}\\left(\\sigma \\|\\curl u\\|_{L^2}^2\\right)_t +\n\\frac{\\mu }{2}\\si'  \\|\\curl u\\|_{L^2}^2\n-\\mu \\sigma  \\int \\curl u \\cdot (\\na u^i \\times \\na_i  u)  dx \\\\&\\quad+ \\frac{\\mu}{2}\\si\n\\int |\\curl u|^2\\div udx \\\\\n& \\le  -\\frac{\\mu }{2}\\left(\\sigma \\|\\curl u\\|_{L^2}^2\\right)_t + C\n\\|\\na u\\|_{L^2}^2 +   C\\sigma^2 \\|\\na u\\|_{L^4}^4  . \\ea \\ee\n\nNow, substituting \\eqref{b2}, (\\ref{m1}),  and  (\\ref{m2}) into (\\ref{m0}), we obtain  (\\ref{an1}) after choosing $\\de$ suitably small.\n\\fi\n\nNow, we will prove  (\\ref{ae0}).\n\n\nFirst, one can rewrite $(\\ref{a1})_2$  as\n\\be\\la{a111}\\ba\\n \\dot u&=\\na G- \\mu \\na \\times \\curl u,\n\\ea\\ee\nwith $G$ defined in \\eqref{hj1}.\nFor $m\\ge 0,$ operating $ \\si^m\\dot u^j[\\pa\/\\pa t+\\div (u\\cdot)]$ to $ (\\ref{a111})^j$ and integrating the resulting equality over $\\Omega$ by parts lead to\n\\be\\la{m4} \\ba\n& \\left(\\frac{\\sigma^m}{2}\\int\\rho|\\dot{u}|^2dx \\right)_t -\\frac{m}{2}\\sigma^{m-1}\\si'\\int\\rho|\\dot{u}|^2dx\\\\\n &=  \\int_{\\p \\O}  \\sigma^m \\dot{u} \\cdot n G_t dS - \\int  \\sigma^m [\\div \\dot{u} G_t + u \\cdot \\na \\dot u \\cdot \\na G]dx \\\\\n&\\quad- \\mu \\int\\sigma^m\\dot{u}^j\\xl[(\\na \\times \\curl u)_t^j + \\div (u (\\na \\times \\curl u)^j)\\xr] dx  \\triangleq\\sum_{i=1}^{3}N_i.\n\\ea \\ee\n\nNoticing that\n\\be\\la{pzw1} u\\cdot\\nabla u\\cdot n=-u\\cdot\\nabla n\\cdot u ~~\\quad \\mbox{on}~\\p \\O\\ee\ndue to $u \\cdot n|_{\\p \\O}=0$,\none can deduce from \\eqref{h1} and \\eqref{pzw1} that\n\\be \\la{bz8}\\ba\nN_1&=- \\int_{\\p \\O}  \\sigma^m (u \\cdot \\na n \\cdot u) G_t dS\\\\\n&=- \\left(\\int_{\\p \\O} \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t + m \\si^{m-1} \\si' \\int_{\\p \\O}( u \\cdot \\na n \\cdot u) G dS \\\\\n&\\quad+  \\int_{\\p \\O}  \\sigma^m (\\dot u \\cdot \\na n \\cdot u) G dS+ \\int_{\\p \\O}  \\sigma^m ( u \\cdot \\na      n \\cdot \\dot u) G dS\\\\\n&\\quad-  \\int_{\\p \\O}  \\sigma^m G( u \\cdot \\na) u \\cdot \\na n \\cdot u  dS -\\int_{\\p \\O}  \\sigma^m G u \\cdot \\na n \\cdot (u\\cdot \\na )u  dS\\\\\n&\\le - \\left(\\int_{\\p \\O}  \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t + C\\si^{m-1} \\si' \\| \\na u\\|_{L^2}^2\\|\\na G\\|_{L^2} \\\\\n&\\quad+\\de \\si^m \\|\\dot u\\|_{H^1}^2+ C(\\de) \\si^m \\|\\na u\\|_{L^2}^2 \\|\\na G\\|_{L^2}^2\\\\\n& \\quad- \\int_{\\p \\O}  \\sigma^m G( u \\cdot \\na) u \\cdot \\na n \\cdot u  dS -\\int_{\\p \\O}  \\sigma^m G u \\cdot \\na n \\cdot (u\\cdot \\na )u  dS,\n\\ea \\ee\nwhere one has used\n\\be \\ba\\notag\n\\left|\\int_{\\partial \\O}  (\\dot u\\cdot \\na n\\cdot u+ u\\cdot \\na n\\cdot\\dot u) G dS\\right| &\\le C \\|\\dot u\\|_{H^1} \\| u\\|_{H^1} \\|G\\|_{H^1} \\\\&\\le C \\|\\dot u\\|_{H^1} \\|\\na u\\|_{L^2} \\|\\na G\\|_{L^2},\n\\ea \\ee\nand\n\\be\\la{b2}\n\\left|\\int_{\\partial \\O}  ( u\\cdot \\na n\\cdot u) G dS  \\right| \\le C \\|\\na u\\|_{L^2} ^2\\|\\na G\\|_{L^2}.\n\\ee\nNow, we will adopt the idea in \\cite{C-L} to deal with the last two boundary terms in \\eqref{bz8}. In fact, denote $u^{\\perp}\\triangleq-u\\times n$, it follows from\n$u \\cdot n|_{\\p \\O}=0$ that\n\\begin{align}\\notag\nu=u^{\\perp} \\times n~~~~~\\text{on}\\, \\p \\O,\n\\end{align}\nwhich along with \\eqref{g1}, \\eqref{cd}, and integration by parts yields\n\\be \\la{bz3}\\ba &- \\int_{\\partial\\Omega} G (u\\cdot \\na) u\\cdot\\na n\\cdot u dS \\\\&= -\\int_{\\partial\\Omega}  G u^\\bot\\times n \\cdot\\na u^i \\nabla_i n\\cdot u  dS \\\\&= - \\int_{\\partial\\Omega} G n\\cdot ( \\na u^i \\times  u^\\bot)    \\nabla_i n\\cdot u dS\\\\\n&= - \\int\\div( G( \\na u^i \\times  u^\\bot)   \\nabla_i n\\cdot u) dx \\\\\n&= - \\int \\na (\\nabla_i n\\cdot u G) \\cdot ( \\na u^i \\times  u^\\bot)   dx  - \\int \\div( \\na u^i \\times  u^\\bot)    \\nabla_i n\\cdot u   G  dx \\\\\n&= - \\int \\na (\\nabla_i n\\cdot u G) \\cdot ( \\na u^i \\times  u^\\bot)   dx  + \\int  G \\na  u^i \\cdot \\na\\times  u^\\bot     \\nabla_i n\\cdot u     dx \\\\\n& \\le C \\int |\\na G||\\na u||u|^2dx+C \\int |G| (|\\na u|^2|u|+|\\na u||u|^2)dx\n\\\\& \\le C  \\|\\na G\\|_{L^6}\\|\\na u\\|_{L^2}\\|u\\|^2_{L^6}\n+C  \\| G\\|_{L^3}\\|\\na u\\|^2_{L^4}\\|u\\|_{L^6}+C\\| G\\|_{L^{6}} \\|\\na      u\\|_{L^2}\\|u\\|_{L^6}^2\n\\\\& \\le \\de \\|\\na G\\|_{L^6}^2+C(\\de) \\|\\na u\\|^6_{L^2}+C\\|\\na u\\|^4_{L^4}+ C  \\|  \\na G\\|_{L^2}^2 (\\|\\na u\\|^2_{L^2} +1 ).\\ea\\ee\nSimilarly, it holds that\n\\be \\la{bz4}\\ba &-  \\int_{\\partial\\Omega}G  u\\cdot\\na n\\cdot ({u}\\cdot\\na) u dS\\\\& \\le \\de \\|\\na G\\|_{L^6}^2+C(\\de) \\|\\na u\\|^6_{L^2}+C\\|\\na u\\|^4_{L^4}+ C \\|  \\na G\\|_{L^2}^2(\\|\\na u\\|^2_{L^2} +1 ).\\ea\\ee\n\n\n\n\n\n\nNext, it follows from \\eqref{hj1} that\n\\be \\ba \\la{pt11}\nG_t =& (2\\mu+\\lambda)\\div u_t - (P_t-\\bp_t )\\\\\n=& (2\\mu+\\lambda) \\div \\dot u - (2\\mu+\\lambda) \\div (u\\cdot \\na u) - R\\rho \\dot\\te + \\div(Pu) + R\\overline{\\rho \\dot\\te}\\\\\n=& (2\\mu+\\lambda) \\div \\dot u - (2\\mu+\\lambda) \\na u : (\\na u)^{\\rm tr} -  u \\cdot  \\na G + P \\div u - R\\rho \\dot\\te + R\\overline{\\rho \\dot\\te},\n\\ea \\ee\nwhere one has used\n\\be \\ba \\label{jia1}\nP_t=(R\\n \\te)_t=R\\n \\dot{\\te}-\\div (Pu),~~~\\bp_t=R\\overline{\\n\\dot{\\te}}.\n\\ea \\ee\nThen, integration by parts combined with \\eqref{pt11} gives\n\\be\\la{m5} \\ba\nN_2 =&  - \\int  \\sigma^m [\\div \\dot{u} G_t + u \\cdot \\na \\dot u \\cdot \\na G]dx \\\\\n=& - (2\\mu+\\lambda) \\int  \\sigma^m (\\div \\dot{u} )^2 dx +  (2\\mu+\\lambda)  \\int \\sigma^m \\div \\dot{u} \\na u : (\\na u)^{\\rm tr} dx \\\\\n&+\\int  \\sigma^m \\div \\dot{u}  u \\cdot  \\na G  dx - \\int \\sigma^m \\div \\dot{u} P \\div u dx  \\\\\n&+ R \\int \\sigma^m \\div \\dot{u} \\rho \\dot\\te dx -  R \\overline{\\rho \\dot\\te}  \\int \\sigma^m \\div \\dot{u}  dx - \\int  \\sigma^m u \\cdot \\na \\dot u \\cdot \\na Gdx \\\\\n\\le &   - (2\\mu+\\lambda) \\int  \\sigma^m (\\div \\dot{u} )^2 dx\\\\\n& + C \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\na u\\|_{L^4}^2 + C \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\na G\\|_{L^2}^{1\/2} \\|\\na G\\|_{L^6}^{1\/2} \\|u\\|_{L^6} \\\\\n&+ C(\\on) \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\te \\na u\\|_{L^2}+ C(\\on) \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}.\\ea \\ee\n\n\n\nNote that\n$$\\curl u_t=\\curl \\dot u-u\\cdot \\na \\curl u-\\na u^i\\times \\nabla_iu,$$\nwhich together with some straight calculations yields\n\\be\\la{ax3999}\\ba\nN_3 &=- \\mu \\int\\sigma^{m}|\\curl\\dot{u}|^{2}dx+\\mu \\int\\sigma^{m}\\curl\\dot{u}\\cdot(\\nabla u^i\\times\\nabla_i u) dx \\\\&\\quad+\\mu \\int\\sigma^{m} u\\cdot\\na  \\curl u \\cdot\\curl\\dot{u} dx  +\\mu \\int\\sigma^{m}    u \\cdot \\na \\dot{u}\\cdot (\\nabla\\times\\curl u) dx  \\\\\n&\\le - \\mu \\int\\sigma^{m}|\\curl\\dot{u}|^{2}dx+\\de  \\sigma^{m}(\\|\\na \\dot u\\|_{L^2}^2 +\n\\|\\na \\curl u\\|_{L^6}^2)\\\\&\\quad +C(\\de) \\sigma^{m} \\|\\na u\\|_{L^4}^4 +C(\\de)  \\sigma^{m}\\|\\na  u\\|_{L^2}^4\\|\\na \\curl u\\|_{L^2}^2.\\\\\n\\ea\\ee\n\n\n\nFinally, it is easy to deduce from  Lemmas  \\ref{le4} and \\ref{uup1} that\n\\be \\la{bz5} \\|G\\|_{H^1}+\\|\\curl u\\|_{H^1}\\le C(\\|\\n \\dot u\\|_{L^2}+\\|\\na  u\\|_{L^2}), \\ee\nand that\n\\be \\la{bz6}\\ba  \\|\\na G\\|_{L^6}+\\|\\na\\curl u\\|_{L^6} +\\| \\dot u\\|_{H^1}\n &\\le C(\\|\\n \\dot u\\|_{L^6}+\\|\\na  u\\|_{L^2})+\\| \\dot u\\|_{H^1}\n\\\\&\\le C(\\on)  (\\|\\na \\dot u\\|_{L^2}+  \\|\\na  u\\|_{L^2}+  \\|\\na  u\\|_{L^2}^2).\\ea \\ee\n\n\\iffalse\nCombining   \\eqref{m5} with \\eqref{bz8}, \\eqref{bz3}, and \\eqref{bz4}, one obtains after using \\eqref{bz5}, \\eqref{bz6}, and \\eqref{z1} that\n\\be\\la{ax399}\\ba\nN_1\n\\le &- \\left(\\int_{\\p \\O}  \\sigma^m u \\cdot \\na n \\cdot u G dS\\right)_t - (2\\mu+\\lambda) \\int  \\sigma^m (\\div \\dot{u} )^2 dx \\\\\n&+ C(\\on) \\de \\si^m \\|\\na \\dot{u}\\|_{L^2}^2+C(\\de,\\on,M) (\\si^{m-1} \\si'+\\si^m) \\| \\rho^{1\/2} \\dot u\\|_{L^2}^2\\\\\n&+C(\\de,\\on) \\si^m \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2 + C(\\de,\\on,M)  \\| \\na u\\|_{L^2}^2\\\\\n&+C(\\de) \\si^m \\| \\na u\\|_{L^4}^4 + C(\\de, \\on) \\si^m  \\|\\te \\na u\\|_{L^2}^2.\n\\ea\\ee\n\\fi\n\nHence, submitting  \\eqref{bz8},  \\eqref{m5}, and \\eqref{ax3999} into \\eqref{m4}, one obtains after  using \\eqref{bz3}, \\eqref{bz4}, \\eqref{z1},  \\eqref{bz5}, and \\eqref{bz6} that\n\\be\\la{ax40}\\ba\n&\\left(\\frac{\\sigma^{m}}{2}\\|\\rho^{1\/2}\\dot{u}\\|_{L^2}^2\\right)_t+(2\\mu+\\lambda)\\sigma^{m}\\|\\div\\dot{u}\\|_{L^2}^2+\\mu\\sigma^{m}\\|\\curl\\dot{u}\\|_{L^2}^2\\\\\n&\\le -\\left(\\int_{\\p \\O}  \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t+ C(\\on) \\de \\si^m \\|\\na \\dot{u}\\|_{L^2}^2 \\\\\n&\\quad+ C(\\de,\\on) \\si^m \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2 + C(\\de, \\on, M)(\\si^{m-1}\\si'+\\si^m)  \\|\\rho^{1\/2} \\dot u\\|_{L^2}^2\\\\\n&\\quad+C(\\de, \\on, M) \\|\\na u\\|^2_{L^2} +C(\\de)\\si^m \\|\\na u\\|^4_{L^4}+ C(\\de,\\on) \\si^m  \\|\\te \\na u\\|_{L^2}^2.\n\\ea\\ee\nApplying  \\eqref{tb11} to \\eqref{ax40} and choosing $\\de$ small enough infer \\eqref{ae0} directly.\n\n\n\n\n\n\n\n\n\nFinally, we will prove   (\\ref{nle7}).\n\nFor $m\\ge 0,$\nmultiplying $(\\ref{a1})_3 $ by $\\sigma^m \\dot\\te$ and integrating\nthe resulting equality over $\\Omega $ yield  that\n \\be\\la{e1} \\ba &\\frac{\\ka\n{\\sigma^m}}{2}\\left( \\|\\na\\te\\|_{L^2}^2\\right)_t+\\frac{R\\sigma^m}{\\ga-1} \\int\\rho|\\dot{\\te}|^2dx\n\\\\&=-\\ka\\sigma^m\\int\\na\\te\\cdot\\na(u\\cdot\\na\\te)dx\n+\\lambda\\sigma^m\\int  (\\div u)^2\\dot\\te dx\\\\&\\quad\n+2\\mu\\sigma^m\\int |\\mathfrak{D}(u)|^2\\dot\\te dx-R\\si^m\\int\\n\\te \\div\nu\\dot\\te dx \\triangleq \\sum_{i=1}^4I_i . \\ea\\ee\n\n First,   combining   (\\ref{g1}) and (\\ref{z1}) gives\n\\be\\la{e2} \\ba\nI_\n&\\le C\\sigma^{m}   \\|\\na u\\|_{L^2}\\|\\na\\te\\|^{1\/2}_{L^2}\n\\|\\na^2\\te\\|^{3\/2}_{L^2}  \\\\\n&\\le  \\de\\sigma^{m} \\|\\n^{1\/2}\\dot\\te\\|^2_{L^2}+\\si^m \\left(\\|\\na u\\|_{L^4}^4+\\|\\te\\na u\\|_{L^2}^2\\right) +C(\\de,\\on,M)\\sigma^{m}\n\\|\\na\\te\\|^2_{L^2}   ,\\ea\\ee\nwhere in the last inequality we have used the following estimate:\n\\be  \\la{lop4}\\ba\n\t\\|\\na^2\\te\\|_{L^2} &\\le C (\\on)\\left(\\|\\n^{1\/2}\\dot \\te\\|_{L^2}+ \\|\\na u\\|_{L^4}^2+\\|\\te\\na u\\|_{L^2}\\right),\n\t\\ea\\ee\nwhich is derived from the standard $L^2$-estimate to the following elliptic problem:\n\t\\be\\la{3.29}\\begin{cases}\n\t\t\\ka\\Delta \\te=\\frac{R}{\\ga-1}\\n\\dot\\te +R\\n\\te\\div\n\t\tu-\\lambda (\\div u)^2-2\\mu |\\mathfrak{D}(u)|^2 ,\\\\\n\t\t\\na \\theta\\cdot n|_{\\p\\O \\times (0,T)}=0 .\n\t\\end{cases}\\ee\n\n\n\n\n\nNext, it holds  that  for any $\\eta\\in (0,1],$\n\\be\\la{e3}\\ba I_2 =&\\lambda\\si^m\\int (\\div u)^2 \\te_t\ndx+\\lambda\\si^m\\int (\\div u)^2u\\cdot\\na\\te\ndx\\\\=&\\lambda\\si^m\\left(\\int (\\div u)^2 \\te\ndx\\right)_t-2\\lambda\\si^m \\int \\te \\div u \\div (\\dot u-u\\cdot\\na u)\ndx\\\\&+\\lambda\\si^m\\int (\\div u)^2u\\cdot\\na\\te\ndx \\\\=&\\lambda\\si^m\n\\left(\\int (\\div u)^2 \\te dx\\right)_t-2\\lambda\\si^m\\int \\te \\div u\n\\div \\dot udx\\\\&+2\\lambda\\si^m\\int \\te \\div u \\pa_i u^j\\pa_j  u^i dx\n+ \\lambda\\si^m\\int u \\cdot\\na\\left(\\te   (\\div u)^2 \\right)dx\n \\\\\n\\le &\\lambda\\left(\\si^m\\int (\\div u)^2 \\te dx\\right)_t-\\lambda\nm\\si^{m-1}\\si'\\int (\\div u)^2 \\te dx\\\\& +\\eta\\si^m\\|\\na \\dot\nu\\|_{L^2}^2+C(\\eta)\\si^m\\|\\te\\na u\\|_{L^2}^2+C\\si^m\\|\\na u\\|_{L^4}^4,\\ea\\ee\nand\n \\be \\la{e5}\\ba I_3&\\le 2\\mu\\left(\\si^m\\int\n|\\mathfrak{D}(u)|^2 \\te dx\\right)_t-2\\mu m\\si^{m-1}\\si'\\int\n|\\mathfrak{D}(u)|^2 \\te dx\n \\\\&\\quad+ \\eta\\si^m\\|\\na \\dot\nu\\|_{L^2}^2+C(\\eta)\\si^m\\|\\te\\na u\\|_{L^2}^2+C\\si^m\\|\\na u\\|_{L^4}^4 .    \\ea\\ee\n\nFinally, Cauchy's inequality gives\n \\be\\la{e39}\\ba\n |I_4|    \\le  \\delta \\si^m \\int \\n |\\dot\\te|^2dx+C( \\delta,\\on)\\si^m \\|\\te\\na u\\|_{L^2}^2.  \\ea\\ee\n\n\nSubstituting  (\\ref{e2}) and \\eqref{e3}--(\\ref{e39}) into (\\ref{e1}), we obtain \\eqref{nle7}\nafter using  \\eqref{h3} and choosing $\\de$ suitably\nsmall.\nThe proof of Lemma \\ref{a113.4} is completed.\n\\end{proof}\n\n\n\nWith the estimates \\eqref{an1}--\\eqref{nle7} (see Lemma  \\ref{a113.4}) at hand, we are now in a position to prove the following estimate on $A_3(T).$\n\n\n\n\n\n\\begin{lemma}\\la{le6} Under the conditions of Proposition \\ref{pr1},   there exists a positive constant $\\ve_2$\ndepending only on   $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O,$ and $M$\nsuch that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1})  on $\\Omega\\times (0,T] $ satisfying      (\\ref{z1}) with $K$ as\nin Lemma \\ref{le2}, the following estimate holds: \\be\\la{b2.34}  A_3(T) \\le C_0^{1\/6},\\ee provided $C_0\\le\n\\ve_2.$\n\\end{lemma}\n\n\n\\begin{proof}\n\tFirst, it follows from \\eqref{an2}, (\\ref{h18}), and (\\ref{p}) that\n\t\\be\\ba \\notag B_1(t)\n\t&\\ge  C   \\|\\na u\\|_{L^2}^2- C \\|P-\\bp\\|^2_{L^2}  \\ge C  \\|\\nabla u\\|_{L^2}^2-C(\\on) C_0^{1\/4},\\ea\\ee\n\twhich together with \\eqref{an1} and \\eqref{z1} implies that\n\t\\begin{align}\\label{dd}\n\t\t\\sup_{t\\in(0,T]}\\left(\\si \\|\\na u\\|_{L^2}^2\\right) + \\int_0^T\\int \\si\\n |\\dot u|^2 dx dt \\le C(\\hat{\\rho}, M)C_0^{1\/4}.\n\t\\end{align}\n\n\t\n\tFor $C_2$ as in  (\\ref{ae0}), adding  (\\ref{nle7}) multiplied by $C_2+1$ to (\\ref{ae0}) and choosing $ \\eta$ suitably small give\n\t\\be\\la{e8}\\ba\n\t&\\left(\\sigma^m\\varphi\\right)'(t) + \\sigma^m \\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx\\\\\n\t&\\le - 2\\left(\\int_{\\p \\O}  \\sigma^m( u \\cdot \\na n \\cdot u )G dS\\right)_t + C( \\on, M) (\\si^{m-1}\\si'+\\si^m ) \\|\\rho^{1\/2} \\dot u\\|_{L^2}^2\\\\\n\t&\\quad+ C(\\on,  M) (\\|\\na u\\|_{L^2}^2 + \\|\\na \\te\\|_{L^2}^2)+C\\si^m \\|\\na u\\|^4_{L^4} + C (\\on)\\si^m \\|\\te\\na u\\|_{L^2}^2,\\\\\n\n\t\\ea\\ee\n\twhere  $\\varphi(t)$ is defined by\n\t\\be\\la{wq3} \\varphi(t) \\triangleq   \\|\\n^{1\/2}\\dot u\\|_{L^2}^2+(C_2+1)  B_2(t).\\ee\n\tThen it follows from (\\ref{e6}) that\n\t\\be\\la{wq2}  \\varphi(t) \\ge  \\frac{1}{2}\\|\\n^{1\/2}\\dot u\\|_{L^2}^2 +\\frac{\\ka(\\ga-1)}{2R}\\|\\na\\te\\|_{L^2}^2-C(\\on,  M)\\|\\na u\\|_{L^2}^2, \\ee\n\twhere one has used that for any $\\de\\in(0,1]$,\n\t\\be \\la{2.48}\\ba   \\int\n\t\\te |\\na u|^2dx   & \\le\n\tC\\int|R\\te-\\bp||\\na u|^2dx+C\\bp \\int |\\na u|^2dx\\\\ &\\le C\n\t\\|R\\te-\\bp \\|_{L^6}\\|\\na u\\|_{L^2}^{3\/2}\n\t\\|\\na u\\|_{L^6}^{1\/2}+   C \\|\\na u\\|_{L^2}^2\n\t\\\\ &\\le C(\\on)\\|\\na\\te\\|_{L^2}\\|\\na u\\|_{L^2}^{3\/2}\n\t\\left(\\|\\n^{1\/2} \\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+ \\|\\na\\te\\|_{L^2}+1\\right)^{1\/2}\\\\\n\t&\\quad+   C \\|\\na  u\\|_{L^2}^2\\\\ &\\le \\de\n\t\\left(  \\|\\na\\te\\|^2_{L^2}  + \\|\\n^{1\/2}  \\dot\n\tu\\|_{L^2}^2 \\right) + C(\\de,\\on,M)  \\|\\na\n\tu\\|_{L^2}^2 \\ea\\ee\n\tdue to  \\eqref{key}, (\\ref{pq}), (\\ref{3.30}),   and (\\ref{z1}).\n\t\n\t\n\tNext, it follows from \\eqref{key},  (\\ref{pq}), \\eqref{z1}, and  (\\ref{3.30}) that\n\t\\be \\la{m20}\\ba\n\t\\|\\te\\na u\\|_{L^2}^2\n\t&\\le C\\|R\\te-\\bp \\|_{L^6}^2 \\|\\na u\\|_{L^2} \\|\\na u\\|_{L^6} + C\\bp^2 \\|\\na u\\|_{L^2}^2 \\\\\n\t&\\le C(\\on,M) \\left( \\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right)\\left( \\|\\n^{1\/2}\\dot u\\|^2_{L^2} +\n\t\\|\\na\\te\\|^2_{L^2} +1\\right).\\ea\\ee\n\tAnd, by virtue of (\\ref{h17}), (\\ref{pq}), and (\\ref{z1}), one gets\n\t\\be\\la{ae9}\\ba\n\t&\\|\\na u\\|_{L^4}^4\\\\\n\t&\\le C\\|\\n\\dot u\\|_{L^2}^3(\\|\\na u\\|_{L^2}+\\|P-\\bp\\|_{L^2})+C(\\|\\na u\\|_{L^2}^4+\\|P-\\bp\\|_{L^4}^4)\\\\\n\t&\\le C(\\on)\\|\\n^{1\/2}\\dot u\\|_{L^2}^3\\left(\\|\\na u\\|_{L^2}+1\\right) +C(\\on)\\|\\na\\te\\|_{L^2}^3+C\\|\\n-1\\|_{L^4}^4 +C\\|\\na u\\|_{L^2}^4\\\\\n\t&\\le C(\\on,M)\\left( \\|\\n^{1\/2}\\dot u\\|_{L^2}^3 + \\|\\na \\te\\|_{L^2}^3\\right)  +C(\\on)\\|\\n-1\\|_{L^2}^2+C(\\on,M)\\|\\na u\\|_{L^2}^2,\n\t\\ea\\ee\n\twhich together with (\\ref{z1})  yields\n\t\\be \\la{m22}\\ba\n\t\\si \\|\\na u\\|_{L^4}^4   &\\le C(\\on,M) \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2  + \\|\\na \\te\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2\\right)\n\t+C(\\on)\\si\\|\\n-1\\|_{L^2}^2.\n\t\\ea\\ee\n\n\nThus, taking $m=2$ in \\eqref{e8}, one obtains after  using  \\eqref{z1}, \\eqref{m20}, and \\eqref{m22}  that\n\\be\\la{e25}\\ba\n& \\left(\\sigma^2\\varphi\\right)'(t) + \\sigma^2 \\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx\\\\\n&\\le  - 2\\left(\\int_{\\p \\O}  \\sigma^2 (u \\cdot \\na n \\cdot u) G dS\\right)_t+ C(\\on,M)\\si\\|\\n^{1\/2}\\dot u\\|_{L^2}^2\\\\\n&\\quad +  C(\\on,  M)\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right)+C(\\on)\\si\\|\\n-1\\|_{L^2}^2.\n\t\\ea\\ee\n\t\n\n\t\n\tNow, we  deduce from \\eqref{h19},  \\eqref{z1}, and \\eqref{b2} that\n\t\\be \\ba\\label{jia4}\n\t\\sup_{ 0\\le t\\le T} \\left|\\int_{\\p \\O}  \\sigma^2 u \\cdot \\na n \\cdot u G dS\\right|\n\n\t\\le & C(\\on) \\sup_{ 0\\le t\\le T} (\\si \\|\\na u\\|_{L^2}^2) \\sup_{ 0\\le t\\le T} (\\si \\|\\rho^{1\/2} \\dot u\\|_{L^2})\\\\\n\t\\le &  C(\\on) C_0^{1\/4}.\n\t\\ea \\ee\n\tFurthermore, note that (\\ref{hj1}) is equivalent to\n \\be\\nota\n\\bp(\\n-1)=-G+(2\\mu+\\lambda)\\div u-\\n(R\\te-\\bp),\n\\ee\n this together with \\eqref{key}, \\eqref{z1}, \\eqref{1h19}, \\eqref{pq}, and \\eqref{dd} implies\n\\be\\ba\\la{67}\n&\\int_0^T\\si\\|\\n-1\\|_{L^2}^2 dt\\\\\n&\\le C\\int_0^T\\si(\\|G\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2) dt+C(\\on)\\int_0^T\\|R\\te-\\bp\\|_{L^2}^2 dt\\\\\n&\\le C(\\on)\\int_0^T\\left(\\si\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) dt\\\\\n&\\le C(\\on,M)C_0^{1\/4}.\n\\ea\\ee\n\t\n\tThus, integrating \\eqref{e25} over $(0,T)$, one obtains after using \\eqref{dd}, \\eqref{wq2}, (\\ref{jia4}), (\\ref{67}), and (\\ref{z1}) that\n\t\\be\\ba\\notag\n\tA_3(T)\\le C(\\on,M)C_0^{1\/4}\\le C_0^{1\/6},\n\t\\ea\\ee\n provided\n\t\\be \\ba\\notag C_0\\le\\ve_2\\triangleq \\min\\{1,(C(\\on,M))^{-12}\\}.\\ea\\ee\n\n \n\n\n  The proof of Lemma \\ref{le6} is completed.\n \\end{proof}\n\n\n\n\n\n\nNext, in order to control $A_2(T)$, we first re-establish the basic energy estimate for short time $[0, \\si(T)]$, and then show that the spatial $L^2$-norm of $R\\te-\\overline P$ could be  bounded by the combination of the initial energy and the spatial $L^2$-norm of $\\na \\te$, which is indeed the key ingredient to   estimate   $A_2(T)$.\n\n\\begin{lemma}\\la{a13} Under the conditions of Proposition \\ref{pr1},\n\tthere exist  positive constants $C$ and $\\varepsilon_{3,1}$\n\tdepending only on\n\t$\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O, $ and $M$ such\n\tthat  if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1})  on $\\Omega\\times (0,T] $ satisfying      (\\ref{z1}) with $K$ as\nin Lemma \\ref{le2},\n\tthe following estimates  hold:\n\t\\be \\la{a2.121} \\ba\n\t&\\sup_{0\\le t\\le \\si(T)}\\int\\left( \\n |u|^2+(\\n-1)^2 + \\n(\\te-\\log \\te-1) \\right)dx\\le C C_0,\\ea\\ee\n\nand\n\\be  \\la{a2.17}  \\ba\n\\|(R\\te-\\bp)(\\cdot,t)\\|_{L^2} \\le C \\left(C_0^{1\/2} +C_0^{1\/3}\\|\\na\\te(\\cdot,t)\\|_{L^2}\\right),\n\t\\ea\\ee\nfor all $t\\in(0,\\si(T)]$, provided $C_0\\le\\varepsilon_{3,1}.$\n\t\\end{lemma}\n\\begin{proof}\n\tThe proof is divided into the following two steps.\n\t\n{\\it Step 1: The proof of (\\ref{a2.121}).}\n\t\n\tFirst, multiplying (\\ref{a11}) by $u$, one deduces from integration by parts, \\eqref{a1}$_1$, and \\eqref{cz1} that\n\t\\be \\la{a2.12} \\ba\n\t&\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+R(1+\\n\\log \\n-\\n)\n\t\\right)dx+ \\int(\\mu|\\curl u|^2+(2\\mu+\\lambda)(\\div u)^2)dx \\\\\n\t&=  R \\int \\rho (\\te -1) \\div u dx\\\\\n\t&\\le \\de \\|\\na u\\|_{L^2}^2 + C(\\de, \\on) \\int \\n(\\te-1)^2dx\\\\\n&\\le\\de \\|\\na u\\|_{L^2}^2 + C(\\de, \\on)(\\|\\te(\\cdot,t)\\|_{L^{\\infty}} +1)  \\int \\rho (\\te -\\log \\te -1)dx.\n\t\\ea\\ee\n\nUsing \\eqref{h18} and choosing $\\de$ small enough in \\eqref{a2.12}, it holds tha\n\t\\be \\la{a2.222} \\ba\n\t&\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+R(1+\\n\\log \\n-\\n)\n\t\\right)dx + C_3 \\int|\\na u|^2dx \\\\\n&\\le C(\\on) (\\|\\te(\\cdot,t)\\|_{L^{\\infty}} +1)  \\int \\rho  (\\te -\\log \\te -1)dx.\n\t\\ea\\ee\nThen, adding \\eqref{a2.222} multiplied by $(2\\mu+1) {C_3}^{-1}$  to \\eqref{la2.7}, one has\n\t\\be \\la{a2.22} \\ba\n\t\n\t\\xl((2\\mu+1) {C_3}^{-1}+1\\xr)\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+R(1+\\n\\log \\n-\\n)\\right)dx\n\t\\\\&+ \\frac{R}{\\ga-1} \\frac{d}{dt} \\int \\n(\\te-\\log \\te-1)dx+\\int|\\na u|^2dx\\\\\n\n\t &\\le C(\\on) (\\|\\te(\\cdot,t)\\|_{L^{\\infty}} +1)  \\int \\rho (\\te -\\log \\te -1)dx.\n\t\\ea\\ee\n\t\nNext, we claim that\n\t\\be \\ba\\la{k}\n\t\\int_0^{\\si(T)}\\|\\te\\|_{L^\\infty}dt \\le C(\\on, M).\n\t\\ea \\ee\n\tCombining this with \\eqref{a2.22}, \\eqref{a2.9}, and Gr\\\"onwall inequality  implies \\eqref{a2.121} directly.\n\t\n\tFinally, it remains to prove \\eqref{k}.\n\tTaking $m=1$ in \\eqref{e8} and integrating the resulting inequality, one deduces from \\eqref{wq2}, \\eqref{m20}, \\eqref{m22}, (\\ref{z1}), \\eqref{67}, \\eqref{h19}, and \\eqref{b2} that\n\\bnn  \\ba\n& \\sigma \\varphi +  \\int_0^t \\sigma\\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dxd\\tau\\\\\n&\\le  2\\sigma\\left|\\int_{\\p \\O}   (u \\cdot \\na n \\cdot u )G dS\\right|(t)\n      + C(\\on,M)\\int_0^t (\\|\\n^{1\/2}  \\dot u\\|_{L^2}^2 +\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)d\\tau \\\\\n&\\quad+  C(\\on)\\int_0^t \\si \\|\\n-1\\|_{L^2}^2d\\tau+C(\\on)\\int_0^t \\left(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) \\si\\varphi d\\tau\\\\\n& \\le  C(\\on) (\\sigma \\|\\na u\\|_{L^2}^2 \\|\\rho^{1\/2} \\dot u\\|_{L^2})(t)+ C(\\on,M)\\\\\n&\\quad+C(\\on,M)\\int_0^t \\left(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) \\si\\varphi d\\tau\\\\\n& \\le C(\\on,M)+C(\\on,M)\\int_0^t \\left(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) \\si\\varphi d\\tau.\n\\ea\\enn\nThen Gr\\\"onwall inequality together with (\\ref{z1}) and \\eqref{wq2} yields\n\\be\\ba\\label{ae26}\n\t\\sup_{0\\le t\\le T}\\si \\left(\\int\\rho|\\dot{u}|^2dx+\\|\\na\\te\\|_{L^2}^2\\right)+\\int_0^T\\si \\int\\left(|\\na\\dot u|^2+\\n|\\dot\\te|^2\\right)dxdt\n\\le  C(\\on,M).\\ea\\ee\n\n\n\n\n\n\n\n\n\n\n\t\n\t Next, it follows from (\\ref{lop4}), \\eqref{m22}, \\eqref{m20},  \\eqref{ae26}, (\\ref{z1}), and (\\ref{67})  that\n\n\n\n\n\n\n\t\\be\\ba  \\la{k1}\n\t\\int_0^{T }\\si \\|\\na^2\\te\\|_{L^2}^2dt\n\t&\\le   C(\\on,M)\\int_0^{T } \\left(\\si \\|\\n^{1\/2}\\dot\\te\\|_{L^2}^2+\n\t\\|\\n^{1\/2}\\dot u \\|_{L^2}^2  \\right) dt\\\\\n\t&\\quad+ C(\\on,M)\\int_0^{T } \\left( \\| \\na u\\|_{L^2}^2+\\| \\na\\te\\|_{L^2}^2+\\si\\|\\n-1\\|_{L^2}^2\\right) dt \\\\\n\t&\\le C(\\on,M).\n\t\\ea\\ee\n\tFurthermore, one deduces from \\eqref{g2}, \\eqref{g1}, and \\eqref{pq} that\n\t\\begin{equation}\n\t    \\label{6yue}\\ba\n\t    \\|R\\te-\\bp\\|_{L^\\infty} \\le& C \\|R\\te-\\bp\\|_{L^6}^{1\/2} \\|\\na\\te\\|_{L^6}^{1\/2} + \\|R\\te-\\bp\\|_{L^2}\\\\\n\t    \\le& C(\\hat{\\rho}) \\|\\na \\te\\|_{L^2}^{1\/2} \\|\\na^2 \\te\\|_{L^2}^{1\/2} + C(\\hat{\\n}) \\|\\na \\te\\|_{L^2},\n\t    \\ea\n\t\\end{equation}\n\twhich together with  (\\ref{z1}) and \\eqref{k1}  gives that\n\t\\be\\la{3.88}\\ba\n\t& \\int_0^{\\si(T)}\\|R\\te-\\bp\\|_{L^\\infty}dt \\\\\n\n\t&\\le C(\\hat{\\rho}) \\int_0^{\\si(T)}\\|\\na\\te\\|_{L^2}^{1\/2} \\left(\\si\\|\\na^2\\te\\|^2_{L^2}\\right)^{1\/4}\\si^{-1\/4}dt + C(\\hat{\\rho}) \\left(\\int_0^{\\si(T)} \\|\\na\\te\\|_{L^2}^2 dt\\right)^{1\/2}\\\\\n\t&\\le C(\\hat{\\rho}) \\left(\\int_0^{\\si(T)} \\|\\na \\te\\|_{L^2}^2dt \\int_0^{\\si(T)}\\si\\|\\na^2\\te\\|_{L^2}^2dt\\right)^{1\/4}\n+ C(\\on)C_0^{1\/8}\\\\\n\t&\\le C(\\on,M)C_0^{1\/16}.\n\t\\ea\\ee\n Combining this  with \\eqref{key} yields \\eqref{k} directly.\n\n\n\n{\\it Step 2: The proof of (\\ref{a2.17}).}\n\nDirect calculations together with \\eqref{cz1} lead to\n  \\be\\notag\\ba\n \\te-\\log\\te-1\n\n\\ge \\frac{1}{8} (\\te-1)1_{(\\te(\\cdot,t)>2)\n}+\\frac{1}{12}(\\te-1)^21_{(\\te(\\cdot,t)<3)},  \\ea\\ee with\n$(\\te(\\cdot,t)> 2)\\triangleq \\left.\\left\\{x\\in\n\\Omega\\right|\\te(x,t)> 2\\right\\}$ and  $(\\te(\\cdot,t)< 3)\\triangleq\n\\left.\\left\\{x\\in \\Omega\\right|\\te(x,t)<3\\right\\}.$\n   Combining this with \\eqref{a2.121} gives\n\t\\be \\la{a2.11}\\ba\n\t\\sup_{0\\le t\\le \\si(T)}\\int \\left(\\n(\\te-1)1_{(\\te(\\cdot,t)>2)}+\\n(\\te-1)^21_{(\\te(\\cdot,t)<3)}\\right)dx \\le C(\\hat\\n,M) C_0.\n\t\\ea\\ee\n\t\n\tNext, it follows from \\eqref{a2.11}, \\eqref{a2.121}, and the Sobolev inequality that for $t \\in (0,\\sigma(T)]$,\n\t\\begin{equation}\\la{la2.19}\n\t\\begin{aligned}\n\t &\\|\\te -1\\|_{L^2(\\te(\\cdot,t)<3)}^2 \\\\\n\t &\\le\\int  \\n (\\te-1)^2 1_{(\\te(\\cdot,t)<3)}dx  +  \\left|\\int  (\\n-1) (\\te -1)^2 dx \\right|\\\\\n\t&\\le C(\\on,M)C_0  +  C\\|\\n-1\\|_{L^2} \\| \\te-1 \\|_{L^2}^{1\/2} \\|\\te -1\\|_{L^6}^{3\/2}\\\\\n\t&\\le C(\\on,M)C_0   +  C(\\on,M)C_0^{1\/2}  \\| \\te-1 \\|_{L^2}^{1\/2} \\left(\\| \\te-1 \\|_{L^2}+ \\|\\na \\te\\|_{L^2}\\right)^{3\/2}\\\\\n\t&\\le C(\\on,M)\\left(C_0+ C(\\delta) C_0^{2\/3} \\|\\na \\te\\|_{L^2}^2 + (\\delta+C_0^{1\/2}) \\| \\te-1 \\|_{L^2}^2 \\right),\n\t\\end{aligned}\n\t\\end{equation}\n\tand\n\t\\be\\la{a2.18}\\ba\n\t &\\|\\te-1\\|_{L^2(\\te(\\cdot,t)> 2)}^2\\\\\n\t&\\le \\|\\te-1\\|^{4\/5}_{L^1(\\te(\\cdot,t)> 2)} \\| \\te-1\\|_{L^6}^{6\/5}\\\\\n\n\t&\\le C(\\on,M) \\left(C_0 +C_0^{1\/2} \\|\\te-1\\|_{L^2}\\right)^{4\/5} (\\| \\te-1\\|_{L^2}+ \\|\\na \\te\\|_{L^2})^{6\/5}\\\\\n\t&\\le  C(\\on,M)\\left(C_0+C(\\delta)C_0^{2\/3}\\|\\na \\te\\|_{L^2}^2 + (\\delta+C_0^{2\/5} ) \\|\\te -1\\|_{L^2}^2\\right),\n\t\\ea\\ee\n\twhere in the second inequality one has used\n\t\\be \\notag\\ba\n\t\\|\\te-1\\|_{L^1(\\te(\\cdot,t)>2)} \\le &\\int  \\n (\\te-1) 1_{(\\te(\\cdot,t)>2)}dx  +   \\int  \\left|(\\n-1) (\\te -1) \\right|dx \\\\\n\t\\le & C(\\on,M)(C_0 +C_0^{1\/2} \\|\\te-1\\|_{L^2}).\n\t\\ea\\ee\n\tHence, adding \\eqref{la2.19} with \\eqref{a2.18} together and choosing $\\delta$ small enough in the resulting inequality, one has   for any $t\\in (0,\\sigma(T)],$\n\t\\be\\ba\\notag\n\t\\|\\te-1\\|_{L^2}^2 \\le  C(\\on,M)\\left(C_0+C_0^{2\/3}\\|\\na \\te\\|_{L^2}^2 + C_0^{2\/5}  \\|\\te -1\\|_{L^2}^2\\right),\n\t\\ea\\ee\n\twhich implies that\n\t\\be\\ba\\la{nnn1}\n\t\\|\\te-1\\|_{L^2}^2 \\le  C(\\on,M)\\left(C_0+C_0^{2\/3}\\|\\na \\te\\|_{L^2}^2 \\right),\n\t\\ea\\ee\n\tprovided\n\t\\begin{equation}\\label{31}\n\tC_0\\le \\ve_{3,1} \\triangleq\\min\\left\\{1,(2C(\\on,M))^{-5\/2}\\right\\}.\n\t\\end{equation}\n\t\n\tFinally, note that\n\t\\be \\ba\\notag\n\t\\| R\\te-\\bp \\|_{L^2} &\\le R \\|\\te  -1\\|_{L^2}\t+C|1-\\overline{\\n\\te}|\\\\\n\t&\\le R \\|\\te  -1\\|_{L^2}+C\\left|\\int\\n(1-\\te)dx\\right|\\\\\n\t&\\le C(\\on) \\|\\te  -1\\|_{L^2},\n\t\\ea\\ee\n\tthis together with \\eqref{nnn1} yields (\\ref{a2.17}).\n\t\n\tThe proof of Lemma \\ref{a13}  is completed.\n\\end{proof}\n\n\nNext, with the help of \\eqref{a2.17}, the estimate on $A_2(T)$ will be handled smoothly.\n\n\n\n\\begin{lemma}\\la{le3}\nUnder the conditions of Proposition \\ref{pr1},  there exists a positive constant $\\ve_3$ depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1})  on $\\Omega\\times (0,T] $ satisfying  (\\ref{z1})   with $K$ as in Lemma \\ref{le2}, the following estimate holds:\n\\be\\la{a2.34} A_2(T) \\le C_0^{1\/4},\\ee\nprovided $C_0\\le \\ve_3.$\n\\end{lemma}\n\n\n\n\\begin{proof}\nTo begin with, multiplying (\\ref{a11}) by $u$ and integrating by parts give that\n\\be \\la{a2.225}\\ba\n&\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+\\bp(1+\\n\\log \\n-\\n)\\right)dx\\\\\n&\\quad+ \\int\\left(\\mu|\\curl u|^2+(2\\mu+\\lambda)(\\div u)^2\\right)dx \\\\\n&= \\bp_t \\int  (1+\\n\\log \\n-\\n) dx +  \\int \\rho (R \\te -\\bp) \\div u dx.\n\\ea\\ee\nNext, multiplying $(\\ref{a1})_3$ by $\\bp^{-1}(R\\te-\\bp)$, one obtains after  integrating the resulting\nequality over $\\Omega $ by parts that\n\\be\\la{a2.231} \\ba\n& \\frac{1}{2(\\ga-1)} \\frac{d}{dt}\\int\n{\\bp}^{-1}\\n {(R\\te-\\bp)^2}dx+ {\\ka}R \\bp^{-1}  \\|\\na\\te\\|_{L^2}^2\\\\\n&= -  \\frac{1}{\\ga-1} {\\bp}^{-1} \\bp_t \\int\n\\n {(R\\te-\\bp)}  dx - \\frac{1}{2(\\ga-1)} \\bp^{-2} \\bp_t \\int \\n {(R\\te-\\bp)^2}  dx\\\\\n& \\quad- {\\bp}^{-1} \\int \\n {(R\\te-\\bp)^2} \\div u dx -  \\int \\rho (R\\te -\\bp) \\div u dx \\\\\n&\\quad+ {\\bp}^{-1} \\int  {(R\\te-\\bp)} (\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2) dx.\n\\ea\\ee\nAdding \\eqref{a2.225} and \\eqref{a2.231} together yields that\n\\be\\la{a2.23} \\ba\n&  \\frac{d}{dt}\\int\n\\left(\\frac{1}{2}\\n |u|^2+\\bp (1+\\n\\log \\n-\\n)+\\frac{1}{2(\\ga-1)} \\n \\bp^{-1}{(R\\te-\\bp)^2} \\right)dx\\\\\n&\\quad+ \\mu \\|\\curl u\\|_{L^2}^2 +(2\\mu+\\lambda)\\|\\div u\\|_{L^2}^2+  \\ka R \\bp^{-1}  \\|\\na\\te\\|_{L^2}^2\\\\\n&= -  \\frac{1}{\\ga-1} {\\bp}^{-1} \\bp_t \\int\n\\n {(R\\te-\\bp)}  dx - \\frac{1}{2(\\ga-1)} {\\bp}^{-2} \\bp_t \\int \\n {(R\\te-\\bp)^2} dx\\\\\n& \\quad+\\bp_t \\int  (1+\\n\\log \\n-\\n) dx - \\bp^{-1}\\int  \\n {(R\\te-\\bp)^2} \\div u dx \\\\\n&\\quad+ \\bp^{-1} \\int  {(R\\te-\\bp)} (\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2) dx\\triangleq \\sum_{i=1}^{5} J_i.\n\\ea\\ee\n\nThe terms $J_i \\,(i=1,\\cdots,5)$ can be estimated as follows.\n\nIt follows from \\eqref{key}, \\eqref{511}, \\eqref{z1}, and \\eqref{pq} that\n\\be \\ba\\la{aa1}\nJ_1+J_2 \\le & C|\\overline P_t|\\left(\\| \\n (R\\te-\\bp)\\|_{L^2}+\\| \\n^{1\/2} (R\\te-\\bp)\\|_{L^2}^2\\right)\\\\\n\\le &C(\\on)\\left(C_0^{1\/8} \\|\\na u \\|_{L^2}+\\| \\na u\\|_{L^2}^2 \\right)\\|\\n^{1\/2}(R\\te-\\bp)\\|_{L^2}\\\\\n\\le & C(\\on) C_0^{1\/8} \\left( \\|\\na u\\|_{L^2}^2+\\| \\na \\te \\|_{L^2}^2\\right),\n\\ea \\ee\nand\n\\be \\ba\\la{aa4}\nJ_4 \\le & C   \\|\\n^{1\/2} {(R\\te-\\bp)}\\|_{L^2}^{1\/2} \\|\\n^{1\/2} {(R\\te-\\bp)}\\|_{L^6}^{3\/2} \\|\\na u\\|_{L^2} \\\\\n\\le & C(\\on) A_2^{1\/4}(T) \\|\\na \\te\\|_{L^2}^{3\/2} \\|\\na u\\|_{L^2}\\\\\n\\le & C(\\on,M) C_0^{1\/16} (\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^{2}).\n\\ea \\ee\nFurthermore, by virtue of \\eqref{a2.9}, \\eqref{511}, and \\eqref{a2.112}, we have\n\\be \\ba\\la{aa3}\nJ_3 \\le & |\\overline P_t|\\left|\\int (1+\\n \\log\\n-\\n )dx\\right|\\\\\n\\le & C(\\on)\\left(  C_0^{1\/8}\\|\\na u \\|_{L^2}+\\| \\na u\\|_{L^2}^2 \\right)\\| \\n -1\\|_{L^2}^2\\\\\n\\le & C(\\on) C_0^{1\/4}  \\|\\na u\\|_{L^2}^2 + C(\\on) C_0^{1\/4} \\| \\n -1\\|_{L^2}^2.\n\\ea \\ee\n\n\nNow, we will estimate the term $J_5$ for the short time $t\\in [0, \\si(T))$ and the large time $t\\in [\\si(T),T]$, respectively.\n\nFor $t\\in[0, \\si(T))$, it follows from \\eqref{key}, \\eqref{3.30}, \\eqref{pq}, \\eqref{a2.17}, and \\eqref{z1} that\n\n\\be \\ba\\la{aa5}\nJ_5 \\le& C \\int  |R\\te-\\bp| |\\na u|^2dx\\\\\n\\le & C \\| {R\\te-\\bp}\\|_{L^2}^{1\/2} \\|R\\te-\\bp\\|_{L^6}^{1\/2}  \\|\\na u\\|_{L^2} \\|\\na u\\|_{L^6}\\\\\n\\le & C(\\on)  \\|{R\\te-\\bp}\\|_{L^2}^{1\/2} \\|\\na \\te\\|_{L^2}^{1\/2}  \\|\\na u\\|_{L^2}\\\\\n & \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+\\|\\na\\te\\|_{L^2}+C_0^{1\/24}\\right)\\\\\n\\le & C (\\on) \\| {R\\te-\\bp}\\|_{L^2}^{1\/2} \\|\\na \\te\\|_{L^2}^{1\/2}  \\|\\na u\\|_{L^2}\\|\\n^{1\/2}\\dot u\\|_{L^2}\\\\\n&+C(\\on,M) C_0^{1\/24} (\\|\\na \\te\\|_{L^2}^{2}+  \\|\\na u\\|_{L^2}^2)\\\\\n\\le & C(\\on,M) C_0^{7\/24} \\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 + C(\\on,M)C_0^{1\/24} (\\|\\na u\\|_{L^2}^2 +  \\|\\na \\te\\|_{L^2}^2),\n\\ea \\ee\nwhere we have used following calculations:\n\\be \\ba\\notag\n&\\| {R\\te-\\bp}\\|_{L^2}^{1\/2} \\|\\na \\te\\|_{L^2}^{1\/2}  \\|\\na u\\|_{L^2} \\|\\n^{1\/2}\\dot u\\|_{L^2} \\\\\n&\\le C(\\on,M)(C_0^{1\/4}\\|\\na\\te\\|_{L^2}^{1\/2}+C_0^{1\/6}\\|\\na\\te\\|_{L^2})\\|\\na u\\|_{L^2}\\|\\n^{1\/2}\\dot u\\|_{L^2}\\\\\n&\\le C(\\on,M) C_0^{7\/24} \\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 + C(\\on,M)C_0^{1\/12}  \\|\\na u\\|_{L^2}^2 + C(\\on,M)C_0^{1\/24} \\|\\na \\te\\|_{L^2}^2\n\\ea \\ee\nowing to \\eqref{a2.17}.\n\nFor $t\\in[\\si(T), T]$, it holds that\n\\be \\ba\\la{aa6}\nJ_5\n\\le &C\\|R\\te-\\bp\\|_{L^3}\\|\\na u\\|_{L^2}\\|\\na u\\|_{L^6}\n\\le  C(\\on)C_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2),\n\\ea \\ee\nwhere one has used \\eqref{pq} and the following fact:\n\\be\\notag\\ba \\sup_{0\\le t\\le T}\\xl(\\si\\|\\na\nu\\|_{L^6}\\xr)\n&\\le C(\\on)C_0^{1\/24} \\ea\\ee\ndue to \\eqref{z1} and \\eqref{3.30}.\n\n\nFinally, substituting \\eqref{aa1}--\\eqref{aa6}  into \\eqref{a2.23}, one obtains after using \\eqref{z1}, \\eqref{key}, and \\eqref{jia10} that\n\\be \\ba  \\label{jia9}\n&  \\sup_{ 0\\le t\\le T} \\int \\left(\\frac{1}{2}\\n |u|^2+\\bp (1+\\n\\log \\n-\\n)+\\frac{1}{2(\\ga-1)} \\n \\bp^{-1}{(R\\te-\\bp)^2}\n\\right)dx\\\\\n&+ \\int_{0}^{T} (\\mu \\|\\curl u\\|_{L^2}^2 +(2\\mu+\\lambda)\\|\\div u\\|_{L^2}^2+ {R\\ka}{\\bp}^{-1}  \\|\\na\\te\\|_{L^2}^2)dt\\\\\n&\\le  C(\\on,M) C_0^{1\/24} \\int_{ 0}^{T}  ( \\|\\na u\\|_{L^2}^2+\\| \\na \\te \\|_{L^2}^2 ) dt+ C(\\on) C_0^{1\/4}\\int_{ 0}^{T} \\| \\n -1\\|_{L^2}^2 dt\n\\\\&\\quad + C(\\on,M) C_0^{7\/24} \\int_{0}^{\\si(T)} \\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 dt+C(\\on,\\bt)C_0\\\\\n&\\le  C(\\on,\\bt, M) C_0^{7\/24},\n\\ea \\ee\nwhere one has used\n\\be\\la{xx}\n\\int_{ 0}^{T} \\| \\n -1\\|_{L^2}^2 dt \\le \\sup_{0\\le t\\le\\si(T)} \\|\\n -1\\|_{L^2}^2 + \\int_{\\si(T)}^{T} \\| \\n -1\\|_{L^2}^2 dt \\le C(\\on,M) C_0^{1\/4}\n\\ee\ndue to \\eqref{a2.121} and \\eqref{67}. Thus, one deduces from \\eqref{jia9}, \\eqref{h18}, and \\eqref{key} that\n\\be \\la{kyu1} A_2(T)\\le C(\\on,\\hat{\\theta}, M) C_0^{7\/24}  \\ee which implies \\eqref{a2.34}\nprovided \\be \\ba \\notag C_0\\le \\ve_3\\triangleq\\min\n\\left\\{\\ve_{3,1},  (C(\\on,\\hat{\\theta},M))^{-24}\\right\\},\\ea\\ee\nwith $\\ve_{3,1}$ as in \\eqref{31}. The proof\nof Lemma \\ref{le3} is completed.\n\\end{proof}\n\n\\begin{remark}\\la{r2}\n\tIt's worth noticing that the energy-like estimate $A_2(T)$ is a little subtle, since $A_2(T)$ is not a conserved quantity for the full Navier-Stokes system owing to the nonlinear coupling of $\\te$ and $u$. Thus, further consideration is needed to handle this issue.\n\tMore precisely,\n\t\\begin{itemize}\n\t\t\\item on the one hand, while deriving the kinetic energy (see (\\ref{a2.225})), we need to deal with the following term\n\t\t$$\\int \\rho (R\\te -\\bp) \\div u dx.$$\n\t\tUnfortunately, this term is troublesome for large time $t\\in [\\sigma(T), T]$. In fact, this term could be bounded by  $\\|\\na \\te \\|_{L^2}\\|\\na u \\|_{L^2}$, which will only be  of the same order as $C_0^{1\/4}$ with the help of all a priori estimates (\\ref{z1}). Therefore, we can not handle this term directly. Here, based on careful analysis on system (\\ref{a1}), we find that this term can be cancelled by a suitable combination of kinetic energy and thermal energy, see (\\ref{a2.225})--(\\ref{a2.23});\n\t\t\\item on the other hand, while deriving the thermal energy (see (\\ref{a2.231})), we need to handle the following term\n\t\t$$\\int (R\\te - \\bp) (\\div u)^2 dx.$$\n\t\tNote that for short time $t\\in [0, \\sigma(T))$, the ``weaker\" basic energy estimate (\\ref{a2.112}) is not enough, hence it's necessary to re-establish  the basic energy estimate (\\ref{a2.121}), which is obtained by the a priori $L^1(0,\\sigma(T);L^\\infty)$-norm of $\\te$ (see (\\ref{k})). Consequently, we can obtain (\\ref{a2.17}) as a consequence of (\\ref{a2.121}) and then handle this term for short time $t\\in [0, \\sigma(T))$ (see (\\ref{aa5})).\t\n\t\n\t\n\t\\end{itemize}\n\t\n\tMoreover, it should be mentioned that the uniform positive lower and upper bounds  of $\\bp$  also play a critical role in estimating $A_2(T)$.\n\\end{remark}\n\n\n\n\n\nWe now proceed to derive a uniform (in time) upper bound for the density, which turns out to be the key to obtaining all the higher order estimates and thus extending the classical solution globally.\n\n\\begin{lemma}\\la{le7}\nUnder the conditions of Proposition \\ref{pr1},  there exists a positive constant $\\ve_4$ depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1})  on $\\Omega\\times (0,T] $ satisfying  (\\ref{z1})   with $K$ as in Lemma \\ref{le2}, the following estimate holds:\n\\be \\la{a3.7}\n\\sup_{0\\le t\\le T}\\|\\n(\\cdot,t)\\|_{L^\\infty}  \\le\n\\frac{3\\on }{2},\n\\ee\nprovided $C_0\\le \\ve_4$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, it follows from \\eqref{k1}, \\eqref{6yue}, and \\eqref{z1} that\n\\be\\la{3.89}\\ba\n\\int_{\\si(T)}^T\\|R\\te-\\bp\\|^2_{L^\\infty}dt\n\\le& C(\\on) \\left(\\int_{\\si(T)}^T\\|\\na\\te\\|^2_{L^2}dt\\right)^{1\/2}\\left(\\int_{\\si(T)}^T \\|\\na^2\\te\\|^2_{L^2}dt\\right)^{1\/2} \\\\\n&+ C(\\on) \\int_{\\si(T)}^T\\|\\na\\te\\|^2_{L^2}dt \\\\\n\\le& C(\\on,M) C_0^{1\/8}.\n\\ea\\ee\n\n\nNext, it follows from  (\\ref{h19}), \\eqref{tb90}, (\\ref{ae26}), and (\\ref{z1}) that\n\\be\\la{3.90}\\ba &\\int_0^{\\si(T)}\\|G\\|_{L^\\infty}dt\\\\\n&\\le C\\int_0^{\\si(T)}\\|\\na G\\|_{L^2}^{1\/2} \\|\\na G\\|_{L^6}^{1\/2}dt\\\\\n&\\le C(\\on)\\int_0^{\\si(T)}\\|\\n \\dot u\\|_{L^2}^{1\/2}(\\|\\na\\dot u\\|_{L^2}+ \\|\\na u\\|_{L^2}^2)^{1\/2}dt\\\\\n&\\le C(\\on)\\int_0^{\\si(T)}\\left(\\si\\|\\n \\dot u\\|_{L^2}\\right)^{1\/4} \\left(\\si\\|\\n \\dot u\\|^{2}_{L^2}\\right)^{1\/8} \\left(\\si\\|\\na \\dot u\\|^2_{L^2}\\right)^{1\/4}\\si^{-5\/8}dt\\\\\n& \\quad+ C(\\on) \\int_0^{\\si(T)}\\left(\\si \\|\\n \\dot u\\|_{L^2}\\right)^{1\/2} \\|\\na u\\|_{L^2} \\si^{-1\/2} dt\\\\\n&\\le C(\\on,M)C_0^{1\/48}\\left(\\int_0^{\\si(T)} \\si\\|\\na \\dot u\\|^2_{L^2} dt\\right)^{1\/4}\\left(\\int_0^{\\si(T)} \\si^{-5\/6}dt\\right)^{3\/4} \\\\\n&\\quad+ C(\\on,M) C_0^{1\/24} \\int_0^{\\si(T)}\n\\si^{-1\/2} dt\\\\\n&\\le C(\\on,M)C_0^{1\/48},\n\\ea\\ee\nand\n \\be\\la{3.91}\\ba  \\int_{\\si(T)}^T\\|G\\|^2_{L^\\infty}dt\n  &\\le C\\int_{\\si(T)}^T\\|\\na G\\|_{L^2} \\|\\na G\\|_{L^6} dt\n   \\\\ &\\le C(\\on,M)\\int_{\\si(T)}^T\\left(\\|\\n^{1\/2} \\dot u\\|^2_{L^2}+\n \\|\\na\\dot u\\|_{L^2}^2 +  \\|\\na u\\|_{L^2}^2\\right)dt\\\\ &\\le C(\\on,M)C_0^{1\/6}.\n    \\ea\\ee\n\n\n\nDenoting $ D_t\\n=\\n_t+u \\cdot\\nabla \\n $ and using (\\ref{hj1}), one can rewrite   $(\\ref{a1})_1$  as follows\n\\bnn\\ba\n(2\\mu+\\lambda) D_t \\n&=-\\bp \\n(\\n-1)- \\n^2(R\\te-\\bp)-\\n G\\\\\n&\\le -\\bp (\\n-1)+C(\\on)\\|R\\te-\\bp\\|_{L^\\infty}+C(\\on)\\| G\\|_{L^\\infty},\n\\ea\\enn\nwhich gives\n\\be\\la{3.92}\\ba\nD_t (\\n-1)+\\frac{\\bp }{2\\mu+\\lambda} (\\n-1)\\le C(\\on)\\|R\\te-\\bp\\|_{L^\\infty}+C(\\on)\\| G\\|_{L^\\infty}.\n\\ea\\ee\n\n\n\nFinally, applying Lemma \\ref{le1} with\n$$y=\\n-1, \\quad\\al=\\frac{\\bp}{2\\mu+\\lambda} ,\\quad g=C(\\on)\\|R\\te-\\bp\\|_{L^\\infty}+C(\\on)\\| G\\|_{L^\\infty},\\quad T_1=\\si(T),$$\n we thus deduce from (\\ref{3.92}), (\\ref{3.88}), \\eqref{3.89}--(\\ref{3.91}), (\\ref{2.34}), and \\eqref{key}  that\n \\bnn\\ba\\n\n & \\le \\on+1 +C\\left(\\|g\\|_{L^1(0,\\si(T))}+\\|g\\|_{L^2(\\si(T),T)}\\right) \\le \\on+1 +C(\\on,M)C_0^{1\/48} ,\n \\ea\\enn\n which gives \\eqref{a3.7}\n provided\n  \\be \\notag C_0\\le \\ve_4\\triangleq\\min\\left\\{1,\\left(\\frac{\\hat \\n-2 }{2C(\\on,M) }\\right)^{48}\\right\\}.\\ee\n The proof of Lemma \\ref{le7} is completed.\n\\end{proof}\n\n\n\n\n\n\n\nNext, we summarize some uniform estimates on $(\\n,u,\\te)$ which will be useful for higher-order ones in the next section.\n\\begin{lemma}\\la{le8}\nUnder the conditions of Proposition \\ref{pr1},  there exists a  positive constant    $C $     depending only   on  $\\mu,\\,\\lambda,\\, \\ka,\\, R$, $\\ga,\\, \\on,\\,\\bt,\\, \\O$, and $M$  such that if $(\\rho,u,\\te)$  is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $  satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}, the following estimate holds:\n\t\\be \\la{ae3.7}\\sup_{0< t\\le T}\\si^2\\int \\n|\\dot\\te|^2dx + \\int_0^T\\si^2 \\|\\na\\dot\\te\\|_{L^2}^2dt\\le C.\\ee\nMoreover, it holds that\n\\be\\la{vu15}\\ba\n&\\sup_{0< t\\le T}\\left(  \\si\\|\\na u \\|^2_{L^6}+\\si^2\\|\\te\\|^2_{H^2}\\right)\\\\\n&+\\int_0^T(\\si \\|\\na u \\|_{L^4}^4+\\si\\|\\na\\te \\|_{H^1}^2+\\si\\|u_t\\|_{L^2}^2+\\si^2\\|\\te_t\\|^2_{H^1}+\\|\\n -1\\|_{L^2}^2)dt\\le C.\n\\ea\\ee\n\\end{lemma}\n\n\\begin{proof}\nFirst, applying the operator $\\pa_t+\\div(u\\cdot) $ to (\\ref{a1})$_3 $ and using  (\\ref{a1})$_1$, one   gets\n\\be\\la{3.96}\\ba\n&\\frac{R}{\\ga-1} \\n \\left(\\pa_t\\dot \\te+u\\cdot\\na\\dot \\te\\right)\\\\\n&=\\ka \\Delta  \\te_t +\\ka \\div (\\Delta \\te u)+\\left( \\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)\\div u +R\\n \\te  \\pa_ku^l\\pa_lu^k\\\\\n&\\quad -R\\n \\dot\\te \\div u-R\\n \\te\\div \\dot u +2\\lambda \\left( \\div\\dot u-\\pa_ku^l\\pa_lu^k\\right)\\div u\\\\\n&\\quad + \\mu (\\pa_iu^j+\\pa_ju^i)\\left( \\pa_i\\dot u^j+\\pa_j\\dot u^i-\\pa_iu^k\\pa_ku^j-\\pa_ju^k\\pa_ku^i\\right).\n\\ea\\ee\nDirect calculations show that\n\\be \\ba\\la{bea}\n \\int  (\\Delta  \\te_t + \\div (\\Delta \\te u)) \\dot \\te dx &=  - \\int  (\\na  \\te_t \\cdot \\na \\dot\\te + \\Delta \\te u \\cdot \\na \\dot \\te) dx\\\\\n&= - \\int  |\\na \\dot\\te|^2 dx  + \\int ( \\na(u\\cdot \\na \\te) \\cdot \\na \\dot \\te - \\Delta \\te u \\cdot \\na \\dot \\te) dx.\n\\ea \\ee\nMultiplying (\\ref{3.96}) by $\\dot \\te$ and integrating the resulting equality over $\\O$, it holds that\n\\be\\la{3.99}\\ba\n& \\frac{R}{2(\\ga-1)}\\left(\\int \\n |\\dot\\te|^2dx\\right)_t + \\ka   \\|\\na\\dot\\te\\|_{L^2}^2 \\\\\n&\\le  C  \\int|\\na \\dot \\te|\\left( |\\na^2\\te||u|+ |\\na \\te| |\\na u|\\right)dx+C\\int  \\n|R\\te-\\bp| |\\na\\dot u| |\\dot \\te|dx\\\\\n&\\quad +C(\\on)  \\int|\\na u|^2|\\dot\\te|\\left(|\\na u|+|R\\te-\\bp| \\right)dx+C   \\int |\\na\\dot u|\\n|\\dot \\te| dx \\\\\n&\\quad +C (\\on)  \\int\\left( |\\na u|^2|\\dot \\te|+\\n  |\\dot\n\\te|^2|\\na u|+|\\na u| |\\na\\dot u| |\\dot \\te|\\right)dx \\\\\n&\\le C\\|\\na u\\|^{1\/2}_{L^2}\\|\\na u\\|^{1\/2}_{L^6}\\|\\na^2\\te\\|_{L^2}\\|\\na \\dot \\te\\|_{L^2}+C(\\on)\\|\\na\\te\\|_{L^2} \\|\\na\\dot u\\|_{L^2} \\|\\dot\\te\\|_{L^6}\\\\\n&\\quad+C(\\on)  \\|\\na u\\|_{L^2}\\|\\na u\\|_{L^6}\\left(\\|\\na u\\|_{L^6}+\\|\\na \\te\\|_{L^2}\\right)\n\\|\\dot\\te\\|_{L^6} +C  \\|\\na\\dot u\\|_{L^2} \\|\\n\\dot\\te\\|_{L^2} \\\\\n&\\quad+C(\\on)  \\|\\na u\\|^{1\/2}_{L^6}\\|\\na u\\|^{1\/2}_{L^2} \\|\\dot\\te\\|_{L^6}\\left(\\|\\na u\\|_{L^2}\n+\\|\\n\\dot\\te\\|_{L^2}+\\|\\na\\dot u\\|_{L^2}\\right)  \\\\\n&\\le\\frac{\\ka}{2}\\|\\na\\dot\\te\\|_{L^2}^2+C(\\on)\\|\\na u\\|_{L^2}^2\\left(\\|\\na u\\|_{L^6}^4+\\|\\na\\te \\|_{L^2}^4\\right)+C(\\on,M) \\|\\na u\\|_{L^6}\\|\\na u\\|_{L^2}^2\\\\\n&\\quad+C(\\on,M) \\left(1+\\|\\na u\\|_{L^6}+\\|\\na\\te\\|_{L^2}^2\\right) \\left(\\|\\na^2\\te\\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+\\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2\\right),\n\\ea\\ee\nwhere we have used \\eqref{bea}, \\eqref{g1}, \\eqref{g2}, \\eqref{z1}, \\eqref{pq}, and the following Poincar\\'e-type inequality (\\cite[Lemma 3.2]{feireisl1}):\n\\be \\la{kk}\n\\|f\\|_{L^p}\\le C(\\on)(\\|\\n^{1\/2}f\\|_{L^2}+\\|\\na f \\|_{L^2}),~~~p\\in[2,6],\n\\ee\nfor any $f\\in\\{h\\in H^1 \\left|\\n^{1\/2}h\\in L^2\\}\\right.$.\n\n\nMultiplying (\\ref{3.99}) by $\\si^2$ and integrating the resulting inequality over $(0,T),$\nwe obtain after integrating by parts that\n\\bnn\\ba\n& \\sup_{0\\le t\\le T}\\si^2\\int \\n|\\dot\\te|^2dx + \\int_0^T\\si^2 \\|\\na\\dot\\te\\|_{L^2}^2dt  \\\\\n&\\le C(\\on) \\sup_{0\\le t\\le T} \\left(\\si^2(\\|\\na u\\|_{L^6}^4+\\|\\na\\te\\|_{L^2}^4)\\right)\\int_0^T\\|\\na u\\|_{L^2}^2dt \\\\\n&\\quad + C(\\on,M) \\sup_{0\\le t\\le T} \\left(\\si\\left(1+\\|\\na u\\|_{L^6}+\\|\\na\\te\\|_{L^2}^2\\right)\\right)\\\\\n&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\cdot\\int_0^T\\si\\left(\\|\\na^2\\te\\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+\\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2\\right)dt\\\\\n&\\quad +C(\\on,M) \\sup_{0\\le t\\le T} \\left(\\si\\|\\na u\\|_{L^6} \\right) \\int_0^T\\|\\na u\\|_{L^2}^2dt+C\\int_0^T\\si\\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2dt\\\\\n&\\le  C(\\on,M), \\ea\\enn\nwhere we have used   (\\ref{z1}), (\\ref{ae26}),  (\\ref{k1}), and the following fact:\n\\be\\ba\\la{ong}\\sup_{0\\le t\\le T}(\\si\\|\\na u\\|_{L^6}^2)\\le C(\\on,M)\\ea\\ee\ndue to \\eqref{3.30}, \\eqref{ae26}, and \\eqref{z1}.\n\nNext, it follows from (\\ref{z1}),  \\eqref{ae26}, (\\ref{lop4}), (\\ref{m20}), (\\ref{m22}), (\\ref{ae3.7}), (\\ref{xx}), \\eqref{nnn1}, and  (\\ref{k1}) that\n\\be \\la{vu02}\\ba\n\\sup_{0\\le t\\le T}\\left(\\si^2\\|\\te  \\|^2_{H^2}\\right)+\\int_0^T \\left(\\si\\|\\na u  \\|_{L^4}^4+\\si\\|\\na\\te \\|_{H^1}^2+\\|\\n -1\\|_{L^2}^2\\right)d\n \\le C(\\on,M), \\ea\\ee\nwhich along with (\\ref{z1}), \\eqref{ae26}, (\\ref{k1}), \\eqref{kk}, \\eqref{ong}, and \\eqref{ae3.7} gives\n \\be\\la{vu12}\\ba    \\int_0^T  \\si \\|u _t\\|_{L^2}^2dt\n &\\le C\\int_0^T  \\si(\\| \\dot u \\|_{L^2}^2+\\|u\\cdot\\na  u \\|_{L^2}^2)dt\\\\\n &\\le C(\\hat\\n)\\int_0^T  \\si(\\|\\n^{1\/2} \\dot u \\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+\\|u \\|_{L^\\infty}^2\\|\\na  u \\|_{L^2}^2)dt\\\\\n &\\le C(\\hat \\n,M) ,\\ea\\ee\n\\be\\la{vu11}\\ba\n\\int_0^T  \\si^2 \\|  \\te _t\\|_{L^2}^2dt\n&\\le C\\int_0^T  \\si^2(\\| \\dot \\te \\|_{L^2}^2+\\|u \\cdot\\na  \\te \\|_{L^2}^2)dt\\\\\n&\\le C(\\hat\\n)\\int_0^T  \\si^2(\\|\\n^{1\/2} \\dot \\te\\|_{L^2}^2+\\|\\na\\dot\\te\\|_{L^2}^2+\\|u\\|_{L^6}^2\\|\\na\\te\\|_{L^3}^2)dt\\\\\n&\\le C(\\hat \\n,M) ,\\ea\\ee\n and\n\\be\\la{vu01}\\ba\n\\int_0^T  \\si^2 \\|  \\na\\te _t\\|_{L^2}^2dt\n&\\le C\\int_0^T  \\si^2\\|\\na \\dot \\te \\|_{L^2}^2  dt+ C\\int_0^T  \\si^2\\|\\na(u \\cdot\\na  \\te )\\|_{L^2}^2dt\\\\\n&\\le C(\\on,M) +C\\int_0^T\\si^2\\left(\\|\\na u \\|_{L^3}^2+\\|u \\|_{L^\\infty}^2\\right)\\|\\na^2 \\te \\|_{L^2}^2dt  \\\\ &\\le C(\\on,M).\\ea\\ee\nHence, (\\ref{vu15}) is derived from (\\ref{ong})--\\eqref{vu01} immediately.\nThe proof of Lemma \\ref{le8} is finished.\n\\end{proof}\n\n\nFinally, we end this section by establishing the exponential decay-in-time for the classical solutions.\n\\begin{lemma}\\la{pr2}\nUnder the conditions of Proposition \\ref{pr1}, there exist  positive constants $\\ve_0$, $C^\\ast$, $\\al$, and $\\te_\\infty$ depending only on   $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\, \\bt,\\,\\O,$ and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}, (\\ref{h22}) holds for any $t\\geq 1$, provided $C_0\\le\\ve_0.$\n\\end{lemma}\n\\begin{proof}\nFirst, it follows from \\eqref{a2.23}, \\eqref{z1}, \\eqref{key}, \\eqref{a2.9}, \\eqref{511}, \\eqref{pq}, \\eqref{6yue}, and \\eqref{vu15} that for any $t\\geq 1$,\n\\be\\la{ao}\\ba\n&\\frac{1}{2}W'(t)+ \\mu \\|\\curl u\\|_{L^2}^2 +(2\\mu+\\lambda)\\|\\div u\\|_{L^2}^2+  \\ka R \\bp^{-1}  \\|\\na\\te\\|_{L^2}^2\\\\\n&\\le C|\\bp_t|(\\|R\\te-\\bp\\|_{L^2}+\\|R\\te-\\bp\\|_{L^2}^2)+|\\bp_t| \\|\\n-1\\|_{L^2}^2 \\\\\n&\\quad+C\\|R\\te-\\bp\\|_{L^\\infty}(\\|R\\te-\\bp\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2)\\\\\n&\\le C(C_0^{1\/8}\\|\\na u\\|_{L^2}+\\|\\na u\\|_{L^2}^2)(\\|\\na\\te\\|_{L^2}+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2)\\\\\n&\\quad+C(\\|\\na\\te\\|_{L^2}^{1\/2}\\|\\na^2\\te\\|_{L^2}^{1\/2}+\\|\\na\\te\\|_{L^2})(\\|\\na\\te\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2)\\\\\n&\\le CC_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2),\n\\ea\\ee\nwhere\n\\be\\la{t3}\\ba\nW(t)&\\triangleq\\int\\left(\\n |u|^2+2\\bp (1+\\n\\log \\n-\\n)+\\frac{1}{\\ga-1} \\n \\bp^{-1}{(R\\te-\\bp)^2} \\right)dx\\\\\n&\\le \\hat C_3(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2)\n\\ea\\ee\nowing to \\eqref{z1}, \\eqref{key}, \\eqref{a2.9}, and \\eqref{pq}.\nCombining \\eqref{ao} with \\eqref{h18} and \\eqref{key} yields that\n\\be\\la{t5}\\ba &W'(t)+\\hat C_1(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)\n\\\\&\\le \\hat C_2C_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2).\\ea\\ee\n\n\nNext, rewriting $\\eqref{a1}_2$   as\n\\bnn\\la{t2}\\ba &(\\n u)_t+\\div(\\n u\\otimes u)\\\\&=\\mu\\Delta u+(\\mu+\\lambda)\\na(\\div u)-\\na(\\n(R\\te-\\bp))-\\bp\\na(\\n-1),\\ea\\enn\n  multiplying this by $\\mathcal{B}[\\n-1]$ and using Lemma \\ref{th00}, \\eqref{z1}, and \\eqref{pq},  one gets that for any $t\\geq 1$,\n\\be\\notag \\ba\n&\\bp\\int(\\n-1)^2 dx \\\\\n&= \\left(\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right)_t -\\int\\rho u\\cdot\\mathcal{B}[\\n_t]dx\n  -\\int\\rho u\\cdot\\nabla\\mathcal{B}[\\n-1]\\cdot u dx \\\\\n& \\quad  +\\mu\\int\\p_j u\\cdot\\p_j\\mathcal{B}[\\n-1] dx +(\\mu+\\lambda)\\int(\\rho-1)\\div udx -\\int\\n(R\\te-\\bp)(\\n-1)dx\\\\\n& \\le\\left(\\int\\n u\\cdot\\mathcal{B}[\\n-1]dx\\right)_t+C\\|\\n u\\|_{L^2}^2+C\\|u\\|_{L^{4}}^{2}\\|\\n-1\\|_{L^2}\\\\\n& \\quad  +C\\|\\rho-1\\|_{L^2}\\|\\na u\\|_{L^2}+C\\|\\rho-1\\|_{L^2}\\|R\\te-\\bp\\|_{L^2} \\\\\n& \\leq \\left(\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right)_t+\\frac{\\pi_1}{2}\\|\\n-1\\|_{L^2}^2+C(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2),\n\\ea\\ee\nwhich as well as \\eqref{key} leads to\n\\be\\la{t6}\\|\\rho-1\\|_{L^2}^2\\le\\frac{2}{\\pi_1}\\left(\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right)_t+\\hat C_4(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2).\\ee\nBy virtue of \\eqref{key}, \\eqref{a2.9}, and Lemma \\ref{th00}, it holds\n\\be \\la{t4} \\ba\n\\left|\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right|\n&\\leq  C \\left(\\|\\n u\\|^2_{L^2}+\\|\\n-1\\|_{L^2}^2\\right)\\\\\n&\\le \\hat C_5\\left(\\|\\n^{1\/2} u\\|^2_{L^2}+2\\bp (1+\\n\\log \\n-\\n)\\right).\n\\ea\\ee\nAdding \\eqref{t5} to \\eqref{t6} multiplied by $\\hat C_6$ with $\\hat C_6=\\min\\{\\frac{\\pi_1}{4\\hat C_5},\\frac{\\hat C_1}{4\\hat C_4}\\}$ yields\n\\be\\ba\\la{t1}\n&W_1'(t)+\\frac{3\\hat C_1}{4}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)+\\hat C_6\\|\\n-1\\|_{L^2}^2\\\\\n&\\le \\hat C_2C_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2),\\ea\\ee\nwhere\n$$W_1(t)\\triangleq W(t)-\\frac{2\\hat C_6}{\\pi_1}\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx,$$\nsatisfies\n\\be\\la{t8}\\frac{1}{2}W(t)\\le W_1(t)\\le 2W(t)\\ee\ndue to \\eqref{t4}.\nThus we infer from \\eqref{t1} that\n\\be\\la{t9} W_1'(t)+\\frac{\\hat C_1}{2}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)+\\frac{\\hat C_6}{2}\\|\\n-1\\|_{L^2}^2\\le 0,\\ee\nprovided\n\\be\\la{t7}C_0\\le\\ve_0\\triangleq \\min\\left\\{\\ve_1,\\cdots,\\ve_4,\\left(\\frac{\\hat C_6}{2\\hat C_2}\\right)^{24},\\left(\\frac{\\hat C_1}{4\\hat C_2}\\right)^{24}\\right\\}.\\ee\nThen by  \\eqref{t3}, one derives that for $\\alpha=\\frac{1}{3}\\min\\{\\frac{\\hat C_1}{2\\hat C_3},\\frac{\\hat C_6}{2\\hat C_3}\\}$,\n\\be\\notag W_1'(t)+3\\alpha W_1(t)\\le 0,\\ee\nwhich along with \\eqref{t8},  \\eqref{a2.9}, \\eqref{key}, \\eqref{t3}, and \\eqref{z1} shows that for any $t\\geq 1$,\n\\be\\la{t10}\\|\\n^{1\/2} u\\|_{L^2}^2+\\|\\n-1\\|_{L^2}^2+\\|\\n^{1\/2}(R\\te-\\bp)\\|_{L^2}^2\\le C W_1(t)\\le Ce^{-3\\al t}.\\ee\nMoreover, we deduce from \\eqref{t9} and \\eqref{t10} that for any $1\\le t\\le T<\\infty$,\n\\be\\la{t11}\\int_1^Te^{\\al t}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)dt\\le C.\\ee\n\nNext, multiplying \\eqref{1hh17} by $e^{\\al t}$ and using \\eqref{z1} imply that for $B_1$ defined in \\eqref{an2},\n\\be\\ba \\la{t12}\n&(e^{\\al t}B_1(t))'+ \\frac{1}{2}e^{\\al t}\\int\\rho |\\dot u|^2dx\\le Ce^{\\al t}\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+\\|P-\\overline P\\|_{L^2}^2\\right).\n\\ea\\ee\nNote that by \\eqref{z1}, \\eqref{key}, and \\eqref{t10},\n\\be\\la{t16}\\|P-\\overline P\\|_{L^2}\\le \\|\\n(R\\te-\\bp)\\|_{L^2}+\\bp\\|\\n-1\\|_{L^2}\\le Ce^{-\\al t},\\ee which together with  \\eqref{t11}, \\eqref{t12}, and \\eqref{h18} gives for any $1\\le t\\le T<\\infty$,\n\\be\\la{t13}\\ba\n\\sup_{1\\leq t\\leq T}\\left(e^{\\al t}\\|\\na u\\|_{L^2}^2\\right)+\\int_1^T e^{\\al t}\\|\\n^{1\/2}\\dot u\\|^2_{L^2}dt\\le C.\n\\ea \\ee\n\nFurthermore, choosing $m=0$ in \\eqref{e8},  it follows from \\eqref{m20}, \\eqref{ae9}, and \\eqref{z1} that for any $t\\geq1$,\n\\be\\notag\\ba\n&\\varphi'(t)+ 2\\left(\\int_{\\p \\O}( u \\cdot \\na n \\cdot u )G dS\\right)_t +\\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx\\\\\n&\\le C(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2 + \\|\\na \\te\\|_{L^2}^2+\\|\\n-1\\|_{L^2}^2),\n\\ea\\ee\nwhere  $\\varphi(t)$ is defined in \\eqref{wq3}. Multiplying this by $e^{\\al t}$ along with \\eqref{e6}, \\eqref{wq3}, \\eqref{wq2}, \\eqref{b2}, \\eqref{t10}, \\eqref{t11}, \\eqref{t13}, \\eqref{h19}, and \\eqref{z1} yields that for any $1\\le t\\le T<\\infty$,\n\\be\\la{t15}\\ba\n\\sup_{1\\leq t\\leq T}\\left(e^{\\al t}(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2)\\right)+\\int_1^T e^{\\al t}(\\|\\na\\dot u\\|_{L^2}^2+\\|\\n^{1\/2}\\dot \\te\\|^2_{L^2})dt\\le C.\n\\ea \\ee\nAdopting the analogous method and applying \\eqref{3.99}, \\eqref{vu15}, \\eqref{t10}, \\eqref{t11}, \\eqref{t13}, \\eqref{t15}, \\eqref{lop4}, \\eqref{m20},  \\eqref{z1}, and \\eqref{ae9}, we obtain that\n\\be\\la{t17}\\ba\n\\sup_{1\\leq t\\leq T}\\left(e^{\\al t}(\\|\\n^{1\/2}\\dot\\te\\|_{L^2}^2+\\|\\na^2\\te\\|_{L^2}^2)\\right)+\\int_1^T e^{\\al t}\\|\\na\\dot \\te\\|_{L^2}^2dt\\le C.\n\\ea \\ee\n\nFinally, it remains to determine the limit of $\\te$ as $t$ tends to infinity. Combining \\eqref{pt}, \\eqref{t16}, and \\eqref{t13} shows that for any $t\\geq 1$,\n\\be\\ba\\notag\n|\\bp_t|\\le C(\\|\\na u\\|_{L^2}^2+\\|P-\\bp\\|_{L^2}^2)\\le C e^{-\\al t},\n\\ea\\ee\nwhich implies there exists a constant $P_\\infty$ such that $\\lim_{t\\rightarrow \\infty}\\overline P=P_\\infty$ and\n\\be\\la{t18}|\\overline P-P_\\infty|\\le Ce^{-\\al t}.\\ee\nDenoting $\\te_\\infty\\triangleq P_\\infty\/R$,   we have\n\\be\\la{t19}\\|\\te-\\te_\\infty\\|_{L^2}^2\\le C\\|R\\te-\\bp\\|_{L^2}^2+C|\\bp-R\\te_\\infty|^2\\le Ce^{-\\al t},\\ee\nwhere we have used \\eqref{pq}, \\eqref{t15}, and \\eqref{t18}.\nTherefore, the combination of  \\eqref{t10}, \\eqref{t13}--\\eqref{t17}, \\eqref{t19}, \\eqref{h17}, and \\eqref{p} concludes \\eqref{t7} and finishs the proof of Lemma \\ref{pr2}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{\\la{se4} A priori estimates (II): higher-order estimates}\n\nIn this section, we will derive the higher-order estimates of smooth solution $(\\rho, u, \\te)$ to problem (\\ref{a1})--(\\ref{h1})  on $ \\Omega\\times (0,T]$ with initial data $(\\n_0 ,u_0,\\te_0)$ satisfying (\\ref{co3}) and (\\ref{3.1}).\n\n\nWe shall assume that (\\ref{z1}) and (\\ref{z01}) both hold as well. To proceed,\nwe define $\\tilde g $ as\n\\be \\la{co12}\\tilde g\\triangleq\\n_0^{-1\/2}\\left(\n-\\mu \\Delta u_0-(\\mu+\\lambda)\\na\\div u_0+R\\na (\\n_0\\te_0)\\right).\\ee\nThen it follows from (\\ref{co3}) and (\\ref{3.1}) that\n\\be\\la{wq01}\\tilde g\\in L^2.\\ee\nFrom now on, the generic constant $C $ will depend only  on \\bnn\nT, \\,\\, \\| \\tilde g\\|_{L^2},    \\,\\|\\n_0\\|_{W^{2,q}}  ,   \\,  \\,\\|\\na u_0\\|_{H^1},  \\ \\,\n\\| \\na\\te_0\\|_{L^2} , \\enn\nbesides  $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O,$ and $M.$\n\n\nWe begin with the following estimates on the spatial gradient of\nthe smooth solution $(\\rho,u,\\te).$\n\n\n\\begin{lemma}\\la{le11}\n\tThe following estimates hold:\n\t\\be\\label{lee2}\\ba\n\t&\\sup_{0\\le t\\le T} \\left(\\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 + \\sigma\\|\\rho^{1\/2}\\dot \\te\\|_{L^2}^2 +\\|\\te\\|_{H^1}^2 + \\sigma \\|\\na^2 \\theta\\|_{L^2}^2\n \\right)  \\\\\n\t  &\\quad+\\ia\\left(  \\|\\nabla\\dot u\\|_{L^2}^2  + \\|\\rho^{1\/2}\\dot \\te\\|_{L^2}^2+ \\|\\nabla^2 \\theta\\|_{L^2}^2 +\\sigma \\|\\nabla\\dot \\te\\|_{L^2}^2 \\right) dt\\le C,\n\t\\ea\\ee\n\tand\n\t\\be\\la{qq1}\n\t\\sup_{0\\le t\\le T}\\left(\\|u\\|_{H^2} +\\|\\n\\|_{H^2}\\right)\n\t+ \\int_0^{T}\\left( \\|\\nabla u\\|_{L^{\\infty}}^{3\/2} + \\si \\| \\na^3 \\te\\|_{L^2}^2+\\|u\\|_{H^3}^2 \\right)dt\\le C.\n\t\\ee\n\\end{lemma}\n\n\n\\begin{proof}\nThe proof is divided into the following two steps.\n\t\n{\\it Step 1: The proof of (\\ref{lee2}).}\nFirst, for $\\varphi(t)$ as in \\eqref{wq3}, taking $m=0$ in \\eqref{e8}, one gets\n\\be\\la{ae8}\\ba\n& \\varphi'(t) +   \\int\\left( \\frac{C_1}{2}|\\nabla\\dot{u}|^2   +\\n|\\dot \\te|^2\\right)dx \\\\\n&\\le -2 \\left(\\int_{\\p\\O} G\\left( u\\cdot \\na n \\cdot u \\right)dS\\right)_t+C\\left(\\|\\n^{1\/2} \\dot u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2+\\|\\na    \\te\\|_{L^2}^2\\right)\\\\\n&\\quad +C\\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^3+\\|\\na\\te\\|_{L^2}^3+\\|\\na u\\|_{L^2}^2+\\|\\n-1\\|_{L^2}^2\\right) \\\\\n& \\quad +C \\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right)\\left( \\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2+1 \\right) \\\\\n&\\le -2 \\left(\\int_{\\p\\O} G\\left( u\\cdot \\na n \\cdot u \\right)dS\\right)_t+C   \\left( \\|\\n^{1\/2}\\dot u\\|_{L^2}^2 + \\|\\na\\te\\|_{L^2}^2\\right) (\\varphi+1)+C\n\\ea  \\ee\ndue to   (\\ref{z1}), \\eqref{wq2}, (\\ref{ae9}),  and \\eqref{m20}.\nTaking into account the compatibility condition \\eqref{co2}, we can define\n\\bnn \\ba\n\\sqrt{\\n} \\dot u(x,t=0) \\triangleq\n-\\tilde g,\n\\ea\\enn\nwhich along with    (\\ref{e6}), (\\ref{2.48}), and \\eqref{wq01} yields that\n\\be \\la{wq02}\n|\\varphi(0)|\\le C\\| \\tilde g\\|_{L^2}^2+C\\le C.\n\\ee\nThen, integrating \\eqref{ae8} over $(0,t)$, one obtains after using \\eqref{z1}, \\eqref{b2}, \\eqref{h19}, \\eqref{wq2}, and \\eqref{wq02} that\n\\be\\ba \\la{q1}\n& \\varphi(t)+\\int_0^t\\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2   +\\n|\\dot \\te|^2\\right)dxds\\\\\n&\\le  2\\left| \\int_{\\p\\O} G\\left( u\\cdot \\na n \\cdot u \\right)dS\\right|(t)+C\\int_0^t \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2\\right)\\varphi ds+C\\\\\n&\\le  C (\\|\\na u\\|_{L^2}^2\\|\\n^{1\/2}\\dot u\\|_{L^2})(t)+C\\int_0^t \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2\\right)\\varphi ds+C\\\\\n&\\le  \\frac{1}{2} \\varphi(t) + C \\int_{0}^{t} \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2\\right)\\varphi ds+C.\n\\ea\\ee\nApplying Gr\\\"{o}nwall's inequality to \\eqref{q1} and\n using \\eqref{z1} and (\\ref{wq2}), it holds\n\\be\\label{lee3}\n\\sup_{0\\le t\\le T} \\left(\\|\\rho^{1\/2}\\dot u\\|_{L^2}^2\n+\\|\\na \\te\\|_{L^2}^2 \\right) + \\ia\\int\\left(|\\nabla\\dot\nu|^2+\\n|\\dot\\te|^2\\right)dxdt\\le C,\n\\ee\nwhich together with  \\eqref{key} and \\eqref{pq} implies\n\\be\\la{ff1} \\ba\n\\|\\te\\|_{L^2} &\\le C( \\|R \\te-\\overline P \\|_{L^2} +\\overline P ) \\le C(\\|\\na \\te\\|_{L^2}+1) \\le C.\n\\ea \\ee\n\n\n\nNext,\nmultiplying (\\ref{3.99}) by $\\sigma$ and integrating over $(0,T)$  lead to\n\\be  \\ba \\la{a5}\n&\\sup\\limits_{0\\le t\\le T} \\si \\int \\n|\\dot\\te|^2dx+\\int_0^T \\si \\|\\na\\dot\\te\\|_{L^2}^2dt\\\\\n&\\le\nC\\int_0^T\\left(  \\|\\na^2\\te\\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+ \\|\\n^{1\/2}\\dot\\te\\|_{L^2}^2 \\right)dt + C\\\\\n&\\le  C,\n\\ea\\ee\nwhere we have used (\\ref{lee3}),  (\\ref{z1}), (\\ref{m20}), \\eqref{ae9},   (\\ref{lop4}), and the following fact:\n\\be\\la{w1}\\|\\na u\\|_{L^6}\\le C\\ee\ndue to \\eqref{3.30}, \\eqref{z1}, and \\eqref{lee3}.\nThen, it  follows from (\\ref{lop4}), \\eqref{lee3}, \\eqref{a5}, \\eqref{w1}, (\\ref{m20}), and (\\ref{z1}) that\n\\be \\notag \\sup\\limits_{0\\le t\\le T} \\si\\|\\na^2\\te\\|_{L^2}^2 + \\int_{0}^{T}\\|\\na^2\\te\\|_{L^2}^2 dt \\le C,\\ee\nwhich along with \\eqref{lee3}--\\eqref{a5} gives \\eqref{lee2}.\n\n\n\n{\\it Step 2: The proof of (\\ref{qq1}).}\nFirst, standard calculations show  that for $ 2\\le p\\le 6$,\n\\be\\la{L11}\\ba\n\\partial_t\\norm[L^p]{\\nabla\\rho}\n&\\le C\\norm[L^{\\infty}]{\\nabla u} \\norm[L^p]{\\nabla\\rho}+C\\|\\na^2u\\|_{L^p}\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^{\\infty}}+\\|\\na^2\\te \\|_{L^2}\\right)\n\\norm[L^p]{\\nabla\\rho}\\\\\n&\\quad +C\\left(1+\\|\\na\\dot u\\|_{L^2}+\\|\\na^2\\te \\|_{L^2}\\right), \\ea\\ee\nwhere we have used\n\\be\\ba\\la{ua1}\n\\|\\na^2 u\\|_{L^p} & \\le C(\\|\\rho \\dot u\\|_{L^p} + \\|\\na P\\|_{L^p} + \\|\\na u\\|_{L^2}  ) \\\\\n& \\le   C\\left(1+\\|\\na\\dot u\\|_{L^2}+\\|\\na \\te\\|_{L^p}+\\|\\te\\|_{L^\\infty}\\|\\nabla\\n\\|_{L^p}\\right)\\\\\n&\\le  C\\left(1+\\|\\na\\dot u\\|_{L^2}+\\|\\na^2\\te\\|_{L^2}+(\\|\\na^2\\te \\|_{L^2} + 1)\\|\\nabla\\n\\|_{L^p}\\right)\n\\ea\\ee\ndue to \\eqref{g1}, \\eqref{tb90}, \\eqref{rmk1},  \\eqref{z1},  and  \\eqref{lee2}.\nIt follows from Lemma \\ref{le9}, \\eqref{z1}, and (\\ref{ua1})  that\n\\be\\la{u13}\\ba\n\\|\\na u\\|_{L^\\infty }\n&\\le C\\left(\\|{\\rm div}u\\|_{L^\\infty}+\\|\\curl u\\|_{L^\\infty}\\right)\\log(e+\\|\\na^2 u\\|_{L^6}) +C\\|\\na u\\|_{L^2}+C \\\\\n&\\le C\\left( \\|{\\rm div}u\\|_{L^\\infty } + \\|\\curl u\\|_{L^\\infty }\n\\right)\\log(e+ \\|\\na\\dot u\\|_{L^2 } + \\|\\na^2\\te \\|_{L^2})\\\\\n&\\quad +C\\left(\\|{\\rm div}u\\|_{L^\\infty }+ \\|\\curl u\\|_{L^\\infty } \\right)\n\\log\\left(e  + \\|\\na \\n\\|_{L^6}\\right)+C.\n\\ea\\ee\nDenote\n\\bnn\\la{gt}\\begin{cases}\n\tf(t)\\triangleq  e+\\|\\na\t\\n\\|_{L^6},\\\\\n\th(t)\\triangleq 1+  \\|{\\rm div}u\\|_{L^\\infty }^2+ \\|\\curl u\\|_{L^\\infty }^2\n\t+ \\|\\na\\dot u\\|_{L^2 }^2 +\\|\\na^2\\te \\|_{L^2}^2.\n\\end{cases}\\enn\nOne obtains after submitting \\eqref{u13} into (\\ref{L11}) with  $p=6$ that\n\\bnn f'(t)\\le   C h(t) f(t)\\ln f(t) ,\\enn\nwhich implies \\be\\la{w2}  (\\ln(\\ln f(t)))'\\le  C h(t).\\ee\nNote that by virtue of\n(\\ref{hj1}),  (\\ref{key}), (\\ref{lee2}),  (\\ref{z1}), \\eqref{bz5}, and (\\ref{bz6}),  one gets\n\\be \\la{p2}\\ba\n& \\int_0^T\\left(\\|\\div u\\|^2_{L^\\infty}+\\|\\curl u\\|^2_{L^\\infty} \\right)dt \\\\\n& \\le  C\\int_0^T\\left(\\|G\\|^2_{L^\\infty}+ \\|\\curl u\\|^2_{L^\\infty}+\\|P-\\bp\\|^2_{L^\\infty}\\right)dt \\\\\n&\\le  C\\ia\\left(\\| G\\|^2_{  W^{1,6} } + \\| \\curl u\\|^2_{W^{1,6}} + \\|\\te\\|_{L^\\infty}^2\\right)dt + C \\\\\n& \\le   C\\ia\\left(\\|\\na G\\|^2_{L^6}+\\|\\na\\curl  u\\|^2_{L^6}+\\|\\na^2\\te \\|_{L^2}^2\\right)dt+C \\\\\n&\\le C\\ia(\\|\\na \\dot u\\|^2_{L^2}+\\|\\na^2\\te \\|_{L^2}^2)dt+C \\\\\n&\\le  C,\n\\ea\\ee\nwhich as well as \\eqref{w2} and \\eqref{lee2}  yields that\n\\be \\la{u113} \\sup\\limits_{0\\le t\\le T}\\|\\nabla \\rho\\|_{L^6}\\le C.\\ee\nCombining this with (\\ref{u13}),  \\eqref{p2}, and (\\ref{lee2}) leads to\n\\be \\la{v6}\\ia\\|\\nabla u\\|_{L^\\infty}^{3\/2}dt \\le C.\\ee\nMoreover, it follows from (\\ref{rmk1}), \\eqref{z1}, \\eqref{u113}, and (\\ref{lee2})  that\n\\be\\ba\\la{aa95}\n\\sup\\limits_{0\\le t\\le T} \\| u\\|_{H^2}\n\\le &C \\sup\\limits_{0\\le t\\le T}\\left(\\|\\n\\dot u\\|_{L^2}+\\|\\nabla P\\|_{L^2}+\\| \\na u\\|_{L^2}\\right)\\le C.\n\\ea\\ee\n\n\n\n\nNext, applying operator $\\p_{ij}~(1\\le i,j\\le 3)$ to $(\\ref{a1})_1$ gives\n\\be\\la{4.52}  (\\p_{ij} \\n)_t+\\div (\\p_{ij} \\n u)+\\div (\\n\\p_{ij} u)+\\div(\\p_i\\n\\pa_j u+\\p_j\\n\\p_i u)=0. \\ee\nMultiplying (\\ref{4.52}) by $2\\p_{ij} \\n$ and  integrating the resulting equality over $\\O,$ it holds\n\\be\\la{ua2}\\ba\n\\frac{d}{dt}\\|\\na^2\\n\\|^2_{L^2}\n& \\le C(1+\\|\\na u\\|_{L^{\\infty}})\\|\\na^2\\n\\|_{L^2}^2+C\\|\\na u\\|^2_{H^2}\\\\\n& \\le C(1+\\|\\na u\\|_{L^{\\infty}} +\\|\\na^2\\te \\|_{L^2}^2)(1+\\|\\na^2\\n\\|_{L^2}^2) +C\\|\\na\\dot u \\|^2_{L^2}, \\ea\\ee\nwhere one has used \\eqref{z1}, \\eqref{u113}, and the following estimate:\n\\be\\ba\\la{va2}\n\\| u\\|_{H^3}&\\le C\\left(\\|\\na(\\n\\dot u)\\|_{L^2}+\\|\\n\\dot u\\|_{L^2}+\\| \\na P\\|_{H^1}+\\|\\na u\\|_{L^2}\\right)\n\\\\&\\le C\\|\\na\\n\\|_{L^3}\\|\\dot u\\|_{L^6}+C\\|\\na \\dot u\\|_{L^2}+C\\|\\na \\te\\|_{H^1}\n\\\\&\\quad+C\\||\\na\\n||\\na\\te|\\|_{L^2}+C\\|\\te\\|_{L^\\infty}\\|\\na \\n\\|_{H^1}+C\n\\\\ &\\le  C\\|\\na\\dot u \\|_{L^2}+ C  (1+\\|\\na^2\\te \\|_{L^2})(1+\\|\\na^2\\n\\|_{L^2}) +C\n\\ea\\ee\ndue to (\\ref{rmk1}), (\\ref{tb90}), (\\ref{lee2}), (\\ref{u113}),  and (\\ref{z1}).\nThen applying Gr\\\"{o}nwall's inequality to \\eqref{ua2} and using  (\\ref{lee2}), (\\ref{v6}) yield\n\\be\\la{ja3} \\sup_{0\\le t\\le T} \\|\\na^2\\n \\|_{L^2}  \\le C,\\ee\nwhich together with \\eqref{va2} and \\eqref{lee2} gives\n\\be\\la{ja4} \\int_0^T\\|u\\|_{H^3}^2 dt\\le C . \\ee\n\n\nFinally,\napplying the standard $H^1$-estimate to   elliptic problem (\\ref{3.29}), one derives from \\eqref{z1},  \\eqref{lee2}, \\eqref{u113}, \\eqref{kk}, and \\eqref{aa95} that\n\\be\\la{ex4}\\ba\n\\|\\na^2\\te\\|_{H^1}\n& \\le C\\left(\\|\\n \\dot \\te\\|_{H^1}+\\|\\n\\te\\div u\\|_{H^1}+\\||\\na u|^2\\|_{H^1}\\right)\\\\\n& \\le C\\left(1+ \\|\\na \\dot \\te\\|_{L^2} +  \\|\\rho^{1\/2} \\dot \\theta\\|_{L^2}  + \\|\\na(\\n\\te\\div u)\\|_{L^2}+ \\||\\na u||\\na^2u|\\|_{L^2} \\right) \\\\\n& \\le C\\left(1+ \\|\\na \\dot \\te\\|_{L^2} +  \\|\\rho^{1\/2} \\dot \\theta\\|_{L^2}  +\\|\\na^2 \\theta\\|_{L^2}+\\|\\na^3 u\\|_{L^2} \\right),\n\\ea\\ee\nwhich along with \\eqref{z1}, \\eqref{ja3}, \\eqref{ja4}, \\eqref{u113}--\\eqref{aa95}, and \\eqref{lee2} yields (\\ref{qq1}).\n\n\n\n The proof of Lemma \\ref{le11} is finished.\n\\end{proof}\n\n\\begin{lemma}\\la{le9-1}\n\tThe following estimates hold:\n\t\\be\\la{va5}\\ba&\n\t\\sup\\limits_{0\\le t\\le T}\n\t\\|\\n_t\\|_{H^1}\n\t+\\int_0^T\\left(\\|  u_t\\|_{H^1}^2+\\si \\| \\te_t\\|_{H^1}^2+\\| \\n u_t\\|_{H^1}^2+\\si \\|\\n \\te_t\\|_{H^1}^2\n\t\\right)dt\\le C,\n\t\\ea  \\ee\n\tand\n\t\\be\\la{vva5}\\ba\n\t\\int_0^T \\sigma \\left( \\|(\\n u_t)_t\\|_{H^{-1}}^2+\\|(\\n \\te_t)_t\\|_{H^{-1}}^2\n\t\\right)dt\\le C.\\ea\\ee\n\\end{lemma}\n\n\n\n\\begin{proof}\n\n\n\nFirst, it   follows from  (\\ref{lee2}) and (\\ref{qq1})   that\n\\be\\label{va1}\\ba\n&\\sup_{0\\le t\\le T}\\int\\left( \\n|u_t|^2 + \\si\n\\n\\te_t^2\\right)dx +\\int_0^T \\left(\\|\\na u_t\\|_{L^2}^2+ \\si \\|\\na\\te_t\\|_{L^2}^2\\right)dt\\\\\n&\\le \\sup\\limits_{0\\le t\\le T}\\int \\left(\\n|\\dot u|^2+ \\si\\n|\\dot\\te|^2 \\right)dx\n+\\int_0^T\\left(\\|\\na\\dot u\\|_{L^2}^2+\\si \\|\\na\\dot\\te\\|_{L^2}^2 \\right)dt\\\\\n&\\quad + \\sup\\limits_{0\\le t\\le T}\\int \\n\\left(|u\\cdot\\na u|^2+\\si |u\\cdot\\na\\te|^2\\right)dx\\\\\n&\\quad +\\int_0^T\\left(\\|\\na(u\\cdot\\na u)\\|_{L^2}^2+\\si \\|\\na(u\\cdot\\na \\te)\\|_{L^2}^2\\right)dt\\\\\n&\\le C,\\ea\\ee\nwhich together with (\\ref{lee2}) and (\\ref{qq1}) gives\n\\be\\la{vva1} \\ba\n& \\int_0^T\\left(\\|\\na(\\n u_t)\\|_{L^2}^2+ \\si \\|\\na(\\n\\te_t)\\|_{L^2}^2\\right)dt\\\\\n& \\le  C\\int_0^T\\left(\\|  \\na u_t \\|_{L^2}^2+\\|  \\na \\n\\|_{L^3}^2\\| u_t \\|_{L^6}^2\n+ \\si \\|  \\na \\te_t \\|_{L^2}^2+ \\si \\|  \\na \\n\\|_{L^3}^2\\| \\te_t \\|_{L^6}^2 \\right)dt\\\\\n&\\le C,\\ea\\ee\nwhere we have used\n\\be\\la{w3}\\|\\te_t\\|_{L^6}\\le C\\|\\n^{1\/2}\\te_t\\|_{L^2}+C\\|\\na\\te_t\\|_{L^2\n\\ee\ndue to \\eqref{kk}.\n\nNext, one deduces from $(\\ref{a1})_1$, (\\ref{qq1}), and \\eqref{hs} that\n\\bnn\\ba\\la{sp1}\n\\|\\n_t\\|_{H^1}\\le& \\|\\div (\\rho u)\\|_{H^1}\n\\le  C \\|u\\|_{H^2}\\|\n\\n\\|_{H^2}\n\\le C,\n\\ea \\enn\nwhich as well as (\\ref{va1})--\\eqref{w3} shows    (\\ref{va5}).\n\nFinally, differentiating $(\\ref{a1})_2$ with respect to $t$ yields that\n\\be   \\la{va7}\\ba (\\n u_t)_t\n=-(\\n u\\cdot\\na u)_t + \\left(\\mu\\Delta u+(\\mu+\\lambda)\\na\\div u \\right)_t -\\na P_t.\\ea\\ee\nIt follows from (\\ref{va5}), \\eqref{qq1}, \\eqref{lee2}, \\eqref{lop4}, and \\eqref{w3} that\n\\be   \\la{va9}\\ba  \\|(\\n u\\cdot\\na u)_t  \\|_{L^{2}}\n&=\\| \\n_t u\\cdot\\na u+ \\n u_t\\cdot\\na u + \\n u\\cdot\\na u_t  \\|_{L^{2}}\\\\\n&\\le C\\|\\n_t\\|_{L^6} \\|\\na u\\|_{L^3}+  C\\|u_t\\|_{L^6} \\|\\na u\\|_{L^3}+  C\\|u \\|_{L^\\infty} \\|\\na u_t\\|_{L^2}\\\\\n&\\le C+   C\\|\\na u_t\\|_{L^2},\\ea\\ee\nand\n\\be   \\la{va10}\\ba  \\|\\na P_t  \\|_{L^2}\n&=R\\|\\n_t\\na\\te +\\n \\na\\te_t +\\na\\n_t\\te +\\na\\n\\te_t\\|_{L^2}\\\\\n&\\le C\\left(\\|\\n_t\\|_{L^6}\\|\\na\\te\\| _{L^3}+\\|\\na\\te_t\\|_{L^2}\n+\\|\\te\\|_{L^\\infty}\\|\\na \\n_t\\|_{L^2}+\\|\\na\\n\\|_{L^6}\\|\\te_t\\| _{L^3}\\right)\\\\\n&\\le C+C\\|\\na\\te_t\\|_{L^2}+C\\|\\n^{1\/2}\\te_t\\|_{L^2}.\\ea\\ee\nCombining (\\ref{va7})--(\\ref{va10}) with   (\\ref{va5}) shows\n\\be\\la{vva04}\\ba\\int_0^T \\si \\|(\\n u_t)_t\\|_{H^{-1}}^2dt\\le C.\\ea\\ee\nSimilarly, we have \\bnn\\int_0^T \\si  \\|(\\n \\te_t)_t\\|_{H^{-1}}^2dt\\le C,  \\enn\nwhich combined with (\\ref{vva04}) implies (\\ref{vva5}). The proof of Lemma \\ref{le9-1} is completed.\n\\end{proof}\n\n\n\\begin{lemma}\\la{pe1}\n\tThe following estimate holds:\n\t\\be\\la{nq1}\n\t\\sup\\limits_{0\\le t\\le T} \\si\\left(\\|\\nabla u_t\\|^2_{L^2}+\\|\\n_{tt} \\|^2_{L^2} + \\|u\\|_{H^3}^2\\right)\n\t+ \\int_0^T\\si \\left(\\|\\rho^{1\/2} u_{tt}\\|_{L^2}^2+ \\|\\nabla u_t\\|_{H^1}^2\\right)dt\\le C.\n\t\\ee\n\\end{lemma}\n\\begin{proof}\nDifferentiating  $(\\ref{a11})$  with respect to $t$ leads to \\be\\la{nt0}\\ba \\begin{cases}\n(2\\mu+\\lambda)\\na\\div u_t-\\mu \\na\\times \\curl u_t\\\\\n= \\n u_{tt} +\\n_tu_t+\\n_tu\\cdot\\na u+\\n u_t\\cdot\\na u +\\n u\\cdot\\na u_t+\\na P_t\\triangleq \\tilde f\n,&\\, \\text{in}\\,\\O\\times[0,T],\\\\\nu_{t}\\cdot n=0,\\  \\curl u_{t}\\times n=0,  &\\,\\text{on}\\,\\p\\O\\times[0,T]. \\end{cases}\\ea\\ee\nMultiplying (\\ref{nt0})$_1$ by $u_{tt}$   and integrating  the resulting equality  by parts, one gets  \\be\\la{sp9} \\ba&\n\\frac{1}{2}\\frac{d}{dt}\\int \\left(\\mu|\\curl u_t|^2 + (2\\mu +\\lambda)({\\rm div}u_t)^2\\right)dx\n+\\int \\rho| u_{tt}|^2dx\\\\\n&=\\frac{d}{dt}\\left(-\\frac{1}{2}\\int_{ }\\rho_t |u_t|^2 dx- \\int_{}\\rho_t u\\cdot\\nabla u\\cdot u_tdx\n+ \\int_{ }P_t {\\rm div}u_tdx\\right)\\\\\n& \\quad + \\frac{1}{2}\\int_{ }\\rho_{tt} |u_t|^2 dx+\\int_{ }(\\rho_{t} u\\cdot\\nabla u )_t\\cdot u_tdx\n-\\int_{ }\\rho u_t\\cdot\\nabla u\\cdot u_{tt}dx\\\\\n&\\quad - \\int_{ }\\rho u\\cdot\\nabla u_t\\cdot u_{tt}dx - \\int_{ }\\left(P_{tt}-\n\\ka(\\ga-1)\\Delta\\te_t\\right){\\rm div}u_tdx\\\\\n&\\quad +\\ka(\\ga-1)\\int_{ } \\na\\te_t\\cdot\\na {\\rm div}u_tdx\n\\triangleq\\frac{d}{dt}\\tilde{I}_0+ \\sum\\limits_{i=1}^6 \\tilde{I}_i. \\ea \\ee\n\n Each term $\\tilde{I}_i(i=0,\\cdots,6)$ can be estimated as follows:\n\nFirst, it follows from simple calculations, $(\\ref{a1})_1,$   (\\ref{va5}), \\eqref{qq1}, \\eqref{lee2}, and (\\ref{va1}) that\n\\be \\ba \\la{sp10}\n|\\tilde{I}_0|&=\\left|-\\frac{1}{2}\\int\\rho_t |u_t|^2 dx- \\int \\rho_t u\\cdot\\nabla\nu\\cdot u_tdx+ \\int P_t {\\rm div}u_tdx\\right|\\\\\n&\\le C\\int  \\n |u||u_t||\\nabla u_t| dx+C\\norm[L^3]{\\rho_t}\\norm[L^2]\n{\\nabla u}\\norm[L^6]{u_t}+C\\|(\\n\\te)_t\\|_{L^2}\\|\\nabla u_t\\|_{L^2}\\\\\n&\\le C \\|\\n^{1\/2}u_t\\|_{L^2}  \\|\\nabla u_t\\|_{L^2} +C(1+\\|\\n^{1\/2} \\te_t\\|_{L^2}+\\|\\n_t\\|_{L^3}\\|\\te\\|_{L^6})\\|\\nabla u_t\\|_{L^2}\\\\\n&\\le C (1+\\|\\n^{1\/2}\\te_t\\|_{L^2}) \\|\\nabla u_t\\|_{L^2} ,\\ea\\ee\n\\be \\la{sp11}\\ba\n2|\\tilde{I}_1|&=\\left|\\int \\rho_{tt} |u_t|^2 dx\\right|  \\le C \\|\\n_{tt}\\|_{L^2}\\|u_t\\|_{L^4}^{2}\n \\le  C\\|\\n_{tt}\\|_{L^2}^2+C \\|\\na u_t\\|_{L^2}^4,\n\\ea \\ee\n\\be \\la{sp12}\\ba\n|\\tilde{I}_2|&=\\left|\\int \\left(\\rho_t u\\cdot\\nabla u \\right)_t\\cdot u_{t}dx\\right|\\\\\n& = \\left|  \\int\\left(\\rho_{tt} u\\cdot\\nabla u\\cdot u_t +\\rho_t\nu_t\\cdot\\nabla u\\cdot u_t+\\rho_t u\\cdot\\nabla u_t\\cdot u_t\\right)dx\\right|\\\\\n&\\le   C\\norm[L^2]{\\rho_{tt}}\\norm[L^6]{\\na u}\\norm[L^6]{u}\\norm[L^6]{u_t}\n+C\\norm[L^2]{\\rho_t}\\norm[L^6]{u_t}^2\\norm[L^6]{\\nabla u} \\\\\n&\\quad+C\\norm[L^3]{\\rho_t}\\norm[L^{\\infty}]{u}\\norm[L^2]{\\nabla u_t}\\norm[L^6]{u_t}\\\\\n& \\le C\\norm[L^2]{\\rho_{tt}}^2 + C\\norm[L^2]{\\nabla u_t}^2, \\ea \\ee\nand\n\\be\\ba\\la{sp13}\n|\\tilde{I}_3|+|\\tilde{I}_4|&= \\left| \\int \\rho u_t\\cdot\\nabla u\\cdot u_{tt} dx\\right|\n+\\left| \\int \\rho u\\cdot\\nabla u_t\\cdot u_{tt}dx\\right|\\\\\n& \\le   C\\|\\n^{1\/2}u_{tt}\\|_{L^2}\\left(\\|u_t\\|_{L^6}\\|\\na u\\|_{L^3}\n+\\|u\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}\\right) \\\\\n& \\le  \\frac{1}{4}\\norm[L^2]{\\rho^{{1\/2}}u_{tt}}^2 + C \\norm[L^2]{\\nabla u_t}^2.\\ea\\ee\nThen, by virtue of\n(\\ref{op3}), (\\ref{va5}), \\eqref{va10}, and Lemma \\ref{le11}, it holds\n\\bnn\\ba & \\|P_{tt}-\\ka(\\ga-1)\\Delta \\te_t\\|_{L^2}\\\\\n&\\le  C\\|(u\\cdot\\na P)_t\\|_{L^2}+C\\|(P\\div u)_t\\|_{L^2}+C\\||\\na u||\\na u_t|\\|_{L^2}\\\\\n&\\le  C\\|u_t\\|_{L^6}\\|\\na P\\|_{L^3}+C\\|u\\|_{L^\\infty}\\|\\na P_t\\|_{L^2}\n+C\\|P_t\\|_{L^6}\\|\\na u\\|_{L^3}\\\\\n&\\quad  +C\\|P\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}+C\\|\\na u\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^\\infty}+\\|\\na^2\\te \\|_{L^2}\\right)\\|\\na u_t\\|_{L^2} +C\\left(1+\\|\\na\\te_t\\|_{L^2}+\\|\\n^{1\/2}\\te_t\\|_{L^2}\\right),\n\\ea\\enn\nwhich yields\n\\be\\ba\\la{sp15}\n|\\tilde{I}_5|&=\\left|\\int\\left(P_{tt}-\\ka(\\ga-1)\\Delta \\te_t\\right){\\rm div}u_tdx\\right|\\\\\n&\\le\\norm[L^2]{P_{tt}-\\ka(\\ga-1)\\Delta \\te_t}\\norm[L^2]{\\na u_t}\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^\\infty}+\\|\\na^2\\te \\|_{L^2}\\right)\\|\\na u_t\\|_{L^2}^2\\\\\n&\\quad+C\\left(1+\\|\\na\\te_t\\|^2_{L^2}+\\|\\n^{1\/2}\\te_t\\|_{L^2}^2\\right).\n\\ea\\ee\nNext, combining Lam\\'{e}'s system \\eqref{nt0} with Lemma \\ref{zhle},   \\eqref{qq1}, \\eqref{va5}, and (\\ref{va10}) gives\n\\be\\la{nt4}\\ba\n\\|\\na^2u_t\\|_{L^2}&\\le C\\|\\tilde f\\|_{L^2}+C\\|\\na u_t\\|_{L^2}\\\\\n&\\le C\\|\\n u_{tt}\\|_{L^2}\n+C\\|\\n_t\\|_{L^3}\\|u_t\\|_{L^6}+C\\|\\n_t\\|_{L^3}\\|\\na u\\|_{L^6}\\|u\\|_{L^\\infty}\\\\\n&\\quad  +C\\|u_t\\|_{L^6}\\|\\na u\\|_{L^3}+C\\|\\na u_t\\|_{L^2}\\|u\\|_{L^\\infty}+C\\|\\na P_t\\|_{L^2}\\\\\n&\\le C\\left(\\|\\n u_{tt}\\|_{L^2}+\\|\\na u_t\\|_{L^2}+\\|\\n^{1\/2} \\te_{t}\\|_{L^2}+\\|\\na \\te_t\\|_{L^2}+1\\right),\\ea\\ee\nwhich immediately leads  to\\be \\la{asp16}\\ba\n|\\tilde{I}_6| &=\\ka(\\ga-1) \\left| \\int_{ } \\na\\te_t\\cdot\\na{\\rm div}u_tdx \\right|   \\\\\n& \\le  C \\|\\na^2u_t\\|_{L^2}\\|\\na\\te_t\\|_{L^2}\\\\\n& \\le \\frac{1}{4} \\|\\n^{1\/2} u_{tt}\\|^2_{L^2}+ C\\left(1+\\|\\na u_t\\|^2_{L^2}+\\|\\n^{1\/2} \\te_t\\|^2_{L^2}+\\|\\na\\te_t\\|^2_{L^2}\\right).\n\\ea\\ee\n\nPutting   (\\ref{sp11})--(\\ref{sp15})  and (\\ref{asp16}) into\n(\\ref{sp9}) yields\n\\be\\la{4.052} \\ba\n& \\frac{d}{dt}\\int \\left(\\mu|\\curl u_t|^2 + (2\\mu +\\lambda)({\\rm div}u_t)^2-2\\tilde{I}_0\\right)dx\n+\\int \\rho| u_{tt}|^2dx\\\\\n&\\le  C\\left(1+\\|\\na u\\|_{L^\\infty}+\\|\\na u_t\\|_{L^2}^2+\\|\\na^2\\te \\|_{L^2}^2 \\right)\\|\\na u_t\\|_{L^2}^2 \\\\\n&\\quad +C\\left(1+\\|\\n_{tt}\\|_{L^2}^2+\\|\\n^{1\/2} \\te_t\\|^2_{L^2}+\\|\\na \\te_t\\|_{L^2}^2\\right).\\ea\\ee\nFurthermore, we infer from $(\\ref{a1})_1$, \\eqref{qq1}, and (\\ref{va5}) that\n\\be \\la{s4} \\ba\n\\|\\n_{tt}\\|_{L^2} &= \\|\\div(\\rho u)_t\\|_{L^2} \\\\\n& \\le C\\left(\\|\\n_t\\|_{L^6}\\|\\nabla u\\|_{L^3}+ \\|\\nabla u_t\\|_{L^2}\n+\\|u_t\\|_{L^6}\\|\\nabla \\n\\|_{L^3}+\\|\\nabla \\n_t\\|_{L^2}\\right) \\\\ &\\le C+C\\|\\na u_t\\|_{L^2}.\\ea\\ee\nMultiplying \\eqref{4.052} by $\\sigma$ and integrating the resulting inequality over $(0,T)$, one thus  deduces from \\eqref{nn2}, \\eqref{lee2}, (\\ref{qq1}),  (\\ref{va1}), (\\ref{sp10}),     (\\ref{s4}),\nand Gr\\\"{o}nwall's inequality that\n\\be\\la{nq11}\n\\sup\\limits_{0\\le t\\le T} \\si \\|\\nabla u_t\\|^2_{L^2}\n+ \\int_0^T\\si\\int\\rho |u_{tt}|^2dxdt\n\\le C.\n\\ee\n\nFinally, by Lemma \\ref{le11}, \\eqref{s4}, (\\ref{va2}), \\eqref{nt4}, \\eqref{va1}, and  (\\ref{nq11}), we have\n\\be\\notag\n\\sup\\limits_{0\\le t\\le T}\\si\\left(\\|\\n_{tt}\\|_{L^2}^2+\\|u\\|^2_{H^3}\\right) + \\int_0^T \\si\\|\\nabla u_t\\|_{H^1}^2 dt\\le C,\n\\ee\nwhich along with \\eqref{nq11} gives (\\ref{nq1}).\nWe  complete  the proof of Lemma \\ref{pe1}.\n\\end{proof}\n\n\\begin{lemma}\\la{pr3} For $q\\in (3,6)$ as in Theorem \\ref{th1}, it holds that\n\t\\be\\la{y2}\\ba\n\t&\\sup_{0\\le t\\le T} \\|\\n\\|_{W^{2,q}} +\\int_0^T\n\t \\|\\na^2u\\|_{W^{1,q}}^{p_0}  dt\\le C,\n\t\\ea \\ee\n\twhere  \\be 1< \\la{pppppp} p_0<\\frac{4q }{5q-6} \\in (1,4\/3).\\ee\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst, it follows from  \\eqref{rmk1}, \\eqref{tb90}, and Lemma \\ref{le11} that\n\\be\\la{a4.74}\\ba\n\\|\\na^2u\\|_{W^{1,q}}\n\\le & C\\left(\\|\\n \\dot u\\|_{W^{1,q}}+\\|\\na P\\|_{W^{1,q}}+\\|\\na u\\|_{L^2}\\right)\\\\\n\\le & C\n(\\|\\na\\dot u\\|_{L^2}+\\|\\na(\\n\\dot u)\\|_{L^q}+  \\|\\na^2\\te\\|_{L^2}+ \\|\\te\\na^2\\n\\|_{L^q}\\\\\n& +\\| |\\na\\n||\\na\\te|\\|_{L^q}+  \\|\\na^2\\te\\|_{L^q}+1)\\\\\n\\le & C\\left(\\|\\na\\dot u\\|_{L^2}+\\|\\na(\\n\\dot u)\\|_{L^q}  + \\| \\na^2 \\theta\\|_{L^q}+1 \\right)\\\\\n&+ C(1+ \\|\\na^2 \\theta\\|_{L^2})\\| \\na^2 \\n\\|_{L^q}.\n\\ea\\ee\nNext, multiplying (\\ref{4.52}) by $q|\\p_{ij} \\n|^{q-2}\\p_{ij} \\n$ and  integrating the resulting equality over $\\O,$ we obtain after using  (\\ref{qq1}) and (\\ref{a4.74}) that\n\\be\\la{sp28}\\ba\n&\\frac{d}{dt}\\|\\na^2\\n\\|_{L^q}^q\\\\\n&\\le C\\|\\na u\\|_{L^\\infty}\\|\\na^2\\n\\|_{L^q}^q\n+C\\|\\na^{2} u\\|_{W^{1,q}}\\|\\na^2\\n\\|_{L^q}^{q-1}(\\|\\na\\n\\|_{L^q}+1)\\\\\n&\\le C\\| u\\|_{H^3}\\|\\na^2\\n\\|_{L^q}^q+C\\|\\na^2 u\\|_{W^{1,q}}\\|\\na^2\\n\\|_{L^q}^{q-1}\\\\\n&\\le C\\left(\\| u\\|_{H^3}   + \\|\\na\\dot u\\|_{L^2} +\\|\\na(\\n\\dot u)\\|_{L^q}+\\|\\na^2 \\theta\\|_{L^q}+1\\right)\\left(\\|\\na^2 \\n\\|_{L^q}^q+1\\right).\n\\ea\\ee\n\nNote that Lemma \\ref{le11}, (\\ref{g1}), \\eqref{tb90}, and \\eqref{nq1} imply\n\\be\\la{4.49}\\ba     \\|\\na(\\n\\dot u)\\|_{L^q}\n&\\le C\\|\\na \\n\\|_{L^6}\\|\\dot u \\|_{L^{6q\/(6-q)}}+C\\|\\na\\dot u \\|_{L^q}\\\\\n&\\le C\\|\\dot u \\|_{W^{1,6q\/(6+q)}}+C\\|\\na\\dot u \\|_{L^q}\\\\\n&\\le C\\|\\na u_t \\|_{L^q}+C\\|\\na(u\\cdot \\na u ) \\|_{L^q}+C\\\\\n&\\le C\\|\\na u_t \\|_{L^2}^{(6-q)\/2q}\\|\\na u_t \\|_{L^6}^{3(q-2)\/{2q}}\\\\\n& \\quad+C\\|\\na u \\|_{L^q}\\| \\na u \\|_{L^\\infty}+C\\| u \\|_{L^\\infty}\\|\\na^2 u \\|_{L^q}+C\\\\\n&\\le C\\si^{-1\/2} \\left(\\si\\|\\na u_t \\|^2_{H^1}\\right)^{3(q-2)\/{4q}}\n+C\\|u\\|_{H^3}+C,\\ea \\ee\nand\n\\begin{equation}\\notag\n    \\begin{aligned}\n    \\| \\na^2 \\theta\\|_{L^q} \\le& C \\|\\na^2 \\theta\\|_{L^2}^{(6-q)\/2q} \\|\\na^3 \\theta\\|_{L^2}^{3(q-2)\/2q} +C\\|\\na^2 \\theta\\|_{L^2}\\\\\n    \\le & C \\si^{-1\/2} \\left(\\si\\|\\na^3 \\theta \\|^2_{L^2} \\right)^{3(q-2)\/{4q}} +C \\|\\na^2 \\theta\\|_{L^2},\n\\end{aligned}\n\\end{equation}\nwhich combined with Lemma \\ref{le11} and \\eqref{nq1} shows that, for $p_0$ as in (\\ref{pppppp}),\n\\be \\la{4.53}\\int_0^T \\left(\\|\\na(\\n \\dot u)\\|^{p_0}_{L^q} + \\|\\na^2 \\theta\\|_{L^q}^{p_0} \\right) dt\\le C. \\ee\n\nFinally, applying  Gr\\\"{o}nwall's\ninequality to (\\ref{sp28}), we obtain after using Lemma \\ref{le11} and (\\ref{4.53}) that\n\\bnn  \\sup\\limits_{0\\le t\\le T}\\|\\na^2 \\n\\|_{L^q}\\le C,\\enn\nwhich together with   Lemma \\ref{le11},    (\\ref{4.53}),\nand (\\ref{a4.74}) gives (\\ref{y2}).  We finish  the proof of Lemma \\ref{pr3}.\n\\end{proof}\n\n\\begin{lemma}\\la{sq90} For $q\\in (3,6)$ as in Theorem \\ref{th1}, the following estimate holds:\n\t\\be \\ba\\la{eg17}\n\t&\\sup_{ 0\\le t\\le T}\\si \\left(\\|\\te_t\\|_{H^1}+\\|\\na^2\\te\\|_{H^1}+\\| u_t\\|_{H^2}\n\t+\\| u\\|_{W^{3,q}}\\right)   +\\int_0^T   \\si^2\\|\\na u_{tt}\\|_{L^2}^2 dt\\le C.\\ea  \\ee\n\t\n\\end{lemma}\n\n\\begin{proof}\nFirst,  differentiating $(\\ref{nt0})$ with\nrespect to $t$ gives\n\\be\\la{sp30}\\ba \\begin{cases}\n\\n u_{ttt}+\\n u\\cdot\\na u_{tt}-(2\\mu+\\lambda)\\nabla{\\rm div}u_{tt} +\\mu\\na \\times \\curl u_{tt}\\\\\n= 2{\\rm div}(\\n u)u_{tt} +{\\rm div}(\\n u)_{t}u_t-2(\\n u)_t\\cdot\\na u_t\\\\\\quad -(\\n_{tt} u+2\\n_t u_t) \\cdot\\na u\n- \\n u_{tt}\\cdot\\na u-\\na P_{tt}, & \\text{in}\\,\\O\\times[0,T],\\\\\nu_{tt} \\cdot n=0,\\quad \\curl u_{tt} \\times n=0, &  \\text{on}\\,\\p\\O\\times[0,T].\n\\end{cases}\\ea \\ee\nMultiplying (\\ref{sp30})$_1$ by $u_{tt}$ and integrating the resulting equality over ${\\Omega}$ by parts imply that\n\\be \\la{sp31}\\ba\n&\\frac{1}{2}\\frac{d}{dt}\\int \\n |u_{tt}|^2dx\n+\\int \\left((2\\mu+\\lambda)({\\rm div}u_{tt})^2+\\mu|\\curl u_{tt}|^2\\right)dx\\\\\n&=-4\\int  u^i_{tt}\\n u\\cdot\\na u^i_{tt} dx\n-\\int (\\n u)_t\\cdot \\left(\\na (u_t\\cdot u_{tt})+2\\na u_t\\cdot u_{tt}\\right)dx\\\\\n&\\quad -\\int (\\n_{tt}u+2\\n_tu_t)\\cdot\\na u\\cdot u_{tt}dx\n-\\int   \\n u_{tt}\\cdot\\na u\\cdot  u_{tt} dx\\\\\n& \\quad+\\int  P_{tt}{\\rm div}u_{tt}dx\\triangleq\\sum_{i=1}^5\\tilde{J}_i.\\ea\\ee\nIt follows from Lemmas \\ref{le11}--\\ref{pe1}, (\\ref{va1}),   \\eqref{w3}, and \\eqref{s4} that, for $\\eta\\in(0,1],$\n\\be \\la{sp32} \\ba\n|\\tilde{J}_1| &\\le C\\|\\n^{1\/2}u_{tt}\\|_{L^2}\\|\\na u_{tt}\\|_{L^2}\\| u \\|_{L^\\infty} \\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta) \\|\\n^{1\/2}u_{tt}\\|^2_{L^2},\\ea\\ee\n\\be \\la{sp33}\\ba\n|\\tilde{J}_2| &\\le C\\left(\\|\\n u_t\\|_{L^3}+\\|\\n_t u\\|_{L^3}\\right)\n\\left(\\| \\na u_{tt}\\|_{L^2}\\| u_t\\|_{L^6}+\\| u_{tt}\\|_{L^6}\\| \\na u_t\\|_{L^2}\\right)\\\\\n&\\le C\\left(\\|\\n^{1\/2} u_t\\|^{1\/2}_{L^2}\\|u_t\\|^{1\/2}_{L^6}+\\|\\n_t\\|_{L^6}\\| u\\|_{L^6}\\right)\\| \\na u_{tt}\\|_{L^2}\\| \\na u_t\\|_{L^2}\\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\| \\na u_t\\|_{L^2}^{3}+C(\\eta)\\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\si^{-3\/2}  ,\\ea\\ee\n\\be  \\la{sp34}\\ba\n|\\tilde{J}_3| &\\le C\\left(\\|\\n_{tt}\\|_{L^2}\\|u\\|_{L^6}+\n\\|\\n_{t}\\|_{L^2}\\|u_{t}\\|_{L^6} \\right)\\|\\na u\\|_{L^6}\\|u_{tt}\\|_{L^6} \\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\si^{-1}  ,\\ea\\ee\nand\\be  \\la{sp36}\\ba &\n|\\tilde{J}_4|+|\\tilde{J}_5|\\\\\n&\\le  C\\|\\n u_{tt}\\|_{L^2} \\|\\na u\\|_{L^3}\\|u_{tt}\\|_{L^6}\n+C \\|(\\n_t\\te+\\n\\te_t)_t\\|_{L^2}\\|\\na u_{tt}\\|_{L^2}\\\\\n&\\le  \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta) \\left(\\|\\n^{1\/2}u_{tt}\\|^2_{L^2}\n+\\|\\n_{tt}\\te\\|_{L^2}^2+\\|\\n_{t}\\te_t\\|_{L^2}^2 +\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}^2\\right) \\\\\n&\\le  \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\left( \\|\\n^{1\/2}u_{tt}\\|^2_{L^2}\n+\\|\\na\\te_t\\|_{L^2}^2 +\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}^2+\\sigma^{-2}\\right). \\ea\\ee\nSubstituting (\\ref{sp32})--(\\ref{sp36}) into (\\ref{sp31}), we obtain after using \\eqref{nn2} and choosing $\\eta$ suitably small  that\n\\be \\la{ex12}\\ba\n& \\frac{d}{dt}\\int \\n |u_{tt}|^2dx+C_4 \\int|\\na u_{tt}|^2dx \\\\\n& \\le  C\\si^{-2} +C\\|\\n^{1\/2}u_{tt}\\|^2_{L^2}\n+C\\|\\na\\te_t\\|_{L^2}^2+C_5\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}^2.\\ea\\ee\n\nNext, differentiating \\eqref{3.29} with respect to $t$ infers\n\\be\\la{eg1}\\ba \\begin{cases}\n-\\frac{\\ka(\\ga-1)}{R}\\Delta \\te_t+\\n\\te_{tt}\\\\\n=-\\n_t\\te_{t}- \\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)-\\n\\left( u\\cdot\\na\n\\te+(\\ga-1)\\te\\div u\\right)_t\\\\\n\\quad+\\frac{\\ga-1}{R}\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_t,  \\\\\n\\na \\te_t\\cdot n|_{ \\p\\O\\times(0,T)}=0  .\n\\end{cases}\\ea\\ee\nMultiplying  (\\ref{eg1})$_1$ by $\\te_{tt}$ and integrating the resulting\nequality over $\\Omega$ lead to\n\\be\\la{ex5}\\ba\n& \\left(\\frac{\\ka(\\ga-1)}{2R}\\|\\na \\te_t\\|_{L^2}^2+H_0\\right)_t+ \\int\\n\\te_{tt}^2dx \\\\\n&=\\frac{1}{2}\\int\\n_{tt}\\left( \\te_t^2\n+2\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\te_t\\right)dx\\\\\n&\\quad + \\int\\n_t\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u \\right)_t\\te_{t}dx\\\\\n& \\quad-\\int\\n\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u\\right)_t\\te_{tt}dx\\\\\n& \\quad -\\frac{\\ga-1}{R}\\int \\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_{tt}\\te_t dx\n\\triangleq\\sum_{i=1}^4H_i,\\ea\\ee\nwhere\n\\bnn\\ba H_0\\triangleq & \\frac{1}{2}\\int \\n_t\\te_{t}^2dx\n+\\int\\n_t\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u\\right) \\te_tdx\\\\\n&- \\frac{\\ga-1}{R}\\int\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2 \\right)_t\\te_t dx. \\ea\\enn\nIt follows from  $(\\ref{a1})_1,$   (\\ref{va1}),    \\eqref{w3}, \\eqref{s4}, and Lemmas \\ref{le11}--\\ref{pe1} that\n\\be\\la{ex6}\\ba\n|H_0|\\le & C\\int \\n|u||\\te_{t}||\\na\\te_{t}|dx+C\\|\\n_t\\|_{L^3}\\|\\te_t\\|_{L^6}\\left( \\|\\na\\te\\|_{L^2} \\|u\\|_{L^\\infty}+ \\|\\theta\\|_{L^6} \\|\\na u\\|_{L^3}\\right)\\\\\n&+ C\\|\\na u\\|_{L^3}\\|\\na u_t\\|_{L^2} \\|\\te_t\\|_{L^6} \\\\\n\\le &C  (\\|\\rho^{1\/2}\\theta_t \\|_{L^2}+\\|\\na\\te_t\\|_{L^2})\\left(\\|\\n^{1\/2}\\te_t\\|_{L^2}+\\|\\na u_t\\|_{L^2}+1\\right)\\\\\n\\le &\\frac{\\ka(\\ga-1)}{4R} \\|\\na\\te_t\\|_{L^2}^2+C\\si^{-1},\\ea\\ee\nand\n\\be\\la{ex7}\\ba\n|H_1|& \\le C\\|\\n_{tt}\\|_{L^2}\\left(\\|\\te_t\\|_{L^4}^{2}\n+\\|\\te_t\\|_{L^6}\\left(\\|u\\cdot\\na \\te\\|_{L^3}+\\| \\te\\div u\\|_{L^3} \\right)\\right)\\\\\n&\\le C\\|\\n_{tt}\\|_{L^2}\\left(\\|\\rho^{1\/2} \\te_t\\|_{L^2}^{2} + \\|\\na \\te_t\\|_{L^2}^{2}\n+\\si^{-1}  \\right) \\\\\n& \\le  C(1+\\|\\na u_{t}\\|_{L^2} )\\|\\na \\te_t\\|^2_{L^2}+C\\si^{-3\/2}.\\ea\\ee\nCombining  Lemma \\ref{le11} with (\\ref{w3}) gives\n\\be\\la{eg12}\\ba\n &\\|\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u \\right)_t\\|_{L^2}\\\\\n& \\le  C\\left(\\|u_t\\|_{L^6}\\|\\na\\te\\|_{L^3}+\\|\\na\\te_t\\|_{L^2}+\\|\\te_t\\|_{L^6}\\|\\na u\\|_{L^3}\n+\\|\\te\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}\\right)\\\\\n& \\le  C\\|\\na u_t\\|_{L^2}(\\|\\na^2 \\te\\|_{L^2}+1)+ C\\|\\na \\te_t\\|_{L^2}+C\\|\\rho^{1\/2} \\theta_t \\|_{L^2},\\ea\\ee\nwhich together with \\eqref{lee2}, \\eqref{va5},  (\\ref{va1}), and \\eqref{w3} shows\n\\be\\la{ex9}\\ba\n|H_2|+|H_3|&\\le C\\left(\\si^{-1\/2}(\\|\\na u_t\\|_{L^2}+1)+\\|\\na\\te_t\\|_{L^2}\\right)\n\\left(\\|\\n_t\\|_{L^3} \\|\\te_t\\|_{L^6}+\\|\\n \\te_{tt}\\|_{L^2}\\right)\\\\\n&\\le \\frac{1}{2}\\int\\n\\te_{tt}^2dx+C\\|\\na\\te_t\\|_{L^2}^2+C\\si^{-1 } \\|\\na u_t\\|^2_{L^2}+C\\si^{-1 } . \\ea\\ee\nOne deduces from \\eqref{qq1}, (\\ref{va1}), \\eqref{w3}, and (\\ref{nq1}) that\n\\be\\la{ex10}\\ba\n|H_4|&\\le C\\int \\left(|\\na u_t|^2+|\\na u||\\na u_{tt}|\\right)|\\te_t|dx\\\\\n&\\le C\\left(\\|\\na u_t\\|_{L^2}^{3\/2}\\|\\na u_t\\|_{L^6}^{1\/2}\n+ \\|\\na u\\|_{L^3} \\|\\na u_{tt}\\|_{L^2}\\right)\\|\\te_t\\|_{L^6}\\\\\n&\\le \\de\\|\\na u_{tt}\\|^2_{L^2}+C\\|\\na^2 u_t\\|^2_{L^2}\n+C(\\de)(\\|\\na\\te_t\\|_{L^2}^2+\\si^{-1 })+C\\si^{-2}\\|\\na u_t\\|_{L^2}^2.\\ea\\ee\nSubstituting (\\ref{ex7}), (\\ref{ex9}), and (\\ref{ex10}) into (\\ref{ex5}) gives\n\\be\\la{ex11}\\ba\n& \\left(\\frac{\\ka(\\ga-1)}{2R}\\|\\na \\te_t\\|_{L^2}^2+H_0\\right)_t\n+\\frac{1}{2}\\int\\n\\te_{tt}^2dx \\\\\n&\\le  \\de\\|\\na u_{tt}\\|^2_{L^2}+C(\\de)((1+\\|\\na u_{t}\\|_{L^2}) \\|\\na \\te_t\\|^2_{L^2}+\\si^{-3\/2})\\\\\n&\\quad +C(\\|\\na^2 u_t\\|^2_{L^2} +\\si^{-2}\\|\\na u_t\\|_{L^2}^2).\\ea\\ee\n\n\n\nFinally, for $C_5$ as in (\\ref{ex12}), adding (\\ref{ex11}) multiplied by\n$2 (C_5+1) $ to  (\\ref{ex12}) and choosing $\\de$ suitably small yield that\n\\bnn\\la{ex13}\\ba\n& \\left[ 2 (C_5+1)\\left(\\frac{\\ka(\\ga-1)}{2R}\\|\\na \\te_t\\|_{L^2}^2+H_0\\right)\n+\\int \\n |u_{tt}|^2dx\\right]_t\\\\\n&\\quad + \\int\\n\\te_{tt}^2dx+\\frac{C_4}{2}\\int |\\na u_{tt}|^2dx\\\\\n&\\le C (1+\\|\\na u_{t}\\|_{L^2}^2) (\\si^{-2} +\\|\\na \\te_t\\|^2_{L^2})\n+C\\|\\n^{1\/2}u_{tt}\\|^2_{L^2} + C\\|\\na^2 u_t\\|^2_{L^2}.\\ea\\enn\nMultiplying this by $\\si^2$ and integrating the resulting inequality over $(0,T),$\nwe  obtain after using (\\ref{ex6}),  (\\ref{nq1}),  (\\ref{va5}), and Gr\\\"{o}nwall's inequality that\n\\be \\la{eg10}\n\\sup_{ 0\\le t\\le T}\\si^2\\int \\left(|\\na\\te_t|^2+\\n |u_{tt}|^2\\right)dx\n+\\int_{0}^T\\si^2\\int \\left(\\n\\te_{tt}^2+|\\nabla u_{tt}|^2\\right)dxdt\\le C,\\ee\nwhich together with Lemmas \\ref{le11}, \\ref{pe1}, \\ref{pr3}, (\\ref{nt4}), (\\ref{ex4}),  \\eqref{va1}, (\\ref{a4.74}),\nand (\\ref{4.49}) gives\n\\be\\la{sp20} \\sup_{ 0\\le t\\le T}\\si \\left(\\|\\na u_t\\|_{H^1}\n+ \\|\\na^2\\te\\|_{H^1}+\\|\\na^2u\\|_{W^{1,q}} \\right)\\le C.\\ee\n\nWe thus derive (\\ref{eg17}) from  (\\ref{eg10}),   (\\ref{sp20}), \\eqref{va1}, \\eqref{w3},\nand (\\ref{qq1}). The proof of Lemma \\ref{sq90} is completed.\n\\end{proof}\n\n\\begin{lemma}\\la{sq91} The following estimate holds:\n\t\\be \\la{egg17}\\sup_{ 0\\le t\\le T}\\si^2  \\left(\\|\\na^2\\te\\|_{H^2}+\\| \\te_t\\|_{H^2}+\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}  \\right)\n\t+\\int_0^T\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2 dt\\le C.\\ee\n\\end{lemma}\n\n\\begin{proof}\nFirst, differentiating $(\\ref{eg1})$ with respect to $t$ yields\n\\be\\la{eg2}\\ba \\begin{cases}\n\\n\\te_{ttt}-\\frac{\\ka(\\ga-1)}{R}\\Delta \\te_{tt}\\\\\n=-\\n u\n\\cdot\\na\\te_{tt}+ 2\\div(\\n u)\\te_{tt} - \\n_{tt}\\left(\\te_t+ u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\\\\n\\quad - 2\\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)_t\\\\\n\\quad - \\n\\left(u_{tt}\\cdot\\na \\te+2u_t\\cdot\\na\\te_t+(\\ga-1)(\\te\\div u)_{tt}\\right)\\\\\n\\quad +\\frac{\\ga-1}{R}\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2 \\right)_{tt},  \\\\\n\\na\\te_{tt}\\cdot n|_{\\p\\O\\times(0,T)}=0   .\n\\end{cases}\\ea\\ee\nMultiplying (\\ref{eg2})$_1$ by $\\te_{tt}$ and integrating the resulting equality over $\\Omega$ yield that\n\\be\\la{eg3}\\ba\n&\\frac{1}{2}\\frac{d}{dt}\\int\\n|\\te_{tt}|^2dx +\\frac{\\ka(\\ga-1)}{R}\\int|\\na \\te_{tt}|^2dx\\\\\n&=-4\\int \\te_{tt}\\n u\\cdot\\na\\te_{tt}dx  -\\int \\n_{tt}\\left(\\te_t\n+ u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\te_{tt}dx\\\\\n&\\quad - 2\\int\\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)_t\\te_{tt}dx\\\\\n& \\quad - \\int\\n\\left(u_{tt}\\cdot\\na \\te+2u_t\\cdot\\na\\te_t\n+(\\ga-1)(\\te\\div u)_{tt}\\right)\\te_{tt}dx\\\\\n& \\quad +\\frac{\\ga-1}{R}\\int  \\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_{tt}\\te_{tt}dx\n\\triangleq \\sum_{i=1}^5K_i.\\ea\\ee\n\nIt follows from\nLemmas \\ref{le11}--\\ref{pe1}, \\ref{sq90},  \\eqref{kk}, (\\ref{eg10}), and \\eqref{va1}  that\n \\be \\la{eg4} \\ba\n\\si^4|K_1|&\\le C\\si^4\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}\\|\\na \\te_{tt}\\|_{L^2}\\|u\\|_{L^\\infty}\\\\\n&\\le \\de \\si^4\\|\\na \\te_{tt}\\|_{L^2}^2+C(\\de) \\si^4\\|\\n^{1\/2}\\te_{tt}\\|^2_{L^2} ,\\ea\\ee\n\\be \\la{eg16}\\ba\n\\si^4|K_2|&\\le C \\si^4\\|\\n_{tt}\\|_{L^2}\\|\\te_{tt}\\|_{L^6} \\left( \\|\\te_t\\|_{H^1}\n+\\|\\na\\te\\|_{L^3}+\\|\\na u\\|_{L^6}\\|\\te\\|_{L^6}\\right) \\\\\n&\\le C\\si^2 (\\|\\na \\te_{tt}\\|_{L^2}+\\|\\n^{1\/2} \\te_{tt}\\|_{L^2})\\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2+C(\\de)(\\si^4\\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2+1),\\ea\\ee\n\\be \\la{eg7}\\ba\n\\si^4|K_4|&\\le C\\si^4\\|\\te_{tt}\\|_{L^6}\n\\left( \\|\\na\\te\\|_{L^3}\\|\\n u_{tt}\\|_{L^2}\n+\\|\\na\\te_t\\|_{L^2}\\|u_t\\|_{L^3}\\right)\\\\\n&\\quad+ C\\si^4\\|\\te_{tt}\\|_{L^6} \\left( \\|\\na u\\|_{L^3}\\|\\n \\te_{tt}\\|_{L^2}\n+ \\|\\na u_t\\|_{L^2}\\| \\te_t\\|_{L^3}\\right)\\\\\n&\\quad+C\\si^4\\|\\te\\|_{L^\\infty}\\|\\n\\te_{tt}\\|_{L^2} \\|\\na u_{tt}\\|_{L^2} \\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2\n+C(\\de)\\left(\\si^4 \\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2+\\si^3\\|\\na u_{tt}\\|_{L^2}^2  \\right)+C(\\de),\\ea\\ee\n\\be \\la{eg8}\\ba\n\\si^4|K_5|&\\le C\\si^4\\|\\te_{tt}\\|_{L^6}\n\\left( \\|\\na u_t\\|_{L^2}^{3\/2}\\|\\na u_{t}\\|_{L^6}^{1\/2}+\\|\\na u\\|_{L^3}\\|\\na u_{tt}\\|_{L^2}\\right)  \\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2\n+C(\\de)\\si^4\\left(\\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2+\\|\\na u_{tt}\\|_{L^2}^2  \\right) +C(\\de),\\ea\\ee\nand\n\\be \\la{eg6}\\ba\n\\si^4|K_3|&\\le C \\si^4\\|\\n_t\\|_{L^3} \\|\\te_{tt}\\|_{L^6}\n\\left( \\si^{-1\/2}\\|\\na u_t\\|_{L^2} +\\|\\rho^{1\/2} \\theta_t\\|_{L^2} +\\|\\na\\te_{t}\\|_{L^2} \\right) \\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2+ C \\si^4 \\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2 +C(\\de),\\ea\\ee\nwhere in the last inequality we have used (\\ref{eg12}).\n\nThen, multiplying (\\ref{eg3})  by $\\si^4,$ substituting (\\ref{eg4})--(\\ref{eg6}) into the resulting equality and choosing $\\de$ suitably small, one obtains\n\\bnn \\ba\n& \\frac{d}{dt}\\int\\si^4\\n|\\te_{tt}|^2dx +\\frac{\\ka(\\ga-1)}{R}\\int\\si^4|\\na \\te_{tt}|^2dx\\\\\n& \\le  C\\si^2\\left(\\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2\n+\\|\\na u_{tt}\\|_{L^2}^2  \\right)+C,\\ea\\enn\nwhich together with (\\ref{eg10})   gives\n\\be\\la{eg13} \\sup_{ 0\\le t\\le T}\\si^4\\int  \\n |\\te_{tt}|^2dx\n+\\int_{0}^T\\si^4\\int_{ } |\\nabla \\te_{tt}|^2 dxdt\\le C.\\ee\n\nFinally, applying the standard $L^2$-estimate  to (\\ref{eg1}), one obtains after using Lemmas \\ref{le11}--\\ref{pe1}, \\ref{sq90}, (\\ref{va1}), and (\\ref{eg13}) that\n\\be\\la{eg14}\\ba\n&\\sup_{0\\le t\\le T}\\si^2\\|\\na^2\\te_t\\|_{L^2}\\\\\n&\\le  C\\sup_{0\\le t\\le T}\\si^2\\left(\\|\\n\\te_{tt}\\|_{L^2}\n+  \\|\\n_t\\|_{L^3}\\|\\te_t\\|_{L^6}+\\|\\n_t\\|_{L^6} \\left(\\|\\na\\te\\|_{L^3}+\\|\\te\\|_{L^6}\\|\\na u\\|_{L^6}\\right)\\right)\\\\\n& \\quad +C\\sup_{0\\le t\\le T}\\si^2\\left(\\|\\n^{1\/2} \\te_t\\|_{L^2}+\\|\\na\\te_t\\|_{L^2}+ (1+\\|\\na^2\\te\\|_{L^2}) \\|\\na u_t\\|_{L^2}+\n \\|\\na u_t\\|_{L^6}\\right)\\\\\n& \\le  C.\\ea\\ee\nMoreover, it follows from  the standard $H^2$-estimate  to\n$(\\ref{3.29})$, (\\ref{hs}), \\eqref{nq1}, and Lemma \\ref{le11}   that\n\\bnn\\ba \\|\\na^2\\te\\|_{H^2}\n&\\le C\\left(\\|\\n\\te_t\\|_{H^2}+\\|\\n u\\cdot\\na\\te\\|_{H^2}\n+\\|\\n\\te\\div u\\|_{H^2}+\\||\\na u|^2\\|_{H^2}\\right)\\\\\n&\\le C\\left( \\|\\n\\|_{H^2} \\|\\te_t\\|_{H^2}\n+\\|\\n\\|_{H^2} \\| u\\|_{H^2}\\|\\na\\te\\|_{H^2}\\right)\\\\\n&\\quad+C\\|\\n\\|_{H^2} \\|\\te\\|_{H^2} \\| \\div u\\|_{H^2}+C\\|\\na u\\|^2_{H^2}+C\\\\\n&\\le C\\si^{-1}+ C\\| \\na^3\\te \\|_{L^2}+C\\| \\te_t\\|_{H^2}.\\ea\\enn\nCombining this with  (\\ref{eg17}),  (\\ref{eg14}), and  (\\ref{eg13}) shows (\\ref{egg17}).\nThe proof of Lemma \\ref{sq91} is completed.\n\\end{proof}\n\n\n\n\n\n\n\\section{\\la{se5}Proof of  Theorems  \\ref{th1} and \\ref{th2}}\n\nWith all the a priori estimates in Sections \\ref{se3} and \\ref{se4}\nat hand, we are ready to prove the main results  of this paper in\nthis section.\n\n\n\n\\begin{pro} \\la{pro2}\n\n For  given numbers $M>0$ (not necessarily small),\n  $\\on> 2,$ and $\\bt>1,$   assume that  $(\\rho_0,u_0,\\te_0)$ satisfies (\\ref{2.1}),  (\\ref{3.1}),\nand   (\\ref{z01}). Then    there exists a unique classical solution  $(\\rho,u,\\te) $      of problem (\\ref{a1})--(\\ref{h1})\n in $\\Omega\\times (0,\\infty)$ satisfying (\\ref{mn5})--(\\ref{mn2}) with $T_0$ replaced by any $T\\in (0,\\infty).$\n  Moreover,  (\\ref{zs2}), (\\ref{a2.112}), (\\ref{ae3.7}), and (\\ref{vu15})  hold for any $T\\in (0,\\infty)$ and (\\ref{h22}) holds for any $t\\geq 1$.\n\n\n\n \n\n \\end{pro}\n\n\n\n\n\n\n\\begin{proof}\nFirst, by the standard local existence result (Lemma \\ref{th0}), there exists a $T_0>0$ which may depend on\n$\\inf\\limits_{x\\in \\Omega}\\n_0(x), $  such that the  problem\n (\\ref{a1})--(\\ref{h1})  with   initial data $(\\n_0 ,u_0,\\te_0 )$\n has   a unique classical solution $(\\n,u,\\te)$ on $\\O\\times(0,T_0]$  satisfyinng (\\ref{mn6})--(\\ref{mn2}).\nIt follows from (\\ref{As1})--(\\ref{3.1}) and (\\ref{z01}) that\n\\bnn A_1(0)\\le M^2,\\quad  A_2(0)\\le  C_0^{1\/4},\\quad A_3(0)=0, \\quad  \\n_0<\n \\hat{\\rho},\\quad \\te_0\\le \\bt,\\enn  which implies  there exists a\n$T_1\\in(0,T_0]$ such that (\\ref{z1}) holds for $T=T_1.$\n We set \\bnn \\notag T^* =\\sup\\left\\{T\\,\\left|\\, \\sup_{t\\in [0,T]}\\|(\\n,u,\\te)\\|_{H^3}<\\infty\\right\\},\\right.\\enn  and \\be \\la{s1}T_*=\\sup\\{T\\le T^* \\,|\\,{\\rm (\\ref{z1}) \\\nholds}\\}.\\ee Then $ T^*\\ge T_* \\geq T_1>0.$\n Next, we claim that\n \\be \\la{s2}  T_*=\\infty.\\ee  Otherwise,    $T_*<\\infty.$\nProposition \\ref{pr1} shows   (\\ref{zs2}) holds for all $0<T<T_*,$ which together with \\eqref{z01} yields\n  Lemmas \\ref{le11}--\\ref{sq91} still hold for all  $0< T< T_*.$\nNote here that all  constants $C$  in  Lemmas \\ref{le11}--\\ref{sq91}\ndepend  on $T_*  $ and $\\inf\\limits_{x\\in \\Omega}\\n_0(x)$, and are in fact independent  of  $T$.\nThen,  we claim that  there\nexists a positive constant $\\tilde{C}$ which may  depend  on $T_* $\nand $\\inf\\limits_{x\\in \\Omega}\\n_0(x)$   such that, for all  $0< T<\n T_*,$  \\be\\la{y12}\\ba \\sup_{0\\le t\\le T}\n\\| \\n\\|_{H^3}   \\le \\tilde{C},\\ea \\ee which together with Lemmas \\ref{le11}, \\ref{pe1},  \\ref{sq90},  \\eqref{mn2}, and (\\ref{3.1}) gives\n \\bnn\n \\|(\\n(x,T_*),u(x,T_*),\\te(x,T_*))\\|_{H^3}\n \\le \\tilde{C},\\quad\\inf_{x\\in \\Omega}\\n(x,T_*)>0,\\quad\\inf_{x\\in \\Omega}\\te(x,T_*)>0.\\enn\n  Thus, Lemma \\ref{th0} implies that there exists some $T^{**}>T_*,$  such that\n(\\ref{z1}) holds for $T=T^{**},$   which contradicts (\\ref{s1}).\nHence, (\\ref{s2}) holds. This along with Lemmas \\ref{th0}, \\ref{a13.1}, \\ref{le8},  and Proposition \\ref{pr1},  thus\nfinishes the proof of   Proposition \\ref{pro2}.\n\n\n Finally, it remains to prove (\\ref{y12}). Using  $(\\ref{a1})_3$ and (\\ref{mn6}), we can define\n \\bnn\n \\theta_t(\\cdot,0)\\triangleq - u_0 \\cdot\\na \\te_0 + \\frac{\\ga-1}{R} \\rho_0^{-1}\n\\left(\\ka\\Delta\\te_0-R\\rho_0 \\theta_0 \\div u_0+\\lambda (\\div u_0)^2+2\\mu |\\mathfrak{D}(u_0)|^2\\right),\n \\enn\n which along with  (\\ref{2.1}) gives\n \\be \\la{ssp91}\\|\\theta_t(\\cdot,0)\\|_{L^2}\\le \\tilde{C}.\\ee\n Thus, one deduces from  (\\ref{3.99}), \\eqref{2.1}, \\eqref{ssp91}, and Lemma \\ref{le11} that\n\\be  \\ba \\la{a51}\n\\sup\\limits_{0\\le t\\le T} \\int \\n|\\dot\\te|^2dx+\\int_0^T \\|\\na\\dot\\te\\|_{L^2}^2dt \\le \\tilde{C},\n\\ea\\ee\nwhich together with  (\\ref{lop4}) and Lemma \\ref{le11}  yields\n\\be\\la{sp211}\n\\sup\\limits_{0\\le t\\le T}\\|\\na^2 \\theta\\|_{L^2} \\le \\tilde{C}.\\ee\nUsing  $(\\ref{a1})_2$ and  (\\ref{mn6}), we can define\n\\bnn\nu_t(\\cdot,0) \\triangleq -u_0\\cdot\\na u_0+\\n_0^{-1}\\left( \\mu \\Delta u_0 + (\\mu+\\lambda) \\na \\div u_0 - R\\na (\\n_0\\te_0)\\right),\n\\enn\nwhich along with  (\\ref{2.1}) gives\n\\be \\la{ssp9}\\|\\na u_t(\\cdot,0)\\|_{L^2}\\le \\tilde{C}.\\ee\nThus, it follows from Lemmas \\ref{le11}, \\ref{le9-1},  (\\ref{4.052}),  (\\ref{s4}), (\\ref{a51})--(\\ref{ssp9}), and Gr\\\"{o}nwall's inequality that\n\\be \\la{ssp1}\n\\sup_{0\\le t\\le T}\\|\\na u_t\\|_{L^2}+\\int_0^T\\int \\n |u_{tt}|^2dxdt\\le \\tilde{C},\n\\ee\nwhich as well as (\\ref{va2}), \\eqref{sp211}, and (\\ref{qq1}) yields\n\\be\\la{sp221} \\sup\\limits_{0\\le t\\le T}\\|u\\|_{H^3} \\le \\tilde{C}.\\ee\nCombining this with Lemma \\ref{le11}, \\eqref{ex4}, \\eqref{nt4}, (\\ref{a51}), (\\ref{sp211}), (\\ref{ssp1}),   and  (\\ref{sp221}) gives\n\\be\\la{ssp24} \\ia\\left(\\|\\na^3\\te\\|_{L^2}^2+ \\|\\nabla u_t\\|_{H^1}^2\\right)dt\\le \\tilde{C} . \\ee\nThen, applying (\\ref{rmk1}), \\eqref{hs}, \\eqref{sp211}, \\eqref{ssp1}, \\eqref{sp221}, and Lemma \\ref{le11}, one has\n\\be \\notag\\ba \\|\\na^2 u\\|_{H^2}\n&\\le\\tilde{C}\\left(\\| \\n \\dot u \\|_{H^2}+\\|\\na  P\\|_{H^2} + \\|\\na u\\|_{L^2}\\right)\\\\\n&\\le \\tilde{C}\\left(\\| \\n \\|_{H^2}\\|  u_t \\|_{H^2}+\\| \\n \\|_{H^2}\\|  u \\|_{H^2}\\|  \\na u \\|_{H^2}\\right)\\\\\n&\\quad+\\tilde C\\left(\\|\\na\\n \\|_{H^2}\\|\\te\\|_{H^2}+\\|\\n\\|_{H^2}\\|\\na  \\te\\|_{H^2}+1\\right)\\\\\n& \\le \\tilde{C} (1+ \\|\\na^2  u_t\\|_{L^2}+\\|\\na^3 \\n \\|_{L^2}+\\|\\na^3 \\te \\|_{L^2}),\\ea\\ee\n\nwhich along with some standard calculations leads to\n \\bnn\\la{sp134}\\ba  & \\left(\\|\\na^3 \\n\\|_{L^2} \\right)_t \\\\\n&\\le \\tilde{C}\\left(\\| |\\na^3u| |\\na \\n| \\|_{L^2}+ \\||\\na^2u||\\na^2\n      \\n|\\|_{L^2}+ \\||\\na u||\\na^3 \\n|\\|_{L^2} +\\| \\na^4u \\|_{L^2} \\right)\\\\\n&\\le \\tilde{C}\\left(\\| \\na^3 u\\|_{L^2}\\|\\na \\n \\|_{H^2}+ \\| \\na^2u\\|_{L^3}\\|\\na^2 \\n \\|_{L^6}\n       +\\|\\na u\\|_{L^\\infty}\\|\\na^3 \\n\\|_{L^2} +\\| \\na^4u \\|_{L^2}\\right) \\\\\n&\\le \\tilde{C}(1+ \\| \\na^3\\n \\|_{L^2}+ \\| \\na^2 u_t\\|^2_{L^2}+ \\|\\na^3\\te\\|^2_{L^2}), \\ea\\enn where we have used (\\ref{sp221}) and Lemma \\ref{le11}.\n Combining this with (\\ref{ssp24})  and\nGr\\\"{o}nwall's inequality  yields   \\bnn\\la{sp26} \\sup\\limits_{0\\le t\\le\nT}\\|\\nabla^3  \\n\\|_{L^2} \\le \\tilde{C},\\enn which together with\n(\\ref{qq1}) gives (\\ref{y12}).\nThe proof of Proposition \\ref{pro2} is completed.\n\\end{proof}\n\nWith  Proposition \\ref{pro2} at hand, we are now in a position to prove  Theorem \\ref{th1}.\n\n\n\\begin{proof}[Proof of  Theorem   \\ref{th1}]\n Let $(\\n_0,u_0,\\te_0)$  satisfying (\\ref{co3})--(\\ref{co2}) be the initial data in Theorem \\ref{th1}.  Assume that  $C_0$  satisfies (\\ref{co14}) with\n\\be\\la{xia}\\ve\\triangleq \\ve_0\/2,\\ee\nwhere  $\\ve_0$  is given in Proposition \\ref{pr1}.\n\nFirst, we construct the approximate initial data $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$ as follows. For constants\n\\be\\la{5d0}\nm \\in \\mathbb{Z}^+,\\ \\  \\eta \\in \\left(0, \\eta_0 \\right),\\ \\  \\eta_0\\triangleq \\min\\xl\\{1,\\frac{1}{2}(\\on-\\sup\\limits_{x\\in \\O}\\n_0(x)) \\xr\\},\n\\ee\nwe define\n\\begin{align*}\n\\n_0^{m,\\eta} = \\n_0^{m}+ \\eta,\\ \\  u_0^{m,\\eta}=\\frac{u_0^m }{1+\\eta},\\ \\  \\te_0^{m,\\eta}= \\frac{\\te_0^{m} + \\eta}{1+2\\eta},\n\\end{align*}\nwhere $\\n_0^{m}$ satisfies\n\\begin{align*}\n0 \\le \\n_0^{m} \\in C^{\\infty},\\ \\  \\lim_{m \\to \\infty} \\|\\n_0^{m} -\\rho_0\\|_{W^{2,q}}=0,\n\\end{align*}\n $u_0^m$ is the unique smooth solution to the following elliptic equation:\n\\be\\notag\\begin{cases}\n\t\\Delta u_0^m=\\Delta \\tilde{u}_0^m,&\\text{in}\\,\\, \\O,\\\\\n\tu_0^m\\cdot n=0 ,\\,\\,\\curl u_0^m\\times n=0,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nwith $\\tilde{u}_0^m \\in C^{\\infty}$ satisfying $\\lim_{m \\to \\infty}\\| \\tilde{u}_0^m -{u}_0\\|_{H^2}=0$, and $\\te_0^m$ satisfying $\\int_\\O\\te_0^m dx=\\int_\\O\\te_0 dx$ is the unique smooth solution to the following Poisson equation:\n\\be\\notag\\begin{cases}\n\t\\Delta \\te_0^m=\\Delta \\tilde{\\te}_0^m- \\overline{\\Delta \\tilde{\\te}_0^m},&\\text{in}\\,\\,\\O,\\\\\n\t\\na \\te_0^m\\cdot n=0 ,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nwith $0\\le \\tilde{\\te}_0^m \\in C^{\\infty}$ satisfying $\\lim\\limits_{m \\to \\infty}\\| \\tilde{\\te}_0^m -{\\te}_0\\|_{H^2}=0$.\n\nThen for any $\\eta\\in (0, \\eta_0)$, there exists $m_1(\\eta)\\ge 1$ such that for $m \\ge m_1(\\eta)$, the approximate initial data\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$ satisfies\n\\be \\la{de3}\\begin{cases}(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})\\in C^\\infty ,\\\\\n\t\\dis \\eta\\le  \\n_0^{m,\\eta}  <\\hat\\n,~~\\, \\frac{\\eta}{4}\\le \\te_0^{m,\\eta} \\le \\hat \\te,~~\\,\\|\\na u_0^{m,\\eta}\\|_{L^2} \\le M, \\\\\n\t\\dis u_0^{m,\\eta}\\cdot n=0,~~\\,\\curl u_0^{m,\\eta}\\times n=0,~~\\,\\na \\te_0^{m,\\eta}\\cdot n=0\\,\\, \\text{on}\\,\\p\\O,\\end{cases}\n\\ee\nand\n \\be \\la{de03}\n\\lim\\limits_{\\eta\\rightarrow 0} \\lim\\limits_{m\\rightarrow \\infty}\n\\left(\\| \\n_0^{m,\\eta} - \\n_0 \\|_ {W^{2,q}}+\\| u_0^{m,\\eta}-u_0\\|_{H^2}+\\| \\te_0^{m,\\eta}- \\te_0  \\|_{H^2}\\right)=0.\n\\ee\nMoreover,  the initial norm $C_0^{m,\\eta}$\nfor $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$ which is defined by  the right-hand side of (\\ref{e})\nwith $(\\n_0,u_0,\\te_0)$   replaced by\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$\nsatisfies \\bnn \\lim\\limits_{\\eta\\rightarrow 0} \\lim\\limits_{m\\rightarrow \\infty} C_0^{m,\\eta}=C_0.\\enn\nTherefore, there exists  an  $\\eta_1\\in(0, \\eta_0) $\nsuch that, for any $\\eta\\in(0,\\eta_1),$ we can find some $m_2(\\eta)\\geq m_1(\\eta)$  such that   \\be \\la{de1} C_0^{m,\\eta}\\le C_0+\\ve_0\/2\\le  \\ve_0 , \\ee\nprovided that\\be  \\la{de7}0<\\eta<\\eta_1 ,\\,\\, m\\geq m_2(\\eta).\\ee\n\n\n\n \\iffalse\n First, we choose  $0\\le\\n^m_0\\in C^\\infty$ satisfying\n\\be\\la{5d1}\\n^m_0\\rightarrow\\n_0\\quad \\text{in} \\,\\,W^{2,q}\\quad \\text{as}\\,\\,m\\rightarrow \\infty.\\ee\n Combining this with Sobolev's inequality yields that there exists some  $m_0(\\eta)\\geq1$ such that for any $m\\geq m_0(\\eta)$,\n\\be\\la{5d2} \\|\\n_0^m-\\n_0\\|_{L^\\infty}\\le \\min\\xl\\{\\frac{1}{4}(\\hat\\n-\\sup\\limits_{x\\in \\O}\\n_0(x)),\\eta\\xr\\}\\ee\nwith fixed $\\eta \\in (0, \\eta_0)$. We set $\\n_0^{m,\\eta}=\\n_0^m+\\eta$ .\n\nNext, for the   initial velocity $u_0$,\nthere exists a sequence  $v_0^m\\in C^\\infty$ such that $v_0^m\\rightarrow u_0$ in $ H^2$ as $m\\rightarrow \\infty$.\nFor fixed $v_0^m$, the Lame's system\n\\be\\notag\\begin{cases}\n-\\mu\\Delta w_0^m-(\\mu+\\lambda)\\na \\div w_0^m=0,&\\text{in}\\,\\, \\O,\\\\\nw_0^n\\cdot n=v_0^m\\cdot n,\\,\\,\\curl w_0^m\\times n=\\curl v_0^m\\times n,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nadmits a unique smooth solution $w_0^m$. Then, it follows from the standard $H^2$-estimate to the above elliptic system and Sobolev's inequality that\n $w_0^m\\rightarrow 0$ in $H^2$ as $m\\rightarrow \\infty$.\nDenote $u_0^m=v_0^m-w_0^m$, it holds that $u_0^m\\in C^\\infty$ satisfy\n\\be\\la{5d3} u_0^m\\rightarrow u_0\\,\\,\\text{in}\\,\\,H^2~\\mbox{as}~m\\rightarrow \\infty,~~~\\,\\,u_0^m\\cdot n=\\curl u_0^m\\times n=0\\,\\,\\text{on}\\,\\,\\p\\O.\\ee\nHence, for any fixed  $\\eta \\in (0, \\eta_0),$ there exists $m_1(\\eta)\\geq m_0(\\eta)$\nsuch that for any $m\\geq m_1(\\eta)$\n\\be\\la{5d4}\\|\\na u_0^m\\|_{L^2}\\le(1+\\eta)\\|\\na u_0\\|_{L^2}.\\ee\n We set $u_0^{m,\\eta}= \\frac{1}{1+\\eta}u_0^m$.\n\n Similarly, we will construct  $\\te_0^{m,\\eta}$.\nIt's easy to find a sequence  $0\\le f_0^m\\in C^\\infty$ such that $f_0^m\\rightarrow \\te_0$ in $ H^2$ as $m\\rightarrow \\infty$.\n For any fixed $f_0^m$, considering the unique  smooth solution $h_0^m$ to the Poisson's equation\n\\be\\notag\\begin{cases}\n\\Delta h_0^m=\\overline{\\Delta f_0^m},&\\text{in}\\,\\,\\O,\\\\\n\\na h_0^m\\cdot n=\\na f_0^m\\cdot n,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nwith $\\overline{h_0^m}=0,$  it is easy to deduce that   $h_0^m\\rightarrow 0$ in $H^2$ as $m\\rightarrow \\infty$.\nDenote $\\te_0^{m}= f_0^m-h_0^m$, it holds that $\\te_0^m\\in C^\\infty$  satisfy\n\\be\\la{5d5} \\te_0^m\\rightarrow \\te_0\\,\\,\\text{in}\\,\\,H^2~\\mbox{as}~m\\rightarrow \\infty,~~~\\,\\,\\na \\te_0^m\\cdot n=0\\,\\,\\rm{on}\\,\\,\\p\\O.\\ee\nHence, for any  $\\eta \\in (0, \\eta_0),$ there exists $m_2(\\eta)\\geq m_1(\\eta)$\nsuch that for any $m\\geq m_2(\\eta),$\n\\be\\la{5d6}\\|\\te_0^m-\\te_0\\|_{L^\\infty}\\le\\frac{\\eta}{4}.\\ee\nWe finally set $\\te_0^{m,\\eta}= \\frac{1}{1+\\eta}(\\te_0^m+\\frac{\\eta}{2})$.\n\nTherefore, one deduces from \\eqref{co4} and \\eqref{5d0}-\\eqref{5d6} that\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$ satisfies\n\\be \\la{de3}\\begin{cases}(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})\\in C^\\infty ,\\\\\n\\dis \\eta\\le  \\n_0^{m,\\eta}  <\\hat\\n,~~\\, \\frac{\\eta}{8}\\le \\te_0^{m,\\eta} \\le \\hat \\te,~~\\,\\|\\na u_0^{m,\\eta}\\|_{L^2} \\le M, \\\\\n\\dis u_0^{m,\\eta}\\cdot n  =\\curl u_0^{m,\\eta}\\times n=\\na \\te_0^{m,\\eta}\\cdot n=0\\,\\, \\text{on}\\,\\p\\O,\\end{cases}\n\\ee\nprovided that $\\eta \\in (0, \\eta_0)$, $m\\geq m_2(\\eta)$, and \\be \\la{de03}\n\\lim\\limits_{m\\rightarrow \\infty} \\lim\\limits_{\\eta\\rightarrow 0}\n\\left(\\| \\n_0^{m,\\eta} - \\n_0 \\|_ {W^{2,q}}+\\| u_0^{m,\\eta}-u_0\\|_{H^2}+\\| \\te_0^{m,\\eta}- \\te_0  \\|_{H^2}\\right)=0.\n\\ee\n Moreover,  the initial norm $C_0^{m,\\eta}$\nfor $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$   the right hand side of (\\ref{e})\nwith $(\\n_0,u_0,\\te_0)$   replaced by\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$\nsatisfy \\bnn \\lim\\limits_{m\\rightarrow \\infty} \\lim\\limits_{\\eta\\rightarrow 0}C_0^{m,\\eta}=C_0.\\enn\nTherefore, there exists  an  $\\eta_1\\in(0, \\eta_0) $\nsuch that, for any $\\eta\\in(0,\\eta_1),$ we can find some $m_3(\\eta)\\geq m_2(\\eta)$  such that   \\be \\la{de1} C_0^{m,\\eta}\\le C_0+\\ve_0\/3\\le  \\ve_0 , \\ee\nprovided that\\be  \\la{de7}0<\\eta<\\eta_1 ,\\,\\, m\\geq m_3(\\eta).\\ee\n\\fi\n\n  We assume that $m,\\eta$ satisfy (\\ref{de7}).\n  Proposition \\ref{pro2} together with (\\ref{de1}) and (\\ref{de3}) thus yields that\n   there exists a smooth solution  $(\\n^{m,\\eta},u^{m,\\eta}, \\te^{m,\\eta}) $\n   of problem (\\ref{a1})--(\\ref{h1}) with  initial data $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$\n   on $\\Omega\\times (0,T] $ for all $T>0. $\n    Moreover, one has (\\ref{h8}), \\eqref{zs2}, \\eqref{h22}, (\\ref{a2.112}), \\eqref{ae3.7}, and (\\ref{vu15})   with $(\\n,u,\\te)$  being replaced by $(\\n^{m,\\eta},u^{m,\\eta}, \\te^{m,\\eta}).$\n\n\n Next, for the initial data $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$, the function $\\tilde g$ in (\\ref{co12})  is\n \\be \\la{co5}\\ba \\tilde g & \\triangleq(\\n_0^{m,\\eta})^{-1\/2}\\left(-\\mu \\Delta u_0^{m,\\eta}-(\\mu+\\lambda)\\na\\div\n u_0^{m,\\eta}+R\\na (\\n_0^{m,\\eta}\\te^{m,\\eta}_0)\\right)\\\\\n& = (\\n_0^{m,\\eta})^{-1\/2}\\sqrt{\\n_0}g+\\mu(\\n_0^{m,\\eta})^{-1\/2}\\Delta(u_0-u_0^{m,\\eta})\\\\\n&\\quad+(\\mu+\\lambda) (\\n_0^{m,\\eta})^{-1\/2} \\na \\div(u_0-u_0^{m,\\eta})+ R(\\n_0^{m,\\eta})^{-1\/2} \\na(\\n_0^{m,\\eta}\\te_0^{m,\\eta}-\\n_0\\te_0),\\ea\\ee\nwhere in the second equality we have used (\\ref{co2}).\nSince $g \\in L^2,$ one deduces from (\\ref{co5}),  (\\ref{de3}), (\\ref{de03}), and  (\\ref{co3})  that for any $\\eta\\in(0,\\eta_1),$ there exist some $m_3(\\eta)\\geq m_2(\\eta)$ and a positive constant $C$ independent of $m$ and $\\eta$ such that\n \\be\\la{de4}\n \t\\|\\tilde g\\|_{L^2}\\le \\|g\\|_{L^2}+C\\eta^{-1\/2}\\de(m) + C\\eta^{1\/2},\n \t\\ee\nwith   $0\\le\\de(m) \\rightarrow 0$ as\n$m \\rightarrow \\infty.$ Hence,  for any  $\\eta\\in(0,\\eta_1),$ there exists some $m_4(\\eta)\\geq m_3(\\eta)$ such that for any $ m\\geq m_4(\\eta)$,\n \\be \\la{de9}\\de(m) <\\eta.\\ee  We thus obtain from (\\ref{de4}) and (\\ref{de9}) that\nthere exists some positive constant $C$ independent of $m$ and $\\eta$ such that  \\be\\la{de14}  \\|\\tilde g \\|_{L^2}\\le \\|g \\|_{L^2}+C,\\ee provided that\\be \\la{de10} 0<\\eta<\\eta_1,\\,\\,  m\\geq m_4(\\eta).\\ee\n\n\n Now, we   assume that $m,$  $\\eta$ satisfy (\\ref{de10}).\n It thus follows from (\\ref{de3})--(\\ref{de1}),  (\\ref{de14}), Proposition \\ref{pr1},\n and Lemmas \\ref{le8}, \\ref{le11}--\\ref{sq91} that for any $T>0,$\n there exists some positive constant $C$ independent of $m$ and $\\eta$ such that\n (\\ref{h8}), (\\ref{zs2}),   (\\ref{a2.112}),  \\eqref{ae3.7}, (\\ref{vu15}), \\eqref{lee2}, \\eqref{qq1},  (\\ref{va5}),  (\\ref{vva5}), (\\ref{y2}),  (\\ref{eg17}),\n  and  (\\ref{egg17})  hold for  $(\\n^{m,\\eta},u^{m,\\eta}, \\te^{m,\\eta}) .$\n   Then passing  to the limit first $m\\rightarrow \\infty,$ then $\\eta\\rightarrow 0,$\n   together with standard arguments yields that there exists a solution $(\\n,u,\\te)$ of the problem (\\ref{a1})--(\\ref{h1})\n   on $\\Omega\\times (0,T]$ for all $T>0$, such that the solution  $(\\n,u,\\te)$\n   satisfies  (\\ref{h8}), (\\ref{a2.112}),    \\eqref{ae3.7},  (\\ref{vu15}),  \\eqref{lee2}, \\eqref{qq1}, (\\ref{va5}),  (\\ref{vva5}), (\\ref{y2}),  (\\ref{eg17}), (\\ref{egg17}),\n   and  the estimates of $A_i(T)\\,(i=1,2,3)$ in\n   (\\ref{zs2}). Hence,    $(\\n,u,\\te)$ satisfying  (\\ref{h8}) and \\eqref{h9} refers to \\cite{H-L} for the detailed proof. Moreover, one deduces from Proposition \\ref{pr1} that the desired exponential decay property \\eqref{h11}.\n\n\n\\iffalse\n   $(\\ref{h9})_2,$ $(\\ref{h9})_3 ,$  and for any $0<\\tau<T$,\n   \\be  \\la{sq1}\\rho \\in L^\\infty(0,T; W^{2,q}),\\ \\  u   \\in L^\\infty(0,T;H^2),\\ \\  \\te \\in L^\\infty(0,T;H^1)\\cap L^\\infty(\\tau,T;D^2).  \\ee\n  \nNext,\nwe will prove  \\be \\la{sq2}\\ba \\rho \\in C([0,T]; W^{2,q}),\n\\ u \\in C([0,T];H^1)\\cap C((0,T];D^2), \\ \\te   \\in  C((0,T];H^2).  \\ea\\ee\nIndeed, it follows from   (\\ref{va5}) and (\\ref{sq1})   that\n  \\be  \\la{sq3}\\rho \\in C([0,T];W^{1,\\infty})\\cap C([0,T]; W^{2,q} \\mbox{ -weak}),\\ee and that for all $r\\in [2,6),$ \\be  \\la{sq5}    u \\in C([0,T]; W^{1,r}),\\quad  \\te \\in  C((0,T]; W^{1,r}).\\ee\n\n Since (\\ref{4.52}) holds in $\\mathcal{D}'(\\Omega\\times (0,T))$ for all $T\\in (0,\\infty),$\n one derives from  \\cite[Lemma 2.3]{L2} that for $j_\\upsilon(x)$ being the standard  mollifying kernel of width $\\upsilon\\le\\frac{1}{2}\\rm{dist}\\{\\O',\\p\\O\\}$ with $\\O' \\subset \\subset \\O$,\n$\\n^\\upsilon\\triangleq \\n*j_\\upsilon$ satisfies\n\\be\\la{4.63}\\ba\n(\\p_{ij}\\n^\\upsilon)_t+\\div (\\p_{ij}\\n^\\upsilon u)&=R_\\upsilon-\\div (\\n\\p_{ij} u)* j_\\upsilon-\\div(\\pa_i\\n\\pa_j u\n+\\p_j\\n\\p_i u)* j_\\upsilon, \\ea\\ee\nin $\\O'\\times(0,T)$, where $R_\\upsilon$ satisfies\\be \\la{5.32} \\int_0^T\\|R_\\upsilon\\|_{L^q(\\O')}^{3\/2}dt\\le C\\int_0^T\\|  u \\|^{3\/2}_{W^{1,\\infty}}\\|\\na^2 \\n\\|_{ L^q}^{3\/2}dt \\le C,\\ee\ndue to (\\ref{qq1}) and  (\\ref{y2}).\nLet $\\varphi\\in C_0^\\infty(\\O)$ satisfying $0\\le\\varphi\\le1$ and $\\text{supp} \\varphi\\subset\\O'$. Multiplying (\\ref{4.63}) by $q|\\na^2 \\n^\\upsilon|^{q-2}\\p_{ij} \\n^\\upsilon\\varphi,$\nwe obtain after using integration by parts that\n\\bnn\\ba &\\left(\\int|\\na^2\\n^\\upsilon|^q\\varphi dx\\right)'(t)\\\\\n&=q\\int R_\\upsilon|\\na^2 \\n^\\upsilon|^{q-2}\\p_{ij} \\n^\\upsilon\\varphi dx+\\int  |\\na^2\\n^\\upsilon|^q u\\cdot\\na \\varphi dx\\\\\n&\\quad-(q-1)\\int  |\\na^2\\n^\\upsilon|^q \\div u \\varphi dx-q\\int (\\div (\\n\\p_{ij} u)* j_\\upsilon)|\\na^2 \\n^\\upsilon|^{q-2} \\p_{ij} \\n^\\upsilon\\varphi dx\n\\\\&\n\\quad-q\\int( \\div(\\pa_i\\n\\pa_j u+\\p_j\\n\\p_i u)* j_\\upsilon)|\\na^2 \\n^\\upsilon|^{q-2} \\p_{ij} \\n^\\upsilon\\varphi dx ,   \\ea\\enn\nwhich together with (\\ref{y2}), (\\ref{sq1}),  and  (\\ref{5.32}) yields that, for $p_0$ as in (\\ref{pppppp}),\n\\bnn\\ba&\\sup_{t\\in [0,T]} \\left(\\int|\\na^2\\n^\\upsilon|^q\\varphi dx\\right)(t)\n+\\int_0^T\\left|\\left(\\int|\\na^2\\n^\\upsilon|^q\\varphi dx\\right)'(t) \\right|^{p_0}dt\\\\\n& \\le  C+C\\int_0^T\\left(\\|\\na u\\|_{W^{2,q}}^{p_0}+\\|R_\\upsilon\\|^{p_0}_{ L^q}\\right)dt \\le C.\\ea\\enn\nThis   combined with the Arzel\\`a-Ascoli theorem thus leads to\n\\bnn \\left(\\int|\\na^2\\n^\\upsilon|^q\\varphi dx\\right)(t)\\rightarrow \\left(\\int|\\na^2\\n |^q\\varphi dx\\right)(t)\n\\mbox{ in }C([0,T]),\\mbox{ as }\\upsilon\\rightarrow 0^+,\\enn\nwhich combined with \\cite[Lemma 2.11]{feireisl1} gives that\n \\be \\la{sq4} \\na^2\\rho \\in C([0,T]; L^q). \\ee\n\nTo prove the second part of \\eqref{sq2},\nit follows from  (\\ref{va5})  and   (\\ref{vva5})     that\n  \\be \\la{sq6} \\n u_t,\\n\\te_t \\in\nC((0,T];L^2),\\ee which together with  (\\ref{lames}), (\\ref{sq3}), (\\ref{sq5}),  and\n(\\ref{sq4}) gives\n\\be\\la{sqq1} u\\in C([0,T];H^1)\\cap C((0,T];D^2).\\ee\nCombining this with (\\ref{3.29}), (\\ref{sq6}), (\\ref{sq4}), (\\ref{sq5}), and\n(\\ref{sq3}) leads to\n\\bnn \\te \\in  C((0,T];H^2),\\enn\nwhich as well as    (\\ref{sq3}),  (\\ref{sq4}),   and (\\ref{sqq1})\ngives  (\\ref{sq2}).\n\\fi\n\n\nFinally,  the proof of the uniqueness of $(\\n,u,\\te)$ is similar to that  of \\cite[Theorem 1]{choe1} and will be omitted here for simplicity. The proof of Theorem \\ref{th1} is completed.\n\\end{proof}\n\n\n{\\it Proof of  Theorem   \\ref{th2}. } We will prove  Theorem   \\ref{th2} in two steps.\n\n {\\it Step 1. Construction  of approximate  solutions.} Assume $(\\n_0,u_0,\\te_0)$ satisfying (\\ref{co4}) and \\eqref{dt7} is the initial data in Theorem \\ref{th2} and  $C_0$ satisfies (\\ref{co14})   with  $\\ve $  as in  (\\ref{xia}).   For $j_{m^{-1}}(x)$ being the standard mollifying kernel of width $m^{-1}$, we construct\n\\begin{align*}\n\\hat\\n_0^{m,\\eta} = (\\n_0 1_\\O)\\ast j_{m^{-1}}1_\\O+ \\eta,\\ \\  \\hat u_0^{m,\\eta}=\\frac{u_0^m }{1+\\eta},\\ \\  \\hat\\te_0^{m,\\eta}= \\frac{(\\n_0\\te_0 1_{\\O_m})\\ast j_{m^{-1}} + \\eta}{(\\n_01_{\\O_m})\\ast j_{m^{-1}}+ \\eta},\n\\end{align*}\nwhere  $\\O_m=\\{x\\in\\O| dist(x,\\p\\O)>2\/m\\}$ and $u_0^{m}$ satisfies\n\\be\\notag u_0^m\\in C^\\infty\\cap H^1_\\omega\\ \\ \\text{and}\\ \\ \\lim_{m\\rightarrow \\infty}\\|u_0^{m}-u_0\\|_{H^1}=0.\\ee\n Then for any $\\eta\\in (0, \\eta_0)$ with $\\eta_0$  as in (\\ref{5d0}), there exists $m(\\eta)>1$ such that for $m \\ge m(\\eta)$, the approximate initial data\n$(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta}, \\hat\\te_0^{m,\\eta})$ satisfies\n\\be \\la{dee3}\\begin{cases}(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta}, \\hat\\te_0^{m,\\eta})\\in C^\\infty ,\\\\\n\t\\dis \\eta\\le \\hat \\n_0^{m,\\eta}  <\\hat\\n,~~\\, \\frac{\\eta}{\\hat\\n+\\eta}\\le \\hat\\te_0^{m,\\eta} \\le \\hat \\te,~~\\,\\|\\na \\hat u_0^{m,\\eta}\\|_{L^2} \\le M, \\\\\n\\dis \\hat u_0^{m,\\eta}\\cdot n=0,~~\\,\\curl \\hat u_0^{m,\\eta}\\times n=0,~~\\,\\na \\hat \\te_0^{m,\\eta}\\cdot n=0,\\,\\,\\,\\, \\text{on}\\,\\p\\O,\\end{cases}\n\\ee\nand for any $p\\geq1$,\n \\be \\la{dee03}\n\\lim\\limits_{\\eta\\rightarrow 0} \\lim\\limits_{m\\rightarrow \\infty}\n\\left(\\| \\hat\\n_0^{m,\\eta} - \\n_0 \\|_ {L^p}+\\|\\hat u_0^{m,\\eta}-u_0\\|_{H^1}+\\|\\hat\\n_0^{m,\\eta}\\hat\\te_0^{m,\\eta}-\\n_0\\te_0  \\|_{L^2}\\right)=0\n\\ee\nowing to (\\ref{co4})  and (\\ref{co14}).\n\nNow, we claim that  the initial norm $\\hat C_0^{m,\\eta}$\nfor $(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta},\\hat\\te_0^{m,\\eta}),$  i.e., the right hand side of\n(\\ref{e}) with $(\\n_0,u_0,\\te_0)$  replaced by\n$(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta},\\hat\\te_0^{m,\\eta}),$   satisfies\n\\be \\la{uv9}  \\lim\\limits_{\\eta\\rightarrow 0}\n\\lim\\limits_{m\\rightarrow \\infty}\\hat C_0^{m,\\eta}\\le C_0,\\ee\nwhich leads to that there exists an $\\hat\\eta\\in(0,\\eta_0)$ such that, for any $\\eta\\in\n(0,\\hat\\eta ),$ there exists some $\\hat m (\\eta)\\geq m(\\eta)$ such that\n\\be \\la{uv8}\\hat C_0^{m,\\eta}\\le C_0+\\ve_0\/2\\le \\ve_0 , \\ee\nprovided \\be\\la{uv01}\n0<\\eta<\\hat\\eta  , \\quad m\\geq\\hat m (\\eta).\\ee\nThen if we assume (\\ref{uv01}) holds, it directly follows from Proposition \\ref{pro2}, (\\ref{dee3}) and (\\ref{uv8}) that there exists a classical solution  $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta})  $  of problem (\\ref{a1})--(\\ref{h1}) with  initial data $(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta},\\hat\\te_0^{m,\\eta})$   on $\\O\\times(0,T]$ for all $T>0$.  Furthermore, $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $  satisfies \\eqref{h8}, (\\ref{zs2}), (\\ref{a2.112}), \\eqref{key}, (\\ref{ae3.7}),  (\\ref{vu15}), and \\eqref{h22} respectively  for any $T>0$ and $t\\geq1$ with $(\\n ,u ,\\te )$   replaced by $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta})$.\n\n\nIt remains to prove (\\ref{uv9}). Indeed, we just need to infer\n\\be\\la{uv10}\\lim_{\\eta\\rightarrow 0}\\lim_{m\\rightarrow \\infty}\\int\\hat\\n_0^{m,\\eta}\\left(\\hat\\te_0^{m,\\eta}- \\log\n \\hat\\te_0^{m,\\eta} -1 \\right)dx\\le \\int\\n_0\\left(\\te_0- \\log\n \\te_0 -1 \\right)dx ,\\ee since   the other terms in  (\\ref{uv9}) can be proved in a  similar and even simpler way.\nNote that\n\\bnn\\ba\n&\\hat\\n_0^{m,\\eta}\\left(\\hat\\te_0^{m,\\eta}- \\log \\hat\\te_0^{m,\\eta} -1\\right)\\\\\n&=\\hat\\n_0^{m,\\eta}(\\hat\\te_0^{m,\\eta}-1)^2 \\int_0^1\\frac{\\al}{\\al(\\hat\\te_0^{m,\\eta}-1)+1}d\\al\\\\\n&= \\frac{\\hat\\n_0^{m,\\eta}(j_{m^{-1}}*(\\n_0(\\te_0-1)1_{\\O_m}))^2}{j_{m^{-1}}* (\\n_01_{\\O_m} )+\\eta}\\\\\n&\\quad\\cdot\\int_0^1\\frac{\\al}{\\al j_{m^{-1}}*(\\n_0(\\te_0-1)1_{\\O_m})+j_{m^{-1}}* (\\n_01_{\\O_m} )+\\eta}d\\al \\\\\n&\\in \\left[0, \\, \\hat\\n\\eta^{-2}(j_{m^{-1}}*(\\n_0(\\te_0-1)1_{\\O_m}))^2 \\right],\n\\ea\\enn\nwhich combined with  Lebesgue's dominated convergence theorem yields that\n\\be \\notag\\ba\n&\\lim_{m\\rightarrow \\infty}\\int\\hat\\n_0^{m,\\eta}\\left(\\hat\\te_0^{m,\\eta}- \\log\n \\hat\\te_0^{m,\\eta} -1 \\right)dx\\\\\n&=\\int(\\n_0+\\eta)\\left(\\frac{\\n_0\\te_0+\\eta}{\\n_0+\\eta}\n - \\log\\frac{\\n_0\\te_0+\\eta}{\\n_0+\\eta}-1 \\right)dx\\\\\n&=\\int\\left(\\n_0\\te_0 -\\n_0+(\\n_0+\\eta)\\log(\\n_0+\\eta)\\right)dx\\\\\n &\\quad-\\int\\left(\\n_0\\log(\\n_0\\te_0+\\eta)+\\eta\\log(\\n_0\\te_0+\\eta)\\right)dx\\\\\n &\\le \\int\\left(\\n_0\\te_0 -\\n_0+(\\n_0+\\eta)\\log(\\n_0+\\eta)\\right)dx-\\int\\left(\\n_0\\log(\\n_0\\te_0)+\\eta\\log\\eta\\right)dx\\\\\n & \\rightarrow \\int \\n_0\\left(\\te_0-  \\log\\te_0 -1 \\right)dx,\\quad \\mbox{ as }\\eta\\rightarrow 0.\n\\ea\\ee\nIt thus gives \\eqref{uv10}.\n\n\n\n{\\it Step 2. Compactness results.}\n With the approximate solutions $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $ obtained in the previous step at hand,  we can derive the global existence of weak solutions by passing to the limit  first   $m\\rightarrow \\infty,$ then $\\eta\\rightarrow 0 .$   Since the two steps are similar, we will only   sketch the  arguments for $m\\rightarrow \\infty.$  For any fixed  $\\eta\\in (0,\\hat\\eta)$, we simply denote  $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $\n   by $(\\n^m ,u^m ,\\te^m ).$ Then  the combination of Aubin-Lions Lemma with (\\ref{zs2}), (\\ref{a2.112}),  \\eqref{key},  (\\ref{vu15}),  and Lemma \\ref{le4} yields that there exists some appropriate subsequence $ m_j \\rightarrow \\infty$ of $m\\rightarrow \\infty$ such that, for any $0<\\tau<T<\\infty $, $p\\in[1,\\infty)$, and $\\tilde p\\in[1,6)$,\n\\be \\la{vu4}\nu^{m_j}\\rightharpoonup u  \\,\\,\\mbox{ weakly star in }\\,\\, L^\\infty(0,T; H^1),\n\\ee\n\\be\\la{lk}\n\\te^{m_j}\\rightharpoonup \\te \\,\\,\\mbox{ weakly in }\\,\\, L^2(0,T;H^1),\n\\ee\n\\be \\la{vu1}\n\\n^{m_j}\\rightarrow \\n  \\quad \\mbox{ in }\\,\\, C([0,T];L^p\\mbox{-weak})\\cap C([0,T];H^{-1}),\\ee\n\\be \\la{vu5}\n\\n^{m_j}u^{m_j}\\rightarrow \\n u,\\, \\n^{m_j}\\te^{m_j}\n\\rightarrow \\n \\te   \\, \\mbox{ in }\\,  C([0,T];L^2 \\mbox{-weak})\\cap C([0,T];H^{-1}),\n\\ee\n\\be \\la{vu26} \\n^{m_j} |u^{m_j}|^2\\rightarrow \\n |u|^2 \\,\\, \\mbox{ in }\\,\\, C([0,T];L^3 \\mbox{-weak})\\cap C([0,T];H^{-1}),\\ee\n\\be\\la{ghh}\nG^{m_j}\\rightarrow G,\\,\\curl u^{m_j}\\rightarrow\\curl u\\,\\,\\mbox{ in }\\,\\,C([\\tau,T];H^1\\mbox{-weak})\\cap C([\\tau,T];L^{\\tilde p}),\n\\ee\n\\be\\la{lll}u^{m_j}\\rightarrow u\\,\\mbox{ in }\\,\\,C([\\tau,T];W^{1,6}\\mbox{-weak})\\cap C(\\O\\times[\\tau,T]),\\ee\nand\n\\be \\la{vu18}\n\\te^{m_j}\\rightarrow \\te\\,\\,\\mbox{in}\\,\\,C([\\tau,T];H^2\\mbox{-weak})\\cap C([\\tau,T];W^{1,\\tilde p}),\n\\ee\nreferring to \\cite{H-L} for the detailed proof.\nNow we consider the approximate solutions $(\\n^{m_j},u^{m_j},\\te^{m_j})$ in the weak forms, i.e. \\eqref{def1}--\\eqref{def3}, then  take appropriate limits. Standard arguments as well as (\\ref{dee03}) and (\\ref{vu4})--(\\ref{vu18}) thus conclude that the limit $(\\n,u,\\te)$ is a weak solution  of   (\\ref{a0})   (\\ref{h2})  (\\ref{h1})  in the sense of Definition \\ref{def} and satisfies (\\ref{hq1})--(\\ref{hq4}) and the exponential decay property \\eqref{h11}.\nMoreover, we obtain the estimates (\\ref{hq5})--(\\ref{hq8}) with the aid of (\\ref{zs2}), (\\ref{a2.112}), (\\ref{vu15}),    and (\\ref{vu4})--(\\ref{vu18}). Finally, (\\ref{vu019}) shall be obtained by adopting the same way as in \\cite{H-L}.\nThe proof of Theorem \\ref{th2} is finished.\n\n\n\\section*{Acknowledgements}   The research  is\npartially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS,\nNSFC Grant (Nos.  11688101,  12071200, 11971217),   Double-Thousand Plan of Jiangxi Province (No. jxsq2019101008),  Academic and Technical Leaders Training Plan of Jiangxi Province (No. 20212BCJ23027), and Natural Science\nFoundation of Jiangxi Province (No.  20202ACBL211002).\n\n\\begin {thebibliography} {99}\n\n\\bibitem{adn}\n\n\nS. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. {\\it Commun. Pure Appl. Math.} \\textbf{17}(1)(1964), 35--92.\n\n\n\\bibitem{akm} S. N. Antontsev,  A. V. Kazhikhov,  V. N. Monakhov, {\\it  Boundary value problems in mechanics of nonhomogeneous fluids.} North-Holland Publishing Co., Amsterdam, 1990.\n\n\\bibitem{CANEHS} J. Aramaki, $L^p$ theory for the div-curl system. {\\it Int. J. Math. Anal.}  \\textbf{8}(6)(2014), 259--271.\n\n\\bibitem{bkm} J.T. Beale,  T. Kato, A. Majda,\nRemarks on the breakdown of smooth solutions for the 3-D Euler\nequations. {\\it Commun. Math. Phys.} {\\bf 94}(1984), 61--66.\n\n\\bibitem{bd} D. Bresch,  B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. {\\it J. Math. Pures Appl.} {\\bf 87}(9)(2007), 57--90.\n\n\\bibitem{C-L} G.C. Cai, J. Li, Existence and exponential growth of global classical solutions to the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domains. {\\it Indiana Univ. Math. J.} in press.\n\n\n\n\n\\bibitem{choe1}Y. Cho, H. Kim, Existence results for viscous polytropic fluids with vacuum. {\\it J. Differ. Eqs.} {\\bf 228}(2006), 377--411.\n\n\n\n\n\n\n\\bibitem{feireisl} E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid. {\\it Indiana Univ. Math. J.} {\\bf 53}(2004), 1707--1740.\n\n\n\\bibitem{feireisl1}E. Feireisl, {\\it Dynamics of Viscous Compressible Fluids}. Oxford Science Publication, Oxford, 2004.\n\n\n\\bibitem{F1} E. Feireisl,  A. Novotny, H. Petzeltov\\'{a},  On the existence of globally defined weak solutions to the\nNavier-Stokes equations. {\\it J. Math. Fluid Mech.} {\\bf 3}  (2001), 358--392.\n\n\n\\bibitem{GPG}\nG.P. Galdi,  \\textit{An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems. Second Edition.} Springer, New York, 2011.\n\n\n\n\\bibitem{Hof1} D. Hoff,  Discontinuous solutions of the Navier-Stokes equations\nfor multidimensional flows of heat-conducting fluids. {\\it Arch.\nRational Mech. Anal.}  {\\bf 139}(1997), 303--354.\n\n\n\n\\bibitem{h101}\n X.D. Huang,  J.  Li, Serrin-type blowup criterion for viscous,\ncompressible, and heat conducting Navier-Stokes\nand magnetohydrodynamic flows. {\\it Commun. Math. Phys.} \\textbf{324}(2013), 147--171.\n\n\\bibitem{H-L}\nX.D. Huang,  J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations.  {\\it Arch. Rational Mech. Anal.} \\textbf{227}(2018), 995--1059.\n\n\n\\bibitem{h1x} X.D. Huang, J. Li, Z.P. Xin,\nSerrin type criterion for the three-dimensional compressible flows. {\\it  SIAM  J. Math. Anal.} {\\bf 43}(4)(2011), 1872--1886.\n\n\n\\bibitem{hulx} X.D. Huang, J. Li, Z.P. Xin,  Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic\ncompressible Navier-Stokes equations. {\\it Comm. Pure Appl. Math.} {\\bf 65}(4)(2012), 549--585.\n\n \\bibitem{kato}\nT. Kato, Remarks on the Euler and Navier-Stokes equations in $\\r^2$. {\\it Proc. Symp. Pure Math. Amer. Math. Soc. Providence.} \\textbf{45}(1986), 1--7.\n\n\\bibitem{kazh01} A. V. Kazhikhov,   Cauchy problem for viscous gas equations. {\\it Siberian Math. J.} {\\bf 23} (1982),   44--49.\n\n\\bibitem{Kaz} A. V. Kazhikhov, V. V.  Shelukhin,\nUnique global solution with respect to time of initial-boundary\nvalue problems for one-dimensional equations of a viscous gas.\n{\\it J. Appl. Math. Mech.  } {\\bf 41} (1977), 273--282.\n\n\n\n\\bibitem{lxz}\nS.H. Lai, H. Xu, J.W. Zhang, Well-posedness and exponential decay for the Navier-\nStokes equations of viscous compressible heat-conductive fluids with vacuum.\narxiv: 2103.16332.\n\n\n\\bibitem{2dlx} J. Li, Z.P. Xin,  Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum. {\\it Annals of PDE.}  \\textbf{5}(2019), 7.\n\n   \n\n\\bibitem{L1} P. L. Lions,  \\emph{Mathematical topics in fluid\nmechanics.  Compressible models. Vol. {\\bf 2}.}  Oxford\nUniversity Press, New York,   1998.\n\n\\bibitem{M1} A. Matsumura, T.   Nishida,   The initial value problem for the equations of motion of viscous and heat-conductive gases. {\\it J. Math. Kyoto Univ.} {\\bf 20}(1980), 67--104.\n\n\\bibitem{Na}\nJ. Nash, Le probl\\`{e}me de Cauchy pour les \\'{e}quations diff\\'{e}rentielles d'un fluide g\\'{e}n\\'{e}ral. {\\it Bull. Soc. Math. France.} \\textbf{90} (1962), 487--497.\n\n\\bibitem{Nclm1} C. L. M. H. Navier,  Sur les lois de l\\'{e}quilibre et du mouvement des corps \\'{e}lastiques. {\\it Mem. Acad. R. Sci. Inst. France.} \\textbf{6 }(1827), 369.\n\n\\bibitem{nir} L. Nirenberg, On elliptic partial differential equations.  {\\it Ann. Scuola Norm. Sup. Pisa.} {\\bf 13}(3)(1959), 115--162.\n\n\n\\bibitem{R} O. Rozanova,   Blow up of smooth solutions to the compressible\nNavier-Stokes equations with the data highly decreasing at\ninfinity. {\\it J. Differ. Eqs.}  {\\bf 245} (2008),  1762--1774.\n\n\n\n\\bibitem{jes} W. Rudin, {\\it Real and complex analysis. Third Edition}. McGraw-Hill Book Company, New York, 1987.\n\n\n\\bibitem{se1} J. Serrin, On the uniqueness of compressible fluid motion. {\\it\nArch. Rational  Mech. Anal.} {\\bf 3 }(1959), 271--288.\n\n\\bibitem{Tani}\nA. Tani, On the first initial-boundary value problem of compressible viscous fluid motion. {\\it Publ. Res. Inst. Math. Sci. Kyoto Univ.} \\textbf{13}(1977), 193--253.\n\n\\bibitem{vww}\nW. von Wahl, Estimating $\\nabla u$ by $\\div u$ and $\\curl u$. {\\it Math. Meth. Appl. Sci.} \\textbf{15}(1992), 123--143.\n\n\n\\bibitem{W-C}\nH.Y. Wen, C.J. Zhu,\nGlobal solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data. {\\it SIAM J. Math. Anal.} \\textbf{49}(2017),  162--221.\n\n\\bibitem{wenzhu1} H.Y. Wen, C.J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat-conducting fluids with vacuum. {\\it SIAM J. Math. Anal.} \\textbf{45}(2013),  431--468.\n\n \\bibitem{wenzhu2} H.Y. Wen, C.J. Zhu, Global symmetric classical solutions of the full compressible Navier-Stokes equations with vacuum and large initial data. {\\it J. Math. Pures Appl.}  \\textbf{102}(2014),  498--545.\n\n\n\n\\bibitem{X1} Z. P. Xin,\nBlowup of smooth solutions to the compressible {N}avier-{S}tokes\nequation with compact density. {\\it Comm. Pure Appl. Math. } {\\bf 51}(1998), 229--240.\n\n\\end {thebibliography}\n\n\n\\end{document}\n\n\n\n\n\n\n\n\\iffalse\nThere is a great deal  of literature  on the  well-posedness of solutions to the compressible Navier-Stokes systems.\nFor multi-dimensional case, the local theory is quite plentiful, see \\cite{se1,Na,Tani} for the local existence and uniqueness of classical solutions without vacuum, and \\cite{cho1,K2,sal,choe1,liliang} for the strong solutions with initial vacuum. The first result of global classical solutions was obtained by Matsumura-Nishida \\cite{M1} for initial data close to a non-vacuum equilibrium in some Sobolev space $H^s.$ Later, Hoff \\cite{Hof1,Hof2} established the global weak solutions with strictly positive initial density and temperature for discontinuous initial data.\nWhen the initial vacuum is allowed, some new challenging difficulties arise.\nFor the compressible barotropic flows, the major breakthrough is due to Lions \\cite{L1} (see also  Feireisl \\cite{feireisl1}), where he obtained the global existence of weak solutions, defined as  finite energy solutions, if the adiabatic exponent $\\gamma$ is suitably large.\nRecently,  Li-Xin \\cite{2dlx} and Huang-Li-Xin \\cite{hulx} obtained the global classical solutions to the 2D and 3D Cauchy problem respectively with the initial data which are of small energy but possibly large oscillations.\nMore recently, for initial-boundary-value problem (IBVP) in bounded domains,\nCai-Li \\cite{C-L} first obtained the global classical solutions with initial vacuum, provided that the initial energy is suitably small.\n\n\n\n\n\\fi\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n## A CATERED HALLOWEEN\n\n\"What's that?\" Libby asked, pointing to the woman's head sitting on the first step from the top. It stared up at her, looking incredibly lifelike. It also looked familiar. Very familiar.\n\n\"Shouldn't the head be sitting in a pool of blood?\" Bernie asked.\n\nMark didn't answer her. \"Wait,\" he said instead, and he put out his hand.\n\nLibby stopped.\n\n\"Give me a moment,\" Mark said.\n\nLibby noticed he was frowning. \"Is something wrong?\" she asked.\n\nMark didn't reply. His attention was focused on the head.\n\n\"Well, is there?\" Bernie asked while she watched Mark take another step forward. She had a bad feeling in her gut. \"Is that a hologram?\" she asked. \"Because it looks pretty solid if it is.\"\n\nLibby watched as Mark stretched out one of his feet and gave the head a tentative tap with the toe of his shoe. It began rolling down the steps...bump, bump, bump...and then it kept going until it stopped at Libby's feet.\n\nThis is not a hologram, Libby thought. Holograms do not make noises like that...\n\n## Books by Isis Crawford\n\nA CATERED MURDER\n\nA CATERED WEDDING\n\nA CATERED CHRISTMAS\n\nA CATERED VALENTINE'S DAY\n\nA CATERED HALLOWEEN\n\nA CATERED BIRTHDAY PARTY\n\nPublished by Kensington Publishing Corporation\nA Mystery with Recipes\n\n## A CATERED HALLOWEEN\n\n## ISIS CRAWFORD\n\nKENSINGTON BOOKS  \nwww.kensingtonbooks.com\nTo Mike Ruffo.   \nThanks for listening.\n\n## Acknowledgments\n\nI'd like to thank my family and friends for being there for me when I needed you, and DJM for his suggestions.\n\n## Contents\n\nPrologue\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\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nEpilogue\n\nRecipes\n\n## Prologue\n\nAmethyst Applegate turned the letter over in her hand. Then she put it down on the desk and looked at the envelope it had come in. It told her nothing. The envelope was one of those plain white, self-sealing ones that you could buy at any pharmacy or office-supply store. Her address had been printed out on a computer. There was no return address. As for the postmark, it revealed that the letter had been mailed from Longely two days ago.\n\nShe put the envelope down and read the letter again.\n\nDear Amethyst,\n\nI've decided that it's time to renew old acquaintances. If you don't want everyone to find out what you did at the Peabody School when you were there, meet me in the Pit and the Pendulum Room at six-thirty sharp. I will be waiting for you. We have lots to discuss. Remember there's no statute of limitations on murder. Signed, Bessie Osgood\n\n\"Bessie Osgood. Now there's a laugh.\"\n\nShe remembered Bessie all right. Bessie, with her braces and her pimples and those stupid glasses she used to wear. Bessie, who always used to roll her skirt up around her waist because she thought it showed off her legs.\n\nHer calves were the only thin part of her, that was for sure. Bessie, who'd told the proctor that she'd seen her smoking in the bathroom and that she kept a flask under her bed. She'd almost gotten her thrown out of school for that, almost being the operative word. Fortunately, she'd been able to convince the proctor that she was being led astray by Bessie. Now that had been good. Poor Bessie. She should have taken the hint and gone home. Instead, she'd come up to her and told her she had proof. That she had witnesses that were going to testify against her. She shouldn't have done that. Amethyst tapped her fingers on the letter. But she'd shown her. Yes. She certainly had.\n\n\"Discuss,\" Amethyst muttered. She'd discuss all right. That was ridiculous. You didn't discuss things with a ghost. Ghosts didn't talk. Bessie was dead and gone, and she should know, because she'd been responsible for Bessie's \"accident.\" Poor thing, but those windows were low to the ground, and it was easy to lose your footing if you leaned out too far.\n\nSo the question was, who had written the letter? Who knew about what had happened at the Peabody School, and why was it coming to light now?\n\nAmethyst folded the letter up, put it in the envelope, and slipped it inside her handbag. When it came down to it, she was pretty sure she knew who had written this, almost positive in fact. One other thing she did know: they were going to be very, very sorry. Evidently, they'd forgotten with whom they were dealing.\n\n\"Amethyst,\" her new husband called from downstairs. \"We're going to be late if we don't get a move on pretty soon.\"\n\n\"One second, dear. I'll be right there,\" she trilled.\n\nFor a moment, she toyed with taking her knife along but discarded the idea. If she did, she'd have to change handbags, and that would ruin her outfit. No. She'd get who she thought it was in a not quite so public setting later. Let them think she would go along with them. That would make the surprise that much sweeter.\n\nShe turned around and studied herself in the mirror. Not bad. Not bad at all, even if she did have to say so herself. The body was still good, and after she had a little work done around her neck area, she'd be perfect. As she went down the stairs, she smiled. It had been a while since she had had this kind of fun. She was looking forward to it. She really was.\n\n## Chapter 1\n\nLibby looked at the darkening sky. In her dream the sky had been black with crows. They'd covered the tree limbs, and then they had popped out of her mother's pumpkin pie, and she'd woken up in a cold sweat. It had been a long time since she'd had dreams like that. God, she hoped they weren't coming back after all these years. She was wondering what the dream meant\u2014dreams like that always meant something\u2014when she realized that her sister was talking to her.\n\n\"It's dark out,\" Bernie observed as she muscled the van around a turn.\n\nLibby grunted. The wind had picked up. Leaves skittered across the road. \"A storm's coming in.\"\n\n\"So you're saying it's a dark and stormy night?\"\n\nLibby looked at her sister and shook her head. \"That was bad.\"\n\nBernie grinned. \"But irresistible.\"\n\nLibby smiled despite herself. Her sister could always make her smile. That was one of the things she loved about her. After a moment, Libby went back to looking out. The road they were on twisted its way through the woods as it went up the hill. Every once in a while, she caught sight of their destination floating above the trees. The view was not reassuring.\n\n\"You should have taken the main road,\" Libby said. Their van wasn't really equipped for driving on dirt and gravel.\n\nBernie shrugged. \"It'll be fine. Nervous?\" she asked.\n\n\"About the job?\" The truth was Libby always got a little nervous before a catering job. It was just the way she was constituted.\n\n\"No. About working at the Peabody School.\"\n\n\"Why should I be?\" Libby asked. It was not that she didn't know the answer to Bernie's question; she did. She just didn't want to admit it.\n\n\"After all, the place is haunted.\"\n\nLibby snorted.\n\n\"People have seen them,\" said Bernie.\n\n\"Them?\"\n\n\"The ghosts, Libby. The Peabody School even has a blurb in the book Haunted Houses in New York.\"\n\nSince Libby didn't believe that everything that was in print was true, she felt no need to comment. She wished Bernie would slow down. The van wasn't that stable to begin with, and when it was fully loaded, well. She closed her eyes and tried not to think about the consequences of going off the road. They'd packed the van well, but there was only so much one could do.\n\nBernie took her eyes off the road for a second and looked at her sister. \"Well, I think that having a Halloween Haunted House in a real haunted house is a neat idea.\"\n\n\"I never said it wasn't,\" Libby replied as she did a quick mental recap of the menu they'd be serving.\n\nShe and Bernie had decided to design the menu around the theme of waffles. Somehow they'd seemed right. Then she'd seen a recipe for them in the food pages of the New York Times, and she knew that she and Bernie were on track.\n\nGiven everything that was happening in the world these days, people wanted comfort food, and waffles certainly fit the bill. In addition, you could dress them up or down. They appealed to everyone. And because they would be served in the evening, instead of during the morning, you had that whole fish out of water thing going on. Not only that, but from a business point of view, waffles were cheap to make. The ingredients cost next to nothing, and it took about five minutes to mix up the batter.\n\nLibby was particularly proud of the chocolate brownie batter waffles she'd dreamed up. The finely ground black pepper gave them a particularly nice kick by balancing the sweet and the hot. Of course, the other waffles weren't bad, either. They would serve four kinds: regular, Belgian, the aforementioned chocolate brownie, and pumpkin. The waffles would be garnished with appropriate homemade ice creams and sauces: whipped cream and strawberries for the Belgian waffles, vanilla ice cream and hot fudge sauce for the chocolate brownie waffles, maple sauce and vanilla ice cream for the regular waffles, and a poached apple compote with cinnamon ice cream for the pumpkin ones. Naturally, they were providing maple syrup, apple butter, and homemade apricot and strawberry jam as well. Just thinking about the waffles quieted the butterflies in Libby's stomach.\n\nAs they rounded another turn in the road, flocks of crows in the treetops swirled up in the air; they came down again as the van passed by. The birds' disapproving cawing followed the vehicle around the next bend.\n\n\"There seem to be more and more of them every year,\" Libby said.\n\n\"There are. Someplace in upstate New York has a crow hunt to get rid of them.\"\n\n\"We should try that.\"\n\nBernie laughed, \"In Longley? The PC capital of the world? I don't think so.\"\n\nLibby didn't answer.\n\nBernie looked at her. Something was bothering her sister. Had been for the past couple of weeks. \"Are you all right?\"\n\n\"I'm fine,\" said Libby.\n\n\"Positive?\"\n\n\"Positive,\" Libby repeated.\n\n\"You and Marvin are okay, aren't you?\"\n\n\"I said everything is okay,\" snapped Libby.\n\nBernie resisted the impulse to make a smarty-pants comment. Her sister was clearly lying, and things weren't all right, but now obviously wasn't the time to push. Libby would tell her when she was ready. She always did. Instead, Bernie changed the subject.\n\n\"There it is,\" Bernie said as the roof of the Peabody School came back into view. \"Our home away from home for the next week. Well, just the evenings really.\"\n\n\"Personally,\" Libby said, \"I've always thought that whoever designed this place had a severe case of indigestion.\" She sighed. Boy, did she wish they weren't doing this. Halloween was one of A Little Taste of Heaven's busier times, and being out of the shop during the evening meant that they'd be staying up till two and three in the morning prepping the next day's food.\n\n\"Lots of work,\" Bernie said, echoing her sister's thoughts.\n\n\"But,\" Libby continued, \"the money\u2014\"\n\nBernie finished the sentence for her. \"The money is too good to pass up.\"\n\n\"It certainly is.\" Libby took another gander at her younger sister's outfit. \"I don't see how you can work in that. Isn't it a little...snug?\"\n\nBernie looked down at the skintight black dress she was wearing, took one of her hands off the wheel, and waggled her long bloodred fingernails in front of Libby. \"Don't worry. I'll manage. I always do.\"\n\n\"I know.\"\n\nAnd even though Libby wouldn't say it to anyone, for reasons she couldn't understand, that fact annoyed the hell out of her. By the end of the evening, she would be sweaty, and her clothes would be covered in stuff, but Bernie would look as if she'd just stepped out of the shower. If she were wearing the shoes Bernie was wearing, she'd trip and go right into the food. It wasn't fair. She looked down at her sneakers and sighed, then gazed at the school again.\n\n\"The place does look like a haunted house,\" she conceded.\n\n\"That's what I just said, Libby.\"\n\n\"No. You said it was a haunted house. There's a difference.\"\n\nLibby frowned as the van slid on the wet leaves. They went around another turn. A sign that read WELCOME TO THE LONGELY VOLUNTEER FIREMEN'S HALLOWEEN HAUNTED HOUSE sprang into view.\n\n\"I'll say one thing for Mark Kane,\" Libby said. \"He did a good job remodeling the place.\n\n\"Bree told me he spent almost a million dollars.\"\n\n\"He could toss some our way.\" Libby began digging around in her bag. \"Why is there never any chocolate when you need it?\" she asked as she dumped the contents of her bag on her lap.\n\n\"Because you probably ate it already.\"\n\n\"I didn't.\"\n\n\"If that's what you want to believe, fine with me.\"\n\nLibby continued rummaging through the contents of her bag. She had her wallet, her tissues, her appointment book, business-card holder, Swiss Army knife, cell phone, and a bag of glazed cashew nuts, but no chocolate bar. She didn't remember eating it. Could she have left it on the counter of their shop, A Little Taste of Heaven? Libby began throwing everything back in her bag.\n\n\"I'll tell you one thing, though,\" Libby said to her sister. \"I sure wouldn't have wanted to go to boarding school here.\"\n\n\"Me either,\" Bernie agreed. \"Way too isolated. And Bessie Osgood going out the window...\"\n\n\"The place may not be haunted,\" Libby mused, \"but lots of bad things have happened here.\"\n\n\"Probably bad feng shui.\"\n\nLibby rolled her eyes.\n\n\"Mock me if you want, but if I were Mark Kane, I would have had this building cleansed before I took it over,\" said Bernie.\n\n\"What is The James Foundation for Scientific Reasoning, anyway?\"\n\n\"It's a scientific think tank....\"\n\n\"Whatever that is,\" replied Libby.\n\n\"I think it's a conservative think tank.\"\n\n\"Meaning?\"\n\n\"I'm just telling you what Bree told me,\" said Bernie.\n\n\"She probably doesn't know, either,\" Libby grumbled. \"No reason she should,\" she added as she picked a piece of lint off her denim jacket. \"She just sold him the property and got a nice fat commission.\"\n\n\"We're definitely in the wrong business,\" Bernie observed.\n\n\"You really think that?\" Libby asked.\n\n\"No. I love what we do,\" Bernie said as she swerved to avoid a rock lying in the middle of the road.\n\n\"Me too,\" Libby replied. If she thought about it, she couldn't imagine her life without the shop.\n\n\"So what do you think happened to the Reverend Peabody and his wife, Esmeralda?\"\n\n\"I think he killed her and threw her body in the Hudson and died of a heart attack a year later,\" replied Libby.\n\n\"They both died on Halloween night, one year apart. Don't you find that a little strange?\"\n\n\"Not really. Can we change the subject?\" Libby asked.\n\nBernie shrugged. \"If you want. Okay. What do you want to talk about?\"\n\n\"I'm just wondering if we brought enough waffle machines with us?\"\n\n\"Of course, we brought enough. We did the math, remember?\"\n\n\"We could have miscalculated.\" Libby bit at her cuticle again.\n\n\"You always say that, and we never do. You worry too much,\" said Bernie.\n\nWhich Libby knew was true. After all, waffles weren't the only things they were serving. The van was packed with apple squares, lemon squares, pumpkin bars, brownies, chocolate chip cookies, sticky buns, pear crumb cake, banana bread, pumpkin pie, and apple walnut cake, among other things, as well as gallons of spiced cider and chai. In addition, they had containers for coffee and for hot water for tea in the van, as well as fifteen large jack-o'-lanterns, which they were planning on using for decoration. They should really be fine.\n\n\"We're here,\" Bernie said as she rounded the last turn.\n\nShe put the van in park. It shuddered. Not a good sign. The last time she'd taken it in for an oil change, Sully had told her the transmission was going.\n\nLibby frowned. \"I would never have picked this place to do business out of.\"\n\n\"Me either,\" Bernie agreed as she yanked up her panty hose. \"But Bree said he fell in love with the place. Said it would suit his needs perfectly.\"\n\nA moment later the door to the back entrance swung open, and Mark Kane came bounding out.\n\n\"And here's the man of the hour,\" Libby said to Bernie out of the corner of her mouth as he came closer.\n\nBernie nodded. He looked exactly like he did when he'd first come into their shop: like the successful entrepreneur, the go-to guy.\n\n\"Glad to have you two on board,\" he said. \"I can't wait to have a piece of your apple pie. I went in yesterday, and I had to settle for a slice of cranberry-apple. Not that that was bad. It was amazingly good. I've gained about ten pounds since I've been here, and I put it all down to your shop.\"\n\nLibby laughed. The guy could be charming, she'd give him that, although she never quite trusted men who were like that.\n\nMark gestured toward the mansion. \"You like what we've done decoration-wise?\"\n\n\"Love it,\" Bernie said.\n\n\"You don't think the heads on the spikes and the severed limbs dangling from the window are too much?\" asked Mark.\n\n\"Maybe a tad,\" Bernie allowed.\n\n\"In the ads I stated that no one under twelve would be admitted,\" said Mark. \"There are plenty of other places for young kids to go, and I wanted to do something over the top without having to worry about the little ones' sensibilities.\"\n\n\"Good thinking,\" Libby said. She couldn't imagine the nightmares a kid would have after seeing something like this.\n\n\"And we're having everyone sign a liability waiver,\" Mark said.\n\nBernie giggled. \"Like they did for some of the horror movies in the fifties.\"\n\nMark grinned. It made his face look slightly lopsided. He snapped his fingers. \"I knew I forgot something. A doctor and an ambulance.\"\n\n\"And the nurse,\" said Bernie. \"Don't forget the nurse.\"\n\nMark's grin grew wider. \"I'll see what I can do.\"\n\n\"So it's going to be a good show?\" Bernie asked.\n\nMark rubbed his hands together. \"A good show? It's going to be a great show. The best this town has ever seen. Forget the guy in the sheet jumping out at you and yelling boo and the dime-store skeletons hanging from the door frames. We're going to do much better than that. Come on. Let me show you what we've done since you were here last.\"\n\n\"We'd love to,\" Libby told him, \"but we have to get the batter in the fridge.\" She consulted her watch. \"Plus, we need setup time. We're behind schedule as it is.\"\n\n\"Not to worry,\" said Mark. He removed his phone from the clip on his belt, called someone, and spoke for a few seconds. \"I just told Carl to come out,\" he explained after he ended the call. \"He'll unload the van for you.\"\n\n\"I think I'd rather do it myself,\" Libby said. She really didn't like the idea of someone else handling their stuff. Actually, didn't like was putting it mildly. Hated would be a more accurate term. Who knew what they would do to it.\n\n\"Come on,\" Bernie said to her. \"It'll be fine.\"\n\nLibby could feel herself start to flush. No. It wouldn't be all right. The rule was no one touched their stuff. And Bernie knew it, too.\n\nMark jumped in. \"Carl's worked in restaurants all his life. Anyway, there's no reason why ladies as lovely as yourselves should have to carry heavy things.\" Before Libby could answer, Carl appeared. \"Tell him what to do,\" Mark ordered.\n\nBernie did.\n\nLibby took a deep breath and told herself not to say anything now. She would talk to Bernie later. She forced a smile. \"Let's go,\" she told Mark.\n\nHe rubbed his hands together. \"Good. I'm going to scare you to death.\"\n\n\"I can hardly wait,\" Libby muttered as they went inside.\n\n## Chapter 2\n\nLibby looked around the hallway they'd just stepped into. It had been totally transformed since she'd been there two weeks ago. The walls were now painted a dull gray and festooned with cobwebs.\n\nMeat hooks hanging from the ceiling rattled menacingly. The floor looked old and dusty, except for the splatters of what, Libby decided, was supposed to be blood. The overhead fluorescent lights flickered on and off, painting shadows on the walls and floor. Libby sniffed. The place even smelled musty. The only things that looked modern were the EXIT signs over two doors down the hall.\n\nMark jerked his head in their direction. \"The fire marshall insisted on those. I tried to talk him out of it\u2014I think it spoils the mood\u2014but it was a no go.\"\n\n\"I would think so,\" Libby said as she took a couple of steps forward.\n\nNo doubt about it. Mark was definitely getting on her nerves. She stopped for a moment in front of a sign on one of the walls, which said, THIS WAY TO THE EXECUTION. A black arrow pointed to the door on Libby's right. Then she moved over to the square wooden table pushed against the wall. On it sat a cash box, a bunch of forms, and a black cup filled with pens.\n\n\"The releases I was telling you about,\" Mark said, pointing to the pile of paper.\n\nBernie nodded. A notice on the table stated that the price of admission was thirty dollars.\n\n\"A little high, isn't it?\" Libby commented.\n\nMark shrugged. \"High tech always is.\"\n\nLibby was about to tell him that was why she liked low tech better, but before she could, Bernie was pointing down the hall.\n\n\"I'm a little confused. The kitchen is the last door on the left, isn't it?\" asked Bernie.\n\nMark nodded. \"Correct. The corridor we're using for the Haunted House loops around and ends up in the kitchen and the dining room. This is the back part of the mansion. The oldest part. The part where they say Esmeralda is buried.\"\n\n\"Who says?\" Libby asked. She wished she could have found her chocolate. That way she wouldn't be so grumpy.\n\nMark waved his hands in the air. \"People say.\"\n\n\"Well, then I guess that's okay. I mean, if dogs said it, then it wouldn't be so good,\" replied Libby.\n\nMark shot Libby a puzzled look. Obviously, he hadn't gotten what she was saying. He took a step away from her as he glanced around. Even better. He probably thought she was nutty.\n\n\"God, you wouldn't believe how many rooms this place has,\" Mark said. \"And they all connect with one another in weird ways. Tracing the wiring was a nightmare, and I thought the guy that put in the fiber optics was going to quit on me. Let me tell you, we had a hell of a time getting this place up to code.\"\n\n\"That I believe,\" Bernie said.\n\nMark nodded toward the door that was marked ENTRANCE. Below it was written, ONCE YOU COME IN HERE, THERE'S NO TURNING BACK.\n\n\"Shall we?\" he asked.\n\n\"You know,\" Libby said, thinking of everything they had to do, \"we really are running out of time. Why don't you and Bernie go ahead, and I'll start in the kitchen.\"\n\n\"Nonsense,\" Mark replied. \"This will only take a few minutes. No more than five, I promise. We'll just do a quick walk-through.\"\n\nLibby was about to say they didn't have five minutes when she caught Bernie glaring at her. Even though her sister hadn't said anything, Libby knew what she wanted to say: something along the lines that Mark was new in town, that he was wealthy, and that he could throw lots of business their way. Which he wouldn't do if Libby pissed him off. So Libby just nodded her head and followed Mark through the door.\n\nWhen she stepped through to the other side, Libby felt a puff of ice-cold air play up and down her spine. She jumped in spite of herself. Mark laughed.\n\n\"That's the oldest trick in the book,\" he said. \"I connected a motion detector to a compressed-air tank.\"\n\nLibby looked around. The room was totally dark for a second; then a strobe light began flashing. She could hear a faint moan rising and falling. Then she heard another sound. It sounded like a chain saw. It was a chain saw. The chain saw got louder. Out of the corner of her eye, she saw a woman over in the corner. Her arms were tied to a chain that was suspended from the ceiling. The woman started screaming. A man with a chain saw appeared from the far corner of the room. The woman's screams got louder as the man got closer.\n\nLibby wanted to look away, but she couldn't as the man came nearer. She could feel her heart racing as the man lifted the saw. The flashes of light from the strobe bounced off the blade, making it dance in the light. This is all a trick, Libby told herself. It's an optical illusion. But somehow it didn't help.\n\nThere was a bloodcurdling shriek as the man raised the chain saw and brought it down on the woman's shoulder. Rivulets of something warm and wet ran down the left side of Libby's face. She couldn't help herself. She screamed.\n\n\"It's water,\" Mark said. \"Warm water. It was my idea. Everything is computer controlled. We also have a state-of-the-art sound system with volume controls and directional speakers.\"\n\nLibby jumped. She'd been so focused on the scene in the room that she'd forgotten that Mark was there.\n\n\"God, that looked real,\" Bernie said. She gave a nervous giggle.\n\n\"Holograms,\" Mark said. \"We can adjust the images if we want. We can adjust the screams and the sound of the chain saw. I think the blood splatter is a nice touch, don't you? It gets everyone involved.\"\n\n\"Involved?\" Libby could hear her voice rising. She took a deep breath and told herself to calm down, because what she wanted to do was throttle him. She took a second breath and a third before she regained control.\n\nBernie gestured around her. \"This must have taken months to figure out.\"\n\n\"Not really,\" said Mark. \"I hired a company, FX Productions, that specializes in this sort of thing. It took them a day and a half to set the show up.\" Mark shrugged. \"I'm not good with technical stuff,\" he confessed.\n\n\"Just making money,\" Bernie observed.\n\nMark's grin flashed on and off. \"Well, I've found that if you can do that, everything else falls into place. And this is only the beginning of the tour. Wait till you see what else FX has come up with.\"\n\n\"I'd rather not,\" Libby told him.\n\nMark reached over and took Libby's hands in his. \"But you have to.\"\n\n\"She doesn't like gory stuff,\" her sister explained.\n\nMark patted Libby's hands and then let go. \"The rest of it is just scary. Promise.\"\n\nLibby was about to say that she didn't think that was much better when Bernie interrupted.\n\n\"Is the whole thing movie themed?\" Bernie asked.\n\nMark shook his head. \"Not at all. We have vampires; we have ghosts; we have a little bit of everything.\"\n\nLibby realized she was still breathing hard. What ever happened to the days when going through a haunted house meant being blindfolded and having your hands forced into a bowl of oatmeal and spaghetti and being told that was someone's guts?\n\n\"After you,\" Libby said to Mark. She just wanted to get this over with.\n\nHe laughed. \"My pleasure.\"\n\n\"Enjoying yourself?\" Bernie asked him.\n\n\"I have to confess that I am. I feel like I'm watching my baby take his first steps,\" said Mark.\n\nAnd with that, they opened a door and stepped out into a hallway. Smog rolled around their feet and drifted upwards. A fog machine, Libby thought. She jumped as a bony hand dropped down in front of her.\n\n\"That's Bob,\" Mark said.\n\nNow that Libby looked closer, she could see the outlines of someone's arm.\n\n\"He's one of our actors,\" added Mark.\n\n\"Actors. That's a laugh,\" came a disembodied voice out of the ceiling.\n\n\"Bob Small?\" Bernie asked.\n\n\"How'd you guess?\" came the voice from the ceiling.\n\n\"Your voice,\" said Bernie.\n\n\"When did you\u2014\" Libby began to say, but Bernie kicked her.\n\nBob finished the sentence for her. \"Get out of jail?\"\n\nLibby rubbed her shin. \"Yes.\"\n\n\"About two weeks ago,\" replied Bob.\n\n\"Comfortable up there?\" Mark asked.\n\nBob snorted. \"Yeah. If you like being in a coffin.\"\n\nMark patted his hand. \"Don't worry. We'll have someone come and relieve you in two hours.\"\n\n\"Two hours?\" Bob squeaked. \"What happens if I have to take a leak?\"\n\n\"We already discussed that. You hold it,\" Mark told him as he guided Libby and Bernie toward the center of the room. When he got there, he stopped. \"I like to give people a second chance,\" he practically whispered.\n\n\"Very noble,\" Libby observed.\n\nMark shot his cuffs. \"No. It's just what I call enlightened self-interest. You give a guy like that a second chance and he's yours for life. See, we have Bob to set the mood, and then you come over here and see this.\" He pointed to a big black coffin that was seemingly rising out of the floor.\n\nThere was a creak as the coffin's door began to open. A squeal of laughter came from a skeleton as it sat up. He had an eye patch over his left socket and a long mane of white hair that came to his shoulders. He stared straight at them and shook a bony finger.\n\n\"Soon you'll look just like me,\" the skeleton cackled. \"Just like me. Eat, drink, and be merry. We don't have six-packs in the graveyard.\" Then he lay back down as the coffin door started closing. A minute or two later, the coffin was gone.\n\n\"It comes up through the floor,\" Mark explained. \"People crossing a point on the floor trip a sensor that raises the coffin.\"\n\n\"Sensor?\" Libby asked.\n\n\"Yeah. It works on the same principle as a doorbell,\" Mark replied.\n\n\"Like the one we have in the store, which alerts us when customers come in,\" Bernie added.\n\n\"I know,\" Libby said.\n\n\"Well, I was just explaining it in case you didn't,\" Bernie mumbled.\n\n\"But I do,\" insisted Libby.\n\nMark cleared his throat. \"This is really a throwaway. Something to get you calmed down after the chain-saw scene and before the next thing.\" He paused for effect. \"Because the next thing, as my father used to say, is going to knock your socks off.\" Mark walked to the door marked EXIT, opened it, and said, \"Ladies, welcome to the Pit and the Pendulum. I have to say, I think that Poe would have approved, were he alive today.\"\n\nLibby took a look around. The walls were mirrored, and in the center of the room was a raised platform. Four steps led up to it. On that platform was a long table, draped in a red cloth. Up above the table, a sharp-looking, curved blade swung back and forth, going lower with each swipe.\n\n\"You have to get closer to get the full effect,\" Mark said as he gave Libby a little nudge. She took a few steps. There was a headless body lying on the table.\n\n\"See,\" Mark said. \"You stand in the center and you see your head being chopped off.\"\n\n\"Lovely,\" Libby said. She gritted her teeth and took another step. Never let it be said that she wasn't a good sport. \"What's that?\" she asked, pointing to the woman's head sitting on the first step from the top. It stared up at her. It looked incredibly lifelike. It also looked familiar. Very familiar.\n\n\"Shouldn't the head be sitting in a pool of blood?\" Bernie asked.\n\nMark didn't answer her. \"Wait,\" he said instead, and he put out his hand.\n\nLibby stopped.\n\n\"Give me a moment,\" Mark said.\n\nLibby noticed he was frowning. \"Is something wrong?\" she asked.\n\nMark didn't reply. His attention was focused on the head.\n\n\"Well, is there?\" Bernie asked while she watched Mark take another step forward. She had a bad feeling in her gut. \"Is that a hologram?\" she asked. \"Because it looks pretty solid if it is.\"\n\nLibby watched as Mark stretched out one of his feet and gave the head a tentative tap with the toe of his shoe. It began rolling down the steps...bump, bump, bump...and then it kept going until it stopped at Libby's feet.\n\nThis is not a hologram, Libby thought. Holograms do not make noises like that.\n\nAnd then she had another thought.\n\nThe head was not made of wax. It wasn't made of plaster. It was flesh and blood.\n\nLibby didn't know how she knew. She just did.\n\nAnd then she knew how she knew.\n\nLibby stared at the face staring up at her. She'd recognize those eyebrows anywhere. \"It's Amethyst Applegate,\" she cried.\n\nWhich was when Libby started screaming.\n\n## Chapter 3\n\nSean Simmons took a bite of his pumpkin bar. \"Not bad,\" he remarked. \"Not bad at all.\"\n\nAs he brushed a small piece of the pumpkin bar off his lap, he thought that in the normal course of things, his daughter Libby would have taken those words as fighting words. Tonight she hadn't even blinked. In fact, she hadn't said much since she and her sister had come running in, yelling about what had happened down at the Peabody School.\n\nNot that he was surprised. Some places just had bad karma. Of course, he hadn't said that to Bernie and Libby when they'd told him about this job, because he didn't like to talk about certain things. Now he was thinking that maybe he should have. Then he pushed that thought away. Better to concentrate on the known and leave the rest to all the weirdos out there.\n\n\"You've added a touch more cinnamon, haven't you?\" he said.\n\nLibby's eyes widened fractionally. \"How can you ask me something like that at a time like this?\" she demanded.\n\n\"I thought you liked talking about food,\" Sean commented as he turned his wheelchair slightly so he could look out. The wind had picked up and was blowing the leaves on the street into the air. He could hear the creak of the store sign as it swayed back and forth. It looked as if a cold front was coming through.\n\n\"I do, but I don't want to talk about food now,\" Libby said.\n\nSean turned to face her. \"Well, what do you want me to talk about?\" he asked.\n\nNot that he didn't know. Bernie would say he was being disingenuous, but he'd found in his years as chief of police of Longley that it was better to let the witnesses, especially if they were in a state of shock, introduce the story themselves.\n\n\"We want to hear what you have to say about what happened at the Peabody School, of course,\" Bernie said.\n\n\"But I wasn't there,\" Sean reminded her.\n\nLibby leaned forward and pointed at her sister. \"And we wouldn't have been, either, if it weren't for her.\"\n\nBernie snorted. \"Like I knew what was going to happen?\"\n\n\"If we had been setting up like we were supposed to, we wouldn't have been there. We would have been in the kitchen,\" Libby snapped.\n\n\"We were cultivating a potential customer, which, in my humble opinion, was worth the ten minutes we were going to spend having Mark show us around,\" Bernie countered. \"Besides, it was interesting to see how the place was rigged up.\"\n\n\"Not to me,\" said Libby as she put her hand on her forehead. \"I'm going to have nightmares for years.\"\n\n\"Oh, don't be so dramatic,\" Bernie snapped.\n\n\"It's true. I am,\" Libby wailed.\n\nBernie groaned. \"Which you're going to blame on me, just like you do everything else.\"\n\n\"That is so unfair,\" Libby retorted.\n\n\"Girls,\" Sean said before Bernie could reply.\n\nBoth of them turned to him.\n\n\"Enough,\" said Sean.\n\n\"But,\" Libby began.\n\nSean held up his hand. \"I mean it. This bickering\u2014\"\n\n\"We're not bickering,\" Bernie objected.\n\n\"Fine,\" Sean said. \"Whatever you want to call what you're doing doesn't do no one\u2014\"\n\n\"Anyone...,\" Bernie corrected.\n\nSean glared at Bernie. She'd been like this ever since she'd learned to talk. \"Any good. So do you have to give the retainer back? Is the Haunted House closed down for the duration?\"\n\nLibby and Bernie both shook their heads.\n\n\"It's opening tomorrow afternoon,\" Bernie volunteered.\n\nSean snorted. \"You're kidding.\"\n\n\"Nope. That's what Mark told us.\"\n\n\"This has to be the fastest processing of a crime scene in the history of the town. But then what do I know? I'm an old man,\" said Sean. He tapped his fingers on the side of his chair. At least in his day, they would have kept the crime scene closed off for a couple of days. He'd learned to his cost that if you rushed, you always missed something.\n\nBernie took another sip of her Scotch and put her glass down. \"Well, it is a big fund-raiser for the volunteer fire department.\"\n\n\"And money always wins,\" Sean said after he'd taken another bite of his pumpkin bar. \"You better bake some more of these. You know what people are like. The more gruesome the crime, the more people want to see where it took place and the hungrier they are after they've seen it.\"\n\n\"Yeah,\" Bernie said. \"When business is slow, we should just kill someone.\"\n\nLibby glared at her. \"That's disgusting.\"\n\n\"But true,\" Bernie countered. \"Every time we're involved in a case, the shop is packed.\"\n\n\"I wonder why?\" said Libby.\n\n\"People are nosy,\" Sean said. \"Look what happens when there's an accident. Everyone always slows down.\"\n\nBernie frowned as she thought. Finally, she said, \"Or maybe it's because we miss that kind of stuff on some primordial level. You know, in the old days, they used to have public hangings and torture. Now there's nothing like that. Maybe that's why haunted houses and horror movies are so popular. People like to be scared and disgusted.\"\n\n\"Not me,\" Libby said firmly. \"Not in any way, shape, or form. In fact, I don't think I want to go back there.\"\n\n\"You don't have to if you don't want to,\" Sean told her.\n\nBernie untwisted her legs and stood up. \"Yes, we do. We have a contract.\"\n\nSean took a final bite of his pumpkin bar. \"I think a beheading might count as a cancellation clause, don't you?\" he asked after he swallowed. He'd love a cigarette now, but since the girls didn't know he'd gone back to smoking, he couldn't ask them to get him any.\n\nLibby folded over the empty wrapper of the chocolate bar she'd been eating and creased the line with her thumb. Then she did the same thing again.\n\n\"I'm serious. Maybe I won't go,\" she announced.\n\n\"Why?\" Bernie said. \"What more can happen? Anyway, then we'd have to give back the money, which we can't exactly afford to do.\"\n\nSean watched his eldest daughter run her thumb across the edge of the wrapper again. She didn't say anything.\n\n\"For real,\" Bernie said to Libby. \"What more can happen?\"\n\n\"Someone could cut off our heads,\" Libby replied.\n\nBernie rolled her eyes. \"Please. You can't be serious.\"\n\n\"Of course, I am. Why wouldn't I be?\" said Libby.\n\nSean coughed. The girls turned back to him.\n\n\"You know,\" he said. \"Amethyst has...had...a lot of enemies. I don't think this crime was committed by some nut looking to get his jollies off. I think it was committed by someone looking to kill Amethyst. So you two have nothing to worry about.\"\n\n\"My sentiments exactly,\" Bernie agreed.\n\nLibby stood up. \"But you don't know that for a fact.\"\n\n\"It's true, but that's what my gut tells me,\" said Sean.\n\nLibby plucked at the top button of the shirt she was wearing, then absentmindedly rearranged the magazines on the table over by the wall. \"She wasn't well liked, was she?\"\n\nSean wiped his fingers on the napkin in front of him. \"That's one way of putting it. Look at what she did to Bob Small.\"\n\nBernie put her hand to her mouth. \"I can't believe I forgot to tell you. We saw him today. He was working at the Haunted House, as a skeleton.\"\n\nSean raised an eyebrow. \"Now that's interesting.\"\n\nBernie sat back down and took a sip of her Scotch. \"I bet the police are going to pick him up in a hurry. I'd be surprised if he's there when we go back.\"\n\nLibby shook her head. \"I don't know. I can't see him doing this. He's just not the kind of guy who would chop someone's head off.\"\n\n\"In my experience, you get someone angry enough and you'd be amazed what they can do,\" Sean said. \"Remember Bernard? He weighed what? One hundred pounds, if that? He was so shy he could hardly look you in the face when he talked, and yet he managed to kill his two-hundred-and-fifty-pound girlfriend, drag her out of the house, and put her in the trunk of his car before he was caught, and that was only because he couldn't get the lid all the way down, so he tied it shut. If he'd gone to Boy Scout camp and practiced his knots, he might never have been found out.\"\n\nLibby frowned. \"Bob Small went to jail because he stole stuff.\"\n\n\"No,\" Sean said. \"Bob Small went to jail because he gave Amethyst a top-of-the-line BMW off the lot of the dealer he was working for so she could get to work and back while she had her car fixed. But instead of doing that, she took off to Florida with a guy she picked up at a bar and totaled the Beamer. Bob lost his wife and his job and spent a year in prison because of her little shenanigans. If that's not a motive, I don't know what is.\"\n\nLibby was about to reply when the door buzzer rang downstairs.\n\n\"You think Bob is guilty?\" Bernie asked her dad as Libby went to see who was at the door.\n\nSean shrugged. \"Don't know, but if I were the investigating officer, I'd like Bob for it. He had motive and opportunity. Of course, like I was saying, Amethyst had plenty of enemies in town. Considering all the things she pulled, it's a wonder someone didn't do something like this before.\"\n\n\"She liked to make trouble just because she could,\" Bernie observed as she studied one of her nails. She really needed to get them done. And talk about getting things done. She was developing frown lines. Maybe it was time for a little Botox. Character lines were nice, but no need to go too far.\n\n\"Some people get a real kick out of that,\" Sean said.\n\n\"In this case she got kicked back.\"\n\nSean grunted. His attention was focused on the footsteps running up the stairs.\n\n\"Who the hell is that?\" he asked as two men burst through the door. Libby was right behind them.\n\nShe gave her father an apologetic look. \"They insisted on coming up.\"\n\n\"It's okay.\" Sean could feel himself relaxing. He glanced up at the two men approaching him. \"Well, well, Curtis and Konrad Kurtz,\" Sean said. \"I haven't seen you boys in a while. Still got the same bad haircuts, I see.\"\n\nBoth men stopped. They shuffled their feet. Konrad hugged the tape deck he was carrying closer to his chest.\n\n\"We don't drink anymore,\" Curtis volunteered. \"That night was the last time.\"\n\nSean laughed. \"Yeah. That was quite a night. How many guys did you send to the ER to get stitched up?\"\n\n\"Six,\" Konrad said. \"But like Curtis said, we don't do that no more.\"\n\nCurtis raised his hand. \"I swear. Except for Thursday night bowling, when we have one or two brews.\"\n\n\"Sometimes three,\" Konrad said. \"But that's the max. Honest.\" He nodded toward his stomach with his chin. \"We're hardworking family men now.\"\n\n\"That's a good thing,\" Sean said.\n\nBoth men nodded solemnly.\n\n\"We're working for Mr. Kane,\" said Curtis. \"Maintenance. Before that, we worked for the housing complex off Ridge Road.\"\n\n\"That's very nice,\" Sean said. \"So what brings you boys out this time of night?\"\n\nCurtis adjusted his suspenders. \"It's about what happened at the Haunted House.\"\n\nSean waited. When nothing else was forthcoming, he told them to go on. There was more shuffling of feet; then Konrad spoke.\n\n\"See, we got this other hobby besides bowling and being volunteer firemen. Only it ain't really a hobby. It's more like an avocation. We go all over doing it. Some people think it's silly, but it's serious.\"\n\nCurtis pointed to himself and puffed his chest out. \"We're professional ghost hunters. We even went to school for it. We got certificates to prove it.\"\n\n\"You went to school?\" Bernie asked. \"What school?\"\n\n\"The Vincent Ludovic School for Paranormal Phenomena. You can look it up on the Web if you want to. We are trained professionals,\" said Curtis.\n\n\"I'm relieved,\" Bernie said. \"I thought you might be fakes.\"\n\nSean motioned for her to be quiet. \"Go on,\" he said.\n\nKonrad shot Bernie a reproachful glance. \"Well, we are the real thing. Being twins and all gives us a certain knack for it.\"\n\n\"I didn't know that twinness gave you a leg up in that department,\" Bernie said.\n\n\"It's true,\" Konrad told her. \"Boy Scout oath of honor. Me and Curtis been doin' this for a while now. We done the Perkins Place and the graveyard over at Three Trees. We even done this house over in Parker, PA.\"\n\n\"And how do you ghost hunt?\" Sean asked.\n\nKonrad motioned to his tape deck with a nod of his head. \"We get tape recordings. You can't hear anything if you're just standing there, but if you ask people questions, sometimes you can hear their answers when you play the tape back. It's called EVP Electronic voice processing.\"\n\n\"You mean electronic voice phenomena,\" Bernie corrected.\n\n\"There was a program on TV about that,\" Libby said.\n\n\"That's where we got the idea from,\" Konrad said. \"And like I just said, it turns out we got a real talent for it.\"\n\n\"I take it this has something to do with why you're here?\" Sean asked.\n\nAs Sean watched Curtis and Konrad nod, he noticed that they didn't really look like twins at all. Curtis was blond and skinny, while Konrad was broad and dark. But they had the same mannerisms and dressed alike.\n\n\"Okay. I think it's time you told me what this is about,\" said Sean.\n\nKonrad and Curtis looked at each other. After a few seconds, Konrad said, \"Well, Mr. Kane hired us to be ghost hunters for this haunted house thing, and we jumped at the chance because there are actual ghosts in there, and lots of people have seen them. There's Reverend Peabody; and his wife, Esmeralda; and Bessie Osgood, the kid that died, the one that was related to your wife.\"\n\nLibby turned to her dad. \"You never told me she was related to Mom.\"\n\n\"She was a distant cousin,\" Sean said.\n\n\"Why didn't you mention it?\" Libby asked.\n\nSean shrugged. \"There didn't seem to be any need to.\"\n\n\"Mom never mentioned it at all,\" Bernie said.\n\n\"Well, she wasn't one to mention painful subjects,\" replied Sean.\n\n\"Painful?\" Libby repeated.\n\n\"People said Bessie committed suicide,\" said Sean.\n\n\"That's not what Bessie says,\" Konrad interrupted.\n\nSean raised an eyebrow. \"And you know this how?\" he asked.\n\n\"'Cause, she told us. She told us other things, too. We got it right down on this tape recorder,\" said Konrad.\n\nThere sure was a lot of nuttiness out in the world, Sean decided as he looked at the two men standing before him. \"Fine,\" he said. \"But how does that explain why you're here?\"\n\n\"We're here,\" Curtis said, \"because we know who killed Amethyst.\"\n\n\"So go to the police,\" said Sean.\n\n\"We tried, but they didn't want to listen to our tape,\" said Konrad.\n\n\"And why do you think we will?\" Sean asked.\n\n\"Because,\" Curtis said, taking over, \"you always listened. You listened to our side that night.\"\n\n\"That was different. It was my job to listen,\" said Sean. \"There was a brawl between you and eight other guys, and I wanted to know what started it.\"\n\n\"And you let us go,\" Konrad said.\n\n\"The Myers brothers were punks.\"\n\n\"So you gonna listen, or what?\" Curtis asked.\n\nSean shook his head. He was definitely getting soft in his old age. But what the hell. Why not? It wasn't like he was going out anywhere tonight.\n\n\"Sure,\" he said. \"Play the tape. Let's hear what you got.\"\n\n## Chapter 4\n\nBernie watched Konrad put the machine down on the table in front of her dad's wheelchair. It was one of those old-fashioned reel-to-reel tape recorders.\n\n\"Where did you get that?\" she asked.\n\n\"From my uncle's basement. It's real old. Almost an antique,\" replied Konrad.\n\n\"It looks it,\" Bernie said.\n\n\"But it works better than cassette players or those new voice recording things. It picks up more stuff,\" said Konrad.\n\n\"I used to have one of those in the eighties,\" Sean remarked. \"My wife made me throw it out. Me, I like to keep things like that.\"\n\n\"That's because you're a pack rat,\" Libby observed.\n\n\"No. You just never know when something is going to come in handy,\" said Sean.\n\n\"Like Konrad said,\" Curtis replied, \"this deck works real fine.\"\n\n\"I didn't say it didn't,\" Bernie said. \"I just said it was old.\"\n\nKonrad held up his hand. \"Listen now.\" Everyone fell quiet and leaned forward. \"Here we go,\" he said, and he clicked the switch.\n\nBernie heard someone that sounded like Konrad say, \"Are we on?\" Then Curtis answered, \"We're rolling,\" and then she heard a lot of static and white noise.\n\nAfter a minute Curtis stabbed the air with his finger. \"Did you hear that?\" he asked excitedly.\n\n\"I hear static,\" Bernie said.\n\n\"No. Listen harder. There's Bessie,\" said Curtis.\n\n\"I'm sorry?\" replied Bernie.\n\n\"Focus,\" Curtis said.\n\n\"I'm trying,\" said Bernie. And she leaned forward, closed her eyes, and concentrated. She thought she heard someone say, \"Get out. Get out,\" in a hoarse whisper. A chill went down her spine. She shook her head. Bernie, get a grip, she told herself. This evening was affecting her more than she'd thought. Now her hearing was playing tricks on her.\n\n\"You heard something, didn't you?\" Konrad asked her.\n\n\"I'm not sure,\" Bernie said.\n\n\"That's Bessie,\" Konrad said. \"Here.\" And he stopped the tape and played it again.\n\nThis time Bernie didn't hear anything except a hiss. There were no words.\n\n\"I didn't hear anyone talking this time,\" Bernie said.\n\n\"Sometimes that happens,\" Konrad said.\n\nBernie turned to her father. \"Did you hear anything?\"\n\nSean shook his head, but the way he shook it made Bernie wonder if he had.\n\n\"I know I didn't,\" Libby said, but Bernie noticed her sister had the look she had on her face whenever she had one of her dreams.\n\n\"But you have to have heard it,\" Konrad said. \"I'll make it louder.\" And he turned up the volume. Now the room was filled with earsplitting static.\n\nSean winced. \"It reminds me of someone drawing their nails across a blackboard.\"\n\nAs Curtis leaned over the tape deck, Bernie decided if he got any closer, he'd be in it.\n\n\"See,\" Curtis said, stabbing the air again. \"There's Bessie.\"\n\nBernie shook her head. All she heard was a hiss. Maybe she heard a word. Home? House? No. There was nothing there. She was just hearing things because Curtis was suggesting that she do so. This was a subtle form of hypnosis.\n\nKonrad began pounding his leg with his fist. \"I can't believe you can't hear this. She's telling us she cut Amethyst's head off.\"\n\n\"I don't hear anything,\" Libby said.\n\nBernie straightened up. This was giving her a headache. \"You know,\" she said to Konrad, \"it's been a really bad day, and I want to finish my drink, take a shower, and go to bed.\"\n\nKonrad turned to Sean. \"You heard something, didn't you?\" There was a pleading note in his voice.\n\nSean shook his head. \"Sorry. I can't say that I did.\"\n\n\"Let me rewind it, and we can try again,\" said Konrad.\n\nSean held up his hand. \"Let's not. Instead, why don't you turn it off and tell me what you think Bessie Osgood told you.\"\n\nKonrad looked at Curtis, who shrugged and nodded.\n\n\"Okay,\" Konrad said.\n\nBernie breathed a sigh of relief as Konrad flicked the switch up. The room became blessedly quiet.\n\nKonrad and Curtis exchanged another glance. Then Curtis turned to Sean and said, \"She said she cut off Amethyst's head.\"\n\n\"That's it?\" asked Sean.\n\n\"What do you mean, 'That's it?'\" Curtis demanded.\n\n\"Well, did she say anything else?\" asked Sean.\n\n\"She says why she did it,\" said Curtis. \"Here\"\u2014his hand moved to the switch\u2014\"you can hear for yourself.\"\n\n\"That won't be necessary,\" Sean said quickly. \"Why don't you tell me instead?\"\n\nKonrad shrugged. \"It's simple. She did it because Amethyst threw her out the window, and she wanted to get even.\"\n\n\"I see,\" Sean said.\n\n\"So you don't believe us?\" Curtis cried.\n\nSean grimaced. \"I believe you think you heard that.\"\n\n\"No. We heard it,\" Konrad insisted.\n\nCurtis nodded. \"We did,\" he said. \"Her voice was as clear as a bell jar.\"\n\n\"Bell. The expression is 'clear as a bell,'\" Bernie corrected.\n\n\"That's what I said,\" Curtis told her.\n\n\"No. You said 'bell jar,'\" Bernie repeated.\n\n\"They're the same thing,\" Curtis retorted.\n\n\"No, they're not,\" Bernie said, and she went over and took another sip of her Scotch. As she was putting her glass down, she looked over and saw Libby eating the piece of pumpkin bar her dad hadn't gotten to yet. She glanced away before she caught her sister's eye. She was having a hard enough time keeping a straight face as it was.\n\n\"The problem is,\" Bernie said to Curtis, \"that ghosts are incorporeal beings.\"\n\n\"They don't get diseases,\" Konrad cried.\n\n\"No. Incorporeal, meaning 'without substance.' They don't have hands to grip axes or chain saws or whatever was used to cut off people's heads.\"\n\n\"They have energy,\" Konrad protested. \"They can do amazing things. We learned that in our class. Right, Curtis?\"\n\n\"Right, Konrad,\" said Curtis.\n\n\"I'm sorry, but ghosts don't go around lopping off people's heads, no matter how good the reason they have,\" Bernie told Konrad.\n\nCurtis put his hands on his hips. \"That shows you how much you know. Ghosts can do anything they want. They can move chairs....\"\n\n\"That's a poltergeist,\" Bernie snapped.\n\n\"Poltergeist is just a fancy name for a different type of ghost,\" Curtis told her.\n\nBernie threw up her arms. \"I give up.\"\n\n\"That's because I'm right,\" Curtis replied. Then he turned to Sean. \"She's awful excitable, isn't she?\"\n\n\"I am not!\" Bernie yelled.\n\nSean held up his hand. \"Let's talk about something else for a moment, if you don't mind.\"\n\n\"I don't mind,\" Curtis said.\n\n\"That was a rhetorical question,\" Bernie informed him.\n\n\"The question is,\" Sean said hurriedly, cutting Bernie off, \"isn't Bob Small related to you?\"\n\nCurtis looked at his feet.\n\n\"That's what I thought,\" Sean said. His legs weren't doing too good anymore, but that didn't mean his memory wasn't as good as it ever was.\n\nKonrad drew himself up. \"Maybe he is our cousin, but so what?\"\n\n\"The so what is obvious,\" Bernie retorted.\n\n\"Are you saying my brother and I are lying about this?\" Curtis said. \"That we have ulcerated motives?\"\n\nBernie laughed. \"You mean ulterior motives.\"\n\nCurtis's face began to get red.\n\n\"She's not saying that,\" Sean said quickly. Curtis and Konrad had had bad tempers when they were younger, and Sean was pretty sure that despite what they said, they still had them.\n\n\"Then what is she saying?\" Curtis asked.\n\n\"She's saying you are hearing what you want to hear,\" said Sean.\n\nKonrad lifted up his tape deck. \"Come on, Curtis,\" he said to his brother. \"Let's get out of here. No point in wasting anyone else's time.\"\n\n\"What did you want us to do?\" Sean asked Konrad.\n\n\"We wanted you to prove that Bessie Osgood killed Amethyst Applegate, of course,\" said Konrad.\n\n\"Of course,\" Sean repeated. He could just imagine what his pal Clyde would say when he heard about this one.\n\nCurtis shuffled his feet for a moment, then said, \"We'll pay you.\"\n\n\"Money is not the issue,\" Sean told him as he moved his wheelchair a little to the left so he could watch Mr. Wilson walk his Chihuahua, Merlin. It always amused him to see such a big man with such a little dog. At the moment Merlin was trying to subdue a jack-o'-lantern on someone's doorstep by peeing on it.\n\n\"Then what is?\" Curtis asked.\n\n\"There is no way to prove that Bessie Osgood killed Amethyst Applegate. Even if you had a viable tape, it wouldn't matter,\" Sean said after Mr. Wilson had rounded the corner. \"I've never heard of ghostly testimony being accepted by the DA. And let me go further. The original crime happened over twenty years ago, and if I remember correctly, opinion was divided as to its cause.\"\n\n\"The dead have just as much right to justice as the living,\" Curtis protested.\n\n\"You're going to have to take that up with the judicial system,\" Sean told him. \"I'm sorry, but there it is.\" He sighed. Why did Curtis and Konrad make him feel guilty? They shouldn't, but they did. \"So do you have anything else you want to tell me?\" he asked in the ensuing silence.\n\nCurtis and Konrad looked at each other. They both cleared their throats.\n\n\"We don't think it's fair,\" Konrad blurted out, \"that Amethyst got Bob in trouble before, and she's done it again.\"\n\n\"Bob loaned her the car,\" Sean said. \"She didn't put a gun to his head and force him to.\"\n\n\"That's true. But she sweet-talked him into it. If he hadn't met her, he'd be all right now,\" Curtis said. \"He'd still have his family and his job.\"\n\nThere was no arguing with that, Sean decided. \"Do the police have Bob in custody yet?\" he asked.\n\n\"No,\" Konrad said. \"But they're gonna.\"\n\n\"He's a convicted felon, and he was there,\" Curtis said. \"Of course, they're going to pick him up.\"\n\n\"True,\" said Sean as he spied Mr. Wilson heading back around the corner. He was carrying Merlin in his jacket pocket.\n\n\"The guy needs a break,\" Konrad continued. \"And it's especially frustrating for us because we got the proof, and no one will listen to us.\" He lowered his voice. \"Even our wives think we're a little wacko with this stuff.\"\n\nSean pursed his lips while he thought. \"I'll tell you what. How about if me and the girls look into this?\"\n\n\"That's all we want,\" Curtis said.\n\n\"But if whatever we find leads in the direction of Bob Small, that's the way we're going to go,\" said Sean.\n\n\"I keep telling you that Bessie Osgood did it,\" Curtis said.\n\n\"Maybe she did, and maybe she didn't,\" Sean said. \"Maybe someone else did. That's what we're going to try and find out.\"\n\nKonrad and Curtis nodded. \"We'll leave the tape for you,\" they both said at the same time.\n\n\"Appreciate it,\" Sean said. \"Tell me, how about the Reverend Peabody or Esmeralda? You heard anything from them?\"\n\n\"Dad,\" Bernie cried.\n\n\"I was just asking,\" Sean replied.\n\n\"To answer your question,\" Curtis said, \"we haven't yet. But we intend to try. Mr. Kane said he'd pay us a thousand bucks if we get their voices.\"\n\n\"Dad,\" Bernie repeated after the two men had left.\n\nSean looked up from the newspaper he'd picked up. It was two days old, but that was the way he liked his news. Past its prime. It gave some perspective to it.\n\n\"What?\" he asked.\n\n\"I can't believe you agreed to do that,\" said Bernie.\n\n\"I don't see why not,\" Sean retorted.\n\n\"Because Curtis and Konrad are crazy,\" said Bernie.\n\n\"A little strange maybe, but not crazy. After all, everyone has their private obsessions,\" said Sean.\n\n\"But Dad,\" Bernie continued. \"You just said that you thought Bob Small chopped off Amethyst Applegate's head.\"\n\n\"No,\" Sean corrected. \"What I said was that if I were the police, I would like him for it. That doesn't mean that I think he did it.\"\n\n\"But Bob Small had motive, means, and opportunity,\" Bernie wailed. \"Those are your words.\"\n\n\"I know. But I think he's too obvious,\" replied Sean. \"Over the years I've found that things that come wrapped up in pretty, neat little packages with bows on top of them rarely are what they seem.\"\n\n\"So you're saying that you think that Bob Small was set up?\" Libby asked.\n\n\"I'm saying it's a possibility,\" Sean replied. \"Which is why we should come up with a list of Amethyst's enemies and everyone who had access to the Haunted House and cross-reference them.\"\n\nBernie sighed. \"It's going to be a lengthy list.\"\n\nSean cast a longing glance at his paper. \"I'm aware of that.\"\n\n\"And we're doing this why?\" asked Bernie.\n\n\"Because the case interests me, and because I don't want to see our redoubtable chief of police put the wrong man in jail,\" said Sean.\n\n\"Lucy could do that,\" Libby agreed, referring to the chief of police by his nickname.\n\n\"Lucas Broadbent has done it,\" Sean said. \"Several times.\" He turned to Bernie. \"And while you're at it, see what you can find out about what happened to Bessie Osgood.\"\n\n\"You're kidding, right?\" Bernie asked.\n\n\"Not in the least,\" said Sean. \"My gut tells me she's at the root of this in some way or other.\"\n\n\"Is this the same gut that told you to go west to Mr. Leonida's house when you should have been going east?\" said Bernie.\n\n\"Just humor an old man, will you?\" said Sean.\n\n\"Fine,\" Bernie said. She finished the last of her drink. \"Maybe Curtis and Konrad do have something. After all, Halloween is the time of year when ghosts are supposed to come visit us mortals, the time when the veil between the two worlds is at its thinnest. The Celts thought so two thousand years ago. Who am I to argue?\"\n\nLibby groaned. \"You don't believe in ghosts, do you, Dad?\" she asked.\n\n\"No. But I believe in bad luck places. And I think the Peabody School falls under that category,\" said Sean.\n\nLibby frowned. Maybe the money they were getting wasn't worth it, after all.\n\n\"Oh, come on,\" Bernie said, looking at her sister's face. \"I told you nothing else is going to happen. Trust me on this.\"\n\nLibby got up. \"You know,\" she said, \"whenever anyone tells me that, I've found that the opposite is usually true.\"\n\nThen, before Bernie could reply, Libby went downstairs to bake some more pumpkin bars. Baking always made her feel better. And, anyway, her dad was right. They were going to be swamped tomorrow.\n\n## Chapter 5\n\nBernie looked at the three pumpkins she'd just carved to put in front of the shop's doorway. The first looked like a witch, the second looked like a cat, and the third one she'd carved into the shape of a goblin, complete with a wart on her nose. Thank heavens for how-to books, Bernie thought. What did people ever do before them? She removed her apron and dusted off her shirt. Black parachute silk and powdered sugar definitely didn't mix. But at least for once she hadn't gotten any pumpkin glop on her clothes.\n\nShe looked over at her sister, who was working away in jeans, a flannel shirt, and a T-shirt. Libby's clothes made more sense, especially when one was dusting powdered sugar on apple bars, but Bernie could never bring herself to wear outfits like that. What was the point? They were boring. And she had a reputation to uphold. People expected her to dress impractically now. It was part of who she was.\n\nBernie brushed a speck of powdered sugar off her black Dolce & Gabbana pants and checked her shoes for smudges. They were black suede and three inches high. Definitely not made for the kitchen, but she wasn't planning on being here for much longer. She had stuff to do for her dad.\n\n\"I'm going now,\" she said.\n\n\"I can see that,\" Libby told her sister, but her eyes remained focused on the pie dough she was rolling out. She'd done six pumpkin pies already and she had six more to go for the Haunted House. Her pumpkin pies always went fast. Maybe that was because she started out with real pumpkin puree instead of the canned stuff. The color was prettier, and the flavor and texture more delicate.\n\n\"Do we need anything?\" Bernie asked.\n\nLibby stopped for a moment and thought. \"A couple of gallons of cider and some red food dye. I want to make two more batches of devil cookies for the display case. Everyone seems to be enjoying them. And if you're stopping by Sam's Club, we could use some more chicken breasts for our red ginger chicken.\"\n\n\"But I got some yesterday,\" Bernie protested.\n\n\"We don't have any left.\"\n\n\"Have you looked in the cooler?\"\n\nLibby went back to rolling. \"Check if you don't believe me,\" she told her sister.\n\nBernie repositioned her bobby pin to stop her bangs from falling in her eyes. That was the problem with letting your hair grow in. It just went every which way. But, on the other hand, it did hide her crow's-feet. \"We sold that much?\"\n\nLibby shrugged. \"It would seem so.\"\n\nMaybe they had, Bernie thought. Or maybe someone was stealing the chicken. Except she couldn't believe that Googie or Amber would do something like that. Shrinkage was a definite problem in their business, but Googie and Amber had been with them for years.\n\n\"We should keep better track of our inventory,\" Bernie said.\n\n\"Yes, we should,\" Libby agreed.\n\nBernie realized that they had this discussion every six months or so, and nothing ever came of it.\n\n\"And I vote that you be the one to do it. And be back by three-thirty,\" Libby added. \"Don't forget, we have to be serving by five-thirty at the Haunted House.\"\n\n\"I haven't forgotten,\" Bernie said. \"I don't think this talk will take that long.\"\n\nAt least she hoped it wouldn't. She was going to talk to Felicity Huffer, who used to work as a proctor at the Peabody School. Bernie had spoken to Felicity's daughter earlier this morning and been told that Felicity lived in the Pine Bough Manor, a residential home for older adults. Now there was a euphemism if ever Bernie had heard one.\n\n\"I told her she could stay with me,\" Felicity's daughter had said. \"But she doesn't want to. I'm sure she'll be happy you're coming. She loves talking to people, and I can't get up there until later in the evening. In fact, I'll call her now and make sure it's all right. Sometimes she gets a little grumpy.\"\n\n\"Don't we all,\" her dad had said when Bernie told him what Felicity's daughter had said. \"Of course, if I remember correctly, she always was a bit irascible,\" he'd added.\n\n\"Maybe you should go,\" Bernie had told him. \"After all, you're the one that suggested this.\"\n\nHer dad had waved the suggestion away. \"She always liked your mom better than me. In fact, she never liked me at all.\"\n\nBernie was wondering why Felicity Huffer hadn't liked her dad when her sister put down her rolling pin and wiped her hands off on her flannel shirt.\n\n\"So,\" Libby asked her sister, \"are we really going to do this?\"\n\n\"Investigate Amethyst's death?\"\n\n\"I'm not talking about baking cookies.\"\n\n\"Yes, we are.\"\n\n\"Why?\"\n\n\"I figure Dad's done enough for us. Maybe we should return the favor.\"\n\n\"I guess you're right,\" Libby said doubtfully as she tugged at her bra.\n\n\"I know I'm right, and your sisters are still lopsided,\" Bernie informed her.\n\nLibby tugged on her left bra strap a little more. \"Better?\" she asked Bernie.\n\nBernie nodded. \"You need new bras. In fact, you need new everything.\"\n\n\"After Halloween,\" Libby said. \"When I have a little more free time.\"\n\n\"Why do you hate to shop?\"\n\n\"Why do you like to shop?\" Libby countered, and she turned back to her pumpkin pies.\n\nOn the way out, Bernie stopped and had Amber pack her up a box of pumpkin chocolate chip cookies for Felicity.\n\n\"You know,\" Amber said as she taped the box shut, \"you ought to make sure someone isn't doing a remake of Michael Myers's Halloween at the Haunted House. Maybe there's this homicidal maniac hiding in one of the rooms, with a chain saw, just waiting for you to arrive. You should keep your cell phone out just in case.\"\n\nThis, Bernie thought, is what happens when someone watches too many horror movies.\n\n\"If we get diced up, you'll be the first one I'll call,\" Bernie promised as Amber handed her the box full of cookies. \"In fact, I'll leave my cell on so you can hear every bloodcurdling scream. Now go wait on Mrs. Stein.\" And with that, Bernie walked out the door.\n\nShe stood in the street for a moment and took a deep breath. It was one of those glorious late fall mornings. The air smelled spicy\u2014like cinnamon and cloves. The sun was still warm, and the leaves remaining on the trees were crimson and gold. Most of the houses on the other side of the street had decorations in their windows: there were witches and goblins and black cats. There were tombstones in the yards. There were jack-o'-lanterns on people's porches. In a week the street would be full of parents and children in costume knocking on doors and yelling, \"Trick or treat!\"\n\nHalloween had been her favorite holiday when she was little. She still remembered her best costume ever. Her mom had made it, and she thought it was the prettiest costume she'd ever seen. It was a blue taffeta dress with a sparkly sash and pale blue wings and a wand. And she'd had a crown on her head and ruby slippers on her feet. She'd gone as Glynda the Good Witch from The Wizard of Oz, and she'd gotten so much candy, the pillowcase she'd carried was half full by the time her dad made her come home.\n\nHalloween was still her favorite holiday. Every year she and Libby opened A Little Taste of Heaven and stood in the doorway and gave out homemade candy and cookies. They labeled every bag they gave away so parents would know where the treats had come from, which the parents seemed to agree with. Otherwise, they'd have to serve the prepackaged stuff, which would be a shame.\n\nBernie sighed. It was a pity that they wouldn't be able to do that this year, but they wouldn't be at the shop; they'd be serving at the Haunted House. She'd definitely miss seeing the little kids come parading by, but what could she do. Mark was paying them well, and they needed the money, although she wasn't sure that any amount of money could compensate for what she and Libby had seen last night. Whenever she closed her eyes, she could see Amethyst's head rolling down the stairs. She could still hear the bump, bump, bump that it had made.\n\nBernie shuddered and tried to think of something else, but she couldn't. The image was fixed in her mind. She kept wondering how it had been done. The person couldn't have used a chain saw. That would have made too much noise. So maybe the weapon was an axe. Which meant someone might have found it on the premises. Or someone could have brought it in. It was really impossible to know.\n\nOr someone could have used a fiber-optic laser beam. She'd just seen someone cut glass with one a few weeks ago. Of course, things would be clearer when the coroner did the postmortem. From what Clyde had told her dad, it was scheduled for tomorrow.\n\nShe hadn't looked closely enough to see if the cuts on Amethyst's neck were smooth or jagged. She'd been too shocked. Of course, if she didn't want to wait for the PM, she could always ask Marvin to find out. His dad would know since both parts of Amethyst's body were resting in his funeral home. Bernie took another deep breath and got in the car. Maybe Amber and Libby were right, Bernie thought as she started up her vehicle. Not that she would let either of them know that. Maybe she and Libby shouldn't go back. Maybe there was some crazy person there, waiting to claim another unsuspecting victim, although that was not what her father thought.\n\nIf he did, he would never have allowed them to go back there, no matter what the circumstances. And he was usually right about these kinds of things. Thirty years in law enforcement had given him pretty good instincts. It was a thought Bernie consoled herself with as she drove over to see Felicity Huffer.\n\nThe lobby of the Pine Bough Manor was practically deserted when Bernie walked in. There was a small cluster of people gathered around a bulletin board, and she could hear the tinkle of music and someone exhorting everyone to \"breath in and out and focus on letting your energy go out into the world.\"\n\nA vision of people walking their energy on leashes became lodged in Bernie's mind. She shook her head to clear it and looked around. There was a fountain over by the far wall, with some goldfish swimming in the pond by the base. The dining room stood off to the right.\n\nThree large ficus trees stood in pots over by a large picture window. The floor was carpeted in a pale green tweed, while the furniture was covered in light tans. As Bernie approached the reception desk, she decided the place reminded her of the lobby of a moderately priced hotel.\n\nThe woman at the reception desk smiled when Bernie asked for Felicity, and pointed over to the sofa in the back of the room. \"She's waiting for you,\" the woman said. \"She's very excited. In fact, I believe she has something to give you.\"\n\nBernie wondered what it was as she made her way over to the sofa she'd been directed to. At first, she didn't notice anyone there, and then she saw a small, kid-sized figure dressed in beige. It wasn't until she got closer that she realized that the figure must be Felicity Huffer. If she were taller than four feet eight, Bernie would have been surprised.\n\nFelicity turned as she heard Bernie approaching and smiled. \"You're Rose's daughter, aren't you?\" she asked in a voice that seemed way too loud for the body it was coming out of.\n\nBernie allowed as how she was.\n\nFelicity patted the space next to her. \"Sit,\" she said.\n\nBernie sat.\n\n\"Your mother was a dear woman. Very refined. And an excellent cook. I understand you and your sister have inherited her ability,\" said Felicity.\n\n\"I hope so,\" Bernie said.\n\nFelicity pointed to the box of cookies Bernie was holding. \"Are those for me?\"\n\n\"Yes. Of course. They're pumpkin chocolate chip cookies from our shop, A\u2014\"\n\nFelicity cut her off. \"I know the name. Even though I'm ninety, I haven't lost my mind. Though there are those that would like to think so,\" she finished darkly. \"I'm sure the staff will like them. I can't digest anymore,\" she explained. \"I exist on a diet of rice, bananas, and bread.\"\n\n\"I'm sorry to hear that,\" Bernie told her.\n\nFelicity waved her words of sympathy away. \"It doesn't really matter. At my age, everything tastes the same to me, anyway. So my daughter tells me you want to hear about the Peabody School.\"\n\nBernie nodded. She didn't think she'd ever seen anyone this thin before. It was as if she was looking at an anatomy illustration. For the first time, she actually understood what the term skeletal-looking meant.\n\n\"Your mother was heartbroken about what happened to Bessie, you know,\" Felicity said.\n\n\"She never talked about it to me.\"\n\n\"I'm not surprised. She was a woman who liked to keep unpleasant things to herself. It was a big tragedy,\" Felicity said. \"Bessie was a good girl. Studious too. She could have done something with herself. Not like some other people I could name.\" Felicity frowned.\n\n\"So what happened?\" Bernie asked.\n\n\"She went out the window. I was the one that found her.\" Felicity's voice trembled slightly. \"Terrible. Truly terrible. The windows were very low. Almost French windows. All you'd have to do is lean out of them to fall. One push and there you'd go. It wouldn't even take much. Of course, some people say she jumped. Others say she fell, that what happened was an accident. But I don't believe that for a moment. I didn't then, and I don't now. I think she was killed. I think she was killed out of spite and jealousy.\"\n\nAs Bernie leaned forward to better hear what Felicity was saying, she caught Felicity's musty scent: dry, brittle paper mixed with the aroma of lavender water.\n\n\"So you know who killed her?\" she asked.\n\nFelicity put her hand up and coughed into it. \"I know, but I never told anyone.\"\n\n\"Why not?\"\n\n\"Because,\" she said when she was done coughing, \"I could have lost my job.\"\n\n\"How so?\" Bernie asked.\n\n\"Well, I didn't have any proof, you see. And the girl's family was very rich, very powerful.\" Felicity paused for a moment. \"I needed that job. If I had lost it, I don't think I could have gotten another one. Some people think that Zinnia was the one, but even if she was, it was Amethyst that put her up to it. She was a bad seed. I know it's not fashionable to use that expression anymore, but it's true. That's why she was there, you know. Because her parents were afraid of her. Her mother told me that in one of her visits. Sweet woman, too. Imagine being afraid of your own child. But things happened when Amethyst was around. Bad things.\"\n\n\"Well, this time something bad happened to Amethyst.\"\n\n\"So my daughter told me. Which, of course, is why you're here. You want to know if I can shed any light on the situation.\"\n\n\"Exactly,\" Bernie said.\n\nFelicity laughed. It was more of a rattle, actually. \"Of course, I can.\"\n\nBernie waited. Felicity didn't say anything for a moment.\n\nThen she said, \"Do you like games and puzzles?\"\n\n\"As well as anyone else,\" Bernie lied. In reality she didn't like them at all.\n\n\"Are you good at them?\"\n\n\"Moderately,\" Bernie replied. \"I'm good at crossword puzzles.\"\n\nFelicity made a dismissive noise. \"I'm talking about a real puzzle.\"\n\nAnd she reached behind her and presented Bernie with a brown paper bag. It was crinkled and splotched with grease here and there. This brown paper bag, Bernie thought, had seen better days. It didn't smell too great, either. It smelled as if it had been storing things that Bernie didn't want to think about.\n\n\"Well, look inside,\" Felicity snapped.\n\nBernie did, with a great deal of circumspection.\n\n\"Take everything in there out,\" Felicity ordered.\n\nBernie reached in gingerly and pulled out an old View-Master and a wooden puzzle box that was covered with smudge marks.\n\n\"Thank you,\" Bernie said, not knowing what else to do.\n\n\"Take them with you,\" Felicity instructed. \"And on your way out, see Odella at the reception desk, and tell her it's time for my nap.\"\n\n\"What are they?\" Bernie asked, nodding toward the items that had been in the bag.\n\nFelicity looked at her with an unmistakable expression of annoyance on her face. \"They're the solution to your problem, of course. I just told you what you wanted to know.\"\n\n\"I'm sorry, but I must not have been paying attention.\"\n\n\"The box and the View-Master,\" Felicity told her. \"The answer is in those. Especially the box. The View-Master not so much.\"\n\n\"I don't understand,\" Bernie stammered.\n\n\"What's to understand? I just told you. The answer to the question you want me to answer is in the bag. Solve the puzzle and you'll find it.\"\n\n\"What puzzle?\" Bernie asked. She was at a total loss.\n\n\"The one in the bag,\" Felicity snapped. \"Honestly. These days people expect you to do all their work for them. When I was younger, we had to figure things out for ourselves. Now go. Go.\"\n\n## Chapter 6\n\nSean looked at the items Bernie held out to him. She put them on the table in front of him.\n\n\"Interesting,\" he said, picking up the View-Master. \"I used to have one of these when I was a kid.\" He held it up to the light. \"Have you looked at the pictures?\" he asked.\n\nBernie nodded. \"When I got in the car. They're all pictures of the Peabody School.\"\n\nWhen Sean was done, he put the View-Master down and picked up the wooden box. He studied it for a moment. \"I haven't seen a puzzle box like this in years.\" He played with it for a moment, then put that down as well.\n\n\"So what do you think?\" Bernie asked her dad when he went back to sipping his coffee.\n\n\"Felicity could be playing a joke,\" he mused. \"It would amuse her to think that I was spending my time over these things. That woman always was a pain in the ass, not to mention being a real nut job.\"\n\n\"So why did you send me there?\"\n\nSean gave a slight shrug. \"I thought she might have mellowed. Evidently, I was wrong.\"\n\n\"Do you think there's anything in what she gave me that will help?\"\n\n\"I'll play around with the puzzle box for a while, but I don't think there's going to be anything in there.\" He sighed. \"Boy, that woman can hold a grudge.\"\n\n\"What do you mean?\"\n\n\"She never forgave me for marrying your mom.\"\n\n\"But why?\" Bernie couldn't believe someone wouldn't like her dad.\n\n\"Felicity was your mom's babysitter at one time. She was quite attached to her\u2014probably too attached\u2014if you get my meaning. And she disapproved of me. She didn't think I was good enough for your mom. I don't know if anyone would have been. She thought your mom deserved better then being a policeman's wife. And maybe she did. I thought she would have forgotten about that by now, but evidently, she hasn't.\" Sean frowned. \"Oh well. On to Plan B.\"\n\n\"And what is Plan B?\"\n\nSean looked at his daughter. \"Can't you guess?\"\n\n\"Find out who had a motive for killing Amethyst?\"\n\nSean beamed. \"Exactly.\" He always felt good when he'd managed to teach something to his children.\n\nIt was going to be a long list, Bernie thought as she went into her room to change into her Halloween costume for the Haunted House. Today she was going to go as a witch. Of course, the bottom of witches' dresses traditionally didn't have ruffles lined in pink, but so be it. She wasn't going to be wearing the kind of homespun cloth witches in thirteenth-century Europe would wear. It would be way too scratchy.\n\nLibby was waiting for her sister downstairs. She and Amber had already packed everything and were in the middle of loading the van.\n\n\"Now be careful,\" Amber warned when Bernie appeared. Libby closed the doors of the vehicle. \"Here,\" Amber said, pressing two small medallions into Libby's and Bernie's hands. \"This is for just in case.\"\n\nLibby looked down at the silver hand with an eye drawn on the palm. \"Just in case what?\"\n\n\"It's to ward off the evil eye,\" Amber explained.\n\n\"There is no such thing as an evil eye,\" Libby said. \"That's just a superstition.\"\n\nAmber gave her a reproachful look. \"Don't you know what happens in movies when the hero or heroine refuses to wear the protective amulet?\"\n\n\"They get chopped up into hamburger,\" Bernie replied.\n\n\"Exactly,\" Amber said.\n\n\"Fine,\" said Libby, as she put the amulet around her neck. \"See. Feel better now?\"\n\nAmber nodded.\n\nLibby shook her head as she drove down the street. She liked Amber, but there were some things about her she just couldn't understand\u2014like her fondness for horror movies. \"What did dad say when you showed him the bag?\" Libby asked as they turned onto Ash Place.\n\n\"He said Felicity was always a little crazy. He said that stuff could be her idea of a joke.\"\n\n\"Lovely,\" Libby said. \"Are you planning on wearing that hat this evening?\"\n\nBernie patted her hat. \"Why? What's wrong with it?\"\n\n\"It's just very large.\" Libby could see herself knocking it into the pumpkin pies by accident.\n\n\"Witches always wear large hats.\"\n\n\"I wouldn't know. I've never made an actual study of it.\"\n\nBernie glanced appraisingly at her sister. \"You should get in costume, too, Libby.\"\n\n\"I don't like costumes,\" Libby said. \"You know that.\"\n\n\"You don't have to go as the Cowardly Lion.\"\n\n\"I'm not going as anything,\" Libby said firmly. \"Besides, it'll be hard to serve people.\"\n\n\"Not if you're wearing the right thing. Come on,\" Bernie pleaded. \"It'll be fun.\"\n\n\"No,\" Libby said as she turned onto the road that led to the Haunted House. \"I always feel ridiculous.\"\n\n\"For Halloween night, you can go as...\" Bernie paused for a moment while she thought. \"I've got it. Fruit of the Loom tighty whities.\"\n\n\"That's it,\" Libby yelped.\n\n\"I'm kidding.\"\n\n\"No, you're not.\"\n\n\"You have to admit it would be funny.\"\n\n\"Not to me.\"\n\n\"We'll get you a nice costume,\" Bernie told her soothingly. \"Something sexy.\"\n\n\"I don't want my boobs hanging out.\"\n\n\"But they're nice boobs.\"\n\n\"Yes, they are,\" Libby agreed. Marvin thought so, too. She just didn't think it was appropriate to show them off when she was serving food.\n\n\"We'll do funky.\" Bernie snapped her fingers. \"Wait. I know. You can come as Esmeralda's ghost. That would be really freaky.\"\n\n\"I'll see.\" Libby demurred. She reached over and turned on the radio, signaling that she didn't want to discuss dressing up anymore.\n\nTen minutes later they were at the school. Libby parked as close as possible to the entrance that led to the kitchen.\n\n\"Loading and unloading are the parts of this job that I hate the most,\" Bernie grumbled.\n\n\"Tell me about it,\" Libby agreed as she lifted a carton filled with cider out of the back of the van.\n\nIt took them about twenty minutes to unload the van and about half an hour to set up. They decked the long tables with the orange and black checkered tablecloths that Libby had found at the dollar store. Then Bernie arranged the decorations: five carved jack-o'-lanterns, pots of mums, and three small candles featuring Casper the Friendly Ghost. She stepped back and looked at the results.\n\nMaybe she should forget about Casper. After all, there weren't going to be any young children. But then some adults liked cartoony stuff. After another moment of debate, she left the candles on the table.\n\n\"You know,\" Bernie said as she began laying out the paper cups, napkins, and spoons and forks, \"we should have had bobbing for apples.\"\n\n\"Too messy,\" Libby said firmly as she began setting out the waffle irons. \"There'd be water all over the floor, and then we'd have to worry about someone slipping.\"\n\n\"But it would be fun,\" Bernie protested.\n\n\"Undoubtedly,\" Libby said as she began setting up the workstation for the waffles. \"But that's not the issue.\"\n\nShe could just see it now: someone walking by and slipping on the floor because it was wet, grabbing the table for support, and having the whole thing come down. Then they'd get sued. No. Bobbing for apples was out of the question. She checked her waffle supplies. They had the homemade strawberry and apricot jam, the hot fudge sauce, the apple compote, whipped cream, maple syrup, and strawberries, but no butter. She went through the box again. No. It wasn't there. She explicitly remembered telling Amber not to forget to pack it.\n\n\"Damn,\" Libby muttered. She should have checked everything over herself. This was what happened when she got lazy.\n\n\"What's the matter?\" Bernie asked.\n\n\"Amber forgot the butter.\"\n\n\"No biggie. I'll run back and get it,\" Bernie told her. She looked at her watch. She had a half hour before they opened. Shouldn't be a problem at all.\n\n\"And bring some of the glazed walnuts while you're at it,\" Libby said. They would go well with the maple syrup.\n\nBernie nodded, put on her jacket, and hurried out the door. Libby turned back to arranging the cookie bars on the platters she'd brought. She probably should have brought some of their larger serving plates, but these would have to do for the moment. She'd just put one platter down and was busy taking the pumpkin pecan bars out of their Tupperware container when she heard a door slam.\n\n\"Bernie?\" she called out, even though it was way too early for her sister to have gotten to the shop and returned. Maybe she'd forgotten something.\n\nNo one answered.\n\nSuddenly, Libby was aware of how quiet the room was. And how large. Another door slammed.\n\n\"Bernie?\" Libby repeated.\n\nHer sister's name echoed in the silence.\n\n\"Hello?\" Libby cried.\n\nNo one answered.\n\n\"This is ridiculous,\" Libby said out loud.\n\nObviously, the sounds were being made by people walking back and forth in the hallway. She looked down. She didn't remember doing it, but somehow she'd taken the amulet Amber had given her off of her neck and was clutching it in her hand. This was what came from listening to Amber and those ghost-hunter freaks. Libby shook herself. She hated when she got like this. She was acting the way she had when she was five.\n\nLibby shook herself again, took a deep breath, and went over her to do list, holding it in her mind the way she was holding the amulet in her hand. She had to focus on finishing setting up. She had to finish plating the cookies, pouring the waffle batter into jugs, putting out the pies, slicing up the pumpkin loaves, arranging the drinks, and setting up the urns for hot coffee, tea, mulled cider, and hot chocolate.\n\nShe started humming Bob Dylan's \"It Ain't Me Babe\" to herself. That always helped. She didn't know why, but it did. It had since childhood. She was finishing up overlapping the pumpkin chocolate chip cookies when she felt something funny around her.\n\nIt was like static electricity on her arms, neck, and shoulders. Then she felt a blast of cold air on her cheeks. She felt the hairs on the back of her neck stand up. She didn't know how she knew they were doing that; she just did. It isn't a figure of speech. They really do that, she thought as she noticed the goose pimples on her arms. Suddenly, she was very, very cold. Colder than she'd ever been. The air around her seemed wavy, as if she was looking down the road on a hot summer's day, with the heat beating down on her.\n\nA young female voice whispered in her ear, \"See you later, alligator.\" Then she giggled.\n\nLibby whirled around. For a second, she thought she saw the outline of a girl wearing a button-down oxford shirt and a long, pleated skirt, and then the image vanished. A moment later the prickly sensation on her skin and the coldness surrounding her were gone as well.\n\nLibby told herself she was imagining things. Or that she was going crazy. But she knew she wasn't. Her mother was right. She was weird.\n\nLibby cleared her throat. \"Are you Bessie Osgood?\" she asked.\n\nThere was no answer. Of course not, Libby thought. What had she expected? Some sort of rapping? Two raps for yes, one rap for no?\n\nYeah, right. There had to be another explanation, had to be, even though she knew in her heart there wasn't. Maybe this was some kind of trick. If it was, she didn't think it was one bit funny.\n\nShe went out in the hallway and took a look around. Bob Small, who was wearing his skeleton costume, waved at her as he walked past. Guess they haven't arrested him yet, Libby thought. She wondered why. Her dad had been sure that Bob Small would be behind bars by now, and he usually wasn't wrong about things like that.\n\n\"Did you see anyone out here?\" she asked Bob.\n\n\"The Ghost of Christmas Past. Whoops. Wrong holiday.\"\n\n\"Seriously.\"\n\n\"Nope.\" Bob shook his head and hurried along.\n\nFor a moment, Libby thought about going through the rooms of the Haunted House, but she quickly ruled that out. She didn't have the time, and even if she did, what was she looking for?\n\nShe didn't know. That was the problem.\n\n## Chapter 7\n\nMarvin looked at Libby, who was perched on her bar stool, trying to make a bird out of a cocktail napkin and failing.\n\n\"You actually saw a ghost,\" he said.\n\nLibby smoothed the napkin out and tried again. \"I didn't say that. I said it seemed as if I saw a ghost.\"\n\n\"Seemed?\" Marvin repeated.\n\n\"Seemed,\" Libby said firmly.\n\n\"Well, I'm still envious. I've never come close,\" said Marvin.\n\nBrandon put the beer Marvin had ordered down in front of him and then got Libby and Bernie their Irish coffees. \"Not once?\" Brandon asked after he'd given Bernie a quick hello kiss. His shift was over, but his replacement hadn't come in yet.\n\n\"Never,\" Marvin said. Then he pointed to the napkins with R.J.'S BAR AND GRILL printed on them in green and blue. \"These are new,\" he observed.\n\n\"The owner's son is starting a printing business, and his dad is trying to help him out,\" said Brandon.\n\nMarvin nodded as Bernie turned toward him\n\n\"Has your dad ever seen a ghost?\" she asked.\n\n\"Nope. He doesn't believe in them,\" replied Marvin. \"He says that when you're dead, that's it.\"\n\nBrandon surveyed the bar. Everyone seemed satisfied for the moment, so he asked his next question. \"Do you believe in ghosts?\"\n\n\"Not in the least,\" Marvin said, with a great deal of conviction.\n\n\"He's had plenty of opportunity to see them, too,\" Libby pointed out.\n\nBernie lifted up her glass, changed her mind, and put it back down. \"Not really,\" she said. \"Ghosts tend to haunt places.\"\n\n\"Like funeral homes,\" Brandon said.\n\n\"No. Like places where they died a violent death. Personally, I think they're some sort of leftover energy that's just stuck there,\" said Bernie.\n\nMarvin frowned. \"I agree with my father\u2014for once\u2014on this one. I think when you're dead, you're dead. You go in the ground, and that's the end of the story. You don't go to heaven. You don't get reborn as something else. You just disappear. In all the time I've worked with my dad, I've never seen or felt anything that was vaguely ghostlike.\"\n\n\"How long has your dad been a funeral director?\" Brandon asked.\n\nMarvin took a sip of his Brooklyn Brown and wiped his mouth on one of the napkins on the bar. \"Maybe thirty years. Maybe thirty-five. I'm not exactly sure.\"\n\nLibby toyed with her glass for a moment and then took a sip. She could feel herself begin to relax. Who was it that said that Irish coffee was the perfect mix of fat, sugar, and alcohol? \"I think I was meant to see Bessie Osgood's ghost,\" she blurted out.\n\n\"Obviously,\" Bernie said.\n\n\"No. I mean, I think someone wanted me to see her,\" said Libby.\n\nBernie raised an eyebrow. \"Someone?\"\n\n\"The someone who did this,\" replied Libby.\n\nBernie snorted. \"And who would that be?\"\n\n\"I don't know,\" Libby told her.\n\n\"But why would someone do that?\" Brandon asked Libby.\n\n\"Because,\" she replied, \"Curtis and Konrad are ready to swear that Bessie Osgood killed Amethyst, and my seeing her backs up their story.\" Libby explained what the ghost hunters had told her dad.\n\n\"Those two guys are nuts,\" Brandon scoffed. \"They also believe in UFOs. In fact, one of them offered to hook me up with a ride.\"\n\n\"And you didn't go?\" Bernie asked.\n\nBrandon laughed.\n\n\"I think,\" Libby continued, \"that what I experienced in the kitchen might be an attempt to keep us from investigating any further. Whoever did it is betting that we'll buy into this fantasy they've created.\"\n\n\"From the way you describe it, you have to admit it's a pretty elaborate fantasy,\" Brandon said.\n\n\"Exactly,\" said Marvin as he picked a handful of peanuts out of the bowl and began to shell them. \"The sensations, the cold, the vision. If what you experienced was created by someone, the question becomes, how was it accomplished? Who has the technical know-how to do this?\"\n\n\"That's easy,\" Bernie said. \"FX Productions, the outfit that set up the show.\"\n\n\"What do we know about them?\" Marvin asked.\n\nBrandon whipped out his iPhone. \"I'll Google them and find out.\" A moment later he said, \"Here they are. They seem pretty legit to me.\" He passed the phone to Bernie.\n\n\"Expensive,\" she said after she'd read the company's Web page. \"Lots of references.\" She handed the phone back to Brandon.\n\n\"I bet they can't be too pleased that they're involved in something like this,\" Brandon said. \"There's a contact number. Maybe I should give them a call and see what I can find out.\"\n\n\"Sounds like a plan,\" said Bernie. She tapped her fingernails on the base of the glass holding her Irish coffee. \"And in the meantime, maybe we should talk about who disliked Amethyst.\"\n\n\"Now that,\" said Brandon, \"will be easy.\" He excused himself for a moment to wait on the people down at the end of the bar.\n\nMarvin swept the peanut shells onto the floor and looked around the place. R.J.'s never seemed to change. There was the dartboard over on the right, the pool table by the window, the tables for two shoved up against the wall, the historical pictures of Longely hanging slightly crookedly on the wall, the spindly ficus tree fighting for life in the window, and the peanut shells on the floor.\n\n\"There aren't many people here,\" he observed.\n\n\"Monday nights are always slow,\" Libby replied. \"You know,\" she continued, \"I just realized that they never put up any holiday decorations around here.\"\n\n\"A good idea if you ask me,\" Bernie noted. She sucked her thumb. It was still sore from using the edge of a spoon to dig out the seeds from all the pumpkins she'd carved in the last two days.\n\n\"Our regulars come here to drink, not to be reminded of Halloween,\" Brandon noted as he planted himself in front of Bernie. \"The only decoration they need is a glass.\"\n\n\"I don't understand. How can you not want to be reminded of Halloween?\" Bernie asked.\n\n\"Because they don't,\" said Brandon. \"They want to forget everything when they come in here. That's what serious drinkers do.\"\n\nLibby interrupted. \"Can we get back to Amethyst?\" she asked, annoyed. She felt as if no one was taking this seriously enough.\n\n\"Sure,\" Brandon said. He unscrewed the top of a bottle of water and took a drink. \"I'll tell you who's on the top of my list. Inez Colley.\"\n\nBernie took another sip of her Irish coffee. \"I thought she went off to Arizona.\"\n\n\"She did, but she came back,\" said Brandon.\n\n\"When?\" Marvin asked.\n\nBrandon looked at the ceiling while he calculated his answer. \"About three months ago, give or take a couple of weeks.\"\n\n\"Where's her husband?\" Marvin asked.\n\n\"Still in the monastery in Kyoto. I don't think he's coming back from Japan,\" Brandon replied. \"A guy who works for his former boss told me Kevin was taking the precepts, or doing whatever it is you do to become a Buddhist priest.\"\n\n\"Usually, it's the criminal, not the victim, that finds religion,\" Bernie noted.\n\nMarvin took another drink of his beer. \"Obviously, not in this case. Do we know what happened specifically?\" he asked.\n\n\"No,\" Bernie replied.\n\n\"Yes,\" Brandon said at the same time.\n\n\"How do you know?\" Bernie demanded.\n\n\"Because he told me,\" said Brandon.\n\n\"He did?\" asked Bernie.\n\n\"Yeah. He was drunk off his ass,\" said Brandon.\n\n\"And you never told me?\" said Bernie.\n\n\"A man never reveals what another man tells him when he's under the influence of alcohol,\" Brandon said.\n\nBernie rolled her eyes.\n\n\"Hey,\" Brandon said, \"I have my code of ethics. But I'll tell you now because it's important.\"\n\nBernie patted her chest. \"Be still, my heart.\"\n\n\"Do you want to hear this or not?\" said Brandon.\n\n\"We want to hear it,\" Marvin said.\n\n\"All right, then,\" replied Brandon as he took another sip of water. \"You know that Amethyst was working for Inez's husband, right?\"\n\nLibby nodded. \"As an office manager.\"\n\nBrandon leaned forward. \"Well, according to him, one night, when they were working late, Amethyst slipped something in his drink, had sex with him, and videotaped it.\"\n\nBernie snorted. \"Yeah, right. Poor Mr. Innocent.\"\n\nBrandon shrugged. \"It's possible.\"\n\nBernie rolled her eyes. \"But not likely.\"\n\n\"Okay. Not likely,\" Brandon agreed. \"But I definitely believe what Kevin said happened next, which was that Amethyst threatened to show the tape to his wife unless he paid her off. Which he did by refinancing his house and taking a loan out on his business.\"\n\n\"Stupid,\" Marvin said.\n\n\"It must have been quite a tape,\" Bernie observed.\n\n\"I'd say,\" replied Brandon as he scanned the room again. \"So here was Kevin, thinking the problem was taken care of, when Amethyst came back, asking for more money. Kevin freaked. He wimped out and took off for Japan to become a Buddhist monk, leaving Inez without a pot to pee in.\"\n\n\"Nice guy,\" Marvin commented.\n\n\"But, Brandon, Inez doesn't know about Amethyst, so she can't want to kill her,\" Bernie protested.\n\n\"She does know. Kevin told me he left Inez a note,\" said Brandon. \"Explained everything in it. Asked her forgiveness. Blah. Blah. Blah.\"\n\nLibby blinked. \"Wow. Poor woman.\"\n\nBrandon took another drink of water and screwed the top back on the bottle. \"I saw her at Sam's Club last week. She doesn't look so great.\"\n\n\"I can imagine,\" Bernie said. \"What's she doing now?\"\n\n\"She's on a cleaning crew,\" replied Brandon.\n\n\"You're kidding,\" Bernie cried.\n\n\"Nope,\" said Brandon, shaking his head.\n\nLibby clicked her tongue against the roof of her mouth. This was a woman who used to come into their store and order four hundred dollars' worth of food at a go without batting an eye.\n\n\"Jobs are tight these days, and with her problem, this was the only job she could get, and that's only because Ian felt bad for her,\" Brandon explained. \"My friend Ian White runs AAAPlus Clean.\"\n\n\"Didn't she work in a lab doing something with remote sensing?\" Libby asked.\n\nBernie finished her Irish coffee. \"She did before she got married and became Mr. Rich Man's wife and started drinking and got two DWIs. Now, I'm sure she couldn't get a security clearance to work at Wal-Mart.\" Bernie stretched. \"I'd say she had a motive to kill Amethyst.\"\n\n\"I'd go for the husband,\" Libby said.\n\n\"Yeah,\" Bernie said. \"But Kevin is in Japan.\"\n\n\"Good point,\" Libby said. \"And Inez has never been too stable.\"\n\n\"Just because she threw a tantrum in our shop when she found out we were out of broccoli cole slaw and we had to call the police to get her out?\" Bernie asked.\n\n\"Something like that,\" Libby said.\n\n\"You want me to find out if she's working tonight?\" Brandon asked.\n\n\"So late?\" Libby asked.\n\n\"That's when Ian's crews work,\" said Brandon. And he punched his friend's phone number in and walked to the end of the bar. A few minutes later, he was back. \"Funny thing,\" he said. \"But she's cleaning at the Foundation as we speak.\"\n\n\"Huh,\" Bernie said. \"Curiouser and curiouser. I think we should go have a chat with her.\"\n\n\"Now?\" Libby said.\n\n\"When better?\" Bernie asked.\n\n\"Tomorrow is better. I want to go home and go to bed,\" replied Libby.\n\n\"Then Brandon and I will go,\" said Bernie.\n\n\"We will?\" Brandon asked. \"I thought we had other plans.\"\n\n\"First things first,\" Bernie said firmly.\n\n\"And then we can go to my place,\" said Brandon.\n\nBernie grinned. \"That depends on your performance.\"\n\nBrandon leered. \"My performance is always stellar.\"\n\nBernie laughed and ate a peanut.\n\n## Chapter 8\n\nBernie watched the Peabody School rise in the moonlight as Brandon rounded the bend in the road. She hugged herself as she turned toward Brandon.\n\n\"This would make a great set for a horror movie,\" she said.\n\n\"Yeah,\" Brandon said. \"The only thing it lacks is bats and a belfry.\"\n\n\"It has a colony of bats.\"\n\n\"I was being metaphorical.\" Brandon pointed to the top of the building. \"It also has a widow's walk.\"\n\nFor a moment Bernie thought she saw a faint shape, a large, light spot in the dark, but when she looked again, it was gone. Probably an optical illusion, she told herself.\n\nBrandon headed toward the main entrance. \"Senior year, Ben Altman bet me I couldn't get all the way up there on Halloween night.\"\n\n\"And did you?\"\n\nBrandon shook his head. \"I chickened out before I got to the second floor. I thought I heard voices.\"\n\n\"Maybe you did. Maybe someone else was in there.\"\n\n\"There were no cars in the lot.\"\n\n\"Maybe it was a couple of squatters,\" said Bernie.\n\n\"Maybe. But I wasn't about to stay and find out.\"\n\nBernie rubbed her arms. For some reason, she was cold. \"You think Libby saw something earlier this evening?\"\n\n\"Definitely. Don't you?\"\n\n\"Yeah. I do. She was really freaked out.\"\n\n\"Do you think she really saw Bessie Osgood?\" asked Brandon.\n\n\"Don't tell anyone, but yeah. I think she did.\"\n\n\"Freaky.\"\n\n\"Halloween is a freaky time of year.\"\n\nBrandon turned into the parking lot. \"There's the cleaning van,\" he said, changing the subject.\n\nBrandon parked right beside it, and he and Bernie got out.\n\n\"And no cops,\" Bernie observed. According to Clyde, they'd packed up and left the crime scene a couple of hours ago. \"Did your friend Ian say anything about Inez?\" Bernie asked Brandon as they walked to the front door.\n\nBrandon zipped up his jacket before replying. \"He said she was a mess. He thought she was drinking again.\"\n\n\"No surprise there,\" replied Bernie.\n\n\"I guess not. He told her she had to have a doctor's note if she called in sick one more night,\" said Brandon.\n\n\"Another DWI and she's going to be going to jail for a long time,\" said Bernie.\n\n\"She's going to be going to jail for an even longer time if she killed Amethyst,\" replied Brandon.\n\n\"True,\" Bernie agreed. \"And she could have. After all, she knows how lasers work.\"\n\n\"If that's what was used.\"\n\n\"I'm guessing fiber-optic laser wire,\" Bernie informed him.\n\nBrandon stared at her. \"Where do you get this stuff from?\"\n\n\"Well, I was reading an article on a new piece of work Jacobs is doing.\"\n\n\"Jacobs?\"\n\n\"The sculptor, uncultured one.\"\n\n\"I'm cultured. I like yogurt.\"\n\nBernie faked a groan. \"Anyway, it got me thinking that that would fit the bill. It's light. It's quiet. It's easy to manipulate.\"\n\n\"Why not a samurai sword or a machete?\"\n\n\"Not enough blood.\"\n\n\"Maybe she was killed somewhere else and moved.\"\n\n\"Maybe,\" Bernie agreed. \"But I don't think so.\"\n\n\"Your feminine intuition tells you this?\"\n\nBernie chucked Brandon under the chin. \"Exactly. I could be wrong. Who knows,\" she mused. \"Maybe the person who killed her used a piece of flexible, glass-coated glazier's wire.\"\n\nBrandon shook his head. \"Sometimes you scare me.\"\n\n\"Then don't piss me off.\" And Bernie pulled her turtleneck up till it covered her chin. She definitely should have brought a scarf. \"Did Ian say how many nights a week they cleaned here?\" she asked, changing the subject.\n\n\"Five.\"\n\n\"That's a lot for an office, isn't it?\"\n\n\"Evidently, Kane's a clean freak.\"\n\n\"So they were here last night?\"\n\nBrandon nodded. \"But they don't clean the Haunted House area.\"\n\n\"How come?\"\n\n\"Kane doesn't want them to. He said there was too much delicate equipment in there. Touch the wrong thing and kablamo. There goes one of the exhibits.\"\n\n\"Who does the cleaning then?\" asked Bernie.\n\nBrandon shrugged. \"I assume the volunteers do, but you'll have to ask Kane.\"\n\n\"And Inez was here cleaning the Foundation part?\"\n\n\"That's what Ian says.\"\n\nBy now Bernie and Brandon were at the door. Brandon put his hand on the handle. \"So, what are you going to say to Inez?\" he asked.\n\n\"I'm going to ask her if she killed Amethyst.\"\n\n\"Seriously?\"\n\n\"Yes, seriously. Just like that.\"\n\nBrandon threw back his head and laughed. \"And you expect her to say, 'Yes, I did'?\"\n\n\"No. I just want to see how she reacts.\"\n\nBrandon pulled the door open. \"A waste of time if you ask me.\"\n\n\"Do you have any other suggestions?\"\n\n\"Yeah. Let's go back to my place and talk about this first.\"\n\n\"After,\" Bernie said and walked inside.\n\nBrandon sighed and followed.\n\nThe Foundation was not what Bernie had expected. She'd psyched herself up for a scary mansion, and she'd gotten a generic office instead. For a reason she couldn't explain, she felt oddly disappointed.\n\nThe lights were on in the hallway, and Bernie could see that Kane had done some serious remodeling. Now the walls were white, instead of wallpapered in paisley, and wainscoted, and the floor was a sea of gray carpeting.\n\nThere were metal-framed pictures on the walls, mostly featuring trees and flowers and grass. About twenty feet in was a reception desk, and just after that was a modern black leather sofa and a coffee table with fanned-out magazines. In the background, Bernie could hear Aretha Franklin competing with the hum of a vacuum cleaner.\n\n\"Boy, this has sure changed,\" Brandon observed. \"I remember all this weird wallpaper and dark wood and crystal lamps hanging from the ceiling.\"\n\n\"So it would appear,\" Bernie said as she cautiously opened the nearest door on the right. A small plaque on the door read MS. LONG. \"I guess Kane was telling the truth when he said the site they're using for the Haunted House is the last place they have to remodel.\"\n\nBernie peeked inside. The room was small and generic. It contained a desk; a monitor, keyboard, and mouse on the desk; a standard-issue office chair; a wall full of bookshelves, which appeared to contain reports; and stacks of reading material on the table by the door. The room was devoid of any personality. Looking at it, Bernie decided she could be in any office anywhere in the world. For all intents and purposes, the old Peabody School was gone except for where the Haunted House was. And soon that would be gone, too. It made her sad. She carefully closed the door behind her and started down the corridor.\n\n\"Let's go find Inez,\" she said.\n\nBrandon hurried after her. \"The sooner the better as far as I'm concerned,\" he told her.\n\n\"So you've said,\" Bernie retorted.\n\nThey walked down the corridor. When they got to the end, they followed the music and turned left. Aretha was louder now. So was the vacuum cleaner.\n\n\"How many people are on this cleaning crew?\" Bernie asked.\n\n\"Ian said three,\" Brandon replied.\n\nThey made another left. Bernie began to feel disorientated. The place was like a rabbit warren, all sharp, angled turns. One hallway led to another, and the white walls and gray carpet and overhead lights made everything look the same. Half of the rooms looked as if they were unused. By now the words to \"I Heard It Through The Grapevine\" were pulsing through Bernie's head. She paused in front of the door where the music was coming from.\n\n\"Let's do it,\" Brandon said and pushed open the door.\n\nThey stepped into a paneled conference room. A vacuum cleaner was parked by the far wall. A large, gleaming wooden table sat in the center of the room. The air smelled of furniture polish. A man in coveralls was bent over the table, spraying its surface with Pledge and wiping it down.\n\nBernie went over and tapped him on the shoulder. He jumped and spun around. It took him a minute to focus his eyes. Whatever he was on, it had taken him someplace else.\n\n\"Sorry,\" Bernie shouted. \"We're looking for Inez.\"\n\nThe man blinked. He made an effort to look at Bernie. She smiled encouragingly.\n\n\"Is she around? We need to talk to her,\" Bernie explained.\n\nShe'd expected that he'd ask why, but he didn't. If anything, he seemed annoyed at being interrupted. He shrugged and pointed to the door on the far wall.\n\n\"Through there,\" he told her. \"She's cleaning the bathrooms. At least that's what she's supposed to be doing. Whether she is or not, I can't say.\" He nodded toward Brandon. \"Hey, pal. How's it going?\"\n\n\"It's going, Josh,\" Brandon said. \"It's going.\"\n\n\"You can say that again,\" Josh said and went back to waxing the table.\n\n\"How do you know him?\" Bernie asked when they got outside.\n\n\"His brother was in our class.\"\n\n\"He was?\"\n\n\"Matt Keller.\"\n\n\"That's Matt Keller's little brother?\"\n\nBrandon nodded. \"Yup.\"\n\nBernie shook her head. She remembered him as this blond little pain that was always following Matt around. Now he looked about fifty and was missing half of his teeth.\n\nBrandon must have read her mind, because he said, \"That's what happens when you live on the streets. You want me to talk to him while you go have a chat with Inez and see if he can tell us what she was doing last night?\"\n\nBernie nodded and went off to find Inez. She located her a little while later in the ladies' room. The door was propped open, held in place by a large garbage can. When Bernie walked in, she could see Inez leaning against one of the sinks, smoking a cigarette. Bernie did a double take. Inez must have gained at least fifty pounds since she'd last seen her. Inez's face had that round, puffy look drinkers' faces got. Her waist had vanished, replaced by a layer of fat that ballooned over her belt. Even her hands looked pudgy.\n\n\"What do you want?\" she asked Bernie.\n\nSuddenly, Bernie decided that Brandon was right. This wasn't such a great idea.\n\n\"To talk to you, Inez.\"\n\nInez took another drag of her cigarette and blew the smoke out of her nostrils. Just like in some B movie, Bernie thought.\n\n\"You want to know why I haven't been in your store?\"\n\n\"Not exactly,\" replied Bernie.\n\n\"I don't have time to talk to you. I've got work to do.\"\n\n\"So I see,\" said Bernie, nodding at the cigarette.\n\n\"Well, I'm going to answer your question for you, Miss Girl Detective,\" said Inez.\n\n\"Can you make that Ms. Girl Detective?\"\n\n\"You think you're smart, don't you?\" hissed Inez.\n\n\"Yes, I do. Now what's the question you're going to answer for me?\"\n\n\"The one you were going to ask me about Amethyst. I didn't kill her. The police already questioned me. And let me go. Go talk to Bob Small. He was there.\"\n\n\"So were you.\"\n\n\"No. I was here cleaning.\"\n\n\"That's what I said.\"\n\n\"Shows you how much you know,\" Inez sneered. \"You can't get from here to the Haunted House section of the mansion. They sealed it off when they remodeled. You have to go outside and around.\"\n\n\"And what's to prevent you from having done that?\" asked Bernie.\n\n\"I was with Josh all the time I was here. We were cleaning the offices.\"\n\n\"It takes two people to do that?\"\n\n\"That's the way Ian says to do it, so that's what we do,\" said Inez. She took another drag of her cigarette and flicked it in the toilet.\n\nBernie folded her arms over her chest and leaned up against one of the sinks. \"Maybe you set up some sort of remote device.\"\n\n\"And how would I have gotten Amethyst there?\"\n\nBernie shrugged. \"I don't know. Wrote a note. Called her and set up a meeting. You're smart.\"\n\n\"I'm not smart enough,\" Inez said.\n\n\"Why do you say that?\"\n\nInez pointed to one of the toilets. \"That's what I let my life turn into.\"\n\nBernie didn't say anything, because what Inez said was true.\n\n\"I'm glad Amethyst is dead,\" Inez continued. \"She deserved everything she got. I only wish she had suffered more. She ruined my life. If it weren't for her, I'd still have my husband and my house. Look at me.\" Inez pointed at herself. \"Look at the way I look now. Look what I'm doing. I shop at Wal-Mart, for God's sake. But I didn't kill the bitch. I wish I had, but I didn't. If I had, I wouldn't have been so merciful. Talk to Bob Small. Talk to Zachery Timberland.\"\n\n\"Zachery Timberland?\"\n\nInez laughed through her nose. It was not a nice laugh, Bernie thought.\n\n\"Yeah. Zachery Timberland. Ask him about his daughter Zoe. Ask him what she's doing now.\"\n\nBernie was about to ask Inez what Zoe was doing now when Brandon appeared at the bathroom door.\n\n\"She and Josh were together last night,\" he said as he stepped inside the ladies' room. He looked around. \"I've always wanted to see what one of these looked like on the inside.\"\n\n\"Well, now you know,\" Bernie replied. She turned to Inez. \"You could have both done it.\"\n\nInez snickered. \"If I were going to do something like that, I certainly wouldn't choose a chucklehead like Josh for my partner.\"\n\nBrandon tugged at Bernie's arm. \"Come on. Let's go.\"\n\n\"Listen to your boyfriend,\" Inez said.\n\n\"I'll go when I'm ready,\" Bernie said, even though she knew Brandon was right. She'd gotten as much as she was going to get from Inez this go-around. Staying longer wasn't going to help anything.\n\nShe was turning to go when Inez called out to her.\n\n\"I have a question for you,\" she said.\n\n\"Yeah?\" said Bernie.\n\n\"How the hell do you walk in those?\" She pointed to Bernie's stilettos.\n\n\"Very carefully,\" Bernie said. \"Very carefully, indeed. And I have a question for you.\"\n\n\"What?\" Inez snarled.\n\n\"Have you met Bessie Osgood?\"\n\nBernie watched Inez's mouth wobble. It was as if someone had taken a giant vacuum cleaner and sucked everything out of her.\n\n\"Get out of here,\" Inez cried. \"I'm not talking to you anymore.\"\n\n\"Obviously, I hit a nerve,\" Bernie said to Brandon when they were outside.\n\n\"Obviously, you did.\"\n\n\"I guess my dad is right,\" Bernie said.\n\n\"How do you mean?\" Brandon asked as he and Bernie retraced their steps.\n\n\"He said that Bessie Osgood is at the center of what happened, and I'm starting to think so, too.\"\n\n## Chapter 9\n\nSean carefully seated himself in a chair in front of a long table and laid before him the bag with the View-Master that Felicity Huffer had given Bernie. His idea probably wouldn't come to anything, but he wanted to check it out, anyway.\n\nIt would have been easier for him to use his wheelchair in here since getting up and down was a problem for him. It would have been more comfortable, too, but he hated it, so he'd brought Marvin along to help out instead. Which had its own set of problems.\n\nLibby said he didn't use the chair, because he was too vain, but that wasn't it at all. It was that he saw using it as a sign of giving in, and for him, giving in meant giving up. He'd been that way all his life, and he couldn't change now. Maybe that was because his mom had taught him not to take the easy way out. Ever.\n\nThe chair was a little too deep for him, but he settled back in it as best he could and looked around. The Longely Historical Society was housed in an old Victorian house that had belonged to one of the town's founders. The town board had rescued it from the wrecking ball and had put on a new roof and painted the outside lilac, an act that Sean had never understood, and that had been the extent of the fixing up. The house itself was narrow and cluttered with pictures and objects from the town's past.\n\nRecently, Halloween decorations had been added to the mix. At this time of day, the Longely Historical Society was empty, but then, it usually was. Most people in this town were too busy worrying about whether or not they could afford a Beamer to think much about the past. In fact, he, Marvin, and the librarian, Jeanine Applegate, who was the real reason they were here, were the only people in the place.\n\n\"Sit down,\" Sean hissed at Marvin, who was roaming around like a demented, fidgeting whooping crane. Why he couldn't stay still was quite beyond Sean.\n\n\"I just wanted to look at the pictures on the wall,\" Marvin said.\n\n\"You're distracting me.\"\n\n\"But you're not doing anything,\" Marvin pointed out.\n\n\"I'm thinking.\"\n\nMarvin turned and almost knocked a stack of brochures off one of the shelves.\n\n\"And you're going to break something.\"\n\nMarvin looked hurt, and Sean felt remorseful for a second or two. But it was true; Marvin was a klutz. Plus he was overly sensitive. The kid had to toughen up.\n\n\"Did you know that jack-o'-lanterns come from Ireland and that people used to use turnips?\" Marvin said.\n\nNow this, Sean thought, is what my daughter would call a random comment. \"Yes, Marvin, I know,\" Sean said.\n\n\"Because you're Irish?\"\n\n\"Because Bernie told me.\"\n\n\"That would mean the Irish don't eat roasted pumpkin seeds. That's one of my favorite parts about Halloween.\"\n\nSean took a deep breath and let it out. \"Marvin.\"\n\n\"Turnips don't have seeds.\"\n\n\"Yes. I know.\" And with that, Sean reached out and grabbed the newspaper that was sitting there, held it up to his face, and pretended to read the front page. He'd never had that much patience, but now that he was getting older, he had none at all.\n\nGod, Marvin talked a lot. There was no disputing that. His driving wasn't great, either, but Libby loved him, and basically, he was a good kid, even if he did make Sean crazy. Back in the Ice Age, when he'd been chief of police, he'd learned from the man he'd served under that you had to work with what you had and bring your men up to a higher level of functioning. That was what being a leader meant.\n\nSean thought about that for a moment, and then he went back to thinking about what he hoped to accomplish here. Besides talking to Jeanine, who was Amethyst's second or third cousin, he couldn't remember which, he wanted to look at the old pictures of the Peabody School and read any material they had pertaining to the place. He was also hoping that the Longely Historical Society had some materials on Bessie Osgood, although he kind of doubted it.\n\nHe threw Jeanine his most charming smile as she came toward him, and she smiled back. Honestly, he couldn't imagine two more different people than Amethyst and Jeanine. Jeanine was sweet and low-keyed, while Amethyst had been...What was that word his mother had always used about women she didn't like? He had it...Amethyst had been a mantrap. Nothing about her had been real.\n\n\"Good to see you, Jeanine,\" Sean said when she got close enough.\n\nRather than shake Sean's hand, she nodded, because her hands were full of books and newspaper clippings. Sean noted she was wearing a pleated skirt, a blouse with a rounded collar, and a green cardigan sweater. A large pin of a witch on a broomstick was attached to her collar. He approved of her clothes; they were sensible and pleasant. And they didn't call attention to themselves the way some of the things that his youngest daughter wore did.\n\nJeanine smiled again. \"Good to see you, too, Sean. I'm glad to see you're up and about.\"\n\nSean spread his hands out. Since they were no longer shaking, he could do that now. \"I'm trying.\"\n\n\"Well, I'm glad you are.\" Jeanine put the stack she was carrying down in front of him. \"Here,\" she said. \"I figure this is what you're interested in.\"\n\nSean was amazed. \"How do you know what I'm interested in?\"\n\nJeanine laughed. \"Well, you're not really into local history, so what else could you be here for but the Amethyst thing.\"\n\nSean just stared at her. How had she known, and how had she pulled everything together so fast?\n\n\"I'm not a mind reader,\" Jeanine said, interpreting his silence correctly. \"Actually, I did this before you arrived because I was interested myself. When I saw you coming up the walkway, I ran and got what I'd found.\"\n\n\"And did you find anything?\"\n\nJeanine shook her head. \"Nothing that I didn't know before. Although I'd forgotten that Bessie Osgood was related to your wife.\"\n\nSean shifted his weight around. \"Rose never liked to talk about it.\"\n\n\"She was like that.\"\n\n\"Yes, she was.\"\n\n\"Well, if it's any consolation, my family never liked to talk about Amethyst's possible role in Bessie's death,\" Jeanine said. \"They just pretended it was an accident.\"\n\n\"But you didn't think so?\"\n\n\"No. Absolutely not. And I don't think they thought so, either. They never said anything, but it was just the sense I got.\"\n\nSean nodded toward the chair next to him. \"Why don't you sit down? Unless you have something else to do, that is.\"\n\nJeanine laughed. Sean decided her laugh was like her clothes. Nice.\n\n\"I'm sure my cataloging will wait for me,\" she said as she folded herself into the chair he'd indicated. She fussed with her pin for a moment, and then she said, \"I don't like to talk this way about people, but Amethyst was just bad. She always was. That's why she was in that school, you know. Her mom was afraid of her.\" Jeanine leaned forward and lowered her voice. \"She got mad at her mom one day and killed her cat. Set it on fire, and then she tried to blame my cousin Natalie.\"\n\nMarvin came and stood behind Sean. \"How did everyone know it wasn't Natalie?\" he asked.\n\n\"Because Natalie loved animals. She was one of those people who always brought strays home with her,\" said Jeanine.\n\n\"Then why did Amethyst pick her?\" asked Sean.\n\n\"Probably because she thought it would be funny, you know, Natalie being blamed for something like that. She even planted one of Natalie's books at the scene. And she was so convincing. If you didn't know Natalie, you'd really think she'd done something like that.\"\n\n\"Maybe we could talk to Natalie,\" Sean said.\n\nJeanine's eyes misted over, and then she blinked, and the tears were gone. \"I'm afraid that's not possible. She died in a plane crash a couple of years ago. Amethyst was so nice on the outside, but inside...\" Jeanine shuddered. \"I tried to stay as far away from her as I could when I was growing up. One day she and Inez Colley were baby-sitting me\u2014\"\n\n\"Wait,\" Sean interrupted. \"Inez was friends with Amethyst?\"\n\n\"Oh yes,\" said Jeanine. She wrinkled her brow while she thought. \"Along with Zinnia McGuire and Zachery Timberland and Bob Small. They used to sneak into the school at night and visit Amethyst. I think Bessie Osgood saw them, and she was going to tell.\"\n\n\"Are you sure?\" said Sean.\n\n\"No. I'm not sure about anything. I just remember overhearing my parents talking,\" replied Jeanine. \"Then they saw me outside, listening, and changed the subject.\"\n\n\"Interesting,\" Sean said. \"Maybe one of them killed Bessie.\"\n\n\"Maybe,\" Jeanine replied.\n\n\"I don't suppose you kept in touch?\" Sean asked.\n\n\"With Amethyst?\" asked Jeanine.\n\nSean nodded.\n\n\"Kind of. She called when she needed something.\" Jeanine fingered her pin. \"I spoke to her about four months ago. She wanted to know if I could give her Ed Banks's private phone number.\"\n\n\"The guy who owns Lexus Gardens?\" asked Sean.\n\n\"Yeah.\"\n\nSean remembered that Bernie had tried to get in contact with Ed Banks and had been told by his personal assistant that he didn't talk to people he wasn't familiar with.\n\n\"He's not very friendly.\"\n\n\"He's a recluse.\"\n\n\"Like Howard Hughes?\" asked Sean.\n\n\"Not that bad, but heading in that direction. I wasn't going to give her the number, but then Amethyst called again, and she was so sweet...That was her talent, you know. She made you believe you were her best friend, and even though you knew it was a lie, you still wanted to believe her.\"\n\n\"So you gave it to her?\" asked Sean.\n\n\"Yes, I did.\"\n\n\"I wonder what she wanted it for.\"\n\nJeanine shrugged. \"Your guess is as good as mine.\"\n\n\"And Zinnia? What about her?\"\n\n\"She died a while ago,\" said Jeanine.\n\nSean raised an eyebrow.\n\n\"No, no,\" Jeanine said. \"It was nothing like that. She was in a car accident.\" She pushed the stack of articles in front of Sean. \"And now I'm sure you're anxious to get going on these materials.\"\n\nActually, Sean wasn't at all anxious to get going on his reading. He was having a good time talking to Jeanine, but he smiled and thanked her again for the time she'd taken with him.\n\n\"Nice lady,\" Marvin said after she'd gone back to her office.\n\nSean grunted.\n\n\"I don't think she's seeing anyone,\" Marvin said.\n\n\"Now why would you say that?\" Sean demanded.\n\n\"I don't know,\" Marvin stammered. \"I thought you might be interested.\"\n\n\"Well, I'm not,\" Sean snapped as he went through the papers that Jeanine had brought him. He separated out all the ones with pictures of the Peabody School. The rest he pushed toward Marvin with the tips of his fingers.\n\n\"Read these,\" Sean ordered.\n\n\"What am I looking for?\" Marvin asked.\n\n\"Anything of interest.\"\n\n\"But how will I know what's of interest to you? I mean, I thought that thing about the turnips being jack-o'-lanterns in Ireland was pretty interesting, and you didn't.\"\n\n\"Just read,\" Sean hissed. He didn't know why he was suddenly in such a bad mood.\n\nMarvin opened his mouth, closed it again, and began to do what he was told. As soon as he was settled, Sean took all the pictures of the Peabody School out and spread them on the table. Then he began to compare them with the slides from the View-Master. Half an hour later, he wasn't any better off than he had been before. He had his hand on the small of his back and was stretching when Jeanine came back out of her office.\n\n\"I haven't seen one of those in years,\" Jeanine said, pointing to the View-Master.\n\n\"Me either,\" said Sean. He explained where he'd gotten it from.\n\n\"So what are you hoping to find?\" Jeanine asked.\n\nSean shook his head. \"I have no idea,\" he confessed. \"No idea at all.\"\n\n\"Mind if I take a look?\" Jeanine asked.\n\n\"Be my guest,\" said Sean. He watched as she went through the slides.\n\n\"I don't get it,\" she said.\n\n\"Neither do I,\" replied Sean.\n\n\"These slides are pictures of the Peabody School,\" Jeanine noted.\n\nSean nodded.\n\n\"What are they supposed to show?\" asked Jeanine.\n\nSean shook his head. He hated to admit it, but he didn't have a clue. \"I thought if I could compare some photos with the slides, it might give me an idea, but it hasn't.\"\n\n\"Maybe you should go talk to Felicity Huffer,\" Jeanine suggested.\n\n\"That's whom I got them from. Or rather my daughter did. Felicity Huffer just told her the answer to our problem is there and to go figure it out for herself.\"\n\nJeanine made a face. \"I could see her doing that and it having nothing to do with the solution to your problem.\"\n\n\"She could,\" Sean said, thinking of what she'd been like.\n\n\"Age doesn't necessarily make people nicer,\" said Jeanine.\n\n\"That's for sure,\" Marvin interjected.\n\nSean glared at him, and he went back to reading the papers he'd been given.\n\n\"What do you think the odds of that are?\" Sean asked Jeanine.\n\n\"I'll tell you what,\" Jeanine said after a moment had gone by. \"Felicity's daughter is on the board of the Longely Historical Society. Maybe she can help us. Would you like me to talk to her?\"\n\nSean nodded.\n\n\"All right then,\" said Jeanine. \"And would you mind if I kept the View-Master and looked at the slides again? Maybe something will occur to me.\"\n\nSean could feel himself smiling. \"That would be lovely,\" he said.\n\n\"Good,\" Jeanine said. \"I'll call you either way.\" She gathered up the materials and withdrew to her office.\n\nA few minutes later Sean and Marvin were out the door.\n\n\"Aren't you going to ask me what I found?\" Marvin asked when they were settled in Marvin's black limo.\n\n\"What?\" Sean asked as he fastened his seat belt. He hated riding in this car. It reminded him of riding in a hearse, but as Libby had said to him when he'd complained, \"Beggars can't be choosers.\"\n\n\"I didn't find out anything. All the articles in the papers reported Bessie Osgood's death as an accident.\"\n\n\"That's what I expected,\" Sean said.\n\n\"Why do you say that?\"\n\n\"Because that's what was in everyone's interest to do. After all, if you report a murder at a private boarding\/day school, most people will pull their kids out.\"\n\n\"True,\" Marvin said.\n\n\"Of course, they did, anyway,\" Sean said.\n\nHe and Marvin were silent for a moment.\n\nThen Sean said, \"Bessie's death pretty much closed the school.\"\n\n\"What happened to the owner?\"\n\n\"George Marak killed himself. He'd put all the money he had, plus his wife's money, into the place, as well as borrowing from his family and friends. He couldn't stand the disgrace when it became clear that the school was going to have to close, so he shot himself in the garage. The note he left asked his wife to forgive him for the shame he'd brought on her.\n\n\"It would have been more considerate if he'd killed himself somewhere else, because she and his son found his body when they came back from grocery shopping. PS: The kid was in the front seat, so he saw everything. The wife never got over it. I dare say the kid didn't, either. The wife died in an auto accident six months later. She'd been drinking and ran her car into a tree. Their kid, poor thing, went to live with a relative in Texas or Wyoming, some place like that.\"\n\n\"That sucks,\" Marvin said.\n\n\"Doesn't it, though.\" Sean leaned back in his seat. \"Like I said, the Peabody School has always been a bad-luck place. And now, if you don't mind, I think I'd like to take a drive up to Lexus Gardens and see if we can talk to Ed Banks and find out what Amethyst wanted from him.\"\n\nSean could feel his gut tightening as Marvin zoomed away from the curb without looking or putting his signal light on. He'd faced coked-up guys and guys with loaded rifles in the line of duty, but they weren't as scary as driving with Marvin.\n\n## Chapter 10\n\nBernie took in a breath of fresh air as she drove toward Zachery Timberland's office. It was a beautiful late fall afternoon, and she was glad to be outside driving around instead of inside the shop. She glanced at the clock on the dashboard. It was a little after three, and she figured she didn't have to start up to the Peabody School until four-thirty, which should leave her more than enough time to talk to Timberland.\n\nShe drove slowly, enjoying the sensation of being in the car, imagining what fun a road trip to Vermont with Brandon would be. She was thinking that fall was her favorite season of the year when her cell phone trilled \"Raindrops Keep Falling on My Head.\" It was her dad's ring. She fished her cell out of her bag and answered it.\n\n\"Dad?\" she said.\n\nShe got static.\n\n\"Dad?\"\n\nHe was talking, but his voice sounded like a jigsaw puzzle\u2014each syllable another piece. She couldn't understand a thing he was saying. She tried calling back. Nothing. She'd just noticed she was down to no bars when her phone went dead. That was when she remembered she'd forgotten to charge it last night. Drat and double drat, she thought as she tossed her cell on the seat next to her. She wondered if her dad wanted to tell her something about Zachery Timberland. He was always a big one for armchair quarterbacking. Oh well. Too late now.\n\nShe knew Timberland well enough to say hello to. Her only memory of him was the time, last March, she'd met him at Laura's place. She'd shaken his hand, and it had been unpleasantly clammy.\n\nWhen she'd called around this morning, it turned out that none of her friends knew him either, not surprising when you considered that his family had moved away after the Bessie Osgood incident, and he'd just come back to town a couple of years ago. Couple that with the fact that Timberland had palled around with Amethyst and Inez back then and you had something interesting going on.\n\nAnd then there was the fact that Timberland was a volunteer at the Haunted House, although according to her dad, he hadn't been scheduled to work there when Amethyst had died, not that that meant anything. He could have easily come and gone without being seen.\n\nWhen she'd called him, she'd told him she was interested in shopping around for a new insurance policy. It was a plausible story, but she was pretty sure he hadn't believed her. She couldn't say why she thought that but something in his tone of voice when she'd talked to him on the phone had led her to that conclusion. So, if that was the case, and she was almost certain it was, the question became, Why was he talking to her?\n\nHe probably had a pretty good idea of what she wanted to talk to him about. Or maybe not. After all, she hadn't known that he'd been friends with Amethyst Applegate until Jeanine had told her dad. Or with Bob Small. Or Inez. Or Zinnia. Which got her thinking about flowers for the dining room at the Peabody School. Maybe she should get some more pots of mums to put on the table. In her view, flowers were like diamonds. You could never have too many.\n\nBernie sighed as she made a left onto Avondale Place. Libby really hadn't wanted to go back to the Peabody School by herself, and Bernie couldn't really blame her. If what had happened to Libby had happened to her, she'd be thoroughly freaked out, too. Well, not really. Libby always overreacted to this kind of stuff, even though, according to her mom, she had the \"gift.\" Or maybe that was the reason she did it. Bernie was glad she'd never been blessed that way.\n\nUsually, Bernie thought of Halloween in terms of crunching leaves underfoot, excited children, hot mulled cider, and pumpkin spice cupcakes with cream-cheese icing on top. But after what had happened with Amethyst, her thoughts were darker.\n\nTry as she might, she couldn't get Amethyst's head out of her mind. And even though she'd really disliked her, she wasn't sure Amethyst had deserved to die that way. She wasn't sure that anyone did. Although if they did, Amethyst would be up there on her list. She'd made bad things happen wherever she'd gone.\n\nThat much was not debatable. And there was a good chance she might have killed someone as well. Looked at in that light, whoever had killed Amethyst had done the world a favor. Not that her father or her sister would agree with that thought. And on that note, Bernie turned into the driveway of the house where the Timberland Insurance Company was located.\n\nThe house was a classic wooden, two-story Colonial, painted a boring shade of beige, with white trim. The two front windows had white blinds pulled halfway down. Bernie parked her car, walked up three steps, rang the doorbell, and walked in. The hallway was a different shade of beige. Obviously, Timberland had beige on his mind. Either that or he'd gotten a deal on the paint.\n\nThe receptionist's desk, which was situated in what had been the sitting room of the house, was empty. Judging from the cup that was sitting there, someone had inhabited that desk not too long ago. She was probably beige, too, Bernie reflected. A moment later Zachery Timberland came out. When he looked at her, Bernie decided he resembled a shark. All teeth. She didn't remember so many teeth from the last time she'd seen him.\n\n\"So you want to change your insurance policy, do you?\" he asked.\n\nBernie smiled her charming smile. \"That's why I'm here.\"\n\n\"Life?\"\n\n\"No. Automobile. I told you that.\"\n\nTimberland put his palms outward. \"Everyone makes mistakes. Some people make more then others,\" he said, glancing pointedly in her direction.\n\nGreat, Bernie thought as she kept smiling. \"I'm sure that's true.\"\n\nTimberland took a step closer to her. \"I know it is.\"\n\nBernie remained where she was. She was damned if she was going to move for this guy. She'd been hoping to start with the buying the insurance thing and to gradually work the conversation around to the Peabody School, but things didn't seem to be going that way.\n\n\"So,\" Bernie said, trying again, \"can you give me a quote on a policy for my car?\"\n\nTimberland's smile got bigger. Bernie decided he'd definitely gotten veneers on his teeth.\n\n\"No. But I can give you a quote on a life-insurance policy.\"\n\n\"Really? You think I'm going to need one?\"\n\n\"Everyone needs one,\" Timberland said in a bland voice.\n\n\"Is that the case?\"\n\n\"It certainly is.\"\n\n\"So that's not a threat or anything?\"\n\n\"What a fertile imagination you have. Why would I threaten you?\"\n\n\"I don't know.\"\n\n\"There you go.\" Timberland leered at her. \"I like the shoes.\"\n\nBernie considered them for a moment. They were high brown suede boots with a thin gold chain threaded through the top. She'd gotten them at deep discount last summer, when she'd been looking for a convection oven on the Lower East Side.\n\n\"They do wonders for your ass,\" Timberland said.\n\n\"Nice. You don't want to sell me life insurance, do you?\"\n\n\"You don't want to buy life insurance, do you?\"\n\n\"So I guess that makes us even,\" Bernie observed. \"By the way, those veneers on your teeth. You should have them redone. They look like Chiclets.\"\n\nBernie was happy to see that Timberland's smile was now slightly smaller.\n\n\"You're not here for auto insurance,\" he said.\n\n\"Then what am I here for?\"\n\n\"To ask me questions about my relationship with Amethyst.\"\n\n\"And why do you suppose that?\"\n\n\"A little birdie told me.\"\n\nSomehow Bernie couldn't believe that birdie was Jeanine. Maybe Inez? Most probably Inez.\n\nThe corners of Timberland's mouth turned up at Bernie's evident confusion. \"But actually, it doesn't matter,\" he said.\n\n\"That you and Amethyst hung out together?\" asked Bernie.\n\n\"So what if we did back in the day?\"\n\n\"And that your family moved away right after Bessie Osgood went out the window?\"\n\nTimberland smirked. \"Again, so what? I'm sure if you check, you'll find that lots of people left around that time.\"\n\n\"So maybe that has something to do with Amethyst's death.\"\n\nTimberland's smirk grew bigger. \"I guess you're behind the times.\"\n\n\"Not me. I'm fashion forward,\" said Bernie.\n\nTimberland blinked for a minute, then recovered himself. \"You mean, you haven't heard?\" he asked.\n\n\"Heard what?\" Bernie replied. Now it was her turn to be puzzled.\n\n\"They arrested Bob Small this morning for Amethyst's murder.\"\n\nBernie shifted from one leg to the other. She wondered if that was what her dad had been calling her about. It probably was.\n\n\"So?\" she said. \"Your point is?\"\n\n\"My point is that I want you to keep away from me. You have no legal authority.\"\n\n\"Is that what you brought me here to tell me?\"\n\n\"As a matter of fact, it is. I brought you here to tell you that I'll have you arrested for stalking if you keep bothering me.\"\n\nBernie couldn't help it. She laughed. \"You're kidding me, right?\"\n\n\"Not in the least.\"\n\n\"You could have said no when I called.\"\n\n\"I just saw your name on today's calendar.\"\n\n\"Even so. You could have called and cancelled.\"\n\n\"I wanted to tell you in person.\"\n\n\"How considerate.\"\n\n\"I'm a considerate guy.\"\n\n\"I don't know...,\" Bernie said.\n\n\"That's obvious.\"\n\nBernie talked over him. \"For someone who is in the clear, you're certainly going to a lot of trouble to tell me to lay off. Why is that?\"\n\nTimberland took another step toward her. \"You and your sister have a reputation for causing trouble.\"\n\n\"It depends on your definition of trouble. And, anyway, what could we do to you? You just told me you had nothing to do with any of this.\"\n\nTimberland went on as if she hadn't spoken. \"If you're smart, you'll stay out of my way.\"\n\n\"You want to tell me about your daughter?\"\n\nTimberland flushed. \"You've been warned,\" he growled.\n\n\"I guess you don't like talking about her.\"\n\n\"Get out before I have you arrested.\" Timberland took his cell off the clip on his belt. \"I'm dialing.\"\n\n\"I'm going. I'm going.\"\n\nWell, that had been a complete waste of time, Bernie thought as she got back in her vehicle. She shook her head in disgust at herself. She'd let him lead the conversation from the get-go. Bad. Bad. Bad. The only thing she did know was that the daughter was a definite sore spot. She sighed and looked at her watch.\n\nShe had another hour before she had to meet Libby at the Haunted House. She decided to use it to pick up more potted mums at the garden center. If she recalled, they still had some left. And as long as she was there, she could ask Kathy about Zinnia. They were the same age. Maybe she'd know something about her.\n\nThen if she had any time left over, she'd buy some more napkins. She'd seen some really cute ones with ghosts on them in the dollar store over on Grand Avenue. And maybe they had some other Halloween stuff as well. That was the thing with the dollar store. You never knew what you were going to find. Kind of like life, Bernie decided. She shook her head. She was definitely getting sappy in her old age.\n\nKathy's Garden Shop was located in a mini strip mall three miles off of Longely's main road. The strip mall had been built recently. There was more and more mall sprawl lately, taking up land that, in Bernie's judgment, should have been left alone. After all, how many Home Depots and Staples did you need?\n\nKathy's Garden Shop was located between a drugstore and a place selling chicken wings, but as Kathy had pointed out to Bernie, the rent was cheap, the utilities were fairly low, and there was plenty of parking. Like Bernie and Libby, Kathy managed to compete with the chains by charm, customer service, and interesting merchandise. So far she'd done pretty well for herself.\n\nWhen Bernie got out of her car, she was happy to see there were eight pots of mums sitting outside of Kathy's shop.\n\n\"You want to do a trade?\" Bernie said when she went inside.\n\nHer friend Kathy looked up from behind the register. Today her hair was in cornrows. She had light brown skin and large green eyes. It was a smashing combination.\n\n\"What kind of trade?\"\n\n\"You let me borrow the mums to decorate the tables up at the Haunted House, and I'll give you three apple pies.\"\n\n\"Apple and cranberry and throw in a parsnip pie and you got a deal.\"\n\n\"You are the only person I know, except for me and Libby, that likes parsnip pie.\"\n\n\"That's because you're making my grandmother's recipe.\"\n\n\"No. It's my grandmother's. We've had this discussion before.\"\n\nBernie pointed to the display of orange and lemon trees. \"Those would look nice in my bedroom.\"\n\n\"And I would sell them to you,\" Kathy said, \"if you didn't have a black thumb.\"\n\n\"Gray thumb.\"\n\n\"Black,\" Kathy said firmly.\n\n\"All right. Black.\" It was true. Bernie had yet to keep a plant alive. She simply forgot to water them. \"By the way, do you remember Zinnia McGuire?\"\n\nKathy put her pen down. \"Course, I do. She used to pal around with Zachery Timberland, Bob Small, and Amethyst Applegate. Now there was a nice crew.\"\n\n\"Whatever happened to her?\"\n\n\"She died in an automobile accident down in Coopersville. Hit-and-run, if I remember correctly. Some guy went right through the stop at Elves and Ash and T-boned her. He must have been going really fast, from what I understand.\"\n\nBernie raised an eyebrow. \"And no one saw anything?\"\n\n\"It was three-thirty in the morning. You know what that place is like. It's even worse than Longely. A neighbor called it in. She said the crash woke her up.\"\n\n\"I wonder why Zinnia was there?\"\n\nKathy shrugged. \"She was probably doing the typical druggie high school dropout thing.\"\n\n\"Did Zinnia ever say anything?\"\n\n\"Nope. She died on the way to the hospital. I guess she was in pretty bad shape when the police got there. It's really too bad.\"\n\n\"How's that?\"\n\n\"Because she was getting her act together. She'd given up the stuff she was doing\u2014the coke and the weed\u2014and was talking about going back to school.\"\n\n\"How long after Bessie Osgood died was she killed?\"\n\nKathy thought for a moment. \"Maybe one year. Certainly no more than two.\"\n\n\"Interesting,\" Bernie murmured.\n\n\"You don't think one thing had anything to do with the other, do you?\" Kathy asked.\n\n\"Probably not,\" Bernie said.\n\nNevertheless, she filed the fact away in her head for possible use later.\n\n## Chapter 11\n\n\"Damn cell,\" Sean said as he and Marvin headed toward Lexus Gardens.\n\nMarvin turned his head to look at Sean. \"Can't you\u2014\"\n\n\"Eyes on the road,\" Sean yelled.\n\n\"They are on the road,\" Marvin said.\n\n\"Now they're on the road. Before they weren't. Maybe you shouldn't talk while you're driving.\"\n\nMarvin didn't say anything. Sean could tell he was sulking. Young people didn't take correction well these days. They should be thankful for the help. But when he'd said that to Libby and Bernie, they'd just rolled their eyes.\n\nAfter a moment, Sean said, \"Things are stressful.\" Which was as close to an apology as he was going to get.\n\nMarvin stared straight ahead. Another moment went by before he spoke.\n\n\"There was no one on the road.\"\n\n\"Maybe there wasn't,\" Sean allowed, \"but that's not the issue. The issue is there could have been.\"\n\n\"I don't get it,\" Marvin told him.\n\nSean watched him for a moment and shook his head. Now he was driving like a little old lady, with both hands gripping the steering wheel.\n\n\"Bad habits are bad habits,\" Sean explained. \"The object in life is to develop good ones. That way, when a situation comes along, you don't have to think about what to do. You know what to do.\"\n\n\"I suppose,\" Marvin said.\n\n\"You suppose! That's the basis for military training. It stops people from getting killed.\" Sean nodded at the turn coming up. \"You want to take a right here.\"\n\n\"I know,\" Marvin said. \"Contrary to popular opinion, I can drive.\"\n\n\"I never said you couldn't.\"\n\nThis time when Marvin talked, Sean was delighted to see he kept his eyes on the road. He knew he should feel guilty about yelling at the kid, but he didn't, because he'd accomplished his objective. He couldn't forbid Libby to ride around with Marvin, but he could endeavor to make him the best driver possible. And if yelling was what it took, then so be it. Sean could live with that.\n\n\"So Banks doesn't know that we're coming?\" Marvin asked.\n\n\"No. I can't call him, because I can't get any reception here. Maybe it's better that way,\" Sean said.\n\n\"What if he's not in?\" Marvin asked.\n\nSean waved his hand to indicate the countryside. \"Then we'll have enjoyed a drive in the country.\"\n\nSean was glad to be out riding around. And if Edward Banks wasn't in, they could always speak to him another time. Some people would say that talking to Banks was just a waste of time, but Sean had always found that the more information one could gather, the better. You could never tell what might prove to be important and what wasn't until you got it all together and laid it out on the metaphorical table.\n\nSo why had Amethyst wanted to speak to Banks? Banks was a recluse. Sean had heard that his house and gardens were spectacular, but no one Sean knew had seen them. So maybe he'd see them now. That is, if Banks was in and was willing to talk to them. If not, all he and Marvin had wasted was the fifteen-mile drive into the country. Which wasn't a waste, especially not today.\n\nMaybe they'd go home by way of Orchid Farms, and he'd get some apples. Cortlands were always good. And they had unpasteurized cider, something you couldn't get in the grocery stores anymore. Maybe he'd get a gallon for the house and a quart for Jeanine for all the help she'd given him. Yes. That was what he'd do. He was feeling very pleased with himself when Marvin started talking.\n\n\"What?\" Sean asked.\n\n\"I asked if Bernie knows that Bob Small was arrested,\" Marvin said.\n\n\"Probably not,\" Sean said.\n\nClyde had phoned to tell him after Bernie had gone, and although he'd tried phoning Bernie, he hadn't gotten through. He hadn't even been able to leave a message, because her mailbox was full. Not that it really mattered.\n\nHe'd tell her when he met up with her at the Peabody School. He sat back and took in the scenery while keeping one eye on Marvin's driving. Even if Marvin were the best driver in the world, which he wasn't, Sean had to admit to himself, he wouldn't be able to relax. The truth was he didn't trust anyone to drive except himself.\n\nThey got to Lexus Gardens twenty minutes later. The road up to the estate formed a winding ribbon as it went up to the top of the Altamar Hill. Huge evergreens hugged either side of the road, blotting out the light.\n\n\"I wouldn't want to drive up and down this road in the winter,\" Marvin observed as he made another turn.\n\n\"Me either,\" Sean replied. \"But he probably doesn't have to. He probably spends his winters in the Caribbean. I mean, I would if I had that kind of money.\"\n\n\"Would you really?\" Marvin asked.\n\n\"No,\" Sean said. He hated the heat.\n\nAnother one-eighth mile and they were at the estate. The whole thing was enclosed in a stucco wall, with barbed wire on the top.\n\n\"I guess he doesn't want any visitors,\" Marvin said.\n\n\"I guess not,\" Sean agreed. He was surprised there wasn't an ARMED RESPONSE sign tacked up to the wall.\n\nThe house and the gardens had been featured in some magazine named Shelters. Or something like that. Maybe when Sean gave Jeanine the cider, he'd ask her if she could look that up, too. He directed Marvin to drive up to the gate. There was an intercom mounted on the far wall. Sean told him to lean out the window and push the button. Marvin did. There was a crackling noise.\n\n\"Ask if we can come up and talk to him for a moment,\" Sean ordered. \"Tell him we're involved in a homicide investigation.\"\n\n\"But we're not the police,\" Marvin protested.\n\n\"I'm not saying we are,\" Sean told him. It just sounded that way.\n\n\"If you say so.\"\n\n\"I do,\" Sean said.\n\n\"Okay,\" Marvin said, and he turned and yelled into the intercom. There was no response. He tried again. Nothing.\n\nMarvin turned back to Sean. \"Maybe no one is home.\"\n\n\"Maybe,\" Sean said. \"Try another time.\"\n\nMarvin did, with the same results.\n\nSean looked through the gate. The opening offered a narrow perspective. All he could see was a lawn, a large red maple, the front of the house, and a large blow-up statue of a witch bowing up and down. Sean did a double take.\n\nTalk about something being out of place. You expected to see something like that at Wal-Mart, not someplace like here, a place that was known for its taste and elegance. Another odd thing caught his attention. No lights were on in the house, and he didn't see any movement anywhere. Which was strange because a house this size required a staff to run it. Sean gave a mental shrug. Banks could be off in China, for all he knew.\n\n\"It's awfully quiet,\" Marvin observed.\n\n\"It is, isn't it,\" Sean agreed. All he could hear was the wind murmuring and a flock of geese honking as they flew overhead.\n\nMarvin leaned out and pressed the intercom button again. Just static. \"Hello,\" he cried. \"Anyone home?\"\n\n\"Let's just go,\" Sean said.\n\nMarvin turned back to Sean. \"Maybe he went to the Caribbean, after all.\"\n\n\"He could be anywhere,\" Sean said. One thing was clear. No one was home. Or if they were, they weren't answering the intercom. \"We can stop at the farm stand on our way to the Peabody School.\"\n\nMarvin nodded and turned around. Later Sean would be sorry that he had made that decision. Later he'd be sorry that he hadn't asked Marvin to try the gate. Later he'd tell himself he was turning into a careless old man. But that was later. Right now he wanted to get to the farm stand before it closed.\n\n\"Thank you,\" Libby said as Mark helped her carry the last of her cartons into the kitchen.\n\nShe began unpacking. Mark coughed. Libby looked up. He was standing beside her, shuffling from one well-shod foot to another.\n\n\"Yes?\" she said.\n\nMark coughed again. Libby decided he looked uneasy. Somehow she liked him better that way.\n\n\"Can I help you with anything?\" she asked. She wished he'd come to the point. She had a lot of things she had to do.\n\n\"I heard.\" Mark stopped.\n\n\"Heard what?\" Libby asked, mystified.\n\n\"Er. I heard you saw something the other day.\" He spoke quickly, running his words together. \"Or maybe I should say someone.\"\n\nLibby knew instantly what he meant. \"That's right,\" she said cautiously as she took the jugs with the waffle batter out of the cardboard box she'd carried them in. \"You're talking about Bessie Osgood, aren't you?\"\n\nMark nodded and fidgeted some more.\n\nLibby rested her hand on top of one of the five-quart jugs. \"I still can't believe I did.\"\n\n\"You don't believe in that kind of thing?\" Mark asked.\n\n\"Not at all. I keep thinking she was some kind of optical illusion that someone had created.\"\n\n\"I don't think so,\" Mark said softly.\n\n\"Why not?\"\n\n\"Because I saw her, too,\" he said a moment later. He held up three fingers.\n\n\"Three times?\" Libby asked.\n\nMark nodded. \"I don't believe in ghosts, either.\"\n\n\"Neither do I,\" Libby lied, because in her heart she did. \"But here we are.\"\n\nMark nodded. \"I didn't tell anyone, because I didn't want them to think I'm crazy.\"\n\n\"I'm surprised. I'd think you'd want the publicity.\"\n\n\"I'm a behind-the-scenes kind of guy.\"\n\nLibby unscrewed the top of the five-quart jug of whole-wheat batter and started pouring it into a pitcher. \"I felt so cold,\" she said.\n\n\"I got goose bumps on my arms,\" Mark said.\n\n\"Me too. And I could kind of see a shape.\"\n\n\"Out of the corner of your eye?\"\n\nLibby nodded. \"I had the impression of a teenage girl. But I don't know why I thought that. I couldn't really see her all that well. I could hardly see her at all. I closed my eyes for a second, and when I opened them again, she was gone.\"\n\n\"That's what happened with me. For some reason, I thought she was wearing a pleated skirt and a white shirt.\"\n\n\"She was so hazy,\" Libby said.\n\nMark bobbed his head. \"Hazy is a good word.\"\n\nLibby finished filling the pitcher and put the cap back on the jug. She didn't know what to feel: relieved because someone else had seen what she had or dismayed that what she had seen was apparently real.\n\n\"Maybe we both saw an optical illusion,\" she said.\n\nMark shook his head. \"I don't see how anyone could create something like that.\"\n\n\"FX has created some pretty strange stuff in your haunted house.\"\n\n\"Yeah. But Bessie Osgood is on a totally different plane.\"\n\nLibby had to agree that she was. \"Why are we seeing her?\"\n\nMark shook his head. \"Maybe because we're here. I don't think anyone else has seen her.\"\n\n\"Curtis and Konrad have a tape of her confessing she killed Amethyst Applegate.\"\n\n\"Them.\" Mark ran his hand through his hair. \"All I could hear was static.\"\n\n\"That's all I could hear, too. The police refused to listen.\"\n\n\"Can you blame them?\"\n\nLibby shook her head.\n\n\"Did they tell you I offered them one thousand dollars if they could get her voice on tape? I thought it would be good for the Haunted House. So maybe that's why they're hearing things that no one else can.\"\n\n\"Could be,\" Libby agreed.\n\nMark gave Libby a wan smile. \"I thought it would be cool to have a real ghost moving around, especially considering that it's Halloween and I'm running a haunted house, but it's really not. In fact, it's kind of upsetting.\" He looked at his watch. \"Gotta go. I have to find a replacement for Bob Small. I thought I had someone, and he pooped out on me.\" He shook his head again. \"Talk about upsetting.\"\n\n\"You mean Bob Small killing Amethyst?\"\n\n\"What do you think I'm talking about?\"\n\n\"I thought you might have been referring to finding a replacement.\"\n\n\"Hardly.\"\n\n\"Do you think he did it?\" Libby asked.\n\n\"I'd like to think he didn't,\" Mark said. \"After all, if it wasn't for me giving him that job, maybe he wouldn't have had the opportunity to do what he did. What is it that they say about the road to hell and good intentions? But I have to assume he did it, and all the tapes in the world aren't going to make things any different.\"\n\n\"I'm not so sure,\" Libby said. \"Chopping off someone's head doesn't seem like something Bob would do.\"\n\n\"Who do you know who would do something like that?\"\n\n\"No one,\" Libby admitted.\n\n\"Exactly,\" Mark said.\n\n\"I just can't see him setting something like this up.\"\n\n\"The police think otherwise.\"\n\n\"They've been wrong before.\"\n\n\"Maybe you're right. Maybe I shouldn't be so quick to judge. I have to watch myself with that. It's my besetting sin.\" He glanced down at his watch. \"Oops,\" he said. \"I didn't realize how late it is. You can't believe how crowded we've been since that...thing...with Amethyst.\"\n\n\"I can believe it,\" Libby said, thinking of what happened at the shop whenever a crime was committed that they were connected to in some way or other. \"That's why I brought extra. I figure we're going to be swamped.\"\n\n\"People are amazing,\" Mark said. \"Who would have thought?\"\n\nThen he headed out the door, leaving Libby by herself. She got right to work, but try as she might, she couldn't stop thinking that there was more to Amethyst's death than she was seeing. Maybe her dad would see something that she wasn't. She decided she was glad he was coming. Having him look over the crime scene would make her feel better.\n\n## Chapter 12\n\nSean went into the Pit and the Pendulum Room with the same attitude he'd brought to every crime scene he'd processed when he'd been the Longely chief of police. With dispassion. He went in without expectations\u2014at least as far as it was possible for a human being to do that.\n\nThe first thing he noticed was that the room was dark enough that it took his eyes a moment to accustom themselves to the low light. According to Mark, everything was the same as it had been the afternoon that Amethyst Applegate died. Sean still couldn't believe that they had opened this up in such a short time. But that was Lucy for you. Always in a hurry. Always overlooking things.\n\nHe took in his surroundings. The mirrored walls, the platform, the high table draped in a red cloth, the spotlight on the blade, the speed at which the blade swung back and forth\u2014all worked to focus your attention on one central point. The table. Everyone walking in here would automatically be looking at one thing and one thing only. Whoever had set this up had done an excellent job. He turned to Libby.\n\n\"Tell me what you saw.\"\n\n\"I already told you,\" she protested.\n\n\"I know you did, but I want to hear it again,\" Sean said.\n\nAs he looked at his daughters, he thought that neither of them were happy being here. Marvin wasn't, either. But given the circumstances, who could blame them?\n\n\"It's simple,\" Libby said. \"We walked in the room, and we saw this table on the platform, with the blade swinging back and forth.\"\n\n\"And then?\" Sean prompted.\n\n\"And then I took a couple of steps and saw a headless body on the table, and then I took another step, and that's when I saw it.\"\n\n\"Amethyst's head?\" asked Sean.\n\nLibby nodded. \"It was just sitting there at the top of the stairs. At first, I thought it was part of the exhibit, and then I looked at Mark's face, and I knew it wasn't.\"\n\n\"How did you know?\" asked Sean.\n\n\"I just did. He was staring at it. He had a strange expression on his face,\" replied Libby.\n\n\"What kind of strange expression?\" asked Sean.\n\n\"I don't know how to describe it,\" Libby said.\n\nSean looked at his other daughter. Bernie shook her head.\n\n\"I wasn't paying attention to him. I was mesmerized by the head,\" said Libby. \"And then he went up and tapped it with his foot, and it started rolling....\"\n\n\"Just like at the Bastille,\" Bernie said.\n\n\"I didn't know whether it was real or not, but then, when it stopped in front of my feet...\" Libby shuddered.\n\nSean nodded. \"And then?\"\n\n\"And then I screamed,\" said Libby.\n\n\"Did anyone come in?\" asked Sean.\n\n\"No,\" Bernie said, \"because we ran out.\"\n\n\"I see,\" Sean said as he looked around.\n\n\"Do you need us for anything else?\" Bernie asked after a moment had gone by.\n\n\"No,\" Sean answered absentmindedly. He was already focused on trying to figure out what had happened. \"I just want to stay and walk around a little.\"\n\n\"Good,\" Libby said. \"Then we'll finish setting up.\"\n\nThey turned to go, and Marvin started walking out with them.\n\n\"No, Marvin. You stay here,\" Sean instructed.\n\n\"But...,\" said Marvin.\n\nSean glared at him.\n\nMarvin's shoulders slumped. \"Fine,\" he said.\n\n\"Do me a favor and turn on the lights for me. They're over in the far corner, hidden behind the cloth panel.\"\n\nSean watched Marvin fumble around, but after a moment or so, he found the switch. Suddenly, the room was bathed in light.\n\n\"It looks different,\" Marvin remarked.\n\n\"It certainly does,\" Sean agreed.\n\nThe mirrors looked dirty; the platform was badly made; the steps looked thrown together. Sean slowly walked over to the control panel that was hidden behind the cloth. From that vantage point, he looked across the wall to where the device that sent out the hologram was situated. A person walked in and tripped a circuit that would trigger a digital camera, which would capture their image and transfer it to a hologram that would make it look as if that person's head were being cut off.\n\nMark had explained it, but Sean hadn't understood the explanation. Clyde hadn't, either.\n\nWhat had Clyde said? There's a whole new world out there, Sean, and I don't want to be part of it. Sean had had to say that he agreed. He turned to Marvin. \"Did you understand Mark's explanation?\"\n\nMarvin shook his head. \"I was lost after the second sentence.\"\n\n\"Me too,\" Sean said gloomily. But then Mark had admitted that he really didn't understand how it all worked, either.\n\n\"So what do you think happened?\" Marvin asked Sean.\n\nSean rubbed his hands together. He'd come up with two scenarios. In scenario number one, Bob Small snuck in while Amethyst was lying on the table and cut her head off with a piece of fiber-optic laser wire that he'd taken with him into the ceiling. At least that was the weapon the ME had identified as the cause of death. But then why had she been lying on the table? Why had she stayed still for Bob Small to do this? Why had she agreed to meet Bob Small in the first place?\n\nMore likely, Amethyst walked into the room, and the killer, who was hiding behind the door, took the wire and wrapped it around her neck. After all, if fiber-optic wire could cut through steel, it would definitely cut through flesh and bone. There wouldn't be any blood, because the high heat of the wire would cauterize the wound. Then the murderer positioned the body the way that Bernie, Libby, and Mark had found it and left. But two questions remained: who killed Amethyst, and why?\n\n\"Are you all right?\" Marvin asked.\n\nSean was startled. He realized he must have been staring off into space.\n\n\"I'm fine,\" he said. \"Absolutely fine.\"\n\n\"Maybe this place has secret passageways or something like that.\"\n\n\"That's in books.\"\n\n\"But it's possible,\" Marvin insisted.\n\n\"Anything is possible,\" Sean retorted. In all his thirty odd years on the police force, he'd never come across a secret compartment, much less a secret passageway. \"What are you doing?\" he asked Marvin as he began tapping on the wall.\n\n\"Looking.\"\n\nSean was just about to tell him to stop when Mark stuck his head into the room.\n\n\"I hope you guys are finishing up, because we're going to open the doors to the Haunted House in ten minutes,\" he said.\n\nSean stood up slowly. \"Not a problem. By any chance, do you have the original plans for this place?\"\n\n\"I do, but I won't be able to get them till the a.m.,\" replied Mark. \"What do you need them for?\"\n\nSean massaged the small of his back. Getting old sucked. \"Just checking to see if there's a connection between this room and the main house.\"\n\nMark raised an eyebrow. \"Like a secret passageway?\"\n\nSean shrugged.\n\n\"There isn't one,\" said Mark.\n\n\"How do you know?\" Sean asked.\n\n\"Because I went over the original plans of this place with the architect last week,\" said Mark. \"We're remodeling this section as soon as the Haunted House closes, and I wanted to make sure we didn't run into any unpleasant surprises. Nothing like not having a wall where you expect to find one or the other way around.\"\n\n\"You must have a lot of rich people bankrolling you,\" said Sean.\n\n\"Well, one or two,\" Mark allowed. \"We've been blessed that way. And now I really have to go. If I had known that I was going to have to be here overseeing everything the whole time the Haunted House was open, I would have hired a manager.\" And he walked away.\n\n\"I'd love to know who his backers are,\" Sean said.\n\n\"Why?\"\n\n\"Just curiosity.\"\n\n\"Are you going to ask him?\"\n\n\"Yes, I am,\" Sean said. \"But he doesn't have to answer, and I have a feeling he won't.\"\n\nMarvin cocked his head, and Sean answered his unspoken question.\n\n\"Because he's not going to want me to go annoy them with questions. If I piss them off, there's a good chance he won't get any more money out of them. After all, why do they need this type of aggravation? The answer is: they don't.\"\n\n\"So what are you going to do?\"\n\n\"I'll think of something,\" Sean said.\n\nAs it turned out, that something came sooner than Sean had expected. Sean and Marvin were crossing the parking lot when the Kurtz twins ran up to them.\n\nKonrad grabbed Sean by the arm. \"Have you heard?\" he cried. \"They've gone and arrested Bob. Poor Bob. You gotta do something.\"\n\n\"I'm trying. Let go of my arm,\" Sean demanded. \"Now.\" Sean didn't like being touched.\n\nKonrad looked down. \"Sorry,\" he muttered and took a step back. \"I forgot.\"\n\n\"Well, don't forget again,\" growled Sean.\n\nCurtis looked as if he was going to cry. \"You have to make them listen to the tape.\"\n\n\"I've tried,\" said Sean. Which wasn't true, but Sean wasn't in the mood to hear a lecture on the merits of ESV. Or was it EVC? Whatever the hell it was called.\n\nKonrad absentmindedly jiggled his ring of keys. \"You have to make them understand.\"\n\nSean looked at the keys. Maybe, he thought, this trip wouldn't be a total waste, after all. \"You guys just going to work?\"\n\nCurtis nodded.\n\n\"Where's Inez?\" asked Sean.\n\n\"She called in sick. Again,\" Curtis told him. \"Ian is going to fire her ass for sure if she doesn't look out.\"\n\n\"So it's just you two?\" asked Sean.\n\nKonrad nodded. \"I don't mind. It's easier without her. Quieter.\"\n\n\"Isn't it kind of early for you guys to be cleaning?\" asked Sean.\n\n\"Oh. We're going to do one of our tapings at the Haunted House,\" Konrad said. \"Mr. Kane called and told us to come in.\"\n\nSean was quiet for a moment, and then he said, \"Is everyone at the offices gone?\"\n\nCurtis answered, \"Well, they're usually gone this time of day. No one seems to work very late in there.\"\n\nSean was silent for another moment. Then he said, \"Are you serious about helping your cousin?\"\n\n\"Of course, we are,\" Konrad answered. \"Why?\"\n\n\"Because I'd like to have a look around the offices of the Foundation,\" replied Sean.\n\n\"Are you nuts?\" Marvin hissed.\n\nSean ignored him and watched Konrad and Curtis exchange looks.\n\n\"What do you say, boys?\" Sean asked.\n\nIn answer, Konrad took his keys off the hook that was holding them and gave them to Sean. \"We forgot them, and we had to go home to get them,\" he said.\n\n\"But we have them,\" Curtis protested.\n\n\"We're\u2014\" Konrad began. Then he stopped. \"Forget it,\" he said to Curtis. \"I'll explain in the truck.\"\n\n\"But what about our demonstration?\" Curtis protested.\n\n\"We'll just be a little late,\" said Konrad.\n\n\"I don't know,\" Curtis said.\n\n\"I do,\" Konrad replied.\n\nSean watched while Konrad dragged Curtis away.\n\n\"You can't do this,\" Marvin said when they were gone.\n\n\"Why not? Now is the perfect time. Mark is tied up. Figuratively speaking.\"\n\n\"But what if someone is at the Foundation, working?\"\n\n\"Then we'll say we're sorry and leave.\"\n\n\"I don't think I can do this,\" Marvin protested.\n\n\"Of course, you can,\" Sean said.\n\nMarvin bit his lip. \"I don't know.\"\n\n\"Well, I do.\"\n\n\"Why don't I drive you over and sit in the car and wait?\"\n\n\"Don't be ridiculous. I need you. And drive around to the side entrance. We can leave the car there. I don't want the police to see it on one of their patrols.\"\n\n\"The police,\" Marvin cried.\n\n\"It'll be fine,\" Sean said in as soothing a voice as he could manage. \"Everything will be just fine.\"\n\n\"No, it won't,\" Marvin protested.\n\n\"Just think of everything you're learning,\" Sean told him.\n\nMarvin straightened his back. \"Breaking and entering is not something I want to learn.\"\n\n\"We're not breaking and entering, because we have the keys. At worst, this could be called unlawful trespass.\"\n\n\"That's wonderful,\" Marvin said.\n\nSean decided to take his comment literally. \"I think so. It's a misdemeanor. At most, we'd be looking at community service if we get caught.\"\n\n## Chapter 13\n\nLibby looked at the woman standing in front of her. She was small and plain and had one of those short, mannish haircuts that women tended to get when they didn't want to bother about their looks anymore.\n\n\"Aren't you scared working here?\" she asked Libby as Libby placed a slice of pumpkin cheesecake on a paper plate and handed it to her.\n\n\"You mean because of the murder?\" Libby asked.\n\n\"After all, Mark said you saw Amethyst's head rolling down the stairs.\"\n\nThanks, Mark, Libby thought. \"Frankly,\" she said to the woman, \"I think I'm too tired to be scared.\"\n\nThe woman gave her an odd look and walked away.\n\nBut it was true. Things had finally settled down some, and for that, she was eternally grateful. People were no longer standing out in the hallway, waiting to get in. But her back and her feet were killing her. So were her wrists and arms, for that matter. She couldn't imagine how Bernie's feet were feeling. She was wearing pink ballet slippers, which had no support whatsoever. She and Bernie had been making waffles and dishing out desserts since they'd opened the doors, and she was ready to take a break.\n\nEvidently, the combination of the chance to visit a real live murder scene and go through a haunted house at the same time had proved irresistible to the population of Longley and the surrounding towns. This was better than reality TV. When Libby thought about it, she realized this could be reality TV.\n\nThey'd sold way more Belgian and chocolate brownie waffles than she or her sister had anticipated. Hopefully, they'd have enough to squeak by until the end of the night, which was another couple of hours away. All she could say was thank heavens she'd prepared extra.\n\nThe real winner of the day, though, was the apple compote. People couldn't stop talking about it. Libby had to admit it was pretty darn good. The lemon peel, the touch of rum, and the small amount of apricot jam in it made all the difference.\n\nThey were doing very well. The bad part was that they had to go home and get ready for tomorrow. Hopefully, Amber and Googie had done everything they were supposed to do. Otherwise, Libby and Bernie would be up till three in the morning, doing prep work. Sometimes Libby thought of the public as an insatiable mouth that she and Bernie labored to feed. She shook her head to clear that thought from her mind and got back to the business at hand.\n\n\"How are we doing with the pumpkin bars?\" she asked Bernie.\n\n\"We've got two trays, but the apple and the pumpkin pies are gone. We should bake a few more of those for tomorrow. Did you get a look at the fat woman in the latex suit? That was certainly a mistake. Fetish dressing is definitely for thin people. Don't look at me like that. Half the fun of Halloween is commenting on the costumes people wear.\"\n\n\"See. This is why I'm not getting into a costume.\"\n\nBernie looked hurt. \"I wouldn't let you make a mistake like that.\"\n\nLibby knew Bernie wouldn't, but she still didn't want to wear a costume. She just felt silly. She just didn't know how to explain that to someone who was dressed as Little Bo Peep (\"to balance the energy,\" Bernie had explained). But at least that was better than the witch and the vampire, Libby thought. Bernie wasn't wearing fake nails today. They had driven Libby crazy. She had been eyeing them the past two nights, waiting for one to fall into the waffle batter.\n\n\"How about we just do make-up?\" Bernie said.\n\n\"I'll think about it.\"\n\n\"Come on,\" Bernie cajoled.\n\nLibby threw up her hands. She was too tired to argue. \"Fine,\" she said. \"I give up.\"\n\nBernie grinned. \"You won't regret it.\"\n\nLibby thought she probably would, but at least she'd have a short interval of peace and quiet. She shook her head to clear it and continued to take stock of what they needed.\n\n\"We need more cider,\" she said. \"Would you mind going out to the van and getting the rest of it?\"\n\n\"Why is it still in the van?\"\n\n\"Because I was too tired to bring it all in.\"\n\n\"Makes sense,\" Bernie said. \"I'll do it now.\"\n\nBernie was back two minutes later with a jug in each hand. Libby noticed her sister was frowning.\n\n\"Libby,\" Bernie said, \"didn't Dad and Marvin leave for home?\"\n\nLibby kept stocking the square rattan box she used for the napkins. \"Yes. At least I thought they did.\"\n\n\"Well, I spotted Marvin's car on the far side of the parking lot. I'm going to call and see what they're up to.\"\n\nBernie was reaching for her phone when it rang. She picked it up and checked her caller ID. It was her dad. She listened to him for a moment and hung up.\n\n\"They're at the Foundation,\" she said to Libby.\n\n\"The Foundation? What are they doing there?\"\n\n\"Snooping.\"\n\n\"Snooping?\"\n\n\"That's what I said,\" Bernie replied. Sometimes she thought her sister had a hearing problem.\n\n\"What are they looking for?\"\n\n\"They wanted to find out who the Foundation backers are,\" replied Bernie.\n\n\"But why?\"\n\nBernie shook her head. \"You got me. But they're going back to Ed Banks's house to talk to him. He's one of the big backers. And guess what?\"\n\n\"What?\" Libby said as she ladled pumpkin batter into one of the waffle machines and closed the top.\n\n\"So was Amethyst.\"\n\nLibby's eyes remained focused on the machine. \"She had that kind of money?\"\n\n\"Evidently, she did.\"\n\n\"Interesting,\" Libby said as she opened the machine up. \"Very interesting indeed.\"\n\nShe was about to say something else when Mark appeared by their side. He'd been so quiet, Libby hadn't heard him approach. She wondered if he'd been standing there long.\n\n\"You gals holding up under the onslaught?\" he asked.\n\n\"Oh yes,\" Bernie said, even though she detested being called a gal.\n\n\"Is your dad around?\" asked Mark.\n\nBernie shook her head. \"Not that I know of.\"\n\nMark wrinkled his forehead. \"That's odd, because I thought I saw the car he came in off to the side of the parking lot.\"\n\n\"Must have been a different car,\" Libby lied.\n\n\"Marvin and dad went home a while ago,\" said Bernie, backing her up.\n\n\"Absolutely,\" Libby agreed.\n\n\"Well, someone is over there,\" Mark said. \"I think I'll take a walk around and see what's what. Can't have any more bad things happening here, can we? And, by the way, Bernie, I like your costume.\" He grinned. \"I'd be one of your sheep anytime of the week.\"\n\n\"My pleasure,\" Bernie said.\n\nAs soon as Mark turned and started walking away, Bernie grabbed her cell phone and called her dad to tell him what was happening.\n\n\"Poor Marvin,\" she said to Libby when she was done. \"He's probably having a heart attack.\"\n\nLibby grimaced. \"I was thinking the same thing myself.\"\n\n\"What's going on?\" Marvin asked Sean as Sean clicked off his cell phone.\n\n\"I guess you were right,\" Sean said.\n\n\"Right about what?\"\n\n\"I guess we should have left a little earlier.\"\n\n\"What do you mean?\" Marvin asked.\n\nSean noticed that Marvin's voice was rising in a spiral of panic. \"Bernie just told me that Mark is on his way over.\"\n\n\"Oh my God,\" Marvin said.\n\n\"Relax,\" Sean said. \"It'll be fine.\"\n\n\"That's what you said when we walked in here.\"\n\n\"And I meant it.\"\n\n\"But we have to get out of here.\"\n\n\"It's too late.\"\n\n\"What are we going to do?\" Marvin wailed.\n\n\"Learn from the master,\" Sean told him. \"Now grab my coat, and let's go.\"\n\nHe watched Marvin hurry away. A moment later he was back with Sean's coat. Sean struggled into it; then he and Marvin walked to the door. Sean would have liked to have gone faster, but he couldn't walk at a decent pace anymore. These days he was lucky he could walk at all. He really should have had Marvin bring the wheelchair along. It would have made things so much simpler. They had just gotten to the door and Marvin had reached out to open it when the door swung open and Mark was standing there.\n\n\"Hello,\" Sean said. \"What a pleasant surprise.\"\n\nMark started to stammer.\n\n\"You're probably wondering what we're doing here,\" Sean said.\n\n\"Well\u2014,\" Mark began, but Sean had already cut him off.\n\n\"Trying to find the old plans for the house.\" Sean smiled.\n\n\"But I told you I'd show them to you,\" Mark protested.\n\n\"I know you did, and I'm sorry, but I just couldn't wait. You know how old men get. Impatient. Like in the song lyrics, 'oh the days dwindle down to a precious few.'\"\n\nMark scowled at him.\n\n\"What? You've never heard 'September Song'?\" Sean asked. \"It's Frank Sinatra. At least I think it's Frank Sinatra. I thought the plans would be in the archives....\"\n\n\"We don't have archives,\" Mark said.\n\n\"I finally remembered you'd told me that after I went looking for them,\" said Sean as he rubbed his hands together. \"I'm telling you, getting old is a bitch.\" He smiled at Mark. \"Well, as long as you're here, maybe you can get them for us now?\"\n\n\"No,\" Mark replied. Sean could tell he was very annoyed and anxious to get away from this garrulous old man in front of him. Which was the whole idea. \"I have to go down to the basement to get them. It could take quite some time.\"\n\n\"I'll be back tomorrow afternoon then,\" Sean told him and started to walk off.\n\n\"Wait,\" Mark said.\n\nSean stopped and turned.\n\n\"How did you get in here?\" asked Mark.\n\n\"I used my superpowers. No, really. Some people were going out....\"\n\n\"What people?\" A sharp thinker, Mark.\n\n\"A woman and a man,\" said Sean.\n\n\"Describe them,\" Mark ordered.\n\nSean shrugged and tried to keep from jingling the keys in his pocket that Konrad and Curtis had given him. \"I don't know. Ordinary people. I have trouble identifying people now.\" Sean tried to look contrite. \"I'm sorry if I caused a problem,\" he said.\n\n\"You didn't.\" Mark ran his hand through his hair.\n\nIt looked so perfect, Sean wondered if he dyed it. Or went to some fancy spa.\n\n\"I just don't want anything to happen to you,\" Mark added.\n\n\"That's so nice. Nothing will,\" Sean said firmly.\n\nMark waved his hands in the air. \"I'd feel better if you weren't in this place by yourself.\"\n\n\"I'm not here by myself. I have Marvin,\" replied Sean.\n\nMark's eyes narrowed slightly. \"You know what I mean.\"\n\n\"I do indeed,\" said Sean. \"Tell me, does your insurance kick in if someone gets killed or injured by a ghost, or does that come under the 'act of God' clause?\"\n\nMark smiled. \"You're a funny man,\" he said.\n\n\"I like to think so,\" said Sean.\n\n\"Now I know where Bernie gets it from. I'll have to ask my insurance agent,\" replied Mark.\n\n\"Do you have a good one?\" asked Sean.\n\nMark reached into his pocket, took out a piece of gum, unwrapped it, and popped it in his mouth. \"The company we hired to mount the Haunted House show carries lots of liability, but it doesn't cover homicide.\"\n\n\"Tomorrow then?\" Sean asked.\n\n\"Absolutely,\" Mark said, with what Sean decided was a tad too much enthusiasm.\n\n\"You see,\" Sean explained to Marvin as they walked to Marvin's car. \"You should never explain. You should never run. If you're caught in a situation like this, you walk toward the person and spin them a plausible story.\"\n\n\"I don't think Mark believed you,\" Marvin said as he opened the door of his car.\n\n\"That's the point,\" Sean said once he was comfortably settled in his seat. \"He didn't believe me, but because of the way I was acting, he couldn't call me a liar. Now let's get back to Lexus Gardens.\"\n\n\"But it's late,\" Marvin protested.\n\n\"Precisely. That way there's a better chance that Banks might be in.\"\n\n## Chapter 14\n\n\"Slow down,\" Sean yelled at Marvin as they took one of the turns on the road up to Lexus Gardens. \"It's too damn dark to be going this fast.\"\n\n\"I'm going twenty miles an hour,\" Marvin told him.\n\n\"Then go fifteen.\"\n\nIt was pitch black out, and even with Marvin using his brights, the turns just leaped out. At night the evergreens on either side of the road reminded Sean of huge saws. Or maybe Halloween was getting to him. In any case, Sean hoped there weren't any deer up this way. There didn't used to be, but these days they were everywhere, eating everything. Clyde called them rodents with long legs.\n\n\"I wish I could drive,\" Sean complained as he scrunched up his eyes so he could see better in the dark.\n\n\"I wish you could, too,\" Marvin said.\n\n\"Was that a note of bitterness I detected?\" Sean asked.\n\n\"No,\" Marvin stammered. \"It's just that\u2014\"\n\n\"I criticize you?\"\n\n\"It's more like I never seem to do anything right.\"\n\nSean's hand tightened around the door handle as Marvin took a particularly sharp curve.\n\n\"Of course, you do,\" Sean told him. \"If you didn't, I wouldn't even bother with you.\"\n\n\"Really?\" Marvin said.\n\nSean could hear Marvin's voice brightening. \"Yes, really,\" Sean replied.\n\nThere was a short pause. Then Marvin said, \"My father says I can't do anything right.\"\n\n\"Well, he's wrong,\" Sean said. \"Dead wrong. And speaking of dead, don't take your eyes off the road. Aside from the turns, there could be deer wandering about.\"\n\nMarvin didn't say anything. Sean turned to look at him. He was smiling in a way Sean had never seen him smile before. The two men sat silently for a moment before Marvin spoke.\n\n\"What are you going to ask Banks?\" Marvin said.\n\n\"What the Foundation is about, what Amethyst wanted from him, stuff like that.\"\n\n\"And if he won't answer? It's not as if he has to.\"\n\n\"Oh, he'll answer,\" Sean said, with assurance. \"I'll make him want to.\"\n\n\"How will you make him do that?\"\n\n\"By letting him know his life will be a lot easier if he talks to me.\"\n\n\"I see,\" Marvin said. \"But what if he doesn't want to? What if you can't convince him?\"\n\n\"Then we'll find another approach.\"\n\nOver the years, Sean had found that there was always a way to get the information he needed. It was just a matter of figuring it out. He sat back and watched the road stitch itself up the hill.\n\n\"We're almost there,\" Marvin said.\n\n\"I know,\" Sean replied.\n\nA moment later, the gate appeared before them. It seemed to spring up out of nowhere. The gate was illuminated by two halogen spotlights that bore down on it from above. For a moment, Sean had the ridiculous feeling that any second now border guards would step out from behind the gate and demand to see his passport. I'm watching too many old movies, he said to himself.\n\nAs they got closer, they could see Banks's house. It was lit up; someone was home, unless the house was on a timer.\n\n\"Press the intercom button,\" Sean instructed Marvin as they stopped in front of the gate.\n\nMarvin got the same results as before.\n\n\"Try again,\" Sean instructed.\n\nAs Marvin did, Sean scanned the wall. Now that he was looking more closely, he could see two cameras mounted on either side of the gate. He didn't know whom he was more annoyed with: himself for not seeing the cameras sooner or Banks for not answering.\n\n\"Do you want to go?\" Marvin asked when no one answered.\n\n\"Not yet,\" Sean replied.\n\nIf Banks had answered him, he'd have let the matter go, but ignoring him was something else entirely. Maybe he wasn't the chief of police anymore, but that didn't mean he had to tolerate this type of rudeness. He never had, and he wasn't about to start now.\n\nHe looked at the house again. He thought he saw a flicker of movement in the right window. And then he realized he was seeing something else as well. The gate was slightly ajar. Very slightly. Was it like that before? Sean closed his eyes and tried to remember. He couldn't.\n\n\"Are you okay,\" Marvin asked.\n\n\"I'm fine. Do me a favor and get out and see if you can push the gate open.\"\n\n\"But...,\" Marvin objected.\n\n\"Just do it,\" Sean snapped. This was what he hated. In the old days, pre-illness, he wouldn't have had to ask. He'd have just gotten out and done it.\n\nHe watched Marvin get out of the car. Could he move any slower, Sean wondered.\n\n\"Don't slam your door,\" Sean warned just as Marvin did.\n\nGreat, Sean thought. Now we'll get to see if someone comes out. But no one did. Maybe they hadn't heard. Maybe no one was at the control panel. Maybe the cameras were just for show. No way to know unless he saw the control room. He turned his attention back to Marvin. Marvin was at the gate now.\n\n\"Open it all the way,\" Sean instructed, after rolling his window down. \"We want to drive in.\" Sean paused. \"And, yes, I'm sure,\" he told Marvin even though Marvin hadn't said anything.\n\nFinally, Marvin began to push. The gate slowly opened. When it was open all the way, he got back in the car. \"I expected it to squeak,\" he said.\n\nSean just sighed. \"Drive in,\" he said.\n\n\"Where?\"\n\n\"Up to the front of the house.\"\n\n\"I still don't think this is a good idea,\" Marvin said as he put the car in gear. \"In fact, I think it's a terrible idea.\"\n\n\"Everything will be fine,\" Sean said automatically.\n\n\"That's what you said about the Foundation,\" Marvin protested.\n\n\"And it was, wasn't it?\"\n\nSean guessed Marvin couldn't think of a comeback, because he didn't say anything. As they drove in, Sean kept looking around, expecting to see someone, but there was no sign of movement. Most of the windows in the house were lit up, but Sean didn't see any people. He must have imagined he saw someone before.\n\nThe lawn seemed like a vast expanse of dark sea. Over to the left, he spotted a garage. He could make out what looked like a Jeep parked in front of it, which really didn't mean a whole lot vis-a-vis whether or not someone was home. People that lived in houses like this one usually had multiple cars.\n\n\"Where do you want me to stop?\" Marvin asked.\n\n\"In front of the house,\" Sean told him as he reached in his jacket pocket and got out his cigarettes.\n\n\"Since when did you start smoking?\" Marvin cried.\n\n\"I didn't start smoking. I restarted smoking.\"\n\n\"Libby must be really pissed.\"\n\n\"Libby doesn't know, and you're not going to tell her.\"\n\n\"I don't think I can do that.\"\n\n\"Of course, you can. I'm not going to stop, so why upset her?\"\n\nMarvin digested this piece of information for a moment and then said, \"It's so bad for you.\"\n\n\"Yes, it is, but look at what I have. What difference does it make? Besides, I read recently that smoking may put what I have in remission.\"\n\n\"Really?\" Marvin said.\n\n\"Yes, really,\" Sean replied. \"Now park in front of the house, and help me out of the car.\"\n\nMarvin did as he was told. \"It's a big house,\" Marvin said.\n\nSean took another couple of puffs of his cigarette and threw it down on the grass. \"It's enormous.\"\n\nHe stood there for a moment, taking it in. The building was a two-story Greek Revival affair. Because the drapes weren't closed, Sean had the feeling he was looking at a stage set. The furnishings in the rooms gave him the same feeling. Everything was for show. Banks must live in the other part of the house, Sean decided. No one's dwelling could be that perfect.\n\nHe lifted his hand and rang the bell. He could hear the chime echoing within the house. No one came. He tried again, only this time he left his hand on the buzzer a bit longer. By the third time, his finger was on the buzzer for a full minute. Sean tried the door next. It was locked.\n\n\"He's not here. Let's go,\" Marvin said eagerly.\n\nObviously, Sean thought, the kid could hardly wait to get out of there. But that wasn't going to happen yet.\n\n\"First, let's go around to the side,\" Sean said as they got back in Marvin's car.\n\n\"But why?\" Marvin wailed.\n\n\"To see if the side door is open. Something's wrong here, and I want to check it out.\"\n\nMarvin began tapping his fingers on the steering wheel. \"Then let's call the police.\"\n\n\"A valid suggestion,\" Sean told him, \"but the moment the police arrive, I'm going to be shut out, and I want to see what I can find out before that happens. We'll call them when I'm done.\"\n\n\"Great,\" Marvin muttered, putting his car in gear.\n\nSean pretended he hadn't heard Marvin's last comment as they drove around to the side. After all, the kid was going along with Sean, and that was all that mattered. Marvin parked, and they both got out. He offered Sean his arm in support, but Sean waved him away. Damned if he wasn't going to do this by himself.\n\nThe trick was to take slow, careful steps. The dark made seeing the path clearly harder, and he didn't want to stumble and fall. Then Marvin would tell Libby, and she wouldn't let him out of the house at all. Of course, she wouldn't be too happy when she heard about this, anyway. Oh well. There wasn't much he could do about that.\n\nAs he walked, he debated about what course of action he was going to take if the side door was locked. After all, there was no reason to think that it wouldn't be. But it wasn't. Sean could see the light spilling out from the space between the door and its frame.\n\n\"We should call the police,\" Marvin repeated.\n\nSean nodded absentmindedly as he pulled his jacket sleeve over his hand and pushed the door open.\n\n\"Why are you doing that?\" Marvin asked.\n\n\"So I don't contaminate the crime scene.\"\n\n\"But you don't know it's a crime scene,\" Marvin pointed out.\n\n\"Always assume the worst,\" Sean told him. \"And don't touch anything,\" he warned.\n\n\"I don't think I want to go in,\" Marvin said.\n\n\"Then stay outside,\" Sean snapped as he took a step inside. God, what a pain in the ass that kid was sometimes.\n\n\"I can't. Libby would kill me if anything happened to you.\"\n\n\"And I'm going to kill you if you don't stop talking. I can't concentrate with you chattering away.\"\n\n\"Okay.\"\n\nSean watched Marvin get that hangdog look. He felt a small stab of guilt but managed to stifle it.\n\n\"I guess no one's here,\" Marvin said. Then he realized what he'd done and put his hand to his mouth. \"Sorry.\"\n\n\"I think that's a fair assumption to make,\" Sean said. \"Given the amount of talking we've been doing, if anyone was here, they'd be pointing their rifles at us by now.\"\n\nSean looked around. He was in the mudroom. There were four jackets hanging on the wooden pegs and three pairs of boots sitting on bootjacks. A wicker basket full of hats, scarves, and gloves sat on a bench. He took another step and found himself in the kitchen. Marvin was right behind him.\n\nThe kitchen was huge. The cooking appliances were at one end, and the family room, complete with a flat-screen TV large enough to cover the entire wall, was at the other end. CNN news was on, but there was no sound. Judging from the size of the stove and the fridge, you could feed a platoon in here and still have room for another couple of dozen people.\n\nThe kitchen table had been set for coffee. There was a French press, plates, mugs, and sugar and cream on the table. A platter of pumpkin bars sat in the center. Sean walked over and took a look at the cookies. They looked like A Touch of Heaven's ginger pumpkin bars with ginger icing. Exactly like them. But just to make sure, he picked one up and took a bite.\n\n\"What are you doing?\" Marvin cried.\n\n\"Eating,\" Sean said. Yup. They were Libby's. No one made them like she did. It was the Jamaican ginger that did it. \"You want one?\" he asked Marvin. \"They're Libby's.\"\n\nMarvin shook his head. \"If you don't mind, I'll pass.\"\n\n\"No appetite?\" Sean asked as he ate another one.\n\n\"What happened to contaminating the crime scene?\"\n\n\"I don't think two cookies will make that big a difference in the scheme of things,\" Sean said, wishing he had some milk to wash them down. There was probably milk in the fridge, but that would be going a little too far.\n\nHe wondered who had brought the cookies here. Maybe Amber or Googie would remember, but Sean doubted it. As he dusted the crumbs off his hands, he noticed that the floor by the sink was wet. He walked over. Three or four apples were bobbing in the basin. Very odd. Then he noticed a few brown-red spots on the lip of the basin. Dirt? He looked closer. No. He got that old familiar feeling. He beckoned Marvin over and pointed.\n\n\"What do you think?\" he asked.\n\nMarvin leaned over and studied the spots. He sucked in his cheeks as he concentrated. Finally, he said, \"I think it's blood.\"\n\n\"Me too,\" Sean agreed.\n\nMarvin straightened up. \"Someone could have cut themselves with a knife.\"\n\n\"Yes. They could have.\"\n\n\"But you don't think so.\"\n\n\"No. Do you?\"\n\n\"No.\" Marvin pointed to the apples. \"What about those?\"\n\n\"You got me. I'm going to have to think on that for a bit.\" He motioned for Marvin to follow him. \"Come on. Let's see what else we're going to find.\"\n\n\"I think we might be finding Mr. Banks,\" Marvin said.\n\n\"I think you may be right,\" Sean agreed.\n\n\"Which is why we should call the cops.\"\n\n\"Soon,\" Sean said.\n\n\"You're just placating me,\" Marvin complained.\n\n\"Yup,\" Sean said. \"You got me. That's exactly what I'm doing.\"\n\nSean looked around once more just to make sure he hadn't missed anything. Then he walked straight ahead, with Marvin on his heels. When he got to the corner, he turned down a hallway.\n\nThere were watermarks on the wallpaper every couple of feet or so, as if someone had bumped into it with his shoulder. He kept walking. About two feet farther, he came to a room. From outside the room, he could hear a fan running. The smell told him what he was going to find.\n\n\"When you're right, you're right,\" Sean said.\n\nHe and Marvin stepped inside. A man was slumped over the toilet. His hands were tied behind him.\n\n\"I have a feeling that we've found Banks,\" Sean said to Marvin.\n\n\"Me too,\" Marvin said.\n\nBanks was wearing casual attire: a pair of jeans, sneakers, a white shirt, and a blue crewneck sweater.\n\n\"How long do you think he's been dead?\" Sean asked Marvin. That was one good thing about Marvin, Sean decided. He wasn't squeamish about corpses.\n\nMarvin assessed Banks with a practiced eye. \"Maybe three or four hours, but Wenzel should be able to narrow it down more closely.\"\n\nSean just hoped that they hadn't been standing by the gate when Banks was getting drowned. He watched Marvin take out his cell phone. \"Calling the police?\" he asked.\n\n\"Yes.\"\n\n\"I am a policeman.\"\n\n\"You're a retired policeman.\"\n\n\"It's almost the same thing.\"\n\n\"Not quite.\" Marvin just looked at him.\n\n\"All I'm asking for is fifteen minutes to see if we can find Banks's records. That's it. I swear.\"\n\nMarvin didn't say anything.\n\n\"Well?\" Sean said after a minute had gone by. \"Is it a yes or a no?\"\n\nMarvin let out a long sigh. Then he said, \"I'm only doing this because you're Libby's dad.\"\n\n\"And I can't tell you how much Libby will appreciate this,\" Sean said.\n\n\"That's the point. I'm not sure she will.\"\n\nSean waved away Marvin's objections. \"Come help me look. The sooner we get started, the sooner we'll be done.\"\n\n\"We'll be arrested. That's what we'll be,\" Marvin muttered.\n\nSean ignored him and led the way out.\n\n## Chapter 15\n\nBernie watched her dad's friend Clyde settle back in his usual chair in their living room. He reached over and took another one of Libby's ginger pumpkin bars. He took little bites and chewed slowly so he could savor every mouthful.\n\n\"Wonderful,\" he said as he poured more cream in his coffee. \"Simply wonderful. And that includes the cream. My wife only has skim milk in the house. It turns coffee the most unappetizing shade of gray.\" He took a sip and then held up the ginger pumpkin bar. \"This is the embodiment of Halloween,\" he declared. \"The color, the bouquet of spices, all suggest late fall to me.\"\n\n\"You've been watching the cooking channel again, haven't you?\" Sean asked. This was a man who in his prime consumed cans of cold Dinty Moore stew, and now he was rhapsodizing about flavor bouquets the same way he used to talk about the Playmate of the Month. Old age was a terrible thing.\n\nClyde glared at him. \"So what if I have?\"\n\n\"Nothing. Nothing at all,\" replied Sean.\n\n\"You're also eating the evidence,\" Bernie said.\n\n\"No. That was what I did last night,\" Sean said.\n\nClyde shook his head. \"Good thing Lucy didn't catch you.\"\n\nSean snorted. \"Lucas never appears at crime scenes.\"\n\n\"He did at this one,\" said Clyde.\n\n\"I didn't see him,\" said Sean.\n\n\"That's because he came after you left, and stayed for about a minute and a half.\"\n\n\"Interesting,\" Sean said as he moved his motorized wheelchair a little more toward the window. It was a little after ten on a Tuesday evening, and the street was empty. But it didn't matter. He enjoyed looking at the Halloween decorations in his neighbors' windows. Black cats, witches, ghosts\u2014all were stuck to the windowpanes. They reminded him of when his daughters were young and they had helped decorate. \"That must mean that Banks's murder is important.\"\n\nClyde reached over and took a third ginger pumpkin bar. \"Well, Banks was rich.\"\n\n\"That would do it,\" Sean said. \"What does Lucy think the relationship between the two homicides is?\"\n\nClyde took a bite of the ginger pumpkin bar and swallowed. \"Oh, the chief doesn't think Amethyst's murder and Banks's murder are related.\"\n\nBernie rolled her eyes. \"That's absurd.\"\n\n\"He said they had dissimilar styles,\" Clyde continued. \"And the fingerprints don't match. Not that there were many of them at either crime scene. Ergo, it's just coincidence.\"\n\nSean took a sip of his tea and put the cup down. \"I, myself, have never believed in coincidence.\"\n\n\"Me either,\" Clyde agreed. \"I don't know how the homicides are linked, but they definitely are.\"\n\n\"That's for sure,\" Sean said. \"What are the odds of having two homicides in a town like this in one week and not having them be related?\"\n\n\"It could be a statistical anomaly,\" Bernie suggested. Her dad glared at her. \"Or maybe Bessie Osgood came back to life and traveled over to Lexus Gardens.\"\n\nLibby rolled her eyes. \"Now, why didn't I think of that?\" She pointedly turned to her dad. \"What do you think happened?\"\n\nSean took another sip of his tea and put the cup down. \"On the most literal level, I think Ed Banks had a visitor, and that visitor brought some ginger pumpkin bars from our shop as a gift.\"\n\n\"Ed Banks could have bought them himself,\" Libby pointed out.\n\n\"I don't think so,\" Bernie said. \"We pretty much know everyone that comes into our shop, and when I asked, Amber and Googie said no one unfamiliar came in that day. Given the fact that we've never done business with Ed Banks, I think the conclusion is self-evident.\"\n\n\"Which means whoever brought them is one of our customers,\" Libby observed.\n\n\"Unfortunately,\" Sean said.\n\n\"Well, that narrows the field,\" Bernie said, thinking of the hundreds of men, women, and children who went through the shop each day.\n\n\"Maybe we could put out a sign reading WHOEVER BOUGHT COOKIES FOR ED BANKS COME TALK TO US,\" Libby suggested.\n\nSean laughed. \"That's what I call wishful thinking.\" He turned to Clyde. \"So what do we know about this guy Banks?\"\n\n\"The most obvious fact is that the guy was a recluse,\" Clyde said.\n\nBernie leaned forward. \"But he let his house be photographed. Recluses don't usually do that.\"\n\n\"True,\" Clyde said. \"Maybe he just didn't want to talk to anyone around here.\"\n\n\"Then why buy a house here?\" Bernie asked.\n\nSean waved his hand impatiently. \"Let's come back to that question later. What else do we know about him?\"\n\n\"Really not that much,\" Clyde replied. \"At this point, we know that Banks has no known next of kin. Both parents are deceased. He never married. He didn't seem to have a girlfriend....\"\n\n\"Maybe he had a boyfriend,\" Bernie interjected.\n\n\"He didn't have anyone that we're aware of,\" Clyde said, with a touch of asperity.\n\nBernie shrugged. \"It was just a suggestion.\"\n\nClyde went on. \"Anyway, he was born here, in this town, but his family moved to Hawaii when he was in his teens...Evidently, his dad was some kind of expert on sugarcane...and he only came back recently. He did hedge funds, which is where he got his money. He has a clean record. No priors. And that's about all we know at this moment.\"\n\n\"Who is claiming the body?\" Sean asked.\n\n\"A distant relative in Hawaii. She wants the body flown to Oahu as soon as it's released,\" said Clyde.\n\n\"How about the staff?\" Sean asked. \"Where were they last night?\"\n\n\"There is no live-in staff,\" said Clyde.\n\nLibby took a sip of her mulled cider and asked, \"He lived in that huge house all by himself?\" Her voice was incredulous.\n\nClyde ate the rest of his ginger pumpkin bar before answering. \"Yes, he did. He had a personal assistant that he brought with him. The guy came in six days a week, from nine to six. He had Sundays off.\"\n\n\"Did you talk to him?\" asked Sean.\n\nClyde nodded. \"We managed to track him down. Conveniently for him, he's been on vacation in Maui for the last two weeks. According to the hotel manager, he hasn't left the island.\"\n\n\"Did he sound upset when he found out about his boss?\" Sean asked.\n\n\"Very. They'd been together for a long time. Maybe there's something there, but I don't see it,\" said Clyde. \"More interestingly, however, is the fact that Banks contracted with the same firm that cleans the Foundation to do the cleaning up there.\"\n\n\"Inez?\" Sean said.\n\n\"It's a definite link,\" Clyde said, turning to Libby. \"How's the pumpkin cheesecake?\"\n\nLibby put a piece on a plate and handed it to him. \"Try it and see.\"\n\nClyde took a bite. \"Delicious,\" he said. \"I have to come over to the Haunted House to try your waffles.\"\n\n\"Anytime,\" Libby said. Then she turned to her dad. \"I've been thinking. It must have taken a really big man to hold Banks's head under water like that. And what were the apples about? Are they a symbol of some kind?\"\n\n\"They could be,\" Bernie said. \"The Celts used bobbing for apples in marriage divination ceremonies. The first person to bite the apple was the first person to get married. It was their version of throwing the bouquet.\"\n\n\"Are you saying that Banks was planning on getting married?\" Libby asked her sister.\n\n\"No. I'm just sharing a little information with you,\" replied Bernie.\n\n\"I'm sorry. I just don't see the relevance,\" said Libby.\n\nSean looked away from the street and settled himself in his wheelchair. He had a pack of cigarettes in his pants pocket, which he would have very much liked to light, but he wasn't going to do it and risk the wrath of his daughters.\n\n\"It is relevant, just not in the way that Bernie said,\" Sean interjected.\n\nLibby shook her head. \"I don't understand.\"\n\n\"It tells us what happened,\" said Sean. He held up his hand as Libby started to speak. \"Let's go back to the beginning. Banks and his friend, and I'm putting friend in quotes here, had made an appointment. We know this because Banks had already set the table. I think that they sat around and talked for a while, and then his friend probably casually introduced the topic of bobbing for apples. You know, he said something like, 'I bet you can't bob for apples,' or something to that effect, and Banks took him up on the challenge. So while Banks was bending over, his friend slipped a plastic tie out of his pocket and cuffed him. Easy enough to do.\"\n\n\"Then why didn't he drown him right there? Why take him to the bathroom?\" Clyde asked.\n\nSean thought about the blood on the kitchen sink and the marks on the hallway walls. \"I'm thinking that Banks got away from him, and they had a scuffle. I figure Banks started running, and his friend finally caught up with him in the hallway, near the bathroom. He managed to get Banks in there and hold his head in the toilet.\"\n\n\"You think the toilet was a metaphor?\" Bernie asked.\n\nSean laughed. \"No. I think the toilet was convenient.\" He took another sip of his tea. \"So Lucy sees no connection at all between these two crimes?\" he asked Clyde.\n\nClyde put his fork down. \"If he does, he isn't telling me.\"\n\nSean sat and thought for a moment. \"It's true we have minimal connections between the two events,\" he finally said. \"We only have Jeanine's word that Amethyst wanted to talk to Banks.\"\n\n\"Okay,\" Clyde said.\n\n\"But why would Jeanine lie?\" Libby asked.\n\n\"I'm not saying she did. In fact, I don't think she did. I'm just talking it through,\" Sean said.\n\n\"And we don't know if Amethyst actually went up and talked to Banks,\" Clyde pointed out.\n\nSean nodded. \"Maybe Banks's personal assistant knows.\"\n\n\"I'll find out, but it might take a little while,\" Clyde told Sean. \"When I spoke to him, he was leaving for a sailing trip.\"\n\n\"And even if he did,\" Sean said, continuing with his train of thought, \"we have a very thin line linking Amethyst and Banks. A very thin line. Maybe she wanted to talk to him about some sort of charity affair. Or about opening up a shop of some kind. We really don't know.\"\n\nBernie tucked a strand of her hair behind her ear. \"I'm getting the impression you agree with Lucy that there is no connection between the two homicides.\"\n\nSean leaned back in his wheelchair and folded his hands over his stomach. \"No. I think there is. I just think it has to be ferreted out. In order to find it, we need more information.\"\n\nClyde finished off his piece of cheesecake and looked at the plate wistfully. He sighed. \"I can't eat another thing,\" he said.\n\n\"Not even a sliver?\" Libby coaxed.\n\nClyde shook his head. \"You're a very bad person.\"\n\n\"I know,\" Libby said as she took his plate, cut him a small piece of cheesecake, and handed the plate back to him.\n\n\"Tell me,\" Clyde asked Sean after he'd taken a bite, \"do you still think that Bessie Osgood had anything to do with this mess?\"\n\n\"Without a doubt,\" Sean said. \"All the names that have come up have been linked to her death. I still think that if you find out what happened the night she died, and you'll find out who killed Amethyst and, possibly, Banks.\"\n\nClyde leaned forward. \"The question is, why is all of this happening now?\"\n\n\"That is the question, isn't it?\" Sean said.\n\n\"There has to have been a precipitating incident,\" Clyde mused.\n\nBernie stifled a yawn. \"But what?\"\n\n\"I wish I knew,\" Sean said, and he went back to watching the street. A little girl decked out in a Hello Kitty outfit skipped by, holding her mother's hand. He smiled, remembering how the girls used to wear their costumes around the house during the week before Halloween.\n\n\"You know,\" Bernie said, \"not to change the subject, but Kathy\u2014\"\n\nSean turned away from the window. \"Big Kathy?\"\n\nBernie made an impatient gesture. \"Garden shop Kathy. My friend Kathy.\"\n\n\"What about her?\" asked Sean.\n\n\"She told me that Zinnia was killed by a hit-and-run driver a year after Bessie Osgood died,\" said Bernie.\n\n\"I remember that one,\" Clyde said. He turned to Sean. \"Didn't Porter get that guy?\"\n\n\"Guys,\" Sean corrected. \"Two of them. They'd just robbed the Quick Mart in Oakley when they got McGuire. They said they didn't see her, because she was sitting in the middle of the road. She had enough alcohol in her to embalm an elephant.\"\n\n\"So much for that one,\" Clyde said.\n\nEveryone was silent for a moment. Then Bernie said, \"We still don't know about Timberland's daughter.\"\n\nClyde stifled a yawn. \"Refresh my memory as to why we care about her.\"\n\n\"Because she might furnish a motive for Timberland's animosity toward Amethyst,\" said Sean.\n\n\"Maybe I can find out,\" Libby said.\n\n\"How?\" her dad finally asked.\n\n\"Well,\" Libby stammered, \"I know his sister from yoga class.\"\n\nBernie's eyebrows shot up. \"You're taking a yoga class?\"\n\nLibby straightened her shoulders. \"As a matter of fact, I am.\"\n\n\"Since when?\" asked Bernie.\n\n\"Since last week, if you must know. Why?\" said Libby.\n\n\"I'm just surprised. It doesn't seem like your kind of thing,\" replied Bernie.\n\nLibby put her hands on her hips. \"And why ever not?\"\n\nSean intervened before things got started. \"Bernie, maybe you could talk to Inez. She spends a lot of time at R.J.'s, and I can talk to Jeanine and see if she's figured out the View-Master yet.\"\n\n\"I understand Bob Small is going to be out on bail tomorrow,\" Clyde said.\n\n\"Then I suppose one of us should go talk to him as well,\" Sean said.\n\n\"Not me,\" Clyde said.\n\n\"Obviously,\" Sean shot back.\n\nClyde stifled another yawn. \"Okeydokey. Time to get going.\" He rose. \"After all, another glorious day in Longely's police force awaits me tomorrow morning.\"\n\n\"By the way,\" Sean called out, \"who put up Small's bail?\"\n\n\"Kane,\" said Clyde.\n\n\"Interesting,\" said Sean.\n\n\"It's not that high,\" added Clyde.\n\n\"Still,\" said Sean.\n\n\"While we're on the subject, I found out some more info on Kane,\" said Clyde.\n\nSean leaned forward. \"Such as?\"\n\n\"Nothing that we didn't already know,\" said Clyde. \"He's considered a genius with numbers, which is how he got so rich. He's pretty much a workaholic. No surprise there. He had a minor heart attack a couple of years ago, and his doctors advised him to get a hobby. Hence the Foundation. I also talked to the guys that rigged up the Haunted House show. They said he couldn't even put in a lightbulb without help.\"\n\n\"That's what he said about himself,\" Libby commented.\n\n\"Well, it looks as if he was telling the truth,\" Sean said. \"So there you go.\"\n\nAs Clyde headed down the stairs, Sean reflected on how nice it was to be working with him again. That was probably the thing he missed most from his days on the force.\n\n## Chapter 16\n\nLibby walked into the class, hung up her jacket, and looked around. It was seven-thirty in the morning, and everyone in the class looked disgustingly perky, but then they probably hadn't been up since five making lemon squares and chocolate chip cookies.\n\nAnd then there were the clothes. Everyone was wearing cute little yoga outfits, the kind that cost a couple of hundred dollars or so, while she was in her old stretched-out sweatpants and T-shirt. It was true she could have gone and bought one of those outfits\u2014nothing was stopping her\u2014but she hated spending money on stuff like that. Okay, that was a lie. What she hated to do was wear stuff like that. Even trying it on was painful. One look in the mirror and she wanted to reach for the cookies\u2014not exactly a productive response given the circumstances.\n\nShe was taking this class to tone up, but she didn't know how long she could stand it. The ad had promised results in three sessions. Well, this was her third session, and she had yet to see any results. She surreptitiously pinched the roll of fat on her belly. Yup. It was still there. She felt especially depressed as she scanned the rows of women rolling out their yoga mats. They were all so trim and taut, and she was so...so not. Even their mats looked better than hers.\n\nLibby sighed. Everyone was doing their warm-up stretching. Little tinkling bells chimed in the air. There was incense burning on the front table, where the instructor, an impossibly lithe woman wearing adorable yoga pants and a bralike top that showed off a midriff with no fat at all, was talking to someone. A sign that said, BREATHE! BREATHE! was hanging on the wall.\n\nShe spotted Timberland's sister in the fourth row and reluctantly headed for her. Why couldn't Ramona be in the back? Why did she have to be up front? God. Libby reminded herself she was here to do a job and that no one was looking at her\u2014yeah, right\u2014as she unrolled her yoga mat and plopped herself next to Ramona.\n\n\"Hi,\" she said.\n\nRamona smiled. She had perfect blond hair and white, white teeth, and was extremely flexible. \"Hi,\" she said, with her head down almost to her right thigh.\n\nLibby, who couldn't even touch her toes and hadn't been able to since high school, hated her.\n\n\"How are things going?\" Libby asked. General questions were always best in situations like this, her dad had taught her.\n\n\"Good. We're all going to the Haunted House tonight. Which waffles would you recommend?\"\n\n\"The pumpkin ones are my favorite,\" Libby replied promptly. \"But lots of people like the chocolate ones.\"\n\nRamona switched to her left thigh and held the stretch for a moment. \"This class makes me feel so alive.\"\n\n\"Me too,\" Libby lied as she followed Ramona's lead. What it really made her feel was sorer than anything she'd ever done.\n\nA moment later Ramona put her left arm up, bent it toward the middle of her back, and grasped her right arm with it. Libby did likewise. She could hear her shoulder pop. She ignored it. All the same, it wasn't a good sign.\n\nRamona looked around the room for a moment, then scooched closer to Libby and whispered, \"Did you really find Amethyst's head?\"\n\nLibby nodded. She had an idea that this wasn't exactly yoga-class discussion material.\n\n\"It must have been horrible,\" Ramona said.\n\n\"It was.\"\n\n\"You know my brother knew her,\" Ramona continued as she stretched out her other shoulder.\n\n\"Yeah. He told me they used to hang out back in the day,\" Libby lied again.\n\nRamona set her mouth in a thin line.\n\nLibby waited. Ramona remained silent.\n\n\"He didn't seem that sorry,\" Libby ventured after it became obvious that Ramona wasn't going to say anything else.\n\nRamona snorted and worked her legs into a lotus position. Libby tried to emulate her and failed. Her calves didn't seem to want to do that.\n\n\"I don't think anyone is that sorry about Amethyst,\" Ramona observed. \"Are you?\"\n\n\"Not really,\" Libby confessed.\n\n\"Exactly my point.\"\n\n\"But I thought she and your brother were friends.\"\n\n\"Zachery thought they were friends, too,\" Ramona said.\n\n\"So what happened?\"\n\n\"What happened?\" Ramona repeated. She took a deep breath and let it out. \"What happened was that Amethyst didn't have any friends. She had people she used, and she had people she was going to use.\"\n\n\"You don't sound like a big fan of hers, either.\"\n\n\"I'm not.\"\n\nLibby started to lean over to ask Ramona why, but the instructor stared at her, brought her hands together, gave a slight bow, and chirped, \"Namaste.\"\n\n\"Namaste,\" the class replied.\n\nLibby took a deep breath. Her questions would have to wait. Class was beginning.\n\nForty-five minutes later it was over, and Libby was still smarting from having gotten stuck in the lotus position. Life was so unfair. She'd finally managed to get herself into the stupid position, and then she couldn't get herself out of it. She closed her eyes and tried to blot out the memory of her tipping over and falling into Ramona. Someone else might have laughed. Ramona hadn't. What she had done was look very annoyed.\n\n\"It will get better in time,\" the instructor murmured in her ear as Libby walked by her. \"You just have to practice, practice, practice. Remember if at first you don't succeed...\"\n\nLibby nodded. Could you get any more clich\u00e9d, she wondered. She didn't feel it necessary to tell the instructor there wasn't going to be a next time. She had four more classes. Maybe Bernie would like to go in her place.\n\nSo much for self-improvement, Libby thought as she walked out of the class. Aside from publicly humiliating herself, she'd learned absolutely nothing about Timberland, and now she was behind schedule at the shop. She was standing by her van, eating a piece of dark chocolate and thinking about the costume Bernie wanted her to wear this evening\u2014she was not going as a bowl of Special K!\u2014when she saw Ramona walk to her car. She was talking on her cell and making angry gestures in the air with her free hand. Okay, Libby decided. Maybe I should give this one more try. After all, what did she have to lose?\n\nEven though Ramona was half turned away from her, as she got closer, Libby could hear Ramona saying, \"Listen, Madison, don't do this. No. I don't have an address to send a card. Neither does your dad. And for heaven's sake, don't ask him. He doesn't need any more aggravation. I mean it, Madison.\"\n\nLibby bit her lip. Madison was the name of Timberland's daughter. Someone had been talking about her recently. As she was trying to remember who it was, she watched Ramona glare at her phone.\n\n\"Great,\" Ramona muttered under her breath as she flipped the phone closed and shoved it in her bag. She gave a little jump as she spotted Libby, but quickly recovered. \"Just be happy you don't have kids,\" she said to Libby. \"Even if they aren't yours, they're an epic pain in the ass.\"\n\n\"Can I ask you a question?\" Libby inquired.\n\nRamona looked at her. \"That depends on what it is.\"\n\nThis time Libby got right down to it. \"It's about why your brother disliked Amethyst.\"\n\nRamona put her hand on the door handle of her Caddy Escalade. \"Everyone disliked Amethyst. We already discussed that.\"\n\n\"But your brother seems to have a special reason.\"\n\nRamona threw Libby what her father would have called a measuring glance and said, \"Go ask him.\"\n\n\"You were going to tell me back in class.\"\n\n\"Maybe I've changed my mind.\"\n\n\"Because you think that your brother had a motive to kill Amethyst?\"\n\nRamona composed her mouth into a shocked O, only Libby wasn't buying it. Too much drama. \"Heavens no. What a terrible thing to say.\"\n\n\"Then why won't you tell me?\"\n\nRamona shrugged again. \"Because it's none of your business.\"\n\n\"True,\" Libby told her. \"It isn't. But the fact that you're not answering me will make other people curious.\"\n\n\"And I should care why?\"\n\nGood question. Libby improvised. \"The police will care,\" she said.\n\n\"You'll have to do better than that,\" Ramona said. \"The police already arrested Bob Small. They're not interested in me.\"\n\n\"How about if we make a deal,\" Libby said.\n\nRamona arched one of her perfectly tweezed eyebrows. \"Which would be?\"\n\n\"You tell me what I want to know, and I'll never come to your yoga class again.\"\n\nRamona burst out laughing. \"Good. But not good enough.\" Ramona's phone started ringing again. She took it out, looked at it, and grimaced. \"Madison,\" she said.\n\nAnd all of a sudden, Libby remembered where she'd heard the name. It was from Amber, one of the kids that worked in her place.\n\n\"She's your niece, isn't she?\" Libby said.\n\nRamona raised her eyebrow again.\n\n\"One of the girls who works for us went to school with her,\" said Libby.\n\nRamona didn't say anything.\n\n\"She was telling me about her. She dropped out of school.\"\n\nRamona sighed. \"It's not that big of a deal. Lots of kids drop out.\"\n\nLibby closed her eyes for a moment and concentrated on remembering the details of what Amber had said. \"But she was at the head of the class. Number one. Editor of the school newspaper. Varsity track. Student council president. Had been admitted to Yale. People like that don't usually do that type of thing.\"\n\n\"Who knows what kids will do these days?\" Ramona said, trying to sound casual and failing, as she opened the door of the Escalade and tossed her yoga mat into the backseat.\n\n\"Still, you have to admit it's pretty unusual.\"\n\nRamona shrugged.\n\n\"Your brother must have had a fit.\"\n\n\"It's true he was very disappointed. Everyone in the family was. He thought she was going to Yale and then on to law school. But what can you do.\"\n\n\"I guess not much,\" Libby said.\n\n\"If she wants to work as a waitress down in the city, that's her business. I'm hoping that eventually she'll come to her senses. And now, if you're through, I have to get back to my house. My cleaning people will be there shortly....\"\n\nLibby raised a finger. \"Just one more thing. Amber said all this happened because your niece had an affair with an older woman, and she dropped your niece in a particularly not nice way.\" Judging from Ramona's expression, Libby knew that what Amber had told her was correct. \"And,\" Libby said, making the logical leap, \"I'm betting the person she had an affair with was Amethyst.\"\n\nRamona blinked.\n\n\"It was, wasn't it?\"\n\nLibby watched Ramona's hand come up and finger the heart-shaped locket she wore around her neck.\n\n\"So what if it was?\" Ramona snapped. \"That's no reason for my brother to kill her. Other kids end up a lot worse off. If you want someone who had a reason to kill Amethyst, talk to Bob Small or Inez.\" With that, Ramona got in her vehicle and drove off, missing Libby by inches.\n\n\"Maybe it's not a great reason,\" Libby said to herself as she walked back to her van, \"but it's good enough.\"\n\nEspecially these days, when more and more parents are living through their children.\n\n## Chapter 17\n\nLibby looked at the clock on the kitchen wall. It read twenty minutes past eleven. Great. Only an hour and a half behind schedule. She had to get the Rogets' birthday cake done before three, as well as the chicken curry for the Mathers' party, which meant she might not be able to make the two apple pies for the Haunted House, which she hadn't gotten to last night.\n\nFortunately, she had a couple of lemon tarts in the freezer. They might not be very Halloweeny, but it was the best she could do. And she had made the black cat cookies, so they were ahead there.\n\nShe wiped her spatula off and went back to the pumpkin chocolate cake she was icing. This time she'd used a different kind of chocolate, one with a lower butterfat content, and the icing was not as shiny as she would have liked.\n\nShe was thinking about complaining to her supplier, who'd told her the results would be the same, when her dad darted in, came up behind her, scooped up a taste from the bowl with the icing in it, and ate it.\n\n\"Stop that,\" Libby ordered.\n\nSean grinned. \"But it's so good, it's hard to resist.\"\n\n\"I wish it looked better.\"\n\n\"It looks fine,\" Sean assured his daughter as he finished licking the teaspoon. \"And it tastes even better. After all, as your mom used to say, 'The proof is in the eating.'\"\n\n\"Yeah. But Mom didn't have the Food Channel to contend with.\"\n\n\"True,\" Sean allowed as he stared wistfully at the icing.\n\n\"You can have the bowl when I'm done,\" Libby told him, and then she told her dad what she'd found out from Ramona.\n\n\"Interesting,\" Sean said. \"Maybe that's not the best motive, but it's certainly up there. If someone had done something like that to you, I'd want to kill them.\"\n\n\"Kill who?\" Bernie asked as she stepped into the kitchen.\n\nLibby explained.\n\n\"I guess it's a motive,\" Bernie said as she poured herself a cup of coffee and added cream and sugar.\n\nLibby drew her spatula across the icing on top of the cake to even it out. \"Amber said Madison's dad was livid when he found out. Timberland wanted to press charges, but his wife told him she'd divorce him if he did. Of course, she left him, anyway.\"\n\n\"And this was how long ago?\" Sean asked.\n\n\"A couple of years,\" said Libby.\n\nSean put the spoon in the sink. \"So once more we come to the question, why now?\"\n\n\"Opportunity?\" Bernie said.\n\nSean shook his head. \"That might apply to Bob Small, but not to Timberland. At least not as far as I can see.\"\n\n\"None of this makes any sense,\" Libby complained.\n\n\"We're missing information,\" Sean said. \"We have too much on the one hand and not enough on the other.\"\n\nBernie flicked a piece of lint from her turtleneck sweater. \"And what about Banks's murder?\"\n\n\"The one that doesn't have anything to do with Amethyst's?\" her dad said.\n\n\"Yes. That one,\" said Bernie.\n\n\"We have even less on that one. Except for the fact that Amethyst was going to go see him, we have nothing to link the two events,\" said Sean.\n\nLibby put her spatula down.\n\n\"It's kind of like trying to put together a jigsaw puzzle blindfolded,\" said Sean.\n\n\"Good analogy,\" Bernie said, and she leaned over and gave her dad a peck on the cheek. \"I'm off to check something at the historical society, and then I'm going to see if I can talk to Inez.\"\n\n\"When will you be back?\" Libby asked.\n\n\"In a couple of hours,\" said Bernie as she eyed her sister's baggy sweatpants and stretched-out T-shirt. \"Is that what you wore to yoga class?\"\n\nLibby put her hands on her hips. \"So what if it is?\"\n\n\"I didn't say anything,\" Bernie mumbled.\n\n\"But you were thinking it,\" said Libby.\n\n\"What was I thinking?\" Bernie asked.\n\n\"That I look like a mess,\" said Libby.\n\n\"You're always so defensive,\" Bernie protested.\n\nThe comment only made Libby more annoyed, because it was true. She was. But Bernie couldn't possibly understand how she felt. She always looked perfect. Then, to make things worse, Bernie reached over and patted her sister on the shoulder. \"We'll find you a nice costume for tonight.\"\n\nLike I'm a little kid, Libby thought resentfully. \"I am not going as a bowl of cereal!\"\n\n\"That was a joke,\" Bernie insisted.\n\n\"It didn't sound like a joke to me,\" Libby told her.\n\n\"Well, it was,\" said Bernie. She took a sip of her coffee and walked toward the door. \"Wish me luck,\" she called.\n\nAs Sean watched her go, he wanted to tell her that he'd changed his mind and she didn't need to go to the historical society, after all. Then he realized that her visit didn't preclude his later in the day. He could ask Jeanine about the stills on the View-Master Bernie had been handed by Felicity Huffer. Somehow he felt better. He was thinking about why that was when a horn beeped outside. His ride was here. It was time to go.\n\n\"That must be Marvin,\" Sean informed his daughter as he put on his rain jacket. \"We're off to talk to Bob Small.\"\n\nLibby reached over and gave her dad two cinnamon spice cupcakes with mocha icing. \"For the road,\" she said.\n\nSean handed one to Marvin as soon as he got in the car. \"Eat it now,\" he instructed.\n\nThe idea of Marvin eating and driving at the same time didn't bear thinking about. When Marvin was done, Sean said, \"Bob Small's house.\"\n\nMarvin turned to him. \"And that would be where?\"\n\n\"I thought you were supposed to look up the address.\"\n\n\"I thought you were.\"\n\n\"How would I know where he's living?\"\n\n\"I thought you knew everything.\"\n\nSean glared at Marvin for a moment. A year ago he would never have said something like that. He was definitely getting entirely too comfortable. When Sean judged that he'd conveyed his displeasure, he reached for his cell phone. Ten minutes later, he'd gotten the address from Clyde. As Marvin pulled away from the curb, Sean realized that he'd forgotten to bring the tapes that Konrad and Curtis had brought him. He was annoyed with himself, but not so annoyed that he was going to go back and get them.\n\nAccording to Clyde, Bob Small was renting a place on the outskirts of Longely, about two blocks away from the train station\u2014which was as slummy an area as Longely possessed. It was damp and chilly in the car, and Sean wished Marvin would put the heat on, but he wasn't going to ask because, one, he was still annoyed with him and, two, that would be admitting he was cold. Instead, he concentrated on the scenery going by. Every other house had a skeleton hanging in the window or a group of tombstones in the yard.\n\nThey looked sad in the rain, Sean thought. The day was dark and gloomy, making them appear as if they were in a netherworld. He was still thinking that when they got to Bob Small's house. The word that occurred to him when he saw it was shack. It had taken Marvin a moment to find it because it was hidden in the alley behind the dry cleaners. The place was a two-story house covered in asphalt shingles. A blue tarp was tied around the front part of the roof, presumably to fend off leaks. If there had ever been paint on the windowsills and the doors, it had vanished a long time ago.\n\nAs Sean studied the place, he couldn't help thinking of Bob's former house. It had been a good-sized Colonial on a half acre of carefully tended lawn, with a swimming pool and a gazebo and a four-car garage. And then there had been the wife and the two kids that had lived in that house, not to mention the two golden retrievers. As Bernie would say, Bob had had the sweet life, and now, because of Amethyst, he had nothing. The woman had cost Bob Small everything that had mattered. If there was a better motive for murder, Sean couldn't think of one.\n\n\"What are you thinking?\" Marvin asked as he brought his car to a stop in front of the house.\n\n\"I'm thinking that if I had to live here, I would want to kill the person who put me here.\" Sean nodded at the van parked by the side of the house. \"It looks as if the Kurtz boys are here.\"\n\n\"You want to come back another time?\" Marvin asked.\n\nSean shook his head. \"No. No. It'll be fine.\"\n\n\"Your call,\" Marvin said as he turned off his vehicle and pocketed the key. \"Tell me, do you believe in Bessie at all?\"\n\nSean laughed. \"People keep asking me that, and I keep saying I don't, but I gotta tell you, the way things are progressing, she could just as easily have done this as anyone else. How about you?\"\n\n\"No ghosts for me.\"\n\nBut there was something about the way Marvin said it that made Sean turn and look at him. \"Is there something you're not telling me?\" he asked.\n\n\"Not at all,\" Marvin said and got out of the car.\n\nSean watched Marvin go around the car to help him out. He wasn't convinced that Marvin was telling him the truth, but he decided to let it go for now. At that moment he had to concentrate on getting from the car to the top of the steps. The ground leading to the steps looked uneven, and it was probably more so because of the rain. In addition, the two steps up to Bob Small's apartment were sagging in the middle, and there was no banister to hang on to.\n\nThis is what he hated more than anything, he reflected as Marvin helped him out of the car. Being dependent on someone. Sometimes things were okay, and other times they weren't, and the hell of it was he never knew what was going to happen. He shook off those thoughts. They were of no use whatsoever. He should take a leaf from Rose's book and look on the bright side of things. Of course, if he could do that, Sean concluded, he wouldn't have been a cop. Cops never looked on the bright side of things. They were paid to be suspicious.\n\nHe was just thinking about where to put his foot on the step when the door flew open. \"See,\" Curtis Kurtz said to Bob Small. \"I told you, you didn't have to call him. I told you he'd be here.\" He turned to Sean. \"We have more recordings. Bessie lied the last time. Remember she said she did it. Only she didn't. She just said that because she wanted to.\"\n\nSean made the second step. \"I didn't know that ghosts lied.\"\n\n\"Ghosts do and feel everything that people do,\" Curtis told him.\n\n\"I didn't realize that. So who did it?\" Sean asked as Konrad Kurtz closed the door and took their rain gear.\n\n\"She's not telling us yet,\" Konrad said and hung the rain gear up on two pegs sprouting out of the hall wall. \"But don't you worry. We have a few tricks up our sleeve. We'll get it out of her. Also, we have her new recordings. You can hear for yourself.\"\n\nSean looked from one brother to the other. \"You know, guys,\" he said, \"what I really need is a large coffee and two chocolate glazed doughnuts from Dunkin' Donuts. It'll help me concentrate better.\"\n\nSean watched Curtis and Konrad exchange glances.\n\n\"That's a ways away,\" Konrad said.\n\n\"Fifteen minutes,\" Sean said.\n\n\"Maybe twenty with construction,\" Curtis corrected.\n\n\"Each way,\" Konrad added. \"That's forty minutes.\"\n\nSean smiled. \"Which would be the amount of time I need to talk to Bob.\"\n\n\"Oh,\" said Konrad. \"Why didn't you say that?\"\n\n\"I just did,\" said Sean.\n\nCurtis frowned. \"So you don't want to listen to the tapes?\"\n\n\"I do,\" Sean said. \"But let's listen to them when you come back.\"\n\n\"Do you really want the coffee?\" Konrad asked.\n\nSean thought for a moment. \"If it's not too much trouble, I do. And change the chocolate glazed to maple frosted doughnuts.\"\n\n\"You got it,\" Konrad said.\n\nAs Marvin and Bob gave their orders to Curtis and Konrad, Sean took a quick look around the living room. The place was just as depressing on the inside as it was on the outside. It smelled of old clothes, garbage that needed to be taken out, and rotting wood mixed with a faint undertone of cat urine. Two of the legs on the sofa by the far wall were broken, making the sofa lean alarmingly toward the floor, while the chair next to it had a broken arm. Sean's eyes moved to the print hanging next to the lamp. It was a print of Van Gogh's Irises.\n\n\"It's the only thing I brought from home,\" Bob said, following Sean's glance. Then he gestured around the living room. \"Nice place, huh?\"\n\nSean couldn't see any place to sit, so he leaned against the wall.\n\n\"Lovely,\" Sean replied. \"You should take the other two legs off the sofa.\"\n\n\"I'm thinking about it. Maybe I will one of these days.\" Bob curled his lips into a bad imitation of a smile. \"Yeah. I can't wait to come home every night.\"\n\n\"I wouldn't either.\"\n\nOut of the corner of his eye, Sean saw Marvin walk into another room. Sean was just about to ask him where he was going when he reappeared, with a kitchen chair. Sean sank into it gratefully. He didn't stand well anymore.\n\nBob went and sat on the lower end of the sofa, while Marvin leaned against the wall.\n\n\"So,\" Bob said, \"how are things going?\"\n\n\"In relation to your case? Not well,\" replied Sean.\n\n\"How come?\" asked Bob.\n\n\"Because you have a motive and you had opportunity,\" said Sean.\n\n\"But I didn't do it,\" Bob protested. \"I'm the fall guy.\"\n\n\"That's not what the DA thinks,\" said Sean.\n\n\"My lawyer thinks the case is circumstantial,\" said Bob.\n\n\"Really?\"\n\n\"Yes. Really.\"\n\n\"He used those actual words?\"\n\n\"Not exactly,\" Bob mumbled.\n\n\"That's what I thought.\" In Sean's experience, the DA didn't usually charge people unless he thought he had a good chance of getting a conviction.\n\n\"I'm so glad you came around,\" Bob said. \"You've really cheered me up.\"\n\n\"Do I detect a note of sarcasm?\"\n\n\"Just a touch.\"\n\n\"Listen, all I'm saying is in order to talk to the DA, we have to present him with evidence that someone else did this, or at least outline a plausible scenario, which I haven't come across so far.\"\n\n\"Everyone else's motive is just as good as mine. Look at what happened to Inez, for example. She lost everything, just like me,\" Bob replied.\n\n\"But she wasn't in the room next to Amethyst's.\"\n\n\"She was in the building.\"\n\n\"She was next door,\" said Sean.\n\n\"There's supposed to be a passageway that runs between the buildings.\"\n\nSean leaned forward. \"Mark says it's been closed off.\"\n\n\"Bessie says it hasn't,\" said Bob.\n\nSean managed to bite his tongue.\n\nBob shifted his weight to try to get more comfortable on the sofa, a feat Sean judged impossible.\n\n\"Anyway,\" Bob continued, \"how could I get out of where I was? I couldn't.\"\n\n\"Marvin and I are going to check that out again after we leave here,\" said Sean.\n\n\"We are?\" Marvin said.\n\n\"Yes, we are,\" Sean told him before turning his attention back to Bob Small. \"Maybe you killed Amethyst before you got into that space.\"\n\n\"I was in that hole in the ceiling for hours,\" said Bob. \"I needed to pee, and I couldn't get down to do that. Go check with Mark.\"\n\n\"Was he there with you all the time?\" asked Sean.\n\n\"No.\"\n\n\"Then he's not a good alibi for you, and for all I know, you may have had an accomplice,\" Sean said, even though he doubted it.\n\n\"Like who?\"\n\n\"I don't know. You tell me.\"\n\nBob glared at him. \"This is ridiculous. Konrad and Curtis said you'd help me,\" he protested.\n\n\"I am,\" Sean insisted.\n\n\"This is your idea of help?\"\n\n\"Yes.\" Sean rubbed his forehead with his hand. \"I need to cover all the possibilities so I can dispute them. Speaking of which, who sprang for your bail?\" It was not that he didn't know; he wanted to hear what Bob had to say.\n\nBob smiled for the first time. \"Mark did.\"\n\n\"Interesting. Did he say how come?\" asked Sean.\n\n\"Because he didn't think I did it.\"\n\n\"He told Libby he thought you had.\"\n\n\"My cousins went to talk to him and changed his mind. I mean, no one else was going to put up money for me. Ever since I got sentenced to Allenwood, no one in my family, except for Curtis and Konrad and a cousin who works as a manager at Burger King, will talk to me.\"\n\nMarvin coughed. Sean glanced at him.\n\n\"You're lucky,\" Marvin said to Bob. \"Not many people would be so nice.\"\n\n\"I know,\" Bob replied. \"He believes in giving people a second chance. Do you want to hear Curtis and Konrad's new tapes?\"\n\n\"Won't they be upset if you monkey around with their machine?\" Sean asked.\n\nBob shook his head. \"I don't see why. I know how to run it. The tapes are pretty interesting.\"\n\n\"So you could hear voices?\" Sean asked.\n\n\"Not voices,\" Bob corrected. \"Bessie.\"\n\n\"Really,\" Sean said.\n\n\"Well, you have to listen real hard,\" Bob allowed, \"but she's there.\" And he got up and went into the other room to get the tapes.\n\nMarvin and Sean looked at each other.\n\n\"He's probably delusional,\" Marvin whispered. \"I understand stress can do that to people.\"\n\nAt last, Sean thought, he and Marvin had found a point they could agree on.\n\n\"At first you don't hear anything,\" Bob said as he turned the machine on, \"except this noise that reminds me of the metal shop. But then\u2014\"\n\n\"You worked in a metal shop?\" Sean interrupted.\n\n\"Yeah. At Allenwood. They taught me how to weld, They've got a really good shop there, with all the latest tools. Why are you asking?\"\n\nSean shrugged. \"No particular reason. Just making conversation.\"\n\n\"I wish I liked doing it better. You can make a lot of money,\" said Bob.\n\n\"I hear it's tough,\" Sean said. \"But then if it wasn't, it wouldn't be paying well.\"\n\n\"True,\" Bob said, and he put a finger on his lips. \"Sssh. Here comes Bessie.\"\n\nSean leaned forward and listened. All he heard was more static. Not that he was listening that carefully. His mind was preoccupied with something else.\n\n\"That was interesting,\" Sean said to Marvin when they were sitting in Marvin's vehicle again, eating the doughnuts and sipping the coffee that Curtis and Konrad had brought back.\n\n\"Are you talking about the tapes?\" Marvin asked as he put the key in the ignition. \"Because I didn't hear anything.\"\n\n\"No. I'm talking about the fact that Bob Small worked in a welding shop.\"\n\n\"So?\" Marvin asked.\n\n\"So that means he probably knows about fiber-optic laser wire. Remember I told you they use it to cut metal.\"\n\nSean sat back and closed his eyes. He thought better that way.\n\n## Chapter 18\n\nBernie parked the van as close to the front of the historical society as she could get, turned off the ignition, and got out of the vehicle. There was no need to lock it, because no one would possibly want to steal it. As she hurried toward the door, she wondered what she'd been thinking when she decided to wear her suede over-the-knee boots on a day like today. Even though she'd put that waterproofing stuff on them that the salesman had recommended, in her experience, it never worked that well, especially not on suede.\n\nThe historical society was nice and warm. Cozy was the word that popped into Bernie's head. Jeanine was sitting by the front desk, staring at the seven bags of candy laid out on it.\n\n\"You got the good stuff,\" Bernie observed as she got close enough to see the kind of candy it was.\n\nShe still remembered getting two large 3 Musketeers Bars plus a Snickers bar from Mrs. Steinberg's mother when she was in the fourth grade. It had been her best score ever, and despite her mother's warning, she'd eaten all three candy bars when she'd gotten home. Much to her mother's dismay, she hadn't gotten sick, either.\n\nJeanine looked up at her and grinned. \"This is what comes from living in a house where my mom gave out apples each year.\"\n\n\"The kids couldn't have been happy.\"\n\n\"They weren't. We got TP'ed a lot. You think this will be enough?\"\n\n\"Depends on the weather.\"\n\n\"I just don't want to be stuck eating it all.\"\n\nBernie laughed. \"The mini bars are the worst.\"\n\n\"Yeah. Every time you go by the bowl, you take just one or two, and by the end of the day, the whole bowl is gone.\"\n\n\"There's always the freezer,\" Bernie suggested.\n\n\"That doesn't help someone with a major sweet tooth.\"\n\nBernie thought of Libby. \"No. I guess it wouldn't.\"\n\n\"So what brings you here? Did your dad want his View-Master back?\"\n\n\"No. He's coming for that himself.\" Was that a smile on Jeanine's face? Bernie wasn't sure. \"I'm here because I know you have a whole mass of old letters, and I'm wondering if I could take a quick look through them just to make sure there's nothing pertaining to Bessie Osgood.\"\n\n\"I'm pretty sure there isn't, but I'll make us some tea,\" Jeanine said. \"And then we'll get started.\"\n\n\"You don't have to help,\" Bernie said.\n\n\"I know I don't, but I'd like to.\" Jeanine gestured around the room. \"It's not as if I'm exactly busy.\"\n\nFive minutes later Jeanine came out, bearing a tray with two bone china teacups, a teapot, sugar and cream, and a plate of gingersnaps.\n\n\"Rosenthal,\" Bernie said, looking at the pattern.\n\n\"My mother's,\" Jeanine said as she poured the tea.\n\nBernie took a sip. It was Indian. Oolong. The two women sat there for a few minutes, savoring their tea, before Jeanine said, \"So how's the case coming?\"\n\nBernie shook her head. \"Could be better.\"\n\n\"That's what I figured.\"\n\n\"Did you find anything with the View-Master?\"\n\nNow it was Jeanine's turn to shake her head. \"If there's anything there, I don't see it. There were just multiple shots from out the second-story windows. It's probably Felicity's idea of a joke.\"\n\n\"Maybe I should go back and ask her,\" Bernie mused, even though talking to Felicity again was not what she wanted to do. Their last conversation had made Bernie feel as if she was standing in front of Mr. Steiffer's math class, and he was saying to her, \"Just look at the board, and tell me the answer. It isn't that difficult.\" But it was to her.\n\nJeanine delicately placed her cup on her saucer. \"You can't ask her. She's in the hospital.\"\n\n\"With what?\"\n\n\"She had a stroke. At this point, she can't use the left side of her body or talk.\"\n\nBernie let out a long sigh. \"My father says getting old sucks.\"\n\n\"It certainly does,\" Jeanine replied. \"Although your dad doesn't seem to be doing too badly in that department.\"\n\nBernie gave her a speculative look. \"He can be a real pain in the ass sometimes.\"\n\n\"I figured,\" Jeanine said.\n\nShe and Jeanine exchanged a woman-to-woman smile.\n\n\"Just so you know,\" Bernie told her while reaching over and grabbing a gingersnap. She took a bite and let the cookie dissolve on her tongue. \"Ready?\" she said when she was done eating.\n\n\"Ready,\" Jeanine answered.\n\nThe two women stood up and moved to the room where the letters\u2014officially called ephemera\u2014were kept. Unlike the other rooms, this one was bare except for the three- and four-drawer file cabinets that lined the wall. A square wooden table sat in the center of the room.\n\n\"The chairs aren't very comfortable,\" Jeanine noted as she went over to the file cabinet on the left. \"But I can't get money to replace them. Everyone wants things done, but they don't want to pay for them.\" She laughed. \"But I'm going to stay off that topic. Otherwise, I'll be talking about it all day. Fortunately,\" she told Bernie, \"I've gone through two of the file cabinets and put them in order, so I know that nothing is misfiled. That leaves us just two to go over.\" And she opened the top drawer of the farthest file cabinet on her left and pulled out three bulging files. \"Here you go,\" she said as she put them in front of Bernie.\n\nBernie eyed the files for a moment before gingerly opening up the first folder. \"Are they in any particular order?\" she asked hopefully.\n\n\"Not that I know of,\" Jeanine said. \"Organization wasn't my predecessor's strong point.\"\n\n\"Wonderful,\" Bernie muttered as she started leafing through the papers.\n\nBy the time she was done, her back was aching, her eyes were burning, and she had a headache. She'd found nothing about Bessie Osgood, but she had found a small item from the local newspaper about the closing of the Peabody School.\n\nBernie sat back in her chair and read it aloud. \"Today, George Marak, headmaster of the Peabody School, has announced that it is with profound regret that he and the Board of Directors of the Peabody School have come to the painful decision to close the school by the end of the semester.\" Bernie looked up. \"There's no mention of Marak killing himself.\"\n\n\"That happened a week later.\"\n\n\"Did the paper do a follow-up?\"\n\n\"I believe they called the suicide an unfortunate accident.\"\n\n\"You could say that.\" Bernie was silent for a moment. Then she said, \"Of course, they called Bessie Osgood's murder an unfortunate accident, too.\"\n\n\"Well, don't forget that in those days people used euphemisms. Suicides were unfortunate accidents, and girls didn't have babies out of wedlock. They went on extended trips to Europe. And married couples slept in separate beds.\"\n\nBernie put the article back in the folder. \"That place really does have bad karma.\"\n\nJeanine stretched. \"So it would seem.\"\n\n\"I wonder why the article is here?\"\n\nJeanine shrugged. \"I guess someone must have thought it was of interest. I really don't know why. As far as I can see, the things in these cabinets are random pieces of stuff, most of which should be thrown out.\"\n\n\"Then why don't you?\"\n\n\"Because I'm afraid that the moment I do, it will turn out that I've thrown out some irreplaceable document.\"\n\nBernie laughed and got up. \"Isn't that always the way.\"\n\n\"Well, it is in my life.\" Jeanine pointed to the folder. \"Here. Take the article with you, and show it to your dad, not that there is anything in it he doesn't already know.\"\n\n\"Are you sure?\"\n\n\"Of course, I'm sure. That's one piece of paper out of here.\"\n\n\"I'll send my dad over with some cupcakes for you,\" Bernie told her.\n\nJeanine patted her hips. \"Just what I need.\"\n\nBernie tucked the article in her bag, thanked Jeanine for all her help, and walked outside.\n\nIt had stopped raining, but the sky was overcast, and the air was cold and damp. Bernie buttoned up her jacket and headed for her vehicle. She'd just gotten behind the wheel when her cell rang. It was Brandon.\n\n\"No. I can't do a matinee,\" she said.\n\nBrandon laughed. \"Don't flatter yourself.\"\n\n\"Very nice. See what you get the next time you ask.\"\n\n\"I'm irresistible. You can't say no to me.\"\n\n\"Talk about misplaced ego.\"\n\n\"Seriously, I'm calling because Inez is here, drinking away.\"\n\n\"I'll be right over,\" Bernie told him.\n\n\"And can you be a sweetie and stop on the way and get me some sugarless gum? I'm all out.\"\n\n\"I'll think about it,\" Bernie said before she clicked off. But, of course, Brandon knew that she would.\n\nIncluding the stop at the gas station to pick up Brandon's gum, it took Bernie fifteen minutes to get to R.J.'s. The place was uncharacteristically empty, except for Inez, who was hunched over her drink at the bar, in the corner farthest from the door.\n\n\"She's been here for the last thirty minutes,\" Brandon told Bernie sotto voce.\n\nBernie nodded as she assessed Inez out of the corner of her eye. She wasn't wearing any make-up, which made the bruise on her jaw stand out. From its purplish color, Bernie judged it was two days old at the most.\n\nThen there was Inez's hair. Bernie shook her head as she contemplated it. It was orange, the obvious result of a bad dye job, and to make matters worse, tufts were sticking out in various directions. It looked as if Inez hadn't taken a comb to it in two or three days. How anyone could do that to themselves, Bernie couldn't imagine. The stained denim jacket over the grey sweatshirt Inez was wearing didn't do anything to help the situation.\n\nBernie turned toward Brandon. \"Well,\" she told him, \"she certainly doesn't look in good shape.\"\n\n\"She's not,\" Brandon said.\n\n\"Is she drunk?\"\n\n\"No. But she's getting there fast. Can I get you anything?\"\n\nBernie grabbed a handful of peanuts and started shelling them. \"A Coke will be fine.\"\n\nA moment later Brandon was back with a Coke. Bernie grabbed it and drifted down to where Inez was sitting.\n\n\"Hi,\" Bernie said as she sat down next to Inez.\n\nInez glared at her. \"What do you want?\" she demanded.\n\nBernie cracked open a peanut and tossed the nut into her mouth. \"Are you okay?\" she asked after she'd swallowed.\n\n\"I'm fine.\"\n\n\"Because it doesn't seem to me as if you are.\"\n\nInez stared into her beer for a moment and then took a drink. \"Well, I am.\"\n\n\"What happened to your face?\"\n\nInez touched the side of her jaw, realized what she was doing, and quickly put her hand back on the bar. \"I fell.\"\n\n\"Are you sure no one hit you?\"\n\nBernie couldn't read the fleeting expression that ran across Inez's face. Was it anger? Sorrow? Pleasure that someone cared enough to ask? All of them? None of them? Bernie didn't know.\n\nThen Inez frowned. \"I got drunk, and I fell off the sofa and hit the edge of the coffee table on the way down. Satisfied?\"\n\nBernie took a sip of her Coke. \"Not really, but if that's what you say happened, then that's what happened.\"\n\n\"That's what I say happened.\" Inez took a gulp of her beer and slammed the glass down on the bar. \"Ian is a putz. You know that?\"\n\nBernie ate another peanut and put the shell on top of a napkin. \"Why is Ian a putz?\"\n\n\"Because he fired me.\"\n\nBernie didn't say anything.\n\n\"I had a doctor's note, too. What am I going to do now?\" Inez demanded. \"Who's going to hire me?\"\n\n\"No one if you don't pull yourself together,\" Bernie told her.\n\nInez gulped down the rest of her beer and signaled Brandon to bring another one. As he went to get it, a man and a woman came in and sat down near the door. The couple was laughing and joking. Inez looked at them with hungry eyes. After a minute or so, she wrenched her eyes away.\n\n\"We used to be like that before Amethyst,\" Inez said wistfully.\n\nBernie knew Inez was talking about herself and her husband.\n\n\"We were really happy together,\" Inez added. She watched while Brandon set her Coors down in front of her and moved on to the new customers. She poured it into her glass and took a big swallow. \"Really happy. We liked each other.\"\n\n\"Is that why you killed Amethyst?\"\n\nInez wiped her lips off with the back of her hand. \"A lot you know. Why would I kill her?\"\n\n\"Because she destroyed your life? You just said as much.\"\n\n\"Yeah. She did. But I was making her pay.\"\n\nInteresting, Bernie thought as she took another sip of her Coke. \"Is that right? And how were you doing that?\"\n\n\"You don't believe me?\" Inez said.\n\n\"Well, it's a little hard to imagine.\"\n\n\"You think she was smarter than I was?\"\n\n\"Well...\"\n\n\"Even with my drinking, I'm still smarter then she was,\" Inez growled. \"Better looking, too.\"\n\n\"I don't believe it,\" Bernie said, egging Inez on.\n\n\"That I was better looking?\"\n\n\"No. That's true.\" Bernie remembered when Inez used to come into the shop.\n\n\"I had a better body.\"\n\n\"That's true.\"\n\n\"And I dressed better.\"\n\n\"I'm not disputing that. It's the intelligence part.\"\n\nInez looked outraged. \"Hey, I have a master's in remote sensing.\"\n\n\"True, but I'm not talking about that kind of intelligence.\"\n\nInez gave Bernie a beery, conspiratorial smile. \"She was a sneaky little bitch, wasn't she?\"\n\nBernie popped another peanut into her mouth. \"In a word, yes.\"\n\n\"But I found out something she didn't want anyone to know.\"\n\n\"Really?\" Bernie hoped that she had conveyed just the right amount of disbelief.\n\n\"Yes. Really. Amethyst was moving, and she didn't want anyone to know. But I knew.\"\n\n\"How did you know?\"\n\n\"I heard her talking.\"\n\nBernie maintained a skeptical expression.\n\n\"It's true. I did,\" Inez insisted.\n\n\"Where? While you were cleaning the bathroom?\"\n\nA crafty expression stole over Inez's face. \"Never you mind.\"\n\nBernie ate another peanut. \"In the supermarket? The garden store? Probably not.\" Bernie contemplated the options some more. \"Did you clean her house?\"\n\nInez looked at her in astonishment.\n\n\"You cleaned her house? I don't believe it,\" said Bernie.\n\n\"Believe it,\" Inez snarled.\n\n\"Given the circumstances, I find that hard to believe.\"\n\n\"I don't want to talk to you anymore,\" Inez said and gulped her drink.\n\n\"Come on,\" Bernie told her. \"You have to give me credit for this. You have to admit I'm good.\"\n\n\"Yes, you are,\" answered Brandon.\n\nBernie was startled. She'd been so engrossed in her conversation with Inez that she hadn't realized Brandon was standing in front of her, with another beer for Inez. She was always amazed at how quietly he moved for a big guy. She smiled at him and turned her attention back to Inez.\n\n\"That must have been hard, working for Amethyst. More than hard given the circumstances.\"\n\nInez didn't say anything.\n\n\"Did you do it so you could get back at her?\"\n\n\"It's none of your business.\"\n\n\"Actually, I think it is.\" Bernie took a sip of her Coke and put it down. \"So after you tried blackmailing her, I'm guessing Amethyst didn't want you working at her place anymore.\"\n\n\"She had no right to do what she did.\"\n\n\"Are you talking about the fact that she broke up your home, the fact that you worked for her, or the fact that she fired you?\"\n\nInez hunched her shoulders and took another swig from her glass. \"A person's got to live, doesn't she?\"\n\n\"So, what did you hear?\" Bernie asked.\n\nA cunning expression stole across Inez's face. \"Maybe it wasn't something I heard. Maybe it was something I saw.\"\n\n\"How much did you ask from Amethyst?\"\n\nInez drained the last drop from her glass and put it down. \"What difference does it make?\" She leaned in toward Bernie. Her mouth twisted itself into a sneer. \"You know what she did? She laughed at me. At me. Told me I was a loser and no one would ever believe me.\"\n\n\"So what did you do?\" Bernie asked.\n\n\"Guess,\" Inez said.\n\n\"Got even?\" Bernie said.\n\nInez laughed.\n\n\"What's so funny?\" Bernie asked.\n\n\"You are,\" Inez replied, and she got up and started for the door.\n\n\"How about if I buy you another drink?\" Bernie said.\n\n\"You think I'm that easy?\" Inez said. \"Well, screw you.\" And she left.\n\n\"That went well,\" Brandon said.\n\n\"Didn't it, though.\" Bernie tapped her fingers on the bar. \"I should have offered her money instead.\"\n\n\"Somehow I don't think that twenty dollars would do it.\"\n\n\"Are you saying I'm cheap?\"\n\n\"No. I'm saying you're broke.\"\n\n\"Amethyst was a real piece of work,\" Bernie said thoughtfully.\n\nBrandon grinned. \"That would be one way of putting it.\"\n\n\"And the other way?\"\n\n\"She was a sadistic little bitch.\"\n\n\"These days people would say she had a personality disorder.\"\n\n\"They'd probably say that about Hitler, too.\"\n\nBernie chuckled. That was one of the things she loved about Brandon. He always made her laugh.\n\n\"You know what?\" she said.\n\nBrandon planted his elbows on the bar and supported his face with his hands. \"What?\" he said.\n\n\"I think I'm going to take a quick look around Amethyst's place.\"\n\n\"And the reason being?\"\n\n\"Because maybe there's something there that we need to know.\"\n\n\"You're thinking about what Inez said.\"\n\nBernie nodded.\n\n\"The police have probably gone through it already.\"\n\n\"It wouldn't be the first time that they missed something.\"\n\n\"You don't have a key,\" Brandon pointed out.\n\n\"I'll use your credit-card trick.\"\n\nBrandon rolled his eyes.\n\n\"You're just jealous that you can't come,\" said Bernie.\n\nBrandon snorted. \"Right. I really want to get arrested. Just don't call me to bail you out.\"\n\nBernie leaned over the bar and kissed him. \"That would never even occur to me.\"\n\n\"Right,\" Brandon repeated.\n\nBernie flashed him her best smile. \"Call me when you're off, and I'll meet you at your place.\"\n\n\"I told you, you'd beg me.\"\n\nShe picked up a peanut and threw it at him. Then she hurried out the door before he could get her back.\n\n## Chapter 19\n\nMarvin and Sean stood in the room of the Haunted House that had the coffin that came out of the floor. No one was there yet, so it was eerily quiet.\n\n\"Should we be here?\" Marvin asked Sean.\n\n\"No one said we couldn't,\" Sean replied.\n\n\"That's because you didn't ask.\"\n\nSean didn't answer. Instead, he stared at the ceiling. It was lower than the ceilings in the other rooms by a good foot or so. Sean was betting it was a dropped ceiling made with two-by-fours and covered over with plaster.\n\n\"You should return those keys,\" Marvin told him.\n\n\"I will,\" said Sean while he continued to study the ceiling.\n\nAnd he would return the keys. He didn't want Konrad and Curtis to get in trouble. He was just going to do it later rather than sooner. Obviously, they didn't need the keys at the moment, or else they would have asked for them by now.\n\n\"Do me a favor,\" he said to Marvin. \"Turn on the lights for me. The control switch is by the door, on your left.\" He paused. \"Left,\" he said to Marvin as Marvin went to the right.\n\n\"You mean this way?\" Marvin asked.\n\n\"Right.\"\n\nMarvin moved toward the right.\n\n\"Left,\" Sean screamed.\n\n\"I did, and you said right.\"\n\nSean took several deep breaths. According to Bernie, this was supposed to help him calm down. It didn't. \"Go to the left,\" he said in as even a voice as he could manage. \"That's correct,\" he told Marvin when he had.\n\n\"You should be clearer,\" Marvin told him in an injured voice.\n\nIn this case Sean decided that silence was a virtue. So, he watched Marvin grope behind the wall hanging. \"Farther in,\" Sean instructed.\n\n\"Got it,\" Marvin told him.\n\nThe lights didn't come on.\n\n\"Obviously, you didn't get it,\" Sean replied.\n\n\"I got something,\" Marvin said.\n\nSean was just about to tell him to try again when out of the corner of his eye, he saw something move. He jumped back as what he now realized was a coffin rose out of the floor. Then he remembered what Bernie had told him about the exhibit, and his heart settled back down.\n\nA moment later the skeleton sat up. It had a patch over one eye. \"Soon you'll look just like me,\" the skeleton cackled. \"Just like me. Eat, drink, and be merry. We don't have six packs in the graveyard.\"\n\n\"I told you I hit the wrong switch,\" Marvin said.\n\n\"See if you can turn the sound off,\" Sean said. It was impossible to think with that thing babbling on.\n\nMarvin went back to fumbling with the wall switches. A moment later the skeleton shut up in mid-cackle.\n\nSean looked at the ceiling. \"Marvin, how high would you say that was?\"\n\n\"About eight feet.\"\n\n\"That would be my guess, too. I want you to do me a favor and stand on the edge of the coffin and see if you can touch the ceiling.\"\n\n\"But...\"\n\n\"You should be used to them.\"\n\n\"That's not the point.\"\n\n\"What is?\"\n\n\"I just don't want to fall in and break the thing.\"\n\n\"You won't,\" Sean said, with more confidence than he felt. After all, if it was possible to break something, Marvin usually did. \"You can hold on to my shoulder for balance.\" Also a bad idea, but the best he could come up with given the circumstances.\n\nMarvin put one foot on the coffin, then stepped up with the other.\n\n\"Steady?\" Sean asked when Marvin was standing on the edge of the coffin.\n\n\"I guess,\" Marvin said.\n\n\"Now lift your hand up, and touch the ceiling.\"\n\n\"Satisfied?\" Marvin said as he made contact.\n\nSean nodded as Marvin started to tilt. He stepped away just as Marvin lurched forward and crashed to the floor.\n\n\"How tall would you say Bob Small is?\" Sean asked Marvin as the younger man picked himself up and dusted himself off.\n\n\"Aren't you even going to ask me how I am?\" Marvin demanded while he rubbed his elbow.\n\n\"No. I'm not asking you, because you're obviously fine,\" Sean said. \"Again. How tall is Bob Small?\"\n\n\"I think he must be almost six feet. I'm five feet ten, and he's about two inches taller than I am. Why?\"\n\n\"Do you have a flashlight in your car?\"\n\n\"No.\"\n\n\"A penlight?\"\n\nMarvin shook his head.\n\n\"Don't you keep an emergency kit in your car?\"\n\nMarvin shook his head again. \"I can go out and buy one.\"\n\n\"No. By the time you get back, there'll be people here.\" Sean thought for a moment. Then he took out his cell phone.\n\n\"Who are you calling?\" Marvin asked.\n\n\"I'm not calling anyone,\" said Sean. He opened up his cell and shined the light on the ceiling, directly above where Marvin had been standing. \"Do you see anything?\" he asked.\n\n\"It would help if I knew what I was looking for.\"\n\n\"A crack in the ceiling.\"\n\n\"Why do we care?\"\n\n\"We just do,\" Sean said. \"Now concentrate.\"\n\nMarvin dutifully put his head back and stared at the ceiling. It all looked like a big expanse of white plaster to him. A moment later he rubbed the back of his neck. \"I'm getting muscle spasms.\"\n\n\"Move down a foot, and stop complaining,\" Sean snapped.\n\n\"I'm not complaining. I'm just telling you I have a hypermobile neck.\"\n\n\"Keep looking, anyway.\"\n\n\"Is this what you mean?\" Marvin asked five minutes later.\n\nSean followed Marvin's finger. There was what looked like a small indentation in the ceiling. He smiled. Today was going to be a good day, after all. \"Yes.\"\n\n\"Good,\" Marvin said. \"Because I don't think I can take much more of this. My neck is killing me. I'll probably have to go see the chiropractor now.\"\n\nIt took a lot of willpower on Sean's side to not say anything, even though he would have liked to. Boy, would he ever. He could just imagine the reaction he would have gotten if he'd said something like that to his dad. He probably would have gotten a kick across the room. But that was what kids were like these days. Whine, whine, whine. How they managed to survive in the world was beyond him. Even though he thought Marvin's dad was an asshole, he was beginning to feel a little more sympathy for him.\n\nSean plastered what he hoped was a pleasant expression on his face and said, \"I just want you to do one more thing for me. I want you to get back on the rim of the coffin and see if you can push the panel forward with your hand.\"\n\n\"What panel?\"\n\n\"The panel that I think is in the ceiling.\"\n\n\"I'm going to fall if I do this,\" Marvin told him.\n\n\"I'll stand next to you. You can lean on me.\"\n\n\"That's what you told me before.\"\n\nSean looked at his watch. They didn't have much time left before people started coming in to prepare for this evening.\n\n\"Please,\" Sean said. \"Try. This is really important.\"\n\n\"For real?\" Marvin asked.\n\n\"Yes. It might help us figure out who killed Amethyst.\"\n\nMarvin thought about it for a moment. Then he got up on the rim of the coffin and spaced his feet apart. Sean moved in so Marvin could lean against him. Marvin raised his hands and pushed. Nothing happened.\n\n\"Try again,\" Sean urged.\n\nMarvin did. \"I think I felt something move.\"\n\n\"Good,\" Sean said encouragingly. \"Just once more. If it doesn't happen, then I'm wrong.\"\n\n\"Okay,\" Marvin said.\n\nSean watched as Marvin rebalanced himself on the rim of the coffin, put his hands up flat against the plaster, and pushed. He could see something moving. Marvin pushed harder. The panel slid back some more.\n\nSean looked up at the eight inches of space Marvin had created. \"There goes Bob Small's alibi,\" he said.\n\n\"Someone else could have done it, too,\" Marvin said.\n\n\"I suppose,\" Sean agreed. \"But it's highly unlikely.\"\n\n\"But it's possible,\" Marvin insisted.\n\n\"It's possible,\" Sean agreed. \"Possible, but not probable.\" He sighed. \"Of course, we won't have proof positive until someone crawls up there and tests my theory out,\" he added as Marvin turned the switch and the coffin disappeared back into the ground.\n\n\"Someone?\" squeaked Marvin as he turned off the lights.\n\n\"Yes. Someone. Obviously, it can't be me,\" Sean told him as they walked out of the room, through the Chain-Saw Massacre Room, and out into the corridor that led to the outside. Sean paused to lock the outside door to the Haunted House. Then he pulled on the door to make sure it was locked. Test and retest. That was his motto.\n\n\"Obviously,\" Marvin said to Sean's back.\n\nSean grunted a reply.\n\n\"I'll be back in a sec,\" Marvin said to Sean, and then he ran to the car, which, on Sean's advice, he'd parked behind the Haunted House. That way no one driving by would see them.\n\nSean looked at his watch. \"Make it fast. People will be here soon,\" he called.\n\nMarvin broke into a trot. He got to his vehicle, put the key in the ignition, started her up, put the car in reverse, and immediately backed into the tree behind him. God. The more nervous he got, the klutzier he became. He jumped out and looked at the damage. Fortunately, it was just a ding in the bumper. He jumped back in his car, put the car in drive, and zoomed off to get Sean.\n\n\"What was that I heard?\" Sean asked him.\n\n\"Nothing,\" Marvin lied. \"Absolutely nothing.\"\n\nA moment later they were driving out of there. And not a moment too soon, Marvin thought as he glanced at the clock on the dashboard. Another couple of minutes and people would be coming up. Thank God they hadn't gotten caught. If Marvin never saw this place again, he'd be ecstatic. As far as he could see, nothing good had ever happened here.\n\nMaybe his dad was right. Maybe a funeral director was a good thing to be. No surprises in that, or at least not the kind of surprises that raised your blood pressure. All he wanted to do now was sit in the kitchen and talk to Libby and eat a slice of her warm apple pie with her homemade vanilla ice cream.\n\nInstead, he was running around with her dad, doing things that could get both of them arrested. And if that happened, it would be his fault. He'd get blamed for it, because he always got blamed for everything that went wrong. That was just the way things were. Marvin sighed and slowed down for the steep curve coming up.\n\nMaybe after Libby was through working tonight, they could go to R.J.'s and have a beer. It wasn't as good as having Libby's pie, but it would be good enough. He had just finished taking another curve\u2014the road to the Haunted House had more twists and turns than a bad soap opera\u2014when he looked over at Sean and saw Sean looking at him, and somehow he knew exactly what Libby's dad was thinking.\n\n\"I don't like heights, and I don't like small spaces,\" Marvin told him.\n\n\"The ceiling isn't that high, and the space isn't that small.\"\n\n\"I'll fall through.\"\n\n\"No, you won't. Ceilings like that, built with drywall and two-by-fours, can support a considerable amount of weight.\"\n\n\"How do you know?\"\n\n\"Because I've put them up.\"\n\nMarvin thought for a moment. \"What happens if I get stuck?\"\n\n\"You won't get stuck.\"\n\n\"But what happens if I do?\"\n\n\"We'll walk away and pretend we never met you.\"\n\nMarvin snuck a quick peek at Sean. He could never tell whether he was kidding or not.\n\n\"I was joking,\" Sean said. \"And you don't have to do this. You've been more than enough help already.\"\n\n\"Then who will?\"\n\nSean shrugged. \"Don't worry about it. I'll find someone. If worse comes to worst, I'll do it myself.\"\n\nMarvin groaned.\n\n\"Well, it's the truth,\" Sean said. \"Or maybe I can get one of my daughters to do it.\"\n\n\"Why don't you twist the knife a little deeper?\" Marvin asked him.\n\nSean grinned. \"I'm good at guilt, aren't I?\"\n\n\"Yes, you are.\"\n\n\"It's a talent I inherited from my mother's side of the family.\"\n\n\"Fine,\" Marvin grumped. \"I'll climb up there. But if anything happens to me, it's on your head.\"\n\nSean laughed. \"I think I can handle that responsibility.\"\n\n\"When is this going to happen?\"\n\n\"Probably tonight after the Haunted House closes.\"\n\n\"What?\" squeaked Marvin.\n\n\"Well, I am going to have to give the keys back to Konrad and Curtis. Or we could go over to the Home Depot in Thompsonville and get them duped. That might give us a little more leeway.\"\n\n\"Leeway is good,\" Marvin said.\n\n\"Always,\" Sean said as he leaned over and turned on the radio. For a moment, both men were quiet as they listened to the music.\n\n\"So what do you think happened?\" Marvin asked once they hit the main road. \"How do you think Bob Small pulled this off?\"\n\nSean sat back in his seat and told Marvin what he thought.\n\n## Chapter 20\n\nBernie still couldn't believe that Amethyst had lived in Stanton as she pulled up in front of Amethyst's flat.\n\n\"Are you sure she lived there?\" she'd asked Bree Nottingham, real estate agent extraordinaire, social arbiter of Longely, and general pain in the butt, when she'd tracked her down yesterday morning. After several false starts, Bernie had finally located her at Kim's Nifty Nails.\n\nBree had fixed her with a gimlet eye. \"Of course, I'm sure,\" she'd snapped.\n\nShe'd been, as was usual these days, dressed in pink from head to toe, up to and including her Prada bag. It was, she'd told Bernie, her signature color. Bernie didn't believe in signature colors\u2014people weren't pens\u2014but she'd never say that to Bree, who threw a fair amount of business their way. Actually, she wouldn't have said it to her face, anyway. However, she did say it to her sister in private on several occasions.\n\nBree had taken her right hand out of the bowl she was soaking it in and had held it out for the Korean girl to dry before speaking. \"I tried to persuade Amethyst to live in town, but she wasn't having any of it,\" she'd told Bernie.\n\n\"But why did she choose Stanton of all places?\" Bernie had wondered out loud. \"That's so strange. I would imagine her somewhere a lot more upscale.\"\n\nBree had frowned. \"Think about it.\"\n\n\"Nothing comes to mind,\" Bernie had said.\n\n\"Well, she said\"\u2014Bree bracketed the she said with her voice\u2014\"that she liked the people there.\" Here Bree had paused meaningfully. \"She said they were more authentic.\"\n\n\"Authentic? Please.\"\n\nBree had nodded encouragingly, pleased that her point had been made.\n\nBernie had frowned. \"Amethyst didn't care about authentic unless the word applied to diamonds and gold. I mean Stanton is made up of Mexicans and Portuguese. I never saw Amethyst look at anyone who wasn't driving at least a Lexus.\"\n\n\"Exactly.\" Bree had nodded at the Korean girl, who had begun filing her nails. \"And this time I want them straight across,\" she'd told her. \"Straight. You understand?\" The girl had bobbed her head and had kept filing. Bree had watched her for a moment to make sure she was doing what she'd asked and then had turned her attention back to Bernie. \"My guess is that Amethyst was living there because she could come and go as she pleased, without anyone knowing her business, which would most definitely not be true if she lived in Longely or another community she socialized in.\"\n\n\"Makes sense,\" Bernie had told Bree.\n\nBree had nodded. \"Of course, it makes sense. The expression 'Don't poop where you eat' comes to mind, which would be especially important to someone of Amethyst's...\" Bree had paused while she hunted for the right word and had finally said, \"To someone of Amethyst's bent. Plus, she kept her overhead low and put her money into what counted\u2014herself. From a business point of view, it was a good decision.\"\n\n\"Not that it helped her,\" Bernie had pointed out.\n\n\"No, it didn't,\" Bree had acknowledged before changing the subject. \"I want you or your sister to call me tomorrow. I'm having a dinner party for fifteen in two weeks, and I want to discuss the menu. I was thinking we could do something retro, something Julia. You do know Julia, don't you?\"\n\n\"Of course, I know Julia Child,\" Bernie had said, incensed. How could someone who loved food not know her? She was an icon.\n\n\"Good. Because I was thinking we could build the menu around beef Wellington, the real one, with a Bavarian cream for dessert. Or maybe some sort of cr\u00eapes flamb\u00e9. Yes. Let's do that. Maybe cr\u00eapes suzette.\"\n\n\"Sounds great,\" Bernie had said.\n\nAnd she'd meant it. She loved traditional French food. She just didn't get a chance to make it anymore, because people were so concerned with their diets and the amount of fat they ingested, but in her mind, there were two kinds of food: regular food and party food. And you should be allowed to eat what you wanted at a party. If you couldn't, what was the point? In fact, when Bernie thought about it, cooking wasn't as much fun as it used to be. Ever since people had started saying things like, \"I need protein\" instead of \"I'd like a nice, fat, juicy steak,\" things had definitely gone downhill.\n\nOn her way to Stanton, Bernie found herself thinking about what kind of first course and appetizers she could serve at Bree's dinner party. She would keep the appetizers on the light side because the meal was going to be heavy. She'd start the ball rolling with Kirs and a good prosecco. They were always good aperitifs. They had just enough alcohol to loosen people up, but not so much that it dulled their taste buds.\n\nAlong with the drinks, she could serve two types of spiced nuts, one with pepper and rosemary, and the other with salt and a dash of anise. People seemed to like those. Then she could serve a selection of olives and tiny toast points with heated goat cheese. That should really be enough. Unless she added cubes of feta cheese marinated in olive oil and garlic.\n\nFor the first course, she'd serve a clear beef broth with one or two small circles of baked custard and a dusting of chopped chives floating in it, or she could do a celery r\u00e9moulade, which was also very nice, and celeriac was in the market these days. Naturally, they'd have to make the mayo for the r\u00e9moulade sauce, but she liked doing that. There was something very meditative about whisking the oil into the egg yolks and watching the emulsion form a light yellow cream. Hellmann's just didn't cut it.\n\nThen, after the beef Wellington, she could serve a salad made up of endive and watercress and arugula in a vinaigrette dressing. The sharpness of the greens would be a nice foil to the richness of the main course. She'd do the cr\u00eapes suzette that Bree had requested for dessert, or maybe something like cr\u00eapes filled with an apple compote and flamb\u00e9ed with a good apple brandy, which would be a little more seasonal.\n\nOr maybe not. The problem with calvados was that you needed a really good one; otherwise, it tasted like diesel fuel. And, thanks to the dollar's weakness, good bottles of calvados were extremely expensive these days, and since they didn't have any on hand in the shop, Bernie would have to purchase a bottle. So she would forget the apple cr\u00eapes and stick with Bree's original suggestion. Unless, of course, Bree was willing to buy a bottle of the stuff, which Bernie was pretty sure she wouldn't be.\n\nBernie was almost at Stanton by the time she'd worked through the pricing of Bree's menu. Between the labor and the ingredients, this was not going to be a cheap meal, but then a good cut of beef, let alone p\u00e2t\u00e9, never was. She was wondering whether or not Bree would want to pay that much when she caught sight of one of the street names on her map. It was, she realized, time to focus on the task at hand.\n\n## Chapter 21\n\nAmethyst's home was four blocks away from the commercial district, in an unprepossessing neighborhood of small houses, smaller yards, and older, dented cars. No BMWs here. According to the numbers painted on the wall, Amethyst lived in the bottom part of a two-family house. The sagging fence posts, the paint peeling on the porch stairs, the empty planter boxes, and the absence of holiday decorations of any kind gave the place a forlorn appearance. It looked as if no one had lived in it or taken care of it for a long time.\n\nThe question, thought Bernie, was how to get in. Now that she was actually here, she wasn't so sure. It was one thing to joke around with Brandon about it and another thing to actually do it, breaking and entering not having been a course in her high school curriculum.\n\nShe parked her car and contemplated her options. She could go the legal or the illegal route. Both had good things going for them. The legal route involved something like knocking on the upstairs neighbor's door, seeing if they had the key to Amethyst's place, then making up a plausible story so she could get it. However, there was a big problem with that approach. The upstairs apartment was for rent. So that took care of that.\n\nWhich left the illegal route. For a moment, she debated the wisdom of what she was about to do, and then she thought, The hell with it. What was the worst that could happen? The neighbors could call the cops, and she'd get arrested, her father would kill her, and she'd never hear the end of it from Brandon or her sister. Which, in the scheme of things, wasn't so terrible. It wasn't as if she was going to get shot or anything.\n\nHere goes nothing, she said to herself as she zipped up her jacket and got out of the car. She strode toward Amethyst's door like she had business being there. As she did, another thought popped into her mind. Inez had cleaned for Amethyst, which meant either Inez had a key and had let herself in, Amethyst had let her in, or Amethyst had left the key for Inez somewhere inconspicuous, like under a potted plant or the welcome mat. Bernie decided to go with the third option because it was the easiest, while the other two left her with nothing to do but break a window or open the door with a credit card\u2014something she wasn't good with. Besides, these days everyone had dead bolts, which required a little more finesse. And if option one didn't work, she could always go to option two or three.\n\nBernie casually strolled around to the back of the house, figuring that if Amethyst had left the key anywhere, it would be there, because there was no place in front to hide a key. There was a slot in the door for the mail and no welcome mat on the porch. If anyone asked what she was doing, she would say she was thinking of renting the upstairs apartment and just wanted to see what things looked like.\n\nLike not much, in her opinion. Maple saplings were pushing up through the cracks in the driveway. Their leaves littered the tarmac. Over by the hurricane fence, black-eyed Susans drooped disconsolately. A sand box sat uncovered. In it, three small, chartreuse, plastic starfish molds sat next to a kiddie-sized pail and shovel. She looked in all of them. No key. Then she addressed the doll whose legs were sticking out of the sand.\n\nBernie reached over and pulled her out. It was a Barbie. She dusted the sand out of her hair and sat her down next to the pail. Then she picked her back up. Poor Barbie. She deserved better than being left out for cats to pee on. Growing up, Libby had never liked Barbie dolls, but Bernie had been a Barbie fanatic, bugging her mom to buy her every new outfit that came on the market. She'd never liked Ken, though. She'd always thought he was a dork.\n\nBernie slipped the Barbie into her bag. She'd donate it to Goodwill, because it was obvious that the little girl that had lived upstairs with her parents wasn't coming back. As Bernie continued on, she wondered who they were and if they would know anything about Amethyst. Somehow she doubted it. She also doubted that she'd be able to locate them, but maybe Bree would know who the realtor was who handled this house. Then she or her sister could call up and see if the family had left a forwarding address. It was a long shot, but the way things were going, any long shot was a shot worth pursuing.\n\nBernie walked up to the back steps and picked up the welcome mat. Nothing. Too obvious. She went back down the steps, and taking care to keep her suede boots out of the dirt, she squatted down in front of what had once been a flower bed but was now a tangle of dead and dying weeds that banked the side of the house.\n\nIt seemed like an unlikely place for Amethyst to hide anything, but she ran her hand through the soil, anyway, because you never knew. As her father always said, \"Do it right the first time so you don't have to go back and do it again.\"\n\nAfter a minute of that, her hands were cold from the dirt. Well, that had definitely been a waste of time, she decided as she stood and brushed the soil off her hands. Not only had she not found anything, she'd ruined her manicure as well. Now she'd have to go back to the salon for a touch-up, something that she didn't have time to do.\n\nThis definitely wasn't turning out well. But the key had to be somewhere. Unless, of course, it didn't exist at all, which was a possibility she wasn't prepared to consider yet. Bernie tapped what was left of her nails against the boards of the house, thinking that maybe the key was behind a loose board. She tried prying a couple of boards out. No go.\n\n\"Okay, Bernie,\" she said aloud to herself. \"If this key is here, it's in some easy, accessible place. Just look around.\"\n\nFirst, she looked at the yard again, but aside from starfish molds and the pail and shovel in the sand box, there was nothing in the yard that could be used to hide a key. No trash cans, no cute little garden gnomes, no faux plastic bunnies. The yard was completely bare. So far she was batting zero for zero, or whatever that expression was. She didn't follow baseball, so she didn't know. Bernie turned and studied the back of the house. She got up on her tiptoes and ran her fingers over the top of the door frame. Then she tried the windows. And that was when she got lucky.\n\nThe key was sitting in a magnetic box on the underside of the window to the right. She hit her forehead with the heel of her hand. \"You are such an idiot,\" she said aloud.\n\nWhy do I always have to make things so much harder than they really are? Bernie thought as she slid the key in the lock and turned it. The door opened with a creak. Then Bernie replaced the key under the window frame, went inside, and locked the door from the inside. Better safe than sorry, as her dad would say. Well, that wasn't what he'd say in this situation, but she didn't want to think about that.\n\n\"Might as well start here,\" she muttered to herself, stepping into a small, oblong-shaped kitchen. Since she didn't know what she was looking for, she couldn't afford to skip anything.\n\nThe first thing that struck her was that the kitchen probably hadn't been touched since the sixties. The walls were painted a greenish yellow, the counters were green Formica, and the floors were some sort of speckled linoleum. The second thing that struck her was that the counters were applianceless. There wasn't even a microwave on them. Who, except for her family, didn't have a microwave these days?\n\nBernie started opening cabinets. Most were empty. There was a set of pots that looked as if it had never been used, a paring knife, seven steak knives, five dishes, a couple of mugs, and a box containing a set of glass tumblers. Bernie had had more stuff in her college dorm than Amethyst had had in her kitchen. Bernie continued on. There were a few cleaning supplies under the kitchen sink, and two boxes of cereal, a box of mac and cheese, a box of Bisquick, and a couple of packets of honey roasted peanuts in the kitchen cabinet across from the stove, as well as a bottle of Wild Turkey, a bottle of good French brandy, and a couple of bottles of not-bad Chilean wine.\n\nThe refrigerator didn't yield much more. There were two bottles of water and a carton of Diet Coke on the top shelf, a lemon with green fuzz on it on the second shelf, and nothing in the vegetable bins. The freezer revealed a bottle of vodka and a bottle of Aquavit. Obviously, Amethyst hadn't done any entertaining here. Or eaten here.\n\nBernie turned and opened up the lid of the garbage can. There was nothing in it but a bottle of Diet Coke, a bridal magazine, and a copy of the Longely newspaper. Bernie reached in and fished out the two publications. The paper was dated three weeks ago.\n\nDid that mean that that was the last time Amethyst was here, or did she simply read and dispose of more recent papers elsewhere? Bernie took a quick look at the paper but didn't see anything that linked it to Amethyst in any way. Then she looked at Modern Bride. There was no mailing label, so Amethyst had bought it off the newsstand. Unless someone else had purchased it and left it here, which, knowing Amethyst, was a more likely scenario. Nevertheless, Bernie tucked both pieces of printed matter under her arm. Maybe her dad would see something in them that she couldn't.\n\nAs she walked into the living room, she was wishing that there was some way of finding out what Inez had found out. If she was telling the truth. If this didn't pan out, Bernie decided that she'd appear at Inez's house with a large bottle of booze and some money. Despite what Brandon had said, she could probably spare fifty bucks, and it wasn't as if Inez didn't need the money. Her mother would have disapproved, but Bernie always felt that there were times that justified the expedient approach.\n\nLooking around, Bernie decided the living room wasn't much better than the kitchen. There was a brown leather sofa, two chintz-covered armchairs, a coffee table that looked as if it was made out of particleboard, and a cheapo television set. The furniture looked as if it had come from one of those rental stores, and the television looked as if it was at least ten years old.\n\nWhat Bernie was really struck by was what wasn't there. No cable box. No provisions for music. No books. No magazines. No pictures on the wall. There didn't seem to be anything personal at all. This apartment looked like it was a place for Amethyst to crash and nothing more. Bernie took another quick glance around and headed for the dining room. She found a bridge table and four chairs set up in the middle of the room, along with a floor lamp with a torn lamp shade.\n\nBernie wondered what she'd find in the bedrooms as she walked down the hall. The walls were covered with the kind of cabbage-rose wallpaper that had been popular in the thirties and forties. She opened the first door she came to. The room was empty. The only thing in it were the shades on the windows. She walked in and opened the closet door. A strong odor of mothballs came out and slapped her in the face. She peered inside. Nothing but some metal hangers, an orange metal container appropriately marked CAMPHOR, and an old navy blue woman's cardigan.\n\nBernie closed the closet door, left the room, and opened the next door. A made-up bed sat in the middle of the room. The coverlet was dark green and matched the curtains on the windows and the skirt on the bed. Amethyst's room, Bernie thought. Had to be. The nightstands were a light ash, as was the dresser standing in the far corner. An old-fashioned doll, the kind with a china head, leaned against the mirror hung over the dresser.\n\nBut what really attracted Bernie's attention was the two photographs hanging on the wall. They were both of preteen girls. Both of them were sitting on beds in what looked like dorm rooms, smiling and waving at the camera. Bernie didn't know how she knew who they were, she just did.\n\nThe titles written on the mats confirmed her guess. The girl with the big smile on her face and the slight buckteeth was Bessie Osgood. The second picture was of Zinnia McGuire. Bernie stared at them for a long time. She decided that Bessie looked a little bit like Libby had at that age. Kind of schlumpy. The sort of kid who always got As and did everything her mother told her to.\n\nOn the other hand, Zinnia looked as if she was going to get in her fair share of trouble. But, of course, she hadn't had the chance. She and Bessie had both been killed, their deaths officially declared accidents. Why did Amethyst have their pictures hanging on her wall? Were they trophies? Expressions of guilt? Reminders of the good old days? Bernie shook her head. She didn't know, and now, since she couldn't ask Amethyst, she never would.\n\nShe turned away and went through the dresser and the closet. The dresser had some underwear\u2014the kind you got at JCPenney on sale\u2014a fake pearl necklace, several T-shirts in different colors, and a couple of stretched-out black turtlenecks. The closet turned out to be a little fuller, but the clothes and shoes in it were old and worn, things that Bernie judged Amethyst wore around the house. In any case, Bernie had never seen her in anything remotely resembling the stuff in her closet. Now more than ever, it looked as if Amethyst had been living somewhere else.\n\nIf she needed any more convincing, the bathroom did it. There were towels hanging on the racks, a shower curtain surrounding the bathtub, and a hamper for dirty clothes. Bernie opened the shower curtain. A nubbin of soap sat in the soap dish, while a bottle of shampoo stood on the side of the tub. Bernie lifted up the shampoo bottle. There was practically nothing left in it. Interesting, Bernie thought as she put it down. She opened the medicine cabinet over the sink. There was a bottle of aspirin, an unopened tube of toothpaste, mascara, and a couple of lipsticks. Bernie opened the mascara. It was practically empty. She put it back and opened the lipsticks. One was a horrendous orange\u2014obviously a fashion error\u2014while the other was a cheapo discount brand Bernie had never heard of. If Bernie remembered correctly, the only brand she had ever seen Amethyst use was Chanel.\n\nBernie put everything back the way it had been. She knew Amethyst. Amethyst had not been a back-to-nature kind of gal. She had used lots of product. Lots of styling gel on her hair, three coats of mascara, eyeliner, eye shadow, blush, not to mention concealer, lip liner, gloss, and lipstick. Basically, she'd applied lipstick like Spackle.\n\nAnd where were the moisturizers, body lotions, hairbrushes, hair spray, and all the other things women like Amethyst couldn't do without? No blow-dryer. No perfume. One thing was clear. Bree's information was outdated. Amethyst hadn't really been living here at the time of her death. Maybe she had in the past, but she'd moved out. The questions were, when and to where?\n\nAnother dead end in a case full of them, Bernie thought. She was closing the medicine-cabinet door when she heard a noise. Bernie froze. It was the sound of the back door opening. Then she heard footsteps and what sounded like a crash.\n\nBernie peered out from the bathroom. The sounds seemed to be coming from the kitchen. There was another crash, followed by a muffled curse. Bernie crept down the hall. The prudent thing, she knew, would be to hide in one of the bedroom closets and wait till whoever it was left, but then she'd never been prudent.\n\nBy now she was at the end of the hallway. She carefully stuck her head out and took a look. No one was in the living room or the dining room. Whoever it was, was in the kitchen. Bernie thought for a moment. If she was very careful, she could get a peek at them. She very slowly inched her way to the door that connected the dining room and the kitchen and looked in.\n\n\"You,\" she cried when she saw who it was.\n\nBernie was gratified to see Inez jump. It was rare that she was able to invoke that kind of reaction. Inez had Amethyst's bottle of Wild Turkey in one hand and the bottle of vodka in the other, while the two bottles of Chilean wine were tucked under her arm.\n\n\"Nice,\" Bernie said as the bottles dropped out of Inez's hands and fell onto the linoleum.\n\nInez swayed for a moment, righted herself, and then, with the deliberation of the very drunk, carefully bent over and started picking the bottles up. \"She owed me money,\" she explained. \"I'm just taking what's mine.\"\n\n\"It's not yours,\" Bernie told her. \"It belongs to Amethyst's estate.\"\n\n\"It's mine,\" Inez insisted.\n\n\"First blackmail and now burglary. You're doing well.\"\n\n\"What about you?\"\n\n\"What about me?\" Bernie repeated coolly.\n\nInez burped. \"You're here, too.\"\n\n\"I was given the key,\" Bernie told her, marveling at how easily she lied.\n\nInez clutched the bottles to her chest and burped again.\n\n\"I guess I'm going to have to call the police,\" Bernie continued.\n\nInez continued clutching the bottles to her chest. Bernie wanted Inez to say something along the lines of \"Oh no, you're not\" so she in turn could say, \"I will unless you tell me what I want to know.\"\n\nBut Inez just held on to the bottles. Bernie sighed. She hated when people didn't \"get it.\" But that was the trouble with alcohol. It just made you stupid.\n\n\"Okay,\" she said. \"Tell me what you know, and I won't call them.\"\n\nInez peered at her suspiciously. \"Like what?\" she asked.\n\nBernie wanted to scream, \"What do you think I'm talking about?\" but she didn't. Instead, she said, \"Remember at the bar you told me you'd heard or seen something that got you fired. What was it?\"\n\nInez's eyes narrowed. She swayed slightly. \"Huh?\"\n\n\"You were trying to blackmail her.\"\n\nInez burped. She nodded at the bottles with her chin. \"Are you going to make me give these back?\"\n\n\"Not if you tell me what I want to know.\"\n\n\"It wasn't anything really. I heard Amethyst talking on her cell phone.\"\n\n\"And?\" Bernie prodded.\n\n\"And she was saying that she couldn't believe it, but it was finally happening.\"\n\n\"What?\"\n\n\"I don't know. I didn't hear that part.\"\n\nBernie looked at Inez.\n\n\"Honestly,\" Inez cried. \"She thought I did, but I didn't.\"\n\n\"Then why didn't you tell her that?\"\n\nInez shrugged and clutched the bottles closer to her chest. \"Because this was more fun. This way I could get back a little of my own.\"\n\n\"I don't believe you,\" Bernie told her. \"I think I'm going to have to take the bottles back.\"\n\n\"Don't,\" Inez cried. \"It's the truth. I wanted to get back at her. She treated me like dirt. We used to be at dinner parties together, and then she ruined my marriage, and I was working for her. Cleaning out her toilets. She asked for me specifically. Did you know that? That's what Ian said. I told him I didn't want to go, but he said I had to. Otherwise, I wouldn't have a job with him. Amethyst had me working for her just for spite. So I thought I could get a little bit back of my own.\" Inez stopped talking.\n\n\"Only things didn't work out that way.\"\n\n\"No, they didn't,\" Inez said. \"But I'm glad she's dead. I'm only sorry I didn't kill her.\"\n\n\"You could have,\" Bernie pointed out.\n\n\"I didn't.\"\n\n\"Someone did.\"\n\n\"It wasn't me.\"\n\nBernie raised an eyebrow.\n\n\"It wasn't,\" Inez insisted.\n\n\"I don't know. Working for Amethyst. That could have been the thing that threw you over the edge.\"\n\n\"That's why I wanted the money,\" Inez cried.\n\n\"Yeah. But killing her would have been so much more rewarding.\"\n\nInez drew herself up. \"Think whatever you want,\" she told Bernie. Then, clutching the bottles to her chest, she stumbled out the door.\n\nA moment later Bernie heard a car engine turn over and realized that Inez shouldn't be behind the wheel of a car. But it was too late. By the time she got outside, Inez was weaving down the road.\n\n## Chapter 22\n\nAmber had finished loading the last of the cartons with the pies snugly nestled in them into the van as Libby hurried toward the vehicle.\n\n\"You have everything in there, right?\" Libby asked.\n\nAmber nodded.\n\n\"You're sure?\"\n\n\"Positive.\"\n\n\"The walnuts? The sprinkles?\"\n\n\"I checked off the list you gave me.\"\n\nLibby noticed she had a Cheshire cat grin on her face.\n\n\"What?\" Libby said. \"Is it the costume I'm wearing?\"\n\n\"No. I think you look very nice.\"\n\nLibby looked down at the blue taffeta number Bernie had convinced her to wear. She couldn't bend without getting stabbed in the side by a piece of plastic. She might as well be wearing her grandmother's corset. How Bernie managed to work in this kind of getup was beyond her. The only saving grace was that the skirt was long enough, so no one could see her shoes. Thank God she was wearing sneakers. If she had to wear heels, she didn't know what she'd do. The truth was, she felt like an idiot. There was just no way around it.\n\n\"Are you sure I don't look ridiculous?\" she asked Amber.\n\n\"Oh yes.\"\n\n\"Then what are you smiling at?\"\n\n\"Nothing,\" Amber told her.\n\n\"You're smiling at something.\"\n\n\"You'll find out,\" Amber told her, and then, before Libby could ask her exactly what it was she was going to find out, Amber turned and headed back in the shop.\n\nUnder different circumstances, Libby would have followed her inside, but she wanted to get going, especially since she was going to have to unload the van by herself. Playing detective was all well and good, she decided as she slid behind the wheel of the van, but then she was always left to do the shop stuff while Bernie went off, running here and there.\n\nShe wondered how Bernie would like it if their positions were reversed. Not at all, Libby would wager. What she wanted to know was, why did she always have to be the responsible one? It really wasn't fair. She was tired of it. And while she was on the subject, she always did favors for Bernie\u2014like wearing this stupid dress. \"It'll just make things more festive,\" Bernie had said. Why had she listened to her? Especially since by now she knew better. The more she thought about everything, the angrier she got.\n\nBy the time she reached the Foundation, she was in a really bad mood, and seeing the fake skeletons dangling from the trees in those stupid cages Mark had rigged up didn't do anything to improve it. It just reminded her that if it hadn't been for Bernie, they wouldn't have taken that tour of the Haunted House, and someone else would have found Amethyst. She was still having nightmares about that. When was the last time Bernie had done something for her? That was what she would have liked to know.\n\nLibby glanced around the dining room she would be serving in. She had never realized how big it was. Cavernous really. So was the kitchen. And quiet. So quiet she could hear the creak of a door shutting somewhere down the hall. Kitchens should be full of people cooking. Otherwise, they were just creepy. Except for A Taste of Heaven's kitchen. That was cozy.\n\nShe looked at the clock on the wall. It was five-fifteen, fifteen minutes before the Haunted House officially opened. The volunteers who were manning the show should have been here by now, which should have made her feel better. Of course, the volunteers manning the show had been here the last time, too, and precious lot of good it had done her then. She'd been on one side of the door, and they'd been on the other.\n\nShe looked out the window on the far side of the room. It was one of those big jobbies, the kind where both sides open up and you can step outside, only someone had sealed them shut, so you couldn't anymore\u2014not that she would want to now. Raindrops were sliding down the panes. The sky had started darkening just as she had brought her last carton of cider in from the van.\n\nWas something hitting the window? Is that the tapping she was hearing? She walked over to get a better look. There was nothing there except the branches from one of the maples. She walked back. From the looks of the tree branches, the wind was picking up. She just hoped the power didn't go off here. Sometimes a strong wind did that. That was all she'd need. She should have really brought a flashlight along, just in case.\n\nLibby shook her head to clear it. She took a deep breath. Several, in fact. Ever since she'd had that encounter, or whatever you wanted to call it, with...with...Libby refused to say her name, because Bernie was right on this score. Naming things called them forth. She didn't like being in this room by herself, which was something she'd never admit to Bernie.\n\nEver.\n\nBecause how could she be nervous if she didn't believe in stuff like that? That was\u2014and here she was using one of Bernie's favorite words\u2014an oxymoron. If she found out, Bernie would never let her forget how she felt. She'd ride her forever about it. It wasn't that Bernie was malicious. It was just that she didn't know when to stop. And things that Libby was sensitive about, Bernie wasn't. It was as simple as that.\n\nThe whole thing was ridiculous, anyway, Libby decided. There had to be another explanation for what had happened to her the other day, but for the life of her, Libby didn't know what it could be. She'd put in a lot of time thinking about it, and she still hadn't come up with anything\u2014unless there was a secret passageway that led in here from somewhere else.\n\nBut in reality, that explanation didn't work, either. The whole thing, for want of a better word, had happened too fast. The...thing...okay...the manifestation couldn't have disappeared in the blink of an eye. And then there was the fact that Mark had said he'd experienced the same thing that Libby had. Three times. Once was enough for her. And Konrad and Curtis had said the same thing. But they were crazy, so they didn't count.\n\nLibby sighed as she rearranged the vanilla cupcakes with the orange icing on the plate and put them toward the front. Then she took the chocolate cupcakes with chocolate frosting that she'd decorated with orange sprinkles and put them where the vanilla cupcakes had been. After that, she took the lemon squares and put them where the chocolate cupcakes had been.\n\nShe knew what she was doing was pointless, but she couldn't help it.\n\nLibby pulled at her skirt. Then she tugged the bodice of her dress up. No matter what she did, her boobs kept popping out. Why, she asked herself again for the second time in as many minutes, had she let Bernie bamboozle her into wearing this? She supposed being a princess was better than being a bowl of cereal, but just barely.\n\nShe wasn't happy in costume. Especially in this one. She'd never liked dressing up, not even as a little girl. She was what she was. Anyway, it was too cold in here to be wearing something like this. She wanted to be back in her jeans and shirt. The only thing that helped was the fact that she didn't have those sleeves that trailed down and got into everything. She could just see green satin and waffle batter. It was not a pretty picture. Bernie had said Marvin would like the dress, but Marvin wouldn't even see it. He was off doing heaven knows what with her dad.\n\nAnd speaking of Bernie, where was she? She should have been here half an hour ago. Just like her, Libby thought crossly. Always late. She never had any consideration for anyone else. She was reaching for her cell to call Bernie when she felt cold. Chilled to the bone, actually. I need to put on a sweater, Libby told herself. Hopefully, she'd left her zip-up fleece in the van. It might not look pretty, but it would definitely do the job. And then she felt something else. Something she remembered from the last time, the sensation of something crawling up and down her skin. She looked down. There were goose bumps on her arms.\n\n\"Oh no,\" she moaned. \"Please not again.\"\n\nShe sensed a crackling around her.\n\n\"Go away,\" she pleaded. \"Just go away.\"\n\nThe crackling increased. She felt a tickling in her left ear. As if someone was breathing in it. She scrunched her eyes shut. This is a bad dream, she told herself. Nothing more. When she opened her eyes again, everything would be all right. It would be...What was her father's word? Copacetic. She closed her eyes tighter. Flashes of light exploded across her retinas. Then, unable to help herself, she opened her eyes again. She saw the room through a wave of energy.\n\n\"Help me,\" a voice said. Only the voice sounded as if it were coming from inside her head.\n\n\"I can't help you,\" Libby cried. Or had she thought it, too? She wasn't sure.\n\n\"You have to.\"\n\nLibby wanted to move, but her legs weren't working properly. Her heart was beating so quickly, she felt light-headed, as if all the air was being sucked out of her lungs.\n\n\"Leave me alone,\" she cried.\n\n\"Find me. Find me.\" Now the voice seemed to be echoing in the air.\n\nI shouldn't have eaten that pint of ice cream for dinner, Libby told herself. That's why this is happening to me. She knew it was ridiculous, but for some reason, she found the thought comforting.\n\n## Chapter 23\n\nJust when Libby thought she was going to pass out, whatever was holding her let go. Or maybe, she thought as she whirled around, she'd done this to herself. She wasn't really sure. Out of the corner of her eye, she caught a fleeting glimpse of a girl dressed in an oxford shirt and pleated skirt. It was the same girl as before. Dressed the same way, too.\n\nOf course, she was. Like she'd have a change of clothes in the great hereafter, Libby thought. They were probably the clothes she died in. Or were they the clothes she was buried in? Libby was trying to decide when the girl vanished. Poof. Just like that. And there was nothing. Libby felt as if she was going to faint again. She closed her eyes and put her hands to her forehead. She was going crazy. She was having hallucinations. Which meant she was a schizophrenic. Or she had a brain tumor. Libby was trying to decide which was worse when she felt a tap on her shoulder. She screamed and spun around.\n\n\"Libby,\" Bernie said.\n\nLibby looked at her sister and struggled to regain her composure. She had no luck. No luck at all.\n\n\"Are you all right?\" Bernie asked her.\n\n\"I'm fine,\" Libby told her.\n\n\"Because you don't look all right.\"\n\n\"Well, I am.\"\n\n\"Who were you talking to?\"\n\nLibby squared her shoulders and tried to stand up straight when what she wanted to do was sit down. Lying down would be even better. She was suddenly exhausted. More than exhausted. Totally fried.\n\nShe moistened her lips with her tongue. \"I wasn't talking to anyone,\" she lied.\n\n\"But I heard you outside,\" Bernie protested. \"You were yelling at someone to leave you alone. That's why I came running in here.\"\n\nWas I yelling that loud? Libby wondered. She didn't think she was, but if she was being honest with herself, she'd have to say she didn't remember. She didn't remember Bernie opening the door, either. A definite brain tumor. She'd be dead inside of a month. Two at the most.\n\n\"So who was it?\" Bernie said when Libby didn't answer. \"Who were you yelling at?\"\n\n\"I wasn't yelling at anyone. You imagined it.\"\n\n\"I see.\" Bernie looked at Libby carefully, something Libby hated. \"You look awfully pale.\" She reached over and took her hands. \"And your hands are freezing.\"\n\n\"That's because I'm freezing in this dress,\" Libby countered while she pulled her hands out of Bernie's grasp and tucked them under her armpits to warm them up. \"I'm going to get pneumonia standing here.\"\n\nBernie studied her sister's face some more. Finally, she said, \"You saw her, didn't you?\"\n\n\"I don't know what you're talking about,\" Libby replied.\n\n\"Yes, you do. What did she want?\"\n\n\"She didn't want anything.\"\n\n\"Then you admit that you saw her.\"\n\nLibby started walking back to the long tables they'd set up the first evening they'd worked here. \"I didn't.\"\n\n\"Why are you lying?\"\n\n\"I'm not lying.\"\n\n\"I told you. I heard you outside.\"\n\nLibby turned to her. \"I don't want to talk about this anymore. Is that clear?\"\n\n\"Yes. But this could be important.\"\n\nLibby started setting up the coffee machines. \"This always has to be about you,\" she told her sister.\n\n\"Now you're making no sense whatsoever.\"\n\n\"So, you're telling me that I'm crazy on top of everything else?\"\n\n\"That's not what I said at all, and you know it.\"\n\n\"And, by the way, I hate this dress,\" Libby told her. Now seemed as good a time as any to share that thought.\n\n\"Then why did you wear it?\"\n\n\"Because you wanted me to.\"\n\nBernie snorted in exasperation. \"You could have said no.\"\n\nLibby shrugged and began measuring out the coffee. She knew that this whole thing with Bernie was about her being scared by\u2014okay, she was going to come out and say it\u2014being scared by Bessie Osgood's ghost. When she got scared, she had a bad habit of covering up by getting angry, and the more upset she became, the further away she pushed people. She closed the coffee bag with a snap and reached for the unleaded stuff. But that was the way she was. She was too old to change now. She knew Bernie was watching her. She wanted to say something to her, but somehow she couldn't bring herself to.\n\nAfter another moment had gone by, Bernie said, \"If that's the way you want to be, fine.\"\n\nBernie walked over to the table and began to stack the paper plates. Then she grabbed a couple of wicker baskets and arranged the napkins in one, a lot of napkins, because people tended to grab a handful of them at a time, and the plastic knives, forks, and spoons in the other. She liked that they were black. It had taken her a while to locate them at a reasonable price, but white would have ruined the ambience. Although in lots of countries, white was the color of death. But not here. Libby would come around when she was good and ready, Bernie knew. She just didn't know why Libby was making such a big deal about something like this.\n\nA ghost appeared to you. So a ghost appeared to you. Especially this time of the year. Cultures all over the world recognized that this was the time of the year when the membrane that separated the living from the dead was at its thinnest. That was just the way it was. It was nothing to be ashamed of.\n\nIn fact, Bernie was slightly miffed that Bessie Osgood hadn't appeared to her. Why pick someone who so obviously didn't want to have anything to do with you? The only reason she could come up with was that Libby had been in the room by herself the two times it had happened and Bernie had always been here with Libby.\n\nFor the next couple of minutes, the two sisters worked in silence. Finally, Libby looked up at the clock.\n\n\"Five minutes till we open,\" she said.\n\nBernie grinned. She recognized a peace offer when she saw one. \"No one ever gets here before six-thirty or seven,\" she countered, extending her own olive branch.\n\nLibby nodded. What Bernie said was true. Most people came after dinner. Actually, their busiest time was between seven-thirty and nine. Then they had lines out the door. But before that, things were pretty dead. Dead was a bad word. Things were quiet.\n\n\"I made extra pumpkin walnut muffins, so we shouldn't run out this time,\" Libby told Bernie as she began slicing up the pies.\n\n\"Good,\" Bernie replied. \"Sorry I was a little late, but I found some interesting things at Amethyst's place.\"\n\nLibby stopped slicing for a moment. \"What were you doing at Amethyst's place?\" she asked. As far as she knew, that wasn't in the game plan.\n\nBernie told her sister about her encounter with Inez at R.J.'s.\n\n\"You're kidding me,\" Libby said when Bernie was done.\n\nBernie shook her head. \"Nope. I'm not.\"\n\nLibby went back to slicing. \"Talk about rubbing salt in the wound, as Mom used to say. Boy, if that isn't a motive for killing someone, I don't know what is.\"\n\n\"I wonder if Inez is telling the truth.\"\n\nLibby put the apple-cranberry pie aside and began on the apple crumb, which was difficult to cut neatly. \"You mean about the whole thing being Amethyst's idea?\"\n\n\"Yeah. Maybe the whole thing was Inez's idea. Maybe she wanted to work for Amethyst so she could get even.\"\n\nLibby bisected the pie and then cut the halves into quarters. \"Could be,\" she said thoughtfully.\n\n\"There's one way to find out,\" Bernie said, and she reached for her cell phone and dialed.\n\nLibby continued cutting pies while her sister talked to Ian.\n\n\"Well, that was interesting,\" Bernie said when she hung up. \"Evidently, Ian didn't even know that Inez was working for Amethyst. She was doing it off the books.\"\n\nLibby put the knife down, reached over, took a half-moon cookie off the table, and began to eat it. \"So we don't know who suggested what.\"\n\n\"And we probably won't know, either. But we do know that Inez was telling the truth about one thing.\"\n\n\"What's that?\"\n\n\"That Amethyst was going away,\" said Bernie. \"All of her clothes are gone. There's no product in the bathroom, and she was definitely a Spackle and spray kind of gal.\"\n\nLibby poured herself a glass of cider to go with the cookie. By their nature, half-moon cookies, even hers, were somewhat dry. She'd never been able to find a recipe that kept their half-moon essence and was moist. It was one of those mysteries that still needed to be solved.\n\n\"I wonder where she went.\"\n\n\"Not that far, obviously.\"\n\n\"Or she went and came back,\" Libby said.\n\n\"Interesting.\" Bernie poured herself a cup of cider and took a sip. \"This is really good,\" she commented. \"Cotter should sell this in the stores.\"\n\n\"He can't, because it's unpasteurized. State law.\"\n\n\"But we can use it, right?\" Bernie asked.\n\n\"Correct.\" Libby finished slicing the apple crumb pie and went to put the knife away in its case. \"Getting married? Libby asked as she moved her sister's jacket and bag off the carton the knife belonged in.\n\nBernie's head shot up. \"What do you mean?\"\n\nLibby lifted up the Modern Bride magazine that had been buried under Bernie's jacket.\n\n\"Oh that.\" Bernie laughed. \"I got that from Amethyst's house, along with a newspaper.\"\n\nLibby looked at the date. It was current. \"You know,\" she said. \"What if Amethyst was planning to get married?\"\n\n\"That's a pretty big jump.\"\n\n\"I agree, but bear with me. You said that Inez overheard Amethyst telling someone it was really going to happen, and she was pretty excited....\"\n\nBernie nodded.\n\n\"And then she packed up everything and left. What if what she was excited about was getting married?\"\n\n\"She could have been moving in with someone,\" replied Bernie.\n\n\"Then why Modern Bride?\"\n\n\"Wishful thinking? A friend getting married?\"\n\n\"Amethyst didn't have friends, and she never struck me as the kind of woman who engaged in wishful thinking.\"\n\nBernie had to admit that was true. \"But she came back.\"\n\n\"Maybe things didn't work out,\" said Libby.\n\n\"That's an understatement if ever I heard one.\"\n\nLibby nibbled on her cuticle for a moment. \"Here's another idea. Maybe she went to speak to Ed Banks to see if she could hold the wedding at Lexus Gardens. It would be a great spot.\"\n\n\"And she did know him through the Foundation.\"\n\n\"Do we know that for a fact?\" Libby asked.\n\n\"No. But it seems that they would have met at a dinner party or something like that.\"\n\n\"Can we find out?\"\n\n\"We can ask Banks's personal assistant when he hits land,\" replied Bernie.\n\nLibby brushed a strand of hair out of her eyes. \"I don't know. You're saying the person who killed Amethyst killed Ed Banks to keep him from talking? That's a definite stretch.\"\n\n\"It's a link.\"\n\n\"A very weak link,\" said Libby.\n\n\"It's possible, but not probable,\" both sisters said together. They laughed.\n\n\"Good old dad,\" Bernie said. She could hear him saying it now.\n\nLibby and Bernie looked at each other.\n\n\"This is all supposition, you know,\" Bernie said.\n\n\"But it makes as much sense as anything else,\" Libby countered. \"It makes more sense than having a ghost kill her.\"\n\n\"Bessie lied about that, remember? She just wanted to take credit for it.\"\n\nLibby took the palm of her free hand and smacked herself on the forehead. \"Excuse me. How could I forget?\"\n\n\"Yeah. Curtis and Konrad are going to come up with the real killer's name now any day.\" Bernie reached over and took a pumpkin bar, broke it in half, and began to nibble on it. \"Maybe Amethyst's husband killed her. You know, she married him under false pretenses, and he realized what he'd gotten himself into.\"\n\n\"An annulment would have been easier.\"\n\n\"But not nearly as satisfying,\" Bernie pointed out.\n\nLibby laughed. \"This is true.\" She finished her half-moon cookie and wiped the crumbs off her hands. \"Do you think we should tell Dad what we're thinking?\"\n\nBernie could just hear her father now. And you think this based on what? A copy of Modern Bride and something a hostile and unreliable witness, a witness who is a suspect in the homicide case, told you? \"No,\" she said. \"I don't think so. At least not yet.\"\n\nLibby tugged at the top of her dress again. \"We could sort of float it by him.\"\n\n\"It would help if we had positive proof that Amethyst got married.\"\n\n\"I could go down to the town hall and check it out,\" said Libby.\n\n\"Go ahead. But the odds are that if she got married, she didn't get married here.\"\n\n\"True,\" Libby agreed. \"She was fairly secretive.\"\n\n\"Fairly? Fairly?\" Bernie opened her eyes so wide, she looked like a Kewpie doll. \"She lived in Stanton, for heaven's sake. She didn't want anyone in Longely to know her business.\"\n\nLibby returned to the subject of Ed Banks. \"That's why Lexus Gardens would fit in so nicely.\"\n\n\"So go check.\"\n\n\"I intend to,\" Libby admitted. \"Although if she did get married, knowing Amethyst, she probably did it somewhere like Palm Springs or Miami Beach....\"\n\n\"Or Paris or Rome.\"\n\n\"Or Morocco,\" Libby added while she tugged at the top of her dress.\n\n\"Here. Let me fix that for you.\" And Bernie went over and pulled the back of Libby's dress up. \"Better?\" she asked.\n\nLibby looked at her boobs. They were back where they should be. \"Much better. Thank you.\"\n\nBernie studied her sister while she tapped her fingers against her chin.\n\n\"Are you still thinking about Amethyst?\" Libby asked.\n\n\"No. I'm thinking that I'm sorry I made you wear this.\"\n\n\"You didn't make me. I could have said no. And, anyway, I think you're right. I think Marvin is going to like this.\"\n\n\"It might even make him jealous,\" Bernie said. \"Which would be a good thing.\" When Libby didn't say anything, Bernie said, \"Trust me on this. I'm the expert.\"\n\nLibby laughed.\n\nBernie stuck out her hand. \"Friends,\" she said.\n\n\"Friends to the end,\" Libby answered. They'd been saying that to each other for as long as Libby could remember.\n\n\"So,\" Bernie said. \"Now that we're pals again, are you going to tell me what happened before, when you were yelling?\"\n\n\"I wasn't yelling.\"\n\n\"Okay. Speaking loudly.\"\n\n\"You just don't give up, do you?\" Libby said.\n\n\"No, I don't. It's the secret to my success.\"\n\n\"Yes, it is,\" Libby agreed.\n\nAnd Libby gave her a blow-by-blow of what had occurred, which, if she was being honest with herself, she knew she was going to do all along, because eventually, she always told her sister everything.\n\n## Chapter 24\n\nLibby looked at the clock on the wall. It was now a little past nine. Half an hour and she and Bernie could pack up and leave. Thank heavens. The traffic had been nonstop. Once the people had started coming in, they had never slowed down. Usually, people came in clumps, so you had a chance to take a break, but she hadn't even had time to run to the bathrooom.\n\nIn addition, her back was killing her from standing, and her wrists hurt from making all those waffles. She hoped she wasn't developing carpal tunnel syndrome. That would be all she'd need. Maybe she should go to the store and get those braces. See if those helped.\n\nAt least her feet weren't killing her. She didn't know how Bernie managed in her three-inch stilettos. Libby didn't know what she'd do if she couldn't wear her sneakers. If she had to wear Bernie's shoes, she'd be going barefoot and the hell with the health codes. She wanted to go home, get out of her dress, take a bath, and go to bed, but instead, she and Bernie were meeting Brandon and Marvin at R.J.'s.\n\nWhen she'd protested, Bernie had said to her, \"First of all, Marvin hasn't seen your dress, and second of all, it's the least you can do to thank him for driving Dad around all afternoon.\"\n\nBernie was right. Marvin was incredibly sweet, and she shouldn't take him for granted, which, she was the first to admit, she had a tendency to do. Still, she hoped it would be an early night.\n\nShe surveyed what was left on the table. The one bright note was that they didn't have much to take back to the shop. All the cider was gone, as were the pies and the fruit breads, except for the last piece of banana bread, which she was going to eat right now. They could store what was left of the apple compote in the fridge in the kitchen adjoining the dining room, though she'd have to make more tomorrow morning. The leftover waffle batter had got tossed, as had the coffee. Libby was thinking that she wished she could find a use for the leftover batter\u2014it pained her to throw anything out\u2014when she realized that Bernie was talking to her.\n\n\"Maybe there's something about you that's attracting Bessie Osgood,\" Bernie was saying.\n\n\"There's nothing about me that's doing that,\" Libby said firmly and dove into her bag for a piece of chocolate. She knew what Bernie was going to say, and she didn't want to go there.\n\nJust the thought of what had happened was enough to give her the chills.\n\n\"You want a piece of chocolate?\" Libby asked her sister. \"I've got French made, estate harvested, unblended, eighty percent dark, or good old Hershey's Milk Chocolate Kisses.\"\n\n\"I'll take the Kisses,\" Bernie told her.\n\n\"Me too,\" Libby said.\n\nYou could say all you wanted about the joys of dark chocolate, Libby thought. You could go on and on about the notes of cinnamon and cranberry present in it, you could feel virtuous because dark chocolate was now considered good for you, but in times of stress, nothing worked like Hershey's Kisses. Somehow when they dissolved on your tongue, they took your worries with them. For a little while. But in this case, a little while was good enough.\n\nDefinitely good enough, Libby thought as she hoisted her bag off the floor and began rummaging through it, although what she really needed was a milk-chocolate IV right now.\n\n\"What in heaven's name do you have in there?\" Bernie asked her.\n\n\"Stuff. You know.\" Libby looked at Bernie's sleek clutch. \"Or maybe you don't.\" And she opened her bag as wide as she could and peered in. \"Where is that bag of Kisses?\" she muttered. \"I know I have it somewhere.\"\n\n\"You have a whole bag?\" Bernie asked.\n\n\"Well,\" Libby said defensively, \"you never know when a chocolate emergency will arise.\"\n\n\"Like the Hudson River will overflow, and we'll be stranded here for weeks.\"\n\n\"Exactly,\" Libby said. \"Or we get caught in a blizzard.\" She started removing things. Out came a pocket knife, a wallet, sunglasses, a bottle of aspirin, a bottle of vitamin B complex, a screwdriver. \"So that's where it was,\" Libby said as she put the screwdriver down on the table. \"I spent twenty minutes looking for this yesterday.\" Her eyes widened. A moment later she drew out a brown paper bag. \"I know I didn't put this in here.\" And then she remembered. \"Amber.\"\n\n\"Amber what?\"\n\n\"Amber said she had a surprise for me.\" Libby felt instantly guilty. Amber was probably waiting for her to call.\n\n\"A surprise from Amber is not necessarily a good thing,\" Bernie observed.\n\n\"Don't make me more nervous than I already am,\" Libby told her. \"It's been a bad enough day already.\"\n\nBernie snorted. \"Such a delicate flower.\"\n\n\"I'm serious. What with everything that's going on, I'm a nervous wreck. I think I need an antianxiety drug.\"\n\n\"No. What you need is to get laid.\"\n\n\"Bernie!\" Libby wailed.\n\n\"What's wrong with what I said?\"\n\n\"That's personal.\"\n\n\"No, it's not. It's true. It's true for everyone. You mean, it's not? You mean, you don't feel more relaxed afterwards?\"\n\nWhen Libby didn't answer, Bernie decided to change the subject. This was another topic Libby didn't like to talk about, and they'd had enough arguments for one day.\n\n\"What's in there?\" Bernie asked, motioning to the brown paper bag with her chin.\n\n\"Let's see,\" Libby said. She gingerly opened the bag and peered inside.\n\n\"Well?\" Bernie demanded.\n\n\"It's the puzzle box Felicity gave you.\" And Libby lifted it out.\n\n\"Why did Amber give it to you?\"\n\nLibby shook her head. \"Don't know. Dad left it downstairs on the kitchen counter. Maybe Amber wanted to make sure it didn't get lost.\"\n\n\"Why should she care?\" Bernie asked.\n\nLibby shrugged her shoulders.\n\n\"I'm going to call her,\" Bernie said, reaching for her cell phone. But her call went straight to voice mail. Amber must have turned off her phone, Bernie decided. So, she left a message to call her and hung up. \"She's probably with that yo-yo she's seeing.\"\n\n\"Her taste in men isn't very good, is it?\" Libby noted as she spotted a piece of paper in the bottom of the bag. Probably a sales receipt, she thought as she took it out.\n\n\"No, it's not,\" Bernie agreed as her sister unfolded the paper.\n\n\"It's a note,\" Libby said. She read it quickly. \"It's instructions for opening up the puzzle box. Amber must have figured it out.\"\n\n\"Impressive,\" Bernie said. She certainly hadn't been able to, and neither had her dad or Clyde.\n\nLibby took the puzzle box and looked at it. It was a little bigger than the palm of her hand. The top was made of three pieces of dark wood; the bottom, of two. The pieces of wood fit together so neatly that you had to look carefully to see where they were joined. Libby smoothed the paper with the edge of her hand and read the instructions.\n\n\"Press on the far left-hand piece once.\" She did. \"Then tap lightly on the far right piece.\" She did that, too. \"Then tap the bottom far right piece of wood, and the box should open.\" Libby looked up at Bernie. \"Here goes.\" She rapped once, and the top flew open.\n\nLibby and Bernie peered in. Inside the box was a piece of paper folded into a tiny square. Bernie lifted it out and began unfolding it.\n\n\"What is it?\" Libby asked as Bernie finished opening it.\n\nBernie flattened the paper out with the side of her hand. The writing was so fine, she was having trouble reading it. She brought the paper closer. \"It's a map,\" she said.\n\n\"A map of what?\"\n\n\"Maybe the old Peabody School. I'm not really sure.\"\n\nLibby took the paper out of Bernie's hand and studied it. \"The ink is brown.\"\n\n\"It's from a fountain pen.\"\n\n\"I know that,\" said Libby.\n\n\"I thought you were asking.\"\n\n\"No. I was commenting.\"\n\nBernie took the map out of Libby's hand and refolded it. \"We should show this to Dad. He'll probably know.\"\n\nLibby was about to agree when Bernie's cell rang.\n\n\"It's Amber,\" Bernie mouthed after she'd answered it. \"Yes,\" she said into her cell. \"We just opened the box. Yes. No. Libby forgot. Well, it was busy from the time she walked in here. I'm sorry your feelings are hurt. No. She doesn't mean to be insensitive.\"\n\n\"Insensitive?\" Libby squeaked. \"I'm insensitive?\"\n\n\"Sssh,\" Bernie whispered and went back to listening to Amber. \"Yes. We were very surprised. And impressed. How did you manage to figure it out?\" She nodded as she listened to Amber's explanation.\n\n\"What is she saying?\" Libby asked.\n\nBernie held up her hand as a signal for silence and went on talking to Amber. \"So you know where the map is pointing to?\" Bernie was silent for a few more moments as she listened to Amber talking. \"Interesting,\" she said as she hung up. Then she turned to Libby as she put her cell back in her clutch and folded it over.\n\n\"We're meeting her and the twins in front of the Foundation in half an hour,\" Bernie told her. \"By that time, everyone should have cleared out from here.\"\n\n\"They'll be gone in ten minutes.\"\n\n\"There are always stragglers. Anyway, it'll take everyone a little time to get here.\"\n\nLibby took the bag of chocolate Kisses out of her bag and poured several into the palm of her hand. She had a feeling she was going to need them.\n\n\"How does Amber know where to go?\" Libby asked.\n\n\"She looked at the old plans for the Peabody School. I guess the twins have them,\" Bernie explained before Libby could inquire.\n\n\"How does she know the twins?\"\n\n\"They live on the same block.\"\n\n\"It figures,\" Libby said morosely. She could see her bath and her bedtime receding further and further into the distance. \"What about Marvin and Brandon?\"\n\n\"I'll call them now. They can meet us there. We'll go for a drink later.\"\n\nLibby unwrapped two chocolate Kisses and popped them into her mouth.\n\n\"What is the matter with you?\" Bernie demanded.\n\n\"I'm tired.\"\n\n\"Aren't you excited to see what we're going to find?\"\n\n\"Of course,\" Libby lied.\n\n\"No, you're not.\"\n\n\"Have it your way.\"\n\nLibby popped another two Kisses into her mouth. It was going to be even colder and wetter out there now than it had been when she'd come in, and she was dressed in this stupid dress. She didn't even have a warm coat with her. Bernie had insisted she use her mom's paisley shawl instead. She'd loved it since she was a little girl, and it was definitely more elegant than her black, puffy coat. But sometimes elegance wasn't everything, although she doubted she could convince Bernie of that.\n\nLibby knew what was going to happen. She was going to get pneumonia, and then she wouldn't be able to work and the shop would close and they'd be out on the street. And on top of that, they wouldn't find anything tonight. She, Libby, would die of pneumonia for nothing.\n\n\"I don't know,\" Bernie said.\n\n\"You don't know what?\" Libby demanded. Her sister was looking at her and shaking her head in that particularly smug, irritating way she had.\n\n\"If you're like this now, I don't know what you'll be like when you're fifty.\"\n\nSince Libby couldn't think of a sufficiently cutting reply, she decided that it was better to say nothing at all.\n\n## Chapter 25\n\nBernie and Libby were five minutes early for their rendezvous, but Amber was even earlier. They immediately spotted her in the parking lot right behind the Foundation. Hers was the only car in the lot, but even if it weren't, it would have stood out like a lighthouse beacon.\n\nWonder Woman was a neon green Beetle with big white flowers on the doors and the words FLOWER POWER printed across the hood. And then there was Amber herself. \"Not exactly inconspicuous,\" Bernie had said to her sister. But she had to admit Amber had style, and that was a good thing. On the other hand, given the circumstances, more circumspection might have been better. Before coming here, Amber had changed out of her work clothes and was now wearing black-and-white striped leggings, motorcycle boots, a red baby-doll dress, and a black leather jacket.\n\n\"I like the hair,\" Libby said to Bernie as Amber came galumphing toward them. Since she'd seen her at the store, Amber had dyed her hair a bright yellow and braided it.\n\nBernie got out of the van. \"Very colorful,\" she commented.\n\nLibby rubbed her arms and drew the shawl around her as tightly as she could. \"At least it's better than the blue she had last month.\"\n\n\"I kinda liked the blue,\" Bernie confessed. \"In fact, I'm thinking of dying my hair that color myself.\"\n\n\"You wouldn't,\" Libby said.\n\n\"Why not?\" Bernie asked. \"I can always change it if I want to.\"\n\nBut before Libby could answer, Amber had reached them. The metal piercings in her nose and eyebrow gleamed in the glow of the security lights.\n\n\"This is so-o-o cool,\" Amber squealed, reminding Bernie of a five-year-old about to go into an ice-cream store. \"I'm so excited.\"\n\n\"I never would have guessed,\" Libby said dryly. If she didn't get pneumonia from being out in this weather, it would be a miracle.\n\nBernie kicked her in the shin.\n\n\"Ouch,\" Libby cried.\n\n\"Sorry about that,\" Bernie said. \"I must have tripped. Where is everyone?\"\n\nAmber pulled on one of her braids. \"They're coming,\" she replied. \"Curtis and Konrad had to stop and get some batteries.\"\n\n\"They're recording this?\" Libby asked.\n\nAmber gave her a look that clearly said that she was too dumb to live. \"Shite, mate. Sodding right they are.\"\n\n\"How silly of me.\" Libby had thought Amber was through with her Brit period, but evidently she was wrong.\n\nA moment later she heard the sound of engines approaching. Then two cars came roaring out from behind the scrim of tall trees that hid the Foundation and headed for Bernie and Libby's van.\n\nFor a second, Libby panicked, because she thought they were security, but then she realized the cars belonged to Brandon and to Konrad and Curtis. They both screeched to a halt at the same time. Marvin and Brandon jumped out of one car, while Konrad and Curtis jumped out of the other.\n\n\"The cavalry has arrived,\" Bernie said to Libby out of the side of her mouth.\n\n\"So it would seem,\" Libby agreed.\n\n\"What's a cavalry?\" Amber asked.\n\nSuddenly, Bernie felt very old. \"I'll explain later,\" Bernie told her.\n\nKonrad beamed at everyone. \"I'm totally psyched,\" he said. Then he dove back in his car and came out with a tape deck, which he handed to Curtis. He went back in again and came out with a camera. \"Infrared,\" he explained.\n\n\"Cool,\" Amber repeated.\n\n\"Very,\" Brandon agreed.\n\nWatching Amber, Bernie thought that she now understood the expression \"grinning from ear to ear.\"\n\nLibby looked at Marvin. She could tell from the expression on his face that he had the same misgivings about this that she had, a fact that comforted her, because everyone else seemed to think that what they were doing was fine.\n\n\"Nice dress,\" Marvin told Libby. \"I especially like the neckline.\" And he moved both eyebrows up and down.\n\nLibby grinned; then she caught Bernie's eye and stopped smiling. Okay. So Bernie was right about a few things, mostly men and clothes. She'd grant her that, but she wasn't going to give her the satisfaction of showing it.\n\n\"Be right back,\" Marvin said, and he trotted off to his car.\n\nA few seconds later, he returned with a jacket under his arm. He handed it to Libby. \"Here,\" he said.\n\nLibby put it on gratefully.\n\n\"So,\" Brandon was asking Amber, \"how did you solve the puzzle?\"\n\nAmber shrugged as everyone gathered around her to hear the answer. \"I just like to fool around with stuff like that, you know.\" She nodded in Bernie's and Libby's direction. \"I asked your dad if I could give it a try, and he told me to go ahead.\"\n\nBernie buttoned up the sweater she was wearing over her she-devil costume. \"Did you tell him that you'd figured it out?\"\n\n\"No. I thought it would be neater if we could show him what we found,\" said Amber.\n\n\"If there's anything there,\" Libby said.\n\n\"Why shouldn't there be anything there?\" Amber demanded.\n\n\"Well, because whatever it is has been...in its place...for a long time. Things happen,\" Libby pointed out.\n\n\"Like what?\" asked Amber.\n\n\"Like stuff,\" Bernie said.\n\nThe sound of a car pierced their conversation.\n\n\"You know what I think?\" Brandon said. \"I think we should get on with it before security comes around.\"\n\n\"He's right,\" Konrad agreed. \"Mark wouldn't be happy if he saw us here.\"\n\n\"Don't be stupid,\" Curtis said. \"Of course, he'd be happy\u2014\"\n\nLibby interrupted. \"So where do we go?\" She had no intention of standing there while they argued.\n\n\"I'll show you,\" Amber said, and she danced off in front of them.\n\nMarvin began walking. \"It must be nice to be young,\" he remarked.\n\n\"You are young,\" Libby said as she walked beside him.\n\nMarvin beamed.\n\n\"Hey, Konrad,\" Brandon called.\n\nKonrad turned slightly.\n\n\"You expect to get anything with that stuff?\" Brandon pointed to the tape deck and the camera.\n\n\"Yeah. Absolutely,\" said Konrad. \"It's almost Halloween. Everyone is out. This is the time when the dead come visiting.\"\n\n\"So Halloween is sort of like New Year's Eve for the dead? Party time,\" Brandon asked.\n\nCurtis laughed. \"Yeah. That's exactly right. And this is something about Bessie, so she'll probably be there, too.\"\n\nBoy, I hope not, Libby thought. She'd had enough of her to last for a long time. Libby gave a short prayer that if Bessie was around, she'd pick on someone else for a change.\n\n\"Where are we going?\" Libby asked.\n\nEveryone shrugged.\n\n\"I'm following Amber,\" Konrad said.\n\n\"Me too,\" Curtis said.\n\nSomehow that didn't make Libby feel any better. After all, Amber was the person who seriously believed in past-life regression and guided dreaming, and based all of her decisions on the I Ching. But then who was she to talk? She'd seen a ghost. She'd done more than see. She'd talked to it. What did that make her?\n\n\"How'd you get the plans?\" Bernie asked Curtis.\n\n\"Mark gave them to us. He said they might help us get...situated vis-\u00e0-vis the whole recording situation. Well, they're not really plans. They're more like sketches we made from the plans, because he wouldn't let us take the plans out of the building, which really is okay, because they're really, really large, and they'd be hard to carry around.\"\n\n\"Are your sketches to scale?\" Marvin asked.\n\n\"What's scale?\" Konrad replied.\n\nMarvin looked at Libby and raised an eyebrow. Libby gave an infinitesimal nod back. Now she was even more dubious about the whole endeavor than she'd been before.\n\n\"How much farther?\" Bernie asked.\n\nAmber turned around and pointed. \"We're almost there.\"\n\nBernie looked around. There was nothing she could see that looked anything like what was on the map. \"I don't get it,\" she said.\n\nAmber asked for the map from the puzzle box; then she laid the map and Konrad and Curtis's sketches side by side on the bench next to the Foundation's front door.\n\n\"It's simple,\" she said. \"See.\" She pointed to the right wing. \"That's the girls' section, and the left wing is the boys'.\"\n\n\"How do you know this?\" Libby demanded. \"It doesn't say that on the map.\"\n\n\"I just know,\" replied Amber.\n\n\"How do you just know?\" asked Libby.\n\nAmber put her hands on her hips and narrowed her eyes. \"I dreamt it, okay?\"\n\nLibby snorted.\n\n\"See,\" Amber cried. \"I knew if I told you, you'd get all weird on me.\"\n\n\"I'm not weird,\" said Libby.\n\n\"But you wouldn't have wanted to come,\" said Amber.\n\n\"You're right,\" Libby told her. \"I wouldn't have, but we're here now, so let's get it over with.\" But not for the reason you think, she silently added.\n\nAmber hesitated for a moment before pointing up to the second floor. \"See the windows?\"\n\nBernie looked up. \"And Bessie Osgood stood there?\"\n\n\"Yes,\" Amber said. \"She liked to watch everyone coming and going.\"\n\nLibby zipped up Marvin's jacket. It was getting colder by the minute. \"And that's where she was pushed from?\"\n\n\"That happened on the other side.\" Amber pointed to one of the sketches. \"That's where the French doors are.\"\n\nBernie leaned in closer. She could vaguely, and she meant vaguely, make out the drawing. \"I don't suppose anyone brought a flashlight?\" she asked.\n\nEveryone shook their heads.\n\n\"Use your cell,\" Brandon suggested.\n\nBernie tried, but the light was minimal, certainly not enough to read by. She flipped her cell phone closed and slipped it back in her clutch.\n\nAmber turned around and pointed over to the maple tree about twenty feet away. \"In my dream, I saw someone burying something right by the base of that tree.\"\n\nKonrad and Curtis ran over to where Amber was pointing.\n\n\"Here?\" Konrad asked.\n\nAmber nodded.\n\nKonrad fumbled with the tape deck for a moment. Then he clicked on the switch. \"Are you here, Bessie?\" he asked.\n\nBrandon moved next to him. \"Is she answering?\"\n\nCurtis shot him a dirty look. \"We won't know that till we play it back.\"\n\n\"I'm surprised you can hear anything on that tape deck. Where did you get that thing?\" said Brandon.\n\nKonrad put his finger to his lips. \"Sssh. You'll scare her.\"\n\n\"How can you scare a ghost?\" Brandon asked.\n\n\"They have feelings just like we do,\" Curtis said.\n\nBrandon was just about to ask him how he knew that when Amber let out a small shriek. Everyone jumped.\n\n\"What is it?\" Libby cried.\n\nAmber put her hand up to her mouth. \"Shite. I gotta go back to Wonder Woman. I forgot the shovel.\"\n\n\"Wonder Woman?\" Marvin asked Libby as Amber dashed away.\n\n\"That's her car,\" Libby explained. She rubbed her arms. Even with Marvin's jacket, she was still cold. \"This is silly. I think we should go.\"\n\n\"Go on,\" Bernie told her. \"I'll stay.\"\n\nLibby sighed. Somehow Bernie telling her to go made it harder to do just that. Go figure. She was still debating with herself what to do when Amber came back with a shovel. Libby looked at it carefully.\n\n\"That looks like the one we keep by the shop's back door,\" Libby observed.\n\n\"That's because it is,\" Amber told her. \"I figured you wouldn't mind.\"\n\nAmber marched over to the tree and started digging. Spadefuls of dirt began to fly. Bernie stepped back to avoid getting hit.\n\n\"Maybe you should slow down,\" Bernie told her as a hole began to appear.\n\nAmber wiped some dirt off her cheek. \"Why?\"\n\n\"Well, whatever is down there might be fragile...,\" said Bernie.\n\nAmber finished the sentence. \"And I don't want to damage it.\"\n\n\"Exactly,\" Brandon said. \"Here. Let me do this.\" And he knelt down next to where she'd been digging and began to scoop the dirt away with his hands while Amber peered over his shoulder.\n\nEveryone moved in closer.\n\nBrandon looked up. \"People,\" he said, \"you have to move back. You're blocking whatever light there is.\"\n\nEveryone moved back half an inch except for Amber. She knelt next to Brandon and began scooping the dirt out with her hands as well.\n\n\"Found something,\" Amber cried after sixty seconds had gone by. Then she corrected herself. \"Sorry. It's a tree root.\"\n\nBrandon grunted and kept digging.\n\n\"Maybe your dream is wrong,\" Libby said to Amber. \"Have you ever considered that?\"\n\n\"I doubt it,\" Amber replied. \"I doubt it very much.\"\n\n\"But how do you know?\" asked Libby.\n\n\"I just do. That's all. Don't you dream?\" said Amber.\n\n\"Not like that,\" Libby lied, trying not to think of her recent dreams.\n\n\"It must be cool to have precognition,\" Bernie observed.\n\n\"Pre...what?\" Amber asked.\n\nBernie was about to explain when she saw Brandon stiffen. Like a hunting dog catching a scent, she thought. \"Did you find something?\" she asked.\n\n\"I think so,\" Brandon replied.\n\nEveryone leaned forward to watch Brandon as he began clearing the dirt away. Amber helped. Their movements became faster. No one spoke, not even Konrad or Curtis. Libby could see their breaths drifting upward in the air. The only sound was the whoosh of the tape deck.\n\n\"I think it's a box,\" Amber said.\n\nBrandon lifted it out.\n\n\"It's definitely a box,\" Bernie said.\n\nAmber got up and brushed the dirt off her leggings. \"Wow,\" she said. \"This is just like a movie. You know, like the Curse of the Maya or something like that. We open the box and the evil spirit comes out and we all die slow, lingering deaths. Wouldn't that be cool?\"\n\n\"Not really,\" Brandon said as he straightened up and carried the box over to the bench in front of the Foundation.\n\n\"But it would be exciting,\" Amber said.\n\n\"Only in the movies,\" Brandon replied as he laid the box down.\n\n\"It looks like an old-fashioned cash box,\" Marvin noted. \"My dad still has one of those in his drawer. He keeps his stamps in it.\"\n\n\"That's exactly what it is,\" said Brandon as he flipped up the latch that held the box closed. He was surprised that it opened as easily as it did. He'd expected he'd need a small crowbar and a can of WD-40 because it would be rusted shut. But it wasn't.\n\nLater, when Bernie recalled the event, she would say, \"It was almost as if Bessie wanted us to open it,\" but right now she was too excited to focus on anything but Brandon opening the box. She leaned in as he opened the lid. A smallish something lay wrapped in a paisley scarf.\n\n\"I guess we're still standing,\" Marvin said. \"So much for the curse.\"\n\nAmber picked a leaf off the hem of her dress. \"Well, there could still be one. Only we won't know about it for a month or so.\"\n\nAmber was about to say more when Libby pointed. \"That scarf looks just like the shawl I'm wearing,\" she cried out.\n\nBernie looked up at her mom's shawl, which was draped over the jacket Libby was wearing, and then back at the scarf in the box. \"You're right. It does,\" she said.\n\nLibby frowned. \"That's very weird.\"\n\n\"It was probably a popular pattern back then,\" Bernie said as she turned her attention back to Brandon while he slowly untied the knots that were holding the scarf together.\n\nThere were three of them.\n\n\"Three is a magic number,\" Amber told everyone.\n\n\"We know,\" Bernie said as Brandon untied the last of the knots. Then he carefully opened up the scarf.\n\nA small leather journal lay in the center.\n\nBernie reached over and took it out of Brandon's hand. It smelled of earth and mold. The brown leather cover was blank. She unwound the cord that bound the book together and gently opened the cover and read the title page. It was printed in a neat hand.\n\n\"This is Bessie Osgood's journal,\" Bernie read. Underneath the title, in larger letters, Bessie had written \"Keep Out\" and had underlined the sentence three times.\n\n\"Wow,\" Libby said to Amber. \"Maybe you were right, after all.\"\n\nAnd then it occurred to Libby that maybe this was what Bessie Osgood had meant when she'd asked Libby to find her. She certainly hoped it was, because then Bessie would leave her alone. Even if Libby hadn't found the diary, she was present when it was found. That had to count for something, right?\n\nBernie put her hand out. \"I felt a raindrop.\"\n\n\"Me too,\" Konrad said.\n\nLibby looked up. The sky was even darker than it had been. She could see the clouds overhead. A raindrop fell on her forehead and ran down her cheek.\n\n\"It's going to pour,\" she predicted.\n\nA second later there was a clap of thunder, and the sky opened up. Everyone ran for their cars.\n\n## Chapter 26\n\nSean sighed as he looked at the tiles he'd just been about to put down on the Scrabble board. The word would have been a twenty-five pointer.\n\n\"Saved by the kids,\" he told Clyde as his daughters, their boyfriends, Amber, Konrad, and Curtis came trooping into the living room. He didn't know what this was about, but it was definitely something big, although getting a twenty-five-point word with three letters was pretty big, too.\n\nClyde snorted and finished the cream-cheese brownie he was eating. \"You'll use any excuse to keep from losing, won't you? I bet you arranged this whole thing.\"\n\nSean leaned forward and tapped the board with his index finger. \"Believe what you want if it makes you feel better,\" he told Clyde as Bernie came over and handed him a leather-bound journal.\n\nHe ran his hand over the cover while everyone crowded around him. They were close enough so that he could smell the damp and the night air on their clothes, which was way too close for him.\n\n\"What is this?\" he asked Bernie. \"You look like the cat that swallowed the canary.\"\n\n\"It's Bessie's journal,\" Bernie said. \"We just dug it up.\"\n\n\"Literally?\" asked Sean.\n\n\"Yes,\" Brandon said and showed Sean his hands. There was dirt underneath his fingernails.\n\nSean raised an eyebrow. \"Did you now?\" he asked Bernie. \"Why don't you fill me in.\"\n\nBernie told him about the puzzle box, the map, and how they dug up the journal.\n\n\"You didn't ask me to come?\" Sean said.\n\nBernie gave an apologetic shrug. \"We didn't want to disturb your game.\"\n\nRight, Sean thought as he turned to Amber, who was looking a little worried about what he was going to say. They just hadn't wanted the old man along. Too much trouble.\n\nBut Sean didn't say that, because that would have been whining. Instead, he told Amber she'd done a good job. But then he went and ruined it all by telling her that she should have come and told him first.\n\n\"But I...,\" Amber protested.\n\nSean held up his hand. \"No need to explain, and I really am very impressed. I couldn't open the box, and heaven only knows I tried.\"\n\nTwo dots of color appeared on Amber's cheeks. She plucked at one of her braids. \"It wasn't anything.\"\n\n\"No, it was,\" Bernie told her.\n\nSean watched as Amber looked down at the floor and dug a little hole in the carpet fibers with her toe. \"Is anything the matter?\" he asked.\n\nAmber paused for a moment.\n\n\"Tell us,\" Sean urged.\n\nAmber looked down and picked a wet leaf off her legging. She hesitated for another moment, and then she blurted, \"The truth is, I didn't figure it out. I got the solution off the Web site of the people who made the puzzle box. I guess I should have told you before, but I kind of liked you thinking I was this really smart person, and I figured that this way you'd let me come along.\"\n\n\"Hey,\" Bernie said. \"You are really smart.\"\n\nAmber studied the toes of her boots. \"No, I'm not. Everyone says that.\"\n\n\"You were smart enough to go online. That was more than any of us did. And we wouldn't have hired you if we didn't think you were intelligent.\"\n\nAmber brightened. \"Really?\" she asked.\n\n\"Really,\" Libby replied.\n\nAmber was about to say something else when Sean interrupted. \"Ladies, enough already with the confidence building.\" He wanted to say \"bullshit\" but managed to restrain himself. \"I want to know what the journal says.\"\n\n\"We don't know,\" Bernie said.\n\n\"What do you mean you don't know?\" Clyde and Sean both said at the same time.\n\n\"Just what I said,\" Bernie snapped.\n\n\"You haven't read it?\" Sean asked. He couldn't believe it. That would have been the first thing he would have done.\n\n\"No, Dad,\" Libby told him. \"We haven't. We wanted to wait till we could read it with you. We figured since you weren't there when we found it, you'd want to be there when we read it.\"\n\nSean ducked his head so no one would see how touched he was by the gesture. \"That's ridiculous,\" he barked. \"You shouldn't have waited for me.\"\n\nBernie grinned. \"I figured that's what you would say.\"\n\nNever could fool her, Sean thought as he handed the journal back to Bernie.\n\n\"Why don't you read it out loud?\" Sean suggested.\n\n\"Good plan,\" Konrad said. \"Then we can play our tapes, too. I bet we got some really good stuff, like maybe Bessie telling us how mad she is that we're reading her journal.\"\n\nNot if I can help it, Sean thought as Marvin said, \"Super.\"\n\nSean glared at Marvin, but Marvin was too busy brushing the water out of his hair to notice.\n\n\"That's what I'm worried about,\" Amber cried out. \"Maybe we should have an exorcism.\"\n\n\"We'll see,\" Libby said as she guided Marvin to the sofa.\n\n\"Because I know an exorcist,\" Amber said. \"I can call him if you like.\"\n\nLibby stole a look at her dad. He didn't look happy. She couldn't imagine what he'd look like if they actually got an exorcist up here. He'd probably shoot him.\n\n\"Maybe later,\" Libby told Amber.\n\nThen Libby sat down next to her father, while Bernie sat in the armchair, and Brandon perched himself on one of its arms. Instead of sitting on one of the wooden spindle-back chairs, Amber plopped herself on the floor, and after a moment Konrad and Curtis sat down next to her.\n\n\"This is pretty exciting,\" Marvin said. \"I wonder what we're going to find.\"\n\n\"The treasure map to the Templar's gold. I'm kidding,\" Sean said quickly before Marvin could say anything else. He nodded toward Bernie. \"Okay. Let's see what this baby has to tell us.\"\n\nEveryone leaned forward as Bernie opened the book and began to read.\n\n## Chapter 27\n\n\"Dear Diary,\" Bernie began. \"Mommy gave this to me because she said it would make things easier. I can put down everything that I would say to her. That way I won't be lonely. I don't want to go to the Peabody School, but Mommy says that Rose says that's a better place for me. She says I'm smart, and I should go to college and do all that stuff that she never had a chance to do. I think I'm going to miss everyone, but Mommy says I'll make lots of new friends. She says that everyone there is really excited I'm coming. I'm taking my teddy with me just in case.\"\n\nBernie stopped reading and looked up at her dad. \"Do you think Rose is Mom?\" she asked.\n\n\"Definitely,\" Sean said immediately. Had to be. It would be too big a coincidence otherwise.\n\nLibby thought back to the scarf the journal had been wrapped in. \"I wonder if Mom gave Bessie that scarf?\"\n\nClyde put down the brownie he'd been eating. \"What scarf?\"\n\n\"This scarf,\" said Libby as she leaned over and took a scarf out of her backpack. \"It has the same pattern as one of Mom's shawls.\"\n\nClyde turned to Sean. \"Any idea on that?\"\n\nSean shook his head. \"I worked a lot in those days. Rose took care of the social end of stuff.\"\n\n\"Did you ever meet Bessie Osgood?\" asked Clyde.\n\n\"I met her mother once,\" said Sean. \"I remember Rose and she were having tea. But that was about it. I was coming back after pulling a fifteen-hour shift.\" He turned to Clyde. \"The Anson robbery.\"\n\nClyde nodded. \"I remember.\"\n\n\"I went straight up to bed,\" said Sean.\n\n\"No wonder Mom never talked about Bessie,\" Bernie mused.\n\nLibby sneezed. She was definitely going to get sick. \"I don't get what you're saying.\"\n\nBernie sighed. \"I'm saying that considering what happened, I bet she felt guilty about being the one that convinced her mother to send her there.\"\n\nLibby bit her cuticle. \"Well, I certainly would have.\"\n\n\"Me too,\" Bernie allowed.\n\nSean nodded at Bernie. \"Read some more.\"\n\nBernie turned back to the book. \"The next entry is dated four days later. Dear Diary, I want to go home.\" Bernie looked up. \"The word home is underlined four times.\"\n\nShe went back to reading. \"No one here is very nice. The food sucks. We've had macaroni and cheese, pineapple, and peanuts four nights in a row. For dessert, we had Jell-O. I asked for ice cream, and the lady in charge of the food said that was a special occasion thing. At home I can have ice cream every night!\n\n\"I called Mom, and she promised to bring me some lasagna and my favorite brownies. The ones she makes with walnuts and tiny marshmallows on top. The only good thing is I like my classes. I get to write stories in English, and we have a class in Greek mythology. I started reading the stories already. The assignment sheet said we should read the first one, but I went ahead and read the first four.\n\n\"My roommate says I'm stupid. But I think she's pretty dumb. She never studies or anything. I guess she doesn't want to be here, either. Amethyst says her mom made her come, too, so that means we have something in common. But I don't think so. She was really mean to the cleaning lady when she came in to wash the floor this afternoon, during free time. I tried to tell her not to do that, but she wouldn't listen to me. I should have told her my mom cleans houses. I don't know why I didn't.\" Bernie looked up. \"I didn't realize they roomed together,\" she said.\n\nSean ate the last of his brownie. \"Me either. Go on,\" he said.\n\nBernie continued on to the next page. \"Not much here,\" she said. \"Bessie's mom dropped by with some homemade brownies and promised she'd be back in two days. They're reading Macbeth in English class, which Bessie loves. She's already memorized act one, scene one.\"\n\n\"I remember doing that,\" Libby said.\n\n\"Me too,\" replied Bernie. \"I guess nothing much has changed. Anyway, she really doesn't like her math teacher, and she has to write a report for social studies on the Revolutionary War. And she says, 'Teddy is feeling very, very sad.' Poor thing. But she doesn't say why.\"\n\nMarvin flicked a drop of water off his sleeve. \"I think I can guess.\"\n\nBernie moved her finger down to the bottom of the page. \"Listen. This is interesting. She says, 'The kids say there are shadow people here.'\"\n\n\"Shadow people?\" Libby echoed.\n\n\"That's another name for ghosts,\" Konrad said.\n\nBernie continued. \"I think I might have seen one, because I saw a lady walking down the hall. She was dressed in funny clothes. The lady that lives in the corner room said not to worry, that it's fine, and that she won't hurt me. I haven't told anyone else, because I don't want them to think I'm crazy.\"\n\n\"She probably saw Esmeralda,\" suggested Curtis.\n\n\"Bad enough to be in boarding school, let alone a haunted one,\" Libby commented.\n\n\"I don't know,\" Bernie commented. \"It could be kind of cool.\"\n\n\"I'm not so sure about the boarding school part,\" Amber said.\n\nLibby watched enviously as Amber straightened her legs out, leaned over and grasped her shoes with her hands, pulled her body down to her knees for a moment, then let go.\n\n\"When my mom was sick, I had to go to my aunt's for a month when I was in high school,\" Amber said. \"It sucked.\"\n\nSean made a hurry-it-along motion with his hand to Bernie. \"Does Bessie say anything else about Amethyst?\"\n\nBernie started leafing through the pages. \"Here's something. I went home for the weekend. I didn't want to leave, so Mom made my favorite cupcakes\u2014white cupcakes with white buttercream frosting and sprinkles on top\u2014to take with me. I put them in my cubby and went to give Mr. Marak, he's the headmaster, the envelope my mom gave me. When I came back, Amethyst and her retarded friends were eating the cupcakes.\n\n\"I told her that wasn't nice, and she started to laugh. I started crying, and she told me she could do whatever she wanted, because no one cares about me. I hate her. Hate her. Hate her. I called my mom and told her, and she said to forget it. She said she'd bake me some new ones.\n\n\"I said I didn't want to stay here anymore, and she said she'd speak to Mr. Marak about getting me another roommate, but I know that's not going to happen, because I heard him say to someone that they were full up. I started crying on the phone and then Mom started crying and that made me feel worse. She said I'd have to make the best of things for now and that I could come home for good if I wanted at Christmas.\"\n\nBernie looked up. \"I wonder what was happening with her mom?\"\n\nSean thought for a moment. \"If I'm not mistaken, I think her dad lost his job. He worked for the power company. And he went to California to help someone he knew put up a house, which meant Bessie's mom had to do more cleaning jobs. Did Bessie say anything to anyone about this?\"\n\nBernie shook her head. \"If she did, it doesn't say so here. Here she says she read The Catcher in the Rye.\" Bernie's finger stopped moving. \"Oh dear. Something happened to teddy.\"\n\n\"What?\" Marvin asked.\n\n\"He disappeared,\" said Bernie. \"Bessie says Amethyst told her the shadow woman took it, but she doesn't believe her....\"\n\n\"Go, Bessie,\" Brandon cried.\n\n\"She thinks Amethyst did it, but she can't prove it,\" said Bernie. \"She says she cried for hours, but she doesn't want to tell her mom, because she doesn't want to upset her even more. Also, she thinks she's fat, and all the other girls are making fun of her because of her clothes. She says, 'Amethyst said I'm fat like a pig and that if I had a circle pin, I would have to wear it on the right side, because no boys want anything to do with me.'\"\n\n\"That's terrible,\" Libby protested, thinking back to her own high school days. No one had said anything like that, but she'd always felt that they'd thought it.\n\n\"Don't kids at boarding schools have to wear uniforms or something?\" Brandon asked.\n\n\"Evidently not here,\" Bernie told him. \"And, anyway, you're missing the point.\"\n\n\"No, I'm not,\" Brandon said. \"I'm just asking a question.\"\n\n\"You know,\" Konrad interjected, \"Bessie said something on tape about how she wanted a white blouse with a Peter Pan collar and a pleated skirt, and how the girls were being mean to her. You want to hear?\"\n\n\"I think we need to finish reading the diary first,\" Sean told him. \"We don't want to mix our media.\"\n\nCurtis frowned. \"What does that mean?\"\n\nSean nodded to Bernie. \"She'll explain.\"\n\n\"It means,\" Bernie said as she marked her place in Bessie's journal with her thumb, \"that we're going to do one thing at a time, and when we're done with this, we'll listen to the tape.\" She looked up and caught her father's eye. \"Tomorrow. Tomorrow we'll listen. We'll be too tired tonight to give it the attention it deserves.\" She went back to reading the journal before Konrad and Curtis could protest.\n\n\"Okay. More trouble with Amethyst. She stole Bessie's favorite pair of socks. They were pink and gray, and her mom had knitted them for her. But she got an A on her math test. She guesses the math teacher isn't so bad. She loves her mythology course and is excited that she's going to take Latin next semester. Her teacher read one of her stories in class and told her she should submit it to the school magazine, which she's going to do. So she's doing well academically, but she still hasn't made any friends, and she really wants to go home.\"\n\nBernie turned the page. \"Here's something.\" And she cleared her voice and began to read. \"We have dance lessons tomorrow in the gym. They're going to teach us all the steps because we have a mixer in two weeks. I don't want to go, because no one is going to want to dance with me, because I'm fat. Amethyst says I'm too clumsy to learn how to dance, anyway. I told the housemother I don't want to go, but she said I had to. Everyone has to. I told her I couldn't learn the steps, and she told me that was ridiculous. Then I told her that no one would dance with me, and she said that that was just plain silly. Lots of the boys liked me, and anyway, everyone had to dance with everyone else. It was the rules. I'm going to run away.\"\n\nBernie stopped reading. \"We never had dance lessons.\"\n\n\"I hated the dances,\" Marvin said. \"My hands used to get all sweaty. It was embarrassing.\"\n\nBrandon stood up and stretched. \"Well, I liked them. I used to sneak out and have a smoke with Daisy Dixon.\"\n\n\"Just a smoke?\" Bernie demanded.\n\nBrandon grinned. \"No.\"\n\n\"That's what I thought,\" said Bernie.\n\nSean cleared his throat. Everyone turned toward him. \"Can we get back to the journal please?\"\n\n\"No problem,\" Bernie said and resumed reading. By now the wind had picked up, and the rain was splattering the windows. \"Let's see.\" She started running her finger down the pages as she scanned them. \"More stuff about classes. She's doing well in everything. More complaints about Amethyst and her friends.\"\n\n\"Do we know who they are?\" Clyde asked.\n\nBernie shook her head. \"So far she hasn't mentioned them by name.\"\n\n\"Pity,\" Clyde murmured as Bernie went back to the journal.\n\n\"Here's another entry about the shadow lady. She says, 'I saw her in the east wing near the kitchen. I'm never going to sneak food again.' That would certainly cut down on midnight snacking,\" Bernie noted as she went on. \"Okay.\" She continued turning pages. \"Same old. Same old. Aha.\" She stopped. \"This is something new.\"\n\nBernie continued reading. \"Dear Diary, We had our first dancing lesson today. I knew I was going to hate it. All the other girls had pretty skirts and blouses on, and I just had my old stuff. Mrs. Richards practically had to push me into the room. And then I kept tripping over my own feet. Why do I have to learn to dance, anyway? No one is ever going to ask me. But the worst was when we had to dance with the boys. We formed two lines, and everyone had to rotate every five minutes or so. All the boys looked unhappy when they had to dance with me.\"\n\n\"Poor thing,\" Libby murmured.\n\n\"And then,\" Bernie continued, \"I danced with the headmaster's son, Ken. He said he had heard a lot about me and that he had wanted to meet me. I thought he was just being polite, but when the dance ended, he said he'd meet me tomorrow by the maple tree outside the girls' wing. He had something he wanted to show me.\"\n\n\"Boy,\" Brandon interjected. \"If that isn't a classic line, I don't know what is.\"\n\n\"Be quiet,\" Bernie said to him. Then she coughed to clear her voice and went on. \"I couldn't sleep all night long. It's probably nothing. His father probably told him to be nice to me. I mean, he's so cute, so why should he pay attention to me?\n\n\"But I couldn't help it, I put on my plaid gray kilt and my good red V-necked sweater and my Sunday loafers and my last pair of clean white kneesocks, anyway. I even used a little bit of Amethyst's rouge on my cheeks. I hope she doesn't find out, but I don't see how she could, because it was just a little dab. Then I tried to get both sides of my hair to turn under the way the cool girls do, but one side kept flipping up. Good grief!\n\n\"I got there five minutes early even though I wanted to get there five minutes late. My mom says it's always best to make the boys wait. Ken wasn't there. I felt so silly standing there that I was going to leave, but then I looked up and saw Ken walking toward me. He was holding something out to me. A book. 'I thought you'd like this,' he said. It was a book of mythology, only this one was about the Celtic people.\n\n\"He said they had lots of stuff about Halloween in there, because that's where it came from. I was so happy, I didn't even say thank you. 'Don't you like it?' Ken asked. I told him I loved it, that that was the best thing anyone had ever given me, and he smiled. He has a great smile. It turns out he likes books, too. He even likes fairy tales, which is fairly weird, but he's going to give me some of the old ones. He says they're different. Also, he said he can help me when I start Latin. We're going to go for a walk tomorrow after dinner. I can hardly wait.\"\n\n\"Looks like things are picking up for our Bessie,\" Marvin said.\n\nBernie grunted as she kept reading. \"Okay. Nothing about Amethyst or being homesick or schoolwork on the next five pages. They're all about Ken. She says, 'Ken gave me a book about Arthur and the Knights of the Round Table today. Ken and I went for a walk, and we found some bright red leaves on the ground. Ken and I held hands. Ken is the cutest guy in the whole school.' Here's one page with Bessie Marak written all over it. Here's another page with little hearts and the initials KM & BO written in them. Looks like our girl is in love. Okay. Here's something. She says, 'Ken kissed me.'\"\n\n\"At last we're getting to the good stuff,\" Brandon said.\n\nBernie clucked. \"Don't you ever think about anything besides sex?\"\n\n\"No. Not really. What does she say about it?\" said Brandon.\n\n\"This. 'Dear Diary, This is the first time I've ever done anything like this except with my cousin Jerome when I was six. I was really worried that Ken's braces and my braces would lock, but they didn't. We kissed for a long time. It was lots better than I thought it would be. We only quit because we heard someone coming. I'm meeting him tomorrow night in the kitchen, and I don't care about the shadow woman, even if she is creepy. I can't believe I made out. I never thought it would happen to me!!!!'\" Bernie paused. \"That was a four-exclamation sentence,\" she noted.\n\n\"Then Bessie goes on and says, 'Amethyst was up when I came back to my room. She was smiling at me, which she never does. Talk about creepy. Anyway, she told me that she thought that Ken was real cute, and that I was very lucky to have a boyfriend like that, and maybe we could all go out together some time. I didn't want to get her angry, so I told her that was a neat idea, even though I don't think it is. I think it's a very bad idea. Then she smiled at me again. I wish I had teddy.'\"\n\n\"Trouble in paradise,\" Libby commented.\n\n\"There always is,\" Bernie observed, thinking back to some of her run-ins with women like Amethyst.\n\n\"Go on,\" Clyde urged. \"Tell us what happened.\"\n\nBernie shook her head to clear it and continued. \"Bessie's next entry takes place the next day. It starts: 'We met in the kitchen. It was really spooky. Lots of shadows, but Ken said I'm just imagining things. He says the shadow people are lots of bunk, so I shouldn't be worried. I'm trying to do what he says, but I keep thinking I'm hearing somebody talking in my ear. I think she's jealous that I'm going to be kissing Ken and she can't kiss anyone. Or maybe shadow people do kiss, and we just don't know it. We did lots of kissing, anyway. Ken says he's going to get me a circle pin, and we're going to go steady. That would be really neat.'\"\n\nBernie looked up. \"However, the next day she says, 'I saw Amethyst talking to Ken at lunch. He was laughing. When I asked him what was so funny, he shrugged his shoulders and said she'd told him a joke and that he didn't think she was as bad as he'd heard she was. I told him she was awful, and he said I was being silly and walked away. When I got back to my room, Amethyst was smirking at me. I told her to leave Ken alone, and she said, good grief, it was a free country, and that she was just talking to him. What was wrong with that? Maybe I'm being silly, but I don't like this at all.'\"\n\n\"I'd say her instincts are pretty good,\" Sean commented.\n\nLibby's stomach rumbled. \"Sorry,\" she said, mortified.\n\nHer dad laughed. \"I think we could all use a little something to eat.\"\n\n\"Cookies would be nice,\" Clyde suggested.\n\n\"More than nice,\" Konrad observed. \"Especially the gingersnaps.\"\n\nThat was the nice thing about baking, Libby thought. It made people happy.\n\nFor the next fifteen minutes, everyone helped themselves to the plates of chocolate chip cookies, gingersnaps, and molasses cookies, and to the decaffeinated tea and coffee that Libby and Amber brought up. Bernie got up, got a bottle of brandy from the bottom shelf of the cabinet, poured a little in her coffee, then passed the bottle around.\n\nKonrad raised his cup. \"A toast to Bessie,\" he cried.\n\n\"To Bessie,\" everyone repeated, and they clinked their cups and drank.\n\n\"To proving my cousin innocent,\" Curtis said.\n\nEveryone drank to that, too, but this time the response was less enthusiastic. Bernie poured another slug of brandy into her cup and passed the bottle around again. Then she took a sip, opened the journal back up to the page she'd marked, and began to speak.\n\n\"Okay,\" she said. \"The next page has more hearts with Ken's and Bessie's initials in them, except down at the bottom, where Bessie's written Amethyst's name and she's drawn a hatchet through it. See?\" Bernie passed the book around.\n\n\"Not a happy camper,\" Clyde commented as he looked at the page.\n\n\"That's because she knows what's coming,\" Amber said, \"and she doesn't know what to do about it.\"\n\nBernie nodded her agreement as Libby handed the journal back to her. Bernie took another sip of her coffee and picked up where she had left off. \"Dear Diary, I hate Amethyst. She says she wants to borrow the book on Norse mythology that Ken gave me. She says it's so interesting, and maybe she and I can discuss it some time. I told her I didn't want to lend it to her, and I guess she told Ken, because when I met him in the kitchen, he asked me why I wouldn't give it to her. I tried to explain, but he said I was just being silly. He said she was just misunderstood, and that she had a really bad mother and lots of problems at home, and I had to be understanding, so I told him about the cupcakes, but he said that just proved his point. We spent all this time talking about Amethyst and almost no time kissing, and when we did kiss, it wasn't very good. I could tell his mind wasn't on what he was doing.\"\n\nBernie turned the page. \"The next day we have, 'I think Ken likes Amethyst better than me. Yesterday I followed them. They didn't see me, because I hid behind the bushes. And they walked down the path that Ken and I walked down. They were both laughing and talking. Amethyst doesn't even like him. He's not her type. And she has lots of boyfriends. She just wants him because I like him. It's so unfair.'\n\n\"New entry. 'Tonight I was supposed to meet Ken and take a walk. I got all dressed up and everything. I even set my hair in the rollers Mom brought me. Then he called on the house phone and told me he was sorry, but he couldn't make it. Amethyst was really upset about her social studies test that was coming up\u2014she didn't understand the chapter\u2014and he had to stay in and help her. I'm so mad I don't know what to do. Amethyst smirked at me when she came in.\n\n\"'My mom says I should just ignore them both. That I'm just playing into Amethyst's hands by doing what I'm doing, and that boys don't like girls that run after them. Then she said that I'm much too young to be in a relationship, anyway, and that I'm at the Peabody School so I can better myself. She just doesn't understand!!!!' That's another four-exclamation sentence.\"\n\nBernie turned the page and read, \"When I got back this afternoon, Amethyst had her friends in our room. They were laughing and drinking and smoking out the window. I think one of them was smoking weed. One of them was even sitting on my bed. I told them they couldn't do that, and they all laughed some more and said they could do whatever they want.\n\n\"I was so mad I stomped out of the room, but on the way, I think I saw something on Amethyst's dresser that said Social Studies final. I'm not sure, but I think I did. I'm going to check when we go to dinner. I was right. I did see it. Oh my God. I bet she stole it. Or maybe Ken gave it to her. That would be too horrible to contemplate. I don't know what I'm going to do! I wish teddy was here.\"\n\n\"Then what?\" Marvin asked as Bernie paused to eat a part of her chocolate chip cookie.\n\nBernie wiped her hands on a napkin before continuing and perused the page. \"Let's see. We have more angst. No more hearts and initials. No more Bessie Marak. Instead, we have, 'Last night Amethyst came in and told me she and Ken kissed in the kitchen, and that they'd done other things too, things that Ken really liked. I started to cry, and she laughed and said he had told her that he went with me because he felt sorry for me because no one else would because I was so fat, and that he doesn't like me at all, and he thinks I smell bad, and all the kids are making fun of me.'\" Bernie stopped reading for a moment and looked up. \"You know, judging by the picture of her hanging on the wall in Amethyst's apartment, Bessie wasn't fat at all. She looked good.\"\n\n\"Yeah,\" Amber interjected. \"But she didn't know that. I bet Ken didn't even say those things. I bet Amethyst made them up.\"\n\n\"Possibly,\" Bernie said. \"No. Make that probably. Poor Bessie.\" Her finger tapped the bottom of the page. \"Her last entry on this page is, 'I called Mom and told her I wanted to come home now, and she said I couldn't, not until Christmas. I told her I had to and she should come and get me right now, and she told me that I had to learn to stand on my own two feet and take care of my own problems, and that I'd thank her later for doing this. Then she hung up. I don't know what I'm going to do.'\"\n\nBernie paused. \"The next entry takes place two days later, which is the night before Halloween,\" she said. \"Here Bessie writes, 'I've thought and thought, and I know what I'm going to do. I'm going to tell the headmaster all about Amethyst. I'm going to tell him about her drinking and smoking and sneaking out and not being in class and about the test and about how I think she stole teddy. Then they'll kick her out, and things will be like they were before. I already told Ken, and I guess he told Amethyst, because she said I'd better not. I told her I would.\n\n\"'Then she said I wouldn't dare. I told her I would do it tomorrow. She was trying on her costume for tomorrow night\u2014she's going as Lucy\u2014and she turned to me and said, \"No, you won't.\" Then she went back to looking at herself in the mirror. I don't know why I'm scared. I'm just being silly. There's nothing else she can do to me. She's already ruined my life, because I'll never get over Ken.'\"\n\n\"She's not going to have the chance,\" Amber said.\n\n\"What does she write next?\" Sean asked.\n\nBernie thumbed through the pages. \"Nothing. She wrote her last entry the evening before Halloween. She died the next day.\"\n\nEveryone fell silent.\n\n## Chapter 28\n\nLibby looked at the clock on the wall. It was a little after eleven, and from the way things were going, she wouldn't get to sleep for quite a long while. She'd been up since six in the morning and had wanted to be in bed an hour ago. The good news was that Konrad, Curtis, and Amber had finally left, which gave everyone a little more breathing space, not to mention giving her ears a rest.\n\n\"I'm glad we don't have to listen to the tape again,\" Libby said. Once was enough. Twice was definitely way too much.\n\n\"I can't imagine why,\" Bernie retorted. \"I love hearing static at ear-piercing levels, don't you?\"\n\n\"Oh yes,\" Marvin said. \"And I'm sure Amber's mom will feel the same.\"\n\n\"Yes, she's going to be very welcoming,\" said Bernie. She could see Amber's mom's prune face now. There was no way that Konrad and Curtis and the tape deck were going to get into Amber's house at eleven o'clock at night.\n\nClyde reached for another chocolate chip cookie. \"A definite case of the emperor's new clothes, if you ask me.\"\n\n\"Well,\" Libby said, stifling a yawn, \"Amber claims she heard Bessie saying, 'That's private. Don't read my journal.'\"\n\nBernie snorted as she slipped off her boots and lined them up next to the sofa. \"First of all, she wouldn't have said journal. She would have said diary. And secondly, you'd think she'd want us to read it.\"\n\nMarvin leaned forward and snagged another cookie from the platter on the coffee table. \"She probably thinks it makes her look bad.\"\n\nLibby turned and stared at Marvin.\n\nTwo dots of color appeared on Marvin's cheeks. \"That's what she'd be feeling if she existed,\" he stammered.\n\n\"But she doesn't,\" Sean said. \"She did, but she doesn't now. Otherwise, we might as well consult the Ouija board. It would be faster.\"\n\n\"My dad has one of those in the attic. I can get it if you want,\" said Marvin.\n\n\"I'm kidding, Marvin,\" said Sean.\n\nSean patted the pocket in his pants where his cigarettes were hiding. He would give anything to have one now. They'd always helped him think. Picked him up and clarified his thoughts. In the old days, he'd tell Rose he was going to answer an emergency call, and then he'd drive a little ways, park on one of the town's side streets, light up, and stare into nothingness. He'd always gotten his best ideas that way. Now, of course, that was impossible, because one or another of the girls was always hovering over him. At least, he thought, I'm well enough now, so I can walk down the stairs if I hold the banister and use a cane. Remission was a beautiful thing, and he was going to do as much as he could for as long as he could.\n\n\"What's the matter, Dad?\" Libby asked.\n\nSean looked at her. \"Why should anything be the matter?\"\n\n\"Well, you were sighing. I just thought you might need something,\" replied Libby.\n\n\"I'm fine,\" Sean lied. Drat those cigarettes. Okay. He knew they were really bad for him, but that didn't help. He could taste them. \"I was just thinking about poor Bessie and wondering who buried the journal.\"\n\n\"I guess she couldn't have,\" Marvin observed.\n\nSean moved his wheelchair so he could be nearer to the cookies. He particularly liked the lace oatmeal ones. They'd been his wife's specialty. At one time they'd been popular, but to his knowledge, his daughters' shop was the only place that made them anymore, mostly as a favor to him, he suspected.\n\n\"No. I don't think Bessie could have,\" Sean remarked. \"Dead people don't usually do things like that.\"\n\nBrandon grabbed another cookie and took Amber's place on the floor. \"So who do you think did, Mr. Simmons?\"\n\n\"My first guess would be Felicity Huffer. I think she found the journal and read it and didn't know what to do with it, so she buried it and drew herself a map so she wouldn't forget where it was.\"\n\nBernie rubbed her feet. She loved the boots she'd been wearing, but if she wore them for too long, they pinched her toes and gave her blisters. \"When I spoke to her, she did say she couldn't tell anyone what had happened, because she was afraid she'd lose her job. She said that Amethyst's parents exerted a lot of influence at the school.\"\n\nMarvin sneezed. \"There's nothing that incriminating in the journal.\"\n\n\"There's enough stuff in there to get Amethyst thrown out of school,\" Bernie countered.\n\n\"Yeah,\" Marvin replied. \"That's true. But I'm talking in the criminal sense.\"\n\n\"Well,\" Clyde said as he wiped cookie crumbs off his leg and into his cupped hand and dumped them in the saucer in front of him, \"I'm not so sure about that. If I had what I thought was an accidental death or a suicide and I read that journal, it would definitely get me thinking in a different direction.\"\n\nMarvin sneezed again. Whenever he got wet, he got sick. \"Then why not just destroy it?\" he asked. \"That would have been lots easier.\"\n\n\"Felicity's conscience probably wouldn't let her,\" Clyde responded. \"This way she could always tell herself that when the time was right, she'd show the journal.\"\n\n\"Which she did,\" Bernie said. \"Never mind that it's a little late to do any good.\"\n\n\"Of course, there's another possibility,\" Sean said. \"Someone else could have buried it. Like Amethyst or Ken Marak.\"\n\n\"And Felicity saw them do it and drew the map,\" Brandon said.\n\nSean nodded. \"Exactly.\"\n\nClyde said, \"This is all very fascinating in the academic sense, and Bessie's journal is very sad, but I don't think it helps us any.\"\n\nBernie looked at her dad. \"What do you say, Dad?\"\n\n\"My gut tells me there's something here,\" said Sean. He pointed to Bessie's journal, which was now resting on the coffee table, next to the cookies. \"I just don't know what.\"\n\n\"I don't see anything that fits with what we already know,\" said Clyde.\n\n\"Clyde, you don't know that for a fact,\" Sean objected.\n\n\"I think I do,\" Clyde insisted. \"The only facts we know for certain are that we have three people with pretty good motives for killing Amethyst, and each one of them had access to the place where she died. We know that the person that killed her had to have had some facility with tools or been some place where they'd seen a fiber-optic laser cutter at work, which actually isn't much of a lead, because they're used at construction sites and in body shops. But most importantly, we know that Amethyst wanted to use Lexus Gardens as the site for a wedding ceremony. Now whether\u2014\"\n\n\"Excuse me,\" Libby interrupted. \"How do we know this?\"\n\n\"I finally got hold of Banks's personal assistant,\" said Clyde.\n\n\"Why didn't you tell us?\" Bernie demanded.\n\n\"We were going to,\" her dad said, \"but you guys came running in here, and we never got the chance.\"\n\n\"Wow,\" Bernie said. \"Married. The ceremony had to be for her.\"\n\n\"That's what I figured,\" Clyde said. \"By the way, Banks refused the request. He didn't want a ceremony, a reception, or anything at his place. He really did like to keep to himself.\"\n\n\"And this ties in with his death how?\" Libby asked.\n\n\"It's a stretch,\" Sean hypothesized, \"but the only connection I can see is that Ed Banks was killed because the person who killed her didn't want it known that Amethyst was getting married.\"\n\n\"That's a big stretch,\" Libby said.\n\n\"It's a huge stretch,\" Sean agreed.\n\n\"Which leaves us in the same position we were in before, with Bob Small as our primary suspect,\" Clyde said.\n\n\"Well, Amethyst certainly wouldn't be marrying him,\" Bernie said. \"He has no money.\"\n\n\"Neither does Inez,\" Brandon observed.\n\n\"Inez is a woman,\" Marvin said.\n\nBrandon just looked at him. \"Really? I hadn't noticed. For your information, Inez goes with whatever moves.\"\n\nMarvin blushed again.\n\n\"Which leaves Zachery,\" Bernie said.\n\n\"But he hated her,\" Libby protested.\n\n\"What better reason to marry and kill her?\" Bernie asked.\n\n\"Someone would have to be really cold-blooded to do that,\" Brandon said.\n\n\"Maybe he is. I've known some psychopaths in my time. They act like everyone else until you get to know them real well,\" said Bernie. She screwed up her face while she thought. \"I think I should go have another talk with him.\"\n\n\"Why? We'd know if they got married,\" Libby said. \"We would have heard.\"\n\n\"Not if it was a secret,\" Sean said. \"Not if they went away and got married somewhere else.\"\n\n\"But we would have heard that they were living together,\" Bernie objected.\n\n\"Maybe they weren't,\" Marvin said. \"Maybe they were living in separate residences.\"\n\n\"Then why bother marrying?\" Libby asked. \"That makes no sense whatsoever.\"\n\n\"It makes as much sense as everything else about this case does,\" said Bernie as she picked up her cup and put it down again. \"Here's another thought. What if Amethyst was getting married to Ed Banks? What if they got married? Then killing him makes a lot more sense. It means that someone needed to kill them both.\"\n\nClyde snorted. \"How do you come up with that?\"\n\n\"What, exactly, did you ask Banks's personal assistant?\" Bernie inquired.\n\n\"First, I asked him if Amethyst Applegate had talked to his employer, and he said she had,\" said Clyde. \"Then I asked him how he knew, and he said she'd come up to the estate, at which point I asked him what the conversation had been about. He said he heard his employer saying that he didn't want the reception held here, and then Banks shut the door, and he couldn't hear anything else.\"\n\nBernie leaned forward and pointed a finger at Clyde. \"So Banks and Amethyst could have been talking about holding a reception for their wedding on the estate grounds. I mean, it is a possibility.\"\n\n\"I guess if you put it that way, then yes,\" Clyde conceded.\n\n\"Would there be a record of the wedding in the town hall or somewhere like that?\" Brandon asked.\n\n\"For the fourth time, not if they didn't get married here,\" Sean said. \"If we're following Bernie's scenario, then it's just as likely that they hopped on a plane to Reno or Vegas, got hitched, and flew back the next day. What do you think, Clyde?\"\n\n\"I think Bernie should have been a lawyer, that's what I think,\" said Clyde.\n\nBernie stood up and took a mock bow.\n\n\"But I'm still sticking with what I said before, and all the fancy logic in the world can't convince me otherwise,\" Clyde told her. \"What do you think, Libby?\"\n\n\"I'm too tired to think,\" replied Libby.\n\n\"Me, too,\" Marvin said.\n\nLibby turned and looked at him. His eyes were like slits. She was just about to tell him he should go home when her father beat her to it.\n\n\"Get some sleep,\" Sean said to Marvin. \"We have a busy day tomorrow, and we need to start bright and early.\"\n\nClyde stood up, too. \"That goes for me, too.\"\n\nLibby and Bernie walked everyone down the stairs and said good night. After Bernie locked up, she turned to Libby and said, \"This is going to sound crazy, but I'm thinking about having a little work done.\"\n\n\"Work done?\" Libby echoed. \"What kind of work?\"\n\n\"Cosmetic surgery kind of work.\"\n\n\"Are you nuts?\" Libby asked.\n\n\"But see\"\u2014Bernie pointed to her forehead\u2014\"I'm beginning to get lines here.\" She pointed to the area between her nose and chin. \"And here. And look at the circles under my eyes.\"\n\n\"You are nuts,\" Libby told her.\n\n\"Look closer.\"\n\nLibby did. \"I still don't see anything.\"\n\n\"That's because the light down here is bad.\"\n\n\"You're having surgery?\"\n\n\"Well, I'm hoping Botox will take care of everything for awhile. And, anyway, it's not surgery. It's maintenance.\"\n\n\"I don't care what you call it. It's still injecting a deadly toxin into your body.\"\n\n\"It's not a big deal. Millions of people do it, and I think there's something new on the market. That's why I have a consult tomorrow morning.\"\n\n\"Tomorrow morning?\" Libby cried. \"Why tomorrow morning?\"\n\n\"Because they had a cancellation. Otherwise, I'd have to wait four months.\"\n\n\"Maybe you should. It'll give you time to think about it.\"\n\n\"I've been thinking about it for almost a year,\" said Bernie.\n\n\"You never mentioned it to me.\"\n\nBernie shrugged. \"That's because I knew what you'd say.\"\n\n\"Do you know how busy we're going to be?\"\n\n\"It's always about you.\"\n\nLibby rolled her eyes. \"I have an idea,\" she said. \"Why don't we grow a little botulinum here, and you can do it yourself.\"\n\nBernie ignored her. \"I've got the first appointment. It's at eight. I'll be in and out of there in half an hour at the most, and then, if there's time, I'm going to drop in on Zachery, and if not, I'll come right back to the shop.\"\n\n\"Have you told Dad about what you're going to do?\"\n\n\"I'm not doing anything. I'm going for a consult,\" said Bernie.\n\n\"Fine, Miss Have-To-Say-It-Exactly-Right. Let me rephrase. Have you told him about what you're thinking of doing?\"\n\nBernie snorted. \"What, are you crazy? He doesn't even like it when I change my hairstyle.\"\n\n\"I bet Brandon won't be too pleased, either.\"\n\n\"He's not going to know. No one is.\" Bernie fixed her eyes on Libby. \"And you're not going to tell them, either. Right?\"\n\nLibby studied the hallway light fixture.\n\n\"Promise me you won't say anything,\" said Bernie.\n\nLibby folded her arms over her chest.\n\n\"I mean it.\"\n\n\"I know you do.\"\n\n\"Well?\" Bernie said after a moment had gone by.\n\n\"Fine,\" Libby said sullenly. \"I swear. Satisfied?\"\n\n\"Sister swear.\"\n\nLibby groaned. Sister swear was an unbreakable oath, the most serious oath there was between them. She might have known Bernie was going to pull this out of the proverbial hat.\n\n\"I mean it,\" Bernie said.\n\n\"All right,\" Libby said. \"I sister swear it. But I think you're making a big mistake.\"\n\nBernie gave her a hug. \"I know.\"\n\nTheir dad barely looked up when Bernie and Libby came upstairs. He was too engrossed in reading Bessie's journal.\n\n\"Do you want me to put the Scrabble game away for you?\" Libby asked him.\n\nSean shook his head. \"No. Leave it. I'm not done with it yet.\"\n\n\"Some more tea?\" asked Libby.\n\n\"No, Libby. I'm fine, honestly,\" said Sean.\n\nBernie leaned over him. \"Are you getting any ideas?\"\n\nSean looked up at her and smiled. \"Actually, I am.\"\n\n\"You want to tell us what they are?\" asked Libby.\n\n\"Not yet. Maybe tomorrow,\" said Sean. He went back to reading.\n\nLibby and Bernie stood there and watched him. After a moment, they both gave him good-night pecks on his cheeks and went off to sleep. Even Bernie was tired. The whole thing with finding Bessie's journal must have taken more out of her than she thought.\n\n## Chapter 29\n\nIsn't this just the way? Libby thought bitterly as she stared at the clock on her nightstand. She desperately needed to go to sleep. She was so tired, she couldn't keep her eyes open, but now that she was in bed, she couldn't fall asleep. All she could do was stare at the numbers on the clock face: 2:30, 2:31, 2:33. She was now down to three hours of sleep before she had to get up. At this rate, she might as well get up and go downstairs and start baking. At least she'd be getting something done.\n\nShe pulled the comforter up and rearranged her pillow. Maybe she needed to get a new pillow. Then she turned over and stared out the bedroom window. The rain was smearing everything. The street lights outside looked blurry, and the houses across the street had disappeared behind a watery film. A branch of the maple tree she and Bernie used to climb up when they were young had twisted itself into a face.\n\nFunny. Why hadn't she noticed that before? It had a mouth and eyes. And a nose of sorts. Nostrils, actually. She blinked. It must be a trick of the light. Then she blinked again, because she recognized the face. It was Bessie's. Bessie was tapping at her window. The strange thing was, she wasn't even scared. She was just annoyed. Very odd.\n\n\"Go away,\" Libby told her. \"Go bother someone else for a change. I've had enough.\"\n\nBessie looked as if she were going to cry. \"But I like you.\"\n\n\"Well, I don't like you.\"\n\nBessie stamped her foot. \"You are so mean.\"\n\nThen, before Libby could answer, Bessie was floating above her, around her. She was everywhere Libby looked. Something wet was falling on Libby's face. Bessie's tears. They were coming faster and faster. She was getting soaked. She felt as if her bed was floating. She looked down. Water was filling her room. It was carrying the bed up toward the ceiling. She was going to drown.\n\n\"I'm sorry,\" Libby cried. \"I didn't mean it. I'm tired.\"\n\n\"You swear?\"\n\n\"I swear,\" Libby said. At least it wasn't sister swearing. All at once the bed fell, and she was on the floor again.\n\n\"We found your journal,\" she told Bessie. \"We dug it up. Isn't that what you wanted to happen? Don't you want everyone to know what's going on?\"\n\nSuddenly, Libby found herself back at the Peabody School. The windows were decorated with cutout witches and pumpkins. Jack-o'-lanterns hung from the trees. Paper skeletons danced on the doors, while students in werewolf masks stalked the hallways. It was the night before Halloween, and she was watching Bessie talk to Ken.\n\n\"I don't understand,\" Bessie was saying. \"I don't understand why you're doing this.\"\n\n\"I'm not doing anything,\" Ken said.\n\nLibby could tell from the way Ken was standing that he was clearly impatient with Bessie. Somehow Libby knew that this discussion had been going on for a half an hour and that Ken was more than ready to leave. And with every word that Bessie said, his interest faded a little bit more.\n\n\"But you are,\" Bessie insisted. \"You're helping Amethyst cheat.\"\n\n\"I didn't know she was going to take the test,\" Ken insisted.\n\n\"Then tell your dad.\"\n\n\"I can't.\"\n\n\"Why not?\" asked Bessie.\n\n\"Because I can't, that's why.\"\n\n\"You can't, because you like her.\"\n\n\"I don't like her at all.\"\n\n\"Yes, you do.\"\n\nJust let it go, Libby wanted to tell Bessie. Just walk away. He still likes you. He likes you better than Amethyst, even though he doesn't know it yet. He'll come back to you. But Libby couldn't get the words out of her mouth.\n\n\"I'm going to tell your dad,\" Bessie declared.\n\n\"Bessie, don't,\" Ken pleaded.\n\n\"I am,\" Bessie said.\n\nWhy does Ken look so familiar? Libby wondered as she watched letters come out of Bessie's mouth and form themselves into words above her head. Just like in the cartoons, Libby thought. She was trying to read them, but the rain kept on getting in her eyes. Now the letters had turned into Scrabble tiles and had fallen to the ground. They were forming themselves into two words. A name? A place? Whatever it was, it was important. It was more than important. It was crucial. Libby leaned over to get a better look, and suddenly, she was falling, down, down, down. The ground was coming up fast, but she was still trying to see the words. Then, just before she hit the ground, a name flashed through her mind.\n\nLibby woke up with a start. She was covered in sweat, and her heart was hammering so hard, she couldn't catch her breath. It took her a second to realize she was in her bedroom. She sat up.\n\n\"It was a dream,\" Libby said out loud. \"Just a dream.\"\n\nBut it hadn't felt like that at all. It had felt real. This was what she got for going along with Amber's nutty schemes and eating too much ice cream. Whenever stuff like this had happened in the past, her mom had always told her it was because of something she'd eaten before bedtime.\n\nShe wanted to believe that now, but even then she had known it wasn't true. Despite what her mom had said, these dreams were different. And Libby thought that her mom had known it, too. She just hadn't wanted to talk about it, the way she hadn't wanted to talk about other stuff. Libby still remembered when she'd had the dream about her grandmother dying right before she did, and she'd gone and told her mom. After the funeral, her mom had told her not to say anything to anybody about stuff like that.\n\n\"People will think you're weird,\" her mom had warned her. And even though Libby had only been nine at the time, she'd understood that her mom had meant \"more weird,\" and she'd never told another living soul about her dreams\u2014except for Bernie\u2014ever again. Eventually, they'd stopped happening. This was the first one she'd had in a long time, and now that she'd had it, she remembered why she'd hated those dreams. They totally freaked her out.\n\nLibby pulled her comforter, which wasn't supplying much comfort, over her head and told herself to go back to sleep. Instead, she spent the rest of the night tossing and turning and trying to remember the name the Scrabble tiles had formed just before she fell, but the name kept skittering away from her, hiding in the recesses of her mind.\n\n## Chapter 30\n\n\"And I thought I had circles under my eyes,\" Bernie said when she saw Libby the next morning.\n\nLibby grunted. She was on her third cup of coffee. Usually, she had just one, but this morning was a notable exception.\n\n\"Maybe you should go as a bag lady tonight,\" Bernie suggested. \"You know, bags under the eyes, bags\u2014\"\n\n\"Ha-ha. I get it. You don't have to explain. You're really going to do this?\" Libby asked.\n\n\"This\" was the cosmetic surgery consult. Bernie looked fine to her. She looked more than fine. The truth was Libby would have killed to look like her sister. Bernie had gotten the good nose, the high cheekbones, and the clear complexion.\n\nBernie nodded. \"You betcha.\"\n\nLibby took another sip of her coffee. Under other circumstances, she would have tried to talk Bernie out of going, but she was too preoccupied with last night's dream to bother arguing that cosmetic surgery was dangerous and a total waste of money.\n\n\"Good luck,\" Libby said and turned back to survey the to-do list she had tacked up on the fridge.\n\nWhen Bernie told Libby she was going to pay a visit to Zachery Timberland after her appointment, Libby merely nodded, because she was trying to figure out, as Julia Child would have said, the order of battle for the day.\n\n\"Are you sure you're all right?\" Bernie asked her. Given everything that had to get done, her sister should be in a frenzy by now.\n\nLibby kept her eyes glued to the list on the fridge. \"Why are you asking?\"\n\n\"Well, ordinarily, I'd think you'd tell me to come right back to the shop. After all, we're going to be pretty busy today.\"\n\nLibby turned and faced her. She had a grim look on her face. \"Why?\" she said. \"Just because it's Halloween night, and we have that dratted Haunted House to deal with, not to mention three large pickup orders between four- and five-thirty, and we're already behind because the chicken hasn't been prepped, and we need to do the onions and the peppers, not to mention hollowing out the pumpkins for the pumpkin and apple soup, as well as all the muffins, pies, cookies, and cakes that still have to be made?\"\n\n\"Something like that. I'll call Amber and Googie.\"\n\n\"I already have,\" Libby informed her. \"They'll be here in half an hour.\"\n\nBernie reached for a semi-stale pumpkin bar, one of several that they'd forgotten to put away last night, broke off a piece, and ate it.\n\n\"You know,\" she said. \"I think I like these better this way. They have more texture. We might even be able to incorporate them into the brownie bars.\"\n\nLibby grunted. Bernie hated when her sister got this way.\n\n\"Well,\" Bernie continued, trying to be positive, \"at least everything will be over tonight.\"\n\n\"Will it?\" Libby asked. \"I don't see how.\"\n\n\"Why won't it be?\" Bernie replied, and then she realized what Libby was really talking about. \"You mean the case?\"\n\nLibby nodded. \"Yes. What were you talking about?\"\n\n\"Our catering gig.\"\n\nLibby reached over and grabbed one of the pumpkin bars and nibbled on it. \"Not bad,\" she conceded. \"As for the Haunted House, if I never see that place again, it will be too soon for me.\"\n\nBernie looked at her carefully. Something was definitely going on behind those bangs and glasses. \"Did something happen last night after I went to bed?\"\n\nLibby shook her head. \"Like what?\"\n\n\"I don't know.\"\n\n\"Then why are you asking?\"\n\n\"Because you seem kinda weird this morning.\"\n\nLibby put her hands on her hips. \"So now I'm weird? Thanks a lot. That's certainly helpful.\"\n\nBernie gave an exasperated sigh. \"You know what I mean.\"\n\n\"No, I don't.\"\n\n\"I mean weird as in off, as in something is bothering you.\"\n\n\"Well, I'm fine.\"\n\n\"Are you sure?\" Bernie asked her, noting that Libby was looking at everything but her.\n\n\"Positive. Aren't you going to be late for your appointment? Can't keep the plastic surgeon waiting, can we.\"\n\n\"It's a consult.\" Bernie ate the rest of the pumpkin bar while she thought. Something was definitely wrong. \"You didn't have a fight with Marvin, did you?\"\n\n\"Marvin and I are fine.\" Libby reached for another pumpkin bar and practically inhaled it.\n\nBernie studied her some more.\n\n\"Stop looking at me,\" Libby told her.\n\nBernie snapped her fingers. \"I know. You had one of your dreams, didn't you?\"\n\nLibby blushed.\n\nBernie scrutinized her for a moment longer. \"You did, didn't you?\"\n\n\"No, I didn't.\"\n\n\"It's not a bad thing,\" Bernie said gently.\n\n\"Maybe the dreams aren't bad to you,\" Libby flung back. \"But that's easy for you to say because you're not the one that has them. You might feel differently if you did.\" She turned back and began studying her list with a great deal of intensity.\n\n\"So you're against miscegenation?\" asked Bernie.\n\nLibby turned back to her. \"What are you talking about?\"\n\n\"You believe there shouldn't be any mixing.\"\n\n\"Mixing?\"\n\n\"You know. The dead should stay in their world, and the living should stay in theirs.\"\n\nLibby burst out laughing. \"I guess I do.\"\n\n\"So you're a deathist.\"\n\n\"Deathist?\"\n\n\"Deathist. Like a racist. You're prejudiced against dead people.\"\n\nLibby laughed harder. \"You're nuts,\" she said when she could talk again.\n\n\"I know.\"\n\nWhen Bernie went out the door, Libby was still laughing, which somehow made Bernie feel better.\n\nIt was so ironic, Bernie thought as she slid into her car. Libby had a gift that some people would give anything to possess, and she refused to have anything to do with it. But wasn't that the way it always was. Bernie, for example, usually wanted men that were totally wrong for her, while she was uninterested in those that were suitable. Fortunately, Brandon was turning out to be the exception to that rule. The operative words here were turning out, because, if truth be told, she still didn't entirely trust him yet.\n\nBernie sighed as she turned on the ignition. It was going to be a long day, and she didn't enjoy starting it this way, but you had to go with what you could get in the appointment department. The drive over to the doc's office took five minutes. Since she was booked for the first appointment of the day, her behind hadn't even had a chance to hit the expensive leather-covered seats, the ones that she'd seen in an architectural magazine and that cost five thousand dollars each, before she was called in. Fifteen minutes later, she left, armed with a price list and an extensively illustrated brochure that described all the procedures Dr. Cornelius Love did.\n\n\"Is that your real name?\" Bernie had asked when she'd walked into his office.\n\nDr. Love had laughed and turned the conversation back to her, which probably meant that it wasn't. Having that last name in school would have been...well, it would have been awful. She was still thinking about Dr. Love as she thumbed through the brochure. The words that came to mind as she did were \"yuck\" and \"blech,\" although \"gross\" was fitting as well. Bernie didn't consider herself squeamish, but looking at the color photographs made her want to throw up.\n\nMaybe Libby's right, Bernie thought as she put her car in drive and headed over to Zachery Timberland's house. Maybe I should forget about this. On the other hand, those lines on her forehead and the folds between her nose and her chin were only going to get worse. It was better to do little maintenance jobs over time than to do a face-lift when she got to fifty. Those really did look awful unless they were done by someone very, very good, and even then you couldn't be sure. Look what had happened to Joan Rivers.\n\nAnd the doc was kinda cute\u2014not that that was a good reason to choose him to do this. But it certainly wouldn't hurt. God. Why did everything have to be so complicated? Maybe she'd try microdermabrasion first, and if that didn't get rid of the lines, she'd go the Botox route. Who knew? By then they might have even come out with a new product.\n\nWith that settled, she gradually started thinking about Bessie's journal again and about Amethyst and whether or not she'd gotten married. She was definitely playing a long shot here, but she was running out of ideas.\n\nShe had half a mind to turn around and go back to the shop and start making the dough for the pies, but the way things were going, Bob Small was going to go to jail for Amethyst's murder for sure\u2014unless she, Libby, or their dad came up with something. And if her dad did what he wanted to do\u2014which was to demonstrate that Marvin could crawl in and out of the ceiling in the Haunted House\u2014that would be another nail in Bob Small's metaphorical coffin. But fortunately, she'd pretty much convinced her dad to leave things alone. For the moment.\n\nWhat she'd said was, \"Why help Lucy?\" And her dad had agreed. Marvin had been incredibly relieved. And she was happy, too, because the more she knew about Bob Small, the more unlikely it seemed to her that Bob had killed Amethyst. Like Libby, she just couldn't imagine him doing something like that. Shooting someone in a fit of rage, yes; but something that took lots of advanced planning, no.\n\nBob Small seemed so clueless, which, of course, was why he'd gone to jail in the first place. If he wasn't clueless, he would have been able to see through Amethyst. Or, to be perfectly accurate, he wasn't so much clueless as thinking with his dick\u2014but maybe that was the same thing.\n\nThis case was like a huge, tangled skein of wool. Every time she, Libby, or her dad thought they were making headway, they ended up in the same place they had been before. Which was why she was going to Zachery Timblerland's house at 8:35 in the morning. He probably wasn't even there. He most likely was on his way to his office. But it was worth a try. As her father always said, \"When in doubt, do something. Don't sit around like a lump.\"\n\nMaybe her visit would shake Zachery up a little. Hopefully. She'd read a statistic somewhere that most homicides were solved within the first seventy-two hours, or they were not solved at all. Well, they'd certainly gone past the seventy-two-hour limit, that was for sure. And with that in mind, Bernie turned onto Smith Street and took that straight to Zachery Timberland's house.\n\nHe lived in an odd little cul-de-sac off of Meadview Drive. The street wasn't marked, because someone had driven into the street sign last year, and the town had never replaced it. You really had to know it was there to be able to find it.\n\nAt one time the cul-de-sac had been the site of a proposed development. Five houses were supposed to have been built on the site, but the builder had gone bankrupt after he'd put up Zachery's. Since Zachery had already purchased his, he was stuck because no one wanted to buy him out. Other developers had bid on the site, but for one reason or another, none of the deals had gone through, so Zachery's house sat in lone splendor on what was now an extremely large vacant lot.\n\nDespite the DO NOT LITTER. VIOLATORS WILL BE FINED TWO HUNDRED DOLLARS sign, piles of leaves, Styrofoam peanuts, and crumpled-up pages of newspaper littered the grass. Three kitchen chairs sat over to the far left, waiting to be claimed by the winter. A little farther on, someone else had disposed of two bags of trash, which were now spilling their guts out onto the ground.\n\nBernie shivered. You could do whatever you wanted here, and no one would ever know. It was not a comforting thought. Even if she had a couple of big dogs, she wouldn't want to be living here, Bernie decided as she turned onto Dewdrop Lane. It was too spooky. She preferred to live close to people rather than to be isolated. She shook her head. Halloween was getting to her. More to the point, she realized, was the fact that there were no neighbors she could talk to.\n\nShe was relieved to see that there were two cars in the driveway of the Timberland residence. The first one, she knew, belonged to Timberland, because she'd seen it parked behind his office. She wondered to whom the second vehicle belonged as she pulled in behind it, got out of her car, and marched toward the front door. This is going to be interesting, she thought while she rang the bell.\n\nThe house itself was a nondescript, generic Colonial, the housing equivalent of a pair of Wal-Mart blue jeans. It was rendered even more so because everything on it was painted beige except for the dark brown front door. Just like his office, Bernie thought. Either the man had a severely limited color palette, or he'd gotten a real good deal on the paint.\n\nAt least the foundation plantings weren't beige. But they were tightly pruned, and the driveway was immaculate. The word constipated came to mind. She rubbed her arms while she waited for someone to come to the door. It had turned out to be way colder than the weatherman had predicted. The kids were going to have to wear jackets over their costumes tonight when they went trick-or-treating.\n\nShe was remembering how much she'd hated that when she heard Timberland say, \"Coming.\" A moment later the door swung open. Bernie could see he was not a thing of beauty in the morning. He had stubble on his chin. His belly was hanging out of his sweatpants, and he had man boobs, a fact his suits had managed to hide. Her mom had always said it was amazing what a decent tailor could do, and she'd been right. Zachery Timberland had a very good one.\n\n\"You,\" he said when he saw her. \"What do you want?\"\n\nThen, before Bernie could reply, she heard another voice. It was a young woman. Probably his girlfriend from the sound of it. Bernie cursed under her breath. It looked as if this trip was going to be a waste of time, after all.\n\n\"Who is it?\" the unknown woman trilled.\n\n\"No one,\" Timberland shot back.\n\n\"No one? How rude,\" Bernie countered. \"I'm sure your company wouldn't like to hear how you treated a potential customer.\"\n\n\"Oh sure,\" Timberland sneered. \"You're just so excited about the prospect of buying insurance from me that you had to come straight to my house.\"\n\n\"And who's to say that isn't the case?\"\n\n\"And I'm the queen of Sheba.\"\n\n\"I didn't know you were a woman. But now that I look more closely, I can see the beginning of boobs. How are the hormones working out?\"\n\nTimberland looked down at his chest, realized what he was doing, and looked back at Bernie. He brought his lips back into something that was supposed to resemble a smile. \"You need to be taught some manners.\"\n\nBernie opened her mouth to reply, but before she could, Timberland's guest came into view. \"Honey, what do you mean no one?\" she asked.\n\nBernie estimated that she was about half Timberland's age. She had long blond hair, which she'd put up in a ponytail, and the kind of glowing, flawless skin that Bernie now realized came from dermabrasion, as well as a killer body. The loose pajama bottoms decorated with pictures of cows emphasized her slender hips, and the forest green cami covered breasts that stood at attention without any visible means of support.\n\nBoob job. Got to be, Bernie thought as she turned her eyes away, but not before the young woman saw her looking.\n\nThe young woman smiled. She had perfect, white, even teeth. She pointed at her chest. \"He bought my boobs for me,\" she chirped, motioning to Timberland with a toss of her head. \"I think the surgeon did a very nice job, don't you? I was going to go down to Mexico to have them done, but Zachy insisted I go somewhere first rate.\"\n\nBernie raised an eyebrow. \"Zachy?\" she repeated.\n\nTimberland glared at her.\n\n\"Well,\" the young woman continued, undaunted, \"if you ever need a little pick-me-up, call and I'll give you my surgeon's name. His office is on Park Avenue.\"\n\n\"Sadie,\" Timberland wailed.\n\nSadie made a little moue with her lips. \"It's not like it's a secret or anything.\" Then, as Bernie watched, she came up next to Timberland and patted him on the shoulder. \"The poor thing is grumpy in the morning without his coffee,\" she confided. She stood on her tiptoes and gave Timberland a peck on his cheek. \"But,\" she said to Bernie, \"he is cute, so I forgive him.\"\n\nBeauty is definitely in the eyes of the beholder, Bernie thought as she watched Timberland open his mouth and close it again. Bernie could see he was delighted with Sadie's attention on the one hand and mortified on the other.\n\nSadie extended her hand to Bernie. \"Hi,\" she said. \"I'm Sadie Palogski. But then you already know that.\"\n\n\"Sadie's a nice, old-fashioned name,\" said Bernie.\n\n\"I think so. Of course, it's not my real name,\" said Sadie. \"My real name is Scarlett. My mom is a huge Gone With the Wind fan. But Scarlett is really lame. All those hoop skirts and that fainting.\" She wrinkled her nose. \"And I don't even like the color.\"\n\n\"I can see why you'd prefer Sadie,\" Bernie told her.\n\n\"I'm going to court to change it,\" Sadie declared. \"Of course, my mom's pissed, but she'll get over it. She always does.\"\n\nTimberland interrupted. \"What do you want?\" he asked Bernie.\n\nBernie smiled. After watching Sadie and Timberland together, she'd decided to tell the truth. She had nothing to lose. Plus, she was curious to see what Sadie's reaction was going to be.\n\nBernie turned to Timberland. \"I came to see if you had married Amethyst.\"\n\nTimberland just stared at her.\n\n\"I guess not,\" Bernie said.\n\nIt took another minute before Timberland recovered himself. \"You're kidding, right? That's the stupidest thing I've ever heard.\"\n\n\"Her being married or married to you?\" Bernie asked.\n\n\"Both,\" said Timberland.\n\nBernie turned to Sadie, who had burst out laughing. Laughing was not a reaction she had expected. \"Pretty silly, huh?\"\n\n\"Well, yeah,\" Sadie said. \"How did you ever get that idea?\"\n\n\"Just sprung into my head,\" replied Bernie.\n\n\"I mean Zach would never marry her, not after what she did with his daughter, and she really wasn't too fond of him,\" said Sadie. She looked at Timberland. \"Well, it's true. She wasn't. She told me you were a real asshole.\"\n\n\"So obviously, you knew her,\" said Bernie.\n\n\"I knew who she was, and we said hello, but that was about it. I didn't know her, if you get my meaning,\" said Sadie. She looked at Bernie, and Bernie nodded to show that she understood. \"I ran into her down in the City when I was getting my consult at this really cool lingerie shop down in SoHo. It has the best things.\" She mentioned the name of the store, one Bernie knew from reading the fashion magazines. \"She told me she was getting married.\"\n\nBernie leaned forward. \"Really?\"\n\nSadie nodded.\n\n\"When?\" asked Bernie.\n\n\"Pretty soon. She was very excited. I've never seen her so excited,\" said Sadie. \"And she was buying all this cool new underwear. Lace bras. Thongs. The whole bit. She showed me her diamond. Five carats.\" Sadie gave Timberland a meaningful look.\n\nTimberland, I hope you have lots of money, Bernie found herself thinking, because you're going to need every cent of it with this one. The thought pleased her.\n\n\"So, Sadie,\" Bernie said. \"Did she happen to mention who she was marrying?\"\n\n\"Yeah, she did. But I've forgotten the name,\" said Sadie. \"I was kinda half listening because I was looking at some really hot teddies at the time. I didn't want to be rude, but I didn't have lots of time, because I was going to have to make the four thirty-seven back to Longely.\"\n\nBernie took a deep breath and told herself to stay calm. \"Do you remember anything?\"\n\n\"Well, it was her old boyfriend. She'd just met up with him again,\" said Sadie.\n\n\"Anything else?\" asked Bernie.\n\n\"Actually, it was more than an old boyfriend. It was the first boy she'd ever kissed,\" replied Sadie. \"Isn't that sweet?\"\n\n\"Very,\" Bernie said. \"By any chance was his name Ken Marak?\"\n\nSadie clapped her hands together. \"That's it,\" she squealed. \"How did you know?\"\n\n\"I'm psychic,\" Bernie replied.\n\nWhen she was back in her vehicle, Bernie got out her cell and hit her dad's speed-dial number. He didn't pick up. Bernie wondered what the point of having a cell was if you always forgot to take it with you. She tried Libby next.\n\n\"You're not calling to tell me you want another two hours, are you, Bernie?\" Libby said when she answered.\n\n\"You could at least say hello. And no, I'm not,\" Bernie replied. \"I'm calling to tell you something much more interesting.\"\n\n## Chapter 31\n\nThe dining room at the Haunted House was draftier than usual, Libby decided as she listened to what her sister was saying. Even with wool socks on her feet, they were beginning to feel like Popsicles.\n\nLibby looked up from the cookies she was arranging on a platter. \"I think you're nuts,\" she said to Bernie when she'd finally paused to take a breath. \"One hundred percent certifiable.\"\n\nBernie dipped the knife she was using to cut the pumpkin cheesecake in a bowl of water, dried it with a towel, and went back to cutting. \"No, I'm not.\"\n\n\"Talk about cobbled together,\" Libby said as she put the platter she'd just finished next to the chocolate cupcakes with orange frosting. \"You don't have a lick of proof. This is all speculation and hearsay.\"\n\n\"How can you say that given your dream?\"\n\n\"That's why I'm saying it. This is like saying that Konrad and Curtis's tapes are real.\"\n\n\"Maybe they are. And, anyway, it's not just the dream. It's the plastic surgery....\"\n\n\"More speculation...\"\n\n\"But it works in conjunction with everything else,\" said Bernie.\n\nLibby sighed and started arranging the cinnamon spiced shortbread cookies on a second platter. It was enough already. She and Bernie had been going at it since this morning. She didn't want to talk about it anymore. At least she thought she didn't, but before she knew it, words were spilling out of her mouth again.\n\n\"I really am sorry I told you about my dream. I should have kept my mouth shut.\"\n\n\"You shouldn't be sorry,\" replied Bernie.\n\n\"Well, I most definitely am.\"\n\nLibby shut her eyes for a moment and opened them again. Nope. Everything was still the same. If she could only keep her mouth shut, half of her problems would be solved. When Bernie had called her on her way back from Timberland's house, she'd been in the middle of writing \"Happy Halloween\" in a nice cursive script on a chocolate frosted yellow cake.\n\nShe'd been admiring the n she'd just made and thinking that her fourth-grade teacher would have been proud of her\u2014she'd always had the best handwriting in the class\u2014when all of a sudden it had hit her. Maybe writing \"Happy Halloween\" had jogged her memory, or maybe it had been hearing the name Ken Marak, or maybe it had been having them both happen at the same time\u2014Who knew?\u2014but she had suddenly seen the letter n that had been in her dream. The a had appeared next. Then she'd seen the rest of the tiles.\n\nShe'd been so excited that when Bernie had walked in the kitchen, she'd blurted the name out before she could stop herself. She should have kept quiet because, as she could have predicted, Bernie had come up with this incredibly stupid, half-baked idea, which Libby had been trying to argue her out of ever since.\n\nBernie dipped the knife blade back in the water and wiped it off again. That was the problem with cheesecake, Libby thought. It was hard to make good, clean cuts, especially when the cheesecake was at room temperature. Cheesecakes were much easier to slice when they were cold, but she hadn't wanted to take the chance of cutting them in the shop and having them fall apart.\n\n\"Okay, Libby,\" she said. \"Then give me a better explanation.\"\n\n\"I can't,\" Libby said. \"But that doesn't mean there isn't one, and that isn't the point.\"\n\n\"It certainly is.\" Bernie made her final cut on the cheesecake, wiped the blade off again, and went on to the pies. They were much easier to slice. The only trick here was to make all the slices even. \"Dad agrees with me. He said my explanation was possible.\"\n\n\"Dad is just saying that to humor you.\"\n\n\"No, Libby. He's not.\"\n\n\"Yes, Bernadine. He is.\"\n\n\"Don't call me that.\"\n\n\"Sorry,\" Libby said, but Bernie could see that she didn't look remotely contrite.\n\n\"What about Bob Small?\" Bernie asked her sister.\n\n\"What about him?\"\n\n\"Don't you care what happens to him?\" asked Bernie.\n\nLibby looked indignant. \"Of course, I do.\"\n\n\"You're not acting that way.\"\n\n\"Now that is a rotten thing to say.\"\n\n\"No, it's the truth. If we don't do something, he's going to go to jail for a long, long time for this.\"\n\n\"Not necessarily,\" said Libby.\n\nBernie banged the knife down on the table and turned to face her sister. \"Yes, necessarily.\"\n\n\"Something could come up,\" Libby countered.\n\nBernie put her hands on her hips. \"Like what?\" she demanded.\n\nLibby remained silent.\n\n\"Exactly my point,\" Bernie said. \"You can't think of anything, can you? And even if some small scintilla\u2014\"\n\n\"Excuse me. What does scintilla mean?\"\n\n\"It means a little bit, a shred.\"\n\n\"So why don't you say that?\" asked Libby.\n\n\"I just did. Anyway, even if a bit of evidence does come to light, no one is going to follow it up or, for that matter, go looking for new leads, and you know it as well as I do. Don't deny it.\"\n\n\"I wasn't going to. It is true,\" Libby conceded.\n\nThat was one point Libby couldn't argue. She knew from her father how the prosecutor's office worked. How could she not? She'd seen them in action. When they had someone they liked for a crime, they didn't go running around, looking for alternate explanations. They stuck with what they had. \"They're like pit bulls,\" her dad used to say. \"Once they hang on, you can't get them to let go.\"\n\nLibby knew that it was the defense attorney's job to sniff out new leads, but a good defense lawyer cost lots of money, a commodity Bob was notably lacking. And who was he going to borrow it from? His wife? His business partner? Not too likely. The Simmons family was all he had.\n\nLibby started filling up the pitchers with waffle batter. \"Fine. I admit that it seems as if Bob is the fall guy for Amethyst's murder. Happy?\"\n\nBernie brushed a crumb off the table and onto the floor. \"Yes, as a matter of fact, I am. At least we're in agreement about that. He's like a custom-made suit. If you'd ordered him up, he couldn't fit this job any better.\"\n\n\"And Ed Banks?\" Libby asked. \"What about him? How does he fit into this mess?\"\n\n\"Simple. He was collateral damage. Killing him was merely a matter of tying up a loose end.\"\n\nLibby stopped to wipe a blob of batter off the rim of the pitcher she'd just filled. \"But once again,\" she continued, \"we come down to the inconvenient fact that you have no proof for any of this.\"\n\nBernie snagged one of the extra lemon cupcakes with a ginger glaze and took a bite. If she had to say so herself, it was pretty good. \"I know,\" she said.\n\n\"And there's no way of getting any.\"\n\n\"That's not true. It's more a question of Bob Small not having the money,\" Bernie replied. \"If he did, we could hire someone to go through the records, but since we don't have a state where the marriage occurred, much less a town, it would be pretty expensive. That's why I've come up with my plan, such as it is.\"\n\n\"Such as it is, is right.\"\n\n\"I wouldn't say that. I do my schtick and stand back and see what the reaction is. It's called stirring the pot.\"\n\n\"I don't think Dad would approve,\" said Libby.\n\n\"That's why we're not telling him.\"\n\n\"I still think we should.\"\n\n\"No, we shouldn't,\" Bernie insisted. \"You know what he's like. He'd be down here in a flash, even if he had to crawl on his hands and knees to get here.\"\n\nLibby couldn't argue that point, either. When it came to her and her sister, her dad was incredibly overprotective. Always had been and always would be. Bernie finished the cupcake, crumpled up the paper wrapper, and threw it in the trash.\n\n\"Okay,\" Libby agreed. \"But what happens if you don't get any reaction?\"\n\n\"Then we'll come up with something else.\"\n\n\"But then he'll know you suspect him, and he'll move away.\" Something else occurred to Libby. \"What happens if he takes out a gun and shoots you?\"\n\nBernie stopped for a moment.\n\n\"Obviously,\" Libby said, \"you haven't considered that possibility.\"\n\n\"Why are you always so negative?\"\n\n\"I'm not negative,\" Libby said, forcing the words out between gritted teeth. \"I'm realistic. And you haven't answered me.\"\n\n\"He won't,\" Bernie said.\n\n\"That's your answer? He won't?\"\n\n\"That's right,\" Bernie said.\n\n\"And why not? Because you wear cute shoes? Because you're carrying a Prada bag?\"\n\n\"Ha-ha. For starters, I don't think he has a gun. If he did, he would have used it on Banks and Amethyst. It would have been a hell of a lot easier than what he did do.\"\n\n\"So he won't use a gun. He'll use a knife. This man has murdered two people,\" said Libby.\n\n\"Yeah. But he did it for revenge.\"\n\n\"Excuse me. According to you, he didn't kill Banks for revenge. He killed him to keep him from talking.\"\n\nThis, Bernie was forced to admit, was true. She clicked her tongue against her teeth while she thought. \"I'll tell you what,\" she finally said. \"How about if I talk to him where there are lots of people about? In five more minutes, this place is going to become extremely crowded. Is that okay with you?\"\n\nLibby nodded reluctantly. It was better than nothing.\n\n\"And if worse comes to worst, Konrad and Curtis can step in.\"\n\n\"Now that's reassuring,\" Libby mumbled.\n\n\"I think they'll be okay. They're going to be pretending they're taping, so if things get funky, I'll signal them and they'll come over.\"\n\nLibby bit on the inside of her cheek. She attempted to think of some more objections and couldn't. She'd pretty much covered them all.\n\n\"Bernie, I still think this is nuts.\"\n\n\"Well, it's not the best plan I've ever come up with,\" Bernie said. \"But then again, it's not the worst. Basically, it's the only thing I can think of to do. I mean, we do have to do something. We can't just stand there and do nothing.\"\n\n\"You're right,\" Libby said softly. \"We do.\"\n\n\"If this doesn't work, at least we will have tried.\"\n\nLibby nodded. There didn't seem to be anything more to say. She surveyed the table. They were almost ready for customers. All she had to do was put out the napkins, knives, spoons, and forks, and they'd be good to go.\n\n\"I almost feel sorry for him,\" Libby mused. \"Amethyst probably deserved what she got.\"\n\n\"I'm sure she did. However, laying the blame on Bob Small isn't very nice.\"\n\n\"No,\" Libby said. \"I suppose it isn't.\"\n\n\"Nor was killing Banks. So are we ready?\" Bernie asked.\n\n\"Ready as we'll ever be. Just be careful.\"\n\n\"I'm always careful,\" Bernie retorted.\n\nLibby snorted. Now that was a big fat lie if she'd ever heard one.\n\n\"Well, I am,\" Bernie flung over her shoulder as she marched out the door.\n\nLibby wondered if her sister was feeling nervous, because she was certainly feeling nervous for her.\n\n## Chapter 32\n\nBernie fought her way through the crowd of people waiting to go into the Haunted House. It used to be that only kids dressed up at Halloween, but that wasn't true anymore. Today everyone did.\n\nBernie looked around the hallway. It was like being at a masked ball. There were witches and goblins and ogres, X-Men, Pillsbury Doughboys, Harry Potters, and Voldemorts. There were people in rhinestone and feathered masks, people with bright purple wigs and false noses, and people who had dressed up like Bush, not to mention all the women in bustiers.\n\nBernie looked down at what she was wearing. It was really pretty lame. Some people wouldn't even think it was a costume. It wasn't sexy or ironic or clever or cute. She wasn't a superhero or a famous person, although why anyone would want to be Paris Hilton for an evening totally eluded Bernie. In fact, she was pretty sure no one would know who she was supposed to be except her target, and she wasn't too sure about that. Maybe she wouldn't get the big response she was hoping for, after all. Oh well. She guessed she'd find out soon enough.\n\nBernie had modeled her clothes on the picture of Bessie hanging on the wall in Amethyst's apartment. She was wearing penny loafers, kneesocks, a pleated skirt, a white oxford shirt, and a cardigan, all of which she'd managed to find in the vintage clothes shop three blocks down from A Little Taste of Heaven. She'd gotten the tortoiseshell frames she was wearing from a costume store, ditto the brown, straight-haired wig.\n\nShe stood off to one side and scanned the crowd. She could see Konrad and Curtis fiddling with their tape deck near the restrooms. She gave them a slight nod, and they waved back. She sighed. Obviously, they weren't totally clear on the concept of being inconspicuous, but there was nothing she could do about it now. Going over and talking to them would only make matters worse.\n\nAfter shaking her head at them, she started studying the crowd in earnest. She knew the person she was looking for had to be here somewhere. She looked around the room. Nope, wasn't here. Maybe he was in costume. She scanned everyone's face again. Still nothing. And then, a few minutes later, she saw her target. No. She locked on her target. She liked that phrase better. Much more military. He'd just come in from the dining room and was standing in front of one of the doors that led to the back portion of the house, the portion that hid the scary stuff.\n\nBernie reflexively patted her shirt pocket. The mini tape recorder she'd bought this morning was in there. That was the nice thing about cardigans and oxford shirts. They hid stuff, unlike her Dolce & Gabbana black Lycra shirt. She figured that as soon as she got close to her target, she'd turn the tape recorder on.\n\nHopefully, it would work better than that huge thing Konrad and Curtis were using, and she'd get an admission of some kind that she could hear. The guy at the shop had said that the model she'd bought would pick up anything, and it had seemed to work pretty well when she tried it out in the shop. But you never knew with this kind of stuff. Just when you needed them the most, things like this tended to poop out.\n\nNot that a tape recording was admissible evidence. In fact, it was illegal to tape-record someone without their knowledge; you could get arrested for it. Bernie wasn't worried about that. All she wanted to get was some sort of admission that she could hand over to Bob's court-appointed lawyer. Hopefully, he would then get the ball rolling on this stuff.\n\nShe clicked on the machine, plastered her best smile on her face, and started walking toward him. He was watching a couple dressed as Humpty Dumpty and All the King's Men, so he didn't see or hear Bernie approach.\n\nHere we go, she said to herself. Then she took a deep breath and tapped Mark Kane on the shoulder. He jumped and spun around.\n\n\"Hi, Ken,\" she said. \"How's life treating you these days?\"\n\nAll the color drained from his face. He opened his mouth and closed it again.\n\nBernie motioned to her clothes with a nod of her chin. \"Like what I'm wearing? I think Bessie would be pleased, don't you? I modeled myself after the picture of her that Amethyst had in her bedroom. Curious that, don't you think? I don't know what to make of it. Do you?\"\n\nKane didn't answer. He was still gulping air. Finally, after another moment had passed, he got hold of himself and spoke.\n\n\"I don't know what you're talking about,\" he said. \"My name is Mark Kane.\"\n\nBernie's smiled widened. \"That's your current name, but before that, your name was Ken Marak. You're the headmaster's son and Bessie Osgood's first true love.\"\n\n\"I've always been Mark Kane.\"\n\nLater, Bernie would tell Brandon that it was the way he looked at her when he said his name and the emphasis he put on the two words that made her realize what they really meant.\n\nShe put her hand to her mouth. \"Oh my God.\" It had been in front of her all the time, and she hadn't seen it. \"Mark Kane. Of course. It's an anagram for Ken Marak, isn't it?\"\n\nKane bit his lip.\n\nBernie realized something else. \"Mark Kane. Mark of Cain. They're homophones. You feel that guilty about having taken up with Amethyst all those years ago?\" Bernie scrutinized his face. \"You do, don't you? No. It's more than that. You were involved in her death, weren't you?\"\n\nKane shook his head.\n\n\"Yes, you were. I can see it on your face. Did you tell Amethyst that Bessie was going to go to your dad? Or did you see Amethyst push Bessie out of the window?\" For a moment, Bernie thought Kane was going to faint. \"You did, didn't you? And you didn't do anything.\"\n\n\"Get out of here,\" Kane growled. \"You're nuts.\"\n\n\"I don't think so,\" Bernie replied.\n\n\"People told me you were crazy, and they're right. You are.\"\n\nBernie watched as he turned and tried the door. It didn't open. He pulled harder. It didn't budge.\n\n\"Looks as if it's locked from the inside,\" Bernie observed pleasantly.\n\nKane ignored her and started walking down the hall. Bernie kept by his side. When she looked around, she could see that Curtis and Konrad were following her.\n\n\"Why did you kill Amethyst?\" she asked Kane. \"Was it revenge because she killed Bessie?\"\n\n\"I don't know what you're talking about,\" Kane snarled as he shoved his way between a couple dressed in look-alike Cowardly Lion costumes.\n\n\"Hey, fella,\" the guy cried. \"Watch where you're going, will you?\"\n\nKane didn't respond. People shouted, \"Hey!\" and \"You can't do that!\" and \"Get in line like everyone else!\" as Bernie followed Kane through the crowd milling around in front of the entrance to the Haunted House rooms.\n\n\"We found Bessie's diary, you know,\" said Bernie.\n\n\"I don't care,\" Kane hissed.\n\n\"You were her first love.\"\n\nBy now they were in the Chain-Saw Massacre Room, with about five other people. The sound of the woman screaming was joined by the sounds of the people in the room going, \"Oh my God, that's terrible.\"\n\n\"You shared her first kiss with her,\" Bernie said as a girl grabbed on to her boyfriend and shrieked.\n\nAnother woman glared at Bernie. \"Will you shut up and let us enjoy ourselves!\" she hissed.\n\nBernie was just about to tell her to get a life when Kane spun around. Even in the dark Bernie could see that his normal affable expression had been replaced by fear and anguish.\n\n\"If I knew who this Bessie Osgood was, I might care,\" Kane said.\n\n\"Oh, you know all right. You gave her a book on Celtic mythology. You introduced her to old fairy tales.\"\n\nKane hurried across the room and opened the next door. The skeleton in the casket was cackling and pointing his finger at people. Kane pushed through the crowd to get to the door after that. A woman dressed as the Statue of Liberty told him to watch where he was going.\n\n\"I'll get the manager and have you thrown out,\" she threatened.\n\n\"You do that. I am the manager. In fact, I'm the owner of this place,\" Kane yelled. He turned to open the next door. Bernie watched his hand freeze on the handle as he realized what was on the other side.\n\n\"Are you sure you want to go in there?\" she asked. \"Amethyst's ghost might be waiting for you.\"\n\n\"I don't believe in ghosts,\" Kane told her. But his hand wavered on the knob, and after a few seconds, he turned and went back the way he had come. By now he was practically running. \"Get out of the way,\" he cried as he plowed through the crowd coming in the opposite direction.\n\nBernie followed him out the door and into the hallway. He looked around for a second and headed outside. As she followed, she tried not to think about the promise she'd made Libby about staying where the people were. Given the circumstances, what other option did she have? The temperature had fallen, and she could see her breath in the air. Bernie rubbed her arms as she followed Kane across to the other house. Her four-ply cashmere sweater was warm, but it wasn't warm enough. She looked back. Konrad and Curtis were nowhere to be seen. She should go back, but she knew she wasn't going to.\n\n\"How did you get Amethyst to marry you?\" she asked Kane.\n\nHe froze for a second, then turned to face Bernie.\n\n\"I know you did,\" she told him. \"We can prove it.\"\n\n\"How?\" Kane asked. His voice was hoarse.\n\n\"Amethyst told somebody, and she told my dad,\" Bernie lied.\n\n\"Who told you? I don't believe it.\"\n\n\"I'm not telling you.\"\n\n\"Because there is nobody,\" Kane hissed.\n\n\"No. I'm not telling you, because I don't want you killing them the way you killed Ed Banks.\"\n\n\"I didn't kill him,\" Kane protested.\n\n\"You most certainly did. I talked to Amber, and she remembers you buying some ginger pumpkin bars that day. You made a big deal of it by telling her you were taking them to a friend and you wanted the best ones possible.\"\n\nKane turned and took a step toward her. \"So if I did everything that you say, how come you're here talking to me? Aren't you afraid I'm going to kill you, too?\"\n\n\"Not really.\"\n\n\"And why is that? Could it possibly be because I'm not a killer?\"\n\n\"Oh no. You are,\" Bernie told him. \"There's no doubt about that. I might not be able to prove it, but it's true.\" And suddenly Bernie knew. She knew that what Kane wanted was a sympathetic ear. Someone to tell him he'd done the right thing. Someone to \"get him,\" as the expression went. \"And she deserved it,\" Bernie continued. \"She deserved everything she got.\"\n\nKane didn't say anything.\n\nBernie went on. \"She did. She was an awful person. She brought a lot of pain and misery to a lot of people. She wrecked lives. She certainly destroyed yours. Your dad killing himself, your mom having that accident.\"\n\nKane turned his face so Bernie couldn't see the expression on it. Then he spoke. \"Running a school was his dream. He'd borrowed all this money from my mom and her family, and from his family, and their friends. Then, when that thing with Bessie happened and everyone started taking their children out of the school, Dad, well, Dad couldn't stand it. He was so ashamed. He couldn't face everyone. And he...I found him, you know.\"\n\n\"No, I didn't know that.\"\n\n\"We'd just turned the corner, too. We were starting to make money. And then Amethyst pushed...\" Kane stopped abruptly. \"Are you recording this?\" he said.\n\nSomething told Bernie not to lie. She took the mini tape recorder out of her pocket and handed it to him. \"I was, but I'm not anymore.\"\n\nKane's hand closed over the tape recorder.\n\nBernie brushed away a snowflake that had fallen on her sleeve. \"You must have been planning this for a long time. You changed your name. You got plastic surgery in case Amethyst recognized you. Did she?\"\n\n\"Not at first. But I think she might have later. But she would have liked that. She was a game player.\"\n\n\"Only this time she lost,\" Bernie said.\n\nKane gave a stiff little bow. \"So it would appear.\"\n\nBernie waved her hand in the direction of the Peabody School. \"And, of course, you spent all this money fixing the place up, but you have it, don't you? I looked you up on the Web. You were one of the partners in the J and K Hedge Fund. You guys made\u2014\"\n\n\"Billions,\" Kane said.\n\n\"So I guess you figured you could do pretty much what you wanted.\"\n\n\"I never said that. You did.\"\n\nBernie nodded her head in assent. \"Where did you go after your mother died?\" she asked.\n\n\"I went out to Dallas to live with some relatives there.\"\n\n\"That must have been very hard.\"\n\n\"No. They were nice.\"\n\n\"That's not what I meant.\"\n\n\"I know what you meant.\" Kane looked at the mini tape recorder in his hand. \"They were nice. But all the time I was there, I just wanted to go home.\"\n\n\"And you finally did.\"\n\nKane nodded. \"I'm sorry about Banks. I never meant for that to happen. I didn't even know Amethyst had asked him about using his garden.\" Kane shrugged. \"And when I heard...I don't know. Something just came over me.\"\n\n\"And you did what you thought had to be done.\"\n\n\"I suppose you could put it that way.\"\n\nBy now they were near the Foundation.\n\n\"Damn,\" Kane said. \"The idiot twins.\"\n\nBernie turned and followed his gaze. Konrad and Curtis were running toward them.\n\n\"Stop,\" Konrad screamed. \"Stay where you are. We've got a gun.\"\n\nBernie cursed as she saw the rifle Konrad was carrying. Kane hesitated for two seconds before he took off and ran toward the Foundation. A couple of seconds after that, he opened the front door and entered the building.\n\n\"What are you doing?\" Bernie yelled at Konrad and Curtis as they rushed by her.\n\n\"We're making sure the son of a bitch doesn't get away,\" Konrad cried.\n\nBernie ran after them. \"He won't.\"\n\n\"Damn right, he won't,\" Konrad said.\n\nBernie grabbed on to the back of Konrad's jacket and pulled. Konrad spun around.\n\n\"I want you to stay here and guard the front door while I go up and bring him down,\" Bernie said.\n\n\"We can't\u2014\"\n\nBernie cut him off. \"You most certainly can.\" She ran off before Konrad could say anything else. She made it to the front door before Konrad and Curtis could and locked it.\n\nA second later Konrad was pounding on the door. \"Hey,\" he yelled. \"Let us in.\"\n\nBernie didn't waste time replying.\n\n\"Kane,\" she called.\n\nThere was no reply, but she heard footsteps to the left of her. She took off after them. She was running down a long, dark hallway. The footsteps were fainter now.\n\n\"Kane, stop,\" she yelled. \"We need to talk.\"\n\nNow she heard nothing. She came to a standstill. She was sweating now. Damn the twins, she thought as she caught her breath. Then she heard a key in a lock.\n\n\"We're coming to get you, Kane,\" Curtis called.\n\n\"Great,\" Bernie muttered to herself.\n\nShe'd forgotten they had keys to this place. Lovely. The way things were going they'd probably shoot her by accident. Friggin' morons. Then she heard something ahead of her again. She strained to listen. Footsteps. Kane's. They were coming from up ahead and over to the right. Bernie followed the sound. She could hear the twins behind her. She ran faster. Now she was in the front hallway. She looked up. Kane was running up the stairs.\n\n\"Wait,\" she cried.\n\nBut Kane just ran faster. She took the steps two at a time. They were now on the second floor, on the side where the French doors were. Kane kept on running. Suddenly, Bernie thought she saw something rectangular\u2014a book maybe?\u2014moving across the floor. She knew Kane wouldn't see it, and he didn't. He tripped. She watched him try and retain his balance. He teetered, swaying back and forth, frantically trying to regain his balance, and then he crashed through the glass and fell to the ground below.\n\n\"Now we're even-steven,\" Bernie could have sworn she heard a voice say. Then she saw Bessie smiling at her. \"See you later, alligator,\" Bessie said. And she was gone.\n\nBernie was still staring at where Bessie had been when Konrad and Curtis reached her.\n\n## Chapter 33\n\nBernie looked at the clock on the kitchen wall. It was a little after one in the morning. Halloween was over for another year. Thank goodness. This one had been a little too intense to suit her. Costumes were one thing, but real live ghosts were another. She thought she knew how Libby felt when she had those dreams. They didn't leave you with a pleasant experience, that was for sure.\n\nShe sighed and poured some milk into a copper pan and put it on the stove to warm up. She was thinking that the kitchen always calmed her down when she heard Libby coming down the stairs.\n\nLibby tapped her on the shoulder. \"How's the hot chocolate coming?\" she asked.\n\nBernie got out six mugs and proceeded to spoon one tablespoon of cocoa powder and two tablespoons of sugar into each of them. \"It's coming.\"\n\n\"Marvin and Clyde want marshmallows in theirs. Brandon and Dad want whipped cream and cinnamon instead, and so do I.\"\n\n\"Is that why you came down?\" asked Bernie.\n\n\"No. I was just wondering if you wanted to tell me what really happened,\" said Libby.\n\n\"I just did up there.\"\n\nLibby looked at her.\n\nBernie hunched up her shoulders. \"Well, I did.\"\n\n\"I told you my dream, remember?\"\n\nBernie sighed and checked the flame under the milk. She didn't want it to boil over.\n\n\"This is just so weird.\"\n\n\"If you're saying that, it must be good.\"\n\n\"And I can't even be sure. I think I imagined the whole thing.\"\n\n\"Kane's tripping and falling?\" asked Libby.\n\n\"But that's the thing. I could have sworn I saw something move across the floor. By itself,\" said Bernie.\n\nThe milk started to bubble. Bernie took it off the flame and poured a tiny bit into each mug. Then she mixed together the ingredients in each mug until they formed a paste, after which she poured the rest of the milk in.\n\n\"And?\" Libby prompted.\n\nBernie took out a tray and began to put the mugs on it. \"I think Bessie did it.\"\n\n\"Did what?\"\n\n\"Moved it. Made it move. I ran up the stairs and looked. Kane had tripped over an old book that was lying on the floor. A book of Celtic mythology.\"\n\n\"Maybe someone left it there?\" Libby suggested.\n\n\"Maybe,\" Bernie said. \"But I don't think so. It had no business being there. And here's the clincher. I opened it up. There was an inscription: To Bessie from Ken. I hope you find this as interesting as I do.\"\n\nLibby stayed silent.\n\n\"Exactly,\" Bernie said as she got the whipped cream out of the fridge. She put two big dollops in four of the mugs and added a sprinkle of cinnamon on top.\n\n\"There's more, isn't there?\" Libby said after a moment had gone by.\n\n\"I saw her. I heard her. She told me, 'Now we're even-steven.'\"\n\n\"Even-steven meaning she and Kane?\"\n\n\"Correct,\" Bernie said.\n\n\"But I thought Amethyst pushed her.\"\n\n\"She did, but I think Bessie blamed Ken. After all, if he hadn't gotten involved with Amethyst...\"\n\nLibby finished the sentence for her. \"This never would have happened.\"\n\n\"Exactly,\" Bernie said. \"And she was smiling. And then she said, 'See you later, alligator.'\"\n\nLibby got out the bag of marshmallows and placed three each in the two remaining mugs. \"She said that to me, too.\"\n\n\"You think I should tell the guys?\" Bernie asked.\n\nLibby put the bag of marshmallows back. \"I think you should stick to the 'he tripped and fell out the window\" story and leave the rest of it alone.\"\n\n\"I think so, too.\" Bernie put the whipped cream back in the fridge.\n\n\"But I believe you,\" Libby said.\n\nBernie grinned. \"You do?\"\n\n\"Yeah. I do. It makes sense in a weird kind of way. But no one else will.\" Libby took a Tupperware container full of gingersnaps down from the shelf and began putting them out on a plate.\n\n\"I guess Bessie finally got her payback,\" Bernie said.\n\n\"So it would seem,\" Libby agreed as she finished arranging the cookies. \"What is it they say about a woman abused?\"\n\n\"What they say is, 'Hell hath no fury like a woman scorned.'\"\n\n\"But she wasn't a woman,\" Libby objected. \"She was a teenager.\"\n\n\"Even worse,\" Bernie said, thinking back to when she was that age. Then she picked up the tray, and she and Libby went upstairs to join the guys.\n\n## Epilogue\n\nKen looked at Bessie. \"How could you do that?\" he demanded.\n\nShe blinked. \"Do what?\"\n\n\"Kill me of course.\"\n\nHe didn't know how he knew he was dead. He just did.\n\n\"Oh that.\" Bessie shrugged. \"You were going to die anyway. At least this way you're here with me.\"\n\n\"But you made me fall.\"\n\n\"You deserved it. You hurt my feelings.\"\n\nKen looked around. He was standing a little way from where he'd gone out the window. He glanced down at himself. He was now wearing the same clothes he had worn at the school. What was that line from one of the Grateful Dead's songs? What a long strange trip it's been? Then a horrible idea occurred to him.\n\n\"Is Amethyst here too?\"\n\nBessie pouted. \"You don't even want to talk to me. All you're interested in is her.\"\n\n\"No. I didn't mean it like that.\"\n\n\"It certainly sounds that way to me.\"\n\n\"Honestly, that's not why I'm asking.\"\n\nBessie looked at him for a moment. Then she said, \"No. She's not here.\"\n\n\"Good.\" Ken heaved a sigh of relief. \"Is anyone else here?\"\n\n\"Like who?\"\n\nHe thought back to when he used to live at the school. \"Like Esmeralda.\"\n\nBessie giggled. \"Nope. It's just us. Isn't that super?\"\n\nKen thought for a moment. Then he took Bessie's hand and gave it a squeeze.\n\n\"Yes, it is,\" he said to her. \"It really is.\"\n\n## RECIPES\n\nI always think of Halloween food as fall food\u2014food made out of apples and pumpkins, pears and cranberries; food seasoned with ginger and cinnamon and cloves; food that says, \"Come in and pull up a chair.\" Here are a few offerings. Two of them are Mexican in origin and are served up in celebration of Mexico's Day of the Dead, one of the recipes comes from the niece of a good friend of mine, and the remaining one comes from my recipe box. I'm not sure where I got it from, but I've been making it for years and can tell you that it's virtually bulletproof.\n\nMaria's Pumpkin Bars\n\n4 eggs\n\n1 2\/3 cups sugar\n\n1 cup vegetable oil\n\n2 cups (16 ounces) canned pumpkin\n\n2 cups flour\n\n2 teaspoons baking powder\n\n2 teaspoons cinnamon\n\n1 teaspoon baking soda\n\n1 teaspoon salt\n\nIcing\n\n8 tablespoons (1 stick) softened butter\n\nTwo 8-ounce packages cream cheese\n\n1 tablespoon vanilla\n\n2 to 3 cups powdered sugar\n\n1 tablespoon milk\n\nBeat all the icing ingredients together in a medium mixing bowl until smooth and creamy. Add more powdered sugar if the icing is too runny. Put aside.\n\nBeat the eggs, sugar, oil, and canned pumpkin in a large mixing bowl. Add the flour and all the other ingredients. Mix well and pour the batter into a greased 10 x 15-inch pan. Bake at 350 degrees for 25 to 30 minutes. Let cool. Apply the icing. Cut in one- to two-inch squares. Keep in the refrigerator.\n\nHere are two recipes that Mexicans serve on the Day of the Dead.\n\nPan de Muerto (Bread of the Dead)\n\n\u00bd cup butter\n\n\u00bd cup milk\n\n\u00bd cup water\n\n5 to 5 \u00bd cups flour\n\n2 (1\/4-ounce) packages active dry yeast\n\n1 teaspoon salt\n\n1 tablespoon whole aniseed\n\n\u00bd cup sugar\n\n4 eggs\n\nGlaze:\n\n\u00bd cup sugar\n\n1\/3 cup fresh orange juice\n\n2 tablespoons orange zest\n\nBring the sugar, the orange juice, and the orange zest to a boil in a small saucepan over medium heat. Boil for 2 minutes. Set the glaze aside.\n\nIn a saucepan over medium flame, heat the butter, milk, and water until they are warm to the touch. Do not boil.\n\nMeasure out 1 \u00bd cups of the flour, and set the rest aside. In a large mixing bowl, combine the 1 \u00bd cups of flour, yeast, salt, aniseed, and sugar. Beat in the warm liquid until well combined. Add the eggs, and beat in another cup of flour. Continue beating in more flour until the dough is soft but not sticky. Knead the dough on a floured board until smooth and elastic.\n\nLightly grease a large bowl, and place the dough in it, cover with plastic wrap, and let rise in a warm place until the dough is double in bulk, about 1 \u00bd hours. Punch the dough down, and shape it into loaves resembling skulls, skeletons, or bones. Let the loaves rise for an hour.\n\nBake the loaves in a preheated 350\u00b0F oven for 40 minutes. Remove the loaves from the oven, and paint on the glaze with a pastry brush.\n\nCalabaza en Tacha\n\nThis Mexican recipe uses pumpkin in an unusual way. The dessert is very sweet, and a little goes a long way.\n\n1 4- to 5-pound pumpkin (Pie pumpkins are best for this.)\n\n2 pounds raw or brown sugar\n\n8 cinnamon sticks\n\nJuice of 1 orange\n\n4 cups water\n\nWith a sharp, heavy knife, cut the pumpkin into 3-inch squares or triangles. Remove the seeds and strings. Cut a diamond design into the pulp.\n\nPut the sugar in a large pot with the cinnamon sticks, orange juice, and water. Boil until the sugar has dissolved.\n\nPlace a layer of pumpkin, pulp side down, in the syrup. Place a second layer of pumpkin on top of the first layer, pulp side up. Cover and simmer. Check every 10 minutes or so. When the pumpkin is ready, the tops of the pumpkin pieces will look glazed and the pulp will be soft and golden brown.\n\nLet the pumpkin cool. Serve the pumpkin with the syrup.\n\nThe recipe for cranberry walnut bread is mine.\n\nCranberry Walnut Bread\n\nThis bread couldn't be simpler to make. It makes one relatively small loaf.\n\n1\/4 cup butter\n\n1 cup brown sugar\n\n1 egg\n\n2 3\/4 cups flour\n\n1 teaspoon baking soda \u00bd teaspoon salt\n\n1 cup canned whole cranberry sauce\n\n3\/4 cup honey or wheat beer (Try different beers to vary the taste.)\n\n2 tablespoons grated orange peel\n\n3\/4 cup chopped walnuts\n\nCream the butter, sugar, and egg in a large mixing bowl. Mix the dry ingredients together in a medium mixing bowl. Combine the cranberry sauce, the beer, and the orange peel. Add the flour mixture and the cranberry mixture alternately to the butter, sugar, and egg mixture. Add walnuts. Bake in an 8-inch loaf pan in a preheated 350\u00b0F oven for 60 minutes. Cool the bread before cutting. This recipe keeps well.\nWhen sisters Bernie and Libby Simmons sign on to cater a prize pooch's birthday bash, they think they're ready for anything. But they haven't bargained for a killer with a bone to pick...\n\nA Little Taste of Heaven catering certainly knows how to feed people. Dogs, however. .? Bernie and Libby will have their chance to impress guests of the four-legged variety when they lay out the spread for Trudy the Pug's birthday luncheon. But this isn't just any doggie 'do. Trudy's owner, Annabel Colbert, is one of the richest women in town\u2014and as mascot of the Colbert toy company, Trudy herself is a bona fide celebrity.\n\nWhen the big day arrives, Trudy and her canine cohorts are ready to dig in to the delicacies\u2014but the first to dip her fangs into the wine is Annabel. Mere moments later, the hostess is shrieking she's been poisoned\u2014and proving it by falling face first into her soup. After two days in a coma, Annabel is dead.\n\nIt seems the woman who had everything also had her share of enemies. In fact, Annabel was cheating, blackmailing, or backstabbing most of the people she knew, including her very own best friend, her very own husband, his personal assistant, and last but not least, Trudy's trainer and kennel owner.\n\nWith so many suspects, sniffing out the truth is rapidly becoming a tricky not to mention risky proposition. Bernie and Libby had better close the oven on this case fast\u2014before they get burned...\n\nPlease turn the page for an exciting sneak peek of A CATERED BIRTHDAY PARTY coming in December 2009!\n\n## Chapter 1\n\nLibby dried her hands on the edge of her apron. She put the spatula covered with brownie batter in the sink before turning to face her younger sister. Then she took a deep breath. When that didn't calm her down, she took a second and a third. Maybe Marvin was right. Maybe she did need to mellow out.\n\n\"You like dogs,\" Bernie said to her in her most soothing voice.\n\n\"Not to the point of making dinner parties for them,\" Libby told her sister.\n\n\"Birthday party,\" Bernie corrected. \"We're making a birthday party.\"\n\nLibby frowned and waved her hand in the air. \"Same thing.\"\n\n\"No it's not.\"\n\n\"It's close enough.\"\n\n\"Let's not get overly semantic.\"\n\n\"You do,\" Libby told her.\n\nBernie decided to ignore the comment and stick to the matter at hand. \"This isn't for any dog,\" Bernie said. \"This is for Trudy, Annabel Colbert's dog.\"\n\n\"I know who Trudy is,\" Libby replied as she studied the toothpick she'd just plunged into the first batch of brownies in the oven. \"Everyone in the world knows who Trudy is.\"\n\nOkay that was an exaggeration, but not by much Libby thought. There might be some obscure tribe living in the Brazilian jungle who didn't know about Trudy but that was about it. Trudy was the model for the Puggables, a group of stuffed toys that were the lynchpin of the Colbert toy empire.\n\nThe collection was composed of Eenie and Minnie, the mom and the dad, plus the three pups, Pagggie, Poogie, and Twinkles, as well as numerous other family members with names too disgustingly cute to mention. Not only that, but they came in a range of annoyingly saccharine pastel colors. However, they had made Annabel and her husband a fortune. Before the Puggables, the Colbert Toy Company had been just another company struggling to survive.\n\nLibby sighed as she turned her thoughts back to the brownies. They were almost done. Five more minutes at the most. That was the trouble with brownies. They were easy to make, but difficult to make well. If you added the chocolate and butter mixture to the flour before it cooled, the bars came out heavy. If you baked them too long, they came out dry.\n\nBernie nodded at the brownies. \"Are these the ones you made out of seventy percent dark chocolate with chili powder?\"\n\nLibby nodded. \"It'll be interesting to see how they sell.\"\n\n\"We need to call them something cool.\" Bernie was a firm believer in the power of names.\n\nLibby shrugged. She wasn't.\n\n\"Dogs can't eat chocolate you know,\" Bernie told her getting back to the matter at hand. \"It gives them heart attacks.\"\n\n\"Then I'm glad I'm not a dog,\" Libby retorted as she watched her sister smooth her shirt down around her waist.\n\nIt was twenty-five degrees out, and Bernie was wearing a long black cashmere sweater, a white and gray stripped scarf, a silk shirt, twill pants, and suede boots. How she did it, Libby didn't know. She herself was wearing a flannel shirt, a hoodie, jeans, and wool socks, and she was still cold.\n\n\"It's really for Annabel Colbert's friends and family,\" Bernie continued. She took up the conversation where she'd left it a moment ago.\n\nLibby slammed the oven door shut. \"And dogs.\"\n\n\"Pugs,\" Bernie corrected. \"Six pugs.\"\n\n\"Wonderful.\"\n\n\"Well, you wouldn't want Trudy to have a birthday party without her friends. That would just be mean.\"\n\nLibby threw the toothpick into the garbage can, stalked over to the cutting board, and began shredding ginger for their special gingered chicken. As Bernie followed her, she reflected that she probably should have talked to her sister about the dinner later in the day. She might have been more receptive then. The fact that they were behind because their counter girl Amber had called and told them she was going to be three hours late this morning hadn't put Libby in a good mood.\n\nAs Bernie peered over Libby's shoulder, she once again marveled at the speed with which her sister's hands moved. \"I'm going to Sam's Club to get napkins, plates, sugar, and salt. Do we need anything else?\"\n\nLibby kept chopping. \"I don't like pugs,\" she informed Bernie.\n\n\"Neither do I,\" Bernie said as she snagged a piece of carrot off the table. \"They wheeze.\"\n\nLibby stopped chopping and turned to face her. \"So why are we doing this?\"\n\nBernie snorted. \"You can't be serious?\"\n\nLibby wasn't. Not really. She new exactly why they were doing this. They were doing this because you don't say no to the wealthiest person in town. At least you don't if you want to stay in business. Libby chewed on the inside of her lip as she extracted a piece of chocolate from the pocket of her shirt, unwrapped it, and popped it in her mouth.\n\nThat wasn't the issue. Not really. The issue was respect. Bernie was always running off and committing them to engagements without asking her first, leaving her to run around like a chicken without its head, to coin one of her mother's expressions. Frankly she was sick and tired of it.\n\n\"You should have discussed it with me first,\" Libby told her.\n\n\"I was going to,\" Bernie protested. \"But you were asleep when I got home.\"\n\nLibby grunted. \"You could have left a note.\"\n\n\"I wanted to talk to you.\"\n\nLibby felt her resolve weakening. This was the problem. She could never stay annoyed with her sister for long. \"When is this event supposed to take place?\"\n\nBernie hemmed and hawed. Libby started tapping her foot.\n\nBernie plastered a grin on her face. \"Friday.\" She gave the word an upward swing.\n\n\"Which Friday?\"\n\n\"Err...this Friday.\"\n\n\"That's two days away!\" Libby yelped.\n\nBernie looked unhappy. \"Well, you know how impulsive Annabel Colbert is. But on the bright side, the Fields canceled their dinner party, so that leaves an opening.\"\n\nLibby's eyes narrowed. \"You didn't tell me that either.\"\n\n\"It was on the answering machine last night. I figured you'd listen to the messages.\"\n\nLibby cursed silently as she strode towards the calendar hanging on the wall. At least they hadn't butter-flied the lamb yet or ordered the scallops for the coquille St. Jacques so that was good. And she had to admit that what Bernie had just said was true. Annabel Colbert was impulsive. And insistent. She never took no for an answer. She had that sense of entitlement the super-rich have. Just saying it made it so for her.\n\nLibby clicked her tongue against the roof of her mouth while she studied the calendar. They were pretty well booked, what with dinner parties, benefits, and bar mitzvahs. But Bernie was correct. The Fields had been the only event they had going for Friday night.\n\n\"She's going to want a deal,\" Libby said. \"She always does.\"\n\n\"No. I'm not,\" a voice behind them trumpeted. \"Nothing is too good for my Trudy.\"\n\nLibby and Bernie spun around. Annabel Colbert was standing in the doorway between the kitchen and the counter area with her pug in her arms. She'd recently gotten a short, spiky hairdo, but instead of making her look punk, it made her appear even more gaunt. Scrutinizing her, Bernie decided that the \"thin\" part of the adage \"you can never be too rich or too thin\" had definite limits, and Annabel Colbert was on the verge of transgressing them. There was thin and then there was just plain bony. Bernie was thinking about what the tipping point was when she noticed that their counter guy, Googie, a.k.a. George Nathan III, was right behind her.\n\n\"I'm sorry,\" he stammered as he wiped his hands on his apron. \"I couldn't stop her. She just barged in.\"\n\nLibby nodded. \"It's okay.\"\n\n\"I...\"\n\n\"Seriously. Go back behind the counter,\" Libby said, and she shooed Googie away before turning her attention to Annabel. A Little Taste of Heaven wasn't packed, but there were ten people out front waiting to be served. Aside from which it didn't do to leave the cash register unguarded. Longely wasn't the type of place where you had to worry about stuff like that, but why take chances? \"I'm sorry but Trudy can't be in the kitchen,\" Libby told Annabel. It would just be their luck to have the health inspector walk in on them.\n\n\"But I'm holding her,\" Annabel protested.\n\nBernie shrugged. \"Health code rules are health code rules.\"\n\nAnnabel scowled. \"She's cleaner than most people.\"\n\n\"I'm sure she is,\" Bernie said as she escorted Annabel out of the kitchen and up the stairs to the Simmons' living quarters.\n\nLuckily her dad was out at the moment, because he wasn't a big fan of the Colbert family.\n\n\"Morons\" was the kindest comment he made whenever their name came up.\n\n\"I like Trudy's collar,\" Bernie said on the way up the stairs. It was thick braided leather with a large gold buckle.\n\n\"Hermes,\" Annabel trilled.\n\nBernie managed to restrain herself from asking how much. But it had to be at least five hundred dollars. Probably more.\n\n\"This is so...so...cozy,\" Annabel commented in that annoyingly arch way she had when she and Bernie walked into the Simmons' living room.\n\nTranslation: small. But then Bernie supposed that when you lived in a place that boasted its own solarium, library, gym, media room, bowling alley, and beauty parlor, as well as a canine fitness center complete with treadmill, the Simmons' flat was small. Annabel plopped herself down on the sofa, put the pug on her lap, and began fingering one of her diamond studs.\n\nIn case I missed them, Bernie thought as she took the armchair opposite her customer. Like that was possible. Missing them would be like missing a flashlight beam in the dark. Despite Annabel's makeup, Bernie noted the dark circles under Annabel's eyes and the fine lines around her mouth. The haircut was definitely not kind to her. Bernie bet it had been expensive, too. Probably six hundred dollars. Maybe even eight. She was just about to ask Annabel who did her hair when Annabel started talking.\n\n\"We have to revamp the menu,\" she announced.\n\nBernie barely managed to suppress a groan. It had taken them three hours yesterday to agree on the one they had.\n\n\"I've decided I want the same menu for the dogs and their humans,\" Annabel continued. \"It'll be much more of a bonding experience that way. And I want the food served on my good Limoges. Naturally the dogs will be seated at the table.\"\n\n\"Naturally,\" Bernie murmured.\n\nAnnabel shot her a look. \"They do it in Paris all the time.\"\n\nBut we're not in Paris, we're in Westchester County, Bernie wanted to say. But didn't.\n\n\"Yes,\" Bernie replied. \"It's wonderful. So humane. When I was in Brussels, I sat next to an extremely well-behaved standard poodle riding on the bus with his owner. There was me, the poodle, and the owner all sitting in a row, all staring straight ahead.\"\n\nFor a moment Annabel looked miffed at having been outdone in the story department, but she rallied. She waved her hand in the air. \"And you don't have to take care of the decorations. I'll do those. I'm thinking of using the whole Puggable family. That will work, don't you think?\"\n\nBernie didn't think it would work at all. She wasn't a fan of huge stuffed animals. On the other hand, she wouldn't have to deal with them. That was a plus. Also, in situations like this, despite what her sister thought, she always went with the client's wishes. Therefore, she lied and told Annabel she thought it would be great.\n\n\"Absolutely,\" Libby added. She'd come up after finishing the chicken and was now standing in the doorway with her arms crossed over her chest. She felt that it behooved her to be in on the planning due to Bernie's well-documented tendency to commit them to things without thinking them through. Witness today. \"Do you have any ideas about what you'd like to serve?\"\n\n\"Ideas? Ideas?\" Annabel repeated. She wrinkled her nose, giving the impression that she found the question puzzling. \"That's why I came to you.\"\n\nLibby flicked a tiny piece of ginger off the sleeve of her hoodie. \"Well, what's your price range?\"\n\n\"Price is no object,\" Annabel snapped. \"I already told you that in the kitchen, isn't that right, Trudy?\"\n\nTrudy yawned, curled her tongue, stretched, and licked Annabel's hand in a desultory manner.\n\n\"See. She agrees.\" Annabel said.\n\nBernie and Libby both managed to not roll their eyes.\n\n\"What about the wine?\" Bernie asked, changing the subject.\n\n\"What about it?\" Annabel said.\n\n\"What are your thoughts? California? Long Island? They've gotten a lot better recently. French? Italian? Australian? Do you want us to get it or...\"\n\n\"No,\" Annabel replied before Bernie could finish her sentence. \"Forget about the wine. Richard will take care of that. He orders it from The Grape, that little shop over by Henley Drive.\"\n\n\"I know where it is,\" Bernie said. Not that she ever bought anything there. They only carried high-end stuff.\n\n\"They get my wine for me there. I'm particularly fond of an estate-bottled Spanish Rioja that's quite hard to come by, but somehow they manage. It's my special wine. In fact, it's the only thing I'm drinking now. In any case, we'll just have wine with the meal; otherwise we'll stick to bubbling water and soda. Richard will take care of that as well. After all, it's Trudy's party so I think it would be nice if there's nothing there she can't have.\"\n\n\"Understood,\" Libby said.\n\nAnnabel went back to fingering her diamond earring. \"And needless to say, I want all the ingredients in this meal to be organic. We don't tolerate anything else in our house. My husband won't allow it. Local would be even better. The less of a carbon footprint we leave, the happier all of us will be. Also, I don't want any black pepper in anything because Trudy's allergic to it. Naturally, I want both of you to set up and serve. It'll be more intimate that way. You won't have to worry about my staff getting in your way. In fact, I've given them the time off.\"\n\n\"Is that it?\" Bernie asked.\n\n\"For the moment,\" Annabel replied as she got up. \"If I think of anything else, I'll let you know.\"\n\nBernie stood up as well. \"We'll call you with the menu.\"\n\n\"That won't be necessary,\" Annabel replied. \"You can fax it to my personal assistant, who will show it to me. If I have any quibbles, I will relay them to her and she will fax you back. Unfortunately, I'm terribly busy with my new projects at the moment. I'm trying to get them squared away, so I can announce them at the birthday party, but I'm sure you girls will do a marvelous job. After all, that's why I hired you.\" She glanced at her watch. \"Can I tell my assistant to be looking for your fax in an hour or so?\"\n\nBernie looked at Libby. Libby looked back at Bernie.\n\n\"That'll be fine,\" Libby said as she saw her plans for the day disappearing over the horizon. Now she'd be even further behind.\n\n\"Good.\" Annabel smiled. \"And by the way, Bernie,\" Annabel added as she got to the doorway, \"you should cut down on the carbs. You're getting a little chubby around the derri\u00e8re.\" And she patted her rear end. \"I hope you don't mind my saying something, but if the positions were reversed I'd certainly want to know.\"\n\nBernie managed to get out a strangled thanks as Annabel walked through the door. \"Am I?\" Bernie asked her sister as soon as she was sure Annabel had left the building. She wasn't going to give Annabel the added satisfaction of overhearing her comments if she could help it.\n\n\"Don't be ridiculous,\" Libby replied.\n\n\"Really?\"\n\n\"Really. Of course, in comparison with her rear end, everyone's is big.\" Big was not the word Bernie wanted to hear at the moment. \"Thanks a heap.\"\n\n\"Oh come on. Annabel just can't stand to see anyone looking good. I would kill to have your body,\" Libby told Bernie as she watched her sister study herself in the mirror hanging on the back of the bathroom door. And she wasn't just being nice. She meant it.\n\n\"Maybe I shouldn't wear these pants anymore.\"\n\n\"Bernie!\"\n\n\"You're right.\" Bernie brushed a lock of hair off her face. \"Why am I listening to someone who could be a stand-in for a famine victim?\" The corners of Bernie's mouth worked themselves into a smile. \"And we are charging her a lot of money.\"\n\nLibby smiled. \"Pots of it. And we're getting three-quarters of it up front or we're not doing it.\"\n\n\"Good,\" Bernie said. If there was one thing she'd learned over her years of catering it was that the rich didn't like to settle their bills. \"And now for the menu. I think we should start with a liver pat\u00e9 on toast points, some cheese stuff, maybe some bacon and peanut butter on crackers...\"\n\nLibby wrinkled her nose.\n\n\"Hey, I know they're not haute cuisine, but dogs and kids like them,\" Bernie said a little defensively.\n\nLibby nodded. It was true. They did.\n\n\"And then,\" Bernie continued, \"we move on to steak and potatoes.\"\n\n\"What about the cake?\"\n\n\"Something vanilla. Maybe an old-fashioned layer cake, light on the frosting, in the shape of a dog bone?\"\n\nLibby pursed her lips. That would do. \"Ice cream?\"\n\nBernie thought for a moment. \"Probably not. That might be overkill in the sugar and dairy department.\"\n\n\"This should work,\" Libby observed after they'd faxed over the menu.\n\n\"Of course it's going to work,\" Bernie said indignantly. \"We designed it, didn't we? Although I'm sure Annabel will have some quibbles.\" Bernie bracketed the word quibbles with her fingers.\n\n\"I'm sure she will,\" Bree Nottingham, real estate agent extraordinaire and social arbiter of Longely, said as she swept into the living room with Rudoph, her six-month-old pug puppy, trailing behind her. They were both wearing pink coats and rhinestone collars. \"Annabel always has quibbles. Of course, when you have that kind of money you can afford to.\"\n\n\"It's not the quibbles I'm worried about,\" Libby replied as she pictured six dogs running up and down the table. \"It's everything else.\"\n\n\"It'll be an interesting event,\" Bree commented as she watched Rudolph sniff the sofa leg. \"I just came by to tell you that Rudolph is allergic to chicken, so don't put chicken on the menu. He's a sensitive soul, the poor dear.\"\n\nBernie looked down at Rudolph, who was currently trying to dig a hole in the carpet. He didn't look sensitive to her. He looked like a miniature Sherman tank.\n\n\"Interesting in what way?\" Bernie asked. She decided to sidestep the whole dog food allergy issue. Bad enough she had to deal with people with food allergies, let alone their canines.\n\nBree smiled brightly. \"In the way that married couples frequently are, dear.\"\n\n\"And that is?\" Libby asked. She'd expected the conversation to go in another direction.\n\nBernie leaned forward slightly. \"Yes. Elucidate for us. Inquiring minds want to know.\"\n\nBut instead of answering, Bree gave the Simmons' sister another of her smiles and said, \"I'm sure you two will do an excellent job. You always do.\" After which she left. Just like the Grand Duchess, Bernie thought.\n\n\"Now what do you think she meant by that?\" Libby asked Bernie as soon as she heard the downstairs door closing.\n\n\"I think she means that the Colberts are getting a divorce. Or one of them is having an affair.\"\n\n\"Seriously?\"\n\nBernie gave her a look. Honestly, sometimes her sister was so na\u00efve. \"What else could it be?\"\n\nLibby shook her head. \"I don't have a clue. All I know is that whatever it is, it's none of our business.\"\n\n\"I suppose,\" Bernie said feigning agreement even though she really didn't believe that, and she didn't think that Libby believed it either. After all, they'd been raised in a house with a mother who had elevated minding other people's business into an art form.\n\n\"I think we should concentrate on planning,\" Libby said.\n\n\"I think you're right,\" Bernie agreed.\n\nThis was not debatable. They had lots to do and not much time to do it in.\nLongely is an imaginary community, as are all its inhabitants. Any resemblance to people either living or dead is pure coincidence.\n\nKENSINGTON BOOKS are published by\n\nKensington Publishing Corp.   \n119 West 40th Street   \nNew York, NY 10018\n\nCopyright \u00a9 2008 by Isis Crawford\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.\n\nKensington and the K logo Reg. U.S. Pat. & TM Off.\n\nISBN: 0-7582-4816-4\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPublished by New Europe Books\n\nWilliamstown, Massachusetts\n\nwww.NewEuropeBooks.com\n\nCopyright \u00a9 2017 by Gazmend Kapllani\n\nEnglish translation \u00a9 2017 by Anne-Marie Stanton-Ife\n\nInterior design by Knowledge Publishing Services\n\nPreviously published in English in 2010 by Portobello Books Ltd,\n\nTwelve Addison Avenue, Holland Park, London.\n\nFirst published in Greece by Livanis Publishers.\n\nThis edition is published by arrangement with Ampi Margini Literary\n\nAgency, London, www.ampimargini.com.\n\nALL RIGHTS RESERVED. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without written permission from the publisher except in the case of brief quotations in critical articles and reviews.\n\nISBN 9780997316988\n\nEbook ISBN 9780997316995\n\nCataloging-in-Publication Data is available from the Library of Congress.\n\nFirst US edition\n\nv4.1\n\na\n_For my mother, Qerime_.\n\n_For my father, Myrteza_.\n\n_For my brother, Artionil_.\n\n# _Contents_\n\nCover\n\nTitle Page\n\nCopyright\n\nDedication\n\nPrologue\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\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nEpilogue\n\nAuthor's Note\n\nNotes\n\nA Conversation with Gazmend Kapllani\n\nOther Titles\n\nAbout the Author\n\nAbout the Translator\n\n# **Prologue**\n\nI'm not crazy about borders; I can't honestly say I hate them, either. It's just that they scare me, that's all, and I always feel uncomfortable when I get too close to one. Let's get something straight: I'm talking about visible borders here, geographical borders, the ones that mark off one country from another, one state from another, one nation from another. Even today, as they become increasingly porous, whenever I cross one I get a very strange feeling, a mixture of deliverance and perplexity. Perhaps it's because of my passport. I've got used to that suspicious look they give me. I return it with one of longing, and impatience to get to the other side; but it's no use, it's almost invariably met with hostility and suspicion. I do try to reassure them, try to persuade them that I represent no danger of any kind, but there always seems to be some pretext or other for rebuffing me, some excuse for not treating me as an equal. This is why I say I suffer from Border Syndrome, and have done for a long time. Border Syndrome is a form of illness that's difficult to describe with precision. Unlike agoraphobia or depression, it doesn't feature on the list of recognized mental disorders. But what I can do, perhaps a little later, is outline some of the main symptoms. I do know that there are other sufferers, a great many of them. Those who have never experienced the urge to cross a border, or who have never experienced rejection at a border, will have a hard time understanding us.\n\nMy difficult relationship with borders goes back a very long way, back to my childhood, because whether or not you end up with Border Syndrome is largely a matter of luck: it depends on where you're born.\n\nI was born in Albania.\n\n# **1**\n\n_R eaching, never mind crossing, the borders of a country under a totalitarian regime like Albania was, until 1991, the equivalent of a miracle\u2014or a mortal sin. Exit visas were few and far between; those who got them were the lucky ones. The rest of us, most of us, that is, looked on the ones who got them as a breed apart, something along the lines of extraterrestrials. We were condemned to speculate endlessly about what lay on the other side of the borders. Either that or we would torture ourselves with the simple conclusion that life went on as normal, even on the other side_.\n\n_Dismissing the possibility that there was life beyond the borders proved quite an effective survival strategy, spiritually as well as physically. At some point, in the subconscious of many of us, the world-beyond-the-borders became more than the continuation in time and space of some common world_.\n\n_As the years went by, and Albania's isolation became absolute, the world-beyond-the-borders gradually assumed the status of a separate planet. For some people that planet was paradise, for others it was a place of terror. But for all that, another planet_.\n\n**Why are you telling us all this?**\n\nYou could ask the immigrant, Why are you telling us all this? The truth is, as an immigrant, especially a first-generation immigrant, your first instinct is to stay silent. Fear; caution; the violent escape, that violent first encounter with the unknown country; that feeling of being uninvited; resentment; longing for and rejection of home, and guilt and rage at the same time, all take root in the immigrant. An immigrant is a confused creature, highly insecure, and that is why he fears confession. All it takes is one gesture from the other side, one sign of denial, or indifference, of the \"What's it to me where you come from and what you've been through?\" kind, for the immigrant to feel ridiculous, vulnerable, and freakish. That's why he doesn't take risks. Instead, he digests his experiences in isolation and before long is convinced that they are of absolutely no interest to anyone. In the final analysis, he thinks: I wasn't made for telling stories, but to fight tooth and nail for my survival. It's not that people can't understand; they just don't want to.\n\nThe alternative, baring all and confessing, is risky. Telling his life story, the painful and contradictory journey of an immigrant. But, if he keeps it all bottled up inside, he is in danger of becoming neurotic and resentful. The most he can hope for is that they will understand, first him, and then all those who cannot speak, who don't know how to speak, who don't have the courage to speak and who bury their narratives deep inside themselves instead. You cannot understand an immigrant if you haven't heard his story first.\n\n# **2**\n\n_T he regime did everything in its power to block all images from the other side of the border by means of controls, arrests, and punishments. I can still remember\u2014I was in the second year of primary school at the time\u2014the day the school's Party secretary walked into our classroom and, among other things, asked us in a ponderous voice (which, with monumental exertion, she tried to soften) if our parents watched anything apart from Albanian state television. With the naivet\u00e9 of a child wanting to impress his classmates, I stuck my hand up and duly announced that my parents enjoyed watching a channel called Savra. There was no such thing; Savra was the name of a village near our town, Lushnj\u00eb. But I didn't know that. My father, alert to the dangers involved, dealt with my unfettered childish curiosity by feeding it with certain pseudonyms and code words for the foreign television stations he watched on the quiet. Ultimately, not even that could save him. That evening, he was summoned to the headmaster's office and asked to account for this unknown television station. You could lose your job over something like that\u2014and that was the least of it. He risked being accused of \"reactionary activities\" and of harboring \"petit bourgeois sympathies\" and risked being tried for disseminating antistate propaganda. After that, he could end up in one of those terrifying jails for political prisoners, or else be sent to one of those villages outside Lushnj\u00eb that were full of exiles. Savra was the best known. The truth was that there were options at their disposal for finishing someone off. My father had to explain himself, and explain why, first of all, he did not restrict his viewing to Albanian state television, and why he moved around in wealthy capitalist-imperialist-revisionist-Titoist-monarcho-fascist etc., etc. circles. Secondly, he was required to account for his choice of pseudonym. Why Savra? Did this imply indirect yet unambiguous sympathy for the enemies of the state exiled in that village? He answered flatly that he did not watch foreign television, adding that if they didn't believe him, they were welcome to come and inspect the antenna on his roof_.\n\n_On the subject of antennas, all citizens were required by the regime to purchase special ones, which had to be installed in such a way as if to renounce any desire to receive images from the world-beyond-the-borders. The truth was that my father had two antennas: one on the roof, for appearances' sake, and another, illegal one, the one referred to inside the family as \"the antlers.\" It was an indoor antenna, used mainly for picking up Italian channels. This double antenna was the perfect symbol for the divided individual living in a totalitarian state. One side of you functioned to appease the terrifying gaze of the regime; the other, more congenial side tried to escape its all-powerful watch by cunning_.\n\n_Fortunately the episode with the antenna did not have further repercussions. I was treated to a stinging slap from my father, sealing my hatred for the Party secretary, and although still a child, I began to realize that I had to stop being one, especially when it came to dealing with Party secretaries. I was so furious with her that I secretly prayed that some terrible fate would befall her, as that, and only that, would assuage my wrath, and only then in part. I wished\u2014fantasized even\u2014about her slipping, falling, and breaking her leg\u2014both legs; that she would fall ill and die a slow, painful death; that a brick would land on her head, killing her on the spot. The fantasy that brought me absolute satisfaction, however, was seeing her lying on the ground, squashed under the wheels of a cart (one of those from the farmers' cooperative, on filthy wheels pulled by emaciated horses that you used to see driving around town)_.\n\n_Time passed and with it my sadistic fantasies. I found out, much later, that my prayers had been answered in part, but not quite in the way I'd imagined: the Party secretary developed a peculiar illness, which made her look permanently cheerful. It was unbelievable considering she was one of the most stony-faced people on the planet. At school she was known as \"the statue.\" Now each time there was a Party convention, she would collapse in fits of laughter, shrieking away, and the more she shrieked, the more idiotic she looked. News of her nervous laughter reached the Party Committee, which ruled that in the interests of ideology, the Party, and the revolution she should go into early retirement. As they pointed out, she was not simply unsuitable but was harming the revolution and the dictatorship of the proletariat through her infirmity. That was the last she was heard of_.\n\n**Leaving your country means breaking with it**.\n\nA true emigrant is an incorrigible, egotistical narcissist. He thinks he's too good for his native land, and does not deserve the poverty, the lack of prospects, the violence, the corruption, the filth, the hypocrisy, or the lack of love. For all these reasons, for him, exile is above all a choice.\n\nLeaving is a choice, a choice to break with the country of his birth. This break follows him for the rest of his life. It will be the source of his sense of guilt and of freedom, rejection and denial, daydreaming and nostalgia, forgetting and melancholy, mood swings and schizophrenia. Only if he makes a success of life abroad, only then can he make peace with his own country again. If he doesn't make it, he will be left hanging, at odds with the world and with the universe. He will make a great show of how much he loves his country of origin for one reason alone\u2014to annoy people in his new country, the country he believed would offer him a better future, but which, in his opinion, has denied him one. In the final analysis, he has rejected his own country in favor of this new one; he has had more faith in this new one than in his own. Shouldn't that in itself be enough for them to hold their doors wide open for me? he asks himself.\n\n# **3**\n\n_T hose antennas will always be connected in my mind with two more people who made a lasting impression on me during my childhood: Uncle Jani and comrade Mete. I had known Uncle Jani almost all my life because we lived in the same block. His flat was upstairs from us, on the fourth floor. He was quite well known around town, mainly because he waged an undeclared war on all enemies of the state and the revolution, the enemy without, certainly, but especially the enemy within. From the conversations of the grown-ups at home and at friends' homes, I learned that Uncle Jani had an enormous logbook, which he kept hidden somewhere in his flat, detailing the works and days of all suspect residents of our town. Woe betide anyone whose name appeared on the list; it meant certain ruin for them and their relatives. Rumors about Uncle Jani's list grew until it acquired mythical status within the community. Some said that it was just a simple notebook; others maintained it was a huge tome containing information relating not only to suspect persons in this town but in the neighboring one, too. Uncle Jani was so committed to hunting down the enemy within that it was said that even his daughter-in-law's name appeared on the list. They all lived under the same roof. One night, he overheard her talking in her sleep, murmuring, \"I don't give a shit about the Party conference....\"_\n\n_Irrespective of how long or how short Uncle Jani's list really was, the fact was that nobody in our block suffered at his hands, except Keme's son, that is. Rolling home drunk one night, he had the misfortune of falling across Uncle Jani, who fixed him with a stern gaze. Keme's son, who worked as a porter, returned Uncle Jani's look with these words: \"They say that drunkards lose all sense of smell\u2014even so, I couldn't fail to sniff out a filthy informer like you!\" before collapsing into fits of laughter. He lost his job because of it, but everyone said he'd got off very lightly and that Uncle Jani had proved that he did have a human side, because if he'd really wanted to, he could have had Keme's son exiled or even thrown into prison_.\n\n_Uncle Jani was not the only one targeting the enemy within. There were others in town doing the very same thing. We knew about some of them, but not all. Comrade Mete, for example, we did not know about. We only found out about him much later, under the most tragic circumstances. Comrade Mete's particular skill and passion was tracking down the enemy within through the precise study of the lie of their television antennas. He left no rooftop in the town uninspected at least three or four times, operating under cover of night to make sure the direction of all antennas conformed to Party regulations. In fact he had compiled a long list of names next to which was recorded the direction of the lie of their antennas. If Comrade Mete discovered an aberration anywhere, the authorities were alerted instantly and the guilty party could find himself digging trenches for years on end. Comrade Mete's legendary list was discovered late one night on the roof of our block. Unfortunately the owner of the list was five floors away from it, lying on the ground at the entrance to the building, having executed the most spectacular fall from the roof, all the way down to the ground below, which was still damp from the recent rain. Comrade Mete's fall was accompanied by a terrible scream that sent a jolt through everyone in our building, waking everyone in the surrounding blocks as well. Comrade Mete died on impact. A tragic, premature death, but a hero's death; he was killed in the line of duty, fighting for socialism and in the class struggle. What caused his fall, however, was never established, and speculation concerning the mysterious circumstances surrounding his demise continued to fuel the imaginations of the people of our town_.\n\n**The Immigrant in the Realm of the Imperative**\n\nYou have to get a job. Any job. You have to survive. You have to find somewhere to live\u2014doesn't matter what it's like as long as it looks vaguely like a home. You have to learn the language, even if you can't understand a single word of it and you get your \"good nights\" and your \"good evenings\" all mixed up. You have to learn to speak more softly, and not shout, because it scares people. You're not back home in the village now, you know. You have to keep out of the way of those Black Marias because you turned up in this country without an invitation, making quite an entrance with that woebegone expression of yours and that primitive haircut. People round here haven't seen anything like that for decades, especially not combined with those clothes, so obviously charity clothes\u2014or maybe you stole them? You have to learn how to walk properly because you've got used to walking too fast, like you've got the Devil on your back. You've got to learn the Highway Code, the sections that apply to pedestrians of course, and you've got to stop looking at all those gorgeous local girls like that, the way Quasimodo looks at Esmeralda in Notre Dame de Paris. You have to, have to, have to...without end or expiry date. Day after day, night after night, week after week, month after month, year after year. Not for you the privilege of wanting\u2014you are condemned to live by the mercilessly cruel claims of \"have to.\" Because you have to make it. Above all, make it.\n\nThis is the immigrant's oath.\n\nJust as doctors are supposed to live by the Hippocratic oath, an immigrant lives by \"I have to succeed.\" This oath is his only real country from now on. He has to make it, not simply because people back home expect something from him; that's the least of it. He has to make it because he cannot go back a failure. The thought of failure makes him tremble like a child afraid of the dark. He has to make it, but how? And this is where the common path of immigrants diverges and they split off into successes and failures, the accepted and the rejected, the lucky and the unlucky. Because immigrants, whatever their superficial similarities, differ from each other in exactly the way that everybody in this world differs from everybody else.\n\n# **4**\n\n_T here were two basic versions of how comrade Mete met his death. The first version circulated in hushed tones and low voices from house to house and caf\u00e9 to caf\u00e9. The story went that comrade Mete, in addition to his weakness for television antennas, was victim of another obsession, much, much stranger than the first. A committed voyeur, he lived to watch couples in the throes of lovemaking, longed for glimpses of women hitching up their skirts to reveal their naked backsides when visiting the toilet or getting into the bath. That fateful night, it was rumored, he was spying on the female judge who lived in the top floor flat in our building, while she was having her bath, and he was completely mesmerized by her ample backside. Completely in the thrall of his passion, and in his efforts not to miss a single ounce of the pleasure this sight brought him, he had balanced himself precariously, giving scant attention to his calculation of the angle. Unfortunately for him, it had been raining, that's why he slid off, losing at once both his view of the judge's backside and his life. That at least is what those loyal to the first version claimed_.\n\n_The second version was the one championed by Uncle Jani. According to him, comrade Mete was the victim of a conspiracy of enemies within. He swore that he would find the enemy behind this treacherous murder, the heroic death of comrade Mete, if it was the last thing he did. One day he announced that his investigations had progressed, and the enemy within was in fact within our own building. His news spread panic through the block. Keme, because his son had been in trouble with Uncle Jani, was worried that Uncle Jani had him on his list as a main suspect, and for that reason sold his television to limit the risk of being accused of watching foreign television stations. Another neighbor, Loni, was in the hospital being treated for severe psychological disorders following a nightmare in which Uncle Jani forced him to dig up comrade Mete's body. Uncle Jani, however, died of a sudden heart attack. His death was greeted by several people in the building with sighs of relief. Behind closed doors, of course_.\n\n**A Hero Verging on the Ridiculous**\n\nThere is something heroic about the way an immigrant abandons his native land. Nevertheless, in his everyday life, he is fragile, confused, and at times ridiculous, like a card player who dreams of that one amazing trick but lacks essential knowledge of the rules of the game. He thought that he had arrived in a place where everything would be easy, where help would be at hand, where people would explain the rules to him, and not only that, would praise him if he managed to beat them. Now he discovers that his idols don't give a damn about him; he discovers something worse, that no one asked him to come, that he is there uninvited, and nobody notices him. An invisible creature, which, on the rare occasions it is noticed, inspires either momentary pity or lasting disgust.\n\nIf only he knew the language, he thinks, he would show all those people who are unworthy to so much as look at him. They need to be shown how much he is worth, and that is a great deal. But he doesn't know the language, and he is terrified, because these locals speak so quickly\u2014it's like listening to a sewing machine. No, there's no way he's going to learn it. They can go hang, the lot of them, and take their language with them. But if he doesn't learn it, how is he going to find a job? How will he survive? Just the basics then, the first ten words. And he does manage to learn them even though he's not quite sure how. That's when he discovers that this language isn't as impossible as he'd thought. It starts to grow on him. He speaks it and is no longer just some mute presence. But this enthusiasm is short-lived. In his hands, the language is not spoken but broken. He doesn't just break it, he butchers it. Feminine names become masculine, masculine feminine, and most nouns are neutered. Better mute than annoying.\n\n# **5**\n\n_A ll of this was going on at a time when the regime decreed Albania the rose garden and solitary beacon of hope in the universe. Our teachers at school told us over and over again that we were something like the chosen people. And, if you number among the elect, two things are required of you: first, you have to hate the nonelect; second, you must at all costs be happy. Happiness in a totalitarian regime is not a question of choice or fortune; it is a duty. Public displays of unhappiness were viewed with suspicion_.\n\n_The point was that we were the only happy people, the only genuinely happy people, the only pure, the only victims, while against us were the miserable, the liars, primitive and polluted predators. Sometimes I used to think that we Albanians had hit something like a Universal Jackpot of Fortune: all the happiness, the authenticity, the purity, of the universe was contained within a mere 28,000 square kilometers, the surface area of Albania_.\n\n_In our minds, the world was divided into two\u2014Paradise (us) and Hell (everyone else), and whoever tried to cross the borders of paradise and escape was automatically branded Hell's accomplice. For that reason, passports were abolished as we had no use for them, and attempts to escape from Albania were considered high treason_.\n\n_One fine day, however, in my childhood innocence, the borders invaded the world of my imagination in the bloodiest way conceivable. The death of my neighbor, Artur, brought them to life for me like the most terrifying ghosts_.\n\n**If you were a tourist...**\n\nIf you were a tourist, your broken Greek would endear you to people. When it comes down to it, this is the real difficulty about difference: when an American speaks broken Greek, he is classed as a \"nice American,\" but when an Albanian speaks broken Greek, he is classed as nothing more than a \"bloody Albanian.\" When an American speaks perfect Greek, he is an \"exceptional American,\" but when an Albanian speaks perfect Greek, all he hears is, \"You'll never be Greek! You'll never be Greek!\"\n\nSince he is nothing more than a pitiful illegal immigrant, his broken Greek gets on the nerves of the locals. And he knows it. Because of this, every time he gets on the bus, he speaks in a low voice, avoids conversation, and sweats with embarrassment when people try to speak to him because as soon as the locals catch his accent, they turn away and he feels like he's suddenly been struck by avian flu.\n\n_How did it happen_ , he asks himself, _why do they look at me with such fear?_ He would like to ask them, \"What are you all so afraid of?\u2014I love this country.\" But he doesn't ask, because he already knows the answer. \"Don't you watch the news? You lot have taken over this country, you lot have slaughtered us in our beds: they should round you all up, the lot of you, and take you back to where you came from. Back to your own bloody country.\" _So soon!_ he says to himself. _They can't send me back so soon. I've only just found a job. And I work like a dog, night and day, in wind and rain. Look at my hands, they look like someone's been hacking them with a sword. I work to spare your hands. I want to buy a car too. No, I'll start off with a big stereo, then I'll get a huge color TV, then a big washing machine, and then, I don't know, we'll see_.\n\nAnd at that moment he realizes that he is upset, that the phrase \"your own bloody country\" has got to him. Of course he's familiar with all the shit there, which is why he left in the first place, but it's not a \"bloody country.\" There are small children there, just like here; there are mothers who love their children there, just like here; there are teenagers falling in love back there, just like here; there are those who hope and those who despair just like here\u2014though the truth is that those who despair now outnumber those who hope.\n\nThe bus suddenly stops and two men get on. They are wearing identical uniforms and identical grim expressions. His blood freezes. \"lllegal immigrants\u2014out, out, out!\" Then \"Down, down, down, down!\" And he gets down, down, down, further down all the time until he can't go any lower. Fuck.\n\n# **6**\n\n_T hat was when the Chinese arrived. It was as though they had fallen from the sky, peacefully invading our town. We woke up one morning and there they were. We saw them and they saw us. There were dozens of them, all dressed in identical blue uniforms, walking hurriedly around, Mao's Little Red Book in hand. Albania was in the throes of a passionate love affair with China. We all looked on, at a loss to understand what they were doing here. Rumors and speculation were rife in the caf\u00e9s and in the conversations of the adults. The townspeople, although they admitted that they could not tell one Chinese person from the next, because they looked like identical twins, even started to learn a few Chinese names and words_.\n\n_Some said that the Chinese had come to open a factory producing military aircraft; and others, blessed with even more fertile imaginations, claimed that they had come to transform Albania into the biggest industrialized nation in Europe. In the end it turned out that they had come simply to build a plastics factory just outside town_.\n\n_As for our teachers, they informed us that we no longer had to fear any enemy, as Mao Zedong had given his word to Enver Hoxha that if anyone dared lay a finger on Albania, one billion Chinese would came running to its defense. In other words, this small nation of three million, given its friendship with China, should now be considered to have a population of 1,300,000,000_.\n\n_The most dramatic scene of our encounter with the Chinese played out one morning as they filled the small square in front of the town's only hotel, where they were all staying. They were making strange movements, something like a slow-motion version of ballet. Each was in his own world; some were shaking their heads, others moving their hands, some were kneeling slowly down, others were slowly raising their knees up to their chests. They were all dressed in blue and looked absolutely mad. We hadn't a clue what they were doing; almost the entire town had gathered around the square to gawk at the Chinese, to the point that the police grew worried at the size of the crowd. Then all of a sudden we were told that they were just taking exercise, which prompted such comments from the crowd as, \"Allah, preserve us from these people.\"_\n\n\u2014\n\n_I mention the Chinese simply because Artur left a few weeks after their arrival, in December, round about New Year's Eve, for a village on the way to Kor\u00e7\u00eb, near the borders. He had an aunt living there and wanted to see in the New Year with her. In those days, in order to get to those border villages, you had to have a special permit from the police, as these villages fell in the designated \"forbidden zone.\"_\n\n_It later turned out that Artur had not gone to his aunt's house with the intention of spending the New Year with her at all; he wanted to escape. He had calculated, at least that's what people said after the funeral, that the soldiers on the borders would be more relaxed and less vigilant over the holiday season than at other times. He was wrong. The soldiers spotted him trying to escape and shot him dead. I later found out that a soldier responsible for killing someone trying to escape was rewarded with a few days' paid leave. The thought of those soldiers sitting around arguing over who was entitled to the time off made my skin crawl_.\n\n_Artur's death shocked me deeply. It also fired my imagination, and gradually the borders took on metaphysical proportions for me. They haunted my sleep, and nothing pleased me more during my waking hours than listening to stories that were in some way connected to borders: arrests, killings, attempted escapes which had mostly failed_.\n\n_To me the few who managed to get out were the strongest, most special people in the entire world. Some were even from our town. Everyone knew about them, not just because they'd escaped, but because of the repercussions and the terror that followed. Their families were sent into exile; their relatives lost their jobs, forever stigmatized for having an \"enemy of the people\" in their family. Of course, their relatives automatically came under suspicion themselves, suspected enemies of the people forever, on a par with the lepers of the Middle Ages, and were favorite targets of the stool pigeons who had infested our lives_.\n\n**Crossing Illegally, Again and Again**\n\nHe will cross back again, go back into Greece. Kor\u00e7\u00eb\u2013Kalabaka, eight days on foot, nights spent under the stars, if they are out, that is. The \"guide,\" who knows his way through all those hidden paths, charges $300 a head. A typical group of illegals ranges from eight to eleven heads. Bargaining starts in Kor\u00e7\u00eb, the guarantor holds the money, and if they're caught by soldiers and sent back, everybody gets their money back. That's the deal, at any rate, but you never know, because frequently the guarantor vanishes into thin air and the illegals are left with nothing more than a bitter taste in their mouths. But he will cross the mountains again, back into Greece. They might catch him again in six months' time in almost exactly the same way. The two men wearing identical uniforms and identical grim expressions will get on the bus again. This will be followed by: Black Maria. Holding cell, crowded: five people in a two by two cell. Prisoners arguing, the stench of piss\u2014a completely shit existence. He will wait a few days, until enough people have been rounded up for transportation up to the borders, and the Black Maria will drive them up to Kakavia. At the border he will get on a truck that will take him back to the village. He'll be greeted with a \"Get caught again, did you?\" from his relatives, while the villagers snidely remark, \"Get caught again, did he? Poor bastard. Not surprising. Takes guts to live so far away from home.\"\n\n# **7**\n\n_A large country\u2014Russia, for example\u2014isolated with hermetically sealed borders is like one endless prison, but tiny Albania, also isolated with hermetically sealed borders, is more like a regular straitjacket_.\n\n_When I was in high school, my friends and I (or at least the few people I dared reveal my thoughts to) used to say that if you wanted to see the borders of the motherland, all you had to do was climb up onto the roof of a block of flats_.\n\n_I remember our school trip to Sarand\u00eb, where at night you can see the lights from the world-beyond-the-borders. They might have been the lights from a village, maybe a town, who knows? We stood there, gawking at them, imagining, speculating, in secret: \"What are people like on the other side?\" We each had our own ideas on the subject, usually the product of rumors, or pictures we'd seen on television, transformed into mythical stories, which we'd use to feed our fantasies as much as we could. We talked about beautiful beaches, swimming pools, color TVs, but most of all about gorgeous girls, hampered neither by excess pounds nor sexual restraint_.\n\n_\"Girls there are liberated; they don't expect you to come on to them, they do all the work,\" one of my friends reliably informed us. \"They're free out there, how can I put it? You're constantly ambushed by sex!\" he continued, placing unbearable pressure on our adolescent fantasies_.\n\n_At that moment, one of the girls in our class, a brilliant but hopelessly na\u00efve student, asked our instructor in Marxist-Leninism to explain why capitalist cities were so well-illuminated when the proletariat living there was supposedly dying of starvation in dark slums. The teacher looked down at her over the top of his spectacles, and let his eyes roam across the entire class. As soon as he'd reassured himself that the question was motivated by nothing more than innocent curiosity, he answered with his customary ease, the style he used when he explained that in the capitalist West, the proletariat works a twenty-hour day, sustained only by a diet of boiled greens. \"It looks to me as if some of you have been up till all hours staring at Greek monarcho-fascist lighting instead of getting a good night's sleep to wake refreshed to face the class enemy. Well, let me tell you, the lights you saw come from the villas of_ the _rich. They live in enormous houses while the workers starve, the workers have nothing. They live in darkness, unlike the happy proletariat in our country. Read over the notes you take from my talks, and you'll sleep better at night, untroubled by reactionary doubts. Understood?\" Of course we \"understood.\" I have the impression that even that na\u00efve straight-A girl got it, because she suddenly went pale, sat down and didn't say another word_.\n\n_In addition to our borders (which were really neither visible nor tangible, and could not be physically crossed), there were the so-called forbidden zones, areas within a radius of between about thirty and forty kilometers from the borders, which you needed a special permit from the police to visit. If you happened to be caught without one, you were automatically suspected of trying to escape. So Albania effectively had two borders: the legal boundaries between legitimate states and the forbidden zone borders, the unofficial borders of the official borders. No matter where you went, you were bound to trip up over one border or another_.\n\n_This being so, the only window you got on the world-beyond-the-borders was through television and radio. We adolescents in particular were desperate to decode the messages transmitted from this alien planet, which was impossible if you didn't understand the language the aliens spoke. So, and this was especially true of the towns, a passion for foreign languages was born. Italian was the most popular_.\n\n_Learning a foreign language was neither a hobby nor a luxury, nor had it any practical value; it was essential for the amazing challenge of traveling through the imagination to places that were strictly off limits, to the world-beyond-the-borders. Our imagination idealized the word-beyond-the-borders, and the more unbearably oppressive the regime at home became, the more idyllic this other world seemed to us. The regime divided the population into two strict categories: absolute good and absolute evil. We did too, but with reverse criteria: absolute evil was to be found at home and absolute good beyond our borders. Just as the regime relied on paranoid xenophobia for its continued survival, we depended on our own special brand of xenomania to resist it_.\n\n**Two Years on One Border**\n\nHe'll spend a few days in that shitty village where nothing ever happens, where nothing ever moves, like corpses in a graveyard, where people only know how to gossip, and to kill each other off with spite.\n\nHe'll set off on exactly the same journey\u2014Kor\u00e7\u00eb\u2013Kalabaka, eight days on foot, because he has to make it, and so he'll cross the borders illegally ten, twelve, nineteen, thirty-six times.\n\nThe route is full of dangers. In summer there are usually soldiers guarding the footpaths who arrest anyone trying to get through illegally. There are just as many armed bandits lurking, too, waiting to pounce on and rob the illegal emigrant of what little he owns. Whoever refuses to empty his pockets gets the thrashing of his life. In winter there are fewer soldiers, fewer bandits. Instead it's a toss-up between dying in the snow or being eaten by wolves.\n\nHe still remembers that young man, just eighteen years old, burning up with fever for two days. He collapsed and died in the snow. They had to bury him and leave him. There was nothing else for it up there in the mountains. They risked getting buried in the snow themselves. They'd got caught in the most almighty snowstorm, and it was each man for himself. They trudged through it like sleepwalkers, day after day, using all their energy just to stay upright, because they knew that if they didn't, they'd be buried alive.\n\nThe one who died in the snow was called Eddy. It was his first time. They'd carried him and another lad on their backs, gasping for breath, for an entire day and night, and kept saying to him, \"Be patient. We'll get there.\" The snow was up to their waists and he was praying to God that some military patrol would find them. But they're never there when you need them, are they?\n\nA few hours before he died in their arms, Eddy had asked them to inform his fianc\u00e9e and ask her to forgive him for not making it. They'd got engaged a couple of weeks earlier, and he'd promised that he'd come back and take her to Greece as soon as he was fixed up with a job and somewhere to live. He never got further than halfway between Kor\u00e7\u00eb and Kalabaka, eight hours on foot.\n\nHe suddenly makes another calculation: thirty-four times in seven years, Albania\u2013Greece\u2013Albania\u2013Greece. If you added on the days spent in detention centers each time they deported him, it was unbelievable: of the last seven years of his life, two of them have literally been spent on the borders.\n\n# **8**\n\n_M y town, Lushnj\u00eb, is close to the Adriatic. We used to spend our summer holidays by the sea. At some point the regime started putting up bunkers by the hundred thousand, the length and breadth of the country, concentrating on remote and coastal areas. We were prepared for the enemy to appear from anywhere but it seemed that the biggest menace was the enemy we waited for, and waited, and waited, and waited to invade us from the sea. The beaches we used to visit as children were suddenly filled with symbols of war. The irony was that these forbidding-looking bunkers became hot favorites for trysting lovers: temples of war rededicated to the goddess of love. Life itself was taking revenge on the paranoia of the regime_.\n\n_Bunkers popped up along the beach, lying in wait for the terrible enemy, but the enemy never did us the courtesy of appearing on the horizon. In place of this terrifying enemy, the sea would wash up several humble objects from the world-beyond-the-borders: disintegrating sacks, empty Coca-Cola bottles, empty cartons of laundry detergent bearing various slogans and brand names_.\n\n_People would often take these things home and decorate their houses with them. I remember a cousin of mine who was ecstatically happy when she found an empty green carton in the sea\u2014in good condition, too. God only knows who had thrown it in from the Italian side, ignorant of the fetish status it would acquire the moment it washed up on an Albanian beach. The name of the manufacturer was printed on it, accompanied by a picture of some silky female underwear \"holding hands\" with its male counterpart. Both looked rather worn out after their long, difficult journey through saltwater. But my cousin was not put off; she and her mother decided that they had to take it home, and gave it pride of place in their sitting room. It could just as well have been an original Van Gogh or a Picasso_.\n\n_There was something indescribably comic as well as unbearably tragic about all this. But if anyone looks beyond appearances, they might see something irreducibly human in it, too: the determination of people to cling on to the slightest contact with the world-beyond-the-borders. Just as the prisoner worships even the most trivial object from the outside because it represents freedom to him, so these people in their own tragicomic way collected evidence of the existence of a world-beyond-the-borders. It was also a way of establishing some kind of relationship with that world, a way to break through their absolute isolation and to promise themselves and others that one day all this madness would end_.\n\n_The fact that the days of the regime were numbered became obvious to us when the ubiquitous statues of the Eternal Leader erected in all public squares and buildings across the country started to fall down at night. Each political system has its own sacred cows, and the sacred cows of totalitarianism are the ubiquitous public statues and portraits of Eternal Leaders, small, medium, and large: in public squares, on facades of public buildings and blocks of flats, inside public buildings, and inside people's homes. As a boy, I used to believe that statues were supposed to represent dead people. Just to make sure, I asked my mother one day, \"Why are there so many statues of Uncle Enver if he's not even dead?\" My mother looked at me with panic and sadness in her eyes, and said, \"One day I'll tell you all about it. But don't you ever go asking your teacher questions like that.\" This left me with the suspicion that behind these statues lurked a terrible secret I would never be able to decipher and was not supposed to know about. Growing older, I started to understand a bit more. I realized that there was a direct analogy between the statues and fear: as the regime intensified the fear and terror it spread among the people, the number as well as the size of the statues it put up grew proportionally_.\n\n_The tyrant's statue essentially symbolized the eternal stasis of tyranny, as the purpose of tyranny is to subject everything to a state of stasis: a stasis of thought, desire, and time itself. It aims at the predictability and stasis of the graveyard. Of course they say that when the statue of a tyrant is pulled down and tyrannical rule comes to an end, people are suddenly liberated and blossom, finding the path to truth and wealth. Sadly, this is not the case. Tyrants are merciless beasts, precisely because they leave behind distorted societies worn down by oppression and above all suffering from an orphan complex. Those who give themselves over to indiscriminate looting and destruction the minute the statues come down are like orphaned children robbing the corpse of a false and terrifying father_.\n\n_After the fall of the tyrant's statue, the tortuous journey to self-knowledge begins: pieces of the statue will live on for a long time to come in the mindsets of the people who grew up and lived under tyranny_.\n\n**Quiet August Days**\n\nAugust. The city is virtually empty. The evening newsreader makes this short announcement: \"Five illegal immigrants have lost their lives, drowning at dawn off the coast of Mytilene when their boat capsized. Among them were three small children.\" This was the last news item before sports. The usual tragic procedure: the survivors will be put up in some kind of reception center with wretched living conditions. The usual procedure will be followed: detention, processing, deportation.\n\nIn all probability most of them will try to repeat the journey, paying dearly for it and risking life and limb in the process. Are they out of their minds? Maybe, but that's just the point. This brand of madness, this doggedness of the wanderer to enter the temple of consumerism, gambling with life itself to get there, fearing neither death nor the police, nor the dehumanizing humiliation of it all. In Morocco illegals are known as _harragas_ from the Arabic verb _haraqa_ , meaning \"to burn.\" In order to avoid deportation, African illegals coming to Morocco burn their papers so that the authorities can't establish their identities. They are undocumented and therefore have no identity. They have wiped their countries off the map. Their homeland is now another country, legal or illegal. And besides, there is something that is more powerful than anything else: death has lost its sting.\n\n# **9**\n\n_I crossed the borders into a foreign country, Greece, on January 15, 1991. It was the first time I'd seen the borders of another country. It was the first time I'd seen the borders of my own country; the outer limits of a world that had perhaps thrown us out of time, out of the world. I thought I'd be escaping on my own, but ended up walking with an entire caravan of human beings. A caravan of human beings moving forward with one single demand: to break through that terrifying taboo otherwise known as the borders. Escape as an end in itself. Escape as illness_.\n\n_I had traveled all night in a truck, hidden behind a mountain of packing cases. At one o'clock in the afternoon we reached Sarand\u00eb. We set off for the borders from somewhere round there half an hour later and arrived at around three. When they saw the human caravan making its way to the border, the driver and the codriver decided to abandon their truck and join it. We covered the final kilometers before the border post on foot. There must have been about sixty of us, mostly young men and all strangers. At the border post on the Albanian side, four soldiers holding Kalashnikovs were waiting for us with their officer, who looked at us with an expression revealing something between contempt and anger_.\n\nOfficer: Where are you going?\n\nVoice: We're leaving....\n\nOfficer: Who said that? [Silence]\n\nOfficer: Have you all got passports?\n\nVoice: Have you given us passports?\n\nOfficer: Who said that? [Silence]\n\nOfficer: You know, things could be worse on the other side.\n\nVoice: Let us through, let us go.\n\nOfficer: OK. Whoever wants to can go through.\n\n_We broke into a run, scared that the soldiers might shoot us in the back with their Kalashnikovs. We got through the barbed wire, which was still intact, meaning that we were the first to cross at this point. We started bashing at it with a stick to see if it was electrified and then we opened a hole in it with some sticks that the drivers had brought from their truck until it was big enough for us to get through, one at a time. In my haste, I ripped the sleeve of my coat, but at that moment my coat was the last thing on my mind. The soldiers, fingers on the triggers, looked on as we tried desperately to cut through the wire, while the officer maintained the cynical smile painted across his face_.\n\n_After getting through the barbed wire we had to cross a small stream. We did so with ease, even though the water was freezing. Then we started running up a hill, looking for the first concrete signs that we had actually fled and reached the other side of the border, the other side of the world. We carried on running until someone shouted, \"We've crossed the border!\" I don't know how he could tell. I can only remember hearing hysterical whoops of joy_.\n\n_There were no soldiers and no policemen on the other side. We had reached the free world. We had arrived in the West. A middle-aged man threw himself to the ground, weeping, pummeling the ground rhythmically with his fists, addressing it as though it were a fellow human being. \"I've paid for you with seventeen years in prison. I've paid for you with seventeen years in prison. I've paid for you with seventeen years in prison, damn it.\" Things calmed down after about half an hour until someone asked, \"Where now?\" Nobody could answer him. Everyone shrugged their shoulders in total bewilderment. We all knew where we had escaped from but not one of us knew anything about where we were going_.\n\n_Some of our number spotted a dirt track and the rest of us decided to follow. A few of us could speak Greek, Greek Albanians who were immediately put in charge of our march into the unknown. As we walked, we started to speculate about what was in store for us. Someone had heard that we were going to be put up for a short while in high-rise blocks and afterward would be asked which Western country we wanted to go and work in. At that point, the two drivers of the truck jumped in, saying that they would put in a request for Germany as they were sure that there was plenty of work there, paying astronomical wages for anyone willing to make the journey from Berlin to Baghdad transporting fuel, especially now as war was about to break out in Iraq. The best news of all came from another man, who said he was sure that the road led to some harbor or other where the American fleet would be waiting for us to take us straight to the States. He almost got into a fight with another man who claimed that he had heard on Greek radio that the ships weren't American but belonged to the UN, and Albanian refugees would be taken to Italy. In the end we agreed that the ships were waiting for us, but that their provenance and destination had yet to be verified. After this, the discussion turned back to the borders. Some argued that the borders would be closed in three days' time; others said ten, while the minority opinion was that they would never close again_.\n\n_In the meantime, we had come across the first houses and first residents along the road. Some Greeks had come out onto their doorsteps, and the pity on their faces was diluted only by the total astonishment they felt. The Greek speakers among us stopped to talk with them, explaining who we were and where we had come from, and asked them where we should go. Some of the old ladies offered to give us water, and that was how I learned my first words of Greek. \"Efharisto,\" meaning \"thank you\" and \"nero,\" \"water,\" and it wouldn't be long before I added a third word to my list: \"douleia\"\u2014\"work.\"_\n\n_After walking and walking we eventually arrived at our destination, a coastal village, at about eight that evening. Someone who was passing through signaled to us where we were supposed to wait for the bus that would take us to the refugee center. We all assembled there and waited. After half an hour, a man approached us, introducing himself as an Australian journalist. He wanted to know if any of us could speak English. With what little English I had managed to master I offered to talk to him. He asked us where we had crossed over from, whether the borders were guarded, where we wanted to go, what part of Albania we had come from, and what the current situation in Albania was. After getting the answers to these questions, he explained that a bus was due to arrive that would take us to a refugee center, where we would be given food, new clothes, one room between two, be taught Greek\u2014and after a few days we would be found work. I translated all this for everyone, our future according to the Australian journalist, and they all started cheering. \"This is what the West is all about!\" one of the drivers called out, at the same time as urging me to ask the Australian if we would be allowed to go and work in a country of our choice. I didn't get the chance to ask either about that or about the ships waiting for us at the unspecified harbor to take us to America, Italy, or wherever else our starved imaginations desired. The Australian had vanished, leaving us alone with our fantasies_.\n\n_A few started allocating the bedrooms we were supposed to be sharing. The two drivers unsurprisingly decided in an instant that they would share. For others it was harder to find a roommate, so the drivers, relieved of that particular anxiety, volunteered to sort everyone out. First we counted the number of couples. None of them were married; this was a problem. We had no idea whether the regulations permitted unmarried couples to stay together, and besides, we didn't even know if men and women would be housed together or in separate quarters_.\n\n_We came up with a temporary solution while we waited for the bus that was to take us to what we all had started calling the refugee center. Some of the group, mainly the younger boys, went and pressed their faces up against the window of a nearby caf\u00e9. They were gawking in astonishment at the color television and the images it transmitted. The patrons of the caf\u00e9 gawked back at the Albanians gawking in astonishment at their color television with a corresponding degree of astonishment. Eventually someone got up and came to the door. He was holding a biscuit tin and started throwing biscuits at us, in the way that you would toss grain to pigeons, or corn to chickens or ducks. Some of us tried to catch the biscuits. And that was when I was reminded that I had a stomach attached to my body and realized that I was hungry_.\n\n_Some of our group just stood there, glued to the TV screen through the caf\u00e9 window. One of them, he must have been about twenty, his face badly worn by the sun and exhaustion, suddenly swung round and whispered in an innocent, stunned tone, \"But there's no sex!\" Perhaps he'd imagined that in the world-beyond-the-borders, just as my old high school friend had claimed, there is a constant ambush of sex and as soon as darkness falls, something along the lines of an all-night orgy begins. This was also part of the fantasy that we had carried with us_.\n\n**The Gender of Borders**\n\nWe don't all cross in the same way. The authority of the borders reminds us all of our very different origins, different classes, different castes, different skin color, and, above all, different genders.\n\nMarch 18. She left that day. She was only seventeen. Everybody was getting out. The country was like a poisoned organism that purged itself of people in its effort to survive. But none of the girls had left town, not until then, at least. She was the first.\n\nPeople were rushing to get out of hell while there was still time. They say that hell is one place, but in reality, hell is private and particular to each one of us. She wanted to escape hers: her father. For as long as she could remember she had wanted to leave, to escape. Escape his shouting, his hateful looks, her mother's screeching, which she still heard in her sleep. She stills shivers today when she recalls that threatening knock on the door. As soon as she heard it, she knew exactly what was going to happen. The same routine over and over again, the same methods. Music; he would always turn the volume up on the radio to smother her screams. That was the only time music was heard in their house. He would pummel her head with his fists like a madman until his hands were sore. Then he'd remove his belt, and whip her entire body with it, until the point where her skin split open and her flesh bled, the same words punctuating the thrashings: \"You ought to be ashamed of yourself!\"\n\nHuman beings have an incredibly high pain threshold and can suffer abject humiliation and degradation if they are convinced that there's no way out. I can't say whether they accept it or just get used to it, but they do learn to live with it, thinking there is no alternative. That was true of this girl; she had just about come to terms with the fact that this is what her life was like and always would be like. I say just about because there was something that was helping her to keep it together, and that was her dream of becoming an actress. That's what she wanted more than anything. Perhaps it was because for as long as she could remember, she had wanted to live the lives of other people and not her own.\n\nShe was in the last year of high school. There was a theater group at school she went along to, behind his back. But he found out about it one day and the curtain went up on the same old show: music, punches, belt, screaming, blood, but a different script this time: \"Actress? Over my dead body!\"\n\nA week later she heard that they'd opened the borders. At first not that many people were leaving, but gradually more and more decided to go. All the men left. She wanted to leave, too. She could think about nothing else. She had to leave. But how\u2014and who with?\n\nShe'd never seen him before. The first time she saw him was when her cousin introduced them. It was raining heavily and she found it hard to see his features clearly. But her desire to leave was so overwhelming that it blurred her vision. \"I came back from Greece three days ago, and I'm off again the day after tomorrow,\" he told her. \"I'm told that you want to leave\u2014I can get you to Athens and help you find work.\"\n\n\"Yes. I do want to leave. If you can take me to Athens, I'll pay you back as soon I get a job.\"\n\n\"That's easy enough,\" he answered. \"See you the day after tomorrow.\"\n\nThey arranged to meet a short distance outside town. Lying awake all night, she got up in the morning, took her school bag, replacing her books with a few clothes, some childhood photos, a few slices of bread, and some olives for the journey. She opened the front door slowly. He had left early, as usual, and gone to work. So had her mother. She stepped out onto the street, stopping occasionally to look back because she had the feeling that he was secretly following her, that he'd suddenly jump out in front of her, just like ghosts do in children's nightmares.\n\nShe arrived on time. Her cousin's contact appeared shortly afterward. She had barely greeted him when a truck pulled up in front of her. \"Jump in,\" he told her, \"before they see us.\" She jumped in and recognized the face of her cousin at the back, surprised to see him, as he hadn't told her that he was planning on coming along. He was very cool toward her and the only thing he said was, \"It was a last-minute thing.\" The truck started. That was important to her. She kept saying to herself, \"I'm not dreaming, I'm leaving. I'm actually leaving!\" They slowly left the villages behind them, the towns, and the people of Albania, but, most importantly, him: her father.\n\nAbout halfway there, sometime in the afternoon, they stopped to eat. Everybody got out of the truck. She was the only girl. She counted the men. Seven. She couldn't eat a thing. She was so anxious her stomach tied itself up in knots. When they finished their food, that character pulled her to one side and said, in a menacing voice, \"You're looking a little thoughtful. If you're having second thoughts, forget it. There's no turning back now.\"\n\nEveryone got back into the truck to continue the journey to the borders. Most of the time nobody said a word. He was sitting close to her now, every now and then \"accidentally\" resting his hand on her leg, her breast, her shoulders. She went numb. Only at that moment did she ask herself, \"Who is this man? Who am I with?\"\n\nShe searched out her cousin's face in a desperate plea for help, but he wasn't even looking in her direction any longer. She could cry for help, but to whom? She started shaking with fear. In her desperation to escape her father's authority she had failed to appreciate that her father was just another link in a chain of violence that had been simmering below the surface for so many years and was waiting to erupt like a volcano.\n\nTwo days later they reached the area outside Gjirokast\u00ebr. The truck stopped. It was dark. The driver announced that they'd be spending the night there: The border was only a couple of hours away on foot, but it was best to cross at dawn. \"You have to cross when they're between shifts because there aren't any guards about then.\" She got out of her seat and climbed down out of the truck. \"Where do you think you're going?\" he asked her threateningly. \"I need to stretch my legs,\" she answered. It was pitch black outside and you could hardly see anything at all. But after a while her vision adjusted to the darkness and she could make out a few houses in a small village in the distance. She started walking, carefully moving away from the truck. She got quite far. She turned back to make sure no one had seen her. She stopped and stared at those small village houses. Everyone would be asleep at this time of night, but what if she tried to run there to ask for help? How long would she need? Ten, fifteen minutes? Of course they'd open up, if they saw a young girl alone begging for help.\n\nJust at that moment she was grabbed violently by the hair and pulled to the ground. \"Whore! Doing a runner, were we, scum?\" He grabbed her hair again and pushed her face into the ground. She couldn't breathe. \"Who exactly do you think you are? Coming across all innocent when you're nothing but an old whore. If you weren't, you wouldn't have left with me, would you?\" With a sudden violent jerk, he turned her over and ripped open her sweater. She started screaming at the top of her lungs, crying for help and lashing at him with all her strength. But she was no match for him, using one hand to punch her and the other to cover her mouth. She couldn't breathe, but still tried to hit him wherever she could. His hand was stuck to her face like some phenomenally strong suction cup until very gradually her breath failed her.\n\nThey say that a short while before you die your entire life flashes in front of you. That's what happened to her. She relived all her life, everything she had left behind: her mother's face, old before its time, her eyes (she'd never been able to understand why they weren't permanently full of tears); her father's screams; the threatening knock on the door; the music; the belt; the punches; the screaming; the blood; prison; hell; the words after every beating, \"You ought to be ashamed of yourself.\" \"Actress? Over my dead body.\"\n\nShe stopped hitting him. She could feel his filthy breath on her and she dug her nails deep into the earth. At that moment her soul left her body, leaving him with the empty shell. I don't know how long it took because, for her, time had come to a standstill, but she collected together whatever remnants of her soul she could find and re-entered the hollow shell. Blood was flowing down her legs. She staggered to her feet, walking in a daze. She could hear his voice calling after her in the dark, mocking her. \"Look! She liked it so much, she's drunk on it!\" Other voices joining in. She struggled a few steps further until she could hear her cousin gloating, \"That's what happens to all of them, all the ones who fancy themselves as artists. Still like the theater, do you?\" She looked like a robot. She pulled herself up into the truck, curled up in a corner like a wounded animal, and slept the deepest sleep of her life.\n\nShe was woken at dawn by voices telling her to get up. They were starting off for the border. Trekking through bushes, her clothes kept snagging on thorns and her jacket got ripped. Suddenly they came to a stream and her shoes got soaked, so she had to continue barefoot. After trudging along for quite some time, they were nearing the Albanian border post. It looked abandoned; no one there. Someone gave the order to run because Albanian soldiers often deliberately kept out of sight. They all started running at once. The soles of her feet were bleeding but she still tried to run as fast as she could. Three gunshots were fired and she felt something burning on her. Her hair had caught fire. She threw herself to the ground and somehow managed to beat out the flames with her hands. The bullets had just skimmed the surface of her skull. She was on her feet again, her knees bloodied, her hands burnt and very painful, but she went on. Nothing could stop her. She had already crossed the border. Back there at Gjirokast\u00ebr.\n\n# **10**\n\n_T he bus did eventually turn up and take us to the so-called refugee center, where all the other Albanian fugitives who'd arrived before us were assembled. Only the bus wasn't a bus but a filthy truck. We climbed in. It was already very dark and I can't remember exactly how long the journey took, only that we were all squeezed in tight next to each other and hardly said a word. We'd suddenly run out of enthusiasm and it was only then, after all those hours, that we started to feel anxious about the unknown_.\n\n_The truck came to a standstill and we realized we were supposed to get out. When we did we saw Greek policemen shouting and brandishing their truncheons in an effort to impose order. A single spotlight illuminated the scene: Greek soldiers and an enormous crowd of Albanian fugitives. From the policemen's gestures, we gathered that they wanted us to stand in a line. We formed a huge line, new arrivals, all of whom had to be registered: name, surname, father's name. I saw the two drivers looking rather nervously at the policemen who were pacing up and down, brandishing their truncheons and bellowing at us. They looked as though they were asking themselves why angry policemen had to carry truncheons in the West as well_.\n\n_After waiting for about an hour in the line, it was our turn. We stood in front of a table while a policeman asked us, one after the other, in rhythmic, broken Albanian:_ \"Ermi? Biemri? Ermi babai?\"\u2014 _Name? Surname? Father's name? I smiled at his efforts. He responded with a look that seemed to say, \"That's the last thing you want to be doing now, you poor idiot!\" I handed him my Albanian identity card, he took down my details, and I stepped aside, waiting for Xhemal to finish. I had met Xhemal on the escape journey. When the policeman asked to see his identity card, Xhemal answered with a phrase that does not exist in any language on earth: \"pashaport kaput,\" which was amplified by a hand gesture indicating decapitation, as though it was his identity card that had had its head severed. The policeman stared at him, as though he had just sighted a UFO, until it dawned on him what Xhemal was trying to say. When he did, he returned to the formula:_ \"Ermi? Biemri? Ermi babai?\" _But even in this Xhemal's originality shone through: apparently his name was Dhimitris Dhimas, son of Vasilis Dhimas_.\n\n_When he finished I asked him what that was all in aid of, and he told me that Greeks look on Christian names in a more positive light than Muslim ones: Muslim names scare them. One of the drivers quickly jumped in, reprimanding Xhemal for his lie. \"This is the West, this is a democracy, all men are equal. And since when were we Albanians religious, anyway?\" he added in the tones of a schoolteacher. The truth is that over the next few days it was hard to judge what precise advantage the name Dimitris yielded Xhemal. He was staying with us, in the same place, in what looked like a warehouse, behind the village soccer field. During the registration process, we discovered that the name of the village was Filiates_.\n\n**You Weren't Invited**\n\nIllegal immigrant. That's your nickname. That's your name. That's your label. After all, you turned up here without an invitation. That's how people used to migrate in the days before World War I, uninvited, without visas. But that was a very long time ago and things are different now. And if you think that by reminding the locals of that time, they'll start feeling sorry for you and accept you, you're living in cloud cuckoo land. You won't just be an illegal immigrant if you go around doing that, you'll be an impudent illegal immigrant.\n\nThe fact that you arrived uninvited makes you feel uncomfortable, and deeply guilty, and you may never get over that feeling. Because apart from everything else, they keep reminding you of the fact. This is your original sin. Each time you try to stand up for yourself, you'll hear it: nobody asked you to come. Each time you try to break out of this obscurity, they'll be there to remind you that you're an unwanted guest.\n\nYou tell them that you want to be legalized, that it's unbearable trembling every time you see a Black Maria and, anyway, who wants to feel like a scared mouse all the time because he hasn't got the right papers in his pocket? I may have arrived without an invitation but I work just like the rest of you do, I pay the same taxes as the rest of you do, and most importantly, my boss, or rather, my bosses, need me. Yes, I do realize that you are feeding me, but let me tell you that I more than repay it. Yes, I am dependent on you for my survival, but you depend on me for your wealth. That's life. Give and take. I have started to build a new life here, I have got used to this city, and who knows, this city might eventually get used to me. So why am I illegal and worse than a stray dog? The city is deaf to your defense. The city is deaf. And on the news, the journalists give voice to the vox pop and want to make sure that you never manage to shake off your nickname, your name, your label: illegal immigrant, illegal life, illegal.\n\n# **11**\n\n_A fter registration we were told to go to the churchyard, where we would be given something to eat. We ate in the same way all hungry people the world over eat. Meanwhile, Albanian fugitives continued to pour into the so-called refugee center, seemingly endlessly. It was as though the entire country had made a unanimous decision to take advantage of the open borders and get out as quickly as possible. The policemen and soldiers looked on in astonishment at the unending human caravan and found it hard to comprehend. \"If this goes on much longer, there won't be anything left in Albania apart from the trees,\" said Xhemal_.\n\n_Since we'd managed to satisfy our hunger a bit with a helping of spaghetti, we started discussing how long we thought it would take for Albania to catch up with Greece. One of the drivers reckoned about five years. The other thought more like fifteen, while Xhemal, who'd been temporarily overcome by some sort of melancholy, had the last word: \"It'll never happen; Albania can't move forward as long as Albanians are running it.\" Later, after our hunger had abated, we started to feel the cold. It was freezing and it was then that we understood that before we could work out how many years it would take for Albania to catch up with Greece, we'd have to decide what was going to become of us that night_.\n\n_The big warehouse where they put us up was full. The few foam mattresses they'd laid out had already been taken over by weary bodies, which had been walking for hours if not days. Whoever found an empty corner of unoccupied foam to lay his head on was very lucky. We slowly realized that the high-rise block of flats, with one room for two people, hot running water, and new clothes that the Australian journalist had promised us might be a long time coming_.\n\n**The Strange Habits of Illegal Immigrants**\n\nThose who cross borders illegally develop strange habits. Laughing too much is one of them. They are overcome by a mood for fun and jokes, as though they have just emerged from some side-splitting revue when in fact danger and the threat of death is all around them. Perhaps it's the imminence of death and the fear of it that provoke this laughter. It's as though they want to cajole and seduce death. Human laughter is the perfect cover, it's like telling death, \"There's no way you're getting us. Look at us\u2014we're laughing, we're not even thinking about you. We love life. We want to live. We want to survive. You'll have to look elsewhere for customers. You won't find any here.\"\n\n# **12**\n\n_I spent the night on the legs of one of my compatriots, and when I woke in the morning, I found another compatriot asleep on my own. I was forced to wake him so that I could get up. Xhemal was already awake. So were the two drivers. They beckoned me over to join them. We had already formed a solid friendship, despite meeting only yesterday. The drivers from Tiran\u00eb, Xhemal from Elbasan, and myself, from Lushnj\u00eb. That's the way people make friends and form groups when they're far from home and miserable. Unpredictable alliances, just like teenagers. The police allowed us to move around freely and go wherever we pleased. The young lad from the previous night who had asked why there wasn't any sex on the color TV in the caf\u00e9 joined us, too. He never said a word and had the look of someone who was searching for something he knew was somewhere close by, but still couldn't find. After the episode outside the coffee shop, we all called him Sex Boy_.\n\n_We left the warehouse and went for a short stroll round the village and ended up in a caf\u00e9. The drivers had ten dollar coins with them, which to us seemed like a small fortune. They were going to treat us to coffee or tea. To my good fortune, the owner of the caf\u00e9 could speak Italian, so I was able to help the drivers change some of their dollars into drachmas. The proprietor took the dollars and examined them closely, first in natural and then artificial light. After satisfying himself that there was nothing suspect about the money, he went ahead with the exchange, and we looked on with almost religious piety. The drivers took the drachmas, most of it in notes, and then gave the rest of us permission to look at them. Xhemal made the first comment. \"There you go. Normal stuff. Not like ours, all full of pictures of soldiers and hammer drills.\"_\n\n_At some point the proprietor spoke to us again, telling me that he'd once been an immigrant in Italy and Germany but had come home a few years ago and opened the caf\u00e9. \"What did you come here for?\" he asked me. I felt a bit uncomfortable. What should I say? That we wanted to cross the border to avoid spending the rest of our lives going insane in a despicable prison? Instead I told him that we were hoping for a better life and to live like the rest of the world did. He listened while he dried a glass on a tea towel, and in the end let a cynical smirk cross his face. \"Poor bastards. You've no idea what you've let yourselves in for,\" and the cynical smirk was slowly replaced by an enigmatic expression, as if he'd just delivered the most profound prophecy in history. \"What do you mean?\" I asked him, trying to understand the first messages from this new planet we had just landed on. The two drivers, Xhemal and Sex Boy, were desperately trying to follow this conversation. The owner eventually resumed his normal expression and started saying that the fate of an immigrant is always really shit, and when you arrive somewhere uninvited, then it's a million times more shit. \"Even the worst country in the world is better than a foreign country,\" he concluded, lit a cigarette and took a sharp intake of breath, as though he wanted to expel something that had lodged itself in his throat_.\n\n_Because I couldn't stand the pressure of my companions, who wanted instant translations of everything, I just told them what I had understood. At first they went all quiet, trying to decipher the meaning of the conversation, and then one of the drivers tried to give his interpretation: \"He wants to sell us something. It's all got to do with these Greeks and their Trojan horses.\" The rest said nothing, at a loss as to what the connection between an ex-immigrant Greek caf\u00e9 owner, the shitty fate of immigrants, and the Trojan horse might be_.\n\n**The Supermarket Door**\n\nEach time he makes the journey from Kor\u00e7\u00eb to Kalabaka, eight days on foot, the illegal immigrant tells a lot of jokes. He always tells the story about the supermarket, which never fails to get even the most morose members of the group laughing.\n\nIt was the first time he'd been to a supermarket. Back home there weren't any. At first he'd lived on diet of souvlaki, two solid months of souvlaki, until he was sick of the sight of it. One evening, when he'd just knocked off work and got his first wages, he decided to go to the supermarket. He got to the entrance, and saw that the door kept opening and closing as people went in and out. But when he tried to go inside, it refused to open. He tried once more, and again. Nothing. He took ten paces back and watched to see how other people were managing to get in. Everybody else seemed to be getting in without any difficulty. He tried again, this time moving more quickly. Nothing. He tried a slower approach; nothing. He'd turned red with embarrassment and frustration. Then he spotted a gentleman reversing through the doors. Interesting. He tried that, too, turning his back to the door. Nothing. He was at a loss. How come all these people were entering and leaving the supermarket so effortlessly, while he, no matter what he tried, failed? Perhaps the door had a thing about Albanians. That might be it. So he tried to do the most obvious thing, trail one of the other customers as they went in. Why hadn't he thought of that earlier, idiot? But at that moment nobody seemed to be going in. \"Just my bloody luck,\" he whispered. Then he saw his salvation; an extremely obese woman, waddling along like a wounded duck, who stood no chance of getting into the supermarket unless both doors opened. He started following her, trying to look casual, from behind, in what was perhaps his last hope of getting in; he was determined not to waste this chance, but in his anxiety he got a bit too close to her, with the result that when he saw the door open, he all but pushed her inside. Realizing that he'd finally crossed the threshold of the supermarket, he apologized to the lady. He was sweating as though he had just finished the 200-meter hurdles. The woman shot him a dirty look, borderline homicidal, and he, embarrassed, quickly moved away to the sound of her muttering. He couldn't be sure but he thought he could make out the word \"Albanian.\"\n\nThe others were listening, laughing, laughing again, some louder than others, the ones making the eight-day journey from Kor\u00e7\u00eb to Kalabaka on foot for the first time. But at that moment someone started shouting, in a strained voice, the voice a hunter uses when he finally tracks down the quarry he's been after for days. \"Quiet. Everyone down. Border patrol.\" Everyone fell silent and dropped to the ground, flush with the earth. One with the earth.\n\n# **13**\n\n_A fter a few minutes' silence I found the courage to ask the caf\u00e9 owner if he could find me a book that would help me learn Greek. He pointed outside through the window and told me I could get them from a bookshop, as those sorts of books were very cheap. Of course I didn't continue the conversation, as I wasn't in a position to buy so much as a stone from the street, let alone an entire book. I asked him if he could give me a job, so I could earn some money and buy some books and learn some Greek. He said he'd think about it and let me know. We sipped our coffee, watching images from the war in Iraq on the TV. They had absolutely no effect on us, though. We had our own battle to fight_.\n\n_Just as we were getting up to leave, a scream was heard, a scream so loud that it made my hair stand on end. It was the owner. He looked ready to kill. I was terrified by this transformation and the only reason for this metamorphosis that I could think of was that he'd discovered that the drivers' dollars were counterfeit. He came rushing over to us, grabbed Sex Boy by the arm, and pulled him off in the direction of the toilets. Sex Boy didn't resist at all. I was completely bewildered. So was everyone else. The owner and Sex Boy disappeared behind the whole door marked \"Toilette\" while we sat there exchanging looks in silence. A minute later the white door opened again and the owner and Sex Boy emerged. The owner seemed a little calmer, and made a gesture as if to say, \"OK, you can all go back to where you came from now.\"_\n\n_Once outside we asked Sex Boy what the hell was going on. \"I pissed and missed,\" he answered, with the naturalness of someone who had just been asked their name. To cut a long story short, some of his urine had splashed onto the toilet seat. \"So what happened?\" we asked him. \"Nothing. He told me to clean it and I did,\" he replied in the same matter-of-fact way. \"And that's why he was behaving as though we'd killed his mother?\" asked one of the drivers. \"Bloody Greek,\" added the other driver, the one who'd earlier volunteered the theory about the Trojan horse and who at that moment was vowing to boycott the caf\u00e9 and told the rest of us that we should, too. Then Xhemal spoke: \"We Albanians don't even know how to piss any more. Fucking communism.\" And, with that, he brought the conversation to a close_.\n\n**Work, Work, Work**\n\nBeing an immigrant can mean a lot of things, but most of all it means work. You don't emigrate so you can play the tough guy, but to save up money. You will do anything to succeed in this: you take on two, maybe three jobs a day, without insurance, of course, undercutting the going rate for the locals; you become a strike breaker, and if need be, you try selling sob stories to your employers to get their sympathy until you realize that sympathy is in short supply; you turn up in Omonia Square at dawn on the off-chance that there's some work available, and you stand there waiting like some filthy municipal statue nobody can be bothered to clean; you live in a cave, with another ten, fifteen, maybe twenty people, blurring the distinction between home and pigsty; you eat bread and salt, or just bread; you always doze off on the bus from exhaustion and lack of sleep; your stale sweat can be smelled a mile off, because you don't have time to wash and you need to economize on electricity. You make even the most tight-fisted look generous. You count out your money with the same precision as the anemic monitors drops of blood during a transfusion. You spend nothing, you buy nothing, you live on only the absolute necessities\u2014only counting money satisfies you, that and the thought of more work, even more work. Suddenly, without warning, your strength starts to fail you, arthritis strikes, you get suspect pains in your kidneys, your back, and your heart. You'll be lucky if you make it into the operating theater. Many don't. They die on the job; some wall or other collapses on top of them, because their bosses don't want the expense of making the site safe. Because everyone knows that immigrants die silently, like flies.\n\n# **14**\n\n_W e went to get the bus to the big town. They told us we might find work in Igoumenitsa. While we waited at the bus stop some passers-by had directed us to, we started talking about cars. Xhemal thought that the price of a car was the equivalent of about ten days' work. The drivers thought about a month. Sex Boy and I said nothing. Xhemal and the drivers went on tirelessly and after a while got into the different makes they were going to drive. Xhemal preferred French manufacturers, the drivers German. And that was when we all found out that cars were Xhemal's great love_.\n\n_He used to drive a car himself, and was more familiar with their design than he was with the insides of his own pockets, as he put it. \"I only worked as a driver for one year. I drove a municipal shitmobile, and in the end they took that away from me, too.\"_\n\n_Standing there in the cold, waiting for the bus, Xhemal started to tell us all about his relationship with the shitmobile, or the yellow car, as he sometimes referred to it on account of its color. The shitmobile was a special municipal vehicle that had something to do with the sewage system. When Xhemal finished his military service, driving the shitmobile was the best job he could get. He found it humiliating at the time, because he felt he was destined for higher things, but need prevailed. Oddly enough, it was while driving the shitmobile that he experienced the best and the worst days of his life. Xhemal narrated with great passion and a touch of theatricality, as though he had been waiting for this moment for years, there at the stop, waiting for the bus that would take us to Igoumenitsa, to pour out his heart. It was all about a girl, who just happened to be one of the most beautiful in town_.\n\n**The Kitchen Dogsbody Generation**\n\nThe first generation of immigrants is the kitchen dogsbody generation. These immigrants get the worst of it: the negativity, the contempt, the fear. To them, all jobs are good jobs, as long as they get their stomachs filled with a little leftover. That little leftover is the reward, their therapy, the Promised Land, the big lie, the promise of return. This is the generation that can't get the language right, or doesn't speak at all, the generation that is terrified of the police, who live in fear of their boss's look, who consume loneliness in silence, who bend right down as far as the dust, who store up as much money as resentment. Resentment is their fortress, it is where their fear resides, the negativity of the locals, the contempt, the pressure and the ridicule of employers. This is the kitchen dogsbody generation. This generation does not have furniture at home, not unless it's been chucked onto the trash heap by some local who has replaced it with something new. For the kitchen dogsbody generation there is no such thing as a holiday. There is only one thing: work. The dream of going home embracing the dream of staying on because that's just it; the kitchen dogsbody generation is schizophrenic at heart.\n\n# **15**\n\n_X hemal hadn't even been driving his shitmobile for two weeks when he first met Alba at a cousin's house. It was love at first sight for both of them, something straight out of a romantic novel. She was in the last year of high school and, being the daughter of a high-ranking army officer, was able to avoid school without any difficulty. She would often get excused from lessons on spurious grounds, and the teachers never minded. Xhemal, on the other hand, would take the day off, usually without making excuses. They would meet on the outskirts of town and drive to the fields outside town in the shitmobile. They often made love in it, too_.\n\n_One day, the sewer in the Party Central Committee building got blocked, and unfortunately for Xhemal, they were expecting a very distinguished visitor to arrive from Tirana that day. They quickly tried to alert him and his shitmobile, but neither vehicle nor driver were anywhere to be found. When he later got back to work, he walked into a state of war, and it was a miracle that they didn't take away his car then and there. He got off with a horrible reprimand and a warning_.\n\n_Less than a week later disaster struck the happy lovers. Alba's father committed suicide. One morning, just like any other, he walked out onto his balcony holding a pistol to his temples, screaming, \"I am no traitor. Long live the Party! Long live our glorious army!\" and pulled the trigger. It was a terrible sight, because Alba's house was right in the center of town, and at that hour everyone was on their way to work. Some said that her father's brains landed a whole kilometer away from their block of flats. But perhaps these calculations can be put down to the usual tendency of the townspeople to exaggeration_.\n\n_At first they said that Alba's father had lost his mind. Later it was discovered that he'd been accused of being a traitor. No one could say what or whom he had betrayed. The official line was the last word from Party Central Committee. They said that he was a member of a group of conspirators inside the army preparing the ground for a foreign invasion of Albania. Opinions were divided on the subject of the ring of conspirators. Some said they were Russian agents; others thought they were American. There was a third version, which had them down as Russian-American double agents. The fourth interpretation was one that terrified everybody. People were used to groups of conspirators periodically coming to light, either in the army, from among teachers or agricultural cooperatives, even from within the ranks of the Party, and sometimes completely random individuals, and such revelations were followed with persecution and imprisonment. That was when people started to be afraid of their own shadows, because any one of them was a potential traitor_.\n\n**Drowning Your Sorrows in Adult Cinemas**\n\nHe imagined that this city would resemble one enormous supermarket where he could get anything he wanted: dreams, happiness, prosperity, even love. That's what it looked like from a distance, anyway, but now, standing in the center, he's feeling dizzy and scared. Because everything around him looks so cold, so fast, so soulless, so indifferent.\n\nIn this city he is suffering from appalling loneliness, so much so that he even envies the stray dogs. He's young: nice things excite him a lot, especially nice-looking girls. At night, when he goes to sleep, he runs his hands across his body in a caress, to stem the hunger of his famished flesh, his loneliness.\n\nLate on Saturday night, when he gets off work, despite his exhaustion, he takes the bus downtown and goes into an adult cinema. It's his only entertainment, the only thing that gets his adrenalin going, his only outlet, the only thing he doesn't count the cost of. He later emerges onto the street full of self-loathing and the loneliness comes back in a crueler form. He walks toward the red-light district, to Metaxourgeio, where he remembered seeing a brothel with a sign outside written in Albanian: GREEKS ONLY.\n\nThe following week, on Saturday night, he will go back to the same cinema. He will walk out onto the pavement after the show, once more full of self-loathing, and his loneliness will intensify and become even crueler. He'll go walking around the red-light district again, and read the same sign in Albanian: Greeks only.\n\nOne day he saw a cousin of his in one of the city's brothels. She was very pale; she looked like a naked ghost. He cried a lot that night without really knowing why: his country, his loneliness, his cousin? Perhaps all three.\n\n# **16**\n\n_A fter the Commandant's suicide, Alba and her mother were exiled for a month. She didn't appear at all during that time, didn't go to school, just sat shut up at home all day. Every day Xhemal would wait outside her school in the shitmobile with the hope that at some point she would come walking out. And then he found out that she'd been exiled. One of the laborers working with him told him. He lived across the road from her. He told Xhemal that a Jeep had turned up, one of those military police ones, accompanied by about a dozen policemen, in the middle of the night. Alba's hair-raising screams had pierced the darkness. \"Don't exile us. We're innocent.\" Nobody dared go to their windows, everyone just stood watching from behind their curtains. So did he, his eyes round with fear. He told Xhemal that they weren't even given time to pack, but he did see Alba carrying a suitcase. \"Even though it was dark, I could make out her face. Ashen white, like a corpse.\"_\n\n_That day Xhemal drank a lot. So much that driving the famous shitmobile around the town center, he lost control and went crashing into a shop front. Thankfully by then it was the middle of the night and the shop was closed, otherwise there would have been bloodshed. The window was shattered, and he escaped with nothing more than a broken arm. But that wasn't the worst of it. Unluckily for him, stretched across the fa\u00e7ade of the building was a gigantic poster of our Great Leader, bearing in enormous letters the caption: \"May you live like the high mountains, comrade Enver.\" It had been there since the Great Leader's birthday celebrations. For that reason, even though he was bent double in agony, he was held for questioning at the station. Fortunately he passed out, so they had no choice but to transfer him to hospital. God knows what would have happened to him if they hadn't_.\n\n_His troubles were not quite over, however. When he was discharged from hospital, he was sent for questioning at an emergency Party meeting where he first heard innuendos about relationships with enemies of the Party, obvious references to Alba. He was lucky in this, too: his father, an old communist, knew some people who managed to get him off with a sacking_.\n\n_\"What about Alba? Did you see her again?\" asked Sex Boy. Xhemal looked down. He took a deep breath and said, \"They found her body a month later. She'd slashed her wrists.\" A profound silence engulfed us, one that nobody dared disturb_.\n\n**First Words**\n\nThe immigrant is a creature surrounded by borders. Conventional borders, the ones that divide one country from another, mean little to him, they are just large, visible borders. There are thousands of invisible borders, however, awaiting him every minute of every day, awaiting his every move, almost, his every desire and ambition. The language: behold the first invisible border. What immigrant has not lived through the initial bewilderment\u2014something resembling first love\u2014that you feel when you pronounce your first sentences in a foreign language, which until the previous day had sounded like hammer blows to you, or like a sewing machine running up a hem? You try to import unusual phrases and expressions into your speech, especially the ones you think will bridge the gulf between you and the native speakers: _re paidi mou_ (look, kid); _oxi re gamoto_ (oh, fuck); _einai moufa_ (crap); _mou ti dineis_ (you get on my nerves); _Christos kai Panagia_ (Jesus and mother of God), as well as the ubiquitous _malaka_ (asshole). All of this you try to say in as convincing an accent as you can, hoping to pass for a native, as far as possible, and convince them that you can become one of them.\n\n# **17**\n\n_I had gone numb. Not from hearing the story about the shitmobile and the suicide of beautiful, tragic Alba, as I couldn't be sure how much truth there was in it and how much was a product of Xhemal's imagination. It was strange, but just as the company immigrants keep is unpredictable, the individuals it is made up of are equally unpredictable. Your flight gives you incredible freedom: suddenly you feel the need to spill the innermost secrets of your heart to complete strangers with an ease that might be seen as mad or just base. Maybe all this happens because when you leave, you also leave behind the weight of commitments. For that reason, you're not really telling your story: it's more an exercise in confession. Whatever the case, real or invented, Alba's story was a terrible one, but by the standards of our towns and cities, not that unusual. I was numb mostly because I felt that something significant had taken place in my life. I had crossed the borders. Was that not one of my biggest dreams? Yes, it was. But I could not, how can I put it\u2014take off. From within the madness of totalitarianism, I had imagined that as soon as I crossed the borders, a completely different life would start, a whole new world. And it was. The world I saw around me was completely different. At least very different from the one I'd been living in until yesterday. I was free in the world-beyond-the-borders. A strange feeling had taken over me, one I couldn't describe. I felt like an orphaned child\u2014completely free but at the same time completely lost. I looked at the road signs and tried to make sense of those incomprehensible letters. \"I will never be able to learn this language,\" said one of the drivers. \"Why don't those Greeks write like everybody else?\" added Xhemal, convinced that the rest of the planet was committed to the Latin alphabet. Meanwhile, the drivers had fixed their gaze on a church near the bus stop we'd been standing at for so long. \"Doesn't that look like a mosque?\" commented one of them, with the confidence of a specialist on Byzantine and Ottoman places of worship. Then, for the first time, Sex Boy revealed his knowledge: \"Those are mosques that were converted into churches when they kicked the Turks out of Greece.\"_\n\n_One more source of confusion had thus been cleared up, this time of a divine rather than a secular nature. The bus eventually appeared and we boarded it. The conductor asked to see our tickets. The drivers decided to handle this. \"Albania fukara; Albania refugiat,\" which loosely translates from this improvised language as \"We are poor Albanian refugees.\" The conductor was angry. Who knows how many Albanians had been using the same line on his buses recently. I understood enough to know that he was warning us that if he caught us again without a ticket, he'd kick us off_.\n\n_We were bunched together at the back of the bus, and all the other passengers had turned round to listen to our exchange with the conductor, which probably looked more like a conversation between the deaf than anything else. I went up to one of the drivers and whispered in his ear that he really should buy us some tickets with the money he'd changed back at the caf\u00e9 because we were making a bad impression on the other passengers. He turned, gave me a filthy look, and after a short pause said, \"I didn't realize you were such an asshole.\" The others seemed to agree. The episode was soon forgotten because, in the last analysis, we were happy to be on our way to a big town, a thoroughbred capitalist town, as Xhemal put it_.\n\n**Remember, Foreigner**\n\nThe more competent you become at the language, the more likely you think you are to make it. _That's it_ , you say to yourself, full of excitement. The natives start asking you how you managed to learn so fast, and you feel their eyes on you, maybe for the first time looking at you with admiration. Or did you just imagine it? You start to believe that the road into their world is now open; even your boss speaks to you differently. It's just that at the end of the week, when it's payday, he remembers that you're a foreigner. And the distance asserts itself again.\n\nAnd as the days go by, the weeks, the months, you realize that speaking the language is not enough. They never miss the chance to remind you that you are foreign: on the news; at the Aliens' Bureau; when you try to rent a flat and read that sign on the door saying, \"No foreigners. No pets.\" And if the color of your skin is different then you have another reason to remember that you are a foreigner, mostly from the curses you hear and the looks you get. The look a white man gives a black man is not hard to decipher.\n\nYou're reminded at work when your boss pays you. You're reminded on the street, during the operations sweep and the identity spot checks. You're reminded on the bus, when the natives shout: \"These foreigners take up our seats,\" or when they warn each other, \"Keep an eye on your wallet\u2014bloody light-fingered Albanians.\" You're reminded at the bar, when the local girls find out where you're from and vanish. And so on and so on, over and over again.\n\n# **18**\n\n_W hen we arrived in Igoumenitsa we got the shock of our lives. The caf\u00e9s were full of people, especially young people. The shops were bright, and still decorated with Christmas lights. Advertisements everywhere. Expensive cars and expensive clothes. All five of us looked on in bewilderment. Without giving it much thought, we decided that this was the perfect place to find work and live. \"You see, you idiots, what we've been missing because of communism?\" asked Xhemal in a tone that suggested that he was talking more to himself than to us. We didn't know where to look and then one of the drivers, who had a bent for theoretical discussion, said the words that ignited yet another heated debate in the biting cold: \"The Greeks are lucky. They always have been. Ever since ancient times.\" And he went on, and an entire reservoir of historical knowledge flooded out, explaining to us how Greece had only escaped communism by the skin of its teeth, thanks to the Americans, and that the Greeks were now living off the fat of Europe and that basically they did nothing except go to the beach in the summer and stay out late having fun. \"Why didn't they get Hoxha instead of us? Bastards,\" he wondered, putting a full stop to his historical marathon. It was just that he addressed this last question to the heavens, as though in communion with a superior being or expecting some kind of concrete answer from God Himself. Then Xhemal, who always entered into these discussions with plenty of zeal, answered, \"Because we're a bunch of assholes. That's why.\" Instead of the skies, Xhemal looked at the driver with the look of a gladiator before a fight. He didn't react, neither did the other driver. Neither did I. Only Sex Boy found the courage to speak up. He said something vague about fate, and that history is nothing but a lousy old whore. It seemed that we'd covered everything in our oral interpretation of history, so we went on_.\n\n_Walking around the town we saw a few building sites. We'd all learned the Greek word for work, \"douleia,\" the night before. Sex Boy and I approached some workmen and tried it out: \"Douleia?\" I said. \"No douleia,\" came the answer from one of them, who seemed to me to be smiling with pity. We abandoned the idea and tried another site. \"Douleia?\" Instead of an answer we received an entire cascade of what was to us meaningless language. Judging by the expression on the man's face, we understood that he was angry with us for speaking to him and were to get out of his sight as quickly as possible_.\n\n_That's what we did. While we were walking, Sex Boy stopped outside a busy caf\u00e9 and looked in at the young people sitting inside. \"I've never seen such beautiful girls in all my life,\" he said innocently. \"Where did you expect to see beautiful girls\u2014in your village? You'd be lucky to see a cow where you come from!\" joked Xhemal with the urbanity of someone who has just stepped off the last flight from JFK. Everyone burst out laughing. But they soon sobered up when they saw a black man kissing a blonde girl. \"You see,\" said one of the drivers, \"that nigger's got cash; he's not broke like us.\" The other driver advised him to forget about the \"nigger\" and concentrate on finding some work. But Xhemal again found a chance to continue the discussion by saying that the wives of rich men are always gagging for black men and immigrants because there aren't any real men in the West because they all sleep with each other so the women get neglected. Sex Boy was following this conversation eagerly, and judging by his expression looked as though his head was literally spinning. One of the drivers, the one who had wondered why Enver Hoxha had been visited on the Albanians and not the Greeks, insisted on completing the puzzle by explaining that economic wellbeing destroys men in the end, making them soft, like women, and gave an original interpretation, according to which only Albanian men, however well they may live, could never become effeminate, because first of all they are all so sex-starved, and secondly, and more significantly, their DNA is far too masculine for that to be a possibility_.\n\n_Sex Boy suddenly sprung to life with one of his unexpected contributions, wanting to tell us a joke. \"This Albanian socialist, a proletarian, arrives in the next world and meets Skanderbeg, the national hero. Skanderbeg asks him how his descendants are getting on, and the proletarian answers that they're not doing that great, really, as they work all the time and are paid a pittance\u2014but worse than that, they're sex-starved and aren't getting any action. They're desperate for women. He explains that the nomenklatura has as much sex as it wants but the proletariat is permanently frustrated. Skanderbeg is sad to hear this and asks if they at least manage to get in some quality masturbation? 'No. Not even that,' replies the proletarian. 'You see, the Party slogan is \"Building Socialism with a gun in one hand and a pickaxe in the other, so we always have our hands full.\"'\"_\n\n**You've Got a Weird Name**\n\nYou train yourself to learn people's names quickly, but they never learn yours. They can never remember it. Or, to be frank, they don't like the fact that you've got that strange, difficult, foreign name. Sometimes, without even asking you, they give you a name of their choosing, to make life easier for themselves, to restore order and harmony. \"I'll call you Giannis,\" the tradesman you're working with announces, \"so that you can have a decent name too, at last, like everybody else.\" And you go along with it, just like that, in the same way that people accept their fate. Because you want to be liked by those around you at any cost and have learned that foreign names are appealing only when tourists have them. You are not a tourist, you are a supplicant. In the final analysis, you say to yourself, that's what people are like round here; this is what they do and I have to adapt. Because if you don't, you'll never make it. And you recall your initial oath: I have to make it. From now on, you might have two names, your given name and your immigrant name. One for here and one for there. And the one for here, and maybe forever, is Immigrant.\n\n# **19**\n\n_I t seemed that everything was going Sex Boy's way that day. We hadn't taken five steps when we found ourselves standing outside a video club. We had never seen anything like it, except on TV. The truth is that even though we were amazed by everything we were seeing, what stunned us more than anything else were the kiosks and the hawkers working the town, bellowing down their megaphones. The kiosks struck us as completely surreal, so much so that Xhemal wondered why nobody robbed them. \"If this was Albania, they'd be empty.\" Anyway, we charged into the video club as though we had come to greet the Messiah. Inside there was only the one assistant, a dark-haired, really gorgeous young woman. When she saw the invasion, all of us dressed in what were effectively rags, black with filth and exhaustion, with uncombed hair, she was terrified. She leapt to her feet and took a step back. She didn't even ask what we wanted, she just stared at us, as though she were waiting for us to attack her or make off with all the videos. Sex Boy was in his element. He had discovered the porn section and was inspecting the covers one by one, back and front, the way that an archaeologist might examine the bones of a dinosaur. If we hadn't literally pulled him out, he could happily have spent at least another week in there_.\n\n_Back outside, Xhemal and the drivers walked a short distance ahead and Sex Boy and I followed them. Sex Boy, transformed by his experience inside the video club, had suddenly become very talkative and started telling me the valium story. Apparently he'd managed to buy a special hi-tech device from a man in his town that allowed you to watch foreign channels, many more than a standard antenna can pick up, including a couple of porn channels that you could get late at night. Unfortunately his house was very small and he couldn't turn on the TV at that hour without waking up his parents, who slept in the next room. Once his father did wake up and saw what his son was watching, all hell broke loose. But since Sex Boy could not resist temptation, he had to come up with a solution to the parent problem: he bought some valium, which he sneaked into his parents' food at dinner time every evening. His father got a double dose, just to be sure that Sex Boy could enjoy an uninterrupted night of pleasure with his favorite films. But his father soon got suspicious about the fact that he and his wife were suddenly sleeping so deeply and were finding it so difficult to wake up in the morning for work. One evening, the father proved himself to be a little more cunning than the son. He pretended to have eaten his dinner, but actually flushed it down the toilet. What happened next is something Sex Boy would rather forget: he'd been caught redhanded; the special hi-tech device was history and he got the thrashing of his life_.\n\n_Through listening to the valium story I also found out what Sex Boy's real name was. He was called Marenglen, a portmanteau name made up of three sainted names: Marx, Engels, and Lenin_.\n\n**It's Worse at Night**\n\nYou change your name. You get baptized. You learn the language, unusual words, colloquial expressions, the former to charm your audience, the latter to prove that you are the same as everybody else. But you still feel foreign, very foreign, extremely foreign, a regular outsider.\n\nIt's worse at night. First news item: A gang of Albanians have raped and murdered a seventy-year-old woman. Second item: horrific crime in the suburbs. Police warn Albanians might be responsible, followed by a statement from a police officer: \"Looking for Albanian criminals is like looking for a needle in a haystack.\" Third item: Masked burglars, most probably Albanians, broke into a house in Peristeri, threatened an elderly couple with axes and knives, and robbed them. And you ask yourself, how could they possibly have identified them, since their faces were hidden under balaclavas? Never mind. And then there's a special report on crime among immigrants, mostly Albanians, with the soundtrack of a Dario Argento film playing in the background, followed by a bit of Wagner. The reporter enumerates all the crimes committed by Albanians, or \"most likely\" committed by Albanians, and in a tone reminiscent of frontline war reporting, cries, \"We are nursing a time bomb that has already been detonated, and will go off any minute. Everybody and everything has been looted and plundered. They murder, rob, rape, spreading fear everywhere. These are extremely dangerous people. But enough is enough. Deport them, now. Please. This is the will of the people. Albanian immigrants are a scourge. They must leave now. They burgle people's country houses, they break into people's homes, take the lead role in most robberies; they'd kill you for your loose change.\" Cut\u2014the reporter goes off the air. So does the Dario Argento music.\n\n# **20**\n\n_B efore Sex Boy had time to finish his story, we found ourselves standing in front of a supermarket. The drivers didn't want to go in. They were keeping their eyes peeled for building sites and opportunities for work. Xhemal, Sex Boy, and I all wanted to go in, and in the end all five of us did. It wasn't a very big supermarket, but it was the first supermarket any of us had ever seen. We didn't have any money to buy anything, so we just walked up and down the aisles, admiring not so much the goods on sale but the bounty and the freedom of choice. We wandered around in speechless admiration, each of us lost in his own world. The other shoppers and the staff looked at us, too, sometimes with curiosity, sometimes with pity and sometimes with fear. At one point I noticed Xhemal at the shampoo and perfume section. I don't know how long we stayed there before we eventually decided to leave_.\n\n_Our exit was somewhat eventful, because on the way out the staff told us to come through the checkout. We told them we hadn't bought anything, at which two men appeared, rounded us up into a corner and told us to wait. Xhemal in particular looked impatient and suggested running for it, but the drivers said the supermarket would have the police on us in no time and we'd be in deep shit if they did. We stood there for a few minutes until a couple of men in uniform arrived. We immediately took them to be policemen. They looked at us with suspicion and told us to put our arms in the air. We all realized that we were about to be searched, and one of the drivers started to protest, but checked himself; most likely because he thought he was dealing with the police. The people in the shop were staring at us as though they were watching a short film about some aliens that had suddenly landed in their local supermarket. The \"policemen\" searched us, stuffed their hands into our jackets and pockets in the hope of finding something, and when they didn't, let us leave_.\n\n_The minute we got outside, the drivers reminded us that they had been right not to want to go in: look what happened\u2014those animals in uniforms humiliated us in public for no reason. I was expecting Xhemal to chip in as usual with some kind of anticommunist remark, but he didn't. Instead he was wearing an expression that was somewhere between cynicism and smugness, and said, \"Back in Albania, my friend, this would have had an entire regiment of women eating out of your hand,\" and held up the bottle of perfume he'd stolen from the supermarket. Before we could work out what had happened back there, one of the drivers starting swearing at Xhemal, telling him that he'd come to Greece to find work, not to shoplift. Things became very heated when Xhemal called him an asshole, arguing that one stolen bottle of perfume wasn't going to put the supermarket owner out of business. One insult followed another until the two of them came to blows. Xhemal started it, kicking the driver when he insulted Xhemal's mother. We tried to separate them, forming a jumble of bodies, but were quickly brought to our senses when Sex Boy shouted, \"Police! Police!\"_\n\n_At a distance of less than 100 meters we could see a patrol car, which had probably seen the fighting, and was coming our way. I don't know who set us off, Xhemal probably, since he had stolen the perfume in the first place and had more to be frightened of than the rest of us\u2014but we all started running for our lives. I can't even be sure that the police car was coming for us. What I do remember is that we ran into a building site and hid, panting from the exercise, terrified that we'd be caught. Then Sex Boy, more impatient than out of breath, turned to Xhemal and asked, \"OK, so how come they didn't find it? Where did you hide it?\"_\n\n**Commercial Break**\n\nTime for a commercial break: beautiful beaches, expensive cars, sexy women in stilettos, toothpaste, toilet paper, fizzy drinks, life insurance, aftershave, women's underwear unfurling in the sky in time to Chopin's Waltz opus 64, No 1. An angelic world, gentle and perfectly proportioned, fast-paced at bargain prices with expensive cars, all easily accessible with amazing mobile phones and perfect English, with multiple personas you change as often as you would your shirt, with late nights, sex and alcohol. In this wonderland, which consumes at a dizzying rate and with dizzying ease, you feel that you and your origins are the only things out of place.\n\nSomething's not quite right, you think, somewhere down the line something has gone wrong, things get tricky, tough and merciless. Why should I have to answer for every last criminal who happens to have been born in the same country as me? I can only answer for myself, maybe for those closest to me, at a pinch: my brother, my parents who haven't got a bad bone in their bodies, who've never harmed anyone. Everybody should be answerable for their own actions, and that's why the cards seem fixed and stacked against you from the start. They want me to feel guilty, a victim at any cost, even though I haven't done anything.\n\nIf you are guilty, you have no rights, only duties and your first duty is to feel inferior, always\u2014and different, grateful that you're allowed to exist at all. You sit there frozen, the remote control in your hand. You turn off the television. You don't know whether to weep or wail. You get the urge to bite, something, anything, the TV screen, the remote, yourself. Then you recall those lyrics you read somewhere, but you can't remember who wrote them: \"I wonder if you remember \/ the broken teeth of love \/ which from sadness and hunger \/ bit the stars?\"\n\n# **21**\n\n_I can't say how long we were hiding out in the building site. The atmosphere was tense, not so much because of the police car\u2014we weren't even sure it was following us\u2014but because of the fight between Xhemal and one of the drivers. We eventually emerged from the site, and our original plan, to return to Igoumenitsa for the night and resume our search for work the following morning, was no longer practical. We were scared of getting into trouble with the police. We were starving, and the drivers, the only ones who had any money, offered to buy everyone something to eat. Even though one of them had been fighting with Xhemal, he was still included in the offer. We got to a shop. The drivers went in and came out carrying sandwiches, which, hungry as we were, seemed to us an irresistible feast, and Xhemal found another opportunity for cursing the mothers of communism and Enver Hoxha, respectively_.\n\n_The cold was getting unbearable, more so after the sandwiches. Because of this, we all agreed to return to the refugee center; it wasn't as if we had any other options. With much difficulty, we found the bus stop and were in for another long wait, as we had no idea when the bus was due. It arrived eventually and we got on\u2014different conductor this time. He asked us to pay for our tickets, and we came back with the same answer, \"Albania fukara, Albania refugiat.\" This conductor also got annoyed, but didn't threaten to kick us off like the previous one had. This time I didn't suggest that the drivers consider paying for our tickets. Now and again I exchanged the odd word with Sex Boy, while the two drivers sat there looking pensive and said nothing. Even Xhemal looked thoughtful. We had set off for Igoumenitsa convinced we'd find work, and were returning with nothing more than a body search in a supermarket, with two of us having got into a fight over a stolen bottle of perfume, which, according to Xhemal, would have had an entire regiment of women eating out of your hand back in Albania_.\n\n_We eventually got back to the warehouse. An almost identical scene to the one we'd left: soldiers, policemen, and Albanian fugitives standing in a line. While we had been wandering around Igoumenitsa in the freezing cold, there were more arrivals, quite a lot of new arrivals, with the result that the population in the warehouse had swollen and it was very difficult to get through. The new ones told us stories, stories that made your hair stand on end. Some were saying that the Albanian soldiers were opening and closing the borders at will and would take pot shots at the fugitives for the fun of it, and at other times would let people through unchallenged. Some were talking about mass murder, while others were saying that the soldiers were no longer taking orders from their superiors, and to avoid murdering people, had started to desert en masse. Some of the new ones had crossed very hazardous terrain, full of snow, wolves, and chasms. They pointed to a group of very young men who had lost one of their friends during the crossing. He was only eighteen. The rest of them had survived by a miracle. There were five or six of them, sitting together, tired and worn out, in silence. Other people were telling stories about wolves attacking exhausted fugitives, tearing them to pieces. The latest piece of information was that a few fugitives had left the refugee center and made their way to Athens, to a big square, where you could find work immediately. We discussed the idea ourselves: Xhemal, Sex Boy, and I were all for it, but the drivers insisted that we should stay put, because we would definitely find work if we did_.\n\n_We relaxed a little when Sex Boy, who had a talent for finding out the latest and most accurate news from the warehouse, told us that one of the new arrivals was someone who was a bit soft in the head, and had become the only light entertainment for the fugitives. The madman was 100 percent sure, and he told everybody this in confidence, that he was George Bush's nephew, and that President Bush was in fact an Albanian, and came from the village of Dardh\u00eb e Vog\u00ebl, or Small Pear, somewhere close to the Greek border. Bush's nephew was promising everyone that he would have a word with his uncle and ask him to send a big airplane that would take everyone to America right away. This was the last piece of information I got that night. I was so shattered I fell asleep without realizing it and still cannot work out how I found any room to lie down amid the rabble_.\n\n**The Small Screen: From Fan to Hostage**\n\nYou learned the language so that you wouldn't stick out any more, but you were only shooting yourself in the foot in the end. Because it is now that you really feel utterly and completely foreign. If you didn't know the language, at least you'd be spared everything they say on the news. Now you understand everything. Watching television has turned into an exercise in masochism, and you watch a lot of television because you're on your own and don't go out much. Lonely people who don't go out much watch a lot of television.\n\nTelevision used to be the pre-eminent source of your fantasies, the only window through which you could escape and travel in your imagination, dreaming of magical, forbidden worlds: the West, freedom, equality, prosperity. You created in your mind a world of harmony and beauty, and with this construct in your head, you crossed the borders.\n\nTelevision is now the source of all your nightmares. It projects a magnified view of your dissonant, repellent, terrifying face. Your great passion is suddenly your most menacing enemy. \"I am not like that,\" you shout, but your voice does not affect the image. The image, of your repulsive face, is all-powerful, it's everywhere, it multiplies with dizzying speed, meeting minimal resistance along the way because it puts it to people in the most simplistic of terms: absolute good versus absolute evil; native versus foreigner. And it sticks, because it does not challenge the human mind, it panders to the most primitive narcissism and speaks to the most primitive fears. And there's nothing you can do about it because in this relentless theater the part allotted to you is that of absolute evil, blackness, the foreigner. There is nothing you can do about it. You have no voice. You are not even a consumer. All those amazing products and goods that are paraded before you after the news are out of your reach: you can't buy them, and that makes you absolutely n-o-t-h-i-n-g. And then the world starts to assume, perhaps for the first time, its real dimensions. Demythologizing begins. Then you start to miss the fresh air, and the tears well up in your eyes, while your brain is banging out that phrase nobody has ever said and nobody is ever likely to come out and say in public: \"You are n-o-t-h-i-n-g, n-o-t-h-i-n-g, n-o-t-h-i-n-g.\"\n\n# **22**\n\n_I woke up the next morning with my head on a stranger's thigh while the head of another stranger was resting on my leg. My entire body was stiff. I was freezing cold and shaking all over. Someone had lit a fire inside the warehouse to get warm. The cold pierced right through your bones. They were taking a risk because if the Greek police spotted it, they would go berserk. I started to move cautiously, trying not to wake either the man above me or the man underneath me. I was partially successful, and moved across to the fire. The men around it seemed to have just arrived and were sleeping in a sitting position. I looked around for a familiar face and eventually spotted Sex Boy. He was awake and told me that he hadn't been able to get any sleep. He had found somewhere to lie down but was complaining bitterly about two men who stank and who weren't just snoring but were braying, so much so, according to Sex Boy, that they could have raised the dead, who would have instantly fled to escape the noise. We both agreed that the warehouse resembled a pigsty more than anything else, and if things went on like this, we'd soon be ridden with fleas and disease_.\n\n_While all of this was going on, I'd woken up with a rather strange idea: I wanted to see what a bookshop in the West looked like. Books were my great passion. I hadn't managed to see a bookshop when we were in Igoumenitsa because the rest of the group had other priorities. I confided this ambition in Sex Boy. \"Why didn't you mention it yesterday? We were walking around the town for hours!\" He finally gave in because he was just as keen as I was to escape the hellish atmosphere of the warehouse and would be hard-pressed to find better company than mine_.\n\n_\"The cold outside has to be better than this\u2014at least there won't be any germs,\" he concluded, so off we went into the village in search of a Western bookshop. As I hadn't the faintest idea where it could be, I asked two young boys in a mixture of Italian and English. They struggled to tell me which general direction we should go in_.\n\n_Before we started our search, we decided to go into a caf\u00e9 to get warm. There were only a few people inside and a waiter who didn't even take the trouble to ask us what we wanted. It looked like he was used to people coming in just to warm up and not order anything. Sex Boy and I talked a bit, in hushed voices, in Albanian, until the waiter came over to us, carrying a tray with two cups of tea on it. He spoke to us in Greek, and pointed across to the other side of the caf\u00e9 at an old gentleman. We worked out that the tea was his treat. We thanked him with the traditional gesture, placing the right hand close to the heart. He responded with a sweet smile and returned the gesture. \"We and the Greeks belong to the same tribe,\" said Sex Boy, sipping his tea. Since we had the privilege of drinking tea we fancied ourselves regular customers, and therefore had the right to stay there a little longer, get a little bit warmer, and to give the sun a chance to warm up a bit as well. And that's what we did. We left with a proud \"efharisto,\" which seemed to please the waiter a great deal, and resumed our search, this time in the best of spirits, for the Western bookshop. Eventually Sex Boy and I did come across something like a bookshop, which to me at that moment was something quite sublime_.\n\n**A Tough Life**\n\nImmigrant, alien, supplicant: your profession is all these things. Your life is a tough one, because you don't want to go back to where you came from, but you're not wanted here, either. It is tough because if you want to find work, you'll have to change your name. It is tough because you'll always be persona non grata. It is tough because when you got sick that day, and were running a fever of 102, and couldn't go to work, your boss sacked you just like that. It is tough because when you ask for a bit more, it's called hubris, but when your employer pays you just one tenth of the basic wage for twelve, thirteen, fourteen hours of working like a dog, he's being kind, he's doing you a favor. It is tough because for you there are three things that are sacrosanct: work, sex, and a residence permit\u2014and you'd happily trade sexual passion for a residence permit. It is tough because the police can catch you whenever they want to, swear at you as much as they want to, and hold you at the station for as long as they want to. Your bitter moniker for the police, the _astynomia_ (civil guard) is _astronomia_ (guards of the stars) because your fate on this earth so often depends on them. A tough life because some of your fellow countrymen commit crimes, and when they do you have to keep out of the way of the Black Marias on Operation Sweep-up, and out of the way of the TV cameras that just love Operation Hatred and Misinformation. One newspaper shouts, \"Albanians out!\" while another declares, \"Albanians are the most disgusting race on earth.\" It is tough because stealing one watermelon is enough for someone to kill you and get away with murder.\n\nIt's a tough life because in many cases the police fire warning shots to scare you and somehow manage to get you in the back of your head with uncanny mathematical precision. Afterward the policeman gets off and nobody loses any sleep over it. It's a tough life because if you want to rent a flat you have to sweat for it much more than the average Olympic gold medalist weightlifter does. You change your name, you watch your accent, give your best linguistic self, and if you still don't make it, you change your nationality and your religion, too. You become a Romanian, a Serb, a Russian, because deep down you are all those nationalities and they are all you. It's a tough life because you are young and full of love, and when the local girls find out you are Albanian they instantly evaporate and you risk (don't laugh) spending your life without a lover. It's a tough life because they have declared you a modern-day cannibal. The zealous cameras zoom in on the crimes you commit (when they aren't fabricating them, that is), but never catch the blatant exploitation that pushed you to crime in the first place, and never notice the benefits you bring to the local economy, and always miss the enormous profits you bring your boss. That's why we hear people talking about you so much but never saying anything nice about you. It's a tough life because there are so many opportunities for becoming neurotic and miserable, and for being consumed by loneliness every single day. Loneliness, as a poet once said, is not missing other people but finding yourself in a big crowd, talking, and not being understood. There are so many opportunities for becoming suspicious and aggressive toward those you have become convinced don't want you. In this way you slide into the underbelly of society, where there is more than enough darkness, and where the greatest danger of all lurks: that you will give in to this darkness.\n\nI'll stop here because you're tired and need your sleep. Another tough day awaits you. But if I've made you smile at any stage at your own sufferings, you know better than anyone that scapegoat humor is only a temporary gift, on loan from heaven to hell.\n\n# **23**\n\n_F ortunately for me there was someone in the bookshop wanting to buy a paper or a book, I couldn't really work out which, who spoke French. We got talking and he showed me where to find the books by Garc\u00eda M\u00e1rquez, there were two or three, as well as one by Borges, which I'd managed to read on the sly in Italian. I was entranced by my conversation with him, and began to feel that I had landed somewhere between the Library of Alexandria and the Sistine Chapel until a hand grabbed me by the collar from behind and demanded, in broken Albanian, \"Ermi babai?\" Everything happened in a matter of seconds. I saw the terrified look on the face of the person I'd been talking to, and the body of a police officer pulling me with tremendous force out of the shop. I was panic-stricken. I started screaming and shouting, begging the man I'd been talking to for help, insisting I was innocent. Quite what I was innocent of he didn't know either, but when a policeman grabs you by the collar, shouting, it's obvious you're about to be accused of something. The officer dragged me outside, still screaming. That's when I saw Sex Boy down on the ground, another policeman bending over him and thrashing him with his truncheon. I also saw the Black Maria they were taking us to. While my Francophone friend tried to talk to the policeman, I carried on screaming and by now the second policeman had ditched his truncheon and was pulling and kicking Sex Boy to his feet. At some point, they threw us into the Black Maria. Sex Boy was sitting next to me, terrified out of his wits, and said to me in a shaky voice, \"I think they're going to kill us.\" I didn't have the slightest idea what was going on. It seemed that Sex Boy was destined to get into trouble: wherever he went he made people angry: the caf\u00e9 owner, the police, supermarket security guards, everyone. I was reassured slightly by the sight of the sports stadium and our warehouse. Policemen and soldiers were screaming. Something very serious must have happened. I looked for Xhemal and the drivers, but it was difficult to walk through that huge crowd of people packed together like sardines. Sex Boy's face was screwed up in pain, from all the truncheon blows he'd received back at the Western bookshop_.\n\n**Resurrecting Memories**\n\nOne of your crimes is that you remind the natives of what life used to be like for them. They had just started to put all that behind them, bury it in the depths of Lethe, all the pain and humiliation of being an immigrant, the worn-out faces, the peasant gait, the heavy stench of sweat and garlic, the fear of hunger and penury, of being the unbearable \"dirty foreigner,\" _dreckiger Ausl\u00e4nder_ in Germany, the sale race in France, dirty Greeks in America, _svartskallar_ in Sweden, and so on. Then you come along looking like something out of a photograph taken over fifty years ago.\n\nOn the subject of photographs, Leon Pantoti is someone I'm sure you've never heard of. He's an immigrant too. Or was. Leon Pantoti is the name he was issued with at Ellis Island, where all new arrivals in America were naturalized. Unlike most Greeks who arrived in America, Pantoti was not illiterate. He also had a trade: he was a photographer. He quickly learned English and opened a studio in San Francisco, photographing Greeks and Americans between 1914 and 1922. Portraits of children, mothers, laborers, rich and poor, bachelors and newlyweds, mail-order brides shipped out from Greece, because the local women were out of reach to the average immigrant. He photographed beautiful foreign girls who married elderly locals, usually for their money or a residence permit, or both. In one such picture taken by Pantoti the bride is showing off her right leg, trying to look flirtatious and give an impression of the comfortable existence she had just embarked on. The photograph has everything in it: dreams, pretense, harsh reality, comedy\u2014because being an immigrant is all those things.\n\nThis pose must seem faintly absurd to the descendants of the bride, American citizens today. Isn't the same thing going to happen when your descendants see a picture of you, when you tell your own child, who will grow up in Greece, all about how you originally furnished your home with the broken chairs and armchairs people chucked out onto the pavement and how gratefully you collected them from the \"immigrants\" supermarket, otherwise known as the trash heap? Your son will say, \"Daddy! You were a beggar\u2014how embarrassing!\"\n\nImmigrant existence goes in cycles. So does the look on its face. It doesn't just have one look, it has several. For the first generation there's that neurotic look of the orphan: \"Which country do I really belong to?\" Then there's the look of the immigrant who makes it: pride and vanity. And then there's the look of the immigrant who doesn't make it, the look of failure and plenty of resentment. The look of the locals looking at immigrants is usually out of range. At first it's full of sympathy, then it becomes puzzled before it turns suspicious and worried when it sees that yesterday's barefooted man wants to become like him, resemble him, and perhaps even do better than him.\n\n# **24**\n\n_T hey eventually let us out of the Black Maria and herded us with the help of their truncheons into the warehouse. We started looking for our friends, and after much difficulty we located the two drivers. We asked them what was going on but even they weren't sure, because some people were saying that some of us had raped a Greek girl in the village while others claimed that a priest had groped an Albanian man who responded by beating him up and shaving him. Another version was that a well-known lout from Tirana had punched a policeman; others insisted that the truth was actually much simpler. It was market day in Filiates and some of our number were strolling through the stalls, demonstrating to the traders there what it means to acquire goods without paying for them. A group stormed into a small shop and literally emptied it of all its Cokes and beers while the shop owner looked on in frozen disbelief. It was said that this event spread fear and panic among local traders and shopkeepers and that the villagers demanded that the police round up the Albanians and establish order_.\n\n_It was impossible to work out what had really happened; the police had turned truly savage. They were swearing, pounding the wire in front of our warehouse, and refusing to let anyone leave. By way of punishment they didn't give us anything to eat that day or the next. Cold, lack of sleep, and now starvation. Some of the villagers would try to throw us a crust or two over the railings but were rewarded with the fury of the police for their pains. The warehouse had been converted into a prison: we were only allowed to relieve ourselves right next to the warehouse, not a step further. Somewhere close to all the shit was the tap we were supposed to drink water out of. Some of us started saying that they were going back to Albania, because at least there was a bed waiting for them at home, but most people insisted that there was no turning back. \"After forty-five years waiting to cross the border it would be a shame to stay for such a short time, wouldn't it?\"_\n\n_Through all this chaos, Sex Boy brought us the main story of the day: in addition to the mad Albanian who was threatening the police that he was going to report them to his uncle, George Bush, there was another guest star, a mute. But he was no ordinary mute. He was not an Albanian mute. He was a Greek mute, unlucky enough to get caught up in the police round-up. Because he couldn't speak, they took him for another Albanian and threw him into the warehouse along with the rest of us. The Albanian crowd had spotted the mistake and took special care of him. I don't how they managed it in this horrendous chaos, but they got hold of a pen and a piece of paper, and somebody who knew Greek was able to communicate with the mute. Before long almost everyone had learned his name. He was called Dimitris. Dimitris was a farmer. He had left home that morning without his ID, which is how he ended up in our warehouse as he had not been able to speak and his face left the police with the unshakeable impression that he was Albanian. Some attempts were made to persuade them that there had been a mistake, but the police were not in a mood to listen that day, and the negotiators were in danger of getting beaten up if they pushed it too much. They eventually gave up, and Dimitris had to spend the night in the warehouse with the rest of us, though his anxiety probably kept him awake most of the night_.\n\n**Even Albanians are Getting Expensive**\n\nThursday. The evening news. The cameras fall on farmland and farm workers, Greeks as well as migrants, somewhere in Larissa, then somewhere in Crete. The news item is brief: \"Albanian immigrant workers demand wage increase.\" The caption flashing across the bottom of the screen reads: \"Even Albanians are getting expensive!\" Just imagine the journalist responsible for that one, sitting in front to his computer screen, trying to squeeze a catchy caption out of his brain. That is truly inspired. Worth its weight in gold, and with it he's finished another tiring day at the network. The truth is, copywriters are just another sector of the workforce struggling to make ends meet, and in all likelihood this one isn't paid or rewarded much for his work. He might even be the Albanian of the media industry. \"Even Albanians are getting expensive,\" much as you'd say, \"Even potatoes have gone up.\" It's that microscopic, seemingly innocuous little \"even\" that does all the work. It establishes the distance between natives and immigrants, between those who have the right to try to improve their lot and those who don't, between yesterday's immigrants and today's immigrants.\n\nThe copywriter probably hadn't given it that much thought. He'd probably never heard of, let alone read the story of his compatriot, the Ludlow martyr, Louis Tikas.\n\nTikas lead the Greek miners' strike in Colorado at the beginning of the twentieth century. Louis Tikas wasn't his real name; he was called Ilias Anastassiou Spantoudakis, but changed it to Tikas to get American citizenship. He was murdered by hired killers put onto him by the bosses because he refused to let Greek immigrants back down from their demands for higher wages and better working conditions.\n\nMuch of the local press back then poured scorn on the Greeks, because up to this point they had been the cheapest labor around, and were sometimes used by their bosses as strikebreakers, and then suddenly, there they were, also wanting to be expensive. They were doused in various epithets such as _\"arrivistes,\"_ \"coffee drinkers,\" \"bloodthirsty,\" and _\"den katalavenees\"_ (he doesn't understand). These young men, most of them from agricultural backgrounds, lived in fear of the landslides inside the mines and lived in fear of amputations, but their greatest fear of all was the fear of failure. They had left home and their country for money, not security. The only education they had received consisted of a few popular sayings, a couple of dozen songs about vendettas and wars with the Turks, the poem \"Erotokritos,\" and the odd folk ballad.\n\nMost of them\u2014the ones that didn't die in the mines, that is\u2014stayed on in America. A few, a tiny minority, returned to Greece. The descendants of these rough, illiterate Greek immigrants did make it: they became American citizens, businessmen, politicians, renowned university professors, and some of them today even spend a lot of time digging up the stories of their ancestors.\n\nWho knows whether after eighty years some descendant of the peasant immigrants of today, from Albania or India, won't be doing the same thing? And perhaps, while looking into his forefathers' experience, he'll stumble across that \"even,\" which might give him an idea for the title of the book he's planning: _Even Albanians Are Getting Expensive_. The irony is that thanks to those immigrants, their faces worn with exhaustion and too much exposure to the sun, the copywriter who penned this catchy legend might just have earned a footnote in the history books.\n\n# **25**\n\n_W e all woke up the next morning hoping that the fury of the policemen had subsided. It hadn't. The police had surrounded the warehouse with even more wire fencing to stop people getting out. The focus of the entire morning was Dimitris, the Greek mute who had been mistaken for an Albanian. He had written down something in Greek on a piece of paper, which we were desperately trying to pass on to the police so that they would know that there was a Greek in our midst. From what I could gather, he'd written his name, surname, and the name of his village, and the words I AM GREEK in larger letters underneath. The police gradually got wind of what had happened, although it took some convincing for them to accept that it wasn't just another Albanian scam. About ten policemen came into the warehouse late in the afternoon looking for Dimitris, the Albanian-looking Greek farmer. They found him easily because the rest of us immediately realized why they'd come. They seized him by the arms and marched him out as though they had just made an arrest. Dimitris was never heard of again. I suspect that the only thing that came of all this was the two-fold satisfaction felt by the residents of the warehouse: first of all from the police blunder and secondly, this episode had proved that a Greek could easily pass for an Albanian, so by extension an Albanian must be able to pass for a Greek_.\n\n_But our biggest concern at that point was our hunger, and the fact that the authorities appeared to have absolutely no intention of feeding us. In a united show of protest, we all started stamping our feet and shouting rhythmically_ , \"duam buk\u00eb, duam buk\u00eb, duam buk\u00eb\"\u2014 _\"we want bread, we want bread, we want bread.\" The chorus of voices was quite impressive, and the warehouse started vibrating from all the sad music. We must have repeated this chant more than ten times, and in addition to expressing our feeling of extreme hunger, this was a civilized way for us to spend our time, as one of the drivers pointed out. Eventually we were heard. After nightfall, ten policemen clutching their truncheons in one hand and large sacks in the other strode into the warehouse. They screamed out orders for us to get down on our knees, which we did. They opened the sacks and started pulling out loaves of bread, and throwing them randomly around while the starving recipients fought like maniacs over them. I saw the two drivers fighting with all their strength over the same loaf, the same two men who only three days earlier had been convinced that they were en route to Germany to work the Berlin\u2013Baghdad route for an astronomical wage. When the policemen left war broke out, a real bread war. In among the fray, a very big, bulky character had, for reasons best known to himself, grabbed George Bush's nephew by the throat, and would have surely killed him had not a few slightly calmer men pulled him off. Fortunately, at some point order restored itself. It was then that I felt someone touching my back. It was the two drivers, ecstatically happy, giving me a flash of their booty, which they'd hidden under a jacket: three whole loaves of bread. This gave me a chance to appease my hunger, as I'd stayed out of the bread war. Naturally the mood of the crowd was as uneven as a rudderless ship travelling through an almighty storm, sometimes disappearing beneath the furious waves, only to reappear later on the crest_.\n\n_Half an hour later, once people had eaten, we heard singing: a love song. A small group at the other end of the warehouse had started it and gradually everybody joined in singing what resembled a compulsory protest song. We'd gone from singing \"We want bread\" to \"Oh! Nightingale in Spring:\"_ Ore bilbil \u00e7'ta kam b\u00ebr\u00eb ben\u00eb \/ vet\u00ebm mos t\u00eb gje\u00e7a mor I pabes\u00eb folen\u00eb \/ do t\u00eb marr\u00eb dhe do t\u00eb hedh n\u00eblum\u00eb \/ se na I le \u00e7upat more \u00e7apk\u00ebn pa gjum\u00eb \/ hidhesh e p\u00ebrdridhesh porsi gjarpri \/ si t'ja b\u00ebj un\u00eb I varf\u00ebri. _Everybody was singing, and those who didn't know the words sang along with a \"na, na, na.\" The fact that such a sweet love song was being sung at that moment by a huge crowd of people in a very aggressive mood made it sound like a war song. I asked myself whether hungry people were capable of singing love songs any other way. The police outside must have thought the Albanians had gone mad. I wanted to share my thoughts with one of the drivers, but he had given himself completely to the mood of the song and there, in the half-light, I could make out the tears running down his cheeks_.\n\n**The Scapegoat of the Poor**\n\nThe rich don't usually mind you. Those who fear you the most are those who read few books but watch plenty of television. Those who send their children to the same school as yours, who live in the same neighborhood, who wait in the same lines as you at the health insurance offices at dawn. Those who worry about sinking to your level, becoming like the Albanians. They latch onto national stereotypes with zeal, any stereotypes for that matter, because they don't want to lose that divine distance between superior and inferior. It's the only thing that makes them feel that they are not at the bottom of the heap. There is nothing worse than the disgust one poor devil shows another. It provides fertile waters for politicians to go fishing for votes as they scream for a country free of immigrants, selling fear and resentment, and spreading the false impression that immigrants are the root of the nation's problems. They will convince voters that if only you had the decency to go back to where you came from, they would be eating from golden spoons. That if you hadn't come in the first place, they would be at work and prosperous, their children wouldn't be at drugs, and they would be able to go to sleep with the windows open at night. And thus, with the promise of open windows, the poor retreat into this simplistic logic, the logic of resentment and xenophobia. It's an age-old game: the rich are responsible for coming up with racist theories, and it's the job of the poor to apply them by pursuing other poor people, and thus the poor get poorer and the rich get richer.\n\n# **26**\n\n_B ecause we felt that the conditions in the warehouse were intolerable and were an insult, some of us took the initiative to organize a kind of general meeting. A self-appointed committee started to summarize the situation of the last few days. \"We are treated like animals even though the European Union has given the Greeks billions for us,\" announced one impassioned member. Everyone around him burst into applause. One of the drivers, who was sitting close to me whispered, \"Look at us now. What big boys\u2014who could guess we've been applauding the Party for the last fifty years?\" I said nothing, just smiled. I followed the scene, feeling like I was watching something straight out of the theater of the absurd. After the first speech, which had whipped everyone into a frenzied fighting spirit, the decision was made to send a letter of protest to the UN. Everyone was convinced that the UN had nothing to occupy itself with other than the terrible fate of Albanian refugees in Greece. They were equally convinced that as soon as our letter reached them, they'd send out economic aid and our troubles would be over once and for all. Initial joy turned to enthusiasm; enthusiasm, to delirium. Everyone agreed, and decided that the protest letter should be written right away and sent with no time to lose. The difficulty was that there was only one pen and no more paper; all the paper had been used for Dimitris. But the most serious obstacle was that nobody had a postal address for the UN, or indeed knew where there was a mailbox in Filiates, and even if we did, the police would never let us go to it. There was no discussion about the language the letter should be written in. And that's how we spent our third night in the warehouse, writing a fantasy protest letter addressed to the UN_.\n\n**The Neurosis of Staying On**\n\nYou used to say: \"As soon as I've saved up enough money, I'll go home.\" Many years later, you're still saying the same thing. You used to say: \"I'll move on to another country, where I'll earn better money and live better.\" You stayed here. The truth is that the idea of leaving has become second nature. You are always somewhere between eternal flight and eternal return. You are, as always, full of contradictions, and you want to compromise where there can be no compromise. That's the way the game goes when you're away from home. It's relentless and only for those with nerves of steel. Anyone who's tried walking along the shifting sands the immigrant walks on knows that. It's the neurosis that comes with staying on. One day you're saying, \"I'm going back, I can't stand being away any more; it's unbearable here,\" and the next day it's, \"I'll stay a bit longer, get a bit more money together,\" and the day after that you're all, \"I have to make it here.\"\n\nEven though people generally do stay in the first country they migrate to, you'll always be a permanent fugitive. So much so that I wonder if it isn't some kind of unconscious strategy, a lifestyle choice made by those who suffer from Border Syndrome; perhaps it's all nothing more than a form of self-defense, because you never know whether you'll make it or not, or whether you'll put down roots here in the country you chose to come to. That's why you keep those two contradictory scenarios in reserve: to remain or to return? Like an emergency exit from insecurity and your fear that perhaps they won't let you put down roots here.\n\nMost likely you'll stay put. You have very few chances of going back, and deep down you know that, because you're beginning to realize that you are more of a foreigner back home than you are here. An immigrant is like a tree: his branches face his homeland, the one he left behind, while his roots grow deeper and deeper into the soil of the country he chose to come to. If you end up being one of a handful that does go back, you'll resemble the ancient immigrant in Ignazio Silone's novel _Bread and Wine_ , the peasant everyone in the village called Sciatap, an Italian corruption of \"shut up\"\u2014apparently the only phrase he'd learned in that country. Shut up. Shut up.\n\n# **27**\n\n_I t had become impossible to sleep at night. There was no room. The burliest men occupied the foam mats, which functioned as beds and had turned black with filth. The weaker ones waited till morning when some space was vacated by the stronger ones. By now hunger had kicked in and I was extremely weak. I looked at the drivers: they had both lost all their initial enthusiasm and optimism for good. I went over to join them and found them in the middle of a tense discussion, the one blaming the other for the decision to cross the border. \"Have you any idea how much the truck we abandoned costs? I was responsible for it, so if they decide to charge me for it, I'll be working it off for the rest of my life,\" the driver shouted at the codriver_.\n\n_They were no longer the two drivers but driver and codriver. The codriver said nothing for a while before retorting, \"I asked you three times if you really had the bottle for it.\" Now it was the driver's turn to go quiet_.\n\n_I asked them if they'd seen Xhemal anywhere. \"Xhemal has escaped from the camp,\" they told me. The refugee center had been referred to as \"the camp\" for the last two days. How, with whom, or where Xhemal had escaped to they couldn't tell me. All they knew was that he was gone, maybe to Athens, maybe he'd found work in Igoumenitsa at some garage, but they really couldn't say. Sex Boy insisted that he had left with some strange characters and headed for Athens. Xhemal hadn't said anything to us. The truth is that after the fight with the driver, he had become a bit distant and had started keeping company with other people in the warehouse. Sex Boy, our very own 007, told us that some of them had knives and one had a pistol. We were all shocked because we hadn't noticed anything of the kind. Apparently these men had taken charge of things inside the warehouse and were offering to help people who wanted to escape, for a fee, which could be paid back at a later date, in dollars, drachmas or lek. Sex Boy also told us that the previous night someone who had challenged them and sworn at them got a bad beating for his pains. That was the kind of company Xhemal had chosen to escape with. \"That man would have sold his own sister if the price was right, mark my words,\" said the driver who had fought with him_.\n\n_Toward the end of the day we had an unexpected visitor, an elderly priest who arrived with an interpreter in tow. Everyone gathered round to listen: buses were arriving the following morning to take us, in alphabetical order, to various destinations around Greece, where we would be housed and given work. The euphoria this news spread through the warehouse is difficult to describe; some were so excited that they grabbed hold of the priest and threw him up in the air repeatedly, the way soccer players do with their coaches after winning the cup. As they tossed him up and caught him, they chanted, \"Papou, papou, papou,\" forgetting that the priest was an old man, and that in the midst of all the chaos and the shouting, things could go horribly wrong. Five or six policeman entered the fray in order to liberate the priest by opening a corridor down the middle of the room with their truncheons. The released him from the grip of his dangerous fans and escorted him outside. He looked like Christ on Golgotha, but in reality bore a closer resemblance to Judas as he had lied to us over one small detail: the buses were scheduled to arrive; we were expected to wait until our names were called and then get on them, but we were not going to \"various destinations around Greece to be housed and given work.\" The buses were taking us straight back to Albania. This we were only to discover much later_.\n\n**Your son doesn't speak broken Greek**.\n\nLife as an immigrant is a net that you get caught up in without realizing it. It is Cyclops and Ithaca, Circe and Penelope, Scylla and Charybdes, all in one. You learn the language and the customs of the locals without quite understanding how. You change, without wanting to. You learn how to drink your coffee in a different way. Your father's view of the world is full of holes. Unfamiliar streets become part of your day-to-day geography. The locals sometimes praise your knowledge of the language, and of course there are times when you're dying to say, \"I didn't learn Greek to please you, I did it for myself.\" But you don't say it.\n\nThis country is slowly beginning to grow on you. There are times when you even love it, without understanding why, and there are other times when you resent it deeply each time you're reminded that you're foreign and each time you're reminded that you're unequal. Each time they refuse to rent out their house to you. But you live here; you work here; your friends are all here and, more to the point, your child is growing up here, growing up thinking himself a native.\n\nHe doesn't worry about being let down by his accent; he doesn't have that kind of problem, because he doesn't speak broken Greek like his father does. When people compliment him on his Greek, he just laughs. He doesn't keep out of sight like you used to\u2014he screams and shouts and asserts his presence. He does not take insults with the same resentment you used to. He is not like you. When you suffered rejection, you would slink away to lick your wounds like an injured dog, because you were a foreigner and felt like a foreigner, but he doesn't feel like that. When he's rejected, he doesn't limp off like an injured dog but like an injured wild animal. You would try to treat your neurosis by saying, \"One day, I'll go back.\" For your child this is home. He has no other. He has no choice but to love it or hate it. Whatever it is he has to win or lose he will win or lose here, in the place where you decided he should be brought into the world.\n\n# **28**\n\n_T he next day the police finally allowed us to go outside onto the soccer field, which had been enclosed by barbed wire all around. Swaggering around, swinging their truncheons, they ordered us to sit down on the steps and wait for our names to be called. We were then to get on the bus that would take us to our destination, where housing and work would be provided. The first fugitives to be called strode up to the bus giving the sign of victory, while the rest of us looked on, green with envy, waiting for our own names to be announced through the megaphone. Two or three busloads left that day; those who weren't called returned to the warehouse disheartened, heads bent and feeling their exhaustion even more keenly. The mysterious glow that had rested on their sullen, tired faces for a few hours had vanished_.\n\n_That evening ten policemen distributed loaves again, sticking to the tried and tested method: random chucking. This time, however, there were fewer of us and the scramble for food displayed some rudimentary aspects of fair play. Now and again the odd \"sorry\" and \"I didn't mean to hit you\" was heard. The drivers had meanwhile become incandescent when they realized their names were not going to be called. Their fury intensified because Sex Boy had made it onto one of the buses, which meant that, according to our collective fantasy, he'd be sleeping in some nicely heated room with an en suite shower. \"Bloody hell, that puppy gets to go, while we two, who grew up in the capital, have to stay here!\" protested one of the drivers, obviously forgetting that I was from the provinces, too. I said nothing. They fell silent, realizing a little belatedly how tactless they'd been. I then remembered how Sex Boy had come up to us, hugged us, and told us that as soon as he was sorted he'd do something for us. He was on the verge of tears and for the first time ever I saw in his eyes something resembling joy, something resembling hope_.\n\n_After a while, with the three of us deep in thought, one of the drivers told us what was at once the funniest and the most tragic story I've ever heard in my life. It was about his cousin, who had been exiled together with her family on account of a fart. Unbelievable but true. It was just after Comrade Enver's death, and the entire nation was obliged to pass by and pay their respects at the tomb of the Eternal Leader. She duly went along with the rest of her collective from the factory where she worked, one of the ones built by the Chinese, in fact. Even though she had a cold that day, she simply had to go, because she couldn't just say she wasn't going at a time when almost all of Albania was making its way to Tirana, even if it meant hanging off those Chinese trains to get there. Compared to the profound sentiment and duty to pay her respects at the tomb of the Eternal Leader, what was a cold if not a sign of bourgeois weakness? The driver's cousin understood only too well that now was not the time to display such infirmities. So she went and waited in the long line, goodness only knows how long for, hours rather than minutes, and when it was eventually her turn, she kneeled down like everyone else, preparing to weep, to sigh and kiss the marble covering the tomb of the Great Leader. However, during the part when she was supposed to emit the customary loud sigh, she instead produced an extremely voluble fart, so distinct and so unmistakable that the person behind her in line heard it and so did the person behind him, both members of the same collective. That was the end of her. They went straight to the Party secretary to report the incident, and the secretary, in view of the solemnity of the occasion, didn't hesitate to call a general meeting to which the cousin was summoned and required to prove that the fart had not been motivated by subversive, hostile aims. The secretary went after her hammer and tongs, bent on proving her guilt. He argued that the enemy would not hesitate to make use of all means available to him in order to insult the memory of the Eternal Leader and so on and so forth_.\n\n_It didn't end there; when they started to sniff around in her past, they discovered that her husband's grandfather had been convicted and executed by the Party in 1948 as a sworn enemy of the Albanian state and of the Party. That was it. How could she possibly hope to convince anyone that her flatulence at the tomb of the Eternal Leader was not somehow connected with the execution of her husband's grandfather? It was a lost cause. Besides, the Party had said it over and over again: water can sleep, enemies never do. Consequently she lost her job and was exiled to a strange village in order to learn how to love the Eternal Leader fittingly in the fields. And thus we passed our fifth night in our warehouse; with the story of the fart. Or was it the sixth? I'd lost track, but at least now we had a bit more room to sleep. I ate a small piece of my loaf and slept like the dead that night_.\n\n**When They Stop Feeling Sorry for You**\n\nThere are times when you don't understand your child. He doesn't understand you, either. He doesn't understand your homesickness, and your homeland is nothing but a tourist destination to him. He doesn't feel guilty like you do for having abandoned it. His homeland is here, where he lives. If he feels stigmatized by his origins, he'll probably become neurotic and incapable of loving himself or of loving any country for that matter.\n\nSometimes your broken Greek embarrasses him; the way you walk embarrasses him; the way you dress embarrasses him; lots of things about you embarrass him. He's not like you, though; he's not an inveterate scrooge desperately saving money in the hope of going back. He wants to dress like the Greek kids, live like them, get on in life like they do, have nice hands like they do, have a career like they do. Heavy manual labor, the kind the Greeks won't touch, is meant for other aliens, the ones who've just arrived. He feels completely at home, which is why he doesn't ask anyone to love him. He knows very well that love is in short supply. He wants his due, that's all: to be treated like an equal, to be judged on his own merits and not on his ethnic background. He wants to make this country his own, because he has no other country to love. If this country rejects him, he'll reject it, too, and he'll be left dangling, much worse off than you because he'll have nowhere to call his own. He'll be a rootless creature ready to embrace an alternative identity\u2014not nostalgia but the ghetto.\n\nYour child doesn't ask to be loved and, above all, he does not ask to be pitied. The pity card that you used to play just to survive is one he despises. Of course he has many more friends than you ever had, and they are gradually accepting him as one of them. But they are slightly more nervous of him than they are of you, mainly because he doesn't inspire pity.\n\nThis is when the trouble really starts, when people are no longer able to read any signs of misery and misfortune in his face, when his is no longer pitiful and different, when he starts acting all familiar and making himself at home.\n\nThat's when the trouble starts, when the immigrant starts demanding equality. Your child is regularly criticized, not for behaving like a foreigner but for behaving like a Greek. Not because he's different but because he isn't. Because that's when it's clear he's here to stay. The immigrant, dear reader, is tolerated as a temporary extra, but feared and reviled when he looks like he's about to move in for good.\n\n# **29**\n\n_I t must have been around midday by the time I woke up. The driver woke me. For some reason the police had not brought us outside to sit on the steps and nobody was reading out any lists of names. \"Something must have happened,\" said the codriver. \"They're not screaming at us, either.\" We went outside and saw that he was right. Something had happened, but we couldn't work out what it was. The policemen had abandoned their grim, truncheon-swinging swagger. Not only that, they had actually cleaned up the filth outside the warehouse, and were now calmly asking us to get in line in readiness for a proper midday meal: half a loaf of bread each and some cheese_.\n\n_Speculation was rife. At first people were claiming that the police had sharpened up after the letter to the UN. Not very likely, considering that the letter, for want of paper, an address and a post office, had never actually been sent. Then it was said that the US ambassador was on his way. Others said it was the British ambassador. By about 2 p.m. it had been agreed that they were both en route but a couple of hours later, there'd apparently been a change of plan: it was a big NATO general, who was recruiting for soldiers to fight in Iraq. George Bush's nephew got into the spirit of things by shrieking every now and then, \"Company march!\" He'd probably got it from some film he'd seen on Italian TV. The news that we were about to be called up for active service in Iraq caused enormous unrest, and the driver and the codriver were equally incensed, insisting that they hadn't come to the West to be packed off to fight but to work. By 5 p.m. people had split off into various groups of roughly the same size, according to which of the above scenarios they found most convincing_.\n\n_The previous day, out there on the field, I'd struck up a conversation with one of the policemen, in English. I wasn't very good\u2014I'd only got as far as \"Essential 2,\" the most common English language learning method followed in Albania, but the policeman knew even less. His name was Pavlos. He seemed different, not like the other policemen. He didn't shout, he didn't go around wielding his truncheon, and was one of few, or rather the only one, who ever smiled. We'd talked about various things, where I'd come from, what I'd studied back in Albania, while he explained to me that he was originally from Sparta, and I told him I'd read all about the ancient Spartans, the war with Athens, Pericles, and the Acropolis. That day they allowed us to go out for a second time to enjoy what little was left of the winter sun, and I saw him again. To be precise, he saw me first and motioned to me to come over. We got talking again. This time he questioned me more closely about why I'd come, and if I wanted to find work in Greece. I told him that I did, and that I intended to go to university eventually, to study history and philosophy. I also told him that I knew French and Italian. Pavlos was impressed and beckoned me to follow him. We walked along for a while until we got to a group of policemen who were gathered around a man who appeared to be their superior, judging by the way they spoke to him_.\n\n_Pavlos started talking to him and pointing in my direction. The others listened in silence. Of course I couldn't understand a word of what he was saying, but I could feel the weight of two or three sardonic pairs of eyes on me. Pavlos's little speech was met with a few moments' silence and then his superior responded. He didn't say much, but the single, short phrase he used, coupled with the look on his face, a mixture of boredom and cynicism, was enough to get the rest of them laughing. The cynical stares falling on me multiplied, and Pavlos fell into an embarrassed silence\u2014I saw him turn red. I hadn't been able to follow anything but it was obvious that he'd been completely humiliated. I realized he must have said something about me, something nice, and had been repaid with the sarcasm of his superior and the ridicule of his fellow police officers. I felt myself blushing, too, as though I'd caught the embarrassment from Pavlos. He signaled to me that we should go back, so I followed him. He explained to me that he wasn't like the rest of them and was waiting to start his new job at the airport, as though the policemen at airports were a breed apart. I had to get back inside with the others. Pavlos shook my hand and wished me luck_.\n\n**The Immigrant's Memory**\n\nThe immigrant is locked in perpetual dialogue with his memory and with his past. More precisely, he is condemned to be in perpetual opposition to them. Some, several, a great many in fact, faced with this, choose Lethe instead, as a survival strategy or as an inevitable rejection of the state of uprootedness and distance. Once he's rid himself of the past, the immigrant feels much lighter about facing the difficult journey ahead, during which he carries both metaphorical and literal burdens. Being in another country can frequently lead to disengagement with your former self. Now you have the chance to disappear, to settle your scores with your origins, to reinvent yourself, to start a new life from scratch. Some manage, some don't. When this strategy fails, memory comes back with a vengeance, more cynical this time; it breaks down all barriers and fills your spirit with screams that replace words, or with words that make you stutter, with ghosts and shadows that haunt your nightmares by night and feed your neurosis by day. Because when you cut loose from your memory, there are only two choices left: either permanently consign it to the deepest depths of Lethe or, if that doesn't work, surrender yourself to this distorted effigy.\n\nAt the other extreme are those who choose to anchor themselves in memory, and turn it into something resembling an old treasure chest, where they keep the clock of their former existence, which stopped ticking long ago, safe. In this case, memory resembles a precious embalmed body: it becomes representative of the golden age, a golden age that never even existed, and never will exist. Those who preserve the past in this way are often regarded as heroes, because they resist change. The truth is that devotees of this brand of memory are nothing more than failed heroes, because they were too frightened or because they simply withdrew from a present that was too painful for them, to seek comfort in an idealized version of the past.\n\nThen there are those immigrants who choose to preserve a paradigmatic relationship with memory. They are not its prisoners but they don't try to eradicate it either. They do not experience their origins as either a stigma or as a protective shield. Memory in their case is not one smooth continual tradition full of nostalgia for roots and scents, a clock stopped and incompatible with the present. This is no marmoreal memory but a part of yourself which is constantly changing. It's the vaulting pole that helps the immigrant live through change with greater honesty and decency and, if possible, with greater wisdom. \"I met pain along the way, pushing me further and further along\"\u2014the lines of a song the Marseilles immigrants sing. This encounter with pain, pushing us forward all the time, is the human condition par excellence. And the way this encounter is narrated is memory itself.\n\n# **30**\n\n_I heard shouting in Albanian, \"The ambassador's here! The ambassador's here!\" People came rushing out of the warehouse. A van drew up close by us. A white-haired, white-bearded, white-mustached man in his fifties with an aristocratic air got out. He was followed by a woman, and then two young men. One was holding a television camera, the other was carrying something else, possibly a microphone. Our speculations were not entirely unfounded, it seemed; some people had arrived, granted, none of them the candidates we had shortlisted earlier. The white-haired man with the aristocratic demeanor was neither the American nor the British ambassador, nor a big NATO general but a Greek filmmaker called Christos_.\n\n_Christos had come to document the tragic influx of Albanian immigrants into Greece. He asked if any of us could speak English, and the driver and codriver immediately grabbed me by the arms and carried me across to him, almost airborne. \"Tell him what we've been through here,\" shouted the Albanians around me. \"Tell him that the police treat us like animals, they don't give us any food, and don't let us do anything, not even take a piss.\" A lot more than that was said but those were some of the few phrases I could pick out from all the noise in this tragic place. I don't know what I told Christos. Even today I can't say for sure what the hell it was I told him. All I can remember is that he took me to one side while his two assistants were filming inside the warehouse. The police, observing the zeal with which they were working, reverted to their former grim expressions_.\n\n_Christos and I started talking about Albania. He was very familiar with Ismail Kadare's work and told me that almost all of Kadare's books had been translated into Greek. Then he looked around at everyone and appeared shocked at the utter misery he saw before him. \"You know, I actually believe in communism,\" he said. I said nothing. There was a pause, and then he said, \"What were you doing for fifty long years? Nothing?\" I looked him in the eye and then motioned at the crowd with a tilt of my head. \"You can see for yourself what we were doing.\"_\n\n_The truth is that if someone had seen us Albanians there in Filiates, they would have thought that we'd just been liberated from a concentration camp: the dull colors of our clothes, the shades of gray and black favored by the regime; the undernourished faces that had something wild about them, exacerbated by the cold; the hunger and exhaustion of the past few days; that vacant stare, the same vacant stare of the orphan or of someone who has failed utterly, been defeated utterly_.\n\n_\"Do you want to come with me to Athens?\" Christos asked me suddenly. \"What would I do in Athens?\" I asked him. \"Find a job\u2014you know so many languages, you're bound to find something, maybe with a paper. You can stay with me till you've learned some Greek.\"_\n\n_I thought I was dreaming. A stranger, someone we'd imagined to be the American ambassador, the British ambassador, and a big NATO general, was offering to put me up in his home. I was desperate to escape the chaos, the misery and the discomfort of the warehouse, and every atom of my being was yearning for a lifeline, some means of moving on now that I had crossed over into the world-beyond-the-borders, a helping hand, just enough to get me on my feet. I said yes at once_.\n\n_Christos told me to wait, and I saw him walk up to the policemen. It was already dark and the big floodlight had already come on. About ten minutes later he was back telling me that they wouldn't allow him to take me to Athens without getting the go-ahead from the mayor. \"I'll be back tomorrow and I'll talk to the mayor,\" he promised. My instinct told me that if I didn't get away that night, he might not find me there the next day. \"Tomorrow will be too late,\" I answered. He took me by the arm and this time we both went over to the policemen to try to persuade them. I could tell that Christos was trying to butter them up, but from what I could make out, he wasn't getting very far. One of them started prodding me in the back of the neck with his truncheon, repeating the question, \"Musulman? Musulman?\" My terrified answer, \"Not Muslim. Atheist,\" which I'd hoped would soften him a little, apparently went over his head because he kept on repeating the word, but this time without a question mark: \"Musulman. Musulman.\"_\n\n_At some point, taking advantage of the relative disturbance being caused, Christos led me off again and took me to one of his assistants, Grigoris. They spoke briefly and Grigoris nodded, as though in response to some secret code, took hold of my arm, and dragged me off. We passed two or three policemen, who at first went for me, but Grigoris managed to get them off by repeating certain words, the same words all the time. He put me in the van and told me to sit at the back and keep out of sight, handed me a blanket (I wasn't sure if this was intended to keep the cold out or to keep me hidden). The rest of the crew soon appeared and hurried into the van and started the engine. We stopped after only a few seconds. More policemen, talking to Christos, but they eventually let us on our way_.\n\n_I soon digested the fact that this was my second escape. I'd escaped from the warehouse at Filiates. Grigoris told me it was OK and I could sit up. We drove through the darkness, far away from the policemen. From the window, I caught sight of a human caravan walking through the night\u2014my fellow countrymen, having crossed the border into Greece. There, surrounded in darkness, they looked like weary shadows walking without direction, like ghosts roaming around in the night, briefly illuminated by the headlights of the van. It was then when I felt a river of tears rushing down my throat and I broke into sobs_.\n\n_It was January 22, 1991, I think. That's more or less how my life in Greece began_.\n\n# **Epilogue**\n\nThis tale is not typical of border stories: it comes to an abrupt halt where most of them just keep going without ever reaching the end. Tales of monstrous, visible borders such as the borders of totalitarianism and tales of the invisible, psychological borders experienced in a foreign country rarely reach a conclusion. It was never my intention to tell you the story of my life; what I wanted to tell you about was my illness, Border Syndrome, a condition you won't find documented in any manual of recognized psychological disorders. It's not like agoraphobia, vertigo, depression. And it's not like any physical disease spread by a virus, but that doesn't make me any less of a carrier\u2014maybe just a carrier with low levels, as the doctors are fond of describing carriers of hepatitis whose organs have developed enough antibodies to keep the deadly march of the virus in check. Nevertheless, Border Syndrome is just as pernicious as the hepatitis virus because you can never truly get rid of it. It just sits there, in a latent state, wedged between time and space, wedged between your body and the gaze of others, ready to strike at any moment and take possession of your memories, your silence, the expression in your eyes, your spleen, your smile, your passion and your life. It's then that you start to experience your body and your face and your origins as a burden. You long to be free of it all, if only for just a second, for as long as it takes to cross the borders\u2014if only for that long. Unlike all those self-satisfied people who scream and shout, asserting \"the right to be different,\" what you crave more than anything is the right to be exactly the same as everyone else. You long to go unnoticed, to be invisible. But you know that can never be more than a fantasy\u2014Border Syndrome is hardly the stuff of fairytales. It tends to affect daydreamers, daydreamers who fight tooth and nail to maintain a grip on reality, who fight tooth and nail to overcome both kinds of borders: the visible and the invisible. To work out how much you are at risk of contracting the Border Syndrome virus, all you need to do is remember which side of the border you were born on.\n\nAs a carrier of Border Syndrome, I have to confess to having a dream: a dream of a world without immigrants. Don't get me wrong\u2014I love traveling; most sufferers of Border Syndrome do. If you never get out from inside yourself, from your body, your ennui, then you can be pretty confident of lifelong immunity from Border Syndrome. I'd simply prefer people to travel in the real sense of the word, to go traveling, like tourists do, like students do, like bohemians do. To travel like people who have lost their way looking for paradise or like those who have found their own Ithaca through some absurd twist of fate\u2014to travel with dignity. I do not want people to travel in the way that so many Germans, Irish, Italians, and Greeks once traveled, and the way so many Albanians, Afghans, Iranians, Somalis, Mexicans, and so many others are forced to travel today. Because being a migrant by definition puts you in a position of weakness. And in this world, the weak are never treated with respect. They might be pitied, but they are never respected. I remember how in the early days in Athens just after I had crossed the borders, I was walking down the road quietly whistling the lyrics of a song I'd just learned: _I walked through the night \/ not knowing a soul \/ and not a soul \/ not a soul \/ knew me_. It's one of Mikis Theodorakis's songs. It wasn't written about immigrants, but it does capture the feelings of all those who have at some point experienced true loneliness and have tasted the fear of defeat\u2014sufferers of Border Syndrome, in other words.\n\nI have a dream\u2014I have a dream of a world in which there are no immigrants. But such a world is not feasible, because a world without immigrants would have to be a world free of tyranny; a world free of poverty; above all a world free of the desire that people have to take control of their own destiny. A world free of immigrants and free of migration would be a much duller world than the world we know today. At the same moment that I yearn for a world devoid of immigrants, I count my blessings that I am an immigrant. In the final analysis, it takes guts; it takes guts to go head to head with borders and start your life again from scratch: with language; street names; people's names\u2014to make this foreign city your own. Being a thoroughbred immigrant means acknowledging the power of the will, and coming to terms with the outrageous tricks of fate and to understand that the greatest human virtue is the ability to adapt and change and has nothing to do with who you are descended from, and to realize that the secret of success is at once simple and complicated: never to tire of life.\n\nThat I am sitting in Berlin writing these lines is due to a simple twist of fate, one that occurred several years after the events that I narrate in this book took place. I'm on Bernauer Strasse, looking across at what's left of the Berlin Wall. A couple of tourists are sitting next to me, experiencing the whole thing on the level of a tourist attraction, as an interesting historical fact. To me it's as though I am looking at a small piece of the skeleton of the monster that once shut me away in his cell, forbidding any contact with the world beyond the borders; I will always carry the ghost of this monster inside me, wherever I go, however widely I travel, however many borders I cross. A mere few meters away from the skeleton of the monster I can see an advertisement for a mobile phone company, urging passersby to enjoy each and every moment of this short life in a world that is getting smaller and smaller every day\u2014a world in which there are no more borders and no more walls. I absentmindedly flick the pages of my passport inside my jacket pocket. The impenetrable borders and the murderous walls of the Cold War are a thing of the past. Borders and walls live on for the most part inside our pockets. They are the passports we carry. I become aware of this each time I stand at a passport control, because at these checks, there are two categories of people: bearers of \"cool passports\" and everybody else\u2014people holding \"bad passports.\" If you've got a \"cool passport,\" you've got nothing to worry about. Borders are nothing more than invisible lines, a trick of the imagination, geographical lines as translucent as the light of the Mediterranean. Having a \"bad passport,\" on the other hand, changes everything. It means you have Border Syndrome, and every crossing you make becomes an unforgettable incident, an event on your existential calendar. And the more borders you encounter, the more determined you become to cross them. I still belong to the \"bad passport\" group. Greek passports once belonged to the \"bad\" group but are now part of the \"cool\" group. I haven't got one. I don't know if I'll ever be given one, so for the time being I'm traveling on my bad one, crossing borders with it. Who knows, it might become \"cool\" one day. I hope so. The face of the immigration officer checking a \"cool\" passport usually looks very relaxed and very human. But the expression of the officer checking a \"bad passport\" is usually very suspicious. He looks at you. He looks at you again. He asks you questions. He asks you outrageous questions. And you give very plausible answers. He asks you the most outrageous questions once again. And you give him very plausible answers once again, waiting for the stamp to fall so that you can get across to the other side of the border, to the other side of the world. When you carry a \"bad\" passport, borders revert to type; they become what they used to be and what they always will be\u2014miserable places. There are some border posts and customs checkpoints in which the immigration officer is himself the son of immigrants. His parents too had crossed on a \"bad passport,\" or had perhaps stowed away on a train or in the darkness of a boat. Perhaps they tore up their passports so that they wouldn't be deported. And he, born here to parents who at some point arrived from somewhere else, is now checking other people's \"bad passports,\" people who perhaps have the same passion to get to the other side of the border, to emigrate, to put down roots here so that someday their own children might be able to carry a \"cool passport\" in their pockets. Carriers of Border Syndrome are constantly changing their place of birth, name, and country. However, the way they keep their gaze fixed on the borders, and the way the borders return their gaze, remain unchanged through the ages. And that's because the world keeps turning and then comes full circle. Sometimes it makes progress, breaking down old borders and establishing new ones, irrespective of which side of the borders you find yourself on. In the final analysis, we are all immigrants, armed with a temporary residence permit for this earth, each and every one of us incurably transient.\n\n# **Author's Note**\n\n_A Short Border Handbook_ is part autobiography, part fiction, its heroes both real and invented. It could not have been written without the help of several people, most of them anonymous, who related their own border experiences to me, especially the women immigrants who were generous enough to share their often tragic stories. During the writing of this book, other writers and their works proved inspirational, particularly Zeese Papanikolas and his _Buried Unsung: Louis Tikas and the Ludlow Massacre;_ Tsvetan Todorov's _Les abus de la memoire_ , as well as newspaper articles by Tahar Ben Jelloun on the subject of immigration. Special thanks go to my mother and brother, to Ilira, to the Internet\u2014and to fortuity.\n\nGazmend Kapllani\n\nAthens, January 2009\n\n# **Notes**\n\n Kor\u00e7\u00eb is a town in southern Albania, close to the Greek border. Kalabaka is a town in the north of Greece, close to Thessaloniki.\n\n Kakavia is the main checkpoint on the Greek-Albanian border.\n\n Gjirokast\u00ebr is a city in the south of Albania.\n\n Zeese Papanikolas, _Buried Unsung: Louis Tikas and the Ludlow Massacre_ , University of Nebraska Press, 1991.\n\n \"Oh! Nightingale, I have vowed \/ To find out your nest, double-crosser \/ Grab you and throw you in the river \/ Drown you there because our girls lie awake at night because of you, you flirt \/ You twist, be\"\n\n# **A Conversation with Gazmend Kapllani**\n\n**You begin (and end) _A Short Border Handbook_ by drawing attention to what you call \"Border Syndrome\"\u2014a hard-to-explain condition not included in any \"list of recognized mental disorders.\" Is this generally understood (especially by immigrants themselves) to be something that many immigrants suffer from, whether named in some manner or not? Did you coin the term or did you encounter it somewhere? Is the condition static or does it change with the times?**\n\nThere are writers who \"owe,\" so to say, their literary obsessions to their imagination. There are others who owe them to their personal experiences. I belong to the second category. Coining the literary metaphor \"Border Syndrome\" was, for me, like giving a name to an obsession that has shaped my entire life. I have lived half of my life under totalitarianism and the other half as an immigrant. I was born during the Cold War, when the Iron Curtain divided Europe in two. By the way, I believe that the tragedies of the twentieth century in Europe cannot be fully understood without knowing the history of its borders, fences, and walls.\n\nI was born in Stalinist Albania, a tiny country that was isolated from the rest of the world for half a century by means of terrifying borders and walls. For me and my generation borders and walls mean madness, xenophobia, and terror. And they also mean a longing to cross them.\n\nWhen communism collapsed I crossed the borders to Greece as a refugee. There I had to deal\u2014this time around\u2014with invisible borders: the foreign language I wanted to conquer, having my papers in order to be recognized as a human being. I was seen as an undesirable when I was a penniless foreigner and I faced hostility and envy when I became a successful immigrant.\n\nNo wonder, then, that borders constitute one of my main literary obsessions.\n\n**You write, \"Those who have never experienced the urge to cross a border, or who have never experienced rejection at a border, will have a hard time understanding us.\" In writing this book, were you motivated more, at least initially, by a desire to broaden public understanding of Border Syndrome or to chronicle your own experience and those of people you knew?**\n\nThe voice of the storyteller in my book is rather timid. He belongs to the so-called \"first generation\" of immigrants. He has experienced borders under totalitarianism and he has faced a lack of understanding and rejection as an immigrant. He is not simply telling a story: he is making a public confession. And he's afraid that many out there, in the imaginary audience, won't be able to understand what he is talking about.\n\nAs a writer, though, I wrote this book mostly for those who have never experienced a rejection at the border or have never felt that strange state of mind called the \"Border Syndrome.\"\n\n**_A Short Border Handbook_ is both the story of one immigrant's personal journey and a handbook of sorts to describe the general experience shared by all immigrants. You balance these two elements deftly\u2014devoting a portion of each chapter to the story of the protagonist (you, with fictional elements woven in) and another to the immigrant experience as a whole. Why did you opt for this narrative approach?**\n\nLet me tell you the story behind this book\u2014which is also my first novel, even if its dual nature leads publishers to categorize it as nonfiction. I started writing it as a long essay that would be strictly based on my personal memories of crossing the border after the collapse of communism. In the course of writing, though, fictional characters popped up in my mind and they drove me to places and meditations that I could have never predicted. They set me free, in the sense that I could travel beyond the narrow horizon of my personal experiences. My personal memoirs were vital to the writing of this book, but at the same time, thanks to my fictional characters\u2014\"Sex Boy,\" for example\u2014I went way beyond my personal story. That's the moment that the book evolved into these two parallel voices. The first voice tells the story of border-crossing, of the journey itself. The second voice is a meditation upon the immigrant journey.\n\n**In the second half of many of the chapters, you use the second person perspective\u2014addressing the reader as \"you\"\u2014to identify the \"immigrant.\" Was the purpose of this choice to allow the reader to some extent imagine him or herself in the place of the immigrant?**\n\nAs I said before, the parallel narrative is a meditation upon the immigrant journey. I feel that the second voice, far more than triggering empathy for the immigrant, gives the book its integrity. It is, above all, an honest effort to make sense of what the story means\u2014that messy, confused, exciting, painful, and comical experience of crossing borders, of encountering a foreign sky, a foreign language, a foreign culture, as a refugee and an immigrant. It was like putting a mirror in front of myself and my readers at the same time.\n\n**Borders play both a literal and metaphorical role in this story. It seems that once immigrants pass into the \"world-beyond-the-borders,\" there are many more metaphorical borders to cross in order to be truly accepted into society\u2014including such difficulties as language and economic barriers. Would you say that there truly isn't a \"world-beyond-the-borders\" for the immigrant?**\n\nWe humans live surrounded by borders and boundaries, both visible and invisible. We are engaged in an eternal game of creating borders and boundaries and, at the same time, of inventing the means to subvert them. That's human nature, I guess. But in the case of my storyteller there's an overwhelming, tragicomic presence of borders in his life. He carries within him these two kinds of experiences, which are apparently very different from each other: those of living under a totalitarian regime and of crossing borders as an immigrant. He starts his immigrant journey while the slogan \"Tear down that wall!\"\u2014from Ronald Reagan's legendary speech in Berlin\u2014is still echoing in his ears; and once he does cross that scary wall, he finds himself surrounded by a crowd that's shouting \"Build that wall!\"\n\n**You also briefly reflect on the history of Greek migration to America\u2014how those immigrants often had to change their names, how they faced challenges in learning a new language, how they experienced xenophobia, and so on. The parallels between that history and the more contemporary history of the Albanian migration to Greece is shocking. Do you think that if more citizens in various countries were aware of their own particular cultural, national, and family histories regarding migration, this might lead to more acceptance regarding immigrants today? It has been noted, after all, that the vast majority of those in America, at any rate, who have called most stridently for curbs on immigration come from families that, not too many generations ago, arrived on that country's shores as refugees or immigrants, in many cases undocumented and unwelcome**.\n\nA Frenchman who knew America well, Alexis de Toqueville, said that history is a gallery of pictures in which there are few originals and many copies. That quote has returned to my mind pretty often while reading a lot about the history of immigration and interviewing immigrants from all over the world. A Greek immigrant at the beginning of the twentieth century in America and an Albanian immigrant at the end of the twentieth century in Greece have many more things in common that they have differences.\n\nWhen I became a well-known journalist in Greece I sometimes highlighted these similarities in my articles. What I saw surprised me: many of my readers were troubled by the comparison. Greece had (and still has) some of the most drastic levels of emigration of any European country. Why, then, did people get so troubled when they were reminded about their past as immigrants, especially by an immigrant like me? I tried to offer some answers. First of all, when Greece became a rich country and people stopped immigrating massively to other countries, there was a cultural tendency to bury into oblivion the painful history of their own immigration. It didn't match the image of the nouveau-riche self.\n\nAs you asked though about parallels with America today, I would add something else. Time after time I received messages from some Greeks who had read my articles and who had been themselves immigrants, telling me that comparing Greek immigrants with Albanian immigrants or other refugees was totally wrong. According to them, \"the Greek immigrants were different.\" What they meant was that they were \"good immigrants and refugees\" as opposed to Albanians and other immigrants in Greece who were considered \"bad, violent, lazy immigrants.\" When I replied to those readers by talking about the historical facts of racism against Greek immigrants in America or elsewhere, where Greek immigrants were seen as violent and lazy, some of these readers would become irrational in their reactions.\n\nIt might sound very simple, but at the root of this division\u2014\"we, good immigrants\" versus \"them, bad immigrants\"\u2014lies the refusal to recognize the same humanity in the newcomers. New immigrants are considered to be less human or in the worst case, dehumanized.\n\nLast, let me say that the history of immigration, both in Europe and America, is full of examples of old immigrants who turn against the new immigrants, because they feel or fancy that their privileged place in the society or the economy is under threat. This situation becomes explosive, though, if a demagogue tries to turn that hostility into a political movement. It might lead to a sort of \"civil war\" between different groups of immigrants, between old and new immigrants.\n\n**In the epilogue, you refer to a dream the protagonist has in which there would be a world without immigrants. Is a world without immigrants the same thing as a world without borders? If not, what would a world without borders look like?**\n\nIn my work I see migration as an essential part of the human adventure on this planet, as old as traveling, religion, and poetry. At the same time, I'm not trying to embellish or turn the immigrant's experience into folklore. Being an immigrant often means having an inferior status to natives; it means exposing yourself to the extreme games of fate. As an immigrant you might lose the best of a country and you often have to rely on the kindness of strangers. On the other hand, it takes guts to be an immigrant, to go head to head with borders and start your life from scratch\u2014with language, street names, people's names\u2014to make a foreign city your own. \"I met pain along the way, pushing me further and further along\"\u2014these are the lines of a song immigrants in Marseilles sing.\n\nI'm not sure how a world without borders would look. What I can say is that my life-long experience with borders has taught me that every time we talk about building strong borders and walls and fences it means that we are angry or frustrated or have badly lost our self-confidence. Before being built on the ground, borders and walls are erected in our minds and souls.\n**ALSO AVAILABLE FROM NEW EUROPE BOOKS**\n\nIn this piercing and resonant debut novel, a young Bosnian Muslim refugee finds a new home in America\u2014until the aftermath of 9\/11 tests him as never before.\n\n978-0-9973169-0-2\n\n\"Powerful, eye-opening reading for everyone.\" \u2014 ** _Library Journal_**\n\n\"Madrygin's harrowing, compelling debut will live long in the reader's memory....Deeply informative and moving, it will spark debates regarding American foreign policy.\"\n\n**_\u2014Booklist_**\n\n\"In Madrygin's gripping debut, the horrors of war give way to the challenges of carving out a life in a hostile country.\"\n\n**_\u2014Publishers Weekly_**\n**ALSO AVAILABLE FROM NEW EUROPE BOOKS**\n\nThe critically acclaimed history of Eastern Europe\u2014now in a second edition\n\n978-0-9850623-2-3\n\n**Praise for the first edition**\n\n\"Displays an ease and familiarity with cultural minutiae while briskly covering intense topics of genocide, religion, and Communist implosion.\"\n\n**_\u2014Publishers Weekly_**\n\n\"A commendable feat.\"\n\n**_\u2014Library Journal_**\n\n\"A brief, concise, and informative introduction to the less-known part of the old continent. No country is left out. Accessible and stimulating reading.\"\n\n**_\u2014CHOICE_**\n**ALSO AVAILABLE FROM NEW EUROPE BOOKS**\n\n\"Heady, dizzying writing....\n\nA master class in how to tell a war story.\"\n\n**_\u2014Kirkus Reviews_ (starred review)**\n\n978-0-9900043-2-5\n\n\"These extraordinary stories are searingly truthful.\"\n\n**_\u2014Independent_ (UK)**\n\n\"Each tale is a rich, poetic slice of life from places you might never go....The stories all describe experiences alien to Westerners and skillfully explore material about which readers are curious....J\u00e1szber\u00e9nyi is a gifted writer, this book is to be savored and relished.\"\n\n**_\u2014Library Journal_**\n**OTHER TITLES**\n\n_Ballpoint: A Tale of Genius and Grit, Perilous Times, and the Invention that Changed the Way We Write_. 978-0-9825781-1-7\n\n_The Essential Guide to Being Hungarian: 50 Facts & Facets of Nationhood_. 978-0-9825781-0-0\n\n_The Essential Guide to Being Polish: 50 Facts & Facets of Nationhood_. 978-0-9850623-0-9\n\n_Illegal Liaisons_. 978-0-9850623-6-1\n\n_Keeping Bedlam at Bay in the Prague Caf\u00e9_. 978-0-9825781-8-6\n\n_Once Upon a Yugoslavia_. 978-0-9000043-4-9\n\n_Petra K and the Blackhearts_. 978-0-9850623-8-5\n\n_The Wild Cats of Piran_. 978-09900043-0-1\n\n_The Upright Heart_. 978-0-990043-8-7\n\n_Voyage to Kazohinia_. 978-0-9825781-2-4\n\n**Williamstown, Massachusetts**\n\nFind our titles wherever books are sold, or visit www.NewEuropeBooks.com for order information.\n**ABOUT THE AUTHOR**\n\n**Gazmend Kapllani** , the author of three books, teaches creative writing and European history at Emerson College and was previously a fellow at Harvard University's Radcliffe Institute and a writer-in-residence at Wellesley College. He has held presentations at numerous colleges and universities including the University of Michigan, Columbia University, Brown University, Harvard University, Wellesley College, and Bennington College. Born in 1967 in Albania, he crossed the mountainous border into Greece on foot in 1991. In Greece he worked as a builder, a cook, and a kiosk attendant while earning a doctorate in political science and history at Athens University. For more than ten years he was a columnist for the leading Greek daily _Ta Nea. A Short Border Handbook_ , inspired by his own experience as an immigrant and written in Greek, was a bestseller in Greece and translated into several languages. Kapllani's other books include the novels _My Name is Europe_ (2010, Greece; 2013, France) and _The Last Page_ (2012, Greece; 2015, France).\n**ABOUT THE TRANSLATOR**\n\n**Anne-Marie Stanton-Ife** is a translator of Modern Greek and Norwegian fiction and theater. Notable translations include Henrik Ibsen's _John Gabriel Borkman_ and _When We Dead Awaken_ (Penguin Classics, 2014) and a biography of Vidkun Quisling (CUP, 1999). Among the Greek authors she has translated are Sergios Gakas, Andreas Staikos, and Vangelis Hatziyannidis. Her translation of Hatziyannidis's _Four Walls_ (Marion Boyars, 2006) was shortlisted for the Independent Foreign Fiction Prize, 2007, while her translation of Gazmend Kapllani's _A Short Border Handbook_ (Portobello, UK, 2009) was runner-up in the Society of Authors 2012 translation prize for Greek.\n\n## Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. Contents\n  6. Prologue\n  7. Chapter 1\n  8. Chapter 2\n  9. Chapter 3\n  10. Chapter 4\n  11. Chapter 5\n  12. Chapter 6\n  13. Chapter 7\n  14. Chapter 8\n  15. Chapter 9\n  16. Chapter 10\n  17. Chapter 11\n  18. Chapter 12\n  19. Chapter 13\n  20. Chapter 14\n  21. Chapter 15\n  22. Chapter 16\n  23. Chapter 17\n  24. Chapter 18\n  25. Chapter 19\n  26. Chapter 20\n  27. Chapter 21\n  28. Chapter 22\n  29. Chapter 23\n  30. Chapter 24\n  31. Chapter 25\n  32. Chapter 26\n  33. Chapter 27\n  34. Chapter 28\n  35. Chapter 29\n  36. Chapter 30\n  37. Epilogue\n  38. Author's Note\n  39. Notes\n  40. A Conversation with Gazmend Kapllani\n  41. Other Titles\n  42. About the Author\n  43. About the Translator\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. \n  10. \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  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137.\n\n  1. Cover\n  2. Cover\n  3. Title Page\n  4. Table of Contents\n  5. Start\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by The Online Distributed Proofreading Team at\nhttp:\/\/www.pgdp.net (This file was produced from images\ngenerously made available by The Internet Archive)\n\n\n\n\n\n\n                           Transcriber Notes\n\n Obvious typos and missing punctuation fixed. Archaic spelling and\n   inconsistencies in hyphenation retained.\n Unclear text in the ads in the original has been clarified by review of\n   the same ads printed more clearly in other issues.\n The table of contents has been created and added by the transcriber.\n Italics are represented by underscores surrounding the _italic text_.\n Small capitals have been converted to ALL CAPS.\n\n------------------------------------------------------------------------\n\n                   Life of Stephen H. Branch.       1\n\n                   Supervisor Blunt, and Paul       2\n                     Julien\u2014My Last Interview\n                     with Madame Sontag.\n\n                   James Gordon Bennett's           3\n                     Editorial Career.\n\n                   _For the Alligator._             3\n\n                   NEW YORK, June 15, 1858.         4\n\n                   Advertisements.                  4\n\n------------------------------------------------------------------------\n\n\n\n\n[Illustration: STEPHEN H. BRANCH'S ALLIGATOR.]\n\n    ----------------------------------------------------------------\n\n   Volume I.\u2014No. 10.]    SATURDAY, JUNE 26, 1858.    [Price 2 Cents.\n\n\n       Entered according to Act of Congress, in the year 1857, by\n                           STEPHEN H. BRANCH.\n\n       In the Clerk's Office of the District Court of the United\n             States for the Southern District of New York.\n\n\n\n\n                       Life of Stephen H. Branch.\n\n\nWestport, Connecticut,\u2014that he boarded at No. 24 Bleecker street, with\nMrs. Mallory, and that he was a clerk for Perkins, Hopkins, and White,\nin Pearl street, near Hanover Square. I carried some beautiful books to\nhis place of business, and requested him to accept them. He sweetly\nsmiled, and opened the books, and warmly thanked me, and said he would\nbe pleased to receive them, but that as I was a stranger, he would\nrather I would see his guardian, Morris Ketchum, a Banker in Wall\nstreet, and give him my name and address, and if he were satisfied with\nmy references, and approved of his acceptance of the generous gift, he\nwould be most happy to receive the books. I was fascinated with his\nmodesty, and caution, and I took the books, and repaired to the Banking\nHouse of Mr. Ketchum, to whom I briefly imparted what had transpired,\nand left my references and departed, and called again, when Mr. Ketchum\nsaid that he had inquired respecting my character, and that young Jesup\nwas prepared to receive my books, which I soon placed in his hands, and\nour acquaintance began under the most favorable auspices. I soon invited\nhim to dine with me at Mrs. Tripler's, when all the boarders were\nenchanted with his beautiful person, and pleasing manners, and highly\ncultivated mind; and I shall never forget how proud I was, as he sat\nbeside me. After dinner, I invited him to my room, where I gave him cake\nand lemonade, and filled his pockets with delicious oranges. I then\nplayed \"Washington's March,\" \"Yankee Doodle,\" and \"Hail Columbia,\" for\nhim on the piano, and he departed for his place of business. He went\nwith me to Niblo's Garden, then in its glory, and as we strolled\narm-in-arm in the meandering paths, and inhaled the exhilarating perfume\nof the flowers, I was charmed with his chaste society, and enraptured\nand inspired, and I breathed the music of language in his ears, and we\nboth were invested with the purest and loftiest and happiest emotions.\nIn a week from that joyous evening, he was seized with bleeding of the\nlungs, caused by excited feelings, during his enthusiastic efforts to\nplease his employers, in the sleepless business season of early autumn.\nHe was borne to his mother's abode in the country, where he soon calmly\nresigned his soul to the Saviour, whose sacred virtues he had always\nstrove to imitate. Although I had briefly enjoyed the pleasure of his\nsociety, yet his premature demise created a void in my bosom that made\nthe world a desolation. His mother soon removed to New York, and\noccupied No. 39 Bond street, where I gratuitously taught her children in\nEnglish and the classics. But the invisible germ of consumption has\nborne to the grave her pure, intelligent, and lovely Caroline, Charles,\nRichard, and Frederick, and Morris, Arthur, Samuel, and Sarah anticipate\nthe same remorseless destiny. And may God cheer and bless their mother\nin her loneliness and tears. The father of this interesting and\nunfortunate family, was prostrated in the commercial crash of 1837, and\nhis depressed and spotless soul fled for refuge to the bosom of his God.\nMorris Ketchum was his early business associate and friend, and has\neducated his children, procured them lucrative clerkships, afforded them\nfacilities to visit nearly every nation, for health and general culture,\nestablished them in houses of commerce, and has clung to them, in sun\nand storm, like Pythias to Damon, and like Washington to his country. At\nthis period of my eventful career, I taught colored and Irish servants,\nand those of all countries, in their kitchens in the evening, and\nsometimes by daylight. Some paid me one shilling a lesson, and some two,\naccording to their wages and generosity. I taught the servants of the\nReverend Doctor Wainright, the Reverend Doctor Orville Dewey, Daniel\nLord, James T. Brady, Mr. Bowen, of Brooklyn, (of the firm of Bowen &\nMcNamee, of New York,) and the servants of other distinguished citizens.\nI obtained scholars by going from door to door, in the basement, and\nasking the servants if they would like to learn to spell, read, write,\nand cipher. My health had been miserable since I left Columbian College,\nand I often expected to fall dead in the street, or suddenly expire in\nthe presence of my pupils. For a long period after young Jesup died, I\nwas very gloomy, and became utterly helpless and bed-ridden, and called\noftener on my father for money than I desired, to pay for board and\nmedical attendance. I got better, and crawled out into the open air, and\nwent in pursuit of scholars in a snow storm. I began at the Battery, and\napplied at every door, until I came to No. 70 Greenwich street, when I\nwas asked to come in and warm myself, by a daughter of the lady of the\nhouse, who kept boarders. After a long conversation, by a cheerful fire,\nI was engaged to teach the daughter in the English branches, for my\nbreakfast and tea, and a very small dark room as a place of lodging,\nwhich I could not conveniently occupy without a candle in the day time.\nHumble as were to be my accommodations, my feelings were extremely\nbuoyant, and my ghastly form trembled with delight at my unexpected\nresurrection from the depths of indigence and despair. Mr. Ditchett,\n(subsequently a very efficient Captain of the Fourth Ward Police, and a\nbrave fireman, and an honest man,) had just married the eldest daughter,\nwhose sister was to be my pupil. I was kindly treated, and remained\nuntil the first of May, when I went to Dey street, and afterwards to the\nGraham House, at No. 63 Barclay street, where I saw the lean Horace\nGreeley, one of the founders of the Graham System. The boarders were\nmostly skeletons, and several were limping about the house, like frogs\nor lizzards or grasshoppers, and among the limpers, was Horace Greeley,\nwho had what the Grahamites called a boiling crisis, or crisis of boils,\nwhich was the result of youthful indiscretion, shower bathing, and\neating heartily of bran bread, mush and molasses, squashes, turnips,\nbeets, carrots, parsneps, and onions, for a long term of years. Although\nI had been a miserable invalid a large portion of my days, yet I fancied\na speedy restoration to health, by eating unbolted wheat bread and\nvegetables, and frequent bathing. I entered into a spirited conversation\nwith Greeley, who was reclining on the sofa, and in a loquacious mood,\nwho told me that he expected to be quite smart after the disappearance\nof a large number of boils then all over his person, which he attributed\nto salt rheum, that he inherited from his father, and which was recently\ndriven to the surface of his skin by a rigid adherence to the Graham\nSystem, and three shower baths a day; and he advised me to begin to\nbathe immediately, and to eat nothing but Graham bread for one month,\nwith warm water, milk, and sugar. I asked Greeley if he was sure it was\nthe secondary or inherited salt rheum that had come to the surface of\nhis snowy flesh in the form of boils, and he said he was quite sure it\nwas, as his father had it from his boyhood. I asked him if his secondary\nor inherited salt rheum ever itched, and he said yes, sometimes, but he\nwas sure it was not the secondary itch, as he never had the first itch.\nI then looked him dead in the eye, and asked him if he was positively\nsure his boils were not the result of itch, and he asked me what I meant\nby such severity of scrutiny. I replied, that I once had the itch, and\nread many books on the subject, and knew all about it, and that his\nboils (he had two on his pale nose) looked very much like secondary itch\nblossoms. He cast searching glances, and sat in paralytic silence, save\nwhen he scratched his boils, and the bell summoned me to my first Graham\ndinner, and Greeley hopped to the table on one leg, and sat near Mrs.\nGoss at the head of the Graham festive board. About forty skeletons were\npresent, and among them were Sylvester Graham (Bread,) himself, on a\nlecturing tour from his country seat at Northampton; John McCracken, of\nNew Haven; Ralph Waldo Emerson; Abby Kelly; Fred Douglas and lady;\nFrancis Copcutt, mahogany dealer, who used to eat raw oats, and ride 30\nmiles a day on a hard trotting horse for dyspepsia; Jeremiah O.\nLanphear, tailor, and now first deacon and missionary of the Fulton\nstreet Dutch Presbyterian Church, who had a gravel nearly as large as\nGeneral Winfield Scott's, which was the largest that ever emanated from\na human bladder; Mrs. Farnham, the accomplished lady and genuine\nphilanthropist, and wife of the noble and famous California traveler,\nwho was the rival of Fremont as a mountaineer; Mrs. Anna Stephens, the\nfertile and genial authoress; the celebrated Doctor Shew and lady; Mrs.\nStorms, of Troy, and long a writer and foreign correspondent of the _New\nYork Sun_, and now of Texas; poor MacDonald Clark, the poet; Galutia B.\nSmith; Matthew B. Brady, the daguerreotypist, who married his sweetheart\nat the Graham House, and the room being crowded, I saw the exercises\nthrough the key hole; Mrs. Travis; Albert Brisbane, a moonlight dreamer;\nMrs. Andrews, a strong Unitarian, (ninety-eight years old,) and her\ngrandson, Albert L. Smith, a nervous and catarrhal gentleman, who now\nkeeps a Graham House and Water Cure Establishment in West Washington\nPlace; Dr. John Burdell, brother of Dr. Harvey Burdell, who was\nassasinated at No. 31 Bond Street; Leroy Sunderland, a Mesmeriser and\nPathetic lecturer; John M. King; George Foss; Dr. Henry W. Brown; E.\nGould Buffum, and his brother, William Buffum, now Consul at _Trieste_;\nMrs. Horace Greeley; Mr. Clutz; Mrs. Van Vleet; Messrs. Tyler, Bennett,\n(a tailor), Otis, and Ward; Mrs. Gove; C. Edwards Lester; Mr. Danforth,\na spurious reformer; the brothers Fowler, phrenologists; father Miller,\nthe Millenium impostor; Mr. Seymour, a journeyman hatter at Beebe's, who\ngot among the noisy methodists, who frightened him into a dangerous\nnervous affection, and in bed one night, poor Seymour felt cold and\nstrange and numb, and pinched himself in the arms and legs, and it\ndidn't hurt him, and he thought he was dead, and he got up, and kindled\na match, and lit a candle, and looked in the glass to see whether he was\ndead or alive, and when he saw his eyes roll, and his jaws open and\nclose, he got into bed, and went to sleep. This was the gang at table,\nand for dinner, we had bran bread and crackers, bean soup, roast apples\nand potatoes, and boiled squash and carrots, but not a particle of meat,\ngrease, nor spices. All grabbed violently at the Graham viands, and\nbrought their teeth together like swine, and with similar grunts and\nsqueals. I calmly surveyed the motley and hungry group, and saw many\nsmall piercing gray eyes, hollow cheeks, and sharp chins and noses, and\nthe voices of nearly all were husky and fearfully sepulchral. The themes\ndiscussed were Anti-Slavery and Grahamism, and I soon perceived it\nextremely perilous to breathe a word against the ultra views of the\nsusceptible and long-haired Graham spectres, who seemed united to a\nghost on these prolific themes. So, I listened and breathed not a\nsyllable in opposition to the crazy views advanced. I took a stroll\nafter dinner, and returned at sunset, and seated myself for my evening\nmeal, when we had Graham-bread-coffee, milk porridge, apple sauce,\nGraham mush, and boiled rice, sparingly saturated with molasses and\nliquid ginger. I ate and drank freely of this light food, and arose from\nthe table in excellent spirits, though I belched frequently. My belly\nsoon began to swell, and I got alarmed, and I asked Mr. Goss, the Graham\nhost, what it meant. He seemed perfectly cool, and said that his\nboarders were often affected in that way, in passing suddenly from\ngreasy meats to the pure food of Grahamites, which was chiefly of a\nvegetable and somewhat of a gassy and flatulent character. Goss then\nleft me. I thrice paced the parlor hurriedly, and began to feel choleric\nand crampy, and went down stairs into the kitchen, and asked Goss to\nsend for a physician immediately, which he declined to do, as he thought\nI was only a little spleeny, which would soon pass away, and advised me\nto go to bed. He got me a Graham candle, and up we went, and did not\nstop until we reached the roof, where he put me in a little room, with\ntwo cots, on which there was a straw mattress, and a straw bolster, and\nscanty covering. He said good night, and shut the door, and I got into\nbed, and strove to sleep. I squirmed like an eel for about two hours,\nand could endure my pains no longer, and arose and awoke my room-mate,\nand asked him to escort me to the sleeping apartment of Mr. Goss. He did\nso, and I knocked at his door, and out he came in his nightcap and white\napparel. I told him that I had cramps, and had an awful quantity of\nfrantic wind in my stomach, and felt as though my belly would burst\nbefore morning, and that I was deathly sick, and asked him what on earth\nI had eaten at his table to give me such violent cramps and flatulence\nand diarrh\u0153a, and nauseous and strange emotions. He told me that I was\nnervous, and not accustomed to Graham food, but that I soon would be,\nand urged me to again retire, and strive to sleep. He spoke these words\nwith kindness, and they soothed me, and I shook his hand, and off I went\nup stairs to bed again. But in about ten minutes, I had a severe spasm,\nwith choking sensations, and I leaped from my nest like a man in his\nlast gasp, and unconsciously cast myself on the cot of my room-mate, who\ninstantly emerged from a profound sleep, and sprang like a tiger from\nhis bed, and threw me severely to the floor, and cried murder to the\npinnacle of his voice, and began to pelt me in the most brutal manner,\nleveling the most savage random blows at my head and stomach. Goss and\nthe spectral boarders rushed into the room, and Greeley soon came\nlimping in, and they searched in vain for knives, revolvers, and human\nblood. And they soon learned the cause of the cry of murder, and raised\nme from the floor, and put me into bed, with a bloody nose and dark eye,\nthat my room-mate gave me, who apologised for his blows on the ground\nthat he always slept soundly, and was only partially awake when he beat\nme. I accepted his apology, and Goss and Greeley, and half-a-dozen\nattenuated Grahamites left me, for their beds again, and my chum took a\nseat by my cot, and strove to soothe me. But the cramps returned, and I\nbecame faint and giddy, and began to vomit profusely. I soon filled\nbasins, pitchers, spit boxes, hats, and boots, and deluged every thing\nwe had in the room, and my chum got a pitcher and basin in the next\nroom, and I soon flooded them, and I vomited until I thought I felt my\nentire bowels struggling at my throat to get out, which nearly strangled\nme. At last an enormous chunk came out, which proved to be the core of a\nstewed apple, and the crust of Graham bread combined into a sort of\npetrified substance, and I began to breathe again, and slowly improved\ntill daylight,\n\n                When I embraced a sweet repose,\n                And snored like thunder through my nose.\n\n                  (To be continued to my last scream.)\n\n------------------------------------------------------------------------\n\n\n\n\n                     Stephen H. Branch's Alligator.\n\n         ------------------------------------------------------\n\n                   NEW YORK, SATURDAY, JUNE 26, 1858.\n\n         ------------------------------------------------------\n\nSTEPHEN H. BRANCH'S \"ALLIGATOR\" CAN BE obtained at all hours, (day or\nnight,) at wholesale and retail, at No. 128 Nassau Street, Near Beekman\nStreet, and opposite Ross & Tousey's News Depot, New York.\n\n                  ------------------------------------\n\n\n\n\nSupervisor Blunt, and Paul Julien\u2014My Last Interview with Madame Sontag.\n\n\nWhen I taught Alderman Orison Blunt the English branches at his elegant\nresidence in Murray street, I gave instruction to Paul Julien, the\njuvenile Paganini, and to Rocco, and also to Madame Sontag in elocution,\nin anticipation of her appearance in English Opera at Niblo's, on her\nreturn from Mexico. At the close of a long and interesting lesson,\nSontag opened her great heart to me, and disclosed her career from her\nearliest recollection. Her narrative was eloquent and exciting, and as\nshe sat before me at the parlor lattice, in alternate tears and smiles,\nwith the moon rolling like a ball of silver through the air, she seemed\ntoo pure and beautiful for earth. Her tears were the very soul of\nsorrow, and none could resist their overwhelming influence,\u2014her smiles\nwere irresistibly enchanting,\u2014her voice in conversation was full of\nentrancing melody,\u2014her cavern dimples were the emblems of purity and\ncharity, and her entire expression was divine. And as her blood warmed,\nand her bosom rose and fell, and her voice trembled and darted from the\nfaintest whisper to its highest intonation, her glorious eyes reflected\ngorgeous temples in her soul, filled with sinless angels, breathing\nsweet music to millions of her species. And the beauteous Sontag told\nme, as we sat together in our last communion as human pilgrims, that her\nchildhood, and girlhood, and early womanhood were all devoted to the\ncultivation of music for the enjoyment of the world more than herself,\nwhich rendered her early years an utter sacrifice, and had deprived her\nof the pastimes enjoyed by all her sex in the morning of life; that from\nthe hour she was called \"_The little Daughter of the Danube_,\" there was\nno happiness for her; that she was early beset by lovers from nearly\nevery nation of Europe; that kings and queens lavished their choicest\ntreasures upon her; that princes besought her affections in tearful\nsupplications; that all France prostrated herself at her feet; that amid\nthe flattery and adulations of all classes and kingdoms, she was\ninduced, in a thoughtless hour, to cast herself into the eternal\nembraces of a being who proved a jealous and savage tyrant, and a\nheartless gamester; that ere her emergence from the brief hours of bliss\nthat should follow the marriage vow, he became odious in her eyes, and\nshe beheld a life of misery in all her future; that after years of\ntorture in his demon fangs, and after he had squandered her splendid\nfortune of four millions of dollars, he dragged her from the sacred\nprecincts of private life, and from the pleasing society of her\nchildren, into the public arena, to toil for his subsistence; that he\nforced her to exchange hemispheres, and leave her tender offspring, when\nthey most required a mother's protection; that he often brandished a\ndagger in her eyes, when she refused to fill his purse for bibbling and\ngaming purposes; that she was in fear of his poignard throughout her\nlong confinement in his hideous clutches; that for his traduction and\npersecution of Alboni in her early years, she resolved to pursue her to\nAmerica to annoy, and, if possible, ruin her, for his sake, by singing\nagainst her in the leading cities; that on the very day she publicly\nannounced her intention to visit America, Alboni went to the Cathedral,\nand knelt at the altar, and swore that she would pursue her through all\nlatitudes, and cut the grass beneath her feet, to avenge herself on\nCount Rossi, who strove to blight the buds and blossoms of her youth and\nindigence; that she kept her oath, and followed her through city, town,\nand village, and allured her choristers, through extravagant salaries\nand donations, and sang on the evenings of her Concert and Opera\nentertainments, and greatly reduced her receipts; that Rossi seized her\nfunds, as they accrued, and deposited them in banks unknown to her; that\nher children often wrote in vain for means to defray their domestic\nexpenses; that Rossi, and Maretzek, and Ullman received all the benefit\nof her arduous labors; that her lovely daughters were in the care of\nstrangers in Europe, and exposed to all the snares of life; that their\neducation was fatally neglected in her absence; that she was a slave to\nRossi, Maretzek, and Ullman, all of whom she thoroughly despised, and\nthat she had very seriously contemplated suicide. And thus did this\ncelestial being breathe her pensive music in my soul, and bathe my\nvision with nature's hallowed waters. And amid our mutual tears, and\nsmiles, and cheerful tones, and lingering glances, she enters the dismal\ncars, and the bell proclaims the parting signal, and she penetrates the\ndeep perspective, until she is forever buried from my melancholy view.\nShe gives concerts on the borders of the northern lakes, and visits\nCincinnati, and quarrels and separates from Ullman, and goes to New\nOrleans, and performs in Opera, and enters Mexico, amid the revengeful\nmaledictions of Ullman, who, as Rocco told me, dug her early grave, by\narousing the fearful jealousy of Rossi, to whom Ullman wrote from New\nYork, that he would find letters in her trunk from Pozzolini, the young\nand fascinating tenor; that Rossi did find letters in her trunk from\nPozzolini, (filled with the most enthusiastic love,) which Rocco said\nwere doubtless placed there by Ullman, prior to her departure for\nMexico, to revenge himself on Sontag, for her refusal at Cincinnati to\ngive more Concerts under his direction; that Rossi belched words of\nfire, and threatened her with instant death; that herself and Pozzolini\nwere seized with violent pains, on their return from the Mexican\nfestivals; that during her confinement, Rocco daily called, but was not\npermitted to see her; that Rossi paced the balcony as a sentinel for\ndays and nights, and would let no one visit her; that he permitted Rocco\nto enter her apartment only one hour before she died, when he found her\nin the wildest delirium. And Rocco told me that Sontag and Pozzolini\nwere doubtless poisoned by Count Rossi, and that Ullman was the\ninstigator. Rossi artfully attributed their sudden death to cholera, but\nthe rumor flew on the wings of lightning, that Rossi was their murderer,\nand he fled for his life to New York, with all her jewels, and went to\nEurope. And Rocco sorely grieved to see her borne to her sepulchre\nwithout kindred mourners in a far distant land; and when he saw her form\nexhumed, and borne through mud and stones, and deposited as luggage in\nthe filthy suburbs of Vera Cruz, and exposed for weeks to the heat and\nrain of those withering latitudes,\u2014when he gazed at the remains of a\nbeing who had been the pride and glory and adoration of all civilised\nnations, and who had long been his own dear friend, poor Rocco\nprostrated himself beside her coffin, and wept for hours in loneliness\nand utter desolation. And now, dear Sontag, I can see thy pure and\ngenial spirit in its happy home, beyond the pretty stars. And while I\nindite these melancholy words, thy sweet face smiles upon me from my\nparlor wall, as you appeared in the immortal _Somnambulist_. It is the\nlikeness you gave me at our final interview, and represents _Amina_, in\nthe joyous bridal scene with _Elvino_, among her native cottagers in the\nmountains. All! Sontag! I often think of thee, and my highest solace is\nin gazing at thy bewitching smile, and laughing eyes, and lovely\ndimples, and even teeth, and classic temples, as depicted in thy\nlikeness, which I shall keep while I linger in the dreary paths of\nearth. And I will part with fame and fortune and with life itself, ere I\nwill separate from the precious picture of my adored Sontag. And my last\nprayer to God shall be, that I may join my Parents and Kindred and\nSontag in the realms of eternal bliss.\n\n                  ------------------------------------\n\n\n\n\n                James Gordon Bennett's Editorial Career.\n\n\n                       BENNETT'S OFFICE IN 1835.\n\n                          _Enter John Kelly._\n\n_Bennett_\u2014Well, my lad, I have borrowed a pair of old shoes for you from\nmy bed-fellow in Cross street. They may be rather large, but you must\ncontrive to wear them until Saturday, when I will get you a new pair, if\nI have the money to spare. Sit down, Johnny, and try on the shoes.\n\n_John_ (puts them on)\u2014They are much too large, aint they?\n\n_Bennett_\u2014Well, yes, but if you put some pieces of newspaper in them,\nyou can lessen their size.\n\n_John_ (stuffs them in the heels and toes and sides with fragments of\nthe _Herald_ of the preceding day)\u2014There, sir, I guess I can wear them\nnow, and I am truly obliged to you for borrowing them for me.\n\n_Bennett_\u2014Not at all, John, for you did more than that for me yesterday,\nin obtaining my papers from Mr. Anderson.\n\n_John_ (in hurriedly walking across the office, steps out of one of the\naged shoes, but steps in again before Bennett's keen eyes perceived that\none foot had stepped out)\u2014That was a great pleasure, sir, and I hope you\nwill have the same good luck to-day.\n\n_Bennett_\u2014I sold very few papers yesterday, and I have very little\nmoney, and Anderson has my watch, and I fear he will not let me have the\npapers until I redeem it, and pay him for the _Heralds_ of to-day.\n\n_John_\u2014I will do all in my power to obtain them for you.\n\n_Bennett_\u2014I know you will, my dear little friend. But come\u2014we will go\nand try to get the papers. (They arrive at Anderson & Ward's, in Ann\nstreet. Anderson is absent, and Ward is partially drunk and asleep on\nthe counter, and Bennett arouses him.)\n\n_Ward_\u2014What are you about? (rubbing his eyes and garrping.) What do you\nwant (hic) so early in the morning, you vagabonds? hic, hec, hoc.\n\n_Bennett_\u2014I want my papers.\n\n_Ward_\u2014You can't (hic) have them without the money, (hoc.)\n\n_Bennett_\u2014Please let me have them.\n\n_Ward_\u2014Where's your (hic) watch?\n\n_Bennett_\u2014I let Mr. Anderson have it yesterday.\n\n_Ward_\u2014Don't you (hic-a-che-a-che-Horatio-darn it, how I sneeze) sell\nany _Heralds_ now-a-days? a-che-a-che-a-che-Horatio\u2014O, Jerusalem! will I\nnever stop sneezing?\n\n_Bennett_\u2014It stormed yesterday, and I did not sell many, but it is\npleasant this morning, and I think I shall sell a large number.\n\n_Ward_\u2014Well, I'll not be (hic, hic, hic,) too hard with you, old fellow.\nThere, take your papers, and try hard (hic) to sell (hic) them to-day,\nand (hic-a-che) bring a whole lot of money to (hic) morrow.\n\n_Bennett_\u2014I will, Mr. Ward, and I'll always remember you with gratitude\nfor your generosity to-day. Good day, sir.\n\n_Ward_\u2014Farewell, old boy. And just shut the door alter you. I have been\n(hic) on a spree all night, (hec,) and I don't want anybody else to come\nin and bother (hic) me, until I finish my nap.\n\n_Bennett_\u2014I'll lock the door outside, and put the key in the window.\n\n_Ward_\u2014Do so, old (hic) boy, do so. (And he goes to sleep, and Bennett\nand John wend their way to Wall street.)\n\n_Bennett_\u2014Now, John, this is the last chance I shall have. If I fail to\nsell my papers to-day, I am ruined for ever.\n\n_John_\u2014Had I not better go into the stores, and try to sell the papers.\n\n_Bennett_ (kisses him in Nassau street)\u2014My dear boy, if you will do\nthat, I will love you next to my God. My great trouble has been to get\nhonest boys to sell my paper, and return the money to me, instead of\ngoing to the Theatre and eating peanuts with my funds. Now, you take\nsome, and I'll take some, and you take one side of the street and I the\nother, and let us toil for our lives (until the sun goes down) to sell\nthese papers, and, if we fail, my fate is sealed for time, and perhaps\nfor eternity!\n\n_John_\u2014What! You won't commit suicide?\n\n_Bennett_\u2014God only knows what I shall do.\n\n_John_\u2014Well, I see there's no time to be lost. So, give me some papers,\nand I'll go into the first store on this side, and you take the other\nside of the street. (They separate, John going into every store on his\nside, and Bennett into every store on the other side, until they arrive\nat Wall street, when they go into Bennett's office, in the old rat hole\nat No. 20 Wall street, where they count their pennies, and find that\nthey have sold quite a large number of _Heralds_. They then drink some\nwater and eat some ginger nuts, for their breakfasts, and go down Broad\nstreet, and enter every store on either side, and meet with great\nsuccess. John then takes South street, and Bennett Front street, from\nthe Battery to Fulton street, and afterwards take Water and Pearl\nstreets, and then they canvass either side of Wall street, and sell all\ntheir _Heralds_, and go to a Restaurant and get something to eat, and\nseparate in the afternoon in high spirits. John then got some boys in\nthe Fourteenth Ward to sell the _Herald_, and in ten days Bennett had\nabout $40 surplus, and begins to put on aristocratic airs, and domineer\nover Johnny Kelly.)\n\n                           (To be continued.)\n\n                  ------------------------------------\n\n\n\n\n                          _For the Alligator._\n\n\n                Wide-mouth shocking Alligator!\n                I wish you were a Boa Constrictor!\n                And crush within your awful fold,\n                The villains with our pilfered gold,\n                Who, with sanctimonious face,\n                Steal with such a pious grace:\n                They dance and dress and call it good,\n                Because it gives the hungry food.\n                But hold your mirror to their face,\n                And show them their sad black disgrace:\n                One robs the City's golden coffers,\n                And then a mighty Fabric offers,\n                And tries to court a worldly fame,\n                Out of such an impious shame.\n                The temple thus to science rears,\n                That he may surely soothe his fears,\n                Lest his ignorance should be known,\n                And lack of knowledge shown,\n                And so the starving, suffering poor,\n                He drives them fainting from his door;\n                And tells them: (Oh! how very strange!)\n                The Mansion's taken all his change!\n                And in his high, majestic wrath,\n                He kicks a female down to earth!\n                The mansion he will never give,\n                While one heir of his shall live.\n                See how this modern Simon Magus,\n                Blinds our eyes, and then deceives us.\n                Soon we shall see how very funny,\n                He'll make his \"Union\" yield him money:\n                He finds it is so very pretty,\n                To have a Mayor made of putty,\n                That he can mould him at his will,\n                To make his son an office fill.\n                But lest Columbia prove too new,\n                He lays a wire the ocean through,\n                That he all Europe may invite,\n                To bask in his resplendent sight.\n                Oh! most happy England Queen,\n                When she can say: \"I've Peter seen!\"\n                Now see him cringe, and jump for fame,\n                To reach the scroll, to write his name:\n                But as he lives alone for fame,\n                My verse will sure preserve his name.\n\n                                                       PETER PIPER PICT.\n\n\n\n\n                                                NEW YORK, June 15, 1858.\n\n\nSTEPHEN H. BRANCH:\n\nSIR:\u2014Permit me the privilege of making a few brief passing remarks,\nasking a few questions, and respectfully suggesting a few hints as to\nyour weekly publication, the ALLIGATOR. Please to attribute any\nintrusive errors in this communication as emanating from an inefficient\nmethod of expressing my sentiments, as my heart is with you whole and\nentire in spirit, and, with a few exceptions, to the very letter, in\nyour laudable endeavor to bring to light before the open day the hidden\nvillainies of the many detestable tyrants that have risen from the very\nscum of poverty and criminal degradation, and who now so unaccountably\nhold despotic sway _under the garb of honorable industry_ in every\nbranch of society, to the unjust injury and oppression of the poor,\nhumble, but honest man.\n\nI am rejoiced to find the ALLIGATOR creeping its way to the literary\ntables of almost every respectable News Depot in this and the adjacent\ncities, piercing its deadly fangs into the very vitals of every\ninfluential thief and scoundrel, and that the business public are now\navailing themselves of the opportunity in patronising it as an\nadvertising medium, and I sincerely wish you every success.\n\nWherever I have an opportunity, I endeavor, indirectly, to pave the way,\nto introduce the merits of the ALLIGATOR, and, as a matter of course,\nhave to give and take in the various opinions expressed as to the\ncarniverous propensities of that astonishing animal, and the choice\nobjects it pitches into for its daily food. The opinions and ideas\nexpressed on the subject are as varied as the colors in the rainbow. Any\nman whose past misdeeds trouble his conscience, dreads the animal, as he\nwould a drawn sword, lest its brutal tusks should tear open to public\ngaze what he had secretly hoped was unknown to mortal being.\n\nIf the crawling reptiles you select to satisfy the craving appetite of\nthat amphibious animal (with such extended jaws continually gaping) are\nreally of such an abhorrent and loathsome nature as represented by you\nin such bold relief, I should never cease lashing their diseased and\nulcerated carcases with whips of poisoned scorpions, till I purged and\npurified their polluted system with wholesome antidotes. It strikes me\nthat your gormandising hydra-headed monster can never be satisfied with\ncommon carrion: it seeks for something more nutritious for its\nsustenance. It appears he is like Pharoah's lean kine\u2014the more he\ndevours, the thinner he gets, and his rapacity increases, and what seems\nso singular is, that he has abundance of choice prey for ever at his\nside, which he selects indiscriminately, and an untold amount laid up in\nhis store houses for ages to come.\n\nNothing do I admire more than the free use of strong and emphatic\nlanguage to express our approbation or disapprobation of men's actions\npublic or private, and from the general tenor of your style, and the\npeculiar advantages you possess as a scholar, and the unlimited\ninformation you have treasured up as a man of experience, with regard to\npublic characters and measures, I feel confident that you can convert\nevery tooth of the Alligator into a poisoned arrow that will deal death\nand destruction into every particle of air whereever it wings its\nflight, and you can more effectively hit your mark with surer certainty\nby avoiding the use of such terms and phrases as would be looked upon by\nthe general class of readers, as rather coarse or vulgar; although I\nmyself consider your style as purely hieroglyphic, and that your\nsarcastic way merely emanates from a proud, manly, straightforward, bold\nand independent above board kind of a spirit than that of malice, with\nthe view to convey the sentiments of your mind, in order to express your\nstrong feeling of detestation and abhorrence of every unprincipled\nscoundrel, against whom your fiery shafts of indignation may happen to\nbe turned, cutting to the very heart's core like a two edged sword.\n\nThe body of the ALLIGATOR is too small by a long shot. It would greatly\nenhance its usefulness by being more liberal. Increase its pages, extend\nits columns, devote a space to correspondents, and, if need be, stretch\nits stomach so as to afford an opportunity to others to open their\nstore-houses, and contribute their quota of similar wholesome food to\nthe hungry cannibal, in order the better to assist in the process of\ndigestion.\n\n                                                  Yours Respectfuly,\n\n                                                          ANTI-TYRANT.\n\n------------------------------------------------------------------------\n\n\n\n\n                    Advertisements\u201425 Cents a line.\n\n\nCredit\u2014From two to four seconds, or as long as the Advertiser can hold\nhis breath! Letters and Advertisements to be left at No. 128 Nassau\nstreet, third floor, back room.\n\n    ----------------------------------------------------------------\n\n\nHERRING'S PATENT CHAMPION FIRE AND BURGLAR Proof Safe, with Hall's\nPatent Powder Proof Locks, afford the greatest security of any Safe in\nthe world. Also, Sideboard and Parlor Safes, of elegant workmanship and\nfinish, for plate, &c. S. C. HERRING & CO.,\n\n                                                         251 Broadway.\n\n    ----------------------------------------------------------------\n\n\nJAMES MELENEY, (SUCCESSOR TO SAMUEL Hopper,) Grocer, and Wholesale and\nRetail Dealer in Pure Country Milk. Teas, Coffee, Sugars & Spices.\nFlour, Butter, Lard, Cheese, Eggs &c. No. 158, Eighth Avenue, Near 18th\nStreet, New York. Families supplied by leaving their address at the\nStore.\n\n    ----------------------------------------------------------------\n\n\nBOOT & SHOE EMPORIUMS. EDWIN A. BROOKS, Importer and Manufacturer of\nBoots, Shoes & Gaiters, Wholesale and Retail, No. 575 Broadway, and 150\nFulton Street, New York.\n\n    ----------------------------------------------------------------\n\n\nDR. SMITH'S ELECTRIC OIL, CURES PAIN IN A few moments. Dr. Smith's\nElectric Oil gives almost instant relief in all nervous diseases. Acute\nrheumatic pains need only a few applications. Dr. Smith may be consulted\nat the Smithsonian House, and at 91 Hudson Street. Try it.\n\n    ----------------------------------------------------------------\n\n\nMCSPEDON AND BAKER'S STATIONERY WAREHOUSE and Envelope Manufactory, Nos.\n29, 31, and 33, Beekman Street, New York.\n\nENVELOPES of all patterns, styles and quality, on hand, and made to\norder for the trade and others, by Steam Machinery. Patented April 8th,\n1856.\n\n    ----------------------------------------------------------------\n\n\nCOZZENS' HOTEL COACHES,\u2014STABLE, Nos. 34 and 36 Canal Street, New York.\n\nI will strive hard to please all those generous citizens who will kindly\nfavor me with their patronage.\n\n                                                     EDWARD VAN RANST.\n\n    ----------------------------------------------------------------\n\n\nJ. W. MASON, MANUFACTURER, WHOLESALE and Retail dealers in all kinds of\nChairs, Wash Stands, Settees, &c. 377 & 379 Pearl Street, New York.\n\nCane and Wood Seat Chairs, in Boxes, for Shipping.\n\n    ----------------------------------------------------------------\n\n\nBENJAMIN JONES, COMMISSION DEALER, IN Real Estate. Houses and stores and\nlots for sale in all parts of the city. Office at the junction of\nBroadway, Seventh Avenue, and Forty-Sixth Street.\n\n    ----------------------------------------------------------------\n\n\nFULLMER AND WOOD CARRIAGE Manufacturers, 239 West 19th Street, New York.\n\nHorse-shoeing done with despatch and in the most scientific manner, and\non reasonable terms.\n\n    ----------------------------------------------------------------\n\n\nW. E. KNAPP'S NEWS DEPOT, 279 BLEEKER ST., near Barrow street.\nSubscriptions for Dailies, Weeklies, and Monthlies, which will be served\nas soon as issued.\n\n    ----------------------------------------------------------------\n\n\nCHEAP PERIODICAL AND PAMPHLET BINDERY, No. 50 Ann street, N. Y. F. S.\nPittman, successor to H. H. Randall. Mr. Gouverneur Carr and N. S.\nPutnam have purchased an interest in the concern.\n\n    ----------------------------------------------------------------\n\n\nAUG. BRENTANO, SMITHSONIAN NEWS DEPOT, Books and Stationery, 608\nBROADWAY, corner of Houston street.\n\nSubscriptions for American or Foreign Papers or Books, from the City or\nCountry, will be promptly attended to.\n\nForeign Papers received by every steamer. Store open from 6 A. M. to 11\nP. M. throughout the week.\n\n    ----------------------------------------------------------------\n\n\nP. C. GODFREY, STATIONER, BOOKSELLER, AND General News dealer, 831\nBroadway, New York, near 13th street.\n\n       At Godfrey's\u2014Novels, Books, &c., all the new ones cheap.\n       At Godfrey's\u2014Magazines, Fancy Articles, &c., cheap.\n       At Godfrey's\u2014Stationery of all kinds cheap.\n       At Godfrey's\u2014All the Daily and Weekly Papers.\n       At Godfrey's\u2014Visiting Cards Printed at 75 cents per pack.\n       At Godfrey's\u2014Ladies Fashion Books of latest date.\n\n    ----------------------------------------------------------------\n\n\nC. TYSON, CORNER OF NINTH STREET & SIXTH AVE. Has for sale all the late\nPublications of the day, including all the Daily and Weekly Newspapers.\n\n    ----------------------------------------------------------------\n\n\nSEE \"JOBSON'S RED FLAG,\" OF THIS DAY, FOR interesting news. Published at\nNo. 102 Nassau Street.\n\n    ----------------------------------------------------------------\n\n\nJOHN B. WEBB, BOAT BUILDER, 718 WATER STREET. My Boats are of models and\nmaterials unsurpassed by those of any Boat Builder in the World. Give me\na call, and if I don't please you, I will disdain to charge you for what\ndoes not entirely satisfy you.\n\n                                                         JOHN B. WEBB.\n\n    ----------------------------------------------------------------\n\n\nSAMUEL SNEDEN, SHIP & STEAMBOAT BUILDER.\u2014My Office is at No. 31 Corlears\nstreet, New York; and my yards and residence are at Greenpoint. I have\nbuilt Ships and Steamers for every portion of the Globe, for a long term\nof years, and continue to do so on reasonable terms.\n\n                                                        SAMUEL SNEDEN.\n\n    ----------------------------------------------------------------\n\n\nALANSON T. BRIGGS\u2014DEALER IN FLOUR BARRELS, Molasses Casks, Water, and\nall other kinds of Casks. Also, new flour barrels and half-barrels; a\nlarge supply constantly on hand. My Stores are at Nos. 62, 63, 64, 69,\n73, 75, 77 and 79 Rutger's Slip; at 235, 237, and 239 Cherry street;\nalso, in South and Water streets, between Pike and Rutger's Slip,\nextending from street to street. My yards in Williamsburgh are at Furman\n& Co.'s Dock. My yards in New York are at the corner of Water and\nGouverneur streets; and in Washington street, near Canal; and at Leroy\nPlace. My general Office is at 64 Rutger's Slip.\n\n                                                    ALANSON T. BRIGGS.\n\n    ----------------------------------------------------------------\n\n\nFULTON IRON WORKS.\u2014JAMES MURPHY & CO., manufacturers of Marine and Land\nEngines, Boilers, &c. Iron and Brass Castings. Foot of Cherry street,\nEast River.\n\n    ----------------------------------------------------------------\n\n\nBRADDICK & HOGAN, SAILMAKERS, No. 272 South Street, New York.\n\nAwnings, Tents, and Bags made to order.\n\n                                                    JESSE A. BRADDICK,\n                                                        RICHARD HOGAN.\n\n    ----------------------------------------------------------------\n\n\nWILLIAM M. SOMERVILLE, WHOLESALE AND Retail Druggist and Apothecary, 205\nBleeker-st., corner Minetta, opposite Cottage Place, New York. All the\npopular Patent Medicines, fresh Swedish Leeches, Cupping, &c.\nPhysicians' Prescriptions accurately prepared.\n\n                                                    WM. M. SOMERVILLE.\n\n    ----------------------------------------------------------------\n\n\nA. W. & T. HUME, MERCHANT TAILORS. No. 82 Sixth Avenue, New York. We\nkeep a large and elegant assortment of every article that a gentleman\nrequires. We make Coats, Vests and Pants, after the latest Parisian\nfashions, and on reasonable terms.\n\n                                                      A. W. & T. HUME.\n\n    ----------------------------------------------------------------\n\n\nTHE WASHINGTON, BY BARTLETT & GATES, No. 1 Broadway, New York. Come and\nsee us, good friends, and eat and drink and be merry, in the same\ncapacious and patriotic halls where the immortal Washington's voice and\nlaugh once reverberated.\n\n                      O come to our Hotel,\n                      And you'll be treated well.\n\n                                                     BARTLETT & GATES.\n\n    ----------------------------------------------------------------\n\n\nJ. N. GENIN, FASHIONABLE HATTER, 214 Broadway, New York.\n\n    ----------------------------------------------------------------\n\n\nGENIN'S LADIES' & CHILDREN'S OUTFITTING Bazaar, 513 Broadway, (St.\nNicholas Hotel, N. Y.)\n\n    ----------------------------------------------------------------\n\n\nEDWARD PHALON & SON, 497 and 517 Broadway, New York\u2014Depots for the sale\nof Perfumery, and every article connected with the Toilet.\n\nWe now introduce the \"BOUQUET D'OGARITA, or Wild Flower of Mexico,\"\nwhich is superior to anything of the kind in the civilized world.\n\n                                                  EDWARD PHALON & SON.\n\n    ----------------------------------------------------------------\n\n\nEXCELSIOR PRINTING HOUSE, 211 CENTRE ST., IS furnished with every\nfacility, latest improved presses, and the newest styles of type\u2014for the\nexecution of Book, Job and Ornamental Printing. Call and see specimens.\n\n    ----------------------------------------------------------------\n\n\nCHARLES FRANCIS, SADDLER, (ESTABLISHED IN 1808,) Sign of the Golden\nHorse, 39 Bowery, New York, opposite the Theatre. Mr. F. will sell his\narticles as low as any other Saddler in America, and warrant them to be\nequal to any in the World.\n\n    ----------------------------------------------------------------\n\n\nH. N. WILD, STEAM CANDY MANUFACTURER, No. 451 Broadway, bet. Grand and\nHoward streets, New York. My Iceland Moss and Flaxseed Candy will cure\nCoughs and Sneezes in a very short time.\n\n    ----------------------------------------------------------------\n\n\nJAMES GRIFFITHS, (Late CHATFIELD & GRIFFITHS,) No. 273 Grand st., New\nYork. A large stock of well-selected Cloths, Cassimeres, Vestings, &c.,\non hand. Gent's, Youths' and Children's Clothing, Cut and Made in the\nmost approved style. All cheap for Cash.\n\n    ----------------------------------------------------------------\n\n\nJ. AGATE & CO., MEN'S FURNISHING GOODS and Shirt Manufacturers, 256\nBroadway, New York.\n\nShirts made to order and guaranteed to fit.\n\nJ. AGATE.    F. W. TALKINGTON.\n\n    ----------------------------------------------------------------\n\n\nBILLIARD TABLES.\u2014PHELAN'S IMPROVED BILLIARD Tables and Combination\nCushions\u2014Protected by letters patent, dated Feb. 19, 1856; Oct. 28,\n1856; Dec. 8, 1857; Jan. 12, 1858. The recent improvements in these\nTables make them unsurpassed in the world. They are now offered to the\nscientific Billiard players as combining speed with truth, never before\nobtained in any Billiard Table. Sales-rooms Nos. 786 and 788 Broadway,\nNew York. Manufactory No. 53 Ann Street.\n\n                             O'CONNOR & COLLENDOR, Sole Manufacturers.\n\n    ----------------------------------------------------------------\n\n\nS. L. OLMSTEAD, IMPORTER, MANUFACTURER and Jobber of Men's Furnishing\nGoods, No. 24 Barclay Street, corner of Church, New York.\n\n    ----------------------------------------------------------------\n\n\nC. B. HATCH, HILLER & MERSEREAU, Importers and Jobbers of Men's\nFurnishing Goods and Manufacturers of the Golden Hill Shirts. 99\nChambers Street, N. E. corner Church Street, New York.\n\n    ----------------------------------------------------------------\n\n\nL. A. ROSENMILLER, DRUGGIST, NO. 172 EIGHTH Avenue, New York. Cupping &\nLeeching. Medicines at all hours.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Stephen H. Branch's Alligator Vol. 1\nno. 10, June 26, 1858, by Stephen H. Branch\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n# **Grow a Hummingbird Garden**\n\n**Dale Evva Gelfand**\n\n## **CONTENTS**\n\n**About Hummingbirds**\n\n**Food**\n\n**Water**\n\n**Perches**\n\n**Shade and Shelter**\n\n**Nesting Sites**\n\n**A Hummingbird Gallery**\n\n**Identifying Hummingbirds**\n\n**Designing a Hummingbird Garden**\n\n**Choosing Plants**\n\n**Laying out Your Garden**\n\n**Cultivating Wildflowers**\n\n**Flowers for Hummingbirds**\n\n**Vines, shrubs, and trees**\n\n**Vines**\n\n**Shrubs**\n\n**Trees**\n\n## **About Hummingbirds**\n\nIf asked to name the most ethereal and captivating creature to be found on our planet, most of us would be hard pressed to top the hummingbird. Their minute size (some species weigh only a fraction of an ounce), their beauty (iridescent plumage courtesy of special structures in their feathers), their incredible aerobatic ability (hummingbirds can fly in any direction, including backward), their extraordinary appetites (hummingbirds eat half of their weight in sugar every day), even their aggression (hummingbirds are fierce protectors of their territory and nests) add up to a singularly fascinating family. A little planning of the design of and the plants in your garden is all it takes to make your yard the kind of environment that will readily attract these flying jewels.\n\nA perfect hummingbird habitat is also a relatively simple one: food sources, water for bathing (liquid intake is generally provided by nectar), lookout perches, shade and shelter, and nesting sites and materials. You should be able to provide these requirements in even the smallest of gardens.\n\n### **_Food_**\n\nHummingbirds are extremely active creatures with an astonishing metabolic rate that requires high caloric intake of both nectar and insects \u2014 which means hundreds of food forays during their waking hours, primarily to flowers. Obviously those flowers that produce lots of nectar and also attract tiny insects are more beneficial, and therefore more appealing, to hummingbirds.\n\nThis attraction forms part of one of Nature's many interdependent relationships: Hummingbirds, while stopping for a meal, act as inadvertent pollinators for the plants they rely upon. Depending on their particular shape, the deep-tubed blossoms that are most suitable to a hummingbird's long tongue deliver pollen from the anthers \u2014 the male organs \u2014 onto hummingbirds' heads or bills or chins, and when the nectar-sipping visitors go on to probe neighboring blossoms of the same variety, that pollen is delivered to the stigmas \u2014 the female organs \u2014 for fertilization. In fact, hummingbirds are some of the most essential bird pollinators for flora in North America.\n\nEqually as important as the nectar that the flowers provide are the tiny insects living within them. Insects are an indispensable source of protein in the hummingbird diet and, like the hummingbirds that prey upon them, insects are attracted to the nectar. When hummingbirds aren't catching them on the wing (or, in the case of small spiders, snaring them in their own webs), they find them inside flowers. Happily for gardeners and other outdoors lovers, these insects include many garden pests such as aphids and gnats.\n\n### **_Water_**\n\nLike all birds, hummingbirds need to bathe, and they are resourceful in employing water sources that match their tiny size \u2014 such as beads of water left on leaves after a rain or the fine spray from a waterfall. Hummingbirds prefer moving water, and are fond of flying through the spray of lawn sprinklers. If you want to set up a sprinkler for a hummingbird bath, use a nozzle that gives off a continuous fine spray. In a pinch, hummingbirds will also use birdbaths, provided they're shallow enough for hummingbirds to stand in. Adding various-size rocks to a birdbath creates different depths for different-size bird species. Hummingbirds prefer not to come to ground for their water, so if you can help them avoid this, they may well use your accommodations.\n\n### **_Perches_**\n\nHaving a place from where they can survey their territory is essential to hummingbirds. (For creatures so tiny to be so protective of their domain may seem peculiar, but considering how important ample food sources are to hummingbirds' survival, the ability to oversee their territory and quickly defend it from intruders is crucial.) Hummingbirds prefer a perch that directly overlooks the flowers on which they feed. The male partner of one ruby-throated hummingbird couple that takes up residence in my garden every summer prefers the slender bottom branch of an enormous hemlock tree that anchors the shady end of my garden. From there it's but a short flight to all of the flowers and to the nearer of the two hummingbird feeders, as well. It's likely that his mate also perches on the hemlock, but I haven't seen her there, as females generally prefer to remain sheltered within the interior of a tree, shrub, or vine.\n\n### **_Shade and Shelter_**\n\nAlthough we think of hummingbirds as being constantly on the wing, in fact they spend about equal amounts of time locating food and resting between meals. Males will rest anywhere from exposed branches to clotheslines to TV antennae, but females and immature birds seek the shade and protection of foliage. Dense foliage for shelter from the elements \u2014 evergreens are excellent windbreaks \u2014 and for roosting at night is also important to these tiny birds. The ideal balance for a hummingbird habitat is about one-half full sun and one-quarter each shade and partial shade.\n\n### **_Nesting Sites_**\n\nIt's doubtful that you'll ever see a hummingbird nest \u2014 or if you do see one, you may not realize what it is. These tiny, usually lichen-covered structures are only 1 to 1\u00bd inches (2\u00bd\u20134cm) in diameter, and are so effectively camouflaged that you would assume they're but a knot on a branch. Hummingbirds nest in both deciduous and evergreen trees, anywhere from 4 to 50 feet (1.2\u201315m) up but most frequently in the 10- to 20-foot-high (3\u20136m) range, often bordering or over a stream. Although there are no guarantees in tempting a hummingbird pair to set up housekeeping in your yard, by planting both food sources and those trees and shrubs that provide nesting materials, you'll make it more appealing for them to do so. Primary among the latter is downy plant material, which makes up the bulk of hummingbird nests. An excellent \u2014 and favored \u2014 source of downy fibers are the filaments of willow seeds; additionally, willow flowers provide both nectar and the insects attracted to it. Consider planting shrub willow in your garden if your yard has a favorable site for this moisture-loving tree. Other trees that provide nesting material for hummingbirds include cottonwoods, aspens, and sycamores.\n\n_The exterior of the nest is often covered with lichens._\n\n_The interior of the next is packed with downy plant material._\n\n## **A Hummingbird Gallery**\n\nHummingbirds are strictly a New World phenomenon, of Central American origin. North America is home to eight common species of hummingbirds and several more rare species. Hummingbirds follow a migratory pattern and are seen in the United States mainly during the summer months, although several species are known to take up permanent residence in coastal California, coastal Oregon, southern Arizona, and southern Florida. In some areas of the country, the ranges of certain species may overlap, although if you live anywhere east of the Mississippi, identification is usually easy: The ruby-throated hummingbird is the only regular visitor to that area.\n\n**A Q UICK GEOGRAPHICAL STUDY**\n\n**Eastern and Midwestern Hummingbirds**\n\n* * *\n\nRuby-throated\n\n**Southeastern and Gulf Coast Hummingbirds**\n\n* * *\n\n**Rocky Mountain Region Hummingbirds**\n\n* * *\n\n**Southwestern Hummingbirds**\n\n* * *\n\n**West Coast Hummingbirds**\n\n* * *\n\n## **Identifying Hummingbirds**\n\nListed below are brief descriptions of the hummingbirds that can typically be seen in the United States (those listed under \"Common\" on page 6). For more comprehensive information on common and uncommmon hummingbirds \u2014 identification, territories, and behavior \u2014 consult a hummingbird identification reference book.\n\nAllen's hummingbird **_(Selasphorus sasin)_**\n\nAllen's hummingbirds are usually about 3\u00be inches (9.5cm) in length. An adult male Allen's has rufous (reddish) sides and tail; its gorget (throat) is a deep orange-red bordered below by white. Its green-bronze back distinguishes it from the otherwise similarly colored male rufous hummingbird, which generally has a red back. An adult female Allen's is similarly marked but lacks the gorget \u2014 its throat is marked with a small patch of iridescent orange-red feathers and rows of small green dots.\n\nAnna's hummingbird **_(Calypte anna)_**\n\nAnna's hummingbirds are about 4 inches (10cm) in length. The adult male Anna's has an iridescent rose-colored gorget and crown; its sides are washed with green and its tail is generally dark. The female Anna's has a green-bronze back and a gray-green tail with a dark band; it lacks the gorget but may have instead a patch of iridescent rose-colored feathers in the center of its throat.\n\nBlack-chinned hummingbird **_(Archilochus alexandri)_**\n\nBlack-chinned hummingbirds are typically 3\u00be inches (9.5cm) long. The adult male has a black gorget bordered below by an iridescent purple band; it has a dull green crown and a deeply notched black tail. The adult female has an all-white breast and throat and a green back.\n\nBlue-throated hummingbird **_(Lampornis clemenciae)_**\n\nBlue-throated hummingbirds are usually about 5 inches (12.5cm) in length. The adult male has a bronze-green crown and back and a white streak that extends back from its eyes, creating a masked look. Its gorget is an iridescent deep blue that is sometimes bordered below by a white stripe. The adult female is similarly marked but lacks the iridescent gorget, having instead a patch of gray on the throat and a white breast. Both males and females have white-tipped tails.\n\nBroad-billed hummingbird **_(Cynanthus latirostris)_**\n\nBroad-billed hummingbirds are typically 4 inches (10cm) in length. The vibrantly colored adult male has an iridescent bronze-green back and crown, an iridescent blue throat that melds into green along the breast and belly, and a coral bill tipped with black. The adult female has a bright green crown and back, a subdued gray breast and belly, scattered green spots along the sides, and a black bill with bright orange or pink at its base\n\nBroad-tailed hummingbird **_(Selasphorus platycercus)_**\n\nBroad-tailed hummingbirds are usually about 4 inches (10cm) in length. The adult male has a deep rose\u2013colored gorget, a green crown and back, and a white breast. The adult female lacks the gorget, having instead a small patch of rose-colored feathers in the center of its throat. Both males and females have a long, squared tail.\n\nBuff-bellied hummingbird **_(Amazilia yucatanensis)_**\n\nBuff-bellied hummingbirds are typically 4\u00bc inches (10.8cm) long. The adult male has a bronze-green back, a brighter green throat and breast, and a buff-colored belly. Its bill is coral with a dull-colored tip, and it has a narrow buff-colored band encircling its eyes. Females are similar in appearance, althought not as brightly colored.\n\nCalliope hummingbird **_(Stellula calliope)_**\n\nCalliope hummingbirds are typically 3\u00bc inches (8.3cm) long, making them the smallest birds found in the United States or Canada. The adult male Calliope has a vibrant gorget composed of streaks of iridescent rose-purple; the adult female lacks this gorget. Both males and females have iridescent blue-green backs and white breasts.\n\nCosta's hummingbird **_(Calypte costae)_**\n\nCosta's hummingbirds are typically 3\u00bd inches (8.9cm) in length. The adult male has an iridescent rose-purple gorget and crown, a dark green back, and a white breast. The adult female is similarly marked but lacks the gorget and crown; it may occasionally have a small patch of purple feathers marking its throat.\n\nMagnificent hummingbird **_(Eugenes fulgens)_**\n\nMagnificent hummingbirds, also known as Rivoli's hummingbirds, are typically 5\u00bc inches (13.3cm) in length. The adult male has an iridescent blue-violet crown, an iridescent green gorget, and a bronze-green back. Its breast and belly are dark, almost black, but shade to grayish along the edges. The adult female lacks the iridescent crown and gorget and is pale gray along its belly.\n\nRuby-throated hummingbird **_(Archilochus colubris)_**\n\nRuby-throated hummingbirds are typically 3\u00be inches (9.5cm) in length. The adult male has an iridescent ruby gorget, bordered below by a white breast, and an iridescent green crown and back. The adult female is similarly marked but lacks the gorget.\n\nRufous hummingbird **_(Selasphorus rufus)_**\n\nRufous hummingbirds are about 3\u00be inches (9.5cm) in length. The adult male has an orange-red gorget bordered below by a white breast and rufous back, sides, and tail. The adult female has green on its back with some rufous on the sides and tail; a small patch of iridescent orange-red feathers marks its throat, accompanied by speckles of dull green dots.\n\n## **Designing a Hummingbird Garden**\n\nYes, it's true, many hummingbirds have a preference for red flowers, and for a very good reason: They have learned through experience that red flowers frequently have more nectar than others. Ruby-throated hummingbirds in particular have the strongest attraction to red, while this characteristic is less pronounced in many of the western species. However, hummingbirds also feed at pink, orange, purple, yellow, and even white blossoms. In short, there are innumerable tubular-shaped flowers, both cultivated and wild, that can justly be called hummingbird flowers.\n\nWith so many flowers to choose from, narrowing down your selection will take some work. Obviously, not all flowers that hummingbirds are attracted to will be hardy to your area, but if you're among those lucky enough to live in a zone suitable to a wide and varied range of plant life, your selections will be limited only by personal preference and space considerations.\n\n### **_Choosing Plants_**\n\nWhen choosing plants for your hummingbird garden, select varieties with overlapping periods of bloom. You can select combinations of annuals, perennials, flowering shrubs and vines, and even some vegetables and herbs. Plant species that are native to your region, especially wildflowers. These will be much better nectar producers than nonnative plants and cultivars.\n\nTo ensure an adequate supply of nectar at all times, plant a variety of flower-producing plants in sizable numbers each. Nectar production in some plants can slow or stop altogether when it's too hot, too cold, too wet, or too dry. Having an assortment of plants blooming in your garden at any one time should minimize this problem.\n\nMost gardens have a combination of perennials (herbaceous plants that live at least three years, although most will happily bloom for far longer than that) and annuals (which, as the name implies, bloom for one season only, although many will self-seed for the following year). However, although perennials will reappear year after year, they bloom for only three or four weeks in any one season, whereas many annuals will produce flowers all summer long. Bear in mind, however, that a zone 8 perennial, such as the nasturtium, will be available at a zone 4 garden shop as an annual simply because the plant can't survive that region's cold winters. And some plants, such as begonias, that are designated annuals in colder climes can be dug up in autumn, potted, enjoyed through the winter as houseplants, and then put back in the garden the following spring, after the last frost.\n\nAlthough not every flower in a hummingbird garden needs to be red, hummingbirds know that red flowers usually offer good nectar. Therefore, to attract hummingbirds to your garden, whether during their summer sojourns or during their long migratory flights, a patch of bright red blossoms will be both highly visible and a good signpost that your garden is an excellent source of food \u2014 which is especially important to a tiny bird with limited energy reserves.\n\n### **_Laying Out Your Garden_**\n\nOnce you have a good idea of the hummingbird-attracting plants that will be appropriate for your region, zone, and soil conditions, you're ready to begin breaking ground for your garden. Remember that hummingbirds prefer areas with trees and\/or shrubs that will provide perching locations, sheltering foliage, and protected roosting sites. The ideal hummingbird habitat has about one-half full sun and one-quarter each full shade and partial shade.\n\nGroup the plants in your garden so that their blossoms are easily accessible to hummingbirds, with ample room for hovering and flight maneuverability. Flowers are visually more interesting when they're tiered, with the tallest ones in back and the shorter ones in front, and a tiered design will also provide better hummingbird access to all of the blossoms in your garden.\n\nDon't overlook your decks and porches \u2014 or even apartment balconies \u2014 as potential hummingbird garden sites. Being fearless creatures, hummingbirds will visit container plants and hanging baskets just as often as they do more traditionally embedded flora. In fact, a hanging basket of fuchsia set outside after wintering indoors may well be the only nectar source around during the early warm days of spring, when many hummingbirds return from a winter sojourn in Central America.\n\n## **_Cultivating Wildflowers_**\n\nIt's likely that hummingbird-attracting wildflowers are already established either in or near your yard. Search them out, and if you find any, help them to thrive by eliminating some of their competitors. Thin or prune surrounding trees to let in more light, and provide the wildflowers with water during dry spells. In doing so you'll not only be  helping native plants to get a stronger foothold, but you'll also be providing a richer source of nectar for your winged visitors (wildflowers generally produce more nectar than their cultivated cousins), and you'll be saving money to boot.\n\n_Note:_ The letters \"spp.\" following the Latin name mean that many of the species in that genus attract hummingbirds.\n\n**E ASTERN WILDFLOWERS**\n\n* * *\n\n**W ESTERN WILDFLOWERS**\n\n* * *\n\n## **Flowers for Hummingbirds**\n\nWhen planting, consider some of the following flowers for your hummingbird garden, depending on your planting zone and sun and soil conditions. The zone listings specify the zones in which each plant may be grown as a perennial \u2014 most plants can be grown as annuals in any zone. The specified heights include flower stalks as well as foliage, which may often be very short by itself. This is followed by the period in which the plant is in flower. For more comprehensive information on any of the plants referenced here, consult a gardening manual.\n\n_Beard-tongue_\n\n**Beard-tongue** _(Penstemon barbatus)_  \nZones:3\u20139 Height: 8\"\u201336\" (20\u201390cm)  \nBlooming season:Late spring to mid-summer\n\nThe individual blossoms of this easy grower are reminiscent of foxglove, although its loose clusters of flower heads are more like annual phlox. Plant in moist, well-drained soil and full sun or light shade.\n\nBee balm _(Monarda_ spp.)  \nZones: 4\u20139  \nHeight: 2'\u20134' (60\u2013120cm)  \nBlooming season: Summer\n\nHummingbirds, bees, and butterflies alike all love the shaggy blossoms of bee balm. Most often seen in red, bee balm also blooms in purple, pink, and even white. Deadhead the spent blossoms regularly to keep these easy growers blooming, and plant in moist soil in full sun or very light shade.\n\nBegonia ( _Begonia_ spp.)  \nZones:6\u201310  \nHeight: 6\"\u201324\" (15\u201360cm)  \nBlooming season: Summer through autumn\n\nPlant begonias in shades of red, pink, or white in your garden or in containers. Give them rich, moist soil in sun where it's cooler and partial shade in hotter climes.\n\n_Bee balm_\n\nBlazing star _(Liatris_ spp.)  \nZones: 3\u201310  \nHeight: 2'\u20136' (.6\u20131.8m)  \nBlooming season: Summer to early autumn\n\nThese rocket-shaped, pink-purple beauties (also known as gayfeather and snakeroot) will perk up any garden. Although preferring a sandy, rich soil, Liatris will tolerate even poor, dry soil, as long as their planting medium is very well drained. They prefer full sun to light shade.\n\nBleeding heart ( _Dicentra spectabilis_ )  \nZones: 3\u20139  \nHeight:2'\u20133' (60\u201390cm)  \nBlooming season: Mid- to late spring\n\nThis traditionally favorite plant with fernlike foliage and sprays of heart-shaped pink and white blooms at the tips of long, slender stems will impart elegance and charm to your garden. Plant in partial sun or partial shade \u2014 more shade is required in hotter areas \u2014 in rich, moist, well-drained soil.\n\nBugleweed ( _Ajuga_ spp.)  \nZones:3\u201310 Height: 4\"\u201310\" (10\u201325cm)  \nBlooming season: Mid-spring to midsummer\n\nCarpet bugleweed _(Ajuga reptans),_ a fast-spreading ground cover that's wonderful under trees or other places too shady for grass to grow, sprouts a forest of purple-blue flowers on a carpet of green and bronze-purple leaves. In rock gardens and mixed shade plantings, try Geneva bugleweed _(Ajuga genevensis)_ and upright bugleweed _(Ajuga pyramidalis),_ which are less invasive. This plant thrives in either sun or shade and just about any soil, even dry, poor soil, provided it drains well (note that in dry soil, bugleweed needs to be shaded).\n\nCalifornia fuchsia _(Zauschneria californica)_  \nZones: 9\u201310  \nHeight: 12\"\u201324\" (30\u201360cm)  \nBlooming season: Late summer to October\n\nSpikes of vivid red, tubular flowers are a good reason that this plant also goes by the name hummingbird flower. Plant in a warm, sunny spot in light, well-drained soil.\n\nCardinal flower _(Lobelia cardinalis)_  \nZones: 2\u20139  \nHeight:3' (90cm)  \nBlooming season: Summer\n\nThe dazzling red spikes of the cardinal flower are a surefire magnet for hummingbirds. Lobelias do best if given afternoon shade, moist soil (they thrive near running water), and good drainage for regular waterings.\n\nCentury plant _(Agave americana)_  \nZones: 6\u201310  \nHeight: Varies; can grow as high as 40 feet (12m)  \nBlooming season: Summer\n\n_California fuchsia_\n\nThese bold succulents with fleshy, sword-shaped leaves and giant flower spikes are highly prized desert hummingbird plants. Plant them in well-drained sandy soil in full sun.\n\nColumbine _(Aquilegia_ spp. _)_  \nZones: 3\u20139 Height: 6\"\u201348\" (15\u2013120cm)  \nBlooming season: Mid-spring to early summer\n\nThese beautiful spurred flowers are hummingbird favorites. The red-and-yellow-flowered wild columbine native to the East, Aquilegia canadensis, will freely reseed itself \u2014 be careful it doesn't become a pest. In the West, plant crimson columbine, Aquilegia formosa, native to California, Oregon, and Nevada. All columbines like moist soils with light to moderate shade. Columbines do well in rock gardens.\n\nCoralbells _(Heuchera_ spp. _)_  \nZones: 3\u20139  \nHeight: 12\"\u201336\" (30\u201390cm)  \nBlooming season: Late spring to early autumn\n\nDelicate sprays of small tubular, red, white, or pink flowers rise on tall, wiry stems out of compact clumps of scalloped foliage. Keep this plant neat by removing the flowering stems once they've bloomed. Grow in light, well-drained soil in full sun in northern zones and partial shade in warmer climes.\n\nCreeping phlox _(Phlox stolonifera)  \n_ Zones: 3\u20139 Height: 6\"\u20138\" (15\u201320cm)  \nBlooming season: Early to late spring\n\nThese spreading perennials form beautiful beds of star-shaped flowers in varying shades from white to purple to blue, although the bright, hot pink ones will attract hummingbirds most readily. Phlox prefer a rich, well-drained, moist soil and will grow in either full sun or partial shade.\n\nDelphinium _(Delphinium_ spp. _)_  \nZones: 3\u20139  \nHeight: 18\"\u201384\" (.5\u20132m)  \nBlooming season: Late spring through autumn, depending on species\n\nThese statuesque spires of blossoms \u2014 some single flowered, some double \u2014 are beautiful in any garden. Also known as larkspur, delphiniums bloom in blue, purple, pink, yellow, and white. Plants over 18 inches (46cm) tall will need to be staked to keep them upright. Although short-lived, fading out after only two or three years, delphiniums can be propagated by cuttings from new spring growth. Grow these beauties in rich, moist, well-drained soil.\n\nFire pink _(Silene virginica)_  \nZones: 4\u20139 Height: 10\"\u201324\" (25\u201360cm)  \nBlooming season: Late spring to early summer\n\nThese masses of dark pink and crimson blooms will stay ablaze for quite some time. Plant in well-drained, humus-rich soil in full sun or light shade (more shade in hotter climates).\n\nFour-o'clock _(Mirabilis Jalapa)  \n_ Zones: 8\u201310  \nHeight: 24\"\u201348\" (60\u2013120cm)  \nBlooming season: Midsummer to late autumn\n\nComing from the tropics, these lovely blooms, in shades from rosy purple or red to white and yellow, are perennials only in the warmest climates. Elsewhere cultivate as container plants in well-drained soil.\n\n_Fire Pink_\n\nFoxglove _(Digitali_ s spp. _)_  \nZones: 3\u20139  \nHeight: 24\"\u201360\" (60\u2013152cm)  \nBlooming season: Late spring to late summer, depending on species\n\nThe tall spires of these plants look majestic in any setting \u2014 and depending on the species and your conditions (they prefer cool, moist climates), they'll grow just about anywhere, from full sun to full shade (the hotter the summer, the more shade necessary). Foxgloves come in creamy yellow, pink, salmon, and orange. All have freckled interiors. Give these beauties moist, rich soil, and they'll happily self-seed.\n\nFuchsia _(Fuchsia_ spp. _)_  \nZone: 10  \nHeight: 1'\u20136' (30\u2013180cm)  \nBlooming season: Late spring to autumn\n\nWith its pendulous blossoms, this plant is much esteemed for hanging baskets and comes in shades of bright red, fiery pink, purple, white, and various combinations of them all. It does best in full or half shade \u2014 I hang my fuchsia basket from a sturdy lower pine branch several feet away from a feeder \u2014 in a rich, moist soil.\n\nGeranium _(Geranium_ spp. _)_  \nZones: 3\u201310  \nHeight: 4\"\u201336\" (10\u201390cm)  \nBlooming season: Spring through late summer, depending on species\n\nWe usually associate the name \"geranium\" with the familiar velvet-leafed houseplant with bright red flowers, but those belong to the Pelargonium genus. Geraniums are hardy, carefree plants with lovely lacy leaves and red, pink, or purple blooms, depending on the species and cultivar. They'll grow in average, well-drained soil, but when planted in rich soil, they'll spread rapidly. Plant in full sun in colder climates and in partial shade in warmer southern climates.\n\nHollyhock _(Althea_ spp. _)_  \nZones: 5\u20139  \nHeight: 2'\u20136' (60\u2013180cm)Blooming season: Summer to early autumn\n\nThis wonderful, old-fashioned, upright plant is available in just about every color. Use the older, taller species as back border plantings, along a fence, or against a cottage wall; the newer, shorter species can go anywhere. Although favoring rich, well-drained, moist soil, they'll tolerate dry soil, too \u2014 but not excessive heat and humidity.\n\nImpatiens _(Impatiens_ spp. _)_  \nZones: Can be grown in any zone as an annual  \nHeight: 12\"\u201324\" (30\u201360cm)  \nBlooming season: Late spring through first frost\n\nImpatiens are grown as annuals in most of the country since they are unable to survive frost. With their vibrantly colored, long-lasting blossoms, they are ideal hummingbird flowers. Mass them under trees or in containers under your porch overhang, since most species prefer full shade. They also like a rich, moist soil.\n\nLily _(Lilium_ spp. _)_  \nZones: 3\u20139  \nHeight: 2'\u20136' (60\u2013180cm)  \nBlooming season: Late spring through autumn, depending on species\n\n_Hollyhock_\n\nLilies have large and exquisite, usually multiple, trumpet-shaped flowers on long, strong stems \u2014 eye-catching either singly or when massed together in any garden. Lilies require moist, well-drained, usually slightly acidic soil in full sun or light shade.\n\nLupine _(Lupinus_ spp. _)_  \nZones: 4\u20138  \nHeight: 18\"\u201360\" (46\u2013152cm)  \nBlooming season: Spring to midsummer\n\nLupines give any garden a boost, even when there are only a few. When massed, the effect of the pink, purple, blue, and red flower spikes is spectacular. Many species thrive in areas with relatively cool summers, although others will tolerate hotter climes. Give them a rich and moist but well-drained soil in either full sun or light shade.\n\nMammillaria _(Mammillaria setispina)_  \nZones: 9\u201310  \nHeight: Various; ball-like  \nBlooming season: Late spring to summer\n\nThis member of the cactus family produces dark red flowers beyond its spines, which makes it a fine place for desert hummingbirds to sup. If you don't live in the Southwest, the mammillaria makes an excellent houseplant that can summer outdoors in half shade. As with most cacti, plant in a light and porous yet stable soil with excellent drainage (a mixture of sandy and clay soils works well).\n\nMonkey flower _(Mimulus_ spp. _)_  \nZones: 3\u20139  \nHeight: 8\"\u201336\" (20\u201390cm)  \nBlooming season: Spring to autumn, depending on species\n\nDepending on the species suitable for your location, these trumpet-shaped, lipped flowers come in all sizes and will want either full sun or partial shade. Check a reference guide for more detailed information. Coming in hues of red, orange, pink, and yellow, these plants grow best in moist, rich soil.\n\nMontbretia _(Crocosmia_ spp. _)_  \nZones: 5\u20138  \nHeight: 12\"\u201348\" (30\u2013120cm)  \nBlooming season: Summer\n\nSpectacular branches of funnel-shaped flowers in red, orange, or yellow emerge from fans of stiff, swordlike foliage. Like the iris the foliage resembles, these plants grow from corms and spread to form clumps. Grow in sun to partial shade in average, well-drained soil.\n\n_Monkey flower_\n\nNasturtium _(Tropaeolum majus)_  \nZones: Can be grown in any zone as an annual  \nHeight: 12\"\u201348\" (30\u2013120cm)  \nBlooming season: Summer to late autumn\n\nThese annuals have bold and beautiful yellow, orange, and red spurred blossoms that are a wonderful contrast to their rounded green leaves. Nasturtiums prefer summers that are mild and dry with cool nights. They do well even in poor soil so long as it's well drained and slightly acidic. Plant in full sun to partial shade.\n\nObedient plant _(Physostegia_ spp. _)_  \nZones: 2\u20139  \nHeight: 12\"\u201348\" (30\u2013120cm)  \nBlooming season: Midsummer to autumn\n\nThe pink, purple, or white blooms on these vigorous, easy-care plants \u2014 also known as false dragonhead \u2014 grow in long, regimented rows. Physostegia enjoy any good garden soil in either full sun or partial shade.\n\nPetunia _(Petunia_ spp. _)_  \nZones:Can be grown in any zone as an annual  \nHeight: 12\"\u201318\" (30\u201346cm)  \nBlooming season: Late spring to frost\n\n_Montbretia_\n\nPetunias are one of the most widely grown annuals due to their tireless (and seemingly endless) blooming period. The cheery trumpet-shaped flowers are borne on long stems that work well in hanging baskets. To make them bushier, pinch back young plants. When planted in the garden, put them in moist, well-drained soil in full sun or very light shade.\n\nRed-hot-poker _(Kniphofia Uvaria)_  \nZones: 5\u20139  \nHeight: 2'\u20134' (60\u2013120cm)  \nBlooming season: Midsummer\n\nAlso known as torch lily, Kniphofia (knee-FOE-fia) has blossoms of fiery red, yellow, and orange on thickly clustered spikes that are dramatic in any setting. Plant in rich, moist, and well-drained soil. Give full sun in moderate climes, partial shade where summers are very hot.\n\n**_Scarlet sage_** _(Salvia splendens)_  \nZones: Can be grown in any zone as an annual  \nHeight: 1'\u20133' (30\u201390cm)  \nBlooming season: Summer to autumn, depending on species\n\nScarlet sage is a tender perennial most often grown as an annual. Its usually bright red spikes are wonderful for creating splashes of color in your garden. Give it well-drained soil and full sun or partial shade.\n\nSpider flower _(Cleome hassleriana)_  \nZones: Can be grown in any zone as an annual  \nHeight: 3'\u20134' (.9\u20131.2m)  \nBlooming season: Summer through late autumn\n\nAlthough annuals, these odd-looking plants will self-seed with wanton regularity, so be careful where you plant them. Give these long bloomers average soil and lots of sun. If your summers are very hot, give them half shade.\n\nSweet William _(Dianthus barbatus)_  \nZones: Can be grown in any zone as an annual, or sometimes a biennial  \nHeight: 6\"\u201324\" (15\u201360cm)  \nBlooming season: Late spring to summer\n\nChoose the crimson variety of this plant with its cushiony, tightly packed blooms. Sweet William self-sows readily and prefers rich, well-drained slightly alkaline soil in the sunny or lightly shaded part of your garden.\n\nVerbena _(Verbena_ spp. _)_  \nZones: 3\u201310  \nHeight: 4\"\u201324\" (10\u201360cm)  \nBlooming season: Late spring to frost\n\nThese showy plants with tightly packed flower heads are rather short-lived and are best grown as annuals. Choose from pink, white, purple, and red blossoms. Plant in full sun in any well-drained soil.\n\nYucca _(Yucca_ spp. _)_  \nZones: 3\u201310  \nHeight: 3'\u201330' (.9\u20139m)  \nBlooming season: Early summer to autumn, depending on species\n\nAn impressive spike of white, bell-shaped flowers stands like a sentinel over the yucca's straplike evergreen leaves. This very tough, drought-resistant plant likes full sun and average or sandy well-drained soil. Some species need protection in winter in colder climates.\n\n## **Vines, Shrubs, and Trees**\n\nSome of the most favored blossoms of hummingbirds are those of vines, shrubs, and trees, which also provide shelter for resting, roosting, and nesting. If you can, make room in your garden for one or more of these beauties.\n\n### **_Vines_**\n\nMany of these vines, having densely packed foliage, are ideal homes for hummingbirds.\n\nMorning-glory _(Ipomoea_ spp. _)_  \nZones: 5\u201310  \nBlooming season: Midsummer through autumn\n\nThese twining vines with their blue, lavender, pink, or white flowers will wrap themselves around just about anything \u2014 and will also grow in just about anything, including poor, dry soil. Grow in full sun.\n\nScarlet runner bean _(Phaseolus coccineus)_  \nZone: 10  \nBlooming season: Midsummer to frost\n\nAnnuals everywhere but tropical climates, scarlet runner beans make everyone happy \u2014 hummingbirds for the nectar in the clusters of flowers, humans for the beans. Grow in full sun against a trellis to appreciate fully the brilliant red flowers.\n\nTrumpet creeper _(Campsis radicans)_  \nZones: 4\u20139  \nBlooming season: Midsummer through fall\n\nThis vigorous, twining perennial vine with its bold, trumpet-shaped (hence the name) scarlet flowers prefers full sun and a rich, moist, well-drained soil. It will need strong supports in your garden.\n\nTrumpet honeysuckle _(Lonicera sempervirens)_  \nZones: 3\u20139  \nBlooming season: Early to late summer\n\n_Morning-glory_\n\nRich orange, scarlet, or yellow clumps of blossoms punctuate this semi- to evergreen vine (depending on your climate); in areas subject to much frost it needs protection. Grow in cool, semishady areas, in porous, well-drained, fertile soil.\n\n### **_Shrubs_**\n\nAzalea _(Rhododendron_ spp. _)_  \nZones: 5\u20138  \nBlooming season: Early spring\n\nIn colder climes, grow this decorative species of the classic rhododendron as a houseplant and put it outdoors after the last frost. Give it a rich, moist soil \u2014 an azalea needs lots of water \u2014 and shade.\n\n**Butterfly bush** _(Buddleia Davidii)_  \nZones: 5\u20139  \nBlooming season: Midsummer to autumn\n\nAlso known as summer lilac, butterfly bush \u2014 which obviously also attracts butterflies \u2014 has profuse white, pink, red, or purple blossoms in long clusters. Grow this and other Buddleias in any soil in full sun.\n\nCape honeysuckle _(Tecomaria capensis)_  \nZones: 9\u201310  \nBlooming season: Year-round\n\nThose in more tropical climes can enjoy the scarlet blossoms of this evergreen all year long. This beautiful shrub, which will reach 8 feet, should be grown in fertile soil in full sun.\n\nFlowering quince _(Chaenomeles_ spp.)  \nZones: 4\u20139, depending on species  \nBlooming season: Spring\n\nThe showy flowers on these deciduous shrubs, not the fruit, are the main attraction for both hummingbirds and humans \u2014 although if the yield is bountiful, you can make preserves from the fruit. Plant in fertile soil in a sunny spot.\n\nHibiscus _(Hibiscus_ spp. _)_  \nZones: 5\u201310, depending on species  \nBlooming season: Mid- to late summer\n\nHibiscus are often thought of as strictly tropical shrubs, but they also include some hardier species such as rose-of-Sharon (Hibiscus syriacus). Grow in a sheltered, sunny spot in any well-drained soil and enjoy masses of pink, purple, orange, yellow, or scarlet blossoms, depending on the species.\n\nHoneysuckle _(Lonicera_ spp. _)_  \nZones: 4\u20139  \nBlooming season: Late spring to early summer\n\nThese bushy shrubs have tubular blossoms in creamy white, pale yellow, pink, and even hot pink, depending on the species. Honeysuckles generally prefer well-drained soil and full sun to partial shade.\n\nLantana _(Lantana Camara)_  \nZones: 8\u201310  \nBlooming season: Year-round\n\nThis 6-foot-tall evergreen shrub produces compact clusters of orange-yellow, orange, red, or red-and-white flowers. Grow in any well-drained soil in full sun.\n\n_Hibiscus_\n\nMealberry _(Arctostaphylos Uva-ursi)_  \nZones: 2\u201310  \nBlooming season: Mid- to late spring\n\nThe drooping clusters of blossoms on this dwarf evergreen shrub give way to brilliant red berries. This species prefers soil that is sandy or acidic, but will grow in almost any type of soil. Prefers full sun but will tolerate some shade.\n\nRosemary _(Rosmarinus officinalis)_  \nZones: 6\u201310  \nBlooming season: Early spring\n\nThe stems of the fragrant herb we get at our greengrocers are actually branches of an evergreen shrub that grows to 7 feet tall \u2014 yet another plant that satisfies the needs of both humans and hummingbirds. The latter appreciate the lovely lilac-blue flowers. Rosemary grows best in light, well-drained soil in a warm, sunny spot.\n\nWeigela _(Weigela florida)_  \nZones: 4\u20139  \nBlooming season: Spring\n\nThis hardy shrub is generally easy to cultivate. It has tubular or funnel-shaped flowers in all shades of pink from deep carmine to almost white. Weigela thrives in any moist, well-drained soil in full sun.\n\n### **_Trees_**\n\nTrees don't usually come to mind when we're thinking of flowers for hummingbirds, but many species have spectacular blossoms.\n\nChinaberry _(Melia Azedarach)_  \nZones: 7\u201310  \nHeight: 30'\u201340' (9\u201312m)  \nBlooming season: Late spring\n\nThis member of the mahogany family produces lovely sprays of lilac-colored flowers. A good shade tree, it grows best in full sun. Chinaberry is often considered a weed species, as its wood breaks easily during heavy storms.\n\nCockspur\/Cry-baby tree _(Erythrina crista-galli)_  \nZone: 10 Height: 15' (5m)  \nBlooming season: Late summer\n\nThis small to average-size tree produces clusters of dark red, waxy flowers. It prefers moist but well-drained soil. Plant in full sun.\n\nEucalyptus _(Eucalyptus_ spp. _)_  \nZones: 9\u201310  \nHeight: 30'\u2013100' (9\u201330m)  \nBlooming season: Winter to summer\n\nEucalyptus provide both nectar and shelter for hummingbirds. These Australian imports have flowers that range from white to showy red and for the most part prefer full sun in moist soil.\n\nFlowering crabapple _(Malus_ spp. _)_  \nZones: 2\u20136  \nHeight: 15'\u201325' (5\u20138m)  \nBlooming season: Mid- to late spring\n\nThis is another species that benefits both hummingbirds (nectar) and humans (fruit for preserves \u2014 if the blue jays don't eat them first). Flowers range in color from white to pink to bright red. Grow in any well-drained, fertile soil in full sun.\n\nPoinciana _(Caesalpinia_ spp. _)_  \nZone: 10  \nHeight: 15'\u201320' (5\u20136m)  \nBlooming season: Summer\n\nThis fast-growing small tree (or shrub) is a member of the pea family. Beautiful clusters of white or yellow pealike flowers with prominent red stamens are framed by delicate fronds. Grow in full sun in well-drained, dryish soil.\n\nRed horse chestnut _(Aesculus x carnea)_  \nZones: 3\u20137  \nHeight: 30'\u201340' (9\u201312m)  \nBlooming season: Late spring to early summer\n\nAs its name implies, the large clusters of blooms on this spectacular tree have rose red panicles that can reach 6 to 8 inches in length in the spring. Grow in moist, well-drained soil in full sun to light shade.\n\nSiberian peashrub _(Caragana arborescens)_  \nZones: 2\u20137  \nHeight: 15'\u201320' (5\u20136m)  \nBlooming season: Spring\n\nLovely yellow flowers dress up this tree's feathery leaves. It does well in all soil types and prefers full sun.\n\n**Silk tree** _(Albizia Julibrissin)_  \nZones: 6\u20139  \nHeight: 20'\u201335' (6\u201311m)  \nBlooming season: Late spring to late summer\n\nThis small tree, also called mimosa tree, produces beautiful bristly clusters of flowers \u2014 looking like so many bottle brushes \u2014 that are white at the base and bright pink at the tips. Grow this member of the sun-loving pea family in full sun; it does well in all soil types.\n\n**Tulip poplar** _(Liriodendron Tulipifera)_  \nZones: 4\u20139  \nHeight: Can grow to 200 feet (60m)  \nBlooming season: Spring to early summer\n\nThis tall, stately tree, also known as tulip tree, produces \u2014 no surprise \u2014 tuliplike greenish yellow flowers lined with orange. Grow in rich, moist soil in full sun.\n_The 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 Nancy W. Ringer  \nCover illustration by Alison Kolesar  \nCover design by Carol J. Jessop (Black Trout Design)  \nIllustrations by Mallory Lake, except pages 4, 5, 11, 14 (top), and 17 by Alison Kolesar  \nText design and production by Heather Clemow\n\n\u00a9 1997 by Storey Publishing, LLC\n\nAll rights reserved. No part of this bulletin 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 bulletin be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopying, recording, or other without written permission from the publisher.\n\nThe information in this bulletin 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 and bulletins are available for special premium and promotional uses and for customized editions. For further information, please call 1-800-793-9396.\n\nPrinted in the United States by Excelsior\n\n**Library of Congress Cataloging-in-Publication Data**\n\nGelfand, Dale, 1944\u2013\n\nGrowing a hummingbird garden \/ by Dale Gelfand  \np. cm. \u2014 (A Storey country wisdom bulletin; A-167)  \nISBN 978-0-88266-713-3 (alk. paper)\n\n1. Gardening to attract birds\u2014North America. 2. Bird attracting\u2014  \nNorth America. 3. Hummingbirds\u2014North America I. Title. II. Series\n\nQL676.57.N7G44 1997  \n635.9'67\u2014dc21\n\n96-53551  \nCIP-10:\n\n##  **M ORE COUNTRY WISDOM  \nBULLETINS YOU WILL ENJOY**!\n\nThese and other Storey Country Wisdom Bulletins are available for $3.95 at your local bookstores, garden centers, farm stores, and gift shops. Use the order numbers listed under each bulletin to make your requests. You can also order directly from Storey Publishing by writing to us at 210 MASS MoCA Way, North Adams, MA 01247 or by calling 1-800-441-5700. 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+{"text":"\n\n## CONTENTS\n\nCover\n\nAbout the Book\n\nAbout the Author\n\nAlso by Milton Crawford\n\nTitle Page\n\nDedication\n\nChew. Slurp. Lick. Nibble.\n\nRelationship Questionnaire\n\nA Note on Unsexy Foods\n\nSOMETHING TO SLURP ON\n\nFrozen Watermelon Margarita\n\nBasil Martini\n\nAlmond Milk White Russian\n\nBloody Mary with Clam Juice\n\nStrawberry Bellini\n\nNIBBLES AND TIT-BITS\n\nSpiced Honeyed Almonds\n\nCaviar Blinis\n\nPuglia Broad Bean Dip\n\nOyster Fritters with Rocket and Wasabi Mayonnaise\n\nChaat Masala Watermelon with Mint\n\nSMOOTH AND SLIPPERY\n\nOysters with Chilli, Ginger and Lime\n\nSalmon Tikka Skewers with Dill and Pomegranate Raita\n\nMilton's Moules\n\nLobster with B\u00e9arnaise Sauce, Home-made Oven Chips and Broccoli\n\nScallops with Brown Butter, Capers and Lemon\n\nFLESH\n\nVenison and Liver Pies\n\nBeef Carpaccio with Parmesan, Rocket and Truffle Oil\n\nYakitori-style Chicken Heart Skewers\n\nSteak Tartare with Confit Egg Yolk and A\u00efoli\n\nHerb-crusted French Rack of Venison\n\nBeef Fillet, Ceps, Marsala Sauce and Roasted Shallots\n\nTHE SPICE OF LIFE\n\nButternut Soup with Nutmeg and Toasted Pumpkin Seed Oil\n\nCrispy Sea Bass with Ginger and Spring Onions\n\nPaneer Tikka Masala\n\nSmoked Garlic Dauphinoise\n\nSalt and Pepper Baby Squid with Garlic Mayonnaise\n\nSaffron Roast Chicken with Lemon and Rosemary\n\nSHAPELY VEG\n\nCrunchy Fennel and Green Apple Salad\n\nAsparagus with Crab, Poached Eggs and Lemon Vinaigrette\n\nAvocado, Pancetta and Toasted Pumpkin Seed Salad\n\nPommes de Terres Sarladaises with Truffles\n\nRed Mullet, Artichoke Hearts and Lemon\n\nGETTING FRUITY\n\nWatermelon Salsa with Black Pepper Goat's Cheese\n\nGrilled Honey Figs, Goat's Cheese and Walnut Salad\n\nPork and Lychee Curry\n\nSalt Caramel and Rum Banana Cake\n\nPoached Quince with Mascarpone and Stilton Cream\n\nSWEET BITS\n\nCranachan with Figs, Candied Walnuts, Honey and Meringue\n\nChocolate Chilli Fondant\n\nRed Wine Poached Pears with Stem Ginger Cr\u00e8me Fra\u00eeche\n\nVanilla and Saffron Baguette and Butter Pudding with Gooseberry-Ginger Jam\n\nRaspberry and Rose Pudding\n\nAcknowledgements\n\nCopyright\n\n## About the Book\n\nThis book offers a unique culinary slant on dinner \u00e0 deux with fun and adventurous recipes that will help love to blossom, tease palates and arouse the senses... and make all the difference between a firecracker of an evening and a damp squib:\n\n**Something to Slurp on** \u2013 get the juices flowing with a **Basil Martini** or a **Bloody Mary with Clam Juice**  \n**Nibbles and Tit-bits** \u2013 get down to some fun foreplay with **Spiced Honey Almonds** or **Caviar Blini**  \n**Shapely Veg** \u2013 get forking Asparagus with **White Crab Meat** or **Pommes Sarladaises with Truffles**  \n**Smooth and Slippery** \u2013 serve **Milton's Moules** or **Salmon Tikka Skewers with Dill and Pomegranate Raita** for a boost in the bedroom  \n**Flesh** \u2013 wrap your mouth around **Pot-roast Haunch of Venison** and **Beef Fillet, Ceps and Marsala Sauce**  \n**The Spice of Life** \u2013 get hot and spicy with **Saffron Roast Chicken** or **Paneer Chilli**  \n**Getting Fruity** \u2013 with **Pineapple and Pork Curry** or **Poached Quince with Mascarpone and Gorgonzola Cream**  \n**Sweet Bits** \u2013 down and dirty desserts, from **Chocolate Chilli Fondant** to **Raspberry Rose Pudding**\n\nRead on and let Milton Crawford share his secret potions of love.\n\n_'Cooking is like love: It should be entered into with abandon or not at all'_ **Harriet van Horne**\n\n## About the Author\n\nMilton Crawford was born somewhere north of the Zambezi and west of the Rift valley in a small town in the middle of Africa. He has travelled the world in search of good liquor, fine food and game women. He is an author and journalist, and in keeping with the most honourable traditions of the writing profession, a bon viveur and lover of distinction. Author of the bestselling _The Hungover Cookbook_ and _The Drunken Cookbook_ , his previous books have been published under a more sober alias.\nAlso by Milton Crawford\n\nThe Hungover Cookbook  \nThe Drunken Cookbook\n\nFor Lovers Everywhere\nDisclaimer\n\nI worry about why all my books have to have disclaimers. I said no, but the lawyers insisted.\n\n_So, here we go:_\n\nThis book is intended for adults, has adult themes and contains some explicit references to sex. Although the recipes contained within these pages may suggest that they will lead to some form of romantic conquest, I cannot guarantee that this will happen. Similarly, none of these recipes claim to cure any medical conditions and, if you believe that you are suffering from a medical condition, then I recommend that you see a medical professional. These recipes are in no way a substitute for any treatment you may need to receive.\n\nI hope you succeed in your amorous attempts, but I accept no responsibility should you fail. Please do not try to contact me for romantic advice; my past is littered with plenty of romantic failures so I do not profess to be an authority on the subject.\n\nGood luck. Love is a wonderful thing. I hope that this culinary adventure helps you on your quest for romance.\n\nDo aphrodisiacs really work? And, even if they do, isn't the concept of eating foods specifically to boost libido and mutual pleasure a bit old-school? After all, these days both men and women can pop a pill to produce the desired effect. So this book is not so much concerned with the direct impacts of aphrodisiacs, which are not always quantifiable, as with the idea of making food in the name of love. And of course there is the hope that, at some point, the effort of making food may gain some form of reward. If you know what I mean.\n\nBut why, you may ask, would you not simply go to a romantic restaurant with your love interest and choose a suitably sublime menu? That is surely just as romantic as cooking for them? I beg to differ. For one thing, it is more of a gift to create something from scratch; to conceive, buy the ingredients for, create and serve a meal for someone you love (or at least have designs on) is a telling gesture. It creates the impression that you're considerate; that you care; that you're prepared to make an effort. I think you get my drift. It also shows your potential partner that you are resourceful and skilful. More and more people seem to be spending more and more time watching people cooking on screen and less and less time actually doing it in the kitchen. You can demonstrate that you have this valuable skill, which, when done well, shows off many of the same skills you also need in the bedroom: finesse, a sure touch, excellent judgement and a willingness to experiment.\n\nAnd, talking of 'doing it in the kitchen', creating a meal at home does offer far greater \u2013 and more immediate \u2013 opportunities for post-prandial love-making than at least most forms of restaurant that I have ever been to. Put simply: you may start sitting at the dining table; you may end up lying on top of it.\n\nFood and sex are intimately connected in all kinds of ways: the similar noises that we make while we're engaged in these two acts, the words we use to describe them, even the feelings that both can engender.\n\nSex, like eating, can be both mundane and remarkable. Both acts are fundamental to human existence and there's nothing new about either: the associations between food, sex and sin have existed at least since Adam and Eve. Don't be prudish about this; as I hope you already realise, I'm not going to be, despite my highly repressed British sexuality.\n\nWe use our hands and our mouths for both food and sex. We make similar noises, not just the chewing, sucking and slurping, but also the expressions of delight, groans of appreciation or \u2013 perhaps most tellingly \u2013 the concentrated silence of genuine enjoyment.\n\nThen there is our language. 'Tasty' is often used to describe an attractive potential partner. Depending on your age, social class, gender and where you grew up, you might also use 'yummy', 'scrummy', 'luscious', 'delicious', 'dishy' or 'hunky' (like a hunk of meat). And it's not unknown for the expression 'I want to eat you up' to pass between two lovers (in a generally metaphorical sense, unless, that is, you have cannibalistic tendencies and probably live somewhere in Germany). We talk about sexual appetite, cravings, being sex-starved or sated. Post-coital bliss and post-prandial satisfaction share much in common. The associations run deep.\n\nBut we're not just talking about sex here \u2013 that purely biological function of procreation \u2013 we're talking about love. I've always thought that romantic love needs to involve both a sense of fun and of adventure and this book offers a unique culinary slant on it, with fun, adventurous recipes that should help romance to blossom.\n\nIn conventional terms, the classic romantic encounter is a dinner date, where food, wine and conversation create an atmosphere of intimacy. Such a meal is the focus of this book. The choices we make about what to eat during this window of opportunity may make all the difference between a firecracker of an evening and a damp squib. Certain foods rich in the right vitamins and minerals, and that are suggestive in colour, shape and texture, may lead to a blossoming of mutual desire.\n\nThe science of aphrodisiacs is hotly disputed and many claims made about certain ingredients are certainly dodgy, perhaps, historically, due to overzealous traders keen to talk up their wares and imbue them with extra benefits that don't really exist. I am not a scientist and this is not a scientific guide. But there is no doubt that food changes our mood and, even if we don't experience the same extreme physiological effects that are produced by Viagra and the like, the subtle changes that the right foods have on us can make the world of difference. For me, the folkloric associations of certain foods with sex make ample case that they do have some effect. There is usually some truth to myths that grow up around food and, as a storyteller myself, I am more inclined to follow this kind of narrative than scientific studies, the quality of which are hard to determine. Oysters contain high levels of zinc, but is it this that makes them so sought after as aphrodisiacs, or is it more to do with their evocative appearance, texture, flavour and the tactile way in which they are consumed? These are things that are hard to determine using science alone.\n\nSo let me guide you through the delightful world of being a lover cooking for your lover. This book is a homage to the pleasures of giving; a culinary odyssey that delves deeply into the relationship between nutritional and emotional sustenance. In short, it is a foray into food and fornicating.\n\nI have used ingredients that are legal, generally accessible and widely regarded as good to eat. There will be no rhino horn or tiger penis, no whale sperm or Spanish fly. Above all, I would like you and your dinner partner to enjoy your meal, as I believe that the warm glow of shared enjoyment is as great an aphrodisiac as anything.\n\nAll recipes are for two. Modern 'couples' consisting of three or more parties should multiply ingredients by the appropriate factor.\n\nAnd a final word of advice: gluttony is the enemy of desire. In other words, don't eat too much!\n\nWhat type of relationship are you in \u2013 or would like to be in? This is important when it comes to choosing an appropriate menu for your romantic assignation. Every couple has a dominant mood or character. So I have classified six of these to provide an idea of what type of dishes may appeal to you and your partner.\n\nThis is not an exact science \u2013 it possibly only describes the six types of relationship that I have ever been in, though I hope that it is more universal than that. But you may feel that your relationship fits into more than one of these categories \u2013 or possibly even none.\n\nThere is, despite I'm sure some protestations to the contrary, no such thing as the perfect relationship. Different people want different types of relationship and have personalities that are suited to different types of relationship.\n\nFor some, a life of dedicated, monogamous commitment is the nearest thing to hell. For others, romantic love is all about loyalty and mutual respect. Not that monogamy occupies the moral high ground when it comes to mutual respect. Ooh, relationships: they're such a minefield aren't they?\n\nIn this book, different recipes are coded to show that they are particularly suitable for different types of relationship. So, answer the following questions to find out what type of relationship you're in \u2013 and help to create a bespoke menu that suits you and your partner.\n\n## The Results\n\nIf you answered mainly (a) then you are either in, or most suited to, an intense relationship.\n\nIf you answered mainly (b) you fall into the slow-burning category.\n\nIf you answered mainly (c) then you are in the quick & messy relationship category.\n\nAnswering mainly (d) means that your relationship is wholesome & nutritious.\n\nIf your answers were mainly (e) it means that you are most probably in a light & frothy relationship.\n\nAnd if your answers were mainly (f) then you fall into the heavy & profound group.\n\n  **Intense**\n\nThis type of relationship is characterised by violent passion, intense physicality, outbursts of jealousy, occasional arguments, quarrels, breakages, spillages and enthusiastic making up afterwards.\n\n  **Slow-burning**\n\nThe antithesis of the intense relationship, the slow-burner is all about gentle, ardent, respectful and growing love. Consideration and loyalty are key words here.\n\n  **Quick & messy**\n\nA hedonistic form of relationship that is non-committal and concerned primarily with the pleasures of sex. Swipe left and move on.\n\n  **Wholesome & nutritious**\n\nThis couple probably likes country walks, visits to art galleries, learning foreign languages together and eating healthily. Domestic duties are shared. Croissants are a weekend treat. Camomile tea is a nightly routine.\n\n  **Light & frothy**\n\nThis relationship is founded on fun and friendship and a refutation of all things serious. Life is for living, so let's enjoy ourselves. Sunday lunch in the pub with friends. Coffee and cake after work.\n\n  **Heavy & profound**\n\nExistentialism is erotic. I orgasm, therefore I am. Cue long nights discussing Kierkegaard, Dostoyevsky and Nietzsche in dimly lit rooms, writing love letters, packing suitcases and meeting in second-hand bookshops.\n\nFoods that actively dissuade us from sexual acts are called anaphrodisiacs. I have given details of a few of these below and you may want to think about avoiding them in a romantic meal. But this is also a subjective category: foods that one person finds delicious and sensuous another may find disgusting. Not everyone likes oysters, or steak tartare. So it is worth doing your research if you are cooking for someone, to see what foods they love and which they dislike.\n\nChasteberry\n\nYou're hardly likely to consume chasteberry, but it is worth a mention simply for the name. If you're in the market for chastity, this is clearly the herb for you.\n\nCherries\n\nOne study suggested that the mere smell of cherries reduces blood flow to the vagina by up to 18 per cent, which is hardly likely to help with female sexual pleasure. This is all the more surprising when you consider some of the associations of the fruit, especially 'popping the cherry'. Probably best to keep them off the menu if you're planning to cook for a lady.\n\nLettuce\n\nThis apparently neutral food in fact exerts a quite subtly powerful effect on us: it is a sedative. Identified by the Roman author Pliny as good for combatting sexual desire, lettuce is best eaten when you desire a good night's sleep rather than a night of frenetic activity.\n\nLiquorice\n\nThis root diminishes desire in both men and women, wreaking all kinds of negative hormonal changes and reducing sex drive. Avoid.\n\nTofu\n\nIt's actually the soy bean that is the culprit here (although soy sauce is all right because it has been fermented), but it's perhaps hardly surprising that something with the texture of soggy cardboard is unlikely to inspire ardour.\n\n## Key\n\nDifficulty ratings\n\n  Part-time Lover\n\n  Baby Love\n\n  Justify My Love\n\n  Love Takes Time\n\n  Whole Lotta Love\n\nRelationships types\n\n  Intense\n\n  Slow-burning\n\n  Quick & messy\n\n  Wholesome & nutritious\n\n  Light & frothy\n\n  Heavy & profound\n\nArouseometer ratings\n\n1 _(lip-moistening)_ to 5 _(raging horn)_\n\n## The recipes\n\nSOMETHING TO SLURP ON\n\nFrozen Watermelon Margarita\n\nBasil Martini\n\nAlmond Milk White Russian\n\nBloody Mary with Clam Juice\n\nStrawberry Bellini\n\nNIBBLES AND TIT-BITS\n\nSpiced Honeyed Almonds\n\nCaviar Blinis\n\nPuglia Broad Bean Dip\n\nOyster Fritters with Rocket and Wasabi Mayonnaise\n\nChaat Masala Watermelon with Mint\n\nSMOOTH AND SLIPPERY\n\nOysters with Chilli, Ginger and Lime\n\nSalmon Tikka Skewers with Dill and Pomegranate Raita\n\nMilton's Moules\n\nLobster with B\u00e9arnaise Sauce, Home-made Oven Chips and Broccoli\n\nScallops with Brown Butter, Capers and Lemon\n\nFLESH\n\nVenison and Liver Pies\n\nBeef Carpaccio with Parmesan, Rocket and Truffle oil\n\nYakitori-style Chicken Heart Skewers\n\nSteak Tartare with Confit Egg Yolk and A\u00efoli\n\nHerb-crusted French Rack of Venison\n\nBeef Fillet, Ceps, Marsala Sauce and Roasted Shallots\n\nTHE SPICE OF LIFE\n\nButternut Soup with Nutmeg and Toasted Pumpkin Seed Oil\n\nCrispy Sea Bass with Ginger and Spring Onions\n\nPaneer Tikka Masala\n\nSmoked Garlic Dauphinoise\n\nSalt and Pepper Baby Squid with Garlic Mayonnaise\n\nSaffron Roast Chicken with Lemon and Rosemary\n\nSHAPELY VEG\n\nCrunchy Fennel and Green Apple Salad\n\nAsparagus with Crab, Poached Eggs and Lemon Vinaigrette\n\nAvocado, Pancetta and Toasted Pumpkin Seed Salad\n\nPommes de Terres Sarladaises with Truffles\n\nRed Mullet, Artichoke Hearts and Lemon\n\nGETTING FRUITY\n\nWatermelon Salsa with Black Pepper Goat's Cheese\n\nGrilled Honey Figs, Goat's Cheese and Walnut Salad\n\nPork and Lychee Curry\n\nSalt Caramel and Rum Banana Cake\n\nPoached Quince with Mascarpone and Stilton Cream\n\nSWEET BITS\n\nCranachan with Figs, Candied Walnuts, Honey and Meringue\n\nChocolate Chilli Fondant\n\nRed Wine Poached Pears with Stem Ginger Cr\u00e8me Fra\u00eeche\n\nVanilla and Saffron Baguette and Butter Pudding with Gooseberry-Ginger Jam\n\nRaspberry and Rose Pudding\n\nAll recipes are for two people, unless otherwise stated. All recipes call for sea salt and freshly ground black pepper, to taste, unless otherwise stated.\n\n## RECIPE LIST\n\nFROZEN WATERMELON MARGARITA\n\nBASIL MARTINI\n\nALMOND MILK WHITE RUSSIAN\n\n[BLOODY MARY  \nWITH CLAM JUICE](chapter004.xhtml#list4)\n\nSTRAWBERRY BELLINI\n\nGiven that I have previously written The Hungover Cookbook and The Drunken Cookbook, and that I was once Beer Writer of the Year, it probably comes as no surprise that drink should figure prominently in this book, too. Other than complete incompatibility, inhibition is probably the biggest single barrier to us getting it on with one another. Let's face it: we were born to fornicate as much as we were to do anything else, yet we have such disabling neuroses about it. Alcohol can help to pull the barriers to one side and make everything seem so much more... simple. I suppose it is alcohol's ability to do this that makes it such a problem for certain religions, which feel it leads to licentiousness and depravity.\n\nI am not necessarily advocating debauchery, though William Blake's words about the road of excess leading to the palace of wisdom have long been a personal motto of mine. Alcohol, as we are constantly reminded, should be used responsibly: it is, after all, a powerful drug. Plying someone with booze to get them drunk and compliant is not what I am suggesting. But I do think alcohol has a significant place in creating an atmosphere in which feelings can be expressed in a more honest way, and in which many of the potentially negative aspects of sobriety \u2013 such as guardedness, prudishness and the need to feel always in control \u2013 can be fleetingly banished.\n\nYet both men and women \u2013 but men in particular \u2013 should be wary that too much drink may, to paraphrase Shakespeare, provoke desire, but take away performance. The porter in Macbeth tells us that,\n\n_'... much drink may be said to be an equivocator with lechery. It makes him, and it mars him; it sets him on, and it takes him off; it persuades him, and disheartens him; makes him stand to and not stand to; in conclusion, equivocates him in a sleep, and, giving him the lie, leaves him.'_\n\nHere are five alcoholic drinks to get the juices flowing. In a sense. Just don't drink too much of them or you, too, could end up desiring to do something but failing to live up to your own expectations, let alone those of your partner.\n\n## FROZEN WATERMELON MARGARITA\n\njuice of 1 lime, plus 1 more lime for the glass rims ~ finely crushed sea salt, for the glass rims ~ 300g watermelon, deseeded and chopped into 2.5cm dice, frozen, plus 2 wedges of watermelon to serve ~ 100ml good-quality aged tequila ~ 25ml Cointreau ~ 2 handfuls of ice\n\nWatermelon appears three times in recipes in this book and not just because it is the colour of flushed skin, roses and lipstick. It contains citrulline, which is sometimes prescribed as a supplement for erectile dysfunction. It has the effect of improving circulation by dilating veins \u2013 vasodilation \u2013 via the production of nitric oxide. In one study, citrulline 'increased hardness by 50 per cent'. Now I doubt very much that the small amount of watermelon needed to make this delicious drink contains anything amounting to a therapeutic dose, but it can certainly do no harm.\n\nAnd, regardless of the impact of the watermelon, I've always found tequila (in modest quantities) to be effective in promoting concupiscence. The chilled quality of this drink adds an extra frisson of excitement... it should be served very cold. You will need martini glasses, or tall narrow tumblers.\n\nMilton's Method   Rub lime around the rims of two martini or Tom Collins glasses, then roll the outer part of the rim in the crushed salt (you don't actually want salt in the drink).\n\nPut all the ingredients for the margarita into a blender (not forgetting the lime juice) and blend thoroughly.\n\nPour into the glasses and serve immediately with a watermelon wedge on each.\n\n## BASIL MARTINI\n\nlarge handful of ice cubes (optional) ~ 8 basil leaves, plus more to serve ~ 4 mint leaves ~ 30ml sugar syrup (see recipe introduction) ~ 120ml gin ~ 30ml dry vermouth\n\nBasil has an ancient reputation for perking up flagging libidos. Pairing it with the martini \u2013 one of the more potent cocktails known to man (and woman) \u2013 seems like the perfect way to begin an evening of high-octane romance. For an insight into the possible consequences of this approach, Dorothy Parker wrote that:\n\n'I like to have a martini,\n\nTwo at the very most.\n\nAfter three I'm under the table,\n\nafter four I'm under my host.'\n\nKnowing when to stop is a key skill when drinking martinis. So be warned.\n\nThere are more ways to make a martini than there are ways to shake a (cocktail) stick, so feel free to amend this basic recipe. I have used a little sugar syrup to bring out the flavour of the basil (made by mixing two parts of water to one of caster sugar, then dissolving the sugar in a saucepan over a medium heat). You will need cocktail-making equipment \u2013 a shaker, a strainer and a muddle stick \u2013 as well as two martini glasses.\n\nMilton's Method   Chill two martini glasses by either placing them in a freezer for 10 minutes or filling them with ice cubes. The glasses should be as cold as possible.\n\nMuddle the basil, mint and sugar syrup thoroughly in a cocktail shaker with the muddle stick, then add the rest of the ingredients and stir for 30 seconds.\n\nEmpty the ice out of the glasses (if you used that method to chill them) and strain the martini into them. Decorate each with basil leaves and serve immediately.\n\n## ALMOND MILK WHITE RUSSIAN\n\n90ml vodka ~ 90ml Kahlua ~ 60ml almond milk ~ large handful of ice\n\nAlmonds have been a well-known aphrodisiac throughout history. Samson wooed Delilah with the branches of almond trees. And the ovoid shape of the almond resembles both the egg and a rather sensitive area of male anatomy.\n\nThere is also some science that explains how almonds help to improve libido: they are packed with various minerals, vitamins and healthy fats. Men who eat monosaturated fats of the type found in almonds have been found to have the highest levels of testosterone. In addition, the nuts contain zinc and the amino acid arginine, both of which play an important role in sexual function.\n\nThe smooth, silky texture and rounded flavour of almond milk makes it ideal in this enormously simple and delicious cocktail, with the added bonus that it can be rustled up in a matter of seconds. You will need a cocktail shaker.\n\nMilton's Method   Mix all the ingredients in a cocktail shaker with ice. Shake thoroughly for around 30 seconds, then serve in two Old Fashioned glasses (short tumblers).\n\n## BLOODY MARY  \nWITH CLAM JUICE\n\n500ml Clamato (tomato and clam juice) ~ 100ml vodka ~ 1 tsp lemon juice \u00bd tbsp Worcestershire sauce ~ splash of Tabasco sauce, or to taste ~ 1 tbsp tomato ketchup ~ pinch of hot smoked paprika ~ celery salt, or sea salt if you don't have it ~ freshly ground black pepper ~ ice cubes\n\nTomatoes, chillies, paprika and clam juice all stake a claim to be aphrodisiacs and, as they all make an appearance here, this can rightfully be seen as a lover's super-drink.\n\nNot everyone may appreciate clam juice in their drink. It's safe to say that it splits opinion. But by incorporating it, the cocktail is even richer in libidinous nutrients, as they offer various amino acids that have been shown to increase sex hormone production in rats. It seems that the same study has not yet been conducted on humans, but the reputation of bivalves such as oysters, mussels and clams seems to indicate that there might be something in it.\n\nThis is my favourite bloody Mary recipe that I have used for many years, to great success. Please feel free to tweak it to your preference.\n\nMilton's Method  Mix all the ingredients very well to make sure that everything is combined, especially the ketchup and paprika. Adjust the seasoning to taste and serve over ice in tall glasses.\n\n## STRAWBERRY BELLINI\n\nhandful of strawberries ~ 2 tsp caster sugar ~ 1 tsp lemon juice ~ 1 mint leaf ~ 1 bottle of champagne\n\nI would recommend champagne on its own as an aphrodisiac, but what kind of recipe would that be? Despite it being on the late polemicist Christopher Hitchens's list of the most overrated things in life (along with lobster, picnics and anal sex), I think there is nothing better than drinking champagne with those you love... or for whom you may yet form a passion. People may say they prefer prosecco, cava, or even English sparkling wine these days, but there is still nothing like the real thing.\n\nOpening a bottle of champagne in such a way that the foam pours out is, when you think about it, an act with plenty of erotic symbolism. The convention is to mute the pop and it is true that this helps to avoid spillages and maintains the atmosphere in those restaurants where 'muted' is seen as a good thing. But the child in me prefers the sense of theatre of a good loud bang.\n\nStrawberries add gentle sweetness, a romantic pink hue and a suggestive shape.\n\nMilton's Method   Begin by macerating the strawberries: put them in a small bowl with the sugar and lemon juice and leave at room temperature for around 20 minutes. The berries will release some of their juice and the sugar will be absorbed into the lemon\/strawberry juice mixture.\n\nPour the strawberries and juice into a blender with the mint leaf. Quickly pulse-blend, then pass through a fine sieve, pushing it with the back of a spoon into the bowl to remove the seeds. Chill until you are ready to drink.\n\nPlace 1 tablespoon of the strawberry pur\u00e9e in each of two champagne flutes and carefully top up with champagne. Serve immediately.\n\n## RECIPE LIST\n\nSPICED HONEYED ALMONDS\n\nCAVIAR BLINIS\n\nPUGLIA BROAD BEAN DIP\n\n[OYSTER FRITTERS  \nWITH ROCKET AND WASABI MAYONNAISE](chapter005.xhtml#list9)\n\n[CHAAT MASALA WATERMELON  \nWITH MINT](chapter005.xhtml#list10)\n\nJust as sex usually begins with some form of foreplay, your erotic culinary journey should also start with small, gentle steps that serve to excite the appetite, rather than sating it. This comes in the form of small bites that aim to tease and arouse palates and minds. The very beginning is one of my favourite parts of any evening, because of the anticipation of delights that are yet to be served up. It is an opportunity for lightness and fun, before the serious business gets under way. There is something to be said, sometimes, for diving straight in, disregarding all niceties and tucking straight into the main course before even a drink has been poured, but playfulness and build-up should not be under-estimated. If all the main action is over too soon, what is there left to do?\n\nThe ability of ingredients to have an aphrodisiac effect is contentious, and there is certainly much more to creating an erotic dining experience than merely using the right foods. It is about conjuring up an atmosphere for romance and, above all, showing that you are an attentive lover. 'Genius,' according to Thomas Carlyle, 'is an infinite capacity for taking pains.' My plan is to turn you into a genius lover... when it comes to making dinner. With these first steps you will hopefully begin not only to arouse your partner with food, but also demonstrate that you know how to press more than just the most obvious buttons.\n\nHere are five nibbles to begin your erotic culinary experience.\n\n## SPICED HONEYED ALMONDS\n\n\u00be tsp cayenne pepper ~ \u00bd tsp smoked paprika ~ 1 tbsp olive oil ~ 400g raw almonds ~ 75ml good-quality honey ~ 30g raw sugar ~ \u00be tsp sea salt\n\nThis simple and delicious pre-dinner snack combines almonds with honey and spices, all of which have some aphrodisiac qualities. Sweet, salty, spicy and crispy, this little dish pleases the senses in lots of different ways and is a great accompaniment to cocktails.\n\nMilton's Method  Preheat the oven to 180\u00b0C\/gas mark 4. Place the spices, olive oil and almonds in a large bowl and mix thoroughly.\n\nGently heat the honey in a heavy-based non-stick pan until it is thin and runny. Add the honey to the nut mixture and mix again so all the nuts are well coated.\n\nLine a baking tray with greaseproof paper and empty the nuts out on to the tray, spreading them out so they form a single layer. Place in the oven for 10 minutes.\n\nRemove the almonds from the oven and allow to cool for 5 minutes, then place in a serving bowl, sprinkle with the sugar and salt, mix once more and serve.\n\n## CAVIAR BLINIS\n\n**For the blinis:** _40g strong white bread flour ~ 60g wholemeal buckwheat flour ~ \u00bd tsp fine salt ~ 1 tsp caraway seeds ~ 100ml milk ~ 1 tsp dried 'quick' or 'fast action' yeast ~ 1 large free-range egg, lightly beaten ~ 70g sour cream ~ unsalted butter, to fry_ **For the topping:** 230g sour cream ~ handful of chives ~ caviar\/roe of your choice\n\nCaviar is a prestige food and its association with that most effective of aphrodisiacs \u2013 money \u2013 is often, on its own, enough to produce the required response. Scarcity is not its only appeal: caviar's texture and flavour are sublime, and it is also incredibly rich in nutrients. 'Wit ought to be a glorious treat like caviar,' said Noel Coward; 'never spread it about like marmalade.'\n\nThere is no greater fertility symbol than the egg and, in each mouthful of caviar, you may consume a hundred or more of them. Assuming that your budget may not quite stretch to the real thing, or at least not on a regular basis, there are alternatives at a more realistic price, which still contain a similar amount of key aphrodisiac components such as selenium, iron and magnesium as well as the amino acids arginine and histidine. So consider pink salmon roe, or black lumpfish caviar. They are less delicate and refined than the real deal, it is true, but also much cheaper.\n\nOligarchs and royalty may consume caviar by the mother-of-pearl spoonful, but a blini \u2013 a Russian buckwheat pancake \u2013 is a perfectly adequate receptacle for the rest of us. And wheeling out a dozen blinis topped with sour cream and fish eggs always creates a sense of occasion. Feel free to buy your blinis from the shop; however, if you want to impress your guest, try making them from scratch.\n\nMilton's Method   For the blinis, tip both types of flour, the salt and caraway seeds into a mixing bowl. Place the milk in a small pan and heat it gently until it begins to bubble, then immediately take it off the heat and stir in the yeast with a balloon whisk, followed by the egg and sour cream, making sure all the ingredients are well incorporated. Slowly pour the milk mixture into the bowl of flour, stirring, until you have a smooth batter. Cover with cling film and leave for an hour or two. The mixture should thicken and some bubbles begin to appear.\n\nWhen you are ready to cook the blinis, heat a large, heavy-based frying pan over a medium heat and melt the butter in it. Use a teaspoon to gently place small circles of batter in the pan, being careful not to crowd them. Cook for around 2 minutes until golden, then flip and cook the other sides until they, too, are golden. Allow to cool on a wire rack while you cook the rest of the batter.\n\nDress each blini with sour cream, chives and caviar and serve on a platter.\n\n## PUGLIA BROAD BEAN DIP\n\n350g dried split broad (fava) beans ~ 2 large old potatoes, peeled and chopped into large cubes ~ sea salt and freshly ground black pepper ~ 2 garlic cloves, very finely chopped or crushed ~ 125ml extra virgin olive oil ~ juice of \u00bd lemon\n\nBroad beans, called fava beans in many places in the world, are perhaps among the more unlikely \u2013 if not outright unsexy \u2013 of aphrodisiacs. Well known across the Mediterranean countries as a peasant food, they historically provided a cheap source of protein. However, they also contain fortifying iron and a chemical called levodopa, which helps to produce dopamine in the brain, a hormone that drives us to seek out pleasure. Whether or not the beans contain enough of this to make a substantial impact is debatable, but its reputation suggests that it does. And, as a dip, this dish can satisfy any Freudian oral-stage fixations that you may have, rehearsing the same compulsive behaviours that you may wish to repeat a little later in the evening.\n\nIn the heel of Italy's boot, in the province of Puglia, broad beans have traditionally been turned into a version of this pur\u00e9e, commonly served as part of a spread of antipasti, or as a dish in its own right. You could use fresh broad beans to produce a green pur\u00e9e, but I have stuck with the classic Puglian version.\n\nMilton's Method  Tip the broad beans and potatoes into a large saucepan and cover with cold water. Bring to the boil, though do not stir, and simmer over a medium heat for around 10 minutes. Now drain and refill the pan with clean water and 1 teaspoon of salt. Return to the boil and simmer for a further 25\u201335 minutes, until the beans have broken down in texture. Drain and, using a hand-held blender, blitz them to a smooth consistency.\n\nIn a separate pan, gently saut\u00e9 the garlic in 1 tablespoon of the olive oil and, when it softens, add it to the bean mix. Over a very low heat, gradually drizzle in the rest of the olive oil, while stirring the mixture fairly vigorously in order to prevent it from splitting. When all the oil has been incorporated, add the lemon juice and pepper and adjust the seasoning to taste.\n\nPlace in a bowl and allow to cool before serving with carrot batons and breadsticks.\n\n## OYSTER FRITTERS  \nWITH ROCKET AND WASABI MAYONNAISE\n\n**For the mayonnaise:** _1 tsp wasabi paste ~ 2 medium free-range egg yolks, at room temperature ~ pinch of sea salt ~ 250ml sunflower oil ~ 25ml extra virgin olive oil ~ 1 tbsp lemon juice ~ handful of rocket, roughly chopped_ **For the fritters:** _150g plain flour ~ pinch of fine salt ~ 1 free-range egg, lightly beaten ~ 150ml beer (ale) ~ 12 very fresh oysters ~ sunflower oil, to deep-fry_\n\nI will assume that you know about oysters being among the most potent of libido-boosters, and I examine their reputation in more detail on here. Perhaps less well known is that both rocket and wasabi have also been ascribed aphrodisiac properties.\n\nWhereas lettuce has since ancient times been known to cause lethargy, rocket is its counterbalance. In a poem ascribed to Virgil, the poet declares that 'rocket excites the sexual desire of drowsy people'. This is perhaps not surprising, given rocket's pungent, peppery flavour that stimulates the taste buds and, by extension, other bits of the body.\n\nThe same basic principle applies to wasabi, a Japanese plant which tastes somewhere between horseradish and hot mustard. Unlike chilli, it targets the sinuses more than the lips and tongue. And also unlike chilli, wasabi's heat quickly diminishes and can be washed away with water. But by then it has already increased the heart rate and dilated the blood vessels enough to achieve a potentially libidinous effect.\n\nMilton's Method   Begin by preparing the mayonnaise. In a large mixing bowl, whisk the wasabi and egg yolks together with the salt until everything is well-blended and the salt has dissolved. Add the sunflower oil in a very thin continuous stream, whisking the whole time. Be careful at this stage: the mixture needs to emulsify and thicken as you go. As the mixture gets thicker, you can increase the rate at which you pour in the oil until you have the amount of mayonnaise you need, finishing with the extra virgin oil, for a bit of flavour. Continue whisking for a further minute to get a stiff consistency and glossy sheen. Then add the lemon juice, which will thin the mayonnaise slightly, and whisk it in gradually. Finally, fold in the rocket with a spoon.\n\nNow for the fritters. Combine 100g of the flour and the salt in a large bowl. In another bowl, mix the egg and the beer. Gradually add the wet ingredients to the dry, whisking until you have a smooth batter. Cover and place in the fridge for 20 minutes.\n\nCarefully shuck the oysters (see here), discarding any with open shells. In a wok or deep-fat fryer, heat sunflower oil to about 180\u00b0C, if you have a thermometer (if you don't, a scrap of the batter should immediately sizzle when thrown into the oil). When you are ready to fry, roll the oysters in the remaining 50g of flour, then dip them in the batter and very carefully place them into the hot oil using a slotted spoon.\n\nCook for 2\u00bd-3 minutes, checking carefully to see that they do not overcook. When they are golden brown, remove them from the oil with the slotted spoon and place on kitchen paper. Dab off the excess oil and serve immediately with the mayonnaise.\n\n## CHAAT MASALA WATERMELON  \nWITH MINT\n\nhandful of mint leaves, finely chopped ~ 1 tbsp chaat masala ~ 2 tsp caster sugar ~ \u00bd small watermelon ~ juice of 1 lime ~ cocktail sticks, to serve\n\nWith a watermelon you will inevitably, at some stage or another, slurp. Juices will run down your chin. You will need to wipe your mouth with the back of your hand. It is, in these respects, a sensual fruit. Yet it has further surprises up its sleeve. For a food that contains such an overwhelming percentage of water, its remaining active ingredients are packed full of libido-boosting goodies. In particular, it contains citrulline, an amino acid that plays an important role in promoting sexual arousal.\n\nTo this recipe I have also added a little spice \u2013 in the form of a tangy powder called chaat masala \u2013 and mint, both of which pair nicely with watermelon and make this pre-dinner tit-bit an exotic, tantalising treat.\n\nMilton's Method  Combine the mint, chaat masala and sugar in a bowl.\n\nCut the watermelon into 2cm-thick wedges, then peel off the skin and cut the wedges into large tapered chunks, each about 3cm long.\n\nSqueeze lime juice over the watermelon chunks, then cover one of the tapered sides of each piece with the chaat masala mixture. Spear with a cocktail stick, arrange on a plate and chill in the fridge before serving.\n\n## RECIPE LIST\n\n[OYSTERS  \nWITH CHILLI, GINGER AND LIME](chapter006.xhtml#list11)\n\n[SALMON TIKKA SKEWERS  \nWITH DILL AND POMEGRANATE RAITA](chapter006.xhtml#list12)\n\nMILTON'S MOULES\n\n[LOBSTER  \nWITH B\u00c9ARNAISE SAUCE, HOME-MADE OVEN CHIPS AND BROCCOLI](chapter006.xhtml#list14)\n\n[SCALLOPS  \nWITH BROWN BUTTER, CAPERS AND LEMON](chapter006.xhtml#list15)\n\nCasanova was a great lover of oysters. To my mind that sentence doesn't prove anything and contains two great implied and unsubstantiated clich\u00e9s: first, that Casanova is the greatest lover that has ever lived; second, that oysters constitute the ultimate aphrodisiac. Casanova did, however, have a nice suggestion for how to eat the briny bivalve: 'I placed the shell on the edge of her lips and after a good deal of laughing, she sucked in the oyster, which she held between her lips. I instantly recovered it by placing my lips on hers.' And various forms of seafood do seem to have a deserved reputation for making us frisky. So perhaps the clich\u00e9s are well founded after all.\n\nOysters are not the only type of seafood to be addressed in this chapter, but they are the most famous of all aphrodisiacs. They have in common with other shellfish a passing, though perhaps not entirely complimentary, resemblance to female genitalia and it is the textural qualities of seafood \u2013 the smooth and slippery title of this chapter \u2013 that make them an aphrodisiac as much as any nutritional qualities. According to one report, the late art critic Brian Sewell was at pains to suggest that oysters were not as feminine as many people suggest. When a colleague told him that she didn't like oysters, he replied, 'Oh. Well, I daresay you ought not to try fellatio.' However, I should probably also point out that Sewell had something of a reputation as a misogynist.\n\nThe sensuous nature of seafood and the tactile way in which we eat it, the high levels of zinc and amino acids contained within it, and its cultural legacy as pre-pharmaceutical Viagra, means these recipes are the closest you will get to sure-fire aphrodisiacs.\n\n## OYSTERS  \nWITH CHILLI, GINGER AND LIME\n\n12 very fresh oysters ~ bag of ice ~ small piece of root ginger, peeled and finely grated ~ 1 medium-hot red chilli, deseeded and very finely chopped ~ juice of \u00bd lime ~ 1 kaffir lime leaf, very finely chopped ~ 1 tsp brown sugar ~ 3 tbsp rice wine vinegar\n\nPurists will no doubt insist that at most a little lemon juice, shallot vinegar and \u2013 at a stretch \u2013 a couple of drops of Tabasco sauce are the most that a good oyster will ever need. As variety is the spice of life, I wanted to present you with this eastern take on serving oysters.\n\nMuch has been made of the aphrodisiac properties of the oyster. The fact that they are very high in zinc is often cited as the primary reason. But I think it is the uniquely tactile way we eat oysters that puts us in mind of affairs of the heart (and other parts of the body). With the oyster we literally kiss the sea, then swallow it.\n\nMilton's Method  Wash and carefully shuck the oysters. Discard any that are open, or partially open. Hold the oyster on a flat surface using a tea towel, curved side down, and insert a short knife with a short blade into the crack at the base of the hinge. Use a combination of twisting and levering to prise the top shell off, then slice gently along the underside of the top shell to remove the oyster. Place carefully on a platter of crushed ice.\n\nCombine all the remaining ingredients to make a dressing and sprinkle over the oysters. Serve within 30 minutes.\n\n## SALMON TIKKA SKEWERS  \nWITH DILL AND POMEGRANATE RAITA\n\n**For the salmon:** _1 tbsp tikka spice paste ~ 4 tbsp full-fat Greek-style yogurt ~ squeeze of lime juice, plus lime wedges to serve ~ 200g salmon fillet ~ a little sunflower oil_ **For the raita:** \u00bd pomegranate ~ 4 tbsp full-fat Greek-style yogurt ~ 1 tbsp roughly chopped dill fronds, plus more to serve ~ sea salt ~ \u00bc tsp cayenne pepper\n\nThis makes a quick and simple starter, packed with omega-3 fatty acids that are said to improve libido. Add spices and pretty jewels of pomegranate to colour the raita pink... and the result is an enticingly sexy beginning to your dinner.\n\nMilton's Method   Mix the spice paste, yogurt and lime juice in a bowl until well blended. Cut the salmon fillet into 2.5cm cubes, add to the bowl and gently stir into the mixture. Set aside.\n\nTake the seeds out of the half pomegranate. If it is a good, juicy fruit, you will also get at least 1 tablespoon of juice from it, too. Add almost all the seeds (reserve a few) and 1 tablespoon of juice to a bowl with the yogurt and add the dill and two pinches of salt. Sprinkle the cayenne pepper over the top.\n\nPreheat your grill to maximum. Taking great care, thread the salmon pieces on to skewers. Lay some foil over a grill pan and brush it with oil where the skewers will sit. Place the skewers on the foil, brush with oil and put them under the grill. Wait until they colour and char a little at the edges, then extremely carefully (the salmon gets flakier as it cooks) rotate the skewers, brushing with oil once more, so the fish gets cooked all the way through; you will probably only need to do this once.\n\nRemove the skewers and slide the salmon pieces on to warmed plates. Spoon a good portion of raita next to them and dress it with the reserved pomegranate seeds and a little dill. Squeeze a little lime juice over the salmon, add a pinch of salt and serve immediately with lime wedges.\n\n## MILTON'S MOULES\n\n1.5kg very fresh mussels ~ large knob of unsalted butter ~ 6 rashers of smoked streaky bacon, finely chopped ~ 2 small shallots, finely sliced ~ 250ml dry cider ~ 75ml double cream ~ 1 tbsp chopped tarragon leaves ~ freshly ground black pepper ~ crusty bread, to serve\n\nThere are many tasty ways to prepare mussels, but my favourite is to use smoky bacon, cider, cream and tarragon to make a luxuriant sauce that works perfectly with seafood. I think that a dry cider is also great to drink with many types of seafood.\n\nMake sure you thoroughly check and clean all your mussels; a success the night before won't mean much if there's failure hours after. A little care will ensure that only good things will come of your bowl of moules.\n\nMilton's Method   Check and clean your mussels. Discard any that are open and refuse to close when tapped sharply on the edge of the sink (they are probably dead) or too heavy (they are probably full of sand). Remove the beards and scrub the shells so they look beautiful enough to serve to someone you love.\n\nSet a large saucepan or casserole dish over a medium heat. Add the butter, let it bubble a little, then add the bacon. When the bacon has coloured a little, add the shallots, cook for 2 minutes, stirring occasionally, then pour in the cider. Let it come to the boil, then throw in the mussels and put a lid over them.\n\nThe mussels only need 2 or 3 minutes, and you should shake the pan well halfway through. Keep an eye on them and, when they have opened, remove them immediately into warmed bowls with a slotted spoon. Discard any that stubbornly refuse to open. Reduce the heat under the pan to low and add the cream, tarragon and a little black pepper. Stir well, let the sauce cook for a minute or two, then pour over the mussels and serve straight away with crusty bread and a glass of cider or dry white wine.\n\n## LOBSTER  \nWITH B\u00c9ARNAISE SAUCE, HOME-MADE OVEN CHIPS AND BROCCOLI\n\n**For the lobster, chips and broccoli:** _3 large Maris Piper potatoes ~ sea salt and freshly ground black pepper ~ 2 tbsp olive oil ~ 2 sprigs of rosemary ~ 2 whole live lobsters ~ 100g tenderstem broccoli_ **For the b\u00e9arnaise sauce:** _150g unsalted butter ~ 2 tbsp white wine vinegar ~ 2 echalion (banana) shallots, chopped ~ 1 tbsp chopped tarragon leaves, plus 1 tbsp whole leaves ~ 2 large free-range egg yolks ~ 2 tsp lemon juice_\n\nDespite having recently become less expensive and more commonly served in restaurants, lobster remains one of life's great culinary indulgences. Full of zinc, B-vitamins and amino acids, it is also one of Aphrodite's envoys from the sea, come (unwittingly, it is true) on to land to promote love and raunchiness between humans. As David Foster Wallace urged, I ask that you both respect and 'consider the lobster' and deal with them in the most humane way possible. Rather than giving a tutorial on how to do this within these pages, I suggest you check out one of the numerous online video tutorials for the best method.\n\nThe b\u00e9arnaise sauce in this recipe is a little bit of a shag to make, but I am assuming that you are going all out to impress your fellow diner and will go the extra mile. And a half. Actually it's not that bad and it's well worth it: the buttery tarragon flavour works brilliantly with lobster.\n\nAs for the oven chips? Well, how many people do you know with a deep-fat fryer in their kitchen? This method is easier, less messy and healthier... and produces pretty fine chips.\n\nMilton's Method   Sort out the chips first. Preheat the oven to 220\u00b0C\/gas mark 7 and put a baking tray inside.\n\nPeel the potatoes and cut them, lengthways, into 1cm-thick chips. Place in a saucepan with cold water, give the pan a good shake, stir the potatoes with your hands, then drain the water. Refill with cold water, add 1 tsp of salt, clap on the lid and place over a medium-high heat. When the water comes to the boil, leave for 2 minutes, then drain and wrap briefly in a clean tea towel to dry them out.\n\nPut the blanched chips in a bowl with 1 tablespoon of the olive oil and a generous pinch of salt, then take the hot baking tray out of the oven and place on the side. Empty the chips out onto the hot baking tray so they form a single layer. Splash the remaining tablespoon of oil over them, throw on the rosemary and place in the oven for 35 minutes, until crispy and golden, shaking once or twice.\n\nTo make the b\u00e9arnaise sauce, start by clarifying the butter: place a heavy-based saucepan over a low heat and put the butter in it. Once all the butter has melted and created a foam, turn the heat off and leave for a couple of minutes. Skim the foam off the top with a teaspoon (discard it), then pass the remaining butter through a fine sieve into a bowl, leaving the milky solids at the base of the pan behind (discard them, too). Set aside.\n\nAdd the vinegar, shallots, chopped tarragon and a pinch of salt to a small pan. Set over a medium heat and reduce the volume of the liquid by half. Strain and set aside to cool.\n\nGently beat the egg yolks with 1 teaspoon of water and add to the cooled vinegar with 1 teaspoon of the lemon juice. Add this mixture to a heatproof bowl suspended over a pan of simmering water (make sure the bowl does not touch the water). Whisk the mixture vigorously for 2-3 minutes until it thickens enough to coat the back of a spoon. Remove from the heat and gradually pour in the clarified butter, still whisking all the time so that the mixture emulsifies. Keep whisking once all the butter has gone in, until glossy and well mixed. Add the whole tarragon leaves, the remaining 1 teaspoon of lemon juice and some black pepper and pour into a small jug.\n\nMeanwhile, bring a very large pan of water to the boil, then reduce the heat to a steady simmer. Humanely dispatch the lobsters as you see fit, then separate the claws and the tail meat and poach in the water: the tail needs 4-6 minutes, the claws 6-8 minutes. Scoop out of the water with a slotted spoon and dunk immediately into cold water, to arrest the cooking. Drain well. Take the flesh from the tail and crack the claws.\n\nSteam the broccoli for 3 minutes, drain, then place everything on warmed plates, pouring some of the b\u00e9arnaise over the lobster tails and serving the remaining sauce in individual ramekins. Season the broccoli and chips with sea salt and serve immediately.\n\n## SCALLOPS  \nWITH BROWN BUTTER, CAPERS AND LEMON\n\n50g unsalted butter ~ 6 very fresh scallops ~ 1 tbsp rapeseed oil ~ 150ml dry white wine ~ 1 tbsp capers, drained ~ finely grated zest and juice of \u00bd unwaxed lemon ~ sea salt and freshly ground black pepper ~ French bread, to serve\n\nIn legend, Aphrodite, goddess of love, emerged from a scallop shell... and the sweet flesh of these treats from the sea is a sensual delight. The smooth and slippery texture of seafood that I allude to in the title of this chapter is uniquely suggestive. Choose the freshest, hand-dived scallops; here, their sweetness pairs nicely with sharp, salty capers, nutty butter and a little lemon juice. The secret is all in the cooking: overcooked scallops are rubbery; undercooked scallops are unappealing; a perfectly seared scallop is sublime.\n\nMilton's Method   First make the brown butter by melting it in a heavy-based saucepan over a medium-low heat; ideally, use a pan with a white or stainless-steel base, so you can see the change in colour. It should foam, then gradually turn brown. Keep a close eye on it, as you don't want it to burn (if it does, you'll have to start again). When it is nut-brown, remove it immediately from the heat and pour into a bowl so it stops cooking.\n\nPrepare the scallops by shucking them from the shells, discarding the frilly 'skirt', black stomach and rubbery foot. I prefer to cook the orange-pink coral with the scallop itself rather than discarding it, but this is up to you. Give the scallops a good wash under cold running water, but do not soak them, then pat dry with kitchen paper or a clean tea towel.\n\nHeat a heavy-based frying pan and add the oil. When it is hot, place the scallops gently in the pan. Cook for two minutes on one side, then turn and cook for around a minute on the other side, until each scallop has a slight spring to it when pressed, but is not too firm.\n\nRemove the scallops from the pan, add the white wine, stir and reduce the liquid by half, then add the capers, lemon zest and juice, brown butter and a little salt and pepper. Stir well and remove from the heat immediately. Plate the scallops on warm (but not hot) plates and drizzle the sauce over the top. Serve with a little French bread and glasses of very cold dry white wine.\n\n## RECIPE LIST\n\nVENISON AND LIVER PIES\n\n[BEEF CARPACCIO  \nWITH PARMESAN, ROCKET AND TRUFFLE OIL](chapter007.xhtml#list17)\n\nYAKITORI-STYLE CHICKEN HEART SKEWERS\n\n[STEAK TARTARE  \nWITH CONFIT EGG YOLK AND A\u00cfOLI](chapter007.xhtml#list19)\n\nHERB-CRUSTED FRENCH RACK OF VENISON\n\nBEEF FILLET, CEPS, MARSALA SAUCE AND ROASTED SHALLOTS\n\nAs any writer worth their salt knows, the only two subjects really worth writing about are sex and death. They form the basis of all great comedy and all true tragedy. Flesh \u2013 or meat \u2013 symbolises both sex and death: the word carnal, which relates to sexual appetite, comes from the Latin for 'flesh'. Sexual sins are 'sins of the flesh'. Meat-eaters who have sex are carnal carnivores.\n\nThere is something primal about eating a hunk of meat \u2013 red meat in particular \u2013 and it is no surprise that the act of eating it may arouse animal passions in us, too.\n\nAtavism is a powerful force: by subconsciously suggesting a time in prehistory before we were bound by the laws and institutions of modern civilisation, it liberates us and allows us to follow our more primal urges. Now, I'm not saying that eating a steak will make you act like an unreconstructed caveman or woman \u2013 and nor should it \u2013 but there is definitely something about tearing into meat that subtly reinforces the notion of our animal selves. Meat contains plenty of useful nutrition for lovers, but it is by emphasising the hunter rather than the gatherer in us that our lusts are truly awakened.\n\nHere are six meaty recipes to wrap your mouth around.\n\n## VENISON AND LIVER PIES\n\n**For the pastry:** _225g plain flour ~ fine salt ~ 100g unsalted butter, finely chopped ~ 1 free-range egg yolk, lightly beaten, plus 1 beaten free-range egg for brushing_ **For the filling:** _2 tbsp rapeseed oil ~ 350g venison, diced ~ 6 rashers of smoked streaky bacon ~ 1 onion, finely chopped ~ 1 carrot, finely chopped ~ 1 bay leaf ~ 1 tbsp plain flour ~ 250ml red wine ~ 350ml beef stock ~ sea salt and freshly ground black pepper ~ 1 sprig of rosemary ~ 1 tsp redcurrant jelly ~ 200g venison or calf's liver ~ a little milk_\n\nA little pie on a plate is a lovely thing. My original plan for this recipe was for venison liver on its own, but, after some reflection, I realised that the only person I know who gets a perverse kick from eating slabs of liver is Hannibal Lecter. It is hardly the stuff \u2013 by itself, at any rate \u2013 of a romantic dinner _\u00e0 deux_. However, liver is a wonderfully nutritious food with iron-rich properties offering fortification for lovers, if fortification be needed. So I decided to work venison liver into these lovely little shortcrust venison pies that make a perfect romantic treatment for a cold winter's day in front of a roaring log fire. If you cannot get hold of venison liver, use calf's liver instead.\n\nYou will need two little pie dishes, each about 12cm across. These are big enough to have a plate to themselves; serve the vegetables on a separate plate.\n\nMilton's Method   Make the pastry by sifting the flour into a large bowl with a pinch of salt. Work in the butter with your fingertips until you have a breadcrumb consistency. Add the egg yolk and 3 tablespoons of water and stir in using a knife. You should now be able to make a ball of dough. If the mixture is still too dry, add a little more water until this is possible. Wrap in cling film and chill in the fridge for at least 20 minutes.\n\nNow for the filling. Pour half the oil in a heavy casserole dish, place over a high heat and seal the venison on all sides. Remove with a slotted spoon and wipe the pan clean.\n\nReturn the pan to the heat and add the remaining oil. Add the bacon, cooking until it is beginning to crisp up, then add the onion, carrot and bay leaf. Stir a few times and cook for around 5 minutes, then reduce the heat to medium, stir in the flour until well mixed, then pour in the red wine. Stir in and reduce a little, then add the stock, salt and pepper and rosemary. Bring to the boil, return the meat (but not the liver) to the pan and reduce the heat to a simmer. Leave to simmer for 80 minutes with the lid on, then add the redcurrant jelly, stir well and check the seasoning, adjusting it if necessary. Leave to cool and thicken a little, then remove the rosemary twig and the bay leaf.\n\nPreheat the oven to 210\u00b0C\/gas mark 6\u00bd.\n\nMake the pastry cases by rolling out the pastry and cutting out four discs for the pie lids and bases. Place a round of the pastry in each pie dish, tucking it snugly in with your fingers. Press the pastry firmly around the rim of the dish and trim any excess with a knife, then fill with the venison using a slotted spoon. Cut the liver and add this to the mix, topping up with a little of the cooking liquid (but reserve some to make the gravy).\n\nBrush a little beaten egg onto the pastry rim of each pie and add the pie lids, seal and crimp the edges, and brush with the rest of the egg. Cut a small hole in the centre of each lid and bake for around 25 minutes, or until golden.\n\nStrain the remaining venison cooking liquid into a small saucepan over a medium heat for a few minutes, stirring occasionally, allow the gravy to reduce and thicken a little, then pour into a jug.\n\nServe the pies with the gravy, creamy mash and red cabbage.\n\n## BEEF CARPACCIO  \nWITH PARMESAN, ROCKET AND TRUFFLE OIL\n\n**For the carpaccio:** _1 tsp black peppercorns ~ 1 tsp fennel seeds ~ 200g aged fillet steak ~ knob of unsalted butter ~ slug of flavourless oil ~ 2 large handfuls of rocket ~ 50g Parmesan cheese_ **For the balsamic dressing:** _1 tbsp aged balsamic vinegar ~ 4 tbsp extra virgin olive oil ~ sea salt_ **For the truffle oil dressing:** _1 tsp lemon juice ~ 4 tsp truffle oil_\n\nBeef is seldom a delicate commodity, but this dish combines finesse with simplicity to create a harmonious whole. Carpaccio is very finely sliced raw meat and can be made using any number of different meats or fish, but the most well-known form uses beef. The home cook has to rely on a different set of tricks and skills than the professional chef. In this case, it is essential to source the best ingredients you possibly can. There is no point in using second-rate beef. It has to be the finest, aged fillet steak that will melt in your mouth.\n\nAnd having said that this is a simple dish, achieving a perfect fine carpaccio slice is slightly tricky. You will need a very sharp knife but, rather than trying to slice the beef too finely and ending up with tatters, I recommend making sure you cut it thickly enough to get a complete slice, then flatten it between sheets of cling film to achieve the fineness you require. I like to sear the outside of the fillet with a peppercorn and fennel seed crust to give it a bit of texture. This is a perfect starter.\n\nMilton's Method   In a mortar and pestle, grind the peppercorns and fennel seeds a little until you have a coarse, nubbly texture. Using your hands, pick up the steak and rub the outer edges (but not the faces) with the fennel mix; it will stick well. Add the butter and oil to a hot non-stick pan and, still holding the fillet in your hand, sear the outer edges very briefly, turning, perhaps for a minute in total. Let the beef cool a little, then wrap it tightly in cling film to preserve its shape and chill in the fridge for around 20 minutes to make it easier to slice.\n\nIn the meantime, prepare the dressings. Pour the balsamic vinegar into a small bowl, then vigorously whisk in the olive oil a little at a time until the mixture is emulsified. Add a pinch of salt.\n\nIn a second bowl, combine the lemon juice, truffle oil and another pinch of salt until emulsified.\n\nIn a larger bowl, combine the rocket with the balsamic dressing. Toss well, then place on two serving plates. Shave the Parmesan cheese using a vegetable peeler and sprinkle it over the rocket.\n\nRemove the beef from the fridge and cut into roughly four slices with a sharp knife. Place each between sheets of cling film and flatten using the back of the knife and pressure from your hand.\n\nArrange the slices with the rocket on the plates and leave to come up to room temperature before serving, sprinkling the beef with the truffle dressing.\n\n## YAKITORI-STYLE CHICKEN HEART SKEWERS\n\n1 tsp finely chopped root ginger ~ 1 tsp finely chopped garlic ~ \u00bd red chilli, finely chopped ~ 1 tsp soy sauce ~ 1 tsp caster sugar ~ 3 tsp mirin (rice wine) ~ 24 chicken hearts ~ a little flavourless oil\n\nI have been studiedly coy about my own romantic affairs in this cookbook, not just because I am in this respect an Englishman of the old school, but because there is probably nothing so off-putting to good eating than hearing about other people's sex lives. However, the reason that I have included this slightly macabre recipe is that once, on a remote island near Borneo with a very special person by my side, we ate a rather similar dish, cooked over flames under the stars, and felt very much in love. When you think of the number of chickens that had to die for this to be created it may make you feel a little less romantic, but I prefer to dwell on the fact that I have done my bit to ensure nothing has gone to waste. Getting hold of chicken hearts is easy enough if you pre-order them from your butcher.\n\nMilton's Method   Prepare the marinade by bashing the ginger, garlic and chilli to a paste in a mortar and pestle. Put this in a bowl, add the soy sauce, sugar and mirin and combine well.\n\nWash the hearts and thread them on to skewers. Brush with the marinade and leave for 10 minutes to allow them to come up to room temperature. It is best to cook these on a barbecue, to get a smoky flavour, but you can also use a hot griddle pan, or place under a hot preheated grill. Regardless of the method, make sure that you brush a little oil beneath where the skewers will cook so that they don't stick. Cook, turning regularly and brushing with the marinade as you go. They will be done in about 5 minutes and should be a uniform colour throughout, perhaps a little pink in the middle but certainly not bloody. Don't overcook them, or they will be tough. And no one likes hard hearts.\n\nServe with sticky rice and salted edamame beans.\n\n## STEAK TARTARE  \nWITH CONFIT EGG YOLK AND A\u00cfOLI\n\n**For the egg yolk confit:** _100ml extra virgin olive oil ~ 2 fresh, good-quality free-range egg yolks_ **For the steak tartare:** _200g aged fillet steak ~ \u00bd small shallot, finely chopped ~ 1 tbsp small nonpareille capers ~ 1 tbsp finely chopped cornichons ~ 1 tsp finely chopped chives ~ 1 tsp finely chopped flat leaf parsley leaves ~ 2 tsp Dijon mustard ~ 1 tsp Worcestershire sauce ~ 6 drops of Tabasco sauce ~ \u00be tsp fine salt ~ freshly ground black pepper_ **For the a\u00efoli:** _\u00bd garlic clove, crushed or minced ~ 1 egg yolk ~ 200ml extra virgin olive oil ~ pinch of sea salt ~ 1 tsp lemon juice_ **To serve:** _2 handfuls of rocket ~ extra virgin olive oil ~ squeeze of lemon juice ~ 2 slices of light rye or sourdough bread_\n\nThere is a blog that details why it is best to leave this particular dish to the professionals, complete with photos of beef tartare prepared with pre-minced beef, big chunks of onion and in various other revolting permutations. The recipe is indeed a tricky proposition, largely because it requires precise execution or else it will end up looking like a dog's dinner. And a dog's dinner is almost certainly not going to achieve amorous success. So take care in your preparation and aim for perfection.\n\nI have enjoyed steak tartare a number of times, yet for some people, I realise, it is a somewhat intimidating dish. I don't believe it should be, as when it is done well it tastes sublime, like a delicious p\u00e2t\u00e9. And, like p\u00e2t\u00e9, it is best served with crisp toast. To make it a little less scary, I have decided to confit an egg yolk (a very simple thing to do) rather than serve it raw, so that, when it is broken, it spreads a comforting warmth through the fantastic flavours beneath.\n\nBuy the best beef and the finest, freshest eggs you can.\n\nMilton's Method   To make the egg yolk confit, preheat the oven to 65\u00b0C\/gas mark \u00bc (or as low as it will go), divide the extra virgin olive oil between two ramekins and gently place an egg yolk into each. Heat in the oven for 30 minutes.\n\nMeanwhile, prepare the other ingredients. The steak should come straight from the fridge, so it is firm and easy to slice. Some people recommend slicing off a thin layer of the exterior of the steak in order to avoid the possibility of bacterial infection. If you do choose to do this, use a separate knife and chopping board. Then, holding it carefully, cut lengthways, then widthways, to achieve a fine mince. You should be aiming to achieve a consistent result rather than the finest mince possible. Place in a bowl with all the other ingredients for the tartare and gently combine. Place a round pastry cutter on a serving plate and push half the mix inside it. Gently press down with the back of a spoon to achieve a firm circle and, once you are confident of the mixture's firmness, remove the pastry cutter. Get another serving plate, and repeat the process above, so that you have two tartares, each on their own plate.\n\nNow make the a\u00efoli. Combine the garlic and egg yolk in a bowl. Drizzle in the olive oil very slowly at first, whisking all the time so that the mixture emulsifies, then gradually adding the oil more quickly until the mayonnaise becomes thick and glossy. Add the salt and lemon juice, whisk again and arrange a precise dollop on each plate.\n\nPlace the rocket in a bowl, dress with a pinch of salt, a splash of olive oil and a squeeze of lemon juice. Toss and place on the plates. Toast the bread, cut diagonally and place on the plates.\n\nFinish by removing the ramekins from the oven and gently removing the yolks with a slotted spoon. Place each yolk on top of a steak tartare and serve immediately.\n\n## HERB-CRUSTED FRENCH RACK OF VENISON\n\n1 small French-trimmed rack of venison ~ fine salt and freshly ground black pepper ~ 2 tbsp Dijon mustard ~ 1 tsp honey ~ 3 tbsp softened unsalted butter ~ 3 garlic cloves, finely chopped ~ 4 slices of white bread, crumbed in a food processor ~ 2 tbsp finely chopped rosemary leaves ~ 6 tbsp finely chopped flat leaf parsley leaves\n\nAnyone who has heard the lusty roar of a stag across a moor on a misty autumn morning knows that this is one of the ultimate expressions of primal sexuality there is. As a beautifully flavoured, lean red meat, venison is also an excellent ingredient for promoting primal passion between lovers.\n\nThis is a sophisticated yet simple recipe that requires a little care and, depending on where you buy your venison, a little home butchery as well. My venison comes direct from source, but you may well be able to ask your butcher to separate the rack from the chine bone so you can separate the individual cutlets easily after cooking. Otherwise you will need to do this yourself with a hacksaw, which may somewhat alarm any potential partner who is arriving for dinner...\n\nMilton's Method   Preheat the oven to 200\u00b0C\/gas mark 6.\n\nIf your venison has not already been French-trimmed by a butcher, remove any excess fat or silvery bits of meat with a short, sharp knife. Trim the bones so they are clean. It is a bit fiddly, but worth persevering with to make the final visual effect all the more impressive.\n\nScore the thin layer of outer fat with a light criss-cross pattern so the crust will stick. Season the whole rack well with salt and pepper and place in a baking tray. Mix all the remaining ingredients in a bowl to make a crust, then apply it to the meat by hand, using some pressure to ensure the mixture sticks.\n\nRoast in the oven for 20\u201325 minutes for medium-rare meat, then rest in a warm place for 10 minutes. Serve on warmed plates with saut\u00e9ed potatoes and green vegetables.\n\n## BEEF FILLET, CEPS, MARSALA SAUCE AND ROASTED SHALLOTS\n\n2 \u00d7 150g fillet steaks, at least 21-day-aged ~ fine salt and freshly ground black pepper ~ 8 echalion (banana) shallots ~ a few sprigs of thyme ~ olive or rapeseed oil ~ large knob of unsalted butter ~ 2 medium ceps, sliced lengthways ~ 100ml medium-sweet Marsala ~ 250ml beef stock\n\nThis rich, indulgent main course features some beautiful ingredients that complement one another perfectly. And the cep (also known as the porcini) is as worthy of attention here as the fillet steak. This is truly the king of mushrooms. When I am foraging in the forest and I see a cep it sends a quiver of excitement through me. The Romans attributed aphrodisiac qualities to the cep and \u2013 with its firm, bulbous stem thrusting powerfully from the earth \u2013 there is something innately sexual about it. Unlike many other fungi it is not slippery or delicate, but dry and robust. This is matched by its flavour, a fantastic accompaniment to a good, well-aged fillet steak. Depending on the season, you may not be able to find fresh ceps, but it is possible to buy them frozen from online retailers.\n\nThere are only three ways to cook this steak: rare, medium-rare or ruined. I would opt for one of the two former options. The dish goes well with Pommes de Terre Sarladaises with Truffles (see here). You could also have Crunchy Fennel and Green Apple Salad (see here) as a side dish to lighten the meal, or just some greens such as spinach or Swiss chard.\n\nMilton's Method   Preheat the oven to 160\u00b0C\/gas mark 3. Rub the steaks well with salt and pepper and set aside.\n\nChop the ends off the shallots but leave the skins on. Place on a small baking tray with the thyme and 1 tablespoon of oil. Give the tray a shake to make sure all the shallots are coated in oil, then cook for 35 minutes.\n\nWhen the shallots are almost ready, heat a frying pan over a medium-high heat and add most of the butter and a glug more oil. When the butter is bubbling, place in the fillet steaks and press down slightly so they sizzle. Fry for 2-3 minutes on each side. When you turn the steaks, add the ceps to the pan with a little more butter. Cook these evenly on both sides so they colour a little.\n\nOnce cooked, remove the steaks and ceps and leave in a warm place to rest while you make the sauce. Remove the shallots from the oven, squeeze them from their skins and reserve half of them. Use a fork and a sharp knife to finely chop the other half. Add the Marsala to the pan over a medium heat to deglaze it, then add the stock and chopped shallots. Allow to reduce by half, then strain into a jug through a fine sieve.\n\nPut the steaks and ceps on to warm plates. Add the reserved shallots with a drizzle of oil and a pinch of salt and pour on the sauce. Serve immediately.\n\n## RECIPE LIST\n\n[BUTTERNUT SOUP  \nWITH NUTMEG AND TOASTED PUMPKIN SEED OIL](chapter008.xhtml#list22)\n\n[CRISPY SEA BASS  \nWITH GINGER AND SPRING ONIONS](chapter008.xhtml#list23)\n\nPANEER TIKKA MASALA\n\nSMOKED GARLIC DAUPHINOISE\n\n[SALT AND PEPPER BABY SQUID  \nWITH GARLIC MAYONNAISE](chapter008.xhtml#list26)\n\n[SAFFRON ROAST CHICKEN  \nWITH LEMON AND ROSEMARY](chapter008.xhtml#list27)\n\nSpices can scintillate not just our taste buds but our minds and libidos, too. By spices, I am referring to a wide category of plant-based pungent or aromatic substances, including garlic, onions, ginger and chilli, or saffron, nutmeg, pepper and paprika, to name a paltry few. Many of these substances are categorised by strict adherents to Hinduism as having undesirable effects on the body and mind. Garlic and onions, for instance, are both said to stimulate the nervous system to such a degree that they are incompatible with vows of celibacy. Garlic, in particular, is said to exert a powerful effect and is credited with invigorating a waning sexual appetite.\n\nMy theory about spices is that they help to enliven food and, as they do so, they also enliven us. An appetite for spice is an appetite for life. Spices pep us up. They can be fiery, fragrant or intense, or all three, and offer infinite variety in the way they can be used. And, for westerners at least, there is still a sense of exciting exoticism surrounding many spices, with their power to transform a dish into something redolent of steamy, far-away climes.\n\nSpices need to be handled with care. Too much chilli, for instance, is definitely a passion killer; you'll end up panting and sweating for all the wrong reasons. And if you are handling chilli and expecting a romantic encounter later on, you ought to take special precautions: one dinner of mine ended in painful disappointment because of my inattention in this respect. If you don't find them to be too much of a passion killer (or too sinister), perhaps invest in some latex gloves for chopping your chilli?\n\nSpice up your sex life with these recipes.\n\n## BUTTERNUT SOUP  \nWITH NUTMEG AND TOASTED PUMPKIN SEED OIL\n\nglug of olive oil ~ 1 tbsp fennel seeds ~ 2 onions, chopped ~ 1 fennel bulb, outer leaves and core removed, chopped ~ 3 garlic cloves, finely chopped ~ 1 butternut squash, peeled and chopped into 2.5cm dice ~ 1 tsp hot smoked paprika ~ 450ml chicken or vegetable stock, plus more if needed (optional) ~ sea salt and freshly ground black pepper ~ 1 tsp freshly grated nutmeg ~ toasted pumpkin seed oil, to serve\n\nThis soup includes a number of aphrodisiacs. Nutmeg has been used as an aphrodisiac in Asia. And in Germany there was, I have read, a rather dubious folk tradition involving a woman swallowing and passing out a whole nutmeg in order to snare a lover, that I shan't share fully with you here for fear of putting you off your food.\n\nA drizzle of shiny, green-black toasted pumpkin-seed oil on top makes this bowl of soup look as good as it tastes.\n\nMilton's Method   Add the oil to a large, heavy-based saucepan. Place over a medium heat. Add the fennel seeds and cook for 1 minute. Then add the onions, fennel and garlic and stir well. Put the lid on and cook for 5 minutes, stirring occasionally, until the fennel and onions have softened a little. Add the squash and the paprika and increase the heat a little to medium-high. Stir and let the onions colour a little. Be careful not to let them burn. Add the stock, season and reduce the heat to medium. Simmer for 15\u201320 minutes or until the squash is tender, but not falling apart.\n\nLadle the mixture into a blender and whizz until well liquidised. Return to the pan, check and adjust the seasoning. If the soup is too thick, add a little more liquid (water or stock). Stir well and serve in warm bowls with crusty bread, with the nutmeg grated on top and drizzled with toasted pumpkin seed oil.\n\n## CRISPY SEA BASS  \nWITH GINGER AND SPRING ONIONS\n\n4 sea bass fillets, around 125g each ~ sea salt and freshly ground black pepper ~ glug of sunflower, vegetable or rapeseed oil ~ thumb-sized piece of root ginger, peeled and cut into fine matchsticks ~ 2 garlic cloves, finely chopped ~ 6 spring onions, finely shredded lengthways ~ 2 red chillies, deseeded and very finely chopped ~ 1 tbsp soy sauce ~ 1 tbsp bought crispy fried shallots ~ Malaysian hot chilli sauce, to serve\n\nOne of my favourite street food places in London is the Malaysian Pit Stop Caf\u00e9 on Berwick Street in Soho. Every lunch time, long queues of office workers line up for Singapore noodles, Malaysian curries and other delicious treats. This is my favourite dish.\n\nWarming, gently spicy ginger helps with circulation and as such is a kind of natural Viagra, though a rather understated version. Soho is increasingly criticised for losing its authenticity, but there are still plenty of shops where you can buy non-culinary Viagra and plenty of places where it is then possible to avail yourself of its benefits. I prefer the gentler pleasures of the Pit Stop Caf\u00e9, and this is my attempt to recreate their iconic dish.\n\nMilton's Method   Slash the skin of the sea bass fillets diagonally two or three times, so that they don't curl up in the pan. Season well with salt and pepper.\n\nAdd the oil to a non-stick frying pan over a medium-high heat. When the oil is hot add the fish fillets, skin-side down. Cook for about two minutes on the skin side, until golden and crispy, then turn and cook for another minute on the other side, until the fish is cooked all the way through. Place on warm serving plates.\n\nWorking quickly, add the ginger, garlic, spring onions and chillies to the same pan and cook for a minute or two over a high heat. Add the soy sauce, then spoon the mixture over the fish with the crispy shallots. Serve with the chilli sauce and some noodles and eat with chopsticks.\n\n## PANEER TIKKA MASALA\n\n**For the paneer tikka:** _1 tbsp tikka paste ~ 4 tbsp Greek-style yogurt ~ squeeze of lemon juice ~ 300g paneer cheese, cut into 2.5cm cubes ~ 1 red onion, quartered, then split up into layers ~ 1 red pepper, cut into 2.5cm sections ~ a little vegetable or sunflower oil_ **For the masala sauce:** _2 tbsp sunflower oil ~ 1 onion, sliced into fine half rings ~ generous 1cm piece of root ginger, peeled and finely grated ~ 3 garlic cloves, finely chopped ~ \u00bd tsp turmeric ~ 2 tsp ground coriander ~ \u00bd tsp cayenne pepper ~ 1 tsp paprika ~ 2 tbsp Greek-style yogurt ~ \u00bd \u00d7 400g can of chopped tomatoes ~ 250ml vegetable stock ~ large pinch of sea salt ~ handful of coriander leaves, chopped_\n\nThere is a lot of protein \u2013 an important substance for the human reproductive organs \u2013 in this book, and this delicious dish is both protein-packed and also vegetarian. The important aphrodisiac ingredients are the chilli, ginger and garlic, but there are also a pleasing mix of textures and flavours: the creaminess of the cheese contrasts with the tang of tomato; while the crispy surface of the paneer is a great foil for the voluptuous sauce.\n\nThe quantities here are for a main course.\n\nMilton's Method   Preheat the grill to high. Mix the tikka paste and yogurt together with the lemon juice in a large bowl. Add the paneer cheese, onion and pepper pieces and toss to coat. Once they are fully coated, thread them on to skewers, brush with a little oil and place under the hot grill. Once the cheese has turned golden on top, turn and repeat, until the skewers are fully cooked. This should take about 10 minutes. Set the skewers aside while you make the sauce.\n\nPut a non-stick frying pan over a medium heat and add the oil. When the oil is hot, add the onion and cook for around 6 minutes until it begins to turn golden at the edges. Then add the ginger and garlic, stir for a minute until you can smell them, then mix in the turmeric, ground coriander, cayenne pepper and paprika. Add the yogurt, stirring it in well, then the tomatoes. Pour in the stock and add the salt. Bring to the boil, then reduce the heat and simmer for 10\u201315 minutes to thicken the sauce, stirring now and again.\n\nCheck the seasoning, then remove the paneer, onion and pepper from the skewers and put them into the sauce. Sprinkle with the chopped coriander leaves and serve straight away with naan bread and basmati rice.\n\n## SMOKED GARLIC DAUPHINOISE\n\nlarge knob of unsalted butter ~ 400g waxy potatoes ~ 1 smoked garlic clove, finely chopped ~ 200ml double cream ~ 100ml whole milk ~ \u00bd tsp fine salt ~ freshly ground black pepper ~ freshly ground nutmeg\n\nAlthough not technically a spice, I include garlic in this chapter because it has such a pungent, unmistakable flavour. Its aphrodisiac properties are well known and have ancient heritage: Aristotle included it on his list of aphrodisiacs. For many, though, its stimulant effect is offset by the lingering odour it leaves behind. One story suggests that the cheating wives of Greek soldiers ate garlic before their husbands came home, so they would not suspect them to be guilty of infidelity with such foul-smelling breath. I have never had any aversion either to eating or, unless in egregious excess, to smelling garlic, though I understand that some people do. A reasonable rule of thumb is that if you eat the same amount as the person you are eating with, you are unlikely to be offended.\n\nOf all the many ways to prepare potatoes, this is probably the most decadent, though Pommes de Terres Sarladaises with Truffles (see here) comes a close second. It goes especially well with Beef Fillet, Ceps, Marsala Sauce and Roasted Shallots (see here). Using smoked garlic, easy to buy online, works brilliantly.\n\nMilton's Method   Preheat the oven to 150\u00b0C\/gas mark 2.\n\nButter the base and sides of an ovenproof dish (mine is around 20cm in diameter) with half the butter.\n\nSlice the potatoes evenly and thinly with a sharp knife or, preferably, a mandolin to no more than \u00bccm thick. Layer in the dish so that they overlap one another, making sure that the top layer is neatly arranged.\n\nIn a saucepan, heat the remaining butter over a medium-low heat and, as soon as it has melted, add the garlic. Cook very gently for 2 minutes, stirring occasionally, then add the cream and the milk. Stir well and do not allow to come to the boil. Season with the salt and pepper and then very gently pour the mixture evenly over the potatoes. Leave standing for 10 minutes, then grate the nutmeg over the top and place in the oven for 65 minutes, or until golden brown and cooked through. Serve immediately.\n\n## SALT AND PEPPER BABY SQUID  \nWITH GARLIC MAYONNAISE\n\n**For the garlic mayonnaise:** _1 free-range egg yolk ~ 1 garlic clove, finely minced ~ 150ml sunflower oil ~ pinch of sea salt ~ 1 tbsp lemon juice_ **For the squid:** _400g baby squid ~ 1 tbsp peppercorns ~ 1 tsp coarse sea salt ~ 75g self-raising flour ~ 75g cornflour ~ 1 free-range egg, lightly beaten ~ vegetable oil, to deep-fry ~ 1 medium-hot red chilli, deseeded and finely shredded ~ 4 spring onions, shredded lengthways ~ lime wedges, to serve_\n\nOf the everyday aphrodisiacs (of which there are a remarkable number; worth investigating if you feel unexpectedly sexy and don't know why), black pepper is perhaps one of the most startling in terms of how commonly it is used. Yet is it really so surprising when you consider all that grinding and those enormous, phallic pepper mills that used to be so _de rigeur_ in Italian restaurants and pizza joints? A recipe in the _Karma Sutra_ , no less, suggests applying a concoction including black pepper and honey to the genitals to enable you to 'utterly devastate your lady.' Now, although I enjoy black pepper in my food, I think applying it to my genitals may be a step too far... but feel free if you would like to experiment (and I refer you to the disclaimer on here). In the same manner as chilli, garlic or rocket, black pepper adds a stimulating heat to dishes and is effective with a large range of ingredients from steak to pasta, from chicken to seafood. The advantage of using baby squid is that they are tender and require less preparation than their bigger brothers.\n\nMilton's Method   Make the mayonnaise first. Beat the egg yolk with the garlic. Very gradually pour in the sunflower oil in a very fine stream, whisking all the time. Once it is emulsifying, you can add the oil more quickly, until you have a thick, shiny mayonnaise. Whisk in the salt and lemon juice to loosen it a little. Set aside.\n\nPrepare and clean the squid (or get the fishmonger to do this for you.) Slice the bodies into rings and keep the tentacles intact.\n\nIn a mortar and pestle, combine the peppercorns and salt and crush until the mixture is nubbly but not too fine. Mix with both types of flour in a large plastic food bag. Put the egg in a shallow dish.\n\nDip the squid pieces first in the egg, then drop into the bag of seasoned flour until well coated.\n\nPour enough oil into a wok to give a depth of at least 5cm. Place over a high heat until the oil is very hot, but not smoking: you're looking for a temperature of around 175\u00b0C if you have a cooking thermometer, or when a scrap of bread will sizzle immediately when dropped in. Very carefully place the pieces of squid in the oil and cook for 60\u201390 seconds, in batches if necessary so that you don't crowd the pan. Lift out with a slotted spoon and place on kitchen paper to blot off excess oil.\n\nPat dry the chilli and spring onions with kitchen paper so that they don't spit when you put them in the fryer. Fry for a few seconds in the wok, then remove with a slotted spoon and drain on more kitchen paper. Serve the squid immediately, sprinkled with the chilli and spring onions and accompanied by the garlic mayonnaise and lime wedges.\n\n## SAFFRON ROAST CHICKEN  \nWITH LEMON AND ROSEMARY\n\n50g unsalted butter, at room temperature ~ 2 tsp saffron threads ~ 1 good-quality smallish (1.3\u20131.5kg) free-range chicken ~ sea salt and freshly ground black pepper ~ 1 lemon, halved ~ glug of extra virgin olive oil ~ 3 sprigs of rosemary ~ 1 garlic clove, skin on, lightly crushed\n\nThere is nothing quite like the beautiful aroma of a chicken roasting in the oven and when using these wonderfully aromatic ingredients \u2013 saffron, lemon, rosemary and garlic \u2013 it is an even more enjoyably sensuous experience. The impact of fragrance is vital, both for our appetite for food and in how we choose partners. Both men and women spend about as much time trying to smell good as they do trying to look good. Great aromas can be seductive; they are enticing and intriguing, and smell is the sense that has the most impact on our emotional memory, connecting us with times when we have felt happy, sad, or turned on. The aromas given off by your chicken as it cooks can help you in your strategy for seduction.\n\nSaffron is said to give women stronger orgasms. Rosemary, meanwhile, was used in ancient times as an emblem of fidelity between lovers, as it is said to enhance the memory. So as well as smelling fantastic, this recipe should also help you in your quest for romantic and physical satisfaction and help implant new \u2013 and hopefully happy \u2013 emotional memories in your brain.\n\nHowever, this recipe is not a gimmick. I believe it is the best roast chicken recipe there is and saffron works especially well here.\n\nMilton's Method   Preheat the oven to 200\u00b0C\/gas mark 6.\n\nGently mash the butter and saffron together with the back of a fork and, using your hands, spread it over the bird. Place in a medium roasting tray, season with salt and pepper and squeeze over the juice of half the lemon, pouring on the olive oil, too. Insert both lemon halves inside the cavity of the bird along with the rosemary and the garlic.\n\nPut the chicken into the oven and cook for 15 minutes. Reduce the oven temperature to 180\u00b0C\/gas mark 4 and cook for a further 40 minutes if your bird is the weight that I have specified in the ingredients list. Check the thickest part of the thigh with a skewer to check that the juices run clear with no trace of pink or, using a meat thermometer, ensure that the meat's temperature has reached 65\u00b0C. The skin should be bronzed and crispy.\n\nRest the chicken for about 10 minutes, so it is nice and tender. I generally do this by leaving it in the roasting tin, in the oven, with the oven door open.\n\nThe roasting tin should contain plenty of scented juices from the bird. Use them as a thin gravy. Serve the carved chicken with roast potatoes, bread sauce and carrots.\n\n## RECIPE LIST\n\nCRUNCHY FENNEL AND GREEN APPLE SALAD\n\n[ASPARAGUS  \nWITH CRAB, POACHED EGGS AND LEMON VINAIGRETTE](chapter009.xhtml#list29)\n\nAVOCADO, PANCETTA AND TOASTED PUMPKIN SEED SALAD\n\n[POMMES DE TERRES SARLADAISES  \nWITH TRUFFLES](chapter009.xhtml#list31)\n\nRED MULLET, ARTICHOKE HEARTS AND LEMON\n\nOur antipodean cousins use earthy phrases such as 'rooting' and 'forking' to describe the act of sexual congress. I'm not sure where this comes from. Perhaps from rooting around \u2013 like Shakespeare's 'delving for trout in mysterious rivers' \u2013 or perhaps the way in which a hard root spears its way through soft soil. Once we start thinking like this it's easy to find a sexual connotation in everything, as I'm sure Sigmund Freud would agree.\n\nThere is something primitive and earthy about many vegetables, and often something rather suggestive about their shapes, too. Consider the phallic thrust of asparagus spears, the protruding knobs of carrots, the labial folds of the globe artichoke or the testicular tubers of the humble spud. This type of thing appeals to the pantomime British sense of humour: the age-old pun on meat-and-two-veg; the episode of Blackadder with a turnip shaped 'exactly like a thingie'. We know that vegetables are good for us. We occasionally find them smutty and hilarious. But can we ever really think of them as being sexy?\n\nI think we can. I am a great lover of vegetables and I think that, used in the right way, they can create dishes that are not simply nutritious but erotic and enticing, too.\n\nA note: although this is a chapter about vegetables, not all of these recipes are suitable for vegetarians.\n\n## CRUNCHY FENNEL AND GREEN APPLE SALAD\n\n**For the salad:** _1 tbsp fennel seeds ~ 1 fennel bulb ~ 1 green apple ~ large handful of rocket_ **For the dressing:** _1 tsp Dijon mustard ~ 1 tsp honey ~ 1 tbsp unfiltered cider vinegar ~ 4 tbsp extra virgin olive oil ~ pinch of sea salt_\n\nThe apple has long been used as a symbol of sex, romantic love, carnal knowledge and sin. In a biblical context, it is the forbidden fruit. Venus, the Roman goddess of love, is often pictured holding an apple. Aphrodite, her Greek counterpart, gave the fruits as gifts. A recent study discovered that women who regularly eat apples have better sex than those that don't. I'm not sure that is conclusive proof that apples are aphrodisiacs, but perhaps it more pertinently relates to the adage that 'an apple a day keeps the doctor away'... and hence to the fact that a physically healthy person is more likely to have a better sex life. Fennel may not be so conspicuous in ancient legend, but it was a surprising ingredient of Dionysian orgies.\n\nThis simple, refreshing salad can be served as a side dish with a variety of different meaty or fishy main courses.\n\nMilton's Method   Toast the fennel seeds in a dry pan over a medium heat until they darken just a touch. Remove from the pan and set aside.\n\nRemove the hard core at the base of the fennel bulb and slice it lengthways as thinly as possible. Quarter and core the apple and slice it thinly, too. Place these and the rocket in a salad bowl and mix together.\n\nNow, working quickly before the salad ingredients turn brown, combine all the ingredients for the dressing in a separate bowl and whisk until emulsified. Add the dressing to the salad, toss well, then sprinkle the fennel seeds on top and serve.\n\n## ASPARAGUS  \nWITH CRAB, POACHED EGGS AND LEMON VINAIGRETTE\n\n**For the vinigarette:** _\u00bd tsp finely grated unwaxed lemon zest, plus 2 tbsp lemon juice ~ \u00bd tsp Dijon mustard ~ \u00bd tsp honey ~ 4\u20135 tbsp extra virgin olive oil ~ sea salt and freshly ground black pepper_ **For the rest:** _12 in-season asparagus spears ~ a little olive oil ~ white bread, torn into rough crouton shapes ~ 4 free-range eggs ~ 2 tsp white wine vinegar ~ 6 tbsp fresh white crab meat ~ \u00bd tsp sea salt_\n\nBoth asparagus and eggs are aphrodisiacs, while seafood contains high levels of zinc and selenium, making this a lover's recipe _par excellence_. Just think: thrusting spears of phallic fecundity lying amongst the most potent symbol of fertility of all \u2013 the egg \u2013 with the fresh ocean flavour of crab sparked into life by a sharp lemon vinaigrette that brings everything together. This makes an excellent starter or a weekend brunch\/lunch in preparation for a spot of frisky afternoon fun.\n\nAsparagus, apart from looking the part, also contains lots of useful vitamins that are helpful for boosting libido and general vim and vigour. The one well-known side effect of asparagus that does make it a bit of a turn-off (smelly wee) means that you should open the window after you have urinated.\n\nMilton's Method   Start by making the vinaigrette. In a small bowl, whisk the lemon zest and juice, mustard and honey together with a small whisk or fork. Drizzle in the olive oil slowly so the mixture emulsifies, then season to taste. Set aside.\n\nUsing the freshest asparagus you can find, trim off any tough part of the stem at the base, wash and place in a steamer. Steam for around 3 minutes until just tender, but still with some bite. Do not overcook: limpness is not a good look.\n\nHeat some regular olive oil in a frying pan and fry the bread until golden. Drain the croutons on a plate lined with kitchen paper.\n\nMeanwhile, crack open the eggs and place each into a cup. Put the vinegar into a large pan of water and heat it until there are small bubbles rising to the surface. Reduce the heat to medium-low and swirl the water round with a spoon. Place the eggs in this whirlpool one at a time and cook for 3\u20134 minutes, or until the whites are set. Gently drain the eggs and run cold water over them for a few seconds so they stop cooking. This should be the very last thing you do, as the eggs need to be still warm when they are eaten.\n\nOn a single warmed platter, or two large warmed plates, roughly scatter the asparagus, seasoning it with the \u00bd tsp of salt, then add the crab meat. Place the eggs on top, sprinkle over the vinaigrette and croutons and serve immediately.\n\n## AVOCADO, PANCETTA AND TOASTED PUMPKIN SEED SALAD\n\n10 very thin slices of pancetta ~ 1 tbsp white wine vinegar ~ 4 tbsp toasted pumpkin seed oil ~ pinch of sea salt ~ 2 handfuls of baby spinach ~ 2 perfectly ripe Haas avocados ~ 2 tbsp pumpkin seeds\n\nPumpkin seeds are packed with zinc, an important mineral for amorousness, and are included here in two ways: both sprinkled on top of this simple salad and as a key ingredient of the dressing. Toasted pumpkin seed oil is absolutely delicious, but expensive and not widely available, though you can find it in health food shops. Make sure you only use perfectly ripe avocados. Packed with nutrition, these are also well known as aphrodisiacs.\n\nMilton's Method   Put the pancetta under a hot grill and cook for around 4 minutes, watching carefully to make sure it doesn't burn, turning once, and removing when crisp.\n\nMake the dressing by combining the vinegar, pumpkin seed oil and salt in a bowl and whisking until well blended.\n\nPut the spinach in a bowl and add half the dressing, tossing thoroughly. Place on two plates.\n\nPrepare the avocados by cutting in half, peeling and removing the stones. Cut into 1cm-thick slices and place them gently on the spinach. Arrange the crispy pancetta on top.\n\nToast the pumpkin seeds in a dry frying pan for around 3 minutes until they begin to colour and pop. Sprinkle over the salad, then drizzle the remaining dressing over the top.\n\n## POMMES DE TERRES SARLADAISES  \nWITH TRUFFLES\n\n3 large potatoes ~ 1 tbsp goose or duck fat ~ sea salt and freshly ground black pepper ~ 1 tbsp finely chopped flat leaf parsley leaves ~ 10g black truffle, shaved, or truffle salt\n\nA favourite way of cooking potatoes in the south west of France, the secret here is to use goose or duck fat rather than butter or oil. The dish is sometimes combined with another favourite of French cooking: truffle. A byword for indulgence, the truffle has an aroma and flavour unlike anything else, a rich, perfumed earthiness that has for centuries perked up flagging sex lives. The black P\u00e9rigord truffles that are most commonly used are expensive, and you may wish instead to use a good-quality truffle salt.\n\nThis is a simple recipe \u2013 just saut\u00e9ed potatoes, really \u2013 that goes brilliantly with grilled meats.\n\nMilton's Method   Peel the potatoes, then slice into thinnish round slices.\n\nPut the goose fat in a large, heavy-based frying pan over a medium heat. When it has melted, add the potatoes and make sure all the slices get coated with fat. Cook for around 15 minutes, turning regularly, until crisp and golden.\n\nSeason with salt, pepper, parsley and shaved truffle, if using, or sprinkle evenly with a little truffle salt instead.\n\n## RED MULLET, ARTICHOKE HEARTS AND LEMON\n\n6 fresh globe artichokes, or 6 pre-cooked frozen artichoke hearts ~ finely grated zest and juice of 1 unwaxed lemon, plus \u00bd lemon to prepare and cook the artichoke ~ good-quality extra virgin olive oil ~ 1 garlic clove, finely chopped ~ 1 tbsp thyme leaves ~ sea salt and freshly ground black pepper ~ 4 fillets of red mullet\n\nThe globe artichoke can be eaten in the most tactile of ways with a lover: boil or steam a fresh, firm artichoke for a good while then, together, lovingly peel off the outer leaves, dip them in butter, vinaigrette or hollandaise sauce and scrape the flesh from them with your teeth. This is best done outside, on a warm summer's evening, at sunset, with a glass of chilled muscat and someone special by your side, as I have often done. In the centre of the mass of overlapping leaves is the true delight of an artichoke \u2013 the heart \u2013 and, when perfectly cooked, the heart is one of the most delicious vegetables it is possible to eat.\n\nI have paired the hearts here with red mullet, as they complement one another well, but this artichoke recipe can also be used on its own as a side dish to grilled chicken.\n\nA relative of the thistle, the artichoke promotes good liver health... and good livers make better lovers.\n\nMilton's Method   If preparing fresh artichokes, peel off the outer leaves, then use a knife to remove the inner leaves until you get down to the heart in the middle. Remove the hairy, fibrous choke in the centre by scooping it out with a teaspoon. Squeeze a little lemon on to each artichoke heart to prevent discoloration and place in a saucepan of cold water. Squeeze the remaining lemon juice from the half lemon into the water and place the lemon half in the pan, too. Bring to the boil, then reduce the heat and simmer over a low heat, with a lid on, for around 30 minutes, or until just tender but still with a little bite. (If using frozen artichoke hearts, they only need to be heated through briefly in boiling water.)\n\nDrain, allow to cool a little, then slice lengthways. In a heavy-based frying pan, heat the olive oil over a medium-high heat and saut\u00e9 the artichoke slices for 2 minutes, then add the garlic, thyme, lemon zest and juice and salt and pepper. Cook for a further 2 minutes, then place on warm plates.\n\nQuickly season the fish fillets with a little salt and pepper and cook in the same pan, skin-side down at first, for 2 minutes on each side, until just cooked. Serve immediately with the artichokes, accompanied by saut\u00e9ed potatoes and a rocket salad.\n\n## RECIPE LIST\n\n[WATERMELON SALSA  \nWITH BLACK PEPPER GOAT'S CHEESE](chapter010.xhtml#list33)\n\nGRILLED HONEY FIGS, GOAT'S CHEESE AND WALNUT SALAD\n\nPORK AND LYCHEE CURRY\n\nSALT CARAMEL AND RUM BANANA CAKE\n\n[POACHED QUINCE  \nWITH MASCARPONE AND STILTON CREAM](chapter010.xhtml#list37)\n\nFrom the suggestive shape of bananas to the delicate, feminine fragrance of quince, fruit can be a surprisingly effective aphrodisiac. I've always thought of fruit in general as an intensely sensual form of food. At its erotic, chin-dribbling, slurp-making, mouth-wiping best, fruit offers treats that are sweet yet wholesome and healthy yet indulgent... a little like sex should be, really.\n\nThe idea of fruit as a treat tends to be a little overlooked these days in the developed world, where fruit is seen in general as less of an indulgence and more about reaching a daily quota of dietary intake. We are primates and our cousins in the animal world go wild for the bright colours and sugar found in fruits. To a large extent, in our sugar-rich society, we have lost that same excitement. But, for me, fruit is always a pure and happy pleasure: the crunchy bite of an apple; blackberry-stained fingers; strawberries and ice cream in the summer.\n\nThe apple has long been used as a symbol of fertility, love and eroticism. It is associated with Aphrodite, the goddess of love, and in Ancient Greece it was customary for couples to eat apples on their wedding night. Read on to discover how the apple \u2013 and other fruits \u2013 can take you from the boughs of the apple tree all the way down to a comfortable spot in the long grass beneath it.\n\n## WATERMELON SALSA  \nWITH BLACK PEPPER GOAT'S CHEESE\n\n200g soft fresh goat's cheese ~ 1 tbsp olive oil ~ 1 tbsp freshly cracked black pepper ~ \u00bd small watermelon, deseeded, cut into 1cm dice ~ \u00bd cucumber, peeled, cut into 1cm dice ~ juice and finely grated zest of 1 lime ~ 1 small red onion, very finely sliced into crescents ~ handful of mint leaves, roughly chopped ~ 1 medium-hot red chilli, deseeded and finely chopped ~ 1 tbsp extra virgin olive oil ~ \u00bd tsp sea salt ~ crackers or crispbread, to serve\n\nMiraculous attributes are ascribed to the watermelon, which is perhaps surprising given that many other aphrodisiac ingredients are pungent in nature whereas the watermelon is rather... well... watery? Being so mellow itself in flavour, however, it works well with spices, salt and herbs, some of which have their own stimulating properties. This could be a starter, an unusual cheese course or a light lunch.\n\nMilton's Method   Blend the goat's cheese, regular olive oil and black pepper with a whisk until you have a lovely, creamy texture. Place an artful dollop on each plate.\n\nCombine all the remaining ingredients in a bowl to make the salsa and mix well. Leave for 5 minutes for the flavours to infuse, then spoon on to each plate. Serve with crackers or crispbread.\n\n## GRILLED HONEY FIGS, GOAT'S CHEESE AND WALNUT SALAD\n\n**For the salad:** _8 round slices from a small baguette ~ olive oil ~ 4 perfectly ripe figs ~ 1 tbsp honey ~ 2 pinches of sea salt ~ 8 \u00d7 1cm-thick round slices of goat's cheese ~ 2 large handfuls of mixed salad leaves ~ handful of walnuts_ **For the dressing:** _1 tsp Dijon mustard ~ 1 tsp honey ~ 1 tbsp white wine vinegar ~ 4 tbsp extra virgin olive oil_\n\nNudity is probably never far from our minds when the word 'fig' is mentioned, as we reach in our minds for a metaphorical fig leaf... Though please don't misunderstand me: I have never liked the idea of naked dining. There is a time and place for everything and nudity at the dinner table strikes me as the wrong time and the wrong place, at least while food is being served. It's a little like breakfast in bed: why would you want big flaky croissant crumbs \u2013 or sharp, nasty toast crumbs \u2013 pricking you on your smooth sheets?\n\nFigs are a luscious fruit and a well-known aphrodisiac. In ripe, tender splendour they are a deep, priapic shade of purple. They go well with sharp, creamy goat's cheese and this simple dish makes an excellent starter or light lunch.\n\nCut the figs in half and place on the grill pan, cut sides up. Loosen the honey with 1 tsp of hot water, add a pinch of salt, then brush the figs with the honey mixture and place under a medium grill for 5 minutes. Set aside.\n\nMilton's Method   Brush each of the pieces of baguette on both sides with a little olive oil and place under a medium grill for around 2 minutes on each side, until golden and crisp. Set aside.\n\nPlace the slices of goat's cheese on the slices of crispy bread. Put under the grill (now medium-hot) for a few minutes, until the cheese colours on top.\n\nMeanwhile, place the leaves in a salad bowl. Make the dressing by combining the mustard, honey and vinegar in a separate bowl, whisking until well mixed, then pour in the olive oil in a steady stream until emulsified. Add three-quarters of the dressing to the leaves, toss well and place on plates. When the cheese is done, place on top of the salad leaves with the figs in between. Toast the walnuts in a dry pan and sprinkle on top, drizzle the remaining dressing over and serve immediately.\n\n## PORK AND LYCHEE CURRY\n\n3 garlic cloves, roughly chopped ~ thumb-sized piece of root ginger, peeled and roughly chopped ~ 2 lemon grass stalks, roughly chopped ~ handful of coriander, torn, plus more to serve ~ 2 long red chillies, roughly chopped ~ 2 tbsp tamarind pur\u00e9e ~ 4 kaffir lime leaves ~ flavourless oil ~ sea salt and freshly ground black pepper ~ 400g diced pork shoulder or chump end ~ 1 onion, finely chopped ~ 2 tsp curry powder ~ 2 star anise ~ 250ml chicken stock ~ 400ml can of coconut milk ~ 1 tsp Thai fish sauce ~ \u00bd \u00d7 400g can of lychees, drained (or 8 fresh lychees, peeled and pitted) ~ juice of \u00bd lime ~ pomegranate seeds, to serve\n\nThis is my version of a curry I once had in Singapore, which in a way represents the racial diversity that can be found there. Singapore is populated by populations from South East Asia, China and India and this curry combines influences from all three. The lychee is considered an aphrodisiac in China and its subtle fruitiness combines well with pork in this fragrant, slow-cooked dish.\n\nMilton's Method   Make a curry paste by blending the garlic, ginger, lemon grass, coriander, chillies, tamarind, lime leaves and 1 teaspoon of oil until everything is finely whizzed into a paste. You may need to squash the ingredients down a couple of times in order to achieve this consistency.\n\nLightly season the pork and place a heavy-based casserole dish over a medium-high heat with a good glug of oil. Sear the pork briefly on all sides, then remove with a slotted spoon and set aside. Immediately deglaze the pan with a splash of water. Give the pan a stir, then add the onion and a little more oil if necessary. Cook over a medium heat for 6 or 7 minutes, until the onion begins to colour slightly.\n\nAdd the blended curry paste and cook for 2 minutes, stirring occasionally and making sure that it doesn't stick to the pan. Add the curry powder and star anise, stir and add the chicken stock. Bring to the boil and allow to simmer for 5 minutes until the volume of the liquid has reduced by half. Now add the coconut milk and fish sauce and gently bring to the boil. Return the pork pieces to the pan, return to the boil once more, then reduce the heat to very low, put on a lid, and simmer for an hour and a quarter, checking occasionally to make sure it is not catching on the bottom.\n\nWhen the time has elapsed, give everything a good stir and add the lychees. Remove the lid and cook for a further 20 minutes over a medium-low heat, allowing the sauce to thicken a little. Finish by adding the lime juice and adjusting the seasoning to your preference.\n\nServe immediately on hot plates with rice or noodles, sprinkled with coriander leaves and pomegranate seeds.\n\n## SALT CARAMEL AND RUM BANANA CAKE\n\n**For the salt caramel:** _100g unsalted butter ~ 100g soft light brown sugar ~ 100g caster sugar ~ 100g golden syrup ~ 200ml double cream ~ 1 tsp sea salt_ **For the cake:** _100g softened unsalted butter, plus more for the tins ~ 100g caster sugar ~ 100g demerara sugar ~ 3 large free-range eggs, separated ~ 3 mashed bananas, plus 1 banana to decorate ~ 200ml whole milk ~ 1 vanilla pod, split ~ 50ml dark rum ~ 300g self-raising flour, sifted ~ 1 tsp baking powder_ **For the buttercream:** _110g unsalted butter ~ \u00bd tsp vanilla extract ~ 120g icing sugar_\n\nThe suggestiveness of bananas hardly requires elaboration on my part. Here they are mashed into a truly rich, indulgent, moist and sweet dessert tempered by a little saltiness that will serve around eight people and works wonderfully well with vanilla ice cream. I have borrowed Nigella Lawson's method for making the salt caramel.\n\nMilton's Method  Start by making the salt caramel. Melt the butter, both types of sugar and the golden syrup in a small heavy-based pan for 4 minutes, stirring occasionally to make sure it doesn't stick and letting it darken a little. Then add the cream and the salt, whisking as you do so, until fully incorporated. Remove from the heat and put in a bowl or jug.\n\nTo make the cake, preheat the oven to 160\u00b0C\/gas mark 3 and butter two 23cm circular cake tins.\n\nIn a large bowl, using either a hand-whisk or a stand mixer, cream the butter and both types of sugar together until fluffy, then beat in the egg yolks one at a time until well incorporated.\n\nIn a separate bowl, whisk the egg whites until they are stiff.\n\nGradually beat the mashed bananas into the butter mixture, then add the milk, scrape in the seeds from the vanilla pod and add the rum. Fold in the flour until everything is incorporated. Add the baking powder and 150ml of the salt caramel and stir it in with a wooden spoon, then fold in the stiff egg whites.\n\nPour the batter evenly into the prepared cake tins. Pour another 50ml of salt caramel equally into each cake tin and use a knife to make it into a swirly pattern. Bake for around 45 minutes. Make sure the middle is cooked by using a skewer; when inserted into the cake it should emerge clean. Set aside to cool on a wire rack while you make the buttercream.\n\nIn a large bowl, using either a hand-whisk or a stand mixer, whisk the butter, 110ml of the caramel sauce and the vanilla extract until creamy. Then, still whisking, add the icing sugar bit by bit until the buttercream is nice and light.\n\nNow you can assemble the whole thing. Add a layer of buttercream to the first tier of cake and drizzle a little of the remaining salt caramel over the top. Add the top layer, ice this with buttercream and pour more salt caramel mixture on top of that. Add slices of bananas and serve with vanilla ice cream.\n\n## POACHED QUINCE  \nWITH MASCARPONE AND STILTON CREAM\n\n1 large quince ~ 200g caster sugar ~ juice of \u00bd lemon ~ 100g Greek yogurt ~ 100g mascarpone ~ 150g good Stilton cheese ~ 1 tsp honey\n\nThe quince was the equivalent of Viagra in Henry VIII's day, but for some reason it is not so widely used today, either as an aphrodisiac or merely as a fruit. Membrillo \u2013 a thick, slightly grainy quince paste most commonly served with Manchego cheese \u2013 is popular, and quince jelly is the preserve of WI ladies all over the country, though what they actually do with the hundreds of jars I regularly see at country f\u00eates is a mystery to me. I sometimes use it to sweeten game sauces instead of redcurrant jelly, but the pairing with cheese is the best.\n\nThis sort of dessert-cum-cheese course matches the dulcet tones of quince with the robust saltiness of Stilton. Try it with a glass of Sauternes or Pedro Xim\u00e9nez sherry.\n\nMilton's Method   Peel the quince and slice it in half lengthways.\n\nMeasure a litre of water into a saucepan, add the sugar and lemon juice and bring to the boil.\n\nCarefully place the quince halves in the pan, cover with a lid and simmer for 30\u201335 minutes or until the flesh is tender (check with a skewer or knife point).\n\nMeanwhile, blend the Greek yogurt and mascarpone in a bowl. Crumble in the Stilton and stir well. Add the honey, mix thoroughly and place a large dollop of this cream on each plate.\n\nWhen the quince halves are cooked, remove with a slotted spoon and put one on each plate, pouring over 1 tbsp of their poaching liquid to serve.\n\n## RECIPE LIST\n\n[CRANACHAN  \nWITH FIGS, CANDIED WALNUTS, HONEY AND MERINGUE](chapter011.xhtml#list38)\n\nCHOCOLATE CHILLI FONDANT\n\n[RED WINE POACHED PEARS  \nWITH STEM GINGER CR\u00c8ME FRA\u00ceCHE](chapter011.xhtml#list40)\n\n[VANILLA AND SAFFRON BAGUETTE AND BUTTER PUDDING  \nWITH GOOSEBERRY-GINGER JAM](chapter011.xhtml#list41)\n\nRASPBERRY AND ROSE PUDDING\n\nAmbrosial sweetness is inherently sensual. Smearing honey, chewing chocolate and licking ice cream, for instance, are all completely erotic activities, even when you're doing them on your own. Add a potential sexual partner into the mix and the sweet ending to your meal could be the start of something equally delicious and \u2013 possibly \u2013 ambrosial, too.\n\nIt is not just the sweetness of desserts that makes them so sexy, but their textures and flavours: sticky and gooey, fruity and chocolatey. They should be platefuls of fun and frolics, indulgence and hedonism, the cause of delight and surprise. They can either be smoulderingly hot or thrillingly frosty, but they must make us either smile or swoon.\n\nChocolate is known as one of the great aphrodisiacs and has been used as such for centuries. But there are many other ingredients that can be used at the end of a meal to signify that the serious business of eating is done and the fun can begin. In this chapter, I will show you how spices, rose petals and honey can also be part of the sensual smorgasbord that pudding can bring to the table. Try them at this crucial stage in the evening to see if they produce the desired result.\n\nAnd, as you savour the last lingering sweetness on your lips, perhaps now is the right moment to lean forward and give your partner an equally lingering kiss.\n\n## CRANACHAN  \nWITH FIGS, CANDIED WALNUTS, HONEY AND MERINGUE\n\nhandful of dried figs ~ 2 tbsp honey ~ 300ml double cream ~ 50g good-quality porridge oats ~ large knob of unsalted butter ~ handful of walnuts ~ 1 tbsp granulated sugar ~ 1 heaped tbsp broken meringue pieces ~ 50ml Scotch whisky 200g raspberries\n\nThis has always seemed to me a slightly straight-laced Scottish dessert, but with the addition of a few fun bits and pieces it can \u2013 like a repressed Englishman (or Scotsman) on holiday \u2013 break free from its shackles.\n\nMilton's Method   Chop the figs, removing their stalks, and put them in a bowl. Stir in 1 tablespoon of the honey and 1 tablespoon of hot water from the kettle. Leave to soak for 30 minutes.\n\nWhip the cream in a large bowl until it forms soft, billowing peaks.\n\nToast the oats in a small, non-stick pan over a medium heat for 3 or 4 minutes, taking care not to burn them. Set aside.\n\nIn the same pan, add the butter, walnuts and sugar, stir well to combine and stir for 3\u20134 minutes, making sure that the sugar doesn't burn. Once the sugar has caramelised, remove the nuts from the pan straight away, place them on a sheet of greaseproof paper and separate them from one another. Allow them to cool.\n\nFold most of the nuts and all the remaining ingredients into the cream, and that extra tablespoon of honey, and reserving a few raspberries and walnuts to serve. Serve in small bowls or glasses, sprinkling with raspberries and candied walnuts.\n\n## CHOCOLATE CHILLI FONDANT\n\n60g good-quality dark chocolate ~ 60g unsalted butter, plus more for the moulds ~ \u00bd medium-hot red chilli, deseeded and finely chopped ~ 2 free-range eggs, lightly beaten ~ 75g caster sugar ~ 25g plain flour ~ icing sugar, to dust ~ a very little cayenne pepper, to dust (optional)\n\nIt may be time for dessert, but, for you lovers, I hope that the 'main course' of your dinner date is still to arrive. With this in mind, the food at the end of your meal should still be focused on exciting you rather than rocking you to sleep, titillating your palate rather than soothing it. Hence chocolate and chilli. The spice here should be subtle rather than explosive, but enough to leave a lingering heat on your tongue, a flush in your cheeks and a light sweat on your brow, as though you are already mildly aroused. You will need two dariole moulds.\n\nMilton's Method   Break the chocolate into small pieces and put into a heatproof bowl suspended over simmering water (don't allow the bowl to touch the water). Add the butter to the bowl and melt with the chocolate together. Once both are fully melted, stir once to combine, remove from the heat and add the chilli. Set aside to cool slightly.\n\nButter two dariole moulds and preheat the oven to 200\u00b0C\/gas mark 6.\n\nIn a separate bowl, whisk the eggs and sugar together until well combined and fluffy. Gradually add the chocolate and butter mixture, stirring as you go, then fold in the flour. When everything is mixed, gently pour the batter into the prepared moulds, to reach about three-quarters of the way up.\n\nBake for around 7 minutes, or until there is a crust on the outside but the puddings are still molten in the middle (test with a skewer).\n\nLoosen the fondants very carefully by squeezing a palette knife down the sides, then carefully tip on to serving plates. Dust with a little icing sugar and a very little cayenne pepper, if you like (though do go carefully and check your partner enjoys heat first, as too much could render them inedible) and serve with vanilla ice cream.\n\n## RED WINE POACHED PEARS  \nWITH STEM GINGER CR\u00c8ME FRA\u00ceCHE\n\n**For the pears:** _500ml good-quality fruity red wine, such as Italian Primitivo ~ 100ml fresh orange juice ~ 1 tbsp finely grated orange zest ~ 150g caster sugar ~ 1 vanilla pod, split ~ 2 star anise ~ 1 cinnamon stick ~ 4 cardamom pods, gently crushed ~ 2 ripe Comice pears, peeled and cut at the base so they can stand up ~ 50g unsalted butter, chopped_  \n **For the ginger cr\u00e8me fra\u00eeche:** _1 tbsp stem ginger syrup from the jar ~ 150g cr\u00e8me fra\u00eeche ~ 6 balls of stem ginger, roughly chopped_\n\nThere is something rather sensuous and beautiful about a pear poached in red wine, it looks magnificent and smells delicious. The aromatics used to achieve the flavour in this dish include vanilla, cinnamon, cardamom and star anise. This subtle, delicate spicing, along with the depth of flavour of good red wine, creates an intoxicating blend that should hopefully prove irresistible to your dinner partner. Cr\u00e8me fra\u00eeche cut through with pieces of stem ginger \u2013 another aphrodisiac \u2013 provides the perfect accompaniment.\n\nMilton's Method   Pour the wine and orange juice into a large lidded saucepan or casserole and add the zest, sugar and spices. Place over a medium heat, stirring until the sugar has dissolved. Add the pears so they are standing in the pan and reduce the heat to low. Place the lid on the pan and simmer for 45 minutes to 1 hour, or until the pears are soft but not falling apart (test with a knife).\n\nMake the cr\u00e8me fra\u00eeche by gently combining the syrup, cr\u00e8me fra\u00eeche and stem ginger in a bowl. Spoon this into the centre of serving bowls. When the pears are ready, remove them from the pan and stand them in the bowls, squidging them down so the cr\u00e8me fra\u00eeche comes up around the base of each.\n\nBring the red wine sauce to the boil and reduce it until it is thick. Then remove from the heat, whisk in the butter pieces until it thickens slightly and serve warm, sprinkling a little over the pears and offering the rest in a jug on the side.\n\n## VANILLA AND SAFFRON BAGUETTE AND BUTTER PUDDING  \nWITH GOOSEBERRY-GINGER JAM\n\n30g unsalted butter, plus more for the dish ~ 2 tbsp gooseberry and ginger jam, plus more to glaze ~ \u00bd large baguette (about 150g) ~ 1 large free-range egg, plus 1 large free-range egg yolk ~ 25g caster sugar ~ \u00bd vanilla pod, split ~ 1 generous pinch of saffron threads ~ 100ml whole milk ~ 200ml double cream ~ 2 tbsp dark rum ~ 40g sultanas ~ 1 tbsp demerara sugar\n\nBread and butter pudding may not seem the most romantic of desserts, but by adding two exotic spices, a hit of rum and some gooseberry and ginger jam, it is elevated into a heady, fragrant dish with an indulgent texture.\n\nSaffron threads are the female sex organs \u2013 the stigma \u2013 of crocuses. That on its own may or may not be important, but it certainly helps to set out its stall as one of the most well-known aphrodisiacs, used by Greek gods, no less, in their pursuit of female company. Zeus seduced the princess Europa by breathing saffron from his mouth. Cleopatra, it is said, bathed in large quantities of the spice before making love.\n\nSaffron also happens to be remarkably expensive, by virtue of the fact that it is so labour-intensive to harvest, and perhaps this also adds to its reputation.\n\nMilton's Method   Preheat the oven to 180\u00b0C\/gas mark 4. Butter a medium-sized rectangular ovenproof dish. Spread the jam over the base of the dish.\n\nSlice the baguette thinly and butter both sides, then overlap the slices in the dish.\n\nPrepare a cold custard: beat the egg, egg yolk and caster sugar in a large bowl until thickened slightly and bubbly. Scrape the seeds from the half vanilla pod into the bowl, add the saffron, then mix in the milk, cream and rum. Mix well, then slowly pour over the bread. Sprinkle the sultanas over the pudding and leave for 30 minutes so the custard soaks into the bread.\n\nPlace the dish in a larger baking tray and pour boiling water around it. Sprinkle with the demerara sugar and, taking great care that the hot water doesn't spill, bake in the oven for around 35 minutes until golden on top.\n\nRemove from the oven. Melt a little more jam in a small saucepan with a splash of hot water to loosen, then brush this glaze over the pudding. Leave to stand for 10 minutes before serving.\n\n## RASPBERRY AND ROSE PUDDING\n\n**For the pudding:** _1 tbsp cornflour ~ pinch of fine salt ~ 1 free-range egg yolk ~ 250g raspberries ~ 50g caster sugar ~ juice of \u00bd lemon ~ 150ml double cream ~ 1 tsp edible rose water_ **To serve:** _1 tbsp white chocolate curls ~ 1 tbsp edible dried rose petals ~ 1 tbsp lightly crushed freeze-dried raspberries ~ 4 chocolate wafer sticks_\n\nI remember as a child sniffing deep red roses and thinking that they also smelt of raspberries. I discovered, while researching this book, that rose is a noted aphrodisiac... and its associations with romance, of course, are as enduring as they are clich\u00e9d. So the first thing that sprung to mind when I thought of using flowers in this book was to make a dessert out of raspberries and roses. And it works. The raspberries really shine through at first taste and the rose lingers a little (but not cloyingly) at the end.\n\nMilton's Method   Mix the cornflour and salt in a large bowl, then slowly pour in 125ml of water, whisking all the time until smooth. Add the egg yolk and whisk until fully incorporated.\n\nPut the raspberries, sugar and lemon juice in a blender and liquidise until smooth. Strain the mixture through a sieve to get rid of the seeds and pour into a heavy-based saucepan. Heat the mixture, stirring all the time, until it comes to the boil. Reduce the heat to its lowest, then take a couple of tbsp of the warm raspberry pur\u00e9e and whisk it into the bowl with the cornflour mixture. When it is well blended, pour back into the saucepan with the rest of the raspberry pur\u00e9e, whisking all the time, until the mixture has thickened and changed in colour from dark red to pink. Keeping the pan over a low heat, return to the boil, simmer for a couple of minutes to cook out the cornflour, whisking as you go, then set aside.\n\nIn a separate bowl, whisk the double cream until it forms billowing soft peaks but is not stiff, then whisk in the rose water. Once the raspberry mixture has cooled a little, fold in the cream until it is completely blended. Spoon the pudding immediately into serving dishes \u2013 I suggest either martini glasses, ramekins or small bowls \u2013 cover with cling film and place in the fridge. Leave for at least 2 hours to chill.\n\nWhen you are ready to serve, sprinkle with white chocolate curls, rose petals and crushed freeze dried raspberries, placing a couple of chocolate wafer sticks on the side.\n\n## Acknowledgements\n\nThis book would not have been written without a number of important people. Rosemary Davidson at Square Peg inspired the idea and Susannah Otter has been an organised and enthusiastic editor who has kept everything on track against some challenging deadlines. Lucy Bannell is a fantastic copy editor who has pointed out many important issues with a number of recipes and Marion Moisy tightened the text up still further at the proofreading stage. As ever, my brilliant agent Andrew Gordon has helped work out the details between everyone, and his assistant David Evans has been a great help in sorting out contracts and the like.\n\nThe dynamic duo of Matt Baxter and Graeme Rodrigo have played an instrumental role, not just in their respective fields of illustration and design, but in coming up with ideas for the whole concept. Katie Morgan's beautiful watercolour illustrations are also a vital part of the book.\n\nI would also like to thank Tom Charman for the fantastic venison and Price's Butchers for their excellent meat. Thanks to Bridget, Gilbert and Rafe for trying out many of the recipes, to Sue Telfer for testing the banana cake and Hilary Chandler for testing the fried oyster recipe.\n\nIt is patently absurd (they can't even read!) but I would also like to thank my three hens Delhi, Naggar and Leh, who have provided me with a plentiful supply of eggs during the writing of a book that has used vast quantities of them.\n\nDespite help from all these people (and chickens), if there are any errors or omissions, they are mine alone.\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 unauthorized 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\nEpub ISBN: 9781473545175  \nVersion 1.0\n\n1 3 5 7 9 10 8 6 4 2\n\nVINTAGE  \n20 Vauxhall Bridge Road,  \nLondon SW1V 2SA\n\nSquare Peg is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com.\n\nCopyright \u00a9 Milton Crawford 2015\n\nMilton Crawford has asserted his right to be identified as the author of this Work in accordance with the Copyright, Designs and Patents Act 1988\n\nFirst published by Square Peg in 2015\n\nwww.vintage-books.co.uk\n\nA CIP catalogue record for this book is available from the British Library\n\nISBN: 9781910931226\n\n### CONTENTS\n\n  1. Cover\n  2. About the Book\n  3. About the Author\n  4. Also by Milton Crawford\n  5. Title Page\n  6. Dedication\n  7. Chew. Slurp. Lick. Nibble.\n  8. Relationship Questionnaire\n  9. A Note on Unsexy Foods\n  10. Something to Slurp on\n    1. Frozen Watermelon Margarita\n    2. Basil Martini\n    3. Almond Milk White Russian\n    4. Bloody Mary with Clam Juice\n    5. Strawberry Bellini\n  11. Nibbles and Tit-Bits\n    1. Spiced Honeyed Almonds\n    2. Caviar Blinis\n    3. Puglia Broad Bean Dip\n    4. Oyster Fritters with Rocket and Wasabi Mayonnaise\n    5. Chaat Masala Watermelon with Mint\n  12. Smooth and Slippery\n    1. Oysters with Chilli, Ginger and Lime\n    2. Salmon Tikka Skewers with Dill and Pomegranate Raita\n    3. Milton's Moules\n    4. Lobster with Bearnaise Sauce, Home-Made Oven Chips and Broccoli\n    5. Scallops with Brown Butter, Capers and Lemon\n  13. Flesh\n    1. Venison and Liver Pies\n    2. Beef Carpaccio with Parmesan, Rocket and Truffle Oil\n    3. Yakitori-Style Chicken Heart Skewers\n    4. Steak Tartare with Confit Egg Yolk and Aioli\n    5. Herb-Crusted French Rack of Venison\n    6. Beef Fillet, Ceps, Marsala Sauce and Roasted Shallots\n  14. The Spice of Life\n    1. Butternut Soup with Nutmeg and Toasted Pumpkin Seed Oil\n    2. Crispy Sea Bass with Ginger and Spring Onions\n    3. Paneer Tikka Masala\n    4. Smoked Garlic Dauphinoise\n    5. Salt and Pepper Baby Squid with Garlic Mayonnaise\n    6. Saffron Roast Chicken with Lemon and Rosemary\n  15. Shapely Veg\n    1. Crunchy Fennel and Green Apple Salad\n    2. Asparagus with Crab, Poached Eggs and Lemon Vinaigrette\n    3. Avocado, Pancetta and Toasted Pumpkin Seed Salad\n    4. Pommes de Terres Sarladaises with Truffles\n    5. Red Mullet, Artichoke Hearts and Lemon\n  16. Getting Fruity\n    1. Watermelon Salsa with Black Pepper Goat's Cheese\n    2. Grilled Honey Figs, Goat's Cheese and Walnut Salad\n    3. Pork and Lychee Curry\n    4. Salt Caramel and Rum Banana Cake\n    5. Poached Quince with Mascarpone and Stilton Cream\n  17. Sweet Bits\n    1. Cranachan with Figs, Candied Walnuts, Honey and Meringue\n    2. Chocolate Chilli Fondant\n    3. Red Wine Poached Pears with Stem Ginger Creme Fraiche\n    4. Vanilla and Saffron Baguette and Butter Pudding with Gooseberry-Ginger Jam\n    5. Raspberry and Rose Pudding\n  18. Acknowledgements\n  19. Copyright\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n#\n\n\"Walters builds the tension masterfully.\"\n\n\u2014 _The Wall Street Journal_\n\n\"Quite simply, a tour de force.... It succeeds magnificently at almost every level.\"\n\n\u2014 _The Observer_ (London)\n\n\"A fascinating, page turning experience that is a cross be tween an Agatha Christie whodunit and a Thomas Harris thriller.\"\n\n\u2014 _The Denver Post_\n\n\"Almost impossible to put down.... A stunner.\"\n\n\u2014 _Booklist_\n\n\"Minette Walters asks as much of her readers as many literary novelists, and yet she offers them a book as gripping as any thriller.\"\n\n\u2014 _The Times Literary Supplement_ (London)\n\n#\n\nMinette Walters's bestselling novels have been published in more than thirty countries. She has won the John Creasey and Golden Dagger Awards from the Crime Writers Association of Great Britain, and the Edgar Allan Poe Award from the Mystery Writers of America, as well as the Grand Prix des Lectrices d'Elle in France. She lives in Dorset with her husband and two children.\n\nwww.minettewalters.co.uk\n\n# Also by MINETTE WALTERS\n\n_The Chameleon's Shadow_\n\n_The Devil's Feather_\n\n_The Tinder Box_\n\n_Disordered Minds_\n\n_Fox Evil_\n\n_Acid Row_\n\n_The Breaker_\n\n_The Echo_\n\n_The Dark Room_\n\n_The Scold's Bridle_\n\n_The Sculptress_\n\n_The Ice House_\n\nFor John, Henry  \nand Frank\n\nWith thanks to:\n\nAdrian, Alec, Andrew, Annika, Beverley, Caroline, James, Jane, Michael, Philippa, Richard, Sharon and Susanna.\n\nAnd with special thanks to Nick Godwin, Allen Anscombe, Rachel Harris and Ruth Wild for their professional help and advice.\n_Tics are categorized as Motor or Vocal, Simple or Complex.... Complex symptoms include: Body jerking, Skipping, Hitting, Walking on toes, Talking to oneself, Yelling, Coprolalia\u2014vocalizing obscene or other socially unacceptable words or phrases.... Tics increase as a result of tension or stress._\n\n**K** ANSAS **C** ITY **C** HAPTER  \nOF THE **T** OURETTE  \n **S** YNDROME **A** SSOCIATION\n\n_Unhappiness has a habit of being passed around._\n\n**M** ARGARET **A** TWOOD,  \n **BBC** **R** ADIO 4'S ** _B OOK **C** LUB_**, **M** AY **9, 1999**\n\n# one\n\nI could never decide whether \"Mad Annie\" was murdered because she was mad or because she was black. We were living in southwest London at the time and I remember my shock when I came home from work one wet November evening to find her collapsed in the gutter outside our house. It was 1978\u2014the winter of discontent\u2014when the government lost control of the trade unions, strikes were an everyday occurrence, hospitals ceased to cater to the sick and uncollected rubbish lay in heaps along the pavement. If I hadn't recognized her old plaid coat I might have ignored her, thinking the bundle in the gutter was a heap of discarded clothes.\n\nHer real name was Ann Butts and she was the only black person in our road. She was a well-built woman with a closed expression and a strong aversion to social contact, who was known to enjoy a drink, particularly Caribbean rum, and was often to be found sitting on the pavement in the summer singing gospel hymns. She had acquired the \"Mad\" label because she made strange faces and muttered to herself as she scurried along in a bizarre dot-and-carry trot that suggested a child playing \"Ride a cock horse to Banbury Cross.\"\n\nLittle was known about her circumstances except that she had inherited her house and a small independent income upon her mother's death and, apart from a menagerie of stray cats that had taken up residence with her, she lived alone. It was said that her mother had been madder than she was and that her father had abandoned them both because of it. One of the long-term residents of Graham Road swore blind that Mrs. Butts senior used to shout obscenities at passersby and twirl like a dervish when the mood was on her, but as Mrs. Butts had been dead some time the story had undoubtedly grown in the telling.\n\nI didn't believe it any more than I believed the rumors that Annie kept live chickens inside her house, which she killed by boiling, feathers and all, for her own and her cats'supper. It was nonsense\u2014she bought her meat, dead, at the local supermarket like everyone else\u2014 but her close neighbors talked about rats in her garden and a terrible smell coming from her kitchen, and the story of live chickens was born. I always said she couldn't have rats _and_ cats, but no one wanted to hear the voice of reason.\n\nThe same neighbors made life difficult for her by reporting her regularly to the local council, the RSPCA and the police, but nothing came of their complaints because the council couldn't force her out of her own house, the cats were not ill-treated and she wasn't mad enough to be committed to an institution. Had there been family and friends to support her, she might have taken her harassers to court, but she was a solitary person who guarded her privacy jealously. At various times health visitors and social workers made unsuccessful attempts to persuade her into sheltered accommodations, and once a week the local vicar knocked on her door to make sure she was still alive. He was always cursed loudly for his trouble from an upstairs window, but he took it in stride, despite Annie's refusal to go anywhere near his church.\n\nI knew her only by sight because we lived at the other end of the road but I never understood why the street took against her so strongly. My husband said it had to do with property values, but I couldn't agree with him. When we moved onto Graham Road in 1976 we had no illusions about why we could afford it. It had a Richmond postcode but it was very definitely \"on the wrong side of the tracks.\" Built for laborers in the 1880s, it was a double row of two-up, two-down terraced housing off the A316 between Richmond and Mortlake, and no one who bought a house there expected to make a fortune overnight, particularly as council-owned properties were seeded among the privately owned ones. They were easily identified by their uniform yellow doors and were looked down on by those of us who'd bought our houses because at least two of them held problem families.\n\nPersonally, I thought the way the children treated Annie was a better barometer of the adults' feelings. They teased her mercilessly, calling her names and aping her dot-and-carry trot in a cruel demonstration of their right to feel superior, then ran away with squeals of fear if their pestering irritated her enough to make her raise her head and glare at them. It was a form of bearbaiting. They goaded her because they despised her, but they were also afraid of her.\n\nIn retrospect, of course, I wish I'd taken up cudgels on her behalf but, like everyone else who stayed silent, I assumed she could take care of herself. Certainly the children weren't alone in finding her intimidating. On the one occasion when I made an attempt to speak to her, she rounded on me angrily, calling me \"honky,\" and I didn't have the courage to try again. Once in a while afterward I would come out of my front door to find her staring up at our house, but she scampered off the minute she saw me and my husband warned me not to antagonize her any further. I told him I thought she was trying to say \"sorry,\" but he laughed and said I was na\u00efve.\n\nOn the night she died a freezing rain was falling. The hunched trees that lined the pavements were black and sodden with water and made the street look very grim as I turned onto it from the main road. On the other side, a couple paused briefly under one of the few lampposts, then separated, the man to walk ahead, the woman to cross on a diagonal in front of me. I pulled up my coat collar to shield my face from the stabbing rain before stepping off the curb to run through sheets of water toward my house.\n\nI found Annie lying on the edge of the yellow lamplight in a space between two parked cars and I remember wondering why the couple hadn't noticed her. Or perhaps they had chosen to ignore her, believing, as I did, that she was drunk. I stooped to rock her shoulder but the movement caused her to cry out and I stepped back immediately. She lay with her head cradled in her arms, her knees drawn up tight against her chin and I assumed she was protecting herself from the rain. She smelled powerfully of urine, and I guessed she'd had an accident, but I shrank from the responsibility of cleaning her up and told her instead that I was going home to call for an ambulance.\n\nDid she think I wouldn't come back? Is that what persuaded her to uncradle her poor head and lift her pain-filled eyes to mine? I have no idea if that was the moment she died\u2014they said afterward it probably was, because her skull was so badly fractured that any movement would have been dangerous\u2014but I do know I will never experience such an intense intimacy with another human being again. I felt everything she felt\u2014sorrow, anguish, despair, suffering\u2014most poignantly, her complete bewilderment about why anyone would want to kill her. \"Was I unlovable?\" she seemed to be asking. \"Was I unkind? Was I less deserving because I was different?\"\n\nMany hours later, the police questioned my incoherent ramblings. Did Miss Butts actually say these things? No. Did she accuse someone directly? No. Did she speak at all? No. Did you see anyone run away? No. So there's no evidence to support your assertion of murder other than a puzzled look in her eyes? No.\n\nI couldn't blame them for being skeptical. As they pointed out, it was unlikely I could have interpreted Annie's look with any accuracy. Sudden death was always difficult to come to terms with because the emotions surrounding it are complex. They tried to convince me it was heightened imagination following my shock at finding her and offered me post-traumatic stress counseling to get over it. I refused. I was only interested in justice. As far as I was concerned, whatever residual shock I felt would vanish the minute Annie's murderer or murderers were caught and convicted.\n\nThey never were.\n\nThe coroner's verdict, based on the postmortem results and witness statements taken during a two-week police investigation, was death by misadventure. He painted a picture of a woman whose hold on reality was poor even when she was sober but who, on the night in question, had been drinking heavily. Her blood showed a high level of alcohol and she was seen staggering around the road by passing motorists and several neighbors. One said he had tried to persuade her to go home but gave up when she started swearing at him. Her injuries\u2014in particular the fractures to her skull and broken left arm\u2014were consistent with a glancing blow from a heavy vehicle, probably a truck, which had thrown her between the stationary cars and against the lamppost as it passed. Due to the heavy rainfall that night, it was unsurprising that no blood, hair or tissue traces were discovered on the lamppost.\n\nThe fact that no driver had come forward to admit liability was not considered significant either. It was dark, pouring with rain, parked vehicles restricted visibility and the street lighting was inadequate. With a critical reference to council officers who allowed badly lit streets in poorer areas to become rat-runs for heavy traffic, the coroner endorsed the police view that Miss Butts had stumbled off the pavement into the side of a passing truck without, in all probability, the driver being aware of the contact. It was impossible to establish when the accident occurred, although due to the seriousness of Miss Butts's injuries it was doubtful she could have survived more than fifteen to thirty minutes.\n\nIt was a sad case, the coroner said, which highlighted the need to have an element of compulsion in the treatment of vulnerable people in a modern society. There was clear evidence\u2014the squalid state of her house when the police entered it the day after her death; her alcohol dependency\u2014that she was unable to look after herself properly, and it was his opinion that if social services and health workers had been able to force Miss Butts to accept help, she would still be alive today. The witness who found Miss Butts's body had alleged a racist campaign against Miss Butts by her neighbors, but there was no evidence to support this and the coroner accepted that her neighbors' actions had been prompted only by concern for her welfare. In conclusion and despite the same witness's emotional insistence that Miss Butts had been deliberately pushed in front of an oncoming vehicle, the coroner's verdict was unequivocal. Death by misadventure. Case closed...\n\nI fell ill shortly afterward and took to my bed for several days. I told the doctor who came to see me that I had the flu, but he diagnosed depression and prescribed tranquilizers, which I refused to take. I became frightened of the telephone; every sound from the road had me jumping out of my chair. My husband, Sam, was initially sympathetic but soon lost interest when I began sleeping in the spare room and talking about rats in the downstairs lavatory. Thereafter, I developed mild agoraphobia and found it harder and harder to go to work. I was a teacher at a local comprehensive school and my overburdened colleagues were even less sympathetic than Sam when I said I felt stifled by the way the children crowded around me in the corridor. After a few weeks I ceased going in altogether.\n\nThe whole episode\u2014from Annie's death to the loss of my job\u2014 caused a rift between me and Sam, who walked delicately around me for weeks, then took to speaking to my mother for hours on the telephone. He was careful to close the door but I could still hear most of his conversation through the paper-thin walls on the rare occasions when I bothered to listen. The phrases repeated most often were \"impossible to live with...\" \"having a nervous breakdown...\" \"got a thing about rats...\" \"stupid fuss over a bloody black woman...\" and \"divorce...\"\n\nSome time in February my parents drove up from Hampshire where they were then living. Sam had moved out three weeks previously to sleep on a friend's sofa, and our marriage was effectively over. Wisely, my father refused to get involved, but my mother couldn't resist taking Sam's part. She comes from that generation of women who believe that marriage is the key to a woman's happiness and she told me in no uncertain terms that if I was determined to reject Sam then I needn't look to her and my father for support. As she pointed out, I'd been deserted by my friends because my behavior was so peculiar.... I was rapidly becoming anorexic.... I had no job... worse, no prospect of a job while I remained firmly closeted in the house. What was I planning to do? Where was I planning to go?\n\nI expressed only mild annoyance that she believed everything Sam told her, and suggested she question a man's honesty for once in her life. It was like a red flag to a bull. We couldn't talk about sex\u2014or lack of it, Sam's real bone of contention with me\u2014because it was a taboo subject between us, so instead she lectured me on the way I was letting myself go, my failure to cook proper meals for my hardworking husband, my lackluster approach to cleaning the house and inevitably my absurd obsession with the death of a colored person.\n\n\"There might be some sense in it if she'd been one of us,\" she finished tartly, \"but she wasn't even English... just another wretched immigrant living off benefit and clogging up the health service with imported diseases. Why we ever let them in, I can't imagine, and for you to jeopardize your marriage\u2014\" She broke off abruptly. \"Can't you see how ridiculous you're being?\"\n\nI couldn't, but it wasn't something I was prepared to discuss with her. Predictably, my silence persuaded her she'd won the argument when all she'd really achieved was to prove to me how little I cared about anyone's opinion but my own. In an odd way her complete lack of sympathy was more liberating than distressing because it made me realize that control rests with whomever worries least about being seen to exercise it, and with cold deliberation I agreed to mend fences with my husband if only to keep a roof over my head.\n\nThree months later, Sam and I moved abroad.\n\nCORONER'S COURT\n\nMedical Report on Miss Ann Butts, submitted to Mr. Brian A. Hooper, coroner, on December 12, 1978, by Dr. Sheila Arnold, GP, FRCP, from the Howarth Clinic, Chicago, Illinois, U.S.A. (Formerly a partner in the Cromwell Street Surgery, Richmond, Surrey.)\n\n**(Dr. Arnold left for a twelve-month sabbatical in America on September 10,1978, and was absent at the time of Miss Butts's death. Although Miss Butts had been assigned to one of Dr. Arnold's partners for the duration of the sabbatical, Miss Butts died before the partner had time to meet and assess her. It was therefore agreed that Dr. Arnold would submit the following report from America. A full set of medical notes relating to Miss Butts has been made available to the Coroner by the Cromwell Street Surgery.)**\n\nAnn Butts was my patient from June 1969 until my departure for America on September 10, 1978. She suffered from Tourette's syndrome, a neuropsychiatric disorder characterized by recurrent muscle tics and involuntary vocalizations. It was an inherited condition from her mother who had a complex form of the disorder, which manifested itself as _coprolalia_ , a compulsion to utter obscenities. Ann, who cared for her mother for many years until her death in 1968, had a good understanding of Tourette's syndrome and had learned to manage her own condition successfully. Ann's most noticeable symptoms were 1) motor tics in the face and shoulders; 2) a compulsion to talk to herself; and 3) obsessive behavior, particularly in relation to home and personal security.\n\nI referred her in December 1969 to Dr. Randreth Patel (Middlesex Hospital), who took a particular interest in Ann and was sympathetic toward her firmly held views against the taking of psychoactive drugs, which she felt had worsened her mother's condition rather than improved it. While no one has yet discovered a cure for Tourette's syndrome, the disorder tends to improve with age and Ann was no exception to this. My understanding is that her tics were a great deal more pronounced when she was a teenager (DOB\u201412.3.36). As a result, she suffered considerable teasing and unkindness from her peers and had few social skills following an early withdrawal from formal education. In recent years Ann's symptoms had been comparatively mild although she was inclined to exacerbate them from time to time through an overindulgence in alcohol. She had an average IQ and had no difficulty leading an independent life, although her obsession with home and personal security meant she shunned the company of others. I made a point of visiting her every six to eight weeks and on my last visit\u2014September 8, 1978\u2014she was in good health, both physical and mental.\n\n_Sheila Arnold Gp, FRCP_\n\nSheila Arnold GP, FRCP\n\n**_Family correspondence prior_**  \n ** _to the Ranelaghs' return to England\u2014dated 1999_**\n\nCURRAN HOUSE\n\n_Whitehay Road_\n\n_Torquay_\n\n_Devon_\n\nPS Dear M, personally, thought the \"dodgy ticker\" was wonderful and loved to hear Sam laugh at the other end. What a marvelous relationship you and he have with your boys and what a blessing they've been these last few months. I'm much looking forward to sharing some of the Ranelagh Jr. fun, even if it means driving two hours to experience it! Tell Luke I have every intention of having at least one go on a surfboard even if I do go \"arse over tit\" in the process. I may be an old codger, but I'm not in my grave yet.\n\nDad\n\nX X X\n\nCape Town\n\n5 June\n\nDear Mother,\n\nWritten in haste. Sorry about the screaming but the line was bad. I enclose a photocopy of the farmhouse details. I have taken up references and am reliably informed that \u00a3650 is a good price. It would be considerably more, apparently, if it weren't a property of \"character,\" which appears to be real estate-agent shorthand for \"somewhat dilapidated.\" However, Sam and the boys are looking forward to slumming it as much as I am. All being well we should be there by the first week in July and will expect you and Dad at the end of the month. I'll ring to confirm a weekend as soon as we're installed. We're all fine and send our love to you both.\n\n# two\n\nI recognized Dr. Arnold as soon as I opened the door to her, although there was no answering smile of recognition from her. I wasn't surprised. We were both twenty years older, and I had changed a great deal more than she had after two decades abroad. She was silver-haired and thinner, late fifties, I judged, but she still had the same rather searching gray eyes and air of unassailable competence. On the only other occasion I'd met her, I'd found her thoroughly intimidating, but today she gave me a sisterly pat on the arm when I told her my husband was complaining of chest pains.\n\n\"He says it's a pulled muscle,\" I said, leading the way up the stairs of our rented farmhouse, \"but he had a coronary six months ago and I'm worried he's about to have another one.\"\n\nIn the event, Sam was right\u2014it was a pulled muscle from too much digging in the garden the day before\u2014and I concealed my total lack of surprise behind an apologetic smile. Dr. Arnold reproved him for scoffing at my concern. \"You can't take chances,\" she told him, folding her stethoscope, \"not when you've had one close shave already.\"\n\nSam, whose memory for faces was almost as bad as his memory for names, buttoned his shirt and cast an irritable glance in my direction. \"It's a ridiculous fuss about nothing,\" he complained. \"I said I'd go to the surgery but she wouldn't let me... just takes it into her head to start treating me like a blasted invalid.\"\n\n\"He's been biting my head off all morning,\" I told Dr. Arnold. \"It's one of the reasons I thought it might be serious.\"\n\n\"Goddamit!\" Sam snapped. \"What's the matter with you? All I said was, I had a small twinge in my side... which isn't surprising in view of the number of weeds I hauled out yesterday. The garden's a mess, the house is falling down. What am I supposed to do? Sit on my hands all day?\"\n\nDr. Arnold poured oil on troubled waters. \"You should be grateful you have someone who still cares enough to make the phone call,\" she said with a laugh. \"I had a patient once whose wife left him to writhe in agony on the kitchen floor while she downed half a bottle of gin to celebrate her imminent widowhood.\"\n\nSam wasn't the type to stay angry for long. \"Did he survive?\" he asked with a grin.\n\n\"Just about. The marriage didn't.\" She studied his face for a moment, then looked curiously toward me. \"I feel I know you both but I can't think why.\"\n\n\"I recognized you when I opened the door,\" I said. \"It's an extraordinary coincidence. You were our GP in Richmond. We lived in Graham Road from '76 to the beginning of '79. You came to our house once when Sam had a bout of bronchitis.\"\n\nShe nodded immediately. \"Mrs. Ranelagh. I should have recognized the name. You're the one who found Annie Butts. I've often wondered where you went and what happened to you.\"\n\nI looked casually from her to Sam, and was relieved to see surprised pleasure on both their faces, and no suspicion...\n\nSam landed a job as overseas sales director for a shipping company, which took us in turn to Hong Kong, Australia and South Africa. They were good times, and I came to understand why black sheep are so often sent abroad by their families to start again. It does wonders for the character to cut the emotional ties that bind you to places and people. We produced two sons who grew like saplings in the never-ending sunshine and soon towered over their parents, and I could always find teaching jobs in whichever school was educating them.\n\nAs one always does, we thought of ourselves as immortal, so Sam's coronary at the age of fifty-two came like a bolt from the blue. With doctors warning of another one being imminent if he didn't change a lifestyle that involved too much traveling, too much entertaining of clients and too little exercise, we returned to England in the summer of '99 with no employment and a couple of boys in their late teens who had never seen their homeland.\n\nFor no particular reason except that we'd spent our honeymoon in Dorset in '76, we decided to rent an old farmhouse near Dorchester, which I found among the property ads in the _Sunday Times_ before we left Cape Town. The idea was to have an extended summer holiday while we looked around for somewhere more permanent to settle. Neither of us had connections with any particular part of England. My husband's parents were dead and my own parents had retired to the neighboring county of Devon and the balmy climate of Torquay. We enrolled the boys at college for the autumn and set out to rediscover our roots. We'd done well during our time abroad and there was no immediate hurry for either of us to find a job. Or so we imagined.\n\nThe reality was rather different. England had changed into New Labor's \"Cool Britannia\" during the time we'd been abroad, strikes were almost unknown, the pace of life had quickened dramatically and there was a new widespread affluence that hadn't existed in the '70s. We couldn't believe how expensive everything was, how crowded the roads were and how difficult it was to find a parking space now that \"shopping\" had become the Brits' favorite pastime. Hastily the boys abandoned us for their own age group. Garden f\u00eates and village cricket were for old people. Designer clothes and techno music were the order of the day, and clubs and theme pubs were the places to be seen, particularly those that stayed open into the early hours to show widescreen satellite feeds of world sporting fixtures.\n\n\"Do you get the feeling we've been left behind?\" Sam asked glumly at the end of our first week as we sat like a couple of pensioners on the patio of our rented farmhouse, watching some horses graze in a nearby paddock.\n\n\"By the boys.\"\n\n\"No. Our peers. I was talking to Jock Williams on the phone today\"\u2014an old friend from our Richmond days\u2014\"and he told me he made a couple of million last year by selling off one of his businesses.\" He made a wry face. \"So I asked him how many businesses he had left, and he said, only two but together they're worth ten million. He wanted to know what _I_ was doing so I lied through my teeth.\"\n\nI took time to wonder why it never seemed to occur to Sam that Jock was as big a fantasist as he was, particularly as Jock had been trumpeting \"mega-buck sales\" down the phone to him for years but had never managed to find the time\u2014 _or money?_ \u2014to fly out for a visit. \"What did you say?\"\n\n\"That we'd made a killing on the Hong Kong stock market before it reverted to China and could afford to take early retirement. I also said we were buying an eight-bedroom house and a hundred acres in Dorset.\"\n\n\"Mm.\" I used my foot to stir some clumps of grass growing between the cracks in the patio, which were symptomatic of the air of tired neglect that pervaded the whole property. \"A brick box in a modern development more likely. I had a look in a real estate agent's window yesterday and anything of any size is well outside our price range. Something like this would cost around \u00a3300,000 and that's not counting the money we'd need to spend doing it up. Let's just hope Jock doesn't decide to visit.\"\n\nSam's gloom deepened at the prospect. \"If we'd had any sense we'd have hung on to the house on Graham Road. Jock says it's worth ten times what we paid for it in '76. We were mad to sell. You need to keep a stake in the property market if you want to trade up to something reasonable.\"\n\nThere were times when I despaired of my husband's memory. It was a peculiarly selective one that allowed him to remember the precise details of past negotiating triumphs but insisted he forgot where the cutlery was kept in every kitchen we'd ever had. It had its advantages\u2014he was easily persuaded he was in the wrong\u2014but once in a while it caught me on the raw. At the very least, he ought to have remembered the weeks of abuse that followed the inquest into Annie's death...\n\n\"It was _my_ choice to leave,\" I said flatly, \"and I don't care if we end up living in a caravan, it's one decision I'll never regret. You might have been able to stay on Graham Road.... I certainly couldn't... not once the phone calls started anyway.\"\n\nHe eyed me nervously. \"I thought you'd forgotten all that.\"\n\n\"No.\"\n\nThe horses kicked up their heels for no apparent reason to canter to the other side of the field, and I wondered how good their hearing was and whether they could pick up vibrations of anger in a single word. We watched them in silence for a moment or two, and I put money on Sam backing away as usual from the period in our lives that had brought us to the brink of divorce. He chose to follow a tangent.\n\n\"In purely financial terms Jock's probably right, though,\" he said. \"If we'd kept the house and let it, we'd not only have had an income all these years but we'd have made a 1,000 percent increase on our capital to boot.\"\n\n\"We had a mortgage,\" I told him, \"so the income would have gone straight into paying it off and we'd never have seen a penny of it.\"\n\n\"Except Jock says...\"\n\nI only half-listened to Jock's views on the beneficial effects to borrowers of the galloping inflation of the late '70s and early '80s and how the Thatcher revolution had freed up entrepreneurs to play roulette with other people's money. I hadn't had much time for him when we lived in London and, from Sam's reports of the conversations he'd had with him via the international phone network over the years, I could see no reason to change my opinion. Theirs was a competitive relationship, based on vainglorious self-promotion from Jock and ridiculous counterclaims from Sam, which anyone with an ounce of intelligence would see straight through.\n\nI roused myself when Sam fell silent. \"Jock Williams has been lying about money since the first time we met him,\" I murmured. \"He latched on to us in the pub for the sole purpose of getting free drinks because he claimed he'd left his wallet at home. He said he'd pay us back but he never did. I didn't believe him then and I don't believe him now. If he's worth ten million\"\u2014I bared my teeth\u2014\"then I've got the body of a twenty-year-old.\"\n\nI was doing Sam a kindness although he couldn't see it because it would never occur to him that I might know more about Jock than he did. How could I? Jock and I had had no contact since our strained farewells on the day Sam and I left London. Yet I knew exactly what Jock was worth, and I also knew that the only person likely to lose sleep over it was Jock himself when his braggadocio lies finally came home to roost.\n\nSam's gloom began to lift. \"Oh, come on,\" he said. \"Things aren't _that_ bad. The old bum's spread a bit, admittedly, but the tits still hold their shape.\"\n\nI gave him an affectionate cuff across the back of the head. \"At least I've still got all my hair.\"\n\n**POLICE WITNESS STATEMENT**\n\n**Date:** 16.11.78\n\n**Time:** 18:27\n\n**Officer in charge:** PC Quentin, Richmond Police\n\n**Witness:** Sam Ranelagh, 5 Graham Road, Richmond, Surrey\n\n**Incident:** Death of Miss A. Butts in Graham Road on 14.11.78\n\nOn Tuesday, 14.11.78, I reached Richmond station at about 7:30. My friend, Jock Williams, who lives at 21 Graham Road, was on the same train and caught up with me as I passed through the ticket barrier. It was raining heavily, and Jock suggested we make a detour to the Hoop and Grapes in Kew Road for a pint. I was tired and invited him back to my house instead. My wife, a teacher, was at a parents'evening and was not due home until 9:30. The walk along the A316 takes approximately 15 minutes, and Jock and I turned into Graham Road at around 7:45.\n\nI have lived in Graham Road for two years and knew Ann Butts well by sight. On several occasions in the last six months I have come across her outside our house, staring in through the windows. I have no idea why she did this although I believe she may have been trying to intimidate my wife, whom she called \"honky.\" In view of the bad weather, I was surprised to see her there again on Tuesday night (14.11.78). She moved away as we rounded the corner. She was clearly drunk and when I pointed her out to Jock we both used the word \"paralytic\" to describe her. We were reluctant to approach her because she seemed to have a strong dislike of white people. We crossed the road behind her and let ourselves into my house.\n\nJock remained with me for approximately one-and-a-half hours, and we spent most of that time in the kitchen. The kitchen is at the back of the house and the door to the corridor was closed. At no point did we hear anything from the road that would suggest an accident had occurred. Jock left at approximately 9:15 and I accompanied him to the front door. I had completely forgotten seeing Ann Butts earlier and it did not occur to me to look for her again. I watched Jock turn right out of our gate toward his own house before going back inside.\n\nI was shocked when my wife came rushing in fifteen minutes later to say Mad Annie had collapsed in the gutter and looked as if she was dying. I ran out with a torch and found her body between two parked cars outside number 1. It seemed obvious to me that she was already dead. Her eyes were open and there was no pulse in her neck or her wrist. I made an attempt at mouth-to-mouth resuscitation but gave up when there was no response. An ambulance arrived shortly afterward.\n\nI regret now that I made no attempt to assist Ann Butts back to her house at 7:45, although I am convinced she would have rejected the offer.\n\n**Signed:**\n\n_Sam Ranelagh_\n\n**In the presence of:**\n\n_A. Quann_\n\n**_Letter from Libby Williams\u2014-formerly  \nof 21 Graham Road, Richmond\u2014dated 1980_**\n\n_39a Templeton Road_\n\n_Southampton_\n\n_Hampshire_\n\n_UK_\n\nMay 20, 1980\n\nM'dear!\n\nYou could have knocked me down with a feather when your letter came through my door. And what great news about the baby. Seven months old, eh? Conceived in England and born in Hong Kong. Has to be lucky! Of course we must remain friends. God knows, I didn't spend hours listening to your heartache in the wake of Annie's death to abandon you the minute you move abroad. I'm just so glad you got in touch because the way things are\u2014i.e., Jock and I aren't speaking. AT ALL!\u2014I didn't know how to contact you. Of course I'll help you in any way I can, although I'm a little worried that your letter seems to imply Jock and Sam had a hand in Annie's death. Much as I loathe the two-timing maggot I married, I don't think he's vile enough to kill anyone and certainly not someone he hardly knew. As for Sam! Do me a favor!\n\nOkay, so Sam got drunk one night and admitted they lied to the police about where they were and now refuses to have Annie's name mentioned. Well, trust me, sweetheart, I don't think you should read too much into it even if I do understand how angry you must feel. Sam had no business to lie for Jock however \"good\" the cause. Still, that's men for you. They stick to each other like glue, but cast off their women whenever it suits them!\n\nRe your questions:\n\n1) Did I tell the police that Jock had been with Sam? Yes. As you know, they started knocking on everyone's doors the day after the event, wanting to know if we'd seen or heard the accident. I said I'd been alone at home watching telly and hadn't heard a thing, so they promptly asked me what my husband had been doing and I said, \"Having a drink with Sam Ranelagh at number 5.\"\n\n2) Did Jock volunteer the information when he got home or did I ask for it? I asked him the night of the 14th. The little toerag came rolling in half-cut as usual and I said, \"Where the hell have you been?\" \"Round at Sam's having a beer,\" he came back quick as a flash. I should have known he was lying! He always used Sam as a way out of a crisis.\n\n3) What time did Jock get home that night? Nine-fifteenish. Can't recall exactly. I'm sure the nine o'clock news was still on.\n\n4) Have I any idea when Jock spoke to Sam in order to concoct the alibi? Knowing Jock, he would have phoned Sam at work the next morning and told him he was on the spot and had to think up a lie on the spur of the moment. \"If anyone asks, I was with you. So don't let me down, will you?\" That kind of thing.\n\nIn passing, I doubt very much if Jock had been gambling again, whatever he may have told Sam. He had a floozy in Graham Road, a bleached vampire called Sharon Percy, who was little better than a prostitute. He claims he was having an affair with her but my solicitor forced him to produce his bank statements and it looks as if he was making regular payments to her every Tuesday in return for sex. He's denying the payments at the moment (but not the affair\u2014he seems quite proud of that!) but my solicitor's confident we can drag the truth out of him if he refuses to make a reasonable settlement and we end up in court.\n\nAnyway, the point is, Annie died on a Tuesday and I suspect Jock was rogering Sharon rather than gambling! For all I know it may have been the first time because he never bothered to explain his lateness on a Tuesday again. Or any other day for that matter! You're right. The prospect of imminent divorce is a great relief and I have every intention of taking him to the cleaners if I possibly can. He only produces documents after my solicitor applies thumbscrews, and he explains his purchase of a house in Alveston Road (v. swank \u00a3.70,000 five-bedroom job within spitting distance of Richmond Park\u2014complete with live-in blonde bimbo!) as a \"long-term, heavily mortgaged investment.\" This, on the back of the paltry \u00a3.10,000 he took as his 50 percent share of 21 Graham Road. Do me a favor! Do the sums add up, or do they? The best I could afford was this two-room flat in Southampton.\n\nFeel free to ask for any help I can give you. It would never occur to me for a minute that talking about Annie might bring on \"a fit of the vapors\" and how very old-fashioned of Sam to come up with such an expression. No woman I know goes in for such idiocies, and I doubt if they ever did. It was yet another invention of man to undermine the onward march of female supremacy. Yes, I'm bitter, and... yes, the whole male sex can go fuck itself as far as I'm concerned... I've taken a leaf out of your book and have come to Southampton to train for a career in teaching. Dammit, girl, if you can make money tutoring Chinamen in Hong Kong, I can surely make money tutoring brats over here!\n\nLove,\n\n_Libby_ ,\n\n# three\n\n\"It blew over very quickly,\" Sheila Arnold told me as we went downstairs, \"except that Annie's house stood empty for about three years. She hadn't made a will and no one knew if there were any living relatives. In the end, everything was appropriated by the government and the property was released for sale. It was bought by a builder who renovated it before selling it to a young couple with two small children.\"\n\n\"A white couple presumably,\" I said with poorly veiled sarcasm.\n\nShe ignored the remark although her mouth lifted in a faint smile. \"I visited the house soon after they moved in when the younger child fell ill,\" she went on, \"and the place was unrecognizable. The builder had gutted the whole of the downstairs and redesigned it as one huge open-plan room with patio doors to the garden.\" There was reservation in her voice as if she wasn't sure that an open plan was an improvement.\n\n\"Didn't you like it?\"\n\nShe paused by the door. \"Oh, it was splendid enough, but I couldn't help remembering what it was like in Annie's day. Did you ever go inside while she was living there?\" I shook my head. \"It was like an Aladdin's cave. She and her mother were both hoarders. The front room was packed with West Indian and Central American artifacts, all brought back to England by Annie's father during the '40s and '50s. Some of them were quite valuable, particularly the gold pieces. I remember there was a little statuette on the mantelpiece which had emeralds for eyes and rubies for lips.\"\n\n\"I didn't know there was a Mr. Butts,\" I said in surprise. \"I always assumed the mother had been left holding the baby.\"\n\n\"Good Lord, no. Her father died of lung cancer some time in the late '50s. I never knew him but one of my partners remembered him fondly. His name was George. He was a retired merchant seaman with a fund of anecdotes about his travels round the world. He married Annie's mother in Jamaica in the '30s and brought her and Annie home to live in Graham Road soon after the war.\" She smiled again. \"He said he couldn't bring them back while his parents were alive because they wouldn't have approved of a black daughter-in-law.\"\n\nI shook my head in amazement, realizing how many gaps there still were in my knowledge of a woman I had never spoken to. Did Annie's neighbors know she was half white? I wondered, and would it have made a difference if they had? I thought \"no\" to both questions. They had been even later arrivals to the street than Sam and I... and Annie had been too dark-skinned to pass as anything other than black. \"I didn't know any of this,\" I told Sheila. \"I certainly didn't know her father was a white man. Why didn't someone come forward to claim her estate? She must have had English relations, surely?\"\n\n\"Apparently not. My colleague said George had a younger brother who was torpedoed in the North Atlantic, but other than that\u2014\" She broke off on a shrug. \"It's tragic but not unusual. Whole families were wiped out during the two world wars, particularly those with sons and no daughters.\" She glanced reluctantly at her watch and stepped outside. \"I really must go. I've two more patients to see.\" But she moved slowly as if she didn't want to break this link with the past. \"Do you still think she was murdered?\"\n\n\"I know she was.\"\n\n\"Why?\"\n\nI led the way down the path. \"I can't explain it. I tried once, but everyone just thought I was as mad as she was. Now I don't bother.\"\n\n\"I meant, why would anyone have wanted to murder her?\"\n\nIt was the great imponderable. \"Because she was different,\" I suggested. \"Perhaps they'd have left her alone if she'd been mad but not black... or black but not mad... Sometimes I think they despised her for her color, other times I think they were afraid of her.\"\n\nWe halted beside her car. \"Meaning you think one of her neighbors killed her?\"\n\nI didn't say anything, just gave a small shrug which she could interpret as she liked.\n\nShe watched me for a moment, then opened the rear door of her car and put her bag on the backseat. \"She wasn't mad,\" she said matter-of-factly \"She had Tourette's syndrome, which caused her to grimace and talk to herself, but she was as normal as you or I in every other respect.\"\n\n\"That's not the impression the coroner gave at the inquest.\"\n\nDr. Arnold nodded unhappily. \"The man was an idiot. He knew nothing at all about TS and wasn't interested in finding out. I've always felt very badly about not giving evidence in person, but I left for a twelve-month sabbatical in the U.S. before she died and had no idea he would effectively ignore Annie's medical records.\" She saw the sudden hope in my face. \"The verdict would have been the same,\" she said apologetically. \"There was no evidence to suggest it was anything other than an accident but I was very angry afterward to discover how her reputation had been destroyed.\"\n\nI thought cynically that the anguish I'd seen in Annie's dying eyes had had nothing to do with concern for her reputation. \"Did you read the pathologist's report?\"\n\nShe nodded. \"I was sent a copy with the inquest verdict. It was very straightforward. She caught a glancing blow from a truck and was thrown against a lamppost. Frankly, it was a tragedy waiting to happen\u2014they should never have let Graham Road be used as a shortcut\u2014 but I always thought a child would be hit, not someone as mindful of safety as Annie was.\"\n\nI nodded. \"She was wearing a dark coat the night she died, and the weather was frightful... rain like stair rods. I only saw her because I almost stepped on her as I was crossing the road.\" I put a hand on Dr. Arnold's arm as she prepared to open the driver's door. \"You said you were angry about her reputation being destroyed. Did you follow that up?\"\n\nA faraway look crept into her eyes as if she were searching some distant horizon. \"Not for three years. It may sound callous but I forgot all about her while I was in the States, and it wasn't until I saw what the builder had done to her house that it occurred to me to ask what had happened to the contents.\"\n\n\"Presumably they were sold off.\"\n\nShe went on as if she hadn't heard me. \"People had a very false impression of Annie because of the way she dressed and the way she behaved, but she wasn't a poor woman by any manner of means. She once showed me a list of valuations that a dealer had put on some of the artifacts, and my recollection was that the total came to over \u00a350,000. That was a fair sum in the 1970s.\"\n\n\"The police must know what happened to it all,\" I said. \"Did you ask them?\"\n\nShe gave a theatrical shudder. \"Not _them,\"_ she countered tartly, \"just the one. A man by the name of Sergeant Drury\u2014-Joseph Stalin's younger, ruder and more aggressive brother. It was his case so I wasn't allowed access to anyone else.\"\n\nI laughed. \"I knew him. It's a good description.\"\n\n\"Yes... well, according to him, Annie was destitute. They took some RSPCA inspectors in the day after the accident in order to remove her cats, and Drury said there was nothing of value in the house. Worse, he described the conditions inside as little short of a cesspit.\"\n\nI nodded again, remembering. \"It was mentioned at the inquest. The Coroner said the RSPCA should have taken her animals away from her when the neighbors first complained about the smell.\"\n\n\"Except squalor was alien to her nature,\" said Dr. Arnold, folding herself behind the steering wheel. \"I used to visit her regularly and it was a battle to stop her jumping up every ten minutes to wash her hands. She had a thing about germs\u2014it's a common symptom of Tourette's syndrome\u2014along with a compulsion to check the bolts on the front door hourly. Of course Drury didn't believe me. It was three years down the road and he decided I was confusing her house with someone else's.\" She reached out to shut her door, apparently under the impression that I understood what she was talking about.\n\nI held it open. \"What didn't he believe?\"\n\nShe blinked in surprise. \"Well... obviously... that Annie's house had been ransacked and everything of value stolen.\"\n\nIn the past, Sam had always shied away from discussions about Annie. I remember his embarrassment when I tackled a chief superintendent at a party in Hong Kong and pinned him to a wall with an hour-long diatribe on the inequities of the Richmond Police. Sam had hauled me away eventually and by the time we reached home his embarrassment had turned to fury. \"Have you any idea how idiotic you sound when you start talking about that bloody woman?\" he had demanded angrily. \"You can't lecture total strangers on garbage about the eyes being the windows of the soul if you want to be taken seriously. You're my wife, for Christ's sake, and people are starting to avoid the pair of us because they think you're as mad as she was.\"\n\nTwo decades on, and once he'd ruminated at length on the bizarre coincidence of having Sheila Arnold as our GP for a second time\u2014 \"You've got to admit, it's pretty damn spooky.... It's only a couple of days since Jock reminded me of Graham Road\"\u2014he was surprisingly interested in what Sheila and I had been talking about. I thought I knew why. He was never inclined to believe anything I said, but he rolled over to have his tummy scratched by doctors... particularly female ones.\n\n\"Does she agree with you? Does she think Annie was murdered?\"\n\n\"I'm not sure,\" I answered. \"All she said was the house had been ransacked.\"\n\nHe ruminated some more. \"When? Before Annie died, or after?\"\n\n\"What difference does it make?\"\n\n\"If it happened afterward,\" he said reasonably, \"then it means somebody knew she was lying in the gutter and grabbed the opportunity to break in.\" He gave his jaw a thoughtful scratch. \"Which in turn means she was probably out there a damn sight longer than the coroner said she was.\"\n\n\"That's one way of looking at it,\" I agreed, before wandering off to the kitchen to prepare some lunch. Old habits die hard, I find, and the subject of Annie had been taboo for so long between us that she wasn't easily resurrected from her grave.\n\nSam pursued me. \"And if it happened _before_ she died,\" he went on, \"that might explain why she drank herself into a stupor. It must have been a terrible shock to go home and find all her treasures gone. Poor woman, I've never understood why she did that. I mean, we saw her pretty tipsy on occasion but never so paralytic that she didn't know what she was doing.\" He flicked me an apologetic smile. \"I've always had a problem believing one of her neighbors pushed her under a lorry. Okay, some of them were shits, and some of them made her life miserable by complaining about her, but that's a far cry from committing cold-blooded murder.\"\n\nI opened the fridge door and wondered what sort of meal I could make out of half a can of tomatoes, some staggeringly old cheese and iceberg lettuce. \"She was five-foot-nine-inches tall and weighed fourteen stone,\" I murmured, \"and was exactly fifteen milligrams over the legal driving limit\u2014the equivalent of five shots of spirits or five pints of lager. By any stretch of the imagination that does not amount to drinking herself into a stupor.\" I pulled out the tin and inspected it for mold. \"In fact the chances were she wasn't even tipsy because she was a practiced drinker and could probably consume twice as much as the rest of us before showing any signs of drunkenness.\" I smiled at him. \"Look at yourself, if you don't believe me. You're a stone lighter and two inches taller, and you can put away eight pints of lager before you become embarrassing.\"\n\nHe retreated into his shell immediately because it was one thing for him to introduce the subject, and quite another for me to challenge the facts from superior knowledge. \"Everyone said she was paralytic,\" he said huffily\n\n\"Even if she was,\" I went on, \"what makes you think one of her neighbors didn't give in to a spur-of-the-moment temptation to shove her into the road? It was dark.... It was raining.... She was mad as a hatter... deeply irritating... the street was empty... and there was a truck coming. One quick shove and, hey presto, problem solved. No more blacks on the road and property values rise immediately.\" I lifted an amused eyebrow. \"No one ever said her murder was planned, Sam.\"\n\nTwo or three days later, a folder of photocopied documents arrived in the post from Sheila Arnold with \"Annie Butts\" written on the front. \"Thought the enclosed might interest you,\" she wrote on a compliments slip. \"It's not much, I'm afraid, because I gave up when I realized I was beating my head against a brick wall! PS How delightful to run into you both again.\"\n\nBy coincidence, it was the same day that Sam and I went for lunch in Weymouth, and Sam took against a man who seemed to be staring at me. We had chosen a pub overlooking the harbor, with tables outside, which allowed us to bathe in the sun and watch the yachts drift in and out of the marina every time the swing bridge lifted. It was a pretty place to while away a couple of hours, with eighteenth-century houses lining cobbled wharfs and battered trawlers unloading crates full of monkfish and crab, but Sam started bellyaching about the landlord who kept coming to the door to look at me, and my pleasure in the peaceful scene evaporated. I was wearing dark glasses and I studied the man secretly from behind my lenses. He was as lean and hungry as he had always been and undoubtedly as vicious. But he was better looking than Joseph Stalin... or Joseph Stalin's brother...\n\n**INCIDENT REPORT**\n\n**Date:** 15.11.78 **Time:** 11:15\n\n**Officer in charge:** PS Drury, Richmond Police\n\n**Incident:** Authorized entry to 30 Graham Road, Richmond, following the death of the owner, Miss Ann Butts. Neighbors had reported a number of cats trapped inside the property. No next of kin was available. **Officers attending:** PS Drury, PC Andrew Quentin. Also in attendance: RSPCA Inspectors: John Howlett, Tony Barrett\n\nEntry to the property was gained via the front door with a Yale key that Miss Butts was wearing on a string around her neck at the time of her death. The house was very cold with no central-heating provision. There were two gas fires in each of the downstairs rooms, but neither was lit at the time of entry. None of the windows was open, although a small casement in the back cloakroom was insecurely fastened due to a broken fixing.\n\nThe officers had been informed by neighbors that there were at least twenty cats inside the premises, and the smell of cats'urine in the front hallway was overpowering. The conditions inside the house were squalid and untidy\u2014in particular the downstairs cloakroom and the upstairs bathroom where neither toilet had been flushed and soiled paper was lying on the floors. Human feces was found in the two downstairs rooms. Boxes of empty vodka bottles were stacked against the kitchen walls.\n\nThe smell of cats'urine in the kitchen was particularly bad. Litter from a tray had been scattered across the linoleum floor and used indiscriminately by the animals. The RSPCA inspectors expressed concern that Miss Butts had wedged a small chest of drawers against the cat flap, which she had agreed to install after a previous visit from them. Numerous food and water bowls lined the edges of the floor, but all were empty.\n\nA search of the cupboards in the kitchen revealed inadequate provisions, both for Miss Butts herself and the colony of cats in her care. There was little in the way of tinned or boxed food, although there were fifteen pints of milk in her fridge and some raw meat, mostly chicken. \"Reduced price\" stickers suggested Miss Butts had acquired the goods cheaply, although subsequent inquiries at a local supermarket provided information that she picked over \"binned\" goods in order to obtain cat food free.\n\nThere were two downstairs rooms, excluding the kitchen and cloakroom, and the doors to both were open. These rooms had also been fouled by the animals, although to a lesser extent than the kitchen. In the front sitting room, three dead cats were found under a pile of cushions in a corner. In the opinion of John Howlett (RSPCA) all three had been dead for a minimum of four days. Two of them, both toms, had little or no fur on their faces and severe scratches to their bodies, and appeared to have died from untended wounds inflicted during a fight. The third, also a tom, had lost most of its fur and had died from a broken neck. Two more dead tomcats were discovered upstairs in Miss Butts's bedroom, wrapped in towels and placed in the wardrobe. Both were severely undernourished, shorn of fur and had broken necks.\n\nThe doors to all the upstairs rooms were closed. Five live cats, all toms, were found trapped in the back bedroom. The animals were in a distressed state, and had clearly been fouling the room for several days. All had bite and scratch wounds from fighting. There were bowls on the floor, which may have contained water or food, but were empty at the time of entry to the house. Apart from the dead toms in the wardrobe, the front bedroom contained four live queens and two neutered toms that were also in a distressed state.\n\nThe total number of cats removed from the premises by the RSPCA inspectors was twenty-one, five of which were dead.* Their full report (attached) concludes that the tomcats had suffered the worst neglect while the condition of the queens and neutered toms gave less cause for concern. It is their view that Miss Butts had been allowing the animals to foul the interior for some time\u2014in particular the tomcats whose odor is powerful and distinctive. They also pointed to clear evidence of cruelty in the shorn fur, broken necks and apparent willingness to allow the animals to fight \"to the death,\" and drew attention to the fact that it was the male animals that appeared to have been singled out for torture. The inadequate supplies in the kitchen and the estimated time of death of the dead cats suggests that Miss Butts had ceased to provide proper care and attention some five to seven days before her death.\n\nA superficial search of the house produced nothing to indicate the names and\/or addresses of Miss Butts's next of kin. A box of papers was removed from a cupboard in the front sitting room for later examination.**\n\nThe general impression received by those attending was that Miss Butts had been living in conditions of extreme poverty for some time. There were no carpets in either of the downstairs rooms, much of the furniture was broken and dilapidated and there were few adornments. The temperature in the house was cold, yet the gas supply in the cupboard under the stairs had been turned off. In addition, several fuses had been removed from the electrical circuits, although the main supply was still on. When an attempt was made to flush the toilets, it was discovered that the stopcock under the sink had also been turned off. One explanation is that Miss Butts had become worried about paying her utility bills. This may have been influenced by a dependence on alcohol.\n\n* Veterinarian postmortems of the five dead cats concurred with John Howlett's assessment at the time of entry to the premises. Two had died of untreated wounds following a fight; three had died after having their necks broken. All showed clear evidence of cruelty, specifically: fur ripped from their faces\u2014probably from the application of cellophane tape, parcel tape or carpet tape, which was then torn off. Also, two of the dead animals appeared to have had superglue applied to their lips and lids, residues of which remained on scraps of fur around their mouths and eyes. Estimated time of death: four to seven days prior to the bodies being found. Allowances were made for the cold conditions in the house, which had slowed decomposition.\n\n** All the papers were official: various bills\u2014some paid, some (gas and electricity) unpaid; a check book and bank statements; a building society book (Abbey National), showing \u00a315,340.21 in an interest-bearing account; TV license; and tax and rate receipts. There was an envelope containing assorted photographs of a woman (black) and man (white) with \"Mum\" and \"Dad\" or \"Elizabeth\" and \"George\" written on the back, but nothing else of a personal nature. Subsequent inquiries at Miss Butts's bank produced the deeds to the house, various share certificates and a current bank-account statement showing \u00a34,324.82 in credit. (NB: Miss Butts's bank manager said she was \"regularly abusive toward the cashiers and had bees in her bonnet about them stealing her money.\" He also said it wouldn't surprise him if she was confused about whether she could afford to pay her bills because she wasn't \"quite with it.\")\n\n**_Correspondence between Dr. Sheila Arnold and  \nInspector John Howlett, of the RSPCA\u2014dated 1983_**\n\n**39 LYVEDON AVENUE, RICHMOND, SURREY**\n\nInspector John Howlett\n\nRSPCA\n\nGuardian House\n\nTwickenham\n\nSurrey\n\nFebruary 22, 1983\n\nDear Mr. Howlett,\n\nI am making inquiries about a visit you and your colleague, Tony Barrett, made to 30 Graham Road, Richmond, just over three years ago on November 15, 1978. The house belonged to a woman called Ann Butts who died in a road accident, and you were asked to accompany the police to her house the following day in order to rescue her cats. I have a copy of the police report of the incident, but the report that you and your colleague made had been omitted. Do you still have a copy on file and, if so, would it be possible for me to read it?\n\nI was Miss Butts's general practitioner for several years and I have a number of concerns relating to the way the police described her living conditions. \"Squalid\" and \"living in conditions of extreme poverty\" does not accord with my memories of Miss Butts and\/or her house. With reference to her cats, my recollection is that they were always sleek, deeply loved and well cared for. In addition: my understanding is that, following complaints from Miss Butts's neighbors, you made several visits to 30 Graham Road in 1978, but that none of the complaints was upheld.\n\nFrom your memory of those visits, can you describe any of the West Indian or Central American ornaments or artifacts that she had in her front room? I am puzzled that the police have no record of their existence, particularly as Ann was proud of them and often told me how valuable they were.\n\nI should be most grateful for any information you can give me.\n\nYours sincerely,\n\n_Dr. Sheila Arnold_\n\n_Dr. Sheila Arnold_\n\nDr. Sheila Arnold\n\n_White Cottage  \nLittlehampton  \nNr Preston  \nLancashire_\n\nDr. Sheila Arnold  \n39 Lyvedon Avenue  \nRichmond  \nSurrey\n\nMarch 7, 1983\n\nDear Dr. Arnold,\n\nI regret to say that I retired from the RSPCA in June 1980, and, while my colleagues were kind enough to forward your letter to my new address in Lancashire, I no longer have access to files and cannot supply you with a copy of the report you requested. Nevertheless, I do recall the case quite well and am happy to pass on what I remember of Miss Butts.\n\nIn fact you're correct in saying I made several visits (four in total) to 30 Graham Road in the months prior to Miss Butts's death. You are also correct in saying that none of the neighbors' complaints was borne out by the evidence. Her cats were well looked after and in excellent condition. However, on none of those visits did I find more than seven cats in residence (six on the last occasion, following the death of one which affected Miss Butts deeply) nor was there any evidence to suggest there were more cats to find.\n\nI made two recommendations on my first visit in March 1978: 1) that she install a cat flap in the kitchen door to allow the animals free access to and from the garden; 2) that she consider having her tomcats neutered in order to mitigate her neighbors' complaints of offensive odor. She followed both recommendations and, despite continued complaints against her, I had no reason to suspect her of neglect and\/or cruelty to her animals. Indeed, I went further and suggested to the police that the complaints were malicious and would warrant investigation. However, I have no idea if anything was done about it.\n\nWhat my colleague and I found in her house on November 15, 1978, was a different matter altogether. Between my last visit\u2014some time in August 78\u2014and that November morning she seemed to have acquired another fifteen cats. If you have a copy of the police incident report, you will know that we found five tomcats dead, and another five severely distressed and injured behind the closed door of the back bedroom. Not to put too fine a point on it, the dead ones had either killed each other or had their necks broken, and the live ones had been so badly tortured and neglected that they were skin and bone, and covered in scratch and bite marks from fighting each other. A decision was made to put three of them down immediately and the other two died within forty-eight hours. Of the remaining eleven cats, all were either neutered or queens, six of which I was able to identify as the cats I had seen on my previous visits.\n\nIn my opinion, the police description of the house as \"squalid\" was an understatement. In truth, it was disgusting. The cat flap in the kitchen had been blocked by a piece of furniture with the result that the animals had been fouling the interior for several days. Miss Butts's own sanitary arrangements were appalling with reeking unflushed toilets and dirty paper and feces on the floors. I cannot stress too highly that I was horrified by what we found, although I have no idea why her circumstances had deteriorated so badly between August and November. There was evidence that she had been drinking heavily\u2014as I recall the police discovered in excess of fifty empty spirit bottles about the house\u2014and this may have contributed to her decline.\n\nI regret I cannot give any clear descriptions of West Indian or Central American ornaments from my previous visits. I do remember that Miss Butts had a number of interesting and colorful displays in her front room but I was never allowed to inspect them long enough to be precise about what they were. Sadly, she viewed me with suspicion because of my uniform and preferred to talk to me in the kitchen. I recall some vivid paintings on the wall opposite the sitting-room doorway and a display of peacock feathers in a brass artillery shell beside the front door; also, a pair of matching silhouette pictures in the hall which she told me represented her grandparents. However, the house was bare of ornamentation on November 15, and I assume she sold everything off to pay for her drinking habit.\n\nWith particular reference to the twenty-plus cats we found in the house, I can only speculate that she began to attract strays to the property following my last visit in August and panicked when the tomcats fought. It seems to me significant that: 1) there was evidence the toms'mouths had been taped, which implies she was trying to find a way to stop them from biting each other; 2) the cat flap was wedged closed with the intention, presumably, of trying to stop more strays from coming in, although why she chose to contain the ones already there remains a mystery. The male cats had received the worst treatment, which I found disturbing\u2014evidence perhaps that Miss Butts had developed some sort of obsession against men in general?\u2014and I do wonder if she chose to contain them for fear of giving her neighbors the proof of neglect and cruelty of which they habitually accused her.\n\nIn conclusion, I have always been sorry that her life ended in the way it did. She wasn't an easy woman to deal with as I'm sure you know. However, despite the official nature of my visits, I believe she looked on me as a friend and I'm saddened that she did not think to call me when my help might have made a difference.\n\nYours sincerely,\n\n**39 LYVEDON AVENUE, RICHMOND, SURREY**\n\nJohn Howlett, Esq.  \nWhite Cottage  \nLittlehampton  \nNr Preston  \nLancashire\n\nMarch 23, 1983\n\nDear Mr. Howlett,\n\nThank you for your letter of March 7.I should tell you that I visited Annie in her house two months before she died and there was nothing then to indicate a deterioration in her circumstances. I am not a cat lover myself, so did not take particular notice of the ones I saw that day. However, had there been more than normal I'm sure I would have registered the fact. Certainly there was no question of the house smelling.\n\nOne of the reasons for my visit was to tell her that I was going away for twelve months. Annie became extremely agitated at the news, which I had been expecting. Sufferers from Tourette's syndrome dislike change, so I sat with her for an hour in her sitting room talking about the colleague who would be taking over in my absence. I had ample opportunity, therefore, to evaluate the room and what was in it. Before I left, she said she wanted to give me a going-away present and invited me to choose something. We spent a further fifteen minutes examining the many treasures she had\u2014most of them quite small\u2014and I can say with absolute certainty that, on that day\u2014 September 8\u2014that room was full of ornaments.\n\nUnfortunately I am having huge difficulty persuading the police that the most likely explanation for the house being \"empty\" nine weeks later is that she was robbed. I have shown your letter to Sergeant James Drury\u2014one of the officers who accompanied you that day\u2014and he tells me that unless I can find someone who saw the interior during the week before her death, he must conclude, as you did, that she sold her possessions herself in order to buy drink. This was his most helpful contribution! Rather less helpful was his suggestion that my memory is at fault or, worse, that I am deliberately lying in order to gloss over my failure to safeguard the health of a patient. Neither is true. I cannot repeat often enough that the last time I saw Annie she was in good mental and physical health. There was no indication that she was drinking more than usual and certainly no evidence of incontinence.\n\nAt the time of her death, I assumed that the only privileged knowledge I had about her was her medical history. However, I now realize I also had privileged information about the inside of her house because I was among the handful of people who were allowed beyond the front door. Even the vicar was made to stand on the doorstep because she mistrusted his perceived friendship with her neighbors. I have located a social worker who was shown into the sitting room in 77 but her description of it, although it accords with mine, has been ruled too out of date to be of value. Sergeant Drury dismisses your recollection of \"vivid paintings,\" \"peacock feathers\" and \"silhouette pictures\" for the same reason\u2014i.e., your last visit was in August 78\u2014on the basis that three months was quite long enough for her to have disposed of the items herself.\n\nI won't bore you with my extreme irritation (and anger!) at having both my memory and my professional expertise questioned by a policeman who is clearly reluctant to reopen an old case, but I do wonder if you could try to picture to yourself what was on the right-hand side of the mantelpiece in the sitting room. The going-away present that Annie gave me came from there and, as I still have it, it would be very helpful indeed if I could demonstrate to Sergeant Drury that in this respect at least I am not \"imagining things.\" A positive and unprompted memory from one of her few \"friends\" would be invaluable.\n\nIt's only fair to tell you that I am extremely unimpressed by both Sergeant Drury and by the coroner, both of whom seem to have paid lip service to their responsibility for investigating Annie's death. While I wouldn't go so far as to say she was murdered\u2014as I understand one of her neighbors did\u2014I certainly believe she was reduced to a state of extreme anxiety by having her house invaded and her treasured possessions stolen. This, in turn, may have led to the deterioration in her circumstances and the overindulgence in alcohol, which was a contributory factor in her death.\n\nYours sincerely,\n\n_Dr. Sheila Arnold_\n\nDr. Sheila Arnold\n\n_White Cottage_  \n _Littlehampton_  \n _Nr Preston_  \n _Lancashire_\n\nDr. Sheila Arnold  \n39 Lyvedon Avenue  \nRichmond  \nSurrey\n\nMarch 24, 1983\n\nDear Dr. Arnold,\n\nI regret I cannot remember the mantelpiece at all, nor what was on it, but my wife has reminded me that one of the pictures in the sitting room was a framed mosaic of an Aztec god\u2014Quetzalcoatl\u2014 otherwise known as the plumed serpent or feathered snake. My wife is a lover of D. H. Lawrence's work and apparently I told her after one of my visits to Graham Road that Miss Butts owned an extraordinary mosaic of \"The Plumed Serpent.\" Sadly, I can barely remember either the picture or the conversation, but my wife is adamant that it was \"the mad black woman with the cats\" who had the Quetzalcoatl on her wall.\n\nTrusting this will be of help,  \nYours sincerely,\n\n**_Correspondence between Dr. Sheila Arnold and  \nRichmond Police\u2014dated 1983_**\n\n**39 LYVEDON AVENUE, RICHMOND, SURREY**\n\nSergeant J. Drury  \nRichmond Police Station  \nRichmond  \nSurrey\n\nMay 25, 1983\n\nDear Sergeant Drury,\n\nRe Miss Ann Butts, 30 Graham Road, Richmond, Surrey\n\nFollowing numerous conversations with you, both in person and on the telephone, I have become increasingly angry about your refusal to investigate the possible theft of property from Miss Ann Butts prior to her death on 14.11.78. In the absence of any other explanation, I am forced to conclude that Richmond Police are as indifferent to Miss Butts today as they clearly were at the time of her death.\n\nIt is unacceptable to say, as you did on the telephone this morning, that \"anyone as mad as Mad Annie could easily have blown a fortune on drink over a nine-week period.\" As your own report from the time states, she had \u00a34,000 in her bank account and \u00a315,000 in a building society. Therefore there was no necessity for her to sell her prized possessions as you claim she must have done. Nor can I stress too strongly that Tourette's syndrome is not a form of insanity, but rather an inability to control certain motor functions, and the fact that Miss Butts made faces and muttered to herself in no way affected her intelligence.\n\nI am now convinced that her extraordinarily rapid decline must have been due to her house being ransacked in the week before her death. I have said to you many times that an invasion of her property would have triggered extreme anxiety because of her compulsive\u2014therefore uncontrollable\u2014obsession with home and personal security, and it is pointless to keep arguing that she would have called the police if such an invasion had happened. All strangers worried her, including officials in uniform (cf. John Howlett's letter, dated March 7, '83) and if you and your colleagues treated her while she was alive with the sort of indifference you seem to be demonstrating now, she would have had no reason to trust you. In this one respect\u2014trusting strangers\u2014Ann's behavior could be described as irrational but only because obsessive behavior is compulsive. In all other respects her behavior was normal.\n\nI hesitate to say that your indifference amounts to contempt, although I am angry enough to believe that that is what it is. Yes, Ann suffered a neuropsychiatric disorder, and yes, she was black, but neither fact should influence your decision to pursue belated justice on her behalf.\n\nOf course it's true\u2014and I am quoting your own words\u2014that the cost of pursuing her alleged robbers will far outweigh any benefit to the taxpayer from the recovery of her possessions, but since when did justice have anything to do with cost? Justice is, and should be, impartial, yet your remark suggests that the police are selective in how, when and for whom they enforce the law.\n\nYours sincerely,\n\n_Sheila Arnold_\n\nDr. Sheila Arnold\n\nc.c. Police Superintendent Hathaway, Richmond Police Rt. Honourable William Whitelaw, home secretary\n\n**From the office of:**\n\n**Police Superintendent A. P. Hathaway** ,\n\n**Metropolitan Police, Richmond**\n\nDr. Sheila Arnold  \n39 Lyvedon Avenue  \nRichmond  \nSurrey\n\nJune 21, 1983\n\nRef: APH\/VJ\n\nDear Dr. Arnold,\n\nRe Miss Ann Butts, 30 Graham Road, Richmond\n\nThank you for the copy of your letter of May 25 to PS Drury, together with photocopies of correspondence and notes of telephone conversations, all of which I have read with interest. I have since discussed the case at some length with Sergeant Drury and, while I have some sympathy with your contention that Miss Butts was robbed prior to her death, I also agree with Sergeant Drury that no purpose would be served by investigating it.\n\nSergeant Drury admits that the inquiry in November 1978 did not take into account the possibility of robbery, however he stresses that at _no time_ was it suggested to him that the situation found inside Miss Butts's house was unusual. Quite the reverse. There was considerable evidence, already on record following complaints from her neighbors, that the house was overrun with cats; that there was a continuous unpleasant smell from the premises; and that her living conditions were unhygienic and squalid. In these circumstances I do not consider that Sergeant Drury was either indifferent or negligent in his handling of the case.\n\nThe incidence of theft and burglary in England and Wales is rising at over 15 percent per annum, with few successful convictions resulting from police investigations. These figures are a matter of public record, and politicians from all parties are now demanding tougher sentencing policies and increased funding for police forces in order to stem what has effectively become a crime epidemic.\n\nIn such a climate it would be unreasonable to order an investigation into a burglary that may or may not have happened five years ago; where the alleged victim is no longer alive to give evidence; where there is no accurate inventory of what was in her house; and where the chances of successful closure are zero. While I realize this is not what you want to hear, I hope you will understand the reasoning behind this decision. It would be different had there been any question marks over the manner of Miss Butts's death, but the inquest verdict was unequivocal.\n\nIn conclusion, let me assure you that Richmond Police take their responsibilities to all members of the public extremely seriously, irrespective of race, color, creed or disability.\n\nYours sincerely,\n\nPolice Superintendent A. P. Hathaway\n\n# four\n\n\"One of your letters to the RSPCA inspector mentions a going-away present that Annie gave you,\" I said to Sheila Arnold when she and her husband came to lunch the following Sunday. \"What was it?\"\n\nShe extended an arm. \"A jade bracelet,\" she said, turning a pale green bangle on her slender wrist. \"There was a set of them on her mantelpiece and she chose this one for me because she thought it suited my coloring. I had red hair in those days.\"\n\n\"I remember,\" I said.\n\nHer husband, Larry, a tall, soft-spoken American, stirred in his seat. \"In fact it's jadeite,\" he said, \"which is the most expensive of the jades. We had it assessed in '83 so that Sheila could demonstrate to the police that she wasn't imagining the value of what was in Annie's house.\" He circled the bracelet with a finger and thumb. \"It's of Mexican origin... probably eighteenth century... worth in excess of \u00a3200. Considering Sheila thinks there were ten in the set, it gives you some sort of starting point for estimating Annie's wealth.\"\n\nSam gave a low whistle. \"No wonder you wanted the police to investigate.\"\n\nSheila sighed. \"I still feel I should have pushed a bit harder... at the very least forced Drury to face a disciplinary hearing. He was appallingly negligent. Worse, a racist. He just assumed a black woman would be living in squalor.\"\n\nLarry clicked his tongue impatiently. \"That's twenty-twenty vision speaking. I agree the man was an asshole but he was correct about one thing... no one suggested there was anything odd about the house... even John Howlett, the RSPCA inspector, didn't challenge the conditions.\" He spoke with surprising firmness as if the subject were a touchy one between them. \"And there weren't enough hours in the day for you to commit any more time to Annie's cause, not with your practice and two kids to bring up. Also,\" he went on, turning to us, \"the Superintendent made sense when he talked about zero success rates. Sheila made a list of the things she remembered but it was very vague on detail and, as the police pointed out, there was no hope of a prosecution if she couldn't be more positive in her descriptions. In the end it seemed pointless to go on.\"\n\nWe were sitting outside on the terrace under the shade of a worn umbrella which had had most of its color bleached out by long summers of sunshine. The garden fell away at the back of the house and some sensible person in the distant past had had the foresight to construct a raised platform out of Portland stone, which gave a glorious view of the other side of the bowl-shaped valley in which we lived. It seemed strange to me how the English climate had changed during the years we'd been away. I had always thought of it as a green and luscious place, but the garden, paddocks and fields had turned brown in the heat, and the drought-starved flowers drooped their heads. Sheila and Larry were sporting matching panama hats and they made an elegant couple, she in a primrose-colored cotton dress, he in white shirt and chinos. I guessed him to be about ten years older than she was, and I wondered where they'd met and when they'd married, and whether the two children he'd mentioned were his or a previous husband's.\n\nI leaned across the table to refill their wine glasses while I thought lazily about going inside to fetch lunch, a simple affair of cold meat, salad and French bread. \"If it was one of her neighbors who robbed her,\" I said idly, \"they might have kept some of the pieces, particularly if they weren't of any value. The peacock feathers in the artillery shell, for example... the one John Howlett described. When I read his letter I couldn't help thinking they were the kind of things someone would hang on to, if only because feathers could never be specifically identified as Annie's.\"\n\nSheila eyed me curiously. \"You seem to have quite a down on her neighbors,\" she remarked. \"Why is that?\"\n\nSam answered for me. \"The whole damn street took against her after she labeled them racists at the inquest. They plagued us for weeks with abusive phone calls. It's the reason we left England.\"\n\n_Liar!_ I thought.\n\n\"No wonder you hate them,\" said Larry sympathetically.\n\nIt was a throwaway line, which Sheila, with a questioning lift of her eyebrows, invited me to expand upon. Instead I stood up and said it was time for lunch. I had learned to talk about threatening phone calls without becoming strident...\n\n... but _hate?_ That was a different matter entirely.\n\nSheila and I walked down to the paddock after lunch and leaned on the rail to watch the horses nibble halfheartedly at the withered vegetation. \"Larry and I always assumed it was professional thieves,\" she told me. \"I don't think it ever occurred to us that it might have been someone closer to home.\"\n\n\"How would professionals know what was in the house?\" I asked. \"You said yourself she never let anyone inside.\"\n\n\"That's equally true of her neighbors,\" she pointed out reasonably. \"She was more suspicious of them than she was of strangers.\"\n\n\"They used to look through her windows,\" I said, remembering how I'd come across a gang of young thugs making faces at her through the glass. \"The children were the worst. They thought it was funny to frighten her.\"\n\nSheila caught at the brim of her hat as a warm breeze blew across the field. \"Larry's convinced it was whoever did that valuation she showed me. He thinks it was a scam\u2014someone knocking on doors and posing as an art or antiques expert in order to find out which houses were worth robbing.\"\n\n_It makes sense_ , I thought.\n\n\"But I don't agree with him,\" she went on. \"I'm almost certain it was a Sotheby's valuation because I remember thinking that the figures must be right if a bona fide auction house had come up with them.\" She sighed. \"And now I'm furious with myself that I didn't question it at the time. I mean, the whole episode was very odd. What prompted her to get a valuation? And how on earth did she gear herself up to letting a stranger loose on her treasures?\" She shook her wrist and rattled the jade bracelet against her watch. \"When she asked me to choose a present she wouldn't let me touch anything. I had to choose by sight, not by feel.\"\n\n\"When did she show you the valuation?\"\n\n\"Sometime during the summer. I remember she was particularly difficult that day. One minute she wanted me to read it, the next she snatched it away as if she thought I was going to steal it. She used to get caught in mental loops that made her repeat the same words and actions over and over until something new pushed her on to another track. She could be very tiresome when she was in that sort of mood, which is probably why I didn't question what the valuation was for.\"\n\n\"An insurance condition?\" I suggested. \"No valuation, no insurance.\"\n\nShe gave an exasperated sigh. \"That's what the police said and it made me boiling mad. 'You can't have it both ways,' I told them. Either she was a mindless cretin who let cats and drink destroy her life, or she was so switched on that she was able to organize insurance for herself. It might have helped if I could have talked to her bank manager but by the time I got around to thinking about it he was long gone. Someone told me he was working in Saudi Arabia but I never followed it up.\"\n\n( _I_ had, and I could remember the man's answer verbatim down a crackly line from Riyadh. \"I can't help you, I'm afraid. Unfortunately Miss Butts decided I was stealing her money, so I passed her account to my deputy, who died five years ago.\")\n\n\"Did you think of contacting Sotheby's to find out if they still had a copy of the valuation and why she wanted it?\" I asked.\n\n\"No, but it wouldn't have made any difference if I had,\" she said with a dry laugh. \"Larry started getting stroppy about the amount of time I was wasting so I put the husband and children first and let Annie go.\"\n\nI thought about Sam's fury over the policeman in Hong Kong. \"It's very irritating, isn't it?\"\n\n\"What?\"\n\n\"Doing your duty.\"\n\n\"Yes.\" She pulled a wry smile. \"The worst is yet to come, though.\"\n\n\"How do you mean?\"\n\n\"Larry's older than I am and he's here under sufferance until I reach pensionable age... and that's only two years away. Then we retire to his condo in Florida.\"\n\n\"Why?\" I asked curiously.\n\n\"It's the bargain we made when he took on me and the children.\" She read my expression as criticism. \"We don't have the same sort of marriage as you and Sam. The plan was to return to the States when Larry retired, but he agreed to wait after I was offered this job in Dorset. He said he could tolerate another few years as long we weren't in London.\" She sighed. \"It's a long story... full of compromise.\"\n\n\"It sounds it,\" I said sympathetically. \"Do you want to live in Florida?\"\n\n\"No,\" she said honestly, \"but I want lonely old age even less. I've seen too much of it to consider it as an option.\"\n\nIt was a salutary warning, coming from a doctor. \"What makes you think Sam's and my marriage is any different?\"\n\nShe shrugged. \"He wouldn't abandon you if you gave him an ultimatum.\"\n\nI was about to point out that Sam had done it once and there was no reason to imagine he wouldn't do it again. But I realized she was probably right. Somewhere along the line our roles had changed and it was Sam who feared ultimatums now. \"He's more afraid of loneliness than I am,\" I said slowly, \"which means I hold the cards in our relationship...just as Larry does in yours.\"\n\nShe glanced at me in surprise. \"That's a very calculated way of looking at it.\"\n\n\"Born out of experience,\" I said lightly. \"I think real loneliness is to be abandoned _inside_ a relationship... to find yourself questioning your worth all the time. I know what that's like, and I know I can survive it. I imagine the same is true of Larry. He's been there, done it, got the T-shirt... and you haven't. Neither has Sam. It puts you both at a disadvantage.\"\n\n\"Larry wouldn't know what loneliness was if it smacked him in the face,\" she protested. \"He's the most gregarious creature I know. It drives me to distraction sometimes. I'm constantly being hauled out to social functions when all I want to do is sleep because I'm dog-tired from pandering to the ill all day.\"\n\nI smiled at her. \"That's the whole point. You're leading a fulfilled existence and Larry isn't. He has to go outside to find a sense of purpose. Yours is so strong you just fall asleep and prepare for the next day's challenges.\"\n\nShe propped her arms on the fence and stared across the field. \"Is this your way of telling me Annie was your sense of purpose?\"\n\n\"Part of it.\"\n\n\"You had babies,\" she said. \"Didn't they fill a gap?\"\n\n\"Did yours?\"\n\n\"No, but then I had a career. In any case I'm not remotely maternal. I can cope with my patients being totally dependent on me... but not my children. I expect my children to fend for themselves.\"\n\nI wondered if she was listening to herself, and whether she'd asked Larry how he felt about the professional\/private divide. \"Mine just added to the general anxiety,\" I said, joining her at the fence. \"My elder one did, anyway. We moved to Hong Kong while I was pregnant and a child was about the last thing I needed at that stage.\"\n\n\"How did Sam take it?\"\n\n\"Blindly.\"\n\nSheila gave a snort of laughter. \"What's that supposed to mean?\"\n\n\"He had a son,\" I said dryly. \"He was thrilled... just so long as someone else looked after it.\" We stood in companionable silence for a few moments, understanding each other. \"Do you still have a copy of the list you made of Annie's possessions?\" I asked her next.\n\n\"Isn't it in the file?\"\n\n\"No.\"\n\nShe looked doubtful. \"I'll have a look when I get home... the problem is we threw so much out when we moved down here seven years ago. The other thing that's missing is the correspondence I had with the social worker. I remember she wrote a long letter describing the interior of Annie's house, but none of it was in the file when I photocopied it for you. I'm afraid it must have come adrift during the move.\"\n\nI wondered what else had come adrift and indulged in a few dark thoughts about Larry, who clearly wasn't above a little sabotage to ensure that his needs came first. _Shades of Sam?_ \"Could you make another list?\"\n\n\"I can try. It won't be as detailed as the first. What do you expect to find?\"\n\n\"Nothing valuable,\" I said. \"Little things that someone might have kept.\"\n\n\"Like the peacock feathers?\"\n\nI nodded.\n\n\"They could never be used as evidence.\"\n\n\"I know but...\" I hesitated, afraid of sounding ridiculous. \"It's a stupid idea really but supposing you put on your list the peacock feathers, the silhouette pictures of her grandparents and... well, other things of little or no value... a wooden statue, say....\" I ran out of ideas. \"I just thought that if I found someone with a similar combination in their house, I'd at least feel I was on the right track.\"\n\nShe threw me a startled glance. \"Does that mean you're going to look?\"\n\nI shrugged self-consciously.\n\n\"But where would you start, for goodness'sake?\"\n\n\"Graham Road? There must be someone left who was there in 1978. If I knock on a few doors I might come up with something.\" I spoke only to give her an answer, not because I had any intention of taking such a scattergun approach. I watched her expression change to one of skepticism.\n\n\"But why? It'll just be a lot of hard work for nothing. Larry was right when he said there'd be no prosecution.\"\n\n\"I wouldn't be looking for a prosecution for theft, Sheila; I'd be looking for a prosecution for murder. As the chief superintendent said in his letter to you, it would be different if there were question marks over Annie's death.\" I smiled. \"Well, there are... and I intend to prove it.\"\n\nShe searched my face intently for a moment. \"What really happened between you and Annie that night?\" she asked abruptly. \"Drury showed me your statement, but you said she never spoke to you.\"\n\n\"She didn't.\"\n\n\"Then... _why?\"_\n\n\"I've got nothing better to do at the moment.\"\n\nIt wasn't much of an explanation but it seemed to satisfy her. \"I doubt many of her neighbors will still be there,\" she warned. \"Most of them had moved on even before we left.\"\n\n\"What about the vicar?\" I asked. \"He was always visiting people in Graham Road.\"\n\nShe pulled on the brim of her hat to shade her eyes from the sun. \"I don't think he's there anymore.\"\n\nI lifted one shoulder in a relaxed shrug. \"His successor at St. Mark's ought to be able to tell me where he is. Do you know his name?\"\n\n\"The new vicar? No.\"\n\n\"What about the one who knew Annie?\"\n\nShe didn't answer immediately, and I turned to look at her. Her expression was impossible to read because her eyes were still in shade, but the set of her jaw was very grim. \"Peter Stanhope,\" she said.\n\n**_Letter from Libby Williams\u2014-formerly  \nof 21 Graham Road\u2014dated 1982_**\n\n# five\n\nAn immediate pall fell over our little party when Sheila told Larry that I was planning to look up Peter Stanhope to see if he knew what had become of Annie's possessions. Neither of them seemed remotely interested that he had never been inside her house, and couldn't possibly know what possessions she had. His name alone spelled depression.\n\nLarry didn't like the idea at all and watched me warily from behind his wine glass, while Sam flicked worried glances around the three of us, clearly wondering who Peter Stanhope was and why his name should cause Larry concern. Sam became rather loud as a result\u2014he always hated finding himself at a disadvantage\u2014and, in an unkind way, I took pleasure from his embarrassment. He had only himself to blame, after all, for it was he who had imposed a silence on the whole subject.\n\nI spent half an hour that evening trying to locate the Reverend Peter Stanhope through directory inquiries, but no one of that name was listed in Richmond and the operator refused to look for Rev P. Stanhopes in other parts of England. Nor was there a listing for St. Mark's Church and, as I didn't know the name of the present vicar, I couldn't obtain the number of the vicarage either. It would all have been a great deal easier if Sam hadn't stood over me while I did it\u2014I could have suggested the operator try Stanhopes in Exeter, but I wasn't ready to show my hand quite so blatantly. In the end, and half-jokingly, I suggested Sam phone Jock Williams, a confirmed atheist, and ask him to drive to St. Mark's Church from his house on the other side of Richmond to see if the new vicar's name was printed on the board outside. To my surprise he agreed.\n\n\"He wants to know what's up,\" Sam said on his return to the kitchen where I was doing the washing up.\n\n\"What did you tell him?\"\n\n\"That the boss would have my guts for garters if I didn't help her track down 'Mad Annie's' missing valuables.\" He gave a quirky grin. \"He thought you were 'round the bend twenty years ago. Now he thinks we've both lost it. He asked me why anyone would think an old tramp like Annie had valuables.\"\n\nI propped a plate on the drain board. \"What did you say?\"\n\n\"Repeated what Larry told us about jadeite. It gave him a bit of a shock, as a matter of fact... said he didn't think Annie had two brass farthings to rub together.\"\n\n\"I expect he'd have been nicer to her if he'd known,\" I said tartly. \"Jock always responds better to the chink of money.\"\n\n\"Mm, well, he's now advising me to put my huge gains from Hong Kong into some offshore fund he's operating out of the Isle of Man. He's got a wheeze for avoiding tax and he's prepared to cut me in on the deal if I'm interested.\"\n\n\"Knowing Jock, it'll be illegal.\"\n\n\"Unethical certainly,\" said Sam cheerfully, \"but then he doesn't believe in a welfare state. Says it's against Darwin's theory of evolution. The sick, the lame and the poor are supposed to die. That's how natural selection works.\"\n\nI held up a fork to examine the prongs. \"He'll get his comeuppance one day,\" I said. \"Arrogant, self-serving bastards always do. That's the _unwritten_ law of natural selection\u2014old bulls die painfully.\" I eyed him suspiciously. \"I hope you told him where to stuff his tax wheeze.\"\n\n\"Not likely,\" he said. \"The only reason he's driven off to St. Mark's on a Sunday evening is because he thinks I'm going to swell his coffers with megabucks.\" He straddled a chair. \"How come you and Jock know each other so well? As I remember it, you used to avoid him as much as possible.\"\n\nThe question took me by surprise. \"What kind of 'knowing'are we talking about?\"\n\n\"I've no idea. That's why I'm asking.\"\n\nI tried unsuccessfully to hide a smile. \"Are you hinting at the biblical kind?\"\n\n\"Maybe.\"\n\nI snorted laughter through my nose. \"That's funny.\"\n\n\"Why?\"\n\n\"He's a boring little squirt with a power complex,\" I said. \"Even his wife didn't fancy him, so I don't know what makes you think I would.\"\n\n\"It was just a question,\" he said huffily\n\n\"What brought it on?\"\n\n\"He wasn't surprised when I told him you'd taken up Annie's cause again. He said he'd been expecting it.\"\n\n\"So?\" I asked curiously.\n\n\"He seems to know you better than I do. I thought you'd forgotten all about her. You haven't mentioned her name in twenty years.\"\n\n\"You asked me not to.\"\n\n\"Did I?\" he said with a puzzled frown. \"I don't remember.\"\n\nI wasn't sure how genuine the frown was, so I changed the subject. \"You shouldn't believe everything Jock tells you,\" I said. \"He's needling you, just as he's needling you about how much money he has. He enjoys making you squirm.\"\n\n\"Why?\"\n\nI shook my head at his na\u00efvet\u00e9. The trouble with my husband, I sometimes thought, was that he was too ready to take people at face value. It should have been a disadvantage to him in his career, but oddly enough it worked the other way because people responded positively to his easy acceptance of the image they wanted to present. When I first knew him, I thought he was using a peculiarly sophisticated form of reverse psychology but, as time passed, I came to understand that he genuinely had no conception of the sides that exist in most people's natures. It was his most attractive quality.... It was also the most irritating....\n\n\"Jock's a stirrer,\" I said lightly. \"He resents other people's happiness... particularly where relationships are concerned. He's only ever known disasters... divorced parents... a brother who killed himself... a failed marriage... no children.\" I pointed a pan scourer at Sam's heart. \"He wouldn't be needling you at all if you'd told him about your coronary and hadn't lied about how much money you've made. As far as he's concerned you've got everything. Health. Wealth. Happiness. Early retirement. A faithful wife. _And_ sons.\"\n\nSam laced his fingers behind his head and stared at the ceiling. \"He never got over his brother's death,\" he said.\n\n\"So you always say, but you never explain why.\" \"I didn't want you jumping to conclusions.\" I frowned at him. \"How did the brother kill himself?\" \"Hanged himself from a tree one day. There was no suicide note so the police thought it was murder and most of the suspicion fell on Jock because he took some money from the kid's bedroom after he was dead. In the end the coroner accepted that the boy had been depressed about his parents'divorce and came down on the side of suicide, but it wrecked the entire family, according to Jock. They all ended up blaming each other.\"\n\n\"That's sad,\" I said, meaning it. \"How old was he?\" \"Sixteen. Three years younger than Jock.\" \"God, that really _is_ sad. What happened to the parents?\" \"Jock lost all contact with them after the divorce. I don't think he even knows where they are anymore... whether they're alive... or whether they still care about him. He claims it doesn't worry him, then spends every waking hour trying to prove he's a man to be reckoned with.\" Sam lowered his eyes to look at me. \"It doesn't stop him being an arrogant, self-serving bastard but it probably explains the reasons behind it.\"\n\nIt explained a lot, I thought, as I promised to make a point of being pleasant when Jock came back with the name of St. Mark's vicar. What it didn't explain was where Jock had found the extra money that had allowed him to trade up his share in 21 Graham Road for a more impressive, and expensive, property near Richmond Park.\n\nIt was Wednesday before I was able to speak to Peter Stanhope in person. My previous calls had been answered by a recorded message and it seemed unreasonable to fill his tape with long explanations of who I was and why I wanted to talk to him. His new parish was in Exeter, about sixty miles west of Dorchester, and I was about to begin a letter to him when he answered the phone on Wednesday morning.\n\nI had spoken to him only once when we lived in Richmond, and I wasn't confident that he'd remember me as well as I remembered him. I gave him my name and said I wanted to talk to him about Annie Butts, \"the black woman who was run over by a truck.\"\n\nThere was a long pause and I had time to recall Libby's description of him as \"a fat little bloke with sweaty palms.\" I was beginning to wonder if the reason for the silence was because the phone had slipped from his hand, when he suddenly barked, \"Did you say Ranelagh? Any connection with the woman who claimed Annie was murdered?\"\n\n\"That's who I am,\" I said. \"I didn't realize the name would mean anything to you.\"\n\n\"Oh, goodness me, yes! You were quite famous for a while.\"\n\n\"For all of fifteen minutes,\" I agreed dryly. \"They weren't the most pleasant fifteen minutes of my life.\"\n\n\"No, I don't suppose they were.\" A pause. \"You had quite a tough time of it afterward.\"\n\n\"Yes.\"\n\nHe clearly didn't like one-word answers and sought for a change of subject. \"Someone told me you and your husband went abroad. Did that work out all right?\"\n\nI guessed it was his polite way of asking me if I was still married, so I assured him I was, gave him a thumbnail sketch of our twenty years away, mentioned my two boys, then asked him if I might pay him a visit. \"To talk about Annie's neighbors,\" I explained, wishing I could put a little more enthusiasm into my voice at the prospect of seeing him again. I was relying on his sense of duty to agree to the meeting, but I didn't believe he had any more relish for it than I did.\n\nA more noticeably guarded edge crept into his voice. \"Is that wise?\" he asked. \"Twenty years is a long time, and you seem to have done so well for yourselves... stayed together... made a family... put the unpleasantness behind you.\"\n\n\"You remember our little chat then?\" I murmured. \"I didn't think you would.\"\n\n\"I remember it well,\" he said.\n\n\"Then you'll understand why I want to talk about Annie's neighbors.\"\n\nI heard his sigh down the wire. \"What good will it do to rake over dead ashes?\"\n\n\"It depends what you find,\" I said. \"My father put a log on the fire once and a gold sovereign dropped out of it as it burnt. Someone had obviously hidden it in the tree and a couple of centuries later my father reaped the reward.\"\n\nAnother pause. \"I think you're making a mistake, Mrs. Ranelagh, but I'm free on Friday afternoon. You're welcome to come any time after two o'clock.\"\n\n\"Thank you.\" It was my turn to pause. \"Why am I making a mistake?\"\n\n\"Revenge is an unworthy ambition.\"\n\nI stared into a gilt-edged mirror that was hanging on the wall in front of me. It was old and cracked and, standing where I was then, it produced a lengthened image that made my face look thin and cruel. \"It's not revenge I'm after,\" I said with studied lightness. \"It's justice.\"\n\nThe vicar gave an unexpected laugh. \"I don't think so, Mrs. Ranelagh.\"\n\nI had no intention of taking Sam to Exeter so I told him it was pointless for the pair of us to go when the lawn needed mowing and the flower beds tidying. He seemed happy enough although I caught him looking at me rather strangely over breakfast. \"What's the matter?\" I asked.\n\n\"I was just wondering why everyone seems to be moving to the west country,\" he said.\n\nPeter Stanhope's parish was in the St. David's area of Exeter. I arrived too early and sat at the end of the road for an hour, watching the world go by through my windshield. I was on the edge of the university campus, and most of the pedestrian traffic seemed to be students\u2014groups of boys and girls carrying books, or young couples clamped at the hip and shoulder like Siamese twins. I found myself envying them, particularly the skimpily clad girls in bottom-hugging skirts and crop tops, who swung along in the sunshine and radiated the sort of confidence I had never had.\n\nThe original vicarage was an impressive Victorian mansion, hidden behind high hedges, with a real estate agent's board outside, advertising a \"desirable penthouse flat\" for sale. The new vicarage was a cheaply constructed cube, across the road from the church, lacking both charm and character. As I parked outside at exactly two o'clock, I was beginning to wish I'd had the sense to spend the last hour in a pub. Dutch courage would have been better than no courage at all. A part of me thought about driving away with my tail between my legs, but I noticed a net curtain twitch in a downstairs window and knew I'd been seen. Pride is always a stronger motivator than courage.\n\nThe door was opened by a tall, cadaverous-looking woman with a beak of a nose, shoulder-length grey hair and the speed of delivery of a machine gun. \"You must be Mrs. Ranelagh,\" she said, taking my hand and drawing me inside. \"I'm Wendy Stanhope. Peter's running late. It's his morning at the shelter. Battered wives, poor souls. Come into the kitchen. He told me you were driving from Dorchester. Are you hungry? What about a drink? Chardonnay do you?\"\n\nI followed her across the tiny hall. \"Thank you.\" I looked around the white melamine kitchen which was mind-numbingly uniform and hardly big enough to swing a cat. \"This is nice.\"\n\nShe thrust a glass into my hand with long, bony fingers. \"Do you think so?\" she asked in surprise. \"I can't stand it myself. I much preferred the one we had in Richmond. The church doesn't give you much choice, you see. You have to make do with whatever pokey little kitchen they give you.\" She took a breath. \"But there you go,\" she went on cheerfully, \"I've only myself to blame. No one forced me to marry a vicar.\"\n\n\"Has it been a good life?\"\n\nShe filled her own glass and tapped it against mine. \"Oh, yes, I don't have many regrets. I wonder sometimes what it might have been like to be a lap dancer, but I try not to dwell on it.\" Her eyes twinkled mischievously. \"What about you, my dear?\"\n\n\"I don't think I've got the body for it,\" I said.\n\nShe laughed happily. \"I meant, has life been good to you? You're looking well, so I assume it must have been.\"\n\n\"It has,\" I said.\n\nShe waited for me to go on and, when I didn't, she said brightly, \"Peter tells me you've been living abroad. Was that exciting? And you've two boys, I believe?\"\n\nThere was so much blatant curiosity in her over-thin face that I took pity on her\u2014it wasn't her fault that her husband was late\u2014and talked enthusiastically about our years abroad and our children. She studied me over the rim of her glass while I spoke, and there was a shrewd glint in her eyes that I didn't much like. I wasn't used to having people see straight through me, not after so many years of growing an impenetrable skin.\n\n\"We've been lucky,\" I finished lamely.\n\nShe looked amused. \"You're almost as good a liar as I am,\" she said matter-of-factly \"Most of the time I can contain my frustration, but every so often I drive to a wide-open space, usually a cliff top, and scream my head off. Peter knows nothing about it, of course, because if he did he'd think I was mad and I simply couldn't bear to have him fussing round me.\" She shook her Lear-like locks in grotesque parody of a lap dancer. \"It's quite absurd. We've been married forty years, we have three children and seven grandchildren, yet he has no idea how much I resent the utter futility of my existence. I'd have made an excellent vicar, but my only choice was to play second fiddle to a man.\"\n\n\"Is that why you scream?\"\n\nShe refilled my glass. \"It's more fun than having a hangover,\" she said.\n\n**_Psychiatric report on_**  \n ** _Mrs. M. Ranelagh\u2014dated 1979_**\n\n**QUEEN VICTORIA HOSPITAL**\n\n_Hong Kong_  \n _Dept. of Psychiatry_\n\nA consultation was requested for Mrs. M. Ranelagh of 12 Greenhough Lane, Pokfulam, Hong Kong, by her general practitioner, Dr. J. Tang, querying postpartum depression after the birth of her son, Luke (DOB 20.10.79). According to her husband, she has been suffering from depression for some time. She refuses all medication. Mrs. Ranelagh had a two-hour consultation with Dr. Joseph Elias on December 19, 1979.\n\n(The following extracts are taken from Dr. Elias's report, which was released to Mrs. Ranelagh in February 1999.)\n\n... Mrs. Ranelagh was a difficult patient. She insisted on making it clear from the outset that her only reason for attending was to prove once and for all that she wasn't suffering from depression. She was uncooperative and angry. She expressed considerable hostility toward \"men in authority\" and \"people who throw their weight about,\" and referred to \"coercion,\" \"bullying\" and \"intimidation\" on a number of occasions. When I suggested to her that, far from persuading me to give her a clean bill of health, such statements were leading me to question the existence of a paranoid disorder, she agreed to cooperate.\n\n... She admits to feelings of emotional turmoil following various events that happened at the end of last year and the beginning of this in London. She refused to discuss these in any detail for fear of confirming my suspicion of paranoia, however, she touched on three\u2014two of a highly personal nature\u2014to explain her \"anger.\" She produced a number of newspaper clippings as evidence that the first incident had occurred\u2014the death of a black woman\u2014but was unable to support her other allegations. Without independent confirmation, I cannot say whether the subsequent incidents a) happened or b) are a construct to validate her sense of injustice re the black woman's death.\n\n... The main focuses of her resentment are her husband (resident with her in Hong Kong) and mother (resident in England), whom, for various reasons, she feels betrayed her. This has resulted in a \"coldness\" toward them, which she \"needs time to overcome.\" She describes her pregnancy as \"ill-conceived\"\u2014(pun intentional?)\u2014pointing to the difficulties of starting a new life abroad while carrying a baby to term. She talks lovingly about the child, calling it \"my baby,\" while blaming her husband for \"exposing her to an unplanned pregnancy.\" She retains a close bond with her father (resident in England), whom she contacts regularly by telephone and who is her only confidant. In addition, she listed a number of related problems: a dislike of being touched; feelings of insecurity when alone in her house; an obsession with hygiene; dislike of certain sounds\u2014i.e. doorbells, London accents, rats scratching(?).\n\n... I advised her against forging alliances\u2014particularly with her father, who is \"conducting some research\" for her\u2014which her husband will almost certainly see as betrayal if he finds out. I also pointed to the potential danger of making an ally of her son as he matures. She concurred on both counts, but remains adamant that her marriage will end tomorrow if she forces another confrontation with her husband. This is not what she wants. She rejected my offer of a joint session with herself and Mr. Ranelagh, as she believes that neither of them would be able to talk honestly without causing the immediate separation referred to above. Her feelings for her husband are confused. She seems to re tain a close bond with him despite her resentment and believes her decision earlier this year to stay in the marriage was the right one. Nevertheless she is intent on punishing him for sins of \"omission and commission.\"\n\n... Mrs. Ranelagh presents herself as an intelligent, self-aware woman who is trying to come to terms with some extremely unpleasant, and as yet unresolved, issues in her life. Once satisfied that she had persuaded me she was not a \"depressive\"\u2014a view I encouraged\u2014 she talked at length about her intention to seek \"closure,\" although she is clearly ambivalent about what sort of closure she wants. In simple terms, she prefers the more anodyne description of \"justice\" for her black friend to the rather more accurate one of \"revenge\" for herself.\n\n... When I warned her that prolonged internalized anger, be it well-founded or capricious, could lead to the sort of paranoid disorder\u2014persecutory, delusional, phobic\u2014that she was so determined to dissociate herself from, she said the damage had already been done. \"I'm between a rock and a hard place, Dr. Elias. I'm a coward if I give in and a neurotic bitch if I fight back.\"\n\n... In conclusion, I can find no evidence of depression in this patient. She is obsessive and extremely manipulative, but is also well in command of herself. I found her rather frightening....\n\n# six\n\nIn the end I exchanged less than twenty words with Peter Stanhope. He bustled in half an hour late, full of apologies for the delay, only to be sidetracked almost immediately by a telephone call. Pausing only to say it was important, he disappeared into his study and left his wife to murmur courtesies into the receiver until he picked up the extension. It hardly mattered. Wendy was a mine of information, and I was fairly sure they weren't the sort of facts I would ever have had from her husband since most of it was gossip, and some of it was scurrilous.\n\nWhile waiting for Peter's return we had transferred to the sitting room, where Wendy had tried to relieve me of my small single-shoulder rucksack, not realizing that it was held by a buckle across my chest. She was surprised by how heavy it was and how reluctant I was to let it go. I relented enough to unbuckle the chest strap and lower it to the sofa beside me\u2014but if she wondered why I needed to bring the kitchen sink on my travels she was too polite to say anything. I was clearly an enigma to her, for whatever picture she had in her mind of a crusading zealot it certainly wasn't me.\n\nShe made a small moue as she replaced the receiver, and I wondered how often she was left to hold the fort and how accommodating Peter would be if their roles were reversed and she were the vicar and he the helpmeet. My expression must have been more revealing than I realized.\n\n\"Has he let you down, my dear?\" she said into the silence.\n\n\"Not at all,\" I assured her. \"I wanted to talk about Annie's neighbors on Graham Road and I think you probably know more about them anyway.\"\n\nShe fixed me with her all-seeing eyes. \"I meant in the past,\" she said gently. \"Did he let you down before?\"\n\n\"In a way,\" I said, looking about the room to avoid looking at her. \"He told me I was hysterical when I wasn't.\" Wendy was apparently a collector of porcelain figures because every surface seemed to have them. She had a fine array of white Dresden ladies along her mantelpiece and some tiny hand-painted birds in a small glass cabinet on the wall. Photographs were her other love, with pictures of her family everywhere, and a huge blown-up snapshot of seven laughing children on one wall. \"Who are they?\" I asked, nodding in their direction.\n\nShe accepted the change of tack without demur. \"My grandchildren. It was one of those rare moments when they all looked their best.\" She gave a little chuckle. \"Usually one of them can be relied on to scowl.\"\n\n\"Who took it?\"\n\n\"I did.\"\n\n\"It's brilliant,\" I said truthfully. \"Forget being a vicar, you should have been a professional photographer.\"\n\n\"I was for a time... well, semi-professional. I used to do the weddings at St. Mark's, particularly for the couples who didn't have much money to spend.\" She pulled open a drawer in a desk to one side of the fireplace and produced a bulging photograph album. \"I think this might interest you. Most of Annie's neighbors are in here somewhere.\"\n\nShe passed it across to me and I flicked my way through a pictorial history of weddings, christenings, funerals and feast-day services at St. Mark's. The pictures from the '70s made me smile because the fashions were so dated\u2014men in suits with flared trousers, frilly shirts and chunky identity bracelets; women with big hair, wearing empire-line dresses and slingback shoes. There was even a picture of me at Annie's funeral, twenty-four years old and desperately self-conscious in a brand-new black maxi-overcoat which hadn't fitted properly and gave me the look of an orphan in someone else's cast-offs. I recognized very few of the faces because they weren't all from my era, but some I remembered.\n\n\"Why did you take so many?\" I asked Wendy. \"You can't have been paid for all of them.\"\n\n\"I thought it would be interesting for future generations,\" she said. \"I wanted to leave copies with the parish register so that when people came looking for information about their families, there'd be a visual record as well as a written one.\" She laughed. \"It wasn't a very good idea. There was so much time and paperwork involved in cross-referencing pictures with written entries that I got snowed under very quickly. After that I went on doing it for fun.\"\n\n_She does a lot of things for fun_ , I thought, warming to her. I even began to wonder if I could excuse what I was doing in the same way. Would anyone accept that I was asking questions about Annie's death because I was bored? I touched a finger to a picture of a family group. \"The Charleses,\" I said. \"They lived next door to us at number 3.\"\n\nWendy moved across to sit next to me on the sofa. \"Paul and Julia, plus two children whose names I can't remember. Peter christened one of them and it howled nonstop throughout the service. These were the christening photographs.\"\n\n\"Jennifer,\" I told her. \"She used to cry all night. Sam went 'round once to read the riot act because we couldn't sleep for the row that was going on, but Julia was so exhausted she burst into tears on the doorstep and he couldn't bring himself to do it. After that we took to wearing ear plugs. Jennifer's about twenty-four now and working as a solicitor in Toronto. The whole family emigrated to Canada in 1980.\"\n\n\"Goodness me! You _are_ well informed.\"\n\n\"I recognize this man's face,\" I said, pointing to another picture.\n\n\"Derek Slater,\" she told me. \"He was a horrible brute... used to beat his wife and children when he was drunk. The poor creature was always taking refuge with us because she was so frightened of him.\" She turned a page and pointed to a dark-haired woman holding a toddler in her arms. \"That's her... Maureen Slater. She had four children by him\u2014two boys and two girls\u2014all of whom got thrashed at one time or another. Derek was always being arrested... usually for drunk and disorderly... although I believe he had theft convictions, too.\" She placed a finger on the toddler's face. \"Derek certainly spent time in prison because this little chap came long after the other three. As far as I know Maureen's still living on Graham Road, but goodness knows where Derek went. There was a terrible fight some time in 1979 or '80 when his elder son finally found the courage to take a baseball bat to him and told him to leave.\"\n\n\"That would be Alan?\"\n\n\"Yes. Did you know him?\"\n\n\"I taught him English for a year... a tall, heavily built child with hands the size of dinner plates. They lived next to Annie at the end of the terrace. Number 32. Do you have a picture of Alan?\"\n\n\"I think so... but he wasn't in church when I took it. As far as I recall the only time he ever set foot in St. Mark's was to see if there was something worth stealing.\" She tut-tutted to herself. \"He was a frightful thief, stole my mother's brooch from under my nose when I offered Maureen sanctuary one day, and I've never forgiven him for it. Mind you, all her children were thieves... only to be expected, I suppose, with a father like Derek. It's very sad the way the sins of the fathers are visited on the next generation.\"\n\n\"Did you report the theft?\"\n\nShe sighed. \"There was no point. He'd just have denied it. And it was my fault, anyway. I should have been more careful. After that I made sure everything was locked away whenever they came to the house.\"\n\nI wondered what else Alan had got away with. \"He tried to steal from me, too,\" I told her. \"I left my bag on my desk while I went to collect some notes from the staff room, and when I came back he was going through my wallet. I didn't report him either.\" I tapped a finger against my lip where a tiny tic of hatred pulsed and throbbed beneath the skin. \"I'd never have let my own children get away with it.\"\n\n\"No,\" she said slowly, watching me with her sharp eyes, \"but I don't suppose you liked Alan much so you overcompensated.\"\n\nI didn't answer.\n\n\"I'd forgotten you were a teacher,\" she said to break the silence.\n\nI nodded. \"For my sins.\" I ducked my head down for a closer look at Derek Slater's face. He had long, dark hair and a pleasant smiling face and appeared anything but a wife-beater. \"What did Derek go to prison for?\"\n\n\"I've no idea. Theft? Assault?\"\n\n\"On his wife?\"\n\n\"A woman certainly. I don't think he was brave enough to pick fights with men.\"\n\n\"Who's this?\" I asked, touching a picture of a heavily made-up blonde, simpering at the camera from beneath a wide-brimmed hat.\n\n\"Sharon Percy,\" said Wendy, turning her mouth down at the corners. \"Mutton dressed as lamb. She wasn't far off forty when that was taken but most of her bosom's hanging out and her skirt barely covers her knickers. You must remember her. She lived next to Annie on the other side from the Slaters and was forever complaining about her.\" She heaved a sigh. \"Poor Annie. She was sandwiched between the two worst families in the street\u2014a thieving violent family, the Slaters, on one side and a tart with an out-of-control son on the other.\"\n\nSharon Percy\u2014aka Jock's floozy and Libby's \"bleached vampire,\" I thought with amusement. \"I don't believe I ever saw her,\" I said, \"or if I did I don't remember. I taught her son, Michael... at the same time I was teaching Alan Slater, but I don't think she ever came near the school.\"\n\n\"She was a dreadful woman,\" said Wendy tartly, \"little better than a prostitute... entertained a different man in her house every night... but she still thought she was superior to a black woman... made Annie's life a misery with her endless complaints to the council.\"\n\nI studied the young-old face with interest and recalled some of the rednecks we'd met in South Africa. \"It's the 'poor white' syndrome,\" I said slowly. \"The lower you are in the pecking order the more important it is to have someone beneath you.\"\n\n\"Mm, well that was certainly true of Sharon.\"\n\nIt seemed a very unchristian attitude and I wondered what the woman had done to make Wendy dislike her. \"How do you know so much about her?\" I asked curiously. \"Was she a regular churchgoer?\"\n\n\"Oh, yes. Regular as clockwork as long as Peter was willing to give her an hour a week to discuss her problems. Hah!\" she snorted suddenly. _\"Alleged_ problems, I should have said. Called him _Father_ Stanhope because she knew it would appeal to his vanity. It was only when she started putting her hand on his thigh that he realized what she was up to and told her he wouldn't see her again unless I sat in on the discussion. After that she never set foot inside the church again.\"\n\nI hid a smile. For all her declared frustration with her marriage, she could still feel jealousy. \"Did she ever marry?\"\n\n\"Not when we knew her. I couldn't even say who Michael's father was, and I don't suppose Sharon could either. The poor child was always getting into trouble with the police and Peter would be dragged out at midnight to stand in loco parentis because his mother was flat on her back somewhere.\"\n\n\"Turned fourteen in '78,\" I said, remembering. \"Dark-haired, rather adult-looking... always wore white T-shirts and blue jeans.\"\n\nShe nodded. \"He wasn't a bad lad, just hopelessly out of control. He was very bright and very articulate\u2014the complete opposite to Alan Slater, who could hardly speak without uttering an obscenity. I was rather fond of him, as a matter of fact, but he wasn't the type to give his affection easily.\" A wistful expression crossed her face. \"I read in the newspaper about six years ago that a Michael Percy had been sentenced to eleven years for armed robbery. The age was right but the photograph was very different from the boy I remembered.\"\n\nI couldn't bring myself to shatter her illusions. \"Does Sharon still live at number twenty-eight?\"\n\n\"Presumably. She was certainly there when we left in '92.\" She took the album from me and leafed through the pages until she came to a picture of a gray-haired man with a pointed, raddled face like a tortoise. \"Geoffrey Spalding,\" she said. \"Married to a woman called Vivi-enne who died of breast cancer in '82. Poor creature\u2014she fought a long battle against it\u2014nearly five years in all. I took this at her funeral. They lived across the road from Sharon, and it was one of the big scandals that, while his wretched wife was dying, Geoffrey spent more time in Sharon's house than he did in his own. He moved in for good about six months after Vivienne's death.\" She sighed again. \"The whole business upset Geoffrey's children terribly. He had two teenage daughters who refused to acknowledge that Sharon even existed.\"\n\n\"Did they move in with her as well?\"\n\n\"No. They stayed across the road and took care of themselves. It was all very sad. They had virtually no contact with Geoffrey afterward except to post the gas and electricity bills through his door. I think they blamed him for their mother's death.\"\n\n\"I suppose we all lash out when we're hurt,\" I said, thinking of Jock and his parents. \"It's human nature.\"\n\n\"They were very quiet girls... rather too quiet, I always thought. I can't ever remember seeing them laugh. They started caring for their mother when they were much too young, of course. It meant they were never able to make friends with their own age group.\"\n\n\"Do you remember their names?\"\n\n\"Oh Lord, now you're asking.\" She pondered for a moment, then shook her head. \"No, dear, I'm sorry. They were pretty girls with blonde hair and blue eyes... always reminded me of Barbie dolls.\"\n\n\"You said they were teenagers when their mother died. Late teens or early teens?\"\n\n\"I think the elder was fifteen and the younger thirteen.\"\n\nI did some mental arithmetic. \"So they'd have been eleven and nine when Annie died?\"\n\n\"More or less.\"\n\n\"They were called Rosie and Bridget,\" I said. \"They used to walk to school every morning, hand-in-hand, wearing beautifully ironed uniforms and looking as if butter wouldn't melt in their mouths.\"\n\n\"That's right,\" said Wendy. \"What a wonderful memory you have.\"\n\n_Not really_ , I thought. Before Annie's death the two girls and I had been friends. We would greet each other with smiles and hellos, I on my way to one school, they on their way to another. Then, for no reason that I ever understood, everything changed in the months following Annie's death. Their wide smiles vanished and they avoided looking at me. Once upon a time Bridget had had pigtails like her sister until someone cut them off and posted them in long, blond strands through our letter box. At the time I didn't know their surname or which house they lived in. All I knew was that Rosie grew paler and thinner, while nine-year-old Bridget's hair was long one day and short the next. But I had no idea why the ends were sent to me or what their significance was.\n\n\"I didn't know their mother was ill,\" I said sadly. \"I used to think what a nice woman she must be because they were so sweetly behaved in contrast to some of the others.\"\n\nMore sighs. \"They were very lost after she died. I tried to help them but Geoffrey became appallingly belligerent and told me to stop interfering. There's only so much you can do, unfortunately... and Geoffrey made them suspicious of me by saying I was trying to have them put into care. It wasn't true, but they believed him, of course.\" Her mouth turned down at the memory. \"He was a beastly little man... I never did like him.\"\n\n\"Are either of the girls still on Graham Road?\" I asked.\n\nShe looked troubled. \"No, and the awful thing is I've no idea where they went or what happened to them. I believe Michael was living with them at one point, but he was in and out of juvenile prison so much it was difficult to keep track. I asked Geoffrey once what had happened to them but he brushed me aside as if I were an irritating gnat. A most pernicious creature. I've always felt he and Sharon deserved each other.\"\n\nI brought her back to Rosie and Bridget. \"Did the girls marry?\"\n\nShe shook her head. \"I couldn't say, my dear. If they did it wasn't in St. Mark's.\" She paused for reflection. \"Mind, the report on the armed robbery\u2014the one about Michael Percy\u2014mentioned a wife called Bridget\u2014and I thought at the time\"\u2014she pursed her lips into a tight little rosebud\u2014\"well, well! All those children were close. They used to run around in a gang together... couldn't prize them apart most of the time.\"\n\nI wasn't there to score points with superior knowledge so I searched for a photograph of Jock Williams instead. Predictably, I couldn't find one. He vaunted his atheism as loudly as a born-again Christian vaunts Jesus's love, and he wouldn't set foot inside a church if his soul depended on it. There was a picture of Libby talking to Sam and me at Annie's funeral, and I pointed it out to Wendy and asked her if she'd ever met the husband. \"His name was Jock Williams. They lived at number 21.\"\n\n\"What did he look like?\"\n\n\"Late twenties... about five years older than Libby... dark-haired, quite good-looking, five foot ten.\" Another shake of her head. \"He and Libby divorced eighteen months after Annie died. Libby took herself off to Southampton but Jock moved to a three-story town house in Alveston Road.\"\n\nWendy smiled apologetically. \"To be honest I wouldn't have known who this woman was if you hadn't told me. Is it important?\"\n\n\"Probably not.\"\n\nShe watched me for a moment. \"Meaning it is,\" she declared. \"But why?\"\n\nI concentrated on a small figurine on a side table which was the same shade as Sheila Arnold's bracelet. \"Most people have to settle for smaller houses when they get divorced,\" I said mildly, wishing I knew more about jade. \"Jock moved into a bigger one.\"\n\nShe was clearly puzzled by my interest. \"It was the way we were living then. People took absurd risks on property after Margaret Thatcher came to power. Sometimes it worked and sometimes it didn't. I remember one of our parishioners saddled himself with a mortgage of nearly \u00a3200,000 and doubled his investment within five years. Another bought just as the market peaked and within a few months found himself owing more than the house was worth. Your friend was lucky.\"\n\nI nodded agreement. \"What about Maureen Slater's and Sharon Percy's houses?\" I asked her. \"If they're still living on Graham Road does that mean they remained as council tenants or did they exercise their right to buy?\"\n\n\"Oh, they bought, of course,\" she said sourly. \"Everything in public ownership was sold off within the first two or three years. It was laughably cheap... no one in their right mind would have turned their backs on an offer like that. Sharon paid for hers outright, I believe, and Maureen opted to stagger her payments. Now, of course, they're quids in. Their houses are worth about \u00a3200,000... and they paid an absolute pittance for them because the wretched taxpayers subsidized the sales.\"\n\nI smiled. \"You don't approve.\"\n\n\"Why would I?\" she countered crossly. \"Every time I see a homeless person lying in a doorway I think how criminal it is that there's no housing left for the genuinely deprived.\"\n\n\"Some might say Maureen Slater was genuinely deprived,\" I murmured. \"She took a lot of punishment from her husband.\"\n\n\"Yes, well, Maureen's different,\" she admitted grudgingly. \"Her brain was turned to mush by that brute. Peter used to describe her as 'punch drunk' from all the beatings, but if I'm honest I think real drunkenness probably had more to do with it. She was just as addicted to alcohol as Derek was... though with rather more justification.\" She caught my surprised expression. \"Anesthesia,\" she explained. \"It must have been very painful to be used as a sparring partner.\"\n\n\"Still...\" I said slowly, \"if her brain was turned to mush, how could she have afforded to buy her house? Presumably she couldn't work, so what did she use for money... even if it was only a pittance that she had to find?\"\n\nThere was a long silence.\n\n\"What aren't you telling me?\" demanded Wendy finally.\n\nI took time to consider my answer, but in the end I decided to be straight with her. \"I met Sheila Arnold recently... Annie's doctor. She told me Annie was robbed. Now I'm wondering who robbed her, how much they made on the deal and what the money was used for.\"\n\n\"Oh, dear, dear,\" said Wendy with genuine concern. \"I really don't think there's any truth in that story. Sheila only came up with it when she was accused of neglecting another patient\u2014and that was three or four years after Annie died. She wasn't remotely interested until her own interests were compromised.\" She tapped the tips of her fingers together in agitation. \"It was all a bit strange. Not a word said for ages... then suddenly Sheila expects us to believe that, far from being the vulnerable soul we thought she was, Annie was a wealthy woman, living in comfort, until shortly before she died. It all became very unpleasant very quickly... insults being thrown about... everyone accusing everyone else of lying.\"\n\nI didn't say anything, and she seemed to think she'd upset me.\n\n\"Are you disappointed?\" she asked. \"I'm so sorry. Peter told me what a shock Annie's death was to you.\"\n\n\"Please don't apologize.\" I wondered what else Peter had divulged. \"I'm not disappointed.\" I opened my rucksack to reveal a six-inch-thick file, then took out an envelope of press clippings and flicked through the pile till I came to June 1982. \"Is this the story you mean?\" I asked her, handing her the \"Local doctor denies neglect\" report.\n\n\"Yes,\" she said slowly, glancing up from the yellowed paper. \"How long have you had it?\"\n\n\"Sixteen years. It was the fifth time Annie's name was mentioned in the press since the publicity over her death. These\"\u2014I removed the remaining clippings from the envelope and flicked my thumb down the guillotined edges\u2014\"are all the other references. Her case is generally cited to illustrate the dangers of allowing vulnerable people to fend for themselves.\" I smiled slightly at Wendy's expression. \"Various friends save articles for me. Also, I pay my old university library to monitor the local and national press for any mention of Ann Butts,\" I explained.\n\n\"Good gracious!\"\n\n\"And for any mention of the two police officers who investigated her death,\" I went on, removing another envelope. \"These are the articles that refer to them. One, PC Quentin, died in a car crash seven years ago. The other, PS Drury retired from the police in 1990 to take over a pub in the Radley brewery chain. There are also clippings about anyone mentioned in previous articles... for example, there's a reference to Dr. Arnold's move to Dorchester... and one about you and your husband leaving St. Mark's to take up a parish in the west country.\"\n\nShe looked at the piece on Sheila's alleged negligence. \"The previous reference to us being Peter's quote at the end, I suppose?\"\n\nI nodded. \"He didn't pull any punches either. 'There's no excuse for this kind of neglect. Lessons should have been learnt... so that the same mistakes could not happen again.' \" My eyes strayed toward the jade figurine. \"Did he know what he was talking about? Had he ever been inside Annie's house?\"\n\nWendy shook her head. \"She wouldn't give him the time of day because she knew Maureen took refuge at the vicarage.\"\n\n\"Then he had no business to talk about 'this kind of neglect,'\" I said lightly. \"It suggests an informed comparison, which he couldn't make, and it was hardly surprising that Sheila was upset about it.\"\n\n\"I know,\" she agreed unhappily. \"The only good thing is, he didn't mention her by name.\"\n\nI shrugged. \"He didn't need to. It's perfectly clear who he's talking about. In any case, the newspaper probably edited it out to avoid a libel suit. The whole article's carefully constructed to record Sheila's _denials_ of neglect without ever actually accusing her of it.\"\n\nWendy gave yet another heartfelt sigh. \"It was my fault really. I'm the one who reminded Peter about Annie, and he promptly rushed off in high dudgeon to talk to the press. Sheila never forgave him for it and it made life very difficult afterward.\"\n\n\"I can imagine\"\u2014I extracted \"Doctor cleared by BMA\"\u2014\"particularly as Sheila was exonerated. Mr. Potts wasn't even her patient.\"\n\n\"It was too late by then. The damage had been done. Peter did try to apologize but Sheila was having none of it.\" She paused. \"But it wasn't entirely his fault, you know. Sheila was spreading some frightful counteraccusations against him, saying the reason Annie distrusted him so much was because he'd supported the neighbors' attempts to get rid of her from the street. She even suggested he was a racist.\"\n\n\"Is he?\"\n\nI thought she might be angry, but she wasn't. \"No. He has many faults but racism isn't one of them. Sheila knew it, too. It was an unkind thing to say.\"\n\n\"Not much fun for any of you,\" I murmured.\n\n\"Terrible!\"\n\n\"But it doesn't mean Sheila was wrong to say Annie was robbed,\" I pointed out.\n\n\"It just seems so unlikely,\" said Wendy. \"No one thought Annie had a house full of treasures while she was alive. Did _you_ think she had?\"\n\n\"No,\" I admitted, \"but Sheila does have evidence to support her story. Letters from the RSPCA inspector, for example, who went in to check on her cats. And if it _is_ true that Annie was robbed, then it's also true that the police investigation into her death was flawed because it failed to take into account that someone took a small fortune off her either before or after she died.\"\n\n\"But who, for goodness'sake?\"\n\n\"That's what I'm trying to find out,\" I said, putting the press clippings back into their envelopes. \"Someone fairly close to home is my guess... someone who knew what was in there.\"\n\nShe canted her head to one side to study me closely with her bright, perceptive eyes. \"What's your husband's view?\"\n\n\"He doesn't have one,\" I said slowly. \"The subject hasn't been mentioned in our house for twenty years.\"\n\nShe put a gentle hand on my shoulder. \"I'm sorry.\"\n\n\"No need to be,\" I told her gruffly. \"This is my project, not his.\"\n\nDid she think \"project\" was an inappropriate word? \"It's not your fault Annie died,\" she said with sincerity. \"You've nothing to feel guilty about.\"\n\n\"I don't.\"\n\nPerhaps she didn't believe me. Perhaps she saw a contradiction between my apparent composure and the evidence of obsession in my lap. \"No one escapes justice,\" she said, dropping her hand to pick up one of mine and rubbing it gently between hers. \"It may not be a justice we can see or understand, but the punishment is always appropriate.\"\n\n\"I expect you're right,\" I agreed, \"but I'm not interested in abstract punishment. I want the kind I can _see..._ the eye for an eye... the pound of flesh.\"\n\n\"Then you'll be disappointed,\" she told me. \"There's no joy in causing pain... however worthy the motive.\"\n\nI had no answer to that except to return the pressure of her fingers. It was acknowledgment of a sort and to that extent it mollified her, but worry remained etched around her eyes until I left.\n\n**_Family correspondence\u2014dated 1999_**\n\n_CURRAN HOUSE_\n\n_Whitehay Road_\n\n_Torquay_\n\n_Devon_\n\nWednesday, July 28 1999\n\nDearest M,\n\nIf I can give you any advice at all\u2014and of course there's no reason why you should take it\u2014it's that you talk things through with Sam before your mother and I come for our visit on Saturday She's still very unhappy about your move to Dorchester and will, I fear, put pressure on the boys to supply answers if she can't get them from you. Sam has told her the farmhouse was the only property you could find at short notice\u2014which is clearly what he believes\u2014and she's now convinced \"something's going on\" as she says her tame real estate agent faxed you a list of suitable lets in Devon at the beginning of June.\n\nSorry to be a nuisance but the old adage\u2014\"of two evils choose the lesser\"\u2014is a good one, I find. You know what your mother's like when she gets the bit between her teeth, and I fear Sam would be very hurt to learn the truth from his children after a quizzing from their grandmother! It won't be easy \"coming clean\"\u2014secrecy is frighteningly addictive, as I've discovered myself since I realized how much closer you and I have become through our shared crusade, my dear, and how jealously I want to guard that closeness\u2014but I think the time has come for honesty. I know you would never hurt Sam unnecessarily.\n\nLove,\n\nDad\n\nX X X\n\n# seven\n\nThe house was full of young people when I arrived back that evening to find an impromptu barbecue taking place on our terrace. \"Another end-of-term celebration,\" explained my younger son en route from the kitchen with a tray of spare ribs. He dropped me a mischievous wink. \"Luke and I were voted the people most likely to throw a good party.\" He had a pretty girl draped off his elbow whose hair was almost as long and blonde as his own. \"Georgie,\" he offered by way of introduction. \"Mum.\"\n\nThe girl was too besotted to look at me for long. \"It's nice of you to invite me,\" she said.\n\nI nodded, wondering how Luke and Tom had managed to become the center of attention so quickly At their age I had hidden behind a fringe, longing to be noticed and invariably overlooked, while Sam had followed in the wake of the Jock Williamses of this world, acquiring girlfriends courtesy of their friends'superior pulling power. The boys would say it was their height, surfers'good looks and neat bums, but I thought it had more to do with taking jobs as checkout cashiers in the local Tesco's, which seemed to be the modern equivalent of the village pump. In the end all paths meet across a supermarket trolley.\n\nWith a promise to put in an appearance as soon as I'd changed, I retreated to the bedroom where I found Sam laid out on the bed and glaring at the ceiling. \"It's bedlam down there,\" he said crossly. \"Why didn't you tell me the boys were planning to invite half of Dorchester to eat us out of house and home?\"\n\n\"I forgot,\" I lied.\n\n\"Well, for your information,\" he growled, \"I was sunbathing in the nude when they all came piling round the corner of the house. It was bloody embarrassing.\"\n\nSmiling, I flopped down beside him. \"Is that why you're hiding up here?\"\n\n\"No,\" he said, jutting his chin toward some boxes in the corner of the room. \"I'm guarding my wine. I found a girl in the kitchen trying to open a bottle of Cloudy Bay because she thought it looked like cheap plonk, so I gave her a lecture on the quality of New Zealand viniculture and she burst into tears.\"\n\n\"I'm not surprised if you had no clothes on. She probably thought you were a rapist.\"\n\n\"Ha-bloody-ha!\"\n\n\"I suppose you shouted at her?\"\n\nHe rolled over to face me, propping himself on one elbow. \"I told her I'd have her guts for garters if she didn't learn to tell the difference between liebfraumilch and a priceless sauvignon blanc. Matter of fact, I nearly asked for her birth certificate in case we got raided. She didn't look more than twelve.\"\n\nHe had a pleasant face, my husband, with laughter lines raying out around his eyes and mouth, and I thought how well he wore his years and how little he had changed in the quarter of a century I'd known him. He had the kind of temperament that people felt comfortable with because he was slow to anger and quick to pacification, and his face was the mirror of his geniality. Most of the time, anyway.\n\nNow, he eyed me thoughtfully. \"How was your day? Did the Reverend Stanhope tell you anything useful?\"\n\nI shook my head. \"I hardly spoke to him.\"\n\n\"Then why so late back?\"\n\n\"I talked to his wife,\" I explained. \"She kept a photographic record of their time at St. Mark's and she's lent me pictures of some of the people who were living on Graham Road in '78.\"\n\nHe studied me for a moment. \"That was lucky.\"\n\nPerhaps I should have seized the opportunity to be honest, but as usual I couldn't decide if then was the right time. Instead I just nodded.\n\n\"I suppose she knew all their names?\"\n\n\"Most of them,\" I agreed.\n\n\"And could tell you every last thing about them?\"\n\n\"Bits and pieces.\"\n\nHe pushed a strand of hair off my forehead with the tips of his fingers. \"There can't be many vicar's wives who photograph their husband's parishioners.\"\n\nI shrugged. \"She was semiprofessional, used to cover the weddings of the poorer couples. It grew out of that. She's rather good actually. If she was forty years younger, she'd have made a career out of it.\"\n\n\"Even so\"\u2014he let his hand drop to the counterpane\u2014\"you could have driven all the way to Exeter to find some dumpy little homebody who'd never done anything more interesting than bake cakes for the Mothers'Union. Instead you come up with David Bailey. That's pretty amazing, don't you think?\"\n\nI wondered what was bugging him. \"Not really. At the very least I knew she must have some photographs of Annie's funeral. Don't you remember her taking a picture of us with Libby Williams? She's a very striking woman, tall and gaunt... like a vulture... rather difficult to miss.\"\n\nHe shook his head. \"How did you know she was the vicar's wife and not a press photographer?\"\n\n\"Julia Charles told me. Apparently, Wendy\u2014Mrs. Stanhope\u2014took pictures of Jennifer's christening so Julia knew her quite well.\" I paused as he shook his head in unhappy denial. \"What's the matter?\" I asked.\n\nHe swung his legs off the bed and stood up, disbelief crackling out of him like small electric charges. \"Larry came to see me this afternoon. He says you're stirring up a hornets'nest by asking questions about Annie. He wants you to stop.\"\n\n\"I hope you told him it was none of his business.\"\n\n\"Quite the reverse. I sympathized with him. Apparently Sheila nearly had a breakdown the last time she got involved. She was hauled before the BMA after your precious vicar accused her of neglect. It was all rubbish, of course\u2014she was cleared immediately\u2014but Larry doesn't want a repeat.\"\n\nHe walked to the window where sounds of laughter drifted up from the terrace. I kept my fingers crossed that Tom wouldn't choose that moment to power up his sound system to full volume, which was the one thing guaranteed to drive his father 'round the bend.\n\n\"What else did Larry say?\" I asked.\n\n\"He wanted to know what brought us to Dorchester. Claims he's not much of a believer in coincidences.\" He frowned at me in hurt recrimination. \"I told him he was wrong... that it was a coincidence... that there was no way we could have known in advance where Sheila was working. So he accused me of being na\u00efve. Your _wife_ knew, he said. She went into the surgery the day after you moved here to register specifically with Dr. Arnold, then asked for a copy of Sheila's work roster so that she could be sure of getting her.\"\n\nI frowned back. \"Where would he get a story like that?\"\n\n\"He asked Sheila's receptionist if Mrs. Ranelagh had known in advance which doctor would respond to her request for a home visit.\"\n\nI sat up and crossed my legs. \"I thought that kind of information was confidential,\" I murmured.\n\nHe waited for me to go on and when I didn't he stabbed a finger at me. \"Is that it?\" he demanded. \"You make me look a complete idiot, then talk about confidentiality.\"\n\nI gave an indifferent shrug. \"What do you want me to say? Yes, I knew this house was in Sheila's practice, and that's why we're renting it.\"\n\n\"Why didn't you ask me?\"\n\n\"Ask you what?\"\n\n\"If I was happy about it.\"\n\n\"I did. You said Dorchester was as good a place as any.\"\n\n\"You didn't tell me there was a hidden agenda, though, did you?\" He was managing to keep his voice under control but I could feel the tremors of a major tantrum building inside him. And that, of course, is the trouble with people with equable temperaments\u2014once they lose it, they lose it big time. \"I might have felt differently if you'd told me you were planning to resurrect Annie Butts. _Jesus!_ Don't you think we went through enough bloody misery at the time?\"\n\nI suppose everyone has a pet subject that triggers their anger\u2014with me it was my mother's wicked talent for stirring, with Sam it was his fear of Mad Annie and everything her death represented: the mask of respectability that overlaid the hatreds and the lies. He always hoped, I think, in a rather free interpretation of the karma principle, that if he refused to look beneath a surface then the surface was the reality. But he could never rid himself of the fear that he was wrong.\n\nI took a moment to reply. \"It wouldn't have passed the 'so what'test, Sam. I'd have come anyway.\"\n\nA look of incomprehension crossed his face. \"Without me?\"\n\n\"Yes.\"\n\n\"Why?\"\n\nIt was such a little word but its interpretations were endless. Why would I think of deserting him? Why was I being so devious? Why didn't I trust him enough to tell him the truth? If he cared to, of course, he could answer those questions rather better than I could as he'd had a great deal longer to think about them. Admittedly, I'd never challenged him with them directly, but there must have been occasions in the small hours when he prepared his explanations in case I did.\n\nI answered straightforwardly. \"I chose Dorchester because I guessed Sheila had more information than anyone else,\" I explained, \"though to be honest, it wouldn't have mattered where we went. The diaspora from Graham Road has been so widespread that we'd have had this conversation whether we'd come here or\"\u2014I gave another shrug\u2014 \"Timbuktu. Paul and Julia Charles are in Canada... Jock and assorted others are still in London... Libby remarried and lives happily with her second husband and three children in Leicestershire... the Stanhopes are in Devon... the coroner retired to Kent... John Howlett, the RSPCA inspector, is in Lancashire... Michael Percy, the son of Annie's immediate neighbor, is in prison on Portland... Bridget Percy, n\u00e9e Spalding\u2014one of the girls who lived opposite Annie\u2014works in Bournemouth...\" I ran out of names and turned to plucking the dowdy candlewick bedspread which was part of the fixtures and fittings and filled me with loathing every time I looked at it.\n\nI'd shocked him to the core. \"How do you know all this?\"\n\n\"The same way you know that Jock lives in Alveston Road. I kept in touch. I have a file of correspondence from my father, who's been writing letters on my behalf for years, and Julia and Libby drop me a line every six months or so to keep me informed about people's movements.\"\n\nHe looked horrified. \"Does Jock know you've been talking to Libby?\" He used the sort of tone that suggested I'd been party to a nasty piece of treachery. _Which was pretty rich, all things considered..._\n\n\"I doubt it,\" I said. \"They haven't spoken since the divorce.\"\n\n\"But he's always believed we were on his side. Dammit, I told him we were.\"\n\n\"You were half-right then,\" I said, absorbed in teasing the bedspread with my fingernails. _\"You've_ always been on his side.\"\n\n\"Yes, but...\" He paused, clearly grappling with some new and unpleasant thoughts. \"Does your mother know your father's been writing letters for you?\"\n\n\"No.\"\n\n\"She'll go apeshit,\" he said in alarm. \"You know bloody well she thought the whole damn mess was dead and buried twenty years ago.\"\n\nI yanked a particularly large tuft out of the candlewick and poked it back again when I realized I'd made a hole. I wondered if he remembered that my parents were coming to stay with us the following day, or whether like all the other disagreeable things in his life he had pushed it out of his mind. \"I wouldn't worry about it,\" I murmured. \"She won't be angry with you... only with me.\"\n\n\"What about your father?\" he demanded, his voice rising in pitch. \"She'll tear strips off him for going behind her back.\"\n\n\"There's no reason why she should ever find out.\"\n\n\"But she will,\" he said pessimistically. \"She always does.\"\n\nI thought of my father's advice on the lesser of two evils. If nothing else, Sam's inability to hide his feelings would start my mother on a hunt for hidden secrets. \"She might be cross for a day or two,\" I said, \"until she persuades herself it's all my fault. She's programmed never to blame men for anything. As far as she's concerned, Eve corrupted Adam\"\u2014I held Sam's gaze\u2014\"when she ought to know that Adam almost certainly took Eve without permission.\"\n\nHe had the grace to blush. \"Is that what this is all about? Getting your own back?\"\n\nI didn't answer.\n\n\"Couldn't you have told me?\"\n\nI sighed. \"Told you what? That I was pursuing something that was important to me? As I recall, the last time I used those words to you, you accused me of being a neurotic bitch and said if Annie's name was ever mentioned in your presence again, you'd divorce me.\"\n\nHe waved a despairing hand. \"I didn't mean it.\"\n\n\"Yes, you did,\" I said flatly, \"and if I'd been half as confident then as Tom and Luke are now, I'd have told you where to stuff your pathetic little divorce. I only stayed with you because I had nowhere else to go. My mother banned me from going home and none of my friends wanted a loony parked in their spare bedroom.\"\n\n\"You said you wanted to stay.\"\n\n\"I was lying.\"\n\nSam lowered himself gingerly on to an unopened box of wine. \"I thought this was all over a long time ago. I thought you'd forgotten about it.\"\n\n\"No.\"\n\n\"Jesus,\" he muttered, dropping his head into his hands and lapsing into a long silence. He roused himself finally. \"Have you _ever_ loved me?\" he asked bitterly.\n\nI wanted to tell him it was a childish question, that if he didn't know the answer after twenty-four years then nothing I could say would make a difference. Did he think anyone could live indefinitely with someone they didn't love? _Could he?_ But outside on the terrace Tom's boom box roared into sudden life, causing the walls and floors of the old farmhouse to thump in sympathy, and I was spared the necessity of replying.\n\nI went into the bathroom to change and left my rucksack on the bed for Sam to explore. It was a cowardly way to impart information but I didn't feel badly about it. As the old adage has it\u2014you reap what you sow\u2014and Sam's harvest was well overdue.\n\n**_E-mail from Mrs. Julia Charles, formerly next-door neighbor to the Ranelaghs at 3 Graham Road, Richmond\u2014now living in Toronto, Canada\u2014dated 1999_**\n\n**M. R**.\n\n**From:** Julia Charles (juliac@cancom.com)\n\n**Sent:** 11 February 1999 18:50\n\n**To:** M. Ranelagh\n\n**Subject:** The Slater children!\n\nYou won't believe the trouble we've had just to locate one of the Slater children! Not the one you wanted, I fear\u2014being the youngest (Danny)\u2014but he may be the most amenable to persuading his mother to answer your letters! I won't bore you with full details of the swings and roundabouts\u2014suffice it to say that Jennifer's preschool chum at number 6 (Linda Barry) kept in touch with another preschool chum (Amy Trent) who was at art school with Danny and still keeps in touch with him. We really did bust a gut to find Alan but got nowhere, I'm afraid. Word has it that he married six or seven years ago and is living somewhere in Isleworth, but I don't know how accurate that is. It might be worth trying international directory inquiries to see if there's an A. Slater in that area, but it's a common name and you may come up with several.\n\nAnyway, Danny Slater is living somewhere in Brixton (no address or phone number) and teaching graphic design at a college there. The name and address of the college is: Freetown Community Center, Brixton, London. However, the really good news is that he has an e-mail address\u2014michelangelo@rapmail.com\u2014and collects his messages regularly via an Internet caf\u00e9 near Waterloo station. Jennifer's game to make the contact if Luke and Tom don't want to, but it would speed things up, I think, if you approach him direct. NB: Your idea to say it's an IT project using e-mail and Internet only is a good one and worked well with Linda and Amy.\n\nSo glad to hear Sam's on the mend. I know what a shock it must have been for you!\n\nSpeak soon, love\n\nJulia\n\n**_Part of the e-mail correspondence between Luke Ranelagh and Danny Slater during the first six months of 1999_**\n\nLuke Ranelagh\n\n**From:** Danny Slater (michelangelo@rapmail.com)\n\n**Sent:** 20 February 1999 20:50\n\n**To:** Luke Ranelagh\n\n**Subject:** IT Project\u2014Database: Graham Road\n\nListen, mate, anyone who wants to build a database 'round a black hole like Graham Road needs his head examining. Okay, so you're on the other side of the world and you don't know shit about the UK. It's an excuse of sorts\u2014and I guess I can accept it on that basis\u2014but do me a favor and send me some pictures of babes in bikinis. I'm an artist, for Christ's sake! I have an aesthetic appreciation of beautiful women. Word pictures will do if you don't have access to a scanner. The truth is I am DESPERATE to forget I ever lived in Graham f*****g Road. If you'd met my mother you'd understand! Cheers. Danny.\n\nDanny Slater\n\n**From:** Luke Ranelagh (beachbum@safric.com)\n\n**Sent:** 22 February 1999 15:12\n\n**To:** Danny Slater\n\n**Subject:** Babes in bikinis\n\nHow about these? Mine's the blonde babe on the right. Word pictures of Graham Road will do if you don't have access to a scanner. I'm an ex-pat, for Christ's sake! I have an aesthetic appreciation of all things English. Cheers. Luke.\n\n**_Extracts from an educational psychologist's report on Alan Slater, 32 Graham Road, Richmond. Requested by his head teacher\u2014re: permanent exclusion from school\u2014dated April 1979_**\n\n**... Alan shows a pattern of bullying behavior. He employs his strength to intimidate others through unprovoked violence, and uses abusive language toward children of different ethnic groups. He has a history of discipline problems and reacts aggressively toward teachers who attempt to control him\u2014particularly the females...**\n\n**... His academic performance across all subjects is poor and this has resulted in feelings of inadequacy and low self-esteem. He regards himself as isolated from his peer group and becomes enraged by seemingly minor slights. He feels rejected by family, peers and teachers, and seeks further rejection through disruptive behavior in order to provide himself with a reason for why no one likes him. There is evidence of violence at home. He speaks about his hatred of his father and refers to his mother as \"a vicious bitch.\" He has a close bond with Michael Percy, a near neighbor and classmate, whom he regards as similarly disaffected....**\n\n**... In conclusion, I have real concerns about Alan's dangerous sense of alienation, which may already have led to criminal behavior. I believe rapid intervention is required to prevent matters from becoming worse. There are problems at home andat school, but permanent exclusion is not a solution. He requires intensive \"special needs\" teaching to improve his self-esteem, and he should be encouraged to form strong and positive bonds with adults\u2014either inside the school environment or in the broader community. He needs to feel valued: Only then will he have the necessary motivation to correct his aggressive and antisocial attitudes....**\n\n# eight\n\nI found Luke, my elder son, straddled across a chair in the kitchen. \"Your man's outside smoking a spliff,\" he shouted into my ear over the cacophony of sound from the terrace. \"I told him not to make it too obvious in case Dad saw him, so he's lurking behind a hedge at the bottom of the terrace steps.\" He handed me a can of lager, then stood up to steer me toward the French windows. \"He's a bit of a whinger,\" he warned. \"Keeps saying we must be loaded to afford a place like this, then goes on and on about how he's never had any luck in his life.\"\n\nI nodded.\n\n\"So where's Dad?\"\n\n\"Upstairs,\" I shouted back.\n\nLuke smiled guiltily. \"He's not still angry about his Cloudy Bay, is he?\"\n\n\"No, but he's going ballistic about the noise.\"\n\n\"Okay.\" He pushed his way through the crowd and turned the volume down to bearable proportions. When he came back he had a wiry, dark-haired man in tow, about age twenty-five, with a nervous frown on his face. \"Danny Slater,\" he said, introducing us. \"He's one of the guys who's been giving me gen on Graham Road... teaches art at a community center in Brixton. He's on Portland for the summer learning to carve stone at a workshop in Tout Quarry. I couldn't believe it when we end up in a house just a stone's throw away... seemed like a good opportunity to get acquainted.\"\n\nLuke spoke for Danny's benefit rather than mine. It was hardly tactful, as he'd pointed out several times, to spend months making friends with a bloke, only for him to guess the first time you meet him that there was a hidden agenda behind the friendship and that the reason you're living less than ten miles from his holiday hideaway is because you want to get close to his parents. \"I'd be sodding mad if it happened to me,\" he'd told me firmly, \"so we take a bit of trouble. Okay? I like him... he's cool... and his e-mails are funny.\"\n\nDid I feel guilty about making an ally of my son? Yes. Did I remember Dr. Elias's words of warning about Sam's sense of betrayal when he found out? Yes. Would it have stopped me using Luke? _No_. I had enough faith in my husband to believe he would never blame his children for something they had done for their mother.\n\n_This patient... is obsessive... manipulative... and...frightening..._\n\nDanny wasn't the most attractive young man I'd ever seen, but I put on my best smile and shook his hand warmly as Luke took his leave and wandered over to the barbecue. \"You won't remember me,\" I said, \"but my husband and I used to live at number 5 Graham Road. You can't have been more than three or four at the time, but I knew your elder brother very well... Alan... I was his English teacher at King Alfred's.\"\n\nHe shook his head. \"It won't have been my brother,\" he answered. \"Alan's thirty-five. You're thinking of someone else.\"\n\n\"No,\" I assured him. \"It was certainly Alan. I taught him in '78 when he was fourteen. He was a bit of a handful,\" I finished with a laugh, \"but I expect he's calmed down by now.\"\n\nDanny examined me closely for a moment, before pulling a packet of cigarettes from his pocket. \"You must have had an easy life then,\" he said, more in criticism than compliment. \"My mum's not much over fifty but she looks a damn sight older than you do.\"\n\nI smiled. \"It depends whether you think teaching is easy. _I_ don't, but then I've never taught art. Perhaps that's less stressful than trying to force Shakespeare down the throats of reluctant adolescent boys.\"\n\nHe rose immediately to the bait and I listened with patience to five minutes of complaint about the intolerable necessity of an artist having to earn a regular income... about the wear and tear on the nerves caused by the arrogant egotism of students who hadn't a creative bone in their bodies... about how, if he'd been lucky enough to live in a country where culture was valued, he'd have been given a grant to make his own art instead of teaching brain-dead morons how to make theirs...\n\nI nodded sympathetically when he drew breath. \"And I suppose your family isn't in a position to help you?\"\n\n\"I'm not married.\"\n\n\"I meant your parents. I remember your father quite well.\" I thought of the photograph of Derek Slater which Wendy Stanhope had lent me. \"Dark-haired, rather good-looking. Very like you, as a matter of fact.\"\n\nHe wasn't easily flattered. \"There's only my mother,\" he said, \"and she's on invalidity benefit.\" He offered me a cigarette and lit one for himself when I shook my head. \"Dad abandoned us years ago... can't even remember what he looked like anymore.\"\n\n\"I'm sorry.\"\n\nHe shrugged. \"It was for the best,\" he said unemotionally. \"He took his belt to all of us at one time or another. Alan worst of all. Dad used to beat him about the head when he tried to protect Mum, and Alan's still got the scars to prove it.\"\n\n\"I did wonder,\" I said, equally unemotionally. \"More often than not he was sporting a black eye at school, but he always told me he'd been in fights with boys from rival gangs. 'You should see the other guys,'he used to say.\"\n\nFor the first time Danny smiled. \"He was a good kid. He took a hell of a lot of punishment till he got to fifteen and slammed a baseball bat into Dad's face. That's when Dad took off.\" Another shrug. \"I don't remember him but everyone says he was a right bastard. He got in touch with one of my sisters a few years ago but nothing much came of it. He was only after money. Sally tried to persuade Alan to help him out, but he refused and we haven't heard from him since.\"\n\n\"Do you know where he is now?\"\n\nThere was a small hesitation. \"Somewhere in London, I think.\"\n\n_Prison?_ I wondered. \"And what of Alan?\" I asked in the sort of reassuring tone that said I was more interested in my ex-pupil than I was in his father. \"How's he getting on? Is he married?\"\n\nDanny nodded. \"He's got a couple of kids, a girl and a boy. Never raises his voice to them... won't even give them a smack.\" He sucked moodily on his cigarette. \"It fucks my head to visit him. He lives in this great little house in Isleworth and his wife's brilliant. She's called Beth... plain as a pikestaff and wide in all the wrong places... but every time I go there I think, this is how families are supposed to be, with everybody loving each other and the kids feeling safe. It makes you realize what you missed.\" His eyes strayed toward Luke and Tom, who were arguing over which CD to put on next. \"I'd say your sons are pretty lucky, too.\"\n\nI realized suddenly how vulnerable he was, and felt ashamed of the way I was using him. Until that evening he had been a name on a computer screen, an unremembered child from twenty years ago who had responded to an e-mail in the innocent belief that he was helping a lad in Cape Town complete a thoroughly trivial IT project. Yet he had no responsibility for Annie's death, and I wondered if he even knew that a black woman had died in Graham Road in '78. Certainly the name Ranelagh meant nothing to him, which suggested that both Annie and I had been long forgotten by the time Danny was old enough to understand that one woman had died on his road and another had accused her neighbors of racially motivated murder.\n\nI followed his gaze. \"Luke and Tom might argue that you're the lucky one,\" I said.\n\n\"How do you make that out?\"\n\n\"Because their upbringing means they will never have your creativity or your commitment to proving yourself. Internalized pain is always a stronger motivator than security and contentment. Contented people take happiness for granted. Anguished people struggle to find it through self-expression. At least you have a chance of greatness.\"\n\n\"Do you honestly believe that?\"\n\n\"Yes.\"\n\n\"Then why aren't you making your son's lives hell?\"\n\nThe question was simplistic enough to bring a smile to my face. At the very least, it was predicated on the assumption that parental love can be switched on and off according to circumstance... although perhaps for him that was the reality of childhood. \"Shouldn't you ask me first if I think greatness is a sensible ambition for a mother to want for her children?\"\n\n\"Why wouldn't it be?\"\n\n\"Because the odds are stacked against it. Anguish doesn't guarantee success, it merely offers the possibility. After that it's down to genius. In any case, as far as Luke and Tom are concerned, I'm guided entirely by selfishness. I want them to like me.\"\n\nHe was unimpressed. \"Everyone's motivated by selfishness,\" he said, \"including Luke and Tom. They behave the way you expect because they think they'll get something in return. Alan used to kowtow to my father to avoid a thrashing, but I'll bet Luke and Tom only kowtow for money.\"\n\nI nodded. \"More often than not.\"\n\n\"Alan's kids are the same. They're barely out of nappies but they've got him wound 'round their little fingers.\" He dropped his cigarette butt on to the terrace and ground it out under his heel. \"All they have to do is burst into tears and say they want ice cream and he starts emptying his pockets. I told him he's making an ass of himself but he's so fucking paranoid about the way Dad treated us that he won't listen to reason.\"\n\nI wondered if Danny realized how confused his views on parenting were and what he meant by \"reason.\" Spare the rod and spoil the child, presumably, although why, like so many people, he believed harshness was a better educator than kindness was a perennial mystery to me. \"How does your mother feel about it?\"\n\n\"Christ knows. She's a Prozac junkie,\" he said bitterly, \"so it depends what mood she's in at any given moment. It's a good day if she can drag herself out of bed... as for having an opinion on something...\" He fell silent, staring at the ground.\n\n\"I'm sorry,\" I said again.\n\n\"Yeah, it's a mess.\" He gave a mirthless laugh. \"I guess you're pretty disappointed.\"\n\n\"About what?\"\n\n\"That a type like me responded to Luke's e-mails. You were probably hoping for something better.\"\n\n\"I never make those kinds of judgments,\" I replied truthfully. \"If I did, I'd have to wear a label 'round my own neck, and that's not something I'm prepared to do. In any case, I'm not sure what type you think you are.\"\n\nHe kicked at a flagstone, refusing to meet my eyes. \"Fucking useless,\" he muttered. \"The last I heard of my dad he was banged up in the Scrubs for assault, but we've all been there at one time or another. I got six months for twocking\u2014that's taking cars without consent. Alan got four years in juvenile for dealing... both my sisters have done time for shoplifting. We're bad news. Poor old Mum used to get the cold shoulder every time she left the house because of the stuff her kids did.\" He lapsed into a brief, unhappy silence. \"I guess that's why she doesn't get out of bed anymore.\"\n\nThe admission clearly wounded him, and I wondered if he hadn't looked for us\u2014or people like us, uninfected by anti-Slater bias\u2014just as assiduously as we had looked for him. _Yet, if that were true, why had he confessed to his family's failings so readily?_ The sly glance he gave me when he raised his head persuaded me it was a cynical test of my refusal to label him, and my sympathy waned a little. I guessed he enjoyed holding grudges and sought rejection for the purpose of fueling them... and I wondered which of us was the more manipulative.\n\n\"I thought you were going to classify yourself as a struggling artist,\" I said with a small laugh. \"I hadn't bargained on 'fucking useless.'Does that mean I'll be wasting my time if I visit you at the sculpture workshop?\"\n\nHe gave me an unwilling smile. \"No. I'm a good sculptor.\"\n\n\"You ought to be,\" I told him. \"Your brother had real talent at fourteen.\"\n\nHe looked surprised. \"Alan?\"\n\nI nodded. \"I've still got a little wooden figure he carved for me. It's in the shape of a snake with feathers 'round its head.\"\n\n\"That's probably right,\" said Danny. \"He's got this thing about an Aztec god who was half snake, half bird. It's a load of crap, but Alan reckons the bastard was an alien who came to earth to create a lost civilization in Mexico.\"\n\n\"Quetzalcoatl?\" I suggested.\n\n\"That's the one. He's got a mosaic of him on his sitting-room wall.\"\n\nI learned nothing further about Alan's picture that evening because Danny was more interested in pouring scorn on his brother's belief in extraterrestrials than he was in discussing his taste in art. I clung to my dwindling patience in order to listen to the hoary old arguments on both sides, and was somewhat relieved when a six-foot-tall brunette with legs up to her armpits seduced him away with a cigarette.\n\nI watched them perform the opening moves of a courtship dance\u2014 an awkward affair of wriggling shoulders and pretended casualness as they dipped their heads to the cigarette lighter\u2014and was about to go back inside when Sam appeared at my elbow with a peace offering. \"It's Cloudy Bay,\" he said gruffly, shoving a glass of wine into my hand. \"I was going to drink the lot to drown my sorrows, then I thought, to hell with it, it's not your fault Larry got me fired up.\"\n\nIt wasn't a white flag exactly but I could always recognize a truce when I saw one. I responded with a chink of glasses and a smile, while wondering if Sam had used the opportunity I'd given him to find out who Danny Slater was and why he was there. If not, I feared the truce would be of short duration. It was one thing for his wife and his father-in-law to keep secrets from him... quite another for his sons to do it as well.\n\nHe might have read my thoughts. \"Who's the dark-haired chap you were talking to?\" he asked, nodding in Danny's direction. \"I was watching from the window. He seemed to have a lot to say to you.\"\n\n\"His name's Danny Slater,\" I told him. \"He's working up at the sculpture park on Portland.\"\n\n\"Any relation to Derek Slater?\"\n\n\"His son,\" I said evenly. \"Do you remember Derek?\"\n\n\"No. I've been going through your rucksack.\" He hunched his shoulders like a boxer preparing to defend himself. \"And don't give me any heartache over it because if you didn't want me looking, you shouldn't have left it on the bed.\"\n\n\"My fault,\" I agreed, hoping he'd had the sense to go through everything. Ignorance had kept him happy for years; _partial_ ignorance would eat away at him like a rotten worm.\n\n\"You were right about the Rev's wife. She took some useful photographs. This lad's the spitting image of his father twenty years ago.\"\n\n\"There's a lot of his mother in him,\" I demurred.\n\n\"That would be Maureen Slater?\"\n\nI nodded.\n\n\"Mm, well, I didn't recognize her. In fact I didn't recognize any of them except Julia Charles and Libby Williams. There's a blonde woman who came into the pub occasionally, I think, but other than that\"\u2014he shook his head\u2014\"they were all strangers.\"\n\nI wondered how much of the correspondence he'd read and how much he thought I'd withheld. If he knew the truth, he'd be devastated.\n\nHe flicked an abstracted glance across the field of heads in front of the house, looking for Luke and Tom. \"That's quite a file the boys have collected on Graham Road. How long have they been doing it?\"\n\n\"Since your coronary.\"\n\nHe smiled slightly. \"On the principle that you'd be coming home whether I lived or died?\"\n\n\"Something like that.\"\n\nHe paused before his next question, as if considering the wisdom of asking it. He knew as well as I that bridges were best left unburnt, but his need for reassurance was stronger than his caution. \"Did you tell them I walked out on you?\"\n\n\"No. I told them Annie was murdered and that I was trying to get the investigation reopened. Nothing else.\"\n\nHe stared into his wine glass, his mouth working strangely as if trying to formulate unaccustomed words. But in the end, all he said was: \"Thank you.\"\n\n**_Statement made by Mrs. M. Ranelagh in 1979 re: an alleged assault by Derek Slater of 32 Graham Road, Richmond_**.\n\n**INCIDENT REPORT**\n\n**Date:** 25.01.79\n\n**Time:** 10:32\n\n**Officer:** PS Drury, Richmond Police\n\n**Witness:** Mrs. M. Ranelagh, 5 Graham Road, Richmond, Surrey\n\n**Incident:** Alleged assault on Mrs. Ranelagh at 15:00 approx. on 24.01.79\n\nMrs. Ranelagh states: I went to the shops yesterday afternoon because there was no food in the house and I had had nothing to eat for three days. I thought it would be safe because it was still light. As I turned onto Graham Road, a man came up behind me and pushed me into the alleyway at the back of the even-numbered houses. I was unable to call out because he put a hand over my mouth and clamped my arms to my sides in a bear hug before slamming me face first into a fence and using his weight to hold me there. It all happened very fast and there was nothing I could do to break free. I couldn't see his face because he was behind me, but his breath smelled of drink and his clothes smelled unclean. I was wearing trousers and could feel something being pushed between my thighs. I thought it was the man's penis. He had his face pressed against the side of my head and whispered \"slag,\" \"bitch\" and \"cunt\" into my ear. He also said he'd \"do for you proper\" if I didn't keep my \"filthy, nigger-loving mouth shut.\" He was very strong and I was frightened because I thought he intended to rape me. I believe that is what he wanted me to think. Before releasing me, he forced me to my knees and pushed my head into the mud at the bottom of the fence. He said if I reported what had happened to the police I wouldn't \"get away so lightly next time.\" I raised my head to watch him turn the corner into the main road. He was dressed in a dark jacket, blue jeans and sneakers. It was Derek Slater who lives in the neighboring house to where Ann Butts used to be. I know him by sight, although I have never spoken to him. He had disappeared by the time I found the courage to go back onto Graham Road. I saw no one else and went straight home.\n\n**MEMO**\n\n**To:** Police Superintendent Hathaway\n\n**From:** PS Drury\n\n**Date:** 29.01.79\n\n**Subject:** Advice re cautioning Mrs. Ranelagh against wastin police time\n\nSir,\n\nAs you know, Mrs. Ranelagh has made a number of accusations against Derek Slater, including: 1) harassing and murdering Ann Butts; 2) making abusive telephone calls to the Ranelagh household in the middle of the night and; 3) attempting to keep Mrs. Ranelagh a prisoner in her own home by loitering outside her front door. None of these accusations stands up to investigation. 1) The inquest verdict on Ann Butts was unequivocal. 2) The Slaters have no telephone\u2014nor do they have access to the Ranelaghs' new ex-directory number. 3) Mrs. Charles at 3 Graham Road\u2014next-door neighbor and friend of Mrs. Ranelagh\u2014denies ever seeing Derek Slater at their end of the street.\n\nThere is no evidence that the above incident took place other than Mrs. Ranelagh's word. The clothes she claims to have been wearing are unmarked and unstained\u2014i.e., there are no muddy marks on the knees of her trousers and no semen staining between the thighs. Despite the aggressive way in which she says she was held \"in a bear hug\" and slammed against a fence, her face and arms are unmarked. (N.B.: She waited nineteen hours to report the incident and claimed to have cleaned herself up.)\n\nMrs. Ranelagh admitted to me that her husband has left her. She is clearly disturbed by Mr. Ranelagh's desertion. She says she phoned him to tell him about the alleged assault and was upset when he accused her of lying. \"He said I'd invented it to make him jealous. I sometimes think the only thing he thinks about is sex.\" (N.B.: Mrs. Ranelagh has lost a lot of weight and appears to be anorexic. Also, her behavior is irrational\u2014she breaks off in the middle of a conversation to listen for rats.)\n\nI spoke to Mr. Ranelagh by telephone. He claims his wife is \"bored with being a teacher and is reveling in her fifteen minutes of fame.\" According to him, nothing she says can be believed at the moment.\n\nI have questioned Derek Slater and he denies being anywhere near Graham Road at 15:00 hours on 24.01.79. He says he was at Kempton Park Races until the early evening and has a ticket stub to support this. He has supplied names and telephone numbers of three friends who were with him\u2014one supports the alibi; two yet to be checked.\n\nPlease advise. My personal view is that Mrs. Ranelagh is pursuing a vendetta against Derek Slater because she believes him to be responsible for the death of Ann Butts. I consider this vendetta to be: a) an invention; b) paranoid; and c) strongly linked to shock and\/or the failure of her marriage. I strongly recommend an official caution against wasting police time.\n\n# nine\n\nWe survived the party wreckage on the terrace the following morning with sore heads and mixed feelings. The boys were savoring last night's success, while Sam and I peered into a black hole as I reminded them all that my parents were due at noon. Luke and Tom, who were both on the afternoon shift at Tesco's, took the news in stride. Forget lunch, they said cheerfully, but as long as dinner was late they'd make an effort to get back for it. Sam, by contrast, crumpled dramatically as if he'd been axed by a pole.\n\n\"It's been on the calendar for ages,\" I said unsympathetically, handing him a cup of black coffee as he slumped into a chair, \"so don't blame me if you never bother to read it.\"\n\n\"I don't feel well.\"\n\nThe boys were immediately solicitous, worried that \"not feeling well\" had more to do with Sam's coronary than too much to drink the night before. They fussed about him, staring anxiously into his face and patting him encouragingly on the shoulder as if that would somehow prevent another attack. Sam eyed me with sudden mischief, as if seeing a way out of a nightmare weekend, so I gave him the Ranelagh glare.\n\n\"Don't even think about it,\" I warned, massaging my hangover. \"You know my mother. Nothing prevents her turning up. And do not _dream_ of disappearing off to bed. It's your job to charm her until the boys get back.\"\n\n\"Oh God!\" he groaned theatrically, sinking his head into his hands. \"She'll _kill_ me. I've told her at least ten times that it was chance that brought us to Dorchester.\"\n\nLuke and Tom eyed him curiously, wondering at this sudden reversal in their father's usually sanguine, if never very thrilled, acceptance of his mother-in-law.\n\n\"What's up?\" asked Luke.\n\n\"Nothing,\" I said. \"Dad's looking for trouble where it doesn't exist.\"\n\n\"We could call in sick,\" said Tom helpfully. \"I quite like Gran.\"\n\n\"Only because you've never seen her breathing fire,\" muttered Sam.\n\n\"She's even more scary than your mother when she's angry\"\u2014another mischievous glance in my direction\u2014\"probably because there's so much more of her.\"\n\nI handed Tom a black plastic bag to start clearing the mess on the terrace. \"Your father's being ridiculous. Granny adores him. He only has to smile and she's putty in his hands.\"\n\nIt didn't work out like that, of course. Nothing ever does. My father had taken his own advice\u2014 _of two evils choose the lesser_ \u2014and had tucked a magazine article on racially motivated murder into his overnight bag, which my mother unearthed and read when she decided unilaterally to repack their clothes into one large case. Dad swore it was an accident, but I didn't believe him any more than he would have believed that Sam had read my files \"by accident.\" I remarked to him afterward that it was a damn good thing I hadn't ignored his letter of warning, otherwise we'd have had a repetition of the mother\/son-in-law alliance of twenty years ago, but Dad just laughed and said Sam wasn't the kind of man to make the same mistake twice.\n\nThe article in question had been written in the wake of the official inquiry into the murder in London in 1993 of a young middle-class black man called Stephen Lawrence. The inquiry\u2014not held until 1999\u2014 had condemned the police for \"institutionalized racism\" following the shoddy and lackluster investigation into Stephen's murder by a gang of youthful white supremacists, all known to the police, who escaped conviction because of the culture of legal carelessness that existed in regard to the deaths of black people. My mother might have thought it was a general interest story if my father hadn't taken the trouble to highlight a paragraph and make this note to me in the margin: _M. Some good points here. Suggest you contact the journalist re police apathy and violent treatment of offenders. N.B.: River of Blood speech, 1968\u2014Annie Butts's murder, 1978_.\n\nThe paragraph read:\n\n_By definition, to describe anything as \"institutionalized\" means the tradition is a deep-seated one, and this suggests that Stephen Lawrence's murder isn't the only investigation to be bungled by a predominantly white police force, long riddled with apathy and indifference toward black victims. In the thirty-one years since Enoch Powell M.P. predicted war between the races in his notorious \"River of Blood\" speech, little has been done by police and government to address the issue of racially motivated attacks on Afro-Caribbeans and Asians. Indeed, many in these communities point to the number of black people who have died while in police custody or while resisting arrest, and argue that some of the worst treatment they receive is at the hands of the very people whose duty it is to protect them_.\n\nMy mother sniffed a conspiracy immediately, and set out to prove it by berating my father nonstop all the way from Devon. By the time they reached our house she had worked herself into a fine fury, made worse by my father's stubborn refusal to comment. He hoped, I think, that good manners would prevail once they reached the farmhouse, but he had forgotten how much she enjoyed confrontation, particularly where her daughter was concerned. She assumed\u2014with some justification\u2014that Sam was as much in the dark as she was and, all too predictably, the full weight of her moral outrage descended on me.\n\nShe cornered me in the kitchen. \"It's the deceit I can't stand,\" she said. \"All your life you've been saying one thing and doing another. I wouldn't mind so much if you didn't involve other people in your lies. I remember the time you and that beastly little friend of yours... Hazel Wright... swore you'd spent the night at her house when the reality was you were both passed out, drunk, on the floor of some boy's bedroom.\" She clenched her fists at her sides. \"You _promised_ us,\" she declared aggressively. \"'A new start,'you said. No more recriminations. No more dragging the family down with your dreadful fantasies. And what do you do? Break your word at the first opportunity, then manipulate your father into helping you.\"\n\nI put some glasses on a tray. \"Is Dad still on the pink gins?\" I asked her, searching the larder for Angostura bitters.\n\n\"Are you listening to me?\"\n\n\"No.\" I raised my voice to reach the open French windows which led directly from the quarry-tiled kitchen on to the Portland flags of the terrace. \"Sam! Find out if Dad wants his gin pink, will you?\"\n\n\"He does,\" came the shout back. \"Do you need a hand?\"\n\n\"Not at the moment,\" I called, taking a lemon from the fruit bowl and cutting it in half.\n\n\"I'll talk to Sam if you insist on ignoring me,\" my mother warned. \"I've already given your father a piece of my mind. God knows what he thought he was doing, encouraging you like this.\"\n\nI watched her for a moment, wishing I hadn't inherited so many of her features. She was a good-looking woman, although she rarely smiled because of worries about wrinkles, but I'd done my damnedest in twenty years to wipe out the similarities between us\u2014slimmed down, changed my hair color, forced a permanently cheerful expression to my face\u2014but it was all just window dressing. Every time I saw her, I was seeing myself thirty years on, and my smile would become a little more fixed and my resolve not to leap to critical judgments a little more determined. It made me wonder who I really was, and whether I had any substance beyond a childish desire to prove I was a better person than she was. I recalled my father telling me once\u2014as if it were something that needed saying\u2014that my mother _did_ love me, and I answered, \"Of course she does, as long as I agree with her. Not otherwise.\"\n\n\"You're her proudest achievement,\" he had said simply. \"If you reject her views, you reject her.\"\n\nI turned one of the lemon halves on its side and sliced into the oozing flesh. \"You look as if you've been sucking one of these,\" I murmured, \"and if the wind changes you'll be stuck with that sour expression forever.\"\n\nHer mouth turned down even further. \"That's not funny.\"\n\n\"You found it funny when you said it to me.\"\n\nThere was a short silence.\n\n\"You have a cruel streak in you,\" she said. \"You don't mind who you hurt, just as long as you can have your petty little revenges. I've often wondered where you get it from. There's no forgiveness in your nature. You brood over people's mistakes in a way that neither I nor your father has ever done.\"\n\nI gave a laugh of genuine amusement. \"My God! And this from elephant-brain who's just been quoting Hazel Wright at me. I was thirteen years old, Ma, and Hazel and I drank two shandies each before falling asleep on Bobby Simpkin's bed.\" I shook my head. \"You wouldn't let it rest. I don't know what you thought we'd been doing but from that moment on, I had nothing but lectures on how no decent man would take on shop-soiled goods.\"\n\n\"There you go again,\" she snapped. \"Always blaming others, never yourself.\"\n\nI shrugged. \"I was merely pointing out that my cruel streak, assuming it exists, comes from you.\"\n\n\"Have I ever broken my word? Do I lie?\"\n\n_Maybe not_ , I thought, _but I might prefer a few white lies and broken promises to the painful recognition that she would rather I had been a son_. \"The only promise I made,\" I reminded her, \"was never to mention Annie Butts in front of you or Sam again, and the fact that you're now interpreting my keeping of that promise as deceit is hardly my fault.\"\n\n\"Then how did your father get caught up in it?\"\n\n\"In what?\"\n\n\"Whatever it is you're doing... the reason you chose to come here despite the trouble I went to to find you a house in Devon.\"\n\n\"I didn't make the promise to Dad,\" I said, \"and he wouldn't have accepted it if I had. He offered to help me before Sam and I left England, and he's been a tower of strength ever since. As a matter of fact, he's the one who spotted the ad for this place in the _Sunday Times_ and phoned me in Cape Town to suggest we rent it for the summer.\"\n\nAnother silence, rather longer this time. She wanted to ask me why\u2014much as Sam had done last night\u2014but she was embarrassed to admit just how far she'd been excluded from our lives and decisions. Instead she adopted an injured air. \"I hope you haven't turned Sam's sons against him as well,\" she said. \"That really would be unforgivable.\"\n\n\"I haven't turned anyone against him,\" I answered, searching the cupboards for a jug.\n\n\"Oh, for goodness'sake!\" she said sharply. \"Don't be so na\u00efve. When you persuaded your father to take your side against your husband's, you effectively set them at each other's throats.\"\n\n\"It was never a question of taking sides,\" I said, finding a glass carafe, \"only a question of research. In any case you took Sam's side against mine, so Dad thought it reasonable for at least one of my parents to redress the balance.\"\n\n\"I did it for your own good. You were behaving like a spoiled child.\"\n\n\"How odd,\" I said with a laugh. \"That's exactly what Dad said about Sam.\"\n\n\"That's nonsense. Your father and Sam used to get on like a house on fire until you insisted on jeopardizing your marriage over that wretched negro.\" She paused. \"Dad's worked hard to restore their relationship, which is why it's so unkind of you to persuade him to go behind Sam's back like this.\"\n\nI cocked an ear to the rumble of relaxed conversation outside. \"They're certainly not at each other's throats yet, so let's hope you're worrying unnecessarily.\"\n\n\"For how much longer? You can't have forgotten how upset Sam was in the wake of that woman's death. What on earth induced you to raise the whole sorry business so soon after his coronary? Do you want to cause another one?\"\n\nI filled the carafe with water and put it on the tray. \"It doesn't seem to have worried him so far,\" I said mildly, \"but feel free to ask him yourself if you don't believe me.\" I lifted the tray. \"That's everything, I think. Could you bring the lemon?\"\n\nWe talked about everything under the sun except Annie Butts, yet her presence was powerfully felt\u2014in my father's refusal to meet my mother's eyes, in Sam's obvious discomfort every time the subject of Dorchester was raised, in my mother's dreadful attempts at flirtatious-ness to reestablish a hold on her menfolk. When it became obvious that I was _de trop_ as far as she was concerned, I took the hint and vanished inside to make lunch. Ten minutes later a monumental row erupted on the terrace. I caught it only in snatches but so much heat was generated, particularly between my parents, that their rapidly rising voices carried through to the kitchen.\n\nIt does me no credit to say I enjoyed every minute of it. But I did. It was the first of my petty little revenges and I raised a silent cheer when my father told my mother it was a pity her life was so bereft of interest that her only joy came from stirring up trouble within her family.\n\nThe silence that followed my reappearance on the terrace with trays of salad was interminable. I remember thinking there was a multitude of wasps that summer. I watched them drone in their black and yellow stripes around the spirit-sugared glasses, and wondered if there was a nest nearby that needed destroying. I also remember thinking that wasps were less harmful than people, and that a sting was a bagatelle compared with the poison of a long-suppressed grievance.\n\n\"Why does your father stay with her?\" Sam asked me in bed that night.\n\n\"Once he signs up to something he always sees it through.\"\n\n\"Is that the only reason they're still together? Because your father has a sense of duty?\"\n\nI shook my head.\n\n\"What else is there?\"\n\n\"Love,\" I said. \"He's a very affectionate man and he never gives up on anyone.\"\n\n\"Like father like daughter then?\"\n\nI turned to look at him. \"Is that how you see me?\"\n\n\"Of course. How else would I see you?\"\n\n**_Letter from Dr. Joseph Elias, psychiatrist_**  \n ** _at the Queen Victoria Hospital, Hong Kong\u2014dated 1980_**\n\n**QUEEN VICTORIA HOSPITAL**\n\n_Hong Kong_\n\n_Dept. of Psychiatry_\n\n**Mrs. M. Ranelagh  \n12 Greenhough Lane  \nPokfulam**\n\n**February 14, 1980**\n\n**Dear Mrs. Ranelagh** ,\n\n**Thank you for your letter of July 3. I'm sorry you feel that a follow-up visit would be of no benefit, particularly as your reference to \"a new calm\" suggested that our previous conversation had been valuable. However, as you so rightly point out, there is no compulsion on you to attend further sessions**.\n\n**I have pondered deeply on the question you posed toward the end of our session. Why should your husband escape punishment for raping you? And I pass on some wisdom I received as a child in Auschwitz concentration camp when I asked a rabbi if the Germans would ever be forgiven for what they were doing to the Jews. \"They will never forgive themselves,\" he said. That is their future and also their punishment**.\n\n**Should you not have asked, however, whether it was right for Sam to escape your punishment? Andare you so free of guilt yourself, Mrs. Ranelagh, that you feel comfortable standing in judgment on your husband?**\n\n**With best wishes,  \nYours sincerely**,\n\nDr. J. Elias\n\n**_Letter from Betty Hepinstall in answer to a request for information about animal cruelty in the UK\u2014dated 1999_**\n\n**THE CHESHIRE CAT HOSPITAL**\n\n**Cheadle Hulme, Cheshire, UK**\n\nMrs. M. Ranelagh  \nJacaranda  \nHightor Road  \nCape Town  \nSouth Africa\n\nDecember 3, 1998\n\nDear Mrs. Ranelagh,\n\nIn response to your detailed inquiry about the ill treatment of cats in the UK, I enclose a copy of a leaflet we produced last year to boost interest in a fund-raising drive. As you will see, it makes grim reading, but I make no apology for the contents. The work we do is costly and time-consuming and would be entirely unnecessary were it not for the terrible cruelty that is regularly inflicted on defenseless animals.\n\nI have no difficulty in believing that someone would put superglue in a cat's mouth and tape its muzzle with Elastoplast or parcel tape to stop it from eating or crying. In the past, we have seen cats with their paws dipped in quick-drying cement to prevent them walking; cats with their back legs paralyzed by broken spines; cats with their claws and teeth pulled out by pliers; cats blinded with red-hot pokers; and cats with rubber bands wound so tightly round their muzzles that the flesh of their mouths had closed over the band. And all, apparently, to the same purpose: to stop them from catching birds and mice.\n\nI would like to be able to tell you that a person who pursues this sort of vendetta against cats is easily identifiable, but I'm afraid I can't. There is considerable evidence-largely through behavioral-science studies in the U.S. and the UK-to indicate that cruelty to animals in childhood leads to sociopathic behavior in adulthood. However, cruelty is far more common in adults than it is in children, and such cruelty is usually the result of an obsessional dislike of certain animals or an uncontrollable temper-often drink-related-which lashes out at anything it finds irritating.\n\nSadly, I cannot say with any certainty that because Miss Butts treated her own cats with kindness she would not have inflicted cruelty on strays intruding into her house. I can only draw parallels with people, and people are notoriously unwilling to show the same charity to foreigners as they show to their family and friends.\n\nYours sincerely,\n\nBetty Hepinstall\n\n# ten\n\nThe following day I drove my mother to Kim-meridge Bay on the Isle of Purbeck. It was a beautiful summer morning with puffs of white cloud dotted across the sky, and we climbed the cliff path to the Clay Tower on the eastern arm of the bight. Larks sang in the air above us, and the occasional walker passed us by, nodding good day or pausing to look at the bizarre folly behind us that some long-dead person had built as a permanent sentinel to guard the ocean approaches. Mother and I conversed with the strangers but not with each other and, in the silences between, we stared as resolutely across the channel at the tower, unwilling to speak in case we started another argument, locked in mutual ignorance despite the genes and experiences we shared.\n\nIn the end I mentioned a vicar's wife I knew who drove to clifftops whenever the pressures of life became too much and screamed her frustrations to the heavens. I suggested my mother had a go. She refused. It wasn't her sort of thing, she told me. Nor could she understand why a vicar's wife would want to do something so common. What sort of woman was she?\n\n\"Eccentric,\" I murmured, as I watched the seagulls float effortlessly over the sea like fragments of tissue paper. \"Very thin and gaunt... hates being married to a vicar... likes her booze... fancies being a lap dancer... looks like a vulture.\"\n\n\"That explains it then,\" said my mother.\n\n\"What?\"\n\n\"The screaming. Thin people are always more high-strung than fat ones.\"\n\nIt sounded reasonable, but then much of what my mother said sounded reasonable. Whether it was true or not was another matter. I decided she was being snide because _she_ was plump and _I_ was thin but, for once, I chose to ignore the bait. \"I've been wondering if it works,\" I said idly. \"My screams are always silent ones that circle 'round my head for days until they run out of steam and die naturally.\"\n\n\"It's pure affectation to scream at all. You should learn to deal with your problems quietly instead of making a song and dance about them.\" I gave a weary sigh as I thought to myself that that was precisely what I had done, and she cast me a suspicious glance. \"I suppose that's why you brought me here? So you can scream at me?\"\n\n\"Not at _you,\"_ I corrected her. \"At the wind.\"\n\n\"You'll only embarrass yourself,\" she said. \"Someone's bound to appear up the path at the wrong moment.\"\n\n\"Perhaps that's the point,\" I murmured reflectively. \"A double whammy Physical and mental adrenaline all in one shot.\" I watched a dinghy, full of divers in wetsuits, motor out of the bay and head toward the southwest. \"Would it embarrass you?\"\n\n\"Not in the least.\" She perched her behind on the edge of a rock. \"If I wasn't embarrassed twenty years ago when you were behaving like a madwoman then I'm hardly likely to start now.\"\n\n_She has a short memory_ , I thought, as I lowered myself to the ground to sit cross-legged in front of her. Her embarrassment had been colossal. I concentrated my attention on a clump of pink thrift that had rooted itself tenaciously in a crevice. \"I wasn't mad, Ma, I was exhausted. We were kept awake night after night by the phone ringing nonstop, and even when we changed our number the calls just kept on coming. If we took the damn thing off the hook, we had mud thrown against our windows or constant hammering on the front door. We were _both_ suffering from sleep deprivation, _both_ behaving like zombies, yet for some reason you decided that everything Sam told you was true while everything I said was a lie.\"\n\nShe examined the distant horizon where the blue of the sea met the blue of the sky, and I remembered her telling me once that the difference between a woman and a lady was that a woman spoke without thinking while a lady always considered what she was about to say. \"You screamed and yelled about rats in your downstairs lavatory,\" she said at last. \"Are you saying _that_ wasn't true? You poured gallons of bleach down the loo in order to kill them, then became hysterical because you said they'd moved into the sitting room.\"\n\n\"I'm not denying I said some strange things, but they weren't lies. I kept hearing scratching sounds and I could only think of rats.\"\n\n\"Sam didn't hear them.\"\n\n\"He most certainly did,\" I contradicted her. \"If he told you he didn't, he was lying.\"\n\n\"Why would he want to?\"\n\nI thought back. \"For a lot of very complicated reasons... mostly, I imagine, because he didn't like me much at the time and thought that everything was my fault. He said I was making the noises myself to get attention and was damned if he was going to pander to any more of my childishness.\"\n\nShe frowned. \"I remember him saying he called in the rat catcher to try to persuade you it was all in your imagination.\"\n\nI shook my head. \"It was me who called in the rat catcher, and for exactly the opposite reason. I wanted proof that there _were_ rats.\"\n\n\"And were there?\"\n\n\"No. The man said there was no evidence of rodent infestation, no nests, no indication that any food had been eaten, and no droppings. He also said that if we had rats then our neighbors would be complaining as well.\" I ran my finger lightly over the thrift and watched the pink heads shiver. \"The next day Sam phoned you to tell you I was going 'round the bend and he wanted a divorce.\"\n\nShe didn't say anything for several moments, and I raised my head to look at her. There was a perplexed expression on her face. \"Well, I'm completely lost. If you and Sam both heard it but it wasn't rats and it wasn't you, then what was it?\"\n\n\"I think it might have been cats,\" I said.\n\n\"Oh, for goodness'sake!\" she declared crossly. \"How could there be cats in your house without your noticing?\"\n\n\"Not _in,\"_ I said, _\"under_. It took me a long time to work it out because I didn't know the first thing about building houses. I couldn't even change a plug when I married, let alone get to grips with the importance of underfloor ventilation.\"\n\nHer mouth thinned immediately. \"I suppose that's a sly dig at me and your father.\"\n\n\"No,\" I said with an inward sigh, \"just a fact.\"\n\n\"What does it have to do with cats?\"\n\n\"Houses have holes in their walls below ground level to allow a free flow of air under the floorboards. It prevents the wood from becoming rotten. They're usually constructed out of airbricks, but the houses in Graham Road were built in the 1880s and in those days they used wrought-iron grills to make a design feature out of them. Before he left, the rat catcher mentioned that one of ours was missing from the back of the house. It happened all the time, he said, because there was quite a market for them in architectural salvage. It wasn't a problem because someone had wedged a metal bootscraper over the hole, but he suggested we get it replaced at some stage if we didn't want trouble in the future. He kept calling it a ventilation grill, and I assumed he was talking about something that was attached to the extractor fan in the upstairs bathroom because that was the only ventilation I knew about.\"\n\nI fell silent and she made an impatient gesture with her hand, as if to say, \"Get on with it.\"\n\n\"I wasn't very with it at the time\u2014all I wanted was confirmation that rats existed\u2014so it went in one ear and out the other because whatever was missing didn't seem to stop the extractor fan working. Then, one day in Sydney, I watched our neighbor's Jack Russell dig a hole in the flower bed beside our house and vanish through a hole into the crawl space beneath the house, and I realized the rat catcher had probably been talking about underfloor ventilation. He was telling me we had a hole in our back wall at ground level, and probably quite a sizeable one if a wrought-iron grill had been hacked out.\"\n\n\"And because of that you think cats got in?\"\n\n\"Yes.\"\n\n\"Didn't you say the rat catcher said it wasn't a problem because a boot scraper was wedged over the hole?\"\n\n\"Yes.\"\n\n\"Then how did they get in?\"\n\n\"I think someone carried them down the alleyway at the back of our house, pushed them in and covered the hole afterward.\"\n\nShe gave a snort of incredulity. \"That's too absurd. The rat catcher would have heard them. They'd have been yowling their heads off. And why cats? Why not dogs? You said it was a Jack Russell that went into your crawl space in Sydney.\"\n\n\"Because Annie's house was full of cats.\"\n\n\"Now you really _are_ being ridiculous! The woman had been dead for weeks by that time. They can't possibly have been hers.\"\n\n\"I'm not suggesting they were,\" I said, \"just that cats are more likely than dogs in the circumstances. My guess is they were pushed in under our floorboards to die because there was no convenient cat flap in our back door. If there had been, I think I'd have found them dying in my kitchen. I called out the gas board twice because I thought I could smell gas, but each time they said there was nothing wrong. One of the men said it smelled like a dead mouse, but I said it couldn't be because we didn't have any.\"\n\nI could feel the weight of her disbelief bearing down on my bent head. \"You'd have known if something had died. The smell of death is terrible.\"\n\n\"Only when it's warm. This happened in winter\u2014a particularly cold winter\u2014and we had fitted carpets over all the floors.\"\n\n\"But\u2014\" She broke off to marshal her thoughts. \"Why didn't you hear them? Tomcats make a terrible noise when they yowl.\"\n\n\"It depends what was done to them first.\" I shook my head. \"In any case, I think they must have died of hypothermia very quickly.\"\n\nAnother pause. \"What on earth could be done to a cat to stop it crying?\"\n\nI hunched my shoulders as I thought of the chilling research I'd done on the subject. \"At a guess, they had superglue pumped into their mouths and eyes and Elastoplast wrapped tightly 'round their faces so they couldn't see, eat, drink or cry. Then they were pushed under our house to try to scratch their way out with the only things left to them... their claws.\"\n\nMy mother drew a disgusted breath, although whether her disgust was leveled at me for making the suggestion or at the suggestion itself, I couldn't tell. \"What sort of people would do a thing like that?\"\n\nI reached into my pocket for a copy of the police report describing the entry into Annie's house the day after her death and passed it across to her. \"The same people who tortured cats for Annie's benefit,\" I said. \"The only difference is, they pushed the wretched creatures through her cat flap so she could see what was happening to them.\"\n\nShe glanced at the report but didn't read it. \"Why? What was the point?\"\n\n\"Any reason you like. Sometimes I think it was done to cause fear, other times I think it was done for pleasure.\" I turned my face to the wind. \"In a perverted sort of way, I ought to feel flattered. I think the assumption was I was cleverer than Annie and could work out for myself that animals were dying in dreadful agony under my house. And the fact that I wasn't... and didn't... must have been a disappointment.\"\n\nIf my mother asked me why once, she asked it a hundred times on our journey home. Why hadn't Annie gone to the police? Why hadn't Annie phoned the RSPCA? Why would anyone feel confident about tormenting me in the same way they'd tormented Annie? Why weren't they afraid I'd go to the police? Why _hadn't_ I gone to the police? Why would anyone want to reinforce my suspicions about Annie's death? Why risk getting Sam involved? Why risk getting the rat catcher involved? Why hadn't I questioned the RSPCA findings at the inquest? Why... ? Why... ? Why... ?\n\nWas she finally beginning to understand how betrayed I'd felt when she hadn't believed me at the time? Or was I being cynical in my absolute conviction that it was only her recognition of my father's tireless support of me that had shamed her into asking any questions at all?\n\nIn any case, I had few answers for her, other than to say no one believes a mad woman. \"But why assume there was a logical thought process at work,\" I asked her finally, \"when whoever tortured the cats was clearly unbalanced?\" It was done for the pleasure of inflicting pain, not because it was possible to predict how Mad Annie or I would react to having mutilated animals left on our doorsteps.\n\n**_Family correspondence\u2014dated 1999_**\n\nCURRAN HOUSE\n\n_Whitehay Road_\n\n_Torquay_\n\n_Devon_\n\nPS I know she's barking up the wrong tree but do give her credit for trying. She's very \"down\" at the moment because she feels we ganged up on her and can't understand why I said she should have expected it\u2014i.e., what goes around comes around\u2014but she doesn't want to be reminded of how she ganged up on you all those years ago. It would be tactful, my dear, to avoid saying \"I told you so,\" however strong the temptation. I would think less of you if you did!\n\nDad\n\nX X X\n\n# eleven\n\nPortland Peninsula was under assault from a blustery southwest wind the following Wednesday when Sam and I drove up from Chesil Beach in search of the sculpture park. Given the choice, I'd rather have gone on my own. There was too much that still needed explaining\u2014my more-than-passing interest in Danny, for example\u2014but I balked at telling Sam his presence would only exacerbate the problem when, like my mother, his way of making up for past indifference was a belated wish to be involved.\n\nI had made a halfhearted attempt the previous day to talk about the three weeks at the end of January and beginning of February '79 that I spent alone on Graham Road, but my habit of silence was so ingrained that I gave it up after a few minutes. I found I couldn't talk about fear without becoming cruel, and I couldn't become cruel without turning on Sam because he had abandoned me when I needed him most. In the end, as so often in my life, I took the fatalistic view that whatever would be would be. Sam was a grown man. If he couldn't learn to live with the truth, irrespective of how it was revealed to him, then nothing I did or said would make a difference.\n\nThe Isle of Portland, a tilted slab of limestone four miles long and one mile wide, forms a natural breakwater between Lyme Bay to the west and the sweep of sheltered water between Weymouth and the Isle of Purbeck to the east. Its precipitous cliffs rise out of the sea to a high point of nearly five hundred feet, with only the hardiest of vegetation surviving the mercurial English weather. As Sam and I wound our way up its spine, I thought how bleak it was, and how unsurprising that successive governments had claimed it both as a fortress against foreign invasion and as a colony for prisoners.\n\nIn 1847 the Admiralty had employed convict labor awaiting transportation to Australia to construct a mighty harbor on Portland's eastern shores, which remained the preserve of the Ministry of Defense until the government abandoned it in the early 1990s. It seemed fitting somehow, in view of the convicted men who had toiled to create the anchorage, that the most prominent feature in Portland harbor that Wednesday was a gray prison ship that had been imported from America some four years previously to deal with the chronic overcrowding in Her Majesty's inland gaols.\n\n\"Is Michael Percy being held there?\" Sam asked me.\n\n\"No. He's in the adult prison here on the island. It's called the Verne. It's off to our left somewhere.\" I pointed to a sprawling Victorian building ahead of us which dominated the skyline. \"That's the young offenders'institution. It was built to house the convicts who worked on the harbor.\"\n\n\"Good God! How many prisons are there?\"\n\n\"Three, including the ship.\" I laughed at his expression. \"I don't think it means Dorset's a hive of criminal activity,\" I said, \"just that desolate lumps of rock make good holding pens for society's rejects. Think of Alcatraz.\"\n\n\"So what did Michael do?\"\n\nI thought back to the press cuttings of his trial which had arrived toward the end of 1993. \"Went into a village post office in leathers and a crash helmet, and pistol-whipped an elderly customer until the postmaster agreed to open his security door and hand over what was in his till.\"\n\nSam whistled. \"A bit of a bastard then?\"\n\n\"It depends on your viewpoint. Wendy Stanhope would say it was his mother's fault for letting him run out of control. Her name was Sharon Percy. She's the blonde you saw in the pub occasionally.\"\n\nHe made a wry face. \"The prostitute? She used to haunt the flaming place looking for customers. She tried to hit on me and Jock once so I gave her a piece of my mind. Jock was furious with me afterward. He said Libby was giving him a hard time, and he'd have been up for it like a shot if I hadn't queered his pitch.\"\n\n\"Mm. Well, at a guess he was double-bluffing you in case you got suspicious about her approach. According to Libby, he was paying out thirty quid a week to Sharon for most of '78. They didn't bother to keep it much of a secret either, except from the people who mattered... like you and me and his long-suffering wife.\" I watched him out of the corner of my eye. \"Paul and Julia Charles worked out what was going on because Paul saw Jock coming out of Sharon's house one evening and put two and two together.\"\n\nHe threw me a startled glance. \"You're joking!\"\n\n\"No. She charged twenty for straight sex, thirty for a blow job, and Jock visited her every Tuesday for months.\" I was amused. \"You can work out for yourself which service he was getting.\"\n\n\"Shit!\" He sounded so shocked that I wondered if \"Tuesday\" had registered as the day Annie died, and if he was now trying to remember the details of the alibi he'd given Jock. \"Who told you?\"\n\n\"Libby.\"\n\n\"When?\"\n\n\"A year or so after we left. It all came out in court when Jock decided to contest the divorce settlement. Libby hired a hotshot solicitor who demanded an explanation for the _\u00a330_ withdrawals from the joint bank account every Tuesday, along with an explanation for the numerous other bank accounts he'd set up without Libby's knowledge. He wasn't very good at hiding his peccadillos and the judge took him to the cleaners for it.\" I pointed to a sign for Tout Quarry. \"I think this is where we need to turn off.\"\n\nHe flicked his indicator. \"Where did they do it?\"\n\n\"In her house. Sharon used to smuggle her clients down the alleyway at the back as a way of protecting her reputation... such as it was.\"\n\n\"What about her kid?\"\n\n\"Michael? I'm not sure he was there very much. Wendy said he was always in trouble with the police so I imagine he was made to roam the streets.\"\n\n\"Jesus!\" said Sam in disgust, as he drove onto a rough unmade track that led down to the sculpture park. \"No wonder he went to the bad.\" He drew the car to a halt and switched off the engine. \"How was he caught for the post-office job?\"\n\n\"He confessed to his wife three months later and she promptly turned him in. She gave the police a black leather jacket which she said Michael was wearing on the day of the robbery. It still had blood spots round the cuffs which matched the customer's in the post office.\" I thought back. \"Michael pleaded guilty but it didn't do him much good. The judge commended Bridget for the brave assistance she'd given the police, and said he was sending her husband down for eleven years as a result of her efforts.\"\n\n\"And this is the Bridget who lived on Graham Road?\"\n\n\"Mm. She was at number 27... opposite Annie's house. Her father, Geoffrey Spalding, shacked up with Michael's mother when Bridget was thirteen, leaving her and her older sister, Rosie, to fend for themselves. I don't know what happened to Rosie, but Bridget and Michael married sometime in 1992, just after Michael finished a long sentence for aggravated burglary and ten counts of breaking and entering. He stayed out of trouble for about six months then robbed the post office. All in all, he and Bridget have spent less than a year living together as a married couple.\"\n\n\"And now they're divorced?\"\n\n\"No. The last I heard she was working in Bournemouth and making a monthly trip to Portland to visit Michael. That's why he was moved down here... because no one visits him except his wife. She said at the trial that she still loved him\u2014said she can't rely on anyone the way she relies on Michael because they've known each other since they were children\u2014and the only reason she turned him in was because she was afraid he was going to kill somebody. I thought how brave she was,\" I said dryly. \"His mother's a coward by contrast\u2014that's Sharon\u2014won't go near him... hasn't done for years because of the shame he's brought on her. She's been respectable ever since Bridget's dad moved in with her and she was able to give up the game.\"\n\n\"She sounds a right bitch,\" he said grimly\n\n\"She's not much of a mother, that's for sure.\"\n\nSam leaned his arms on the steering wheel and stared thoughtfully out of the window. \"Were all the kids as bad?\" he asked. \"What about the Charles children next door?\"\n\n\"The oldest was only five,\" I said, \"and Julia never let them out of her sight. It was really only Michael and the Slaters who ran wild... in both cases because their mothers had given up on them. Sharon didn't care... and Maureen was so brutalized by Derek that she spent most of her time getting stoned in her bedroom.\"\n\n\"Did you know all this in '78?\"\n\n\"No. Most of it came from Libby after we moved. I knew Alan Slater was getting into fights because he had so many bruises, but I didn't realize it was his father who was hitting him. I talked it over with the head on one occasion, but he just said it would do Alan good to be thrashed by his own peer group because he was a bully himself. As for Michael\"\u2014I gave a small laugh\u2014\"I always thought how mature he was for his age. He wrote me a couple of love poems and left them on my desk, signed: The Prisoner of Zenda.\"\n\n\"How did you know they were from him?\"\n\n\"I recognized his handwriting. He was an incredibly bright child. If he'd come from a different background, he'd have an M.A. from Oxford by now instead of a ten-page criminal record. The trouble was he was a persistent truant so he only ever attended one class in three.\" I sighed. \"If I'd been a little more experienced\u2014or less intimidated by the bloody headmaster\u2014I could have helped him. As it was, I let him down.\" I paused. \"Alan, too,\" I added as an afterthought.\n\n\"Did Jock know Michael was truanting?\"\n\nI reached for my door handle. \"I shouldn't think so,\" I said bluntly. \"He was paying to have his dick sucked, not listen to stories about Sharon's only child.\"\n\nIt was years before I understood that Michael's poems were more about loneliness than love. At the time I swung between suspicion that he had plagiarized them, possibly from song lyrics, and admiration that a fourteen-year-old could write so poignantly. Either way, I decided he had an unhealthy crush on me and made a point of keeping him at arm's length to prevent him becoming a nuisance.\n\n_If I was older. If I was wise.  \nYou'd look at me with different eyes  \nand love me_.\n\n_If I was handsome. If I was strong.  \nNo one would say that you were wrong  \nto love me_.\n\n_It always makes me sad to see  \na weed grow where a flower should be.  \nSo I think of flowers when I think of you.  \nIt always makes me sad to hear  \nthe deathly silence in the air.  \n_\n\n_So I think of music when I think of you.  \n_\n\n**_Letter from Libby Garth\u2014ex-wife of Jock Williams,  \nformerly of 21 Graham Road, Richmond\u2014  \nnow resident in Leicestershire_**\n\n**Windrush  \nHenchard Lane  \nMelton Mowbray  \nLeicestershire**\n\n**December 4, 1989**\n\n**M'dear** ,\n\n**Happy Christmas! I'd send a card if I didn't think Sam would go ballistic at the idea. It still hurts, you know, that he took Jock's side without ever bothering to hear mine. I know you say it's not in his nature to think ill of anyone\u2014 let alone a close friend\u2014but he must think ill of me if you can't even tell him we're still in touch. It's one of the horrible truisms that divorce doesn't just divide property, but friends as well. That being said, it's probably better this way if he's still shying away from the whole subject of Annie's death**.\n\n**Have you ever worked out why that is? I know you say he has a habit of forgetting anything he doesn't want to remember\u2014like your frigid spell, your near-divorce, your \"fits of the vapors,\" your police caution, etc.\u2014but surely Annie doesn't hold any fears for him now? He can't possibly have killed her because he's not the type to push people under trucks! Surely, that had to be Derek Slater? He was the only man on Graham Road who was vicious enough**.\n\n**Jim and the girls are fine. At the moment I'm resistingJim's blandishments for one more try to see if we can make a boy. I keep telling him a three-year-old, a nine-month-old and a teaching job are more than enough to occupy anyone, but he seems to think I'm Superwoman. I don't know how you managed without a nanny. The only thing that keeps me sane is to get into my car every morning and spend the day with my alternative \"family\" at school, though I'm still trying to work out how to persuade fourteen-year-old gorillas with twice as much testosterone as brain that learning is a \"good thing.\" I leave every class feeling as if I've been raped and ravaged by their revolting imaginations. Did that contribute to your agoraphobia after Annie died? I've often wondered. I remember you telling me that you couldn't stand the way Alan Slater and Michael Percy looked at you**.\n\n**Apropos, I enclose two cuttings. One about Michael, who goes from bad to worse, which is only to be expected of the tart's son. Yes, I'm being beastly, but I'd have to be a saint to view the \"bleached vampire\" and her progeny with anything other than hatred since they received a rather more regular income from Jock than I ever did! The second is about the policeman, Sergeant Drury\u2014the one you had a yen for at the beginning. (Looked like a shorter-haired version of Patrick Swayze in Dirty Dancing\u2014have you seen the movie? It's to die for!) I might have fancied him myself if he hadn't turned out to be such a shit. It was unforgivable of him to \"kiss and tell\" to Sam. Have you considered that that might be Sam's problem with the Annie saga? It was certainly his problem the night before he took himself off for three weeks. Have you forgiven him for forcing you yetIt was a shabby and beastly way to treat you when you were struggling with agoraphobia and depression. But that's men for you\u2014act first, think later! I bet he regrets it now, especially if you managed to persuade him that Drury was lying**.\n\n**Anyway, Drury's taken early retirement, though by the way the piece is worded the implication is he was given the boot for whacking a seventeen-year-old Asian boy**.\n\n**Keep smiling** ,\n\n**All my love** ,\n\n_Libby_\n\n# twelve\n\nTout Quarry, home to the sculpture park and once a source of hand-quarried Portland stone, had been long worked out and abandoned. It was a wild and wonderful place. A man-made labyrinth of tangled gorges and wide-open spaces like amphitheaters where stunted shrubs and trees grew among blocks of half-excavated sedimentary rock. It looked as if a giant hand had rummaged in the belly of the earth and stirred the stone into a chaotic, tumbling dance.\n\nSam was fascinated by the intermittent sculptures which had been carved in situ and introduced subtle shape to the craggy landscape. Antony Gormley's _Still Falling_ \u2014a cameo figure plunging down a rocky cliffside. Robert Harding's _Philosopher's Stone_ \u2014an intricate layer of cut stones perched between V-shaped rocks. A crouching man with his chin resting on his knees. Footsteps. A tulip, prized from the rock to lie in reflection on the ground. \"Is anyone allowed to have a go?\" he asked, examining a fossil in a slab and trying to work out if it was a real ammonite or a simulated one.\n\n\"I think you have to be invited.\"\n\n\"Pity,\" he said wistfully. \"I quite fancy leaving my mark for posterity.\"\n\nI laughed. \"Which is probably why people like you aren't allowed to do it. You'd get bored after a while and carve out 'Sam woz 'ere, 1999,'then the whole place would be desecrated with graffiti.\"\n\nWe heard the sculpture workshop before we saw it. A constant rata-tat of hammers on chisels, overlaid by the whistle of wind through a polythene canopy that had been rigged above the sculptors'heads. It was a scene of intense industry because everyone was there for a purpose, to learn how to work in three dimensions. White stone chippings littered the ground, and a fine dust clung to arms, hair and clothing like baker's flour. It might have been a Renaissance atelier in Italy but for the polythene canopy, the uniform prevalence of T-shirts and jeans, and the fact that half the sculptors were women.\n\nIt was situated in a sheltered gulley, and Danny stood out from the rest of the group, not just because he'd positioned himself near the entrance but because his block of stone was three times the size of anybody else's. It was also a great deal more advanced. Where most of the others were still working to establish basic form, Danny had already released a bespectacled head and upper torso from the limestone's grip and was using a claw chisel to give grained texture to the skin of the face.\n\nHe looked up as we approached. \"What do you think?\" he asked, stepping back and letting his hands fall to his sides, unsurprised that we'd come to admire his work. His physique interested me. I was amazed by how well developed his shoulders and arms were without a jacket to hide them.\n\n\"Excellent,\" observed Sam with the overdone bonhomie that he reserved for men he didn't know very well. \"Who is it? Anyone we know?\"\n\nA scowl of irritation narrowed Danny's eyes.\n\n\"Mahatma Gandhi,\" I said, casting a quick verifying glance at the drawings and photographs on the ground beside him. I didn't need to. The likeness was there, even if more reliant on intuition than reality. \"It's an ambitious subject.\"\n\nThat didn't please him either. \"I can tell you're a teacher,\" he said witheringly glancing toward the canopy where instructors were passing on advice and help to the other students. \"That's what they keep telling me.\"\n\nI eyed him curiously. \"Why don't you take it as a compliment?\"\n\nHe shrugged. \"Because I know a put-down when I hear it.\"\n\n\"You're too sensitive,\" I said. \"In my case it's a spur to keep you going. You're obviously the star here\u2014head and shoulders above the others\u2014and unless you're blind and stupid you must recognize the fact.\"\n\n\"I do.\"\n\n\"Then stop bellyaching and prove you can cope with an ambitious subject.\" I ran a finger along the larger-than-life spectacles which grew at a forty-five-degree angle from the wrinkled stone cheeks. \"How did you do these?\"\n\n\"Carefully,\" he said, more serious than ironic.\n\nI smiled. \"Weren't you afraid of knocking them off?\"\n\n\"I still am.\"\n\n\"There's a bronze statue of Gandhi in Ladysmith in South Africa. It commemorates the ambulance corps that he set up there during the Boer war. It's the only other one I've ever seen of him.\"\n\n\"How does it compare?\"\n\n\"With this one?\"\n\nHe nodded. I might have mistaken his question for arrogance if the muscles in his shoulders had been less rigid or his scowl less ferocious. _He is preparing to defend himself again_ , I thought.\n\n\"It's a thoroughly professional life-size representation in bronze of a tiny little man who did his duty by the Empire after accepting British citizenship,\" I said. \"But that's all it is. It gave me no sense of his greatness, no sense of the extraordinary effect his humility had on the world, no sense of inner strength.\" I moved my fingers to touch the rough limestone face. \"Gandhi was a giant with no pretensions. For myself, I'd rather have him larger-than-life and rough-hewn in stone than realistically small and neatly polished in bronze.\"\n\nHis scowl relaxed. \"Will you buy it?\"\n\nI shook my head regretfully.\n\n\"Why not? You just said you liked it.\"\n\n\"Where would I put it?\"\n\n\"In your garden.\"\n\n\"We don't have a garden. We're only renting the farmhouse for the summer. After that\"\u2014I shrugged\u2014\"who knows? If we're lucky we may be able to afford a brick box with a tablecloth for a garden and a few roses 'round the border... and, frankly, a bust of Mahatma Gandhi in the middle of it would look very out of place.\"\n\nHe was disappointed. \"I thought you were loaded.\"\n\n\"Sadly not.\"\n\nHe pulled out his cigarettes. \"Just keeping up appearances, eh?\"\n\n\"Something like that.\"\n\n\"Ah, well,\" he said with resignation, bending his head to shield his lighter from the wind. \"Maybe I'll give him to you for free.\" He blew smoke through his nose. \"It'll cost me an arm and a leg to get him back to London, and the chances are the specs'll get knocked off in the process. You can start a collection... put him next to Alan's Quetzal-coatl... make the Slaters famous for something other than drugs, burglary and wife-beating...\"\n\nI suggested we treat Danny to lunch at the Sailor's Rest in Weymouth but Sam wasn't keen. \"The food's good,\" he admitted, \"but the landlord's an asshole.\"\n\n\"I think you already know him,\" I told Danny as we made our way back to the car. \"He's the policeman who got Alan sent down. I thought it might amuse you to see him in different surroundings.\" I interrupted the silence that followed this remark to point at the wreck of a Viking long ship that was creatively cast upon some rocks to our left. \"That's a clever use of materials,\" I murmured.\n\n\"What's his name?\" asked Danny.\n\n\"James Drury He was a uniformed sergeant in Richmond until he was forced to take early retirement and took himself off to train as a pub manager for Radley's Brewery. They started him off in Guildford, then moved him to the Sailor's Rest in '95.\"\n\nDanny eyed me with understandable suspicion. \"How do you know he's the one who nicked Alan?\"\n\n\"A neighbor of ours in Graham Road told me,\" I explained. \"Libby Williams?\" He shook his head. \"She knew I was interested in anything Mr. Drury did, particularly if it concerned an ex-pupil.\" I tucked my hand companionably into the crook of Sam's elbow to soften the blows of revelation. \"I had several encounters with him before we moved abroad. He's probably the most corrupt person I've ever met... a thief, a liar, a bully... and a racist. Quite the wrong sort of man to be given a police uniform.\"\n\nA mirthless laugh escaped Danny's mouth. \"He sure as hell stitched Alan up. Okay, I'm not saying my brother was an angel, but he was no drug pusher. A user, maybe\u2014never a pusher.\"\n\n\"What happened?\"\n\n\"I don't know the exact details... I was just a kid at the time... but Ma said Drury nicked him in a pub one night, and dropped four ounces of hash into his pocket while he was slapping on the handcuffs. He was a right bastard. If he couldn't get you for one thing, he'd get you for another.\"\n\n\"What had Alan really done?\"\n\nDanny made fists of his hands and punched the knuckles against each other. \"Couldn't stay out of a fight, particularly when he was drunk. Took the whole damn police force on one evening and laid about him like a good 'un, never mind he was only fifteen.\" A reminiscent smile curled his mouth. \"He got five thousand quid compensation for it.\"\n\n\"That's some trick,\" said Sam.\n\n\"Not really. Al's injuries were much worse than the coppers' Three broken ribs... boot marks all over him from the kicking they gave him... internal bleeding. You name it, he had it. The only problem is\"\u2014Danny sent a stone flying with a well-aimed toecap\u2014\"Drury had it in for the Slater kids from that moment on. He arrested us all at one time or another. He rubbed his arm in tender recollection. _\"And_ gave us a damn good thrashing whenever he got the chance.\"\n\n\"So what was Alan actually convicted of?\" I asked curiously. \"Possession or assault on the police?\"\n\nDanny frowned. \"Dealing, I think,\" he said vaguely. \"But it was a stitch-up whichever way you look at it. They reckoned he was a bad influence on the rest of us, so Drury had him put away till he calmed down. He's been straight ever since... so I guess it worked.\"\n\nI wondered if any of it was true, or if it was a story the family had invented for public consumption.\n\nSam turned to me with a puzzled look. \"And this Drury is the man who was staring at you?\"\n\nI nodded. \"I think he was trying to work out who I was.\"\n\n\"Well, he'll know by now. I paid by credit card.\"\n\n\"Yes,\" I agreed. \"That's why we went there.\"\n\nHe looked away, racing to put the pieces of his own personal jigsaw puzzle together. \"So what's the plan?\" he asked as we approached the car. \"Do we walk up to this thug and confront him? Or do we behave with civilized disdain?\"\n\n\"We behaved with civilized disdain last time,\" I reminded him.\n\n_\"You_ may have done,\" he snapped irritably, inserting his key into the car door. \" _I_ didn't know him from Adam. All I saw was a middle-aged prick ogling my wife.\" He frowned at us across the roof. \"If you're intending to talk to him about the robbery you won't get anywhere. Larry said he wasn't remotely interested when Sheila tried to raise it with him. He just became extremely offensive and reduced Sheila to a nervous wreck.\"\n\nI exchanged a quick glance with Danny and saw only curiosity in his eyes. \"I want to unsettle him a little,\" I said. \"Make him wonder what three ex-residents of Graham Road are doing in his pub.\"\n\nSam shook his head, clearly unimpressed. \"Yes, but _why?_ What are you expecting to achieve? There's no reason to think you'll be any more successful than Sheila, and I'm not keen to end up in the middle of a screaming match in public.\"\n\nDanny spoke before I could answer, after shoving his hands into his pockets to protect whatever was there. _Cannabis?_ I wondered. \"I've done my best to put distance between me and Mr. Drury these last ten years,\" he grumbled, \"and I'd be well pleased if he thought I was dead.\"\n\nI gave a noncommittal shrug. \"Okay, we'll go somewhere else. I always planned to confront him on my own anyway. He doesn't scare _me..._ or not as much as he'd like to think.\"\n\nI was lying of course.\n\nSam took up the gauntlet as I hoped he would, albeit reluctantly since he obviously thought I was planning a scene, while Danny muttered that the issue had nothing to do with being scared and everything to do with common sense. He asked me if we intended to drive him back to the sculpture park afterward, and when I said we would, he brightened visibly and tucked something down between the cushions of the back seat before we left the car.\n\nWhen we reached the Sailor's Rest, Sam chose a table near the harbor wall and eyed the other customers warily to see if he recognized anyone. \"Just try to keep your voice under control,\" he muttered irritably. \"You get very strident when you talk about Annie.\"\n\n\"Not anymore,\" I said, switching my attention to Danny and asking him to come inside with me. \"Sam can guard the table,\" I told him, \"while you and I sort out the drinks.\"\n\n\"What you mean is, you want the ferret to see the rabbit,\" Danny muttered despondently as I led the way across the cobbles to the front door of the pub.\n\nI smiled, liking him a great deal. \"Rabbits plural,\" I said. \"We're both in the same boat. But there's strength in numbers... any rabbit will tell you that.\"\n\n\"So who's this Annie you get strident about?\" he asked as we paused on the threshold to let our eyes adjust from brilliant sunshine to the Stygian gloom inside.\n\n\"Annie Butts,\" I told him. \"She was living next door to you in Graham Road while Sam and I were there. Your mother would probably remember her. She was a black lady who was killed by a truck shortly before we left. Her death was one of the reasons Mr. Drury and I came to blows.\"\n\nHe shook his head. \"Never heard of her.\"\n\nI believed him. He seemed to have no memories of his early childhood\u2014perhaps because it was too painful and he had chosen to blank it, just as I had done with some of my more disturbing memories\u2014and I was grateful for his ignorance. If nothing else, it meant my conscience could ride a little lighter. \"There's no reason why you should,\" I said. \"People die every day, but they're usually only remembered by their families.\"\n\nHe looked toward the bar where Drury was standing. \"So why did you and him come to blows over her?\"\n\nIt was a good question. \"I don't know,\" I answered honestly. \"It's something I've never understood. But I'll get an explanation one day... assuming there is one.\"\n\n\"Is that why we're here?\" he asked in an unconscious echo of my mother three days before. In a way it was flattering. They both assumed I knew what I was doing.\n\n**_Correspondence from Michael Percy, son of Sharon\u2014  \nconvicted of armed robbery and serving out his sentence at  \nthe Verne Prison, Portland\u2014-formerly of 28 Graham  \nRoad\u2014dated 1999_**\n\n# thirteen\n\nI don't know if it's enough to say I wanted revenge on Drury because I hated him. One should have reasons for hatred, not just a visceral antipathy that causes a red mist before the eyes at the mere mention of a name. Dr. Elias had asked me several times why I bothered to invest so much emotion in a man I had known for only a matter of weeks, but I could never bring myself to answer for fear of sounding paranoid.\n\nHe had changed very little in twenty years except that his hair was grayer and his eyes darker and more impenetrable. He was the same age as Sam, but he'd always been tougher, stronger and more attractive. He was a type that women invariably fell for and invariably wished they hadn't when the hard-man image\u2014a thin disguise for misogyny\u2014 proved to be an immutable reality.\n\nHe studied us with amusement as we approached. \"Mrs. Ranelagh.\" He gave an ironic nod in Danny's direction. \"You're scraping the bottom of the barrel with this one, aren't you? What is he? Toy boy or minder?\"\n\nI had to run my tongue around my mouth to stimulate some saliva. \"Moral support,\" I replied.\n\nHis smile broadened. \"Why would you need it?\"\n\n\"Because you won't like these,\" I said, taking some photographs from my pocket and laying them on the bar.\n\nHe reached out a hand to pick them up but Danny was there before him. \"Is this the black lady you were talking about?\" he asked.\n\n\"Yes.\"\n\n\"She looks as if she's been hit with a baseball bat,\" he said, laying them back on the counter.\n\n\"She does, doesn't she?\" I put my finger on the top picture and pushed it aside to fan out the five others underneath. None of them was pleasant. Each showed Annie in death, bruised and battered about the face, and with a discolored right arm where blood had seeped under the skin to form an extended hematoma from shoulder to wrist. \"Mr. Drury decided all these injuries came from a single glancing blow from a truck, which resulted in death within thirty minutes... but I can't find anyone who agrees with him. These pictures were taken during her autopsy in 1978. I've had them examined by two independent pathologists and they both say the bruising to the arm points to severe physical trauma some hours before she died.\"\n\n\"What's that in English?\"\n\n\"Annie was murdered.\"\n\nThe irritation from across the counter heightened abruptly, and I wondered why Drury thought I was there. A desire to renew an old friendship? Lust?\n\n\"Jesus wept!\" he growled. \"Don't you ever give up? It's like listening to a skipping record. Haven't you anything better to do with your life than make a martyr out of a miserable black who couldn't hold her drink?\" He lifted the top picture and turned it over to inspect the back for an official stamp. \"Where the hell did you get these?\"\n\n\"PC Quentin sent them to me.\"\n\n\"Andrew?\"\n\nI nodded.\n\n\"He's been dead seven years,\" he said dismissively \"Died in a car crash after chasing a joyrider at high speed for three miles.\"\n\n\"I know. He sent them to me shortly after we left England. I wrote and asked him for copies because I knew he was unhappy with the inquest verdict.\"\n\nDrury gave a grunt of irritation. \"What would he know? The guy was still wet behind the ears. He had a half-assed degree in sociology, and he reckoned it gave him an edge over a home office pathologist and a beat copper with ten years'graft on the streets.\"\n\n\"He was right, though,\" I said. \"This kind of bruising\"\u2014I touched one of the photographs\u2014\"takes time to develop. It also suggests more than one contact. If her arm was hit in several places, the individual hematomas would have spread out, darkening the skin from shoulder to wrist.\"\n\n\"A photograph proves nothing. She was black. You can't say what's a bruise and what's not.\"\n\n\"These are color photos,\" I pointed out mildly, \"so unless you're blind you can certainly see the bruising.\"\n\nHe shook his head angrily. \"What difference does it make? The accepted version was given by the man who performed the postmortem and he said her injuries were caused by a glancing blow from a truck.\"\n\n\"But not fifteen to thirty minutes before I found her. Two or three hours _perhaps_. And that means the people who say they saw her staggering about the road were probably looking at someone with severe head injuries.\"\n\nHis eyes flickered unwillingly toward the pictures again, as if he were both repelled and fascinated by them. \"Even if that's true, you can't blame them for assuming she was drunk.\"\n\n\"I don't.\"\n\n\"Then what the hell is this in aid of?\"\n\nI licked the inside of my treacherous mouth again. \"I'm going to have the case reopened,\" I said. \"I want the way you handled it investigated. I want questions asked about why a rookie cop with a half-assed degree in sociology could see that something was wrong... but _you_ couldn't. I want to know why, when he tried to raise it with you, you had him thrown off the case.\"\n\nHe tore the photographs in half and tossed the pieces across the bar to flutter at my feet. \"Problem sorted. And if that's all you've got to show for the last twenty years then you've been wasting your time.\"\n\nDanny stooped to retrieve the bits. \"You don't want to let him get to you,\" he said as he handed them back to me. \"He's a bully. It's the only way he knows how to control people. He's busting a gut to change the subject rather than explain why he did fuck all about this poor black lady having her face smashed in.\"\n\nDrury stared him down. \"What would you know about it, shithead? You were still in nappies.\" He jerked his chin at me. \"And you're backing the wrong horse if you back her. It was your dad she wanted locked up... your dad she accused of murder. No one else.\"\n\nThere was a long silence.\n\nDanny cast me an uncertain glance. \"Is that true?\"\n\n\"No,\" I said honestly. \"Mr. Drury asked me if I knew of anyone who had a grudge against Annie, so I named your father, mother and Sharon Percy. I never at any point suggested they'd murdered her. That was Mr. Drury's interpretation.\"\n\nDrury laughed. \"You were always good at twisting the facts.\"\n\n\"Really? I thought that was your speciality.\"\n\nHe held my gaze for a moment, searching for chinks in my armor, then crossed his arms and turned to Danny. \"Ask yourself why she brought you here and why she wanted you to see those photographs. She's planning to use you to get at your family, preferably by turning you against them first. It's what she's good at\u2014manipulating people.\"\n\nDanny hunched his shoulders unhappily as if all his worst fears had been confirmed, and my son's voice echoed uncomfortably in my ear. _I'd be sodding mad if it happened to me...._\n\n\"Your father had an alibi from five o'clock until midnight,\" I told him, \"and it was Mr. Drury who established it. He knows as well as I that Derek couldn't have killed Annie.\"\n\n\"Then why am I here?\"\n\n\"Because Mr. Drury lied about me to your family. He told your parents I was saying things that I wasn't... and I need you to pass on to your mother and brother that all I ever accused them of was racism. And that was true, Danny. They _were_ racists\u2014probably still are\u2014and they weren't ashamed of it.\"\n\nI touched a hand to his shoulder by way of apology because it was cruel to associate him with his family's hate when he'd stated so often in his e-mails to Luke that he didn't approve of white people living in South Africa. \"But my argument isn't with the Slaters,\" I told Drury, \"it's with _you.\"_ I stirred the torn photographs with the point of my finger. \"Because when I accused you and your colleagues of the same thing, it frightened you so badly that you manipulated every piece of evidence to support the theory that Annie had died in an accident. And I'd like to know why you did that.\"\n\nDid I imagine the flicker of fear in his cold, reptilian eyes or was it real? \"We didn't have to manipulate anything,\" he said sharply. \"We accepted the inquest verdict... accidental death after stumbling under a truck some fifteen to thirty minutes before you found her.\"\n\n\"But you didn't know what the verdict was going to be when you began the investigation into Annie's death.\"\n\n\"So?\"\n\n\"So you can't claim it as justification for your refusal to make proper inquiries. The only evidence you put forward was a description of Annie's house after she was dead, but it didn't stop you weighing in with a conclusion that she was a chronic drunk, an abuser of animals and a mental incompetent who neglected herself. I even remember your words. You said that in view of 'Mad Annie's'numerous problems your only surprise was that she'd lived as long as she had.\"\n\n\"Which was a view endorsed by everyone except you.\"\n\n\"Her doctor didn't endorse it.\"\n\nHe looked beyond me toward the door. \"Your husband did,\" he murmured. \"He and Mr. Williams described Annie as paralytic outside your house when they came home an hour and a half before you did. They also implied it wasn't unusual.\"\n\nI followed his gaze to where Sam was hovering uncertainly in the doorway. We'd tarried too long, I thought. In the end everyone's patience ran out, even the guilty's. \"They were lying,\" I said flatly.\n\n\"So you kept saying in '78.\"\n\n\"It's the truth.\"\n\n\"Why would they want to? If anyone was going to back you it ought to have been the man you married.\"\n\nOnce upon a time that had been my view, too, but only because I'd believed that truth was simple. \"He was trying to protect his friend,\" I said carefully. \"The two people I saw under the street lamp that night were Jock Williams and Sharon Percy. I suppose Jock was afraid I'd seen him... and didn't want his wife finding out he'd been with a prostitute. So he and Sam concocted their story about going back to our house for a beer.\"\n\nDrury glanced toward the door again, but Sam had disappeared. \"Why didn't you tell me this twenty years ago?\"\n\n\"I did. I gave you Jock's name as the man I thought I saw.\"\n\n\"But that's the point,\" he said sarcastically. \"You only _thought_ you saw him... and you didn't say he was with Sharon Percy.\"\n\n\"At the time I didn't know who she was.\"\n\nHe gave a dismissive shake of his head. \"Sharon had an alibi and Mr. Williams was ruled out when your husband vouched for him.\"\n\n\"But you never even questioned him,\" I said, \"just accepted Sam's word against mine. But why? Wasn't a woman's word as good as a man's?\"\n\nHe leaned his hands on the counter and shoved his face close to mine. \"You were 'round the bend, Mrs. Ranelagh. Nothing you said was believable. Everyone agreed with that... even your husband and mother. And they should know because they had to live with you.\"\n\nIf I'd had a gun at that moment, I'd have killed him. _Bang! Straight between the eyes_. How dare he quote my family at me when he had been the cause of their distrust? But hatred is a futile emotion which damages the hater more than the hated. Yes, he'd have been dead... but so would I... to everything that mattered to me. Perhaps my expression said more than I realized because he straightened abruptly.\n\n\"Sam and Jock invented their story to conform with what you told Jock's wife the next morning,\" I said evenly. \"You told Libby Williams, and anyone else who was interested, that Annie had been seen staggering about the road an hour before she died, you also mentioned the outside time she could have stumbled in front of the lorry. All Sam and Jock did was recycle that information to give you what you wanted\u2014a stupid, drunk nigger lurching around from 7:45\u2014and the fact that none of it was true didn't bother you one little bit.\"\n\n\"Why would your husband and Mr. Williams do that?\"\n\nI shrugged. \"It was easier for everyone if she died in an accident. For the police, too. It meant no one had to address the issue of racism.\"\n\nHe stared at me for a moment, his brows furrowed in what looked like genuine perplexity. \"When did your husband tell you this?\"\n\n\"Six months after we left England.\"\n\nIt was in the wake of the Hong Kong policeman debacle. Sam had drowned himself in whisky while stomping about the room, lecturing me on my behavior. Most of it\u2014the issue of how my \"madness\" was affecting his career and social life\u2014washed over me. Some of it did not, particularly when he started to feel sorry for himself at three o'clock in the morning. He was missing England... and it was my fault. What the hell had induced me to go spouting off to the police about murder... ? He could hardly switch horses midstream... not when poor old Jock was caught between a rock and a hard place. Half the bloody road had seen the stupid woman roaring around like a bear with a sore head. All he did was agree with them...\n\nI fancied I could hear Drury's brain whirring.\n\n\"You told me your husband was lying as soon as I read his statement to you. How could you know that if he didn't admit it until six months later?\"\n\n\"There were no beer cans in the rubbish bin,\" I said.\n\nDanny took a swig of Radley's draught lager and eyed Sam suspiciously across the table as he wiped the froth from his lips. \"How come you didn't recognize Mr. Drury when your missus brought you here the other day?\" he demanded. \"I haven't seen him in years but he hasn't changed that much.\"\n\nSam went on the defensive immediately. \"I only met him a couple of times. As far as I remember, I was more interested in what he was saying than what he looked like.\"\n\n\"Sam's not very good with faces,\" I offered by way of mitigation.\n\nDanny ignored me. \"How about when you made your statement? He must have interviewed you first. Didn't you look at him then?\"\n\n\"It wasn't Drury who took it. It was a constable. And, no, I was never interviewed... just asked to write out where I was and what I was doing.\" He raised his eyes briefly to mine. \"The statement ended my involvement. I wasn't even required to appear at the inquest.\"\n\nDanny was unimpressed. \"Yeah, but you don't walk away when your family's in trouble,\" he said. \"You should have insisted on being there whenever your missus was questioned. Christ! I wouldn't let my lady go through Drury's wringer on her own.\"\n\nSam cupped his own glass in his hands but made no move to drink from it. \"You're describing a different scenario. My wife wasn't facing charges, she was the one who was asking for charges to be brought.\"\n\n\"I don't blame her. That poor black lady looks as though she had the shit beaten out of her. It doesn't make any difference anyway. Your wife is family. You should have been there for her. That's the way it works.\"\n\nSam buried his face in his hands, and I had to harden my heart to his pain because there was no avoiding the issue that my husband was part of the problem... not part of the solution...\n\n\"It wasn't that simple,\" he muttered wretchedly.\n\n\"Sure it was,\" said Danny scathingly. \"Trust me. I know this stuff backward. Families pull together... rats jump ship.\"\n\n**_Letter from Danny's mother, Maureen Slater, dated 1999_**\n\n**_Letter to Sergeant James Drury\u2014dated 1999_**\n\n**Leavenham Farm  \nLeavenham  \nNr Dorchester  \nDorset DT2 XXY  \n**\n\nThursday, August 5, 1999\n\nDear Mr. Drury,\n\nFollowing our conversation yesterday, I enclose a copy of a letter I received in 1985 from a colleague of Dr. Benjamin Hanley, the pathologist who performed the postmortem on Ann Butts. In view of your confidence in Dr. Hanley's findings, you may find it interesting reading. The colleague's name was Dr. Anthony Deverill and he worked with Benjamin Hanley from 1979 until Hanley's compulsory retirement on medical grounds in 1982.\n\nYours sincerely,\n\nPS: Following the investigations referred to in (3) of Anthony Deverill's letter, both cases (believed at the time to be murders) were referred back to the Court of Appeal and the convictions against two innocent men were overturned. The evidence provided by Dr. Hanley was deemed \"unsafe\" and the deaths of the alleged \"victims\" were subsequently ruled to have occurred from \"natural causes.\"\n\nPPS: I have several sets of the postmortem photographs.\n\n**Dr. Anthony Deverill, MRCPath  \n25 Avenue Road  \nChiswick  \nLondon W4**\n\nMrs. M. Ranelagh  \nP.O. Box 103  \nLangley  \nSydney  \nAustralia\n\nFebruary 6,1985\n\nDear Mrs. Ranelagh,\n\nThank you for your letter of January 10, together with the enclosed postmortem photographs of Miss Ann Butts and the written report from Professor James Webber. As you so rightly say, I have met Professor Webber on several occasions and have a high regard for his judgment. Indeed, after studying the photographs myself, I have no reason to disagree with his detailed assessment that Miss Butts received the injuries to her face and arm some hours before her death.\n\nYour specific request was for information on my predecessor Dr. Benjamin Hanley, who conducted the postmortem in November '78. You say that both you and your father made unsuccessful attempts to contact him over the years, and that the only response either of you had was when his secretary admitted to your father over the telephone in 1982 that the file relating to Miss Butts's postmortem was \"missing.\" Unfortunately, a search of the archive files appears to confirm this last statement, as the only evidence that Dr. Hanley conducted a postmortem on Miss Butts is an entry beside his name on the work schedule for 15.11.78\u2014\"10:30 A.M. Butts. RTA. Report requested by PS Drury, Richmond.\"\n\nYou may be interested to learn that Miss Butts's file is not the only one we have been unable to locate. Of the 103 entries against Dr. Hanley's name on the '78, '79 and '81 rosters, nine are currently \"missing.\"\n\nRe your specific inquiries:\n\n  1. As you already know, Dr. Hanley was compulsorily retired on medical grounds in 1982 and died of liver failure eighteen months later. However, the compulsory element to his retirement related to a deterioration in his work and performance over a twelve-month period and not to a diagnosed medical condition since he refused to consult a doctor. This is not unusual among pathologists who deal with death every day and can forecast their own prognoses. In simple terms, Dr. Hanley was a chronic alcoholic who became increasingly incapable of doing the job assigned to him. The \"medical\" tag was attached to the retirement order to allow him to keep his pension, but the cirrhosis that killed him was not discovered until shortly before his death when he was admitted to hospital. These facts are a matter of public record, and I betray no confidences by passing them on to you.\n\n  2. I worked alongside Dr. Hanley for two and half years\u2014 from September '79, to March '82, when he retired\u2014and I am sorry to say that I had serious reservations about his competence from the beginning. It is, of course, impossible for me to comment on a postmortem that a) took place before I joined the team and b) has no supporting documentation on file; however, it is my considered opinion that Dr. Hanley's alcoholism would certainly have affected his judgment in November '78.\n\n  3. I have no precise knowledge of Dr. Hanley's relationship with PS Drury of Richmond Police, nor can I validate your contention that: \"Dr. Hanley may have taken direction from PS Drury and produced a report that suited Richmond Police.\" However, I expressed concern on several occasions that Dr. Hanley was compromising the independence of his department by writing postmortem reports that appeared to mimic the police version of affairs. Two of these incidents are now under official investigation. In defense of Dr. Hanley, I do not believe there was any malicious intent behind his actions, simply a recognition that he could no longer cope with the demands of his job and a compensatory willingness to place too much confidence in the \"hunches\" of certain police officers. I should say that in most cases this would not be a cause for concern\u2014most of the deaths we see are \"natural\"\u2014but clearly it could create problems where facts are disputed.\n\n  4. I can say with absolute certainty that Dr. Hanley would have had no racist motive for ignoring evidence of murder in the case of Miss Butts. I am black myself and never experienced any sort of prejudice at his hands. He was a kindly man who had no interest in politics and clearly found his job distressing, particularly when he was obliged to open the chest cavities of women and children, which he began to see as an \"unnecessary mutilation.\"\n\n  5. In the absence of a file, I am afraid there is very little assistance I can give you other than to support Professor Webber's interpretation of the photographs. As mentioned above, nine sets of case notes seem to be missing and there is some evidence that Dr. Hanley destroyed them himself prior to leaving the department. In view of his long service record, a decision was taken to allow him to \"work out\" a three-month notice and we believe he used that time to remove any files that he believed to contain questionable findings. Sadly, he appeared to become deeply confused about the \"examiner's\" role in society and consistently questioned the value of \"righteous judges.\"However, there is no proof of this and any such speculation could never be used in court.\n\nIn conclusion, I am happy to give my permission for this letter to be used as supporting evidence for the deterioration in Dr. Hanley's performance and standard of work during the years I worked with him, all of which is already in the public domain. Beyond that, I can only advise you to gather as much supporting evidence as you can, from whichever source, in order to present a tight and compelling argument for a reopening of the investigation into Miss Butts's death. Trusting this is of help, With all best wishes,\n\nDr. Anthony Deverill\n\n# fourteen\n\nI took the train to London on my own the following Monday. It caused a row because I refused to tell Sam where I was going or what I was planning to do, and he drove off in a huff after dropping me at Dorchester South station at eight o'clock in the morning. His mood had been depressed since Danny's throw-away line about rats jumping ship\u2014 _It wasn't like that.... I needed time to get my head together....Jock was on my back all the time trying to persuade me to make you take those flaming tranquillizers.... He said you needed help...he said you'd flipped...he said... he said_ \u2014and his temper was not improved by my sour comment that if Jock was such a guru he should be talking to him and not to me.\n\nI didn't keep tabs on him, so when I set out on Monday morning I had no idea if he'd taken my advice or not. I thought it unlikely. Sam wasn't the type to poke a sleeping dog unnecessarily, particularly when he was the most afraid of being bitten.\n\nI found Graham Road changed beyond recognition that August morning. It had become a one-way street with speed bumps down the center. Parking was restricted to permit holders only, and trucks were banned. The houses were smarter than I remembered, the pavements wider, the sunlight brighter and more diffuse. It had lived for so long in my memory as a dark, foreboding place that I found myself wondering what else my mind had poisoned over the years. Or perhaps it wasn't my memory that was at fault? Perhaps Annie's death had actually achieved something?\n\nI glanced at number 5 as I passed and was put to shame by its natty appearance. Someone had lavished the love and care on it that we should have done. Window boxes splashed the front with brilliant color, a new stained wood door had taken the place of our elderly blue one and the tiny front garden, barely three feet deep, boasted a neat brick wall, tubs of scarlet petunias and a semicircle of clipped green grass beside the path to the door. Nor was it alone. Here and there, untidy front gardens and peeling paintwork spoke of residents who were unable or unwilling to conform, but for the most part the road had moved decisively upmarket and made sense of Jock's statement that property prices had skyrocketed.\n\nI guessed that some of that was due to the sale of the council-owned properties, which had stood out like sore thumbs twenty years before because of their uniform yellow doors. Now it was impossible to distinguish them from those that had always been in the private sector, and I wondered how many of them were still owned by the council tenants who had bought at rock-bottom prices. If Wendy Stanhope was to be believed, most of them had sold up within a year to achieve a 100 percent return on their investment, but the wiser ones had stayed and watched their investments grow.\n\nI crossed the road and paused beside Sharon Percy's gate. Her house was almost as natty as ours, with Austrian blinds in the windows and a clump of Pampas grass in the front garden, but I couldn't believe she hadn't cut and run the minute she saw a profit. I knew she'd bought the house because Libby's letters had ranted on for months about how Jock's thirty quid a week had paid for Sharon's bedroom, but I found it difficult to equate the new subdued classiness of number 28 with the simpering peroxide blonde in Wendy's photograph.\n\nI looked into her downstairs window\u2014more curious than expectant\u2014and was taken aback when her flour-white face, slashed red lips and heavily outlined eyes appeared briefly behind the pane. I recalled Libby's nickname for her, \"the bleached vampire,\" but she looked more pathetic than predatory that morning. An aging woman trying to paint away the ravages of time. Was Geoffrey Spalding still with her? Or had his infatuation died along with her sex appeal? I felt an absurd desire to raise my hand in greeting before I remembered that we'd never spoken and that, if she'd known me at all twenty years ago, she certainly wouldn't recognize me now.\n\nI barely glanced at Annie's house as I moved on to number 32. Even when I'd stood in front of her boarded-up house in the months following her death, her ghost had never troubled me, and I certainly didn't expect to be bothered by it now.\n\nIn the end, the only ghosts that lingered here were lonely mothers....\n\nMaureen Slater opened her door before I could knock, and thrust out a miniature hand to pull me inside. \"I don't want anyone to see you,\" she said.\n\n\"They won't know who I am.\"\n\n\"They'll guess. Everyone talks.\"\n\nI wondered why that mattered when there was no one left who remembered Annie, and decided that by \"everyone\" she meant Sharon. I thought it would be counterproductive to say I'd already been seen, and followed her down the corridor to the kitchen, catching glimpses of the two ground-floor rooms as I passed their open doors.\n\nThe sitting room looked as though it was rarely used, but the dining room had been converted into a comfortable den with brightly colored bean bags littering the floor, a cushioned sofa along one wall and a wide-screen television in the corner. It was already switched on, showing a daytime magazine program, and the rumpled duvet on the sofa and the fug of smoke in the room suggested Maureen had either been watching it all night or had started early. She closed the door as we passed to deaden the volume.\n\nEven though Maureen's was an end-of-terrace house, the layout in side was identical to ours, as indeed was every alternate house in the row: sitting room and dining room on the right with the corridor running past the stairs on the left-hand side toward the kitchen at the back. The in-between houses were built in mirror image so that corridors adjoined corridors on one side, and living space adjoined living space on the other. Upstairs, in precisely the same way, it was either the bedrooms or stairwells that adjoined. In order to allow for a window in the dining rooms, the kitchens were offset against the ends of the houses and shared a party wall with the people on the corridor side. As none of the structures was built to modern soundproof standards, the inevitable result was that we all came to know our neighbors rather better than we would have liked.\n\nIndeed, it had been Sam's permanent complaint that we should have done some \"noise\" research before we bought number 5. On the corridor side, number 7\u2014the side that acted as a sound buffer\u2014lived an elderly couple, who rarely spoke above whispers even when they were in their kitchen. On the living-space side, number 3\u2014the side that acted like a huge echo chamber divided by a thin vibrating wall\u2014were the Charles children, whose nighttime screaming had kept us awake. One day, in a spirit of optimism, Sam invited both couples in for drinks and suggested they swap houses so that peace could break out all 'round, but Paul Charles took exception to some of the things Sam claimed to have heard through the wall and treated him with hostility from that moment on.\n\nI had often wondered if a similar situation had existed with Annie, although, of the many complaints made against her, the question of noise had never been mentioned. In fact it was more likely she had been a victim of it and had suffered in silence while her life was made a misery. Certainly Michael Percy and Alan Slater had taken great pleasure from teasing her in public, and I couldn't believe they hadn't continued the sport in private by shouting insults at her through the party walls.\n\n\"Danny phoned last night,\" said Maureen, pulling out a chair in the kitchen and pressing me on to it. \"You seem to be making quite a hit with him.\" She had a trace of the Midlands in her voice, which showed itself in the hard \"g\" she added to \"ing,\" but whether she had been born there or whether she had learned it from her parents I didn't know.\n\nLike everything else about her it set my teeth on edge, and I had to glue a smile to my face to mask my dislike. Whatever Wendy Stanhope may have said about her brutal treatment at the hands of her husband, I'd always thought there was something evil about Maureen Slater, perhaps because I held her responsible for the hate campaign against Annie. I'm sure she knew what my real feelings were, but for the moment she was prepared to go along with the pretense of friendship.\n\n\"The feeling's mutual,\" I assured her. \"Danny's a nice fellow.\"\n\nShe busied herself with cups and saucers. I had written to her many times over the years, seeking answers, but the only response I had ever received was the one a week ago, agreeing to this meeting. I assumed it was my contact with Danny that had persuaded her to change her mind, and I wondered how far she suspected that I had sought him out deliberately and how far she was worried about what he had been telling me. There was, after all, so much that she wouldn't want me to know.\n\n\"You're the only person who thinks so,\" she said, filling a kettle at the sink. \"Danny's been in and out of trouble since he was ten... fighting... stealing cars... he started shooting heroin when he was twelve.\" She paused, waiting for an answer. When I didn't give one, she went on a little tartly, \"Not the type most mothers want hanging around teenage boys. He says he's been out drinking with your lads.\"\n\n\"He has. They've met up with him on Portland a couple of times.\"\n\n\"You know he smokes cannabis.\"\n\n\"Yes.\"\n\n\"He's probably offering it to your kids,\" she said with a touch of malice, as if the idea pleased her.\n\n\"Then he isn't the first, and he won't be the last.\"\n\nShe eyed me suspiciously. \"You're pretty laid back about it. You must have a lot of faith in your boys.\"\n\nI gave a noncommittal smile. \"I'd be more worried if Danny was still on heroin.\"\n\n\"No chance of that.\" She plugged in the kettle. \"It's the one good thing Mr. Drury did for me... caught the stupid little bastard at it one day and put the fear of God into him so he'd never go near a needle again.\"\n\n\"How did he do that?\"\n\n\"He gave him a choice: punishment now, or a care order imposed by the juvenile court. Danny chose punishment now.\" She laughed. \"I think he thought Drury was going to slap him about a bit... didn't reckon on honest-to-God sadism.\" The idea seemed to amuse her.\n\n\"What did he do?\"\n\n\"He broke off the needle and pressed it into Danny's arm with the edge of his handcuffs, then told him if he went to a doctor to get it removed there'd be that many questions asked he'd find himself in care so quick his legs wouldn't touch the ground. It was two days before Danny could find the courage to cut down deep enough to pull the needle out with a pair of tweezers. He's never been able to look at a syringe again without turning green.\"\n\n\"That sounds like Mr. Drury's style,\" I murmured. \"Brutal but effective. Did you report him for it?\"\n\n\"Did I hell!\" She spooned coffee into the mugs. \"In any case I was grateful. The last thing I wanted was one of my children dead of an overdose.\"\n\nA silence fell while we waited for the kettle to boil. I had no idea what kind of background she came from, but Drury's parting shot to Danny\u2014\"How's that downtrodden slut of a mother of yours? Still on the booze?\"\u2014was uncomfortably close to the mark. My mother would say it was breeding (or lack of it)\u2014a scientist would say it was genes\u2014 I would say it was poor education and low self-esteem. If she cared about anything at all, I thought, it was probably her benefit checks and whether they would buy enough smokes and alcohol to last her through the week.\n\nHer windowsill was lined with empties, testimony to the drinking habit she hadn't been able to kick. An unopened bottle of vodka stood beside the salt and pepper on the table like an unearned reward. But if she was drunk or stoned on Prozac that day it wasn't noticeable. Indeed in some ways the sharp, assessing glances she kept flicking in my direction reminded me of Wendy Stanhope, although there was no kindness in them, only suspicion.\n\n\"Thank you,\" I said when she put a mug of coffee in front of me. Out of habit, she had added milk and sugar, neither of which I could stand, but I sipped enthusiastically as she sat herself in the chair opposite and lit a cigarette.\n\n\"Do you want one?\" she asked.\n\nI shook my head. \"I never got hooked, thank God. If I had, I'd be a sixty-a-day woman by now.\"\n\n\"How do you know?\"\n\n\"I have an addictive personality. Once started, I can't stop.\"\n\n\"Like this thing with Annie?\"\n\n\"Yes.\"\n\nMaureen gave a baffled shake of her head. \"You wouldn't have liked her, you know. That's what makes this all so... _stupid_. If anyone else had found her, there'd have been no fuss, she'd have been quietly buried and we could all have got on with our lives.\" She paused to draw pensively on her cigarette. \"You, too,\" she added, watching me through the smoke.\n\n\"I haven't done badly so far.\"\n\nShe dropped ash into her saucer. \"Except you can't let her go, and that's not healthy.\"\n\nI might have answered that Annie was the least of my obsessions but I didn't want to put her on her guard. Instead I asked, \"Why wouldn't I have liked her?\"\n\n\"Because she wouldn't have liked _you_. She didn't like any white people. We were all 'white trash'to her. She used to chant it through the kitchen wall whenever Derek raised his voice. 'White trash... white trash...'On and on for minutes on end. It used to drive him mad.\"\n\n\"Is that why he hated her?\"\n\nShe nodded.\n\n\"Perhaps he didn't like hearing the truth?\" I remarked dryly.\n\nA wary expression crept into her eyes. \"We never claimed to be anything we weren't.\"\n\nThe pretense at friendship began to unravel at speed. \"You were known as 'the family from hell,'Maureen. When you and Derek weren't screaming at each other, your children were running riot in the street. I've never known a group of people make their presence felt so rapidly in so short a time. Alan's favorite occupation was to practice his jump kicks against other people's front fences. He flattened Annie's within a month of you being rehoused here... and ours within three months.\"\n\nShe bridled immediately. \"He wasn't the only one. Michael Percy was just as bad.\"\n\n\"I agree.\"\n\n\"But it was always my Alan who got the blame.\"\n\nI shook my head in disagreement. \"Michael faced up to what he'd done. Your son never did. Alan used to run away the minute trouble appeared and leave Michael to take the flak.\"\n\n\"Only because he knew his father would give him a larruping if he got caught.\"\n\n\"But it was all right for Michael to be given a larruping?\"\n\nHer mouth thinned immediately. \"It never happened. Who was going to give it to him? Sharon? He'd have thumped her first. He was a nasty piece of work that Michael... a bad influence on all the kids 'round here. He was the one got my lad into trouble, never the other way 'round.\"\n\nI wondered if Sharon saw it that way, or if she cared. \"I watched a man hurl him head first into a brick wall once,\" I said idly. \"It all happened very quickly and I was too far away to stop it. The wretched child was only fourteen\u2014and he wasn't very big for his age\u2014so he went down like a sack of potatoes.\"\n\n\"Serves him right,\" said Maureen balefully \"He almost killed somebody not so long ago... got eleven years for his troubles. That should tell you the sort of boy he was. It makes me sick the way we got dumped on all the time, when it was him and his tart of a mother caused all the problems in the street.\" A sly expression crept into her eyes. \"Annie had their measure all right. She called Sharon a 'whore'and Michael a 'son of a bitch.'\"\n\n\"Did she call her 'white trash'?\"\n\n\"Nn-nn. 'Whore'... 'ho'... 'cunt.'Annie'd get going at the top of her voice every time she was with a client. It was pretty funny.\"\n\nThere was a time, I remembered, when she and Sharon had been thick as thieves, and I wondered what had happened to make them fall out. Something to do with money, I guessed, as it was the single passion they both shared. \"So it was just the Slaters who were 'white trash'?\" I murmured.\n\nMaureen studied the end of her cigarette. \"Think what you like,\" she said.\n\n\"Do you know who the man was who knocked Michael down?\" I asked her.\n\nShe gave an indifferent shrug.\n\n\"It was your husband,\" I told her. \"He was fighting drunk and caught Alan and Michael trying car doors to see if any of them were unlocked. Alan took to his heels but Michael stood his ground and ended up with a bloody face. I wanted to report Derek, but Michael said he'd take his anger out on you if we grassed him up. 'Mr. Slater's a right bastard,'he told me. 'He beats up on his missus every time his kids get the better of him.'\" I watched her for a reaction, but there was none. \"So I let Derek get away with it, and took Michael to my house instead of a police station. It was three hours before his nose stopped bleeding.\"\n\nShe stubbed out her cigarette, refusing to meet my eyes. \"You can't blame me for that. Most of the time I didn't even know where Derek was, let alone what he was doing.\"\n\nIt sounded like the beginnings of a defense. \"I'm not blaming you.\"\n\n\"Sure you are. You're like everyone else. It's Maureen's fault her kids were out of control. Maureen's fault she married a lousy husband. Well, maybe it was and maybe it wasn't. But who the fuck ever cared about me? Tell me that.\"\n\n\"The vicar and his wife?\"\n\nAnger sparked briefly in her eyes. \"They were more interested in the nigger than they were in me.\"\n\nI looked away to hide my anger, recalling what Wendy Stanhope had said. _The poor woman was always taking refuge with us...._ \"I understood they took you in whenever Derek became violent.\"\n\n\"Only out of charity, never out of liking.\"\n\n_It was something she resented_ , I thought.\n\n\"The vicar knocked next door once a week. He never did that for me. I had to go looking for help.\"\n\n\"Perhaps he felt Annie had more to put up with.\"\n\n\"No more than we did. You should have heard her cursing and swearing at us through the wall.\"\n\n\"You said she only did it when you made a noise.\"\n\n\"Not always. Sometimes it was hard to say which came first... her or us. She had a mouth like a sewer. When it wasn't 'white trash,'it was 'honkies'or 'scum.'It used to rile us up something rotten.\"\n\n\"She couldn't help herself,\" I said. \"She suffered from a neuropsychiatric disorder called Tourette's syndrome. Sometimes it manifests itself as _coprolalia_ , which is a compulsion to utter obscenities. Her mother was far more prone to it than Annie, but maybe Annie resorted to it when she was stressed.\"\n\n\"Then she should have been in a loony bin.\"\n\n_Does she believe that?_ I wondered. Or was it something she repeated like a mantra to excuse what she did? \"A more sensible solution would have been for the council to rehouse you and your family somewhere else,\" I suggested. \"To be honest, I never understood why they didn't. You lived entirely on benefit, had more social workers allocated to you than anyone else in the street, yet for some reason the pressure was always on Annie to move and never on you. That always seemed grossly unfair to me when she was a householder, paying rates, and you were paying nothing.\"\n\n\"That wasn't our fault. Derek was out of work. Would you have liked it better if we'd starved?\"\n\nI refused to be sidetracked. \"Why did the council take your side against Annie's, Maureen? It must have been clear to them that she wasn't getting on with her neighbors.\"\n\n\"Why would it? She never complained.\"\n\n\"She called you 'white trash.'What's that if it's not a complaint?\"\n\nShe lit another cigarette and shook her head at my stupidity. \"I meant she didn't complain to the council.\"\n\nI had to make a conscious effort to stop my mouth dropping open. I had imagined any number of conspiracy theories to account for why the Slaters and the Percys had been allowed to wage a terror campaign against Annie, but it had never entered my head that the explanation was so simple. \"Are you saying that, despite all the complaints you and Sharon made against her, she never once retaliated?\"\n\nMaureen nodded.\n\n\"Why not?\"\n\nShe didn't answer and another silence developed between us. She wore her hair in a tight ponytail and kept running the flat of her hand across her crown as if to check that the elastic was still in place. She seemed to be debating with herself whether after twenty years there was anything to be gained by telling the truth, although I guessed that her real concern\u2014indeed the only reason we were having this conversation\u2014was to find out how much I knew and what I was planning to do about it.\n\n\"She was too afraid of Derek,\" she admitted suddenly.\n\n\"To make an official complaint?\"\n\n\"Yes.\"\n\n\"What did he do to stop her?\"\n\nAnother silence, longer this time, before she gave an embarrassed shrug. \"Killed one of her cats and said he'd kill the others if she ever spoke out against us. The thing is\"\u2014she wriggled her shoulders uncomfortably, knowing that nothing could excuse her husband's behavior or her complicity in it\u2014\"we'd been moved three times in three years, and we didn't want to move again. We sure as hell didn't want to go back to a high rise.\"\n\n\"No,\" I said slowly. \"I don't suppose you did.\"\n\n\"It was only a cat.\"\n\n\"Mm.\" I paused to glance along the corridor. \"It was quite a bargain when you think about it... a cat for a house.\"\n\n\"There you are, then.\"\n\n\"Oh, no.\" I gave a small laugh. \"Don't you dare bracket me with a sadist. If Derek had been married to me, he'd never have got near a cat. I'd have beaten his brains out with a sledgehammer the minute he lifted a finger against one of my children. Why were you such a coward? Why didn't you fight back?\"\n\nHer malice intensified. \"You don't know what it was like. You didn't go in fear of your life every day. What do you think he'd have done to me and the kids if I'd tried to stop him?\"\n\n\"Why didn't you go to the police?\"\n\nShe shook her head scornfully as if the question weren't worth answering, and in fairness, it probably wasn't. Domestic violence was a low priority in 1978. As was harassment of black people.\n\n\"How did he kill the cat?\" I asked, reverting to what interested me.\n\n\"Strangled it,\" she said irritably. \"They kept coming into our garden, and he'd already warned her he wasn't going to stand for it. He chucked the body back over the fence with a note tied to its collar so she'd get the message.\"\n\n\"What did the note say?\"\n\n\"I don't know, for sure. Something like he'd nail the next one to the fence. He didn't tell me about it till afterward.\" She watched me slyly through her lashes while she cooked up another defense. \"I like cats. I'd have stopped him if I could. The children were all over them when we first came here... they kept asking where the marmalade one was.\"\n\n\"When did it happen?\"\n\n\"About two months before she died.\"\n\n\"September '78?\"\n\n\"Probably.\"\n\nI recalled John Howlett's letter to Sheila Arnold. _I made two recommendations on my first visit in March 1978: 1) that she install a cat flap in the kitchen door to allow the animals free access to and from the garden...._ \"After you'd set the RSPCA inspector on to her then?\"\n\nMaureen tapped the glowing end of her cigarette against her saucer and watched a curl of ash deposit itself against the side. \"I can't remember.\"\n\n\"His first visit was in March. He ordered her to put a cat flap in her door because you and Sharon kept complaining about the smell coming from her house.\"\n\nShe lifted a shoulder in a careless shrug.\n\n\"Weren't you worried she'd show Derek's note to him the next time he came?\"\n\n\"She wouldn't have dared. She was almost as frightened of the RSPCA as she was of Derek.\"\n\n\"How did she let the cats out before she had the flap installed?\"\n\n\"She never did. That's why the house stank.\"\n\n\"That's not true,\" I said bluntly. \"You just told me how your children were all over the cats when you first came here. How could they have had any contact if there was no way for the animals to get out until the flap was installed?\"\n\nA stubborn note crept into Maureen's voice. \"Maybe she didn't bother to close her back door.\"\n\n\"Well, did she or didn't she? You must have known. Your kitchens were next door to each other.\"\n\n\"Most of the time it was open.\" Her eyes caught mine, then slid away to hide their cunning. \"That's what made us think she had chickens in there. The smell that came out of it was disgusting.\"\n\n\"Oh, for Christ's sake!\" I said wearily. \"The only stink 'round here was your family's body odor. God knows if you ever gave Alan a bath or washed his clothes, but no one wanted to sit next to him at school. Poor kid. He was always the first to be checked for head lice... and _always_ had them. Always the first to have his locker searched for missing games kit... and _always_ had it. The PE teacher asked him once what his problem was, and he said he liked things that smelled clean.\"\n\n\"It wasn't my fault,\" she said again, her voice rising to an irritating whine. \"We didn't have a washing machine.\"\n\n\"Neither did we. I used the laundrette on the main road.\"\n\n\"You didn't have kids.\"\n\n\"Two machines take the same time as one.\"\n\n\"The bags were too heavy.... I couldn't abandon Danny.... In any case, I never had any money. Derek spent it all on drink.\"\n\nI looked at the vodka bottle on the table. \"He wasn't the only one.\" I rode roughshod over her attempt at a retort. \"Why didn't you do the washing by hand in the bath? You weren't working. You had all day to devote to your children. The one thing you could have done was keep them clean.\"\n\n\"I did my best.\"\n\nI'd waited so long to get this off my chest that caution gave way to honesty. \"Then you should be ashamed of yourself,\" I said flatly. \"I've seen women in Africa do better than that when all they had was a tub of cold water. You did nothing for your children, and the only reason Danny's a nice kid now is because somewhere along the line someone took an interest in him. I suspect it was Alan's wife\"\u2014I could see from her expression that I was right\u2014\"because it certainly wasn't you. You were in a drunken stupor most of the time... like your husband.\"\n\nShe was surprisingly indifferent, as if she'd heard the same accusations many times before. \"You do what you can to get by,\" she said, \"and it wasn't always like that. Some days were better than others. In any case, you don't feel the pain so much when you're drunk. You should try having your face smashed into a brick wall once in a while and see how you like it.\"\n\n**_Letter from Ann Butts to Councillor J. M. Davies,  \nRichmond\u2014dated 1978_**\n\n**_Carbon copy of Councillor J. M. Davies's reply_**\n\nPendlebury\n\nDuke's Avenue\n\nRichmond\n\nSurrey\n\n01-940-0000\n\nJune 20,1978\n\nDear Miss Butts,\n\nThank you for your letter of June 12, 1978.I am deeply disturbed by what you say, however, there is little I can do without more information. You did not give me Morin's surname, nor the name of her husband, nor indeed did you say where she lives. As I'm sure you appreciate, it will be difficult for me to raise the matter with the appropriate authorities without these details.\n\nIf you wish me to pursue the matter, please write again or telephone me on the above number. Alternatively, you may prefer to attend one of my \"surgeries\" at the above address, which would give you a chance to discuss your concerns in person. They take place between 9 A.M. and midday on the first Saturday of every month and do not require an appointment.\n\nYours sincerely,\n\n# fifteen\n\nI stared into my coffee. \"How did she stop her cats coming into your garden after Derek killed the marmalade one? It was long after the cat flap had been installed.\"\n\n\"She propped a board in front of it so they couldn't use it, then let them out one at a time to do their business. It was quite funny watching her. She used to run up and down flapping her arms to stop them coming anywhere near our fence. We reckoned she'd have lost a couple of stone with all the exercise if she hadn't had her face in the trough all the time. You should have heard her... right noisy she was. Gobble, gobble, gobble. It made us sick just to listen.\"\n\nMy expression must have given away more than I intended because she dropped her eyes immediately. I thought what a vile little woman she was and how injurious her poison must have been to her family.\n\n\"You asked.... I told you,\" she muttered. \"Don't blame me if you don't like the answers.\"\n\nI caught at the edges of my anger and drew it back inside. \"How do you know she used a board?\"\n\n\"The kids used to climb over the fence at night and push the flap open to make the board fall on the floor.\"\n\n\"That must have frightened her.\"\n\n\"It did. She used to wail her head off.\"\n\n\"Why didn't she fix the board to the door?\"\n\n\"Because she didn't want the RSPCA to know she was blocking the flap. She'd keep the inspector waiting at the door while she scurried around trying to find somewhere to hide the stupid thing.\"\n\n\"Is that why you and Sharon kept pestering the RSPCA? So they'd catch her out?\"\n\nShe blew a smoke ring in my direction, then stabbed it through the heart with the point of her cigarette. \"Maybe.\"\n\nI gave my coffee cup a violent shove and watched it slop across the table. \"You had her in a vise. On the one hand Derek was threatening to kill her cats if they _did_ run free; on the other, the RSPCA was telling her she could face prosecution if they _didn't_ run free.\"\n\nShe took to smoothing her hair again.\n\n\"What was she supposed to do?\"\n\n\"Leave,\" she said matter-of-factly, \"and take her cats with her.\"\n\n\"Just because she was black?\"\n\n\"Why not? We didn't want a coon for a neighbor.\" She retreated rapidly as she saw my expression. \"Look, it wasn't my idea.... I'd have done it differently if I could. But Derek wanted rid of her... he had this thing about nig\u2014\" She corrected herself\u2014\"blacks... really hated them. In any case, she had her chance. The social workers told her she only had to say the word and they'd rehouse us. But she said no, she was happy the way things were.\"\n\n\"She had no choice. Derek knew where she lived. Her cats were never going to be safe from him.\"\n\n\"Right, and she got so scared of him in the end, we reckoned she'd leave before Christmas.\" She paused. \"Then the silly cow falls under a flaming truck,\" she finished lamely, \"and the cops find she's been killing the cats herself.\"\n\nI rested my chin in my hands and studied her with grim curiosity. \"They were already half-dead when they were pushed through her flap,\" I told her. \"Someone thought it was funny to catch strays and bind their mouths with superglue and parcel tape so they'd either starve to death or have most of the fur ripped out of their heads if Annie tried to save them. I think she killed the weakest ones when the others started attacking them, but it was done out of kindness, not out of cruelty.\" I favored her with a crooked smile. \"So whose bright idea was that? Yours? Or your husband's?\"\n\nShe squashed her cigarette into the ashtray, mashing it to shreds with nicotine-stained fingers. \"It weren't nothing to do with us,\" she said flatly, while apparently agreeing with the facts. \"We weren't like that.\"\n\n\"Oh, come on!\" I said sarcastically. \"You've just told me Derek strangled one cat and threatened to nail another to a fence. And all for what? Because he was thick as pig shit and had to terrorize women to give himself a sense of authority.\"\n\nShe didn't like the way the conversation was going, and licked her lips nervously. \"I don't know anything about that.\"\n\n\"What? The way he liked to terrorize women?\"\n\nShe recovered quickly. \"All I know is what he did to me and the kids. But he was more talk than action. Most of the time he never followed through.\"\n\n\"Maybe not when Annie was alive,\" I agreed, \"but he certainly made up for it after she was dead. He was far more violent when he knew there were no witnesses.\"\n\nI recalled my visit to her in hospital. It was a wet afternoon at the end of November, and I'd dripped water all over the vinyl floor beside her bed while I tried not to show my shock at Derek's handiwork. I couldn't believe how small she was, how damaged she was, and how panic-stricken her eyes were. It was a wasted journey in terms of gathering information because she was too suspicious of me to answer questions. I listened to her dreary insistence that, far from Derek using her as a punchbag, she'd been alone in the house and had missed her footing at the top of the stairs, while saying in her next breath that she'd be dead if Alan hadn't been there to call an ambulance for her. It was a ridiculous story because her broken cheekbone and blackened eyes looked too much like Annie's death mask for anyone to believe that either of them had suffered an accident; but all too belatedly I was given a glimpse of the walls of terrified silence that protect violent men.\n\n\"What are you talking about?\"\n\n\"Derek putting you in hospital two weeks after Annie's death. Didn't you ask yourself why that happened? He'd never hit you so hard before that you went into a coma and had to rely on your children calling an ambulance for you.\" I jerked my head toward the party wall. \"Your protector was dead. Her house was empty. Derek was free to break every bone in your body if he wanted to, then dump you in the road somewhere and claim you were run over by a truck...\"\n\nMaureen rejected my suggestion that Annie had been her \"protector.\" It was rubbish, she protested\u2014Annie hated her. I repeated what she'd said herself, that Annie had wailed every time Derek raised his voice. \"You asked me earlier who had ever cared about you,\" I reminded her. \"Well, Annie did. I know it isn't what you want to hear, but it _is_ the truth.\" I took two letters from my rucksack and pushed them across the table. \"The top one's a copy of a note she wrote to your then councillor, J. M. Davies, in June '78. The one underneath is his reply. She obviously didn't know how to spell your name and, because she was incoherent when she tried to raise the matter over the telephone, he put the whole thing down to maliciousness.\"\n\nMaureen looked uncomfortable reading Annie's bold handwriting as if, even in reproduction, it had the power to summon her into the room with us. \"Perhaps it _was_ malicious,\" she said, laying the letters aside. \"Perhaps she was just trying to cause trouble for me and Derek.\"\n\n\"Oh, for God's sake!\" I sighed impatiently. \"If that was her intention, she'd have made a better job of it. She'd have written a barrage of letters, almost certainly unsigned, and she'd have accused Derek of killing animals instead of hurting them. Can't you see her concern was for _you?_ She says, 'something should be done for Maureen,'not, 'something should be done about the white trash next door because they keep stealing from me.'\"\n\nShe fumbled nervously at her cigarette packet. \"She'd have been lying if she had.\"\n\nI shook my head in contradiction. \"Alan gave me a tiny wooden statuette which he told me he'd carved himself from an old table leg as an end-of-year present. I believed him because it's very primitive, and looks like something a child might do, but I'm certain he stole it from Annie.\"\n\n\"You can't prove that.\"\n\n\"No,\" I agreed, \"but I can certainly prove he never carved it. It's been analyzed by an expert. It's a representation of an Aztec god, called Quetzalcoatl, and was cut from pine, probably around the turn of the century, in a style common to natives of Central America. Annie's father made a collection of Central American artifacts during the '30s and '40s, so the circumstantial evidence suggests that the Quetzalcoatl in my possession once belonged to her. The only question is, did she give it to Alan, or did he steal it?\"\n\nMaureen leaped at the bait. \"She gave it to him.\"\n\n\"How do you know?\"\n\nShe thought for a moment. \"He ran an errand for her... it was her way of saying thank you. Matter of fact, I was the one who made him pass it on to you. He kept on about what a nice lady you were, and how you'd kept quiet about the time you caught him thieving from your wallet. 'One kindness deserves another,'I said, 'and Mrs. Ranelagh's more likely to appreciate a wooden statue than you are.'\"\n\n\"Why did he tell me he'd carved it?\"\n\nShe caught my eye briefly. \"I expect he wanted to impress you.\"\n\nI laughed. \"I'd have been more impressed if he'd told me he earned it running errands for Mad Annie. He used to shout 'daft nigger'after her in the street. She turned on him once with a growl and grabbed at the sleeve of his jacket. He was so terrified he took to his heels and left the jacket in her hand.\" I paused. \"She'd never have asked him to run an errand for her. And, even if she had, she'd have cut off her right hand before she rewarded him with one of her treasures. She disliked him even more than she disliked Derek. The little brute never left her alone.... He was always on the lookout for her....\" I fell silent before anger made me strident.\n\n\"That's lies. You're inventing things to suit yourself. All you're saying is that Alan played in the street a lot. It doesn't mean he was on the lookout for Annie.\"\n\n\"He was an abused and neglected child, Maureen, who was too frightened to take on his father and saw Annie as easy meat. He learned that bullying worked and put it into practice on the most vulnerable person he could find.\" I gave a humorless laugh. \"I wish I'd known how you and Derek were treating him. I wish I'd had him prosecuted when I had the chance. Most of all, I wish he'd been taken away from you and taught some decent values when it mattered.\"\n\n\"You were just as responsible as us,\" she muttered. \"You were his teacher. Why didn't you say something to him when he called her a 'daft nigger'?\"\n\nIt was a good question. Why hadn't I? And what sort of excuse was it to say I was frightened of a fourteen-year-old? But I was. Alan was a huge child for his age, tall and heavily built, with a low IQ and little understanding of anything except aggression, which both emboldened and scared him. Had there been no Michael Percy to take the flak, then I think Alan's problems would have been more obvious and he might have attracted sympathy instead of dislike and disgust. As it was, most people avoided him and, in the process, turned a blind eye to the way he and his gang terrorized Mad Annie. It seemed an even contest, after all. She was bigger than they were, crazier than they were, older, bulkier and perceptibly more aggressive\u2014particularly when she'd been drinking\u2014and she had no compunction about lashing out when their teasing became intolerable.\n\n\"I've spent twenty years regretting my silence,\" I told Maureen. \"If I'd been a little braver, or a little more experienced\"\u2014I gave an uneasy laugh\u2014\"maybe I wouldn't feel so guilty now.\"\n\nShe shrugged. \"I wouldn't fret about it. Alan wouldn't have listened to you even if you had taken him to task. The only person he paid any mind to was his father.\"\n\n\"Until he turned on him with a baseball bat.\"\n\n\"It was bound to happen one day,\" she said indifferently. \"Kids grow up. It was Derek's fault anyway. He didn't realize Alan wasn't up for a thrashing anymore.\"\n\nI looked again at the cluster of empty bottles on her windowsill. \"Do _you_ ever feel guilty, Maureen?\"\n\n\"Why should I?\"\n\nI gave her a copy of Michael Percy's letter, detailing how her children had stolen trinkets from Annie. She was more amused than fazed by it for, as she said herself, I'd have a job proving it. \"No one's going to believe Michael,\" she pointed out, \"and he wouldn't talk to the police anyway, not while he's in prison. It's more than his life's worth to be known as a grass.\"\n\n\"They might believe Alan,\" I suggested.\n\n\"He'll just deny it. He's got a family now... doesn't want something he did as a kid coming back to haunt him this long afterward. And Danny doesn't even remember his dad, let alone who lived next to us twenty years ago. He asked me down the phone what Annie was like and why I've never mentioned her.\"\n\n\"What did you say?\"\n\n\"That she was a fat bitch who made our lives hell, and he didn't ought to believe anything you said because you had about as many screws loose as she did.\"\n\nI smiled at her as I pulled a manila envelope from the bottom of my rucksack and put it on the table in front of her. \"He'll probably believe this, though. I made this copy for you. When you've read it, give me a call. My number's on the front.\"\n\n\"What is it?\"\n\n\"An affidavit from a jeweler in Chiswick who bought several items off a woman called Ann Butts. It took me and my father about two hundred letters to find him after Michael suggested you'd sold the ring you took off Bridget. We started with jewelers and pawnbrokers in Richmond and radiated out until we hit pay dirt in Chiswick. He's still in business and keeps a record of every item that passes through his hands... together with the name of the seller and purchaser.\"\n\nShe dropped the envelope on to the table as if it were a red-hot coal.\n\n\"He's an honest trader and pays an honest price, so he requests proof of identity and ownership in order to be sure that the goods aren't stolen. He also records the type of proof that's offered. In the case of Ann Butts, it was a bank card and supporting statement, and a Sotheby's valuation of a list of items, including the jewelery, which were viewed on site at 30 Graham Road, Richmond. I presume you don't still have it?\" I said with a lift of my eyebrows. \"You wouldn't have been that stupid, would you?\"\n\nShe reached for another cigarette but I took the packet away from her and flattened it under my heel as I stood up.\n\n\"The really interesting fact,\" I finished, leaning my hands on the table, \"is that the first item wasn't sold until June '79, and my jeweler friend is positive that the Ann Butts he dealt with was a small white woman with a Brummy accent.\"\n\nShe had a quick mind for a Prozac junkie and a drinker. \"Just like half a million others then,\" she said.\n\n\"My phone number's on the envelope,\" I reminded her. \"Call me if you want to trade. If you don't, I'll give the affidavit to the police.\"\n\n\"Trade what?\"\n\n\"Information. I want to know who murdered Annie, Maureen... not who stole from her.\"\n\nSharon Percy refused to open her door beyond the burglar chain. \"I'm not going to talk to you,\" she said. \"You thought I wouldn't recognize you, but I watched you go into Maureen's so it didn't take much guessing.\"\n\nA tortoise head loomed behind her in the hallway. \"First you pester us with bloody letters,\" Geoffrey spat at me, \"now you turn up in the flesh. Why don't you just bugger off and leave us alone?\"\n\n\"I would have done if you'd written back,\" I said.\n\n\"What's to say?\" he growled. \"We don't know anything. Never did.\"\n\n\"Then why did you lie in your statements to the police?\"\n\nThere was a look of panic on both their faces before the door was slammed against me. As I hadn't expected anything else, I set off on the two-mile walk to Jock Williams's house.\n\n**_Letter from Libby Garth\u2014ex-wife of Jock Williams,  \nformerly of 21 Graham Road, Richmond\u2014now resident in  \nLeicestershire\u2014dated 1997_**\n\n**Windrush  \nHenchard Lane  \nMelton Mowbray  \nLeicestershire**\n\n**June 19, 1997**\n\n**M'dear** ,\n\n**Written in haste before I start cooking supper for the hungry horde. Would you believe Jock's moved yet another bimbo into that mansion of his! He seems to replace them every few months, yet he's hardly sex on wheels, for God's sake! How on earth does he attract them? I know he makes money from time to time but it's not as if he holds on to it for very long**.\n\n**His new project, \"Systel\"\u2014something to do with mobile phones\u2014looks optimistic, but if it goes the way of the others he'll be looking for a huge injection of cash within a year or so. Word has it (the new bimbo) he has such a lousy reputation with venture capitalists he's now looking at loans secured against the house. He needs his head examined if he does because he'll end up without a roof over his head if he overreaches himself. Heh! Heh!**\n\n**God, I'm a bitch! And why am I still doing this? Perhaps I'm a** _voyeur manque?_ **If so, I blame you for it. You should never have encouraged me to keep tabs on him, because it is so addictive chatting up his \"crumpet.\" It must be a \"comfortthing.\" I feel better knowing I wasn't the only one who couldn't make a relationship work with him**.\n\n**All love** ,\n\n# sixteen\n\nJock kept me standing on the step for several minutes before he opened the door, and I took the time to catch my breath after my hike from Graham Road to the rather grander street between Queen's Road and Richmond Hill where he was now living. The area had been developed in the wake of rail travel when the middle classes first began to exploit the benefits of living at a distance from their workplaces in noisy city centers; and the houses, though still in terraces, were more substantial than their humbler counterparts in Mortlake, with a third story to accommodate servants. A hundred years ago, each house would have had a walled front garden with trees and shrubs for privacy, but since the advent of the two-car family the gardens had been opened and paved to provide off-street parking.\n\nTo one side of Jock's frontage was an elderly black Mercedes with worn leather seats, and I was peering through the windscreen wondering if it was his when the door to the house snapped open and he appeared at my side. \"You're half an hour early,\" he said irritably. \"I thought we agreed two o'clock.\"\n\nI had expected age, divorce and thwarted ambition to have mellowed him a little, but attack, I saw, was still his favored form of defense. I was surprised by the sense of pleasurable recognition I felt, as of an old friend, and offered my cheek for a kiss. \"Hello, Jock,\" I said. \"How are you?\"\n\nHe gave me a quick peck. \"Where's Sam?\"\n\n\"Didn't he phone you?\"\n\n\"No.\"\n\n\"He got tied up at the last minute and couldn't make it,\" I lied with convincing regret, \"so I'm on my own.\" There was a beat of time while I pretended not to see the swift look of relief that crossed Jock's face. \"I didn't know you were into classic cars,\" I said mischievously, patting the Mercedes's hood. \"You always hankered after the newest model in the old days. I remember how rude you were when Sam and I bought that secondhand Allegro estate.\"\n\nHe made a dismissive gesture. \"I keep the Merc as a runabout. The Jag's in a lockup garage down the road.\"\n\n\"A Jag!\" I exclaimed. \"My God! Sam will be green with envy. He's been wanting an XK8 since they came out.\" I looked past him into the shadows of the hall where a coin-operated telephone hung off the wall. \"Don't let me stop you if I've interrupted something,\" I said. \"I'm in no hurry.\"\n\nHe pulled the door to. \"I've some e-mails to answer.\"\n\n\"I can wait.\" I hitched a buttock onto the bonnet of the Mercedes and lifted my head to look at the house. It was an attractive building with sandstone bays containing the high, wide windows of which Victorian architects were so fond. According to Libby the house had cost him \u00a370,000 in 1979 and, according to a local real estate agent, it was now worth upward of three-quarters of a million. \"Nice place,\" I murmured when he made no move to return inside.\n\nHe nodded. \"I like it.\"\n\n\"So what was wrong with it when you bought it? Sitting tenants? Subsidence? Dry rot?\"\n\nHe looked surprised. \"Nothing.\"\n\n\"You're joking! How on earth did you afford it? I thought you got a pittance out of the divorce settlement.\"\n\nHe recoiled slightly as if I'd just revealed teeth. \"Who told you that?\"\n\n\"Libby.\"\n\n\"I didn't know you were still in touch.\"\n\n\"On and off.\"\n\n\"Well, she's wrong,\" he said warily. \"She thought she could clobber me by hiring an expensive solicitor but he never came close to finding the investments that mattered.\"\n\n_It is odd_ , I thought, _how the memory plays tricks_. In my mind, I had likened him for so long to a weasel that his rather charming face surprised me. \"It must have been a first then,\" I said with a small laugh. \"You never managed to hide anything else from your wife.\"\n\n\"What else has she told you?\"\n\n\"That you moved a blonde in here before the ink was dry on the divorce. 'Young enough to be his daughter,'she said, 'but old enough to recognize a sucker when she sees one.'\"\n\nAnother flash of relief. \"That's her jealousy talking,\" he scoffed.\n\nI laughed again, amused by his cocky expression. \"You always were a hopeless liar, Jock. It used to irritate me... now it amuses me... probably because I know so much more about your business affairs than Sam does.\"\n\nHis expression soured. \"Like what?\"\n\n\"Like you have an outstanding loan on this place of \u00a3500,000 which you took out to keep Systel afloat and now can't repay.\"\n\nThere was a short silence while he considered his response. \"Is this something else Libby told you?\"\n\nI nodded.\n\n\"Well, it's a lie,\" he said curtly. \"She doesn't know a damn thing about my finances. Hell, she didn't even know what they were twenty years ago, so she certainly wouldn't know now. I haven't spoken to her since the divorce.\" He waited for me to say something and, when I didn't, he ratcheted up his aggression. \"I could sue you both for slander if you repeat it to anyone else. You can't go 'round destroying people's reputations just because you hold a grudge against them.\"\n\nI was tempted to say such considerations hadn't stopped him twenty years ago from helping Sam to destroy _my_ reputation. Instead I said mildly: \"I'm always happy to be put straight, Jock. So which is the lie? That you don't have an outstanding loan, that you didn't plow it into Systel and lose it or that you can't repay it?\"\n\nHe didn't answer.\n\n\"Perhaps you should have been a little more selective in your girlfriends,\" I suggested. \"According to Libby the blonde bimbette was the first of many, and none of them knew how to keep her mouth shut.\"\n\n\"What's that supposed to mean?\"\n\n\"Libby's been prying information out of them for years while you were out at work, and even she couldn't believe how indiscreet some of them were. All she had to say was that she was conducting research for a hosiery manufacturer, then offer them a dozen free packs of luxury tights in return for twenty minutes of their time answering lifestyle questions, and the floodgates would open.\"\n\nHe frowned. \"Why the hell would she do a thing like that?\"\n\nIt was a good question, but not one I was ready to answer. I needed him off balance if I was ever to get at the truth. \"She wanted to know how much you ripped her off in the divorce settlement.\"\n\n\"None of my exes could have told her that,\" he said confidently.\n\n\"No,\" I agreed, \"but she never asked anything so direct. She was far more subtle\"\u2014I smiled\u2014\"and _very_ patient. She built on what she already knew about you.\" I thought of the regular lists Libby had sent me with updated information on Jock. \"Do you or your partner own the house? Was the value of the house more or less than \u00a350,000 at the time of purchase? Is the present value of the house more or less than \u00a3100,000... \u00a3200,000, et cetera? Is your partner self-employed? Does he earn more or less than \u00a350,000... \u00a3100,000, et cetera? Does he have a mortgage? Is it more or less than \u00a350,000, et cetera, et cetera?\" I laughed unkindly. \"She never got a straightforward yes or no. One of your girlfriends even fished out your bank statements so that the figures would be accurate.\"\n\n\"That's illegal.\"\n\n\"Undoubtedly.\"\n\n\"You're lying,\" he said with more certainty than his expression sug gested he felt. \"Why would she keep on with something like that? It doesn't make sense.\"\n\nI smiled ruefully as if I agreed with him.\n\n\"What answers did she get?\"\n\n\"That your mortgage went from \u00a320,000 to \u00a3500,000 in fifteen years, and you worked your way through seven girlfriends in the process. Two of your start-up businesses failed and the half million you made on the one you sold last year went straight into staving off bankruptcy. The only reason you're still here\"\u2014I nodded toward his front door\u2014 \"is because the capital value of the house exceeds the loan and the bank's allowing you to make interest-only repayments while you look for a job with a six-figure salary. You're not having much success because you're almost fifty and your track record is far from impressive. You're fighting the bank's pressure to sell the house because you're afraid you'll only walk away with \u00a3200,000 once the bills have been paid, and that's barely enough to buy back your old place in Graham Road.\"\n\nHe looked devastated, as if I'd just torn his life apart and tossed the pieces to the winds. I felt no remorse. In a small way he was beginning to understand what he had once done to me.\n\n\"If it's any consolation,\" I went on amiably, \"Sam's been just as economical with the truth. We didn't make a killing in Hong Kong, there's no eight-bedroom mansion on the horizon, and the farmhouse we're renting is falling down. In fact we're not much better off than you are, so it seems rather pointless to spend the next half-hour trying to impress each other with our nonexistent fortunes.\"\n\nHe sighed\u2014more in resignation than anger, I thought\u2014and gestured toward the door. \"You'd better come in, though I warn you I'm pretty much confined to my study these days. The rest of the house is let out to foreign students as the only way to cover the bills. Matter of fact, I was planning to take you to the pub so you wouldn't find out, but it's a hell of a sight easier this way.\" He led me across the hall toward a room at the back. \"Have you told Sam any of this?\" he asked, opening the door and ushering me in.\n\n\"No. He still believes everything you tell him.\" I took stock of the room, which had barely enough space to maneuver. It was packed to the scuppers with sealed boxes, piles of books and pictures hung in tiers on every wall, and if anything of Annie's was in here, it was stubbornly invisible. \"My God!\" I said, unbuckling my rucksack and dropping it to the floor. \"Where the hell did all this come from? You haven't taken to burglary, have you?\"\n\n\"Don't be an idiot,\" he said tetchily \"It's the stuff I'm keeping from the lodgers. If they don't steal it they'll break it. You know what they're like.\"\n\n\"No,\" I assured him. \"I haven't met them.\"\n\n\"I meant foreigners in general.\"\n\n\"Ah!\" I gave a snort of laughter, enjoying the irony of Jock sharing his house with strangers. \"Are we talking _black_ foreigners, Jock?\"\n\n\"Arabs,\" he said crossly. \"They're the only ones with any money these days.\"\n\n\"Is that why you're sleeping in here?\" I asked, looking at the bed in the corner. \"To guard your possessions from dusky predators?\"\n\n\"Ha-bloody-ha!\" He took the swivel seat in front of his desk, leaving the armchair for me. \"Only when the other rooms are full. It's a bit hand-to-mouth but it's tiding me over.\"\n\nHe had grown a beard since the last time I saw him and his dark hair was going gray, but it suited him and I decided he must thrive on adversity because he had none of the worry lines that characterized Sam's face. \"You look good,\" I said, settling myself in the chair. \"Sam's lost most of his hair and is very sensitive about it. He'll be upset to hear you've kept all yours.\"\n\n\"Poor bugger,\" he said with surprising sympathy. \"He was always paranoid about going bald... used to count the hairs in his comb every day.\"\n\n\"He still does.\" I transferred my attention to a tortoiseshell cat that was curled up on a padded footstool in the corner of the room. \"I didn't know you liked cats.\"\n\nHe followed my gaze. \"This fellow's grown on me. One of the exes stormed out when I refused to pay her credit-card bill and poor old Boozey got abandoned in the rush.... Either that, or he went to ground the minute the estrogen started to fly. He's more interested in me than my wallet so we rub along pretty well together.\"\n\n\"Do you have a girlfriend now?\"\n\n\"You mean Libby hasn't told you?\" he asked sarcastically. \"I thought she knew everything.\"\n\n\"She gave up calling when foreigners started answering the phone.\"\n\n\"Why wasn't she worried about me picking it up?\"\n\n\"You did,\" I told him, \"several times. She always pretended to be an old lady phoning the doctor's surgery. You were very patient with her. Kept telling her to correct the number in her book so she wouldn't get it wrong again.\"\n\n\"Godammit! Was that Libby? It didn't sound like her.\" He looked impressed, as if I'd just said something laudable about a nonexistent daughter instead of the wife who'd cast him aside nearly a quarter of a century before.\n\n\"She's good at putting a tremble in her voice.\" I paused. \"Do you miss her?\"\n\nIt was a question he hadn't expected and he stroked his beard pensively while he considered his answer. \"Sometimes,\" he admitted. \"Where is she now? I know she remarried because one of her friends told me, but I've no idea where she went.\"\n\n\"Melton Mowbray in Leicestershire. She did a postgraduate course in Southampton after you and she split up and now she's head of the history department at a comprehensive in Leicester. Her husband's a bank manager called Jim Garth. They have three daughters. The eldest is thirteen and the youngest seven.\"\n\nHis lips twisted in a regretful smile. \"She always said she could do better without me.\"\n\n\"She wanted an identity of her own, Jock\"\u2014I leaned forward, clamping my hands between my knees\u2014\"and if you'd encouraged her to train as a teacher while you were still married... who knows? Maybe you'd still be together.\"\n\nHe didn't believe that any more than I did. \"Hardly. We weren't even on speaking terms by the end.\" His eyes narrowed as he looked at me, and I guessed he had as much pent-up distrust of me as I had of him. \"I've always blamed you for the divorce, you know. Libby didn't have a problem till you came along, all she wanted was babies... then _you_ move into the street and, suddenly, babies aren't good enough. She _has_ to have a career, and it _has_ to be teaching.\"\n\n\"I didn't know she was so easily influenced.\"\n\n\"Oh come on! Every idea she had was recycled from the last person she spoke to. That's probably why she became a history teacher,\" he said sarcastically. \"You don't have to think so much when your subject's been chewed over for centuries by other people.\"\n\n\"That's rubbish, Jock. Libby knew exactly what she wanted out of life... also what she _didn't_ want.\"\n\n\"Yes, well, I could always tell when she'd been with you. She was a hell of a sight more belligerent about her rights when she'd had a dose of Ranelagh left-wing feminism.\"\n\n\"Maybe it's a good thing you never introduced her to Sharon then,\" I said dryly. \"Or you'd have had a prostitute for a wife.\"\n\nHe wouldn't look at me\u2014afraid, I think, of what I might read in his eyes\u2014but his neck flushed an angry red. \"That's a stupid thing to say.\"\n\n\"No more stupid than you trying to blame me for your divorce,\" I said evenly. \"Nothing I said or didn't say could alter the fact that Libby was sick to death of your gambling. She wanted some stability in her life, not a roller-coaster ride of wins one day and losses the next. It was bad enough when it was just the stock market, but when you admitted to losing three thousand quid on a poker game....\" I shook my head. \"What did you expect her to do? Pat you on the back?\"\n\n\"It was _my_ money,\" he said sulkily.\n\n\"It was also your money when you won,\" I pointed out, \"but you never shared your winnings with her, only your losses. You put Libby through hell every time you lost and used your winnings to buy blow jobs off Sharon.\"\n\nIt began to dawn on him just how much Libby had told me and he retreated into an offended silence, punctuated only by the regular ticking of a pendulum clock on the mantelpiece. I made no effort to break it. Instead I glanced about the study, trying to imprint what I could see on my memory. It was an impossible task, so I looked for what wasn't there: silhouette pictures of Annie's grandparents, mosaics of Quetzal-coatl, items of jade, artillery shells and peacock feathers...\n\nThere was a fine seascape in a gilded frame on the wall opposite showing a ship under full sail battling with a storm-tossed sea, and I could just make out the words on the small plaque screwed to the bottom of the frame. _Spanish Privateer in Great Storm off Kingston, Jamaica, 1823_. I was so absorbed in trying to decide whether the date represented the year the storm happened or the year the picture was painted, that it was a while before I realized Jock was watching me.\n\n\"What the hell's going on?\" he asked suspiciously, following my gaze. \"Has Libby got some crazy idea that she can get more money out of me?\"\n\nI shook my head. \"I came to ask you about the night Annie Butts died.\"\n\nHe gave an exasperated sigh. \"So why drag Libby into it? Why not be upfront at the beginning?\"\n\nIt was an obtuse remark from a man who always attacked first and asked questions later. \"Sorry,\" I said apologetically.\n\n\"You could have talked to me on the phone,\" he said, warming to his grievance. \"I've always answered your questions in the past. I even drove round to St. Mark's Church the other day to find out the vicar's name for you.\"\n\n\"That was kind,\" I agreed.\n\n\"Then what's the big deal?\"\n\nI pulled a wry expression. \"Nothing really. I'm just not very good at this. I was afraid you'd clam up if I dived straight in with questions about where you were and who you were with that night.\"\n\nHe looked surprised. \"You already know all that. It was in my statement. I was with Sam at your place. We had a couple of beers and then I went home.\"\n\n\"Except it was a Tuesday,\" I reminded him, \"and Libby told me Tuesday was your fellatio day.\"\n\n\"God almighty,\" he growled angrily, hating the whole subject, \"I went to Sharon first. Okay? I came out at about half-seven, bumped into Sam and went back to his place for a beer.\"\n\n\"Sam said you bumped into each other at the tube station.\"\n\nHe shifted uncomfortably. \"It was twenty years ago. You can't expect me to remember every wretched detail.\"\n\n\"Why would you be at the tube if you'd just left Sharon? I thought you had sex in her house.\"\n\n\"What the hell difference does it make? Annie was alive and well when we passed her in the street.\"\n\nI shrugged. \"The reason Sam knows he met you at the station is because you were on your way home from a poker game.\"\n\nHe was taken by surprise. \"A poker game?\" he echoed. \"Where on earth did that spring from?\n\n\"It's what Sam said.\"\n\n\"Not in his statement, he didn't.\"\n\n\"No, it was the explanation he gave me afterward,\" I lied. \"He said he took you home for a drink because you were in a blue funk about how to tell Libby you'd lost another fortune.\"\n\nSurprise was abruptly replaced by irritation. \"You didn't pass that on to Libby, did you?\"\n\n\"No. I didn't hear about it until after we'd left England.\"\n\nHe pondered for a moment. \"Maybe Sam didn't want to say I'd been with Sharon.\"\n\n\"Did he know about her?\"\n\nHe gave a half-hearted nod.\n\n\"But who could have told him, Jock? _You?\"_ I said in amazement when he didn't answer. \"God! I'd have put money on you keeping that a secret. It wasn't something to be proud of, was it?\"\n\nHis mouth thinned. \"Give it a rest, all right? None of this has anything to do with Annie's death.\"\n\nI shook my head. \"It has everything to do with it, Jock. She died because she was beaten half to death some hours before she managed to drag herself down to our end of the street for me to find, yet you just said she was alive and well when you came out of Sharon's house at a quarter to eight.\" I lifted copies of the autopsy photographs from the front pocket of the rucksack and spread them on my lap. \"Look at the bruising. It's too extensive to have come from injuries inflicted fifteen to thirty minutes before she died.\" I isolated a close-up of Annie's right arm. \"This is a classic picture of a multitude of defense wounds sustained hours before death. The probability is that she curled herself in a ball to try to protect her head and, instead of a few individual bruises, which is what you would expect to see if a truck had flung her against a lamppost minutes before she died, all the bruises have merged over a period of hours to produce one massive hematoma from shoulder to wrist.\"\n\nHe stared at the photographs with shocked fascination but instead of expressing revulsion over Annie's bludgeoned face, he offered a poignant non sequitur. \"I'd forgotten how young she was.\"\n\n\"Younger than you are now,\" I agreed, \"and very strong, which is why she took so much punishment before she passed out. This bruising at the top of her thighs\"\u2014I turned a picture of Annie's torso toward him\u2014 \"suggests massive internal injuries from being kicked or beaten around her abdomen, causing blood to seep into the tissue of her legs. It's what's usually described as a 'frenzied attack'and it almost certainly happened in her own house because anywhere else would have been too public.\"\n\nHe took time to assimilate what I was saying. \"I thought she was wearing her coat. Why would she wear a coat in the house?\"\n\nIt was a question I'd asked myself many times because she certainly wouldn't have been in any condition to put it on after she'd been attacked. \"I can only guess that someone pushed in behind her when she came home from the pub and attacked her before she had time to remove it.\"\n\nHe began to look worried. \"The police would have found some evidence,\" he protested. \"There'd have been blood on the walls.\"\n\n\"Not if most of her injuries were internal. In any case, there _was_ evidence. The police recorded it themselves. Broken furniture which suggested a fight... absent floor coverings which suggested she _did_ bleed and the rugs were removed... human waste in the hall, which is a classic fear response from intruders. She stank of urine when I found her, Jock, which suggests they pissed on her as well.\"\n\nHe turned away to fiddle with the pens on his desk. \"That's disgusting.\"\n\n\"Yes.\" I gave a tired shrug. \"And if you and Sam hadn't lied about seeing her at a quarter to eight, then maybe the police would have interpreted the evidence properly instead of condemning her as a tramp.\"\n\nHe licked his lips nervously. \"Does Sam say we lied?\"\n\nI nodded, carefully squaring the photographs in my lap. \"He was feeling homesick one night in Hong Kong and started blaming me for the fact that we'd had to leave England. It all came out at about three o'clock in the morning... how you'd phoned him and begged him to give you an alibi... how I'd made his life impossible by telling the police it was murder... how the choice between supporting me or his closest friend had been one of the hardest he'd ever had to make.\" I shrugged. \"I've not had much sympathy for you since. You put me through hell and I've never forgiven you for it.\"\n\n\"I'm sorry,\" he said awkwardly.\n\nI couldn't help admiring his loyalty. It was more than Sam deserved but it said much for their friendship, which had stayed healthy through their regular exchange of phone calls, faxes and e-mails. \"It's only a matter of time before the police reopen the case,\" I told him, \"and the first thing they're going to look at is where people were in the hours before Annie's death. She died shortly after 9:30,\" I reminded him. \"So if you spent thirty to forty minutes in Sharon's house and left at half-seven, then you were there within the timespan that bruises like this\"\u2014 I tapped the photographs\u2014\"need to develop.\"\n\nHis eyes flicked toward my lap.\n\n\"And that means you must have heard what was going on next door,\" I went on matter-of-factly \"or you joined Sharon shortly after she heard it. Either way you'd have noticed something. You don't get good sex off a woman who's just listened to her neighbor being clubbed into unconsciousness.\" I eyed him curiously. \"But Sharon's bound to claim your story's bullshit anyway because, according to her inquest testimony, she was in the pub from 6 till 9:15.\"\n\n\"This is crazy,\" he said, his eyes straying toward the telephone on his desk. \"What does Sam say?\"\n\n\"Nothing much... except that he's adamant he didn't know about Sharon and refuses to take the blame if you lied to him about why you needed an alibi.\"\n\nIt was the accusation of lying to Sam that seemed to goad him toward honesty. Either that or his resentment at being made everybody's scapegoat finally boiled over. \"Sam knew better than anyone that I didn't have the bottle to go near another card game,\" he said bitterly. \"I may be a risk taker but I'm not a bloody fool. I was taken to the cleaners by some pros the first time and I wasn't going to give them a second chance.\" He squeezed the bridge of his nose between finger and thumb. \"And there's no way Sharon was an issue. I could have paraded half the whores in London in front of Libby without her turning a hair. The marriage had been dead for months... it was just a question of which one of us was going to pack our bags first.\"\n\n\"Then why did you lie in your statement?\"\n\nHe saw in my eyes that I knew the answer. \"Do you really want it spelled out? It was dead and buried before you even left England.\"\n\n\"For Sam maybe,\" I said. \"Not for me. That's why I'm here. I've waited a long time to find out who he was with that night... and what they were doing...\"\n\n**_E-mail from Libby Garth\u2014ex-wife of Jock Williams,  \nformerly of 21 Graham Road, Richmond\u2014now resident in  \nLeicestershire\u2014dated 1999_**\n\n**M. R**.\n\n**From:** Libby Garth (liga@netcomuk.co)\n\n**Sent:** 05 May 1999 14:37\n\n**To:** M.R.\n\n**Subject:** Re coming home at last!\n\nM'dear, this is such marvellous news! I truly believed you were gone for good! I suppose Sam's coronary accounts for it\u2014so every cloud does have a silver lining then? Anyway, I can't wait to see you again. Perhaps you can persuade Sam to visit Jock one weekend while you and the boys come up to visit us in Leicestershire? I can't imagine Sam will want to buddy up to Jim for fear of betraying his old friend\u2014and Jim would be v. nervous to have a suspicious mate of Jock's about the place.\n\nTalking about suspicious mates (ha! ha!) are you planning to confront Jock after you get back? As you know, I've never managed to find out how he bought that house in Alveston Road, although I met a university friend of his at a party a while back who made an oblique reference to the effect that Jock's parents had helped him\u2014viz.: \"Jock's never been known to miss a trick. He told me once he'd screwed a small fortune out of the pair of them because each one thinks he doesn't talk to the other, and they can't check because they haven't exchanged a word since the brother died.\" Could that be where the money came from? It sounds like Jock's modus operandi, whatever he may say about being a \"self-made man.\"\n\nHave I ever told you how impressed I am by all your efforts? Who'd have thought the little teacher from Graham Road would have turned into such a tigress! Poor old Sam must wonder what's hit him. You say he still won't talk about the night Annie died, but perhaps that's only to be expected. The longer you remain married the harder it must be for him to admit that he was prepared to put a friend before his wife.\n\nYou're too wise not to put something that happened twenty-plus years ago into perspective. Let's face it, we all make mistakes, and in fairness to Sam you did go a little crazy afterward\u2014classic posttraumatic stress reaction for which you should have had counseling\u2014 and neither he, Jock, nor anyone else had reason to doubt the police view that Annie died in a tragic accident. I know you'll say it doesn't pass the \"so what\" test, but I can't help feeling it would put your marriage under unnecessary strain to keep reminding Sam of his \"failure\" when all the police will require is a straightforward admission that they didn't see Annie that night.\n\nRe the Slaters, Percys and Spaldings. Do be careful how you approach them as there is no doubt in my mind that they'll be extremely hostile to answering questions. Hate groups are notoriously violent\u2014they're too low on the pecking order to be anything else\u2014 and I really don't want to read in my newspaper that your body's been fished out of the Thames! The burning fiery cross is a frightening reality, my darling, not a figment of KKK imagination. They believe in terror because terror gives them status. (It probably gives them orgasms as well because they're all sadists, but they would never admit to that!) Anyway, I do wonder if you shouldn't leave the Slaters to the police, particularly as you've amassed so much evidence already of their petty thieving.\n\nSpeak soon,\n\nAll my love,\n\nL\n\n# seventeen\n\nIt was an involved tale about a pretty little secretary in Sam's office who had ensnared him during the August of '78 while I was in Hampshire dog-sitting for my parents, who were on holiday. It was a brief madness, Jock assured me, a fatal attraction that went sour almost as soon as it started. Sam wanted to dump her the minute I came home, but the girl was having none of it. If she'd worked anywhere else it wouldn't have been a problem, but Sam was worried about how it would affect his career if she turned on him out of spite. It was the early days of sexual harassment cases, and this was a girl who knew what she was doing.\n\nSam strung her along for a couple of months, then made an attempt to end it on the night I was due to stay late at school for a parents'evening. By malign chance, it was also the night Mad Annie died. \"Sam was way out of his depth,\" said Jock. \"He had this crazy idea that if he wined and dined his mistress first, then told her he planned to do the decent and honorable thing and stay with his wife, the girl would accept it. Instead she went ballistic... screamed and yelled at him in the restaurant... poured wine down his suit... and, what with one thing and another, he was in a pretty dire state by the time he reached home.\n\n\"He passed Annie in the gutter,\" said Jock. \"She was under the lamppost so he could hardly miss her, but she reeked of drink so he left her. He knew you'd be back any minute and his first priority was to get out of the suit, clean himself up and pretend he'd been in all evening.\" A glint of humor flashed in his eyes. \"Then you come running in fifteen minutes later to ring for an ambulance and the silly sod promptly shoots himself in the foot.\"\n\nI frowned. \"He was watching television. I never even questioned where he'd been.\"\n\n\"You told him Annie Butts was dying in the road outside and he said, 'No, she's not, she's paralytic.'\"\n\n\"So?\"\n\n\"Why would he say that if he hadn't seen her?\"\n\nI bit back a laugh. \"Are you telling me you lied to the police because of some half-assed remark he made while I was screaming down the telephone for an ambulance? He could have told me she was standing on her head and waggling her legs in the air for all the notice I paid him. I wouldn't have remembered afterward.\"\n\nJock shrugged. \"That's exactly what I said, but he didn't believe me. He reckoned you had a memory like an elephant. He said it would be easier all round if we supported the police version that Annie was staggering drunk at a quarter to eight. I mean it wasn't as if we were the only ones to say it... _everyone_ was saying it. We thought it was the truth.\"\n\n\"There were only five other people who claimed to have seen her,\" I reminded him. \"One was Geoffrey Spalding who lived opposite Annie at number 27. He's the man who said at the inquest that he tried to persuade Annie to go home but gave up when she started cursing him. His estimate of the time was between 8 and 8:30. Two were an elderly couple at number 8, Mr. and Mrs. Pardoe, who went up to bed at approximately nine o'clock because they were cold and saw her from their upstairs window, but decided against doing anything because she was clearly drunk and the last time they'd tried to help her she spat at them. And the remaining two were a man and a woman in a car who were using Graham Road as a shortcut and said they had to slam on the brakes when a bulky figure in a dark coat suddenly lurched in front of them, screaming abuse. They decided she was 'an aggressive drunk'and drove on to avoid a confrontation. They couldn't be accurate about the time, but thought it was shortly after nine o'clock.\"\n\nHe looked at the photographs that were still in my lap. \"You've just destroyed your own argument,\" he said. \"Why would any of those people lie about seeing her?\"\n\n\"I don't think they did,\" I answered slowly, \"except possibly Geoffrey Spalding, and he may only have been lying about the time. You see, timing's important. One of the reasons the police estimated that she received her injuries fifteen to thirty minutes before I found her was because the Pardoes and the couple in the car both said she was on her feet at or around nine o'clock. If she was dead by 9:30, then ipso facto something must have hit her during those thirty minutes.\"\n\n\"Then how can you expect anyone to believe she was beaten to death hours earlier?\"\n\n\"I said she was beaten unconscious, Jock, I didn't say she was dead. There _is_ a difference... particularly when you're talking about someone as well-built and powerful as Annie.\" I ran an exploratory finger across her celluloid face as if it could tell me something. \"I think she came 'round inside her house and managed to get herself out in search of help. The miracle is she had enough strength left to try to stop a passing car. A doctor would probably say it was impossible because her skull was so badly fractured, but it's the only explanation for why she was in the road and why she appeared drunk.\"\n\n\"Or the police were right all the time,\" suggested Jock. \"I remember reading the inquest report. It said there was a high level of alcohol in her blood.\"\n\nI shook my head. \"It was ninety-five milligrams per hundred millilitres of blood\u2014or fifteen milligrams above the legal driving limit. That's the equivalent of four or five shots of rum... a drop in the ocean for someone who drank as much as Annie. Sam and I can manage that with no trouble at weekends... you, too, I expect... but it doesn't make us stagger about like zombies.\" I gave a weary shake of my head. \"She was labeled a road-traffic accident, so the pathologist routinely recorded her as 'unfit to drive,'which the police and the coroner then interpreted as a 'high concentration of alcohol.'In fairness they had witness statements that described her as 'paralytic'and the police found cases of empty vodka bottles in her house, but if the pathologist had done his job properly he would have questioned whether ninety-five milligrams was enough to cause staggering in a fourteen-stone woman with a known alcohol habit.\"\n\n\"You really have done your homework, haven't you?\"\n\n\"Yes.\"\n\n\"What do the police say?\"\n\n\"Nothing yet. I want my evidence so watertight that they'll be forced to reopen the case whether they like it or not.\" I paused. \"I'll need you and Sharon to admit you were the couple I followed into Graham Road that night,\" I told him.\n\nHe shrugged. \"That won't worry me. It might worry her, though.\"\n\n\"Why?\"\n\n\"She lied at the inquest. She didn't get to the William of Orange till 9:15. We usually met up about half-eight, had a quick drink, then cut down the alleyway at the back of her house, but she was dropped off by a taxi that night, high as a kite, and totally uninterested in making any more money. So I walked her along the A316 and split away from her when we turned into Graham Road.\" He went on before I could ask the obvious question. \"She said she'd been at a hotel with another client. I assume it was true because she was dressed up like a dog's dinner and stank of fags.\" He gave a small shake of his head as he recalled the memory. \"She certainly didn't give the impression that she'd come from her house. Rather the reverse, in fact. Kept saying she wanted to get back to it because she was sick as a dog from all the champagne she'd drunk.\"\n\n\"But if Tuesday was _your_ day, why would she go with somebody else?\"\n\n\"She was a pro,\" he said sarcastically. \"Someone else offered her more money.\"\n\n\"Did she say who it was?\"\n\n\"She didn't give me a name... just said it was another regular whom she couldn't afford to disappoint.\"\n\n\"Geoffrey Spalding was one of her clients,\" I said slowly. \"His wife was dying of breast cancer and he didn't want her or his daughters to know he was paying for sex. He took Sharon to a hotel once a month.\" I laughed at his expression. \"No, it wasn't Libby who told me. It was Sharon's son, Michael. I've been writing to him in prison.\"\n\n\"Jesus! Rather you than me then,\" he said dryly. \"He was a right little sadist when I knew him... used to pluck the whiskers out of Annie's cats just for the fun of it. Do you know why he's in prison?\" I nodded. \"Then you ought to be careful. His mother was shit-scared of him. And with reason. He had a real temper when he was roused.\"\n\nI watched the cat lick itself drowsily in the afternoon sun. \"You know the one thing that's always puzzled me, Jock... why neither you nor Sharon stopped to find out if Annie was alive. You must have seen her. Sharon virtually had to step over her to cross the road.\"\n\n\"We truly didn't,\" he said. \"I asked Sharon about it afterward and she went white as a sheet... kept begging me to keep my mouth shut in case we got accused of being involved in some way.\"\n\nThere seemed little else to say, but I couldn't find the energy to rise from the chair. The journey home held few attractions and, like the cat, I wanted nothing better than to curl into a ball and forget that life was complicated. Perhaps Jock felt the same because the shadows lengthened noticeably before he spoke again.\n\n\"You've changed,\" he said finally.\n\n\"Yes,\" I agreed.\n\nHe smiled. \"Aren't you going to ask me how?\"\n\n\"There's no point.\" I leaned my head against the back of the chair and stared at the ceiling. \"I know what you're going to say.\"\n\n\"What?\"\n\n\"I'm more relaxed than I used to be.\"\n\n\"How did you know?\"\n\n\"It's what Sam always says.\"\n\n\"You used to get pretty hyped-up in the old days,\" he said. \"I remember going into your house one day and having to duck a flying saucepan.\"\n\nI turned my head to look at him, laughing at the memory. \"Only because you and Sam came home plastered at some god-awful hour in the morning and got me out of bed with the row you were making downstairs. The minute you saw me you started demanding food, so I tossed the saucepan in your direction and told you to cook it yourselves. You were supposed to catch it, not duck it.\"\n\n\"Is that right?\" he asked dryly. \"Then how come most of the crockery ended up on the floor as well?\"\n\nI thought back. \"I was hopping mad, particularly as we had a school inspection the next day. In any case, I never liked those plates. Sam's mother gave them to us.\"\n\nHe grinned at me. \"We were so damn legless we probably thought you'd be thrilled to see us. And at least we never did it again. As Sam said, you'd probably start hurling knives the next time.\"\n\nWe exchanged smiles. \"I never did find out where you'd been,\" I murmured lazily. \"You swore it was the pub, but it can't have been because pubs closed at 11.\"\n\nThere was the smallest of hesitations before he answered. \"A strip club in Soho,\" he said. \"Sam didn't think you'd approve.\"\n\nI gave a noncommittal shrug. \"Was the pretty little secretary with you?\" I asked. \"It was October-time, so she must have been around.\"\n\nHe shook his head. \"Sam wouldn't take a woman to a strip club.\"\n\nI leaned forward to tuck the photographs of Annie into my rucksack. \"Did you ever meet her, Jock?\"\n\n\"No,\" he admitted.\n\n\"So you've only Sam's word that she existed?\"\n\nThere was real surprise in his voice when he answered. \"Of course she existed! You can't hate someone who isn't real. He told me that night that strangling was too good for her, and trust me... I was _there_.... I _heard_ him. He meant every word. That's why I took him to the club in the first place... to try to get his mind on to something else. He was terrified she was going to come to you with the sordid details... either that or blackmail him. I'd just about persuaded him to come clean and tell you about it\"\u2014he gave a dispirited sigh\u2014\"then we walk through the door and you start throwing bloody saucepans at us.\"\n\nI smiled at his innocence, thinking it was no wonder Sam loved him as a guru. Pupils always preferred a teacher they could manipulate. \"Sorry,\" I said without contrition, \"but if it's any consolation there's no way he was going to own up to it. I'm not questioning the affair, Jock, only the conveniently streetwise secretary. He invented her for your sake. He's always been useless at keeping secrets and you were bound to get suspicious if he started saying he was too busy to have a drink with you. I think you'll find he was performing closer to home.\"\n\nHe rubbed his head ferociously. \"I don't understand.\"\n\n\"Oh, come on, it's not that hard to work out.\" I started gathering my bits and pieces together. \"What do you think Libby was doing the night Annie died? Darning your socks?\"\n\nHe wouldn't accept it. \"She can't have been with Sam,\" he said. \"Hell, I'd have known if she'd been out. She had my supper waiting, and all the laundry done, for Christ's sake.\"\n\n\"There was a perfectly good bed in your house,\" I murmured. \"What makes you think they didn't use that?\"\n\nHe stared at me with a look of bewildered hurt on his face, and I was reminded of my own devastation as I listened to Sam's drunken ramblings that night in Hong Kong. _It's your fault we're here.... If you hadn't left me in the lurch none of this would have happened.... Women are crooked.... They do one thing and say another.... Why the hell did you have to ask people what they were doing that night? Did you expect them to be honest?_\n\n\"I could have walked in at any moment,\" protested Jock, clutching at straws.\n\n\"It was a Tuesday,\" I said, \"and you never got home before 10 on a Tuesday.\"\n\n\"But...\" His bewilderment increased. \"Was anything Sam told me true?\"\n\n\"I think it was true that it started during the two weeks I was away. I remember him telling me over the phone that Libby had offered to do his washing for him, but when I asked him later if he'd taken her up on it he became incredibly tetchy and said he hadn't seen her. At the time I thought he was cross because she'd let him down, but now I think he was just frightened of giving too much away...\"\n\nI watched resentment steal into Jock's face like a thief, and was surprised at how hollow my little victory felt.\n\n\"I think it's also true that he wanted to end it,\" I went on, \"and was terrified of making an enemy of her. Personally, I doubt Libby would ever have confessed to it herself-\u2014she didn't want to give you ammunition for a divorce\u2014but Sam certainly believed she would.\" I smiled slightly. \"The irony is, I suspect he was far more worried about you finding out than he was about me. He says your friendship is important to him.\"\n\n\"He's a bloody hypocrite.\"\n\nI didn't disagree. \"Why do you care?\" I asked. \"As you said yourself it was dead and buried years ago.\"\n\nBut Jock didn't want to be reminded of his own mealymouthed platitudes. \"He got me to lie for him.\"\n\n\"You were happy to do it,\" I pointed out.\n\n\"I might have felt differently if I'd known he was with Libby.\"\n\nI lifted a shoulder in a half-shrug. \"Who's the hypocrite now?\"\n\nHe turned away, pulling a handkerchief from his pocket.\n\n\"In any case,\" I went on, \"I'm betting it was Libby who pushed him into it. The police were asking everyone in the street if they'd seen or heard anything at the time of the accident, and I think she was afraid someone would say they'd spotted Sam leaving your house around nine o'clock. It was safer all 'round if he could deny it and say he was with you at our place.\"\n\nThe steps from bewilderment to hate were short and ugly and could be measured in their passage across his face. I had taken those steps myself and recognized the signs. Yet the object of his hatred was not the man who'd betrayed him, but the woman. \"She loved making a fool of me, you know. She's probably been wetting herself for years knowing I was the one who covered their tracks.\"\n\nI shook my head. \"You shouldn't dwell on it. If Sam had been anything more to Libby than a stopgap lover, you'd have been out on your ear and he and I wouldn't still be married.\"\n\n\"I was out anyway,\" he said angrily. \"I never had a chance.\"\n\n\"You had the same chances I did,\" I said coolly. \"If either of us had known what was going on, then both our marriages would have ended in divorce. Because we didn't, yours held together a little longer and mine survived. But yours was on the rocks already, Jock, and you can't blame Sam for that. He was a symptom, not a cause.\"\n\nHe began a rambling defense of his own part in that long-dead relationship. Did I have any idea what it was like to be rejected by someone I loved? Why would he have taken up with Sharon if Libby had shown the slightest bit of interest in him? What did I think it did to a man's self-esteem to have to pay for sex? Of course he hadn't told Sam about her. No man in his right mind would want his friends laughing at him behind his back....\n\nListening to him expose his heartache in that room stuffed with hidden secrets, I was more amused than sympathetic. Was he so blind to his own duplicity that double standards held no embarrassment for him? And why did he think he could trust me with his pain, when mine was older, more monstrous and a great deal crueller? Like Sam, he saw himself more sinned against than sinning, and, like Sam, his belligerence grew as his own guilt paled before the guilt of others.\n\nWhen he finally ran out of steam, I stood up and pulled on my rucksack. \"I wouldn't waste any more time on it if I were you,\" I said kindly. \"It won't change anything, just make you angry.\"\n\n\"If that's what you wanted you should have left me in ignorance.\" He watched morosely as I checked to see I'd left nothing behind. \"Why didn't you?\"\n\n\"I didn't think it was fair.\"\n\nHe gave a mirthless laugh. \"Well, maybe I don't put such a high price on fairness as you do. Did you think about that? Sam and I go back a long way. Maybe I'd have been happier not knowing.\"\n\nI was sure he was right. It was truly said that what you don't know can't hurt you, and Sam and he could have gone on forever, the one lying about his stalwart support of his friend, the other lying about his success. It was also truly said that misery loves company and I laid a quiet bet with myself that Jock\u2014a man not given to suffering in silence\u2014would pick up the telephone after I left and offload some of his misery onto my husband.\n\nIt seemed eminently fair to me\u2014-justice demands a penalty\u2014but whether they would ever speak again was questionable. I wasn't troubled by it. I had waited a very long time for my pound of flesh.\n\n**_Family correspondence\u2014dated 1999_**\n\nCURRAN HOUSE\n\n_Whitehay Road_\n\n_Torquay_\n\n_Devon_\n\nFriday\n\nDearest M,\n\nI can't help feeling Libby is right, and you should rethink these visits on Monday particularly the one to Alan's house. I know Danny's told you Alan won't be there\u2014but do at least consider how he's likely to react when his wife tells him you've taken photographs of what's there. Are you sure it wouldn't be more sensible to involve the police? I know I don't need to remind you of what Alan and his father did to you\u2014it distresses me to see you washing your hands all the time\u2014but I'm not as confident as you that just because Alan's brother doesn't seem to know about his past, his wife won't either.\n\nLove,\n\nDad\n\nX X X\n\n# eighteen\n\nMy last port of call that day was a small 1930s semi in Isleworth with pebble-dashed walls and lattice-style windows. It was too far to walk so I took a taxi from Richmond station and asked the driver to wait in case there was no one at home or the occupants refused to speak to me. I heard a dog bark as I rang the bell, then the door was flung open by a small curly haired boy and a Great Dane came bounding out to circle 'round me, growling. \"Mu-mmy!\" the child screamed. \"Satan's going to bite a lady. Mu-mmy!\"\n\nA plump blonde in a baggy T-shirt and leggings appeared behind him and sent the dog back inside with a click of her fingers. \"Don't worry,\" she said comfortably. \"His bark's worse than his bite.\"\n\nI smiled weakly \"How do you know?\"\n\n\"I'm sorry?\"\n\n\"How many people has he bitten?\"\n\n\"Oh, I see!\" She giggled. \"None. _Yet..._ No, I'm joking. Actually, he's a big a softy. Mind you\"\u2014she ruffled her son's hair\u2014\"how many times do I have to tell you not to open the door, Jason? Not everyone's as easy 'round dogs as this lady and if Satan did bite someone we'd have the police here in no time flat.\" She turned him 'round and steered him toward a door to her right. \"Go and watch Tansy for me. I don't want her sticking her fingers in the sockets again.\" The corners of her mouth lifted in a questioning smile. \"So what can I do for you? If you're a Jehovah's Witness you'll be wasting your time. That's why Satan's called Satan... to scare off the God squad.\"\n\nShe was like a gust of fresh air after the watchful suspicion of Maureen Slater, and I wasn't remotely surprised that Danny preferred her company to his mother's. \"That would be Alan,\" I said.\n\n\"That's right.\"\n\n\"And you're Beth?\"\n\nShe nodded.\n\n\"Alan knew me as Mrs. Ranelagh,\" I said, holding out my hand. \"My husband and I used to live down the other end of Graham Road from his parents when he was a child. I was one of his teachers.\"\n\nShe looked surprised as she returned my handshake. \"Are you the lady Danny was on about? He phoned a couple of nights ago and said he'd met someone who used to teach Al.\"\n\n\"Yes.\"\n\nShe glanced past me toward the taxi. \"He said you were in Dorset.\"\n\n\"We're renting a farmhouse there for the summer. It's about ten miles from where Danny's staying. I'm in London today because there were some people I needed to see\"\u2014I didn't think she'd accept that I'd dropped in on a whim\u2014\"one of whom was Alan.\"\n\nA look of uncertainty crossed her face. \"He went really quiet when Danny mentioned your name... almost like you were Jack the Ripper or something.\"\n\n\"Did he?\" I asked in surprise. \"He always told me I was his favorite teacher. I wouldn't have dreamed of dropping in otherwise.\"\n\nShe looked embarrassed. \"He's not here. He's working on a site out Chertsey way.\" A frown developed. \"I'm surprised Danny didn't tell you. It's one of these executive-type estates... you know, houses with fancy stonework and porches on pillars\u2014and he's been pestering Al for weeks to put his name up for the decorative bits. They're behind with the contract so my poor old boy's working overtime.... most evenings he doesn't get back till 'round 10.\" The frown deepened. \"Anyway, how come you _needed_ to see him? Most of his teachers were glad to be shot of him.\"\n\n\"Me, too,\" I said honestly. \"Most of the time he couldn't be bothered to turn up, and when he did he was so disruptive that I wished he hadn't.\" I smiled to take the sting from my words. \"Then I'd take a deep breath, remind myself of what his father was like, and try again. I couldn't bear to think he'd end up like Derek. And he obviously hasn't if everything Danny's told me about you and the children is true.\"\n\nCuriosity won out, as I hoped it would, because my excuse for being there wasn't enough to persuade her to invite me in. \"I never met his dad,\" she said, a gleam of interest sparkling in her eyes. \"He was long gone before I met Al, but everyone says he was a bastard. Did you know him well?\"\n\n\"Oh, yes. He threatened to rearrange my face once, so I tried to have him arrested.\" I turned irresolutely toward the taxi. \"I asked the driver to wait in case you weren't at home, but I think he's left the clock running.\"\n\n\"Fuck that for a load of bananas,\" she said cheerfully. \"They're all rip-off merchants... charge you an arm 'n'a bloody leg just to look you. Excuse my French. How about I give you a cup of tea and we'll call for a minicab later? If you're lucky, Al might get home early for once. I mean, it's not every day one of his teachers turns up\"\u2014she canted her head to one side\u2014\"though you don't look much like any of the old bats who taught me, and that's a fact.\"\n\nWith a grateful smile for the offer of tea and the compliment\u2014and a silent, earnest prayer that _nothing_ would induce Alan to come home early\u2014I paid off the cab and followed her inside. As I might have predicted, the interior was a reflection of Beth's down-to-earth character. Colors were simple and direct\u2014terra-cotta and straw being her obvious favorites. Floor coverings were practical\u2014sanded floorboards in the hall and cork in the kitchen\u2014and all her furniture was arranged to maximize space and minimize accidents for her children. It worked, and it was attractive, and when I told her so she was pleased but not surprised.\n\n\"It's what I want to do when the kids are both at school,\" she said, \"take over someone's house and make it nice for them. I reckon I've got a talent for it, and it seems a shame to work in a factory if I can make money out of something I enjoy. I do it all myself-\u2014Als too tired to be sanding floors when he gets home\u2014and most of my mates go green when they come visiting. Half of them think women aren't made for this kind of caper, and the other half say they'd be too embarrassed to go to the hire shop for tools like sanding machines and wallpaper strippers because they wouldn't know what to ask for.\"\n\nI skirted cautiously around the Great Dane, which had stretched itself full length on a fluffy rug in front of the cooker. \"What did you do before you married Alan?\" I asked, pulling out a kitchen chair and straddling it. The dog raised its head with a hostile look in its eyes, then, at a click from his mistress's fingers, yawned and went back to sleep.\n\n\"I was a hairdresser,\" Beth said with a laugh, \"and I hated every minute of it. I was supposed to be a stylist but the only styling I ever did was blue rinses for miserable old women who had nothing better to do with their time than whinge about their husbands. And it didn't seem to make much difference whether the poor old bastards were dead or alive, they still got the treatment. Yackety... yackety... yack. He's mean.... He's stupid.... He dribbles on the toilet seat.... _Honestly!_ It didn't half put me off getting old.\"\n\nI laughed. \"It sounds like my mother.\"\n\n\"Is she like that?\"\n\n\"A bit.\"\n\n\"I never knew my mum,\" said Beth, pushing an armful of bangles up to her elbow as she carried the kettle to the sink and turned on the water. \"Not the biological one anyway. She put me up for adoption when I was a baby. My adoptive mum's great... so is my dad... they love Al and they love the kids. They asked me once if I wanted to go looking for my real mum, and I said no chance. I mean, there's no guarantee I'm going to like her\u2014half the people I know can't stand their parents... so why waste time looking for her?\"\n\nI didn't say anything.\n\n\"You think I'm wrong?\"\n\n\"Not at all,\" I said with a smile. \"I was thinking what a levelheaded woman you are and how lucky Alan is to have married you.\" I was also thinking of something that an educational psychologist had once written about Alan.... _He should be encouraged to form strong and positive bonds with adults.... He needs to feel valued...._ \"You've obviously been good for Danny as well.... He certainly talks about you very fondly.\" It was sincerely meant and her cheeks flushed pink with pleasure. \"What about their sisters?\" I asked. \"Do you see much of them?\"\n\nThis must have been a thornier subject because her frown promptly returned. \"The last time we saw them was at Tansy's christening and that's three years ago now. Al said we'd give it one more try, so we invited them, and they promptly lammed into me and Al as per usual... ruined poor little Tan's party... and we thought, to hell with it, life's too short for this kind of aggro.\" She swept some breadcrumbs from the table into her cupped hand and I watched in fascination as the bangles rattled down to her wrist. \"Al says it's jealousy because we're doing okay and they're not\u2014one of them's got four kids and no bloke because he buggered off when she fell pregnant the last time\u2014the other's got five kids by different dads and two of them are in care.\"\n\n\"Where do they live?\"\n\n\"A big estate near Heathrow.\"\n\n\"Together?\"\n\n\"Neighboring blocks. The kids run around in a gang and terrorize the old people living there. I hate to think how many police cautions they've had. Someone told me the other day that the council's planning to take out injunctions to force Sally and Pauline to keep them inside... but I don't know if it's true. The worst thing is, they've been trying to get on the housing list here and I said to Al we'd have to move if it happened because there's no way I'm going to let Jason and Tansy get sucked into trouble by their cousins.\" She poured the tea and, like her mother-in-law, added milk automatically. \"Al says it's not really his sisters'fault,\" she went on, passing me a cup, \"not when you look at the kind of upbringing they had, but as I keep telling him, if that's true then he and Danny should be as bad.\"\n\nShe reminded me of Julia Charles, our neighbor in Graham Road, who had agonized in the same way about the terrible influence Alan Slater and Michael Percy would have on her children if she ever let them play in the street. They're such awful boys, she used to say, and it's not as though it's a \"class\" thing, or not really. It's their parents who're to blame. If their mothers spent more time with the kids and less time on their backs or on the bottle, then the kids would be better behaved. Everyone knew that.\n\n\"It sounds as if they've reversed their roles,\" I said slowly. \"The two girls always seemed quite sensible when they were children. Either that or they were too terrified of their father to put a foot wrong. They used to follow in Alan's wake sometimes, but never further than the end of the road. They were both small and dark like their mother and Sally's the older of the two?\" She nodded. \"They were friendly with two other little girls of about the same age\u2014Rosie and Bridget Spalding\u2014and used to play hopscotch on the pavement. Bridget married Alan's friend, Michael Percy, and moved down to Bournemouth, but I've no idea what happened to Rosie.\" I raised a questioning eyebrow but Beth shook her head as she perched on the edge of a worktop and cradled her cup between her hands.\n\n\"Al hasn't kept up with anyone from Graham Road,\" she said. \"He goes to see his mother once in a while, but never for very long because it makes him so depressed. Left to himself, he wouldn't go at all... but I keep saying, he has to set a good example to Jace and Tan... I mean, I'd die if they never came to see me after they were grown.\" She had pale lashes and pale eyebrows, which gave her face a bland look until she screwed one of her many expressions onto it. Now she made a grimace of irritation. \"She doesn't make it easy, though. All she ever does is complain about how lonely she is and how miserable she is. It's a vicious circle. If she put herself out to be pleasant, he'd probably drop in more often.... Instead, he delays as long as he can, then goes out of guilt.\"\n\n\"Do you and the children visit her?\"\n\nMore grimacing. \"We did till Jason found her Prozac and ended up in the hospital. I was that mad with her. It's not as though she needs the stupid things.... Half the time she doesn't take them.... It's just a way of getting invalidity benefit so she can sit at home and watch the telly all day. I wouldn't mind so much if I hadn't asked her to keep them out of harm's way, but it's like talking to a brick wall. She smokes and drinks around Jace and Tan, takes no notice of what I might feel about it, then has the nerve to tell me she doesn't know what the fuss is about. 'It didn't do _my_ kids any harm,'she says.\"\n\nI laughed. \"I used to get it about disposable nappies. I made the mistake of telling my mother what they cost and she lectured me for months about how I was wasting my money. 'What's wrong with terry toweling?'she kept saying.... 'If it was good enough for you, then it's good enough for your boys.'\"\n\nShe sipped at her tea. \"You don't like her much, do you?\"\n\nThe directness of the question took me aback, if only because it was something I never asked myself. \"Rather more than you like Maureen, I suspect.\"\n\n\"Yes, but Maureen's not my mum,\" she said despondently. \"It doesn't half worry me. I don't like falling out with people, but the way Alan's lot behave we won't be talking to any of them soon. I have nightmares sometimes that it's in the genes, and that my kids'll storm off after a huge row, and me and Al will never see them again.\"\n\n\"I'm sure that won't happen,\" I consoled her. \"If behavior was inherited my two would have upped stumps and left long ago. But they're so laid-back it'll take a stick of dynamite to shift them. Either that or a stunning blonde with a Ferrari.\"\n\nShe eyed me thoughtfully. \"Maybe they got their genes from their father,\" she suggested.\n\n_More likely their grandfather_ , I thought, while thinking it was hardly a good time to remind Beth of the genetic link between her children and Derek. \"Except I agree with Alan that upbringing has more to do with it,\" I answered. \"Jason and Tansy are the sum of their genes and experiences, not the sum of their genes alone, otherwise they'd be virtually indistinguishable from each other. You made the point yourself when you said how different Alan and Danny are from their sisters.\" _And how different Alan seemed now from the boy I once knew_ , I thought wryly.\n\n\"They're also very different from each other,\" she said. \"Danny's a bit of a goer, but Al behaves like he was born middle-aged.\" She giggled and her face lit up immediately. \"Jace said 'fuck'the other day because he heard it at nursery school and Al spent the next two hours worrying about whether it was his fault. I said, 'Don't be so fucking stupid....'Excuse my French.... And he said, 'It's all very well for you to laugh but the only time my dad showed me any attention was when he said, \"Fuck off, bastard.\" ' And now he's hoping he _is_ a bastard and Derek isn't his dad.\"\n\n\"I'd probably feel the same in his shoes,\" I said. \"It's a bit like owning up to having Ivan the Terrible for a father.\"\n\nShe was agog with curiosity. \"You said he threatened you. Why? What happened to make him angry?\"\n\nI was tempted to be honest, and not just because I liked her and felt guilty about using her. She was one of those rare people, irrespective of age, sex or background, whose straightforward, open personality demanded, and deserved, a reciprocal trust. Indeed, if I had any sadness at deceiving her, it was because I knew that in different circumstances I would be only too pleased to have her as an ally.\n\n\"We had a row in the street about the way he was treating Alan and he pushed my arm up behind my back and said if I ever interfered again he'd wipe the smile off my face.\" _It's not a complete lie_ , I thought. The place was wrong, and the threat\u2014which had nothing to do with my smile\u2014was inflicted anyway, but Derek had certainly told me never to interfere again. \"So I did what any sensible person would do and reported him to the police,\" I told her, \"but they didn't believe me and repeated what I'd said to Derek.\"\n\nIf I were telling it as it really was, I would have added that I was betrayed twice in as many days by the same policeman and reaped a double dose of Derek's anger in consequence. But I wanted to woo Beth with amused indifference, not frighten her off with evidence of her father-in-law's savagery.\n\nHer eyes widened. \"What did he do?\"\n\n\"Nothing much,\" I lied. \"He was a typical bully\u2014bluster on the outside and blancmange inside.\" I paused. \"Danny told me he disappeared after Alan gave him a thrashing with a baseball bat?\" I put an upward inflection into my voice, and Beth nodded. \"So where did he go? Does anyone know?\"\n\n\"Al doesn't talk about him much, except to say he doesn't want him anywhere near our kids. I know he went to prison because Sally got his address off a con who'd done time with him. It was before Tan's christening and she kept pestering us to invite him. She said he was back in London and wanted to meet up with his family again\"\u2014she shrugged\u2014 \"but Al said if he showed his face here he'd get another thrashing, and that's what caused most of the aggro at the party. Sally and Pauline said Derek was skint and needed help, and Al said he could die of starvation before he'd lift a finger for him.\"\n\n\"Weren't you worried he'd come anyway?\"\n\nShe glanced at the dog. \"That's why Al got Satan. He wanted a Rottweiler but I said it'd be too dangerous with kids. Mind, I thought it was money down the drain at the time.\" She flexed the muscles in her right arm. \"I reckon I'd be able to deal with Derek... no trouble... if he dared push his way in here, but Satan's sort of grown on us and I wouldn't be without him now.\"\n\nI wanted to warn her against complacency, but instead I murmured, \"Still... it's not surprising Alan was worried, particularly if Derek was close.\"\n\n\"Not _that_ close. Sally said he was shacked up with some bird out Whitechapel way.\"\n\n\"He was lucky to find someone.\"\n\n\"Too right. I said the bird needed her bloody head examined\u2014unless, of course, he hadn't bothered to tell her he was a wife-beater\u2014and Sally got miffed and said I shouldn't spread rumors about people I'd never met. So I said, 'I'll remind you of that when he clobbers this one.'\"\n\nI smiled. \"What did Maureen say?\"\n\nBeth grinned back. \"Said it was a pity Derek hadn't died of drink years ago and that the two girls deserved everything they got if they let him swan back into their lives just because he was family. She got well worked up, said he'd done his best to ruin their lives when they were children and if they had any sense they'd keep well away from him now.\"\n\n\"Better late than never, I suppose,\" I said dryly. \"She didn't do much to protect them when they were living with him.\"\n\nA thoughtful expression creased Beth's forehead and I wondered if I'd displayed my prejudice a little too blatantly. \"Except I reckon she was just as bad. It was her who bought the baseball bat, you know... not to use against Derek... but to beat her kids about the head whenever they annoyed her.\"\n\n\"How do you know?\"\n\n\"Danny was teasing Al one day about being slow on the uptake. He said it was because his mum had addled his brains with the bat.\"\n\n\"Was she strong enough?\" I asked doubtfully.\n\n\"According to Danny she was. He said she was like a wild animal when she was in a temper, and you either scarpered or locked yourself in the toilet till she calmed down.\" She watched my frown of disbelief and gave a small shrug. \"I couldn't swear on oath that it's true\u2014Danny's always telling fibs\u2014but it was pretty convincing at the time. Let's put it this way, Al didn't deny it\u2014just told me never to lift a finger against Jace and Tan or I'd be answering to him. So I said, 'You've gotta be joking! Since when have I ever raised my hand in anger to anyone?'\" She grinned suddenly. \"And I told him straight, it's fucking rich coming from a bloke who calls his dog Satan and reckons the only way to discipline him is to take a rolled-up newspaper to his backside.\" She blew an air kiss toward the animal, which raised its head immediately and thumped its tail on the floor. \"I mean, what kind of way is that to train a dog when he'll do anything as long as you give him a biscuit?\"\n\nSatan and I examined each other cautiously. \"He's a good guard dog,\" I murmured. \"I wouldn't fancy my chances much if I were Derek.\"\n\n\"He'd rip his throat out soon as look at him,\" said Beth. \"I used to tie Satan to the pram outside shops when the kids were smaller. He growled at anyone who came closer than six yards, which meant I could do my shopping in peace without worrying about someone stealing my babies.\"\n\n\"Amazing! And he does all that for a biscuit?\"\n\nHer grin broadened. \"Don't knock it,\" she said. \"It's a damn sight more effective than beating the poor brute with a newspaper. That just made him vicious.\"\n\n\"Mm.\" Ripping throats out seemed fairly vicious to me, and I wondered how he'd react if I stood up unexpectedly. I glanced at my watch. \"I really ought to be going,\" I said, putting reluctance into my voice. \"It's a long journey back to Dorchester and Sam will be wondering where I am.\"\n\n\"Al'll be sorry to have missed you.\"\n\nI nodded. \"It's a shame. I'll phone first another time.\" I finished my tea and stood up. \"Can I say good-bye to the children?\"\n\n\"You surely can. They're in the sitting room. I'd be interested what you think of it.\" She pointed to the floor as a growl rose in Satan's throat and he subsided immediately.\n\n\"So when does he get the biscuit?\" I asked, following her into the hall.\n\n\"When I feel like it. That's why he does what I tell him. He's never too sure when the moment will come.\"\n\n\"Does it work with husbands and children as well?\"\n\nShe flattened her palm and made a rocking gesture. \"It depends what the treat is. Biscuits don't work so well on Al. He likes basques and black stockings better.\" She grinned as I gave a splutter of laughter. \"The kids are in here,\" she said, opening a door. \"You'd better like it because it took me two months to finish. I'll phone for a cab while you're looking.\"\n\nI did like it, although it was totally out of keeping with a 1930s semi. It can't have been bigger than five meters square, but it was decorated in Mexican style with an arched ceiling, tessellated floor, roughly stuccoed walls and an ornate bronze candelabra hanging from the ceiling. French windows opened out onto a tiny patio, and a huge rococo mirror, with a myriad of tilted facets in a scrolled gilt frame, reflected the light in indiscriminate dazzling shafts across every available surface. Even the fireplace had been transformed into something that would have been more at home on a ranch than in a back street in Isleworth, with a brass artillery shell laden with silk flowers standing in the hearth. I wondered why she'd done this room so differently from the rest.\n\n\"It's all fake,\" said Jason from the corner where he and his sister were watching television. \"Mum just painted it to make it look real.\"\n\nI tapped my foot on the tessellated floor and listened to the hollow ring of wood. \"She's a clever lady,\" I said, touching a hand to the rough stucco and feeling the smoothness of plaster. \"Did she make the mirror as well?\"\n\n\"Yup. And the candy-laber.\"\n\n\"What about the picture?\" I asked, gazing at the Quetzalcoatl mosaic on the wall.\n\n\"That's Dad's.\"\n\n\"The sofa and chairs?\"\n\n\"Ten quid, job lot, from a junk shop,\" said Beth proudly behind me. \"And five quid for the patchwork throws. I begged, borrowed and stole material... dresses... old curtains... tablecloths... whatever... from everyone I knew. The five quid went on the reels of cotton to do the sewing. What do you think?\"\n\n\"Brilliant,\" I said honestly.\n\n\"But a bit OTT for Isleworth?\"\n\n\"A bit,\" I agreed.\n\n\"That's what Al thinks, but all I'm doing is setting out my stall. I can create any image you want, and I can do it for peanuts. This whole room cost under three hundred quid. Okay, it doesn't count my time, but you wouldn't believe how many of my friends say they'd pay me a tenner an hour to do it in their houses.\"\n\n\"I bet they would,\" I said dryly. \"They're probably paying their cleaners as much just to Hoover their floors.\"\n\nShe looked crestfallen. \"Al doesn't want me to do it at all, says he won't even think about it unless I ask a hundred per hour minimum.\"\n\n\"He's right.\"\n\n\"Except none of my friends can pay a hundred quid per hour.\"\n\nI gave her hand a quick squeeze. \"It's a bad mistake to work for friends,\" I said. \"You should photograph each room and put a portfolio together, then go out and sell yourself.... Get some fliers printed... Take out ads in the local newspaper. You're way too good to work for \u00a310 an hour.\" I patted my rucksack. \"If you like, I'll take some photographs now and send them to you. I've got my camera with me, and I'd love my husband to see what you've done. We're toying with buying the farmhouse we're renting, and you never know\"\u2014 _How can you be such a bitch?_ I asked myself\u2014\"maybe I can persuade Sam that you're the interior decorator we need.\"\n\nHer face flushed pink with pleasure again. \"If you're sure.\"\n\n\"Of course I'm sure.\" I squatted down beside Jason and Tansy. \"Would you two like to be in the pictures?\" They nodded solemnly. \"Then how about we turn off the telly and you sit on Mum's sofa, one at each end? It might be better if you stood behind me,\" I told Beth as I sat cross-legged in front of the windows and lined up the shot. \"You're blocking the mirror.\"\n\nShe scurried onto the patio. \"I hate having my picture taken. I always look so fat.\"\n\n\"It depends how they're done,\" I said, as I snapped off half a dozen shots of the sofa-side of the room before zooming in on the Quetzl-coatl. \"Why don't you sit on one of the chairs with the kids on your lap, and I'll see if I can get a view of the fireplace with the three of you to the left?\"\n\nI should have choked on my own duplicity, instead I marveled at how easy it was to cajole her into letting me make a record of everything in the room, including the bangles on her wrist and a collection of small china cats at one end of the mantelpiece. \"Who's the cat lover?\" I asked, as I tucked my camera back into my rucksack, when the doorbell rang to announce the arrival of the minicab.\n\n\"Al. He bought them at a jumble sale years ago.\" She jumped the children off her lap and stood up. \"You never said why you needed to see him,\" she reminded me, as we went back into the hall.\n\n\"I wanted to talk to him about Michael Percy,\" I lied, dredging up the only excuse I'd been able to come up with. \"But you've already told me they lost touch\"\u2014I gave a rueful shrug\u2014\"so he wouldn't have been able to help me anyway.\"\n\n\"What did you want to talk about?\"\n\n\"Whether Michael's as bad as he was painted in the newspapers,\" I said, pulling open the front door and nodding to the cab driver to say I was coming. \"I'm thinking of visiting him in prison\u2014he's just down the road from us on Portland\u2014but I'm not sure if it's a sensible thing to do. I rather hoped Alan could give me some advice.\"\n\nIt sounded so weak to my ears that I expected suspicion to bristle out of her like hackles, but she seemed to find it reasonable. \"Well, if it's any help, Al said it was well out of character for him to hit that woman. He reckons Michael was a lot less violent than he was when they used to hang out together. They had a fight before they fell out, and Al said Michael took a beating because he wouldn't defend himself.\"\n\n\"What were they fighting about?\"\n\n\"That girl you mentioned\u2014Bridget. It was when they were in their late teens. Al was so crazy about her he wanted to marry her, then he walked in one day and found her in bed with Michael. He went berserk... broke Michael's jaw and God knows what else... even attacked the policemen who arrived to break it up. It was mayhem, apparently. Bridget was screaming in the hall, Michael was half out the window and it took four policemen to get Al off him. He ended up in juvenile detention for it.\"\n\n\"Goodness!\"\n\n\"He's been straight ever since,\" she assured me.\n\n\"I should hope so.\"\n\nBeth laughed. \"It all worked out for the best. He wouldn't be married to me if he'd stuck with her.\" A wistful note entered her voice. \"But he's never broken anyone's jaw for me... so I guess I'm not as attractive as Bridget.\"\n\nI gave her an impulsive hug before heading for the cab. \"Just don't test him,\" I warned over my shoulder. \"I have a nasty feeling he'd break more than jaws if he found you in bed with someone else.\"\n\nI spoke lightly, but the warning was sincere.\n\n**_Letter from Dr. Joseph Elias, psychiatrist  \nat the Queen Victoria Hospital, Hong Kong\u2014dated 1985_**\n\n**QUEEN VICTORIA HOSPITAL**\n\n_Hong Kong  \nDept. of Psychiatry_\n\n**Mrs. M. Ranelagh  \n12 Greenhough Lane,  \nPokfulam**\n\n**June 12, 1985**\n\n**Dear Mrs. Ranelagh** ,\n\n**I am sad to hear you're leaving Hong Kong. I have enjoyed your letters and those all-too-rare occasions when you have consented to talk to me in person! You will like Sydney. I spent two years there from '72 to '74 and it was a delightful experience. Australia has the enthusiasm and vigor that comes from a mix of different cultures and I'm confident you will enjoy a polygeny where class divides are nonexistent and success depends on merit and not labels. You see, I have come to understand you**.\n\n**You mentioned in your last letter that you and Sam have reached a fine understanding where the past remains buried in England. You also tell me he's an excellent father. You do not, however, say you love him. Am I supposed (like Sam?) to take that for granted? My friend the rabbi would say that nothing thrives in a desert. He would also say thatwhatever lies buried in England will resurface the minute you go home. But perhaps that's the plan? If so, you are a patient woman, my dear, and a little cruel, too, I think**.\n\n**With best wishes for your future wherever it may be**.\n\n**Yours affectionately** ,\n\n**Dr. J. Elias**\n\n# nineteen\n\nSam was sitting in the car outside Dorchester South station when I finally reached it at ten o'clock that evening. I wondered how long he'd been waiting because I hadn't phoned to say which train I was catching, and I feared it couldn't have done his temper much good if he'd been there any length of time. My intention had been to take a taxi home and face the inevitable row behind closed doors but, if his bleak expression when he got out of the car at my approach was anything to go by, he planned to have it in public.\n\n\"Jock phoned,\" he said tersely.\n\n\"I thought he might,\" I murmured, opening the back door and dumping my rucksack on the seat.\n\n\"He told me you left him at about four o'clock. What the hell have you been doing? Why the hell didn't you phone? I've been worried sick.\"\n\nI showed my surprise. \"I said I'd make my own way home.\"\n\n\"I didn't even know if you were _coming_ home.\" He stalked angrily round the bonnet to open the passenger door for me, but it was so out of character that I stepped back automatically, assuming he was opening it for himself. \"I'm not going to hit you,\" he snapped, gripping me by the arm and pressing me clumsily into my seat. \"I'm not a _complete_ bastard.\"\n\nHe slid in behind the steering wheel and we sat for several minutes in silence. The tension in the confined space was palpable, although I had no idea if it was due to anger over my perfidy or concern at my late return. The station was virtually empty at that time of night, but one or two people peered curiously through our windows as they passed, presumably wondering why the two dimly seen occupants were sitting so stiffly and refusing to look at each other.\n\n\"Aren't you going to say anything?\" he asked at last.\n\n\"Like what?\"\n\n\"Explain?\" he suggested. \"I still can't believe you'd talk to Jock, and not to me. Why didn't you tell me Annie was beaten up? You know I'd have come clean if I'd realized how serious it all was.\"\n\n\"When?\"\n\n\"What do you mean, when?\"\n\n\"When would you have come clean?\" I asked evenly. \"I told you at the time what PC Quentin said about the bruising\u2014but you just said we were talking bollocks. As I recall your comment was, 'Since when did a neurotic bitch and a disgruntled policeman know the first damn thing about pathology?'You could have told me the truth then and given me and Andrew Quentin a fighting chance against Drury... but you didn't.\"\n\nHe dropped his head into his hands. \"I thought you were wrong,\" he muttered. \"I was pretty stressed out at the time, and you didn't make it easy for me.\"\n\n\"Fine. Then you've nothing to feel guilty about. You were saving me from myself. No one's going to blame you for that.\" I looked impatiently at my watch. \"Can we go now? I'm hungry.\"\n\n\"You're not making this very easy for me,\" he said. \"You must know how awful I feel.\"\n\n\"Actually, I don't,\" I said honestly. \"You've never felt awful before. That year, 1978, was one of those little unpleasantnesses\u2014like where the cutlery drawer is and how to boil eggs\u2014that you manage to erase so successfully from your memory. I've always envied you for it, and if you're troubled now it's probably just a reaction to knowing you've been rumbled. It'll pass. It usually does.\"\n\nHe tried a different tack. \"The boys are twitched as hell,\" he said. \"They keep asking me what I've done that's so bad you'd want to run away.\"\n\n\"Oh, for Christ's sake!\" I said bluntly. \"If you want to make me angry, then hiding behind your children is the surest way to do it. Luke and Tom know damn well I don't run away from things. They also know I wouldn't abandon them unless I was on a life-support machine somewhere. In any case, I told them I wouldn't be home until late so I imagine they're lying in front of the telly, as per normal, wondering why their father has suddenly gone 'round the bend.\"\n\n\"We had a row,\" he admitted. \"I told them they were unfeeling bastards.\"\n\nI didn't bother to comment because I wasn't in the mood to massage his bruised ego. \"Look,\" I said, tapping my watch, \"I haven't had anything to eat all day and I'm starving, so can we either go home or get a takeaway? Have you and the boys eaten?\"\n\n\"Tom made some spag bol for him and Luke, but I wasn't hungry.\"\n\n\"Good, then we'll have a curry.\"\n\n\"Why didn't you eat on the train?\"\n\n\"Because it was trolley service,\" I said crossly, \"and the only thing left to eat by the time it reached me was a packet of dry biscuits. So I had some wine instead... and now I'm fighting mad and in no mood to play silly buggers with you or anyone else.\"\n\n\"I don't blame you,\" he began self-pityingly as he fired the engine. \"I just wish there was something I could do or say\u2014\"\n\nI cut him off. \"Don't even think of apologizing,\" I said. \"As far as I'm concerned you can grovel to me for the rest of your life. And it won't make a blind bit of difference. It'll make a difference to Jock, though. The sorrier you are the happier he'll be, and you'll be back in each other's pockets before you know it.\"\n\nHe mulled this over quietly as we turned on to the main road. \"I've already apologized to Jock.\"\n\n\"I assumed you would.\"\n\n\"He calmed down pretty quickly as a matter of fact, once I'd explained what a mistake the whole damn thing had been.\"\n\n\"Okay.\"\n\n\"It didn't mean Jack shit, you know... just something that happened while you were away. The trouble is, Libby took it more seriously than I did. She and Jock weren't getting on too well at the time and it sort of ran out of control.\" He paused, inviting me to say something. When I didn't he went on, \"Jock understands that. He's been there himself, knows what it's like to be caught between a rock and a hard place.\"\n\n\"Okay.\"\n\n\"Does that mean you understand?\"\n\n\"Of course.\"\n\nHe flicked me an uneasy glance as he turned left at a pelican crossing. \"You don't sound as if you do.\"\n\nI sighed. \"I'm your wife, Sam, and I've known you since I was twenty. If I don't understand you by now then I doubt I ever will.\"\n\n\"I didn't mean, do you understand _me_. I meant, do you understand how the thing with Libby happened? What a fucking disaster it was? How sorry I was afterward?\"\n\nI gave a small laugh. \"The thing? Do you mean your _affair?_ The time you rogered your best friend's wife because your own wife was away and you hadn't had sex for twenty-four hours?\"\n\n\"It wasn't like that,\" he protested.\n\n\"Of course it wasn't,\" I agreed. \"It was Libby's fault. She caught you at a low ebb, plied you with drink, then persuaded you into a quickie on the kitchen floor. Afterward, you found yourself in an impossible position. _You_ regretted it intensely and hoped it was a one-off. _She_ loved every minute of it and looked on it as the beginning of a great love affair.\" I watched him for a moment. \"I should imagine Libby's version is a little different\u2014 _you_ seduced _her_ in other words\u2014but the truth probably lies somewhere in the middle.\"\n\n\"I knew you'd be angry,\" he said unhappily. \"That's why I never told you.\"\n\n\"Now you're flattering yourself,\" I said. \"It's probably a huge disappointment but the only emotion I have ever experienced re you and Libby is indifference.\" Of course I was lying... but he _owed_ me _... I_ had honored my promises... and _he_ hadn't. \"If I'd been able to work up the energy to feel angry, I think you'd have realized something was wrong. Certainly Libby would, but then she's a woman and women are better at picking up vibes.\"\n\nHe pulled up in front of the Indian restaurant. \"Wasn't it her who told you about us?\"\n\n\"No. I suspect she's even more embarrassed than you are. We're hardly talking Abelard and Heloise in all conscience.\"\n\nHe clamped down on his anger. \"Who then?\"\n\n\"You.\" I smiled at his expression. \"One night in Hong Kong. Not in so many words.... You weren't that drunk... but you said enough for me to put two and two together. It was quite a relief, actually. I remember thinking, _So that's what this has all been about\u2014a grubby little affair with Libby Williams_. I even laughed about it afterward. I kept picturing you and her working up a sweat in Jock's bed while he was out getting blow jobs off the Graham Road tart. There was such a sweet irony about it\u2014you being piggy in the middle of a couple of predators. It explained everything. Your unpleasantness... your lies... your dash to leave England. I even felt sorry for you in a funny kind of way because it seemed so obvious you'd sold your soul to the devil for something you hadn't enjoyed very much.\"\n\nHe shook his head in bewilderment. \"Why didn't you say something?\"\n\n\"I couldn't see the point. We were on the other side of the world. All I'd have been doing was closing the stable door after the horse had bolted.\"\n\nSam wasn't designed to remain humble for long. \"Do you know what this feels like? It feels like I'm married to a stranger. I don't even know who you are anymore.\" He propped his elbows on the steering wheel and ground his knuckles into his eyes. \"You always tell people what a great marriage we have... what great kids we have... what a great father I am. But it's all just crap... one huge pretense at happy families when the truth is you hate my guts. How could you _do_ that? How could you be so bloody devious?\"\n\nI reached for the door handle. \"The same way you did,\" I said lightly. \"Closed my eyes to what a bastard you'd been and pretended none of it had ever happened.\"\n\nHe agonized over my indifference while we waited for the curry, almost as if I'd thrown doubt on his manhood by refusing to take his infidelity seriously. For myself, I was wondering when he was going to realize that the bone of contention was Annie, not Libby and how he would explain that when he did. We took seats in a corner and he muttered away in an undertone, afraid of being overheard, although my refusal to lighten his burden with sympathetic comments meant his tone became increasingly\u2014 _and to my ears sweetly_ \u2014strident.\n\nHe didn't want me to get the wrong impression.... It wasn't true that he'd tried to pretend nothing had happened... more that he'd been terrified of losing me.... Of course he'd have admitted to it if I'd asked but it seemed more sensible to let sleeping dogs lie.... He knew I probably wouldn't believe him, but he _was_ drunk the night Libby seduced him and the whole thing _did_ turn into a total nightmare.... It was absolutely correct to describe Libby as a predator.... She was one of those women who thought the grass was always greener on the other side.... He remembered how shocked he'd been when he realized how jealous she was of me and how determined she was to bring me down to her size....\n\n\"When I told her I wanted to end it, she said she was going to tell you what a rat you'd married,\" he said grimly \"I know it's not much of an excuse but I honestly think I'd have killed her if she'd actually done it. I loathed her so much by then I couldn't be in the same room with her without wanting to strangle her.\"\n\nI believed him, not just because I wanted to but because he'd never been able to mention Libby's name without prefacing it with \"that bitch Jock married.\" There was a brief period when I wondered if he said it out of regret because he, too, had been rejected but I soon realized that the antipathy was real and that Libby was as irrelevant to him as the women he'd slept with before we married. That's not to say I wouldn't have clawed his eyes out if I'd known about the affair at the time\u2014objectivity needs time and distance to develop\u2014but to come across it when the ashes were cold was a reason for private grief only, and not for a fanning of the embers.\n\n\"You don't need to do this,\" I said, glancing toward a nearby customer who had one ear cocked to everything he was saying. \"Not unless you insist on washing your dirty linen in public. Libby's a dead issue as far as I'm concerned.\" I lifted one shoulder in a careless shrug. \"I've always assumed that if you'd loved her you'd still be with her.\"\n\nHe brooded for a moment in offended silence, his gaze fixing abstractedly on the eavesdropper. \"Then why tell Jock about it? Why get everyone worked up if it's all so unimportant to you?\"\n\n\"Not all of it, Sam. Just Libby. I couldn't give a shit what you did to her... but I _do_ give a shit what you did to Annie. You left her to die in the gutter then labeled her a drunk in case anyone accused you of neglect. That's the issue. And, as usual, you're busting a gut to avoid it.\" I paused. \"I know you saw her there\u2014and not just because Jock confirmed it this afternoon\u2014but because you always get so angry every time her name's mentioned.\"\n\nHe wouldn't meet my eyes. \"I thought she was drunk.\"\n\n\"What if she was? It was freezing cold and pouring with rain and she needed help, whatever state she was in.\"\n\n\"I wasn't the only one,\" he muttered. \"Jock and that woman ignored her, too.\"\n\nIt was hardly an answer but I let it go. \"They never got as close as you did,\" I said. \"I was watching them.\"\n\n\"How do you know how close I got?\"\n\n\"Jock said you told him Annie was reeking of drink, but I didn't smell anything until I stooped down to rock her shoulder.\" I watched him curiously for a moment. \"And it wasn't drink I smelled either, it was urine, and I don't understand how you could mistake that for alcohol.\"\n\n\"I didn't. All I told Jock was that she reeked to high heaven. He assumed it was alcohol.\"\n\n\"Did you recognize it as urine?\"\n\n\"Yes.\"\n\n\"Oh, my God!\" I slammed my palms on to the table. \"Do you know that every time I told Drury to question why her coat was reeking of piss he told me her neighbors said it was _normal..._ that she was filthy and disgusting and _always_ stank.\"\n\nAbruptly he dropped his head into his hands. \"I thought it was funny,\" he said wretchedly. \"Your good cause for the year... Mad bloody Annie... wetting herself on your doorstep because she was too drunk to control her bladder. I went into the house and spent the next ten minutes laughing about it until I realized you were the most likely person to find her. Then I knew you'd bring her inside and clean her up and I thought, this is the day my marriage goes down the drain.\"\n\n\"Why?\"\n\nHe breathed hard through his nose. \"She knew about Libby\u2014I think she must have seen us together at some point because she kept sneaking up behind me in the road and calling me 'dirty man.'\" He forced out the words as if his life depended on it. \"'Have you been fucking the tart today, dirty man.''Is that the tart I can smell on you, dirty man?''What do you want with trash, dirty man, when you've got a pretty lady at home?'I loathed her for it because I knew she was right and when I smelled her in the gutter\"\u2014he faltered painfully\u2014\"when I smelled her in the gutter, I kicked her and said, 'Who's dirty now?'\"\n\nI watched a tear drip through his fingers onto the table.\n\n\"And I've been in hell ever since because I so much wanted to take it back, and I've never been able to.\"\n\nI watched a waiter come out of the kitchen and hold up a shopping bag to signal that our curry was ready, and I remember thinking that fate was all about timing. If I hadn't been at a parents'evening that night... if Jock had abandoned the pub at 8:30 when Sharon didn't show... if food didn't arrive at inopportune moments...\n\n\"Let's go home,\" I said.\n\nTwo days later, Maureen Slater phoned. She was angry and suspicious because Alan had told her I'd taken photographs of his house, and she demanded to know what my side of the trade was going to be. I repeated what I'd told her on Monday, that if she wasn't prepared to tell me what she knew, I would pass the Chiswick jeweler's affidavit to Richmond Police... and, for good measure, the shots of the Mexican artifacts in Alan's sitting room. \"No one would doubt they were thieves,\" I said. The only question would be, were they murderers, too? She told me some of what I wanted to know, but rather more interesting was what she chose to leave out.\n\n**_Letter to Sergeant James Drury\u2014dated 1999_**\n\n**Leavenham Farm**\n\n**Leavenham**\n\n**Nr Dorchester**\n\n**Dorset DT2 XXY**\n\n4:30 A.M.\u2014Friday, August 13, 1999\n\nDear Mr. Drury,\n\nOne of the downsides of finding Annie dying was that my sleep patterns were shot to pieces, and I count myself lucky now if I manage a four-hour stretch without waking. I've always hoped that an uneasy conscience has kept you similarly awake over the years, but I suspect it's misplaced optimism. To have a conscience at all means a man must question himself occasionally, and even in my wildest dreams I've never been able to picture you doing that.\n\nI already know you will be absent when I leave this letter and enclosures at the Sailor's Rest, but it seems only fair you should have time to consider your response to the outstanding issue between us. I have, after all, had twenty years to consider mine.\n\nYours sincerely,\n\n# twenty\n\nDrury was watching for me when I came through the door of the Sailor's Rest at half past ten that evening. Being a Friday night in summer, the pub was crowded with holidaymakers and yachtsmen from the boats in the marina, and I felt a small satisfaction when I saw the flicker of apprehension in his eyes as I approached.\n\nHe came out from behind the bar before I could reach it. \"We'll go through to the back,\" he said curtly, jerking his head toward a door in the corner. \"I'm damned if I'll have this conversation in public.\"\n\n\"Why not?\" I asked. \"Are you afraid of witnesses?\"\n\nHe made an angry movement as if to grab my arm and manhandle me in the direction he wanted me to go, but the curious glances of his other customers persuaded him to change his mind. \"I don't want a scene,\" he muttered, \"not in here and not on a Friday night. You said you wanted to be fair... so _be_ fair. This is my livelihood, remember.\"\n\nI smiled slightly. \"You could have me arrested for making a nuisance of myself, then tell your customers I'm mad,\" I suggested. \"That's what you did last time.\"\n\nHe resolved the problem by heading toward the door and leaving me to follow or not as I chose. I followed. The \"back\" was a scruffy office full of dusty filing cabinets and a gray metal desk, covered in used polystyrene coffee cups and piles of paper. It was a smaller, dirtier version of Jock's office and, as Drury motioned me toward the typist's chair in front of the desk and perched himself on a stack of boxes in the corner, I wondered why men always seemed more comfortable when surrounded by the trappings of \"work.\"\n\nHe watched me closely, waiting for me to speak. \"What do you want?\" he demanded abruptly. \"An apology?\"\n\nI dropped my rucksack to the floor and used the tip of my finger to push a half-filled cup of congealed coffee away from me. \"What for?\"\n\n\"Whatever you like,\" he said curtly, \"just as long as it gets you off my back.\"\n\n\"There'd be no point. I wouldn't accept it.\"\n\n\"What then?\"\n\n\"Justice,\" I said. \"That's all I've ever been interested in.\"\n\n\"You won't get it... not this long afterward.\"\n\n\"For Annie or for myself?\" I asked curiously.\n\nHe placed the flat of his hand on the opened brown envelope, which was sitting to one side of the desk. \"Neither,\" he said confidently.\n\nI wondered if he was aware of what he was saying because his words suggested he knew there was justice to be had. For Annie _and_ me. \"That envelope contains twenty-one years of patient research, showing Annie was murdered,\" I said lightly.\n\n\"And it's a bunch of crap.\" He leaned forward aggressively. \"For every pathologist you produce, saying the bruises were inflicted hours before Annie's death, the Crown Prosecution Service will produce five who agree with the original postmortem finding. It's a budgeting exercise\u2014always has been. Prosecutions are expensive and taxpayers get stroppy about funding failures. You're going to need a damn sight more than that to get the case reopened.\"\n\nHe was uncomfortably close and I sat back to get away from him, repelled by the energy that flowed out of him in waves. It was a far cry from twenty years ago when the same energy\u2014authoritative, capable, comforting\u2014had given me the confidence to talk more freely than I might otherwise have done. It's one of the great truisms that we only learn from our mistakes and, like Annie, I had since developed an abiding distrust of men in uniform.\n\n\"The climate's changed since the Stephen Lawrence inquiry,\" I said mildly. \"I think you'll find the murder of a black woman will be at the top of the CPS's agenda, however long ago it happened... particularly when it's supported by evidence that the sergeant in charge of the case was a racist.\"\n\nHe pummeled and squeezed one fist inside the other, exploding the joints like tiny firecrackers. \"A letter from a WPC claiming sexual and racial harassment, which wasn't upheld at the time?\" he sneered. \"That won't stand. And neither will Andy Quentin's log. The guy's dead, for Christ's sake, and he had an ax to grind because he blamed me for his career going nowhere.\"\n\n\"With reason,\" I said. \"You never had a good word for him.\"\n\n\"He was a creep.\"\n\n\"Yes, well, he didn't have much time for you either.\" I opened the envelope and removed Andy's log of Drury's stop-and-search arrests of Afro-Caribbeans and Asians between 1987 and 1989, giving details of the derogatory language Drury had habitually used. \"What difference does it make if he did have an ax to grind?\" I asked curiously. \"It's a straightforward account, which you're perfectly entitled to challenge if it contains errors.\"\n\n\"He hasn't logged the names of the whites I stopped and searched.\"\n\n\"He's given comparative figures. Your ratio of black to white was way higher than anyone else's in Richmond at the time.\" I shrugged. \"But it's all on record so it's easily proved. If Andy's figures are wrong then you'll be vindicated. If they aren't, his conclusion that you used your stop-and-search powers as a form of racist sport will carry weight.\"\n\n\"Not true,\" he snapped. \"I was doing my job like everyone else. You can twist figures to fit any conclusion you like. I can just as easily demonstrate that his motive in producing that list was vindictive. There was a known history between us.\"\n\n\"What about the seventeen-year-old Asian boy whose cheek you fractured?\"\n\nHis jaw worked angrily. \"It was an accident.\"\n\n\"The police paid undisclosed damages.\"\n\n\"Standard procedure.\"\n\n\"So standard,\" I murmured sarcastically, \"that you were put on sick leave for the duration of the internal inquiry, then took early retirement immediately afterward.\" I unzipped the front pocket of my rucksack and removed a folded piece of paper. \"I left this out of the envelope. It was the last thing Andy sent me. It's the confidential assessment made of you by your superior officer. Among other things, he describes you as 'a violent individual with extreme racist views who has no place in the Metropolitan Police Force.'\"\n\nHe snatched the paper from my grasp and shredded it to pieces, the muscles of his face working furiously. He was the opposite temperament to Sam. A man who brooded over long-held grudges. A man who saw loss of face as weakness.\n\nI stirred the pieces with my toe, thinking I'd be safer poking a viper's nest. \"Is that how you always deal with evidence you don't like? Tear it up?\"\n\n\"It's inadmissible. The slate was wiped clean as part of my retirement package. You'd be prosecuted just for having it in your possession. Quentin, too, if he were still alive.\"\n\n\"Well, maybe I think a prosecution's worth it,\" I murmured, \"just to get it into the public domain. I can fire off a thousand copies tomorrow and throw so much mud at you that there'll be no one left who won't question your motive for wanting Annie's death ruled accidental.\"\n\n\"You'll be seen for what you are,\" he warned, \"a bitter woman with a personal vendetta against the police.\"\n\n\"One policeman, possibly,\" I agreed, \"but not the police in general. I was given too much help by Andy for anyone to think I tar you all with the same brush. In any case, who's going to tell them it's a personal vendetta? _You?\"_ I smiled at his expression. \"How do you plan to explain why I'd want to pursue one?\"\n\nHe screwed his forefinger into his temple. \"It's all in your statements,\" he said. \"You were a head case... persecution complex... mother complex... anorexia... agoraphobia... sexual fantasy... What was I supposed to do? Sit beside your bed and hold your hand while you bawled your eyes out?\"\n\n\"You might have questioned your own judgment,\" I suggested.\n\n\"You've only yourself to blame for that,\" he retorted sharply. \"Maybe if you'd backed off once in a while, I'd have taken you more seriously. I don't like people in my face all the time.\" As if to prove the point, he slammed his back against the wall and stared at me through half-closed eyes.\n\nI looked away. \"Then why didn't you let someone else take over? Why wasn't I allowed to talk to Andy? Why did you get him pushed off the case?\"\n\n\"He was more trouble than he was worth. He believed everything you told him.\"\n\nWe both knew that wasn't the real reason, but I let it go. \"Because everything I said was true.\"\n\n\"You mean like this.\" He jerked his chin at the brown envelope. \"There's no evidence of murder in there. Just different opinions.\"\n\n\"That's only a fraction of what I've got,\" I said. \"You didn't think I'd show my whole hand, did you?\" I took the photographs of Beth and Alan Slater's house from my rucksack. \"There's plenty of evidence that Annie was robbed.\" I passed the pictures across to him. \"Maureen Slater admits most of this stuff was sitting in her house for months after Annie died... claims you saw it and even went back on one occasion, offering to buy the Quetzalcoatl mosaic. Which means you should have treated Annie's house as a scene of crime, if only because it must have been obvious to you that the Slaters had robbed her.\"\n\nHe gave the pictures a perfunctory glance. \"Maureen said she bought it in a junk shop,\" he said dismissively \"I had no reason to think otherwise.\"\n\n\"She couldn't afford to go to the laundrette. How could she afford to buy paintings?\"\n\n\"Not my problem. None of the stuff had been reported stolen.\"\n\n\"You must have recalled the Quetzalcoatl when Dr. Arnold started asking questions about Annie's possessions.\"\n\n\"No,\" he said bluntly. \"It was four years later. How many houses do you think I'd entered in that time? I couldn't describe a picture I'd seen a week before, let alone one from way back.\"\n\n\"You offered Maureen twenty quid for it,\" I reminded him, \"so it obviously made an impression on you.\"\n\nHe shrugged. \"I don't remember.\"\n\n\"I didn't think you would,\" I said with a small laugh. \"Any more than you'll remember Maureen giving you a gold statuette with emeralds for eyes and rubies for lips. She said you had no intention of buying the Quetzalcoatl.... All you wanted was something valuable as a quid pro quo for not asking awkward questions. What did you do with it? Keep it? Sell it? Melt it down? It must have scared you rigid when Sheila Arnold described it as one of the artifacts Annie had on her mantelpiece.\"\n\n\"Maureen's lying,\" he said bluntly.\n\n\"She's prepared to make a statement about it.\"\n\nA glint of amusement sparked in his eyes. \"You think anyone's going to believe her about something that happened twenty years ago? And why wouldn't I want to ask awkward questions of the Slaters? I had a reputation for being tough on the whole damn family.\"\n\n\"Not just tough,\" I said casually. \"According to Danny, you were quite happy to frame them as well. He says you planted some cannabis in Alan's pocket and got him sent down for dealing.\"\n\nDrury shook his head pityingly. \"And you believe him, of course.\"\n\n\"Not necessarily. No one seems to know what Alan actually did. Danny says dealing, but Alan told his wife he was sent down for assaulting Michael Percy.\"\n\n\"Why am I not surprised?\" he said with irony.\n\n\"Well?\" I prompted when he didn't go on.\n\n\"She wouldn't have married him if she'd known the truth.\"\n\n\"Why is it such a secret?\"\n\nHe pointed an accusing finger at me, as if Alan's crime were my responsibility. \"He was always going to get off lightly. He was fifteen and couldn't be named, and neither could his victim. It's a bloody stupid rule in my book. All a kid has to do is see out his sentence, lie through his teeth, put a bit of distance between himself and what he did and he gets off virtually scot-free.\" He started popping his knuckles again. \"Maureen kept it quiet because she was scared stiff of what people would say.\"\n\n\"What did he do?\"\n\n\"Work it out for yourself. The victim was a woman.\"\n\n\"Rape,\" I suggested.\n\nHe nodded. \"Took himself off to the other side of London where he thought he could get away with it. Dragged the woman into a car park behind some houses and proceeded to beat her up. But she managed to scream and one of the occupants called the police. Alan was caught in the act, pleaded guilty and did four years before he was let out.\"\n\n\"Anyone could have predicted it,\" I said unemotionally. \"He was appallingly abused as a child, both physically and mentally.\"\n\nBut Drury wasn't interested in bleeding-heart excuses. \"On that basis Danny would have become a rapist as well.\"\n\nI stared at my hands. \"Danny has no memories of his childhood. He was so young when his father left that he can't even remember what he looked like... and if he heard his mother being thrashed in the bedroom, he wouldn't have understood the connection between sex and violence.\" I raised my head to look at him. \"It makes a difference. All poor Alan ever learned from his parents was that reducing a woman to a shivering wreck would result in an orgasm.\"\n\n# twenty -one\n\nDrury's gaze veered away from mine but not before I saw the quickly veiled flash of intelligence that told me he knew what I was talking about. It was a powerful revelation because, despite everything, I had never been certain of how much knowledge he had. For the moment I let it go. \"Did Alan get into trouble again after the rape conviction?\" I asked then.\n\n\"Not that I know of. He moved into a bedsit out Twickenham way and took laboring jobs. We kept an eye on him but he was wary of coming into Richmond or seeing anyone he knew.\"\n\nI had no reason to disbelieve him. \"So why did Danny tell me Alan received \u00a35,000 compensation for being beaten up by the police?\"\n\nAmusement brought a gleam to Drury's eyes. \"Because the guys who arrested him didn't much like what he'd done to the woman. His solicitor bellyached about police brutality until he saw the state of the victim, then settled on five thousand and told Alan to be grateful they hadn't killed him. I'd say it was cheap at the price.\"\n\nI nodded. \"Did Derek ever get done for rape?\"\n\n\"That would suit you, wouldn't it?\"\n\n\"Why?\" I asked mildly. \"I never accused him of rape.\"\n\n\"All but. You said he shoved his penis between your legs.\"\n\n\"I said he put something between my legs which I _thought_ was his penis, and as a result I _thought_ he was going to rape me. I also told you that's exactly what he wanted me to believe. He was giving me a demonstration of how bad things would get if I didn't keep my nigger-loving mouth shut. It was your choice to tell him I'd accused him of attempted rape... your choice to put me in danger... even though you'd already agreed with Andy's assessment that the worst Derek could be charged with was threatening behavior.\"\n\n\"We couldn't charge him with anything,\" he said dismissively \"He had an alibi. In any case, I thought the guy had a right to know what the latest accusation was. You weren't exactly stinting yourself on the Derek Slater front... and sexual assault was a damn sight more serious than heavy breathing on the end of a phone.\"\n\n\"His alibi was a joke,\" I said. \"You didn't follow it up until three days later.\"\n\n\"It makes no difference. It was watertight.\"\n\n\"Oh, come on!\" I said impatiently. \"A Kempton Park ticket stub which he could have picked out of the gutter the next day? The course is only a few miles outside Richmond in all conscience. And a telephone conversation with one of his friends? You didn't even bother to check on the remaining two.\"\n\n_\"You_ didn't bother to report the incident until the day after it happened,\" he countered sarcastically.\n\nI fingered my lip to quell the tic that was leaping and jumping beneath the skin. I couldn't stand the idea that he might interpret it as fear. \"It took me twenty-four hours to pluck up the courage,\" I said matter-of-factly \"Half of me wanted to let the whole thing drop, the other half recognized that Derek wouldn't be terrorizing me if I wasn't right in what I was saying. I was very na\u00efve, of course. It never occurred to me you'd bend over backward to protect a man you described as scum... just because he was white.\"\n\n\"That's not true, and you know it.\"\n\n\"Then why have you consistently protected the Slaters from questioning about Annie's death?\"\n\n\"I haven't.\"\n\n\"Why didn't you follow through when Dr. Arnold told you Annie had been robbed? You must have realized then where the Quetzalcoatl mosaic came from.\"\n\n\"I didn't. I remember some bits of rubbish in the Slaters'sitting room, but I couldn't describe it now and I certainly didn't link it with anything Dr. Arnold said later.\"\n\nI could almost believe him, if only because the death of a black woman had meant so little to him. \"The children had been stealing from Annie for months,\" I said, \"but they weren't very good at hiding what they were taking, and Maureen beat the truth out of Bridget Spalding when she spotted her wearing a ring that obviously hadn't come from Woolworth's. That's when she began to realize Annie might be sitting on a gold mine.\"\n\nDrury flicked his hand dismissively \"The police can't act if a crime isn't reported.\"\n\nI went on as if he hadn't spoken. \"Annie was such an easy target. She wouldn't let people into her house, she distrusted anyone who spoke to her neighbors, thought council officials and men in uniform were against her, made an enemy of her bank manager. In fact the only person who came close to being a friend was her GP.\" I watched his face for a reaction, but it remained impassive. \"Annie was fairly safe while Sheila was making regular visits because even Derek wasn't stupid enough to make a move while her doctor was taking an interest. Then Sheila left for America and everything changed.\"\n\n\"You can't blame me for that.\"\n\n\"More to the point, after Sheila's departure there was no one who could say what Annie did or didn't have.\" I held his gaze. \"And you never bothered to ask because it suited you to believe a black woman would live in a slum.\"\n\n\"You're forgetting how many empty bottles we found. The conditions inside the house had nothing to do with the color of her skin, they were the result of a drink habit.\"\n\n\"They were vodka bottles,\" I said.\n\nThere was a tiny flicker of doubt in his eyes. \"So?\"\n\n\"She didn't drink vodka.\" I took a sheaf of papers from my rucksack. \"Andy sent me a list of every landlord and off-license manager in Richmond in 1978. My father managed to locate just over half of them. Two of the off-license managers remember Annie well. They both say she was a regular and that she only bought Jamaican rum. And the landlord of the Green Man says he kept a stock of it just for Annie Butts because she used to get agitated if he ran out.\" I thrust the pages into his hand.\n\nDrury frowned as he flicked through them. \"It doesn't prove she wasn't buying vodka from a supermarket,\" he said.\n\n\"No,\" I agreed.\n\n\"Then it's not evidence.\"\n\n\"Not on its own, perhaps, but if you look at the last two pages you'll see that several off-license managers remember Maureen Slater as a vodka drinker. One of them describes how she used to come in after picking up her benefit money and buy half a dozen bottles at a time. He says he refused to serve her after she slapped one of her children\u2014 probably Alan\u2014when he said he needed new shoes.\"\n\n\"So? All that proves is Maureen bought vodka; it doesn't prove Annie didn't. What are you trying to say, anyway? That the Slaters put their bottles in Annie's kitchen?\"\n\n\"Yes.\"\n\n\"When?\"\n\n\"After she was dead.\"\n\n\"Why?\"\n\n\"To make you think what you did: that she was a chronic drunk who lived in a tip and neglected herself. That's why they turned off the main supplies and took away all the food that she'd bought for the cats.\"\n\n\"Oh, come on,\" he growled impatiently. _\"Everyone_ said she was drunk, not just the Slaters.\" He smacked the paper with the back of his hand. \"In any case, Derek was as thick as two short planks. He couldn't have followed through on a plan like that. He'd have given himself away the minute we started asking questions.\"\n\n\"Not Derek, maybe, but Maureen certainly could. All she had to do was play on your prejudices.\" I quoted his own words at him. \"You'd never believe a 'downtrodden slut'could outmaneuver you, and 'a miserable black who couldn't hold her drink'was bound to soil her own floor and piss on herself. And why would you question the kind of bottles you found in Annie's house when the mere fact of their existence confirmed everything Maureen wanted you to believe?\"\n\n\"There was no reason to question them. No one told us she didn't drink vodka.\"\n\nI handed him another piece of paper.\n\n\"What's this?\"\n\n\"A copy of Sharon Percy's statement. Your name's at the top as the interviewing officer. The first half deals with where she was during the evening\u2014none of which is true as a matter of interest\u2014the second half is her description of what Annie was like. Somewhere in the last paragraph it says: 'She used to get drunk on rum and start insulting everyone. She took swipes at the kids with the empty bottles. I kept reporting it but nothing was done.'\"\n\nImpatiently, he tore this page, too, into shreds and dropped it to the floor. \"You're clutching at straws,\" he said. \"You can muddy the waters as much as you like, it doesn't alter the fact that there was no reason to question anyone's statement at the time... and that includes your husband's. The pathologist's findings were unambiguous\u2014Ann Butts died because she walked in front of a truck.\"\n\n\"Which is what you told him to say.\"\n\n\"You can't prove it. If Hanley's files are missing, there's nothing to show which of us said what first.\"\n\nI gave a small laugh. \"He didn't do you any favors by getting rid of them. At the moment, the only document supporting your accident theory is the one-page report Hanley submitted to the coroner, and that has so many mistakes it's a joke. He spelled Annie's name wrong, referred to bruising on her left arm instead of her right, and completely ignored the lividity in her thighs, which is very pronounced in the photographs.\"\n\nI was amazed to see him run a nervous tongue across his lips. \"I don't think that's right.\"\n\n\"It is,\" I assured him. \"Hanley was so incompetent by that time he was taking dictation from whichever police officer presented a body for inspection. I assume you got muddled over the arms because I told you she was lying left side uppermost with her back to the lamppost.\"\n\nHe had to think about his answer. \"Not my responsibility. He had his job.... I had mine. Let him take the flak.\"\n\nI reached for my rucksack and zipped up the pockets. \"Reporters don't hound dead people,\" I told him. \"Only the living. And there's more human interest in a racist policeman who refused to investigate a black woman's murder than a troubled pathologist who killed himself with drink because he couldn't stand the unnecessary mutilation of corpses. Radley's won't keep you on,\" I went on dispassionately, \"not once you're plastered across the front of the newspapers. All your decent trade will vanish overnight to be supplanted by thugs from the National Front.\"\n\nSmall beads of sweat dampened his forehead. \"Tell me what you came for,\" he said, \"because we both know this has nothing to do with Annie.\"\n\n_Was he right?_ I honestly didn't know anymore. \"It was two years before I learned to trust myself again,\" I said slowly, \"and another two before I dared trust anyone else. I still have nightmares... still run to the basin to wash myself... still check the bolts on the door... still jump out of my skin every time I hear a sound I don't recognize.\" I pushed back my chair and stood up, hooking my rucksack over my shoulder. \"I'd say this has everything to do with Annie. The only difference between us is that she had the courage to stand and fight... and I ran away.\" I moved to the door. \"Which is why she's dead and I'm alive.\"\n\n**_Letter from Dr. Joseph Elias, psychiatrist  \nat the Queen Victoria Hospital, Hong Kong\u2014dated 1999_**\n\n**QUEEN VICTORIA HOSPITAL**\n\n_Hong Kong  \nDept. of Psychiatry_\n\n**Mrs. M. Ranelagh  \nJacaranda  \nHightor Road  \nCape Town  \nSouth Africa**\n\n**February 17, 1999**\n\n**Dear Mrs. Ranelagh** ,\n\n**Goodness me! So it's home to England at last. I shall await your news with bated breath. Yes, despite my incredibly advanced years, I still have a small consultancy in the hospital, but only because my patients seem to prefer the devil they know to the devil they don't**.\n\n**And what of your devils, my dear? Somehow I doubt that justice for Annie will be enough for you. But who am I to criticize when my friend the rabbi would say: To win the peace you must first fight the war?**\n\n**As requested, I enclose the notes I made in 1979. Yours fondly** ,\n\n# twenty -two\n\nDrury couldn't leave it alone, as I knew he wouldn't. For all his protestations about hating having people in his face, he disliked it even more when they walked away. I turned, left the pub and went about fifty yards toward the trawler moorings before I heard his footsteps behind me. Lights from the buildings along the quay shone a quiet glow across the cobblestones and, far ahead, tiny beacons bobbed upon the water like multicolored jewels, showing safe navigation for incoming yachtsmen. I had a moment to wish I could enjoy the scene for what it was\u2014something beautiful\u2014 before his fingers closed about my arm.\n\n\"This is crazy,\" he said, jerking me 'round to face him. \"You say you want to get even. Well, how? Destroying me isn't going to produce justice for you or for Annie. Are you asking me to deliver Derek Slater on a plate? Is that what this is about?\"\n\nI tried to pull away. \"People are watching,\" I said.\n\n\"Let them watch,\" he growled. \"I want this sorted.\"\n\n\"Fine. So when I decide to scream\u2014which I certainly will if you don't let go\u2014there'll be a hundred witnesses to confirm your superintendent's assessment that you're a violent man.\"\n\nHe released me immediately.\n\nI smiled cynically as I rubbed my arm. \"It's not so much fun when the boot's on the other foot, is it? The way things are at the moment, you'd crawl on your belly over these cobbles in exchange for a promise to burn what's in my rucksack. Am I right?\"\n\n\"Don't push your luck,\" he said in an undertone. \"I'm in no mood for games. All you'll achieve by going public is to make me a scapegoat, and that's not going to put Derek behind bars... not after all this time. Is that the kind of justice you want?\"\n\n\"It's better than nothing.\"\n\nHe grasped one writhing fist inside the other as if afraid he wouldn't be able to control them. \"If it was me you wanted, you wouldn't have put me on my guard,\" he said reasonably.\n\n\"Perhaps I like watching you sweat,\" I murmured.\n\n\"How about I break your fucking neck?\" he said through gritted teeth.\n\n\"You wouldn't get very far. My two sons are standing right behind you.\"\n\nThe words made no sense to him\u2014he didn't associate me with children\u2014and he stared at me in baffled fury like a tired bull trying to work out how to defeat a matador. \"What the hell are you talking about now?\"\n\n\"Protection.\" I nodded to Luke and Tom. \"I come better prepared these days.\"\n\nIt took a second or two for his brain to catch on, but he spun 'round eventually to discover I was telling the truth. Perhaps he was expecting something younger\u2014 _or smaller?_ \u2014but whichever, he was suitably impressed. \"Shit!\" he said. \"What the fuck's going on?\"\n\n\"Sam's waiting for us in the car,\" I explained. \"I'd like him to hear what you're going to say next.\"\n\nDrury glanced nervously at the boys. \"Which is what?\"\n\nI made him the same offer I'd made Maureen. \"A trade?\" I suggested. \"You see you're right about one thing. The kind of justice I'm looking for is a little more\"\u2014I sought for a word\u2014\"basic than making you take the blame for everything that happened.\"\n\nI didn't think he'd follow me, particularly as the boys returned to the pub as soon as I moved away. But perhaps he misunderstood what I wanted Sam to hear... or what I meant by basic justice... The car was parked beyond the trawler moorings, facing out over the water, and as we approached Sam opened the door and climbed out. In a spirit of mischief, I introduced them to each other as I lowered my rucksack onto the bonnet. \"Mr. Ranelagh. Mr. Drury.\" They nodded to each other like a couple of wary Rottweilers, but didn't shake hands. \"You asked me if I was expecting you to deliver Derek on a plate,\" I reminded Drury, \"but I don't see how you can do that unless you suppressed evidence at the time.\"\n\nHe looked tight-lipped at Sam, aware that anything he said now would be heard by a witness. \"There was no suppression of evidence,\" he said sharply, \"merely question marks over where Derek was at nine o'clock. He claimed he was having a drink with the local tart who'd been touting for custom since the place opened.\"\n\n\"Sharon Percy?\"\n\nHe nodded. \"It was straightforward stuff-\u2014the two of them were regulars\u2014and the publican agreed they were both in there that night although he disputed the timing when we first questioned him. He remembered seeing Sharon at nine o'clock but he didn't think Derek came in until later.\" He shrugged. \"He backed off when we asked for a statement... said one day was much like another, and he couldn't swear he wasn't confusing two different occasions.\"\n\n\"This being the William of Orange,\" I said. \"The pub Annie was banned from because she was black.\"\n\nHe gave an impatient shake of his head. \"She was banned because she couldn't hold her drink and swore at the other customers. The publican was within his rights to refuse to serve her.\"\n\nI looked questioningly at Sam.\n\n\"It was known locally as the Orange Free State,\" he told Drury. \"There was a sign on the door saying 'no dogs'and the 'd'had been crossed out and changed to a 'w.'It was a popular pub\u2014a fair number of policemen used it\u2014but you never saw any blacks in there.\"\n\n\"If it offended you, you should have reported it.\"\n\n\"It didn't,\" said Sam honestly. \"I never even questioned it.\"\n\n\"Then why expect me to?\"\n\n\"Because it was your job. I'm not saying I'd have given you any medals for it\u2014hell, the last thing I wanted was to have Mad Annie swearing at me over a pint\u2014but the laws on discrimination were clear and anyone who put 'no wogs'on their front door ought to have been prosecuted.\" He paused to exchange a glance with me, clearly wondering how far he should or could go. \"The landlord was cock-a-hoop after the accident,\" he went on abruptly. \"Kept telling anyone who cared to listen that we had a truck driver to thank for making the streets cleaner.\"\n\n\"Not in front of me he didn't,\" said Drury so quickly that I guessed he'd had to answer that question before, probably at the time of his \"retirement.\"\n\n\"So did you bother to challenge Derek about his alibi?\" I said dryly. \"Or was that when you decided to take him aside and tell him that _I_ was the problem, and the best solution for everyone would be to shut me up? And how did you put it exactly? Do us all a favor, Derek, and teach that nigger-loving bitch a lesson because your alibi stinks and you'll be in trouble if you don't. Or did you drop hints to Maureen when you were looking at the bits of junk in her sitting room?\"\n\nI watched him flick a wary glance at Sam, but he took confidence from Sam's obvious ignorance of what I was talking about. \"Of course I challenged him,\" he said bullishly \"He stuck to his story... so did Sharon. They both said they'd been there all evening. We didn't believe them, but there was nothing we could do if no one was willing to contradict them.\"\n\n\"Did you ever find out what they were really doing?\"\n\nHe shrugged. \"Our best guess was that Sharon had been on her back somewhere and Derek was out thieving. They both had convictions\u2014 Sharon for prostitution; Derek for assault and theft.\"\n\n\"Sharon was with Geoffrey Spalding,\" I said. \"He lived at number 27 and used to meet her at a hotel once a month because he didn't want his wife and daughters finding out what he was doing. He's the one who said he saw Annie in the street around a quarter past eight and tried to persuade her to go home.\"\n\n\"I remember him.\"\n\n\"I think he was lying about the time,\" I went on. \"According to Jock Williams, Sharon arrived in a taxi at the William of Orange shortly after nine. He said she was high as a kite and had obviously been with another client, and I'm betting the client was Geoffrey and the same taxi dropped him off at the top of Graham Road before taking Sharon on to the pub. Which means, if Geoffrey talked to Annie at all, it must have been an hour later than he said it was.\"\n\nHe refused to accept it. \"I spoke to him in front of his wife and she didn't question that he was home by 8:30.\"\n\n\"She wouldn't have known. She was on chemotherapy for breast cancer and would have been asleep whatever time he came in. Where did he say he'd been?\"\n\nDrury thought back. \"Late at work. Nothing to raise any eyebrows over.\"\n\nI turned to Sam. \"I've always thought he must have passed the Williams'house as you came out... otherwise you and Libby wouldn't have needed an alibi.\"\n\n\"Someone did,\" he admitted, \"but I've no idea who it was. To be honest, I can't even be sure it was a man. It could have been a total stranger taking a shortcut, but Libby went apeshit and said tongues would start wagging\u2014\" He pressed his thumb and forefinger to the bridge of his nose. \"I'm sorry,\" he said after a moment. \"Is this the man you think killed Annie?\"\n\n\"I don't know,\" I answered slowly, \"but I've never understood why he said he spoke to Annie unless it actually happened. It was an unnecessary lie. He could have done what you and Jock did and said he saw her on the other side of the road.\"\n\n\"People embroider all the time,\" said Drury. \"It makes them feel important.\"\n\nI shook my head. \"She was seen by two different couples at around nine o'clock. The Pardoes at number 8 who watched her from their bedroom window, and the couple in the car who say she lurched out in front of them. They all said she was on her feet... but by the time Sam passed her at 9:15 she was collapsed in the gutter.\"\n\n\"That's not what Mr. Ranelagh said at the time.\"\n\n\"His revised statement was in the envelope,\" I said impatiently, \"so I know you've read it. The question is, was Annie on her feet when Geoffrey Spalding passed her? And if she was, did she speak to him? I think she was\u2014and did\u2014and that whatever she said made him so angry that he pushed her into the road. It would explain why he advanced the time by an hour... it would also explain why Sharon was prepared to give Derek an alibi. If she told you she'd been with a customer\u2014and you found out who it was\u2014you'd have worked out PDQ that Geoffrey was the last person to speak to Annie.\"\n\nDrury frowned. \"And?\"\n\n\"You'd have come to the same conclusion he did... that he killed her.\"\n\nHe gave a grunt of irritation. \"Half an hour ago you were producing pathology reports saying she was beaten up several hours in advance of her death, now you're saying Geoffrey Spalding murdered her. When are you going to make up your mind, Mrs. Ranelagh?\"\n\nSam roused himself. \"She's not saying Spalding killed her,\" he said reasonably, \"just that he _thought_ he did. If it comes to that, I've spent twenty years worrying that I did the same. And maybe I did. Maybe that fifteen minutes I left her to lie in the gutter was the difference between life and death.\"\n\n\"Then you should have cleared your conscience by telling us the truth at the time,\" said Drury with a far from friendly smile, \"instead of contaminating the investigation because you couldn't keep your hands off your friend's wife.\"\n\n_He would have been wiser not to mention Libby_ , I thought with private amusement as I watched an angry flush stain Sam's cheeks. Guilt was the one thing guaranteed to fire my husband's temper.\n\n\"You told us there wasn't going to be an investigation,\" he snapped. \"I remember it very distinctly. You came to the house the next day to explain the postmortem findings. Unequivocal, you said... a clear-cut accident... no hint of foul play. I also remember you saying that if there had been any question marks over the death, the whole matter would have been turned over to the CID.\"\n\n\"There _were_ no question marks, Mr. Ranelagh. It might have been different if you hadn't lied, but we could only work with the information we had.\"\n\nSam smoothed a hand across his bald patch, staring past Drury to the lights on the other side of the water. \"Jock and I didn't offer any information until the Thursday evening when we were asked to make voluntary statements in support of what Libby had told you the day before, namely that Jock was in my house.\"\n\n\"So now you're blaming Mrs. Williams?\"\n\n\"No, merely pointing out that you'd made up your mind it was an accident a good twenty-four hours before Jock or I said anything.\" He stared thoughtfully at Drury as if he were fundamentally reassessing some previous judgment. \"Would it have made any difference if we _had_ told the truth? Wouldn't you just have claimed she was hit by a truck between the couple in the car seeing her and my finding her?\"\n\nDrury's silence was an answer in itself.\n\n\"You telephoned me several times at work,\" Sam went on, \"telling me that my wife was suffering a classic response to stress and needed psychiatric help. You said you'd seen that sort of reaction before and it always led to more and more wild accusations.\"\n\n\"You agreed with every word, Mr. Ranelagh, including the necessity for an official caution.\"\n\nMy husband folded his arms and stared fixedly at the cobbles as if certainty lay within their uneven surface. \"Did I have any choice?\" he asked. \"You read out a catalog of complaints against her... wasting police time... making false accusations against Derek Slater... reporting imaginary sex attacks to win sympathy... plaguing you with telephone calls and visits because she had an unhealthy obsession with you.\" He lifted his head. \"You were a policeman. I had to accept you were telling me the truth.\"\n\n\"It must have tallied with your own opinion,\" said Drury persuasively, \"otherwise you'd have argued your wife's case.\"\n\nSam made a troubled gesture with his hand. \"I was in no position to argue. I hadn't seen her for nearly three weeks, and on the one occasion she phoned she was hysterical. I couldn't make head or tail of what she was saying so I called her parents and asked them to help her.\" He paused, trying to marshal facts in his head. \"But you'd already persuaded my mother-in-law that an official caution in front of her family was the best way to deal with the situation. 'She needs shaming to stop her wasting any more police time,'was the way you put it.\"\n\nThere was a short silence.\n\n\"It worked then,\" I said lightly. \"I'd have slit my throat rather than say another word to Mr. Drury... or to you and Ma, Sam. You both stood by and watched this bastard bully me into keeping my mouth shut\"\u2014I jerked my chin at Drury\u2014\"and then shook his hand at the end as if he'd done something fine. The only person who refused to go along with the charade was my father, yet he knew no more at that stage than you did. He just had faith in the woman he knew me to be rather than a pathetic, disturbed creature who was resorting to sexual fantasy to prolong her fifteen minutes of fame.\"\n\n\"You were never described in those terms or treated with anything other than courtesy,\" said Drury curtly. \"Your husband knows that. That's why I asked him to be present, so that you wouldn't be able to rewrite the history afterward.\"\n\n\"You could be as courteous as you liked,\" I said, \"because you knew I wouldn't argue with you. Not after the _unofficial_ caution you arranged for me the night before. You should have joined the party,\" I told him. \"I imagine it was a great deal more exciting than hammering a needle into a twelve-year-old's arm or pounding at a black face until the cheekbone snapped.\"\n\nThe muscles along the man's jawline tightened. \"Now you're slandering me in front of a witness.\"\n\n\"Then sue me. Give me my day in court. It's all I've ever wanted. But you'll be on thin ice... I've another copy of your assessment in my rucksack.\"\n\nHe took an abrupt step forward, swinging his fists at his sides. I thought he was going to hit me and dodged away 'round the bonnet of the car, but he snatched up the rucksack instead and tossed it into the water beyond the harbor wall. There was a second of silence before it hit with a splash, and Drury stared after it with a look of satisfaction on his lean face.\n\nHe flung off the nervous hand that Sam laid on his arm. \"Leave it,\" he warned. \"This is between me and your wife.\"\n\n\"You always were a shithead,\" I hissed angrily as I thought of my wallet and credit cards sinking into the sludge at the bottom of the river. \"That's the only solution you've ever had to anything. Get rid of the evidence before your crimes find you out.\"\n\nHe laughed at my anger. \"It's not so much fun when the boot's on the other foot, is it?\" he taunted, resting his palms on one side of the bonnet and staring me down.\n\nI did the same from my side, thrusting my face toward his and raking him with furious eyes. \"Do you know what pisses me off the most? Not what you did to me\"\u2014I lifted a finger and stabbed it at his chest\u2014 \"I learned to deal with that. It's the fact that you had the nerve to underestimate me... and _still fucking well_ do.\" I could feel the stridency roaring in my voice and, for once, didn't give a damn how it sounded. If the truth be told, I'd always been closer to loudmouthed fishwives than effete Victorian ladies who gave way to the vapors. \"How _dare_ you think I'm so stupid as to carry a master file with me? How _dare_ you think I'd give you an opportunity to outflank me?\"\n\n\"You talked about a trade,\" said Drury aggressively.\n\n\"I want justice first,\" I flung at him. _\"Then_ I'll trade.\"\n\n\"What sort of justice?\"\n\n\"The eye-for-an-eye kind. The same kind you believe in. You pumped a Neanderthal full of lies, then told him I'd be gagged the next day. What did you think he was going to do? Send me a bunch of flowers?\"\n\nHe looked edgily toward Sam. \"I don't know what you're talking about.\"\n\n\"Yes, you do. You got more and more angry every time I accused you of racism. That's why you made my official caution so public... so that even a moron like Derek Slater knew he could have a free run at the nigger-lover without any fear of my reporting it.\"\n\n\"You're inventing things again, Mrs. Ranelagh. If a crime was committed you had a perfect opportunity to give us the details the next morning.\"\n\n\"You mean in the middle of an official caution for wasting police time? In front of a husband and mother who didn't believe a word I said because they had more faith in a corrupt policeman than they had in me?\" I flung out my arms and caught him across the chest with the backs of my hands. \"How _dare_ you suggest I had an unhealthy fixation on you? How dare you imagine for one minute that I'd be interested in someone who thought a woman's place was under a man... preferably bound and muzzled so that he wouldn't have to listen to her criticizing his performance.\"\n\nHe retreated warily but didn't say anything.\n\n\"I had nothing but contempt for you,\" I said. \"I saw you as a _little_ man... a pygmy in uniform... someone who was allowed to strut his stage because his superiors were too inept to see how incompetent he was... and the only reason I spoke to you at all was because I wanted justice for Annie. But I never thought of you as anything other than a reptile.\" I looked deep into the dead black of his eyes. \"And that was my mistake, wasn't it? If I hadn't made it so obvious that the mere sight of you made my flesh crawl, you wouldn't have set Derek on me. Because it wasn't me who fancied you, you bastard, but _you_ who fancied _me.\"_\n\nI felt Sam move behind me.\n\n\"You're crazy,\" said Drury.\n\n\"You'd better believe it,\" I agreed, slithering 'round the bonnet of the car. \"I haven't been sane since Derek did your dirty work for you. He knew I'd never let him into my house, so he sent Alan in first, blubbing about how his father had been hitting his mother again. The child was twice as big as me, and I was stupid enough to put an arm 'round him while I turned to shut the door.\" I gave a hollow laugh. \"He had me flat on my back before I knew what was happening, and used his weight to hold me down while his filthy great hands yanked my hair out by the roots every time I moved my head.\" I halted in front of the offside headlight so that he wouldn't retreat any further. \"They couldn't mark me,\" I went on, \"because you'd told Derek I'd be at the police station the next morning. And they couldn't rape me because they didn't want to leave any incriminating evidence inside me.\" I tapped two fingers against my mouth. \"So I got a mouthful of Derek Slater's urine instead.\"\n\nI caught a glimpse of Sam's strained, white face out of the corner of my eye. \"He pissed over my mouth and nose while his son held me down\"\u2014I glanced at the harbor's edge\u2014\"and it's like drowning. You can't breathe\u2014so you drink. And the legacy is that you wash your mouth out every hour of every day as long as you live.\" I lifted my lips in a wolfish snarl. \"They swapped places while I was choking to give Alan his turn\u2014but he was too excited and couldn't control himself....\" I fell silent as Sam moved 'round behind the car.\n\nDrury made a half-turn so he could keep an eye on Sam as well. \"No one's going to believe you,\" he said, \"not if there's no record of such a crime being committed. And why focus your anger on me, anyway? Why not blame your husband for abandoning you? If he'd had any guts he'd have stood by you instead of protecting his tart.\"\n\nI had time to think that Drury was a shocking judge of character before\u2014with one galvanizing charge\u2014Sam launched at him, head down, and shunted him into the estuary after my rucksack.\n\n# twenty-three\n\nSam doubled up and backed away from the edge, roaring obscenities from an overdose of adrenaline, but I stayed to watch Drury rise. Luke had assured me that the westerly tidal stream in Weymouth harbor would carry a floating body toward the pontoons, but I had a small twinge of concern about how good a swimmer Drury was. When his face bobbed to the surface, we stared at each other for a moment before I gave him a one-fingered salute and turned away. _Gotcha!_\n\n\"We ought to call the police,\" said Sam, taking deep breaths to calm himself while he watched the man swim to safety.\n\n\"He can do it himself if he wants to. He knows our address.\" I walked back to the car. \"But he won't. He'll bury his head in the sand and hope this counts as an eye for an eye.\"\n\n\"And does it?\" he asked, following me.\n\n\"No chance,\" I said cheerfully, opening the passenger door. \"He still has to answer for Annie, and he'll only do that when his name's plastered across every newspaper in this country with 'racist'attached to it.\" I slid on to the seat. \"Come on,\" I called, buckling my seat belt, \"let's shift. He'll be after your blood if I know anything about anything. Not reporting you to the police doesn't mean he won't break your jaw at the first opportunity.\"\n\nSam scrambled in beside me and fired the engine, twisting 'round to reverse the car out on the road. \"I should have seen to him twenty years ago,\" he said as he spun the wheel. \"I would have done, too, if I hadn't believed him.\"\n\n\"About Annie?\"\n\n\"No,\" he growled, \"about you stalking him. I know it sounds absurd now but at the time it seemed to make sense. The way you went off me after Annie died... the hours you spent at the police station... the fact that you were prepared to talk to him and not me.\" He eased the car forward and pulled out onto the road. \"I started to think he was more your type than I was.\"\n\n\"That figures,\" I said sarcastically, reaching across him to buckle his belt. \"I mean he had everything I wanted in a man: _hair_ , a uniform, not to mention an enormous dick, which he kept permanently erect for the purposes of rogering every bit of totty that crossed his tracks.\"\n\nHe gave me a sheepish grin. \"Actually, I'm being serious. I was incredibly jealous but I didn't think I had much of a leg to stand on after Libby Then you got pregnant, and I thought, _Shit, is the baby mine or Drury's?..._ and I was so bloody churned up that when you agreed to try to make a go of it, all I could think about was getting away, burying the whole bloody saga and starting again.\"\n\nI was so surprised that I felt as if my jaw had just hit the floor. \"You thought Luke was Drury's?\"\n\nHe nodded.\n\n\"Good God! What on earth gave you that idea?\"\n\nHe took his foot off the pedal and the car slowed to a crawl. \"Because the only time we had sex throughout that whole miserable period,\" he said with a sigh, \"was when I forced myself on you and you told me you never wanted to see me again. You really hated me that night... and I couldn't believe that something that was done with so much viciousness could produce something so grand.\"\n\nI shook my head in amazement. \"Why didn't you say something?\"\n\n\"Because it didn't matter,\" he said simply. \"I always thought of Luke as mine whether he was or not.\"\n\nI was humbled. If our roles were reversed\u2014if Libby had given birth to Sam's child\u2014I could never have been that generous. \"But of course he's yours,\" I said, touching the back of my hand to his cheek. \"You should never have doubted it for a minute.\"\n\nHe leaned his head to one side, trapping my hand against his shoulder. \"I haven't for a long time... not since Tom was born, anyway, because they looked so alike.\" He gave an abrupt laugh. \"Then you insisted on bringing me here for lunch so that Drury could leer at you, and I thought, _Is this the first step to telling the sod that my son is really his?\"_\n\nI snatched my hand away. \"You said you didn't recognize him.\"\n\nHe speeded up again. \"I never forget the faces of men who make me jealous.\"\n\n\"There haven't been any.\"\n\n\"That's what you think.\" He leaned forward to wipe mist from the screen. \"Where are we picking up the boys?\"\n\n\"Beyond the swing bridge.\"\n\n\"Well, be prepared for some embarrassed silences,\" he warned matter-of-factly \"I spotted them creeping in behind one of the other cars, so I think the chances are they heard every word.\"\n\n\"Damn!\" I said with sudden weariness, leaning my head against the seat. \"I told them to make themselves scarce.\"\n\n\"Mm, well, I suspect curiosity won out. You can't blame them. We've both been behaving very oddly lately. It could makes things difficult with Danny,\" he warned again. \"And I'll have to come clean about Libby... why I lied... why I ignored Annie. It's only right they should hear the truth from me.\"\n\n\"It's not what I wanted, Sam,\" I said with a sigh. \"It was supposed to be just you who heard it because I didn't think you'd believe it if I told it to you cold.\"\n\n\"You should have trusted me,\" he said lightly. \"I stopped being a bastard twenty years ago.\"\n\n\"I know.\" I felt tears prick behind my eyes. \"But I could never find the right time to tell you. I'm sorry.\"\n\n\"Well, I'm not,\" he declared with sudden boisterous good humor. \"You've got more balls than an entire rugby team, my girl, and it's about time the boys found out what an amazing mother they have.\" He slapped his hands against the steering wheel. \"I keep thinking of this Chinese proverb Jock quoted to me the other day. It's a variation on the theme of 'everything comes to him who waits'\"\u2014he turned to me with another grin\u2014\"and it's peculiarly apt in the present circumstances.\"\n\n\"How does it go?\"\n\n\" 'If you sit by the river long enough the bodies of all your enemies float by.'\"\n\nI thought I knew the man I married before that night, but now I know I could live to be a hundred and still not understand the twists and turns of human nature. I don't know what he said to the boys but whatever it was made them treat me like a valuable antique for twenty-four hours, until I started effing and blinding out of pure frustration, and normal service was resumed. They carefully avoided any reference to the Slaters, all of them understanding that it is one thing to reveal the presence of a scar, quite another to have it split open under the pressure of constant examination.\n\nNevertheless, it wasn't a subject that could be avoided for ever and, after much shuffling of feet on Saturday night, Tom confessed they were supposed to be meeting Danny Slater for a drink but weren't sure whether they should. Sam and I said in unison that Danny bore no responsibility for what his father and brother had done and that it wouldn't be fair to tell him. Leave him in ignorance was our advice.\n\n\"Has Dad told you he's thinking of letting Danny use the barn as a studio?\" Tom asked me. \"Assuming we buy the place, of course.\"\n\n\"It's just an idea at the moment,\" said Sam, \"but I'd like him to know that we aren't just fair-weather friends.\"\n\n\"He'd have to slum it,\" put in Luke, \"because Dad won't let him smoke dope in the house. But he can clean out the tack room and make it reasonably habitable. There's electricity down there and the loose boxes are big enough to work in. All he'd need to do then is beg some stone off one of the quarries, and he could have a bash at being a sculptor without having to bankrupt himself in the process.\"\n\nThree eager faces turned toward me. What did I think?\n\nI nodded and smiled and said it was a grand idea. But I knew it wouldn't happen. Danny would never forgive me for what I was about to do to his family.\n\nThe following Monday I visited Michael Percy in prison on Portland. It was a troubling experience because I was constantly reminded that his life was in limbo. Perhaps the extraordinary setting of the Verne, built inside an old citadel overlooking the harbor and standing alone at the end of a series of hairpin bends, added to my sense of unfulfilled promise and waste. Certainly, I felt its isolation very strongly and wondered if the same feeling was shared by the inmates.\n\nThe weather had turned blustery again and the wind plucked at my hair and clothes as I scurried from my car to the main entrance in the wake of a huddle of similarly windblown visitors. I hung behind them, following their lead, unwilling to show my ignorance in front of older hands who, by their relaxed expressions, had queued at reception a hundred times to present their visiting orders.\n\nI thought of Bridget repeating this process month after month, year after year, and wondered if it was a cause for depression or happiness that at the end of it she would see her husband. For myself, I was overcome by a frightening regression to the agoraphobia of twenty years ago when I hadn't been able to leave my house for fear of being watched. Perhaps it had something to do with the officers'uniforms\u2014or being touched during the searches\u2014or having to sit at a table, twiddling my thumbs until Michael was brought to me\u2014certain that everyone's gaze was upon me, even more certain that their gazes were hostile.\n\nWhichever the case, his arrival was a relief, and I watched him walk toward me with an intense\u2014and pleasurable\u2014recognition. _There is no accounting for taste_ , I thought. He was as bad\u2014 _if not worse_ \u2014than Alan, but, like Wendy and Bridget and every other woman he'd ever met, I imagine, he had won a place in my affections. He gave me a shy smile as he shook my hand. \"I wasn't sure you'd come.\"\n\n\"I said I would.\"\n\n\"Yeah, but not everyone does what they say.\" He dropped into the chair on the other side of the table and scrutinized my face. \"I wouldn't have recognized you if they hadn't said it was Mrs. Ranelagh.\"\n\n\"I've changed a bit.\"\n\n\"That's for sure.\" He tilted his head to one side to examine me, and I became very aware suddenly that the fourteen-year-old didn't exist anymore and this was a thirty-five-year-old man with a troubled background and a history of violence. \"Any reason why?\"\n\n\"I didn't much like that person,\" I said honestly.\n\n\"What was wrong with her?\"\n\n\"Too complacent by half.\" I smiled slightly. \"I decided to try lean and hungry instead.\"\n\nHe grinned. \"I bet it made your husband sit up and take notice.\"\n\nI wondered if he'd known about Sam and Libby or if his intelligence was even more acute now than when I'd known him in school. \"It helped,\" I agreed, scrutinizing him in return. \"You haven't changed a bit, although Mrs. Stanhope, the vicar's wife, claims not to have recognized you from the photograph in the newspaper. She's still hoping it was a different Michael Percy who robbed the post office.\"\n\nHe ran the flat of a hand across his closely cropped hair. \"Did you tell her?\"\n\n\"I didn't need to. I'm sure she knows.\"\n\nHe sighed. \"She was pretty decent to me when I was a kid. I bet she was wrecked to find out I got done for pistol-whipping a lady.\"\n\n\"I doubt it. She has no illusions about you.\"\n\n\"She offered to adopt me one time, you know, and I said, You've gotta be joking.'It'd be like going from the ridiculous to the gorblimey On the one hand there was Mum who couldn't give a shit if I never came home... on the other there was the vicar who kept giving me lectures about how Jesus could change my life. The only one who was halfway sensible was Mrs. S.... but she kept wanting to hug me, and I didn't much fancy that.\" He leaned forward to create an enclosed space for us among the intrusive hubbub of conversation around us. \"I wouldn't have minded _you_ giving me a hug,\" he said with an amused up-from-under look, \"but you never showed much inclination.\"\n\n\"I'd have been sacked on the spot.\"\n\n\"You weren't sacked when you gave Alan Slater a hug.\"\n\n\"When did I give Alan a hug?\"\n\n\"When he bawled his eyes out because the nurse found lice in his hair again. You put your arm 'round his shoulder and said you'd give him some shampoo to get rid of them. You never did that for me.\"\n\nI had no recollection of it\u2014as far as I knew I'd only put my arm 'round Alan once\u2014and I wondered if Michael was confusing me with another teacher. \"Did you ever have lice? You always looked so spick and span while, most days, poor Alan smelled as if he'd emerged from a sewer.\"\n\n\"He was a slob,\" said Michael dismissively \"I used to nick Prioderm out the chemist for him but he never bothered to use it until the nurse spotted eggs in his hair.\" He favored me with a crooked smile. \"It bugged me that everyone thought I was a neat little kid with clean clothes and felt sorry for Alan because he came from a shit background. I started washing my own stuff myself when I was six years old, but it was only ever Mum who got credit for it.\"\n\nI wondered fleetingly if the hug I gave Alan, and the hug I hadn't given Michael, had resulted in one settling down and the other doing fifteen years. \"Most people thought she was a better mother than Maureen,\" I told him, \"but it wasn't much of an endorsement. On a scale of one to ten, Maureen scored nought.\"\n\n\"At least she wasn't a prostitute,\" he said bitterly. \"It does your head in to have a slag for a mother. Did you know that's what she was at the time?\"\n\n\"I didn't know anything, Michael. I was very na\u00efve and very stupid, and if I had my life over again I'd do things differently.\" I watched him for a moment. \"You were too sexually aware,\" I said gently. \"I never felt threatened by Alan in the way I felt threatened by you. I didn't think you'd be content with a hug.\"\n\nHis smile became even more crooked. \"Maybe not, but I'd have been too scared to do anything about it.\"\n\n\"That's not how I saw you,\" I said with a small laugh. \"You had a knack of singling out women who were vulnerable... like Wendy Stanhope. She becomes very wistful when she talks about you, so I doubt her feelings were entirely maternal.\"\n\n\"What about yours?\"\n\n\"I don't know. I never tested them.\"\n\n\"But you _did_ like me?\"\n\nI wondered why that was important. \"Oh yes.\"\n\n\"What about Alan? Did you like him?\"\n\n\"No,\" I said flatly, wondering how much he knew.\n\n\"He had a crush on you,\" he said. \"Used to talk about how you couldn't keep your hands off him and how the only reason you refused to get the police involved when you caught him nicking from your handbag was because you were afraid he'd spill the beans about the sex he'd had with you.\" He examined my face closely and seemed to find the reassurance he wanted. \"I knew it was a load of crap but it used to bug me the way you put yourself out to be nice to him.\"\n\nI didn't say anything.\n\n\"And you're wrong about him not being sexually aware,\" he went on. \"He was so damn big he had tackle the size of an elephant's by the time he was ten. Sex was the only thing he thought about. He used to nick porno mags and wank himself stupid over the pictures. It was pretty funny till he started doing it for real. He got hold of Rosie, Bridget's sister, and said he wanted to do it with her, and when she told him to fuck off he pushed her to the ground and said he was going to do it anyway. Poor little kid, she was only twelve and she didn't stop bleeding for weeks.\" His mouth thinned angrily at the memory. \"But she was too frightened to tell anyone except me. Her Mum was ill and her dad was never around. So it was down to me to do the business. I beat the shit out of Alan and said if he ever did something like that again, I'd rip his head off.\"\n\n\"How old were you?\"\n\n\"Fifteen. It wasn't long after you left.\"\n\n\"Did he do it again?\"\n\nMichael shrugged. \"If he did, I never got to hear about it. He turned on his dad with a baseball bat a week or so later... almost as if his brain caught up with his size and a bubble came out of his head saying, 'I'm big enough to take on guys.'After that, he didn't seem so interested in sex.\"\n\nI tried to get a grip of the timing. \"His wife told me you and he came to blows over Bridget.\"\n\nHe shook his head. \"We only fought the once, and that was over Rosie.\"\n\n\"She told me Alan was besotted with Bridget until he found her in bed with you... then he beat you half to death and spent time in juvenile prison for it.\"\n\n\"In his dreams maybe.\" He pulled a puzzled frown. \"Bridget never gave him a second look after what he did to her sister, so why pretend otherwise? Who's he trying to con?\"\n\n\"Beth?\" I suggested. \"His wife.\"\n\n\"Why?\"\n\nIt was my turn to shrug.\n\n\"Stupid bugger. It's always better to be honest\"\u2014he smiled as he listened to himself\u2014\" _after_ you've been caught anyway. Nothing remains secret very long in this kind of environment.\"\n\nI looked around the room, which was packed with prisoners and their families\u2014all talking, all listening, all under observation from prison officers\u2014and I thought I could easily believe it. There was no privacy in a goldfish bowl. And I wondered what sort of control Maureen Slater exercised over her family that no hint of Alan's viciousness had ever leaked out.\n\n**_Letter from John Howlett\u2014RSPCA inspector who entered  \nAnn Butts's house on the morning after her death\u2014  \nnow resident in Lancashire\u2014dated 1999_**\n\n_White Cottage  \nLittlehampton  \nNr Preston  \nLancashire_\n\nMs. M. Ranelagh  \nLeavenham Farm  \nLeavenham  \nNr Dorchester  \nDorset DT2 XXY\n\nAugust 11, 1999\n\nDear Ms. Ranelagh,\n\nMay I say first how heartened I am by what you wrote. I have always been troubled by what we found in Miss Butts's house, and I feel so much happier to be asked to view it from a different perspective. As you so rightly suggest, I never had any reason to believe Annie was cruel until after she was dead.\n\nDr. Arnold was of the opinion that Annie had been robbed in the days before her death and suggested this was the cause of the rapid decline in her circumstances which we found on 15.11.78. While I had some sympathy with that view, I never felt it adequately explained the number and\/or condition of the cats. The police \"take\" on the matter was that Annie was a difficult and disturbed woman who was clearly unable to look after herself and whose behavior had given rise to numerous complaints. What we found in her house, therefore, merely confirmed this belief. It's worth mentioning here that PS Drury told me an hour in advance of entering the house that there were in excess of twenty cats on the premises in order to ensure I brought enough cages to accommodate them. When I questioned this figure, saying that in my experience there had never been more than seven, he said it was based on information received from neighbors.\n\nI blame myself now for not asking how her neighbors could be so exact about numbers, but it's easy to be wise with hindsight. At the time, my colleague and I were so shocked by what was there that all our efforts went into assessing and rescuing the animals. It would have been different had Annie still been alive because we would certainly have sought to prosecute on the grounds of cruelty, but her death meant that we effectively handed the responsibility for asking questions to Sergeant Drury. I know that Dr. Arnold had severe reservations about his handling of the case\u2014and it would seem from your letter that you do, too\u2014but in fairness I should stress that he was as shocked as we were by the conditions in the house and said several times, \"I should have believed them.\" By this I assume he was referring to her neighbors, whom he described constantly as \"low-life.\" I say this only to remind you that he, and we, were dealing with a situation that, even if it was unexpected, did in fact bear out everything that had been said about Annie for the last twelve months.\n\nWith respect to your specific questions: Annie said her \"marmalade\" cat had died of \"heart failure.\" She was extremely distraught about it and asked me several times if I thought cats felt pain in the same way we did. I said I didn't know.\n\nMost of the live cats were malnourished\u2014except the six I was able to identify as hers. Several of the strays had bald patches 'round their muzzles, but in almost every instance the fur was beginning to grow back. I'm afraid there was no evidence that \"efforts had been made to help them.\" Rather the reverse, sadly, as the only sensible help would have been a visit to the vet. However, if your premise that the cats'mouths were taped by someone other than Annie, then clearly the removal of the tape and the purchase of chicken and milk, etc., were an indication of \"efforts to help.\" Her own cats were in noticeably better health than the rest.\n\nI'm afraid it's impossible to say how much time had elapsed since the tomcats'mouths were taped, simply because their condition when we found them was so appalling. However, I take on board your suggestion that Annie was unlikely to render them helpless only to release them again.\n\nIf I accept your premise that it wasn't Annie who brutalized the animals, then I can also accept your premise that the reason we found sick cats shut into the back bedroom was because she wanted to protect the vulnerable cats from the rest. However, and sadly, I can recall no evidence from the postmortems to prove this, as we had no way of telling if the cats were confined after being bitten and scratched, or before.\n\nAssuming the above premises to be true, then it is certainly possible that the healthy cats killed the sick ones and that the ones with broken necks were the result of \"mercy killings.\" However, if Annie confined the sick toms to protect them from the others, they may well have turned on each other within the confines of the room. I agree that Annie may have chosen to confine the cats inside the house\u2014despite their fouling the floors\u2014in order to protect them from a greater danger outside.\n\nIn conclusion, I am a great deal happier with the suggestion that Annie was a savior of cats rather than a tormentor of them, though I fear you will have difficulty proving it.\n\nWith best wishes for a successful campaign,\n\n# twenty-four\n\nI asked Michael when he last saw Alan. \"We stopped hanging around together after he hurt Rosie,\" he said, stroking his jaw in thoughtful reminiscence. \"If I remember right, I didn't see hide nor hair of him from about '80 on... but I was in and out of the nick myself on a pretty constant basis which probably accounts for it.\" He shook his head. \"It's pretty bad when you think about it.\"\n\n\"What?\"\n\n\"That there were only two families in that whole road that couldn't keep out of trouble. The Percys and the Slaters. We had the same chances as everyone else, but never used them. Do you realize we must have done over twenty years in prison between us\u2014what with Derek and me, and whatever it was Alan did?\"\n\n\"Habits are hard to break,\" I said.\n\n\"Yeah, like Rosie's.\"\n\n\"What happened to her?\"\n\n\"OD'd on smack in a squat in Manchester about five years ago,\" he said bitterly. \"Some idiot dealer was selling it uncut around that time so it was probably accidental and not deliberate. Bailiffs found her body under a mattress the day after her mates vacated the place. The police reckoned she'd been dead three days, but no one did a thing about it... just left her there while they packed their bags and scarpered.\"\n\n\"I'm sorry.\"\n\nHe nodded. \"It was pretty sad. Bridget kept trying to get treatment for her, but Rosie couldn't hack life without it. She always said she'd die of an overdose, so I guess she wouldn't have minded too much if she knew what was happening to her.\"\n\n\"What did her father say?\"\n\n\"Zilch. I'm not sure he even knows she's dead. The girls stopped talking to him after he shacked up with Mum.\"\n\n\"Couldn't you have told him?\"\n\n\"No way. He kicked me out when he moved in. That's when I started living with Rosie and Bridget.\" He jammed his hands between his knees, shoulders hunched in sudden anger. \"He really hates me... persuaded Mum I was no good,\" he said resentfully, \"even though I was the one that looked out for her when it mattered.\"\n\n\"When was that?\"\n\nHe turned away so that I couldn't see his expression. \"It's not important.\"\n\nI was sure it was, but couldn't see the point of pursuing it as he clearly didn't want to tell me. \"What did you do to make Geoffrey dislike you so much?\"\n\n\"Told Rosie and Bridget he was one of Mum's clients. He was a two-faced bastard... kept making out what a saint he was to have given up his job to look after his dying wife... while all the time he was 'round at our place. It was the girls who did everything for their ma. Geoff did sod all except complain when his dinner was late. Vivi-enne was a nice lady. I used to sit with her most afternoons, and it pissed me off to hear her talking about Geoff as if he'd been good to her.\"\n\n\"Did she ever find out about your mother?\"\n\n\"I don't think so. She died with a smile on her face, so I reckon he fooled her to the end. Me and the girls never told her anyway. It didn't seem kind.\"\n\nA small silence fell while I wondered what to say next and immediately unwanted sounds crowded in on us\u2014raucous seagull cries from the skylights above our heads, laughter, a baby's cry from the children's play area\u2014and I found myself blurting out the one question I had been determined to avoid: \"What on earth are you doing here, Michael? How can a man who's kind enough not to tell a dying woman that her husband's cheating on her attack an innocent stranger in a post office? It doesn't make sense.\"\n\n\"I needed the cash,\" he said simply, \"and it seemed like a good idea at the time.\"\n\n\"And now?\"\n\nHe gave a mirthless laugh. \"Now I reckon it's the dumbest thing I ever did. I was only planning to frighten her... hold the pistol to her head... but she started screaming and shouting... and I went crazy.\" He fell silent, contemplating some private darkness. \"She reminded me of Alan's mum,\" he said abruptly, \"so I smashed her ugly face in. I really hated that bitch. It was her who used to get everyone worked up.\"\n\n\"How?\"\n\n\"Just stuff,\" he said before lapsing into another longer silence.\n\nI changed the subject by asking him what he'd meant in one of his letters when he said Bridget had posted her hair through my letter box as a \"sacrifice.\" \"Sacrifice for what?\" I asked.\n\nHe was more comfortable talking about Bridget. \"All the bad things that were happening to you,\" he said. \"You told her once that you wished you had hair like hers, so she thought if she gave it to you the bad stuff would stop.\" He smiled at my expression. \"Okay, it was a bit wacky but she always did have weird ideas. She put a load of raw onions into her mother's room one time because she read somewhere that onions absorb disease, but the smell was so bad that Vivienne couldn't sleep.\"\n\n\"I think they're supposed to work on colds,\" I said abstractedly, while pondering the rest of what he'd said. \"What made Bridget think bad things were happening to me?\"\n\n\"You looked so scared all the time,\" he said matter-of-factly \"It stood to reason there was some lousy shit in your life.\"\n\n\"Did you know what it was?\"\n\nA flicker of emotion crossed his face. \"We guessed they were doing to you what they did to Annie.\"\n\n\"Who?\"\n\n\"The Slaters. I saw Alan's dad try to barge you off the pavement one day... and his mum used to call you a nigger-lover. She said you'd be lynched for the things you were saying if we lived in America.\"\n\n\"What about _your_ mother? Did she agree with Maureen?\"\n\nHe looked away again, as if the subject of his mother was something he found hard to deal with. \"I don't know,\" he said curtly. \"We never talked about it.\"\n\n\"Did you talk about Annie's death?\"\n\n\"No.\" Even more curt.\n\n\"Why not?\"\n\n\"What was to talk about? Hell, we were glad to see the back of her. It meant Mum could take in more clients without having abuse bellowed at them through the wall. And that's all she was interested in,\" he finished bitterly, \"making money out of saps.\"\n\n\"It was a vicious circle,\" I told him. \"Every time you or the Slaters ratcheted up your aggression, Annie got worse. She might have been able to control her language if you'd left her alone, but she hadn't a hope in hell's chance once you started invading her space and making her afraid.\"\n\nHe shrugged. \"Mum always said she should be in a loony bin.\"\n\n\"Only to give herself something to feel superior about,\" I murmured. \"She didn't like being called a 'whore'... because that's what she was. The Slaters didn't like being called 'trash'... because that's what they were.\"\n\nHe gave a surprised whistle as if the comfortable image he had of me had suddenly been shattered. \"That's a bit harsh.\"\n\n\"Do you think so?\" I asked mildly. \"I've always thought how generous Annie was. Had I been her, I'd have come up with something far stronger to describe low-grade scum who got their rocks off torturing cats.\"\n\nHe flinched perceptibly.\n\n\"Was it you and Alan who did it?\" I asked. \"It's the kind of brutality I can imagine you enjoying... inflicting pain on something smaller and weaker... then pushing the sad little remains on to Annie to see how she'd react. Was it Derek killing the marmalade cat that gave you the idea or was Maureen lying about that to protect Alan?\"\n\n\"Jesus!\" he said with a spurt of anger. \"And you wonder why I hate the bitch? Talk about fucking twisted. Alan used to say her brains were shot because his dad knocked the sense out of her, but I'd say it was the other way 'round. The bitch was born twisted and that's the reason the poor sap went for her.\" He leaned forward aggressively. \"It was Maureen killed the cat, and she did it because it made her feel good. She got Alan to hold it down on the kitchen table while she beat its brains out with a baseball bat, and when Alan started blubbing because he really liked animals, she took the bat to him instead and said, if he ever told on her, she'd nail the next one to the fence and make him watch it while it died.\"\n\nIt was like a floodgate opening. Once Michael started on his hatred of Maureen he couldn't stop. He talked about her lousy parenting, her drinking, her vilification of him and his mother. \"It makes me sick what she's got away with,\" he finished angrily. \"It makes me even sicker that she's on the out and me and Derek are stuck inside.\"\n\n\"What would she have been charged with?\"\n\n\"Assault and battery of her kids... drunk and disorderly... you name it.\"\n\n\"Killing Annie?\"\n\nHe didn't answer immediately. \"All I know,\" he said then, \"is what I told you in my letter. That I came home from the arcade to hear that the stupid cow had died in the street from some sort of accident.\"\n\nI nodded as if I believed him. \"Did you know the Slaters went into the house later and robbed it?\"\n\n\"Rosie sussed it when the police described old Annie as living in poverty,\" he admitted. \"She reckoned we ought to say something but I didn't want have to explain how any of us knew what was in there.\"\n\n\"Did Alan not mention it?\" I asked curiously. \"You were inseparable at the time. I'd have expected him to boast about how clever they'd been.\"\n\n\"No.\"\n\n\"Because it _was_ clever, Michael,\" I said idly. \"Way too clever for Derek and Alan alone. It was the odd little extras like turning off the mains water... and soiling the floors to give the impression of self-neglect and poor hygiene. I've always wondered why that was necessary. Unless the smell of human urine was stronger than cats'urine and needed explaining.\"\n\nHe shook his head, but whether in denial that he knew what I was talking about or in refusal to answer the question, I couldn't tell. From the way he started to look for an officer to rescue him from me, it was clear the whole subject made him as uncomfortable as talk of his mother.\n\nI plowed on determinedly. \"You said it makes you sick that Derek's in prison,\" I reminded him. \"Does that mean he's in at the moment?\"\n\n\"He got two years in February '98. A guy on my wing shared a cell with him in Pentonville before he got shipped down here. He reckons Derek's dying. His liver's packed up with the drinking, and the one brain cell he has left can just about remember his name... and fuck all else.\"\n\n\"When's he due for release?\"\n\nHe made a quick calculation in his head. \"He'll have served half so he'll be out by now... assuming he's not dead already.\"\n\n\"What was he convicted of?\"\n\n\"Burglary,\" said Michael dispassionately. \"It's what he gets done for every time.\"\n\n\"Why does that make you sick?\"\n\nHe gave an unexpected sigh. \"Because he needs an education, not endless bloody punishment. Him and me were on the same landing in the Scrubs when I was on remand for this one. He's completely illiterate... just about manages a 'd'and 'e'for his signature but can't get to grips with the 'r'or the 'k.'I wrote some letters for him to his kids, but the only one who ever answered was Sally, and then only because she thought he might have some dosh hidden away somewhere. It pissed me off, it really did. The poor sod was only trying to tell them he loved them, but as far as they were concerned he didn't exist.\"\n\nI was surprised. \"You used to hate him when you were a child.\"\n\nMichael shrugged. \"It doesn't mean I can't feel sorry for him. I got to realizing how limited a guy's life is if he can't read and write. It's pretty mind-blowing when you think about it. I mean, you can't apply for a job if you can't sign your name to a form... and people sure as hell look down on you if they think you're an ignorant jerk. I reckon it's what made Derek violent. The only way he could get people to respect him was to slap 'em about and make 'em afraid of him.\"\n\n\"Is that his excuse?\"\n\n\"No. He's not into excuses. Maybe that's why I feel sorry for him. He told me a bit about his childhood... how he got dumped in institutions because his mum didn't want him, then legged it to live on the streets till he was nicked for shoplifting and sent to Borstal. That's why he's illiterate, never stayed in school long enough to learn basic skills. It makes you realize how important love is to a kid. If his mum had wanted him\"\u2014he pulled a rueful face\u2014\"maybe he'd have been one of the good guys.\"\n\nI guessed he was talking as much about himself as he was about Derek. \"Everyone has to deal with rejection at some point in their lives,\" I said.\n\n\"Worse when you're a child, though,\" he said bleakly. \"There has to be something wrong with you if even your mum doesn't like you.\" He fell silent, squeezing his fists in an echo of Drury \"Derek reckoned he only married Maureen because she reminded him of his mum,\" he said suddenly. \"He had this black and white photo of her, and it was the spitting image of Maureen... skinny and slitty-eyed... called her a sidewinder.\"\n\n\"As in snake?\"\n\nHe nodded.\n\n\"Why?\"\n\n\"Because she never looked him in the face... just stabbed him in the back. It sounded halfway reasonable till I realized he feels like that about all women. 'They're all snakes,'he said, and snakes have shapes. If you can't recognize the poisonous ones you're a dead man.\"\n\n\"How did Maureen stab him in the back?\"\n\n\"Gee-ed up Alan to take him on. It was like a war zone in there, went on for months. If we had our windows open we could hear the fights all the way past Annie's empty house... the screaming and yelling... bodies being slammed against walls. It was like the minute Annie died all hell let loose.\"\n\n\"Why? What changed?\"\n\nMichael shook his head. \"Mum reckoned they reverted to type. They were bullies and bullies need someone to hit on... so while Annie was alive they hit on her and when she was dead they hit on each other.\"\n\nIt made sense, I thought. People never pull together so well as when they have a common enemy. \"How many times did Maureen end up in the hospital?\"\n\n\"Two or three. But it wasn't Derek who put her there, it was Alan. He was well out of control. It was around the time he raped little Rosie. Derek kept him in check for as long as he could but by the time Alan got to fifteen he was two inches taller than his dad and twice as heavy, and there wasn't much Derek could do to stop him.\"\n\n\"Did they know about Rosie's rape?\"\n\nHe shook his head. \"Not unless Alan told them. Rosie was paranoid about her mum finding out\u2014thought it'd kill her quicker than the cancer\u2014so we kept quiet.\"\n\nI tried to make sense of the chronology. \"And this all happened in '79?\"\n\nHe nodded.\n\n\"Was it Alan who attacked Maureen while I was still living there?\" I thought back. \"Sometime during the February of '79?\"\n\nAnother nod. \"She was drunk one day and started slapping him about when he answered her back. He went for her like a maniac.\"\n\n\"Who called the ambulance?\"\n\n\"Derek. He came in about an hour later and found her on the floor with little Danny trying to clean up the blood. Alan was blubbing in the garden because he thought he'd killed her. Derek had to run to the nearest phone box.\"\n\nI eyed him curiously. \"Did you know this at the time, or did Derek tell you about it afterward?\"\n\n\"Derek told me,\" he admitted, \"but it made sense when I thought about what Alan did to Rosie.\"\n\n\"Except Maureen said Derek did it,\" I murmured.\n\n\"Yeah, well, she's a liar. She snapped little Danny's arm across her knee one time, then swore to the doctors he'd fallen off his bike. Us kids knew it wasn't true because she did it in front of us.\" His lips thinned to threads. \"She was a scary woman, and if we hadn't been such fucking cowards\u2014\" He broke off to stare at the table. \"Derek was right pissed off when I told him about it. That's why he wanted to write letters to his kids. He really cared about them.\" He lifted his eyes to mine. \"I know what you're thinking. Michael's not as bright as I thought. He spends a couple of months talking to a man he despises and ends up getting conned by him. Well, that might be true\u2014I wouldn't go to the wall for it\u2014but the one thing I _do_ know is that Derek's so damn stupid even a moron could run rings around him. Sure, he was a bully and, sure, he used his fists, but he had to be _told_ to do it. He was like a guided missile. Point him in the right direction and give him an instruction and\u2014 _wham!_ \u2014he did the business.\"\n\n**_E-mail from Dr. Joseph Elias, psychiatrist  \nat the Queen Victoria Hospital, Hong Kong_**\n\n**M. R**.\n\n**From:** Sarah Pyang (spyang@victorhos.com)\n\n**Sent:** 15 August 1999 14:19\n\n**To:** mranelagh@jetscape.com\n\nSent as from: Dr. Elias\n\nSuch are the wonders of modern technology! My secretary tells me she received your e-mail yesterday (Saturday) and you wish me to reply by return. Well, I'm happy to do so but I wonder if answers given in haste are wise.\n\nYou pepper me with questions. Who is more to blame: the architect of a crime or the one who carries it out? Should a whole police force be smeared because of one bad apple? Can justice be selective? Can the damage done by a mother to her child be mended? Can rapists be cured? Can children be evil? Is any crime excusable? Should the sins of a father be visited on his family? Should the sins of a mother?\n\nIn a poor attempt at wisdom might I suggest that, if you are honestly seeking justice for your friend, then you arrogate too much authority merely by thinking such things? These are not your decisions to make, my dear. Justice is impartial. Only revenge is prejudiced.\n\nBut isn't prejudice what you've been fighting all these years?\n\nAll best wishes,\n\nJoseph\n\n# twenty-five\n\nIt was three o'clock by the time I drove down to the main road with my brain worrying away at what Michael had said like a tongue at a sore tooth. Each time I negotiated a hairpin bend the panoramic view of Weymouth Bay and Chesil Beach was spread out below me, but I was too absorbed in thoughts on motherhood to notice it. I wondered sometimes if my rush to judgment of the Sharon Percys and Maureen Slaters of this world was a way of punishing my own mother\u2014and by extension myself. For everything I did as a parent was either in mimicry of her\u2014or in defiance\u2014 and I had no idea which was right and which was wrong.\n\nI had few feelings for Sharon beyond contempt for abandoning her son out of embarrassment the minute she acquired a modicum of respectability after Geoffrey moved in with her. Yet I couldn't understand why Michael had seemed so worried every time her name was mentioned when anger would have been a more normal reaction. He'd been angry enough with Maureen. Did Sharon's shying away from society's censure of her son's violence really make her ipso facto incapable of murder? And did Maureen's willingness to keep Alan's violence under wraps, together with my absolute certainty that she was the instigator of the hate campaigns against Annie and myself, make her ipso facto capable?\n\nI was tired, and a little depressed, and I hadn't intended to see Danny that afternoon, but when I reached the T-junction at the bottom of Verne Common Road I took an abrupt decision to turn left toward Tout Quarry. He was still at work on Gandhi when I turned into the gulley fifteen minutes later. \"How's it going?\" I asked.\n\nHe dropped his hands to his sides, resting the chisel and hammer against his thighs. \"Okay,\" he said with a pleased smile. \"How about you?\"\n\n\"I've been to see Michael Percy. He sends his regards, says if you're bored he'll be happy to entertain you for an hour in the visitors'room.\"\n\nDanny grinned. \"A bit of a comedian, eh?\"\n\n\"He has his moments,\" I agreed.\n\nDanny laid his tools on the ground and brushed dust from his arms. \"What would we talk about? I was just a snotty-nosed kid to him.\" He took his cigarettes from his pocket and perched on a rock beside Gandhi. \"He gave me a lecture once when he caught me sniffing glue behind the church.\"\n\nI sat next to him. \"Did it do any good?\"\n\n\"It did as a matter of fact. He was pretty decent about it, said he understood why I was doing it, then gave me a graphic description of what it's like to die of suffocation. He told me I had more going for me than to peg it in a graveyard with a nose full of glue fumes.\" He flicked me a sideways glance full of amused self-deprecation. \"So I tried heroin instead.\"\n\nMy disillusionment must have shown. \"Meaning Mr. Drury's terror tactics were more effective than Michael's lecture?\"\n\nDanny's smile widened. \"I never liked glue-sniffing anyway... and as for heroin\"\u2014he gave a sudden laugh\u2014\"I'd been sitting on the bog for half an hour trying to pluck up courage to stick the bloody needle in before Mr. Drury caught me. I'd always hated the damn things.\"\n\nI eyed him affectionately. \"You were going to give up anyway?\"\n\n\"Sure... injecting at least. I went on smoking it for a while, then I thought, to hell with it. I don't need this. I prefer cannabis. You keep a better grip on things with dope.\"\n\n\"Why didn't you tell your mother that at the time instead of letting Drury take the credit?\"\n\n\"Because she wouldn't have believed me.\" He turned his cigarette in his fingers. \"You wouldn't either. I was a pretty wild kid and it's not easy getting people to change their opinion of you when all you do is let them down.\"\n\nI nodded. I'd seen it myself many times during my teaching career. Give a dog a bad name and he was hanged forever afterward. It was the sort of unforgiving prejudice I hated\u2014as Dr. Elias had so pointedly reminded me. \"What did Michael mean when he said he understood why you were sniffing glue?\"\n\n\"He knew what it was like for me at home. There was only me and Mum and we loathed each other's guts. Most of the time she was passed out drunk\"\u2014he shook his head\u2014\"and when she wasn't, she'd lam into the first person she saw\u2014usually me. It was pretty depressing. She's got real problems but she won't do anything about them... just locks the door and sinks into a stupor.\"\n\n\"Has she ever said what her problems are?\"\n\n\"You mean apart from the physical dependency?\"\n\nI nodded.\n\n\"The same as any other addict I guess,\" he said with a shrug. \"Fear of living... fear of pain... fear of having to look at yourself too closely in case you don't like what you see.\"\n\nI wondered if he was right. \"She seemed all right when I saw her.\"\n\n\"Only because she knew you were coming,\" he said dismissively \"but you can bet she was back in front of the telly with her fags and her booze within five minutes of you leaving. She can put on an act for a while... but she's too lazy to want to make it permanent. It makes me sick.\"\n\n\"Do you ever see her?\"\n\n\"No. Last time was at Tansy's christening. I phone her once in a while just to let her know I'm still alive but the only one of her kids she wants to hear from is Alan. He's always been her favorite. She'd forgive him anything... but not me or my sisters.\"\n\nI nodded. \"What stopped you from being wild?\"\n\nHe thought about it. \"Getting sent down at sixteen for nicking and driving cars,\" he said with a grin. \"Remember I told you I spent time in prison? It was the best thing that ever happened to me. Got me out of Graham Road. Made me think about what I wanted in life.\" He tilted the tip of his cigarette toward Gandhi. \"There was an art teacher who showed me I had a talent for this kind of thing... he was a good bloke... got me a place at art school... even let me live with him and his wife for a while till I found somewhere of my own.\"\n\nPerhaps I'd been wrong to tell Maureen that Beth had worked a change in Danny when it seemed to have been an unknown art teacher who had influenced his life. \"Prison can work then?\"\n\n\"Only if you want it to.\"\n\n\"Did Alan want it to work? Is that how he turned himself 'round?\"\n\nHe shrugged. \"He had a bad time... got bullied because he wasn't too bright... made him scared to go back. Then he met Beth and reckoned he had a future even though she strung him along for ages before she agreed to marry him.\" Another shrug\u2014more dismissive this time. \"Prison doesn't seem to have done Michael much good.\"\n\n\"Or your dad,\" I said slowly, thinking about Alan being bullied and the truism that most bullies are cowards. \"Michael told me he and your dad were in the Scrubs together five years ago.\"\n\n\"Lucky Michael,\" said Danny sarcastically.\n\n\"He said your dad's illiterate... can't even manage his own name. So Michael wrote some letters for him. He said there was one to you which you didn't answer.\"\n\n\"He's lying,\" said Danny bluntly. \"I could be dead for all that bastard cares.\"\n\n\"I don't think so.\"\n\n\"Where did he send it?\"\n\n\"To your mum's house.\"\n\n\"She'd have torn up anything with a prison logo on it. What did it say?\"\n\n\"That he cared about you.\"\n\nDanny gave a snort of derision. \"He doesn't even know what I look like.\"\n\n\"Mm,\" I agreed.\n\n\"I expect he was feeling guilty about abandoning us.\"\n\n\"Mm,\" I said again.\n\nDanny frowned. \"What else did Michael say?\"\n\n\"That you had a broken arm when you were a child. Do you remember that?\"\n\nHe cast an involuntary glance at his right hand. \"Sort of. I know I was in plaster once, but I thought it was something to do with my wrist. It aches sometimes.\"\n\n\"Do you know how it happened?\"\n\n\"I fell off my bike.\"\n\n\"Is that something you remember or something you've been told?\"\n\nA difference in my tone\u2014too much curiosity perhaps\u2014made his brows draw together in a puzzled frown. \"Why so interested? All kids break bones at some time or another.\" I didn't answer and he seemed irritated by my silence. \"Probably something I was told,\" he said curtly. \"I don't remember much before I was six or seven.\"\n\n\"Neither do I,\" I said equably. \"It's odd. Some people have very clear memories of their early childhood, but I have none at all. I used to think the stories my parents told me were real memories, but I've come to the conclusion now that if something is repeated often enough it acquires a reality.\" I paused to watch one of the student sculptors chip nervously at a small block of stone which had so little shape I wondered why he was bothering. \"Michael said he doesn't remember seeing Alan after your dad left,\" I said next. \"Is that the time he went to prison for dealing?\"\n\nDanny appeared to be on safer ground with this question. \"Sure. It's the only sentence he's done. He told me about it once, said it did his head in something chronic.\" He leaned forward to pick a stone from the ground. \"He didn't come home afterward. I think they reckoned he was a bad influence on the rest of us, or vice versa.\" He polished the stone with the ball of his thumb. \"I only found out what he looked like when I skived off school one day to go wandering 'round Twickenham. It was when I was about thirteen, and this big guy stops me in the street and says, 'Hi, I'm Alan, how you doing?'He'd have been about twenty-four by then\"\u2014he gave a hollow laugh\u2014\"and I hadn't a clue who he was. I knew I had a brother somewhere but it was a bit of a shock to find he was only four miles away. He said he'd been keeping an eye on me from a distance.\"\n\n\"Did you tell your mother you'd seen him?\"\n\n\"No chance. She used to get really wound up every time his name was mentioned, then she'd hit the bottle and start breaking furniture. I always thought she blamed Al for making my dad leave until Al turned up out of the blue a year later and she wept all over him and said how much she'd missed him.\"\n\n\"Why did he come?\"\n\n\"Wanted to see her, I guess.\"\n\n\"No, I meant, why _then?_ Why wait so long?\"\n\nHe looked interested, as if it were something he'd never considered before. \"It was after Mr. Drury retired,\" he said. \"I remember Mum saying there was no one left who'd recognize him\u2014\" He broke off abruptly. \"She probably just meant he wouldn't be picked on anymore.\"\n\n\"Is Alan fond of her?\" I asked, remembering what Beth had said about Alan's depression every time he visited Maureen.\n\n\"Maybe. He's the only one who bothers to see her.\"\n\n\"But?\" I prompted when he didn't go on.\n\nHe stretched his right arm and dropped the stone, staring at his hand in absorbed fascination as he flexed his fingers. \"He's scared of her,\" he answered abruptly. \"That's the only reason he goes... to keep her from turning on him.\"\n\nWe went for a walk through the sculpture park, diving down little alleyways between craggy walls of stone. We squeezed through a cleft into a cave where a pink blanket and a pile of empty cans suggested someone had taken up residence or a couple of lovers had found a private retreat.\n\n\"Maybe I should take it over,\" said Danny, \"and sneak out at night to carve the stones by moonlight.\"\n\n\"Do you enjoy it that much?\"\n\nHe made a rocking motion with a hand. \"Not all the time\u2014it can be bloody frustrating when it doesn't go right\u2014but it's what I want to do.\"\n\n\"Sam's willing to let you work in the barn at the bottom of our garden,\" I said, leading the way back out again. \"It'll mean slumming it in the tack room and working with the doors open if you want any light\"\u2014I shrugged\u2014\"but it won't cost you anything. If you can scrounge some stone and don't mind sleeping rough for a bit... it's free and available.\"\n\nHe was less than appreciative. \"I'd freeze to death in the winter.\"\n\n\"Mm,\" I agreed, \"and Sam'll have your hide if he catches you smoking cannabis.\"\n\n\"What about you?\"\n\n\"I never argue with my husband in public so if you come... and he catches you... you're on your own.\" I turned to look at him. \"Think about it, anyway. You won't get a better offer today.\"\n\nHe became very quiet as we approached the car. \"Why would you want to help me?\" he asked, taking the keys from my hand to unlock the door.\n\n\"Think of it as an investment in the future.\"\n\nHe held the door open. \"You'd never make a penny,\" he said gloomily. \"I haven't that much talent.\"\n\nI gave him a quick hug. \"We'll see.\" I lowered myself on to the seat. \"But it's not a financial investment, Danny, more a loan of goodwill that you can repay with interest to someone who deserves a similar chance at another time.\"\n\nHe wouldn't meet my eyes. \"What do you want in return?\"\n\n\"Nothing,\" I said honestly, reaching for the door. \"There are no strings attached. The barn's there if you want to take us up on it. If you don't, no hard feelings.\"\n\nHe shuffled his feet on the gravel. \"Alan's phoned a few times wanting to know what you've been saying about him,\" he said abruptly. \"He's really twitched even though I keep telling him you're only interested in what happened to the black lady.\"\n\nI didn't answer.\n\n\"What did he do to you?\" he asked me.\n\n\"What makes you think he did anything?\"\n\n\"Your face goes blank every time his name's mentioned.\" He put his hand on the door to stop me closing it. \"I'd never go against him,\" he said painfully. \"He's my brother.\"\n\n\"I wouldn't expect you to,\" I said as I started the engine. \"But the offer of the barn has nothing to do with Alan, Danny. If you're happy to come, we're happy to have you. I hope you'll remember that... whatever happens...\"\n\nMy last visit that day was a prearranged one to Sheila Arnold in her office. She and Larry had been away the previous week on a whistlestop visit to the Florida condo\u2014\"keeping Larry happy\" had been her wry description over the telephone\u2014and this was my first opportunity to show her the photographs of Beth and Alan's house. She had agreed to see me at the end of afternoon surgery and was updating some patients'notes on her computer when I dropped into the chair beside her desk. She gave me a quick smile, then pushed her keyboard away and turned to face me.\n\n\"Well?\"\n\nI'd had more copies made from the negatives after Drury's fit of pique with my rucksack, and I produced these from my pocket and spread them across her desk.\n\n\"My God!\" she declared in amazement. \"I thought you were exaggerating when you said you'd found the mother lode.\"\n\nI tapped the bangle on her wrist, then pointed to a close-up of Beth Slater's forearm. \"Snap?\" I suggested. \"She has four of them, and I think she wears them all the time because she pushes them up her arm every time she goes near the sink. I doubt she has any idea they're valuable or even that they're jade. She probably thinks they're plastic or resin.\"\n\nSheila studied a picture of Beth with her children. \"She has a nice face.\"\n\n\"Yes,\" I agreed.\n\n\"You liked her.\"\n\n\"Very much,\" I said with a sigh, \"which makes it difficult to know what to do next. I don't think she has any idea these things were stolen. She told me Alan bought the Quetzalcoatl in a junk shop then started collecting other Mexican pieces because he believes the Aztecs were an alien civilization. Her children were full of it while I was taking the photographs\u2014they think their dad's a genius because he knows more about aliens than anyone else\u2014and it seems rather pointless to make them unhappy just for the sake of proving he was a thief twenty years ago.\"\n\nSheila lifted each picture in turn and studied it closely. \"I remember some of these things,\" she said finally, \"but I couldn't swear to all of them. Also, apart from the bangles and the mosaic, there doesn't seem to be much of any value. What happened to the gold and silver pieces, for example?\"\n\n\"Alan's mother sold them to buy her house,\" I said, \"but I've very little proof to back that up.\" I showed her the Chiswick jeweller's affidavit. \"The description of the woman fits Maureen\u2014along with half a million others who can manage a Birmingham accent\u2014but it's only five items and accounts for less than.\u00a31,000.\"\n\n\"How much did the house cost her?\"\n\n\"About \u00a315,000 in total. She claims it all came from a win on the football pools which is why she didn't have to declare it.\" I lifted an amused eyebrow. \"The house is now worth upwards of \u00a3200,000 and increasing every day as the housing boom takes off.\"\n\n\"My God!\" said Sheila in disgust. \"We didn't get much more than that for our four-bedroom job seven years ago.\"\n\n\"I know. It's sickening.\" I isolated a wide-angled shot of the sitting room. \"Maureen stuffed most of this into the cupboard under her stairs because she didn't think it had any value\"\u2014I smiled ironically\u2014\"and it was still there when you were trying to persuade Drury Annie had been burgled. In fact, as Alan didn't retrieve it until a good ten years later Drury could have found it if he'd bothered to investigate.\"\n\nShe looked annoyed. \"And I would have been vindicated?\" I nodded. \"I'll never forgive Peter Stanhope for accusing me of neglecting her, you know. He said I'd only invented her wealth to make myself look better.\"\n\n\"I know.\" It obviously still rankled with her, I thought, and decided to keep to myself that Drury had known about the Quetzalcoatl long before Sheila had reported it stolen. I wanted an objective opinion, not one given in anger. \"The worst of it is,\" I said, showing her the picture, \"that poor Beth did all this decorating herself to make a Mexican setting for the artifacts... and it seems cruel to take them away just to prove a point. No one else is going to appreciate them as much as she and Alan do.\"\n\nSheila propped her chin in her hands and regarded me solemnly. \"Is this a way of asking me to forget that I ever said Annie was robbed?\"\n\n\"I don't know,\" I sighed. \"I keep wondering if it's right to destroy innocent children's lives over a crime that was committed twenty years ago.\"\n\n\"Except I seem to remember you telling me that if you found Annie's thief you'd also find her murderer. Were you wrong?\"\n\nI studied a close-up of the brass artillery shell that stood in Beth's fireplace with colorful silk flowers fanning out of it like peacock feathers. \"Does it matter?\" I asked her. \"Doesn't the same principle apply whatever the crime? Wouldn't I be choosing the lesser of two evils if I left Annie's death as an accident?\"\n\nShe eyed me thoughtfully. \"It depends how two-faced you want to be,\" she said bluntly. \"That was probably Sergeant Drury's excuse as well... yet you've spent twenty years trying to prove him wrong.\"\n\n**_Correspondence re: a meeting on 20.08.99_**\n\n**Leavenham Farm**\n\n**Leavenham**\n\n**Nr Dorchester**\n\n**Dorset DT2 XXY**\n\nAlan Slater\n\n12 Peasmont Road\n\nIsleworth\n\nSurrey\n\nTuesday, August 17, 1999\n\nDear Alan,\n\nI shall be at your mother's house at midday on Friday, August 20. Please ensure that you and she are both there, otherwise I shall carry out my threat to go to the police despite the pain this will cause your wife, children and brother. You should be aware that I am writing to Sharon Percy and Geoffrey Spalding to insist that they, too, attend.\n\nYours sincerely,\n\nFrom: Mrs. Wendy Stanhope, The Vicarage,  \nChanters Lane, St. David's, Exeter\n\n# twenty-six\n\nIt wasn't until we approached the Kew Road intersection on the outskirts of Richmond that Sam asked me if I knew what I was doing. The drive from Dorchester had taken over three hours and he was remarkably restrained throughout with only the odd bout of swearing at other drivers to betray his anxiety. We had discussed tactics the previous day while sitting over a glass of wine in the sunshine, and the plan had seemed reasonable then\u2014 perhaps plans always do under the influence of alcohol\u2014but the friendly, rolling hills of Dorset were a far cry from the congested rat-runs of London's link roads, and the idea of taking on four potentially violent people in the most anonymous city in the world began to seem dangerously flawed.\n\nEven then, I might have abandoned the whole project if Sam hadn't agreed with Sheila's view. The story wasn't mine to control anymore. And it wasn't a question of the lesser of two evils, he said. More of a Pandora's box. I'd opened the lid and the secrets were out. Danny for one\u2014Michael Percy for another\u2014would start asking questions: of Alan, of their mothers, even of Derek if they could find him. And it wasn't fair for the innocent to be tarred with the same brush as the guilty.\n\nI laid an affectionate hand on his arm as he drew up at the traffic lights. \"Thank you,\" I said.\n\n\"For what?\"\n\n\"Holding back. I know how worried you are, but it has to be more sensible to take an open-minded woman with me than an angry husband who's likely to lose his temper.\"\n\n\"We can still go to the police.\"\n\nI shook my head. We'd been through this a dozen times. \"They wouldn't do anything... certainly not today... probably never. It took Stephen Lawrence's parents seven years to win an inquiry, so I can't see myself walking into Richmond Police Station out of the blue and being believed.\" I sighed. \"I tried that twenty years ago and all it achieved was to persuade everyone I was a head case.\"\n\nHe nodded.\n\n\"In any case I really do want the truth this time, and Wendy was the only person I could think of. Sheila's too conservative to work outside the rules\u2014and Larry wouldn't have let her come, anyway.\"\n\n\"Could he have stopped her?\" asked Sam in surprise.\n\n\"She'd have insisted on it,\" I said cynically. \"She uses him as her get-out card whenever her involvement becomes too onerous.\" I recalled Sheila's horrified refusal when I invited her to confront the Slaters\u2014 _Good God, I couldn't possibly. Larry would never allow it for a minute_ \u2014and I thought how wrong I had been to think that Annie's doctor would be the best support I could have. If I'd had any sense I'd have realized how passive she was when she admitted abandoning Annie's cause at the first sign of irritation from Larry, but I'd been seduced by her sympathetic report on Annie to the coroner and her gutsy defense of herself against accusations of negligence. The real irony, of course, was that I need never have upset my mother by moving the family to Dorchester if only I'd known in advance that an eccentric vicar's wife in Devon had more courage and crusading spirit in her little finger than Sheila Arnold would ever have. \"And, apart from my ma,\" I went on with a sigh, \"I couldn't think of anyone other than Wendy who had the guts to come with me.\"\n\nSam gave an abrupt laugh. \"Did I hear right? Did you seriously consider asking your mother? Is this progress... or what?\"\n\n\"Actually, she was the first person I thought of,\" I said with a wry smile, \"until I realized she'd handbag the lot of them and leave me worse off than when I started.\" I gave an indecisive shrug. \"But it _is_ odd... maybe it's true that blood's thicker than water.\"\n\nHe sobered rapidly as we approached the station. \"Well, just keep that in mind when you're talking to Alan Slater,\" he advised. \"Unless he's a complete idiot, he's bound to realize that the best way to keep his children in ignorance is to stand by his mother....\"\n\nWe were fifteen minutes early but I refused to let Sam hang around to meet Wendy. I was afraid he'd be shocked by her age and her thinness\u2014 in his mind I believe he thought of her as a larger-than-life person, a mighty Valkyrie come to conduct me through a battlefield\u2014and I had visions of him putting his foot down about the whole enterprise when he was presented with the reality. It was worse than I feared. Wendy's early start and long journey from Exeter had exhausted her and, away from the secure confines of the vicarage, the impressive vulture had given way to something with about as much substance and solidity as a stick insect.\n\n\"Oh dear,\" she said cheerfully as she crossed between the taxis on the forecourt in response to my wave, \"do I look that bad?\"\n\n\"No,\" I lied, giving her a warm hug, \"but are you sure you want to go through with it? There'll be four of them and two of us,\" I warned, \"and it could get very rough.\"\n\nShe nodded. \"Nothing's changed then. You made that all very clear on the phone the other day. But don't forget I have the advantage of knowing a few of their secrets\"\u2014she gave a little chuckle\u2014\"so if all else fails I ought to be able to shame them into behaving well.\"\n\n_Or fire them up even worse_ , I thought worriedly. \"It just seems more real now,\" I said lamely.\n\nShe tucked her hand through my arm and turned me firmly in the direction of Graham Road. \"If you'd wanted someone to tan their hides, you'd have invited your husband and your sons to go with you,\" she pointed out. \"Instead you invited me. Now, I can't promise not to let you down\u2014I may fold at the first huff and puff-\u2014but I have no intention of giving up before we've even tried.\"\n\n\"Yes, but\u2014\"\n\nShe rapped me sharply over the knuckles. \"You haven't come this far to walk away at the last hurdle, so let's have no more argument about it.\"\n\nSharon and Geoffrey were standing in their open doorway when we came abreast of their house, but they made no move to come out. \"This is pure bloody blackmail,\" snapped Geoffrey angrily. \"And what's _she_ doing here?\" he demanded, catching sight of Wendy at my side. \"What the hell business is it of hers? She was always poking her long nose in where it wasn't wanted.\"\n\n\"Hello, Geoffrey,\" said Wendy with an amiable nod. \"I see your temper hasn't improved much since I left. You really ought to have your blood pressure checked, my dear.\" She switched her attention to the woman. \"And how are you these days, Sharon? You're looking well.\"\n\nA tight little smile thinned Sharon's lips, as if she suspected the compliment was insincere, although as she'd taken so much trouble with her appearance\u2014 _Intent on putting Maureen in the shade_ , I thought\u2014Wendy had spoken only the truth. \"We're not coming,\" she said. \"You can't make us.\"\n\nI shrugged. \"Then the Slaters can say whatever they like about you and I'll have to accept it because this is the only opportunity you'll have to set the record straight before I go public.\"\n\nThey stared at me with fear in their eyes.\n\n\"Look, I know you were together that night until nine o'clock and that because of it Geoffrey was the last person to speak to Annie,\" I said bluntly. \"And I'm guessing that if I could work that out then so could Maureen.\" I watched their fear increase. \"So what did she do? Demand money?\" I shook my head impatiently as I saw from their expressions that I was right. \"And you have the nerve to accuse _me_ of blackmail?\"\n\n\"You're no different,\" said Geoffrey, clenching his fists. \"Sending us threatening letters... on our backs all the time... trying to ruin our lives.\"\n\n\"If you'd been honest at the time,\" I said wearily, \"I wouldn't have had to write any letters at all. You weren't responsible for Annie's death, Geoffrey, any more than my husband was. He passed her _after_ you\u2014also thought she was drunk\u2014and also did nothing to help her. You were both guilty of unkindness but neither of you killed her.\" I watched his eyes widen in shock and smiled unkindly. \"But I'm glad you've spent so long thinking you did. You deserve to pay some sort of penalty for lashing out when she begged you for help. That's what you did, isn't it? Knocked her down, then panicked when you thought you must have pushed her into the path of the traffic?\"\n\nHe put a nervous hand on the door, but whether to steady himself or slam it in my face it was hard to say. Whatever his intention, Sharon thrust him away and wedged her foot against the bottom of the door. \"Go on,\" she told me tightly.\n\n\"Whoever killed Annie attacked her in her house three or four hours before Geoffrey passed her in the street and those are the injuries she died of. She was beaten so savagely that she passed out... but she came to some time later and found the strength to stagger out into the street to look for help. The most likely time for the assault was around six o'clock but, as far as I've been able to discover, neither of you was in Graham Road at that time, so I can't see what you have to fear by telling the truth.\"\n\nGeoffrey wasn't easily convinced. \"How do we know you're not lying?\" he asked.\n\n\"To what purpose?\"\n\n\"To catch us out... make us say what you want.\"\n\n\"Oh, for heaven's sake!\" said Wendy in sudden exasperation. \"I had no idea you were such a stupid man, Geoffrey. Is the truth really so frightening that you must keep Sharon a prisoner to it?\" Her eyes sparkled angrily. \"Mrs. Ranelagh's trying to help you\u2014though, goodness me, I'm not sure you deserve it\u2014but you'll be tying her hands if you can't find the courage to stand up to Alan and Maureen.\"\n\n\"It's not just them though, is it?\" he said unhappily. \"They've got Derek in there as well.\"\n\nI felt like a rag doll that had just lost all the sawdust out of its knees, and, from the way Wendy clutched at the gatepost, I clearly wasn't the only one.\n\nI should have considered the size of Maureen's sitting room before I picked her house as the meeting place. Barely ten feet square, it was too small to allow each of us the amount of space we wanted, and we grouped ourselves in uncomfortable proximity according to our fragile alliances. This meant the Slaters sat rigidly on a sofa against the internal wall while Wendy, Sharon, Geoffrey and I faced them on hard-backed chairs in front of the window. It was reminiscent of trench warfare during the First World War\u2014and I began to wonder if the outcome would be as futile.\n\nI had been swept with nausea from the moment I saw Derek, and I struggled to contain it as the sour smell of him\u2014more remembered, I think, than real\u2014filled my nostrils. I kept asking myself why it hadn't occurred to me that Maureen would confront me with him, when instilling fear was what she was best at. I tried to speak and found I couldn't.\n\n\"Go on then,\" she said, gloating over my discomfort. \"Say what you have to say, then get out.\"\n\nIt was a strange moment. The anger and bitterness inside me had been through a number of evolutions over the years\u2014from a savage desire to kill, through apathy and a wish to forget\u2014to this, my final position. Most of the time I could delude myself that I was pursuing justice for Annie\u2014indeed I believe that most of the time that's what I was doing. But every so often I recognized that Dr. Elias and Peter Stanhope were right and my motives were based on revenge. If Maureen had kept her mouth shut, I might have been able to persuade myself forever that it was justice I was seeking... but such a surge of hatred shot through me in that moment that I was back where I started.\n\nIf Derek was dying, as Michael had suggested, it wasn't immediately obvious. He was thinner than I remembered, and his hands had the permanent tremor of alcoholism, but he still held his head like a boxer, watching for any opening, and he still radiated an illiterate's aggression. As for Alan, he was just an older, broader version of his brother and I couldn't look at him without thinking of Danny. I had pictured him for half my life as a muscular giant with a child's brain, but the reality was a nervous man with grimy fingernails and a beer gut who strove to keep as much distance from his parents as a three-seater sofa would allow.\n\nIn the end it was Derek who spoke first. His voice had changed very little\u2014hard vowels and glottal stops\u2014and it grated on my ears as it had twenty years ago. \"You can't blame the boy,\" he muttered, putting a cigarette between his lips and lighting it. \"He only did what I told him to do.\"\n\n\"I know.\" I looked at Alan's bent head. \"I've never blamed him.\"\n\n\"Then you'll drop the rest of the stuff if I admit to it? That's what you've come for, isn't it? My head in a noose.\"\n\n\"Not just yours.\"\n\nHis eyes glinted dangerously. \"You brought it on your own head,\" he ground out. \"You shouldn't have set Drury on me... shouldn't have accused me of murdering the nigger.\"\n\nI swallowed bile. \"I didn't,\" I answered, forcing my voice to remain steady. \"Mr. Drury asked me to give him the names of anyone I thought might have a grudge against Annie, so I named Maureen, Sharon and you. But he was only interested in you\u2014probably because you had convictions for assault\u2014and asked me what your grudge was. I said you were a drunken bully who made no secret of your racist views, that you had low self-esteem, a negligible IQ and a 'poor white'mentality. I also told him you were in the habit of punching and kicking anyone who annoyed you, and cited the time you thrashed Michael Percy because he stood up to you after your own son ran away. At no point did I accuse you of murdering Annie.\" I held his gaze for a moment. \"In fact the only accusation I ever made was that you threatened me with what might happen if I didn't keep my mouth shut.\"\n\nHe stabbed a trembling finger at me. \"You lied about that.\"\n\nI shook my head. \"If you'd read my statement you'd have known what I said. But you couldn't read, so you accepted Mr. Drury's interpretation.\" I smiled slightly. \"The funny thing is, I don't even blame _you_ very much either. It's your nature to piss on anything you don't understand, so to condemn you for doing it is about as senseless as blaming a rat for spreading disease\"\u2014I looked at Maureen\u2014\"or a snake for being venomous.\"\n\nThe woman's eyes narrowed immediately. \"Don't drag me into this,\" she snapped. \"It was none of my doing.\"\n\nThere was a short silence while she and I stared at each other with our mutual hatred written strong on our faces. \"But at least you know what Derek and I are talking about,\" I said evenly. \"Which no one else does\"\u2014I gestured to right and left\u2014\"except Alan of course. You see, I've always wanted to know who planned it. It was too\"\u2014I sought for a word\u2014 _\"subtle_ for either of these morons to work out alone.\"\n\n\"Whatever they did, they did off their own bat. Ask them if you don't believe me.\"\n\n\"There'd be no point,\" I said with an indifferent shrug. \"You've already persuaded Derek to take the blame. Just as you always did.\"\n\n\"And how would I do that, Miss High-and-Mighty?\" she demanded with a sneer. \"He's a man, isn't he? He does what he wants.\"\n\nIt was interesting to watch Alan's reactions. He sat between his parents, leaning forward, elbows on knees, staring at the floor, but every time his mother spoke his body leaned perceptibly closer to his father's.\n\n\"I don't know,\" I said honestly. \"Probably by frightening Alan into paying him off. It has to be worth a try. Alan's got so much to lose. A wife and children who love him... a home... happiness.\"\n\nAlan's knuckles squeezed into white knots as I spoke. \"You said you didn't blame me,\" he muttered.\n\n\"I don't,\" I answered, \"but I will if you insist on supporting your mother's lies. I came for explanations, Alan, not to have your father made the scapegoat. Why did I have to be threatened, anyway? Drury had lost interest in the whole subject by that time.... All he wanted was to shut me up because I kept accusing him of racism.... That's the only reason he got Derek fired up.\"\n\nMaureen's lip curled in a sneer. \"You were no better than the nigger,\" she said. \"You called my man a 'poor white'and types like him don't take kindly to insults. Particularly not from a jumped-up schoolteacher who fancied herself way above us. Why wouldn't he want to shut you up?\"\n\nThe depressing part was, I was sure she was telling the truth, at least where Derek was concerned. A woman's sneer was the only motivation he would ever need to assault her. I looked at him. \"Did you piss on Annie, too?\" I asked him. \"Is that why she reeked of urine?\"\n\nHe stared at me through unintelligent eyes.\n\n\"When did you do it?\" I went on. \"Before or after she lost consciousness?\"\n\nHe turned irresolutely to his wife, looking for an answer.\n\n\"None of us touched her,\" she snapped angrily. \"She was in the morgue by the time we thieved her stuff. I've already told you that.\"\n\nIt was such an open admission\u2014and so unrepentant\u2014that you could have heard a pin drop in the silence that followed. And I remember thinking to myself, _This would all be so much easier if I didn't believe her_.\n\n# twenty-seven\n\nAlan stirred unhappily to life. \"Mum's telling the truth,\" he said doggedly. \"Okay, I'm not saying we're perfect\u2014and I'm not saying we didn't go into Annie's house after we heard she was dead\u2014but we're not murderers.\"\n\n\"Then why was her coat stinking of urine when I found her?\" I asked him.\n\n\"She always smelled,\" flashed Maureen sharply. \"And how do you know it was on her coat, anyway? Maybe she pissed her pants after she was hit.\"\n\n\"The smell was too strong and she was curled up in a ball to protect herself. In any case, she must have been saturated in it otherwise the rain would have washed it away.\" I turned back to Alan. \"I think it was a practice run for what you did to me two months later... just as _I_ was a practice run...\" I hesitated, all too aware that Rosie Spalding's father was sitting next to me\u2014\"for the cause of your bust up with Michael Percy.\"\n\nHis eyes flickered involuntarily toward Geoffrey before he dropped his forehead into his hands to mask his expression.\n\n\"That was Michael's doing,\" retorted Maureen so fast that my blood ran cold. _My God! Had she known about Rosie's rape and done nothing about it? She didn't stop bleeding for weeks, Michael had said...._ \"Michael lost his temper for no good reason and went berserk. He's always been dangerous... look at what he's in prison for now.\" She flicked a spiteful glance at Sharon. \"If it's a murderer you want, then concentrate on him\u2014even better, his mother's fancy man. Try asking who was the last person to speak to Annie. That'll give you the answers you want.\"\n\nGeoffrey half rose from his seat, his face purpling with anger, but Wendy laid a restraining hand on his arm and held him back. \"Don't let Maureen set the agenda, my dear. Can't you see she's trying to start a fight by provoking that peppery temper of yours? It's really most interesting. She doesn't want Derek and Alan to answer Mrs. Ranelagh's questions and I'm intrigued to know why.\"\n\nMaureen's mean little eyes slid across to look at her. \"What's it got to do with you?\"\n\n\"Quite a lot considering I was one of your victims. You admit to theft so casually, Maureen, as if it's something to be proud of but your children broke my heart when they stole my mother's brooch. It was quite irreplaceable\u2014the only thing I had of hers\u2014but completely worthless, of course, as you must have discovered the minute you tried to sell it.\"\n\n\"Nothing to do with us. It was Michael who took that.\"\n\nWendy shook her head. \"No,\" she said firmly \"I know exactly when it went. You came seeking shelter, as usual, and kept me talking in the kitchen while your children looked for what they could steal. I blamed myself, of course, as you knew I would. I should have locked all the doors the minute you came into the house. It wasn't as though I had any illusions about you.\"\n\nThe woman smiled unpleasantly. \"Too right. You treated us like dirt.\"\n\n\"Not at all,\" said Wendy firmly \"I made a point of extending the same courtesy to you and your family as I did to everyone else.\"\n\n\"Yeah, well maybe you made that a bit obvious. You never liked us, that's for sure.\"\n\nWendy nodded immediately. \"Yes, that is certainly true,\" she confessed. \"In fact, it was a lot worse. I couldn't _bear you..._ couldn't bear your children... couldn't bear to have you in my house. My heart used to sink every time you came knocking on our door because I knew I'd face a struggle between the complete revulsion you all inspired in me and my duty as a Christian.\"\n\nThe directness of this response took Maureen aback, as if she believed vicar's wives should deal only in euphemism. \"There you are then,\" she said doubtfully. \"That proves you treated us like dirt.\"\n\n\"Oh, I don't think so,\" murmured Wendy, \"otherwise you wouldn't be so surprised to hear me agree with you. I said I _struggled_ with my revulsion, not that I gave into it. Our door was never closed to you, Maureen, not even after the theft of my brooch. We gave you and your children every assistance even though you were quite the most unpleasant family we'd ever had dealings with.\"\n\nI watched Alan's head sink deeper into his hands.\n\n\"What about Michael Percy?\" demanded Maureen belligerently. \"He was a thief same as mine, but you couldn't do enough for him... always turning out to hold his hand while the tart\"\u2014she jerked her chin at Sharon\u2014\"was otherwise engaged. But your pet ends up pistol-whipping old ladies and my lad comes good. So how did that happen, eh? Explain that.\"\n\nWendy shook her head. \"I don't claim to know the answers, Maureen. All I can do is tell the truth as I see it.\" She, too, looked at Alan. \"In any case, it's Alan you should be asking, not me. He's the only one who knows his story.\"\n\n\"Yeah, well, maybe I was a better mother than you thought I was,\" said Maureen triumphantly. \"How do you like that for an explanation?\"\n\n\"You were no better than me,\" said Sharon in a tight little voice. \"The only difference between us was that yours were frightened of you, and mine wasn't.\"\n\n\"More fool you then,\" retorted Maureen, her eyes glinting to have lured the woman into the open. \"Look where it's got you. Your Michael's such an embarrassment to you, you haven't spoken to him in years... or that bitch of a wife who shopped him.\" She gave a harsh laugh. \"Not that I blame you. He was a wrong 'un through and through. Do you think my kids would have thieved if he hadn't shown them how? Do you think Annie would have been stinking of piss if he hadn't found her and done the honors?\" She pointed her cigarette at Sharon's heart. \"That makes you sit up and take notice, doesn't it? You didn't even know he was in her house that night, let alone used her as a piss pot.\"\n\nI glanced irresolutely at Sharon and was shocked by her terrible pallor. \"Are you suggesting Michael killed her?\" I asked Maureen.\n\n\"Helped her on her way maybe. He told Alan he got home about 8:30, saw that her door wasn't properly shut and went in to see if he could nick something. He found her lying on the rug in her sitting room, reckoned she was drunk and thought it'd be funny to piss on her.\" She broke off on a laugh. \"The place stank of cats so he didn't reckon she'd notice when she came 'round.\"\n\n\"What happened?\"\n\nShe gave a careless shrug. \"He said she started moaning so he got the hell out in case she went for him. But the chances are he's lying through his teeth, and he gave her a kicking as well. It's what he liked doing.\"\n\nI glanced at Alan's bent head. \"Was Alan with him?\"\n\n\"'Course he wasn't,\" snapped Maureen. \"He's already told you he never touched her. But you'd rather believe it of him than Michael, wouldn't you? You're like _her\"_ \u2014she cast a baleful look at Wendy\u2014 \"always think well of one and ill of the other.\"\n\nWendy leaned forward, propping her elbows on her knees and examining Maureen curiously. \"Why is it so important that Alan wasn't there?\" she asked.\n\nA ferocious frown gathered on Maureen's face. \"What's that supposed to mean?\"\n\n\"You seem so determined to push the blame on to Sharon's son but, if I've understood correctly, it was _your_ son who performed the same disgusting act on Mrs. Ranelagh a few weeks later. Yet none of you seems too worried about that.\"\n\n\"So?\"\n\n\"It suggests that something worse happened to Annie than happened to Mrs. Ranelagh... something you don't want Alan associated with.\"\n\nWas it my imagination or was Maureen scared? Certainly Alan was\u2014if his head dropped any lower, I thought, it would be touching his knees.\n\n\"Michael told us about it afterward... gave us the idea,\" said Derek suddenly. \"Seemed only fair to do to the nigger-lover what was done to the nigger. They both thought they could badmouth us and get away with it.\"\n\n\"That's right,\" said Maureen. \"But it was Michael did it first, just like always. He was a bad influence, that boy. Everything evil in this street started with him and his mother, but it was always us got pilloried for it.\"\n\n\"What about rape?\" I said cynically. \"Whose idea was that? Because it certainly wasn't Michael's. He thrashed Alan within an inch of his life when he did it to Rosie. Doesn't that count as something evil?\"\n\nIt was a form of words\u2014spoken in angry defense of someone who wasn't there to defend himself-\u2014yet time stood still as soon as I'd uttered them. No one moved on the sofa. It was as if they believed that stillness could somehow freeze us all in time and space and leave my knowledge forever unspoken. My first reaction was surprise that Derek seemed to understand what I was talking about until I remembered Michael saying that Alan hadn't come to blows with him until after Rosie's rape.\n\nMy second reaction was entirely physical as the reason for their petrified expressions dawned on me. _Alan had raped Annie, too...._ Oh, dear God! Forget control. Forget justice. Forget _revenge_. Twenty years of reasoned evolution were overturned in a second, and I regressed to a primeval desire to kill.\n\nI leaped on Alan like a tigress\u2014my revulsion\u2014my fear\u2014my hatred\u2014 _everything_ \u2014racing in a torrent through my blood. \"You _FUCKING_ little _SHIT!\"_ I roared, slamming his head against the wall. \"She was _DYING_ , for Christ's sake. How _DARE_ you violate a dying woman?\"\n\nHe cringed away from me. \"I never... only in her mouth...\"\n\nOut of the corner of my eye I saw Maureen's claws reach out to scratch my face. And with every ounce of hatred that I had I planted my fist between her teeth.\n\nIt would have turned into a free-for-all if Geoffrey hadn't been a pacifist at heart. He pulled me off Maureen by seizing me by the arms and spinning me 'round behind him. \"Enough,\" he said sharply, standing between me and the sofa. \"Control your mother,\" he ordered Alan, \"or I'll ask Mrs. Stanhope to call the police.\"\n\nIt was an unnecessary instruction because Alan was already holding her back with an arm hooked 'round her neck, but mention of the police at least persuaded her to sink back into her seat. She eyed Geoffrey balefully \"You're in no position to play God,\" she spat. \"Your hands are as dirty as ours.\"\n\nHe lowered his head like a terrier after a ferret, and stared fixedly at her. \"Mrs. Ranelagh says Annie was beaten up in her house two or three hours before I passed her\u2014and those are the injuries she died of-\u2014so don't accuse me of having dirty hands. You're the only one 'round here got free with a baseball bat.\"\n\nMaureen's narrowed gaze focused on me. \"You're being fed a pack of lies. A week ago this bitch was saying it was Derek gave Annie a thrashing before dumping her in the street.... Now she's trying to put the blame on me. Well, how could I have got the fat cow through her front door? Tell me that.\"\n\n\"She got herself out,\" I said, breathing deeply through my nose to try to quell the shudders that were jolting my body with electric shocks. \"She had a fractured skull... a broken arm... she'd been unconscious for God knows how long with your filthy son all over her... but she still had enough will to live to stagger out into the street and look for help.\" I made another lunge forward, only to be blocked by Geoffrey. \"And no one gave it to her because they thought she was drunk.\"\n\n\"Your husband being one,\" she snarled.\n\nI pressed a finger to the tic of hate that throbbed beneath my lip. \"I think she came to my end of the street because she knew I was the only person who would help her. I even think she may have knocked on my door\u2014and I feel so damn guilty I wasn't there, because I was sitting at school waiting for parasites like you and Derek to come and talk to me about your children's progress.\" I dropped abruptly on to my chair again, energy spent. \"Some joke, eh? We all knew the only direction your children were ever going was toward prison.\"\n\n\"Don't you call us para\u2014\" began Derek.\n\nBut Geoffrey cut him short. \"What did you do to Rosie?\" he demanded of Alan.\n\n\"Don't answer him, son,\" snapped Maureen, spitting blood. \"Just because that bitch of a teacher tells lies about us, it doesn't mean we have to start explaining ourselves.\"\n\n\"It damn well does,\" said Geoffrey belligerently. \"If he raped my Rosie I want to know about it. He ought to be locked up.\"\n\n_\"Your_ Rosie?\" demanded Maureen, dashing the blood from her mouth with the cuff of her sleeve. \"That's rich, that is. How come she's yours all of a sudden when you couldn't get shot of her quick enough to move in with the tart?\"\n\n\"More to the point, Maureen,\" said Wendy forcefully, \"when did Alan tell you about his part in all of this? And why did you do nothing to stop it getting any worse?\"\n\nShe shrank into the back of the sofa. \"You should be asking Derek that,\" she said mutinously. \"He's already said it was him told Alan what to do. What could I have done except take a beating myself... which is what happened every time Derek reckoned I was interfering.\"\n\nBut Derek gave an angry shake of his head. \"I said I'd take the blame for the schoolteacher,\" he muttered. \"Nothing else.\"\n\n\"There _is_ nothing else,\" she snapped angrily. \"All we ever did was thieve a few things off the nigger and teach Miss High-and-Mighty here a lesson in manners. All the rest is lies.\"\n\nI looked up. \"What about the cats?\" I asked coldly. \"Were they a lesson in manners, too?\"\n\nShe dropped her eyes immediately and fumbled for a cigarette.\n\n\"You were too precise about the numbers inside Annie's house. It's not a figure you'd have known if you hadn't notched up each sad little stray as you tortured it.\"\n\nWhy should this be the key that unlocked Alan? Was a cat's death more dreadful than a woman's? A cat's humiliation harder to forget? A cat's cries more poignant? _Apparently so_. Annie could die.... I could be humbled.... Rosie could weep... but an animal must be loved. His anguish was frightening for, as I watched him battle with tears for those long-dead creatures, I found myself wondering if he was still as dissociated from human pain as he so obviously had been then. If so, I had little hope for Beth and her children.\n\nIt would be impossible to relate what he said in the way he said it. Once released, his emotions were a river in spate, sweeping aside everyone's sensibilities but his own and given in stuttered sentences which were barely comprehensible at times. We became party to his mother's hatred of sex, his father's brutal taking of her whenever he wanted it, their drunkenness, their violence toward each other and their children. But, more than anything, he dwelt on Maureen's slaughter of the marmalade cat, repeating over and over that when he tried to stop her she turned the baseball bat on him.\n\nI asked him why she'd done it and, like Michael, the only explanation he could offer was that it made her feel \"good.\" She laughed when its brains went everywhere, he said, and she wished it had been the nigger's head she'd smashed.\n\n\"What about the other cats?\" I asked him. \"Why did she go on with it?\"\n\n\"Because it sent Annie 'round the bend to have them put through her flap. She took to wailing and hollering all the time and behaving like a crazy woman, and Mum reckoned if she didn't pack up and go of her own accord, it was a dead cert she'd be taken out in a straitjacket.\"\n\n\"But if hurting animals upset you so much, why did you help?\"\n\n\"I wasn't the only one,\" he muttered. \"We all did it\u2014the girls, Mike, Rosie, Bridget. We used to go out looking for strays and bring them home in boxes.\"\n\nI wondered sadly if that was the real explanation for Bridget's sacrifice of her hair. \"But why, if you knew what was going to happen to them?\"\n\n\"It wasn't as bad as having their heads split open.\"\n\n\"Only if you believe a quick death is worse than a slow one.\"\n\n\"They didn't all die.... Annie saved most of them... and that's what we reckoned would happen.\" He pressed his forehead into his hands. \"It was better than having Mum kill them straight off, which is what she wanted to do. It was them dying that got Annie worked up.\"\n\n\"The ones you put under my floorboards died,\" I said, \"because I didn't know they were there.\"\n\nHe raised his head with a look of bafflement in his eyes, but didn't say anything.\n\n\"And if you'd refused your mother,\" I pointed out, \"none of the cats need have died. Surely Michael was bright enough to work that out even if you couldn't.\"\n\n\"Us kids wanted rid of Annie, too,\" he said sullenly. \"It wasn't right to make us live next door to a nigger.\"\n\nI don't know what was going through Maureen's mind while he spoke. She made one or two halfhearted attempts to stop him but I think she realized it was too late. The odd thing is I believe she was genuinely ashamed of her cruelty\u2014perhaps because it had been the one crime she committed herself. More interestingly, she had eyes only for Sharon when Alan admitted that he and Michael had entered Annie's house together around 8:30 on the night she died.\n\n\"It was Mike spotted the door was ajar,\" he said. \"We were going into his place to watch telly because we knew his mum was out, and he says to me, 'The coon's left her door open.'The place was black as the ace of spades... no lights... nothing... and he says, 'Let's do a prowl before she gets back.'So we creep into the front room and damn near fall over her. It was Mike started it,\" he insisted. \"He turns on the lamp on the table... reckons she's drunk as a skunk and pulls out his dick\u2014\" He broke off, refusing to go any further.\n\n\"Did she speak to you?\"\n\nHe raised his eyes briefly to Sharon's. \"Kept saying the tart had hit her... so Mike goes apeshit and kicks her till she shuts up. After that we went down the arcade, and Mike says he'll kill me if I ever breathe a word about his mum... and I say, 'Who cares? It's good riddance, whoever did it....'\"\n\n\"I told you it wasn't us,\" jeered Maureen with a gloating smile on her face. \"'Look to the tart,'I said. It was her and her son did it between them.\" She jabbed two fingers in the air at Geoffrey. \"That's why you shoved the mad cow in the gutter\u2014because she told you who'd hit her.\"\n\nI felt physically sick. Even though I'd suspected Michael had known how Annie had died, I'd always hoped he hadn't been involved. _But could a \"kicking\" at 8:30 have caused the sort of blood-seepage into Annie's thighs that was so obvious in the photographs?_ I looked at Sharon. _\"Stand up for your son,\"_ I wanted to shout at her. _Tell them how small he was for his age... and how murderous kicks like that must have come earlier...from someone who was stronger..._\n\n\"Is that true, Geoffrey?\" asked Wendy in shocked tones.\n\n\"No,\" he muttered, looking at Sharon in sudden disbelief. \"She didn't say anything... just kept grabbing at my sleeve, trying to hold herself up... so I pushed her away...\" His voice petered into silence as he began to question how many lies Sharon had fed him. \"No wonder you let me think it was my fault,\" he said resentfully. \"Who were you protecting? Yourself or that bloody son of yours?\"\n\nBut Sharon's only response was a tiny gesture of denial as the last vestiges of color fled from her face.\n\n\"If she faints she'll hurt herself,\" I warned.\n\n\"Let her,\" said Maureen spitefully. \"It's no more than she deserves.\"\n\n\"Oh, for God's sake,\" I sighed wearily, standing up to help Wendy support the limp body. \"If you believed that then why didn't you tell Mr. Drury the truth at the time?\"\n\nBut it was a stupid question, which she didn't bother to answer. She had no regrets over Annie's death. Indeed her only goal had been to steer retribution well away from herself so that she could indulge her spoils to good advantage. And if that meant exploiting men's baser instincts to instill terror in women, then so be it. In a bizarre sort of way I could even admire her for it, for hers was a vicious world, where greed\u2014be it material or sexual\u2014was a way of life, and by her own standards she had made a success of it. Certainly she was the only person in that room who owned her house through the quickness of her mind.\n\nI touched a hand to Sharon's peroxide hair and it felt dry and dusty beneath my fingers. \"The worst thing this lady ever did to Annie was pour a bucket of water over her head and make a few complaints to the council,\" I told Geoffrey, \"but if you can't believe that then you should bugger off and give her a chance to get her son back. Wendy's right. All you've ever done is make her a prisoner to the truth.\"\n\n\"But\u2014\"\n\n\"But what?\" I snapped. \"Would you rather trust Maureen's version of events? Mine come free, remember, and hers come at a price.\" I grabbed his elbow and forced him to look at Sharon. \"This woman has stood by you for over twenty years\u2014how much longer do you need to know her before you trust her? Or must she always be judged by the rotten standards that you\"\u2014I gestured toward the sofa\u2014\"and this vermin over here choose to live by?\" I spoke as much for myself as I did for Sharon, for I knew all too well the pain of living in an atmosphere of disbelief and distrust. You sink or swim... fight or give in... and whichever route you choose, you take it alone.\n\nGeoffrey gave an uncertain shake of his head.\n\nI knelt down abruptly in front of Sharon and took her hands in mine. \"Sell your house and move,\" I urged her. \"Cut this man out of your life and start again. Make friends with Bridget... help Michael go straight. He needs his mother's love just as much as he needs his wife's... and you owe him that much. He thought you were a murderer, Sharon... but he protected you... and he doesn't understand why you were so quick to abandon him. Fight for him. Be the mother he wants you to be.\"\n\nShe was too dazed to grasp what I was talking about and stared helplessly from me to Geoffrey, her subservience to men so ingrained that she would do whatever he told her to do.\n\nMaureen's triumphant voice came at me from the sofa. \"There's only ever been one tart in this street, and she's gone down like a sack of potatoes because she's been found out. So go tell that to the police and see if they care about the few bits of trash we pilfered.\"\n\nI wanted to kill her. I wanted to squeeze her scrawny throat between my fingers and choke the venom out of her. Instead, I stood up with a sigh and reached for my rucksack. \"Annie never called Sharon a 'tart,'Maureen, she called her a 'whore.'You told me that yourself.\"\n\nHer mouth dropped open, for once unable to find words, because she knew I was right. I longed to be strident... to scream and yell... to stamp my feet... to roar my frustration to the winds. I had hoped for the miracle that would prove me wrong, but instead I just felt desperately sad and desperately weary.\n\n\"And I wouldn't rely on the police letting you get away scot-free if I were you,\" I went on with commendable steadiness, exercising the sort of ladylike control that would have brought a smile of approval to my mother's face. \"The only protection you've ever had was other people's silence. As long as they had secrets to hide you were safe.\" I shrugged. \"But there aren't any secrets anymore, Maureen. So where does that leave you?\"\n\nDerek gave an unexpected laugh. \"I told her you'd never give up,\" he said, \"but she wouldn't listen. Said schoolteachers were too prissy to get up off their knees and fight.\" Maureen pursued Wendy and me to the door, demanding answers, which I refused to give. Who did it if it wasn't Sharon? How much was I going to tell the police? What proof had I of anything? Her lip had fattened from the punch I'd delivered and she caught at my sleeve to hold me back, threatening me with prosecution if I didn't give her some _\"fucking_ explanations.\"\n\nI pulled away from her. \"Go ahead,\" I urged. \"I'll even tell you where I'm going\u2014I'll be with Mr. Jock Williams at 7 Alveston Road, Richmond\u2014so by all means send the police 'round to arrest me. It'll save me having to call them. And, as for giving you answers\"\u2014I shook my head\u2014\"no chance. What you don't know can't help you, and I'm damned if I'll be party to Derek and Alan telling any more lies for you.\" I raised my eyes to where Alan was standing in the shadows of the hall. \"I have every reason to hate and despise you,\" I told him, \"but I think your wife is the one woman in a million who can rescue you from your mother. So my best advice is, go home now and take your father with you. If Beth hears the truth about you from Derek, then she may understand and forgive. If she hears it from your mother, she won't.\"\n\n\"Goodness me!\" Wendy gasped, patting her fluttering heart as we walked away. \"That's the first time I've seen her afraid.\"\n\n\"Are you all right?\" I asked in concern, reaching out to support her under the elbow.\n\n\"Absolutely not. I've never had so many shocks in my life.\" She lowered her bottom on to the garden wall of number 18. \"Just let me get my breath back.\" She took some deep breaths, then wagged a finger at me as she began to recover. \"Peter would counsel you strongly against this obsession with revenge, my dear. He'd say the only path to heaven is through forgiveness.\"\n\n\"Mm,\" I agreed. \"That's the advice he gave me when I told him about Derek and Alan.\"\n\nShe tut-tutted crossly. \"Is that the time he let you down?\"\n\nI watched a car negotiate the speed bumps in the road. \"He didn't do it on purpose,\" I demurred. \"He was like everyone else.... He thought I was hysterical.\" I looked toward Maureen, who still hovered by her gate. \"I think I know why now. I never remained objective long enough to keep my voice under control. And that worries people.\"\n\n\"But why Peter?\" she asked curiously. \"Didn't you have anyone else to talk to?\"\n\n_Only Libby..._ \"It was the church more than Peter,\" I said non-committally \"I couldn't think of anywhere else to go.\"\n\n\"Oh, my dear, I'm so sorry. You really _were_ let down then.\"\n\nI shook my head. \"Rather the opposite actually. I went in weepy and pathetic, looking for sympathy, and came out like an avenging angel.\" I gave an abrupt laugh. \"I kept thinking, if I _ever_ forgive, it'll be on my terms and not on the say-so of a fat, sweaty bloke in a dress who thinks I'm lying.\" I sobered just as suddenly. \"I'm sorry. I didn't mean to be rude.\"\n\nWendy squared her thin shoulders and stood up. \"It's a good description of Peter,\" she said tartly. \"He's an actor at heart so he's only really happy when he's in costume. He thinks it lends authority to what he's saying.\"\n\n\"I was pretty peculiar at the time,\" I said by way of apology, \"and he did try to be kind.\"\n\n\"He's got no fire in his belly, that's his problem. I keep telling him his sermons are ridiculously PC. He's supposed to be addressing evil, not offering a policy statement on behalf of liberals.\"\n\nI chuckled. \"You'd be a thunderbolts-and-lightning vicar then?\"\n\n\"It's the only kind to be,\" she agreed cheerfully. \"A whiff of brimstone and sulphur puts sin to flight quicker than anything. _And_ it's more dramatic. The fires of hell and damnation are a great deal more exciting than the bliss and majesty of heaven.\"\n\nI adored her... for her openness... her steadfastness... _God save me, her similarity to my mother..._ but I could see she was too exhausted to take another step. I persuaded her to sit back down while I fished in my rucksack for the mobile I'd borrowed off Luke that morning for the purpose of calling a minicab. A car drew up beside us before I could find it.\n\n\"Do you want a lift?\" asked Alan gruffly through the open passenger window as he leaned across to fasten his father's seatbelt. \"We'll be passing Alveston Road.\"\n\nI was too startled to answer, and looked at Wendy.\n\n\"Thank you, my dears,\" she said, rising graciously to her feet. \"That's most generous of you.\"\n\nNothing more was said until Alan stopped his car in front of Jock's house. Derek and Wendy were content to appreciate the silence, while Alan kept darting me worried glances in the rearview mirror, his mouth working to frame a form of words that would be acceptable to me.\n\nBut it was only when he drew to a halt in Alveston Road that he found the courage to take a chance. He turned 'round. \"It's probably a bit late\"\u2014he faltered\u2014\"and I wouldn't blame you if said no\u2014but I wish I'd never\u2014 Did Danny tell you I've been straight these fifteen years?\"\n\nI stared him down. \"If you want to say sorry, Alan, then say it. Don't spoil it by making excuses.\"\n\nHe ducked his head in a scared nod, an echo of the schoolboy I had caught thieving from my purse. \"I'm sorry.\"\n\n\"Me, too.\" I held out my hand to him. \"I didn't help you when I had the chance and I've always regretted it.\"\n\nHis hand was warm and sweaty in mine\u2014and I can't say my flesh didn't crawl at the contact\u2014but it felt like closure. For both of us. I toyed with warning him against interpreting it as a reason not to be honest with Beth, but Derek's presence was an optimistic sign and I held my peace. In the event I was glad I did.\n\n\"Just so you know,\" he said, as I assisted Wendy out of the car, \"it wasn't us who put cats under your floorboards.\"\n\nI frowned. \"Does that mean there were no cats? Or someone else put them there?\"\n\nHe jerked his head at Jock's front door. \"There was nothing Mr. Williams didn't know.... He used to watch everything us kids did from Sharon's window... and the only reason he kept quiet was because the nigger called him a 'faggot.'He hated her for it 'bout as much as we did for being called 'trash.'\"\n\nI closed my eyes for a moment. \"I _will_ be going to the police, Alan,\" I said sadly. \"You do understand that, don't you?\"\n\n\"Yeah.\"\n\n\"Then do yourself a favor,\" I said with a heartfelt sigh, \"and drop 'nigger'from your vocabulary because I will take you apart piece by piece if you ever refer to Annie in that way again.\"\n\nHe nodded obediently as he engaged his gears. \"Whatever you say, Mrs. Ranelagh.\"\n\nWendy rapped sharply on Derek's window. \"What about you?\" she asked. \"Are you going to apologize, too?\"\n\nBut he looked at her as if at an irrelevance before gesturing to his son to drive on.\n\nWe stood on the pavement, looking after them until they turned on to the main road. \"I think you've just been suckered,\" said Wendy with a small laugh. \"What's the betting they head straight for the nearest cash-point so that Alan can drop Derek a hundred quid to vanish off the face of the earth?\"\n\n\"Oh, ye of little faith,\" I said, escorting her between our car and a mud-splattered Renault Espace that was parked beside it in Jock's drive. I wondered briefly where the elderly Mercedes was before it occurred to me that Jock, forever a prisoner to truth, would have hidden it away in order to continue his pretense that he had an XK8 in a lockup garage.\n\n**_E-mail correspondence with Libby Garth, formerly  \nLibby Williams of 21 Graham Road\u2014dated 1999_**\n\n**M. R**.\n\n**From:** Libby Garth (liga@netcomuk.co)\n\n**Sent:** 17 August 1999 20:17\n\n**To:** M. Ranelagh\n\n**Subject:** Re Meeting on Friday at Jock's house\n\nDear M\u2014written in haste before I rush out to collect Amy from her friend's house. You say it's water under the bridge and that none of us needs be embarrassed after so long, but I am MORTIFIED! How can I look any of you in the face, particularly you and Jock? I know you've asked me not to explain or apologize, but I do feel badly about it. AND I AM SORRY. Please believe me, whatever Sam and I had going all those years ago\u2014it was dead as a dodo before you left England.\n\nI know you say this meeting on Friday is important, but I truly can't face it. You and Jock must feel very raw to have Sam confess after twenty years\u2014you, in particular, must hate me for my hypocrisy. You probably think I was only pretending friendship by helping you over Annie, but it honestly wasn't like that. I was pleased to help, even more pleased that we went on being friends in spite of everything. The truth is I let myself believe that Sam would never tell\u2014more because he remained so close to Jock, I think, than because he thought you wouldn't be able to take it\u2014and it wasn't as if it was terribly important where he was that night, just so long as he and Jock admitted they didn't see Annie at 7:45. In all these years, the only lies I have ever told you were to do with that wretched alibi\u2014God knows, I wish we'd been honest at the time\u2014but it seemed so unimportant compared with the hurt you'd feel to learn about the affair. Of course, it was wrong, but I couldn't see any harm in it. Sam and Jock obviously had nothing to do with Annie's death\u2014as I kept telling you\u2014and I didn't want you thinking badly of me.\n\nAnyway, the upshot of all this is that I have written out a statement, with precise times and details of all movements in and out of number 21 that night\u2014insofar as I remember them\u2014which I am sending as an attachment. I think you'll find they agree with what Jock and Sam say. I will, of course, do it formally when the time comes. Meanwhile, I will end with love, and pray you can still accept it.\n\nLove,\n\nL\n\nPS: Apart from anything else, I can't just abandon the girls to their own devices for the day, and poor old Jim would be deeply alarmed if I told him I was about to have a reunion with my ex-husband! He'd want to know why... and then I'd have to tell him about Sam... and how I'd wronged my best friend. I'm sorry, m'dear. I hope you understand.\n\n**Libby Garth**\n\n**From:** M.R. (mranelagh@jetscape.com)\n\n**Sent:** 18 August 1999 12:42\n\n**To:** Libby Garth\n\n**Subject:** Re Friday\n\nMy dear Libby, attachment received and understood and love accepted in the spirit it was given! Believe me, I've always valued the help you've given me re Annie's \"cause\"\u2014I wouldn't have known half as much as I do about her hateful neighbors if it hadn't been for you! Unfortunately, there are one or two discrepancies between your statement and Sam's\u2014i.e., he says you were doing the laundry when he arrived, while you say you were watching TV. Also, he says you'd had a bath before he arrived, and you say you'd been cooking. I know they're only small things\u2014but I really do need the accounts to match before I hand them to the police. You know I wouldn't ask if it wasn't important. And there really is nothing to worry about. Jock and Sam are reconciled\u2014if a little cool\u2014while Sam and I are like Darby and Joan. We've been together so long we're tied at the hip and can only walk in step these days. However, if you can't manage to come down here because of leaving the girls and worrying about Jim, the three of us can\u2014and will\u2014come to Leicester.\n\nLove,\n\nM\n\n# twenty-eight\n\nI had asked Jock to leave his front door ajar so that Wendy and I could let ourselves in when we arrived, and as we walked through the hall toward the kitchen I saw Libby before she saw me. She was sitting on a hard-backed chair, her face in profile, and I had a second or two to take stock before the sound of our footsteps alerted her to our presence. _Oh, sweet revenge!_ Gone was the startling brunette in her mid-twenties who had flaunted her looks and figure to good advantage, and in her place was a scrawny, beak-nosed woman with a sagging chin and newly dyed hair that was too dark for her complexion.\n\nMy strongest memories of her were the impatient gestures and petulant expression that had spoken volumes about her frustration with her life in 1978, and I was amused to see she still had them. Indeed, the impression she gave was that all the water that should have passed beneath her bridge had merely piled behind a fragile dam... which was on the point of bursting.\n\n\"I've had enough of this,\" she was saying, stabbing angrily at her watch as Wendy and I approached. \"She told me 12:30, and if she doesn't come in the next five minutes\u2014\"\n\nShe broke off as Sam and Jock looked up in relief to see us in the doorway.\n\n\"Hello, Libby,\" I said with a bright smile. \"You're looking well.\" She took similar stock of me but there was no answering smile. \"You're late,\" she snapped.\n\nPerhaps I should have been surprised by her lack of cordiality after the numerous letters, faxes and e-mails she'd sent over the years professing support, friendship and... _love..._ but I wasn't. Her saccharine sweetness had been conditional on my continued ignorance of her affair with Sam because that made me a fool. But Sam and Jock had obviously told her, as I had asked them to do, that I'd known about it since before my first letter to her\u2014which made _her_ the fool. And that was the one thing she had never been able to tolerate... being a laughing stock.\n\n\"I know and I'm sorry,\" I said cheerfully. \"It took longer than I thought. Do you remember Wendy Stanhope, the vicar's wife? Wendy... Libby... Jock... Sam.\" I raised inquiring eyebrows at the men as they stood up to shake Wendy's hand. \"Did you get the sandwiches? Because _we\u2014are\u2014starving!\"_\n\nJock pulled open his fridge door with a flourish. \"All here,\" he said, removing plates to the table, and handing a bottle of chilled Chardon-nay to Sam.\n\n\"We were reliably informed this was your favorite,\" said Sam, filling a glass and handing it to Wendy. \"I should think you've earned it, haven't you?\"\n\nShe chuckled happily as she took a huge swig. \"Goodness me, no! I was just the chorus to your wife's dazzling _coloratura_. You must be very proud of her, Sam.\"\n\n\"Oh, I am,\" he said, handing me a glass before shepherding Wendy to a chair. \"She's a bit of a cracker, too, don't you think?\" He dropped me a sly wink. \"Just as beautiful as the day we married.\"\n\nI watched Libby's mouth turn down as she rejected the glass Sam tried to offer her and wondered how much of this she would be able to take before she sunk her talons into my cheeks. \"I'm driving,\" she said curtly.\n\n\"What do you think of Jock's beard?\" I asked, stationing my back against a worktop from where I could look at her. \"It suits him, don't you think?\"\n\n\"She hates it,\" said Jock, giving it a stroke. \"Says it makes me look seedy.\"\n\nLibby gave an irritated smile. \"We've been there, done it. Also Sam's baldness... Dorchester... Leicester... the weather...\" She drummed her fingers impatiently on the table. \"You _promised_ me 12:30 so that I could be back on the motorway before the Friday rush hour begins,\" she said sharply. \"You _knew_ I wanted to be home before Jim.\"\n\n\"Call him and tell him you're going to be late,\" I said reasonably.\n\n\"That's what we've been suggesting,\" murmured Jock.\n\n\"I can't. I don't want him knowing I've left the girls on their own.\"\n\n\"Couldn't they have gone to friends?\"\n\n\"Not without questions being asked,\" she snapped, \"and I really didn't want to go into long explanations about why this ridiculous meeting was necessary. Can we just get on with it?\"\n\nI ignored the request. \"You should have let us come to Leicester,\" I said disingenuously.\n\n_Ah, me! If looks could kill..._\n\n\"It's not as though Jock's about to stake a prior claim or anything,\" I went on, reaching for his hand and swinging it lightly at my side, cementing alliances, ranging my troops. \"He prefers them younger and blonder these days.\"\n\nJock gave a snort of laughter. \"Too bloody right,\" he agreed unkindly. \"And _never_ with marriage in mind. That's one mistake I don't plan to repeat.\"\n\nIt was cruel but I have no conscience about it. If I'd known of the affair at the time, I'd have slapped the smile off her face before nailing my husband's bollocks to the wall. But a slow revenge is just as satisfying. I was sure it would drive her to distraction if she was forced to make banal small talk with her ex-lovers\u2014her nature was too impatient and too self-centered for anything else\u2014and neither Sam nor Jock was equipped to deal with a frustrated woman. They had failed dismally in the past, and I couldn't believe much had changed in the meantime.\n\nHer lips thinned. \"It's got nothing to do with Jock,\" she said tightly. \"Jim thinks Amy's too young to look after her sisters. But she's not. She's almost fourteen.\"\n\n\"It's only natural,\" said Wendy idly, protruding long fingers like forceps to select a tuna and cucumber sandwich. \"An untended nest and a hungry brood suggests to the male that his partner has flown.\" She smiled at Libby \"I suppose he's found it empty before, has he?\"\n\nThere was a minor hiatus while Libby looked daggers at her and Wendy bit into her sandwich. The rest of us buried our noses in our wine glasses. To be honest, I wasn't remotely surprised that she was still a player but it was a shock to the men who both assumed, na\u00efvely that her passionate nature could be tamed by motherhood and a career. They lowered their heads to stare at their feet, and it was so perfect an example of the double standards that operate between men and women that I couldn't help smiling to myself.\n\nOf course Libby saw it. I was her only real enemy so I was bound to be the focus of her attention. She bridled immediately. \"You think you know it all, don't you?\" she flashed.\n\n\"No,\" I murmured. \"I was completely wrong about you. I thought you had more dignity than to go sniffing after other people's husbands.\"\n\n\"Oh, _please!\"_ she said scathingly. \"Any sniffing that was done was done by Sam. He couldn't get unzipped quick enough when the opportunity arose. Or is that forgiven and forgotten because he's served twenty years of your downcast looks and injured pride?\"\n\nSam stepped forward angrily but I shook my head at him. This was my fight and I'd waited a long time for it. \"If you want a slanging match, Libby, then I'm happy to oblige... Sam and Jock, too, I should imagine. But if you're as desperate to get away as you say then I suggest we sort these statements.\"\n\nShe hated her position of weakness, but she had the sense to force a smile. \"All right. What do you want to know?\"\n\n\"Which is correct? That you'd had a bath and were doing the laundry when Sam arrived? Or that you'd done the cooking and were watching television?\"\n\nShe shook her head in convincing perplexity. \"I honestly don't know,\" she said slowly. \"It's so long ago I've forgotten most of the details. I just wrote down what I normally did at that time\u2014cooking then catching the news\u2014but if Sam's positive\u2014?\" She broke off to look at him. \"Do you remember it that well?\"\n\n\"Yes.\"\n\nShe was disconcerted by the bluntness of his answer. \"I don't see how you can. It's not as if it was the only time you came to the house looking for sex.\"\n\n\"No,\" he agreed, \"but it was the _last_ time... and I'd told you it was going to be the last time over the phone that afternoon. I said I wanted to talk to you about ending the affair without destroying everyone in the process. And I was furious when you draped yourself all over me the minute I came through the door, saying you'd had a bath in my honor and were washing sheets so you'd be able to replace our dirty ones on the bed before Jock came home. You can't have forgotten that, Libby You told me I was frightening you because I said I'd do you some damage if you didn't take your hands off me immediately.\"\n\nShe gave a small laugh. \"Oh, well... if that's how you want to play it... it's no skin off my nose. What does it matter what I was doing, anyway?\" She shifted her gaze back to me. \"We'll go with Sam's version. Does that make you happy?\"\n\nI nodded.\n\n\"Then you're a fool.\"\n\n\"Maybe.\" I crossed my arms and studied the point of my shoe, in no hurry to go on.\n\n\"Is that all there is?\" she said indignantly. \"Did you make me come all this way just so you could feel better about your husband's cheating?\"\n\n\"Not quite,\" I said without rancor. \"There's a major question-mark over the time of Sam's arrival. He says 7:45, you say 6:30.\"\n\nShe frowned, as if trying to remember. \"Okay, split the difference,\" she said helpfully. \"Make it 7. Neither of us can be that precise after twenty years.\"\n\n\"Sam can,\" I countered mildly. \"He's worked out his timing rather more accurately than you have... and there's no way he could have reached you before a quarter to eight. If you calculate his walk from the office to the tube, the average time of the train journey, plus the walk from Richmond station to Graham Road, it's impossible for him to have done that trip in under an hour and a quarter. Which means 7:45 has to be the agreed time because he didn't leave work until 6:30.\"\n\nHer hands moved impatiently in her lap. \"How do you know that? Why should Sam's memory of the time he left his office be any better than mine of the time he arrived?\"\n\n\"Because I'm not going by Sam's memory,\" I told her. \"I was so suspicious of him after he and Jock made their statements that I checked with his office. I hoped I could get some proof that he was lying about the time he reached Graham Road because I knew the security guard clocked everyone out at the end of the day to make sure the building was empty before he locked up. I persuaded him to let me have a photocopy of the register for 14.11.78.\" I nodded toward the rucksack at my feet. \"It's in there with 18:30 against Sam's name.\"\n\nHer eyes dropped immediately to the bag but she didn't say anything.\n\n\"So we're agreed that 7:45 was the time Sam arrived?\" I repeated.\n\nShe made a dismissive gesture with her hand. \"I can't see what difference it makes. All we did was talk.\"\n\n\"Yes, that's what you both say. Your version is that you talked for two and a half hours. His is that you talked for an hour.\"\n\nShe shrugged. \"I didn't keep track.\"\n\n\"But you disagree over how the conversation went. Sam says he gave you an ultimatum\u2014either the affair had to end or he'd come clean with me that night. You say it was you who delivered the ultimatum.\"\n\nShe cast a malicious glance in Sam's direction. \"He can't say anything else,\" she said, \"not if he wants you to believe I draped myself all over him when he came through the door.\"\n\nI smiled slightly. \"But that's the whole point, Libby After the show you put on when he arrived, Sam expected you to be difficult... but you weren't. You said you'd leave him alone... no more hanging around outside his office... no more demands on his time... and the only quid pro quo was that he keep his mouth shut so that Jock wouldn't have an excuse to divorce you.\"\n\n\"Which suggests it was me who delivered the ultimatum, doesn't it?\"\n\n\"If that were true, why was Sam so keen to accept it?\"\n\nHer eyes narrowed warily as she tried to see the point I was making. \"What makes you think he was?\"\n\nI shrugged. \"Because he couldn't sign up to your fabricated alibi quick enough. He was even happy to rope Jock into the lie if it meant he could distance himself from you. Not that your husband minded,\" I said with an ironic glance in Jock's direction, \"because he didn't want his Tuesday evenings with Sharon made public. But why would _Sam_ go along with it unless he had something to gain? There were any number of reasons he could have given for being in your house that night\u2014 none of which were remotely suspicious. Looking for Jock, being one.\"\n\n\"Why ask me?\" she demanded. \"Sam's the one who lied. All I did was tell the truth, which was that I'd been at home all evening, waiting for my husband. And I didn't have to pretend I was alone either because the police made that assumption themselves. It's not my responsibility if Sam decided to sign a statement saying he was at your place when he wasn't.\"\n\n\"Except he says you didn't give him any choice. According to him, you phoned him at his office the next morning to say the police were asking about people's movements the previous night because they were looking for anyone who'd seen Annie. You then told him you'd dug him out of a hole by saying he and Jock had been at our house from 7:45 and it was down to him to persuade Jock to support the story. You said I'd never suspect he'd been with you if it was your husband who gave him an alibi. And you were right, I didn't.\"\n\n\"This is Sam's version, presumably?\" she murmured sarcastically.\n\n\"Yes.\"\n\nShe glanced at my rucksack again. \"And there's no statement from an earwigging telephone operator to back it up?\"\n\n\"No.\"\n\n\"Then _you_ can believe what _you_ like, and the police can believe what _they_ like,\" she said indifferently. \"Sam's always going to put his own gloss on it\u2014he wouldn't be human if he didn't\u2014but he's the one who lied and I'm the one who told the truth. And I'm damned if I'll let him put the blame for his perjury on to me.\"\n\nI nodded as if I agreed with her. \"Fair enough, but you'll need to be ready for police questions about who proposed what and when because Sam's revised statement says the ideas came from you\u2014in particular his and Jock's alleged sighting of Annie at 7:45.\" I paused. \"According to Sam, that was your suggestion. You told him the police wanted proof that she was staggering about in the road earlier in the evening, and if he gave it to them they'd call it an accident and the whole bloody mess would go away.\"\n\nI was lying, of course\u2014Sam had never denied that the reason he mentioned Annie was to get himself out of the hole he'd dug with me when he told me she was drunk\u2014but Libby didn't have a monopoly on invention, and it was fascinating to see how rapidly her control deserted her when she was accused of something she hadn't done. In a horrible sort of way, she reminded me of Maureen as she hissed and spat her furious denials. We were all shits... ganging up on her because we didn't like her... making Sam out to be the victim... trying to shove responsibility on to her...\n\n\"Why would I have suggested anything so bloody stupid?\" she finished. \"Supposing the police hadn't believed Sam and Jock? Supposing we'd all had to admit what we'd really been doing that night? Why would I tell him to say he'd seen Annie just before the one period in the whole evening when we both had a cast-iron alibi? It's ridiculous. They'd think we were in collusion to cast suspicion away from ourselves. I'd never saddle myself with anything so unnecessary.\"\n\nI studied her for a moment. \"But why would you even worry about collusion?\" I asked curiously. \"Surely all you knew when you phoned Sam the next morning was that Annie had died outside our house at 9:30? How does that make mention of her stupid and unnecessary?\"\n\nShe sobered rapidly. \"Sam told me you were saying it was murder.\"\n\n\"Not true,\" Sam countered fiercely. \"I was so ashamed of leaving the poor woman in the gutter that I steered clear of the whole blasted subject. All you and I discussed that morning was how to avoid saying that I'd been with you.\"\n\nShe gave an angry smile. \"Then maybe I'm talking with hindsight, but it's hardly the point at issue. You're accusing me of inventing an absurd lie when anyone who focused attention on themselves by saying they'd seen Annie that night was a fool... particularly if they were trying to hide an affair. _You_ may be that kind of fool, Sam, but _I'm_ certainly not.\"\n\n\"That's very true,\" I said before Sam could fire off again. \"I've always thought how clever you were to keep your story simple, claim absolute ignorance and offer no alibi at all. All you had to say was: _'I can't help you... I was home alone from five o'clock... didn't hear anything... didn't see anything... didn't go anywhere.'_ You could repeat that till you were blue in the face because there was no one to contradict you except Sam. And once you'd muzzled him, you were safe as houses, because if the police _had_ caught you out in a lie, you'd have shrugged and said, you were only trying to keep the affair secret.\"\n\n\"I didn't need an alibi,\" she said.\n\n\"No,\" I agreed, \"but only because no one saw you with Annie at 6:30. I presume you bumped into each other in the road, and she started calling you a 'dirty tart'again. But why the hell did you have to go out at all, Libby? What was it for? To buy some booze in the hopes of putting Sam in a better mood? Or maybe you needed it yourself because you were boiling mad about being given the elbow? Is that why you lost your temper with Annie so quickly? Because you were angry that Sam had made it clear he'd rather stay with his wife than play stud to a bored tart who hadn't got the gumption to get up off her backside and find an identity for herself that didn't involve exploiting men? Why couldn't you stay in your sordid little bed and weep for your own inadequacies instead of killing Annie because she dared to point them out to you?\"\n\nCaution smoothed the planes of her face turning it into a practiced mask. \"Don't be ridiculous,\" she said. \"What's 6:30 got to do with anything?\"\n\nI took a printout of her e-mailed statement from my pocket. \"It's the time you gave in here, so presumably it's important.\"\n\nShe made another dismissive gesture. \"I've already said I'll go with Sam's version, not mine. Are you going to crucify me for making a mistake?\"\n\n\"Your worst mistake was to have a bath and start washing your clothes,\" I said, \"but I suppose you had her blood on you. The postmortem photographs prove you went for her like a madwoman.\"\n\n\"Oh, for God's sake!\" she said wearily. \"I assumed Sam and I were going to make love, so of course I had a bath. And it wasn't my clothes I was washing\u2014it was sheets.\"\n\nI tapped the e-mail. \"Then why didn't you put that in here? Why pretend otherwise?\"\n\nShe managed a creditable laugh. \"Because I forgot. In any case, I wouldn't have let Sam in at all if I'd had anything to hide.\"\n\n\"You couldn't afford not to. He'd already told you over the phone that he was going to confess everything to me that evening if you didn't agree to end it.\"\n\n\"It was over anyway. Why should I care?\"\n\nI looked at Sam. \"Because you were afraid he'd tell me Annie knew about the affair. He says she was always accosting you in the street calling you a 'dirty tart.'\" I touched my toe to the rucksack. \"There's a letter in here from Michael Percy, describing how you lashed out at her with your shopping bag and ended up on the ground, arse over tit. And you wouldn't want me adding you to the list of people with grudges against Annie,\" I finished, \"not if you'd just left her for dead in her house.\"\n\n\"I never set foot in that tip,\" she said in a remarkably steady voice, \"then, or at any other time.\"\n\n\"Oh, yes, you did,\" I told her. \"You pushed in behind her as she unlocked her door because she'd had the bloody nerve to call you what you were\u2014a cheap tart.\" I took the photograph of the brass artillery shell in Beth Slater's sitting room from my pocket. \"Is this what you used?\" I asked, showing it to her. \"It's the first thing that would have come to hand because Annie kept it in her hallway. What did you do? Yank out the peacock feathers and bring it down on the back of her head with two hands so that she collapsed on her sitting-room floor? Then what? You lost your rag completely and beat her and kicked her until she lost consciousness? Do you dream about that, Libby? Do you wake up in a sweat every time you remember it?\"\n\nShe stood up abruptly, sending her chair flying. \"I don't have to listen to this,\" she said, reaching for her handbag.\n\nSam raised his head. \"I'm afraid you do,\" he said in a surprisingly gentle voice, \"because it won't go away, Libby. Not this time. No one's prepared to support your lies anymore.\"\n\nShe turned to look at him. \"I haven't told any, Sam, or not deliberately anyway. You know that... and so does Jock.\"\n\nHe watched her for a moment. \"You primed Jock to tell me Sergeant Drury was getting his leg over in my house. Wasn't that a lie?\"\n\nShe flicked a triumphant glance in my direction. \"Of course it wasn't. Anyone with an ounce of sense could see what was going on. Your trouble is you're so full of guilt yourself, you assume everything this sanctimonious little bitch says must be true. But why should she be any more faithful than you were?\"\n\nThere was a short silence before my husband answered. I felt his hand creep into mine, and I felt it tremble, but whether from hatred of Libby or hatred of himself I couldn't tell. \"She believes in keeping her promises,\" he said simply. \"Unlike you and me, Libby, who broke ours the minute it suited us.\"\n\nMy one-time friend flicked me another glance, this time full of loathing. \"You're such a child, Sam,\" she said scathingly. \"Don't you know by now how vindictive she is? She was always going to pay me back for stealing you... even if it meant accusing me of murder...\"\n\n**_Official correspondence with the Metropolitan Police\u2014  \ndated 1999_**\n\n**From the office of the Commissioner  \nMetropolitan Police  \nNew Scotland Yard**\n\nMrs. M. Ranelagh  \nLeavenham Farm  \nLeavenham  \nNr Dorchester  \nDorset DT2 XXY\n\nOctober 5, 1999\n\nDear Mrs. Ranelagh,\n\nRe The death of Ann Butts, 30 Graham Road, Richmond\u201414.11.78\n\nThe Commissioner has asked me to keep you informed on matters relating to the above. I can now confirm that a full series of interviews has been conducted, with the exception of Mr. Derek Slater, whose present whereabouts are unknown.\n\nI can also confirm that the following charges have resulted from these interviews. Mr. Alan Slater\u2014burglary at 30 Graham Road at or around 02:00 on 15.11.78. Mr. Alan Slater and Mr. Michael Percy\u2014 indecent assault and actual bodily harm of Miss Butts at or around 20:30 on 14.11.78. Mrs. Maureen Slater\u2014obtaining money by deception from Smith Alder, Jewelers, Chiswick, between 06.06.79 and 10.11.79. In addition, RSPCA officers are looking at the issue of animal cruelty, although as Miss Butts almost certainly contributed to the cats'distress and deaths by failing to report incidents and\/or seek veterinary advice, a prosecution is unlikely.\n\nThe commissioner is aware that these charges may fall short of your expectations. However, he asks me to remind you that the burden of proof in criminal cases is an onerous one, which is not made easier with the passage of time. Indeed, the only reason any charges have been brought is because Mr. Alan Slater, Mr. Michael Percy and Mrs. Bridget Percy have cooperated fully with the investigators. No such cooperation has been forthcoming from Mrs. Maureen Slater, Mr. James Drury or Mrs. Libby Garth, all of whom vigorously deny the allegations made against them.\n\nMr. Drury refutes your allegation that he saw stolen articles in Mrs. Slater's house following Miss Butts's death. He also refutes any suggestion that he accepted a bribe from Mrs. Slater to \"turn a blind eye.\" Without confirmation from Mrs. Slater that these allegations are true, there is no evidence that Mr. Drury was negligent in failing to treat Miss Butts's house as a \"scene of crime.\" Mrs. Slater categorically denies that she ever suggested to you that Mr. Drury had accepted a bribe and further denies any collusion with him, either at the time of the original investigation or more recently.\n\nMrs. Slater also denies that she had any advance knowledge of the crimes her husband and son committed. She admits being told about the burglary afterward, but claims the articles were taken away by her husband and son and subsequently displayed in Mr. Alan Slater's house where you photographed them. She further denies being the woman who sold the rings in Chiswick. Nor is it likely that Mr. Alan Slater's assertion that it was his mother who \"ordered\" the burglary will stand up to cross-examination as he was adjudged during a trial in 1980 to \"be seeking to lay the blame for his worst excesses on his mother.\" This is a matter of public record, and Mrs. Slater has quoted it several times in her defense during interviews. Investigations continue into how she was able to afford the premises at 32 Graham Road. To date there is no evidence to disprove her statement that she won the money on the football pools as records are regularly destroyed.\n\nMrs. Libby Garth has been interviewed on a number of occasions and refutes all suggestions of having any involvement in the death of Miss Butts or trying to persecute you through the making of telephone calls to your house, the writing of poison-pen letters and inflicting cruelty on animals. She denies that the various \"supportive\" conversations she had with you following Miss Butts's death were \"fishing expeditions\" to discover how much you knew and whether your husband was beginning to waver over his alibi. She further denies any knowledge of the Slaters'harassment of Miss Butts in the months prior to her death, refuting absolutely any allegation that she similarly harassed you in order: 1) to focus your suspicions on the Slaters; and 2) to drive a wedge between you and your husband.\n\nIn conclusion, the commissioner has asked me to tell you that the file on the death of Miss Butts remains open, even though, on the evidence to date, it is doubtful that the Crown Prosecution Service will agree to prosecute Mrs. Garth for Miss Butts's murder.\n\nYours sincerely,\n\nFor: the commissioner, Metropolitan Police\n\n**Leavenham Farm  \nLeavenham  \nNr Dorchester  \nDorset DT2 XXY**\n\nAlisdair Fielding\n\nThe Office of the Commissioner  \nLondon Metropolitan Police  \nNew Scotland Yard\n\nOctober 7, 1999\n\nDear Alisdair Fielding,\n\nPlease inform the Commissioner that, not only do the charges mentioned in your letter fall short of my expectations, but I had already foreseen three of them when I encouraged Alan Slater and Michael Percy to be honest with the police. Both men were fourteen years old in 1978, therefore any charges now amount to little more than a technicality unless you intend to try them as adults in a juvenile court. The charge against Maureen Slater is equally valueless, as it will depend on the jeweler's identification of her after twenty years.\n\nI presume the commissioner is offering these charges by way of a sop to keep me quiet for another few months while his officers continue the pretense of investigating Ann Butts's murder. If so, he has dangerously underestimated my commitment to justice for my friend. I repeat what I wrote at the beginning of the report I submitted in September: _Ann Butts was murdered because a regime of racialhatred and contempt for handicap was allowed to fester unchecked in Graham Road_.\n\nI have no intention of letting this rest. Unless you come back to me within a week with more positive news, I will approach the press.\n\nYours sincerely,\n\n# epilogue\n\nIt was an unsettled autumn in Dorset with southwesterly winds piling in from the channel and whipping the trees into a frenzied dance around the farmhouse. Sam and I spent days raking the leaves into russet piles, only for them to be blown away again as soon as the wind returned, but it didn't seem to matter. It was so long since we'd enjoyed the glorious turning colors of an English autumn that just being outside brought contentment.\n\nThe boys settled into local college life in order to prepare for university the following year. They were older than their contemporaries, particularly Luke, but they preferred the idea of a year's adjustment to diving in at the deep end. Sam and I appreciated it, too. None of us was quite ready to see them go their separate ways when we were still trying to put down roots. I had one or two anxieties as we signed away our fortune to buy the farmhouse. Would the roof blow off before we had time to repair it? Was the wet rot under the floorboards as bad as it looked? But Sam was indomitable and gave us all confidence.\n\nMy father took the boys to the highlands of Scotland during the half-term break to give them a taste of the true Ranelagh homeland, and in return Sam and I had my mother to stay. My father's somewhat Machiavellian intention was that we should all get to know each other a little better\u2014and in a way we did\u2014because Mother had a fine old time interfering with Sam's renovation work and telling me how frightful my taste in curtain material was.\n\nIt would be an exaggeration to say our relationship improved. The dynamics of competitiveness and mutual criticism had existed too long between us to vanish overnight. I was still a poor wife to Sam, ignoring his coronary, encouraging him to do too much, not cooking his meals on time... and the boys, though absent, were still too free and easy in their manners and still needed haircuts. As for her... well... she would always be a control freak, offering advice that wasn't wanted and dominating everyone while pretending to play the role of martyred slave. But the sparks flew a little less regularly, so perhaps we were making progress.\n\nShe had a residual jealousy of Wendy Stanhope, whose visits had been rather more frequent than hers. I introduced them on one occasion but it was a mistake. They were too alike, both of them strong-minded women with decided views, though with little prospect of their minds ever meeting. Wendy admired youth and longed to give it space, while my mother wanted only to corral and discipline it. Wendy would never be so rude as to pass a comment afterward, but Mother, with no such restraints, told me it didn't surprise her at all that the silly woman was in the habit of screaming from clifftops. \"Why?\" I asked. \"Because she was unable to make friends with her own age group,\" was the barbed answer.\n\nOne of the reasons for Wendy's frequent trips was to visit Michael in prison and drive on to Bournemouth to see Bridget. Wendy and I made the round journey together the first time, but on subsequent occasions she went alone. In between whiles I visited Michael myself. I asked him once if he thought Wendy still wanted to adopt him. He grinned and said she only ever gave him lectures these days because she'd transferred her affections to Bridget and was acting like his mother-in-law. Was that a good thing or a bad thing? Good, he told me. It would be harder to let his wife down in the future if he had a fire-breathing dragon on his back. He added somewhat wistfully that it was a pity Mrs. S. hadn't taken that tack before. And, by implication, me too.\n\nFor myself, I wondered why my more intelligent pupil had to struggle with the concept of good behavior being its own reward, while Alan, the Neanderthal, had put it into practice and accepted it. In the end I accepted Sam's analysis\u2014a strong-minded woman is a man's best friend.\n\nI had an angry letter from Beth Slater midway through September in response to one from me, which had set out to explain how committed I was to Annie's cause and why it was necessary to involve Alan. But she couldn't be persuaded, and her anger saddened me. She hated people who pretended one thing and did another. She hated the police, who had stripped their house of everything, even the things Alan could prove he'd bought himself. She hated Derek, who was a bastard, and Maureen, who was a bitch. And was it surprising Alan had gone off the rails when he had been so abused as a child? But nothing could excuse what I had done. Did I not realize that by destroying Alan I'd destroyed Danny as well?\n\nShe ended by saying she never wanted to hear from me again. However, I remained optimistic because I'd learned a great deal about the healing powers of time\u2014and I was sure she knew how much I admired her.\n\nTo my relief, Danny turned up like a bad penny toward the end of October. He had a filthy hangover. He was irritable and tetchy, and laid down rules and regulations about his private space and what he was allowed to do in it. \"Like what?\" asked Sam.\n\n\"Chill out... smoke a joint now and then...\" He needed peace and quiet to get his head straight, and we owed him that much for setting his family at each other's throats.\n\nSam, equally relieved, backed him against a wall. \"What about my wife's head?\" he demanded. Didn't his family owe me something for what his father and brother had done to me? Danny was scornful. How could the Slaters compensate his missus? What did they have that she wanted? Hell, she was in a different league. That's why he'd come. He reckoned she could teach him a thing or two... about internalized pain... and how he could use it to exploit his genius.\n\nSheila Arnold and I remained friends, but at a distance. We greeted each other warmly when we met in the street but recognized we had little in common. In the end I preferred the anarchy of clifftop screaming to the elegant conformity of matching panama hats. She agreed grudgingly to allow me to use parts of her correspondence in press releases, but insisted that I make it clear she was unavailable for interviews. \"Larry would never approve,\" she said.\n\nJock arrived for a long weekend in November and helped us re-felt and re-tile the western end of the roof above the attic. He and I did most of the heaving of materials while Sam straddled the gable and shouted orders. Then, come the evenings, we dropped into armchairs and threw cushions at Sam until he agreed to pour us enormous glasses of wine and make our supper. I came to wonder why I had ever disliked Jock, and what had persuaded me that Sam might choose his friends unwisely.\n\nJock disappeared into the barn every so often to share spliffs with Danny and give him the benefit of his wisdom on money and women but, fortunately, most of it went in one ear and out the other. More sensibly, he bought the first, and rather fine, sculpture that Danny carved at Leavenham Farm. It was a folded figure of a woman with her head resting on her knees, entitled _Contemplation_ , and was a huge leap forward from the _Gandhi_ on my terrace. But I wouldn't have swapped _Gandhi_ for the world.\n\nOn his first evening, Jock produced a copy of Richmond's local newspaper, featuring an article on Annie's death with the headline: \"Accident or murder?\" He asked us if we'd seen it, and showered me with new respect when Sam laughed and said I'd written it. Of course it had been heavily edited, but I'd tried to re-create the atmosphere in London during 1978's winter of discontent when society was at war with itself in the months leading up to a vote of no confidence in Parliament and the dramatic fall of the Labor government. I asked how in such a climate there could be any certainty that the death of a black woman had been properly investigated. I went on to describe the racial hatred that had been allowed to flourish in Graham Road, citing the catalogue of unsubstantiated complaints against Annie by \"benefit scroungers\" which went unchallenged by the authorities, and the vicious bullying and harassment of a vulnerable woman by a \"hate group\" that was never questioned by the white policeman in charge of the investigation. They allowed his name to stand, Sergeant James Drury, together with his subsequent \"forced retirement\" for a racist assault on an Asian youth. Publish and be damned, they said! But, for me, the most satisfying part of the article was an unflattering photograph of Maureen Slater, caught in the act of closing her front door, with the caption: \"Benefit recipient denies orchestrating hate campaign.\" _They've done me proud_ , I thought.\n\nI made Sam swear he wouldn't mention Libby There was too much pain involved. Jock had lingering sympathies for her because he felt himself partly to blame.... Sam had lingering guilt for the same reason... while I swung between a sense of triumph at my vindication, and an ongoing sadness for what I was doing to her children. But somewhere along the line I was outvoted and, at Sam's instigation, Jock brought me up to date over the dinner table on the last night of his stay\n\nThe word from \"mutual friends,\" he told me, was that Libby's husband had kicked her out and imposed a restraining order to prevent her having access to her children. Apparently her fuse was so short these days\u2014\"too many police asking too many questions\"\u2014that she'd taken a steel rod to her eldest daughter and the child had ended up in hospital. More disturbingly, the girls had revealed that beatings had been commonplace whenever Libby's frustrations had reached the boiling point, and now she faced prosecution for child abuse and the inevitable loss of her teaching job.\n\nJock said she was showing her true colors and he wouldn't blame me for crowing. But Sam just reached for my hand under the table and held it companionably while I pictured myself beside a river... watching the bodies of Annie's enemies float by...\n\n**_Note from Ann Butts, which was pushed through the  \nRanelaghs' letterbox at number 5 Graham Road the day  \nbefore she died. It was addressed to the \"Pretty Lady. \"_**\n\n_Copyright \u00a9 2001 by Minette Walters_\n\nVintage is a registered trademark and Vintage Crime\/Black Lizard and colophon are trademarks of Random House, Inc.\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. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nLibrary of Congress Cataloging-in-Publication Data  \nWalters, Minette.  \nThe shape of snakes \/ Minette Walters.  \np. cm.  \n1. West Indians\u2014England\u2014Fiction. 2. Women cat owners\u2014Fiction.  \n3. London (England)\u2014Fiction. 4. Women, Black\u2014Fiction. I. Title.  \nPR6073.A444 S48 2001  \n00-065319  \n823'914\u2014dc21\n\n**eISBN: 978-0-307-49784-0**\n\nwww.vintagebooks.com\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCONTENTS\n\nTITLE PAGE\n\nINTRODUCING . . .\n\nRULE 1\n\nRULE 2\n\nRULE 3\n\nRULE 4\n\nRULE 5\n\nRULE 6\n\nABOUT THE AUTHOR\n\nCOPYRIGHT PAGE\n\nWhen I grow up, I'm going to be a _Performer Extraordinaire!_ So I do singing on Mondays, ballet on Tuesdays, acrobatics on Wednesdays, tap on Thursdays and drama on Fridays!\n\nDrama is my favourite! **FACT!** Each week our drama teacher, Miss Oscar, who is only the most amazing-est drama teacher _in the world_ , gives us a scene from a famous play or musical to perform. Her **RULE** is that you have to learn to walk in your character's _shoes_ before you can perform that character.\n\nMiss Oscar says walking in a character's shoes is also called 'method acting' or 'going method'. She says going method is _crucial_ if you want to be a **PERFORMER EXTRAORDINAIRE!**\n\nYou can't just _pretend_ to be that character \u2013 you have to _be_ them. You have to do what they do, and think like they think, and feel what they feel.\n\nSo that is what I do, every week! But enough about me ...\n\n**'Mum! Dad!** You will not **BELIEVE** what we have to do. It's so uncool! So ultra-blah! _So babyish!_ I'm almost ten!' Lily D storms into the kitchen, drops her schoolbag and slumps on a chair at the kitchen table. Lily's bead bracelets, which run all the way up one arm and all the way down the other, clack on the tabletop like castanets.\n\n'Miss O must think I am a little _kid_!' she groans.\n\n'What are you talking about, Lily-Pilly?' asks Dad. 'Here, I made some fairy bread,' he says, holding out a plate.\n\n'Aaaargh!' yelps Lily. 'Don't mention fairies!'\n\nDad is a chef and makes the **BEST SNACKS**. But he has no idea that no-one eats fairy bread anymore. Once you turn nine, you eat sprinkle sandwiches \u2013 **totally different!**\n\n'I thought you liked fairies, Lily?' says Mum from the other end of the table. She is scrawling little dots and dashes all over a bunch of papers with a red pen. Mum is a writer, and she's in the middle of editing her newest book.\n\nTo Lily D, editing looks like scribble. Funny, if Sonny or Lily drew all over _their_ work it would be naughty, but when Mum does it, it's 'editing'. Weird.\n\n**'BLURGH!'** Lily scrunches up her face. 'Fairies are, like, so grade two. I'm in grade four now, Mum! Compleeeeetely different! I do _not_ like fairies.'\n\nUp in her room, Lily still has a pair of gloriously glittery fairy wings. They are hidden in the back of her cupboard. Sometimes she puts them on in secret. Lily doesn't want to throw them out \u2013 just in case \u2013 but she doesn't need Mum to know that!\n\nMum scratches her head with the red pen. When she reaches for her cup of tea, the pen stays there, tangled in her hair. Parents can be so weird!\n\n'So why all this fairy talk all of a sudden?' Mum asks.\n\n'Drama class,' sighs Lily. 'Next week Miss Oscar wants us to do scenes from this totally strange musical called _Into the Woods_.'\n\n**'Oooooooh!'** squeals Mum. **'I LOVE that play!'**\n\n'What's it about?' asks Dad.\n\n'Fairytales! Heaps of them,' moans Lily, stuffing another piece of fairy bread in her mouth. Whoops \u2013 sprinkle sandwich. 'Cinderella is in it. So is Little Red Riding Hood and that wolf, and Jack from _Jack and the Beanstalk_ and even Rapunzel! Well, the play is actually the story of a baker and his wife. They want to have a baby, but a witch has put a curse on them. So they have to go into the woods and collect all these things to break the curse. And all the fairytale characters are mixed up in the play.' Lily stops to take a breath. 'It's all very **_com-pli-cated_** ,' she says. Lily feels quite grown-up using such a big word. 'But fairytales... well, they are so **_babyish!_** '\n\n'Infantile,' corrects Mum.\n\nLily frowns. 'No offence, Mum, but this is kind of serious. It's not a good time to practise speaking French!'\n\n'Infantile means exactly the same thing as babyish, except it's a real word,' explains Mum in a half-stern, half-giggle voice.\n\n**'WIKE FFFWAT?'** asks Lily around a mouthful of sprinkle sandwich.\n\n'Like manners,' says Mum, in an actual stern voice.\n\nIt's the kind of voice that makes Lily chew politely, with her mouth closed. Lily knows Mum has superpowers when it comes to words. She can tell if a word is used incorrectly, or spelled incorrectly \u2013 even if you just _say_ it with the wrong spelling. But Lily didn't know Mum understood 'talking with your mouth full'!\n\n'Fairytales were invented to scare kids so much that they behaved themselves, right?' says Dad.\n\n'In a way,' says Mum. 'But more importantly, fairytales teach children about the world, and how to face their fears.'\n\n'Even Cinderella?' asks Lily. 'What's Cinderella scared about? Doesn't she just mop the floor, have some mean stepsisters, ride in a carriage and wear glass slippers?'\n\n'Yes, definitely even Cinderella!' says Mum. 'What do you think she might be afraid of?'\n\nLily thinks. 'I guess Cinderella might be scared because she lost her mum.' Lily pauses for a moment. **WHOA.** Losing your mum _would_ be totally scary, and super-sad. 'And her wicked stepmother is ultra-mean and kind of freaky. That would be scary too.'\n\n'Exactly!' says Mum. 'But Cinderella confronts those fears and goes to the ball, and that changes her life for the better!'\n\n'Yeah, but with the help of her _fairy_ godmother,' mutters Lily, rolling her eyes. **'Not another fairy!'**\n\n'Not in _Into the Woods_ ,' Mum says. 'That Cinderella is based on the original story, where a magical tree growing above her mother's grave gives Cinderella the shoes and the dress for the ball.'\n\n'Phew, one less fairy!' sighs Lily. The sprinkles are scratching at the inside of her throat, so she pushes her plate away. Lily is feeling a bit yucky actually.\n\nSonny, Lily D's little brother, wanders into the kitchen wearing some kind of alien overlord cape. 'Oooh! Fairy bread! Yes!'\n\n'Sprinkle sandwiches!' Lily thinks. But Sonny is only six, so she doesn't say anything.\n\nSonny grins at Lily and then stuffs three pieces into his mouth at once. **Little brothers... gross!**\n\n'So,' Lily asks, 'fairytales are about people facing their fears, and then waiting for a prince to rescue you?'\n\n'Well, some might see it that way,' Dad says with a grin. 'But really I think fairytales teach you to face your fears so you can rescue yourself.'\n\nLily smiles. She thinks that sounds **pretty crazy-cool.** But she can't really see how a fairytale could ever help her. She can't think of one single thing that scares her. Lily D is positively _fearless_.\n\n'So which character are you playing, Lily-Pilly?' asks Dad.\n\n'Not sure yet,' Lily whispers. Her throat is starting to tickle, but not in a funny, makes-you-laugh way. 'We're supposed to read the whole play, and prepare a scene with the character we _most identify_ with.'\n\n'I could help you with the baker,' says Dad with a grin. 'I can show you how to bake a chocolate cake!'\n\n'Thanks, Dad... but no thanks.'\n\nDad looks worried. 'Are you okay, Lil? You never say no to chocolate cake.'\n\n'I'm fine,' she replies. But she's not feeling so sure. 'It's just that there are so many fairytale characters all kind of mixed up in the play,' Lily adds. 'I don't know which one to choose.'\n\n'Maybe you could play a mixed-up fairytale character!' giggles Sonny.\n\n'You mean like from _Rapunzel and the Beanstalk_?' asks Dad, laughing.\n\n'Or _The Three Little Cinderellas_ ,' suggests Mum.\n\n'Or _Little Princess Snow Bean Cinder Hood_!' shouts Sonny.\n\nNow everyone is laughing, holding their sides like they might split.\n\n'I like that one.' Lily smiles weakly. **'Little Princess Snow Bean.'**\n\nFairytale character mash-ups are actually kind of funny. But after all those giggles, Lily's head is feeling fuzzy. She feels pretty yucky all over.\n\n'Are you sure you're okay, honey?' Mum asks, coming over to press her hand to Lily's forehead. 'You do feel a bit warm.'\n\n'I'm okay,' mutters Lily, pulling her head away. 'I'm just going up to my room for a bit.'\n\nLily stomps up the stairs to her room a little more slowly than usual. She is going to work on her 'Going Method Rules' notebook.\n\nBut as she lies on her bed with her pen ready to make some ultra-important notes, Lily realises she isn't feeling okay. Her tummy is all squirmy, and her face feels hot and steamy and 'like you could fry an egg on it', as Grandad would say.\n\nWorst of all, Lily is starting to feel worried. How can she 'go method' this time, when she isn't scared of anything? Will she be able to give an amazing performance of a fairytale character facing their fear, when she, Lily D, has no fears to face?\n\nLily just feels like crying, which is so babyish \u2013 so _infantile_ \u2013 it's just embarrassing. 'Focus, Lily, focus!' she tells herself.\n\n'What does it feel like to be truly scared?' she wonders, tapping her swirly-rainbow pencil with the crystal love-heart on the end against her chin. Time to make some character notes...\n\n**'N** o-one can beat Hudson the Great, strongest man in the world.'\n\nLily cracks open one eye. Bright sunlight is pouring through her polka dot curtains and something sharp is sticking into her cheek. **FACT!**\n\nLily props herself up on her elbows to see what it is. It's her notebook. She must have fallen asleep writing her final 'going method' rule. Just as she's pondering what the rest of   might be, Lily notices she's still wearing her school uniform and that her little brother is standing in the doorway.\n\nSonny is dressed in his cape and speaking in a big, deep voice. **'And YES! I can fly too!'** he announces, swishing his cape from side to side. Brothers! _So annoying!_\n\nLily D flops onto her stomach. She feels super- _weird_... like she's Sleeping Beauty, waking from a **_looooong_** sleep.\n\nShe goes to yell her brother's name, but no sound comes out! Just a squeaky 'So\u2014'. Then comes the pain... **OUUUUUCCCCHHH!**\n\nLily's throat burns. It burns like she has smooshed up a mountain of red-hot chillies and drunk the juice.\n\nEvery swallow feels like when you don't chew up a cracker properly before you swallow it, but multiplied by a _gazillion_! It feels as though Lily has pricked all her fingers on one hundred spinning wheels, stubbed all her toes, jammed all her fingers in a door and banged both her funny bones, all at the same time... and then shoved all that pain between two bits of bread and eaten it. Yep, Lily has definitely chomped down on a **pain sandwich** , a **hurt-burger**... and now her throat throbs so bad Lily doesn't think she can ever eat again!\n\n'Hey, Lily,' says Sonny. He has his serious battle face on. 'I know you need to practise fighting your fears for drama class, but what about fighting Hudson the Great to defend your castle first?'\n\nLily wants to tell him that she is busy enough trying to choose a character from _Into the Woods_ to play without fighting a battle, but she only has a tiny little voice left. 'What time is it?' she manages to squeak.\n\n'Huh? Breakfast time, silly. But only if you can get past Hudson the Great!' says Sonny, twirling his light-up plastic sword.\n\nLily is seriously confused. It's morning?! So Lily really did have a Sleeping Beauty-style looooong sleep! Did she sleep through dinner?\n\n'You slept through dinner last night,' says Sonny. 'Mum said to let you rest because you weren't feeling well.'\n\n'Lily!' At that moment, Mum bustles through Lily's bedroom door, like one of the good fairies from _Sleeping Beauty_. She is carrying a mug and a little bottle of medicine. 'How are you feeling?' Mum asks gently.\n\nLily just shakes her head. It hurts too much to talk.\n\n'Oh, dear,' sighs Mum. 'Open up, let's have a look.'\n\nLily opens her mouth wide.\n\n'Well,' says Mum, with a great big sigh that goes from her head to her toes. 'It's back again.'\n\nLily knows exactly what's back, and it isn't Christmas, the Royal Show, or that neighbourhood cat that liked to sit on the front fence and meow in the middle of the night. It's Lily's own **EVIL FAIRY. It's tonsillitis.**\n\nLily has had tonsillitis three times already this year. And five times last year!\n\nLast time, Dr Rodriguez had told Lily in a super-serious voice, 'If it comes back again we will have to do something about those tonsils.'\n\nLily isn't exactly sure what that meant, but she guesses Dr Rodriguez wasn't talking about giving Lily's tonsils detention or making them do extra chores. Whatever it is Dr Rodriguez had planned, **it couldn't be good...** even if Dr Rodriguez is the nicest, gentlest doctor in the entire galaxy and always remembers to warm up the stethoscope before listening to your chest.\n\n'I'm fine, Mum!' Lily croaks.\n\n'No, you're NOT fine!' insists Mum. 'Now drink this while I call Dr Rodriguez.' She hands Lily the mug filled with warm honey and lemon juice.\n\nAs Lily sips, she thinks about how 'doing something about your tonsils' probably means having them **removed**. Lily's friend Missy said it was kind of a teensy bit awesome getting her tonsils removed, because she got to eat nothing but ice-cream for a _whole week_! But before the all-you-can-eat-ice-cream bit, there's the having-the-tonsils-removed bit. And that means doctors, and hospital... and _needles_!\n\nLily sits up straight so fast her warm honey and lemon spills a little bit on her polka dot doona cover.\n\n**NEEDLES!** Of course! Just like Sleeping Beauty and spinning wheels. That is **Lily's greatest fear!** Needles are the _opposite_ of ultra-coolness.\n\nLily lies back on her pillows. This time she thinks even harder. Maybe this is her chance to 'go method' and follow her Little Princess Snow Bean rules. If she faces her fear of needles, then maybe she can learn to be a fairytale character. If Lily can learn to be brave, she can have her own happily ever after! **FACT!**\n\nLily D finishes her honey and lemon juice and then swings down from her bed. She rummages through her top drawer, pulling out her big box of beads.\n\nRunning her pointer finger through the rainbow of beads, Lily selects a handful that she thinks look tough, courageous and ultra-cool. She then chooses a bunch of beads with tiny red letters painted on one side. Cutting a length of thin elastic, Lily threads her beads on before tying the two ends together and slipping her brand-new bracelet onto her wrist. It joins the bracelets that run all the way up one arm and all the way down the other. Lily spins the beads around so the letters are on top. **BE BRAVE.**\n\nShe feels a bit more fearless already! Maybe she can become a fairytale character _and_ fight her fear of needles at the same time.\n\nBut needles... Lily keeps her bracelet on but spins the **BE BRAVE** beads under her wrist and out of view. On second thought, Lily D _likes_ hanging out with her tonsils... and she is determined they will live together happily ever after. **FACT!**\n\nDr Rodriguez puts down the giant icy pole stick she was using to poke around in Lily's mouth.\n\n'Yes, my dear,' she says. 'I'm afraid those tonsils of yours need to be removed as quickly as possible.'\n\nLily grips Mum's hand, hard. Right next to her is a trolley with little plastic drawers. Lily feels sure those drawers are full of _needles_.\n\n'I'm going to give you a super fast-acting antibiotic,' Dr Rodriguez tells Lily, scribbling on her prescription pad. 'I'll also book Lily in at Chester Hill General Hospital for Friday,' Dr Rodriguez says to Mum.\n\n'B-b-but I have DRAMA CLASS on Fridays! I'm performing _Into the Woods_!' croaks Lily. She knows Dr Rodriguez only wants to help her, not put her to sleep for one hundred years. But what will Miss Oscar say if she misses drama class? Lily D has to perform!\n\nDr Rodriguez clasps Lily's hands between her own. 'Just you wait,' she says. 'Once those tonsils are out, you'll feel like a new girl!'\n\n'O-kaaaay, it's just that\u2014' mumbles Lily forlornly. She is actually pretty happy being Lily D, V.A.P. just the way she is \u2013 tonsils and all. She wants to explain, but at that moment Dr Rodriguez gently takes hold of Lily's wrist.\n\n'Beautiful bracelet. What does it say?' she asks.\n\n'BE BRAVE,' reads Lily, twisting the beads towards Dr Rodriguez before giving them a little shake, one quick round of applause for courage.\n\nLily _loves_ beads. When you give them a shake, you have an instant audience clapping! Everyone needs a round of applause sometimes, so why not applaud yourself?!\n\n'Be brave,' repeats Dr Rodriguez. 'Very clever. I couldn't have said it better myself.' She gives Lily a crinkly little wink that makes her smile twinkle.\n\nLily gives a teeny little half-smile in return. Brave or not, she is still determined to hold onto her tonsils, be they pink, green or tangerine. But at least she has discovered she has a fear... that has to mean she is one step closer to becoming Little Princess Snow Bean!\n\n**'I** have three words for you, Lily,' says Missy, waggling three fingers in front of Lily's face. 'Jelly, custard, ice-cream.'\n\n'That's four words,' says Lily with a grin.\n\nLily, Missy, Kat and Jolly are doing some very gentle gymnastics on the oval at lunchtime. The girls have marked out a performance floor using their lunch boxes and water bottles.\n\nDr Rodriguez gave Lily a medical certificate, so she could have stayed at home all week. But after a few days spent cataloguing her slime collection, googling spooky stories to freak Sonny out with and studying fairytales, Lily was bored. **She missed her friends.**\n\nBesides, thanks to the antibiotics, she is feeling okay! So okay, in fact, that Lily is certain she won't need her tonsils removed after all.\n\n'You'll have to get them out,' says Kat, while walking on her hands. 'If you don't, you'll just keep getting sick!'\n\n'Hmm,' says Lily. It's kind of hard to take Kat seriously when she's upside down, but she does have a point. 'But why does it get worse?'\n\n'Dunno.' Kat tries to shrug, which makes her collapse. But, being such a bendy gymnast, Kat twists into an elegant roll and is back on her feet in a second. Awesome!\n\n'Maybe your tonsils will, like, fall off!' suggests Jolly, who is stretching on the grass.\n\n**'Or maybe your WHOLE HEAD will fall off!'** giggles Kat.\n\n'Phew, lucky I've had my tonsils out then!' breathes Missy, ultra-seriously. 'Oh, sorry Lily!'\n\n'That's okay,' Lily says, half-giggling, half-seriously. 'Was it horrible having your tonsils removed? Were there any... ummm... _needles_?'\n\n'Nah!' says Missy, waving her hand like she's shooing flies. 'It was nothing! I mean, I was only five so I can't _totally_ remember it, but I do remember the ice-cream, jelly and custard, and all the teddies my grandparents gave me.'\n\n'Still, I really think I'm too busy to go to hospital this week... I mean, I really need to focus on _Into the Woods_.' Lily decides the best way to take her mind off her tonsils is to concentrate on being Little Princess Snow Bean and fighting her fears. Even though dealing with her tonsils means hospital, which means needles, which are her **BIGGEST fear!** Secretly, Lily is hoping the whole tonsillitis thing will go away and she'll be on stage Friday afternoon, not in a hospital gown.\n\n'Oh! Is _Into the Woods_ the one about fairytales?' Jolly asks, dropping out of a backbend.\n\n'Yeah,' mutters Lily. 'Not that I like fairytales or anything.'\n\n'Yeah, no way!' says Jolly quickly. 'They are so babyish.'\n\n**'INFANTILE,'** confirms Lily.\n\n'Is that how you say \"fairytale\" in Latin?' asks Jolly.\n\nKat says nothing and kicks into a handstand.\n\n'Fairytales aren't so bad,' says Missy, matter-of-factly. 'Sometimes they are really useful. Like for Katherine when she has her insulin shot.'\n\nMissy's little sister, Katherine, was born with a thing called diabetes. She needs to have insulin every day, so her body can process the sugar she eats in food. Lily knows the insulin comes in a **(GULP)** _needle_!\n\n'How does it help?' asks Kat, pausing with her arms outstretched, ready to do a cartwheel.\n\n'Well,' says Missy. 'Katherine loves the _Little Red Riding Hood_ story, so Mum started telling Katherine that the little needles were the wolf trying to have a nibble on her finger.'\n\n**'That sounds scary!'** says Jolly.\n\nMissy grins. 'Then Mum bought Katherine a red cape and told her that as long as she has her red cape on \u2013 like Red Riding Hood \u2013 the wolf can never really eat her up!'\n\nLily crunches her nose up, thinking hard about what Missy said. Nose scrunching always helps when you have a tricky question to answer, or a problem to solve.\n\nShe finds all this information a bit spooky-weird. Is it possible that other kids 'go method' too? Lily always thought she was the only one.\n\nAs the end-of-lunch bell rings, Lily is thinking about Katherine 'going method' as Little Red Riding Hood, to give her the courage to face the needles. If Katherine can use fairytales to face her fear of needles, then Lily can too... maybe!\n\n**'T** ry this one. It's totally not too sour.' Lily passes Kat a bright red lolly covered with sparkly sugar.\n\nLily isn't allowed to go to tap class after school because of her tonsillitis, so Kat invited her over to her place. On the way, they stopped at the supermarket and bought a little bag of funny spicy, sour and sweet lollies. You never knew which was which until you popped it in your mouth, but Lily's lolly bag from Amy's party had had a red one in it. So Lily knew _exactly_ what the red ones tasted like.\n\nKat takes the red lolly from Lily and pops it in her mouth. Her eyes go wide like a goldfish, and her mouth stretches into a giant O. She flaps her hands and breathes in and out super-fast.\n\n**'Ooooooooh! HOT!'** She sticks her tongue all the way out and fans it with her hands. **'Not sweet!'**\n\n'Gotcha!' Lily laughs hard. She hands Kat her drink bottle, and Kat guzzles the cool water.\n\n'Ish ma thongue burnt?' Kat asks, poking her tongue out and crossing her eyes, trying to check out the damage.\n\n'No!' Lily laughs. 'But let's go get some ice cubes. You can make me eat two sour ones at the same time to get me back!'\n\nLily and Kat are at Kat's front door, so they tuck their lollies away deep in their schoolbags before they go inside. Kat has five brothers, and her youngest brother, Joe, gets sick a lot, so Kat's mum only allows lollies in the house on **Very Special Occasions**. Which is kind of confusing because, when you are a kid, any time you have a lolly is a Very Special Occasion. **Weird!**\n\nAs Lily and Kat step through the front door, Joe comes barrelling down the hallway and smacks right into them.\n\n**'JOE!'** Kat roars.\n\n'Sorry!' puffs Joe. 'Escaping the wolf ... he's not coming in! Not by the hairs of my chinny-chin-chin!' Joe is huffing heavily, gasping each breath.\n\nJust then, Kat's mum appears at the end of the hallway, following Joe with an inhaler. Joe immediately darts away.\n\n'Hi Lily! Hi little chika-dee.' Kat's mum does a funny little chicken dance down the hallway and then kisses Kat's forehead. Lily thinks Kat's mum is seriously cool. Kat just rolls her eyes.\n\n'I just need to get Joe to use his inhaler,' Kat's mum says, waving the inhaler with a flourish. 'His lungs are playing up.'\n\nLily notices Kat's mum's eyes change when she says that. Like the sky changes on a summer day when that first rain cloud passes over. 'Go into the kitchen and I will make you both a snack in just a minute.'\n\nLily notices Kat has gone very quiet and is standing very straight and stiff. Kat just nods and Lily follows her to the kitchen.\n\n'Is Joe okay?' Lily asks, as she and Kat sit up at the kitchen bench.\n\nKat sighs. 'You know how Joe was born **reeeally** early?'\n\nLily nods. Kat's brother was teensy tiny and super-fragile when he was born. 'Not quite cooked' Kat's mum always says. Ever since then, Joe gets sick a lot and is in and out of hospital.\n\n'Well,' Kat continues. 'He gets bad lung infections now, and he needs to use an inhaler, but he really doesn't like it.'\n\n'C'mon, sweet little piggy,' Kat's mum is saying to Joe in the lounge room. 'One more huff and puff and you will be made of bricks again!'\n\nLily raises her eyebrows at Kat. **Piggy?!** Lily doesn't want to be rude and ask a whole heap of questions, but her eyebrows sure do! Missy's little sister Katherine with her Red Riding Hood cloak and the needle wolf, now Joe becoming bricks, and Kat's mum huffing and puffing! Had Lily wandered into some magical fairytale parallel universe? Or was everyone 'going method' along with her? No wonder Lily's eyebrows had so many questions! Spooky-freaky, but also ultra-coolness!\n\nKat grabs an apple from the bowl on the bench and turns it around between her fingers.\n\n'From when Joe was about three years old,' Kat explains, 'whenever he got sick, Mum read him fairytales.' Kat speaks in her ultra-quiet voice. 'His favourite is _The Three Little Pigs_ and Mum would tell him to imagine that when he was sick he was made of sticks. But, when he came to hospital, took his medicine and let the doctors do their job, all this helped change his body into a house of bricks.'\n\n'Ahh...' says Lily. 'So the wolf couldn't blow him down?'\n\n'Exactly! If Joe imagined his illness was the wolf and imagined his body was getting stronger, built out of bricks, he seemed to get better much quicker.'\n\nKat does a little cough, like she's swallowed a piece of apple too fast. But she hasn't taken a bite of her apple yet. Lily wonders if it's one of those little balls of sadness that gets stuck in your throat when you talk about something that makes your heart hurt.\n\n**'Whoa!'** breathes Lily. 'That is incredible... so he will be okay if he uses his inhaler?'\n\nKat nods. 'Totally!' she says. 'Joe's seven now, and he is a bit smaller than the other kids, but he'll catch up. Plus, he is super-smart, and much better than me at gymnastics! And he... he...' Kat stops talking and looks at the apple again, her cheeks burning bright red.\n\n**Oh no!** Poor Kat! Lily can tell Kat is still super-scared Joe might get sick again. Lily wants to make Kat feel better.\n\n'Hey, Kat...' Lily says, with the dreamy, faraway voice she sometimes uses when a big idea pops into her head. 'Do you realise Joe was totally \"going method\" when he was imagining himself as the house of bricks? Just like I do for drama class. Just like a **Very Amazing Performer!** And when he became that house of bricks he learned to face his fears. Now he isn't scared of anything. Miss O would be ultra-impressed!'\n\nLily's friends sometimes get their own faraway look in their eyes when Lily starts talking about 'going method', but not today!\n\nKat smiles a big, juicy, beaming smile. 'And he's never going to be a house of sticks ever again!'\n\nLily grins at her friend. That's when she realises the truth, like when Little Red Riding Hood saw the wolf from behind the disguise of Grandma. Fairytales really can help people. Lily can see that now... and maybe they aren't so babyish after all... **FACT!**\n\n**'A** re you feeling scared about tomorrow?' asks Dad, passing Lily a mug of his famous hot chocolate.\n\nMum is upstairs helping Sonny with his reader before bed.\n\nLily shrugs her shoulders, then nods. But she doesn't say a word. It looks like there's no way to escape having her tonsils removed now.\n\n'And are you disappointed you won't get to perform your scene from _Into the Woods_?' Dad asks.\n\nLily is super-disappointed she won't be performing, and **super-dooper** nervous about going to the hospital tomorrow. But she's scared that if she opens her mouth to say anything, a whole bunch of tears might start squirting out of her eyes. Lily gives her beads a little shake \u2013 just a tiny round of applause for bravery.\n\nDad just smiles. 'So... which character from _Into the Woods_ did you choose anyway? Must be pretty sick of those babyish fairytales by now!'\n\nLily thinks for a moment, taking a long sip of hot chocolate. **It is _sooo_ good.** Sipping hot chocolate even makes her grumpy tonsils feel a bit happier. 'I still haven't chosen a character,' Lily confesses. 'But now I do think fairytales can be quite useful.'\n\n'Really? How so?' asks Dad. He takes a long sip of hot chocolate too. When he places his mug back on the table, Lily sees that he has a chocolate moustache! Dad is so funny, especially when he isn't meaning to be.\n\nLily giggles and then puts down her mug, looking serious.\n\n'Well, like Mum said, fairytales are for learning about the world, and facing your fears. So, when a kid is scared, they can imagine themselves in a fairytale. They can go method like I do when I become a character. They can be that strong house of bricks or be clever Red Riding Hood or whatever part of the fairytale that makes them feel a little bit braver than they thought they were. Do you know what I mean? It's a bit complicated, Dad.'\n\nDad nods. He is grinning so wide his chocolate moustache stretches from ear to ear. 'That makes perfect sense, honey. And you know what? It doesn't just work for _kids_ who are feeling a bit scared.'\n\nLily is confused. 'Huh? You mean dogs like fairytales too?'\n\nDad laughs. 'I don't know, I don't speak Dog, so I've never asked. I mean _adults_ can use fairytales too!'\n\n**'Seriously?!'** gasps Lily.\n\n'Wanna hear a secret?' Dad asks.\n\nLily nods, her eyes wide. She loves secrets.\n\n'Okay,' begins Dad. 'Years ago, when you were little, and Sonny was still just a bump in Mum's belly, your mum and I were busy building the restaurant\u2014'\n\n'Oh yeah! I remember that!' Lily had been about three years old, and Dad would leave first thing in the morning, and come home way past Lily's bedtime. Lily remembers that he always looked really tired, and always had paint splattered in his hair from painting the walls.\n\n'Well, it was really hard work, and every day when I went in, I would tell myself I was a bit like Jack from _Jack and the Beanstalk_.'\n\n'Huh? You met a giant?' Lily laughs. **'That is ultra-coolness!'**\n\n'No! I mean that I had to take some risks, like Jack did selling his only cow for some beans. And sometimes, Lil, I worried I had made a big mistake. But building the restaurant was like Jack climbing that beanstalk. I had to climb all sorts of hurdles to get the restaurant up and running. There wasn't a golden goose at the top of the beanstalk, but now the restaurant pays for us. It lets us take holidays, and buy stuff and have pizza on the weekends!'\n\nLily grins. She likes the idea of Dad's restaurant being on top of a giant beanstalk. If it grows giant beans, even better! Yum!\n\n'So, now, when I feel worried about something, or there is a problem, I still imagine I am Jack and I'm chopping down that beanstalk, chopping down my problems!'\n\nLily scrunches up her nose, thinking. By imagining himself as Jack, chopping away at his problems, Dad was totally 'going method'. Then a funny picture pops into Lily's head.\n\n'So...' says Lily, 'if the restaurant is the giant's castle, and you are Jack, is Mum like the giant?' Lily giggles.\n\n**'Hahaha!'** Dad slaps the table, making both mugs of hot chocolate jiggle. 'Only when she can't find her keys and stomps around the house in a bad mood!'\n\nLily laughs. Mum does turn into a grumpy, stompy giant when she loses her keys! If she loses her glasses, she gets even more stompy! **FACT!**\n\n'No,' Dad says. 'Mum is my golden goose, and you guys are the most precious things in my life. It's our job in this fairytale to keep you all happy and safe, forever after!'\n\nSonny pops up from behind the couch. ' _Muuuum!_ Dad said you are a goose!' he calls up the stairs. 'Oooh! Hot chocolate!'\n\nDad pulls out a chair for Sonny to join them at the table and then pours him a mug of steaming chocolate.\n\n'Is Daddy telling you guys his _Jack and the Beanstalk_ story?' asks Mum, coming downstairs with Lily's PJs and fluffy bunny slippers.\n\n'Dad said building the restaurant was kind of like climbing the giant's beanstalk,' blurts Lily.\n\n'That's right,' says Dad, smiling at Mum in the special way that makes Lily all squirmy with embarrassment, but all shiny with joy at the same time.\n\n'Do you think it was worth risking everything on those magic beans?' Dad asks Mum.\n\n'Absolutely.' Mum nods, looking back at Dad in _her_ special way.\n\nFor the first time all week, Lily feels really, really good. Maybe fairytales aren't so ultra- _infantile_. Everyone wants a happily ever after and, looking at her family, Lily knows that everyone is scared of something, but that doesn't mean your dreams can't come true.\n\n**'T** hat's right, Lily's nearly ten.' Mum is busy at the hospital admissions desk, filling in paperwork and answering questions.\n\nLily D is busy squirming in her waiting-room seat. A part of her wants to just run away, and the rest of her is generally freaking out! Outside it's still dark. It seems like the whole world is asleep, except for Lily D, Mum, the admissions desk lady and the girl sitting two seats down from Lily.\n\nThe girl looks about Lily's age, and she's wearing the **MOST AWESOME BEANIE** Lily has ever seen. It's all the colours of the ocean: blues, greys, greens and white. The girl has her nose deep in a book, but keeps scratching her head underneath her beanie now and then, without once taking her eyes off the page. Lily is about to suggest she get checked for nits, like they do twice a year at school, when suddenly the girl whips her beanie off \u2013 and there is no hair underneath!\n\nLily stares... then looks at her hands, looks at Mum at the admissions desk, then looks at the girl. **AARGH!**\n\n'Stop it, Lily!' she thinks to herself. 'Don't stare!' So, Lily looks at the clock on the wall, then checks her bag to see if she had packed PJs and a toothbrush, and then looks at the girl.\n\nThis time, the girl _is looking back at her_!\n\n**'Hi!'** Lily says, with an awkward little wave and a lopsided smile.\n\nThe girl waves back, smiling her own version of a half-smile, before returning to her book.\n\n'What are you reading?' Lily asks. 'Ah, _Rapunzel_ ,' she says, reading the cover of the book the girl holds up to her. 'Do you like fairytales?'\n\n'Ummmmm,' the girl looks kind of startled that Lily is speaking to her. 'Nah, I don't like all fairytales, but I always read _Rapunzel_ when I come into hospital for my treatments.'\n\nSuddenly the girl blushes and looks down at her beanie, before quickly pulling it on over her head. 'Sounds kind of dumb, I know... but this story makes me feel, um, braver?'\n\n'Totally,' Lily says, nodding seriously. 'Fairytales help lots of kids face their fears.'\n\nThe girl smiles. 'Yeah! And it makes me feel better about this too,' she says, pointing to her head.\n\n'Your beanie?' Lily asks. 'Oh! Coz it's itchy, right? Hmm... does Rapunzel have an itchy beanie? I don't remember that bit,' Lily jokes. 'Maybe she just has nits!'\n\nThis makes the girl laugh hard. She has an awesome, honking laugh that is so full of joy it makes Lily feel tingly all over.\n\n'No! Reading _Rapunzel_ makes me feel better about my hair!' the girl explains. 'The treatments made it start falling out so we had the hairdresser shave it all off.'\n\n**'Oh... whoa!'** says Lily. Now it is her turn to feel a bit embarrassed. 'I just thought YOUR parents were really cool and let you shave your head. I begged Mum and Dad to let me shave my hair a while ago, so I could put tattoos all over my head. I wanted to practise being a character that looked different to everyone else, but they wouldn't let me.'\n\n'Well, I sure know what that feels like,' grins the girl.\n\n'What?!' Lily says, her eyes wide. **'Have you tattooed your head?'**\n\n**'NO!'** laughs the girl, and there is that totally awesome laughter again. Lily feels so tingly she giggles too.\n\n'I mean,' the girl starts, 'wherever I go, my shaved head makes me look different, and I am smaller than the other kids my age so that makes me different too. All that difference kind of makes me feel bad... like I'm a bit ugly, so no-one will like me.'\n\nThe girl is facing Lily, but her eyes seem to be looking somewhere far, far away. Seeing things Lily can never imagine. Feeling things Lily doesn't understand.\n\n'That's why I love Rapunzel... because I am kind of like her. Being in hospital all the time is kind of like being locked away in a tower. It makes me different.' The girl pauses, and takes a deep breath. 'Rapunzel is beautiful, with amazing hair, but even after the evil witch cuts it off, she still gets her happily ever after. Well, I don't think I'm beautiful, and all my hair is gone.' The girl's voice gets a bit croaky, and she takes off her beanie again and rubs her hand over her shiny scalp. Her words make Lily's heart hurt. 'But I am DETERMINED I am going to have my happily ever after,' the girl whispers, like it is a very special secret. 'And when I look in the mirror, I see that I am getting better, and I see how hard I have worked to get better... and so I guess I like what I see.'\n\nThe young girl with the big sad eyes and the tinkly, honky laugh rubs her head again. She obviously feels a bit weird without her beanie on.\n\nLily's heart feels like a wet sponge that someone is wringing out to dry. Looking at the girl beside her, all Lily can see is someone who is incredibly brave and more deserving of a happy ending than any character in a fairytale.\n\nLily really wants to make this girl feel good, feel happy, but she is no fairy godmother, and **she has no magic wand.** But then she remembers what Mum tells her and Sonny every time they pass someone in the street who isn't as happy, healthy and fortunate as them: 'Show everyone kindness, and treat them with respect. That is the greatest gift you can give in this world'.\n\nLily sticks out her hand. **'Hey, I'm Lily D, V.A.P.'**\n\nThe girl takes Lily's hand, and gives a firm shake that makes Lily's bracelets applaud.\n\n'Samara Waterhouse... with no letters after my name. What do your letters mean?'\n\n'Pleased to meet you, Samara. The letters usually stand for Very Amazing Performer, but today they stand for **Very Anxious Patient!** I'm having my tonsils out, and I am totally freaking out!'\n\nSamara laughs her tingly, honky laugh.\n\nLily leans in close to Samara's ear and whispers, 'Don't tell anyone, but I am sooooo scared of needles!'\n\nSamara leans in, and whispers in Lily's ear in reply. 'Don't you tell anyone, but I am soooo scared of clowns. Sometimes in here, they bring clowns in to entertain the kids, and I am all like **aaaaaarGHHHHH!** '\n\nLily laughs so hard she snorts.\n\nJust then, a woman and a man walk up. They are wearing big smiles that are a mixture of happy and sad, and pride mixed with a bit of fear. Lily is wearing the exact same smile. Samara is too.\n\n'That's Mum and Dad. I better go...' says Samara.\n\nLily nods as Samara collects her book and turns to face her parents. The girl pulls her beanie on and gives her head a quick scratch as she walks away.\n\n'Hey, Samara!' Lily calls. Samara turns back. 'Your beanie is totally the amazing-est beanie I have ever seen, but if it makes you feel itchy, just don't wear it. You look different, but\n\nSamara smiles a huge smile.\n\n'Hey, while we are in here together, we should hang out,' suggests Lily. 'I could get my mum to bring in my special make-up from drama class, and we could tattoo your head!'\n\nSamara laughs again, big and loud and tinkly. 'That would be cool... and Lily, needles are nothing. If you have to have one, try saying the alphabet backwards, and I guarantee you won't even notice it being done. Even better, imagine the doctor with underpants on their head!'\n\nLily giggles. 'Thanks, Samara! Good advice! See you around.'\n\n**A** lone again, that squeezed-sponge feeling returns and grips Lily's heart.\n\nSuddenly, although she isn't sure why, Lily begins to cry.\n\nMum returns from the admissions desk to find Lily shaking, with tears streaming down her cheeks.\n\n'Sweetie, what's wrong?' she asks.\n\nLily shakes her head. She can't talk.\n\nMum pulls Lily into her arms and makes the gentle _shh shh shh shh sssshhhhhh_ sound she's used on Lily and Sonny since they were tiny babies.\n\nSoon Lily calms down, and stops shaking.\n\nMum tilts Lily's face up and looks into her eyes. 'I'm so proud of you, my wondrous, exceptional daughter. **The amazing-est girl I have ever met!'**\n\n'Mum!' gasps Lily. 'You just said _amazing-est_!'\n\nMum chuckles. 'I know. Don't tell your father! I feel like I need to wash my mouth out with soap!' Mum cuddles Lily closer.\n\nIt feels nice just to be near each other.\n\n'Hey, we have a bit of a wait, still,' Mum says after a little while. 'Why don't you perform the piece you've been learning from _Into the Woods_ for me?'\n\n'Okay.' Lily stands up, feeling a bit nervous. She's learned lots of bits from _Into the Woods_ so she could be ready to play any character, but right now, there's one that seems perfect. So Lily D closes her eyes, and very quietly starts singing the chorus song introducing the woods themselves.\n\nLily loves the part of the song that says even when the woods are dark and full of scary things \u2013 like wolves and witches \u2013 you just have to keep going through the scary parts. You have to face your fears!\n\n**'That was beautiful!'** says a strange voice, as a pair of hands start clapping. Lily's eyes fly open to find a nurse in a pale blue uniform, smiling so wide it looks like she has stuck a banana in her mouth sideways!\n\n'That's from _Into the Woods_ , isn't it?' says the nurse in a happy, giggly, squeaky voice. 'I love that musical. I think it's because **I LOVE fairytales!'**\n\n'Yeah!' Lily grins. 'I love them too!'\n\n'Now,' says the smiley nurse, 'you're Lily D, aren't you?'\n\nLily nods.\n\n**'Ready to head down to your room?'** asks the nurse.\n\nLily starts to follow her, but notices Mum is still sitting, frozen! Mum looks totally freaked out!\n\nLily walks back to the chairs and puts her hand on her mum's. 'Mum, it's cool. You don't have to worry, I've got this!' she whispers. 'I am totally in the zone. I'm more than method. I am Little Princess Snow Bean right now. I could even be Snow White! I'm not scared at all. Well... not much anyway.'\n\nMum looks confused. 'So, now you want to be a princess?'\n\n'No! What I mean is, every time I get tonsillitis it stops me from performing, so it stops me from doing what I love.' Lily smiles her bravest smile, and gives her beads a dramatic shake. Joyful bead-applause rings through the hospital corridor.\n\n'So these doctors and nurses, even the yucky... um... the yucky needles, I guess they are kind of like my fairy godmother, or my magic tree, or my poisoned apple, or beanstalk, or _whatever_.'\n\nLily scrunches her nose, trying to find the right words to explain that after a week of going 'fairytale method', and seeing how people she cares about and respects use fairytales to fight their own fears, she is ready to face hers. Lily is **DETERMINED** to show Mum that like all the brave characters she has studied this week, she is no longer scared.\n\n'I am going to wake up after this operation to see everyone I love... like Snow White! **But I don't need a prince to save me.** I am going to find my own happily ever after. Because I will be me, a healthy, happy, tonsil-free Lily D.'\n\nLily grabs Mum's hand, and together they follow the nurse past the admissions desk and into the hospital.\n\nLily is determined to walk out of the hospital a better V.A.P. than ever. Not a _Very Attractive Princess_ \u2013 **NO WAY!** Lily will be a **Very Amazing Performer**.\n\nAnd after hospital, Lily has a whole week of ice-cream to look forward to! Now _that_ is a happy ending! **FACT!**\n\nOnce upon a time, there was a little girl named Madeleine. Madeleine **LOVED** to tell stories because she believed **EVERYONE** has a story and that story deserves to be told.\n\nThat little girl grew up to become a **Very Ambitious Performer** on stage, film and TV, and now, she is **Virtually an Author Person**. **FACT!**\n\nMadeleine also has six kids, and she believes it is very important for children to experience what it is like to walk in someone else's shoes. Why? Because the greatest gift we can give is treating others with respect and kindness, no matter who they are, what they do or where they come from.\n\nSo Madeleine has created Lily D, V.A.P., a character actor with a difference, and hopes that you have enjoyed reading about her in the book.\n\nMadeleine can also do backflips with just the occasional injury, and dreams of one day owning a chocolate-coated castle made entirely of peanut butter... or Nutella even. Madeleine isn't fussy.\n\n**'OMG!** Wow! You are **TOTALLY** not going to believe this! _Guyzzzzz_ , you are seriously going to _flip_!'\n\n'Lily, it's \"guyS\" with an _s_ , not a _z_ ,' says Mum, who is working on her laptop. Mum is a writer and she knows everything about words. She can even tell if you're spelling them wrong when you _say_ them!\n\n'Hmm? What's going on, Lily-Pilly?' Dad asks. He's chopping vegetables so fast his hands look all blurry.\n\nSonny, Lily's little brother, just keeps stacking his building blocks. His real name is Hudson, but Lily only calls him that when she's cross. Sonny is six, and can be seriously annoying. _But_ , when he pretends to be a zombie it's so scary it makes your skin tingle in the spookily-cool way. So Lily thinks he is okay ... sometimes.\n\n**'You guySSSSSS!'** Lily exclaims.\n\n'Why are we going to flip, Lil?' asks Mum.\n\n' _Well!_ Next week in drama class we are going to perform scenes from _Annie_! It's a _really famous_ musical about this orphan girl called Annie. She is all alone in the world. She lives in a horrible orphanage but then finds a new home, and it's really sad, then really happy.' Lily takes a breath. 'It's _totally_ cool!'\n\nLily shakes her arms to make her beads rattle. Lily D wears heaps of coloured bead bracelets all up one arm and down the other.\n\nSonny rolls his eyes. 'Mum! Lily's applauding herself again!'\n\nIt's kind of true \u2013 when Lily's bracelets rattle, it sounds just like an audience clapping. **INSTANT APPLAUSE!** Just what every performer extraordinaire needs!\n\n'And will you be playing Annie?' asks Dad, changing the subject.\n\n**'You got it, Daddy Warbucks!** That's Annie's new dad's name, by the way. Well, when she gets a new home, that is.'\n\n'That's marvellous, Lil. We are very proud.' Mum actually stops typing on her keyboard!\n\n'So all this week I have to practise being Orphan Annie,' Lily explains. 'I'm going method. That means I have no mum, and no dad and no stinky little brother.'\n\n**'LILY!'** Mum says in her _stern_ voice.\n\n'Sorry, Sonny,' says Lily. 'Anyway, this week I have to do everything for myself. I have to get my own food, clothes, _everything_. Those are the rules... the ORPHAN RULES! **I'M TOTALLY ORPHAN ANNIE** and you can't do anything for me. Okay, Mum and Dad? I mean \u2013 _Clara and Sam_?'\n\n'Mr and Mrs D will do,' Dad says. He tries to do a _stern_ voice too, but he isn't as good at it as Mum. He needs to go to drama class with Lily D to learn how!\n\n'I guess you won't be having any of the chocolate cake I made for dessert then,' Dad says.\n\nLily's dad is a chef, and he makes the _best_ chocolate cake. He makes the best _everything_.\n\n**GGGGGGGRRRRRR!** Lily's tummy grumbles.\n\n**To you ultra-cool captains of chaos, my super-dooper 7. Every day you show me the true meaning of bravery: laugh loud, dream big and never let fear stop you from being the best you can be.**\n\n**\\- M.W.**\n\n_Lily D, V.A.P: Little Princess Snow-Bean_\n\nfirst published in 2018 by\n\nHardie Grant Egmont\n\nGround Floor, Building 1, 658 Church Street\n\nRichmond, Victoria 3121, Australia\n\nwww.hardiegrantegmont.com\n\neISBN 9781743585955\n\nText copyright \u00a9 Madeleine West, 2018\n\nArtwork copyright \u00a9 Joanie Stone, 2018\n\nDesign by Kristy Lund-White\n\nAll rights reserved\n\nCataloguing-in-Publication data is available from the National Library of Australia\n\nWe welcome feedback from our readers. All our ebooks are edited and proofread vigorously, but we know that mistakes sometimes get through. If you spot any errors, please email info@hardiegrantegmont.com.au so that we can fix them for your fellow ebook readers.\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers and copyright holders.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nText copyright \u00a9 2009 by Gena K. Gorrell\n\nPublished in Canada by Tundra Books,  \n75 Sherbourne Street, Toronto, Ontario M5A 2P9\n\nPublished in the United States by Tundra Books of Northern New York,  \nP.O. Box 1030, Plattsburgh, New York 12901\n\nLibrary of Congress Control Number: 2009928409\n\nAll rights reserved. The use of any part of this publication reproduced, transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, or stored in a retrieval system, without the prior written consent of the publisher - or, in case of photocopying or other reprographic copying, a licence from the Canadian Copyright Licensing Agency - is an infringement of the copyright law.\n\n**Library and Archives Canada Cataloguing in Publication**\n\nGorrell, Gena K. (Gena Kinton), 1946-  \nSay what? : the weird and mysterious journey of the English language \/ Gena K. Gorrell.\n\neISBN: 978-1-77049-079-6\n\n1. English language\u2013History\u2013Juvenile literature. I. Title.\n\nPE1075.G675 2009 j420.9 C2009-902917-0\n\nWe acknowledge the financial support of the Government of Canada through the Book Publishing Industry Development Program (BPIDP) and that of the Government of Ontario through the Ontario Media Development Corporation's Ontario Book Initiative. We further acknowledge the support of the Canada Council for the Arts and the Ontario Arts Council for our publishing program.\n\nv3.1\nThis book is dedicated to my father,  \nJack Muir, who was always my best editor,  \nand who taught me from the beginning  \nthat \"Well, you know what I mean\" is  \njust  \nnot  \ngood  \nenough.\nLanguage is the memory of the human race.\n\n\u2014 WILLIAM HENRY SMITH\n\n# Contents\n\n Introduction: The Puzzle of English\n\n1 The Mother Tongue\n\n2 The Glory That Was Greece\n\n3 Echoes of Ancient Greek\n\n4 Britannia, Provincia Romana\n\n5 Amicus, Amice, Amicum\n\n6 Angles and Saxons and Vikings\n\n7 Alfred and the Vikings\n\n8 Riddles, Hymns, and Tales of Battle\n\n9 The Defeat of the English\n\n10 Tales of Cnihts and Fair Ladies\n\n11 How to Spell It? How to Say It?\n\n12 Making the \"Good Book\" Better\n\n13 Gloriana and the Bard\n\n14 Greener Pastures\n\n15 The War between English & English\n\n16 Britannia's Heyday\n\n17 Words, Words, Words\n\n18 Looking Forward\n\n Timeline\n\n Appendix\n\n More about Sanskrit\n\n More about Greek\n\n More about Latin\n\n More about Old English\n\n Source Notes\n\n Selected Bibliography\n\n Picture Sources\n\n Acknowledgments\n\n# Introduction\n\n# THE PUZZLE OF ENGLISH\n\nWhere on earth did the English language come from? Over a million words, and such weird spellings! Aren't there any rules? Some words are spelled the same but said differently (\"The **wind** made the flag **wind** around the pole\"). Others are spelled differently but pronounced the same (\"Did you bring **enough stuff**?\") How did English get so complicated?\n\nThe answer is that English isn't just the speech of one nation; it's the memory of thousands of years of history. It tracks the places people came from and the places they went to; the adventures they had; the friends and enemies they made; the battles they won and lost. It marks the days when legions of Roman soldiers stomped onto England's shore \u2013 the days when Viking warriors sailed their dragon boats across the sea to plunder British villages \u2013 the day a king of England was felled by an arrow on the battlefield and a foreigner seized his throne. As English changed and grew, it became a jumble of sounds and words and rules from countless languages and lands. And it's still changing, still growing, every day.\n\nIf you want an example, look at Tolkien's novel _The Hobbit_. When Bilbo Baggins first meets the wizard Gandalf, the hobbit exclaims:\n\nGandalf, Gandalf! Good gracious me!... Not the fellow who used to tell such wonderful tales at parties, about dragons and goblins and giants and the rescue of princesses...?\n\nThat looks like plain English, doesn't it? But **gracious** and **princess** both come from Latin \u2013 _gratia_ , \"kindness,\" and _princeps_ , \"leader.\" **Dragon** and **giant** are from Greek _drakon_ and _gigant_. **Fellow** and **wonderful** come from the Norse spoken by those ancient Viking warriors. As for **Gandalf** , it's Norse for \"wand-elf\"!\n\nEnglish has words from languages so old that we barely know they existed. It has words from prehistoric tribes, and from today's languages \u2013 from Hindi, Arabic, and Chinese; from the Choctaws in North America, the Guaran\u00eds in South America, and the Aboriginals in Australia. More than seventy percent of all English words were born someplace other than England. Spin a globe and put your finger down on almost any patch of land; English has words that come from there.\n\nThat's why the language can be so confusing and inconsistent. That's why, for every rule about English, there's a fistful of exceptions. And that's why this is the richest, most international, most versatile language in the world.\n\n#\n\nWhere did language start in the first place? It must have been long before there were humans. Imagine primitive ape-creatures huddled together in a cave; they'd have to communicate somehow. They'd want some way to say, _Give that back, it's mine!_ or _Move over, you're standing on my foot!_ And the more they lived together, the more they'd need to say. _Where'd you find those blueberries? Chase that rabbit over here and I'll grab it_. As they evolved into humans, and learned farming and other skills, their language would become more complicated: _I'll trade you half my honey next week for one of those catfish today_.\n\nWhile early humans were developing languages, they were also migrating from place to place. Sometimes they were driven by changes in the weather, or by natural disasters like floods and forest fires. Sometimes they were moving away from hostile neighbors, or following the animals they hunted. For many reasons, groups of people were leaving one home territory and looking for another.\n\nBy comparing certain languages and tracing their roots, finding similar words that seem to come from the same ancient, mysterious source, scholars have concluded that in the far distant past \u2013 maybe six or eight thousand years ago \u2013 some people were speaking a language that has now been lost. They had no writing (that we've found) so we have no records, and we know very little about who they were, or where and how they lived. They seem to have had words for **snow** and **wolf** but not for **ocean** , so their homeland was probably cold and far from the sea \u2013 likely somewhere in Central Europe. But that's about all we can say. That, and one other thing: for some reason, small bands of these people migrated in different directions. They carried their language across Europe and Asia and even up into Scandinavia.\n\nOnce their communities became separated, their ways of speaking slowly began to differ, creating a whole family of languages that we describe as Indo-European. One version, Celtic Indo-European, is the source of modern Welsh, and of the Gaelic spoken in Ireland and Scotland. Farther south, Greek Indo-European developed into ancient and then modern Greek. The Indo-Iranian form is the foundation of most languages of India, Pakistan, and Iran. Italic Indo-European gave rise to Latin, which gave rise to Italian, French, and Spanish. The Slavic version is the basis of Russian, Bulgarian, and Polish. Germanic Indo-European is the root of Dutch, most Scandinavian languages, German and Yiddish \u2013 and English. And these are only a few examples. Altogether, the roots of the languages spoken by about half the people in today's world reach back to the Indo-European tongue of those unknown people so long ago.\n\nHow much Sanskrit is hiding in your kitchen? The ancient Hindus cooked with _pippali_ (berry), _sarkara_ (grit), and _crngaveram_ (powdered horn-body, an antler-shaped root). The Greeks called these seasonings _peperi_ , _sakkharon_ , and _ziggiberis_. We know them as pepper, sugar, and ginger. (Do you see where the artificial sweetener saccharin gets its name?)\n\nOne of the earliest Indo-European languages that we have in written form is Sanskrit. In India, Hindu scriptures called the Veda were written in Sanskrit about 3,500 years ago. ( _Veda_ is Sanskrit for \"knowledge.\") Sanskrit was the language of India's court and literature 2,500 years ago, and although it's not used in daily speech any more, it's still part of Hindu religious ceremonies.\n\nAlthough modern Indo-European languages have been developing separately for thousands of years, and now seem very different from one another, it's intriguing to track down the similarities. For example, the Sanskrit words for **one** , **two** , **three** , are pronounced _eka_ , _dvi_ , _tri_. Compare those ancient words to these:\n\nAnd look at the \"family tree\" for this word:\n\nSanskrit and its descendants \u2013 such as Latin, Ancient Greek, and early French and Germanic languages \u2013 gave English more than just vocabulary. They also left fragments of their grammar, ghosts that lurk behind our sentences like scraps of faded photos in a family album. These odds and ends can be puzzling, even frustrating, but they deserve respect. They come from the dawn of the English language, and long before.\n\nOne of the greatest Hindu gods, Shiva ( _SHEE-vah_ ), appears in Sanskrit texts over two thousand years old. Shiva can take various forms, some kind, some cruel and dreadful. Here he appears as the four-armed Lord of the Dance, surrounded by a fiery halo representing the cycle of creation, destruction, and rebirth, and trampling a demon to symbolize the end of evil.\n\n#\n\nGreece is a smallish country, about the size of England \u2013 and like England, it was once the heart of an empire. At its height, starting around 500 BCE (Before the Christian Era), Greece boasted magnificent architecture, painting, and sculpture. Its poets and playwrights were renowned across Europe. Education was greatly prized, and orators (public speakers) who were logical and eloquent were celebrated as heroes.\n\nAs the Greeks sailed their ships along the shores of the Mediterranean Sea, setting up colonies to trade their wine, olive oil, and pottery for products such as wheat and metals, their language became known as far away as Spain and North Africa.\n\nIn the mid-300s BCE, the brilliant military campaigns of the young king Alexander the Great spread Greek civilization and culture to even more distant lands \u2013 down the Nile River through Egypt, and east into India.\n\n(photo credits 2.1)\n\nThe Greek language was used in government as well as in fashionable, artistic circles. It was also handy as a lingua franca \u2013 a common language understood by traders and travelers from many different places.\n\nOne reason for the success of the language was the Greek alphabet. Earlier cultures had found ways to write words, but their systems were generally very complicated and required a lot of fancy symbols, usually based on pictures. All those symbols were hard to learn, and they weren't easy to copy out, either.\n\nThis pillar marks the tomb of a high-ranking priest-scribe at an Egyptian pharaoh's court more than four thousand years ago. The Ancient Egyptians had about seven hundred hieroglyphic signs, each representing a consonant or a mix of consonants. (They didn't write vowel sounds.) The symbols inside the cartouche (oval line, upper left and middle right) represent a flowering reed, a feather, a hare, and water, and spell out the pharaoh's name. Hieroglyphs were elegant and decorative, but only a few people \u2013 mostly priest-scribes \u2014 could read them. (photo credits 2.2)\n\nBut another ancient civilization \u2013 the Phoenicians, who lived about where Lebanon is now \u2013 had had the brilliant idea of using a handful of simple signs to represent consonants. (The Phoenicians and the Egyptians didn't bother writing vowels.) The Greeks had adapted this system sometime around the 800s BCE, adding a few vowels and other sounds and ending up with about twenty-four letters. And there it was: a quick, efficient way to write a letter or a shopping list or a poem, and so easy that a schoolchild could learn it. And this set of letters began: _a_ , _b_ \u2013 alpha, beta.\n\nWhat word can you make out of A-B-C-D? Abecedarian, meaning something related to the alphabet, or someone learning it. \"Isn't little Sally clever? She's only three, and already she's an abecedarian.\"\n\nThe Greeks were seen as highly cultured, and experts in life's little pleasures \u2014 good food, fine wine, music, and romance. They decorated pottery bowls and vases with scenes from daily life, historical events, and the endless shenanigans of their gods. (photo credits 2.3)\n\nAfter Alexander died suddenly when he was only thirty-three, his empire began to collapse. In time, the remnants of Greece were swallowed up by the Roman Empire. But the Romans themselves had been nourished by the culture of Greece. The Roman alphabet was derived from the Greek alphabet. The Roman language, Latin, had been influenced by Greek. Roman literature was patterned after the works of Greece. Educated Romans were expected to be fluent in Greek; for less privileged people, Greek plays, poems, and speeches were translated into Latin. Children of wealthy Roman families studied under Greek tutors, and often completed their education by traveling in Greece.\n\n\"The art of speaking enchants the soul,\" declared Plato (427?-347? BCE), a writer and philosopher, and one of the most famous Greeks. He also pointed out that \"An orator's virtue is to speak the truth.\" (photo credits 2.4)\n\nThe Greek gods lived on Mount Olympus and led soap-opera lives of passion, deceit, jealousy, and revenge. Nemesis was a goddess who punished those who were evil, arrogant, or vain. Today, a nemesis is a supreme enemy or destroyer. (Harry Potter is Lord Voldemort's nemesis.)\n\nIn Greek mythology, when Halcyone found her drowned husband's body on the shore, she was so heartbroken that she drowned herself. Halcyone and her husband were turned into kingfishers, birds that were thought to nest on a calm sea. That's why calm and peaceful times are called halcyon ( _HAL-see-on_ ) days.\n\nIn the centuries that followed, the writings of the ancient Greeks \u2013 myths, fables, plays, and philosophy \u2013 shaped European society. Well into the 1800s, people could hardly claim to be educated unless they read at least a little Greek.\n\n#\n\nThe Greek alphabet often shows up in our maths and sciences: **beta** blockers are a class of medications, **gamma** rays are a kind of electromagnetic radiation, and **pi** (\u03c0, the Greek letter _p_ ) is a ratio used in math involving circles. The leaders of an animal pack are the **alpha** female and **alpha** male. A triangular airplane wing, or the triangle of a spreading river-mouth, is a **delta** , after the capital Greek _D_ , \u2206. (You can find the whole Greek alphabet in \"More about Greek,\" at the end of this book.)\n\n_Iota_ was the name of the Greek _i_ , the smallest letter in their alphabet. In English an **iota** is the smallest possible amount (\"She hasn't got an iota of talent!\") Our word **jot** comes from _iota_ , and means the same thing. And when we \"jot down a note,\" we are writing as little as possible.\n\nWe borrow Greek words to create all kinds of new meanings. The Greek for **circle** , _kuklos_ , shows up in **bicycle** , **cyclone** (a circular storm), and **recycling** (using over and over). From _naus_ , **ship** , we get **nautical** (related to ships and sailing), navigable (passable by ships \u2013 \"Is that river navigable?\"), and astronaut (\"star-sailor\"). From _hudor_ (water) come **dehydrated** (deprived of water), **hydraulic** (powered by water), and **hydroponics** (growing plants in water without soil). Knowing a little Greek can help you figure out a lot of English words, even if you've never seen them before.\n\nThe Greeks had two versions of the letter o. One was _omicron_ (o), or o-little (as in \"microscope,\" for seeing little things). The other was _omega_ (\u03a9), or o-big (as in \"megacity,\" big city). Since omega was the last letter of the Greek alphabet, \"My dog knows every trick from alpha to omega\" means Rover knows all the tricks from A to Z - in other words, absolutely everything.\n\nIn Ancient Greece, male athletes practiced their exercises naked _(gumnos)_. We still call a room for athletics a gymnasium.\n\nThe Greek alphabet may have been simple, but the language certainly wasn't. It was highly \"inflected\" \u2013 that is, words changed depending on how they were used. A noun could have many forms, because if you wrote \"This crocodile stole the fish from that crocodile and gave it to another crocodile,\" the ending on \"crocodile\" was different each time, depending on what role that particular crocodile played in the action. If you said \"These crocodiles\" you needed a whole different set of endings, for the plural. And if you talked about one lion ( _leon_ ) or two lions instead of one or two crocodiles, you needed two more sets of endings, because words ending in _os_ (like _krokodilos_ ) didn't have the same endings as words ending in _on_ (like _leon_ ) \u2013 in the singular, or in the plural!\n\nIn Ancient Greek, _kroke_ means \"stones\" and _drilos_ means \"worm\" \u2014 so a worm-shaped animal that liked to sunbathe on stony shores was called a _krokodilos_. (photo credits 3.1)\n\nAnd that's only the beginning. All nouns also had gender. **Grasshopper** was masculine but **flea** was feminine. **Soup** was masculine but **dessert** was neuter (neither masculine nor feminine). So if you described some animal as large ( _megas_ ), the ending on _megas_ depended on how many animals you meant, and whether they were masculine or feminine or neuter, and what role they played in the action!\n\nAnd verbs were no easier. We say **come** and **coming** and **came** , and we add a few auxiliary verbs (such as **was** , **will** , or **had** ). A Greek verb could have as many as _two hundred_ different forms, depending on the tense (past, present, future, etc.), the voice (active or passive), how many people were involved, and so on.\n\nHow did schoolkids ever learn all those endings? _Memorize, memorize, memorize!_\n\nEuphemisms (from Greek _eu pheme_ , \"fine-speaking\") are polite ways of not saying what we really mean \u2014 and we work them overtime when it comes to certain bodily functions. Our word \"toilet\" is from French _toilette_ , dressing room. The Romans went to the _necessarium_ ; we go to the washroom, but not just to wash; to the bathroom, but not to bathe; to the powder room, but not to powder anything! We go to the ladies' and gents', the little room, the bog (wet like a swamp), the privy (private place), the john. In England we pop out to the WC (water closet), lavatory (from French _laver_ , to wash,) or loo (perhaps from French _l'eau_ , \"water,\" or _lieu_ , \"place\"). And if we need to \"spend a penny,\" we're recalling the days when many public toilets in England were behind a one-penny turnstile.\n\n#\n\nWe don't know when humans first arrived in England, or how they lived. We do know that by about 2,500 years ago, they were not only using stone and copper tools; they also were mixing copper and tin to make bronze, for tools and weapons that were stronger and lasted longer. But over in Europe, some tribes called Celts were even more advanced. They had learned to make tools and weapons out of iron, which was stronger than bronze. They had mastered the art of riding horses, and harnessing them to chariots.\n\nSometime around thirty-five hundred years ago, people in Britain built Stonehenge and other monuments out of giant, roughly carved stones. We're still not sure what these constructions were used for, and we don't know what language people spoke as they worked together on these huge projects. (photo credits 4.1)\n\nBy 700 BCE, small bands of Celts were sailing from Europe to England as traders or settlers. Larger and larger groups followed. The native Britons, the Stonehenge-builders, tried to resist the invaders, but they were defeated again and again.\n\nMore Celts flooded in, building great earthen castles on hilltops, and fortifying them with high walls and deep ditches. They were ruled by priests and by fierce warriors who fought on horseback or from chariots, cloaked in bronze armor and brandishing iron weapons. Celtic artisans crafted ornate jewelry, wove fine fabrics that they colored with vegetable dyes, and made fragrant soaps and lotions and perfumes. Fleets of Celtic merchant ships carried goods to distant kingdoms, to be traded for exotic novelties.\n\nCeltic weapons and armor were often decorated with superbly crafted metal and enamel, and precious stones like coral and amber. This bronze shield, with its pattern of elegant curlicues, is studded with disks of red glass. (photo credits 4.2)\n\nThe Celts' language, religion, and folklore spread across much of Britain with them. Celtic was not a written language, but Celtic bards (storytellers) knew endless ballads of adventure and romance to enthrall their listeners. For hundreds of years Britain was dominated by the Celts, yet only a few English words come from Celtic \u2013 including **_Britain_** , from _Prydein_.\n\nBut while the Britons and Celts were learning to live together, sometimes fighting, sometimes intermarrying, a new empire was creeping toward their shores. The citystate of Rome had overgrown its boundaries and spilled out into the rest of Italy. Rome's armies were on the march, fighting their way across one land after another, building an empire. The well-armed Roman legions became renowned for their discipline and organization. Building a great navy, Rome spread its rule into France, Spain, and Africa. In 146 BCE, Rome conquered Greece. A hundred years later, Roman territories circled the entire coast of the Mediterranean. And wherever the Romans went, they took Latin as the language of law and government, and Greek as the language of the arts.\n\nIn 55 and 54 BCE, Julius Caesar led the first two Roman invasions into Britain. Some ninety years later \u2013 in 43 AD \u2013 forty thousand Roman troops braved the sea crossing for a massive attack. By 67 AD, four legions (about twenty thousand soldiers) were posted in Rome's new province of Britannia.\n\nCaius Julius Caesar ( _1_ 00?-44 BCE) was not only a Roman general but a brilliant statesman, writer, and orator. He was also a military genius, and he expanded the empire so spectacularly that later Roman emperors also called themselves Caesar. Various forms of the name now mean \"emperor\" in other countries: Kaiser in Germany, and czar (or tsar) in Russia. (photo credits 4.3)\n\n_Castra_ is Latin for army camp. Today, the names of many English towns \u2013 Lancaster, Cirencester, Winchester \u2013 mark them as places where Roman soldiers lived almost two thousand years ago.\n\nIn time, Rome took over much of Britain, well up into what is now Scotland. Roman troops were kept busy suppressing rebellions from within the province, repelling cross-border raids by Celtic tribes on the north and west, and fending off raids by seafaring Germanic tribes along the south and east.\n\nBut the Romans were a practical, businesslike people. Once they more or less controlled Britannia, they focused on matters of trade and government (especially tax-collecting!) The result was the Pax Romana, or Roman Peace (27 BCE to 180 AD), a fairly calm and prosperous interval after so many years of warfare.\n\nRoman soldiers were tough, highly trained professionals who used expert tactics against their many enemies. By locking their shields into this tanklike _testudo_ (\"tortoise\") formation, for example, they could stay protected while attacking an enemy position. (photo credits 4.4)\n\nThe Romans built fortified towns as refuges for times of rebellion, warehouses to protect trade goods, safe harbors for their ships. They laid out long, straight, solidly built roads, with stone bridges and distance-markers, so their armies and supplies could travel swiftly. Then they turned their attention to the details of everyday life: aqueducts to bring water to towns and garrisons, and sewers to take it away; temples and statues to please the many Roman gods; theaters and amphitheaters for entertainment; public toilets and bathhouses; lavish villas for the wealthy. Towns grew into cities, and little villages blossomed into towns.\n\n\"Great Caesar's ghost!\" exclaimed Superman's boss at the _Daily Planet_ newspaper, when he was excited. Most of us use exclamations that don't mean anything except that we're astonished or angry or we've stepped on a wasp. Each culture has its own exclamations, many of them not too polite. When religion dominates daily life, expressions like damn or hell are forbidden (people get around that with inventions like dang or darn or heck). At other times, quite innocent body parts (legs, for example) have been seen as offensive. Today, various body parts and functions are considered unmentionable \u2013 so we won't mention them!\n\nThe Romans did their best to spread Latin and the Roman culture in the province. Roman-style schools were set up, to teach elite British boys. It was difficult to succeed in business, at least in the towns and cities, without speaking passable Latin. Britons enlisted in the Roman legions, to serve at home and abroad; aging Roman soldiers acquired land in Britannia, and built themselves retirement estates. One way and another, Latin became part of the Britons' life, especially around Londinium (London). But in the countryside, and farther north and west, Celtic remained the language of the land.\n\nThe Romans had countless gods and goddesses, and they are with us still. Cereals are named for Ceres, goddess of grain harvests. Flora and fauna, the plants and animals of a particular area, are named for Flora, goddess of flowers, and Fauna, sister of Faunus, a god of woods and forests. And vulcanizing - treating something (like a rubber tire) at high temperatures to make it hard and durable - is named for Vulcan, god of fire. His name also gives us volcanoes, and volcanology or vulcanology, the science of volcanoes.\n\nMeanwhile, things were happening back in Rome. All those colorful, quarrelsome gods were being challenged by the single god of a new religion, Christianity. Rome itself was being attacked by primitive tribes from the east. Weakened by corruption, bad government, and a host of other problems, the Roman Empire split into two sections, and its power declined. The troops occupying Britannia were withdrawn and sent to fight elsewhere. In 410 AD, Rome officially gave up its occupation of Britannia. By the late 400s, the days of the Ancient Roman Empire were over.\n\n#\n\nOn the other side of the English Channel, in the Roman provinces of Gaul and Iberia (France and Spain), Latin had gradually replaced most Celtic languages. (Italian, French, and Spanish are known as Romance languages because they developed from the language of the Romans.) But although Latin was Britain's language of law and government for some four hundred years, it never caught on among the common people. They continued to speak Celtic, with a sprinkling of handy Latin words.\n\nWhat was Latin like? The Romans had adopted the Greek alphabet, dropping some letters, changing others, and rearranging the sequence. They ended up with pretty much the alphabet we use today.\n\nLike Greek, Latin was highly inflected. Nouns, and the adjectives that described them, changed their endings depending on their number and gender and case (the role played by the word). We often use a preposition to indicate the role: \" _to_ a friend, _from_ a friend, _about_ a friend;\" the Romans could just change the ending, like this:\n\n**nominative** ( _subject of verb_ ) My friend ( _amicus_ ) came.\n\n**vocative** ( _one spoken to_ ) \"Hey, friend ( _amice_ )!\"\n\n**accusative** ( _direct object_ ) I gave my friend to the frog ( _amicum_ ).\n\n**genitive** ( _possessive_ ) The father of my friend ( _amici_ ) came.\n\n**dative** ( _indirect object_ ) I gave the frog to my friend ( _amico_ ).\n\n**ablative** ( _other indirect connection_ )\n\nMy frog is from my friend ( _amico_ ).\n\nSalt \u2013 mined from the ground or recovered from sea-water \u2013 used to be a precious commodity; that's why we call good people the salt of the earth. Salt seasoned many dishes people ate, and kept food from spoiling during the long, hungry months of winter. Domestic animals like cows and goats couldn't survive without it. The soldiers of the Roman armies were sometimes paid in salt ( _sal_ in Latin), giving us our word salary, and we still say that people who earn their pay are worth their salt. Greens with salt and other seasonings make a salad, and preserved meats are salami. All of this from salt!\n\nAs in Greek, if the friend was a girl, or there were several boys, or several girls, or the word's nominative (subject) case didn't end in _us_ (like _amicus_ ), the endings were different.\n\nCompare this to English. Do we change endings for number, gender, and case? We usually change plurals, one way or another ( **frog** , **frogs** ; **mouse** , **mice** ). We occasionally change endings to show gender ( **actor** , **actress** ; **waiter** , **waitress** ), although this is going out of fashion. And we don't change adjectives at all \u2013 whether we're talking about one _rude_ girl or ten _rude_ boys!\n\nWhen you study Latin, you learn how to \"decline\" the nouns \u2013 how to change them according to their use. Here's how you decline the word for \"pig.\"\n\nNow, here's how we decline the word in English: pig pig pig pig's pig pig; pigs pigs pigs pigs' pigs pigs.\n\nIt's no wonder you don't find a lot of people speaking Latin these days!\n\nWe don't use all those case endings for nouns, but we do say, \"My _friend's_ glove,\" or the plural, \"My _friends' house_.\" The apostrophe tells us that \"friend\" is in the possessive case. In fact, we can use three systems to show the role of a noun: that _s_ , if it's possessive; a preposition (like **to** or **from** ); or simply the order of the words. We can say, \"I showed Jason's bike _to_ Ali,\" or \"I showed Ali Jason's bike,\" putting Ali first. (Because \"I showed the cat the bird\" is not the same as \"I showed the bird the cat\"!)\n\nPeople sometimes confuse the possessive s with the plural s. This mistake so often appears on shop signs that it's known as the greengrocer's apostrophe. Another common mistake is to put an apostrophe in the possessive _its_. Remember: the possessive his, hers, and its have _no_ apostrophe. It's is short for it is. (photo credits 5.1)\n\nWe still see Latin as a language of scholarship. Those spells that Harry Potter struggles to learn at Hogwarts are Latin, or at least mock-Latin \u2013 _Ferula! Lumos! Expelliarmus!_ When Harry forgets his Latin cases and cries, _\"Expecto patron o\"_ instead of \" _Expecto patron um_,\" his patronus doesn't appear. (In Latin, a _patronus_ is someone who protects people \u2013 like Harry's patronus, a dazzling silver stag that appears out of nowhere to drive away his enemies.)\n\nEven today, some Latin words appear, unchanged, in English. _Campus_ means \"flat field.\" _Victor_ means \"winner.\" _Veto_ (\"I forbid\") now means a rejection: \"Your parents will **veto** your plan to go sky-diving.\" _Recipe_ meant **take** ; now, as well as telling you what ingredients to use and how to cook them, a **recipe** is a prescription for medicine (the \"Rx\" you see in drugstores is a short form for **recipe** ). _Alibi_ (\"in another place\"), _exit_ (\"he, she, or it leaves\"), _ignoramus_ (\"we don't know\") \u2013 see how much Latin you already know?\n\n#\n\nAfter the Romans abandoned Britannia, life there changed for the worse. Without the Roman merchant fleet, international trade collapsed, and workshops ground to a halt. Without constant maintenance by the Roman army, roads and aqueducts fell into disrepair. With fewer jobs available, and not much law enforcement, there was more crime. Towns dwindled as people moved back to the countryside, to make a living any way they could.\n\nThe most serious threat, though, was from overseas. The Romans had fought vigilantly against pirates, raiders, and invaders from all sides. They had manned a chain of forts and watchtowers along the borders, and kept a fleet of warships nearby. Now, Britain's coast lay undefended.\n\nFewer jobs? Less jobs? Which is right? Fewer refers to things counted in whole numbers; less is for things you might have just part of. You wash _fewer_ than three teacups but you drink _less_ than three cups of tea (maybe just two and a half). You go trick-or-treating at fewer than twenty houses (it must be raining!) and bring home less than two bags of loot (the second bag isn't full).\n\nBy 440 CE, immigrants from Germanic tribes were beginning to arrive in Britain from northern Europe. Although they were expert seafarers and skilled farmers, they were not as advanced as the Britons had become under the Romans. And they were pagan, worshipping a host of Scandinavian gods of war and nature. In the last years of the Roman occupation, Rome had adopted Christianity, and the new faith had swept through Britain as well.\n\nAt first, the Britons and the new arrivals got along fairly peacefully, despite their differences. But then more and more immigrants came flooding in \u2013 Angles, Saxons, Jutes, and others \u2013 and began spreading through the southeast. The Britons fought many desperate battles, trying to drive the invaders out, but they failed.\n\n(photo credits 6.1)\n\nAlthough the borders of Anglo-Saxon Britain were constantly changing, we generally divide it into seven kingdoms: Essex, Wessex, Sussex, Kent, Mercia, Northumbria, and East Anglia. (photo credits 6.2)\n\nBy the mid-600s, there were Germanic communities through much of southern and eastern Britain. These grew into little kingdoms that were constantly fighting one another, killing each others' kings, merging territories and then splitting them up again. We call these people the Anglo-Saxons. They called themselves Angles, and their land was Engla-land. Their Germanic dialects, mixed with Celtic and Latin, became the early form of English we call Old English.\n\nAs the Anglo-Saxons took over, many Celts fled back across the sea to Europe, or deeper into the hinterlands of the west and south. The Anglo-Saxons dismissed them as foreigners, Wealas \u2013 the root of our word **Welsh**. (To this day Wales and Cornwall, in the far southwest, are strongholds of Celtic history and language.) The Wealas built tiny chapels and erected stone crosses, and clung to their Christianity. But while their faith remained strong, their memory of Latin slipped away. They repeated the rites and ceremonies of their church, but many of them barely understood the words they were saying.\n\nThe years after the fall of Rome are sometimes called the Dark Ages, because so much knowledge was lost, or at least temporarily misplaced. As the Roman Empire disintegrated into small territories, learning and culture took a back seat to rivalry and war. But Europe still had some important centers of scholarship, mostly in religious communities \u2013 especially in Rome, where the Catholic Church had grown wealthy and powerful.\n\nIn 597, the pope sent a party of monks, led by a learned monk named Augustine, to King Ethelbert of Kent, who was married to a Christian. Augustine persuaded Ethelbert to become Christian, and the religion began to spread through the Anglo-Saxon kingdoms. The monks built a church and monastery in Ethelbert's capital city, Canterbury. Augustine became the first Bishop of Canterbury, and later Saint Augustine of Canterbury. (Today, the Archbishop of Canterbury is still the senior clergyman of the Church of England.)\n\nAt first, many Anglo-Saxons resisted Christianity fiercely. Monks and nuns were murdered; churches were robbed and burned. But the upper classes were gradually converted by scholars like Augustine, while country folk were won over by humble traveling priests. The pagan gods were set aside. The spring festival of the pagan goddess Eastre became Easter, and the pagan winter festival became the Christ-mass.\n\nThese days, with scanners and photocopiers and faxes, it's hard to imagine the time and labor and mental concentration needed to reproduce a book by hand. Yet texts were not only copied meticulously, but embellished with illustrated initials and ornamental borders, sometimes with real gold. Notice all the birds and animals lurking in this capital P. And what's that funny little face in the top left? (photo credits 6.3)\n\nBy the end of the 600s, most of England was once again Christian. New churches were built, and adorned with \"ivories and jewelled crucifixes, golden and silver candelabra... superbly embroidered vestments, stoles and altar-cloths. _\"_ Convents and monasteries sprang up, where ordinary people, as well as monks and nuns, could learn to read and write. The spread of literacy brought back basics of society like record-keeping and accounting and report-writing.\n\nMeanwhile, the Anglo-Saxon kings were still fighting for supremacy. By the late 700s, King Offa of Mercia had more or less taken control of all the kingdoms, by various means (such as beheading the king of East Anglia). Offa called himself Rex Anglorum \u2013 Latin for \"King of the English.\" But when he died in 796, his little empire fell apart. In the early 800s King Egbert of Wessex likewise managed, for a time, to control all the Anglo-Saxon domains, and he is sometimes counted as the first king of England.\n\nBut while the Anglo-Saxons continued their rivalry, some Scandinavian tribes to the east, across the sea, had plans of their own. By the late 700s, these bold Norsemen (\"north-men,\" also called Vikings or Danes) were sailing to Britain and making fast, brutal raids \u2013 looting unprotected churches and monasteries of their gold and silver treasures, burning precious manuscripts, smashing tombs, plundering villages, stealing horses, and slaughtering people or seizing them as slaves. \"'From the fury of the Norsemen,' prayed the peasants in their churches, 'good Lord, deliver us!'\"\n\nThe Vikings (Danes) had learned to be the world's best shipbuilders and sailors, because the rough northern seas were often the only road between their villages. In the 800s and 900s they were not only raiding Britain and northern France, but building their own settlements there. Sometime around the year 1000, their square-sailed, dragon-headed longboats, powered by oarsmen, even traveled as far as North America. (photo credits 6.4)\n\nBefore long, the Danes were not only raiding and going home again. They were beginning to take over the country for themselves \u2013 just as the Anglo-Saxons had done a few centuries earlier. Year after year the Vikings attacked Britain's coasts, and year after year they claimed more land as their own. Sometimes the Anglo-Saxons bribed them to go away, with payments called Danegeld (Danes' gold) \u2013 but sooner or later they always came back.\n\nIn 851, a fleet of Viking dragon boats moored in the Thames River, and the raiders burned London and Canterbury. In 865 they crossed Northumbria, looting and burning; they seized the city of York, and destroyed the school and library. In 869 they murdered the ruler of East Anglia when he refused to give up Christianity. In 872 the King of Mercia abandoned his crown and fled to Europe.\n\nNow only one Anglo-Saxon dominion stood against the Danes \u2013 Wessex. It was ruled by a young king named Alfred.\n\nThe religious life offered women a refuge from difficult circumstances. For a few it was also a rare chance for a professional career. Princess Hilda of Northumbria (614-80) became a Christian when she was thirteen, and took her vows as a nun twenty years later. Revered for her wisdom and piety, she founded a church and \"double convent\" (for both men and women) at Whitby, on a cliff overlooking the sea. Hilda is now a saint.\n\n#\n\nWhen Alfred became King of Wessex in 871, the Vikings (Norsemen) controlled all of England north of the Thames \u2013 including the city of London \u2013 and they were attacking Wessex. Unable to defeat them right away, the new king played for time, paying a large bribe of Danegeld in exchange for a few years' peace. Meanwhile, he started building a fleet of warships to fight off the raiders. By 878 his army was able to win a decisive victory, and England was divided into Alfred's kingdom, and a region to the north and east (called the Danelaw) that was still held by the Norse.\n\nAlfred trained and equipped a strong army to defend the kingdom, and fortified many of the towns. He took back London, and rebuilt it. He compiled a code of laws, reformed the justice system, and reorganized finances. He built monasteries and convents, and brought in foreign scholars to make the Church once again a center of learning. He even invented some handy household gadgets, including a candle-lantern \"clock\" that measured time by the burning-speed of specially made candles. While he was doing all this, he was also fending off fresh attacks by the Vikings (Norse).\n\nAt the age of forty he learned Latin, and began translating important writings \u2013 including some by Saint Augustine \u2013 into Old English. Before then, English had been used for practical purposes like record-keeping and law-writing, and for poetry, but almost everything else was in Latin. Now, both translations and new works began to appear in Old English \u2013 Englisc, as it was called then.\n\nAlfred the Great (849-899) is remembered as much for his wisdom and scholarship as for his political and military feats. \"There will be more wisdom,\" he wrote, \"the more languages we know.\" (photo credits 7.1)\n\nThe Old English verb _witan_ meaning \"know\" has fallen out of use, but it still casts a shadow. Today wit means cleverness or intelligence. Unwittingly means without knowing (\"He unwittingly revealed the secret.\") Wise people, witnesses, and witches and wizards are all named for their knowledge.\n\nThere were further clashes between the English and the Norse after Alfred's death in 899. In 1012, the Archbishop of Canterbury was slaughtered by drunken Vikings. For a short while, from 1016 to 1042, Norse kings once again ruled much of England. But Alfred was the turning point. He brought the Anglo-Saxons together, restored Christianity, and showed his people that they could be one great nation, with a language and literature of their own. For all of this, history remembers him as Alfred the Great.\n\nMany of our most basic day-to-day words \u2013 **land** , **cow** , **sheep** , **dog** , **plow** , **bread** \u2013 come from Old English. **This** and **that** come from _this_ and _thaet_. Our comparisons of **long** , **longer** , **longest** come from Old English _lang, lengra, lengest_. But as many of the invaders adopted Christianity, and the two peoples lived together and intermarried, their language \u2013 Old Norse \u2013 affected Old English. About seven percent of our words \u2013 **take** , **get** , and **keep** ; **sky** , **skin** , and **skirt** \u2013 come from the Vikings. **He** is from Old English, but **they** is from Old Norse.\n\nEtheldrida was a princess of East Anglia. As a girl she loved to wear fancy necklaces, but later she became deeply religious. When she developed a tumor on her neck, she saw it as God's punishment for her youthful vanity. After her death she became Saint Audrey, and the cheap, badly made lace sold in her honor \u2014 \"Saint Audrey's lace\" \u2014 gave us our word tawdry (t'audrey), meaning gaudy but worthless.\n\nAs Norse words passed into English, people could choose between an English-based term and a Norse-based one: **wish** or **want** , **craft** or **skill** , **hide** or **skin**. Over time, those similar words picked up slightly different meanings, different associations, and the English language grew more subtle and versatile. We still have distinctions like this, even if we don't think about them. For example, **Nativity** and **Yule** both mean \"Christmas\" \u2013 but **Nativity** (\"birth,\" from a Latin root) suggests the religious side of the season, while **Yule** \u2013 from Old Norse _jol_ , a twelve-day pagan festival \u2013 suggests a jolly good party. (Yes, **jolly** is also from _jol_.)\n\nOld English nouns and pronouns varied depending on their case, but the changes were not as complicated as they had been in Latin and Greek. In fact, many pronouns looked very similar to ours:\n\nIn Old Norse, _by_ meant \"town.\" Today, bylaws are town laws (such as parking laws), and town names like Grimsby, Whitby, and Rugby echo those long-ago Viking invasions.\n\nAlthough these forms haven't changed much in the past thousand years, people still get confused about the cases of pronouns. **I** or **me**? **He and she** , or **him and her**? There's an easy trick that can help you keep them straight.\n\nAre these sentences right or wrong?\n\nDad saw John and I.  \nMom wants she and Chris to go.  \nHarry thinks Lily and us are right.  \nKids like you and I know a lot!\n\nIf you're not sure (and here's the BIG SECRET), drop one of the people. Drop **John and** , **and Chris** , **Lily and** and **you and**. See how wrong they all sound now? So if you're ever not sure which pronoun to use, you can either stop and think about the case \u2013 is it the subject or the object of the verb? \u2013 or you can try dropping that confusing other person!\n\nOld English had some letters we don't have:   ( _th_ as in \"cloth\"),   ( _th_ as in \"clothe\"), and   (a _w_ sound). A and E joined together (  or  ) sounded like _a_ in \"cat.\"\n\nBut if Old English and Old Norse were both inflected, with words changing their form depending on how they were used, why don't we have all those endings in modern English? As the Norse and Anglo-Saxons struggled to understand each other, they probably gave up most of the pesky endings, just to save trouble.\n\nAfter all, if you were talking to someone who didn't speak much English, would you say, \"Might I trouble you to pass the salt?\" Or \"Pass the salt, please\"?\n\n#\n\nMuch of what we know about Anglo-Saxon times comes from Saint Bede, who wrote \u2013 by hand, of course \u2013 countless works of history, religion, medicine, astronomy, even poetry and grammar. Bede was born about two hundred years before Alfred's reign, in the days when most prose was still written in Latin. Other works \u2013 poems, word games, and long, elaborate riddle-verses \u2013 were sometimes written in Old English. What creature shoots from the stomach and can \"serve no master when unstrung\"? What complains that \"Everybody lifts me, grips me, and chops off my head\"? (See end of chapter.) But most of these writings have been lost, and the ones that survive are often anonymous.\n\nWhen Bede (673-735) was just seven years old, he was sent to the northern monastery of Jarrow. He spent almost all his life there, studying in the monastery's library and filling it with his own books \u2013 including a history of the English people (in Latin) ever since the days of Julius Caesar. Because of his intelligence and tireless scholarship, he's often called the Venerable (much-revered) Bede. (photo credits 8.1)\n\nThe earliest English poet we know by name was a man called Caedmon, who lived around 670. In those days, before TV or radio or even printed books, poets recited works by memory, to entertain and inspire their audiences. According to Bede, Caedmon was an uneducated stablehand at the abbey in Whitby. One night a vision came to him and he was suddenly able to make up a hymn praising the _weorc Wuldor-Faeder_ (\"work of the Glory-Father,\" meaning God). Caedmon remained in Whitby as a lay brother (not a monk), composing other works.\n\nIn 1898 a sandstone cross was erected at Whitby with this picture of Caedmon; notice the horses in the stable. (The image of Saint Hilda in Chapter 6 is from the same cross.)\n\nAlthough Old English looks hard to read, the sounds suggest modern English. Speaking of God, Caedmon says, _He aerest sceop eorthan bearnum heofon to hrofe_. Try reading that out loud (pronouncing _sc_ as _sh_ ) and comparing it to this translation: \"He first shaped for earth's bairns [children] heaven as a roof...\" (The next thing God shaped, according to Caedmon, was _middangeard_ \u2013 middle earth.)\n\nThe most famous Old English work is _Beowulf_ , an epic poem of over three thousand lines celebrating the feats of a \"prince of warriors,\" and his courage and honor. The story comes from the myths and history of Scandinavia, where the adventures are set. We don't know when or where the poem was first recited, but the version that has come down to us is from a manuscript more than a thousand years old.\n\nBeowulf defeats a gruesome monster, Grendel, ripping off Grendel's arm with his bare hands. Grendel's mother later attacks Beowulf in vengeance, and Beowulf dives to the bottom of a haunted lake and slays her and cuts off her head. Beowulf becomes king, but many years later he has to defend his people against a fiery dragon. The dragon is defeated but Beowulf dies in the battle, and is mourned as a noble and beloved leader.\n\nLike Caedmon's poem, _Beowulf_ is built out of short half-lines that build up into sentences. The lines are held together not by rhyme, which the English rarely used in those days, but by powerful rhythm and lots of alliteration (words beginning with the same sound). Try reading these phrases out loud, pausing at the break; imagine how dramatic they would have sounded to an audience sitting in a great hall lit only by a fire, surrounded by darkness.\n\n_Grendel gongan Godes yrre baer...  \n(Grendel walked in, bearing God's ire [anger])...._\n\n_on fagne flor feond treddode...  \n(on the fine floor the fiend trod)...._\n\nHere's the opening of _Beowulf_ , in Old English written around 1000. This manuscript \u2013 the only one surviving from ancient times \u2013 was badly damaged in a fire in 1731, before any copies had been made, so some bits have been lost forever. The word order differs from ours, but the beginning says, \"Truly we have heard of the glory of the Spear Danes' kings  _cyninga_ ] in yore-days [ _in gear-dagum_ ].\" _Gar_ is an old word for spear; the Danes are called _Gar dena_. ([photo credits 8.2)\n\nLate in Alfred's reign, once Old English had been established as a language for serious writing, some monks began compiling a historical record called _The Anglo-Saxon Chronicle_. They filled in past events from whatever sources they could find, and kept adding to the book as the years went by. Here's some of the description of the desperate, bloody year when Alfred became king:\n\n**871:** Then King \u00c6thered fought the troops of the [Viking] kings, and King Barsecg [of the Vikings] was killed; and \u00c6thered's brother Alfred fought the troops of the Viking earls,... and many thousand were slain, and they were fighting until nightfall. About a fortnight later, King \u00c6thered and Alfred his brother fought the invading force at Basing, and there the Vikings were victorious. Two months after that, King \u00c6thered and Alfred his brother fought at Merton, and there was great slaughter on both sides, but the Vikings controlled the battlefield.... Over Easter, King \u00c6thered died; he had ruled for five years.... Then Alfred son of \u00c6thelwulf came to the throne of Wessex. About a month later, King Alfred fought with a small company against the whole Viking force.... In that year, nine great battles were fought against the invading Vikings in the kingdom south of the Thames....\n\nJ.R.R. Tolkien was a professor of English with a special interest in early languages. When he wrote _Lord of the Rings_ he borrowed words from Old English: an _orc_ is a demon, an _ent_ is a giant, and the name of Rohan's King Theoden means \"ruler.\" As for the dread land of Mordor, _morthor_ is Old English for \"death, destruction;\" it's the root of our word murder.\n\nAt last, English was the accepted language of prose and poetry, of science and history, of law-writing and account-keeping. Priests and scholars might still work in Latin and Greek, but English had finally won its place as the language of England.\n\nOr so it seemed.\n\nANSWERS\n\n#\n\nEngland was not the only land invaded by the Vikings. During the 800s, the Norsemen had sent their dragon boats up the rivers of northern France, raiding and burning Paris and other cities. Then they had moved in and begun building settlements. In 911, to put an end to their raids, the French king had granted them land on the north coast of France. The Norse leader had converted to Christianity and had been made a duke. Now these Vikings were called Normans, their territory was Normandy, and they came to speak their own Norman version of French.\n\nBy 1042, England was ruled by a religious but weak king named Edward the Confessor. Edward was descended from Alfred the Great, but his mother was from Normandy and he had grown up in the Norman court. Many people in England thought he was much too friendly with the Normans. Apparently Edward even promised the English throne to their leader, William, Duke of Normandy, since Edward himself had no heir.\n\nIn 1064, a rich and powerful English nobleman \u2013 Harold, Earl of Wessex \u2013 was shipwrecked on the coast of Normandy. It seems that he vowed to support the duke's claim to the throne of England. (Harold may have been forced to make this promise.) But two years later, when Edward the Confessor died, it was Harold who took over the throne.\n\n(photo credits 9.1)\n\nWilliam of Normandy's victory is portrayed in the Bayeux Tapestry, an ancient strip of embroidered linen that is 230 feet (70m) long. The Latin shown here says, _Harold Rex Inter-fectus Est_ \u2014 \"King Harold is killed.\" According to legend, Harold was shot in the eye with an arrow, but experts aren't sure whether he's the victim on the left, or the one on the right \u2013 or both! (photo credits 9.2)\n\nWilliam of Normandy was furious. He loaded troops onto a fleet of ships and sailed for England, to claim the country for himself. The two armies met in battle near the town of Hastings in October 1066, shooting rafts of arrows, hurling spears, dueling face to face with swords, clubs, and battle-axes. The Normans probably had no more than five thousand soldiers, but they had one great advantage: their knights rode horses specially bred and trained for combat. The English used horses for transportation, but they had never before faced an enemy fighting from horseback.\n\nBy the end of the day King Harold was dead, and so were many of his noblemen. William of Normandy \u2013 who was clever and practical, but merciless \u2013 led his army across southern England and seized the city of London. On Christmas Day he was crowned England's new king \u2013 William the Conqueror.\n\nThe Normans expanded their power throughout England, building massive stone castles with moats and drawbridges, and manning them with soldiers. One by one, the English lords were killed or driven from their lands, and Norman barons took their place. Each baron had to keep a host of armed knights ready to do battle whenever the king needed them. Norman French was suddenly the language of law and government, the language of the ruling class, the language of fashionable literature.\n\nWhen William of Normandy landed in England in 1066, he set up camp inside the walls of Anderida, a sea fort built by the Romans eight hundred years earlier to hold off the Vikings. After defeating King Harold, the Normans built Pevensey Castle on Anderida's remains, and used it for about two hundred years. Notice the mighty keep (refuge) at the back, and the twin towers of the gatehouse in front, where the bridge crosses the moat. In the 1500s, when Spain sent a vast \"Armada\" of warships against England, Pevensey was refortified to resist the attack. In World War II, machine-gun posts were hidden in Pevensey's walls in case of a German invasion. (photo credits 9.3)\n\nAlthough most English people continued to speak their own language, and Old English was still used for some writing, French words inevitably filtered into English \u2013 especially words connected with government or fine living. The English we speak today still has traces of the social gap between the victorious Normans and the conquered English. **Crown** is from French _couronne;_ **parliament** is from French _parlement_ ; **royal** is from Old French _roial_. The cow and pig down in the stable are from Old English _(cu_ and _pigga);_ the beef and pork up in the dining hall are from Old French _(boef_ and _porc)_. Plain straw and wool: Old English _streaw_ and _wull;_ elegant lace and velvet: Old French _laz_ and _veluotte_. Most English words ending in _ion, or, ment_ , or _ence_ (like **mention** , **honor** , **compliment** , or **patience** ) came from French.\n\nWe still use French (or French-based) expressions to sound sophisticated. It's _chic_ to have good _etiquette_ , to be _debonair_ (with a _bon aire_ , \"good air\") and _nonchalant_ (not _chal\u00e9_ , worried). We'd like enough _savoir-faire_ to know how to behave, enough _savoir-vivre_ to know how to enjoy life, and enough _sang-froid_ (cool blood) to stay calm in the face of danger. The French even have a name for what they can't name \u2013 it's _a je ne sais quoi_ (\"I don't know what\").\n\nBack in the 1890s, only a few rich people had cars, and it took a while to get them going. The employee who was sent out ahead of time, to _chauffer_ (warm up) the engine so it would run properly, was called a chauffeur.\n\nAnd look how French dominates fine dining! At a _restaurant_ or _caf\u00e9_ we are greeted by the _ma\u00eetre d'_ and we order a _gourmet_ dinner _\u00e0 la carte_ \u2013 an _ap\u00e9ritif_ , _p\u00e2t\u00e9_ , or cheese _souffl\u00e9_ as an _hors d'oeuvre_. \"And for the _entr\u00e9e_ , the _chef_ suggests the _quiche_ or _fondue_ , or a _casserole_. And then perhaps some _\u00e9clairs_ , _meringue_ , _sorbet_ , _cr\u00e8me br\u00fbl\u00e9e_?\"\n\nThe Norman Conquest also brought more Latin into English. Although Latin was still used in church, it had pretty much disappeared from England's daily life after the Romans went home. But French had grown from Latin, and now Latin roots (and Latin ways of organizing a sentence) were back, buried in Norman French. On top of that, much of the history and record-keeping that had been done in English was switched back to Latin. After 1154, there are no more entries in the _Anglo-Saxon Chronicle_. One in three words in English today comes from Latin \u2013 directly from the Romans, or through the church, or through the Normans, or through more modern French.\n\nWhich is correct \u2014 \"one in three _is\"_ or \"one in three _are_ \"? This looks confusing because \"three\" is closer to the verb \u2014 don't we say \"three _are_ \"? But it's the actual subject of the verb that has to match the verb. That subject is \"one\" \u2014 \"in three\" describes it but doesn't change the number \u2014 so we say \"one in three is,\" just as we'd say \"the one in trouble is.\" When people let a subject and verb disagree in number, it's often because something in between confused them. They say, \"Each of the lions have their own den\" \u2014 _each have?_ What they should say is \"Each of the lions has its own den.\" They say, \"The turtle is one of the oldest animals that lives in the zoo\" \u2014 _animals that lives?_ \"The turtle is one of the oldest animals that live in the zoo\" is correct.\n\nJust remember whether the subject of your sentence is singular or plural, and make the verb match. Simple!\n\nEnglish had already lost some of its complications in the days when the Anglo-Saxons and Vikings were trying to understand each other. Now, as English people tried to cope with Norman French and Latin as well, their language lost yet more of its endings, and became even simpler. And once again, as foreign words slipped into the vocabulary, English became more subtle and versatile at expressing different shades of meaning.\n\nThe Norman Conquest of 1066 turned out to be another of history's great turning points, and its effect lives on in both Britain and the English language. Next time you see a picture of handsome young Prince William, grandson of Queen Elizabeth II and likely a future king, remember that one of his ancestors is William the Conqueror, that heir to the Vikings, who crossed the Channel to seize England's throne almost a thousand years ago.\n\n_\"How stinky! Well, not stinky, exactly, but....\"_ We're lucky that English gives us so many ways to say what we mean. When we can't think of exactly the right word, we can reach for a thesaurus \u2014 which sounds like some kind of dinosaur, but is really a book of words grouped by general meaning ( _thesauros_ is Greek for treasure). If you look up stink in the index of one popular thesaurus, you're directed to a section labeled Fetor \u2014 which offers choices like stench, rank, rancid, musty, fusty, skunky, fetid, putrid, and (oddly!) garlic. The thesaurus is a handy tool when you're writing, but remember: all these words have slightly different senses. It's up to you to decide which one is just right.\n\n#\n\nThe Norman Conquest of 1066 was followed by years of unrest, chaos, even civil war. King after king struggled to crush rebellions in England, and to subdue the Welsh, Scots, and Irish. Being as ambitious and warlike as their Viking ancestors, the Norman kings were also fighting to win more territory in France.\n\nCrusaders brought home stories of distant lands and exotic beasts. Fire-breathing dragons were a popular feature, especially if they were guarding fair maidens or fabulous treasures. Beowulf had killed a dragon, after all, and so had Saint George, England's patron saint. Since Christians often portrayed dragons as symbols of the Devil, killing them was noble and virtuous. (photo credits 10.1)\n\nWhen they weren't doing battle with the French, the Norman English were riding beside them as allies on religious crusades \u2013 military campaigns to take back Palestine, the Holy Land of Christians, Jews, and Muslims, from the Muslims. Throughout the 1100s and 1200s, kings, noblemen, and knights rode off to seek victory in the Middle East. Often they stayed away for years. Richard Lion-Heart (the king in the Robin Hood story) ruled England for ten years but spent only six months in the country; most of the time he was away on a crusade.\n\nThe Holy Land wasn't the only place the Europeans wanted to reclaim. Back in 711 \u2013 in the days of the Anglo-Saxon kingdoms, before the Vikings began their raids \u2013 an Arab army had crossed the Mediterranean from Africa and seized much of Spain. It took the Europeans hundreds of years to drive the Muslims out.\n\nWith all this warfare between Arabs and Europeans, there was a constant exchange of ideas and language, and many Arab words were transformed into English \u2013 like _suffah_ (sofa), _sukkar_ (sugar), _ghul_ (ghoul), and _makhazin_ (magazine). A _matrah_ was a place where you put down your mattress; your _rahat_ was the palm of your hand, which holds your tennis racket; an orange was a _naranj_.\n\nNotice how the _n_ of _naranj_ drifted off the noun and attached itself to the indefinite article, **a**. Sounds often floated one way or the other, back in the days when most people couldn't read or write, and just repeated whatever they thought they'd heard. **An apron** used to be **a napron** ( **napron** is related to **napkin** ); a newt was **an ewt** ; **a nickname** was **an ekename** (eke meant \"extra\"); **an umpire** was **a noumpere** , a \"non-peer\" rather than one of the players.\n\nLong ago, alchemy was the search for a way to turn cheap metals into silver and gold. The unknown liquid that would make this happen was called the elixir (from the Arabic _al-iksir; al_ is Arabic for \"the\"). Alchemy and chemistry are from the Arabic word _al-kimiya_ , \"the art of transforming metals.\" These days an elixir is a medicine, perhaps one promised to cure all ills and help you live forever.\n\nEarly European scholars believed that everything was made up out of four elements \u2014 earth, water, air, and fire \u2014 plus a \"fifth essence,\" in Latin _quinta essentia_. These days, the quintessence of something is the most perfect or most complete example of it. In _The Lord of the Rings_ , Sauron is the quintessence of evil.\n\nKohl is a black powder once used as eye makeup. The Arabic word, _al-kohl_ , came to mean any powder, and later came to mean a purified liquid. Eventually it also meant wine \u2014 alcohol.\n\nWhy are some verbs so easy \u2013 I walk, I walked, I have walked \u2013 and some so bizarre? Look at eat ate eaten, freeze froze frozen, slay slew slain, see saw seen, sing sang sung. Verbs that have different forms for the past tense come from the Scandinavian roots of English, not Latin or French. There used to be about three hundred and fifty of them, but many have gradually lost their irregular forms. Of the sixty or so odd verbs that remain, some are half gone. You'll hear people say, \"She sung\" instead of \"She sang,\" or \"I rung the bell\" instead of \"I rang.\"\n\nOld English was now evolving into Middle English (spoken from 1100 to 1500, more or less). Most nouns started taking an _s_ in the plural, though a few continued to change their vowel ( **goose\/geese** ) or to take an _en_ ending ( **ox\/oxen** ), or didn't change at all ( **fish\/fish** ). Endings that showed the case of a noun were being replaced by prepositions: **of** , **to** , **from** , and so on. Letters like **_\u00fe_** ,   and \u03b8 gradually disappeared. Words like _cnawan_ and _cniht_ ( **know** and **knight** ) traded their _c_ for a _k_ , and in time the _k_ became silent. Many of our silent letters today \u2013 the _g_ in **gnaw** , the **l** in **would** \u2013 are whispers of letters that were pronounced in those days.\n\nLiterature was changing too. The Normans brought French styles of poetry to England \u2013 including _chansons de geste_ (\"songs of deeds\") that celebrated the adventures of gallant, gentlemanly knights who risked death to please God (or some fair lady). Though the first _chansons_ were in French, some were soon translated for those who spoke only English, and then original English works began to appear.\n\nMany romantic poems were about King Arthur and his Knights of the Round Table, and his court at Camelot. (Arthur had, of course, killed a dragon.) Although his story is still being told in books, songs, plays, and movies, Arthur remains a mystery. Was he real? Or is his story just a myth? (photo credits 10.2)\n\nThe most famous writer of the time is Geoffrey Chaucer. His best-known work, _The Canterbury Tales_ , tells of a group of people \"high and low, old and young, male and female, lay and clerical, learned and ignorant, rogue and righteous, land and sea, town and country,\" riding to Canterbury together on a religious pilgrimage, and passing the time by telling stories. Chaucer includes humorous details about each of the travelers \u2013 merchant, miller, housewife, nun \u2013 and each story imitates that traveler's own speech, painting us a picture of how these people would have \"cursed and complained and questioned and told each other jokes,\" six hundred years ago. Unlike Caedmon's hymn, this poem depends more on rhyme than on alliteration. Here's how he introduces a \"parfit gentil knight.\" (To make the rhythm work, pronounce the underlined letters as extra syllables.)\n\nA Knight ther was, and that a worthy man,  \nThat fro the time that he first bigan  \nTo riden out, he loved chivalrye,  \nTrouthe and honour, freedom and curteisye.\n\nChaucer started this poem in 1386, but we don't know exactly how he wrote it, because the manuscripts from his time (written by hand) have been lost. All we have now are later copies.\n\nAs a youth, Chaucer (1340?-1400) served as a pageboy to royalty, and learned courtly manners as well as Latin and French. Later, as a diplomat and civil servant, he was sent to France and Italy. His writings are rich in words, ideas, and story twists that he brought back from his travels. Unlike the dragon-slaying heroes in tales of chivalry, his characters are as real and down to earth as anyone you might meet on the road. (photo credits 10.3)\n\nBut here's another poem \u2013 an anonymous work called \"I Have a Young Sister\" \u2013 that shows us what English looked like around 1300. It's similar to the riddle-verses the Anglo-Saxons wrote hundreds of years earlier. The poet announces that his young sister has sent him a cherry with no stone and a dove with no bones.\n\nHow sholde any cherye\n\nBe withoute stoon?\n\nAnd how sholde any dove\n\nBe withoute boon?...\n\nWhan the cherye was a flowr,\n\nThanne hadde it no stoon;\n\nWhan the dove was an ey [egg],\n\nThanne hadde it no boon.\n\nWhat did the poet's sister send him? (If you find this hard to understand, try reading it out loud. There's a modern version at the end of the chapter.)\n\nEnglish was flourishing in all kinds of ways. Although the aristocrats still spoke French and boasted of their French blood, they had to speak English to their servants, their staff, and the farmers on their estates. By the 1300s, Parliamentary proceedings were again conducted in English. By 1350, English had replaced Latin as the teaching language in most schools. In 1362, English replaced French in the law courts, although legal records were still in Latin. In 1399, Henry IV became the first king since the 1066 Norman Conquest to make his coronation speech in English. The offficials of the kingdom followed his example, doing more and more of their work in English.\n\nBy the mid-1400s, the Normans had lost almost all their territory in France. They had become English, and anglophone (English-speaking), whether they liked it or not. English was once again the language of the land.\n\n**_Modern version of \"I Have a Young Sister\"_**\n\n_How could a cherry not have a stone [pit]?_\n\n_And how could a dove not have any bones?..._\n\n_When the cherry was a flower, it had no stone;_\n\n_When the dove was an egg, it had no bones_.\n\nOne reason for the decline of Latin was the Black Death, a terrible epidemic of bubonic plague that reached England in 1348. Spread by fleas on the rats that infested wooden merchant ships, the disease killed thirty to forty percent of England's population. By the time that wave of plague was over, there weren't enough teachers left to educate children in French, let alone Latin, so most schools taught in English. The plague would flare up again and again in the centuries to come.\n\n#\n\nWhile fewer people were learning French and Latin, more were learning to read and write English. The old cathedral schools staffed by monks and priests had grown into universities. (Oxford and Cambridge \u2013 probably the most famous universities in the world \u2013 both date back to the 1200s.) Smaller schools and private tutors were much in demand. But there was one huge problem: creating a book still meant writing it out by hand. One hard-working scribe \u2013 often a monk or nun in a writing hall called a scriptorium \u2013 might complete as little as a hundred pages in a year. So books remained expensive, and most people couldn't afford them.\n\nBut all that was about to change. In the 1400s, a few people in Europe realized that if they molded little letters backwards \u2013   \u2013 and grouped them into a block of words, they could create the reverse image of a page. Then the block could be covered with ink and pressed against paper to make a real page. (People in China and Korea had been doing this for years, but it seems the Europeans hadn't heard about it.) After the individual letters (called movable type) had been used to print enough copies of that page, they could be taken apart and assembled differently to spell out another page, and another.\n\nA scribe works on the laborious process of copying a book, in the early 1500s. (Scribe is from Latin _scribo_ , \"I write.\") (photo credits 11.4)\n\nIn 1455, a German named Johannes Gutenberg printed a Bible in Latin, using movable type. Copies were soon on sale across England and Europe. Then an Englishman named William Caxton went to Germany to learn the procedure, and created the first English book ever printed. Around 1476, Caxton went back to England, built himself a wooden printing press, and went to work. He produced books of his own writings, and translations of foreign writings (including Aesop's fables), and Chaucer's _Canterbury Tales_ , and _Le Morte D'Arthur_ , a long English-language account of King Arthur and his knights.\n\nA page from Chaucer's _Canterbury Tales_ , in Caxton's version of 1484, with an illustration of the \"parfit gentil knight.\" Caxton (1422-1491) printed over ninety books in all, seventy-four of them in English. It troubled him that the handwritten books he copied could be full of mistakes, but he took pains to work from the most accurate, most original text he could lay hands on. (photo credits 11.5)\n\nOne of the problems Caxton had to resolve was how to spell the works he translated. He wanted his books to be read all across England, but different regions pronounced and spelled words differently, and even used different words for the same thing. Which version should he choose? To explain his difficulty, Caxton told the story of a hungry traveler who asked a countrywoman for _egges_. She had no idea what the word meant. Then another man asked her for _eyren_ , and she gave him \u2013 eggs. \"Loo,\" wondered Caxton, \"what sholde a man in thyse dayes now wryte, egges or eyren?\" After much consideration, he decided to reproduce (more or less) the words and spellings used by officials in London. After all, their English was similar to the English used at Oxford and Cambridge. How could it be wrong?\n\nBooks printed before 1501 \u2013 when printing was still in its infancy \u2013 are extremely rare and precious. They're known as incunabula, Latin for \"baby's cradle.\"\n\nNow that publishing was so much faster and cheaper, a snowstorm of books, pamphlets, and other writings swept across the country. In 1525, the first English translation of the Bible's New Testament was printed. A complete English Bible was published ten years later, another version in 1537, and two more in 1539. By this time, perhaps half the population could read, and books (especially the Bible) were becoming part of their daily life. The words and spellings chosen by Caxton and other printers became the normal, customary forms. Eggs were _egges_ , and if you called them _eyren_ , well, you must be ignorant.\n\nIn Caxton's day, many scholarly books were still in Latin. But soon, more and more were printed in English. As late as 1653, _Grammatica Linguae Anglicanae_ \u2013 a book explaining the grammar of the English language \u2013 was written in Latin. But by the late 1600s, even the most important academic and scientific works were being published in English. And as scholars used English to explore all areas of human thought, no matter how complex, their vocabulary had to expand. Now, English had to have the power and precision of classical Latin and Greek.\n\nA printing shop in the early 1500s. Workers at the back are assembling letters of backwards type into a page-size block. Those at the front are rubbing ink onto the page block, and pressing sheets of blank paper against the block to print a stack of pages. Our expression \"the press,\" meaning newspapers and magazines, comes from this pressing of paper against inky type.\n\nMeanwhile, something very strange was happening to speech \u2013 something known as the \"great vowel shift.\" Starting around 1400, people changed the way they pronounced certain vowels.\n\nWe have five real vowels ( _a_ , _e_ , _i_ , _o_ , and _u_ ), and _y_ , which is sometimes a vowel and sometimes not. We have various ways of pronouncing them. They can be short (bit, mat) or long (bite, mate). They can be silent, or run together into a single sound (cream), or one can slide into another (coin). Sometimes they stand side by side but we say them separately, as in No\u00ebl. (The two dots over the _e_ tell you that the vowels are pronounced separately.)\n\nMany old words have fallen out of use, but some of them still pop up inside other words. _Tadde_ used to mean \"toad,\" and _pol_ meant \"head.\" (Today, a poll is a count of one vote per person \u2013 one vote per _head_.) A toad that was mostly head \u2013 that wiggly, legless larva that grows into a toad or frog \u2013 used to be a _taddepol_ (\"toad-head\"). That's why we call it a tadpole.\n\nDo you say _tomayto_ or _tomahto_? Do you call pigeons _brrds, bards, beds, buhds_ , or _boyds_? The way we pronounce vowels varies from place to place, and even from person to person. In the movie _My Fair Lady_ , Eliza, a ragged flower-seller who's learning to talk like a fine lady, has to work desperately hard to say that rain in Spain (not _rine in Spine_ ) falls mainly on the plain (not _minely on the pline_ ).\n\nPronunciation also changes over time. In Chaucer's day, **fine** rhymed with **bean** ; **see** rhymed with **say** ; **to** rhymed with **no**. But in the great vowel shift, many long vowels changed dramatically, to the pronunciations we use today. That's one reason why old poetry like Chaucer's sometimes doesn't seem to rhyme.\n\nWhich is right: \"a historic moment\" or \"an historic moment\"? \"A herb\" or \"an herb\"? Some people omit the _h_ (\"aitch\") on these words, so they need \"an\" as the indefinite article (\"an 'istoric,\" \"an 'erb\") \u2013 but if you pronounce the aitch, drop the _n_ (\"a historic,\" \"a herb\"). Poor Eliza struggles to pronounce her aitches in _My Fair Lady_ , huffing them into a flame \u2013 \"hurricanes hardly happen\" \u2013 to make it flicker.\n\nHave you ever wondered why **police** and **polite** don't have the same _i_ -sound? It's because **polite** is an old word, and the sound of the _i_ changed during the great vowel shift. **Police** came into English after the shift, so it never changed.\n\nAs for spelling, while some people find it difficult, there are tricks to make it easier. If you think about what the word means, what other words it resembles, and what language it comes from, you are more likely to remember the spelling. For example, _terra_ is Latin for **earth** , **land** , **ground**. If you can spell **terrier** (a dog that burrows into the ground), you know how many _r_ s there are in **subterranean** (underground), **Mediterranean** (a sea in the middle of the land), **terrestrial** (on earth), **terraforming** (transforming another planet to resemble earth), and so on.\n\nKids who compete in spelling bees (contests) have an amazing ability to remember how thousands of words are spelled. When the judge asks them to spell a word, they're allowed to ask what the word means; if it sounds like \"tericolous\" (with one _r_ ) but it means \"living on the earth,\" they can guess that it's **terricolous** , from _terra_. They can ask what part of speech the word is (noun? verb?); the noun **callus** (a hard spot on skin) is spelled differently from the adjective **callous**. They can ask what language it comes from; a word that sounds like Latin _terra_ might actually come from Greek _pteron_ , \"wing,\" with that tricky silent _p_.\n\nTwo good movies show what a spelling bee is like. _Akeelah and the Bee_ tells a story about a girl who loves to spell and ends up in a bee. _Spellbound_ follows a real-life spelling bee as, one after another, contestants flub their words, until only the winner is left. Both movies show how much training and practice and hard work it takes to be a winner \u2013 and how scary it is to be up on that stage, spelling words you've never even heard of.\n\nMany people have argued that we should simplify our spelling, and write words the way they sound. But others treasure our weird spellings as a mini-history of the English language. These seemingly strange words are an echo of all those long-ago voices, from so many different cultures and places, that left their mark on this language called English.\n\n#\n\nIn the early days of printing, England was still a Catholic country. It's difficult now to imagine the power that the Church (and the Pope in Rome) had back then, before science improved our lives and helped us understand the world around us. The winters were long and cold, and the typical house was a bare, dark, drafty hut with no heating but a fire. There were few remedies for small miseries like fleas and lice, or aches and pains. Once autumn was past, most people had to survive on whatever food could be dried or salted. When illness or disaster struck, it seemed to come out of nowhere, for no earthly reason.\n\nBut churches, abbeys, and cathedrals, with their lofty pillars and soaring spires, lifted people out of all this. Altars glittered with treasures of silver and gold. Massive stonework echoed the music from organs, choirs, and pealing bells. Statues and paintings and stained-glass windows told stories of virtue triumphant. Feasts and pageants and processions broke up the weary work year. There was drama; there was mystery; above all, there was the promise that those who obeyed God's will during their lifetime would enjoy a glorious eternity.\n\nThe church was the center of the community, consoling people for hard times and giving them hope for the future. Overflowing with beauty, glowing with color, it was an earthly promise of heavenly joy. (photo credits 12.1)\n\nThe cloak ( _cappa_ in Latin) of Saint Martin of Tours, who lived in the 300s, was preserved in a church in a special shrine called the _cappella_. In time, any small area dedicated to worship became known as a _cappella_ , or chapel. ( _Cappella_ is also connected to cobra, the snake that looks as if it's wearing a hood.)\n\nPeople depended on priests to tell them what it was God wanted, and to help them atone for any sins they committed, so they could avoid eternal damnation. Most Bibles and other religious books were still in Latin, and church services were conducted in Latin, even though the congregation didn't understand the language. Indeed, a lot of country priests didn't know much Latin, especially after so many perished in the Black Death. They just followed the rites and rituals they had been taught.\n\nWhen the Church approved a book for publication, it decreed, _\"Imprimatur\"_ \u2013 Latin for \"Let it be printed.\" Today, imprimatur means any official approval \u2013 \"The new dog park bylaws have the mayor's imprimatur.\"\n\nBut the immense wealth and influence of the Church led to problems. One was the conflict between the power of local rulers and the decrees of Church authorities. The Church even decided which books people could read and which were forbidden. Another problem was corruption. While many priests were faithful to their duties, some \u2013 especially high-ranking officials \u2013 lived in pomp and luxury, and abused their position to fill their pockets. Among other things, they sold indulgences \u2013 free pardons \u2013 that let rich people \"buy forgiveness\" for bad behavior.\n\nBack in the 1300s some people had called for reform, saying that the Church's wealth should be put to better use, and that common people should be able to read the Bible for themselves. But while the Bible had been translated into English, at that time it still had to be copied by hand, and there weren't enough books to go around.\n\nBy the 1500s, though, the protest movement was growing stronger. Those protesting \u2013 the \"Protestants\" \u2013 wanted everyone to have access to the Bible. They argued that people should learn to live by their own judgment and conscience, and find salvation through their personal faith \u2013 not through favors bought from the Church.\n\nAnd then along came printing, and the Church no longer controlled the Bible.\n\nIn 1525 that first English printing of the New Testament appeared, translated and printed by an English priest named William Tyndale. Church officials opposed Tyndale, knowing that he supported the protest movement. In any case, it was considered heresy (a crime against God) to translate the Bible into English without permission. Tyndale had to print his Bibles in Europe and smuggle them into England, hidden in bales of cloth and other wares. When copies were found, they were seized and tossed into a fire. People who had them were imprisoned, even tortured and burned alive \u2013 just for daring to own a Bible.\n\nTyndale (1494?-1536) was an exceptional scholar, reading Spanish, Italian, German, and Hebrew as well as English, French, Latin, and Greek, but he paid dearly for his beliefs. He went into hiding in Europe and kept on translating, but English authorities tracked him down and arrested him. He was tied to a cross and strangled, and his body was burned.\n\nBut the Church soon had something new to worry about. King Henry VIII was desperate for a son who could inherit his throne. (After all, no woman could be trusted to rule England!) He wanted to end his marriage and take a new, younger wife. The Pope refused to let him do it.\n\nIn 1534, Henry broke away from the Roman Catholic Church and declared himself head of a new Church of England (the Anglican Church). He closed Catholic monasteries, destroyed shrines, and took over the lands and treasures of the old churches. People who got in Henry's way found their heads on the chopping-block. Enough of this interference from Rome; the English monarch was now the leader of the English church.\n\nWith Rome out of the picture, a number of English-language Bibles were soon on the market. In 1539, a \"Great Bible\" was published under Henry's authority. People across England were reading their Bibles and other books about religion, and asking questions, and making up their own minds about God and their faith. Biblical phrases were becoming everyday expressions: \"my brother's keeper,\" \"the writing on the wall,\" \"an eye for an eye,\" \"casting pearls before swine.\"\n\nAs for Henry VIII, he did finally get a son, but the boy-king died just six years after Henry's death in 1547. Henry's older daughter, Mary, who had remained a devout Catholic, became queen and married King Philip of Spain. In the fifty-odd years since Christopher Columbus had found America in 1492, Spanish armies had defeated and ransacked the empires of the Aztecs in Mexico and the Incas farther south. Shiploads of gold, silver, and gems from America had made Spain immensely wealthy, and a powerful ally for England.\n\nIn her determination to turn England Catholic again, Queen Mary (1516-1568) persecuted Protestants and had almost three hundred people burned at the stake, including the Archbishop of Canterbury. She's remembered as Bloody Mary. (photo credits 12.2)\n\nBut Mary died after only five years on the throne. Her successor \u2013 Henry's second daughter, Elizabeth \u2013 would reign for forty-five years. Although religious conflict continued through much of that time, Elizabeth was firmly Anglican, and England remained Anglican too.\n\nBy the time of Elizabeth's successor, James I, Church authorities felt there were too many different Bibles around; they wanted a new official version. Some fifty scholars worked for several years, revising and retranslating, and depending greatly on William Tyndale's work a hundred years earlier. Their goal was to produce a Bible that was beautifully written, yet accurate and easy to understand. If a passage described carpentry, they asked carpenters what words to use; if trees were the subject, they consulted gardeners.\n\nBishops of the Anglican Church disapproved of some of the attitudes expressed in various Bibles. In 1568 they published their own version, the \"Bishops' Bible,\" and made it the official Bible of the Anglican Church. (The word \"bible\" comes from Greek _biblos_ , \"scroll\" or \"book\" \u2013 also the root of bibliophile, \"book-lover.\") Here's the beginning of the Book of Genesis, opening with a decorative initial: _In the beginning God created the heaven and the earth. And the earth was without form, and was void, and darkness was upon the face of the deep, and the spirit of God moved upon the face of the waters_. In the third line, \"the\" has been shortened to _y e_. In the second-last line, \"spirit\" begins with a long form of _s_ , like an _f_ with one arm missing. (photo credits 12.3)\n\nDo you notice something funny about these personal pronouns?\n\nWhat happened to all those other forms? Why do we always use you?\n\nIn many languages, the singular form is considered informal \u2013 all right for talking to children or friends, but not for strangers or people senior to you. In Chaucer's time, the landowner called the peasant \"thou\" but the peasant respectfully called the land-owner \"you.\" (The French have a word for this; to call people _tu_ is to _tutoyer_ them.) But as the gap between classes shrank, people began calling everyone \"you\" to avoid labeling one person as inferior to another.\n\nIf the distinction seems odd, consider this: we call children and friends by their first names (\"Hey, Chris!\") but we often use last names to show respect for other people (\"Hi, Mrs. Jacobs\"). Back when rich people had lots of servants, they might call the maid \"Alice\" but they called the cook \"Mrs.\" even if she wasn't married. Servant or not, they had to treat her with respect if they valued their dinner!\n\nTry reading this speech of Ruth's out loud, listening to the flow of the language. (Ruth's husband has died, and her mother-in-law has said that Ruth should go back to her own people.)\n\nIntreat me not to leave thee, or to return from following after thee: for whither thou goest, I will go; and where thou lodgest, I will lodge: thy people shall be my people, and thy God my God....\n\n#\n\nWhen Elizabeth Tudor was still a toddler, her father, Henry VIII, had her mother's head cut off. Being Anglican, Elizabeth was lucky not to lose her own head in the years when her Catholic half-sister, Queen Mary, was trying to lead England back to the \"true\" faith.\n\nWhen the English weren't cutting off people's heads, they were hanging them, especially from the dreaded gallows on Tyburn Hill, near London. Big cranes called derricks, used to hoist cargo onto ships, take their name from a Mr. Derrick, a hangman who \"hoisted\" convicts there some four hundred years ago.\n\nWhen Mary died and Elizabeth became queen, she found her new realm in sad shape. England was weak and nearly bankrupt, torn by religious violence, riddled with crime, and threatened by richer and stronger Catholic neighbors, France and Spain.\n\nBut Elizabeth was highly intelligent, well educated, and skilled in six languages. She was shrewd and practical, and too sensible to waste her resources on war if she could possibly avoid it. During her long reign, she fended off foreign attacks by dallying with at least fifteen marriage proposals. (Even the King of Spain \u2013 Queen Mary's widower \u2013 hoped to marry Elizabeth and recover England as an ally.) Along the way, Elizabeth almost died of smallpox, and ducked endless conspiracies to exploit her, kidnap her, even murder her.\n\nThough she looks calm and confident, Elizabeth I (1533-1603) had to fight hard to keep her throne. One of the gravest dangers was her Catholic, half-French cousin, Mary, Queen of Scots (not to be confused with Elizabeth's half-sister, Bloody Mary), who kept plotting to take the throne, bring back Catholicism, and make England an ally of France. After years of protecting her troublesome cousin, Elizabeth reluctantly agreed to have Mary, Queen of Scots executed. But after Elizabeth died childless, it was Mary's son who ascended the English throne as James I \u2013 the Anglican who put his name on the King James Bible. (photo credits 13.1)\n\nTo defend the country, Elizabeth made sure the army and the great navy her father had begun were rebuilt and modernized. While Europe was suffering terrible religious wars and atrocities, she did her best to create tolerance and cooperation among the English. The resulting peace and security brought a boom in trade and industry. Elizabeth symbolized a prosperous golden age, and her subjects \u2013 many of them, at least \u2013 adored her. Gloriana, they called her. A few even built their homes in the shape of an E, to show their devotion.\n\nIn Elizabeth's day, some writers felt they had to use classical languages if they wanted their thoughts preserved for eternity (which of course they did). But others deliberately wrote in English, borrowing from Greek and Latin when necessary, to make their own language as rich and versatile as any other. Indeed, some _detested_ the remnants of Latin and Greek, and wanted them rooted out of the English language. One bishop argued that instead of calling something \"penetrable\" (from Latin _penetrare_ ), we should call it \"gothroughsome\" (from Old English _gan-thurh-sum_ ).\n\nNow that they had time and money to spend on themselves, many people turned to entertainment, especially plays. Small companies of actors sprang up. Before that, plays had been staged in any available space, but now real theaters were built, where even poor laborers could stand and watch for just a penny. (Whenever the plague returned, the theaters had to shut down for months or years, until it was over.) By 1592, one of the people working in London theater \u2013 acting, writing, perhaps directing \u2013 was William Shakespeare.\n\nShakespeare was a country boy, born in Stratford-on-Avon ( _avon_ is a Celtic word for **river** ). We don't know much about his early life. His father was a leather-worker, and later a town official. Shakespeare seems to have had a reasonably good education, although he didn't go to university.\n\nLike Chaucer, Shakespeare created lively characters of all types and classes who thought and talked just the way they should. Then he wove them into brilliant tales that felt true to life, no matter how far away or long ago they were set. Shakespeare's language is quite similar to what we speak today. There are some outdated words, and many references from that age of horse-carts and sailing ships are unfamiliar to us now. But for the most part, this is the English we know.\n\nThere were romantic comedies like _The Taming of the Shrew_ , in which Petruchio plans to win the heart of hot-tempered Kate by acting like a lunatic. (To \"rail\" is to complain angrily.)\n\nSay that she rail, why then I'll tell her plain   \nShe sings as sweetly as a nightingale.   \nSay that she frown, I'll say she looks as clear   \nAs morning roses newly washed with dew....   \nIf she do bid me pack, I'll give her thanks   \nAs though she bid me stay by her a week.\n\nThere were tragedies like _King Lear_ , in which an aged king gives his kingdom and his power to his two evil daughters and then discovers that they both despise him. He is so furious that he can barely talk:\n\nI will have such revenges on you both   \nThat all the world shall \u2013 I will do such things \u2013   \nWhat they are, yet I know not; but they shall be   \nThe terrors of the earth....\n\nThis is thought to be a portrait of Shakespeare (1564-1616), although we can't be sure. He wrote almost forty plays, and probably acted in most of them. He also wrote beautiful poems, especially love sonnets. His work is so brilliant that some people refuse to believe he wrote it; they insist that the writer must have been someone with a nobler family and a better education. (photo credits 13.2)\n\nDr. Thomas Bowdler appreciated Shakespeare's genius, but thought his writing was spoiled by indecent words, scenes, and even plots. In 1807 Bowdler published a new collection of Shakespeare, \"improving\" the parts he disapproved of. Today, bowdlerizing means rewriting someone else's work, changing the parts you find unacceptable.\n\nHistorical dramas were often designed to please and flatter Queen Elizabeth. (It was always wise to stay on her good side.) In _Richard II_ , Shakespeare tactfully supported Elizabeth's claim to the throne by giving an earlier king a speech that made him sound like a coward, a liar, and a heartless killer:\n\nIs there a murderer here? No. Yes, I am....   \nO no! Alas, I rather hate myself   \nFor hateful deeds committed by myself.   \nI am a villain. Yet I lie, I am not....\n\nWith Shakespeare and others doing so much writing, thousands of new words entered the English language. It's impossible to say how many were actually invented by Shakespeare, but as far as we know, the following (and many others) first appear in his writing: **countless** , **courtship** , **critical** , **excellent** , **frugal** , **horrid** , **leapfrog** , **lonely**.\n\n\"Striped shorts and a polka-dot T-shirt! He must pick his clothes willy-nilly.\" Willy-nilly (\"any old way\") comes from \"will I, nill I,\" meaning \"whether I'm willing or not willing.\" In _The Taming of the Shrew_ , Petruchio teases Kate that \"will you, nill you, I will marry you.\" We seem to enjoy expressions that rhyme, or at least repeat sounds. Infants learn to say bowwow, din-din, choo-choo; later we speak of a mishmash, flip-flops, hanky-panky, a hodgepodge, and so on. The parts may not make sense (what's a mish? what's panky?) but the sounds are fun.\n\nHe also put together wonderful expressions. He had such a fine ear for the perfect way to say something that we quote him more than any other author \u2013 often mindlessly, without hearing the literal meaning of the words. But take a fresh look at these Shakespearean phrases, and see how vividly an image can sum up an idea in just a few words:\n\n**barefaced  \ncrack of doom   \ndead as a doornail   \neyesore   \nflesh and blood   \nfoul play   \nlily-livered   \ntongue-tied   \ntower of strength   \nvanish into thin air**\n\nMost of Shakespeare's plots were not original; he borrowed from real life, ancient myths, and tales from all over Europe. But nobody had told these stories so vividly and powerfully. His works have been translated into innumerable languages. Scholars re-examine them obsessively, writing thousands of books about them, counting how many different words the \"Bard of Avon\" uses (as many as thirty thousand, compared to just eight thousand in the King James Bible), even counting his commas (138,198). The plays are presented over and over around the world, sometimes as movies and musicals, sometimes with a twist: Julius Caesar as a businessman in a suit and tie, Romeo and Juliet as New York teenagers separated by warring street gangs.\n\nWhat's the difference between these two sentences?\n\n**1) Alice will go to the zoo if she can**.\n\n**2) Alice would go to the zoo if she could**.\n\nDoesn't the second sentence tell you that she can't go?\n\nIn the past, different \"moods\" of verbs were used to show whether something was just doubtful (sentence 1), or actually contrary to fact (sentence 2). English has lost many of these distinctions; they were too complicated, and the moods got confused with other forms. But they can still help you make it clear that what you're describing is _not_ the way things are. The three examples below describe imaginary situations that are contrary to fact. Can you change them so they mean that you simply _don't know_ whether the situation is true or not?\n\n1) If you had five dollars he could bake you a cake.\n\n2) The warthog might still be out there, if the lion hadn't chased it away.\n\n3) If I were going I would have to take a present (but I'm staying home and keeping it for myself!)\n\nANSWERS\n\nIn _Macbeth_ , Shakespeare describes three witches as \"weird sisters.\" The witches foretell Macbeth's fate (death \u2013 though he misunderstands them), and weird comes from the Old English _wyrd_ meaning fate, \"what will be.\" But over the years, audiences watching these eerie characters have given the word a new meaning: \"strange and bizarre.\"\n\nSeven years after Shakespeare died, most of his plays were published in a collection called the First Folio. In this First Folio copy of _Macbeth_ , the spelling is not quite like ours. The letters u and v are the same ( _V_ is the capital), and the long _s_ (similar to _f)_ replaces our _s_ several times in the opening speech by one of the three weird sisters. (photo credits 13.3)\n\nYet if Shakespeare had lived before the invention of the printing press, which could turn out hundreds or thousands of scripts, all his plays and poems might easily have been lost forever.\n\n#\n\nSome of Elizabethan England's new prosperity was coming from South America, through the back door. While Spain had claimed most of that continent, English sea captains were committing outrageous acts of piracy against the Spanish galleons as they lumbered home with their loads of gold and silver and precious gems. Queen Elizabeth pretended to disapprove of the raids, but much of the loot ended up in her treasury. The bold captains were celebrated as heroes, and richly rewarded.\n\nWhat England really needed, though, was more farmland to feed its growing population. In North America, France was taking over the territory around the St. Lawrence River and the Great Lakes. Spain dominated the areas farther south and west. So the English hoped to start settlements along the east coast, between the two.\n\nIn 1607 an English colony was successfully established at Jamestown. In 1620 another group of settlers sailed from England to Cape Cod, in \"New England,\" on a little ship called the _Mayflower_. Some of them were \"Puritans\" \u2013 people who wanted to \"purify\" the English religion of its politics and its fancy rituals. In England, Puritans were being persecuted for their beliefs. Those who sailed to America, in hopes of creating a more virtuous \"Kingdom of God\" where they could live simple, pious lives, are known as the Pilgrims.\n\nFrance held much of the northeastern continent, Spain had territories in the south and west, and before long a chain of English fur-trading posts was spreading through the northwest. As European interests expanded, more and more native peoples were driven off their land. (photo credits 14.1)\n\nThe _Mayflower_ settlers suffered illness and starvation during the winter of 1620, but with help from the local native people the colony survived. After the harvest of 1621, the colonists and natives joined in a feast of thanksgiving. Their dinner probably wasn't as elegant as this romantic painting suggests, but it did include four wild turkeys, and probably cranberries too \u2013 something to think about, next Thanksgiving Day. (photo credits 14.2)\n\nWe often create a new word by naming something for the place it comes from. Those little raisins called currants were once \"raisins of Corinth\" because they came from around Corinth, in the south of Greece. The island of Cyprus, south of Turkey, gave its name to copper ( _cyprium_ is Latin for \"of Cyprus.\") Denim began as cloth _de N\u00eemes_ \u2013 cloth from N\u00eemes, a town in France.\n\nAnd then more shiploads of English-speaking settlers arrived, and they started more farms and built more towns. As they spread out across North America and down into the Caribbean, they needed words for all the unfamiliar things around them. Sometimes they stuck together words they already knew: a frog that roared like a bull became a **bullfrog** ; a snake that rattled its tail became a **rattlesnake**. Sometimes they borrowed (and mangled) names from native North American languages: _segongw_ became **skunk** ; _aroughcun_ became **raccoon** , _mus_ became **moose** , and other native words became **chipmunk** , **pecan** , **squash** , **toboggan** , and so on.\n\nFor a few decades in the 1600s, there were Dutch settlements along the east coast, around where New York is today. Some Dutch words were also adapted into English \u2013 including **cookie** ( _koekje_ , little cake), **coleslaw** ( _koolsla_ , cabbage salad), **waffle** , **sleigh** , and (since this was the time of the great \"Dutch Masters\" painters) the painting terms **landscape** and **easel**.\n\nIn 1763, a treaty ending a war in Europe gave England almost all France's land in North America, and Spain's Florida territory as well. Some eighty years later, the once-Spanish territories of California and Texas would join English-speaking America. The language of that little island on the other side of the Atlantic \u2013 the language that had muddled together the words and rules of so many invaders \u2013 was taking over this faraway continent.\n\n**Another way to create new words is to add onto old ones. A** **_prefix_** **turns think into rethink; a** **_suffix_** **turns it into think ing; with both prefix and suffix we can make unthinkable. Most prefixes and suffixes come from Latin or Greek, and they're handy to know. They can help you figure out words you've never seen before. Look at these prefixes, and see if you can answer the questions below**.\n\n  * 1)If elocution is clear speaking, what's circumlocution?\n\n  * 2) If a utopia is an ideal land, what's a dystopia?\n\n  * 3) If a tripod is a stand with three feet, what's a polypod?\n\n  * 4) If zoology is the study of animals, what kind of animal is an epizoon?\n\n  * 5) If your epiglottis is the little tab that hangs above your tongue, what's a polyglot?\n\n  * 6) If euphoria is a general sense of feeling good, what is dysphoria?\n\n  * 7) If oxygen is a gas we need to live, what's hypoxia?\n\n  * 8) If your domicile is where you live, what's a word for a building where people live together? (This is trickier!)\n\nANSWERS\n\nMany flowers are named after people who discovered or grew them: Kamel's camellia, Dahl's dahlia, Mag-nol's magnolia, Poinsett's poinsettia. Others take their names from ancient words that suggest their shape. The aster has points like a star (Greek _aster_ ); part of the delphinium suggests a dolphin (Greek _delphin_ ); a flower with a leaf like a Roman soldier's sword (Latin _gladius_ ) is a gladiolus. As for the nasturtium, it can make your nose itch ( _nasum torqueat_ is Latin for \"irritates the nose\")!\n\nBut what kind of English did the settlers speak? It depended on where they came from, and what level of society. It also depended on when they migrated, for they took with them the speech of that time. From then on, their English and Britain's English would develop in different ways. For example, the Canadian island of Newfoundland is isolated by the long winters and cold, rough waters of the North Atlantic. Fishing folk from Ireland and western England began settling there way back in Shakespeare's day. Because the island was so cut off, their speech developed in its own direction, keeping or inventing words like **brewis** for stew; **slims** for pancakes; **totties** for dandelions. They called a bumblebee a **dumbledore** , your two front teeth were your **butter teeth** , and if you weren't a **livyer** (a real Newfoundlander), you must be from **upalong** \u2013 meaning just about anyplace else in Canada or the United States.\n\nTo the Aztecs of Mexico, coyote and ocelot were _coyotl_ and _tlalocelotl_ , and tomato was _tomatl_. Our word chocolate comes from _chocolatl_ , but the hot chocolate the Aztecs brewed was dark and bitter, almost coffee-like \u2013 no milk, no sugar, no marshmallows!\n\nThe way people spoke also depended on where they moved to. Perhaps there was already a local population with a certain way of speaking. Perhaps there was a strong Spanish or French influence. Perhaps there were Africans who had been shipped in to labor as slaves in the fields and homes. All these would influence the speech of new arrivals, and their children and grandchildren. In the Caribbean particularly, the jumble of languages \u2013 from native people, early explorers, sailors, Africans, and settlers \u2013 grew (and is still growing) into versions of English that sound almost like whole new languages.\n\nFrom Africa we get banana, chimpanzee, tote, and voodoo. From the Spanish of the American West we get bonanza, ranch, rodeo, plaza, stampede, canyon, bronco, and tornado. English also borrowed words the Spanish had adopted from native people farther south: barbecue, canoe, and potato from Haiti; condor and puma from Peru; hammock, hurricane, manatee, and tobacco from the Caribs, who gave their name to the Caribbean Sea.\n\nWhat about their writing? The first colonial writers were driven by their religion, and their determination to found a better and fairer society. They wrote sermons and prayers and lectures, not thrilling dramas and romances that might waste people's time and lead them into temptation. Cotton Mather, a solemn scholar and historian, was a Puritan minister around 1700, and a grandson of two Puritan ministers who had fled England to find religious freedom. This is from his \"Christian\" account of how the colonists triumphed over the native people (or \"Salvages,\" as he calls them):\n\nThe Flame of War then Raged thro' a great part of the Country, whereby many whole Towns were Laid in Ashes, and many Lives were Sacrificed. But in little more than one years Time, the United Colonies... bravely Conquered the Salvages. The Evident Hand of Heaven appearing on the Side of a people whose Hope and Help was alone in the Almighty Lord of Hosts, Extinguished whole Nations of the Salvages at such a rate, that there can hardly any of them now be found... upon the face of the Earth.\n\nJust after Mather died in 1728, a young American named Benjamin Franklin began an extraordinary career as a writer and editor \u2013 and scientist and inventor and statesman and many other things as well. Franklin disliked pompous language \u2013 \"Here comes the Orator!\" he wrote, \"with his Flood of Words, and his Drop of Reason.\" His writing combined Puritan values like honesty and hard work with humor, common sense, and immense intelligence and learning. When he was twenty, he resolved to set out a plan for his life:\n\nI am now entering upon a new [life]: let me, therefore, make some resolutions, and form some scheme of action, that, henceforth, I may live in all respects like a rational creature.\n\nHis resolutions included being \"extremely frugal\" till his debts were paid, being truthful and sincere \"in every word and action,\" being industrious in his work, and never speaking ill of anyone, \"not even in a matter of truth.\"\n\nBen Franklin (1706-90) dropped out of school at age ten to help his father make soap and candles, became an apprentice in journalism, and then published his own magazines \u2013 including \" _Poor Richard's_ \" _Almanack_ , written under the pen name of Richard Saunders. Franklin started a circulating library, a debating club, a hospital, and an academy that grew into a university. He studied science, especially the mystery of electricity, and proved that lightning was electric. He traveled widely, learned foreign languages, worked in various levels of government, reorganized the post office, invented bifocal glasses and the Franklin woodstove, and was a key contributor to the American Declaration of Independence and the Constitution of the United States.\n\nWhile more and more settlers headed west from Britain, hoping for a better life in the New World, ships were sailing in the opposite direction for a very different reason. They carried not pilgrims but prisoners, men and women convicted of various crimes. Britain's prisons were desperately overcrowded and full of disease, and the government wanted to transport some of the inmates to a far-off land where they would make no more trouble. In 1787, eleven convict ships set out for Australia. Over the next eighty years, many more would follow.\n\nThose convicts who were lucky enough to survive the years of their sentence sometimes remained in Australia, and law-abiding settlers began moving there as well, to raise sheep for wool, or to start other businesses. Since they were even more isolated than Newfoundlanders, they developed their own words and ways of saying them; once again, the English language was shifting shape. Some words were adopted from the native Aboriginal people, such as **boomerang** and many plant and animal names: **kiwi** , **kangaroo** , **wombat** , **kookaburra**. Others were adapted or invented: **outback** (the barren interior), **bushman** (an expert traveler in the outback). With jokey suffixes, barbecue and tin can and Christmas became **barbie** and **tinnie** and **Chrissy**.\n\nIn some colonial areas, native people invented their own twist on English, known as \"pidgin.\" (The name is a mangling of \"business,\" since it lets people work together even if they all speak different languages.) Pidgin languages use a minimum number of words \u2013 some English, some not \u2013 and very basic grammar. In Tok Pisin \u2013 the pidgin talk of Papua New Guinea, north of Australia \u2013 anything large is _bigpela_ (\"big fellow\"); possession is shown by _bilong_ (\"belongs to\"); _meri_ (from \"Virgin Mary\") means \"woman,\" _na_ means \"and.\" So _gras bilong fes_ is a beard (face-grass), and a _bigpela haus buk bilong ol man na meri_ is a big house of books for all men and women \u2013 a public library. Pidgin may seem roundabout, but it covers a host of meanings as simply as possible. It's a language of its own.\n\n**Can you guess what these Tok Pisin phrases mean?  \nListen to the sounds of the words. Here are two hints:**   \n**_kaikai_** **means \"food\" and** **_pikinini_** **means \"child.\"**\n\n  * 1) Mi laikim kaikai bilong moningtaim.\n\n  * 2) Man nogut i stilim kar.\n\n  * 3) Hamas yias bilong yu?\n\n  * 4) Nek bilong mi i drai.\n\n  * 5) Mi lukim nambawan pikinini bilong Misis Kwin.\n\nANSWERS\n\n#\n\nAs England's colonies south of the Great Lakes grew larger and richer and stronger, many people there resented being ruled (and taxed!) by a small country far across the Atlantic. In the 1770s these colonies went to war, breaking away from Britain and creating an independent America. Thousands of Loyalists \u2013 people who preferred to remain British \u2013 moved north into Canada, which still belonged to Britain.\n\nNoah Webster (1758-1843) was a schoolteacher, and also a magazine publisher, a lawyer, a lecturer, and an outspoken opponent of slavery.\n\nEven after winning their independence, many Americans still resented all things English. A few thought Americans should all speak French \u2013 or even Greek \u2013 to teach England a lesson. But most people realized that it would be ridiculous to give up the language altogether. Instead, they would start speaking and writing English in their own way. They would make it simpler. More sensible. And better, of course.\n\nOne of these anti-English Americans, Noah Webster, saw the new country as a utopia of liberty and virtue. In 1789 \u2013 not long after the American states became independent \u2013 he predicted that their English would become a different language altogether. It would even use a different alphabet, with added letters and accents, to spell every word the way it sounded. **Health** , **neighbor** , and **tongue** would become **helth** , **nabor** , and **tung**. \"There iz no alternativ,\" he declared. \"A _national language_ is a brand of _national union_.\" In the name of independence and patriotism, Americans should speak American.\n\n**How good is your** **_English_** **English? Can you translate these British words into North American?**\n\n  * 1) Hold the baby for a moment, I'll get a nappy from the boot.\n\n  * 2) The way that lorry is parked, it's blocking the dustmen.\n\n  * 3) I didn't go and get the post because the lift was broken.\n\n  * 4) There's a coach up there on the flyover.\n\n  * 5) Do you want an ice lolly for afters?\n\n  * 6) I'm going to the chemist's. Play noughts and crosses till I get back.\n\nANSWERS\n\nJust a few decades earlier, some people back in England had also been dissatisfied with their language. They complained that English was not properly defined, the way Latin and Greek were. Words were used in too many different ways. Grammar and spelling were inconsistent. Meanings kept changing, and sloppy expressions were slipping in. It was time for some expert to write a rule book everyone could follow, so that English would be preserved for ever. After all, if the language kept changing, future generations wouldn't be able to read the treasures of English literature. In 1722, Jonathan Swift \u2014 who was in the midst of writing _Gulliver's Travels_ \u2013 published an impassioned letter addressed to Britain's Lord High Treasurer, asking him how any author could be expected to go on working\n\nwith Spirit and Chearfulness, when he considers, that he will be read with Pleasure but a very few Years, and in an Age or two shall hardly be understood without an Interpreter? This is like employing an excellent Statuary [sculptor] to work upon mouldring Stone.\n\nDefining a language that had grown out of so many sources, over so many centuries, was a huge challenge. But one man stepped up to it \u2013 a brilliant writer and scholar named Samuel Johnson. It was time, he said, to rid English of its impurities \u2013 its \"barbarous corruptions, licentious idioms and colloquial barbarisms.\" He and a few assistants spent nine years writing an authoritative English dictionary. Johnson came up with definitions for over forty-three thousand words, indicating how each word was said, the part of speech, and the derivation. By including one hundred and eighteen thousand quotations from well-known authors, he showed various ways the words could be used, and how their meanings had changed over the centuries. For good measure, he threw in a history of the English language, and a guide to grammar.\n\nSamuel Johnson (1708-1784) had problems with his vision and hearing, and suffered ill health for most of his life. He was generally gloomy and depressed, and often irritable, even rude. His father owned a bookshop, and Johnson grew up reading English, Latin, and Greek. He went to Oxford University for a while but couldn't afford to stay, so he dropped out to work as a bookseller, teacher, and journalist. But after his dictionary was published, the government gave him a pension, Oxford University gave him a doctorate, and he became known as \"Dictionary Johnson,\" the foremost expert on the English language. (photo credits 15.1)\n\nIn the 1700s there was a kind of writing paper that had a watermark (faint pattern) like a fool (a court jester) or a fool's cap. Three hundred years later, we still call legal-sized paper foolscap.\n\nThe result, published in 1755, was the first great English dictionary. For a hundred years it was the standard reference book that families trusted and relied on. It also gave future lexicographers (dictionary writers) a foundation to build on, to create their own dictionaries. Even today, many of Johnson's definitions appear in modern dictionaries.\n\nThere had been dictionaries before then, but most had been guidebooks to rare or difficult words. Johnson was the first lexicographer to include all kinds of day-to-day words. (He defined **lexicographer** as \"A writer of dictionaries, a harmless drudge....\") He also made some general decisions. For example, what was the right spelling of words like **color** \/ **colour** and **center** \/ **centre**? Some people argued that words from French should have French endings and words from Latin should have Latin endings. But since English came from both Latin and French, and since French itself came from Latin, who could say which words began where? Resolving to settle the matter, Johnson decided on the _our_ ( **colour** ) and _re_ ( **centre** ) endings.\n\nThe title page of Johnson's dictionary, published in 1755, \"in which the WORDS are deduced from their ORIGINALS, and ILLUSTRATED in their DIFFERENT SIGNIFICATIONS by EXAMPLES from the best WRITERS.\" But a greater dictionary was yet to come. In 1928, after seventy years of work by numerous editors and thousands of volunteers, _The Oxford English Dictionary_ was completed. It filled twelve immense volumes \u2014 \"fat, heavy, shelf-bendingly huge\" \u2014 with more than four hundred thousand words, and almost two million quotations. One of the many assistants who worked on it was Tolkien, author of _Lord of the Rings_ ; he was assigned to the letter _w_ , and wrestled with words like wallop, walrus, and wasp.\n\nAs for Noah Webster, over in America, by 1800 he had published several schoolbooks, including a wildly popular spelling book that eventually sold some _400 million_ copies. He began working on his own dictionary, using Johnson's book as a base and noting his additions and changes in the margins. He included some specifically American words, and defined an American style of spelling and pronunciation. For twenty-five years Webster labored alone, compiling entries for seventy thousand words and writing them all by hand. His dictionary, published in 1828, cut many double consonants down to singles, and used the _or_ and _er_ endings that Johnson had rejected. That's why Americans today have **jewelers** with **honor** , the British have **jewellers** with **honour** , and other English-speaking countries have to pick one side or the other.\n\n**We take dictionaries for granted these days, but stop and think for a moment. Isn't it difficult to think of a brief, clear explanation of even a basic word? How would you define** **_to walk_** **? And** **_to laugh_** **?**\n\nANSWERS\n\nDictionaries are handy for telling us what words mean, and how to spell and pronounce them. But why do we need them for short, simple words \u2013 like **run** , from Old English _rinnan_ , \"to set in motion\"? The word is easy to say and easy to explain, isn't it? Running is like walking, only faster.\n\nBut running is also what a motor does when it's working. It's what water does when it's flowing. It's what color does when it leaks out of fabric. You can run a movie or run a business or run a crook out of town. **Run** can be a noun: there are baseball runs and toboggan runs, stocking runs and runs on the stock market. And then there's the present participle, **running**. (Participles are verbs used like adjectives, to describe nouns.) There are running boards, running lights, running mates, and running stitches. If you check a dictionary, you'll see that **run** is used in a lot of other ways as well. And that's just one small word!\n\nHave you ever meant one word, and said another instead? This can happen in a moment of brain-scramble, when we're tired or not paying attention. Other times, we use a word we have heard, maybe to sound clever, and \u2013 oops! \u2013 it doesn't mean what we thought it did.\n\n**See if you can fill in some other runs**.\n\nTry a prefix in front:\n\n1) A movie you've already seen is a_________.\n\n2) Somebody faster can_________you.\n\nTry a suffix after:\n\n3) Butter left in the sun gets_________.\n\n4) A small motorboat is a_________.\n\n5) Someone who won't give you a straight answer is giving you the_________.\n\nNow try adding an adverb or a preposition:\n\n6) If you're slow, time can run_________.\n\n7) Pour too fast, and your cup will run_________.\n\n8) A car can run_________a lamp post.\n\n9) To practice a piano piece, you run_________it many times.\n\n10) A flag is run_________the flagpole.\n\n11) An unwound clock will run_________.\n\n12) The cost of something can run_________twenty dollars.\n\n13) Throw a ball, and your dog will run_________it. (Maybe.)\n\nANSWERS\n\nSpoonerisms are another mistake people fall into when they speak carelessly. Reverend Spooner, a dean at Oxford University around 1890, was famous for mixing up the beginnings of his words, saying \"a scoop of Boy Trouts\" meaning \"a troop of Boy Scouts,\" or \"a well-boiled icicle\" for \"a well-oiled bicycle.\" It's not clear how many spoonerisms really came from the reverend \u2013 mischievous students may have invented some of them \u2013 but Spooner buries the claim \u2013 sorry, carries the blame. Pea soup? Be careful how you say it!\n\nOver two hundred years ago, the Irish playwright Richard Brinsley Sheridan invented a character called Mrs. Malaprop who does exactly this. (Her name comes from French _mal \u00e0 propos_ , \"out of place.\") Straining to speak elegantly, Mrs. Malaprop says \"illegible\" when she means \"ineligible,\" confuses a \"pineapple\" with a \"pinnacle,\" and imagines a scaly \"allegory\" sunning itself on the banks of the Nile.\n\nImagine a malapropist trying to impress a high muck-a-muck: \"Oh please, your exultation, not a syllabub of apoplexy. I would never ask someone of your extinction to defenestrate himself for a person as incontinent as myself.\"\n\n1) What does this garbler mean to say?\n\n2) What is really coming out?\n\nANSWERS\n\nDo you remember Alice, who visited Wonderland and then met Humpty Dumpty in the land behind the looking-glass (mirror)?\n\n\"When _I_ use a word,\" Humpty Dumpty said in rather a scornful tone, \"it means just what I choose it to mean \u2013 neither more nor less.\"\n\n\"The question is,\" said Alice, \"whether you _can_ make words mean so many different things.\"\n\nThe more a word is misused, the more mixed up people get about what it's supposed to mean. Eventually both meanings, old and new, end up in dictionaries, because most dictionaries report what people are actually saying, even when the meaning has become muddled. And pretty soon, when you hear or read the word, you can't be sure which meaning is intended. English grows and changes over time, of course. But careless, sloppy mistakes damage the language, instead of making it richer and stronger.\n\nHere are a few examples of words being mixed up these days:\n\n**imply** (suggest) and **infer** (deduce)\n\n_\"No, I'm not implying that you stole my watch. But from the watch strap hanging out of your pocket, I infer that you stole somebody's.\"_\n\n**flout** (refuse to obey) and **flaunt** (show off)\n\n_Brittany flouted the dress code to flaunt her new miniskirt_.\n\n**fortuitous** (by chance) and **fortunate** (lucky)\n\n_Our fortuitous meeting with the karate teacher turned out to be fortunate; a few minutes later, a bully demanded our lunch money_.\n\n**literally** (actually) and **figuratively** (so to speak)\n\n_My dad says he's literally drooling for dinner. That makes him sound like the dog! I hope he means \"figuratively.\"_\n\n**problem** and **dilemma** (a choice between two options when you don't like either one)\n\n_If I tell my dad I lost my math problems, he won't let me go to the movies \u2013 but if I tell my mom, she'll make me clean my room. What a dilemma!_\n\n**disinterested** (unbiased) and **uninterested** (not interested)\n\n_We offered Ajay a dollar to act as a disinterested umpire, but he had just got his allowance so he was uninterested_.\n\n**mano a mano** (by fighting; Spanish for \"hand to hand\") and **man to man**\n\n_Phil and Franco were threatening to settle their argument mano a mano, but eventually they talked it out man to man_.\n\nEven the stress on a word \u2013 the syllable we say more loudly \u2013 can be confusing. How do you say **produce** , **rebel** , **record** , **research** , and **address**? This is a trick question, because all these words can be either a noun or a verb. In Shakespeare's day they were all stressed on the second syllable \u2013 proDUCE, reBEL \u2013 but since then, the stress on the nouns has tended to shift forward to the first syllable. A farmer proDUCEs PROduce. A REbel reBELS. You addRESS an envelope with an ADDress, preSENT a PREsent, and reCORD a REcord.\n\nOr maybe you don't \u2013 because these words are still changing, and how you say them depends on who you are and where you live. Will other words shift their stress in the same way? Will we someday misTAKE our MIStake and rePEAT a REpeat? Who knows?\n\nWith so many words to learn, how can you keep them straight? Think of Samuel Johnson and Noah Webster, and reach for a dictionary. It's not just for spelling. It's a user's guide to the English language.\n\n#\n\nBefore the 1800s, trips to distant lands were slow, uncomfortable, and often dangerous. The wealthy might travel for amusement, taking heaps of luggage and an army of servants. Some adventurers headed off alone, daring the risks and hardships and hoping for the best. But most people only left home if they had to; otherwise, they lived out their lives in the neighborhoods where they had been born.\n\nIn any case, Europe was a dangerous place for a holiday. After the French Revolution in 1789, and the beheading of the king and queen, Napoleon had taken control of France, and had crowned himself emperor. His armies were on the march as he extended his empire west into Spain, south into Italy, and east across Europe and into Russia.\n\nBut Napoleon was defeated in 1815, and much of his empire collapsed, while the British empire was flourishing. The British flag flew over territories not just in the Americas and Australia, but also in the Mediterranean, the East Indies, the Far East, Africa, and countless islands scattered around the globe. The British boasted that the sun never set on their empire; \"Our tongue is known in every clime, our flag on every sea.\"\n\nBy this time, travel was not only safer; it was easier than ever before. Thanks to the inventions of the Industrial Revolution, railways whisked passengers over iron bridges spanning deep valleys, and through tunnels blasted under mountains. Ships chugged along on steam power, no longer waiting for wind to fill their sails. By the 1850s you could even buy a package tour, with the tedious details of tickets and hotels and meals arranged in advance.\n\nTourists were drawn to Italy by ancient Roman ruins and by spectacular buildings like St. Mark's Basilica, in Venice \u2014 so grand, so elaborate, so excitingly foreign. (photo credits 16.1)\n\nTourism \u2013 especially the \"grand tour\" of major cities and antiquities \u2013 became the rage. Travelers returned home entranced by foreign styles of art, architecture, landscaping, cooking, clothing, music, and dance. Wherever the English went, their language went too \u2013 and brought home new words in its baggage.\n\nIn 1869 the Suez Canal opened, linking the Mediterranean and the Red Sea so that ships could head east without making the long, dangerous trip around the bottom of Africa. The voyage to India, once an ordeal of many months, could be done through the canal in just seventeen days. This was vital for the British, because by now much of India was part of their empire.\n\nIndia was a thrilling destination, with its elephants and tigers, its snake charmers, its elaborate temples and romantic palaces. In the evening, travelers could return to a British-style hotel, order roast beef and Yorkshire pudding, and settle down to read a London newspaper, though it might be a few weeks old. (photo credits 16.2)\n\nBack in 1600, in the days of Elizabeth I, a group of merchants had formed the East India Company, to send out great sailing ships (called East Indiamen) that would bring back exotic goods like pepper and other spices, and gems, and indigo (blue) dye. By 1700 the East India Company owned territories around India, and was shipping tea and coffee and printed fabrics. By 1800 the company had its own army and navy. British soldiers, government officials, teachers, missionaries, and merchants lived there, many with their families. Indian servants cleaned their houses, cooked their dinners, and cared for their children. Inevitably, words from Hindi and other Indian languages crept into English.\n\nOur word **guru** (teacher, expert) is a Hindi word, and **pundit** (also meaning expert) comes from _pandit_ , Hindi for scholar; Jawaharlal Nehru, one of India's prime ministers, was known as Pandit Nehru. **Bungalow** began as _bangla_ , \"Bengal-style\" \u2013 light one-story houses must have been common in the steamy Bengal region of India. **Chintz** (flowered cotton fabric) was called _chints_ in Hindi. The small rowboat we call a **dinghy** started out as a _dingi_ , **jungle** comes from _jangal_ , **thug** from _thag_. **Veranda** began as _varanda_ , **bangle** as _bangri_ , **loot** as _lut_ , and **shampoo** as _chhampo_. The British in India developed a taste for hot **curry** ( _kari_ in Tamil) and **chutney** ( _chatni_ ). Even the British nickname for their homeland \u2013 **Old Blighty** \u2013 came from Hindi; _bilayati_ meant \"foreign.\"\n\nHave you ever heard of a juggernaut? It's something immense and almost unstoppable \u2014 like an eighteen-wheeler barreling down the highway. In the Hindu religion, Jagannath is an avatar (form) of the great god Vishnu. (His name comes from Sanskrit _jagat nathas_ , \"world protector.\") Jagannath's statue is paraded through the streets of India in a heavy cart. It's said that some fervent worshippers used to hurl themselves under the cart's giant wheels. (photo credits 16.3)\n\nBy the early 1800s, English tourists were flocking to the cities and health spas of Europe's German-speaking states \u2013 Prussia, Bavaria, Austria, and many others. After all, the English and the Germans were almost cousins. Queen Victoria, who reigned from 1837 till 1901, was partly German; her husband was German; their oldest daughter married the crown prince of Germany. German literature was greatly admired, while German music \u2013 Beethoven, Schubert, Schumann, Wagner \u2013 was unparalleled. How could one be cultured without speaking at least a smattering of German?\n\nSince the English weren't the only ones bringing home new vocabulary, some words ended up with a tangled history. For example, workers in Iraq, in a part of Baghdad called _attabiy_ , used to make glossy striped silk. Italians called the cloth _tabi_. The English turned _tabi_ into tabby \u2013 and pretty soon, glossy striped cats were called tabbies.\n\nGerman looks daunting at first, partly because the words can be stuck together like beads on a string. This creates phrase-words that are useful for combining ideas that really don't fit in one word \u2013 like _Kindergarten_ , \"child-garden.\" (Unlike English nouns, German nouns still begin with capital letters.) Here are some other German constructions that have slipped into English:\n\n_Bildungsroman_ (education-novel): a novel about someone growing up. The Harry Potter books add up to a **bildungsroman**.\n\n_Schadenfreude_ (harm-joy): pleasure from someone else's misfortune; \"After Terry stole my toffee, I felt a thrill of **schadenfreude** when he pulled out one of his fillings.\"\n\n_Doppelganger_ (double-goer): a stranger who looks exactly like someone else. When a **doppelganger** appears in a story, it usually means something spooky is going to happen!\n\nWith many words run together like this, German offers less choice about the order that words are put in. Often an important word like the main verb comes right at the end of the sentence:\n\n_Der Unfall war in zwei Sekunden geschehen_.   \nThe accident had in two seconds happened.\n\nIn English we have a lot of freedom about how we arrange our words. We can choose a certain order because it makes the sense clearer, or it sounds better, or it's less boring. But it's generally wise to keep modifiers close to whatever they describe. Look what can happen when you don't:\n\n_I'm going to say I'm leaving tomorrow_. (Are you _leaving_ tomorrow? Or are you _saying_ tomorrow that you'll leave some other time?) The adverb, **tomorrow** , should be close to the verb it modifies \u2013 but is that **leaving**? Or **going to say**?\n\n_I took my puppy to the vet, who has droopy ears and a waggly tail_. Oops! The modifying clause, \"who has droopy ears and a waggly tail,\" should be close to **puppy**.\n\nWhat's a dangling participle? Remember that a participle is a form of verb that, like an adjective, modifies a noun \u2013 \"the escaped rhinoceros,\" \"the fleeing crowd.\" If the participle is too far from what it modifies, we say it's dangling; it's not attached to the correct word, so it gloms onto whatever is nearby:\n\nAfter chomping on the dragonfly, we saw the frog swallow. (Yuck! The participle, chomping, should be close to frog.)\n\nDressed in her tutu, the cat and the ballerina looked lovely. (Miaow!)\n\nSometimes, whatever the participle was supposed to describe isn't even in the sentence:\n\nGetting up next morning, the house was cold.\n\nDriving into the forest, the mountain disappeared.  \n(How far up did the house get? Does that mountain have a driver's license?)\n\nWhile some English words come from the depths of history and some are from foreign languages, others were deliberately invented. Lewis Carroll, the creator of _Alice in Wonderland_ and _Alice through the Looking-Glass_ , invented many words, including **slithy** and **mimsy**. Meaning what? \"Well,\" Humpty Dumpty explains to Alice, \"'slithy' means 'lithe and slimy'... You see it's like a portmanteau \u2013 there are two meanings packed up into one word.\" ( **Portmanteau** is an old name for a suitcase, from the French for \"carry-coat.\") \"Well then,\" he goes on, \"'mimsy' is 'flimsy and miserable' (there's another portmanteau for you.)\"\n\n**Galumph** blends **gallop** and **triumphant** (\"My dog finally found her ball, and came **galumphing** back to me.\" A guesstimate is an estimate based on guesswork, and **smash** is part **smack** and part **mash**.\n\n**Do you know these portmanteau words? If not, can you figure them out?**\n\n1) What's the name of the tunnel under the English Channel?\n\n2) What's a bit like breakfast and a bit like lunch?\n\n3) What's halfway between a snort and a chuckle?\n\n4) What kind of dog is part Labrador and part poodle?\n\n5) What's a cross between broccoli and cauliflower?\n\n6) What's partly a skirt and partly shorts?\n\n7) How do we describe a day that's gray and drizzly?\n\n8) What are we breathing when fog mixes with smoky pollution?\n\nanswer\n\nSometimes we create a language so that people _won't_ understand us. Kids try to fool their parents by speaking pig Latin \u2013 not Latin at all, but English mangled so that \"hide the book\" becomes \"ide-hay e-thay ook-bay\" (though their parents probably spoke pig Latin when they were young). Each generation invents its own slang, its own private vocabulary, especially for common meanings like **excellent** or **disgusting** or **boring**. Can you think of special words that you and your friends use? Words you don't expect to hear from your parents or grandparents? They probably had their own slang words, when they were your age.\n\n**Can you guess the meanings of these slang expressions from not so very long ago?**\n\n1) groanbox\n\n2) have a cow\n\n3) pucker paint\n\n4) make like a boid (bird)\n\n5) the bee's knees\n\n6) the cat's pajamas\n\nANSWERS\n\nAlthough most biscuits are baked only once, biscuit is derived from Latin _biscoctu_ , \"twice-cooked.\" Those long, thin Italian cookies called biscotti are baked twice \u2013 first in a loaf, and then again in slices. So is zwieback, a crisp bread baked till it's brittle all the way through \u2013 the name comes from German _zwei backen_ , \"twice baked.\"\n\nFrench street slang reverses a word's syllables, or at least its sounds. _Caf\u00e9_ becomes _f\u00e9ca_ , _bonjour_ becomes _jour-bon_ , _fou_ (\"crazy\") becomes _ouf_. _L'envers_ , meaning \"reverse,\" becomes _vers-l'en_ (the _s_ is silent), or _verlan_ , the name of this slang.\n\nWe know now that the tropical disease called malaria is spread by mosquitoes. In the past, though, people thought you caught malaria by breathing infected air \u2013 \"bad air\" \u2013 in Italian, _mal'aria_.\n\nTrickiest of all is the rhyming slang of the Cockneys (people from a region of London). It works like this: take the word you mean (such as **phone** ) and replace it with an expression that rhymes ( **dog and bone** ) \u2013 \"Quiet, I'm on the dog and bone.\" And if you want to be really mystifying, _drop the part that rhymes_. Then **phone** becomes **dog** \u2013 \"Quiet, I'm on the dog _\"_ \u2013 but unless you already know that, good luck figuring it out!\n\nFor Britons making the grand tour, Italy was also a must-see destination. Besides, the language was far easier to learn than German. Because Italian is closely related to Latin, many words resemble their English relatives.\n\nThe years of easy travel were not to last. The German states, already joined in a confederation to protect their mutual interests, were drawing closer together. By 1871, after a war with Prussia and a war with France, Germany declared itself an empire. The daughter of Queen Victoria was now, awkwardly, the Empress of Germany. Europe was seething with rivalry and resentment.\n\nIn 1914, the age of world wars began.\n\n#\n\nWar has been a frequent visitor in our history, and it too has added to the English language. The Romans had hundreds of military words \u2013 for ranks, weapons, armor, strategies \u2013 and some of them remained in Britain after Rome's armies went home. A _cohors_ was a unit of several hundred soldiers, a _legio_ was ten _cohortes, militia_ was military service. To us, **cohorts** are people working together; **legion** means \"many;\" a **militia** is a group of civilians who are trained to fight in time of war.\n\nThe ancient Greeks and Romans invented countless ways of hurling things at their enemies, including the ballista (from Greek _ballein_ , \"to throw\") \u2013 a heavy wooden contraption, similar to a crossbow, that fired a rock or spear. Since a missile (from Latin _missum_ , \"sent\") is anything thrown or shot, many early \"ballistic missiles\" were just big rocks. (photo credits 17.1)\n\nWhen William the Conqueror ferried his Norman horsemen ( _chevaliers_ , from French _cheval_ , \"horse\") across the English Channel, he brought along the language of cavalry. A **cavalcade** used to be a horse parade, but now it's a special series like \"a cavalcade of films.\" A **cavalier** was once a gallant gentleman like a knight, but today being cavalier means being rather arrogant and inconsiderate: \"I said Max could taste my birthday cake, but he's pretty cavalier to take three slices.\"\n\nBomb and bombard are \"imitative\" words; they sound like what they mean. Many imitative words begin with a sibilant sound like _s_ : screech, spit, splash, squawk, squeal. ( _Sibilus_ is Latin for \"hissing.\") There are heavy words (bump, clump, thump), swingy words (swirl, sway, sweep), glowing words (gleam, glare, glitter). We can whoop or whistle; we can whack, wham, or whomp. Other words imitate natural reactions: eek, ugh, blah, pooey, yum, _OW!_\n\nAncient sea battles depended on primitive tactics like ramming or hurling stones, trying to punch holes in the enemy's boats. But sometime in the 1300s, Europeans learned about the explosive force of gunpowder (probably from the Arabs, though it was first invented in China). When the fleet of the Spanish Armada set sail in 1588, it was bristling with **cannons** (from Latin _canna_ , \"tube\"), and the goal was to blast England's ships to smithereens. With gunpowder came words like **detonate** , \"explode loudly\" (related to Latin _tonare_ , \"thunder\"), and **bomb** and **bombard** (from Greek _bombos_ , \"booming\"). Originally, a bombard was a kind of cannon. Today it's a persistent attack, but not necessarily with guns \u2013 \"That new kid bombarded me with questions.\"\n\nBy the time of the twentieth century and the two world wars, combat had spread up into the skies, and down into the ocean. Once again, new words were needed. Some came from German: **snorkel** from _Schnorchel_ , the tube that lets a submarine breathe without surfacing; **blitz** from _Blitzkrieg_ , \"lightning war.\" Today a blitz can be as trivial as an advertising campaign or a football play; in World War II it meant a campaign of sudden, deadly attacks.\n\nOther military words came from Greek: **helicopter** from _helikos pteron_ , \"whirling wing,\" and **periscope** \u2013 a tube that lets a submarine crew see without surfacing \u2013 from _peri skopein_ , \"around-look.\"\n\nSome complex inventions just couldn't be defined in one word. Instead of sticking words together, as the Germans did, anglophones turned to acronyms, made from the beginnings of words. **Radar** detects objects like planes or ships, and determines their distance (range), by bouncing radio waves off them; the name comes from \"radio detecting and ranging.\" **Sonar** locates underwater objects by using sound waves: \"sound navigation and ranging.\" We say **flak** meaning angry complaints (\"Tanesha got a lot of flak for bringing her pet porcupine to school.\") but it began as a wartime acronym from the German for \"anti-aircraft gun,\" _Fl iegerabwehrkanone_ (flight-defence-cannon).\n\n**Acronyms** are pronounced as words (NATO, UNICEF), unlike **initialisms** , which are spelled out (FBI, DVD). (\"Acronym\" comes from Greek _akros onoma_ , \"at the top [of the] name.\") Acronyms and initialisms are both handy shortcuts, but it's easy to forget what they stand for. Do you know the words behind these?\n\n  * 1) OPEC\n\n  * 2) AWOL\n\n  * 3) navy SEAL\n\n  * 4) op-ed page\n\n  * 5) scuba\n\n  * 6) RSVP\n\n  * 7) a.m.\n\n  * 8) RIP\n\nANSWERS\n\nIn the 1900s, anti-Semitism drove many Jewish people out of Europe, to England and other anglophone lands. Most spoke Yiddish as their everyday language, and Yiddish words spread into English. Yiddish \u2013 the name comes from German _judisch_ , \"Jewish\" \u2013 is similar to German but it's written in Hebrew characters. (Hebrew, Israel's official language, is used for religion and scholarship.)\n\nMany people think SOS \u2013 the Morse code message sent by telegraph when a ship is in desperate trouble \u2013 means \"save our souls.\" In fact, it was chosen because in Morse code, S-O-S \u2013 _di-di-dit, dah-dah-dah, di-di-dit_ \u2013 is easy to remember and recognize. When ships communicate in English, the emergency message is mayday, mayday, from French _m'aidez, m'aidez_ , meaning \"help me, help me.\"\n\nBy the 1950s and the Cold War \u2013 the armed standoff between the Communist dictatorships of the east and the democracies of the west \u2013 the race for power reached out into space, as both sides built rockets and satellites. Along with the space race came computer technology, with words for things that are unimaginably tiny and fast. Science in general was booming, coming up with new products, new processes, new ideas \u2013 and words and more words. Back in the early 1900s, about a thousand new words entered English every year. By the end of that century, the number added each year was close to _twenty thousand_.\n\nA nanosecond \u2013 a billionth of a second \u2013 takes its name from Greek _nanos_ , \"dwarf.\" A googol is the number you get when you put a hundred zeroes after \"1,\" and it was apparently named by the nine-year-old nephew of a mathematician. He defined an even bigger number, a googolplex, as \"one, followed by writing zeroes until you get tired\" \u2013 so many zeroes that if you turned the whole universe into paper and ink, you still couldn't write them all down.\n\nBecause scientists cooperate internationally, it makes sense to derive new words from Greek and Latin, so they're familiar in many languages. Besides, classical words often sound so grand: **stratosphere** and **troposphere** , **intermolecular** and **intergalactic** , **supersonic** and **supernova**. Need a name for a scientist who studies fossils of prehistoric spiders? Reach for the Greek: _palaios_ (\"ancient\"), _arakhne_ (\"spider\"), and _logos_ (\"word, talk, discussion\") add up to **paleoarachnologist**.\n\nStill, many of our technical terms are old-fashioned English words with new meanings. Look at computer vocabulary: we **boot** up, use a **mouse** and a **menu** , put up a **firewall** so nobody can **hack** into the system, track down a **bug** , **surf** the **net** , set a **bookmark** , and delete any **cookies**. We call one of the great mysteries of space a **black hole** , and dream of finding a shortcut across the universe through a **wormhole** \u2013 using words that date back to King Alfred and beyond: _wyrm, blaec, hol_.\n\nBig words, small words, new words, old words, Anglo-Saxon words, imported words \u2013 the richness of the English language gives us so many ways to say pretty much the same thing. How do you go about speaking and writing well? Here's a good place to start: _Think about exactly what you mean, and then say it, simply and clearly_.\n\nAs the King of Hearts explains to the White Rabbit in _Alice in Wonderland:_\n\nBegin at the beginning, and go on till you come to the end; then stop.\n\n**The Greek word for \"far away\" is** **_tele_**. **What invention lets you**\n\n  * 1) have a closer look at the moon\n\n  * 2) watch a baseball game in another city\n\n  * 3) talk to a friend in another country\n\n  * 4) use a distant computer\n\n  * 5) photograph a woodpecker on a faraway tree\n\n  * 6) beam yourself up to the spaceship (at least in movies!)\n\nANSWERS\n\n#\n\nClose to four hundred million people speak English as their native language. As many again learned it as a second language. Another seven or eight hundred million have studied the language. All in all, perhaps a quarter of the people on earth have some ability to speak and understand English.\n\nAnd many more would like to. After all, English opens the door to travel, to the Internet, to everything from leading-edge science to the latest Hollywood blockbusters.\n\nWhat will happen to it in the years to come?\n\nWill English continue to grow and spread, displacing all other languages, as people around the globe become more tightly linked by instant communications? Will other languages wither away as nations become more dependent on this common speech of technology and trade?\n\nOr will English fragment into many different languages, the way Latin split into Italian and Spanish and French? Is this happening already?\n\nConsider the Caribbean island of Jamaica, colonized first by Spain and later by England. Around 1640, the first shiploads of Africans were taken there to work as slaves in the sugar plantations. Over the centuries, a Jamaican form of English has developed. The words are mostly English but they are said differently, and linked by African-style grammar. Some words are specifically Jamaican \u2013 _winjy_ for \"sickly,\" _duppy_ for \"ghost,\" _braata_ for \"a little extra,\" _bangarang_ for \"a big fuss,\" _peeny wally_ for \"firefly.\" The result is almost incomprehensible to outsiders. In colonial days, this local speech was dismissed as ignorant and inferior. But today Jamaica is an independent country, and many people say this is their true national language. Will other new languages grow out of English in the same way? Will they eventually replace it?\n\nPerhaps the upset will be more dramatic. Perhaps English will be swept aside by the preferred language of some new superpower. Will it be Chinese, already spoken by a billion or so people? Or Spanish, the language of hundreds of millions in Europe and the Americas? Or will it be some other language that we can't yet imagine being a competitor? Will the world's anglophones scramble to master this vital new language, and end up speaking English only to their children and their dogs?\n\nThe growth of English has depended on so many twists of fate: Rome's invincible armies, the rise of Christianity, the Vikings in their dragon boats, the shocking victory of William and his Norman cavalry, the Black Death, the religious conflict that drove the Pilgrims to North America. Who knows what events will shape it in the future? Or what triumphs or catastrophes will drive other languages?\n\nBut for the past thousand years and more, through conquest and through defeat, English has grown amazingly strong and subtle and versatile. The power of the language, and our ability to use it, affects our whole lives. Language \u2013 any language \u2013 is not just a tool for getting through the day, for chatting and arguing and reading the TV listings. It's the foundation of society, the bond between us that lets us work together and cooperate and share. \"Speech is civilization itself,\" said a German novelist. \"The word, even the most contradictory word, preserves contact.\"\n\nThe way we speak shapes the very way we think. \"Language is the dress of thought,\" said Samuel Johnson. \"Words are absolutely necessary for thinking,\" noted another writer, \"and with a minimum of words there is a minimum of thought.\"\n\nOur speech also determines our ability to change the thinking of others. Whatever we have in our hearts and minds, we need language to express it. \"Axes and atom bombs can change the material world,\" declared a British scientist, \"but only language can alter human ideas.\"\n\nBut let's leave the last word to _My Fair Lady_ 's cranky Professor Higgins, as he urges Eliza the flowerseller to speak English well, to honor its beauty and power and its wealth of history:\n\nRemember that you are a human being with a soul and the divine gift of articulate speech; that your native language is the language of Shakespeare and Milton and the Bible; and don't sit there crooning like a bilious pigeon.\n\n(photo credits 18.1)\n\n# TIMELINE\n\n_(most dates are approximate)_\n\n**BCE** |   \n---|---  \n**2500s** | early Britons build Stonehenge  \n**700s** | Celts begin migrating from Europe to England  \n**55** | Julius Caesar leads first of two expeditions to Britain  \n**CE** |   \n**43** | Romans invade Britain, bringing Latin language  \n**400s** | Romans withdraw from Britain; Angles, Saxons, and others invade, bringing Germanic languages  \n**600s** | Anglo-Saxons are spread across much of Britain  \n**800** | Vikings (Norse, Danes) begin raiding Britain and France  \n**865** | Vikings begin settling in England  \n**871** | Alfred the Great becomes king, unites Anglo-Saxon England  \n**911** | Vikings in France win their own dukedom, Normandy  \n**1000** | earliest surviving copy of _Beowulf_ is written  \n**1066** | Normans conquer England, bringing Norman French  \n**1340** | birth of Chaucer (d. 1400)  \n**1348** | Black Death sweeps Britain (and returns later)  \n**1384** | Wycliffe's hand-written English-language Bible is published  \n**1400s** | English is again the official language of England  \n**1476** | Caxton sets up England's first printing press  \n**1525** | Tyndale publishes New Testament in English  \n**1564** | birth of Shakespeare (d. 1616)  \n**1600s** | England begins gaining power in India  \n**1607** | Jamestown, first permanent English settlement in America, is founded  \n**1755** | Johnson publishes his dictionary in England  \n**1776** | thirteen American colonies rebel against England  \n**1787** | first convict ships sail to Australia  \n**1800s** | British Empire extends around the world  \n**1828** | Webster publishes his dictionary in America\n\n\"Let him read a book.\"\n\n## MORE ABOUT SANSKRIT\n\nSanskrit is the classical language of ancient India. An early version was spoken some thirty-five hundred years ago; long before Indo-Europeans learned to write, Hindu religious texts called the Veda were recited and sung from memory. By 400 bce, classical Sanskrit was the language of Hindu culture and romantic literature, including lengthy poems about love and adventure, and fables and fairy tales. ( **Sanskrit** comes from _samskrta_ , \"perfected.\" _Veda_ is Sanskrit for \"knowledge,\" from the same root as Old English _witan_ , \"know.\")\n\nSanskrit uses a much larger alphabet than English, and a lot of special marks to show how words are pronounced. The language is complex and highly inflected (words change depending on how they're used). It has eight cases for nouns, three genders (masculine, feminine, and neuter), and three numbers (not only singular and plural but also dual, for talking about just two things or two people).\n\nToday a simpler language, Hindi \u2013 which comes from Sanskrit \u2013 is India's official language. Sanskrit is used mostly in the temples. It's often written in a Hindi alphabet called Devanagari, in which the letters of a word are spelled from left to right, as in English, but joined together. Sanskrit numbers look similar to ours:   but the words for **one two three** , pronounced _eka, dvi, tri_ , are written:   .\n\n\"The art of speaking enchants the soul.\"\n\n## MORE ABOUT GREEK\n\nLike Sanskrit, Greek (or something close to it) was spoken at least thirty-five hundred years ago. Greek writing probably developed about five hundred years later. Again like Sanskrit, classical Greek is highly inflected, with words changing according to how they're used. But the Greek uses a much simpler alphabet of about twenty-four letters (a few letters came and went over the years). In fact, the Greek alphabet is much like ours, although some of the letters look a little different.\n\nHere are some Greek words from chapter 4, in English spelling. Can you match up the English spelling with the Greek? (Hint: the letter _e_ has two forms, depending on how it's said.)\n\nGreek does have some tricky bits \u2013 _ph_ and _th_ are combined into single letters, the letter _h_ is shown by punctuation ('\u0454V, above, is pronounced \"hen,\") the _r_ looks like our _p_ , the _p_ looks like a little table, and so on \u2013 but it doesn't take long to learn the alphabet. Here are some harder words from chapter 3 \u2013 can you match them up?\n\nThe ancient Greeks also had capital letters, though they're a bit harder to recognize. As well, they used their letters in their numbering system. Here's the whole alphabet.\n\nModern Greek looks very similar to the Greek of classical times, but the pronunciation has changed, as it has in English.\n\nIn the old days, Greek could be written left to right, right to left, or weaving back and forth in a style called \"boustrophedon\" \u2013 meaning the way an ox ( _bous_ , or ) plows a field, from one side to the other and then back again:\n\nan ox plows a field from   \nrehto eht ot edis eno   \nand then back again\n**VIRTUS MILITUM FUIT ADMODUM LAUDANDA**|    \n---|---  \n|\n\n\"The excellence of our soldiers was most laudable.\"\n\n## MORE ABOUT LATIN\n\nAbout twenty-five hundred years ago, Latini people lived around Roma, in Italy. As these Romans conquered their neighbors, their language, Latin, spread through Italy and beyond. But there were two versions of Latin \u2013 the classical kind, for writing and fine speeches, and a simpler \"vulgar\" kind (from Latin _vulgus_ , \"common people\"), which most people spoke. Sometimes the two were quite different. For example, **cat** was _feles_ in classical Latin but _cattus_ in common Latin; borrowing from both versions, we can talk about either \"cats\" or \"felines.\" Over time, common Latin disintegrated into Romance languages like French and Spanish. Classical Latin was preserved in the works of great writers across Europe, although it hasn't been anyone's mother tongue for over a thousand years.\n\nThe Romans used an alphabet of twenty-three capital letters (Latin was always written in capitals at that time):   \n**A B C D E F G H I K L M N O P Q R S T V X Y Z**\n\nThey had no _J_ , which is a variety of _I_ \u2013 our word \"eject\" comes from _i actare_, \"to throw.\" They had no _W_ , and _U_ and _V_ were the same letter (notice that our letter shaped like two _V_ s is called \"double- _U_ ,\" not \"double- _V_ \").\n\nRoman Latin was highly inflected, but it didn't have as many versions of nouns as Greek or Sanskrit. There were no dual forms, for one thing \u2013 just singular and plural. But some verbs changed dramatically depending on how they were used, and studying Latin means memorizing four \"principal parts\" for every verb. For the Latin word meaning \"give,\" those parts are _do_ , _dare_ , _dedi_ , _datus_ :\n\nI give  | _do_ (the root of English **donate** )  \n---|---  \nto give  | _dare_  \nI have given  | _dedi_  \nhaving been given  | _datus_ (the root of English **date** )\n\nSo English \"donate\" and \"date\" come from Latin _dare_ , and look different because the verb is inflected \u2013 yet our verb \"to dare\" doesn't come from _dare_ at all, but from an Old English word, _durran_.\n\n\"There will be more wisdom, the more languages we know.\"\n\n## MORE ABOUT OLD ENGLISH\n\nEnglish evolved in a long, slow process, but for convenience we give the name \"Old English\" to the language spoken from early Anglo-Saxon times up to about 1150. This is the language of _Beowulf_ and Caedmon and Alfred the Great. Some of it sounds a bit like English, if we read it out loud, but the writing looks foreign to our eyes, as you can see in chapter 8.\n\nBy 1150, English had expanded and borrowed a lot from Norman French. The language from 1150 until about 1500 is called Middle English, and this is what Chaucer used \u2013 more like the language we know, but still not easy to read.\n\nWhen printed books appeared around 1500, they helped pull the language together into Modern English \u2013 the language of Shakespeare's plays and the King James Bible, which are readable and familiar even though they were written hundreds of years ago.\n\nThe earliest Anglo-Saxon writings we have are in futhark, an alphabet of signs called \"runes.\" The runes are made almost entirely of straight lines, and futhark was probably used mostly for carving short inscriptions \u2013perhaps a few words on a coin or a stone, or the blade of a sword. (The name comes from the first six letters \u2013 _th_ is a single letter \u2013 just as our alphabet is named for alpha, beta.) Almost all manuscripts written in Old English use a later alphabet. It's much like the Roman alphabet, with a few letters missing and a few extras added. _A_ and _E_ could be written together _(AE)_ in a sign called a ligature, to mean a sound like the _a_ in \"cat.\"   (eth) and \u00fe (thorn) stood for the _th_ sound in \"cloth\" or \"clothes.\"   (wynn) is confusingly similar to thorn, but it represents a _w_ sound. The alphabet looked something like this:\n\nAs time went by, spellings gradually changed: _ac_ and _bat_ became **oak** and **boat** , _scip_ became **ship** , _ecg_ became **edge**. The letters _h_ and _w_ might change places _(hwael_ became **whale** , _hwil_ became **while** ); an _h_ before a consonant might be dropped _(hrof became_ **roof** , _hlot_ became **lot** ); and so on.\n\nWhile the verbs could still be complicated, Old English was less inflected than Sanskrit or Greek or Latin. Nouns had only four cases, though they still came in singular and plural, and masculine, feminine, and neuter. As time went by, English-speakers were more and more likely to use prepositions to show exactly what they meant \u2013 \"to my friend, from my friend\" \u2013 instead of changing the word endings.\n\nIn grammar as well as vocabulary, English was growing close to the language we know today.\n\n# SOURCE NOTES\n\nPublisher and date are given where a book is first referred to, unless the book is a classic or is included in the Selected Bibliography. All Shakespeare quotations are from two volumes of _The Complete Pelican Shakespeare: The Comedies and the Romances_ and _The Tragedies_ (London: Penguin, 1969). I have relied extensively on the _Oxford Dictionary of English Etymology_ , the _Canadian Oxford Dictionary_ , and _The Oxford Companion to English Literature_.\n\nThe early history of Britain and the English language is still in dispute, and sources often differ. I have tried to speak generally where details are contested.\n\nThe epigraph is from _Thorndale_.\n\n**Introduction: The Puzzle of English**   \nBilbo's speech is from Chapter 1 of _The Hobbit, or There and Back Again_ , by J.R.R. Tolkien (London: Unwin, 1970). Bryson notes in _The Mother Tongue_ that the number of English words depends on whether **mouse** and **mice** are two words, whether computer **mouse** and field **mouse** are one word, and so on. The homophonic spelling quizzes are based on _The Story of English_ , by McCrum, Cran, and MacNeil, and _Alpha Beta_ , by John Man.\n\n**Chapter 1 The Mother Tongue**   \nThe table of numbers is drawn partly from _Inventing English_ , by Lerer.\n\n**Chapter 2 The Glory That Was Greece**   \nThanks to Roberta Shaw at the Royal Ontario Museum for correcting the caption about the priest Khenu's tomb at Sakkara, Egypt. The Plato quotes are from _Phaedrus_ , sec. 271, and _Apologia of Socrates_ , sec. 18, respectively.\n\n**Chapter 5 Amicus, Amice, Amicum**   \nThe magic spells are from chapter 12 of _Harry Potter and the Prisoner of Azkaban_ , by J.K. Rowling (Vancouver: Raincoast, 1999).\n\n**Chapter 6 Angles and Saxons and Vikings**   \nThe quotes \"ivories and jewelled crucifixes\" and \"'From the fury'\" are from _Set in a Silver Sea_ , by Bryant.\n\n**Chapter 7 Alfred and the Vikings**   \n\"There will be more wisdom\" is from the prose preface of King Alfred's translation of Pope Gregory the Great's _Pastoral Care_. The comparison of English and Norse words is from _The Year 1000_ , by Lacey and Danziger.\n\n**Chapter 8 Riddles, Hymns, and Tales of Battle**   \nThe riddle-verses are numbers 23 and 65 in _The Exeter Book Riddles_ , translated by Crossley-Holland. The _Beowulf_ translation has been tweaked to emphasize resemblances to English words, for the sake of comprehension. (The meaning of _fagne_ is closer to \"decorated\" \u2013 in this context, probably with blood!) The quote for the year 871 in _The Anglo-Saxon Chronicles_ is based on a translation by Professor Harvey De Roo.\n\n**Chapter 9 The Defeat of the English**   \nThe thesaurus referred to is _Thesaurus of English Words and Phrases_ (revised), by Peter Mark Roget (London: Longman's, 1959).\n\n**Chapter 10 Tales of Cnihts and Fair Ladies**   \nThe story of kohl is drawn from _The Meaning of Tingo_ , by Jacot de Boinod. The quote \"high and low\" is from Professor Nevill Coghill, cited in _English Life in Chaucer's Day_ , by Hart. The quote \"cursed and complained\" is from _The Meaning of Everything_ , by Winchester.\n\n**Chapter 11 How to Spell It? How to Say It?**   \nThe remarks on Caxton, and the Caxton quote, are drawn from _The Story of English_ , by McCrum, Cran, and MacNeil. _The Mother Tongue_ , by Bryson, attributes Caxton's story to Caxton's preface to _Eneydos_ (1490), and is also the source of the comment on _Grammatica Linguae Anglicanae_.\n\n**Chapter 12 Making the \"Good Book\" Better**   \nThe Ruth quote is from Ruth, 1:16.\n\n**Chapter 13 Gloriana and the Bard**   \nThe bishop advocating \"gothroughsome\" was Reginald Pecock. Some details about Shakespeare are drawn from _Shakespeare_ , by Bryson. Petruchio's words are from Act II Scene i. Lear's words are from Act II Scene iv. Richard's words are from Act V Scene iii. References to _Macbeth'_ s \"weird\" witches are in Act II Scene i, Act III Scene i, and Act IV Scene i. The number of different words used by Shakespeare is disputed.\n\n**Chapter 14 Greener Pastures**   \nNative American words could be similar across several languages, so it's often not clear which version was adapted into English. Much of the Newfoundland section is from _The Dictionary of Newfoundland English_ , edited by G.M. Story and others (Toronto: University of Toronto Press, 1982). The Mather quote is from _Decennium Luctuosum_ , cited in _Narratives of the Indian Wars_ , 1675-1699, edited by Charles H. Lincoln (New York: Barnes & Noble, 1913). All Franklin quotes are from _Benjamin Franklin: A Biography in His Own Words_ , edited by Joseph L. Gardner, in the Founding Fathers series (New York: Newsweek\/ Harper & Row, 1972). Some of the Tok Pisin is drawn from _Pidgin Phrasebook: Pidgin Languages of Oceania_ , by Trevor Balzer and others (Hawthorn, Australia: Lonely Planet, 1999). Tok Pisin is now gaining a place as a first language, becoming a creole rather than a pidgin language.\n\n**Chapter 15 The War between English and English**   \nThe Webster quotes, \"There iz no alternativ,\" from an essay, and \"A _national_ language,\" from a 1786 letter, are cited in _The Long Journey of Noah Webster_ , by Rollins. The Swift quote is from _A Proposal for Correcting, Improving and Ascertaining the English Tongue_..., addressed to the Earl of Oxford and Mortimer (London: Benj. Tooke, 1712). The Johnson quote, \"barbarous corruptions,...\" is cited in _Blooming English_ , by Burridge. Some remarks on the _Oxford English Dictionary_ , including the quote \"fat, heavy,\" are from _The Meaning of Everything_ , by Winchester. The Humpty Dumpty quote is from chapter 6 of _Alice through the Looking-Glass_ , by Lewis Carroll.\n\n**chapter 16 Britannia's Heyday**   \n\"Our tongue is known\" is from \"Old England Is Our Home,\" by Mary Howitt. The Humpty Dumpty quote is from chapter 6 of _Alice through the Looking-Glass_. The example of rhyming slang is from \"The New Pig's Ears,\" by Soraya Roberts, in the _National Post_ , January 4, 2007.\n\n**Chapter 17 Words, Words, Words**   \nThe statistics on new words per year are extrapolated from _The New York Times_ , April 3, 1989, cited in _The Mother Tongue_ , by Bryson. The origin of \"googol\" and \"googolplex\" is from Wikipedia; Milton Sirotta was the nephew of Edward Kasner. The King of Hearts quote is from chapter 12 of _Alice's Adventures in Wonderland_ , by Lewis Carroll.\n\n**Chapter 18 Looking Forward**   \nRemarks on the dominance of English are based partly on _Empires of the Word_ , by Ostler. Much of the Jamaican material is drawn from _The Story of English_ , by McCrum, Cran, and MacNeil. The Johnson quote is from his _Lives of the Poets: Cowper_. The quote from German novelist Thomas Mann is cited in _The Miracle of Language_ , by Lederer, as are those from \"another writer,\" Aubrey A. Douglas, and British scientist H.C. Lonquet-Higgins. Professor Higgins's speech is from Act I of _Pygmalion_ , by George Bernard Shaw, and is also in the film _My Fair Lady_.\n\n**More about Sanskrit, Greek, Latin, Old English**   \nAll these languages evolved over many centuries, across their various regions. The Julius Caesar quote is from _Commentaries of Caesar on the Gallic War_ , Book V (New York: David McKay, 1963), author's translation.\n\n# SELECTED BIBLIOGRAPHY\n\n* suggested reading\n\nAbrams, M.H. (editor), and others. _The Norton Anthology of English Literature_ , Volume 1, fourth edition. New York: Norton, 1962.\n\nBarber, Katherine, editor. _Canadian Oxford Dictionary_ , second edition. Don Mills, Ontario: Oxford University Press, 2004.\n\n*Branigan, Keith. _Roman Britain: Life in an Imperial Province_. London: Reader's Digest Association, 1980.\n\nBryant, Arthur. A History of Britain and the British People, Volume 1: _Set in a Silver Sea_. London: Grafton, 1985.\n\n____. A History of Britain and the British People, Volume 2: _Freedom's Own Island_. London: Grafton, 1987.\n\nBryson, Bill. _The Mother Tongue: English and How It Got That Way_. New York: William Morrow, 1990.\n\n____. _Shakespeare: The World As Stage_. (Eminent Lives.) New York: HarperCollins, 2007.\n\nBurridge, Kate. _Blooming English_. Cambridge: Cambridge University Press, 2004.\n\nChickering, Howell D., Jr., translator. _Beowulf: A Dual-Language Edition_. New York: Doubleday, 1977.\n\nCrossley-Holland, Kevin, translator. _The Exeter Book Riddles_ (revised). London: Penguin, 1993.\n\nDrabble, Margaret. _The Oxford Companion to English Literature_ , fifth edition. Oxford: Oxford University Press, 1985.\n\nFatsis, Stefan. _Word Freak: Heartbreak, Triumph, Genius, amd Obsession in the World of Competitive Scrabble Players_. New York: Penguin, 2002.\n\nGaur, Albertine. _A History of Writing_. New York: Cross River Press, 1992.\n\n*Hart, Roger. _English Life in Chaucer's Day_. London: Wayland, 1973.\n\nHitchings, Henry. _The Secret Life of Words: How English Became English_. New York: Farrar, Straus and Giroux, 2008.\n\n*Humez, Alexander and Nicholas. _Alpha and Omega: The Life and Times of the Greek Alphabet_. London: Kudos and Godine, 1983.\n\n*____. _Latin for People: Latina pro Populo_. Boston: Little, Brown, 1976.\n\n*Jacot de Boinod, Adam. _The Meaning of Tingo and Other Extraordinary Words from around the World_. New York: Penguin Press, 2006.\n\nKurlansky, Mark. _Salt: A World History_. Toronto: Knopf, 2002.\n\n*Lacey, Robert, and Danny Danziger. _The Year 1000: What Life Was Like at the Turn of the First Millennium_. New York: Little, Brown, 1999.\n\nLass, Abraham H., David Kiremidjian, Ruth M. Goldstein. _Dictionary of Allusions_. London: Sphere, 1989.\n\n*Lederer, Richard. _The Miracle of Language_. New York: Pocket Books, 1991.\n\nLerer, Seth. _Inventing English: A Portable History of the Language_. New York: Columbia University Press, 2007.\n\n*McCrum, Robert, William Cran, Robert MacNeil. _The Story of English_ (revised edition). London: Faber and Faber, 1992.\n\n*Man, John. _Alpha Beta: How Our Alphabet Changed the Western World_. London: Headline, 2000.\n\n*Manguel, Alberto. _A History of Reading_. Toronto: Knopf, 1996.\n\nMichaelson, O.V. _Words at Play: Quips, Quirks & Oddities_. New York: Sterling, 1997.\n\nMitchell, Bruce, and Fred C. Robinson. _A Guide to Old English_. Toronto: University of Toronto Press, 1982.\n\nOnions, C.T., editor. _Oxford Dictionary of English Etymology_. Oxford: Oxford University Press, 1996.\n\nOstler, Nicholas. _Empires of the Word: A Language History of the World_. New York: HarperCollins, 2005.\n\nRollins, Richard M. _The Long Journey of Noah Webster_. Philadelphia: University of Pennsylvania Press, 1980.\n\n*Ross, Val. _You Can't Read This: Forbidden Books, Lost Writing, Mistranslations and Codes_. Toronto: Tundra, 2006.\n\n*Savage, Anne (translator). _The Anglo-Saxon Chronicles: The Authentic Voices of England, from the Time of Julius Caesar to the Coronation of Henry II_. Godalming, UK: CLB, 1997.\n\n*Winchester, Simon. _The Meaning of Everything: The Story of the Oxford English Dictionary_. Oxford: Oxford University Press, 2003.\n\n# PICTURE SOURCES\n\nEvery reasonable effort has been made to trace the ownership of copyright materials. Any information allowing the publisher to correct a reference or credit in future will be welcomed. Pictures not attributed are from the author's collection.\n\n**MVM** : original art by Margrieta Vandelis-Muir  \n **TRL** : courtesy of the Toronto Reference Library\n\n2.1, MVM; 2.2, photo courtesy of W. Peter Gorrell; 2.3, MVM; 2.4, TRL; 3.1, 4.1, 4.2, MVM; 4.3, TRL; 4.4, 5.1, 6.1, 6.2, MVM; 6.3, TRL; 6.4, MVM; 7.1, 8.1, 8.2, TRL; 9.1, MVM; 9.2, TRL; 9.3, MVM; 10.1, 10.2, 10.3, 11.4, 11.5, TRL; 12.1, MVM; 12.2, TRL; 12.3, by kind permission of David Betts; 13.1, 13.2, 13.3, TRL; 14.1 top, MVM; 14.2 bottom, Library of Congress, LC-USZ62-15195; 15.1 left, MVM; 16.1, by kind permission of the artist, Thomas E. Brown; 16.2, 16.3, photos courtesy of W. Peter Gorrell; 17.1, 18.1, MVM.\n\n# ACKNOWLEDGMENTS\n\nMy thanks to Kathy Lowinger, publisher of Tundra Books, and Carolyn Jackson, my editor, for their patience and fortitude; to Professor Harvey De Roo for all his help and enlightenment regarding the Old English material; and to John Elmslie of the Toronto Reference Library and Susan Meisner for their special expertise. I am indebted to Margrieta Vandelis-Muir for the time and care she put into the maps and other original artwork. Professor Andy Orchard, of the Centre for Medieval Studies at the University of Toronto, very generously read the manuscript and pointed out my lapses into ignorance; any remaining errors are, of course, entirely my own fault.\n\n\u2013 G.K.G.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nEditor\n\nMaura J. Mills\n\nGender and the Work-Family ExperienceAn Intersection of Two Domains\n\nEditor\n\nMaura J. Mills\n\nDepartments of Psychology and Management, Hofstra University, Hempstead, New York, USA\n\nISBN 978-3-319-08890-7e-ISBN 978-3-319-08891-4\n\nDOI 10.1007\/978-3-319-08891-4\n\nSpringer Cham Heidelberg New York Dordrecht London\n\nLibrary of Congress Control Number: 2014947532\n\n\u00a9 Springer International Publishing Switzerland 2015\n\nThis work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher's location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.\n\nThe use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.\n\nWhile the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.\n\nPrinted on acid-free paper\n\nSpringer is part of Springer Science+Business Media (www.springer.com)\n\nFor my parents, Chris and Ellen, and for my husband Matthew, each of whom have shown me different aspects of the work\u2013family interface throughout varying phases of my life thus far. And most especially for David, my beautiful son \u2014 my biggest work\u2013family challenge to date, but also my most rewarding.\n\nForeword\n\nEllen Ernst Kossek, PhD\n\nIt is an exciting time for work and family research! The field has been exploding with growing numbers of scholars identifying themselves as work\u2013family researchers. Unlike previous decades, work\u2013family conflict and positive work\u2013life spillover are now core constructs used across many fields of research. A recent Google search in April 2014 of the terms \"work family\" yielded 2 billion six hundred and thirty million results! Growing appreciation of the importance for work\u2013life well-being is recognized and valued in many societies around the globe. Many countries around the world from the USA to the UK to Sweden to Singapore are starting to create work\u2013family research and practice groups. As an example, as the first formally elected President of the Work Family Researchers Network, I helped convene its second ever international conference. Over 1000 scholars and policy makers from over 33 countries met in June 2014 to share work\u2013family research in New York, USA. Work\u2013family scholars build on the work of WFRN Founder Jerry Jacobs of the University of Pennsylvania, the Alfred P. Sloan Foundation, and the seminal work of thought leaders such as Rosabeth Moss Kanter (my former dissertation committee member, now at Harvard University), and many other luminaries.\n\nDespite this progress and growing interest in the work\u2013family field, many questions remain regarding work and family research. Despite thousands of studies, the work\u2013family field is still in its relative infancy compared to other scholarly research domains, and faces epistemological challenges. Common measures and ways of conceptualizing work and family are still under discussion (Kossek et al. 2011). Disciplines continue to dispute how best to understand this growing phenomenon. A recent Wall Street Journal article reports that use of workplace flexibility policies seems to have leveled off and stigmatization still faces those who work in different ways from the mainstream, even if strong performers. Such debate suggests that core knowledge and ways of knowing and understanding the work\u2013family nexus are still in flux. More importantly, organizations, individuals, and societies are continually discussing how to best address the continuing chasm between work and family, as solutions are unclear. Despite the increase in availability of work and family policies and the acceptance of work and family in our popular zeitgeist, reports show a vast majority of members of society\u2014regardless of whether they are male or female, single or married, old or young, working or not\u2014reporting increasing work\u2013life stress. Having access to workplace flexibility is another growing aspect of work inequality, as occupation, class, and gender stratification remain potential hurdles to flexibility and work\u2013life supports.\n\nI believe that a key reason for these persistent issues is that theory and research on gender and feminism are generally not well-integrated with traditional psychological work and family research. As an illustration, many work\u2013family researchers under-examine how gendered workplace and societal experiences inherently relate to work and family relationships. Yet it is undisputed that those who grew up in poverty are likely to have been children raised by single mothers. Experiencing a glass ceiling, limited career opportunities, and self-limiting career ambition are all linked to gender discrimination. Women continue to face stigmatization for a host of reasons related to maternity, patriarchal-based stereotypes, and lack of access to flexible jobs that allow caregiving and breadwinning to be aligned. Theories of gender, gender role norms, and gendered work and societal cultures are also under-discussed in the work\u2013family field. When gender is discussed in many work\u2013family studies, it is typically framed mainly as a variable reflective of biological sex differences or a quantitative moderator of an outcome. However, such a narrow view of gender overlooks cultural and social science underpinnings.\n\nMaura Mills' exciting book, Gender and the Work \u2013 Family Experience: An Intersection of Two Domains , begins to address these gaps by assembling an interesting collection of papers. Several cross-cutting themes that help advance the integration between gender and work\u2013family research were apparent across contributors' works.\n\nTheme 1: Increase the range of gender diversity in samples and tailor research to specify the gendered nature of contexts.\n\nTheme 1: Increase the range of gender diversity in samples and tailor research to specify the gendered nature of contexts.\n\nA number of contributors argued that our existing research of work, family, and gender linkages is limited theoretically and empirically by not studying a wider range of gender-diverse samples in specified cultural contexts.\n\nFor example, Sawyer, Thoroughgood, and Cleveland (Chap.  ) build on intersectionality theory (Crenshaw 1989) to examine the ways in which multiple forms of role identity such as LBGT and work and family identities interact and combine to create unique social experiences. They argue that LGBT discrimination fosters gaps between the work and family domains, forcing a separation boundary management strategy for LGBT employees. They also make the interesting case that current measures of work\u2013family conflict do not fully capture unique LGBT needs such as identity-based conflict.\n\nNomaguchi and Milkie (Chap.  ) argue that most work\u2013family conflict research overlooks individuals in the context of the quality of their coupled relationships, which are shifting in gendered expectations. Using a national sample of dual-earner heterosexual couples, they examine how wives' and husbands' inaccuracy in estimating the other partner's level of work-family conflict may shape couples' relationship quality. They find that that over half of couples overestimate or underestimate each other's work\u2013family conflict based on how they believe women or men \"should\" feel. Specifically, men's overestimating their female partners' work-family conflict is related to better relationship quality, while women's underestimating their male partners' conflict is related to poorer relationship quality.\n\nMoving to the cross-cultural level, Rajadhyaksha, Korabik, and Aycan (Chap.  ) note the lack of systematic variation in cross-cultural and institutional values regarding gender ideology. They contend that more research is needed that includes broader cultural views of gender and linkages between the work\u2013family interface. Scholars need to go beyond examining mere biological sex differences to examine cross-cultural value patterns in gender role attitudes. Micro-level frameworks of individual gender and gender role ideology should be meshed with macro-level cultural gender values assessing gender egalitarianism and gender inequality in order to foster multi-level research.\n\nRosiello and Mills (Chap.  ) observe that shiftwork is another understudied occupational context that is gendered and has work\u2013family implications that differ by gender. While it remains disputed whether more men than women engage in shiftwork, as this may vary by occupation, men are more likely to work overtime and weekends than are women. Rosiello and Mills also note that shiftwork has negative work\u2013family consequences, limiting time to dedicate to family or caregiving, have a social life, and increased likelihood of experiencing negative health problems linked with schedule unpredictability and lack of schedule control.\n\nMitchell, Eby, and Lorys (Chap.  ) encourage researchers to examine gender as a \"downstream\" antecedent of the ways in which negative spillover from work to family unfolds in gender-unequal domestic contexts. They argue that the mechanisms underlying within and between sex and gender differences in emotion and behavior must be examined in specified contexts such as the persistent inequality in domestic household division of labor. Gender-unequal contexts shape the mechanisms and unfolding of the work\u2013family spillover process, emotions, and response. For example, women in unequal contexts may adopt different coping behaviors, choices, or goals, and may have access to varying job and family demands and resources. Mitchell and colleagues also make the interesting argument that demonstrating negative emotions related to work and family is not always dysfunctional but rather may mobilize the use of effective coping strategies to reduce negative spillover such as pursuing greater domestic household equality.\n\nTheme 2: Develop new frames to focus on within- and between-group gender differences in work\u2013life inequality.\n\nSeveral contributors pointed to the need to develop new constructs to examine work\u2013family inequality, despite progress. Work\u2013family experiences provide a window into the persistence of work\u2013life inequality across gender and other minority subgroups in society and the workplace.\n\nCleveland, Fisher, and Sawyer (Chap.  ) develop the notion of work \u2013 life equality, which they see as a critical form of equality for well-being. They identify the paradox that while educational attainment and labor force participation have evened out across genders, women continue to not only devote more time to family caregiving but also to housework, family, social, and other meaningful functions than do men.\n\nFrevert, Culbertson, and Huffman (Chap.  ) point to another form of work\u2013life inequality; the growing linkage between work\u2013family experiences, race, and class. Noting the persistence of existing research to overemphasize the work\u2013family experiences of professionals and white women, they highlight the double jeopardy of gender and race in regard to work and family experiences. Minority women's work\u2013family experiences significantly differ from those of white women largely due to continued lack of equal opportunity access to managerial and professional job structures, which have greater job autonomy and access to work\u2013family supports systematically available in professional jobs.\n\nStanfors (Chap.  ) takes a national institutional comparative perspective to understand occupational inequality for women in fast track professions. Using data from Sweden and the USA, Stanfors shows that although the Swedish public policy goals are to provide supports to alleviate work\u2013family conflict for all workers, this goal is not achieved equally across professions and genders. For instance, she found that even though the USA and Sweden have very different levels of public supports for work and family, women doctors in both the USA and Sweden are more likely to have higher fertility rates than academics and lawyers in their countries. This suggests common occupational tradeoffs transcending variation in national cultural values and institutions.\n\nGiven the growing feminization of poverty, Odle-Dusseau, McFadden, and Britt (Chap.  ) conduct an integrative review of the poverty, gender, and work\u2013family literatures. They note that due to the sex segregation of lower income service jobs, women are more likely to fill these jobs and experience work\u2013life inequality. Lower wage secondary market sector jobs have more demands and fewer resources, thereby impeding one's ability to advance to better jobs and garner more work\u2013family resources for effective management of work and family demands. This gap leads to greater likelihood of experiencing work\u2013family conflict, lower levels of work\u2013family enrichment, and ultimately poorer health outcomes and quality of life for low income workers and families.\n\nLucas-Thompson and Goldberg (Chap.  ) look at generational differences in social orientation toward work\u2013family egalitarianism in division of labor in the home, and a trend reverting back toward gender inequality in young adults' work\u2013life behaviors. They note the paradox that despite possessing more gender-egalitarian values, the younger generation entering adulthood is behaving in ways that are inconsistent with their professed values about gender. Their explanations for the gap in aligning ideas and behaviors related to gender are persistent societal ambivalence about maternal employment, the stigmatization of fathers reducing or ending work to stay home with children, and the continued scarcity of resources to help new mothers jointly manage employment and caregiving. They cite data from the Pew Research Center (2013) showing that while marriage and parenthood limit women's time spent in paid work and career advancement, these same factors are associated with greater work involvement.\n\nTheme 3: Adopt new language, constructs, and refined frames to advance more nuanced understanding of gendered images of work and family.\n\nSeveral authors proposed new terms to refine gendered images of work and family. Clark, Belier, and Zimmerman (Chap.  ) focus on the interesting sample of \"women workaholics.\" They encourage work\u2013family researchers to apply the notion of competing devotions coined by Blair Loy (2003) to help reframe work\u2013family conflict as a moral dilemma involving competing work and family devotions.\n\n\" On demand jobs \" is another new term that is being suggested to replace face-time as a form of virtual accessibility. Grotto (Chap.  ) discusses how executive, managerial, and professional jobs have the paradox that while they have job autonomy built into their positions, they are facing a new form of face time\u2014being available \"on demand.\" On demand jobs are defined as jobs that necessitate individuals to be constantly available and accessible to employers and clients during nonwork hours. Moreover, responding on demand is a virtual visibility strategy as a means to demonstrate loyalty.\n\n\" Work \u2013 family guilt \" is explored by Korabik. In Chap.  , she argues that women always feel as though they are cheating their husbands, children, and themselves. Korabik believes that women are more likely to feel and express work\u2013family guilt than are men.\n\nTheme 4: Increase the incorporation of men, masculinity, and masculine work contexts in work\u2013family studies.\n\nCalling for deeper study of men's experiences in work\u2013family samples, Munn and Greer (Chap.  ) observe the paucity of studies on men's work\u2013family experiences, particularly in the USA, and observe that the prevalent conceptualization of the \"ideal worker\" is gendered. Our post-industrial society seems to see women as in greater need to overcome the ideal worker stigma. Research on work\u2013family breadwinning stereotypes needs to incorporate how work\u2013family conflicts affect whether men and women are seen as \"ideal\" workers. Interestingly, however, studies show that men who sought to use workplace flexibility practices are likely to be deemed poor \"organizational citizens,\" less committed to work, and possessing \"undesirable\" feminine traits. Such a reaction is unfortunate as men and especially fathers are increasingly more committed to family not just for financial roles, but also emotional and caretaking roles.\n\nSprung, Toumbeva, and Matthews (Chap.  ) examine how gender influences awareness of, access to, use of, and outcomes associated with work\u2013family policies. In general, women are more aware, have greater access to, and use of policies. Both men and women face stigma for using policies, but contexts and processes may differ. For example, a higher proportion of women than men are more likely to experience positive outcomes when using flexibility. However, supervisors may differentially encourage or tolerate men's versus women's usage of such flexibility policies. As such, Sprung et al.'s interesting chapter suggests that gender is very important for understanding work\u2013family processes and outcomes related to workplace flexibility policy usage.\n\nHuffman, Culbertson, and Barbour (Chap.  ) develop the notion of \"gendered occupations,\" which can be defined as an occupation that society associates with a particular gender. In doing so, they focus on one of the most stereotypically masculine workplace contexts in which to examine work\u2013family relationships\u2014the military. They develop an interesting model and discuss how work\u2013family conflict may differ in the military as opposed to in the civilian context due to the dominant gender of the personnel and more importantly due to the arguably gendered job demands. Examples of the latter include going into combat and hazardous work zones, high schedule unpredictability, lack of flexibility and schedule control, and heavy travel demands and deployment away from family, thereby limiting opportunity for family involvement while increasing work\u2013family stress.\n\nJean, Payne, and Thompson (Chap.  ) focus on another gendered occupational context: that of science, technology, engineering, and mathematics (STEM) fields. STEM jobs are often embedded in organizational cultures that are more supportive of men and masculinity. Although many factors create a \"leakier pipeline\" for women as compared to men in STEM, work\u2013family challenges remain an under-addressed critical factor impacting the recruitment and retention of women in such domains. National and individual initiatives countering gender stereotypes and barriers are needed to address this STEM gender gap.\n\nOverall, Mills' edited book includes many interesting chapters that break new ground and offer new perspectives on linkages between gender, work, and family. It is a creative collection of perspectives that will enhance scholarly and practical understanding of gender and the work\u2013family nexus.\n\nEllen Ernst Kossek, PhD\n\nBasil S. Turner Professor of Management\n\nResearch Director, Susan Bulkeley Butler Center for Leadership Excellence, Purdue University, West Lafayette, Indiana, USA\n\n## References\n\nCrenshaw, K. (1989). Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory, and antiracist politics. University of Chicago Legal Forum, 139\u2013167.\n\nKossek, E. E., Baltes, B. B., & Matthews, R. A. (2011). How work-family research can finally have an impact in the workplace. Industrial and Organizational Psychology: Perspectives on Science and Practice, 4, 352\u2013369.\n\nPew Research Center (2013). On pay gap, millenial women near parity-for now. Washington, DC: Pew Research Center.\n\nPreface\n\nMaura J. Mills, Ph.D.\n\nAs I begin writing this preface, I am sitting at the hospital waiting for my husband to emerge from day surgery. Such is the life of a work\u2013family juggler\u2014and we all do it. The irony of this has emerged at every corner throughout the development of this book, beginning when I submitted the book proposal the night before leaving for my honeymoon\u2014which was already a year delayed as a result of inflexible post-wedding work schedules. The challenge of being a successful work\u2013family balancer becomes even heavier once one begins a book on the topic\u2014and yet perfect success in both domains still always seems fleeting to me\u2014and to most.\n\nThis almost ubiquitous struggle was highlighted by the plethora of enthusiastic and positive responses I received from across the globe when I initially announced plans for this book and was soliciting chapter proposals. While many submitted chapter proposals, many more took the time to contact me about how very much needed\u2014and long overdue\u2014this book was, and to communicate their deep hopes for its impact on research and practice, as well as on general societal awareness of the issues at hand. Further evidencing this interest in and relevance of the book's topics, when I accessed my university's library to find the 'competitive' books of similar topics while researching for the book proposal, I was informed that all but one of the books were checked out, indicating the high demand for information regarding work\u2013family issues, and pointing toward the universal tensions that we all feel between these domains.\n\n## Why This? Why Now?\n\nAs suggested by the above anecdotes, the current zeitgeist\u2014both in the field and also in popular culture\u2014is ripe for a book thoroughly addressing the intersection of these domains. In light of the recent media coverage garnered by Yahoo CEO Marissa Mayer's controversial announcement regarding work\u2013life policy changes at the company, in addition to the recent publication of Lean In by Facebook's Sheryl Sandberg, this seems the ideal time for a comprehensive, research-based, and interdisciplinary work addressing the various aspects of the intersection of the work\u2013family and gender 1 domains.\n\nAlthough work\u2013family research and policy are becoming more widespread, comparably little research has examined possible gender differences or similarities between male and female employees in terms of work\u2013family conflict, its antecedents, and its outcomes. In an attempt to account for this research gap, the chapters comprising this book explore various aspects of work\u2013family conflict for both genders, as well as offering comparisons between the two in terms of career and gender perceptions, the conflict experience itself, and the consequences of such conflict, among other considerations. This is crucial because with gender-stereotypical ideologies shifting, women are taking on more demanding work roles, and men are taking on increased home responsibilities. This is the case not only in so-called traditional families, but also as divorce rates increase and as homosexual couples adopt children, forming nuclear families of their own. As such, work\u2013life conflict arguably becomes as much an issue for male employees as it has traditionally been for female employees.\n\nNevertheless, despite these shifting gender roles and family constitutions, many organizations\u2014not to mention national policies\u2014are not adapting accordingly. Consequently, male employees with substantial home responsibilities may not be receiving the support they need from their organizations, and likewise female employees\u2014who are increasing their participation in paid work but also still absorb the majority of household duties\u2014may not be receiving sufficient support either. In response, this book, in part, responds to criticism suggesting that neither research nor practice has kept up with these changing gender roles insofar as examining or managing work\u2013family conflict for males as well as for females, and for shifting family structures. In an attempt to contribute toward filling this gap, this book incorporates various chapters which collectively impact how work\u2013family research considers employee gender as the field moves forward. Each chapter is grounded within the work\u2013family research literature as well as gender-role literature, and each addresses a unique but related consideration of work\u2013family conflict in regard to employee gender and\/or gendered jobs.\n\n## For Whom?\n\nI like to think that the issues explored herein are relevant to everyone, to varying extents, as well as to society as a whole. Nevertheless, the book is likely to be more relevant for some individuals or purposes than for others. For instance, this book is ideal for use as a text or reader in an upper-level undergraduate or graduate-level seminar-style course. Beyond more traditional textbooks that focus on outlining definitions and the like, this edited book takes a critical and in-depth approach to a representative variety of issues surrounding the work\u2013family\/gender intersection, thereby yielding opportunities to spur students' critical thinking for class discussions, debates, and dialogues. Further, this book is also likely to be of interest to researchers in the fields of work\u2013family and\/or gender, who I hope will find it to be a thorough and representative consideration of issues surrounding the intersection of these fields of interest. As such, this book serves as potential fodder for future research ideas and recommendations, as well as giving a comprehensive, research- and theory-informed discussion of various issues surrounding these domains.\n\n## The End of the Beginning\n\nAs should be clear by now, given the increasing interest in these issues as of late, in addition to new and important zeitgeist shifts in the field and in society as a whole, there is a very real need for an up-to-date, comprehensive book evaluating them from various perspectives. The unique and comprehensive collection of chapters included herein together offer an updated assessment of these topics in light of their various facets and with an eye toward both depth and breadth. My greatest hope for this volume is that it will serve as an updated, interdisciplinary, and comprehensive resource in these domains, giving both a voice and a research-based justification to those currently entrenched in the struggle, and in the joys, of the work\u2013family interface.\n\nAs I finish writing this preface, I am 9 months pregnant and furiously trying to tie up as many work-related loose ends as possible before this little one makes his appearance. Nothing could be more fitting, and the irony of it has not been lost on me throughout my work on this book. Indeed, it is a funny thing, loving both your children and your job in a maternal, protective, enjoyable\u2014and yet frantically overworked\u2014type of way. Both bring some of life's greatest joys, proudest moments, and most frustrating struggles. Neither lets up during or makes way for the other. And yet somehow, someway, we do it. And we love it. Because this is the life we have made and chosen for ourselves, and we would not have it any other way (despite what we may think when we're awake in the wee hours of the morning consoling a crying child while sketching out the upcoming day's work commitments in our foggy minds). Keep at it, mothers and fathers and employees everywhere. This is life.\n\nAcknowledgments\n\nIt goes without saying that an edited book does not come to fruition through one person alone. It took the hard work and dedication of many wonderful people who believed in the importance of this work, who saw value in the cause, and who each viewed the topic from a distinctive and worthwhile perspective, contributing something uniquely theirs. My editing of this book is merely the vessel through which their hard work is disseminated. As such, I would like to thank all of my chapter authors for their continued commitment to this project and their intrinsic belief in the importance of this compiled work. It very literally could not have been done without them.\n\nIn a similar vein, I would like to thank the two very wonderful women at Springer Publishing with whom I have had the pleasure of working throughout this process. It is worth emphasizing that this book would not exist without Sharon Panulla, Executive Editor at Springer. Were it not for her attending one of my conference symposia on this topic, and approaching me afterward with the request that I undertake this venture, there would be no book at all. Subsequently, Sylvana Ruggirello, Assistant Behavioral Sciences Editor at Springer, was indispensable in navigating the publication process and ensuring that my chapter authors and I had all the information we needed in order to produce a quality work. Both of these women have made this process a pleasure, and I thank them for that.\n\nFinally, I would be remiss in a book about work\u2013family were I not to acknowledge my own family. Thank you to my mother, who sacrificed the frequency of our phone calls throughout my work on this book, and to my father, who instinctively knew that what I needed most was simply space and time, and who quietly gave me both without my having to ask. Thank you to my husband for setting up a wonderful home office for me, complete with dreamy extra-wide double screens, which greatly facilitated those times when I \"needed\" five or six windows open at the same time. And finally, many thanks to my baby for sacrificing many hours of prenatal yoga and the like so that I may complete this venture (and similarly to my dog, who endured many substantially abridged walks during my work on this book!).\n\nMy greatest appreciation goes to each and every one of the aforementioned individuals who contributed to making this book a reality. It exists because of you.\n\nContents\n\nPart I SOCIETAL INFLUENCES & ENTRENCHMENT\n\n1 Gender Ideology and Work\u2013Family Plans of the Next Generation\n\nRachel G. Lucas-Thompson and Wendy A. Goldberg\n\n2 Beyond the \"Ideal\" Worker:\u200b Including Men in Work\u2013Family Discussions\n\nSunny L. Munn and Tomika W. Greer\n\n3 Gender, Poverty, and the Work\u2013Family Interface\n\nHeather Odle-Dusseau, Anna C. McFadden and Thomas W. Britt\n\n 4 Exploring the Double Jeopardy Effect: The Importance of Gender and Race in Work\u2013Family Research\n\nTonya K. Frevert, Satoris S. Culbertson and Ann H. Huffman\n\n5 Challenging Heteronormative and Gendered Assumptions in Work\u2013Family Research:\u200b An Examination of LGB Identity-Based Work\u2013Family Conflict\n\nKatina B. Sawyer, Christian N. Thoroughgood and Jeanette N. Cleveland\n\n6 Gender, Gender-Role Ideology, and the Work\u2013Family Interface:\u200b A Cross-Cultural Analysis\n\nUjvala Rajadhyaksha, Karen Korabik and Zeynep Aycan\n\nPart II CONSIDERATIONS FROM THE HOMEFRONT\n\n7 Feeling Work at Home:\u200b A Transactional Model of Women and Men's Negative Affective Spillover from Work to Family\n\nMelissa E. Mitchell, Lillian T. Eby and Anna Lorys\n\n8 The Intersection of Gender and Work\u2013Family Guilt\n\nKaren Korabik\n\n9 Gender, Accuracy About Partners' Work\u2013Family Conflict, and Relationship Quality\n\nKei Nomaguchi and Melissa A. Milkie\n\n10 Work\u2013Life Equality:\u200b The Importance of a Level Playing Field at Home\n\nJeanette N. Cleveland, Gwenith G. Fisher and Katina B. Sawyer\n\n11 On-Demand:\u200b When Work Intrudes upon Employees' Personal Time\u2014Does Gender Matter?\u200b\n\nAngela R. Grotto\n\nPart III CAREER & ORGANIZATIONAL CONSIDERATIONS\n\n12 Family-Friendly Organizational Policies, Practices, and Benefits through the Gender Lens\n\nJustin M. Sprung, Tatiana H. Toumbeva and Russell A. Matthews\n\n13 Shiftwork as Gendered and Its Impact on Work\u2013Family Balance\n\nRyan M. Rosiello and Maura J. Mills\n\n14 Gender Roles in a Masculine Occupation:\u200b Military Men and Women's Differential Negotiation of the Work\u2013Family Interface\n\nAnn H. Huffman, Satoris S. Culbertson and Joseph Barbour\n\n15 Women in STEM:\u200b Family-Related Challenges and Initiatives\n\nVanessa A. Jean, Stephanie C. Payne and Rebecca J. Thompson\n\n16 Examining the Work\u2013Family Experience of Workaholic Women\n\nMalissa A. Clark, Angela A. Beiler and Lauren M. Zimmerman\n\n17 Family Life on the Fast Track?\u200b Gender and Work\u2013Family Tradeoffs Among Highly Educated Professionals:\u200b A Cross-Cultural Exploration\n\nMaria Stanfors\n\nIndex\n\nContributors\n\nZeynep Aycan\n\nDepartments of Psychology and Management, Ko\u00e7 University, Sariyer, Istanbul, Turkey\n\nJoseph Barbour\n\nDepartment of Psychology, Northern Arizona University, Flagstaff, AZ, USA\n\nAngela A. Beiler\n\nDepartment of Psychology, University of Georgia, Athens, GA, USA\n\nThomas W. Britt\n\nDepartment of Psychology, Clemson University, Clemson, SC, USA\n\nMalissa A. Clark\n\nDepartment of Psychology and Institute for Women's Studies, University of Georgia, Athens, GA, USA\n\nJeanette N. Cleveland\n\nDepartment of Psychology, Colorado State University, Fort Collins, CO, USA\n\nSatoris S. Culbertson\n\nDepartment of Management, Kansas State University, Manhattan, KS, USA\n\nLillian T. Eby\n\nDepartment of Psychology and Owens Institute for Behavioral Research, University of Georgia, Athens, GA, USA\n\nGwenith G. Fisher\n\nDepartment of Psychology, Colorado State University, Fort Collins, CO, USA\n\nTonya K. Frevert\n\nOrganizational Science PhD Program, University of North Carolina at Charlotte, Charlotte, NC, USA\n\nWendy A. Goldberg\n\nDepartment of Psychology and Social Behavior, University of California, Irvine, Irvine, CA, USA\n\nTomika W. Greer\n\nDepartment of Human Development and Consumer Sciences, College of Technology, University of Houston, Houston, TX, USA\n\nAngela R. Grotto\n\nDepartment of Management and Marketing, Manhattan College, Riverdale, NY, USA\n\nAnn H. Huffman\n\nDepartments of Psychology and Management, Northern Arizona University, Flagstaff, AZ, USA\n\nVanessa A. Jean\n\nDepartment of Psychology, Texas A&M University, College Station, TX, USA\n\nKaren Korabik\n\nCentre for Families, Work and Well-Being, University of Guelph, Guelph, ON, Canada\n\nAnna Lorys\n\nDepartment of Psychology, University of Georgia, Psychology Building, Athens, GA, USA\n\nRachel G. Lucas-Thompson\n\nDepartment of Human Development and Family Studies and Colorado School of Public Health, Colorado State University, Fort Collins, CO, USA\n\nRussell A. Matthews\n\nDepartment of Psychology, Bowling Green State University, Bowling Green, OH, USA\n\nAnna C. McFadden\n\nDepartment of Psychology, Clemson University, Clemson, SC, USA\n\nMelissa A. Milkie\n\nDepartment of Sociology, University of Toronto, Canada, ON, USA\n\nMaura J. Mills\n\nDepartments of Psychology and Management, Hofstra University, Hempstead, NY, USA\n\nMelissa E. Mitchell\n\nDepartment of Psychology, University of Georgia, Psychology Building, Athens, GA, USA\n\nSunny L. Munn\n\nDepartment of Leadership Studies, Dwight Schar College of Education, Ashland University, Ashland, OH, USA\n\nKei Nomaguchi\n\nDepartment of Sociology, Bowling Green State University, Bowling Green, OH, USA\n\nHeather Odle-Dusseau\n\nDepartment of Management, Gettysburg College, Gettysburg, PA, USA\n\nStephanie C. Payne\n\nDepartment of Psychology, Texas A&M University, College Station, TX, USA\n\nUjvala Rajadhyaksha\n\nDepartment of Business Administration and Economics, St. Mary's College, Notre Dame, IN, USA\n\nRyan M. Rosiello\n\nDepartment of Psychology, Hofstra University, Hempstead, NY, USA\n\nKatina B. Sawyer\n\nDepartment of Psychology, Villanova University, Villanova, PA, USA\n\nJustin M. Sprung\n\nDepartment of Psychology, Luther College, Decorah, IA, USA\n\nMaria Stanfors\n\nCentre for Economic Demography, Lund University, Lund, Sweden\n\nRebecca J. Thompson\n\nUniversity of Baltimore, Baltimore, MD, USA\n\nChristian N. Thoroughgood\n\nDepartment of Management and Organizational Development, Northeastern University, Boston, MA, USA\n\nTatiana H. Toumbeva\n\nDepartment of Psychology, Bowling Green State University, Bowling Green, OH, USA\n\nLauren M. Zimmerman\n\nDepartment of Psychology, University of Georgia, Athens, GA, USA\n\nFootnotes\n\n1\n\nIt should be noted that, for simplicity's sake, throughout this book the term gender is used to refer to biological, anatomical sex, as opposed to the gender with which one psychologically identifies.The latter is usually, although not necessarily, redundant with anatomical sex, a consideration overlooked by assumptions of biopsychological equivalence.\n\n# Part I  \nSOCIETAL INFLUENCES & ENTRENCHMENT\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_1\n\n# 1. Gender Ideology and Work\u2013Family Plans of the Next Generation\n\nRachel G. Lucas-Thompson1  and Wendy A. Goldberg2\n\n(1)\n\nDepartment of Human Development and Family Studies and Colorado School of Public Health, Colorado State University, Campus Delivery 1570, Behavioral Sciences Building Room 419, 80523 Fort Collins, CO, USA\n\n(2)\n\nDepartment of Psychology and Social Behavior, University of California, Irvine, 4564 Social and Behavior Sciences Gateway, 92697 Irvine, CA, USA\n\nRachel G. Lucas-Thompson (Corresponding author)\n\nEmail: lucas-thompson.rachel.graham@colostate.edu\n\nWendy A. Goldberg\n\nEmail: wendy.goldberg@uci.edu\n\nKeywords\n\nGender ideologyGenderWork\u2013familyGenerationAgePerceptionStereotypeEmerging adultAdolescentDemographics\n\n## 1.1 Introduction\n\nOver the last several decades, the gender disparity in the work force has been declining, as evidenced by findings that women now make up approximately half of the work force (Bureau of Labor Statistics 2010) . Women's employment is vital to the national economy and to the well-being of their families (Joint Economic Committee Majority Staff 2010). In addition, the gender gap in education has not only narrowed but reversed: in the USA, women now receive almost 60 % of 4-year college degrees (Becker 2010). Most Americans (62 %) endorse the idea that both husband and wife should contribute to family income (Pew Research Center 2010), but ambivalence remains about the effects of mothers working, particularly while children are young (Sylvester 2001). Most Americans think that men should be economic providers for their families: Two-thirds of Americans say that in order to be ready for marriage, a man should be able to financially support his family, while only one-third say that about women (Pew Research Center 2013). Due to increases in egalitarianism over time, American attitudes toward gender roles were expected to widely embrace egalitarianism by this time; instead, changes in gender-role attitudes appear to have leveled off, or stalled, since the mid-1990s (Cotter et al. 2011). The stalled attitudes correspond to the slowed pace for maternal employment since the 1990s, the persistent though narrowed wage gap (Cotter et al. 2011; Joint Economic Committee Majority Staff 2010), and the less-than-equal amount of time that dual-earner married men and women spend on housework (Bianchi et al. 2006).\n\nFollowing the birth of children, it is rare for men to reduce or end their work outside the home; in the USA, only 3 % of families with children younger than 15 have stay-at-home fathers (e.g., Kreider and Elliott 2009; Townsend 2002). In the USA, most women with children are employed outside the home (Bureau of Labor Statistics 2012) ; 34 % of employed mothers are their families' sole wage-earners, and over half of mothers with children under 18 work full-time (51 %; Joint Economic Committee Majority Staff 2010). However, in dual-earner families, fathers spend more time on paid work while mothers spend more time on housework and childcare (Pew Research Center 2013). Moreover, many married women in the USA delay their return to work following childbirth and\/or reduce their work hours to balance work and family (e.g., Kaufman and Uhlenberg 2000).\n\nLonger parental leave policies would help dual-earner and single parents spend more time at home before re-entering the labor force . Federal parental leave policies in the USA provide up to 12 weeks of job-protected, unpaid leave (see the 1993 Family and Medical Leave Act), which extends to up to 24 weeks of combined unpaid parental leave for two-parent families (Ray et al. 2009). Although some states provide paid leave , parental leave policies in the USA are modest compared to the extended, paid leave policies found in the European Union and in most industrialized nations around the world (Ray 2008). Extended paid maternity leaves sound like an ideal policy for employed women but Dex (2010) suggests that maternity leave policies can \"give with one hand\": By reinforcing traditional gender divisions, generous maternity leave policies can impede women's long-term career potential by encouraging them to be out of the labor force , which in turn limits their career earnings and advancement opportunities. Cross-nationally , longer leaves are associated with a lower proportion of women in the labor force (Hegewisch and Gornick 2011). As such, there are serious financial and advancement costs to women and families when women curtail their employment to raise children (e.g., Barker 1993; Drobnic and Witting 1997; Hochschild 1989; Machung 1989). A number of countries (e.g., Australia, the Scandinavian nations) have taken steps to encourage men to share primary caregiving of infants by stipulating provisions for paternity leave and\/or parental leave that can be divided (Ray 2008).\n\nWhile strides are being made in the sharing of child care, the equitable division of housework has lagged behind (Parker and Wang 2013). Many theorists have argued that division of labor in the home is a particularly important context for understanding gendered behavior (e.g., Brines 1994; Coltrane 1989). From the perspective of gender theorists, the allocation of housework is a \"symbolic enactment of gender relations\" (Drobnic 2010, p. 241). Within the home, even adults who endorse gender equality continue to divide domestic labor and decision-making in traditional ways (e.g., Bianchi et al. 2000; Franco et al. 2004; Himsel and Goldberg 2003; Hochschild 1989). The household division of labor remains gendered (Parker and Wang 2013) even though men are doing somewhat more housework and women are spending less time on housework than in the past (Bianchi et al. 2000, 2006).\n\nWhen women work outside the home and carry the weight of the \"second shift\" (Hochschild 1989), they often experience negative mental health consequences (e.g., Moen and Yu 2000). At work, the status of motherhood may elicit negative evaluations for women workers, particularly those in high-powered occupations (Ridgeway and Correll 2004), thus giving pause to young women who plan to combine a demanding career with family. Considering the career advancement and mental health ramifications of inconsistencies between gender role attitudes and behavior, coupled with stalled behavioral change in the domestic sphere, insight into whether these gendered patterns are likely to continue can be revealed by studying the views of the generation transitioning to adulthood.\n\nIn the current chapter, we review past research on the plans and strategies that adolescents and emerging adults (defined as the age period roughly from 18 to 25, Arnett 2004, 2007) have for managing work and family responsibilities, and what factors account for individual variability in these plans. In the face of empirical and theoretical arguments that ideas about gender change in important ways across adolescence and young adulthood (e.g., Galambos 2004; Hill and Lynch 1983), and that it is in adolescence that individuals first begin to decide how to balance work and family (e.g., Corder and Stephan 1984), we also present new data from two studies that examine these issues in two age groups , one in mid-adolescence and the other college-aged, i.e., in emerging adulthood. Attention in these studies is given to examining the work\u2013family plans and expectations of the participants. The theories of planned behavior (e.g., Ajzen and Fishbein 1977) and reasoned action (e.g., Ajzen 1991) include intentions as one of the few and primary predictors of later behavior, and support for these theories has been demonstrated across a wide range of behaviors (see for reviews: Ajzen 1991; Eagly and Chaiken 1993). Although expectations of future plans can be criticized for being a far from perfect representation of future behavior (Fishbein and Ajzen 1975), there are several lines of research indicating that expectations do predict later behavior in terms of educational attainment, risky behaviors, and childbirth (e.g., Harris et al. 2002; Sewell and Hauser 1980).\n\nIn the first study, the Adolescent Study, participants were 15-year-olds (n = 101; 49.5 % female) from one site of the NICHD Study of Early Child Care and Youth Development, a sample that was primarily (66 %) European American with significant Hispanic American representation (18 %); adolescents came from families that were mostly middle-to-upper-middle class. Adolescents' families were recruited at childbirth from selected hospitals based on a conditional random-sampling plan that was designed to ensure the sample reflected the economic, educational, and ethnic diversity of the area. Adolescents provided the information described below during a visit to their home, and received payment for participation in the larger study during which these data were collected.\n\nIn the second study, the Emerging Adult Study, participants were undergraduates (n = 343; 69 % female) from three diverse colleges\/universities across the country; a small liberal arts institution in Wisconsin (n = 92), a commuter university in Chicago, IL (n = 76), and a large research university in California (n = 175). As a result, the college sample was heterogeneous geographically and in terms of socioeconomic status and ethnicity (42 % European American, 30 % Asian American, 12 % Hispanic American, and 16 % other or multiple ethnicities). College students averaged 20.35 years of age (SD = 1.59; range of 17\u201324 years). Students provided the information described below during a visit to their classrooms, and received extra credit for participating.\n\nAcross studies , participants provided background\/demographic information and completed a 24-item scale that assessed their gender ideology (Wenzel and Lucas-Thompson 2012). Their expectations for work\u2013family roles were reported in terms of how much\/whether they planned to work after marriage but before children (\"I will not work outside the home,\" \"I will work part-time,\" and \"I will work full-time\") and after the birth of children (\"I will not work outside the home,\" \"I will work part-time,\" and \"I will work full-time\"). Participants also indicated how they anticipated dividing childcare responsibilities (\"I will be the primary care-taker,\" \"My partner will be the primary care-taker,\" and \"My partner and I will share care-taking equally\").\n\nIn data analysis for both studies, gender differences in work and caregiving aspirations were examined; in addition, using bivariate correlations, chi-square analyses, and ordinary least squares regression analyses, associations were examined among gender ideology , work aspirations, and plans to divide caregiving responsibilities . Before presenting the results of these studies, we review prior research on the work\u2013family plans of the generation coming to adulthood.\n\n## 1.2 Adolescents' and Emerging Adults' Plans for Managing Work and Family\n\nResearch focusing on how young individuals expect to manage work and family has primarily focused on work aspirations. Social changes that picked up steam during the 1970s and 1980s have led to many young women planning to continue to work after having children (e.g., Baber and Monaghan 1988; Boushey 2008; Dennehy and Mortimer 1993; Goldin and Katz 2008; O'Connell et al. 1989), but many other young individuals plan to replicate the gendered pattern of their parents' generation in terms of men, but not women, planning full-time employment when they have young children (Goldberg et al. 2012). More recent cohorts of women expect future employment (Goldin et al. 2006) but typically both young men and women expect that wives will dramatically reduce (or end) their work outside the home when children are born (de Valk 2008; Weinshenker 2006). In addition, by high school, concerns about balancing work and family are already on the minds of young women, including choosing occupations that will allow a work\u2013family balance that they desire (McDonald et al. 2011; Hardie and Hayford 2012). In contrast, young men do not engage in the same planning process (Hardie and Hayford 2012), or report expecting that family life will be a far second to work life in their adult years (McDonald et al. 2011).\n\nExpectations about work before and after children are clearly important to include when investigating how emerging adults expect to deal with gender issues. However, other expectations need to be considered as well, including how individuals transitioning to adulthood expect to divide or share child-rearing, and how and whether work expectations are related to expectations about child-rearing (e.g., do emerging adults who expect to share child-rearing expect both partners to work equal hours?). For instance, a robust finding is that adults are often inconsistent in terms of behaving in line with their gender ideologies, in that even those with egalitarian ideals continue to divide household tasks in traditionally gendered ways (e.g., Bianchi et al. 2000; Franco et al. 2004; Himsel and Goldberg 2003; Hochschild 1989). Therefore, it is likely that emerging adults may also display inconsistencies in their expectations about balancing work and family.\n\nTraditionally, work\u2013family arrangements are most likely to become more gender-typed after the birth of children (e.g., Goldberg et al. 1985). Following the birth of children, it is rare for men to reduce or end their work outside the home (e.g., Kaufman and Uhlenberg 2000; Townsend 2002); indeed, emerging adults who become fathers plan to work more hours once the baby is born (Goldberg 2014). However, in the USA, most women with children under six years are employed outside the home (61.6 %; U.S. Census Bureau 2011) , and over half of mothers with children under 18 work full-time (51 %; Joint Economic Committee Majority Staff 2010).\n\n### 1.2.1 Findings from the Adolescent and Emerging Adult Studies\n\nResults from our new studies confirm many of the findings from past studies, but also extend knowledge about how contemporary adolescents and emerging adults think about balancing work and family in the future. For example, among those adolescents and emerging adults who reported that they want to get married, men and women had similar aspirations for paid work after marriage and before children, with most expecting to work full-time (see Figs. 1.1 and 1.2). However, among those who hoped to have children someday, when asked to think about work aspirations after children are born, plans differed between males and females, \u03c72s(3) > 25.34, ps < 0.001. Young men by and large expected to be employed full-time when they become parents, whereas a sizable number of women did not expect to be employed or expected to be employed part-time (see Figs. 1.1 and 1.2). Interestingly, however, the majority of both male and female individuals reported that they planned to share caregiving equally with their partner (although more women than men reported that they planned to be the primary caregiver of their children; see Fig. 1.3).\n\nFig. 1.1\n\nGender differences in work aspirations before and after children are born. (Adolescent Study)\n\nFig. 1.2\n\nGender differences in work aspirations before and after children are born. (Emerging Adult Study)\n\nFig. 1.3\n\nGender differences in plans to divide childrearing responsibilities\n\nAn interesting difference between adolescents and emerging adults was revealed in the links between employment aspirations and child-rearing plans. Specifically, for adolescent girls, their plans to share caregiving equally with future partners were not related in predictable ways to their expectations about work status after children were born, \u03c72(4) = 3.40, p = 0.49. Among girls who indicated that they will share caretaking equally with their partner , 23 % reported that they will not work for pay outside the home, 51 % reported that they will work part-time, and 26 % reported that they will work full-time. However, among emerging adult women, these work and family aspirations were significantly related to each other, \u03c72(6) = 18.10, p = 0.006. College women who planned on being the primary caregiver were more likely to report not planning to work outside the home (42 %) or planned to work part-time (38 %) than to report planning to work full-time (21 %). In contrast, college women who planned to share childrearing responsibilities equally with their partners were more likely to report plans to work full- (37 %) or part- (51 %) time than to report planning not to work outside the home (11 %). Compared to adolescents, emerging adults may be displaying plans that are more consistent across work\u2013family domains because they are more cognitively mature, because they are further along in the process of identity exploration, because they are more likely to have a steady partner, and\/or because they have had more time to think about work\u2013family issues. This pattern, though, is in line with suggestions that increasing age predicts having more family- and work-related goals (relative to education- and friend-related goals; Salmela-Aro et al. 2007).\n\nOverall, these new results combined with past studies suggest that a large number of adolescents and emerging adults expect to manage work\u2013family commitments in much the same traditional, gendered way as the generations that have come before them in terms of the general pattern of women continuing to assume more housework and family care responsibilities than men (e.g., Dennehy and Mortimer 1993; O'Connell et al. 1989) and less paid work involvement after having children (Kaufman and Uhlenberg 2000; Townsend 2002). Despite changes in rates of employment and education that reflect increasing gender equality (e.g., U.S. \u00adDepartment of Labor 2010) , college students still seem to hold stereotyped notions about the work competence of mothers (Fuegen et al. 2004) and exaggerated stereotypes about the negative consequences of maternal employment for children (Goldberg and Lucas-Thompson 2014). Therefore, the pattern of mothers being employed but working less than full-time (Kaufman and Uhlenberg 2000) may continue to be a common strategy to manage work\u2013family commitments , despite more equal participation of men and women in the workforce (U.S. Department of Labor 2010) .\n\nSimilar to results of studies with adults (Bianchi et al. 2000; Franco et al. 2004; Goldberg et al. 2012; Himsel and Goldberg 2003; Hochschild 1989), adolescents and young men and women are also demonstrating idealized expectations for gender equality. Across gender and age , most participants reported wanting to share childcare equally (although a small sub-set of women plan to be the primary caregiver), but the majority of young women plan to work part-time or not at all after having children. The expectation seems to be one of sharing child-rearing while following a traditional gendered employment pattern , the \"transitional\" pattern that echoes findings of adults; endorsing gender equality while behaving \u00adunequally (Hochschild 1989). However, unequal does not equate to unfair in the eyes of \u00admarried women of different gender ideologies using differing comparative referents (i.e., \"relative deprivation theory\"; Greenstein 1996) .\n\nIf the future behavior of these adolescents and emerging adults matches their expectations, the next generation may perpetuate some of the gender inequality in the workforce that persists today, despite the greater equality in labor force participation overall (U.S. Department of Labor 2010) that includes a narrowed wage gap (Pew Research Center 2013). Reduced labor force involvement when women become mothers can mean longer time to promotion and lower lifetime pay as well as economic hardship for their families (Drobnic and Witting 1997; Machung 1989). Despite these potential costs, some researchers have commented that this pattern may still be \"the most rational plan for those who value both career and family\" (Weinshenker 2006, p. 885), given the meager institutional and policy support given to parents. What is perhaps most troubling, though, is the potential that these young adults will make decisions to reduce or end employment without being aware of the long-term consequences. Indeed, the finding in the current study that young women expect to share child-rearing equally but follow a traditionally gendered employment pattern suggests that their plans may not be fully informed or \"rational.\" However, these new data suggest increasing consistency in work\u2013family plans once youth reach college-age. To sort out which factors explain this apparent developmental shift, longitudinal studies are needed, as is a closer look at individual differences in work\u2013family plans.\n\n## 1.3 Individual Differences in Plans to Manage Work and Family\n\n### 1.3.1 Demographic Differences\n\nThere is consistent evidence that women's employment rates as well as their work\u2013family values differ based on culture and ethnicity (Orbuch and Eyster 1997): Caucasian and African American women typically return to work sooner after the birth of a child than do Asian and Hispanic women (Han et al. 2008; Laughlin 2011). From a developmental perspective, parental ideas about gender and work\u2013family balance are shaped, in part, by ethnic and cultural values, and parental ideas in turn influence children's expectations, aspirations, and values (Eccles 1994). It is not surprising, then, that there is some evidence that ethnic differences are reflected in adolescent and emerging adult work\u2013family balance plans. For instance, research in the Netherlands has found that ethnic differences extend to the work\u2013family plans of adolescent males and females (de Valk 2008) and research in the USA has found differences in the plans of Asian and Caucasian college students (Goldberg et al. 2012). Other family characteristics also appear to shape the types of expectations that adolescents have for balancing work and family in their future. For instance, coming from a religious family, or a family with lower levels of parental education, is related to more traditional work\u2013family balance plans, particularly for girls (de Valk 2008) .\n\nTheoretically, one of the most important characteristics shaping children's ideas about balancing work and family is whether or not their own mothers worked while they were young. Working mothers provide positive role models for combining work and motherhood (Greenberger et al. 1988), and role theory predicts that children, particularly girls, of working mothers will be more likely to make nontraditional career choices and have greater levels of labor force attachment , and research generally supports this argument (Alessandri 1992; Eccles and Hoffman 1984; Hoffman 1980; Weinshenker 2006). There is some evidence that same-sex parental models are most important for adolescents in terms of expectations about their own work behavior, and other-sex parents are important for adolescents' expectations about their future partners' work behavior (Wiese and Freund 2011). Young women who had working mothers have more positive stereotypes about the effects of \u00ad(particularly full-time)employment on several domains of child development, differences that appear explained, in part, by differences in gender ideology , attitudes about the benefits and costs of maternal employment, and work values (\u00adGoldberg and Lucas-Thompson 2014). Less studied are young men's experiences of having an employed mother, but these childhood experiences are relevant for them, too. For example, greater participation in household chores has been observed among married men whose own mothers were employed during their childhood (Gupta 2006).\n\n#### 1.3.1.1 Findings from the Adolescent and Emerging Adult Studies\n\nResults from the new studies presented here suggest that there are many ways in which having a working mother does not appear related in predictable ways to the work\u2013family plans of adolescents and emerging adults. For instance, for college-aged men and women, plans about dividing or sharing caregiving did not appear related to maternal work history . In terms of one's own employment plans, adolescent boys were more likely to report plans to work part-time after the birth of children if their own mothers had stayed home rather than worked for pay while they were children, an unexpected finding that deserves further investigation. More expected was the finding that adolescent girls of mothers who did not work for pay during their childhood were much less likely to report planning to work full-time (0 %) than part-time or not at all, particularly after the birth of children. In addition, female (but not male) college students were more likely to report planning to work full-time (41 % vs. 18 %) after children if their own mothers had been employed when they were young. Therefore, although sharing of childcare was not related to one's own mother's employment, it appears that adolescent girls and emerging adult men and women are more likely to report plans to reduce work hours after the birth of children if their own mothers were not employed. Other research indicates that college students' beliefs about the costs and benefits of maternal employment for children are linked to whether or not their own mothers were employed during their childhood (Goldberg et al. 2012). In line with role theory , it seems likely that a mechanism of transmission of gendered ideas, gendered behavior , and inconsistency between the two may be in part due to what children observe their own parents do.\n\n### 1.3.2 The Importance of Gender Ideology\n\nSocial role theory was developed as a means to explain behavioral similarities and differences between men and women (Eagly 1987; Eagly et al. 2000). From this perspective, beliefs that people have about the genders are formed from observations of the roles that men and women occupy. Gender differences in personality and social behavior reflect the gendered division of labor around paid work and domestic work and the gender hierarchy for power , status, and resources (Eagly et al. 2000). Importantly, social role theory views gender roles as dynamic and open to change as the typical work and family roles change in a given culture (Eagly et al. 2000).\n\nConsistently, gender ideology has emerged as an important contributor to the nature of the plans that young individuals make for their future work\u2013family balance . Gender ideology refers to a set of attitudes about the appropriate roles, rights, and responsibilities of men and women in a given society; traditional ideologies ascribe distinctive work and care roles for the genders whereas egalitarian ideologies endorse equal provider and care roles for the genders (Hochschild 1989; Kroska 2007). Gender ideologies can be thought of as \"factors that frame the strategies of action an individual sees as possible\" (e.g., Davis and Pearce 2007, p. 250); they reflect the extent of an individual's support for gendered and separate spheres of paid work and family work (Davis and Greenstein 2009). Gender ideology has been linked with a range of gender-relevant behaviors across the lifespan, including marriage, timing of first birth, division of household labor , educational attainment, and labor force participation (see Davis and Greenstein 2009 for a review). Furthermore, considering gender ideology can inform our understanding of the work and family decisions that young adults make (Davis and Greenstein 2009).\n\nResearch indicates that egalitarian and traditional gender ideologies formed before adulthood are associated with decisions about future occupation and education . Young men are frequently reported to hold more traditional attitudes toward work and family roles than do young women, but the mean differences can be small and there is substantial variability within each gender (Kerpelman and Schvaneveldt 1999). Adolescents, particularly young girls, who endorse more egalitarian gender ideologies also espouse more ambitious educational and career aspirations (Davis and Pearce 2007; Eccles 1994). In addition, a more egalitarian maternal (but not paternal) gender ideology is associated with adolescents' less gendered work\u2013family balance plans (Weinshenker 2006).\n\nThere are several mechanisms through which gender ideology early in life can exert its influence on future plans. They include girls' and boys' internalization of societal norms about their abilities and the value placed on the expected roles for their gender (Correll 2001; Eccles 1994), and adolescents' attitudes about how men and women should balance work and family (Davis and Pearce 2007). This evidence suggests that young men and women with more traditional ideas about gender will anticipate work\u2013family arrangements that are more gendered than those of less traditional men and women. However, much of the data concerning young peoples' extent of work expectations and gender ideology were collected in the 1970s (e.g., Corder and Stephan 1984). Constructions of gender have dramatically changed since that time (e.g., Bolzendahl and Myers 2004; Thornton and Young-DeMarco 2001), as has the demographic composition of the USA. Therefore, an updated examination of the relations between gender roles and work expectations is needed in a culturally diverse sample.\n\n#### 1.3.2.1 Findings from the Adolescent and Emerging Adult Studies\n\nConsistent with past research (e.g., Kerpelman and Schvaneveldt 1999), the results from our new studies suggest that in both adolescence and emerging adulthood, young women were more egalitarian in their gender ideology than were young men. Interestingly, gender ideology was associated in ways that theory would predict with the work\u2013family aspirations of emerging adults, but only very rarely with the work\u2013family aspirations of adolescents. For female (but not male) college students, those who reported more egalitarian gender ideologies were more likely not to want to get married, r = \u2212 0.14, p = 0.031, but were no less likely to report wanting to have children, rs < \u25020.13\u2502, ps > 0.11. This finding suggests that marriage and having children are not as intertwined for contemporary young adults as they once were. In addition, gender ideology was associated with plans about dividing caregiving responsibility for both male and female emerging adults: a more egalitarian gender ideology was associated with plans to share caregiving responsibilities , associations that remained after adjusting for ethnicity and income. In addition, for women , there was a significant association between gender ideology and work aspirations after children, controlling for demographic variables: more egalitarian ideologies were associated with plans to work full-time relative to part-time.\n\nFor adolescents, particularly adolescent boys, gender ideology was largely unrelated to work\u2013family aspirations. More specifically, gender ideology was not related to adolescents' plans to get married or have children, rs < \u25020.06\u2502, ps > 0.67. For boys, gender ideology was not related to plans about work after children or dividing caregiving responsibilities. For adolescent girls, a more traditional gender ideology was associated with more traditional expectations about who will raise children (e.g., women being the primary caregiver). This association remained significant after controlling for ethnicity and family income.\n\nBy definition, gender ideologies reflect whether individuals support gendered separate spheres or shared spheres of paid work and family work (Davis and Greenstein 2009); however, for young adolescent and emerging adult men and women, there were many ways in which work\u2013family role behavior appears independent of reported ideas about gender roles. There was some evidence that work\u2013family expectations were related to gender ideology , particularly for adolescent girls and for the sample of college students. Some researchers expect the liberalization of gender role attitudes to continue apace (e.g., Bolzendahl and Myers 2004; Brooks and Bolzendahl 2004), but others claim a plateau or reversal since the mid-1990s (Brewster and Padavic 2000; Cotter et al. 2011; Thornton and Young-DeMarco 2001). Our findings with circa 2007\u20132009 data can be seen as consistent with the latter position and with studies that found women to be more egalitarian than men (e.g., Bolzendahl and Myers 2004; Brewster and Padavic 2000; Galambos 2004; Goldberg et al. 2012; Steil 2000). In addition, for female adolescents and college students, a more traditional gender ideology was associated with expectations for more traditional caregiving arrangements. Across developmental periods, then, gender ideology seems to be particularly important for ideas about who will take primary responsibility for children.\n\nThe stronger associations between gender ideology and work aspirations in the college student sample compared to the adolescent sample could be due to several factors. First, the comparatively smaller sample of adolescents had less adequate power; therefore, the non-significant associations between gender ideology and work aspirations could be due to limited power to detect small effects rather than non-existent relations. Second, it is possible that the college students\u2014for whom decisions about balancing work and family are more imminent and who have had more time to explore their identities\u2014may display behavior that is more consistent with their ideals (Arnett 2007). They may also be more realistic about the \u00adchallenges of managing work and family.\n\n## 1.4 Conclusion\n\nOn a societal level, we are witnessing the reversing of the gender gap in higher education, but a stall in the progress toward egalitarianism in the home and workplace. Looking at the plans and behavior of adolescents and emerging adults, we do not find seeds of renewed vigor in the move toward egalitarian gender roles. Why is the generation now entering adulthood behaving in ways that are inconsistent with their ideas about gender? One potential reason is the difficulty of successfully aligning ideas and behavior related to gender (Sjoberg 2010), particularly with societal ambivalence about maternal employment (Sylvester 2001), disapproval of fathers reducing or curtailing work to stay home with children (Nock 1998; Townsend 2002), limited parental leave policies (Ray et al. 2009), and scarce allocation of resources to help new mothers reconcile employment and caregiving. Marriage and parenthood exact a toll on women's time spent on paid work activities, including career interruptions , whereas for men, marriage and parenthood are associated with more work involvement (Pew Research Center 2013).\n\nAlthough the new studies presented here provide important current information about gender equality and gendered behavior , there are several limitations of note in addition to the power issue mentioned above. The adolescent sample was also mostly middle- to upper-middle-class and from one region of the USA; however, the college student sample was diverse geographically and in terms of socioeconomic background. Demographic differences between the adolescent and college samples (e.g., proportion Asian American, socioeoconomic status) could have contributed to differences attributed to age cohorts. Utilizing samples of different age cohorts is important for gender and work\u2013family issues (e.g., Corder and Stephan 1984; Galambos 2004; Hill and Lynch 1983), and along with samples composed of diverse cultures\/ethnicities, strengthen the generalizability of our findings.\n\nThe new data presented in this chapter were from cross-sectional studies. Needed are longitudinal studies that could illuminate the factors that might explain the apparent developmental shift from adolescence to college such as greater experience with romantic relationships, more time spent contemplating work\u2013family balance , and more mature cognitive and long-term planning abilities. In the current study, participants reported their plans for future employment; however, future research would benefit from providing more specific trajectories for employment plans (e.g., 'employment in the first year after a child is born' rather than 'after a child is born'). Despite these limitations , these two studies make important contributions to our knowledge about how the next generation of potential worker-parents intends to balance work and family, and suggest that a traditionally gendered pattern will likely persist at least into the near future.\n\nReferences\n\nAjzen, I. (1991). The theory of planned behavior. Organizational Behavior and Human Decision Processes, 50, 179\u2013211.CrossRef90020-T)\n\nAjzen, I., & Fishbein, M. (1977). Attitude-behavior relations: A theoretical analysis and review of empirical research. Psychological Bulletin, 84, 888\u2013918.CrossRef\n\nAlessandri, S. M. (1992). Effects of maternal work status in single-parent families on children's perception of self and family and school achievement. Journal of Experimental Child Psychology, 54, 417\u2013433.\n\nArnett, J. J. (2004). Emerging adulthood: The winding road from the late teens through the twenties. Oxford: Oxford University Press.\n\nArnett, J. J. (2007). Emerging adulthood: What is it, and what is it good for? Child Development Perspectives, 1, 68\u201373.CrossRef\n\nBaber, K. M., & Monaghan, P. (1988). College women's career and motherhood expectations\u2013New options, old dilemmas. Sex Roles, 19, 189\u2013203.CrossRef\n\nBarker, K. (1993). Changing assumptions and contingent solutions: The costs and benefits of women working full- and part-time. Sex Roles, 28, 47\u201371.CrossRef\n\nBecker, G. (2010). The revolution in the economic empowerment of women. http:\/\/\u200bwww.\u200bbecker-posner-blog.\u200bcom\/\u200b2010\/\u200b01\/\u200bthe-revolution-in-the-economic-empowerment-of-women-becker.\u200bhtml. Accessed 3 June 2012.\n\nBianchi, S. M., Milkie, M. A., Sayer, L. C., & Robinson, J. P. (2000). Is anyone doing the housework? Trends in the gender division of household labor. Social Forces, 79, 191\u2013228.CrossRef\n\nBianchi, S. M., Robinson, J. P., & Milkie, M. A. (2006). Changing rhythms of American family life. New York: Russell Sage.\n\nBolzendahl, C., & Myers, D. J. (2004). Feminist attitudes and support for gender equality: Opinion change in women and men, 1974\u20131998. Social Forces, 83, 759\u2013790.CrossRef\n\nBoushey, H. (2008). \"Opting out?\" The effect of children on women' employment in the United States. Feminist Economist, 14, 1\u201336.CrossRef\n\nBrewster, K. L., & Padavic, I. (2000). Change in gender ideology, 1977\u20131996: The contributions of intracohort change and population turnover. Journal of Marriage & the Family, 62, 477\u2013487.CrossRef\n\nBrines, J. (1994). Economic dependency, gender, and the division of labor at home. American Journal of Sociology, 100, 652\u2013688.CrossRef\n\nBrooks, C., & Bolzendahl, C. (2004). The transformation of U.S. gender role attitudes: Cohort replacement, social-structural change, and ideological learning. Social Science Research, 33, 106\u2013133.CrossRef00041-3)\n\nBureau of Labor Statistics, U.S. Department of Labor. (Dec 2010). Women in the labor force: A databook. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200bwlf-databook-2010.\u200bpdf. Accessed 2 June 2012.\n\nBureau of Labor Statistics, U.S. Department of Labor. (April 2012). Employment characteristics of families summary. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bnews.\u200brelease\/\u200bfamee.\u200bnr0.\u200bhtm. Accessed 2 June 2012.\n\nColtrane, S. (1989). Household labor and the routine production of gender. Social Problems, 36, 473\u2013490.CrossRef\n\nCorder, J., & Stephan, C. W. (1984). Females combination of work and family roles-adolescents aspirations. Journal of Marriage and the Family, 46, 391\u2013402.CrossRef\n\nCorrell, S. J. (2001). Gender and the career choice process: The role of biased self-assessments. American Journal of Sociology 106(6), 1691\u20131730.CrossRef\n\nCotter, D., Hermsen, J. M., & Vanneman, R. (2011). The End of the Gender Revolution? Gender Role Attitudes From 1977 to 2008. American Journal of Sociology, 117, 259\u2013289.CrossRef\n\nDavis, S. N., & Greenstein, T. (2009). Gender ideology: Components, predictors, and consequences. Annual Review of Sociology, 35, 87\u2013105.CrossRef\n\nDavis, S. N., & Pearce, L. D. (2007). Adolescents' work-family gender ideologies and educational expectations. Sociological Perspectives, 50, 249\u2013271.CrossRef\n\nde Valk, H. A. (2008). Parental influence on work and family plans of adolescents of different ethnic backgrounds in the Netherlands. Sex Roles, 59, 738\u2013751.CrossRef\n\nDennehy, K., & Mortimer, J. T. (1993). Work and family orientations of contemporary adolescent boys and girls. In J. C. Hood (Ed.), Men, work, and family (pp. 87\u2013107). Newbury Park: Sage.\n\nDex, S. (2010). Can state policies produce equality in housework? In J. Treas & S. Drobnic (Eds.), Dividing the domestic: Men, women, and housework in cross-national perspective. Palo Alto: Stanford University.\n\nDrobnic, S. (2010). Concluding thoughts on the societal context of housework. In J. Treas & S. Drobnic (Eds.), Dividing the domestic: Men, women, and household work in cross-national perspective. (pp. 241\u2013251). Palo Alto: Stanford University.CrossRef\n\nDrobnic, S., & Witting, I. (1997). Part-time work in the United States of America. In H. P. Blossfield & C. Hakim (Eds.), Between equalization and marginalization: Women working part-time in Europe and the United States of America (pp. 289\u2013314). New York: Oxford University.\n\nEagly, A. (1987). Sex differences in social behavior: A social-role interpretation. John M. MacEachran memorial lecture series; 1985. Hillsdale: Erlbaum.\n\nEagly, A. H., & Chaiken, S. (1993). The psychology of attitudes. Orlando: Harcourt Brace.\n\nEagly, A. H., Wood, W., & Diekman, A. B. (2000). Social role theory of sex differences and similarities: A current appraisal. In T. Eckes & H. M. Trautner (Eds.), The developmental social psychology of gender (pp. 123\u2013174). Mahwah: Erlbaum.\n\nEccles, J. (1994). Understanding women's educational and occupational choices: Applying the Eccles et al. model of achievement-related choices. Psychology of Women Quarterly, 18, 485\u2013609.CrossRef\n\nEccles, J. S., & Hoffman, L. W. (1984). Sex roles, socialization, and occupational behavior. Child development research and social policy: Harold W. Stevenson & Alberta E. Siegel.\n\nFishbein, M., & Ajzen, I. (1975). Belief, attitude, intention, and behavior: An introduction to theory and research. Reading: Addison-Wesley.\n\nFranco, J. L., Sabattini, L., & Crosby, F. J. (2004). Anticipating work and family: Exploring the associations among gender-related ideologies, values, and behaviors in Latino and White families in the United States. Journal of Social Issues, 60, 755\u2013766.CrossRef\n\nFuegen, K., Biernal, M., Haines, E., & Deaux, K. (2004). Mothers and fathers in the workplace: How gender and parental status influence judgements of job-related consequences. Journal of Social Issues, 60, 737\u2013754.CrossRef\n\nGalambos, N. L. (2004). Gender and gender role development in adolescence. In R. M. Lerner & L. Steinberg (Eds.), Handbook of Adolescent Psychology (pp. 233\u2013262). Hoboken: Wiley.\n\nGoldberg, W. A. (2014). Father time: The social clock and the timing of fatherhood. London: Palgrave Macmillan.CrossRef\n\nGoldberg, W. A., & Lucas-Thompson, R. (2014). Stereotype accuracy: Do college women hit or miss the mark when estimating the \"impact\" of maternal employment on children's achievement and behavior? Psychology of Women Quarterly, 38, 1\u201313.\n\nGoldberg, W. A., Kelly, E., Matthews, N. B., Kang, H., Li, W., & Sumaroka, M. (2012). The more things change, the more they stay the same: Gender, culture, and college students' views about work and family. Journal of Social Issues, 68, 814\u2013837.CrossRef\n\nGoldberg, W. A., Michaels, G. Y., & Lamb, M. E. (1985). Husbands' and wives' adjustment to pregnancy and first parenthood. Journal of Family Issues, 6, 483\u2013503.\n\nGoldin, C., Katz, L. F., & Kuziemko, I. (2006). The homecoming of American college women: The reversal of the college gender gap. Journal of Economic Perspectives, 20, 133\u2013156.CrossRef\n\nGoldin, C., Katz, L. F. (2008). Transitions: Career and Family Lifecycles of the Educational Elite. American Economic Review: Papers and Proceedings 98, (2), 363\u2013369.\n\nGreenberger, E., Goldberg, W. A., Crawford, T. J., & Granger, J. (1988). Beliefs about the consequences of maternal employment for children. Psychology of Women Quarterly, 12, 35\u201359.CrossRef\n\nGreenstein, T. N. (1996). Gender ideology and perceptions of the fairness of the division of household labor: Effects on marital quality. Social Forces, 74, 1029\u20131042.CrossRef\n\nGupta, S. (2006). The consequences of maternal employment during men's childhood for their adult housework performance. Gender & Society, 20, 60\u201386.CrossRef\n\nHan, W. J., Ruhm, C. J., Waldfogel, J., & Washbrook, E. (2008). The timing of mothers' employment after childbirth. Monthly Labor Review, 131, 15\u201327.PubMedCentralPubMed\n\nHardie, J. H., & Hayford, S. R. (2012). Opting In: Adolescent girls' work and family plans. paper submitted at the inaugural work and family researchers network conference. New York, USA.\n\nHarris, K. M., Duncan, G. J., & Boisjoly, J. (2002). Evaluating the role of \"nothing to lose\" attitudes on risky behavior in adolescence. Social Forces, 80, 1005\u20131039.CrossRef\n\nHegewisch, A., & Gornick, J. C. (2011). The impact of work-family policies on women's employment: A review of research from the OECD countries. Community, Work, & Family, 14, 119\u2013138.CrossRef\n\nHill, J. P., & Lynch, M. E. (1983). The intensification of gender-related role expectations during early adolescence. In J. Brooks-Gunn & J. Petersen (Eds.), Girls at puberty: Biological and psychosocial perspectives (pp. 201\u2013228). New York: Plenum.CrossRef\n\nHimsel, A. J., & Goldberg, W. A. (2003). Social comparisons and satisfaction with the division of housework: Implications for men's and women's role strain. Journal of Family Issues, 24, 843\u2013866.CrossRef\n\nHochschild, A. R. (1989). The second shift: Working parents and the revolution at home. New York: Viking.\n\nHoffman, L. W. (1980). The effects of maternal employment on the academic attitudes and performance of school-age children. School Psychology Review, 9, 319\u2013336.\n\nJoint Economic Committee Majority Staff, U. S. C. (Dec 2010). Invest in women, invest in America: A comprehensive review of women in the U.S. Economy. http:\/\/\u200bjec.\u200bsenate.\u200bgov\/\u200bpublic\/\u200bindex.\u200bcfm?\u200bp=\u200bReports1&\u200bContentRecord_\u200bid=\u200b11dd6449-8d54-41ff-9bb2-87d98cbf66c2&\u200bContentType_\u200bid=\u200befc78dac-24b1-4196-a730-d48568b9a5d7&\u200bGroup_\u200bid=\u200bc120e658-3d60-470b-a8a1-6d2d8fc30132. Accessed 2 June 2012.\n\nKaufman, G., & Uhlenberg, P. (2000). The influence of parenthood on the work effort of married men and women. Social Forces, 78, 931\u2013947.CrossRef\n\nKerpelman, J. L., & Schvaneveldt, P. L. (1999). Young adults' anticipated identity importance of career, marital, and parental roles: Comparisons of men and women with different role balance orientations. Sex Roles, 41, 189\u2013217.CrossRef\n\nKreider, R. M., & Elliott, D. R. (2009). America's families and living arrangements 2007. Current Population Reports, P20-561. Washington, DC: U.S. Census Bureau.\n\nKroska, A. (2007). Gender ideology and gender role ideology. In G. Ritzer (Ed.), The Blackwell Encyclopedia of Sociology (Vol. 4, pp. 1867\u20131868). Malden: Blackwell.\n\nLaughlin, L. (2011). Maternity leave and employment patterns of first-time mothers: 1961\u20132008. Current Population Report, P70-128. Washington, DC: U.S. Census Bureau.\n\nMachung, A. (1989). Talking career, thinking job-Gender differences in career and family expectations of Berkeley seniors. Feminist Studies, 15, 35\u201358.CrossRef\n\nMcDonald, P., Pini, B., Bailey, J., & Price, R. (2011). Young people's aspirations for education, work, family and leisure. Work, Employment and Society, 25, 68\u201384.CrossRef\n\nMoen, P., & Yu, Y. (2000). Effective work\/life strategies: Working couples, work conditions, gender, and life quality. Social Problems, 47, 291\u2013326.CrossRef\n\nNock, S. L. (1998). Marriage in men's lives. New York: Oxford University.\n\nO'Connell, L., Betz, M., & Kurth, S. (1989). Plans for balancing work and family-life-Do women pursuing nontraditional and traditional occupations differ? Sex Roles, 20, 35\u201345.CrossRef\n\nOrbuch, T. L., & Eyster, S. L. (1997). Division of household labor among Black couples White couples. Social Forces, 76, 301\u2013332.CrossRef\n\nParker, K., & Wang, W. (2013). Modern parenthood: Roles of moms and dads converge as they balance work and family. Pew Research: Social & Demographic Trends. http:\/\/\u200bwww.\u200bpewsocialtrends.\u200borg\/\u200b2013\/\u200b03\/\u200b14\/\u200bmodern-parenthood-roles-of-moms-and-dads-converge-as-they-balance-work-and-family\/\u200b. Accessed 5 Sep 2013.\n\nPew Research Center. (2010). The decline of marriage and rise of new families. Washington: Pew Research Center.\n\nPew Research Center. (2013). On pay gap, millenial women near parity-For now. Washington: Pew Research Center.\n\nRay, R. A. (2008). A detailed look at parental leave policies in 21 OECD countries. Washington: U.S. Center for Economic and Policy Research.\n\nRay, R. A., Gornick, J. C., & Schmitt, J. (2009). Parental leave policies in 21 countries: Assessing generosity and gender equality. Washington: U.S. Center for Economic and Policy Research.\n\nRidgeway, C. L., & Correll, S. J. (2004). Unpacking the gender system: A theoretical perspective on gender beliefs and social relations. Gender & Society, 18, 510\u2013531.CrossRef\n\nSalmela-Aro, K., Aunola, K., & Nurmi, J. E. (2007). Personal goals during emerging adulthood: A 10-year follow up. Journal of Adolescent Research, 22, 690\u2013715.CrossRef\n\nSewell, W. H., & Hauser, R. M. (1980). The Wisconsin longitudinal study of social and psychological factors in aspirations and achievements. Research in Sociology of Education and Socialization, 1, 59\u201399.\n\nSjoberg, O. (2010). Ambivalent attitudes, contradictory institutions ambivalence in gender-role attitudes in comparative perspective. International Journal of Comparative Sociology, 51, 33\u201357.CrossRef\n\nSteil, J. M. (2000). Contemporary marriage: Still an unequal partnership. In C. Hendrick & S. Hendrick (Eds.), Close relationships: A sourcebook (pp. 125\u2013152). Thousand Oaks: Sage.\n\nSylvester, K. (2001). Caring for our youngest: Public attitudes in the United States. (Review). The Future of children\/Center for the Future of Children, the David and Lucile Packard Foundation, 11, 52\u201361.PubMedCrossRef\n\nThornton, A., & Young-DeMarco, L. (2001). Four decades of trends in attitudes toward family issues in the United States: The 1960s through the 1990s. Journal of Marriage and the Family, 63, 1009\u20131037.CrossRef\n\nTownsend, N. W. (2002). The package deal: Marriage, work, and fatherhood in men's lives. Philadelphia: Temple University.\n\nU.S. Census Bureau (2011). The 2011 statistical abstract. http:\/\/\u200bwww.\u200bcensus.\u200bgov\/\u200bcompendia\/\u200bstatab\/\u200bcats\/\u200blabor_\u200bforce_\u200bemployment_\u200bearnings.\u200bhtml.\n\nWeinshenker, M. N. (2006). Adolescents' expectations about mothers' employment: Life course patterns and parental influence. Sex Roles, 54, 845\u2013857.CrossRef\n\nWenzel, A. J., & Lucas-Thompson, R. G. (2012). Authenticity in college-aged males and females, how close others are perceived, and mental health outcomes. Sex Roles, 67, 334\u2013350.CrossRef\n\nWiese, B. S., & Freund, A. M. (2011). Parents as role models: Parental behavior affects adolescents' plans for work involvement. International Journal of Behavioral Development, 35, 218\u2013224.CrossRef\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_2\n\n# 2. Beyond the \"Ideal\" Worker: Including Men in Work\u2013Family Discussions\n\nSunny L. Munn1  and Tomika W. Greer2\n\n(1)\n\nDepartment of Leadership Studies, Dwight Schar College of Education, Ashland University, 128 Dwight Schar, 401 College Ave., 44805 Ashland, OH, USA\n\n(2)\n\nDepartment of Human Development and Consumer Sciences, College of Technology, University of Houston, 110 Cameron Building, 77204 Houston, TX, USA\n\nSunny L. Munn (Corresponding author)\n\nEmail: smunn2@ashland.edu\n\nTomika W. Greer\n\nEmail: twgreer@central.uh.edu\n\nKeywords\n\nMalesMenTraditional familyIdeal workerContemporary employeeGenderWork\u2013family\n\n## 2.1 Introduction\n\nTraditional workplaces are based on the \"ideal worker\" (Williams 2000)\u2014\u00adidentifying a man as the earner of the primary paycheck within a family\u2014and coincide with gender ideology that perpetuates the norm of the \"ideal parent\" (Sperling 2014), which names a woman as the primary caretaker of the family. The ideal worker's life is organized in a way that accommodates the demands of the employer, while the ideal parent's life accommodates the demands of family.\n\nThe ideal worker is the preferred employee in both white collar and blue collar contexts. This employee is willing to work swing shifts as a blue-collar worker and able to travel on short notice or work 80-hour weeks in white-collar positions. This employee is not distracted by the demands of children or commitments outside of work. The ideal worker is\u2014nearly without exception\u2014characterized as a male. Alternatively, the ideal parent provides unlimited support at home and handles all childcare responsibilities. The expectation is that the ideal parent is female and maintains this role regardless of whether or not she is working outside of the home. For employed women, their home responsibilities are frequently referred to as the \"second shift\" (Hochschild 1989).\n\nFemale employees are, therefore, thought to need work\u2013family friendliness in their workplace more so than do male employees. Accordingly, organizations are more likely to offer work\u2013family initiatives when the percentage of women employed in the organization is high (Konrad and Mangel 2000). Although alternative work arrangements are becoming increasingly possible, men are less likely to take \u00adadvantage of organizational work\u2013family initiatives (Fried 1998; Kirby and Krone 2002). This is likely influenced by unspoken organizational norms and the attitudes of coworkers which perpetuate negative attitudes toward men who use work\u2013\u00adfamily benefits. Furthermore, men's use of work\u2013family initiatives challenges traditional perceptions of masculinity (Vandello et al. 2013). Employees who are most likely to use such initiatives are married, female, and\/or have children (Thompson et al. 1999). As women are considered to be the ideal parents, there is little effort on behalf of organizations to offer work\u2013family benefits to men , to accept that they might need to use such benefits, or to encourage a culturally responsive organization that allows men to do so.\n\nIn this chapter, we point to evidence that contemporary men do not always fit the ideal worker stereotype. Instead, men are taking on more responsibilities in the home and may struggle with the collision of work and family demands (Kaufman 2014)\u2014a struggle that is stereotypically reserved for women. Even the gender neutral terms \"work\u2013family\" and \"working parent\" are often interpreted as being primarily applicable to women (Levine and Pittinsky 1997), as well as excluding individuals without children.\n\nWorking men have real needs for work\u2013family initiatives and balance just as working women might. According to data from the 1997 National Study of the Changing Workforce, collected by the Families and Work Institute, employed fathers with children under the age of 18 years reported 48.5 hours of work per week (Hill 2005). However, these working fathers also reported 24.7 hours in child care and 21.2 hours doing household chores each week (Hill 2005). Increased involvement with childcare and household responsibilities is one reason why \"work\u2013life research on men is necessary in order to challenge the norm of the 'ideal worker'\" (Sav et al. 2013, p. 673) and the ideal parent. As a result, there is a need to consider men in work\u2013family discussions and make workplace adjustments that consider the work\u2013family interface for male employees.\n\n## 2.2 From Ideal Worker to Contemporary Employee\n\nOver the past four decades, the US workforce has experienced a general declining trend of men's prominence in the workforce as more women seek employment outside of the home (U.S. Bureau of Labor Statistics 2013) . In the 1970s, men comprised about 61 % of the US workforce. Into the 1980s, that number dropped to about 56 % of the workforce. In the 1990s, men only made up 54 % of the workforce. This trend continued into the 2000s as men comprised 53 % of the workforce. Currently, women comprise about half of the workforce as the gap continues to close (U.S. Bureau of Labor Statistics 2013).\n\nAs women's presence in the workforce is equalizing with men's presence, there have been shifts in the traditional breadwinner status that men historically held in American society. Since the late 1980s, the number of wives who earn more than their husbands in dual-earner families has steadily increased (U. S. Bureau of Labor Statistics 2013). In 1987, there were 29.7 million dual-earner couples and 17.8 % of the wives earned more than their husbands. By 2011, there were 31.4 million dual-earner families. This number represents a relatively modest increase in dual-earner families over the preceding 24 years. However, 29.2 % of the wives were earning more than their husbands in 2011, suggesting that nearly one-third of dual-earner couples no longer fit the male breadwinner family structure.\n\nIn conjunction with women taking on more participation and responsibilities in the paid labor force , men have increased their engagement in matters of the home and family. Although men's increased involvement in housework and childcare did not mirror women's entrance into the workforce during the 1960s and 1970s (Coverman and Sheley 1986), a marked increase in men's household involvement has been documented (Coltrane 2000).\n\nUltimately, as traditional male-role expectations are changing, more American men have started to transition from the breadwinner role to the involved family man. As a result, balancing work and family roles has become an important issue for working men just as it has been for women (Yonetani et al. 2007). In fact, in a recent study , Kaufman (2014) classified fathers into three types: old, new, and superdad . The \"old\" dads are the traditional breadwinners; \"new\" dads are a mixture between breadwinner and caregiver; and \"superdads\" are those whose priority is caregiving as opposed to work.\n\nThe changes in the contemporary definition of family also contribute to the need to consider work\u2013family interactions for men . \"While the definition of family can be interpreted widely (Powell et al. 2012) within the work\u2013family literature, it is typically used to refer to married and dual-earner couples or those with children\" (Munn 2013, p. 6). The work\u2013family literature overwhelmingly fails to define the meaning of family, most frequently creating the assumption that the family structure is \"traditional.\" The traditional family is comprised of an employed father and an unemployed mother (Schneer and Reitman 1993), thus perpetuating the notion of the ideal worker and the ideal parent. As evidenced by the rise in women's employment and the realization that 54 % of married-couple families are dual-earner families (U.S. Bureau of Labor Statistics 2013) , and 13.6 million families are separated or unmarried (U.S. Census Bureau 2012) , the traditional family including the ideal worker and parent do not accurately reflect the lives of many Americans.\n\nInstead, a \"post-traditional\" family can include dual-earner couples and children who require adult supervision and care. In this family, it may be the case that mothers pick up a \"second-shift\" by becoming active labor force participants and remaining primary caregivers at home (Bailyn et al. 2001; Damaske 2011; Hochschild 1989, 1997). Additionally, fathers may also share more household responsibilities and\/or earn less income than mothers.\n\nThe post-traditional family can also be a single-parent family. The prevalence of single-parent families has been on the rise in the United States since the 1960s. In 1960, there were less than 300,000 households with minor children that were headed by a single father ; in 2011, that number had risen above 2.6 million single father families (Livingston 2013). The number of single mother households has risen even more sharply in the same time period\u2014from 1.9 million households in 1960 to 8.6 million households in 2011 (Livingston 2013). Therefore, the ability of single parents\u2014including fathers\u2014to balance work and family responsibilities is increasingly important.\n\nIn this post-traditional family era, the \"new\" father (Kaufman and Uhlenberg 2000) differs from the traditional breadwinner in that he spends more time with his children, although he may be torn between spending quality time and working more to provide better options for his family (Kaufman 2014; Roy 2005). Still, the work\u2013family literature continues to perpetuate the ideal worker by differentially including women and men in work\u2013family discussions according to stereotypical views of traditional gender roles. For example, in western societies, work\u2013family research has focused on the experiences of married, white, educated women in professional\/managerial job positions (Casper et al. 2007; Sav et al. 2013). Within this population, researchers have studied the relationships between working mothers and child outcomes , marital satisfaction, and issues of self-efficacy. This narrow focus is likely an outcome of the persistence of the ideals of the traditional family structure where women are thought to be plagued with higher demands to maintain equilibrium between work and the rest of the life (Guest 2002) due to the child- and home-care responsibilities not regularly expected of men. Researchers have also studied the effects of women in the workplace, organizational needs, and family needs to help shape work\u2013family policies and practices for women and their children.\n\nAlternately, compared to women, men are minimally studied in work\u2013family research. When men are included, the research tends to be more concerned with work outcomes than family outcomes. However, we contend that a more thorough understanding of men's work\u2013family issues is warranted to develop equitable work\u2013family initiatives and make concessions for changing gender role expectations and the contemporary US workforce (Hill 2005). The myth of the ideal worker and ideal parent is now obsolete. Instead, we find men in the workforce who have demanding work and family responsibilities and a desire to perform well in both roles. Accordingly, the aim of this chapter is to portray a holistic view of men's work\u2013family experiences and advocate for the value of consistently considering men in work\u2013family discussions.\n\n## 2.3 Theory of the Work\u2013Family Interface: Conflict, Enrichment, and Balance\n\nThe work\u2013family interface is often studied in terms of work\u2013family conflict and work\u2013family enrichment. Work\u2013family conflict occurs when work or family demands hinder a person's performance in the alternate role. Work\u2013family conflict follows when the responsibilities of work and family are incompatible (Greenhaus and Beutell 1985). Conversely, work\u2013family enrichment ensues when work or family experiences enhance performance in the other domain. Enrichment occurs when resources are gained in one domain that can be used to improve performance or affect in the other domain (Carlson et al. 2006). Work\u2013family conflict is more often studied than work\u2013family enrichment. However, researchers advocate for studying both conflict and enrichment: The positive side of the work\u2013family interface has been shown to predict work and nonwork outcomes over and above the effects of work\u2013family conflict alone (van Steenbergen et al. 2007).\n\nWork\u2013family conflict and work\u2013family enrichment have been theorized and empirically supported as bidirectional constructs (Carlson et al. 2006; Greenhaus and Powell 2006). The conflict can originate in either the work domain or the family domain. The conflict that originates in the work domain has been called work-interference-with-family (WIF) . The conflict that originates in the family domain has been called family-interference-with-work (FIW) . WIF is thought to be more prevalent than FIW among both men and women (Allard et al. 2011; Kinnunen and Mauno 1998). The work\u2013family enrichment characterized by experiences in the work domain positively influencing the quality of life in the family domain has been called work-to-family enrichment (WFE). The enrichment characterized by experiences in the family domain that positively influence the quality of life in the work domain has been called family-to-work enrichment (FWE).\n\nPrevious studies have led researchers to believe that work\u2013family conflict and work\u2013family enrichment are independent constructs. Work\u2013family conflict is not necessarily the opposite of work\u2013family enrichment (Powell and Greenhaus 2006). For many, the elusive notion of \"work\u2013family balance\" could exist in the case of minimized work\u2013family conflict and maximized work\u2013family enrichment. There is no single agreed-upon definition of work\u2013family balance, but in general there is an agreement that work\u2013family balance occurs when an employee is satisfied with the amount of time and effort spent in each life domain with as little conflict as possible (Clark 2000; Greenhaus et al. 2003; Grzywacz and Carlson 2007).\n\n### 2.3.1 Differences Between Men's and Women's Work\u2013Family Experiences\n\nIn 1992, Higgins and Duxbury published one of the earliest identified studies that looked specifically at men's work\u2013family conflict amidst changing family structures . The primary premise of their study recognized that the woman's work\u2013family conflict had been thoroughly studied as her role expectations were changing from homemaker to working mother while work\u2013family conflict for men who found themselves in dual-earner families was drastically understudied. Higgins and Duxbury (1992) sought to compare the antecedents and consequences of work\u2013\u00adfamily conflict for men in single-earner families compared to men in increasingly popular dual-earner families. Their results confirmed differences between the two groups of men in their model of work\u2013family conflict.\n\nThe results of the Higgins and Duxbury (1992) study suggested that as men's roles in relation to work and family have evolved, so have their experiences with work\u2013family conflict. A similar case can be made regarding work\u2013family enrichment. The results of this study suggested that what has been discovered about work\u2013family interactions based on women's experiences is not necessarily generalizable to contemporary men. It is, therefore, necessary to discuss men's work\u2013family experiences separately from women's experiences because the expectations of men in their work and family roles differ from the expectations of women in similar roles. For instance, mothers spend more time in the presence of their children, dealing with daily hassles, and performing caregiving tasks as compared to fathers (Lee et al. 2003).\n\nAlthough there is some evidence to support the idea that there are no significant differences between men and women in work\u2013family conflict (Anafarta and Kuru\u00fcz\u00fcm 2012; Kinnunen and Mauno 1998), there is also research that suggests that men experience less work\u2013family conflict and more work\u2013family enrichment than women (e.g., Figueroa et al. 2012; Hill 2005; van Steenbergen et al. 2007). Such differences point to gender as an embedded factor in how employees negotiate their work and family role demands (Emslie and Hunt 2009). Indeed, working fathers have reported less individual stress , and greater family satisfaction , marital satisfaction , and life satisfaction than working mothers (Hill 2005). Even among self-employed men and women, women report more intrusions on their work from their family than do self-employed men (Loscocco 1997). Alternately, work intrudes more on family for self-employed men than for self-employed women (Loscocco 1997).\n\nMuch of the research around men's work\u2013family experiences compared to women's experiences provides a foundation for continuing research in this area. There is clearly not enough research on men in this regard. However, there is also evidence that men's work\u2013family experiences have different antecedents and consequences than women's work\u2013family experiences. In the following sections, we explore the extant research on the antecedents and consequences of men's work\u2013family conflict and work\u2013family enrichment.\n\n## 2.4 Understanding Men's Work\u2013Family Conflict\n\n### 2.4.1 Antecedents of FIW and WIF Conflict\n\nFor both men and women, FIW is best explained by family domain variables, such as the number of children living at home (Kinnunen and Mauno 1998). Other findings indicate that FIW is likely to increase when men have more advanced job tenure (Anafarta and Kuru\u00fcz\u00fcm 2012), and for those that are married or cohabiting and have eldercare responsibilities (Barrah et al. 2004). Furthermore, spousal support is negatively related to FIW (Aycan and Eskin 2005).\n\nCompared to FIW, there is much more research on the WIF experiences of men. WIF is best explained by work domain variables (e.g., full-time job, poor leadership relations) among women, as well as by high education and number of children living at home (Kinnunen and Mauno 1998). For men, one study indicated that higher workloads and more work hours are associated with increased WIF (Sav et al. 2013). However, organizational support \u2014including supervisory support , work\u2013family policies, and time flexibility\u2014is associated with less WIF for men (Aycan and Eskin 2005). Another study found that WIF is negatively correlated with men's education and job tenure (Anafarta and Kuru\u00fcz\u00fcm 2012). Additionally, for men, the frequency of family intrusions and total role involvement predict time-based WIF (Loerch et al. 1989). In another recent study, male senior leaders reported more WIF than female senior leaders (Mills and Grotto 2012). Although gender was not significant, this study also found that satisfaction with flexible work practices and organizational work\u2013life support was important for reducing WIF .\n\nAnother important factor for men is relationship status . In general, men's marital status has been positively correlated with WIF (Anafarta and Kuru\u00fcz\u00fcm 2012). Employed fathers who are not partnered have reported greater work\u2013family conflict and lower family-to-work enrichment than partnered fathers (Janzen and Kelly 2012). Men who are married with an employed spouse experience more work\u2013family conflict than men in single earner families or who are partnered with a homemaker (Crowley 1998; Higgins and Duxbury 1992). Men who view themselves as adequate breadwinners reported a greater sense of work\u2013family role overload than those who view themselves as inadequate breadwinners (Crowley 1998).\n\nGender role assignments remain prevalent within work and family situations. A recent LA Times (2013) article noted that women are more likely to break the gender divide by entering more male dominated fields such as medicine, business, or law while men are much less likely to take on traditional female-oriented employment opportunities like teaching, nursing, or administrative work. In fact, less than 1 % of fathers were stay-at-home dads in 2013 while 24 % of mothers were stay-at-home moms, according to U.S. Census data . Previous research findings suggest that while stereotypical gender-role attitudes tend to increase work\u2013family conflict for women, they actually decrease work\u2013family conflict for men (Izraeli 1993). Spending time with family on the weekends and being concerned about work performance contributes to more work\u2013family conflict in men than in women (Izraeli 1993). For example, fathers who are also managers might have increased access to flexible work practices yet still tend to experience high levels of work\u2013family conflict. However, for these same dads in more gender egalitarian families, access to flexible working practices creates less work\u2013family conflict and also sets a positive example, encouraging other employees' use of such benefits (Allard et al. 2007).\n\n### 2.4.2 Consequences of FIW and WIF Conflict\n\nThere is significant research demonstrating the impact of FIW and WIF conflict on women and children. However, much less is known about the impact on men and fathers. For instance, when fathers bring stress from work into the home, mothers are more likely to experience decreased work\u2013family balance , thus increasing conflict for mothers (Fagan and Press 2008). Additionally, mothers tend to experience higher levels of work\u2013family balance when fathers have more flexibility at work and are more involved at home with childcare (Fagan and Press 2008).\n\nHigher levels of emotional exhaustion are common for both men and women experiencing work\u2013family conflict. Work\u2013family conflict was associated with poorer health among women and drinking problems among men (Leineweber et al. 2013). Another study found that work\u2013family conflict was strongly related to problem drinking among both women and men (Roos et al. 2006). Aycan and Eskin (2005) found that work\u2013family conflict had a negative relationship with psychological well-being , marital satisfaction , and parental role performance for men. Additionally, FIW was positively related to men's guilt (Livingston and Judge 2008).\n\n## 2.5 Understanding Men's Work\u2013Family Enrichment\n\nAlthough work\u2013family conflict is more prominently studied in the literature as compared to work\u2013family enrichment, there is a small amount of research related to work\u2013family enrichment for men, which is summarized here.\n\n### 2.5.1 Antecedents of Men's Work\u2013Family Enrichment\n\nA qualitative study of Australian Muslim men revealed interesting experiences of work and family interactions (Sav et al. 2013) . Results of the study suggest that these Muslim men experienced both conflict and enrichment, however enrichment appeared to dominate their experiences. The researchers in this study suggested that the high reports of work-to-family and family-to-work enrichment were likely due to the religious beliefs of the Muslim men, who view paid employment as a religious obligation and value the breadwinner role (Sav et al. 2013). The study concludes that religion can influence men's experience of work\u2013family enrichment. Furthermore, these Muslim men also engaged in flexible work hours and perceived a supportive workplace, which is also likely to improve work\u2013family enrichment (Sav et al. 2013) .\n\nInterestingly, in a representative sample of working fathers in the United States, work\u2013family conflict was more prevalent among the employed fathers than work\u2013family enrichment (Hill 2005). Moreover, as an additional contrast to the Australian Muslim men, supportive organizational culture was negatively related to family-to-work enrichment, suggesting that with a more supportive organization, American employed fathers are less likely to experience positive gains from the family to the work domain. In the same study, organizational commitment was also negatively associated with family-to-work enrichment (Hill 2005). So, when working fathers experienced family-to-work enrichment, they were less committed to remaining employed in their organizations. However, the more time these employed fathers spent on childcare , the less family-to-work enrichment they experienced (Hill 2005) .\n\nThe findings from the studies of Australian Muslim men and the American working fathers highlight how context-specific experiences of work\u2013family interactions can be. However, in both cases, the men appear to value their work and their family roles, which is consistent with the idea of contemporary men who are not conforming to the ideal worker stereotype.\n\n### 2.5.2 Consequences of Men's Work\u2013Family Enrichment\n\nWork-to-family enrichment has been positively related to job satisfaction and life satisfaction, and negatively related to individual stress (Hill 2005). Family-to-work enrichment has been positively related to marital satisfaction , family satisfaction , and life satisfaction, and negatively related to organizational commitment (Hill 2005).\n\nOthers took a more granular approach to looking at the consequences of men's work\u2013family enrichment by measuring its distinct facets (van Steenbergen et al. 2007). Specifically, they looked at enrichment as being energy-based , time-based , behavioral , and psychological in addition to being bidirectional. For men, energy-based work-to-family enrichment was a significant predictor of life satisfaction and job satisfaction (van Steenbergen et al. 2007) . This finding suggests that when men perceive that their work gives them additional energy to perform at home, they are more satisfied with both their work and nonwork roles. Higher psychological work-to-family enrichment predicted better job performance and lower job search behavior, suggesting that these men were performing well and less likely to be looking for a job change. Furthermore, energy-based work-to-family enrichment predicted higher affective commitment for men. It is important to note that many of these significant findings are different between men and women, reinforcing the notions that the outcomes of men's and women's work\u2013family interactions are experienced differently .\n\n## 2.6 Practical Implications\n\n\"The way work itself is organized\u2014around stereotypical [heterosexual white] male employees with no substantial responsibilities outside of the workplace\u2014is often a major obstacle for people trying to combine work and family\" (Rapoport et al. 2002, p. 183). Yet, there is often an unconscious disconnect between the existence and the use of work\u2013family benefits because the \"ideal worker\" (Bailyn et al. 2001; Williams 2010) (also called a \"zero-drag\" worker ; Hochschild 1997) is most accurately depicted as the traditional male worker, while work\u2013family benefits were created to assist female employees with children (Lewis 2001). The \"ideal worker\" is traditionally male and works full-time. He is able to keep the spheres of work and family separate thus not \"needing\" to use work\u2013family policies (Allen 2001; Bailyn et al. 2001; Campbell 2001; Thompson et al. 1999; Williams 2010; Williams et al. 2006). However, while work may still be organized this way, the way individuals work and interact with family have changed over time\u2014especially for fathers. Unfortunately, organizational expectations and the actions of male employees who are also fathers have not caught up with one another. \"Opportunities and [organizational] rewards go to those who most closely conform to the \"ideal worker\" by compartmentalizing work and family caregiving so that these are separate spatially, temporally, and psychologically\" (Winfield and Rushing 2005, p. 58). The desire to fulfill this ideal worker norm causes many employees to fear and, therefore, avoid using any policies that may provide \"family-friendly\" benefits (Bailyn et al. 2001; Blair-Loy 2001; Kirby and Krone 2002; Williams et al. 2006; Winfield and Rushing 2005). Similarly, Berdahl and Moon (2013) found that men who take on childcare responsibilities, thus acting outside of the role of an ideal worker and traditional father, are typically mistreated at work via criticisms of their masculinity . This suggests that adjustments need to be made regarding how we conceptualize work and fatherhood\u2014adjustments which allow for a new definition of fatherhood that goes beyond the provider role to also include the caretaker role. In other words, the \"new dad\" and \"superdad\" conceptualizations should also be perceived as strengths of a man's masculinity.\n\nWhile research has shown that men also value flexibility and work\u2013family balance, they tend to seek out opportunities for improving such balance less frequently than do women (Vandello et al. 2013), perhaps because of the fear of stigmatization and ridicule (Berdahl and Moon 2013). Fathers are more dependent on spousal support than organizational support (Hill 2005), sustaining research demonstrating men's underuse of work\u2013family benefits (Kirby and Krone 2002; Lewis 1997). With the presence of a more supportive organizational culture, supervisor-supported flexibility, and greater use of family-oriented benefits, mothers experience less work\u2013family role strain (Warren and Johnson 1995). Winfield and Rushing (2005) found, for both men and women, that in organizations with \"family-friendly\" policies and jobs that provide autonomy \"employees are more likely to perceive their supervisors as supportive of interactions in the workplace that bridge the borders between work and family life\" (p. 56).\n\nFrequently, companies have cultural norms that keep work and family separate from one another. When a family-supportive organizational culture is not well developed, fathers tend to experience more work\u2013family conflict. However, when a family-supportive culture exists within the organization, work and family are more easily combined (Allard et al. 2011, Galinsky et al. 2013). A common perception applied to working mothers is the \"mommy track\" , which includes lower earnings and less challenging work for women with children who seek more flexible work schedules . Recently, it was found that fathers who choose the \"daddy track\"\u2014opting for more flexibility for family purposes\u2014also incur decreased earnings over the course of their careers because they are seen as less focused (Coltrane et al. 2013). Similarly, managers are more likely to accommodate men who request flexible work practices for career advancement rather than family (Brescoll et al. 2013).\n\nEmployees who believe their managers support the use of family-friendly policies and programs are more likely to use available options (Allen 2001). When employees do not perceive informal support from their managers and peers, they are less likely to take advantage of family-friendly interventions. Although Hill et al. (2003) found that fathers were less likely than mothers to have used corporate programs to help find harmony between work and family life, fathers frequently chose options that provided flexibility as to when and where work was done. Overall, use of work\u2013family programs such as flexible work practices by fathers were found to be work\u2013family adaptive strategies that predicted greater work\u2013family fit (Hill et al. 2003).\n\nA very basic work\u2013family policy offered in many countries is some type of national parental leave policy (Block et al. 2013). The United States has no national paid maternity or paternity leave policies, thereby leaving this decision to each individual employer. Even without the existence of a national paternity leave policy, which would provide fathers with the ability to take paid time off work after the birth or adoption of a child, many fathers still find a way to take leave for varied amounts of time. This might be through the use of sick days, vacation time , flexible work practices, or even unpaid time off such as the Family and Medical Leave Act. Use of leave, regardless of type, is influenced by fathers' employment characteristics, such as type of employer, tenure of position, level of earnings, etc. The fact that fathers are taking leave regardless of policies or payment indicates that they want and need to spend time with their children and partners after this significant life event. If leave after the birth of a child can increase a father's work\u2013family balance , then it is likely that work commitment and productivity will also be greater due to reduced stress and conflict levels at home. Similar situations exist in the UK and Australia, where paternity leave is either not available or is difficult to use due to cultural barriers and stigmatization of appropriate family roles (Fox et al. 2009; Whitehouse et al. 2007).\n\nOverall, we know that work\u2013family balance programs in the workplace have received great attention, but typically focus on assisting mothers with work\u2013family issues, thus excluding fathers and individuals without children (Hill 2005). Yet, we know that taking leave and working shorter hours are related to fathers being more involved with their infants (Nepomnyaschy and Waldfogel 2007; O'Brien et al. 2007). Employee autonomy at work has also been shown to impact the relationship between work hours and time spent with children (MacDonald and Almeida 2004), as well as decrease work\u2013family conflict (Galinsky 2013). Available work\u2013family policies have the ability to impact fathers' behavior both at home and on the job. Therefore, policies that provide parental leave, shortened work hours, or flexible work practices are likely to increase father involvement (Tanako and Waldfogel 2007). For the employer, the goal is to create and successfully implement programs which not only assist the worker but also maintain or increase productivity of their employees on the job.\n\nSimilarly, developing work\u2013family initiatives that specifically aim to benefit fathers and those without children has the potential to increase productivity, retention , and company loyalty. For example, through the existence of maternity leave but not paternity leave , the policies imply that a mother's responsibility is to childcare and a father's responsibility is financial support , thereby reinforcing the gendered ideal parent and worker stereotypes . \"Social policies supporting men's care\u2014particularly parental leave dedicated to fathers\u2014are needed to enhance gender equality and work\u2013family reconciliation for men and for women\" (Fox et al. 2009, p. 313).\n\n## 2.7 Opportunities for Future Research\n\nFuture research regarding men and their work\u2013family experiences is very necessary, and there are ample research streams to pursue (Hill 2005) . For instance, there are relatively few published research studies that analyze men separately from women. As demonstrated earlier in this chapter, there are some differences in the ways that men and women experience work\u2013family conflict and enrichment. By not considering men separately from women in work\u2013family research, researchers send the message that they do not recognize the need to understand men's work\u2013family interactions distinctly from women's. Omitting such research diminishes our understanding of the work\u2013family interface and disenfranchises men, especially fathers .\n\nIn addition to analyzing men's experiences separately from women's experiences, future work\u2013family research should also consider more cross-cultural studies of men. This should include fathers and men without children, sexual minorities, and others with responsibilities outside of the family, such as community. Other countries appear to be ahead of the USA in terms of understanding men's work\u2013family experiences. For instance, Canada (Doucet 2004; Konrad and Yang 2012), Belgium (Doucet and Merla 2007), Sweden (Allard et al. 2011), and Australia (Sav et al. 2013; Whitehouse et al. 2007) have been represented in men's work\u2013family research. However, this research is not necessarily generalizable to American men, given the differences in work\u2013family accommodations at national levels among the various countries and the role that culture can play in perceptions of work\u2013family experiences (Sav et al. 2013) .\n\nAdditionally, cultural differences based on race , ethnicity, and country of origin are likely to provide further insight into men's work\u2013family experiences, especially in the USA, where cultural diversity is prevalent. For example, one study found differences in the work\u2013family experiences of Hispanics versus Blacks and Whites. Hispanics are more likely to demonstrate an increased difference between men and women, likely because of differences in culture and work experiences (Roehling et al. 2005). Future research is needed to continue to build understanding of different cultures and perceptions of gender in terms of work\u2013family research.\n\nWhile we agree that paternity leave allowance is an important starting point for accommodating the evolution of men's work and family involvement, there is a need for future research to be devoted to further understanding the determinants and consequences of paternal leave-taking (Nepomnyaschy and Waldfogel 2007). Researchers have reported that taking advantage of parental leave policies may have a negative stigma within an organization that generally discourages employees from taking advantage of such policies. When women take maternity leave , they may find themselves perceived as less devoted to their work than either before having children and\/or in comparison to their male coworkers. For many women, their maternity leave initiates their entrance onto the \"mommy-track\" where they have fewer upward mobility options within their organization, regardless of whether or not they are on the mommy-track by choice. Future research should be devoted to determining whether such consequences also await men who take advantage of paternity leave options. We surmise that because the length of maternity leave is generally longer than a paternity leave, men may not necessarily face the same consequences as women, but understanding what consequences men will face is critical to fully exploring paternity leave policies as a consideration for men's changing roles in both the work and family domains .\n\nSimilarly, women usually take the maternity leave that is offered to them because of the physical complications associated with giving birth, in addition to the desire to bond with their new child (which would also apply to adoption). However, men's motivations for engaging in paternity leave are somewhat less obvious than women's reasons for taking maternity leave . Future research into the determinants of men's use of paternity leave will be important for determining how best to devise paternity leave policies that actually meet the needs of employed fathers.\n\nFinally, future research should also consider the roles of occupational status and\/or differences in place of employment as potential explanatory variables for differential work\u2013family experiences between men and women (Izraeli 1993). Since the work\u2013family literature is primarily applicable to professional\/managerial employees, little is known about the work\u2013family experiences of employees who are employed in blue-collar positions, earn hourly wages, and\/or are not eligible for benefits such as the paternity leave policies for which we advocate. For this segment of the population, there may be fewer differences between men and women with regard to work\u2013family experiences. This speculation is based on the assumption that lower wage workers have fewer occupational options than their counterparts in higher status positions. So, both women and men in the lower status demographic may feel like they need their jobs to support their families, whereas managerial\/professional employees may have the option of reducing work hours, working a flexible schedule, working from home, or utilizing benefits to help them manage work and family demands with less fear of losing their livelihood. Such options, of course, will vary from one organization to the next, based on the organizational culture and \"work\u2013family friendliness\" .\n\n## 2.8 Conclusion\n\nWhile we are moving away from the notions of the \"ideal worker\" and the \"ideal parent\" in contemporary society, they have not gone without penalizing men who desire more flexibility in the workplace or more time with family. Two recent studies found that men who sought to use workplace flexibility practices were deemed poor \"organizational citizens,\" seen as uncommitted to their workplace, and were characterized as portraying more \"undesirable\" feminine traits (Rudman and Mescher 2013; Vandello et al. 2013). Such a reaction is unfortunate, as men\u2014especially fathers\u2014are increasingly committed to family at emotional and caretaking levels. As Kaufman (2014) demonstrates, an increasing number of fathers are filling the roles of \"new\" or \"super\" dads, as described earlier, in an effort to contribute more equally with their female partners. One step to including men in the work\u2013family discussion is first to recognize that they\u2014like women\u2014have work\u2013family needs. We can begin to be inclusive of men by more actively pursuing research which investigates men's work\u2013family issues independently from those of women. It is time to recognize that everyone, not just mothers, have work\u2013family needs. As such, we start by including men, specifically fathers, in the conversation.\n\nReferences\n\nAllard, K., Hass, L., & Hwang, C. P. (2007). Exploring the paradox: Experiences of flexible working arrangements and work-family conflict among managerial fathers in Sweden. Community, Work, and Family, 10, 475\u2013493.CrossRef\n\nAllard, K., Haas, L., & Hwang, C. P. (2011). Family-supportive organizational culture and fathers' experiences of work-family conflict in Sweden. Gender, Work, and Organization, 18, 141\u2013157.CrossRef\n\nAllen, T. D. (2001). Family-supportive work environments: The role of organizational perceptions. Journal of Vocational Behavior, 58, 414\u2013435.CrossRef\n\nAnafarta, N., & Kuru\u00fcz\u00fcm, A. (2012). Demographic predictors of work-family conflict for men and women: Turkish case. International Journal of Business and Management, 7, 145\u2013158.CrossRef\n\nAycan, Z., & Eskin, M. (2005). Relative contributions of childcare, spousal support, and organizational support in reducing work-family conflict for men and women: The case of Turkey. Sex Roles, 53, 453\u2013471.CrossRef\n\nBailyn, L., Drago, R., & Kochan, T. A. (2001). Integrating work and family life: A holistic approach. A report of the Sloan work-family policy network. MIT: Sloan School of Management.\n\nBarrah, J. L., Shultz, K. S., Baltes, B., & Stolz, H. E. (2004). Men's and women's eldercare-based work-family conflict: Antecedents and work-related outcomes. Fathering, 2, 305\u2013330.CrossRef\n\nBerdahl, J. L., & Moon, S. H. (2013). Workplace mistreatment of middle class workers based on sex, parenthood, and caregiving. Journal of Social Issues, 69, 341\u2013366.CrossRef\n\nBlair-Loy, M. (2001). Cultural constructions of family schema: The case of women finance executives. Gender and Society, 15, 687\u2013709.CrossRef\n\nBlock, R. N., Park, J. Y., & Kang, Y. H. (2013). Statutory leave entitlements across developed countries: Why US workers lose out on work-family balance. International Labour Review, 152, 125\u2013143.CrossRef\n\nBrescoll, V. L., Glass, J., & Sedlovskaya, A. (2013). Ask and Ye shall receive? The dynamics of employer-provided flexible work options and the need for public policy. Journal of Social Issues, 69, 367\u2013388.CrossRef\n\nCampbell, C. S. (2001). Work cultures and work\/family balance. Journal of Vocational Behavior, 58, 348\u2013365.CrossRef\n\nCarlson, D. S., Kacmar, K. M., Wayne, J. H., & Grzywacz, J. G. (2006). Measuring the positive side of the work-family interface: Development and validation of a work-family enrichment scale. Journal of Vocational Behavior, 68, 131\u2013164.CrossRef\n\nCasper, W. J., Eby, L. T., Bordeaux, C., Lockwood, A., & Lambert, D. (2007). A review of research methods in IO\/OB work-family research. Journal of Applied Psychology, 92, 28\u201343.PubMedCrossRef\n\nClark, S. C. (2000). Work\/family border theory: A new theory of work\/family balance. Human Relations, 53, 747\u2013770.CrossRef\n\nColtrane, S. (2000). Research on household labor: Modeling and measuring the social embeddedness of routine family work. Journal of Marriage and family, 62, 1208\u20131233.CrossRef\n\nColtrane, S., Miller, E. C., DeHaan, T., & Stewart, L. (2013). Fathers and the flexibility stigma. Journal of Social Issues, 69, 279\u2013302.CrossRef\n\nCoverman, S., & Sheley, J. F. (1986). Change in men's housework and child-care time, 1965-1975. Journal of Marriage and Family, 48, 413\u2013422.CrossRef\n\nCrowley, M. S. (1998). Men self-perceived adequacy as the family breadwinner: Implications for their psychological, marital and work-family well-being. Journal of Family and Economic Issues, 19, 7\u201323.CrossRef\n\nDamaske, S. (2011). For the family? How class and gender shape women's work. Oxford: Oxford University Press.\n\nDoucet, A. (2004). \"It's almost like I have a job, but I don't get paid\": Fathers at home reconfiguring work, care, and masculinity. Fathering, 2, 277\u2013303.CrossRef\n\nDoucet, A., & Merla, L. (2007). Stay-at-home fathering: A strategy for balancing work and home in Canadian and Belgian families. Community, Work, and Family, 10, 455\u2013473.CrossRef\n\nEmslie, C., & Hunt, K. (2009). 'Live to work' or 'work to live'? A qualitative study of gender and work-life balance among men and women in mid-life. Gender, Work, and Organization, 16, 151\u2013172.CrossRef\n\nFagan, J., & Press, J. (2008). Father influences on employed mothers' work-family balance. Journal of Family Issues, 29, 1136\u20131160.CrossRef\n\nFigueroa, A. J., Aburto, M. A. C., & Acevedo, R. Z. (2012). Work-family conflict, parental efficacy and perceived parenting styles in fathers and mothers from the city of Talca, Chile. Acta Colombiana de Psicolog\u00eda, 15, 57\u201365.\n\nFried, M. (1998). Taking time: Parental leave policy and corporate culture. Philadelphia: Temple University Press.\n\nFox, E., Pascall, G., & Warren, T. (2009). Work-family policies, participation, and practices: Fathers and childcare in. Europe. Community, Work, and Family, 12, 313\u2013326.CrossRef\n\nGalinsky, E., Aumann, K., & Bond, J. T. (2013). Times are changing: Gender and generation at work and at home in the USA. In S. Poelmans, J. Greenhaus, & M. Las Heras Maestro (Eds.), Expanding the boundaries of work-family research: A vision for the future (p. 279). London: Palgrave Macmillan.\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGreenhaus, J. H., & Powell, G. N. (2006). When work and family are allies: A theory of work-family enrichment. Academy of Management Review, 31, 72\u201392.CrossRef\n\nGreenhaus, J. H., Collins, K. M., & Shaw, J. D. (2003). The relation between work-family balance and quality of life. Journal of Vocational Behavior, 63, 510\u2013531.CrossRef00042-8)\n\nGuest, D. E. (2002). Perspectives on the study of work-life balance. Social Science Information, 41, 255\u2013279.CrossRef\n\nGrzywacz, J. G., & Carlson, D. S. (2007). Conceptualizing work-family balance: Implications for practice and research. Advances in Developing Human Resources, 9, 455\u2013471.CrossRef\n\nHiggins, C. A., & Duxbury, L. E. (1992). Work-family conflict: A comparison of dual-career and traditional-career men. Journal of Organizational Behavior, 13, 389\u2013411.CrossRef\n\nHill, E. J. (2005). Work-family facilitation and conflict, working fathers, and mothers, work-family stressors and support. Journal of Family Issues, 26, 793\u2013819.CrossRef\n\nHill, E. J., Hawkins, A. J., M\u00e4rtinson, V., & Ferris, M. (2003). Studying 'working fathers': Comparing fathers' and mothers' work-family conflict, fit, and adaptive strategies in a global high-tech company. Fathering, 1, 239\u2013261.CrossRef\n\nHochschild, A. R. (1989). The second shift. New York: Avon Books.\n\nHochschild, A. R. (1997). The time bind: When work becomes home and home becomes work. New York: Holt.\n\nIzraeli, D. N. (1993). Work\/family conflict among women and men manager in dual-career couples in Israel. Journal of Social Behavior & Personality, 8, 371\u2013385.\n\nJanzen, B. L., & Kelly, I. W. (2012). Psychological distress among employed fathers: Associations with family structure, work quality, and the work-family interface. American Journal of Men's Health, 6, 294\u2013302.PubMedCrossRef\n\nKaufman, G. (2014). Superdads: How fathers balance work and family in the 21st century. New York: New York University Press.\n\nKaufman, G., & Uhlenberg, P. (2000). The influence of parenthood on the work effort of married men and women. Social Forces, 78, 931\u2013947.CrossRef\n\nKinnunen, U., & Mauno, S. (1998). Antecedents and outcomes of work-family conflict among employed women and men in Finland. Human Relations, 51, 157\u2013177.\n\nKirby, E. L., & Krone, K. J. (2002). \"The policy exists but you can't really use it\": Communication and the structuration of work-family policies. Journal of Applied Communication Research, 30, 50\u201377.CrossRef\n\nKonrad, A. M., & Mangel, R. (2000). The impact of work-life programs on firm productivity. Strategic Management Journal, 21, 1225\u20131237.CrossRef21%3A12<1225%3A%3AAID-SMJ135>3.0.CO%3B2-3)\n\nKonrad, A. M., & Yang, Y. (2012). Is using work-life interface benefits a career-limiting move? An examination of women, men, lone parents, and parents with partners. Journal of Organizational Behavior, 33, 1095\u20131119.CrossRef\n\nLee, M., Vernon-Feagans, L., Vazquez, A., & Kolak, A. (2003). The influence of family environment and child temperament on work-family role strain for mothers and fathers. Infant and Child Development, 12, 421\u2013439.CrossRef\n\nLeineweber, C., Baltzer, M., Hanson, L. L. M., & Westerlund, H. (2013). Work-family conflict and health in Swedish working women and men: A 2-year prospective analysis (the SLOSH study). European Journal of Public Health. http:\/\/\u200beurpub.\u200boxfordjournals.\u200borg\/\u200bcontent\/\u200bearly\/\u200b2012\/\u200b06\/\u200b08\/\u200beurpub.\u200bcks064.\u200bfull.\u200bpdf+html. Accessed 15 Jan 2014.\n\nLevine, J. A., & Pittinsky, T. L. (1997). Working fathers: New strategies for balancing work and family. Reading: Addison-Wesley.\n\nLewis, S. (1997). 'Family friendly' employment policies: A route to changing organizational culture or playing about at the margins? Gender, Work and Organization, 4 (1), 13\u201323.\n\nLewis, S. (2001). Restructuring workplace cultures: The ultimate work-family challenge? Women in Management Review, 16, 21\u201329.CrossRef\n\nLivingston, B. A., & Judge, T. A. (2008). Emotional responses to work-family conflict: An examination of gender role orientation among working men and women. Journal of Applied Psychology, 93, 207\u2013216.PubMedCrossRef\n\nLivingston, G. (2013). The rise of single fathers: A ninefold increase since 1960. Pew Research Center: Washington, D.C. (http:\/\/\u200bwww.\u200bpewsocialtrends.\u200borg\/\u200bfiles\/\u200b2013\/\u200b07).\n\nLoerch, K. J., Russell, J. E., & Rush, M. C. (1989). The relationships among family variables and work-family conflict for men and women. Journal of Vocational Behavior, 35, 288\u2013308.CrossRef90031-6)\n\nLoscocco, K. A. (1997). Work-family linkages among self-employed women and men. Journal of Vocational Behavior, 50, 204\u2013226.CrossRef\n\nMacDonald, D. A., & Almeida, D. M. (2004). The interweave of fathers' daily work experiences and fathering behaviors. Fathering, 2, 235\u2013251.CrossRef\n\nMills, M. J., & Grotto, A. R. (2012). Work-life and the upper echelon: Perceptions, outcomes, and gender differences in senior leadership. Paper presented at the Academy of Management Conference. Boston, Massachusetts, USA.\n\nMunn, S. L. (2013). Unveiling the work-life system: The influence of work-life balance on meaningful work. Advances in Developing Human Resources, 15, 401\u2013417.CrossRef\n\nNepomnyaschy, L., & Waldfogel, J. (2007). Paternity leave and fathers' involvement with their young children. Community, Work, and Family, 10, 427\u2013453.CrossRef\n\nO'Brien, M., Brandth, B., & Kvande, E. (2007). Fathers, work, and family life. Community, Work, and Family, 10, 375\u2013386.CrossRef\n\nPowell, B., Bolzendahl, C., Geist, C., & Steelman, L. C. (2012). Counted out: Same-sex relations and Americans' definitions of family. New York: Russell Sage.\n\nPowell, G. N., & Greenhaus, J. H. (2006). Managing incidents of work-family conflict: A decision-making perspective. Human Relations, 59, 1179\u20131212.CrossRef\n\nRapoport, R., Lotte, B., Fletcher, J. K., & Pruitt, B. H. (2002). Beyond work-family balance: Advancing gender equity and workplace performance. San Francisco: Jossey-Bass.\n\nRoehling, P. V., Jarvis, L. H., & Swope, H. E. (2005). Variations in negative work-family spillover among White, Black, and Hispanic American men and women: Does ethnicity matter? Journal of Family Issues, 26, 840\u2013865.CrossRef\n\nRoos, E., Lehelma, E., & Rahkonen, O. (2006). Work-family conflict and drinking behaviours among employed women and men. Drug and Alcohol Dependence, 83, 49\u201356.PubMedCrossRef\n\nRoy, K. M. (2005). Transitions on the margins of work and family life for low-income African-American fathers. Journal of Family and Economic issues, 26, 77\u2013100.CrossRef\n\nRudman, L. A., & Mescher, K. (2013). Penalizing men who request a family leave: Is flexibility stigma a femininity stigma? Journal of Social Issues, 69, 322\u2013340.CrossRef\n\nSav, A., Harris, N., & Sebar, B. (2013). Work-life conflict and facilitation among Australian Muslim men. Equality, Diversity, & Inclusion, 32, 671\u2013687.CrossRef\n\nSchneer, J. A., & Reitman, F. (1993). Effects of alternate family structures on managerial career paths. Academy of Management Journal, 36, 830\u2013843.CrossRef\n\nSperling, J. H. (2014). Reframing the work-family conflict debate by rejecting the ideal parent norm. American University Journal of Gender, Social Policy & the Law, 22, 47\u201390.\n\nTanako, S., & Waldfogel, J. (2007). Effects of parental leave and work hours on fathers' involvement with their babies. Community, Work, and Family, 10, 409\u2013426.CrossRef\n\nThompson, C. A., Beauvais, L. L., & Lyness, K. S. (1999). When work-family benefits are not enough: The influence of work-family culture on benefit utilization, organizational attachment, and work-family conflict. Journal of Vocational Behavior, 54, 392\u2013415.CrossRef\n\nU.S. Bureau of Labor Statistics. (2013). Women in the labor force: A databook. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200bwlf-databook-2012.\u200bpdf. Accessed 29 March 2014.\n\nU. S. Census Bureau. (2012). Profile America factors for features: Unmarried and single Americans week Sept. 16\u201322, 2012. http:\/\/\u200bwww.\u200bcensus.\u200bgov\/\u200bnewsroom\/\u200breleases\/\u200bpdf\/\u200bcb12ff-18_\u200bunmarried.\u200bpdf.\n\nVandello, J. A., Hettinger, V. E., Bosson, J. K., & Siddiqi, J. (2013). When equal isn't really equal: The masculine dilemma of seeking work flexibility. Journal of Social Issues, 69, 303\u2013321.CrossRef\n\nvan Steenbergen, E. F., Ellemers, N., & Mooijaart, A. (2007). How work and family can facilitate each other: Distinct types of work-family facilitation and outcomes for women and men. \u00adJournal of Occupational Health Psychology, 12, 279\u2013300.PubMedCrossRef\n\nWarren, J. A., & Johnson, P. J. (1995). The impact of workplace support on work-family role strain. Family Relations: An Interdisciplinary Journal of Applied Family Studies, 144, 63\u2013169.\n\nWinfield, I., & Rushing, B. (2005). Bridging the border between work and family: The effects of supervisor-employee similarity. Sociological Inquiry, 75, 55\u201380.CrossRef\n\nWhitehouse, G., Diamond, C., & Baird, M. (2007). Fathers' use of leave in Australia. Community, Work, & Family, 10, 387\u2013407.CrossRef\n\nWilliams, J. C. (2000). Unbending gender: Why work and family conflict and what to do about it. Oxford: Oxford University Press.\n\nWilliams, J. C. (2010). Reshaping the work-family debate: Why men and class matter. Cambridge. Massachusetts: Harvard University Press.\n\nWilliams, J. C., Manvell, J., & Bornstein, S. (2006). How the press covers work\/family conflict: The untold story of why women leave the workforce. The Center for Worklife Law, University of California, Hastings College of the Law.\n\nYonetani, Y., Watanabe, S., & Kanazawa, Y. (2007). On the relationships among organizational family supportiveness, work-family conflict, and turnover intention: Evidence on Japanese men. International Journal of Human Resources Development and Management, 7, 319\u2013334.CrossRef\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_3\n\n# 3. Gender, Poverty, and the Work\u2013Family Interface\n\nHeather Odle-Dusseau1  , Anna C. McFadden2  and Thomas W. Britt2\n\n(1)\n\nDepartment of Management, Gettysburg College, 300 North Washington Street, Box 395, 17325 Gettysburg, PA, USA\n\n(2)\n\nDepartment of Psychology, Clemson University, 418 Brackett Hall, 29634 Clemson, SC, USA\n\nHeather Odle-Dusseau (Corresponding author)\n\nEmail: hodle@gettysburg.edu\n\nAnna C. McFadden\n\nEmail: amcfadd@clemson.edu\n\nThomas W. Britt\n\nEmail: twbritt@clemson.edu\n\nKeywords\n\nWork\u2013familyGenderPovertyJob demands-resourcesSegmented labor markets\n\n## 3.1 Introduction\n\nThe work\u2013family interface cannot be discussed without taking into consideration the role that gender plays in work and family systems. Yet much of our understanding of gender and the work\u2013family interface has been through the lens of middle- and upper-class, as well as professional-level, employees' experiences. As such, the work\u2013family experiences of low-income individuals and those living at or below the poverty line are less understood. Given that women are more likely to be in poverty than men, leading to the coined phrase \"feminization of poverty\" (Pearce 1978), it is important to consider gender and work\u2013family interactions within the context of poverty. Drawing from multiple disciplines (e.g., psychology, sociology, gender studies, and economics), this chapter examines how gender and the work\u2013family interface interact for individuals who are working in low-wage jobs , and are often found to be living at or below the poverty line. Combining the theoretical frameworks of segmented labor markets and the Job Demands-Resources (JD-R) model (Demerouti et al. 2001), our goal is to integrate the themes of gender, poverty, and work\u2013family experiences, and to develop avenues for future research .\n\nIn order to guide our discussion of these issues, in Fig. 3.1 we provide a descriptive model for how gender interacts with employment in low-wage jobs to affect work\u2013family dynamics, which then affect the health and well-being of women. We begin this chapter by first addressing descriptions and explanations of the sex segregation of jobs, or the trend that certain jobs are more likely to be filled by men (e.g., \u00adconstruction workers) while other jobs are more likely to be filled by women (e.g., child care providers). This leads into a discussion of the phenomenon of the \"feminization of poverty,\" examining the organizational, economic, and labor structures that contribute to the fact that women are more likely to live in poverty than men, which can be directly linked to the sex segregation of jobs.\n\nFig. 3.1\n\nModel of the integration of gender, poverty, and the work\u2013family interface\n\nWe then apply the framework of the Job Demands-Resources (JD-R) model (Demerouti et al. 2001) to help describe how different types of jobs are also associated with different demands, resources, and subsequent outcomes for males and females. Specifically, we describe how low-wage jobs , which are more likely to be filled by women, often have higher demands and fewer resources (e.g., wages, flexibility, autonomy) for managing family responsibilities. The lack of resources then contributes to a higher potential for work\u2013family conflict, a lower potential for work\u2013family enrichment, and subsequently compromised levels of health and well-being. We also discuss public programs aimed at curbing poverty and its correlates , relative to the interaction of gender, poverty, and the work\u2013family interface. Our overall model addresses how the sex segregation of jobs contributes toward a context of poverty that is experienced at different rates for men and women, subsequently creating situations in which men and women differentially experience job demands and resources. These differential demands and resources then influence work\u2013\u00adfamily experiences and subsequent work and well-being outcomes.\n\n## 3.2 Sex Segregation of Occupations and the Feminization of Poverty\n\nThe number of low-income working families in the United States reached 10.4 million in 2011 (an increase from 10.2 million in 2010), according to the Working Poor Families Project (Roberts et al. 2012). Across the globe, and especially in the United States, women are more likely to live in poverty than men; in 2011, the poverty rate for women was 14.6 %, while the poverty rate for men was 10.9 % (National Women's Law Center 2012). Given that this difference can be largely attributed to the fact that there are higher proportions of women found in low-wage jobs (Litchenwalter 2005), we attempt to further explore the work\u2013family interface within these contexts.\n\nSingle mothers and their children are particularly vulnerable to poverty, often due to limited access to employment and the cost associated with raising children (Misra et al. 2012). When looking at poverty rates of female-headed households with children, the proportion of those women living in poverty jumps (from 14.6 % overall) to 40.0 % (National Women's Law Center 2012). Because women are often the main caregivers for children, and because of the risk of poverty associated with women and children, it is imperative to examine how this systemic, dynamic interaction of work, family, gender, and poverty influences the work\u2013family interface.\n\nOccupations can be sex segregated, meaning that some occupations are held by a higher proportion of men, while others are held by higher proportions of women. Powell (2011) used the terms \"female-intensive\" and \"male-intensive\" to describe those jobs that have two-thirds or more of the workforce made up by males or females, such that those jobs where women make up at least 66.7 % of the workforce are considered \"female-intensive.\" Scholars have described various mechanisms to explain why this differentiation exists, with many explanations being traced back to societal categorizations of sex and the prescribed gender roles that form the expectations of male and female roles in society. One set of these expectations is the division of labor , which outlines the paid work domain as mainly the responsibility of men, while the nonwork, home, and caregiver domains are the obligation of women. Although women have increased their labor force participation to approximately 47 % (Powell 2011), there still persists a division of labor in the form of sex segregation of occupations.\n\nThere are a myriad of inter- and intraorganizational reasons that help explain why sex segregation takes place, such as the sex composition of the applicant pool, job search techniques of applicants, job assignment, and promotion practices (Reskin and Bielby 2005). While some of these reasons lie with individuals' choices, there are organizational structures and practices that also contribute to the segregation. Kmec (1995) proposed that sex segregation can arise from both the job applicants' techniques for attaining jobs, as well as the employer's organizational practices and policies and their subsequent discretion in hiring decisions. Similarly, Reskin and Bielby (2005) suggested that there are two specific ways that organizations create sex segregation. First, contributing to trends at the societal level, by not hiring a woman for a particular organization, the chance that she will go to another firm that is more likely to hire women increases. Stemming from psychological biases, although some organizational decision-makers may not consider sex when making the initial hiring decision, sex can be a consideration when making job assignments (Reskin and Bielby 2005). For example, in a factory, males may be more likely to be assigned a job that requires heavy lifting, while females are more likely to be assigned to support staff jobs that do not require heavy lifting. So while sex segregation across organizations may be lowered by hiring practices that explicitly, and legally, disregard biological sex , segregation within an organization could still exist and be reinforced through organizational structures, polices, and practices (Reskin and Bielby 2005).\n\nNoting the shifts in our country's economic structure over the past 40 years, Reid and Rubin (2003) recommended Dual Labor Market Segmentation Theory to help describe potential explanations of the wage inequality among men and women. This framework of segmented labor markets , which became popular in the 1960s and 1970s, is often used to describe two types of labor markets: (1) the primary sector that consists of jobs requiring high levels of skills , education , and training , and rewards employees with high wages and benefits, (2) the secondary sector consisting of low-wage jobs that require little training and education, low levels of skills, and pay low wages with little opportunity for promotion (Reid and Rubin 2003). There is an obvious parallel between these two sectors and the differences seen between the jobs that are male-intensive and female-intensive . Not surprisingly, jobs in the secondary sector tend to be female-intensive. Thus, segmented labor theories provide us with a framework for describing the sex segregation of occupations and jobs.\n\nAlthough not without its critiques (see Reid and Rubin 2003) , segmented labor market theories can aid in describing sex segregation of occupations for several reasons. As outlined by Reid and Rubin (2003), wage inequality, and the place of human capital in work outcomes, is of interest to scholars. Additionally, women and minorities continue to disproportionately make up low-wage occupations and do not as often occupy jobs where there are opportunities to develop new skills and \"move up\" in the organization. Reid and Rubin (2003) note that although intraorganizational processes do influence wage inequality for gender and race , labor market and industrial sectors are also important influences .\n\n### 3.2.1 The Feminization of Poverty\n\nAs seen in Fig. 3.1, the sex segregation of jobs contributes to the feminization of poverty, a term coined in 1978 by Diana Pearce in reference to the concentration of poverty among women as compared to men. When describing this trend, it is important to understand that it is too simplistic to designate the feminization of poverty as that which describes poor women in need of financial assistance . Rather, beyond the struggle to obtain basic living wages for food and housing, the feminization of poverty also refers to the lack of opportunity to move out of poverty and obtain a quality of life that considers more than basic needs. These effects on quality of life then permeate to the family structures in which these women live.\n\nThe disproportionate number of women in poverty is seen despite increased workforce participation by women and increased education levels . Litchenwalter (2005) analyzed data from the 2000 census in order to examine the contributors to the gender-poverty disparity in the 70 largest cities in the United States. Noting the long-standing sex differences in paid and unpaid labor (i.e., women are usually more responsible for unpaid labor like childrearing and domestic work), the shift toward more single female-headed families, and organizational structures that are themselves hindrances to working women and mothers, Litchenwalter (2005) found that women were more likely than men to be in poverty when in cities where women work more low-wage jobs and where there were a high proportion of women heading single parent homes. In addition, there was no significant correlation of poverty with education. When using regression analysis, which included the predictors of overall earnings, female representation in low- and high-wage jobs, labor force participation, and proportion of female-headed households, only the proportion of females in low-wage jobs remained a significant predictor of poverty disparity between men and women. What is more, it explained a substantial 41 % of the variance in that disparity (Litchenwalter 2005) .\n\nThe feminization of poverty is not unique to the United States, although it is more prominent in the United States. For example, the risk of poverty for single mothers is below 2 % in Sweden, but is 35 % in the US (Misra et al. 2012). Casper et al. (1994) compared the correlates of the gender-poverty gap in the US to those of other Western industrialized countries. The results revealed that the US has the largest gender\u2013poverty gap, with women being 41 % more likely to be living in poverty than men. In a more recent analysis, Brady and Kall (2007) examined the pattern of men and women's poverty in 18 countries across 31 years. They found that important correlates of poverty for both men and women were economic growth, manufacturing employment , social security transfers (e.g., social security pensions, family allowances, unemployment) , public health spending, and labor force participation. However, the feminization of poverty was most related to social security transfers, single motherhood, and labor force participation. In other words, the experiences of work and family are different for women in poverty. We consider these factors when discussing the implications of the feminization of poverty for the work\u2013family interface .\n\nA related issue when discussing the feminization of poverty is that of the motherhood wage penalty , or the occurrence of overall loss in wages and benefits that mothers experience as compared to fathers and to women in the labor force who do not have children. Budig and Hodges (2010) examined whether the motherhood penalty affected women across income levels similarly. They compared low-wage, middle-wage, and high-wage white1 female workers. High-waged women were more likely to have access to resources that assist in work\u2013family management than their lower-waged counterparts, as well as hold more family-friendly jobs (e.g., have workplace policies that assist in managing family responsibilities) .\n\nThere are various contexts the authors analyzed to better understand how the motherhood penalty may impact those differently across wage levels , such as spouse's earnings, welfare receipt, age of children at home, and timing of motherhood. Although human capital (e.g., education and experience) plays a part, it is small in that the motherhood penalty decreases (but does not go away) as education increases, yet the motherhood penalty increases with skill levels. What is more, the authors found that low-wage workers experienced the largest motherhood wage penalty, and that the penalty for low-wage women with young children is five times higher than that of high-wage women with young children. Budig and Hodges (2010) concluded that mothers who are most in need of stable wages (low-income mothers) are the ones most likely to experience the motherhood penalty .\n\nIn addition, Glauber (2012) examined if the wage penalty is larger for women in female-intensive jobs or for those in male-intensive jobs. She pointed out that both men and women pay a wage penalty when working in female-intensive jobs, but the penalty is greater for women. Glauber outlined some of the arguments for the production of sex segregation, such as differences in human capital (i.e., women lose job experience and skills after a child is born while they take time off), and discrimination against hiring and promoting mothers. Yet even when controlling for job characteristics, part-time work , education, and experience, the motherhood wage penalty still exists. Based on data collected longitudinally from a national sample, Glauber (2012) found that women in female-intensive jobs were paid less, were more likely to be married and to have children, and were also more likely to work part-time than were women working in male-intensive or sex-neutral jobs. In addition, the wage penalty for women working in female-intensive jobs increased with the number of children a mother had (7 % for one or two children, and 15 % for three or more), while those women in neutral or male-intensive jobs only paid a penalty when the number of children reached three or more (10 %). Overall, women in female-intensive jobs \"start out with lower wages and pay a large wage penalty for each child they have\" (p. 126). What is more, this penalty does not appear to be offset or compensated for by benefits in female-intensive jobs , such as job satisfaction , access to health insurance , vacation time , flexible scheduling, or maternity leave .\n\n### 3.2.2 Feminization of Poverty and the Work\u2013Family Interface\n\nResearch has provided evidence that low-wage workers experience work and family responsibilities differently, and when we consider gender, these negative effects become even more robust. Low-income families have different experiences when managing work and family domains compared to the more commonly studied middle- to upper-class families. Perry-Jenkins (2005) interviewed 153 dual-earner , working-class couples during their transition to parenthood. The parents were working in occupations that would be classified in the \"secondary sector\" based on the level of education and experience required, as well as the wages paid. Participants described experiencing unstable work hours, the need to \"piece together\" sick and personal time for maternity leave (thereby leaving limited or no time to take off from work when a child was sick or had a doctor's appointment), limited health insurance coverage, and rarely any child care assistance or benefits. Relatedly, Offer (2012a) points out that recent research undermines prior thinking that low-income individuals rely more on social support networks for childcare than high-income individuals, finding that in terms of total financial, instrumental , and emotional support , low-income families tend to have fewer support networks when compared to middle and upper-class families.\n\nWhen limiting analyses to female-headed households in low-wage families, the picture becomes even bleaker. Offer (2012b) examined how low-income mothers, who often rely on social support for childcare , also face barriers to achieving that support. These barriers include poor health, which is experienced at higher rates among low-income individuals than middle- and upper-class individuals (Burton et al. 2005), as well as \"adverse life events\" such as domestic violence incidents and illegal activities . Offer (2012b) provided data suggesting that both poor physical and psychological health serve as barriers to social support , including child care support, emotional support , instrumental support, and financial support . Specifically, participants who reported low health at Time 1 reported low perceived support at Time 2. This pattern was also observed for women who experienced domestic violence. Moreover, there appeared to be a mediating effect of poor health on the relationship between domestic violence and low support, in that the experiences of domestic violence lead to poor health, which in turn leads to low levels of perceived support .\n\nIn summary, sex segregated occupations have been demonstrated to contribute to the higher proportion of females in poverty as compared to males. This propensity to live in poverty is even higher when considering the presence of children in a household. What is more, individuals in low-wage jobs have very different experiences when managing work and family domains. In the next section, we utilize the Job Demands-Resources (JD-R) model to provide a framework for explaining the effects that the sex segregation of jobs and poverty may have on the work\u2013family experiences of women in low-wage jobs. Components of this model are highlighted in Fig. 3.1.\n\n## 3.3 Applying the JD\u2013R Theory to Gender, Poverty, and the Work\u2013Family Interface\n\nThe JD-R model (Demerouti et al. 2001) contends that job demands deplete energy , and therefore have negative effects on employees, leading to burnout . Alternatively, job resources can activate motivational processes, thereby having positive effects that lead to job engagement . This theory has been successfully applied to the work\u2013family interface, in that resources have also been associated with positive outcomes in the context of work\u2013family experiences. For example, perceptions of family-supportive work environments have been associated with employee well-being (e.g., Odle-Dusseau et al. 2013; Thompson and Prottas 2006).\n\nIn considering how demands and resources differ for low-wage individuals, and how these differences might contribute to problems managing the work\u2013family interface, we turn to an analysis of how poverty affects individuals, as provided by Leana et al. (2012). Specifically, these authors view the effect poverty may have on low-income individuals as developing for three potential reasons: categorical explanations, whereby individual-level factors are seen as the source (e.g., less training can mean fewer skills to take with them to the next job or to deal with specific work stressors, causing an increase in negative personal and work-related outcomes); compositional explanations, whereby the structure of the environment (such as the organization) is the focus (e.g., low-wage workers may not be well integrated into the organization with those who earn higher wages, which can limit opportunities for advancement); and relational explanations, whereby individual relationships and social connections are the focus (e.g., low-wage workers may have a more difficult time engaging in social or networking relationships with higher wage earners within the organization). Each facet may exert a negative influence when explaining how lower income individuals may lack the resources necessary to appropriately combat demands at work, resulting in negative work\u2013family experiences.\n\nUnderstanding why women in single-parent households are more likely to experience poverty and other negative consequences of low-wage work can be considered within the JD-R framework. Specifically, children tend to create more demands on a household, and are not able to provide equivalent resources to offset these demands (Brady and Kall 2007). In the same vein, childcare (a potential resource) is not available in poor communities at the same rate that it is in more affluent communities, making childcare for low-wage mothers even more difficult to obtain (Budig and Hodges 2010). Thus the demands of having children in single-parent households, in combination with a lack of resources to manage these responsibilities, can contribute to the poverty of single mothers and their children.\n\nThe JD-R model can also assist us in understanding how sex segregation of occupations influences different work and family experiences. Reskin and Bielby (2005) point out that segregation, whether at the organizational level or the job level, can result in unequal rewards for men and women. Therefore, inequalities between men and women can be traced to sex compositions of organizations and jobs, not actual differences between men and women (i.e., their biological sex) . Some researchers suggest that this can be partly attributed to the devaluation of jobs that are female-intensive (e.g., Rapoport et al. 2002). Importantly, the inequalities are not due to the amount of effort put into work by men and women. Past research has contradicted this suggestion, finding women to report just as much or more effort compared to men with similar household responsibilities (Bielby and Bielby 1988). As this chapter highlights, there are more valid explanations of these inequalities.\n\nWhen considering why resources may differ for jobs that are lower wage and are more likely to be filled by women, Glauber (2011) compared the opportunities for flexibility in male-intensive, female-intensive, and integrated\/neutral (i.e., neither male- nor female-intensive) jobs. Based on the theory of compensating wage differentials , female-intensive jobs should provide employees more access to flexibility as a trade-off for the lower wages. However, Glauber's (2011) results did not support this\u2014individuals in integrated occupations reported more flexibility access than individuals in female-intensive jobs. These results suggest that dual labor market theory is a more appropriate framework for understanding how resources differ among male- and female-intensive jobs, in that female-intensive jobs tends to be those in the secondary sector, thus having fewer benefits like flexibility.\n\nAs seen in Fig. 3.1, the sex-segregation of jobs has the effect of increasing the number of demands for female employees, while decreasing the resources available for offsetting those demands. The differential experience of demands and resources has implications for the work\u2013family interface of employees in low-wage, female-intensive occupations . Yet, while there has been some research on the work\u2013family experiences of low-wage individuals, as well as research on how work\u2013family experiences can differ for men and women, the research incorporating all three components is sparse. As such, we turn first to research that assessed how work\u2013family experiences differ for low-wage workers, followed by research showing how males and females can differentially experience work and family interactions. We then discuss how to integrate these findings to fit within the larger context of our proposed model.\n\nRecently, Sinclair et al. (2013) suggested that income or economic status may have an important relationship with work\u2013family conflict. Relatedly, Leana et al. (2012) suggested that poverty is a \"strong situation\" that affects how individuals process information and respond to their environment . Thus differences in economic standing have the potential to affect how low-income individuals differ on perceptions of various work and family experiences, as well as work-related outcomes. The negative impact of work\u2013family conflict on work, health, and well-being outcomes may be reduced for those individuals in high-earning jobs because of the increased resources and control over work. Looking specifically at the part one's income plays on the relationships among work demands and work\u2013family experiences, Ford (2011) tested models of the cross-domain effect of work-to-family conflict and family-to-work conflict, hypothesizing that work-to-family conflict would mediate the relationship between work demands and family strain, while incorporating income as a moderator into the models. Finding income to have an important moderating role, Ford (2011) concluded that while high levels of income did not appear to lower the rates of work\u2013family conflict, higher income did seem to provide resources to lower the influence of work\u2013family conflict onto family strain. If gender is taken into account, this effect perhaps could be exacerbated for women, given the evidence that women are more likely to live in poverty.\n\nOther empirical research has shown a direct impact of perceived economic hardship on work and family outcomes. Schieman and Young (2011) hypothesized that the stress from difficulty in obtaining basic necessities due to economic hardship would spill-over into other domains (namely the work domain) and might result in family-to-work conflict. Relying on the premises of the JD-R framework, Schieman and Young (2011) expected that more demanding work conditions would intensify the relationship between economic hardship and family-to-work conflict, but conditions which acted as resources would buffer the negative association between the two. The study found that economic hardship was predictive of family-to-work conflict, and that work conditions also acted as a moderator. For example, those individuals in jobs with less job authority and more job demands experienced a stronger relationship between economic hardship and family-to-work conflict. Although not including gender in their analyses, these findings become relevant as women may be more likely to work in demanding situations with low autonomy , based on the segmented labor theories (Reid and Rubin 2003), therein perhaps increasing their experiences of family-to-work conflict.\n\nTurning to the part that gender plays in the links among poverty and work\u2013family experiences, it is difficult to find research that specifically addresses these relationships simultaneously. The experience of women in professional and managerial-levels jobs has been studied to a much larger extent. For example, in a study of white collar, dual-earner workers, Batt and Valcour (2003) found that women reported marginally higher levels of work\u2013family conflict than did men, but significantly less control over managing work and family conflict. So how do these experiences translate for low-wage women? Unfortunately, the influence that income may have on these relationships is not well understood. As seen in Fig. 3.1, we would hypothesize that the higher levels of work\u2013family conflict experienced by employees in low-wage jobs (that are more likely to be occupied by a greater percentage of women) have the potential to more negatively affect the work, health, and well-being of the employees.\n\nThe negative work outcomes brought on by lack of resources when facing a stressor can be numerous, and in the work\u2013family literature, the outcomes experienced by those facing work-to-family conflict have been shown to include lower performance , job satisfaction (Bruck et al. 2002), and organizational commitment (Aryee et al. 2005), as well as higher turnover intentions and actual turnover (Shaffer et al. 2001), and absenteeism (Kirchmeyer and Cohen 1999). Work\u2013\u00adfamily conflict has also been linked with job performance , with meta-analyses finding that such conflict is related to decreased employee performance. In a review of work-to-family conflict and performance outcomes, Kossek and Ozeki (1999) found that, on average, increases in both work-to-family conflict and family-to-work conflict were associated with decreased performance metrics. This finding was echoed by Hoobler et al. (2010), who reported both work-to-family conflict and family-to-work conflict were negatively related to manager-rated performance (r = \u2212 0.19 for work-to-family conflict; r = \u2212 0.16 for family-to-work conflict), and self-rated performance (r = \u2212 0.03 for work-to-family conflict; r = \u2212 0.22 for family-to-work conflict). In addition, the perception of family support from one's supervisor (a potential resource) has been linked across time with increased perceptions of work\u2013family enrichment, and to subsequent supervisor ratings of performance (Odle-Dusseau et al. 2012). If, as outlined by our model, we expect resources to be fewer for those working in low-wage jobs , we would then expect less likelihood of increases in work\u2013family enrichment and any consequent effects on job \u00adperformance.\n\nAdvancement in one's career is another work-related outcome that stands to be differentially influenced by income, demands, and resources for women. The \"glass ceiling effect,\" or the seemingly invisible barriers women face in advancing their careers, was the focus of Hoobler et al.'s (2009) research. When asked to rate their subordinates' family-to-work conflict and performance , managers tended to rate females lower in performance . These results were obtained even while controlling for a number of variables that can be related to family-to-work conflict (i.e., caring for a dependent, number of children, and marital status) . Additionally, managers also perceived those subordinates high in family-to-work conflict as having less fit with the organization, a variable that was also shown to have a significant impact on subordinate promotability. These results show that women in particular are perceived as experiencing more family-to-work conflict, regardless of how much conflict they may actually experience. In turn, these perceptions predicted how employees were rated for performance and fit with the organization, ultimately affecting the promotions an employee may receive.\n\nHealth and well-being outcomes are also important factors to consider when examining the outcomes of work\u2013family conflict. The JD-R model (Demerouti et al. 2001) provides an explanation as to why individuals experiencing work\u2013family conflict may subsequently experience other negative outcomes, such as depression (Kossek et al. 2006), hypertension, poor physical health , and heavy alcohol use (Frone et al. 1997), decreased well-being (Karimi et al. 2011), and decreased life satisfaction (Greenhaus et al. 2003). Individuals experiencing the demand of work\u2013family conflict may use up their resources attempting to deal with the demand, leading to negative personal outcomes. Additionally, individuals in low-wage jobs may be at a further disadvantage in that they also likely have fewer resources to buffer the negative consequences of work\u2013family conflict, again leading to negative personal and work-related outcomes. For certain, the research examining the intersection of work\u2013family outcomes, gender, and poverty is limited. Additional research is clearly needed in this area in order to understand more fully how contextual variables such as poverty and gender can help to explain differences in work\u2013family experiences.\n\n## 3.4 The Effects of Public Programs on Managing the Work\u2013family Interface\n\nWe have established that low-income workers, who are more likely to be female, are more likely to experience negative work and family interactions. Public policy has been developed to address some of the correlates of these patterns, in the spirit of decreasing the negative outcomes on one's health and well-being . One important assumption of work\u2013family policies on a societal level is that people should not be at risk of falling into poverty because of family care demands (Misra et al. 2007). Although there are public policies in place in the United States that are aimed to assist low-income individuals, Casper et al. (1994) noted that some feminist analyses have demonstrated that welfare programs are often unsuccessful at increasing gender equality , but rather reward households comprised of heterosexual, married couples . In other words, single mothers are often unable to become financially independent with current welfare programs.\n\nIt is important to understand how public policy influences all types of families so that the effectiveness of such policies can be evaluated holistically. Hennessy (2009) proposed that societal expectations of the roles held by men and women are exacerbated by the current policies that are created to assist low-income women, creating a moral dilemma for these women who need to provide financially for their families, but are also expected to be the primary caregiver to their children. Hennessy (2009) described how low-income women are expected to fulfill work responsibilities because they should be responsible for pulling themselves out of the need for welfare . At the same time, a mother's commitment to her family, rooted in societal expectations, is one associated with intense mothering, and is based on the experiences of white, middle-class women. Specifically, work and family conflict arises not only from the need to provide for families, but also from \"cultural and moral contradictions that arise out of constraints on poor mothers' ability to do so\" (p. 560). The part that public policy plays to reinforce societal expectations that perpetuate scenarios for women to remain in poverty is an important component of this discussion.\n\nLooking directly at how work\u2013family policies (i.e., parental leave and child care policies) influence poverty outcomes via support for employment, Misra et al. (2012) noted that we need to be aware of how individual and societal factors intersect to influence the risk of poverty. Misra et al. (2012) examined two types of policies across countries: those that provide state support (such as financial allowances), and those that provide employment opportunities (such as parental leave policies). Based on a sample of women with children, across countries in 11 developed welfare states (i.e., wealthy Western Europe and North American countries, as well as Australia, Israel, and former socialist nations in Eastern Europe), Misra et al. (2012) found that single mothers were at a higher risk of poverty than mothers with partners, and that this probability of poverty was increased when there were fewer family allowances, less generous parental leave policies, and fewer child care provisions. For example in Sweden, where mothers have a low probability of falling into poverty, there are moderate levels of government financial allowance along with generous child care provisions, and high levels of women working full-time. In the United States, where women have a high risk of falling into poverty, there are high levels of women working full-time, but no government allowance for paid leave and low amounts of childcare provisions. Thus, policies that support single mothers outside the home are just as important as supporting them inside the home when it comes to reducing the risk of poverty .\n\nRelatedly, Ray et al. (2009) noted the need to consider the effect that generosity (of time) of family leave has on gender equality . These authors looked at the degree to which family leave policies, across 21 countries, promote an even distribution of time devoted to child care by a mother and father. Interestingly, policies that were more generous to the mother actually created less gender equality, in that the women receive more incentive to take time off of work, therein decreasing the ease with which they could re-enter the workforce after giving birth because of the amount of time she is away from the workplace. With more generous family leave time, mothers end up working even less compared to fathers. What's more, there is also evidence that policies requiring fathers to take parental leave or \"lose it\" (i.e., not transfer it to the mother) increases gender equality.\n\nYet, a lack of family paid leave is also harmful. The United States is the only industrialized country to not offer paid leave when a child is born or adopted into a family. Although the United States does not currently provide paid leave, there is some protection of one's job if time is taken off work when a child is born or adopted, as well as due to the illness of an individual (or one of their family members). The Family and Medical Leave Act (FMLA) protects jobs up to 12 weeks for those employees who have been working for at least 1250 hours over the past year at an organization that has 50 or more employees. However, there are several limitations of the FMLA. For one, there is a large number of workers not covered\u2014only 11 % of work establishments are eligible (Kelly 2008). In addition, those who are eligible also tend to have higher levels of education and income. Put another way, the poorest individuals and families have the least eligibility (Ray et al. 2009), yet these are the ones who need it the most. Finally, the lack of pay makes it hard to take time off. As a result, men are less likely to use FMLA , because they tend to be paid higher amounts than women, and people tend to take shorter leaves so as not to lose out on pay (Kelly 2008). Given that those who fall into low-income categories are in greater need of such a resource, it becomes clear that improvements are necessary. By comparison, Sweden provides 47 weeks of paid leave, the United Kingdom provides 31 weeks, Canada provides 28 weeks, Japan provides 26 weeks, and Italy provides 25 weeks of paid leave (Ray et al. 2009).\n\nAs of January 2014, there is a proposal in the United States Senate that would pass the Family and Medical Insurance Leave Act , otherwise known as the FAMILY Act . The FAMILY Act would allow both full-time and part-time employees, regardless of the size of their employer or tenure with their current employer, the opportunity to take partially paid leave in the event of pregnancy or childbirth , a personal serious health condition, or a serious health condition of a child, parent, spouse, or domestic partner , or for specific military leave purposes. Paid leave would be available for 12 weeks at 66 % of the average monthly income, with a cap of $ 1000 per week (Center for American Progress 2013). The ability to take paid leave from work would assist low-wage workers in managing health issues and would be particularly important for women, to whom the majority of caregiving responsibilities (not to mention childbirth medical needs) fall.\n\nPolicy makers, therefore, need to consider the information gained through research on the influence of work\u2013family policies on those families they are intended to help. It would be advantageous for policy researchers and work\u2013family researchers to work together to assess the intricate interactions that influence the success of such policies for individuals, organizations, and society, so that policy makers have the information that is necessary to help those who are most in need.\n\n## 3.5 Conclusion\n\nIn sum, it becomes apparent that sex (the biological characteristics that make an individual male or female) and gender (the societal expectations of how women and men are to behave\/stereotypes) are critical considerations when having \u00adconversations surrounding the experiences of low-income individuals and the work\u2013family interface. Based on this review and integration of information on poverty, gender, and work\u2013family experiences, we make several conclusions:\n\n1.\n\nWomen are more likely to fill lower-wage jobs, a result of sex segregation of jobs. One of the correlates of this sex segmentation of jobs is the fact that women are more likely to live at or below the poverty line.\n\n2.\n\nThese low-wage, secondary market sector jobs have more demands and fewer resources, including the ability to move into higher levels of the organizational hierarchy (that are also paid higher incomes), as well as resources that allow for the successful managing of family demands in conjunction with work demands .\n\n3.\n\nThe high level of demands and low amount of resources can create a higher probability of work\u2013family conflict and lower levels of work\u2013family enrichment.\n\n4.\n\nThis interaction of gender, poverty, and work\u2013family interactions therein produce a greater likelihood of lower well-being and greater numbers of health symptoms for women compared to men when balancing work and family responsibilities in the context of poverty.\n\nWe see these interactions as a cycle wherein women and others in low-paying jobs have difficulties in managing the work\u2013family interface that place them at greater risk for well-being and health-related problems, which hurts the quality of life for themselves and their families. As such, there is a systematic nature of poverty that includes feedback loops, with a myriad of predictors and outcomes. Work\u2013family researchers have recently begun to incorporate a systems approach to understanding the work\u2013family interface (e.g., Hammer and Zimmerman 2011), allowing for a more holistic evaluation of work and family systems. This approach incorporates the importance of reciprocal relationships, which clearly become important when poverty and gender are incorporated into the overall systemic nature of work and family domains.\n\nTo add to the understanding of the interactions among these systems, we encourage future research that examines how workplace demands and resources differentially impact work\u2013family conflict, work\u2013family enrichment, and subsequent work, family, health, and well-being outcomes for low-wage workers compared to the more traditionally-studied professional and managerial employees. In addition, a better picture of how policy change could improve the ability of low-wage employees, especially women and single mothers, to move out of poverty and into positions where their overall life quality can improve is warranted. We also recommend using a systems-based approach for modeling these variables and their dynamic, interrelated contexts and processes.\n\nWe have proposed a descriptive model (Fig. 3.1) to understand the part that poverty plays on the relationship between gender and work\u2013family experiences. We see poverty as a moderator of this relationship, in that poverty differentially impacts women's work\u2013family experiences compared to men's work\u2013family experiences. This is especially important to consider given the societal role expectations that women are assumed responsible to fill. The restrictions poverty imposes on resources available to manage the work\u2013family interface is pronounced among women in comparison to men. However, we also see poverty as part of the process that explains how sex impacts work\u2013family experiences. Specifically, we see sex as a predictor of poverty, which in turn predicts work\u2013family experiences. As such, we also view poverty as a mediating mechanism explaining how sex can predict work\u2013family experiences. In sum, poverty creates a context for understanding these differences between men and women in work and family domains, and as a result, should be included in discussions of gender and work\u2013family experiences.\n\nReferences\n\nAryee, S., Srinivas, E. S., & Tan, H. H. (2005). Rhythms of life: Antecedents and outcomes of work-family balance in employed parents. Journal of Applied Psychology, 90, 132\u2013146.PubMedCrossRef\n\nBatt, R., & Valcour, P. M. (2003). Human resources practices as predictors of work-family outcomes and employee turnover. Industrial Relations: A Journal of Economy and Society, 42, 189\u2013220.\n\nBielby, D. D., & Bielby, W. T. (1988). She works hard for the money: Household responsibilities and the allocation of work effort. American Journal of Sociology, 93, 1031\u20131059.CrossRef\n\nBrady, D., & Kall, D. (2007). Nearly universal, but somewhat distinct: The feminization of poverty in affluent Western democracies, 1969\u20132000. Social Science Research, 37, 976\u20131007.CrossRef\n\nBruck, C. S., Allen, T. D., & Spector, P. E. (2002). The relation between work-family conflict and job satisfaction: A finer-grained analysis. Journal of Vocational Behavior, 60, 336\u2013353.CrossRef\n\nBudig, M. J., & Hodges, M. J. (2010). Differences in disadvantage: Variation in the motherhood penalty across white women's earnings distribution. American Sociological Review, 75, 705\u2013728.CrossRef\n\nBurton, L. M., Lein, L., & Kolak, A. (2005). Health and mothers' employment in low-income families. In S. M. Bianchi, L. M. Casper, & R. B. King (Eds.), Work, family, health, and well-being (pp. 493\u2013509). Mahwah: Lawrence Erlbaum.\n\nCasper, L. M., McLanahan, S. S., & Garfinkel, I. (1994). The gender-poverty gap: What we can learn from other countries. American Sociological Review, 59, 5949\u20135605.CrossRef\n\nCenter for American Progress. (2013). The FAMILY Act: Facts and frequently asked questions. http:\/\/\u200bwww.\u200bamericanprogress\u200b.\u200borg\/\u200bwpcontent\/\u200buploads\/\u200b2013\/\u200b12\/\u200bFamilyActFactshe\u200bet-FAQs1.\u200bpdf. Accessed Jan 2014.\n\nDemerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The job demands-resources model of burnout. Journal of Applied Psychology, 86, 499\u2013512.PubMedCrossRef\n\nElmelech, Y., & Lu, H. (2004). Race, ethnicity, and the gender-poverty gap. Social Science Research, 31, 158\u2013182.CrossRef00044-9)\n\nFord, M. T. (2011). Linking household income and work-family conflict: A moderated mediation study. Stress and Health, 27, 144\u2013162.CrossRef\n\nFrone, M. R., Russell, M., & Cooper, M. L. (1997). Relation of work-family conflict to health outcomes: A four-year longitudinal study of employed parents. Journal of Occupational and Organizational Psychology, 70, 325\u2013335.CrossRef\n\nGlauber, R. (2011). Limited access: Gender, occupational composition, and flexible work scheduling. The Sociological Quarterly, 52, 472\u2013494.PubMedCrossRef\n\nGlauber, R. (2012). Women's work and working conditions: Are mother's compensated for lost wages? Work and Occupations, 39, 115\u2013138.CrossRef\n\nGreenhaus, J. H., Collins, K. M., & Shaw, J. D. (2003). The relation between work-family balance and quality of life. Journal of Vocational Behavior, 63, 510\u2013531.CrossRef00042-8)\n\nHammer, L. B., & Zimmerman, K. L. (2011). Quality of work life. In S. Zedeck (Ed.), APA handbook of industrial and organizational psychology, Vol 3: Maintaining, expanding, and contracting the organization (pp. 399\u2013431). Washington, DC: American Psychological Association.CrossRef\n\nHennessy, J. (2009). Morality and work-family conflict in the lives of poor and low-income women. The Sociological Quarterly, 50, 557\u2013580.CrossRef\n\nHoobler, J. M., Wayne, S. J., & Lemmon, G. (2009). Bosses' perceptions of family-work conflict and women's promotability: Glass ceiling effects. Academy of Management Journal, 52, 939\u2013957.CrossRef\n\nHoobler, J. M., Hu, J., & Wilson, M. (2010). Do workers who experience conflict between the work and family domains hit a \"glass ceiling?\": A meta-analytic examination. Journal of Vocational Behavior, 77, 481\u2013494.CrossRef\n\nKarimi, L., Karimi, H., & Nouri, A. (2011). Predicting employees' well-being using work-family conflict and job strain models. Stress and Health, 27, 111\u2013122.CrossRef\n\nKelly, E. L. (2008). Work-family policies: The United States in international perspective. In M. Pitt-Catsouphes, E. E. Kossek, & S. Sweet (Eds.), The work and family handbook: Multi-disciplinary perspectives and approaches (pp. 99\u2013123). Mahwah: Lawrence Erlbaum.\n\nKirchmeyer, C., & Cohen, A. (1999). Different strategies for managing the work non-work interface: A test for unique pathways to work outcomes. Work & Stress, 13, 59\u201373.CrossRef\n\nKmec, J. A. (1995). Setting occupational sex segregation in motion: Demand-side explanations of sex traditional employment. Work and Occupations, 32, 322\u2013354.CrossRef\n\nKossek, E. E., & Ozeki, C. (1999). Bridging the work-family policy and productivity gap: A literature review. Community, Work & Family, 2, 7\u201332.CrossRef\n\nKossek, E. E., Lautsch, B. A., & Eaton, S. C. (2006). Telecommuting, control, and boundary management: Correlates of policy use and practice, job control, and work-family effectiveness. Journal of Vocational Behavior, 68, 347\u2013367.CrossRef\n\nLeana, C. R., Mittal, V., & Stiehl, E. (2012). Organizational behavior and the working poor. Organization Science, 23, 888\u2013906.CrossRef\n\nLitchenwalter, S. (2005). Gender poverty disparity in US cities: Evidence exonerating female-headed families. Journal of Sociology and Social Welfare, 32, 75\u201396.\n\nMisra, J., Moller, S., & Budig, M. J. (2007). Work-family policies and poverty for partnered and single women in Europe and North America. Gender and Society, 21, 804\u2013143.CrossRef\n\nMisra, J., Moller, S., Strader, E., & Wemlinger, E. (2012). Family policies, employment and poverty among partnered and single mothers. Research in Social Stratification and Mobility, 30, 113\u2013128.CrossRef\n\nNational Women's Law Center (2012). Poverty rates among women, men, and children, 2011, 2010, 2000. http:\/\/\u200bwww.\u200bnwlc.\u200borg\/\u200bresource\/\u200bsummary-table-poverty-rates-among-women-men-and-children-2011-2010-2000. Accessed Dec 2013.\n\nOdle-Dusseau, H. N., Britt, T. W., & Greene-Shortridge, T. M. (2012). Work resources as predictors of job attitudes and performance: The process of work-family enrichment. Journal of Occupational Health Psychology, 17, 28\u201340.PubMedCrossRef\n\nOdle-Dusseau, H. N., Herleman, H. A., Britt, T. W., Moore, D. D., Castro, C. A., & McGurk, D. (2013). Family-supportive work environments and psychological strain: A longitudinal test of two theories. Journal of Occupational Health Psychology, 18, 27\u201336.PubMedCrossRef\n\nOffer, S. (2012a). Barriers to social support among low-income mothers. The International Journal of Sociology and Social Policy, 32, 120\u2013133.CrossRef\n\nOffer, S. (2012b). The burden of reciprocity: Processes of exclusion and withdrawal from personal networks among low-income families. Current Sociology, 60, 788\u2013805.CrossRef\n\nPearce, D. (1978). The feminization of poverty: Women, work, and welfare. Urban and Social Change Review, 11, 28\u201336.\n\nPerry-Jenkins, M. (2005). Work in the working class: Challenges facing families. In S. M. Bianchi, L. M. Casper, & R. B. King (Eds.), Work, family, health, and well-being (pp. 453\u2013472). Mahwah: Lawrence Erlbaum.\n\nPowell, G. N. (2011). Women and men in management (4th ed.). Thousand Oaks: Sage.\n\nRapoport, R., Bailyn, L., Fletcher, J. K., & Pruitt, B. H. (2002). Beyond work-family balance: Advancing gender equity and workplace performance. San Francisco: Jossey-Bass.\n\nRay, R., Gornick, J. C., & Schmnitt, J. (2009). Parental leave policies in 21 countries: Assessing generosity and gender equality. The Sloan Work and Family Research Network News, 11.\n\nReid, L. W., & Rubin, B. A. (2003). Integrating economic dualism and labor maker segmentation: The effects of race, gender, and structural location on earnings, 1974\u20132000. The Sociological Quarterly, 44, 405\u2013432.CrossRef\n\nReskin, B. F., & Bielby, D. D. (2005). A sociological perspective on gender and career outcomes. Journal of Economic Perspectives, 19, 71\u201386.CrossRef\n\nRoberts, B., Povich, D., & Mather, M. (2012). The working poor families project: Policy Brief Winter 2012\u20132013. The Working Poor Families Project. www.\u200bworkingpoorfamil\u200bies.\u200borg. http:\/\/\u200bwww.\u200bworkingpoorfamil\u200bies.\u200borg\/\u200bwp-content\/\u200buploads\/\u200b2013\/\u200b01\/\u200bWinter-2012_\u200b2013-WPFP-Data-Brief.\u200bpdf Accessed Sept 2013.\n\nSchieman, S., & Young, M. (2011). Economic hardship and family-to-work conflict: The importance of gender and work conditions. Journal of Family and Economic Issues, 32, 46\u201361.CrossRef\n\nShaffer, M. A., Harrison, D. A., Gilley, K. M., & Luk, D. M. (2001). Struggling for balance amid turbulence on international assignments: Work-family conflict, support and commitment. Journal of Management, 27, 99\u2013121.CrossRef\n\nSinclair, R. R., Probst, T., Hammer, L. B., & Schaffer, M. M. (2013). Low income families and occupational health: Implications of economic stress for work-family conflict research and practice. In A. Antoniou & C. Cooper (Eds.), The psychology of the recession on the workplace (pp. 308\u2013323). Cheltenham: Edward Elgar.CrossRef\n\nThompson, C. Q., & Prottas, D. J. (2006). Relationships among organizational family support, job autonomy, perceived control, and employee well-being. Journal of Occupational Health Psychology, 10, 100\u2013118.CrossRef\n\nFootnotes\n\n1\n\nOf course, left out of this discussion is the important issue of race and ethnicity that intersects with gender and class in terms of income inequality. See Frevert, Culbertson, and Huffman (Chap. 4 in this book), and Elmelech and Lu (2004) for an insightful analysis of gender, race, ethnicity, and poverty.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_4\n\n# 4. Exploring the Double Jeopardy Effect: The Importance of Gender and Race in Work\u2013Family Research\n\nTonya K. Frevert1  , Satoris S. Culbertson2  and Ann H. Huffman3\n\n(1)\n\nOrganizational Science PhD Program, University of North Carolina at Charlotte, 9201 University City Boulevard, 28223 Charlotte, NC, USA\n\n(2)\n\nDepartment of Management, Kansas State University, 110 Calvin Hall, 66506 Manhattan, KS, USA\n\n(3)\n\nDepartments of Psychology and Management, Northern Arizona University, P.O. Box 15106, 86011 Flagstaff, AZ, USA\n\nTonya K. Frevert (Corresponding author)\n\nEmail: tfrevert@uncc.edu\n\nSatoris S. Culbertson\n\nEmail: satoris@ksu.edu\n\nAnn H. Huffman\n\nEmail: ann.huffman@nau.edu\n\nKeywords\n\nGenderRace\/ethnicityDouble jeopardyMultiple identitiesIntersectionalityWork\u2013family\n\n## 4.1 Historical Background\n\nThe passage of the Civil Rights Act in 1964 created significant economic, political, and social changes within the USA. Title VII of this historic legislation changed the structure of the labor market in the United States by outlawing some forms of discrimination in the workplace, most notably (at least for discussion of work\u2013life issues) in terms of gender and race . For example, the ratio of women to men in the workforce has moved to near equal numbers (Domenico and Jones 2006). As the USA has slowly implemented and adapted to these changes in workforce demographics , researchers have responded in kind, by examining how these changes differentially impact women versus men (e.g., England 2010; Kmec 2005; Padavic and Reskin 2002) and, in a separate but related research stream, how these changes differentially impact individuals from different racial\/ethnic groups, such as blacks versus whites (e.g., Collins 1997; Pager et al. 2009) or blacks versus Latinos (e.g., Zamudio and Lichter 2008). While these separate streams of research certainly inform us, there is a pressing need to reframe research such that we categorize individuals according to their multiple identities (e.g., black women, Asian men, upperclass Hispanics) to allow for a better understanding of how experiences differ not only between groups (e.g., women vs. men) but also within groups (e.g., black women vs. white women).1 For example, whereby some have suggested that all women are already disadvantaged in the workplace (e.g., Hakim 2004), women of color are considered to be in \"double jeopardy\" for belonging to two disadvantaged status categories\u2014gender and race (King 1988). As such, given their status, it is reasonable to suspect that the work\u2013family needs, resources, and experiences of women of color may be significantly different from the professional white women who have been the primary source of data for work\u2013family researchers. Thus, our goal in this chapter is to demonstrate why using a multiple identities approach is crucial to future work\u2013family research.\n\n## 4.2 Multiple Identities in Work\u2013Family Research: The Intersection of Gender and Race\n\nThere appears to be a paucity of work\u2013family research that examines the experiences of people of color, as the majority of research has been conducted on white employees who live in the USA (Casper et al. 2007). Furthermore, in studies directed at women, the emphasis has been on professional white women to the exclusion of women of color. Given the extant research combined with the beliefs espoused in the popular press that women are getting the short end of the work\u2013family stick (so to speak), researchers have attempted to examine the needs of women in particular as they attempt to fit into a work structure that remains best suited for men (Sandler and Rao 2012). One emergent criticism of this research stream, however, is that the existing research has tended to emphasize work\u2013family experiences of professional white women (typically in management positions; Bianchi and Milkie 2010; Blair-Loy 2003), while work\u2013family experiences of women of color have been largely overlooked (for exceptions, see Glauber 2008; Glauber and Gozjolko 2011; Grzywacz et al. 2007; Han et al. 2008). The advancement of theoretical and empirical work\u2013family research rests on developing a meaningful understanding of women of color's work\u2013family experiences that are situated within the complex power relations affecting their daily lives. In the following section, we discuss the possible reasons for this oversight, both in the academic research and in the popular press .\n\nAlthough all women are linked by their gender (or more specifically, their biological sex) , race separates women of color from white women and thus their access to power, such that black women largely remain in the lowest-paid and lowest-status occupations (Branch 2011). Their position at the bottom of the labor market is one potential reason why women of color have been overlooked in work\u2013family research. Holvino (2010) put forth that \"women of color have always worked and been seen as workers\" (i.e., throughout history as slaves, indentured servants, and domestic service workers) and \"have been generally confined to secondary labor markets and to positions at the bottom of the organizational hierarchy\" (p. 252). Holvino further posits that during the early stages of the women's movement , the contentious division between the private and the public spheres primarily reflected the reality of white women. While white women desired for both the role of housewife or mother to be recognized as work and also for access to the paid workplace, women of color considered \"being able to stay at home and being supported by a husband's paycheck...a luxury that only affluent white women have\" (p. 252). In contrast to their white peers, women of color\u2014traditionally confined to secondary labor markets and jobs at the bottom\u2014desired better working conditions and more job opportunities. Arguably, all women continue to encounter the effects of sexism . The differences in the daily realities between black women and white women, however, reflect the privileges and deficits that stem not only from sexism but also from the simultaneous intersection with racism .\n\nThe omission of women of color has been reflected not only in scholarly research, but also within the contemporary popular press , where the ongoing debate of whether women can \"have it all\" still assumes that the question is most germane to professional white women. For example, while the 2013 bestselling book Lean In by Sheryl Sandberg (2013) was widely lauded as helpful career advice for women, dissenters criticized it for being applicable to only a very small percentage of the female workforce (Rottenberg 2013). Sandberg is a Harvard graduate who once worked as Chief of Staff to the US Secretary of the Treasury and has been Facebook's Chief Operating Officer since 2008 (CNBC.com 2012). Critics argue that, while she is certainly credentialed, Sandberg sits in an extremely high position of privilege that precludes her from grasping the day-to-day hurdles that the majority of women face in simply trying to survive in a rough economy (Dowd 2013) where choices and access are often limited. Books such as Lean In may help women who have fought their way into the C-suite (i.e., corporate officers and directors), but they do not address the needs and realities of women (regardless of race) working in retail\/service\/low-wage jobs that typically have low autonomy , low power , and few career ladders . Further, it assumes that all women have the freedom and resources to make choices as women in Sandberg's position do, when the reality for many women is a job that is not due to choice, but merely due to survival. That is not to say that the work\u2013family experiences among all women (and for that matter men) do not share commonalities (e.g., needing time off from work to handle family matters), but the resources available to manage both are far from universal (e.g., actual time-off policies), and thus it behooves work\u2013family researchers to investigate how and when these differences and similarities occur. Ultimately, such research should thus help inform organizations' work\u2013family policies to fit the needs and realities of their workforces, at all levels of the hierarchy.\n\n### 4.2.1 Gender in Work\u2013Family Research\n\nThe role of gender in work\u2013family research has dramatically changed over the years. Historically, within family settings, men were seen as the breadwinners whereas women were seen as the caretakers, responsible for fulfilling domestic duties (Diekman and Goodfriend 2006; Eagly et al. 2000). These beliefs about where men and women belong\u2014men at work and women at home\u2014persisted long after women joined the paid workforce en masse and began performing alongside their male counterparts. Indeed, whether it is a case of art imitating life or life imitating art, the 1983 comedy-drama film Mr. Mom was arguably successful because it depicted what was seen as a seemingly absurd situation at the time: A married man (portrayed by Michael Keaton) takes on the role of a stay-at-home dad while his wife (portrayed by Teri Garr) becomes the primary breadwinner.2 Thirty years later, the demarcation line persists as men's and women's work and family roles remain relatively divided, particularly at home. Advertisements for cleaning products typically feature women (Lindner 2004) and qualitative research suggests that men who act as primary caregivers to their children are often viewed as an anomaly, despite the practice becoming increasingly common (Rochlen et al. 2008). However, there has been a slow shift away from this traditional framework in which men work outside the home while women work within it, toward a more contemporary one , in which men and women work outside of the home in approximately equal numbers (Friedman and Greenhaus 2000).\n\nDespite the proportion of men and women in the workforce being relatively even in 2013, gender differences persist in terms of family responsibilities as well as perceptions of women who work. The playing field remains far from even, with research demonstrating that women, unlike their male counterparts, face several penalties in both domains. For example, within the workplace, women suffer lower wages compared to men, and remain underrepresented in managerial positions and, in particular, the C-suite (O'Neil et al. 2008). Women also appear to be penalized within the workplace for choices they make outside of the work domain, such as having children. Women with children often suffer a \"motherhood penalty\" 3 in the workplace whereby their careers stagnate, they are seen as less serious workers, and they are not promoted because people assume they prioritize family over the job, career, or firm (Budig and England 2001). Along with penalties in the workplace, women also face penalties at home. Although men have become more involved as caregivers, women are by and large still the ones that are primarily responsible for childcare (Bianchi et al. 2012), even when they are working outside of the home just as much as or more than their partners. As Duxbury and Higgins (1991) aptly noted over two decades ago, and which seems to hold true today, \"...there have been very few changes in society's perception of gender-specific work- and family-role responsibilities over the past few decades,\" and that \"...women have fewer options than men for achieving control over competing role demands\" (p. 71).\n\nThe \"gender perspective posits that caregiving responsibilities reflect cultural prescriptions about who should perform these tasks\" (Maume 2011, p. 414; emphasis in original). For example, when a woman has a child, she is often expected by both her coworkers and her family and friends to reduce her work hours or quit her job, although this assumption is not made about men (Nsiah et al. 2013). In fact, even when women earn more than their husbands, they will often leave their job to care for their children (e.g., Blair-Loy 2003). Moreover, once a woman does leave the workplace for family reasons, research suggests that it is very difficult for her to regain her footing and pick up where she left off in her career (Budig and England 2001).\n\nOf course, it is not just the academics who have noticed these gender discrepancies. In the popular press , a 2012 article in The Atlantic generated much controversy when its female author argued that women \"cannot have it all\"\u2014career and family\u2014given today's economic and social structures (see Slaughter 2012)4 and must often choose one or the other (Hewlett 2002). Yet men do not face the dilemma of choosing between career or family\u2014it is assumed they will have both. A 2001 survey of ultra-achieving career professionals (i.e., those earning more than $ 100,000) by economist Hewlett (2002) found that 49 % of the women were childless , contrasted with 19 % of the men. Although ambitious careers make high demands of all individuals pursuing them, men often do not have to make tradeoffs, as their likelihood of finding a spouse and having children increases as they become increasingly successful. The opposite is true for women and thus those pursuing careers often put off childbearing until later in life or drop out of the career track if they do have children5 . It is clear that differences persist between men and women in terms of the work and family expectations that they encounter both in the labor market and in society-at-large.\n\n### 4.2.2 Race in Work\u2013Family Research\n\nThe role of race in work\u2013family research has also been examined, but to a much lesser extent than gender. In 2000, Perry-Jenkins et al. (2000) compiled a review of work\u2013family research conducted in the 1990s, noting that the focus was almost entirely placed on white families who were typically middle-class professionals. An updated review of work\u2013family research conducted from 2000 to 2010 by Bianchi and Milkie (2010) noted that national changes in the economy and the demographic composition of the US population has driven an expansion in the scope of work\u2013family research. For example, increased immigration from Latin America and Asia has led to increased research on families of color (e.g., Goldberg et al. 2012; \u00adGrzywacz et al. 2007; Olson et al. 2013; Roehling et al. 2005). While this expansion is most welcome, it still largely fails to address the experiences of the United States' historically oppressed and largest racial\/ethnic minority: black Americans.\n\nIt is worth noting at this point that although we are attempting within this chapter to remedy some of the oversight in the work\u2013family literature for not substantially addressing the needs of people of color, the scant research that does exist makes it difficult to paint a precise picture across all racial or ethnic groups (i.e., combining all people of color into a \"nonwhite\" category), let alone between them (i.e., comparing blacks to Asians to Hispanics\/Latinos). Although we do highlight some work\u2013family research studies representing both approaches, our primary frame is comparing and contrasting the experiences of black women to white women. We take this approach for two reasons: (1) although the work\u2013family literature is lacking in its inclusion of all women of color, the parallel inequality literature from sociology does inform our understanding of the experiences of black women (but to the large exclusion of Hispanic\/Latina women and Asian women) and (2) it has been hypothesized that colonized groups (i.e., in the US, blacks) experience greater prejudice and discrimination than immigrant groups (i.e., in the US, Asians and Hispanics\/Latinos; Healey 2012). For example, Asian Americans who were schooled in the United States have recently shown an approximate parity in the labor market with whites (Sakamoto et al. 2009). However, while much more empirical research is needed to truly understand these differences, methodological considerations when employing a multiple identities approach pose some hurdles, given the variability that occurs within racial\/ethnic groups (as we discuss later). Further, the United States' tumultuous history regarding race relations suggests that the \"answers\" are embedded within a network of complex relationships at the individual, familial, organizational, and larger social, political, or economic levels that will take some effort to unravel.\n\nStructural inequalities in terms of race and gender stem from centuries ago, when white males colonized the New World and people of color and women had little to no rights. While there certainly have been some significant moves toward equality following the US Civil War and in the 150 years that subsequently followed, these old threads of race and gender inequality unfortunately persist, despite legislation and efforts otherwise. As history marches on, old patterns of discrimination are reinforced\u2014often without our awareness\u2014as they are so deeply embedded in our social, political, and economic structures. Complex power relations shape the experiences of people of color (Wight et al. 2013), as argued by contemporary scholars such as Patricia Hill Collins, Kimberl\u00e9 Crenshaw, bell hooks, and Charles W. Mills. Because sexism and racism remain embedded in our social structures, those individuals who belong to a dominant social group (i.e., whites) have different experiences than those individuals who do not belong to a dominant social group (i.e., people of color). To a vast extent, work\u2013family research has acknowledged the complex power relations of sexism by exploring how women manage the work\u2013family interface6, but at the same time has failed to acknowledge the complex power relations of racism.\n\nOver the course of the last several decades since the passage of the Civil Rights Act , research has consistently demonstrated that women have faced penalties in both the work and home domains. In parallel to women's penalties, individuals in minority racial\/ethnic groups have also been penalized in both the work and home domains. For example, upon comparing whites and blacks (without consideration of other racial minority groups for the sake of simplicity and brevity), black individuals have been, and continue to be, at a disadvantage within the workplace. An experimental study found that\u2014all else being equal in terms of education and experience on job applicants' resumes\u2014white applicants were twice as likely than black applicants to get a callback or job offer (Pager et al. 2009). This finding suggests that employers still discriminate against black workers in favor of white workers. Although many blacks have moved into management positions, they remain underrepresented at executive management levels (Collins 1997). Further, despite years of legislation and organizations' well-intentioned attempts to diversify their workforces, affirmative action policies and diversity training programs have been shown to be largely ineffective due to larger structural forces that reinforce discriminatory beliefs and attitudes (Kalev et al. 2006). For example, an individual manager or supervisor may want to promote a black employee, but unspoken barriers within some organizations (e.g., a black person has never been in a position of authority in that organization) prevent career advancement from occurring.\n\nAlong with workplace changes following the Civil Rights Act , the subsequent decades also saw shifts in family structures, such as the increased prevalence of single parents (McLanahan and Percheski 2008). Indeed, over half of today's children are projected to spend some time living with a single-parent family before the age of 18. What effect this will have on shaping their life chances is beginning to be explored by researchers, as family structure is argued to be a key link between inequality and intergenerational mobility. Although some previous research found that married black couples divide housework and paid work more equally than do married white and Latino couples (Glauber 2008), recent trends indicate that more black households are headed by women and that blacks have also seen a decline in marriage rates (Branch 2011). As such, increases in single motherhood are more prevalent among black women as compared to white women (McLanahan and Percheski 2008). Additionally, black women have lower education levels relative to white women and black women are overrepresented in low-wage occupations such as service work (e.g., home health aides)\u2014in 2008, 25 % of black women held such jobs (Branch 2011). This overrepresentation is problematic, because individuals in service work (regardless of race and gender) face higher poverty levels. We must make clear, however, that we are not reporting these recent trends to deem single parenthood as \"immoral\"\u2014rather, we report these recent trends to demonstrate that single parents are often penalized economically, particularly if the single parent is a woman of color. With fewer economic resources available, it stands to reason that single parents will face high levels of work\u2013family conflict. Furthermore, for all individuals who live in precarious economic conditions, the stress of such daily living conditions (often correlated with poverty, crime, and substandard housing) translates into greater disparities in physical and mental health outcomes for those individuals (Jackson et al. 2010). Again, this suggests the potential for work\u2013family conflict to be higher among single parents, especially for women of color.\n\nAs empirical research on the work\u2013family experiences of blacks is scant, however, such conclusions are merely speculative and may not be empirically supported. For example, blacks may tap into noneconomic resources to buffer against these stressors, such as social support from extended-kin networks and the larger community (McLanahan and Percheski 2008). Until work\u2013family research is extended beyond the experiences of professional white women, however, nuanced differences will remain unknown. As we have outlined above, gender inequality and racial inequality have been traditionally examined as separate complex power relations. In the past few decades, however, the recognition that individuals exist in the social structure in multiple categories that differentially affect their individual experiences (e.g., not just as women, but as white women or black women) has spurred researchers to study multiple identities through the intersection of gender and race .\n\n## 4.3 The Consequences of Multiple Identities: The Double Jeopardy Effect\n\nOne particularly salient concept for work\u2013family research is the double jeopardy effect, which occurs when race and gender intersect in ways that disadvantage individuals in the social structure (King 1988). For example, white men have historically held the most advantages in US society, seconded by black men and white women (often on parallel trajectories), with black women suffering the most disadvantages at the bottom (Stainback and Tomaskovic-Devey 2012). Since 1964, when institutionalized white male privilege began to be officially dismantled, gains in equal opportunity have spread to black men, white women, and black women. However, these gains have been uneven and inconsistent, tempered by varying trends and varying enforcement during each subsequent decade. For example, the strength of the civil rights movement in the 1960s yielded gains primarily for black men. When legal enforcement of Title VII began in earnest in the 1970s, both white women and black women made progress into better jobs. Indeed, as a negative response to affirmative action by some groups, black women were falsely accused of getting a \"double advantage\" at work, as they fulfilled both race and gender requirements for employers seeking to be compliant with the new laws (Sokoloff 1992). This advantage, however, was a myth. As the civil rights movement declined in the 1980s and corporate human resource practices grew in response to legal oversight, pressures on employers began to wane (Stainback and Tomaskovic-Devey 2012). The women's movement was able to maintain pressure on employers until the 1990s, although white women were the primary beneficiaries . For example, although wage inequality between black women and white women had decreased in the 1970s, black women's wages lagged behind white women's wages by 14 % by 1991 (Branch 2011). Additionally, by 1999, the unemployment rate for black women was double that of white women.\n\nStainback and Tomaskovic-Devey (2012) argue that without pressures from both the government and affected constituencies today, many organizations are slipping back toward resegregation in terms of race and gender (see also Sokoloff 1992). If this regressive trend persists, black women, holding minority status for both race and gender\u2014double jeopardy \u2014are thus most at risk for deleterious treatment within the workforce. For example, black women experience more overall harassment in the workplace than white women (Berdahl and Moore 2006). Overall, research supports the hypothesis that black women are most disadvantaged in the workplace, but how that plays out between the work\u2013family domains remains unclear. As we discuss below, the scant work\u2013family research including women of color has yielded inconsistent findings.\n\n## 4.4 Work\u2013Family Research Including Gender and Race\n\nStudies considering both gender and race are infrequent in work\u2013family research, although the few studies that have done so categorized participants differently such that patterns are hard to discern. For example, in a study using employed women (those working at least 30 hours per week) from a variety of occupations and income ranges, Bernas and Major (2000) found that women of color (defined as nonwhite) experienced significantly less work-to-family conflict and less job stress compared to the women in their sample who were white. Such findings are contradictory to the expectations posited by the double jeopardy effect, but may be due to grouping all women of color into a nonwhite group, as findings may be different when people of color are distinguished between racial\/ethnic groups. Further, Bernas and Major make a point of mentioning that they \"found recruiting a sufficient number of working women with families to be quite challenging and, as a result, relied on several convenient sources\" (pp. 175\u2013176). We further address methodological considerations and limitations below.\n\nOther studies, however, have distinguished between racial\/ethnic groups. For example, Glauber (2008) posits that black women face a second penalty at work beyond the motherhood penalty that all women face. Citing an earlier study by Kennelly (1999)\u2014which found that while employers view all women workers as mothers and less capable in their jobs, they also view all black women as single mothers\u2014Glauber argues that black women are thus perceived as having children out of wedlock, which exacerbates the motherhood penalty . Yes, recent data indicate that more black women are single mothers than white women (Branch 2011; McLanahan and Percheski 2008), but anti-discrimination laws prohibit employers from making decisions rooted in both sex-based and race-based stereotyping (Equal Employment Opportunity Commission 2011) . As men often get a double bonus for both marriage and fatherhood in the workplace, this benefit stands in stark contrast to black women who are often penalized in the workplace not only for motherhood, but also for a perceived lack of marriage. This unequal treatment provides further evidence of the double jeopardy effect for black women.\n\nCompared to their white counterparts, however, African American women who are married and have children are more likely to be employed (Han et al. 2008). Recent research conducted by Glauber and Gozjolko (2011) suggests that wives of white men who hold traditional views (e.g., \"a woman's place is in the home\") are less likely to work outside of the home compared to wives of white men who hold egalitarian views. In contrast, wives of African American men are equally likely to work outside of the home, regardless of their husbands' beliefs about whether this is acceptable or not. One would imagine that tension would arise for women who are employed and whose husbands are against this. If this is indeed the case, it would not be surprising to discover that African American women experience greater levels of work\u2013family conflict than do white women.\n\nReflecting Bianchi and Milkie's (2010) earlier finding that since 2000, work\u2013family research has responded to rising immigration rates from Asia and Latin America, studies have started to reflect this growth among their samples. For example, Grzywacz et al. (2007) quantitatively and qualitatively examined work\u2013family conflict and health experiences of immigrant Latino men and Latina women, and found that work\u2013family conflict was greater for the women than it was for the men. As the authors noted, \"women described clear examples of work-to-family conflict in their daily life, whereas men saw little connection between their work and their family\" (p. 1125). Grzywacz et al., however, did not examine any non-Latino\/a individuals in their study. In a study that did contrast work\u2013family conflict between Hispanic Americans and whites, Hispanic individuals reported more strain-based family interference with work (Olson et al. 2013). In another study that included whites, blacks, and Hispanics, larger gender differences in family-to-work and work-to-family spillover were found among Hispanics than among whites or blacks (Roehling et al. 2005).\n\nDue to the limited research on the intersection of gender and race in work\u2013\u00adfamily research, we now turn to recent labor force data to highlight larger trends in the differences between white women and black women that may inform future work\u2013family research.\n\n### 4.4.1 Labor Force Characteristics of White and Black Women\n\nProgress toward race and gender equality in the workplace is best advanced by increasing the number of women and people of color in management and professional occupations (Stainback and Tomaskovic-Devey 2012). When management and professional positions become increasingly diverse, the odds increase that women and people of color in these positions will help \"pull-up\" other women and people of color into these positions. If women and people of color remain underrepresented in these positions, however, opportunities for advancement stall and patterns of work inequality perpetuate; gains are made, but at a very slow pace. Furthermore, these gains are made disproportionately in terms of race . While women are advancing into management and executive positions, white women are gaining over black women. While women's wages are getting closer to those of men, white women are gaining over black women. While women are participating in the labor force at an increasing rate, white women are gaining over black women.\n\nRecent labor force data from the US Bureau of Labor Statistics (BLS 2011) supports this state of affairs. Among white women, 42 % were employed in management, professional, and related occupations\u2014the highest paying major job category\u2014compared to 34 % of black women. This difference is very important to work\u2013family research, as the emphasis on management\/professional participants leads to more white women participating in said research, thus reflecting their experiences more than others. Ideally, work\u2013family research should reflect the experiences of all workers at all levels of the labor market, which would certainly include those in management\/professional occupations. If the default remains set to management\/professionals occupations, however, we are missing the needs of workers in other jobs and occupations who have work\u2013family issues to tackle too. For example, as black women are more likely than white women to work in service occupations (28.3% vs. 20.1 %), it is crucial that we do not leave service workers out of work\u2013family research. We recognize, however, that this limitation is partly due to access to and representation of people of color across the labor force , as discussed above. However, researchers can overcome this limitation by examining more occupational and industrial sectors so as to include those on the fringes and lower tiers of the labor market.\n\nThe need for work\u2013family research to incorporate racial diversity is also reflected in labor force data about families and mothers (BLS 2011), as previously discussed. \"In 2010, nearly one-half (45 %) of black families were maintained by women without a spouse present\" (p. 2), compared to about 15 % of white women. Black women are three times as likely as white women to be \"going it alone.\" If we consider Hobfoll's conservation of resources theory (1989) as it pertains to work\u2013family research in terms of resource allocation , black women are thus at a greater disadvantage than white women in terms of resources and support, often lacking a second income and emotional support from a partner.\n\nAlthough women as a group overall earn less than men, full-time black female employees typically earn about 85 % of what their full-time white female counterparts earn (BLS 2011). The 2008 recession also hit black women harder than white women, as labor force participation for white women fell 2.3 % from 2008 to 2010 (56.3%\u201354.0%), but fell 4.1 % for black women (55.8%\u201351.7%) during the same time period. Further, \"among mothers with children under 18, black mothers are more likely to be in the labor force than white mothers\" (75 % vs. 71 %; p. 2). For households with children under six years old\u2014arguably when demands for flexible work\u2013family policies are at their highest\u2014black mothers are more likely to be in the labor force than white mothers (70 % vs. 63 %). In today's global 24\/7 economy where the boundary between work and home is diminishing, all employees need tools, skills , and resources to successfully manage their work and personal lives . Since work\u2013family research helps inform the policies that organizations implement for their employees' benefit, knowing the needs of the factory worker on the line are just as important as knowing the call center manager's needs. If we do not know how those needs differ\u2014not only within the organization but also across occupations\/industries\/locations\u2014and how those needs are compounded by multiple identities that differentially locate individuals within the overarching social structure, then we are only serving a limited subset of the population. Some might argue that policies implemented in one sector will trickle down to another, but only empirical research can answer our questions and tell us if those needs have been met or are yet to be met.\n\nThus far, we have contrasted the similarities and differences between white women and women of color in terms of their work and family experiences. Next, we address some of the methodological considerations and limitations that arise with intersectional models.\n\n## 4.5 Methodological Considerations\n\nBecause the racial divide in the United States has historically fallen along a black-white dichotomy, there is scant social science research that explicitly includes other racial\/ethnic groups, such as Hispanics\/Latinos, Asians, or Native Americans. These groups are frequently lumped together with blacks into a \"nonwhite\" category and findings generalized to minorities as a whole (Ruggs et al. 2013). While this approach is an attempt to include other racial\/ethnic groups in research, it fails to not only consider the differences between these racial\/ethnic groups (e.g., social class, education, family norms) , but also the differences within the various racial\/ethnic groups (e.g., outcomes for light-skinned blacks vs. dark-skinned blacks; Marira and Mitra 2013). That said, it is important to remember that researchers rarely have free, easy, and open access to whatever data they desire. As Thompson et al. (2013) point out, researchers are often limited by (1) numerical representation of minority groups, such as Native Americans comprising 1.7 % of the general population and (2) methodological issues that may violate assumptions of sampling, power, and generalizability in such targeted approaches. Nevertheless, given that the overall population of people of color in the United States is projected to become the numerical majority of the total US population by 2043 (Ortman 2013), these limitations should start to ease. Further, as scholarly research usually reflects the larger culture, a greater number of studies will likely give way to more diverse participants in time. If work\u2013family researchers want to be ahead of the curve and not stuck in the past or scrambling to catch up, however, it is incumbent upon us to remedy this oversight sooner rather than later.\n\nA second methodological consideration for intersectional research is that it should be employed as a multiplicative model, not an additive one. One of the \u00adfrequent assumptions of measurement in data analyses is that social identities are additive (e.g., black + woman) versus intersectional (e.g., black woman; Bowleg 2008). An additive approach, however, contradicts the central tenet of intersectionality : Social identities and inequality are interdependent for groups such as black women, not mutually exclusive. They do not act independently (i.e., one is never just a person of color without also being of one sex or the other). Intersectionality theory requires that we conceptualize social categories in terms of the stratification that arises through the practices of individuals, institutions, and cultures rather than merely as individual characteristics (Cole 2008). For example, the life experiences of a black woman are a product of the intersection of sexism with racism such that her simultaneous social identity both as a woman and as a person of color marginalizes her in both socially structured categories (Crenshaw 1991).\n\nMoreover, as \"psychologists aim to simplify models for parsimony, either by omitting variables or by statistically controlling for membership in categories other than the one of interest\" (Cole 2009, p. 170), work\u2013family researchers should move forward by considering \"the meaning and consequences of social categories\" (p. 176). Cole (2009) urges researchers to ask three pertinent questions during the research process (Who is included within this category?, What role does \u00adinequality play?, and Where are the similarities?), so that we move away from empirical models in which self-evident demographic variables shape how participants are categorized and thus move toward empirical models that employ intersectional frameworks. Intersectional frameworks may increase our understanding of the complex, intertwined relationships that differentially locate individuals within the overarching social structure. That is not to say that work\u2013family researchers must do away with all methodological practices and turn all of their focus onto underrepresented groups . Rather, work\u2013family researchers can use an intersectional framework\n\n> to look for causes of human behavior both upstream and downstream, to notice and hypothesize about the multiple paths that may lead individuals to the same or similar outcomes, and to understand the ways that different social categories depend on each other for meaning and, thus, mutually construct one another and work together to shape outcomes. (Cole 2009, p. 179)\n\n## 4.6 Broader Impacts and Future Research\n\nAlthough overt prejudice and discrimination have been substantially tempered over the past 50 years, covert prejudice and discrimination persist. For example, women and people of color still face significant barriers to getting a seat at the boardroom table, and those barriers are fortified by inequalities of place, education , and opportunity. As these inequalities come to a head, however, they are on a collision course with changing definitions of family as well as changing roles and role expectations that men and women have traditionally held. The en masse entry of women into the workforce in the 1960s and 1970s coincided with major shifts in work arrangements in the post-Fordist era (e.g., the disappearance of career ladders; Rubin 1995). As men and women negotiated new institutional arrangements in both the work domain and the family domain, their race\/ethnicity rode along with their gender. The long crusade toward workplace equality cannot be achieved with a sole gender lens or a sole racial\/ethnic lens\u2014a proper set of glasses includes both lenses.\n\nTo that end, it behooves researchers to expand their populations of study so that the larger conversation taking place among the popular press , public policy makers, and business leaders is not exclusionary. As such, another factor for consideration is differences by occupational and industrial sector. Not only do different occupations have their own unique stressors, but they are also gendered and\/or racialized. For example, women of color account for an above-average percentage of representation in the field of nursing aides (Alonso-Villar and del Rio 2013). Research has shown that the job of nursing aide\u2014a job that does not require a college degree\u2014has many characteristics that can lead to high stress. Nursing aides primarily focus on caregiving, are more likely to have shift work, and have lower than average pay. But this pattern also exists for jobs that require college degrees. The field of social work, for example, also has a disproportionate number of employees who are women of color. Similar to nursing aides, social work is also considered a high stress job (Wooten et al. 2011). Work\u2013family research has consistently shown that high demand and low resource jobs are especially susceptible to work\u2013family stress , and thus simply examining different occupations may be one way to diversify work\u2013family research.\n\nDifferences in family characteristics should also be considered. The age of becoming a mother for women of color is younger than other women. A report by the Centers for Disease Control and Prevention (2013) showed that women of color have their first child, on average, at the age of 22.7, compared to white women at the age of 26.0. Research has suggested that this age difference could have several implications. Starting a family at a younger age could introduce some additional stressors for women of color. For example, younger parents are likely to be less financially secure and less mature to handle this responsibility . On the other hand, research has shown that the work\u2013family interface can vary over the life course (Huffman et al. 2013), with the least work\u2013family stressors occurring early in life and then again late in life. This inverted u-shaped curve could mean that individuals who have children early might have fewer stressors since they are not also burdened with extensive job demands. Whatever the case, future research should examine the implications of younger parenting for women of color and their associated work\u2013family stressors.\n\nWe suggested earlier that the numerical representation of people of color makes it methodologically more challenging to assess work\u2013family issues based on both gender and race\/ethnicity . To overcome these challenges, the work\u2013family community needs to find strategies that will allow these groups to be included in work\u2013family research. For example, qualitative methodologies allow us \"to generate new conceptualizations and interpretations of culture that will enable us to make sense of increasingly complex cultural phenomena\" (Birkinshaw et al. 2011, p. 574), an approach that seems particularly suited to the complexity of intersectional research. Additionally, archival data sets exist (e.g., General Social Survey, National Study of the Changing Workforce, Midlife Development in the United States, Panel Study of Income Dynamics) that tap into work\u2013family issues and provide demographic information that would allow researchers to answer questions related to the intersection of gender and race . These extensive data sets may provide work\u2013family researchers with the necessary respondents and variables to answer research questions specific to women of color.\n\nWork\u2013family researchers have used different theories to describe race and\/or gender differences in work\u2013family experiences (e.g., stress resource perspective, organizational leadership theory; Bernas and Major 2000). Work\u2013family researchers need to continue to build on these different theories to develop a comprehensive model of how the characteristics associated with being a woman of color affect the work\u2013life interface. For example, the demands-resource paradigm has become very relevant in trying to explain work and family experiences. The job demands\u2013resources model (Bakker and Demerouti 2007) suggests that every occupation has its own specific risk factors and resources that are associated with the job, and when the worker experiences high risks\/demands and low resources, the individual is likely to experience negative outcomes. Bakker and Demerouti (2007) suggest that this comprehensive model could be applied to various occupational settings. We extend this idea and suggest that this could also be applied to individuals with different personal characteristics. For example, women of color might experience additional demands due to discrimination they endure at work compared to white males (Sokoloff 1992), and these demands that women of color experience may result in increased distress. Although this is just one example of using one theory, we propose that there are others across the social sciences that can provide researchers with a more comprehensive understanding of the workplace experiences of women of color.\n\nAlthough our chapter focuses on the two minority characteristics of gender and race\/ethnicity (specifically women of color), we should also note that there are many other minority characteristics that could further affect women of color. For example, women of color who are lesbian, gay, bisexual, or transgender (LGBT) experience additional stressors. Potentially, a lesbian woman of color who has children could face \"quadruple jeopardy\" since she is dealing with four minority status characteristics (i.e., gender, race\/ethnicity, sexual orientation, and lesbian parent) that not only carry each of their own disadvantages, but also intersect to compound those disadvantages (Huffman et al. 2012).\n\n## 4.7 Conclusion\n\nThe negotiation between work and family is not a woman's problem; it is a problem that belongs to all of us, as we all interact in the larger social structure that encompasses our work days, our home lives, and the interface between the two. An individual's success in negotiating the increasingly fuzzy work\u2013family boundary depends upon not only her gender, but also upon her race\/ethnicity . Work\u2013family researchers have done a commendable job helping organizations understand why it is in their best interest to offer policies that help workers manage their work and home lives. However, because of structural inequalities that keep women of color out of management\/professional positions, not everyone has benefitted from these efforts. As such, if work\u2013family researchers can inform the leaders of work organizations as well as public policy makers about what needs to be done to eradicate these inequalities, then we must ensure that work\u2013family research reflects the experiences of everyone.\n\nReferences\n\nAlonso-Villar, O., & del Rio, C. (2013). The occupational segregation of black women in the United States: A look at its evolution from 1940 to 2010. Number 304, Working Papers, ECINEQ, Society for the Study of Economic Inequality.\n\nBakker, A., B., & Demerouti, E. (2007). The job demands-resources model: State of the art. Journal of Managerial Psychology, 22, 309\u2013328.CrossRef\n\nBerdahl, J. L., & Moore, C. (2006). Workplace harassment: Double jeopardy for minority women. Journal of Applied Psychology, 91, 426\u2013436.PubMedCrossRef\n\nBernas, K. H., & Major, D. A. (2000). Contributors to stress resistance: Testing a model of women's work-family conflict. Psychology of Women Quarterly, 24, 170\u2013178.CrossRef\n\nBianchi, S. M., & Milkie, M. A. (2010). Work and family research in the first decade of the twentyfirst century. Journal of Marriage and Family, 72, 705\u2013725.CrossRef\n\nBianchi, S. M., Sayer, L. C., Milkie, M. A., & Robinson, J. P. (2012). Housework: Who did, does or will do it, and how much does it matter? Social Forces, 91, 55\u201363.CrossRef\n\nBirkinshaw, J., Brannen, M. Y., & Tung, R. L. (2011). From a distance and generalizable to up close and grounded: Reclaiming a place for qualitative methods in international business research. Journal of International Business Studies, 42, 573\u2013581.CrossRef\n\nBlair-Loy, M. (2003). Competing devotions: Career and family among women executives. Cambridge: Harvard University Press.\n\nBowleg, L. (2008). When black + lesbian + woman \u2260 black lesbian women: The methodological challenges of qualitative and quantitative intersectionality research. Sex Roles, 59, 312\u2013325.CrossRef\n\nBranch, E. H. (2011). Opportunity denied: Limiting black women to devalued work. New Brunswick: Rutgers University Press.\n\nBudig, M. J., & England, P. (2001). The wage penalty for motherhood. American Sociological Review, 66, 204\u2013225.CrossRef\n\nBureau of Labor Statistics. (Aug 2011). Labor force characteristics by race and ethnicity, 2010. Report 1032. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200bcpsrace2010.\u200bpdf. Accessed 25 Sep 2013.\n\nCasper, W. J., Eby, L. T., Bordeaux, C., Lockwood, A., & Lambert, D. (2007). A review of research methods in IO\/OB work-family research. Journal of Applied Psychology, 92, 28\u201345.PubMedCrossRef\n\nCenters for Disease Control and Prevention. (21 Nov 2013). Births and natality. http:\/\/\u200bwww.\u200bcdc.\u200bgov\/\u200bnchs\/\u200bfastats\/\u200bbirths.\u200bhtm. Accessed 18 Jan 2014.\n\nCNBC.com. (22 May 2012). Sheryl Sandberg: Facebook IPO. http:\/\/\u200bwww.\u200bcnbc.\u200bcom\/\u200bid\/\u200b47386998.\n\nCole, E. R. (2008). Coalitions as a model for intersectionality: From practice to theory. Sex Roles, 59, 443\u2013453. Accessed 9 Jan 2014.CrossRef\n\nCole, E. R. (2009). Intersectionality and research in psychology. American Psychologist, 64, 170\u2013180.PubMedCrossRef\n\nCollins, S. M. (1997). Black mobility in white corporations: Up the corporate ladder but out on a limb. Social Problems, 44, 55\u201367.CrossRef\n\nCrenshaw, K. W. (1991). Mapping the margins: Intersectionality, identity politics, and violence against women of color. Stanford Law Review, 43, 1241\u20131299.CrossRef\n\nDiekman, A. B., & Goodfriend, W. (2006). Rolling with the changes: A role congruity perspective on gender norms. Psychology of Women Quarterly, 30, 369\u2013383.CrossRef\n\nDomenico, D. M., & Jones, K. H. (2006). Career aspirations of women in the twentieth century. Journal of Career and Technical Education, 22, 1\u20137.\n\nDowd, M. (23 Feb 2013). Pompom girl for feminism. The New York Times. http:\/\/\u200bwww.\u200bnytimes.\u200bcom\/\u200b2013\/\u200b02\/\u200b24\/\u200bopinion\/\u200bsunday\/\u200bdowd-pompom-girl-for-feminism.\u200bhtml?\u200b_\u200br=\u200b2&\u200b.\n\nDuxbury, L. E., & Higgins, C. A. (1991). Gender differences in work-family conflict. Journal of Applied Psychology, 76, 60\u201374.CrossRef\n\nEagly, A. H., Wood, W., & Diekman, A. (2000). Social role theory of sex differences and similarities: A current appraisal. In T. Eckes & H. M. Trautner (Eds.), The developmental social psychology of gender (pp. 123\u2013174). Mahwah: Erlbaum.\n\nEngland, P. (2010). The gender revolution: Uneven and stalled. Gender & Society, 24, 149\u2013166.CrossRef\n\nEqual Employment Opportunity Commission. (8 Feb 2011). Enforcement guidance: Unlawful disparate treatment of workers with caregiving responsibilities. http:\/\/\u200bwww.\u200beeoc.\u200bgov\/\u200bpolicy\/\u200bdocs\/\u200bcaregiving.\u200bhtml.\n\nFaludi, S. (2007). The terror dream: Fear and fantasy in post-9\/11 America. New York: Metropolitan Books.\n\nFriedman, S. D., & Greenhaus, J. H. (2000). Work and family-allies or enemies? New York: Oxford University Press.CrossRef\n\nGilbert, N. (2008). A mother's work: How feminism, the market, and policy shape family life. New Haven: Yale University Press.\n\nGlauber, R. (2008). Race and gender in families and at work: The fatherhood wage premium. Gender and Society, 22, 8\u201330.CrossRef\n\nGlauber, R., & Gozjolko, K. L. (2011). Do traditional fathers always work more? Gender ideology, race, and parenthood. Journal of Marriage and Family, 73, 1133\u20131148.CrossRef\n\nGoldberg, W. A., Kelly, E., Matthews, N. L., Kang, H., Li, W., & Sumaroka, M. (2012). The more things change, the more they stay the same: Gender, culture, and college students' views about work and family. Journal of Social Issues, 68, 814\u2013837.CrossRef\n\nGrzywacz, J. G., Arcury, T. A., Mar\u00edn, A., Carrillo, L., Burke, B., Coates, M. L., et al. (2007). Work-family conflict: Experiences and health implications among immigrant Latinos. Journal of Applied Psychology, 92, 1119\u20131130.PubMedCrossRef\n\nHakim, C. (2004). Key issues in women's work: Female diversity and the polarisation of women's employment. London: The Glass House.\n\nHan, W. J., Ruhm, C. J., Waldfogel, J., & Washbrook, E. (2008). The timing of mother's employment after childbirth. Monthly Labor Review, 131, 15\u201327.PubMedCentralPubMed\n\nHealey, J. F. (2012). Race, ethnicity, gender, and class: The sociology of group conflict and change. Thousand Oaks: Sage.\n\nHewlett, S. A. (April 2002). Executive women and the myth of having it all. Harvard Business Review, Reprint R0204E, 1\u201311.\n\nHewlett, S. A. (2003). Creating a life. New York: Hyperion.\n\nHewlett, S. A. (2009). Top talent: Keeping performance up when business is down. Cambridge: Harvard Business Press.\n\nHobfoll, S. E. (1989). Conservation of resources. American Psychologist, 44, 513\u2013524.PubMedCrossRef\n\nHolvino, E. (2010). Intersections: The simultaneity of race, gender and class in organization studies. Gender, Work & Organization, 17, 248\u2013277.CrossRef\n\nHuffman, A. H., Culbertson, S. S., Henning, J., & Goh, A. (2013). Work-family conflict across the lifespan. Journal of Managerial Psychology, 28, 761\u2013780.CrossRef\n\nHuffman, A. H., King, E., & Goldberg, A. E. (2012). Valuing lesbian and gay parenting in the workplace. In M. Paludi (Ed.), Managing diversity in today's workplace (pp. 163\u2013185). Westport: Praeger.\n\nJackson, J. S., Knight, K. M., & Rafferty, J. A. (2010). Race and unhealthy behaviors: Chronic stress, the HPA axis, and physical and mental health disparities over the life course. American Journal of Public Health, 100, 933\u2013939.PubMedCentralPubMedCrossRef\n\nKalev, A., Dobbin, F., & Kelly, E. (2006). Best practices or best guesses? Assessing the efficacy of corporate affirmative action and diversity policies. American Sociological Review, 71, 589\u2013617.CrossRef\n\nKennelly, I. (1999). \"That single-mother element\": How employers typify Black women. Gender & Society, 13, 168\u2013192.CrossRef\n\nKing, D. K. (1988). Multiple jeopardy, multiple consciousness: The context of black feminist ideology. Signs, 14, 42\u201372.CrossRef\n\nKmec, J. (2005). Setting occupational sex segregation in motion. Work and Occupations, 32, 322\u2013354.CrossRef\n\nLindner, K. (2004). Images of women in general interest and fashion magazine advertisements from 1955 to 2002. Sex Roles, 51, 409\u2013421.CrossRef\n\nLundberg, S., & Rose, E. (2000). Parenthood and the earnings of married men and women. Labour Economics, 7, 689\u2013710.CrossRef00020-8)\n\nLundberg, S., & Rose, E. (2002). The effects of sons and daughters on men's labor supply and wages. Review of Economics and Statistics, 84, 251\u2013268.CrossRef\n\nMarira, T. D., & Mitra, P. (2013). Colorism: Ubiquitous yet understudied. Industrial and Organizational Psychology: Perspectives on Science and Practice, 6, 103\u2013107.CrossRef\n\nMaume, D. J. (2011). Reconsidering the temporal increase in fathers' time with children. Journal of Family and Economic Issues, 32, 411\u2013423.CrossRef\n\nMcLanahan, S., & Percheski, C. (2008). Family structure and the reproduction of inequalities. Annual Review of Sociology, 34, 257\u2013276.CrossRef\n\nNsiah, C., DeBeaumont, R., & Ryerson, A. (2013). Motherhood and earnings: Wage variability by major occupational category and earnings level. Journal of Family and Economic Issues, 34, 224\u2013234.CrossRef\n\nOlson, K. J., Huffman, A. H., Leiva, P. I., & Culbertson, S. S. (2013). Acculturation and individualism as predictors of work-family conflict in a diverse workforce. Human Resource Management, 52, 741\u2013769.\n\nO'Neil, D. A., Hopkins, M. M., & Bilimoria, D. (2008). Women's careers at the start of the twentyfirst century: Patterns and paradoxes. Journal of Business Ethics, 80, 727\u2013743.CrossRef\n\nOrtman, J. M. (7 Feb 2013). U.S. population projections: 2012 to 2060. U.S. Census Bureau presentation for the Federal Forecasters Consortium Brown Bag Seminar Series on Forecasting at George Washington University, Washington, D.C.\n\nPadavic, I., & Reskin, B. (2002). Women and men at work. Thousand Oaks: Pine Forge.\n\nPager, D., Western, B., & Bonikowski, B. (2009). Discrimination in a low-wage labor market: A field experiment. American Sociological Review, 74, 777\u2013799.PubMedCentralPubMedCrossRef\n\nPerry-Jenkins, Repetti, R. L., & Crouter, A. C. (2000). Work and family in the 1990s. Journal of Marriage and the Family, 62, 981\u2013998.CrossRef\n\nRochlen, A. B., Suizzo, M., McKelley, R. A., & Scaringi, V. (2008). \"I'm just providing for my family\": A qualitative study of stay-at-home fathers. Psychology of Men & Masculinity, 9, 193\u2013206.CrossRef\n\nRoehling, P. V., Jarvis, L. H., & Swope, H. E. (2005). Variations in negative work-family spillover among White, Black, and Hispanic American men and women: Does ethnicity matter? Journal of Family Issues, 26, 840\u2013865.CrossRef\n\nRottenberg, C. (2013). The rise of neoliberal feminism. Cultural Studies, 28, 418\u2013437.CrossRef\n\nRubin, B. A. (1995). Shifts in the social contract. Thousand Oaks: Pine Forge.\n\nRuggs, E. N., Hebl, M. R., Law, C., Cox, C. B., Roehling, M. V., Wiener, R. L., et al. (2013). Gone fishing: I-O psychologists' missed opportunities to understand marginalized employees' experiences with discrimination. Industrial and Organizational Psychology: Perspectives on Science and Practice, 6, 39\u201360.CrossRef\n\nSakamoto, A., Goyette, K. A., & Kim, C. H. (2009). Socioeconomic attainments of Asian Americans. Annual Review of Sociology, 35, 255\u2013276.CrossRef\n\nSandberg, S. (2013). Lean in: Women, work, and the will to lead. New York: Knopf.\n\nSandler, J., & Rao, A. (2012). The elephant in the room and the dragons at the gate: Strategising for gender equality in the twentyfirst century. Gender & Development, 20, 547\u2013562.CrossRef\n\nSlaughter, A. (July\/Aug 2012). Why women still can't have it all. The Atlantic. http:\/\/\u200bwww.\u200btheatlantic.\u200bcom\/\u200bmagazine\/\u200barchive\/\u200b2012\/\u200b07\/\u200bwhy-women-still-cant-have-it-all\/\u200b309020\/\u200b. Accessed 25 Sep 2013.\n\nSokoloff, N. J. (1992). Black women and white women in the professions. New York: Routledge.\n\nJohn, S. W. (20 May 2002). The talk of the book world still can't sell. The New York Times. http:\/\/\u200bwww.\u200bnytimes.\u200bcom\/\u200b2002\/\u200b05\/\u200b20\/\u200bus\/\u200bthe-talk-of-the-book-world-still-can-t-sell.\u200bhtml?\u200bpagewanted=\u200ball&\u200bsrc=\u200bpm.\n\nStainback, K., & Tomaskovic-Devey, D. (2012). Documenting desegregation: Racial and gender segregation in private-sector employment since the Civil Rights Act. New York: Sage.\n\nThompson, R., Bergman, M., Culbertson, S. S., & Huffman, A. H. (2013). Yes, we're fishing-in rough waters for hard-to-find fish. Industrial and Organizational Psychology: Perspectives on Science and Practice, 6, 61\u201365.CrossRef\n\nWight, V. R., Bianchi, S. M., & Hunt, B. R. (2013). Explaining racial\/ethnic variation in partnered women's and men's housework: Does one size fit all? Journal of Family Issues, 34, 394\u2013427.PubMedCentralPubMedCrossRef\n\nWooten, N. R., Kim, H., & Fakunmoju, S. B. (2011). Occupational stress in social work practice. In J. Langan-Fox & C. L. Cooper (Eds.), New horizons in management. Handbook of stress in the occupations (pp. 71\u201387). Northampton: Edward Elgar.\n\nZamudio, M. M., & Lichter, M. I. (2008). Bad attitudes and good soldiers: Soft skills as a code for tractability in the hiring of immigrant Latina\/os over native Blacks in the hotel industry. Social Problems, 55, 573\u2013589.CrossRef\n\nFootnotes\n\n1\n\nFor a scholarly debate about why and how organizational scientists and practitioners should study workers' multiple identities, see Ruggs et al. (2013) and its twelve associated commentaries in Industrial and Organizational Psychology: Perspectives on Science and Practice.\n\n2\n\nArguably, the plot device in this film was not only a man being a stay-at-home father or a woman being the primary breadwinner, but also from the difficulties that each faced in his\/her new role, which speaks even more to prevailing social norms that these were not \"typical\" or \"natural\" roles for these individuals because of their gender. Further, the film presents the switch as forced: the husband is laid off and unable to find a new job, thus the wife dusts off her college degree and gets a job in advertising. At the film's end, he is called back to work and she happily quits her job, insinuating the role reversal was an anomaly and life has returned to \"normal.\"\n\n3\n\nThe persistence of traditional gender norms is further demonstrated by the sharp contrast between the expectations of mothers at work and the expectations of fathers at work, as men appear to be a given a wage premium for fatherhood (Lundberg and Rose 2000, 2002). The \"fatherhood premium,\" however, seems to advantage white and Latino males, but not necessarily black males (Glauber 2008).\n\n4\n\nAs of 2013, Slaughter's article is the most widely read piece in The Atlantic's history (Rottenberg 2013).\n\n5\n\nIndeed, Hewlett once suggested that women should start having babies in their twenties or risk ending up being childless and sad (see Creating a Life 2003). Her controversial message was trumpeted by news media outlets as a dire warning to women to mind their biological clocks (St. John 2002), yet was criticized by the feminist community (Gilbert 2008), attributed to the \"baby panic\" narrative (Faludi 2007), and lampooned on Saturday Night Live (Hewlett 2009).\n\n6\n\nFor a review on men and work\u2013family research, see Chap. 2.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_5\n\n# 5. Challenging Heteronormative and Gendered Assumptions in Work\u2013Family Research: An Examination of LGB Identity-Based Work\u2013Family Conflict\n\nKatina B. Sawyer1  , Christian N. Thoroughgood2  and Jeanette N. Cleveland3\n\n(1)\n\nDepartment of Psychology, Villanova University, St. Mary's Hall, Suite 119, 800 Lancaster Avenue, 19085 Villanova, PA, USA\n\n(2)\n\nDepartment of Management and Organizational Development, Northeastern University, 360 Huntingdon Avenue, 112 Hayden Hall, 02115 Boston, MA, USA\n\n(3)\n\nDepartment of Psychology, Colorado State University, 228 Behavioral Sciences Building, 1876 Campus Delivery, 80523 Fort Collins, CO, USA\n\nKatina B. Sawyer (Corresponding author)\n\nEmail: katina.sawyer@villanova.edu\n\nChristian N. Thoroughgood\n\nEmail: c.thoroughgood@neu.edu\n\nJeanette N. Cleveland\n\nEmail: jeanette.cleveland@colostate.edu\n\nKeywords\n\nWork\u2013family conflictLGB identitiesHeterosexual biasDisclosure disconnects\n\n## 5.1 Introduction\n\nA vast body of research on work\u2013family conflict (WFC) exists across a variety of disciplines, with studies accumulating rapidly since the formalized inception of the term in 1985. WFC refers to a form of interrole conflict in which role-related pressures from the work and family domains are incompatible in some respect- that is, participation in the work role is made more difficult by virtue of participation in the family role and vice versa (Greenhaus and Beutell 1985). However, despite many important contributions made by scholars over time, studies of WFC continue to suffer from certain heteronormative assumptions that organize, make coherent, and privilege heterosexuality by building it into the fabric of organizational cultures and structures (Berlant and Warner 1998). This chapter seeks to highlight how these assumptions may inform our conceptualization of the family as a unit of study, which constrains the extent to which we capture the experiences of WFC for lesbian, gay, and bisexual (LGB) individuals. Specifically, we suggest that WFC may be experienced differently for LGB employees due to the exclusion of LGB families within traditional conceptualizations of the family structure.\n\nBy expanding our definition of the family, and by implication, WFC, more inclusive measures that are less susceptible to heterosexual bias are made possible. In turn, such work promises to provide an understanding of the underlying biases that may exist within the current WFC framework, as well as a greater awareness of the ways in which WFC operates across certain nontraditional family structures. In contrast, by ignoring heterosexual bias in our conceptualizations of WFC, we run the risk of perpetuating potential construct contamination and deficiency in our measures and limiting the construct validity of the WFC construct more broadly. Even more importantly, we may risk limiting the explanatory power of our theories, as well as silencing and rendering invisible members of LGB families in future WFC studies. Finally, because LGB individuals make up an increasingly greater proportion of the workforce (Day and Schoenrade 2000), with estimates of anywhere between 4 and 17 % (Ragins et al. 2007), employers stand to gain from work\u2013family policies that effectively attract and retain top LGB talent.\n\nIn order to highlight previously unexamined heterosexual biases within the WFC literature, we discuss how conceptualizing WFC as an issue of time, strain, or behaviors only, without considering issues related to identity, may restrict our definition of what a family can be and how families are envisioned in the workplace. Specifically, we examine how the current framing of WFC creates a heteronormative view of the family by overlooking the potentially unique identity-related experiences of LGB individuals (e.g., conflicting pressures between partners and coworkers to come out at work, reluctance to ask for family insurance and other benefits) that may contribute to tensions between their work and family lives.\n\nLGB experiences of WFC may differ dramatically from those of heterosexuals, and thus have implications for the effectiveness of potential work\u2013family policies. Given that LGB individuals may not be accepted at work (Croteau 1996; Foldy and Creed 2003; Ragins 2004; Ragins et al. 2007; Ragins and Cornwell 2001), WFC may occur when individuals have families who fall within the LGB domain (i.e., when they are members of same sex couples\/families). Examining WFC for members of same sex couples allows for the examination of unique WFC issues which have not previously been considered, and calls for more inclusive WFC policies within organizations.\n\nTo provide a theoretical foundation for our propositions, we take an interdisciplinary perspective by drawing on the field of women's studies and, in particular, Intersectionality Theory (Crenshaw 1989). Intersectionality refers to the ways in which multiple aspects of identity, including race, gender, class, and sexual orientation , combine and interact with one another to form qualitatively different meanings and social experiences (Warner 2008). Intersectionality allows us to examine experiences and outcomes for members of more specific identity groups (e.g., black women instead of women more broadly, or with an additional layer, black gay women instead of all black women). Given that previous work has not considered LGB identity as an additional layer that might affect the work\u2013family interface, intersectionality offers a useful tool for examining how LGB identities might impact relationships between work and family. Thus, the current work extends our understanding of WFC beyond a singular focus on traditional gendered and sexuality-based experiences and toward the inclusion of more diverse populations. This added layer may alter some of the prevailing assumptions within the WFC literature and expand our conceptualization of WFC moving forward.\n\nTo organize our discussion, we first briefly discuss how WFC has been traditionally conceptualized and assessed in the literature. Second, we discuss the origins and prevalence of LGB discrimination in the workplace, and the implications of prejudice on LGB employees' decisions to \"come out\" at work, in order to provide a basis for understanding how LGB workers may experience WFC differently from their heterosexual peers. Third, based on our discussion, we suggest that current measures of WFC may suffer from heterosexual bias in that such scales consist of items that may be uniquely interpreted by LGB individuals . Furthermore, we argue that an additional dimension (identity-based conflict) may be necessary to better capture the WFC experiences of LGB employees. Fourth, we suggest how to proceed with developing a more inclusive measure of WFC, highlight the potentially unique antecedents of LGB identity-based WFC, and describe several other important avenues for future research. Finally, we conclude with practical implications.\n\n## 5.2 Defining Work\u2013Family Conflict\n\nAs noted earlier, WFC broadly refers to the ways in which aspects of one's work life interfere with one's family life and vice versa. A number of researchers have called attention to the need to distinguish between two separate constructs\u2014one that focuses on the ways in which work demands impede on the family (WFC) and the other examining the ways in which family interferes with work (family-work conflict, FWC) (Frone et al. 1992, 1997; MacEwen and Barling 1994; O'Driscoll et al. 1992; Williams and Alliger 1994). Both WFC and FWC have been traditionally conceptualized and measured using time, strain, and behavior-based dimensions (Carlson et al. 2000; Greenhaus and Beutell 1985; Greenhaus et al. 2006).\n\nTime-based conflict occurs when time devoted to one role makes it difficult to devote time to another role, reducing one's performance in the latter role. For example, time-based conflict may arise when a family member's birthday dinner falls on an evening when an employee is required to stay late at work. Strain-based conflict exists when stress experienced in one role spills over into another role, causing strain symptoms (e.g., anxiety, irritability) and reducing one's ability to perform in the second role. People may experience strain-based conflict, for instance, when work-related stress leads to a conflict with a family member or members. Finally, behavior-based conflict occurs when behaviors required to be effective in one role are incompatible with those required to be effective in another role. For example, behavior-based conflict might occur when an individual is lacking authority at work, but is then required to act as an authoritative figure at home.\n\nIt is important to note that we use the term \"family\" to refer more specifically to \"families of choice\" , a distinction that captures the experiences of individuals with LGB family structures (Cherlin 2004) . Families of choice do not necessarily include biological or legal relationships, but nonetheless encompass relationships that are considered as close as those in which blood or legal ties are present (Weeks et al. 2001). As such, we define \"family\" as those individuals that one identifies as family members, without the necessity of blood or legal ties, allowing us to categorize those in same-sex relationships as families.\n\n## 5.3 Gender and Sexuality: Intersecting Identities and the Roots of Heterosexism\n\nIn order to better understand why LGB individuals might experience WFC differently than their heterosexual counterparts, it is important to first consider the influence of gender and gender norms in creating heterosexist attitudes within society. While sex is determined by one's biology, gender refers to a more general term that encompasses \"all social relations that separate people into different gendered statuses\" (Lorber 1994, p. 3). As such, gender is not solely a reflection of biology, but rather is a product of a complex and interwoven pattern of social interactions, which constitutes and reconstitutes what it means to be \"male\" and \"female\" at a societal level.\n\nButler (2004) further focused on the link between gender and sexuality by questioning the extent to which both categories are purely performative. Specifically, she argued that gender, as well as sexuality, should not be thought of categorically, but rather as a continuous state of being that can fluctuate over time. Butler (2004) also suggested that our construction of gender assumes that part of being male and part of being female is implicated in who we choose as a sexual partner (i.e., someone of the opposite sex). That is, once one is labeled as male or female in terms of their gender, the individual is assumed a priori to have a partner of the opposite sex. In this way, gender and sexuality are inextricably linked within our current heteronormative societal framework.\n\nSimilarly, our performance of sexuality assumes that there exists a \"natural\" sexual orientation (heterosexual) and that any other orientation reflects merely a poor imitation (Butler 2004). In turn, binaries between heterosexuality and all other forms of sexuality are created, along with a false sense of naturalness for heterosexuality and an assumption of abnormality for any other form of sexuality . Given that gender norms lay the framework for the assumption that people should prefer romantic associations with those of the opposite sex, the link between the false naturalization of gendered performance and heterosexuality is made (Butler 2004). Consequently, LGB individuals may be perceived negatively at a societal or individual level because they simultaneously break with deeply rooted gender norms and demonstrate assumedly \"unnatural\" sexual behavior. In the current work, we call for work\u2013family research that begins to challenge these heteronormative assumptions and examines the intersections of gender and sexuality in a less biased way, by placing a greater emphasis on LGB identities in experiences of WFC.\n\n## 5.4 LGB Discrimination in the Workplace\n\nMaking clear the discrimination that LGB employees experience at work contributes to a better understanding of why they may view the workplace as particularly insensitive, or even hostile, toward their LGB identity and family status, in turn contributing, as we argue below, to differential experiences of WFC compared to their heterosexual peers. It has been estimated that between 25 and 66 % of LGB employees have been discriminated against in the workplace, based upon their sexual orientation (Croteau 1996). Further, Ragins and Cornwell (2001) found that one-third of gay and lesbian professionals had been verbally or physically harassed at work and a little over a third had been harassed just because they were assumed to be gay or lesbian. Nearly 12 % found the harassment to be so severe that they decided to leave their jobs entirely. Because sexual orientation is not yet a legally protected class , organizations are not required to include sexual minority status in nondiscrimination statements or to be LGB-friendly. Thus, LGB workers can be fired without any legal protection (Van Den Bergh 2004). Employees are particularly vulnerable to discrimination if they are not living in one of the 21 US states (or the District of Columbia) which have made LGB discrimination illegal at the state level. Even these state laws , however, can be overturned at the federal level, demonstrating a lack of real legal protection for LGB individuals at work.\n\nIn addition to interpersonal prejudice , certain structural features of organizations, including policies and practices related to work\u2013family issues, play a major role in the level of discrimination that LGB employees experience (Ragins and Cornwell 2001; Button 2001). Many organizations do not provide domestic partner benefits and will not extend the Family and Medical Leave Act (FMLA) to domestic partners (Van Den Bergh 2004). The lack of policies that are sensitive to the experiences of LGB individuals and their families is problematic given that their absence has been previously shown to be the best predictor of frequency and severity of discrimination at work (Button 2001; Ragins and Cornwell 2001). Because LGB employees are frequently not able to benefit from inclusive antidiscrimination policies, inconsistent and potentially harmful experiences for LGB employees are more likely to occur across their careers. Further, organizations that do not endorse a friendly climate toward LGB employees create spaces in which it is both frightening and detrimental for individuals to reveal their \"family of choice\" (Van Den Bergh 2004). Although it has been demonstrated that participation in diversity training , which includes information about sexuality , may produce more positive attitudes toward LGB individuals (Probst 2003), diversity training sessions, especially those including sexuality, may not be offered within organizations (Day and Schoenrade 2000).\n\nThus, as a result of prejudice toward LGB employees and their families, LGB individuals may experience unique WFC when compared to heterosexual employees and their families. The pathologizing of homosexuality and the belief that LGB individuals suffer from mental illness or that they make the choice to be LGB (Sedgwick 1988) allows individuals to feel as if discrimination against LGB individuals is warranted. If the general population believes that LGB individuals are inherently \"sick\", then fully incorporating LGB individuals into the workplace is not seen as a given, but rather as a choice that should be made on an individualized basis. Thus, LGB individuals are not granted the same rights as other minority groups, who have identities which are viewed as immutable and psychologically \"normal\".\n\nIndeed, Colgan et al. (2000) found evidence for many people's restricted definitions of \"family\" with regard to LGB families versus heterosexual families. The authors highlighted the extent to which stereotypical markers of a heterosexual family (in this instance, children), may lead to confusion when an employee reveals an LGB identity. Thus, based on stereotypically heterosexual family indicators, coworkers may make assumptions that fellow employees must also be heterosexual. In the same vein, Williams et al. (2009) highlighted the ways in which organizations privilege heterosexual families through policy, reinforcing heterosexual relationship norms, and by implicitly encouraging employees and clients to normalize heterosexuality in their interactions with one another. Overall, it appears that (in many instances) LGB families are rendered invisible and impossible (Foucault 1992) within organizations. However, because LGB individuals are able to choose whether or not to reveal their LGB identity, these individuals have the option to either remain silent or to become visible within their organization with regard to their LGB identity. Thus, LGB individuals have a unique WFC vantage point\u2014their partner or family identity must be actively revealed in order for individuals at work to be aware of its existence.\n\n## 5.5 Implications of LGB Discrimination on \"Coming Out\" at Work\n\nBecause LGB employees may be well aware of the risks inherent in being LGB at work, they may choose not to reveal their LGB identity in the workplace. Ragins et al. (2007) found that fear of disclosure predicted a variety of important workplace outcomes in addition to employee strain, over and above actual disclosure. Thus, fear of disclosure is an important consideration when estimating the effects of acceptance (or lack thereof) for LGB employees in the workplace. Further, Croteau (1996) found that, across nine studies, fear of disclosure is a major concern for LGB employees. Results also demonstrated that LGB employees were more likely to be discriminated against when they were \"out\" at work. This is pertinent to WFC because LGB individuals may be unlikely to disclose their LGB identity and, at the same time, their family identity in order to avoid this discrimination. In this way, the fear of being LGB at work causes LGB individuals to monitor their behavior at all times, self-policing and managing their identity constantly, similarly to the way in which Foucault (1992) described the watchtower mentality. Even when an authoritative figure is not physically present, individuals continue to monitor their identity, fearing the repercussions that might befall them if they are \"discovered\" as their true selves. This constant experience of invisibility and the impossibility of truly \"being\" at work may be both mentally and physically exhausting for LGB employees (Colgan et al. 2008; Ellis and Riggle 1996; McDermott 2006). Thus, if individuals actively wish to be out of the closet and are unable to do so because of a hostile work environment (c.f., Croteau 1996; Ragins et al. 2007; Ellis and Riggle 1996), the experienced push and pull between identity preferences and situational reality may lead to increased conflicts between work life and home life. We focus here on the relationships between being closeted and WFC stressors that may arise as a result.\n\n## 5.6 Disclosure Disconnects and LGB Work Family Conflict\n\nGiven the impact of actual and perceived discrimination on LGB employees' decisions to \"come out\" at work, it is easy to imagine the various pushes and pulls between the work and family domains on such individuals. Ragins (2004) noted that LGB individuals may be \"out\" to varying degrees in both their work and personal lives. Indeed, research suggests that LGB employees disclose at different rates to parents, spouses, friends, schoolmates, and coworkers (Schope 2002). This variation in \"outness\" across groups produces a \"disclosure disconnect\" (Ragins 2008), which creates stress for LGB employees. Hill (2009), for instance, noted that LGB employees are often expected to leave their family lives at home, while heterosexual employees may not have to worry about creating separation between work and family life.\n\nThis creates an important difference in the WFC experienced by heterosexual employees versus LGB employees. It may be the case that for heterosexual individuals the separation between work and family leads to lower WFC, given that heterosexual employees are able to freely choose the extent to which they disclose various aspects of their family lives at work. It may also be the case that, when convenient, heterosexual employees are also able to effectively fuse, or reduce the space between, their work and family lives through various work\u2013family policies (e.g., on-site childcare, bringing family to work events). Thus, the impetus to separate work and family domains by choice or increase the ways in which work and family life can be combined may benefit heterosexual employees who can freely select between a number of viable WFC solutions. In contrast, the option to mix work and family may not be available for LGB employees, resulting in the forced construction of barriers between the work and family realms.\n\nThus, within organizations that do not accept LGB families as legitimate, the active separation between the work and family domains is not done by choice, such that even speaking about one's LGB relationship\/family at work is considered taboo. Consequently, it may be that the separation of the work and family domains is an issue of having options\u2014an employee (no matter their sexual identity) benefits from the personal choice to leave work at the office and family at home when it is convenient. However, for LGB employees this decision is made a priori\u2014the family domain is deemed unacceptable within the work domain. The conclusion is that, while attempting to extricate the work and family domains from one another may reflect a plausible solution for WFC among heterosexual employees, this very act may produce WFC for LGB employees. The implicit message sent when organizations allow for selective separation of work and family time is that the organization is cognizant of the heterosexual family only, and thus creates policies that make it possible for certain employees to keep family time \"sacred.\"\n\nHowever, when LGB relationships\/families are invisible at work (thereby forcibly separating the work and family spheres), it fosters a belief that the organization views LGB families as nonexistent. This belief is propagated by the lack of fair and inclusive workplace policies and practices for LGB employees. In effect, the experience of being \"allowed\" to keep work and family life separate or integrated is quite different from the experience of forcible separation of work and family realms through fear and a lack of acceptance, especially when this separation is not favorable or preferred. Consistent with this assertion, Day and Schoenrade (2000) found that higher levels of disclosure were related to less perceived WFC.\n\nOverall, the persistence of LGB discrimination seems to reflect a continued social stigma related to being gay, thereby leading to fear, ostracism, disregard, or even disgust toward LGB employees in the workplace (Embrick et al. 2007). Given that WFC is a major issue at work, and given that members of LGB families may be discriminated against with respect to work\u2013family policies, it is vital that we study the ways in which our assessments of and preventative measures regarding WFC may be biased toward heterosexism. A necessary first step is to examine how current conceptualizations and measures of WFC may be contaminated or deficient, potentially leading to more inclusive work\u2013family studies and policies.\n\n## 5.7 Heterosexual Bias in Measures of Work\u2013Family Conflict\n\nWhile measurement bias and unfair testing practices remain a consistent issue of concern to psychologists, the impact of heterosexual bias on many popular survey measures is still understudied . Heterosexual bias is a belief system that places more value on heterosexuality and views heterosexuality as a more \"natural\" form of sexuality than homosexuality (Morin 1977). Heterosexual bias is reinforced by systems and methodologies, which assume heterosexuality as a given (heteronormativity) , privileging heterosexual views and rendering others unnecessary or invisible (Morin 1977). Indeed, most surveys are constructed and tested within populations that are assumed a priori to be heterosexual. As a result, heterosexuality tends to be the normative lens through which researchers study \"natural\" phenomena. However, given that LGB individuals make up between 4 and 17 % of the workforce (Gonsiorek and Weinrich 1991), questions remain regarding the extent to which various surveys used in studies of organizational phenomena function equally across participants and capture certain populations' unique experiences.\n\nGiven our discussion in the preceding sections, we suggest that it is important to consider the possibility that LGB workers may experience WFC differently given real and perceived prejudicial barriers that prevent them from discussing their LGB identities and families at work. More specifically, we argue that current WFC measures contain items that may be differentially interpreted across heterosexual and LGB groups, introducing construct contamination and inaccuracy into studies of WFC. We also suggest the possibility that the WFC construct may be deficient in that it may not fully capture the ways in which work and family can conflict with one another, particularly for LGB individuals. In the following section, we examine the ways in which LGB members may uniquely interpret WFC items from existing scales.\n\n### 5.7.1 Differential Item Interpretation Among LGB Members\n\nFor the purposes of the present effort, we discuss Carlson et al.'s (2000) measure as a potential example of heterosexual bias in measures of WFC. Carlson et al.'s measure is comprised of six dimensions: time (WFC and FWC) , strain (WFC and FWC), and behavior-based (WFC and FWC) conflict . As such, it conforms to the popular three-dimensional framework introduced by Greenhaus and Beutell (1985), while also acknowledging more recent distinctions made between WFC and FWC. Carlson et al. (2000) selected nonredundant items across 25 previously existing scales in the work\u2013family literature, resulting in their inclusion of 31 items from 8 scales. This initial pool of items was then factor-analyzed and narrowed down to a final set of 18 items, which were then validated against various work\u2013family criteria. Because the measure represents a synthesis of 25 existing measures, it is the most comprehensive to date, and therefore provides a useful starting point for identifying potential heterosexual bias inherent in various WFC scales. (See Carlson et al. 2000 for the full measure, although sample items are discussed below.)\n\n#### 5.7.1.1 Time-Based WFC and FWC\n\nBeginning with time-based work interference with family, from a heterosexual viewpoint, items such as \"My work keeps me from my family activities more than I would like\" clearly relate to time . However, from an LGB perspective, this item may be differently interpreted. Being kept from family activities because of work could be due to a variety of circumstances, which may have little to do with time. For example, if one works in a small town and does not feel inclined to reveal his\/her LGB identity to an employer, then work may serve to keep LGB employees away from taking part in family activities where it is possible they will be observed by a coworker (e.g., going to a local park with one's LGB family, grocery shopping with one's LGB partner), even if there is ample time to perform such activities. Further, being able to bring a spouse or significant other to work events has been noted as a major determinant of satisfaction at work for LGB individuals (Ragins and Cornwell 2001). Those who are unable to bring partners or other family members to work events may feel that they are missing family time given that they are unable to combine work and family time, even when they are \"off the clock.\"\n\nAuthors have noted that those who hide their sexual orientation at work may find it stressful merely discussing summer vacations, weekend activities, or other occurrences outside of work (Preston et al. 2006; Gedro 2007). As such, the fear of revealing one's sexual identity may keep employees from participating in family activities through which they may be \"outed.\" This is likely to vary according to one's level of outness , however. Those who are \"out\" at work are unlikely to feel that they need to shield themselves from being seen at events outside of the workplace. However, for those who are \"out\" to only a few or to none, the consequences of being seen with one's partner or family outside of work are much greater . The other two items within time-based WFC are more clearly worded to be about time specifically and, as such, are unlikely to pose a problem with respect to heterosexual bias . As such, we believe that current measures of time-based WFC may suffer from heterosexual bias such that LGB individuals will interpret certain scale items differently than heterosexual individuals .\n\nMoving from WFC to FWC, time-based family interference with work also contains another potentially problematic item\u2014\"The time I spend with my family often causes me not to spend time in activities at work that could be helpful to my career.\" When read in the context of a heterosexual family , the item reflects an issue of time\u2014if you spend a lot of time with your family, there may not be enough time to complete work tasks. Yet those in LGB families may interpret spending time with their family as inextricably linked with being a part of a family structure which is not well accepted at work . As such, for those who are \"out\" at work, this item might read very differently, such that the interpretation may be, \"Because I am a part of an LGB family, I am passed over when opportunities arise at work.\" Further, if a closeted LGB employee is unable to engage in social gatherings where employees are expected to bring their partners, or becomes involved in conversations surrounding weekend plans, vacations, etc., he\/she may feel disconnected from coworkers and less likely to receive job rewards . Thus, deciding whether to engage in work activities that could result in rewards may have nothing to do with whether one has time to complete such activities and more to do with whether one feels comfortable engaging in them .\n\nIt is important to mention that one of the most popular interventions to WFC stems from the time-based dimension: flexible working arrangements (FWAs). FWAs provide the ability to vary working hours and scheduling around personal obligations and family needs (Rau and Hyland 2006). For example, Shockley and Allen (2007) found that FWAs were negatively related to WFC for women with high levels of family responsibility . FWAs are becoming increasingly popular in organizations as effective interventions for WFC. However, FWAs have only been demonstrated to have beneficial outcomes for those who experience high levels of WFC as it is traditionally measured (Rau and Hyland 2006). Therefore, time reflects only one cause of work\u2013family constraints and while a lack of time may be a primary facet of WFC for LGB employees, similar to their heterosexual counterparts, there may be additional layers of WFC that LGB individuals experience. Following this logic, it is possible that current measures of time-based FWC may suffer from heterosexual bias such that LGB individuals will interpret certain scale items differently than heterosexual individuals .\n\n#### 5.7.1.2 Strain-Based WFC and FWC\n\nStrain-based items focus in on a different facet of WFC, such that their interpretation may be similar across heterosexual and LGB participants, yet their antecedents may differ. For example, the strain-based WFC scale includes the items, \"When I get home from work, I am often too frazzled to take part in family activities\/responsibilities,\" \"I am often so emotionally drained when I get home from work that it prevents me from contributing to my family\", and \"Due to pressures at work, sometimes when I come home I am too stressed to do the things I enjoy.\" Here, the root cause for being \"frazzled\", \"emotionally drained\", or \"stressed\" may vary for LGB employees. Higher levels of stress and strain at home may reflect not only job-related demands but also a need to conceal one's LGB identity\/family at work. Thus, LGB employees may score higher on these items than their heterosexual peers for reasons that reflect their LGB identities and family situations, rather than simply job demands. Thus, current measures of strain-based WFC may suffer from heterosexual bias such that LGB individuals may interpret certain scale items differently than heterosexual individuals .\n\nSimilarly, items within the strain-based FWC scale that read, \"Due to stress at home, I am often preoccupied with family matters at work\", \"Because I am often stressed from family responsibilities, I often have a hard time concentrating on my work\", and \"Tension and anxiety from my family life often weakens my ability to do my job\", may elicit higher endorsement from LGB workers. Stress, tension, and anxiety at work arising directly from one's family life may stem from the stress and strain of daily living (e.g., conflicts with a spouse, caring for a disabled child), but can also be exacerbated by the inability to reveal one's LGB family structure at work. Such items may be assessing a different kind of strain from an LGB standpoint, given that stress and anxiety can arise when employees are discriminated against for coming out at work or are \"closeted\" and find themselves in heterosexist work environments. Further, some individuals may experience conflict at home about whether or not to come out at work. For example, it may be the case that one's partner is out at work, while they are not . The partner who is out might put pressure on the closeted partner to reveal his\/her identity. In this way, being a part of an LGB family may add to tension and stress at work given there may be more complicated boundaries to navigate between partners when only one person in a couple is willing or able to reveal the existence of the other. Thus, we may find a higher level of endorsement (reflecting contamination) and a different set of antecedents for such items. As discussed earlier, all of this may be due to overflow from other unmeasured domains of WFC (i.e., identity-based) for LGB relationships and families, potentially leading to criterion deficiency.\n\nFor example, Button (2001) found that LGB individuals who experienced greater workplace discrimination were more likely to use strategies such as counterfeiting (pretending to be heterosexual) or avoiding (acting as if one has no sexuality at all) at work. The use of such strategies may be stressful for LGB employees, especially if they are used in order to avoid discrimination. While there are various reasons that one may wish to remain closeted (and power may often reside in the closet) (Brown 2000; Ferfolja 2009), for those who do not wish to be closeted, yet feel they have no other choice, choosing strategies of silence may create anxiety . Indeed, there is evidence to suggest that LGB individuals who work in gay-friendly environments and who are out in the workplace experience higher job satisfaction and lower stress and anxiety (Day and Schoenrade 2000; Driscoll et al. 1996; Griffith and Hebl 2002). Therefore, current measures of strain-based FWC may suffer from heterosexual bias such that LGB individuals will interpret certain scale items differently than heterosexual individuals .\n\n#### 5.7.1.3 Behavior-Based WFC and FWC\n\nHeterosexist bias may also exist in the behavior-based WFC items: \"The problem solving behaviors I use in my job are not effective in solving problems at home\", \"Behavior that is effective and necessary for me at work would be counterproductive at home\", and \"The behaviors I perform that make me effective at work do not make me a better parent and spouse.\" For example, the behaviors that may make an LGB individual more \"effective\" at work (e.g., concealing one's family status, pretending to be heterosexual) would not contribute to being an effective member of an LGB family. While these behaviors are obviously not the only behaviors that contribute to on-the-job performance, they reflect a subset of activities that are not required of heterosexual employees and thus expand the requirements for what it takes to be \"successful\" at work as an LGB employee. Further, while faking or passing behaviors are not equivalent to actual job-related tasks (typing, answering calls), they must be constantly monitored, and thus may infringe upon or affect behaviors that are directly job-related. As such, it is possible that LGB individuals may be aware of the necessity of these behaviors in the workplace and interpret items in light of this range of seemingly \"required\" on-the-job behaviors. This suggests that current measures of behavior-based WFC may suffer from heterosexual bias such that LGB individuals will interpret certain scale items differently than heterosexual individuals .\n\nSimilarly, the behavior-based FWC items highlight similar themes: \"The behaviors that work for me at home do not seem to be effective at work\", \"Behavior that is effective and necessary for me at home would be counterproductive at work\", and \"The problem solving behavior that works for me at home does not seem to be as useful at work.\" Such items may be differently interpreted given that behaviors that are necessary at work and behaviors that \"work\" at home may not coincide for individuals with LGB families. In a heterosexist workplace, behaviors that LGB individuals enact at home (displaying affection toward one's partner [while at a company party, for example], talking freely about one's LGB family structure and other LGB-related issues or concerns in the presence of coworkers) may be damaging when performed at work and are most likely limited in coworker discussions on the job. Finally, in line with the previously discussed dimensions , it may be the case that current measures of behavior-based FWC suffer from heterosexual bias such that LGB individuals will interpret certain scale items differently than heterosexual individuals .\n\nFigure 5.1 illustrates the hypothesized overlap between WFC as it is currently assessed and a more inclusive measure of the construct. The figure suggests that preexisting scales may suffice for LGB and heterosexual individuals with respect to certain items (e.g., \"The time that I spend at work keeps me from family responsibilities more than I would like\"). Other items, however, may be contaminated (e.g., \"The behaviors that work for me at home don't seem to be as effective at work\" may be interpreted quite differently for LGB individuals) and deficient, such that individuals cannot talk about their family at work but do not have a place within present measures to report that experience. Current measures may contain possible contamination and construct overlap, but we have not yet explored their deficiencies .\n\nFig. 5.1\n\nConceptual model of current work-family measures\n\n## 5.8 LGB Identity-Based Work-Family Conflict: A New Dimension of WFC\n\nBased on our discussion above, it is also possible that we are omitting an entire dimension of WFC for LGB individuals, which stems directly from their LGB family identity. Given that LGB individuals may face discrimination that leads to a lack of \"outness\" at work (Ragins and Cornwell 2001; Waldo 1999), it may be the case that WFC stems directly from one's LGB family identity. For example, if employees are not out at work, or are out only to some individuals, they may experience difficulty asking about LGB family-friendly benefits or taking advantage of other work\u2013family policies within the organization that might be useful to them. Moreover, given that LGB individuals may feel uncomfortable coming out at work, they may be less likely to bring their partner to company events or display pictures of their partner on their desks at work. These conflicts stem more from one's LGB family status and are not currently being captured in the work\u2013family domain. While some of these items have been captured in other measures (e.g., Griffin 1992; Ragins 2004), it seems that these issues are more clearly tied to work and family domain conflicts than to other constructs within the psychological domain. As such, we suggest that an additional dimension of WFC, identity-based conflict , may be necessary in order to more fully capture the WFC experiences of LGB populations. Here, we define identity-based conflict as occurring when one's relationship\/family identity is in conflict with the range of acceptable\/recognized relationship\/family identities at work. Thus, we propose that an additional dimension of WFC, identity-based conflict, is necessary in order to more fully capture the unique WFC experiences of LGB individuals.\n\n### 5.8.1 Potential Antecedents of LGB Identity-Based Work\u2013Family Conflict\n\nIn order to properly evaluate the conditions under which LGB individuals may experience WFC to a greater or lesser degree, it is also important to consider potential antecedents of LGB identity-based WFC . That is, while the larger WFC literature has explored various predictors of WFC in the general population, we still know relatively little regarding the potentially unique antecedents of WFC within LGB samples. Greater insight into the links between LGB-specific antecedents and LGB identity-based WFC is further necessary given that different predictive models of WFC may exist for LGB individuals as compared to their heterosexual counterparts.\n\n#### 5.8.1.1 Level of \"Outness\" at Work\n\nAs noted earlier, while some LGB employees may derive a certain degree of power from remaining closeted and freely choose to conceal their LGB identities for strategic purposes (Brown 2000), others may wish to come out but do not believe that they have the choice if the workplace is believed to be discriminatory toward them and their LGB families. As noted by Ragins (2004), one's level of \"outness\" across different life domains, such as work and family, tends to produce disclosure disconnects that are stressful for LGB individuals. We argue that lower levels of \"outness\" in the workplace lead to higher levels of identity-based WFC for LGB employees, given that disclosure disconnects are likely to be associated with perceptions that the organization is imposing barriers between the work and family domains and restricting the choice to freely intermix elements of work and family when convenient. It may be the case, then, that lower levels of \"outness\" at work among LGB individuals may be associated with higher levels of LGB identity-based WFC .\n\n#### 5.8.1.2 Fear of Disclosure\n\nAlthough one's fear of disclosure is likely to be negatively related to his\/her level of \"outness\" in the workplace (Ragins et al. 2007), the two constructs may nonetheless be unique predictors of LGB identity-based WFC given that decisions to remain closeted may reflect other factors outside of such fear (e.g., self-esteem, introversion, locus of control). Fear of disclosure may stem from real and perceived discrimination at work, which causes LGB employees, in most cases, to hide their sexual orientation and family status from supervisors and colleagues. In turn, they are likely to experience a clash between the work and family realms, given that they are unable to freely select among the range of work\u2013family solutions that are more accessible to heterosexual employees. As a result, we predict that fear of disclosure may be positively related to higher levels of identity-based WFC for LGB individuals within the workplace.\n\n#### 5.8.1.3 Social Support\n\nA number of studies point to the buffering effects of perceived social support on experienced levels of WFC (Carlson et al. 2000; Greenhaus et al. 1987; Schaubroeck et al. 1989) . Within a sample of 99 LGB individuals, Huffman et al. (2008) found that supervisor support was positively associated with job satisfaction , coworker support was positively correlated with life satisfaction , and organizational support was positively linked to being \"out\" at work. It may be that when LGB employees perceive their organization, supervisor , and colleagues as supportive and sensitive to LGB-related issues, they may feel more comfortable disclosing information regarding their LGB identity and family status. Thus, perceptions of support may play a vital role in shaping an organizational culture where LGB families are valued and accepted, thereby removing disclosure disconnects that forcibly separate the work and family domains for LGB employees. When social support is present, LGB employees, like their heterosexual counterparts, may feel free to choose the extent to which they separate or mix aspects of their work and family lives, such that higher levels of social support may be associated with lower levels of LGB identity-based WFC .\n\n#### 5.8.1.4 Organizational Policies and Practices\n\nIn addition to social support, organizational policies and practices that are LGB-friendly, including same-sex partner benefits, LGB antidiscrimination policies, and LGB resource support groups, are likely to promote a perception of the organization as supportive and sensitive to LGB-related issues, which in turn may eliminate disclosure disconnects and socially imposed boundaries between the work and family lives of LGB employees. In contrast, organizations that do not institute such policies and practices are likely to promote, whether implicitly or explicitly, an assumption that they do not value or recognize LGB families. Perceptions of discrimination contribute to disclosure disconnects and in turn to higher levels of identity-based WFC among LGB employees. As noted earlier, an organization's espoused policies regarding LGB-related issues have been shown to be the most powerful predictor of perceived discrimination and various work-related attitudes (Button 2001; Ragins and Cornwell 2001). Overall, then, LGB-friendly organizational policies and practices may be associated with lower levels of LGB identity-based WFC .\n\n#### 5.8.1.5 Centrality of Sexual Identity\n\nIt is also likely that the more strongly one sees their sexual identity as being central to their overall identity, the more likely they will report experiencing LGB identity-based WFC. For people who do not attach aspects of their sexuality as strongly to their core identity, the desire to \"come out\" at work may not be as much an issue as it is for those who place a greater focus on their sexual identity. Such individuals may not experience the same tension between their work and private lives and may be more comfortable keeping these life domains separate. Moreover, centrality of identity has been found to relate to identity salience (Sellers et al. 1998) for race (Griffith and Hebl 2002; Law et al. 2011). Thus, it is possible that it holds for sexuality as well. As a result, greater centrality of sexual identity among LGB individuals may be associated with higher levels of LGB identity-based WFC .\n\n#### 5.8.1.6 Gender\n\nAlthough patterns of WFC differ by gender in heterosexual samples (Duxbury and Higgins 1991), this same pattern may not hold true for LGB couples. Gay and lesbian couples are found to split household chores and responsibilities for child care in a more egalitarian manner than heterosexual couples (Solomon et al. 2005). In fact, in a study of same-sex couples in civil unions and heterosexual couples in Vermont, Solomon et al. (2005) found that the strongest predictor of couples' division of household labor was sexual orientation , which predicted over and above income differences. Given these findings, it may be the case that LGB couples experience smaller gender discrepancies in identity-based WFC compared to heterosexual couples.\n\n## 5.9 Implications for Research\n\nBased on our discussion, there are several areas that future research should address. First, in order to begin assessing whether LGB employees experience WFC differently, a first step might be to explore LGB individuals' WFC experiences via qualitative interviews, as well as quantitative analyses of the potentially unique antecedents of LGB identity-based WFC . Such work would lend initial insight into whether or not current measures \"work\" across LGB populations or whether there are substantive differences in the WFC experiences of LGB individuals. More specifically, participants might be asked how they define and experience WFC and how they interpret items comprising existing scales, as well as to report the extent to which various presumed antecedents of WFC play a role in their experiences of WFC.\n\nSecond, if qualitative differences in LGB WFC experiences appear to exist, a necessary second step would be to generate a set of items, based on earlier interviews, that capture the content domain of identity-based conflict and to examine whether these new items and those comprising existing scales demonstrate content validity in representative samples of LGB employees; that is, can LGB individuals adequately classify the new identity-based items and preexisting WFC scale items into their respective dimensions? Consistent with best practices outlined in the scale development literature (e.g., Conway and Huffcut 2003; Ford et al. 1986; Hinkin 1995), it would be necessary to then examine and confirm the underlying dimensionality of this more inclusive measure and determine the extent to which it offers predictive validity over and above existing measures of WFC for LGB individuals.\n\nThird, beyond the work\u2013family domain, our discussion points to a broader need for researchers to examine the common assumptions that may underlie surveys of organizational phenomena. Thinking more inclusively about who constructs a given scale, which groups were included in samples that were used to validate the scale, and which groups the validation process did not include allows researchers to assess various measures from unique and more inclusive perspectives. Indeed, our analysis suggests that LGB family issues have not been considered to the extent that they should be within the existing work\u2013family literature, despite the long-standing history of work\u2013family research. Similarly, other researchers have called attention to the need to reassess many common scales for potential bias. Cole and Sabik (2009), for example, pointed out that racial bias in the measurement of women's self-esteem about appearance may be driving differences between white and ethnic minority women. While the need to re-examine our surveys from a minority perspective is not a difficult concept to grasp, substantive work in this area lags in psychology. However, there are significant gains to be made by thinking about \"for whom\" our measures work and for whom they do not if we are to understand the complexities underlying the phenomena we study. As such, we hope that this chapter spurs future work that reexamines long-standing constructs in I\/O psychology and organizational behavior and asks whether or not these constructs hold in more diverse populations.\n\nFourth, our discussion highlights the need to examine alternative family structures in greater depth. It may be the case that families who possess other stigmatized identities (disability, size, religion , socioeconomic status , single parent status, etc.) may experience similar identity-based WFC. Examining unique family experiences may lead to a more complete and nuanced conceptualization of the work\u2013family domain. Our definition of identity-based WFC may relate to a range of families with stigmatized identities. However, the examination of LGB families is a logical first step since there appears to be both a stigma attached to an LGB identity (Ragins et al. 2007) and also a visceral negative reaction associated with the realization of this identity for some individuals (Stacey 1998). As such, there may be something particularly unique about an LGB identity that affects the work and family domains in a more negative fashion than other \"alternative\" families might experience because an LGB identity is often viewed as \"controllable\". It is possible that this difference produces a greater level of conflict because the stakes for \"coming out\" about one's stigmatized identity might be higher. Thus, starting with LGB families might provide insight into the more extreme cases in which work and family conflict on the basis of identity, but should also be complemented with work on other stigmatized family identities.\n\nFifth, while we examine WFC in the present effort, it may be of interest in the future to examine the ways in which work and lifestyle conflict for those LGB individuals who do not consider themselves a part of a family. For example, for single LGB individuals who are not a part of an LGB family, the survey might be administered with the words \"life\" or \"lifestyle\" in place of \"family.\" In turn, this may permit researchers to compare and contrast responses of those who are currently in an LGB family\/relationship with those who are not. Although much of the literature talks about work-life balance conceptually , scale items still reflect a family focus [as demonstrated in Carlson et al.'s (2000) meta-measure]. While only those who are partnered or consider themselves to be a part of an LGB family are the focus of this chapter, nonpartnered individuals may be included in future research to determine whether or not it is necessary to consider additional issues when measuring LGB identity-based work-life conflict as opposed to LGB identity-based WFC .\n\nSixth, this chapter provides an impetus for researchers to assess sexuality as part of a standard set of demographics measured within psychological research. While measures of gender or race are staples in most surveys' demographics sections, measures of sexual orientation are not commonly collected. To understand LGB experiences at work, we must begin assessing sexuality in a more standardized way. In particular, and consistent with Intersectionality Theory (Crenshaw 1989), it is possible to examine family identity as a layered construct with multiple pieces. Instead of assuming a unified definition for the term \"family\", we open the door to realizing how multiple layers of family identity may influence outcomes of interest in unique ways. By beginning to expand our definition of family and becoming cognizant of the multitude of ways that individuals might define family [i.e., \"family of choice\" (Cherlin 2004)], we are able to better understand how multiple family identities might combine and then, as a result, have a different impact on variables that are important to families, their quality of life and their quality of work.\n\nFinally, we highlight the need for gender researchers to become more heavily invested in examinations of sexuality-based bias within the work\u2013family literature. Because gender and sexuality-based discrimination are inherently intertwined, open dialogue between researchers interested in gender and researchers interested in sexuality will help to develop future work in a more holistic way. By focusing on the intersections of identity and resulting discriminatory or silencing mechanisms, we create more accurate models of the work\u2013family experiences of diverse individuals with potentially diverse or alternative family structures.\n\n## 5.10 Implications for Practice\n\nThe literature reviewed above suggests that organizations may be overlooking the experiences of LGB employees, a significant omission given gay and lesbian employees are estimated to comprise anywhere between 4 to 17 % of today's workforce (Gonsiorek and Weinrich 1991; Ragins and Cornwell 2001). It behooves organizations to create better LGB organizational policies and practices (i.e., same-sex partner benefits, LGB partner-friendly social events, LGB resource support groups, and LGB antidiscrimination policies), given that these policies may have the strongest effects on gay and lesbian employees' perceptions of discrimination (Ragins and Cornwell 2001). Thus, by actively assessing LGB identity-based WFC , organizations may allow employers to implement more inclusive work\u2013family policies which address the needs of a historically silenced population within the workplace. These policies may not only prove effective in increasing the perception of acceptance and lowered discrimination, but may also impact important business metrics such as job satisfaction , commitment, and turnover intention (Ragins and Cornwell 2001).\n\nKnowledge of the potentially unique WFC experiences of LGB employees may further serve to inform relevant LGB affinity groups, who might utilize such work to lobby for more effective work\u2013family policies within organizations, such as those related to LGB family benefits, same-sex partner recognition, and more inclusive, LGB-friendly work environments. Including organizational identity politics within a work\u2013family framework may be an effective means for affinity groups to work toward more inclusive organizational policies and practices for LGB families in a way that fits with existing organizational goals.\n\nFinally, because improving WFC may improve job satisfaction (Bruck et al. 2002; Grandey et al. 2005; Kossek and Ozeki 1998), examining LGB identity-based WFC may provide the tools for creating the \"business case\" for HR professionals interested in decreasing the conflicts between work and family for employees. Creating awareness of the positive effects of decreasing LGB identity-based WFC among those responsible for talent management and human capital programs and solutions within organizations may help to spearhead inclusive work\u2013family policies and practices on a broader scale.\n\n## 5.11 Concluding Remarks\n\nThe present chapter has brought to light the role that heterosexual bias has played thus far in our conceptualizations of WFC. Because gender stereotypes also encompass our assumptions about sexual preferences (i.e., that men are attracted to women and that women are attracted to men), questioning the gendered nature of WFC should also prompt us to question the heteronormative assumptions that exist within the current work\u2013family literature. If our current measures do not allow for or consider the presence of LGB employees and their families, we run the risk of silencing these populations within the WFC domain, both academically and practically. However, urging researchers and employers to consider the plight of LGB individuals when examining and remedying WFC may allow us to collectively create more inclusive studies, measures, and interventions for addressing the needs of LGB families in the future. Decreasing the deficiencies in our current conceptualizations of WFC can only lead to a more nuanced and accurate understanding of the various forms of WFC moving forward\u2014a goal which is preferable for both scientists and practitioners alike.\n\nReferences\n\nBerlant, L., & Warner, M. (1998). Sex in public. Critical Inquiry, 24, 547\u2013566.CrossRef\n\nBrown, M. P. (2000). Closet space: Geographies of metaphor from the body to the globe. London: Routledge.\n\nBruck, C. S., Allen, T. D., & Spector, P. E. (2002). The relationship between work-family conflict and job satisfaction: A finer grained analysis. Journal of Vocational Behavior, 60, 336\u2013353.CrossRef\n\nButler, J. (2004). Undoing gender. New York: Routledge.\n\nButton, S. B. (2001). Organizational efforts to affirm sexual diversity: A cross-level examination. Journal of Applied Psychology, 86, 17\u201328.PubMedCrossRef\n\nCarlson, D. S., Kacmar, K. M., & Williams, L. J. (2000). Construction and validation of a multidimensional measure of work-family conflict. Journal of Vocational Behavior, 56, 249\u2013276.CrossRef\n\nCherlin, A. J. (2004). The deinstitutionalization of American marriage. Journal of Marriage and Family, 66, 848\u2013861.CrossRef\n\nCole, E. R., & Sabik, N. J. (2009). Repairing a broken mirror: Intersectional approaches to diverse women's perception of beauty and bodies. In M. T. Berger & K. Guidroz (Eds.), The intersectional approach: Transforming the academy through race, class and gender (pp. 173\u2013192). Chapel Hill: The University of North Carolina Press.\n\nColgan, F., Creegan, C., McKearney, A., & Wright, T. (2008). Lesbian workers: Personal strategies amid changing organisational responses to 'sexual minorities' in the UK workplaces. Journal of Lesbian Studies, 12, 31\u201345.PubMedCrossRef\n\nConway, J. M., & Huffcutt, A. I. (2003). A review and evaluation of exploratory factor analysis practices in organizational research. Organizational Research Methods, 6, 147\u2013168.CrossRef\n\nCrenshaw, K. (1989). Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory, and antiracist politics. University of Chicago Legal Forum, 140, 139\u2013167.\n\nCroteau, J. M. (1996). Research on the work experiences of lesbian, gay, and bisexual people: An integrative review of methodology and findings. Journal of Vocational Behavior, 48, 195\u2013209.CrossRef\n\nDay, N. E., & Schoenrade, P. (2000). The relationship among reported disclosure of sexual orientation, anti-discrimination policies, top management support, and work attitudes of gay and lesbian employees. Personnel Review, 29, 346\u2013363.CrossRef\n\nDriscoll, J. M., Kelley, F. A., & Fassinger, R. E. (1996). Lesbian identity and disclosure in the workplace: Relation to occupational stress and satisfaction. Journal of Vocational Behavior, 48, 229\u2013242.CrossRef\n\nDuxbury, L. E., & Higgins, C. A. (1991). Gender differences in work-family conflict. Journal of Applied Psychology, 76, 60\u201373.CrossRef\n\nEllis, A. L., & Riggle, E. D. B. (1996). The relation of job satisfaction and degree of openness about one's sexual orientation for lesbians and gay men. Journal of Homosexuality, 30, 75\u201385.CrossRef\n\nEmbrick, D. G., Walther, C. S., & Wickens, C. M. (2007). Working class masculinity: Keeping gay men and lesbians out of the workplace. Sex Roles, 56, 757\u2013766.CrossRef\n\nFerfolja, T. (2009). State of the field review: Stories so far: An overview of the research on lesbian teachers. Sexualities, 12, 378\u2013395.CrossRef\n\nFoldy, E. G., & Creed, W. E. D. (2003). Action learning, fragmentation, and the interaction of single, double, and triple-loop change: A case of gay and lesbian workplace advocacy. In R. J. Ely, E. G. Foldy, & M. A. Scully (Eds.), Reader in gender, work, and organization (pp. 242\u2013257). Oxford: Blackwell.\n\nFord, J. K., MacCallum, R. C., & Tait, M. (1986). The application of exploratory factor analysis in applied psychology: A critical review and analysis. Personnel Psychology, 39, 291\u2013314.CrossRef\n\nFoucault, M. (1992). The history of sexuality Vol. 2: The use of pleasure. London: Penguin.\n\nFrone, M. R., Russell, M., & Cooper, M. L. (1992). Antecedents and outcomes of work-family conflict: Testing a model of the work-family interface. Journal of Applied Psychology, 77, 65\u201378.PubMedCrossRef\n\nFrone, M. T., Yardley, J. K., & Markel, K. S. (1997). Developing and testing an integrative model of the work-family interface. Journal of Vocational Behavior, 50, 145\u2013167.CrossRef\n\nGedro, J. (2007). Conducting research on LGBT issues: Leading the field all over again! Human Resource Development Quarterly, 18, 153\u2013158.CrossRef\n\nGonsiorek, J. C., & Weinrich, J. D. (1991). The definition and scope of sexual orientation. In J. C. Gonsiorek & J. D. Weinrich (Eds.), Homosexuality, research implications for public policy (pp. 161\u2013176). London: Sage.CrossRef\n\nGorman, E. H., & Kmec, J. A. (2007). We (have to) try harder: Gender and required work effort in Britain and the United States. Gender and Society, 21, 828\u2013856.CrossRef\n\nGrandey, A. A., Cordiero, B. L., & Crouter, A. C. (2005). A longitudinal and multi-source test of the work-family conflict and job satisfaction relationship. Journal of Occupational and Organizational Psychology, 78, 305\u2013323.CrossRef\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Journal, 10, 76\u201388.\n\nGreenhaus, J. H., Bedeian, A. G., & Mossholder, K. W. (1987). Work experiences, job performance, and feelings of personal and family well-being. Journal of Vocational Behavior, 31, 200\u2013215.CrossRef90057-1)\n\nGreenhaus, J. H., Allen, T. D., & Spector, P. E. (2006). Health consequences of work-family conflict: The dark side of the work-family interface. In P. L. Perrewe & D. C. Ganster (Eds.), Research in occupational stress and well-being (Vol. 5, pp. 61\u201398). Amsterdam: JAI\/Elsevier.\n\nGriffin, P. (1992). From hiding to coming out: Empowering lesbian and gay educators. In K. M. Harbeck (Ed.), Coming out of the classroom (pp. 167\u2013196). Binghamton: Harrington Park.\n\nGriffith, K. H., & Hebl, M. (2002). The disclosure dilemma for gay men and lesbians: Coming \"out\" at work. Journal of Applied Psychology, 87, 1191\u20131199.PubMedCrossRef\n\nHill, R. J. (2009). Incorporating queers: Blowback, backlash, and other forms of resistance to workplace diversity initiatives that support sexual minorities. Advances in Developing Human Resources, 11, 37\u201353.CrossRef\n\nHinkin, T. R. (1995). A review of scale development practices in the study of organizations. Journal of Management, 21, 967\u2013988.CrossRef\n\nHuffman, A. H., Watrous-Rodriguez, K. M., & King, E. B. (2008). Supporting a diverse workforce: What type of support is most meaningful for lesbian and gay employees? Human Resource Management, 47, 237\u2013253.CrossRef\n\nKossek, E. E., & Ozeki, C. (1998). Work-family conflict, policies, and the job-life satisfaction relationship: A review and directions for organizational behavior-human resources research. Journal of Applied Psychology, 83, 139\u2013149.CrossRef\n\nLaw, C. L., Martinez, L., Ruggs, E. N., Hebl, M. R., & Akers, E. (2011). Trans-parency in the workplace: How the experiences of transsexual employees can be improved. Journal of Vocational Behavior, 79, 710\u2013723.CrossRef\n\nLorber, J. (1994). Paradoxes of gender. New Haven: Yale University.\n\nMacEwen, K. E., & Barling, J. (1994). Daily consequences of work interference with family and family interference with work. Work & Stress, 8, 244\u2013254.CrossRef\n\nMcDermott, E. (2006). Surviving dangerous places: Lesbian identity performances in the workplace, social class, and psychological health. Feminism Psychology, 16, 193\u2013211.CrossRef\n\nMiller, S. L., Forest, K. B., & Jurik, N. (2003). Diversity in blue: Lesbian and gay police officers in a masculine occupation. Men and Masculinities, 5, 355\u2013385.CrossRef\n\nMorin, S. F. (1977). Heterosexual bias in psychological research on lesbianism and male homosexuality. American Psychologist, 32, 629\u2013663.PubMedCrossRef\n\nO'Driscoll, M. P., Ilgen, D. R., & Hildreth, K. (1992). Time devoted to job and off-job activities, interrole conflict, and affective experiences. Journal of Applied Psychology, 77, 272\u2013297.CrossRef\n\nPreston, R. C., Fredericksen, R. J., & Herrschaft, D. (2006). The need for change: Bridging employers and business. In M. D. Shankle (Ed.), The handbook of lesbian, gay, bisexual, and transgender public health: A practitioner's guide to service (pp. 317\u2013332). Binghamton: The Haworth.\n\nProbst, T. (2003). Changing attitudes over time: Assessing the effectiveness of a workplace diversity course. Teaching of Psychology, 30, 236\u2013239.CrossRef\n\nRagins, B. R. (2004). Sexual orientation in the workplace: The unique work and career experiences of gay, lesbian and bisexual workers. Research in Personnel and Human Resources Management, 23, 35\u2013120.CrossRef23002-X)\n\nRagins, B. R. (2008). Disclosure disconnects: Antecedents and consequences of disclosing invisible stigmas across life domains. Academy of Management Review, 33, 194\u2013215.PubMedCrossRef\n\nRagins, B. R., & Cornwell, J. M. (2001). Pink triangles: Antecedents and consequences of perceived workplace discrimination against gay and lesbian employees. Journal of Applied Psychology, 86, 1244\u20131261.PubMedCrossRef\n\nRagins, B. R., Singh, R., & Cornwell, J. M. (2007). Making the invisible visible: Fear and disclosure of sexual orientation at work. Journal of Applied Psychology, 92, 1103\u20131118.PubMedCrossRef\n\nRau, B. L., & Hyland, M. (2006). Role conflict and flexible work arrangements: The effects on applicant attraction. Personnel Psychology, 55, 111\u2013136.CrossRef\n\nSchaubroeck, J., Cotton, J. L., & Jennings, K. R. (1989). Antecedents and consequences of role stress: A covariance structure analysis. Journal of Organizational Behavior, 10, 35\u201358.CrossRef\n\nSchope, R. D. (2002). The decision to tell: Factors influencing the disclosure of sexual orientation by gay men. Journal of Gay and Lesbian Social Services, 14, 1\u201322.CrossRef\n\nSedgwick, E. K. (1988). Epistemology of the closet. Berkeley: University of California.\n\nSellers, R. M., Smith, M. A., Shelton, J. A., Rowley, S. A. J., & Chavous, T. M. (1998). Multidimensional model of racial identity: A reconceptualization of African-American racial identity. Personality and Social Psychology Review, 2, 18\u201339.PubMedCrossRef\n\nShockley, K. M., & Allen, T. (2007). When flexibility helps: Another look at the availability of flexible work arrangements and work-family conflict. Journal of Vocational Behavior, 71, 479\u2013493.CrossRef\n\nSolomon, S. E., Rothblum, E. D., & Balsam, K. F. (2005). Money, housework, sex, and conflict: Same sex couples in civil unions, those not in civil unions, and heterosexual married siblings. Sex Roles, 52, 561\u2013575.CrossRef\n\nStacey, J. (1998). The neo-family-values campaign. In R. Lancaster & M. di Leonardo (Eds.), The gender\/sexuality reader (pp. 453\u2013470). New York: Routledge.\n\nVan Den Bergh, N. (2004). Getting a piece of the pie: Cultural competence for GLBT employees at the workplace. Journal of Human Behavior and the Social Environment, 8, 55\u201373.CrossRef\n\nWaldo, C. R. (1999). Working in majority context: A structural model of heterosexism as minority stress in the workplace. Journal of Counseling Psychology, 46, 218\u2013232.CrossRef\n\nWarner, L. R. (2008). A best practices guide to intersectional approaches in psychological research. Sex Roles, 59, 454\u2013463.CrossRef\n\nWeeks, J., Heaphy, B., & Donovan, C. (2001). Same sex intimacies: Families of choice and other life experiments. London: Routledge.CrossRef\n\nWilliams, K. J., & Alliger, G. M. (1994). Role stressors, mood spillover, and perceptions of work-family conflict in employed parents. Academy of Management Journal, 37, 837\u2013868.CrossRef\n\nWilliams, C., Giuffre, P. A., & Dellinger, K. (2009). The gay-friendly closet. Sexuality Research and Social Policy, 6, 29\u201345.CrossRef\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_6\n\n# 6. Gender, Gender-Role Ideology, and the Work\u2013Family Interface: A Cross-Cultural Analysis\n\nUjvala Rajadhyaksha1  , Karen Korabik2  and Zeynep Aycan3\n\n(1)\n\nDepartment of Business Administration and Economics, St. Mary's College, 46556 Notre Dame, IN, USA\n\n(2)\n\nCentre for Families, Work and Well-Being, University of Guelph, N1G2W1 Guelph, ON, Canada\n\n(3)\n\nDepartments of Psychology and Management, Ko\u00e7 University, Rumeli Fener Road, 34460 Sariyer, Istanbul, Turkey\n\nUjvala Rajadhyaksha (Corresponding author)\n\nEmail: urajadhy@saintmarys.edu\n\nKaren Korabik\n\nEmail: kkorabik@uoguelph.ca\n\nZeynep Aycan\n\nEmail: zaycan@ku.edu.tr\n\nKeywords\n\nGender-role ideologyGenderWork\u2013familyCross-culturalInternationalEgalitarianismEquality\n\n## 6.1 Overview\n\nThe current chapter examines the effects of gender and culture on work\u2013family conflict (WFC) and work\u2013family enrichment (WFE) . We first review the literature on WFC and WFE. We then look at the impact of different aspects of gender (including physical gender, gender-role ideology, gender egalitarianism , and gender inequality) 1 on the work\u2013family (W\u2013F) interface. Throughout the chapter we highlight the similarities and differences in relationships between the various aspects of gender and WFC and WFE that have been found in international studies and cross-cultural comparisons. The chapter concludes with a discussion on the agenda for future research.\n\n## 6.2 Work\u2013Family Conflict\n\nWFC has become the reality in the lives of many employed adults all over the world. Several factors are associated with the ever-increasing challenge of juggling work and family responsibilities. Some of these include longer work days and international business travel as a result of globalization (Westman et al. 2008); increased participation of women in the workforce and its concomitant impact on gender roles and ways of doing 'work'; changing family structures characterized by a rising number of dual career couples, single-parent families, families with aging parents requiring care; and the growth of a sandwich generation of workers who have to simultaneously care for dependent elders and children (Casper and Bianchi 2002).\n\nWFC arises due to \"simultaneous pressures from both work and family that are mutually incompatible in some respect\" (Greenhaus and Beutell 1985, p. 77). W\u2013F research has been dominated by Role Theory whereby conflicting expectations from multiple roles create psychological tension and conflict. Underlying the notion of WFC is the 'scarcity hypothesis' \u2014the belief that an individual's time and energy resources are in short supply, and therefore that any resource drain or depletion in one sphere will have negative ramifications for the other sphere (Rothbard 2001). WFC is generally operationalized as a bidirectional and multi-dimensional construct with the possibility of work-interfering-with-family conflict (WIF) and family-interfering-with-work conflict (FIW) occurring along the three dimensions of time, strain, and behavior (e.g., Frone et al. 1992a; Gutek et al. 1991). Further, boundaries between work and family are asymmetrically permeable\u2014the incidence of WIF tends to exceed that of FIW (Frone et al. 1992b; Pleck 1977).\n\nResearch on WFC includes many investigations of integrative models that examine how it is related to its antecedents and outcomes (e.g., Frone et al. 1992a). A number of meta-analyses and path analytic meta-analyses of the literature have been carried out (Amstad et al. 2011; Bryon 2005; Ford et al. 2007; Michel et al. 2011; Shockley and Singla 2011). The results have indicated that work domain variables have the potential to impact and create conflict in the work domain as well as cross over boundaries and create conflict in or affect variables in the family domain. Similarly, family domain variables can create conflict in or affect variables in the family domain as well as in the work domain. However, within-domain relationships tend to exceed cross-domain relationships in magnitude (Amstad et al. 2011; Shockley and Singla 2011).\n\nThe majority of studies on WFC have been conducted in the context of Anglo English speaking countries (Casper et al. 2007). This is a notable limitation, and pleas have been made for more WFC research to be performed in international contexts (Poelmans et al. 2005). Studies on WFC have been carried out in many areas of the world beyond the United States, including Western Europe and Scandinavia (e.g., Kinnunen and Mauno 1998), Eastern Asia (e.g., Aryee 1992; Aryee et al. 1999; Matsui et al. 1995), and Southern Asia\/Africa (e.g., Ayo et al. 2009; Tabassum 2012). However, most of the research conducted outside North America consists of single-country studies that have not specifically accounted for cultural variables or facilitated cross-cultural comparisons (Aycan 2008; Gelfand and Knight 2005; Shaffer et al. 2011). Recently, there have been more cross-cultural investigations of the W\u2013F interface (e.g., Billing et al. 2014; Joplin et al. 2003; Lapierre et al. 2008; Spector et al. 2004, 2007; Yang et al. 2000), but these have not explicitly focused on the role of gender.\n\n## 6.3 Work\u2013Family Enrichment\n\nThe notion of WFC implies a negative spillover between the work and family domains. This negative thrust has been critiqued with researchers asking for more examination of positive relationships between the work and family domains (Greenhaus and Powell 2006). Positive W\u2013F relationships have been variously referred to as positive spillover , enrichment, enhancement, or facilitation (Greenhaus and Powell 2006; Grzywacz and Marks 2000; Wayne et al. 2006). Essential to all these terms is the idea that the role exchange between work and family is not always detrimental and can have enhancing effects. Following Shockley and Singla (2011), we use the term work\u2013family enrichment (WFE) to refer to positive W\u2013F relationships. Unlike WFC, which is based on the scarcity hypothesis , WFE is based on Role Accumulation Theory , which suggests that participation in multiple roles could produce positive outcomes through opportunities and resources that individuals can use to promote growth and better functioning in different life domains. Similar to WFC, WFE has been found to be bi-directional in nature with work\u2013family enrichment (WFE) occurring along with family-work enrichment (FWE) . Relative to WFC, research on WFE is sparse (Frone 2003). While some WFE studies have been conducted outside of the Anglo context (e.g., Baral and Bhargava 2011; Bosmat 2013; Choi and Kim 2012; Ho et al. 2013; Karatepe and Azar 2013; Nasurdin et al. 2013), there are few cross-cultural studies that specifically examine gender and WFE.\n\nMeta-analyses have indicated that both WFE and FWE are related to a number of positive outcomes including affective commitment, physical and mental health , and job and family satisfaction . For example, Haar and Bardoel (2008) looked at positive spillover in 420 Australian employees and found WFE was negatively associated with psychological distress and turnover intentions, while FWE was negatively linked with psychological distress and positively linked with family satisfaction. However, WFE is more strongly related to work-related variables, whereas FWE was more strongly related to non-work related variables (McNall et al. 2010; Shockley and Singla 2011). This is somewhat resonant of the stronger impact of same domain relationships observed in WFC literature (Michel et al. 2011).\n\n## 6.4 Gender and Work\u2013Family Conflict\n\nThe examination of gender is integral to W\u2013F research\u2014both its positive and negative elements\u2014given that gender roles influence how men and women distribute resources across work and family domains. Interestingly, although the boost to work and family research has come from the increasing presence of women in the workforce, few studies have specifically focused on W\u2013F and gender (Parasuraman and Greenhaus 2002). Even when gender has been included as a variable, its treatment has been atheoretical and gender as operationalized by whether someone is a man or a woman has often been used as a proxy for other aspects of gender (Korabik et al. 2008). One drawback of treating gender as a biologically determined variable is that it does not consider the possibility that there may be greater within-gender variation than across-gender variation in roles. Gender is actually a multidimensional phenomenon. Korabik et al. (2008) proposed that rather than considering only physical gender, research should consider intrapsychic aspects of gender that influence not only individuals' identities, but also their behaviors, the roles they choose to enact and how they choose to enact them. One such variable that is pertinent to W\u2013F research is gender-role ideology (GRI) (Korabik et al. 2008). In addition, sociological or cultural conceptions of gender, like gender egalitarianism and gender inequality , can be studied in an effort to better understand the W\u2013F interface (Lyness and Kropf 2005; Powell et al. 2009).\n\n### 6.4.1 Gender and WFC: Theoretical Approaches\n\nTwo perspectives are frequently used in the literature to explain the relationship between gender and WFC. One is the rational viewpoint (Gutek et al. 1991), which posits that the more the hours one spends in a domain, the more potential there is for conflict to occur. This theory predicts that men should experience more WIF than women because they spend more time at work, whereas women should experience more FIW than men because they spend more time at home. A competing theory is Pleck's (1977) gender-role theory. It postulates that family demands will more likely affect work roles for women, whereas work demands will more likely spill over into the family for men. This gender-asymmetrical impact of work and family roles is due to the differential importance given to these roles by men and women. This affects their perceptions of WIF and FIW so that additional hours spent in one's prescribed gender-role domain (family for women and work for men) are not seen as an imposition as much as additional hours spent in the domain associated with the other gender. According to the gender-role perspective, women should report more WIF than men even when they spend the same number of hours in paid work, and men should report more FIW than women even when they devote the same number of hours to family activities.\n\nIn their review of WFC studies, Eby et al. (2005) concluded that there is mixed evidence as to whether men and women report different levels of WFC. Their meta-analysis mentions that some research has found no gender differences in WFC (Duxbury and Higgins 1991; Eagle et al. 1997), some studies have found men to report higher WFC than women (Parasurman and Simmers 2001), and other studies have found women to experience higher WFC than men (Frone et al. 1992b) or higher levels on some dimensions of WFC (e.g., Gutek et al. 1991; Loerch et al. 1989).\n\nRegarding Pleck's (1977) hypothesis of gender differences in the permeability of W\u2013F boundaries , support has also been mixed. Gutek et al. (1991) reported more WIF for women than men, but no gender difference in FIW . Other studies have found no significant gender differences at all, though family boundaries were consistently more permeable than work boundaries (Eagle et al. 1997; Frone et al. 1992a, b). Overall it appears that gender has a near zero relationship to WIF and is only weakly related to FIW , suggesting that men and women experience similar levels of interference in both domains (Bryon 2005).\n\n### 6.4.2 Gender Differences in Antecedents and Consequences of WFC\n\nAs with research examining mean differences, studies of gender differences in the antecedents of WFC have produced conflicting results. In an investigation of family-related antecedents of WFC, Loerch et al. (1989) found that family role conflict was a significant predictor of strain- and behavior-based WFC for both genders, and a unique predictor of time-based conflict for men, but not women. Furthermore, family intrusions and total role involvement predicted time-based conflict for women, whereas family intrusions were a unique predictor of strain-based conflict for men.\n\nFocusing on work domain antecedents, Wallace's (1999) study of lawyers found that work involvement variables were not predictive of either time- or strain-based WFC among women, though higher work motivation and working more hours were associated with men's strain- and time-based conflict, respectively. Furthermore, work overload and being in a profit-driven environment had similar effects on time-based WFC for men and women lawyers, while the impact of these antecedents on strain-based WFC was stronger for women as compared to men. Wallace also found that work context was important\u2014working in a law firm setting was associated with greater time- and strain-based conflict for women though not for men. Finally, as the percentage of women lawyers in the workplace increased, men reported greater time-based conflict, but women reported more strain-based conflict. Duxbury and Higgins (1991) examined gender differences in work and family domain antecedents of WFC. They found that work involvement was a stronger predictor of WFC for women, whereas family involvement was a stronger predictor for men. Work expectations , on the other hand, were a more significant predictor of WFC for men, while family expectations were a stronger predictor of family conflict for women. Gutek et al. (1991) examined the effect of hours spent in the home and work domain and found that for women, the number of hours spent on the job was related to WIF . For FIW , no gender effect of hours spent in family work was found. This was contrary to Huffman et al.'s (2003) study of military personnel where time demands measured by the number of hours worked were more strongly related to WFC for men than for women. This gender difference disappeared, however, when time demands were measured via perceptions of workload . McElwain et al. (2005) used a time demands measure that was a composite of work hours and perceived job demands and found no significant gender differences for the relationships between work time demands and WIF. However, there was a significant gender difference in the relationship between family demands and FIW. Specifically, women tended to experience high levels of FIW when they had high family demands as compared to men. Men's levels of FIW, however, were not dependent on the amount of family demands they had. It would seem that control over work hours plays a significant role in determining women's versus men's WFC rather than work hours per se. In support of this notion, Gr\u00f6nlund (2007) found that in Sweden women in jobs with high demands and high control did not experience more WIF than men, even when working the same hours.\n\nSocial support is another important antecedent associated with reduced WFC. Aycan and Eskin (2005) found gender differences in the type of support used by men and women in Turkey. For women, spousal support was associated with reduced WIF and FIW , whereas for men spousal support was associated with reduced FIW and organizational support was associated with reduced WIF. Their finding supported earlier literature suggesting that men needed more social support than women in coping with WFC, because although changing gender roles prescribe men to be more involved in their families, this is still not tolerated by most organizations (cf., Cinnamon and Rich 2002). Furthermore, men do not have as many role models as women do to show them how to cope with WFC, and they do not exhibit help-seeking behavior as much as women do (Addis and Mahalik 2003).\n\nTurning to the outcomes of WFC, although Frone et al. (1996) and Frone et al. (1993) established that higher WFC resulted in poor physical and psychological health, as well as heavy alcohol use, their findings did not support the hypothesized gender differences in the relationship between WFC and outcomes. By contrast, Duxbury and Higgins (1991) found that women were more likely than men to report low quality of work life when they had high WFC, whereas men were more likely than women to report low quality of family life when they had high WFC. In addition, Burley (1995) found that the relationships of WFC to spousal support and marital adjustment were significantly more negative for men than for women.\n\nThe results with regard to the moderating effect of gender on the relationships of WIF and FIW to satisfaction are extremely inconsistent. Bagger et al. (2008) found that family identity salience acted as a buffer such that higher FIW was related to higher job distress and lower job satisfaction more so for women than men. By contrast, McElwain et al. (2005) found the negative relationship between FIW and job satisfaction to be stronger for men than for women, but that there was no significant gender difference in the relationship between WIF and family satisfaction . Ford et al.'s (2007) path analytic meta-analysis indicated that the negative correlation between job stress and family satisfaction was less strong for women than for men. However, Shockley and Singla's (2011) meta-analysis showed that the negative correlation between WIF and family satisfaction was stronger for men than women, whereas the negative correlation between FIW and family satisfaction was stronger for women than men.\n\n## 6.5 Cross-Cultural Studies of Gender Interacting with WFC and WFE\n\nSeveral studies of gender and WFC have been conducted in individual countries outside of the North American context, including Malaysia (e.g., Hassan et al. 2010), India (e.g., Ramadoss and Rajadhyaksha 2012), Iran (e.g., Namayandeh et al. 2011), Hong Kong (e.g., Fu and Shaffer 2001), Finland (e.g., V\u00e4\u00e4n\u00e4nen et al. 2004), and Turkey (e.g., Yavas et al. 2008). Some studies have demonstrated within-gender differences in WFC based on ethnicity (e.g., Mahpul and Abdullah 2011), whereas other single-country studies have supported both a 'gender similarity' as well as a 'gender difference' model of WFC depending on the database used for drawing the sample (e.g., Keene and Quadagno 2004). We will not review the results of this research here since a substantial cross-cultural literature exists.\n\nThe results of cross-cultural comparisons on gender and WFC have been characterized by the same inconsistency found in the North American literature on gender and WFC. Some studies have revealed no significant gender effects. For instance, in a cross-cultural test of the W\u2013F interface in 48 countries, Hill et al. (2004) found little support for a culture-specific or gender-specific model of WFC. A few of their observed gender differences included marital status being more likely to reduce FIW for women than for men, parental status being associated with significantly more FIW for women than for men, and perceptions of a lack of fit being more strongly related to WIF for women than for men. However, there were no gender differences across cultures. Shaffer et al. (2011) examined 219 studies that used non-USA samples and also found no differences due to gender. Their results indicated that work demands predicted both WIF and FIW in their Anglo, Asian, Nordic, and Western European country clusters. Family stressors were positively related to WFC for countries in the Anglo cluster, to both WIF and FIW for countries in the Asian cluster, and to FIW for countries in the Western European cluster. WIF and FIW were negatively related to work-related affective reactions and positively related to withdrawal outcomes in all country clusters. WIF and FIW were often, but not always, negatively related to family attitudes (e.g., satisfaction) in the Anglo, Asian, and Nordic, East, and West European country clusters.\n\nBy contrast, other studies have found gender differences in WFC, but these have not always been consistent. Steibler (2009) examined dual-earner couples in 23 European countries and found that after work hours were controlled, women reported both more time- and strain-based WIF than men. In a study of eight European countries, Simon et al. (2004) found that WIF was higher than FIW in all countries, supporting the notion of asymmetrical boundaries between work and family (Pleck 1977). However, men experienced greater WIF than women in Italy, whereas the reverse was the case in the Netherlands. Mortazavia et al. (2009) conducted a cross-cultural study using samples from Iran, Ukraine, and the USA to examine interactions between gender, nationality, and the cultural values of horizontal individualism\/collectivism measured at the national and individual levels (idiocentrism and allocentrism) on W\u2013F demands and WFC. They found that men reported more WIF than women in all three countries, but there was no significant gender difference in FIW . In a study comparing the USA and China, Yang et al. (2000) reported that men experienced higher levels of WFC than women did in China, whereas a gender difference did not exist in the USA sample. Using data from Project 3535 2, Korabik et al. (2012) found that men reported higher WIF than women in all countries except India, Taiwan, and Turkey where WIF was higher for women than men. Conversely, women reported higher FIW than men in all countries except Australia, China, and the USA, where FIW was higher for men than women. In addition, greater satisfaction with organizational policies was associated with lower WIF for both genders in Australia, Canada, Indonesia, and the USA. This relationship also held for men in Israel and women in Spain and Turkey. Greater satisfaction with organizational policies was also significantly associated with lower turnover intent for both genders in every country except Turkey. Greater satisfaction with organizational policies was significantly associated with higher life satisfaction for both genders in Canada, Indonesia, Israel, Turkey, and the USA. This relationship also held for men in China and India and women in Australia and Taiwan. Greater satisfaction with government policies was significantly associated with higher life satisfaction for both genders in Canada, China, India, and Indonesia. This relationship also held for men in Australia and the USA and for women in Israel. Satisfaction with government policies was less consistently related to intent to turnover, with the strongest correlations evident for men in the USA, women in Spain, and both men and women in Israel (Korabik et al. 2012).\n\nGender has been largely overlooked in the literature on WFE (Greenhaus and Powell 2006). Studies do show, however, that women tend to perceive higher levels of FWE than men (Grzywacz and Marks 2000; Rothbard 2001). Although in a three-year cross-lagged panel study carried out in Finland (Hakanen et al. 2011) gender was found to be related to WFE the same way for men and women, Shockley and Singla's (2011) meta-analysis indicated that WFE had a greater effect on satisfaction outcomes for women than for men. Moreover, using a sample from Spain, Boz et al. (2009) found that FWE was related to life satisfaction and mental and physical well-being in a different manner for women and men. Their results also showed that women were able to experience FIW and FWE concurrently, while men were more likely to separate their family from their work roles. In addition, when women reported experiencing both FIW and FWE, the positive effect of FWE outweighed the negative consequences of FIW on well-being .\n\nData from Project 3535 (Korabik et al. 2012) have indicated that women reported higher WFE than men in all countries except China, Spain, and Indonesia where WFE was higher for men than women. Moreover, greater satisfaction with organizational policies was associated with higher WFE for both genders in Canada and Turkey. This relationship also held for men in China and women in Australia, Israel, Taiwan, and the USA (Korabik et al. 2012).\n\n## 6.6 Gender-Role Ideology\n\nTypically, GRI is conceptualized as falling on a unidimensional continuum ranging from traditional to nontraditional or egalitarian (Gibbons et al. 1997). Individuals with a traditional GRI believe that women should give priority to family responsibilities and men to work responsibilities (Gutek et al. 1991). By contrast, nontraditional or egalitarian individuals believe in a more equal role distribution for men and women. The conceptualization of GRI does not make the assumption of bio-psychological equivalence , i.e., both men and women can have either traditional or egalitarian attitudes (Korabik et al. 2008).\n\nResearch indicates that individuals' GRI can change both between and within generations (Moen et al. 1997; Wentworth and Chell 2005), thus supporting the notion that GRI is shaped by environmental influences. The GRI of individuals also can vary between countries. In a comparative study of gender-role attitudes among the North American countries of Mexico, Canada, and the USA, Harris et al. (2006) found that in general Mexicans were slightly more likely to exhibit traditional attitudes and gender-role behavior that was congruent with their biological gender. Additionally, being a man was slightly more likely to predict traditional GRI than being a woman in Canada and the USA, as compared to Mexico. Goldberg et al. (2012) found that GRI varied between ethnic groups within a country. They studied a large culturally and ethnically diverse sample of students at a US university and found that Asian Americans, especially men and those less acculturated , were more likely to hold more conservative GRI and support gender-role segregation and maternal nonemployment when children are young. Their mothers' employment and own employment status, however, was associated with more positive views about maternal employment overall. Additionally GRI can be shaped by national cultural and economic variables. For instance, Parboteeah et al. (2008) found that managers' traditional gender-role attitudes were positively related to a nation's cultural values of power distance and uncertainty avoidance while being negatively related a nation's cultural values of gender egalitarianism , as well as to degree of regulation of the economy and degree of educational attainment.\n\nWomen's exposure to the labor force and to education can foster more egalitarian GRI on account of role expansion (Smith-Lovin and Tickamyer 1978). Individuals in industrialized countries have been found to hold more egalitarian GRIs with more favorable views of married women's labor force participation (Treas and Widmer 2000). In fact, with more women entering the workforce, individuals' attitudes in many societies are experiencing a change from traditional to more egalitarian (Friedman and Weissbrod 2005). However in some transitional societies men's GRI could become more traditional as a backlash. For instance, in a study conducted in urban China, Pimentel (2006) found that husbands had become more traditional in their GRI over cohorts while wives' GRI had stayed constant and was more egalitarian than that of husbands. Pimentel explained this difference in terms of increasing expectations of equality among women and declining job opportunities for men in reformist China, resulting in declining levels of marital satisfaction over time. In a comparison of GRI of women in Taiwan and coastal China, Tu and Liao (2005) found that urbanization had a more significant impact on women's attitudes than on men's attitudes and this effect was more significant in coastal China which has seen more marked changes in development in recent decades, as compared to Taiwan. Also, cohort-effect differences in gender-role attitudes were found to be more significant in coastal China and more so for men than for women.\n\nThere can be gender differences in GRI particularly for working individuals. Husbands' and wives' GRIs within working couples can influence each other (Kulik 2004). In a study of dual-earner couples in Sri Lanka, Kailasapathy and Metz (2008) found that spouse's GRI affected the ability to negotiate within the home to reduce WFC. Women's egalitarian GRI in dual-earner marriages could cause them to experience a sense of unfairness when they feel that they do more than their spouse (Frisco and Williams 2003). On the other hand, the more egalitarian the husband's GRI relative to the wife's, the weaker the relationship between job-role quality and distress tends to be (James et al. 1998). The more egalitarian the GRIs of husband-wife pairs, the less likely they are to adopt gender-stereotypic decisions regarding relocation for jobs, i.e., with the wife hesitating from capitalizing on a job move because of the loss that it would mean to the husband (Bielby and Bielby 1992). Along with gender, GRI can moderate the link between spousal support and marital quality. For instance, emotional spousal support has been found to better predict marital satisfaction and less marital conflict for traditional women and egalitarian men, while emotional and instrumental spousal support has been found to better predict marital satisfaction for egalitarian women and traditional men (Mickelson et al. 2006).\n\n### 6.6.1 GRI and the W\u2013F Interface\n\nStudies have examined relationships between GRI and broad work and family variables. For example, Davis and Pearce (2007) found that adolescents' work\u2013family gender ideologies were linked to their educational attainments. For girls and boys, having more egalitarian views of gendered work and family roles made them more likely to desire a college education and a graduate or professional degree, although the relationship was stronger for girls than for boys. Schwarzwald et al. (2008) in a study of Izraeli married couples found that GRI moderated the relationship between a couple's usage of power tactics and marital satisfaction .\n\nStudies linking GRI and the W\u2013F interface specifically defined as WFC and WFE are relatively few. GRI has been found to be a better predictor of WFC than physical gender (Velgach et al. 2006) which might help explain many of the inconsistent findings in the gender and WFC literature. To the extent that attitudes affect behavior, GRI is likely to impact WFC by influencing the degree to which individuals resort to traditional versus nontraditional division of labor . Thus, it would be expected that working women with traditional GRI would experience more WIF than would working women with egalitarian GRI because their traditional attitude would influence them to exert most of their efforts into their home roles. Conversely, working men with traditional GRI should experience more FIW than working men with egalitarian attitudes as their attitudes would dictate that they should spend their time providing for their family through paid work (Gutek et al. 1991; Korabik et al. 2008).\n\nEmpirical examinations of the relationship between GRI and WFC have produced mixed results. No significant differences between those with traditional and egalitarian GRI were found on either WIF or FIW in a study by Chappell et al. (2005) with a sample from Canada or by Drach-Zahavy and Somech (2004) using Project 3535 data from Israel. By contrast, Ayman et al. (2005) found significant negative relationships between GRI and strain-based WIF, time-based WIF, and time-based FIW, indicating that egalitarian individuals experienced lower WFC than traditional individuals on these dimensions. In a sample of working men and women from India gathered as part of Project 3535, Rajadhyaksha and Velgach (2009) found that those with traditional GRI experienced more of both WIF and FIW than those with egalitarian GRI. Moreover, there was an interaction effect between gender and GRI. Women 'traditionals' experienced more FIW than men 'traditionals'. Using Project 3535 data from Israel, Somech and Drach-Zahavy (2007) found an interaction effect between gender and GRI with regard to the relationship between coping strategies and WIF and FIW.\n\n### 6.6.2 Cross-Cultural Studies of GRI and the W\u2013F Interface\n\nThere have been few cross-cultural examinations of the relationship between GRI and WFC, and almost no studies have examined the relationship between GRI and WFE. Steibler's (2009) study of dual-earner couples in 23 European countries showed that men with more egalitarian gender-role attitudes had lower levels of WIF. Using preliminary data from five of the countries (i.e., Canada, India, Taiwan, Spain, and the USA) involved in Project 3535 , Poelmans et al. (2006) conducted a cross-cultural analysis of the relationship between GRI and WFC. After controlling for job sector and job level, it was found that in all countries, those with traditional GRI reported greater WIF and greater FIW than those with egalitarian GRI. Somech and Drach-Zahavy (2011) used the data from all ten Project 3535 countries and found that individual coping strategies were related to WIF and FIW as a function of GRI and individualism\/collectivism .\n\n## 6.7 Cross-Cultural Studies of Cultural Gender Egalitarianism, Gender Inequality, and the W\u2013F Interface\n\nThere have been relatively few studies that have investigated the impact of cultural gender egalitarianism or societal\/national gender inequality on the W\u2013F interface. In a study of managers and professionals from 20 European countries, Lyness and Kropf (2005) found that both national gender equality (measured using the United Nations Gender Development Index) and the proportion of women in senior management positions predicted perceptions of supportiveness of organizational culture which in turn was related to employees' perceptions of greater work\u2013life balance. They suggested that future studies of the W\u2013F interface recognize the importance of the larger context and a nation's standing with regard to gender equality . \u00d6un (2012) looked at data from four Nordic countries and found that overall, women reported higher WFC than men. However, greater gender equity at the household level was related to lower WIF . Further, WFC was higher in Finland compared to the other three Nordic countries of Denmark, Sweden, and Norway. This difference was explained in terms of the relatively greater gender inequity within Finland with lesser integration of women in to the workforce. Beham et al. (2012) conducted a study of how utilization of W\u2013F resources affected work interference with family (WIF) and satisfaction with W\u2013F balance (SWFB) among professional and non-professional service employees across five Western European countries. They found that the utilization of organizational W\u2013F resources varied across welfare state regimes and levels of national gender equality, being highest in Sweden and the Netherlands and lowest in Portugal. Across the five countries, professional employees experienced higher levels of WIF and lower levels of SWFB than non-professional employees and the gap between professionals and non-professionals varied across countries for WIF, but not for SWFB. It was significantly larger in the UK and in Sweden than in the other three countries. Finally, a family-supportive supervisor and a family-supportive organizational culture differentially affected the WIF of professional and non-professional workers, with a family-supportive supervisor being more beneficial to non-professionals and a family-supportive organizational culture being more beneficial to professional employees.\n\nMany national policies that affect working conditions, especially of female workers, play a role in determining women's socioeconomic status and therefore gender inequality (e.g., Guan et al. 2011; Ray et al. 2010). In a recent study conducted across 12 industrialized nations that examined the relationship between WFC and national paid leave policies, Allen et al. (2014) found that of the three forms of paid leave examined\u2014sick, annual, and parental\u2014paid sick leave had a small but significant negative relationship with WFC, while paid parental leave and paid annual leave had very little impact on WFC. Also, the impact of national leave policies on WFC was moderated by employees' perceptions of the family-supportiveness of their organization as well as their supervisor, such that the impact was more beneficial when employees perceived more rather than less support . Clearly, it appears that national gender inequality has the potential to impact the W\u2013F interface directly as well as indirectly through the utilization and\/or availability of organizational resources and national policies that affect conditions of work.\n\nCultural gender egalitarianism has also been found to affect the W\u2013F interface. Steibler (2009) examined 23 European countries and found that those in countries with higher levels of gender egalitarianism as a cultural value had higher levels of strain-based WIF. In a comparison of work\u2013life balance (WLB) perceptions of employees in Indian and American multinational corporations, Chandra (2012) found that within the Indian context, gender socialization led WLB to be perceived as more of a women's issue that needed to be solved at the individual level, and there was greater emphasis on employee welfare schemes to balance work and family. By contrast, within American companies the emphasis of WLB programs was on flexible work arrangements.\n\nA recent study conducted by Lyness and Judiesch (2014) examined the interplay between gender and gender egalitarianism on the W\u2013F interface in 36 countries. Their study used multi-source data as well as multilevel analyses and found that supervisors' perceptions of employees' WLB differed by ratee gender and country context. Supervisors rated women lower than men in WLB in low egalitarian countries, but similar to men in high egalitarian countries, and only appraisals of women (but not men) varied depending on egalitarian context. Societal gender inegalitarianism explained the majority of variation in supervisors' appraisals of women's WLB, whereas women's self-reported balance was linked to objective gender inequalities.\n\n## 6.8 Conclusion\n\nIn a call for more culture sensitive theories of the W\u2013F interface, Powell et al. (2009) suggest gender egalitarianism, among other cultural variables, for inclusion in future research studies. They also refer to several ways in which culture could be examined in W\u2013F research\u2014namely, culture-as-nations, culture-as-referents, and culture-as-dimensions. Poelmans (2003) recommended that we consider the micro-layer, the meso-layer, the macro-layer, and the micromacro-layer while studying the antecedents and consequences of WFC. Ollier-Malaterre et al. (2013) \u00adrecommended that future W\u2013F research should consider positive and negative aspects of the W\u2013F interface and examine interactions between institutional factors such as national policy contexts in addition to cultural dimensions on individual and organizational levels of analyses.\n\nOur review of the literature indicates that gender-based examinations of the W\u2013F interface built on the notion of 'biological' gender provide a somewhat incomplete understanding of the phenomena of WFC and WFE by yielding mixed results. Future W\u2013F studies should incorporate wider conceptions of gender that account for differences in the ways that gender is enacted in diverse contexts by men and women. GRI could be one such variable that operates between biological and sociological conceptions of gender. Our review of the literature also indicates that there are several interactions and complex relationships between gender and GRI (micro variables) and gender egalitarianism and gender equity (macro variables covering culture and nation). Exploring the relationships between these variables and WFC\/WFE may turn out to be the most likely way to plug the many existing lacunae in W\u2013F research.\n\nReferences\n\nAddis, M. E., & Mahalik, J. R. (2003). Men, masculinity and the contexts of help seeking. American Psychologist, 58, 5\u201314.PubMed\n\nAllen, T. D., Lapierre, L. M., Spector, P. E., Poelmans, S. A. Y., O'Driscoll, M., Woo, J., et al. (2014). The link between national paid leave policy and work-family conflict among married working parents. Applied Psychology: An International Review, 63, 5\u201328.\n\nAmstad, F. T., Meier, L., Fasel, U., Elfering, A., & Semmer, N. (2011). A meta-analysis of work-family conflict and various outcomes with a special emphasis on cross-domain versus matching-domain relations. Journal of Occupational Health Psychology, 16, 151\u2013169.PubMed\n\nAryee, S. (1992). Antecedents and outcomes of work-family conflict among married professional women: Evidence from Singapore. Human Relations, 45, 813\u2013837.\n\nAryee, S., Luk, V., Leung, A., & Lo, S. (1999). Role stressors, interrole conflict, and well-being: The moderating influence of spousal support and coping behaviors among employed parents in Hong Kong. Journal of Vocational Behavior, 54, 259\u2013278.\n\nAycan, Z. (2008). Cross-cultural approaches to work-family conflict. In K. Korabik, D. S. Lero, & D. L. Whitehead (Eds.), Handbook of work-family integration: Research, theories, and best practices (pp. 353\u2013370). San Diego: Elsevier.\n\nAycan, Z., & Eskin, M. (2005). Childcare, spousal, and organizational support in predicting work-family conflict for females and males in dual-earner families with preschool children. Sex Roles, 53, 453\u2013471.\n\nAyman, R., Velgach, S., & Ishaya, N. (April 2005). Multi-method approach to investigate work-family conflict. Poster presented at the meeting of the Society for Industrial and Organizational Psychology, Los Angeles, California, USA.\n\nAyo, H. T., Henry, A. S., & Taiwo, A. (2009). Psychosocial variables as predictors of work-family conflict among secondary school teachers in Irele local government area, Ondo State, Nigeria. Pakistan Journal of Social Sciences, 6, 11\u201318.\n\nBagger, J., Li, A., & Gutek, B. A. (2008). How much do you value your family and does it matter? The joint effects of family identity saliency, family-interference-with-work and gender. Human Relations, 61, 187\u2013211.\n\nBaral, R., & Bhargava, S. (2011). Examining the moderating influence of gender on the relationships between work-family antecedents and work-family enrichment. Gender in Management: An International Journal, 26, 122\u2013147.\n\nBeham, B., Drobnic, S., & Pr\u00e4g, P. (2012). The work-family interface of service sector workers: A comparison of work resources and professional status across five European countries. Applied Psychology: An International Review, 63, 29\u201361.\n\nBielby, W. T., & Bielby, D. (1992). I will follow him: Family ties, gender-role beliefs, and reluctance to relocate for a better job. American Journal of Sociology, 97, 1241\u20131268.\n\nBilling, T. K., Bhagat, R. S., Babakus, E., Krishnan, B., Ford Jr, D. L., Srivastava, B. N., Rajadhyaksha, U., Shin, M., Kuo, B., Kwantes, C., Setiadi, B., & Nasurdin, A. M. (2014). Work\u2013family conflict and organisationally valued outcomes: The moderating role of decision latitude in five national context. Applied Psychology: An International Review, 63 (1), 62\u201395.\n\nBosmat, S. (2013). Combining multiple roles among high position women in Israel, as seen by the woman, her husband and a child. Journal of International Women's Studies, 14, 69\u201393.\n\nBoz, M., Martinez, I., & Munduate, L. (2009). Breaking negative consequences of relationship conflicts at work: The moderating role of work family enrichment and supervisor support. Revista de Psicolog\u00eda del Trabajo y de las Organizaciones, 25, 113\u2013121.\n\nBryon, K. (2005). A meta-analytic review of work-family conflict and its antecedents. Journal of Vocational Behavior, 62, 169\u2013198.\n\nBurley, K. A. (1995). Family variables as mediators of the relationship between work-family conflict and marital adjustment among dual-career men and women. Journal of Social Psychology, 135, 483\u2013497.PubMed\n\nCasper, L. M., & Bianchi, S. M. (2002). Continuity and change in the American family: Anchoring the future. Thousand Oaks: Sage.\n\nCasper, W., Bordeaux, C., Eby, L., Lockwood, A., & Lambert, D. (2007). A review of research methods in IO\/OB work-family research. Journal of Applied Psychology, 92, 28\u201343.PubMed\n\nChandra, V. (2012). Work-life balance: Eastern and western perspectives. The International Journal of Human Resource Management, 23, 1040\u20131056.\n\nChappell, D., Korabik, K., & McElwain, A. (June 2005). The effects of gender-role attitudes on work-family conflict and work-family guilt. Poster presented at the meeting of the Canadian Psychological Association, Montreal, Quebec, Canada.\n\nChoi, H., & Kim, Y. T. (2012). Work-family conflict, work-family facilitation, and job outcomes in the Korean hotel industry. International Journal of Contemporary Hospitality Management, 24, 1011\u20131028.\n\nCinnamon, R., & Rich, Y. (2002). Gender differences in the importance of work and family roles: Implications for work-family conflict. Sex Roles, 47, 531\u2013541.\n\nDavis, S. N., & Pearce, L. P. (2007). Adolescents' work-family gender ideologies and educational expectations. Sociological Perspectives, 50, 249\u2013272.\n\nDrach-Zahavy, A., & Somech, A. (July 2004). Emic and etic characteristics of coping strategies with WFC: The case of Israel. Paper presented at the International Congress of Cross-Cultural Psychology, Xi'an, China.\n\nDuxbury, L. E., & Higgins, C. A. (1991). Gender differences in work-family conflict. Journal of Applied Psychology, 76, 60\u201373.\n\nEagle, B. W., Miles, E. W., & Icenogle, M. L. (1997). Interrole conflicts and the permeability of work and family domains: Are there gender differences? Journal of Vocational Behavior, 50, 168\u2013184.\n\nEby, L. T., Casper, W. J., Lockwood, A., Bordeaux, C., & Brinleya, A. (2005). Work and family research in IO\/OB: Content analysis and review of the literature (1980\u20132002). Journal of Vocational Behavior 66, 124\u2013197.\n\nFord, M. T., Heinen, B. A., & Langkamer, K. L. (2007). Work and family satisfaction and conflict: A meta-analysis of cross domain relations. Journal of Applied Psychology, 92, 57\u201380.PubMed\n\nFriedman, S. R., & Weissbrod, C. S. (2005). Work and family commitment and decision-making status among emerging adults. Sex Roles, 53, 317\u2013325.\n\nFrisco, M. L., & Williams, K. (2003). Perceived housework equity, marital happiness, and divorce in dual-earner households. Journal of Family Issues, 24, 51\u201373.\n\nFrone, M. R. (2003). Work-family balance. In J. C. Quick & L. E. Tetrick (Eds.), Handbook of occupational health psychology (pp. 143\u2013162). Washington: American Psychological Association.\n\nFrone, M. R., Russell, M., & Cooper, M. K. (1992a). Antecedents and outcomes of work-family conflict: Testing a model of the work-family interface. Journal of Applied Psychology, 77, 65\u201378.PubMed\n\nFrone, M. R., Russell, M., & Cooper, M. L. (1992b). Prevalence of work-family conflict: Are work and family boundaries asymmetrically permeable? Journal of Organizational Behavior, 13, 723\u2013729.\n\nFrone, M. R., Russell, M., & Cooper, M. L. (1993). Relationship of work-family conflict, gender, and alcohol expectancies to alcohol use\/abuse. Journal of Organizational Behavior, 14, 545\u2013558.\n\nFrone, M. R., Russell, M., & Barnes, G. M. (1996). Work-family conflict, gender, and health-related outcomes: A study of employed parents in two community samples. Journal of Occupational Health Psychology, 1, 57\u201369.PubMed\n\nFu, K. C., & Shaffer, M. A. (2001). The tug of work and family: Direct and indirect domain-specific determinants of work-family conflict. Personnel Review, 30, 502\u2013522.\n\nGelfand, M. J., & Knight, A. P. (2005). Cross-cultural perspectives on work-family conflict. In S. A. Y. Poelmans (Ed.), Work and family: An international research perspective (pp. 401\u2013414). Mahwah: Erlbaum.\n\nGibbons, J., Hamby, B., & Dennis, W. (1997). Researching gender-role ideologies internationally and cross-culturally. Psychology of Women Quarterly, 21, 151\u2013170.\n\nGoldberg, W. A., Kelly, E., Matthews, N., Kang, H., Li, W., & Sumaroka, M. (2012). The more things change, the more they stay the same: Gender, culture, and college students' views about work and family. Journal of Social Issues, 68, 814\u2013838.\n\nGreenhaus, J., & Beutell, N. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGreenhaus, J. H., & Powell, G. N. (2006). When work and family are allies: A theory of work-family enrichment. Academy of Management Review, 31, 72\u201379.\n\nGr\u00f6nlund, A. (2007). More control, less conflict? Job demand-control, gender and work-family conflict. Gender, Work & Organization, 14, 476\u2013497.\n\nGrzywacz, J. G., & Marks, N. F. (2000). Reconceptualizing the work-family interface: An ecological perspective on the correlates of positive and negative spillover between work and family. Journal of Occupational Health Psychology, 5, 111\u2013126.PubMed\n\nGuan, J., Rajadhyaksha, U., & McElroy, J. (2011). The relation between work conditions and women's socio-economic status: A global exploratory study. Journal of Research in Peace, Gender and Development, 1, 271\u2013285.\n\nGutek, B. A., Searle, S., & Klepa, L. (1991). Rational versus gender-role explanations for work family conflict. Journal of Applied Psychology, 76, 560\u2013568.\n\nHaar, J. M., & Bardoel, E. A. (2008). Positive spillover from the work-family interface: A study of Australian employees. Asia Pacific Journal of Human Resources, 46, 275\u2013287.\n\nHakanen, J. J., Peeters, M., & Perhoniemi, R. (2011). Enrichment processes and gain spirals at work and at home: A 3-year cross-lagged panel study. Journal of Occupational and Organizational Psychology, 84, 8\u201330.\n\nHarris, R. J., Firestone, J. M., & Bryan, P. J. (2006). A comparative analysis of sex-role ideology in Canada, Mexico and the United States. Advances in Gender Research, 10, 97\u2013124.\n\nHassan, Z., Dollard, M. F., & Winefield, A. H. (2010). Work-family conflict in East vs Western countries. Cross Cultural Management: An International Journal, 17, 30\u201349.\n\nHill, E. J., Yang, C., Hawkins, A. J., & Ferris, M. (2004). A cross-cultural test of the work-family interface in 48 countries. Journal of Marriage and the Family, 66, 1300\u20131316.\n\nHo, M., Xuefei, C., Cheung, F. M., Liu, H., & Worthington, E. (2013). A dyadic model of the work-family interface: A study of dual-earner couples in China. Journal of Occupational Health Psychology, 18, 53\u201363.PubMed\n\nHuffman, A. H., Payne, S., & Castro, C. (April 2003). Time demands, work-family conflict and turnover: A comparison of men and women. Poster presented at the Annual meeting of the Society for Industrial and Organizational Psychology, Orlando, Florida, USA.\n\nJames, B., Barnett, R., & Brennan, R. T. (1998). The psychological effects of work experiences and disagreements about gender-role beliefs in dual-earner couples: A longitudinal study. Women's Health: Research on Gender, Behavior and Policy, 4, 341\u2013368.\n\nJoplin, J. R. W., Shaffer, M. A., Francesco, A. M., & Lau, T. (2003). The macro environment and work-family-conflict: Development of a cross-cultural comparative framework. International Journal of Cross-Cultural Management, 3, 305\u2013328.\n\nKailasapathy, P., & Metz, I. (2008). Work-family conflict in Sri Lanka: Negotiations of exchange relationships in family and at work. Journal of Social Issues, 68, 790\u2013813.\n\nKaratepe, O., & Azar, A. K. (2013). The effects of work-family conflict and facilitation on turnover intentions: The moderating role of core self-evaluations. International Journal of Hospitality and Tourism Administration, 14, 255\u2013281.\n\nKeene, J. R., & Quadagno, J. (2004). Predictors of perceived work-family balance: Gender difference or gender similarity. Sociological Perspectives, 47, 1\u201323.\n\nKinnunen, U., & Mauno, S. (1998). Antecedents and outcomes of work-family conflict among employed women and men in Finland. Human Relations, 51, 157\u2013177.\n\nKorabik, K., McElwain, A., & Chappell, D. (2008). Integrating gender-related issues into research on work and family. In K. Korabik, D. S. Lero, & D. L. Whitehead (Eds.), Handbook of work-family integration: Research, theory, and best practices (pp. 215\u2013232). San Diego: Elsevier.\n\nKorabik, K., Lero, D. S., Aycan, Z., & Bardoel, A. (Dec 2012). HR policies to enhance work-family balance: Fostering organizational sustainability in global context. Paper presented at the Research Colloquium on Creating Sustainable Organizations: The Human and Social Dimensions, Deakin Management Centre, Victoria, Australia.\n\nKulik, L. (2004). Predicting gender role ideology among husbands and wives in Israel: A comparative analysis. Sex Roles, 51, 575\u2013588.\n\nLapierre, L. M., Spector, P. E., Allen, T. B., Poelmans, S., Cooper, C. L., O'Driscoll, M. P., Sanchez, J. I., Brough, P., & Kinnunen, U. (2008). Family-supportive organization perceptions, multiple dimensions of work-family conflict, and employee satisfaction: A test of model across five samples. Journal of Vocational Behavior, 73, 92\u2013106.\n\nLoerch, K. J., Russell, J. E., & Rush, M. C. (1989). The relationships among family domain variables and work-family conflict for men and women. Journal of Vocational Behavior, 35, 288\u2013308.\n\nLyness, K. S., & Judiesch, M. K. (2014). Gender egalitarianism and work-life balance for managers: Multisource perspectives in 36 Countries. Applied Psychology: An International Review, 63, 96\u2013129.\n\nLyness, K. S., & Kropf, M. B. (2005). The relationships of national gender equality and organizational support with work-family balance: A study of European managers. Human Relations, 58, 33\u201360.\n\nMahpul, I. H., & Abdullah, N. A. (2011). The prevalence of work-family conflict among mothers in peninsular Malaysia. International Journal of Humanities and Social Science, 1, 154\u2013161.\n\nMatsui, T., Ohsawa, T., & Onglatco, M. L. (1995). Work family conflict and the stress buffering effects of husband support and coping behavior among Japanese married workingwomen. Journal of Vocational Behavior, 47, 178\u2013192.\n\nMcElwain, A., Korabik, K., & Rosin, H. M. (2005). An examination of gender differences in work-family conflict. Canadian Journal of Behavioural Science, 37, 269\u2013284.\n\nMcNall, L. A., Nicklin, J. M., & Masuda, A. D. (2010). A meta-analytic review of the consequences associated with work-family enrichment. Journal of Business and Psychology, 25, 381\u2013396. doi:10.1007\/s10869-009-9141-1.\n\nMichel, J. S., Kotrba, L. M., Mitchelson, J. K., Clark, M. A., & Baltes, B. B. (2011). Antecedents of work-family conflict: A meta-analytic review. Journal of Organizational Behavior, 32, 689\u2013725.\n\nMickelson, K. D., Claffey, S. T., & Williams, S. L. (2006). The moderating role of gender and gender role attitudes on the link between spousal support and marital quality. Sex Roles, 55, 73\u201382.\n\nMoen, P., Erickson, M., & Dempster-McClain, D. (1997). Their mother's daughters? The intergenerational transmission of gender attitudes in a world of changing roles. Journal of Marriage and the Family, 59, 281\u2013294.\n\nMortazavia, S., Pedhiwala, N., Shafiro, M., & Hammer, L. (2009). Work-family conflict related to culture and gender. Community, Work & Family, 12, 251\u2013273.\n\nNamayandeh, H., Yaacob, S. N., & Juhari, R. (2011). The effect of gender-role orientation on work interference with family (WIF) and family interference with work (FIW) among married female nurses in Shiraz-Iran. Asian Culture and History, 2, 165\u2013171.\n\nNasurdin, A., Ahmad, N. H., & Zainal, M. (2013). Comparing work-family conflict and facilitation among male and female entrepreneurs in Malaysia. International Journal of Business and Society, 14, 149\u2013162.\n\nOllier-Malaterre, A., Valcour, M., Den Dulk, L., & Kossek, E. E. (2013). Theorizing national context to develop comparative work-life research: A review and research agenda. European Management Journal, 31, 433\u2013447.\n\n\u00d6un, I. (2012). Work-family conflict in the Nordic countries: A comparative analysis. Journal of Comparative Family Studies, 43, 165\u2013184.\n\nParasuraman, S., & Greenhaus, J. H. (2002). Toward reducing some critical gaps in work family research. Human Resource Management Review, 12, 299\u2013312.\n\nParasurman, S., & Simmers, C. A. (2001). Type of employment, work-family conflict and well-being: A comparative study. Journal of Organizational Behavior, 22, 551\u2013568.\n\nParboteeah, K. P., Hoegl, M., & Cullen, J. B. (2008). Managers' gender-role attitudes: A country institutional profile approach. Journal of International Business Studies, 39, 795\u2013813.\n\nPimentel, E. E. (2006). Gender ideology, household behavior, and backlash in urban China. Journal of Family Issues, 27, 341\u2013365.\n\nPleck, J. H. (1977). The work-family role system. Social Problems, 24, 417\u2013427.\n\nPoelmans, S. A. Y. (2003). The multi-level 'fit' model of work and family. International Journal of Cross Cultural Management, 3, 267\u2013274.\n\nPoelmans, S. A. Y., O'Driscoll, M., & Beham, B. (2005). An overview of international research on the work-family interface. In S. A. Y. Poelmans (Ed.), Work and family: An international research perspective (pp. 3\u201346). Mahwah: Lawrence Erlbaum.\n\nPoelmans, S. A. Y., Ayman, R., Korabik, K., Rajadhyaksha, U., Huang, T. P., Lero, D. S., & Pande, D. T. (July 2006). How far is too far?: Comparing Spain, Taiwan, India, US & Canada on work-family conflict. Paper presented at the 18th Congress of the International Association for Cross-Cultural Psychology, Spetes, Greece.\n\nPowell, G. N., Francesco, A. M., & Ling, Y. (2009). Toward culture-sensitive theories of the work-family interface. Journal of Organizational Behavior, 30, 597\u2013616.\n\nRajadhyaksha, U., & Velgach, S. (Aug 2009). Gender, gender-role ideology and work-family conflict in India. Paper presented at the annual meeting of the Academy of Management, Chicago, Illinois, USA.\n\nRamadoss, K., & Rajadhyaksha, U. (2012). Gender differences in commitment to roles, work-family conflict and spousal support. Journal of Social Sciences, 33, 227\u2013233.\n\nRay, R., Gornick, J. C., & Schmitt, J. (2010). Who cares? Assessing generosity and gender equality in parental leave policy designs in 21 countries. Journal of European Social Policy, 20, 196\u2013216.\n\nRothbard, N. P. (2001). Enriching or depleting: The dynamics of engagement in work and family roles. Administrative Science Quarterly, 46, 655\u2013684.\n\nSchwarzwald, J., Koslowsky, M., & Izhak-Nir, E. (2008). Gender-role ideology as a moderator of the relationship between social power tactics and marital satisfaction. Sex Roles, 59, 657\u2013670.\n\nShaffer, M. A., Joplin, J., & Hsu, Y. (2011). Expanding the boundaries of work-family research: Review and agenda for future research. International Journal of Cross-Cultural Management, 11, 221\u2013268.\n\nShockley, K., & Singla, N. (2011). Reconsidering work-family interactions and satisfaction: A meta-analysis. Journal of Management, 37, 861\u2013886.\n\nSimon, M., K\u00fcmmerling, A., & Hasselhorn, H. M. (2004). Work-home conflict in the European nursing profession. International Journal of Occupational and Environmental Health, 10, 384\u2013391.PubMed\n\nSmith-Lovin, L., & Tickamyer, A. R. (1978). Nonrecursive models of labor force participation, fertility behavior, and sex role attitudes. American Sociological Review, 43, 541\u2013557.PubMed\n\nSomech, A., & Drach-Zahavy, A. (2007). Strategies for coping with work-family conflict: The distinctive relationships of gender-role ideology. Journal of Occupational Health Psychology, 12, 1\u201319.PubMed\n\nSomech, A., & Drach-Zahavy, A. (July 2011). Coping with work-family conflict from a cross cultural perspective. Paper presented at the meeting of the International Association of Cross-Cultural Psychology, Istanbul, Turkey.\n\nSpector, P. E., Allen, T. D., Poelmans, S., Lapierre, L. M., Cooper, C. L., O'Driscoll, M., et al. (2007). Cross-national differences in relationships of work demands, job satisfaction and turnover intentions with work-family conflict. Personnel Psychology, 60, 805\u2013835.\n\nSpector, P. E., Cooper, C. L., Poelmans, S., Allen, T. D., O'Drsicoll, M., Sanchez, J. I., Siu, O. L., Dewe, P., Hart, P., Lu, L., Moraes, L. F. R., Ostrognay, G. M., Sparks, K., Wong, P., & Yu, S. (2004). A cross-national comparative study of work-family stressors, working hours, and well-being: China and Latin America versus the Anglo world. Personnel Psychology, 57, 119\u2013142.\n\nSteibler, N. (2009). Reported levels of time-based and strain-based conflict between work and family roles in Europe: A multilevel approach. Social Indicators Research, 93, 469\u2013488.\n\nTabassum, A. (2012). The impact of social support on work-to-family and family-to-work conflict: An analysis on the female primary school teachers of Bangladesh. International Journal of Research Studies in Management, 1, 67\u201378.\n\nTreas, J., & Widmer, E. D. (2000). Married women's employment over the life course: Attitudes in cross-national perspective. Social Forces, 78, 1409\u20131436.\n\nTu, S., & Liao, P. (2005). Gender differences in gender-role attitudes: A comparative analysis of Taiwan. Journal of Comparative Family Studies, 36, 545\u2013566.\n\nV\u00e4\u00e4n\u00e4nen, A., May, K., Ala-Mursula, L., Pentti, J., Kivim\u00e4ki, M., & Vahtera, J. (2004). The double burden of and negative spillover between paid and domestic work: Associations with health among men and women. Women & Health, 40, 1\u201318.\n\nVelgach, S., Ayman, R., Antani, A., & Ishaya, N. (July 2006). Impact of GRI on WFC: The case of a health organization in the USA. Paper presented at the International Congress of Applied Psychology, Athens, Greece.\n\nWallace, J. E. (1999). Work-to-nonwork conflict among married male and female lawyers. Journal of Organizational Behavior, 20, 797\u2013816.\n\nWayne, J. H., Randel, A. E., & Stevens, J. (2006). The role of identity and work-family support in work-family enrichment and its work related consequences. Journal of Vocational Behaviour, 69, 445\u2013461.\n\nWentworth, D. K., & Chell, R. M. (2005). Gender identity at home: Comparing the role of househusband to housewife. In J. W. Lee (Ed.), Psychology of gender identity (pp. 113\u2013126). Hauppauge: Nova Biomedical.\n\nWestman, M., Etzion, D., & Gattenio, E. (2008). International business travels and the work-family interface: A longitudinal study. Journal of Occupational and Organizational Psychology, 81, 459\u2013480.\n\nYang, N., Chen, C., Choi, J., & Zou, Y. (2000). Sources of work-family conflict: A Sino-US comparison of the effects of work and family demands. Academy of Management Journal, 43, 113\u2013123.\n\nYavas, U., Babakus, E., & Karatepe, O. M. (2008). Attitudinal and behavioral consequences of work-family conflict and family-work conflict: Does gender matter? International Journal of Service Industry Management, 19, 7\u201331.\n\nFootnotes\n\n1\n\nIn this chapter we discuss the impact of several distinct, yet interrelated aspects of gender. We use the term gender to refer to the physical aspect of gender or whether someone is a man or a woman. Gender-role ideology is defined as whether individuals' attitudes about the roles of men and women are traditional versus egalitarian. In contrast to these two individual-level variables, gender egalitarianism and gender inequality are cultural or societal level variables. In cultures with traditional values men are seen primarily as breadwinners and women primarily as caregivers, whereas in cultures with egalitarian values there is less adherence to this traditional division of labor. The term gender inequality is used to refer to differences in power and status between men and women in different countries.\n\n2\n\nProject 3535 is a collaborative investigation of the W\u2013F interface among organizationally employed married\/cohabiting parents in ten countries (i.e., Australia, Canada, China, India, Indonesia, Israel, Spain, Taiwan, Turkey, and the US). The contributions of the members of the Project 3535 research team to this chapter are gratefully acknowledged. The team consists of: Drs. Zeynep Aycan, Roya Ayman, Anne Bardoel, Tripti Desai, Anat Drach-Zahavy, Leslie B. Hammer, Ting-Pang Huang, Karen Korabik, Donna S. Lero, Artiwadi Mawardi, Steven Poelmans, Ujvala Rajadhyaksha, Anit Somech, and Li Zhang. The project website is www.\u200bworkfamilyconfli\u200bct.\u200bca.\n\n# Part II  \nCONSIDERATIONS FROM THE HOMEFRONT\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_7\n\n# 7. Feeling Work at Home: A Transactional Model of Women and Men's Negative Affective Spillover from Work to Family\n\nMelissa E. Mitchell1  , Lillian T. Eby2  and Anna Lorys1\n\n(1)\n\nDepartment of Psychology, University of Georgia, Psychology Building, 30602 Athens, GA, USA\n\n(2)\n\nDepartment of Psychology and Owens Institute for Behavioral Research, University of Georgia, 510 Boyd GSRC, 30602 Athens, GA, USA\n\nMelissa E. Mitchell (Corresponding author)\n\nEmail: melmitch@uga.edu\n\nLillian T. Eby\n\nEmail: leby@uga.edu\n\nAnna Lorys\n\nEmail: lorysa@uga.edu\n\nKeywords\n\nWork\u2013familySpilloverAffectEmotionAppraisalCopingGender\n\n## 7.1 Introduction\n\nAffect is central to individuals' daily experiences at work and with family. Affect is fundamentally tied to those things that are most important to us\u2014our goals, values, and beliefs\u2014and is salient in virtually all of our interactions with others (Lazarus 2006). The most important and memorable moments of our lives are characterized by high emotion, and our attitudes toward our relationships and our jobs are, at least in part, determined by our affective reactions within those domains (Levenson and Gottman 1985; Weiss and Cropanzano 1996). Given the importance of affect in daily experience, it is critical to understand how affective states transfer across life domains. Specifically, how does affect generated at work transmit into the family, who is most likely to experience this transmission, and what are the ways in which the transmission of affect can be regulated? Through the lens of spillover theory (Staines 1980), this chapter examines these questions with a focus on the transmission of negative affect from work to family.\n\nSpillover theory is a dominant paradigm in the work\u2013family literature. The central premise of this theory is that experiences in the work domain spill over into the family domain, and vice versa (e.g., Staines 1980). Several studies find that affect in the work domain is related to affect in the family domain (e.g., Ilies et al. 2009; Judge and Ilies 2004; Song et al. 2008; Sonnentag and Binnewies 2013). Given the importance of the family domain to overall life satisfaction and well-being (e.g., Heller et al. 2004; Proulx et al. 2007), and findings suggesting that negative mood mediates associations between job stressors and family behavior (Story and Repetti 2006), it is crucial for researchers to understand what processes may exacerbate or ameliorate the transfer of negative affect from work to family. However, the existing literature lacks a comprehensive understanding of how the emotion process, including cognitive appraisal and coping , is involved in the transfer of affect from work to family. In addition, it is critical to explore the person and environment factors that make certain work events and emotion processes more or less likely. Thus, the purpose of this chapter is to explore the role of the emotion process in the transfer of work experiences into family life. Guided by Lazarus and Folkman's (1984) seminal transactional model of emotion , we conceptualize affective responses as resulting from the dynamic interplay between characteristics of the person and the environment . In doing so, we respond to Judge and Ilies' (2004) call for an expansion of affective events theory (AET; Weiss and Cropanzano 1996) to include the effects of work events on affective processes with family, and to Allen et al.'s (2012) call for expanded theoretical models of how personal and situational variables interact to predict outcomes in the work and family domains.\n\nThroughout this review, we consider gender as a key antecedent that has downstream effects on the work-to-family spillover process. We conceptualize gender as both a person and an environment variable consisting of gender role expectations , individual differences in masculinity and femininity , and biological sex differences. We suggest that these aspects of gender influence work events, as well as the appraisal and coping processes. By placing gender front and center in the proposed model, we address Story and Repetti's (2006) call for an approach to affective spillover that focuses on cognitive processes and gender.\n\nAlthough we recognize that there is a growing body of literature on positive affective spillover and family-to-work spillover, this chapter focuses on negative affective work-to-family spillover for several reasons. First, research finds that gender differences are most apparent under conditions of stress (Schulz et al. 2004; Vogel et al. 2003). Second, there is evidence that individuals experience stronger work-to-family spillover than family-to-work spillover (Brotheridge and Lee 2005; Evans and Bartolom\u00e9 1984; Williams and Alliger 1994). Third, research suggests that negative spillover is more enduring than positive spillover (Sonnentag and Binnewies 2013). However, we note that it is also important to examine the role of gender and the emotion process in positive affective spillover in order to gain a complete understanding of the spillover process. Although we focus primarily on negative affective spillover in our review, we believe that our model can be extended to capture the unique processes involved in positive affective spillover. We think it would be fruitful for future scholars to examine the person and environment factors that give rise to benign-positive appraisals of work events, and the behaviors that tend to generate and sustain positive affect across domains. In addition, our review focuses on within-person affective spillover rather than between-person crossover (i.e., the transfer of affect from one person to another). Although crossover processes are important to consider, it is excluded from the current review in order to provide an in-depth treatment of within-person spillover.\n\nWe begin our review by briefly introducing the transactional model of emotion . Next, we discuss how negative affective work-to-family spillover fits into the transactional framework and present our conceptual model. Then, we discuss person and environment factors that may play roles in the emotion process, and consequently in affective spillover , specifically focusing on how these factors may differ for men and women. This leads to a discussion of the emotion process as a moderator of the association between affective states at work and affective states at home. We conclude our review with suggestions for future research.\n\n### 7.1.1 Transactional Model of Emotion\n\nTransactional theories of emotion posit that all affective experience is mediated by appraisal: \"an evaluation of the personal significance of a particular encounter between the person and the environment\" (Lazarus 1991, p. 820) . Individuals engage in two types of appraisal when evaluating an event as stressful (Lazarus and Folkman 1984). In primary appraisal , people evaluate whether the event is harmful or beneficial for their well-being . In secondary appraisal , people evaluate their ability to cope with the event. Appraisals are influenced by both person and environment factors, that is, within a particular encounter, characteristics of the person and the environment interact to make certain appraisals more or less likely. Appraisals then give rise to coping behaviors , defined as efforts to manage environmental demands (problem-focused coping) and the emotions generated from those demands (emotion-focused coping). Coping behaviors, in turn, give rise to reappraisals as individuals encounter new information; these reappraisals engender additional coping behaviors . Thus, appraisal and coping reciprocally influence each other throughout an encounter (Lazarus and Folkman 1984) .\n\n### 7.1.2 Transactional Model of Work-to-Family Affective Spillover\n\nAffect is an umbrella term for both mood and emotion; following previous emotion theorists (Lazarus 1991; Weiss and Cropanzano 1996), we conceptualize emotion as distinct from mood in that emotions or affective states arise in response to a specific event, whereas moods are not directed at a particular object or event. A number of studies have found that negative affect at work is related to negative affect at home (e.g., Ilies et al. 2009). Moreover, research suggests that the spillover of negative work experiences into the family domain is mediated by the affective strain resulting from those experiences (Barling and Macewen 1992). Additionally, as discussed above, transactional theories of emotion (e.g., Lazarus and Folkman 1984) suggest that all affective states are mediated by cognitive appraisal . Taken together, prior theory suggests that work events give rise to appraisal processes, which generate negative affective states if (1) the person appraises the event as threatening to their well-being and (2) the person evaluates their coping resources as inadequate for managing the event. We propose that these affective states may spill over into the family domain through the reciprocal processes of appraisal and coping. These associations between work events, the emotion process, and affective states at work and with family are depicted in Fig. 7.1.\n\nFig. 7.1\n\nA transactional model of affective work-to-family spillover\n\nOur conceptual model follows AET's (Weiss and Cropanzano 1996) tenet that work events lead to affective states at work. However, in accordance with transactional theories of emotion, we conceptualize the emotion process as the critical mediator required for a work event to produce an affective state. In addition, we follow AET's recognition of person and environment factors as causes of work events; however, consistent with transactional theories of emotion, we assert that person and environment factors have interactive effects on the occurrence of work events and appraisals of those events. For example, imagine that Leslie, a local government employee with high career goals, has just finished an important project proposal when her supervisor informs her that the funding for the project has been cut due to government overspending. Because of her high career goals, Leslie appraises this event as highly threatening to her personal well-being , causing her to feel anxious and angry. This example illustrates how person (career goals) and environment factors (government overspending) interact to trigger a work event (loss of funding for valued project), which is appraised as threatening (appraisal), resulting in negative affective states at work (anxiety and anger) . Our transactional spillover model acknowledges that work events, appraisals, and resulting affective states derive from a complex interplay of the person and the environment .\n\nA novel contribution of the model is our framing of the emotion process as not just a mediator of the association between work events and affective states, but also as a moderator of the association between affective states at work and affective states at home (see Fig. 7.1). In so doing, we contend that certain appraisals and coping behaviors make affective work-to-family spillover more or less likely, and that these processes have a dynamic relationship within a particular transaction. For example, imagine that Ron, an engineer who has low self-esteem , receives negative feedback from a supervisor, which causes him to feel anger and shame at work. Throughout the day, Ron ruminates about the feedback, which only exacerbates his negative emotion. By the time he returns home, Ron is extremely upset and attempts to further cope with his anger by venting to his partner . Thus, the use of rumination as a coping strategy moderates the association between negative affect at work and negative affect at home, such that it exacerbates Ron's negative emotion, and thus his spillover of negative emotion from work to home. This example illustrates the dynamic interplay between coping and appraisal; appraisals influence coping, which in turn influence reappraisals . As such, we propose that these emotion processes hinder or facilitate the transfer of negative affect from work to home.\n\nAlthough appraisal and coping occur at the individual level, transactional theories of emotion acknowledge that characteristics of the social system also influence (1) the environmental conditions groups of people are exposed to and (2) characteristic ways of reacting to the environment, including appraisals and coping (Lazarus and Folkman 1984; Lazarus 1999). While acknowledging the considerable intragroup and intraindividual variability in appraisal and coping, we suggest that gender plays a role in the stress process by making certain person and environment factors more or less probable for men and women (Lazarus 1999). We suggest that these factors, in turn, have downstream effects on the emotion process and affective spillover from work to home. We view gender as a multifaceted construct composed of socially constructed norms and beliefs about the roles of men and women (i.e., gender role expectations), individual differences in masculinity and femininity , and biological sex differences. Throughout this chapter, we explore the role of gender in the emotion process, specifically as it relates to negative affective spillover from work to family. We suggest that gender role expectations present individuals with social demands (social pressure to act in accordance with gender roles and to hold gendered attitudes) and constraints (social pressure to not act in certain ways in accordance with gender roles) that shape both personal and environmental factors (see Fig. 7.1). These differentiated gender roles are facilitated by biological sex differences, including those resulting from physical specialization (e.g., pregnancy , breastfeeding) and hormones (Wood and Eagly 2002), and individual differences in gender role (Rosenkrantz et al. 1968). Through gender's effects on person and environment factors, we suggest that gender has reverberating effects on the affective spillover process. Thus, we take the position that gender differences in spillover are largely the result of differences in the meaning of men's and women's experiences. These meanings, we argue, are derived through the interdependent processes of appraisal, coping, and reappraisal . We now turn our attention to a discussion of specific person and environment factors that affect the emotion process, and consequently negative affective spillover from work to family.\n\n## 7.2 Person Factors\n\n### 7.2.1 Dispositional Traits\n\nThere are a number of dispositional traits that influence affective experience. However, most research on negative affect at work, and negative spillover more specifically, focus on affective disposition as a predictor of work and nonwork outcomes. A large body of research demonstrates that trait negative affectivity (NA) , the tendency to experience negative emotional states (Watson and Clark 1984), consistently influences appraisals of negative events. Research finds that people higher in NA tend to experience negative work events more negatively than those who are lower in NA, and NA has been found to influence mood and interpretations of events at work (Judge and Ilies 2004). Additionally, people who are higher in NA may trigger aversive work and family events through their negative appraisals of the environment (Weiss and Cropanzano 1996). Individuals who are higher in NA and trait anxiety also are more likely to engage in potentially problematic coping behaviors, such as confrontation, disengagement, and avoidance, than those lower in these traits (Carver et al. 1989; O'Brien and DeLongis 1996). Thus, there is considerable support for the idea that NA affects how individuals appraise and cope with life and work events.\n\nThere is also evidence that trait-based negative affect plays a role in negative affective work-to-family spillover. For example, Repetti and Wood (1997) found that the associations between workload and parental withdrawal and irritability were particularly strong among mothers with more symptoms of depression and higher trait anxiety , suggesting that individual tendencies to experience negative emotion can have downstream effects on negative affective spillover from work to family. Grzywacz and Marks (2000) also found that negative affective spillover was higher among more neurotic individuals. Similarly, Wang et al. (2011) found that neuroticism exacerbated negative spillover among men, such that the positive association between job stress and negative emotion display with family was stronger among more neurotic men. Collectively, these studies indicate that negative affective spillover is higher among individuals who tend to experience negative emotions, particularly anxiety.\n\nResearch suggests that women score higher on negative affectivity and neuroticism than men (e.g., Costa et al. 2001), and report feeling negative emotions more intensely and for longer durations than men (Brody and Hall 2008). Thus, to the extent that negative affectivity is related to greater negative affect, we would expect women to experience more spillover than men. However, the literature in this area is inconsistent. For example, Wang et al. (2011) found that neuroticism exacerbated negative emotion spillover among men, but not women. By contrast, Grzywacz and Marks (2000) found no gender differences in the association between neuroticism and negative spillover , although they found that women experienced more negative spillover than men. Other research finds that men and women may differ in specific emotional reactions to work stress when in the home. For example, Schulz et al. (2004) found that more negative workdays were associated with more angry marital behavior for women and more withdrawn behavior for men. These results suggest that both men and women experience negative affective spillover from work to family, although their specific affective reactions may differ.\n\nAlthough women tend to score higher on negative affectivity than men, there is also evidence that women score higher in positive affectivity (Grandey et al. 2002). This suggests that women may be more emotionally reactive than men, regardless of valence (Brody and Hall 2008). Consistent with this idea, Campos et al. (2013) found that mothers displayed both more positive and negative emotion than fathers per day, and Williams and Alliger (1994) found that women experienced both higher negative and positive affective spillover than men. Other researchers have suggested that women's affective range may be broader than men's (Fujita et al. 1991); therefore, women may experience spillover of a broader range of emotions from work to home. In support of this idea, Matjasko and Feldman (2006) found that mothers experienced spillover of happiness, anger , and anxiety from work to home, whereas fathers only experienced anxiety spillover. These findings indicate that women may experience higher levels of negative and positive spillover than men as well as spillover of a greater range of emotions. These gender differences in emotional reactivity may be the result of gender role socialization . In childhood, boys learn to minimize, suppress, and control their expression of vulnerable emotions, whereas girls' emotions are accepted and attended to by others (Brody and Hall 2008). As a result, women learn to express a greater range of emotion across domains than men. As a result, women may be more likely than men to experience spillover of both positive and negative emotion from work to family.\n\n### 7.2.2 Goal Commitments\n\nAccording to the transactional model (Lazarus and Folkman 1984; Lazarus 1999), appraisals are influenced by a number of person factors, most importantly goal commitments\u2014defined as those things that are valued, meaningful, and important to an individual . Goal commitments determine the significance of a particular experience to an individual's well-being . Lazarus and Folkman (1984) describe several ways that goal commitments influence appraisal. First, goal commitments affect the activities and situations in which individuals choose to be involved. For example, individuals who are highly committed to work goals will likely spend a lot of time involved in work activities and much of their time thinking about and discussing work (Greenhaus and Beutell 1985). Second, people are more sensitive to those things that are relevant to their goal commitments . For example, an individual who is highly committed to work goals may be more aware of and attentive to potential threats to these goals, and thus more likely to appraise ambiguous work events negatively. Third, goal commitments influence the extent to which individuals are vulnerable to threat and harm, such that the more a person is committed to a particular goal, the greater the potential threat. For example, individuals who are highly committed to work goals may appraise negative work events as particularly distressing. In support of these propositions, Song et al. (2008) found that negative affective work-to-home spillover was higher among those who were higher in work orientation (a strong commitment to the work role).\n\nHistorically, it was assumed that men were more likely to be committed to work goals, whereas women were more likely to be committed to family goals due to men and women's differential socialization into sex-stereotypic roles (Pleck 1977). From this perspective, men may be expected to have higher negative work-to-family spillover than women because they are more likely to attend and react to negative work events, given their higher commitment to work goals. Thus, men may experience greater negative affect at work, which may then spill over into the family . However, although some previous research has found that men are more likely to report stressful episodes at work than are women (Folkman and Lazarus 1980; Kendler et al. 2001), and that women tend to be more involved with family and men with work (e.g., Byron 2005), gender differences in work and family goals are often small. Additionally, recent research suggests that as more women have entered the workforce, gender roles have eroded somewhat (Wood and Eagly 2002), and as a result, men and women have become more equally committed to both work and family roles (Barnett and Hyde 2001; Barnett et al. 1993). For example, more than a third of men and women in Cinamon and Rich's (2002) sample viewed both work and family roles as highly important. Additionally, recent research finds that men and women who hold similar jobs report similar levels of job demands (McElwain et al. 2005), suggesting that earlier findings of gender differences in commitment may have had more to do with the differing characteristics of men's and women's jobs than men's greater prioritization of the work role . Moreover, research suggests that gender identity (i.e., description of the self in terms of masculine or feminine traits) may be a better predictor of commitment to the work and family roles than biological sex (Powell and Greenhaus 2010). One potential implication of men and women's greater commitment to goals in both domains is an increase in affective spillover for both genders. This is in line with Williams et al.'s (1991) suggestion that the effects of role juggling may be more severe for those who are highly committed to both their work and family roles, as a compromise in either domain threatens an important role. Given these findings, we suggest that researchers who are interested in affective spillover also carefully consider respondents' commitment to work and family goals .\n\n### 7.2.3 Personal Resources\n\nAccording to Lazarus (1999), as well as other prominent theories of job stress (e.g., Job Demands-Resources, Demerouti et al. 2001; Conservation of Resources Theory, Hobfoll 1989) , individuals and groups experience stress when the demands of the environment are greater than their resources. Although researchers often include dispositional factors such as self-esteem and locus of control in definitions of personal resources (e.g., Lazarus and Folkman 1984), we define personal resources more narrowly as external sources of support that may be engaged under stress. From the view of transactional emotion theories, although individuals may have \"objective\" personal resources, these resources only influence affect through cognitive appraisal ; that is, although individuals' appraisals are generally congruent with reality, it is the appraisal of these resources that determines affective reactions rather than the objective situation (Lazarus and Folkman 1984). Thus, individuals who hold the same or similar personal resources may appraise the availability of resources differently. Research has found that personal resources, such as perceived social support, are negatively related to psychological strain (e.g., Vermeulen and Mustard 2000) and are positively related to adaptive coping (Holahan and Moos 1991). Given their effects on the emotion process, personal resources may also influence the extent to which negative affect from work spills over into the family .\n\nDespite advances in women's status in the workplace and society, considerable research finds that men have more financial resources than women. American women working full-time earned 77 % of men's earnings in 2012 (DeNavas-Walt et al. 2013). Women's comparatively lower income may increase women's likelihood of experiencing negative affective spillover from work to family. This is because women\u2014particularly those in single-earner households, who earn considerably less money than those in other household structures (DeNavas-Walt et al. 2013)\u2014may appraise work events that threaten their earnings as particularly distressing, resulting in more negative affect at work. This negative affect may be exacerbated to the extent that women are confronted with additional financial demands in their family life. Thus, to the extent that women have fewer financial resources than men, we would predict that women experience greater negative affective spillover from work to family.\n\nAlthough women may have less access to financial resources than men, women tend to have broader social support networks that may mitigate negative affective spillover from work to home. There is consistent evidence that men and women differ in the desire to affiliate with others during stress. Taylor et al. (2000) argued that women's stress response may be better characterized as \"tend-and-befriend\" rather than \"fight-or-flight\"; that is, women tend to seek support during times of stress more than men, particularly from other women. Seeking support may have several benefits, including an expanded network of potential supporters with greater exchange of resources , and increased recovery from stress (Taylor et al. 2000). There is also evidence that social support has a larger impact on negative affect for those with more feminine gender roles; for example, Beehr et al. (2003) found that individuals higher in femininity had a greater decline in psychological strain when their supervisors were supportive than those lower in femininity. Thus, women may have greater access to social support resources than men, potentially mitigating the threat of negative work events, and consequently reducing negative affect .\n\nHowever, there are also drawbacks to receiving social support. For one, involvement of a supportive network may necessitate both the receipt and provision of support; the provision of social support may be draining and may undermine one's own recovery from stress (Shumaker and Hill 1991). Second, affiliation may increase the likelihood of problematic social coping strategies such as corumination (Haggard et al. 2011). Considerable research has found that women are more likely to focus on and discuss negative affective states (i.e., ruminate), which tends to sustain or amplify negative affect (Nolen-Hoeksema 1987) and cortisol and immune reactivity (Denson et al. 2009). This ruminative tendency may be fostered by intimate affiliative groups, and may paradoxically sustain negative affect rather than ameliorate it. Finally, high social contact also increases the likelihood that one will encounter stressful social situations (Taylor et al. 2000). For example, Kendler et al. (2001) found that women reported higher rates of interpersonal problems within their proximal networks than did men, and that the relationship between experiencing interpersonal problems within one's network and the onset of major depression was stronger among women than men. Clearly, these complex gender differences in personal resources need to be taken into account in further comparisons of men and women's spillover experiences .\n\n## 7.3 Environmental Factors\n\n### 7.3.1 Job Demands and Resources\n\nAccording to AET (Weiss and Cropanzano 1996), features of the work environment may trigger the occurrence of certain work events, leading to specific affective reactions. Considerable research supports the idea that the combination of high job demands and low job resources predict psychological strain (e.g., Demerouti et al. 2001). Thus, to the extent that individuals are employed in jobs that are associated with high job demands and low job resources, we would expect greater appraisal of stress and subsequently greater transfer of negative emotion from work to family. Men and women may differ in the types of jobs they are employed in and therefore in the types of workplace stressors to which they are exposed. These differences may impact appraisal and coping , and consequently influence affective spillover from work to family. That is, gender differences in job demands and resources may have downstream effects on men and women's negative affective spillover from work to family.\n\nMuch research has focused on the additional job demands faced by women in organizations. For example, women are more likely to experience gender discrimination and harassment at work than are men (Rotundo et al. 2001). In addition, occupational sex segregation makes certain work demands more salient for women than men. For example, many authors have argued that emotional labor \u2014managing affective displays as part of a work role (Hochschild 1979)\u2014is more commonly expected of women than men (e.g., Guy and Newman 2004). Research finds that surface acting, a component of emotional labor involving faking emotions as part of a work role (Hochschild 1979), is associated with negative affective states at work (Scott and Barnes 2011), which in turn are associated with work\u2013family conflict (Yanchus et al. 2010). Given these associations, it is likely that emotional labor has subsequent effects on work-to-family affective spillover, and that this job demand is faced disproportionately by women. As discussed below, this effect may be compounded given women's arguably greater emotional work in the home domain compared to men (Wharton and Erickson 1993).\n\nMen and women may also differ in job resources. For example, there is some evidence that women have greater access to work\u2013family resources at work. Research suggests that women perceive greater availability of work\u2013family benefits than men (Thompson et al. 1999) and use work\u2013family benefits more than men (Allen 2001). This may be indicative of strong gender norms that discourage men from using the work\u2013family resources that are available to them. For example, Allen and Russell (1999) found that men who took an extended family leave received fewer organizational rewards than those who did not. Similarly, Butler and Skattebo (2004) found that raters viewed men experiencing family conflict as lower performers and gave them lower reward recommendations than those who did not experience family conflict, whereas ratings of women were unaffected by family conflict. This may suggest that men may be less likely to use family supportive resources in order to protect themselves from negative career perceptions and\/or outcomes associated with family conflict. Women's greater access to support for work\u2013family benefits may decrease women's negative affective spillover through its effects on appraisal of coping resources; that is, women may perceive greater ability to cope with work stressors due to their greater perceived and actual availability of work\u2013family support. However, although women may receive greater work\u2013family and social support at work, they may also be more affected when they perceive a lack of support. For example, Grzywacz and Marks (2000) found that low social support at work was more strongly associated with negative work-to-family spillover for women than for men.\n\nOther gender differences in job demands and resources may influence how men and women cope with work events. For example, Folkman and Lazarus (1980) found that men were more likely than women to use problem-focused coping (i.e., attempting to change the environment in order to cope with a stressor) at work. They interpreted this difference as resulting from men and women's different job contexts, such that the men in their sample likely had higher-level jobs and thus more opportunities to change their environment through problem-solving than women. It is likely that these gender differences have weakened over time as women's status in the workplace has improved.\n\n### 7.3.2 Family Demands and Constraints\n\nResearch consistently finds that women face greater demands in the home than men. Hochschild and Machung (1989) argue that women face a \"second shift\" after work that involves housework , childcare , and other domestic duties. As a result of this greater household burden, women may not have the same opportunities as men to recover from job stress through after-work coping behaviors such as withdrawal . As a result, women may experience sustained negative affect from work within the family domain. In addition, several studies have found that women experience less physiological recovery from work compared to men, as indicated by women's higher stress hormone excretion after work and on rest days (e.g., Frankenhauser et al. 1989; Pollard et al. 1996). Thus, due to their higher demands in the home, women may be less able than men to recover from negative affect from work, resulting in women experiencing increased spillover of negative affect from work to home.\n\nWomen are the primary providers of socioemotional support in the family domain. This emotion work can be considered an additional household demand that primarily falls on women (Erickson 2005). Considerable research finds that men are more comfortable seeking social support from and disclosing emotion to women than to men, particularly their spouses (Brody and Hall 2008; Erickson 1993). As a result, female partners are often the primary sources of social support for their male partners. As discussed above, providing social support confers both costs and benefits; however, an additional cost of support provision within the family is emotional contagion. Research finds that men's negative emotions tend to crossover to other family members to a greater extent that do women's (Larson and Almeida 1999); one explanation for this finding is that women are expected to respond to their male partners with empathy and support, and therefore, are more likely to \"catch\" their partners' negative emotions (Brody and Hall 2008). Thus, women are expected to manage emotions both at work and with family to a greater extent than men. This additional burden may exacerbate women's negative affective spillover from work to home.\n\n## 7.4 The Emotion Process as a Moderator of the Work-to-Family Spillover Process\n\nThroughout this chapter, we have argued that gender has direct effects on person and environment factors, which in turn influence the emotion process. Further, we argued that this process has downstream effects on affective work-to-family spillover. We also contend that affective spillover from work to home is moderated by the emotion process\u2014specifically, the appraisals and coping strategies involved in the response to a work event. We have already discussed how gender influences the person and environment factors that give rise to work events and appraisals of those events. To the extent that individuals appraise work events as more threatening to well-being and taxing to their resources, we would predict greater negative affective work-to-family spillover based on higher levels of negative affective states at work. However, we also argue that some coping strategies are likely to lead to further negative reappraisals , and thus greater negative spillover from work to family. Specifically, certain coping behaviors, and subsequent reappraisals resulting from coping behaviors , may moderate the association between affective states at work and affective states at home, such that affective spillover is ameliorated or enhanced depending on the strategy used.\n\nOne common strategy for coping with negative work events is withdrawal . Although both men and women tend to withdraw at home after stressful workdays (Repetti 1989; Repetti and Wood 1997), some research suggests that this strategy is more common among men. For example, Schulz et al. (2004) found that husbands, but not wives, were more likely to withdraw from their partners in the evening after feeling negative affect at the end of the workday. In the short term, withdrawal behavior may be beneficial for well-being (Repetti 1992). Research finds that psychological detachment from work attenuates negative affective spillover (Sonnentag and Binnewies 2013); thus, withdrawing or detaching may be one way that individuals use to minimize the extent to which negative affect from work spills over into the home.\n\nHowever, although withdrawal may be an adaptive coping strategy in the short-term, such behavior may be problematic for family relationships. Women may be particularly distressed by their male partners' withdrawal behavior due to their greater preference for affiliation and emotional engagement (Levenson and Gottman 1985). This is consistent with Schulz et al.'s (2004) finding that wives whose husbands tended to withdraw in the evening had lower marital satisfaction than wives whose husbands did not withdraw. Since women tend to prefer to cope with strain in affiliative ways by discussing problems with others (Taylor et al. 2000), husbands' tendencies to withdraw from marital interaction may result in wives' feeling unsupported and rejected. Among heterosexual dual-earner couples, men's tendencies to withdraw to cope with negative work events may be problematic when combined with women's desire to affiliate. These gendered patterns of coping with stress may not only undermine both partners' ability to minimize negative affective spillover, but may also trigger additional negative events (e.g., arguments) within the family domain.\n\nOther ways of coping may lessen negative affective spillover from work to home. For example, Wiese et al. (2010) found that individuals who were told to think about a positive family-related experience after thinking about a job-related failure showed faster emotional recovery than those told to think about a positive job-related experience. These results suggest that negative spillover can be ameliorated to the extent that individuals cope with negative work events by reflecting on positive experiences in their family lives. As discussed earlier, other coping behaviors , such as rumination and co-rumination, may amplify negative spillover by sustaining negative affective states (Nolen-Hoeksema 1987). Future research is needed to examine the specific coping strategies individuals can use to change their appraisals of negative work events and reduce negative affective spillover .\n\n## 7.5 Conclusions and Directions for Future Research\n\nThroughout this chapter, we emphasized the importance of the emotion process to negative affective work-to-family spillover. Although research demonstrates the affective links between the work and family domains, more research is needed to examine how specific discrete emotions experienced at work are associated with discrete emotions at home. For example, an initial anger reaction at work may change over the course of the day and interactions with others, and may develop into a different emotion as the individual engages in coping behaviors and reappraises the work event. Thus, affective states at work may develop into quite different affective states at home. An expanded focus on the spillover of discrete emotions is consistent with Lazarus' (1999) view of emotions as distinct \"stories\" of how a person appraises and copes with a particular transaction. However, a focus on discrete emotions requires more complex methodologies, particularly when it comes to physiological measures. Recent research suggests that typical physiological measures of stress, such as cortisol, differ depending on the specific emotion experienced and the coping strategy used (Denson et al. 2009). Thus, future research is needed to examine spillover processes at the discrete emotion level.\n\nAnother direction for future research is an examination of how family characteristics affect the spillover process. The research on spillover consistently finds that the quality of family life can ameliorate or exacerbate negative affective spillover. For example, relationship satisfaction can protect against negative emotions resulting from work events (Campos et al. 2013). Likewise, Story and Repetti (2006) found that positive associations between workload and anger were strongest in conflictual family environments. Extending these findings, it would be interesting to explore the types of behaviors that spouses and children engage in to help reduce their family members' negative affect, and the short- and long-term consequences of these behaviors on relationship satisfaction and stability.\n\nFollowing Judge et al. (2006), we also encourage more research on the functionality of negative emotions in the work and family domain. Negative affective spillover may have adaptive value as a signal to family that something is wrong and that caretaking behaviors should be mobilized. For example, research indicates that preschool children behave in a less aversive and more positive manner on days when mothers are rated as more withdrawn (Repetti and Wood 1997). In addition, Repetti (1989) found that husbands were more withdrawn when wives engaged in more supportive behavior, perhaps suggesting that wives' support enabled men to psychologically detach from work. Likewise, negative affective spillover may motivate employees to engage in more effective coping strategies to reduce negative spillover, such as negotiating housework with a partner or exercising. These findings underscore the need to study affective spillover from a family systems perspective by focusing on the ways that family members inhibit and encourage negative affective spillover.\n\nAlthough the focus of this chapter was negative affective spillover, it is important for researchers to examine the roles of gender and the emotion process in positive affective spillover. Recent research finds that positive affect at work is related to positive affect at home (e.g., Judge and Ilies 2004; Ilies et al. 2009; Song et al. 2008); however, less research has examined the mechanisms and antecedents involved in the positive affective spillover process. Aspects of our model may be applicable to positive spillover. For example, personal factors that encourage appraisal of stressors as challenges rather than threats\u2014such as resilience , optimism, and self-efficacy beliefs\u2014may encourage the spillover of positive affect from work to home through the emotion process. Job resources, such as supportive supervision or mentoring support, may also result in positive affective spillover from work to home. This would be in line with previous research documenting that autonomy , learning opportunities, respect, and meaningful work are positively related to work\u2013family facilitation (Voydanoff 2004). It would be interesting to examine the implications of such job resources for the emotion process; perhaps when job resources are high, environmental events are appraised benevolently, generating positive affect. There is also some evidence that experiencing stressors can bolster subsequent positive affect; for example, Bolger et al. (1989) found that mood was more positive on days following a stressful event than it would have been if a stressful event was not experienced. This suggests that there may be some time-lagged affective gains associated with successful coping with stress. Future research should examine how individual differences in protective traits like resilience and optimism may encourage gains in positive affect following stressful work events.\n\nAlthough coping is discussed as a response to threat appraisals in the transactional model, there may be analogous behaviors that sustain positive affect across work and family domains. For example, savoring can be considered an analog to rumination for positive events, whereby individuals recall and reminisce about positive events in ways that tend to sustain and amplify positive affect (Sonnentag and Grant 2012). To the extent that individuals savor positive work experiences throughout the day, we would expect them to experience greater positive affective spillover from work to home. Additionally, capitalization \u2014sharing positive events with others (Gable et al. 2004)\u2014can be considered an analog to co-rumination about positive events. Previous research has found that capitalization is associated with increases in positive affect beyond experiencing the event itself (Lambert et al. 2013; Langston 1994), suggesting that positive affective spillover may be enhanced to the extent that individuals share positive work events with their family members. This is consistent with research finding that capitalization about work events strengthens the positive association between daily work engagement and work\u2013family facilitation (Culbertson et al. 2012), and that capitalization about work events is positively related to life satisfaction and positive affect (Ilies et al. in press). Future research should also explore the role of gender in these cognitive and social processes; it may be that behaviors such as capitalization are more common among women, given their greater preferences for affiliation (Taylor et al. 2000). This propensity may lead women to experience greater spillover of positive and negative emotions, depending on how work events are appraised. This is consistent with evidence that women experience higher levels of both work\u2013family conflict and facilitation compared to men (Grzywacz and Marks 2000; Powell and Greenhaus 2010; van Steenbergen et al. 2007).\n\nConsistent with previous critiques of the use of gender as an explanatory variable, we encourage researchers to measure the proposed mechanisms underlying sex and gender differences using strong theoretical rationale. Some potential mechanisms proposed in this chapter include differences in goal commitment, personal resources , job and family demands and resources, and choice of coping behaviors. With this said, it is important to note that there is usually greater variability within gender than between genders, and that gender differences in emotion and behavior often depend on the context (Brody and Hall 2008). We suggest that researchers focus on understanding these contextual features when examining gender and the spillover process. Wood and Eagly (2002) suggest that gender differences in behavior are driven by a gendered division of labor ; thus, to the extent that work and family responsibilities become more equitable across genders, we should expect gender differences to become smaller. Meta-analytic work on gender differences in work attitudes and work\u2013family conflict across time would be a welcome addition to the often inconsistent literature in this area.\n\nWe acknowledge that given the reciprocal associations within the emotion process, depicted in Fig. 7.1, and our focus on work events as primary causes of affective reactions, our proposed model will be difficult to fully test. We support Lazarus' (1999) suggestion that researchers shift their measurement of emotion from self-report surveys to narratives ; narrative methodologies allow individuals to describe how a particular transaction unfolds in terms of appraisal, coping, and reappraisal , while drawing on their past experiences, goal commitments, and resources. In combination with experience sampling methodologies, narratives of emotion would be a fruitful approach to a person-centered understanding of affective spillover.\n\nIn conclusion, given the importance of affect and the work\u2013family interface to individuals' everyday lives, an approach to spillover that focuses on the how characteristics of the person and the environment jointly affect the emotion process is important and necessary. Throughout this chapter, gender was discussed as an important antecedent variable that has downstream effects on negative affective spillover from work to family. It is hoped that the proposed model offers a theory-based framework for additional empirical and conceptual work on the intersection of gender, the emotion process, and affective spillover from work to family.\n\nReferences\n\nAllen, T. D. (2001). Family-supportive work environments: The role of organizational perceptions. Journal of Vocational Behavior, 58, 414\u2013435.CrossRef\n\nAllen, T. D., & Russell, J. E. (1999). Parental leave of absence: Some not so family-friendly implications. Journal of Applied Social Psychology, 29, 166\u2013191.CrossRef\n\nAllen, T. D., Johnson, R. C., Saboe, K. N., Cho, E., Dumani, S., & Evans, S. (2012). Dispositional variables and work\u2013family conflict: A meta-analysis. Journal of Vocational Behavior, 80, 17\u201326.CrossRef\n\nBarling, J., & Macewen, K. E. (1992). Linking work experiences to facets of marital functioning. Journal of Organizational Behavior, 13, 573\u2013583.CrossRef\n\nBarnett, R. C., & Hyde, J. S. (2001). Women, men, work, and family. American Psychologist, 56, 781\u2013796.PubMedCrossRef\n\nBarnett, R. C., Marshall, N. L., Raudenbush, S. W., & Brennan, R. T. (1993). Gender and the relationship between job experiences and psychological distress: A study of dual-earner couples. Journal of Personality and Social Psychology, 64, 794\u2013806.PubMedCrossRef\n\nBeehr, T. A., Farmer, S. J., Glazer, S., Gudanowski, D. M., & Nair, V. N. (2003). The enigma of social support and occupational stress: Source congruence and gender role effects. Journal of Occupational Health Psychology, 8, 220\u2013231.PubMedCrossRef\n\nBolger, N., DeLongis, A., Kessler, R. C., & Wethington, E. (1989). The contagion of stress across multiple roles. Journal of Marriage and the Family, 51, 175\u2013183.CrossRef\n\nBrody, L. R., & Hall, J. A. (2008). Gender and emotion in context. In M. Lewis, J. M. Haviland-Jones, & L. Feldman Barrett (Eds.), Handbook of emotions (pp. 395\u2013408). New York: Guilford.\n\nBrotheridge, C. M., & Lee, R. T. (2005). Impact of work\u2013family interference on general well-being: A replication and extension. International Journal of Stress Management, 12, 203\u2013221.CrossRef\n\nButler, A. B., & Skattebo, A. (2004). What is acceptable for women may not be for men: The effect of family conflicts with work on job performance ratings. Journal of Occupational and Organizational Psychology, 77, 553\u2013564.CrossRef\n\nByron, K. (2005). A meta-analytic review of work\u2013family conflict and its antecedents. Journal of Vocational Behavior, 67, 169\u2013198.CrossRef\n\nCampos, B., Wang, S. W., Plaksina, T., Repetti, R. L., Schoebi, D., Ochs, E., et al. (2013). Positive and negative emotion in the daily life of dual-earner couples with children. Journal of Family Psychology, 27, 76\u201385.PubMedCrossRef\n\nCarver, C. S., Scheier, M. F., & Weintraub, J. K. (1989). Assessing coping strategies: A theoretically based approach. Journal of Personality and Social Psychology, 56, 267\u2013283.PubMedCrossRef\n\nCinamon, R. G., & Rich, Y. (2002). Gender differences in the importance of work and family roles: Implications for work\u2013family conflict. Sex Roles, 47, 531\u2013541.CrossRef\n\nCosta Jr, P., Terracciano, A., & McCrae, R. R. (2001). Gender differences in personality traits across cultures: Robust and surprising findings. Journal of Personality and Social Psychology, 81, 322\u2013331.PubMedCrossRef\n\nCulbertson, S. S., Mills, M. J., & Fullagar, C. J. (2012). Work engagement and work\u2013family facilitation: Making homes happier through positive affective spillover. Human Relations, 65, 1155\u20131177.CrossRef\n\nDemerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The job demands-resources model of burnout. Journal of Applied Psychology, 86, 499\u2013512.PubMedCrossRef\n\nDeNavas-Walt, C., Proctor, B. D., Smith, J. C. (2013). Income, poverty, and health insurance coverage in the United States: 2012. Resource document. U.S. Census Bureau. www.\u200bcensus.\u200bgov\/\u200bprod\/\u200b2013pubs\/\u200bp60-245.\u200bpdf.\n\nDenson, T. F., Spanovic, M., & Miller, N. (2009). Cognitive appraisals and emotions predict cortisol and immune responses: A meta-analysis of acute laboratory social stressors and emotion inductions. Psychological Bulletin, 135, 823\u2013953.PubMedCrossRef\n\nErickson, R. J. (1993). Reconceptualizing family work: The effect of emotion work on perceptions of marital quality. Journal of Marriage and Family, 55, 888\u2013900.CrossRef\n\nErickson, R. J. (2005). Why emotion work matters: Sex, gender and the division of household labor. Journal of Marriage and Family, 67, 337\u2013351.CrossRef\n\nEvans, P., & Bartolom\u00e9, F. (1984). The changing pictures of the relationship between career and family. Journal of Organizational Behavior, 5, 9\u201321.CrossRef\n\nFolkman, S., & Lazarus, R. S. (1980). An analysis of coping in a middle-aged community sample. Journal of Health and Social Behavior, 21, 219\u2013239.PubMedCrossRef\n\nFrankenhauser, M., Lundberg, U., Fredrikson, M., Melin, B., Tuomisto. M., Myrsten, A. L., et al. (1989). Stress on and off the job as related to sex and occupational status in white-collar workers. Journal of Organizational Behavior, 10, 321\u2013346.CrossRef\n\nFujita, F., Diener, E., & Sandvik, E. (1991). Gender differences in negative affect and well-being: The case for emotional intensity. Journal of Personality and Social Psychology, 61, 427\u2013434.PubMedCrossRef\n\nGable, S. L., Reis, H. T., Impett, E. A., & Asher, E. R. (2004). What do you do when things go right? The intrapersonal and interpersonal benefits of sharing positive events. Journal of Personality and Social Psychology, 87, 228\u2013245.PubMedCrossRef\n\nGrandey, A. A., Tam, A., & Brauburger, A. (2002). Affective states and traits in the workplace: Diary and survey data from young workers. Motivation and Emotion, 26, 31\u201355.CrossRef\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGrzywacz, J. G., & Marks, N. F. (2000). Reconceptualizing the work\u2013family interface: An ecological perspective on the correlates of positive and negative spillover between work and family. Journal of Occupational Health Psychology, 5, 111\u2013126.PubMedCrossRef\n\nGuy, M., & Newman, M. A. (2004). Women jobs, men's jobs: Sex segregation and emotional labor. Public Administration Review, 64, 289\u2013298.CrossRef\n\nHaggard, D. L., Robert, C., & Rose, A. J. (2011). Co-rumination in the workplace: Adjustment trade-offs for men and women who engage in excessive discussions of workplace problems. Journal of Business and Psychology, 26, 27\u201340.CrossRef\n\nHeller, D., Watson, D., & Ilies, R. (2004). The role of person versus situation in life satisfaction: A critical examination. Psychological Bulletin, 130, 574\u2013600.PubMedCrossRef\n\nHobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. American Psychologist, 44, 513\u2013524.PubMedCrossRef\n\nHochschild, A. R. (1979). Emotion work, feeling rules, and social structure. American Journal of Sociology, 85, 551\u2013575.CrossRef\n\nHochschild, A., & Machung, A. (1989). The second shift: Working parents and the revolution at home. New York: Viking.\n\nHolahan, C. J., & Moos, R. H. (1991). Life stressors, personal and social resources, and depression: A 4-year structural model. Journal of Abnormal Psychology, 100, 31\u201338.PubMedCrossRef\n\nIlies, R., Wilson, K. S., & Wagner, D. T. (2009). The spillover of daily job satisfaction onto employees' family lives: The facilitating role of work\u2013family integration. Academy of Management Journal, 52, 87\u2013102.CrossRef\n\nJudge, T. A., & Ilies, R. (2004). Affect and job satisfaction: A study of their relationship at work and at home. Journal of Applied Psychology, 89, 661\u2013673.PubMedCrossRef\n\nJudge, T. A., Ilies, R., & Scott, B. A. (2006). Work\u2013family conflict and emotions: Effects at work and at home. Personnel Psychology, 59, 779\u2013814.CrossRef\n\nKendler, K. S., Thornton, L. M., & Prescott, C. A. (2001). Gender differences in the rates of exposure to stressful life events and sensitivity to their depressogenic effects. American Journal of Psychiatry, 158, 587\u2013593.PubMedCrossRef\n\nLambert, N. M., Gwinn, A. M., Baumeister, R. F., Strachman, A., Washburn, I. J., Gable, S. L., et al. (2013). A boost of positive affect: The perks of sharing positive experiences. Journal of Social and Personal Relationships, 30, 24\u201343.CrossRef\n\nLangston, C. A. (1994). Capitalizing on and coping with daily-life events: Expressive responses to positive events. Journal of Personality and Social Psychology, 67, 1112\u20131125.CrossRef\n\nLarson, R. W., & Almeida, D. M. (1999). Emotional transmission in the daily lives of families: A new paradigm for studying family process. Journal of Marriage and Family, 61, 5\u201320.CrossRef\n\nLazarus, R. S. (1991). Progress on a cognitive-motivational-relational theory of emotion. American Psychologist, 46, 819\u2013834.PubMedCrossRef\n\nLazarus, R. S. (1999) Stress and emotion: A new synthesis. New York: Springer.\n\nLazarus, R. S. (2006). Emotions and interpersonal relationships: Toward a person-centered conceputalization of emotions and coping. Journal of Personality, 74, 9\u201346.PubMedCrossRef\n\nLazarus, R. S., & Folkman, S. (1984). Stress, appraisal, and coping. New York: Springer.\n\nLevenson, R.W., & Gottman, J. M. (1985). Physiological and affective predictors of change in relationship satisfaction. Journal of Personality and Social Psychology, 49, 85\u201394.PubMedCrossRef\n\nMatjasko, J. L., & Feldman, A. F. (2006). Bringing work home: The emotional experiences of mothers and fathers. Journal of Family Psychology, 20, 47\u201355.PubMedCrossRef\n\nMcElwain, A. K., Korabik, K., & Rosin, H. M. (2005). An examination of gender differences in work\u2013family conflict. Canadian Journal of Behavioural Science, 37, 283\u2013298.CrossRef\n\nNolen-Hoeksema, S. (1987). Sex differences in unipolar depression: Evidence and theory. Psychological Bulletin, 101, 259\u2013282.PubMedCrossRef\n\nO'Brien, T. B., & DeLongis, A. (1996). The interactional context of problem, emotion- and relationship-focused coping: The role of the big five personality factors. Journal of Personality, 64, 775\u2013813.CrossRef\n\nPleck, J. H. (1977). The work\u2013family role system. Social Problems, 24, 417\u2013427.CrossRef\n\nPollard, T., Ungpakorn, G., Harrison, G., & Parkes, K. (1996). Epinephrine and cortisol responses to work. Annals of Behavioral Medicine, 18, 229\u2013237.PubMedCrossRef\n\nPowell, G. N., & Greenhaus, J. H. (2010). Sex, gender, and the work-to-family interface: Exploring negative and positive interdependencies. Academy of Management Journal, 53, 513\u2013534.CrossRef\n\nProulx, C. M., Helms, H. M., & Buehler, C. (2007). Marital quality and personal well-being: A meta-analysis. Journal of Marriage and Family, 69, 576\u2013593.CrossRef\n\nRepetti, R. L. (1989). Effects of daily workload on subsequent behavior during marital interaction: The roles of social withdrawal and spouse support. Journal of Personality and Social Psychology, 57, 651\u2013659.PubMedCrossRef\n\nRepetti, R. L. (1992) Social withdrawal as a short-term coping response to daily stressors. In H. S. Friedman (Ed.), Hostility, coping, and health (pp. 151\u2013165). Washington, D.C.: American Psychological Association.CrossRef\n\nRepetti, R. L., & Wood, J. (1997). Effects of daily stress at work on mothers' interactions with preschoolers. Journal of Family Psychology, 11, 90\u2013108.CrossRef\n\nRotundo, M., Nguyen, D., & Sackett, P. (2001). A meta-analytic review of gender differences in perceptions of sexual harassment. Journal of Applied Psychology, 86, 914\u2013922.PubMedCrossRef\n\nRosenkrantz, P., Vogel, S., Bee, H., Broverman, I., & Broverman, D. M. (1968). Sex-role stereotypes and self-concepts in college students. Journal of Consulting and Clinical Psychology, 32, 287\u2013295.PubMedCrossRef\n\nSchulz, M. S., Cowan, P. A., Cowan, C., & Brennan, R. T. (2004). Coming home upset: Gender, marital satisfaction, and the daily spillover of workday experience into couple interactions. Journal of Family Psychology, 18, 250\u2013263.PubMedCrossRef\n\nScott, B. A., & Barnes, C. M. (2011). A multilevel field investigation of emotional labor, affect, work withdrawal, and gender. Academy of Management Journal, 54, 116\u2013136.CrossRef\n\nShumaker, S. A., & Hill, D. R. (1991). Gender differences in social support and physical health. Health Psychology, 10, 102\u2013111.PubMedCrossRef\n\nSong, Z., Foo, M. D., & Uy, M. A. (2008). Mood spillover and crossover among dual-earner couples: A cell phone event sampling study. Journal of Applied Psychology, 93, 443\u2013452.PubMedCrossRef\n\nSonnentag, S., & Binnewies, C. (2013). Daily affect spillover from work to home: Detachment from work and sleep as moderators. Journal of Vocational Behavior, 83, 198\u2013208.CrossRef\n\nSonnentag, S., & Grant, A. M. (2012). Doing good at work feels good at home, but not right away: When and why perceived prosocial impact predicts positive affect. Personnel Psychology, 65, 495\u2013530.CrossRef\n\nStaines, G. L. (1980). Spillover versus compensation: A review of the literature on the relationship between work and nonwork. Human Relations, 33, 111\u2013129.CrossRef\n\nStory, L. B., & Repetti, R. (2006). Daily occupational stressors and marital behavior. Journal of Family Psychology, 20, 690\u2013700.PubMedCrossRef\n\nTaylor, S. E., Klein, L. C., Lewis, B. P., Gruenewald, T. L., Gurung, R. A., & Updegraff, J. A. (2000). Biobehavioral responses to stress in females: Tend-and-befriend, not fight-or-flight. Psychological Review, 107, 411\u2013429.PubMedCrossRef\n\nThompson, C. A., Beauvais, L. L., & Lyness, K. S. (1999). When work\u2013family benefits are not enough: The influence of work\u2013family culture on benefit utilization, organizational attachment, and work\u2013family conflict. Journal of Vocational Behavior, 54, 392\u2013415.CrossRef\n\nvan Steenbergen, E. F., Ellemers, N., & Mooijaart, A. (2007). How work and family can facilitate each other: Distinct types of work\u2013family facilitation and outcomes for women and men. Journal of Occupational Health Psychology, 12, 279\u2013300.PubMedCrossRef\n\nVermeulen, M., & Mustard, C. (2000). Gender differences in job strain, social support at work, and psychological distress. Journal of Occupational Health Psychology, 5, 428\u2013440.PubMedCrossRef\n\nVogel, D. L., Wester, S. R., Heesacker, M., & Madon, S. (2003). Confirming gender stereotypes: A social role perspective. Sex Roles, 48, 519\u2013528.CrossRef\n\nVoydanoff, P. (2004). The effects of work demands and resources on work-to-family conflict and facilitation. Journal of Marriage and Family, 66, 398\u2013412.CrossRef\n\nWang, S. W., Repetti, R. L., & Campos, B. (2011). Job stress and family social behavior: The moderating role of neuroticism. Journal of Occupational Health Psychology, 16, 441\u2013456.PubMedCrossRef\n\nWatson, D., & Clark, L. A. (1984). Negative affectivity: The disposition to experience aversive emotional states. Psychological Bulletin, 96, 465\u2013490.PubMedCrossRef\n\nWeiss, H. M., & Cropanzano, R. (1996). Affective Events Theory: A theoretical discussion of the structure, causes and consequences of affective experiences at work. In B. M. Staw & L. L. Cummings (Eds.), Research in organizational behavior: An annual series of analytical essays and critical review (Vol. 18, pp. 1\u201374). New York: JAI.\n\nWharton, A. S., & Erickson, R. I. (1993). Managing emotions on the job and at home: Understanding the consequences of multiple emotional roles. Academy of Management Review, 18, 457\u2013486.\n\nWiese, B. S., Seiger, C. P., Schmid, C. M., & Freund, A. M. (2010). Beyond conflict: Functional facets of the work\u2013family interplay. Journal of Vocational Behavior, 77, 104\u2013117.CrossRef\n\nWilliams, K. J., & Alliger, G. (1994). Role stressors, mood spillover, and perceptions of work\u2013family conflict in employed parents. Academy of Management Journal, 37, 837\u2013868.CrossRef\n\nWilliams, K. J., Suls, J., Alliger, G. M., Learner, S. M., & Wan, C. K. (1991). Multiple role juggling and daily mood states in working mothers: An experience sampling study. Journal of Applied Psychology, 76, 664\u2013674.PubMedCrossRef\n\nWood, W., & Eagly, A. H. (2002). A cross-cultural analysis of the behavior of women and men: Implications for the origins of sex differences. Psychological Bulletin, 128, 699\u2013727.PubMedCrossRef\n\nYanchus, N. J., Eby, L. T., Lance, C. E., & Drollinger, S. (2010). The impact of emotional labor on work\u2013family outcomes. Journal of Vocational Behavior, 76, 105\u2013117.CrossRef\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_8\n\n# 8. The Intersection of Gender and Work\u2013Family Guilt\n\nKaren Korabik1\n\n(1)\n\nCentre for Families, Work and Well-Being, University of Guelph, N1G2W1 Guelph, ON, Canada\n\nKaren Korabik\n\nEmail: kkorabik@uoguelph.ca\n\nKeywords\n\nGenderWork\u2013familyGuiltEmotionFaultResponsibilityBlameAttitudeMeasurement\n\n## 8.1 The Problem\n\n> I always feel as though I am failing in some way, as though I am cheating my children, my husband, and myself. The guilt is very difficult to deal with. (McElwain 2008)\n\nAs this quote from a working mother illustrates, work\u2013family (W\u2013F) guilt is not only pervasive, but it can also be extremely detrimental to the well-being of workers and their families. As illustrated by the following quote, there is a common perception that women are more likely to suffer from W\u2013F guilt than men.\n\n> I do think men and women experience guilt differently. I believe...women feel more guilt about the work\/family balance\u2014perhaps it's innate...perhaps it's because of the traditional roles men and women have had in the home. I know my husband is sad that he doesn't get to spend as much time with his family as he would like, but I don't think he struggles with the same degree of guilt. (McElwain 2008)\n\nGuilt arising from attempting to balance work and family has been a frequent topic of interest in the media and popular press (Bort et al. 2005; Chapman 1987), as well as in the organizational behavior literature. Despite this, until recently, research on W\u2013F guilt has been very limited (Seagram and Daniluk 2002). In this chapter, I review the empirical evidence pertaining to the intersection of gender and W\u2013F guilt.\n\n## 8.2 Definition and Conceptualizations of W\u2013F Guilt\n\nMost theory and research on guilt has focused on guilt in general rather than on guilt as it applies specifically to the W\u2013F interface. In the general guilt literature, guilt is viewed as a negative emotion that arises when individuals violate their internalized standards about what constitutes proper behavior (Kubany 1994). Thus, when individuals believe they should have thought, felt, or acted differently, it can result in feelings of guilt (Kubany 1994; Zahn-Waxler et al. 1990). Guilt has been conceptualized as consisting of a cognitive component, which consists of the recognition that one has harmed another; an affective component, which refers to the unpleasant feelings that are experienced; and a motivational component, which is the desire to undo the harm that has been caused (Hoffman 1982). Kubany et al. (1996) propose that more guilt will be experienced to the extent that individuals act in a way that violates their values, feel their actions are unjustified, feel responsible for what happened, and\/or believe that they could have foreseen and prevented the outcome.\n\nMany different definitions of W\u2013F guilt can be found in the literature (McElwain and Korabik 2004). Most of them emphasize one or more of the aspects of general guilt discussed above. Thus, W\u2013F guilt is often seen as resulting from having to make a choice between work and family (Conlin 2000; Pollock 1997), allowing work to interfere with family life (Glavin et al. 2011), or failing to adequately balance work and family roles (Napholz 2000). Similarly, W\u2013F guilt has been defined as a discrepancy between one's preferred and actual level of role participation at home versus at work (Hochwarter et al. 2007). Another view is that W\u2013F guilt arises from the perceived failure to adequately fulfill prescribed gender-role norms (Livingston and Judge 2008; Simon 1995). Finally, some authors have speculated that W\u2013F guilt stems from attempts to deal with the double standards that are placed on women as compared to men (Banarjee 2003; Bui 1999).\n\n## 8.3 The Measurement of W\u2013F Guilt\n\nResearch into W\u2013F guilt has been hampered by a lack of reliable and valid measurement instruments (McElwain and Korabik 2004). As a result, many studies in this area have been qualitative in nature. Some quantitative investigations exist, but mostly these have relied on single-item (e.g., \"In the past seven days, how many days have you felt guilty?\") or multi-item measures of general guilt.\n\nRecent meta-analytic results have demonstrated that W\u2013F-specific support constructs are more strongly related to W\u2013F conflict than are the more general constructs of supervisor support and perceived organizational support (Kossek et al. 2011). It is likely that this is also the case for guilt. However, hardly any measures of guilt specific to the W\u2013F context exist and many of these consist of single items (e.g., \"I feel guilty that I don't spend enough time with my family\").\n\nAmong the few multi-item measures that are specific to W\u2013F guilt is the Feelings of Guilt about Parenting Scale (Martinez et al. 2011). This 14-item measure was developed and validated in Spain. It assesses situations that could evoke guilt in employed parents (e.g., \"Playing with my child for less time than I would like\") on a scale ranging from 1 = not at all guilty to 4 = very guilty. The Spanish version of the scale has excellent internal consistency reliability, and confirmatory factor analysis has indicated that it appears to be unidimensional.\n\nAnother measure is an employment-related guilt scale constructed based on the results of interviews and focus groups conducted in Turkey (Aycan and Eskin 2005). The items (e.g., \"I feel guilty for not being able to spend as much time as I wish with my children\") are rated on a scale ranging from 1 = strongly disagree to 5 = strongly agree. The internal consistency reliability of the English version is excellent. A similar employment-related guilt scale (Hochwarter et al. 2007) has three items (e.g., \"I feel guilty about the time that I am unable to spend with my family due to work\") that are rated on a scale ranging from 1 = strongly disagree to 5 = strongly agree. The measure has good internal consistency reliability.\n\nThe W\u2013F guilt scale (WFGS; McElwain 2008; McElwain et al. 2005a) consists of seven items. It differs from the previously discussed measures in that, analogous to the W\u2013F conflict literature, W\u2013F guilt is viewed as being bidirectional. Thus, four of the items assess work interference with family guilt (e.g., \"I regret not being around for my family as much as I would like to\") and three assess family interference with work guilt (e.g., \"I feel bad because I frequently have to take time away from work to deal with issues happening at home\"). The items are rated on a scale ranging from 1 = strongly disagree to 6 = strongly agree. The measure and its subscales have excellent internal consistency reliability and very good test-retest reliability over a 3-month interval. A confirmatory factor analysis established the existence of a two-factor structure with work interference with family guilt (WIFG) and family interference with work guilt (FIWG) as separate dimensions. The WFGS has excellent convergent and discriminant validity. The scale was included in the survey for Project 3535 1. Measurement equivalence for culture was established across all ten countries (Korabik and van Rhijn 2014). Measurement equivalence for gender has also been found for the Canadian subsample, indicating that men and women attribute the same meaning to the scale items (McElwain 2008).\n\nMcElwain (2008) also created a 24-item faceted version of the W\u2013F guilt scale (WFGS-R). It has six subscales assessing physical, emotional, and psychological WIFG and FIWG . Physical W\u2013F guilt refers to guilt from one's inability to be physically present to attend to both work and family duties, e.g., \"I regret missing family (work) events because of work (family) responsibilities\". Emotional W\u2013F guilt refers to the negative feelings experienced due to W\u2013F conflicts, e.g., \"I regret when I take out my frustrations from my work (family) on my family (at work)\". Psychological W\u2013F guilt refers to the psychological spillover from one role to the other, e.g., \"I feel guilty for having my family (work) on my mind while at work (spending time with my family)\". Items are rated on a scale ranging from 1 = never to 7 = always. Thus far, the WFGS-R has only been validated on a sample of women. Despite this, the results look very promising. The internal consistency reliability was excellent. A confirmatory factor analysis verified a structure with two higher order factors (WIFG and FIWG), each with three lower order factors (physical, emotional, and psychological guilt). The measure also has excellent content, convergent, and discriminant validity in the contexts in which it was evaluated (McElwain 2008). There is a high positive correlation between the WIFG subscale from the WFGS-R and the WIFG subscale from the WFGS; however, there is only a moderate positive correlation between the FIWG subscale from the WFGS-R and the FIWG subscale from the WFGS (McElwain 2008).\n\n## 8.4 Guilt and the W\u2013F Interface\n\nSeveral studies have shown that W\u2013F conflict and W\u2013F guilt are interrelated. Aycan and Eskin (2005) found that work interference with family (WIF), but not family interference with work (FIW), was positively correlated with employment-related guilt. Similarly, data from Canada indicate that for both the WFGS and the WFGS-R, significant positive correlations exist between the WIF and WIFG , as well as between the FIW and FIWG subscales . Despite this, scores on W\u2013F guilt do not correlate so highly with those on W\u2013F conflict as to indicate redundancy between these constructs (Korabik and Lero 2004; McElwain 2008; McElwain et al. 2005a, b). As this research is correlational in nature, however, it cannot be determined whether W\u2013F guilt is an antecedent or an outcome of W\u2013F conflict.\n\nEmployment-related guilt has been found to be associated with a variety of negative consequences including time inflexibility, depression , and lower satisfaction with life, organizational policies, parenthood , and time spent with children (Aycan and Eskin 2005). Hochwarter et al. (2007) examined whether the ability to manage resources at work could enhance personal control and help to reduce the negative effects of work-induced guilt. They conducted two studies using business school students and public employees as participants, respectively. They found that work-induced guilt had detrimental effects on job and life satisfaction when individuals did not have the ability to manage resources. However, the unfavorable effects of work-related guilt on job and life satisfaction were neutralized when there was an ability to manage resources.\n\n## 8.5 Physical Gender and W\u2013F Guilt\n\nGender is a multidimensional construct consisting of physical and demographic gender,2 as well as a range of socialized gender-role characteristics such as gender-role orientation , attitudes, and behaviors (Korabik et al. 2008b). The vast majority of studies into the effects of gender on the W\u2013F interface have focused solely on physical or demographic gender.\n\nIn terms of general guilt proneness, gender differences in guilt have been found as early as 33 months. Zahn-Waxler and Kochanska (1988) reported that preschool-aged girls exhibited guilt-related behaviors when they either observed or committed a transgression. Furthermore, girls have been found to be more affected by their wrongdoings than boys (Kochanska et al. 2002). Many other studies have indicated that general guilt levels are higher in women than men (e.g., Kubany and Watson 2003). It is believed that this may stem from the stereotype that women are more interpersonally sensitive than men (Zahn-Waxler et al. 1990), making them more vulnerable to guilt feelings (Zahn-Waxler and Kochanska 1988).\n\n### 8.5.1 Qualitative Research\n\nMost of the research on guilt as it specifically applies to the W\u2013F interface has been qualitative in nature, and much of it has employed samples consisting solely of women. For example, Napholz (2000) interviewed eight employed Native American women. They were asked to offer explanations for the feelings of guilt that arose due to their multiple roles. One participant stated that women felt obligated to fix everything and felt guilty when they couldn't handle all of the demands placed on them. Another respondent said that guilt had driven her to spend more time with her family because she constantly felt like she should be doing more. She admitted to being so preoccupied with guilt that she was unable to properly complete the tasks on which she was working. Some of the women tried to make amends for the guilt by being a \"cool\" mother or by spending more time with their children.\n\nIn another study, Elvin-Nowak (1999) examined W\u2013F guilt in 13 working mothers in Sweden. Participants' feelings of guilt stemmed from their perceived failure in responsibility toward caring for others. This was due primarily to their lack of control over circumstances which resulted in their inability to balance the demands from their different life spheres. While the mothers felt the most guilt in regard to their children, they also expressed guilt feelings toward their husbands, parents, friends, and coworkers. For example, because these women felt that they should be responsible for mothering their children, they felt guilty when delegating this to someone else (e.g., a babysitter). The women also reported experiencing aggression and anger in reaction to the burden of their guilt feelings. They tried to alleviate the guilt by devising strategies that would allow them to justify putting their own needs first.\n\nGuendouzi (2006) investigated women teachers in the UK, finding that W\u2013F guilt stemmed from trying to achieve an ideal balance between personal and social needs. In addition, she found that guilt resulted from the social pressure placed on mothers to be constantly available and accessible (i.e., the intensive mothering norm). On a related note, Seagram and Daniluk (2002) studied maternal guilt in eight mothers of preadolescent children. They found that the mothers felt a connection or bond with their children and a sense of responsibility for their children's well-being . This manifested itself both in feeling that they needed to prepare their children for life's challenges and in fearing their children might come to harm. The participants' maternal guilt resulted in a sense of inadequacy and emotional depletion (e.g., feelings of anger, frustration, exhaustion, and resentment) .\n\nOne component of Project 3535 was to collect qualitative data. The findings revealed that women from a variety of countries mentioned W\u2013F guilt during focus group discussions about W\u2013F balance. These included Australia (Bardoel 2004), Canada (Korabik and Lero 2004; McElwain et al. 2005a), India (Desai and Rajadhyaksha 2004), Indonesia (Mawardi 2004), Israel (Drach-Zahavy and Somech 2004), Taiwan (Huang 2004), and the USA (Velgach et al. 2005). Although there were many similarities, women in different countries tended to emphasize somewhat different themes when speaking about what made them feel guilty. Women in Australia and India were most likely to report experiencing guilt due to their inability to be superwomen (Bardoel 2004; Korabik 2005). Women in the USA mentioned feeling guilty about having to put their jobs before their families and their inability to be in two places at one time (Velgach et al. 2005). By contrast, women in India appeared to experience guilt more when they ignored the academic achievement of their children (Desai and Rajadhyaksha 2004). Women in Indonesia, Taiwan, and the Arab women in Israel spoke about feeling guilty for not fulfilling their traditional gender roles (Korabik 2005). Four of the seven Jewish Israeli women, however, felt that they had moved from guilt to positive spillover as a function of their life stage (Drach-Zahavy and Somech 2004).\n\nMore recently, there has been recognition that fathers are not immune from experiencing guilt due to having to deal with the stresses of balancing work and family life (Daly 2001; Martinez et al. 2011). As a result, some qualitative studies have included both men and women as participants. My colleagues and I obtained data from online focus groups with both male and female parents employed by Canadian organizations (Korabik and Lero 2004; Korabik et al. 2007; McElwain et al. 2005a, 2007). When asked if they felt W\u2013F guilt, all participants except one man overwhelmingly indicated that this was a common occurrence. The majority of both men and women said that although they felt guilt both toward balancing their work roles and their family roles, the guilt was strongest in regard to their family responsibilities. Moreover, respondents reported feeling more guilt about their children than their spouses\/partners due to a greater sense of responsibility regarding the well-being of their children. Respondents also reported experiencing guilt when their coworkers made them feel that they were not \"pulling their weight\" at work.\n\nThe men and the women were very similar to one another both in the extent to which they admitted experiencing W\u2013F guilt and in the things that made them feel guilty. Despite this, most participants, both men and women alike, believed that there were gender differences in W\u2013F guilt, such that women were more prone to feelings of guilt than men. Some respondents felt that men and women experienced W\u2013F guilt differently either because women were more emotionally sensitive than men or because women were more able to verbalize their feelings than men were. Other participants felt that these gender differences stemmed from societal expectations that men and women should fulfill traditionally prescribed gender roles. Both men and women felt that higher expectations were put on women than on men. Although some men realized that they were currently \"getting off easy\" compared to women, they did worry that they might have to pay a price later in life for their current neglect of their families.\n\nIn another study from Canada, Daly (2001) investigated men and women in both dual-earner and single-parent families. The participants reported feeling guilty about what they were unable to do. The contradiction between their ideal aspirations and the reality of their lives resulted in chronic guilt. More specifically, both women and men felt guilty for working too much, not spending enough time with their children, leaving their children with babysitters, and taking time for themselves. Many had given up on trying to rid themselves of the guilt and had focused on how to live with it.\n\nSimon (1995) compared 40 men and women in dual-earner couples who were employed full-time and had a child under the age of 18 living at home. Eighty-five percent of the women reported feeling guilty because their work took time away from their families and made them feel like they were neglecting their children and spouses. Men, on the other hand, did not report feeling guilty or a feeling of being pulled in different directions. Men's guilt appeared to stem more from their inability to fulfill their breadwinner role.\n\nIn summary, it appears that in many countries around the world women report experiencing W\u2013F guilt. Although there are some cultural differences , their stories have many similar themes. There also seem to be few differences between the extent to which men and women report feeling guilty. Despite this, both men and women hold the stereotype that women are more prone to W\u2013F guilt than are men.\n\n### 8.5.2 Quantitative Research\n\nThe quantitative research on physical\/demographic gender and W\u2013F guilt is still quite sparse. In an early study with a sample of all women, Nevill and Damico (1977) found that those between the ages of 25 and 39 reported significantly higher levels of guilt due to the stress experienced from competing role demands than women at other ages. They suggested that because older women had older children who required less supervision, they were more able to pursue their individual goals without feeling as though they were doing so at the expense of others.\n\n#### 8.5.2.1 Mean Gender Differences in W\u2013F Guilt\n\nAmong the few studies that have examined whether W\u2013F guilt is more prevalent in men or women, the results have been mixed. Aycan and Eskin (2005) found that women reported significantly higher levels of employment-related guilt than did men. However, there was still a significant relationship between W\u2013F conflict and guilt for men. By contrast, Hochwarter et al. (2007) found no correlation between gender and employment-related guilt in either of their samples. Martinez et al. (2011) found that mothers and fathers of preschoolers in Spain were similar in reporting high levels of guilt on the Guilt About Parenting Scale. Both mothers and fathers felt guilty that they could not pay as much attention to their children as they wanted and because they had to delegate some parenting tasks to others. However, mothers' guilt was more related to their fear that they were not being a \"good\" mother, whereas fathers' guilt was more related to their conflict between their desire to be involved with their children and their need to be breadwinners.\n\nData on the WFGS from Project 3535 indicated no significant overall main effect for gender or gender by country interaction for either WIFG or FIWG . However, in the Canadian subsample men scored significantly higher than women on WIFG (McElwain 2008), whereas in the Israeli subsample women scored significantly higher than men on FIWG.\n\n#### 8.5.2.2 Gender Differences in How W\u2013F Guilt Relates to Antecedents and Outcomes\n\nMy colleagues and I conducted research examining the antecedents and outcomes of W\u2013F guilt (Korabik and McElwain 2011; Korabik et al. 2009). We collected data from two samples of employed parents in Canada. The first consisted of 171 employed women who completed both the WFGS and the WFGS-R along with measures of antecedent (i.e., demands, W\u2013F conflict) and outcome (e.g., satisfaction, turnover intent) variables on one occasion. The second sample consisted of 264 men and 180 women who were employed and who were married\/cohabiting parents. They completed the WFGS and measures of the antecedent and outcome variables. Data on the outcome variables were also collected three months later from a subsample of this group (122 men and 138 women).\n\nWe carried out a variety of structural equation modeling analyses on the data to try to understand what was driving the effects (i.e., the gender composition of the sample, the version of W\u2013F guilt scale used, or use of a cross sectional versus prospective design). The results can be found in Table 8.1. Column 1 displays the effects for the all-women sample on the WFGS-R using a cross-sectional design, whereas Column 2 displays the effects for the same sample and design, but using the WFGS as the measure. Column 3 displays the effects for the mixed gender sample (N = 444) on the WFGS using a cross-sectional design. Column 4 displays the effects for the mixed gender subsample (N = 260) on the WFGS using a prospective design.\n\nTable 8.1\n\nResults of analyses with different samples, measures, and designs | Results of analyses\n\n---|---\n\nRelationships among variables | 1 | 2 | 3 | 4\n\nJob demandsa \u2192 WIF conflict | + | + | + | +\n\nFamily demandsa \u2192 FIW conflict | + | + | + | +\n\nWIF conflict \u2192 WIFG | + | + | + | +\n\nFIW conflict \u2192 FIWG | + | + | + | +\n\nWIF conflict \u2192 Job satisfaction | \u2212 | \u2212 | \u2212 | \u2212\n\nWIF conflict \u2192 Turnover intent | + | + | + | +\n\nFIW conflict \u2192 Family satisfaction | \u2212 | \u2212 | \u2212 | \u2212\n\nWIFG \u2192 Job satisfaction | \u2212 | \u2212 | \u2212 | ns\n\nWIFG \u2192 turnover intent | + | + | + | ns\n\nFIWG \u2192 Family satisfaction | ns | ns | ns | ns\n\nJob satisfaction \u2192 Life satisfaction | + | + | + | +\n\nFamily satisfaction \u2192 Life satisfaction | + | + | + | +\n\nWIF conflict \u2192 Psychological distress | N\/A | N\/A | + | +\n\nFIW conflict \u2192 Psychological distress N\/A | N\/A | + | +\n\n|\n\nWIFG \u2192 Psychological distress | N\/A | N\/A | + | +\n\nFIWG \u2192 Psychological distress | N\/A | N\/A | ns | ns\n\naJob and family demands were used as predictors only in models where turnover intent was the outcome variable; Column 1 = 171 women, WFGS-R, cross-sectional design; Column 2 = 171 women, WFGS, cross-sectional design; Column 3 = 448 men and women, WFGS, cross-sectional design; Column 4 = 360 men and women, WFGS, prospective design\n\nAs can be seen, except for the fact that higher FIWG did not significantly predict a lack of family satisfaction or greater psychological distress, most of the relationships were significant in the expected direction. Moreover, most of the relationships were consistent across samples, measures, and designs. The exception was that higher WIFG did not predict lower job satisfaction or higher turnover intentions when a prospective design was used. This pattern of effects indicates that the gender composition of the sample (all women versus mixed gender) did not have an impact on the results. This may have been due to the fact that many extraneous factors that often produce spurious gender effects had been controlled (Korabik et al. 2008b). For example, in the mixed gender sample, the men and women were similar to one another in that they were all employed parents who were married\/cohabiting. However, although the results appear similar for men and women, additional analyses on the mixed gender sample data specifically testing whether the men's and women's models differ significantly from one another are necessary before firm conclusions can be drawn regarding a lack of gender differences.\n\n#### 8.5.2.3 Gender in Context\n\nShields (2013) has argued that we need to move away from studying whether men and women are different from one another and toward studying gender in context or what factors accentuate or diminish gender differences. In this regard, Glavin et al. (2011) examined the impact of work interruptions outside of work hours. In their large sample of employed American adults, they found that women reported higher levels of general guilt than men. Moreover, among women, but not men, the frequency of work interruptions outside of work hours was positively correlated with guilt, and guilt mediated the association between work contact outside of work hours and psychological distress even after W\u2013F conflict was controlled. In a similar study, Offer and Schneider (2011) examined mothers and fathers in dual-earner families in the USA. They found no gender differences on their single-item family time guilt measure. However, for mothers, but not fathers, greater frequency of W\u2013F multitasking at work and in public places was associated with more family time guilt. Dealing with work interruptions outside of work hours and engaging in W\u2013F multitasking are circumstances that are characterized by low control coupled with high demands\/overload. In the literature on W\u2013F guilt, issues related to demands, overload , and control have often been mentioned as being important (Aycan and Eskin 2005; Elvin-Nowak 1999; Hochwarter et al. 2007; Napholz 2000). Thus, the relationship between gender and W\u2013F guilt may be impacted by contextual variables such as overload and control.\n\nTo address this possibility, my colleagues and I used the WFGS with samples of employees from the USA (Ishaya et al. 2013) and Canada (Ewles et al. 2013). We examined the effects of gender, work and family overload, and job and family control on WIFG and FIWG , respectively. For both men and women in Canada, higher work overload predicted higher WIFG. However, in the USA, WIF moderated this relationship. A lack of job control was predictive of higher WIFG for both genders in the USA, but was not a significant predictor in the Canadian sample. In both the USA and Canada, greater family overload was related to higher WIFG for both genders. A lack of family control predicted higher WIFG for both genders in the USA, but this held true only for women in Canada. Finally, in both the USA and Canada, higher family overload was associated with greater FIWG for men, but not for women. It is difficult to reconcile the disparate findings about how gender interacts with overload and control to impact W\u2013F guilt. This is because the various studies carried out thus far differ as to their samples, their designs, and the measures of W\u2013F guilt used.\n\nSocial support has been found to help alleviate the negative effects of W\u2013F conflict (Ayman and Antani 2008). Aycan and Eskin (2005) found that for women, but not men, supervisory support and emotional spousal support were associated with lower work-induced guilt. We (Ewles et al. 2014) used the WFGS to examine gender differences in how received social support from work and nonwork sources was related to W\u2013F guilt. Support from work-related sources (supervisors and coworkers) for work-related issues was not significantly related to WIFG , but it was associated with significantly lower FIWG for those of both genders. In terms of support for family-related issues, when supervisors provided support for household tasks, men were more likely than women to report higher WIFG. By contrast, the more the supervisors and coworkers provided encouragement and appreciation regarding the family, the lower the FIWG for both men and women. Greater support for work-related duties by spouses\/partners, neighbors, relatives, and friends was associated with greater WIFG for both genders. Conversely, for both men and women there was no significant association between the support received from nonwork sources for work-related issues and FIWG. However, when their parents\/in-laws provided support for both work and family issues, women were significantly more likely than men to report higher WIFG. Furthermore, when they received appreciation from their spouses\/partners regarding their work or family life or from their children regarding their family life, men reported significantly lower WIFG than women. Finally, for both genders, the more often children, neighbors, relatives, and friends listened to and discussed family-related problems, the lower the FIWG . These results indicated that the effects of gender and social support on W\u2013F guilt are complex and depend upon the source, domain, type of support, and the direction of W\u2013F guilt examined.\n\n## 8.6 Gender-Role Attitudes\/Ideology and W\u2013F Guilt\n\nGender-role ideology (GRI) refers to the attitudes or beliefs an individual holds about the proper roles of men and women in society. It is generally conceptualized as a unidimensional construct with traditional attitudes at one pole and egalitarian attitudes at the other. Chappell et al. (2005) studied a small sample of dual-earner parents from Canada using the WFGS. They found that men with egalitarian GRI reported experiencing more FIWG than traditional men. Egalitarian men also reported higher levels of FIWG than egalitarian women. No significant results were found for WIFG . The Project 3535 data, however, showed a different pattern of results. Those with egalitarian attitudes had lower WIFG than those with traditional attitudes in every country except China (where there was no significant difference) and Turkey (where traditionals had lower WIFG than egalitarians). For FIWG, those with egalitarian attitudes were also lower than those with traditional attitudes in every country except Spain (where there was no significant difference) and Turkey (where traditionals had lower FIWG than egalitarians). Over all countries, both men and women with egalitarian attitudes reported lower FIWG than those with traditional attitudes . However, for WIFG there was an interaction between gender and GRI such that egalitarian men scored higher than traditional men, whereas traditional women scored higher than egalitarian women.\n\nThe Project 3535 data from India, Indonesia, and Taiwan were examined by Rajadhyaksha et al. (2011). They established that their GRI scale had measurement equivalence for culture , and found no differences in the gender-role attitudes of men and women in each country. In structural equation models , GRI was treated as an antecedent variable that impacted WIFG via work overload and WIF and impacted FIWG via family overload and FIW. GRI predicted W\u2013F guilt in the same way in each country such that more traditional GRI was associated with higher WIFG and FIWG.\n\nLivingston and Judge (2008) looked at how traditional\/egalitarian attitudes and W\u2013F conflict were related to general guilt. They found that there was a stronger positive relationship between WIF and guilt for those with more egalitarian gender role attitudes than for those with more traditional attitudes. By contrast, there was a stronger positive relationship between FIW and guilt for those with traditional gender-role attitudes than for those with egalitarian gender-role attitudes. However, there was an interaction between gender and FIW. Traditional men reported the highest levels of guilt and this was exacerbated when FIW was high. By contrast, egalitarian men reported the lowest levels of guilt, and this was particularly so when FIW was high. Traditional and egalitarian women reported moderate levels of guilt.\n\n## 8.7 Gender-Role Orientation and W\u2013F Guilt\n\nGender-role orientation refers to those personality characteristics in the instrumental\/ agentic and expressive\/communal domains that are acquired through gender-role socialization . McElwain et al. (2004) used the WFGS to examine W\u2013F conflict and W\u2013F guilt with respect to gender-role orientation as assessed by the Extended Personal Attributes Questionnaire. Participants were dual-earner employed parents from Canada. W\u2013F guilt was significantly related only to the instrumentality aspect of gender-role orientation such that those lower in instrumentality (feminine and undifferentiated individuals) had higher levels of FIWG than those higher in instrumentality (masculine and androgynous individuals). There were no significant differences between those in the different gender role groups in their levels of WIFG.\n\n## 8.8 Summary and Critique of the Literature\n\nResearch on gender and W\u2013F guilt is still in its infancy. Despite a tremendous interest in the issue, empirical research on the subject is very limited. Mirroring the literature on gender and emotions (Shields 2013), the overall picture is currently one of few gender differences and many inconsistent results. Where differences have been found, results are often congruent with the stereotype of women as being more guilt-prone than men. What's more, such findings have been more frequent when global, nonspecific measures of self-reported guilt have been used and when women have been sampled from the general population regardless of their employment, marital, or parental status. This may be because under such circumstances societal gender stereotypes are more likely to influence self-construals to produce stereotyped gender differences on self-reports about emotions (Shields 2013).\n\nResearch in this area may also be characterized by a lack of meaningful and consistent results because it suffers from a number of other methodological limitations. First, many studies on W\u2013F guilt have employed qualitative methodologies, and most of these have used samples comprised solely of women. Not only can conclusions about gender differences not be drawn from such data, but an implicit assumption behind such research is that W\u2013F guilt is a woman's problem that does not pertain to men.\n\nSecond, when comparative research has been carried out, it has consisted almost exclusively of atheoretical studies of physical\/demographic gender. This is particularly problematic when it pertains to gender and emotions. If a theoretical reason for why gender differences exist is not provided, the categories of man\/woman can become reified and used as the explanation (Shields 2013). Moreover, this can produce conceptual confusion and result in physical gender being employed as a proxy for some other variable, something that introduces confounds into the research (Korabik et al. 2008b). For example, physical gender is frequently used as a proxy for gender role variables like gender-role orientation . In this case, findings regarding W\u2013F guilt can be mistakenly attributed to whether someone is a man or woman instead of to the degree of instrumentality and expressivity in their personality. Moreover, in our society, physical gender is a status marker. Because of this, when research fails to control for status-related variables (e.g., job level), differences in W\u2013F guilt may be mistakenly attributed to whether someone is a man or a woman rather than to gender-related status differences (Korabik et al. 2008b).\n\nA third problem is that W\u2013F guilt has often been assessed with one-item measures of unsupported reliability and validity or with measures of general guilt, which are only moderately correlated with W\u2013F specific measures like the WFGS (McElwain 2008). Fourth, a wide variety of different operationalizations, measures, samples, and designs have been used, making it very difficult to compare the results of different studies to one another.\n\n## 8.9 Future Directions\n\nClearly, much more research on the topic of gender and W\u2013F guilt is necessary. This should include studies that focus not only on physical gender, but also on gender role constructs. Researchers should be careful to articulate their theory about why they expect gender differences to exist and to use gender-related constructs that are appropriate to the underlying processes that they are studying (i.e., intrapsychic, interpersonal, or social structural) (Korabik et al. 2008b). In addition, more attention needs to be paid to making sure that confounding variables are controlled and to examining gender in context (Korabik et al. 2008b; Shields 2013).\n\nFuture research should employ multi-item bidirectional measures created specifically to assess W\u2013F guilt. For example, the WFGS has been shown to be psychometrically sound and to evidence measurement equivalence across a number of cultures . It is also essential to establish the measurement equivalence for gender of any W\u2013F guilt measure before going on to draw any conclusions regarding gender differences. If this is not done, any differences found may be due to the different meaning that men and women attribute to the scale items rather than to actual gender differences.\n\nThus far, most quantitative research on gender and W\u2013F guilt has been correlational in nature, making it impossible to draw strong inferences about causality. Furthermore, W\u2013F guilt has been variously conceptualized as an antecedent to W\u2013F conflict, as an outcome of W\u2013F conflict, and as a mediator of the relationships between WIF and FIW and outcomes. Longitudinal data is necessary to understand the place of W\u2013F guilt in the larger context of the W\u2013F interface.\n\n## 8.10 Conclusion\n\nW\u2013F guilt is frequently discussed in the literature on W\u2013F conflict as being an important phenomenon. Gender-related effects in this area appear to be complex and dependent upon participant characteristics (e.g., marital, parental, and employment status), culture , and the direction of W\u2013F guilt assessed (WIFG versus FIWG). In addition, it appears that variables such as the degree of control, overload , and social support in the work and family domains may be important moderators .\n\nWhen it comes to gender, it is crucial that we do not fall into the traps of viewing W\u2013F guilt as primarily a woman's concern or of accepting gender-related stereotypes as reality (Shields 2013). In addition, it is important not to overgeneralize results so as to emphasize between-gender differences at the expense of within-gender variability (Shields 2013). For example, we must recognize that not all mothers are the same, nor are all fathers, and that many different types of families exist. Likewise, the potential impact of race and culture cannot be underestimated.\n\nDealing with the combined pressures of work and family is a major issue for employees in today's global workforce (Korabik et al. 2008a). The ensuing stress can result in W\u2013F guilt, a negative emotional state that is associated with a variety of consequences detrimental to individual workers as well as to their organizations, including decreased job and life satisfaction and increased psychological distress, depression , and turnover intentions . Understanding what makes people feel guilty, why they feel guilty, under what circumstances they feel guilty, and the gender dynamics underlying these processes will assist us in designing interventions that will improve the well-being of individual workers and the bottom line of organizations.\n\nReferences\n\nAycan, Z., & Eskin, M. (2005). Relative contribution of childcare, spousal, and organizational support in reducing work\u2013family conflict for males and females: The case of Turkey. Sex Roles, 53, 453\u2013471.CrossRef\n\nAyman, R., & Antani, A. (2008). Social support and work\u2013family conflict. In K. Korabik, D. S. Lero, & D. L. Whitehead (Eds.), Handbook of work\u2013family integration: Research, theory, and best practices (pp. 287\u2013304). San Diego: Elseiver.CrossRef\n\nBanarjee, S. (1\u20133 Nov 2003). Double standards. Businessline.\n\nBardoel, A. (April 2004). Work family conflict in Anglo-Saxon and European cultural context: The case of Australia. Paper presented at the Society for Industrial and Organizational Psychology, Chicago, Illinois, USA.\n\nBort, J., Pflock, A., & Renner, D. (2005). Mommy guilt: Learn to worry less, focus on what matters most, and raise happier kids. New York: American Management Association.\n\nBui, L. S. (1999). Mothers in public relations: How are they balancing career and family? Public Relations Quarterly, 44, 23\u201326.\n\nChapman, F. S. (Feb 1987). Executive guilt: Who's taking care of the children? Fortune, 115 (4), 30\u201337.\n\nChappell, D. B., Korabik, K., & McElwain, A. (June 2005). The effects of gender-role attitudes on work\u2013family conflict and work\u2013family guilt. Poster presented at the Canadian Psychological Association, Montreal, Quebec, Canada.\n\nConlin, M. (2000). The new debate over working moms: As more moms choose to stay home, office life is again under fire. Business Week, 3699, 102\u2013104.\n\nDaly, K. J. (2001). Deconstructing family time: From ideology to lived experience. Journal of Marriage and Family, 63, 283\u2013294.CrossRef\n\nDesai, T. P., & Rajadhyaksha, U. (April 2004). Work family conflict in the Asian cultural context: The case of India. Paper presented at the Society for Industrial and Organizational Psychology, Chicago, Illinois, USA.\n\nDrach-Zahavy, A., & Somech, A. (April 2004). Work family conflict in the Middle Eastern cultural context: The case of Israel. Paper presented at the Society for Industrial and Organizational Psychology, Chicago, Illinois, USA.\n\nElvin-Nowak, Y. (1999). The meaning of guilt: A phenomenological description of employed mothers' experiences of guilt. Scandinavian Journal of Psychology, 40, 73\u201383.PubMedCrossRef\n\nEwles, G., Korabik, K., & Lero, D. S. (June 2013). Work\u2013family guilt: The role of overload and control. Poster presented at the Canadian Psychological Association, Quebec City, Quebec, Canada.\n\nEwles, G., Korabik, K., & Lero, D. S. (June 2014). Social support and work\u2013family guilt: The role of gender differences. Poster presented at the Canadian Psychological Association, Vancouver, British Columbia, Canada.\n\nGlavin, P., Scheiman, S., & Reid, S. (2011). Boundary spanning work demands and their consequences for work and psychological distress. Journal of Health and Social Behavior, 52, 43\u201357.PubMedCrossRef\n\nGuendouzi, J. (2006). \"The guilt thing\": Balancing domestic and professional roles. Journal of Marriage and Family, 68, 901\u2013909.CrossRef\n\nHochwarter, W. A., Perrew\u00e9, P. L., Meurs, J. A., & Kacmar, C. (2007). The interactive effects of work-induced guilt and ability to manage resources on job and life satisfaction. Journal of Occupational Health Psychology, 12, 125\u2013135.PubMedCrossRef\n\nHoffman, M. L. (1982). Development of prosocial motivation: Empathy and guilt. In N. Eisenberg (Ed.), The development of prosocial behaviour. New York: Academic.\n\nHuang, T. P. (Aug 2004). Work\u2013family conflict of employees in business organizations in Taiwan. Paper presented at the International Association of Cross-Cultural Psychology, Xi'an, China.\n\nIshaya, N., Ayman, R., & Korabik, K. (April 2013). Why so much guilt? Investigating how overload hurts, and why control may help. Paper presented at the Society for Industrial and Organizational Psychology, Houston, Texas, USA.\n\nKochanska, G., Gross, J., Lin, M., & Nichols, K. (2002). Guilt in young children: Development, determinants, and relations with a broader system of standards. Child Development, 73, 461\u2013482.PubMedCrossRef\n\nKorabik, K. (July 2005). Alleviating work\u2013family conflict for women managers in a global context. Paper presented at the Eastern Academy of Management, Cape Town, South Africa.\n\nKorabik, K., & Lero, D. S. (Aug 2004). A cross-cultural research project on the work\u2013family interface: Preliminary findings. Paper presented at the International Association of Cross-Cultural Psychology, Xi'an, China.\n\nKorabik, K., & McElwain, A. (April 2011). The role of work\u2013family guilt in work\u2013family conflict. Paper presented at the Society for Industrial and Organizational Psychology, Chicago, Illinois, USA.\n\nKorabik, K., & van Rhijn, T. R. (May 2014). Examining the cross-cultural measurement equivalence of work\u2013family interface measures. Paper presented at the Society for Industrial and Organizational Psychology, Honolulu, Hawaii, USA.\n\nKorabik, K., McElwain, A., Warner, M., & Lero, D. S. (July 2007). The impact of coworker support and resentment on work\u2013family conflict. Paper presented at IESE Conference on Work and Family, Barcelona, Spain.\n\nKorabik, K., Lero, D. S., & Whitehead, D. L. (Eds). (2008a). Handbook of work\u2013family integration: Research, theory, and best practices. San Diego: Elseiver.\n\nKorabik, K., McElwain, A., & Chappell, D. B. (2008b). Integrating gender-related issues into research on work and family. In K. Korabik, D. S. Lero, & D. L. Whitehead (Eds), Handbook of work\u2013family integration: Research, theories, and best practices (pp. 215\u2013232). San Diego: Elsevier.CrossRef\n\nKorabik, K., McElwain, A., & Lero, D. S. (Nov 2009). Does work\u2013family guilt mediate relationships between work\u2013family conflict and outcome variables? Paper presented at the Work, Stress and Health Conference, San Juan, Puerto Rico.\n\nKossek, E., Pichler, S., Bodner, T., & Hammer, L. (2011). Workplace social support and work\u2013family conflict: A meta-analysis clarifying the influence of general and work\u2013family-specific supervisor and organizational support. Personnel Psychology, 64, 289\u2013313.PubMedCentralPubMedCrossRef\n\nKubany, E. S. (1994). A cognitive model of guilt typology in combat-related PTSD. Journal of Traumatic Stress, 7, 3\u201319.PubMedCrossRef\n\nKubany, E. S., Haynes, S. N., Abueg, F. R., Manke, F. P., Brennan, J. M., & Starhura, C. (1996). Development and validation of the Trauma-Related Guilt Inventory (TRGI). Psychological Assessment, 5, 428\u2013444.CrossRef\n\nKubany, E. S., & Watson, S. B. (2003). Guilt: Elaboration of a multidimensional model. The Psychological Record, 53, 51\u201390.\n\nLivingston, B. A., & Judge, T. A. (2008). Emotional responses to work\u2013family conflict: An examination of gender role orientation among working men and women. Journal of Applied Psychology, 93, 207\u2013216.PubMedCrossRef\n\nMartinez, P., Carrasco, M. J., Aza, G., Blanco, A., & Espinar, I. (2011). Family gender role and guilt in Spanish dual earner families. Sex Roles, 65, 813\u2013826.CrossRef\n\nMawardi, A. (April 2004). Work\u2013family conflict in an Asian cultural context: The case of Indonesia. Paper presented at the Society for Industrial and Organizational Psychology, Chicago, Illinois, USA.\n\nMcElwain, A. (2008). An examination of the reliability and validity of the Work\u2013family Guilt Scale. (Unpublished doctoral dissertation), University of Guelph, Guelph, Ontario, Canada.\n\nMcElwain, A., & Korabik, K. (2004). Work\u2013family guilt. In M. Pitt-Catsouphes & E. Kossek (Eds.). Work and Family Encyclopedia Online. Chestnut Hill, MA: Sloan Work and Family Research Network. Retrieved February 16, 2009, from http:\/\/\u200bwfnetwork.\u200bbc.\u200bedu\/\u200bencyclopedia_\u200bentry.\u200bphp?\u200bid=\u200b871&\u200barea=\u200bAll.\n\nMcElwain, A., Korabik, K., & Chappell, D. B. (Aug 2004). Beyond gender: Re-examining work\u2013family conflict and work\u2013family guilt in the context of gender-role orientation. Paper presented at the International Society for the Study of Work and Organizational Values, New Orleans, Louisiana, USA.\n\nMcElwain, A., Korabik, K., & Chappell, D. B. (June 2005a). The work\u2013family guilt scale. Poster presented at the Canadian Psychological Association, Montreal, Quebec, Canada.\n\nMcElwain, A., Korabik, K., & Chappell, D. B. (June 2005b). The impact of work\u2013family conflict on work\u2013family guilt. Poster presented at the Canadian Psychological Association, Montreal, Quebec, Canada.\n\nMcElwain, A., Korabik, K., & Lero, D. S. (June 2007). Coping mechanisms in work\u2013family conflict. Paper presented at the Canadian Psychological Association. Ottawa, Ontario, Canada.\n\nNapholz, L. (2000). Balancing multiple roles among a group of urban midlife American Indian working women. Health Care for Women International, 27, 255\u2013266.CrossRef\n\nNevill, D., & Damico, S. (1977). Developmental components of role conflict in women. The Journal of Psychology, 95, 195\u2013198.CrossRef\n\nOffer, S., & Schneider, B. (2011). Revisitng the gender gap in time use patterns: Multitasking and well-being among mothers and fathers in dual-earner families. American Sociological Review, 76, 809\u2013833.CrossRef\n\nPollock, E. J. (10\u201313 March 1997). Work and family (a special report): Regaining a balance\u2014This is home; This is work. Wall Street Journal (Eastern Edition).\n\nRajadhyaksha, U., Huang, T. P., Mawardi, A., & Desai, T. P. (July 2011). Gender-role ideology, work\u2013family overload, conflict and guilt: Examining a path analysis model in three Asian countries. Paper presented at the International Association of Cross-Cultural Psychology, Istanbul, Turkey.\n\nSeagram, S., & Daniluk, J. C. (2002). It goes with the territory: The meaning and experience of maternal guilt for mothers of preadolescent children. Women and Therapy, 25, 61\u201388.CrossRef\n\nShields, S. A. (2013). Gender and emotions: What we think we know, what we need to know, and why it matters. Psychology of Women Quarterly, 37, 423\u2013435.CrossRef\n\nSimon, R. W. (1995). Gender, multiple roles, role meaning, and mental health. Journal of Health and Social Behaviour, 36, 182\u2013194.CrossRef\n\nVelgach, S., Ishaya, N., & Ayman, R. (April 2005). A multi-method approach to investigate work\u2013family conflict. Paper presented at the Society for Industrial and Organizational Psychology, Los Angeles, California, USA.\n\nZahn-Waxler, C., & Kochanska, G. (1988). The origins of guilt. In R. A. Thompson (Ed.), Nebraska Symposium on Motivation: Vol. 36. Sociometric Development (pp. 183\u2013258). Lincoln: University of Nebraska Press.\n\nZahn-Waxler, C., Kochanska, G., Krupnick, J., & McKnew, D. (1990). Patterns of guilt in children of depressed and well mothers. Developmental Psychology, 26, 51\u201359.CrossRef\n\nFootnotes\n\n1\n\nProject 3535 is a collaborative investigation of the W\u2013F interface among employed married\/cohabiting parents in ten countries (i.e., Australia, Canada, China, India, Indonesia, Israel, Spain, Taiwan, Turkey, and the USA). The contributions of the members of the Project 3535 research team to this chapter are gratefully acknowledged. The team consists of: Dr. Zeynep Aycan, Dr. Roya Ayman, Dr. Anne Bardoel, Dr. Tripti Desai, Dr. Anat Drach-Zahavy, Dr. Leslie B. Hammer, Dr. Ting-Pang Huang, Dr. Karen Korabik, Dr. Donna S. Lero, Dr. Artiwadi Mawardi, Dr. Steven Poelmans, Dr. Ujvala Rajadhyaksha, Dr. Anit Somech, and Dr. Li Zhang.\n\n2\n\nPhysical gender (whether someone is, or considers themselves to be, a man or a woman) refers to the psychological ramifications of biological sex (whether someone is biologically male or female). This terminology avoids assumptions of biopsychological equivalence (equating sex and gender with one another) and biological essentialism (the belief that behavior is solely attributable to biological causes). In self-report studies, physical gender is not directly observed, rather it is assessed by proxy as a demographic category. For simplicity's sake, in this chapter the term gender is used to refer to physical and demographic gender in contrast to the term gender-role which is used to refer to gender-role orientation, attitudes, ideology, etc.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_9\n\n# 9. Gender, Accuracy About Partners' Work\u2013Family Conflict, and Relationship Quality\n\nKei Nomaguchi1  and Melissa A. Milkie2\n\n(1)\n\nDepartment of Sociology, Bowling Green State University, 213 Williams Hall, 43403 Bowling Green, OH, USA\n\n(2)\n\nDepartment of Sociology, University of Toronto, 725 Spadina Avenue, M5S 2J4 Toronto, ON, Canada\n\nKei Nomaguchi (Corresponding author)\n\nEmail: knomagu@bgsu.edu\n\nMelissa A. Milkie\n\nEmail: melissa.milkie@utoronto.ca\n\nKeywords\n\nCultureCouplesSpouseGenderRelationship qualityWork\u2013family\n\nAuthor Note: Earlier versions of this chapter were presented at Pilot Data Conference, National Center for Family and Marriage Research at Bowling Green State University, Ohio, August 4, 2011, and the Work and Family Researchers Network Inaugural Conference, New York City, June 14\u201316, 2012. The data were collected with support from a grant to the National Center for Family and Marriage Research, Bowling Green State University from the U.S. Department of Health and Human Services, Office of the Assistant Secretary for Planning and Evaluation (5 UOI AEOOOOOI-03). This research was supported in part by the Center for Family and Demographic Research, Bowling Green State University, which has core funding from the Eunice Kennedy Shriver National Institute of Child Health and Human Development (R24HD050959-07).\n\n## 9.1 Overview\n\nFor employed women and men in a dual-earner marriage or partnership, every day is truly a balancing act\u2014of the regular demands that are related to two paid work positions, of an often complex relationship with each other, and for many pairs, the endless dance of emotions and labor that relate to owning a home or raising children. Two jobs means that both adults have a multitude of day-to-day issues at home to deal with\u2014mundane but vital things like having clean clothing to wear to work, having meals ready, and making sure that the household and any children stay afloat during the days, weeks, and months, as dishes, laundry, repairs, and bills pile up seemingly endlessly. At the same time, demands and strains from the job, such as an overload of tasks or hours, coworker problems, and unreasonable clients, often \"spill over\" into the home domain (Pearlin and McCall 1990). Not surprisingly, then, many employed adults feel that their work and family lives interfere with one another (Bellavia and Frone 2005; Nomaguchi 2009; Schieman et al. 2009; Young et al. 2014). Accordingly, scholars have focused a great deal of attention on this topic (for reviews see Bellavia and Frone 2005; Bianchi and Milkie 2010).\n\nPartners are at the epicenter of conflicts. As Pearlin (1983) noted, families have a \"uniquely pivotal position\" in the stress process because they not only generate their own stressors, but are also the backdrop for problems that enter from the outside world. And yet, how a partner's problems with balancing work and family come into the awareness of the other partner may be anything but straightforward. Of course, there are often clear work\u2013family conflicts displayed\u2014for example, he may openly discuss how being passed over for a promotion at work disrupts his sleep ; she may be visibly upset about an absence from an important school event that other parents are present for. These provide some clues about how the partner's world outside the home conflicts with his or her home life. But what a partner understands from the other partner is likely to be incomplete. This is because some conflicts are not easily observable given that the partners' respective places of work are typically separate. Sometimes partners may purposely try to segregate job stresses and concerns from their partner in order to protect their spouse, or perhaps because they have not received supportive feedback in the past (Pearlin and McCall 1990). Moreover, within relationships, even the same reality\u2014of a wife's or husband's low or high level of work\u2013family conflict\u2014may be viewed quite differently. As Bernard (1972) insightfully observed years ago, there is not \"a\" marriage, but \"his\" and \"hers\" marriages. She notes that even many questions about basic components of life together, such as how long partners have been together, or who did the dishes yesterday, generate very different responses from spouses. Thus, in terms of work\u2013family conflict, even things that can be directly observed by partners\u2014perhaps exasperation or frustration at work demands spilling over\u2014are not likely to be fully appreciated or understood in the same way by the other spouse.\n\nEspecially when information is incomplete, a gender perspective (Ridgeway 2011) posits that a partner's views about the amount and kinds of conflict that a wife or husband experiences is filtered through a gendered lens about what that spouse \"should\" be expected to feel. Thus, what a husband does observe of the problems his wife is experiencing in balancing work and family\u2014and what a wife observes about her husband's conflicts\u2014sits against a cultural backdrop of work and family ideology that remains highly gendered (Correll et al. 2007; Cotter et al. 2011; Milkie 2010; Milkie and Peltola 1999). In an era of blurred boundaries and insecurities in work and family roles, then, an important and under-examined question about couples' work\u2013family conflict centers on how accurate people are about their partner's level of conflict and how patterns may be gendered.\n\nDo inaccuracies matter for relationship quality? The question of the consequences of inaccuracies in assessing how the partner is balancing work and family is important to investigate. If misunderstandings cause dissatisfaction in relationships, then it becomes crucial to try to increase understanding, especially as partners make work decisions that affect one another. To maximize both partners' ability to meet workplace and family obligations, dual-earner couples employ various adaptive and coping strategies (Moen and Wethington 1992) and these decisions and strategies about their combined work and family roles tend to be made at the couple level (Becker and Moen 1999). Understanding each other's work\u2013family conflict accurately may help couples make decisions about their adaptive strategies that both partners find to be fair and to enhance the relationship.\n\nIn this chapter, we ask two questions: First, how accurately do members of US dual-earner heterosexual couples assess their partner's level of work\u2013family conflict? As part of this question, we ask whether inaccuracies are systematically gendered, i.e., do men tend to overestimate the level of conflict wives feel whereas women tend to underestimate men's conflicts? Second, is a partner's inaccuracy related to relationship quality ? How? For each question, we develop a framework for examining the question, provide a brief review of prior empirical studies, and state our predictions. Then we assess our predictions with dual-earner couples in The Married and Cohabiting Couples 2010 Study (see the Appendix for methodological notes) .\n\n## 9.2 Gender and Inaccuracy in Estimating Partners' Work\u2013Family Conflict\n\nWhy might dual-earner couples inaccurately estimate each other's work\u2013family conflict? A gender perspective suggests that individuals' perceptions are largely influenced by cultural meanings of what men and women \"should\" do and be like (Correll et al. 2007; Milkie 1999; Ridgeway 2011). In the area of work and family responsibilities, a series of qualitative studies have emphasized that although dual-earner relationships have become the norm, cultural scripts of the division of labor remain gendered. Women are assumed to be better and more interested in care work\u2014i.e., the tasks and mental activities in which a person meets the needs of others\u2014and to feel greater family responsibilities than men (Blair-Loy 2003; Hays 1996), whereas men are believed to feel greater breadwinning responsibilities than women (Townsend 2002). Because employed wives may feel responsible for overseeing what is happening at home, perhaps they are believed to feel a strong domestic pull, and thus feel a high level of work\u2013family conflict whenever they are employed. Employed husbands are seen to \"help\" their employed wives around the house and with childcare, but rely on their wives for the overall responsibility of the home and thus may not feel a substantial degree of work\u2013family conflict (Doucet 2006; Hays 1996; Hochschild 1989).\n\nIn contrast, recent quantitative research has shown that husbands' and wives' work\u2013family conflicts are relatively similar, indicating that the cultural images of stressed women and slacker men may be somewhat inaccurate. Bianchi et al. (2006) found that there was little difference between married mothers' and married fathers' total work time (i.e., time for market work and nonmarket work). Additional studies show few or no gender differences in work\u2013family conflict and balance (Grzywacz and Marks 2000; Gutek et al. 1991; Milkie and Peltola 1999; Schieman et al. 2009). Notably, it may be that women may not report more work\u2013family conflict than men in part because they may be pushed toward employing various strategies or trade-offs in order to balance work and family responsibilities (Milkie and Peltola 1999), as unfriendly workplaces and cultural expectations degrading them as \"bad mothers\" push them to reduce employment hours (Correll et al. 2007; Ridgeway 2011; Stone 2007). For men, unlike the cultural image, the sense of work\u2013family conflict has increased in recent cohorts. Nomaguchi (2009) found that work\u2013family conflict increased significantly more among men in dual-earner marriages than among women in dual-earner marriages between 1977 and 1997.\n\nWe argue that the discrepancy between qualitative and quantitative findings suggests the possibility that there may be an increasing gap between the gendered cultural script of work\u2013family conflict and actual work\u2013family conflict that men and women are experiencing. Specifically, we expect that men are more likely to overestimate their partners' work\u2013family conflict due to cultural scripts that employed women \"should\" feel a pull toward the home, whereas their female partners may not feel as much conflict as those cultural scripts suggest they should. In addition, we expect that women are more likely to underestimate their partners' work\u2013family conflict due to cultural scripts that men's \"hearts\" are in breadwinning and they do not feel primarily responsibility toward family, whereas their male partners actually feel more work\u2013family conflict than the cultural scripts suggest. Thus, on the basis of a potential gap between gendered cultural scripts and reality, we predict that male partners are more likely to overestimate than underestimate female partners' work\u2013family conflict, and female partners are more likely to underestimate than overestimate male partners' work\u2013family conflict.\n\nTo date, only a handful of studies have examined accuracies of couples ' perceptions of each other's work\u2013family conflict. Using a sample of 191 dual-earner parents collected in the Netherlands, Demerouti et al. (2005) found that the average ratings that women provided for their partners' work\u2013family conflict were higher than the average ratings that their partners reported as their own work\u2013family conflict (2.06 vs. 1.98 ranging from 1\u20135), suggesting women overestimated their partners' work\u2013family conflict, a finding opposite from our prediction. They found no difference between men's average rating of their partners' work\u2013family conflict and their partners' average self-rating of conflict (1.76 vs. 1.76). In contrast, using 224 dual-earner couples (married, cohabiting, or in a serious relationship) from a 1989 random sample of residents in Erie County, New York, Streich et al. (2008) found that the average scores that women provided for their partners' work\u2013family conflict were lower than the average scores that their partners provided as their own work\u2013family conflict (2.45 vs. 2.72 with a range from 1\u20135), indicating that women were underestimating their partners' work\u2013family conflict, as we predict. There was little gap between men's reports of their partners' work\u2013family conflict and their partners' self-reports of work\u2013family conflict (2.47 vs. 2.41).\n\nThese studies measured the inaccuracies in partners' perceptions of work\u2013family conflict at the aggregated individual level, not by matching a pair of partners and calculating the differences at the couple level. Thus, it is not clear what percentage of men and women overestimate or underestimate their partners' work\u2013family conflict. In addition, because the US study by Streich et al. (2008) used a regional sample, it is not clear to what extent the findings could be generalized. In this regard, The Married and Cohabiting Couples 2010 Study is useful because it offers couple-level data\u2014that is, data were collected from each partner of the couple\u2014from a \u00adnational sample of married and cohabiting couples in the USA. The couple-level data allow us to more clearly examine how accurately US dual-earner couples perceive each other's work\u2013family conflict.\n\nIn The Married and Cohabiting Couples 2010 Study, each partner rated his\/her own work\u2013family conflict by responding to the question, \"How much conflict do you face in balancing your paid work and family life?\" (1 = not at all, 2 = not too much, 3 = some, 4 = a lot, 5 = a great deal). The mean scores for men and women's self-report of work\u2013family conflict were very similar\u20142.3 and 2.2 (range 1\u20135), respectively. Each partner also rated his\/her partner's work\u2013family conflict via the question, \"How much conflict does your partner\/spouse face in balancing his\/her paid work and family life (1 = not at all 2 = not too much, 3 = some, 4 = a lot, 5 = a great deal)?\" On average, men rated perceptions of their partners' work\u2013family conflict higher than did women (2.4 vs. 2.2).\n\nTaking advantage of these two pieces of information, we assessed whether couples accurately perceive each other's work\u2013family conflict. Table 9.1 shows two cross-tabulations, one for male partners' report of female partners' work\u2013family conflict by female partners' self-report of work\u2013family conflict, and the other for female partners' report of male partners' work\u2013family conflict by male partners' self-report of work\u2013family conflict. These tabulations show how we created measures of inaccuracies of partners' work\u2013family conflict. First, male partners' accuracies or inaccuracies of female partners' work\u2013family conflict was measured by taking men's perceptions of their partners' work\u2013family conflict and subtracting their partners' self-report of work\u2013family conflict. We created three categories. If the value was less than 0, men were underestimating their partners' work\u2013family conflict. If the value was more than 0, men were overestimating their partners' work\u2013family conflict. If the value was 0, men were accurately estimating their partners' work\u2013family conflict. For example, if a husband reports that his wife experiences a \"great deal\" of conflict (5), whereas the wife herself reports \"some\" (3), (5\u22123 = 2; 2 > 0), he would receive a score of \"1\" for the dummy \"overestimating,\" a score of 0 for \"underestimating,\" and a score of 0 for \"accurately estimating.\" Likewise, the same three groups were created for female partners' underestimating, overestimating, and accurately estimating partners' work\u2013family conflict.\n\nTable 9.1\n\nCross-tabulations between respondents' perception of partner's work\u2013family conflict (WFC) and partner's self-report of WFC for men and women (%). (Source: The Married and Cohabiting Couples 2010 Study (N = 545))\n\n|  |\n\nMale partner's report of female partner's WFC\n\n---|---|---  \n|  |\n\nNot at all | Not too much | Some | A lot | Great deal\n\n|  |\n\n(1) | (2) | (3) | (3) | (4)\n\nFemale partner's self-report\n\n|\n\nNot at all (1) | 13.3 | 7.6a | 5.3a | 0.4a | 0.2a\n\n|\n\nNot too much (2) | 7.0b | 13.4 | 13.0a | 1.6a | 0.0a\n\n|\n\nSome (3) | 2.0b | 6.7b | 14.2 | 4.7a | 0.1a\n\n|\n\nA lot (4) | 0.6b | 0.9b | 3.6b | 2.9 | 0.5a\n\n|\n\nGreat deal (5) | 0.0b | 1.0b | 0.5b | 0.3b | 0.2\n\n|  |\n\nFemale partner's report of male partner's WFC\n\n|  |\n\nNot at all | Not too much | Some | A lot | Great deal\n\n|\n\n(1) | (2) | (3) | (4) | (5)\n\nMale partner's self-report\n\n|\n\nNot at all (1) | 10.4 | 8.0c | 1.9c | 0.0c | 0.2c\n\n|\n\nNot too much (2) | 11.0d | 18.9 | 10.8c | 0.6c | 0.8c\n\n|\n\nSome (3) | 3.4d | 9.1d | 14.2 | 3.3c | 0.3c\n\n|\n\nA lot (4) | 0.8d | 0.9d | 3.2d | 1.1 | 0.4c\n\n|\n\nGreat deal (5) | 0.0d | 0.3d | 0.3d | 0.1d | 0.4\n\naMale partner overestimating female partner's WFC = 33.4 %\n\nbMale partner underestimating female partner's WFC = 22.6 %\n\ncFemale partner overestimating male partner's WFC = 26.1 %\n\ndFemale partner underestimating male partner's WFC = 28.9 %\n\nThe percentage distributions of these three groups\u2014overestimating, underestimating, and no discrepancy\u2014are presented in Fig. 9.1. Less than half of male partners (44.1 %) and female partners (45.0 %) were accurate in their report of their partners' work\u2013family conflict. One third (33.4 %) of male partners overestimated female partners' work\u2013family conflict, whereas 22.6 % underestimated it. About one fourth (26.1 %) of female partners overestimated male partners' work\u2013family conflict, and slightly more female partners (28.9 %) underestimated male partners' work\u2013family conflict. As expected, male partners were more likely to overestimate than underestimate female partners' work\u2013family conflict, while female partners were only slightly more likely to underestimate than overestimate male partners' work\u2013family conflict. Differences between male partners and female partners in these distributions were statistically significant.\n\nFig. 9.1\n\nPercentage distributions for overestimating or underestimating partners' work\u2013family conflict. (Source: The Married and Cohabiting Couples 2010 Study)\n\n## 9.3 Does Inaccuracy Matter for Relationship Quality?\n\nHow are accuracies or inaccuracies of partners' estimates of each other's work\u2013family conflict related to their relationship quality ? Prior work has shown that individuals' work\u2013family conflict is related to family-relationship qualities, such as lower family satisfaction (Frone et al. 1994) and lower marital satisfaction (Coverman 1989). However, we do not know much about how couples' inaccuracies of each other's work\u2013family conflict are related to relationship quality, nor of the influence of actual level of work\u2013family conflict on relationship quality. We are particularly interested in the patterns wherein partners' estimates are biased in the direction that reflects the possibility that there has been an increasing gap between the gendered cultural script of work-family conflict and actual work-family conflict that men and women experience, i.e., male partners expecting more work\u2013family conflict for female partners and female partners expecting less work\u2013family conflict for male partners.\n\nOn the basis of prior research on relationship quality , we expect that a male partner's overestimating of his female partner's work\u2013family conflict may be linked to both the male partner's and female partner's perceptions of better relationship quality, because it is indicative of an acknowledgment of and openness to her difficulties. For example, Thompson (1991) found that husbands' appreciation and understanding of wives' juggling paid work and housework is important for wives' sense of fairness and relationship happiness. In contrast, women's underestimating their partners' work\u2013family conflict may be linked to their partners' perception of poorer relationship quality, because male partners may feel as though their female partners do not understand or appreciate their sense of being torn between paid work and family life. Wilkie et al. (1998) found that a higher sense of being understood by one's partner was related to a higher level of marital satisfaction for husbands. Underestimating their partners' conflicts may also relate to women's own perception of poorer relationship quality , because women may perceive it as unfair that their husbands work at their paid jobs without (from their perspective) feeling guilty or torn. Other research has shown that a sense of fairness in the division of labor is related to a higher level of marital satisfaction for wives (Frisco and Williams 2003; Wilkie et al. 1998).\n\nTo our knowledge, there have been no published studies that have examined the associations between partners' inaccuracies in understanding each other's work\u2013family conflict and their relationship quality with empirical data. For example, \u00adStreich et al. (2008), mentioned earlier, examined organizational commitment as an outcome of inaccuracies, but not relationship quality. Thus, our assessment using data from dual-earner couples in The Married and Cohabiting Couples 2010 Study is one of the first studies to examine this question. We focused on three aspects of couples' relationship quality\u2014emotional support , enchantment, and global relationship happiness (for measurement specifics, see the Appendix). These three aspects of relationship quality are known as strong indicators of divorce (Booth et al. 1985; Bradbury et al. 2000; Huston et al. 2001). In general, we found that couples tended to highly rate each of these aspects of relationship quality. The average \u00adrating for emotional support was 4.5 for male partners and 4.4 for female partners (range 1\u20135); average rating for enchantment was 4.2 (range 1\u20135) for male partners and 4.2 for female partners; average rating for relationship happiness was 8.7 (range 1\u201310) for male partners and 8.6 for female partners.\n\nFirst, we examined whether male partners' overestimating or underestimating of female partners' work\u2013family conflict was related to male partners' and female partners' reports of relationship quality respectively. As shown in Panel A of Table 9.2, compared to male partners who accurately estimated female partners' work\u2013family conflict, male partners who overestimated female partners' work\u2013family conflict reported higher ratings of enchantment and global quality. However, male partners' underestimating of their female partners' work\u2013family conflict, compared to their accurately estimating it, was not related to their own report of relationship quality. Panel B, which predicted female partners' reports of relationship quality, indicates that male partners' inaccuracies about female partners' work\u2013family conflict\u2014either overestimating or underestimating\u2014were not related to female partners' reports of relationship quality.\n\nTable 9.2\n\nUnstandardized coefficients from ordinary-least-squared (OLS) regression models predicting the associations between male partner's underestimating or overestimating female partner's work\u2013family conflict (WFC) and male partner's and female partner's reports of relationship quality. (Source: The Married and Cohabiting Couples 2010 Study) | Emotional support | Enchantment | Relationship happiness\n\n---|---|---|---  \n|\n\nb | SE\n\n|  |\n\nb | SE\n\n|  |\n\nb | SE\n\n|\n\nPanel A. Male partner's reports of relationship quality\n\nMale partner overestimating female partner's WFC | 0.109 | 0.056\n\n|  |\n\n0.230 | 0.073 | ** | 0.426 | 0.144 | **\n\nMale partner underestimating female partner's WFC | \u2212 0.099 | 0.060\n\n|  |\n\n\u2212 0.028 | 0.076\n\n|  |\n\n\u2212 0.113 | 0.150\n\n|\n\nMale partner accurately estimating female partner's WFC | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nControls\n\n|  |  |  |  |  |  |  |  |\n\nMale partner's WFC | \u2212 0.061 | 0.028 | * | \u2212 0.133 | 0.036 | *** | \u2212 0.219 | 0.070 | **\n\nMale partner's report of female partner's WFC | \u2212 0.165 | 0.029 | *** | \u2212 0.243 | 0.037 | *** | \u2212 0.379 | 0.073 | ***\n\nMale partner's age | \u2212 0.006 | 0.002 | ** | \u2212 0.007 | 0.003 | * | \u2212 0.010 | 0.006\n\n|\n\nMale partner nonwhite | \u2212 0.008 | 0.051\n\n|  |\n\n\u2212 0.114 | 0.066\n\n|  |\n\n0.024 | 0.129\n\n|\n\nMale partner's relationship type\n\n|  |  |  |  |  |  |  |  |\n\nFirst marriage | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nRemarriage | 0.244 | 0.068 | *** | 0.272 | 0.088 | ** | 0.591 | 0.173 | ***\n\nCohabiting | \u2212 0.092 | 0.064\n\n|  |\n\n\u2212 0.100 | 0.083\n\n|  |\n\n\u2212 0.240 | 0.162\n\n|\n\nHave children < 18 (male partner's report) | \u2212 0.098 | 0.048 | * | \u2212 0.024 | 0.062\n\n|  |\n\n\u2212 0.089 | 0.122\n\n|\n\nMale partner college educated | 0.053 | 0.055\n\n|  |\n\n-0.005 | 0.069\n\n|  |\n\n0.055 | 0.137\n\n|\n\nFemale partner college educated | 0.057 | 0.053\n\n|  |\n\n0.203 | 0.068 | ** | 0.127 | 0.133\n\n|\n\nMale partner self-employed | \u2212 0.029 | 0.067\n\n|  |\n\n\u2212 0.146 | 0.087\n\n|  |\n\n\u2212 0.140 | 0.171\n\n|\n\nFemale partner self-employed | 0.147 | 0.068 | * | 0.119 | 0.089\n\n|  |\n\n0.316 | 0.174\n\n|\n\nIntercept | 5.257 | 0.132 | *** | 5.316 | 0.171 | *** | 10.335 | 0.338 | ***\n\nR 2 | 0.146*** | 0.185** | 0.129***\n\nN | 530 | 520 | 536\n\nPanel B. Female partner's reports of relationship quality\n\nMale partner overestimating female partner's WFC | \u2212 0.010 | 0.066\n\n|  |\n\n\u2212 0.051 | 0.079\n\n|  |\n\n\u2212 0.009 | 0.154\n\n|\n\nMale partner underestimating female partner's WFC | \u2212 0.036 | 0.080\n\n|  |\n\n0.139 | 0.095\n\n|  |\n\n0.054 | 0.186\n\n|\n\nMale partner accurately estimating female partner's WFC | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nControls\n\n|  |  |  |  |  |  |  |  |\n\nFemale partner's WFC | \u2212 0.202 | 0.038 | *** | \u2212 0.274 | 0.046 | *** | \u2212 0.376 | 0.089 | ***\n\nFemale partner's report of male partner's WFC | 0.034 | 0.037\n\n|  |\n\n0.003 | 0.044\n\n|  |\n\n\u2212 0.077 | 0.086\n\n|\n\nFemale partner's age | \u2212 0.003 | 0.003\n\n|  |\n\n\u2212 0.007 | 0.003 | * | \u2212 0.008 | 0.006\n\n|\n\nFemale partner nonwhite | 0.019 | 0.069\n\n|  |\n\n-0.109 | 0.081\n\n|  |\n\n0.064 | 0.159\n\n|\n\nFemale partner's relationship type\n\n|  |  |  |  |  |  |  |  |\n\nFirst marriage | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nRemarriage | \u2212 0.166 | 0.061 | ** | \u2212 0.226 | 0.073 | ** | \u2212 0.527 | 0.143 | ***\n\nCohabiting | 0.145 | 0.083\n\n|  |\n\n0.288 | 0.098 | ** | 0.452 | 0.193 | *\n\nHave children < 18 (female partner's report) | \u2212 0.095 | 0.084\n\n|  |\n\n\u2212 0.148 | 0.100\n\n|  |\n\n\u2212 0.368 | 0.196\n\n|\n\nMale partner college educated | 0.042 | 0.069\n\n|  |\n\n0.115 | 0.082\n\n|  |\n\n\u2212 0.126 | 0.160\n\n|\n\nFemale partner college educated | 0.203 | 0.068 | ** | 0.108 | 0.079\n\n|  |\n\n0.314 | 0.156 | *\n\nMale partner self-employed | \u2212 0.030 | 0.082\n\n|  |\n\n\u2212 0.200 | 0.098 | * | \u2212 0.344 | 0.191\n\n|\n\nFemale partner self-employed | \u2212 0.001 | 0.085\n\n|  |\n\n\u2212 0.052 | 0.102\n\n|  |\n\n0.117 | 0.199\n\n|\n\nIntercept | 4.833 | 0.173 | *** | 5.117 | 0.200 | *** | 10.061 | 0.393 | ***\n\nR 2 | 0.121*** | 0.155*** | 0.131***\n\nN | 528 | 516 | 532\n\n*p < 0.05; **p < 0.01; ***p < 0.001\n\nSecond, we examined whether female partners' overestimating or underestimating of male partner's work\u2013family conflict was related to their own as well as their partner's reports of relationship quality respectively. Panel A in Table 9.3 shows that female partners who underestimated male partners' work\u2013family conflict were more likely than those who accurately estimated male partners' work\u2013family conflict to report lower emotional support. However, female partners' overestimating male partners' work\u2013family conflict was not related to their own perceptions of relationship quality. As shown in Panel B, female partners' underestimating of male partners' work\u2013family conflict was related to male partners' lower rating of enchantment. Female partners' overestimating of male partners' work\u2013family conflict was not related to male partners' report of relationship quality. In sum, our findings indicate that male partners' overestimating female partners' work\u2013family conflict was related to their own\u2014but not their female partners'\u2014perceptions of better relationship quality. Female partners' underestimating male partners' work\u2013family conflict was related to their own and male partners' reports of poorer relationship quality.\n\nTable 9.3\n\nUnstandardized coefficients from ordinary-least-squared (OLS) regression models predicting the associations between female partner's underestimating or overestimating male partner's work\u2013family conflict (WFC) and female partner's and male partner's reports of relationship quality. (Source: The Married and Cohabiting Couples 2010 Study) | Emotional support | Enchantment | Relationship happiness\n\n---|---|---|---  \n|\n\nb | SE\n\n|  |\n\nb | SE\n\n|  |\n\nb | SE\n\n|\n\nPanel A. Female partner's reports of relationship quality\n\nFemale partner overestimating male partner's WFC | \u2212 0.019 | 0.077\n\n|  |\n\n\u2212 0.010 | 0.092\n\n|  |\n\n0.017 | 0.177\n\n|\n\nFemale partner underestimating male partner's WFC | \u2212 0.174 | 0.072 | * | \u2212 0.091 | 0.086\n\n|  |\n\n\u2212 0.295 | 0.165\n\n|\n\nFemale partner accurately estimating male partner's WFC | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nControls\n\n|  |  |  |  |  |  |  |  |\n\nFemale partner's WFC | \u2212 0.202 | 0.035 | *** | \u2212 0.233 | 0.042 | *** | \u2212 0.357 | 0.080 | ***\n\nFemale partner's report of male partner's WFC | 0.000 | 0.044\n\n|  |\n\n\u2212 0.013 | 0.052\n\n|  |\n\n\u2212 0.169 | 0.101\n\n|\n\nFemale partner's age | \u2212 0.003 | 0.003\n\n|  |\n\n\u2212 0.007 | 0.003 | * | \u2212 0.011 | 0.006\n\n|\n\nFemale partner nonwhite | 0.019 | 0.070\n\n|  |\n\n\u2212 0.084 | 0.082\n\n|  |\n\n\u2212 0.011 | 0.160\n\n|\n\nFemale partner's relationship type\n\n|  |  |  |  |  |  |  |  |\n\nFirst marriage | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nRemarriage | 0.160 | 0.082\n\n|  |\n\n0.272 | 0.098 | ** | 0.498 | 0.190 | **\n\nCohabiting | \u2013 0.124 | 0.085\n\n|  |\n\n\u2013 0.190 | 0.102\n\n|  |\n\n\u2013 0.404 | 0.196 | *\n\nHave children < 18 (female partner's report) | \u2013 0.160 | 0.062 | * | \u2013 0.207 | 0.074 | ** | \u2013 0.568 | 0.142 | ***\n\nMale partner college educated | 0.063 | 0.069\n\n|  |\n\n0.110 | 0.081\n\n|  |\n\n\u2013 0.080 | 0.157\n\n|\n\nFemale partner college educated | 0.213 | 0.068 | ** | 0.106 | 0.080\n\n|  |\n\n0.365 | 0.155 | *\n\nMale partner self-employed | \u2013 0.021 | 0.082\n\n|  |\n\n\u2013 0.209 | 0.099 | * | \u2013 0.332 | 0.190\n\n|\n\nFemale partner self-employed | 0.027 | 0.085\n\n|  |\n\n\u2013 0.050 | 0.102\n\n|  |\n\n0.201 | 0.197\n\n|\n\nIntercept | 4.958 | 0.175 | *** | 5.082 | 0.207 | *** | 10.395 | 0.401 | ***\n\nR 2 | 0.133*** | 0.143*** | 0.149***\n\nN | 526 | 514 | 530\n\nPanel B. Male partner's reports of relationship quality\n\nFemale partner overestimating male partner's WFC | \u2212 0.029 | 0.057\n\n|  |\n\n\u2212 0.018 | 0.072\n\n|  |\n\n\u2212 0.101 | 0.145\n\n|\n\nFemale partner underestimating male partner's WFC | \u2212 0.038 | 0.059\n\n|  |\n\n\u2212 0.239 | 0.076 | ** | \u2212 0.193 | 0.150\n\n|\n\nFemale partner accurately estimating male partner's WFC | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nControls\n\n|  |  |  |  |  |  |  |  |\n\nMale partner's WFC | \u2212 0.065 | 0.032 | * | \u2212 0.084 | 0.041 | * | \u2212 0.211 | 0.081 | **\n\nMale partner's report of female partner's WFC | \u2212 0.122 | 0.025 | *** | \u2212 0.189 | 0.033 | *** | \u2212 0.268 | 0.065 | ***\n\nMale partner's age | \u2212 0.006 | 0.002 | ** | \u2212 0.005 | 0.003\n\n|  |\n\n\u2212 0.007 | 0.006\n\n|\n\nMale partner nonwhite | \u2212 0.007 | 0.052\n\n|  |\n\n\u2212 0.130 | 0.067\n\n|  |\n\n0.002 | 0.131\n\n|\n\nMale partner's relationship type\n\n|  |  |  |  |  |  |  |  |\n\nFirst marriage | \u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|  |\n\n\u2013 | \u2013\n\n|\n\nRemarriage | \u2212 0.107 | 0.048 | * | \u2212 0.039 | 0.061\n\n|  |\n\n\u2212 0.115 | 0.122\n\n|\n\nCohabiting | 0.233 | 0.068 | *** | 0.242 | 0.088 | ** | 0.502 | 0.175 | **\n\nHave children < 18 at home (male partner's report) | \u2212 0.092 | 0.065\n\n|  |\n\n\u2212 0.103 | 0.084\n\n|  |\n\n\u2212 0.229 | 0.164\n\n|\n\nMale partner college educated | 0.078 | 0.055\n\n|  |\n\n0.008 | 0.069\n\n|  |\n\n0.104 | 0.138\n\n|\n\nFemale partner college educated | 0.040 | 0.053\n\n|  |\n\n0.176 | 0.068 | ** | 0.101 | 0.134\n\n|\n\nMale partner self-employed | \u2212 0.050 | 0.067\n\n|  |\n\n\u2212 0.188 | 0.086 | * | \u2212 0.202 | 0.171\n\n|\n\nFemale partner self-employed | 0.167 | 0.068 | * | 0.171 | 0.088\n\n|  |\n\n0.397 | 0.174 | *\n\nIntercept | 5.198 | 0.137 | *** | 5.155 | 0.175 | *** | 10.172 | 0.349 | ***\n\nR 2 | 0.132*** | 0.181*** | 0.116***\n\nN | 526 | 516 | 531\n\n*p < 0.05; ** p < 0.01; *** p < 0.001;\n\n## 9.4 Discussion\n\n### 9.4.1 Do Men and Women Differ in Inaccurately Estimating Their Partners' Work\u2013Family Conflict?\n\nWith increasing complexities and insecurities related to work and family life, and blurring boundaries between these spheres, knowing a partner's work\u2013family conflicts may be fraught with difficulty. How much do American spouses know about their \"other half,\" particularly their wives' or husbands' level of felt work\u2013family conflict? The findings from The Married and Cohabiting Couples 2010 Study discussed in this chapter suggest that more than half of partners are at least somewhat inaccurate in their estimates of their partners' work\u2013family conflict. It is not an easy or even \"normal\" process to see things exactly as a partner does, especially when one realm (the partner's workplace) is most often outside of the purview of the other's vision. Consistent with the predictions of a gap between gendered cultural scripts and reality in men's and women's work\u2013family conflict, our findings indicate that male partners are more likely to overestimate than underestimate their female partners' work\u2013family conflict and, although to a smaller degree, female partners are more likely to underestimate than overestimate their male partners' conflict. We are cautious in our conclusions, however, because we recognize that one third of male partners and less than one third of female partners show these patterns. A majority of couples' reports do not reflect the patterns that the thesis about a gap between cultural scripts and reality suggests. Of this group, most agree with each other, but there were also some who are inaccurate but in the direction opposite from predictions. Investigation of these patterns is warranted in future research.\n\n### 9.4.2 Do Inaccuracies Matter for Relationship Quality?\n\nOur evidence suggests that partners' over- or under-estimating the others' work\u2013family conflict level is related to couples' relationship quality (and notably, higher levels of one's own actual work\u2013family conflicts are detrimental to relationship quality too). Although future research should address how over- and under-estimating relate to relationship qualities with more refined measures, we provide some speculative interpretations. Male partners' overestimates of female partners' work\u2013family conflict seems to be related to their own perceptions of better relationship quality. It may be that when male partners see their female partner as quite torn between work and family responsibilities, they may see her as trying to fulfill as many home responsibilities as she can, as cultural lore suggests, giving her the benefit of the doubt in home investment level. In these cases, male partners report better relationship quality, perhaps signaling their appreciation for what the woman does. On the flip side, female partners' underestimates of male partners' work\u2013family conflict relates to their own and male partners' reports of poorer relationship quality . In essence, when a female sees her partner in the traditional light of breadwinning masculinity , believing that he is less conflicted between work and family worlds than he really is, both the female and her partner feel worse about their relationship. Our findings are consistent with the findings of prior research, which show that a sense of fairness in the division of labor , a partner's empathy, and a greater sense of being understood by a partner on difficult issues, are related to higher relationship satisfaction (Wilkie et al. 1998).\n\n### 9.4.3 Future Directions\n\nThe Married and Cohabiting Couples 2010 Study offered insight into understanding inaccuracies of couples ' perceptions of each other's work\u2013family conflict and their implications for relationship quality. Yet, there are several questions that we were unable to examine with this data set that future research should address. First, we examined the global measure of work\u2013family conflict that does not distinguish differences between work-to-family conflict and family-to-work conflict. As such, future research using more detailed measures of the bidirectional nature of such conflict is warranted. Second, The Married and Cohabiting Couples 2010 Study did not include detailed information about conditions of paid work, such as hours of employment, job autonomy, and workplace flexibility, which are related to work\u2013family conflict (Bakker and Guerts 2004; Schieman and Young 2010). Third, this data was cross-sectional and therefore precluded causal assessments of the associations between partners' inaccuracies in each other's work\u2013family conflict and relationship quality . Although we argue that being inaccurate in underestimating difficulty about a partner's work\u2013family conflict can lead to worse relationship quality, it is also possible that having a poorer relationship at the outset makes one less aware of the partner's (difficult) subjective experience across work\u2013family domains. Likewise, having better relationship quality could influence male partners to misestimates that skew, in a sense, in a \"positive\" way; perhaps male partners who are more empathetic and loving are more aware of the employed women's \"plight\" and thus believe their partners have it harder than they do in reality. While the current data preclude adjudication among these possibilities, we hope to propel future research to address such nuanced questions about couples' work\u2013family conflicts and the meanings attached to gender, work, and family.\n\n## 9.5 Conclusion\n\nThe landscape of work\u2013family conflict among dual-earner couples is changing\u2014in many ways. Work worlds may be becoming more complex, as globalization and technology may make some workers' time and space more fluid, and boundaries as to when work ends and when \"family life\" begins are in flux. Gender spheres, too, are changing, as the cultural landscape regarding what men and women, husbands and wives, fathers and mothers \"should do\" responds to and challenges workplace policies and practices, and may create new exigencies for future generations. Assessing a partner's level of work\u2013family conflict, and what it means for relationship quality, may be even more complex in such a future. Research in this nascent area is warranted as gender schemas shift, and as dual-earner couples attempt to negotiate their work and family lives in a complex and changing societal context.\n\n## Appendix: Methodological Notes on The Married and Cohabiting Couples 2010 study\n\n### Data and Sample\n\nThe Married and Cohabiting Couples 2010 Study (MCC2010) is a web-based household survey that was obtained through a collaboration between the National Center for Family & Marriage Research (NCFMR) at Bowling Green State University and Knowledge Networks (KN). KN maintains a national panel of potential respondents, called KnowledgePanel (KP), who were selected by using random digit dialing sampling and address-based sampling methodology. Among the KN, individuals who do not already have Internet access are provided free Internet access and a laptop computer. Those who already have Internet access are given points redeemable for cash as incentives for their participation. KP consists of about 50,000 adult members (ages 18 and older) and includes persons living in cell phone only households as well as persons who have a landline phone. The KP members completed a demographic profile that determined eligibility for inclusion in specific studies. When selected, members receive a notification email letting them know there is a new survey available for them to take (Knowledge Networks 2010).\n\nFor the MCC2010 study, a nationally representative sample of US heterosexual married and cohabiting adults aged 18\u201364 was selected from active KP members with a supplement of cohabiting adults aged 18\u201364 from an opt-in panel (n = 1075). The survey was conducted from July to October 2010. The data and a field report that describes the sampling design are publicly available through the Inter-university Consortium for Political and Social Research (ICPSR 2013). For this chapter, we selected dual-earner couples (N = 545), including 391 married and 154 cohabiting couples. Using household ID numbers and gender of respondents, we created couple-level data. KN provides study-specific post-stratification weights to adjust the data to the distributions provided by the Current Population Survey for male partners and female partners respectively. We used these weights in our statistical analyses. Appendix Table 9.4 shows sample characteristics.\n\n### Measures of Relationship Quality\n\nEmotional support was an average of four questions (\u03b1 = 0.85): (a) \"My spouse\/partner shows love and affection toward me\"; (b) My spouse\/partner encourages me to do things that are important to me\"; (c) \"My spouse\/partner will not cheat on me\"; and (d) \"My spouse\/partner listens when I need someone to talk to\" (1 = strongly disagree to 5 = strongly agree).\n\nEnchantment was a scale created using 11 items of \"marital disillusionment scale\" (Huston et al. 2001) (\u03b1 = 0.95): (a) \"My marriage\/relationship hasn't gone quite as perfectly as I thought it might\"; (b) \"I'm beginning to see my spouse\/partner in a somewhat more negative light\"; (c) \"I'm beginning to see my marriage\/relationship in a somewhat more negative light\"; (d) \"Marriage\/Life together is not as enjoyable as I had expected it to be\"; (e) \"Our relationship has changed for the worse\"; (f) \"I no longer really like my spouse\/partner as a person\"; (g) \"My marriage\/relationship is no longer as important to me as it used to be\"; (h) \"I am very disappointed in my marriage\/relationship\";(i) I feel tricked, cheated, or deceived by love\"; (j) \"I feel no longer quite as positively about my spouse\/partner as I once did\"; and (k) \"If I could go back in time, I would not marry my spouse\/live with my partner again\" (1 = strongly agree to 5 = strongly disagree). Each item was reverse-coded and we averaged the scores of 11 items to create an enchantment scale.\n\nGlobal relationship happiness was measured by one question, \"Taking all things together, how satisfied are you with your relationship with your spouse or partner?\" (1 = very dissatisfied to 5 = very satisfied).\n\n### Controls\n\nIn our analyses of the associations between inaccuracies in partners' perceptions of each other's work\u2013family conflict and relationship quality, we took demographic and socioeconomic characteristics into account, such as age , race\/ethnicity (white vs. nonwhite), education (whether they have a college degree or not), self-employment , relationship status (first marriage, remarriage, or cohabiting), and parental status (whether they had at least one child under age 18 living in the household), because these characteristics are related to gender ideology (Thornton and Young-DeMarco 2001), work\u2013family conflict (Milkie and Peltola 1999; Schieman et al. 2009), and relationship quality (Amato et al. 2003). We also accounted for the effects of the levels of partners' own work\u2013family conflict and their perceptions of the other partner's work\u2013family conflict while we focus on how discrepancies between partners' perceptions are related to relationship quality. These characteristics were included as controls in regression models (see Tables 9.2, 9.3).\n\nTable 9.4\n\nMeans (SD) or % distributions for variables for dual-earner couples in The Marriage and Cohabiting Couples 2010 Study (N = 545) | Male partners | Female partners\n\n---|---|---\n\nAge (M) | 42.7 | (10.7) | 42.2 | (11.1)\n\nNonwhite (%) | 28.1 | 23.0\n\nRelationship status (%)\n\n|  |  |  |\n\nFirst marriage | 69.5 | 69.3\n\nRemarriage | 13.8 | 15.3\n\nCohabiting | 16.6 | 15.3\n\nHave children < 18 (%) | 45.1 | 41.4\n\nCollege education (%) | 33.6 | 39.1\n\nSelf-employed (%) | 13.9 | 13.4\n\nWork\u2013family conflict (WFC) (M, range: 1\u20135)\n\n|  |\n\nSelf | 2.3 | (0.9) | 2.2 | (0.9)\n\nPartner | 2.4 | (0.9) | 2.2 | (0.9)\n\nRelationship Quality (M):\n\n|  |  |  |\n\nEmotional support (1\u20135) | 4.5 | (0.5) | 4.4 | (0.7)\n\nEnchantment (1\u20135) | 4.2 | (0.8) | 4.2 | (0.8)\n\nRelationship happiness (1\u201310) | 8.7 | (1.3) | 8.6 | (1.5)\n\nData are weighted\n\nReferences\n\nAmato, P. R., Johnson, R., D., R., Booth, A., & Rogers, S. J. (2003). Continuity and change in marital quality between 1980 and 2000. Journal of Marriage and Family, 65, 1\u201322.CrossRef\n\nBakker, A. B., & Geurts, S. A. E. (2004). Toward a dual-process model of work-home interference. Work and Occupations, 31, 345\u2013366.CrossRef\n\nBecker, P. E., & Moen, P. (1999). Scaling back: Dual-earner couples' work-family conflict. Journal of Marriage and the Family, 61, 995\u20131007.CrossRef\n\nBernard, J. (1972). The future of marriage. New York: Bantam.\n\nBellavia, G. M., & Frone, M. R. (2005). Work-family conflict. In J. Barling, E. K. Kelloway, & M. R. Frone (Eds.), Handbook of work stress (pp. 113\u2013147). Thousand Oaks: Sage.CrossRef\n\nBianchi, S. M., & Milkie, M. A. (2010). Work and family research in the first decade of the 21st century. Journal of Marriage and Family, 72, 705\u2013725.CrossRef\n\nBianchi, S. M., Robinson, J. P., & Milkie, M. A. (2006). Changing rhythms of American family life. New York: Russell Sage.\n\nBlair-Loy, M. (2003). Competing devotions. Cambridge: Harvard University Press.\n\nBooth, A., Johnson, D. R., White, L. K., & Edwards, J. N. (1985). Predicting divorce and permanent separation. Journal of Family Issues, 6, 331\u2013346.PubMedCrossRef\n\nBradbury, T. N., Fincham, F. D., & Beach, S. R. H. (2000). Research on the nature and determinants of marital satisfaction: A decade in review. Journal of Marriage and the Family, 62, 964\u2013980.CrossRef\n\nCorrell, S. J., Benard, S., & Paik, I. (2007). Getting a job: Is there a motherhood penalty? American Journal of Sociology, 112, 1297\u20131338.CrossRef\n\nCotter, D., Hermsen, J. M., & Vanneman, R. (2011). The end of the gender revolution? Gender role attitudes from 1977 to 2008. American Journal of Sociology, 117, 259\u2013289.CrossRef\n\nCoverman, S. (1989). Role overload, role conflict, and stress: Addressing consequences of multiple role demands. Social Forces, 67, 965\u2013982.CrossRef\n\nDemerouti, E., Bakker, A. B., & Schaufeli, W. B. (2005). Spillover and crossover of exhaustion and life satisfaction among dual-earner parents. Journal of Vocational Behavior, 67, 266\u2013298.CrossRef\n\nDoucet, A. (2006). Do men mother? Toronto: University of Toronto Press.\n\nFrisco, M. L., & Williams, K. (2003). Perceived housework equity, marital happiness, and divorce in dual-earner household. Journal of Family Issues, 24, 51\u201373.CrossRef\n\nFrone, M. R., Barnes, G. M., & Farrell, M. P. (1994). Relationship of work-family conflict to substance use among employed mothers: The role of negative affect. Journal of Marriage and the Family, 56, 1019\u20131030.CrossRef\n\nGrzywacz, J. G., & Marks, N. F. (2000). Reconceptualizing the work-family interface: An ecological perspective on the correlates of positive and negative spillover between work and family. Journal of Occupational Health Psychology, 5, 111\u2013126.PubMedCrossRef\n\nGutek, B. A., Searle, S., & Klepa, L. (1991). Rational versus gender role explanations for work-family conflict. Journal of Applied Psychology, 76, 560\u2013568.CrossRef\n\nHays, S. (1996). The cultural contradictions of motherhood. New Haven: Yale University Press.\n\nHochschild, A. R. (1989). The second shift. New York: Quill.\n\nHuston, T. L., Caughlim, J. P., Houts, R. M., Smith, S. E., & George, L. J. (2001). The connubial crucible: Newlywed years and predictors of marital delight, distress, and divorce. Journal Personality and Social Psychology, 80, 237\u2013252.CrossRef\n\nICPSR. (2013). Study no. 31322. Retrieved from http:\/\/\u200bwww.\u200bicpsr.\u200bumich.\u200bedu\/\u200bicpsrweb\/\u200bDSDR\/\u200bstudies\/\u200b31322\/\u200bversion\/\u200b1. Accessed 17 March 2012.\n\nKnowledge Networks. (2010). Knowledge panel design summary. Retrieved from http:\/\/\u200bwww.\u200bknowledgenetwork\u200bs.\u200bcom\/\u200bknpanel\/\u200bdocs\/\u200bKnowledgePanel(R)-Design-Summary-Description.pdf. Accessed 29 June 2011.\n\nMilkie, M. A. (1999). Social comparisons, reflected appraisals and mass media: The impact of pervasive beauty images on black and white girls' self-concepts. Social Psychology Quarterly, 62, 190\u2013210.CrossRef\n\nMilkie, M. A. (2010). The stress process: Some family-level considerations. In W. R. Avison, C. S. Aneshensel, S. Schieman, & B. Wheaton (Eds.), Advances in the conceptualization of the stress process: Essays in honor of Leonard I. Pearlin (pp. 93\u2013108). New York: Springer.\n\nMilkie, M. A., & Peltola, P. (1999). Playing all the roles: Gender and the work-family balancing act. Journal of Marriage and the Family, 61, 476\u2013490.CrossRef\n\nMoen, P., & Wethington, E. (1992). The concept of family adaptive strategies. Annual Review of Sociology, 18, 233\u2013251.CrossRef\n\nNomaguchi, K. M. (2009). Change in work-family conflict among employed parents between 1977 and 1997. Journal of Marriage and Family, 71, 15\u201332.CrossRef\n\nPearlin, L. (1983). Role strains and personal stress. In H. B. Kaplan (Ed.), Psychosocial stress: Trends in theory and research (pp. 3\u201332). New York: Academic.CrossRef\n\nPearlin, L. I., & McCall, M. E. (1990). Occupational stress and marital support: A description of microprocesses. In J. Eckenrode & S. Gore (Eds.), Stress between work and family (pp. 39\u201360). New York: Plenum.CrossRef\n\nRidgeway, C. L. (2011). Framed by gender. Oxford: Oxford University Press.CrossRef\n\nSchieman, S., & Young, M. (2010). The demands of creative work: Implications for the stress in the work-family interface. Social Science Research, 39, 246\u2013259.CrossRef\n\nSchieman, S., Milkie, M. A., & Glavin, P. (2009). When work interferes with life: The social distribution of work-nonwork interference and the influence of work-related demands and resources. American Sociological Review, 74, 966\u2013988.CrossRef\n\nStone, P. (2007). Opting out? Why women really quit careers and head home. Berkeley: University of California Press.\n\nStreich, M., Caspter, W. J., & Salvaggio, A. N. (2008). Examining couple agreement about -family conflict. Journal of Managerial Psychology, 23, 252\u2013272.CrossRef\n\nThompson, L. (1991). Family work: Women's sense of fairness. Journal of Family Issues, 12, 181\u2013196.CrossRef\n\nThornton, A., & Young-DeMarco, L. (2001). Four decades of trends in attitudes toward family issues in the United States: The 1960s through the 1990s. Journal of Marriage and Family, 63, 1009\u20131037.CrossRef\n\nTownsend, N. W. (2002). The package deal: Marriage, work, and fatherhood in men's lives. Philadelphia: Temple University Press.\n\nWilkie, J. R., Ferree, M. M., & Ratcliff, K. S. (1998). Gender and fairness: Marital satisfaction in two-earner couples. Journal of Marriage and the Family, 60, 577\u2013594.CrossRef\n\nYoung, M., Schieman, S., & Milkie, M. A. (2014). Spouse's work-to-family conflict, family stressors, and mental health among dual-earner mothers and fathers. Society and Mental Health, 4, 1\u201320.CrossRef\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_10\n\n# 10. Work\u2013Life Equality: The Importance of a Level Playing Field at Home\n\nJeanette N. Cleveland1  , Gwenith G. Fisher2  and Katina B. Sawyer3\n\n(1)\n\nDepartment of Psychology, Colorado State University, 225 Behavioral Sciences Building, 1876 Campus Delivery, 80523 Fort Collins, CO, USA\n\n(2)\n\nDepartment of Psychology, Colorado State University, 228 Behavioral Sciences Building, 1876 Campus Delivery, 80523 Fort Collins, CO, USA\n\n(3)\n\nDepartment of Psychology, Villanova University, St. Mary's Hall, Suite 119, 800 Lancaster Avenue, 19085 Villanova, PA, USA\n\nJeanette N. Cleveland (Corresponding author)\n\nEmail: jeanette.cleveland@colostate.edu\n\nGwenith G. Fisher\n\nEmail: gwen.fisher@colostate.edu\n\nKatina B. Sawyer\n\nEmail: katina.sawyer@villanova.edu\n\nKeywords\n\nGenderEqualityWork\u2013familyHouseworkWorkforce participationDivision of laborSecond shift\n\n## 10.1 Introduction\n\nMen and women increasingly participate at similar rates in the workforce (U.S. Bureau of Labor Statistics 2012a), with women holding half of managerial and professional jobs . Yet little appears to have changed at home in the USA as women continue to perform more housework than men. The division of household labor is important as women's participation in the workforce continues to increase, with 53.6 % of women in the USA working in 2010 (U.S. Current Population Survey 2011). The division of household chores is one of the greatest sources of conflict and dissatisfaction among married couples (Kluwer et al. 1996, 2000; Kluwer 1998). For example, almost a quarter of women and men report the division of household chores as a key factor in divorce; currently, nearly double the percentage reported in the 1950s and 1960s (de Graaf and Kalmijn 2006). Importantly, the gender gap in the reporting of division of labor problems has increased, with women reporting greater problems than men (de Graaf and Kalmijn 2006). Therefore, whether paid or unpaid, both inside and outside of the home, the division of housework has important implications for working couples.\n\nThe goal of this chapter is to demonstrate the way in which couples divide paid work and housework is significantly guided by gender norms . Further, the division of paid work and housework has significant implications for the upward mobility of women in the workforce while placing tension on and jeopardizing the health and well-being of individuals and families. First, we highlight the importance of housework as an important work\/life issue and describe it in the context of extant work\/life research. Second, the methodological issues complicating this research area are identified, including construct deficiency and measurement bias . Next, we discuss gender specialization and describe various theories related to household division of labor . Finally, we conclude the chapter with a discussion of individual-, couple-, and organizational- level strategies that can level the playing field at home as well as suggestions for future research.\n\n## 10.2 Work\/Life Conflict\n\nAt the heart of the issue of work\/life equality is the issue of work\/life conflict, or interference between work and nonwork roles (Greenhaus and Beutell 1985). Work\/life conflict is important for both individuals and organizations because there are many negative attitudinal, behavioral, and other outcomes of work\/life conflict including lower-levels of job satisfaction and organizational commitment (Ernst Kossek and Ozeki 1998), \u00adhigher levels of absenteeism and turnover and lower levels of job performance, higher levels of stress and burnout (Anderson et al. 2002; Wayne et al. 2004), lower levels of marital satisfaction , poor physical health (Allen et al. 2000), and cognitive problems including poor concentration and low alertness (MacEwen and Barling 1994).\n\n### 10.2.1 Work\/Life Criterion Deficiency\n\nThe construct of work\/life conflict, as currently measured, may be deficient in a number of ways. The vast majority of research that has examined work and nonwork roles has (1) focused on family in the nonwork domain, and (2) examined conflict between these roles. Although the notion of family is unquestioningly important to most workers, the topic of household labor in the nonwork domain extends the construct domain beyond caregiving roles. Although the work\/life literature has contributed a great deal to our understanding of work and family issues, as well as the role of gender in determining participation in work and family spheres, Tetrick and Buffardi (2006) highlighted a number of measurement problems within the work\u2013family domain. One issue in particular is that work\/life conflict has focused heavily on family within the nonwork domain, without considering the potential nonfamily related conflicts which may arise within the more general life sphere. Fisher et al. (2009) developed a measure of work\/nonwork interference based on the notion that, although family is important, traditional work\/family measures may be too narrow for measuring the interface between work and nonwork roles. To more accurately understand the impact of work on nonwork roles and vice-versa, important aspects of life at home may be missed or ignored if too much emphasis is placed on only family roles (e.g., with child care being predominant). Additional research is needed to articulate more specific nonfamily roles in the nonwork domain to improve our understanding of how work relates to nonwork roles in a more thorough manner.\n\nWhile this chapter will primarily focus on the ways in which couples divide housework, including child care, it is also possible that the burden may fall on women to perform a variety of life-related (but not family-related) tasks as well. For example, women may be expected to purchase gifts for friends, keep track of social schedules , or participate more heavily in exercise and grooming activities. As such, future research should examine the impact of gender on the division of life-related activities outside of the family sphere.\n\n## 10.3 Importance of Housework and Workforce Participation\n\nAlthough historically, the division of housework was of little academic interest outside the field of home economics (Ehrenreich and English 1978), the dramatic increase in working women especially among married, divorced, and separated women has brought this issue into more prominent view. For example, in 2010, 53.6 % of women and 63.7 % of men overall were working (Current Population Survey 2011). The proportion of working women is even higher among women who are married (57.4 %), divorced (61.1 %), or separated (56.5 %). This is a dramatic increase compared to previous generations; in 1970, only 40.8 % of women were employed (Current Population Survey 2011). In addition, women continue to work largely in gendered jobs with lower salary , benefits, and flexibility. For example, women are more likely than men to work in nonmanagement positions (26.9 % women vs 17.9 % men), service occupations (21.3 % women vs 14.5 % men), and in education and health services (36 % women vs 11 % men; Current Population Survey 2011). Of the only 17 % of board positions in Fortune 500 companies, fewer than 25 are Fortune 500 CEOs, and only 18 % have been elected to senior legislative positions (Catalyst 2012) . Further, more women hold multiple jobs than men do (5.4 % vs 4.5 %) (Current Population Survey 2010) and since 1970, women on average have increased their contributions to the family income from 2.6 % to 37.1 % (Current Population Survey 2011). Finally, during the last 20 years, the proportion of wives who earn more than their husbands (among dual-earner couples) has increased from 17.8 % in 1987 to 28.9 % in 2009 (Current Population Survey 2011).\n\n### 10.3.1 Why Focus on Housework?\n\nIn spite of the changes in recent decades, men are typically the breadwinners. Women's time spent doing housework is related to paid work involvement (Bianchi et al. 2000). The popular press , including recent books such as Lean In (Sandberg 2013) have also highlighted continued gender gaps at work and home in spite of the narrowed gap between men and women in labor force participation rates. Gender equality theorists argue that these differences in gender roles underlie inequality, placing women at a distinct disadvantage at home and at work (Budig 2004). Even though both women and men report that they want to spend more time with their children, women are the ones who are more likely to actually scale back on their work time to prioritize family (Becker and Moen 1999). Women are then penalized for this prioritization of family over work, evidenced by lower earnings and having primary responsibility for the bulk of tasks related to child rearing (Budig and England 2001). For example, only 9 % of working mothers between the ages of 25 and 44 spend more than 50 hours per week at work (Williams 2013). Although this increases to 13.9 % when only considering those with at least a college degree, this is still a small percentage overall (Williams 2013). Ely and Padavic (2013) indicated that the low proportion of women at the highest levels is due to the long hours required of such jobs and that many women (mothers) are unwilling to work in jobs that require such long hours. An alternative explanation is that women are willing to work in jobs with long hours yet cannot when they disproportionately bear the burden of housework responsibility .\n\n#### 10.3.1.1 Childcare and Division of Labor\n\nWhile it is important to compare men and women overall when discussing household labor division, it is also important to focus directly on those with children. The presence of children adds to the level of necessary housework , particularly laundry, cleaning, and meal preparation (Jacobs and Gerson 2004). Two trends have emerged regarding time in relation to parental caregiving responsibilities . First, the amount of parents' primary childcare time has increased since 1965 among western industrialized countries (Jacobs and Gerson 2004). Second, gender variations in parental care have narrowed. Specifically, fathers' time spent attending to childcare responsibilities has increased significantly, with proportionately smaller increases among mothers' childcare time (Bianchi et al. 2006; Sayer et al. 2004). Fathers have shifted some of their time from sleep and leisure to spend more time with their children while mothers have reduced some of their housework time and increased time spent attending to childcare responsibilities (Gauthier et al. 2004; Sayer 2005).\n\nThis is an encouraging pattern due to the increase in time spent on childcare overall. Yet the increased participation of men in the childcare domain may not make significant changes in the amount of time women are spending in childcare related activities. For this reason, it is important to consider the increased time commitments to the home domain experienced by parents (and particularly mothers, who are also more likely to be responsible for the bulk of home duties) in comparison with those who do not have childcare responsibilities.\n\n## 10.4 Housework: Actual and Perceived Division of Labor\n\nWomen continue to do more housework or household labor than men (Liss 2013). Even when employed full-time outside of the home, women continue to spend more hours doing housework than their husbands (Blair and Lichter 1991). Further, in addition to this 'double shift' of housework and paid employment, women spend more time on tedious house tasks in addition to bearing the bulk of hours spent in child rearing. Although since 1965, the proportion of time that women contribute to housework has declined while men's has increased (Bianchi et al. 2000), women continue to do more regardless of how the division of household labor is measured.\n\nIn order to properly understand the nuances associated with the division of household labor, it is important to understand how these constructs are determined and assessed methodologically, as well as how they are perceived by those individuals completing the tasks. In this section, we briefly review the types of tasks which are performed in the household and by whom, methodological issues regarding how the division of housework is assessed, including perceptions of hours spent in household labor, and the role of fairness in housework labor division.\n\n### 10.4.1 Measuring Household Work\n\nA number of studies have sought to quantify the amount of housework performed by men vs. women. Interview studies suggest that women work 15 hours more than men each week when paid and unpaid work is combined (Hochschild and Machung 1989). More recent diary studies show that when paid and unpaid labor are combined, men and women work equal number of hours (Sayer et al. 2009). Employed mothers with young children, however, work more total hours than men do. Using time diaries where people report what they are doing with their time, generally women report less leisure time per day than men (Mattingly and Liana 2006), averaging approximately 30 minutes less. On the other hand, when researchers actually followed dual-earner couples who worked more than 30 hours per week, they found that men spent significantly more time engaged in leisure activities than women and women spent significantly more time on housework (Saxbe et al. 2011). These results highlight important differences in methodologies used to measure household work.\n\nThere are two key methodological issues regarding the measurement of housework and division of housework within couples. The first issue involves defining the construct: what exactly is the domain of housework tasks? Some research has highlighted the blurry boundaries between routine and leisure activities (Coltrane 2000). As indicated previously, although women perform more housework than men, the degree of imbalance among such work varies by the type of housework tasks (Blair and Lichter 1991). According to Twiggs et al. (1999), housework tasks are gendered and there is a hierarchy within the gender-typed tasks. For example, women are largely responsible for routine and repetitive tasks such as laundry or washing dishes. These tasks take up the majority of housework time. Further, although both laundry and washing dishes are considered routine female housework activities, men are less likely to do the laundry than the dishes. Housework assessments then reflect not only the amount of time spent but also responsibility for a given (historically, often gendered) task and is measured either through estimates of the actual time spent or as a report on who is more frequently responsible for the task.\n\nThe second issue pertains to quantifying the amount of time spent performing housework . For example, while individuals may engage in one task at a time, they may also multitask by performing two or more tasks at once (e.g., laundry and picking up around the house), making the reporting of time spent in household labor difficult to estimate. Relatedly, there is the potential for recollection problems for participants as housework is so routine to daily life that they may not be as salient as nonroutine events. In other words, participants may not be consciously aware of or be able to accurately recall how much time they actually spent doing these tasks.\n\nThree approaches to measuring housework and division of domestic labor have been compared by Juster et al. (2003) and Lee and Waite (2005). The most frequently used measure is respondents' retrospective reports of housework. Specifically, either one or both partners are asked to report time spent on total housework over a specific time period (e.g., last week, month, etc.) or about the time spent on one specific activity. The measure is easily implemented and therefore popular. However, combining housework time across all tasks into one overall measure may not capture the nuanced differences in the types of housework completed or in linking the completion of specific types of housework tasks to assessments of well-being. The second measurement method is use of a \"time diary\" in which respondents are asked to record their actual time use for all activities (usually measured with reference to the previous day). Such measures are viewed as more accurate than retrospective report survey questions, yet they are also more time-consuming and expensive to implement. In addition, the time diary approach assumes that the previous day is an accurate representation of how participants usually spend their time. Results using both survey questions and time diary measures do correlate closely especially when tasks are those that are performed regularly and are externally structured (Juster et al. 2003).\n\nA third measure related to the time diary approach is the experience sampling method (ESM; Lee and Waite 2005). In ESM, respondents report their primary and secondary activities eight times a day, often at random intervals when prompted to do so by a device. Compared to survey measures, ESM-based measures produce lower housework time estimates. Further, the gender gap between husbands and wives reports of activities is smallest based on ESM estimates (for primary activities only) and largest based on wives' retrospective survey reports (Lee and Waite 2005). Thus, it is important to examine the extent to which gender discrepancies exist as a function of the measurement tool used to capture housework hours.\n\n### 10.4.2 Gender Discrepancies in Reporting Housework\n\nMost research on housework relies on retrospective survey-based measures or diary studies in which only one person reports about the division of housework . When information is collected from both partners, there is often a discrepancy (Coltrane 2000). Although both men and women in the USA tend to inflate their reports of time spent, men do so more than women (Coltrane 1996, 2000; Press and Townsley 1998). There are a number of reasons for reporting differences. First, social desirability may influence the reporting depending upon the extent to which the task may be considered more gendered. Men may overestimate their contribution in order to reflect a more supportive husband or due to lack of information on how much the partner is contributing (Kamo 2000). Finally, wives' and husbands' differences in reporting may reflect domestic harmony or disharmony or couple marital conflict. In other words, differences in reporting may be more apparent when members of a couple do not agree on how work is being done or should be done.\n\n### 10.4.3 Fairness Perceptions\n\nFairness perceptions are important because they have implications for a range of emotions and well-being and, additionally, whether one feels that they are being fair (or unfair) to their partner (Lively et al. 2008). However, fairness is a subjective experience and differs from reports of equal division of labor . Men report that division of housework is fair if they contribute 36 % or more, while women perceive housework as fair if they do 66 % or less (Lennon and Rosenfield 1994). Women are more likely to feel that the division of housework is unfair to them while men are more likely to report the distribution of tasks as fair (DeMaris and Longmore 1996).\n\n## 10.5 Gender Specialization: The Role of Separate Spheres\n\nTraditional gender stereotypes create circumstances in which women are expected to perform nearly all of the household duties, while men are expected to perform most of the paid work. Thus, when women contribute little to the household in terms of income, it may be an indicator of traditional gender roles operating within the family unit (Bittman et al. 2003; Brines 1994; Greenstein 2000; Prince-Cooke 2006). While \"gender specialization\" is decreasing in prevalence, it was viewed historically as beneficial to both men and women, because it resulted in mutual dependence within couples (Prince-Cooke 2006). Specifically, when women specialize in the home domain and men specialize in the work domain, both partners depend upon each another because neither partner is able to perform the necessary tasks and duties required for success in the opposite domain. This gendered division of labor has been labeled the two \"separate spheres,\" one of work (which is male) and one of the home (which is female) (Lundberg and Pollak 1993). When husbands and wives are responsible for separate and distinct sets of work activities (Lundberg and Pollak 1996), they no longer must coordinate or compromise with one another. This arrangement is assumed to create optimal conditions for both husbands and wives, because each individual is able to complete their activities independently and without negotiation.\n\nHowever, the viability of gendered division of labor appears to be diminishing significantly within our current economy . Demographic data suggest that traditional gendered arrangements occur less frequently today because both men and women increasingly participate in the labor force (U.S. Bureau of Labor Statistics 2012b) . As a result, women and men participate in both work and family spheres. As such, couples increasingly need to negotiate with one another about the extent to which they participate in the work domain versus the nonwork domain, creating a new set of circumstances for them to consider when navigating the work\u2013life interface. Further, research shows that when either member of a couple perceives unfairness in arrangements either inside or outside of the home, greater levels of marital conflict and lower marital satisfaction are reported, and may contribute to divorce (Lavee and Katz 2002; Prince-Cooke 2006). Thus, a key factor to maintain healthy relationships is for couples to successfully negotiate or communicate with each other to arrive at a consensus regarding their work and family arrangements.\n\nWhile it is important to investigate perceptions of and measurements related to the amount of paid work and family labor completed by males and females, it is also necessary to examine the underlying theoretical frameworks which may be driving these perceptions and their related antecedents and consequences. Further, we review a sampling of theories which help to shed light on the mechanisms through which work and family negotiations may occur. While this is not an exhaustive review, the theories outlined below are particularly relevant when considering how paid work and family labor are split among partners, allowing for a closer examination of the role of gender and household politics in determining levels of work\u2013life conflict within couples . As such, we review a sample of theories that highlight the key drivers of work\u2013life negotiations between partners, both at home and at work.\n\n## 10.6 Theoretical Approaches to Household Division of Labor\n\n### 10.6.1 Exchange Theory: The Division of Paid Work and Housework\n\nOne underlying explanation for gender specialization is exchange theory (Bittman et al. 2003; Molm and Cook 1995). According to this theory, one partner (e.g., economically dependent) within a couple contributes to the relationship by providing nonmonetary goods (e.g., household labor, child care) and, in return, receives monetary goods (e.g., income, property) from the economically independent partner. The dependent partner relies on the independent partner for economic resources and thus, is unlikely to leave the relationship as a result. Further, because the dependent partner is financially tied to the relationship, the independent partner has little reason to contribute additional nonmonetary resources in exchange for their contributions. Given that women often make smaller economic contributions in relationships compared to men (as evidenced by the gender pay gap and the lower representation of women in high powered positions in the workplace, U.S. Bureau of Labor Statistics 2012a, b) , women are more likely to occupy the dependent partner role. Further, because the independent partner does not feel the need to increase their contributions to the relationship, women may feel pressured to contribute increasingly to the household domain, while receiving static levels of assistance in exchange.\n\n### 10.6.2 Bargaining Theory: The Division of Paid Work and Housework\n\nA related theoretical perspective regarding family and work negotiation is bargaining theory. Bargaining theory describes the exchanges between partners as primarily based around \"threat points\" (Bittman et al. 2003; McElroy 1990) . A threat point is determined by the resources that each individual in the relationship independently acquires and can use at his\/her discretion in the event that the relationship ends. Thus, threat points are constantly balancing in a relationship, in anticipation of a potential divorce. For example, if a woman does not contribute monetarily to a relationship, yet her partner does, then (in terms of resources) her partner does not need to fear that the relationship might end. He would, in essence, have more bargaining power over his partner, and therefore, more power to create the conditions in the marriage that are more favorable to him. In this way, the \"threat point\" is used to exert power over the more dependent member of a relationship because the dependent member lacks personal resources available for use in the event of a divorce. Again, because the less economically powerful person in a relationship is more likely to be a woman (as measured by income differentials and power differentials in the workplace; U.S. Bureau of Labor Statistics 2012a, b) , women are at greater risk for potential oppression and power misuse than men. However, when a woman is employed in paid work, she assumes a better exchange\/bargaining position with her husband. Thus, there is an incentive for women to join the labor force: to gain valuable resources, and maintain threat point balance .\n\n### 10.6.3 Understanding Equitable Exchanges\n\nThe challenges associated with the division of paid and household work may become easier as partners view each other as either contributing wholly to one sphere (work or the home) or as equal contributors to the paid work sphere (spending similar hours and\/or earning similar salaries in the paid workforce). Research shows that women who work fewer hours than their partner outside of the home are likely to do more household work (Brines 1994; Greenstein 2000) and also report greater fairness in the distribution of household labor (Lennon and Rosenfield 1994). Women who perform all of the housework while their husband performs all of the work outside of the home report the highest levels of fairness (Lennon and Rosenfield 1994). These women may perceive the gender specialization of tasks (Bittman et al. 2003) as fair (e.g., husband takes care of work outside of the home, while wife takes care of work inside of the home). Thus, perceptions of fairness, in this case, are not based on actual hours worked in the home, but rather how these hours are perceived in light of the overall contribution of both partners to the marriage. However, there is evidence that as couples increasingly equally share the division of paid work, they begin to divide the household labor more equally as well (Bittman et al. 2003; Brines 1994; Greenstein 2000; Prince-Cooke 2006). This implies that as partners begin to contribute equally in terms of income and hours worked outside of the home, these equitable practices are also maintained inside the home.\n\nTo this end, equity theory , which states that unfair exchanges decrease satisfaction for both parties (Klumb et al. 2006), is a theoretical framework which enables us to better understand the psychological undercurrents between partners when they perceive that they are completing either more or less than their equitable share. Similar to exchange and bargaining theories, equity theory focuses on the exchange of or bargaining for paid labor versus household labor between partners. Yet, equity theory diverges from exchange and bargaining theories and predicts that neither person in the relationship will be happy when exchanges are inequitable, regardless of the amount of power one holds.\n\nEqually sharing work, on the other hand, may be associated with lower distress for both parties. While exchange and bargaining theories revolve around the maintenance of power within relationships, the elimination of this power struggle may alleviate tension for couples engaged in this struggle. In fact, according to exchange, bargaining, and equity theories, equality between individuals may be beneficial for both partners' well-being , particularly because it is associated with a lower risk of divorce (Prince-Cooke 2006). Further, perceptions of unfairness in the division of household labor are linked with marital conflict and marital dissatisfaction (Lavee and Katz 2002).\n\n### 10.6.4 Gender Role Attitudes: The Effects of Traditional and Nontraditional Attitudes on the Division of Labor\n\nWhile equity theory may serve to better understand outcomes associated with the division of housework, there may be limits to its usefulness in prediction. When members of a couple strongly subscribe to traditional gender roles, tenets of bargaining and exchange theories may be more valid. Research demonstrates that among couples , adherence to traditional gender roles is associated with a more traditional division of labor inside the home (Lu et al. 2000). Further, traditional gender ideology has been found to predict the unequal distribution of labor in the home to a greater degree than the wife's nonparticipation in the paid workforce (Pina and Bengston 1993). Husbands' gender ideology also has been shown to be a predictor of the division of household labor, with wives completing a greater percentage of housework when husbands adhered to traditional norms (Pyke and Coltrane 1996). In conjunction, women with traditional gender ideologies report greater role clarity both inside and outside of the home because they feel that well-developed roles (lack of participation in paid labor, high levels of participation in household labor) are available to them (Buunk et al. 2000).\n\nFurther, the ways in which women or men identify themselves outside home can influence the division of household labor . For example, husbands and wives that consider themselves \"co-breadwinners\" are more likely to share labor in the household (Deutsch 2001). On the other hand, couples who identified more strongly with traditional masculine and feminine traits continued to maintain traditional gender identities by reporting that the male was the breadwinner (Deutsch and Saxon 1998) even though they actually had nontraditional work arrangements. The women who reported employment in paid work due solely to financial pressure continued to report that being a mother was their primary role (Deutsch and Saxon 1998). Therefore, gender ideology influences the ways in which individuals divide and also perceive their labor contributions within a couple, even if objective information about income and hours worked indicates that they are equally sharing paid labor.\n\n### 10.6.5 Gender Deviance Neutralization and the Division of Household Labor\n\nIncorporating the influence of gender ideology, Brines (1994) and Greenstein (2000) proposed two related theories: gender display theory and deviance neutralization theory, respectively, as explanations for the ways that couples divide household labor. According to these theories, couples divide routine household activities consistent with their adherence to gender norms . For example, among couples with more traditional gender views and similar relative employment hours, husbands spend less time doing female sex-typed household tasks, such as laundry or washing dishes (Atkinson and Huston 1984). Similarly, the opposite pattern would be found for those with less traditional gender norms. Therefore, Brines (1994) and Greenstein (2000) assert that gender ideology may be one guiding mechanism behind the division of household labor . The context of their research focused upon couples in which the women were the primary breadwinners (e.g., either the sole income or earning a higher income than her spouse). Within such couples (woman as breadwinner), the predictions based on equity theory , bargaining theory , and exchange theory do not appear to be supported.\n\nAccording to equity theory, when the wife is the primary economic contributor, the husband should perform more housework . However, the evidence shows the opposite pattern (Perry-Jenkins and Folk 1994; Van Willigen and Drentea 2001). Specifically, husbands with wives who work full-time do no more housework than husbands with wives who are full-time homemakers. Men's work inside the home increases as the relative proportion of hours worked outside the home decreases, until it reaches equity with their partner. This finding is consistent with equity theory predictions. However, as men's paid work hours continue to become proportionately less than their wives', men's participation in the household also decreases. Therefore, when the number of hours spent in the labor force by a woman proportionally exceeds that of a male, the increase in paid work hours and income by females are associated with lower levels of help from males in the household.\n\nOther research (Hochshild and Machung 1989) showed that one-fifth of husbands who earn more than their wives share the housework (although husbands were more likely to do male sex-typed chores, such as mowing the lawn, car maintenance, and cleaning the garage). Within couples where the husband and wife earn the same amount, one third of the husbands shared the housework. However, in couples where husbands earn less than their wives, none of the husbands performed the majority of housework. Brines (1994) suggested that this finding among female breadwinner couples is associated with men's unwillingness to be further feminized by participating in housework when they are not the major economic provider. Further, with wives complicit in the process, women and men may attempt to neutralize the nontraditional gender behaviors that occur outside of the home (e.g., wife as breadwinner) by engaging in more traditional gendered behaviors within the home and through housework (Greenstein 2000). In addition to their responsibility as the primary provider outside the home, female breadwinners with traditional gender ideologies are expected to and likely perform the majority of household tasks. This pattern predicts an increased risk for role and stress overload for female breadwinners. Additionally, the finding is inconsistent with the predictions of exchange theory or bargaining theory because, according to these theories, the female breadwinner should have greater resources and greater bargaining power, and therefore, a greater opportunity to negotiate preferable household arrangements. Yet, these women (who work more hours and\/or are paid more than their husbands) contribute more hours to housework than their husbands.\n\nWhile exchange, bargaining, or equity theories cannot fully account for these findings, the role of gender norms may explain the phenomenon which occurs within female breadwinner couples (Brines 1994; Greenstein 2000). As women increase hours spent in the workforce relative to their husbands, traditional gender roles may be threatened. Because the notion of a female breadwinner does not fit with traditional gender stereotypes , women and men who hold more traditional gender views may feel that they must counteract the perceived violation of traditional gender norms outside of the home. This \"gender deviance neutralization\" (Bittman et al. 2003) predicts that women will perform the majority of housework in order to feel as though they are, in a sense, still fulfilling their \"womanly duties.\" Similarly, husbands of female breadwinners may believe that they have violated or failed in their masculine role by not being the primary economic provider. Thus, if husbands of female breadwinners do not believe they are adequately fulfilling their masculine role outside the home, they may preserve their masculinity by opting out of household duties, thereby diffusing feelings of gender deviance.\n\nOverall, exchange , equity , and bargaining theory can explain instances where traditional gender roles are enacted or the situation is gender neutral. However, when gender roles are violated, gender deviance neutralization appears to provide a more complete or nuanced explanation (Bittman et al. 2003; Brines 1994; Greenstein 2000) for the division of household labor . Neutralizing gender deviance does not necessarily account for the potential negative effects that female breadwinners may experience as a result of completing more than their fair share of housework , as well as working long hours outside of the home. The finding that female breadwinners often shoulder the bulk of the labor, both inside and outside of the home, is indicative of the influence that traditional gender roles may have on the division of paid and household labor (Weber 1998). Taken together, these theories help to better explain the outcomes associated with the division of labor between partners, as well as the effects of traditional gender roles on the interplay between males' and females' participation in the paid and household labor domains. Thus, it is important to consider the role of traditional gender norms in determining which spheres partners have historically specialized in, the exchanges and bargains which may exist to hold these spheres in place, and the outcomes related with violations of these norms, which may have serious implications for burdens placed on working women. These competing pressures may add to work\u2013family conflict for women in the workforce, compared to their male counterparts, which we discuss in greater detail below.\n\n## 10.7 Addressing Perceptions and Leveling the Playing Field\n\nIn 1996, the National Committee on Pay Equity established Equal Pay Day in order to emphasize the fact that women must work more than 15 months in order to earn the same salary that a man earns in 1 year (York 2013). As this chapter and other chapters in this book attest, the correlates or causes of this gender gap in pay are many, including differences in occupational distribution (with women clustered in lower paying occupations), differences in the human capital accumulation, and both intentional and unintentional gender discrimination. In work organizations, there have been significant legal and organizational changes that have begun to level the field of work among women and men. These changes have been crucial and necessary, yet they are not sufficient. We continue to find that women do not rise to the top of organizations nor seek or secure significant leadership positions. Most often these findings are attributed to women themselves, as either due to their lack of qualifications, motivation, or interest in such positions, or because these positions require long hours. However, even if all of these work issues were eliminated or managed, women (and men) would still face at least one societal or cultural issue that significantly affects male\u2013female differences in work experiences and outcomes\u2014i.e., that women spend a greater number of hours doing household (and caregiving) work, which in turn decreases the time and energy they can devote to work for pay (York 2013).\n\nSimilar to the workplace changes that resulted in greater diversity awareness and inclusion, enduring and pervasive changes in the 'private' sphere of home and housework must be implemented at multiple levels (e.g., individual, critical pairs or groups, organizations, communities) and in formal as well as informal ways (e.g., observable task changes, norms, power sharing, etc.) to provide a truly more level playing field for men and women at work and home. As we have noted throughout this chapter, work\u2013life tension and scarce time and economic resources are at the heart of work and nonwork gender bias and discrimination. Drawing from this literature, and extending it to the domain of household labor , there are a number of ways that work\u2013life conflict or interference might be addressed via the division of household labor. Using a framework of target and\/or source of change and formality of change (e.g., 3 \u00d7 2), we describe some suggestions for formal and informal changes at a number of levels.\n\n### 10.7.1 Individual-Level Changes\n\nIn order to begin to engage in the process of improving the gendered nature of work and family participation levels, we believe that it is important to focus on potential formal as well as informal changes at the individual level. We will discuss the implications of both of these individual-level changes below.\n\n#### 10.7.1.1 Informal Changes\n\nMost of us are guided by societal norms , including gender norms, interpreted and conveyed to us by our families and social context. However, both women and men as individuals must consciously examine and re-examine the relevance and usefulness of specific norms as they apply to themselves and within their evolving couple or family structures. Both women and men benefit by making clear choices and identifying what beliefs guide their behavior solely or largely by tradition or gender norms\/stereotypes , and whether or not they feel comfortable continuing to engage in those behaviors or, alternatively, changing these norms. For example, men are still socialized into breadwinner roles and women into caretaker roles. If a child or an elderly parent becomes ill and someone needs to take a day off from work, the gendered assumption is that the woman will make this sacrifice, as opposed to the man. For some individuals, this might work very well, but it is certainly worthwhile for both men and women to consider whether this is the best way for them individually to handle this situation.\n\nWe assert that the process of examining gender norms and expectations begins with individual self-awareness, and this process can be initiated well before formally entering the workplace full-time (e.g., during high school and college, if not before). This, of course, begs the question: Does this fall within the domain of work psychologists? We believe that it is absolutely a workplace psychology issue, and can be accomplished through assessments of vocational interests and assumptions, career counseling and self-assessments and self-awareness exercises. Work psychologists are well-positioned to design and develop tools to assist individuals in understanding the nature and the implications of their beliefs, assumptions, and norms regarding the roles of men and women at work and at home.\n\n#### 10.7.1.2 Formal Changes\n\nAt a more systemic level, there would be real and immediate benefits that could be realized by training women and men in effective decision-making , setting priorities, time management, and negotiation and conflict management, as they apply in work and nonwork settings. For example, it would be useful for both men and women to develop individual skills to navigate household labor , community service, and other roles, and to develop the ability to translate or apply work skills to the home domain. In particular, women and men can be trained to listen, depersonalize a situation, and step back (or empathize) to identify and address the important and unimportant household tasks and potential configurations for the division of these tasks. It would be useful to train men and women on how to develop methods and divide up tasks based on skill level, liking\/enjoyment, and avoiding the assignment of repetitive, time consuming tasks to one person (for example, by rotating these tasks or delegating them to a third party) . Men and women will also benefit from developing the perspective that household work division will not be a fixed phenomenon, but rather is likely to change as work or life situations change. For example, the demands on men's and women's time and energy will vary depending on developmental stages of children, or changes in the frequency of certain tasks (e.g., less laundry, more eating out, etc.).\n\n### 10.7.2 Couple-Level Changes\n\nIn addition to individual-level changes, there are a variety of informal and formal changes at the couple-level that can contribute to a more even distribution of work and nonwork responsibilities. We outline these couple-level changes below.\n\n#### 10.7.2.1 Informal Changes\n\nPrior to engaging in a formal relationship, couples can benefit substantially by discussing the division of tasks in a gender neutral way, particularly if the members of a couple have interests or preferences that might be accommodated by divisions of responsibility that are not strictly traditional (e.g., many men like to cook, but this is often stereotyped as a female activity). Couples can benefit by developing superordinate goals together which consider the division of paid and unpaid work as contributions to the well-being of the pair, and eventually the family, with the presence of children or expanded family needs. As children or elder care needs increase, couples can revisit the division of household labor ; gender roles often firmly reassert themselves once children (or elderly parents) arrive.\n\nIn considering how to develop more equitable roles for women, it is important to recognize that this sort of change creates special challenges for men, particularly if they have been socialized to cultures that emphasize rigid gender norms. A critical issue to negotiate is the best way to mitigate men's feelings of loss of a favored gender role (i.e., the more powerful role).\n\nIt is very useful to identify the tasks that can be completed with flexibility in terms of time and place (paying bills, etc.), to identify nonessential tasks, and to lengthen time between task completion (bi-weekly house cleaning vs weekly; weekly market shopping vs daily or thrice weekly, etc.). It also helps to identify core times that all family adults meet to divide or delegate household work. Secondly, it is also very useful for men and women to communicate openly and clearly about task roles and responsibilities, rather than making assumptions regarding what one or the other will do or is expected to do especially if these assumptions are based on traditional gender stereotypes.\n\n#### 10.7.2.2 Formal Changes\n\nDrawing from work\u2013life interventions in organizations, skills developed in paid work can be applied to navigating nonwork domains as well, especially in terms of seeking support and collaboration from one's spouse during potentially stressful times, much in the same way one might do with a supervisor at work. Supervisory support as well as coworker support are critical facets in mitigating work\/life conflicts and strains (Cleveland et al. 2006). It is important to develop skills to identify when one needs spousal support in managing housework and how to effectively obtain this support, as well as planning a course of action and a timeframe for accomplishing tasks.\n\n### 10.7.3 Work and Organization-Level Changes\n\nChanges at both the formal and informal level in organizations can complement individual- and couple-level changes to provide a balanced workplace. If we begin to address the systemic issues related to gender, work, family, and community, we can create a society which is more equitable for men and for women alike.\n\n#### 10.7.3.1 Informal Changes\n\nWork group members have beliefs and assumptions about work and family roles that can substantially influence the success or failure of attempts to create an equitable balance. It is important to recognize that these beliefs and assumptions are likely to vary, and that no one set of beliefs is necessarily right or correct. Coworkers can be a surprising source of resistance to efforts to change the balance of work and nonwork responsibilities. For example, suppose a husband increases his nonwork responsibilities and is not able to be present or engaged at work quite as often as in the past. Coworkers, particularly those who are socialized to more traditional work role expectations , might resent this and feel that they will have to pick up the slack. It is important for men and women who are trying to make changes in their relative responsibilities to recognize that their coworkers' concerns are not necessarily unreasonable, and to address them in realistic and respectful ways. Much in the same ways that the members of a couple often have to negotiate roles and responsibilities, members of work groups are likely to benefit from explicit and honest discussions of how changes in one person's workload might affect others he or she works with, and how challenges that might arise from these changes can be mitigated.\n\n#### 10.7.3.2 Formal Changes\n\nOrganizations are increasingly confronted with requests for flexibility. A great deal of empirical research has demonstrated that flexible work practices (e.g., offering flexibility in when and\/or where one works) offer many benefits (Baltes et al. 1999; Halpern 2005). Yet the simple implementation of flextime may lead to unintended consequences such as perceptions of inequity or complications in negotiating work vs. nonwork roles. For example, technological advances, increases in flexible work policies, and increases in telecommuting have culminated in the blurring of boundaries between work and home life. This is relevant to the division of labor in the household as gender role expectations may interact with time and location of work. Individuals who work at home may encounter situations in which \"being\" physically in their home for work results in their spouse or partner expecting that they will perform more domestic labor. In addition, among dual-career couples , the individual with a more flexible schedule may experience more family interference with work, because it is assumed that this member will have more time to complete household tasks. It is often the case that an individual with more flexibility is more likely to miss work and stay home to deal with family demands (e.g., a sick child). Hammer et al. (2005) conducted a longitudinal study to examine the role of using family-friendly workplace supports in relation to work and family outcomes. Their results indicated that women's use of workplace supports was related to higher levels of family-to-work conflict. Therefore, we suggest that although flexible work arrangements can be beneficial, they can also have a disproportionately negative effect in some situations.\n\nAnother limitation of flexible work arrangements is that they are only available to a minority of the workforce (Bulger and Fisher 2012; Golden 2001). Golden (2001) indicated that availability of flexible work practices varies in important ways by demographic, work, and job characteristics. Individuals in lower-level jobs, who are more likely to be women and ethnic minorities and less likely to be educated, are less likely to have flexibility. Men and women in higher level professional jobs may have more flexibility, but with higher level jobs come longer work hours and\/or have higher levels of responsibility such that it may be more difficult to use benefits.\n\nPart-time work would seem to offer a good solution to many of the problems of balancing work and nonwork responsibilities. It seems intuitive that working fewer hours should be associated with lower levels of work-to-family conflict and that one potential solution is to reduce work hours. However, organizations, particularly in the United States, do not offer many part-time options, particularly for professional occupations. Further, Barnett and Gareis (2002) suggest that \"success\" of part-time work in improving marital quality depends on the amount of control one has over nonwork tasks. Thus, part-time jobs may only help to deflate work\u2013family conflict if one has more control within the family sphere.\n\nSupporting the idea that work and family domains may be dependent on one another in terms of overall time spent and the outcomes associated with time spent in either domain, Duxbury and Higgins (1991) found that the hours an individual works are not just those recorded on a time sheet, but rather reflect a range of responsibilities at the individual and family levels. Unfortunately, the demands of individuals, couples, families and organizations are not always in sync. At work, there are teams, committees, and departments with coworkers and others who depend on individual employees (and upon whom the individual employee depends as well). Therefore, there are times when \"face-time\" (being physically present at work) is simply essential, and other times where it is a norm or more subtle expectation but not a genuine requirement. Similarly, in nonwork settings some activities may be put off or shared with other family members, but other activites must be completed by a given family member at a given point in time (e.g., mothers engaging in breastfeeding).\n\nOne of the best ways to approach potential conflicts between the demands of different spheres is with frequent, tempered and informed communication. Many conflicts can be defused by working through questions of whether the circumstances that drive conflict come about because of what is essential vs. what is customary. For example, suppose staff meetings have always been held at 3 pm. If some workers take on responsibility for school-age children, this can be a source of time-based conflict, because it coincides with the time many children are dismissed from school. If this meeting time is customary, but another can be reasonably found, it might be possible to reduce or even eliminate the conflict by changing the meeting time. However, it is worth recognizing that asking people to change, even when there is little real cost to making a change, may feel like an imposition. Thus, these feelings should be recognized and respected.\n\n## 10.8 Future Research\n\nBased on the issues and empirical research described in this chapter, we offer a few recommendations for additional research . First, we issue a call for more work-life research to investigate nonwork roles and responsibilities in addition to family caregiving. Much of the extant research has focused on family roles, but seems criterion deficient in understanding the true scope of the impact of work on workers' nonwork lives, as well as how nonfamily-related issues may interfere with work. It may be helpful to understand the specific household tasks or chores in which work is more likely to interfere, or that may be more likely to interfere with work .\n\nSecondly, additional research is needed among workers in nonmanagerial and professional occupations. This is due to the fact that a preponderance of work\/family research focuses on workers in managerial and professional occupations. Workers in hourly and nonprofessional settings are likely to have less flexibility in time and location of work, which may serve to exacerbate problems navigating the division of household labor . Although some work-life research has studied workers in nonprofessional jobs (e.g., Grzywacz et al. 2002, who found higher rates of conflict among workers in service jobs; Grzywacz et al. 2006, who studied work-family issues among Mexican migrant farmworkers; and Grandey et al. 2007, who studied male blue-collar hourly workers), additional work is needed .\n\nNext, the majority of work\/life research has focused on individuals. In the past decade, work\/family research has expanded a great deal to study couples or dyads (e.g., spillover from one spouse to another; Westman 2001; Westman and Etzion 2005). The issues of work and the household division of labor point to a critical need to study households as a unit. Therefore, it is important to consider multiple units or levels of analysis when posing our research questions.\n\nAs previously described, empirical evidence points to the efficacy of flexible work arrangements as being beneficial to help individuals manage work and life demands. However, workers with more flexibility may end up managing more domestic and caregiving responsibilities . Additional research is needed to increase our understanding about boundary conditions regarding when flexibility helps vs. hinders. The meta-analysis by Baltes et al. (1999) indicated that too much flexibility about flexible work practices is associated with more negative outcomes than more rigid or structured flexible work arrangements. Additional research is needed to examine the impact of flexible work arrangements within the home and, specifically, with the division of household responsibilities. Systematic evaluation of organizational interventions designed to reduce work-to-life conflict and life-to-work conflict are needed. The issues discussed in this chapter point to the need for attention to more than just family caregiving roles, but household tasks as well .\n\nFinally, additional research should be conducted to identify individual differences and processes that are related to successfully managing and negotiating household division of labor. What are the factors that make for equality and peace on the home front? Learning this may be fruitful for helping to offer additional recommendations on narrowing the gender gap at home .\n\n## 10.9 Conclusion\n\nIs the glass half full (progress) or half empty (gender inequalities and disparities)? Although we have begun to level the playing field between men and women with regard to educational attainment and labor force participation, research still highlights inequality on the home front. Women are more likely than men to spend more time caring for other family members, in addition to performing more household tasks. The trend is moving in a positive direction toward more balance or equity between men and women, but there is still a great deal of room for improvement. Time is a finite resource. As long as women have more caregiving responsibilities and spend more time involved in domestic work than men, it is unlikely that the gender discrepancies will disappear.\n\nReferences\n\nAllen, T. D., Herst, D. L., Bruck, C. S., & Sutton, M. (2000). Consequences associated with work-to-family conflict: A review and agenda for future research. Journal of Occupational Health Psychology, 5, 278\u2013308.PubMedCrossRef\n\nAnderson, S. E., Coffey, B. S., & Byerly, R. T. (2002). Formal organizational initiatives and informal workplace practices: Links to work-family conflict and job-related outcomes. Journal of Management, 28, 787\u2013810.\n\nAtkinson, J., & Huston, T. L. (1984). Sex role orientation and division of labor early in marriage. Journal of Personality and Social Psychology, 46, 330\u2013345.CrossRef\n\nBaltes, B. B., Briggs, T. E., Huff, J. W., Wright, J. A., & Neuman, G. A. (1999). Flexible and compressed workweek schedules: A meta-analysis of their effects on work-related criteria. Journal of Applied Psychology, 84, 496\u2013513.CrossRef\n\nBarnett, R. C., & Gareis, K. C. (2002). Full-time and reduced-hours work schedules and marital quality: A study of female physicians with young children. Work and Occupations, 29, 364\u2013379.CrossRef\n\nBecker, P. E., & Moen, P. (1999). Scaling back: Dual-earner couples work-family strategies. Journal of Marriage and the Family, 61, 995\u20131007.CrossRef\n\nBianchi, S. M., Milkie, M., Sayer, L. C., & Robinson, J. (2000). Is anyone doing the housework? Trends in the gender division of household labor. Social Forces, 79, 191\u2013228.CrossRef\n\nBianchi, S. M., Robinson, J. P., & Milkie, M. A. (2006). Changing rhythms of American life. New York: Russell Sage.\n\nBittman, M., England, P., Sayer, L., Folbre, N., & Matheson, G. (2003). When does gender trump money? Bargaining and time in household work. American Journal of Sociology, 109, 186\u2013214.CrossRef\n\nBlair, S. L., & Lichter, D. (1991). Measuring the division of household labor: Gender segregation of housework among American couples. Journal of Family Issues, 12, 91\u2013113.CrossRef\n\nBrines, J. (1994). Economic dependency, gender, and the division of labor at home. The American Journal of Sociology, 100, 652\u2013688.CrossRef\n\nBudig, M. (2004). Feminism and the family. In J. Scott, J. Treas, & M. Richards (Eds.), The Blackwell companion to the sociology of families (pp. 416\u2013434). Oxford: Blackwell.\n\nBudig, M., & England, P. (2001). The wage penalty for motherhood. American Sociological Review, 66, 204\u2013224.CrossRef\n\nBulger, C. A., & Fisher, G. G. (2012). Ethical imperatives of work-life balance. In N. P. Reilly, M. J. Sirgy, & C. A. Gorman (Eds.), Work and quality of life: Ethical practices in organizations (pp. 181\u2013200). New York: Springer.CrossRef\n\nBuunk, B. P., Kluwer, E. S., Schuurman, M. K., & Siero, F. W. (2000). The division of labor among egalitarian and traditional women: Differences in discontent, social comparison, and false consensus. Journal of Applied Social Psychology, 30, 759\u2013779.CrossRef\n\nCatalyst. (2012). 2012 Catalyst census: Fortune 500 women board directors. http:\/\/\u200bwww.\u200bcatalyst.\u200borg\/\u200bknowledge\/\u200b2012-catalyst-census-fortune-500. Accessed 4 Jan 2014.\n\nCleveland, J. N., McCarthy, A., & Jones, A. (September 2006). The influence of managerial support and family structure on work-family outcomes. Presented at the Irish Academy of Management conference, Cork, Ireland.\n\nColtrane, S. (1996). Family man: Fatherhood, housework and gender equity. Oxford: Oxford University.\n\nColtrane, S. (2000). Research on household labor: Modeling and measuring the social embeddedness of routine family work. Journal of Marriage and the Family, 62, 1208\u20131233.CrossRef\n\nCurrent Population Survey. (2011). Laborforce statistics from the currently population survey. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200b. Accessed 15 Dec 2013.\n\nde Graaf, P. M., & Kalmijn, M. (2006). Divorce motives in a period of rising divorce: Evidence from a Dutch life-history survey. Journal of Family Issues, 27, 487\u2013505.CrossRef\n\nDeMaris, A., & Longmore, M. A. (1996). Ideology, power, and equity: Testing competing explanations for the perceptions of fairness in household labor. Social Forces, 74, 1043\u20131071.CrossRef\n\nDeutsch, F. M. (2001). Equally shared parenting. Current Directions in Psychological Science, 10, 25\u201328.CrossRef\n\nDeutsch, F. M., & Saxon, S. E. (1998). Traditional ideologies, nontraditional lives. Sex Roles, 38, 331\u2013362.CrossRef\n\nDuxbury, L. E., & Higgins, C. A. (1991). Gender differences in work-family conflict. Journal of Applied Psychology, 76, 60\u201374. doi:10.1037\/0021-9010.76.1.60.CrossRef\n\nEhrenreich, B., & English, D. (1978). For her own good: Two centuries of the experts' advice to women. New York: Random House.\n\nEly, R., & Padavic, I. (2013). Work-family conflict is not the problem: Overwork is. Blog for The Huffington Post and Stanford University's Michelle R. Clayman Institute for gender research in conjunction with the latter's redesigning, redefining work summit (November, 7). http:\/\/\u200bgender.\u200bstanford.\u200bedu\/\u200bnews\/\u200b2013\/\u200bwork-family-conflict-not-problem-overwork.\n\nErnst Kossek, E., & Ozeki, C. (1998). Work-family conflict, policies, and the job-life satisfaction relationship: A review and directions for organizational behavior-human resources research. Journal of Applied Psychology, 83, 139\u2013149.CrossRef\n\nFisher, G. G., Bulger, C. A., & Smith, C. S. (2009). Beyond work and family: A measure of work\/non-work interference and enhancement. Journal of Occupational Health Psychology, 14, 441\u2013456.PubMedCrossRef\n\nGauthier, A. H., Smeeding, T. M., & Furstenberg, F. F. (2004). Are parents investing less time in children? Trends in selected industrialized countries. Population and Development Review, 30, 647\u2013671.CrossRef\n\nGolden, L. (2001). Flexible work schedules: Which workers get them? American Behavioral Scientist, 44, 1157\u20131178.CrossRef\n\nGrandey, A. A., Cordeiro, B. L., & Michael, J. H. (2007). Work-family supportiveness organizational perceptions: Important for the well-being of male blue-collar hourly workers?. Journal of Vocational Behavior, 71, 460\u2013478.CrossRef\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources and conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGreenstein, T. N. (2000). Economic dependence, gender, and the division of labor in the home: A replication and extension. Journal of Marriage and the Family, 62, 322\u2013335.CrossRef\n\nGrzywacz, J. G., Almeida, D. M., & McDonald, D. A. (2002). Work-family spillover and daily reports of work and family stress in the adult labor force. Family Relations, 51, 28\u201336.CrossRef\n\nGrzywacz, J. G., Quandt, S. A., Early, J., Tapia, J., Graham, C. N., & Arcury, T. A. (2006). Leaving family for work: Ambivalence and mental health among Mexican migrant farmworker men. Journal of Immigrant and Minority Health, 8, 85\u201397.PubMedCrossRef\n\nHalpern, D. F. (2005). How time-flexible work policies can reduce stress, improve health, and save money. Stress and Health, 21, 157\u2013168.CrossRef\n\nHammer, L. B., Neal, M. B., Newsom, J., Brockwood, K. J., & Colton, C. (2005). A longitudinal study of the effects of dual-earner couples' utilization of family-friendly workplace support on work and family outcomes. Journal of Applied Psychology, 90, 799\u2013810.PubMedCrossRef\n\nHochschild, A. R., & Machung, A. (1989). The second shift: Working parents and the revolution at home. New York: Penguin.\n\nJacobs, J. A., & Gerson, K. (2004). The time divide: Work, family, and gender equality. Cambridge: Harvard University Press.\n\nJuster, F. T., Ono, H., & Stafford, F. P. (2003). An assessment of alternative measures of time use. Sociological Methodology, 33, 19\u201354.CrossRef\n\nKamo, Y. (2000). \"He said, she said\": Assessing discrepancies in husbands' and wives' reports on the division of household labor. Social Science Research, 29, 459\u2013476.CrossRef\n\nKlumb, P., Hoppmann, C., & Staats, M. (2006). Division of labor in German dual-earner families: Testing equity theoretical hypotheses. Journal of Marriage and the Family, 68, 870\u2013882.CrossRef\n\nKluwer, E. S. (1998). Responses to gender inequality in the division of family work: The status quo effect. Social Justice Research, 11, 337\u2013357.CrossRef\n\nKluwer, E. S., Heesink, J. A. M., & Van de Vliert, E. (1996). Marital conflict about the division of household labor and paid work. Journal of Marriage and the Family, 58, 958\u2013969.CrossRef\n\nKluwer, E. S., Heesink, J. A. M., & Van de Vliert, E. (2000). The division of labor in close relationships: An asymmetrical conflict issue. Personal Relationships, 7, 263\u2013282.CrossRef\n\nLavee, Y., & Katz, R. (2002). Division of labor, perceived fairness, and marital quality: The effect of gender ideology. Journal of Marriage and the Family, 64, 27\u201339.CrossRef\n\nLee, Y., & Waite, L. J., (2005). Husbands' and wives' time spent in housework: A comparison of measures. Journal of Marriage and the Family, 67, 328\u2013336.CrossRef\n\nLennon, M. C., & Rosenfield, S. (1994). Relative fairness and the division of housework: The important of options. American Journal of Sociology, 100, 506\u2013531.CrossRef\n\nLiss, M. (2013). Inequality in the division of household labor and child care: Cause, consequences, and how to change. In M. A. Paludi (Ed.), Psychology for business success: Juggling, balancing, and integrating work and family roles and responsibilities (pp. 19\u201342). Denver: Praeger.\n\nLively, K. J., Powell, B., Geist, C., & Steelman, L. C. (2008). Equity among intimates: Applying equity theory to the family. Advances in group processes, 25, 87\u2013115.CrossRef25005-8)\n\nLu, Z., Maume, D. J., & Bellas, M. L. (2000). Chinese husbands' participation in household labor. Journal of Comparative Family Studies, 31, 191\u2013215.\n\nLundberg, S., & Pollak, R. A. (1993). Separate spheres bargaining and the marriage market. The Journal of Political Economy, 101, 988\u20131010.CrossRef\n\nLundberg, S., & Pollak, R. A. (1996). Bargaining and distribution in marriage. Journal of Economic Perspectives, 10, 139\u2013158.CrossRef\n\nMacEwen, K. E., & Barling, J. (1994). Daily consequences of work interference with family and family interference with work. Work & Stress, 8, 244\u2013254.CrossRef\n\nMattingly, M., & Liana, C. S. (2006). Under pressure: Gender difference in the relationship between free time and feeling rushed. Journal of Marriage and the Family, 68, 205\u2013221.CrossRef\n\nMcElroy, M. (1990). The empirical content of Nash-bargained household behavior. Journal of Human Resources, 25, 559\u2013583.CrossRef\n\nMolm, L., & Cook, K. (1995). Social exchange and exchange networks. In K. Cook, G. Fine, & J. House (Eds.), Sociological perspectives on social psychology (pp. 209\u2013235). Needham Heights: Allyn & Bacon.\n\nPerry-Jenkins, M., & Folk, K. (1994). Class, couples, and conflict: Effects of the division of labor on assessments of marriage in dual-earner families. Journal of Marriage and the Family, 56, 165\u2013180.CrossRef\n\nPina, D. L., & Bengston, V. L. (1993). The division of household labor and wives' happiness: Ideology, employment, and perceptions of support. Journal of Marriage and the Family, 55, 901\u2013912.CrossRef\n\nPress, J., & Townsley, E. (1998). Wives' and husbands' housework reporting: Gender, class, and social desirability. Gender and Society, 12, 188\u2013218.CrossRef\n\nPrince-Cooke, L. (2006). \"Doing\" gender in context: Household bargaining and risk of divorce in Germany and the United States. American Journal of Sociology, 112, 442\u2013472.CrossRef\n\nPyke, K., & Coltrane, S. (1996). Entitlement, obligation, and gratitude in family work. Journal of Family Issues, 17, 60\u201382.CrossRef\n\nSandberg, S. (2013). Lean in: Women, work, and the will to lead. New York: Random House.\n\nSaxbe, D. E., Repetti, R., & Graesch, A. (2011). Time spent in housework and leisure: Links with parent's physiological recovery from work. Journal of Family Psychology, 25, 271\u2013282.PubMedCrossRef\n\nSayer, L. C. (2005). Gender, time and inequality: Trends in women's and men's paid work, unpaid work and free time. Social Forces, 84, 289\u2013303.CrossRef\n\nSayer, L. C., Bianchi, S. M., & Robinson, J. P. (2004). Are parents investing less in children? Trends in mothers' and fathers' time with children. American Journal of Sociology, 110, 1\u201343.CrossRef\n\nSayer, L. C., England, P, Bittman, M., & Bianchi, S. M. (2009). How long is the second (plus first) shift? Gender differences in paid, unpaid, and total work time in Australia and the United States. Journal of Comparative Family Studies, 40, 523\u2013545.\n\nTetrick, L. E., & Buffardi, L. C. (2006). Measurement issues in research on the work-home interface. In F. Jones, R. J. Burke, M. Westman (Eds.), Work-life balance: A psychological perspective (pp. 90\u2013114). New York: Psychology Press.\n\nTwiggs, J. E., McQuillan, J., & Ferree, M. M. (1999). Meaning and measurement: Reconceptualizing measure of the division of household labor. Journal of Marriage and the Family, 61, 712\u2013724.CrossRef\n\nU.S. Bureau of Labor Statistics. (2012a). http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bopub\/\u200bted\/\u200b2012\/\u200bted_\u200b20120110.\u200bhtm. Accessed 30 Dec 2013.\n\nU.S. Bureau of Labor Statistics. (2012b). http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bopub\/\u200bted\/\u200b2009\/\u200bted_\u200b20090807.\u200bhtm. Accessed 30 Dec 2013.\n\nVan Willigen, M., & Drentea, P. (2001). Benefits of equitable relationships: The impact of sense and fairness, household division of labor, and decision making power on perceived social support. Sex Roles, 44, 571\u2013597.CrossRef\n\nWayne, J., Musisca, N., & Fleeson, W. (2004). Considering the role of personality in the work-family experience: Relationships of the big five to work-family conflict and facilitation. Journal of Vocational Behavior, 64, 108\u2013130.CrossRef00035-6)\n\nWeber, L. (1998). A conceptual framework for understanding race, class, gender, and sexuality. Psychology of Women Quarterly, 22, 13\u201332.CrossRef\n\nWestman, M. (2001). Stress and strain crossover. Human Relations, 54, 717\u2013751.CrossRef\n\nWestman, M., & Etzion, D. (2005). The crossover of work-family conflict from one spouse to the other. Journal of Applied Social Psychology, 35, 1936\u20131957.CrossRef\n\nWilliams, J. C. (2013). Why men work so many hours. Harvard Business Review. HBR Blog Network. http:\/\/\u200bhbrorg\/\u200b2013\/\u200b05\/\u200bwhy-men-work-so-many-hours\/\u200b. Accessed 4 Jan 2014.\n\nYork, A. (2013). Why men should share equally in housework. Special to CNN. http:\/\/\u200bwww.\u200bcnn.\u200bcom\/\u200b2013\/\u200b04\/\u200b08\/\u200bopinion\/\u200byork-equal-housework\/\u200b\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_11\n\n# 11. On-Demand: When Work Intrudes upon Employees' Personal Time\u2014Does Gender Matter?\n\nAngela R. Grotto1\n\n(1)\n\nDepartment of Management and Marketing, Manhattan College, De La Salle Hall 507, 4513 Manhattan College Parkway, 10471 Riverdale, NY, USA\n\nAngela R. Grotto\n\nEmail: angela.grotto@manhattan.edu\n\nKeywords\n\nGenderWork\u2013familyOn-demand workExecutivesManagementProfessionalTechnologyCommunicative technologiesBoundary managementRole identity\n\n## 11.1 Introduction\n\nExecutive, managerial, and professional jobs (EMPs) are often accompanied by high earnings, job autonomy , and responsibility. Not surprisingly, the average weekly work hours for EMPs exceeds that of other employees. According to the Current Populations Survey (CPS) (2012), the average weekly work hours for both men and women in managerial and professional occupations was the highest compared to all other occupations, at 45 and 42 hours per week, respectively. Not only do EMPs work the most weekly hours, but they also have excessive work hours, that is, 49 plus hours per week (Jacobs and Gerson 2004; Kuhn and Lozano 2008). In fact, the 2012 CPS shows that 21 % of men and 14 % of women in managerial and professional jobs worked 49 plus hours per week, compared to just 14 % for men and 7 % for women in all other occupations. Furthermore, when comparing 2012 CPS data to 2010 CPS data (2010), the weekly work hours of both male and female EMPs have increased over the past 10 years.\n\nThe high level of responsibility associated with these occupations certainly requires long work hours, but that alone cannot account for such excessive levels. Are EMPs choosing to work long hours? If so, what motivates them and allows them to do so? Or is the changing nature of their work forcing them to work more? What other external conditions might be to blame? Many companies encourage a culture of \"face-time\" \u2014an organizational norm for visibility in the workplace with employees' physical presence indicating their true level of commitment or contribution to organizations (Bailyn 1993; Perlow 1995). Employees working for these time-demanding companies may feel pressure to put in long hours at the office. However, EMP work is no longer limited to the physical workplace (i.e., the office). Radical advancements in communicative technologies (CT; e.g., smartphones) have dramatically increased the flexibility of jobs but also allow work to intrude on employees' personal time, making face-time far less relevant. EMPs who use technology on a daily basis essentially work \"boundaryless\" jobs (Voydanoff 2009) with no limits, which can lead to excessive work hours.\n\nA study conducted by the Center for Creative Leadership (CCL) revealed that managers and professionals who use CT extend their work beyond normal work hours (Deal 2013). Managers and professionals who carry smartphones reported interacting with work 13.5 hours each work day and 72 hours per week (Deal 2013). Among managers and professionals working 50 plus hours per week with their smartphones, 15 % reported working 7 days a week, with just over half of men (52 %) and 42 % of women working on weekends (Deal 2013).\n\nDespite the fact that flexible jobs may help employees balance their work roles with their various personal roles (e.g., parent, spouse, friend, community member; e.g., Kossek et al. 2009), the CCL survey results clearly demonstrate that job flexibility can be quite dysfunctional (Clark 2000). Smartphones allow for relentless \"boundary violations\" at home, as work constantly breaches the boundaries surrounding employees' personal lives (Kreiner et al. 2009). The use of CT in jobs that are already very flexible exacerbates organizational norms for EMPs to work long hours, morphing face-time into something far worse. Presently, EMPs are working in \"on-demand\" jobs\u2014jobs that require employees to be constantly available and accessible to their employers and clients during nonwork hours via technology . Visibility at the office seems no longer sufficient for EMPs to demonstrate their loyalty and thus advance and be rewarded. On-demand is the new face-time.\n\nThe current chapter dissects the factors giving rise to on-demand jobs and whether these factors differ for men and women in EMP jobs. The focus is placed on EMPs because of the demanding nature of their work and their excessive work hours. Other employees experience work intrusions during their personal time via technology, but the pressure to respond to these intrusions is not nearly as great as that of EMPs. Also discussed are the consequences of being on-demand and whether there are gender differences. On-demand jobs can be likened to being on-call. Regardless of whether the employee is actually called in or not, the mere possibility of being disturbed by work-related matters during personal time has been linked with negative consequences such as negative moods (Bamberg et al. 2012). Similarly, when work calls and e-mails intrude on evenings and weekends, it may significantly undermine employees' personal lives (e.g., emotional exhaustion) and work lives (e.g., decreased productivity) . This chapter argues that the strains of on-demand jobs transcend gender, and so both men and women suffer the consequences. As individuals can only do so much to manage their on-demands jobs, companies must intervene to help ease the pressure.\n\n## 11.2 The Weakening of Borders Surrounding Work and Personal life\n\nThe changing boundaries surrounding the worlds of work and nonwork play a critical role in the rise of on-demand jobs. Three different types of boundaries exist around work (Ashforth et al. 2000; Clark 2000; Nippert-Eng 1996). Spatial boundaries define where work behavior takes place (e.g., work duties are performed at the office). Temporal boundaries define when the work role is performed (e.g., jobs are performed between 9:00 a.m. and 5:00 p.m.). Psychological boundaries dictate when and where work thoughts, behaviors, and emotions are appropriate (e.g., assertiveness is appropriate at the office but not at home).\n\nThe flexibility and permeability of a work boundary determines its strength. Flexible work boundaries allow jobs to be to be carried out in various settings at various times, while inflexible boundaries constrain when and where work can be performed (e.g., Beach 1989; Ritchter 1992). For instance, most nurses perform their jobs within the boundaries of hospitals, and manufacturing workers perform their jobs within factories. Permeable borders surrounding various nonwork domains allow an individual to be involved in the work role while physically located in a nonwork domain (e.g., the home; Ashforth et al. 2000; Nippert-Eng 1996). For instance, a permeable home boundary allows an individual to receive job-related phone calls or e-mails while at home. Together, flexible and permeable (i.e., weak) boundaries allows work to permeate other life domains.\n\nEconomic, organizational, and job-related changes are responsible for the weakening of employees' work and nonwork boundaries. First, the flexible nature of jobs coupled with sophisticated CT give employees the ability to be on-demand. Second, EMPs who are heavily invested in their jobs are willing to use CT to be on-demand. Lastly, economic instability, employers' high expectations, and intense work pressures compel EMPs to be on-demand. I argue that the on-demand experiences of men and women in EMP roles are more similar than different. Not only are both male and female EMPs equally exposed to economic, organizational, and job-related pressures, but they also tend to hold egalitarian attitudes toward gender roles, use CT as part of their jobs, and highly value their jobs. These attributes make EMPs more willing and able to be on-demand, regardless of their gender.\n\n## 11.3 On-Demand Jobs: Willing and Able\n\n### 11.3.1 The Ability to be On-Demand\n\nTwo key aspects of work have weakened job borders and thereby give EMPs the ability to be on-demand. Job flexibility (i.e., control over where and when one works) and advanced CT and have made EMP jobs substantially more flexible. EMPs have always had a considerable amount of autonomy over how they get their jobs done (Voydanoff 1988, 2004). But many EMPs have newer forms of autonomy: place autonomy (control over work locations) and time autonomy (control over work scheduling; Kossek et al. 2006). Place and time autonomy give EMPs the ability to work in various settings (e.g., home, car, outdoors) and at various times (e.g., evenings, weekends, during vacation). Changes in the nature of work and CT are largely responsible for these newer forms of autonomy.\n\nMany jobs are now knowledge-based, requiring complex mental skills (Kelloway and Barling 2000). Jobs that require the use of equipment or materials at the worksite (e.g., manufacturing jobs) and work that involves the use of technical or manual skills in addition to mental skills (e.g., health care) are typically restricted to the workplace (Finegold and Frenkel 2006). However, work that primarily involves cognitive skills can easily be performed outside of the workplace and beyond normal work hours (Ezzedeen and Swiercz 2007). The work of EMPs tends to be highly knowledge-intensive. For instance, the strategic nature of executive jobs requires very complex mental proficiencies, such as the ability to forecast. Professionals possess high-levels of cognitive skills in specialized areas of knowledge, such as information systems data analysts. These skills can be used anywhere and at any time. Since knowledge work is not restricted to the workplace, EMPs can be involved in their work while in other life domains (e.g., home) during nonwork hours (e.g., evenings). For instance, the information systems data analyst can use his laptop to perform his job while sitting on the couch with his family during movie night.\n\nMajor advancements in CT have radically enhanced the flexibility of EMP jobs (e.g., Jackson 2002). For example, while at home, EMPs can connect to a virtual network to access work files. Also, most EMPs today have smartphones that allow them to be constantly connected to their work e-mail accounts. So CT makes jobs mobile, allowing employees to perform their work in various nonwork locations (e.g., home, cars, airplanes, gyms, restaurants, hotels, etc.) during any hour of the day (e.g., early mornings, late nights, weekends; e.g., Wikstrom et al. 1998). Indeed, 60 % of EMPs who use a smartphone for business work between 13.5 and 18.5 hours per day, while only 29 % of those who do not use a smartphone are connected with work that many hours on average (Deal 2013). This significant job flexibility also means that EMPs can be on-demand to their employers and clients via CT. In summary, the growth of knowledge work and advancements in CT have drastically increased the flexibility of EMP jobs, thereby creating on-demand EMP jobs.\n\n### 11.3.2 Gender Differences in On-Demand Abilities\n\nWhen considering employees at all job levels, men report more job autonomy than women (Day et al. 2012; Park and Jex 2011). But are there gender differences at the highest ranks? Given that common job features impinge on the lives of both men and women in similar ways (Jacobs and Gerson 2004), female EMPs should possess the same level of job autonomy as their male counterparts. Indeed, past research has shown that male and female EMPs have the ability to work long hours beyond the workplace (e.g., Mennino et al. 2005; Schieman and Reid 2008). These findings suggest that men and women in EMP jobs have similar levels of autonomy that give them the ability to be on-demand.\n\nSimilarly, CT gives all EMPs the ability to push the limits of their work roles , regardless of their gender. Although there is evidence suggesting that men are more likely to use job-related CT while at home (e.g., Day et al. 2012; Park and Jex 2011), none of the extant research has focused on EMPs exclusively. Given the broad responsibilities and high demands of EMP jobs, men and women may be equally likely to use CT to perform their EMP roles beyond the workplace. In fact, the results of a CCL survey of managers and professionals showed no gender differences in job-related smartphone use (Deal 2013). Together, extensive job autonomy and frequent CT use during off-hours enable both male and female EMPs to be on-demand.\n\nSome may argue that despite technology and job flexibility, the demands that women face at home may prevent them from being on-demand to their jobs. However, employment tends to be associated with egalitarian attitudes toward gender roles at home among women (Fan and Marini 2000). In fact, employed women who are highly paid and well-educated are more likely to equally share household labor with their partners, as they have more power in their personal relationships (Cunningham 2007; Mannino and Deutsch 2007; Parkman 2004). So, rather than being the primary caretaker, these women are more likely to equally share their domestic roles with their partners. When both partners are unwilling to follow gender-normed behavior, it is even more likely that housework and child care duties are shared (Donald and Linington 2008). This team-based approach at home means that female EMPs can devote time and energy to their valued work roles during nonwork hours, which includes responding to work-related phone calls and e-mails. Nevertheless, although male and female EMPs are both able to be on-demand, their willingness to do so may vary, based on personal preferences (Park et al. 2011).\n\n### 11.3.3 The Willingness to be On-Demand: Role Identities\n\nAs work is able to exceedingly intrude on personal life, employees must decide whether to allow these intrusions or not. Some employees make conscious efforts to keep work separated from nonwork through the creation and maintenance of boundaries (e.g., Lambert 1990). For example, an accountant who prefers to keep her work separated from her home life might choose to turn off her e-mail notifications during the weekends. She may also have a personal rule that she is always home by 6:00 p.m. on work days. She might even avoid thinking about work while at home (a psychological boundary). This active maintenance of boundaries can help employees uphold order in their interdependent work and nonwork lives (e.g., Weick 1979).\n\nWho might keep work separated from their personal life? Based on an individual's beliefs and values about what roles do and do not belong in a particular domain, one may idiosyncratically construct boundaries (Douglas 1975). For instance, individuals who highly value their time with family may believe that work does not belong at home and therefore devise strategies to prevent job contacts while at home. For example, these individuals might leave their company-provided smartphones at the office over the weekend, or they might send messages to their coworkers letting them know that they will be unavailable over the weekend. Indeed, a recent study showed that employees who preferred to keep their work lives separated from their home lives were more likely to psychologically detach from their work roles during off-hours compared to those who preferred to integrate their roles (Hahn and Dormann 2013).\n\nConversely, individuals who prefer to integrate work with their personal lives do not tie their jobs to the workplace (Ashforth et al. 2000; Nippert-Eng 1996). Instead, these individuals may consider it acceptable for work to interrupt their personal time and thus are quite willing to be on-demand during evenings and weekends. Who might prefer to integrate work and nonwork roles? Individuals who strongly identify with their work roles choose to be highly involved in their work, meaning they are cognitively preoccupied with, engaged in, and concerned with their jobs (Paullay et al. 1994). These individuals are internally motivated to extend their workdays by spending more effort on work-related activities and working longer hours (Fenner and Renn 2004). In fact, research has shown that employees who are highly involved in their jobs were more likely to use CT after work hours to perform work-related activities and allow work intrusions at home (Hecht and Allen 2009; Major et al. 2002), and they allowed work to permeate into other areas of their lives (Olson-Buchanan and Boswell 2006). Given the high-status of their work roles and their investments in their careers, EMPs should highly identify with their work roles, making them very willing to use CT during nonwork hours and be involved in their jobs at all hours of the day.\n\n### 11.3.4 Gender Differences in Work Role Identity Among EMPs\n\nAre there gender differences in the extent to which male and female EMPs identify with their high-status work roles? Women have historically prioritized family over work, while men often prioritized their jobs over others aspects of their lives (Blair-Loy 2009; Kelly et al. 2010; Moen and Roehling 2005) . Many women today still assume the role of primary caretaker, whether for children or the elderly, even if they are working full-time (Milkie et al. 2002). This perspective suggests that men are more willing to be available to work during their personal time than women. Yet, individuals in high-status careers, regardless of gender, tend to highly identify with their work roles, suggesting that female EMPs are just as willing as male EMPs to be on-demand. Research in this area is contradictory, with some studies showing that women prefer to integrate roles more than men (Kossek et al. 2009) and others showing women with a greater desire for role segmentation (Rothbard et al. 2005). But no studies have focused on EMPs, so it is not clear whether men and women in high-status positions differ in their role identities and thus their willingness to integrate work and nonwork roles.\n\nGender roles may be more relevant than gender when considering the relationship between role identity and on-demand behavior among EMPs. Egalitarian individuals tend to believe that men and women should identify equally with their work and personal life roles, which includes devoting equal time to paid work and housework (Hochschild 1989). As employed women are likely to hold egalitarian attitudes toward gender roles, particularly those who are highly paid and highly educated, it is plausible that female EMPs are just as willing as male EMPs to engage in on-demand behavior during nonwork hours.\n\n### 11.3.5 The Willingness to be On-Demand: Organization-Based Self-Esteem\n\nExecutives, managers, and professionals may be willing to be on-demand during nonwork hours based on their individual needs. Organization-based self-esteem (OBSE) is a specific form of self-esteem that considers employees' self-evaluations specifically within the context of work . Employees with high OBSE believe that they can satisfy their needs by participating in roles within the organization (Pierce et al. 1989), and they also believe in their self-worth and competence as organizational members (Bowling 2010). Individuals in high-status positions at work may have higher levels of OBSE than others. Given the autonomy , complexity, and challenge of EMP jobs, these individuals are likely to believe that they are competent, valuable, and capable members of their organizations (Pierce et al. 1989). The work roles of EMPs not only fulfill basic needs (e.g., my work role serves an economic need), but satisfy higher-level needs (e.g., being a psychologist is fulfilling; Ashforth et al. 2000). As the work role serves as a source of esteem and fulfills higher-level needs, EMPs with high OBSE may be willing to invest a lot of their personal time into their jobs.\n\nCompensation may be another source of OBSE for EMPs, as it can communicate to an individual his\/her worth or value to the organization (Thierry 2001). Moreover, pay-level might influence an employee's beliefs about his\/her worthiness as an organizational member. In a study that took into account employees' length of company tenure and changes in pay, pay-level was significantly related to OBSE (Gardner et al. 2004). EMPs are at a higher pay-level than most other employees, and these high pay-levels might trigger feelings of self-worth that motivate them to invest personal time in their work roles.\n\nOn-demand behavior can be very reinforcing for EMPs. After-hour job contacts can make them feel needed, engendering feelings of self-worth (Bowling 2010). Each time an EMP is contacted after-hours for help, his\/her self-esteem gets a boost and thus she\/he is further motivated to invest in the work role. Also, in a recent study, employees who perceived intruding work matters as important reported high-levels of OBSE (Grotto 2013). When EMPs attend to important work matters after-hours, the interactions may boost their self-esteem (Bowling 2010), further reinforcing the on-demand behavior.\n\n### 11.3.6 Gender Differences in OBSE Among EMPs\n\nDo male EMPs have stronger beliefs in their self-worth and competence compared to their female counterparts? If so, then men in EMP roles may be more willing to fulfill their work-related needs through on-demand behavior. But gender differences in self-esteem seem to diminish with age (Reitzes and Mutran 1994). Rather than gender, commitment to various life roles (e.g., worker, parent, and spouse) has been related to self-esteem for employed men and women. Additionally, income, education , and occupational status have positive effects on self-esteem for both men and women. This research suggests that aspects of the work role are more important for self-esteem than gender. There is an abundance of evidence indicating no gender differences in OBSE levels (e.g., Bowling 2010; Grotto 2013; Lapointe et al. 2011). Perhaps the prestige associated with high-status positions within organizations helps women overcome any doubts in their self-worth or competence. If so, then both men and women in EMP roles are expected to have high-levels of OBSE and thus will want to expend great effort in their work roles. Being on-demand is one way that they can demonstrate their efforts and contribute to their organizations. Therefore, gender may be less of a factor when looking at employees specifically in the upper-ranks of organizations.\n\n### 11.3.7 The Willingness to be On-Demand: Perceptions of On-Demand Work\n\nThe perceived legitimacy of the work that intrudes on personal time is another underlying factor that may influence someone's willingness to be on-demand. Illegitimate work tasks are not part of an individual's core work role requirements and thus constitute \"identity stressors\" that violate one's professional identity (Thoits 1991). Given someone's role within an organization (e.g., director of finance), there are certain tasks that one may expected to do (e.g., approve budgets) and tasks that one may not be expected to fulfill (e.g., write the budget). Being asked to perform a work task that an individual considers to be incompatible with his\/her work role may be perceived as an identity-threatening stressor and an \"offense to the self\" (Semmer et al. 2007). Although EMP employees are very eager to be on-demand for reasons discussed earlier, they may not be willing to perform work tasks during nonwork hours that threaten their identity rather than boost their self-esteem . Furthermore, EMPs may not be willing to take on illegitimate work tasks in an effort to conserve their time and energy for more legitimate work matters during off-hours.\n\nWork tasks that are considered unreasonable or unnecessary may also be perceived as illegitimate (Semmer et al. 2010). Unreasonable tasks are beyond the scope of someone's responsibility, such as when an expert is asked to perform a task that is far below his\/her ability level, or when a leader is asked to perform a task that is not commensurate with his\/her rank. Likewise, EMPs may not be willing to perform unreasonable tasks during their personal time. For example, a senior software engineer may consider checking someone else's code as beneath her ability and think her time is better spent developing new code. A director of finance who approves departmental budgets may consider drafting the budget an unreasonable task for himself.\n\nAn unnecessary task is one that does not have to be performed. EMPs may also not be willing to perform unnecessary tasks during their personal time. For instance, if a CEO calls her director of human resources (HR) at 8:00 p.m. on a Thursday night asking him to join an 11:00 p.m. call with the China team, but the HR director does not believe his attendance on the call is necessary given the agenda, he may not be willing to participate.\n\nNot all work tasks are of equal value, and employees may consider some work tasks a \"waste of time\" or low-value work (Galinsky et al. 2004). Employees want to do work that will make a difference, especially when it is on their personal time. Low-value work may not be perceived as urgent or important enough to expend limited personal resources during their nonwork time, so individuals may be less willing to perform this type of work during nonwork hours. Similar to the perceived legitimacy of the work tasks, the use of scare personal resources during one's personal time must be well-justified, especially for EMPs whose time is quite scarce due to their demanding jobs.\n\n### 11.3.8 Gender Differences in Perceptions of On-Demand Work\n\nThere is little prior research on gender differences in the perceptions of work tasks. In a study of Swedish government sector managerial workers, women reported more illegitimate work tasks than men (Bj\u00f6rk et al. 2013). However, it is unclear whether this relationship would hold for employees working in private companies, which are not as bureaucratic as government agencies. One study that examined employees at different job levels from various companies and industries did not find any gender differences in the perceived importance, urgency, or necessity of work intrusions during personal time (Grotto 2013).\n\nThis is clearly a promising area for future research . The time of EMPs is very valuable, regardless of their gender. EMPs embedded within very time-demanding organizations must be very selective about how much of their scare time, energy , and attention is expended on work-related tasks during their precious nonwork hours. It is highly expected that both men and women in EMP roles would not be willing to take the time to respond to work e-mails or telephone calls during their off-hours if it involves illegitimate, unnecessary, or low-value work. Therefore, the nature of the work that is intruding into the personal lives of EMPs may affect their willingness to be on-demand. But there are other factors to consider. Perhaps there are certain clients who are known for intruding on a manager's personal time with unnecessary questions and illegitimate requests, and so the manager purposely does not make herself available to this client. Conversely, the manager's boss might frequently contact her during the evenings with pressing questions about urgent client deliverables, and in this case the manager is willing to accept these interruptions. When taking into consideration the nature of the intruding work task, we start moving away from factors that are under the individual's control to external circumstances that may create expectations for EMPs to be on-demand to their employers and clients 24\/7.\n\n## 11.4 The Pressure to be On-Demand\n\nDespite most managers and professionals showing appreciation for the flexibility a smartphone affords them (Deal 2013), certain aspects of their jobs pressure them to be available during nonwork hours and respond to technological work intrusions. Within a global service economy , intense jobs are embedded in high-performing companies that are constantly restructuring, making employees feel incredibly insecure about their jobs. Thus, several workplace conditions may pressure employees to be on-demand.\n\n### 11.4.1 Workplace Structures, Culture, and Rewards\n\nThe global service economy has grown tremendously over the past few decades, spurring intense competition for customers. In response to the need to provide 24\/7 service to clients, many companies have shifted to global structures with global job roles (Presser 2003, 2005). Global roles increase time pressures for employees by changing the temporal nature of their work (Kanter 1999). Traditional 9:00 to 5:00 work schedules used to dictate when employees were expected to work. But EMPs with global roles must work in different time zones and manage multiple clocks, depending on where their team members and clients are located (Moen et al. 2013). They frequently work during evenings and weekends to manage global teams, be available to global clients, participate in global meetings, and meet deadlines in other time zones. Accordingly, global EMPs must now decide when they are not working (Goodrick and Reay 2011; Moen et al. 2013). This \"always on\" mentality requires EMPs to be continually connected to work outside of normal work hours, regardless of their willingness to do so.\n\nThe economic recession has forced companies to cut costs. Many employees have had first-hand or vicarious experiences with downsizing , which can create concerns about their future with a company (Sverke et al. 2002). Perceived or anticipated layoffs have been linked with feelings of job insecurity (Kalimo et al. 2003; Sverke and Hellgren 2002; Taris et al. 2006). In fact, during a past economic recession, 25 % of workers in the USA have reported at least \"moderate\" risk for job loss because of company actions to manage finances in the economic downturn (Probst et al. 2002).\n\nJob insecurity is a source of stress (Ashford et al. 1989; Sverke et al. 2002) that can engender low-levels of work effort and withdrawal behaviors such as turnover (e.g., low-levels of organizational commitment , employee absences, and employee turnover; Cheng and Chan 2008; Sverke et al. 2002). However, the relationship of job insecurity with actual employee behaviors is complex within the present economic context. When employees perceive few alternative job opportunities, this creates an economic need to work (e.g., Brockner et al. 1992). So employees who feel at risk for job loss may actually work more in an effort to avoid job loss (Brandes et al. 2008; Brockner et al. 1992). These work efforts might extend beyond the workplace to on-demand behavior.\n\nEven employees who have survived layoffs may feel insecure about their jobs. Layoff survivors tend to feel that their work roles are threatened (Brockner et al. 1992; Sverke and Hellgren 2002). In fact, a 2008 study found that, following a layoff, employees perceived their work environments as insecure, which was related to increased work efforts among employees (Brandes et al. 2008). Thus, the threat of job loss may compel employees to stay connected to work via CT during their personal time, to show their commitment and dedication. This is especially expected among managers who may be concerned about whether layers of management will be eliminated as a result of downsizing.\n\nDownsizing also has widened job scopes and increased responsibilities for many employees (Presser 2003, 2005), contributing to work extensification , or work that is highly demanding, fast paced, and requires long hours (Currie and Eveline 2011; Jarvis and Pratt 2006; Lu 2009). Many workers accept the extensification of work as status quo (Moen et al. 2013), perhaps because they see it as the only way to keep pace with their heavy workloads and avoid falling behind in their careers. Being on-demand to their jobs is one way for EMPs to persevere.\n\nCertain aspects of a company's culture may also pressurize employees to be on-demand. In Slaughter's (Slaughter 2012) popular article in The Atlantic, \"Why Women Still Can't Have It All,\" she blames the persistence of a time-demanding culture on the relentless competition to work harder in a global and unstable economy , which translates into working earlier, later, and longer. Indeed, organizational expectations to work long hours tend to result in long work hours (Kossek and Ozeki 1999; Thompson et al. 1999). This type of workplace culture takes advantage of EMPs whose additional work hours do not cost additional wages for employers. Essentially, companies have little incentive to limit the work hours of EMPs (Jacobs and Gerson 2004), but plenty of reasons to overwork them (e.g., stay ahead of the competition).\n\nEmployees are most likely to be subjected to an \"always on\" culture when embedded within a high-commitment culture. Based on the ideas of social exchange theory (Blau 1964), there is an inherent psychological contract in high-commitment workplaces. Employees are given a great deal of job autonomy , and in return are expected to demonstrate commitment by making work a central part of their lives (Moen et al. 2013). So the perks of job autonomy and flexibility signal to employees that their employers are invested in them, compelling them to prioritize work over personal life and work long hours (Bailyn 1993; Blair-Loy and Wharton 2002, 2004; Schor 1991). Thus, there is a growing lack of clarity on what is considered \"normal working hours\" (Moen et al. 2013). Employees working in these environments may feel as though they owe it to their employers be on-demand 24\/7.\n\nHigh-commitment workplaces further enable an \"always on\" culture by providing employees with smartphones, which not only creates pressure to respond to work interruptions during nonwork hours, but to respond immediately (Dikkers et al. 2007). In return for their high-status, job flexibility, and autonomy , EMPs working in high-commitment workplaces may feel particularly driven to show their commitment by being on-demand. So although high-commitment workplaces give EMPs considerable flexibility, the perks come with a price.\n\nEvidence suggests that the extension of acceptable work hours is often viewed unfavorably by workers (Harrison et al. 2004). Yet many employees still succumb to the pressure because they are motivated by the anticipated rewards (Lobel 1991) as well as fear of negative consequences (Thompson et al. 1999). For instance, EMPs may respond to work interruptions after-hours because of incentives such as career advancement (e.g., promotions) and financial rewards (e.g., bonuses; Jacobs and Gerson 2004). In high-commitment companies, the exchange of long work hours and high-work efforts for advancement and compensation reinforces the belief that employees are appreciated by their employers (Osterman 2000; Konrad and Mangel 2000). As most EMPs are typically generously rewarded for their experience, expertise, and dedication, they may be more willing than others to invest their personal time in their jobs. Basically, EMPs believe that if they are accessible via CT during their personal time, then they will have even more opportunities to contribute to their organizations, thereby further increasing their rewards .\n\nFor individuals who do not give in to on-demand pressures, there is a fear of being left behind. Two decades ago when face-time served as the primary indicator of an employee's commitment, research showed that a lack of time spent at the office did indeed lead to being passed over for promotions (Schor 1991). Another study showed that advancement in management was in fact associated with working long hours (Judge et al. 1995). Employees may not want to risk ignoring e-mails or job-related phone calls during the evenings or weekends, as it might lessen their chances of being promoted or translate into smaller bonus payments. The repercussions for deviating from the norm to be on-demand may be strongest for EMPs (e.g., Blair-Loy 2003; Stone 2007; Xie and Shauman 2003). With incredibly flexible jobs and company smartphones that allow them to be on-demand, EMPs may strive to differentiate themselves from those who simply put in their time only when at the office.\n\n### 11.4.2 The \"Stress of Higher Status\" Perspective\n\nThe nature of EMP work itself creates on-demand expectations. According to the \"stress of higher status\" perspective (Schieman et al. 2009), the autonomous nature of EMP jobs increases the permeability of the borders surrounding personal life. So the great degree of schedule control afforded to these individuals may be both a blessing and a curse. Although job control allows EMPs to work whenever they want, it also engenders expectations of undivided dedication to one's career and organization (Blair-Loy 2003; Schieman et al. 2009). The time-demanding nature of EMP work is not limited to the physical workplace, but spills over to other life domains via CT. To prove their commitment, EMPs may feel pressure to prioritize work over their personal responsibilities and be available 24\/7. Ample evidence suggests that the nature of EMP work creates expectations to be on-demand, as job authority, excessive work pressures, schedule control, and decision-making latitude have all been linked with blurring of work and nonwork roles (Schieman et al. 2009).\n\n### 11.4.3 Gender Differences in Workplace and Job Pressures\n\nAre men or women more likely to find themselves in time-demanding organizations and intense jobs? In general, males report higher workloads , more feelings of work overload , and higher expectations to be available during their personal time (Day et al. 2012). But are there differences among men and women in EMP positions with respect to the demands of their organizations and the intensity of their jobs?\n\nGiven that job features encroach equally on men and women (Jacobs and Gerson 2004), the pressures of a high-commitment workplace and the stress of high status are likely to affect men and women in similar ways. Female EMPs in high-commitment workplaces are likely to confront the same excessive time demands and pressures as their male counterparts. Likewise, male EMPs are just as likely as female EMPs to work in fast paced, demanding jobs with long work hours. It is the nature of their high-ranking and specialized positions that engenders on-demand behavior. Research does suggest that the corporate work conditions and intense job demands associated with high-commitment workplaces and high-status jobs make it difficult for EMPs of both genders to lead balanced lives (Blair-Loy and Wharton 2004; Wharton and Blair-Loy 2006).\n\nYet gender may be a factor when taking into consideration the lack of women in leadership positions (Cha 2013). In companies with fewer women in the higher ranks, there tends to be a lack of support for work\u2013life balance. A study by Wharton and Blair-Loy (2006) showed that women reported higher-levels of work-to-family conflict within global financial service firms, which have some of the lowest percentages of females in the C-suite . Also, women in companies that value overwork are more likely to leave their jobs than men (Stone 2007). Parental status may compound the situation in male-dominated industries that do not support work\u2013life balance. In fact, mothers in male-dominated industries were more likely to leave their jobs and the labor force when they experienced conflicting time demands between work and family (Cha 2013). So it is important to consider the responsibilities of men and women beyond their intense jobs.\n\nEven though many EMP men and women may have selected themselves into highly demanding companies and jobs and may even thrive in these environments, responsibilities outside of work can complicate matters. In particular, the job of being a parent is 24\/7. So whether the EMP is a man or a woman, the extent of involvement in the parental role may influence their on-demand behavior. Combining such an intense and time-demanding personal life role with an intense and time-demanding work role may push some employees to their limits.\n\n## 11.5 The Consequences of On-Demand Jobs\n\nResearch suggests that employees who are constantly connected to work during their nonwork hours are vulnerable to a number of health consequences. Much of this prior research has focused on the consequences of blurring the work role with other life roles and a lack of psychological detachment (i.e., mental distance) from work during nonwork hours. By remaining psychologically attached to the work role during their personal time, employees drain their personal resources (e.g., energy; Edwards and Rothbard 2000) and do not give themselves a chance to recover from their work day demands (Meijman and Mulder 1998). If employees remain on-demand for long periods of time, psychological health problems such as chronic stress could develop (Geurts and Sonnentag 2006). If recovery remains insufficient for an extended period of time, psychological health problems can develop into serious physical health risks, such as increased risk of cardiovascular problems (von Thiele et al. 2006).\n\nIndeed, a number of psychological health problems have been reported by employees who have not had the chance to fully recover from their work day demands. Emotional exhaustion is commonly reported by employees who blur their work and nonwork roles and inadequately detach from work during nonwork hours (de Jonge et al. 2012). Employee interviews revealed that specific episodes of work intruding on personal time (e.g., an unexpected phone call from a boss during the weekend) engendered negative feelings, such as anger (Grotto 2013). A potential psychological outcome of on-demand jobs are feelings of work interfering with other life roles (e.g., Schieman and Young 2013). Employees who reported demands to use CT and persistently used CT during nonwork hours also reported poor psychological well-being (Chesley 2005) and feelings of strain, stress, and job burnout (Day et al. 2012). On-demand employees may also be at risk for poor physical health. When specific work events conflicted with family, employees have reported subsequent increases in heart rate (Shockley and Allen 2013). Also, employees who reported work intrusions or a lack of detachment from work also reported physical health complaints (de Jonge et al. 2012; Schieman and Young 2013), psychosomatic complaints (e.g., feelings of dizziness and fatigue; Sonnentag et al. 2010), and a need for recovery (e.g., feeling too tired to for personal activities; Sonnentag et al. 2010).\n\nAll of these health consequences can negatively impact work behavior and attitudes. For instance, employees who used CT during nonwork hours and then reported feelings of work interfering with personal life also reported lower-levels of work satisfaction (Diaz et al. 2012). Also, continual lack of recovery can eventually lead to less effort at work and poor job performance (Fritz and Sonnentag 2006; Sonnentag et al. 2010). Yet some individuals still choose to be on-demand to their jobs, perhaps for personal reasons that were discussed earlier or because they are rewarded for this behavior. However, given the potential for negative consequences, there may be a limit to how much one can and should be on-demand. Perhaps attachment to the work role during nonwork hours is beneficial to individuals, but only up to a certain point, regardless of personal preferences or how the intrusions are perceived (e.g., important or not).\n\nIndeed, the sheer number of work intrusions reported during nonwork time was linked with self-reported negative health complaints, despite many of the intrusions being perceived as important by individuals (Grotto 2013). So although employees may believe that prioritizing work over other life roles leads to a successful career (Moen and Roehling 2005), it may be a very unhealthy approach to work. Given the ample evidence suggesting how on-demand employees may suffer, success at work may come at the expense of a healthy personal life (Amichai-Hamburger 2009). Even the benefits of intrinsic personal rewards (e.g., feelings of accomplishment) and extrinsic work rewards (e.g., promotions) that result from employees' endless involvement in work may have its limits once it takes a toll on physical and mental health .\n\n### 11.5.1 Gender Differences in the Consequences of On-Demand Jobs\n\nWhen a 24\/7 job is integrated with other demanding life roles, the health consequences may be severe. Who has more demanding roles at home\u2014men or women? The 24\/7 job of being a parent still primarily falls on the shoulders of women, as men's careers continue to take precedence (Milkie et al. 2002). But over time, men, particularly those with egalitarian attitudes, are sharing the burden of unpaid work at home with their partners (Bulanda 2004; Cunningham 2007). Also, women in EMP roles tend to more equally divide household labor with their partners (Cunningham 2007; Mannino and Deutsch 2007). It seems that dual-earner couples veer away from gender-normed behavior (Donald and Linington 2008) and work as a team at home (Marks 2006). So gender may be somewhat irrelevant when deciding on the share of household labor. This approach can be advantageous for on-demand employees who are either willing or feel pressured to invest much of their personal time in work-related activities.\n\nEven employees who share domestic responsibilities with their partners may suffer from on-demand work . EMPs of both genders who are exposed to high-work expectations and intense time demands have reported feeling imbalanced or conflicted in life (Blair-Loy and Wharton 2004; Schieman et al. 2006; Wharton and Blair-Loy 2006). Moreover, those who combine such intense and demanding work roles with demanding personal roles (e.g., parent) may be pushed to their limits, even if the negative effects are not realized immediately. For instance, among employees with children living at home, high-job demands and an inability to withdraw from work during their personal time were associated with fatigue six months later (von Thiele 2011).\n\nEMPs are a unique group. Given their high-status at work and their high-commitment to their workplaces, these individuals are very likely to be on-demand to their employees and clients, making them vulnerable to a number of health consequences. Their egalitarian approach to household labor might help buffer some of these negative effects. However, if these employees have children, the situation is more complicated. Since the parent role is very time-demanding, even EMPs who are willing to be on-demand may eventually see their health suffer.\n\n## 11.6 Conclusions and Recommendations\n\nThere is much reason to believe that increasingly flexible jobs and readily available CT are responsible for the rise of on-demand jobs. However, the intense manner in which EMPs are working and the way they are using CT are the real underlying issues, and these issues similarly affect men and women. There is a strong need to limit how much the job and CT dictate how employees live their lives (Jacobs and Gerson 2004). Employees in high-commitment workplaces are given abundant freedom to work when and where they want with the aid of CT and are rewarded for their on-demand behavior. But they clearly lack a different type of control.\n\nBased on the job demands-control model (Karasek and Theorell 1990), it is not the level of job demands that create stress or strain, but whether employees feel they can exert control over the demands they confront. EMPs do have considerable control over their jobs, but they have less control over the intense conditions of their work (Allvin et al. 2011). So even though EMPs are willing and able to invest their personal time in their work roles, most lack control over the work demands that are constantly intruding into their personal lives via technology. How much EMPs work during their personal time is heavily affected by what is happening at work (e.g., client deadlines, crises, needy clients, demanding bosses). Although EMPs can decide whether or not to respond to work interruptions during nonwork hours, they have much less control over the interruptions themselves. Even if one has the authority to ignore a work e-mail during nonwork hours, lacking control over the intrusion can still be problematic. In support of this, a recent study found that the more frequently work intruded into employees' personal time, the more they reported poor physical health complaints (Grotto 2013). Therefore, communicative technologies, time-demanding employers and clients, and intense jobs give EMPs very little power over when and how often their work demands invade their personal life.\n\nRealistically, we will never return to the old ways of work, with the job being left at the office once the employee leaves for the day, especially for EMPs. Perhaps intertwined work and nonwork roles would be somewhat manageable if the world still operated with traditional gender roles, with men as the primary wage earners and wives dedicated to domestic work. Then men could continue to focus on their work roles even while at home. But many households today are comprised of dual-earner couples , with men and women participating in multiple life roles, some of which can be very demanding (e.g., parent). Yet, many corporate cultures continue to operate as if employees have no commitments outside of work (Moen et al. 2013; Thompson et al. 2006).\n\nSome may argue that employees are at fault . Many workers seem to accept unreasonable demands for their time because they value the rewards they get in return, including job perks (e.g., onsite child care, free gym memberships), career advancement opportunities, and pay raises. But these rewards may not matter if employees burn out, under-perform, or leave. As companies continue to encroach on employees' personal time in an effort to keep up with competitors in the short-term, they may be hurting long-term performance . Corporations are slowly chipping away at their most highly valued workers. So what can be done about it?\n\nGiven the potential for negative impacts on employee and organizational health, on-demand work is a very public issue, yet companies expect employees to manage it on their own (Cha 2013). Accordingly, many individuals have devised their own strategies. Some scholars claim that individuals act as active agents, setting their own boundaries around their personal life (Allvin et al. 2011). The maintenance of these boundaries can be beneficial for employees' psychological feelings of role interference (Park et al. 2011). For example, employees who keep their work roles separated from other life domains are able to psychologically detach from their work role (Park et al. 2011), which contributes to their well-being (Hahn and Dormann 2013). But this is not always realistic, especially for EMPs.\n\nGiven the lack of control that today's employees have over their work demands , employee are actually active reactors who are forced to respond to and manage their work demands rather than actively controlling or preventing them (Moen et al. 2013). These strategies involve making personal efforts to allow or not allow a particular work intrusion during nonwork time (Flaherty 2003). However, employees using these strategies have only so much control, as the structural conditions of their jobs and the temporal norms of their companies constrain their options. Thus, the burden is ultimately on employees to react and adjust to their on-demand jobs. This approach does not necessarily give employees control over their job demands. These employees rarely have the chance to fully detach from their work roles and recover from the day's work demands . For example, EMPs might block 2 hours during the evening to focus on family time, but this timeframe is only a temporary break from work (Moen et al. 2013).\n\nIndividual strategies may not be sufficient to mitigate the negative effects of on-demand work (Butler et al. 2009). Corporations need to pitch in as well. Some time-demanding companies help employees maintain a sense of work\u2013life balance by providing benefits at the worksite, such as onsite childcare . But only the employees who are fortunate enough to work for these supportive companies have access to these resources. There are no national mandates to provide work\u2013life resources to all employees. Also, resources and support provided by companies or the government do not address the root issue\u2014intense time-demands. It is apparently not healthy for employees to work 24\/7, which warrants company and national policies that dictate acceptable work hours for EMPs.\n\nThere are many benefits to giving employees time to psychologically detach from work, such as positive effects on health and creativity (de Jonge et al. 2012). So high-commitment companies need to reevaluate their cultures and strongly consider reshaping their norms to encourage mental disengagement from work during off-hours, including interventions training employees on how to do so. But broad organizational policies may not be practical for some EMPs. In these cases, employers should reconsider the legitimacy of the on-demand work , as the cognitive appraisal of a work task can be a strong factor in how employees are impacted by work intrusions (Lazarus and Folkman 1984). An event is stressful when the demands of the environment are perceived as exceeding one's resources and thus endangering one's well-being. When work intrudes on personal time, employees assess their relative job demands and the availability of their resources (e.g., time, energy) . If the job demands exceed their available resources, then on-demand work may negatively impact them. If employers can minimize illegitimate work during employees' personal time, then employees would have more time to keep up with the most important work matters (Galinsky et al. 2004). For instance, when a boss sends an e-mail, she can be clear how important or urgent the work issue is so that the receiver can decide whether to protect his\/her personal time or not.\n\nIt is critical to re-examine the need for on-demand jobs, as a lack of recovery from work during personal time may not only negatively impact employees' personal lives and health, but also their task performance, personal initiative, and work effort; so organizational performance can suffer as well. Individual strategies and organizational policies and interventions can help address these issues, and future exploration of the causes and consequences of on-demand work will be important for employees' quality of life and organizational health.\n\nReferences\n\nAmichai-Hamburger, Y. (2009). Technology and well-being: Designing the future. In Y. Amichai-Hamburger (Ed.), Technology and psychological well-being (pp. 260\u2013278). New York: Cambridge University Press.\n\nAllvin, M., Aronsson, G., Hagstrom, T., Johansson, G., & Lundberg, U. (2011). Work without boundaries: Psychological perspectives on the new working life. Oxford: Wiley-Blackwell.\n\nAshford, S. J., Lee, C., & Bobko, P. (1989). Content, causes, and consequences of job insecurity: A theory-based measure and substantive test. Academy of Management Journal, 32, 803\u2013829.\n\nAshforth, B. E., Kreiner, G. E., & Fugate, M. (2000). All in a day's work: Boundaries and micro role transitions. Academy of Management Review, 25, 472\u2013491.\n\nBamberg, E., Dettmers, J., Funck, H., Kr\u00e4he, B., & Vahle-Hinz, T. (2012). Effects of on-call work on well-being: Results of a daily survey. Applied Psychology: Health and Well-Being, 4, 299\u2013320.\n\nBailyn, L. (1993). Breaking the mold: Women, men, and time in the new corporate world. New York: Free Press.\n\nBeach, B. (1989). Integrating work and family life. Albany: State University of New York Press.\n\nBj\u00f6rk, L., Bejerot, E., Jacobshagen, N., & H\u00e4renstam, A. (2013). I shouldn't have to do this: Illegitimate tasks as a stressor in relation to organizational control and resource deficits. Work & Stress, 27, 262\u2013277.\n\nBlair-Loy, M. (2003). Competing devotions. Cambridge: Harvard University Press.\n\nBlair-Loy, M. (2009). Work without end? Scheduling flexibility and work-to-family conflict among stockbrokers. Work and Occupations, 36, 279\u2013317.\n\nBlair-Loy, M., & Wharton, A. S. (2002). Employees' use of work-family policies and the workplace social context. Social Forces, 80, 813\u2013846.\n\nBlau, P. (1964). Exchanges and power in social life. New York: Wiley.\n\nBlair-Loy, M., & Wharton, A. S. (2004). Organizational commitment and constraints on work-family policy use: Corporate flexibility policies in a global firm. Sociological Perspectives, 47, 243\u2013267.\n\nBowling, N. A. (2010). Effects of job satisfaction and conscientiousness on extra-role behaviors. Journal of Business and Psychology, 25, 119\u2013130.\n\nBrandes, P., Castro, S. L., James, M. S. L., Martinez, A. D., Matherly, T. A., Ferris, G. R., & Hochwarter, W. A. (2008). The interactive effects of job insecurity and organizational cynicism on work effort following a layoff. Journal of Leadership & Organizational Studies, 14, 233\u2013247.\n\nBrockner, J., Grover, S., Reed, T. F., & DeWitt, R. L. (1992). Layoffs, job insecurity, and survivors' work effort: Evidence of an inverted-U relationship. Academy of Management Journal, 35, 413\u2013425.\n\nBulanda, R. E. (2004). Paternal involvement with children: The influence of gender ideologies. Journal of Marriage and Family, 66, 40\u201345.\n\nButler, A. B., Bass, B. L., & Grzywacz, J. G. (2009). Job demands, spousal support, and work-family balance: A daily analysis of the work-family interface. In D. R. Crane & E. J. Hill (Eds.), Handbook of families and work (pp. 9\u201330). Lanham: University Press of America.\n\nCha, Y. (2013). Overwork and the persistence of gender segregation in occupations. Gender & Society, 27, 158\u2013184.\n\nCheng, G. H., & Chan, D. K. (2008). Who suffers more from job insecurity? A meta-analytic review. Applied Psychology: An International Review, 57, 272\u2013303.\n\nChesley, N. (2005). Blurring boundaries? Linking technology use, spillover, individual distress, and family satisfaction. Journal of Marriage and Family, 67, 1237\u20131248.\n\nClark, S. C. (2000). Work\/family border theory: A new theory of work\/family balance. Human Relations, 53, 747\u2013770.\n\nCunningham, M. (2007). Influences of women's employment on the gendered division of household labor over the life course: Evidence from a 31-year panel study. Journal of Family Issues, 28, 422\u2013444.PubMedCentralPubMed\n\nCurrent Populations Survey. (2010). U.S. Bureau of Labor Statistics, unpublished data. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200baa2010\/\u200baat23.\u200btxt. Accessed 17 Dec 2013.\n\nCurrent Populations Survey (CPS). (2012). U.S. Bureau of Labor Statistics, unpublished data. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200baa2012\/\u200bcpsaat23.\u200bhtm. Accessed 20 Dec 2013.\n\nCurrie, J., & Eveline, J. (2011). E-technology and work\/life balance for academics with young children. Higher Education, 62, 533\u2013550.\n\nDay, A., Paquet, S., Scott, N., & Hambley, L. (2012). Perceived information and communication technology (ICT) demands on employee outcomes: The moderating effect of organizational ICT support. Journal of Occupational Health Psychology, 17, 473\u2013491.PubMed\n\nde Jonge, J., Spoor, E., Sonnentag, S., Dormann, C., & van den Tooren, M. (2012). \"Take a break?!\" off-job recovery, job demands, and job resources as predictors of health, active learning, and creativity. European Journal of Work and Organizational Psychology, 21, 321\u2013348.\n\nDeal, J. J. (2013). Always on, never done? Don't blame the smartphone. http:\/\/\u200bwww.\u200bccl.\u200borg\/\u200bleadership\/\u200bpdf\/\u200bresearch\/\u200bAlwaysOn.\u200bpdf. Accessed 16 Dec 2013.\n\nDiaz, I., Chiaburu, D. S., Zimmerman, R. D., & Boswell, W. R. (2012). Communication technology: Pros and cons of constant connection to work. Journal of Vocational Behavior, 80, 500\u2013508.\n\nDikkers, J. S., Geurts, S. A., Den Dulk, L., Peper, B., Taris, T. W., & Kompier, M. A. (2007). Dimensions of work-home culture and their relations with the use of work-home arrangements and work-home interaction. Work & Stress, 21, 155\u2013172.\n\nDonald, F., & Linington, J. (2008). Work\/family border theory and gender role orientation in male managers. South African Journal of Psychology, 38, 659\u2013671.\n\nDouglas, M. (1975). Implicit meanings: Essays in anthropology. London: Routledge and Paul.\n\nEdwards, J. R., & Rothbard, N. P. (2000). Mechanisms linking work and family: Clarifying the relationship between work and family constructs. Academy of Management Review, 25, 178\u2013199.\n\nEzzedeen, S. R., & Swiercz, P. M. (2007). Development and initial validation of a cognitive-based work-nonwork conflict scale. Psychological Reports, 100, 979\u2013999.PubMed\n\nFan, P. L., & Marini, M. M. (2000). Influences on gender-role attitudes during the transition to adulthood. Social Science Research, 29, 258\u2013283.\n\nFenner, G. H., & Renn, R. W. (2004). Technology-assisted supplemental work: Construct definition and a research framework. Human Resource Management, 43, 179\u2013200.\n\nFinegold, D., & Frenkel, S. (2006). Managing people where people really matter: The management of human resources in biotech companies. International Journal of Human Resource Management, 17, 1\u201324.\n\nFlaherty, M. G. (2003). Time work: Customizing temporal experience. Social Psychology Quarterly, 66, 17\u201333.\n\nFritz, C., & Sonnentag, S. (2006). Recovery, well-being, and performance-related outcomes: The role of workload and vacation experiences. Journal of Applied Psychology, 91, 936\u2013945.PubMed\n\nGalinsky, E., Bond, J. T., Kim, S. S., Backon, L., Brownfield, E., & Sakai, K. (2004). Overwork in America: When the way we work becomes too much. http:\/\/\u200bfamiliesandwork.\u200borg\/\u200bsite\/\u200bresearch\/\u200breports\/\u200bOverWorkInAmeric\u200ba.\u200bpdf. Accessed 17 Dec 2013.\n\nGardner, D. G., Van Dyne, L., & Pierce, J. L. (2004). The effects of pay level on organization-based self-esteem and performance: A field study. Journal of Occupational and Organizational Psychology, 77, 307\u2013322.\n\nGoodrick, E., & Reay, T. (2011). Constellations of institutional logics: Changes in the professional work of pharmacists. Work and Occupations, 38, 372\u2013416.\n\nGrotto, A. R. (2013). Unexpected work intrusions into employees' personal lives: Investigation, measure development, and exploration of causes and consequences. Unpublished doctoral dissertation. Baruch College and the Graduate Center at the City University of New York, NY: New York.\n\nGeurts, S., & Sonnentag, S. (2006). Recovery as an explanatory mechanism in the relation between acute stress reactions and chronic health impairment. Scandinavian Journal of Work, Environment & Health, 32, 482\u2013492.\n\nHahn, V. C., & Dormann, C. (2013). The role of partners and children for employees' psychological detachment from work and well-being. Journal of Applied Psychology, 98, 26\u201336.PubMed\n\nHarrison, A., Wheeler, P., & Whitehead, C. (2004). The distributed workplace. London: Spon Press.\n\nHecht, T. D., & Allen, N. J. (2009). A longitudinal examination of the work-nonwork boundary strength construct. Journal of Organizational Behavior, 30, 839\u2013862.\n\nHochschild, A. R. (1989). The second shift: Working parents and the revolution at home. New York: Viking Penguin.\n\nJacobs, J. A., & Gerson, K. (2004). The time divide: Work, family, and gender inequality. Cambridge: Harvard University Press.\n\nJackson, M. (2002). What's happening to home? Balancing work, life and refuge in the information age. Notre Dame: Sorin Books.\n\nJarvis, H., & Pratt, A. C. (2006). Bringing it all back home: The extensification and 'overflowing' of work, the case of San Francisco's new media households. Geoforum, 37, 331\u2013339.\n\nJudge, T. A., Cable, D. M., Boudreau, J. W., & Bretz, R. D. (1995). An empirical investigation of the predictors of executive career success. Personnel Psychology, 48, 485.\n\nKalimo, R., Taris, T. W., & Schaufeli, W. B. (2003). The effects of past and anticipated future downsizing on survivor well-being: An equity perspective. Journal of Occupational Health Psychology, 8, 91\u2013109.PubMed\n\nKanter, R. M. (1999). Change is everyone's job: Managing the extended enterprise in a globally connected world. Organizational Dynamics, 28, 7\u201323.\n\nKarasek, R. A., & Theorell, T. (1990). Healthy work. New York: Basic Books.\n\nKelloway, E. K., & Barling, J. (2000). Knowledge work as organizational behavior. International Journal of Management Reviews, 2, 287\u2013304.\n\nKelly, E., Ammons, S., Chermack, K., & Moen, P. (2010). Gendered challenge, gendered response: Confronting the ideal worker norm in a white-collar organization. Gender & Society, 24, 281\u2013303.\n\nKonrad, A., & Mangel, R. (2000). The impact of work-life programs on firm productivity. Strategic Management Journal, 21, 1225\u20131237.\n\nKossek, E. E., & Ozeki, C. (1999). Bridging the work-family policy and productivity gap: A literature review. Community, Work & Family, 2, 7\u201332.\n\nKossek, E. E., Lautsch, B. A., & Eaton, S. C. (2006). Telecommuting, control, and boundary management: Correlates of policy use and practice, job control, and work-family effectiveness. Journal of Vocational Behavior, 68, 347\u2013367.\n\nKossek, E. E., Lautsch, B. A., & Eaton, S. C. (2009). \"Good teleworking\": Under what conditions does teleworking enhance employees' well-being? In Y. Amichai-Hamburger (Ed.), Technology and psychological well-being (pp. 148\u2013173). New York: Cambridge University Press\n\nKreiner, G. E., Hollensbe, E. C., & Sheep, M. L. (2009). Balancing borders and bridges: Negotiating the work-home interface via boundary work tactics. Academy of Management Journal, 52, 704\u2013730.\n\nKuhn, P., & Lozano, F. (2008). The expanding workweek? Understanding trends in long work hours among U.S. men, 1979\u20132006. Journal of Labor Economics, 26, 311\u201343.\n\nLambert, S. J. (1990). Processes linking work and family: A critical review and research agenda. Human Relations, 43, 239\u2013257.\n\nLapointe, E., Vandenberghe, C., & Panaccio, A. (2011). Organizational commitment, organization-based self-esteem, emotional exhaustion and turnover: A conservation of resources perspective. Human Relations, 64, 1609\u20131631.\n\nLazarus, R. S., & Folkman, S. (1984). Stress, appraisal, and coping. New York: Springer.\n\nLobel, S. A. (1991). Allocation of investment in work and family roles: Alternative theories and implications for research. Academy of Management Review, 16, 507\u2013521.\n\nLu, J. L. (2009). Effect of work intensification and work extensification on women's health in the globalized labour market. Journal of International Women's Studies, 10, 111\u2013126.\n\nMajor, V. S., Klein, K. J., & Ehrhard, M. G. (2002). Work time, work interference with family, and psychological distress. Journal of Applied Psychology, 87, 427\u2013436.PubMed\n\nMannino, C. A., & Deutsch, F. M. (2007). Changing the division of household labor: A negotiated process between partners. Sex Roles, 56, 309\u2013324.\n\nMarks, S. R. (2006). Understanding diversity of families in the 21st century and its impact on the work-family area of study. In M. Pitt-Catsouphes, E. E. Kossek, & S. Sweet (Eds.), The work and family handbook: Multi-disciplinary perspectives and approaches (pp. 41\u201365). Mahwah: Lawrence Erlbaum.\n\nMeijman, T. F., & Mulder, G. (1998). Psychological aspects of workload. In P. J. D. Drenth, H. Thierry, & C. J. de Wolff (Eds.), Handbook of work and organizational, Vol. 2: Work psychology (pp. 5\u201333). Hove: Psychology Press.\n\nMennino, S. F., Rubin, B. A., & Brayfield, A. (2005). Home-to-job and job-to-home spillover: The impact of company policies and workplace culture. Sociological Quarterly, 46, 107\u2013135.\n\nMilkie, M. A., Bianchi, S. M., Mattingly, M. J., & Robinson, J. P. (2002). Gendered division of childrearing: Ideals, realities, and the relationship to parental well-being. Sex Roles, 47, 21\u201338.\n\nMoen, P., & Roehling, P. (2005). The career mystique: Cracks in the American dream. Boulder: Rowman & Littlefield.\n\nMoen, P., Lam, J., Ammons, S., & Kelly, E. L. (2013). Time work by overworked professionals: Strategies in response to the stress of higher status. Work and Occupations, 40, 79\u2013114PubMedCentralPubMed\n\nNippert-Eng, C. (1996). Calendars and keys: The classification of 'home' and 'work'. Sociological Forum, 11, 563\u2013582.\n\nOlson-Buchanan, J. B., & Boswell, W. R. (2006). Blurring boundaries: Correlates of integration and segmentation between work and nonwork. Journal of Vocational Behavior, 68, 432\u2013445.\n\nOsterman, P. (2000). Work reorganization in an era of restructuring: Trends in diffusion and effects on employee welfare. Industrial and Labor Relations Review, 53, 179\u2013196.\n\nPark, Y., & Jex, S. M. (2011). Work-home boundary management using communication and information technology. International Journal of Stress Management, 18, 133\u2013152.\n\nPark, Y., Fritz, C., & Jex, S. M. (2011). Relationships between work-home segmentation and psychological detachment from work: The role of communication technology use at home. Journal of Occupational Health Psychology, 16, 457\u2013467.PubMed\n\nParkman, A. M. (2004). Bargaining over housework: The frustrating situation of secondary wage earners. American Journal of Economics and Sociology, 63, 765\u2013794.\n\nPaullay, I. M., Alliger, G. M., & Stone-Romero, E. F. (1994). Construct validation of two instruments designed to measure job involvement and work centrality. Journal of Applied Psychology, 79, 224\u2013228.\n\nPerlow, L. A. (1995). Putting the work back into work\/family. Group & Organization Management, 20, 227\u2013239.\n\nPierce, J. L., Gardner, D. G., Cummings, L. L., & Dunham, R. B. (1989). Organization-based self-esteem: Construct definition, measurement, and validation. Academy of Management Journal, 32, 622\u2013648.\n\nPresser, H. B. (2003). Working in a 24\/7 economy. New York: Russell Sage.\n\nPresser, H. B. (2005). Embracing complexity: Work schedules and family life in a 24\/7 economy. In S. M. Bianchi, L. M. Casper, & B. R. King (Eds.), Work, family, health, and well-being (pp. 43\u201348). Mahwah: Lawrence Erlbaum.\n\nProbst, T. M., Brett, J. M., & Drasgow, F. (2002). The impact of job insecurity on employee work attitudes, job adaptation, and organizational withdrawal behaviors. In J. M. Brett & F. Drasgow (Eds.), The psychology of work: Theoretically based empirical research (pp. 141\u2013168). Mahwah: Lawrence Erlbaum.\n\nReitzes, D. C., & Mutran, E. J. (1994). Multiple roles and identities: Factors influencing self-esteem among middle-aged working men and women. Social Psychology Quarterly, 57, 313\u2013325.\n\nRitchter, J. (1992). Balancing work and family in Israel. In S. Zedeck (Ed.), Work, families, and organizations. San Francisco: Jossey-Bass.\n\nRothbard, N. P., Phillips, K. W., & Dumas, T. L. (2005). Managing multiple roles: Work-family policies and individuals' desires for segmentation. Organization Science, 16, 243\u2013258.\n\nSchieman, S., & Reid, S. (2008). Job authority and interpersonal conflict in the workplace. Work and Occupations, 35, 296\u2013326.\n\nSchieman, S., & Young, M. C. (2013). Are communications about work outside regular working hours associated with work-to-family conflict, psychological distress and sleep problems? Work & Stress, 27, 244\u2013261.\n\nSchieman, S., Whitestone, Y., & Van Gundy, K. (2006). The nature of work and the stress of higher status. Journal of Health and Social Behavior, 47, 242\u2013257.PubMed\n\nSchieman, S., Milkie, M. A., & Glavin, P. (2009). When work interferes with life: Work-nonwork interference and the influence of work-related demands and resources. American Sociological Review, 74, 966\u2013988.\n\nSchor, J. B. (1991). The overworked American: The unexpected decline of leisure. New York: Basic Books.\n\nSemmer, N. K., Jacobshagen, N., Meier, L. L., & Elfering, A. (2007). Occupational stress research: The \"Stress-As-Offense-to-Self\" perspective. In J. Houdmont & S. McIntyre (Eds.), Occupational health psychology: European perspectives on research, education and practice (pp. 43\u201360). Maia: ISMAI Publishing.\n\nSemmer, N. K., Tschan, F., Meier, L. L., Facchin, S., & Jacobshagen, N. (2010). Illegitimate tasks and counterproductive work behavior. Applied Psychology: An International Review, 59, 70\u201396.\n\nShockley, K. M., & Allen, T. D. (2013). Episodic work-family conflict, cardiovascular indicators, and social support: An experience sampling approach. Journal of Occupational Health Psychology, 18, 262\u2013275.PubMed\n\nSlaughter, A. M. (2012). Why women still can't have it all. The Atlantic. http:\/\/\u200bwww.\u200btheatlantic.\u200bcom\/\u200bmagazine\/\u200barchive\/\u200b2012\/\u200b07\/\u200bwhy-women-still-cant-have-it-all\/\u200b309020\/\u200b. Accessed 20 Dec 2013.\n\nSonnentag, S., Kuttler, I., & Fritz, C. (2010). Job stressors, emotional exhaustion, and need for recovery: A multi-source study on the benefits of psychological detachment. Journal of Vocational Behavior, 76, 355\u2013365.\n\nStone, P. (2007). Opting out?: Why women really quit careers and head home. Berkeley: University of California Press.\n\nSverke, M., & Hellgren, J. (2002). The nature of job insecurity: Understanding employment uncertainty on the brink of a new millennium. Applied Psychology: An International Review, 51, 23\u201342.\n\nSverke, M., Hellgren, J., & Naswall, K. (2002). No security: A meta-analysis and review of job insecurity and its consequences. Journal of Occupational Health Psychology, 7, 242\u2013264.PubMed\n\nTaris, T. W., Beckers, D. G. J., Verhoeven, L. C., Geurts, S., Kompier, M., & van der Linden, D. (2006). Recovery opportunities, work-home interference, and well-being among managers. European Journal of Work and Organizational Psychology, 15, 139\u2013157.\n\nThierry, H. (2001). The reflection theory on compensation. In M. Erez, U. Kleinbeck, & H. Thierry (Eds.), Work motivation in the context of a globalizing economy (pp. 149\u2013166). Mahwah: Lawrence Erlbaum.\n\nThoits, P. A. (1991). On merging identity theory and stress research. Social Psychology Quarterly, 54, 101\u2013112.\n\nThompson, C. A., Beauvais, L. L., & Lyness, K. S. (1999). When work-family benefits are not enough: The influences of work-family culture on benefit utilization, organizational attachment, and work-family conflict. Journal of Vocational Behavior, 54, 392\u2013415.\n\nThompson, C. A., Beauvais, L. L. & Allen, T. D. (2006). Work and family from an industrial\/organizational psychology perspective. In M. Pitt-Catsouphes, E. E. Kossek, & S. Sweet (Eds.), The work and family handbook: Multi-disciplinary perspectives and approaches (pp. 283\u2013307). Mahwah: Lawrence Erlbaum.\n\nWeick, K. E. (1979). The social psychology of organizing (2nd ed.). Reading: Addison-Wesley.\n\nWharton, A. S., & Blair-Loy, M. (2006). Long work hours and family life: A cross-national study of employees' concerns. Journal of Family Issues, 27, 415\u2013436.\n\nWikstrom, T., Palm Linden, K., & Michelson, M. (1998). Hub of events or splendid isolation: Teleworkers and the home. Stockholm: Swedish Transport and Communications Research Board.\n\nvon Thiele, U. (2011). Inability to withdraw from work as related to poor next-day recovery and fatigue among women. Applied Psychology: An International Review, 60, 377\u2013396.\n\nvon Thiele, U., Lindfors, P., & Lundberg, U. (2006). Self-rated recovery from work stress and allostatic load in women. Journal of Psychosomatic Research, 61, 237\u2013242.\n\nVoydanoff, P. (1988). Work role characteristics, family structure demands, and work\/family conflict. Journal of Marriage & the Family, 50, 749\u2013761.\n\nVoydanoff, P. (2004). The effects of work demands and resources on work-to-family conflict and facilitation. Journal of Marriage & Family, 66, 398\u2013412.\n\nVoydanoff, P. (2009). The interaction of work and family demands and resources: Linking mechanisms and boundary-spanning strategies. In D. R. Crane & E. J. Hill (Eds.), Handbook of families and work (pp. 141\u2013171). Lanham: University Press of America.\n\nXie, Y., & Shauman, K. A. (2003). Women in science: Career processes and outcomes. Cambridge: Harvard University Press\n\n# Part III  \nCAREER & ORGANIZATIONAL CONSIDERATIONS\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_12\n\n# 12. Family-Friendly Organizational Policies, Practices, and Benefits through the Gender Lens\n\nJustin M. Sprung1  , Tatiana H. Toumbeva2  and Russell A. Matthews2\n\n(1)\n\nDepartment of Psychology, Luther College, 700 College Drive, 52101 Decorah, IA, USA\n\n(2)\n\nDepartment of Psychology, Bowling Green State University, 43403 Bowling Green, OH, USA\n\nJustin M. Sprung (Corresponding author)\n\nEmail: spruju01@luther.edu\n\nTatiana H. Toumbeva\n\nEmail: ttoumbe@bgsu.edu\n\nRussell A. Matthews\n\nEmail: ramatth@bgsu.edu\n\nKeywords\n\nWork\u2013familyGenderPracticesPoliciesBenefitsOrganizational cultureStigmaSupervisor support\n\n## 12.1 Introduction\n\nConcern for employees' family and leisure time is by no means a new point of interest. Recently, however, with almost 80 % of workers reporting an increased desire for workplace flexibility (Galinsky et al. 2004), organizational leaders are increasingly realizing the need for providing appealing and effective family-friendly benefits. This amplified interest in work\u2013family issues has been brought about, in part, due to the increase in numbers of full-time working women and dual-earner couples in the past few decades (Hammer and Zimmerman 2011). Although the percentages of working men and women in the USA today are roughly equal, with women comprising 46.9 % of the workforce (Bureau of Labor Statistics 2013) , gender still seems to influence a variety of factors within the work environment. One of the most publicized and contentious gender differences in the workplace concerns perceptions and usage of family-friendly benefits. In fact, the Journal of Social Issues recently dedicated an entire special issue to this topic\u2014\"The Flexibility Stigma\" (Williams et al. 2013)\u2014exploring how gender and social class contribute to the stigma surrounding flexible work arrangements. Therefore, although traditional gender norms are thought to be fading, they still appear to actively influence today's workforce, especially when it comes to the use of, and access to, family-friendly benefits.\n\nOur goal in the present chapter is to summarize how gender contributes to the overall efficacy of family-friendly benefits by focusing on two primary aims. First, we review the literature on gender differences in employee utilization and awareness of available family-friendly benefits, followed by a discussion of why these discrepancies may exist. Second, we briefly summarize the literature on the general effectiveness of benefit availability and utilization for improving important employee outcomes. Given the major role that the supervisor plays in the lives of workers (Kossek and Distelberg 2009), our main focus in this section centers around the interplay between supervisor gender and perceived supervisor supportiveness of employee family life. To help further contextualize the effect of gender, we consistently address underlying individual characteristics that may contribute to the existence of gender differences within our two areas of focus. We conclude the chapter by documenting several areas for future inquiry.\n\n## 12.2 Family-Friendly Benefits\n\nAlthough there are a variety of family-friendly benefits offered within the USA, the 1993 Family and Medical Leave Act (FMLA) represents the only legal obligation that US employers have regarding work\u2013family issues . The FMLA stipulates that employers with 50 or more employees must provide up to 12 weeks of unpaid leave employees in order for them to deal with a family or medical issue, as long as the employee has worked at the organization for a minimum of 1250 hours (United States Department of Labor 2013) . Because of the limited amount of federal legislation in this area, the majority of family-related benefits are typically provided at the employer's discretion and generally considered a component of an organization's plan for implementing strategic human resource (HR) practices (Bloom et al. 2011).\n\nOutside of the FMLA, the amount and types of discretionary family-friendly benefits vary across organizations, but typically include some form of flexible work arrangements and\/or dependent care support. Flexible work arrangements (FWA) come in a variety of forms, but share the common objective of allowing employees to have enhanced control over when and where they complete their work (Major and Germano 2006). Commonly recognized FWAs include telecommuting , flextime , a compressed work week , part-time schedules, and job sharing (for an in-depth review of different types of FWAs, see Kossek and Michel 2011). Whereas FWAs provide flexibility in work scheduling, dependent care support programs offer financial and\/or instrumental assistance (e.g., on-site childcare, caregiving referrals, paid leave) to employees for their childcare or eldercare responsibilities (Major and Germano 2006). Both FWAs and dependent care support programs are tools intended to assist employees in balancing their work and family lives. These discretionary benefits are often viewed as a sign of the organization's commitment toward creating a family-friendly workplace (Grzywacz and Butler 2008).\n\nAccording to a recent study conducted by the Society for Human Resource Management (SHRM) , 57 % of organizations offer some form of telecommuting (either part- or full-time) whereas 53 % offer flextime opportunities. In addition, 17 % of organizations provide childcare referral services and 10 % offer eldercare referral services (SHRM 2012). Although family-friendly benefits have become increasingly common in the USA, the actual use of these benefits has lagged behind (Galinsky et al. 2008). Multiple studies demonstrate that employees may not perceive these benefits as a viable option due to informal regulations and\/or a fear of negative consequences as a result of their use (e.g., Eaton 2003; Thompson et al. 1999). In addition, even though these benefits are expected to promote gender equality and workplace inclusion, there is mixed evidence as to whether they are achieving that goal. In fact, some scholars have even suggested that the availability and usage of these benefits may be exacerbating the very problems they were expected to prevent (e.g., Ryan and Kossek 2008; Rudman and Mescher 2013). We can gain a better understanding of how and why these benefits may be having counterintuitive effects by examining their associated gender differences.\n\n## 12.3 Benefit Usage\n\nDespite the increased availability of family-friendly benefits, simply offering these benefits does not necessarily result in their use . For example, in 2012, only 16 % of all employees who were eligible for FMLA leave actually took it (Klerman et al. 2013). Similarly, several studies examining discretionary family-friendly benefits reveal rather low overall usage rates, regardless of their increased availability (e.g., Blair-Loy and Wharton 2002; Galinsky et al. 2008). For both the FMLA and discretionary family-friendly benefits, gender has been a primary point of interest regarding usage rates of available benefits .\n\nSince the origin of the FMLA in 1993, men have consistently taken leave less often and tend to take significantly shorter leaves than women (Armenia and Gerstel 2006; Gerstel and Armenia 2009; Waldfogel 2001). According to the most recent FMLA survey conducted by the Department of Labor , women (21 %) were significantly more likely than men (16 %) to take parental leave within the past 18 months. Although the gender gap in leave-taking still exists, it has slightly decreased over time (7 % in 1995 to 5 % in 2012; Klerman et al. 2013), suggesting that slow progress is being made toward gender equality in the prevalence of leave-taking. In addition, there is preliminary evidence to suggest that although men are less likely than women to take leaves for newborns, men are nearly as likely as women to take leaves for seriously ill children or parents, and just as likely to take leave to care for a sick spouse (Armenia and Gerstel 2006). However, more research is necessary to make any firm conclusions regarding gender equality (and the closing of the gender usage gap). Even if men and women are approaching a similar prevalence rate in leave-taking, women still tend to take significantly longer leaves regardless of the reason for leave (Klerman et al. 2013). Thus, there is a consistent gender gap in leave-taking and leave length , especially when the reason for leave is childbirth .\n\nThese discrepancies (i.e., leave-taking and leave length) may be partially due to the fact that women on maternity leave are more likely to receive some form of replacement pay than men on paternity leave (Galinsky et al. 2008; Matos and Galinsky 2012). At the same time, however, the opportunity for at least some form of pay during parental leave has remained rather low for both men and women in the USA over the past 20 years (Van Giezen 2013). As such, other forces are likely influencing this discrepancy. Men's lower use may also be due to traditional gender norms , as there seems to be a larger social stigma associated with men taking paternity leave than there is for women taking maternity leave (Berdahl and Moon 2013), an issue we will discuss later in the chapter. In addition, economic factors may come into play. Specifically, excluding cases of adoption or surrogacy, women must take at least some time off in order to physically have the child. If the husband also takes leave, the likelihood of there being any money coming in for that family is, based on the data available, low. Thus, regardless of any social stigma , this economic reality can have a large impact on the decisions that people make and should not be ignored by researchers examining this issue .\n\nAs for gender differences in usage rates of discretionary family-friendly benefits, the evidence is even less clear. Consistent with popular perception, the majority of studies provide evidence suggesting that women use flexible schedules, work from home, and make use of childcare referral more often than men (e.g., Galinsky et al. 2004; Hammer et al. 2005; Hill et al. 2003; Singley and Hynes 2005). However, recent findings suggest that there may be no clear gender difference in the usage of discretionary family-friendly benefits as a whole. According to McMenamin (2007), men and women use flexible benefits equally, with the percentage of men and women using flexible schedules remaining comparable over the past 20 years. The difference between men and women may, however, lie in the type of flexibility that they prefer .\n\nFor example, a recent popular press article suggested that men and women do not differ in usage of flexible work options that permit face time within the organization (i.e., flexible arrival and departure times), but women tend to be more likely to telecommute than men (Fondas 2013). Thus, women may be more willing to use benefits in which they would sacrifice face time, while men are less willing to do so. This is congruent with empirical research suggesting that telecommuting is an arrangement that is preferred by women (e.g., Raghuram et al. 2003; Singh et al. 2013). This is also consistent with traditional gender roles regarding work and family responsibilities which, in turn, may help to contribute to the career penalties that women face as a result of using family-friendly benefits .\n\nIt should be noted, however, that the majority of research on family-friendly benefits focuses on perceptions about the availability and perceived outcomes of these benefits rather than gender differences in their actual use. Accordingly, it is difficult to draw firm conclusions regarding gender differences in the usage of discretionary family-friendly benefits. Due to the mixed findings, and paired with the fact that usage rates tend to be low for both men and women, further research would be beneficial to form a clear picture regarding the actual usage of discretionary family-friendly benefits and whether men and women prefer different forms of benefits. Another specific area for future research may be to examine potential gender differences in eldercare assistance given the growing number of employees with both child and eldercare responsibilities .\n\n## 12.4 Benefit Awareness\n\nOne of the most proximal barriers to employee utilization is employees' awareness (or lack thereof) of available benefits . The FMLA , in particular, suffers from ignorance of its provisions. According to the Department of Labor , only 66.2 % of employees have heard of FMLA in 2012, which is only 7 % higher than the year 2000 (Klerman et al. 2013). Given that men tend to use FMLA benefits significantly less than women do, it should come as no surprise that men are also less aware of the benefits available to them. In fact, gender has been suggested as one of the strongest predictors of employees' knowledge and awareness of FMLA regulations, with women purported as being 3\u20136 times more likely than men to be aware of their eligibility for FMLA parental leave (Kramer 2008). Similarly, evidence based on a nationally representative sample indicated that among eligible employees, men were significantly less likely than women (72 % compared to 91 %, respectively) to report that they had access to FMLA parental leave regardless of various sociodemographic factors (i.e., race, marital status, children, firm size, and authority level; Baird and Reynolds 2004). Consequently, this lack of awareness among male workers may contribute to the gender discrepancy in the usage of FMLA leave .\n\nRegarding discretionary family-friendly benefits, although the evidence is mixed in terms of gender differences in actual utilization, studies assessing employee awareness consistently provide similar findings to results within the FMLA literature. Specifically, multiple studies demonstrate that women are more likely to have knowledge and awareness of discretionary family-friendly benefits offered by their employers (e.g., Martenengo et al. 2010; Prottas et al. 2007; Villablanca et al. 2011). Therefore, a clear gender gap emerges from the available evidence with men being less aware of both legal and discretionary family-friendly benefit options .\n\nThis lack of awareness may be a contributing factor to why a large majority of people do not use the benefits available to them. A promising area for future research may be an identification of effective strategies for increasing benefit awareness among employees. Klerman et al. (2013) state that the most common sources of knowledge for the FMLA are posters and communication from the HR department; however, no evidence is presented as to whether certain strategies are utilized to a greater extent\u2014or more effective\u2014for men compared to women. It would be interesting to examine whether HR professionals put forth more effort in advertising benefits to female employees, and whether these strategies are actually (possibly unintentionally) contributing to the gender discrepancy in awareness. Regardless of gender differences, raising employee awareness would likely be beneficial for all eligible employees .\n\n### 12.4.1 Intentions of Using Benefits\n\nMen's general lack of awareness of family-friendly benefits, compared to women, seems to suggest that improved awareness might lead to increased usage of the available benefits. Multiple studies assessing intentions to seek out and\/or use family-friendly benefits, however, suggest that this may not be the case. For example, Gerstel and McGonagle (1999) found that women (25 %) were more likely than men (16 %) to report that they need to take a leave from a job for reasons covered by the FMLA , regardless of whether they actually took the leave. Similarly, Klerman et al. (2013) reported that women were nearly twice as likely as men to report a need for FMLA-related leave that was unmet . Therefore, even though men tend to be using leaves less often, evidence from the FMLA literature suggests that men may still be having their needs met more-so than women. As such, men may be less aware of the FMLA simply because they do not seek out family friendly-benefits as often.\n\nOn a related note, a recent survey of college students indicated that although men and women placed equal value on work flexibility and work\u2013life balance, men were significantly less likely to seek flexible arrangements upon entering the workforce, primarily due to traditional gender stereotypes (Vandello et al. 2013). This is congruent with Butler et al.'s (2004) findings demonstrating that even when men report that they have positive work outcome expectancies associated with family-friendly benefits, this does not increase their intentions of using them. Thus, although both men and women are motivated to achieve work\u2013life balance, men seem reluctant to use family-friendly benefits to achieve that goal. Therefore, factors other than awareness (i.e., gender norms) may be more important to consider regarding the gap between policy availability and utilization for both men and women.\n\n## 12.5 Benefit Accessibility\n\nSome have proposed that gender, in and of itself, contributes to the accessibility of family-friendly benefits, yet, empirical evidence is mixed . Some findings indicate that women have greater access to FWAs (e.g., Glauber 2011; McNamara et al. 2012) whereas others suggest that men have greater access (e.g., Golden 2009; McCrate 2005). Taking the previous literature together as a whole, accessibility to discretionary family-friendly benefits seems much closer to \"gender neutral\" than benefit awareness. For example, even though Glauber (2011) reported that men are less likely to have access to formal flexible work scheduling than women overall, the discrepancy (52 % compared to 56 %) was not very large and may be due to other factors (i.e., size of organization). Moreover, a factor that makes this research less straightforward than the awareness literature is the fact that some studies examined formal flexibility policies alone, whereas other studies include informal flexibility offered on a case-by-case basis .\n\nTaking into account the \"formality\" of FWAs, there is more consensus regarding gender differences. For example, although Golden (2009) and Glauber's (2011) overall conclusions regarding gender and access to FWAs contrast, both acknowledge that women may be more likely to have access to formal FWAs offered by the organization while men tend to be more likely to benefit from informal flexibility given on a case-by-case basis. Therefore, many organizations may selectively implement FWAs as a function of formalized discretion of managers to grant or deny requests based on a case-by-case system rather than a universal policy (Kelly and Kalev 2006). Due to this gender discrepancy between formal versus informal accessibility, it has been suggested that implementation of more formal programs\u2014rather than the case-by-case system\u2014would likely promote more gender equity in access to family-friendly benefits (Golden 2009). To this end, we discuss the importance of the supervisor in the perceived availability, implementation, and effectiveness of family-friendly benefits later in this chapter given the integral role supervisors play in this process .\n\nAlthough the evidence regarding who has greater access to family-friendly benefits is mixed, some workers may be at a clear disadvantage when it comes to accessing these benefits regardless of gender. For example, employees that are less privileged (i.e., lower wages, lower education levels, and hourly work) have restricted access to family-friendly benefits compared to their more privileged counterparts (Galinsky et al. 2011; Swanberg et al. 2005). In addition, men and women who are married, have children, and work in managerial positions appear to have somewhat greater access to flexibility in work scheduling (Golden 2009). There has also been speculation that race plays a role, with white employees proposed as having greater access to family-friendly benefits than African\u2013Americans (e.g., Golden 2008, 2009). Though a full review of this area is beyond the scope of this chapter, it is important to note that many factors other than gender are likely to play a critical role in the perceived accessibility and subsequent usage of family-friendly benefits .\n\n### 12.5.1 Organizational Gender Composition\n\nA seemingly common assumption is that jobs with a greater composition of women employees offer more family-friendly benefits. This assumption is supported by many studies (e.g., Davis and Kalleberg 2006; Gerstel and Aremenia 2009; Lowen and Sicilian 2009). Recent evidence on this topic, however, is not so clear cut. For example, Glauber (2011) suggests that gender-integrated occupations (40\u201360 % female) have the greatest access to flexible scheduling, with male-dominated occupations (greater than 70 % male) having the least access and female-dominated occupations (greater than 70 % female) falling between the two. Furthermore, in the 2012 National Study of Employers , Matos and Galinsky (2012) suggested that employers that were larger, nonprofit, and have more women and racial\/ethnic minorities in top positions were more likely to offer flexibility and dependent care assistance to their employees. Therefore, access may not necessarily be dependent on the gender composition of the organization as a whole but the size of the organization and gender composition of the employees who are in a position of power .\n\nInterestingly, organizational size may have a differential impact on men and women. Glauber (2011) found that men tend to have the greatest access to flexible scheduling when they work in small organizations whereas women have the greatest access when they work in large organizations. This may be due to women's access to formal benefits in larger organizations and men's access to informal benefits in smaller organizations (Glauber 2011). This is consistent with a previous study suggesting that women may have greater access to formal benefits whereas men may have greater access to informal benefits (Golden 2009), and further illustrates the importance of supervisors allowing all employees the same opportunity for FWAs regardless of whether the benefits are formally or informally offered.\n\n## 12.6 Gender Norms and the Flexibility Stigma\n\nPerhaps the most important influence on whether or not employees view family-friendly benefits as a viable resource is the perceived accessibility and usefulness of these resources. Although it is debatable whether firm conclusions can be made regarding gender differences in actual accessibility and usage of family-friendly benefits, it is clear that the overall usage rates of family-friendly benefits are low, compared to their availability, regardless of gender (Galinsky et al. 2008, 2011). Whereas awareness and formal accessibility may influence actual usage of available benefits, many informal and unspoken norms play a large role as well. One of the most powerful contributors to perceived accessibility is traditional gender norms .\n\nAlthough gender equality has become an overarching societal goal, gender stereotypes continue to serve as a powerful influence on human behavior (Prentice and Carranza 2002). For example, traditional gender norms and expectations (i.e., male as provider, female as housekeeper) are still often prescribed today; men are commonly expected to be dominant, career-focused, and independent while women are often perceived as more caring, interdependent, and modest than men. When men or women do not conform to these stereotypes, they often suffer from social and professional consequences (e.g., Moss-Racusin et al. 2010; Rudman and Phelan 2008). Experiencing \"backlash\" as a result of gender norm-breaking reinforces typical gender norms. In turn, both men and women may be prevented from breaking these norms, thereby maintaining the gender status quo (Rudman and Fairchild 2004). This maintenance of the gender status quo appears to be playing a key role in employees' perceptions and usage of family-friendly benefits.\n\nEven though formal benefits may recognize the growing needs of today's families, there are many informal regulations and processes that determine the extent to which these benefits are perceived as available to the employees (Eaton 2003). In fact, these informal practices may, knowingly or unknowingly, stigmatize workers who take advantage of family-friendly benefits. This is referred to as the flexibility stigma , and is often based on typical gender stereotypes (Bornstein 2013). Both men and women can suffer from stigmatization associated with perceptions and usage of family-friendly benefits, albeit in different ways. As described below, the flexibility stigma for men is based on gender non-conforming behavior whereas the stigma for women stems from gender-conforming behavior (Williams et al. 2013).\n\n### 12.6.1 Stigma for Women\n\nMultiple studies highlight the negative outcomes that working women with children encounter. It has been well established that working mothers face a \"maternal wall\" when it comes to various workplace decisions. The reason for this may be that female employees, especially low-wage workers, are penalized for motherhood through perceived incompetence and reduced consideration for promotions regardless of their actual work qualifications (e.g., Crosby et al. 2004; Heilman and Okimoto 2008; Dodson 2013). This stigmatization of mothers stems from the ideal worker norm, which states that ideal workers place more emphasis on work obligations than unpaid family responsibilities (Kelly et al. 2010). As a result, mothers are probably more likely to be seen as a \"risk\" for making use of a family-related benefit due to the stereotypical gender expectation that they will be preoccupied with family care responsibilities. Although the option of family-friendly benefits may be available to help women balance work and family, Williams et al. (2013) suggest that making use of a family-friendly benefit may be interpreted by other professionals, as well as the individual employee herself, as a form of moral violation for not living up to one's work demands and expectations. As a result, because women are traditionally thought of as the primary caregiver, their careers may be negatively impacted when they have to fulfill family responsibilities.\n\nWomen may be discounted not only within the workplace but outside of it as well. Working women may be viewed as less effective parents than non-working mothers (e.g., Eataugh and Folger 1998; Gorman and Fritzsche 2002), especially when working in a stereotypically male-gendered occupation (Berdahl and Moon 2013; Okimoto and Heilman 2012). Therefore, although women may suffer career penalties for being a mother, they are also likely to suffer social consequences (i.e., being viewed as a bad mother) if they are not the primary caregiver. Herein lies the conundrum for women: if they have children or family responsibilities, they are expected to take leave and\/or use available benefits, for which their careers are penalized. At the same time, if they choose to focus on their careers by avoiding the use of family-friendly benefits, they may be ostracized for not conforming to what a woman \"should\" do. In terms of using family-friendly benefits, women seem to potentially be at a disadvantage either way.\n\n### 12.6.2 Stigma for Men\n\nOn the other hand, it has also been well established that men who are seen as less masculine are respected less and suffer more negative consequences than men judged as more masculine (e.g., Funk and Werhun 2011; Heilman and Wallen 2010; Judge et al. 2012). Thus, although simply having family responsibilities does not result in as many negative consequences for men as it does for women (Fuegen et al. 2004), the act of caring for children and\/or publicizing their family responsibilities may result in stigmatization. For men, requesting a family leave or some form of flexibility to care for a dependent is violating their traditional gender norm of placing work as their primary priority, and may be seen as a sign of weakness or low status. Multiple studies suggest that men experience more negative reactions from others than women when asking for a family leave (e.g., Allen and Russell 1999; Wayne and Codeiro 2003), experiencing family conflict (Butler and Skattebo 2004), and working fewer hours in order to balance family responsibilities (Vinkenburg et al. 2012). Unlike women who get penalized for conforming to their gender role, men get penalized for not conforming to their traditional gender role. Thus, though the mechanism is different, both men and women's careers may be penalized for making use of available benefits.\n\nSimilar to women, men are also socially stigmatized for breaking gender norms , as demonstrated in two recent studies. Berdahl and Moon (2013) found that men who were more involved in domestic-related tasks and caregiving for their children experienced more harassment (i.e., being seen as \"not man enough\") than men who spent less time caring for their children. Similarly, Rudman and Mescher (2013, p. 325) demonstrated that male employees who request family leave suffer from a femininity stigma and are punished for \"acting like a woman.\" Interestingly, both men and women were equally likely to feminize male leave requesters, viewing them as weaker than men who did not ask for leave. This finding is congruent with previous research suggesting that men who act modestly (rather than dominantly) tend to suffer more prejudice from both men and women due to a perceived violation of their gender-prescribed behavior (Moss-Racusin et al. 2010). Therefore, even though women tend to report more acceptance of gender norm-breaking (e.g., Coleman and Franiuk 2011; Goldberg et al. 2012), men who utilize family-friendly benefits are not only subjected to harassment by other men but women as well. As such, traditional gender norms regarding benefit use are being perpetuated by both men and women.\n\n### 12.6.3 Gender Norm Summary\n\nAs can be seen, both men and women suffer career and social consequences, albeit in different ways, as a result of the flexibility stigma . In fact, Berdahl and Moon (2013) suggest that gender performance may drive workplace mistreatment more than work performance . Unfortunately, being perceived as a lesser man and\/or a lesser worker for using family-friendly benefits or violating gender norms is not all that uncommon (cf., Bornstein 2013; Stone and Hernandez 2013). These stereotypes , in turn, may undermine worker's use of available benefits: men may be less likely to use benefits to maintain their status of masculinity whereas women may be hesitant to utilize benefits for fear of being stigmatized as less competent (Correll et al. 2007). Therefore, the potential for improving work\u2013life balance among all employees continues to be limited by persisting traditional gender norms.\n\nAlthough there is some evidence that gendered perceptions regarding the flexibility stigma may be changing for the better among young adults (e.g., Coleman and Franiuk 2011), other evidence suggests that perceptions regarding gender roles and work\u2013family responsibilities has remained fairly constant since the 1980s (Goldberg et al. 2012). Poelmans (2012) notes that these norms are deeply rooted in several layers of socio-cultural values (i.e., dyadic, organizational, societal, and cultural), and will take a tremendous amount of time and effort to change. Given that we know gender stereotyping and traditional gender norms still persist when it comes to family-friendly benefits, future research should examine how to potentially reduce the negative perceptions associated with breaking gender norms and will likely result in a better understanding of how to encourage utilization of family-friendly benefits.\n\n## 12.7 Outcomes Associated with Family-Friendly Benefits\n\nThe negative career and social penalties, the flexibility stigma , and the perpetuation of traditional gender norms represent some of the unintended negative consequences associated with family-friendly benefits. It has been well established that both men and women may forego utilizing family-friendly benefits for fear of negative career consequences and\/or due to financial necessity (e.g., Klerman et al. 2013). However, the overarching goal of these benefits is to help employees balance work and family responsibilities and to improve worker satisfaction. In this section, we will provide a brief overview of the effectiveness of family-friendly benefits at promoting positive outcomes , and whether benefit effectiveness differs between men and women. Given the importance of the supervisor in determining employee outcomes\u2014as well as the fact that supervisor support may serve as an important family-friendly benefit in and of itself\u2014the majority of this section focuses on the worker-supervisor interaction in contributing to positive employee outcomes.\n\n### 12.7.1 General Effectiveness\n\nOne of the most immediate effects of offering family-friendly benefits concerns the attractiveness of the organization for potential and current employees. Both men and women value work-life balance (Vandello et al. 2013). Not surprisingly, organizations that offer family-friendly benefits are often perceived as more supportive and desirable among job applicants of both genders (e.g., Bourhis and Mekkaoui 2010; Wayne and Casper 2012). Therefore, the organization's reputation concerning family-friendly policy availability serves as a beneficial recruiting tool, irrespective of gender.\n\nIn terms of positive individual and organizational consequences, family-supportive benefits have been shown to be associated with a variety of favorable outcomes for employees and the organization as a whole. For example, multiple studies have demonstrated that the availability and use of family-friendly benefits tends to be associated with positive work attitudes (i.e., commitment, job satisfaction , and intentions to stay in the organization; e.g., Allen 2001; Breaugh and Frye 2007; Casper and Harris 2008), increased performance (e.g., Bloom et al. 2011; Jones and Murrell 2001), and reduced conflict between the work and family domains (e.g., Butts et al. 2013; Kossek et al. 2011; Thompson et al. 1999). Although not all studies find beneficial effects associated with family-friendly benefits (e.g., De Menezes and Kelliher 2011), a review of the literature suggests that family-friendly benefits can, and often do, benefit employees.\n\nStudies that have examined gender differences, however, suggest that the magnitude of these positive effects may not be universal. For example, Butts et al. (2013) report in their meta-analysis that in samples with more women, the relation between policy availability and job satisfaction, as well as the relation between policy use and work\u2013family conflict (i.e., a form of inter-role conflict characterized by incompatibility between the work and family domains; Greenhaus and Beutell 1985), were weaker compared to samples with more men. The authors speculated that available benefits may be more beneficial to men because they have fewer caregiving responsibilities whereas women's needs are less likely to be met due to their greater caregiving workload . However, Carlson et al. (2010) found that women benefit more from using flexible schedules than men, as women experienced less work\u2013family conflict. Still, Hammer et al. (2005) found that usage of family-friendly benefits may not reduce work\u2013family conflict and may actually lead to increased conflict for women, reasoning that women's usage may lead to the expectation of taking on even more family responsibilities.\n\nIn addition, there is evidence suggesting that men and women may possibly judge the utility of family-friendly benefits based on different criteria. For example, Sharpe et al. (2002) found that women were more likely to use FWAs to reduce work\u2013family conflict whereas men were more likely to report using the benefits to enhance personal productivity (i.e., structure and routine in work and personal life). This implies that men may be more likely to judge the effectiveness of FWAs based on their job performance whereas women may be more likely to judge their effectiveness based on the ability to reduce work\u2013family conflict. Future research would be beneficial in determining whether this is indeed the case.\n\nThe majority of recent studies on the effectiveness of family-friendly benefits mention supervisor support as a crucial factor in whether these benefits will be effective and\/or perceived as useful. For example, Breaugh and Farabee (2012) noted that telecommuting and other FWAs are not quick fixes, and require supervisor support to be effective. In terms of performance , although Bloom et al. (2011) found that family-friendly benefits were positively related to organizational performance, this relation disappeared once the authors controlled for the quality of management practices. In fact, some have even questioned whether work\u2013life balance is actually a matter of choice for the employee. Gregory and Milner (2009) suggest that work\u2013life balance may be constrained by the culture created within the organization due to the enabling of traditional gender roles; in turn, the organizational culture may limit the employees' perceptions of what benefits they can actually use. Accordingly, to better understand the effectiveness of family-friendly benefits for producing positive employee outcomes , it is important to also understand the supervisors' role in the process of administering these benefits.\n\n## 12.8 Importance of Supervisor\n\nIn an attempt to enhance employees' ability to manage the work\u2013family interface, many organizational initiatives involve the development and maintenance of a family-supportive workplace. Thomas and Ganster (1995) identify family-supportive supervision as one of the two components comprising a family-supportive workplace environment, with the other component being family-friendly benefits. Despite the importance of offering formal workplace supports such as competitive family-friendly benefits, informal supports (i.e., direct supervisor support) are perhaps even more critical to the overall success of family-supportive workplace initiatives. For example, Behson (2005) demonstrated that supportive supervision explained more variance in job satisfaction , work\u2013family conflict, and turnover intentions than formal support . Furthermore, unsupportive supervisors can act to undermine the effect of family-friendly benefits and offset any potential positive impact of benefit utilization (Kossek 2005). Despite the importance of supervisor support , the role of family-friendly benefits should not be downplayed. Foley et al. (2006) found that family-supportive organizations are a boundary condition under which family-supportive supervision can occur. In other words, supervisors are less likely to provide family-specific support to employees in the absence of a family-supportive organizational culture , as evidenced by formal family-friendly benefits.\n\nAs already noted, scholars have offered multiple explanations for the counterintuitive evidence surrounding organizational family-friendly benefits, such as underutilization due to lack of awareness about what is available and concerns about possible negative repercussions associated with their use (e.g., Kossek 2005; Neal and Hammer 2007). Consistently, however, the role of direct supervision has been presented as a critical determinant of benefit usage and efficacy. Immediate supervisors are in a unique position to facilitate employees' management of work and family roles, especially given that the simple presence of family-friendly benefits is not sufficient to alleviate the experience of work\u2013family conflict (Allen 2001; Kossek 2005).\n\nA great deal of evidence exists that links supervisor support and the experience of work\u2013family conflict (e.g., Byron 2005; Hammer et al. 2007), and the negative impact of work\u2013family conflict has been demonstrated on an array of individual and organizational outcomes (e.g., job and family satisfaction , job performance, turnover intentions , organizational commitment , physical and mental health ; Allen 2001; Allen et al. 2000; Ford et al. 2007; Grzywacz and Bass 2003; Hammer et al. 2005). Employees who perceive their supervisors as being supportive of personal work\u2013family management needs are likely to experience reduced levels of work\u2013family conflict (cf., Kossek et al. 2011). Family-supportive supervisors engage in behaviors such as provision of emotional support (e.g., listening, empathizing), instrumental support (e.g., accommodating flexible work schedules) , role modeling, and brainstorming creative work\u2013family management strategies with employees to help them manage work and non-work demands (Hammer et al. 2009). Supervisors who exhibit such family-supportive supervisor behaviors (FSSB) tend to be perceived as more accommodating of employees' non-work responsibilities than supervisors who do not exhibit these behaviors. In a recent study, Matthews et al. (2014) provided preliminary evidence that FSSB explain incremental variance in employee work engagement above and beyond perceptions of general managerial effectiveness and organizational support. In addition, work engagement was found to mediate the relationship between employee perceptions of FSSB and subjective wellbeing.\n\nWays in which family-supportive supervisors can help employees reduce the experience of work\u2013family conflict include: fostering a high-quality exchange relationship with their subordinates (Liden and Graen 1980), demonstrating family-supportive behaviors (Hammer et al. 2009), championing the use of family-friendly benefits (Hammer et al. 2007), and helping create or maintain a family-friendly organizational culture (Thompson et al. 1999). Supervisors are typically viewed as representatives of organizational practices and top leadership , and as such, they can influence employee attitudes and outcomes (e.g., wellbeing, affective organizational commitment , turnover intentions , work\u2013family conflict; Allen 2001; Major and Lauzun 2010; Thompson et al. 1999). As such, the power of the supervisors should be harnessed to enhance the utilization and efficacy of family-friendly benefits. For an expanded discussion on the role of supervisor support in the management of the work\u2013family interface, see Matthews et al. (2013).\n\n### 12.8.1 Supervisor Support and Gender Differences\n\nGiven the importance of supervisor support in helping employees manage the work\u2013family interface, it is surprising that the supervisor\u2013employee dyad has not been examined in greater depth in the literature. Specifically, as it pertains to benefit utilization, supervisor gender is a key contextual factor that can help further explain why family-friendly benefits do not always have the intended positive results.\n\nTo address this gap in the literature, researchers have begun to examine the issue of supervisor gender as it relates to work\u2013family management. Anecdotally, female supervisors are typically considered as more supportive of work\u2013family management needs than male supervisors. Research seems to support this general belief (Hopkins 2002; Parker and Allen 2002). For example, supervisors often provide differential support to employees based on gender, with female supervisors offering higher levels of family-specific support (i.e., making referrals to employee assistance programs) to female employees (Gerstein et al. 1993). Perhaps one reason for this comes from research by Foley et al. (2006) who found that employees feel more comfortable requesting family-related help from same-gender, compared to cross-gender supervisors. Interestingly, Foley et al. noted this gender effect did not weaken in the presence of a highly family-supportive organizational culture. Although the gender effect they found explained less variance in family-supportive supervision and family-supportive culture, gender differences present a critical issue that should not be overlooked in future research .\n\nConsistent with Foley et al.'s (2006) findings, Hopkins (2002) showed that female employees are more likely to seek assistance for family and personal problems from same-gender supervisors. Hopkins also provided support for the interaction between employee and supervisor gender with respect to important family- and work-related outcomes. Specifically, female supervisors were more likely to provide formal support referrals to male employees. Interestingly, however, Hopkins also noted that female employees perceived their male supervisor to be more supportive (i.e., understanding and flexible) than did female employees with female supervisors. A potential caveat that could help explain this seemingly contradictory finding is the gender composition of the organization. Specifically, in organizations where female employees are underrepresented relative to male employees, female employees perceive their female supervisor as less supportive than their male supervisors (Ryan et al. 2012). This finding is inconsistent with most anecdotal and scholarly evidence, and highlights the need to always consider important contextual and organizational environments within which supervisor\u2013employee dyads are nested.\n\nFurther highlighting the role that supervisor gender plays in widening the gender gap associated with supervisory support, recent findings by Raghuram et al. (2012) demonstrated that the difference in stress levels for male versus female immigrants was twice as large when the supervisor was perceived as unsupportive. Taking a more granular approach, Winfield and Rushing (2005) examined whether gender influenced the type of support provided and found that when the direct supervisor was of the same gender, employees were likely to receive interactional but not instrumental support.\n\nLi and Bagger (2011) extended the issue of supervisor\u2013employee dyad similarity to encompass similarity in not only gender but also childcare responsibilities. Specifically, the authors suggested that within cross-gender dyads, provision of family-specific support was dependent on the amount of supervisor childcare responsibilities such that supervisors with higher levels of childcare responsibilities relative to their employees provided higher levels of family-supportive supervision. These effects, however, were not supported for same-gender supervisor\u2013employee dyads, perhaps due to the unconditional provision of support within same-gendered dyads regardless of any differences (or similarities) in childcare responsibilities. As such, in addition to consideration of gender, dependent care responsibilities should be examined as another factor that may play a role in perpetuating the gender gap.\n\nThese findings indicate that supervisors may in fact be key perpetrators of the development and maintenance of a gender gap as it pertains to utilization and efficacy of family-friendly benefits. Well-known social psychology phenomena such as favoritism, interpersonal attraction, and similarity bias may be the primary culprits for this gender-driven differential provision of support (Foley et al. 2006). Future research should continue to devote efforts to further delineate how supervisor demographic characteristics such as gender can act to facilitate or impede the success of family-friendly benefits and employees' ability to effectively manage the work\u2013family interface.\n\n### 12.8.2 I-Deals and the Gender Gap\n\nIdiosyncratic deals (i.e., i-deals) have been widely promoted as a best practice in strategic management (e.g., Hornung et al. 2008). Major and Lauzun (2010) extended the i-deal framework to the work\u2013family context by drawing on previous qualitative research demonstrating that many of the work\u2013life needs and requests employees make of supervisors were highly idiosyncratic in nature (e.g., childcare, eldercare , flextime ; Morganson et al. 2007). The work\u2013family i-deal is defined as a \"voluntary, personalized agreement in which an aspect or aspects of the employment arrangement are negotiated between an individual employee and the employer expressly for the mutual benefit derived from reduced work\u2013family conflict\" (Major and Lauzun 2009, p. 6). Major and Lauzun argue that when supervisors foster high-quality leader-member exchange (i.e., LMX; unique relationships ranging from low to high-quality based on exchange of resources; Liden and Graen 1980) with their employees, they are more likely to engage in constructive negotiation of the work\u2013family i-deal that would address specific employee work\u2013family needs.\n\nEncouraging supervisors to negotiate i-deals with each of their employees is one way to reduce the gender gap that occurs with respect to utilization and effectiveness of family-friendly benefits. Furthermore, and in light of Kossek et al.'s (2011) meta-analytic findings suggesting that perceptions of family-supportive supervision are more important in helping to alleviate employee work\u2013family conflict compared to perceptions of generally supportive supervision, i-deals could be harnessed to match the type of provided supervisor support to the unique employee work\u2013family situation, irrespective of gender. A \"one-size-fits-all\" approach in the provision of family-specific supervision should be discouraged as it is considered ineffective work\u2013family management (Major et al. 2008). As such, and extending previous recommendations for supervisor training programs (e.g., Major and Lauzun 2009), future intervention work should aim to educate supervisors on how to effectively negotiate i-deals with both male and female employees. Specifically, supervisors should become aware of relevant gender differences, biases, and stereotypes and learn how to overcome them, and as a result, become well-equipped to recognize the idiosyncrasies specific to each individual employee.\n\n### 12.8.3 Importance of Supervisor Summary\n\nThere is irrefutable evidence that supervisors are crucial to the success of family-friendly benefits and the employees' ability to effectively manage the work\u2013family interface. Although studies on the supervisor\u2013employee dyad are limited, empirical and theoretical support exists for the effect of supervisor\u2013employee gender similarity on the experience of supportive supervision and various employee outcomes. Given that supervisors can serve as a communication channel connecting various levels within an organization, they should be encouraged to champion family-friendly benefits and provide fair and idiosyncratic family-specific support that is free of gender biases and stereotypes.\n\n## 12.9 Future Directions\n\nBecause gender plays a large role in the utilization and effectiveness of family-friendly benefits, we can greatly benefit from future research on the subject. Although the intersection of gender and family-friendly benefits has been widely researched of late, there are many areas for future inquiry. First of all, further examination of employer practices regarding advertisement of family-friendly benefits is warranted. In particular, we suggest exploration into whether certain practices are advertised on a gendered basis, i.e., are some benefits advertised more to women than men, and vice versa. Second, assessing potential gender differences in strategies to obtain work\u2013life balance is another valuable avenue to investigate. For example, as previously noted, the evidence concerning gender preferences for particular benefits is mixed. Researchers should aim to examine whether men and women use these benefits for different reasons, and whether men and women differ in their tendency to seek out formal versus informal benefits . Due to the stigmatization associated with these benefits, employees may be covert in seeking out support for their family life.\n\nSimilarly, it would be constructive to examine whether men and women judge the effectiveness of family friendly-benefits based on different criteria. As Sharpe et al. (2002) noted, men and women may put more emphasis on performance and work\u2013family conflict, respectively. Is this indeed the case, or are men and women simply reporting in a way that is congruent with traditional gender stereotypes? This leads to perhaps the most significant area for future investigation\u2014how to reduce the negative perceptions associated with breaking gender norms . There is a desperate need for future research in this area, as gendered stigmatization is evident for both men and women. Finally, a topic that can work in concert with that mentioned previously is future investigation into the supervisor\u2013employee dyad. Preliminary evidence suggests the importance of considering gender as a large contributor to employee perceptions and supervisor decisions related to family-friendly benefits. Given that supervisors may reinforce traditional gender norms (Kelly 2010)\u2014thereby discouraging some employees from utilizing benefits while possibly encouraging others\u2014future research should continue to explore this dynamic.\n\n## 12.10 Conclusion\n\nGender norms and expectations are still alive and well in the workplace today, particularly when it comes to family-friendly benefits. As a result, both men and women are at risk for social and professional stigmatization. Although many organizational benefits and procedures advertise gender equality, these claims appear to be superficial due to the persistent ideals of how men and women \"should\" act in their work and family domains. Until employees, supervisors, and organizations put forth a genuine effort to enforce gender equality by overcoming traditional gendered expectations, in both formal and informal procedures, family-friendly benefits will continue to show a mix of positive and negative outcomes. It is hoped that this chapter will stimulate interest in the topic and therefore encourage progress toward the goal of gender equality.\n\nReferences\n\nAllen, T. D. (2001). Family-supportive work environments: The role of organizational perceptions. Journal of Vocational Behavior, 58, 414\u2013435.\n\nAllen, T. D., & Russell, J. E. (1999). Parental leave of absence: Some not so family-friendly implications. Journal of Applied Social Psychology, 29, 166\u2013191.\n\nAllen, T. D., Herst, D. E. L., Bruck, C. S., & Sutton, M. (2000). Consequences associated with work-to-family conflict: A review and agenda for future research. Journal of Occupational Health Psychology, 5, 278\u2013308.PubMed\n\nArmenia, A., & Gerstel, N. (2006). Family leaves, the FMLA, and gender neutrality: The intersection of race and gender. Social Science Research, 35, 871\u2013891.\n\nBaird, C. L., & Reynolds, J. R. (2004). Employee awareness of family leave benefits: The effects of family, work, and gender. Sociological Quarterly, 45, 325\u2013353.\n\nBehson, S. J. (2005). The relative contribution of formal and informal organizational work-family support. Journal of Vocational Behavior, 66, 487\u2013500.\n\nBerdahl, J. L., & Moon, S. H. (2013). Workplace mistreatment of middle class workers based on sex, parenthood, and caregiving. Journal of Social Issues, 69, 341\u2013366.\n\nBlair-Loy, M., & Wharton, A. S. (2002). Employees' use of work-family policies and the workplace social context. Social Forces, 80, 813\u2013845.\n\nBloom, N., Kretschmer, T., & Van Reenen, J. (2011). Are family-friendly workplace practices a valuable firm resource? Strategic Management Journal, 32, 343\u2013367.\n\nBornstein, S. (2013). The legal and policy implications of the \"flexibility stigma\". Journal of Social Issues, 69, 389\u2013405.\n\nBourhis, M., & Mekkaoui, R. (2010). Beyond work-family balance: Are family-friendly organizations more attractive? Industrial Relations, 65, 98\u2013117.\n\nBreaugh, J. A., & Farabee, A. M. (2012). Telecommuting and flexible work hours: Alternative work arrangements that can improve the quality of work life. In N. P. Reilly, M. J. Sirgy, & C. A. Gorman (Eds.), Work and quality life: Ethical practices in organizations (pp. 251\u2013274). New York: Springer.\n\nBreaugh, J. A., & Frye, N. K. (2007). An examination of the antecedents and consequences of the use of family-friendly benefits. Journal of Managerial Issues, 19, 35\u201352.\n\nByron, K. (2005). A meta-analytic review of work-family conflict and its antecedents. Journal of Vocational Behavior, 67, 169\u2013198.\n\nBureau of Labor Statistics. (2013). Labor force statistics from the Current Population Survey. Annual average data. Retrieved from http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200btables.\u200bhtm.\n\nButler, A., Gasser, M., & Smart, L. (2004). A social-cognitive perspective on using family-friendly benefits. Journal of Vocational Behavior, 65, 57\u201370.\n\nButler, A., & Skattebo, A. (2004). What is acceptable for women may not be for men: The effect of family conflicts with work on job-performance ratings. Journal of Occupational and Organizational Psychology, 77, 553\u2013564.\n\nButts, M. M., Casper, W. J., & Yang, T. S. (2013). How important are work-family support policies? A meta-analytic investigation of their effects on employee outcomes. Journal of Applied Psychology, 98, 1\u201325.PubMed\n\nCarlson, D. S., Grzywacz, J. G., & Kacmar, K. M. (2010). The relationship of schedule flexibility and outcomes via the work-family interface. Journal of Managerial Psychology, 25, 330\u2013355.\n\nCasper, W. J., & Harris, C. M. (2008). Work-life benefits and organizational attachment: Self-interest utility and signaling theory models. Journal of Vocational Behavior, 72, 95\u2013109.\n\nColeman, J. M., & Franiuk, R. (2011). Perceptions of mothers and fathers who take temporary leave. Sex Roles, 64, 311\u2013323.\n\nCorrell, S. J., Benard, S., & Paik, I. (2007). Getting a job: Is there a motherhood penalty? American Journal of Sociology, 112, 1297\u20131339.\n\nCrosby, F. J., Williams, J. C., & Biernat, M. (2004). The maternal wall. Journal of Social Issues, 60, 675\u2013682.\n\nDavis, A. E., & Kalleberg, A. L. (2006). Family-friendly organizations? Work and family programs in the 1990s. Work and Occupations, 33, 191\u2013223.\n\nDe Menezes, L. M., & Kelliher, C. (2011). Flexible working and performance: A systematic review of the evidence for a business case. International Journal of Management Reviews, 13, 452\u2013474.\n\nDodson, L. (2013). Stereotyping low-wage mothers who have work and family conflicts. Journal of Social Issues, 69, 257\u2013278.\n\nEataugh, C., & Folger, D. (1998). Perceptions of parents whose work and parenting behaviors deviate from role expectations. Sex Roles, 39, 215\u2013223.\n\nEaton, S. C. (2003). If you can use them: Flexibility policies, organizational commitment, and perceived performance. Industrial Relations: A Journal of Economy and Society, 42, 145\u2013167.\n\nFoley, S., Linnehan, F., Greenhaus, J. H., & Weer, C. H. (2006). The impact of gender similarity, racial similarity, and work culture on family-supportive supervision. Group & Organization Management, 31, 420\u2013442.\n\nFondas, N. (2013). How women and men use flexible work policies. The Atlantic. Retrieved from http:\/\/\u200bwww.\u200btheatlantic.\u200bcom\/\u200bsexes\/\u200barchive\/\u200b2013\/\u200b07\/\u200bhow-women-and-men-use-flexible-work-policies-differently\/\u200b277954\/\u200b.\n\nFord, M. T., Heinen, B. A., & Langkamer, K. L. (2007). Work and family satisfaction and conflict: A meta-analysis of cross-domain relations. Journal of Applied Psychology, 92, 57\u201380.PubMed\n\nFuegen, K., Biernat, M., Haines, E., & Deaux, K. (2004). Mothers and fathers in the workplace: How gender and parental status influence judgements of job-related competence. Journal of Social Issues, 60, 737\u2013754.\n\nFunk, L. C., & Werhun, C. D. (2011). \"You're such a girl!\" The psychological drain of the gender-role harassment of men. Sex Roles, 65, 13\u201322.\n\nGalinsky, E., Bond, J. T., & Hill, E. J. (2004). A status report on workplace flexibility: Who has it? Who wants it? What difference does it make? New York: Families and Work Institute.\n\nGalinsky, E., Bond, J. T., & Sakai, K. (2008). 2008 national study of employers. New York: Families and Work Institute.\n\nGalinsky, E., Sakai, K., & Wigton, T. (2011). Workplace flexibility: From research to action. The Future of Children, 21, 141\u2013161.PubMed\n\nGerstein, L., Moore, D., Duffy, K., & Daina, C. (1993). The effects of biological sex and ethnicity on EAP utilization and referrals. Consulting Psychology Journal, 45, 23\u201327.\n\nGerstel, N., & Armenia, A. (2009). Giving and taking family leaves: Right or privilege? Yale Journal of Law & Feminism, 21, 161\u2013184.\n\nGerstel, N., & McGonagle, K. (1999). Job leaves and the limits of the Family and Medical Leave Act: The effects of gender, race, and family. Work and Occupations, 26, 510\u2013534.\n\nGlauber, R. (2011). Limited access: Gender, occupational composition, and flexible work scheduling. The Sociological Quarterly, 52, 472\u2013494.PubMed\n\nGoldberg, W. A., Kelly, E., Matthews, N. L., Kang, H., Li, W., & Sumaroka, M. (2012). The more things change, the more they stay the same: Gender, culture, and college students' views about work and family. Journal of Social Issues, 68, 814\u2013837.\n\nGolden, L. (2008). Limited access: Disparities in flexible work schedules and work-at-home. Journal of Economic Issues, 29, 86\u2013109.\n\nGolden, L. (2009). Flexible daily work schedules in U.S. jobs: Formal introductions needed? Industrial Relations: A Journal of Economy and Society, 48, 27\u201354.\n\nGorman, K. A., & Fritzsche, B. A. (2002). The good-mother stereotype: Stay at home (or wish that you did!). Journal of Applied Social Psychology, 32, 2190\u20132201.\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGregory, A., & Milner, S. (2009). Work-life balance: A matter of choice? Gender, Work, and Organization, 16, 1\u201313.\n\nGrzywacz, J. G., & Bass, B. L. (2003). Work, family, and mental health: Testing different models of work-family fit. Journal of Marriage & Family, 65, 248\u2013261.\n\nGrzywacz, J. G., & Butler, A. B. (2008). Work-family conflict. In J. Barling & C. L. Cooper (Eds.), The Sage handbook of organizational behavior (pp. 451\u2013468). Los Angeles: Sage.\n\nHammer, L. B., Kossek, E. E., Yragui, N. L., Bodner, T. E., & Hanson, G. C. (2009). Development and validation of a multidimensional measure of family supportive supervisor behaviors (FSSB). Journal of Management, 35, 837\u2013856.PubMedCentralPubMed\n\nHammer, L. B., Kossek, E. E., Zimmerman, K., & Daniels, R. (2007). Clarifying the construct of family-supportive supervisory behaviors (FSSB): A multilevel perspective. In P. L. Perrew\u00e9, D. C. Ganster, P. L. Perrew\u00e9, & D. C. Ganster (Eds.), Exploring the work and non-work interface (pp. 165\u2013204). Bingley: Emerald.\n\nHammer, L. B., Neal, M. B., Newsom, J. T., Brockwood, K. J., & Colton, C. L. (2005). A longitudinal study of the effects of dual-earner couples' utilization of family-friendly workplace supports on work and family outcomes. Journal of Applied Psychology, 90, 799\u2013810.PubMed\n\nHammer, L. B., & Zimmerman, K. L. (2011). Quality of work life. In S. Zedeck (Ed.), American Psychological Association handbook of industrial and organizational psychology (pp. 391\u2013431). Washington, DC.: American Psychological Association.\n\nHeilman, M. E., & Okimoto, T. G. (2008). Motherhood: A potential source of bias in employment decisions. Journal of Applied Psychology, 93, 189\u2013198.PubMed\n\nHeilman, M. E., & Wallen, A. S. (2010). Wimpy and undeserving of respect: Penalties for men's gender-inconsistent success. Journal of Experimental Social Psychology, 46, 664\u2013667.\n\nHill, E. J., Hawkins, A. J., Martinson, V., & Ferris, M. (2003). Studying \"working fathers\": Comparing fathers' and mothers' work-family conflict, fit, and adaptive strategies in a global high-tech company. Men's Studies Press, 1, 239\u2013261.\n\nHopkins, K. M. (2002). Interactions of gender and race in workers' help seeking for personal\/family problems: Perceptions of supervisor support and intervention. Journal of Applied Behavioral Science, 38, 156\u2013176.\n\nHornung, S., Rousseau, D. M., & Glaser, J. (2008). Creating flexible work arrangements through idiosyncratic deals. Journal of Applied Psychology, 93, 655\u2013664.PubMed\n\nJones, R., & Murrell, A. J. (2001). Signaling positive corporate social performance. An event study of family-friendly firms. Business & Society, 40, 59\u201378.\n\nJudge, T. A., Livingston, B. A., & Hurst, C. (2012). Do nice guys-and gals-really finish last? The joint effects of sex and agreeableness on income. Journal of Personality and Social Psychology, 102, 390\u2013407.PubMed\n\nKelly, E. L. (2010). Failure to update: An institutional perspective on noncompliance with the Family and Medical Leave Act. Law & Society Review, 44, 33\u201366.\n\nKelly, E. L., Ammons, S. K., Chermack, K., & Moen, P. (2010). Gendered challenge, gendered response: Confronting the ideal worker norm in a white-collar organization. Gender & Society, 24, 281\u2013303.\n\nKelly, E. L., & Kalev, A. (2006). Managing flexible work arrangements in US organizations: Formalized discretion or 'a right to ask'. Socio-Economic Review, 4, 379\u2013416.\n\nKlerman, J. A., Daley, K., & Pozniak, A. (2013). Family and medical leave in 2012: Technical report. Washington, D.C.: Abt Associates.\n\nKossek, E. (2005). Workplace policies and practices to support work and families. In S. Bianchi, L. Casper, & R. King (Eds.), Work, family, health & well-being (pp. 97\u2013116). Mahwah: LEA Press.\n\nKossek, E. E., & Distelberg, B. (2009). Work and family employment policy for a transformed work force: Trends and themes. In N. Crouter & A. Booth (Eds.), Work-life policies (pp. 1\u201351). Washington, D.C.: Urban Institute Press.\n\nKossek, E. E., & Michel, J. S. (2011). Flexible work schedules. In S. Zedeck (Ed.), American Psychological Association handbook of industrial organizational psychology (pp. 535\u2013572). Washington, D.C.: American Psychological Association.\n\nKossek, E. E., Pichler, S., Bodner, T., & Hammer, L. B. (2011). Workplace social support and work-family conflict: A meta-analysis clarifying the influence of general and work-family-specific supervisor and organizational support. Personnel Psychology, 64, 289\u2013313.PubMedCentralPubMed\n\nKramer, A. (2008). Unions as facilitators of employment rights: An analysis of individuals' awareness of parental leave in the National Longitudinal Survey of Youth. Industrial Relations, 47, 651\u2013658.\n\nLi, A., & Bagger, J. (2011). Walking in your shoes: Interactive effects of child care responsibility difference and gender similarity on supervisory family support and work-related outcomes. Group & Organizational Management, 36, 659\u2013691.\n\nLiden, R. C., & Graen, G. B. (1980). Generalizability of the vertical dyad linkage model of leadership. Academy of Management Journal, 23, 451\u2013465.\n\nLowen, A., & Sicilian, P. (2009). \"Family friendly\" fringe benefits and the gender wage gap. Journal of Labor Research, 30, 101\u2013119.\n\nMajor, D. A., Fletcher, T. D., Davis, D. D., & Germano, L. M. (2008). The influence of work-family culture and workplace relationships on work interference with family: A multilevel model. Journal of Organizational Behavior, 29, 881\u2013897.\n\nMajor, D. A., & Germano, L. M. (2006). The changing nature of work and its impact on the work-home interface. In F. Jones, R. Burke, & M. Westman (Eds.), Work-life balance: A psychological perspective (pp. 13\u201338). London: Psychology Press.\n\nMajor, D. A., & Lauzun, H. M. (July 2009). An i-deals approach to coping with work-family conflict: Antecedents and consequences. Paper presented at the III international conference of work and family, Barcelona, Spain.\n\nMajor, D. A., & Lauzun, H. M. (2010). Equipping managers to assist employees in addressing work-family conflict: Applying the research literature toward innovative practice. The Psychologist-Manager Journal, 13, 69\u201385.\n\nMartenengo, G., Jacob, J., & Hill, E. (2010). Gender and the work-family interface: Exploring differences across the family life course. Journal of Family Issues, 31, 1363\u20131390.\n\nMatos, K., & Galinsky, E. (2012). 2012 national study of employers. New York: Families and Work Institute.\n\nMatthews, R. A., Bulger, C. A., & Booth, S. M. (2013). Managing the work-family interface to the benefit of both subordinates and supervisors: Looking beyond supervisor support and focusing on leadership behaviors. In M. Paludi (Ed.), Psychology for business success: Juggling, balancing, and integrating work and family roles and responsibilities (pp. 153\u2013170). Santa Barbara: Praeger.\n\nMatthews, R. A., Mills, M. J., Trout, R. C., & English, L. (2014). Family-supportive supervisor, behaviors, work, engagement, and subjective, well-being: A contextually dependent mediated process. Journal of Occupational Health Psychology, 19, 168\u2013181.PubMed\n\nMcCrate, E. (2005). Flexible hours, workplace authority, and compensating wage differentials in the US. Feminist Economics, 11, 11\u201339.\n\nMcMenamin, T. M. (2007). Time to work: Recent trends in shift work and flexible schedules. Monthly Labor Review, 130, 3\u201315.\n\nMcNamara, T. K., Pitt-Catsouphes, M., Brown, M., & Matz-Costa, C. (2012). Access to and utilization of flexible work options. Industrial Relations: A Journal of Economy and Society, 51, 936\u2013965.\n\nMorganson, V. J., Major, D. A., & Green, A. P. (Aug 2007). Seeking work-life balance: Employees' requests and managers' responses. Poster presented at the annual convention of the American Psychological Association, San Francisco, CA, USA.\n\nMoss-Racusin, C. A., Phelan, J. E., & Rudman, L. A. (2010). When men break the gender rules: Status incongruity and backlash against modest men. Psychology of Men and Masculinity, 11, 140\u2013151.\n\nNeal, M. B., & Hammer, L. B. (2007). Working couples caring for children and aging parents: Effects on work and well-being. Mahwah: Erlbaum.\n\nOkimoto, T. G., & Heilman, M. E. (2012). The \"bad parent\" assumption: How gender stereotypes affect reactions to working mothers. Journal of Social Issues, 68, 704\u2013724.\n\nParker, L. B., & Allen, T. D. (April 2002). Factors related to supervisor work\/family sensitivity and flexibility. Paper presented at the Society for Industrial and Organizational Psychology Conference, Toronto, Canada.\n\nPoelmans, S. (2012). The \"triple N\" model: Changing normative beliefs about parenting and career success. Journal of Social Issues, 68, 838\u2013847.\n\nPrentice, D. A., & Carranza, E. (2002). What women and men should be, shouldn't be, are allowed to be, and don't have to be. The contents of prescriptive gender stereotypes. Psychology of Women Quarterly, 26, 269\u2013281.\n\nProttas, D. J., Thompson, C. A., Kopelman, R. E., & Jahn, E. W. (2007). Work-family programs: Factors affecting employee knowledge and accuracy. Personnel Review, 36, 163\u2013189.\n\nRaghuram, S., Weisenfeld, B., & Garud, R. (2003). Technology enabled work: The role of self-efficacy in determining telecommuter adjustment and structuring behavior. Journal of Vocational Behavior, 63, 180\u2013198.\n\nRaghuram, A., Luksyte, A., Avery, D. R., & Macoukji, F. (2012). Does your supervisor stress you out? How support influences sex differences in stress among immigrants. Journal of Career Development, 39, 99\u2013117.\n\nRudman, L. A., & Fairchild, K. (2004). Reactions to counterstereotypic behavior: The role of backlash in cultural stereotype maintenance. Journal of Personality and Social Psychology, 87, 157\u2013176.PubMed\n\nRudman, L. A., & Mescher, K. (2013). Penalizing men who request a family leave: Is flexibility stigma a femininity stigma? Journal of Social Issues, 69, 322\u2013340.\n\nRudman, L. A., & Phelan, J. E. (2008). Backlash effect for disconfirming gender stereotypes in organizations. Research in Organizational Behavior, 28, 61\u201379.\n\nRyan, K. M., King, E. B., Adis, C., Gulick, L. M. V., Peddie, C., & Hargraves, R. (2012). Exploring the asymmetrical effects of gender tokenism on supervisor-subordinate relationships. Journal of Applied Social Psychology, 42, E56\u2013E102.\n\nRyan, A. M., & Kossek, E. E. (2008). Work-life policy implementation: Breaking down or creating barriers to inclusiveness? Human Resource Management, 47, 295\u2013310.\n\nSharpe, D. L., Hermsen, J. M., & Billings, J. (2002). Gender differences in use of alternative full-time work arrangements by married workers. Family and Consumer Sciences Research Journal, 31, 78\u2013111.\n\nSingh, P., Paleti, R., Jenkins, S., & Bhat, C. R. (2013). On modelling telecommuting behavior: Option, choice, and frequency. Transportation, 40, 373\u2013396.\n\nSingley, S. G., & Hynes, K. (2005). Transitions to parenthood: Work-family policies, gender, and the couple context. Gender & Society, 19, 376\u2013397.\n\nSociety for Human Resource Management (SHRM). (2012). 2012 Employee benefits: The employee benefits landscape in a recovering economy. Retrieved from http:\/\/\u200bwww.\u200bshrm.\u200borg\/\u200bResearch\/\u200bSurveyFindings\/\u200bArticles\/\u200bDocuments\/\u200b2012_\u200bEmpBenefits_\u200bReport.\u200bpdf.\n\nStone, P., & Hernandez, L. A. (2013). The all-or-nothing workplace: Flexibility stigma and \"opting out\" among professional-managerial women. Journal of Social Issues, 69, 235\u2013256.\n\nSwanberg, J. E., Pitt-Catsouphes, M., & Drescher-Burke, K. (2005). A question of justice: Disparities in employees' access to flexible schedule arrangements. Journal of Family Issues, 26, 866\u2013895.\n\nThomas, L. T., & Ganster, D. C. (1995). Impact of family-supportive work variables on work-family conflict and strain: A control perspective. Journal of Applied Psychology, 80, 6\u201315.\n\nThompson, C. A., Beauvais, L. L., & Lyness, K. S. (1999). When work-family benefits are not enough: The influence of work-family culture on benefit utilization, organizational attachment, and work-family conflict. Journal of Vocational Behavior, 54, 392\u2013415.\n\nUnited States Department of Labor. (2013). Family and Medical Leave Act: Overview. Retrieved from http:\/\/\u200bwww.\u200bdol.\u200bgov\/\u200bwhd\/\u200bfmla\/\u200b.\n\nVandello, J. A., Hettinger, V. E., Bosson, J. K., & Siddiqi, J. (2013). When equal really isn't equal: The masculine dilemma of seeking work flexibility. Journal of Social Issues, 69, 303\u2013321.\n\nVan Giezen, R. W. (2013). Paid leave in private industry over the past 20 years. Pay and benefits, 2, U.S. Bureau of Labor Statistics. Retrieved from http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bopub\/\u200bbtn\/\u200bvolume-2\/\u200bpaid-leave-in-private-industry-over-the-past-20-years.\u200bhtm.\n\nVillablanca, A. C., Beckett, L., Nettiksimmons, J., & Howell, L. P. (2011). Career flexibility and family-friendly policies: An NIH-funded study to enhance women's careers in biomedical sciences. Journal of Women's Health, 20, 1485\u20131496.PubMedCentralPubMed\n\nVinkenburg, C. J., van Engen, M. L., Coffeng, J., & Dikkers, J. S. E. (2012). Bias in employment decisions about mothers and fathers: The (dis)advantages of sharing care responsibilities. Journal of Social Issues, 68, 725\u2013741.\n\nWaldfogel, J. (2001). Family and medical leave: Evidence from the 2000 surveys. Monthly Labor Review, 124, 17\u201323.\n\nWayne, J. H., & Casper, W. J. (2012). Why does firm reputation in human resource policies influence college students? The mechanisms underlying job pursuit intentions. Human Resource Management, 51, 121\u2013142.\n\nWayne, J. H., & Codeiro, B. L. (2003). Who is a good organizational citizen? Social perception of male and female employees who use family leave. Sex Roles, 49, 233\u2013246.\n\nWilliams, J. C., Blair-Loy, M., & Berdahl, J. L. (2013). Cultural schemas, social class, and the flexibility stigma. Journal of Social Issues, 69, 209\u2013234.\n\nWinfield, I., & Rushing, B. (2005). Bridging the border between work and family: The effects of supervisor-employee similarity. Sociological Inquiry, 75, 55\u201380.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_13\n\n# 13. Shiftwork as Gendered and Its Impact on Work\u2013Family Balance\n\nMaura J. Mills2  and Ryan M. Rosiello1\n\n(1)\n\nDepartment of Psychology, Hofstra University, 135 Hofstra University, 11549 Hempstead, NY, USA\n\n(2)\n\nDepartments of Psychology and Management, Hofstra University, 135 Hofstra University, 11549 Hempstead, NY, USA\n\nMaura J. Mills (Corresponding author)\n\nEmail: maura.mills@hofstra.edu\n\nRyan M. Rosiello\n\nEmail: rrosie3@pride.hofstra.edu\n\nKeywords\n\nWork\u2013familyGenderGendered occupationsShiftworkWork scheduleFlexibilityFatigueHealth\n\n## 13.1 The Shift to Shiftwork\n\n### 13.1.1 Standard and Nonstandard Shifts\n\nThroughout history, the most common representation of a work shift has been the fixed day shift in which employees' work hours fall between the hours of 8:00 am and 5:00 pm, Monday through Friday (Perrucci et al. 2007; Presser 2006) . Since this shift has been most common throughout history, it is often referred to as a 'standard' shift or schedule (Presser 2006). Although the standard shift has traditionally been the most common, there also exist other types of shifts worth exploring. In fact, it is because of these other shift types that the common fixed day schedule is considered to be a work shift and not just a typical work day. Any shift that operates outside of standard work hours or days is defined as a nonstandard shift (Albertsen et al. 2008; Barthe et al. 2011; Perrucci et al. 2007; Presser 2003; Wittmer and Martin 2010). Some types of work that are representative of nonstandard work schedules are part-time work , contract work, and self\u2013employment (Presser 2003). Furthermore, nonstandard schedules are more likely to occur in service sector occupations such as protective community services or health care than in nonservice sectors with more predictable and less time-sensitive demands (Tuttle and Garr 2012). For employees in the former fields, likely subjected to shiftwork, the 40-hour (or more) work week is undoubtedly still manageable, but how it is managed by the employee and facilitated by the organization can make all the difference .\n\nUnlike standard shifts, which are characterized by one type of work schedule, nonstandard shifts can take several different forms . One type of nonstandard shift is fixed evening schedules, which involve working most days between 4:00 pm and midnight (Presser 2006). Another type is the fixed night schedule which involves working most days between midnight and 8:00 am (Presser 2006). In contrast to fixed schedules\u2014and perhaps more problematically in terms of work\u2013family needs\u2014there is another type of nonstandard shift, the rotating shift , in which an employee's work hours change periodically from evenings or nights to days, and\/or the employee has rotating days off between pay periods (Barthe et al. 2011; Perrucci et al. 2007; Presser 2006). Other types of nonstandard shifts include early-morning, late-evening, split-workday, and weekend work (Barthe et al. 2011).\n\nResearch has found that a substantial\u2014and growing\u2014percentage of Americans are working nonstandard shifts. Colligan et al. (1979) wrote that 26 % of the American workforce was involved in shiftwork at that time, and Fenwick and Tausig (2001) noted that by the early 1990s this number had risen to 30 %. A more recent estimation by Williams (2010) indicated that the number of people involved in shiftwork had further increased to 40 %. More specifically, Wittmer and Martin (2010) noted that 36 % of Americans are required to work weekends at least occasionally . Understandably, there is a higher incidence of shiftwork in some industries than in others, as well as in blue-collar work versus white-collar work, thereby potentially creating socioeconomic and\/or racial considerations insofar as shiftwork's effect on work\u2013family conflict.1 For instance, Root and Wooten (2008) noted that 50 % of employees working nonstandard hours work in restaurants and bars, while 25 % work in hospitals and manufacturing .\n\n### 13.1.2 The Forces Behind Shiftwork\n\nThere are some major forces behind the increasing number of employees working nonstandard schedules , including industry, demographic, and technological considerations (Presser 2006; Wittmer and Martin 2010). First, the recent shift in the industrial and economic landscape has influenced this. The economy in the 1960s appeared much different than it does today in a variety of ways, including that at that time many jobs were in large manufacturing organizations (Presser 2006; Wittmer and Martin 2010). Such jobs provided employees with greater job security and a more rigid, standard work schedule. Gradually, however, as new types of businesses started to emerge over time, the job market in the US started to shift from a primarily manufacturing economy to a more service-based economy. As opposed to manufacturing-based industries which are focused on making products for consumers, the service-based industries are more focused on providing services. Consequently, some of these services are required at all times of the day and night, such as medical care and transportation. Therefore, in order to meet the increasing demands of consumers and to effectively compete with other companies, organizations have been forced to provide services at all hours in what has been labeled as the 24\/7 economy . In order to accommodate this, organizations implemented varied shifts to staff all needed days and hours while not overburdening individual employees' quantity of work hours (Perrons et al. 2006; Presser 2006).\n\nAnother major force behind employees working nonstandard schedules is a change in workplace demographics . Over the past 50 years, there has been an enormous increase in the number of women who have entered the workforce (Barthe et al. 2011; Presser 2006; Rapoport et al. 2002). One of the reasons for this is the increasing availability of service-based jobs as a result of the changing economy (Perrons et al. 2006). Many of these jobs are subject to nonstandard scheduling since such services are needed either continuously or unpredictably. As such, with women spending an increasingly substantial proportion of their time working outside the home, the number of hours they have in the home to complete daily housework has declined. As a result, there is an increased need for services in order to help with daily in-home tasks, such as pre-made meals and household cleaning services. Some of these services may be required at nonstandard times, in turn further increasing the demand for jobs with nonstandard work schedules (Presser 2006).\n\nLastly, the increase in quality and quantity of readily-available technology is yet another major force behind employees working nonstandard schedules. The technology has made it possible for consumers to reach services they need at any time. For instance, smartphones and other mobile technologies have provided people with the opportunity to order goods or services online or over the phone at almost any hour throughout the day or night (Presser 2006). For those who work in a standard day shift, having the opportunity to obtain access to such services during the evening may be very convenient and therefore boost revenue for the providing service organization. As a result, the number of jobs requiring nonstandard hours has increased in order to meet this need.\n\n### 13.1.3 The Impact of Shiftwork\n\nThe type of shift that an employee works can have a major impact on their life, including exacerbating typical work stressors as well as introducing new ones. Employees who work nontraditional and\/or inconsistent shifts may experience a variety of negative outcomes, as explicated below.\n\n#### 13.1.3.1 Shiftwork and the Family\n\nOne of the major negative outcomes experienced by those working nonstandard shifts is increased work\u2013family conflict (Akanni 2011; Haines et al. 2008).2 For instance, employees working nonstandard night and evening shifts may be on opposite sleep\/wake schedules , thereby augmenting time-based work\u2013family conflict. Additionally, their schedules are likely to cause fatigue and potentially resultin moodiness and\/or stress (\u01feamil et al. 2013), which may lead to increased strain-based conflict (Haines et al. 2008). Indeed, various researchers have found that employees who work night, split, and rotating shifts report higher levels of work\u2013family conflict than those who work day shifts (e.g., Liu et al. 2011; Staines and Pleck 1984). Also, employees may be more likely to experience work\u2013family conflict when they have to unexpectedly cover alternate shifts at short notice (Estryn-B\u00e9har and Van der Heijden 2012). Interestingly, some initial research has found that it is not so much one's actual shift hours that lead to work\u2013family conflict so much as it is one's perception of the fit between work and family that influences how shifts are related to outcomes (Tuttle and Garr 2012). It has also been found that shiftwork negatively impacts how employees balance their work and family roles. For instance, some initial research has suggested that employees' work\u2013family balance may be more negatively affected by night and evening work than by rotating shifts (Albertsen et al. 2008). However, this is increasingly debated as rotating shifts are recognized as less predictable, and therefore more of a hindrance to day-to-day family planning considerations as well as employees' effective establishment of a consistent sleep routine . For example, Bambra et al. (2008) indicated that realignment to a new shift schedule can take between 8 and 12 days to adjust effectively, therefore leading to challenges when shifts are constantly rotating without consideration for such adjustment periods.\n\nIn addition to work\u2013family conflict overall, shiftwork can also have serious negative repercussions on family relationships specifically. One of the major ways shiftwork can impact the family is by decreasing the degree of satisfaction a couple may feel in their marriage3 (Perrons et al. 2006; Tuttle and Garr 2012). When partners work competing shifts, they may have little waking time to spend together. Therefore, spouses may no longer have time to partake in the activities that they would normally do together, such as eating meals together or participating in family or leisure activities with one another. Over time, this can cause partners to lose touch with one another and therefore feel more dissatisfied with their marriage (Albertsen et al. 2008; Presser 2006; Williams 2010).\n\nAnother way shiftwork can impact the family is by affecting parents' relationships with their children. Since shifts can occur at many different times, it may be possible for parents to adjust their schedules so that one parent cares for the child while the other is at work\u2014often referred to as tag-teaming (Williams 2010). Tag-teaming can lead to some financial benefits and may make it easier for families to cope with their work and family responsibilities (Estryn-B\u00e9har and Van der Heijden 2012; Lewis 2009; Williams 1999). Also, families may view this strategy as a way to avoid the cost and\/or perceived risk of placing their children in daycare (Williams 2010). However, despite these benefits there may also be negative effects from a relationship standpoint. For instance, when children are passed from one parent to the other, they only get to experience being with one parent at a time. Therefore, the child does not obtain the same family experience that they would if both parents and the child were all together at the same time (Lewis 2009). Furthermore, when parents work late afternoon and early evening shifts they often miss many of their children's extracurricular activities, which often take place during this time period (Presser 2006; Root and Wooten 2008). Notably, not only are they likely to miss the events themselves but they are correspondingly likely to miss opportunities to share their children's emotions immediately before and after the event. As a result, parents and children may feel less connected to each other and the quality of their relationship may decline. Lastly, when parents work different shifts, children may experience various mental and emotional issues. Some studies have shown that when parents work shifts, children are more likely to have poorer cognitive stimulation and increased odds of suffering from behavioral and emotional issues (Albertsen et al. 2008). However, although shiftwork may have a negative influence on children in the short term, a longitudinal study by White and Keith (1990) suggests that shiftwork may decrease child-related problems over time.\n\nBeyond marital and parental roles, shiftwork can also have a major impact on the family dynamic as a whole. For instance, La Valle et al. (2002) found that when family members worked nonstandard shifts they had less time to spend together sharing evening meals, family holidays, and other family activities. These activities are similar to team-building in that they help to build and reinforce a sense of unity and togetherness (Presser 2006). Also, other negative implications of shiftwork on the employee (e.g., energy depletion, lack of quality and\/or duration of sleep, moodiness) are likely to spillover to negatively affect the overall family dynamic. Such considerations have been supported by Grosswald (2003), who found that employees working nonstandard shifts tend to be less satisfied with their family lives than are employees working standard shifts.\n\nFinally, while a step removed from family relationships, an employee's general social life and nonfamilial relationships are also likely to be compromised by shiftwork. That is, such an atypical work schedule is likely to present a challenge for employees in trying to find a common time to commiserate with friends, the majority of whom are statistically likely to work more standard schedules. As such, the shift worker's ability to participate in various social activities is threatened, therefore limiting time spent with friends and, often consequently, serving as detrimental to both the number and the quality of friendships. For instance, working nonstandard hours may make it more difficult for individuals to attend important life events such as weddings, baby showers, funerals, graduations, and the like (Fenwick and Tausig 2001). As a result, shift workers may experience compromised satisfaction with their social relationships in general, both within as well as outside of the nuclear family.\n\n#### 13.1.3.2 Shiftwork and the Experience of Work\n\nAlthough shiftwork can have a major impact on an employee's familial and other social relationships, it can also have a major impact on the employee's day-to-day experience at work. For instance, such employees may have difficulty utilizing standard work\u2013family policies or benefits, as they are typically framed with the traditional worker in mind. Furthermore, nonstandard schedules have been linked to decreased job satisfaction (Barthe et al. 2011) as well as to increased job-related burnout (Estryn-B\u00e9har and Van der Heijden 2012). Moreover, those working rotating shifts may continually work with different people depending on the shift, which may prevent them from developing strong relationships at work and may impact their perceived (and actual) levels of coworker support , particularly during times of need or change, thereby negatively affecting the degree to which their needs for relatedness and affiliation are satisfied. Moreover, employees working the late night or \"graveyard\" shift, even on a consistent basis, may experience little customer interaction, again negatively affecting the satisfaction of their relatedness need in the workplace context.\n\nLinked to this idea of the quality of employees' experiences at work is shiftwork's possible impact on absenteeism , although the research has been mixed in this regard depending on shift type. For instance, Colligan et al. (1979) found that nurses who worked the night shift took more sick leaves than those who worked afternoon shifts. Similarly, Sergean and Brierly (1968) found that British metal processers working the night shift were more likely to get sick than those working day and afternoon shifts. In contrast, Thiis-Evensen (1958), while dated, performed a longitudinal study of Danish employees over a 10-year period and found that fixed day shift workers were absent from work more frequently than those who worked in nonstandard hours. Other studies have found that while employees with rotating schedules get sick more often and develop more serious health problems, they generally take fewer sick leaves than those who work in fixed shifts, regardless of whether those shifts are day or night (Colligan et al. 1979; Costa 1996).\n\n#### 13.1.3.3 Shiftwork and Health\n\nAs implied above, shiftwork can impact the quality of employees' lives through its negative impact on both physical and mental health (Estryn-B\u00e9har and Van der Heijden 2012; Fenwick and Tausig 2001; Wittmer and Martin 2010). Perhaps most obviously is shiftwork's detrimental impact on sleep . Indeed, major health issues can occur as a result of the employee sleeping during the day or on a variable schedule (Costa 1996; Fenwick and Tausig 2001). Evolutionarily, it is often suggested (e.g., Costa 1996) that humans have historically been conditioned to sleep during the nighttime (e.g., hunting was harder during darkness) and be awake during the daytime (e.g., without manufactured light, the natural light of daytime provided more opportunities for hunting, gathering, social interaction, etc.). Further, when we sleep at irregular times, we disrupt the production of melatonin, the hormone that regulates sleep cycles and strengthens the immune system. Consequently, when we force ourselves to sleep at such times, we do not sleep as long or as soundly (Costa 1996). Indeed, \u01feyane et al. (2013) found nighttime shift workers to be significantly more likely to suffer both insomnia and fatigue than employees working in traditional shifts, a finding also suggested by Estryn-B\u00e9har and Van der Heijden (2012). In addition, sleeping during the daytime is subject to more disturbances from typical daytime noises (e.g., telephone, doorbell, traffic, household noises, etc; Costa 1996). To compound these sleep problems, it is not unusual for shift workers to purposefully deprive themselves of sleep in order to spend more time with others who are normally awake during daytime hours (Presser 2006). Furthermore, when workers sleep at irregular or inconsistent times, they disturb their circadian rhythms and this can have many negative effects on their bodies (Albertsen et al. 2008; Costa 1996; Fenwick and Tausig 2001), a problem particularly relevant for employees working in biologically-insensitive shift rotation schemes. Some of the sleep-related symptoms that employees may suffer from working in these types of schedules include fatigue, insomnia, lethargy, poorer agility, decreased reaction time, and compromised productivity, performance, and safety-related behaviors (Costa 1996; Fenwick and Tausig 2001; Lindsey 2007; Pierce and Dunham 1992).\n\nBeyond the sleep challenges that are perhaps the most obvious physical symptom associated with shiftwork, employees with nonstandard work hours may also experience gastrointestinal disorders resulting from variable eating habits and their bodies' difficulties adjusting to that inconsistency (Bambra et al. 2008; Costa 1996; Fenwick and Tausig 2001). Shift workers may have to eat in a hurry or away from home, and unless they plan and prepare healthy meals ahead of time, the types of food available at later hours is often of lower quality (e.g. fast food) (Costa 1996). These gastrointestinal challenges may be particularly pertinent for employees with rotating shifts , as Wyatt and Marriot (1953) found, with results indicating that both appetite and digestion were greatly disrupted when employees changed shifts from days to nights and vice versa (Colligan et al. 1979).\n\nAnother major health issue that employees working nonstandard schedules may experience is cardiovascular disorders, including heart attacks (Wilson 2002). In fact, Fenwick and Tausig (2001) found that the likelihood of coronary heart disease increased more than 200 % after 6 years of working a rotating night shift. It is thought that one of the reasons for this is the increased stress that shiftwork often causes (Lewis 2009; Williams 1999, 2010). That is, shiftwork\u2014particularly rotating shifts\u2014may make it difficult for employees to plan ahead from week to week, resulting in undue stress linked to perceived inability to successfully manage typical mundane chores and plans. For instance, an employee may experience increased stress if he or she schedules a doctor's appointment 6 months in advance only to find out that he or she is unable to keep the appointment due to work conflicts\u2014a not uncommon situation that is further complicated by appointment cancellation fees as well as by the difficulty in trying to reschedule appointments with limited notice. This serves as only one example of why shiftwork can lead to increased stress, and others can be found throughout this chapter, in particular throughout the discussion of work\u2013family challenges in particular.\n\nMoreover, research has further suggested that shift workers are also likely to be at an increased risk for other physical health issues , although the evidence for these is less conclusive. For instance, a study by the World Health Organization (2007) found that an increased incidence of cancer might be linked to night shift work, and Fenwick and Tausig (2001) suggested that the same may be true for diabetes, elevated levels of uric acid, and high cholesterol. Most recently, Zhang et al. (2014) found that extended or inconsistent wakefulness may even lead to brain damage, as was evident in their study that subjected rats to inconsistent sleep schedules mirroring those of shift workers. Specifically, even after periods of recovery sleep , cognitive functioning and alertness remained compromised on a long-term basis, indicating damage to the area of the brain responsible for these functions\u2014the locus coeruleus\u2014via neuron loss that persists over time and, Zhang et al suggested, is likely irreversible. While this recent research obviously needs to be replicated with humans for more generalizable findings, Zhang et al.'s initial findings suggest that the overall health effects of shiftwork could be even more detrimental than previously believed.\n\n## 13.2 Shiftwork and Gender\n\nThe literature presents some mixed results in regard to the proportions of men and women participating in shiftwork. According to Beers (2000) and Presser (2003), men are more likely to work in nonstandard shifts than women because the former generally choose occupations where shiftwork is found more frequently. Beers further notes that men are more likely to work nonstandard shifts than women, even when members of both sexes are working the same job type. In contrast, Tuttle and Garr (2012) more recently noted that shiftwork has risen in lower-paying service sector jobs that tend to be female-dominated, thereby yielding more shiftwork to women. Overall, however, the majority of the literature supports the notion that shiftwork tends to remain dominated by males, even if the pendulum is beginning to shift slightly to decrease that gap.\n\nOne of the major ways that men and women may differ in regard to shiftwork is the type of shift each gender is more likely to perform. Although in general, men are slightly more likely to work in nonstandard hours than women (Presser 2006), Estryn-B\u00e9har and Van der Heijden (2012) found that female nurses generally work extended shifts more so than do men, while slightly more male nurses worked overtime than did female nurses. Other research (Presser 2006) has found that higher proportions of men work rotating hours and fixed nights schedules than women. Moreover, although few employees, regardless of gender, work exclusively weekend shifts, men are more likely than women to work weekend hours in addition to their weekly shifts. Women, on the other hand, are more likely to work only in weekdays but less than 5 days a week. Barthe et al. (2011) found that all women with heavier responsibilities reported childcare being a major reason for their choice of shift, with two-thirds of women choosing their shift to accommodate familial responsibilities and homecare duties.\n\nMen and women may also differ in their perceptions of gendered shiftwork. Marsh (1979) found some interesting results in how women and men view shiftwork and how others view them doing shiftwork; women were generally not in favor of working in the more committed types of shiftwork such as double-day shifts or weekend work. One of the reasons they tended to reject the notion of shiftwork is that they felt their lives were too complicated already and shiftwork would only make them more complicated and less manageable. In regard to how others view women working in nonstandard hours, Marsh noted that a majority of respondents in the study generally believed that women should have the opportunity to work evening shifts and overtime , especially if they are single. However, interestingly, a large majority of respondents believed that married women should not work night shifts, while a majority of male respondents saw no problem with working nonstandard schedules . Overall, Marsh noted, respondents indicated that male employees in particular should work as much as they desired, and that any issues arising from that decision are simply problems to be resolved within their respective families.\n\n## 13.3 Gender and Work\u2013Family Conflict\n\nDespite growing similarities, there remain some important distinctions between how men and women may experience work\u2013family tensions. For instance, women may arguably experience more work\u2013family conflict as a result of their increased participation in the paid workforce. That is, there is reason to believe that in the past, when women spent most of their time solely looking after household and family responsibilities, they had fewer time constraints in accomplishing those duties. However, once women started to enter the paid workforce at increasing rates, they acquired extra work responsibilities in addition to their existing familial and household responsibilities. Compounding this problem is the consideration that despite women's increasing involvement in the paid work sphere, and despite men's increase in household involvement, the two have not increased at the same rate, with women still bearing the burden of the majority of the housework (Lewis 2009). This has kept women's workload in the family sphere at a very high level despite the additional burden of external employment, thereby increasing the likelihood and severity of felt work\u2013family conflict.\n\nAlthough research suggests that women tend to experience high levels of work\u2013family conflict, there is also a research to suggest that men are also experiencing such conflict at increasing rates. Gerson (2009) reported that the amount of work\u2013family conflict that men experience has risen by over 10 % since 1977, and when considering only men in dual-earner relationships , this increases to over 25 %. Nevertheless, although men are experiencing high levels of work\u2013family conflict, they are also likely to experience such conflict somewhat differently than do women. In the past, when work and family-care responsibilities were separated by gender, being able to provide economically for the family, or breadwinning , defined males' identity insofar as their familial role. However, as more women entered the workforce and as cultural values shifted, men started to place a higher stake on their involvement in the home sphere in addition to their preexisting provider role.\n\nIndeed, Root and Wooten (2008) reported that fathers have gone from spending about 30 to 45 % as much time with their children as did mothers in the 1970s and 1980s to about 65 % as much time on the weekdays and 87 % as much time on the weekends in more recent years. Interestingly, Gottz\u00e9n and Kremer-Sadlik (2012) suggested that many fathers may prefer to be involved in their children's lives in ways that are different than mothers. While mothers generally become involved in their children's lives through general childcare, many fathers may prefer to be involved with their children's youth sports. This allows fathers to become more involved with their family without performing traditionally feminine activities, therefore not compromising their stereotypically masculine identity. However it is worth noting, as Gottz\u00e9n and Kremer-Sadlik (2012) did, that since many after-school activities such as sports tend to occur in the afternoon or early evening, work obligations may often prevent fathers from being able to attend their child's sporting events, thereby threatening fathers' ability to care for their children and elevating their work\u2013family conflict. This is certainly a consideration relevant to the varied temporal requirements of shiftwork, in addition to speaking to the necessity of gender-blind work\u2013family policies available to allow for diminished conflict.\n\n### 13.3.1 The Threat of Career Penalties\n\nOne of the major factors contributing to gender differences in work\u2013family conflict is workplace policies and procedures. Such policies, such as parental leave and flexible scheduling, are typically designed to allow employees more control over how they balance their work and family lives with the idea of reducing work\u2013family conflict (Gerson 2009; Lewis 2009). However, instead of increasing balance, many times such policies may be ineffective because they are often associated with substantial career penalties. That is, when an employee has less of a presence at work (either physically, mentally, or both) they may appear less committed than other employees who are regularly seen by key organizational members, a physical benefit commonly referred to as 'face-time.' Therefore, management may be less inclined to offer workplace benefits or career advancement opportunities\u2014such as raises and promotions \u2014to employees who are less visible within the organization. Gerson (2009) wrote that although all employees may be subject to these types of career penalties, women may be more likely to accept them\u2014voluntarily or otherwise\u2014for the sake of their family. Thus, women may experience lower levels of job satisfaction as a result of being less likely to reach their desired career goals. Furthermore, even though men may be less likely to accept career penalties, they may still experience substantial work\u2013family conflict since they miss out on using flexible policies which would help them spend more time with their family and on other nonwork commitments.\n\n### 13.3.2 The Influence of Gender Norms\n\nAnother major reason why these policies may be ineffective is that there are often strong tacit norms against men using these policies. Williams (2010) writes that there is a stigma attached to men who make known their involvement in family care. Men who take time off for family purposes risk being viewed by others in the workplace as lacking in commitment, they may be openly ridiculed, and their 'manliness' may even be questioned (Levine and Pittinsky 1997; Williams 2010). Men who take even short family leaves may be penalized with both fewer rewards and lower performance ratings, as noted above. In addition, fathers who opt to serve as stay-at-home parents for any period of time are often viewed more negatively than fathers who are employed outside of the home. As a result, men are less likely to acknowledge and speak out about their need for work\u2013family accommodations. It has even been reported that men are more likely than women to be fired because they had major work-life conflicts and were reluctant to seek employer help in managing them (Williams 2010). To cope with their reluctance toward utilizing official work\u2013family policies, men often use a combination of vacation and sick leave to meet family demands (Levine and Pittinsky 1997). However, such reluctance can also lead to women experiencing heightened work\u2013family conflict, since the more time men feel compelled to spend at work, the less availability they have to assist with home- and childcare responsibilities.\n\n## 13.4 Gender, Shiftwork, and Work\u2013Family Conflict\n\nCompounding the various aforementioned research indications warrants consideration of how shiftwork may differentially impact each gender's perceptions and\/or reality of work\u2013family conflict. Indeed, there is evidence to suggest that there may be differential effects of work\u2013family conflict for individuals working nonstandard schedules depending on gender.\n\n### 13.4.1 Women, Shiftwork, and Work\u2013Family Conflict\n\nSeveral studies suggest that women who work nonstandard shifts experience greater work\u2013family conflict than do men working such shifts (Fenwick and Tausig 2001; Haines et al. 2008; Tuttle and Garr 2012). Likewise, supplemental evidence supports the consideration that mothers' time is more affected by nonstandard hours than is fathers' time (Barnes et al. 2006; Craig and Powell 2011). One potential reason for this differential sense of conflict rests upon the fact that women are generally expected to shoulder more of the family responsibilities, therefore the same work complication (e.g., shift) may cause greater conflict with home roles for women than for men (Fenwick and Tausig 2001; Presser 2003). Furthermore, women are more likely to revolve their shift scheduling around their husband's shift, whereas the reverse is less likely (Presser 2003). Therefore, more women may get 'stuck' working shifts that they might perceive as conflicting with their family life because they have adopted such shifts based upon their husband's work schedule rather as opposed to their own preference.\n\nWomen may also experience greater health problems when they work nonstandard hours in comparison to men. Costa (1996) noted that women may experience greater stress due to time pressures that occur when they work at irregular times. Also, when women who are married and have children work nonstandard hours, they generally experience poorer quality sleep and more fatigue than do men (Costa 1996). Lastly, numerous studies have reported that problems of the female reproductive system\u2014including increased menstrual cycle problems and miscarriages\u2014can also be exacerbated by working nonstandard shifts (Axelsson et al. 1989; Barthe et al. 2011; Costa 1996).\n\nHowever, although research suggests that women who work nonstandard hours generally experience higher work\u2013family conflict than men working such shifts, when in the right situation, women may also experience greater work\u2013family benefits. For example, Scheller (2011) noted that a bus driving profession allowed mothers to be home with their children at the times when their children needed them the most. However, this too rests primarily on the assumption that women may inherently feel (or be societally expected to feel) more responsibility to the home- and childcare domains of family life than do men, which is certainly a broad statement that fails to consider individual differences and other such contextual considerations.\n\n### 13.4.2 Men, Shiftwork, and Work\u2013Family Conflict\n\nAlthough women may be experiencing greater work\u2013family conflict than men as a result of working nonstandard schedules, men working such shifts may nonetheless experience relatively high levels of conflict . Perhaps one of the reasons it is reported that women experience higher levels of work\u2013family conflict is that women are often more vocal about the conflicts they are experiencing, and those conflicts are more societally accepted from a woman than from a man. In fact, when Levine and Pittinsky (1997) interviewed men about their work and family conflicts, they found that men were experiencing more conflict than people thought but were reluctant to speak about it. One of the forces pressuring men to be more private and reserved about their work\u2013family issues might be the negative, informal repercussions discussed earlier from using organizational policies to improve work\u2013family balance .\n\nRegardless of how open men are about their work\u2013family issues, one of the reasons that they experience high levels of work\u2013family conflict when working in nonstandard schedules is that many have expanding parenting responsibilities and these responsibilities may be more difficult to handle with a nonstandard work schedule. For instance, it was mentioned earlier that men often experience work\u2013family conflict as a result of not being able to be involved in their children's lives to the extent that they would like. Since many fathers may prefer to be involved in their children's lives through their extracurricular activities, which often occur in the late afternoon or early evening, they may experience increased work\u2013family conflict when they are assigned a shift during this time period. Moreover, men's responsibilities within the home are broadening as compared to past decades, so this alone is likely to yield such conflict, which is then exacerbated by the lack of societal acceptance of such responsibilities.\n\n## 13.5 Recommendations for Improving Work\u2013Family Balance for Shift Workers\n\n### 13.5.1 Methods of Improvement\n\nIn summary, both men and women are likely to experience heightened work\u2013family conflict as a result of working nonstandard shifts . However, there are some administratively controllable methods by which organizations can facilitate a reduction in such conflict for their employees. In particular, increasing the amount of flexibility and control that employees have over their hours, ensuring workers receive the support they need, and developing policies to support those who work nonstandard hours are all effective steps that organizations can take toward facilitating employees' improved work\u2013family balance in the face of shiftwork.\n\n### 13.5.2 Flexible Work Hours\n\nOne of the ways that organizations can help both men and women working nonstandard hours to improve their work-life balance is by offering them greater flexibility over when and how they perform their shifts. Both Levine and Pittinsky (1997) and more recently Gerson (2009) have suggested that allowing employees greater flexibility over their work hours may be an effective strategy for reducing work\u2013family conflict. In fact, Williams (1999) noted that flexible work policies were some of the most preferred workplace benefits. There is also some evidence that mothers who have flexible work hours are significantly less likely to voluntarily turnover from their jobs. Lastly, Gerson (2009) suggests that flexibility is absolutely essential with nonstandard shifts because it helps families overcome interpersonal and economic uncertainties. Vila et al. (2002) discussed the importance of employee involvement in shift scheduling in relation to law enforcement officers in particular, a highly male-dominated shiftwork position. Vila et al. determined that excluding officers from scheduling and shift assignment decisions can trigger a vicious cycle that contributes toward employees' feelings of powerlessness in some of the most important aspects of their lives, as well as exacerbating the various other effects discussed herein, including fatigue, stress, and performance decrements.\n\nRegardless of the extent to which employees are involved in scheduling decisions, if some flexibility is offered in work hours, employees can alternatively have control over how they utilize such options. Although some nonstandard shifts involve either fixed or rotating hours in the evening or at night, the employees may be allowed to use flexibility in order to vary their start and end times by an hour or two (Lewis 2009; Williams 1999). This may be especially helpful to those working in nonstandard shifts as it may allow them to take care of an errand, attend an event, or care for another when needed. Another way in which employees can utilize flexibility in their nonstandard hours is by telecommuting , which allows them to perform some or all of their work at home, although such an option is highly job-dependent. Some employees may alternately prefer to use flexibility to compress their work schedules , whereby they work for only 3 or 4 days a week but for longer-than-usual hours. Job sharing is another flexible work method whereby two part-time employees split the hours and the responsibilities of one full-time job. It is recommended that, to the extent possible, organizations allow employees one or more of these flexibility options to tailor to their needs, which will inevitably vary both within-person (over time), as well as between-person (wherein different employees will have different flexibility needs).\n\nAlthough shiftwork may be viewed by some as a flexible work option in itself, the flexibility we are suggesting here is different in that it allows for some degree of employee control over the work demands , thereby reducing the sense of helplessness that can often exacerbate work\u2013family conflict (Barthe et al. 2011; Fenwick and Tausig 2001; Gerson 2009). In fact, Lewis (2009) proposes that flexibility is only beneficial for nonstandard shift workers when the employee has some level of control over their hours. This is because employees who work in nonstandard shifts are often assigned to their shifts on short notice, therefore inhibiting the ability to attend preplanned appointments and events without using vacation time to do so. Thereby, having more control may be instrumental in helping parents who work in nonstandard shifts to be more consistently involved in their children's lives (Levine and Pittinsky 1997). For example, if a father who works a fixed, late afternoon shift can adjust his hours, he will have more ability to attend his child's extracurricular activity when his attendance is most desired (e.g., a game, recital, etc.). In addition, having more control over the scheduling of their nonstandard shift may help to reduce the anxiety that both mothers and fathers may feel over getting their child to and from daycare or other commitments (Levine and Pittinsky 1997). Having control over work hours would also help parents who opt not to use daycare: As Lewis (2009) notes, this is particularly likely to be the case for parents who are unable to afford the sharply rising and extremely high cost of full-time (or even part-time) daycare, an issue often pertinent to shift workers since nontraditional shifts tend to be more prevalent in lower-paying jobs.\n\nAs one might imagine, there exist differing viewpoints about how providing employees with flexibility options may differentially affect men versus women. Williams (1999) noted that there is evidence of flexibility options being utilized more by female employees than by male employees. Specifically, she suggests that because women may request flexibility more so than do men, flexible hours may reinforce the negative stereotype of women and caregiving. This stereotype may in turn make it more difficult for women to achieve work success if they are utilizing such flexibility options. In addition, Perrons et al. (2006) noted that flexible scheduling allows women to be able to better manage work and family responsibilities since they are the ones using the policies. However, this does not necessarily result in improved work\u2013family balance for women because the responsibilities are not necessarily being shared with their spouses. Therefore, women are left with the same extent of nonwork responsibilities, they just have more control over when they can attend to them. In opposition to the idea that men do not make use of flexibility over their work hours when available, Levine and Pittinsky (1997) suggest that men may be more likely to use flexible schedules than other types of accommodations because such flexibility does not necessarily compromise family income, as opposed to other policies such as paternity leaves , which typically go unpaid. A third opinion suggests that giving both male and female employees the ability to have more flexible work schedules will allow them to coordinate their schedules so that they can better share household responsibilities. As a result, we firmly suggest that if increased schedule control is offered to employees, efforts must be made not only to make such policies available to both men and women, but also to encourage their usage equally amongst employees of both sexes.\n\nAlthough organizations may view incorporating flexibility into shift workers' schedules with some skepticism as to its effects on organizational outcomes, there is some evidence to suggest that organizations may actually benefit from doing so (Lewis 2009; Williams 1999). Specifically, this added flexibility can allow employers to operate for longer hours without having to hire additional employees. For instance, if employees vary their start and\/or end times by a few hours outside of normal business hours, it allows the organization to stay open longer without having to create new shifts, staffed by additional employees, to operate those hours. Lastly\u2014but perhaps most importantly for long-term organizational success\u2014flexibility (and its perceived usability) has been found to facilitate employee loyalty and commitment, as well as to enhance safety-related behaviors (via decreased fatigue and increased cognitive alertness) and overall performance (Eaton 2003; Kelliher and Anderson 2010; Ng et al. 2006).\n\n### 13.5.3 Support of Employees\n\nAnother way that organizations can help shift workers to improve their work\u2013family balance is by fostering an environment in which they are highly supported. Supportive environments can help employees because they can provide them with extra resources such as someone to cover a shift, help with extra work, or comfort them emotionally. These extra resources may compensate for those lost in working a nonstandard shift. Estryn-B\u00e9har and Van der Heijden (2012) suggest that one of the ways that organizations can improve the support they offer to employees is to organize work schedules in a way that allows for social support from employees with similar familial responsibilities. A more revolutionary way that organizations can develop a more supportive environment is by fostering informal caretaking networks in which other employees can help each other with childcare responsibilities. These networks may allow employees who have different shifts to cover some childcare activities at times when other employees are working (Gerson 2009). Yet another strategy that organizations can employ to be supportive of employees' childcare needs is to provide on-site childcare which operates 24\/7 so that employees working at all hours of the day are provided with a legitimate childcare option (Barthe et al. 2011).\n\nOn a much broader scale, organizations can also help to foster informal coworker support networks. As noted by Root and Wooten (2008), coworkers can serve as an extended kinship network to help provide each other with emotional support and instrumental assistance in coping with work responsibilities and family needs. For instance, coworkers can help to accomplish certain tasks when one employee needs to leave early to take care of a family responsibility. Of course, this type of support would have to be developed with reciprocity in mind so that employees do not take advantage of each other or create unnecessary interpersonal conflict within the organization.\n\nA third type of support that organizations can foster is supervisor support, increasingly recognized in the literature as family-supportive supervisor behaviors (FSSB) . Supervisors have a substantial influence over their subordinates in that they often serve as a gatekeeper to effective work\u2013family management. Supervisors may choose to be extremely accommodating to an employee's desire for flexibility, or they can be very restrictive. As such, a lack of supervisor support can effectively render the aforementioned coworker support systems inoperable if they refuse to facilitate or allow for such work modifications. In addition to gatekeeping responsibilities, supervisors can also serve as an important source of emotional support for employees. As a result, organizations can foster supportive environments (and perceptions) a great deal though their supervisors, who serve as the 'front-line' contact for employees to organizational culture and policy.\n\nThere is some evidence to suggest that providing support as a work\u2013family improvement may have differential effects dependent upon employee gender. For instance, some research has suggested that women may experience greater benefits from social and emotional support than men (Forbes and Roger 1999; van der Pompe and Heus 1993). However, other research has suggested that while women may receive more social support from nonwork sources than do men, they may not necessarily obtain more support from work sources (van der Pompe and Heus 1993). Hopkins (2002) suggested that a reason for this is that both men and women tend to have male supervisors, possibly leading female employees to feel less comfortable confiding in them than would male employees, who may enjoy more of a gender-fueled camaraderie, although this is certainly arguable. As a result, organizations would do well to consider gender in establishing and bolstering their social support programs so that both male and female employees are more likely to reap the benefits equally.\n\n### 13.5.4 Strengthening Improvements Through Policies\n\nRegardless of how organizations decide to facilitate work\u2013family balance for employees working nonstandard shifts , it is imperative that they ground such improvements within formal organizational policies. Such formal policies facilitate employees' understanding of organizational culture and accepted accommodations. However, policies alone are grossly insufficient. As seen throughout this chapter, both men and women are often reluctant to utilize organizational policies in support of work\u2013family balance for fear of negative perceptions and\/or career implications. Therefore, it is crucial that any formal organizational policies be supported by upper management, that such support is evidenced through the trickle-down effect to middle- and lower-level managers who reinforce such a culture , and that employees are freely given the necessary information and encouragement to utilize such policies when necessary.\n\nOne of the most effective ways to assure employees of a supportive culture beyond formal policies is likely to be through the aforementioned FSSBs. Comprised of four different dimensions \u2013 emotional support , role modeling, instrumental support , and creative work\u2013family management (Hammer et al. 2013)- FSSBs are expected to positively impact employees of both genders insofar as their comfort level in using work\u2013family policies when they are available. In addition FSSB has been shown to have a variety of positive effects on work\u2013family outcomes, thereby enhancing the likelihood that such accommodations would enhance employees' work\u2013family balance (Hammer et al. 2011; Odle-Dusseau et al. 2012).\n\n## 13.6 Conclusion\n\nIn sum, the intersection between gender and the work\u2013family domains, while intriguing, is widely complex, with a variety of supplemental issues affecting how the interplay between these domains presents itself in a variety of situations and contexts. Insofar as work structures go, the presence or absence\u2014and the extent\u2014of shiftwork for males versus females is one such contextual factor. This chapter has explored the various considerations underlying shiftwork's influence in this regard, with a hope toward sparking more extensive applied considerations of this work constraint and how it can best be managed in order to facilitate a more fulfilling experience of work\u2013family balance for both male and female employees alike.\n\nReferences\n\nAkanni, A. (2011). Shiftwork, job satisfaction and family life: An empirical study among nurses in Osun State Nigeria. Lambert: Academic.\n\nAlbertsen, K., Rafnsd\u00f3ttir, G., Grimsmo, A., T\u00f3mas-son, K., & Kauppinen, K. (2008). Workhours and worklife balance. SJWEH Supplements, 5, 14\u201321.\n\nAxelsson, G., Rylander, R., & Molin, I. (1989). Outcome of pregnancy in relation to irregular and incovenient work schedules. British Journal of Industrial Medicine, 46, 393\u2013398.PubMedCentralPubMed\n\nBambra, C., Whitehead, M., Sowden, A., Akers, J., & Petticrew, M. (2008). A hard day's night? The effects of compressed working week interventions on the health and worklife balance of shift workers: A systematic review. Journal of Epidemiology & Community Health, 62, 764\u2013777.CrossRef\n\nBarnes, M., Bryson, C., & Smith R. (2006). Working atypical hours: What happens to family life? London: National Centre for Social Research.PubMedCentralPubMedCrossRef\n\nBarthe, B., Messing, K., & Abbas, L. (2011). Strategies used by women workers to reconcile family responsibilities with atypical work schedules in the service sector. Work, 40, S47\u2013S58.PubMed\n\nBeers, T. M. (2000). Flexible schedules and shift work: replacing the '9-to-5' workday? Monthly Labor Review, 123, 33\u201340.\n\nColligan, M. J., Frockt, J. I., & Tasto, D. L. (1979). Frequency of sickness absence and worksite clinic visits among nurses as a function of shift. Applied Ergonomics, 10, 79\u201385.PubMedCrossRef90058-9)\n\nCosta, G. (1996). The impact of shift and night work on health. Applied Ergonomics, 27, 9\u201316.PubMedCrossRef00047-X)\n\nCraig, L., & Powell, A. (2011). Non-standard work schedules, work-family balance and the gendered division of childcare. Work Employment Society, 25, 274\u2013291.CrossRef\n\nEaton, S. C. (2003). If you can use them: Flexibility policies, organizational commitment, and perceived performance. Industrial Relations: A Journal of Economy & Society, 42, 145\u2013167.\n\nEstryn-B\u00e9har, M., & Van der Heijden, B. M. (2012). Effects of extended work shifts on employee fatigue, health, satisfaction, work\/family balance, and patient safety. Work, 41, 4283\u20134290.PubMed\n\nFenwick, R., & Tausig, M. (2001). Scheduling stress: Family and health outcomes of shift work and schedule control. American Behavioral Scientist, 44, 1179\u20131198.CrossRef\n\nForbes, A., & Roger, D. (1999). Stress, social support and fear. British Journal of Health Psychology, 4, 165\u2013180.CrossRef\n\nGerson, K. (2009). The unfinished revolution: Coming of age in a new era of gender, work, and family. New York: Oxford University Press.\n\nGottz\u00e9n, L., & Kremer-Sadlik, T. (2012). Fatherhood and youth sports: A balancing act between care and expectations. Gender & Society, 26, 639\u2013664.CrossRef\n\nGrosswald, B. (2003). Shift work and negative work-to-family spillover. Journal of Sociology and Social Welfare, 4, 31\u201356.\n\nHaines, V. Y., Marchand, A., Rousseau, V., & Demers, A. (2008). The mediating role of work-to-family conflict in the relationship between. Work & Stress, 22, 341\u2013356.CrossRef\n\nHammer, L. B., Kossek, E. E., Anger, W. K., Bodner, T., & Zimmerman, K. L. (2011). Clarifying work-family intervention processes: The roles of work-family conflict and family-supportive supervisor behaviors. Journal of Applied Psychology, 96, 134\u2013150.PubMedCentralPubMedCrossRef\n\nHammer, L. B., Kossek, E. E., Bodner, T., & Crain, T. (2013). Measurement development and validation of the family supportive supervisor behavior short-form (FSSB-SF). Journal of Occupational Health Psychology, 18, 285\u2013296.PubMedCrossRef\n\nHopkins, K. M. (2002). Interactions of gender and race in workers' help seeking for personal\/family problems: Perceptions of supervisor support and intervention. Journal of Applied Behavioral Science, 38, 156\u2013176.CrossRef\n\nKelliher, C., & Anderson, D. (2010). Doing more with less? Flexible working practices and the intensification of work. Human Relations, 63, 83\u2013106.CrossRef\n\nLa Valle, I., Arthur, S., Millward, C., Scott, J., & Clayden, M. (2002). Happy families? Atypical work and its influence on family life. Bristol: Policy Press\n\nLevine, J. A., & Pittinsky, T. L. (1997). Working fathers: New strategies for balancing work and family. Reading: Addison-Wesley.\n\nLewis, J. (2009). Work-family balance, gender and policy. Northampton: Edward Elgar.CrossRef\n\nLindsey, D. (2007). Police fatigue. FBI Law Enforcement Bulletin, 76, 1\u20138.\n\nLiu, H., Wang, Q., Keesler, V., & Schneider, B. (2011). Non-standard work schedules, work-family conflict and parental. Social Science Research, 40, 473\u2013484.CrossRef\n\nMarsh, A. (1979). Women and shiftwork: The protective legislation survey carried out for the Equal Opportunities Commission. Office of Population Censuses and Surveys: Social Survey Division.CrossRef\n\nNg, T. H., Butts, M. M., Vandenberg, R. J., DeJoy, D. M., & Wilson, M. G. (2006). Effects of management communication, opportunity for learning, and work schedule flexibility on organizational commitment. Journal of Vocational Behavior, 68, 474\u2013489.CrossRef\n\n\u01feyane, N. M. F., Pallesen, S., Moen, B. E., \u00c5kerstedt, T., & Bjorvatn, B. (2013). Associations between night work and anxiety, depression, insomnia, sleepiness and fatigue in a sample of Norwegian nurses. PLoS ONE, 8 (8), 1\u20137.\n\nOdle-Dusseau, H. N., Britt, T. W., & Greene-Shortridge, T. M. (2012). Organizational work-family resources as predictors of job performance and attitudes: The process of work-family conflict and enrichment. Journal of Occupational Health Psychology, 17, 28\u201340.PubMedCrossRef\n\nPerrons, D., Fagan, C., McDowell, L., Ray, K., & Ward, K. (2006). Gender divisions and working time in the new economy: Changing patterns of work, care, and public policy in Europe and North America. Northampton: Edward Elgar.CrossRef\n\nPerrucci, R., MacDermid, S., King, E., Tang, C., Brimeyer, T., Ramadoss, K., et al. (2007). The significance of shift work: Current status and future directions. Journal of Family and Economic Issues, 28(4), 600\u2013617. doi:10.1007\/s10834-007-9078-3.CrossRef\n\nPierce, J. L., & Dunham, R. B. (1992). The 12-hour work day: A 48-hour, eight-day week. Academy of Management Journal, 35, 1086\u20131098.CrossRef\n\nPresser, H. B. (2003). Race-ethnic and gender differences in nonstandard work shifts. Work and Occupations, 30, 412\u2013439.CrossRef\n\nPresser, H. B. (2006). Employment in a 24\/7 economy: challenges for the family. In Perrons, D., Fagan, C., McDowell, L., Ray, K., & Ward, K. (Eds.)Gender divisions and working time in the new economy: Changing patterns of work, care, and public policy in Europe and North America. (pp. 35\u201357). Cheltenham, UK: Edward Elgar Publishing.PubMed\n\nRapoport, R., Bailyn, L., Fletcher, J. K., & Pruitt, B. H. (2002). Beyond work-family balance: Advancing gender equity and workplace performance. San Francisco: Jossey-Bass.\n\nRoot, L. S., & Wooten, L. P. (2008). Time out for family: Shift work, fathers, and sports. Human Resource Management, 47, 481\u2013499.CrossRef\n\nScheller, L. (2011). How important is the liaison between professional work and family work? The case of women bus drivers. Work, 40, S5\u2013S14.PubMed\n\nSergean, R., & Brierly, J. (1968). Absence and attendance under non-continuous three-shift systems of work. Nature, 219, 536.PubMedCrossRef\n\nStaines, G. L., & Pleck, J. H. (1984). Nonstandard work schedules and family life. Journal of Applied Psychology, 69, 515\u2013523.CrossRef\n\nThiis-Evensen, E. (1958). Shift work and health. Industrial Medicine and Surgery, 27, 493\u2013497.PubMed\n\nTuttle, R., & Garr, M. (2012). Shift work and work to family fit: Does schedule control matter? Journal of Family and Economic Issues, 33, 261\u2013271.CrossRef\n\nvan der Pompe, G., & de Heus, P. (1993). Work stress, social support, and strains among male and female managers. Anxiety, Stress & Coping: An International Journal, 6, 215\u2013229.CrossRef\n\nVila, B., Morrisson, G. B., & Kenney, D. J. (2002). Improving shift schedule and work-hour policies and practices to increase police officer performance, health, and safety. Police Quarterly, 5, 4\u201323.CrossRef\n\nWhite, L., & Keith, B. (1990). The effect of shift work on the quality and stability of marital relations. Journal of Marriage and the Family, 52, 453\u2013462.CrossRef\n\nWilliams, J. (1999). Unbending gender: Why family and work conflict and what to do about it. New York: Oxford University Press.\n\nWilliams, J. (2010). Reshaping the work-family debate: Why men and class matter. Cambridge: Harvard University Press.\n\nWillis, T. A., O'Connor, D. B., & Smith, L. (2008). Investigating effort-reward imbalance and work-family conflict in relation to morningness-eveningness and shift work. Work & Stress, 22, 125\u2013137.CrossRef\n\nWilson, J. L. (2002). The impact of shift patterns on healthcare professionals. Journal of Nursing Management, 10, 211\u2013219.PubMedCrossRef\n\nWittmer, J. L., & Martin, J. E. (2010). Emotional exhaustion among employees without social or client contact: The key role of nonstandard work schedules. Journal of Business and Psychology, 25, 607\u2013623.CrossRef\n\nWorld Health Organization. (2007). Carcinogenicity of shift-work, painting, and fire-fighting. The Lancet Oncology, 8, 1065\u20131066.CrossRef70373-X)\n\nWyatt, S., & Marriott, R. (1953). Night work and shift changes. British Journal of Industrial Medicine, 10, 164\u2013172.PubMedCentralPubMed\n\nFootnotes\n\n1\n\nFor a more detailed discussion of socioeconomic and racial considerations in regard to work\u2013family conflict, see chapters 3 and 4, respectively.\n\n2\n\nIt should be noted, however, that any such negative effects that nontraditional shiftwork may have on work\u2013family balance are likely to be affected by the employee's degree of 'morningness' or 'eveningness', as found by Willis et al. (2008).\n\n3\n\nFor a more detailed discussion of the impact of work\u2013family conflict on marital relationship quality, see chapter 9.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_14\n\n# 14. Gender Roles in a Masculine Occupation: Military Men and Women's Differential Negotiation of the Work\u2013Family Interface\n\nAnn H. Huffman1  , Satoris S. Culbertson2  and Joseph Barbour3\n\n(1)\n\nDepartment of Psychological Sciences and WA Franke College of Businesss, Northern Arizona University, P.O. Box 15106, 86011 Flagstaff, AZ, USA\n\n(2)\n\nDepartment of Management, Kansas State University, 110 Calvin Hall, 66506 Manhattan, KS, USA\n\n(3)\n\nDepartment of Psychological Sciences, Northern Arizona University, 5 East McConnell Drive, P.O. Box 15106, 86011 Flagstaff, AZ, USA\n\nAnn H. Huffman (Corresponding author)\n\nEmail: ann.huffman@nau.edu\n\nSatoris S. Culbertson\n\nEmail: satoris@ksu.edu\n\nJoseph Barbour\n\nEmail: jeb57@nau.edu\n\nKeywords\n\nWork\u2013familyGenderMilitaryMasculineGendered occupationGender normsStigmaHomosexualityCombat\n\n## 14.1 Introduction\n\nGender plays a key role in the attitudes and behaviors exhibited by individuals in both their work and family domains. Just as individuals can lie anywhere on a continuum from masculine to feminine, occupations can be similarly gender typed. For example, some occupations, such as construction, mining, and forestry, are often seen as highly masculine in nature, while others, such as nursing, early childhood education, and cosmetology are seen as highly feminine in nature, in part due to the sheer prevalence of men and women in the respective occupations. We argue that norms related to masculinity (vs. femininity) can influence one's self-selection into an occupation as well as one's work attitudes and behaviors once associated with an occupation. Since men and women associated with masculine occupations have unique work and family expectations, they also experience and negotiate the work\u2013family interface differently than do individuals in more gender-neutral organizations.\n\nIn this chapter, we use the military as an example of a gendered (masculine) occupation. We provide a theoretical framework to guide an understanding of how the gender of employees in a gendered occupation can beget a unique set of work and family norms. We first provide a summary of the research on gendered occupations and the work\u2013family interface, followed by a theoretical overview of the proposed framework. We then provide a brief discussion of special populations (e.g., lesbian, gay, and bisexual [LGB] individuals) that may have diverse experiences within a gendered organization. Finally, we close with a section on application, emerging issues, and areas for future research.\n\n## 14.2 Gendered Occupations\n\nThe term \"Gendered Occupation\" typically refers to an occupation that, for one reason or another, is comprised almost entirely of individuals of the same sex or individuals who self-identify as the same gender (Blackburn and Jarman 2006). Although gender (i.e., socially constructed) and sex (i.e., biological) are not synonymous in meaning or concept, a \"gendered occupation\" may be referring to either one depending on the specific context. An occupation might be considered gendered if it is comprised of a certain percentage of male or female employees, but it might also be considered gendered based on the degree of masculinity or femininity that society associates with it. Sex and gender are inevitably intertwined, and any attempt to completely tease them apart for the purposes of this chapter will surely prove futile. Because, however, the intent here is to eventually discuss work\u2013family issues surrounding these gendered occupations, occupational segregation by sex (rather than gender) will be the primary focus. As such, \"gendered occupations\" in this chapter will be measured primarily by their percentage of biologically male and female employees.\n\nInfantry soldiers in the United States military, for example, would certainly be considered an occupation segregated by sex, because women are not, as of now, permitted to hold this job, thus making the occupation 100 % male1. Similarly, it would probably be safe to refer to medical transcriptionists as a gendered occupation, as 98 % of the jobs are held by women (Bureau of Labor Statistics 2012). The case is slightly less clear, however, for many jobs. For example, detectives and criminal investigators are primarily male, with 75.2 % of jobs held by men (Bureau of Labor Statistics 2012) . This last example highlights the ambiguous lower limit for what might be considered a gendered occupation when measured by percent male and female. Currently, there does not seem to be a unanimously agreed-upon threshold for determining gendered occupations on the basis of sex prevalence.\n\n### 14.2.1 Gendered Occupations: Theoretical Perspectives\n\nWhy is it that certain occupations are comprised almost entirely of one gender? Do men prefer different kinds of work than women? Do societal expectations for employment differ between the sexes? Do people, at some stage of development, learn that certain occupations are for men or women only (or primarily)? The following section will address these and other questions by discussing some of the leading theories on why gendered occupations exist, and how they have changed over time.\n\nResearch investigating the potential sources of gendered occupations has focused heavily on gender role socialization and theories concerning learned social roles and norms (Aziz and Kamal 2012; DiDonato and Strough 2013; Feliciano and Rumbaut 2005; Jiang et al. 2010; Schuette et al. 2012; Vervecken et al. 2013; Wilbourn and Kee 2010; Wootton 1997). Nevertheless, there is certainly not a consensus on the issue. Multiple additional theories have been proposed that attempt to shed light on exactly why certain types of employment are so profoundly gendered; most notably, preference theory (Hakim 2000), social dominance theory (Sidanius and Pratto 1999), ascriptive inequality or biased hiring (Reskin 2003), professional role confidence and\/or assessment of abilities (Cech et al. 2011), and workplace climate (i.e., certain workplaces seeming cold or unwelcoming toward either males or females; Sandler et al. 1996).\n\nIt is perhaps not surprising that such a large portion of the existing research and literature available on the subject is focused on gender role socialization. After all, from a very young age , males and females are treated differently by those around them, including parents, siblings, friends, and teachers. They are dressed differently, given different toys with which to play, introduced into different sports, and perhaps even spoken to differently (Thorne 1993). Moreover, children are capable, from a very early age, of picking up on the gender roles assumed by their parents. Children who observe their parents practicing an egalitarian division of labor are less likely to gravitate toward heavily gendered occupations than gender-neutral ones. Children who observe a more traditional and\/or gendered division of labor are more likely to equate certain occupations and household duties with gender, and thus, choose their life course accordingly (Aziz and Kamal 2012). It is probably safe to state that the gender stereotypes to which a person is exposed, and the gender roles that they observe and assume in their youth, play a part in their vocational trajectory ; the question is how much of a part do they play?\n\nAlong with children of all ages being exposed to certain stereotyped gender roles and expectations, many additional factors contribute to the segregation of occupations by sex. Preference theory (Hakim 2000) stresses, from a historical perspective, that women are faced with having to make very practical decisions regarding work and family, and that individual preference in those areas will largely determine the type and extent of work that a woman chooses to pursue. Social dominance theory (Sidanius and Pratto 1999), in this context, is referring to the notion that society at large is shaped by the gender stereotypes that we have created, and that subtle messages or wording in advertisements and job descriptions can be gender biased (Gaucher et al. 2011). Reskin's (2003) ascriptive inequality model posits that those in hiring positions within organizations may bias their decisions due to their own beliefs of performance and ability based on gender or the expectations of their profession.\n\nProfessional role confidence refers to the propensity of members of a particular gender to gravitate toward occupations in which they feel confident in their abilities to perform duties adequately (Cech et al. 2011). Historically, this source of occupational segregation by gender has been evident in the science , technology, engineering , and math (STEM) fields, where women have avoided employment largely based on their lack of confidence in their ability to perform in those jobs (Cech et al. 2011). Of course, any lack of confidence that they may have is at least in part the result of either gender role socialization or social dominance, not to mention the chilly climate experienced by women often associated with those traditionally male-held occupations. Development of an unwelcome or intolerant organizational climate , the last potential cause for gendered occupations that will be discussed here, refers to gendered segregation in occupations due a feeling on the part of one gender that they are unwelcome (Schneider et al. 2013). Each of these theories and possible explanations represent a piece to the puzzle that is occupational segregation by sex, and each must be considered when attempting to determine why a particular area of employment has evolved to become dominated by one sex or the other.\n\n### 14.2.2 Masculine Occupations Gender Role Model\n\nNow that the \"what\" and \"why\" of gendered occupations have been introduced, two new questions must be addressed: How do gendered occupational environments differentially affect the men and women who work in them, and how might these environments influence the work\u2013family interface for women and men? In the following section we propose a comprehensive model to explain the gender disparity related to the work\u2013family interface for occupations in which women represent the extreme minority. This Masculine Occupations Gender Role Model (see Fig. 14.1) provides an illustration of the cyclical nature of gender role influences within the masculine organization, beginning and ending with the high prevalence of men. This male prevalence not only affects organizational culture , but is also fueled by the consequences related to the organizational culture. The model suggests that certain, already existing, facets of masculine organizations combine to create a masculine culture. Further, the details of that culture encourage attitudes, beliefs, and experiences, such as perceived and actual work\u2013family conflict, that are different for women than they are for men. Finally, these factors subsequently influence attitudes and behaviors such as organizational commitment, work performance , advancement, and retention. The Masculine Occupations Gender Role Model is explained in four distinct phases, and each is elaborated upon thoroughly in the following sections.\n\nFig. 14.1\n\nMasculine occupations gender role model\n\nPhase I\u2014Prevalence of Gender\n\nBy definition, a masculine occupation is traditionally comprised of men. We propose two reasons why masculine organizations have a high prevalence of men, and are slow to change to more gender-neutral organizations. First, according to social role theory (Eagly 1987), men and women develop different skills, behaviors, and traits associated with their paid work roles. These skills , behaviors, and traits become the norm for the gender, and are adopted by society as indicative of what men and women are expected to do. These societal expectations perpetuate the presumptions that men play a more important role than women in masculine organizations. As such, men are more likely than women to join and stay with masculine organizations. Second, although change in gender prevalence is possible, change is slow due to the strength of attitudes that can affect organizational change. Attitudes toward social groups are key components in the development of discrimination (Fiske 1998; Glick and Fiske 2007). Whereas attitudes can indirectly influence group membership through factors such as an unsupportive supervisor, discrimination is an actual behavior (e.g., passing over for promotion, not allowing admittance to a specific job) that can directly affect the prevalence of a particular gender within a group. Unfortunately, \"the power of gender stereotypes derives from their pervasiveness\" (Eagly and Sczesny 2009, p. 24), with one of the most robust signaling cues for classification being simply whether someone is male or female.\n\nThe prevalence of one gender in an organization, of course, is not unique to a specific industry or occupation (Cabrera et al. 2009). In fact, noted researchers Carli and Eagly (1999) proposed that historical norms of women in domestic labor and men in paid work explain currently held gender stereotypes. The resulting consequence of this pattern of norms is gender typing , or the development of attitudes, behaviors, and interests that correspond to stereotypical masculine and feminine social roles (Liben and Bigler 2002). Gender typing is affected by both the prevalence of one particular gender in an occupation or job, and by the gender norms related to the tasks associated with the job (Lyness and Heilman 2006). Gender typing is the crux of gendered organizations, and can have effects on the minority sex through widespread organizational beliefs such as the organization's culture.\n\nPhase II\u2014Organizational Culture\n\nA high prevalence of men in an organization can create a general organizational culture not always conducive to the professional or personal needs and desires of female employees (and vice versa, although the focus here is on women en masse). According to Schein (1990), organizational culture is \"(a) a pattern of basic assumptions, (b) invented, discovered, or developed by a given group, (c) as it learns to cope with its problems of external adaptation and internal integration , (d) that has worked well enough to be considered valid and, therefore (e) is to be taught to new members as the (f) correct way to perceive, think, and feel in relation to those problems\" (p. 111). As evidenced by this definition, organizational culture encompasses all aspects of an organization and is a major influencer on organizational and employee behavior. The possibility should be noted that, because of the magnitude of effects of a majority male occupation, this culture is not one that is likely to adapt to make itself truly gender neutral, despite any expressed intentions to do so (Rothrauff et al. 2004). This is partly due to the fact that cultural beliefs are deeply ingrained (O'Reilly 1989), and therefore difficult to change (Haas and Hwang 2007).\n\nResearch has suggested that a high prevalence of one gender (in this case male) in an organization can affect the culture of the organization. Gender is a substructure of an organization's culture (Acker 1998), considered to be a foundational element of the organization's working structure. Research has shown that men and women value different qualities associated with an ideal organizational culture (Wicks and Bradshaw 1999), with women finding characteristics such as acceptance, emphasis on relationships, and cooperation as valued components of a healthy organizational culture. Conversely, men value objectivity, rationality, and problem-solving, and place less emphasis on friendliness and acceptance of authority in the workplace (Wicks and Bradshaw 1999). Based on social role theory (Eagly 1987), if people's views are not congruent with (fit) the organization's culture, then they are more likely to experience negative work outcomes. With conflicting values determining the culture, this is very likely the case for women in masculine organizations.\n\nPhase III\u2014Work\u2013Family Conflict\n\nOne negative work outcome that is common to employees is work\u2013family conflict (Morgan and Milliken 1992). Work\u2013family conflict is \"a form of inter-role conflict in which the role pressures from work and family domains are mutually incompatible in some respect\" (Greenhaus and Beutell 1985, p. 77). Work\u2013family conflict is bidirectional, meaning that the work domain can interfere with the family domain and the family domain can interfere with the work domain (Greenhaus and Beutell 1985).\n\nHaas and Hwang (2007) suggest that an organization's gendered substructure affects the degree to which an organization values the employee's family life, thus affecting the work\u2013family interface. They introduce two related ideologies that help explain why organizational culture might affect fathers differentially than mothers. We put forth that this can be extended to understanding why organizational culture can be related to work\u2013family conflict. The first ideology is the concept of separate spheres , which is based on the idea that work and family are two independent domains of influence. This assumption allows work organizations to run their businesses without concern for the role that organizations might have on the well-being of employees' families. Additionally, it devalues the skills and behaviors that are used in the family sphere, an area that is more likely to be filled by women. The second ideology is masculine hegemony , which is the idea that men should be in the primary position of power at work. Masculine hegemony helps to maintain the \"maleness\" in masculine organizations (Billing 2000), and sustains the ideas associated with the separate spheres ideology.\n\nThe existence of the ideologies of separate spheres and masculine hegemony manifests itself in the experiences of the minority sex, which can lead to work\u2013family conflict. For example, because masculine organizations are less likely to adapt to meet female needs, women must make the choice of whether family or work will take priority in their lives. Some, of course, will choose work and some will choose family according to their individual needs and desires, and the result, inevitably, will be some degree of work\u2013family conflict.\n\nWith regard to the work\u2013family experiences for the minority gender in gendered occupations, there has been a fair amount of literature dedicated to the topic. The general consensus is that the minority gender will experience greater work\u2013family conflict than their majority gender counterpart in their own workplace, and greater work\u2013family conflict than they would if they worked in a gender-neutral setting (Cook and Minnotte 2008). With women in particular, striving for work\u2013family balance in male-dominated professions is quite challenging, and can cause high attrition of the already few women in these jobs, as well as deter new women from seeking jobs within the organization (Glass and Estes 1997). Male-dominated professions are less likely to adjust their work schedules and expectation to fit the unique needs of females. This may be even more the case for women who are pregnant, nursing, or bearing the majority of the household labor burden, as male-dominated occupations have, over the years, demanded of their employees longer work hours and less time off, forcing women to either delay or forego childbearing in order to stay in their career, or leave for a more female-friendly work environment (Harris and Giuffre 2010). A particularly interesting and pertinent finding from Maume and Houston (2001) showed that longer work hours and tokenism (i.e., the perception that one is only present to fill a quota or represent some required or expected demographic criterion) for women in a male-dominated white collar profession were each positively associated with increased perceptions of work\u2013family conflict.\n\nPhase IV\u2014Work and Family Outcomes\n\nOne of the main concerns associated with work\u2013family conflict is that it has been associated with many negative work and family outcomes (Allen et al. 2000). This relationship can be explained using the conservation of resources theory (Hobfoll 1989), which puts forth that \"people strive to obtain, retain, foster and protect\" (Hobfoll 2011, p. 117) those things in life that are valued most by them. These \"resources,\" which can be personal, material, social, or energy resources , provide the foundation of support that both bolsters a person's positive affective state and sense of well-being and protects against negative psychological and health outcomes (Hobfoll 1989). The amassing of these resources will have positive outcomes for a person, while the loss of resources carries the potential for a cascade of negative outcomes ranging from depression to declined physical health and functioning. Moreover, resource loss spirals (Hobfoll et al. 2012) can occur, such that the pool of resources and support on which a person could normally rely to help reverse such a downslide is depleted, which of course is the cause of the downslide in the first place. Increased occupational stress alone can act as a catalyst for slow or possibly even rapid loss of resources, but when combined with additional sources of stress (e.g., marital conflict, parental needs, family time management issues) the potential for rapid resource loss is exponentially increased (Hobfoll et al. 2012)\n\nWomen in masculine occupations who experience increased work\u2013family conflict can experience distress in the work domain, the family domain, or both. When the conflict in either or both domain builds to a certain point, the result is often the slow or rapid loss of centrally valued resources, which in turn may cause a mental or physical regression in either or both domains. This \"breakdown\" can vary widely in severity and may come in the form of increased depression and anxiety (Voydanoff and Donnelly 1989), often accompanied by decreased organizational commitment and increased turnover intentions (Jones and Butler 1980; Kelley et al. 1994).\n\nThe cyclical nature of the Masculine Occupations Gender Role Model culminates with the diminished performance , retention, confidence , and attitudes of distressed female employees, feeding back into the first phase and perpetuating the cycle. Recall that the cycle started with an organization built on a foundation of traditionalist views and gender roles and consisting of nearly all males. This primary phase is further exasperated by negative work and family outcomes, while the primary phase also feeds into subsequent phases. It does not appear advantageous to stay in an organization that makes it so difficult to manage one's work\u2013family conflict. In the military , for example, the percentage of females in the military grows, yet grows very slowly (Mackenzie 2012). This slow growth of female service members can be attributed, in part, to the combined effects of resource loss outcomes and the cascade of obstacles deriving from the attitude and behaviors of male service members in a male-dominated profession. Additionally, the sense of military tradition as an all-male force, and the conscious or unconscious belief that women will not prosper in this setting (McSally 2011), is perpetuated as males in uniform notice the difficulty women experience trying to manage work\u2013family conflict, and the wide-spreading ripple effects of this difficulty.\n\nIn the following section we build on the military example. First, we provide a context for the military organization as a gendered occupation. We then use the Masculine Occupations Gender Role Model to explain how employees in one specific masculine occupation (the military) deal with the challenges and demands associated with the work\u2013family interface.\n\n## 14.3 The Military as a Gendered Occupation\n\nThe US military organization provides an obvious example of a gendered occupation. As of 2011, women comprise roughly 15 % of the armed forces of the United States (Army, Navy, Air Force, Marines, and Coast Guard2), with 214,098 women in active duty components, 118,781 women in reserve components, and 72,790 women in the National Guard (Military Women in Service for America Foundation 2011). The fact, however, that over 80 % of the current military is male only partially explains the gendered nature of military service. In order to fully grasp the heavily masculine, male-dominated culture that is the military, one must consider the military's historic and contemporary goals and how it has evolved over time, both in general and with regard to gender. Only then can the depth of the military as a gendered occupation be fully appreciated.\n\nWith the US armed forces being only 15 % women, one could consider this as constituting occupational segregation by sex. Certainly a woman entering such an occupation cannot ignore the obvious demographic minority of which she has become a part. More important than the current numbers, however, is the lingering sense of tradition and gender expectations experienced by women in an institution that once allowed no females whatsoever (McSally 2011). Although women have aided the armed forces in varying capacities since the American Revolution, only since the turn of the twentieth century have they been allowed to serve in uniform; and then only in very limited occupations already deemed socially acceptable for women (McSally 2011).\n\nThe military is grounded in a sense of tradition, history, and pride. This sense is intentionally fostered by leaders who know that tradition strengthens morale during times of trial and tribulation and that, should tradition give way to pragmatism in the minds of service members, the ranks will falter (Schading 2007). Although some traditions have been identified as unfair, immoral, or unjust, the process of correcting certain immoral traditions while keeping all others intact while keeping moral high is a delicate issue. Hence, since 1901, when women were first allowed to serve in the Army and Navy Nurse Corps, women in uniform have been forced to work in a culture of tradition that seems disparagingly dominated by male-based pride and prejudice (McSally 2011). The rules have changed, and continue to change, but the climate remains arguably chilly for women.\n\nThough the precise mission and scope of the military and its branches has changed and evolved over the course of the USA's history, the purpose of its very existence remains the same. The US military exists, first and foremost, as a fighting force, meant to defend the freedom and safety of its citizens. And whereas most jobs in the military are not of a combatant nature, they all exist to support the elements of the force created to make war (Schading 2007). Women are, at the moment, still not able to hold many of the more combat-related occupations within the military (Schading 2007). This point, in and of itself, highlights the gender inequality that must be perceived by most females in the military , regardless of whether or not they agree with the reasoning behind it. To be allowed to serve in the military in a support capacity, but not in the manner for which the military is, at its core, intended, may very likely promote perceptions of unwantedness or tokenism .\n\nVirtually independent of the traditional atmosphere and subsequent gender-typed attitudes instilled in many young male service members is the strong possibility of an already existing overrepresentation of gender-biased males in the military (Young and Nauta 2013). Recall that gender role socialization in childhood and adolescence is theorized to be among the leading causes of occupational segregation by sex (Aziz and Kamal 2012; DiDonato and Strough 2013; Feliciano and Rumbaut 2005; Jiang et al. 2010; Schuette et al. 2012; Vervecken et al. 2013; Wilbourn and Kee 2010; Wootton 1997). The military , as an occupation, is likely more attractive to those who have been raised to believe that men should work in certain settings and women in others (Matthews et al. 2009; Young and Nauta 2013). The result is a force comprised largely of service members who believe that certain jobs are socially acceptable for women, and that the military is not one of them. Compounding that problem, of course, is the fact that, though they may be ever-so-slowly opening new doors for women, the military does not exactly discourage this traditional sex role manner of thinking. Rather, subtle and blatant forms of encouragement for this type of thinking and behaving are abundant, and evidence of social dominance theory can be found throughout the military (McSally 2011).\n\n### 14.3.1 Work\u2013Family Issues of Military Personnel\n\nOf the nearly 2.2 million service members in the US military, roughly 53 % have dependents; a much higher percentage than service members in the past or current civilians in the same age brackets (Blaisure et al. 2012). Having a family (of creation, rather than a family of origin), as defined by the military, can mean one of several situations, and according to 2009 statistics regarding active military personnel, the precise familial breakdown is as follows: dual-military with children (2.8 %), dual-military without children (3.8 %), single with children (5.3 %), married to civilian [no children] (13.7 %), married to civilian [with children] (35.6 %), single [no children] (38.8 %) (Blaisure et al. 2012). The percentage breakdown for the reserve and National Guard components do not differ substantially from these figures and, for all components of the military, the majority of personnel comprising the single [no children] category fall in the junior enlisted (E-1 to E-4) and junior officer (O-1 to O-3) ranks (Blaisure et al. 2012).\n\nBoth men and women in the military commonly experience work\u2013family conflict (Adams et al. 2006) since meeting the needs of the military as well as the needs of the family is quite challenging, particularly given the unrelenting \"greediness\" of both (Segal 1986). The military, being more of a lifestyle than an occupation, can be particularly unforgiving toward outside competitors for a service member's time, energy , focus, and dedication. The family, as in most cases regardless of occupation type, demands much time and devotion. The military has made, and continues to make, huge steps in the direction of a family-friendly force (e.g., family time, paternity leave; Adams et al. 2006), especially considering the core reasons for the very existence of these occupations. What has not changed throughout history, and likely never will, is that in the military the mission comes first meaning nothing supersedes the duties and obligations that these men and women have to their occupations (Adams et al. 2006). When it comes to properly caring for a spouse or family, this mentality affects men and women in uniform in ways large and small, long term and short term.\n\nWork\u2013family conflict in the military can potentially differ from work\u2013family conflict in the civilian world in several key ways, not the least of which is the moderating role of the gender of military personnel. Other principal contributors to this difference include, but are not limited to, extreme lack of flexibility and\/or predictability in work schedule; the possibility of being called into work at random and inconvenient times; frequent moves that may be inconvenient, stressful, or traumatic for service members, spouses, and children; hazardous duty assignments and unaccompanied tours of duty; long lengths of time away from family due to training or combat deployments; and the work-related stress that often accompanies both garrison life and combat (Adams et al. 2006; Blaisure et al. 2012). These represent some of the more pronounced influences on work\u2013family conflict that the military can provide, and they are experienced by men and women alike.\n\nExactly how is work\u2013family conflict influenced by these military-specific antecedents? Unfortunately there is no simple answer to this question due to the incredibly diverse nature of military work, the rank structure, and countless other variables which blur the picture. It can be stated with certainty, though, that the aforementioned points are largely absent from civilian occupations, and are present in most military family situations (Adams et al. 2006; Blaisure et al. 2012). There are certain special circumstances in which a service member might not experience some or any of these added stressors, but, generally speaking, they are part of life for most members.\n\nSo is work\u2013family conflict actually different for women in the military than it is for men? After a thorough review of all of the pertinent literature (not military specific), Frone (2003) concluded that gender was not a significant (statistically or otherwise) moderator in the work\u2013family interface. Indeed, this may be the case when the full spectrum of occupations is taken into account. Nevertheless, when the focus is on a gendered occupation, particularly one as deeply rooted in traditional sex roles as the military, work\u2013family differences by gender become apparent. Would a stay-at-home wife and a husband as the sole breadwinner create the same sort of resentment , sexual confusion, or tension? Would having a second child have been as difficult? Would a male have needed to worry about his career by discussing his family with his chain of command? Would he have felt the need to discuss it? Only in 2008 was the first woman in the history of the US military promoted to the rank of General (Military Women in Service for America Foundation 2011). Advancement for women in the military has been and continues to be a hard-fought battle, and it is precisely the advancement dilemma that exacerbates the already present work\u2013family conflict issue for women. As illustrated in the Masculine Occupations Gender Role Model, these increases in the potential for work\u2013family conflict for women, as the minority gender, are directly and indirectly driving work and family distress, decreased functioning and performance, low retention, and the perpetuation of a male-centric organization.\n\n## 14.4 Emerging Issues in the Gendered Military\n\nOur discussion thus far of the Masculine Occupations Gender Role Model and the military as a gendered occupation has been tailored toward traditional familial situations, and has intentionally disregarded any subpopulations. In the following section we introduce some emerging issues and unique populations that require special attention. Many of these issues (e.g., LGB marriage in the military ; women in combat) are related to policies that are currently, or have very recently been passed, and we do not know yet the unintended consequences (both positive and negative) associated with these changes. We do believe that it is beneficial, however, for future researchers to investigate work\u2013family conflict with regard to these emerging issues, and the relationship that the military, as a gendered occupation, has with them.\n\n### 14.4.1 Lesbian, Gay, and Bisexual (LGB) Populations\n\nWith approximately 4\u201317 % LGB employees in the civilian workforce (4\u201317 %, Gonsiorek and Weinrich 1991; 10\u201314 %, Powers 1996), one can estimate that the military has a similar prevalence of LGB military personnel. These numbers are difficult to confirm with the military's longstanding policy of Don't Ask Don't Tell (DADT) , which essentially forced LGB soldiers into the closet. In late 2010, DADT was repealed (Rich et al. 2012), and LGB service members were, for the first time ever, allowed to serve openly in the US armed forces. Even more recently, the government has taken steps to ensure that the spouses of homosexual and bisexual service members are eligible for full benefits, as is the case with any other military marriage (Secretary of Defense 2013). These are clearly large steps for occupational equality with regard to sex and gender, and the stamping out of discrimination in the armed forces.\n\nTo fully understand how work and family outcomes of LGB military personnel are affected by the gendered military organization, we need to examine processes related to the Masculine Occupations Gender Role Model. We noted earlier that a strong masculine culture leads to less acceptance of nontraditional gender identities. We propose that this masculine culture can stigmatize an LGB service member. A stigma is a very powerful phenomenon that has two dimensions\u2014concealability and perceived controllability (Crocker et al. 1998). Individuals with concealable stigmas (such as LGB individuals) have to make a conscious decision regarding the degree of identity disclosure. Individuals with stigmas that are perceived to be controllability (such as being LGB3) can experience negative reactions if others believe it is their choice to have the stigmatized characteristics (in this case LGB status; Goffman 1963).\n\nIndividuals who have a stigma are also potentially exposed to minority stress (Meyer 2003). LGB individuals usually live and work among non-LGB individuals (Waldo 1999), especially in the military environment. The LGB military personnel will experience the typical stressors that all military personnel might experience (e.g., separation from family, deployments), but will also experience additional stressors that are unique to their minority status (e.g., fear of being \"outed\", Ragins et al. 2007; discrimination, Kerrigan, 2012; identity management, Kerrigan, 2012) . This accumulation of stress can affect both mental and physical health (Hobfoll, 1989), and could further exacerbate the consequences of work\u2013family conflict (King et al. 2013).\n\nFuture researchers should consider two points when delving into work\u2013family conflict in the military for LGB populations. The first is that recent research is showing evidence that the repeal of DADT is having an opposite effect on some service members, in that rather than allowing them the opportunity to serve openly, it has forced them further into the closet, and in some instances increased the level of workplace stress and anxiety with which they live (Rich et al. 2012). Researchers need to further probe this issue to understand why this effect might be occurring, and what the military can do to offset this unintended consequence. The second point worthy of consideration is the finding that LGB-specific social support can decrease negative work outcomes for employees (Huffman et al. 2008). The military organization (and other masculine organizations) might want to develop training programs that will teach supervisors how to provide support for this group. This is especially important since it is only recently that the military has formally recognized LGB personnel.\n\n### 14.4.2 Women in Combat\n\nIn 2013, the Department of Defense announced that women would, for the first time in the history of the US armed forces, be allowed to serve in combat-related (combat arms) occupations (e.g., infantry, artillery, cavalry; Roulo 2013) . Although the policy change has already been announced, the change itself will not take effect until 2016 at the earliest, due to the time it will take to reorganize the force to allow for such a drastic alteration to its established operating system. How will it change the nature of the military as a gendered occupation? And how will the inclusion of women in all jobs in the Army (and eventually every branch of the military) influence work\u2013family conflict? To address these questions we can first return to our model. Although women make up approximately 14.4 % of military personnel, currently and historically, they have been entirely absent from combat arms jobs (Department of Defense 2011) . This exclusion likely reinforces stereotypical views, such as beliefs that women are weak and are ill-suited for combat jobs (Young and Nauta 2013). Furthermore, men who are in combat arms jobs are less likely to have daily exposure to women, particularly within the workplace. As such, there are likely fewer opportunities for these stereotypes to be changed. Yet with the impending changes in rules concerning women in combat arms, some of these stereotypes may lessen. Pettigrew and Tropp (2011) suggest that intergroup contact by itself can decrease stereotypes. With the inclusion of women in combat arms jobs, the prevalence of women in the armed forces might increase overall, thereby allowing for more intergroup contact. This heightened contact could decrease stereotypes, and, with time, change the culture of combat arms units and the military as a whole.\n\nYet, we also have to examine the introduction of women into combat in the context of potential additional stressors for women, which could lead to work\u2013family conflict and the resultant negative job outcomes. Although prevalence of women might introduce a more healthy culture, the exposure to combat could increase the likelihood of additional job stressors. To understand how women in combat might affect work\u2013family conflict issues, we can extrapolate from research on military personnel in combat and its effect on such conflict . Vinokur et al. (2011), for example, provide some insight into the role that combat plays on work\u2013family conflict. They assessed the exposure to war on military personnel's well-being. Their results showed that exposure to combat was related to increased conflict between work and family domains. Similarly, Hobfoll et al. (2012) examined exposure to trauma (which could occur for both combat arms and noncombat arms jobs) and work\u2013family conflict as a source of stress and found that, together, these stressors led to lost resources, which led to outcomes such as posttraumatic stress disorder and perceived health outcomes. When taken together, the aforementioned theories would seem to predict a different, but not necessarily improved work\u2013family conflict scenario for women in these combat arms occupations. The overall gendered culture of the military will have likely changed for the better, which, in theory, would create a decrease in work\u2013family conflict for women within the military. However, for those women specifically in these combat arms professions, work\u2013family conflict may not change, or may even increase, due to the ultra-male environment and more extreme gender-based beliefs of those who often work there .\n\n### 14.4.3 Gender Stigmas for Injured Military Personnel\n\nThe subject of gender stigmas with regard to injury is particularly germane to today's military because of the recent wars and sheer number of physical and psychological injuries that have resulted from them (Fisher 2013). Although men are more likely to suffer combat-related injuries, women are significantly more likely to be injured in training , and both genders sustain injuries while off duty that then may impact their ability to perform their occupational duties and\/or their readiness to deploy to combat (Tiesman et al. 2007). Additionally, research has suggested that women are more susceptible to the nonvisible wounds of battle (e.g., PTSD) than are men (Lutwak 2012). Recent research investigating the roots of this gender disparity in PTSD prevalence has identified combat related stressors, prior interpersonal victimization, and predeployment concerns about life and family disruptions during deployment as being differentially associated with PTSD for men and women (Polusny et al. 2014). Kline et al. (2013) further contributes that lower scores for women on predeployment military preparedness (i.e., lack of combat-specific training) and unit cohesion also predict the postdeployment PTSD gender disparity . Whether the injuries sustained are physical or psychological, the question then is whether these combat and noncombat-related injuries affect military men and women in the same way, and whether any differential treatment and\/or expectations exist between the genders based on the gendered nature of military service.\n\nDespite a growing body of literature exploring the individual and societal implications of traumatic brain injury , PTSD, and various other mental and physical injuries commonly found in those returning from deployments to Iraq and Afghanistan (Morissette et al. 2011), very little research has attempted to explore these issues through the lens of gender expectations. Given what is known regarding the gender-typed thinking in the military, it is perhaps not unreasonable to assume that those women who suffer physical and psychological injuries in the military do not feel comfortable talking about or displaying them in a manner that might otherwise be acceptable, for fear of being seen as weak or unsuitable for military work (Young and Nauta 2013). This places injured women in a rather precarious position because, while it is traditional \"male\" behavior to downplay the severity of physical and psychological hardship, as noted by Glick and Fiske (2001), when women in male-dominated occupations do not conform to traditional sex gender roles they may encounter various forms of resistance and negative attitudes form their male colleagues. Dealing with an injury in a masculine or feminine manner, it would appear, may bring additional distress to a female service member. This is an important area to consider and further explore when considering the implications, work\u2013family related and otherwise, of work in a gendered occupation.\n\n## 14.5 Conclusion\n\nThe United States military is simply one illustrative example of a male-gendered occupation. In every gendered occupation, the minority gender may face challenges that are unique to that specific occupation or organization (e.g., deploying to combat as a female), or challenges that are common to all occupations that are segregated by sex (e.g., tokenism) . As such, some of the antecedents of work\u2013family conflict mentioned in our example of the military as a gendered occupation may be more or less generalizable to other male-dominated professions. The same would be true if any other gendered occupation had been used as an example. We explicitly note this so that little doubt is left as to our intention with this chapter: to provide general knowledge concerning the potential for increased work\u2013family conflict for those in the minority gender of a gendered occupation, and to provide specific examples from one of the most glaringly gendered organizations.\n\nWe proposed a model that can be applied to the military , or any other male-gendered profession, to help explain the cycle of gender segregation and the propensity for increased work\u2013family conflict for women as the minority gender. In order to lay the groundwork for our model, a theoretical basis for the very existence of gendered professions was established to help bring to the forefront the reality of biological and social phenomena combining to form less than perfectly equitable occupational environments . Our model revealed how and why an occupation such as the military can be so heavily dominated by males (Phase I). Specifically, this was done by emphasizing the very facts and theory that provide support to the points that make certain occupations male-gendered. This prevalence of men create an occupational culture that is, both consciously and unconsciously, both actively and passively, supportive of the personal and professional needs of one sex (men), but neglectful to the needs of the other (women; Phase II). The translation of an unsupportive culture to the needs of women leads to the potential for increased work\u2013family conflict (Phase III). Finally, conservation of resources theory highlights how increased work\u2013family conflict for women in these male-dominated professions can instigate adverse psychological and physiological repercussions which, in turn, bring negative outcomes to both the work and family spheres (Phase IV). The cyclical nature of the model culminates, perpetuating the military, or any other male-gendered profession, to be dominated by men.\n\nStress or distress from the work domain spilling into and adversely affecting the family domain, and vice versa, is a possibility for any professional, of any gender, in any organization or industry. It is important to recognize, however, that certain groups and categories of people working in certain occupations may be naturally disadvantaged in this regard. Although women, as a minority group in the male-dominated military, provided an example of this phenomenon here, men could also experience minority status in another occupation (e.g., nursing). Remaining cognizant of the potential for inequitable conditions in and around work\u2013family conflict should be a priority for researchers and employers alike.\n\nReferences\n\nAcker, J. (1998). The future of \"gender and organizations\". Gender, Work and Organization, 5, 195\u2013206.CrossRef\n\nAdams, G. A., Jex, S. M., & Cunningham, C. J. L. (2006). Work\u2013family conflict among military personnel. In C. A. Castro, A. B. Adler, & T. W. Britt (Eds.), Military life: The psychology of serving in peace and combat (pp. 169\u2013192). Westport: Praeger.\n\nAllen, T. D., Herst, D. E. L., Bruck, C. S., & Sutton, M. (2000). Consequences associated with work-to-family conflict: A review and agenda for future research. Journal of Occupational Health Psychology, 5, 278\u2013308.PubMedCrossRef\n\nAziz, S., & Kamal, A. (2012). Gender role attitudes and occupational aspirations of Pakistani adolescents. FWU Journal of Social Sciences, 6, 89\u201398.\n\nBilling, Y. (2000). Organizational cultures, families and careers in Scandinavia. In L. Haas, P. Hwang, & G. Russell (Eds.), Organizational change and gender equity. Thousand Oaks: Sage.\n\nBlackburn, R. M., & Jarman, J. (2006). Gendered occupations. International Sociology, 21, 289\u2013315.CrossRef\n\nBlaisure, K. R., Saathoff-Wells, T., Pereira, A., MacDermid Wadsworth, S., & Dombro, A. L. (2012). Serving military families in the 21st century. New York: Routledge.\n\nBureau of Labor Statistics. (2012). Employed persons by detailed occupation, sex, race, and Hispanic or Latino ethnicity. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bcps\/\u200bcpsaat11.\u200bpdf. Accessed 1 Sep 2013.\n\nCabrera, S. F., Sauer, S. J., & Thomas-Hunt, M. (2009). The evolving manager stereotype: The effects of industry gender typing on performance expectations for leaders and their teams. Psychology of Women Quarterly, 33, 419\u2013428.CrossRef\n\nCarli, L. L., & Eagly, A. H. (1999). Gender effects on social influence and emergent leadership. In G. N. Powell (Ed.), Handbook of gender and work (pp. 203\u2013222). Thousand Oaks: Sage.CrossRef\n\nCech, E., Rubineau, B., Silbey, S., & Seron, C. (2011). Professional role confidence and gendered persistence in engineering. American Sociological Review, 76, 641\u2013666.CrossRef\n\nCook, A., & Minnotte, K. (2008). Occupational and industry sex segregation and the work\u2013family interface. Sex Roles, 59, 800\u2013813.CrossRef\n\nCrocker, J., Major, B., & Steele, C. (1998). Social stigma. In D. Gilbert, S. T. Fiske, & G. Lindzey (Eds.), Handbook of social psychology (pp. 504\u2013553). Boston: McGraw-Hill.\n\nDepartment of Defense. (2011). Profile of the military community: 2010 Demographics. Washington, D.C.: Office of the Deputy Under Secretary of Defense (Military Community and Family Policy).\n\nDiDonato, L., & Strough, J. (2013). Do college students' gender-typed attitudes about occupations predict their real-world decisions? Sex Roles, 68, 536\u2013549.CrossRef\n\nEagly, A. H. (1987). Sex differences in social behavior: A social-role interpretation. Hillsdale: Erlbaum.\n\nEagly, A. H., & Sczesny, S. (2009). Stereotypes about women, men, and leaders: Have times changed? In M. Barreto, M. K. Ryan, & M. T. Schmitt (Eds.), The glass ceiling in the 21st century: Understanding barriers to gender equality (pp. 21\u201347). Washington, D.C.: American Psychological Association.CrossRef\n\nFeliciano, C., & Rumbaut, R. G. (2005). Gendered paths: Educational and occupational expectations and outcomes among adult children of immigrants. Ethnic & Racial Studies, 28, 1087\u20131118.CrossRef\n\nFiske, S. T. (1998). Stereotyping, prejudice, and discrimination. In D. T. Gilbert, S. T. Fiske, & G. Lindzey (Eds.), Handbook of social psychology (pp. 357\u2013411). New York: McGraw-Hill.\n\nFrone, M. R. (2003). Work\u2013family balance. In J. C. Quick & L. E. Tetrick (Eds.), Handbook of occupational health psychology (pp. 143\u2013162). Washington, D.C.: American Psychological Association.CrossRef\n\nGaucher, D., Friesen, J., & Kay, A. C. (2011). Evidence that gendered wording in job advertisements exists and sustains gender inequality. Journal of Personality and Social Psychology, 101, 109\u2013128.PubMedCrossRef\n\nGlass, J. L., & Estes, S. B. (1997). The family responsive workplace. Annual Review of Sociology, 23, 289\u2013313.CrossRef\n\nGlick, P., & Fiske, S. (2001). Ambivalent sexism. In M. P. Zanna (Ed.), Advances in experimental social psychology (pp. 115\u2013188). Thousand Oaks: Academic.\n\nGlick, P., & Fiske, S. T. (2007). Sex discrimination: The psychological approach. In F. J. Crosby, M. S. Stockdale, & S. A. Ropp (Eds.). Sex discrimination in the workplace: Multidisciplinary approaches (pp. 155\u2013188). Malden: Blackwell.\n\nGoffman, E. (1963). Stigma: Notes on the management of spoiled identity. Englewood Cliffs: Prentice Hall.\n\nGonsiorek, J. C., & Weinrich, J. D. (1991). The definition and scope of sexual orientation. In J. C. Gonsiorek & J. D. Weinich (Eds.), Homosexuality: Research implications for public policy (pp. 1\u201312). Newbury Park: Sage.CrossRef\n\nGreenhaus, J. H., Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nHaas, L., & Hwang, C. P. (2007). Gender and organizational culture. Gender & Society, 21, 52\u201379.CrossRef\n\nHakim, C. (2000). Work-lifestyle choices in the 21st century: Preference theory. Oxford: Oxford University Press.\n\nHarris, D., & Giuffre, P. (2010). \"The price you pay\": How female professional chefs negotiate work and family. Gender Issues, 27, 27\u201352.CrossRef\n\nHobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. American Psychologist, 44, 513\u2013524.PubMedCrossRef\n\nHobfoll, S. E. (2011). Conservation of resource caravans and engaged settings. Journal of Occupational & Organizational Psychology, 84, 116\u2013122.CrossRef\n\nHobfoll, S. E., Vinokur, A. D., Pierce, P. F., & Lewandowski-Romps, L. (2012). The combined stress of family life, work, and war in Air Force men and women: A test of conservation of resources theory. International Journal of Stress Management, 19, 217\u2013237.CrossRef\n\nHuffman, A. H., Watrous-Rodriguez, K., & King, E. (2008). Supporting a diverse workforce: What type of support is most meaningful for lesbian and gay employees? Human Resource Management, 4, 237\u2013253.CrossRef\n\nJiang, X., Wang, P., & Wang, M. (2010). Relationship between college students' gender, gender-role identity and occupation-sex stereotype. Chinese Journal of Clinical Psychology, 18, 366\u2013368.\n\nJones, A. P., & Butler, M. (1980). A role transition approach to the stresses of organizationally induced family role disruption. Journal of Marriage and the Family, 42, 367\u2013376.CrossRef\n\nKelley, M. L., Herzog-Simmer, P. A., & Harris, M. A. (1994). Effects of military-induced separation on the parenting stress and family functioning of deploying mothers. Military Psychology, 6, 125\u2013138.CrossRef\n\nKerrigan, M. F. (2012). Transgender discrimination in the military: The new don't ask, don't tell. Psychology, Public Policy, and Law, 18, 500\u2013518.CrossRef\n\nKing, E., Huffman, A., & Peddie, C. (2013). LGBT parents and the workplace. In A. E. Goldberg & K. R. Allen (Eds.), LGBT-parent families: Possibilities for new research and implications for practice (pp. 225\u2013237). New York: Springer.CrossRef\n\nKline, A., Ciccone, D. S., Weiner, M., Interian, A., St. Hill, L., Falca-Dodson, M., Black, C. M., & Losonczy, M. (2013). Gender differences in the risk and protective factors associated with PTSD: A prospective study of National Guard troops deployed to Iraq. Psychiatry: Interpersonal & Biological Processes, 76, 256\u2013272.\n\nLiben, L. S., & Bigler, R. S. (2002). The developmental course of gender differentiation: Conceptualizing, measuring, and evaluating constructs and pathways. Monographs of the Society for Research in Child Development, 67, vii\u2013147.\n\nLutwak, N. (2012). The negative effects of MST and combat injuries among female soldiers: A major concern. Journal of Psychiatric Research, 46, 691\u2013723.PubMedCrossRef\n\nLyness, K. S., & Heilman, M. E. (2006). When fit is fundamental: Performance evaluations and promotions of upper-level female and male managers. Journal of Applied Psychology, 91, 777\u2013785.PubMedCrossRef\n\nMackenzie, M. H. (2012). Let women fight. Foreign Affairs, 91, 32\u201342.\n\nMatthews, M. D., Ender, M. G., Laurence, J. H., & Rohall, D. E. (2009). Role of group affiliation and gender on attitudes toward women in the military. Military Psychology, 21, 241\u2013251.CrossRef\n\nMaume, D. J., & Houston, P. (2001). Job segregation and gender differences in work\u2013family spillover among white-collar workers. Journal of Family and Economic Issues, 22, 171\u2013189.CrossRef\n\nMcSally, M. E. (2011). Defending America in mixed company: Gender in the U. S. armed forces. Daedalus, 140, 148\u2013164.CrossRef\n\nMeyer, I. H. (2003). Prejudice, social stress, and mental health in lesbian, gay, and bisexual populations: Conceptual issues and research evidence. Psychological Bulletin, 129, 674\u2013697.PubMedCentralPubMedCrossRef\n\nMilitary Women in Service for America Foundation. (2011). Statistics on women in the military. http:\/\/\u200bwww.\u200bwomensmemorial.\u200borg\/\u200bPDFs\/\u200bStatsonWIM.\u200bpdf. Accessed 1 Sep 2013.\n\nMorgan, H., & Milliken, F. (1992). Keys to action. Human Resource Management, 31, 227\u2013248.CrossRef\n\nMorissette, S. B., Woodward, M., Kimbrel, N. A., Meyer, E. C., Kruse, M. I., Dolan, S., & Gulliver, S. B. (2011). Deployment-related TBI, persistent postconcussive symptoms, PTSD, and depression in OEF\/OIF veterans. Rehabilitation Psychology, 56, 340\u2013350.PubMedCrossRef\n\nO'Reilly, C. (1989). Corporations, culture, and commitment: Motivation and social control in organizations. California Management Review, 31, 9\u201325.\n\nPettigrew, T. F., & Tropp, L. R. (2011). When groups meet: The dynamics of intergroup contact. New York: Psychology Press.\n\nPolusny, M. A., Kumpula, M. J., Meis, L. A., Erbes, C. R., Arbisi, P. A., Murdoch, M., Thuras, P., Kehle-Forbes, S., & Johnson, A. K. (2014). Gender differences in the effects of deployment-related stressors and pre-deployment risk factors on the development of PTSD symptoms in National Guard soldiers deployed to Iraq and Afghanistan. Journal of Psychiatric Research, 49, 1\u20139.PubMedCrossRef\n\nPowers, B. (1996). The impact of gay, lesbian, and bisexual workplace issues on productivity. In A. Ellis & E. Riggle (Eds.), Sexual identity on the job: Issues and services. New York: Haworth.\n\nRagins, B. R., Singh, R., & Cornwell, J. M. (2007). Making the invisible visible: Fear and disclosure of sexual orientation at work. Journal of Applied Psychology, 92, 1103\u20131118.PubMedCrossRef\n\nReskin, B. F. (2003). Including mechanisms in our models of ascriptive inequality. American Sociological Review, 68, 1\u201321.CrossRef\n\nRich, C., Schutten, J. K., & Rogers, R. A. (2012). \"Don't drop the soap\": Organizing sexualities in the repeal of the U.S. military's 'don't ask, don't tell' policy. Communication Monographs, 79, 269\u2013291.CrossRef\n\nRothrauff, T., Cable, S. M., & Coleman, M. (2004). All that you can be: Negotiating work and family demands in the military. Journal of Teaching in Marriage & Family, 4, 1\u201325.CrossRef\n\nRoulo, C. (2013). Defense department expands women's combat role. U.S. Department of Defense News. www.\u200bdefense.\u200bgov\/\u200bNews\/\u200bNewsArticle.\u200baspx. Accessed 1 Sep 2013.\n\nSandler, B. R., Silverberg, L. A., & Hall, R. M. (1996). The chilly classroom climate: A guide to improve the education of women. Washington, D.C.: The National Association for Women in Education.\n\nSchading, B. (2007). A civilian's guide to the U.S. military: A comprehensive reference to the customs, language and structure of the armed forces. Cincinnati: Writer's Digest Books.\n\nSchein, E. H. (1990). Organizational culture. American Psychologist, 45, 109\u2013119.CrossRef\n\nSchneider, B., Ehrhart, M. G., & Macey, W. H. (2013). Organizational climate and culture. Annual Review of Psychology, 64, 361\u2013388.PubMedCrossRef\n\nSchuette, C. T., Ponton, M. K., & Charlton, M. L. (2012). Middle school children's career aspirations: Relationship to adult occupations and gender. Career Development Quarterly, 60, 36\u201346.CrossRef\n\nSecretary of Defense. (2013). Extending benefits to same sex domestic partners of military members. http:\/\/\u200bwww.\u200bdefense.\u200bgov\/\u200bnews\/\u200bsame-sexbenefitsmemo.\u200bpdf. Accessed 1 Sep 2013.\n\nSegal, M. (1986). The military and the family as greedy institutions. Armed Forces & Society, 13, 9\u201338.CrossRef\n\nSidanius, J., & Pratto, F. (1999). Social dominance: An intergroup theory of social hierarchy and oppression. New York: Cambridge University Press.CrossRef\n\nThorne, B. (1993). Gender play: Girls and boys in school. Piscataway: Rutgers University Press.\n\nTiesman, H. M., Peek-Asa, C., Zwerling, C. S., Sprince, N. L., & Amoroso, P. J. (2007). Occupational and non-occupational injuries in the United States Army: Focus on gender. American Journal of Preventive Medicine, 33, 464\u2013470.PubMedCrossRef\n\nVervecken, D., Hannover, B., & Wolter, I. (2013). Changing (s)expectations: How gender fair job descriptions impact children's perceptions and interest regarding traditionally male occupations. Journal of Vocational Behavior, 82, 208\u2013220.CrossRef\n\nVinokur, A. D., Pierce, P. F., Lewandowski-Romps, L., Hobfoll, S. E., & Galea, S. (2011). Effects of war exposure on air force personnel's mental health, job burnout and other organizational related outcomes. Journal of Occupational Health Psychology, 16, 3\u201317.PubMedCentralPubMedCrossRef\n\nVoydanoff, P., & Donnelly, B. W. (1989). Work and family roles and psychological distress. Journal of Marriage & Family, 51, 923\u2013932.CrossRef\n\nWaldo, C. R. (1999). Working in a majority context: A structural model of heterosexism as minority stress in the workplace. Journal of Counseling Psychology, 46, 218\u2013232.CrossRef\n\nWicks, D., & Bradshaw, P. (1999). Gendered organizational cultures in Canadian work organizations: Implications for creating an equitable workforce. Management Decision, 37, 372\u2013383.CrossRef\n\nWilbourn, M., & Kee, D. (2010). Henry the nurse is a doctor too: Implicitly examining children's gender stereotypes for male and female occupational roles. Sex Roles, 62, 670\u2013683.CrossRef\n\nWootton, B. H. (1997). Gender differences in occupational employment. Monthly Labor Review, 120, 15\u201324.\n\nYoung, L. M., & Nauta, M. M. (2013). Sexism as a predictor of attitudes toward women in the military and in combat. Military Psychology, 25, 166\u2013171.CrossRef\n\nFootnotes\n\n1\n\nThis exclusion of women in combat jobs is changing in 2016 with new Department of Defense policies that will allow women to hold combat positions.\n\n2\n\nStatistics on the Coast Guard were included in this discussion of the military, though they are no longer a branch of the Department of Defense.\n\n3\n\nWe are not saying here that being LGB is a choice. We are merely noting that some individuals may see it as such, and therefore stigmatize LGB individuals accordingly.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_15\n\n# 15. Women in STEM: Family-Related Challenges and Initiatives\n\nVanessa A. Jean1  , Stephanie C. Payne1  and Rebecca J. Thompson2\n\n(1)\n\nDepartment of Psychology, Texas A&M University, 77843 College Station, TX, USA\n\n(2)\n\nUniversity of Baltimore, Baltimore, 21201, Maryland, MD, USA\n\nVanessa A. Jean (Corresponding author)\n\nEmail: vjean@tamu.edu\n\nStephanie C. Payne\n\nEmail: scp@tamu.edu\n\nRebecca J. Thompson\n\nEmail: rthompson@ubalt.edu\n\nKeywords\n\nSTEMScienceTechnologyEngineeringMathAcademiaWork\u2013familyUnderrepresentationBiological clockOrganizational climateMotherhood penalty\n\nThe authors express their appreciation to Jos\u00e9 Berm\u00fadez and Nate Keiser for their assistance with the childcare data and to Mindy Bergman, Laura Lomeli, Kathi Miner, Xiaohong (Violet) Xu, and Carla Zimmerman for their feedback on earlier versions of the chapter. This chapter is based upon work supported by the National Science Foundation under NSF Cooperative Agreement No. HRD-1008385.\n\n## 15.1 The Problem\n\nWomen represent more than 50 % of the population and capacity for innovation in the USA, but they earn less than 25 % of the doctoral degrees in engineering, computer science, and physics, and are underrepresented in almost all science technology, engineering, and math (STEM) careers (National Science Foundation 2013). Despite relatively equal proportions of boys and girls enrolled in STEM courses during grade school, women are significantly underrepresented in STEM degrees and occupations around the world (Hill et al. 2010). Advances in STEM are critical to our health and security, and diversification of the STEM workforce is believed to enhance creativity, innovation, and quality of STEM contributions (Burke 2007).\n\nIt is important to increase the representation of women in STEM for various reasons. First, women have relevant knowledge and skills to contribute which has the potential to extend our accomplishments in these areas. Second, women offer additional perspectives, which can facilitate problem solving and generation of solutions. More perspectives tend to challenge preexisting assumptions, thereby leading to innovation. Third, having more women in STEM creates more role models and mentors for future generations of STEM workers.\n\nExtensive research has been conducted on sex differences focusing particularly on the underrepresentation of women in STEM (for a review see Ceci and Williams 2007). This research examines genetic differences, environmental influences throughout the lifespan beginning even before birth (i.e., prenatal), and the combination of the two. Although relatively equal proportions of boys and girls enroll in STEM courses during grade school, the distribution is not equal thereafter. The phrase \"leaky pipeline\" has been used to describe statistical trends concerning women in STEM reflecting a \"sex-based filter that removes one sex from the stream and leaves the other to arrive at the end of the pipeline\" (Blickenstaff 2005, p. 369).\n\nIn the current chapter, we highlight the influence that family-related challenges have on STEM academic and nonacademic careers. The objectives of this chapter are fourfold: (1) briefly review gender-related theories used to explain underrepresentation of women in STEM, (2) summarize some of the empirical evidence of work\u2013family challenges that STEM women in academia and industry experience, (3) present some new data about the importance of childcare to STEM women's careers, and (4) identify various initiatives, including family-friendly policies and practices, that have been implemented to alleviate work\u2013family challenges and in turn retain women in STEM.\n\n## 15.2 Theories About Underrepresentation of Women in STEM\n\nThere are a number of theories that have been offered to explain why women are underrepresented in STEM. We briefly review gender-related theories, focusing particularly on the theories that explain why women disproportionally experience family-related challenges. These include social role theory (Eagly 1987; Eagly et al. 2000), role congruity theory (Eagly and Karau 2002), stereotype threat (Steele and Aronson 1995), and implicit bias (Greenwald and Banaji 1995).\n\nOne theory proposed to explain the underrepresentation of women in STEM is social role theory (Eagly 1987; Eagly et al. 2000). Gender roles are common beliefs about what men and women tend to do (descriptive norms) as well as what men and women are supposed to do (injunctive norms; Eagly 1987). Historically, men have been the breadwinners, providing financial support for the family, while women have been cooks, homemakers, and the caretakers of the children and the elderly. Further, STEM fields are historically male-dominated (Kanter 1977). Thus, history and tradition contribute to gender-based stereotypes about who should be scientists, technology professionals, engineers, and mathematicians.\n\nThese normative expectations influence boys and girls from the day they are born\u2014through the toys people give them to play with and the reactions people had when they expressed their preferred interests and career-aspirations. These stereotypes continue to have an impact on men and women well beyond their developmental years, as stereotypes influence many people who play important roles in their lives. For example, family members frequently expect women to adopt caretaking roles and responsibilities in the home. Organizational decision makers may be more inclined to hire men for STEM positions (e.g., Steinpres et al. 1999) and give men more developmental opportunities, raises, and promotions than women. Students may take gender roles into consideration when evaluating professors in the classroom (Bennett 1982).\n\nGender roles not only influence how people perceive others, but they also influence how people perceive themselves. Correspondingly, gender roles can influence women's identity and the decisions they make concerning both their personal and professional lives. Beyond the decision to pursue a career in STEM, gender roles can influence decision making about taking a break from one's career, bearing children, and prioritization of a partner's career.\n\nA related theory concerning women's perceptions of themselves, decisions that they make, and their performance is stereotype threat, or a fear of confirming stereotypes about one's groups (e.g., \"girls aren't good at math\"; Steele and Aronson 1995). This theory has been used to explain lower levels of performance on math tests by women than men (e.g., Spencer et al. 1999).\n\nA third theory that has been used to explain sex differences in STEM is role congruity theory . This theory extends social role theory to explain what happens when women (or men) violate gender roles (Eagly and Karau 2002) or when there is an incongruity between characteristics of an individual and the requirements of another role they are taking on. It is tempting to discount traditional gender roles as old-fashioned and question how much they still exist today. Unfortunately, a number of recent studies suggest that such views are still perpetuated in the media (e.g., Collins 2011) and in work-related decisions (e.g., Correll et al. 2007). For example, researchers at Ohio State University recently manipulated the authors listed next to abstracts of scientific studies and asked graduate students to rate the quality of the study. Sadly, the abstracts of studies with male authors were rated as higher quality than identical abstracts of studies with female authors (Knobloch-Westerwick et al. 2013).\n\nSome people react negatively to violations of gender roles (Eagly and Karau 2002) especially if those people have developed a bias, implicit or explicit , in favor of people conforming to traditional gender roles. Implicit biases are unconscious and automatic preferences that people make based on stereotypes (Greenwald and Banaji 1995). Implicit bias is not the same as explicit bias , which is likely to be manifested in more blatant discrimination. Implicit bias is much more subtle, unintentional, and unconscious. It influences how we perceive people and the decisions that we make.\n\nAt first, implicit biases may not seem to make much of a difference, but scientists argue that subtle slights accumulate over time and result in larger disparities (i.e., a multiplier effect) : As Valian (1999) noted, \"Mountains are molehills piled one on top of the other\" (p. 4). This is evidenced by the relatively small number of women who achieve some of the highest ranks within STEM within education, academia , and industry. While implicit biases are unconscious, they still have negative consequences. For example, implicit biases have been revealed in hiring recommendations based on gender stereotypes conveyed in letters of recommendation (Madera et al. 2009; Trix and Psenka 2003). Disadvantages accrue and have a major impact on salary, promotion , prestige, and advancement (Martel et al. 1996).\n\n## 15.3 Work\u2013Family Challenges in STEM Occupations\n\nMany interventions designed to increase the percentages of women in STEM fields focus on supporting and educating undergraduate women on the virtues of these majors. There have been some increases in women in these majors, the number of women in STEM occupations has not increased correspondingly (Fouad et al. 2011; Preston 2004). While women and men are earning undergraduate degrees in STEM fields at similar rates, consistent with others (e.g., Preston 2004), we posit that family-related challenges pose significant hurdles for STEM women's careers. Fundamentally, the more roles people take on, the more opportunities there are for role conflict. Work\u2013family conflict (WFC) is broadly defined as the interrole conflict that occurs when strains from the work and family domains are in some way mutually incompatible, making involvement in each role more challenging as a result of the other (Greenhaus and Beutell 1985). The following section describes six family-related challenges faced by many women in STEM: (1) overlap of biological clock and the timeline for career advancement , (2) pregnancy and childbirth, (3) quality childcare , (4) eldercare challenges, (5) dual-career couple challenges, and (6) balancing critical career experiences with family demands.\n\n### 15.3.1 Overlap of Biological Clock with the Timeline for Career Advancement\n\nOne family-related challenge to a STEM career is overlap of the biological clock with the timeline for career advancement. A woman's biological clock conveys the time period in which it is biologically optimal for her to conceive and reproduce. During this time she is fertile\u2014regularly releasing eggs and also physically capable of bearing children. For American women, this time period is between the ages of 15 and 44, with the most babies in the USA born to women between 20 and 24 years of age in both 2007 and 2009 (Sutton et al. 2011). This timeframe directly overlaps with the critical timeframe for career advancement.\n\nIn academic occupations, the timeline for career advancement is somewhat prescribed. Postsecondary teachers or college\/university professors are typically required to have a doctorate (Bureau of Labor Statistics 2012) . A 4-year degree must be obtained prior to an approximately 6-year doctorate. Thus, a direct path to an academic career means most tenure-track professors start the tenure track around 28, which is already beyond the most common age range for human reproduction in the USA (20\u201324 years; Sutton et al. 2011). Tenure can take up to 7 years to obtain, making most women 35 by the time they achieve tenure. If they deviate from a direct path, they could be well into their 40s by the time they achieve the milestone of tenure. In fact for many STEM-related fields, earning a doctorate is followed by a postdoctoral position that lasts anywhere from 1 to 7 years and sometimes longer, further extending time devoted to training for one's career.\n\nThe tenure clock schedule coincides with the biological clock of a woman, such that a woman who chooses to not have a child because she is striving for tenure may not be able to physically have a child after completion of the tenure process due to risk factors associated with age . Although the age when fertility begins to decline is different for each woman, the chance of conceiving drops considerably after age 35 and even more prominently after age 40 (National Center for Biotechnology Information 2013). The rates of miscarriage and fetal chromosomal abnormalities increase with maternal age after 35 years, as do the risks of developing complications during pregnancy , such as gestational diabetes (Cleary-Goldman et al. 2005).\n\nWhile most women working in nonacademic STEM careers do not face the specified career goal of tenure, their career advancement frequently overlaps with family demands (e.g., new projects, promotions). In a US study of women in mid-level information technology (IT) occupations, nearly one-third of women reported delaying their career goals in order to have children (Simard et al. 2008). Additionally, mid-level women in IT reported being more likely than their male colleagues to sacrifice marriage\/partnership and\/or having children altogether in order to accomplish their career goals. While this group was in the minority, these career pressures were also coupled with stress-related health issues (Simard et al. 2008), revealing additional difficulties experienced by women in STEM occupations when trying to have a family.\n\nMany STEM organizations are either unaware or uninterested in the extent to which family demands hinder career advancement. In a study of German IT firms, Menez et al. (2001) found that roughly 60 % of the firms interviewed did not identify obstacles to the professional advancement of women, despite the fact that only 27 % of the employees in IT firms were women. \"Temporary interruptions of the career because of family obligations are the most important obstacle\" for women in IT fields (Menez et al. 2001, p. 8). The authors attributed these findings to the domain-specific knowledge which often evolves quite rapidly in STEM fields, making any leave, however short, particularly difficult on an individual's career (Menez et al. 2001). In other words, when women take leaves to have children, they are often penalized simply by the constantly progressing nature of the work of which they are a part. Additionally, because of the manner in which STEM professionals are trained, many have few alternative career options if and when they choose (or are forced) to leave.\n\n### 15.3.2 Pregnancy and Childbirth\n\nA second family-related challenge to a STEM career is pregnancy and childbirth . Working in some STEM disciplines can be dangerous, and this may be problematic for women who are planning to conceive in the near future, women who are pregnant, or women who are breastfeeding a newborn. As an example, chemists and chemical engineers could be exposed to toxic chemicals and substances that could be detrimental to a pregnancy or a newborn via breast-feeding. Further, some fields, like anthropology, are physically demanding, requiring women to crawl around in tight spaces or endure miles of outdoor hiking for fieldwork. Thus, some women may not be able to do all of their job-related tasks when they are pregnant. In addition, physicians may impose additional restrictions (e.g., bed rest) during the pregnancy due to the pregnancy being classified as \"high risk\" or because of other medical complications. These job requirements and restrictions could interfere with the woman's ability to make adequate progress in her career during the pregnancy. Further, independent of other medical complications, travel is usually prohibited in the late stages of pregnancy.\n\nIn order to maximize their time with their newborns, women may feel compelled to try to time their deliveries around peak times in their annual work schedules . In fact, \"May babies\" are common enough among female faculty to have warranted this label (Armenti 2004). Some female professors intentionally attempt to schedule their pregnancies such that they can give birth to a child in the month of May allowing them to recuperate and care for their newborn child over the summer, or even hide their pregnancy from others if they are pretenure (Armenti 2004). Unfortunately, women have valid reasons for believing that they may be discriminated against if they disclose their pregnancy to others in the workplace. For example, in a recent study investigating discrimination toward pregnant women, researchers found that pregnant job applicants in the USA experienced more interpersonal hostility compared to nonpregnant applicants and this was attributed to stereotypic beliefs about their commitment to the job and flexibility (Morgan et al. 2013).\n\nHaving May babies typically provides female faculty with a reprieve from their teaching load following the birth or adoption of a child because classes are out for the summer, although it is common for professors to conduct research during the summer months (Armenti 2004). In universities nationwide, academics can be found mentoring graduate students, attending meetings, conducting research, and sometimes teaching classes during the summer months. Thus, a professor's workload during the summer months is not likely to change dramatically from the traditional school year. A lack of participation in these activities due to childbirth and childcare would be noticed because summer is often about catching up rather than resting.\n\n### 15.3.3 Quality Childcare\n\nA third family-related challenge to a STEM career is obtaining quality childcare. Reliable and affordable childcare is essential to any working parent, as the ability to manage and focus on work demands is greatly hindered when childcare is unavailable and\/or inadequate. This is particularly salient for female faculty, who have a short window of time to prove themselves to the academic community. Female STEM faculty may be further disadvantaged by the university's geographical location. It is common for top tier research universities to be located in rural areas of the country and for faculty members to relocate to the university town. As a result, faculty may live far from their families and have less support for childcare than those in other fields. Further, childcare options are not always optimal or convenient to faculty in rural areas.\n\nIn order to determine how childcare issues affect STEM women in academia , we recently conducted a brief childcare needs analysis of all faculty at a large, southern US university. We asked the following open-ended question: \"In what ways have good, reliable childcare facilitated your career and\/or in what ways have inadequate, unreliable childcare interfered with your career?\" A number of themes emerged in the 93 responses received to this question from STEM female faculty1 and were systematically coded into one or more of the categories that emerged. Most respondents reported more than one theme in their narrative answer. Three themes emerged the most frequently: (1) childcare is critical to one's career, (2) childcare permits peace of mind\/psychological focus on work, and (3) childcare is difficult to find\/insufficient childcare options available. It is interesting to note that although our sample consisted of only STEM women in academia (faculty), none of these themes appear to be unique to academic careers. The frequency with which these themes were mentioned and an example quote for each of them is displayed in Table 15.1.\n\nTable 15.1\n\nThree most common responses by STEM women faculty as to how childcare has impacted their career\n\nFrequency | Type of impact on career | Description\/definition | Example respondent quotes\n\n---|---|---|---\n\n44 | Inadequate\/difficult to find | Childcare reported to be inadequate in terms of convenience, quality, or availability | I was on the waitlist for (the campus childcare center) for 1 year. I struggled to find good daycare in the town, but that information was difficult to obtain (i.e., there are no lists of recommended daycare providers). The daycare resulted in added time commuting. In addition, the daycare days of operation were not always consistent with the university schedule\n\n30 | Critical to career | Affordable, adequate childcare is a key element to have successful career | If my kids are not properly taken care, work cannot be done.\n\nI could not have gone back to work\u2014period\u2014without good, reliable childcare as I do not have family anywhere in [this state]. So having a reliable, good daycare has been critical\n\n9 | Psychological focus\/peace of mind | Childcare provides peace of mind and freedom from worry, allowing for focus on job tasks | High quality, reliable childcare allows me to concentrate on my job when I am at work so that I am not distracted worrying about my child.\n\nIt is clear that the STEM female faculty in our sample experienced considerable challenges locating childcare options. This theme emerged in the responses from STEM women faculty more than any other theme. The comments coded for this theme included concerns related to the unavailability of infant care, after hours care, and care for sick children. There were also comments pertaining to the difficulty they experienced trying to find convenient childcare . Additionally, STEM women faculty reported that reliable, adequate childcare is critical to their careers, with some noting that they are unable to work if it is not attainable. Comments classified under this theme indicated that childcare is a career necessity, not an optional benefit. Lastly, many STEM women faculty reported that having adequate childcare allows them to focus on their work, freeing them from worrying extensively about their children during their work time. As highlighted in our brief needs analysis, affordable, convenient, quality childcare is imperative to the careers of female faculty in STEM fields. Future research should directly examine the influence of childcare challenges on the leaky pipeline to both academic and nonacademic careers.\n\nOngoing family obligations have been frequently identified as an obstacle for women in nonacademic STEM careers as well (Preston 2004). In a recent national study examining why women in the US leave engineering, Fouad and Singh (2011) identified that one quarter of women left their engineering career because they wanted more time with family. Caretaking obligations, particularly for children, are consistent concerns for STEM women in nonacademic careers (Fouad et al. 2011). \"Predictably, women with childcare responsibilities experienced greater interference between their work and nonwork roles than those without such responsibilities\" (Fouad and Singh 2011, p. 42). It appears that the lack of adequate childcare is a considerable obstacle to the advancement and retention of women in STEM careers.\n\n### 15.3.4 Eldercare Challenges\n\nA fourth family-related challenge to a STEM career is eldercare. Due to the aging population of baby-boomers, eldercare presents a challenge for many employed caregivers. The number of persons older than 85 years of age in the USA will grow to more than 18 million by 2050 (U.S. Census Bureau 2009). Further, some employees serve as caregivers to both children and elders, a cohort of individuals sometimes referred to as the \"sandwich generation\" (Hammer and Neal 2008). Women caring for elders could be perceived as being less committed to their work than their male counterparts if the caregiving interferes with work (Morgan and Milliken 1993) .\n\nAdditionally, according to a recent report from the Sloan Center on Aging and Work that examined nine US workplaces as part of their Age & Generations study (Pitt-Catsouphes et al. 2009), the experiences of women with eldercare responsibilities differed significantly from those with other dependent care responsibilities, such as childcare. Specifically, women who provided care to an elder reported having less access to flexible work options, receiving less support from their supervisors, and having less job security than did those who provided childcare or dependent care other than eldercare. Although there are similarities between childcare and eldercare and both present challenges, eldercare has several differences that lead to unique effects in the workplace. As noted by Wagner (1991), childcare is more predictable than eldercare, mainly because the intensity and duration of eldercare needs vary. Eldercare-giving may result in more work interruptions because sudden medical issues such as strokes, heart attacks, or other episodic conditions are more prevalent in older adults than young children. To our knowledge, no research has systematically examined the influence of eldercare responsibilities or how supportive STEM organizations are of these challenging responsibilities on STEM women's careers. Future research is needed on this differential responsibility to determine the extent to which eldercare contributes to the leaky pipeline .\n\n### 15.3.5 Dual-Career Couple Challenges and Benefits\n\nA fifth family-related challenge to a STEM career is dual-career considerations. There is an upward trend of dual-earner couples in the USA such that in 2010, 58 % of married-couple families with children under 18 years of age were classified as dual-earner status (Bureau of Labor Statistics 2010) . It has been estimated that 83 % of female scientists in academia in the USA are partnered with another scientist in academia (Schiebinger et al. 2008). Likewise, women in nonacademic STEM occupations experience similar partnership dynamics. For example, a study of midlevel IT employees in the USA found that nearly 70 % of women who have full- or part-time working partners have partners working in IT. In contrast only one third of men had partners in the same field (Simard et al. 2008).\n\nDual-career couples have the unique challenge of finding equally satisfying positions in the same geographical location, which has been referred to as the \"two-body problem\" (Li 2009) and the \"co-location problem\" (Wolf-Wendel et al. 2004). The dual-career situation often leads to a compromise, with one partner putting his\/her career on hold or settling for a less prestigious position for the benefit of the other's career. According to the Cornell Couples and Careers study, 12 % of men and 39 % of women regard their spouses' career as more important than their own (Moen et al. 1999). In fact, a study of 13 leading research universities in the USA found that 40 % of academic women refused outside offers due to their partner not receiving an adequate offer in the area, making it the top reason why women refused, compared to only 27 % of academic men (Schiebinger et al. 2008). Based on these findings, STEM women faculty who are married or partnered to STEM men may be more inclined to sacrifice their careers for their spouse or partner, which further contributes to the leaky pipeline effect.\n\nIn a study investigating the \"two-body problem\" at one university in the USA, researchers found support for their hypothesis that joint hire faculty members (both the primary hire and partner) in STEM fields are just as statistically productive than their colleagues, if not more so, thus alleviating concerns about the productivity of joint hires in STEM (Woolstenhulme et al. 2012). This hypothesis was based on the premise that couples in academia will likely accept positions at less prestigious universities in order to be in the same location together. Additionally, joint hires in STEM were found to be 22 % more likely than their colleagues to obtain a grant, and the joint hire in STEM produced on average 0.73 more publications than their mean nonjoint hire colleague (Woolstenhulme et al. 2012).\n\nAlthough women in STEM fields with families face numerous challenges when juggling work and family, there are many benefits associated with having a supportive family. For instance, one study in the western USA found that female university employees with supportive spouses or partners experienced significantly less WFC than unmarried or unpartnered employees, and this effect was five times larger for women than men (Elliott 2003). This WFC should lead to women staying in STEM fields as opposed to leaking from the pipeline. Additionally, married faculty members have the benefit of similar yearly work schedules, which makes planning vacations and sabbaticals less complicated.\n\nIn another study of dual-career academic couples in the USA, the couples reported they were \"more successful\" due to their partner and cited several reasons for their increased success (Schiebinger et al. 2008). Fifty-eight percent of the dual-career academic couples in the sample said that they benefit from shared professional networks, colleagues, and friends, compared to only 25 % or less of the faculty members with spouses employed elsewhere or stay-at-home spouses. Interestingly, 35 % of the faculty in the sample with spouses who were nonacademic or stay-at-home reported gains in research productivity due to their partner, whereas 44 % of the dual-academic career faculty reported research gains due to their partner (Schiebinger et al. 2008).\n\n### 15.3.6 Balancing Critical Career Experiences with Family Demands\n\nA sixth family-related challenge to a STEM career is simultaneously managing critical career experiences and family demands. Attending conferences is important for professors to disseminate their research and establish their reputations in their fields. It fosters networking with colleagues with similar interests who could become collaborators on research projects and grant applications and facilitates new relationships with potential students and postdoctoral candidates as well as senior colleagues who may serve as external reviewers of tenure packages. Similarly, attending conferences and professional development workshops is advantageous for individuals in nonacademic STEM careers in order to learn about up-to-date knowledge and best practices in their field as well as to facilitate networking opportunities with potential clients and\/or business associates.\n\nA potential effect of having children and breastfeeding on women in STEM is the reduced availability and comfort with extended conference travel , particularly international travel. In a study of presenters at the European Society for Evolutionary Biology (ESEB) Congress 2011, female scientists declined invitations to speak half of the time, compared to only about a quarter of the time for men (Schroeder et al. 2013). Women were significantly underrepresented as presenters at the symposia (15 %), as well as among the oral presenters (41 %), plenary speakers (25 %), and all presenters (46 %), although all abstracts submitted were accepted. The authors of this study surmised that this disparity is due to a number of reasons, including childcare responsibilities (Chesley and Moen 2006; Schroeder et al. 2013). Taken together, these findings demonstrate that further research should examine the possibility that family and childcare responsibilities preclude women from having the availability to travel to speak at conferences and other important networking opportunities. Presenting at and attending conferences is essential, particularly during the first few years of career establishment, and this lack of availability could stymie the success of women in STEM fields.\n\nA recent study by Casadevall and Handelsman (2014) examined symposia at two meetings held by the American Society for Microbiology and found that having at least one woman on the convening team was associated with reduced likelihood of having an all-male presentation by approximately 70 %. When organized by all men (who were perhaps influenced by gender roles), an average of 30 % of the sessions were held by all-male speakers, compared to an average of only 8.9 % when organized by a team that included at least one woman. Although this study is correlational and does not identify the cause of these findings, it nonetheless supports the importance of having women on STEM conference committees as a way to promote the inclusion of women on conference programs.\n\nAlong those lines, grants are extremely important and necessary in STEM fields as evidenced by some university policies that require evidence of a history of extramural funding prior to consideration for hire and when awarding tenure. Not attending conferences may also result in lack of access to funding opportunities that are imperative to employment at a research university or national laboratory. A recent study found that among married tenure-track faculty with young children, women are 21 % less likely than men to have their work partially or entirely funded by federal grants or contracts (Goulden et al. 2009). Additionally, among married tenure-track women, those with young children were found to be 26 % less likely than those without young children to have their work funded partially or completely by federal grants or contracts. These small setbacks early on can accumulate to form a larger disadvantage that lingers for the length of the career.\n\n## 15.4 Organizational Barriers to the Success of STEM Women with Family Demands\n\nSeveral organizational barriers can also impede STEM women's success. Specifically, contextual factors such as an unsupportive or \"chilly\" climate, historical underrepresentation of women in STEM careers, long-term career consequences for those who choose to have children, and a lack of successful female role models who effectively manage work and family demands have all been linked to the lack of women in STEM careers. These barriers are discussed in the next section.\n\n### 15.4.1 Chilly Organizational Climates\n\nAnother fundamental explanation for the leaky pipeline is believed to be the lack of work\u2013family support for individuals in these job domains (Fouad and Singh 2011) . As these occupations have been historically male, the culture surrounding STEM careers has developed in a manner that is not conducive to the careers of individuals also managing extensive family-demands. As a result, STEM organizations offer fewer programs and policies that enable schedule control or the ability to take breaks in one's career trajectory than non-STEM organizations (Fouad and Singh 2011). Similarly, the culture that has evolved is one in which attending to family obligations (or even considering this) is not supported or encouraged. Fouad and Singh (2011) found that women engineers in the USA with high levels of work-demands working in environments with negative climates for women experienced high-levels of WFC. Finally, some STEM occupations (e.g., IT) require employees to be available during nontraditional work hours as well as have unpredictable schedules (Major et al. 2003). While STEM careers have become renowned for innovation in numerous arenas, due to the stagnation of family-supportive leadership, as well as lack of initiatives to enable employees to manage competing role demands, women frequently leave STEM careers .\n\nA recent meta-analysis of family-friendly policies examined how the availability of policies and family-supportive organizational perceptions (FSOP; Allen 2001) influence work attitudes and found that FSOP yielded stronger relationships with positive job attitudes and a stronger negative relationship with work-to-family conflict than the availability of policies (Butts et al. 2013). This study indicates that perceptions of family-supportiveness matter, and simply having family-friendly policies does not guarantee their effectiveness at improving employee attitudes and decreasing work\u2013family conflict .\n\nFurther, a recent study of faculty members at a Tier I research institution in the USA found that women reported significantly less FSOP than men (Jean et al. 2014). In addition, both male and female STEM faculty reported less FSOP compared to non-STEM faculty. These results showed that STEM departments were lagging behind non-STEM departments in family-friendliness, which may contribute to women being less attracted to or ultimately leaving positions in STEM fields. Some researchers have pointed to the \"old-boy network\" defined as \"managers' preference to interact with other managers of similar status rather than with line employees, and intentional exclusion by managers who have negative stereotypes about women's and minorities' abilities, motivation , and job preferences\" (Noe 2013, p. 437) as a possible explanation for the \"chilly climate\" experienced in STEM departments.\n\nSome researchers argue that initiatives designed to increase the family supportiveness of work cultures may ultimately be of limited effectiveness, as work cultures do not reduce the family demands women experience due to their primary caregiver role (Fouad and Singh 2011). Programs that provide women with the ability to successfully manage family demands around work demands and vice versa may be more effective at increasing the retention of women in the STEM workforce. For many women with resources, this means outsourcing some household chores including cleaning, laundry, and cooking .\n\n### 15.4.2 Motherhood Penalty\n\nData from the Perceptions of Women in Academic Science study demonstrate that US female scientists have fewer children than desired compared to their male counterparts, and they are more likely to attribute having fewer children than desired to their careers as scientists (Ecklund and Lincoln 2011). A number of researchers have speculated about the effects of parenting on careers. Unfortunately, researchers have consistently demonstrated a \"motherhood penalty\" in which women experience more negative effects of parenting than men and sometimes men actually benefit (Correll et al. 2007). This has been demonstrated when examining salaries, hierarchical positions within academia (non-tenure track vs. tenure track and associate vs. full), and retention in academia with even stronger effects for women in STEM (Mason et al. 2013). The motherhood penalty also occurs outside of academia. For example, Budig and England (2001) found a 5 % wage penalty per child after controlling for job experience in a longitudinal study of a US sample of 5000 women employed in a wide range of occupations. In addition, women in nonacademic STEM careers describe experiencing a \"motherhood penalty\" or \"family penalty\" as an important obstacle to their career success (Simard et al. 2008) .\n\n### 15.4.3 Underrepresentation of STEM Women in Leadership Positions\n\nFamily-related challenges discouraging women from remaining in STEM careers have led to a dearth of women in STEM leadership positions. Since Valian's (1999) influential work examining gender inequities in various occupations, little has changed with regard to the underrepresentation of women in executive leadership roles in organizations, particularly STEM companies. Although gender schemas have begun to change, STEM careers and organizations have been particularly slow to accept women's roles in management as well as recognize how family supportive organizational culture can influence the attraction and retention of capable female employees (Soe and Yakura 2008).\n\nAs STEM domains have been historically male-dominated (Kanter 1977), the culture of STEM workplaces has been heavily defined by issues of gender inequality and sexism (Fouad et al. 2011). STEM job domains and organizations have remained predominantly populated by men and thus sustained cultural norms and expectations of what is an \"ideal\" STEM employee. Because the need to balance career and family remains a primary factor pushing women out of STEM careers, relatively few women have attained senior or leadership positions in STEM careers (Rossler and Taylor 2009). Sociological examinations of the culture of engineering have identified that stereotypically \"masculine\" characteristics of technical obsession, territorialism, and aggressive self-promotion were distinguishing features of engineering cultures (Fouad et al. 2011; Robinson and McIlwee 1991). While both men and women associate characteristics of their own sex as relating to managerial ability, men do not identify female traits with managerial characteristics (whereas women classify traits from both sexes as managerial characteristics; Schein and Mueller 1992). Thus, leadership in STEM fields has perpetuated a schema of masculine characteristics associated with leadership (Eagly and Karau 2002). Correspondingly, the individuals who are most likely to succeed are those who display traits typically associated with maleness, leading to the marginalization and underrepresentation of women in STEM leadership roles. As a result, women in STEM careers have relatively few examples to look toward when trying to navigate their career paths while simultaneously maintaining family obligations.\n\nThese findings suggest that a lack of women in organizational leadership , particularly in STEM industries, has led to the perception that the best model of leadership is the male model. While many STEM leaders have families, this model suggests that in order to succeed, one cannot be the primary caregiver. As a result, relatively few examples of executive women leaders exist, and even fewer who have managed to effectively balance work and family demands. Therefore, a lack of female role models in STEM industries appears to be an additional barrier facing women seeking successful careers in STEM. While both career and psycho-social (i.e., social support) mentorship have been associated with career mobility in the USA (Scandura 1992), relatively few studies have examined how psychosocial mentorship or specifically having a mentor provide guidance on how to manage work and family demands can benefit women, particularly women in STEM careers. Thus, more research is needed examining the ways in which organizations can provide guidance for women regarding how to be successful in managing a STEM career and family demands.\n\n## 15.5 Initiatives to Facilitate the Promotion and Retention of Women in STEM Careers\n\nDespite the upsetting trends in STEM academic and nonacademic jobs, many promising initiatives have recently launched in order to assess the underlying causes of the leaky pipeline effect and implement programs to help women overcome the obstacles to success in these fields. These initiatives are targeted at women of all age groups, including young children. Trends supported by federal government research grants alongside organizational efforts suggest that women in STEM disciplines and careers will have brighter futures.\n\nCompanies have begun actively attempting to change the perceptions of what STEM fields look like by encouraging girls and young women to become and remain interested in STEM careers (Starnes 2013). One particularly exciting effort to facilitate the increase of interests in STEM in young girls is by a company called Goldie Blox based in the USA. These construction sets are designed to increase girls' interest in STEM subjects. The goal behind these and similar efforts is to engage girls in STEM at an early age , potentially mitigating the societal gender expectations of these career interests. Another noteworthy partnership began in 2010 between the Fortune 500 company 3M and the University of Minnesota's STEM Education Center to train elementary and secondary school teachers to successfully integrate STEM subjects into their courses in order to increase students' interest in STEM disciplines and careers (University of Minnesota 2013). Through these efforts, the University of Minnesota has trained thousands of graduates to go on to work at 3M. Numerous similar efforts are underway across the country, many of them funded by the National Science Foundation (NSF) and by corporations. In fact, each year Catalyst , a leading nonprofit organization with a mission to expand opportunities for women and business, gives the Catalyst Award to one or more companies that demonstrate innovative organizational approaches to the recruitment and advancement of women (Catalyst.org).\n\nThe NSF's ADVANCE-IT program provides funding to academic institutions with the goal of promoting research and interventions to improve outcomes of STEM women faculty members and to better understand the unique issues they face (National Science Foundation n.d.). The NSF has awarded over $ 130 million to over 100 institutions in support of proposed research and initiatives aimed at increasing and retaining faculty in STEM disciplines. As a result, research and programs developed by these institutions have shed light on the experiences of women in STEM fields and the factors that lead to the leaky pipeline effect. Additionally, many tools have been developed that can be utilized to support similar objectives in other academic and nonacademic settings. For example, Rice University developed a career workshop for postdocs and late stage PhD students from underrepresented minority groups in STEM disciplines to help prepare them for academic careers (Rice University n.d.). Another notable ADVANCE initiative is Michigan's Strategies and Tactics for Recruiting to Improve Diversity and Excellence (STRIDE) training for recruitment and retention (University of Michigan 2013). The NSF also has a career-life balance initiative and the National Institute of Health (NIH) has a number of family-friendly initiatives including parental leave stipends, developmental supplements following time off to attend to family, and the requirement to identify childcare resources when hosting a conference funded by NIH.\n\nAlthough women in STEM careers have relatively few examples to look toward when trying to navigate their career paths, there are some resources available that may help them succeed in leadership roles while meeting family demands. For example, a recent book describes the experiences and success of 10 women US university presidents, 9 of whom have children (Madsen 2008). Additionally, a book of personal narratives was recently published that focuses on the ways in which female scientists can be successful in both their careers and motherhood, and underscores the need for organizations to assist with and adapt to the needs of scientists with children (Monsson 2008). Likewise, other compilations of success strategies for STEM women have been published as well (e.g., Pritchard 2006).\n\nRecognizing the overlap between the biological and tenure clocks, universities have attempted to accommodate by allowing faculty members to take extension(s) on their tenure clock in order to provide care for a newborn. Flexible tenure clock policies, also referred to as \"stop clock policies,\" are defined as policies that provide tenure-track faculty with the option of postponing their tenure review if events occur that are likely to negatively impact their research productivity . Extensions are usually granted in increments of 1 year (Manchester et al. 2013) and were originally adopted to attract women to academia , to develop equity, and to facilitate work\u2013family balance (Manchester et al. 2010). For these reasons, tenure clock extensions are common, particularly amongst female faculty for the purpose of childbirth or childcare. Nonetheless, some recent research revealed a wage penalty associated with extending the tenure clock for family reasons as opposed to nonfamily reasons (Manchester et al. 2010). This finding highlights the persistent disadvantage associated with the alignment of tenure and biological clocks for women. Certainly some female faculty choose to have children before earning tenure and given the ticking biological clock, one can certainly appreciate why.\n\nIn addition to fostering a family-supportive climate, there are specific work\u2013family support policies which are aimed at providing employees the ability to balance work and nonwork role demands that are being adopted with increasing prevalence (e.g., tenure clock extensions, flextime ; Matos and Galinsky 2012). Work\u2013family support policies are defined broadly as policies organizations implement that are designed to reduce WFC (Butts et al. 2013). While at least some work\u2013family support policies exist at most universities and organizations, the climate at each university\/organization, and even within each department, may not support their use, thereby rendering them less useful to employees. Therefore, it is important to have both a supportive climate and formal policies in place.\n\nWorkplace flexibility has been cited as an organizational tool enabling employees to navigate between work and family roles while meeting the needs of employers and clients (Kossek et al. 2014). In a study of best practices of IT supervisors in the USA, Major et al. (2003) identified that high-performing supervisors offered some form of flexible scheduling to their employees due to the highly demanding nature of IT work schedules, despite the fact that the policies were often informal and on case-by-case bases. Correspondingly, because of the detrimental role that WFC plays in STEM women's careers (Fouad and Singh 2011), researchers have identified the importance of family-friendly policies for the retention of women in STEM (Villablanca et al. 2011).\n\nIn support of these trends, in 2009, President Obama signed an Executive Order establishing the White House Council of Women and Girls to identify and confront issues of inequality and resulting challenges for women of all ages (whitehouse.gov n.d.). In response, many STEM companies, groups, and agencies have initiated efforts to actively recognize examples of women in STEM careers as well as target and address issues that lead girls and women to drop out of the STEM pipeline. The National Aeronautics and Space Administration (NASA) launched the Women@NASA website, celebrating the success of women throughout NASA's history through a collection of 64 videos and essays describing the many roles that women play across the agency (NASA 2013). In addition, the National Institutes of Health (NIH) supports work\u2013life balance by providing grantee institutions with policies and practices to support family-friendly environments (NIH n.d.). Similarly, the NSF launched the Career-Life Balance Initiative to promote family-friendly policies with the goal of clearing the path for women in STEM from graduate education to careers in STEM (White House 2011).\n\n## 15.6 Conclusion\n\nIn summary, there are a number of family-related challenges that likely contribute to underrepresentation of women in STEM, especially at the higher ranks. Overall, there is considerable overlap between the time periods in which critical career-related development and experiences are expected and required to occur (e.g., academic tenure) and when women are at the optimal age to conceive and bare children. Additional challenges include dependent care (both child and elder) and dual-career considerations. Theoretically, many of these challenges come about because traditional gender roles and stereotypes result in (implicitly) biased decision making by women themselves as well as many important others. Fortunately, there are many national and organizational\/university initiatives directed at repairing the leaky pipeline , as well as individual strategies. Continued efforts to educate key decision makers and alleviate these challenges will hopefully result in fewer disparities between men and women in STEM careers.\n\nReferences\n\nAllen, T. D. (2001). Family-supportive work environments: The role of organizational perceptions. Journal of Vocational Behavior, 58, 414\u2013435.CrossRef\n\nArmenti, C. (2004). May babies and posttenure babies: Maternal decisions of women professors. The Review of Higher Education, 27, 211\u2013231.CrossRef\n\nBennett, S. K. (1982). Student perceptions of and expectations for male and female instructors: Evidence relating to the question of gender bias in teaching evaluation. Journal of Educational Psychology, 74, 170\u2013179.CrossRef\n\nBlickenstaff, J. C. (2005). Women and science careers: Leaky pipeline or gender filter? Gender and Education, 17, 369\u2013386.CrossRef\n\nBudig, M. J., & England, P. (2001). The wage penalty for motherhood. American Sociological Review, 66, 204\u2013225.CrossRef\n\nBureau of Labor Statistics. (2010). Employment characteristics of families. http:\/\/\u200bwww.\u200bbls.\u200bgov\/\u200bnews.\u200brelease\/\u200barchives\/\u200bfamee_\u200b03242011.\u200bpdf. Accessed 1 April 2014.\n\nBureau of Labor Statistics. (2012). Postsecondary teachers. http:\/\/\u200bwww.\u200bbls.\u200bgov.\u200booh\/\u200beducation-training-and-library\/\u200bpostsecondary-teachers.\u200bhtm. Accessed 1 April 2014.\n\nBurke, R. J. (2007). Women and minorities in STEM: A primer. In R. J. Burke & M. C. Mattis (Eds.), Women and minorities in science, technology, engineering and mathematics: Upping the numbers (pp. 3\u201327). Northampton: Edward Elgar.CrossRef\n\nButts, M. M., Casper, W. J., & Yang, T. S. (2013). How important are work-family support policies? A meta-analytic investigation of their effects on employee outcomes. Journal of Applied Psychology, 98, 1\u201325.PubMedCrossRef\n\nCasadevall, A., & Handelsman, J. (2014). The presence of female conveners correlates with a higher proportion of female speakers at scientific symposia. mBio, 5, 1\u20134.CrossRef\n\nCeci, S. J., & Williams, W. M. (2007). Why aren't more women in science? Top researchers debate the evidence. Washington, DC: American Psychological Association.CrossRef\n\nChesley, N., & Moen, P. (2006). When workers care: Dual-earner couples' caregiving strategies, benefit use, and psychological well-being. American Behavioral Scientist, 49, 1248\u20131269.CrossRef\n\nCleary-Goldman, J., Malone, F. D., Vidaver, J., Ball, R. H., Nyberg, D. A., Comstock, C. H. et al. (2005). Impact of maternal age on obstetric outcomes. Obstetrics & Gynecology, 105, 983\u2013990.CrossRef\n\nCollins, R. L. (2011). Content analysis of gender roles in media: Where are we now and where should we go? Sex Roles, 64, 290\u2013298.CrossRef\n\nCorrell, S. J., Benard, S., & Paik, I. (2007). Getting a job: Is there a motherhood penalty? American Journal of Sociology, 112, 1297\u20131338.CrossRef\n\nEagly, A. H. (1987). Sex differences in social behavior: A social-role interpretation. Hillsdale: Erlbaum.\n\nEagly, A. H., & Karau, S. J. (2002). Role congruity theory of prejudice toward female leaders. Psychological Review, 109, 573\u2013598.PubMedCrossRef\n\nEagly, A. H., Wood, W., & Diekman, A. B. (2000). Social role theory of sex differences and similarities: A current appraisal. In: T. Eckes & H. M. Trautner (Eds.), The developmental social psychology of gender (pp. 123\u2013174). Mahwah: Erlbaum.\n\nEcklund, E. H., & Lincoln, A. E. (2011). Scientists want more children. PLoS One, 6(8), e22590.PubMedCentralPubMedCrossRef\n\nElliott, M. (2003). Work and family role strain among university employees. Journal of Family and Economic Issues, 24, 157\u2013181.CrossRef\n\nFouad, N. A., & Singh, R. (2011). Retention and attrition: The impact of workplace climate on women in the engineering field. Marine Technology, 48, 9\u201311.\n\nFouad, N., Fitzpatrick, M., & Liu, J. (2011). Persistence of women in engineering careers: A qualitative study of current and former female engineers. Journal of Women and Minorities in Science and Engineering, 17, 69\u201396.CrossRef\n\nGoldie Blox. (n.d.). http:\/\/\u200bwww.\u200bgoldieblox.\u200bcom\/\u200b. Accessed 1 April 2014.\n\nGoulden, M., Frasch, K., & Mason, M. A. (2009). Staying competitive: Patching America's leaky pipeline in the sciences. Berkeley: Berkeley Center on Health, Economic, & Family Security and the Center for American Progress.\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGreenwald, A. G., & Banaji, M. R. (1995). Implicit social cognition: Attitudes, self-esteem, and stereotypes. Psychological Review, 102, 4\u201327.PubMedCrossRef\n\nHammer, L. B., & Neal, M. B. (2008). Working sandwiched-generation caregivers: Prevalence, characteristics, and outcomes. The Psychologist-Manager Journal, 11, 93\u2013112.CrossRef\n\nHill, C., Corbett, C., & St. Rose, A. (2010). Why so few? Women in science, technology, engineering, and math. Washington, DC: AAUW.\n\nJean, V. A., & Payne, S. C., & Thompson, R. J. (June 2014 ). Are STEM departments family-supportive work environments? Paper presented at the work family researchers network conference. New York, USA.\n\nKanter, R. M. (1977). Men and women of the corporation. New York: Basic Books.\n\nKnobloch-Westerwick, S., Glynn, C. J., & Huge, M. (2013). The Matilda effect in science communication: An experiment on gender bias in publication quality perceptions and collaboration interest. Science Communication, 35, 603\u2013625.CrossRef\n\nKossek, E., Hammer, L., Thompson, R., & Burke, L. 2014. Leveraging workplace flexibility: Fostering engagement and productivity. Alexandria: SHRM Foundation report.\n\nLi, J. (2009). Academic couples and the economics of the co-location problem (Unpublished doctoral dissertation). University of Minnesota, Minneapolis, Minnesota.\n\nMadera, J. M., Hebl, M. R., & Martin, R. C. (2009). Gender and letters of recommendation for academia: Agentic and communal differences. Journal of Applied Psychology, 94, 1591\u20131599.PubMedCrossRef\n\nMadsen, S. R. (2008). On becoming a woman leader: Learning from the experiences of university presidents. San Francisco: Wiley.\n\nMajor, D. A., Davis, D. D., Germano, L. M., Fletcher, T. D., Sanchez-Hucles, J., & Mann, J. (2003). Managing human resources in information technology: Best practices of high performing supervisors. Human Resource Management, 46, 411\u2013427.CrossRef\n\nManchester, C. F., Leslie, L. M., & Kramer, A. (2010). Stop the clock policies and career success in academia. American Economic Review, 100, 219\u2013223.CrossRef\n\nManchester, C. F., Leslie, L. M., & Kramer, A. (2013). Is the clock still ticking? An evaluation of the consequences of stopping the tenure clock. ILR Review, 66, 3\u201331.\n\nMartel, R., Lane, D. M., & Willis, C. (1996). Male-female differences: A computer simulation. American Psychologist, 51, 157\u2013158.CrossRef\n\nMason, M. A., Wolflinger, N. H., & Goulden, M. (2013). Do babies matter? Gender and family in the ivory tower. New Brunswick: Rutgers University Press.\n\nMatos, K., & Galinsky, E. (2012). 2012 National study of employers. New York: Families and Work Institute. http:\/\/\u200bfamiliesandwork.\u200borg\/\u200bsite\/\u200bresearch\/\u200breports\/\u200bNSE_\u200b2012.\u200bpdf. Accessed 1 April 2014.\n\nMenez, R., Munder, I., & T\u00f6psch, K. (2001). Personnel recruitment and qualification in the IT \u00adsector\u2013first results of the on-line study BIT-S. In Proceedings of the conference \u00abInnovations for an e-society: challenges for technology assessment, ITAS Karlsruhe.\n\nMoen, P., Harris-Abbott, D., Lee, S., & Roehling, P. V. (1999). The Cornell couples and careers study. Cornell Employment and Family Careers Institute.\n\nMonsson, E. (2008). Motherhood, the elephant in the laboratory: Women scientists speak out. Ithaca: Cornell University Press.\n\nMorgan, H., & Milliken, F. J. (1993). Keys to action: Understanding differences in organizations' responsiveness to work-and-family issues. Human Resource Management, 31, 227\u2013248.CrossRef\n\nMorgan, W. B., Walker, S. S., Hebl, M. R., & King, E. B. (2013). A field experiment: Reducing interpersonal discrimination toward pregnant job applicants. Journal of Applied Psychology, 98, 799\u2013809.PubMedCrossRef\n\nNASA. (2013, November). Women and girls initiative. http:\/\/\u200bwomen.\u200bnasa.\u200bgov\/\u200b. Accessed 1 April 2014.\n\nNational Center for Biotechnology Information. (2013). Infertility: Inability to conceive; Unable to get pregnant. http:\/\/\u200bwww.\u200bncbi.\u200bnlm.\u200bnih.\u200bgov\/\u200bpubmedhealth\/\u200bPMH0002173\/\u200b. Accessed 1 April 2014.\n\nNational Institute of Health. (n.d.). NIH family-friendly initiatives. http:\/\/\u200bgrants.\u200bnih.\u200bgov\/\u200bgrants\/\u200bfamily_\u200bfriendly.\u200bhtm. Accessed 1 April 2014.\n\nNational Science Foundation. (n.d.). ADVANCE at a glance. http:\/\/\u200bwww.\u200bnsf.\u200bgov\/\u200bcrssprgm\/\u200badvance\/\u200b. Accessed 1 April 2014.\n\nNational Science Foundation, National Center for Science and Engineering Statistics. (2013). Women, minorities, and persons with disabilities in science and engineering: 2013. Special Report NSF 13-304. Arlington, VA. http:\/\/\u200bwww.\u200bnsf.\u200bgov\/\u200bstatistics\/\u200bwmpd\/\u200b. Accessed 1 April 2014.\n\nNoe, R. A. (2013). Employee training and development. New York: McGraw Hill.\n\nPitt-Catsouphes, M., Matz-Costa, C., & Besen, E. (2009). Age and generations: Understanding experiences at the workplace (Research Highlight 6). Chestnut Hill: The Sloan Center on Aging and Work at Boston College.\n\nPreston, A. E. (2004). Leaving science: Occupational exit from scientific careers. New York: Russell Sage.\n\nPritchard, P. A. (Ed.). (2006). Success strategies for women in science: A portable mentor. Boston: Elsevier.\n\nRice University. (n.d.). Rice University's future faculty workshop. http:\/\/\u200badvance.\u200brice.\u200bedu\/\u200bfuturefacultywor\u200bkshop\/\u200b. Accessed 1 April 2014.\n\nRobinson, J. G., & McIlwee, J. S. (1991). Men, women, and the culture of engineering. The Sociological Quarterly, 32, 403\u2013421.CrossRef\n\nRossler, S. V., & Taylor, M. Z. (2009). Why are we still worried about women in science? Academe. American Association of University Professors. http:\/\/\u200bwww.\u200baaup.\u200borg\/\u200barticle\/\u200bwhy-are-we-still-worried-about-women-science#.\u200bUvjq7vldXzg. Accessed 1 April 2014.\n\nScandura, T. A. (1992). Mentorship and career mobility: An empirical investigation. Journal of Organizational Behavior, 13, 169\u2013174.CrossRef\n\nSchein, V. E., & Mueller, R. (1992). Sex-role stereotyping and requisite management characteristics: A cross-cultural look. Journal of Organizational Behavior, 13, 439\u2013447.CrossRef\n\nSchiebinger L., Henderson, A. D., & Gilmartin, S. K. (2008). Dual-career academic couples: What universities need to know. Stanford: The Michelle R. Clayman Institute for Gender Research.\n\nSchroeder, J., Dugdale, H. L., Radersma, R., Hinsch, M., Buehler, D. M., Saul, J., et al. (2013). Fewer invited talks by women in evolutionary biology symposia. Journal of Evolutionary Biology, 26, 2063\u20132069.PubMedCrossRef\n\nSimard, C., Henderson, A. D., Gilmartin, S. K., Schiebinger, L., & Whitney, T. (2008). Climbing the technical ladder: Obstacles and solutions for mid-level women in technology. Michelle R. Clayman Institute for Gender Research, Stanford University, Anita Borg Institute for Women and Technology, Stanford, CA.\n\nSoe, L., & Yakura, E. K. (2008). What's wrong with the pipeline? Assumptions about gender and culture in IT work. Women's Studies, 37, 176\u2013201.CrossRef\n\nSpencer, S. J., Steele, C. M., & Quinn, D. (1999). Stereotype threat and women's math performance. Journal of Experimental Social Psychology, 35, 4\u201328.CrossRef\n\nStarnes, A. (2013, Jun). Eight organizations promote women in STEM. http:\/\/\u200bstemwire.\u200borg\/\u200b2013\/\u200b06\/\u200b04\/\u200beight-organizations-promote-women-in-stem\/\u200b. Accessed 1 April 2014.\n\nSteele, C. M., & Aronson, J. (1995). Stereotype threat and the intellectual test performance of African-Americans. Journal of Personality and Social Psychology, 69, 797\u2013811.PubMedCrossRef\n\nSteinpres, R. E., Anders, K. A., & Ritzke, D. (1999). The impact of gender on the review of the curricula vitae of job applicants and tenure candidates: A national empirical study. Sex Roles, 41, 509\u2013528.CrossRef\n\nSutton, P. D., Hamilton, B. E., & Mathews, T. J. (2011). Recent decline in births in the United States, 2007\u20132009 (NCHS data brief, no. 60). Hyattsville: National Center for Health Statistics.\n\nTrix, F., & Psenka, C. (2003). Exploring the color of glass: Letters of recommendation for female and male medical faculty. Discourse and Society, 14, 191\u2013220.CrossRef\n\nUniversity of Michigan. (2013). Committee on strategies and tactics for recruiting to improve diversity and excellence (STRIDE). http:\/\/\u200bsitemaker.\u200bumich.\u200bedu\/\u200badvance\/\u200bSTRIDE_\u200bCommittee. Accessed 1 April 2014.\n\nUniversity of Minnesota. (2013, October). Future STEM innovators. Discover. http:\/\/\u200bdiscover.\u200bumn.\u200bedu\/\u200bnews\/\u200bteaching-education\/\u200b3m-supports-us-stem-education-efforts-prepare-future-leaders. Accessed 1 April 2014.\n\nU.S. Census Bureau. (2009). Projections of the population by selected age groups and sex for the United States: 2010 to 2050. Washington, DC: U.S. Census Bureau, Population Division.\n\nValian, V. (1999). Why so slow? The advancement of women. Cambridge: MIT Press.\n\nVillablanca, A. C., Beckett, L., Nettiksimmons, J, & Howell, L. P. (2011). Career flexibility and family friendly policies: An NIH funded study to enhance women's careers in biomedical sciences. Journal of Women's Health, 20, 1485\u20131496.PubMedCentralPubMedCrossRef\n\nWagner, D. L. (1991). Eldercare: A workplace issue. In B.B. Hess & E.W. Markson (Eds.), Growing old in America (pp. 377\u2013387). Piscataway: Transaction Publishers.\n\nWhite House (2011). The White House and National Science Foundation announce new workplace flexibility policies to support America's scientists and their families. Retrieved August 5,2014 from: http:\/\/\u200bwww.\u200bwhitehouse.\u200bgov\/\u200bthepress-office\/\u200b2011\/\u200b09\/\u200b26\/\u200bwhite-house-and-national-science-foundation-announce-new-workplace-flexi. Accessed 1 April 2014.\n\nWolf-Wendel, L., Twombly, S. B., & Rice, S. (2004). The two-body problem: Dual-career-couple hiring practices in higher education. Baltimore: Johns Hopkins University Press.\n\nWoolstenhulme, J. L., Cowan, B. W., McCluskey, J. J., & Byington, T. C. (2012). Evaluating the two-body problem: Measuring joint hire productivity with a university. Unpublished manuscript.\n\nFootnotes\n\n1\n\nIn order to keep the assessment brief, respondents were asked to indicate their college but not their department. As such, STEM classification was determined by college rather than department. Because two of the colleges include both STEM and non-STEM departments as defined by the National Science Foundation, our STEM classification is likely over-inclusive, containing some responses from faculty belonging to non-STEM departments.\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_16\n\n# 16. Examining the Work\u2013Family Experience of Female Workaholics\n\nMalissa A. Clark1  , Angela A. Beiler2  and Lauren M. Zimmerman2\n\n(1)\n\nDepartment of Psychology and Institute for Women's Studies, University of Georgia, Psychology Building, 30602 Athens, GA, USA\n\n(2)\n\nDepartment of Psychology, University of Georgia, 30602-3013 Athens, GA, USA\n\nMalissa A. Clark (Corresponding author)\n\nEmail: clarkm@uga.edu\n\nAngela A. Beiler\n\nEmail: aabeiler@uga.edu\n\nLauren M. Zimmerman\n\nEmail: lzimm@uga.edu\n\nKeywords\n\nWorkaholismWork\u2013familyWomenOvertimeOverloadWorkloadWork demands\n\n## 16.1 Introduction\n\nEvidence from a variety of sources suggests that employees in the USA are working increasingly long hours, both in terms of annual hours worked (Murphy and Sauter 2003; Schor 2003) and average hours worked per week (Brett and Stroh 2003; Kuhn and Lozano 2008). Several factors may be contributing to these trends, including advances in technology (e.g., smart phones) which have the potential to blur the boundaries between work and home (Wajcman et al. 2008) and facilitate working outside the traditional 9\u20135 workday. Furthermore, there have been steady increases in women's participation in the workforce over time (Jacobs and Gerson 2004; U.S. Department of Commerce 2011), the percentage of dual-earner households (Galinsky et al. 2009), and the number of households where mothers are the sole or primary provider (Wang et al. 2013). These trends have caught the attention of researchers, leading to a number of studies examining the impact of increased work involvement on work\u2013family conflict.\n\nWork\u2013family conflict is a form of interrole conflict that occurs when role pressures from the work and home domains are mutually incompatible in some respect (Greenhaus and Beutell 1985). This conflict can be bidirectional in that one's work may interfere with one's family or home (work-to-family conflict; WFC) and one's family or home may interfere with one's work (family-to-work conflict; FWC) . This interference may manifest in a number of forms, including time-based (when time spent in one domain makes it difficult to fulfill responsibilities in the other domain), strain-based (when strain experienced in one domain hinders performance in the other domain), or behavior-based (when behaviors performed in one domain are incompatible with behaviors in the other domain). Examinations of overall work\u2013family conflict have strongly supported a relationship between work hours and increased work\u2013family conflict (Adkins and Premeaux 2012; Hughes and Parkes 2007; Matthews et al. 2012). For example, Adkins and Premeaux (2012) examined the relationship between work hours and both WFC and FWC at a variety of organizational levels, as well as examining potential moderators . They found a positive linear relationship between hours worked and WFC, and a curvilinear relationship between hours worked and FWC. These findings suggest an ability to adjust family roles to minimize interference with work after conflict exceeds a certain point, although that same flexibility does not exist in the opposite direction for work roles. Additionally, the study found that number of children moderates this relationship, such that individuals with more children experience greater levels of WFC, yet families with fewer children experience a stronger relationship between the number of hours worked and levels of WFC. Hughes and Parkes (2007) also found a positive relationship between work hours and work\u2013family conflict, as well as support for the moderating role of work-time control. Individuals with higher work-time control experienced lower levels of conflict, which in turn was negatively related to family satisfaction outcomes.\n\nAlthough the relationship between work hours and work\u2013family conflict has been examined in a number of contexts, conspicuously absent from the work\u2013family literature is an examination of the role of workaholism (a dispositional form of heavy work investment; see Snir and Harpaz 2012). Given that a key characteristic of workaholics is that they often spend excessive time at work or thinking about work (Scott et al. 1997), it is not surprising that research has found that workaholics tend to experience a variety of negative outcomes, such as greater work\u2013family conflict (e.g., Bakker et al. 2009) and poorer psychological and physical well-being (Andreassen et al. 2007; Taris et al. 2005). While workaholism may yield negative outcomes overall, the basic premise of this chapter is that workaholic women may experience a greater number of negative work\u2013family outcomes compared to their male counterparts, due to such factors as societal pressures suggesting that women should be more heavily invested in their families (Blair-Loy 2001) and the fact that women typically have more family responsibilities than men (Carli and Eagly 2012).\n\nIn this chapter, we first review the concept of workaholism, as well as its relationship with work\u2013family conflict. Next, we review the literature on gender roles and societal norms , and how these will uniquely impact work\u2013family outcomes of workaholism for women. Specifically, we propose that workaholic women face extra pressures compared to workaholic men, and may in fact suffer even greater negative consequences as a result of their workaholic behavior. These extra pressures are largely due to the conflict that women experience between their inner drive to work and traditional gender role expectations, which expect women to be primarily devoted to their families. We then present empirical data which tests some possible emotional mediators in the relationship between workaholism and work\u2013family conflict for working women. Finally, we discuss overall conclusions and an agenda for future research .\n\n## 16.2 Workaholism\n\nWorkaholism can be defined as \"an internal drive to work, in which an individual is consumed with thoughts and feelings about work, and he\/she spends excessive time working and\/or thinking about work (even when it is not required of them) to the exclusion of other important life roles\" (Clark et al. 2014). Workaholics are driven to work because of internal factors such as an obsession with working or feelings that one \"should\" be working (Oates 1971; Schaufeli et al. 2008a), rather than external factors such as organizational requirements or financial needs (Sussman 2012). Contrary to what is often discussed in the popular press , workaholics are not driven to work because of an internal passion or love of working (Clark et al. 2014; Mudrack 2006). That particular group of individuals is more appropriately called engaged workers (Schaufeli et al. 2008b) or devoted workers (Snir and Harpaz 2012). Instead, we (and others) argue that it is more appropriate to conceptualize workaholics as individuals who are essentially addicted to work, and that are consumed by feelings of guilt that one \"should\" be working as opposed to working because of a love or passion for work (Clark et al. 2014; McMillan and O'Driscoll 2006; Schaufeli et al. 2008b). Workaholics often continue working even when their behavior leads to negative outcomes (Sussman and Sussman 2011). They often miss family events, work evenings, and blur the lines between work and home by bringing their work home with them or choosing recreational activities that advance or complement their work (Bonebright et al. 2000; Ng et al. 2007) .\n\nGiven that workaholics tend to spend an exorbitant amount of time at work at the expense of their family obligations (Scott et al. 1997; Sussman and Sussman 2011), it is not surprising that workaholism has been linked to greater work\u2013family conflict. This relationship between workaholism and work\u2013family conflict can be explained by both resource drain theory as well as spillover theory (see Edwards and Rothbard 2000). Resource drain theory suggests that when individuals expend personal resources (e.g., time, energy) in one domain (e.g., work), this depletes the available resources in another domain (e.g., family). Thus, workaholics may be more likely to experience WFC because they are expending a large amount of personal resources at their job. For example, a workaholic parent who spends a large amount of cognitive resources focusing on work tasks and projects during the day may be too exhausted to help their children complete their homework in the evening. Spillover theory, on the other hand, suggests that one's mood, skills , values , and behaviors experienced in one domain may spill over and impact (either positively or negatively) one's functioning in the other domain. Thus, negative emotions felt at home (e.g., guilt felt by workaholics when they are not working) may spill over and negatively affect them at work . Conversely, negative emotions felt at work (e.g., anxiety due to a workaholic's compulsive nature) may spill over and negatively affect them at home. Several studies have supported these ideas that workaholism is related to greater work\u2013family conflict (e.g., Aziz et al. 2010; Bonebright et al. 2000; Taris et al. 2005). Furthermore, the positive relationship between workaholism and work\u2013family conflict has been found in many different countries which have varied cultural norms and expectations toward work, including the USA (Aziz et al. 2010), Norway (Andreassen et al. 2013), the Netherlands (Bakker et al. 2009), Spain (del Libano et al. 2012), Italy (Guglielmi et al. 2012), and Japan (Shimazu et al. 2011) .\n\nWorkaholism has also been shown to negatively affect the family unit. Specifically, research has shown that workaholism results in work\u2013family crossover, where stress experienced by one individual crosses over to his or her spouse (Westman 2001). For example, Bakker et al. (2009) found that partners of workaholics reported receiving less social support, which in turn was related to lower partner relationship satisfaction. Additionally, a recent meta-analysis found that workaholism is related to lower family satisfaction and functioning and greater marital estrangement (Clark et al. 2014). Taken together, these studies suggest that not only does the workaholic individual suffer negative consequences, but the family of the workaholic also suffers.\n\nRecent meta-analytic evidence (Clark et al. 2014) as well as primary studies (Burke 1999; Graves et al. 2012) have found little to no evidence for gender differences in self-reported workaholism. In other words, men and women may be equally likely to be workaholics. However, it is interesting to note that gender has been found to moderate the relationship between age and workaholism , suggesting that men and women may have different lifespan trajectories of workaholic behaviors (Clark et al. 2014). Specifically, using meta-analytic data and assessing gender as the proportion of women in the sample, Clark et al. found that in samples with the highest proportions of women, there was a positive correlation between age and workaholism , whereas in samples containing the largest proportions of men, there was a negative relationship between age and workaholism. This suggests that for men, workaholic behavior may decrease over one's lifespan, while for women, workaholic behavior may increase over one's lifespan. However, a direct test of this has not been conducted to date. Unfortunately, there has been little research on gender differences in workaholism , and no known studies have examined the relationship between gender role and workaholism.\n\nIn the next section, we propose that because of existing societal pressures suggesting that women should be more heavily invested in their families (Blair-Loy 2001), coupled with the fact that women typically have more family responsibilities than men (Carli and Eagly 2012), women will experience more negative work\u2013family outcomes from their workaholic behavior than will men. We now turn to theory supporting this proposition .\n\n## 16.3 Gender Roles and Societal Norms\n\nTo date, studies of workaholism have focused primarily on how someone's workaholic nature impacts various outcomes (e.g., personal, family, work). However, a basic premise of this chapter is that men's and women's experiences as a workaholic may differ in consequential ways. Specifically in regard to work\u2013family conflict, we argue that the research to date has focused exclusively on how the noticeable characteristics of workaholism (e.g., the excessive number of hours worked or the persistent compulsion to work) influence work\u2013family conflict. To our knowledge, one factor that has not been considered to date in the workaholism and work\u2013family literature is whether a workaholic's perceptions and experiences of work\u2013family conflict may be due to pressures from gender role expectations (Blair-Loy 2001).\n\nAccording to traditional gender role expectations, men are responsible for acting as the main financial contributors for the family, allowing for a greater amount of attention to be paid directly to the work role rather than splitting time and resources between work and family roles. For men, being largely devoted to work may be viewed as an expected behavior and part of the \"good provider\" role, wherein men work to provide a stable income for the family (Bernard 1981). Because paid work has been traditionally viewed as men's main contribution to the family unit, a high investment in work is not likely to be viewed by men as having a strong impact on their perceived levels of work\u2013family conflict (Gutek et al. 1991). Women, on the other hand, face unique struggles as they strive to balance their inner drive to be heavily involved in work, while also conforming to traditional gender role expectations that women should be intensely committed to their families. Further support for these differing gender expectations has been offered by Groysberg and Abrahams (2014), whose surveys of executives found that\u2014regardless of gender\u2014these employees felt that the responsibility of handling tensions between work and home to be primarily a women's problem. In addition to the internal struggle between their devotion to work and the pressures of gender role expectations , women may also be chastised by family, friends, and individuals in the community for their heavy involvement in work at the expense of their families (Gutek et al. 1991; Ridgeway 2011). All of these factors combined may lead to higher perceived levels of work\u2013family conflict for women.\n\nIn her book, Competing Devotions: Career and Family among Women Executives, sociologist Mary Blair-Loy (2003) presents a fascinating description of the struggles faced by executive women, framing their struggles in terms of a clash between cultural work and family devotion schemas. According to Blair-Loy, cultural work and family devotion schemas are frameworks that individuals utilize to view, filter, understand, and appraise the present reality. Through time, schemas are developed by societies and become internalized within individuals to shape their personal goals, identities, and desires. In addition to fostering individual aspirations, cultural schemas are collectively shared to shape social structures within organizations and families. Specifically, Blair-Loy's (2001) work and family devotion schemas represent \"ideal types\" of individuals that influence how individuals spend their time, energy , and passion, that is particular either to their work or family.\n\nThe devotion to work schema emerged during the twentieth century when the pressure of capitalism encouraged men to immerse themselves in their professional careers, in order to secure a good income that allowed their wives to stay home and devote themselves to caring for the family. In addition to providing financial rewards , the work devotion schema promised individuals a sense of competence, identity, belonging, and meaning from their work (Blair-Loy 2003). On the other hand, the family devotion schema ascribes women to care for the home and family, while demanding complete allegiance to the role of being a mother. Additionally, this schema mandates a heterosexual marriage in which wives rely on their husbands for social status and financial sustenance, while husbands rely on their wives for physical and emotional support for themselves and their children (Blair-Loy 2001). In return for complying with these mandates, the family devotion schema provides women with a sense of fulfillment, meaning, creativity, and intimacy from raising their children. At the work and family devotion schemas' inception, these schemas were complementary; they prescribed husbands to care for the family financially through an intense devotion to their work, which allowed women to embrace the family devotion schema by staying at home with the children, and since women cared for the home and children, it also enabled husbands to fully embrace the work devotion schema (Blair-Loy 2003). However as more women embrace the work devotion schema by dedicating themselves to their work, the work and family devotion schemas have shifted from complementary devotion schemas to competing devotion schemas (Blair-Loy 2003).\n\nAs the number of dual-earner (Jacobs and Gerson 2004) and female-as-breadwinner (Wang et al. 2013) households continue to steadily increase, women are increasingly likely to experience competing work and family devotions. Specifically, competition between the work and family devotion schemas arises when women embrace the traditionally masculine work devotion schema to pursue professional careers, which is in opposition to the traditionally feminine family devotion schema that women are expected to embrace. This illustrates a violation of Eagly's (1987) social role theory of sex differences, which asserts that individuals tend to engage in activities that align with their culturally defined gender roles, due to external social pressures and individual internalized cultural gender expectations that intrinsically motivate individuals to engage in gender role congruent behaviors. However, when women immerse themselves in their work, they are engaging in behaviors that are incongruent with traditional gender role behaviors requiring women to devote all their time to their children in order to be \"good\" mothers.\n\nOne implication of women investing heavily in the work devotion schema is that they will likely experience higher levels of both WFC and FWC due to expectations that they are active and perform effectively in both work and family roles. Their male counterparts, however, may not experience the same pressure to take on responsibilities in the family domain, allowing them to invest more heavily in their work role. From a gender role perspective, women's experience of work\u2013family conflict can be conceptualized at a deeper moral level as the conflict between what it means to be a good employee and what it means to be a good parent (Blair-Loy 2003). Specifically, WFC is likely to arise when workaholic women first have child(ren), which evokes the family devotion schema to fully devote oneself to caring for the home and child(ren), even though they have previously embraced the work devotion schema due to their inner drive to work (Ridgeway 2011). However, men who are workaholics are not as likely to experience the same level of WFC as women because upon the arrival of children, the masculine work devotion schema of providing financially for the family is reemphasized, which mandates a contribution to the family by working as opposed to physically caring for the children (Ridgeway 2011). Accordingly, studies have shown that men tend to work more hours upon the arrival of children, while women tend to work fewer hours once they have children (Paull 2008). Therefore, as workaholic women juggle competing work and family demands, their long work hours and persistent inner drive to work may not allow them to fulfill all of their responsibilities in their family roles, resulting in WFC.\n\nIn addition to experiencing higher levels of WFC in comparison to men, workaholic women are also likely to experience increased levels of FWC. As noted by Ridgeway (2011), working women encounter \"the maternal wall\" in the workplace when they have children and are no longer viewed by the organization as the \"ideal\" worker who is agentic and assertive. Instead, they are now identified also as a mother who is communal and sensitive, which clashes with the \"ideal\" worker. One result of the \"mother\" image in the workplace is that working mothers tend to be viewed by organizations as lower-status, less valued, and less competent employees, due to the demands of motherhood impeding their effort to perform at work. Therefore, women experience inner conflict between conforming to the ideal worker, who is fully committed to and available for work, and the ideal mother, who is completely dedicated to and always there for her family and children. At the same time, women are also being held to higher standards by organizations to prove their ability and commitment to their work, since they are undermined as working mothers (Ridgeway 2011). Once again, men who are workaholics are not subject to opposition in the ideal worker and father roles, because the father role prescribes men to care financially for the family by working (Ridgeway 2011). Collectively, these experiences that women encounter, particularly the higher commitment and ability standards set by organizations, may hinder workaholic women's abilities to fulfill all of their responsibilities in their work roles since they have to work harder to meet the higher standards, leading to greater FWC.\n\nAn additional contributing factor to workaholic women's heightened experiences of WFC and FWC in comparison to men emanates from the increasing prevalence of organizations' \"gendered\" cultures (Cahusac and Kanji 2014; Lovejoy and Stone 2011). As more organizations promote the traditional linear career model by encouraging continuous work and upward mobility, organizational cultures are becoming more gendered with greater advantages for men. Specifically, men typically ascribe to the traditional linear career model due to their stable devotion to work, even upon the arrival of children. This consistency provides men with greater power in the organization in comparison to women, who may forgo the traditional linear career model by taking a break from employment to have children. Men's greater organizational power is reflected in increased opportunities for advancement, expansive social networks, and up-to-date skills due to their continual work, while women who have taken employment breaks (e.g., due to children) may have fewer advancement opportunities, smaller social networks, and greater skill deterioration, contributing to this deficit in power (Cahusac and Kanji 2014). Furthermore, as organizations encourage their employees to adopt the traditional linear career model that emphasizes long work hours to attain success, women experience conflict between the organization's push to work countless hours and the gender role expectation for women to devote time to the family. Therefore, women experience greater WFC and FWC due to the push and pull between work and family expectations as organizations promote the traditional career model that discourages women's employment breaks and results in diminished organizational power as women seek to return to work.\n\nIn addition to the inner conflict that workaholic women experience from their own competing work and family devotion schemas, these women are also likely to experience opposition from family, friends, and individuals in the community for their devotion to work at the (perceived) expense of their families. These external pressures arise when other individuals hold gender beliefs that are incongruent with the employee's gender beliefs and they express their disapproval (Ridgeway 2011). For example, even when someone has a supportive partner, individuals outside of the immediate family unit may still hold traditional work and family devotion schemas (Blair-Loy 2001), which prescribe men to care for the family by working and women to care for the family by managing the home and raising the children. Therefore, if a workaholic woman leaves her husband in charge of caring for the children in the evening so that she can complete a project for work, individuals outside of the immediate family unit may chastise her (whether confrontationally or more subtly) because it violates traditional work and family devotion schemas. Thus, workaholic women not only experience pressure and conflict from their internal struggles with work and family roles, but they also experience external pressure from other individuals.\n\nWorkaholic women face general societal norms that encourage greater participation in the home rather than work domain, potential inner conflict regarding their role as a \"good parent\" versus a \"good employee\", and possible outside pressure from individuals who do not agree with their decision to immerse themselves in the work environment . For these reasons, we propose that workaholic women will experience higher levels of negative outcomes than workaholic men, including greater work\u2013family conflict. These negative outcomes may include both self-inflicted outcomes (via guilt) and negative outcomes inflicted by others (including guilt, societal pressures, etc.). In order to further understand the emotional experiences of workaholic women, we conducted an empirical study exploring the discrete emotions experienced by women both at work and at home, and how these discrete emotions mediate the relationship between workaholism and work\u2013family conflict.\n\n## 16.4 An Empirical Examination of Mood Spillover for Workaholic Women\n\nRecall that work\u2013family mood spillover theory (see Edwards and Rothbard 2000) suggests that the negative emotions that individuals experience at work or home may spill over and negatively impact functioning in the other domain. For example, after a frustrating day at work, this negative mood may influence one's mood at home and in turn negatively affect one's ability to interact in a positive and fulfilling way with one's spouse or children. In addition to traditional sources of negative moods at work or home (e.g., job demands, angry customers, family demands, conflict with spouse), we propose that workaholic women may experience additional stressors (e.g., pressure from her family members to devote more of her time and energy to family, self-imposed expectations of conforming to societal expectations of what it means to be a good mother), which may trigger a variety of negative emotions that her male counterparts may not experience to the same degree. For example, a woman who is a workaholic may experience a great amount of guilt and anxiety while she is at work because she is choosing not to adhere to the family devotion schema (Blair-Loy 2003). Men, on the other hand, are encouraged to contribute to the family via financial support. Therefore, they may not experience these same emotions when they are working.\n\nTo test these ideas, we conducted an empirical study involving 215 working women that explores the mediating role of negative emotions in the relationship between workaholism and work\u2013family conflict. Specifically, we tested a model which examines whether negative emotions (i.e., guilt, anxiety, anger, and disappointment) at work and home mediate the relationship between workaholism and WFC\/FWC.\n\nThe women involved in this study had an average age of 45 years (SD = 10.1) and worked an average of 37.7 hours (SD = 8.1) per week. Ninety-two percent of the women were married or in a serious committed relationship and 84 % had at least one child. Turning to the results of the two models, we first discuss the WFC model in which women's experience of negative emotions at work (i.e., guilt, anxiety, anger , and disappointment) mediate the relationship between workaholism and WFC (Fig. 16.1). Overall, this model fit the data very well [\u03c7\u00b2 (9) = 14.30, p = 0.11; CFI = 0.99; TLI = 0.96; RMSEA = 0.05; SRMR = 0.03] and accounted for a significant amount of the variance in WFC (R 2 = 0.38, p < 0.001).\n\nFig. 16.1\n\nMultiple-mediation model of workaholism on work-to-family conflict through negative emotions at work\n\nNext, we discuss the specific relationships between workaholism, negative emotions at work, and WFC. First, there was a positive relationship between women's workaholic behaviors and WFC (\u03b2 = 0.23, p < 0.001). Therefore, women with higher levels of workaholism experienced higher levels of WFC as compared to women with lower levels of workaholism. Next, we examined the mediating role of women's negative emotions experienced at work in the relationship between workaholism and WFC. Mediation results revealed that workaholism was positively related to work guilt (\u03b2 = 0.17, p < 0.05), work anxiety (\u03b2 = 0.31, p < 0.001), and work disappointment (\u03b2 = 0.30, p < 0.001); however, the relationship between workaholism and anger experienced at work was not significant (\u03b2 = 0.13, p = 0.07). Only work anxiety (\u03b2 = 0.23, p < 0.001) was positively related to WFC. Additionally, specific indirect effect analyses indicated that work anxiety (indirect effect = 0.08, CI0.95 = 0.03, 0.13) mediated the relationship between workaholism and WFC. Therefore, women's experience of anxiety at work is one mechanism through which workaholism can spill over and negatively impact women's functioning at home.\n\nMoving on, we now discuss results from the FWC model, in which women's experiences of negative emotions at home (i.e., guilt, anxiety, anger, and disappointment) mediate the relationship between workaholism and FWC (Fig. 16.2). In general, this model fit the data well [\u03c7\u00b2 (9) = 20.65, p < 0.05; CFI = 0.97; TLI = 0.91; RMSEA = 0.08; SRMR = 0.04] and accounted for a significant amount of the variance in FWC (R 2 = 0.29, p < 0.001). There was a positive relationship between workaholism and FWC (\u03b2 = 0.27, p = 0.001), which suggests that women exhibiting higher levels of workaholism experience higher levels of FWC, compared to women exhibiting lower levels of workaholism.\n\nFig. 16.2\n\nMultiple-mediation model of workaholism on family-to-work conflict through negative emotions at home\n\nWe next examined the possible mediating effects of negative emotions in the relationship between workaholism and FWC. Results show that workaholism was positively related to home anxiety (\u03b2 = 0.17, p = 0.05), home anger (\u03b2 = 0.18, p = 0.01), and home disappointment (\u03b2 = 0.16, p = 0.01); however, workaholism was not significantly related to home guilt (\u03b2 = 0.12, p = 0.14). Only home anger (\u03b2 = 0.21, p < 0.05) was positively related to FWC, indicating that the greater a woman's anger at home, the higher her levels of FWC. However, the indirect effect for home anger was not significant (indirect effect = 0.04, CI0.95 = \u22120.01, 0.09).\n\nCollectively, these results provide important insights into the relationships between workaholism, negative emotions experienced at work and home, and work\u2013family conflict. In regard to the work-to-family model, women's workaholism exhibits a direct and positive relationship with WFC. Furthermore, when examining women's experiences of negative emotions at work as an underlying mechanism of this relationship, women's anxiety is an important mediator. Therefore, as women engage in more workaholic behaviors they tend to experience more anxiety at work, which in turn relates to greater amounts of WFC, possibly through mood spillover. Specifically, negative moods experienced in one domain can spill over and negatively influence performance and produce negative moods in another domain (Edwards and Rothbard 2000). Women may feel anxiety while at work because they are not adhering to the family devotion schema (Blair-Loy 2003). However, because we assessed anxiety at work in general, rather than anxiety specifically about being at work and not at home, future research is needed to further understand this relationship.\n\nSimilarly, results from the family-to-work model indicated a direct and positive relationship between women's workaholism and FWC, while the inclusion of women's experience of negative emotions at home also served as a mechanism underlying this relationship. Specifically, as women display higher levels of workaholism they tend to experience more anger at home, which then relates to higher levels of FWC. One reason for our findings regarding anger at home could be that workaholic women are more severely impacted by resource drain than their male counterparts, due to the fact that women have more responsibilities in the home than men (Carli and Eagly 2012). This in turn may lead to resentment and anger , possibly toward one's spouse, who does not have as many family obligations, or anger about societal expectations that the majority of family responsibilities falls firmly on the woman's shoulders. Again, future research is needed in order to further examine this finding. One surprising finding from the family-to-work model is that women's workaholism was not significantly related to their experience of guilt at home, which is a prominent characteristic of workaholism (Ng, Sorenson, and Feldman 2007). However, it is important to note that the model examined the role of each mediator in the context of the other mediators in the model. Examinations on the bivariate level revealed that workaholism was significantly related to higher levels of guilt experienced at home (r = 0.15, p < 0.05), which suggests that it is still important to consider the role of guilt when investigating the emotional experiences of workaholic women .\n\n## 16.5 Conclusion\n\nIn this chapter, we have proposed that workaholic women experience specific and unique work\u2013family outcomes as compared to workaholic men. Specifically, we argue that workaholic women face extra pressures and may in fact suffer from even greater negative consequences as compared to workaholic men, due to the conflict between their inner drive to work and traditional gender role expectations . Men, on the other hand, are expected to be devoted to work and the providers for their families based on traditional gender role expectations. Thus, a man's experience as a workaholic is likely to be very different than a woman's experience. The current empirical study examined the relationships between workaholism , discrete emotions, and work\u2013family conflict with a sample of women. Our findings provide evidence that workaholism directly impacts work\u2013family conflict, and that this relationship is mediated by anxiety at work and, to some extent, anger at home. Additionally, this is the first known study to specifically examine the emotional mechanisms through which workaholism influences work\u2013family conflict specifically for women. However, additional research is needed to fully understand the internal and external factors underlying these negative emotions. In a society where working hours are extending and dual-earner family dynamics are becoming more prevalent, it is important for future research to more closely examine these relationships to minimize the unique negative impact felt by workaholic women.\n\nThere are a number of future research directions that could aid in the study of workaholic women. More specifically, qualitative studies or experience sampling (ESM) studies specifically targeting the experiences of women who are workaholics could allow for much greater insight into their perceptions and experiences by examining the dynamic nature of their experiences over time. Additional research could parse out specific examples of self-inflicted (e.g., personal guilt) and other-inflicted (e.g., gender and societal norms) pressures to see how the type of pressure influences the manner in which workaholic women experience negative emotions.\n\nIn addition to pursuing future research on the unique experiences of workaholic women, it is vital to address the ways in which these women may cope with their workaholic behaviors, especially in regard to reducing work\u2013family conflict. From a general managerial perspective, Haas (1991) urged managers to assist their employees in altering their workaholic behaviors by referring them to employee assistance or recovery programs for treatment, which includes encouraging workaholics to delegate their work and take scheduled breaks from work. As workaholic women endure the tug-of-war between work and family demands, as well as gender role and societal pressures, we encourage women to seek out support groups for women balancing work and family. These specialized groups for women should expand beyond Workaholics Anonymous groups (e.g., www.workaholics-anonymous.org) by providing women with a network of support that targets the unique pressures women experience as they strive to balance their work and family. Another avenue that workaholic women may pursue to reduce work\u2013family conflict is family involvement and counseling, to determine if family life is contributing to their workaholic behaviors (Burke 2000). Furthermore, family initiatives may enhance satisfaction with time spent with the family, thus reducing work\u2013family conflict.\n\nIn sum, more research is needed in order to understand the relationship between workaholism and work\u2013family conflict, particularly research that focuses on the unique experiences of workaholic women. Research on workaholic women and their experiences of work\u2013family conflict is particularly important given that both work hours and levels of work\u2013family conflict are on the rise (e.g., Brett and Stroh 2003; Siegel et al. 2005), coupled with the increase in dual-earner households and households with women as the sole or primary provider (Galinsky et al. 2009; Wang et al. 2013). We encourage researchers to apply some of the sociological theories discussed in this chapter to the study of work and family, including (although certainly not limited to) thinking about work\u2013family conflict as a moral dilemma involving competing work and family devotions (see Blair-Loy 2003). This alternative conceptualization of work\u2013family conflict is relevant not only to understanding the specific experiences of workaholic women, but also to understanding the experiences of any working woman.\n\nReferences\n\nAdkins, C. L., & Premeaux, S. F. (2012). Spending time: The impact of hours worked on work family conflict. Journal of Vocational Behavior, 80, 380\u2013389.CrossRef\n\nAndreassen, C. S., Ursin, H., & Eriksen, H. R. (2007). The relationship between strong motivation to work, \"workaholism\", and health. Psychology and Health, 22, 615\u2013629.CrossRef\n\nAndreassen, C. S., Hetland, J., & Pallesen, S. (2013). Workaholism and work\u2013family spillover in a cross-occupational sample. European Journal of Work and Organizational Psychology, 22, 78\u201387.CrossRef\n\nAziz, S., Adkins, C.T., Walker, A.G., & Wuensch, K.L. (2010). Workaholism and work\u2013life imbalance: Does cultural origin influence the relationship? International Journal of Psychology, 41, 72\u201379.CrossRef\n\nBakker, A. B., Demerouti, E., & Burke, R. (2009). Workaholism and relationship quality: A spillover-crossover perspective. Journal of Occupational Health Psychology, 14, 23\u201333.PubMedCrossRef\n\nBernard, J. (1981). The good-provider role: Its rise and fall. American Psychologist, 36, 1\u201312.CrossRef\n\nBlair-Loy, M. (2001). Cultural constructions of family schemas: The case of women finance executives. Gender and Society, 15, 687\u2013709.CrossRef\n\nBlair-Loy, M. (2003). Competing devotions: Career and family among women executives. Cambridge: Harvard University Press.\n\nBonebright, C. A., Clay, D. L., & Ankenmann, R. D. (2000). The relationship of workaholism with work\u2013life conflict, life satisfaction, and purpose in life. Journal of Counseling Psychology, 47, 469\u2013477.CrossRef\n\nBrett, J. M., & Stroh, L. K. (2003). Working 61 plus hours a week: Why do managers do it? Journal of Applied Psychology, 88, 67\u201378.PubMedCrossRef\n\nBurke, R. J. (1999). Workaholism in organizations: Gender differences. Sex Roles, 41, 333\u2013345.CrossRef\n\nBurke, R.J. (2000). Workaholism among women managers: Personal and workplace correlates. Journal of Managerial Psychology, 15, 520\u2013530.CrossRef\n\nCahusac, E., & Kanji, S. (2014). Giving up: How gendered organizational cultures push mothers out. Gender, Work, and Organization, 21, 57\u201370.CrossRef\n\nCarli, L. L., & Eagly, A. H. (2012). Leadership and gender. In J. Antonakis & D. Day (Eds.), The nature of leadership (2nd ed., pp. 417\u2013476). Thousand Oaks: Sage.\n\nClark, M. A., Michel, J. S., Zhadnova, L., Pui, S., & Baltes, B. B. (2014). All work and no play? A meta-analytic examination of the correlates and outcomes of workaholism. Journal of Management. Advance online publication. doi: 10.1177\/0149206314522301\n\ndel Libano, M., Llorens, S., Salanova, M., & Schaufeli, W. B. (2012). About the dark and bright sides of self-efficacy: Workaholism and work engagement. The Spanish Journal of Psychology, 15, 688\u2013701.PubMedCrossRef\n\nEagly, A.H. (1987). Sex differences in social behavior: A social-role interpretation. Hillsdale: Erlbaum.\n\nEdwards, J. R., & Rothbard, N. P. (2000). Mechanisms linking work and family: Clarifying the relationship between work and family constructs. Academy of Management Review, 25, 178\u2013199.\n\nGalinsky, E., Aumann, K., & Bond, J. T. (2009). Times are changing: Gender and generation at work and at home. New York: Families and Work Institute.\n\nGraves, L. M., Ruderman, M. N., Ohlott, P. J., & Weber, T. J. (2012). Driven to work and enjoyment of work: Effects of managers' outcomes. Journal of Management, 38, 1655\u20131680.CrossRef\n\nGreenhaus, J. H., & Beutell, N. J. (1985). Sources of conflict between work and family roles. Academy of Management Review, 10, 76\u201388.\n\nGroysberg, B., & Abrahams, R. (2014). Manage your work, manage your life: Zero in on what really matters. Harvard Business Review, 92(3), 58\u201366.\n\nGuglielmi, D., Simbula, S., Schaufeli, W. B., & Depolo, M. (2012). Self-efficacy and workaholism as indicators of the job demands-resources model. Career Development International, 17, 375\u2013389.CrossRef\n\nGutek, B. A., Searle, S., & Klepa, L. (1991). Rational versus gender role explanations for work\u2013family conflict. Journal of Applied Psychology, 76, 560\u2013568.CrossRef\n\nHaas, R. (1991). Strategies to cope with a cultural phenomenon\u2014workaholism. Business and Health, 36, 4.\n\nHughes, E. L., & Parkes, K. R. (2007). Work hours and well-being: The roles of work-time control and work family interference. Work & Stress, 21, 264\u2013278.CrossRef\n\nJacobs, J. A., & Gerson, K. (2004). The time divide: Work, family, and gender inequality. Cambridge: Harvard University Press.\n\nKuhn, P., & Lozano, F. (2008). The expanding workweek? Understanding trends in long workhours among U.S. men, 1979\u20132006. Journal of Labor Economics, 26, 311\u2013343.CrossRef\n\nLovejoy, M., & Stone, P. (2011). Opting back in: The influence of time at home on professional women's career redirection after opting out. Gender, Work, and Organization, 19, 631\u2013653.CrossRef\n\nMatthews, R. A., Swody, C. A., & Barnes-Farrell, J. L. (2012). Work hours and work\u2013family conflict: The double-edged sword of involvement in work and family. Stress and Health, 28, 234\u2013247.PubMedCrossRef\n\nMcMillan, L. H. W., & O'Driscoll, M. P. (2006). Exploring new frontiers to generate an integrated definition of workaholism. In R. J. Burke (Ed.), Research companion to working time and work addiction (pp. 89\u2013107). Northampton: Edward Elgar.\n\nMudrack, P. E. (2006). Understanding workaholism: The case for behavioral tendencies. In R. J. Burke (Ed.), Research companion to working time and work addiction (pp. 108\u2013128). Northampton: Edward Elgar.\n\nMurphy, L. R., & Sauter, S. L. (2003). The USA perspective: Current issues and trends in the management of work stress. Australian Psychologist, 38, 151\u2013157.CrossRef\n\nNg, T.W.H., Sorensen, K.L., & Feldman, D.C. (2007). Dimensions, antecedents, and consequences of workaholism: A conceptual integration and extension. Journal of Organizational Behavior, 28, 111\u2013136.CrossRef\n\nOates, W. E. (1971). Confessions of a workaholic: The facts about work addiction. New York: World Publishing.\n\nPaull, G. (2008). Children and women's hours of work. The Economic Journal, 118, F8\u2013F27.\n\nRidgeway, C.L. (2011). Framed by gender. New York: Oxford University Press.CrossRef\n\nSchaufeli, W. B., Taris, T. W., & Bakker, A. B. (2008a). It takes two to tango: Workaholism is working excessively and working compulsively. In R. J. Burke & C. L. Cooper (Eds.), The long work hours culture: Causes, consequences and choices (pp. 203\u2013225). Bingley: Emerald.\n\nSchaufeli, W. B., Taris, T. W., & van Rhenen, W. (2008b). Workaholism, burnout, and work engagement: Three of a kind or three different kinds of employee well-being? Applied Psychology: An International Review, 57, 173\u2013203.CrossRef\n\nSchor, J. (2003). The (even more) overworked American. In de Graaf, J. (Ed.), Take back your time: Fighting overwork and time poverty in America (pp. 6\u201311). San Francisco: Berrett-Koehler.\n\nScott, K. S., Moore, K. S., & Miceli, M. P. (1997). An exploration of the meaning and consequences of workaholism. Human Relations, 50, 287\u2013314.CrossRef\n\nShimazu, A., Demerouti, E., Bakker, A. B., Shimada, K., & Kawakami, N. (2011). Workaholism and well-being among Japanese dual-earner couples: A spillover-crossover perspective. Social Science & Medicine, 73, 399\u2013409.CrossRef\n\nSiegel, P. A., Post, C., Brockner, J., Fishman, A. Y., & Garden, C. (2005). The moderating influence of procedural fairness on the relationship between work\u2013life conflict and organizational commitment. Journal of Applied Psychology, 90, 13\u201324.PubMedCrossRef\n\nSnir, R., & Harpaz, I. (2012). Beyond workaholism: Towards a general model of heavy work investment. Human Resource Management Review, 22, 232\u2013243.CrossRef\n\nSussman, S. (2012). Workaholism: A review. Journal of Addiction Research and Therapy, S6, 4120.\n\nSussman, S., & Sussman, A. N. (2011). Considering the definition of addiction. International Journal of Environmental Research and Public Health, 8, 4025\u20134038.PubMedCentralPubMedCrossRef\n\nTaris, T. W., Schaufeli, W. B., & Verhoeven, L. C. (2005). Workaholism in the Netherlands: Measurement and implications for job strain and work\u2013nonwork conflict. Applied Psychology: An International Review, 54, 37\u201360.CrossRef\n\nU.S. Department of Commerce (2011). Women in America: Indicators of social and economic well-being. Prepared by the U.S. Department of Commerce Economics and Statistics Administration, for White House Council on Women and Girls.\n\nWajcman, J., Bittman, M., & Brown, J. E. (2008). Families without borders: Mobile phones, connectedness and work\u2013home divisions. Sociology, 42, 635\u2013652.CrossRef\n\nWang, W., Parker, K., & Taylor, P. (2013). Breadwinner Moms. Pew Research Center. http:\/\/\u200bwww.\u200bpewsocialtrends.\u200borg\/\u200b2013\/\u200b05\/\u200b29\/\u200bbreadwinner-moms\/\u200b.\n\nWestman, M. (2001). Stress and strain crossover. Human Relations, 54, 717\u2013751.CrossRef\n\u00a9 Springer International Publishing Switzerland 2015\n\nMaura J. Mills (ed.)Gender and the Work-Family Experience10.1007\/978-3-319-08891-4_17\n\n# 17. Family Life on the Fast Track? Gender and Work\u2013Family Trade offs Among Highly Educated Professionals: A Cross-Cultural Exploration\n\nMaria Stanfors1\n\n(1)\n\nCentre for Economic Demography, Lund University, P. O. Box 7083, 220 07 Lund, Sweden\n\nMaria Stanfors\n\nEmail: maria.stanfors@ekh.lu.se\n\nKeywords\n\nFast-track professionsGenderWork\u2013family conflictFertilityFlexibility\n\n## 17.1 Introduction\n\nDuring recent decades, the progress women have made in education and the labor market has been comprehensive throughout most industrialized nations . Not least does this apply to their inroads into the professions. Today, both men and women are attaining high-status degrees and entering fast-track professions such as law, medicine, and academia. This has changed the conditions for both family formation and pursuing a career. Generally, high levels of education and career orientation among women are associated with delayed and reduced involvement in family life , indicating that professional gains may be offset by familial losses (cf. Bertrand et al. 2010). One reason for this is that higher education, getting a degree, and starting a career takes time. Also, childbearing and career breaks are considered costly for the highly educated. Perhaps more important; however, is that while women have made substantial inroads into the professions, the career paths of some jobs have changed little. They are still inflexible and are arguably therefore largely incompatible with having a family, especially for women who tend to take on more caregiving responsibilities than men do.\n\nAlthough there is an extensive literature documenting the negative relationship between education and fertility, there is evidence of change. A recent study on the case in Sweden showed that couples where both partners were highly educated professionals were more likely to continue childbearing and less likely to separate than were other couples (Dribe and Stanfors 2010). Sweden has been in the forefront internationally when it comes to gender equity and work\u2013family policies targeting men as well as women, so there are clearly more opportunities to combine career and family in Sweden than in many other countries. However, there is also evidence from the USA indicating that highly educated women nowadays have more children (Shang and Weinberg 2013). This does not extend to the same degree to all the highly educated (Herr and Wolfram 2011; Wolfinger et al. 2010), and reports that have attracted a lot of media attention have suggested that highly educated women are \"opting out\" due to work\u2013family conflicts, choosing family over career (Belkin 2003; Stone 2008; Story 2005).1\n\nThe current chapter focuses on how fast-track professionals fare with respect to family life (more specifically continued childbearing). Given the long-term increasing trends in women's education and labor force participation, fast-track professionals are highly relevant to study from a work\u2013family perspective since they put work\u2013family support systems to the test. The career structures of many fast-track jobs were designed for men with stay-at-home wives and few family responsibilities, but these have not changed much over time. It should be noted that the work\u2013family issues that men and women in fast-track professions face are different from those faced by men and women in other jobs. For many fast-trackers, family formation coincides with defining moments on the job with respect to career growth, so the question arising is whether and how to build a family. As parents, the work\u2013family conflicts facing this group relate to scheduling and shortage of time rather than money. It is also relevant to document whether gender has an impact on family life among fast-trackers, because it puts into perspective the investments in education and career that women make nowadays. The demands of a professional career, the asymmetries of male\u2013female relationships, and the difficulties of conceiving later in life are supposed to undermine the combination of career and family for women but not for men. In this chapter, evidence from Sweden is explored and contrasted with the experiences from the USA. Sweden has been a precursor in terms of various aspects of gender equality, partly through its longstanding, strong orientation toward work\u2013family policies that target men as well as women. While Sweden has introduced extensive policies alleviating parents from work\u2013family conflicts, such policies are limited in the USA. If there is a certain pattern emerging in the combination of career and family among highly educated professionals, Sweden is one of the places where this should be noticed first, while indicating what might be emerging elsewhere.\n\n## 17.2 Background\n\n### 17.2.1 Women's Changing Economic Roles\n\nDuring the twentieth century, women became increasingly engaged in labor market activities, with the most dramatic increase in maternal employment taking place during the 1970s (Goldin 1990; Stanfors 2003). Today the dual-earner household model has overtaken the traditional male breadwinner model in Europe as well as in the USA (Bianchi and Raley 2005 on the USA; Drew et al. 1998 on Europe). Although female employment rates and the time spent by women, especially mothers, in paid work have increased substantially, it has been argued that their time in unpaid work has not declined enough to compensate for this (Bianchi 2000; Gauthier et al. 2004). Men have increased their time in unpaid activities, but this has also failed to compensate for the changing employment status of women since this increase over time has been small and has started from a very low level (Coltrane 2000; Gershuny and Robinson 1988). Quite commonly, it is argued, women are doing a \"second shift\" of unpaid work and they are more often short of time and have less of it to devote to leisure than do men (Berk 1985; Hochschild 1989; Sayer 2005). Another view has challenged the second shift, claiming that men's and women's work patterns have converged over the past 40 years, with women now performing more paid work and less unpaid work than before, and men performing more unpaid work and less paid work (Bianchi et al. 2006; Gershuny 2000). These changes have been facilitated by household technology and various services that have reduced unpaid domestic activities. The proponents of convergence have argued that the gender revolution set off in the 1970s has not stalled at all. Quite contrary, change will, according to this view, go on because younger men and women, especially those with a higher education, are likely to adopt more gender-equal ideals and adapt to a less traditional division of labor . The predictions as to whether experiences of work\u2013family conflict will extend to men are, however, not clear.\n\n### 17.2.2 Work\u2013Family Conflict\n\nAlthough many men and women had experienced conflicts between work and family life before 1970, the rise in dual-earner households at this time sparked substantial interest in this question, both academically and politically, putting work\u2013family issues on both the research and political agendas in many countries. That work interferes more with family commitments than vice versa may reflect the idea of the \"ideal worker,\" who prioritizes work over family (cf. Acker 1990; Hochschild 1997; Williams 2000). The image of the ideal worker stems from the era of the male breadwinner model but still lives on in the minds and visions of employers and a variety of organizational features and practices sustains the notion that work should trump other commitments (Ely and Meyerson 2000). It also lives on in the minds of many workers, especially in the fast-track professions where career structure rigidity and work demands such as long working hours compete with home demands. Having a fast-track career and a family is thus often seen as incompatible, not least for women.\n\nWork\u2013family conflicts occur when both work and family demand commitment, time, and energy . Quite commonly, periods when demands are the strongest in each domain\u2014peaks of family formation and career growth\u2014may coincide. Reconciling the demands of work and home is thus a challenge for many, but experiences of work\u2013family conflict are certainly most common among men and women belonging to dual-earner couples with (young) children at home (Bellavia and Frone 2005; Moen and Yu 1999). Parenthood is often considered a major cause behind gender differences in time allocation, especially between paid work and housework (Bianchi 2000; Sayer 2005 on the USA; Hallberg and Klevmarken 2003 on Sweden). The extent to which men's and women's time allocation is affected by parenthood varies according to individual, household, and contextual factors (Craig and Mullan 2010; Neilson and Stanfors 2014).\n\nWork\u2013family conflict mainly occurs in three respects stemming from time demands , scheduling demands , and work\u2013family spillover (Greenhaus and Powell 2003). To a certain extent work\u2013family conflict apply to all workers, but in some respects more to fast-track professionals than to others. Many workers feel stressed when they try to accommodate the demands of both work and family. A common strategy for solving this problem among women is to limit work demands by, for example, working part-time. Part-time work is, however, seldom compatible with having a career. Moreover, time shortage does not only occur as a result of work demands; people also face time demands at home because the basic upkeep of the household and the maintaining of personal relationships demand time.\n\nScheduling is another problem, which arises when a worker needs to be in two places at the same time. Late meetings or overtime when the child care center is closed are such examples, but activities relating to children, household maintenance, and personal care that occur during standard working hours also create a conflict. Lack of control over work schedules is a source of stress among working parents, especially mothers, and for this reason they seek flexible work schedules (Barnett and Rivers 1996; Goldin and Katz 2013). Scheduling demands are also the result of inflexible child care arrangements (Hofferth 1999).\n\nOne way of reconciling work and family demands is to bring work home. This has become more prevalent with the erosion of the male breadwinner model. In the heyday of the male breadwinner model, men were responsible for earnings and women were responsible for the family and home, and the concerns of each domain tended to stay separate. Today, both men and women have responsibilities in both the work and family domains. Advances in information technology have increased the ability of some workers to work from home, providing them with the flexibility to accommodate child care schedules. While many professionals find this option an advantage (Epstein et al. 1999), especially when they are parents with young children at home, it is double-edged in that it has increased the pressure on them to be constantly available outside the office (Glass 2000; Hochschild 1997; Hymowitz and Silverman 2001).\n\n### 17.2.3 Responses to Work\u2013Family Conflict\n\nCombining work and family is a pertinent issue. There is considerable variation across countries in terms of policies and institutions that affect the opportunities of combining work and family. When it comes to policy, attempts to resolve work\u2013family conflicts tend to center on the practical problems of combining work and family, particularly child care. More recently there have been attempts through collective agreements and the provision of technical services to facilitate flexibility and work at home for many groups of workers. Institutions differ between countries when it comes to parental leave schemes, income replacement, the working conditions of parents, the provision and pricing of child care, and whether families or individuals are the subjects of benefits and taxation. All these components are important, but most likely it is the combined effect of all these factors taken together that determines the degree of work\u2013family compatibility.\n\nAlmost every country provides its own specific set of solutions for the issue of combining work and family. In many countries policies only deal with the reconciliation of women's double roles, whereas in other countries, like Sweden, institutional arrangements have a broader scope and address gender equality issues alongside the work\u2013family balance for all parents (Kimball 1999; McDonald 2000). Sweden is often seen as a forerunner when it comes to family and work-related policies (Ruhm and Teague 1997). Part of the reason why countries differ in their responses to work\u2013family conflict is ideological.\n\nIn Sweden and the other Nordic countries, work\u2013family reconciliation is a public concern. There is a strong belief that society bears the responsibility for family well-being in that it will gain from investing in the next generation. At the other ideological endpoint, in the USA, work\u2013family reconciliation is a private concern; the responsibility for family well-being and finding ways to accommodate both work and family roles has first and foremost fallen on individuals. Thus Nordic policies are universalistic and general, and are in contrast to the USA where government-sponsored programs are rare and minimalistic and employer-sponsored programs only apply for some groups of employees, which does not solve work\u2013family conflict for most workers. Despite the fact that managers and professionals are among those privileged workers who most commonly have access to family benefits, assistance with child care, and flexible scheduling, they seem to have more problems combining work and family than do their Nordic counterparts (Jacobs and Fanning Madden 2004). Due to its universalistic and general design, all workers potentially benefit from the Swedish welfare state and its opportunities to combine work and family. In reality, however, the combination of work and family is a result of choices concerning both career and childbearing that have been made at the individual and couple level. For example, highly educated couples are more likely to make use of publicly provided child care and other services provided by the welfare state than are other couples (Stanfors 2003). This may be explained by their stronger work orientation, better information, and the fact that highly educated individuals have more gender-equal attitudes (Bernhardt and Goldscheider 2006). Parental leave, child benefits, and subsidized child care reduce the negative price effect of parents' and especially of women's wages on fertility, making it independent of education, and instead boost the income effect. In Sweden, public child care is of a high quality and fees are greatly subsidized, especially for high income earners.2 Higher income also enables people to buy goods and services in order to cope with both career and family (cars, housing, home services, etc.).\n\n## 17.3 Work\u2013Family Tradeoffs among Highly Educated Professionals\n\n### 17.3.1 Education and Fertility\n\nMuch interest has been devoted to the association between education and fertility, especially educational attainment and fertility (see Brand and Davis 2011, for a review). Typically, a negative relationship between education and fertility has been assumed. This assumption is based on standard human capital theory, which predicts that better-educated women will have their first births later in life and ultimately fewer children compared to less educated women. Women with a higher education postpone family formation because enrollment in education and earning a degree is demanding and not compatible with childbearing . After finishing their degrees, women with a higher education face higher opportunity costs of childbearing than do women with a lower education because they usually have better jobs, career prospects, and wages (Becker 1991; Mincer 1963). They are also, and more often than is the case with women with less education, in occupations with steeper earnings profiles and potentially faster depreciation rates (Polachek 1981). High opportunity costs in both real economic and career terms depress fertility among highly educated women, especially among those in fast-track professions. Whereas the opportunity cost of childbearing is supposed to outweigh the income effect of earnings potential among women, this is not the case among men. As long as fathers are not expected to give up the same amount of their working time for child care as mothers are, men's education and professional careers will not conflict with childbearing to the same extent as women's; instead they are expected to be positively rather than negatively related to fertility.\n\nWhereas several studies have shown that women's educational attainment has a negative effect on various childbearing outcomes, others have more recently found positive relationships between educational attainment and fertility after first birth (e.g., Dribe and Stanfors 2010; Joshi 2002; Kravdal 2001, 2007; Kreyenfeld 2002). This challenges the idea that opportunity costs outweigh a positive income effect among highly educated women. The emergent positive association between education and fertility may be explained by a strengthening of the income effect in more gender-equal contexts, where childbearing and employment are made more compatible, women are not primarily caregivers but workers, and highly educated women contribute substantially to household income. The income effect may be reinforced by assortative mating based on education. A positive association between education and higher order births may also be a result of self-selection when modeling birth intervals separately (Kravdal 2001).\n\n### 17.3.2 Occupation and Fertility\n\nIt has been shown that educational orientation may be an even more important determinant of fertility than educational level (Hoem et al. 2006; Lappeg\u00e5rd and R\u00f6nsen 2005; Stanfors 2003; Van Bavel 2010). Educational orientation works through occupational choice since these are closely related, especially for the professions. Should there be systematic differences when it comes to career tracks, work demands , and perceived work\u2013family conflict between occupations, the investment in different kinds of higher education will not only determine professional life but also choices with respect to personal life.\n\nFew studies have considered how career choice affects fertility. Existing studies deal mainly with the USA. Examples are studies of gender differences in childbearing among academics (Mason and Goulden 2004) and doctors (Boulis 2004) that conclude that women in the professions analyzed are less likely to have children than are other women or their male peers. Cooney and Uhlenberg (1989) analyzed the situation in 1980 for women only, finding that doctors felt more of a family commitment and were more likely to have children than were lawyers and post-secondary teachers. Wolfinger et al. (2010) built on Cooney and Uhlenberg by using data from the 2000 Census for both men and women. They confirmed that physicians had the highest rate of birth events, followed by attorneys, and academics. Men had, within each profession, more birth events than did women. The factors that explain fertility variation by profession for men (i.e., marital status, income, and spousal employment) only partially account for differences by profession for women. This may be related to the more complex relationship between family and career for women as documented by many (e.g., Avellar and Smock 2003; Budig 2003; Goldin and Katz 2008; Hewlett 2002; Noonan and Corcoran 2004).\n\n### 17.3.3 Career and Family Compatibility\n\nThe reason why a career and family may be seen as incompatible, at least for women, is because they both demand commitment, time, and energy , thereby inherently implying a tradeoff. Although women have increased their education and labor force participation, they have largely retained responsibility for child care and domestic work. For women on the fast track, family formation is often associated with taking up a second career as \"supermom\" and assuming the main responsibility of housework . For men, on the other hand, both paid and family work is usually less influenced by their role as fathers. However, during the last decades the effects of parenthood on time allocation in Scandinavia have started to converge between men and women (Dribe and Stanfors 2009). As seen above, there exists considerable variation across countries when it comes to policies and institutions which may affect the combination of work and family. On an aggregate level, it seems as if the institutional arrangements in Sweden, and its Scandinavian neighbors, are most favorable for dual-earner households (Blossfeld and Drobnic 2001) and are more supportive of combining employment and childbearing (e.g., Billari and Kohler 2004) than are conservative and liberal models with more traditional approaches to family and gender roles. In a universalistic and general welfare state, like Sweden, all parents potentially benefit from the opportunities to combine work and family. Parental leave, child benefits, and subsidized quality child care reduce the negative price effect of parents'\u2014especially women's\u2014wages on fertility , making it independent of education, and instead they boost the income effect. Once parenthood and employment are no longer alternatives at odds with one other but are instead possible to combine, the negative effects of women's education, labor force participation, and wages are expected to be greatly reduced, and even reversed, to produce a positive income effect. Thus, women's as well as men's education and earnings may impact positively on continued childbearing, although perhaps not equally across the professions.\n\nIn some occupations the career structure is rigid and the job inflexible (cf. Goldin and Katz 2011). In these the workload is high, hours are long, and production in a given number of (early career) years is most important. Moreover, skill depreciation is more of a problem in some occupations than in others. Expertise, firm-specific, and technological skills tend to depreciate faster than general skills, and this aggravates career interruptions . In some occupations the career structure is less rigid and working conditions are more flexible. This implies that those who work longer hours and produce more will be promoted at a faster pace than others, but it is possible to adjust the workload and reduce hours through part-time work , or even stop the \"career-clock\" in order to take time off for caring. Public sector work is generally more flexible than private sector employment and provides greater employment security. Steeper earnings profiles among private sector employees exacerbate the negative effects of career breaks on lifetime earnings in relation to the public sector. Thus, in order to cope, (potential) parents may choose occupations and careers that accommodate family responsibilities and reduce earnings punishment (cf. Polachek 1981).\n\nCareer structure rigidity and work inflexibility are likely to affect men and women in the professions differently. They are the legacies of organizational norms that were developed in times when men, married to full-time homemakers, were the only ones to be fast-track professionals. The norms that establish which time in life specific career goals should be achieved rest upon expectations about work-related productivity applying to a worker who does not have the responsibility to care for a home and a family. If different social norms and practices apply for men and women with respect to parenting, putting more pressure on women to be primary caregivers, and to take on more responsibility for housework than men, women will face more difficulties in combining a career and family.\n\n### 17.3.4 Differences Between Professions\n\nThe current chapter contrasts the patterns of continued childbearing among lawyers with those of medical doctors and PhDs working as post-secondary teachers and researchers in academia (i.e., higher education institutions). They are all high-status professions that demand higher education, specific degrees, and long training programs. By tradition, these professions have been male-dominated but are now becoming increasingly feminized. They are, to varying degrees, examples of a career model that is not conducive to childbearing if that implies assuming more childrearing responsibilities and greater involvement in parenting. There is thus the anticipation that occupational differences in childbearing will be stronger for women than for men. Primary evidence from Sweden is explored and contrasted with experiences from the USA, as documented by Wolfinger et al. (2010). While Sweden has launched extensive policies alleviating parents from work\u2013family conflicts, these are limited in the USA. There is thus the anticipation that occupational differences in childbearing will be more articulated in the USA than in Sweden.\n\nA few words are warranted here regarding the professional development of lawyers, doctors, and academics, and the extent to which there are differences between the professions and between Sweden and the USA.\n\nIn Sweden, law school is a 4-year program and medical school comprises 5 years. Entry into these vocational programs is highly restricted. After graduation, an essentially compulsory 2-year training period as a clerk or an intern follows. It is often seen as a defining moment in terms of career opportunities as to when and where this internship takes place. The majority of law and medical school graduates are in the age range of 25\u201329, the average age being slightly higher for men than for women, which means that they are around 30 when they finish as clerks or interns.\n\nA law graduate has several career options, one being to join a law firm and spend 5\u201310 years to make partner, which is challenging. It implies a less secure position in the company hierarchy. Failure to make partner implies a significantly slower career development and perhaps a change of law firms. Not all law graduates aspire to such a career, choosing instead alternatives routes such as to work in administration elsewhere. One alternative is the career track in the national courts administration with the aim of becoming a district attorney or judge. This is competitive, as are all professional careers, but different in character because of a more bureaucratic and standardized schedule for promotions long-standing in the public sector. There is no up-or-out decision. It is possible to slow down or stop one's career for a period of time when, for instance, caregiving demands are pressing. This means that mothers and fathers may become district attorneys and judges at a later point in life than colleagues that do not have similar family responsibilities, but they are not excluded from the competition. There are, however, wage differences that follow with the career choices , with law practitioners in the private sector earning considerably more than those working in the public sector.\n\nMedical school graduates who finish their internships may follow additional specialist training or work as internists. All doctors\u2014even interns\u2014are regularly employed during their residencies, the overwhelming majority being in publicly run hospitals or health care centers. Wages improve with each step up the hierarchical ladder. Due to the limited supply of doctors and a high demand for their competence, medical graduates can expect to rank among the top earning professional groups, not only among those employed in the public sector but also more generally. Medical school graduates achieve a fairly secure position with respect to both career and income at around age 30, which is conducive to childbearing . Job stability is good, although terms of contracts may vary. Irregular hours and night calls may be demanding, especially for parents with young children, but are solved partly through the provision of child care that is available for parents who work at odd hours of the day.3\n\nBecoming an academic is different from the other professional tracks. Doctoral-level education follows bachelor's and Masters degrees,4 which means that PhDs are older than lawyers and doctors when they graduate. PhD candidates are formally employed on fixed-term contracts. After receiving a PhD, they may either stay in academia (commonly on fixed-term contracts) or seek work elsewhere. Since the Swedish academic system has no tenure track, there is great variability among academic careers, and the challenge to publish or perish within a limited time period does not apply to all (cf. Wolfinger et al. 2010). It may take time until a PhD is in a position that is conducive to childbearing. For many, graduate school seems to be a better time to have children than the post-doctoral period, given work load and income stability . Academics receive, on average, less pay than lawyers and doctors. On the other hand, academics enjoy more flexibility when it comes to working hours and the opportunity to work at home than the other professions do. Like other public sector employees, there is no up-or-out decision with respect to an academic career in Sweden.\n\nThe most obvious differences between the professions studied in Sweden and the USA relate to academics. The career structure of academia in the USA offers no good time for childbearing . Academics who want to reduce working hours get reduced pay and generally resort to lower status jobs. Only a few universities offer employer-sponsored programs with a beneficial scheme for prospective parents. It seems that in the USA both medicine and law offer more opportunities for flexibility than academia (Goldin and Katz 2011, 2013; Wolfinger et al. 2010). Another obvious difference between Sweden and the USA is that many more fast-trackers are in public sector employment in Sweden, and generally this sector offers more family-friendly working conditions than does the private sector.\n\nBased on differences in working conditions, career structures, and professional demands between the professions, doctors are expected to be the most likely to have a second or a third birth once they have started childbearing, irrespective of gender. In Sweden, they are expected to be followed by academics and law practitioners in the national courts system (i.e., public sector employees). Lawyers working in the private sector are expected to be the least likely to continue childbearing, especially if they are women. Differences with respect to type of profession are expected to be stronger among women than among men, especially in professions where the career structure is unpredictable and not very flexible, and where there is more likely to be an \"up-or-out\" decision than in other professions. Differences according to profession and gender are thus expected to be fewer among medical doctors and greater among lawyers.\n\n## 17.4 Continued Childbearing Among Highly Educated Professionals\n\nIn order to establish whether there are differences by profession in continued childbearing among highly educated men and women, multivariate analyses have been performed. The focus is set on what happens after the birth of the first child among individuals who, already at the time of their first birth, have a degree and have started a career in medicine, law, or academia.5 The analysis is thus conditional on having a career and on the partnerships formed whereby at least one child has been born. The fast-track professionals studied have all made considerable career investments, yet follow professional routes that are not equally compatible with family life. The scope of the analyses is to determine which fast-track professions facilitate higher order childbearing for men and women.6\n\n### 17.4.1 Data\n\nThe data analyzed comes from the Swedish population registers maintained by Statistics Sweden.7 From a dataset consisting of all the individuals in the birth cohorts 1942\u20131989 who resided in Sweden at any time from 1961 onwards, individuals in heterosexual couples (married or cohabiting without being formally married) who were in their first partnership have been selected and followed from the birth of the first child, beginning in 1989 and up to the time of either the woman reaching the age of 45, the dissolution of the partnership, emigration, or the end of the study period in 2009. The analytical sample is limited to include individuals in couples who were in their first partnership and wherein either the man or the woman belonged to one of the professions studied. The sample is also truncated at 8 years since the previous birth. The analytical sample consists of 13,334 focal persons (7,428 men and 5,906 women).\n\nThe professions of interest are identified using a combination of educational and occupational codes (highest educational level attained the field of that education and branch of employment). Profession is a variable divided into lawyer working in a private law firm, law practitioner employed in the national courts and taxation system, medical doctor; and PhD in academia (e.g., post-secondary teacher and\/or researcher belonging to any academic field).\n\nThe dependent variable in all analyses is a higher order (more specifically a second and third) birth event. Independent variables are used to explain the relationship between profession and a higher order birth. Although the register data cover the entire population and are very reliable, they mostly include crude measures of demographic and economic indicators, few social and job-related indicators, and do not include any attitudinal variables. Independent variables used here include age, union status, partner's educational status and professional orientation, income,8 duration since last birth, age of woman at first birth, cohort, country of birth, and urbanicity.\n\nMore than half of the men and women studied were medical doctors. It is more common for men to be PhDs employed in academia. Among law professionals, men are equally distributed between private law and the national courts and taxation system, while women are to a larger extent employed in the courts administration than in private law firms.\n\nAs for the empirical analysis, logit models are estimated separately for men and women at each birth interval (i.e., second and third births). The results from the multivariate analysis are reported as odds ratios and displayed graphically, net of controls, for an intuitive yet statistically correct interpretation.9 It should be noted that there is a very strong two-child norm in Sweden, which is reflected in the data where a large majority of the births studied are second births.\n\n### 17.4.2 Empirical Findings\n\nEmpirical results displayed in Fig. 17.1 establish that doctors have the highest chances of having a higher order birth in relation to the other professional groups. This is true for both second and third births, and for women as well as for men. The patterning of second or third births by profession is stable for both men and women, yet there is a difference in pattern by profession and gender. Among women, lawyers in private law firms are the least likely to experience a second and third birth. For women in law, employment in the public sector seems to be more conducive to continued childbearing , all else equal. This is not the case for men who work as law professionals. Women in academia also have less chance of continued childbearing, having almost equally low odds ratios as women working in private law firms. Men, too, who work in academia have lower odds ratios of experiencing a second or a third birth as compared to both doctors as well as to lawyers in the private sector. The members of the group of \"job changers\", which consists of professionals who belonged to one of the four professional categories at the time of their first birth but have since then changed professional status, have quite consistently medium-low odds of experiencing a birth at both intervals, among men as well as among women. There is generally more variation according to profession among women than among men, but the main result is that doctors as a group stand out in relation to the other professional groups.10 The pattern by profession observed for second births is attenuated, particularly among women, for third births.\n\nFig. 17.1\n\nThe patterning of experiencing a second or third birth among fast-track professionals in Sweden 1991\u20132009, by gender. Odds ratios from logit estimates are indicated on the y-axis (1 = reference category). a Men. b Women. (Note: regressions control for own profession, partners education and professional orientation, own age, partner's age, their square terms, age of woman at first birth, couple income and its square term, woman's share of couple's income, time since last birth, cohort, country of birth for both partners, civil status, and place of residence; Source: Statistics Sweden, see text)\n\nOne interesting finding to note is that the relationship between professional orientation and continued childbearing is not driven by income. This is made visible in Fig. 17.2. Should the impact of professional status be felt through income; i.e., the fact that physicians as a group are more likely to have a second and a third birth because they earn more money, and thus can afford more children, then the exclusion of the income controls would make the impact of own profession a stronger one. The results indicate that virtually no differences in continued childbearing among fast-track professionals relate to income. When the odds ratios for second and third births of two separate models (one including income controls and one without such controls) are compared, there is only a slight change in odds ratios for academics and lawyers working in the national courts administration. This is understandable since academics and lawyers in the public sector earn on average less than medical doctors. Accordingly, they have somewhat lower chances of experiencing a second and a third birth than do physicians. The patterns are highly robust, so the main explanation for the observed patterns according to (1) profession and (2) profession and gender combined seems to come from underlying differences between the professions regarding, for example, career structures and flexibility\n\nFig. 17.2\n\nModel comparisons of the patterning of experiencing a second or third birth among fast-track professionals in Sweden, 1991\u20132009. Odds ratios from logit estimates of experiencing a birth, with and without controlling for couple income and woman's share of couple income. (1 = reference category). a Men, Second births. b Men, Third births. c Women, Second births. d Women, Third births. (Note: Based on model estimates controlling for all other variables, cf. Fig. 17.1; Source: See Fig. 17.1)\n\nThese results may be compared with results reported for the USA by Wolfinger et al. (2010), who use data from the 2000 Census to examine the likelihood of a birth event (defined as having a child younger than 2 years in the household) for men and women belonging to the same professions as studied above. In the USA too, doctors have the highest rate of birth events, followed by lawyers and academics. The patterning is similar for men and women, but within each profession men have more birth events than do women. Although the patterning of childbearing among fast-track professionals in Sweden and the USA is similar, there are interesting differences relating to couple characteristics. While doctors in Sweden are very likely to belong to a dual-earner household, not seldom consisting of two medical doctors, male doctors in the USA more often have spouses who are out of the labor force. They therefore have higher fertility in part because they have wives who see to child care and domestic work. Male professionals whose wives are academics do not have higher fertility, which is an indication that the low fertility among female academics can affect the fertility of men in these professions.\n\n## 17.5 Concluding Discussion\n\nThere are obvious differences with respect to continued childbearing within the group of highly educated professionals, not only in Sweden, but also in a contrasting context such as that in the USA. Doctors emerge as being more likely to go on to having a second or third child once they have started a family compared to law professionals and academics. Doctors are clearly different compared to the rest, a result that holds for both men and women. There are, however, differences in the variation across professions by gender, indicating that public sector employment is conducive to Swedish women's continued childbearing. Moreover, the tendency among the other professional groups to be less likely to go on to a second birth compared to doctors is attenuated when it comes to third births, especially among women.\n\nThe evidence at hand gives general support to the hypothesis that differences in the working conditions, career structures, and professional development that follow upon occupational choice generate differences in continued childbearing within the professions studied. In line with expectations, those who are in a rigid fast-track hierarchy, with the constant pressure of producing measurable results and output, obviously face more difficulties combining work and having more than one child than do those who work in organizations where it is possible to step out of the rat race temporarily, and who enjoy greater job protection and fewer career penalties. In line with this reasoning, it is found that, in the case of women, lawyers working in the private sector are the least likely to have either a second or a third birth once they have started childbearing , when compared to doctors. Women in academia are, however, not very different from lawyers working in the private sector when it comes to their chances of continued childbearing, whereas women working in the courts administration are somewhat more likely to experience higher order births than are the two aforementioned groups. Yet they are all less likely to continue childbearing in relation to physicians, all else equal.\n\nIt is perhaps not surprising that the results indicate a sharper tradeoff between career and continued childbearing for women working in private law firms than for women working as doctors and in the national courts system, but it is somewhat more challenging to understand why academics have similarly low chances of experiencing a second and third birth in relation to physicians. In the light of the career structure rigidity and work demands that are important in all fast-track professions, the lower chances of continued childbearing among academics could be understood in the sense that, even though there is no formal tenure track in Swedish academia , PhDs have invested to a high extent in their careers through undergoing a lengthy training period and facing a competitive environment after graduating. This is especially the case if they are research-oriented and aim for an academic career and not just a teaching position. There is a de facto tenure track for the career-oriented that puts pressure on Swedish academics to perform in a way similar to the US system (cf. Mason and Goulden 2004; Wolfinger et al. 2010). There is also a long-standing shortage of academic positions, in stark contrast to the great demand for physicians, which adds to the competitive pressure. This puts academics in a relatively disadvantaged position compared to other professional groups, not least doctors. When it comes to family building, these disadvantages seems to net out the advantages of flexibility that the academic profession is characterized by.\n\nAnother explanation could be that academics systematically have different preferences regarding family building as compared to doctors; i.e., that there is occupational selectivity with respect to personality, meaning that career-oriented individuals choose certain professions such as academia and law, whereas more family-oriented individuals choose other professions such as medicine. If there is a selection of individuals into different professions due to different work\u2013family orientations, the result will be more of continued childbearing among those in professions that are more compatible with childrearing . If there is a selection of individuals at risk of having a second and a third child, those who have started childbearing will have already revealed a preference for having children, which may make them more willing to have an additional child. Individuals in professions where it is more difficult to combine parenthood and caregiving responsibilities with a career, who nevertheless did become parents, might be a select group with strong family preferences. The result then would mean lower bound estimates of differences according to profession.\n\nWith this in mind, the results indicate that continued childbearing is facilitated among doctors who corroborate selection into profession. The results also indicate that continued childbearing is facilitated among public sector employees, confirming selection into family-friendly public sector employment of which Sweden is well-known. Yet continued childbearing is not facilitated to the same extent across the professions, not even in Sweden. Since a similar pattern is observed for both men and women, selectivity with respect to personality and career-family orientation seems to be gender-neutral, although this is not indicated by other labor market outcomes.\n\nThe pattern observed also indicates that medical doctors are different compared to other professionals irrespective of gender and country context. Physicians are more likely to continue childbearing in both Sweden and the USA, although the countries offer very different employment conditions and work\u2013family support systems. Overall the opportunities to combine work and family are good in Sweden compared to elsewhere, particularly the USA. Parents have statutory rights to take time off on parental leave for more than a year, to work part-time until the child is 12, and they receive generous compensation during their parental leave, even if they belong to the upper tail of the earnings distribution. Quality child care services are available at low cost through government subsidies, which facilitates the combination of work and family for all parents. Yet, despite its many pros, the Swedish model has limitations in some respects, especially for fast-track professionals who may need more flexible solutions in order to accommodate work demands . The lack of alternatives may be more of a compatibility issue in the Swedish context, especially for fast-track professionals, and so may be the lack of home care services that make people devote more time and energy to housework and maintenance on a do-it-yourself basis and thereby spend less time and effort on market work and careers. Lately, there has been a tendency among private companies to accommodate workers who are also parents with family-friendly policies and flexible solutions. This does not, however, eliminate the fact that fast-track jobs in rigid hierarchies are inflexible; the volume of work produced is still vital for achieving promotion , and the common response to this is longer work hours. There are norms regarding the time in life when certain career goals should be achieved, and these are designed not around the primary caregiver but rather around the full-time breadwinner with support from within the home. All else equal, workers who spend more hours on the job and devote more effort to their careers will produce more, while mothers (and fathers) who have family responsibilities, and perhaps also a partner who has a demanding profession, will be disadvantaged.\n\nCompatibility issues are found in the workplace and the career structure of the professions investigated in this study, but in many senses the degree of career-family compatibility also depends on the relationship between partners. Contrary to other contexts, couples in Sweden where both partners have a higher education are more likely to continue childbearing compared to couples where one partner has a lower education. In line with this, and in contrast to the US experience, it is also the case that couples where both partners belong to one of the professional groups studied here are more likely to continue childbearing compared to others. It thus seems as if the assumed disadvantages and pressures of pursuing a professional career are mediated by shared interests and networks and a mutual understanding of each other's situation. The higher rate of continued childbearing among professional couples can probably also be explained by more egalitarian attitudes and practices toward the household division of labor made by these couples. In dual-career households, in which both spouses have a high level of career involvement, both partners have access to economic resources that serve as a basis for negotiation and therefore mutual goals can be achieved. Moreover, competing preferences and stress can be mediated through having a higher degree of understanding for each other's workload , and this may facilitate work\u2013family issues, especially for women. When it comes to childrearing , couples with higher educational status and skills share more of the total parental leave benefits than do those with lower educational status and skills, and they make more use of public child care.\n\nTo conclude, in the current chapter it has been made clear that continued childbearing varies by profession and also by profession and gender. These differences are smaller in Sweden than in the USA, but cannot be explained by income and the fact that some professionals, including doctors earn more than others, such as academics. Rather, the results indicate that working conditions and career structure contribute to making it easier for some groups than for others to combine a professional career and children. The patterns that emerge for men and women reflect the facts that they are not equally involved in childbearing and are not equally sensitive to career structure, especially not those hierarchies presenting them with up-or-out decisions implying a tradeoff between career and children at an early stage in their career. The results presented indicate that the way professional careers are structured and how flexible jobs are matter a great deal for the combination of work and family. Among fast-track professions, there still exist differences as to whether jobs are flexible (at least temporarily during childbearing years) , even in a supposedly family-friendly country like Sweden where the same rights to leave schemes and part-time work apply to all workers. These differences make the combination of work and family more difficult for some workers than others. On the other hand, flexibility on the job often comes with sacrifices in regard to both salary and career advancement\u2014prices that, up until now, more women than men have been willing to pay. This puts policies promoting work\u2013family compatibility for all into perspective, especially as increasingly many women and men are opting for higher education and careers and when the prestigious professions are feminizing.\n\nReferences\n\nAcker, J. (1990). Hierarchies, jobs, bodies: A theory of gendered organizations. Gender & Society, 4, 139\u2013158.CrossRef\n\nAvellar S., & Smock, P. J. (2003). Has the price of motherhood declined over time? A cross-cohort comparison of the motherhood wage penalty. Journal of Marriage and Family, 65, 597\u2013607.CrossRef\n\nBarnett, R. C., & Rivers, C. (1996). She works\/he works: How two-income families are happier, healthier, and better off. New York: Harper Collins.\n\nBecker, G. S. (1991). A treatise on the family. Cambridge: Harvard University Press.\n\nBelkin, L. (October 26 2003). The opt-out revolution. New York Times Magazine, p. 42.\n\nBellavia, G., & Frone, M. (2005). Work-family conflict. In J. Barling, E. K. Kelloway, & M. R. Frone (Eds.), Handbook of work stress (pp. 113\u2013147). Thousand Oaks: Sage.\n\nBerk, S. F. (1985). The gender factory: The appointment of work in American households. New York: Plenum.CrossRef\n\nBernhardt, E., & Goldscheider, F. (2006). Gender equality, parenthood attitudes, and first births in Sweden. In Vienna yearbook of population research (pp. 19\u201339). Vienna: Vienna Institute of Demography.\n\nBertrand, M., Goldin, C., & Katz, L. F. (2010). Dynamics of the gender gap for young professionals in the financial and corporate sectors. American Economic Journal: Applied Economics, 2, 228\u2013255.\n\nBianchi, S. (2000). Maternal employment and time with children: Dramatic change or surprising continuity? Demography, 37, 410\u2013414.CrossRef\n\nBianchi, S. M., & Raley, S. (2005). Time allocation in working families. In S. M. Bianchi, L. M. Casper, & R. B. King (Eds.), Work, family, health, and well-being (pp. 21\u201342). Mahwah: Lawrence Erlbaum.\n\nBianchi, S. M., Robinson, J. P., & Milkie, M. A. (2006). Changing rhythms of American family life. New York: Russell Sage.\n\nBillari, F. C., & Kohler, H.P. (2004). Patterns of low and lowest-low fertility in Europe. Population Studies, 58, 161\u2013176.CrossRef\n\nBlossfeld, H. P., & Drobnic, S. (Eds.), (2001). Careers of couples in contemporary societies: From male breadwinner to dual earner families. Oxford: Oxford University Press.\n\nBoulis, A. (2004). The evolution of gender and motherhood in contemporary medicine. Annals of the American Academy of Political and Social Science, 596, 172\u2013206.CrossRef\n\nBrand, J. E., & Davis, D. (2011). The impact of college education on fertility: Evidence of heterogeneous effects. Demography, 48, 863\u2013887.PubMedCentralPubMedCrossRef\n\nBudig, M. (2003). Are women's employment and fertility histories interdependent? An examination of causal order using event history analysis. Social Science Research, 32, 376\u2013401.CrossRef00012-7)\n\nColtrane, S. (2000). Research on household labor: Modeling and measuring the social embeddedness of routine family work. Journal of Marriage and the Family, 62, 1208\u20131233.CrossRef\n\nCooney, T., & Uhlenberg, P. (1989). Family-building patterns of women professionals: A comparison of lawyers, physicians, and post-secondary teachers. Journal of Marriage and the Family, 51, 749\u2013758.CrossRef\n\nCraig, L., & Mullan, K. (2010). Parenthood, gender and work-family time in USA, Australia, Italy, France and Denmark. Journal of Marriage and Family, 72, 1344\u20131361.CrossRef\n\nDrew, E., Emerek, R., & Mahon, E. (1998). Women, work and the family in Europe. London: Routledge.CrossRef\n\nDribe, M., & Stanfors, M. (2009). Does parenthood strengthen a traditional household division of labor? Evidence from Sweden. Journal of Marriage and Family, 71, 33\u201345.CrossRef\n\nDribe, M., & Stanfors, M. (2010). Family life in power couples: Continued childbearing and union stability among the educational elite in Sweden, 1991\u20132005. Demographic Research, 23, 847\u2013878.CrossRef\n\nEly, R. J., & Meyerson, D. E. (2000). Theories of gender in organizations: A new approach to organizational analysis and change. In A. Brief & B. M. Staw (Eds.), Research in organizational behavior (pp. 103\u2013151). New York: Elsevier.\n\nEpstein, C. F., Seron, C., Oglensky, B., & Saut\u00e9, R. (1999). The part-time paradox: Time norms, professional lives, family, and gender. New York: Routledge.\n\nGauthier, A., Smeeding, T., & Furstenberg, F. (2004). Are parents investing less time in children: Trends in selected industrialized countries. Population and Development Review, 30, 647\u2013671.CrossRef\n\nGershuny, J. (2000). Changing times: Work and leisure in post-industrial societies. Oxford: Oxford University Press.\n\nGershuny, J., & Robinson, J. P. (1988). Historical changes in the household division of labor. Demography, 25, 537\u2013552.PubMedCrossRef\n\nGlass, J. (2000). Envisioning the integration of family and work. Contemporary Sociology, 29, 129\u2013143.CrossRef\n\nGoldin, C. (1990). Understanding the gender gap: An economic history of American women. New York: Oxford University Press.\n\nGoldin, C. (15 March 2006). Working it out. New York Times, p. A27.\n\nGoldin, C., & Katz, L. (2008). Transitions: career and family life cycles of the educational elite. American Economic Review: Papers and Proceedings, 98, 363\u2013369.CrossRef\n\nGoldin, C., & Katz, L. F. (2011). The cost of workplace flexibility for high-powered professionals. Annals of the American Academy of Political and Social Science, 638, 45\u201367.CrossRef\n\nGoldin, C., & Katz, L. F. (2013). The most egalitarian of all professions: Pharmacy and the evolution of a family-friendly occupation. NBER Working Paper No 18410.\n\nGreenhaus, J. H., & Powell, G. N. (2003). When work and family collide: Deciding between competing role demands. Organizational Behavior and Human Decision Processes, 90, 291\u2013303.CrossRef00519-8)\n\nHallberg, D., & Klevmarken, A. (2003). Time for children: A study of parents' time allocation. Journal of Population Economics, 16, 205\u2013226.CrossRef\n\nHerr, J. L., & Wolfram, C. (2011). Work environment and \"opt-out\" rates at motherhood across high education career paths. NBER Working Paper No 14717.\n\nHewlett, S. A. (2002). Executive women and the myth of having it all. Harvard Business Review, 80, 66\u201373.PubMed\n\nHochschild, A. R. (1989). The second shift: Working parents and the revolution at home. New York: Penguin.\n\nHochschild, A. R. (1997). The time bind: When work becomes home and home becomes work. New York: Metropolitan Books.\n\nHoem, J. M., Neyer, G., & Andersson, G. (2006). Educational attainment and ultimate fertility among Swedish women born in 1955\u201359. Demographic Research, 14, 381\u2013403.CrossRef\n\nHofferth, S. (1999). Child care, maternal employment, and public policy. The Annals of the American Academy of Political and Social Science, 563, 20\u201338.CrossRef\n\nHymowitz, C., & Silverman, R. E. (16 January 2001). Can work stress get worse? Wall Street Journal, p. B1.\n\nJacobs, J. A., & Fanning Madden, J. (2004). Mommies and daddies on the fast track: Success of parents in demanding professions. Special issue of The Annals of the American Academy of Political and Social Science, 596.\n\nJoshi, H. (2002). Production, reproduction and education: Women, children and work in a British perspective. Population and Development Review, 28, 445\u2013474.CrossRef\n\nKimball, G. (1999). 21st century families. Chico: Equality.\n\nKravdal, \u00d8. (2001). The high fertility of college educated women in Norway: An artefact of the separate modelling of each parity transition. Demographic Research, 5, 187\u2013216.CrossRef\n\nKravdal, \u00d8. (2007). Effects of current education on second- and third-birth rates among Norwegian women and men born in 1964: Substantive interpretations and methodological issues. Demographic Research, 17, 211\u2013246.CrossRef\n\nKreyenfeld, M. (2002). Time-squeeze, partner effect or self-selection? An investigation into the positive effects of women's education on second birth risks in West Germany. Demographic Research, 7, 15\u201348.CrossRef\n\nLappeg\u00e5rd, T., & R\u00f6nsen, M. (2005). The multifaceted impact of education on entry into motherhood. European Journal of Population, 21, 31\u201349.CrossRef\n\nMason, M. A., & Goulden, M. (2004). Marriage and baby blues: Redefining gender equity in the Academy. Annals of the American Academy of Political and Social Science, 596, 86\u2013103.CrossRef\n\nMcDonald, P. (2000). Gender equity in theories of fertility transition. Population and Development Review, 26, 427\u2013439.CrossRef\n\nMincer, J. (1963). Market prices, opportunity costs, and income effects. In Christ, E., et al. (Eds.), Measurement in economics. Stanford: Stanford University Press.\n\nMoen, P., & Yu, Y. (1999). Having it all: Overall work\/life success in two-earner families. In T. Parcel (Ed.), Research in the sociology of work (pp. 107\u2013137). Greenwich: JAI.\n\nNeilson, J., & Stanfors, M. (2014). It's about time! Gender, parenthood and household divisions of labor under different welfare regimes. Journal of Family Issues, 35, 1066\u20131088.\n\nNoonan, M. C., & Corcoran, M. E. (2004). The Mommy track and partnership: Temporary delay or dead end? Annals of the American Academy of Political and Social Science, 596, 130\u2013150.CrossRef\n\nPolachek, S. W. (1981). Occupational self-selection: A human capital approach to sex differences in occupational structure. Review of Economics and Statistics, 63, 60\u201369.CrossRef\n\nRuhm, C., & Teague, J. L. (1997). Parental leave policies in Europe and North America. In F. D. Blau & R. G. Ehrenberg (Eds.), Gender and family issues in the work place (pp. 133\u2013156). New York: Russell Sage.\n\nSayer, L. (2005). Trends in women's and men's paid work, unpaid work and free time. Social Forces, 84, 285\u2013303.CrossRef\n\nShang, Q., & Weinberg, B. A. (2013). Opting for families: Recent trends in the fertility of highly educated women. Journal of Population Economics, 26, 5\u201332.CrossRef\n\nStanfors, M. (2003). Education, labor force participation and changing fertility patterns. A study of women and socioeconomic change in twentieth century Sweden. Stockholm: Almqvist and Wiksell International.\n\nStone, P. (2008). Opting out? Why women really quit careers and head home. Los Angeles: University of California Press.\n\nStory, L. (20 September 2005). Many women at elite colleges set career path to motherhood. New York Times, pp. A1, A18.\n\nVan Bavel, J. (2010). Choice of study discipline and the postponement of motherhood in Europe: The impact of expected earnings, gender composition and family attitudes. Demography, 47, 439\u2013458.PubMedCrossRef\n\nWilliams, J. (2000). Unbending gender: Why family and work conflict and what to do about it. New York: Oxford University Press.\n\nWolfinger, N. H., Mason, M. A., & Goulden, M. (2010). Alone in the ivory tower. Journal of Family Issues, 31, 1652\u20131670.CrossRef\n\nFootnotes\n\n1\n\nThese reports have; however, been contested by scholars like Goldin (2006), who claims that facts speak against the \"opting out\" story and that greater fractions of college-educated women today are combining family and career than ever before.\n\n2\n\nChild care fees are based on the total gross income of the household in which the child resides. Fees are only income-dependent up to the equivalent of a gross monthly income of 42,000 SEK (in 2009; equivalent to 5,250 USD at the time in question). After that, the same fee applies to all households, irrespective of income, which is especially beneficial for high income earners. Coverage is extensive\u2014about 85 % of all children aged 1\u20136 were in public day care in 2005\u2014and therefore dual-career couples are able to spend more of their income on other goods and services that may help them combine career and family.\n\n3\n\nMen and women who opt for a PhD are generally 25\u201329 years old. Only about 10 % of all PhDs in Sweden are awarded to people under 30. In 2007, 41 % of all male PhD recipients and 37 % of all female PhD recipients were in the age range of 30\u201334.\n\n4\n\nThus there is per definition not much of a problem of reverse causality.\n\n5\n\nThe analyses do not allow for any strictly causal inferences and do not deal with selection, whether this be into professions based on family orientation or into larger family sizes.\n\n6\n\nThe data derive from the multigenerational register ( Flergenerationsregistret), which contains information on biological and adopted children to all index persons in the sampling frame (all individuals in birth cohorts 1942\u20131989 who resided in Sweden at some point in time after 1960). Due to frequent cases of missing information on adoption dates, only biological children are included in the analysis.\n\n7\n\nTotal income includes wages for employees and self-employed and benefits paid in connection to work (i.e., parental leave, pensions, unemployment benefits, and payment from sickness insurance). To enable comparisons over time, annual income is related to the so called price base amount (hereafter simply called base amount) of the year. The base amount is set for each year on the basis of changes in the consumer price index (CPI). Its main purpose is to adjust different kinds of public benefits (pensions, student aid, sickness insurance, etc.) to account for inflation. A variable measuring the income share earned by the woman proxies her relative position in the partnership.\n\n8\n\nAn odds ratio is a measure of effect size. It describes the strength of association between data values (for example x and y). If the odds ratio is greater than 1, then there is an association in the sense that having some properties, such as x) (relative to not having x) increases the odds of having y. It should be made clear that we are talking about associations since the causal link has not necessarily been established.\n\n9\n\nThis is corroborated by sensitivity tests where the model is estimated with different professions as reference categories.\n\n10\n\nThe results hold up for a number of sensitivity tests including stepwise modeling with a gradual inclusion of independent variables, and the estimation of logit models of the probability of having a second or a third birth within 5 years, to see if the results are driven by a subgroup with a specific temporal fertility behavior. Also, logit models where the covariates refer to the situation at the time of the previous birth are estimated in order to obtain a counterfactual that renders similar results. Since the original analysis is based on a select group consisting of men and women who have attained their professional status and started their careers at the time of their first birth, logit models are estimated for individuals who ever attain the professional status in question. None of these sensitivity checks alters the pattern observed.\nIndex\n\nSymbols\n\n3M\n\n24\/7 economy\n\nA\n\nAbsenteeism\n\nAcademia\n\nAcculturation\n\nAdvancement See Promotion\n\nAffect\n\nnegative affectivity (NA)\n\npositive affectivity (PA)\n\nAffective Events Theory (AET)\n\nAffirmative action\n\nAge\n\nAggression\n\nAllocentrism\n\nAnger\n\nAnxiety\n\nApplicant behavior\n\njob search techniques\n\nAppraisal\n\naffective\n\ncognitive\n\ncoping behaviors\n\ninfluences on\n\nprimary\n\nreappraisal\n\nsecondary\n\nArchival data\n\nAscriptive inequality\n\nAttitudes\n\ncontemporary\n\ncoworkers\n\nliberalization\n\nplateau\n\nreversal\n\ntraditional\n\nAttrition See Turnover\n\nAutonomy\n\nB\n\nBaby panic\n\nBargaining theory\n\nBehaviour\n\nsocial\n\nBenefits\n\naccessibility\n\nawareness\n\nbeneficiaries\n\neffectiveness\n\nformal\n\ngendered\n\ninformal\n\nLGB-friendly\n\noffered\n\norganizational\n\noutcomes\n\nperceptions of\n\nsatisfaction with\n\nusage\n\nBias\n\nexplicit\n\nimplicit\n\nBiological\n\nclock\n\nsex\n\nBio-psychological equivalence\n\nBisexual\n\nBlame\n\nBlue-collar\n\nBoundaries\n\nflexibility of\n\nmanagement of\n\npermeability of\n\npsychological\n\nspatial\n\ntemporal\n\nviolation of\n\nBreadwinning\n\nBreastfeeding\n\nBuffer\n\nBureau of Labor Statistics (BLS; U.S.)\n\nBurnout\n\nC\n\nCapitalization\n\nCareer advancement\n\nCareer choices\n\nnontraditional\n\ntraditional\n\nCareer clock\n\nCareer interruptions\n\nCareer ladder See Promotion\n\nCareer orientation\n\nCaregiving responsibilities\n\nCatalyst\n\nCensus\n\nCenter for Creative Leadership (CCL)\n\nChanging family structures\n\nChild developmental outcomes\n\nChildbearing\n\nChildbirth\n\nChildcare\n\nChildless employees\n\nChildrearing\n\nCivil Rights Act\n\ntitle VII\n\nCivil War (U.S.)\n\nCognition\n\nCollectivism\n\nCo-location problem See Two-body problem\n\nCombat\n\nwomen in\n\nComing out\n\nbehavior\/self-policing\n\nbenefit availability\n\ndisclosure disconnect\n\ndiscrimination\n\nDon't Ask Don't Tell (DADT)\n\nfear of disclosure\n\nharassment\n\nhostile work environment\n\nidentity management\n\nimplications for conflict\n\noutness\n\nrepercussions\n\nseparation of work and family\n\nCommunicative technologies (CT)\n\nCompetence See Performance\n\nConfidence\n\nConservation of Resources Theory\n\nContext\n\nControl See Autonomy\n\nCoping\n\nCounterfeiting\n\nCouples\n\nCross-cultural studies See Cross-national studies\n\nCross-domain effect\n\nCross-national studies\n\nCulture\n\nD\n\nDaycare See Childcare\n\nDecision-making\n\nDelegation\n\nDemand\n\nnonwork\n\nscheduling\n\ntime\n\nwork\n\nDemographics\n\nDepartment of Defense\n\nDepartment of Labor\n\nDepression\n\nDetachment\n\nDisability\n\nDisappointment\n\nDiscrimination\n\nDispositions See Individual differences\n\nDiversity\n\ntraining\n\nDivision of labor\n\nDomestic violence\n\nDouble advantage\n\nDouble jeopardy\n\nDouble shift See Second shift\n\nDouble standards\n\nDownsizing See Layoffs\n\nDual Labor Market Segmentation Theory\n\nand gendered occupations\n\ncriticisms\n\nprimary sector\n\nsecondary sector\n\nDual-career See Dual earner\n\nDual-earner\n\nE\n\nEconomic hardship See Poverty\n\nEconomic status See Wage level\n\nEconomy\n\nEducation\n\nEgalitarianism\n\nEldercare\n\nchallenges\n\nEmotion See Affect\n\nEmotional exhaustion\n\nEmotional labor\n\nEnergy\n\nEngagement\n\nEngineering See STEM\n\nEnvironment\n\nEqual Employment Opportunity Commission\n\nEqual Pay Day\n\nEquity theory\n\nEthnicity\n\nExchange theory\n\nExecutive jobs\n\nExpectations\n\nfamily\n\nwork\n\nExtensification See Workload\n\nExtracurricular\n\nF\n\nFace-time\n\nFairness\n\npreceptions\n\nFamilies of choice\n\nFamily and Medical Insurance Leave Act (FAMILY Act)\n\nFamily and Medical Leave Act (FMLA)\n\nFamily satisfaction\n\nFamily-interference-with-work (FIW)\n\nFamily-supportive organizational perceptions (FSOP)\n\nFamily-supportive supervisor behaviors (FSSB)\n\nFamily-supportive work environments (FSWE)\n\nFast-track professions See Professional jobs\n\nFatigue\n\nsleep\n\nFault\n\nFeedback\n\nFemale employment\n\npercentage\n\nFemininity\n\nFertility See Biological clock\n\nFit\n\nFlexibility\n\nCompressed work week (CWW)\n\nflexibility stigma\n\nFlexible Work Arrangements (FWA)\n\nflextime\n\njob sharing\n\ntelecommuting\n\nFortune 500\n\nFuture research\n\nopportunities for\n\nG\n\nGender deviance neutralization\n\nGender disparity\n\nGender display theory\n\nGender divide\n\nGender equality\n\nineqality\n\ninequality\n\nGender gap\n\neducation\n\nworkplace\n\nGender ideology\n\nGender role ideology (GRI) See Gender ideology\n\nGender role theory See Role Thoery\n\nGender roles\n\norientation\n\nsocialization\n\nGender specialization\n\nGender typing\n\nGendered assumptions\n\nGendered behavior\n\nGendered employment pattern\n\nGendered occupations\n\nfemale-intensive\n\nmale-intensive\n\nwage differentials\n\nGeneration\n\nGlass ceiling\n\nGlobalization\n\nglobal roles\n\nglobal service economy\n\nGoal commitment\n\nGoldie Blox\n\nGovernment policies See Public programs\n\nGuilt\n\nantecedents\n\nconsequences\n\nFamily interference with work guilt (FIWG)\n\ngender differences\n\nGuilty\n\ncultural differences\n\nsources of\n\nwork interference with family guilt (WIFG)\n\nH\n\nHarassment\n\nHealth\n\nmental\n\nphysical\n\nHealth insurance\n\nHeteronormativity\n\nHeterosexual\n\nbias\n\nHigh-powered occupations See Professional jobs\n\nHigh-status professions See Professional jobs\n\nHiring See Selection\n\nHomosexual\n\nHousehold labor\n\nHousework\n\nHuman capital\n\nI\n\nIdeal worker\n\nIdentity\n\ncentrality\/salience\n\nconflict\n\nmanagement\n\nmultiple\n\nrole\n\nsexual\n\nsocial\n\nIdiocentrism\n\nIdiosyncratic deals (i-deals)\n\nIllegal activities\n\nImmigration\n\nIncome See Wage level\n\nIndividual differences\n\nIndividualism\n\nIndustrialized nations\n\nInitiatives See Policies\n\nInjury\n\nIntegration\n\nInternational studies See Cross-national studies\n\nInterruptions\n\nIntersectionality Theory\n\nIntrusions See Interruptions\n\nInvolvement\n\nfamily\n\nwork\n\nJ\n\nJob (in)security\n\nJob Demands-Resources (JD-R) Model\n\nJob satisfaction\n\nJoint hire\n\nL\n\nLabor force\n\nLabor force attachment See Work-role salience\n\nLabor market\n\nLayoffs\n\nLeader-member exchange (LMX)\n\nLeadership\n\nLeaky pipeline\n\nLeave\n\nand gender equality\n\ncross-nationally\n\nFAMILY act\n\nFMLA\n\nleave-taking\n\nlength\n\nmaternity leave\n\npaid\n\npaternity leave\n\nunpaid\n\nLegal protection\n\nlaws\n\nlegally-protected class\n\nLeisure\n\nLesbian, gay, and bisexual (LGB)\n\nLesbian, Gay, Bisexual, or Transgender (LGBT)\n\nLife satisfaction\n\nLimitations\n\nLinear career model\n\nLongitudinal studies\n\nLow-wage jobs\n\nM\n\nMale-dominated fields See Gendered occupations\n\nManagerial positions\n\nManufacturing\n\nMarital satisfaction\n\nMarital status See Relationship status\n\nMasculine hegemony\n\nMasculinity\n\nMaternal employment\n\nconsequences\n\nMaternal wall\n\nMath See STEM\n\nMay babies\n\nMeasurement\n\nbias\n\ncontamination\n\ndeficiency\n\ndifferential item interpretation\n\ndimensions\n\ndiscrepancy\n\nequivalence\n\nexperience sampling method\n\nnarratives\n\nscales\/surveys\n\nsocial desirability\n\nMedia\n\nMediators\n\nMentor\n\nMilitary\n\nMinorities\n\nsexual\n\nModels\n\nModerators\n\nMommy track\n\nMorality\n\nMotherhood penalty\n\nMotivation\n\nMultiple identities\n\nMultiplier effect\n\nMultitask\n\nN\n\nNational Aeronautics and Space Administration (NASA)\n\nNational Committee on Pay Equity\n\nNational Institute of Health (NIH)\n\nNational Science Foundation (NSF)\n\nADVANCE\n\ncareer-life balance initiative\n\nSTRIDE\n\nNational Study of Employers\n\nNeuroticism\n\nNorms\n\ngender\n\ntraditional\n\nNumber of children\n\nO\n\nOccupational distribution See Gendered occupations\n\nOn-demand work\n\nOrganizational climate\n\nchilly\n\nOrganizational commitment\n\nOrganizational culture\n\nOrganizational hierarchy\n\nOrganizational norms\n\nOrganization-based self-esteem (OBSE)\n\nOverload\n\nOverload See also workload\n\nOvertime\n\nOvertime See also Workload\n\nP\n\nPartner\n\nPart-time work\n\nPay level See Wage level\n\nPerception See Appraisal\n\nPerceptions of Women in Academic Science study\n\nPerformance\n\nactual\n\nevaluations\n\nPolicies See Benefits\n\nPopular press\n\nPost-Fordist era\n\nPost-traditional family\n\nPosttraumatic Stress Disorder (PTSD)\n\nPoverty\n\ncorrelates\n\ncross-nationally\n\nfeminization of poverty\n\ngender-poverty disparity\n\nrisk\n\nPower\n\ndistance\n\nPreference theory\n\nPregnancy\n\nPrejudice\n\nPressure\n\nPrivilege\n\nProductivity\n\nProfessional jobs\n\nC-suite\n\nEMPs\n\nProject 3535\n\nPromotion\n\nPsychological contract\n\nPsychological well-being See Health, Mental\n\nPublic programs\n\ncross-nationally\n\nemployment opportunities\n\nstate support\n\nfinancial\n\nQ\n\nQuality of life\n\nQuota\n\nR\n\nRace\n\nRacism\n\nRational viewpoint\n\nRecession\n\nRecovery\n\nRelationship quality\n\nRelationship satisfaction See Marital\/family satisfaction\n\nRelationship status\n\nRelationships\n\nRelative deprivation theory\n\nReligion\n\nReligiosity See Religion\n\nResentment\n\nResilience\n\nresource drain theory\n\nResources\n\nallocation\/distribution\n\ndepletion\n\neconomic\/financial\n\nenergy\n\nexchange\n\ngender differences in access to\n\nloss spirals\n\nnoneconomic\/nonfinancial\n\nnonwork\n\npersonal\n\nwork\n\nResponsibility\n\nRetention See Turnover\n\nRewards\n\nRole Accumulation Theory\n\nRole commitment\n\nfamily\n\nwork\n\nRole congruity theory\n\nRole expansion\n\nRole expectations\n\nRole model\n\nRole Theory\n\nfor males\n\nmaternal work history\n\nobservation\n\nsame-sex parent\n\nRumination\n\nS\n\nSalary See Wage level\n\nSandwich generation\n\nSavoring\n\nScarcity hypothesis\n\nSchedules\n\nSchema\n\nScience technology, engineering, and math (STEM)\n\neducation center\n\nScience See STEM\n\nSecond shift\n\nSegmentation\n\nSelection\n\njob assignments\n\nletters of recommendation\n\nSelf\n\nself-selection\n\nSelf-employment\n\nSelf-esteem\n\nSeparate spheres\n\nService sector\n\nSexism\n\nSex-segregated occupations See Gendered occupations\n\nSexual orientation\n\nSexuality\n\nShiftwork\n\nnonstandard\n\nrotating\n\nstandard\n\nSingle parent\n\nsingle father\n\nsingle mother\n\nSkill\n\nSleep\n\nSocial changes\n\nSocial dominance theory\n\nSocial exchange theory\n\nSocial role theory\n\nSocietal norms\n\nSociety for Human Resource Management (SHRM)\n\nSocioeconomic status\n\nSpillover\n\naffective\n\nmodel\n\nnegative\n\npositive\n\nSpillover theory\n\nSpillover See also work\u2013family conflict\/enrichment\n\nSpouse See Partner\n\nStay-at-home parents\n\nfathers\n\nmothers\n\nStereotypes\n\nStigma\n\nStress\n\nSuperdad\n\nSupermom\n\nSuport\n\nsocioemotional\n\nSupport\n\nbarriers\n\nby income status\n\nchildcare\n\ncoworker\n\nemotional\n\nfinancial\n\nformal\n\ninformal\n\ninstrumental\n\norganizational\n\nperceived\n\nsocial\n\nspousal\n\nsupervisor\n\nT\n\nTag-teaming\n\nTechnology\n\nTenure\n\nstop clock policies\n\nTheory of Planned Behavior\n\nTheory of Reasoned Action\n\nThreat\n\nTiming of parenthood\n\nTokenism\n\nTraditional family\n\nTraining\n\nTraits See Individual differences\n\nTransactional model of emotion\n\nTransgender\n\nTravel\n\nTurnover\n\nactual\n\nintensions\n\nTwo-body problem\n\nU\n\nUncertainty avoidance\n\nUnderrepresented groups\n\nUnderstudied populations\n\nUnemployment\n\nUnited Nations Gender Development Index\n\nUnpaid work\n\nUrbanization\n\nV\n\nVacation time\n\nValues\n\nVocational trajectory\n\nW\n\nWage gap\/inequality\n\nWage level\n\nWage penalty See Motherhood penalty\n\nWelfare\n\nWell-being\n\nWhite-collar\n\nWhite House Council of Women and Girls\n\nWithdrawal\n\nWomen's movement\n\nWorkaholism\n\nage\n\nconsequences\n\ncultural\n\nexperiences\n\ngender differences\n\nWorkaholics Anonymous\n\nWork competence of mothers\n\nWork-family balance\n\nWork\u2013family conflict (WFC)\n\nantecedents\n\nbehavior-based\n\nconsequences\n\nfamily-to-work conflict (FWC)\n\nLGB WFC\n\nstrain-based\n\ntime-based\n\nWork\u2013family enrichment (WFE)\n\nantecedents\n\nbehavioral\n\nconsequences\n\nenergy-based\n\npsychological\n\ntime-based\n\nWorkforce participation\n\nWork-interference-with-family (WIF)\n\nWork-life\n\nWorkload\n\nWork orientation, See Work-role sailence\n\nWork-Role Salience\n\nWork values\n\nWorld Health Organization (WHO)\n\nZ\n\nZero-drag worker\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n**Books by Richard Leviton**\n\n_Seven Steps to Better Vision_\n\n_The Imagination of Pentecost: Rudolf Steiner and Contemporary Spirituality_\n\n_Brain Builders_\n\n_Weddings by Design_\n\n_Looking for Arthur: A Once and Future Travelogue_\n\n_Physician: Medicine and the Unsuspected_\n\n_Battle for Human Freedom_\n\nCopyright \u00a9 2001\n\nby Richard Leviton\n\nAll rights reserved, including the right to reproduce this work in any form whatsoever, without permission in writing from the publisher, except for brief passages in connection with a review.\n\nCover design by Steve Amarillo\n\nCover art by Index Stock Imagery\n\nFor information write:\n\nHampton Roads Publishing Company, Inc.\n\n1125 Stoney Ridge Road\n\nCharlottesville, VA 22902\n\n434-296-2772\n\nfax: 434-296-5096\n\ne-mail: hrpc@hrpub.com\n\nwww.hrpub.com\n\nIf you are unable to order this book from your local bookseller, you may order directly from the publisher.\n\nCall 1-800-766-8009, toll-free.\n\nLibrary of Congress Catalog Card Number: 00-105253\n\nISBN 1-57174-209-3\n\n10 9 8 7 6 5 4 3 2 1\n\nPrinted on acid-free paper in Canada\n\nwww.redwheelweiser.com\n\nwww.redwheelweiser.com\/newsletter\n\n# _Dedication_\n\n**_To Judith A. Lewis\u2014  \nvisionary artist, sculptor,   \nhealer, and partner._**\n\n# _Table of Contents_\n\n**Acknowledgments**\n\n**The 70 Healthy Living Space Detoxifiers at a Glance**\n\n**Notes on Products, Services, and Organizations**\n\n**Introduction**\n\nWhy Toxicity Should Concern Us\n\n**P ART ONE The Health Perils of the Toxic Body and Home**\n\n**C HAPTER 1 Toxicity and Illness**:\n\n**_How Your Body and Home Became Toxic_**\n\n250 Chemical Contaminants in Every Human Body \u2022 Healthy Living Space Detoxifier #1: Fill Out Your Toxicity Self-Assessment Questionnaire \u2022 How Unrelieved Toxicity Can Compromise Your Health \u2022 Being Slowly Poisoned Throughout a Lifetime \u2022 Healthy Living Space Detoxifier #2: Maintain a Household Toxic Exposure Inventory \u2022 How Toxic Is Our Living Space, Both Body and World? \u2022 Drinking Water Contaminated with Pesticides and Toxic Residues \u2022 When Prozac, Viagra, and Sunscreen Flow through Your Kitchen Tap \u2022 Air Pollution-Trigger for Allergies, Asthma, and Respiratory Symptoms \u2022 Organochlorines\u2014Toxic Chlorine Blankets the Planet \u2022 A Radioactive Cloud Enshrouds an Entire Generation \u2022 Tainted Food\u2014Glutamates, Additives, Antibiotics, and POPs \u2022 The Toxic Living Space and You\u2014What It All Means\n\n**C HAPTER 2 Why You Need to Detoxify**:\n\n**_Augmenting the Body's Natural Detoxification System_**\n\nThe Body's Burden\u2014Total Toxic Load and Toxic Stress \u2022 The Oxidation-Reduction Cycle at the Heart of Detoxification \u2022 The Body's Two-Phase Natural Process of Detoxification \u2022 Immunotoxicity and How the Detoxification Pathways Get Blocked \u2022 Neurological and Endocrinal Damage from Environmental Toxins \u2022 Damaging Synergistic Effects Extending into the Next Generation \u2022 Previewing the Scope and Advantages of Deliberate Detoxification \u2022 The Spiritual and Global Implications of Detoxification\n\n**P ART TWO Creating the Healthy _Body_ Living Space**\n\n**C HAPTER 3 How Toxic Are You?**\n\n**_Testing for Toxicity and Detoxification Ability_**\n\nThe Healthy Living Space Expert Interview: Peter Holyk, M.D., Contemporary Health Innovations \u2022 Healthy Living Space Detoxifier #3: Get a Clear Idea of Your Nutritional Status \u2022 Healthy Living Space Detoxifier #4: Assess Your System's Ability to Detoxify \u2022 Healthy Living Space Detoxifier #5: Determine Your Body's Total Toxic Load \u2022 Healthy Living Space Detoxifier #6: Find Out if You're Suffering from Hidden Food Allergies\n\n**C HAPTER 4 Detoxification Preliminaries**:\n\n**_Start Your Internal Cleansing with Dietary Changes and by Minimizing Your Exposure_**\n\nHealthy Living Space Detoxifier #7: Rehydrate Your Body with Nature's Best Detoxifying Substance \u2022 Healthy Living Space Detoxifier #8: Get the Chlorine Out of Your Drinking and Shower Water \u2022 Healthy Living Space Detoxifier #9: Don't Fluoridate Your Teeth\u2014Your Body Doesn't Need Fluorides \u2022 Healthy Living Space Detoxifier #10: Cleanse Your Produce of Toxic Chemicals \u2022 Healthy Living Space Detoxifier #11: Whenever Possible, Eat Organic Foods \u2022 Healthy Living Space Detoxifier #12: Minimize Your Consumption of Foods Containing Trans-fatty Acids \u2022 Healthy Living Space Detoxifier #13: Do a 3-Day Water or Juice Fast to Rest and Regenerate Your Cells \u2022 Healthy Living Space Detoxifier #14: Do a Media\/Image Fast to Cleanse Your Mind and Energy Field \u2022 Healthy Living Space Detoxifier #15: Make a Lifestyle Out of Nonstop Stress Reduction with Healing Hot Water Soaks \u2022 Healthy Living Space Detoxifier #16: Enhance Your Relaxation by Floating in 800 Pounds of Epsom Salt \u2022 Healthy Living Space Detoxifier #17: Use Music Therapy to Deepen Your Sense of Melting Relaxation\n\n**C HAPTER 5 Safe and Effective Cleansing Programs for the Liver and Intestines**\n\nThe Healthy Living Space Expert Interview: Jacob Farin, N.D., Naturopathic Physician \u2022 Healthy Living Space Detoxifier #18: Do a One-Week Elimination Diet and Liver Cleansing Program \u2022 Healthy Living Space Detoxifier #19: Use Natural Foods and Herbs to Aid Your Liver in Cleansing Itself \u2022 Healthy Living Space Detoxifier #20: Do a Natural Liver Flush to Improve Your Internal \"Plumbing\" \u2022 Intestinal Cleansing: Emptying Out a Storage Space the Size of a Tennis Court \u2022 Healthy Living Space Detoxifier #21: Use Six Natural Substances to Help Cleanse Your Intestines \u2022 Healthy Living Space Detoxifier #22: Remove the 50+ Feet of False Mucoid Lining from Your Intestines \u2022 Healthy Living Space Detoxifier #23: Purge the Parasites from Your Intestines \u2022 Healthy Living Space Detoxifier #24: Get Proactive with Probiotics\u2014Plant Food for the Intestines \u2022 Healthy Living Space Detoxifier #25: Fertilize Your Bifidobacteria with Prebiotics\n\n**C HAPTER 6 Whole\u2014Body Cleansing Routines**:\n\n**_Extending the Detoxification into Everyday Activities_**\n\nHealthy Living Space Detoxifier #26: Bounce on a Trampoline to Improve Your Lymphatic Drainage \u2022 Healthy Living Space Detoxifier #27: Dry Skin Brushing\u2014Tone Up Your Body's Largest Elimination Organ \u2022 Healthy Living Space Detoxifier #28: Exercise Regularly to Enhance Blood Circulation and Toxin Removal \u2022 Healthy Living Space Detoxifier #29: Sauna Detox: How Heat Therapy Can Help Your Body Release Toxins \u2022 Healthy Living Space Detoxifier #30: A Purifying Bath: Whole-Body Detoxifying Soaks in Herbs, Salts, Oils, and Other Natural Substances\n\n**C HAPTER 7 The Poisons in Your Mouth**:\n\n**_Specialized Dental Detoxification Programs_**\n\nA Gallery of Possible Toxic Consequences of Modern Dentistry \u2022 Healthy Living Space Detoxifier #31: Make Your Mouth a Mercury-Free Zone by Having Your Mercury Fillings Correctly Replaced \u2022 Healthy Living Space Detoxifier #32: Have Your Cavitations Cleaned Out and Reduce Possible Jawbone Infections \u2022 Healthy Living Space Detoxifier #33: Think Twice before Digging Another Root Canal in Your Mouth \u2022 Healthy Living Space Detoxifier #34: Determine Your Biocompatibility with Dental Materials before Putting Any More in Your Mouth\n\n**C HAPTER 8 Guidelines for Emotional Detoxification**:\n\n**_Unresolved Emotional Issues Can Become Toxic to the Body_**\n\nHow Unresolved Emotions Can Contribute to Illness \u2022 The Healthy Living Space Expert Interview: Patricia Kaminski, Flower Essence Practitioner \u2022 Healthy Living Space Detoxifier #35: Remedying the Four Basic States of Emotional Toxicity with Flower Essences\n\n**C HAPTER 9 Guidelines for Spiritual Detoxification**:\n\n**_Cleansing the Space around the Body_**\n\nPoisons in Our Invisible Atmosphere \u2022 The Outer and Inner Aura in Disease \u2022 Healthy Living Space Detoxifier #36: Salting Your Energy Centers \u2022 Charting the Subconscious Mind in the Aura \u2022 Healthy Living Space Detoxifier #37: Request a Cleansing from Above \u2022 The Healthy Living Space Expert Interview: Rev. Leon S. LeGant, Psychic Healer \u2022 Healthy Living Space Detoxifier #38: Do a 30-Minute Spirit-Detoxifying Bath \u2022 Healthy Living Space Detoxifier #39: Shake Yourself Free of Toxic Energies \u2022 Healthy Living Space Detoxifier #40: Scrub Your Aura Clean with a Golden Sponge \u2022 Healthy Living Space Detoxifier #41: Turn Your Body into Transparent Glass \u2022 Healthy Living Space Detoxifier #42: Cutting the Cords and Ties That Bind\u2014Psychic Plumbing \u2022 Healthy Living Space Detoxifier #43: Ground Yourself to the Center of the Earth \u2022 Healthy Living Space Detoxifier #44: Protect Your Space by Connecting to a Higher Source \u2022 Healthy Living Space Detoxifier #45: Cleanse and Protect Your Aura with Pomanders and \"Air Conditioners\" \u2022 Spiritual Detoxification Is a Lifelong Activity\n\n**P ART THREE Creating the Healthy _Home_ Living Space**\n\n**C HAPTER 10 Cleaning Up the Home's Indoor Environment**:\n\n**_Overcoming the Effects of Sick Building Syndrome_**\n\nSick Building Syndrome: The Disease of Modern Architecture \u2022 Indoor Pollutants\u2014A Matter of \"Genuine Concern\" \u2022 The Healthy Living Space Expert Interview: Michael Riversong, Design Ecologist and Environmental Assessor \u2022 Healthy Living Space Detoxifier #46: Get the Electropollution Out of Your Home \u2022 Healthy Living Space Detoxifier #47: Benefits of Nontoxic Carpets\u2014Don't Let Your Carpets Kill You Softly \u2022 The Trouble with Indoor Air\u2014Worse Pollution Indoors than Out \u2022 Healthy Living Space Detoxifier #48: Filter Your Home's Indoor Air to Remove Toxins and Allergens \u2022 Healthy Living Space Detoxifier #49: Install an \"Indoor Waterfall\" to Add Negative Ions to Your Air \u2022 Healthy Living Space Detoxifier #50: Use Houseplants to Filter Out Toxins from Your Indoor Air \u2022 Healthy Living Space Detoxifier #51: Get the Radon Out of Your Basement \u2022 Healthy Living Space Detoxifier #52: Take the SADness Out of Your Indoor Space with Full Spectrum Lighting \u2022 Healthy Living Space Detoxifier #53: Go Green with Your Household Products\n\n**C HAPTER 11 The Feng Shui of a Healthy Home**:\n\n**_Detoxifying the Energy Aspects of Your Living Space_**\n\nCharting the Movement of Life Force Energy \u2022 Healthy Living Space Detoxifier #54: Evaluate the Energy Flow in Your Home with the _Ba-Gua_ \u2022 The Healthy Living Space Expert Interview: Stanley Aaga Bartlett, Feng Shui Master \u2022 Healthy Living Space Detoxifier #55: Tidy Up All the Cluttered Areas in Your Home \u2022 Healthy Living Space Detoxifier #56: Repaint All the Interior Walls with Your Color of Happiness \u2022 Healthy Living Space Detoxifier #57: Clear Your Space of All Predecessor Energy \u2022 Healthy Living Space Detoxifier #58: The Health Benefits of Ghost-Busting \u2022 Healthy Living Space Detoxifier #59: Eliminate Poison Arrows and Cutting _Qi_ \u2022 Healthy Living Space Detoxifier #60: Pacify Your Neighborhood with Feng Shui Mirrors \u2022 Healthy Living Space Detoxifier #61: Optimizing the Mouth of _Qi_ at Your Front Door \u2022 Healthy Living Space Detoxifier #62: Setting Up a Feng Shui-Friendly Bedroom \u2022 Healthy Living Space Detoxifier #63: Setting Up a Feng Shui-Friendly Bathroom \u2022 The Health Costs of Having an Energy-Imbalanced Home\n\n**C HAPTER 12 Cleansing the Home's Exterior Environment**:\n\n**_Neutralizing the Effects of Geopathic Stress_**\n\nDisease Can Be a Problem of Location \u2022 Healthy Living Space Detoxifier #64: When Nothing Else Heals, Try Moving Your Bed \u2022 Healthy Living Space Detoxifier #65: Scan YourHome and Environment for Signs of Geopathic Stress \u2022 The Healthy Living Space Expert Interview: Patrick MacManaway, M.D., Ch.B., Geomancer \u2022 Healthy Living Space Detoxifier #66: Take Your Energy Body Out of the Curry-Hartmann Grid \u2022 Healthy Living Space Detoxifier #67: Five Simple Techniques for Detoxifying a Room or Environment \u2022 Healthy Living Space Detoxifier #68: Put in an Energy Drainage Pipe from Your House to the Center of the Earth \u2022 Healthy Living Space Detoxifier #69: How to Take the Stress Out of a Geopathic Zone \u2022 Healthy Living Space Detoxifier #70: Selecting Energy Wells in a Geography of Enlightenment\n\n**Afterword**\n\nLife in Healthy Living Space\n\n**Endnotes**\n\n**Index**\n\n# _Acknowledgments_\n\nThe author wishes to thank the generous contribution of time, insights, and protocols from the following experts in detoxification: Peter Holyk, M.D., Jacob Falin, N.D., Patricia Kaminski, Rev. Leon S. LeGant, Michael Riversong, Stanley Aaga Bartlett, and Patrick MacManaway, M.D. Thanks to Stephanie Marohn for her excellent editing.\n\n# The 70 Healty Living Space Detoxifiers at a Glance\n\n**_1_** Fill Out Your Toxicity Self-Assessment Questionnaire\n\n**_2_** Maintain a Household Toxic Exposure Inventory\n\n**_3_** Get a Clear Idea of Your Nutritional Status\n\n**_4_** Assess Your System's Ability to Detoxify\n\n**_5_** Determine Your Body's Total Toxic Load\n\n**_6_** Find Out if You're Suffering from Hidden Food Allergies\n\n**_7_** Rehydrate Your Body with Nature's Best Detoxifying Substance\n\n**_8_** Get the Chlorine Out of Your Drinking and Shower Water\n\n**_9_** Don't Fluoridate Your Teeth\u2014Your Body Doesn't Need Fluorides\n\n**_10_** Cleanse Your Produce of Toxic Chemicals\n\n**_11_** Whenever Possible, Eat Organic Foods\n\n**_12_** Minimize Your Consumption of Foods Containing Trans-fatty Acids\n\n**_13_** Do a 3-Day Water or Juice Fast to Rest and Regenerate Your Cells\n\n**_14_** Do a Media\/Image Fast to Cleanse Your Mind and Energy Field\n\n**_15_** Make a Lifestyle Out of Nonstop Stress Reduction with Healing Hot Water Soask\n\n**_16_** Enhance Your Relaxation by Floating in 800 Pounds of Epsom Salt\n\n**_17_** Use Music Therapy to Deepen Your Sense of Melting Relaxation\n\n**_18_** Do a One-Week Elimination Diet and Liver Cleansing Program\n\n**_19_** Use Natural Foods and Herbs to Aid Your Liver in Cleansing Itself\n\n**_20_** Do a Natural Liver Flush to Improve Your Internal \"Plumbing\"\n\n**_21_** Use Six Natural Substances to Help Cleanse Your Intestines\n\n**_22_** Remove the 50+ Feet of False Mucoid Lining from Your Intestines\n\n**_23_** Purge the Parasites from Your Intestines\n\n**_24_** Get Proactive with Probiotics\u2014Plant Food for the Intestines\n\n**_25_** Fertilize Your Bifidobacteria with Prebiotics\n\n**_26_** Bounce on a Trampoline to Improve Your Lymphatic Drainage\n\n**_27_** Dry Skin Brushing\u2014Tone Up Your Body's Largest Elimination Organ\n\n**_28_** Exercise Regularly to Enhance Blood Circulation and Toxin Removal\n\n**_29_** Sauna Detox: How Heat Therapy Can Help Your Body Release Toxins\n\n**_30_** A Purifying Bath: Whole-Body Detoxifying Soaks in Herbs, Salts, Oils, and Other Natural Substances\n\n**_31_** Make Your Mouth a Mercury-Free Zone by Having Your Mercury Fillings Correctly Replaced\n\n**_32_** Have Your Cavitations Cleaned Out and Reduce Possible Jawbone Infections\n\n**_33_** Think Twice before Digging Another Root Canal in Your Mouth\n\n**_34_** Determine Your Biocompatibility with Dental Materials before Putting Any More in Your Mouth\n\n**_35_** Remedying the Four Basic States of Emotional Toxicity with Flower Essences\n\n**_36_** Salting Your Energy Centers\n\n**_37_** Request a Cleansing from Above\n\n**_38_** Do a 30-Minute Spirit-Detoxifying Bath\n\n**_39_** Shake Yourself Free of Toxic Energies\n\n**_40_** Scrub Your Aura Clean with a Golden Sponge\n\n**_41_** Turn Your Body into Transparent Glass\n\n**_42_** Cutting the Cords and Ties That Bind\u2014Psychic Plumbing\n\n**_43_** Ground Yourself to the Center of the Earth\n\n**_44_** Protect Your Space by Connecting to a Higher Source\n\n**_45_** Cleanse and Protect Your Aura with Pomanders and \"Air Conditioners\"\n\n**_46_** Get the Electropollution Out of Your Home\n\n**_47_** Benefits of Nontoxic Carpets\u2014Don't Let Your Carpets Kill You Softly\n\n**_48_** Filter Your Home's Indoor Air to Remove Toxins and Allergens\n\n**_49_** Install an \"Indoor Waterfall\" to Add Negative Ions to Your Air\n\n**_50_** Use Houseplants to Filter Out Toxins from Your Indoor Air\n\n**_51_** Get the Radon Out of Your Basement\n\n**_52_** Take the SADness Out of Your Indoor Space with Full Spectrum Lighting\n\n**_53_** Go Green with Your Household Products\n\n**_54_** Evaluate the Energy Flow in Your Home with the _Ba-Gua_\n\n**_55_** Tidy Up All the Cluttered Areas in Your Home\n\n**_56_** Repaint All the Interior Walls with Your Color of Happiness\n\n**_57_** Clear Your Space of All Predecessor Energy\n\n**_58_** The Health Benefits of Ghost-Busting\n\n**_59_** Eliminate Poison Arrows and Cutting _Qi_\n\n**_60_** Pacify Your Neighborhood with Feng Shui Mirrors\n\n**_61_** Optimizing the Mouth of _Qi_ at Your Front Door\n\n**_62_** Setting up a Feng Shui-Friendly Bedroom\n\n**_63_** Setting up a Feng Shui-Friendly Bathroom\n\n**_64_** When Nothing Else Heals, Try Moving Your Bed\n\n**_65_** Scan Your Home and Environment for Signs of Geopathic Stress\n\n**_66_** Take Your Energy Body Out of the Curry-Hartmann Grid\n\n**_67_** Five Simple Techniques for Detoxifying a Room or Environment\n\n**_68_** Put in an Energy Drainage Pipe from Your House to the Center of the Earth\n\n**_69_** How to Take the Stress Out of a Geopathic Zone\n\n**_70_** Selecting Energy Wells in a Geography of Enlightenment\n\n# _**Note on Products, Services, and Organizations**_\n\nThere are many products, services, and organizations listed in this book. Their inclusion in the text does not constitute an endorsement by the author. In some cases, the author has personally tested or used the products or services, but in other cases, not. Further, the author has no financial or business connections with any of the products, services, or organizations listed.\n\nThe inclusion of this information is meant solely to help the reader find a point of orientation in what is perhaps a new and overly large field. References to specific products, such as water or air filters, for example, are only meant to serve as a starting point from which readers can launch their own more extended research and vetting. They are not meant to be the result of exhaustive research into these specific categories; rather, they are the results of a preliminary and provisional research of the field, principally through the Internet.\n\nAll health information in this book is based on sound medical and scientific research and\/or the clinical protocols of credentialed health care practitioners. However, this is no substitute for qualified medical guidance and advice if there is any question about the appropriateness or applicability of any recommendation in this book to individual readers and their unique health conditions. Readers concerned about the suitability of specific techniques or substances to their specific health condition should consult a qualifed health care practitioner before undertaking any of the detoxification steps.\n\n# _INTRODUCTION_\n\n**_Why Toxicity Should Concern Us_**\n\nEver since Rachel Carson's shocking indictment of industrial pollution in _Silent Spring_ , the concept of poisons in the environment has been with us. Carson gave us the basic idea: too many toxic substances in the environment sicken, even kill, wildlife and natural systems, and they eventually make us sick too. As a society we haven't forgotten the concept of toxicity introduced by her book in 1962, yet to a large measure it has remained a concept and not a living reality, not something we know to be true in our own lives.\n\nIn other words, we don't quite see how toxicity is the order of the day for most of the world, in terms both of the landscape and the body, the planet and the home. Yet as this book will show, toxicity\u2014the long-term health impact of long-term exposure to toxic substances\u2014has crept up on us and now to a large extent _defines_ our physical reality.\n\nWhat is our physical reality? One way to describe it is to call it a living space. We, as conscious human beings, live in a body and in a home, be it studio apartment, condominium, house, or mansion. The body _and_ home are our living space. Both should be considered together in any discussion of health, toxicity, or illness because they exist in a remarkable feedback system; both can make us well and both can sicken us. If we want to be healthy, we need to concern ourselves with both. A healthy living space is a body and home free of toxins.\n\nBut most of us don't have a healthy living space. We live in a toxic world, and it is steadily sickening us; in fact, so pervasive is its toxic effect that we almost don't see it. Yet scientific studies now show that nearly every corner of the planet is toxic and every human being carries residues of dozens of toxic chemicals in their cells. Every element of our physical world\u2014air, water, earth\u2014carries a toxic burden, is poisoned, and is poisoning us.\n\nAs we well know, the incidence of chronic and degenerative diseases, including cancer, is ever increasing. Toxicity is slowly draining th e life out of us, yet so pervasive and effective is this systematic poisoning that we are barely, if at all, aware of it in our everyday living. We think the way we feel is \"normal,\" the way a human is supposed to feel, even though we may suspect we're a little off.\n\nWe may not personally feel \"sick\" and we may not have been officially diagnosed as having a \"condition,\" but we vaguely sense a degree of dysfunction in our body. We don't quite have the energy we used to; we get colds more often than before; allergies seem more prevalent; we're bothered by various minor symptoms that come and go but never really leave; our sense of vitality and immunity seem off, a bit compromised.\n\nIncreasingly, our homes have become toxic too. This is a sad and shocking idea to contemplate: the home is supposed to be the heart of our domestic life, a refuge, a haven\u2014how can such a place designed to nourish life make us sick? Yet it does. Scientific studies show that most of our modern building and home furnishing materials\u2014paints, plywood, glues, varnishes, carpets, upholstery, household cleaning products, bedding materials, even our clothes-are laced with toxic chemicals that are slowly released from the products into the indoor air of our increasingly airtight or inadequately ventilated homes. Headaches, rashes, allergies, respiratory problems, frequent colds, irritability, depression, immune system problems\u2014these are among the numerous documented symptoms attributed to the effect of our exposure to toxic building materials in our homes.\n\nSo how do you get a healthy living space? You detoxify. Detoxification is a term generally understood in reference to substance abuse treatment programs, but it actually refers to a much more generalized process of ridding the body of all toxic substances, not just freeing it from dependency on alcohol or other addictive substances. Detoxification is a natural, routine activity of the body, carried out every day by the liver, lymphatic system, kidneys, and intestines. The body can handle a fair degree of toxic materials and survive, and in fact remain healthy.\n\nBut there is a limit, and for many of us, we reached that limit, and surpassed it, years ago. Now our natural detoxification processes are hindered; they don't work at full capacity; they have a tremendous backlog of toxic substances to process, and without our help at this point, they may never catch up. Not catching up with its detoxification load means the body will start to get poisoned by its ever-increasing reservoir of unprocessed toxins, and it will start to show signs of impaired health.\n\nIn many traditional societies, regular detoxification was part of the lifestyle. Once or twice a year, individuals would deliber ately take steps to help their body rid itself of the toxic burden of the preceding months. People would fast, drink copious amounts of water, do a sweatlodge, take special herbs, practice specific exercises, and give their bodies a chance to cleanse themselves.\n\nEven in healthy times, when people lived in a relatively toxic-free environment, it was a good idea to periodically give the body a break from the endless routine of eating, digesting, assimilating, and eliminating, and let it fully catch up and \"clean house.\" With a clean house\u2014a healthy living space\u2014people felt better in themselves; their emotions were clearer, their minds were more unfettered, and they felt more attuned with the world.\n\nIn our day, with the environment and the body intensely polluted, the need for regular detoxification is acute. The only way to survive and _flourish_ in such an environment is to undertake a comprehensive program of internal _and_ external cleansing. The abiding assumption in this book is that detoxification, the basis of true and lasting health, must address both aspects of one's living environment\u2014the body and the home. Body and home are reflections of each other and must be addressed simultaneously.\n\nIt is essential that we detoxify _both_ expressions of our living space at the same time: our body and our home. If you cleanse your body but live in a toxic living space, your home will eventually sicken you; if you detoxify your home but do nothing to purge your body of its toxic burden, the healthy home will not prevent the health consequences of bodily toxicity. Your internal toxins will gradually sicken you.\n\nA clean (detoxified) house and a clean (detoxified) body complement each other, leading to a state of harmony, balance, and deep health. The goal is a healthy body and home\u2014a healthy living space\u2014and the means to get there are the seventy practical steps presented in this book. _The Healthy Living Space_ provides instructions on how to regain a healthy body and home based on safe, proven, nontoxic, noninvasive methods from the field of alternative medicine.\n\nThis is a how-to book on alternative medicine applications for better health through systematic detoxification, or cleansing, of both our internal and external living environments. The essence of this book is self-care: you can do it yourself. All of the procedures recommended in the book can be done by the average nonmedically trained person, although in some cases professional consultation is recommended as a complement to one's own informed initiative.\n\nThe procedures\u2014I call them Healthy Living Space Detoxifiers\u2014are supported by scientific research and the clinical experience of numerous physicians, health-care practitioners, and detoxification experts, seven of whom I interviewed specifically for this book. For each of the seventy Detoxifiers, I explain the reason for doing them\u2014that is, the nature of the toxicity addressed\u2014and the steps to follow to do them. In many cases, information about practitioners, and sources of products, services, or more information are provided to help you get started on detoxification.\n\nDo you need to do all seventy steps? Yes, if you want a thorough detoxification of body and home. No, if you're willing to settle for less. Regarding the detoxification of the body, as a minimum it is advisable to do the steps that address the liver and intestines, in the order they're presented in Chapter 5. But the steps described in chapters 4 and  complement the approach that focuses on the liver and intestines, as a kind of before and after framework.\n\nYou not only want to purge your body of its accumulation of toxic substances (Chapter 5), but you should reduce as much as possible your further intake of toxic substances through food, air, water, and your exposure to toxic products and environments (chapters 4 and ). If you don't change some of the aspects of your lifestyle that put you in contact with toxic substances, you may end up replenishing the supply of poisons, even though you removed them from your body with liver and intestinal detoxification.\n\nAt a bare minimum, you should seriously consider trying the steps in Chapter 10 which address the sources of toxicity in your physical indoor environment. But as this book emphasizes, toxicity is a broad spectrum of pollution. It affects not only the body and our physical living space, but our emotions, mind, and energy field\u2014all the aspects of being human.\n\nSimilarly, toxicity in the home can affect it, as a living space, in all aspects, which in turn affects your experience of the space\u2014how it feels to be in it, what its energy is like. You will see that, remarkably, placing your bed in the wrong spot in your bedroom can be just as toxic to you as breathing the outgassed fumes from a new synthetic carpet. It poisons you on a different level.\n\nThe 70 Healthy Living Space Detoxifiers work from the more obvious physical aspects of our individual being and living environment into the more interior aspects (emotions and thoughts) and beyond this into the energy field in which our body and home reside. Ultimately, the _healthy_ living space is one that is healthy for consciousness, for our lives as awake, aware human beings living in a body that dwells in an indoor living space set in a living external environment.\n\nWe don't know what it feels like to be truly healthy; our great grandparents might have, but we don't. How much of our view of the world and ourselves is conditioned by being filled with toxic substances from our industrialized world? It's a question we can answer only from the inside, after we have detoxified and taken a fresh look around at things, at our body, home, and world-a prospect that might serve as an inducement to start detoxifying.\n\nDetoxification is not merely a medical issue, a route to better health. It isn't something to do only if you're \"into\" being healthy. Whether we are toxic or nontoxic defines every aspect of our physical existence, our feelings, thoughts, how our body works, how we see the world, how consciousness lives in matter. Toxicity deals with the foundations of our life, with how every cell, molecule, and organ functions, or dysfunctions. With how we think, or don't think; how we feel, or don't feel. With how our thinking and feeling are distorted or compromised or not fully available to us. A great deal is at stake.\n\nThe extent of toxicity in our environment today\u2014body, home, and world\u2013is shocking and distressing, but the situation is not hopeless. There is _much_ an individual person can do to reverse this, both at home and in the world. Individual actions and choices count; they have an impact globally. If you start detoxifying and change your consumer habits, this starts to send little shock waves through the marketplace. You start thinking, feeling, and acting differently; you no longer are conditioned or weighed down by toxicity, and you see your responsibility in the world in a new light, or perhaps for the first time. Life after detoxification can be refreshing, even liberating.\n\nHere is a brief survey of the ground we'll be covering in _The Healthy Living Space_ , a roadmap for the way to detoxification. The book is divided into three parts. Part one, \"The Health Perils of the Toxic Body and Home,\" discusses the nature of toxicity, defining a wide range of toxins from chemicals to energy influences (Chapter 1). It documents the injurious biological effect of toxins, their prevalence in the world, and how they compromise immunity and produce illness. The biochemical processes of detoxification (through the liver, kidneys, and intestines) are explained in Chapter 2, showing how you can work with your internal detoxification system.\n\nThe toxins discussed range from estrogen-mimicking industrial chemicals and irradiated and genetically modified foods to radon, stagnant life force energy, electromagnetic pollution, and harmful energies from the Earth itself, among many others. The book also explores the effects of two levels of toxins not ordinarily considered in this kind of discussion: unresolved emotional toxicity and auric toxicity, which means deleterious energies in the field surrounding the human body (the aura).\n\nPart two, \"Creating the Healthy _Body_ Living Space,\" addresses the safe, systematic detoxification of the body using natural remedies and alternative medicine practices. First, nineteen laboratory tests are outlined in Chapter 3, each of which provides essential (and low-cost) information about your specific level of toxicity and biochemical ability to withstand toxicity and disease.\n\nNext, in Chapter 4, we look at practices and lifestyle changes for internal cleansing, such as dietary modifications, the need for pure water, the avoidance of contaminated foods and beverages, and the benefits of short-term fasting, among others. It isn't all avoidance. There are some positive things to do, such as taking healing baths to reduce stress, using music to melt your tension, bouncing on a trampoline to move your lymph, or sitting in a sauna, as explained in chapters 4 and . These are the before and after aspects of detoxification, how to prepare for it, how to sustain the benefits.\n\nIn Chapter 5, you'll get detailed instructions on how to detoxify your liver, kidneys, lymphatic system, and intestines, and information on why you should regard your teeth (and what dentists have done to them) as potential sources of toxicity. Chapter 7 will tell you how to address and resolve that dental toxicity.\n\nTraumatic emotional experiences act as toxins when left unresolved in the system and have been shown by medical research to significantly contribute, to illness. So in Chapter 8, the effective role of flower remedies is discussed in conjunction with a leading practitioner. In a similar manner, unsuspected injurious energies and presences in the energy field, or aura, surrounding the human act as toxins, contributing to states of ill health and numerous physical and mental symptoms. In chapter 9, you'll find techniques for diagnosing and removing auric pollution, as well as a fascinating explanation of where it comes from.\n\nPart three, \"Creating the Healthy _Home_ Living Space,\" looks at the layers of toxicity in the typical indoor living environment, and how to remove them or minimize their negative health effects. Chapter 10 discusses what is called \"sick building syndrome,\" a field of research that documents the numerous ways in which structures and contents of a house or apartment contribute to ill health. The emerging discipline of Baubiologie (the \"biological, living house\") is detailed, along with its many practical tips on detoxifying the physical aspects of one's home. Some of the tips include the use of pollution-absorbing houseplants, negative ionization, nontoxic cleaning products, air filtration, and full spectrum lighting.\n\nBuilding on this, the book moves to the next level of household toxicity, the realm described by feng shui, the increasingly popular Chinese art of placement. The shapes of rooms, the orientation of a house, the placement of furniture in a room, and many other similar factors affect the flow of energy through the house. This flow, in turn, either supports or hinders the health of the occupants. In Chapter 11, feng shui experts provide numerous practical tips for improving energy flow and removing the places of stagnation or energy toxicity in the house that contribute to ill health.\n\nIn Chapter 12, we look at how harmful energies from the Earth itself or the land around the house can permeate your home and make you toxic. Such energies include aberrant electromagnetic fields, deriving from electrical power lines and\/or underground water currents.\n\n_The Healthy Living Space_ will show you how your body and home form a living system in constant interaction, feedback, and communication. Lasting health can be had when you consider both in the health equation and when you address the toxic burden of both at the same time. You live in two places: your body and your home. Both can be toxic, both can be healthy, but given the current state of the world, neither are likely to be healthy without your active participation. You already know what a toxic living space feels like; all of us in the Western industrialized part of the world know this today. But what might a detoxified living space feel like?\n\n# _Part One_\n\n# _The Health Perils of the Toxic Body and Home_\n\n# CHAPTER 1\n\n# _Toxicity and Illness: How Your Body and Home Became Toxic_\n\nIt is an unfortunate fact that most of us living in the industrialized world today are toxic. We have been poisoned by the chemicals in our environment, and our bodies have become a limited toxic waste dump, a contaminated landfill, a biohazard zone. This is not an exaggeration to make a point: _it is_ the point, and there is sufficient data now available to substantiate what might otherwise be viewed as a hysterical assertion.\n\nOur bodies are overfull of poisons, not the kind that immediately make you ill, but the slow-acting kind, the ones that sicken, and even kill, you over time. They have a cumulative effect such that the more you amass in your body and the older you get without removing them, the more likely it is they will interfere with your health. Allergies, frequent colds and flus, unexplained headaches and muscle cramps, immune system disorders, hyperactivity, attention deficit disorder, chronic fatigue, environmental illness, multiple chemical sensitivities, asthma, infertility, arthritis, Alzheimer's, multiple sclerosis, heart disease, cancer\u2014all can be attributed in large measure to unrelieved toxicity. Many more health conditions could be added to this list of toxin-generated discomfort and disease.\n\nIn fact, the spectrum of health effects ranges from chronic minor problems, such as allergies, to persistent major problems, degenerative disease, cancer, and shortened life span. The alarming fact is that there is a near certainty that right now both your body and your home are toxic. Your body is loaded with toxic substances, and your home and its furnishings, from carpets to cleaning agents, acts as a source for your continuous toxic exposure.\n\n**_They're called xenobiotics\u2014industrially produced foreign chemicals, synhetic organic chemicals that are not native to our natural environment, either within our bodies or in nature, hence the appropriate term, \"xeno.\" As foreign substances in our natural ecosystems, most xenobiotics confound the ability of nature orour bodies to get rid of them, or even to neutralize them_**.\n\n### 250 Chemical Contaminants in Every Human Body\n\nThey're called xenobiotics\u2014industrially produced foreign chemicals, synthetic organic chemicals that are not native to our natural environment, either within our bodies or in nature, hence the appropriate term, \"xeno.\" Think of plastics, pesticides, preservatives, aluminum, and thousands of other modern substances and products you take for granted, or don't even know about. That's where you'll find xenobiotics. They're in your foods and in your home. As foreign substances in our natural ecosystems, most xenobiotics confound the ability of nature or our bodies to get rid of them, or even to neutralize them.\n\nThough they may seem to make our world run more efficiently, their hidden cost is that they unfailingly make us sick. And this buildup of toxins in our external and bodily environment has been steadily expanding since about 1800 and the onset of the Industrial Revolution.\n\nXenobiotics do not observe national borders. It is not only Americans or Canadians who are growing steadily more toxic. Toxicity is global. \"Virtually anyone willing to put up the $2,000 for the tests will find at least 250 chemical contaminants in his or her body fat, regardless of whether he or she lives in Gary, Indiana, or on a remote island in the South Pacific.\"\n\nThis sweeping\u2014and alarming\u2014statement was made only a few years ago by Theo Colborn and her colleagues in _Our Stolen Future_ , a startling documentation of our toxic environment and its ramifications for worldwide health. Synthetic chemicals are everywhere and you cannot escape them, Colborn concludes. They are in the Arctic and Polynesia, the Swiss Alps and mother's milk. In just six months of breast-feeding, an infant in the United States or Europe gets \"the maximum recommended lifetime dose of dioxin,\" one of the world's most toxic substances, a byproduct of pesticides. This killer chemical \"rides through the food web\" from plants to animals to humans and can end up anywhere on the planet.\n\nIn fact, the Arctic, says Colborn, may be the most polluted environment on the planet in terms of its concentration of volatile persistent chemicals, such as PCBs, or polychlorinated biphenyls. Canadian researchers found that residents on a remote Arctic island had the highest observed levels of PCBs of any human population excepting those specifically exposed through an industrial accident. PCBs, created as coolants and insulators for electrical equipment, are world travelers, migrating through ecosystems and over great distances, lodging in the fat cells of all kinds of living organisms, Colborn observes.\n\nPersistent chemicals such as PCBs are characterized by \"extreme stability, volatility, and a particular affinity for fat.\" That means they persist, do not easily decay or biodegrade into nontoxic forms, are easily excited or activated and vaporized, and tend to lodge in fat cells of living biological organisms (such as humans) from which they are hard to remove. Because they lodge in fat cells, they are almost exempt from the body's natural detoxification system, which requires toxins to be water-soluble, and cannot easily deal with fat-soluble toxins.\n\nIntroduced in 1929, PCBs were banned in the United States in 1976, but by then an estimated 3.4 billion pounds of PCBs had been produced worldwide and released into the global environment. And they're still there, in the body fat of almost every living creature in the world, states Colborn. PCBs are only one class of _many_ synthetic organic chemicals produced and unleashed into the global environment during the period from 1900 to 2000, which in retrospect surely ought to be called the toxic century.\n\nOne name for this global family of industrial synthetic chemicals is POPs, persistent organic pollutants. These are carbonbased chemical compounds\u2014such as PCBs, DDT, chlordane, dieldrin, HCB, aldrin, dioxins\u2014that persist in the environment (resisting degradation), are semi-volatile (evaporating easily), and have a low water solubility (preferring fat cells). As POPs move up the food chain, from fish to human, their concentrations can increase by factors of thousands or even millions.\n\nHere's another rivetting example. You would think living mostly outdoors and close to nature in ultra-rural Greenland amidst vast amounts of snow and no industry would be a healthy lifestyle. The opposite is true. Researchers at a major hospital in Quebec City, Quebec, found that the Inuits of Greenland (a people native to the country) had the world's highest body burden of organochlorines (toxic chlorine-derived compounds) due to environmental exposure. The Inuits depend largely on sea mammal fat for their food supply; unfortunately, the sea mammals are intensely polluted with fat-loving POPs. The researchers found evidence of twenty-five different pesticides and POPs in the fatty tissues of the Inuits examined, and they found that the native Greenlanders had concentrations that were three to thirty-four times higher than found in a population in urban, industrialized Quebec City,\n\nAlthough neither Greenland nor the Arctic are still pristine, surely the rarefied peaks of high mountain ranges, such as the Swiss Alps must still be relatively toxin free. Not so, says Roland Psenner, a professor at Innsbruck University in Innsbruck, Austria. Ironically, the Alps act as a magnet for toxic pollutants in the atmosphere, Psenner reports. In lakes above the tree line (typically at 8,000 feet) the fish are contaminated with DDT applied in the tropics against malaria. \"We found that fish in the most contaminated lakes have 1,000 times more DDT than lower-lying lakes,\" Psenner states.\n\nHow can toxins used in the tropics end up on an alpine snow peak? Ecologists call it global distillation, to denote a kind of \"chemical nomadism.\" The subzero temperatures around peaks in the Alps cause atmospheric DDT, collected over Africa and India as evaporation, to humidify and fall as precipitation on the peaks. Chemicals used in Southeast Asian rice paddies vaporize and drift across the planet and condense on the bark of Arctic trees. Global distillation explains why \"the bodies of seals in Siberia's Lake Baikal\u2014the world's oldest and deepest lake\u2014contain the same two contaminants as the alpine soil of New Hampshire's Mount Moosilauke.\"\n\n\"There is no safe, uncontaminated place,\" Colborn concludes. Our body, home, neighborhood, state, nation, planet\u2014all have become toxic, an unhealthy living space. All are contaminated with numerous xenobiotics that are capable of making us sick, and are in the process of doing so.\n\nThe concept of global distillation gives us an image of the planet uniformly blanketed in poisons\u2014globally toxified. Organochlorines do not observe national boundaries; hydrocarbons do not stay put; pesticides hunger for the entire world. The use of a single toxic substance has planetary repercussions. When you spray dandelions in your front yard with a convenient pesticide dispenser you bought at the hardware store, you are potentially sending toxics to the far ends of the planet: You could be poisoning polar bears at the North Pole.\n\nBut once you _get_ the relationship between local and global toxics, you get something bigger, better. When you realize that your actions have global effects, you start thinking globally. It may be a shocking moment when you see the reciprocity, the mirroring, that exists between you and the planet\u2014both overflowing with toxins. You see that you and your home are embedded in the planet's own body, its life, which we call its ecosystem. The load of toxins you and the planet share is the negative aspect; the positive aspect is that your willingness to detoxify also has world consequences. You can start saving the polar bears from your garage. If you can't tolerate dandelions in your lawn, it is a global act to reach for the weeding spade instead of the herbicide.\n\nThe volume of toxic substances regularly added to the world environment is staggering to contemplate. In one year alone: 550 million pounds of industrial chemicals were dumped into public sewage; 1 billion pounds of chemicals were released into the ground; 188 million pounds of chemicals were discharged into surface waters; 2.4 billion pounds of particulates were sent into the air. The estimated total of toxic chemical pollutants released into the environment in one year (1989) was 5.7 billion pounds. \"That is enough to fill a line of semi-trailers parked bumper to bumper, and having a cargo capacity of 45,000 pounds each, stretching from downtown Los Angeles to Des Moines, Iowa.\"\n\nThe purpose of this chapter is not to scare, however, but to inform and motivate. It is necessary to fully grasp the _fact_ of toxicity and the degree to which it has permeated our environment, our bodies and home living spaces, not to mention the planet's ecosystem itself. It is vital to understand what unchecked toxicity is doing to our bodies and our world. But this alarming knowledge should become the _foundation_ upon which we build practical, effective steps to detoxify ourselves, to deliberately remove the myriad toxins from our bodies and homes, and eventually, from the world as well.\n\nWhen we start detoxifying, we inevitably become more aware\u2014grippingly so, usually\u2014of the state of the global body, our planetary living space. We see how it, too, is toxic. We become aware of what we're buying, using, consuming, ingesting, and, with shock, realize we have been steadily, if unknowingly, poisoning ourselves\u2014and the environment\u2014for years. As our consumer choices change, as our lifestyle puts a premium on health, and as our political priorities shift as a result, our personal commitment to detoxification starts to have global ramifications. But first, let's get a visceral sense of what being toxic feels like. Are we toxic and don't know it?\n\n## HEALTHY LIVING SPACE DETOXIFIER #1 Fill Out Your Toxicity Self-Assessment Questionnaire\n\nUnless we are obviously, uncomfortably sick, ill to the extent that we are conspicuously disabled, it seems unlikely\u2014even illogical\u2014to think we might be toxic. If I'm so toxic, why aren't all these terrible poisons making me really ill, laying me low with a serious health problem, we reason. While it may seem logical, the reasoning is flawed.\n\nThe truth is that we are all being slowly poisoned, because although the toxins are potent, we are rarely exposed to a lethal or even sublethal, but dangerous, dose. We are instead _routinely_ exposed to very small doses of many toxic chemicals, which together overwhelm our body's natural detoxification system. Notice the key words here: routinely, small doses, many. It's the _frequent_ exposure to _microdoses_ of _multiple_ toxic chemicals that sickens us slowly. Very few of us are seriously poisoned through a single toxic exposure to a large amount of one chemical.\n\nOver time, these many toxic chemicals to which we are routinely exposed in the _normal_ course of living sicken us, at first generating dozens of \"hidden\" symptoms, then creating serious debilitating illnesses that conventional medical analysis does not link to a lifetime of toxic exposure. The symptoms are \"hidden\" only because we are not trained to look for them. Once you know what they are, toxicity becomes fairly easy to spot. Various laboratory tests, detailed in Chapter 3, can confirm presumed toxicity and document its extent and nature. It may seem overwhelming to open this Pandora's box, but the guidelines presented throughout the rest of the book will show you how to deal with this toxicity effectively on your own.\n\nYou don't have to be a physician to deal with detoxification; in fact, don't expect your doctor necessarily to be on the same page as you once you wake up to the toxicity factor. Conventional doctors are still resistant to this concept and are often not even especially well educated in its intricacies and seemingly indirect causal links. On the other hand, practitioners of alternative medicine, who base diagnosis on a careful observation of the whole person, on all the body's systems and their interactions, are much more likely to credit unrelieved systemic toxicity as a prime contributing factor in numerous health conditions.\n\n**_Practitioners of altematiue medicine, who base diagnosis on a careful observation of the whole person, on all the body's systems and their interactions, are much more likely to credit unrelieved systemic toxicity as a prime contributing factor in numerous health conditions._**\n\n\"Most people who think they are tired are actually toxic,\" observes Sherry Rogers, M.D., a physician based in Syracuse, New York, who for several decades has been an outspoken and well-informed advocate of the systemic toxicity hypothesis of illness. An alarming number of people, says Dr. Rogers, are \"not functioning with all oars in the water.\"\n\nThey pass their annual physicals yet cannot deny they feel \"dreadful\" a fair amount of the time. They have lots of vague symptoms and never feel quite well, but it's not enough to constitute a diagnostic category, so they're dismissed as hypochondriacs. They do not wake up energized and vivacious in the morning. They rely on stimulants or relaxants to suppress the strange, persistent signals coming from their body, as if the body, in its own baffling language, is _insisting_ that something is not quite right.\n\n\"Because environmental illnesses are apparently characterized by patterns of multiple symptoms in many parts of the body, including the central nervous system, it is understandable that such patients have often received diagnoses of neurasthenia, hysteria, somatization disorder, and various other psychosomatic disorders,\" explains Iris Bell, M.D., a psychiatrist at the University of California-San Francisco School of Medicine, and author of _Clinical Ecology: A New Approach to Environmental Illness?_ 8\n\nBut there is no such thing as a hypochondriac, asserts Dr. Rogers. When a doctor fails to understand that the clinical picture presented by a patient with multiple, vague, or ill-defined symptoms adds up to systemic toxicity, it teaches the patient to ignore or tune out the various body symptoms until they become much worse, unbearable, or even dangerous, says Dr. Rogers. Yet these same persistent, wide-ranging, \"soft, subtle symptoms\" have a biological function: they're meant to serve as \"early warnings of worse symptoms to come if we ignore or mask (cover them up) them with medications.\"\n\nConsider then how many of the following soft, subtle symptoms you have experienced in the last thirty to sixty days, listed according to the particular body system or organ involved. If you get any of these symptoms at a rate between occasionally and frequently, it will be worth your while to start thinking about your health in terms of possible toxicity:\n\n  * _Brain_ : Headaches, faintness, dizziness, insomnia.\n  * _Digestive Tract_ : Nausea, vomiting, diarrhea, constipation, indigestion, bad breath, bloated after eating, belching, passing gas, heartburn, intestinal or stomach pain, unusually smelly stools, offensive smelling urine.\n  * _Eyes_ : Itchy, bloodshot, or watery eyes, swollen, reddened, or sticky eyelids, dark circles or bags under the eyes, blurred or tunnel vision, poor night vision, inflamed or swollen eyelids.\n  * _Muscles and Joints_ : Aches and pains in the joints, arthritis, stiffness, movement limitations, muscle aches and pains, sense of weakness or tiredness, tight or stiff neck, backaches, joints sore upon waking, joint pain after minor exertion or after eating certain foods, numbness or tingling.\n  * _Ears_ : Itchiness, infections or aches, drainage, ringing, hearing loss.\n  * _Nose_ : Sinus congestion, stuffmess, hay fever, sneezing attacks, unusual amount of mucus, runny nose.\n  * _Weight_ : Overweight, underweight, anorexia, food cravings, compulsive or binge eating, water retention.\n  * _Mouth and Throat_ : Persistent coughing; frequent need to clear the throat; sore throat; hoarseness; voice loss; discoloration or swelling in tongue, lips, and gums; canker sores; swollen lymph glands; diminished sense of taste; tongue with a grayish-white, yellow, or thick film; or tongue that is shiny, swollen, or shrunken.\n  * _Skin_ : Acne; dry skin; scaly skin; boils; rashes; hives; hair loss; flushing; unusual amount of perspiration; bruise easily; slow or incomplete wound healing; pale, sallow, greyish, or slightly yellow skin color; loose or flabby skin; skin with a sharp, sour odor; facial blemishes; too many wrinkles or too much skin sagging for your age.\n  * _Heart_ : Irregular, rapid, missed, or pounding heartbeat; circulatory deficits; angina pectoris or other forms of chest pain.\n  * _Lungs_ : Shortness of breath, chest congestion, asthma, bronchitis, wheezing.\n  * _Mind_ : Memory deficits, confusion, poor comprehension, difficulty concentrating, learning disabilities, slurred speech, stuttering, stammering, indecisiveness, mental fogginess, unusually slow or fuzzy mental processes.\n  * _Energy Level_ : Fatigue (persistent or extreme), sluggish feeling, apathy or lethargy, hyperactivity, restlessness, sleepiness, exhausted after ordinary activities, tired after a long night's sleep, feeling hung-over without having consumed alcohol.\n  * _Emotions_ : Sudden, uncontrollable mood swings, unaccountable anger, irritability, aggressiveness, nervousness, fear, anxiety, or depression, persistent negativity in outlook.\n  * _Other Problems_ : Frequent or urgent urination; frequent colds, flus, and other short-term illnesses; a generalized itchiness; pronounced environmental sensitivity (unpleasant reaction to substances around you); unrelieved, intense allergies; immune weakness; unaccountable fever or chills; strange mucus discharges from eyes, ears, nose, and\/or throat; general feeling of congestion, even long after a cold or flu is over; tendency to get a cold or flu when the seasons change.\n\nYou can also assess your possible toxicity level by evaluating your lifestyle for evidence of certain traits or behaviors known to lead to toxicity:\n\n  * _Medical Care_ : Do you have a history of antibiotic use? Do you frequently or continuously use prescription medicines? Take pain-killers and\/or anti-inflammatory drugs such as aspirin, cortisone, and ibuprofen? Do you rely on allergy shots to get through allergy season? Do you have a dozen or more mercury-silver amalgam dental fillings? Several root-canalled teeth?\n  * _Dietary_ : Do you drink more than two strong cups of caffeinated coffee per day? More than one glass of wine, beer, or spirits per day? Smoke cigarettes or get routinely exposed to secondhand tobacco smoke? Consume very little fresh fruits and vegetables or whole grains? Eat a lot of fried, fast, junk, or processed foods? Depend on sugary, high-fat, high-calorie foods? Eat irradiated foods?\n  * _Environment_ : Do you work or live in a brand new facility with poor indoor air circulation and windows that do not open? Are you exposed to new synthetic carpets or wood products that contain formaldehyde? Do you routinely apply garden pesticides or indoor insect fumigants? Do you work or live under conventional fluorescent lights? Do you drink chlorinated, fluoridated water? Do you live or work near high-tension electrical utility poles?\n\nIf any of these lifestyle practices (among the many that could be listed and which are discussed later in the book) characterize your situation, this may indicate that you are toxic to some degree. What kinds of effects can unchecked chronic toxicity have on us? That's the next subject we need to examine.\n\n### How Unrelieved Toxicity Can Compromise Your Health\n\nThe shocking fact is that toxicity may already be interfering with the health of millions. A University of California study in 1996 reported that an estimated 100 million Americans have a chronic illness or disability. Of this vast number, only a few (1.5 million) were sufficiently sick to require treatment in a nursing home or personal care facility. That means the other 98.5 million were among those \"not functioning with all oars in the water,\" as Dr. Rogers pithily commented above. They are in the world, per haps working, perhaps taking a lot of time off, not well, but not acutely sick either, suffering from a condition that does not get better or heal, but which persists and gradually worsens-the definition of chronic.\n\nThe study further documented that the 100 million chronically ill\u2014this is more than one-third of the U.S. population\u2014took eighty-three percent of the prescriptions written and comprised eighty percent of the hospitalizations recommended, sixty-six percent of consultations with doctors, fifty-five percent of emergency room calls, and ninety-six percent of professional home care. It's not just elderly men and women who have chronic illnesses. The study demonstrated that twenty-five percent of men and women under eighteen have at least one chronic health complaint, while thirty-three percent of those between eighteen and forty-four and sixty-six percent of those between Iorty-five and sixty-four have a chronic condition. In other words, the elderly account for only about one-quarter of the number of people with chronic health conditions, while the non-elderly (those aged eighteen to sixty-four and presumed to be in the prime of their lives) represent sixty percent of the cases.\n\nWhen you say chronic illness, you might as well say chronic toxicity. The links between the two are becoming increasingly obvious and empirically documented in the field of alternative medicine where this kind of etiology is understood and expected. One of the prime long-term results of unrelieved toxicity is the formation of a debilitating or degenerative condition. Not only is the link between chronic exposure to toxic substances and the incidence of chronic and degenerative disease becoming ever more evident, but in many respects the typical symptom picture for these illnesses match the list of symptoms associated with chronic toxicity. Perhaps there is only one disease, at the root of all others, and that disease is toxicity.\n\nFor instance, some alternative medicine physicians now speak in terms of \"toxic triggers\" for common and serious health problems. Along these lines: Alzheimer's can be triggered by aluminum, lead, and pesticides; allergies by formaldehyde and various common foods; anemia by lead poisoning and alcohol consumption; arthritis by allergenic foods; asthma by lead, milk products, allergenic foods (foods that produce allergic reactions), and exposure to mildew; autoimmune disease by mercury and silicone breast implants; cancer by pesticides, radiation, and low-frequency electromagnetic fields; cataracts by steroids and industrial pollutants; fibromyalgia (chronic muscle pain) by pesticides; headaches by mercury and allergenic foods; hypertension by lead, cadmium, and mercury; Parkinson's by pesticides and tobacco smoke.\n\nFurther, in a well-regarded book on cancer and alternative medicine treatments, of the thirty-three documented contributing causes to all cancers listed by the two physician authors, about half were directly related to toxicity.\n\nMany toxic substances are more than toxic; they are carcinogens, capable of starting a cancer process in a human body.\n\nThe term carcinogen is an important one in any discussion of toxicity. A carcinogen is something you should avoid whenever possible. Due to the ubiquity of toxic substances in our contemporary environment, you will probably never succeed in avoiding exposure to all carcinogens; your goal should be rigorous minimization of your contact. A fair number of carcinogens have been studied and characterized by health authorities, including the U.S. government, but a larger number remain unstudied and at large in the environment.\n\nAs part of the National Toxicology Program, the Environmental Health Information Service (EHIS, part of the United States Department of Health and Human Services) pub lished \"Substance Profiles\" for forty-one \"agents, substances, mixtures, or exposure circwnstances known to be human car cinogens\" in their _9th Report on Carcinogens, 2000._ On this list you will find aflatoxins, asbestos, benzene, dyes, environmental tobacco smoke, radon, tamoxifen, vinyl chloride, and other substances you may have heard of. The EHIS also profiled 146 substances (or categories of substances) \"reasonably anticipated\" to be carcinogenic. Among these poisons you will find chloroform, diesel exhaust particles, formaldehyde, urethane, and a great number of other unfamiliar agents with long, chemical names.\n\nThat's 187 chemical substances you should avoid, which is not as easy as it sounds, because these agents and mixtures are used throughout the industrial world in thousands of products. And 187 toxic substances represent only a fraction of the poisons out there\u2014what about the hundreds that remain unstudied, not yet officially tagged as carcinogens?\n\nIt's estimated that eighty percent (or 48,000) of the 60,000 industrial chemicals in use have been untested for their carcinogenicity, or their ability to damage the immune or central nervous system (CNS). As a variation on this statistic, another report says only ten percent of 70,000 commercially used chemicals have been tested for their toxic effects on the nervous system. Further, at least 50,000 deaths per year are due to toxic exposures in the workplace, while another 350,000 new cases of illness occur as a result of workplace-related exposures every year.\n\nThese deaths and illnesses include lung cancers from asbestos; bladder cancer from dye exposure; leukemia and lymphoma in workers exposed to benzene and ionizing radiation; chronic bronchitis from dust exposure; numerous CNS disorders from exposure to pesticides and solvents; and cardiovascular disease from carbon monoxide exposure. Bear in mind, these are conservative figures, based on 1992 data, so you can safely assume the numbers are now higher.\n\nYou can make the case for the health-damaging effects of long-term unchecked toxicity by looking at the change in morbidity (disease prevalence) and mortality statistics since 1980. Between 1980 and 1992, the United States death rate due to infectious diseases increased by fifty-eight percent. A fair amount of this was due to AiDS-related deaths, but there was also a measured increase in the death rate from respiratory infections (up twenty percent) and from blood poisoning (septicemia, up eighty-three percent); and mortality from infections increased by twenty-five percent among the elderly (sixty-five and older). In 1980, death by infectious disease was not even in the list of the top four killers, but by 1992, it had become number three. Why?\n\nConventional doctors point to the failure of standard antibiotics to kill the old germs, now known as antibiotic-resistant microbes. Perhaps. But the more astute medical researchers are pointing to sustained damage to the human immune system due to our constant exposure to toxic chemicals. For example, cases of asthma increased by fifty-eight percent between 1970 and 1992, a trend that led the National Academy of Sciences to state that the various known common air pollutants (nitrogen dioxide and ozone) \"interact with allergens to increase the frequency and severity of asthma attacks.\"\n\nIn other words, pollutants plus a predisposition to allergenicity equals more asthma. Once you damage or compromise the function of the immune system, you become vulnerable to more illness, more infectious disease. When the immune system is damaged by chemical exposure, this can result in heightened allergic reactions to specific or, in many cases, to any chemicals.\n\nThe more pronounced chronic illnesses, such as chronic fatigue syndrome (CFS), environmental illness (El), and multiple chemical sensitivities (MCS), point acutely, even dramatically, to the way unrelieved toxicity can generate a host of unpleasant, debilitating, _seemingly_ unrelated symptoms sufficient to seriously incapacitate a person. But the symptoms _are_ related: they are the \"fruit\" of unchecked toxicity. But this is, without qualification, poisoned fruit.\n\nOf these three major conditions, EI and MCS tend to be ignored, denied, misdiagnosed, or willfully marginalized by conventional physicians. At the same time, worldwide membership in MCS support groups increases at an average of 132% per year, while a 1995 survey by the Chemical Injury Information Network of White Sulpher Springs, Montana, revealed that MCS is reported (by patients) as a health problem in 36 countries, including France, South Africa, Brazil, and Croatia.\n\nOne recent study showed that sixty-seven percent of patients diagnosed with CFS and fibromyalgia (chronic muscle pain) also had MCS; while medical professionals estimate 0.5% to 1% of the total population suffers from MCS, MCS support groups place that figure higher, at 10% to 15%. Among the top three chemical offenders most commonly listed in MCS cases are pesticides, formaldehyde, and solvents. (For more on MCS, see Chapter 10.)\n\nConsider the typical or composite symptom list for a person suffering from environmental illness (EI). The symptoms run the gamut of neurologic, immunologic, respiratory, endocrine, cardiovascular, and genitourinary dysfunction. EI symptomatology is staggering (see figure 1-1).\n\nIt doesn't take too much imagination to see that the symptom picture of EI (or CFS, MCS, or even, to some extent, fibromyalgia) is highly similar to the standard list for toxicity symptoms. The symptom lists for chronic fatigue, multiple chemical sensitivities, and fibromyalgia are also suggestively similar. In these intensely debilitating conditions, you can see the body-wide effects of unchecked toxicity.\n\n**Figure 1-1. The Staggering List of Symptoms for Environmental Illness**\n\nThe body's natural detoxification systems\u2014liver, kidneys, intestines, and lymph and immune system\u2014are ouerwhelmed by the toxic load. They are so swamped by the poisons they can barely function and the body falls apart, system by system, leaving the person vulnerable and reactive to nearly everything. For good reason, the popular press has dubbed environmental illness \"20th century disease,\" and characterized its sufferers as being \"allergic to the 20th century.\"\n\n**_The body's natural detoxification systems\u2014liver, kidneys, intestines, and lymph and immune system\u2014are overwhelmed by the toxic load. They are so swamped by the poisons they can barely function and the body falls apart, system by system, leaving the person vulnerabie and reactive to nearly everything._**\n\n### Being Slowly Poisoned Throughout a Lifetime\n\nSome people are so toxic they are \"an entire organic chemistry lab\" in themselves, says William J. Rea, M.D., president of the Environmental Health Center in Dallas, Texas, which he founded in 1974 as a facility to detoxify patients of chemical toxicity. Like Dr. Rogers, Dr. Rea is one of the leading medical authorities in forwarding this new understanding of systemic toxicity and its relationship to chronic illness.\n\nEarly on, Dr. Rea's research showed that our industrial chemical pharmacy is evident even in the newborn. When he analyzed the blood of new mothers, and the chemical composition of the umbilical cord blood of their infants, he found numerous chemicals present in both, including toxins such as acetone, benzene, styrene, carbon tetrachloride, and methopropylketone. Often the levels were higher in the babies than in their mothers, suggesting \"bioconcentration in the newborn.\"\n\nIn another study, Rea tested the blood levels of pesticides in 200 chemically sensitive patients. Among the highest concentrations of chemicals, Rea found DDT and DDE in sixty-two percent of the samples, Hexachlorobenzene in fifty-seven percent, Heptachlor epoxide in fifty-four percent, Beta-BHC in thirty-four percent, and Endosulfan 1 in twenty-four percent. In another study of volatile organic chemicals in 114 patients, Rea found tetrachloroethylene in eighty-three percent, toluene in sixty-three percent, xylene in fifty-nine percent, and six other major chemicals in a range of twenty-two to fifty percent of patients tested. Rea next tested for aliphatic hydrocarbon solvents (found in glue, cement, adhesives, paint thinner, plastics, gasoline) in eighty-five chemically sensitive patients. He also discovered seventy-two patients had detectable blood levels of solvents, and eighty-five percent had at least three solvents present.\n\nIn a different test of 200 patients Rea found that ninety-nine percent had residues at or above 0.05 parts per billion of chlorinated hydrocarbon pesticides, while the mean was 3.4 pesticides per patient. \"These data suggest that chlorinated hydrocarbon pesticides are extremely common in the patient population investigated ... (any) of which may be capable of inducing a variety of biochemical changes at low levels.\"\n\nNot only can multiple toxic chemicals present in the body produce numerous physiologically based symptoms, they can also generate many of the symptoms normally associated with \"mental illness.\" In a clinical study of forty-two subjects under a month's treatment for hypersensitivities and allergies, Rea found that a variety of psychological states\u2014depression, mental blankness, anger, a sense of physical detachment, anxiety, psychotic behavior, emotional disruption, fatigue, and concentration deficits\u2014have been linked to allergic exposures to chemical substances, fumes, and industrial products.\n\nHe also proved the case in reverse. When these chemically toxic patients were detoxified, they showed \"a significant measure of relief from such states as depression, alienation, suspiciousness, and misspent emotional\/physical energy, and an increase in effortful cognitive processing.\" They also had \"substantial gains in essentially all spheres of activity.\" Depression is one of the cardinal signs of an adverse reaction to environmental toxins, Rea stated.\n\nSometimes it's easier to understand how long-term, low-dose toxicity works and what effects it has on the body by considering what happens when a person has an acute reaction to a larger than typical exposure to common toxic elements such as paint or drycleaning materials. Study a case of pronounced, short-term clinical toxicity and you'll see a condensed version of what it's like to be slowly poisoned throughout a lifetime. It's the slowness that makes us often miss the body's indications that a toxic state exists.\n\n\"Most persons 'enjoy' good health, and have a tendency to dismiss 'minor' symptoms, attributing them to a 'cold,' the 'flu,' or some temporary inconvenience,\" observes Janette D. Sherman, M.D., a physician based in Alexandria, Virginia, who specializes in toxicology. Because of this masking and misinter-pretation of body symptoms, illness gets established and is often far advanced before a person typically seeks medical help.\n\nMost people, says Dr. Sherman, ignore the early symptoms\u2014a change in bowel, sexual, or bladder function; shortness of breath during mild exercise; confusion and memory loss\u2014but when their inconvenient symptoms become too significant to discount, _then_ they seek a physician. If they're lucky, they might find one who understands the links between toxicity and symptoms. Here are some cases from Dr. Sherman's practice that illustrate these links:\n\n  * **Case: Toxicity from Paint Exposure.** A man worked in the paint industry as a mixer for almost thirty years. He routinely handled heavy leaky bags of pigments and was exposed to ure thane acid and vinyl paints; he also handled various toxic chemicals such as toluene, benzene, and others, and said his shoes and socks were often soaked from having absorbed the floor spills of these chemicals. When he was fifty-two, he reported chest pain and soon after developed cancer. In the next three years, he became so short of breath he had to use an oxygen mask when he slept. By the time he was fifty-five, he was \"permanently and totally disabled,\" says Dr. Sherman. Thirty years of daily toxic exposure had nearly killed him, and what life was left him he would spend as a cripple.\n  * **Case: Toxicity from Dry-Cleaning Solvent Exposure.** A teenager worked in a dry-cleaning shop. One day she had to mop up some thirty gallons of dry-cleaning solvent that had leaked all over the floor. Two days later she was in the hospital, complaining of headache, shortness of breath, and dizziness. Five days after this, she was back in the emergency room reporting chills, fever, and nausea. A year later, she was still having headaches, as well as vision problems, memory deficits, and a lack of interest in socializing. An examination revealed other neurological problems, including lack of muscle coordination, emotional disturbances, and mental deterioration. She was so sick that at eighteen, she was unable to remain in school. Her exposure to the dry-cleaning solvent\u2014most likely 1, 1, 1-trichloroethane, says Dr. Sherman\u2014produced \"undoubtedly irreversible\" brain damage.\n  * **Case: Toxicity from Coffee Filter Exposure.** This case involved a man who worked in a factory that made paper coffee filters. One day he went to his doctor complaining of headaches over his right eye and ghost images; he said he'd been suffering daily with the ghost images for six months, but from the headaches for two years. Various clinical tests (X-rays, CAT scans, and others) were performed, but the results came back normal. Over the next few years, as his symptoms did not abate, the man was put on a roster of different drugs, and received more tests and procedures. After five years of headaches, by then age forty-five, he was no better, and he was given even stronger drugs. Within the next year, he sustained three heart attacks and was sicker than ever.\n\nThis unfortunate man became \"a walking pharmacopoeia and suffered an irreversible side effect from his medication, while the _cause_ of his illness went unquestioned and unrecognized,\" comments Dr. Sherman. The medical mystery of this man's case was that no doctor had thought to ask him about his work history and possible exposure to toxic substances. Instead, they chased after his symptoms with various medical \"magic bullets,\" failing to help him because they never understood the cause of his sickness. It turned out the coffee filters were permeated with a resin containing formaldehyde, the vapors of which were released in the process. The man inhaled this gas every day. Formaldehyde is a well-known and widely documented toxic substance, capable of producing a great number of symptoms.\n\n  * **Case: Toxic Effects on the Central Nervous System**. A healthy, vigorous woman, sixty-eight, worked part-time doing lawn maintenance, a seemingly innocuous employment: One day she mowed the lawn, raked the clippings, and bagged them, as she usually did. About ten days later, she had a persistent headache and a prickly sensation in her legs. Two weeks later, her legs had become so weak she couldn't climb the stairs in her house, and she tended to lose her balance and fall backwards. Two months after the original incident, she was unable to work at all. She was experiencing intense pain and numbness in her legs and had a chronic rash on her upper torso. A clinical evaluation showed she had an acute case of nerve toxicity.\n\nIt turned out the lawn she had been maintaining had been sprayed with herbicides to kill the weeds. The specific chemicals in the pesticide formula have been linked with neurotoxicity, cancer, skin disorders, and birth defects. In mowing, raking, and bagging the grass clippings, the woman inhaled a strong dose of these killer chemicals.\n\n  * **Case: Toxic Effects on the Respiratory System.** A man, aged thirty-three, was in good health and physically fit. One weekend a pesticide was sprayed inside the office building where he worked, but during its application, the ventilation system was not running, which meant the pesticide fumes remained inside the building. When he reported to work on Monday, the man detected an offensive odor in the building, and soon noticed dizziness and burning sensation in his chest. He began to feel listless. In a few days, he ended up in the hospital with a respiratory infection and remained there for almost two weeks. Nine months later, the office building was sprayed again. This time, within three hours of being exposed to its indoor air, the man developed cold symptoms; a few days later at work, he got nausea and chills, then fainted. He lost six weeks of work from a chest obstruction; when he returned to work, he got sick immediately, with the same symptoms.\n\nHis was a case of acute pesticide toxicity, says Dr. Sherman. The specific pesticide used (a semisynthetic pyrethroid mixed with aromatic hydrocarbons) is known to be an allergenic chemical. It is also known that once you are exposed to it and manifest allergic symptoms, the next time it will take a much smaller dose to produce the same, or worse, reaction. You may also start reacting to products chemically similar to the original offending substance.\n\nThese cases represent a problem of increasing prevalence, says Dr. Sherman; she calls it \"chemical assault.\" A healthy, previously nonreactive person is exposed to a toxic pesticide or other chemical agent without his or her knowledge or consent, and sickens immediately. Worse, a previous ability to tolerate some degree of toxic exposure is now removed. Once the chemical sensitivity has been induced, the person becomes \"symptomatic after exposure to levels of agents that previously would not have bothered him.\" The chemical assaults are often not minor or temporary, Dr. Sherman adds. They can result in \"lifelong adverse effects.\"\n\nThe discerning reader may query: But most of us are never exposed to such high or prolonged doses, so how can this be relevant? These cases of acute toxicity are a kind of time-lapse photograph of the toxic process. It's true, their effects are exaggerated in terms of what most people are likely to experience, but the symptoms and consequences are right on point for what unrelieved toxicity will eventually produce in anyone over a lifetime, or sooner. These people were sickened fairly quickly and noticeably; the rest of us just have weaker doses of the same poisons, so we will grow ill more slowly. But the life-long adverse effects and induced chemical sensitivity are still the outcome, and they are what we must contend with as a result of merely living in the industrialized world today.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #2 Maintain a Household Toxic Exposure Inventory_\n\nThe life of the typical North American routinely exposes one to a stunning variety of toxic substances (in the guise of familiar, presumed \"safe\" consumer products) that can slowly poison us. But we assume their safety at our peril. Among the familiar consumer products that we must now regard as potentially toxic are: solvents for cleaning carpets, furniture, and clothing; caustic cleaning agents for the bathroom and kitchen; household or garden pesticides; aerosol spray cans (they can leak other chemicals besides the intended ones, such as the chemicals used to pressurize the cans); cosmetics (hair sprays, dyes, shampoos, nail polish); detergent soaps; and plastic food and prescription drug containers.\n\nConsider the matter of household pesticides. It is estimated that eighty percent of U.S. homes have an average of three to four pesticide products in use, such as pest strips, bug bombs, bait boxes, flea collars, pesticidal pet shampoos, and others. It is further estimated that about 70 million U.S. households perform 4 billion pesticide applications per year\u2014in other words, putting on a new flea collar, applying anti-cockroach material, or spraying bugs directly. This averages out to fifty-seven pesticide applications annually per household.\n\nSeen from a different angle, the non-agricultural use of pesticides is burgeoning and, with it, the release of ever more toxic substances into the environment and human body. The global market for home and garden pesticides is worth $7 billion annually, and is growing by four percent a year, with U.S. householders accounting for forty percent of the market for household pesticides. To get the full picture, you have to add to this staggering data the amount of pesticides used as \"turf care,\" for lawns and golf courses. In 1999, for example, 125 new golf courses were created in Europe and 40 in Japan; Japanese golf courses now account for about 33% of all commercially applied non-agricultural pesticides in the world.\n\nThe United States Environmental Protection Agency (EPA) reports that about thirty-nine percent of households use insecticides because bugs are a major problem, but thirty-seven percent of households use them even when there is no major infestation other than perhaps a few ants strolling through the kitchen. There are lots of consumer choices when it comes to household pesticides: about 20,000 different pesticide products for the homeowner, bearing 300 active ingredients and 1,700 inert ingredients (these are not listed on the labels nor are they regulated). Are the inert ingredients worth worrying about? Yes, according to the EPA, which stated in 1991 that 300 inert ingredients were \"generally recognized as safe,\" 68 were \"potentially toxic,\" 56 were definitely toxic, and as to the possible toxicity of another 1,300, they didn't know.\n\nGiven the fact that the majority of conventional physicians still do not recognize\u2014or understand, perhaps\u2014the link between toxic exposure and symptom manifestation, Dr. Sherman recommends that individuals keep a log of their toxic exposures. In effect, she says, pretend you are an industrial worker in your own home and wherever you go, and maintain a daily inventory of toxic substances you are exposed to, including all household items (as indicated in the partial list cited above) with which you are routinely in contact. Dr. Sherman calls it a Consumer Product Use Record.\n\nHere's how to do it. In one column, record the name of the product, including its manufacturer; for example, dog flea powder, irethane pillows, hair spray, carbon paper. In the middle column, record the date you used it. If you can't remember the day and month, at least record the year. In the third column, on the right, keep a log of any symptoms observed as a result of using the product.\n\nFor example, under oven cleaner, which you use monthly, you might be recording coughing in user and wheezing in young child who observed the oven cleaning; or for vinyl paint to which you were exposed ten years ago, you remember you got skin and eye redness soon after using the product; or after using carbon paper over a period of four years as part of a job, you developed itching on your face and hands; or the diarrhea and wheezing that came on after you used that insect spray.\n\nAs you gain in knowledge regarding the great number of potentially toxic substances, you will find yourself adding more items to your list, such as prepared foods, nonorganic foods (ones produced with synthetic fertilizers and sprayed with pesticides), synthetic clothing, furniture, fluorescent lights (non-full-spectrum)\u2014indeed, _many_ things you always took for granted to be harmless.\n\nThe inventory of toxic exposures should include substances you are exposed to on the job as well as in your home, or in any public facility you might visit. There are several benefits to this simple task. First, as you record your own reactions to familiar products you realize how slow-acting, low-dose poisons actually affect your system. Your cough, headache, mild little skin rash\u2014suddenly they have a traceable cause. Second, if you develop anomalous symptoms (such as allergies, headaches, rashes, aches and pains) for which your conventional doctor cannot find a satisfactory cause, you can present this list as evidence of prolonged, low-dose toxic exposure.\n\nThird, as you maintain the list on a daily basis, you become more _aware_ of the consumer products with which you surround yourself, and you begin to realize that maybe some substitutions with less toxic products might be a good idea. You begin to gradually lessen your exposure to toxic substances. This is a very important element of detoxification: reduce your exposure, and minimize your toxic dose.\n\nFourth, you become a more discerning, ecologically sensitive, \"green\" consumer, less easily swayed by the claims of advertising and custom. Convenience often has a toxic price\u2014your health. Green always has a consumer benefit\u2014your health. \"A bland acceptance of advertising and failure to appreciate the ramifications of chemical manufacture, use, and disposal allow unecological and unhealthy practices to be promoted and to flourish without critical assessment,\" comments Dr. Sherman.\n\n### How Toxic Is Our Living Space, Both Body and World?\n\nThe answer is straightforward: intensely toxic. In 1986, the Environmental Protection Agency gave the scientific world a preliminary glimpse at the degree to which xenobiotics had infiltrated the tissues of most Americans. As part of their National Human Adipose Tissue Survey launched in 1976, EPA studied human tissue samples for residues of fifty-four different environmental chemical toxins, or xenobiotics.\n\nThey found that 5 of these were present in 100% of the samples, 9 ranged between 91% and 98% percent of samples, and 6 were present in between 76% and 89% of samples. Not only were a wide range of industrial chemicals present in human tissue, but their concentrations in those tissue cells were high, sufficient to give every person a total toxic burden of between 57.4 to 6,350ng (ng = 1 nanogram) of toxins per gram of fat. Benzene, a widely used industrial solvent, was present in ninety-six percent of samples; toluene, a chemical used widely in industrial processes, was found in ninety-one percent of sampl es; PCBs in eighty-three percent. In total, twenty different toxic compounds were detected in seventy-six percent of human tissue cells tested. Further, seventy-six percent of the people tested carried 25,704ng of total toxic compounds per gram of fat.\n\nAnother U.S. government study from the late 1980s examined 5,994 people aged 12-74 for toxic residues and found that 99.5% had blood levels of p, p-DDE (derived from a pesticide) equal to or greater than 1 part per billion (ppb), with a range of 1 ppb to 379 ppb. Autopsies of deceased elderly Texas residents showed traces of five pesticides or pesticide breakdown products in 100% of the tissue samples studied. Another study of Michigan school children aged four showed that seventy percent had residues of DDT, fifty percent of PCBs, and twenty-one percent of PBBs. \"These ongoing assessments have shown quite clearly it is not a question of if we are carrying a burden of toxic xenobiotic compounds; it is a question of how much and how they affect our health.\n\nOften contamination comes from common, everyday products we take most for granted, such as cosmetics, solvents, detergents, plastics, lubricating oils, and wood finichses. Scientists at the Centers for Disease Control (CDC) and Prevention's National Center for Environmental Health examined urine samples of 289 adults, aged 20-60 years, for traces of metabolites of seven toxic substances called phthalates. This chemical family represents yet another toxic substance commonly contaminating humans at this time.\n\nThese are chemicals commonly used in the products mentioned above (although mainly as plasticizers in PVC products); as a result, they are described as \"ubiquitous industrial chemicals\" with \"multiple exposure routes\" for humans, according to the scientists at CDC. Humans may be exposed to these chemicals orally, through the skin, through breathing, or intravenously. When these substances are broken down in the body, their presence can be traced by way of their metabolites, the new chemical form in which they exist and can exert toxic effects inside the body.\n\nAll seven of the phthalates have a wide range of chemical and toxic characteristics, the CDC scientists noted. They are suspected to be cancer-causing and they are associated with reproductive disorders based on animal studies. Not only were all the test subjects found to be contaminated with residues of these chemical substances, but the levels detected of each of the seven were high. The highest levels of exposure were to diethyl phthalate, dibutyl phthalate, and benzyl butyl phthalate, ranging from 137 to 3,750 parts per billion.\n\nMore troubling still was the discovery that women of childbearing age (20-40) had \"significantly higher levels\" of monobutyl phthalate, known to be a reproductive toxicant to laboratory rodents. The levels of this chemical in women of this age were higher than in any other age or gender group. Further, most of the highest exposure levels to monobutyl phthalate were found in women of the twenty to forty age group, portending possible future damage to their reproductive capacities.\n\nThe release of toxic chemicals into the world environment has certainly not abated since the 1980s. So we can reasonably expect the general level of toxicity, in terms of toxic compound residues detectable in human fat cells, to be most likely higher today. In many respects, our bodies and the planet's ecosystem have been forced to deal with many toxic substances not formerly present in either context.\n\nThousands of new, synthesized organic chemicals\u2014hydrocarbons (from petroleum), chlorine-based chemicals, and other potentially injurious substances such as heavy metals (mercury, aluminum, cadmium)\u2014have been applied in every aspect of our world and its industries.\n\nWe also have to contend with nuclear radiation residues in our air and soil, left over from atomic blasts in the 1950s, and in our foods irradiated without our consent for longer supermarket shelf life. Our municipal water supplies are laced with chlorine and fluorine, both of which have toxic resumes, and our air and soil are contaminated with multiple toxic agents. The toxic world is too much with us; we have ingested it; its poisons live within us.\n\nThe _fact_ of pollution is so deeply entrenched in public awareness that we almost screen it out, figuring our body will deal with it somehow. It does deal with it, as best it can, and for as long as it can. But eventually, our body's detoxification system gets overwhelmed, perhaps a little at first, then gradually more so, until the load of toxins outstrips our body's ability to eliminate them.\n\n\"One reason that so many people are walking toxic dumps is that we humans are at the top of the food chain,\" comments one physician. Toxins bioaccumulate in the food chain, from plants, to animals, to humans. Eat a plant and you may get pesticides and synthetic fertilizer residues; eat a chicken that ate the plant and that has also been fed hormones, antibiotics, and other additives, and you get all these toxins as well in your herbgrilled chicken breast.\n\n### Drinking Water Contaminated with Pesticides and Toxic Residues\n\nConsider the matter of pesticide and herbicide residues potentially found in fruits and vegetables in the United States. In terms of sales, pesticides are a $30 billion industry, with 62% of sales in the United States so it's not the kind of toxic substance you can easily legislate off the market.\n\nCurrently about 400 pesticides are licensed for use on America's agricultural crops, and in 1995, 1.2 billion pounds were applied to farm lands, forests, lawns, and fields\u2014a 100 million pound increase from 1993. It is estimated that about half of pesticide use is for nonagricultural purposes\u2014on golf courses, as previously discussed, but also in parks, on roadsides, on school grounds and in the buildings, along railroad tracks and under utility lines, in airplanes, in hospitals, in mass transit areas, in swimming pools and hot tubs, in hotels, in paper mills, and in building materials and food containers.\n\nHere is a graphic way to conceive of the enormous volume of pesticides released into the world environment every year, which is estimated to be one billion pounds. \"If put in 100-pound sacks and laid end to end, they would encircle the planet.\" Further consider the fact that since the 1940s, approximately 15,000 different chemical compounds and 35,000 different chemical formulations have been used as pesticides. Add to this the fact that there are about 630 active ingredients present in pesticides, but that these can be combined with other chemicals present in the formulas\u2014the inert ingredients\u2014to make several thousand more toxic formulations.\n\nThe Environmental Protection Agency has not tested all the pesticides in commercial use to determine their safety levels and maximum exposure recommendations for humans. Many may have been discontinued but their residues are long-lived and remain in the environment and possibly in human fat cells. Just because a seriously toxic pesticide such as DDT is banned in the United States does not mean another country will not use it on vegetables, fruits, and even coffee it ships to U.S. markets.\n\nWhen it comes to pesticides, as California goes, so goes the nation, except in this case the trend is to increasing application of toxic pesticides. California agriculture accounts for twenty-five percent of total U.S. pesticide use (and five percent of world use) such that 6.5 pounds of pesticides are used per person each year in California, about double the national rate of 3.1 pounds per capita. And total pesticide application in California is rising, up 31% between 1991 and 1995 to 212 million pounds; pesticides used only for agriculture increased by 37% during this same period.\n\nFurther, the use of the most toxic pesticides, ones known to produce cancer, increased by 129%, so that this type of supertoxic pesticide accounts for 11% of California agricultural pesticide use. Use of pesticides known to be acute toxic nerve poisons increased by fifty-two percent, and use of supposedly restricted pesticides climbed by thirty-four percent. It was as if there was a frenzy to use ever more toxic sprays to insure the California crops. \"The total volume of carcinogens, reproductive hazards, endocrine disruptors, Category I highly acute systemic poisons, Category II nerve toxins, and restricted use pesticides increased 32% between 1991 and 1995, and now comprise 72 million pounds, or 34% of total reported pesticide use in the state.\"\n\nIt's not that Californian farmers are planting more acreage; acres devoted to agriculture remained constant during this period, but the amount of pesticides dumped on each acre grew by thirty-five percent, from eighteen to twenty-five pounds per acre. Which California crops got the biggest dose of poison? Strawberries and grapes.\n\nIn 1995 alone, 59 million pounds of pesticides were sprayed on California grapes, while their strawberries were doused with 300 pounds of pesticide per acre. California fruits and nuts are dosed with seven times the amount of pesticides per acre as other crops; vegetables and melons get four times the pesticide per acre as other crops; and carrots are the seventh most heavily treated crop, receiving a disproportionately high amount of total pesticides as well as some of the most toxic. Also high on the pesticide list are tomatoes (fresh and for processing), almonds, oranges, dates, pears, lemons, and cabbage.\n\nSo if you're buying \"fresh\" fruits and vegetables in the winter, you might wish to keep these figures in mind because every grape, strawberry, and most vegetables will enter your kitchen bearing residues of this vast volume of pesticides. From the toxic fields of California to your overworked liver.\n\nIf you live in California, your potential toxic exposure to pesticides is even greater. Consider first the ramifications of \"pesticide drift.\" When a pesticide is applied to a field, a certain portion of it drifts off in the air currents and is blown elsewhere. But something else happens, and that is even more troublesome. After application, pesticides emit a large amount of gas. This is a reactive organic gas, the components of which are known collectively as volatile organic chemicals, or VOCs, and which contribute to smog. These VOCs can cause cancer, birth defects, and damage to the nerves, heart, and kidneys. Pesticide use\u2014and drift\u2014in California represents \"the tip of a 100-million pound iceberg of hazardous chemicals emitted statewide each year,\" according to the Environmental Working Group (EWG), an ecological advocacy group in Washington, D.C.\n\nDuring a two-year period, from 1996 to 1998, EWG collected 100 air samples from eight agricultural counties in central California. About two thirds of the samples contained toxic pesticides. In another test of fifty-five air samples, EWG found that fifty-three percent tested positive for traces of carcinogenic pesticides or ones known to produce birth defects and\/or damage the brain. In a test of thirty-nine air samples, eighty percent had traces of methyl bromide, a fumigant that causes nerve damage. EWG estimated that statewide about 98.9 million pounds of VOCs are emitted from pesticides applied for agricultural land in California. In the San Joaquin Valley, one of the state's prime farming regions, 34 million pounds of VOCs are emitted each year from pesticides, representing 13% of VOCs from all possible sources in that region.\n\nConsider the data published by the World Resources Institute in 1996 on the public health risks of pesticides and their effects on the human immune system. \"Hundreds of millions of people are significantly exposed to pesticides every day, either directly in farm and garden use or in residues in water, air, and food.\" Children in rural areas and breastfeeding infants whose mothers have been exposed can receive \"substantial doses\" of pesticides. Insofar as pesticides can suppress the immune system, the risk is greatest with infants and children, as well as the elderly or those chronically ill or malnourished.\n\nIn fact, the Institute reported that based on studies from Canada and Russia, children and adults exposed to pesticides \"suffer similar immune system alterations and higher rates of infectious disease\" than the general population. A study in Moldova showed that eighty percent of children living in agricultural districts, where pesticide application was heavy, had suppressed immunity as a result of pesticide toxicity. They were also three times more likely to have an infectious disease of the digestive tract, and two to five times more likely to be ill with a respiratory illness than children living in non-farming districts. Even more striking was the data from India showing that Indian factory workers chronically exposed to pesticides registered a sixty-six percent decline in white blood cell counts; after three months' vacation away from the offending factory, their white blood cell counts returned to a normal level.\n\nExposure to pesticides is now being linked to Parkinson's disease, according to a new study from Stanford University's School of Medicine. Researchers interviewed 496 patients diagnosed with Parkinson's regarding their pesticide exposure (through home and garden pesticide use). They found that people exposed to in-home insecticides were seventy percent more likely to develop Parkinson's than those who had not been exposed. The average amount of time for exposure was about seventy-seven days. The study concluded that exposure to garden insecticides brought a fifty percent increased risk of developing Parkinson's. The amount of time people spent using pesticides was directly related to their disease risk. People who handle herbicides for up to thirty days (total exposure) had a 40% increased risk; people who were exposed for 160 days, had a 70% heightened risk.\n\nYet another source of toxicity through agriculture is the practice of incorporating industrial toxic waste materials into fertilizer used by farmers. Among the most prevalent toxic substances found in hazardous-waste-derived fertilizers are lead, cadmium, chromium, nickel, zinc, copper, sulfuric acid, and dioxin. According EWG, each year millions of pounds of toxic waste materials are used to make agricultural fertilizers. The toxic sludge is typically laden with metal and chemical impurities, steelmill smokestack ash, air pollution scrubber brine, and other industrial byproducts. These byproducts become basic materials for \"a substantial portion\" of all fertilizers used in U.S. agriculture.\n\nEWG states that more than 600 companies in 44 states shipped 270 million pounds of toxic wastes to farms and fertilizer companies between 1990 and 1995. California was in the top three both for states that shipped hazardous wastes and for those that received them for fertilizer material.\n\nAlthough, in _theory_ , says EWG, fertilizers are subject to federal toxic chemical contamination standards, in _practice_ \"there is almost no monitoring of fertilizer or soil contamination levels\" and they may be \"much higher\" than the allowable limits. The implications are obvious and upsetting: the toxic substances get absorbed by the plants, which get consumed by either people or animals.\n\nFurther, one of the chief hazards of this intense and widespread use of pesticides and herbicides is that invariably some portion of the volume applied runs off into public water supplies. World Resources Institute states that eighty-five percent to ninety percent of agriculturally applied pesticides get dispersed into the air, soil, and water, and not on the intended plants, thereby guaranteeing environmental pollution.\n\nFor example, municipal and private well water sources in farming states tend to have high levels of pesticide residues. Traces of at least one pesticide were detected in forty percent of wells tested in Minnesota; twenty-eight out of seventy wells had one or more pesticides in Iowa; and in northeast Iowa alone, at least seventy percent of the wells tested had pesticide contamination. In thirty-eight states, traces of as many as seventy-four different agricultural pesticides were detected sources.\n\n**_One of the chief hazards of this intense and widespread use of pesticides and herbicides is that invariably some portion of the volume applied runs off into public water supplies._**\n\nIn California alone, researchers found evidence of fifty different pesticides in the groundwater. As of late 1999, more than one million residents of the state's Central Valley (where much ofthe agriculture happens) were consuming water contaminated with DBCP, a pesticide regarded as one of the most potent carcinogens (it produces testicular cancer) presently available. DBCP contaminates were found in the tap water of 38 communities in nine counties and at amounts far above the recommended safety level. In 19 communities, bottle-fed babies, by the time they turned one year old, had already received a lifetime's \"safe\" dose of DBCP through the infant formula mixed with tap water. The fact that DBCP was banned nationwide in 1979 has not stopped the toxic substance from showing up in the tap water of California's Central Valley twenty years later.\n\nIn the farming-intensive American Midwest, one of the most toxic pesticides, atrazine, is now showing up in the tap water of nearly 800 communities. Approximately 10.5 million people are affected by this atrazine-polluted tap water. \"Some tap water is so contaminated that infants get their lifetime limit of atrazine before they are four months old,\" and in forty midwestern towns, they get it by their first birthdays.\n\nEPA data indicates that 34 million Americans in 6,900 communities drink tap water contaminated with arsenic at levels that pose a health risk. Several national surveys found arsenic traces in three percent to thirty-nine percent of all drinking water samples studied, with concentrations averaging 10 ppb (parts per billion) or less. What's a safe level for arsenic? The World Health Organization (WHO) says 10 ppb, EPA says 50 ppb, but the Natural Resources Defense Council (NRDC) says it should be only 3 ppb. NRDC is contemplating a lawsuit against EPA to force the agency to revise the 58-year-old arsenic standard. A different study conducted by the U.S. Geological Survey showed that 10% of the 18,850 samples tested (private wells and public water supplies) had arsenic at levels that exceeded WHO's safety standards of 10 ppb.\n\nSo what happens if you drink arsenic-tainted water for a fair bit of time? You could get heart disease or prostate cancer, and\/or die sooner than expected. Researchers for the EPA studied the relationship between various arsenic levels in public and private drinking water (wells) in towns in one county in Utah. The arsenic levels ranged from 14 to 166 ppb\u2014in other words, from well under the EPA's safety limit to more than three times it. EPA researchers found that sustained arsenic ingestion led to a higher rate of deaths from heart disease, prostate cancer, and kidney inflammation in men; in women, more died from all forms of heart disease.\n\nFinnish researchers found that even very low exposure levels of arsenic were associated with an increase in bladder and kidney cancer. They studied the water consumption patterns of 275 subjects over a 14-year period as well as 61 cases of bladder cancer and 49 cases of kidney cancer. The lesson here is that _low_ exposures to a _single_ toxic substance _sustained_ over time can still produce serious health consequences.\n\nToxic substances also enter public drinking water sources from nonagricultural sources, such as heavy industry. In May 2000, the EPA reported that U.S. total toxic pollution was three times worse than previously estimated. The EPA stated that in 1998, 7.3 billion pounds of toxic materials were dumped into the environment, an amount three times higher than the previous measure, obtained a few years earlier. According to the EPA, sixty-three percent of the increase in toxics comes from the mining and electric utility industries. The mining industry alone dumped 3.5 billion pounds of toxics in 1998, mostly in the form of heavy metals and caustic materials applied to ores to leech out minerals. U.S. electric utilities dumped 1.1 billion pounds of toxic substances (including hydrochloric acid, sulfuric acid, and hydrogen fluoride, all emitted as airborne pollutants) into the U.S. environment in 1998. Bear in mind, this data describes toxic releases for only one year from only two sources of pollution.\n\nWhen you talk about contaminated water, inevitably the subject of chlorine comes up. If you are a public health official, your attitude about chlorine is that it's a chemical hero; but if you're versed in nutrition, biochemistry, and toxicology, you think of chlorine as a substance to avoid. Ever since 1908 when the first public water supply was chlorinated (in Boonton, New Jersey), Americans have been drinking, cooking with, showering, and bathing in chlorinated water such that today the water of seventy percent of Americans is chlorinated. The subject of chlorinated water and how to minimize your exposure to it is discussed in detail in Chapter 4, but there is an interesting observation to make here.\n\nYou would logically think drinking chlorinated tap water, versus bathing or showering in it, would be the less healthy of the two activities. But it's not. \"The simple, relaxing act of taking a bath turns out to be a significant route of exposure to volatile organics,' as Sandra Steingraber commented in _Living Dournstream:\"_ 52 Volatile organics are carbon-based compounds that vaporize more readily than water. Chlorine is one. A recent study showed that elevated levels of volatile organic compounds, including chlorine, can be detected in the exhaled breath of people who have just showered. From the shower spout to your cells, as it were.\n\nTaking a ten-minute shower or thirty-minute bath can give you a higher internal dose of volatile compounds (including chlorine) than drinking a half-gallon of chlorinated tap water. One reason a bath chlorinates you more fully than a glass of tap water is that the latter goes essentially from your mouth to your liver, where it is broken down (metabolized) before its components enter the bloodstream. But when you shower or bathe, your skin is enveloped in the subtle chlorinated fumes, your lungs get a direct infusion through your inhalations, and still more volatile-compounded water may enter your system through your body's orifices.\n\nConsider this strange yet revealing example. A 1984 study in Illinois showed that 150 private wells and one municipal well were intensely polluted with volatile organochlorine solvents. Five years later, a follow-up study ofthe indoor air quality of the homes serviced or affected by those same toxic wells found elevated levels of the organochlorines in the blood of the residents. However, their blood levels were more in accord with the organochlorine level found in the indoor air rather than in the contaminated water.\n\nResearchers found that the air levels were more or less matched with \"shower run times\"\u2014how long the residents ran a shower. These studies \"support the notion that inhalation contributes more significantly to overall body burden of volatile organic compounds than drinking\u2014even when water contamination is dramatic,\" said Steingraber.\n\n### When Prozac, Viagra, and Sunscreen Flow through Your Kitchen Tap\n\nThere is another reason to be circumspect about wells and public water supplies. Evidence is increasingly accumulating that groundwater and municipal water supplies may be contaminated with a new class of pollutants: residues of pharmaceutical drugs given to humans and animals. Traces of substances such as antibiotics, hormones, pain killers, tranquilizers, and even chemotherapy agents are being measured in drinking water. German researchers announced in 1998 that they had discerned thirty to sixty drugs in a typical water sample, some of which were at concentrations comparable to the level at which pesticides are found in water (parts per billion).\n\nThe German researchers specifically detected clofibric acid, a drug used to reduce blood cholesterol levels. It wasn't only German water that was polluted with this prescription drug. It turns out the entire North Sea contains a measurable quantity of this drug, specifically, one to two parts per trillion (ppt). It doesn't sound like much until you calculate that this means the sea contains forty-eight to ninety-six tons of clofibric acid. Tap water in Berlin was found to have 165 ppt, and both the Danube and Po Rivers had traces of the drug.\n\nHow did all this cholesterol-reducing drug get into the public water? Through human excretion, then seepage into the groundwater through sewage sludge. It seems that for humans and animals, anywhere from fifty percent to ninety percent of most drugs are excreted without any chemical change.\n\nThe danger of clofibric water pollution is just one example out of many that could be cited. It's more instructive to take the larger view. Two scientists\u2014Christian G. Daughton, an environmental chemist with EPA, and Thomas A. Ternes, an environ mental chemist with the ESWE-Institute for Water Research and Water Technology in Wiesbaden-Schierstein, Germany\u2014provided just such a view in a landmark article in _Environmental Health Perspectives_ in 1999.\n\nFirst, they state that the quantity of pharmaceuticals (prescription drugs, diagnostic agents) and personal care products (fragrances, sunscreen agents, and others) entering the environment each year is about equal to the amount of pesticides used in a year. A large portion of a given drug and residues of the personal care products enter the environment unused or unmetabolized (not chemically broken down and neutralized) by the human body. More crucially, they continuously enter the environment; there is no seasonality to their discharge, as with agricultural pollutants.\n\nThey get excreted through the urine or feces, enter the sewage system, and eventually find their way into the waterways, and still later, into the bodies of animals and people drinking that water. Or they might get collected in toxic sludge used for fertilizer and eventually be absorbed by plants fed to animals or humans.\n\nThe trouble is that the biochemical action of these substances is at best only poorly understood at this point. What happens when they combine unnaturally by virtue of their proximity in a given water source is unknown. Further, most drugs and personal care products were never tested for their environmental impact, only for their effects on the human user. This is highly risky because the number of pharmaceuticals and personal care products is escalating, \"adding exponentially to the already large array of chemical classes\" that can potentially pollute the environment.\n\nDaughton and Ternes list sixty-six classes of pharmaceuticals (which include 200 ofthe most popular prescription drugs), all of which are designed to have powerful, deep-set biological effects on living organisms. In fact, the drugs are designed to change the functioning of the immune, endocrine, or nerve signalling systems in the body. That is acceptable if you are the intended receiver\u2014it's your prescription. But with the presence of these pharmaceuticals in our environment, most of us are ingesting samplers from an \"enormous array\" of prescription drugs that were not meant for us.\n\nSo consider this, state Daughton and Ternes. \"A major unaddressed issue regarding human health is the long-term effects of ingesting via potable waters very low subtherapeutic doses of numerous pharmaceuticals multiple times a day for many decades.\" Put differently, when you pour a glass of water from the tap for your young children, you could be feeding them low levels of birth control pills, athlete's foot remedies, sunburn creams, Viagra, Prozac, Valium, and any of dozens more prescription drugs or personal care product residues. Bear in mind that low and very low do not mean inactive. For almost all of these substances, a very low dose is all that's needed to produce the intended result.\n\nThe residues of the drugs and products remain in the water for a long time, as most of them are highly persistent and resistant to breakdown. The profound danger here is not necessarily obvious, however. There is a high possibility of \"continual but undetectable or unnoticed effects\" on the various life forms living in the water, and the effects produced could \"accumulate so slowly that major change goes undetected\" until suddenly it starts to \"cascade to irreversible damage.\" This damage can include human disease, even genetic mutation. You can't get much more toxic than that.\n\nEven that is not the end of the problem. It is likely to get worse. The \"enormous array\" of pharmaceuticals will continue to diversify and grow as the human genetic blueprint (the human genome) is completed, observe Daughton and Ternes. Currently, prescription drugs work because they are targeted at some 500 distinct biochemical receptors, but as the human genome is mapped, many more receptors will be identified, such that the total number could grow by twenty times, yielding 3,000 to 10,000 drug targets. This means of course that hundreds of more drugs will be available to potentially pollute the environment and toxify animal and plant life and humans.\n\n### Air Pollution\u2014Trigger for Allergies, Asthma, and Respiratory Symptoms\n\nAir pollution is a matter so often discussed as to be almost taken for granted or accepted as an inconvenience of modern urban life in especially toxic air zones such as Los Angeles and Mexico City. We've heard the term \"smog\" so often we tend to become oblivious to its meaning. Yet it remains a vital concern and is a source of heavy toxicity.\n\nBy now, each of the six major air pollutants (primarily released from automotive emissions) has its own well-researched toxic resume. The six are ozone, lead, carbon monoxide, sulfur dioxide, nitrogen dioxide, and particulate matter.\n\nOzone is a respiratory irritant and allergen and can produce coughing, shortness of breath, chest tightness, mucus membrane irritation, allergies, asthma, and respiratory infections. Lead can cause behavioral problems and lowered IQ in children. Carbon monoxide reduces the blood's ability to transport oxygen to the body's cells, thereby affecting breathing, producing dizziness and headaches, and causing strokes or heart attacks. The two dioxides seriously irritate the lungs, producing coughs, chest tightness, respiratory illness, or even respiratory paralysis in high doses.\n\nParticulate matter, found in smog, gets inhaled into the deepest part of the lungs where it produces lung problems. With the exception of lead, all of these air pollutants interfere with oxygen delivery. If a person is already asthmatic, exposure to any of these pollutants will worsen the condition, making them a serious health risk.\n\nLet's be sure we understand what smog is. It is defined as a sickening photochemical mixture produced in the lower atmosphere when sunlight strikes various pollutants in the air, most notably, industrial pollutants and particulates from automotive exhaust. The largest component of smog is ground-level ozone (a form of oxygen). Compounds called nitrogen oxides (produced when fuels are burned) mix with volatile organic compounds (VOCs, including organochlorines, released when liquid solvents, fuels, and organic chemicals evaporate) in the atmosphere. As most people who have been to Los Angeles know, smog is at its worst on hot, sunny summer days because smog formation is dependent on temperature and sunlight. Just about any kind of combustion-driven vehicle contributes to smog, from motor scooters to jumbo jets.\n\nSmog is bad news because it is a strong and irritating pollutant\u2014irritating to the mucus membranes of the respiratory tract, especially the lungs. Ironically, on a high smog day, if you're out exercising, thinking you're doing your mind and body some good by stretching the muscles and inhaling deeper, you may get these benefits, but at the cost of an increased ingestion of smog pollutants. Typical symptoms of smog toxicity include aching lungs, wheezing, eye irritation, shortness of breath, nausea, coughing, and headache.\n\nPublic health estimates state that about 15 million residents of the South Coast Basin, a 12,000-square-mile area that includes Los Angeles and three neighboring counties\u2014it's the nation's second most populous urban area\u2014inhale dirty air about 33% of the time. And when the smog index is high, it's not unusual to find hospital admissions higher than usual. Researchers at the Rancho Los Amigos National Rehabilitation Center in Downey, California, found that between the years 1992 and 1995, regardless of type of patient (such as age or ethnic group, excepting those older than sixty-five), atmospheric stagnation with a high degree of air pollution increased the risk and incidence of hospitalization for lung and heart problems (see figure 1-2).\n\nEven if you don't experience the obvious symptoms of the \"smog complex,\" as it's called, you may still have smog residues in your body. Medical researchers at the University of Southern California examined the bodies of 152 deceased young people aged fifteen to twenty-five who had died from accidents or homicide. In lung autopsies of 100 of these subjects, seventy-five percent had a slight degree of lung inflammation while twenty-seven percent had signs of severe lung tissue damage. All of the bodies autopsied showed some degree of airway inflammation, thirty-nine percent had severe illness lodged in the bronchial glands, and twenty-nine percent had disease in the bronchial linings. In all, fifty-four percent of the youths\u2014more than half of a group of young people randomly examined\u2014had signs of at least one severe lung illness.\n\n**Figure 1-2. The 33 Top Toxic Substances Found in Urban Air**\n\nParticulate matter, which in large cities is comprised of thirty-five percent to eighty percent diesel exhaust particulates, may be a trigger for allergies. Japanese scientists noted that allergic diseases are more prevalent in polluted areas and speculated this higher incidence must be due to environmental factors. They found that exposure to diesel exhaust particulates in the air produces allergic reactions in the immune system such that they felt confident in attributing some of the increase in allergies and allergic diseases to this air pollutant. It gets even stranger because the airborne pollutants can make natural allergenic materials, such as plant pollens, even more irritating to the human nervous system and lungs, producing a greater degree of inflammation.\n\nFurther, if you already have asthma, exposure to air pollution is likely to make your symptoms worse. A study of almost 4,000 Southern California school children (fourth to tenth grade) from twelve different communities showed that children with asthma were more likely to develop \"persistent\" lower respiratory tract symptoms (including bronchitis) when exposed to air pollution. Among the rest of the children, exposure to air pollution resulted in a significant increase in phlegm production (mucus in the nose and throat). Among the air pollutants, exposure to nitrogen dioxide produced the greatest number of respiratory symptoms.\n\n**_Particulate matter, which in large cities is comprised of thirty-five to eighty percent diesel exhaust particulates, may be a trigger for allergies. Japanese scientists noted that allergic diseases are more prevalent in polluted areas and speculated this higher incidence must be due to environmental factors._**\n\nResearchers in the former East Germany obtained similar results when they studied the respiratory health of 2,470 school children, aged 5 to 14. Some had lived in one of two industrially polluted counties (from mining and smelting), and others came from a neighboring county without any sources of industrial pollution. The health differences were striking. Children in the polluted county had a fifty percent increased lifetime prevalence for developing allergies, eczema, and bronchitis, and fifty percent more respiratory symptoms, such as wheezing, shortness of breath, and coughing than those in the pollution-free county. Further, children from the toxic counties had a lifetime higher rate of sensitization (a chronic allergic response, such that you react to increasingly smaller doses of an allergen) and respiratory disorders.\n\nVery young children seem to be especially sensitive to airborne pollutants, especially particulate matter. A study of medical visits by children to physicians in Santiago, Chile, over a two-year period showed that as the particulate concentration in the air increased, so did the doctor visits. For children under two, respiratory symptoms requiring medical visits increased by four percent to twelve percent as the air pollution increased, and for children three to fifteen years old, the rate of respiratory tract symptoms rose by three percent to nine percent.\n\nAir polluted with fine particulate matter may shorten a person's lifespan. A sixteen-year study conducted by scientists at Harvard University involving 8,111 residents of six U.S. cities showed that air pollution (specifically particulate matter content) was strongly associated with an unusual number of deaths from lung cancer and heart disease, even in people who did not smoke. The researchers also concluded that sustained exposure to particulate air pollution can reduce your lifespan by two years. Incidentally, all six cities had air pollution levels _lower_ than the federal health permissible standard\u2014in other words, the air pollution was at a supposedly safe level.\n\nA study from Central Europe found similar results, namely, more deaths as the air pollution index rose. German researchers studied mortality rates in a highly polluted region of the Czech Republic for the years 1982 to 1994. They found a 3.8% increase in the number of deaths in association with measurable increases in particulate air pollution. They also found that for each day the particulate matter concentration increased by 100 micrograms per cubic meter (l\u03bcg = 0.00000lg), there was a 9.8% rise in deaths. However, the researchers found \"no evidence for an association\" between the number of deaths and particulate air pollution in the rural nonindustrialized area in Germany near the Czech border used for a comparison.\n\n### Organochlorines\u2014Toxic Chlorine Blankets the Planet\n\nAir pollution and its toxic effects are increasingly a global health concern, as numerous scientific studies show. We could also focus on a single family of chemicals to deepen our impression of the extent of environmental pollution: organochlorines.\n\nThese are synthetic compounds composed of chlorine and organic substances\u2014chlorinated organic substances, to put it simply. According to Joe Thornton, a biologist at Columbia University's Center for Environmental Research and Conservation in New York City, and author of the landmark study, _Pandora's Poison_ : _Chlorine, Health, and a New Environmental Strategy_ , \"Everyone on Earth now eats, drinks, and breathes a constantly changing and poorly characterized soup of organochlorines, including dozens of compounds that cause severe health damage at low doses.\"\n\nThey're found everywhere, but in plastics especially; anytime you bum some plastic, you release chlorine gas that can form new organochlorines. In fact, disposing of plastic waste without producing organochlorines is regarded as nearly impossible. Ever since the large-scale manufacture of plastics began in the 1940s, \"a witch's brew of toxic, persistent pollutants has come to blanket the entire planet, from suburban backyards to the deep oceans, from our own bodies to snow at the North Pole.\"\n\nWe have reached the state of universal exposure, says Thornton. Laboratory analysis of samples of human fat cells, blood, breath, semen, urine, and mother's milk all bear traces of organochlorines; you don't have to have been living near a site of conspicuous contamination to be organochlorinated. Anywhere you check in the Northern Hemisphere, according to Thornton, you will find the air contains a baseline concentration of 200 parts per trillion of trichloroethane, an organochlorine.\n\nThe term organochlorines refers to a chemical family with some 11,000 different forms now commercially produced, not to mention the thousands more formed as byproducts. When chlorine gas produced by the chemical industry meets organic matter in industrial processes it forms an organochlorine. When chlorine gas is employed in the manufacture of plastics, paper, pesticides, solvents, bleach, sewage disinfectants, and numerous industrial chemicals, or in the treatment of water, the combination generates organochlorines, such as dioxins, PCBs, and thousands of other lesser-known though seriously toxic substances. You will find organochlorines in chlorinated public water, in solvents and dry-cleaning solutions, PVC plastics, and bleached paper, among many other items.\n\nThe trouble is that almost none of the 11,000 organochlorines are native to the natural world. It is true that nature produces a small number\u2014estimates vary from several hundreds to one thousand\u2014mainly through algae, fungi, sponges, coral, and microorganisms. But no organochlorines are known to occur naturally in animal and human tissue as a result of this generation; they don't get deposited in the fat cells of living organisms. Organochlorines, says Thornton, appear to be \"completely foreign\" to all vertebrates, including humans, mammals, birds, reptiles, and amphibians.\n\nOrganochlorines may have excellent chemical and industrial properties, but these same qualities make them toxic to the environment and all living organisms. Chlorine gas is highly reactive, combining quickly, indiscriminately, even _randomly_ , with any organic matter it encounters, leading to numerous _incidental_ byproducts of unknown but presumed toxicity. For example, chlorinating public water supplies leads to at least seventy identified organochlorine byproducts, a fair number of which are known carcinogens. This number, says Thornton, represents only twenty-five percent to fifty percent of all the organochlorines present in the water; the rest simply have not been characterized yet.\n\nOrganochlorines are highly stable, which means they persist in the environment for a long time, in some cases for centuries, it is speculated. Nearly all of the world's persistent organic pollutants (POPs, discussed previously) are organochlorines. Thus their concentration in the air, water, and soil progressively accumulates over time.\n\nIndustry likes organochlorines because they enable organic chemicals to dissolve in oil, but this same trait makes organochlorines favor the fatty tissues of biological organisms, including people. Fat-soluble toxic substances are much harder for the body's natural detoxification mechanisms to flush out than those that dissolve readily in water. Further, organochlorines, as they move up the food chain to humans, can reach tissue concentrations \"many millions of times greater\" than those found in the outside environment, says Thornton. A 1990 Japanese study showed that striped dolphins had PCB levels 13 million times higher than the levels found in their ambient environment, the North Pacific Ocean, and DDT residues 37 million times higher.\n\nIn the 1990s, several hundred organochlorines were finally subjected to toxicology studies and were found to have serious health effects on almost every aspect of human biological functioning, including disruption of the endocrine system with effects across the generations. Even more alarming, some of these chemicals can produce effects at extremely tiny doses, in the parts per trillion range. How small a dose is this? It's a ratio \"equivalent to one drop in a train of railroad tank cars ten miles long,\" says Thornton.\n\nAt least 190 different organochlorines (including dioxins, PCBs, and DDT) have been detected in the fatty tissues (as well as mother's milk, blood, breath, semen, saliva, and urine) of people living in the United States and Canada, and now comprise a considerable portion of their total toxic load. \"Absolutely everyone\" now carries a toxic body burden of these 190 organochlorines, says Thornton; since humans are at the top of the food chain, \"our own bodies have become the ultimate dumps\" for the products of the chlorine chemistry industry. Ironically, one way of getting some of these organochlorines out of the body is through breast milk.\n\nThe breast milk of women in the United States and Canada has been found to contain sixteen different pesticides, eighteen dioxins and furans, sixty-five types of PCBs, and thirty other toxic, industrially derived substances. The obvious consequence is that nursing infants are exposed to excessive amounts of toxic substances virtually at birth and, in many cases, even before birth as many of these substances are known to cross the placental barrier. In the case of dioxin, for example, one study showed that in one year an infant was likely to receive a dose of dioxin representing a lifetime cancer risk 187 times above the government's \"acceptable\" level.\n\nAt least 100 organochlorines are carcinogens, and many others are believed to be mutagenic, that is, capable of causing DNA to mutate.\n\nOrganochlorines are known to suppress the immune system, reduce sperm count, produce infertility, contribute to cancer, create learning disabilities in children, and cause many other serious conditions. They are regarded as neurotoxins (poisons to nerve and brain cells) and reproductive toxicants (poisons to the male and female reproductive systems).\n\nFor example, chlorinated hydrocarbons (CHCs) may be responsible for some miscarriages. Researchers found that twenty percent of eighty-nine women who had sustained repeated miscarriages had traces of at least one CHC in their blood at a range that exceeded the acceptable level. As another example of reproductive poisoning, exposure to dioxins may contribute to endometriosis in women. When scientists examined the blood of forty-four women with endometriosis (a menstrual disorder in which monthly bleeding occurs from multiple sites outside the uterus), they found they were eight times more likely to have measurable levels of dioxin than women of the same age without endometriosis.\n\nOrganochlorines disrupt the basic mechanisms by which the body regulates itself, making them \"incompatible with basic physiological functions,\" says Thornton. For example, the level of dioxin the average human bears in his fatty tissues (clinically called the average body burden) already is enough to generate reproductive, developmental, and immunological problems in laboratory animals. In many cases, it's proven to be enough to produce those problems in people, too.\n\nSo pervasive and toxic are organochlorines that Thornton likens chlorine chemistry, the industrial process that spawned the entire class of chlorine-based chemicals, to splitting the atom. Like nuclear power, chlorine chemistry is \"an inherently dangerous technology of great power,\" an unprecedented \"human intervention in the structure of matter,\" he says, warning that we may harness it, but we will never be able to completely control it. In fact, organochlorines may be a major contributing factor to the \"global toxic pollution\" under way, which is contributing to \"a slow, worldwide erosion\" of human health, Thornton concludes.\n\n### A Radioactive Cloud Enshrouds an Entire Generation\n\nAnother toxic element that is so widespread in North America as to be judged ubiquitous is nuclear radiation. You can be exposed by living near a nuclear reactor or through contact with the residual fallout from atomic testing in the 1950s in the continental United States. Of the 3,053 counties in the United States, 1,321 (or more than 33%) are nuclear counties. This means the residents in those counties live within 100 miles of a reactor and are regularly exposed to radiation through the air and water.\n\nResearch by Jay Gould, a former EPA scientist and author of _The Enemy Within_ , indicates that the 1,321 nuclear counties account for fifty percent of all deaths from breast cancer; that four more women per 100,000 die of breast cancer in nuclear counties than in non-nuclear ones; and that in the 14 U.S. counties that house the seven oldest reactors, there was a 37% increase in breast cancer deaths between 1950 and 1989 compared to a 1% rise everywhere else.\n\nNearly all of the continental United States has been irradiated with nuclear fallout. Bear in mind, 184 atomic bombs have been exploded in the air over Nevada since 1945, and their fallout drifted east across the country. In fact, the combined fallout from all nuclear testing performed above ground by the United States and former Soviet Union is equivalent to 40,000 Hiroshima bombs. It doesn't matter where you live in the United States. Your soil has been visited by toxic nuclear fallout in the last fifty years.\n\nThe radioactive clouds from the earliest atomic blasts in Nevada in 1951 were tracked as they drifted across the United States, heading out over the Atlantic Ocean after passing over New England. In some instances, due to the speed with which the radioactive clouds were blown by the prevailing easterly winds, various northeastern cities \"reported levels of radioactivity as high as those reported close to the test site,\" observes Gould. As everyone knows, many radioactive isotopes take a very long time to decay; their half-lives are reckoned in the thousands of years.\n\n**_The Baby Boomers are the first U.S. population group to have been uniformly poisoned with nuclear radiation, uniformly exposed_ in utero _to strontium-90 (a radioactive isotope generated by nuclear weapons testing) in the atmosphere making them toxic even before birth._**\n\nMedical research shows that radiation tends to affect the immune system and the activity of the thyroid gland, an important endocrine gland in the neck. Nuclear radiation is also linked to low birth weights in newborns as well as breast cancer. In fact, depressed thyroid gland function, fairly widespread already in the general adult population, is now showing up more in newborns.\n\nAccording to Gould's research, in fifteen states (accounting for forty-four percent of the U.S. population), the number of hypothyroid live births (babies born with underactive thyroid glands) increased by seven percent from 1986 to 1992. This was linked with releases of radioactive iodine and strontium into the environment during that time (such as by the Chernobyl nuclear reactor accident in 1986). \"Hypothyroidism among newborns indicates damage to the fetal thyroid from radioactive iodine (I-131) ingested by the mother,\" says Gould. Between 1986 and 1987 (the year after Chernobyl), the incidence of people having underactive thyroid function rose by 8.4% in 31 states.\n\nSeen in another way, an entire generation\u2014the Baby Boomers, bom between 1945 and 1964\u2014are the first U.S. population group to have been uniformly poisoned with nuclear radiation. Baby Boomers were uniformly exposed _in utero_ to strontium-90 (a radioactive isotope generated by nuclear weapons testing) in the atmosphere making them toxic even before birth. During that twenty-year span, says Gould, atmospheric levels of strontium-90 increased fiftyfold; at the same time, there was a fifty percent increase in the percentage of underweight newborns.\n\nGould observes that the true cost of victory in World War II\u2014achieved by the atomic blasts in Japan\u2014was \"an uncontrollable epidemic rise in radiation-induced illnesses.\" In fact, Gould contends that Baby Boomers reaching the age of forty-five in 1995, were \"contracting and dying of both breast and prostate cancer at rates unprecedented in medical records going back to 1935.\" Among the major reasons for this increase in cancer rates, Gould says, were the toxic effects of the Three Mile Island (1979) and Chernobyl (1986) nuclear disasters, compounded by continuing radioactive emissions from the numerous nuclear power reactors online throughout the United States.\n\nEvidence supportive of this last supposition was provided in a recent study. Between 1987 and 1998, twelve nuclear power reactors in the United States were closed down. One of them was at Rancho Seco, a densely populated area near Sacramento, California. After the reactor ceased operations, public health officials observed \"significant decreases\" in mortality and cancer incidence in fetuses, infants, and small children. This data contrasted with the worsening of infant health after the Rancho Seco plant opened in 1974 and started releasing low levels of radionuclides into the local environment.\n\n\"The data suggest that a relationship between nuclear emissions and adverse health effects exists, especially since fetuses and newborns are most sensitive to radiation,\" concluded Joseph J. Mangano, director of the Radiation and Public Health Project in Brooklyn, New York, and author of the study. The plant's closing reduced local levels of dietary radioactivity, as measured in the local pasteurized milk supply, added Mangano.\n\nGould also provides evidence to support the point. Why are cancer rates so high in the Hamptons, the tony eastern end of Long Island, New York? Rates for breast and prostate cancer were twenty percent higher from 1978 to 1987 in the Hamptons than in the rest of Long Island, according to the New York Cancer Registry. The reason, says Gould, is probably that for fifty years that end of Long Island has received sustained toxic doses of both pesticides from agriculture and strontium-90 from nuclear reactors (one on Long Island, and three in Connecticut, a few miles across the Long Island Sound), both of which made their way into the public water and food supply. In particular, cancer rates in five towns, just south of the four nuclear reactors, had cancer rates thirty percent higher than the county average (sixty-two towns in total).\n\n### Tainted Food\u2014Glutamates, Additives, Antibiotics, and POPs\n\nYou will encounter yet another family of toxic substances in foods. Not foods as they originate in nature, but as they have been processed by the food industry. This family is known as excitotoxins, and includes the food additives MSG (monosodium glutamate, often added to Chinese food to amplify its flavor), hydrolyzed vegetable protein (HVP, concentrated soybean protein, both an additive and a processed food), and aspartame (the artificial sweetener found in NutraSweet), all of which have been called into question in recent years. These three substances are everywhere in our prepared food system, present in almost all processed foods: as natural flavorings, flavor enhancers, spices, yeast extracts, textured protein, soy protein extracts, and in soups, gravies, and diet soft drinks.\n\nAccording to neurosurgeon Russell L. Blaylock, M.D., these excitotoxins are best avoided. Ingested in prepared foods or beverages, they stimulate a body's neurons. More specifically, they excite nerve and brain cells (neurons) because of their chemical similarity to neurotransmitters, whose job is to excite (or inhibit) neuronal cell activity. But this neural stimulation by a foreign substance is not a desirable occurrence. Excitotoxins, present in the brain at unnaturally and therefore dangerously high concentrations are \"neurological time bombs,\" says Blaylock.\n\nResearch suggests excitotoxins can play a critical role in the development of neurological disorders including migraines, strokes, seizures, hypoglycemia, infections, abnormal neural development, learning disorders, Alzheimer's, Parkinson's, and possibly another eight serious health conditions.\n\nPart of the problem has to do with glutamate, a natural amino acid. It is the glutamate in monosodium glutamate and other excitotoxins that is the toxic substance. This at first seems strange since glutamate is a naturally occurring amino acid (a protein building block) found in many foods. In whole foods, glutamate is not dangerous at all; but when it becomes free\u2014isolated from its natural context\u2014and used on its own as a chemical substance, then it becomes a risk. When bound in foods, it is slowly digested and absorbed by the tissues, especially muscles, before it can build up to a toxic concentration. But when you take it straight, in MSG, HVP, or any of the other excitotoxin forms, its effects are more pronounced and it becomes toxic.\n\nNormally, glutamate is found in large but appropriate concentrations in the human brain. In fact, it is the most commonly used brain neurotransmitter, but it exists in the extracellular fluid in very small concentrations. When the body takes in additional\u2014or excessive\u2014amounts of glutamate through food additives and taste enhancers, the brain's natural and correct balance of neurochemicals that excite and inhibit activity is disturbed. In effect, the brain gets overstimulated, or too excited\u2014toxically so. When the brain's concentration of glutamate rises above a certain specific level, brain cells start to fire abnormally. The brain cells start a process of slow, delayed death called excitotoxicity\u2014they are excited (overstimulated) to death. Excitotoxins entering the body through prepared foods thus set in motion \"a cascade of destruction\" that leads to \"serious distortions\" of the nervous system.\n\nThe effects of excitotoxicity are not always immediate or obvious, but play out over time, and can easily be missed or misdiagnosed by clinicians, or attributed to other causes. While excitotoxins may not _cause_ neurodegenerative conditions such as Alzheimer's, Parkinson's, and ALS (amyotrophic lateral sclerosis), they are believed to _precipitate_ these disorders and probably _worsen_ their pathology, says Blaylock. As mentioned earlier in this chapter, they can serve as triggers. Exposure to excitotoxins in prepared foods might be what it takes to push a person with a propensity to develop a neurodegenerative disease into progressing to a full-blown disorder.\n\nThe problem of glutamates and excitotoxins is actually a subset of a much larger problem, namely, that of food additives in general. Additives and their potential dangers have been topics of discussion since the 1970s in some circles in the United States, but today, additives and processed foods are virtually a fact of life that we've come to take for granted. While the natural foods and organic agriculture industry continue to grow, the bulk of the American food supply is still adulterated with some 3,000 additives and other chemical substances added to foods for various reasons.\n\nWhat is a food additive? Think sulfites, nitrates, nitrites, salicylates, food colors and dyes, monoglycerides and diglycerides, sugar substitutes (aspartame, acesulfame K, saccharin), BHA, BHT, polysorbate 60, olestra, emulsifiers, thickening agents, calcium (or sodium) propionate, citric acid, EDTA, and many more.\n\nNearly all food additives have a toxic resume to some degree. For example, the yellow dye known as tartrazine and the preservative benzoate are known to cause hives in people of all ages, and hyperactivity and abnormal behavior in children. On its own, tartrazine can generate symptoms within ninety minutes of ingestion, including asthma, generalized swelling, headaches, and behavioral changes. Nitrites (in the form of sodium salts, used as preservatives) can combine with amino acids in the body and produce nitrosamines, which are potentially carcinogenic. Olestra, the notorious \"fake fat\" introduced into the U.S. food supply in the mid-1990s, can cause diarrhea and loose stools, abdominal cramps, and flatulence. Cochineal extract (or carmine dye), a colorant derived from insects and used in fruit drinks, candy, yogurt, and other foods, can produce life-threatening allergic reactions, from mild hives and itchy skin to dangerous anaphylaxis (toxic allergic shock).\n\nNutritionist and food researcher Carol Simontacchi takes the subject of neurotoxins and toxic food additives a step further, declaring that our modern Western diet of prepared and processed foods is \"quite literally, driving us crazy.\" They are \"crazy makers,\" she explains in a new book of the same title, destroying our brains and the neurochemical foundations of our intelligence. The bulk of the foods most Americans regularly eat are devoid of essential nutrients and loaded with too many toxic chemicals, notably pesticides, says Simontacchi. They have too few of the essential organ-building nutrients and too many of the organ-damaging industrial chemicals.\n\nThe result is a near epidemic of nutrition-related illnesses such as anorexia nervosa, bulimia, deficient cognition, depression, panic and anxiety attacks, mood imbalances, various forms of mental dysfunction or illness, and antisocial, even violent behavior, among others. Our host of new, manufactured foods, Simontacchi argues, have changed our bodies and brains, altered the brain's ability to think, and even altered the structure and function of the brain itself. \"When the brain malfunctions, sending bizarre messages to our mouths or limbs, we never wonder if the toxic thought is the product of a toxic lunch.\"\n\nForget proper nutrition; forget even poor nutrition. Simontacchi says our dietary approach is an expression of \"antinutrition.\" Through our ill-advised food choices\u2014either nutrient-deficient or neurotoxic\u2014we accelerate the damage to our brains and nervous systems with the \"very foods (or nonfoods) that we are stuffing into our mouths.\" We dig our graves with our forks, she says. We poison ourselves with nearly every mouthful.\n\nSimontacchi argues that infants, adolescents, and adults alike consume diets that lead directly to diminished brain capacity because the necessary brain-building nutrients are lacking and toxic additional elements are present. For example, she reports that ninety percent of children under age five are routinely exposed to thirteen different neurotoxic insecticides in their food; that for infants aged six to twelve months, commercial baby food is the primary source of unsafe levels of organophosphates; and that laboratory tests of eight brands of baby food (in 1995) showed residues of sixteen different pesticides. This is no way to build a healthy, optimally functioning young brain, says Simontacchi.\n\nOne class of food additives that we should be especially concerned about are antibiotics given to \"factory farm\" animals. These are cattle, chicken, and pigs raised in large lots and intended for human ingestion. Some 750 different drug products are approved for use in food animals, and fifty percent of the total antibiotic use in the United States is on food animals. The farm animals who receive this infusion of antibiotics are not necessarily sick. In many cases, \"farmers feed livestock a low-level diet of antibiotics to attack bacteria that might require the animals' body to expend energy to kill off.\" Farmers would rather the animals use that energy to fatten up quickly.\n\nThe farmers may make their quick profit return, but the rest of us pay the price. The antibiotic residues are passed on to humans who consume the animals; aside from the additional toxicity this presents to the human liver, antibiotic drug use in food animals actually contributes to the development of multiple drug-resistant bacteria and thereby a failure of antibiotics to work when administered to humans. In fact, the connection between factory farm antibiotic use and the failure of antibiotics to work in humans is becoming increasingly evident.\n\nIn 1974, an antibiotic called Virginiamycin was approved for livestock use. In 1999, a closely related antibiotic called Synercid was approved for human use. Soon after, researchers started to find bacteria that were resistant to Virginiamycin (it didn't kill them anymore) in fifty percent of chicken, turkey, and pork found in the supermarket. Since the two antibiotics are closely related, concern was raised that the effectiveness of Synercid might now be seriously diminished.\n\nThe Synercid-Virginiamycin example is only one among many. A medical report published in May 1999 stated that infections by antibiotic-resistant bacteria increased eightfold between 1992 and 1997. Further, the resistance of one strain of the bacteria salmonella (a microorganism that produces food poisoning) to five different antibiotics grew by thirty-four percent between 1980 and 1996. In other words, in about one third of the cases, five different antibiotics will have no effect against this bacteria. Chickens are a prime source for two types of food poisoning bacteria: an estimated twenty percent of U.S. broiler chickens are contaminated with salmonella and eighty percent with campylobacter.\n\nIt's all insidious: you eat some supermarket-bought chicken (laced with antibiotics); some months later you develop a serious respiratory infection and are given antibiotics to save your life, and they don't\u2014because they can't. Ironically, the antibiotics, when overused, also become less effective on the farm animals. As one commentator quipped, \"Low doses don't kill bacteria\u2014they just make them mad.\"\n\nFurther, the introduction of yet more antibiotics into the human system (most people have antibiotic residues in their cells and intestines from various medical treatments earlier in their life) gives the liver, the body's prime detoxifying organ, yet more work, and it disrupts the delicate balance of microflora in the intestines, tipping the scale towards the toxic microorganisms. This in turn provides a \"seed bed\" for the development of numerous forms of bodily discomfort and disease.\n\nFactory-farmed food animals are also laced with hormones (six are approved for use), growth hormone (bovine somatotropin growth hormone, BGH), and other toxic substances that make their way into the human body when we consume the meat, eggs, or dairy products from these animals.\n\nEarlier in this chapter, I mentioned the term persistent organic pollutants, or POPs, the toxic \"family\" of industrial chemicals found almost everywhere in the modern environment, including our foods. In fact, \"the major route of exposure to POPs in humans is through consumption of food,\" most notably, high-fat foods such as meat, fish, and dairy products, although POPs are also found in vegetables, fruits, and cereals. Ironically, once POPs enter the human intestinal tract, ninety percent of them are absorbed, which is a rate higher than a fair number of other nutrients and natural foodstuffs. Further, with infants, absorption is \"nearly complete.\"\n\nThe United States Food and Drug Administration (FDA) conducted an analysis of 155 various common foods found in the U.S. marketplace during the years 1991 to 1997 for traces of the top twelve POPs. The so-called \"Dirty Dozen\" include eight insecticides (aldrin, chlordane, DDT, dieldrin, endrin, heptachlor, toxaphene, and mirex), one fungicide (hexachlorobenzene), two byproducts of organochlorine production (dioxins and furans), and PCBs.\n\nOf the twelve POPs, FDA found evidence of seven of them in the various foods tested. They found DDT residues in 136 food items, dieldrin in 100, and 14 to 34 foods contaminated with chlordane, heptachlor, hexachlorobenzene, or toxaphene. Consider this carefully: 136 out of 155 foods were contaminated with DDT (remember, this was banned in the United States years ago), and 100 with dieldrin. This means that about eight out of ten foods you are likely to buy in the supermarket will have traces of DDT, and six out of ten will be contaminated with dieldrin.\n\nA Greenpeace study on POPs in our food supply revealed that the average daily dietary intake (ADI) of organochlorines in the United States is as follows, based on 1990 statistics: 1.56 Ug\/person\/day of DDTs; 0.12 \u03bcg of HCH (hexachlorocyclohexane); 0.1 \u03bcg of dieldrin. It's worse in other countries, such as China. There the ADI is 20.47 \u03bcg for DDTs and 5.04 \u03bcg for HCH. Vietnam is pretty toxic too: intake of DDTs is 19 \u03bcg and for HCH, 5.4 \u03bcg. India has by far the worst intake averages, with ADI of 48 \u03bcg for DDTs, 155 \u03bcg for HCH, and 19 \u03bcg for aldrin and dieldrin.\n\nResearch by the FDA and the United States Department of Agriculture also showed that pregnant and nursing women tend to eat \"many\" more POP-contaminated foods than men. For example: both butter and pickles have been found to contain chlordane residues; DDT has been found in butter, American and cheddar cheese, sour cream, hamburger, peanut butter, and milk; dieldrin in dill pickles, pumpkin pie, butter, peanut butter, and American and cheddar cheese; heptachlor in plain bagels and cracked wheat bread. Further, according to the Environmental Working Group (EWG), which presented the government data on their website, a pregnant or nursing woman tends to eat sixteen of the fifty most POP-contaminated foods, a rate much higher than for the typical U.S. male. Among the top sixteen most POP-toxic foods, EWG lists: American cheese, butter, cheddar cheese, cheeseburgers, dill pickles, hamburgers, lasagna with meat, peanut butter, peanuts, pepper-oni pizza, potato chips, pumpkin pie, raisins, sour cream, vanilla ice cream, and whole milk.\n\nYet another term used to describe these ubiquitous environment chemicals is PBTs, or persistent, bioaccumulative, and toxic pollutants, according to a report released by the EPA in September 2000. With all of these terms\u2014VOCs, POPs, PBTs\u2014there is a fair amount of overlap, but at the same time, they provide confirmation and public\u2014in this case, governmental\u2014recognition that toxic substances in the environment are a serious human health problem. Chief among the PBTs, according to the EPA, are methylmercury (mainly from tainted fish and hospital waste incineration), polychlorinated biphenyls, or PCBs, and dioxin (from pesticides).\n\nIn a report on their new initiatives to control and reduce levels of PBTs in the environment, the EPA acknowledged that PBTs concentrate in the food chain to toxic levels, and have an uncanny ability to travel long distances and to linger for generations in people and the environment. They noted that dioxin levels have now reached the level associated with adverse non-cancer effects, which is probably one-step away from being a widespread carcinogen; and that twenty-five percent of children and nine percent of adults in the United States are exposed to unacceptable levels of methylmercuiy. Levels of PBTs in the Arctic are expected to rise significantly due to the distribution phenomenon called global distillation (described previously) as a result of \"increased local and southeast Asian industrialization.\"\n\n### The Toxic Living Space and You\u2014What It All Means\n\nIt is not the intent of this chapter to provide complete documentation of every source of environmental pollution and bodily toxicity. To do this subject justice would take a book in itself. It is more important here to gain an understanding of the extent of pollution and its health consequences. You have gotten a sense of the players\u2014POPs, VOCs, PBTs, pesticides, phthalates, organochlorines, nuclear radiation, excitotoxins, \"crazy-maker\" foods\u2014and how they permeate our bodies and the modern environment. And you have come to see that to some extent everyone has toxic substances within their fatty tissues and that every corner of the planet carries toxics in its soil, water, plant, and animal life.\n\nThis knowledge can seem at first overwhelming, intimidating, even depressing. Part of the reason it might seem too much to know is that, as a culture, we have known so little, and have never been encouraged to ask. So it is shocking to suddenly expand our awareness, to start digesting the awesome details of global toxicity and its permeation of our living space, both body and home. But what else would motivate us to take the needed steps to start detoxifying ourselves and to start petitioning for ecological awareness and pollution reform in society? Taking a long hard look at the extent of the damage, and the risk of still more damage, is sobering, but it gives us a sound information base upon which to build a practical program of detoxification and minimization of further exposure to toxic substances.\n\nAs discussed throughout this chapter, many of the persistent organic pollutants are very stable, very persistent, and very deeply entrenched in human flesh. It is unlikely that even the most comprehensive detoxification program will flush them all out of your system. It is also unlikely that you will be able to live in such a way that you avoid all contact with toxic sources.\n\nWhat is likely and therefore prudent is to _reduce_ your exposure, _minimize_ your contact, and _maximize_ your body's excretion of the toxic substances you can't avoid. The goal of this book is to show you how (and why) to adopt many practical ways of helping your body detoxify itself of as much of its toxic load as it can and at the same time to build up your body's natural resistance to toxic substances.\n\nOne of the chief tools you need in this effort to clean up your living space is awareness. Pollutants are so prevalent, so taken for granted, so invisible in our world today that it is very easy to get exposed many times during a single day without being aware of it. Here are a few examples:\n\nI visited some friends in another state recently. One of their neighbors was an avid gardener, and had an acre devoted to a lovely flower garden, with statuary, fountains, a fish pond, contemplation benches, and many flowering bushes and plants. On the day of my visit, she was using a small bottle with a pump to spray pesticide on dandelions growing by the roadside at the edge of her garden. The device hardly made a sound, and the liquid looked like water, but it was a poison and it was emitting little molecular clouds of toxins. I stepped back so as not to breathe these pesticide fumes, but my friend showed no such concern.\n\n**_What is likely and therefore prudent is to reduce your exposure, minimize your contact, and maximize your body's excretion of the toxic substances you can't avoid._**\n\nAnother friend works in an art gallery and reported this incident to me. The gallery is a small, rectangular space with no cross ventilation from front door to back door. An artist supervising the installation of his show asked if anyone on the staff\u2014there were five people present\u2014minded if he sprayed a \"fairly nasty paint\" on the wall behind his sculptures. Nobody minded, except my friend, who opened the front door and stood outside while the toxic paint was sprayed on the walls. Everyone else regarded the paint as if it were as routine as secondhand tobacco smoke, and therefore to be viewed as benign (an erroneous conclusion in both cases).\n\nA third friend made a routine shopping visit to a small mall in a medium-size college town. When she arrived in the parking lot, she noticed a few police blocking entry of traffic at the far end, while others ran around, shouting and waving things. She parked, walked a few paces, and was about to enter a store when she smelled the foulest chemical odor she had ever experienced. She had no idea what it was, but noticed that already some of her throat muscles felt stiff, as if they had turned to wood. She left the mall and drove home.\n\nWhen she called one of the stores in the mall, she was told a medical waste truck had leaked diesel and this had combined with recent acid rain to produce the smell. She was also told that officials from the Environmental Protection Agency had been called in. No diesel she had ever encountered had smelled like that, so she didn't believe the story. It took a day for her neck to feel normal again, and for all numbness to go away. But as to the health effects on all the store personnel and shoppers who were exposed to those obviously toxic fumes all that day and who were not kept from the mall one can only guess.\n\nYou can see from these three simple examples\u2014presented to me in the space of five days\u2014how crucial is awareness, being alert to when you are about to get a low-dose toxic exposure and avoiding it or minimizing your exposure time.\n\nSome of the toxic agents mentioned in this chapter will be discussed in greater detail in later chapters, along with practical steps for detoxification and lifestyle arrangement to help you avoid or minimize further exposure to them. In the next chapter, we look at the body's natural detoxification system and see what happens when it gets overwhelmed by environmental toxins and can no longer rid the body of toxic substances.\n\n# CHAPTER 2\n\n# _Why You Need to Detoxify: Augmenting the Body's Natural Detoxification System_\n\nThe essential point in this chapter is that while your body has innate detoxification mechanisms, the degree to which the average person is now toxic often limits the ability of these systems to work. In other words, your body may be so overloaded with toxins that it cannot do its naturally appointed job of toxin removal, of detoxification. This failure leaves you less than optimally fit, sluggish, tired, allergic, aching, sore, subject to any of dozens of symptoms, and eventually suffering a chronic health problem, or worse, a degenerative condition, such as cancer.\n\nWhile the solution is straightforward\u2014deliberately, responsibly detoxifying to unclog and catalyze the body's own internal cleansing systems\u2014the question that must be answered is this: why is the body's natural detoxification system ineffective? What are the consequences of being toxic and unable to fully detoxify ourselves of poisons we get from our environment? How should the detoxification system work if it is in top condition? Why should we strive to return it to optimal efficiency? These are the questions we'll consider in this chapter.\n\n### The Body's Burden\u2014Total Toxic Load and Toxic Stress\n\nPhysicians who are alerted to the necessities of healthy detoxification often speak of the body in terms of toxic overload. They may use clinical terms such as total toxic burden, total toxic load, and toxic stress to indicate a condition of dysfunctional, inefficient, even incompetent detoxification. In nonmedical, layperson's terms, this is like a drain that is so clogged with debris it lets only ten percent of the grey water flow down, trapping the other ninety percent and creating a stagnant pool of toxic water.\n\nNaturopaths Peter Bennett, N.D., and Stephen Barrie, N.D., have come up with an amusing yet accurate term for our life under the influence of toxins: we are in- _toxin_ -cated, they declare. It's like being inebriated, except it isn't alcohol we consumed, but a long list of toxins from substances in our environment; even worse, we don't even remember the \"party\" at which we imbibed the poisons. Drs. Bennett and Barrie, like many alternative-minded practitioners sensitive to the importance of detoxification, use the image of an overflowing barrel\u2014a perhaps more apt analogy than the clogged drain.\n\n**_Think of your body and its organs and systems as a barrel into which there is a slow, steady drip of toxic water. Eventually, unless you drain the barrel (detoxify), the barrel will overflow, and toxic water will spill over everything, will permeate parts of your body ill-equipped to handle the deluge._**\n\nThink of your body and its organs and systems as a barrel into which there is a slow, steady drip of toxic water. Eventually, unless you drain the barrel (detoxify), the barrel will overflow, and toxic water will spill over everything, will permeate parts of your body ill-equipped to handle the deluge. \"That toxic overflow is the beginning of disease and disability,\" explain Bennett and Barrie. They add that if your detoxification system (your liver, kidneys, intestines, and immune and lymphatic systems) is overworked (which includes being nutritionally depleted or deficient, as explained below), any new exposure to toxins from foods or environmental substances and products is even harder to handle and creates an even greater toxic problem for the components of that system\u2014your liver, intestines, and immune system. In other words, if your \"barrel\" is full, even a minimal toxic exposure can have the health impact of a large one.\n\nWhere does the excess toxicity go when it overflows into other parts of your body? A great deal of it accumulates in the spaces between the cells, in a place called the extracellular connective tissue matrix. \"It is, in a sense, a living matrix\u2014a place where many systems intersect and are coordinated,\" explains Rudolf Ballentine, M.D., in _Radical Healing._ This space between the cells is the background, the field or ground supporting the major organs; for this reason, conventional medicine tends to overlook this area as a focal point for toxicity, says Dr. Ballentine. But they shouldn't. It is crucial to the body's functioning.\n\nThis matrix is an all-purpose dump site and part of the drainage system within the body. It takes cellular and lymphatic system waste and holds it in temporary storage. The fluids in this space can get saturated with toxins and start to leak the excess into neighboring connective tissue. When the storage capacity of this extracellular space is exceeded and leakage occurs, you begin to develop signs of toxicity. For example, you may feel stiffness or pain in the body; these symptoms result from the irritation produced by these deposits in the connective tissues.\n\nThe extracellular matrix also comprises \"an intricate communications network\" running alongside that of the nervous and endocrine system. But if the matrix is overburdened with toxic storage, the communications don't work properly and the energy channels get compromised. At that point, you will start feeling ill, with symptoms such as general tiredness, dullness of the mind and senses, bad breath, heaviness, lack of strength, indigestion, constipation, coated tongue, and foul-smelling urine and\/or feces, says Dr. Ballentine. Then you start moving \"from congestion to disease.\"\n\nHere is another analogy by which to appreciate an overburdened detoxification system. Think of your immune system as a kettle, suggests Kenneth Bock, M.D., an alternative-minded physician who directs the Rhinebeck Health Center in New York State. Like a barrel, a kettle can hold only a finite amount of toxins. But there are many categories of toxins that can get into our immune system kettle, explains Dr. Bock.\n\nThese include psychosocial factors (such as emotional stress); environmental pollution (toxins and heavy metals out in the world); tobacco smoke; radiation and electromagnetic field exposure; prescription and recreational drugs; hormonal imbalances; infectious agents (microorganisms); allergies and other sensitivities (to chemicals, foods, plant substances); dietary stresses (excesses or deficiencies in nutrients, too many synthetic chemicals); and our genetic blueprint, which is to say, the balance of strengths and weaknesses and our general predisposition to certain types of health problems.\n\n\"Any one part of our immune system load can put us 'over the top' of the kettle if our accumulated burden is high,\" says Dr. Bock. In other words, we can tolerate perhaps six categories of toxins or toxic loads, but the seventh will make the immune system kettle overflow, and we get sick from the spillage. (Note that individual tolerance varies; some people might experience overflow at four, or even fewer, toxic categories.)\n\nHere is a vivid example provided by Dr. Bock, citing a case originally described in _The Wall Street Journal_. A man who was vigorous and athletic mowed his lawn and collected the grass clippings, then almost fell down dead. He got dizzy, nauseous, and developed a terrible headache. Ten days later he was just as sick, and getting worse, and five months later, he got testicular cancer. As the years passed, he had seizures, walking difficulties, and other health deficits. What had happened?\n\nThe man was poisoned by a lawn pesticide sprayed on his grass; he inhaled the fumes and they sickened him. His liver was unable to detoxify the chemical because it was already burdened with detoxifying the components of Tagamet, a drug for peptic ulcers, on top of the environmental pollutants to which we are all exposed. The constant demand to detoxify the Tagamet left him vulnerable to carcinogens and defenseless against their effects once they entered his body.\n\n\"How many of us have toxins accumulating in our bodies?\" queries Dr. Bock. \"The scary truth is, we all do.\" Many of us, shockingly, are \"walking toxic dumps,\" he adds. It is almost impossible for anyone in the Western world today (and perhaps anywhere in the world at all now) to avoid building up some degree of immune toxic load, Dr. Bock explains, but he adds that the problem is also the doorway to the solution. Anyone can purposefully lower their immune load\u2014drain the kettle\u2014by a careful program of detoxification.\n\n### The Oxidation-Reduction Cycle at the Heart of Detoxification\n\nBefore we look at how we can clear our detoxification system\u2014empty the kettle or at least pour out some of the water\u2014it's important to get a picture of the biochemical struggle constantly under way in the human body, and meet its players: the toxins, called free radicals; and the anti-toxins, called antioxidants.\n\nFirst, there are two types of toxins, or free radicals. _Exotoxins_ are those that enter the body from the environment, from products, substances, medicines, additives, foods, the air and water we consume. _Endotoxins_ are produced inside us, as natural byproducts of digestive activity or through illness. Sources of free radicals include pesticides, industrial pollutants, smoking, alcohol, viruses, most infections, allergies, stress, even certain foods and excessive exercise. Toxins of either type can bioaccumulate by being stored in the body's fatty tissues, including cell membranes; once the toxins are fat-soluble (as opposed to water-soluble), it is harder for the body to detoxify them.\n\nSecond, there is a natural chemical process that takes place continuously inside the body that is the key to the toxicity-detoxification situation. It's called oxidation and refers to the way oxygen combines with a substance to release energy. A fire is \"fueled\" or driven by oxidation, and so are the body's digestive and metabolic processes. The body uses oxygen, drawn from the outer environment, to \"burn\" fats and carbohydrates from ingested foods to generate energy to run a myriad of cellular processes. The body generates energy for itself by oxidizing the foods we consume in a controlled way. That's why we eat food in the first place: to provide this cellular fuel released by the chemical processes known as metabolism.\n\nHowever, this process of cellular oxidation has some natural byproducts\u2014smoke, so to speak, if this were wood we were burning\u2014including carbon dioxide and water, which are harmless, and free radicals, which are harmful, highly reactive molecules\u2014\"molecular marauders,\" says Dr. Bock. All chemical reactions in the body involve the transfer of electrons, as a kind of \"currency\" exchange. In this case, the exchange is for energy to run the body. The metabolism of food, or oxidation, removes electrons one by one from the foods. Think of oxidation as a controlled fire that releases energy but also sends off sparks. In this analogy, the sparks are free radicals, or unpaired, unstable, chemically reactive electrons.\n\nA free radical is a toxic molecule of oxygen with an unpaired electron. A free radical tries to remedy its instability by stealing an electron from another molecule; when it does, a new free radical is produced, and this leads to a chain reaction of ever multiplying free radicals and unstable molecules. In a sense, it's the instability\u2014the unpairedness\u2014that creates all the damage. The proliferation of free radicals as a result of oxidation is called oxidative stress. The degree of oxidative stress can be taken as a measure of the body's capacity, or incapacity, to handle toxins and perform its own detoxification (see Chapter 3).\n\nFree radicals are formed when molecules within cells react with oxygen (oxidize) as part of normal metabolic processes. Free radicals then begin to break down cells, especially the cell membranes, often in a matter of minutes to an hour. A single free radical can destroy a cell. Their work is enhanced if there are not enough free-radical quenching nutrients, such as vitamins C and E, which are normally present in the cell. A free radical attacks the cell's membranes, eventually rupturing it; this allows the egress of bacteria, viruses, and other pathogenic microorganisms into the cell and the leakage of essential nutrients out of the cell.\n\nOnce inside the cells, free radicals can change the cell's physical structure, making it rigid and inflexible. Free radicals harmfully alter important molecules, such as proteins, enzymes, fats, even DNA. They can commandeer the cell's genetic apparatus, as it were, rewriting or destroying essential genetic programming. This in turn can lead to cellular mutations, which is only one step away from cancer.\n\nFat cells (the unsaturated lipid molecules, specifically) are especially vulnerable to free radical disturbance. When the fat components of cells are damaged by free radicals, lipid peroxides result. The quantity of lipid peroxides in your blood can be taken as an index of your risk for what's called oxygen free radical pathology and the chance of developing a degenerative disease as a result of widespread and continuous cell damage. High blood levels of lipid peroxides mean your system has _already_ been damaged by free radicals.\n\nFree radical pathology in many ways is the cornerstone of the model that explains how toxins make you sick. The means by which other toxic substances or procedures, such as irradiation, heavy metals, solvents, pesticides, and prescription drugs, produce pathology is by generating free radicals in the body. As free radicals can potentially damage every aspect of cell function including the integrity of its membrane, a long list of degenerative illnesses can be set in motion once free radicals start damaging your cells.\n\nSo central to the dynamics of health is the concept of free radical damage that practitioners of alternative medicine now speak in terms of free radical pathology as \"a unified cause of chronic illness.\" The term mentioned above, oxidative stress, is pivotal to this discussion of toxicity and illness. A person's degree of oxidative stress depends on the balance between the rate at which free radicals produce damage and antioxidants repair it, or the rate at which free radicals proliferate and the antioxidants quench them. If free radicals predominate, the degree of oxidative stress increases, and eventually you get sick, and still later, start to develop degenerative disease.\n\n\"Unfortunately, many people run an abnormally high level of oxidative stress that could increase their probability of early incidence of age-related diseases,\" states a consumer information booklet from Genox Corporation, a clinical laboratory in Baltimore, Maryland, that tests for oxidative stress. Evidence now suggests that even individuals who seem healthy may have a much higher than average (and therefore undesirable) level of oxidative stress and may already be \"well on their way towards developing an oxidative stress-related disease,\" states Genox.\n\nAccording to Genox, oxidative-stress-induced dysfunctions and diseases are many but can include accelerated aging processes; heart disease (stroke, heart attack); disorders in the gastrointestinal tract (including diabetes and pancreatitis); macular degeneration, cataracts, and other eye problems; kidney and skin disorders; nervous system problems (Parkinson's, Alzheimer's, muscular dystrophy, multiple sclerosis); lung cancer and emphysema; and serious problems with red blood cells, the liver, and immune system (rheumatoid arthritis). As early as 1988, scientists were prepared to attribute at least sixty disorders to free radical damage, and one expert in the field declared that few people reach their maximum life span but rather die prematurely from diseases, the vast majority of which are free-radical-related diseases.\n\nThe theory connecting free radical toxicity with aging and disease was formulated in 1954, however, and one of its leading proponents since then has been Denham Harman, M.D., of the University of Nebraska College of Medicine in Omaha. After age twenty-eight, says Dr. Harman, \"the inborn aging process\" is the major risk factor for disease and death; this inborn aging process is strongly (but not exclusively) determined by free radical reactions in the body and the body's ability to neutralize them. Such free radical reactions, a growing consensus maintains, \"are a major cause of aging, possibly the only one,\" states Dr. Harman. He adds that average life expectancy at birth can be increased by five to ten years and a \"healthy active life span\" obtained \"by nutritious low caloric diets supplemented with one or more free radical reaction inhibitors [antioxidants].\"\n\n**_Early detection of high levels of oxidative stress gives you the chance to stop the progression of oxidative stress to disease. The earlier you can find out if you are oxidatively stressed, the sooner you can take corrective action in the form of a carefully planned detoxification program and antioxidant supplementation._**\n\nHarman's research makes it clear that _early_ detection of high levels of oxidative stress (through laboratory testing, as explained in Chapter 3) gives you the chance to stop the progression of oxidative stress to disease. The earlier you can find out if you are oxidatively stressed, the sooner you can take corrective action in the form of a carefully planned detoxification program and antioxidant supplementation.\n\nOn the other side of the toxicity picture are the antioxidants, the natural elements that work against oxidative free radicals. An antioxidant (meaning \"against oxidation\") is a natural biochemical substance that protects living cells against damage from harmful free radicals. Antioxidants work against the process of oxidation\u2014the robbing of electrons from substances. Antioxidants in the body react readily with oxygen breakdown products and free radicals, and neutralize them before they can damage the body. Dr. Bock uses the analogy of an indoor fireplace spewing sparks that could ignite the carpet; the sparks are free radicals, and the screen that you place preventively before the burning logs comprises the antioxidants.\n\nThe antioxidants render free radicals harmless by a process known as reduction: they add back the missing electron by giving up a molecule from themselves to the unpaired electron in the free radical. Thus oxidation and reduction form a system (sometimes referred to by the abbreviation \"redox\"). In oxidation, an electron is lost, producing a free radical; in reduction, the electron is given back by an antioxidant; even simpler: oxi-dation=electron loss; reduction=electron gain. When the redox system runs in balance, you stay healthy because your natural detoxification system works efficiently; when it runs in a state of imbalance, you get toxic and eventually, you get sick.\n\nAdequate nutrient intake is vital to your body having the biochemicals it needs to run the antioxidant-reduction pole of the redox cycle. Antioxidant nutrients include vitamins A, C, and E, beta carotene, selenium, coenzyme Q10, pycnogenol (pine bark extract), L-glutathione, superoxide dismutase, and bioflavonoids. Plant antioxidants include _Ginkgo biloba_ and garlic. When antioxidants are taken in combination, the effect is stronger than when they are used individually.\n\n### The Body's Two-Phase Natural Process of Detoxification\n\nThe human body is well equipped to handle a normal load of toxins and to neutralize and remove them from its interior. Of course in this discussion, the key term is \"normal load\" because, as discussed in Chapter 1 and earlier in this chapter, most of us no longer carry a normal toxic burden, able to be handled by the body's innate detoxification system. Rather, many of us have far too many toxins in our bodies, a situation that tends to compromise and limit the body's natural ability to detoxify itself.\n\nIn effect, many of us are too toxic for our bodies' detoxification mechanisms to work properly, and we end up depositing toxins in various fatty tissues as \"seeds\" for imminent health problems. To understand how our natural detoxification pathways get blocked and dysfunctional, let's first study how the healthy body is meant to deal with toxicity through what physiologists call the liver's two phases of detoxification.\n\nThe liver is the prime receptacle of and processing center for all the toxins our body ingests or produces internally as part of its standard biological processes. As noted earlier, the kidneys and intestines are the other principal players in detoxification, but only about twenty-five percent takes place in the intestines; the rest happens in the liver. In brief, in Phase I detoxification various toxic biochemicals are modified to make them an easier mark for the detoxification enzymes that will neutralize them in Phase II. The key concept is that liver-mobilized enzymes neutralize toxic chemicals so the body can excrete them, and this process depends on adequate body reserves of key nutrients.\n\n### Phase I Detoxification\n\nThe goal of this phase, which scientists label \"biotransformation,\" is to chemically alter in the liver any substances the liver identifies as toxins (\"xenobiotics,\" or foreign life) and to change them into an intermediate form. You might say this is like taking the stinger out of the bee without killing it.\n\nGenerally speaking, the liver has to deal with toxins that have a preference for fat (lipophilic, or \"fat-loving\") and that are not water soluble, that is, they don't mix with or dissolve in water. These fat-soluble toxins also are nonpolar, which means they have no electrical charge. Phase I detoxification will make them water-soluble and electrically charged, both of which changes favor easier excretion from the body. This biotransformation is accomplished in part by the addition of an atom (or several) to the toxin, such that it loses electrons (oxidation) or gains electrons (reduction), and the removal of a specific component of its makeup, such as taking away a group of sulfur or halogen molecules.\n\nThe toxins can include pollutants that enter the body through the air, water, and foods we consume, or through medicines and drugs. They also include toxic chemicals produced naturally\u2014that is, for a biological reason\u2014but which, if allowed to build up internally, are toxic to the body. These include various hormones and inflammatory chemicals such as histamine, involved in the allergic reaction.\n\nVarious enzymes are required to perform Phase I detoxification, and these are known as mixed function oxidative enzymes, collectively called cytochrome P450. It's estimated that about 50-100 enzymes comprise this group. But let's stop for a moment: What is an enzyme?\n\nAn enzyme is a specialized living protein fundamental to all living processes in the body, necessary for every chemical reaction and the normal activity of our organs, tissues, fluids, and cells. Enzymes, of which there are hundreds of thousands, are essential for the production of energy required to run all the functions of cells. Enzymes enable the body to digest and assimilate food. There are special enzymes for digesting proteins, carbohydrates, fats, and plant fibers. Enzymes also assist in clearing the body of toxins and cellular debris, in addition to assisting the liver in its detoxification processes.\n\nThe competence of the cytochrome P450 system can vary considerably among individuals, due apparently to genetics, one's level of toxic exposure, and one's nutritional status. For example, it is known that, to work optimally, cytochrome P450 requires vitamins C, B2, B3, B6, and B12; copper; magnesium; zinc; folic acid; and flavonoids. If your diet has been nutritionally inadequate and you are exposed to a high degree of toxins\u2014say air pollution in a big city such as Los Angeles\u2014your detoxification capabilities are already set a notch or two lower than ideal. An Italian study of chemical plant factory workers, for example, showed that those with an underactive cytochrome P450 system were more vulnerable to carcinogens (substances known or suspected to contribute to the emergence and spread of cancer).\n\nWhat happens when the liver's enzyme team encounters a xenobiotic, or toxin? It tries to 1) change the substance into a less toxic form; 2) neutralize it by rendering it water-soluble so it can be easily excreted in the urine; or 3) convert it to a more chemically active form, making it an easier target for the Phase II detoxification (this process is called bioactivation). It is ironic that the very process of changing, neutralizing, and converting toxins itself produces free radicals, which also must be neutralized or removed from the system. Every time a foreign toxin is metabolized (chemically acted upon\u2014\"digested\" in a sense), a free radical results. Each new free radical generated during detoxification can in turn harm the liver and compromise its continuing detoxification abilities. Incidentally, eating foods to which we have allergies also produces free radicals as well as the histamine and other inflammatory products related to an allergic reaction. All of these further strains the liver's detoxifying capacity.\n\nNormally, the liver is able to handle the temporary increase in free radicals via its highly competent antioxidant protein, glutathione, which comes into play in Phase II detoxification. However, if the liver is flooded with toxins in Phase I and has to work overtime to neutralize the free radicals, this uses up the glutathione reserves, leaving none for Phase II. This highlights a tricky part of Phase I. In converting toxins to make them easier targets for Phase II, the new forms, known as \"activated intermediates,\" can be much more toxic than the originals.\n\nThe goal is to get these activated intermediates dealt with as quickly as possible in Phase II and removed from the body. In a balanced detoxification process, the activated intermediates are generated at a rate that Phase II can keep up with; in an unbalanced process, Phase II gets overloaded with the activated intermediates and your detoxification ends up being pathological, itself producing poisons. People with this problem are known clinically as \"pathological detoxifiers,\" and they tend to be excessively reactive to externally contacted poisons because their total toxic burden has already been reached and their Phase II abilities are dangerously compromised.\n\nBiologists have discovered that certain foods, herbs, nutrients, and industrial or human-made substances can activate Phase I while others can inhibit its action. Among those that can trigger Phase I are alcohol, nicotine (tobacco smoke), phenobarbital, steroids, niacin, riboflavin, vitamin C, sassafras, caraway and dill seeds, oranges, tangerines, cabbage, broccoli, Brussels sprouts, and charcoal-broiled meats.\n\nIn a complementary way, specific nutrients can help the Phase I process: choline, fatty acids, lecithin, methionine (an amino acid), milk thistle (an herb), beta-carotene (a vitamin A precursor), vitamin Bl, vitamins C and E, copper, iron, magnesium, manganese, molybdenum, sulfur, and zinc.\n\nThe practical aspect of this information is immediate: if you are deficient in any of these nutrients, your Phase I detoxification may be hindered; if you are undergoing a deliberate detoxification program and you supplement with these nutrients, the process may proceed at a far better rate of competence than otherwise.\n\nBut there is an equally long list of substances that can block Phase I from getting started, and these include various antidepressants, antihistamines, and other drugs; chemicals found in grapefruit juice; turmeric; red chili pepper; and clove oil. Scientists found that naringenin, a substance naturally occurring in grapefruit juice, can decrease your levels of cytochrome P450 by thirty percent.\n\n### Phase II Detoxification\n\nChemically speaking, what happens in this second part of your body's natural detoxification process is called conjugation. This means various liver enzymes affix chemicals to the toxin, which has been oxidized or partially changed from Phase I activity, either to neutralize it or to make it easily excreted through the urine or bile. The goal is to prepare (alter) the toxic substance so it can be excreted from the body. There are six pathways in Phase II, each involving specific ways of neutralizing toxins; we might say that each of these six detoxification pathways specializes in dealing with a certain family or type of toxin.\n\nOne of the main pathways is glutathione conjugation. A liver enzyme called glutathione S-transferase (glutathione is made of three amino acids, the building blocks of proteins) combines (or conjugates) sulfur with the toxic substance making it water soluble and easily excreted. Glutathione is the body's main defense against free radicals and a key neutralizer of the free radicals generated during Phase I. \"The combination of detoxification and free radical protection results in glutathione being one of the most important anticarcinogens and antioxidants in our cells, which means that a deficiency is devastating,\" comments naturopathic educator and physician Joseph Pizzorno, N.D., former president of Bastyr University in Seattle, Washington.\n\nThe glutathione detoxification (or conjugation) pathway generates up to sixty percent of the toxins that get excreted in the bile. These include industrial toxins such as PCBs and a fair number of carcinogenic substances. Glutathione is also a potent antioxidant, collecting and neutralizing free radicals in the blood. The trouble is if your system gets overloaded by deadly toxins\u2014by an intense one-time or prolonged exposure to toxins, such as in a chemical plant or nuclear power plant\u2014your glutathione reserves get used up and you become essentially defenseless against the powerful free radicals. You then become more susceptible to serious illnesses including cancer because your immune defenses have been dangerously weakened.\n\nAnother important detoxification pathway is called amino acid conjugation, which involves five amino acids that combine with and neutralize specific targeted poisons. These poisons include at least four classes of drugs and five types of xenobi-otics, or foreign toxic chemicals. Let's stop for a moment to be sure we understand the essential term, amino acid.\n\nAmino acids are the basic building blocks of the 40,000 different proteins in the body, including enzymes, hormones, and the key brain chemical messenger molecules called neurotransmitters. Eight amino acids cannot be made by the body and must be obtained through the diet; others are produced in the body but not always in sufficient amounts. The body's main \"amino acid pool\" consists of: alanine, arginine, aspargine, aspartic acid, carnitine, citrulline, cysteine, cystine, GABA, glutamic acid, glutamine, glycine, histidine, isoleucine, leucine, lysine, methionine, ornithine, phenylalanine, proline, serine, taurine, threonine, tryptophan, tyrosine, and valine. The amino acids involved in amino acid conjugation are glycine, taurine, glutamine, arginine, and ornithine.\n\nHere is an interesting fact about one aspect of the amino acid conjugation pathway. If you consume a lot of soft drinks (sodas) that contain benzoates, this substance binds with your body's reserve of glycine, one of the five conjugation amino acids, and makes it unavailable to the body. This then makes it hard for your system to detoxifiy toluene, an organic solvent found in many industrial products and substances one encounters on an almost daily basis. Similarly, aspirin can grab up glycine reserves in the liver, thereby slowing down the glycine detoxification pathway. These limitations in glycine detoxification are of course reversible by supplementing with glycine, along with the other amino acids.\n\nAnother Phase II detoxification pathway is called glucuronidation, which entails the binding of glucuronic acid with foreign toxins. This pathway is responsible for neutralizing at least fifteen known types of prescription drugs, five xenobiotics, and various substances of dietary origin (free radicals naturally generated in the body during the metabolism of foods).\n\nSulfation, still another detoxification pathway, is often defective in people due to inadequate intake of dietary sulfate. This means if you are exposed to toxins that are supposed to be detoxified by the sulfation pathway and you are deficient in sulfate, your system will be unable to neutralize these substances and they remain free to produce illness and various health problems such as Alzheimer's, Parkinson's, autism, rheumatoid arthritis, food sensitivity, and multiple chemical sensitivity, among others.\n\nWhen it's working optimally, the sulfation pathway will use sulfur-containing compounds to handle the neutralization of neurotransmitters, steroid hormones (including excess thyroid and estrogen), drugs, industrial chemicals, various compounds derived from benzene (as used in plastics, disinfectants, and drugs), and toxins from the environment and our own intestines. According to naturopathic educators Peter Bennett, N.D., and Stephen Barrie, N.D., this is often \"the weakest pathway in most people, from a dietary standpoint.\"\n\nThe preceding examples make it clear how important satisfactory nutrition is to successful and efficient detoxification, and show what happens when you are deficient in only one element, such as glutathione or sulfate. These facts also make a strong case for undergoing some kind of diagnostic screening and testing, as described in the next chapter, to document your actual biochemical status and degree of detoxification competency.\n\nAfter the liver finishes its two-phase detoxification process, the bile carries the toxic metabolites (the neutralized residues or converted substances) to the intestines for excretion. The liver produces about one quart of bile every day as a carrier for toxins to be excreted from the intestines (to which it is sent), but also as a way of emulsifying fats and fat-soluble vitamins in the intestines, and as the primary excretion pathway for cholesterol and unneeded calcium.\n\nAs with Phase I, this second aspect of the liver's detoxification process requires sufficient internal levels of key nutrients or can be enhanced by the ingestion of specific substances, including cysteine, glycine, and taurine, garlic, L-glutathione, N-acetyl-cysteine (NAC, a cysteine precursor), folic acid, vitamins B 1, 2, 3, 5, 6, and 12, germanium, magnesium, manganese, molybdenum, selenium, sulfur, and zinc.\n\n### Immunotoxicity and How the Detoxification Pathways Get Blocked\n\nWe have just reviewed the liver's two-phase detoxification system as it is meant to work, but for many people, this optimum operation is only a physiological ideal and not a bodily reality. For many, certain elements of the detoxification pathway are blocked, compromised, or dysfunctional, leading to a state of chronic internal stagnation and immunotoxicity.\n\nWhen the detoxification pathways in the liver are blocked or inefficient, the toxic products build up in the bloodstream and intestines, producing an extra load of \"foreign\" substances the immune system must deal with. The immune system may get overloaded and overstimulated; it will start to show signs of this condition in allergies, headaches, chronic sinus conditions, inflammatory states (muscle soreness and intestinal irritation), infections (colds and flus) that keep coming back, swollen lymph glands, and from here on to more serious disorders. In other words, through your system's inability to completely detoxify itself, you are slowly but definitely \"growing\" illness from within. Eventually, assuming you do not actively detoxify, the toxicity will have \"sprouted\" some noxious growth inside your body and you will be sick in one form or another.\n\n**_Through your system's inability to completely detoxify itself, you are slowly but definitely \"growing\" illness from within. Eventually, assuming you do not actively detoxify, the toxicity will have \"sprouted\" some noxious growth inside your body and you will be sick in one form or another_**.\n\nIn addition to considering the state of the liver's detoxification pathways, we must also pay attention to the intestines. While the issue of chronic constipation and colon cleansing is discussed thoroughly in Chapter 4, a quick overview is worthwhile here. You must appreciate the astonishing fact that the human intestines, large and small, comprise a twenty-six-foot-long internal organ. That leaves a lot of room for storage; daily bowel motions do not indicate that a person is not chronically constipated. Most people are, because the intestines are so, literally, accommodating. Early signs of intestinal toxicity include headaches, sinus problems, allergies, sluggishness, mental torpor, irritation, mood swings, concentration problems, weight gain, and others.\n\nIf unrelieved, these symptoms provide a platform for the next layer of complications to arise. The intestinal wall might start to \"leak,\" producing a symptom physicians call \"leaky gut syndrome\" or intestinal permeability. Intestinal toxins migrate out of the intestines into the bloodstream and into general systemic circulation. This is not desirable. The immune system goes on alert and soon will become hyperstimulated. It tags all the toxins floating in the blood as foreign proteins and has to work overtime to deal with them.\n\nThe immune system treats these foreign substances or antigens as invaders, causing antibodies to form and couple with them. This antigen and antibody combination is known as a circulating immune complex, or CIC. In a healthy person, CICs are neutralized, but in someone with a compromised immune system, they tend to accumulate in the blood where they burden the detoxification pathways or initiate an allergic reaction. If too many CICs accumulate, the kidneys cannot excrete enough of them through the urine. The CICs are then stored in soft tissues, causing inflammation and bringing stress to the immune system. The overload can lead to a variety of chronic health conditions, including autoimmune disorders and cancer.\n\nWhen environmental pollutants and our own internally generated free radicals are not neutralized and removed from the body, they start having serious adverse effects inside us. They become toxic to the immune system\u2014hence, immunotoxins. Numerous symptoms can result, but among those related to unrelieved toxicity is the strange cluster of symptoms that began to show up in people in the 1990s. This cluster was subsequently labeled environmental illness. In a sense, environmental illness represents the exaggerated, almost surreal result of unrelieved toxicity, demonstrating how it affects numerous body functions, throws out multiple symptoms, and generally baffles conventional medical thinking. However, the thinking in the world of alternative, natural medicine is that conditions like El are the result of aggravated systemic toxicity.\n\nAs discussed in Chapter 1, El is similar in many respects to multiple chemical sensitivity (detailed in chapter 10) and chronic fatigue syndrome. It is a multispectrum dysfunctional state believed to be produced by excessive exposure to environmental pollutants and a corresponding inability of the body to detoxify them. Dismissed as an imaginary illness and given other pejorative labels by conventional medicine, El found confirmation initially among the ranks of practitioners of environmental medicine, formerly called clinical ecology.\n\nIn fact, it was in the late 1940s when pioneering clinical ecologist Theron Randolph, M.D., first identified the health deteriorating effect of environmental toxins and promulgated the medical understanding that many conditions, such as allergies and other chronic problems, could be produced by the glut of unprocessed toxins in the human body.\n\nDr. Randolph had been building on earlier work that linked food allergies with chronic diseases of unknown origin (rheumatoid arthritis, ulcerative colitis, migraines) and common symptoms (headache, fatigue, hyperactivity, depression). He proposed that synthetic environmental chemicals (particularly in air pollution) could cause the same allergic symptoms as did reactions to food. Randolph reported 1,000 cases of patients who showed psychotic behavior directly related to exposures to specific toxic substances.\n\nThe prime \"cause\" of this growing list of illnesses was the \"relatively new chemical environment,\" Dr. Randolph claimed. This environment itself is ill, he said, and the aim of clinical ecology is to \"identify, minimize, or neutralize the impingement of specific environmental exposures.\" In the 1980s, other progressive physicians, such as William J. Rea, M.D., of the Environmental Health Center, in Dallas, Texas, extended and further documented Randolph's model of toxicity-mediated illness and developed clinical protocols for detoxifying patients who were dangerously toxic. Today his center is one of the nation's prime detoxification facilities, offering a pollution-free environment in which chemically toxic patients can flush the poisons out of their systems under medical supervision.\n\nAmong its other contributions, Dr. Rea's ongoing research has dissolved the dismissive label of \"hypochondriasis\" from the complaints of numerous patients that they have been poisoned by their environment. It's not in their minds; they are not making it up, says Dr. Rea. \"Susceptibility to environmental inciters such as air, food and water components is becoming an increasingly recognized health problem,\" he observed in 1991. The adverse reactions patients sustain can include \"a spectrum of symptoms\" that affect the smooth muscles, mucus membranes, and collagen in various body systems to do with breathing, digesting, and excreting. Their sensitivities and allergic reactions are \"actually due to reactions to foods and chemicals found in the patient's home and work environments,\" says Dr. Rea. (See Chapter 10 for more details.)\n\nDr. Rea based this conclusion on solid laboratory work. He took fifty patients aged twenty-one to sixty-one who reported chemical sensitivities and who had various symptoms including asthma and arthritis, and partially detoxified\u2014that is, desensitized\u2014them in a less chemically polluted indoor environment. They went on a partial fast for three to four days, staying clear of all suspected allergens. Then Dr. Rea tested them, challenging their systems with small doses of inhaled toxic chemicals to see if they produced the same toxic symptoms again. They did. He used phenol, petroleum-derived ethyl alcohol, formaldehyde, chlorine, and a pesticide in doses they were likely to encounter in the outside environment. This study demonstrated that chemical sensitivity \"clearly does exist\" and may be blamed for a considerable number of symptoms.\n\nAs the discipline of clinical ecology, or environmental medicine, developed in North America, its essential principles became clarified. The first is the concept of total environmental load, or total toxic burden, as mentioned previously. This states that the sum of all toxic exposures\u2014all potential stressors, such as food intolerances, inhalant allergies, and psychosocial stress, all of which are cross-reactive\u2014is predictive of health. Second, chronic low-level exposure produces damage to the immune system and initiates flawed physiological adaptation. Third, because many toxic chemicals are lipophilic (fat-soluble and stored in body fat) and remain in the body for decades, low-level chemical exposures have a cumulative effect.\n\nAnother prominent physician practicing environmental medicine in the spirit of Randolph and Rea is Sherry Rogers, M.D., a Diplomate of the American Academy of Environmental Medicine, the author of ten patient-centered medical books linking toxicity to health conditions, and director of the Northeast Center for Environmental Medicine in Syracuse, New York. Since 1977, she has specialized in otherwise baffling medical conditions like environmental illness and has written extensively about the blocked detoxification pathways involved in conditions like El. Her own debilitating bout with chemical sensitivity\u2014for years she had arthritis-like chronic back pain due to unrelieved chemical exposure, including formaldehyde; she called herself \"a severe universal reactor\"\u2014led her to investigate the biochemistry of detoxification and the factors that blocked it.\n\nAs we learned above, there are complex biochemical processes in the human organism whereby, under conditions of good health, foreign toxic substances are filtered out, neutralized, and removed from the body. El patients, for example, do not operate under conditions of normal health; their detoxification pathways are blocked. El patients of course are the extreme example of faulty detoxification, but the extremity of this dysfunction helps us understand how problems in detoxification can affect us at lesser levels of blockage and how these blockages can become major problems.\n\nDue to toxic overload or nutritional deficit, or both, many people have \"a maladaptation of the detoxification system,\" explains Dr. Rogers. We are the first generation to be exposed to a tremendous number of toxic chemicals, and the process by which the body tries to detoxify our contact with these pollutants produces enormous stress and nutrient deficiencies. The maladaptation of the detoxification system that results helps generate numerous chronic and serious health problems, explains Dr. Rogers.\n\nTo illustrate the concept of maladaptation, let's look at the aldehyde detoxification pathway. Aldehydes (the basis of formaldehyde) are extremely toxic to humans; if the aldehyde pathway is bottlenecked, aldehydes accumulate dangerously in the system. The blocked aldehyde detoxification pathway can explain the whole \"spreading phenomenon\" of chemical sensitivity (the expanding web of interlinked, multiple symptoms, in which symptoms multiply and everything worsens) including \"toxic brain syndrome.\"\n\nWhen chemicals like aldehydes can't be degraded, they get metabolized into chloral hydrate in the brain. Chloral hydrate was once infamous as the prime ingredient in the \"Mickey Finn\" knockout drops used by gangsters wanting to dispatch uncooperative colleagues. When somebody slipped you a Mickey, you were out for hours. In the brain, chloral hydrate produces \"bizarre fluctuating cerebral symptoms\u2014feeling spacey, dopey, dizzy, an inability to concentrate, mood swings, irritability, exhaustion, numbness, depression,\" says Dr. Rogers. The average physician could easily misdiagnose multiple chemical sensitivity as mental illness at this point.\n\nWhat worsens the impact of the environmentally overloaded detoxification system is nutritional deficiency. Most people aren't detoxifying with \"a full deck\" of nutrients, says Rogers. Nutritionally poor diets and depleted agricultural soil fail to provide adequate supplies; meanwhile, the detoxification process itself depletes vital stores of minerals and vitamins (notably, glutathione). Even healthconscious adults may be nutrient-deficient. Magnesium, for example, is a component of over 300 enzymes and metabolic pathways, present in nearly every phase of the detoxification pathways; it is also the focus of the most under-recognized electrolyte disorder in the United States, claims Dr. Rogers.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Sherry Rogers, M.D., contact:\n\nNortheast Center for\n\nEnvironmental Medicine,\n\nP.O. Box 2716\n\n2800 West Genessee Street,\n\nSyracuse, NY 13220;\n\ntel: 315-488-2856.\n\nFor her books and a medically referenced monthly newsletter, _Total Wellness_ , contact:\n\nPrestige Publishing,\n\nP.O. Box 3068,\n\nSyracuse, NY 13220;\n\ntel: 800-846-6687 or 315-455-7862.\n\nFor William J. Rea., M.D.:\n\nEnvironmental Health Center\u2014Dallas,\n\n8345 Walnut Hill Lane, Suite 220, Dallas, TX 75231;\n\ntel: 214-368-4132;\n\nfax: 214-691-8432;\n\nwebsite: www.ehcd.com.\n\nFor Walter J. Crinnion, N.D.,\n\n13401 NE Bel-Red Road, #A4, Suite A-4, Bellevue, WA 98005;\n\ntel: 206-747-9200;\n\nfax: 206-747-7567.\n\nUnrelieved toxicity may proceed to other categories of symptoms besides allergies, environmental illness, and multiple chemical sensitivity. Immuno-toxicity can lead to autoimmune disease and cancer. First, the capacity of your immune system to withstand infections and the constant creation of individual cancer cells (which happens even in healthy individuals) diminishes. At the same time, your sensitivity (or reactivity) to allergens (substances producing allergies) intensifies, and you react to a greater number of substances than ever before. You may eventually develop autoimmunity, in which your immune system reacts against itself\u2014against your body\u2014somehow now perceiving it as foreign. Various autoimmune diseases such as rheumatoid arthritis and lupus erythematosus may result.\n\nThe documentation of immune damage produced by the pesticide DDT is extensive. Even though it was banned in the United States decades ago, it is present in the environment in residual amounts and in agricultural products deriving from countries in which the poison has not been banned. DDT toxicity was linked with numerous functional changes in the immune system and its key cells, and in damaging changes in various immune-related organs such as the spleen, thymus, and lymph glands. Serious health effects of a similar nature have been noted with human exposure to chlordanes (once used in termite pesticides, then banned, but still found in certain crops and seed treatment) and HCBs (hexachlorobenzene, a chlorinated pesticide, found in fungicides and dry-cleaning solvents).\n\nStudies conducted at Dr. Rea's Environmental Health Center showed that people with two or more OCCs (organochlorine compounds) in their blood have some degree of immuno-toxicity. As naturopath and detoxification expert Walter J. Crinnion, N.D., of Bellevue, Washington, comments, we are now struggling under a burden of multiple environmental toxins. \"Some individuals appear to be less able to clear the daily chemical exposure from the body than others, leading to a total load of toxins that exceeds the ability of the body to adapt.\"\n\nWhen this happens, the organ systems typically affected are the immune, neurological, and endocrine systems, Dr. Crinnion says. Further, the resulting immunotoxicity may be the prime cause of the increasing incidence of asthma, allergies, cancers, and chronic viral infections. A great number of symptoms can result from this toxic residue, including mood and cognitive disorders as well as problems with reproduction, libido, menstruation, metabolism, and the ability to cope with stress. Dr. Crinnion cites research in which blood analysis revealed at least four major immune cell imbalances in 298 patients who had been exposed to industrial chemicals.\n\nChronic fatigue syndrome (CFS), like environmental illness and multiple chemical sensitivity, is a baffling illness that starts to make sense when you figure environmental toxicity as a probable contributing cause, says Dr. Crinnion. In 1998, he tested this idea by taking a group of people with CFS through a detoxification program, \"designed to reduce their body burden of environmental toxins.\" Ninety-three reported \"good or great improvement\" as a result of the program.\n\nIt is becoming more apparent that a variety of industrial chemicals can produce many of the chief CFS symptoms, adds Dr. Crinnion. Exposure to any one of at least sixty-six different chemical compounds can produce fatigue, and trichloroethylene, a common airborne solvent found in urban air, can produce excessive fatigue. Chlorinated pesticides, lead toxicity, and tissue residues of DDE (the breakdown product of DDT), hexa-chlorobenzene, and organophosphates have been associated with CFS. In fact, organophosphates cause \"an identical set\" of symptoms to that of CFS, as well as similar disturbances in the nervous and endocrine systems, says Dr. Crinnion.\n\nThe correlation of unrelieved toxicity and the presence of specific carcinogens with the onset of cancer is well-documented. For example, women with breast cancer have been shown to have elevated levels of OCCs compared to women who are free of cancer. Among the specific chemicals found in the fatty tissue of the breast cancer patients were traces of DDT, DDE, PCBs, and lindane (also known as HCH or BHC, used to kill lice). Further, levels of these chemicals were found to be higher in the malignant tissues than elsewhere in the cancer patient's body. Researchers reporting in the _Journal of the National Cancer Institute_ estimated that higher than average levels of DDE and PCB in the blood can represent a _fourfold_ increase in the risk of breast cancer.\n\nCorrelations have also been reported between exposure to toxic chemicals found in household pesticide products and the onset of childhood cancers, such as brain cancer. Specifically, these products included pesticide \"bombs,\" treatments for termites, flea collars for pets, and various pesticide sprays for the garden. Other common household pesticide products or weed killers have been associated with soft tissue sarcomas (a form of cancer) and leukemia in children, while among adults, the use of 2, 4-D herbicides have been linked to higher rates of lung and stomach cancer, leukemia, and both Hodgkin's lymphoma and non-Hodgkin's lymphoma. The risk of developing soft tissue sarcomas due to exposure to 2,4-D can be as much as five to seven times higher, and even, according to one study, forty times higher.\n\nExposure to polyvinyl chloride (PVC) has also been linked with the development of a form of testicular cancer. In a study of 148 cases of cancer of the testicles and 314 healthy men, scientists at the Orebro Medical Center in Sweden correlated the patients' work history and lifetime toxic exposure. They found a sixfold increase in the risk for seminoma, a type of testicular cancer, among workers in the plastics industry who were exposed to PVC.\n\n### Neurological and Endocrinal Damage from Environmental Toxins\n\nThe neurological effects of toxicity are evident in a rising incidence of childhood behavior and cognitive function disorders as well as development delays. A report by the Greater Boston Physicians for Social Responsibility published in May 2000 put the matter of childhood and toxic effects in bold relief. The intent was to study the various toxic environmental threats to child development, specifically in the areas of learning, behavioral, and developmental disabilities.\n\nThe report, entitled _In Harm's Way: Toxic Threats to Child Development_ , documents the \"epidemic\" of developmental, learning, and behavioral disabilities now evident in the United States. For example, seventeen percent of children under age eighteen suffer from one of these above-mentioned problems; attention deficit hyperactivity disorder affects an estimated three percent to six percent of schoolchildren, but the percentage may be as high as seventeen percent; learning disabilities affect five percent to ten percent of children in school; between 1977 and 1994, the number of children in special education programs for learning disabilities increased by 191%; autism exists in 2 out of 1,000 children, with a doubling of its prevalence from 1966 to 1997.\n\nNext, the report demonstrates the proven relationship between exposures and body levels of certain toxic substances and cognitive defects. These toxic chemicals include lead, mercury, manganese, nicotine, dioxins, PCBs, pesticides, and solvents. The scientific research now available makes a strong case for a link between exposure to each of these substances (or categories of substances) and specific developmental or learning disabilities, including hyperactivity; impaired stamina, coordination, and memory; reduced IQ; impulsive behavior; decreased social adaptability; attention deficits; impaired speech and language skills; mental retardation; gait and visual disturbances; aggression and delinquent behavior, and others.\n\nConsider what lead can do to a child. A 1998 estimate proposed that nearly one million American children younger than age five are believed to suffer from low-level lead poisoning, derived mostly from leaded gasoline residues (as a fine toxic dust in the soil and air) and lead paint in old buildings (which also flakes, peels, and disintegrates into a fine powder). Low-level lead toxicity can cause permanent cognitive problems, such as learning disabilities, hyperactivity, reduced IQ, poor motor coordination, hearing loss, and diminished stature (failure to grow). Lead-toxic children may also have a hard time paying attention and controlling their impulses, and in some cases, may be prone to delinquency and violence.\n\n**_Lead is rightly blamed for the \"dumbing down\" of schoolchildren. A 1993 study concluded that lead exposure diminishes a child's mental abilities and that the relationship between lead levels and IQ deficits was \"remarkably consistent\"_**\n\nThese negative results have been observed from exposures to lead considerably below the so-called \"acceptable\" exposure level put forth by the U.S. government. The developing nervous system of a child is especially vulnerable to lead toxicity, even at the \"low\" and \"acceptable\" exposure (or ingestion) rate of ten micrograms (mcg\/dL) per day. There is so much lead in the environment today\u2014an estimated 10 million metric tonnes were introduced into the environment during the twentieth century\u2014that there is now \"an almost unbelievably large 'sink' of toxic powder available in soil and in house dust, waiting to cause brain damage in toddlers.\" Even more alarming is the estimate by the National Research Council in 1993 that the average American has a typical body burden of lead 300 to 500 times higher than that found in our prehistoric ancestors.\n\nLead is rightly blamed for the \"dumbing down\" of school-children. A 1993 analysis by the American Academy of Pediatrics of eighteen medical studies on lead toxicity concluded that lead exposure diminishes a child's mental abilities and that the relationship between lead levels and IQ deficits was \"remarkably consistent.\" They further reported that for every increase in blood lead levels of 10 mcg\/dL there was an average lowering of IQ in children by 4 to 7 points. The effect of losing five IQ points means that fifty percent more children fall into the IQ 80 category, which is barely in the realm of normal intelligence.\n\nLead may also be a key factor in the outbreak of violence among children and adolescents, reveals research from Dartmouth College. According to the \"neurotoxicity hypothesis of violent crime,\" lead and manganese toxicity can result in loss of control over impulsive, antisocial, and\/or violent, aggressive behavior. It is believed these elements can alter brain chemistry (their inevitable action as neurotoxins) in such a way that there is a breakdown in the \"inhibition mechanism\" that normally holds us back from becoming assaultive.\n\nA 1996 report from the University of Pittsburgh showed that young boys\u2014the study examined 301 males, aged 7 and 11\u2014with more lead in their bones than the expected average \"consistently\" had more incidents of aggressive and delinquent behavior. The leaded boys exhibited ever more aggressive, delinquent behavior as they moved through adolescence. Studies linking lead and aggressive behavior have been appearing since the 1970s such that the case against lead is very persuasive.\n\nRoger D. Masters, who headed the Dartmouth College research, also showed a correlation between high lead rates, criminal behavior, and the use of silicofluoride to fluoridate public water supplies. He noted that there was a doubling of the crime rate in areas where silicofluoride was in the water. This substance appears to increase the amount of toxins that enter the bloodstream from the intestines. In places where the public water contained silicofluoride, five times as many children had high lead levels compared to other communities. One clear effect is that toxic chemicals can destroy inhibitory systems and cause violence, Masters stated.\n\nThis is only a pr\u00e9cis of a small portion of the available research on lead toxicity and child development. Similar documented cases can be made against numerous other toxic industrial chemicals regarding their injurious effect on intelligence, development, and other aspects of growing up.\n\nThis makes it clear why substances such as lead and the others cited above are labeled \"developmental neurotoxicants,\" according to the authors of _In Harm's Way._ They use this term to indicate how the toxic chemicals poison the nerve cells to such an extent that they impair the normal growth processes in a child. \"These chemicals, which the child ingests through foods, environmental exposures, and in utero from the mother, may be directly toxic to cells or interfere with hormones (endocrine disrupters), neurotransmitters, or other growth factors.\" The authors further state that our scientific understanding of the effect on humans of toxic exposures is not advanced enough to \"accurately predict the impact of toxicants\" and thereby protect children from unhealthy exposures and their long-term effects.\n\nThe most chilling consequence of toxicity\u2014acting as it does deep within the body's core life systems and even spanning generations\u2014is the way it disrupts the endocrine system. Evidence now exists that chemicals can disturb the endocrine system on both a short- and long-term basis (see figure 2-1).\n\nHere is a review of some aspects of the short-term side of endocrine disruption due to environmental chemicals. According to Dr. Walter Crinnion, there are five principal ways in which toxins can damage the endocrine system. This system consists of the various endocrine glands: the testicles, ovaries, pancreas, adrenals, thyroid, parathyroid, and pituitary. All of these glands and their hormones are central to the regulation and normalization of the body's complex, interconnected systems, from metabolism and heat production to spermatogenesis and uterine preparations for pregnancy.\n\nThe endocrine glands accomplish these important functions by virtue of the numerous critical but highly sensitive hormones they secrete. Hormones\u2014at least 100 different ones have been identified\u2014are the chemical messengers of the endocrine system that impose order through an intricate communication system among the body's estimated 50 trillion cells. Central among the body's hormones are the \"male\" sex hormone (testosterone), the \"female\" sex hormones (estrogen and progesterone), melatonin (produced by the pineal gland), growth hormone (by the pituitary gland), thyroxin (by the thyroid gland), and DHEA (by the adrenal glands).\n\nToxic damage to the endocrine system, says Dr. Crinnion, shows up in sleep disturbances, shifts in energy level or mood; changes in weight, appetite, or intestinal function; a change in sexual interest and function, and for women, in menstrual changes; alterations in how we sense temperature and the degree to which we flush or perspire; and unaccountable changes in skin texture and hair growth. These symptoms, with the exception of the reproductive ones, typically manifest after problems in the immune and neurological systems are already evident, Dr. Crinnion says.\n\nOne endocrine organ that is especially sensitive to toxicity is the thyroid gland, located in the front of the neck. It is not surprising to find that a growing number of North Americans have imbalanced thyroids, being either underactive (hypothyroidism, the more common problem) or overactive (hyperthyroidism). At least seven industrial chemicals, including calcium-channel blockers, steroids, and chlorinated hydrocarbons, have been shown to reduce the secretion of a key thyroid hormone called T4.\n\n**Figure 2-1. The Endocrine System**\n\nIn a laboratory experiment, when rats were exposed to soil, dust, and air extracts taken from a toxic landfill known to harbor at least three dangerous pollutants including PCBs, their levels of thyroid hormone dropped; and when laboratory animals were fed fish harvested from the toxic, polluted Great Lakes, their thyroid's ability to regulate body functions became compromised. Thyroid function is also weakened when the system is exposed to lead, carbon disulfide, and polybrominated biphenyls, or PBBs (used as fire retardants, in plastic parts subject to heating, and in equipment such as televisions, radios, business machines, and hand tools). Research published in 1994 showed that exposure to even very low levels of PCBs and dioxin could damage thyroid function in the mother and her developing fetus and result in neurological damage to the child, producing learning disabilities, hyperactivity, and problems with attention.\n\nPolychlorinated biphenyls, or PCBs, a toxic compound related to PBBs, are structurally similar to a thyroid hormone (an example of the hormone mimicking effect of environmental toxins). PCBs are used for insulating and cooling electrical equipment, in hydraulic fluids, as plasticizers, and in inks, dyes, adhesives, and certain pesticides. Because they mimic a thyroid hormone, once inside the human body, PCBs can either help or hinder thyroid regulation and thus all the body processes that the thyroid controls, which include body temperature and metabolism. They can further damage the way a thyroid hormone (in this case, T3) is moved through the bloodstream and how it is produced within the cells. If an infant is exposed to PCBs, \"this can have a devastating effect on neurological and anatomical development,\" observes Dr. Crinnion.\n\nThe long-term endocrine disruption is especially accomplished by what are known as environmental estrogens. These are industrial, human-made chemicals (often petrochemicals, made with a petroleum base) that are widely distributed throughout the environment, having permeated the water and plant and animal foods. They enter the human system from the environment as pollutants; once inside the body, they mimic estrogen, naturally present in both women and men, and overload the system with this one hormone.\n\nIt is alarming to realize that the body, so competent and well-designed, can be overwhelmed by environmental chemicals and _mistake_ a human-made chemical for a natural, essential hormone. It also highlights how serious the matters of environmental pollution and human physiological toxicity have become. Not only can the body no longer keep up with the toxic burden, it can no longer distinguish, in some instances, an unwanted toxin from a desired hormone.\n\nLet's have a closer look at this subject. Environmental estrogens, also called xenoestrogens, are present primarily in \"greenhouse gases,\" herbicides, and pesticides such as DDT, and industrial byproducts from the manufacture of plastics and paper, as well as from the incineration of hazardous wastes.\n\nEnvironmental estrogens often cause an imbalance of estrogen relative to progesterone, another key \"female\" hormone. When a woman's body has too much estrogen (a condition called estrogen dominance), a variety of health problems can result, including breast cancer, fibroids, and endometriosis. According to some researchers, environmental estrogens also affect men, and may contribute to testicular cancer, urinary tract disorders, and low sperm count, or infertility.\n\nResearchers in the past decade have been studying links between environmental estrogens and prostate and testicular cancer, reduced sperm counts and sperm volume, undescended testes in newborn males, urinary tract abnormalities, a 400% increase in ectopic pregnancies (the fetus forms outside the uterus), and a serious rise in breast cancer. Hormone-disrupting chemicals are also linked with miscarriages and endometriosis. One type of breast cancer that is on the rise is occurring in women past the age of menopause in the form of estrogen-responsive tumors. This means the cancerous tissue in the breast is rich in estrogen receptors and will grow when exposed to additional estrogen.\n\nResearch now supports the theory that exposure to estrogenic organochlorines (found in pesticides) may increase the risk and incidence of breast cancer. Tissue samples from 240 women who developed breast cancer between 1976 and 1993 were studied along with samples from 477 cancer-free controls. Scientists found that one organochlorine in particular, dieldrin\u2014they also looked for traces of DDT and polychlorinated biphenyls\u2014was associated with a \"significantly increased dose-related risk\" of breast cancer, meaning, the more dieldrin the woman was exposed to, the greater the chances she had breast cancer.\n\nEndocrine-disrupting chemicals may have a special preference for women, new research suggests. It is often noted that women are more prone to autoimmune diseases, such as lupus erythematosus and rheumatoid arthritis. Why is that? One theory, backed by preliminary research, is that estrogens promote autoimmune diseases mediated by certain immune cells. Male sex hormones, called androgens, tend to inhibit the onset of these conditions. In other words, it seems that estrogens regulate autoimmunity, so if women have more than their fair share of toxic exposure to environmental estrogens, they will be more prone to develop estrogen-related health problems.\n\nToxic exposures can also upset other hormone-dependent processes. Here is an example, substantiated by scientific research, of what pesticide residues can do to sexual maturation in children. Scientists at the University of Granada School of Medicine in Spain took tissue fat samples from children living in farm areas in southeastern Spain. They found traces of 14 pesticides, and of the 113 samples studied, 43 had traces of at least one pesticide, such as lindane, heptachlor, aldrin, or diel-drin. Some of the chemicals have estrogenic activity. The scientists also found a correlation between exposure to estrogenic chemicals in pesticides and sexual maturation problems in boys, specifically, a failure for testicles to descend in some cases.\n\nGiven the fact that estrogenic chemicals are found not only in pesticides, but are used in the food industry, in plasticizers, and in dental sealants, \"concerns are warranted.\" Even more so because \"the number of new substances that mimic the action of endogenous [within the body] estrogens is increasing rapidly.\"\n\n\"Unequivocal evidence\" now exists, state researchers at Brunel University in Uxbridge, England, that the waters of the world contain a wide variety of endocrine-disrupting chemicals. So extensive is this pollution and so unknown the results that \"this makes predicting possible effects difficult.\" Scientists do know that strange sexual changes and other \"adverse effects\" are happening to marine life living in the chemically polluted waters: female mollusks are becoming masculinized, and male fish are getting feminized. In other words, basic gender differentiation is starting to blur in marine life under the influence of environmental estrogens.\n\nAnother alarming fact about hormone-disrupting chemicals is how little it takes to produce biological damage. Hormones are powerful, which means the body deals with them in very small, even infinitesimal, doses. Estrogen is biologically active even at only a few parts per trillion. While most environmental estrogens are less powerful than the hormone estrogen, they tend to be present in body tissues in concentrations of several parts per billion, or million\u2014in other words, they are more pervasive in the body than natural estrogen.\n\nFurther, synthetic estrogens may be much more active in the body than natural estrogen because they do not bind to proteins as carrier molecules and are therefore more mobile. The result is that because the human body is used to dealing with only tiny amounts of hormones, high doses of synthetic estrogen \"can swamp the receptors and actually shut down any response.\"\n\nThere is yet another factor showing how environmental estrogens can easily damage the body and disrupt its endocrine system, especially during fetal and infant development. There are \"windows of extreme susceptibility\" during the development of the fetus or baby. Get exposed to a synthetic estrogen during these small windows and your system is in serious trouble. For example, only those girls who as fetuses were exposed to the toxin DES during the first sixteen weeks of gestation developed clear cell cancer. Girls exposed to DES after the sixteenth week did not develop this disease.\n\nOne toxin can do a lot of damage to the endocrine system, but we are rarely exposed to only one environmental estrogen. Since 1991, fifty different synthetic industrial chemicals capable of disrupting the human endocrine system have been identified; they can also interfere with hormonal activities in birds, fish, mammals, and other species. These hormone-disrupting chemicals are found in common, frequently encountered products, such as plastics, detergents, and pesticides. Dr. Crinnion notes that when you factor the _combined_ estrogenic effect of the organochlorines we are likely to be exposed to, it _increases_ our exposure rate by a factor of 1,600 times. This has been demonstrated in the case of exposure to two or three common pesticides at low levels.\n\n**_One toxin can do a lot of damage to the endocrine system, But we are rarely exposed to only one environmental estrogen. Since 1991, fifty different synthetic industriai chemicals capable of disrupting the human endocrine system have been identified._**\n\nFurther, estrogenicity (the state in which nonestrogen chemicals act _as_ estrogen, thereby multiplying the body's reserves of this one hormone) can activate other compounds, not normally acting like estrogens, to become estrogenic, thereby adding to the total. It produces a synergistic effect, yielding toxic results greater than the sum of individual chemical actions. Chlordane, which on its own does not disrupt human hormones, greatly enhances the estrogenic effect of other chemicals already present, thus magnifying the total hormone disruption.\n\n### Damaging Synergistic Effects Extending into the Next Generation\n\nTo say that fifty chemicals have been identified as estrogenic, or hormonally disruptive, must be measured against the staggering number of chemicals now in use: 70,000, plus the 1,000 or so new ones introduced every year. Most of these haven't been tested to see if they interfere with hormonal activity, so the consequences of our routine or occasional exposure is unknown. However, based on the evidence accumulating for the first fifty, the chance that many of these chemicals are hormone-disrupters is high.\n\nBut that's only part of the picture. To accurately assess their potential effect, you would have to test what happens when any number of these chemicals are combined in a human physiology; the number of possible permutations is sufficient to stagger the statistical imagination. If you wanted to test the 1,000 most common toxic chemicals in unique combinations of three, this would take 166 million different experiments; if each experiment took an hour and 100 laboratories worked 24\/7, it would take 180 years to finish the work. Meanwhile, millions more men, women, and children are being slowly poisoned by exposure to toxic environmental chemicals, the health impacts of which have never been adequately tested.\n\nThis is especially unfortunate because children are particularly sensitive to pesticide exposure and are routinely exposed to them in their school environment, both indoors and outdoors. Given their smaller size, a single exposure to pesticides results in a larger dose of toxins than for adults. Children's internal organs, still developing, are more vulnerable to toxins, and their systems generally are less competent at detoxifying than an adult system is. \"Low levels of pesticide exposure can adversely affect a child's neurological, respiratory, immune, and endocrine system,\" noted Kagan Owens and Jay Feldman in a comprehensive review article on the problem, \"The Schooling of State Pesticide Laws,\" in 1998.\n\nThey report that chloropyrifos (brand name: Dursban) is one of the insecticides most widely used in schools. It \"poisons\" children by diminishing the body's output of cholinesterase, an enzyme needed to transmit nerve impulses. This enzyme deficit in turn produces various symptoms including nausea, dizziness, headaches, aching joints, disorientation, and concentration problems, state Owens and Feldman. Another widely used class of insecticide, synthetic pyrethroids, overstimulates nerves producing hypersensitivity and asthma; still others are known to negatively affect the immune system, resulting in an increased incidence of allergies, asthma, hypersensitivity to chemicals, and a generally reduced immune defense against infections and cancer. While about thirty states have taken some action to limit pesticide exposure of children while at school, the combined regulatory (and preventative) action is still \"limited.\"\n\nOf considerable concern in this matter is the way environmental estrogens present in a mother can be transferred to and produce health damage in her offspring. According to the World Resources Institute of Washington, D.C., an advocacy group that monitors environmental toxins and their health impact, these chemicals interfere with the development of babies before (prenatally) or soon after birth. \"Estrogen mimics are extremely potent in part because unlike most natural estrogen, they cross the placental barrier, exposing the fetus to greater than normal levels of hormone.\" These exposures in turn can disturb the fetus' hormonal balances, thereby throwing potentially everything biological in its young life into jeopardy, \"from fertility to gender itself.\"\n\nResearch shows that in most cases the effects of environmental estrogen exposure are more pronounced in the offspring than in the exposed parents, although the extent of the effects on the latter should not be minimized by any means. In other words, if you, as a female, are toxic with environmental estrogens and do not detoxify before pregnancy, you may be inadvertently passing these poisons on to your baby in the womb. You may also transmit them through your milk if you breastfeed your infant. It has further been noted that the effects of exposure in many cases are delayed in the next generation, such that they do not fully manifest until the offspring is an adult.\n\nIt is not only the fact that hormone-disrupting chemicals have a long-term, multi-generational impact. They interfere with deep-set, essential life processes, with the way chemical messages are transmitted throughout the body. In many respects, this is more serious than the fact that they are carcinogens as well, capable of initiating a cancer process. According to Theo Colborn and her co-authors in _Our Stolen Future_ , an excellent review of the health consequences of environmental estrogens, hormonally active synthetic chemicals are \"thugs on the biological information highway that sabotage vital communication.\"\n\nThey jam and scramble the messages; they spread biochemical disinformation; they seriously damage the endocrine system at its critical level of development, says Colborn, including sexual differentiation and brain organization. It is a \"toxic assault\" that reaches into the unborn generation and poisons its members even before they're born. Colborn also says that hormone-disrupting chemicals, when they affect the fetus, \"can diminish individuals without making them sick,\" producing \"impaired function\" but not necessarily overt disease.\n\nDevelopment can be derailed; cognitive unfolding can be stunted; and the toxic etiology can be hard to trace using conventional medical models. For example, says Colborn, a boy who gets exposed to toxic estrogenic chemicals in utero (from his mother) could have health problems throughout his life. As a child his testicles do not descend; at puberty, he has a low sperm count; at middle age, he gets cancer of the testicles. The fact of \"hand-me-down\" poisons means potentially three generations receive the negative health impact of environmental estrogens, says Colborn. In other words, the grandchildren of a toxic grandmother can also be poisoned by her toxic exposures, transmitted across two generations.\n\n### Previewing the Scope and Advantages of Deliberate Detoxification\n\nA great deal more evidence could be marshaled to make the case for an alarming degree of widespread toxicity among humans in our time. But the facts and extrapolations reviewed here should be enough to motivate us to think about detoxification, that is, voluntarily assisting our internal detoxification system to catch up on processing its toxic burden and ridding our bodies of a lifetime accumulation of toxins. Getting yourself involved in a carefully planned and responsibly staged detoxification program could be one of the smartest things you ever do for your health and well-being.\n\nA chemical detoxification program capable of ridding the body of its toxic burden must address five key physiological functions of the body, states Stephen B. Edelson, M.D., director of the Edelson Center for Environmental and Preventive Medicine in Atlanta, Georgia. Dr. Edelson has for many years been a leading proponent of and expert in the means of chemical detoxification as a way of treating numerous serious health conditions, including cancer. He has formulated a multidisciplinary medical approach he calls clinical molecular medicine as an effective way of treating and preventing disease.\n\nDr. Edelson says you must correct the functioning of the liver (help its two phases to work optimally); the blood circulation (it must be \"free flowing\" so it can carry and adequate oxygen supply to the cells); the adrenal and thyroid glands (their hormones influence enzyme activities and metabolism); the kidneys (so they can flush out toxins through the urine); and the intestines (accelerated \"transit time\" so that old and new fecal matter as well as neutralized toxins and bile are excreted frequently). So how do you do this?\n\nAccording to Dr. Edelson, a detoxification program should include a reduction in your body's burden of environmental chemicals and toxins, a lowering of its overall stress, and an enhancement of its antioxidant mechanisms. Some of the ways these goals can be accomplished include eating a wholesome nontoxic diet, normalizing your nutritional status, exercising, \"upregulating\" liver detoxification (giving the liver all it needs to run its two-phased detoxification), and using sauna and\/or heat treatment to sweat toxins through the skin.\n\nFor cases of serious heavy metal toxicity, a process called biodetoxification is required, says Dr. Edelson; here a chelating (binding- up) substance is infused intravenously into the patient's blood to gather up heavy metals such as mercury from within the body and enable the system to excrete it through the urine. In another process called heat depuration, you get into a sauna that can produce an indoor temperature of 150\u00b0F, a level of heat judged necessary to mobilize chemicals stored deep in body tissues. For biodetoxification and heat depuration you need to resort to a clinic (Edelson offers a Biodetoxification Center as part of his medical offices), but the recommendations cited above can be undertaken at home or on your own. These will be taken up in practical detail in various chapters in this book.\n\nDetoxification is a medical term that refers to a wide range of practices, from the simple to the complex. \"Anything that supports our elimination can be said to help us detoxify,\" comments detoxification authority Elson M. Haas, M.D., director of the Preventive Medical Center of Marin in San Rafael, California. \"Doing nothing more than drinking an extra quart of water a day will usually help us eliminate more toxins.\"\n\nAs a therapy, Dr. Haas regards detoxification (including fasting and internal cleansing) as \"one of the most powerful\" healing approaches he has encountered in his more than twenty-five years of medical practice. In fact, detoxification is \"the missing link in Western medicine and a key to the health and vitality of our civilization,\" he states. The detoxification program reduces your body's intake of toxins and enhances its removal of those already present. A detoxification program, says Dr. Haas, should help the body's five cleansing systems to work better: these are the respiratory (lungs, throat, sinuses, nose), gastrointestinal (liver, gallbladder, colon), urinary (kidneys, bladder), skin (sweat and tear glands), and lymphatic (the lymph glands, channels, and organs, such as the spleen).\n\nDr. Haas also defines toxins broadly. They include any substance that \"creates irritating and\/or harmful effects\" in the body such that these effects undermine our health or put stress on our biochemical processes. As such, toxins can include the obvious physical poisons from our environment, but they can also include more subtle influences. \"Negative 'ethers,' psychic and spiritual influences, thought patterns, and negative emotions, all can be toxins as well, both as stressors and by changing the normal physiology of the body and possibly producing specific symptoms.\" (See Chapter 9 for more information about these subtler toxins.)\n\nAs to _why_ we should detoxify, Dr. Haas groups the answers into two categories. First, in terms of disease prevention and treatment, internal cleansing can reduce symptoms, treat existing disease, help the internal organs rest, purify, and rejuvenate all the body's systems, facilitate weight loss, improve skin quality, slow the aging process, improve flexibility, and enhance fertility. Second, in terms of gaining new benefits, detoxification can help us become more organized, creative, motivated, relaxed, energized, clear in the head, inwardly focused, environmentally attuned, and can generally fine-tune the senses, says Dr. Haas.\n\nThese last two benefits of detoxification tend to be underacknowledged by most medical practitioners but should be given much more attention. Once the body is relatively purified and freed to conduct its physiological processes of digestion, assimilation, detoxification, and elimination _unimpeded_ by a toxic overload, it can actually support our awareness. It can support our state of mind, our processes of consciousness, even the degree to which we are conscious, or shall we say, freely aware of our inner and outer environment.\n\n### The Spiritual and Global Implications of Detoxification\n\nLet's put it simply: detoxification can enable us to have more _sustained awareness_ at our disposal. Dr. Haas explains that during detoxification, clarity of mind, emotional cleansing, and profound spiritual awareness may result. \"We are more sensitive to subtle vibrations of people, situations, emotions, and nature.\"\n\n**_The processes of detoxification, including the three levels of physiology, emotions, and energy field can teach us a lot about ourselves. All we have to do is observe. Detoxification is an excellent way to gain self-knowledge._**\n\nThe processes of detoxification, including the three levels of physiology, emotions, and energy field (the realm of Dr. Haas' \"psychic and spiritual influences\") can teach us a lot about ourselves. All we have to do is observe. Detoxification is an excellent way to gain self-knowledge.\n\nWe can see, for example, the effect of a lifetime of dietary neglect, inattention, or abuse as we purge our intestines and liver. We may find ourselves saying: Who would have ever thought my colon could hold that much old matter and in such a toxic state? We may observe with a small measure of awe as we unravel buried emotional patterns, untie very old emotional knots, and see our physical and psychological symptoms start to ease up and go away. Again we may marvel to ourselves: I never would have thought my childhood or adolescent feelings (or whatever: you fill in the appropriate time period and content) could have had that kind of powerful effect on my body.\n\nFinally, as we discern the effects of subtle psychic and spiritual influences on our bodies and energy fields (or auras), derived from other people, toxic environments, and energy imbalances in our homes or work spaces, we may step back and reflect. How could I have gone these many years oblivious to this realm of influence, when now I can see what a mess it can make of my mind, emotions, and body? \"I cleanse because it makes me feel more vital, creative, and open to emotional and spiritual energies,\" comments Dr. Haas.\n\nIn an unexpected but tangible sense, detoxification is a spiritual activity. Certainly it can produce numerous physical benefits, such as increased energy, reduced allergies and congestion, and a cessation of many negative health symptoms, and it can lead to heightened mental focus, sharpness, and performance as well as emotional clarity and stability. But there is more.\n\nDetoxification enables you to get into a new, wakeful, conscious relationship with your body, your prime living space in the material world. Most of the time we take our physical bodies for granted; in fact, we take our existence for granted\u2014the fact of our existence, the fact that we are awake and aware inside biological human forms. Illness can rivet our attention, for a while, on this sheer physical fact, but most of the time we try to avoid this awareness.\n\nDuring detoxification, you can _watch_ the changes in how you feel, think, and feel as you live in your body, as you purify your bodily living space. You realize\u2014you see the irrefutable proof\u2014that what you eat affects how your body runs, how it is, how you are. You come to understand that healthy, organic, wholesome foods and substances produce one kind of biological result, and fast foods, junk foods, and processed, toxic foods and substances produce another, and that this other kind of result is unpleasant, undesirable, and sickening.\n\nYou come to see that where you live, what you surround yourself with, in terms of furnishings and architectural shapes and energies, also have marked effects on all aspects of your body, mind, and spirit. You begin to see your body and home as one environment, as one extended living space that is either healthy or toxic. You begin to consciously inhabit your body and home and to appreciate how both are nested in the larger world. Even better, you see that it's your choice whether you reside in a healthy or a toxic living space.\n\nHere is the practical payoff from this growing environmental awareness. If you choose to have a healthy living space, start a detoxification program, and develop a lifelong detoxification awareness, all of your consumer choices start to shift. Coming to terms with the elements in your diet and home that either support or destroy your health, you form a new relationship to your body, your home, and to the world\u2014to the planet itself, as the ultimate source of nutrients and furnishings. You see your body and home as a single living space, as a place for you as a conscious being to reside in, but which is dependent on the \"outside\" world for supplies.\n\nAs you eliminate the toxic substances and reach for organic, wholesome, \"green\" products, your choices have an effect in the marketplace. They have _global_ implications because through your consumer choices in service of your detoxification (itself in service of your desire to have a healthy living space, both body and home), you are energizing the nontoxic world economy and removing your support from the toxic world economy.\n\nConsider one example. Say you choose to buy only organically grown soybeans and soybean products as part of your detoxification dietary change. What are the global implications of this single consumer choice?\n\nFirst, you are saying no to the input of numerous pesticides routinely sprayed on soybeans, which are absorbed to an extent by the beans themselves and which eventually enter the human food chain.\n\nSecond, you are saying no to the use of genetically modified (GM) soybeans with their resident pesticide and to the interests of world agribusiness concerns that want to convert the world to GM foods using seed stocks they own.\n\nThird, you are saying no to the standard use of synthetic agricultural fertilizers, which deplete the soil and run off into public water systems, residues of which eventually enter the body through municipal water.\n\nFourth, you are saying no to the soil-depleting practices of monocropping or the single crop rotation farming cycle of soybeans and corn, a process that in part has reduced the total number of farms and farmers and concentrated food production in the hands of agribusiness.\n\nFifth, you are saying no to the use of soybeans as a food additive in the form of concentrates, isolates, and hydrolyzed vegetable protein, products widely found in processed, fast, and junk foods that contribute to toxicity and intestinal stagnation.\n\nSixth, you are saying no to the mass distribution of processed foods through conventional supermarkets that lack ecological awareness and support the interests of world agribusiness and its steady despoliation of the environment.\n\nSeventh, you are saying no to the use of hormones and antibiotics in the meat industry (which get into human bodies through ingestion of the animal) and to the squandering of plant resources in raising animal protein versus consuming plant proteins directly.\n\nThese are some of the global implications of your decision to buy only organically raised soybeans. What are the implications of saying yes to only organically grown beans?\n\nFirst, you are affirming the ecologically sound practices of organic agriculture, with its emphasis on soil husbandry and abstention from pesticides and synthetic fertilizers.\n\nSecond, you are saying yes to all the green initiatives under way around the planet, in support of the environment and its responsible stewards.\n\nThird, you are saying yes to the decentralization of food production, to the resurgence of regional and community agriculture, and to the preservation of natural seed stocks and the natural delineation between species. (Genetically modified foods mix genes from different species.)\n\nFourth, you are saying yes to foods that are relatively easy to digest and which, due to their purity, can contribute to a calmer, clearer state of mind.\n\nFifth, you are contributing to a rise in the general level of awareness about the relationship between health, food, consciousness, and the environment\u2014expressed in part by the Green movement. Spread across the Western industrialized nations, this awareness could eventually produce a radical shift in culture and industry.\n\nSixth, you are saying yes to a nutritionally based, dietary approach to health maintenance, symptom reduction, and, in some cases, to disease treatment. This in turn has ramifications for drug-based conventional medicine and its health-care infrastructure.\n\nSeventh, you are assisting in reducing global warming insofar as it is due in part to deforestation, especially in the rain forests, clearcut for grazing of cattle required by the world animal protein market.\n\nAll of this may seem far removed from the benefits you reasonably expect from a few days of colon cleansing, liver flushing, and a couple of hours sweating in a sauna. Yet it is true and will probably become evident to you upon reflection, especially _after_ you have detoxified. In any event, now that we have identified the numerous toxins in our environment, seen what they can do to our health, and appreciated the need to deliberately support the body's natural but probably overburdened detoxification system, the next step is to assess your individual level of toxicity. The best and most reliable way to do this is through a laboratory test of your blood, urine, feces, saliva, or breath, a topic we explore in the next chapter.\n\n# _Part Two_\n\n# _Creating the Healthy_ Body _Living Space_\n\n# CHAPTER 3\n\n# _How Toxic Are You?: Testing for Toxicity and Detoxification Ability_\n\nAt this point, we are sufficiently alerted to the possibility of toxicity in our bodies and homes and what it can do to our health. But how can we tell if we are toxic, if our bodies actually harbor a fair share of the toxic residues known to exist\u2014unnaturally\u2014in the human body in the twenty-first century. Being sick is one kind of feedback, though not an ideal one. Having a chronic problem, such as frequent colds, susceptibility to the flu, or chronic fatigue, multiple chemical sensitivity, or environmental illness, is another indication, but still imprecise.\n\nThe most reliable way to find out if your body is unduly toxic, meaning so toxic your body can no longer naturally cope with the toxic load, is to have one or more laboratory tests on your blood, urine, stool, or hair, looking for _traces_ of toxic materials, such as heavy metals, and looking for _absences_ of key nutrients necessary for the body's natural detoxification processes. You will observe here a definite interactive system: the more toxins your body carries, the less likely it is to have the detoxification nutrients required to purify itself, _and_ the more deficient your system is in detoxification nutrients, the easier it is for toxins to accumulate and place an almost unbearable toxic burden on the body.\n\nGradually, as real and true medical knowledge of the human system develops, more health-care practitioners are including laboratory testing of patients for toxic burden and nutrient status as a routine part of their initial diagnosis and case workup. I say \"real and true\" because in recent decades conventional medicine's dependence on powerful drugs and procedures that mask, suppress, or eradicate symptoms has tended to eclipse an accurate understanding of the real complexity of human physiology.\n\n**_But before you detoxify, in many cases it is helpful to have a clear idea of what toxins you're dealing with and what the body's detoxification capabilities are._**\n\nThe concept of broadscale toxicity as a primary causative factor for many states of illness is not a new idea, only one Western medicine has forgotten. Toxicity has always been the conceptual foundation of all natural medical practices, especially naturopathy, which developed and flourished in the nineteenth century and is experiencing a resurgence today\u2014for the same reason it was successful when it first appeared on the medical scene. It understands the role of toxicity and it knows how to remove it. But before you detoxify, in many cases it is helpful to have a clear idea of what toxins you're dealing with and what the body's detoxification capabilities are. This is where laboratory tests come into the picture, the focus of the present chapter.\n\n### The Healthy Living Space Expert Interview: Peter Holyk, M.D., Contemporary Health Innovations\n\nA good example of the new trend to focus on general toxicity as a primary factor in illness is the medical practice of Peter Holyk, M.D., a board certified ophthamologist and founder of Contemporary Health Innovations, P.A., in Sebastian, Florida. Founded in 1997, Dr. Holyk's clinic offers a variety of toxicity-oriented services including testing for heavy metals, toxins, and micronutrient status; nutritional supplementation programs; allergy elimination techniques; diet and lifestyle counseling; hormonal rebalancing; and EDTA chelation.\n\nDr. Holyk, originally trained in conventional medicine, was \"converted\" to the more broadly focused toxicity-conscious viewpoint by direct experience. His wife got very sick; in fact, so did he. In dealing with both illnesses and finding ways to reverse them, Dr. Holyk came to appreciate the importance of toxicity in health and illness.\n\nHis illness was about as bad as it gets: brain cancer. When he was thirty-five, his physician discovered a tumor in his brain the size of a baseball. Forecasts of lifelong quadriplegia were handed him; doubts were cast on whether he would even survive the brain surgery. In other words, he got the usual \"doom and gloom\" world view of conventional medicine in the face of an extreme example of unchecked toxicity. Surgery was successful in reducing the size of the tumor to that of a thimble. Since chemotherapy and radiation were offered him with only limited hope of success after surgery, he refused them. Dr. Holyk got his life back, but as the months passed, he began to suspect \"something was missing\" from the medical world view in which he been trained and treated.\n\nHow come conventional medicine had had so few options to offer him for the cancer, he wondered. And what had produced the cancer in the first place? \" _Now_ I've figured out why that problem occurred,\" says Dr. Holyk, fifteen years later. \"I didn't get a brain tumor because the sun was on the wrong side of the moon. I got a tumor because I was exposed to enough toxins and did not have the right micronutrients to help support my immune system to keep the cancer at bay.\"\n\nEvery day, every person has at least one cancer cell arise in their body in the natural course of cell division, but in most people, the immune system is alert and strong enough to neutralize these rogue cells before they can do any damage. \"This is why 100% of the population does not get terminal cancer\u2014because their body is strong enough to kill the cancer cells.\"\n\nNow Dr. Holyk knows that what makes this immune vigilance possible is a high essential nutrient status, yet this nutrition-supported immune vitality is decreasingly present in Westerners, says Dr. Holyk. One of the key immune agents that deals with cancer as well as other \"foreign\" proteins and materials in the body is a group of cells called natural killer cells. Unfortunately, research suggests that across the general population, natural killer cell function is declining at \"an alarming rate\" because \"we don't have the nutrition to support it.\" Dr. Holyk adds, \"You may be exposed to the same level of toxins as the next person who doesn't get sick, but you do. That's because if you're deficient in your immune defenses, you are going to get more sick.\"\n\nAs we saw in Chapter 2, toxicity is not only prevalent, it is ubiquitous, around us and inside us. The average person in a Western industrialized nation is now routinely exposed to potentially 70,000 different chemicals every day, and the average person carries trace amounts in their body of an estimated 500 different toxins at one time, explains Dr. Holyk. \"If your immune defenses aren't good, you're in deep trouble.\" Further, the average person is typically deficient in vital micronutrients, certain trace elements such as vanadium or lithium, among others, whose presence, however minute, is essential for other biochemical processes necessary to health.\n\nBut Dr. Holyk didn't appreciate this all at once. The picture of toxicity, immune response, and nutrient status emerged over a period of years as he delved into the subject. In fact, for many years, despite his experience with the brain tumor, he remained in a state of \"absolute skepticism\" about the role of toxicity and nutrition, as he admits today. \"I was so busy being a doctor that I didn't pay attention to the facts in front of me.\" He would read reports and scientific studies attesting to the interplay of toxins and nutrients, but discard it as \"nonsense.\" \"I dismissed a lot of it, but then, all of a sudden, I couldn't dismiss it anymore. I had to start paying attention to it.\"\n\nHis wife's long illness\u2014gallbladder removal followed by chronic fatigue\u2014shook him out of his skepticism. First she had her gallbladder removed when she was only in her late twenties, an unusually young age to develop a dysfunctional gallbladder. Then she developed chronic fatigue syndrome, such that she was unable to get out of bed at all on some days. Eventually the possibility of severe toxicity became apparent to Dr. Holyk, and his wife underwent laboratory testing for heavy metals. They were specifically looking for mercury and lead, two of the most prevalent toxic heavy metals to show up in the human body.\n\nTheir hunch was sound because it turned out she had severe lead and mercury poisoning and that these toxic metals were contributing mightily to her chronic illness. Where did the lead and mercury come from? Toxic residues of lead are almost ubiquitous in the modern environment, from old water pipes and lead-based paint to leaded gasoline residues in the dirt by the roadsides. The chief source of mercury, besides contaminated fish and pesticides, is dental amalgam fillings in which mercury typically comprises about fifty percent of the different components of the filling.\n\nA growing body of research shows that not only can mercury leach from a dental filling, but it can migrate throughout the body, producing (or contributing to) a long list of mild to serious health problems, including chronic fatigue. Mercury is released from the fillings as a vapor that is absorbed through the nasal cavity and then lungs, then transported through the bloodstream to be deposited in fatty tissues, including in the brain and nervous system, says Dr. Holyk.\n\nIn both cases, lead and mercury toxicity can be a cumulative development; you usually don't have a single, intense exposure to a heavy metal with resulting obvious signs of poisoning. Rather, you get poisoned slowly, in tiny doses that steadily accrete within your system, continuously polluting your internal environment and contributing to its dysfunction. In his wife's case, carefully removing the mercury and lead from her system (in that order) helped restore her to health and productivity. It also counselled Dr. Holyk in the vital importance of screening patients for toxicity, such as in the following case history he offers from his medical files.\n\nA woman in her mid-forties, named Penelope, developed sudden blindness in one eye. Her conventional physician forecast that she might develop multiple sclerosis (MS), as this kind of seemingly inexplicable blindness is a precursor of MS in twenty percent of cases. Penelope easily could have slipped into the MS diagnosis had she not had the good fortune to contact Dr. Holyk. In taking Penelope's health history, he learned that three weeks before the sudden onset of blindness, she had sustained a major break in one of her mercury-based dental fillings. Instead of having the tooth capped, she had the amalgam filling replaced.\n\nThis dental procedure actually doubled her mercury exposure: first, the broken mercury filling meant that mercury vapors and tiny particles of the metal were free to enter her bloodstream; second, removing the broken tooth (without specialized dental precautions required for _safe_ mercury removal) further mobilized free mercury so it could enter her system; then installing another mercury filling introduced more mercury that could potentially leach into her blood, along with that amount steadily liberated from the other numerous mercury-based fillings in Penelope's mouth.\n\nRegrettably, it is still the case that the majority of American physicians and dentists do not make this well-proven connection between mercury fillings and mercury toxicity\u2014to their patients' peril. Ironically, \"studies performed on dentists who have occupational exposure to mercury have shown diminished fine motor coordination and slowed response time to various stimuli,\" states Dr. Holyk.\n\nDr. Holyk, with good evidential grounds for suspecting mercury toxicity as the prime cause of Penelope's blindness, had her go through two simple medical tests: a hair analysis, which has a good reputation for revealing heavy metal toxicity (described below), and a DMPS challenge. This latter procedure involves the introduction by injection of a nontoxic substance (DMPS) that binds up mercury in the system, which is then excreted in the urine; the urine is then evaluated for mercury levels. Both tests confirmed that Penelope had a high level of mercury toxicity.\n\nBased on this accurate medical information, Dr. Holyk next laid out a treatment program. He prescribed an intensive two-week supplemental program of nutrients and micronutrients (minerals needed in minute or trace levels) and he arranged to have Penelope's mercury-based dental fillings removed by a _qualified_ dentist. Qualified here means that the dentist understands the extreme need to protect the patient from mercury mobilization and migration once the fillings are in the process of being removed. (See Chapter 7 for more on mercury toxicity and safe amalgam removal.)\n\nIf done inexpertly, a person can end up far sicker after amalgam removal than before, because the procedure itself liberates a comparatively vast amount of mercury into the bloodstream, compared to the much smaller, though toxic, amounts slowly released with the fillings in place. \"It's very dangerous to take out amalgams if you don't know how to do it correctly,\" comments Dr. Holyk. \"I've had people walk into my office after an amalgam removal with partial paralysis, partial blindness\u2014you name it. So beware.\"\n\nPenelope had her mercury amalgams removed under carefully controlled conditions and Dr. Holyk continued to boost her nutritional status for several weeks. She also received DMPS (by injection) to clear free mercury from her bloodstream, enabling it to pass out of her body in her urine. Her visual acuity now became a kind of daily feedback report on the ongoing state of detoxification. In the troubled eye, her eyesight had diminished from an original 20\/20 to 20\/60, and from there to perceiving only light, but no detail or color in that eye. Yet twenty-four hours after receiving a DMPS injection, that eye registered 20\/25, the abatement of symptoms showing the dramatic effect of mercury removal.\n\nOver the next weeks, Penelope's visual acuity fluctuated a bit, dropping off to 20\/40, then coming back to 20\/25 with another DMPS injection, says Dr. Holyk. While Penelope continued with the nutrients and DMPS detoxification, her eyesight stayed at around 20\/25 for a year; after a while, she let some of the elements slide, and her eyesight dropped back to around 20\/70. These fluctuations reflected ongoing detoxification. The amount of time it takes for complete mercury removal and health improvement can vary considerably among patients. \"Sometimes we see improvement in patients right away; other patients have so much mercury in their system that we're still pulling out large amounts after a year of treatment,\" says Dr. Holyk.\n\n\"Mercury toxicity is one of the very biggest problems that I encounter, followed by lead,\" says Dr. Holyk. But that's only the start of the list. Generally, Dr. Holyk tests for fifteen toxins, following a startling report by the Environmental Protection Agency in the 1980s stating that five heavy metals (mercury, cadmium, lead, antimony, and beryllium) showed up in 100% of the human fat samples tested in the study. \"I can tell you if you routinely tested _all_ patients for heavy metals, you would find almost everyone has a problem,\" confides Dr. Holyk.\n\nWhen he says \"all\" patients, Dr. Holyk doesn't mean only those who crawl into a medical practice chronically ill, fatigued, or hypersensitive to numerous chemicals, but also people who are not obviously sick, who may not even suspect toxicity, yet who perhaps notice a vague sense of sluggishness or compromised function or ability. The thing about toxicity is that there is only a quantitative difference between \"a vague sense\" and fullblown toxicity.\n\nThis is why for Dr. Holyk two matters come immediately to mind in assessing a new patient's symptoms: micronutrient status and toxicity. Is this person deficient in key vitamins or minerals necessary for normal detoxification processes, and is this patient carrying a toxic burden of cumulative heavy metal, solvent, or pesticide poisoning?\n\nThe relationship between mercury toxicity and nutrient status is even more interwoven, says Dr. Holyk. There are two ways of making this point. First, certain substances such as mercury can be mistaken for magnesium by the body. \"If you don't have enough magnesium in your body, your body will suck in a molecule of mercury to take its place.\" With the magnesium slots filled with the impostor, mercury, there is now no room for real magnesium, and you become magnesium deficient. The toxin now blocks your system from absorbing the required real nutrient, explains Dr. Holyk. You may want to fortify yourself nutritionally, but your body cannot absorb the desired nutrients because mercury molecules are taking their rightful place in your cell chemistry. That's the first complication.\n\n**_For Dr. Holyk two matters come immediately to mind in assessing a new patient's symptoms: micronutrient status and toxicity. Is this person deficient in key vitamins or minerals necessary for normal detoxification processes, and is this patient carrying a toxic burden of cumulative heavy metal, solvent, or pesticide poisoning?_**\n\nThe second problem is that with all this mercury now in place in the cells, the body's natural detoxification processes are compromised, even blocked. As discussed in Chapter 2, the liver has a two-phase detoxification procedure. Phase I involves a group of enzymes called cytochrome P450, a key to successful liver detoxification. \"Yet if you get mercury in your system, it will bind to specific sites of the cytochrome P450 system and shut it down. This phase of the liver's detoxification will not work. Your liver cannot detoxify because mercury has shut down your detoxification system.\" It doesn't end there, either, adds Dr. Holyk.\n\nCytochrome P450 is crucial to the formation of hormones, such as testosterone and progesterone, the male and female \"sex\" hormones, respectively. \"If mercury binds into the cytochrome P450 system, you cannot make your own hormones any more. You have a real problem. You can see when one toxin binds one specific chemical how much biochemical havoc it creates. What about the effects of all the other toxins we're exposed to? We're lucky we function as well as we do considering the degree to which I find toxic levels and deficient micronutrient status in patients.\"\n\nAt this point in the evolution of enlightened medical practice in the United States, there are many inexpensive but reliable testing options available to both physician and patient. Many of them do not require a physician's prescription other than for the initial drawing of a blood sample. When it comes to tests for toxicity, Dr. Holyk tends to rely on hair analyses, the DMPS challenge for mercury levels, certain specialized forms of blood assays, and kinesiological muscle testing.\n\n\"The best way in my opinion to find out about micronutrients is to look for intracellular nutrients,\" explains Dr. Holyk. By \"intracellular\" he means nutrients found inside blood cells. For this information, he relies on a blood test called a packed red blood cell intracellular analysis. It's a bit complicated, but the idea is this: normally, a blood sample is spun in a centrifuge and the serum (the watery portion of the blood) is examined for a specific nutrient, say magnesium. But magnesium exists mostly _inside_ the cells\u2014it's an _intra_ cellular nutrient. Since the plug of red blood cells was discarded in the centrifuge, the test will not indicate the true level of magnesium in this patient, explains Dr. Holyk. The packed red blood cell intracellular analysis looks at what's inside the cells and therefore provides a more accurate result, he adds.\n\nAs for kinesiological muscle testing, Dr. Holyk asserts that he has found its results to be comparable and consistent with those obtained from blood assays and another diagnostic approach called electrodermal screening (EDS). Chiropractors, for example, often use kinesiological muscle testing to determine if a person is allergic to a substance, will respond well or poorly to a proposed substance, and ask other medical questions necessary to form a clear diagnosis. The principle behind muscle testing is simple: a muscle's strength noticeably wanes in the presence of a toxic or nondesired substance.\n\nIn the case of electrodermal screening, specific surface points on the hands and feet (associated with acupuncture meridians) are probed with a device hooked up to a computer; a single patient can be tested for the suitability of hundreds of substances in a short time using this approach. Specific toxins can be ascertained along with the appropriateness, or not, of proposed remedies.\n\nDr. Holyk also reports good results from a testing method called Biological Terrain Assessment, another computer-assisted evaluation of a patient's cellular environment. It studies the biochemical condition of the blood, urine, and\/or saliva in terms of pH (the ratio of acidity to alkalinity, a general indication of chemical balance or unbalance), oxidation-reduction potential (how much \"oxidation stress\" the body is carrying, a measure of cellular toxicity), and resistivity (a tissue's resistance to the flow of electrical current).\n\nInsofar as food and substance allergies play a large role in toxicity, Dr. Holyk often uses Nambudripad Allergy Elimination Technique (NAET), a system of both diagnosis and treatment that uses acupressure, acupuncture, and allergy desensitization techniques to neutralize the allergenic effect of numerous substances. \"I've almost not found a person who did not have a toxic problem or did not have an allergy problem, or both,\" he states. \"To some degree they may be interdependent, although I think they actually are two independent variables\u2014but they are both extremely common.\"\n\nThe essential point, Dr. Holyk emphasizes, is to _know_ your patient's nutritional status, immune defense capabilities, and level of toxicity. Then you can unravel their health problems, initiate a successful detoxification program, and start rebuilding, or maintaining, optimal health. In other words, reliable nutritional and physiological _information_ is the key to successful detoxification and creating a healthy living space. Reliable diagnostic information about your biochemical and physiological status provides answers not only as to why you may be already sick, but it can also highlight vulnerable areas and tendencies toward developing future illnesses or imbalances. Even better, diagnostic tests such as these can equip you with enough advance information to _prevent_ most future health problems and in effect to treat disease before it ever happens.\n\n  **_The Healthy Living Space_ Info Tip**\n\nTo contact Peter Holyk, M.D.: Contemporary Health Innovations, PA, 600 Schumann Drive, Sebastian, FL 32958;\n\ntel: 561-388-555H;\n\nfax: 561-388-2H10;\n\nwebsite: www.chimed.com.\n\nWhile you most likely will not need to do all of the laboratory tests outlined in this chapter (see figure 3-1), a selection may illuminate certain otherwise baffling aspects of your health condition. Some tests you can order on your own; others require a physician's prescription; and most yield their best results if you work with a qualified health-care practitioner who can interpret the data.\n\n**Figure 3-1. 19 Reliable Tests to Measure Your Possible Toxicity**\n\n## _HEALTHY LIVING SPACE DETOXIFIER #3: Get a Clear Idea of Your Nutritional Status_\n\nAs Dr. Holyk explains, your nutritional status\u2014the levels of specific major nutrients and micronutrients in your body, especially those involved in detoxification\u2014is a key element to both your health and your system's ability to detoxify and mount a strong immune defense.\n\nYou may think you have good, satisfactory, or even excellent nutritional status, yet it is often the case that a person is deficient, or even seriously lacking in a few key elements. It's only when you get sick that this unchecked deficiency reveals itself and hinders your recovery process. The _only_ way to know in advance if you are lacking in a few or many essential elements is to have good testing (such as urine, stool, and hair, described below) before you get sick, or at the start of a deliberate detoxification program. Below are brief overviews of a number of well-known and widely used laboratory tests to determine nutrient status.\n\n### _Packed Blood Cell Elements Profile_\n\nThis is the test referred to by Dr. Holyk above as \"packed red blood cell intracellular analysis.\" Offered by Doctor's Data, an independent reference laboratory that specializes in tests to measure levels of toxic and essential elements in the blood and urine, this test assesses the status of nutrients inside blood cells or in blood cell membranes. According to Doctor's Data, this test is \"an excellent diagnostic tool\" to measure antiinflammatory processes (your body's ability to withstand inflammations, especially internal ones that could lead to arthritis or fibromyalgia), immune function, and problems associated with zinc deficiency (such as poor wound healing and attention deficit disorder).\n\nSpecifically, this test looks for levels of thirteen nutrients, major players such as calcium, magnesium, potassium, phosphorus, zinc, and iron; and several key micronutrients such as vanadium, molybdenum, and boron. It contrasts the results against an established reference range\u2014levels generally considered optimal for health and statistically prevalent in healthy people.\n\nFor example, in a sample laboratory report, a male, aged thirty-five, had calcium at 13 \u03bcg\/g (which means parts per million); the reference range for calcium is 8-31 \u03bcg\/g, which means he was at the very low end of \"acceptable,\" and probably on the verge of being deficient. His iron came in at 968 \u03bcg\/g compared to the reference range for iron of 745-1050 \u03bcg\/g, placing him well within the acceptable, well-fortified range for this nutrient. The test also measures blood levels for five toxic elements: antimony, arsenic, cadmium, lead, and mercury. This man tested 0.076 \u03bcg\/g for mercury compared to the reference range of less than 0.01 \u03bcg\/g; this meant that he had some mercury toxicity in his system as his numbers for mercury were higher than the acceptable range.\n\n### _Whole Blood Elements Profile_\n\nThis is also offered by Doctor's Data and is designed to measure circulating blood levels of toxic and nutrient elements. It is particularly recommended by Doctor's Data to be used before and during a detoxification treatment, and is well suited to identify dietary inadequacies (in terms of nutrient intake) and the function of the gastrointestinal, endocrine, and kidney systems. The test measures levels of twelve nutrients (many of the same nutrients measured in the Packed Blood Cell Elements Profile) and six toxic elements (including bismuth, nickel, and uranium) based on their levels in the whole blood (the watery portion, or plasma, plus the red blood cells). The results are not necessarily the same as in the previous test because the test measures nutrient and toxic status in a larger, more diffuse volume than only packed red blood cells.\n\n### _CellMate Blood Test Report_\n\nThe manufacturer of this test describes it as an \"extension\" to standard blood tests because it compares a person's blood chemistry to a database of medical research. The results for concentrations of individual nutrients are presented in terms of a percent status, that is, how much above or below the normal range they deviate. The report also matches results against tabulated symptom correlations for numerous illnesses; this is called disease pattern matching and enables the physician to make quick recognition of similar patterns. For example, a person may have five results that give them an eighty-three percent match for the blood conditions typically observed in someone with hypersplenism (overactive spleen).\n\nThe Basic Status Report presents your lows and highs in over fifty categories (including key immune cells) as a bar graph, making it easy to see at a glance where your nutrient excesses and deficiencies lie. And the test indicates nutrients that would be useful to take as supplements, drugs and drug interactions to avoid, and supplements to avoid based on your specific biochemistry. The value of this test is that it puts the biochemical information in the context of normal ranges for excesses and deficiencies and with respect to known matching disease patterns.\n\n### _Individualized Optimal Nutrition (ION)_\n\nThe ION test measures your specific nutritional status and provides a specific supplemental program to restore deficiencies. The test studies blood and\/or urine samples to uncover nutritional status and \"metabolic trends,\" by which MetaMetrix, ION's provider, means how well your body processes foods to extract the necessary nutrients.\n\nA blood or urine test, says MetaMetrix, is like taking a \"snapshot in time\" of your nutrient status, showing the effects of what you have been eating, the degree of stress your system has been under (which can result from nutrient deficiencies), and the impact of supplementation. It is very hard to know if you have optimal nutritional status without a test. You may assume you are healthy, or you may feel normal or \"okay,\" but you may still have a suboptimal nutritional state.\n\nThe ION program has five subprofiles including amino acid, B-vitamin\/mineral, antioxidant, fatty acid, and disease risk status. Let's look at each of these and see why this knowledge is valuable for a detoxification program.\n\nAmino acids, which are the building blocks of proteins, have to do with energy, endurance, concentration, mood, and alertness, and play key roles in brain function, immune system operation, tissue repair, regulation of blood sugar levels, and general energy levels. If your diet lacks sufficient amino acids, any of these vital functions may be impaired, providing the foundation for toxicity and the body's inability to detoxify, as well as the development of dysfunction and eventual illness.\n\nSimilarly, B-vitamins and minerals are essential for numerous bodily functions and are, in the words of MetaMetrix, \"the spark plugs necessary for your metabolic engine.\" For example, the mineral magnesium is required for at least 300 different enzyme reactions in the body, but studies show that in fifty percent of hospitalized patients, magnesium levels are low, which suggests the magnesium deficiency might be linked to numerous health problems that lead to hospitalization.\n\nAs discussed in Chapter 2, antioxidants are specific nutrients (such as vitamins A, C, and E) that protect your system against toxins (free radicals) that can lead to disease. As MetaMetrix puts it: \"Normal biological processes and environmental pollutants produce unstable molecules called free radicals that wreak havoc on tissues by setting tiny 'fires' of oxidation.\"\n\nThe unbalanced state of oxidative stress (see below, for another test) can lead to many physiological problems; in this analogy, antioxidants are the \"indispensable molecular fire extinguishers.\" When free radicals damage the fat (lipid) portion of cell membranes, toxic products called lipid peroxides are formed. These are associated with the development of numerous physical problems including heart disease, arthritis, liver disorders, allergic inflammation, and others. Therefore, this component of the ION test measures your level of free radical activity (and oxidative stress) and the defense status of selected antioxidants.\n\nCertain fatty acids\u2014the test measures levels of twenty-five\u2014are essential to optimal body function because, among other things, they are a source of essential hormones and regulate inflammation, which, if unchecked, can lead to arthritis and chronic muscle pain, among other conditions. Typically, we get too many of the wrong kinds of fats from our diet (hydrogenated and trans-fatty acids), which contribute to cellular degeneration and illness, and too little of the \"good\" fats, such as those found in fish, flaxseed, and various oils from vegetables and nuts. An imbalanced fat intake is directly linked with many serious illnesses, including stroke, heart attack, and cancer. If you are deficient in any of the essential fatty acids, various metabolic mechanisms that require them may no longer work properly.\n\nThe last component of the ION test screens the various organs (especially the liver and kidneys) for risk of developing disease, based on levels of certain biochemicals interpreted as disease risk indicators.\n\n### Functional Intracellular Analysis (FIA)\n\nThis test of nutrient status is based on examining a sample of the person's white blood cells, called lymphocytes. The test looks inside your white blood cells to see how well nineteen nutrients and antioxidant systems are functioning based on their level of cell growth. According to SpectraCell Laboratories, providers of the test, the FIA uncovers deficiencies usually missed by serum-based tests.\n\nFIA is called \"functional\" for a good reason. White blood cells are isolated and removed from the blood sample, then grown in culture media, being given the optimal nutrients for cell growth. Then specific nutrients are removed, one at a time from the media, requiring the white blood cells to replenish these nutrients from their own reserves. \"If there are no deficiencies in a particular vitamin, the cells will still grow adequately,\" explains SpectraCell. \"If, however, the cells do not grow adequately, it represents a deficiency of that vitamin.\" In other words, you get an accurate report on your body's functional nutritional status, on how well it performs its job for _your_ system specifically based on the nutrients and nutrient levels available to it.\n\n### _Total Antioxidant Status_\n\nThis test measures levels of all the antioxidants present in a blood sample according to their ability to stop a particular oxidation reaction, which is to say, a targeted free-radical-initiated activity. The test, developed by King James Medical Laboratory, looks at three antioxidant defense systems: primary antioxidants (superoxide dismutase, glutathione peroxidase, and metal-binding proteins), which prevent new free radical groups from arising; secondary antioxidants (vitamins E and C, beta-carotene, uric acid, bilirubin, and albumin), which collect free radicals, preventing them from working together to produce toxic chain reactions; and third-level antioxidants (DNA repair enzymes), which repair \"biomolecules\" that have been damaged by free radicals.\n\n### _Pantox Profile_\n\nPantox Laboratories provides a blood test called the Pantox Profile, which \"interprets the biochemical defense system\" by measuring the blood concentration of twenty-five substances central to the body's antioxidant response and compares these results with values obtained from other people of the same gender and age. Each substance analyzed for a given person is then plotted as a percentile, expressed as a vertical bar graph for easy comprehension. When the blood concentrations of these twenty-five substances are interpreted, they serve as \"very early indicators\" of potential health problems. The test results don't show the presence of disease; rather they demonstrate your ability to resist it.\n\nThe key point here is that early recognition of nutritional deficiencies or imbalances known to be associated with various illnesses leads to correction and prevention. This is why Pantox recommends this test as a \"regular biochemical checkup.\" Pantox emphasizes that aging and age-dependent degenerative disease (which encompasses a great number of illnesses) are the \"byproducts of normal metabolism.\"\n\nTechnically, \"normal\" metabolism involves the production of free radicals; it should also include their deactivation by antioxidants, but when this doesn't happen, your imbalanced metabolism slowly sickens you. A test of this type is useful in terms when a person needs to know if their nutritional defense systems are in optimal shape.\n\n### _Biological Terrain Assessment_\n\nThis test provides an in-depth view of the elements of an individual's chemistry and the underlying causes of an imbalance, thereby enabling the practitioner to determine an individualized program for the patient. The test looks at biochemical conditions within the cells, specifically in three categories. These are pH (the balance between acidity and alkalinity), oxidative stress (oxidation-reduction potential, or redox, indicating electron potential and enzymatic activity), and resistivity (molecular ion movement and gross mineral concentration).\n\n**_The key point here is that early recognition of nutritional deficiencies or imbalances known to be associated with various illnesses leads to correction and prevention. This is why Pantox likes to recommend this test as a \"regular biochemical checkup.\"_**\n\nThese three factors largely determine what is now being called the cell's internal environment, milieu, or terrain\u2014its overall quality of (cellular) life. You could even think of it as the \"soil\" of the cell; just as in gardening, healthy soil with the right balance of microorganisms and nutrients produces the healthiest, most nourishing vegetables. The \"vegetables\" in this analogy are the biochemical activities of cell life and function. The idea of terrain actually originated with the early nineteenth century physiologist, Claude Bernard, who stated that the cell's environment determines its function and integrity, and thereby the vitality of the entire body.\n\nBiological terrain is assessed with the help of a computerized device called the BTA S-2000, which displays the results\u2014information on \"the body's internal biochemical markers,\" explains the manufacturer\u2014in various charts and graphs on a computer screen, based on its analysis of a sample of the patient's blood, urine, and saliva. \"Many patients who initially undergo Biological Terrain Assessment come into their practitioner's office with reports of 'normal' laboratory values and yet display illness both objectively and subjectively,\" explains Robert C. Greenberg, M.D., the inventor of the BTA S-2000. \"After analyzing their bodily fluids for pH, redox, and resistivity, important data begins to emerge.\"\n\nDr. Greenberg explains that \"often very subtle yet potent influences\" are under way in a person's biochemistry, but are not necessarily detected by standard laboratory tests. Some of these toxic influences can include parasites, viruses, fungi, pollutants, free radicals, nutritional deficiencies, oxygen depletion, and the body's inability to excrete carbon dioxide. If standard tests cannot measure these factors, many patients may remain toxic or sick and their doctors confused as to the true medical condition.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Packed Blood Cell Elements Profile,\n\nWhole Blood Elements Profile:\n\nDoctor's Data,\n\nP.O. Box 111,\n\nWest Chicago, IL 60186;\n\ntel: 800-323-2784 or 630-377-8139;\n\nfax: 630-587-7860;\n\ne-mail: inquiries@doctorsdata.com;\n\nwebsite: www.doctorsdata.com (see website for sample laboratory reports for these tests).\n\nFor CellMate Blood Test Report:\n\nCarbon Based Corporation,\n\nP.O. Box 4549,\n\nIncline Village, NV 89450;\n\ntel: 775-832-8485 or 800-908-0000;\n\nfax: 775-832-8488;\n\nwebsite: www.carbonbased.com.\n\nFor ION: MetaMetrix Clinical Laboratory, Inc.,\n\n5000 Peachtree Industrial Boulevard,\n\nNorcross, GA 30071;\n\ntel: 800-221-4640;\n\nfax: 770-441-2237;\n\ne-mail: nutrition@metametrix.com;\n\nwebsite: www.metametrix.com.\n\nFor FIA: SpectraCell Laboratories, Inc.,\n\n515 Post Oak Boulevard, Suite 830,\n\nHouston, TX 77027;\n\ntel: 800-227-5227;\n\nwebsite: www.spectracell.com.\n\nFor Total Antioxidant Status: King James\n\nMedical Laboratory, Inc., 24700 Center Ridge\n\nRoad, Cleveland, OH 44145;\n\ntel: 800-437-1404;\n\nfax: 440-835-2177;\n\ne-mail: CustomerService@kingjamesomegatech-lab.com;\n\nwebsite: www.kingjamesomegatech-lab.com.\n\n## HEALTHY LIVING SPACE DETOXIFIER #4: Assess Your System's Ability to Detoxify\n\nThe preceding tests will give you a clear idea about your system's nutrient levels and its overall ability to provide the nutritional elements essential for a strong immune defense. Now you need to find out how capable your natural detoxification system is and to what degree its detoxification ability may be already compromised by an excess of toxins.\n\n### _Oxidative Protection Screen_\n\nThis test is offered as a functional assessment of oxidative protection, according to its provider, AAL Reference Laboratories. It looks at the extent of oxidative damage (from free radicals) and your body's ability to protect itself against free radical onslaughts. Free radicals not only come from substances brought into the body, but they are the natural byproducts of metabolic processes and the liver's own detoxification program. As explained in Chapter 2, if your liver detoxifies the body too quickly (as in Phase I) or has blocked pathways for toxin elimination (Phase II), free radicals accumulate. Oxidative stress therefore is a reliable measure of how well your body can handle the stress of free radical oxidation\u2014too many poisons in the system.\n\nThe test generates a Total Oxidative Protection (TOP) Index for each client, which is a measure of your system's total antioxidant ability\u2014in other words, it tells you how well protected you are. The higher the TOP Index, the stronger your immune defense is.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Pantox Profile:\n\nPantox Laboratories,\n\n4622 Santa Fe Street,\n\nSan Diego, CA 92109;\n\ntel: 888-PANTOX-8, or 888-726-8698,\n\nor 858-272-3885;\n\nfax: 858-272-1621;\n\nwebsite: www.pantox.com.\n\nFor Biological Terrain Assessment:\n\nBiological Technologies International,\n\nInc., P.O. Box 560, Payson, AZ 85547;\n\ntel: 520-474-4181;\n\nfax: 520-474-1501;\n\nwebsite: www.bioterrain.com.\n\nThe test also measures levels of lipid peroxides, which indicate the degree to which your system is suffering oxidative stress, or the overall impact of too many free radicals, especially those acting on the fatty acids in cells. According to AAL Reference Laboratories, the higher the lipid peroxide levels, the greater the degree your system has been negatively affected by free radicals, or toxins. A healthy individual will have very low levels of lipid peroxides. Therefore, this assay provides information on your level of toxicity.\n\nThe two results\u2014TOP Index and lipid peroxides levels\u2014are correlated and interpreted together. For example, if the test indicates you have normal lipid peroxides but low TOP Index, it means you are losing your antioxidant protection reserve. If unchecked, this may progress to a loss of oxidative protection. If the test indicates you have high lipid peroxides and normal TOP Index, this means your system is producing free radicals faster than your antioxidant system can cope with. This may be happening because of excess tobacco smoke inhalation, too much exercise, being obese, or having a diet with too many polyunsaturated fats or oils. If you have increased lipid peroxides levels and a low TOP Index, this indicates a state of poor oxidative protection.\n\n### _Oxidative Stress Profile_\n\nGenox Corporation of Baltimore, Maryland, offers an analysis of oxidative stress with four components. The test measures the rate of generation of free radicals; the levels of antioxidants (\"scavengers of reactive oxygen species\") and the rate at which lipid peroxides are reduced; the level of oxidative damage already sustained by the person; and the rate at which oxidative repair is accomplished and the rate of removal of free radicals. Genox obtains these measures by assaying a sample of blood, urine, or breath, although in special cases, a tissue or cell sample from the mouth or bladder may be used.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Oxidative Protection Screen:\n\nAAL Reference Laboratories, Inc.,\n\n1715 E. Wilshire, #715,\n\nSanta Ana, CA 92705;\n\ntel: 714-972-9979 or 800-522-2611;\n\nfax: 714-543-2034;\n\ne-mail: inquire@aalrl.com;\n\nwebsite: www.antibodyassay.com.\n\nFor Oxidative Stress Profile:\n\nGenox Corporation,\n\nl4l4 Key Highway,\n\nBaltimore, MD 21230;\n\ntel: 410-347-7616 or 800-810-5450;\n\nlab: 410-347-7637;\n\nfax: 410-347-7617;\n\ne-mail: info@genox.com;\n\nwebsite: www.genox.com.\n\nFor Detoxification Profile:\n\nGreat Smokies Diagnostic Laboratory,\n\n63 Zillicoa Street,\n\nAsheville, NC 28801;\n\ntel: 800-522-4762;\n\nfax: 828-252-9303;\n\ne-mail: cs@gsdl.com;\n\nwebsite: www.gsdl.com.\n\n### _Detoxification Profile_\n\nIn this test, certain \"challenge\" substances, such as caffeine, acetaminophen, and salicylate (aspirin), are given in tablet form to see how well the body performs each stage of the detoxification process related to these substances. In other words, the test studies how well your body can \"clear\" these substances, passing them out through the urine. The results, as analyzed in the urine and saliva, and sometimes blood, shed light on your system's detoxification ability, in both its Phase I and Phase II stages. As explained in Chapter 2, in liver detoxification, the body's enzymes collect and \"activate\" (alter or neutralize) toxic substances, such that other enzymes can convert these toxins into water-soluble forms that can be eliminated through the urine and stool.\n\nIn effect, this test highlights your liver's ability to detoxify. The test studies the metabolites (organic compounds produced by breaking down the challenge substances), the byproducts of detoxification (such as glutathione, a key detoxifying substance found in the liver), and the toxins (in various changed forms) one would expect to be excreted during a detoxification, such as lipid peroxides. The information provided by this test is valuable because both Phase I and Phase II of your liver's detoxification system must function adequately so that toxins can be neutralized and excreted, and because the two phases must work in balance with each other.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #5: Determine Your Body's Total Toxic Load_\n\nThe preceding set of tests provides information on the functional status of your liver and its two-phase detoxification system. Now it's time to get specific: what is your total toxic load? Which toxic substances is your body storing? Do you have heavy metal toxicity, intestinal imbalances, or parasites? The matter of intestinal balances and parasitic infestation will be discussed at greater length later in the book (Chapter 5), but both are the focus of important laboratory tests. Imbalances in the microbial population of the intestines can result from prolonged, unrelieved toxicity and they also further contribute to this toxicity. Parasites (living microscopic organisms such as various species of worms that don't belong in the human body) can enter the intestines through our food and water and contribute to intestinal imbalance, rob us of crucial nutrients, and generally sicken us.\n\nSimple laboratory tests involving samples of hair, urine, and\/or stool can provide clinically valuable information on your toxic load and possible intestinal imbalances or parasitic involvement.\n\n### _Hair Analysis_\n\nAlthough hair analysis got its start in the 1930s, progressive physicians have turned increasingly to it in recent decades. In fact, over 1,500 peer-reviewed scientific papers have been published on hair analysis. An analysis of a patient's hair sample is considered a reliable window into the level of toxicity present, specifically, heavy metals. A hair analysis typically evaluates the amount (parts per million) in the body of twenty-two elements such as aluminum, mercury, cobalt, cadmium, arsenic, nickel, and others.\n\nThe hair seems to have a remarkable ability to register the concentration of various metals in the body. Typically about one gram of hair is selected, or about two teaspoons, taken from the first one and one-half inches of hair closest to the scalp at the nape of the neck. In fact, the deposition of metals in strands of hair is somewhat like tree rings, which form in accordance with growth spurts. In the case of hair, as a strand grows, its biochemical composition reflects the various metals the individual has been exposed to or ingested. In fact, in the case of some toxic metals, \"hair more closely reflects the body's mineral stores than blood or urine,\" states King James Medical Laboratory, which offers the Omegatech Hair Analysis.\n\nThe Laboratory further explains that \"the hair becomes a stable time record of the exposure and circulation of metals. The metal content of the hair varies along its length relative to the type and amount of each metal the hair cells are exposed to during their growth.\"\n\nIf for example you submitted a hair sample today and compared the analysis results with one you obtained a year earlier, the results could be quite different. They would depend on which metals you were exposed to during the time that hair strand was growing. If you worked around lead paint or had a mercury filling replaced (perhaps not expertly) a year ago but since then had stayed away entirely from all paints and never seen a dentist, your hair analysis would yield different results for lead and mercury.\n\nDoctor's Data, which offers a Hair Elemental Profile, explains that the hair analysis offers a \"temporal record of metabolism\"\u2014how well your digestion and absorption of nutrients are going right now\u2014as well as an exposure index to various toxic metals. The company emphasizes that this test is useful for the early detection of \"elemental aberrations, because deviations often appear in hair prior to overt symptoms.\" Not only do you get information about possible heavy metal toxicity, but you can potentially get this information early enough to serve as an advance warning of possible future (or imminent) health problems.\n\nAccording to Trace Mineral Systems, a testing company in Alexandria, Virginia, which also offers a hair analysis, hair is \"the ideal biopsy material\" and hair analysis is now \"an extremely precise analytical test.\" A blood mineral analysis will show which substances are circulating extracellularly (outside and between the cells), and a urine test will tell you which elements have been excreted, but a hair analysis will highlight a person's individualized mineral absorption during a specified time period, explains Trace Mineral Systems.\n\n### Urine Elements Profiles\u2014Essentials and Toxics\n\nThe value of the hair analysis is that it provides an initial mineral screening of your system. If it indicates significant metal toxicity, you might need a more comprehensive test called a urine analysis to give diagnostic information about various heavy metals based on their presence in the urine. The better urine analyses will collect a patient's urine from a 24-hour period, with samples taken at regular intervals. This gives a more balanced and thereby accurate picture of urine composition, as its contents may vary by the hour according to circadian hormonal secretions, dietary intake, and fluctuating stress levels.\n\nA urine test reveals information about elements both essential to the body's operation and those regarded as toxic. For example, a typical urine test will assess the levels of eighteen \"expected elements\" such as sodium, potassium, calcium, magnesium, and various other key nutrients, and compare their excretion levels against optimal norms and reference ranges. Then a twenty-four-hour urine toxic metals test can show the amounts present of at least fifteen undesirable toxic metals, including mercury, tin, uranium, cadmium, and others, again, comparing them against reference ranges.\n\nFor example, in a sample urine toxic metals test report supplied by Doctor's Data (available on their website) a male, aged 65, tested with 4.7 \u03bcg\/g (parts per million) of cadmium, a toxic metal found in tobacco smoke. The reference range (the safe level) for cadmium is less than 2 \u03bcg\/g, meaning this man had cadmium toxicity. He also had high levels of mercury at 2.4 \u03bcg\/g (reference range: 3.0 \u03bcg\/g) and lead at 13 \u03bcg\/g (reference range: 15 \u03bcg\/g). His levels for mercury and lead may not have exceeded the reference range, but they were close, indicating an excess of both metals in his system and a potential for heavy-metal-mediated illness.\n\nThis same patient had notable imbalances in his essential elements, namely, in calcium, magnesium, sulfur, zinc, and iron, all of which considerably exceeded the reference range. Recall Dr. Peter Holyk's observation, cited earlier, that mercury can be mistaken for magnesium by the body. \"If you don't have enough magnesium in your body, your body will suck in a molecule of mercury to take its place,\" he commented. This patient had significant mercury in his urine and a magnesium level of 261 \u03bcg\/g, compared to the reference range for magnesium of 46-200 \u03bcg\/g.\n\nWas there a magnesium-mercury feedback loop happening in this man? A health-care practitioner, skilled in interpreting this data and understanding that heavy metal toxicity can lead to essential nutrient imbalances, and vice versa, could assemble a useful diagnostic picture of this client, using the data from both aspects of the urine test.\n\n### _Special Provocative Mercury Test_\n\nThe situation may arise in which you or your physician may wish to gather more specific data on suspected high mercury levels in your system. The man referred to above, whose mercury was 2.4 \u03bcg\/g as against the reference range of 3.0 \u03bcg\/g, is a good example of someone who would probably want to quantify his mercury load more precisely.\n\nUsing a metal chelator called DMSA (2,3-dimercaptosuccinic acid), the body is challenged to release some of its mercury store through the urine; DMSA, as a chelator, binds up the mercury stored in tissues and removes it from the body. The DMSA challenge also measures urine levels of lead, cadmium, arsenic, and nickel. You take a 500-mg capsule of DMSA, then collect all your urine during the next six hours, after which you mail the specimen to the laboratory. Analysis of the mercury in the urine gives an indication of the \"mercury burden\" of the body. As a substance, DMSA is an FDA-approved drug used for the treatment of lead toxicity in children, and is widely regarded as a capable binding agent (chelator) for removing mercury from the body, even from the brain.\n\nThere is some disagreement among practitioners on the utility of this test, however. \"According to Dr. David Quegg of Doctor's Data,\" says Dr. Holyk, \"DMSA is a very poor way to do a mercury challenge. I only use (and he only suggests using) DMPS.\"\n\n### _Oral Toxicity Testing_\n\nMercury is not the only toxic element associated with your teeth. Various standard dental procedures, such as root canals and extractions, can generate toxic products over time that gradually sicken the individual and generally remain undetected because physicians aren't looking in the right place for them. This test seeks to identify these somewhat hidden toxins by analyzing what the layperson would regard as saliva samples. As you will learn in Chapter 7, dental toxicity from mercury fillings, root canals, and cavitations, is potentially as major a factor in total toxic body load as that produced by the intestines. \"According to some authorities, it is even worse,\" notes Dr. Holyk.\n\n\"Studies have shown that pathogenic [disease-producing] oral bacteria produce extremely high levels of toxins at sites of active infection,\" states Affinity Labeling Technologies of Lexington, Kentucky, a biotechnology company founded by Boyd Haley, Ph.D., chairman of the chemistry department at the University of Kentucky, and Curt Pendergrass, Ph.D. Their test is called the ALT, Inc., Four Part in Vitro Toxicity Testing of Oral Samples.\n\nThe test analyzes extracts from a patient's oral samples including gingival crevicular fluid, extracted teeth (such as those that bear root canals), and materials from cavitations (\"cavities\" in the jawbones, located underneath the areas where extracted teeth were once set in the jaw) for the presence of bacterial toxins, human inflammatory proteins (such as antibodies and serum albumin), and other undesired toxic elements.\n\nThe company also offers a seven-minute chairside test called TOPAS (Toxicity Prescreening Assay), which dentists can perform (painlessly) on patients during a dental checkup. For this, a specially treated swab is placed against a given tooth at the gumline, and it turns one color or another, depending on the degree of periodontal toxicity present. The test measures levels of protein in the gingival fluid, based on the understanding that dental bacterial infection increases these levels.\n\n### Comprehensive Digestive Stool Analysis\n\nA diagnostic tool whose use is steadily becoming more widespread is the stool analysis, or the biochemical evaluation of a sample of the patient's feces. This test shows how well (or poorly) your digestive system handles the foods you eat, how well the wastes and toxins are eliminated in the stool, and the general state of intestinal microbial balance or imbalance. The technical term for this latter state is intestinal dysbiosis.\n\n**_\"The human gut has been described as a continuous flow microbial growth chamber,\" explains Richard S. Lord, Ph.D., of MetaMetrix Clinical Laboratory, adding that a healthy intesting hatches about three pounds of viable microorganisms every day._**\n\nThe human intestines form an ecosystem with temperature, food source, and moisture at near ideal conditions for the growth of simple life forms, explains Richard S. Lord, Ph.D., of MetaMetrix Clinical Laboratory. \"The human gut has been described as a continuous flow microbial growth chamber,\" he says, adding that a healthy intestine hatches about three pounds of viable microorganisms every day. In other words, _many_ microorganisms are _supposed_ to be resident in the intestinal tract; most are benign and beneficial, but a few are not. Even so, in a healthy person, there is a balanced ecology in which the beneficial and undesirable organisms live together in a specific balance or ratio. When this balance is disturbed, intestinal dysbiosis results, and this sets the stage for numerous gastrointestinal-mediated illnesses and health problems.\n\nDysbiosis leads to an excess of bacterial toxins which in turn irritate the mucosal tissues of the gastrointestinal tract; if their buildup is excessive, they may \"leak\" into the bloodstream and put a significant burden on the liver and immune system. Intestinal dysbiosis favors the overgrowth of yeast, specifically _Candida albicans_ , which causes numerous health problems, including allergies. Dysbiosis also favors (or can be the result of) parasitic infestation\u2014the prevalence of tiny worms and other parasitic organisms in the gastrointestinal tract that leach nutrients and disturb digestive processes.\n\nAccording to Great Smokies Diagnostic Laboratory, which provides the Comprehensive Digestive Stool Analysis (CDSA), this noninvasive tool for analyzing the digestive tract \"helps pinpoint imbalances, provides clues about current symptoms, and warns of potential problems should the imbalances progress.\" The CDSA can identify \"critical imbalances previously unsuspected,\" as well as hidden yeast or bacterial infections, intestinal flora balance, dietary fiber intake, and intestinal immune function. It does this by studying the levels of various marker substances including certain fats (triglycerides), vegetable and meat fibers, various fatty acids, and others (twenty-five markers in all).\n\nDoctor's Data offers a Fecal Metals test to assess the levels of thirteen toxic heavy metals. In a sample report (available at their website) on a woman, aged forty-six, it was shown that many of her heavy metal levels exceeded the reference range. For example, her mercury was high at 0.321 mg\/kg, compared to a reference range of less than 0.05 (for someone without dental amalgams); her antimony was 0.254 mg\/kg against a reference range of less than 0.08; lead was 1.26 mg\/kg (reference range: less than 0.5 mg\/kg); tungsten 0.185 mg\/kg (reference range: less than 0.09 mg\/kg); copper was 90 mg\/kg (reference range: 60 mg\/kg); and bismuth was 0.417 mg\/kg (reference range: less than 0.05 mg\/kg).\n\nBasically, this woman was seriously high in about half the toxic metals tested, and she had significant levels of the other heavy metals. Her overall health status (or presenting symptoms) was not provided with the sample report, but one can assume she had (or will soon have) some significant health problems given this level of heavy metal toxicity. \"For several toxic elements such as mercury, cadmium, lead, antimony, and uranium, biliary excretion of metals into the feces is the primary natural route of elimination from the body,\" says Doctor's Data.\n\n### _Comprehensive Parasitology Profile_\n\nMention parasites in the intestines and most people assume you have to travel in exotic lands before this is something to worry about. However, the medical reality is far different. Many people get infested with gastrointestinal parasites merely from the foods they consume while living in North America. We also get exposed to parasites through raw and undercooked food, treated and untreated water, insects, and household pets, as well as through contact with people who have poor hygiene. Microscopic parasites\u2014protozoa, roundworms, pinworms, hookworms, tapeworms, and flukes\u2014can get transmitted through fecally contaminated foods, water, or other materials.\n\nUndetected, intestinal parasites begin destabilizing the microfloral ecology of the intestines, an imbalance that then leads to numerous other health problems, including generalized toxicity. A parasite growing inside your body, leaching all your nutrients and energy, can weaken your immune system and leave you vulnerable to many diseases. Further, a long-term parasitic infection can alter the mucosal lining of the intestines, leading to \"leaky gut\" or intestinal permeability. Toxins leak through the gut wall into the blood and get distributed throughout the body, producing, at a minimum, allergies and digestive problems. If parasites are present, your body's ability to detoxify its toxic burden is further compromised, and environmental toxins can gain a deeper foothold on your body's systems.\n\nHere's a vivid example of how this works, as related by Gary Kaplan, D.O., an osteopath practicing in Arlington, Virginia. Dr. Kaplan explains that he has observed in a number of patients that parasitic infestation produces symptoms of arthritis and fibromyalgia, or generalized muscle pain, and sometimes asthma. The parasites trigger the immune system, which dispatches antibodies (specialized immune cells that neutralize specific toxins) to deal with the foreign protein in its midst.\n\n  **_Healthy Living Space_ Info Tip**\n\nFor Omegatech Hair Analysis, Special\n\nProvocative Mercury Test:\n\nKing James Medical Laboratory, Inc.,\n\n24700 Center Ridge Road,\n\nCleveland, OH 44145;\n\ntel: 800-437-1404;\n\nfax: 440-835-2177;\n\ne-mail: CustomerService@kingjamesomegatech-lab.com;\n\nwebsite: www.kingjamesomegatech-lab.com.\n\nFor Hair Elemental Profile, Urine Elements Profiles\u2014Essentials and Toxics, Fecal Metals:\n\nDoctor's Data,\n\nP.O. Box 111,\n\nWest Chicago, IL 60186;\n\ntel: 800-323-2784 or 630-377-8139;\n\nfax: 630-587-7860;\n\ne-mail: inquiries@doctorsdata.com;\n\nwebsite: www.doctorsdata.com.\n\nSee also:\n\nTrace Mineral Systems, LC,\n\n127 South Fairfax Street, Suite 141,\n\nAlexandria, VA 22314;\n\ntel: 703-299-9306;\n\ne-mail: info@tracemineralsys.com;\n\nwebsite: www.tracemineralsys.com.\n\nThe parasites generate an \"antigenic reaction,\" which is not unlike an intense allergic attack, only in this case, it's prolonged. An antigen is a foreign protein or particle that does not belong in the body and which is tagged as a toxin by the immune system, and to which an antibody is assigned. It is often the case that as the parasitic infestation proceeds without medical resolution, the person's digestion suffers. One of Dr. Kaplan's patients developed asthma after being infected with a strain of _Giardia lamblia_ , a microscopic organism. Not only did this indicate an antigenic reaction, \"the antigenic reaction manifested as asthma,\" says Dr. Kaplan. After he treated the patient for _Giardia_ infestation, all the asthma symptoms cleared up.\n\nTesting for parasites is often indicated when other more obvious sources and sites for toxicity are not forthcoming in terms of explaining a chronic health problem. The more overt symptoms of parasite infestation are diarrhea and abdominal pain, but many other symptoms have been noted: nausea, vomiting, gas, bloating, foul-smelling stools, anorexia or weight loss, gastritis, fever, chills, headache, constipation, and fatigue. Add to this list other, seemingly unrelated problems including joint and muscle aches, anemia, allergies, chronic fatigue, sleep disturbances, and depression, according to Great Smokies Diagnostic Laboratory.\n\nTheir Comprehensive Parasitology Profile examines stool specimens for a wide range of protozoal parasites, including amoebae, flagellates, ciliates, coccidia, and microsporidia. If parasites are identified, this makes it easier for a qualified health-care practitioner to put together an effective parasite detoxification program.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #6: Find Out if You're Suffering from Hidden Food Allergies_\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Oral Toxicity Testing:\n\nAffinity Labelling Technologies, Inc.,\n\n235 Bolivar Street,\n\nLexington, KY 40508;\n\ntel: 606-388-9445 or 606-388-9345;\n\nfax: 606-388-9645;\n\ne-mail: info@altcorp.com;\n\nwebsite: www.altcorp.com.\n\nFor Comprehensive Digestive\n\nStool Analysis, Comprehensive Parasitology Profile:\n\nGreat Smokies Diagnostic\n\nLaboratory,\n\n63 Zillicoa Street,\n\nAsheville, NC 28801;\n\ntel: 800-522-4762;\n\nfax: 828-252-9303;\n\ne-mail: cs@gsdl.com;\n\nwebsite: www.gsdl.com.\n\nWhen people think of allergies, more often than not it's in terms of pollen and substance allergies and their characteristic symptoms of stuffy nose; itchy, red, and teary eyes; sneezing; face reddening; and the rest. However, just as serious and widespread a health problem is the matter of food allergies, most commonly to wheat, corn, dairy products, eggs, and chocolate. However, the list of possible allergenic foods\u2014that is, foods that trigger an allergic reaction\u2014is enormous, and given the ubiquity of the five substances just mentioned within the hundreds of processed foods available in North America, your chances of encountering a food to which you are allergic are very high.\n\nAccording to James Braly, M.D., an expert in allergies and allergy testing and medical director of Immuno Laboratories in Fort Lauderdale, Florida, a recent poll of 722 U.S. physicians quoted them as estimating that between 8% and 19% of the general population and 8.8% and 21% of infants have food allergies. Dr. Braly contends those numbers, in reality, are probably higher. In fact, he says that along with undernutrition, undiagnosed, or hidden, food allergies are a major health problem in the United States today. For many people, the continuous exposure to environmental toxins either creates or worsens their allergies. The allergies in turn, in the way they provoke the immune system into a state of perpetual alert, add to the body's toxic load and eventually weaken the immune system.\n\nHow can you tell if you have a hidden food allergy? There are many symptoms that typically occur in one who is chronically reactive, and these can be taken as reliable indicators of probable food allergies, according to a questionnaire developed by Immuno Laboratories. Consider whether you have any of the following symptoms: digestive (diarrhea, constipation, bloating, gas, belching); joints and muscles (pain, aches, weakness, stiffness); mouth and throat (coughing, clearing throat, sore throat, canker sores); weight (cravings, binge eating, water retention); energy levels (fatigue, apathy, hyperactivity); ears (itchiness, aches, infections, ringing, draining); eyes (watery, itchy, swollen, blurred vision); emotions (mood swings, anxiety, irritability, depression); mental (confusion, impaired memory and concentration); head (headaches, dizziness); hives, rashes, hair loss, shortness of breath, irregular heartbeat, and many others.\n\nThe problem in identifying food allergies is that in most cases they involve a _delayed_ reaction, taking several days\u2014up to seventy-two hours\u2014for symptoms to appear, whereas substance and pollen allergies typically produce immediate reactions with obvious, characteristic signs appearing within a few hours after contact or ingestion. Not only can food allergies produce delayed reactions, but the symptoms that manifest are not necessarily obviously related to foods.\n\nIt all has to do with immunoglobulins, which are activated as part of the allergic reaction. An immunoglobulin is one of a class of five specially designed antibody proteins produced in the spleen, bone marrow, or lymph tissue and involved in the immune system's defense response to foreign substances. The main types of immunoglobulins, grouped according to their concentration in the blood, are: IgG (80%), IgA (10-15%), IgM (5-10%), IgD (less than 0.1%), and IgE (less than 0.01%).\n\nFood allergies that produce immediate reactions are mediated through IgE, but those that are delayed pertain to IgG, and in some cases, IgA, explains Dr. Braly. Usually with an IgE allergic reaction, only one food is involved, but with delayed IgG-mediated allergies, there can be multiple foods involved, even as many as ten. \"Up to 80 medical conditions\u2014from arthritis, asthma, and autism to insomnia, psoriasis, and insulindependent diabetes\u2014have been clinically associated with IgG food allergy reactions,\" says Dr. Braly. Ironically often the foods that produce the most serious food allergies are those we are most drawn to or for which we feel the strongest craving\u2014the one to which we are addicted, if you will.\n\n### _Immuno 1 Bloodprint_\n\nThe diagnostic problem lies in the fact that the standard blood test or skin prick test for food allergies will not indicate a delayed food sensitivity, or one working through IgG. You have to test a person for IgG activity to accurately pinpoint a delayed food allergy, says Dr. Braly. That is why his company, Immuno Laboratories, developed the Immuno 1 Bloodprint to monitor for levels of IgG antibodies that get activated after a person ingests any of 102 suspected allergenic foods. If these antibodies are detected in the blood, it means the immune system is reacting to food allergens, regarding them as hostile to the body's internal environment.\n\n**_\"Up to 80 medical conditions\u2014-from arthritis, asthma, and autism to insomnia, psoriasis, and insufin-dependent diabetes\u2014have been clinically associated with IgG food allergy reactions,\" says 'Dr. Braly._**\n\nIn fact, this test looks for evidence of antibodies to IgG's subsets, which include IgGl, IgG2, IgG3, and IgG4. Testing for this degree of specificity is important, Dr. Braly stresses, because fifty percent of IgG-mediated food allergies work through IgGl, but forty percent come through IgG4, while ten percent are with IgG2 and IgG3. Unless you test for all these variations, you may still miss a delayed food sensitivity. By not screening for all the subclasses of the IgG immunoglobulin, the resulting allergy treatment could be only sixty percent effective, says Dr. Braly.\n\nIt pays to be specific when it comes to the top five allergenic foods, too. For example, the prime allergen in wheat is gluten, but the component that drives that allergenicity is a protein called gliadin; scientific studies have identified at least eighteen medical conditions directly linked to gliadin sensitivity. In other words wheat (and eggs and dairy protein) have \"subfractions\" that can produce equal, if not worse, allergic reactions than the whole food. This means that to accurately pinpoint food allergies, you need to test the whole and subfractions of the prime allergenic foods, according to Dr. Braly. Dairy milk has twenty-five different proteins, of which five have been consistently linked with allergies.\n\nThe Immuno 1 Bloodprint exposes a sample of the patient's drawn blood to a series of vials containing one of 102 test allergens, reactions are observed, and delayed food allergies are identified. The following are results from a sample report. This patient tested reactive (allergic) to fourteen foods: banana, barley, cheese, clam, egg, oats, oyster, pineapple, rice, rye, sugar cane, wheat, baker's yeast, and brewer's yeast. Of these, she had the strongest reactions to rye, wheat, and baker's yeast (registering a \"three\" out of possible \"four,\" which is the most severe allergic reaction).\n\nBased on the test results, the next step is dietary change, such as only eating the offending foods once every four days (or never, not for a long while, or not for one to four weeks) to reduce the body's \"habit\" of reacting to them as allergens and to give it time to clear out the allergy \"products\" from its system.\n\nDr. Braly cites another case history, this time involving a woman, aged thirty-nine, who was always bloated and cramped and endured other intestinal misery after every meal, even simple ones. This had been her lot for many years. She also had chronic sinus and ear problems, pain in her joints and muscles, mood swings, and hyperactivity, and just before seeing Dr. Braly and immediately after taking some cold medicine with orange juice, she developed rectal bleeding and esophageal reflux (severe heartburn).\n\nThe Immuno 1 Bloodprint showed that she was allergic to 17 foods, or 16.6% of the items on the test list; it also demonstrated she had a serious overgrowth of the yeast-like fungus _Candida albicans_ in her intestines. With the allergens identified, the woman stayed away from the seventeen allergenic foods and all foods containing yeasts, so as not to support further growth of the _Candida._ After implementing these steps, she regained her health in the next few months, even losing twenty-two pounds.\n\n### _Nambudripad Allergy Elimination Technique (NAET)_\n\nWhile the Immuno 1 Bloodprint can identify the various types of hidden food allergies, the standard food allergy treatment essentially involves avoiding the offending foods for a long time, then gradually rotating them with foods better tolerated.\n\nHowever, back in 1984, a California chiropractor named Devi S. Nambudripad, D.C., figured out a way to use acupuncture, chiropractic, and kinesiology to not only diagnose food and substance allergies, but to eliminate them as well. She then publicized her results in a popular book called _Say Goodbye to Illness._ Today Dr. Nambudripad's success is attested to by the fact that there is now a nationwide network of 3,000 NAET-trained physicians and health-care practitioners offering it to their patients (see her website for practitioner referrals).\n\nUsing kinesiology (muscle testing), the NAET practitioner is able to first ascertain if the patient is allergic to a specific substance. Allergens weaken the muscles, which makes it fairly easy to judge allergenicity by using a test muscle, such as the arm. The list of possible allergenic substances is surprising, even bizarre, revealing that you can potentially be allergic to almost anything: cotton underpants, fenugreek tea, leather belts, piano keys, saliva, sewing needles, diamond rings, tap water, and computer keyboards. One woman was allergic to her husband.\n\nThe gist of the technique is that the specific allergenic substance is held by the patient close to one of the acupuncture meridians; the practitioner applies acupressure to a specific acupuncture meridian at the same time; then the substance is avoided entirely for twenty-five hours, slightly longer than the time it takes for energy in the body to cycle through all the meridians. Somehow, the nervous system is desensitized through the acupuncture meridians to no longer react allergically to the substance. The brain is taught to reinterpret its contact with the substance, now regarding it as a neutral, acceptable thing and not a poison. Dr. Nambudripad claims that often only one session is required to \"permanently eliminate an allergy\" and to put the person in a situation in which he or she can actually resume contact with or ingestion of the previously allergenic substance.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Immuno 1 Bloodprint:\n\nImmuno Laboratories,\n\n1620 West Oakland Park Boulevard,\n\nFort Lauderdale, FL 33311;\n\ntel: 800-231-9197;\n\ne-mail: pcssales@immuno.com;\n\nwebsite: www.immunolabs.com.\n\nFor NAET:\n\nDevi Nambudripad, D.C.,\n\nPain Clinic,\n\n6714 Beach Boulevard,\n\nBuena Park, CA 90621;\n\ntel: 714-523-3068 or 714-523-0800\n\nor 71H-523-8900;\n\ne-mail: naet@earthlink.net;\n\nwebsite: www.naet.com or www.allergy-naet.com.\n\nWith the laboratory testing of your blood, urine, stool, saliva, muscles, teeth, and other physiological indicators detailed in this chapter, you can get a clear, quantified picture of your nutritional and immune status and degree of toxicity. Armed with this information, you can now move confidently into the next phase of creating the healthy living body space: practical detoxification protocols that work to clear the intestines, liver, kidneys, and lymph of this residue of toxins and rebuild your cellular health and reestablish proper nutrient levels\u2014the subject of the next four chapters.\n\n# CHAPTER 4\n\n# _Detoxification Preliminaries: Start Your Internal Cleansing with Dietary Changes and by Minimizing Your Exposure_\n\nAs groundwork for the more organ-focused stages of detoxification, you will need to reconsider your many dietary choices, even staples as basic as water. The U.S. food supply today is considerably adulterated, either deliberately in the course of its manufacture, with additives and preservatives, or inadvertently, in the course of its preparation or packaging.\n\nOne of the simplest and most effective steps you can take as a preliminary to colon and liver cleansing is to drastically reduce your intake of toxic substances through your food and water. This in itself is a detoxification step of profound physiological impact, but it is one that will require of you a fair measure of attention and vigilance to what constitutes your daily diet.\n\nAs a complement to reducing your toxic intake, you may find it beneficial to help your mind and nervous system calm down, to step back a bit from the daily onslaught of toxic or stressful energies, thoughts, and emotions. Detoxification itself brings deep relaxation and ease to all the cells of the body. Purposefully relaxing yourself supports the detoxification effort. This chapter incudes a review of a few relaxation techniques.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #7: Rehydrate Your Body with Nature's Best Detoxifying Substance_\n\nOne of the most basic recommendations for detoxification is to drink a great deal of pure water, probably double the amount you presently drink. At least thirty-two ounces daily is a good start, consumed directly as water, which disqualifies the water content of teas, coffee, and other beverages. In fact, naturopathic physicians recommend that during the more advanced stages of detoxification (see Chapter 5) you increase your water intake to sixty-four ounces daily, consuming it in small but regular amounts throughout the day. Some say you should drink that much routinely, even if you're not doing a detoxification regimen.\n\nIt may surprise many people to learn they might be functionally dehydrated. Even though we consume a fair amount of liquid during the day\u2014tea, coffee, sodas, beers, soups\u2014we may still be depriving our bodies and cells of even a minimal water supply. Take coffee, for example.\n\nYou would think that the more coffee you drink in a day, the more water enters your body, thereby fulfilling your body's daily need for fresh water. It's not the case. Naturopathic educators Peter Bennett, N.D., and Stephen Barrie, N.D., observe that dehydration is \"especially prevalent\" in heavy coffee drinkers because coffee is a strong diuretic, which means its chemical action is to flush water out of the body (which you will experience as increased urination), more than the amount contained in each cup of coffee consumed.\n\nChronic low-grade dehydration allows wastes to accumulate in the blood and overtaxes the kidneys, which need adequate water to mix with toxins to excrete them from the body in the urine. In fact, our bodies may be \"crying out\" for water and we are entirely missing the call, according to F. Batmanghelidj, M.D., a physician who practices in Falls Church, Virginia, and who, for the past eighteen years, has made the issue of unsuspected chronic low-grade dehydration the basis of a medical theory and practice. Dr. Batmanghelidj claims that a great many ailments, from indigestion to arthritis, high blood pressure to diabetes, and back pain to depression are actually due, in small or large part, to the body's unsatisfied demands for adequate water intake, that they are \"dehydration-produced disorders.\" This also means that the \"water cure,\" as he calls it, may be effective in reversing a number of otherwise difficult health conditions, such as angina, migraines, colitis, asthma, and high blood cholesterol, as well as the ones mentioned above.\n\nMost bodies live in a condition of water-deprivation stress. You are not sick\u2014you are thirsty, which is why you don't need to treat thirst with medications, but with copious amounts of fresh, pure water, Dr. Batmanghelidj says.\n\nThe simple truth, he states in his book, _Your Body's Many Cries for Water_ , is that \"dehydration can cause disease.\" Every function of the body is dependent on adequate water and the continuous flow of fresh water into the body. It is important to remember that it is not dehydration alone that is the problem. Copious amounts of water are needed every day by the body to flush out toxins from the cells and to complete the body's multifaceted detoxification process. Water is needed for each step of the process. If you are dehydrated, your detoxification processes may be stalled, or certainly compromised, and toxins accumulate in the tissues, leading to disease. Like Drs. Bennett and Barrie, Dr. Batmanghelidj points out that tea, coffee, alcohol, and prepared beverages (sodas, fruit drinks) are counterproductive in terms of hydrating the body. They contain dehydrating agents (and diuretics), so depending on them to satisfy the body's daily needs for water is \"an elementary but catastrophic mistake.\"\n\nComplicating matters is the fact that your body will not necessarily manifest a dry mouth symptom to let you know it is stressed for lack of water, says Dr. Batmanghelidj. This signal, when it occurs, \"is the _last_ outward sign of _extreme_ dehydration.\" The damage occurs, he says, \"at a level of persistent dehydration that does not necessarily demonstrate a 'dry mouth' signal.\"\n\nConsider what restoring adequate water intake can do for a case of dyspepsia\u2014indigestion based in the stomach. Dyspeptic pain (which is experienced mainly as heartburn, duodenitis, or gastritis) is the most important thirst signal the body produces, denoting dehydration, says Dr. Batmanghelidj. The sole treatment for this condition should be increased water intake, he says. He has successfully treated 3,000 cases in this way. \"They all responded to an increase in their water intake, and their clinical problems associated with the pain disappeared.\"\n\nHe cites a dramatic case in which a patient had such severe stomach pain he had to be carried by others into Dr. Batmanghelidj's clinic. The patient experienced relief after he had consumed three glasses (twenty-four ounces) of water. On the average, says Dr. Batmanghelidj, it takes eight minutes to achieve total pain relief from dyspepsia by drinking water. \"A well-regulated and constantly alert attention to daily water intake will prevent the emergence of most of the major diseases,\" he states.\n\nHow much water is enough to satisfy the body and prevent disease? This would be an \"absolute minimum\" of forty-eight to sixty-four ounces daily of fresh, pure water, says Dr. Batmanghelidj. The best times to drink water are thirty minutes before eating, and two and one-half hours after eating, and before going to bed. Another \"water cure\" advocate, Terry Grossman, M.D., who practices in Denver, Colorado, advises six to ten glasses of water daily, which is forty-eight to eighty ounces. His \"water prescription\" is that \"every day you should drink that amount of water in ounces to equal one-half of your body weight in pounds.\" That means, if you weigh 110 pounds, you drink 55 ounces; if you weigh 200 pounds, you drink 100 ounces. Dr. Grossman says it is best to drink the water away from mealtimes so as not to dilute the digestive juices required to digest your meals.\n\n**_How much water is enough to satisfy the body and prevent disease? This would be an \"absolute minimum\" of forty-eight to sixty-four ounces daily of fresh, pure water, says (Dr. Batmanghelidj. The best times to drink water are thirty minutes before eating, two and one-half hours after eating, and before going to bed._**\n\nOne way to monitor your intake is to examine the color and consistency of your urine. If it is dark yellow, dense or strong smelling, or even slightly orange in color, it means your body is dehydrated, says Dr. Batmanghelidj, and your kidneys are stressed. Dark yellow urine is concentrated urine, dense with toxic excretions. Ideally, if you are healthy, urine from a well-hydrated body should be almost colorless to light yellow, he says. This is something you can observe during the day in relationship to both your food and water intake.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #8: Get the Chlorine Out of Your Drinking and Shower Water_\n\nA compelling reason to filter your drinking water is that your body depends on a substantial supply of fresh, pure water to accomplish its continual detoxification. In fact, water itself is an excellent detoxifying substance, flushing out toxins from cells and carrying them out of the body through the urine. During a deliberate detoxification regimen, you will most likely be at least doubling your water intake.\n\n**_It's important to remember that the_ cumulative _effect is the key: the more you ingest a toxic substance, the greater the chance it will have a toxic effect one day. In the same way, unilateral_ reduction _of toxic intake is the order of the day in the field of self-care detoxification._**\n\nIt is excellent advice to drink _lots_ more water, but the trouble with this recommendation is that nearly all tap water is contaminated with trace amounts of pesticides, industrial pollutants, and chlorine byproducts. If you use chlorinated water with all its toxic baggage as your prime water source (especially in a detoxification program), you are continually introducing more toxic substances even though you are trying to flush others out of your body. You won't make as much progress in cleansing your cells as you will if you do not introduce more chlorinated byproducts.\n\nIt's important to remember that the _cumulative_ effect is the key: the more you ingest a toxic substance, the greater the chance it will have a toxic effect one day. In the same way, unilateral _reduction_ of toxic intake is the order of the day in the field of self-care detoxification. Ingest fewer toxic substances of any kind\u2014this is one of the most practical steps you can take.\n\nSo, before you double your water intake, you need to reexamine your source of drinking water and look into nonchlorinated alternatives, such as bottled water and home filtration. First, let's look more closely at the problems with chlorinated water.\n\nChlorination of municipal drinking water has been a fact of life that most people have accepted routinely and uncritically for nearly the entire twentieth century, since 1908 when municipalities began adding chlorine to drinking water to disinfect it. Even though chlorine does a good job at disinfecting water, unfortunately it produces other toxic substances that, in terms of public health issues, may produce worse conditions than the original organic toxicants (mostly bacteria) found in water sources.\n\nThe problem is that chlorine produces toxic \"disinfection byproducts.\" When chlorine is added to water, it naturally reacts with organic matter such as humic and fulvic acids, which are present in the untreated, \"raw\" water. This reaction creates a new category of chlorinated chemicals jointly known as tri-halomethanes (THMs). It was only in the 1970s, some seventy years after water chlorination began on a large scale, that scientists discovered these toxic disinfection byproducts, and that chloroform produced cancer in laboratory rats.\n\nThere are potentially many different kinds of THMs produced when water is chlorinated, but public health officials tend to focus on only four: chloroform, bromoform, bromodichloromethane, and chlorodibromomethane. Technically, for it to be deemed \"acceptable,\" municipal water servicing more than 10,000 people is not supposed to have any of these four THMs at levels greater than 100 parts per billion.\n\nUnfortunately, THMs are not the only pollutants likely to be found in your drinking water and which are not removed or affected by chlorination. These others include pesticides, poly-cyclic aromatic hydrocarbons, volatile hydrocarbons (also known as VOCs, volatile organic chemicals), and phenols. Researchers identified seventy VOCs (notably hydrocarbons\u2014see figure 4-1) that were formed during municipal treatment of drinking water; most of the hydrocarbons passed through the entire water treatment process with \"little change in concentration\"\u2014meaning they were available full strength at the typical kitchen faucet.\n\nLet's look at this more closely. What are we likely to find in our tap water if it is municipally treated, or perhaps regardless of its source? Here is a partial list: trace metals (lead, aluminum, cadmium, chromium), VOCs, THMs, organisms (coliform bacteria, worms, parasites, or protozoa, such as _Cryptosporidium_ and _Giardia 7)_, and numerous chemical contaminants\u2014at least fifty-nine identified contaminants in all. Seen from another angle, these are fifty-nine reasons to consider installing a competent water filter in your kitchen.\n\n**Figure 4-1. Toxic Contaminants Typically Found in Tapwater\u2014A Partial List**\n\n\"There is widespread potential for human exposure to disinfection byproducts (DBPs) in drinking water because everyone drinks, bathes, cooks, and cleans with water,\" stated G.A. Boorman of the National Institute of Environmental Health Sciences of Research Triangle Park in North Carolina in a 1999 review of THMs. Add to this flushing the toilet and washing clothes and dishes\u2014these activities can also introduce THMs into the indoor air of your home. It doesn't stop there either.\n\nIn fact, drinking chlorinated water is not necessarily the prime route for exposure to THMs. Taiwanese researchers measured the inhalation exposure for THMs of household residents during a day. They found that the people inhaled 26.4 mcg\/day during the shower, 1.56 meg during pre- and post-cooking activities, and 3.29 meg during cooking itself. This exposure is in addition to whatever amount the people directly ingested by drinking tap water, which the researchers estimated to be about the same amount, \"indicating that inhalation is an important pathway for THM exposure from drinking water.\"\n\nYou must additionally factor in the chlorine and chlorination disinfection products found in all the commercial beverages you drink, cups of tea or coffee you enjoy in restaurants, and the water used in hundreds of commercially prepared foods. Even though you may be filtering out the chlorine and its byproducts from your home drinking water supply (explained below), it doesn't mean the rest of the world is.\n\nThe principle becomes one of minimization; you probably will not be able to completely avoid chlorine and its byproducts in all the food and water situations you encounter, but you probably can _reduce_ your exposure by a great deal. As with many slow-acting toxic substances to which we are routinely exposed at low but constant doses, they have a cumulative effect. If you cut way back on the amount regularly entering your body, you greatly improve your chances that your body's own detoxification system will be able to handle this amount. You will get even better results if you help your liver out with specific liver-cleansing herbs (see Chapter 5).\n\nThere are several reasons why you should to try to minimize or eliminate your exposure to chlorinated water through drinking and from inhaling the fumes when you shower. Chlorine itself is believed to destroy polyunsaturated fatty acids and vitamin E in the body; it gets stored in the body's fatty tisues where it is slow to degrade; and it can also destroy beneficial intestinal bacteria that aid in digestion and in defense against pathogenic microorganisms. Regarding trihalomethanes, there are far more than four THMs, and they are for the most part unstudied, their toxicity unknown. At least six other THMs have been identified. It is known, however, that THMs can contribute to reproductive disorders, including miscarriage, birth defects, and various types of cancer, such as bladder and colon cancer.\n\n### _Reproductive Effects_\n\nLet's look at miscarriages first. California researchers announced in 1998 that they had developed strong evidence linking the consumption of chlorinated water with an increased risk of miscarriage or spontaneous abortion, based on a study of 5,144 pregnant women drawn from three counties and serviced by 78 municipal water companies. The women, whose average age was twenty-eight, reported their water consumption patterns during their first trimester some years earlier.\n\nWomen who drank at least five cups daily of cold chlorinated tap water with more than 75 ppm of THMs had miscarriages at a rate 15.7% higher (nearly double) the national average. Women who drank less than this amount or whose tap water had less than 75 ppm of THMs had miscarriages at the rate of 9.5% higher. The study also showed that eighteen percent of the women drank tap water with much higher THM levels, up to 157 ppm. In summary, women with a high THM exposure had a likelihood of spontaneous abortion 1.8 times higher than among women with a low exposure.\n\nThe results of this study reinforced conclusions published six years earlier in another evaluation of chlorination and women's health in California. In 1992, scientists at the California Department of Health Services studied 5,000 women in a single county (one of the three counties in the 1998 study). These women had a ten percent to fifty percent higher risk of spontaneous abortion than should be expected if they drank more tap water than bottled water during their first eight weeks of pregnancy. Women who drank at least six cups of chlorinated tap water per day had an average 2.17 times higher risk of miscarriage (the range of risk was 1.1 to 3.9).\n\nThree other studies published that same year (1992) confirmed the results, placing the miscarriage rate at twelve percent to fourteen percent from a four-cup daily consumption of chlorinated tap water. Researchers at the University of California at San Francisco studied pregnancies from 1980 to 1985 in four areas of one California county and found that among women who consumed no chlorinated tap water during their pregnancies (relying on bottled water), no birth defects were noted in 263 births, but for women who drank tap water, 4% of 908 births had defects. The scientists also found \"an overall excess\" of spontaneous abortions among women drinking chlorinated tap water during their first trimester.\n\nThe California Department of Health Services concluded in 1992 that results from four out of five studies of pregnancy outcomes and water consumption patterns \"suggest that women abstaining from tap water or [those] drinking bottled water during the first trimester of pregnancy may be at reduced risk of spontaneous abortion.\"\n\nFurther correlations were demonstrated by researchers at the School of Public Health at the University of North Carolina at Chapel Hill in 1998. They looked at the records of 1,893 live births in Colorado between 1990 and 1993; all the women had consumed water containing THMs during their pregnancies. The scientists found \"a weak association\" between THM exposure during the third trimester and low birth weight and \"a large increase in risk\" for low birth weight at the highest rate of THM exposure. The data was sufficient to indicate \"a potentially important relation\" between a pregnant woman drinking THM-contaminated tap water and the chances of birthing a baby with some kind of defect as a result of \"retarded fetal growth.\"\n\nA similar retrospective study in Halifax, Nova Scotia, revealed that women were 1.66 times more likely to experience stillbirths if their average THM exposure level during pregnancy was 100 mcg\/liter or greater compared to women who ingested only 0-49 mcg\/1 during the same time. The researchers, who studied birth records from 1988 to 1995, found little evidence for birth defects, but they did note \"chromosomal abnormalities,\" which is to say, defects in the infants' DNA. When toxic effects start showing up in chromosomes and genes, you are at the molecular level where cancer (and many other serious health disorders) begins (see page 148).\n\nOther recent research shows a connection between serious birth defects, such as neural tube defects (spina bifida, in which the spinal cord is not properly enclosed by bone), and total THMs consumed in tap water. The research, conducted in New Jersey, reported a doubled risk of neural tube defects among women with the highest THM exposure from drinking water.\n\nA study three years earlier in New Jersey showed the chance of neural tube defects was increased threefold when the pregnant woman was exposed to THMs at rates higher than 80 ppb. These researchers, part of the Agency for Toxic Substances and Disease Registry in the United States Department of Health and Human Services, looked at 80,938 live births and 594 fetal deaths during the period 1985-1988. They were able to tentatively link specific THMs with different birth defects.\n\nFor example, total THM ingestion was linked with infants born small for their gestational age, and with central nervous system and cardiac defects. Carbon tetrachloride intake was associated with low birth weight and various defects in the nervous system, heart, and mouth; benzene was linked to neural tube and cardiac defects. It is possible that chloroform (one of the four main THMs) disrupts the fetus' uptake of vitamin B12, which has been linked to neural tube defects.\n\nNorwegian researchers looked at three aspects of \"reproductive toxicity\" with respect to pregnant women ingesting chlorinated tap water. They found that out of 141,077 births in a two-year period (1993-1995), 1.8% of infants (2,608) were born with birth defects and there was a strong likelihood these birth defects were due in part to THMs. The scientists reported that women with a high chlorination exposure were 1.4 times more at risk of having a baby delivered with some kind of malformation, 1.26 times more likely to birth a baby with a neural tube defect, and 1.99 times more likely to birth an infant with urinary tract defects.\n\n\"This study provides further evidence of the role of chlorination of humic water [raw water containing organic matter] as a potential cause of birth defects, in a country with relatively low levels of chlorination byproducts,\" the researchers noted. In other words, because Norway is not so heavily chlorinated as other industrialized countries, the results are more meaningful, because the average person is routinely exposed to fewer THMs there than elsewhere.\n\nAmong the THMs, bromodichloromethane (BDCM) and chloroform are considered the two most prominent. In rat studies, both produced noticeable damage to the liver and kidneys twenty-four hours after ingestion. After forty-eight hours, BDCM produced \"more persistent liver toxicity\" than chloroform and was \"slightly more toxic\" to the kidneys as well. Bear in mind that the liver and kidneys are two of the body's prime detoxification, filtration, and toxin excretion organs, so if THMs make these organs toxic, their ability to remove other toxic substances from the body is impaired.\n\n### _Carcinogenic Effects_\n\nThe scientific research linking cancer and sustained THM intake through drinking chlorinated tap water is equally strong. In fact, evidence suggesting a link between THMs and cancer was suspected as early as 1974; since that time, at least eighteen studies have appeared strengthening the link between carcinogens in chlorinated drinking water and the onset of various types of cancer, especially bladder and colon cancer.\n\nA study by the National Cancer Institute in 1987 examined the water consumption habits of 2,805 men and women with bladder cancer and 5,258 cancer-free men and women drawn from ten different areas of the United States. They found that people who drank eight cups of chlorinated tap water for forty to fifty-nine years had a forty percent higher risk of getting bladder cancer than those who consumed less than this, or who drank unchlorinated water. Among those in the study, men and women who drank the highest volume of chlorinated tap water for sixty years had an eighty percent increased risk.\n\nBy some strange biochemical twist, nonsmokers had a far higher risk than smokers, even though tobacco smoke on its own can cause bladder cancer; nonsmokers drinking chlorinated tap water for 60 years had a 310% increased risk of bladder cancer. The researchers estimated that of the people studied, twelve percent of the bladder cancer cases could be blamed on chlorinated water; among the nonsmokers in the group, that number was twenty-seven percent.\n\n  * In 1982, researchers acknowledged that studies have \"clearly shown\" that most, possibly all, U.S. drinking water contains chemicals that have mutagenic or carcinogenic activity, meaning they can make healthy human cells change (mutate) or become poisoned (by carcinogens), initiating a cancer.\n  * In 1982, scientists studied cancer deaths in women from twenty-eight Wisconsin counties during the years 1972-1977 and found that colon cancer appeared to be \"related significantly\" to THM levels in drinking water, increasing the risk of colon cancer by 2.8 times.\n  * In 1987, scientists injected laboratory mice regularly for two years with bromodichloromethane, a THM found in chlorinated water. There was \"clear evidence\" of a cancer process in both males and females after two years, particularly in the kidneys and colon of both sexes and ovarian abcesses in the females. Of the four primary THMs, bromodichloromethane caused the \"widest spectrum\" of cancers in rodents, researchers stated.\n  * In 1993, researchers at the National Institutes of Health reported the results of their two-year study with laboratory rodents that had received daily amounts of three THMs, chlorine, and chloramine. They found that the THMs were carcinogenic to the liver, kidneys, and intestines of the rodents studied. They concluded that these are \"the chemicals of greatest concern\" in determining whether chlorinated drinking water is cancer causing.\n  * In 1995, researchers at the Department of Family and Community Medicine at the Medical College of Wisconsin at Milwaukee stated that the byproducts of water chlorination \"possibly\" account for 5,000 cases of bladder cancer and 8,000 cases of rectal cancer each year in the United States.\n  * In 1996, scientists at the Department of Community Health and Epidemiology at Queen's University in Kingston, Ontario, studied 696 cases of bladder cancer and correlated them with the patient's drinking water patterns over a thiry-five-year period. They found that those who drank chlorinated tap water regularly for thirty-five years or more had \"an increased risk\" of bladder cancer compared to those who consumed it for ten years or less. They also reported that those who had a THM intake of 50 mcg\/liter or more for 35 years had an increased risk as well. In other words, \"the risk of bladder cancer increases with both duration and concentration of exposure to chlorination byproducts.\" The general risk to the population of bladder cancer was thereby increased by fourteen percent to sixteen percent, making THMs \"a potentially important risk factor.\"\n  * In 1997, researchers at the School of Public Health at the University of Minnesota at Minneapolis announced the results of their study of 28,237 women from Iowa. Women who drank chlorinated tap water had an increased risk of colon and all types of cancers compared to women who did not drink chlorinated water.\n  * In 1998, researchers at the National Cancer Institute found that among 1,123 cases of bladder cancer compared to 1,983 controls (people without cancer but of the same age and gender) there was an \"overall association\" between chlorinated water intake and the risk of bladder cancer. Specifically, among men, if they were smokers and drank chlorinated tap water, this \"mutually enhanced\" their risk of getting bladder cancer.\n  * In 1998, other scientists at the National Cancer Institute studied 685 cases of colon cancer and 655 cases of rectal cancer, compared to 2,434 controls. They found an association between drinking chlorinated water over a sixty-year period and the incidence of rectal cancer, but not of colon cancer. However, they found that subjects who drank chlorinated water and had a low dietary fiber intake over a period of 40 years increased their risk of rectal cancer by 2.4 times compared to 0.9 times for those who had a high fiber intake and did not drink chlorinated water.\n  * In 1999, ingesting chlorination byproducts was positively linked with brain cancer. A study of 375 brain cancer cases in Iowa and the patients' lifetime water consumption patterns showed an increased risk of brain cancer with exposure to THMs.\n\nThe preceding brief review of some of the available scientific evidence makes it clear that it would be prudent to think about radically reducing your exposure to THMs found in chlorinated tap water. There are two practical steps you can take in your home: install a water filter for your drinking and cooking water; and install a shower and bath-faucet filter.\n\nObviously, using high quality bottled water is a viable solution to contaminated drinking water, an effective remedy that's easy to implement. There is a financial factor, however. Using bottled water for drinking purposes (not cooking) for a year in a two-person household (including the rental or amortized purchase of the dispenser) will cost you about the same as the onetime purchase of a mid-level water purifier for about $500. Using bottled water for cooking purposes adds to the expense. The only subsequent costs of a water purifier are cartridge replacements. In year two, you will start saving a significant amount of money on your water purification.\n\nAt any rate, since the option of using bottled water is already generally well known, there is no need to pursue it further here. Instead, we will focus on home water filtration units.\n\nAs elsewhere in this book, discussion of a specific product is not meant as an endorsement, nor is the listing of one or more products meant to be a definitive listing of what is available, or a ranking as to quality, or even an assurance of quality. Rather, specific products are mentioned to give you an idea of what is available, and a few places to begin your investigation of items best suited to your needs.\n\n### _Water Filters to Remove Chlorine_\n\nThere are many options here, from the inexpensive counter-top activated charcoal filter approach to expensive under-the-counter reverse osmosis units. You can spend $50 or $3000, depending on how thoroughly you want to remove tap water contaminants. Let's look at a few examples:\n\nINEXPENSIVE FILTRATION. The P\u016br line of water purifiers offers a series of pitcher, countertop, or under-sink filtration units in a modest price range, from about $25 to $100. The pitcher unit holds two gallons. The filter lasts about two months (forty gallons), and the unit removes _Giardia_ and _Cryptosporidium_ , and reduces the levels of lead, chlorine, copper, zinc, and asbestos. The faucet mounts reduce levels of some THMs, a broader list of VOC chemicals, and the microorganisms cited above. The countertop\/under-sink units remove the same substances, but have a longer filter life (200 gallons).\n\nCHARCOAL FILTRATION. An inexpensive countertop approach uses a charcoal (carbon) block filter and a separate spout to channel purified water. An example of this method is the TerraFlo CBLX, which uses a solid carbon cartridge (under-the-counter models are available, too) to filter out selected agricultural pollutants and reduce THMs, VOCs (forty-one different ones are cited by the manufacturer), mercury, lead, chlorine, and the levels of certain parasites. The filter lasts for 400 gallons, and the units sell for $100 or less.\n\nMICROFILTRATION. The Seagull IV X-l series of \"microfiltration\" water purifiers made by General Ecology of Exton, Pennsylvania, are countertop models (although they can be installed under the counter) that are effective against three types of water contaminants, according to the manufacturer. These include parasitic cysts, specific chemicals such as pesticides and VOCs, hazardous waste from various industries (such as iron), and THMs.\n\nThe systems employ the Structured Matrix technology that removes viruses, bacteria, protozoa, and many organic and inorganic chemicals at \"the highest 'purification' micron level (0.4 microns absolute),\" says the manufacturer. However, the unit does not remove fluoride (see below for a discussion of why fluoride should be filtered out of drinking water), which is a salt and part of a family not considered contaminants.\n\nThe unit accomplishes its purification through an ultrafine submicron filtration device that collects contaminants as small as 0.1 microns; the cartridge (which does not contain charcoal) needs to be replaced annually, or sooner, depending on use, but generally can handle from 1,000 to 6,000 gallons. The different Seagull models process from one to six gallons of water per minute, and weigh from four to twenty-four pounds. You can install them without a plumber, and they generally cost between $500 and $1000.\n\nAccording to the company's technical reports, the Seagull units are able to remove (to the level of being not detectable) seventeen out of twenty pesticide residues, THMs, and VOCs; and drastically reduce the levels of the other three pesticides. Out of thirteen listed microbiological water contaminants (mostly bacteria), the unit reduces levels of all to nondetectable amounts; it also significantly reduces the iron, cadmium, mercury, and lead content in the finished water.\n\nAnother water purification system that uses filtration is the Everpure line, which consists of eleven different units to choose from. Two\u2014S-200 and QC4-VOC\u2014remove or reduce the levels of many volatile organic chemicals (such as benzene and toluene) and many THMs as well as lead, chlorine, chloramines, molds and algae, _Giardia_ and _Cryptosporidium_ cysts, asbestos fibers, oxidized iron, manganese, and sulfides, as well as 99.9% of all particles 0.5 microns and larger (which the other 9 units do also). The S-200 has a water flow rate of 0.5 gallons per minute and a cartridge life of 300 gallons, while the QC4-VOC has the same flow rate but a cartridge life of 500 gallons; both units are installed under the counter.\n\nREVERSE OSMOSIS. This approach employs a semipermeable membrane that removes particles, salts, and dissolved contaminants, as if molecule by molecule, from raw tap water. In reverse osmosis (RO), water first passes through a 5-micron particle prefilter that takes out large particles of dirt and debris; then the water is squeezed under great pressure through microscopic pores to eliminate about ninety-nine percent of its impurities; finally, any remaining odors or objectionable tastes are removed by a carbon post filter.\n\nOne example of an RO device is the PPW Home Reverse Osmosis System, made by Pure-Pro Water Corporation in Mokena, Illinois. According to the manufacturer, this unit removes ninety-five percent to ninety-nine percent of all dissolved mineral and chemical contaminants from raw tap water. Specifically, the unit can remove ninety percent of the fluoride, ninety-seven percent of detergents, ninety-seven percent of herbicides, insecticides, PCBs, lead, and strontium (a radiation product), among others, including ninety-eight percent of THMs (especially chloroform) and other organic toxic chemicals. The unit can produce about fifty to seventy gallons of RO water per day, and holds three to four gallons at a time in a special tank.\n\nThe RO approach has advantages and disadvantages: it removes salts (the technology was originally designed by the U.S. Navy to remove salt from ocean water to make it drinkable) which means it catches fluoride (undesirable, see below), but also calcium, sodium, and potassium (essential nutrients); it is not always well suited to remove biological pathogens, but it is well equipped to filter out VOCs. If the RO membrane is _small_ enough\u2014that is, smaller than bacteria, viruses, or various parasites\u2014it will remove most pathogenic microorganisms from the water it treats.\n\nOther advantages of RO are daily volume capacity and the thoroughness with which it removes finely dissolved contaminants from the water. According to the manufacturer, once maintenance and water costs are factored in, the unit produces pure water for about five cents a gallon. However, there is a certain degree of water waste with RO. To purify one gallon of water, it will flush it with several gallons (as many as six) of raw water, and these extra gallons get sent back down the drain. So you are paying for water that gets recycled down the drain.\n\nULTRAVIOLET PURIFICATION. In this approach, an ultraviolet (UV) lamp sterilizes tap water and removes or reduces the levels of chlorine, THMs, VOCs, microorganisms, and selected chemicals. For example, the Sun-Pure Water Purifier uses a five-step program to clean the water: carbon block filters remove chlorine and many THMs and VOCs; a lead sorbent matrix mixed with the carbon block filter removes heavy metals; activated tricalcium phosphate removes fluorides; granular activated carbon removes bad tastes and odors as well as chemicals; and the ultraviolet sterilization chamber deactivates the various microbial life forms that could cause disease. The filter cartridges last for 1,300 gallons, and the UV lamp needs replacement annually. The system delivers water at the rate of 30 gallons\/hour (or 0.5 gallons\/minute), and at a cost of $0.05\/gallon daily (about $400 for the unit), according to the manufacturer.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor P r units:\n\nP r Recovery Engineering, Inc.,\n\n9300 North 75th Avenue,\n\nMinneapolis, MN 55428;\n\ntel: 800-787-5463;\n\nwebsite: www.purwater.com.\n\nFor TerraFlo CBLX:\n\nGlobal Environmental\n\nTechnologies, Inc.,\n\nP.O. Box 8839,\n\n1001-1003 S. 10th Street,\n\nAllentown, PA 18105;\n\ntel: 610-821-4901;\n\nfax: 610-821-5507;\n\nemail: getwater@terraflo.com;\n\nwebsite: www.terraflo.com.\n\nOr from Real Goods,\n\n200 Clara Street,\n\nUkiah, CA 95482;\n\ntel: 800-762-7325;\n\nwebsite: www.realgoods.com.\n\nFor more information about\n\nSeagull IV X-l series, contact:\n\nGeneral Ecology, Inc.,\n\n151 Sheree Blvd.,\n\nExton, PA 19341;\n\ntel: 800-441-8166 or 610-363-7900;\n\nfax: 610-363-0412;\n\ne-mail: info@generalecology.com;\n\nwebsite: www.general-ecology.com.\n\nAlso available from:\n\nNirvana Safe Haven,\n\n3441 Golden Rain Road, Suite 3,\n\nWalnut Creek, CA 94595;\n\ntel: 800-968-9355 or 925-472-8868;\n\nfax: 925-938-9019;\n\ne-mail: dailya@nontoxic.com;\n\nwebsite: www.nontoxic.com.\n\n### _Shower Filters to Remove Chlorine_\n\nThe idea behind a shower filter is to remove a fair amount of chlorine and selected heavy metals, and sometimes radon and certain bacteria (such as _Cryptosporidium_ ) from the incoming water, usually by way of an activated charcoal filter. The device is installed in place of the conventional shower faucet. Shower filters do not remove THMs, only chlorine. Let's briefly review a few typical products:\n\nThe Shower Filter from Paragon Water Systems, for example, uses a five-stage filtration system, including activated carbon and a KDF filter. The KDF uses an electrochemical process to reduce the water content of various organic and inorganic elements that affect the water's taste, including some pesticides, organic chemicals, and hydrogen sulfide. Water passes through very fine micron filters before coming out as a spray.\n\nAnother version of the same approach is called the Aqua-Stream Shower Filter. Using a combination of a charcoal filter and a KDF filter, this unit processes 2.2 gallons of water per minute; its cartridge lasts up to two years; and the device weighs four pounds and sells for $150 or less.\n\nA third version, offered by Nirvana Safe Haven, is called the KDF Chlorine-free Shower Unit. The company states that one filter will filter about 45,000 gallons of water (or last about eighteen months), and will remove chlorine (converting it to nontoxic zinc chloride), eradicate algae and fungi, control bacterial growth, and reduce hydrogen sulfide and factors contributing to water hardness.\n\nThe Sprite Slim-Line Shower Filter from Ace Pump Corporation uses a noncarbon-based filtration media (called Chlorgon, specifically designed for the needs of the shower, says the company) to remove both free and combined (bonded with other elements) chlorine, as well as odors, heavy metals, dirt, and sediment from shower water. The cartridge needs to be replaced about every six months.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Everpure S-200 and QC4-VOC:\n\nEverpure, Inc.,\n\n660 Blackhawk Drive,\n\nWestmont, IL 60559;\n\ntel: 630-654-4000;\n\nfax: 630-654-1115;\n\ncustomer service: 800-942-1153;\n\nwebsite: www.everpure.com.\n\nFor PPW Home Reverse\n\nOsmosis System:\n\nPure-Pro USA Corporation,\n\n19425 Everett Lane,\n\nMokena, IL 60448;\n\ntel: 800-993-2933;\n\ne-mail: info@purepro.net;\n\nwebsite: www.purepro.net.\n\nFor Sun-Pure Water Purifier: see Real Goods info on page 154.\n\n### _Get the Chlorine Out of Your Swimming Pool_\n\nRoutine or long-term exposure to chlorinated water in swimming pools can be hazardous to your health. Competitive swimmers have been shown to absorb toxic levels of chlorine and THMs during their training; similar chlorine absorption patterns have been observed in recreational swimming, especially among children. \"This exposure comes from breathing chloroform-laden air near the water's surface and absorbing this and other compounds through the skin.\"\n\nIf you own a swimming pool and want an alternative to chlorine to keep the microorganisms out of your water, a device called the Floatron may be the solution. Using solar ionization, this is a small, floatable device that employs solar power to generate a low electric current; the mineral ions produced (and introduced into the water) effectively stop the growth of algae and bacteria in your swimming water. Essentially, the unit produces mineral water as a way of controlling microbial growth, and it cuts the need for any chlorine by eighty percent. The remaining twenty percent of the normal chlorine (or other oxidizers such as bromine or hydrogen peroxide) amount is required to control water clouding due to suntan lotion, body fluids, and dust accumulation in the water. The Floatron can handle pool volumes of up to 40,000 gallons and sells for about $300.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more information about Shower Filter, see:\n\nPara\u01f5on Water Systems,\n\n14001 63rd Way North,\n\nClearwater, FL 33760;\n\ntel: 800-288-9708 or 727-538-4704;\n\nwebsite: www.paragonwater.com;\n\nalso: www.resourcewater.com.\n\nFor Sprite Slim-Line Shower Filter:\n\nAce Pump Corporation,\n\n57 West 21st Street,\n\nNew York, NY 10010;\n\ntel: 888-242-4500; fax: 212-989-2260;\n\nwebsite: www.acepump.com.\n\nFor Aqua-Stream Shower Filter:\n\nNatural Solutions Environmental, Inc.,\n\n4238 North Arlington Heights Road, #113,\n\nArlington Heights, IL 60004;\n\ntel: 847-577-7000 or 847-675-9200;\n\nwebsite: www.naturalsolutionsl.com.\n\nFor KDF Chlorine-free Shower Unit:\n\nNirvana Safe Haven,\n\n3441 Golden Rain Road,\n\nSuite 3, Walnut Creek, CA 94595;\n\ntel: 800-968-9355 or 925-472-8868;\n\nfax: 925-938-9019;\n\ne-mail: dailya@nontoxic.com;\n\nwebsite: www.nontoxic.com.\n\nThere are at least three other nonchlorine alternatives for cleansing your swimming pool water. These include ozonation (injecting ozone gas [a form of oxygen] into pool water; a small amount of chlorine will still be needed); ultraviolet light (supplemented with a small dose of chlorine); and biguanide, or polyhexamethylene biguanide (Baquacil), a safe and nontoxic chemical disinfectant.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #9: Don't Fluoridate Your Teeth_ \u2014 _Your Body Doesn't Need Fluorides_\n\nWe find another water-related toxicity issue with fluorides. The matter of adding fluoride to municipal water supplies is charged with the same controversy and lack of understanding of the biological consequences as is the chlorination issue. Water fluoridation has been touted for decades as beneficial in preventing dental caries (cavities) in children, but the research supporting this claim is spotty and not definitive. Further, a fair amount of evidence that fluoridation produces minor health consequences such as dental fluorosis (see below) and contributes to serious health disorders such as osteoporosis and cancer has received insufficient attention from medical and water authorities, and is not widely known among the public.\n\nThe recommendation here is straightforward: if possible, reduce or eliminate your intake of fluoridated tap water and fluoridated toothpaste. There is good reason to regard fluoride in the same way as chlorine\u2014a toxic substance best avoided.\n\nToday an estimated 250 million people worldwide and some 130 million Americans in 9,600 communities (or almost fifty percent of the U.S. population) drink fluoridated water, including residents of 41 of the country's fifty largest cities. According to the Centers for Disease Prevention and Control (CDC), 62% of the U.S. population (as of 1996) received fluoridated water, with a target of seventy-five percent set for the year 2000; the CDC planned to achieve this increase of another 30 million fluoridated Americans by way of \"substantial public education and advocacy\" primarily focused on the dental benefits. Fluoridation is an American obsession, as only two percent of Europeans are forced to drink fluoridated water, and most of that takes place in England (ten percent of the population) and Ireland (sixty-six percent).\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Floatron:\n\nReal Goods,\n\n200 Clara Street,\n\nUkiah, CA 95482;\n\ntel: 800-762-7325;\n\nwebsite: www.realgoods.com.\n\nOr:\n\nFloatron Factory Sales and Service,\n\nP.O. Box 51000,\n\nPhoenix, AZ 85076;\n\ne-mail: info@floatron.com;\n\nwebsite: www.floatron.com.\n\nFor a source of nonchlorine pool filtration (and shower filters and water filters):\n\nThe Cutting Edge Catalog,\n\nBefit Enterprises, Ltd.,\n\nP.O. Box 5034,\n\nSouthampton, NY 11969;\n\ntel: 800-497-9516 or 631-287-3813;\n\nfax: 631-287-3112;\n\ne-mail: cutcat@cutcat.com;\n\nwebsite: www.cutcat.com.\n\nSome states and communities in the United States still do not routinely fluoridate their public water, so if you live in such a place, count yourself lucky. The trend, however, is to fluoridate all water sources in the United States, typically at the rate of 1 part fluoride per million parts water, or 1 ppm, although up to 4 ppm is allowed. Why such an urgency to fluoridate?\n\nWater fluoridation in the United States began in 1945 in Grand Rapids, Michigan, when manufacturers of phosphate fertilizer and aluminum found a new place to dump their abundant supplies of their hazardous byproduct: fluoride. Even though a fifteen-year scientific study had been set in place to determine the safety (or not) of fluoridated water, it was suspended after only eighteen months so that widespread fluoridation could begin.\n\nPreviously, fluoride stores had been disposed of in small amounts as components in insecticides and rat poison; then scientists came up with evidence that fluoride ingestion may be related to a lower incidence of dental cavities. The first proposal that the nation's water be fluoridated (1939) came not from a dentist, however, but from a scientist employed by the aluminum industry. \"Now instead of paying for the expense of disposing of their toxic waste products, companies could make money by selling them to municipal water authorities, turning the public into a highly cost-effective toxic waste dump for industry's excess fluoride.\"\n\nAside from the human health impact of consuming fluoridated water, there is a negative environmental impact. To fluoridate water for 100 million people, you need to dump about 20,000 tons of fluoride into municipal water supplies each year. The human body stores about fifty percent of its fluoride intake from fluoridated water in the teeth and bones. Each year, the remaining 10,000 tons of fluoride gets flushed down the toilet, and eventually enters the environment as yet another source of toxic pollution. It is quite likely we get some of this excreted fluoride back (and into our bodies again) when we drink tap water and eat conventionally raised produce and prepared foods. This recycling of fluoride was known as early as 1964, when a study of the fluoride levels in the sewage of fifty-six California cities showed levels equal to or higher than the permissible levels in fluoridated water.\n\n**_The recommendation here is straightforward; if possible, reduce or eliminate your intake of fluoridated tap water and fluoridated toothpaste. There is good reason to regard fluoride in the same way as chlorine\u2014a toxic substance Best avoided._**\n\nIs there evidence supporting the claim that fluoridated water reduces cavities in children? Yes, to some extent, the research supports in a limited sense the basic claim, but other research contradicts it and, as we will see below, other health risks associated with fluoride may far outweigh the modest benefits.\n\nIt is instructive that when fluoridation stops, dental health does not necessarily plummet. A town in Finland stopped fluoridating its water after doing so for 33 years. Scientists found, after examining the dental status of 1,325 children (age 3 to 9) before and after cessation of fluoridation, that there was no increase in dental cavities three years after cessation. In Durham, North Carolina, fluoridation was halted for eleven months in 1990, enabling scientists to assess the effect of fluoride on children's teeth. Examining the dental records of 1,696 children (kindergarten to grade five), researchers found the temporary cessation of fluroidation \"had little effect on caries\"\u2014in other words, there was no measurable increase in cavities.\n\nIn 1990, public health officials in La Salud, Cuba, studied the effects on children's teeth of no longer fluoridating public water (at 0.8 ppm). Seven years after cessation, the level of dental caries for children six to nine was low, and had not risen; for ten to eleven year olds, the numbers had decreased; and in twelve to thirteen year olds, there was a \"significant decrease,\" and the percentage of children this age who were free of cavities had increased from thirty-three percent (during fluoridation) to fifty-five percent, under no fluoridation. However, researchers concluded the fact that schoolchildren did a fluoride-based mouth rinse fifteen times a year at school might have contributed to the improved dental status.\n\n### _Fluoride Can Produce Dental Fluorosis_\n\nThe evidence that ingesting fluoridated water leads to dental fluorosis is compelling. In the early stages of this condition, the enamel on teeth gets mottled, opaque, and sometimes stained. Later on, the teeth pit and get brittle, and they may chip, and develop a yellow, brown, or even black appearance.\n\nThe incidence of fluorosis increases when the concentration of fluoride in public water supplies exceeds 2 ppm; when it reaches 8 ppm, systemic fluorosis can start to appear, involving a bony overgrowth, neurological problems, and even arthritis. Researchers at the College of Dentistry at the University of Iowa concluded in 1999, \"Several studies indicate that primary-tooth fluorosis can be prevalent and severe in areas of very high water fluoride concentrations.\" Typically, the fluorosis happens on primary molars (premolars), but it can also affect incisors.\n\nMore specifically, a study by the School of Public Health at the University of Michigan at Ann Arbor found that fluoride was able to reduce dental cavities best at levels of up to 0.7 ppm, but that at these levels, 13% to 21% of schoolchildren developed dental fluorosis as a result. At fluoridation levels higher than 0.7 ppm, the incidence of dental fluorosis soared to between 29% and 41% (at 1.2 ppm fluoride). Another study showed that children and teenagers (8-16 years old) who consumed fluoridated water containing from 0.3 ppm (considered a low level) to 1 ppm (considered optimal) of fluoride had fluorosis on an average of 15% of their teeth surfaces.\n\nResearchers at Harvard School of Dental Medicine in Boston, Massachusetts, found that in children aged one to seven who had been routinely exposed to fluoridated water, sixty-nine percent had at least \"very mild fluorosis\" while thirteen percent had \"moderate to severe\" fluorosis. In a study of 233 \"optimally fluoridated\" children, aged 10-14, researchers at the University of Connecticut Health Center at Farmington found a \"strong association\" between the use of fluoride toothpaste, fluoride supplement, and powdered infant formula (containing fluoridated water) and mild to moderate fluorosis. Norwegian scientists reported that fourteen percent of children aged five to eighteen who had been lifelong consumers of fluoridated water at a low level had dental fluorosis, while seventy-eight of those who had fluoride at a moderate to high level had the problem.\n\nThe age at which the body is subjected to fluoridated water is important to the onset of fluorosis, researchers at the University of Chile at Santiago found. Children under sixteen months of age who had fluoridated water by way of powdered milk formula were twenty times more likely to develop fluorosis than those who were exposed later; children aged sixteen to twenty-four months were four times more likely. Norwegian researchers came to the same conclusion, stating that \"early mineralizing teeth (central incisors and first molars) are highly susceptible to dental fluorosis if exposed to fluoride\" in the earliest months of life, from birth to twelve months. The fluoride apparently interferes with mineral absorption and utilization needed to produce strong enamel on the teeth.\n\n### _Fluoride Can Contribute to Cancer_\n\nEvidence continues to accumulate linking fluoride ingestion (through tap water, toothpaste, and supplements) with bone, skin, joint, kidney, mouth, and uterine cancer. According to one estimate, fluoride may be linked to as many as 10,000 cancer deaths annually in the United States, but a former chemist with the National Cancer Institute estimated that fluoride caused 61,000 cases of cancer in 1995 and will cause 90,000 cases annually by the year 2015. Researchers at the School of Medicine at the University of the Ryukyus in Okinawa, Japan, reported that, based on their study of twenty fluoridated municipalities, \"a significant positive correlation\" was observed between fluoride concentration in the drinking water and uterine cancer among women drinking it.\n\nEven at supposedly \"safe\" and \"acceptable\" doses of 1 ppm, fluoride can initiate a cancer process (as demonstrated in laboratory mice, whose tumor growth rate increased by twenty-five percent), and it can synergistically increase the carcinogenicity of other toxic substances. One study of cancer rates over a thirty-year period as seen in ten fluoridated cities versus ten nonfluoridated cities showed that after seventeen years of fluoridation, the cancer rate grew by ten percent, mostly in people over age forty-five. At least eight other scientific studies present evidence linking fluoridated water with the onset of bone cancer.\n\n### _Other Health Problems Associated with Fluoride_\n\nFluoride exposure can disrupt the production of collagen in bones and lead to the breakdown of collagen in bones, muscles, skin, cartilage, the lungs, kidney, and trachea. Fluoride depletes the energy supplies of white blood cells, key immune system agents needed to destroy foreign cells; it can also cause premature body aging. Fluoride can also confuse the immune system and cause it to \"attack\" the body's own tissues. Further, it has been shown to depress the activity of the thyroid gland in the neck, an endocrine gland that regulates metabolism and the body's internal temperature.\n\nFluoride consumption may also contribute to the risk of bone fractures in males and females of all ages, and to hip fracture in the elderly, even though, paradoxically, it seems to promote bone density. An English study of 914 cases of hip fracture among the elderly found some degree of risk associated with lifelong fluoridated water intake, at levels ranging from 0.15 to 1.79 ppm.\n\nSince 1986, eight peer-reviewed medical studies have confirmed an increase in the cases of hip fractures in fluoridated communities, ranging from 40% to 100%, depending on the age of the subjects and the fluoride concentration in their water. One study reported that for women seventy-five and older consuming fluoridated water, the risk was double that of women in nonfluoridated communities. Another study stated that the risk of hip fracture increases as the fluoride level climbs above 0.11 ppm.\n\nScientists at the University of Technology in Sydney, Australia, stated in 1997 that their review of the scientific data \"reveals a consistent pattern of evidence\u2014hip fractures, skeletal fluorosis, the effect of fluoride on bone structure, fluoride levels in bones and osteosarcomas [bone cancers]\u2014pointing to the existence of causal mechanisms by which fluoride damages bones.\" These scientists further pointed out that research supports the view that ingesting fluoride, by drinking fluoridated water, has \"negligible\" benefit; what benefit does accrue to the teeth comes from using fluoride lozenges and mouth rinses in which the chemical has direct contact with the teeth.\n\nSustained fluoride ingestion can cause widespread toxicity. Laboratory rats were fed differing amounts of sodium fluoride over a period of ninety-nine weeks. Scientists observed a thirty percent drop in weight and found evidence of fluoride toxicity in the teeth, bones, and stomach; the incidence and severity of these symptoms were directly related to the dose and duration of exposure.\n\nFluoride appears to have deleterious effects on the brain. One critic of fluoridation, referring to the fact that many children use fluoridated toothpaste, queried: \"Brushing your brain with fluoride?\" Evidence now links fluoride with IQ deficiencies in children, and suggests fluoride may produce nerve cell and brain damage _in utero_ , as the mother passes the fluoride on to the fetus when she consumes fluoridated water. Children living in a low-fluoride area had higher IQs than comparable children in a high-fluoride area. Further, low levels of fluoride given to test animals produced damages to brain tissue similar to those changes seen in the brains of humans with Alzheimer's and other forms of dementia.\n\nIn fact, a study with rats showed that the presence of fluoride in the body made aluminum (in the tissues already) more available and more capable of entering the brain. Similarly, fluoride makes lead more \"bioavailable\" in the body, and scientists have demonstrated that children drinking fluoridated water have elevated lead levels in their blood.\n\nVarying levels of fluoride in drinking water (3-11 ppm) can affect the central nervous system and brain without producing noticeable physical malformations. Instead, it can affect functioning, producing symptoms such as attention deficit disorder when ingestion was 100 ppm from sublingual drops. This may seem like a high exposure, but it isn't an unlikely one, because toothpastes often contain 1000-1500 ppm of fluoride, and mouth rinses 230-900 ppm.\n\nAccording to toxicologist Phyllis Mullenix, Ph.D., of the Forsyth Research Institute in Boston, Massachusetts, fluoride accumulates in brain tissues. The central nervous system is \"vulnerable\" to fluoride, and the effects on behavior depend on what age you are when exposed. In other words, not only can fluoride negatively affect mineralizing tissues (as in dental fluorosis in very young people), but it seems to negatively affect the developing brain as well.\n\nDr. Mullenix found evidence of a \"generic behavioral pattern disruption\" in rats dosed with fluoride, and that these disruptions were indicators of likely muscle and coordination problems, IQ deficits, and learning disabilities in humans due to fluoride intake. Dr. Mullenix also reported that the severity of the effect fluoride had on the brain depended on the section of the brain in which it was concentrated. Should we be surprised? Sodium fluoride, which is routinely added to water supplies across the United States, is listed with the Environmental Protection Agency as a rat poison.\n\nAll you have to do is brush your teeth in fluoridated water with a fluoridated toothpaste and rinse your mouth with a fluoridated mouthwash and you will get the high exposure\u2014every day. Plus you need to add to this fluoride ingestion from ambient sources, such as air pollution. At least 155,000 tons of industrial fluoride are released into the atmosphere every year by factories (from production of or involving iron, steel, aluminum, copper, lead, and zinc), and the level of emissions of fluoride into lakes, oceans, and rivers has been placed at 500,00 tons annually. In 1991, the United States Public Health Service estimated that a 110-pound adult living in an \"optimally fluoridated\" city had a total daily fluoride intake, from all sources, as high as 6.6 milligrams. In 1997, the Environmental Protection Agency estimated that the typical American consumed five times more fluoride in that year than in 1971, and this was from foods and beverages only, not fluoridated tap water.\n\nSTEAM DISTILLATION. One technology known to effectively remove fluoride from tap water is called steam distillation. Two others that also perform this function are reverse osmosis and deionization. According to Waterwise, a company that manufactures three types of steam water distillers, the units also remove various heavy metals, selected THMs and VOCs, radon, selected microorganisms including viruses, and most minerals.\n\nThe process is simple to describe: Water is heated in a chamber where it turns to steam; this vapor rises, leaving impurities (dissolved solids, salts, heavy metals) behind in the boiling chamber. The vapor then condenses into water in another chamber, percolating through a coconut shell carbon filter, where more of the VOCs are removed.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about Steam\n\nDistillation Models 4000, 7000, and 9000:\n\nWaterwise, Inc.,\n\nP.O. Box 494000,\n\nLeesburg, FL 34749;\n\ntel: 800-874-9028 or 352-787-5008;\n\nfax: 352-787-8123;\n\nwebsite: www.waterwise.com.\n\nFor Fluoride-X:\n\nNirvana Safe Haven,\n\n344l Golden Rain Road, Suite 3,\n\nWalnut Creek, CA 94595;\n\ntel: 800-968-9355;\n\nfax: 925-938-9019;\n\ne-mail: daliya@nontoxic.com;\n\nwebsite: www.nontoxic.com.\n\nSee also: Environmental\n\nManagement, Inc. (Fluoride-X's manufacturer),\n\n8680 Miralani Drive, Suite 121,\n\nSan Diego, CA 92126;\n\ntel: 858-566-4522;\n\nfax: 858-566-9521.\n\nSteam distillation is a much slower process of preparing high-quality drinking water; it takes about four hours to produce one gallon (Model 9000) at a cost of about $0.25 per gallon, or 24 hours to make 9 gallons (Model 7000). Obviously, this approach will be suitable only if you want a limited amount of high-quality drinking water, and don't plan to use it for cooking as well.\n\nSPECIALTY FLUORIDE FILTER. A special filter called Fluoride-X can be attached to your present drinking water inflow or can be used in conjunction with a water filter. The Fluoride-X (which sells for about $100) is a specialty cartridge containing the ABA2000 series of aluminum oxides that remove fluoride salts from the point-of-use water. According to the unit's distributor, the filter lasts for about two years' worth of household water filtration.\n\n## HEALTHY LIVING SPACE DETOXIFIER #10: Cleanse Your Produce of Toxic Chemicals\n\nAs profoundly important and as rich with impact as your choice in water consumption is your choice of food. As a consumer, you have the option, every time you buy a food, of selecting one that is conventionally produced or one that is organic and\/or natural or, if a prepared food, contains organically produced ingredients. Either way, your choice makes a powerful difference, both to your physical health and to the well-being of the planet. Thinking green is good for your cells as well as for the planet's ecosystem.\n\nIf you cannot find organic produce (or cannot afford it\u2014admittedly, it costs more to get chemically clean foods), yet wish to eat fresh, _clean_ fruits and vegetables, there are some simple measures you can take to cleanse your produce of some of their agricultural pollutants before you consume them. Even if your foods are organic, there is no assurance that tap water will be sufficient to cleanse them of any chemical contaminants they might carry on their surface, picked up in transit from organic field to your kitchen. For this reason, cleaning your produce is a good practice.\n\nConceptually, using a produce rinse is like washing your hands before eating. Further, the rinse can remove germs (pathogenic bacteria, fungal spores, molds, parasites, pinworms) introduced to the produce during processing and handling, as well as chemical contaminants. Washing produce with water alone, or soap and water alone, is not sufficient to remove pesticide residues; this is largely because many pesticides are not water soluble. You need something stronger.\n\n**_If you cannot find organic produce (or cannot afford it_ \u2014 _admittedly, it costs more to get chemicatfy clean foods), yet unsh to eat fresh_ , clean _fruits and vegetables, there are some simple measures you can take to cleanse your produce of some of their agricultural pollutants before you consume them._**\n\nBe careful, however. The availability of nontoxic rinses to cleanse pesticides and residues from produce (thereby making it relatively easy to de-pollute fruits and vegetables) should not reduce our consumer pressure to have the use of pesticides in agriculture curtailed as much as possible.\n\n### _Bleach Your Produce_\n\nIt may surprise you, but an effective method for ridding produce of most of its surface layer of pesticide residue is washing it with conventional bleach, available as Clorox. This approach was originally publicized by Hazel Parcells, N.D., D.C., Ph.D., a natural healing expert who lived to 106 by following her own health protocols, including this one.\n\nAccording to Dr. Parcells, bleach is \"the most effective food cleanser.\" However, household bleach does not mean chlorine; it contains no more free chlorine than table salt (sodium chloride) does. The sodium hypochlorite in Clorox (present in a 5.25% solution) acts as a natural oxygenator, capable of neutralizing fungi and bacteria and removing industrial chemicals. Dr. Parcells called her approach the Parcells Oxygen Soak, and she (and her many patients) used it successfully for forty years to cleanse and even revitalize produce.\n\nAdd one tablespoon of original Clorox bleach to a gallon of nonfluoridated and nonchlorinated water. Separate your foods into different groups, as each group will get a soak of a different length. If you have just returned from shopping, this would be a convenient time to process your produce. Immerse leafy vegetables in the Clorox bleach solution for five to ten minutes; root and heavy fiber vegetables and thick-skinned fruits get ten to fifteen minutes; thin-skinned berries, five minutes; mediumskinned fruits, ten minutes; citrus and bananas, fifteen; eggs, twenty to thirty minutes; thawed meats, ten minutes\/pound; frozen meats, fifteen to twenty minutes\/pound. After the Clorox rinse, place the produce or food item in a fresh water rinse for five to ten minutes; then dry and prepare for storage in the refrigerator, says Dr. Parcells. This last water rinse oxygenates the produce, giving them a new burst of vitality and color.\n\n### _Organiclean Fruit and Vegetable Wash_\n\nThis is a nonsoap spray that comes in an eight-ounce plastic bottle and is made from natural ingredients including coconuts, bilberry, sugar cane, maple sugar, and oranges, lemons, and other citrus fruits. Using other natural cleaning agents (called \"biosurfactants\"), the product \"creates a slippery surface and 'lifting action' for better removal of surface pesticides, chemicals, dirt, and other inorganic pollutants,\" says its manufacturer. All you have to do is spray and rinse off.\n\n### _Fit_\n\nThis is a widely advertised natural rinse solution that contains purified water, vegetable-derived oleic acid and glycerol, ethyl alcohol (from corn), potassium hydrate (from minerals), baking soda, citric acid (from corn starch and molasses), and distilled grapefruit oil. With Fit, you can either soak (broccoli, cauliflower, grapes, lettuce, snow peas) or spray (fruits, carrots, cucumbers, corn, onions, tomatoes) your produce. If you spray, you need to rub the produce for about twenty seconds, or if the produce is hard to rub, let it sit with the Fit spray on its surface for a few minutes; in either case, then rinse with fresh water. According to the manufacturer (Procter & Gamble), there is no aftertaste or smell on the cleansed produce.\n\n### _Mom's Veggiewash_\n\nThe product manufacturer does not provide a list of ingredients, but assures consumers it is made of safe, inert, nontoxic, biodegradable liquid surfactants that contain no animal products or petroleum derivatives. It is based on natural vegetable oils instead. According to Earth Partnership, the manufacturer, Mom's Veggiewash can remove ninety-four percent of oil-based chlorinated hydrocarbon pesticide residues, such as the organochlorines from Endosulfan. Endosulfan is found on \"practically every vegetable and fruit\" in the United States, according to company information sheets.\n\nResearch first published in 1965 demonstrates how washing produce with a \"surfactant\" can drastically reduce the pesticide residue content. Untreated green beans had 31 ppm of toxaphene; after being washed with water, the level was 24.9 ppm; after being washed with 0.1% polyether alcohol, it had dropped to 13 ppm; and after being washed in a 1% nonionic surfactant solution (functionally similar to Mom's Veggiewash, which uses a 5% nonionic surfactant solution from plant sources), the pesticide level was down to 12 ppm, or almost 2.5 times reduced. The results were even more striking with mustard greens. They started off having 233 ppm of toxaphene, then after water washing, 100 ppm; after the alcohol, 23 ppm; and after the surfactant, 20 ppm.\n\nWith this product, place the produce in a colander and spray. It is recommended to treat even those fruits and vegetables which have skins you will not be eating, such as citrus, bananas, watermelon, cantaloupe, and the like, as the contaminants found on their skin can get into your body through peeling the fruit, followed by eating it without washing the hands again. After spraying, let the produce sit for a few minutes, then rinse with clean water.\n\nSome produce may be waxed (such as cucumbers, apples, peppers, tomatoes). In that case, douse the produce in warm water, spray with Veggiewash, brush briskly with a vegetable brush, and rinse. If you see a dusty powder or scale form on the produce, this is a wax residue, and you need to repeat the cleansing.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Organiclean Fruit and Vegetable Wash, contact:\n\nOrganiclean E-Commerce Division,\n\n10877 Wilshire Blvd., Suite 1200,\n\nLos Angeles, CA 90024;\n\ntel: 888-VEGWASH;\n\ne-mail: cservice@organiclean.com;\n\nwebsite: www.organiclean.com.\n\nFor Fit, see:\n\nwebsite: www.cleanproduce.com.\n\nFor Mom's Veggiewash:\n\nEarth Partnership, Inc.,\n\nP.O. Box 31,\n\nOrange City, IA 51041;\n\ntel: 800-259-7006;\n\nfax: 712-737-3714;\n\ne-mail: info@veggiewash.com;\n\nwebsite: www.veggiewash.com.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #11: Whenever Possible, Eat Organic Foods_\n\nChapter 1 acquainted you with the perils of conventionally grown food in terms of pesticide residues, industrial pollutants, processing additives, and other contaminants (see the section entitled \"Tainted Food\u2014Glutamates, Additives, Antibiotics, and POPs\"). Avoiding further exposure to these toxic substances through your foods alone is sufficient reason to consider eating only organically produced foods.\n\nHowever, there are three additional, perhaps even more persuasive reasons, for going organic: genetically modified foods; irradiated foods; and bovine growth hormone (BGH) in dairy products. In most cases, foods containing these substances are not labeled as such.\n\nThus, at present, only if you use foods that are certified and labeled organic can you avoid the health hazards of these three obnoxious, adulterative processes, which are now routinely done to foods in the American marketplace with the endorsement of U.S. government \"health\" authorities. By some bizarre inversion of its public health protection mandate, the FDA does not require food manufacturers (or growers) to label foods or products with the information that they contain genetically modified organisms, have been irradiated, or have traces of bovine growth hormone.\n\n### _Why You Should Avoid Genetically Modified Foods_\n\nThe matter of genetically engineered or modified crops and foods is one of the most intense public information scams of our time. It is as if the government, the food industry, and nearly all the media do not want the public to be aware of what is going on, or to know the health hazards posed by creating transgenic foods.\n\nGenetically modified organisms (GMO) are plants, vegetables, fruits, and animals in which genes from other species have been inserted to improve or alter various targeted characteristics. This is not the same as plant breeding, or manipulating gene patterns within a single species. It is called transgenic because genes (genic) are moved across (trans) species in ways that most critics of GMOs emphatically state nature never intended to have happen. Some people call them \"genfoods,\" but others call them \"frankenfoods.\"\n\n**Figure 4-2. Crops Containing Genetically Modified Organisms (GMOs)**\n\nScientists now regard the gene pool of many species of plants and animals as a kind of smorgasbord of genetic possibilities and desired traits. The once sacrosanct and uncrossable boundaries between species have now been broached, with experimental abandon. For examples, soybeans are crossbred with petunias; tomatoes are interlaced with genes from viruses; and corn, potatoes, soybeans, and cotton now bear pest-resistant genetic material.\n\nThe transgenic combinations are strange: antifreeze genes from flounder have been inserted in tomatoes to protect them from frost damage; chicken genes have been inserted into potatoes to bolster disease resistance; Chinese hamster genes are now in tobacco plants to increase sterol production; and human genes have been inserted in pigs, salmon, trout, and rice. Almost forty common vegetables, dairy products, and hundreds of processed foods now contain genes from all over the place: from viruses, bacteria, flowers, insects, and animals (see figure 4-2). An estimated 3,000 varieties of plants and animals now contain genes from other species, and transgenic experimentation is a burgeoning field.\n\nBeware of potatoes. Monsanto, a major player in GMOs, recently released its New Leaf potato under the oxymoronic \"NatureMark\" label. This potato bears a gene intended to ward off insects. In 1997, nine million U.S. acres were planted to potatoes, corn, and cotton with seed carrying a gene from _Bacillus thuringiensis_ , a synthetic version of a natural insecticide. As of 1998, GMO-based seeds accounted for twelve percent of U.S. com acreage, thirty percent of soybeans, and forty percent of cotton.\n\nWatch out for soybeans, especially if you like soy foods such as tofu. Traces of a major pesticide used on corn and soybeans called Roundup have now been genetically inserted into the soybean seed itself, enabling the resultant plant to tolerate higher dosages of Roundup and other pesticides applied to the fields. The new GMO soybean is called Roundup Ready, and in 1997, it represented ten percent of the U.S. soybean crop. So unless your tofu or tempeh is made from certified organically grown soybeans, you have just added a little pesticide to your vegetarian stir-fry every time you use tofu.\n\nIt's not just GMO-based tofu you should be concerned about. The same problem holds with soymilk and soy-based infant formula. In fact, an estimated sixty percent of prepared, packaged foods contain GMOs, especially if there is a corn or soybean ingredient. A company called Genetic ID Inc., of Fairfield, Iowa, offers a DNA of food items to determine if GMOs are present. \"With soy foods, such as tofu, soymilk, and infant formula, if it is not made from organically raised soybeans, almost always it is contaminated with genetically engineered soy,\" states molecular biologist John B. Fagan, who developed Genetic-ID's test. Dr. Fagan tested the top five brands of soy-based infant formula and found that four (Carnation Alsoy, Similac Neocare, Isomil, and Enfamil Prosobee; Edensoy had no GMOs) had \"measurable levels\" of GMOs, and that three had \"very high levels.\"\n\nThe reasonable person, upon hearing this for the first time, will probably say, yes, it sounds a bit adventurous, if not out of control, but are there any health hazards? This is the shocking part: nobody knows because no long-term studies have been done or were apparently even required. The American public is the experimental subject in this safety test, but the results won't be in for a few years, or a generation. Genetic effects from consuming GMO-based foods could well be on the order of the multigenerational damage produced by endocrine-disrupting hormones (discussed in Chapter 2).\n\nSay you have food allergies and you are carefully avoiding your known allergens, one of which is Brazil nuts. Now you consume a product that contains a Pioneer Hi-Bred GMO soybean, which contains Brazil nut genes. Suddenly you get a food reaction from a food on your nonallergenic list. This scenario is not a supposition, but a summary of a situation that actually happened. In 1996, eight individuals consuming these soybeans developed serious, potentially deadly allergic attacks because of the hidden Brazil nut genes. They reacted to the miniscule amount of Brazil nut in each soybean as if they were eating only Brazil nuts. We can only await news of further outbreaks of unsuspected allergic reactions to other GMOs.\n\nAccording to the Council for Responsible Genetics, a nonprofit organization of scientists and public health advocates concerned with biotechnology and based in Cambridge, Massachusetts, GMOs \"may cause unintended side effects that make foods hazardous for human consumption. Unpredictable gene expression may result in unanticipated toxic effects or allergies.\" In other words, the Brazil nut allergy outbreak may have been only the proverbial \"shot across the bow\" presaging a much more traumatic outbreak of health problems resulting from GMOs.\n\nNot only is the risk of sustaining surprise food allergies heightened by the advent of GMOs, but it is making it harder to guarantee that organically produced crops are truly organic, which is to say, free of GMOs. The problem is now being called transgenic pollution: pollen from a GMO crop blows across the field to an organic crop, cross-pollinating it, making it de facto another GMO crop. A small producer of tortilla chips made from organically produced grains had to throw out 87,000 bags of corn chips worth $147,000 in 1999 when a test revealed their corn contained GMOs. There was no deception involved because the organic farmer was certified, but his neighbor, who planted GMO corn, wasn't. The GMO corn cross-pollinated the organic corn. This same news report stated that GMO crops now occupy 90 million U.S. acres, or about 25% of the country's total croplands.\n\nThere are at least three other identified health hazards with GMOs. That is to say, a few progressive scientists have identified potential problems, but have no idea how to extrapolate these into real-time scenarios. They just know the hazards portend serious trouble.\n\nThe first has to do with DNA uptake by human cells. At least two scientists propose that the viruses used in various GMO foods could be the most dangerous aspect of the genetic modification process itself. Typically, the virus used for gene implantation comes from the cauliflower mosaic virus (CMV), found naturally and harmlessly in many common vegetables. CMV in nature cannot enter human cells because it is biochemically specific to plants, but when it is used as the source for GMO virus genes, its protein coat is removed (which normally alerts mammalian immune systems that it is a foreign protein). This creates \"naked DNA\" enabling the virus to be taken into human and animal cells (because it's no longer recognized as foreign), there to become part of the organism's DNA structure.\n\nThe naked viral DNA is more infectious and thereby more dangerous than intact virus. Nobody knows what this might produce, but the speculations run to the grim type of scenario. Other research has shown that viral DNA resists digestion and can enter the bloodstream and white blood cells, the immune system's key defense agents. \"Because these viruses are capable of recombining and jumping species, we must be aware that we cannot rule out the possibility of their triggering a vast range of public health disasters.\" Because the cauliflower mosaic virus (or viral promoter) is found in almost all GMO-foods, the inclusion could be setting off bizarre, unknown, and\/or hazardous biochemical reactions in the body.\n\nThe second health hazard pertains to the GMO-allergen problem, but at a deeper level. It seems GMOs produce a new allergen called the anti-idiotope allergen. When the body produces an antibody (an immune defense protein) against an allergen (a foreign protein), it also produces an antibody against this antibody, and this is called an anti-idiotope antibody. Now most GMOs contain a gene to resist the action of antibiotics, and these genes produce an enzyme that is similar to an antibiotic and is also an allergen. In other words, antibodies that are allergens get produced, which means most GMOs \"are likely to be allergenic to people sensitive to antibiotics.\" If you are already allergic to antibiotics, you will probably also be allergeric to GMOs.\n\nThe possibilities for new forms of allergic reactions to supposedly common\u2014in truth, GMO-ed\u2014foods could be staggering, and possibly innumerable. \"The allergenic potential of these newly introduced microbial proteins is uncertain, unpredictable, and untestable,\" stated Marion Nestle, Ph.D., in an editorial in the _New England Journal of Medicine._ 75\n\nThe third problem has to do with antibiotic resistance. All GMO-based foods contain antibiotic-resistant marker genes enabling them to resist getting destroyed by antibiotics. It seems that these genes survive the genetic manipulation process and remain intact and functional in the digestive tracts of mammals. They transfer horizontally from the GMO-foods to the bacteria in the digestive track of animals and humans consuming them. The result is that the consumers may be \"incubating ever more virulent pathogens\" as their natural defense systems are compromised. Their systems may be more resistant to the effect of antibiotics given for medical reasons, and it is possible that the antibiotic resistance may be passed also to other dangerous microbes, further worsening the problem of antibiotic-resistant infections.\n\nThere is no indication that the public, to the degree it is aware of this development to include GMOs in many foods, is in favor of it. Nearly all polls since 1995, when public awareness of GMOs began (about a year after their introduction) both in the United States and internationally, register strong majority opposition to GMOs. A _Time_ magazine ongoing Internet poll, which began in October 1999, revealed that by February 2000, of the 154,000 respondents who had logged in to the questionnaire, 55% were \"very concerned\" they were consuming GMO-based foods. An MSNBC poll in January 2000 showed that eighty-one percent of Americans want GMO-foods to be labeled as such and that eighty-nine percent think the U.S. government should require premarket safety testing of GMOs. Twelve other U.S. polls registered an overwhelming majority in favor of GMO restrictions, as did most polls conducted in Europe, Australia, and New Zealand.\n\n**Figure 4-3. What Is Organically Grown Produce?**\n\nIn the meantime, U.S. consumers who wish to avoid or minimize their consumption of GMO-foods must exercise extreme market vigilance and should use certified organic foods when possible (see figure 4-3). According to BioDemocracy and Organic Consumers Association, a public advocacy group based in Little Marais, Minnesota, there are fifteen companies producing GMOs, which they dub the \"Frankenfoods Fifteen\": Kellogg's, Campbell Soup Company, Safeway, Nabisco, McDonald's, Frito-Lay, Nestle, Heinz Foods, Quaker Oats, General Mills, Kraft, Hershey's, Coca-Cola, Starbucks, Procter & Gamble. The inference is that if you want to avoid GMOs, don't use any food products from any of these fifteen food companies (see figure 4-4).\n\nThere are upwards of 30,000 prepared foods now on the market that contain GMOs and must be avoided by the GMO-banning consumer. For example, consider how many products, such as sodas and beverages, including health food versions, contain corn syrup or fructose, both now made from GMO corn. Corn syrup and fructose are also found in yogurt, aspirin, and most sweet products. Then consider these GMO-rich corn-based products: nonorganic corn oil, cornstarch, cornmeal, baking soda, baking powder, and glycose syrup.\n\nYou will need to avoid soy protein in all its forms (unless it is certified organic): soy flour in baked goods, pizza, cookies, and pasta; meat product fillers; vegetarian meat substitutes (tofu hot dogs); soymilk infant formula; diet and protein shakes; protein bars; chocolate and candy bars; margarine; ice cream; soy oil in salad dressings and snack chips; lecithin.\n\n**Figure 4-4. Commercially Prepared Foods Known to Contain GMOs*** :\n\n**_In the meantime, U.S. consumers who wish to avoid or minimize their consumption of GMO-foods must exercise extreme market vigilance. There are upwards of 30, 000 prepared foods now on the market that contain GMOs and must be avoided by the GMO-banning consumer._**\n\nAspartame, found in the artificial sweeteners Equal or NutraSweet, contains a GMO as do most nonorganic cheeses. Food additives such as lactase, catalase, and amylase contain GMOs; most livestock and commercially raised seafood are fed GMO-based feeds. Without argument, the list is dizzying in scope. You can easily end up spending a great deal of your time merely trying to avoid getting poisoned by what you eat and drink.\n\n### _Why You Should Be Wary of Nonorganic Dairy Products_\n\nThe problem with conventionally produced dairy products is that there is a good chance they contain another GMO in the form of a hormone. Specifically, it is called the bovine growth hormone (BGH), or recombinant bovine somatotrophin hormone, in its more technical guise. The result is that standard dairy milk and dairy products may no longer be healthy foods.\n\nAgain, thanks to the FDA in concert with Monsanto, the hormone's manufacturer, there is almost no way to tell which dairy products contain the hormone unless they are certified organic. Not only does the FDA not require dairy products to carry a BGH label, for a while, Monsanto was suing any dairy product manufacturers who labeled their products as BGH-free. Since 1993, BGH has been administered to dairy cows to increase their milk yield, but evidence is already available suggesting it might cause cancer in humans.\n\nBGH contains a hormone called IGF-1 (insulin-like growth factor), found in both cows and people (in the saliva and blood). It causes cells to divide and grow, and this is how the hormone increases cows' milk yield. However, the hormone's penchant for stimulating cell proliferation in cows is what makes it a possible carcinogen in humans.\n\nResearch shows that pasteurization does not destroy BGH and that the human stomach does not break down BGH when it comes into the body in a dairy product; instead, the large intestine absorbs it because it already has receptor sites for IGF-1. Pasteurization actually increases the IGF-1 content of milk as does BGH itself, so when you drink BGH-produced milk, you consume higher than usual levels of this hormone.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor a source of copious information and public action\/advocacy guidelines:\n\nBioDemocracy and Organic\n\nConsumers Association,\n\n6144 Highway 61,\n\nLittle Marais, MN 55614;\n\ntel: 218-226-4164;\n\nfax: 218-226-4157;\n\ne-mail: campaign@organic-consumers.org;\n\nwebsite: www.purefood.org.\n\nFor another advocacy\/public action source:\n\nGenetically Engineered Food Alert,\n\n1200 18th Street, NW,\n\n5th Floor, Washington, D.C., 20036;\n\ntel: 800-390-3373;\n\nfax: 800-390-4751;\n\ne-mail: gefoodalert@emediacy.org;\n\nwebsite: www.gefoodalert.org.\n\nThis organization is a coalition of seven other advocacy groups committed to the testing and labelling of GMOs. For more information about detecting the presence of GMOs in foods:\n\nGenetic ID Inc.,\n\n1760 Observatory Drive,\n\nFairfield, IA 52566;\n\ntel: 515-472-9979;\n\nfax: 515-472-9198;\n\ne-mail: info@genetic-id.com;\n\nwebsite: www.genetic-id.com.\n\nAccording to research published by Samuel Epstein, M.D., chairman of the Cancer Prevention Coalition and professor at the University of Illinois Medical Center, both in Chicago, Illinois, IGF-1 promotes the transformation of normal breast tissue into cancerous tissue and the proliferation of cancer cells in the breast. \"All women will now be exposed to an _additional_ breast cancer risk due to milk from cows treated with recombinant growth hormone,\" he states.\n\nDr. Epstein states that IGF-1 levels in BGH-produced milk can be up to ten times the levels in nontreated milk, and ten times as potent because its chemical form has been truncated. This is especially risky for infants and small children, he adds. European research showed that the amount of excreted IGF-1 in milk may be increased over non-BGH-treated milk by twenty-five to seventy percent.\n\nResearch published in _The Lancet_ in 1998 attributed a sevenfold increase in breast cancer risk among premenopausal women to IGF-1 content of milk. Blood from 32,826 nurses was collected and analyzed (in 1989) for IGF-1 levels; of these women, 397 eventually developed breast cancer. All of these 397 women had elevated IGF-1 levels when they were \"healthy\" and after they developed the cancer. The risk of breast cancer for women under fifty-one with the highest IGF-1 levels was seven times higher than women with lower levels. On this basis, the scientists stated there is \"substantial indirect evidence of a relation\" between IGF-1 levels and breast cancer.\n\nAnother study, this one in _Science_ , linked prostate cancer and IGF-1 in a study involving 152 men with prostate cancer. The researchers found \"a strong positive association\" between IGF-1 levels and prostate cancer risk. Specifically, they found that men with elevated but \"normal\" levels of IGF-1 levels were four times more likely to get prostate cancer. Among men sixty and older, if their IGF-1 levels were high, they were eight times more likely to get prostate cancer than men with the lowest levels of this hormone. The results were sufficiently vivid to \"raise concern,\" the researchers stated, that sustained intake of IGF-1 through BGH-treated milk could increase the risk of prostate cancer. In fact, they suggested that elevated IGF-1 levels in the blood might well be used as an indicator of prostate and breast cancer risk, an early warning signal, if you like.\n\nThe true extent of the health hazard of BGH-treated milk is not known. Yet. Consider it an experiment still in progress. \"With the active complicity of the FDA,\" stated Dr. Epstein, \"the entire nation is currently being subjected to an experiment involving large-scale adulteration of an age-old dietary staple by a poorly characterized and unlabeled biotechnology product.\" The experiment \"poses major potential health risks for the entire U.S. population.\"\n\nAs with GMOs, a majority of consumers aware of the issue are demanding dairy products be clearly labeled if they contain BGH. Of 1,900 polled, ninety-four percent said BGH-milk should be labeled. Another study showed that already ten percent of consumers are taking evasive or preventive action by deliberately buying dairy products from nontreated cows, while seventy-four percent of consumers state they are concerned about the possible\u2014eventual\u2014discovery of negative long-term health consequences of consuming BGH-based dairy products.\n\nUntil more is known about the long-term health impact of increased IGF-1 intake and blood levels in humans from consuming dairy products, it would be prudent for those wishing to minimize their intake of toxic substances to avoid consuming all dairy products not labeled organic (see figure 4-3). For the present, a label of organic on a dairy product is the only guarantee that the milk was not produced with BGH. Sheep or goat's milk products are exempt because BGH cannot be used on them; therefore, goat's and sheep milk products by definition (at present) are BGH-free. Also, certain imported cheeses are BGH-free, specifically from the countries that have banned BGH including Canada, France, Italy, Ireland, Great Britain, the Netherlands, Belgium, Luxembourg, Spain, Portugal, Germany, Austria, Switzerland, Norway, Sweden, Finland, Denmark, Greece, New Zealand, and Australia.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor a list of organic and BGH-free dairy products, see:\n\n\"The Shopper's Campaign\/Mothers' Milk List,\" at:\n\nwww.igc.apc.org\/mothers\/fieldwork\/mo_field_milklist.html.\n\nPrepared by: Mothers & Others\n\nfor a Livable Planet,\n\n40 West 20th Street,\n\nNew York, NY 10011;\n\ntel: 212-242-0010;\n\ne-mail: corevalues@mothers.org.\n\nFor more information and public advocacy action against BGH and other food contamination issues, see:\n\nThe Center for Food Safety,\n\n666 Pennsylvania Avenue, SE, Suite 302, Washington, D.C. 20003;\n\ntel: 202-547-9359;\n\nfax: 202-547-9429;\n\ne-mail: office@centerforfoodsafety.org;\n\nwebsite: www.centerforfoodsafety.org.\n\nFor information about organic standards (and complete document \"International Certification Standards, March 2000\") for agriculture:\n\nOrganic Crop Improvement Association (OCIA),\n\n1001 Y Street, Suite B,\n\nLincoln, NE 68508;\n\ntel: 402-477-2323;\n\nfax: 402-477-4325;\n\ne-mail: info@ocia.org;\n\nwebsite: www.ocia.org.\n\n### _Good Reasons to Avoid Irradiated Foods_\n\nThe irradiation of agricultural produce and animal products poses another health threat for the toxic-substance-conscious consumer. As with GMOs and BGH, food irradiation represents another ill-founded toxic adulteration of our food supply.\n\nThe FDA began approving selective use of irradiation procedures on U.S. foods in the 1960s, starting with wheat, wheat flour, and potatoes; in the 1980s, pork, spices, herb teas, fruits, vegetables, and poultry were added to the list; and in 1997, meat started being irradiated before it reached the supermarket shelves. New foods likely to be approved for irradiation include deli meats, frozen foods, prepared fresh foods, fresh juices, seeds, and sprouts. Presently, thirty-five countries irradiate their foods.\n\nThe ostensible reason for food irradiation has been food safety, because it was believed that the radiation killed pathogenic organisms that might cause produce and meats to sicken consumers or spoil. However, the means to accomplish this goal are toxic. One of three energy sources is typically used: gamma rays from cobalt-60 or cesium-137 (which are radioactive waste products from the nuclear industry); X-rays; or high-energy electron beams.\n\nWhen food irradiation received FDA support in 1987, the agency required all irradiated foods to be labeled as such, but in recent years, that requirement has been considerably relaxed. In 1997, Congress passed legislation allowing food packagers to reduce the label size (the words \"treated with radiation\" accompanied by a specific graphic design) and to exclude the identifying label if all the components of a food product had not been irradiated. In other words, if a product contains fifteen ingredients, and twelve have been irradiated, and three have not, the product does not have to say \"treated with radiation.\" If the irradiated food will be used in a restaurant, deli, salad bar, hotel, airline, hospital, or school food service, labeling is not required.\n\nYou must now be on guard, too, for semantic trickery in labels. The food industry is lobbying FDA to allow deceptive labeling such as \"electronic pasteurization,\" \"cold pasteurization,\" and \"pasteurization with X rays\" to disarm public disapproval (and awareness of) food irradiation.\n\nWhat's wrong with radiation in foods? First, evidence suggests it reduces the nutritional content of foods. Vitamin C levels can be reduced by five to ten percent and vitamin E by twenty-five percent; thiamin, and vitamins A and K are especially susceptible to radiation damage. Some place the range of nutrient depletion between five percent and eighty percent, depending on how the produce is stored. Other research documents a ninety-five percent depletion of vitamin A in irradiated chicken, an eighty-six percent drop in vitamin B in irradiated oats, and a seventy percent depletion of vitamin C in irradiated fruit juices.\n\nEven though irradiation destroys ninety-five percent of all bacteria (including beneficial ones), it also deactivates many essential digestive enzymes found in the produce, requiring the human system to work harder to process the foods.\n\nAlso, because radiation kills all bacteria, you can't smell a food to see if it's spoiled, even if it has, and botulinum, the toxic agent that produces botulism, is not affected by irradiation. This means your risk of unknowingly consuming tainted food is higher with irradiated foods. In addition, radiation is likely to have a mutagenic effect on bacteria and viruses present in the meat or produce, causing them to mutate into radiation-resistant strains. In fact, scientists have already developed a radiation-resistant strain of salmonella, one of the chief pathogens in animal products.\n\nSo you start off with your \"fresh\" vegetables being functionally and nutritionally already cooked. Insofar as most consumers cook their vegetables and thereby leach out further nutrients, nutrient loss due to irradiation adds to the nutritional depletion of our food supply and can contribute over time to selective or chronic nutrient deficiency or even subacute malnutrition.\n\nSecond, the so-called standard safe doses permitted for irradiation are shockingly high when compared to their equivalent in terms of X-ray exposures. For meat, the FDA permits the equivalent exposure of 22.5 million X-rays per piece; for poultry, 15 million X-rays; for spices, 150 million; fruits and vegetables, 5 million. That is an extraordinary number of X-rays.\n\nExtensive research by nuclear radiation experts such as John W. Gofman, Ph.D., M.D., director of the Committee for Nuclear Responsibility, former professor of molecular biology at the University of California at Berkeley, and author of several important books on radiation exposure and cancer, indicates that far less exposure to X-rays can be carcinogenic. In fact, Dr. Gofman states that, over the course of a lifetime, X-ray exposure (including from mammograms) can account for seventy-five percent of all cases of breast cancer. Dr. Gofman found that the longer you are exposed to radiation, the weaker the dose it takes to produce damaging effects. When it comes to X-rays, less is very much more.\n\nAn X-ray is a type of ionizing radiation, which means its high energy waves can rip electrons away from stable molecules. The principal means by which it is believed X-rays and thus all radiation produce biological damage is through the creation of free radicals, unstable molecules that destroy cells (see Chapter 2). Overlooked in the FDA's \"safety\" standards is the fact that \"food irradiation produces 'radiomimetic' foods\u2014they _mimic_ the effects of actual radiation when eaten.\"\n\nThird, even though, technically, irradiated foods are not radioactive as such, studies show that eating irradiated foods is still hazardous because they contain \"radiolytic products,\" other toxic forms or byproducts of nuclear radiation. These byproducts have been associated with cancers, gene damage, birth defects, and other biological abnormalities. The radiolytic products also involve toxic chemicals found in foods, such as benzene, formaldehyde, formic acid, and quinines. In layperson's terms, the bombardment of the radiation shakes everything up in a food item, producing unstable molecules (free radicals) and new, bizarre, probably toxic products containing well-known toxic substances, such as benzene, formaldehyde, and the others.\n\nInsofar as the \"unique radiolytic products\" are new to the food technology world, never seen before food irradiation, nobody can say for sure that they are harmless and that a diet of irradiated foods is safe. Like the long-term effects of GMO-based foods, it is entirely unknown to science what food irradiation will do to us. However, suspicions have been raised as to probable health effects, and they are distressing.\n\nAccording to one estimate, for every 7.5 ounces of irradiated meat you consume, you are ingesting 2,560 potentially carcinogenic or mutagenic radiolytic product (RP) molecules that can affect liver cells. \"Over a long period of time, the RP assault on the liver combined with fewer antioxidants in the diet [irradiation depletes antioxidant levels] will create a 'fertile field for the ultimate growth of cancer cells' and 'almost certainly evolve' to produce liver cancer.\"\n\nThe ramifications of this calculation are that even at one-tenth the concentration of radiolytic products known by the FDA to be formed by food irradiation, \"irradiation of foods in the human diet represents predictably unacceptable risks to the public's health.\" In other words, this one aspect of food irradiation may produce serious health consequences even if your exposure is only ten percent of the \"safety\" level established by the FDA.\n\nCritics of food irradiation state that the technology has never been adequately tested for long-term safety and public health effects. In fact, the FDA approved food irradiation on the basis of only 5 out of 413 animal feeding trials under way at the time in 1982, and the key trial lasted only fifteen weeks. In the FDA's own review in 1982, it admitted that of these 413 studies, 344 were either inconclusive or inadequate and could not demonstrate or refute irradiation safety; 32 showed adverse effects; 37 appeared to support the procedure's safety; only 5 studies (1% of all studies reviewed) \"appeared to support safety.\"\n\nThere is strong evidence of health hazards from food irradiation. For example, in 1979, the Hungarian Academy of Sciences looked at all the available scientific studies on food irradiation. They found that the literature revealed food irradiation was responsible for 7, 191 neutral effects, 185 beneficial effects, and 1,414 negative outcomes. The latter included tumor formation, damage to genes (including chromosomal polyploidy, or extra chromosomes in the genes), kidney and heart disease, and birth defects.\n\nEvidence linking gene damage to consuming irradiated foods was developed as early as 1973 in India. There scientists found that rats and mice fed irradiated wheat had increased levels of cells with abnormalities in their bone marrow chromosomes (called polyploid cells). Monkeys and undernourished children also developed abnormalities in their white blood cells after eating irradiated wheat. The cells became abnormal after only four weeks of the children's feeding program; it took twenty-six weeks for the abnormalities to dissipate after the program stopped. An increased number of polyploid cells is often associated with either rapidly regenerating tissues or rapidly growing malignant tumors, observed the researchers in presenting their data to the United States Congress in 1987.\n\nOpposition to food irradiation continues to mount as consumers become more aware of its potential health risks. The Center for Food Safety, a public advocacy group in Washington, D.C., states that \"the use of food irradiation poses significant human and environmental impacts.\" Another political advocacy group called Public Citizen, also in Washington, D.C., summarizes the negative health effects of food irradiation for which supporting research already exists:\n\n  * Increased chromosomal damage in animals and humans\n  * Formation of unique radiolytic products in the foods\n  * Increased frequency of cell mutations\n  * Formation of mutant bacteria\n  * Decreased nutritional content of foods\n  * Increased levels of carcinogens and other toxins in foods\n  * Increased frequency of tumors, reduced survival rate, and other health problems in animals\n  * Corrupted flavor and texture of foods.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more on the health hazards of X-rays and ionizing radiation and the work of Dr. John Gofman:\n\nThe Committee for Nuclear Responsibility,\n\nP.O. Box 421993,\n\nSan Francisco, CA 94142;\n\ntel & fax: 415-776-8299;\n\nwebsite:\n\n<http:\/\/radical.org\/radiation\/CNR>.\n\nFor information and advocacy\/action positions on food irradiation: Public Citizen,\n\n1600 20th Street, NW,\n\nWashington, DC, 20009;\n\ntel: 202-588-1000;\n\nwebsite: www.citizen.org.\n\nPublic Citizen was founded by\n\nRalph Nader in 1971 to represent consumers' interests.\n\nAs with GMOs and BGH-treated dairy products, the U.S. public is overwhelmingly _not_ behind food irradiation. A Louis Harris poll in 1986 reported that seventy-six percent of Americans considered irradiated foods to be a hazard. In 1997, a CBS News poll indicated that seventy-three percent of the public opposes food irradiation and that seventy-seven percent would not knowingly eat irradiated foods. An April 1999 survey found that eighty-eight percent of consumers support the labeling of irradiated foods.\n\n## HEALTHY LIVING SPACE DETOXIFIER #12: Minimize Your Consumption of Foods Containing Trans-fatty Acids\n\nIn many respects, this next category of food adulteration is easier to pinpoint and to avoid than GMOs, BGH-dairy products, and irradiated foods. It pertains to the fats found in prepared foods, specifically, what are called saturated hydrogenated fats, or trans-fatty acids. Hydrogenated is a chemical term that refers to a fat in a semisolid state, such as margarine.\n\nThe recommendation here is simple: avoid consumption of all foods containing trans-fatty acids, or at least minimize your consumption. Trans-fatty acids are linked to heart disease, depression, and other long-term degenerative health conditions. You will have to read product labels carefully, looking for the words \"hydrogenated fats\" or \"partially hydrogenated fats,\" and avoiding certain categories of prepared foods. But to understand why this is so important, we need to briefly review the chemical facts of fats.\n\nFats and oils are made of basic building blocks called fatty acids. A fatty acid is a chain of carbon atoms with a hydrogen atom at the end. The chains can have anywhere from two to thirty carbon atoms. When a fatty acid has all the hydrogen atoms it can hold in its configuration, it is saturated. Examples of this are animal fats and other fats that are solid at room temperature. If the fatty acid doesn't have all its hydrogen spots filled, it is unsaturated; examples of these are most vegetable oils, which are liquid at room temperature. Unsaturated fats are preferable in the diet and are called essential fatty acids; these include omega-6 oils, such as linoleic acid, and omega-3 oils, such as linolenic acid, all derived from natural plant sources.\n\nA trans-fatty acid is an industrial alteration of the natural shape of oils. Although trans-fatty acids exist in small amounts in nature, that is nothing compared to the scale on which they have entered the food supply The manufacturing process changes the way the hydrogen atoms bond to the carbon atoms in a fat. Hydrogenated fats or trans-fatty acids were originally synthesized and introduced into foods in 1910 (with a big push coming in the 1950s) to extend the shelf life of oil-based consumer foods. The oils were hardened or hydrogenated to prevent rancidity. This change in chemical configuration gives margarines a butter-like consistency and makes shortenings fluffy. However, from the viewpoint of some nutritionists, the introduction of trans-fatty acids into the food supply was not a good idea.\n\n**_\"The production of trans-fatty acids for human consumption is the most devastating, nutritional_ mistake _ever made,\" claims (Patricia Kane, (Ph.D., a nutritional biochemist practicing in Millville, New Jersey._**\n\n\"The production of trans-fatty acids for human consumption is the most devastating, nutritional _mistake_ ever made,\" claims Patricia Kane, Ph.D., a nutritional biochemist practicing in Millville, New Jersey. It's a mistake because all traces of essential fatty acids are \"obliterated\" from processed foods, says Dr. Kane, and trans-fatty acids or hydrogenated oils take their place.\n\nThe result of this substitution of hydrogenated oils for essential fatty acids is first a deficiency of the healthful oils in your body and a corresponding excess of unhealthful hydrogenated fats. This in turn has been identified as a contributing factor in obesity and heart disease. Fats are needed to produce cholesterol in the body, which in turn is needed to produce \"a cascade of crucial hormones,\" explains Dr. Kane. All cholesterol is not \"bad\"; you actually need some to run your body properly.\n\nFatty acids have many key functions: they comprise sixty percent of the brain itself; they contribute to cell membrane fluidity, allowing biochemical \"traffic\" into and out of each cell; they help maintain the integrity of the gastrointestinal tract, enabling it to protect the body against pathogens. Disturbances in the energy activities associated with fatty acids, says Dr. Kane, \"are reflected in every system of the body and can result in many physical and mental disorders, including depression.\" In fact, she states that in the difference between trans-fatty and essential fatty acids you have both the cause and the cure of depression. With a diet based on hydrogenated fats, \"Prozac is only a few more wrong meals away.\"\n\nFatty acid chains are said to be like \"tails\": the tails of essential fatty acids vibrate fast and move quickly, a sign of molecular health, but trans-fatty acid tails are rigid and have no movement, says Dr. Kane. \"A trans-fatty acid sits like a heavy blob inside you, shutting down the fatty acid metabolism within the immune, endocrine, and central nervous systems, and replacing the good, necessary fats (omega-3s and omega-6s) with something harmful to health.\"\n\nYour body burns the trans-fatty acid for calories but can't absorb nutrients from it. The accumulation of trans-fatty acids in the body acts like a systemic poison and generates an abnormal biochemistry within you, Dr. Kane says. She recommends avoiding trans-fatty acids altogether to protect the integrity of your immune, endocrine (hormonal), and central nervous systems.\n\nResearch on trans-fatty acids has started to highlight their health hazards. As early as 1990, Dutch scientists showed that margarine consumption, thought to be healthy because it was not cholesterol-rich like dairy butter, actually increased the risk of coronary heart disease. Now new research links trans-fatty acids with elevated levels of LDL, the so-called \"bad\" cholesterol (HDL being the \"good\" kind). Scientists at Tufts University in Boston studied the effects on thirty-six men and women over the age of fifty of diets in which thirty percent of the calories came from fats. For the study, two-thirds of their fat came from soybean oil, semiliquid margarine, soft margarine, shortening, stick margarine, or butter.\n\nThe best results in terms of lowering LDL-cholesterol levels were obtained with soybean oil and semiliquid margarine (minimally hydrogenated) diets. It is interesting to note that for the most part the levels of trans-fatty acids used in the foods in this experiment were lower than what a typical American consumer is likely to get on a daily basis. For example, the soybean oil shortening contained 9.9 g of trans-fatty acids per 100 g of fat compared to 35-50 g of trans-fatty acids per 100 g of fat found in most commercially produced cakes, cookies, crackers, and doughnuts. The stick margarines used contained 20 g per 100 g of fat compared to 30 g typically found in popular stick margarines.\n\nThose consuming only soybean oil had their LDL-cholesterol levels drop by twelve percent from pre-diet levels. They also found that trans-fatty acids raised the cholesterol levels more than saturated fats did. On the basis of their results, the researchers concluded that the general public and those with high blood cholesterol \"should be encouraged to use vegetable oil in its natural state or after minimal hydrogenation.\" This is especially so since, as mentioned above, many people may consume trans-fatty acids at higher levels than those used as a basis in the study.\n\nHere is a summary of some of the research supporting the undesirability of consuming trans-fatty acids:\n\n  * Lowers levels of desirable HDL cholesterol, and raises levels of undesirable LDL cholesterol.\n  * Reduces the amount of \"cream\" or volume in the breast milk of nursing mothers.\n  * Increases blood levels of insulin, increasing the risk of diabetes.\n  * Decreases the response of red blood cells to insulin, potentially complicating the blood sugar balance in diabetics.\n  * Lowers immune response by affecting how the immune system's B cells respond.\n  * Causes unfavorable changes in the activities of enzymes that deal with carcinogens as part of the liver's detoxification process.\n  * Alters the functional properties of cell membranes such as transport and fluidity.\n  * Decreases levels of the male sex hormone testosterone, increases the level of abnormal sperm, and interferes with gestation in women.\n\nOn a practical level, if you wish to minimize your intake of trans-fatty acids, you will need to avoid the following: fast-food burgers, french fries, and hash browns, as they will probably have been fried in hydrogenated fats; commercially baked cakes, cookies, muffins, doughnuts, corn chips, and pastries, as they probably contain hydrogenated fats; soft-tub and stick margarines and shortenings. French fries are the biggest offenders here, with four ounces containing 2.4 to 3.4 g of trans-fatty acids; doughnuts are close behind with a one ounce serving containing 3.19 g. As healthful alternatives, look for and consume products containing olive, canola, or other unsaturated vegetable oils.\n\nYou will probably find that a great many foods you are accustomed to eating come under the trans-fatty acid ban. The average consumption of trans-fatty acids in grams per day for the United States is estimated to be between 11.4 and 12.1, comprising, respectively, 7.3% to 7.8% of total fat intake. However, for adults aged 20-65, that level is 14. 9 g\/day or 8% of total daily fat intake, and for female teenagers, it is 30 g\/day representing 28% of total fat intake. By comparison, Sweden is much lower at 5 g\/day, Germany is 4.5 to 6.5, Canada is 9.1, England is 12, and the Netherlands is 17.\n\nWhere does all this leave you in terms of what to eat? Essentially, if your sole guideline is to eat only organically produced food this in itself dictates a diet. By definition, an organic diet will eliminate a great deal of the nutritionally inferior, high-fat, sugar-based, additive-laden, preservative-rich, processed and prepared foods that are the carriers of many of the toxic substances you are trying to eliminate from your body. Beyond this, determining the most appropriate diet for an individual is one of the hardest tasks of medicine (and living) because everyone's physiology is specialized to some extent, unique to the individual, or at least uniquely idiosyncratic. You will need to give this matter considerable thought and further research to find the approach that works best to keep you nutritionally fit and minimally toxic.\n\n## HEALTHY LIVING SPACE DETOXIFIER #13: Do a 3-Day Water or Juice Fast to Rest and Regenerate Your Cells\n\nBy this time, you may be so dismayed with the state of the American food supply and the toxic burden it imposes on your body that the idea of fasting, that is, abstaining from solid foods for a limited period, may seem logical if not irresistible. The fact is, even if the food supply (and our air and water) were not so stunningly tainted, brief, recurrent abstention from foods is an old, well-proven remedy for improving one's health. Fasting has been practiced around the world in many traditions of medicine and spirituality.\n\nWhy fast? Think of it as a vacation for your digestive system and its \"support staff,\" a break from business as usual for the gastrointestinal tract. Say you are a forty-year old man or woman and have never fasted. That means your digestive system has been running flat out for 14,600 days without stopping. It is almost impossible for your body to be entirely current with its digestive affairs; in most people, there will be a backlog of unfinished, incomplete, or even abandoned digestion.\n\nAs you will learn in Chapter 5, the intestines, comprising about twenty-six feet of internal organ space, is a remarkably competent _storage_ vessel for incompletely processed foods. Fasting for even a few days gives your intestines a chance to start getting caught up with its unending elimination demands. And it gives all the cells in your body, but especially those in the liver and kidneys, a chance for some rest and regeneration.\n\nAs a benefit of this respite from the ceaseless demands to process, assimilate, and eliminate foods, your body will have a chance to speed up its collection and excretion of toxins, from the colon, liver, kidneys, lymph nodes, bladder, lungs, even the skin. Your mind, too, will reap benefits from this brief abstention from food intake. It is quite likely you will emerge from a brief fast with enhanced clarity, focus, concentration, uplifted spirits, a general calmness, and a sense of heightened energy and vitality.\n\nFor the purposes of the detoxification recommendations offered in this book, the three-day water or juice fast can be undertaken as a preliminary to the more comprehensive intestinal, liver, kidney, and lymph cleansing regimens described in later chapters. You can do the three-day fast several times a year as a kind of routine internal housecleaning, if you wish.\n\nThere are a few precautions to observe regarding your limited fast:\n\nFirst, do _not_ do the fast if you have hypoglycemia (chronic low blood sugar), diabetes, weakened immunity, heart problems (such as low blood pressure), cancer, stomach ulcers, if you are pregnant, lactating, suffering from nutrient deficiencies or generally malnourished, excessively thin or anorexic, or under heavy medication for a serious illness. Or undertake it _only_ under the supervision of a qualified health-care practitioner.\n\nSecond, don't start it tomorrow. Prepare your body for the drastic reduction in food intake it will experience. Over the course of a few days\u2014a week perhaps\u2014gradually reduce your food consumption to one light meal daily. Naturopathic physician Bruce Fife, N.D., author of _The Detox Book_ , recommends following a natural foods diet for three to four weeks before starting the fast if you have been accustomed to a diet of fast, prepared, and convenience foods as that are notoriously deficient in nutrients. The preparatory health foods diet will fortify you nutritionally so that your system can function well during the fast. The natural foods diet Dr. Fife recommends includes only organically produced fruits, vegetables, and protein sources (in limited amounts), whole grains, and no prepared or packaged foods.\n\nThird, try the fast for one day to see how it feels. Then wait a week and, following the same preparation steps, try it for two or three days.\n\nFourth, expect a few transient symptoms to arise during the fast, such as headaches, mild muscle aches and pains, irritability, and old emotions or thoughts.\n\nFifth, after your three days of fasting, do not immediately return to a full-fledged solid foods diet. Build up to it gradually for three days, reversing your preparatory steps from before the fast. For example, you might continue with some of the juices, and add salads or steamed vegetables, followed the next day with well-cooked brown rice or millet.\n\nSixth, when you resume your regular diet, observe if any foods produce allergic reactions. Because your system has somewhat cleared itself of old debris and digestive backlog, hidden food allergies may now become apparent and you may find that wheat, eggs, chocolate, or corn, for example, produce allergic symptoms. If so, reduce or discontinue your intake of the specific food for a few weeks, then reintroduce at perhaps twenty-five percent of the previous intake rate.\n\nDuring the fast you stop consuming solid foods and increase your intake of pure liquids, either in the form of water or juices. These liquids should not include stimulants such as coffee, tea, or alcohol; or milks, such as cow, goat, sheep, soy, rice, almond, or any other. Basically, consume small amounts of the selected beverage frequently throughout the day, perhaps hourly. Don't overtax your kidneys by drinking too much at one time, such as more than eight ounces. There are many options to choose from regarding what your liquid intake will consist of, and we'll review a few of these below.\n\nSome detoxification experts, such as Elson Haas, M.D., director of the Preventive Medical Center of Marin in San Rafael, California, recommends a twice yearly fast, corresponding with the changes in season from winter to spring and summer to autumn. For example, as a general protocol, Dr. Haas advises consuming two to three quarts of liquids daily on the limited fast; these liquids can be fresh, pure water or specially blended vegetable juices.\n\n### _Juices for Seasonal Cleanses_\n\nFor the spring cleanse, Dr. Haas suggests drinking eight to twelve glasses daily of a blend of fresh lemon or lime juice (two tablespoons), maple syrup (one tablespoon), cayenne pepper (one-tenth of a teaspoon), and eight ounces of pure water. This can be supplemented with peppermint or chamomile tea, consumed at different times. For the autumn cleanse, the blend includes pure water (three cups), chopped, blended ginger root (one tablespoon), soybean miso (one to two tablespoons), chopped, blended green onion (one to two stalks), cilantro (a few pinches), cayenne pepper (a few pinches), olive oil (two teaspoons), and lemon juice (from one-half a lemon). Simmer the ginger root for ten minutes in water, stir in the miso, turn off the burner, and add the remaining ingredients; steep for ten minutes, then consume.\n\n### _Cleansing Juices from Fruits and Vegetables_\n\nAmong the fruit juices that can be helpful during the three-day fast are watermelon, grape, papaya, pineapple, lemon, apple, pear, black cherry, and citrus. You can experiment with finding the blend that is most pleasing, but it is prudent to mix no more than three at one time. According to Dr. Haas, you can also make a juice of any of a number of vegetables that are beneficial for cleansing during a fast, including potatoes, radish, dark leafy greens, spinach, parsley, wheat grass, cabbage, beet greens, watercress, comfrey, carrots, beets, celery, cucumber, Jerusalem artichokes, garlic, or radish. Again, don't mix more than three or four vegetables together at one time.\n\nUnless you have a juice bar in your neighborhood (which is possible if you live near a good natural foods store and it has a fresh juice bar), you will have to blend the fruits and vegetables yourself in a juicer. Remember to cleanse them of pesticides and bacteria, as described above in Healthy Living Space Detoxifier #10. Don't use canned, frozen, or bottled juices, as their enzymes will probably be weakened if not destroyed, and you need the living enzymes to spark digestion during the fast. Even though bottled juices from a natural foods company are still wholesome, they may be not fresh enough to best serve your detoxifying purposes.\n\n### _Vegetable Juices, Herb Teas, and Potato Broth_\n\nHere is a tested variation provided by health journalist Lewis Harrison, author of _30-Day Body Purification._ During the three to four days before the fast, reduce your food intake to fresh fruits and vegetables and eight ounces of plain yogurt or buttermilk. Then during the fast, drink freshly pressed vegetable juices, especially those from beets and green leafy vegetables at the rate of one quart daily. Supplement this with an herbal cleansing tea, such as red clover or ginger, at the rate of twenty-four ounces daily and with sixteen ounces of an alkaline potassium broth. This broth can be prepared from potatoes, green beans, celery, and zucchini, says Harrison. Cook them in pure or distilled water, then blend the mixture. Other beneficial herbal teas to consume during the fast are peppermint, chamomile, or rose hips.\n\n### _Detoxifying Green Drink_\n\nPart of your daily liquids intake might include a few glasses of this detoxifying blend of herbs and vegetables, says Jacquelin Krohn, M.D., a practitioner in environmental medicine based in Los Alamos, New Mexico. Dr. Krohn suggests adding any of the following to a liquid base of fresh pineapple juice in a blender: celery, radish tops, comfrey leaves, burdock root, parsley, Swiss chard, plantain, dandelion, sprouts, carrot tops, wheat grass, or raspberry leaves.\n\nDr. Krohn also describes a \"basic vegetable broth fasting drink\" consisting of red potatoes (two large ones with skins intact), celery (three stalks), beets (three), carrots (four), plus small amounts of cabbage, turnip, onions, or beet tops. Chop up the vegetables, boil, simmer for forty-five minutes, then blend into a puree; consume in small amounts throughout the fast.\n\n### _Special Lemonade for a Water Fast_\n\nIf you choose to undertake the water fast for three days, abstaining even from fruit and vegetable juices, here is a recommendation from Dr. Fife. \"Fasting is the primary mechanism nature has given us for cleansing and detoxification,\" he comments. If you are fasting for longer than three days or if you are planning to consume only water during your three day fast, this dilute form of lemonade can be beneficial, advises Dr. Fife. Mix fresh lemon or lime juice (two tablespoons), raw honey or unprocessed maple syrup (one tablespoon), and filtered or distilled water (eight ounces).\n\nConsume this lemonade regularly during your fast as part of your overall liquid intake, which Dr. Fife advises should be at least six glasses daily, or forty-eight ounces. The lemon is \"a good cleanser and astringent that rids tissues of toxins,\" he says, and it stimulates liver detoxification. The honey or maple syrup provides minimal simple carbohydrates to provide energy to the body.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #14: Do a Media\/Image Fast to Cleanse Your Mind and Energy Field_\n\nA character in John Updike's novel _Rabbit Redux_ comments on the oppressive clutter in her mind: \"The things I have in my mind, Hassy, it reminds me of when they clean out a drain. All that hair and sludge mixed up with a rubber comb somebody went and dropped down years ago. Sixty years ago in my case.\"\n\nThis is an excellent description of the mental sludge that builds up in our mind and energy fields over the months and years of being continuously subjected to a barrage of images, words, slogans, and sounds. Thus is our lot as participants in a media-intensive, highly literate, advertising-based society. Consider the sources of daily input: magazines, radio, television, Internet, billboards, books, newspapers, conversations, movies, commercials, advertisements, slogans, and signs, among others.\n\nThe nervous system, mind, and the body itself are bombarded by this array of insistent images and words. If the source is commercial in nature, be assured it is structured to influence you, to persuade you to do something, think something, get something. If the source is intellectual in nature, it is still designed, though at a more subtle level, to have a particular effect on you, such as giving you a certain slant on history or current affairs. Your consciousness, even your sense of self, are perpetually assaulted by a myriad of influences, each vying for your attention and their chance to produce a shift in allegiance in its favor in your thinking and way of being.\n\nSome forms of media, such as television and movies seen in a theater, have an energetic effect on you, even to the extent that sensitives, or psychics, can observe energy residues in the aura surrounding the body of a person who has just watched television or sat through a movie. Like swirling reflections in a soap bubble, images from the television show or movie are still observable in your aura for at least an hour, sometimes longer, following your immersion in the energy field of that media. Everyone is familiar, too, with the queer sensation of lyrics and melodies repeating themselves, unbidden, in our minds, like a radio that cannot be easily shut off. Or perhaps you catch yourself about to automatically repeat something someone else said, perhaps a news commentator, or a salesman for a product advertised on television.\n\nAll of these sensory inputs have an impact on your mind and energy body. Like undigested (or hard to digest) food in the intestines, these media inputs can, if left unprocessed, act like toxins in your system and eventually contribute to an illness process. But we can take this further. What if the inputs are deliberate distortions of the truth, or prejudiced spins on the facts, or mean-spirited commentaries, or lies cast to protect special interests?\n\nUntruths can be toxic substances, too. Our parents, grandparents, siblings, and friends tell us who we are, and this telling can become a mask we wear over our true identities. If we never strip off the mask\u2014 _their_ mask imposed on us\u2014we never know who we are. There are some who say what we take for history is a lie, a fabrication, a fantasy; others point out that what we take as our family history or the nature of our family interactions may be a tissue of untruths; indeed, perhaps much of what we tell ourselves in our interior mental monologues is similarly misleading. From a certain angle of observation, we could say that the total of media\/image input acts as a straitjacket on human consciousness, forcing it into a groove (a rut) in which only one-way movement is possible. Put more simply, we live in a trance state induced by someone else when we weren't looking.\n\n**_Consensus reality, like untruth, is unnatural and can Be a toxic substance._**\n\nSome call this consensus reality, and some say consensus reality is a toxic substance for consciousness. It has guidelines for behavior, perception, self-definition, for how reality works, including what produces illness (germs) and what accomplishes healing (drugs and doctors). As children, we become entranced by the prevailing mode of cultural consensus reality. We develop what Charles Tart, Ph.D., calls '\"consensus consciousness' to reflect the fact that our so-called ordinary state of consciousness, or 'normal consciousness' (which is a culturally relative term of course) means we have actually constructed the habits of our thinking and feeling and perceiving to reflect the consensus of what our culture thinks is important and good.\" Consensus reality is not what consciousness is really like; it is \"a semi-arbitrary construction.\"\n\nConsensus reality, like untruth, is unnatural and can be a toxic substance.\n\n_Recommendation_ : Devote a week, perhaps in conjunction with the food fast and\/or with the relaxation baths as described in the next section, and abstain entirely from media\/image inputs. Don't read anything: books, newspapers, product labels, billboards, signs. Listen to as little as possible: radio, music with words, television, other people's gossip or conversations, your own interior monologue. Stop absorbing other people's view of the world, your family's definitions of roles and identities, even your own constantly reiterated identity statements. Allow your mind to cohere, to collect itself, to facilitate self-awareness, selfobservation: _who are you_? Even if you think you know, even if you have spent years unpeeling the family and societal masks, you may be surprised by what emerges when you turn off the cultural tap.\n\nLet your mind and energy body relax deeply, divesting themselves of a lifetime of media\/image sludge. Consider meditating once daily, just sitting quietly in a comfortable position for a period of time, paying attention only to your breathing, and not chasing after any thoughts that come to mind. Perhaps you would like to go off for a week to a meditation retreat to make this withdrawal from input easier to sustain.\n\nOr imagine that before you, a few feet away from your body, stands a very large red rose with absorbent petals; in fact, it has hundreds of petals, capable of surrounding you. This rose has a root system so deep it goes straight to the center of the Earth. Instruct the rose to absorb from you the energy residues of all the media\/image input you have had in your lifetime. When the rose seems to have absorbed all it can, pluck it from the imaginary ground in which it grows and throw it away. You may need to imagine another rose\u2014and probably many more\u2014and do the same thing when it gets clogged with your energy effluent.\n\nAlternatively, you could picture a large pipe connecting your body to the center of the Earth; it is a drainage pipe perhaps three feet in diameter, specifically suited to your body and energy field. Instruct your body to release the energy residues of all the images, words, songs, ideas, pictures, and other media\/image input you have received in your lifetime. You may experience this as a kind of dark water or mucus flowing out of your body into the drainage pipe; or you may simply feel lighter, calmer, clearer, and see nothing.\n\nYou may wish to perform the meditation, rose, or drainage pipe exercises several times daily during your weeklong media\/image fast. Just as your physical body can start to catch up on unfinished digestive business during the food fast, so can your mind and energy body catch up on processing a lifetime of energized media\/image inputs. The process will free up a lot of your energy that was trapped or held hostage in the gravitational field of these inputs.\n\nOne of the benefits of this media\/image fast is that you may find yourself opening up to new possibilities, to taste, perhaps, an infusion of what is often called the new paradigm, planetary transformation, or evolutionary global change under way. \"Perhaps the most significant event in our lifetimes is the one that is taking place in our consciousness: Our collective view of reality, rooted in millennia of historical experience, is being shattered completely,\" stated the editors of _Cogenesis Journal_ , an online magazine. \"Our foundation of consensus reality is simply being swept away . . . _everything is changing_ ,\"\n\nOur human understanding of how the world works, where we fit in the scheme of creation, our historical framework, our traditional cosmology\u2014\"all are being blown apart,\" add the editors. So the voluntary media\/image fast could put you in synchrony with vaster changes in progress.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #15: Make a Lifestyle Out of Nonstop Stress Reduction with Healing Hot Water Soaks_\n\nIt is important to appreciate that stress is a global\u2014that is, completely body-wide\u2014experience. If your cells are stressed with toxins, it is likely your mind and emotions are stressed as well. If you are not relaxed, it is difficult for your tissues, muscles, and cells to be at ease. Remember, stress is a toxic substance, too. Let's take another tack to make the same point: when was the last time you felt so relaxed you felt you were melting into your chair or bed?\n\n**_If you can achieve that delicious state of meltingness, you are well on your way to relieving your system of stress. Once you have the sense of how to melt, on cue, as it were, you can make stress reduction a continuous part of your general detoxification program._**\n\nIf you can achieve that delicious state of meltingness, you are well on your way to relieving your system of stress. Once you have the sense of how to melt, on cue, as it were, you can make stress reduction a continuous part of your general detoxification program. For the present, here are several practical approaches that may ease you into meltingness.\n\n### _Epsom Salt-Aromatherapy Relaxing Baths_\n\nEpsom salt is a 100% dissolvable salt that when added to warm bath water can relax your muscles, helping them to release accumulated stress, tension, aches, or discomforts. At the same time, your nearly full-body immersion (up to the chin, ideally) in an Epsom-salt-saturated bath can be excellently relaxing for your mind and emotions as well. Epsom salt, which is primarily magnesium sulphate (MgS04), is widely available in supermarkets and pharmacies; it is inexpensive (less than three dollars for a four-pound container of dry salts) and entirely safe. Pour two to four pounds of dry Epsom salt mixture into a tub of hot water and stir until it is fully dissolved.\n\nAromatherapy is a term that refers to essential oils prepared from herbs and plants that are used topically (diluted, applied on the skin as a massage oil), mixed with water in baths, or diffused in the air via special burners. For the most part, the effective therapeutic action of the oil comes from your inhaling its aroma. There are aromatherapy oils, or blends (available in most natural foods stores), for almost any minor physical or mental condition, including anxiety and stress. Add these after the Epsom salt to your hot water bath.\n\n### Relieving Anxiety\n\nThe following essential oils can help relieve general anxiety: clary sage, rose otto, frankincense, bergamot, mandarin, sandalwood, lavender, juniper, patchouli, vetiver, cedarwood, melissa, and Roman chamomile. However, you don't want to use all of these at once; generally, it's preferable to use no more than three different essential oils at one time.\n\nLet's say you have muscle pain, your body feels tense, with aches and soreness. The essential oils helpful for this condition are sandalwood, lavender, clary sage, Roman chamomile, and patchouli. Or perhaps you are feeling a restless anxiety; you have been overactive, you're sweating, perhaps dizzy, and your stomach is upset. Here vetiver, cedarwood, juniper, Roman chamomile, or frankincense might help you. Suppose you are feeling on edge, are irritable, cannot concentrate or sleep well, and you feel exhausted most of the time. The oils to address this type of anxiety include bergamot, melissa, neroli, cedarwood, sandalwood, and rose otto. Or finally, your anxiety is marked by a feeling of apprehension; you're worrying, filled with unease, a sense of foreboding, even paranoia. Try some of these oils: bergamot, lavender, neroli, rose otto, melissa, or geranium.\n\nFor each type of anxiety, choose one to three oils, then use a total of thirty drops of all oils, dividing them into ten drops each, or one at five drops, one at ten, one at fifteen. Don't use any more than thirty drops at first; these oils are strong, both topically and in terms of the emotional response they can provoke. (For some people, thirty drops may be too rich a mixture; start lower, at ten drops total, then gradually increase drops over a few weeks as you get used to their effects.) Dribble the drops into an empty plastic container (such as a quart yogurt container), fill with hot bath water coming out of the tap, then slowly dribble the water plus oils into the down-flowing stream of hot bath water from the tap. This helps disperse the oil droplets into the general bath water.\n\n### _Relieving Stress_\n\nIf your primary sense of discomfort is better labelled \"stress,\" then any of the following ten essential plant oils can help relieve some of it: mandarin, geranium, clary sage, ormenis flower, eucalyptus citriodora, Roman chamomile, sandalwood, ylang ylang, petitgrain, or lavender.\n\nAs with anxiety, there are different types of stress, each requiring a different blend of aromatherapy oils. In _The Fragrant Mind_ , aromatherapy educator and practitioner Valerie Ann Worwood states that there are three basic types of stress. In type one, you feel tired, irritable, achy, and sometimes depressed. To get some relief from this kind of stress, Worwood recommends eucalyptus citriodora, geranium, and lavender, or mandarin, ylang ylang, and petitgrain, a total of thirty drops for all three oils in either formula.\n\nThe second type of stress is characterized by anxiety or depression, food allergies, persistent infections, or chronic background discomfort. Here Worwood advises clary sage, Roman chamomile, lavender, and geranium, or ormenis flower, ylang ylang, petitgrain, and sandalwood. Again, use no more than thirty drops total for all four oils in either formula. If your stress is considerable, marked by stomach pain, suicidal tendencies, fear, despair, or a general retreat from social interactions, Worwood suggests Roman chamomile, clary sage, and ormenis flower, or geranium, eucalyptus citriodora, and mandarin.\n\n### _Progressive Body Relaxation_\n\nWhile immersed in your Epsom salt-aromatherapy-infused tub, why not practice a simple relaxation exercise borrowed (and somewhat adapted) from hatha yoga. Start with your toes, and visualize (or imagine) that all the stress is being released from all the tissues, bones, cells, molecules, and atoms of your ten toes, and that the stress (or toxins) is flowing like effluent into the bath water from your toes. Next do your ankles, then legs, knees, thighs, hips, pelvis, and on, progressively, up to your head, continually releasing tension from each body area.\n\n### _Improve the Ambience_\n\nWhy not infuse the air of your bathroom with some pleasing aromatherapy scents that will contribute to the growing mood of relaxation? British aromatherapist Chrissie Wildwood recommends using one of the four following mixtures in an aromatherapy burner or diffuser (usually sold where aromatherapy oils are found) or water-based electric vaporizer. Dribble three drops of neroli, eight drops mandarin, and three of ylang ylang into a small amount of water; or Roman chamomile (three drops), rose otto (two), and bergamot (ten); or frankincense (three), lemon (six), and juniper (four); or galbanum (one), lavender (six), and petitgrain (six). Within ten minutes or so, the bathroom, and adjacent rooms to an extent, will be suffused with the relaxation-inducing scents of these essential oils.\n\nUse only one blend of aromatherapy oils per bath. Stay in the tub at least twenty minutes, and forty-five minutes if you can. Immerse yourself to your chin, if that's comfortable. When you leave the tub, don't towel off, but lie down immediately on your bed (on a dry towel) and air dry for at least thirty minutes. A fair measure of the relaxation effect of the Epsom salt-aromatherapy oils bath occurs during this critical air-drying time. You may even fall asleep during this time. Arrange beforehand not to be disturbed during this time. As a general stress-relieving technique, you might wish to do this bath once a week, at least; if you have a high stress life, then consider doing it two to three times every week. If you have the time, there is no reason not to do it every day.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor a source of aromatherapy oils and diffusers, as well as a gateway to an aromatherapy web ring (a series of related, interconnected websites called The Aromatic Life Web Ring that will connect you with many more sources of products and information), contact: Bassett Aromatherapy,\n\nP.O. Box 176,\n\nCardiff-by-the-Sea, CA 92007;\n\ntel: 800-738-8678;\n\ne-mail: joanne@aromaworld.com;\n\nwebsite: www.aromaworld.com.\n\n## HEALTHY LIVING SPACE DETOXIFIER #16: Enhance Your Relaxation by Floating in 800 Pounds of Epsom Salt\n\nIf you enjoy the Epsom soak in your bathtub, you may relish this next suggestion. For a profound introduction to relaxation, try an hour in a float tank. Flotation REST (Restricted Environmental Stimulation Therapy), originally developed in the 1950s, has you floating in very high salt concentration water inside an enclosed chamber (like a spa-sized fiberglass bathtub with a canopy) so that all your sensory input is minimized and your relaxation is optimized.\n\nApproximately 700-800 pounds of Epsom salt are added to 170 gallons of water, which are kept heated to 93.5\u00b0 F, or skin temperature. Your body experiences (no doubt unique in its experience) a gravity-free condition because the huge amount of salt keeps you effortlessly suspended on top of about ten inches of water. So you float and relax on a \"bed\" of salt water and in a darkened atmosphere of calm and warmth, free of outside stimulation or distraction.\n\nThe flotation tank shields the body and mind from external environmental stimulation, thereby reducing the body's workload by an estimated ninety percent. This in turn conserves a great deal of energy, both mental and physiological, which can be constructively directed inwards. The result is what medical science recognizes as the \"relaxation response,\" or more technically, the parasympathetic response, one pole of the central nervous system easing into relaxation. \"Tension and stress wear down our bodies, lower our resistance to illness and injury, upset our emotional balance, reduce our ability to think clearly, and even make us more accident prone,\" states a floatation tank manufacturer. \"Flotation releases deep muscular tension (often unconsciously 'locked' into the body) starting a chain-reaction that spreads throughout the body to every organ, tissue, and cell.\"\n\nPsychologists Roderick A. Borrie and Thomas H. Fine (director of the International REST Investigators Society in Toledo, Ohio) concluded that flotation REST \"serves as a powerful relaxation inducer and has clinical potential in working with patients who have stress-related disorders.\" Borrie and Fine measured the changes in key stress-indicating biochemicals in the blood of subjects using the salt flotation tanks. Subjects spent thirty to forty minutes in the tank every third day for a total number of four to twenty sessions. Levels of hormones associated with stress, such as cortisol, ACTH, and epinephrine, decreased during REST sessions, as did hormones associated with the adrenal glands (the \"stress\" organs), such as aldosterone and renin.\n\nFurther, they found that the lower levels of cortisol (a key stress-associated hormone) as well as blood pressure did not immediately reset to normal when the REST session was over, but that there was a significant carry-over effect of the benefits obtained during the immersion in the flotation tank. The researchers also studied the \"subjective\" responses of over 1,000 REST experiencers and found that by their own reports, ninety percent of the subjects found the immersion \"deeply relaxing,\" and that they had experienced an increase in positive emotions (joy, elation, happiness, contentedness) and a corresponding decrease in negative emotions.\n\nOn the basis of this data and that developed by other researchers, Borrie and Fine report that REST has merit as a medical treatment for high blood pressure, muscle tension, headache, anxiety disorders, chronic pain, insomnia, PMS, and rheumatoid arthritis. It can also work as a \"mood elevator\" for cases of depression. \"Uniquely, Flotation REST provides an effortless introduction to deep mental and physical relaxation.\"\n\nThe therapeutic benefits of floating are numerous, and include an increased release of endorphins, the pain-killing brain chemicals that make you feel good. According to research, floatation lowers blood pressure and heart rate, reduces oxygen consumption (an index of cell metabolism); increases concentration, motivation, and energy levels; improves skin quality; helps the body remove harmful stress-related biochemicals such as cortisol from the bloodstream; stimulates an increase in T-cells (key immune cells); and allows every muscle in the body to relax, surely an experience hard to come by under any other circumstances.\n\nAs the mind drifts into a deep state of relaxation, its two hemispheres start to balance out and synchronize, research suggests; creativity and problem-solving abilities can be enhanced. In the words of one manufacturer, floating users report \"a tranquil, carefree, super-relaxed feeling, a light, happy exuberance, attended by deep calmness and a sense of wholeness.\"\n\nOther benefits include fatigue reduction, alleviation of mental and physical stress, improvement in mental clarity, intensification of acuteness of all the senses; further, floating diminishes fear, depression, and anxiety; it speeds up rehabilitation and recovery from illness; it accentuates any attempt at self-hypnosis or auto-suggestion; and generally rejuvenates and revitalizes.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more information about flotation tanks:\n\nOasis Relaxation System, Biofeedback Instrument Corporation,\n\n255 W. 98th Street,\n\nNew York, NY 10025;\n\ntel: 212-222-5665;\n\nfax: 212-222-5667;\n\ne-mail: ac@inx.net;\n\nwebsite: www.biof.com\/oasis;\n\nSamadhi Tank Company,\n\nP.O. Box 2119,\n\nNevada City, CA 95959;\n\ntel: 530-477-1319;\n\nfax: 530-477-1953;\n\ne-mail: float@samadhitank.com;\n\nwebsite: www.samadhitank.com.\n\nAlso see: PathFinder, by ThinkTank International (Singapore) at www.thinktank.com.sg; and Floatworks (London, England) at www.floatworks.com.\n\nFor a partial list of flotation spas and places where you can float, see: www.floatation.com.\n\nAccording to another manufacturer, the flotation tank is designed to keep outside stimulation away, so that you can float in an enclosed, darkened environment with as little sensory input as possible. \"Sight, sound, temperature, gravity, and the presence of other people are minimized. This is the environment to examine your own responses, free of external input.\"\n\nWhy is there so much Epsom salt involved, and why Epsom in the first place? The quantity used is to ensure maximum, effortless buoyancy. The concentration of Epsom salt in the water is so dense it makes the user's body comparatively light, so it floats on top like a cork. Epsom salt (pharmaceutical grade magnesium sulphate) has a strong affinity for carbon and carbon compounds, and can absorb and nullify carbon in the form of harmful bodily waste products. The magnesium in the Epsom salt has the power to draw out various carbon-based wastes (toxic substances) from bodily tissues where they are stored, thereby enabling the body to excrete them through the skin.\n\nFlotation tanks are often available in gyms, spas, health resorts, fitness and sports centers, and can be purchased from several different manufacturers for around $5000. The salt is not changed after every use, but it can be regularly sanitized by hydrogen peroxide, ultraviolet light, or ozone with the result that there can be fewer bacteria remaining in the salt water than in the typical tap water.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #17: Use Music Therapy to Deepen Your Sense of Melting Relaxation_\n\nAll music has an effect on us, on our emotions, nervous system, and state of mind, but some music has been deliberately composed to heighten certain effects, specifically the beneficial effect of relaxation. The focused use of music to produce beneficial physiological and psychological effects is called music therapy, and while it is a large and well-researched discipline, it offers us a practical application in terms of producing deep, effortless relaxation.\n\nOne of the prime innovators in the use of specially composed music for heightening relaxation is Steven Halpern. One day in 1969, while meditating in a redwood grove in Santa Cruz, California, Halpern heard music in his head, fully realized, floating, ethereal, nonrhythmic music with sustained harmonies, overtones, and long pauses between phrases. \"It was the most beautiful, soothing, and comforting music I had ever heard,\" he commented afterward.\n\nThis \"aural vision\" answered all of Halpern's questions about what healing music might sound like. \"It was as if I had tapped into an ongoing concert, a wavelength that was broadcasting 'Music of the Spheres.' That was my quantum leap, a peak experience that changed my life.\" It also launched a very successful career as a composer of a genre of music Halpern variously calls Inner Peace Music, Inner Directed, Anti-Frantic, and Music Rx.\n\nAs a composer with a Ph.D. in music, Halpern deliberately keeps elements out of his music that will provoke a listener into thinking, anticipating, or looking ahead. These activities obstruct the relaxation response. Halpern's compositional strategy is to induce relaxation and inner harmony as fast as possible, \"to go right to the desired outcome without going through the angst and tension build-up\" that characterizes Western music. His \"music of being, not doing\" hovers around a steady tonal center so that the listener feels \"always at home\" and in the present moment, without the need to go somewhere, following a melodic theme to its future unfoldment.\n\n\"People in general not only do not know how it feels to be relaxed, but mistake bodily stimulus and intellectual entertainment as relaxing,\" says Halpern. \"At its most basic, relaxation involves slowing down the heartbeat and breathing rate, as well as shifting from the everyday beta brainwave activity to the more tranquil alpha brainwaves. When practiced regularly, relaxation can help prevent many stress-related diseases such as heart attack, insomnia, migraine, and hypertension.\"\n\nMost music, from Bach to rock, was not composed with the intent to relax its listeners, Halpern comments, but music can be composed that produces \"predictable, positive results\" of relaxation, or what he variously calls \"push button serenity\u2014as close as your CD player\" or \"stress relief at the speed of sound.\" The music creates a \"background ambience\" that in effect surrounds your body, envelops the nervous system and brain, and entrains them to a more relaxed mode of operation.\n\n**_Halpern composes music that produces \"push button serenity\u2014as close as your CD player\" or \"stress relief at the speed of sound.\" The music creates a \"background ambience\" that surrounds your body, envelops the nervous system and brain, and entrains them to a more relaxed mode of operation._**\n\nFor example, in _Higher Ground_ , Halpern creates a \"sonic entrainment matrix\" that quickly induces brain waves into states of relaxation, ease, euphoria, even harmonic resonance with the 7.83 hz electromagnetic field of the Earth itself. As a first-time listener to Halpern, I confess this CD was so richly relaxing, so luxuriously calming, that I felt as if my bones melted, as if my entire skeletal system sighed and dissolved. The mind-body, after all, has an inherent self-healing mechanism; it just needs a little boost. Halpern's intention is to bring a listener into a state of deep relaxation as a foundation for psychological insight, healing, meditative states, and spiritual illumination. \"The body heals itself most effectively in a state of deep relaxation.\"\n\n\"My sense of the true function of art is to bring us back in touch with the divine, with our true Self,\" states Halpern. When we relax, we start hearing with our whole body; one of the reasons we feel so good is that we are tapping into the wholeness of our being, linking up \"with a greater pattern of perfection that has been there all along,\" Halpern says. Halpern's music will relax a listener even if it's played as background, but focused listening is recommended. Then it becomes entrainment, not entertainment. More recent CDs by Halpern continue this focus on deliberately-induced relaxation, such as _Comfort Zone_ , _Sleep Soundly_ , _Serenity Suite_ , and _Music for Sound Healing._\n\nBear in mind that while Halpern may be one of the pioneers in this field, many other musicians have recorded music designed to heighten the relaxation response such that you will have a rich variety to choose from. You can easily get oriented to what is available by doing a web search under \"relaxation CDs.\"\n\n_Recommendation_ : Listen to a relaxation CD while in the Epsom salt-aromatherapy bath, or while air-drying afterwards; or listen to it while lying down in a comfortable place in an environment free from distraction or interruption. Do not listen to music of this type while driving a car or doing anything requiring focused, linear concentration. Remember, the reason you are using music in your detoxification program is to help you deeply relax. Tension and stress impede the detoxification process; in fact, they are toxins in their own right. So as you \"melt\" to the music and Epsom salt, know that your cells are being freed of the energy grip of stress and tension, and are now a little more free to do their detoxifying jobs.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more information, contact:\n\nSteven Halpern's Inner Peace Music,\n\nP.O. Box 2644,\n\nSan Anselmo, CA 94979;\n\ntel: 800-909-0707 or 415-485-0511;\n\ne-mail:\n\ninnerpeacemusic@innerpeacemusic.com;\n\nwebsite: www.stevenhalpern.com.\n\nThis completes our groundwork for detoxification. Now that you have a sense of the role of dietary change and stress reduction in the detoxification process, we can move on to more focused strategies for helping the body purge itself of a lifetime of toxins, starting with the liver and intestines.\n\n# CHAPTER 5\n\n# _Safe and Effective Cleansing Programs for the Liver and Intestines_\n\nYou are now in a position to start a deliberate, planned, and supervised detoxification program, focusing on the liver and intestines. These three qualifiers are carefully chosen to emphasize the key elements in a detoxification program.\n\nYou can make the changes in food and water suggested in Chapter 4 without any complications and with almost guaranteed benefit. But when it comes to relieving the liver and intestines of their toxic burden, it is well advised by detoxification authorities to do so prudently, with good advice, and in a sequential way. While there are many helpful components to a detoxification program, discussed below, they should be used in the context of a medically grounded internal cleansing program. That is, one that has solid clinical and evidential support, and one that you undertake with the supervision of, or at least consultation with, a qualified health-care practitioner.\n\n### _The Healthy Living Space Expert Interview: Jacob Farin, N.D., Naturopathic Physician_\n\nWhen it comes to finding a medically authoritative internal cleansing program, there is perhaps no better source than naturopathy. In this well-established natural medicine discipline, relieving the body of its store of toxins is paramount, and developing ever more effective natural means to do so has occupied the medical attention of naturopathic practitioners and researchers for well over a century. One such practitioner is Jacob Farin, N.D., associate physician at the Center for Traditional Medicine in Lake Oswego, Oregon.\n\nWhen Dr. Farin recommends a one- or two-week detoxification program to a client, he emphasizes that it must be centered around helping the liver, the body's prime detoxification organ. \"You must talk about the liver because the liver is the main organ that detoxifies toxic substances in the body. There are other organs that support the detoxification process, such as the intestines, kidneys, skin, and lungs, and they all help expel toxins, but everything that enters the body needs to pass through the liver, so that is the main organ to support.\"\n\n**_There are several conditions under which a person is likely to ask Dr. Farin for recommendations on detoxification. A person may want to feel better, to optimize their physiological functions. A person may be ill and suspect (or is told by a health-care practitioner) that detoxification may help. Or, more commonly, \"they get stuck.\"_**\n\nThere are several conditions under which a person is likely to ask Dr. Farin for recommendations on a detoxification program, he explains. A person may want to feel better, to optimize their physiological functions. A person may be ill and suspect (or is told by a health-care practitioner) that detoxification may help. Or, more commonly, \"they get stuck.\"\n\nSomething doesn't work for them anymore, or a well-recommended health-care regimen fails to produce the expected results. \"A lot of times with certain protocols we get stuck, and the treatment plan goes nowhere. So I use the detoxification process as a way either to jump-start my protocols or as a way to help patients make necessary lifestyle changes.\" Dr. Farin mentions that detoxification can be particularly helpful to someone who is trying to make lifestyle changes and can't. \"I've noticed that those who initiated a detoxification program had better results and had an increased ability to _sustain_ the lifestyle changes I recommended.\"\n\nWhat kind of lifestyle changes? Mainly food issues, says Dr. Farin. People eat too much, or are too dependent on sugar-based foods, coffee and other sources of caffeine, stimulants, or cigarettes. A detoxification program helps them see what their food issues are and how they use food cravings, dependencies, and indulgences to cover up emotional issues needing attention, says Dr. Farin. \"During the detoxification program, I restrict the types of foods a person eats for a week or two,\" he explains. \"When you restrict a person's foods, you take away their comfort foods, which is to say, their means of coping with stress. And when you take away that coping mechanism, they have to deal with those emotional issues head on. They discover they have been using food as a way to palliate their emotions.\" Thus an effective detoxification program has both physiological and psychoemotional benefits and, in either case, cleansing and supporting the liver holds the key to health.\n\n\"I think everyone has some degree of toxicity in their liver today,\" states Dr. Farin. Given the excessive degree of environmental and bodily pollution documented in the earlier chapters of this book, perhaps Dr. Farin's bold declaration will not strike readers as without foundation. A great number of our contemporary illnesses are the result of toxic overload, Dr. Farin explains, and central to toxic overload is a liver that is so burdened with the demands of detoxification that it becomes unable to keep up.\n\nLike clogged plumbing, the liver develops a backlog of undetoxified substances. An unwholesome feedback loop develops between the liver and intestines, like a sink drain backing up into the sink itself. For example, in most people today you can find colon-derived bacteria in the portal vein, which is a large vein that leads to the liver, Dr. Farin notes. Somehow bacteria leak through the intestinal walls and get picked up by mesenteric veins, then are passed on to the larger portal vein and transported to the liver. This isn't the way toxic substances are supposed to move in the body. But what is more abnormal, he says, is that often the liver's key immune defense agents, called Kupffer cells, are unable to engulf the incoming pathogenic bacteria and keep up with detoxifying the backflow. The condition of colonic backflow, medically referred to as intestinal permeability, \"plays a big role in liver toxic overload,\" says Dr. Farin.\n\nWhy does the liver get backlogged? Nutritional deficiencies account for some of it, and these in turn derive from malabsorption of nutrients, faulty diet (lacking the correct nutrients in their proper amounts and ratios), or from unsatisfied increased nutritional demands, as happens during a chronic illness or sustained athletic exertion. Emotional and psychological stress can also deplete the body of liver-focused nutrients, adds Dr. Farin. These stresses initiate hormonal changes in the body, requiring the body to adapt to the new, unhealthy, or unbalanced conditions. \"If you don't have the nutritional support to produce these hormones to process them, then you put considerable additional strain on the liver.\"\n\nAs a general recommendation, Dr. Farin advises doing the following one-week liver and intestinal cleansing program once a year as a kind of preventive internal housecleaning. On the other hand, if you are pregnant or suffering from an acute or serious illness, the program is not recommended. He further advises doing a detoxification program under the medical attention of a trained health-care practitioner, at least to the extent that you have someone expert on hand to call if problems or uncertainties arise. \"I would definitely consult a physician before initiating a detoxification program to rule out any conditions that would not warrant detoxifying,\" Dr. Farin notes.\n\nWhile cleansing programs that specifically focus on the intestines have merit (and are discussed in detail later in this chapter), Dr. Farin emphasizes the importance of dealing with the liver first. An intestinal cleanse without liver support, or undertaken before the liver has been cleansed and supported, can further flood the liver with toxins and potentially be counterproductive.\n\nThe benefits of a liver cleanse are impressive and easily quantified, Dr. Farin states. \"People have more energy. They feel better. They are happier. They can do more things without getting tired. They are not so fatigued in the afternoon. They find they can live without sugar and coffee, and they have a sense of empowerment that they can quit using these foods.\" Often a person emerges from the one- or two-week detoxification program with the sense of having clarified emotional issues, such as issues to do with anger, jealousy, and sadness, and having touched on uplifting ones such as happiness, radiance, and joy.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #18: Do a One-Week Elimination Diet and Liver Cleansing Program_\n\nDr. Farm's one-week liver cleansing program has three components. There is a dietary, a supplemental, and a self-care introspective element. Here are the procedural steps for each component.\n\n### _Dietary Restrictions_\n\nStart by eliminating from your diet for a week the five top food allergens: glutinous grains, corn, potatoes, eggs, and dairy products. These may comprise your favorite \"comfort\" foods, but they are also well-documented allergenic foods, capable of regularly producing food allergy reactions and symptoms. These allergic reactions in turn burden your liver and intestines with additional toxic substances or incompletely digested food components.\n\nFor the week, you should also curtail your consumption of all members of the nightshade family (eggplants, tomatoes, peppers) because they can, for many people, quietly but persistently aggravate the digestive system. Other foods eliminated during the detoxification program include oranges, strawberries, sugars, red meats, coffee, caffeinated teas, and peanuts.\n\nThere is a sound reason for curtailing allergenic foods for a week. \"Long-term allergies, to foods and substances, can be addressed through a detoxification program,\" explains Dr. Farin. \"These allergies will invariably set someone up or predispose them for illness later. A detoxification program is a great way to clear those allergens from the body.\" To determine if a patient has unsuspected food allergies, Dr. Farin typically submits a patient's blood sample to an immunoglobulin analysis, specifically for IgG and IgE, which are indicators of allergic reactions to foods (see Chapter 3 for more on this).\n\nThus the positive term for a diet that eliminates allergenic foods is \"hypoallergenic diet\"\u2014that is, a diet comprising foods known to produce no allergic reactions. Once you stop the ingestion of irritating foods, then you add liver-supporting ones. These might include vegetables medically known to help liver function, such as the cruciferous vegetables\u2014beets, broccoli, bok choy, Brussels sprouts, cabbage, cauliflower, swiss chard, and collard greens. Dr. Farin recommends eating four cups of fresh vegetables daily.\n\nIronically, for many people, these vegetables are guaranteed producers of intestinal gas and bloating. However, if these symptoms arise, it means the person would benefit from digestive enzymes and perhaps supplementary hydrochloric acid (HC1, the stomach's main digestive \"juice\"), says Dr. Farin. A gaseous reaction to eating cruciferous vegetables generally means the body cannot digest the sulfur found in these vegetables, so enzyme support for the digestive process is indicated.\n\nAs another part of this one-week program, Dr. Farin limits carbohydrate intake although he makes sure his patients get a sufficient intake of protein. In the modified elimination diet, he allows or recommends some meat products such as lamb, chicken, turkey, and fish, but in the vegan hypoallergenic diet (a stricter variation on this approach), he disallows all animal products during the detoxification program. Instead, soybean tofu is relied upon for a protein source.\n\n### _Nutritional and Dietary Supplements_\n\nThe second component of the one-week program calls for specific nutritional support in the form of supplements and specialized nutrient formulas to provide support for the liver. Here Dr. Farin relies on two commercially prepared products: UltraClear and Phyto-Pro, both requiring a health-care practitioner's prescription. They are not drugs or medicines as such; rather, they are powerful formulas precisely designed for use in a detoxification program supervised by a health-care practitioner. Dr. Farin notes that both are free of cascara sagrada, a powerful herb often found in other liver and\/or intestinal cleansing programs. This herb is widely recommended for detoxification, but its irritating effect on the intestines can sometimes be too unpleasant for users.\n\nUltraClear is a \"low-allergy potential nutritionally fortified food-based beverage mix\" designed for use by patients who have food allergies or are environmentally sensitive, according to its manufacturer, Metagenics of Gig Harbor, Washington. \"It can be used as part of a comprehensive elimination or rotation diet program as well as nutritional support for patients undergoing metabolic clearing or detoxification programs,\" says the company.\n\nA blend of rice syrup solids and rice protein concentrates provides forty-four percent of its calories from carbohydrates. It is low calorie, containing only 170 calories per 44g serving. It also offers high levels of various antioxidants needed to help the body reduce its toxic load (oxidative stress) and nutrients to support the liver's two phases of detoxification. Remember that during Phase I, the liver converts toxic compounds into intermediate toxins. In Phase II, the liver converts these intermediate toxins into substances that can be eliminated from the body, delivering them to the colon (through the gallbladder) or bladder for excretion. (For more on the liver's detoxification process, see Chapter 2.)\n\nSpecifically, among the liver-supportive nutrients is L-glutathione. This amino acid is a crucial detoxifier of foreign substances and is found naturally in the liver, but it may be deficient in a condition of toxic overload. Other liver-supportive nutrients found in UltraClear include N-acetylcysteine (an amino acid that helps make L-glutathione), L-cysteine (an amino acid), molybdenum and selenium (both trace minerals, which with cysteine help support the liver's Phase I activities), and vitamins A, C, E, and beta-carotene (all antioxidants). The product also contains numerous other vitamins, minerals, and amino acids needed for detoxification.\n\nPhyto-Pro is a natural vegetarian herb-based (\"phytonutrient\") protein powder complex that contains organic grasses, chlorella, spirulina, beneficial microflora, amino acids, adaptogens (specialized herbs that exert a nonspecific effect on the entire body by increasing resistance to stress and toxins and promoting a balancing or normalizing condition), antioxidants, and \"cleansing botanicals,\" according to NF Formulas, its manufacturer.\n\nDue to its high content of organic grasses and blue-green algae components (chlorella and spirulina), NF Formulas calls this a \"green drink,\" noting it is high in vitamin B12 and protein, and is suitable for providing nutritional support \"for those stressed by toxins in the environment and food.\" They also make a variant in capsule form called Phyto-Pro II; this contains soy sprouts and lecithin, green tea polyphenols, and the other ingredients from Phyto-Pro. Neither version contains gluten, yeast, refined sugar, or eggs.\n\nWhether you use UltraClear or Phyto-Pro, Dr. Farin typically recommends two to three doses per day for the one-week elimination diet program.\n\nIf the patient declines using UltraClear or Phyto-Pro, then Dr. Farin urges them to try an old naturopathic recipe called Bieler's Broth. You can also drink this to ensure an adequate intake of detoxifying vegetables. Bieler's Broth is a special vegetable broth made at home to increase one's nutritional status. \"It's a way to get nutrients into the body in a gentle way and so they are not so hard to digest,\" he comments. Use any or all of these vegetables: celery, zucchini, broccoli, parsley, beets, and cabbage. Chop up two cups of them and steam in two cups of filtered, pure water until they get soft. Then puree vegetables and broth in a blender. This is like a vegetable juice, but even better because you get the juice and the fiber, says Dr. Farin, who advises drinking one batch of this daily.\n\nAs an additional support for the liver during its intensified period of detoxification, Dr. Farin advises a lemon-based liver cleanse. Take the juice of half a lemon and mix with eight ounces of pure water; add, if desired, one teaspoon of flaxseed oil or olive oil, and one clove of crushed garlic (optional). Drink the beverage once daily, preferably first thing in the morning during the days of the detoxification program. The lemon supports the Phase I aspects of the liver's detoxification activities; the oils provide essential fatty acids to the body; and the garlic kills bacteria. Further, Dr. Farin encourages detoxifiers to drink at least thirty-two ounces of pure water daily, and preferably more.\n\n### _Self-Care\/Introspective Activities_\n\nIn this third component of the detoxification program, Dr. Farin recommends journal writing, hatha yoga, regular walks, meditation, prayer, letter writing, and a media fast. \"I ask people to refrain from television, radio, magazines, newspapers, and if possible, to avoid toxic people and toxic relationships.\" (See Healthy Living Space Detoxifier #14 in Chapter 4.) \"This phase helps bring a person's awareness to their emotions and to find appropriate ways to manage their emotional stress.\"\n\n### _Variation: Two-Week Modified Fast and Liver Cleanse_\n\nThis is a more intensive detoxification program, best undertaken after you have first gone through the one-week program and allowed a few months to elapse. For the first week, you follow the same procedures as described in the one-week program. In the second week, you begin a modified fast and take UltraClear or Phyto-Pro five times daily, says Dr. Farin. For three days, your entire intake is only the five doses of either formula, so that during the three days you take a total of fifteen doses, he says.\n\n\"This way, you give the liver all the support it needs. You really get the liver cleansing going by restricting the amount of toxins entering the body.\" In other words, no foods ingested equals no toxins ingested. \"You do not get the same effect from the dietary changes alone unless you are ultra-careful about the foods you consume. Even with organic foods, they can still come in contact with environmental pollutants.\"\n\nThis is different from a juice or water fast, comments Dr. Farin. \"With a juice fast, you get the high carbohydrate intake, but there is no protein ingested to support the liver, and the liver relies heavily on enzymes, which are specialized proteins. With the water fast, you get neither the carbohydrates for energy production nor the protein.\"\n\n### _Typical Results to Expect_\n\nDuring the two-week modified fast, about eighty percent of people experience a headache, ranging in discomfort from mild to severe, says Dr. Farin. There are usually three factors that contribute to the onset of these detoxification headaches: low blood sugar due to the limited calorie intake; caffeine withdrawal from going off coffee, teas, and sodas; and food allergy withdrawal symptoms, from going off allergenic foods such as wheat or com.\n\nAs a palliative, Dr. Farin advises consuming small amounts of nuts or a piece of fruit to raise the blood sugar level, and a few cups of green tea, an herbal tea with a small caffeine content but additional antioxidants to help the liver. For coping with the symptoms of withdrawal from wheat and glutinous grains, Dr. Farin advises increasing your intake of pure water.\n\nOn the positive side, you can expect to move your bowels more often. If you have not been regular, expect to become regular during the program, Dr. Farin explains. \"It's partly because I've increased the amount of vegetables in their diet to four cups daily. For many people, that is a significant increase. If they are experiencing constipation, they will start going more often due to this increased fiber intake. What also occurs due to the increased fiber consumption is that their system binds onto bile more readily. The bile passes into the intestines from the liver and has a beneficial irritating effect. So a person, if they are constipated, will start going more because of the increased presence of bile in the intestines.\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Jacob Farin, N.D.,\n\nCenter for Traditional Medicine,\n\n560 First Street, Suite 204,\n\nLake Oswego, OR 97034;\n\ntel: 503-636-2734;\n\nfax: 503-636-3250;\n\ne-mail: DrFarin@JacobFarin.com;\n\nwww.JacobFarin.com.\n\nFor UltraClear: Metagenics,\n\nP.O. Box 1729,\n\nGig Harbor, WA 98335;\n\ntel: 800-843-9660 or 253-851-3843;\n\nwww.metagenics.com.\n\nFor Phyto-Pro and Phyto-Pro II:\n\nNF Formulas,\n\n9775 SW Commerce Circle, Suite C5,\n\nWilsonville, OR 97070;\n\ntel: 800-547-4891 or 503-682-9755;\n\nfax: 503-682-9529;\n\ne-mail: info@nfformulas.com;\n\nwww.nfformulas.com.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #19: Use Natural Foods and Herbs to Aid Your Liver in Cleansing Itself_\n\nDr. Farin's detoxification programs call for a one-or two-week commitment, but there are dietary changes you can make that will provide detoxifying benefits all year round. Or you can use them as a follow-up to an intensive liver cleansing program. Here are a few recommendations culled from various health authorities and researchers. The idea here is simple: try to find ways to consume many or all of these foods and herbs on a regular basis to make detoxification and liver support a year-round activity.\n\n### _Liver Cleansing Foods_\n\nTry to incorporate some or all of these liver-cleansing foods into your regular diet: dandelion greens, watercress, and mustard greens. These bitter greens are known to stimulate bile flow and therefore to aid digestion and liver cleansing. They are also high in chlorophyll, which nutritionists believe has a cleansing and revitalizing effect on the whole body, but especially for the intestines and liver.\n\nOther whole foods with generalized liver-supporting benefits include carrots, beets, endive, grapefruit, lemons, limes, parsley, and romaine. A salad of bitter greens every day supplemented with a dose of herbal bitters formula (available in natural foods stores) is recommended by some nutritionists as a way to reduce indigestion.\n\nBurdock _(Arciium lappa_ ) is an especially helpful vegetable for the liver, enjoyed as cooked fresh root or as an herbal tea or tincture. Besides its high nutrient content, burdock root \"helps to cleanse the blood through its action on the liver, and its gentle diuretic properties promote the elimination of toxins through increased kidney function.\" It also stimulates the immune system and the production of digestive juices. Burdock seeds contain volatile oils that aid in neutralizing and eliminating toxins from the liver and body. Herbalists regard this plant as nature's best blood purifier; herbalists traditionally used burdock concoctions to treat snakebite. Burdock also stimulates bile flow, is a gentle laxative, and helps cleanse the blood by stimulating the body as a whole to improve its efficiency in removing waste products. Herbalists generally recommend three to four cups of fresh burdock root tea per day.\n\nGinger root _(Zingiber officianale')_ is also highly recommended as a liver-supporting vegetable. According to medical research, fresh ginger root can prevent the liver from destroying (metabolizing) other medicinal herbs, allowing them to remain in the bloodstream to complete their detoxifying activities. Ginger is also noted for its ability to aid the intestine in its absorption of herbs and to move their medicinal elements into the blood. Ginger root can be enjoyed fresh, grated or sliced in salads or cooked vegetable dishes; as a component in a blended vegetable juice; or as an herbal tea. Ginger is regarded as a classic tonic and stimulant for the digestive system; it also keeps the intestinal muscles toned, which in turn makes the movement of substances through the intestinal tract easier and less prone to produce irritation of the intestine's mucosal lining.\n\n**_'Burdock_ (Arciium lappa) _is an especially helpful vegetable for the liver, enjoyed as cooked fresh root or as an herbal tea or tincture. 'Besides its high nutrient content, burdock root \"helps to cleanse the blood through its action on the liver, and its gentle diuretic properties promote the elimination of toxins through increased kidney function.\"_**\n\n### _The Detoxifying Benefits of Beets_\n\nBeets are an excellent fiber source and can thereby work as a natural laxative, helping to stimulate peristalsis and purge the intestines of old fecal matter. According to traditional Chinese medicine, beets can purify the blood, benefit the liver, and moisten the intestines; they can also treat liver stagnancy and liver problems in general and relieve constipation, especially the type that results from fluid dryness. One way to enjoy beets is to mix fresh beet juice with equal amounts of carrot or apple juice; another way is to slice and saut\u00e9 them in a wok. When they are fully cooked, you can season them with soy sauce, Dijon mustard, and nutritional yeast.\n\n### _Hot Lemon Juice Cleanser_\n\nMany experts in natural healing extol the benefits of freshly squeezed lemon juice for helping the system detoxify. Rejuvenation and detoxification expert Helene Silver recommends drinking an eight-ounce glass of hot lemon juice first thing every morning during the detoxification program. Mix the juice of one-half of a squeezed lemon with eight ounces of pure _warm_ water, and add a pinch of cayenne pepper (up to one-eighth of a teaspoon is permissible) or one drop of cayenne tincture, she suggests. \"Lemon juice helps to break up and dislodge the sticky mucus deposits that tend to clog up the system,\" she states.\n\n### _Liver Cleansing Herbs_\n\nHerbalists recommend teas or herbal tinctures made from any of the following liver-cleansing herbs: gentian, barberry, yellow dock, Oregon grape, sarsaparilla, dandelion root, burdock, or milk thistle. As with the bitter greens, it is the bitterness itself that produces the beneficial effects. When the digestive system encounters the striking bitter taste of the greens or herbs, it stimulates digestive secretions throughout the gastrointestinal system, including bile flow in the liver. Anything that sparks increased bile flow benefits the liver.\n\nMilk thistle _(Silybum marianum_ ) and its active constituent, silymarin, is considered by many herbalists \"nature's offering for optimal liver protection.\" In fact, this herb is prominent among effective treatments for chronic liver disease produced by alcoholism and hepatitis. \"It is also emerging as an important supplement for those desiring to optimize liver function and maximize the detoxifying potential of this important organ,\" explains naturopathic physician Donald J. Brown, N.D.\n\nAccording to Dr. Brown, milk thistle extract protects liver cells and even helps regenerate them. Specifically, silymarin acts as an antioxidant (a fighter against free radical damage from toxic substances), helping liver cells produce glutathione (increasing its levels by up to fifty percent), a key agent in liver detoxification. Given milk thistle's multifaceted support for liver function, Dr. Brown states it benefits people exposed to toxins at work or who already have multiple chemical sensitivities. Milk thistle, Dr. Brown says, can provide \"critical support\" for the liver during a detoxification program. In general, he recommends an 8-week course of 420 mg of silymarin in three divided doses per day; after 8 weeks, reduce the dose to 280 mg daily for preventive purposes.\n\nNutrition researcher Laurel Vukovic recommends a strong, dark herbal tea (called an herbal decoction) made of the roots of dandelion and Oregon grape. To two tablespoons each of fresh sliced dandelion and Oregon grape root, Vukovic adds one teaspoon each of ginger and licorice root, simmered in three cups of water for about fifteen minutes. As part of a gentle and generalized detoxification program lasting three to six weeks, Vukovic suggests consuming three cups daily of this \"robust\" tea in between meals.\n\nNumerous liver-cleansing herbs are now available as teabags. For example, you can get roasted dandelion root in convenient teabags at most natural foods stores. Simply steep the teabag for at least three minutes in freshly boiled water then enjoy the strong roasted-grain-like beverage. Try having at least three cups per week as a general liver-supporting activity. Dandelion _(Taraxacum officinale_ ) is specifically recommended for liver conditions. Herbalists regard it as an effective blood and liver purifier. Clinical research shows that it helps the body recover from liver disease including hepatitis, as well as liver congestion.\n\nEven more impressive is the fact that it can increase bile secretion by fifty percent; liver conditions in which insufficient bile secretion is the hallmark are particularly benefited by dandelion. Further, inflammation of the bile duct and\/or gallbladder, obstruction of the bile duct, and jaundice of the liver are all relieved by dandelion. The herb can also alleviate indigestion, encourage the free flow of urine, and improve the functioning of the pancreas, spleen, liver, bladder, and kidneys. It is \"completely free\" of any toxic side effects.\n\n### _Herbal Liver Tonic_\n\nMaster herbalist David Hoffmann recommends a general liver tonic, to be consumed as a tea after each meal. Mix dandelion root (two parts), meadowsweet (two parts), fringetree bark (one part), and goldenseal (one part). Hoffmann recommends using these herbs in their whole, fresh form. Boil the herbs in water, simmer, and then strain to make the tea. Hoffmann states that the liver and entire digestive system will benefit from the use of herbal bitters, especially gentian, goldenseal, and\/or wormwood. These will have a decongestant action on the liver. While taking these bitter herbs (including dandelion root, \"the simplest and most widely applicable one,\" Hoffmann says), take some meadowsweet tincture to help the stomach.\n\n### _Bile Lubricants_\n\nAccording to naturopathic physicians Peter Bennett, N.D., and Stephen Barrie, N.D., four herbs, one vegetable, and one naturally derived substance are excellent bile lubricants. This means they stimulate and enhance the flow of bile from the liver. The herbs include dandelion root, turmeric, milk thistle, and chelidonium; the vegetable is artichoke; and the naturally derived substance is lecithin from soybeans.\n\nRegarding specific recommendations, Drs. Bennett and Barrie suggest dandelion root at the rate of 1 teaspoon of solid extract three times daily or 8 g taken as tea; 4 capsules of tumeric after each meal or 1 teaspoon of extract with water after each meal; and 500 mg of lecithin three times daily. They warn that taking too much lecithin at one time can overstimulate bile flow, producing headaches and pain in the gallbladder.\n\n### _Detoxification Support Nutrients_\n\nA carefully chosen short list of nutrients in supplement form is often recommended for a detoxification program. Bear in mind that during detoxification, your nutrient requirements are higher than usual. If you are deficient in essential nutrients, this can block key biochemical pathways and possibly make you more toxic since your body can't finish its cleanup of toxic substances. Below are a few critical nutrients with intake recommendations by Drs. Bennett and Barrie.\n\n  * _Antioxidants._ These nutrients work against free radicals, the unstable molecules created by toxic substances that damage cells. Chief among the antioxidants is vitamin C, sometimes called a \"universal antitoxin.\" Recommended general dosage: 4,000 to 20,000 mg daily (in divided doses) during detoxification. Ultimate dosage is determined by bowel tolerance; this is the dosage after which (usually within three hours) you spontaneously have one movement of diarrhea. This means your body cannot handle any more vitamin C for a while; wait about twelve hours then resume intake but at a reduced dosage.\n\nThere are two other helpful detoxification support antioxidant nutrients. Vitamin E helps protect cell membranes from damage by lipid peroxides, free radicals sometimes produced by detoxification itself. Recommended general dosage: 200 to 1,200 IU daily. Alpha-lipoic acid is both an antioxidant and a rich source of B vitamins. It helps cells generate energy and it protects the body and liver against poisoning from toxic substances such as heavy metals. Recommended general dosage: 600 mg, twice daily.\n\n  * _Amino Acids._ These are protein building blocks, and they are capable of bonding with toxins in the system and removing them from circulation. N-acetyl-cysteine (NAC) is a precursor to glutathione, a critical liver detoxification nutrient. NAC helps to maintain adequate levels of glutathione, thereby enabling the liver to complete its two phases of detoxification. Taking NAC is considered more effective than taking glutathione. Recommended general dosage: 500 mg, three times daily. Even though glutathione is the major player, Drs. Bennett and Barrie do not recommend taking it as a supplement; rather they contend that ingesting it through foods is preferable, notably fresh fruits and vegetables.\n\nGlycine is another amino acid involved in the liver's detoxification pathways. Like NAC, it is a building block in the production of glutathione. Recommended general dosage: 1,500 to 3,000 mg daily, taken between meals.\n\nMethionine helps remove fat from the liver and contribute essential molecules to the detoxification pathway. Recommended general dosage: 1,000 mg two or three times daily.\n\n  * _Oxystat._ This is a commercially prepared formula (it requires a doctor's prescription) comprising twenty-two natural substances (including the ones listed above) needed to support both phases of the liver's detoxification.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #20: Do a Natural Liver Flush to Improve Your Internal \"Plumbing\"_\n\nOne protocol many naturopaths, herbalists, and acupuncturists often recommend as part of a detoxification program is a liver flush using natural substances to stimulate the liver to flush itself clean of its toxic overload. Generally, the flush takes a few days, and there are a number of variations on the basic theme, which we will review below. The goal is to help your liver unburden itself of excess toxins and thereby enhance its natural ability to detoxify the body. The strategy behind the flush is that when you ingest the substance on an empty stomach, your liver gets stimulated (without the obstruction of other foods) to release stored toxins into the bile so they can be excreted from the body.\n\n**_The strategy behind the flush is that when you ingest the substance on an empty stomach, your liver gets stimulated (without the obstruction of other foods) to release stored toxins into the bile so they can be excreted from the body_**.\n\nWhile several different liver flush protocols are reviewed below, it is prudent to select one and follow _only one_ protocol during your detoxification program. At a future date, try another of the variations.\n\nHowever, all natural health practitioners who recommend or describe liver flushes strongly advise undertaking a flush _only_ under the supervision of a trained health-care expert. While it is a safe, self-care procedure, complications or physiological symptoms may arise, requiring qualified review. For example, the flush may inadvertently dislodge gallstones from the gallbladder; this is a good thing, but it can be painful for a short period as the stones pass through the common bile duct to the stomach. More likely is the possibility of experiencing some odd or transiently uncomfortable symptoms, so having a practitioner on call to explain them can be psychologically reassuring.\n\n### _Garlic and Cayenne Pepper Flush_\n\nNoted author, herbalist, and expert in Chinese medicine Michael Tierra, L.Ac., O.M.D., A.H.G., suggests a four-day flush, optimally undertaken in the spring or summer. It is based on drinking eight ounces of distilled water with eight ounces of apple juice blended with one to four cloves of garlic, a small piece of fresh ginger, and one to four tablespoons of olive oil. This is to be consumed every morning on an empty stomach for four days. Dr. Tierra advises following this immediately with a cup of herbal tea made from dandelion root and fennel seed.\n\nDr. Tierra further recommends eating a generous amount of green vegetables or going on a warm apple juice fast during this four-day period; add a teaspoon of agar flakes to the juice. Take two capsules of cayenne pepper, or one-half teaspoon cayenne powder mixed in a small amount of water, three times daily, he adds. Cayenne pepper (its major component is called capsaicin) elevates the activities of at least two liver enzymes and stimulates the mobilization of lipids from fat tissue, a necessary step in detoxification.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Oxystat, see:\n\nNF Formulas,\n\n9775 SW Commerce Circle, Suite C5,\n\nWilsonville, OR 97070;\n\ntel: 800-547-4891 or 503-682-9755;\n\nfax: 503-682-9529;\n\ne-mail: info@nfformulas.com;\n\nwebsite: www.nfformulas.com.\n\nFor Bupleurum Liver Cleanse and Turmeric Extract Full Spectrum: Planetary Formulas and Planetary Herbology,\n\nP.O. Box 275,\n\nBen Lomond, CA 95005;\n\ntel: 800-717-5010;\n\nfax: 831-336-4548;\n\ne-mail:\n\nherbcourse@planetherbs.com;\n\nwebsite: www.planetherbs.com.\n\nFinally, Dr. Tierra advises taking two liver-supporting commercially prepared supplements during the four-day flush: Bupleurum Liver Cleanse (containing thirteen herbs such as wild yam root, dandelion root extract, ginger root, and milk thistle) and Turmeric Extract Full Spectrum at a dose of two tablets of each formula three times daily. The liver flush, says Dr. Tierra, \"is invaluable for dredging and removing stored toxins and fat accumulated from a sluggish liver and stored throughout the body.\"\n\n### _Organic Apple Liver Flush_\n\nHerbalist and rejuvenation expert Helene Silver recommends this liver-flushing beverage\u2014based on the juice of organic apples\u2014as part of a detoxification program. The malic acid in the apples helps to cleanse the intestines and heal inflammation of its mucus membranes, says Silver, while the other components stimulate the liver to dump toxins. Combine freshly squeezed organic apple juice (one cup), freshly squeezed lemon juice (from two lemons), grated fresh ginger root (one teaspoon), crushed garlic cloves (one to three), cold-pressed saf-flower oil (one tablespoon), cold-pressed olive oil (one tablespoon), and cayenne pepper (a pinch) or cayenne tincture (one drop).\n\nOnce the ingredients are combined, consume the beverage in the morning, and do not eat any foods for two hours afterwards. However, Silver recommends drinking one or two cups of a specially prepared Herbal Purification Tea, consisting of fennel seeds (one-fourth cup), fenugreek seeds (one-fourth cup), licorice root (one-fourth cup), flaxseed (one-half cup), and peppermint leaves (one-half cup). Mix four teaspoons of this dry blend in four cups of pure water, boil, simmer and steep for ten minutes each; strain out the herbs and drink.\n\n### _Citrus and Olive Oil Flush_\n\nAcupuncturist-herbalist Christopher Hobbs, L.Ac., offers this variation on the liver flush, noting, \"I have taken liver flushes for many years now and can heartily recommend them.\" In general, Dr. Hobbs recommends doing this flush in the morning, preceded by some gentle exercise, such as stretching and breathing exercises (hatha yoga, for example). It is best to eat nothing for an hour after taking the flush mixture, he says.\n\nProduce one cup of fresh juice by squeezing fresh lemons or limes, then dilute the juice slightly with some distilled or pure spring water. Add to this the juice of garlic (one to two cloves, freshly pressed) and ginger (freshly grated and squeezed in a garlic press for the juice). Then add one tablespoon of extra virgin olive oil to the citrus, ginger, and garlic juice. Blend or shake the ingredients until all are mixed together, then drink the mixture, says Dr. Hobbs. An hour later, drink two cups of Polari Tea, an herbal blend containing dry portions of fennel (one part), flax (one part), burdock (one-fourth part), licorice (one-fourth part), fenugreek (one part), and peppermint (one part). Simmer all the herbs except the peppermint for twenty minutes, then add the peppermint and steep the entire mixture for ten minutes; then consume, says Dr. Hobbs.\n\nContinue with the liver flush for ten days, taking the flush mixture once every morning, followed, an hour later, by the herb tea, he says. He recommends taking a three-day break, then resuming the flush for another ten days. He further advises doing the flush twice a year, preferably in the spring and fall, two cycles each time.\n\n### _Castor Oil Compress_\n\nMany natural medicine practitioners recommend the use of a compress of castor oil (also known as ricinus oil) applied to the liver during the liver cleansing program. The therapeutic benefits of castor oil compresses were first introduced to Western medicine by Virginia psychic Edgar Cayce in the 1930s, and the practice has been taken up by many people ever since, often for relief of menstrual cramps or pain in the joints.\n\nTo prepare a castor oil pack, lightly heat enough castor oil to thoroughly wet, but not soak, a ten-by-twelve-inch flannel cloth. Immerse the flannel in the hot oil, then fold to make three to four layers and place against the skin over the liver (at the bottom of the rib cage on the right side). The oil helps to draw out toxins, release tension, and improve blood circulation, especially in the lower abdomen. Wrap a heating pad or hot water bottle in a towel and place this over the pack, then cover pack and bottle with another towel to retain heat. Keep in place for one to two hours. After you remove the compress, spend five to ten minutes gently rubbing small amounts of extra virgin olive oil into the skin over the liver. Following the treatment, you can wrap the oil-soaked flannel in plastic and store it in the refrigerator for later use. After the flannel has been used twenty times, discard it.\n\nIt is generally recommended that you apply the castor oil pack three times per week for about three months, but for women, not during menstruation. The therapeutic action behind the castor oil pack is that the oil is absorbed through the skin and stimulates blood flow and thereby produces improvements in sluggish liver (even cirrhosis), gallbladder inflammation, constipation, and numerous other conditions. Cayce stated that the technique would also stimulate lymph circulation, facilitating the drainage of toxic substances collected by the lymph system.\n\n### Intestinal Cleansing: Emptying Out a Storage Space the Size of a Tennis Court\n\nOur intestines are not something we routinely, or perhaps ever, think about, yet when it comes to detoxification, they are crucially important. Here is a fact that may help spark a different attitude about your small intestine. If you were to stretch out smooth and flat the inner convoluted surface of the small intestine, which is about twenty-one feet long in an adult, it would cover the size of a tennis court. Think about your small intestine in this way for a moment: think about a lifetime of food choices, of digestion, assimilation, and elimination, or of indigestion, constipation, gas, cramps, or infrequent elimination. What have you put into your tennis court? Is what you put in still there?\n\nThe small intestine, one inch in diameter and stuffed in coils into a small space in your abdomen, absorbs about ninety-five percent of your dietary fats and ninety percent of the amino acids from your foods. Its inner surface, due to a remarkable design concept, is expanded about 600 times by being folded into millions of villi. These are finger-shaped projections and creases, little corners and in-tuckings, called crypts. Each villus has special cells for nutrient absorption and a minute lymph vessel called a lacteal. Each villus is made of thousands of smaller units, called microvilli. These act like a microscopic brush border, facilitating absorption at the molecular level. The purpose of all this intricate design is to multiply the area available for nutrient absorption. Food typically remains in the small intestines from two to six hours in a healthy individual (see figure 5-1).\n\n**Figure 5.1. Large and Small Intestines**\n\nThe large intestine, or colon, is five feet long and two and one-half inches in diameter. There is less of it and it is more ponderous, forming a kind of fence or border around the small intestine. The large intestine is responsible for reabsorbing water from digested foods, that is, foods processed and moved on by the small intestine. The large intestine organizes the various undigestible cellulose fibers as well as some of the \"friendly\" bacteria (resident in the colon and responsible for maintaining intestinal health) for eventual elimination as feces.\n\nAverage transit time in the colon is about twelve hours. This means the amount of time it takes for food\u2014say yesterday's lunch\u2014to pass through the entire gastrointestinal tract, from the mouth though the twenty-six feet of the small and large intestines, and be excreted as feces. Slow transit time is the same as constipation, a problem that concerns many today, especially the elderly. A normal, or healthy, transit time is less than twenty-four hours, although due to the low-grade toxic condition of many people today, this norm is not widely found any more. Slow transit time means that toxic substances present in the feces\u2014for example, growth hormones and pesticides from animal products\u2014remain longer in the intestines where they can begin working as carcinogens. Fast transit time means they don't have the chance to do any harm. Remember, in a constipated person, the transit time for both intestines can be more like thirty to fifty-five hours, according to medical studies.\n\nSlow transit time means more than constipation, even unnoticed or uncomfortable. It means immune system stagnation. That is because both the large and small intestines are functional parts of the body's immune system. In fact, \"the intestinal mucosa and some of its sub-mucosal structures constitute the largest immunological system of the body,\" explains Serafina Corsello, M.D., an expert in intestinal detoxification and director of the Corsello Centers for Nutritional-Complementary Medicine in Huntington, New York. The mucus layers of the intestines collect cellular debris and pathogenic material as the active, defense front of the immune system.\n\nAside from the skin, the intestines constitute the body's largest organ, thereby being an excellent place to mount a major immune system initiative. According to medical estimates, the intestines account for some eighty percent of the body's immune and lymphatic resources. They are one of the body's first \"screening systems\" against the daily intake of toxic substances, such as bacteria, viruses, and parasites (discussed later in the chapter), that if not dealt with, would move deeper into the body's blood and tissues, creating illness.\n\n\"A healthy intestine is immunologically _vigilant_ against undesirable pathogens and toxins,\" says Dr. Corsello, but one that is \"overburdened and compromised\" cannot be an immune defensive front. Toxic intestines can thereby contribute seeds of illness to the body. When the intestine's immune vigilance falters or is dysfunctional, then our health is in trouble.\n\nThe crucial relationship between intestinal health and overall physical well-being has been recognized for a long time. Almost a century ago, a prominent natural healer named John Harvey Kellogg, M.D., founder of the Battle Creek Sanitarium in Michigan (and the cereal empire named after him), declared that ninety percent of the diseases of civilization are the result of \"improper functioning of the colon.\" In a sense, it's all due to transit time.\n\nAbout one century ago, most Americans had a short intestinal transit time, of about fifteen to twenty hours, but today, it is much more likely to be fifty to seventy hours. That means that today's lunch will sit inside your body, unprocessed, or being _slowly_ processed, for the next three days, or longer. This means there is more time for the stool to putrefy, to spoil inside you, and thereby for harmful intestinal bacteria to flourish and for toxins to develop and enter the bloodstream, taxing the liver. It is a bit like having meat going bad inside your small intestine; in fact, if you are a meat eater and have slow transit time, that is essentially what happens. If you eat low-fiber, high-fat, high-calorie, processed, prepared, or fast foods, especially ones high in mucus (dairy and flour), your transit time is slowed down even more.\n\nSlow transit time and colonic stagnation are the foundation for what physicians call a toxic bowel, and this is the seedbed for numerous illnesses. These can include chronic headaches, allergies, skin blemishes and acne, infections, various autoimmune disorders, in which a confused immune system starts attacking the body's own tissues, and cancer, among many others.\n\nThe colon-cancer connection is scientifically well established, ever since a key study of 88,751 women was published in 1990. Women, aged thirty-four to fifty-nine, without a history of cancer or intestinal pathology, were interviewed in 1980. In 1986, they were interviewed again, and this time, 150 cases of colon cancer were noted. The researchers were able to \"positively\" associate the intake of animal fat and the heightened risk and incidence of colon cancer. In fact, the risk of getting colon cancer was 2.5 times higher among women who ate beef, pork, or lamb as a main protein dish every day. Consumption of processed meats and liver was also \"significantly associated\" with increased colon cancer risk. In contrast, skinless chicken and fish were related to a decreased risk.\n\nThe scientists further noted a correlation between low fiber intake from fruits and vegetables and the risk of colon cancer. \"These prospective data provide evidence for the hypothesis that a high intake of animal fat increases the risk of colon cancer,\" the researchers concluded.\n\nThe essential point for our purposes here is not only the conclusion that animal products in themselves may act as carcinogens (based on their injurious chemical content of hormones, pesticides, and other contaminants), but that they take longer to digest (due to their low fiber content), may be incompletely digested, and are more likely to contribute to delayed transit time.\n\nIn 1990 researchers provided proof that high-fiber diets may help prevent colon cancer. Researchers at the Fox Chase Cancer Center in Philadelphia, Pennsylvania, studied the evidence from thirty-seven studies on the relationship of diet, fiber, and cancer. They reported that the majority of the cases \"gave support for a protective effect associated with fiber-rich diets.\" A related study in Majorca, Spain, of 286 cases of colorectal cancer (compared to 295 cancer-free subjects) showed that not only was an increased risk of colon cancer associated with the consumption of fresh meats, but a high intake of cruciferous vegetables \"afforded protection\" against the onset of cancer.\n\nOnce slowly or incompletely digesting foods, such as animal fats and protein, start to spoil in the intestines, free radicals are produced. These unstable toxic substances are created by the body's metabolism or brought into the body from toxic materials in the environment, including foods. Once produced and distributed, free radicals start to damage and destroy cells; they are also regarded to be carcinogenic.\n\nResearchers at the Biomedical Engineering Center at Purdue University in West Lafayette, Indiana, studied the production of free radicals from bacterial metabolism inside the colon and were amazed to find it corresponded to that which would be produced by over 10,000 rads of gamma radiation per day. They noted a connection between the high free radical levels and high iron concentrations in the feces; the iron interacts with bile pigments to support free radical generation from resident bacteria.\n\n\"Such free radical generation in feces could provide a missing link in our understanding of the etiology of colon cancer,\" the scientists concluded. It also helps explain why most gastrointestinal tract cancer occurs in the rectum and colon, and why a higher incidence of colon cancer is observed in red meat eaters. It is because red meat increases the iron content of the feces, and the excessive animal fat increases the fecal content of bile pigments and \"procarcinogens.\" These are substances (in this case, bacteria) that, once oxidized (by free radicals), can become full-fledged carcinogens. The presence of animal fats in the colon provides the ideal conditions for this.\n\n**_Remember the basic equation: high fiber means faster transit time, which means less chance of fecal stagnation and toxicity, which means less chance of free radical production, which, in turn, means less chance of toxicity-related diseases (such as cancer) developing._**\n\nRemember the basic equation: high fiber means faster transit time, which means less chance of fecal stagnation and toxicity, which means less chance of free radical production, which, in turn, means less chance of toxicity-related diseases (such as cancer) developing.\n\nA toxic bowel almost always means a body-wide state of toxicity exists, in which toxins \"leak\" out of the dysfunctional intestines into the bloodstream, the liver, lymph, and other body tissues. Your body now exists in a state of \"intestinal toxemia and autointoxication.\" Your body is absorbing its own toxic waste products, like a drain or garbage disposal backing up into the sink. \"It is the root cause of many of today's disorders and illnesses,\" explains Bernard Jensen, N.D., a naturopathic educator, author, and detoxification expert. Death begins in the colon, he states.\n\nThe bowel holds on to waste material much longer than people suspect, says Dr. Jensen. He cites the case of an autopsy that revealed a stagnant colon weighing forty pounds; in other words, that body had accumulated nearly forty pounds of uneliminated fecal matter over a lifetime. Dr. Jensen also describes a distended colon that was nine inches in diameter but had only a pencil-width open passage for the movement of feces. The rest of the nine inches was taken up with multiple layers of encrusted mucus and rubber-like feces.\n\n**Figure 5-2. Typical Symptoms and Results of Intestinal Stagnation and Toxicity**\n\nHealth-care practitioners at the Angel Healing Center, a colonics and detoxification center in Las Vegas, Nevada, amplify Dr. Jensen's observation. \"The average American male colon today carries within it about five pounds of putrid, half-digested red meat, plus another five to ten pounds of foul toxic waste impacted for years in the folds of the colon with mucus.\" They also cite Dr. Kellogg as having made this remark: \"Of the 22,000 operations I have personally performed, I have never found a single normal colon.\" What would Dr. Kellogg say one hundred years later about the state of the North American colon?\n\nFollowing Dr. Kellogg's original research, Dr. Jensen claims that if you have only one bowel movement per day, the large intestine will hold the residue of six previous meals; if it holds nine or more meals, you have chronic constipation. Your intestines are healthy, he says, if you eliminate the residue of a given meal fifteen to eighteen hours after consuming it (see figure 5-1).\n\nConstipation is only one of many signs of intestinal toxemia, states Dr. Jensen. According to his research, at least seventy-four different medical conditions can be attributed to improperly functioning intestines, including depression, drowsiness, concentration difficulties, indecision, dry eyes, nausea, bad breath, body odor, insomnia, arthritis, high blood pressure, ovarian cysts, boils, clammy skin, muscle twitches and inflammations, and bladder infections. Intestinal toxemia generally shows up in any of sixteen major symptom categories (see figure 5-2), says Dr. Jensen, who adds, \"All of these conditions have responded to therapy for intestinal toxemia.\"\n\nThere are many effective therapies for purging the intestines of its possibly lifelong backlog of uneliminated fecal matter. Many natural medicine practitioners recommend some form of colon hydrotherapy, more popularly known as colonics. These procedures involve flushing the large intestines with filtered and temperature-regulated pure water to soften the hard, impacted feces; the water is introduced through the rectum with a sanitary plastic tube. The intestines are stimulated to pass a fair amount of the stored waste matter. Sometimes herbs, ozone, and other natural substances are used in the colonics program as well. Colonics require a trained practitioner, which means you will need to visit a facility to have the procedure. It is painless and almost always effective.\n\nWhile colonics is a viable option for intestinal cleansing, it does not necessarily provide a thorough cleanse of the full twenty-six feet of both small and large intestines, and it is not something you can do on your own. The next section reviews six substances you can take on your own as part of a gentle, home-based, self-care detoxification program with modest cost and no risk.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #21: Use Six Natural Substances to Help Cleanse Your Intestines_\n\nJust as herbs, nutrients, and natural substances can be used to stimulate and support liver decongestion, so are there natural substances perfectly suited to get your intestines into a cleansing mode. In large measure, cleansing has a laxative effect. When you stimulate your intestines to cleanse, they will increase their evacuation of old fecal matter. So from an experiential viewpoint, using any of the substances described below will seem as if you have taken a remarkably effective laxative.\n\nHowever, do not use all six substances at the same time. You may wish to try a few at one time, but it is probably most prudent to use _only one_ at a time, depending on the desired effect you wish; there are exceptions to this, which are reviewed at the end of this section. Some can be used for an extended period of time (such as bentonite clay, which might be used productively for a month), but others are meant only for a short time (charcoal and psyllium husks, which shouldn't be used much more than seven to ten days).\n\n### _Activated Charcoal_\n\nActivated charcoal, a reliable and very effective poison absorber, is a traditional remedy with a long history of use in many cultures. The charcoal is prepared by carbonizing organic matter such as wood or peat in a kiln, then activating it with oxidizing gases (steam or air) at high temperatures. One typical source of the carbon matter is coconut shells heated in the absence of air; then the partially burned shells are granulated to a size that maximizes their optimal absorption capacity. The high steam (applied at 800\u00b0 C for 12-24 hours) opens up and enlarges the pores of the charcoal, thereby creating an enormous internal surface area, estimated to be 800 square meters per one gram of charcoal.\n\nIf you suspect food poisoning from tainted restaurant food, taking two to four charcoal tablets (easily obtained at drugstores and natural foods outlets) can help reduce the toxic effect on your intestines and hasten the removal of these toxins from your body. The charcoal absorbs the poisons in the small intestines along with chemicals secreted in the bile, thereby preventing their re-uptake into the liver and blood circulation. It is even more effective when combined with two to four drops of oil of oregano. Charcoal is also effective against flatulence and excessive intestinal gas.\n\nAs a general aid for detoxification, activated charcoal is often recommended to help in the systemic clearance of drugs and intoxicants and is regarded as a \"universal antidote\" to the ingestion of poisonous substances. Activated charcoal helps detoxify the body in several ways: it purifies the six to eight liters of various digestive fluids secreted daily in the body; it absorbs noxious chemicals (both the poisons and their breakdown products, called metabolites) from the bile duct; it absorbs ingested drugs that would otherwise diffuse back into the stomach and small intestine; and it reduces the liver's detoxification burden. Further, charcoal can absorb a fair amount of the toxins produced by a systemic yeast overgrowth of _Candida albicans_ , preventing these toxins from being reabsorbed into the bloodstream. Charcoal is also said to counteract decomposition products from ingested foods (such as beans) that would otherwise produce flatulence, diarrhea, or odorous stools.\n\nIn fact, according to one authority, \"activated charcoal is the best single supplement for enhancing detoxification . . . [understood to be] an ongoing biological process that prevents toxins (from infectious agents, food, air, water, and substances that contact the skin) from destroying health.\" Richard C. Kaufman, Ph.D., recommends twenty to thirty-five grams of activated charcoal as a daily detoxification protocol, divided into two or three doses (morning, midday, and before bed\u2014on an empty stomach in all cases), taken on two consecutive days for one week. For intestinal complaints, he recommends 4-10 g taken twice daily; for _Candida albicans_ , 20-35 g daily until the problem is cleared.\n\nHowever, charcoal also removes nutrients from the intestines, so it should not be used routinely, or if employed during the detoxification program, it should not be used regularly afterwards. Charcoal can produce short-term constipation and, of course, black-gray stools.\n\n### _Bentonite, or Liquid Clay_\n\nThe term bentonite refers to a natural clay deposit found in cretaceous rocks, notably in Fort Benton, Wyoming, and throughout the Great Plains region of the United States, but also in France. In France, the active mineral in bentonite is called montmorillonite and is found in weathered volcanic ash that has become clay over time. A third term, smectite, is sometimes used to describe this same therapeutically active natural substance. Essentially, all three terms are interchangeable.\n\nFor our purposes, the key information is that bentonite or liquid clay\u2014the clay is hydrated with water into the consistency of a very thick milkshake or tasteless pale grey gel\u2014contains minerals that can absorb toxins from and deliver nutrients to the gastrointestinal tract. Since liquid clay is inert (not biologically active), its effects are physical, not chemical, and it passes undigested through the system, bearing its toxic load out of the body through the stool.\n\nLike charcoal, bentonite has an extensive adsorptive surface area. So vast is this area, that one manufacturer claims a quart bottle represents 960 square yards of surface area, or about the size of twelve American football fields. Once inside the body, bentonite adsorbs toxins such as heavy metals, pesticides, and free radicals by attracting them to its generous surface area; then it gathers the poisons up for excretion in a way that has been compared to how a magnet draws filings to it.\n\nThe research supporting the beneficial detoxifying claims for bentonite is solid and impressive. Bentonite expands like a super-absorbent sponge and can adsorb selected pathogenic viruses (reovirus type 3 and a bacteria, coliphage Tl, both common to sewage, aquatic, and land environments), according to a study involving soils and sediments. The study showed that bentonite has a \"specificity\" to adsorbing \"mixed populations\" of these microorganisms. Another study suggested that bentonite clay solids had some ability to collect poliovirus microorganisms from tap water. A Russian study showed that bentonite can effectively adsorb selected species of poliomyelitis type II viruses, monkey disease rotavirus, Coxsackie B1 and B6, and Picornaviridae and Reoviridae.\n\nIt also has been observed to specifically bind up the herbicides Paraquat and Roundup as well as aflatoxin (a toxic mold that infests grains and nuts). A poultry study involving 528 chickens showed that bentonite \"partially neutralizes\" the toxic effects of aflatoxin, a potent developmental poison for animals. Comparable results were shown in a rat study; rats were fed an aflatoxin-infected diet, then given bentonite and another binding substance. None showed any developmental defects, indicating the bentonite and second substance had adsorbed or neutralized the aflatoxin. A pig study showed bentonite added to aflatoxin-contaminated corn feed led to the partial restoration of liver performance and function, again indicating the clay's ability to collect and deactivate the toxins. A cow study showed that bentonite could remove a toxin (lantana, a dairy poison) as effectively (but not as quickly) as activated charcoal, thereby showing \"promise as a cheap alternative\" to activated charcoal for lantana poisoning.\n\nAccording to Ran Knishinsky, a bentonite enthusiast and author of _The Clay Cure_ , the benefits of using bentonite for only two to four weeks are numerous and impressive. They include improved regularity in intestinal evacuation; relief from chronic constipation, diarrhea, indigestion, and ulcers; an increase in physical energy; improved complexion; greater mental alertness; uplifted emotional mood; and increased resistance to infection. In other words, with liquid clay you get a lot of the general benefits expected from detoxification.\n\nKnishinsky contends that taking one to three tablespoons of liquid clay daily, in divided doses, can help eliminate intestinal parasites, provide relief from allergies and hay fever, clear up acne, and turn around anemia (because of its high iron content). Liquid clay manufacturers state that the substance shines as a general internal cleansing agent, because it scrapes and cleans the lining of the intestines at the same time as feeding it with essential and trace nutrients.\n\nOne liquid clay product (Mineral-Rich by Earth Essentials\/ White Rock) offers seventy-one trace and ultra-trace minerals, which the body needs to be able to absorb the bigger nutrients like vitamins. It works as both a mineral supplement and an internal body cleanser. According to this manufacturer, the bentonite begins adsorbing toxic substances as soon as it reaches the stomach (in this case, excess stomach bile); then it moves into the small intestine where its large surface area adsorbs toxins and releases minerals. Users of this product report that a variety of symptoms have abated, including hair loss, acne, insomnia, brittle nails, allergies, joint problems, spastic colon, digestive problems, and low energy.\n\nSeveral brands of liquid clay are available in natural foods stores. A general usage recommendation is to start with one tablespoon daily mixed with juice or water and observe the results for a week. Gradually increase your intake to up to four tablespoons daily, in two divided doses, morning and evening. As part of an intensive detoxification program, manufacturers suggest a _short-term_ dose of one tablespoon per hour through the day. You will probably notice a gentle laxative effect in the range of about a twenty-five percent increase in daily intestinal elimination. Use the bentonite for up to a month, then take a month off, and repeat, if you wish. It is important that you drink extra amounts (at least double your normal intake) of pure water during the time you take liquid clay.\n\n### _Chlorella_\n\nThis single-celled freshwater green algae _(Chlorella pyrenoidosa_ ) is noted for its ability to eliminate toxins from the body through the feces. Chlorella binds to toxic metals and chemicals, thereby facilitating their removal. The substance also has antibacterial and antiviral properties and is believed to stimulate the immune system. According to Earthrise, a chlorella manufacturer in Petaluma, California, the hard cellulose cell wall of chlorella helps to remove impurities from the body, and it may be involved in absorbing toxins from the intestines. Earthrise further states that chlorella \"may have the ability to improve bowel function and stimulate the growth of beneficial [intestinal] bacteria.\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Mineral-Rich:\n\nEarth Essentials,\n\nP.O. Box 910,\n\nSpringville, UT 84663;\n\ntel: 888-328-2529;\n\nfax: 801-489-3175;\n\ne-mail: support@earthessentials.net;\n\nwebsite: www.earthessentials.net.\n\nFor Great Plains Bentonite:\n\nYerba Prima, Inc.,\n\n740 Jefferson Avenue,\n\nAshland, OR 97520;\n\ntel: 800-488-4339 or 541-488-2228;\n\nfax: 541-488-2443;\n\ne-mail: yerba@yerba.com;\n\nwebsite: www.yerbaprima.com.\n\nFor Chlorella:\n\nEarthrise Nutritionals, Inc.,\n\n424 Payran Street,\n\nPetaluma, CA 94952;\n\ntel: 800-949-7473;\n\nfax: 707-778-9028;\n\ne-mail: info@earthrise.com;\n\nwebsite: www.earthrise.com.\n\nFor Triphala Internal Cleanser:\n\nPlanetary Formulas\/Planet Herbs, P.O. Box 275,\n\nBen Lomond, CA 95005;\n\ntel: 800-717-5010;\n\nfax: 831-336-4548;\n\ne-mail: herbcourse@planetherbs.com;\n\nwebsite: www.planetherbs.com.\n\nFor Ultimate Cleanse:\n\nOmni Nutraceuticals,\n\n5310 Beethoven Street,\n\nLos Angeles, CA 90066;\n\ntel: 800-841-8448; fax: 310-306-1790;\n\nproduct information: 888-297-3273;\n\nwebsite: www.naturessecret.com.\n\nChlorella, with a higher chlorophyll content per gram than any other plant and a complete array of essential amino acids, is known as the \"unpoisoner\" because it can detoxify the body of cadmium, lead, and the effects of uranium radiation. One way chlorella might do this is through its antioxidant effects on albumin.\n\nAlbumin, a substance continuously secreted by the liver and the most abundant protein found in human blood, is a major antioxidant. It appears both to transport vital nutrients and remove toxins from cells in the body's connective tissue. Perhaps because of albumin's detoxifying effect, high blood levels of albumin have been linked with longevity, which is to say, low levels of albumin are associated with increased states of toxicity. One estimate has it that chlorella supplementation can raise albumin levels by sixteen percent to twenty-one percent.\n\nA Japanese study demonstrated that chlorella is a first-rate detoxifying agent capable of removing alcohol from the liver. The study showed that taking four to six grams of chlorella before consuming alcohol prevented hangovers in ninety-six percent of cases, even following a night of heavy drinking. Numerous studies demonstrate that chlorella is effective in use against tumors and in reducing the negative effects of chemotherapy and radiation treatments (and against gamma rays in particular). Another chlorella manufacturer states, \"Numerous research projects in the United States and Europe have indicated chlorella can also aid the body in the breakdown of persistent hydrocarbon and metallic toxins such as DDT, PCB, mercury, cadmium, and lead, as well as strengthening the immune system response.\"\n\n**_A Japanese study demonstrated that chlorella is afirst-rate detoxifying agent) capable of remouing alcohol from the liver. The study showed that taking four to six grams of chlorella before consuming alcohol prevented hangovers in ninety-six percent of cases, even following a night of heavy drinking._**\n\nRandal Merchant, M.D., of the Department of Neurosurgery at the Medical College of Virginia provides support for this last claim. Dr. Merchant is conducting clinical trials on chlorella and has summarized much of the key research on chlorella's health and detoxification benefits on a health-related website. Dr. Merchant states that a physician at Kitakyushu City Institute for Environmental Pollution Research in Japan gave thirty patients with PCB exposure daily chlorella doses of 4-6 g for one year, after which time, the key symptoms of PCB toxicity\u2014exhaustion, poor digestion, abnormal bowel movements\u2014had all improved. From his own research he reports that twenty patients with brain tumors took chlorella (20 g in tablet form, and 150 ml in liquid form) during their cancer treatment and experienced fewer than expected respiratory infections and flu-like illnesses. They stated the chlorella helped them maintain their strength and resist common colds and infections prevalent among cancer patients.\n\nChlorella seems to be effective in reducing symptoms in many conditions that involve toxicity. A study at the School of Medicine of West Virginia University showed that chlorella, given to rats, increases the rate of detoxification of a harmful pesticide called chlordecone. Chlorella was shown to remove the toxin from the body twice as fast as the control substance; it reduced the half-life of the toxin from forty to nineteen days. Chlorella is also competent at enhancing the excretion of the heavy metal cadmium in people suffering from cadmium poisoning. Once chlorella binds with cadmium, it does not re-release it to the body while still in the intestinal tract, but removes it definitively from the body.\n\nChlorella given daily to patients with fibromyalgia (chronic, often body-wide, muscle pain) for two months (at a dosage of 10 g of a dry powder and 100 ml of liquid form) produced a twenty-two percent reduction in pain intensity. In a similar study (conducted by Dr. Merchant), 34 fibromyalgia patients took 50 tablets of chlorella and 100 ml of liquid chlorella every day for three months, after which they reported a fourfold decrease in pain and a 25% reduction in anxiety.\n\nAccording to Dr. Merchant, chlorella achieves its detoxifying effects through at least three mechanisms (in addition to the albumin explanation above). First, it stimulates the body's immune system to produce more macrophages, immune cells that \"eat\" toxic substances. Macrophages \"actively clean the blood, body fluids, and cavities of harmful substances,\" explains Dr. Merchant. Second, in the intestines, chlorella stimulates the growth of beneficial microflora, specifically the \"good\" intestinal bacteria called Lactobacillus; this family of bacteria help to destroy foreign and toxic substances and keep the ecology of the numerous microorganisms present in balance. Third, chlorella's thick cell wall adsorbs toxins in the intestine, and in so doing, reduces stool odor.\n\nUnlike activated charcoal, bentonite, and psyllium (discussed below), chlorella can be taken both during the intensive phase of a detoxification program and regularly as a supplement. During the detoxification program, you might wish to take more chlorella than during routine times, due to the increased mobilization of toxins in the body as a result of the various herbs and fibers you are taking to stimulate cleansing and evacuation.\n\n### _Triphala, an Ayurvedic Herbal Cleanser_\n\nTriphala, an Ayurvedic herbal medicine, is the most popular herbal formula in India. Ayurveda, which is India's classical and ancient medical system\u2014still practiced today there and around the world, including to a limited extent in the United States\u2014holds that Triphala is an effective but gentle laxative, capable of supporting the body's strength.\n\nThe name means \"three fruits,\" and refers to the three Indian herbs used in it: _Haritaki (Terminalia chebula), Amalaki (Emblica officinalis)_ , and _Bhibhitaki (Terminalia belerica)._ Each of these herbs is said to rejuvenate one of the three constitutional categories (called _doshas)_ described in Ayurveda, namely, _vata, pitta_ , and _kapha._ Haritaki balances the air element (vata) and the large intestine; Amalaki balances the fire element (pitta) and the liver; and Bhibitaki balances the water element (kapha) and reduces mucus and fats in the circulatory system. \"Triphala uniquely cleanses and detoxifies at the deepest organic levels without depleting the body's reserves,\" explains Michael Tierra, L.Ac., O.M.D. \"This makes it one of the most valuable herbal preparations in the world.\"\n\nDr. Tierra makes an interesting observation about intestinal cleansing herbs and formulas that work as laxatives, explaining that there are two basic types. One type is called purgative; strong bitter herbs, such as senna, cascara, and others stimulate the intestine's peristaltic action by promoting bile secretion by the liver. Triphala is primarily a purgative. The other type is a lubricating bulk laxative, typified by herbs such as psyllium and flaxseed; this approach does not directly affect the liver or bile secretion, but works as an \"intestinal broom\" or sponge, absorbing fluids, says Dr. Tierra. He notes that people who require purgatives have a bowel irregularity caused by congestion in the liver and gallbladder along with some blood toxicity. People who need lubricating bulk laxatives tend to have intestinal dryness caused by metabolic imbalances (including nutritional deficiency).\n\nNotwithstanding these distinctions, Triphala is effective for all types of constipation except those caused by a lack of vital energy, Dr. Tierra states. While Triphala will gently help move the bowels, it will also effectively purify the blood, stimulate bile secretion, promote liver detoxification, clear up stagnation, enhance digestion and nutrient assimilation, and significantly reduce blood levels of cholesterol and fats throughout the body. Indians believe that Triphala is \"able to care for the internal organs of the body as a mother cares for her children,\" says Dr. Tierra.\n\nHe notes that one of his patients who was forty pounds overweight was able to lose twenty pounds in only a month by taking Triphala, without making any major dietary changes. Dr. Tierra also relates the arresting story about a yogic master he once met in California. The man was in his late eighties but could outwalk anyone in terms of speed or distance. His secret: regular meditation, kicharee (mung beans, rice, ghee, and spices), and a daily dose of Triphala.\n\nTriphala is now available in prepared pill form from several manufacturers, although Ayurvedic physicians recommend preparing it from scratch, using the dry herbs. One practitioner suggests pouring boiling water into a cup in which you have put one teaspoonful of Triphala herbs; let it stand overnight; in the morning, strain it, then drink. The taste is \"bitter, sour, pungent, astringent, and, at the end, thankfully, slightly sweet.\" As with chlorella, you can take Triphala routinely to promote regularity and detoxification without causing any weakness or deficiency during your detoxification program. Planetary Formulas, one of the makers of Triphala, recommends taking it once weekly on the same day on a continuing basis, or one to three times daily for \"balanced detoxification.\"\n\n### _Psyllium Husks and Seeds_\n\nNot only do we need to remove intestinal toxins, we need to evacuate the bowels more often and more thoroughly. One of the easiest ways to do this is through increasing our intake of fiber or roughage. Of course this can be accomplished routinely by increasing the vegetable and whole grain content of our diet, but during a deliberate detoxification program, one method frequently recommended by natural health practitioners is the temporary use of psyllium husks _(Plantago ovata_ ).\n\nThink of it as a natural laxative. But it is actually more than that. As a bulking agent, psyllium husks and seeds (taken in powder form) help \"eliminate toxins and xenobiotics [foreign substances] by binding them in the feces so they are not absorbed back into the bloodstream,\" explains one detoxification expert. Psyllium consists of about seventy-five percent to eighty percent dietary fiber, of which sixty-five percent is water soluble and ten percent to fifteen percent is insoluble (meaning it is not digested or absorbed, but passes through the system intact).\n\nOne of the main benefits of taking psyllium is that it speeds up intestinal transit time. A study involving men and women, age sixty-five and older, all of whom suffered from chronic constipation, showed that by taking psyllium daily for one month (at the rate of 24 g per day), transit time was nearly cut in half, from 53.9 hours to 30 hours. The patients also experienced an increase in bowel movements, from 0.8 per day to 1.3, again almost double. \"Fiber supplementation appeared to benefit constipated older patients clinically, and it improved colonic transit time,\" the researchers concluded.\n\nA similar study looked at the effects of giving twelve subjects various daily fiber supplements containing wheat bran, psyllium gum, and\/or a combination of both for two weeks. Fiber supplementation reduced transit time and increased the frequency of bowel movements. While wheat bran reduced transit time the most, psyllium had a greater effect on the amount of water in the stools, which means they were less hard and compacted, and therefore less painful or difficult to pass. One of the causes of constipation is dryness in the stools; they get tight, dense, and compacted, and thus hard for the intestine's innate rhythmic pulsing to move; it's like trying to push a boulder with a whisk broom. In this study, patients who had no fiber supplement said fifty percent of their stools were \"hard,\" whereas those taking fiber said only ten percent of their stools were hard.\n\nA third study further supported these conclusions. This study involved twenty patients, all of whom suffered from chronic constipation and half of whom also had irritable bowel syndrome (which is a kind of spasticity of the colon, leading to frequent, unpleasant bursts of evacuation of irregular consistency and accompanied by abdominal pain). Patients took psyllium daily for one month, after which their weekly rate of bowel motions had increased from an average of two to eight, a fourfold increase, and their transit time was reduced from forty-eight to eighteen hours. Even better, no \"side effects\" or adverse effects such as flatulence were observed.\n\nA fourth study involving twenty-two subjects with chronic constipation reported pronounced improvement after the subjects took psyllium daily for two months. Their stool frequency increased from 2.9 to 3.8 stools per week, stool consistency improved (meaning higher water content and thus easier movement), pain on defecation decreased, and the weight of the average stool increased from 405 to 665 g per motion, which means they had a more thorough elimination.\n\nLest all this sound unpleasantly clinical or too biologically intimate, it's important to see in these numbers and medical reports the nature of chronic constipation. Ideally, the healthy human is meant to have a bowel motion two to three times daily, based on food intake; in the study just summarized, the people had barely three bowel movements a week, and with pain. The mere addition of psyllium was able, over a few weeks, to almost double the output.\n\nPsyllium has another beneficial effect, namely, an ability to reduce cholesterol levels and thereby help regulate a condition called hypercholesterolemia, which means, excess cholesterol. This benefit was demonstrated in a study involving forty-four men and women with this medical condition. They took a psyllium-enriched cereal daily for seven weeks; at the end of this time, they all showed \"significantly lower\" levels of blood cholesterol and low-density lipoprotein (LDL) cholesterol. LDL is regarded as the undesirable form of cholesterol, in contrast to the \"good\" kind, called high-density lipoprotein (HDL) When study subjects took a blend of psyllium husks, pectin, guar, and locust bean gums (various sources of water-soluble dietary fiber) for only four weeks, they had an 8.3% reduction in total cholesterol and a 12.4% decrease in LDL cholesterol.\n\nAgain, you can clearly see the vital importance of intestinal cleansing and purging for health. Merely by increasing the intestine's ability to eliminate fecal matter, you reduce your chances of heart disease from high cholesterol. On the other hand, living with a stagnant colon in which excess dietary cholesterol is not regularly or effectively removed can lead to various life-threatening medical conditions. Either way, you are improving your health by reducing your body's toxic load.\n\nThere are now numerous commercial intestinal cleansing formulas containing psyllium available at natural foods stores or websites. Psyllium is available on its own in a dry powder, or in blends with other detoxifying herbs. In some people, psyllium can produce flatulence, abdominal distension (bloating), and\/or pain. These symptoms are generally transient, but if they persist, it is advisable to use a different bulking agent. While taking psyllium, it is essential to drink generous amounts of pure water.\n\nDosages will vary with the product, but the following is a general recommendation. According to Yerba Prima, a provider of psyllium based in Ashland, Oregon, adults and children over twelve should take one rounded teaspoon of dry psyllium husks one to three times daily. Children between six and twelve should take one-half to one level teaspoon, one to three times a day. Take the psyllium a half-hour before or one hour after eating.\n\nPlan on taking the psyllium once or twice daily\u2014no more than three times in one day\u2014for ten to fourteen days. There is a buildup phase of perhaps four to six days depending on your state of long-term constipation and intestinal stagnation. To an extent, most adults in Western industrialized countries have stagnation in the intestines, so don't worry that you are an exception to the healthy norm. The healthy norm is rarely observed anymore. For perhaps the first two days, you may actually not move your bowels at all, and you may feel bloated, even uncomfortably so. The psyllium is building up a head of steam, so to speak, moving slowly through the intestines, seeping into all the cracks and fissures. Around the third or fourth day, you will start evacuating much more than usual.\n\nYou will probably feel transient symptoms of discomfort, such as headaches, flu-like symptoms, congestion, a sore throat, aches and pains, or a slight fever. These will last for perhaps two days and are called the Herxheimer reaction. When you mobilize toxins in the body\u2014that is, identify and collect them\u2014there is a short-term heightening of toxicity in the system. This is sometimes call the \"die-off' phase of detoxification; toxic substances are neutralized but temporarily remain, like corpses, within the system, making it feel deeply poisoned. It's like sweeping a very dusty floor; at first, there is dust everywhere, and the place seems far messier than before you contemplated cleaning it. Ultimately, the floor will be much cleaner, as will your intestines. The arrival of Herxheimer symptoms is your body telling you that the intestinal cleanse is working. After a few days, your Herxheimer reaction will dissipate.\n\nOne way to palliate feelings of extreme toxicity\u2014you may feel dreadful for an hour now and then\u2014is to take a high dose of vitamin C, something on the order of 3000-6000 mg, perhaps in two divided doses, over a two-hour period. Vitamin C used this way acts the way aspirin does for a headache. Be sure to drink at least sixteen ounces of fresh water with the vitamin C intake any time you feel the Herxheimer effect.\n\nBy the peak of the cleanse, perhaps days six to nine, you may be passing three to four times your normal amount; six or seven evacuations per day for a few days is not unusual if this is the first time you have done an intestinal cleanse. This may not appeal to you, but you will notice (if you look) that around the time of the peak elimination, the bowel movements are strangely shaped, like twisted, moldy old rope. This is the \"old stuff,\" finally being dredged up out of the immense storage space of your intestines for elimination. To a large measure, this is why you are taking psyllium in the first place: to get this old stuff moving and out of your body.\n\n**_One way to palliate feelings of extreme toxicity\u2014you may feel dreadful for an hour now and then_ \u2014 _is to take a high dose of vitamin C, something on the order of 3000-6000 mg, perhaps in two divided doses, over a two-hour period. Vitamin C used this way acts the way aspirin does for a headache._**\n\nBear in mind that taking psyllium should not be a lifestyle, a way of coping with the high-fat, high-sugar, low-fiber fast foods diet that many rely on; rather, it should be a component in an occasional (as in twice yearly) detoxification program. In other words, it is not advisable to take psyllium (or any other high-fiber supplement) routinely and continuously; it will make your intestines lazy and may irritate their mucosal linings as well. A high-fiber, whole-food diet consisting of fresh, organic, living foods is the preferred way to keep a steady intake of roughage. That is all your intestines need to do their nature-appointed task of frequent and thorough elimination of unwanted, toxic matter.\n\n### _Cascara Sagrada_\n\nIn the same family as buckthorn, this herb was once known to Native Americans as \"sacred bark,\" as reflected in the Spanish name given to it by European explorers (botanical name: _Rhamnus purshianus_ ). Used in its dry bark form, it has been, on its own or in combination with other herbs, a popular and successful natural laxative for many years. Its chemical components are effective in stimulating intestinal peristalsis. A particular component (an anthracene derivative) gives cascara its distinctive laxative effect by increasing the motility of the colon. That means it stimulates it to move\u2014to enhance its peristaltic motions. Thus, the feces move faster through the colon and absorb less water.\n\nCascara is perhaps \"the best researched\" of the laxative herbs, notes Daniel B. Mowrey, Ph.D., in _Herbal Tonic Therapies._ Its mildness and nonhabituating qualities are widely noted. In other words, your intestines will not become dependent on its presence for continued bowel motions once you stop taking the herb. It does not lose its effectiveness with use, and you do not have to take increasingly larger doses to get the same laxative results. Cascara is an intestinal tonic, which means it can \"nurture, soothe, stimulate, and condition\" the intestines, says Dr. Mowrey. Cascara can be used in small doses to \"restore and maintain tone\" in the colon and it can be used daily in small doses \"to enhance the health of the liver and other organs.\" However, long-term use can lead to a loss of electrolytes (key minerals).\n\nAlthough cascara sagrada can be taken on its own at the rate of 50-100 mg per dose, depending on the degree of constipation, it is more common to use it in prepared herbal detoxification formulas where it is combined with other detoxifying herbs. It is often the major player in detoxification formulas.\n\nThere are many such detoxification-laxative formulas, but one that has received considerable attention and garnered favorable public and medical opinion is called Ultimate Cleanse, originally designed by a naturopath-nutritionist and produced by Nature's Secret, now a subsidiary of Omni Nutraceuticals of Los Angeles, California. This formula consists of twenty-nine cleansing herbs and thirteen sources of fiber and is designed to cleanse and tone all channels of elimination, including the bowel, kidneys, lungs, skin, and lymphatic system, according to its manufacturer. This formula also includes Siberian ginseng, included for \"extra energy and support against stress while cleansing.\" The program is meant for daily use, in the morning and evening, for about two weeks.\n\nIn closing this section, here are some general usage recommendations for the six substances just reviewed.\n\n  * Activated charcoal: use on its own\n  * Bentonite: use with chlorella, triphala, and psyllium or cascara sagrada (not both)\n  * Chlorella: use with bentonite, triphala, psyllium or cascara sagrada (not both)\n  * Triphala: use with chlorella or bentonite\n  * Psyllium: use with chlorella or bentonite; take at different times\n  * Cascara sagrada: use with chlorella or bentonite; take at different times\n\nIn many respects, it is better to use each substance on its own to gauge its effects. Later, you might consider using several at the same time.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #22: Remove the 50+ Feet of False Mucoid Lining from Your Intestines_\n\nThe use of any of the above-described natural substances will definitely stimulate your intestines to purge a fair amount of their stored waste matter. But there is a more biologically arcane level of old intestinal matter that these herbs will not address. It is called mucoid plaque or false mucoid lining. It \"bizarrely resembles hardened blackish-green truck tire rubber or an old piece of dried rawhide,\" observes naturopathic practitioner and detoxification expert, Richard Anderson, N.D., N.M.D., of Mount Shasta, California.\n\nIn 1986, Dr. Anderson developed (and started to market) a four-phase, four-week, all natural colon cleansing program (called Arise & Shine and the Cleanse Thyself program) that dredges out this mucoid plaque. When the mucoid plaque comes out in an intensive intestinal cleansing program, it is characterized by \"ropelike twists, striations, overlaps, folds, creases\u2014the shape and texture of the intestinal wall,\" says Dr. Anderson.\n\nIt can be hard and brittle; firm and thick; tough, wet, and rubbery; soft, thick, and mucoid; or soft, transparent, and thin; and its color can range from light brown, black, or greenish-black to yellow or grey, and it usually stinks. Even more stunning is the fact that it is not unusual to pass thirty-five to forty-five feet of mucoid plaque during a four-week intestinal cleanse program; some pass more than sixty feet, and at least one of Dr. Anderson's clients passed a record single length that was fifteen feet long.\n\nSo what is mucoid plaque? The term is one of Dr. Anderson's devising. He uses it to indicate a film of mucus, different from the intestine's natural mucosal lining, which is the result of the unhealthy accumulation over time of abnormal mucus matter on the intestinal walls. Medicine knows this plaque as a natural layering of mucin or glycoproteins (comprising twenty amino acids and fifty percent carbohydrates); it is natural in that the intestines secrete it as a protection against acids, salt, heavy metals, parasites, and numerous other toxic substances. In fact, mucoid plaque starts out as mucin, or a layering of glycoproteins, then just about everything else in the intestines gets added: water, electrolytes, epithelial cells (that line the intestinal walls), bacteria, bacterial metabolities and toxins, digested food, blood proteins, bile salts, enzymes from the pancreas, drug residues, antigen-antibody complexes (from allergies), and toxins produced by bacteria, yeasts, and parasites.\n\nMucoid plaque appears to develop in the presence of acids, or in acid pH conditions in the intestines; a large measure of this requisite acidity is provided by the standard American diet, which is acid-producing, says Dr. Anderson. So this \"gel-like, viscous and slimy mucus\" gets deposited in successive layers over the inner lining of the intestines.\n\nWhen the diet is unbalanced and biochemically toxic to the intestines, additional amounts of mucoid plaque are formed, says Dr. Anderson. Even though it is a natural protective mechanism, if you were healthy, your intestine wouldn't need to protect itself with mucoid plaque in the first place, comments Dr. Anderson.\n\nMucoid plaque does not routinely get excreted. Rather, it collects in the folds and crevices of the large intestine, and can remain there in storage for years, even for a lifetime. \"Over time, the mucoid plaque grows thicker, firmer, and more widespread\u2014colonizing, as it were, the tennis court-sized interior of the small intestine,\" says Dr. Anderson. Newly created feces adhere to it and the plaque grows, slowing down peristalsis and harboring intestinal pathogens; it thereby adds to bowel toxicity, emitting toxins continuously into the bloodstream and creating a favorable atmosphere for disease.\n\nThe plaque accumulation weakens overall intestinal function, interferes with nerve meridians, acts as a barrier to nutrient absorption, and may cause protein and carbohydrate intolerance (food allergies). Further, Dr. Anderson explains that prescription antibiotics and an unhealthy or unbalanced diet, comprising sugar, soft drinks, wheat products (allergenic for many people), rancid or spoiled foods, salt, and animal products containing hormones and antibiotics, can help negatively alter and transform the mucoid plaque into a toxic substance and the seedbed for \"various disease states.\" \"The hundreds of testimonies I have received clearly indicate that mucoid plaque contributes towards a high percentage of pathological problems, as well as premature death,\" says Dr. Anderson.\n\nAt the other end of the spectrum, completion of Dr. Anderson's four-week self-care cleansing program yields impressive results, an undoing of numerous latent and full-blown pathological problems. A woman, age fifty-five, had suffered from a skin rash for thirty-five years. No medical treatment had been able to reverse this condition. After the four-week program, ninety-five percent of her rash had disappeared. Another woman, age forty-two, had chronic bronchitis and could not exercise or climb stairs without wheezing. All these problems cleared up after the cleanse. Yet another woman, forty-seven, had suffered with hip pain for seven years and needed a cane for walking. X-rays showed that her hip nerves were pinched and the cartilage of the femur (the tip of the thigh bone that connects to the hip bone) was actually worn off. On the second day of the four-week program, fifty percent of her pain disappeared; on the third day, she could stretch and exercise gently; and over the next several months, her pain continued to diminish.\n\nOther benefits reported by users, says Dr. Anderson, include emotional release and cessation of problems such as menstrual pain, migraines, joint pain, skin problems, colds, sinus congestion, fibrocystic breasts, overweight, and poor concentration. Dr. Anderson explains that often old, unprocessed, negative emotions, states of consciousness, and thinking patterns are held in the body-mind system in association with old fecal matter. As you purge the intestines, you inadvertently purge the emotions. \"I estimate that about 70% of those doing the intensive colon cleanse will experience long-forgotten memories and buried emotions,\" says Dr. Anderson.\n\nIn fact, often emotions surface \"with all their original charge\" and this is followed by a discharge of a section of mucoid plaque, he says. It's as if somehow the old fecal matter held\u2014bound up\u2014the old emotions in place. Dr. Anderson further proposes that, at an energy level, a negative state of consciousness born of these held emotions and thoughts is often at the root of the particular physical problem we have. (See Chapter 8 for more on emotional detoxification.)\n\nThe Arise & Shine Cleanse Thyself program consists of a series of herbal preparations and natural substances taken in specific order over a period of four weeks. The program provides herbal laxatives (including cascara sagrada), herbal nutrition, bentonite, psyllium, probiotics (\"friendly\" bacteria, discussed below), electrolytes, and antioxidants. As you take the various preparations, you alter your diet, and during the intense phase, you severely restrict it to enable the natural substances to produce the maximum benefit.\n\nEach successive stage of the program calls for stricter dietary controls, offers a higher degree of nutrient supplements, and produces a deeper level of cleansing. Usually, it takes three to four weeks to complete the Pre-Cleanse phase of the program. This is followed by the Cleanse phase, which lasts one week. Typically, the mucoid lining comes out in the Cleanse phase, although some may start being eliminated a little sooner.\n\nBecause the last week of the program can be intense\u2014involving the frequent intake of program substances, mostly in the form of \"shakes,\" and no food intake\u2014it might be prudent to schedule this as a vacation period unless you work at home where it is easier to handle the requirements of the Cleanse.\n\nThe mucoid plaque removal program is best undertaken after you have tried some of the gentler intestinal cleansing steps outlined in Healthy Living Space Detoxifier #21. This is especially true if this is the first time you have ever done anything to detoxify your intestines. The steps in #21 will _ease_ you into the practice and experience of detoxification. Although the mucoid plaque removal phase, as presented by Arise & Shine, is a complete program requiring no previous experience or preparation, it can be intense, which is why I recommend some preparatory detoxification before trying it. After completing some of the steps in #21, wait a month or two, then start the mucoid plaque removal program.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Arise & Shine Cleanse Thyself: Arise & Shine Herbal Products,\n\nP.O. Box 1439,\n\nMt. Shasta, CA 96067;\n\ntel: 800-688-2444;\n\nfax (orders): 530-926-8866 or\n\n(main office) 530-926-0891;\n\nuser support: 530-926-8867;\n\nwebsite: www.ariseandshine.com.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #23: Purge the Parasites from Your Intestines_\n\nMucoid plaque is not the only unsuspected and undesirable resident in the twenty-six feet of your intestines. Increasingly, Westerners have been beset by a variety of unwholesome intestinal parasites. Parasites are microorganisms that do not belong in the human intestines, but once there, disrupt the delicate balance of microflora, deplete the body nutritionally, and help to generate numerous symptoms and illnesses.\n\nIt was once thought that a Westerner needed to travel in exotic lands and be subjected to unsanitary conditions, notably food and water, before parasites became a worry. This is no longer the case. In 1993, for example, hundreds of thousands of residents of Milwaukee, Wisconsin, were parasitized by an outbreak of _Cryptosporidium parvum_ through their muncipal tap water. The parasite produced diarrhea in many of those who drank the water. Also, it used to be the case that when someone mentioned the word \"parasite,\" people would think of a tapeworm. This is a gross example of the problem, but it is now no longer the sole or even typical expression of intestinal parasitosis, as parasitic infection is clinically known.\n\nParasitic infestation does not have to be biologically dramatic, producing terrible symptoms and scores of deaths. In most cases, once inside the body, parasites act as silent invaders and relentless sickeners. Because conventional Western physicians still discount the growing prevalence of intestinal parasites in otherwise seemingly healthy individuals who have no record of travel to countries where parasitic infestation is likely, the problem tends to get overlooked. Fortunately, alternative medicine practitioners are starting to become aware of the problem and to develop effective protocols against parasites. To many laypersons, the idea of parasites in the intestines is bizarre, troubling, or unbelievable, but once you understand _how_ they can get in, it begins to make more sense.\n\n**_Parasitic infestation does not have to be biologically dramatic, producing terrible symptoms and scores of deaths. In most cases, once inside the body, parasites act as silent invaders and relentless sickeners._**\n\nThe key idea that makes sense of parasitic infestation in the human intestines is a medical term called intestinal dysbiosis. This term refers to an imbalance in the microflora that populate the intestines. Here the community of living microorganisms\u2014the natural ecology, the _bios_ \u2014is abnormal, disrupted, disturbed, or unfavorable ( _dys)._ Microscopic life in the intestines has become disordered\u2014hence, disordered intestinal life. When a condition of intestinal dysbiosis exists, the healthy ratio between friendly and unfriendly, benign and pathogenic, microorganisms has shifted.\n\nBear in mind that the human body contains an estimated several trillion beneficial bacteria comprising over 400 species; all of these are necessary for health, nutrient assimilation, waste processing, and numerous other essential biological functions. In a healthy person, these friendly microorganisms (such as _Lactobacillus acidophilus_ and _Bifidobacterium bifidum_ ) and unfriendly microorganisms ( _Escherichia coli_ and _Clostridium perfringens_ ) exist in an ecological balance in which the friendly ones predominate.\n\nHowever, a number of factors arising from the typical Western lifestyle are shifting the balance the other way. Among these are overly acidic body conditions, chronic constipation or diarrhea, insufficient stomach acid, diminished output of pancreatic digestive enzymes, dietary\/nutrient imbalances, the excessive use of antibiotics or consumption of animal products containing antibiotics and hormones, poor digestive function, and suboptimal immune function. In a sense, it is a two-way situatipn biologically: intestinal dysbiosis creates conditions favorable for parasites, and parasitic infestation (such as from contaminated tap water) can create intestinal dysbiosis.\n\nIn dysbiosis, the unfriendly or pathogenic bacteria dominate the intestine's beneficial microflora population. In the United States, the most common parasites, apart from head lice, are microscopic protozoans. These include _Giardia lambia_ , a virulent form found in the contaminated waters of lakes, streams, and oceans, and a common cause of traveler's diarrhea; _Entamoeba histolytica_ , which can cause dysentery and injure the liver and lungs; _Blastocystis hominus_ , which is increasingly linked to acute and chronic illnesses; _Dientamoeba fragilis_ , associated with diarrhea, abdominal pain, anal itching, and loose stools; and _Cryptosporidia_ , which is particularly dangerous to those with compromised immune function.\n\nArthropod parasites include mites, fleas, and ticks. These in turn can carry smaller parasites that are also infectious to humans including, for example, the microorganism responsible for Lyme disease. Larger parasites, often referred to as \"worms,\" include pinworms, tapeworms, roundworms, hookworms, filaria (thread-like worms that inhabit the blood and tissue), and flukes (which invade the liver). It is estimated that more than 300 kinds of parasites can live in the human body, where often they reside as what microbiologists call the \"great masqueraders.\"\n\nParasites live off the life and nutrients of another living organism, usually to the host's detriment. Certain types can cause malabsorption of certain nutrients, such as iron, and vitamins A, B12, and beta-carotene. They begin a fermentation process, producing toxic byproducts, such as ammonia, amines, nitrosamines, phenols, cresols, indole, and skatole, which interfere with the normal elimination cycle. While they tend to stay in the intestines, parasites can also migrate elsewhere in the body\u2014the blood, lymph, heart, liver, gallbladder, pancreas, spleen, eyes, and brain. They begin to eat us from the inside out, robbing our systems of nutrients and energy.\n\nThey generate numerous symptoms that are usually, and mistakenly, associated with other illnesses and never with a parasitic infestation. For example, if you have parasites, you may have headaches, back pain, energy loss, spaciness, vomiting, weight loss, colitis, gas, uncontrolled appetite, acne, bloating, diarrhea, constipation, irritable bowel syndrome, aches in the muscles and joints, anemia, allergies, depression, yeast infestations _(Candida albicans_ ), skin problems, sleep disturbances, chronic fatigue, and a weakening of the immune system response.\n\nDysbiosis is also considered a primary cause or major cofactor in the development of many health problems, such as PMS, rheumatoid arthritis, and cancer. You could easily be set to wandering through a labyrinth of unsuccessful treatments for each of the symptoms because the attending physicians never suspect\u2014and therefore never treat\u2014a parasite infestation. On the other hand, those practitioners aware of the principles and practices of natural medicine often report impressive clinical results when they accord the possibility of parasitic infestation some reality. \"Over the years, I have seen a multitude of patients with symptoms of chronic fatigue, hypoglycemia, food allergy, spastic colon, and respiratory disorders get well when parasites were eradicated from their systems,\" states nutritionist Ann Louise Gittleman.\n\nHow can you tell if you have parasites somewhere in your body and if they are slowly but steadily compromising your health? If you suspect contamination, a Comprehensive Digestive Stool Analysis and parasitology profile (explained in Chapter 3) can be very helpful in confirming (or refuting) intestinal parasitosis and in identifying the species. However, sometimes a consideration of one's symptoms helps to clarify the situation.\n\nIn the course of presenting six case studies involving parasites, naturopathic physicians Silena Heron, N.D., and Eric Yamell, N.D., both based in Sedona, Arizona, highlight some of the key presenting symptoms reported by their patients. One woman said she had a heavy feeling at the top of her stomach (where the esophagus empties into it), occasional aching there, and fatigue; a man had sporadic lightheadedness, vertigo, and hemorrhoidal irritation and pain that was worse for poor food choices; a woman had chronic constipation; a man had low energy, nausea, bloating, pain in his stomach and upper abdomen, and night sweats; a woman had lethargy, heat and light sensitivity, diarrhea, and occasional nausea; and a man had allergic rhinitis that was worse during certain seasons, flatulence, bloating, and abdominal pain. Only two of these individuals had been to an \"exotic\" land (India) where parasite infestation might have been expected.\n\nAmong the parasites identified in the preceding case histories were _Cryptosporidium_ species, _Giardia lamblia, B. hominis, D. fragilis_ , and _E. nana._ Also of interest was the discovery that in several of the patients there were low levels of secretory IgA. This is an immunoglobulin, one of the immune system's five such defensive and protective substances. \"In our experience, decreased gut IgA levels are common in patients who are infected with _Cryptosporidium_ and may allow the organism to survive in such patients\u2014this can happen despite normal systemic immune function,\" state Drs. Heron and Yarnell. They further state that low levels of this substance, which is supposed to protect the body against parasites, \"have been shown to predispose people to contacting giardial _[Giardia lamblia]_ infection.\"\n\nHow common is the problem of parasite infection? Shockingly common, according to medical research. One New York physician found that out of 197 consecutively seen patients who complained of gastrointestinal symptoms, 95 of them (almost fifty percent) had giardiasis (an infestation of the parasite _Giardia lamblia_ ), and ninety-eight percent of these reported improvement of their complaints after completing a parasite elimination program. This same physician discovered giardiasis in 61 of 218 patients who sought treatment for chronic fatigue; 42 of these patients experienced partial to complete recovery from their symptoms when treated for parasites. Further, parasites were detected in 20% of 200,000 stool samples analyzed; most notable among the invaders were _Giardia lamblia_ (in 7.2% of samples), _Entamoeba coli_ and _Endolimax nana_ (each 4.2%), _Blastocystis hominis_ (2.6%), hookworm (1.5%), _Trichuris trichura_ (1.2%), _Entamoeba histolytica_ (0.9%), _Ascaris lumbricoides_ (0.8%), and lesser amounts of other parasites.\n\nAdditional evidence supports the conclusion that parasite infestation is a problem that affects all age groups and is both prevalent in but not limited to people living in Third World or developing countries. For example, stool samples were analyzed from 112 physically handicapped residents at a medical institution in Korea. In forty-two percent of the samples, evidence of at least two different parasites was found; three cases of _Trichuris trichiura_ and one case of _Enterobius vermicularis_ infection were found. Further, researchers found that 25% of the adults tested had an infection with _Entameoba coli_ , 21% with _Endolimax nana_ , 1.8% with either _E. histolytica_ or _Iodameoba butschlii_ , and 0.9% with _Giardia lamblia._ It's important to note that these men and women were institutionalized due to their physical handicap, not because of a reported or inexplicable gastrointestinal complaint; the discovery of the various parasites resident in their colons was a surprise to the doctors.\n\nHow parasitized is the Third World? Korean researchers analyzed stool samples from sixty-four children and adolescents in Legaspi City in the Philippines. The overall parasitic infection rate among these children was seventy-eight percent; for those of this group living in rural areas it was ninety-two percent and for those living in cities, fifty-six percent. The most prevalent protozoan cysts were _Trichuris trichiura_ (51%), _Ascaris lumbricoides_ (40%), hookworm (23%), _Iodamoeba butschlii_ (15%), _Endolimax nana_ (14%), _Entamoeba coli_ (9.4%), and _Giardia lamblia_ (7.8%). Among the children, more than half (thirty-three) had multiple parasitic infections, and fifteen children had evidence of three or more parasites.\n\nResearchers at Yarmouk University in Irbid, Jordan, analyzed stool samples from 265 children under age five suffering from acute diarrhea. They found parasites in sixty-six percent of the children, one of which was common to fifty percent of the cases; fifteen percent of the children had multiple pathogens. Most notable among the pathogens (bacteria, viruses, and parasites) found were rotavirus (in 32% of cases), three types of _Escherichia coli_ (5.7% to 12.8%), _Shigella spp._ and _Entamoeba histolytica_ (4.9%), and Salmonella (4.5%). In three provinces of Morocco, stool samples were studied from 1682 individuals representative of the urban and rural population. About sixty-six percent of samples from those living in rural areas and fifty percent of those from urban areas had parasites. Further examples such as these can be found by the dozens.\n\nNaturopathic physician Joseph Pizzomo, N.D., emphasizes that often parasitic infestations can be asymptomatic\u2014you will not necessarily notice any overt or obvious symptoms of their presence. Such an infestation is most easily confirmed by seeing an actual worm of some kind on the stool or when \"chronic inexplicable symptoms\" develop after visiting a Third World country or being in close contact with someone who has, he says.\n\nDr. Pizzorno cites the case of a man who every month for ten years had a two- to three-day bout of an unexplained fever combined with a worsening of his existing symptoms of chronic fatigue and intestinal upset. He had spent over $10,000, fruitlessly, in consulting various specialists for an effective diagnosis and treatment. It turned out he had two kinds of parasites, and the periodicity of his symptoms was due to the reproductive cycle of these parasites, explains Dr. Pizzorno. When they reproduced\u2014once every month during a two- to three-day period, the patient's symptoms were at their worst, exacerbated by the parasitic activities.\n\nIn a related way, a person infested with parasites may seem to be asymptomatic only because the parasites are producing symptoms in an unexpected and therefore unexamined area, namely, personality changes. Researchers at Charles University in Prague in the Czech Republic found that among 230 women who had been diagnosed with acute toxoplasmosis (from the parasite _Toxoplasma gondii_ ) during the preceding 14 years, noticeable and quantifiable personality changes could be documented according to standard personality profile tests. In a related review of fifty-five younger women with latent toxoplasmosis, the scientists correlated various psychological factors with the length of the infection. \"We suggest that the parasite induced the changes in personality profiles of the women because of our observation of an increasingly different personality profile over time between women with latent infection and controls.\"\n\n### _Berberine_\n\nDr. Pizzorno stresses that generally parasites should not be self-treated and their removal requires expert medical care; however, in mild cases or as a preventative, he recommends goldenseal (1,500 mg, three times daily), whose constituent, berberine, can stop the growth of several types of common parasites. There are a few other berberine-containing plants that are helpful against parasites: barberry, Oregon grape, and goldthread. When using these herbs, the dosage should be based on the berberine content at the recommended level of 25-50 mg, three times daily for adults, and for children, 5-10 mg of berberine per kilogram of body weight.\n\nBerberine has been shown to be effective in treating seven different types of parasites in children. In one study, 65 children (younger than 5) were given berberine tannate (25 mg every six hours) for relief of acute diarrhea caused by five different parasites; they responded better than children given conventional antibiotics. In another study with 40 children (ages one to 10) infested with _Giardia_ , the children received either berberine (5 mg per kilogram of body weight each day), the standard drug metronidazole (called Flagyl commercially, at the rate of 10 mg per kilogram of body weight each day), or a placebo of vitamin B.\n\nSix days later, forty-eight percent of the children who took berberine were free of parasitic symptoms, and sixty-eight percent of these were free of all signs of _Giardia_ , based on their stool samples. Thirty-three percent of the children who took the standard drug were free of symptoms, and the stools of all were free of _Giardia._ Berberine was more effective in relieving symptoms at half the dose of metronidazole, but was less effective than it in removing the parasites from the body, the scientists concluded. In a third study involving 200 adults with acute diarrhea, those who had berberine with antibiotics recovered more quickly than those who had the antibiotic treatment alone.\n\n### _Parasite Purge Formula_\n\nBased on their extensive review of the scientific data supporting the herbal constituents involved, Drs. Heron and Yarnell recommend this liquid herbal blend for treatment of parasites. (While their protocol can be done on one's own, it is prudent to do it under the supervision of a qualified health-care practitioner.) Drs. Heron and Yamell suggest mixing two parts each of worm-seed (Epazote), Jamaican guassia, and Sweet Annie with one and one-half parts of gentian; one part each of garlic, black walnut, Oregon grape, ginger, male fern, tansy, and wormwood; and one-half parts each of greater celandine, anise, and Turkey rhubarb.\n\nThey recommend taking one teaspoonful of the mixture three times daily until the problem has cleared up. In addition, they advise taking grapefruit seed extract (ten drops three times daily, mixed in water); the commercially prepared antiparasite herbal tannate formulas Tanalbit (six capsules, three times daily) or Viracin (two capsules, three times daily), and garlic (four fresh cloves, two to three times daily, or extract at the rate of 5,000 mcg of allicin per capsule and eight capsules, two to three times daily).\n\nThe active ingredients of Tanalbit are plant tannins, astringent natural substances that attach themselves to the cells of fungi, yeasts, and bacteria and prevent them from metabolizing and colonizing, and ultimately cause them to die. Viracin is similar but a different strength, and contains extracts of Norwegian maple fruit, Babul bark, and wild rhubarb. Grapefruit Seed Extract (commercially available as Citricidal and made from the bioflavonoids [vitamin C helpers] found in grapefruit seed and pulp) is regarded to be effective against a broad spectrum of pathogenic microorganisms. It is also regaled as a general purpose disinfectant, dental rinse, and skin cleanser.\n\nOver the course of two to four weeks, slowly build the daily dose of these substances to the amount above, Drs. Heron and Yarnell recommend. Then continue with these doses (regarded as the maximum) for about two months, they say. The substances should not be taken with food, and if diarrhea develops or the Herxheimer reaction sets in (due to parasitic die-off inside the body), take one level teaspoon of bentonite once daily. Drs. Heron and Yarnell also suggest that during and after the program you should take some form of probiotics, or \"friendly\" bacteria, such as _Lactobacillus acidophilus_ , discussed below, as a way of replenishing beneficial bacteria in the intestines.\n\n### _Bitter Melon_\n\nThis is a bitter-tasting cucumber-like green vegetable _(Momordica charantia_ ) common to Asian cooking and Ayurvedic and Chinese medicine that can be used as a gentle food-based vermifuge, or agent against intestinal parasites. The fruit is best described as an oblong warty-looking gourd, although it tastes better than it looks. Traditionally, the food is used as, among many other things, a treatment for worms and parasites; in Western herbal medicine, bitter melon is increasingly used in the treatment of diabetes, hypoglycemia, colds, flus, HIV and other viruses, and psoriasis.\n\nScientific research is beginning to substantiate the folk claims for the antihelmintic (worm-killing) actions of this fruit. An herbal preparation made from bitter melon was more effective in treating infestations of _Ascaridia galli_ worms in birds than the conventional drug. Bitter melon is also effective against infectious diseases transmitted by microorganisms. Bolivian researchers reported that extracts of bitter melon were \"moderately active\" against an antibiotic-resistant strain of malaria. Puerto Rican scientists found that out of fifty plants tested for activity against tuberculosis-producing microorganisms _(Mycobacterium smegmatis_ ), bitter melon was one of six that possessed \"inhibitory capacity.\"\n\nBitter melon has also been shown to improve factors necessary for liver detoxification, namely, significantly increasing the levels of glutathione and acid-soluble sulfhydryl, two substances crucial to the liver's two-phase detoxification pathways (see Chapter 2). Finally, Italian researchers have determined that bitter melon, in addition to other herbs used in Ayurveda, contains \"antioxidant principles that can explain and justify their use in traditional medicine in the past as well as the present.\" This would suggest bitter melon is effective against free radicals, such as the metabolites or toxic byproducts of parasites or the intestines' action against them.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Tanalbit and Viracin, contact:\n\nScientific Consulting Service,\n\n1972 Republic,\n\nSan Leandro, CA 94577;\n\ntel: 800-333-7414 or 510-632-2370;\n\nfax: 510-632-8561;\n\ne-mail: intensivenutrition@pacbell.net;\n\nwebsite: www.intensivenutrition.com.\n\nFor Citricidal:\n\nNutriteam,\n\nP.O. Box 71,\n\nRipton, VT 05766;\n\ntel: 800-785-9791 or 802-388-0661;\n\nfax: 815-377-2198;\n\ne-mail: support@gfex.com;\n\nwebsite: www.nutriteam.com.\n\nOne general antiparasite recommendation is to eat two bitter melons (cooked) daily for about ten days. Bitter melon is available in Asian food stores and in dry form as an herbal capsule.\n\n### _Pumpkin Seeds_\n\nAnother vermifuge folk remedy that is gaining scientific acceptance is the ingestion of pumpkin seeds, derived from autumn and crook-neck squash and the Canada pumpkin. This treatment is believed to rid the body of intestinal parasites, especially roundworms and tapeworms. A typical regimen is three doses of 20-150 grams of seeds; other research shows that chewing twelve pumpkin seeds (sometimes in combination with Rangoon creeper fruit seeds, _Quisqualis indicae_ ) in the morning on an empty stomach for two weeks can help the intestines purge themselves of parasites. The treatment is believed to paralyze the parasites, causing them to loosen their grasp within the intestines and enabling their elimination from the body. An amino acid called cucurbitin, found in pumpkin seeds, is believed responsible for the worm-expelling effects.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #24: Get Proactive with Probiotics_ \u2014 _Plant Food for the Intestines_\n\nProbiotics is a term used earlier in this chapter to denote the friendly, beneficial bacteria, or microflora, that inhabit\u2014or ought to inhabit\u2014your intestines. They are _pro_ biotics in that they support and enhance the natural life functions performed by the intestines on behalf of the life of the body.\n\nAmong the two most prominent (and researched) probiotics are _Lactobacillus acidophilus_ and _Bifidobacterium bifidum._ Others include _B. longum_ , a primary resident of the large intestine, where it protects this organ from invading yeasts and bacteria, detoxifies the bile, and helps make B vitamins; _B. infantis_ , the most prevalent probiotic in the intestines of infants; and _Streptococcus thermophilus_ and _L. bulgaricus_ , which are commonly found in yogurt and, although only transiently present in the intestines, produce lactic acid\u2014encouraging the growth of other probiotics\u2014and bacteriocins, natural antibiotic-like substances that kill pathogens. Other probiotics include _L. casei, L. plantarum, L. sporogenes, L. brevis_ , and _Saccharomyces boulardii._\n\nOverly acidic bodily conditions, chronic constipation or diarrhea, dietary imbalances, consumption of highly processed foods, and the excessive use of antibiotics and hormonal drugs can interfere with probiotic function and even reduce the number of these microbes, setting up conditions for illness. One of the factors contributing to slow transit time is a deficiency of friendly bacteria which otherwise stimulate peristalsis.\n\nAt the same time, if you do a deliberate intestinal cleansing program, you must replenish your intestines' supply of friendly bacteria. Some inadvertently get flushed out of the system; others get used up in the struggle to restore proper intestinal microfloral balance; and some you may have been deficient in for a long time due to untreated toxicity.\n\nRecolonizing your intestines with a new population of probiotics is a bit like giving your lawn plant food to make the grass grow better. Introducing probiotics into the intestines enables them to perform their functions better, to counteract long-term antibiotic damage, and retune their immune activities. Think of recolonizing the intestines as a preventive measure against possible future illness. The body of scientific evidence supporting the benefits of _L. acidophilus_ is large and irrefutable, as this brief overview will show.\n\n**_If you do a deliberate intestinal cleansing program, you must replenish your intestines' supply of friendly bacteria. Some inadvertently get flushed out of the system; others get used up in the struggle to restore proper intestinal microfloral balance; and some you may have been deficient in for a long time due to untreated tonicity._**\n\n_L. acidophilus_ is effective in inhibiting the growth of _Helicobacter pylori_ , a pathogen believed to be the cause of acid-peptic disease of the stomach. In fact, it stopped the growth of seven different strains of _C. pylori_ recovered from human gastrointestinal tracts. Several strains of acidophilus had a \"bacteriostatic\" (bacteria-killing or suppressing) effect against the spoilage pathogen _Listeria monocytogenes_ in dairy milk.\n\nRelated to this benefit is the proven ability of _L. acidophilus_ to act as a powerful antioxidant. It can inhibit the peroxidation of linoleic acid (the process by which this essential fatty acid is made into a damaging free radical) by twenty-eight percent to forty-eight percent; it can scavenge between twenty-one percent and fifty-two percent of three dangerous free radicals present in cells; and it has a \"high inhibitory effect\" against a particular cell-killing chemical, reducing its activities by fifty percent. Even better, _Bifidobacterium longum_ demonstrated all these abilities, but was able to curtail ninety percent of the activity of the cell-killing chemical.\n\n_L. acidophilus_ can also inhibit the growth of coliform bacteria (such as _Escherichia coli_ , the food-borne pathogen) and reduce their existing populations, which makes it highly serviceable clinically for people suffering from gastrointestinal distress caused by coliforms. A review of forty-nine studies on probiotics showed consistently that _L. acidophilus_ intake can shorten the duration of diarrhea produced by rotavirus infection by one day. _Lactobacillus_ also exerts a protective effect against induced colon tumors (in rats), reducing the incidence by ten percent to twenty-five percent (depending on the strain of _Lactobacillus_ ), and reducing the tumor mass (or general size of the cancer that emerged).\n\nGiving infants pure cultures of _L. acidophilus_ prevents \"dysbacteriosis\" (intestinal dysbiosis, or the state of being overrun by pathogenic bacteria, notably _Staphylococcus aureus_ ), thereby reducing the incidence of disease in very young children (in their first year) and improving the overall health and immune vitality of the gastrointestinal tract; acidophilus supplementation may contribute to resistance to potential food allergens in later life. In a trial, _L. acidophilus_ given to sixty infants (orally and dabbed in the nose), resulted in sixty percent of the infants leaving the maternity home with \"normally formed intestinal microflora\"; additionally, eighty percent had normally formed nasal microflora and seventy percent normal skin microflora, attesting to the bacteria's ability to reduce opportunistic pathogens in the three areas. Dietary supplementation with three strains of _Lactobacillus_ (in a mice study) also led to an enhancement of several key elements of natural and acquired immunity, such as the activity of white blood cells and the generation of immune cells by the spleen.\n\nIt is also impressive to note that many of the friendly bacteria produce viable natural antibiotics in the colon, all of which have been studied by scientists. For example, _Bifidobacterium bifidum_ produces bifidin; _L. acidophilus_ produces acidolin, acidophilin, lactobacillin, and lactocidin; _L. brevis_ makes lactobrevin; _L. bulgaricus_ produces bulgarican; and _Streptococcus lactis_ makes nisin. Once produced, these natural antibiotics do not so much destroy harmful bacteria, but make it impossible for them to flourish in the colon. Inhibition is better than destruction because the latter would leave toxic debris in the colon, thereby contributing to toxicity. Bulgarican and acidophilin are \"exceedingly active against a wide variety of organisms,\" including pathogens, researchers have reported.\n\nSo the case for taking _L. acidophilus_ and related strains of friendly bacteria is strong, but how best to ingest it? You would think fresh unsugared yogurt would be ideal, but it actually is not because in most instances it primarily contains _L. bulgaricus_ and _S. thermophilus_ , friendly bacteria that only _temporarily_ inhabit the intestines. This means their populations will die off after a short time in the colon. Another problem to overcome is the stomach. You need to get the friendly bacteria alive, active, and intact into your intestines, past the stomach's powerful digestive apparatus. The best way to ensure this is to use enteric-coated _L. acidophilus_ capsules, which means they will not be digested until they reach the intestines.\n\nIt's advisable to take a probiotic supplement during the active detoxification program, as a complement to the liver and intestinal cleansing. But it is better to take it _after_ the intestinal cleansing substances\u2014that is, when you have finished that phase of the detoxification program. The regular intake of friendly bacteria after the program is also a good idea, at least for a month or so, because it offers broad-spectrum protection and disease prevention in the same way as vitamin C.\n\nIn times of illness or medical treatment, taking _L. acidophilus_ can be beneficial, especially if you are taking antibiotics or have recently done so. It can also be beneficial if you use birth control pills or steroids (as these deplete intestinal probiotics); if you have chronic constipation, a yeast infection, or bacterial vaginitis; if you have a gastrointestinal infection or inflammation; or if you are lactose intolerant (cannot digest milk sugar). As a regular dosage, 3 to 7 billion organisms (primarily L. _acidophilus_ and _B. bifidus_ ) daily\u2014most brands of probiotics specify their estimated organism count\u2014is advised.\n\nThere are many brands and formulations of friendly bacteria now on the market, giving consumers many choices. To give you an idea of what is out there, here are two offerings that exhibit interesting features. First, Culturelle, according to its manufacturer, contains 20 billion live and active cells ( _L. acidophilus_ ) per capsule, and it's the same strain as appears in the human intestines, making it the \"ideal probiotic.\" It is capable of surviving stomach and small intestine digestion to arrive in the colon where it implants itself on the cell walls and starts colonizing. Second, Jarro-Dophilus combines six species of friendly bacteria, including four species of _Lactobacillus_ and two of _Bifidobacteria._ According to the manufacturer, each of the species colonizes its \"own separate regional niche\" in the colon, based on appropriate biochemical factors. Each capsule contains an estimated 3.36 billion organisms of the six species specified.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #25: Fertilize Your Bifidobacteria with Prebiotics_\n\nThe idea of prebiotics was developed in Japan in the 1980s around the concept of introducing nutrients that directly feed the friendly bacteria already in place in the intestines, mostly _Bifidobacteria_ and _Lactobacilli._ In effect, prebiotics supply the essential building blocks\u2014the nutrient foreground, if you will\u2014for beneficial intestinal bacteria.\n\nForemost among the prebiotics are natural carbohydrates called fructo-oligosaccharides, or FOS for short. FOS occurs naturally in foods such as garlic, honey, Jerusalem artichokes, soybeans, burdock, chicory root, asparagus, banana, rye, barley, tomato, onions, and triticale, but in minute amounts, too minimal to be of much use therapeutically. In fact, you would have to eat an estimated 429 garlic cloves to derive the same FOS as you get from FOS in supplement form. Made by fermenting sucrose with a fungus called _Aspergillus niger_ , FOS (called neosugar) is now found in over 500 commercial foods in Japan as a beneficial, slightly sweet nutritional additive.\n\nIt's interesting to note that in the infant gastrointestinal tract, _Bifidobacteria_ represent an estimated ninety-five percent to ninety-nine percent of all friendly bacteria, but this amount declines with age. Once introduced into the intestines, FOS, available as a white powder, acts as a kind of fertilizer that selectively feeds and nurtures the friendly bacteria so that their numbers will increase. FOS also helps lower the intestinal pH (the ratio of acidity and alkalinity) to a slightly more acidic condition so that the friendly bacteria will thrive, and the pathogenic bacteria will not.\n\nAccording to information provided by the primary manufacturer, GTC Nutrition Company, FOS has been shown to increase the total count of beneficial bacteria by five to ten times after supplementation for two to four weeks. Other benefits attributed to FOS include relief from diarrhea and constipation, the promotion of bowel regularity, lowering of cholesterol and blood pressure, control of blood sugar levels, enhancement of immune function, improved B-vitamin synthesis and calcium absorption, better digestion of dairy milk proteins, and an easing of the liver's toxic burden.\n\nGTC Nutrition puts forward at least fifteen scientific studies supporting the major claims that FOS (and friendly bacteria in general) assists and protects the immune system, protects the body against the encroachment of pathogenic and putrefactive bacteria such as _E. coli_ and _Clostridium_ that would otherwise produce diarrhea and other intestinal disturbances, and promotes better health. A University of Illinois study showed that fermentable fiber\u2014the study used NutraFlora as a food source\u2014promotes the growth of beneficial intestinal organisms that retard the colonization by the pathogen _Clostridium difficile._ In fact, the scientists could find \"no culturable counts\" of _C. difficile_ after the intestines had been \"modified\" by the introduction of FOS. Another study examined the effect on twelve healthy adults of supplementing the diet for 42 days with 4 g of FOS daily. They found that the FOS altered the fecal microflora in a beneficial way by decreasing the (undesirable) activities of certain enzymes by seventy-five percent to ninety percent, thereby protecting the colon.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Culturelle:\n\nVitamin Research Products,\n\n3579 Highway 50 East,\n\nCarson City, NV 89701;\n\ntel: 800-877-2442 or 775-884-1300;\n\nfax: 800-877-3292 or 775-884-1331;\n\nwebsite: www.vrp.com.\n\nFor Jarro-Dophilus:\n\nJarrow Formulas,\n\n1824 S. Robertson Blvd.,\n\nLos Angeles, CA 90035;\n\ntel: 800-726-0886 or 310-204-6936;\n\nfax: 800-890-8955 or 310-204-2520;\n\ne-mail: info@jarrow.com;\n\nwebsite: www.jarrow.com.\n\nFor a source of FOS:\n\nNutraFlora,\n\nGTC Nutrition Company,\n\n1400 W. 122nd Avenue, Suite 110,\n\nWestminster, CO 80234;\n\ntel: 800-522-4682 or 303-254-8012;\n\nfax: 303-254-8201;\n\ne-mail:\n\ngeneralinfo@gtcnutrition.com;\n\nwebsite: www.nutraflora.com.\n\nFOS is not without its critics, however. It is not digestible in the mouth or small intestine, but only in the colon, where it could alter that organ's metabolic function; it can stimulate the growth of a few unfriendly microorganisms, such as _Klebsiella_ ; it is specific to probiotic strain and species, not to all beneficial bacteria; and if you have a yeast infection _(Candida albicans_ ), as a sugar, it will undesirably feed the yeast.\n\n# CHAPTER 6\n\n# Whole-Body Cleansing Routines: Extending the Detoxification into Everyday Activities\n\nCleansing the liver and intestines represents the intensive phase of self-care detoxification. It is something you need do only once or twice yearly. But during the rest of the year, there are gentler but equally beneficial detoxification activities you can pursue to reinforce, and even deepen, the positive effects you obtained from the liver and intestinal cleansing.\n\nThese additional, regular cleansing routines tend to provide detoxification benefits for the entire body. It's not that liver and intestinal cleanses only benefit those organs; of course their effect is body-wide, but their application is organ specific. In the exercises outlined in this chapter, the application itself encompasses the entire body. For example, in this chapter we'll review the whole-body benefits of stimulating the lymphatic system by bouncing on a trampoline, dry skin brushing, getting regular exercise, taking a sauna, or enjoying an herbal detoxification bath. You may easily engage in any or all of these activities throughout the year as part of a continuing detoxification program\u2014as new components of an evolving healthy lifestyle.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #26: Bounce on a Trampoline to Improve Your Lymphatic Drainage_\n\nThe human body has another detoxification system of equal importance to the liver and intestines in effecting removal of toxic substances. It's called the lymphatic system, but it doesn't get the attention it deserves from conventional medicine, and even in some areas of alternative medicine, it is a whole-body cleansing system that is underemphasized.\n\nThe lymphatic system consists of lymph fluid and the structures such as vessels, ducts, and nodes involved in transporting it from tissues to the bloodstream. Lymph fluid occupies the space between the body's cells and contains plasma proteins, foreign particles, and cellular waste. Lymph nodes are clusters of immune tissue that work as filters or \"inspection stations\" for detecting and removing foreign and potentially harmful substances in the lymph fluid (see figure 6-1).\n\nWhile the body has hundreds of lymph nodes (more than 500), they are mostly clustered in the neck, armpits, chest, groin, and abdomen. The lymph nodes contain different types of scavenger cells that destroy toxic substances. The tonsils, for example, are part of the lymphatic system; so are the adenoids, the thymus gland (behind the sternum in the chest), the spleen (at the bottom of the left rib cage), the appendix, and Peyer's patches in the small intestine. The familiar sensation of having \"swollen glands\" in the throat refers to congested lymph nodes in that area of the body. Regrettably, conventional medicine regards the tonsils, adenoids, and appendix as largely expendable (surgically removable without physiological consequence), not understanding that these structures are key players in the body's lymphatic drainage system.\n\n**Figure 6-1. The Lymphatic System**\n\nThe essential function of the lymphatic system (see figure 6-1) is twofold: to provide immune defense, and to maintain the fluid balance in the intracellular environment. For example, if the system didn't collect this interstitial fluid (the fluid between cells which leaks out of capillaries), the body's tissues would swell up dangerously. The lymphatic system is the body's master drain, collecting and filtering the lymph fluid and conveying it to the bloodstream, thereby clearing waste products and cellular debris from the tissues. But it also delivers food nutrients and oxygen to the cells, fulfilling its unique role as a connecting medium between the body's cells and tissues and the arteries, veins, and capillaries of the cardiovascular system.\n\nAs a fluid, lymph consists of water, inorganic mineral salts, and white blood cells (lymphocytes). Among its many cleansing, immunological, and defense activities, the lymph removes excess proteins resulting from a number of sources (such as dead cells and complexes of antigens and antibodies that form in an allergic reaction); collects protein wastes from the fluids in the regions between cells; removes foreign material and environmental toxins from the tissues; and maintains the fluid balance in body's web of connective tissue.\n\n**_As with the intestines_ , _if the lymphatic system does not work optimally, not only does it help create a condition of vulnerability to disease, it impedes your body's ability to heal itself from the disease._**\n\nLymph system activity increases during illness (such as the flu) when the nodes (particularly at the throat) visibly swell with collected waste products. Some doctors have likened the lymphatic system to the \"metabolic garbage can\" of the body. The nodes and fluid collect the body's dead cells, metabolic wastes, pathogenic bacteria, toxins, and other foreign or injurious substances and remove them from circulation. It is a parallel and complementary detoxification track to the liver-intestines.\n\nOn the other hand, the lymphatic system, if it is clogged, dysfunctional, or sluggish, can be a health problem, participating in and slowly contributing to a condition of body-wide toxicity. As with the intestines, if the lymphatic system does not work optimally, not only does it help create a condition of vulnerability to disease, it impedes your body's ability to heal itself from the disease. Inadequate lymph flow, or a generally stagnated lymphatic system, has been correlated with numerous health problems, such as bursitis in the shoulders, joint stiffness, bunions, spasms in the soft tissues, bad breath, body odor, skin that is dry and flaky, lethargy, depression, and cancer. When the lymphatic system is egregiously blocked, you get the disfiguring disability called lymphedema, in which certain lymph nodes and channels get blocked and filled with water and literally balloon out to grotesque proportions.\n\nIn effect, the lymphatic system is the beginning and end of all disease, of aging, and _premature_ aging, and of general toxicity or toxemia. Typically, as people age, they exercise less; one of the results is that lymph circulation slows down. Proteins and debris do not get flushed out by regular exercise and tend to accumulate in the nodes. The lymph, which should be clear, becomes cloudy and thick, changing its consistency from that of milk to yogurt then cottage cheese, and it begins to stagnate.\n\nThickened, stagnant lymph saturated with toxic wastes is an ideal setup for many illnesses, including cancer. \"When the collecting terminals [lymph nodes] become blocked, it's like a bottleneck: lymph starts backing up in the system creating a toxic, oxygen-deprived environment conducive to degeneration.\" The lymphatic system can store stagnant, toxic lymph fluid just as the intestines store old fecal matter; but just because storage is possible does not mean it's physiologically desirable. It's not.\n\nTo fulfill its function as a detoxification system, the lymphatic system needs a little help from the \"user.\" It does not have a pump\u2014like the heart in the circulating system or peristalsis in the intestines\u2014to move the fluid. The fluid does flow on its own, at the average slow rate of 125 ml\/hour, but this is mostly uphill, that is, against gravity. The natural slow movement of lymph is facilitated by breathing and skeletal muscle contractions.\n\nThere are three manual ways to get the lymph moving faster from the tissues into the blood: muscular contraction from movement and exercise; gravitational pressure; and internal massage of the lymph ducts. Lymph fluid movement is dependent on the body as a whole moving; in practical terms, that means when we exercise\u2014run, walk, jump, swim, or move about with some degree of vigor\u2014then the lymphatic system is primed and starts moving. One of the best ways (and one that is well-researched medically) to decongest the lymph nodes and move the lymph is to jump, and one of the easiest ways to jump is on a trampoline.\n\nYou don't need a gymnasium-grade trampoline to move the lymph. Smaller versions called mini-trampolines or rebounders are available and deliver the same results with less danger of injury from falling off. Typically, a rebounder is about three feet in diameter and about six to eight inches off the floor. Rebounding changes the gravitational forces affecting the body\u2014at the height of the bounce you are for a moment free of gravity; when you land, you hit the trampoline with twice the force of gravity\u2014thereby facilitating greater blood flow.\n\nIn fact, researchers have found that this twice-gravity bounce affects every muscle and cell in the body. It delivers gravitational pressure and an internal massage to the valves of the lymph ducts, enhancing their function. Rebounding also provides a gentle massage of the spinal column through its low-impact bouncing. This in turn enhances the amount of waste products flushed out of cells into the lymph and the movement of the lymph itself to its penultimate destination in the intestines and final excretion. Somehow exercise makes the lymph nodes expand and the flow of lymph becomes active; when you are not exercising, or remain physically inactive for a long time, the lymph flow becomes sluggish.\n\nRebounding is quintessentially a lymphatic exercise, and it satisfies all three physical requirements cited above needed to move the lymph. \"The bouncing motion effectively moves and recycles the lymph and the entire blood supply through the circulatory system many times during the course of the rebounding session.\"\n\nMedical research suggests that during vigorous exercise, such as rebounding, lymph flow rate can increase by as much as fourteen times the resting state. Further, if a 150-pound man exercised for one hour on a rebounder, he would burn 410 calories compared to only 355 calories if he spent that hour running approximately five miles. In fact, for similar levels of oxygen intake and heart rate, the \"magnitude of the biomechanical stimulation is greater with jumping on a trampoline than with running,\" researchers concluded in 1980 based on a study of eight exercising males. The key point is that by increasing your lymph flow, you are moving toxins out of your body that much faster.\n\nMedical research shows that rebounding has numerous health benefits, extending beyond the biomechanical stimulation of lymphatic flow. Rebounding benefits the heart and blood circulation, it can slow the effects of aging, revitalize vision, reduce stress, and help children with cystic fibrosis and learning disabilities. One medical authority has cited thirty health benefits. Rebounding increases the capacity for respiration (oxygen intake); reduces arterial pressure during exertion; reduces the time blood pressure remains abnormal after extreme exertion; strengthens the red bone marrow's ability to produce red blood cells; promotes body growth and repair; stimulates metabolism; enhances digestion and the elimination processes; affords deeper relaxation and easier sleep; and produces better mental performance and keener learning abilities.\n\nIf a practice has therapeutic benefits in helping to reduce or reverse a health problem, it is logical to assume it may produce positive gains in people who don't have those problems. In other words, if rebounding helps those with cognitive deficits, it may enhance the brain function of those who are more or less healthy and normal. For example, researchers writing in the medical journal _Rehabilitation_ reported that trampoline therapy \"is a useful part\" in rehabilitating brain-injured children and adolescents (especially with partial brain paralysis, or hemiparesis) and that it improves standing balance and movement coordination.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more information about rebounding and information about the ReboundAIR rebounder, contact: American Institute of Reboundology, 1240 East 800 North,\n\nOrem, UT 84097;\n\ntel: 888-464-5867 or 801-377-0570;\n\nfax: 801-377-0655;\n\nwebsite: www.healthbounce.com.\n\nFor Needak Soft-Bounce rebounder:\n\nNeedak Manufacturing,\n\nP.O. Box 776,\n\nO'Neill, NE 68763;\n\ntel: 800-232-5762 or 402-336-4083;\n\nfax: 402-336-4941;\n\ne-mail: needak@inetnebr.com;\n\nwebsite: www.needakmfg.com.\n\nA Norwegian study showed that exercising on a rebounder for almost two hours weekly for two months produced positive improvements for children with cystic fibrosis. In this disease, the ducts in the pancreas, sweat glands, and lungs get clogged with thick mucus and cannot work properly. Two children in this study with the most serious lung congestion were able to significantly increase their oxygen intake as a result of rebounding.\n\nThe recommendation for rebounding is simple: _bounce often._ How long? The American Institute of Reboundology in Orem, Utah, recommends a forty-minute moderate workout at least three times weekly. Rebounders for home use are available in the price range of $200 to $300.\n\n## HEALTHY LIVING SPACE DETOXIFIER #27: Dry Skin Brushing\u2014Tone Up Your Body's Largest Elimination Organ\n\nIt sounds a bit peculiar, but brushing your skin with a dry brush can help move the lymph and aid the body in detoxifying through the skin. Don't overlook the benefits of perspiration, or heavy sweating. Throughout a day the skin can eliminate an estimated pound of wastes and toxic products in the form of perspiration through the skin, the body's largest eliminative organ, sometimes dubbed the third kidney. As discussed in Chapter 2, the skin can also absorb substances from the environment\u2014helpful substances such as vitamin D, and toxic ones such as chlorine.\n\nDry skin brushing appears to work partly through gently massaging many of acupuncture's treatment nodes across the body's surface. These nodes are little areas of heightened vital energy (or _qi)_ along the numerous energy channels, or meridians. These nodes are believed to correspond, by way of subtle energy connections, with various organs and organ systems throughout the body.\n\nSo by applying friction to these points (with a brush against the skin), energy is stimulated to move from the node to the organ in question, be it the liver, intestines, or lymphatic system. In its most elementary sense, dry skin brushing opens the skin's many pores, allowing it to \"breathe,\" but it also increases blood flow. This helps to draw out stored wastes and toxic substances that otherwise result in a buildup of cellulite (fat deposits close to the skin surface, especially prevalent in the legs, thighs, and buttocks).\n\nDry skin brushing also encourages lymph fluid to move through the channels to the central place in the chest where it dumps into the blood circulation system, thereby dumping its collected toxins and waste products. Other benefits include the following: nerve endings are stimulated, thereby rejuvenating the entire nervous system; the skin is better toned so it can more efficiently purge itself of toxins; blood circulation improves, facilitating better toxic substance collection and removal; stress is reduced; tissue metabolism is improved, reducing fatigue; the amount of cellulite (toxic material stored in the body's fat cells) is reduced; dead layers of the skin are removed, opening pores.\n\n_Recommendation_ : Use a soft natural vegetable fiber bristle brush with a wooden handle and _gently_ brush all the surfaces of your body for about five to ten minutes daily, until the skin appears rosy. Some doctors suggest brushing the skin daily for three months then cutting back to twice weekly as a standard ongoing detoxification routine. Make sure your skin is dry, which means, do the exercise before your morning shower, not after. Begin at your feet and brush vigorously in a circular fashion, moving slowly up the front of your body to the lower neck, then proceed down the back side to the feet again. Try to brush _toward_ the sternum throughout the exercise. Brush from the hands up the arms to the shoulders. Wash and sanitize the brush every several weeks, because it will accumulate dead material from the skin. You might do the dry skin brushing immediately before a hot shower or a bath.\n\n### _Scrubbing with Sea Salt_\n\nThis complementary, whole-body cleansing scrub is recommended by detoxification expert Helene Silver. Silver says to put four to five handfuls of 100% sea salt in a plastic container; into another container, pour a quart of very hot water. Stand or sit in an empty bathtub and, using a sponge in one hand, daub parts of your body, such as your arm, your leg, and your abdomen with the hot water. Then use your other hand to apply the dry sea salt to those lightly moistened areas, rubbing it in with circular motions.\n\nThe goal is cover your entire body with the salt (avoid the genitals, as the salt will sting and irritate the urethra). Then take the dry skin brush and gently massage the salt into the skin. Relax for ten minutes with the salt in place until your skin begins to tingle. \"As the salt dries, it is drawing the toxins out of your skin,\" says Silver. Finally, rinse off all the salt under the shower, using first hot then cold water.\n\n### _Epsom Salt and Cider Vinegar Scrub_\n\nThis variation is part dry skin brushing and part detoxification bath, as suggested by iridologist-physician Farida Sharan, M.D. Add one tablespoon of an organic vegetable oil to one cup of dry Epsom salts in a plastic container. Rub the salt-oil mixture all over your skin while you stand in the bathtub. In this instance, the tub should already be filled with hot bath water, to which you've added one cup of apple cider vinegar.\n\nThe salt-oil rubbing helps the skin shed its old dead cells and the toxins they carry, leaving the skin smooth and soft, says Dr. Sharan. As you scrub, the salt will fall off your body into the tub. When you have finished the rub, lie down in the hot water and soak for thirty minutes; apply cold washcloths to your forehead if you get too hot. Complete the detoxification routine with a three-minute cold shower. Dr. Sharan adds that it is useful to drink herbal teas that encourage perspiration, such as yarrow, sage, catnip, pleurisy root, peppermint, or spearmint\u2014ideally about thirty minutes before your bath.\n\n## HEALTHY LIVING SPACE DETOXIFIER #28: Exercise Regularly to Enhance Blood Circulation and Toxin Removal\n\nSeemingly all health authorities are constantly urging us to move, to be physically active, to exercise. Usually the reason given is better heart performance, longevity, prevention of muscular stagnation, even the joy of movement. All of these reasons are medically valid, yet there is an additional benefit to regular exercise. It is one of the easiest but most comprehensive forms of body-wide detoxification you can perform. The means are simple: walk, run, swim, play tennis, climb, ski, paddle, roller blade, ice skate, jump rope, bicycle, row, dance, bounce on a trampoline, stride on a treadmill.\n\nPersonally, I like to walk. Writers tend to sit around a lot, and sometimes, when the ideas and words are flowing well, it is easy to convince yourself not to move at all. So I walk two miles a day perhaps five to six days a week. Nothing too athletic, just a brisk, comfortable pace in which I feel muscles get stretched and the lungs work a little harder. In the summer, I supplement this with swimming in a nearby lake; again, nothing too fancy or professionally athletic here, just some informal laps with whatever swimming stroke I feel like or remember. What is important is not so much the vigor with which you exercise or even the duration (although research suggests that you need a minimum of twenty minutes), but the _regularity_ with which you provide this benefit to your body.\n\nWhen exercising, you breathe more oxygen, expand your lungs, stretch and work your muscles, and get your blood moving and your heart rate elevated, and here is the detoxification key. When you move your body during exercise, it stimulates the blood and lymph circulation to flow better, more vigorously, to catch up on unfinished or postponed business, to finally dump those protein wastes or collect the byproducts of those toxins you ingested last week, or last year, or deliver more life-supporting oxygen to the cells so they can continue detoxifying themselves.\n\n**_When you move your body during exercise, it stimulates the blood and lymph circulation to flow better, more vigorously, to catch up on unfinished or postponed business, to finally dump those protein wastes or collect the byproducts of those toxins you ingested last weefa or last year, or deliver more life-supporting oxygen to the cells so they can continue detoxifying themselves_**.\n\nExercising enhances your body's circulation of vital nutrients and the flushing out and elimination of stored toxins from the cells. It also stimulates the peristaltic rhythm in your intestines, encouraging this organ to move its contents along more quickly, thereby reducing bowel transit time and potential constipation. Further, some pathogenic organisms are anaerobic, which means they don't use oxygen to survive and can't exist in an oxygen-rich environment. Breathing more means oxygenating your cells, which in turn means suffocating the non-oxygen-breathing microorganisms. And don't overlook the obvious here: when you exercise you are likely to perspire, and the sweat itself carries toxins out with it through the skin.\n\nThese considerations are of vital importance during the intensive detoxification phase outlined in Chapter 5, when your cells are dumping, collecting, and eliminating toxins at a rate far beyond normal. Exercise and physical movement will greatly facilitate toxin removal at a time when your body most needs it. Also, the exercise helps clear your head; during detoxification, old emotions and stray, scattered, or simply weird thoughts arise like vapors in your mind, and physical movement, even as simple as walking, helps blow the obnoxious mists out of your head.\n\nThe other obvious benefit to regular exercise is weight reduction or ideal weight maintenance. The exercise stimulates the removal or utilization of excess body fat stored in the fatty tissues. Remember that the body's fat reserves are its primary storage site for toxins, so if you start reducing body fat, you are also dumping toxins. If you are in your forties, you may have noticed that your weight is now much more sensitive to excesses or deficiencies in your diet or to the proportion of time during which you are luxuriously sedentary. In other words, if you are of this age, or older, you can't get away with overeating, indulging, or lounging about the way you used to. You will probably start putting on weight and your metabolic, digestive, and eliminative processes may start slowing down noticeably.\n\nScientific studies have shown that even though strenuous aerobic exercise in a person new to this kind of exertion can produce free radical activity and possible muscle injury as a result of this oxidative damage, the same exercise strengthens your body's ability to withstand free radical activity Exercise fortifies your body's antioxidant defense system; you can gain even more benefits from this natural positive activity by taking vitamins C and E as daily supplements. These are both antioxidants (free radical fighters). Studies show that both these vitamins counteract whatever free radicals are produced during exercise, and that they have other specific benefits for the immune system and its specialized defense cells.\n\nThe good news about the \"exercise paradox,\" as fitness experts call it, is that when your body reaches the plateau of being used to the exercise you are doing, you will be able to exercise even more, or perform more strenuous work, without a corresponding rise in oxidative damage. Once you reach this plateau, you can actually get ahead on the paradox, and generate more antioxidants than free radicals. This is why scientists at the West Los Angeles Veterans Administration Medical Center in Los Angeles, California, concluded, after studying the effects of exercise on twenty patients, that \"physical exercise training can reduce potential chronic health effects associated with daily activities by contributing to an overall reduction in exercise-induced free radical production.\"\n\nIf you are among the \"elderly\"\u2014let's say seventy and older\u2014the relationship between the free radicals generated by exercise and the antioxidant strengthening that exercise also produces is more of a crucial factor, and there is a greater risk that too much exercise can tip the scale towards a predominance of oxidative damage. However, all that this means, in practical terms, is that it is essential to keep your antioxidant nutrient intake at an optimal level, and the easiest way to do this is through vitamin and mineral supplements. With this balance in place, the benefits to the elderly of regular exercise are numerous. It reduces age-related lean body mass loss and the concomitant risk for various chronic diseases such as coronary artery disease, high blood pressure, diabetes (non insulin dependent), anxiety, depression, functional decline, and frailty.\n\nOnce you see what the game is and are aware of its stakes, you will probably be inclined to get moving. \"Motion is life. Stagnation is death,\" comments naturopathic physician Bruce Fife, N.D. \"Without movement we deteriorate and head toward disease and death. Physical activity is the closest thing we have to the fountain of youth.\"\n\n_Recommendation_ : Most health experts advise exercising for twenty to thirty minutes per session, preferably every day. If you sweat a lot or sweat easily when you exercise, be sure to replenish your electrolytes\u2014these are essential minerals that are sweated out of your body\u2014by drinking some form of electrolyte-enriched water immediately after exercising. However, try to avoid the sugar-loaded kinds (such as fructose-sweetened drinks and\/or those containing artificial colors and flavors), because these will bring yet _more_ toxins into your system.\n\n## HEALTHY LIVING SPACE DETOXIFIER #29: Sauna Detox: How Heat Therapy Can Help Your Body Release Toxins\n\nA logical activity to take up after your exercise is basking in a dry heat sauna. First, you tax your muscles and cardiovascular system and work up a healthy sweat, then you relax in a hot, dry space. This sequence is logical from the viewpoint of the detoxification process, too.\n\nSitting in a dry or radiant heat sauna produces hyperthermia, or \"heat stress,\" which is an effective way of getting your cells to release fat-stored toxins; it also induces profuse sweating which, as we understood above, is an excellent, natural way for your body to excrete toxins through the skin. Saunas are no longer exotic, \"something Scandinavians do,\" or even hard to find. Many fitness facilities, public gyms, health clubs, and upscale hotels now offer saunas, and relatively inexpensive home sauna units are available as well (see below).\n\nThe therapeutic function underlying heat stress is temporarily to raise the body's internal temperature to 101-103\u00b0F, in effect, to produce a transient fever. Heat stress is actually a fundamental premise of natural medicine, harking back some two thousand years to Hippocrates, the reputed progenitor of Western medicine, who said, in effect: Give me the power to create a fever and I shall cure any disease.\n\n**_Sitting in a dry or radiant heat sauna produces hyperthermia, or \"heat stress,\" which is an effective way of getting your cells to release fat-stored toxins; it also induces profuse sweating, which is an excellent, natural way for your body to excrete toxins through the skin._**\n\nOnce this internal heat has been generated, the cleansing process can begin. The artificial fever, or hyperthermia, works as an immune system stimulant by accelerating the output of white blood cells. There is some discussion among fitness experts and researchers whether you get the most efficient internal heating through a dry or wet sauna approach. Exponents of the wet or moist heat approach (we'll consider an example called aromaSpa below) say the high ambient humidity keeps the perspiration coming out on your skin from evaporating; further, ambient moisture condenses on the skin and acts as a heat insulator, making your body hotter. In dry heat conditions, your perspiration evaporates and you lose body heat and you may have to spend longer in the sauna until you reach the desired internal temperature.\n\nScandinavian researchers were able to measure the actual changes in various physiological aspects in people taking regular saunas for a week. The researchers studied the changes in heart function and metabolism in ten healthy males who were exposed to the dry heat (80\u00b0C and higher) of a Finnish sauna for one hour twice daily for seven days. They found that after each session in the sauna the body temperature of the men rose by almost one degree and the body weight dropped by almost one kilogram.\n\nWhile one aspect of the blood pressure (systolic, when the heart contracts) did not change, the other (diastolic) decreased noticeably; the pulse rate rose from an average of 75-80 to 106116 beats per minute, indicating increased heart rate, although this change was less noticeable after the third day; and metabolic rate increased by twenty-five percent to thirty-three percent after the first day. The researchers noted that the men tended to have lower levels of certain minerals such as potassium, sodium, and iron after the third day, indicating the need to supplement sauna use with electrolyte intake.\n\nGenerally, sauna use is well tolerated and safe for people of all ages, from infancy to old age, and it does not pose any undue risk for people with heart problems such as a history of heart attacks or high blood pressure. As a rule of thumb, many doctors recommend that if you can walk into the sauna, you can walk out of it, and you will derive a measure of its cardiovascular benefits. \"The normal sauna bath, with moderate cool-off phases, increases the cardiac workload about as much as a brisk walk.\" Even patients who have recovered from acute heart attacks (myocardial infarctions) \"can enjoy the sauna without incurring any harmful cardiovascular effects.\"\n\nAnything that enhances cardiac output facilitates detoxification, but even more specific research establishes the sauna's role in chemical detoxification. Scientists in the Ukraine found good reason for advocating the use of a sauna as a means of detoxifying chemical industry workers. \"[The] sauna increased excretion with sweat of toxic substances (lead, thiuram, captax, sulphenamide C) that penetrated the body during work.\" Researchers at the Institute of Physiology in Moscow, Russia, concluded that the sauna is effective for alleviating psycho-emotional stress due to its relaxation effect; this is produced by the alternation of heating and cooling phases and its regulatory effect on the body's autonomic nervous system.\n\n### _Health Mate_ \u00ae _Sauna_\n\nThis is a prefabricated, portable dry heat sauna made of cedar ready to plug into an electrical socket wherever you situate it in your house. According to the manufacturer, PLH Products of City of Industry, California, this unit uses infrared thermal heat, which is felt by the body to be similar to direct sunlight and its heat. Its source in this case is radiant energy, which works directly on the body's tissues rather than having to heat the air around the body (an approach called convection).\n\nThe air temperature in the unit is thus much lower than in a conventional sauna, typically 110\u00b0-130\u00b0F compared to 180\u00b0-235\u00b0F, and as a radiant heat, it tends to penetrate the body by an inch or more, producing two to three times more perspiration than other types of saunas.\n\nOne of the intriguing benefits put forward for this sauna is that spending thirty minutes in its dry heat envelope is an equivalent cardiovascular workout to running six to nine miles. According to medical research, a moderately conditioned person can sweat off 500 g in the sauna, burning 300 calories. Regular use of the sauna \"may be as effective\" as the cardiovascular conditioning and calorie bum-off expected from running or other forms of vigorous exercise. Here's how it works: the body absorbs heat from the sauna then strives to cool itself by sending blood to the skin (the body's periphery); this action raises cardiovascular activity and speeds up metabolism.\n\nAccording to the manufacturer, you will burn about 300 calories if you run 1.5 to 3 miles, but you will bum 900 calories if you spend thirty minutes in the sauna. The unit costs about $4000 for a two-seater. It is not recommended for people with adrenal suppression, lupus, multiple sclerosis, or hemophiliacs, for people prone to hemorrhage, or for women during pregnancy.\n\n### _Finlandia Sauna_\n\nAccording to folk wisdom, the Finnish use of the sauna, which means \"bathhouse,\" goes back about two thousand years to when the Finns invented it. There is an amusing Finnish bromide extolling the benefits of the sauna: \"If the sauna, vodka, or pine tar can't cure your ills, nothing will.\" In the various sauna models made to order per customer specification by Finlandia Sauna in Portland, Oregon, the inside temperature is typically 180\u00b0F (with a range of 175\u00b0 to 195\u00b0F) and the humidity a low twenty-five percent. The Finlandia has rocks placed around and over the heater so as to filter the heat (the sauna user dribbles water over the stones to produce some steam to induce sweating), and the walls are made of soft wood (cedar, redwood, or hemlock) so as to absorb the humidity produced and to keep the interior atmosphere relatively dry.\n\nAccording to Finlandia, the soft heat and low humidity soothe and relax tired muscles, relieve stress, and promote a generally \"wonderful\" feeling of satisfaction. One way the sauna may do this is by stimulating the brain to release additional levels of its own natural pain-killing biochemicals, such as beta-endorphins and norepinephrine. Further benefits to spending a half-hour in the sauna include increasing the rates of blood circulation, respiration (the inhale-exhale cycle), and pulse rate (indicating the cardiovascular system is working a little harder, in this case, usefully so).\n\n**_As the skin pores open up under the influence of the hot steam, the body accelerates its excretion of cellular zvaste products through the skin. In fact, it has been observed that heavy smokers will excrete tobacco toxins through the skin, often leaving a visible yellow stain on white towels used to pat dry the skin._**\n\n### _Aromatherapy Spa_\n\nThis home unit, called the aromaSpa, is a free-standing, plug-in steam heat therapy unit (alternatively called an Aromatic Steam Capsule by its manufacturer) for one user. The unit weighs about seventy pounds, stands sixty-six inches tall, and has one chair inside. It has transparent walls of polycarbonate, the same window substance used in airplanes. It is portable and costs about $1500. The aromaSpa has a steam generator (which circulates one quart of distilled water during a forty-minute session) and an aromatherapy diffuser designed to infuse the aromas of hundreds of essential plant oils (the basis of aromatherapy) into the misty enclosed air.\n\nThis way you get both the heat therapy benefit and an olfactory one by breathing aromatherapy oils chosen for their focused therapeutic benefits, such as relaxation, invigoration, detoxification, mental clarity, and other purposes.\n\nIndependent scientific studies on the unit have shown that it can increase vascular flow (which enhances oxygen, nutrient, and lymph exchange at the cellular level), increase muscle flexibility, decrease the risk of everyday and athletic injuries, improve blood flow and oxygen delivery to the cells, and heighten immune response by stimulating white blood cell production. Moist heat therapy in the aromaSpa can also relieve the symptoms of allergies, hay fever, and sinusitis, according to the product's manufacturer. Research also suggests that the steam bath or moist heat chamber can remove toxic chemicals such as DDE (a byproduct of the pesticide DDT), PCBs, and dioxin (a pesticide breakdown product) from fat cells.\n\nSitting in the aromaSpa for twenty to forty minutes can reduce lactic acid buildup in the muscles following exercise and thereby prevent post-exercise muscle soreness; it may also be able to reduce cellulite deposits (lumpy fat areas in the skin), especially if you use the following aromatherapy oils during the session: rosemary, sandalwood, juniper, geranium, and\/or lemon essential oils. The manufacturer states that these oils, diffused into the enclosed chamber, can produce detoxifying and water-draining effects in only ten minutes.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Health Mate Sauna:\n\nPLH products, 16000 Phoenix Drive,\n\nCity of Industry, CA 91745;\n\ntel: 800-946-6001;\n\nfax: 626-968-0444;\n\nwebsite:\n\nwww.healthmatesauna.com.\n\nFor Finlandia Sauna:\n\nFinlandia Sauna,\n\n14010-B.S.W. 72nd Avenue,\n\nPortland, OR 97224;\n\ntel: 800-354-3342 or 503-684-8289;\n\nfax: 503-684-1120;\n\ne-mail:\n\nfinlandiasauna@worldnet.att.net;\n\nwebsite: www.finlandiasauna.com.\n\nFor aromaSpa:\n\nVariel Health International,\n\n618 Variel Avenue,\n\nChatsworth, CA 91311;\n\ntel: 800-800-7222;\n\nfax: 818-407-0738;\n\ne-mail: info@aromaspa.com;\n\nwebsite: www.aromaspa.com.\n\nAccording to the manufacturer, Variel Health International, the unit can help induce muscle relaxation and detoxification, stimulate the immune system, ease fatigue, revitalize the skin, and generally rejuvenate your system.\n\nAs the skin pores open up under the influence of the hot steam, the body accelerates its excretion of cellular waste products through the skin. In fact, it has been observed that heavy smokers will excrete tobacco toxins through the skin, often leaving a visible yellow stain on white towels used to pat dry the skin. During a forty-minute session (the manufacturer's recommended standard length), the inside temperature of the unit can reach 115\u00b0-120\u00b0F, and temporarily elevate the body temperature from 101\u00b0-103\u00b0F.\n\n_Recommendation_ : Exercise for twenty minutes then spend thirty minutes (or less, if the heat is too intense for you) in a sauna or steam bath. Thoroughly wash yourself in a hot shower afterward to remove toxins that may have come out through the skin. If you can get someone to give you a good rubdown or professional massage following the shower, this is another excellent complement to sauna therapy. If no massage or rubdown is forthcoming, try to rest for twenty minutes after showering, so your body can assimilate the relaxation benefits of the sauna.\n\n## HEALTHY LIVING SPACE DETOXIFIER #30: A Purifying Bath: Whole-Body Detoxifying Soaks in Herbs, Salts, Oils, and Other Natural Substances\n\nOne of the easiest and surely among the most inexpensive ways to detoxify through the skin is through whole-body soaks in special natural substances known for their cleansing effects, such as Epsom salt, seaweed, essential plant oils, apple cider vinegar, and mustard. This section will highlight several whole-body purifying soaks out of the range of possibilities.\n\n### _Seaweed and Eucalyptus_\n\nFor this bath, you need Epsom salts (one cup), baking soda (one cup), powdered kelp (one-half cup; you may have to pulverize it in your blender), eucalyptus essential oil (five to ten drops), and witch hazel (one teaspoon). According to herbalist Laurel Vukovic, eucalyptus oil has an antiseptic effect, and is also a capable respiratory system tonic and stimulant if you suspect a cold or flu coming. Vukovic suggests mixing the eucalyptus oil with the witch hazel and adding it to your bath after the tub is full to prevent it from evaporating under the hot running water. You can add the other ingredients earlier. It's important to stay in the bath at least twenty minutes, and you might try visualizing toxins seeping out of your body as you soak and perspire.\n\n### _Specialized Detoxification Baths_\n\nNoted naturopathic physician and educator Hazel Parcells, N.D., who took her own health advice to live to be 109, recommended using any of four therapeutic bathing formulas, depending on one's suspected toxicity.\n\nFor all four baths, Dr. Parcells recommended not showering or washing off for four hours after the soak; she also suggested doing the soaks just before going to bed at night.\n\nFor exposure to X-rays or environmental radiation: sea salt (one pound) and baking soda (one pound) dissolved in a tub of hot water; drink a mixture of rock salt (one-half teaspoon), baking soda (one-half teaspoon), and warm water while you are having this soak.\n\nFor exposure to heavy metals, carbon monoxide, or pesticides: regular-brand Clorox bleach (one cup) to a tub of very hot water.\n\nFor exposure to low grade radioactive materials in the food or in irradiated foods: baking soda (two pounds) dissolved in a tub of hot water; drink a mixture of baking soda (one-half teaspoon) and warm water while soaking.\n\nDr. Parcells also described a general detoxifier, especially useful for building immunity when you feel you're getting sick, or if you have fatigue, muscle aches and pains, or emotional and mental stress: pure apple cider vinegar (two cups) mixed into the hot water in the bathtub.\n\n### _Bentonite Bath_\n\nIn Chapter 5, we learned about the detoxifying benefits of bentonite clay. Bentonite can also be used as a bath ingredient. Put two to four pounds of bentonite in the water and allow it to sit overnight to mix with the water. Then add another two pounds of bentonite to your bath water and soak in it for one hour, or add four pounds and soak for thirty minutes. The more bentonite you use, the faster the detoxifying effect. Don't use a commercially prepared bentonite because it will be too dilute and you will have to use a great deal (at considerable cost) to get the same effect.\n\n### _Pore-Opening Ginger_\n\nAdd one cup of Epsom salts (or more, as you like, up to four pounds if you're feeling extravagant) and two tablespoons of fresh grated ginger to your bath water. Stay in the tub up to thirty minutes, but no longer. The ginger should make you sweat fairly vigorously after a while, thereby facilitating the release of toxins through the skin. The combination of Epsom salts and ginger helps open skin pores to excrete toxins, and it also helps eliminate pain throughout the body.\n\n### _Mustard Bath_\n\nThis is a traditional detoxifying soak in many parts of the world. Mustard is believed to increase blood circulation, open skin pores, stimulate the sweat glands, and generally draw out impurities and toxins through the skin. Mustard powder (about two large tablespoons per bath) may be supplemented with essential oils of wintergreen, eucalyptus, rosemary, and thyme (three drops each), for their antiseptic, cooling, skin toning, or stimulant effects. Stay in the bath at least twenty minutes.\n\n### _Liquid Needle Soak_\n\nThis bath, based on a commercial product called Liquid Needle\u00ae Rebalancer acts as a kind of liquid acupuncture needle on the body and provides the system with an energy rebalancing, according to its manufacturer, BioPhotanicals. The product is based on a blending of light (photons) and plant energies (botanicals).\n\nBioPhotanicals claims that while you are soaking in a Liquid Needle Body Soak, \"every pore and every acupoint on the skin is stimulated with the highly charged light-activated molecules.\" The pores and some 1,300 acupoints are stimulated simultaneously by \"charged light signals\" in the Liquid Needle mixture to collect and release toxic substances from inside the body. According to BioPhotanicals, reduced energy and energy blockages (addressed by Liquid Needle) are almost always the result of long-term toxic buildup as well as the presence of external or internal scar tissue. The company also states that each of their Body Soaks carries a photonic charge of 170-250 millivolts, which gives them considerable traction in addressing energy blockages or imbalances.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Liquid Needle Body Soaks:\n\nDNR (Developmental Natural Resources), 4193 Englewood Drive,\n\nIndianapolis, IN 46226;\n\ntel: 800-886-6222 or 317-543-4886;\n\nfax: 317-543-4880;\n\ne-mail: main@dnrinc.com;\n\nwebsite: www.dnrinc.com.\n\nThe company offers five types of Liquid Needle Body Soaks: Clear (to maintain vital energy balance affected by emotional stress); Gold (for maximum detoxifying effects); Blue (to gain a sense of increased energy and vitality); Amber (for those severely out of balance due to pain, inflammation, and stress); and Foot Soak (for those who can't use a bathtub).\n\n# CHAPTER 7\n\n# The Poisons in Your Mouth: Specialized Dental Detoxification Programs\n\nThe subject of this chapter takes up relatively little space in your body, yet it can make your entire body sick, even seriously so, if left unattended. I am referring to your teeth and gums. In recent years, increasing attention has been accorded the teeth in the field of alternative and natural medicine. Physicians from various disciplines are starting to realize that the condition of a patient's teeth\u2014more specifically, the nature of the dental materials used in dental work and how they interact with the body's physiology\u2014have a great deal to do with many health conditions.\n\nEvidence continues to mount in support of contention that many so-called standard dental procedures and substances\u2014using mercury-based fillings, root canals, placing crowns, extracting teeth improperly\u2014have toxic consequences, affecting the entire body. This chapter identifies the major toxic factors in modern dentistry and indicates effective ways to eliminate this toxicity from your system.\n\nThis is a crucial element of the detoxification picture. You may faithfully and successfully do all the other detoxifying steps described in this book, and not progress very much _if_ your teeth are toxic themselves. It is by no means a sure thing that they are; the teeth of many people are in satisfactory condition and are not a major source of toxicity for the body. Either your immune system is strong enough to handle whatever toxins are released by the materials or conditions in your mouth, or, providentially, not too many are released. It is probably prudent to assume that some dental toxins are being released all the time, but they may or may not be seriously contributing to body-wide toxicity. One possible indication that your teeth are interfering with your body's ability to purge itself of its toxic burden is if you do not feel healthier and less toxic after performing the detoxifying steps in this book.\n\nUnlike most of the other Healthy Living Space Detoxifier recommendations in this book, the action outlined below requires the services of a qualified biological dentist, namely, one who is trained in the natural medicine protocols of what one dentist has termed \"whole-body dentistry.\" This term refers to the many ways in which events and conditions in your mouth can affect any or all systems elsewhere in your body. Your action input is to _choose_ to do something health promoting about the possible (some experts would say probable) dental poisons in your mouth. But how will you know if a dental matter is the cause of a seemingly remote, unconnected medical problem?\n\nOne way is to examine your symptoms and consider whether they seem to be treatment-resistant. In other words, despite wellindicated treatment, either conventional or alternative, your health problem has not improved. The reason might be your teeth. In 1996, a group called the Toxic Element Research Foundation in Colorado Springs, Colorado, interviewed 1,320 dental patients to seek correlations between dental procedures and substances and reported health symptoms. The subjects had received all kinds of standard dental work, such as root canals, dental restorations, dental implants, mercury fillings, orthodonture, and periodonture.\n\nThe researchers found that the people showed symptoms associated with thirty different diseases across a range of physical, mental, and emotional difficulties. For example, seventythree percent reported unexplained irritability; seventy-two percent constant or frequent periods of depression; sixty-seven percent numbness and tingling in their extremities; sixty-four percent frequent urination; sixty-three percent unexplained chronic fatigue; sixty-two percent cold hands and feet even under conditions of external warmth; sixty percent constant bloating; fifty-eight percent memory deficits; fifty-five percent sudden, inexplicable anger; fifty-four percent regular constipation. Other frequently occurring symptoms in this group of 1,320 patients were tremors, twitches, leg cramps, shortness of breath, recurrent heartburn, itching, skin irritations, jitteriness, metallic taste in the mouth, suicidal tendencies, insomnia, chest pains, joint pain, irregular heart beat, headaches after eating, and more.\n\nHealth-care practitioners who are aware of the dental link realize now that in most cases the connections are not obvious, direct, or even validated by conventional Western medicine. After all, how can something happening in your mouth-say a minor gum infection, a broken mercury amalgam filling, a poorly done root canal-have any effect on other systems in the body?\n\n### A Gallery of Possible Toxic Consequences of Modern Dentistry\n\nThe question is certainly a valid one, but to respond that they can have no effect because conventional medicine has no way of accounting for such a physiological causal chain or any mode of interdependent interaction, is not valid. You just need a broader, more comprehensive model of the body and how it works. This is to say, you need a medical model based on energy relationships in the human organism, and you find such a system primarily in acupuncture, or traditional Chinese medicine.\n\nIn this empirical way of viewing the body, based on many centuries of trained clinical observation, various energy channels running down the body connect the teeth to various organs and organ systems, such as the liver and the digestive system, resp ectively. If you have toxins in your mouth clustered around an upp er back molar, this state of toxicity (which is also an energy imbalance) can get translated (or transported, as it were) to somewhere else in your body and start producing a disease or pathology. Cancer is the most extreme example, but it certainly makes the case vividly.\n\nSwiss physician Thomas Rau, M.D., medical director of the Paracelsus Clinic in Lustmuhle, Switzerland, reports that both breast and prostate cancer can be caused by unaddressed dental problems. The relationship is called a dental focus: as it were, the toxicity in the tooth focuses itself on another part of the body. In fact, Dr. Rau is on record stating that in about ninety percent of breast cancer cases he has treated there is a dental factor. The breast lies on the energy channel acupuncturists call the Stomach meridian. So if you have a dental problem, such as a faulty root canal or a jaw infection (called a cavitation) in a tooth on this meridian, the flow of vital energy (called _qi_ by acupuncturists) through this meridian will be blocked and lead to degeneration further down the energy channel.\n\nIn a sense, brushing your teeth faithfully has very little to do with any of this. A German study showed that only five patients out of sixty examined had sterile, germ-free teeth; fifty-five of those examined had various species of aerobic and nonaerobic bacteria present in their mouth. Each of these fifty-five patients had at least one tooth with a dental focus, that is, so infected with bacteria that it was transmitting toxicity energy signals to other parts of the body. In this particular case, the sixty subjects were tested because they were not responding favorably (or at all) to well-indicated treatments. The conclusion was that the dental foci or energetic disturbances were blocking their bodies' receptivity to the therapeutic efforts.\n\nDr. Rau treated a man, age fifty-five, with prostate cancer. Even though the man's prostate organ had been removed, he still had cancer. A dental X-ray revealed that the man had a halfformed tooth impacted in his upper jaw, above the incisor. This extra tooth was affecting the patient's kidney and bladder energy channels and disrupting its energy flow through the groin, including the prostate. This imbalance was involved in the inception of the man's prostate cancer, concluded Dr. Rau, who promptly extracted the half-formed tooth. After the patient received various alternative medical therapies for his cancer, he regained his health and was cancer free.\n\nDr. Rau also treated a woman, forty-five, who had recently had part of her breast removed due to breast cancer, and she was about to begin chemotherapy. Dr. Rau, in examining the woman's teeth, discovered that one of her upper jaw incisors, which had received a root canal fifteen years earlier, was situated with respect to her kidney and bladder energy channels. A neighboring tooth, also on this channel, was poorly developed. Dr. Rau learned that prior to developing breast cancer, the woman had sustained an ovarian cyst (a noncancerous fluid-filled sac that forms in the ovaries) and had had it surgically removed. It was likely, said Dr. Rau, that the dental problem had initially focused itself in the ovaries, creating the cyst, which appeared two years after the root canal was performed. Again, removal of the two offending teeth, in conjunction with other medical treatments, helped reverse her health problems.\n\nNot only can toxic energies be transmitted through the body, but physical toxins can move outwards from the mouth, too. Dental toxins that leach from the mouth and migrate to other parts ofthe body are now associated with numerous health problems, including immune dysfunction, coma, multiple sclerosis, and leukemia. In the United States, one of the pioneering dentists to promote this concept is Hal Huggins, D.D.S., M.S., who is based in Colorado Springs, Colorado.\n\nHere's how a dental problem can precipitate leukemia, a potentially fatal cancer of the blood-forming cells in bone marrow. A healthy young man of eighteen received two mercury amalgam dental fillings. Two days later he developed a high fever of 105\u00b0F, and the next day he was hospitalized with a diagnosis of leukemia, says Dr. Huggins. Within twenty-four hours of having the two mercury fillings removed, all signs of the boy's dis ease disappeared. Since Dr. Huggins caught the problem very early, it was easily reversible because he understood the true causal connection.\n\nThe mercury, which is a lethal poison on its own and regarded by the Environmental Protection Agency as an environmental hazard, somehow leached from the man's fillings in sufficient levels to destabilize his immune system and white blood cell count and precipitate a catastrophic decline into leukemia. It is ironic\u2014and shocking\u2014that a toxic substance the EPA calls a waste disposal hazard, something a dentist cannot throw out, bury in the ground, or dump in a landfill, is okay to put in a person's mouth. It is even more unbelievable that den tists who decline to do this and instead discuss the dangers of mercury with their patients are subject to professional reprisals, harassment, and legal action by medical authorities.\n\nSometimes it is harder to completely undo dental-caused leukemiaifthe problem has been developing for some time, as in this next example. A dentist, age forty-two, came to Dr. Huggins, after having been exposed daily to mercury amalgam materials for many years in the course of his dental practice. One day the man was exposed to yet another dose of mercury: in helping his father, a farmer, clean out a grain silo, he came into contact with a puddle of liquid mercury on the floor of the grain silo where the toxic heavy metal was used to keep the rats out. Within a week, the man was seriously ill and received a leukemia diagnosis.\n\nOne of the first treatment steps he took was to have all his mercury amalgam fillings removed; otherwise they would continue to slowlybut steadily contaminate his system with mercury vapors and emissions. However, he was resistant to discontinuing his dental use of mercury as fillings for his patients. As a result, his recovery from leukemia was not complete, although it was satisfactory and enabled him to resume his life.\n\nThinking about having your teeth cleaned? Of course it's good for your gums and may prevent periodontal disease, but there is also the risk that it can scratch free some mercury from old fillings and allow a toxic dose to migrate through your body, states Dr. Huggins, who saw it happen. A young woman of twenty-nine who was in good health had a routine teeth cleaning by her dentist. The next day she detected a metallic taste in her mouth; her gums, lips, and mouth started tingling, then went numb; and her ears were ringing. Then things got much worse. She started to have seizures, lost her ability to speak, and slid into a state of near coma, oblivious ofher surroundings. Her doctors expected her to die in a few days, and they had no idea what her medical problem was.\n\nAccording to Dr. Huggins, \"As a result of having her teeth cleaned, Sabina now had a higher level of mercury leaching from her twelve mercury-based fillings because the protective corrosion had been cleaned off.\" Her immune system could not handle this extra load of toxic mercury, and collapsed. It was a tough and delicate operation, but Dr. Huggins and his team removed all of the woman's mercury fillings while she was almost comatose. After the first day, she showed signs of alertness and three days later she was able to speak again; after a week or so, she was back on her feet, well again. This is a startling example of acute mercury toxicity and while, one hopes, relatively rare in occurrence, it shows the awesome health-debilitating effect mercury can have when it manages to seep out of a dental filling and migrate through the body.\n\nHere are two more patient studies that make a compelling case for systemic toxicity caused by mercury dental fillings. The first comes from an acupuncturist named David J. Nickel, O.M.D., L.Ac., of Santa Monica, California. A man, age thirty-two, came to him with a list of debilitating symptoms: fatigue, anger, low blood sugar, prematurely greying hair, back problems, ringing in his ears, nervousness, and problems with concentration. At twenty-eight, he had had a nervous breakdown, and at the age of fifteen, he had developed anxiety and severe fatigue immediately following a root canal procedure.\n\nDr. Nickel did a mercury vapor analysis on the patient's mouth. He discovered that the man's teeth were outgassing mercury vapor from six teeth at levels forty-two times higher than the maximum allowable limit set by the Environmental Protection Agency had ruled. Dr. Nickel did a hair analysis on the patient and found high levels of other heavy metals (aluminum and copper; he also got confirmation of the presence of systemic mercury) and a dangerously slow rate of metabolism. Dr. Nickel advised the patient to have the six mercury fillings and the root-canalled tooth removed, and he put him on a nutritional supplement program to address his low metabolism and blood sugar imbalance.\n\nA month after the corrective dentistry, the man reported his energy levels had returned to the high levels he had enjoyed seventeen years earlier. All his other symptoms had gone away as well, and he said he felt himself now in excellent health. There was no clinical doubt, as far as Dr. Nickel was concerned, that the mercury fillings and root-canalled tooth had produced the multiplicity of symptoms.\n\n\"Of the 90% of my patients with amalgam dental fillings, most have a mercury-induced copper toxicity, high calcium levels, and reduced thyroid and adrenal function,\" he states, outlining some of the secondary problems mercury produces, such as the dangerous link with copper, another toxic heavy metal. Dr. Nickel adds that mercury is linked with 258 different medical symptoms, and copper with at least 100. \"Almost all my patients who have had their mercury-based fillings removed show moderate to dramatic improvement in their heath in usually less than one month.\"\n\nHere is another case from an acupuncturist. This time, a male patient, age thirty-seven, had suffered from seven years of unremitting fatigue. He also complained of swelling in his lymph nodes, headaches, light-headedness, irritability, and pain in his chest and muscles. M.M. van Benschoten, O.M.D., a doctor of Oriental medicine practicing in Reseda, California, found traces of viruses and bacteria in the patient's system at sufficient levels to contribute to the chronic fatigue; but he also detected mercury toxicity from dental fillings at such levels as to suggest it was the \"fundamental underlying cause\" of the health problems.\n\nDr. Benschoten explains that in acupuncture theory, a toxic metal such as mercury interferes with the circulation of vital energy, or _qi_ , through the energy channels of the body (meridians). Dr. Benschoten gave the patient a series of Chinese herbs to help deal with the viral, bacterial, and heavy metal toxicity, and he advised the man to have all fourteen of his mercury amalgams removed, which he did three months later. However, the dentist this patient employed for the mercury removal did not perform the procedure correctly, and the patient's body-wide mercury levels actually increased. This of course is the risk of mercury filling removal. If it isn't done right, it can enable dental mercury to be released into the body, increasing the body's total mercury load. In this case, Dr. Benschoten gave the patient a second round of Chinese herbs to round up the free mercury in the body, and after a few months, the man was symptom free.\n\nAnother potent source of hidden dental infection comes from an incompletely performed tooth extraction, often of a wisdom tooth or an abscessed tooth. This type of dental focus is called a cavitation. A cavitation is an area in the jaw where a tooth has been removed, and the resulting bone lesion or tissue wound becomes infected and inflamed. When the bone of the extraction site is unable to heal properly, it cavitates, or forms a mushy depression (cavity) subject to infection and the eventual death of bone tissue. The dead bone can produce nerve pain, usually in the head, face, neck, or shoulders, but can sometimes create problems elsewhere in the body, such as the lower back, legs, or intestines.\n\nChristopher Hussar, D.D.S., D.O., a dentist-osteopath who practices in Reno, Nevada, relates the case of a woman, fifty-five, who had suffered from severe facial pain for fifteen years. Nobody could figure out what was causing it, and she had consulted an estimated fifty doctors prior to seeing Dr. Hussar. Her dental focus was a deep chronic infection inher lower jaw where the infection had wrapped itself around the major nerve trunk in that area of her mouth, explains Dr. Hussar.\n\nThe problem most likely originated years earlier following a faulty wisdom tooth extraction in which the dentist had failed to thoroughly debride (clean out) all the unhealthy tissue and dead bone from the extraction site. This set in motion the chronic infection and in turn the referred pain, that is, _facial_ pain rather than strictly locatable _dental_ pain. As a secondary dental focus, the woman had a localized, but festering, jaw infection around the site of a root-canalled tooth. Dr. Hussar cleaned out the site of infection in her mouth and irrigated the area with disinfectants; following the two-hour procedure, the woman had no post-operative pain and her facial pain did not return.\n\nDr. Hussar notes that the inside of these dental lesions or cavitations can be quite gross. \"Inside these jawbone cavities you may also encounter viruses, bacteria, yeasts, and parasites, all of which contribute to the harmful dental focal disturbance. The mouth is the filthiest place in the body.\" Dr. Hussar has seen direct clinical evidence linking untreated cavitations with irregular heartbeat, orbital pseudo-tumor, or pain around the eyes, headaches, blindness, hearing disorders, arthritic pain, rheumatological problems, \"and all manner of unexplained pain disorders.\" All of these conditions were reversed when the hidden dental infection was treated correctly, says Dr. Hussar.\n\nWith these examples as a conceptual starting point, let's examine in greater detail the potential toxic contribution of mercury amalgam fillings, root canals, and cavitations, and consider grounds for undergoing corrective dentistry to eliminate these sources of toxicity from the body.\n\n## HEALTHY LIVING SPACE DETOXIFIER #31 Make Your Mouth a Mercury-Free Zone by Having Your Mercury Fillings Correctly Replaced\n\nThe matter of mercury amalgam fillings and their effect on the body is a controversial and politically charged one. Most dentists in the United States are still resistant to the concept that mercury fillings might be hazardous to the patient's health. Many dentists claim they are unfamiliar with the argument, or have never seen evidence supporting the claim, or that the evidence is unsubstantial.\n\nHowever, as a health-care consumer, you must see through the ignorance and duplicity of modern dentistry and consider the facts and theories on your own. Other countries have actually banned the use of mercury amalgam fillings, regarding the substance as too toxic for human contact, while the United States remains highly resistant to not only taking this action or openly discussing the evidence, but even to informing dental patients that such evidence exists.\n\nIt certainly doesn't help matters that in many cases state licensing boards come after dentists and basically punish (legally and fmancially) those who break away from conventional attitudes about dental mercury and start informing their patients that mercury is a hazard and ought to be removed from their teeth at the earliest opportunity. Let's establish the basic facts first.\n\nMercury fillings, or amalgams, have been used in dentistry since the 1820s, and today an estimated ninety-two percent of dentists routinely place them in their patients' mouths. Today's amalgam typically contains an alloy of several metals including mercury (fifty percent), silver (thirty-five percent), tin (nine percent), copper (six percent), and a trace of zinc. An estimated 150 million are placed in mouths every year, representing seventy-five tons of amalgam alloy. Dentists say they like mercury fillings because they are durable, easy to manipulate and install, and don't cost much. However, the disadvantages far outweigh the benefits. Mercury is a toxic heavy metal, a noted carcinogen, an immune system damager, and generally a heavy-duty poison best avoided or handled with extreme caution in all other circumstances other than in your mouth.\n\nMercury as a heavy metal is also a potent free radical, capable of destroying cells and ruining cell membrane integrity, and even affecting DNA processes and producing cell death. Further, mercury interferes with the energy producing function of the cells by causing the mitochondria, the cell's energy \"factories,\" to become dysfunctional. Not only does it increase the number of free radicals in the system, but it reduces the body's defense system, the antioxidants.\n\nAs a free radical, mercury can block the body's detoxification pathways by blocking the action of key enzymes. Specifically, it can have \"devastating effects\" on the glutathione content of the body, which in turn facilitates the increased retention by the body of other environmental toxins. Glutathione is one of the liver's key detoxification substances, so if its levels are diminished, so is the liver's detoxification capability.\n\nIt was not until 1988 that the routine use of mercury raised serious enough questions for the Environmental Protection Agency (EPA) to declare scrap dental amalgam a hazardous waste. Dentists have to dispose of dental mercury with the same care as they would extend for any other highly toxic, lethal hazardous waste. Yet they routinely keep putting it in the mouths of millions of patients. It takes only a modicum of intelligence to ask why if this substance is an EPA-designated hazardous waste it is ever considered safe to store in the human mouth in teeth? Is the mouth a strange new kind of unregulated hazardous waste site?\n\nThe traditional answer has been that once in the amalgam composite fillingthe mercury is fixed fast and will not leach out of the dental filling, and if it did, it would not go anywhere in the body and thereby create a health problem. Medical research is refuting both of these assumptions, showing them to be danger ously incorrect. Mercury does leach and it does migrate through the body.\n\nMercury vapors from the fillings are continuously being out gassed in the mouth, entering the body through the blood or inhaled mouth air. Chewing or grinding your teeth contribute to this steady release of minute amounts of mercury vapor in the form of what chemists call elemental mercury. You don't have to be chewing or grinding your teeth to still be absorbing outgassed mercury vapors; in fact, sometimes just drinking a hot beverage such as coffee can increase the vapor release from the fillings. Scientists estimate that eighty percent of mercury vapor absorption from the teeth happens in the lungs, after which the mercury vapor rapidly enters the bloodstream; once in the body's tissues, it has a preference for lodging in the central nervous system and the kidneys.\n\nPotentially, your dental fillings can outgas mercury twenty four hours a day, every day of your life. How much gets released in a single day? A Norwegian study involving 147 patients (meas uring mercury release in the blood, urine, and exhaled air) found that people with more than 36 restored dental surfaces (36 mercury fillings, even if some were on the same tooth) absorb 10-12 mcg\/l of mercury every day. In 1985, the World Health World Health 17 mcg of mercury every day.\n\nOther studies estimate the daily release to be from 20-150 mcg per filling, although the ultramodern high-copper amalgams can release fifty times more mercury vapor than conventional mercury amalgams. The key point here is that the exact amount of mercury emission is not yet known, though the observed range has been documented, and this alone should concern us-that it is leaking from teeth continuously. A Swiss study showed that the greatest volume of mercury released into the body was during the first 24 hours after the filling was installed, ranging from 17.4 mcg\/l to 34.5 mcg\/l. After the fillings \"annealed\" (hardened and solidilled over time) for two years, the daily mercury emission rate (for eight fillings, as used in the study) was 1.5 meg.\n\nSo while the amounts said to be absorbed vary according to the study, there seems to be agreement that some mercury is definitely absorbed.\n\nA Danish dentist, H. Lichtenberg, D.D.S., undertook his own research to find out how bad the mercury emissions were from teeth. He studied oral mercury vapor emissions from the teeth of 103 patients, whose average age was 47. He noted that each of his patients in this study already had at least fourteen symptoms representative of mercury toxicity; each patient also had an average of twenty-two mercury fillings. The mercury emissions from a single mouth varied from 3 mcg to 329 mcg per cubic meter of air, with an average of 54 mcg, Dr. Lichtenberg reported. He found that people who had twenty-six or more mercury fillings tended to have emissions above fifty, and those who had less than twenty mercury fillings had levels below fifty.\n\nThe number fifty is important here because in Denmark levels above 50 mcg in a workplace are judged unsafe by the government. This means many of Dr. Lichtenberg's patients had oral mercury levels exceeding the legal safe limit for industrial workplaces. Put differently, they had a mercury exposure equivalent to a workplace exposure of 168 hours a week (twenty-four hours a day) _indefinitely_. How do Denmark's mercury exposure standards stack up compared to other countries? In Russia, only 10 mcg are allowed; in the United States, 100 mcg; in Canada, 1 mcg for a person weighing 150 pounds was recently judged a Tolerable Daily Intake. Dr. Lichtenberg's patients were getting a daily mercury exposure fifty times higher than this safety norm.\n\nSo mercury is being released from fillings\u2014is this a lot? According to one study, it takes only one microgram (mcg) of mercury to damage nerve tissue. Further, \"the bad thing about mercury release is that it is cumulative,\" explains Hal Huggins, D.D.S., M.S., an outspoken opponent of mercury amalgams and author of _Uninformed Consent._\n\n\"In the fastest elimination mode, if one microgram is absorbed, it will take 70 days to several months to eliminate half of it.\" The next day you absorb another microgram, then perhaps you get another mercury-based filling, and still another one cracks, doubling the outgassing volume. The math suggests that unless you take major action to detoxify yourself of mercury, you will never get it all out of your body on your own by depending on natural internal detoxifIcation processes. \"You will still increase your total body burden of mercury daily\".\n\nInhalation of mercury vapor from the teeth and its absorption by the lungs is only one of three possible routes by which mercury can pass from your fillings into your body. Once in the intestines, soluble mercury compounds break down into mercuric ions that are easily absorbed and transported from the intestines throughout the body along with nourishing fluids. A third route is through the mouth itself. Mercury outgassed from the fillings can be absorbed by the nerves and tissues beneath the fillings and possibly also in root canals.\n\nOnce in the bloodstream, bacteria convert mercury to methylmercury, the organic form of mercury that is one hundred times more toxic than the form of mercury that starts out in your dental filling. The methylmercury form of the heavy metal is capable of crossing the blood-brain barrier, usually a physiologi cally sacrosanct threshold that keeps toxic substances out of the brain. Methylmercury is also distributed throughout the body. The blood readily absorbs it because it mixes easily with fat molecules; it can potentially be distributed to every cell in the body.\n\nCroatian scientists studied the migration of mercury from amalgams in rats over a two-month period. Rats were exposed to either four mercury amalgams or were fed powdered amalgam in their diet. After two months, the kidneys and brains had \"significantly higher\" concentrations of mercury than other organs for both groups compared to the controls, which had no mercury exposure. Rats with actual amalgams had mercury levels eight times higher than controls and two times higher than those fed the amalgam powder, and they had kidney concentrations of mercury five times that of the control group.\n\nEvidence now shows that mercury amalgams are the major source of mercury exposure for the general public, six times higher than mercury exposure from fish and seafood. A study of the effect of abrasion (chewing) on amalgam surfaces (using real teeth with mercury amalgam fillings but in an artificial mouth to simulate chewing) showed that the \"steady-state\" release of mercury every day from a _single_ filling is 0.03 mcg. Again, you can see the range of estimated levels of emission.\n\nA Danish study of a random sample of 100 men and 100 women showedthat increased blood mercury levels were related to the presence of more than four amalgam fillings in the teeth. As Dr. Morton Walker concludes in _Elements of Danger_ , an indictment of the toxic practices of modern dentistry, \"In my opinion, anyone who allows dental amalgam fillings to remain as part of his or her oral cavity has elected to commit slow but steady suicide by mercury poisoning.\"\n\nMercury toxicity has been shown to have destructive contributory effects on kidney and immunologic function and in cardiovascular disease, neuropsychological dysfunction, reproductive disorders, and birth defects, to name a few. Symptoms of or diseases resulting from mercury toxicity make a very long list: anorexia, depression, fatigue, insomnia, arthritis, Alzheimer's, moodiness, Parkinson's, Lou Gehrig's disease, periodontal disease, irritability, memory loss, nausea, diarrhea, gum disease, swollen glands, headaches, multiple sclerosis, antibiotic resistance, and many more.\n\nResearchers have found that hair analysis of multiple sclerosis (MS) patients reveals significantly higher mercury levels than those found in non-MS patients. They also found that MS patients with mercury amalgams have thirty-three percent more MS exacerbations (flare-ups of debilitating symptoms) in a twelve-month period than MS patients without mercury fillings.\n\nResearchers have also established that MS patients with mercury fillings tend to have a poorer mental health status than those patients with no mercury fillings. When forty-seven MS patients with mercury fillings were polled using standard psychological questionnaires, scientists found that they had higher rates of depression, anger, hostility, psychosis, and symptoms of obsessive-compulsive disorder compared to MS patients with no mercury fillings. Further, MS patients with mercury amalgams had forty-three percent more MS symptoms over a twelve month period. The researchers concluded that the lowered psychological health profile of MS patients may be due to mercury toxicity from dental fillings.\n\nA 1993 study of 1,569 patients drawn from the United States, Denmark, Sweden, and Canada tabulated the correlation of numerous health problems with mercury amalgams. They found that 705 patients reported fatigue, 531 headaches, 347 depression, 343 dizziness, 270 concentration problems, 265 memory loss, 260 metallic taste in the mouth, 231 intestinal problems, 221 allergies, and the list continued.\n\nThe study also correlated improvement in reported symptoms with mercury amalgam removal: eighty-six percent with fatigue reported a cure or improvement in this problem after amalgam removal; eighty-seven percent of those with headaches, ninety-one percent with depression, eighty-eight percent with dizziness, eighty percent with concentration deficits, seventy-three percent with memory loss, ninety-five percent with metallic taste, eighty-three percent with intestinal problems, and eighty-nine percent with allergies. In other words, the causal connection between systemic mercury poisoning produced by mercury amalgams and numerous minor to serious health conditions was well established in this study. The report further showed the dramatic improvement in health conditions with the removal of the mercury fillings.\n\nIn 1991, a large group of dentists submitted themselves to a study on the effects of systemic mercury toxicity resulting from their prolonged professional contact with the heavy metal. At the annual meeting ofthe American Dental Association, 1,502 dentists had their urine analyzed for mercury levels. Of this group, 2%, or 29 dentists, had levels of 29 \u03bcg\/l(micrograms\/liter) and were clas sified as \"exposed.\" Their mercury levels exceeded the safety level of 19 \u03bcg\/l, and it was seven times higher than the mean level measured in dentists. The \"exposed\" dentists placed 50% more mercury fillings per week than nonexposed dentists; they had worked an average of 19 years in the same office, and averaged 2.3 mercury spills in their offices compared to only 0.1 for nonexposed dentists. This data suggested clearly that mercury toxicity is related both to duration of exposure and degree of regular contact.\n\nNext, the dentists volunteered to take a behavior test based on sixty-five different mood descriptors to register mental and emotional function. The results were \"very significantly linked\" to urine mercury levels, the researchers found. The strongest associations between mood and mercury were found in the areas of tension, fatigue, and confusion, as well as reduction in vitality and increased depression. Correlations were also noted between mercury exposure and poor mental concentration, emotional lability (mood fluctuation), and nervous system irritation. The study found \"some evidence of adverse preclinical effects at mercury doses averaging 36 \u03bcg\/l in urine.\" The report suggested the World Health Organization safety standard of 50 \u03bcg\/l should be critically reevaluated, and observed that \"this is the first U.S. dental study to detect potential behavioral deficits at such a low level of exposure.\"\n\nAnother study demonstrated the mood-altering effects of mercury toxicity. Researchers interviewed twenty-five women with mercury amalgams and twenty-five without, using various standard psychological inventories employed by psychologists to gauge and quantify emotional states. They found that women with mercury fillings had higher scores overall (higher meant they were more reactive), with more symptoms of fatigue and insomnia. They had a greater tendency to express anger without provocation; to experience more intense angry feelings and a greater level of general anxiety; to feel less pleasant, satisfied, happy, secure, or steady; and to have a harder time making decisions. Women without mercury in their mouths had an easier time controlling their anger. On this basis, the researchers concluded that mercury toxicity may be a causative factor in states of depression, excessive anger, and anxiety.\n\nA study of the teeth of military personnel showed that the more fillings in the mouth, the greater the body load of leached mercury. Researchers at the National Institute of Dental Research examined the teeth of 1,127 adult male military personnel, average age 52, with an average of 19.9 mercury amalgam fillings (surfaces of the teeth exposed to mercury fillings, meaning one tooth might have multiple fillings). The total and inorganic mercury concentrations in the urine were 3.09 mcg\/l and 2.88 mcg\/l, respectively, while the average blood level for both were 2.55 mcg\/l and 0.54 mcg\/l.\n\nNot only did this study show conclusively that mercury from amalgams leaches and migrates into the body, but the researchers concluded that \"on average, each ten surface increase in amalgam exposure is associated with an increase of 1 mcgll of mercury in urine concentration.\" In other words, for every new filling you can count on there being _at least_ one more microgram of toxic mercury released into your system. One microgram of mercury may be excreted in the urine, but how many micrograms remain in the body? \"Mercury is more effective as a killer of cells (cytotoxicity) than many cancer chemotherapies,\" comments Dr. Walker. \"There is _no_ harmless level of mercury vapor exposure.?\"\n\nFor prospective parents, there is yet another cause for concern for mercury fillings. Research shows that women can pass on mercury to their fetuses in the womb, as mercury can cross the placenta from mother to fetus. Studies have shown that the fetus can store eight times more mercury than the mother's own tissues, where it tends to store in the breasts and get passed on to infants through breast milk.\n\nSummarizing the scientific research and clinical evidence, Dr. Huggins says mercury \"can express its toxicity\" in ten ways (see figure 7-1). It can:\n\n  1. 1) Alter cell membrane permeability. Mercury binds to the cell membranes, affecting how the cell discriminates between \"good\" and \"bad\" substances to let cross its border.\n  2. 2) Change the shape of molecules, affecting their ability to bond with other molecules.\n  3. 3) Alter enzyme function. Mercury binds to sites on enzymes, slowing down their reaction time and ability to react.\n  4. 4) Interfere with nerve impulses. Mercury blocks the synaptic gap between nerve cells, hindering transmission.\n  5. 5) Affect the genetic code. Mercury cleaves DNA, producing genetic defects in fetuses.\n  6. 6) Inhibit DNA repair mechanisms.\n  7. 7) Disrupt endocrine gland activity. One atom of mercury can deactivate an entire hormone molecule.\n  8. 8) Contribute to autoimmune illnesses. Mercury, attached to cells, distorts their shape and confuses the immune system, which judges it to be a foreign and not a host cell.\n  9. 9) Change digestion and absorption functions.\n  10. 10) Contribute to antibiotic resistance. Mercury causes some bacteria to alter their shape, inadvertently enabling them to resist conventional antibiotics.\n\nOn this last point, researchers at the University of Georgia at Athens reported that mercury exposure through dental fillings can produce antibiotic resistant bacteria in the intestinal flora of otherwise healthy and nonmedicated subjects. They studied 640 people, of whom 356 had not had a recent exposure to antibiotics, and found that those with a high prevalence of mercury resistance in their intestinal flora (the \"friendly\" bacteria resident in the intestines) were also likely to resist the action of at least two standard antibiotics. In other words, the mercury changed the internal bacteria (mostly oral _Streptococcus_ organisms) in such a way that they became resistant to the effect of antibiotics designed to kill them.\n\n**Figure 7-1. 10 Ways Mercury Can Interfere with Vital Body Systems**\n\n\"Mercury amalgams are as close as you can get to the center of the illness universe,\" comments Bruce Shelton, M.D., M.D.(H), DLHom., a physician specializing in homeopathy and natural medicine based in Phoenix, Arizona, where he runs The Allergy Center. \"Their use in dentistry has set us up for most of the health problems we see today.\"\n\nFor example, there is a \"domino effect\" that starts with mercury toxicity, explains Dr. Shelton. The overwhelming majority (ninety percent) of the patients he sees who report allergies have an overgrowth of the yeast _Candida albicans_ , and they often have this yeast infestation because of heavy metal toxicity in the body, notably of mercury. Candida is an effective, natural absorber of mercury, so in a perverse but logical move, the body attracts the yeast to it so it will absorb the toxic mercury. The logic is that of a tradeoff: the yeast overgrowth will be less toxic overall than unguarded free IILercury, so the body opts for a _Candida_ _albicans_ infestation to contain the mercury\u2014chronic allergies instead of multiple sclerosis, as it were.\n\nHowever, it's really a bad trade-off, because the yeast overgrowth sets the stage for intestinal permeability, or what's popularly known as leaky gut syndrome. This is a kind of backwash into the blood from clogged, dysfunctional intestines. Food allergies develop and soon the person is allergic to everything; the next step from here is multiple chemical sensitivity, environmental illness, or chronic fatigue syndrome, says Dr. Shelton. \"Environmental illness is then the end-product of a string of health problems that stem from mercury and build, one upon the other.\"\n\n_Candida albicans_ is not the only thing in the body that has an affinity for mercury. Research now shows that nerve endings in the peripheral nervous system voluntarily absorb mercury (that has leached from teeth and migrated into the body at large) even though it is a neurotoxin\u2014toxic to nerve cells. According to Dietrich Klinghardt, M.D., Ph.D., an innovative physician who practices in Seattle, Washington, these nerve cells absorb mercury \"out of curiosity,\" transporting it up the spinal column. It is their natural function to scan their environment continuously and inspect all foreign substances for their threat potential.\n\nThe trouble is that when mercury moves up the spinal column, it destroys a crucial substance called tubulin, which is necessary for moving substances in the nerves. In effect, the mere presence of mercury destroys the nerves and their transport mechanisms, so it is a one-way and deadly \"visit.\" The intent of the peripheral nervous system in examining the mercury, explains Dr. Klinghardt, is to make an antineurotoxin against it to neutralize and eliminate it from the body. But the mercury is too deadly and overpowers the nerve cells.\n\nDr. Klinghardt states that scientific studies establish that within twenty-four hours after injecting a tiny dose of mercury into a muscle anywhere in the body (in this study, it was monkeys), it is detectable in the spinal cord and brain, as well as kidneys, lungs, blood, connective tissue, and adrenal and other endocrine glands. It is also now known that mercury prefers the brain's hypothalamus gland nervous system) and the believed to be the organic seat of the emotions).\n\nThese affinities help explain the range of physical and emotional symptoms attributed to mercury toxicity. \"As soon as anybody has any type of medical illness or symptom, whether medical or emotional, the amalgam fillings should be removed and the mercury residues should be eliminated from the body, especially the brain,\" states Dr. Klinghardt.\n\nBut let's say you don't have an overt or manifest illness at the moment. Should you have your mercury fillings removed? How can you tell if you have mercury in your system at levels beyond what your immune system can handle? In Chapter 3, we reviewed five different laboratory tests that can give you an answer about your body's mercury load. These tests include hair analysis for heavy metals, a urine elements profile, a fecal metals stool analysis, an oral toxicity test called ALT, and the DMSA mercury provocation test.\n\nThe recommendation here is before you have your mercury fillings replaced, find out if you have too much mercury in your body, and do this by employing one or more of the tests just cited and described in detail in Chapter 3. As a complement, get a reading on the vitality of your immune system and your general nutrient status; fmd out if you are nutritionally and immunologically equipped to handle whatever toxic load your teeth may be putting on your body. Intelligent, focused nutrient fortification may enable your system to handle the mercury load for a while longer until you decide what permanent course of action to take, orifyou elect to have a few fillings replaced at a time over a twoyear period, for example.\n\nThe reason for the conservative approach here is that not only is mercury amalgam replacement expensive and not necessarily reimbursed (if at all) by dental insurance plans, it is potentially dangerous if not done correctly. Minute mercury particles can be swallowed and minute amounts of mercury vapor inhaled during the procedure, and you can end up with more mercury in your system than before. If your immune system is already on the edge, at its limit of being able to handle your body's toxic load, this additional toxic input may be too much, and you may get sick as a result, even _very_ sick.\n\nAt a minimum, for safe mercury amalgam removal, your dentist must use a rubber dam (a rubber latex sheet or dental dam) in your mouth to prevent you from swallowing mercury. Your dentist should also use strong air suction to collect vapors, and frequent water suctioning and washing out of the mouth. The patient must be draped so no mercury chips get carried home; ideally, the patient breathes oxygen through a nosepiece during the procedure rather than through the mouth or nose to avoid any possible inhalation of mercury vapors.\n\nPerhaps the following data will help clarify your thinking. A researcher polled sixty patients who had undergone replacement of their mercury fillings, followed up with a nutritional program and heavy metal detoxification using DMPS (see below). Of these people, seventy-eight percent said they were either satisfied or very satisfied with the results. In terms of the patient's own assessment of their medical problems before having the fillings replaced, the most common complaints were memory and concentration deficits; pain in the muscles and joints; anxiety; insomnia; problems in the stomach, intestines, and bladder; depression; food or chemical sensitivities; numbness; tingling; or eye problems, in that order. Of the symptoms that bothered them the most, the patients cited headaches, backaches, fatigue, and memory and concentration problems.\n\nEvidence exists that even with these precautions, \"significant\" new deposits of mercury in the body (lungs, kidneys, endocrine organs, liver, and heart) can occur. This means you need an effective way to bind up (chelate) existing mercury in the body so as to effectively escort it out of the body.\n\n### _Cilantro_\n\nResearch conducted by Chinese acupuncturists found that cilantro _(Coriandum sativum)_ can effectively collect and bind up mercury and other heavy metals in the brain and central nervous system. Another mobilization agent, such as chlorella, is then needed to facilitate the excretion of these toxic substances from the body.\n\nFor patients who were having their dental amalgams replaced, the acupuncturists prescribed cilantro, also known as Chinese parsley, at the oral dose of 100 mg four times daily (in conjunction with other \"drug-uptake enhancement methods,\" e.g., the essential fatty acids EPA and DHA) to collect and remove the mercury from the body, including the new deposits that resulted from the amalgam removal. They recommended taking cilantro before the amalgam removal, and then for another two to three weeks afterwards.\n\nThe Chinese researchers also found that cilantro worked synergistically with antibiotics and natural antiviral agents (EPA and DHA) against several otherwise intractable infectious organisms that had been unresponsive to the antibiotics alone. When cilantro was given in tandem with these other treatments, it \"rapidly reduced the generalized symptom and infection.\" Cilantro \"accelerates\" the excretion of mercury and ead from the body through the urine, they reported. They postulated that the infectious microorganisms used the heavy metals in the body (lead and mercury) to protect themselves from the antibiotics.\n\nCilantro of course is available without a prescription, and you can also take it as a cooked herb (stalks), at the rate of one to two teaspoons daily. Dragon River Herbals offers a blend of cilantro and yellow dock to be taken at the rate of 5-15 drops 2 times daily for the first 5 days; stop for 2 days, then resume at this dosage for 30-120 days.\n\n### _DMSA_\n\nThis substance\u2014dimercaptosuccinic acid\u2014was mentioned above (and in Chapter 3) as a provocative agent to reveal mercury load in the body as measured in the urine. A nontoxic, watersoluble substance, DMSA chelates mercury and excretes it out of the body, giving the physician an indication of the body's mercury burden. But it can also be used as the official chelation agent to remove mercury following mercury amalgam replacement.\n\nDMSA, administered orally, has been used since the 1950s as an antidote to heavy metal toxicity such that today, according to some experts in the field of alternative medicine, it is considered \"the premier metal chelation compound.\" As an example of its \"premier\" abilities, nine children were exposed to vapors of metallic mercury from an external source (not teeth). Their initial urinary mercury levels ranged from 61 to 1,213 mcg\/g, with an average of 214 meg. Immediately after DMSA treatment, the urinary mercury average soared by 268% to 573 mcg\/g, then after six weeks, leveled out to an average of 102 mcg\/g, or only fifty percent of the average level originally detected after the exposure. None of the children experienced any adverse side effects.\n\nSwedish researchers found that when they gave oral DMSA to twenty dental patients (suspected to have mercury toxicity from their fillings) for 14 days at the rate of 20 mg\/kg of body weight, it resulted in an average increase in urinary mercury excretion of 65%. The DMSA also reduced blood mercury levels by 0.04 mcg\/l\/day, The mercury kept coming out of their bodies for three months even though the DMSA was only given for two weeks. The patients also experienced a decrease in their levels of reported fatigue and inertia.\n\nResearchers at Odensk University in Denmark gave fifty patients with environmental-illness-type medical complaints a program of DMSA (30 rug\/kg body weight) for five days. These patients had elevated levels of overall distress, obsessivecompulsive tendencies, depression, anxiety, reduced extroversion, and greater emotional volatility, according to standard psychological profiling. Six weeks later, when they were queried again, the patients said most of their distress symptoms had \"improved considerably.\" The tests showed they had excreted a great deal of mercury and lead in their urine during that time.\n\n### _DMPS_\n\n2, 3-dimercaptopropane-1-sulfonic acid is another favored chelating agent for increasing the removal of elemental mercury from the human body. It can be given orally, intravenously, or intramuscularly, and is useful for people who have been exposed to mercury amalgam through their dental fillings, or for those who show evidence or suspicion of heavy metal toxicity from other sources. DMPS has been used with a good safety record for many years in Russia (since the 1960s) and Germany (since 1978) and is used by some progressive physicians in the United States as part of a post-mercury-amalgam-removal protocol.\n\nLike DMSA, it can also be given as a challenge to see how much mercury the body is carrying. For example, twenty-eight dental personnel (dentists, dental technicians, and office staff) were given DMPS, after which their urinary excretion of mercury was eighty-eight times higher (for the technicians who formulate amalgams), forty-nine times higher (dentists), and thirty-five times greater (office staff).\n\nAmerican researchers found that giving dental patients an oral DMPS dose of 300 mg increased their excretion of mercury through the urine from 0.70 to 17.2 mcg among patients with mercury amalgams and from 0.27 to 5.1 mcg for those without mercury amalgams in only nine hours. The scientists gauged that two-thirds of the excreted urinary mercury in the patients with amalgams came from vapor released by those fillings, the rest from environmental sources of mercury contamination, such as air pollution and fish.\n\nIn Mexico, scientists monitored the urinary excretion of mercury from dental technicians, dentists, and office personnel who had taken DMPS. They compared the rate before DMPSwas given and six hours afterwards. For the technicians, before DMPS, they excreted 4.84 mcg of mercury, but after, 424 meg; for dentists, 3.28 mcg before, and 162 after; and for office staff, 0.7 mcg before and 27 mcg after. The researchers concluded that it is more meaningful to determine how much mercury the kidney is burdened with based on the excretion rate after DMPS is given.\n\nDMPS may be somewhat more effective than DMSA in removing mercury from the kidneys. Researchers at the Mercer University School of Medicine in Macon, Georgia, used both chelating substances to monitor mercury removal from the kidneys of rats. Both substances increased the rate of mercury elimination from the body during the first twenty-four hours after administration, but DMPS was more effective at reducing the \"renal burden\" (the amount of mercury stored in the kidneys).\n\nAt the National Institute of Public Health in Prague in the Czech Republic, researchers studied the effect of DMPS on three groups of subjects: industrial workers exposed to workplace mercury, dentists, and rats injected with mercury. Two doses of DMPS were given, spaced one day apart. After two days, the urinary mercury excretion rate in the workers was 1513 mcg; in the dentists, 132 mcg; and in human controls, 3.78 mcg. Two doses of DMPS decreased the kidney content of mercury in the rats by thirty percent when given orally and fifty percent when injected into the muscles. The scientists found that in the human subjects, two doses of DMPS reduced their kidney mercury content by seventeen to twenty percent (oral dose) and twenty-five to thirty percent (injection) in only forty-eight hours.\n\n### _Alginate_\n\nThis substance derives from the Fucales seaweed, _Ascophyllum nodosum_ , and is accumulating a good track record for its ability to bind up heavy metals and radioactive solutions in the body. The body is not able to break down or digest alginate, so the substance passes through intact, bearing its heavy metal burden. Alginate is commercially available without a prescription under the brand name ProAlgen (from Nordic Naturals), which is comprised of _Ascophyllum nodosum_ and Fucus and Laminaria algae (containing 40 minerals and trace elements) in 600 rugcapsules. You can use alginate following mercury amalgam removal, or even after every meal, to collect any mercury that might have been released into your system through chewing.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor cilantro:\n\nDragon River Herbals,\n\nP.O. Box 74,\n\nOjo Caliente, NM 87549;\n\ntel: 505-583-2348 or 800-813-2118;\n\nfax: 505-583-2339;\n\nwebsite: www.dragonriverherbals.com.\n\nFor ProAlgen, Alginate Detox Supplement: Nordic Naturals,\n\n3040 Valencia Ave., #2,\n\nAptos, CA 95003;\n\ntel: 800-662-2544 or 831-662-2852;\n\nfax: 831-662-0382;\n\nwebsite: www.nordicnaturals.com.\n\nFor information about\n\nbiological dentistry:\n\nHal Huggins, D.D.S., M.S.,\n\nP.O. Box 49145,\n\nColorado Springs, CO 80949;\n\ntel: 888-843-5832 or 719-522-0566;\n\nfax: 719-548-8220;\n\nwebsite: www.hugnet.com.\n\nFor nutritional and homeopathic protocols for mercury detoxification:\n\nDaniel F. Royal, D.O.,\n\nThe Royal Center of Advanced Medicine, 2501 N. Green Valley Parkway, Suite D-132, Henderson, NV 89014;\n\ntel: 888-DANROYAl or 702-433-8800;\n\nfax: 702-433-8823;\n\ne-mail: royal@drroyal.com;\n\nwebsite: www.drroyal.com.\n\n  **_The Healthy Living_ space Info Tip**\n\nFor information for dentists contemplating biological dentistry (protocols and standards): International Academy of Oral Medicine and Toxicology,\n\nMichael E. Ziff, Executive Director,\n\nP.O. Box 628531,\n\nOrlando, FL 32860;\n\ntel: 407-298-2450;\n\ne-mail: mziff@iaomt.org;\n\nwebsite: www.iaomt.org.\n\nFor patients to get referrals for\n\nbiological dentists:\n\nHolistic Dental Association,\n\nP.O. Box 5007,\n\nDurango, CO 81301;\n\ntel: 970-259-1091;\n\ne-mail: had@frontier.net;\n\nwebsite: www.holisticdental.org.\n\nFor more information about biological dentistry, conferences, and research:\n\nAmerican Academy of Biological Dentistry,\n\nP.O. Box 856,\n\nCarmel Valley, CA 93924;\n\ntel: 831-659-5385;\n\nfax: 831-659-2417.\n\nFor information about healthy treatment options for periodontal disease:\n\nInternational Dental Health Foundation,\n\n2414 Black Cap Lane, Suite L-l,\n\nReston, VA 20191;\n\ntel: 703-860-9244 or 800-368-3396;\n\nfax: 703-860-9245;\n\nwebsite: <http:\/\/members.aol.com\/idhf>.\n\n### _Timing Your Amalgam Removals_\n\nIf you decide to have your mercury fillings removed, Dr. Huggins has a tip about finding the best time to do so. Apparently your immune system has a cycle in which it replenishes its supplies of white blood cells or lymphocytes. You don't want to hit that low time with a big challenge to the immune system generated by removing your mercury fillings.\n\nDr. Huggins reports that about seven days after an immune challenge\u2014mercury exposure during amalgam replacement, having an immunization shot, a bad case of the flu, serious emotional trauma from death or accidents\u2014your body's defenses are weakened. The lymphocytes are in the process of being replaced, so there are several hours on day seven when your immune defenses are down and vulnerable. On day fourteen, the lymphocytes are again regrouping, regenerating, and you are immunologically vulnerable. So \"if there is an additional [immune] challenge on day 7, 14, or especially 21, then your genetic weak link is apt to be stretched, and you may become ill,\" says Dr. Huggins.\n\nIn other words, don't have any mercury amalgam work done on days seven, fourteen, or twenty-one. He advises not having dental appointments on the same day of the week during a two-month period, and don't have dental work two days in a row, or procedures that last longer than two hours. In practical terms, he suggests having dental appointments on Monday, Wednesday, and Friday of one week, then Tuesday, Thursday, and Saturday the next week. It is immunologically preferable to have all the amalgams removed in a thirty-day period, if possible.\n\n### _Nutritional Support for Amalgam Removal_\n\nIn conjunction with having your amalgams replaced, and especially as an immediate follow-up, many alternative medicine physicians recommend a targeted nutritional program to help your body cope with the mobilization and removal of mercury.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor detailed information on mercury hazards and mercury-free dentistry:\n\nDAMS (Dental Amalgam Mercury Syndrome),\n\nP.O. Box 64397,\n\nVirginia Beach, VA 23467;\n\ntel: 800-311-6265;\n\nwebsite: www.amalgam.org.\n\nDaniel F. Royal, D.O., an osteopath who practices in Henderson, Nevada, recommends the following: chlorella (1-3 capsules daily); L-glutathione (150 mg daily); kyolic garlic (1 capsule with meals, three times daily); silymarin (1 capsule, twice daily); vitamin C (2,000-8,000 mg daily, depending on tolerance, in divided doses); vitamin B complex ( 25-100 mg daily); DHEA (an adrenal hormone precursor, 5 mg for men, 2.5 mg for women, daily); pregnenolone (a steroid building block made from cholesterol; 10 mg for men, 30 mg for women, once daily); _Mercurius solubilis_ 30C (a homeopathic remedy; thirty drops, 2-3 times weekly, for a few weeks); selenium (50 meg, three times daily); and _Lactobacillus acidophilus_ (one teaspoon daily). Dr. Royal also recommends time in a sauna to sweat out mercury and other toxins through the skin (see Chapter 6).\n\n## HEALTHY LIVING SPACE DETOXIFIER #32: Have Your Cavitations Cleaned Out and Reduce Possible Jawbone Infections\n\nAs explained above, a cavitation is a formal term from dentistry to describe a hole in the upper or lower jawbone following a tooth extraction in which an infectious process is occurring, usually unbeknownst to the person. The cavitation typically is the same size as the tooth that was removed from that site. It is a problem because it is a site of incomplete healing.\n\nWhen a tooth is root-canalled and then later removed, often a little bit is left behind, called the periodontal ligament, half of whose fibers come from the jaw, the other half from the tooth. When the tooth is removed but the ligament is left in place, it starts to release neurotoxic chemicals that slowly seep into the body; these chemicals were originally produced by the harmful bacteria in the root of the decayed, dying tooth.\n\nNow if the root-eanalled tooth is extracted and the ligament left there, a small cap of bone forms over the socket \"leaving a cesspool of these chemicals lining the hole, and sealed within the bone,\" explains Dr. Huggins. The cavitation then serves as a breeding ground for bacteria and the toxic substances they excrete. It also creates osteonecrosis, or bone death, through the progressive impairment of blood circulation through that part of the jawbone. Biological dentists also postulate that the toxic cavitations can block the flow of _qi_ through the acupuncture meridians that pass through the mouth and also transmit the energy signal of toxicity to other organs and organ systems in the body through this same network of energy channels. There is some evidence that cavitations can act as reservoirs for leeched amalgam mercury.\n\nUnfortunately, this setup is not easily detectable with X-rays, so many dentists can overlook it. On close examination of the X-ray, the cavitation may appear like the shadow of a tooth, or a phantom tooth. \"Almost always, this is indicative of a cavitation,\" states Mark A. Breiner, D.D.S., in _Whole-Body Dentistry._ Dr. Breiner notes that the most common site for a cavitation to form is at the extraction site of a wisdom tooth, and the second most common site is a root-eanalled tooth extraction site.\n\nDr. Huggins reports that cavitation in dental patients is \"an _extremely_ common finding.\" The third molars, or wisdom teeth, are the most prevalent site, such that out of 354 extraction sites examined, 88% had cavitations. The second molars had a seventy percent incidence of cavitations, and in the first molars, eighty-two percent had cavitations. For all molars examined, eighty-five percent had cavitations (441 sites out of 517 examined). In nonmolar sites, the incidence of cavitations was much lower, at fifty-five percent. This is a shockingly high incidence of a dental condition (and source of dental toxicity) still largely unrecognized by most dentists.\n\nInsofar as the wisdom teeth have energy correlations through the acupuncture meridians with many of the body's important organs, including the heart, a cavitation at this point\u2014also known as a focal infection, as mentioned earlier\u2014is highly undesirable and a potentially serious health threat. Dr. Breiner recounts how his own episode of an irregular heartbeat was traced to a cavitation in the wisdom tooth area of his jaw. Surgical debridement (cleaning toxins and dead bone material out of the cavitation) resolved the arrhythmia in a few days, he reports. Cavitations can also produce chronic facial pain (trigeminal neuralgia), as well as some headaches and phantom toothaches. Dr. Breiner calls cavitations SICO, for Sickness Inducing Cavitational Osteonecrosis (dead bone material) to highlight their ability to negatively affect many aspects of a person's health.\n\nIn support of Dr. Breiner's sweeping indictment of cavitations is evidence developed by Boyd Hailey, Ph.D., a prominent dental researcher at the University of Kentucky. His research shows that all cavitation tissue samples tested contain toxins that are able to significantly block the activity of one or more of five critical body enzymes involved in the production of energy. The toxins produce health problems both localized (in the jawbone) and systemic (throughout the body).\n\nThere also is a succession of bacterial species in the cavitation. Swiss dentists reported that in advanced root caries ninety percent of the microflora found were bacteria, and of these, in the early stage of tooth infection, _Actinomyces naeslundii_ was significantly higher than other species. In the advanced stage of decay, the predominant bacteria was _Streptococcus mutans._ In other words, there was a succession of bacterial populations observed during the \"destruction process\" of a single tooth.\n\nAs noted earlier in this chapter, Dr. Christopher Hussar has remarked that the mouth is a dirty place. He was not exaggerating. Over 300 different species of bacteria exist in the mouth, creating a population estimated to be several hundred billion. In an insufficiently cleaned mouth, that number can be a thousand billion. These billions of bacteria create their own ecological niches inside the human mouth, reproduce, metabolize, and thereby threaten oral and systemic health.\n\nScientists know the action of these bacteria is associated with the incidence of rheumatoid arthritis, dermatitis, nephritis (bacteria-induced kidney inflammation), bacterial pneumonia, pregnancy troubles, blood circulation problems, and coronary heart disease. Periodontal disease can also lead to bacteremia (bacteria in the blood), diabetes, respiratory disease, prosthetic device infection, and atherosclerosis. Periodontal disease is seven times more likely to be linked with a preterm delivery of a low birth weight infant than other factors, including the mother's age and tobacco and alcohol use.\n\nInfections in the gum and jawbone have been linked to 150350 different anaerobic bacterial species, depending on the type of periodontitis; dental treatment and tooth extraction can cause these toxic microorganisms to move into the bloodstream where they are free to circulate around the body. If a person has a preexisting heart problem (such as an ineffective heart valve or a vascular disease), the circulation of these bacterial species can lead to serious problems, including heart attacks.\n\nIn an odd way, brushing your teeth is potentially bad for your heart. English researchers studied the effect on 150 children of tooth brushing, professional cleaning, and scaling on the release of bacteria into the body. They found that all three methods of cleaning the teeth released at least three species of bacteria implicated in the development of bacterial endocarditis (inflammation of the membrane lining the heart). In other words, the average mouth has these toxic bacteria in residence, and cleaning the teeth mobilizes them and gets them moving through the body.\n\nThe connection between oral focal infections and\/or periodontitis (inflammation and degeneration of the supporting structures of the teeth) and the heart is well established in scientific research. Finnish researchers examined the microbial population in the mouths of fifty patients (average age sixty-five) about to have abdominal aortic surgery. They found that eighty-two percent had evidence of oral infection. On average, most of the patients had only nine teeth, and twenty-one percent of these were potential focal infection sites; further, twenty-six percent of the patients had oral candidiasis (an infection caused by _Candida albicans_ ). \"Oral infectious foci occur frequently in patients needing aortic surgery,\" said the researchers, adding that untreated focal infections may contribute to heart infection.\n\nWhen 261 patients with suspected focal-infection-produced diseases were examined for chronic inflammatory problems in their mouths, researchers found that 83% had periodontal problems, such as dead tissue and inflammation. These oral problems are frequently detected as factors in chronic cardiac disorders, eye problems (uveitis), and skin diseases, the researchers said. It is also possible for oral bacteria to first start a systemic infection, like endocarditis, then indirectly stimulate a brain abscess. Various oral bacteria (including Streptococci species) can cause sinusitis, either acute or chronic.\n\nThese are examples of how an oral focal infection\u2014the human mouth as a focus of infection\u2014can produce a range of specific and systemic diseases throughout the body. This is a medical insight at least one hundred years old, but largely ignored until recently. The term focal infection can encompass many problems in the mouth: cavitations, cavities, extraction sites, infected teeth and pulps. While periodontitis is not the same as a cavitation, it is a potent source of oral focal infection that can exist alongside or in conjunction with a cavitation, adding to the total toxicity present in the mouth.\n\nThe crucial point to keep in mind is that bacterial and viral microorganisms or their toxic products can enter the body tissues through the mouth and thereby produce health problems. Arthritis is a good example. In one study, an antigen (foreign protein) was injected to the mouth and soon after produced a knee inflammation, leading scientists to postulate that a dental focal infection may be involved in arthritis of the knee and arthritis in general.\n\nShould you, if you have extraction sites in your mouth, automatically have your cavitations debrided? Not necessarily, says Dr. Huggins. If you feel well, enjoy good health, and have normal laboratory profiles for nutrient and antioxidant defense status, it probably means the toxins are stable, staying in place in the jawbone, and that your immune system is coping. But if you get sick or develop a chronic problem, then you should consider investigating the possibility that you have cavitations acting as focal infections, and get them treated.\n\n\"Cavitations can progress over time, with more and more adjacent jawbone dying,\" says Dr. Huggins. You may not have a problem this year, but perhaps in five or ten years, you might. As J.H. Meurmann, a dental researcher at the Institute of Dentistry at the University of Helsinki in Finland remarks, \"Chronic dental infections may worsen the conditions of medically compromised patients.\" Dr. Meurmann adds that if a person is acutely or chronically ill, frequent dental checkups to determine possible focal infections in the mouth are highly advised.\n\nWhen a biological dentist cleans the cavitation, the walls of the jawbone socket must be cut out with a dental burr, then the toxic material is scraped out. Since some of the toxic material then enters the lymphatic system as part of the body's natural (and now freed up) detoxification pathway, the patient may feel sick for a few days while the body processes the toxins, Dr. Huggins says.\n\n## HEALTHY LIVING SPACE DETOXIFIER #33: Think Twice before Digging Another Root Canal in Your Mouth\n\nHaving a root canal \"dug\" in one of your teeth is generally regarded as a highly unpleasant experience by most people. It may be a miserable experience, but in 1996, an estimated 25 million were installed in American mouths, and for 2000, the dental industry predicted 30 million.\n\nMost dental patients and dentists regard it as preferable to having the tooth extracted because, depending on the location, this can destabilize the jaw and lead to other dental problems later. In the root canal procedure, the dental pulp, dying (or dead) and infected, is removed and replaced with a wax material called gutta percha. Technically the tooth is now dead, but as biological dentists have known for decades, dead does not mean inactive. In this case, the activity is bacterial, and it is not good for you.\n\nYou may think the dental infection is terminated once the gutta percha is laid into the tooth and the painful abscess goes away, but research suggests it isn't. In fact, serious health consequences may result from a hidden, deep-set infection beneath the root-canalled tooth. According to George Meinig, D.D.S., author of Root Canal Cover-up, \"root canal treated teeth have side effects that cause many disorders.\" He made this statement on the foundation of forty-seven years of clinical practice as a root canal clinician and dentist.\n\nIt is almost impossible for a dentist to clean out all the infection from a decayed root canal; so pockets of infection remain when the canal is sealed up. The result is that bacteria get trapped within the estimated three miles of microscopic dentin tubules inside a single root-canalled tooth. The cementum (the outermost hard surface of the tooth) prevents the bacteria from migrating from the tooth into the bloodstream, but not the toxins produced by these bacteria.\n\nOver the years, these pockets of infection can start spreading their process throughout the body, without the person suspecting it, at least directly. As toxins leak from the root-canalled tooth, they chronically trigger the immune system to react, and eventually the immunological defense system gets weary of the engagement. Infections under root canals are focal infections, which means they continuously disseminate toxins into the body, contributing, undetected, to a myriad of problems throughout the body.\n\nAs early as 1925, the connection between bacterially contaminating root canals and systemic health problems was known to professionals who read the scientific literature. Weston A. Price, D.D.S., reported that many degenerative diseases result from root canals, including endocarditis and other heart ailments, disorders of the kidneys and bladder, rheumatism, arthritis, lung problems, complications of pregnancy, bacterial infections, and even mental illness. Dr. Price demonstrated his point vividly when he implanted actual root-canalled teeth (from human subjects suffering from various diseases) under the skin of rabbits. The rabbits developed the same diseases as the humans whose teeth they were.\n\nDr. Price discovered many changes in blood chemistry as a result of root-canalled teeth. The white blood cell count (lymphocytes) goes up considerably; in rabbits, it increased fifty-eight percent following the implantation of the human teeth. But at the same time, another kind of white blood cell (polymorphonuclear leukocytes) decreased by thirty-three percent in rabbits. These changes indicated both a mobilization of the immune response and a diminution of one of its prime components.\n\nHow can you tell if a root-canalled tooth is acting as a focal infection in your body? Consider undergoing the oral toxicity test developed by Boyd Haley, Ph.D., of Affinity Labeling Technologies in Lexington, Kentucky, as discussed in Chapter 3. In conjunction with this, have your immune vitality tested based on a blood and urine sample, again, as described in chapter 3.\n\nWhether a root-canalled tooth is making you sick depends a great deal on the vitality of your immune system. \"If your immune system is strong, your body may be able to quarantine the toxins by 'walling off' the area [around the tooth],\" says Dr. Breiner. This area might appear as a \"more radiolucent\" area suggestive of an abscess on a dental X-ray, he adds. The radiolucent sign is a good one, indicating the successful immunological containment of toxins. Sometimes, Dr. Breiner adds, a \"drain\" will open up under the root canal tooth and the toxins will slowly, almost imperceptibly drain out into the mouth over time.\n\n### _Gutta Percha Alternative: BioCalex_\n\nOne of the problems with gutta percha, explains Dr. Breiner, is that it contracts over time, creating a gap between itself and the tooth's remaining dentin. This allows bacteria from the bloodstream to enter and exacerbate the infectious process already under way.\n\nAn alternative substance from France, newly introduced among biological dentists (1994), is called BioCalex, made from calcium and zinc oxide. Used in France since 1979, BioCalex seals the root canals more effectively than gutta percha, creating a tight bond and preventing bacterial ingress; it has been shown to penetrate into the most inaccessible of the tiny canals and tubules inside the tooth.\n\nIt is also alkaline (nonacidic), which creates an unfavorable environment for bacteria. This was demonstrated by Australian researchers who studied the changes in root dentin pH (the ratio of acidity to alkalinity on a scale of 0.1-14, with 7 being neutral) over a four-week period after they had been treated with calcium hydroxide. In the inner part of the dentin, the pH peaked at a high alkaline reading of between 9.7 and 10.8; after 2-3 weeks, the outer dentin reached a pH of 9.0 to 9.3. Although the pH change took longer to obtain in the outer dentin of the root, the study showed that calcium hydroxide could diffuse throughout the tooth and change its chemical nature.\n\nIn fact, the calcium oxide composition of BioCalex is very effective at destroying anaerobic bacteria found in teeth, according to BioProbe, its U.S. marketer. Greek researchers tested calcium hydroxide alongside a standard disinfectant (paramono-chlorophenol, PMCP) against anaerobic bacteria isolated from infected root canals, and observed results at time intervals of five, fifteen, thirty, and sixty minutes.\n\nThey found that calcium hydroxide was \"significantly more effective\" than PMCP against the total bacterial population. Further, it was \"quickly and highly effective\" against several microorganisms (B. melaninogenicus, P. gin-givalis, and Actinomyces species) associated with severe clinical symptoms. Preliminary tests show a considerably reduced level of toxicity associated with BioCalex used in root-canalled teeth compared to those treated with gutta percha.\n\nFurther, when calcium oxide combines with water, it forms calcium hydroxide, and this is one of the most biocompatible dental materials available. Biocompatibility will be discussed below, but it means the substance is nonallergenic and nontoxic to the human body; it will not produce overt or hidden allergic or immunological reactions when it comes in contact with human tissue, such as in the mouth.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about BioCalex, BioProbe Newsletter, and scientific documentation about mercury hazards, and for dentists wishing to obtain BioCalex:\n\nBioProbe, Inc.,\n\nP.O. Box 608010,\n\nOrlando, FL 32860;\n\ntel: 800-282-9670 or 407-290-9670;\n\nfax: 407-299-4149;\n\nwebsite: www.bioprobe.com.\n\n## HEALTHY LIVING SPACE DETOXIFIER #34: Determine Your Biocompatibility with Dental Materials before Putting Any More in Your Mouth\n\nMention was made above about biocompatibility of dental materials. This is an important issue with regard to toxicity and allergic reactions to substances, yet it is not discussed enough in medical circles. Dentists routinely place many substances in our mouths\u2014mercury, silver, and tin (in amalgams), gutta percha (in root canals), sealants, gold or chrome (in crowns), porcelain and plastic (bridges), nickel (orthodontic braces), titanium or ceramic (implants)\u2014but who is to say a given dental patient is not allergic, or will not develop an allergy over time to these strange substances not meant by nature to be in the human body. Who is to say (who has tested them?) they are compatible with the immune system?\n\n\"Not too many dental materials are free from toxins,\" comments Dr. Huggins. \"Unfortunately, the stronger materials are the most toxic,\" he adds. This is why he recommends compatibility testing for dental products to assess in advance the potential or likely allergenicity _to you_ of any dental substance that might be placed in your mouth. Bear in mind that, to an extent, allergic reactions are individual, based on your total biochemistry and immune status; this means that if a given substance is said not to produce allergic reactions generally, it may still generate symptoms in you because of your biochemical specificity.\n\nFor example, scientists now know that toxic substances can be leaked from composite dental resins used instead of mercury amalgams and that these \"exudations\" into the mouth can be estrogenic\u2014act like the hormone estrogen. Researchers at the University of Granada in Spain applied a dental sealant based on a substance called bisphenol-A diglycidylether methacrylate (found in dental resins) to the molars of eighteen dental patients. Saliva samples were tested one hour before this application and one hour after it.\n\nAnywhere from 90 to 131 meg of bisphenol-A were detected after the sealant had been applied. Further, the researchers found that certain biochemical changes typical of estrogenic activity were noted in these patients, and on that basis they made this conclusion: \"The use of bis-GMA-based resins in dentistry, and particularly the use of sealants in children, appears to contribute to human exposure to xenoestrogens.\" As you may recall from Chapter 2, a xenoestrogen is a foreign estrogen; it is a synthetic chemical that acts _like_ estrogen in the body, thereby upsetting the hormonal balance.\n\nAnother study showed that formaldehyde can be released by dental composites in your teeth. Researchers at the NIOM-Scandinavian Institute of Dental Materials in Haslum, Norway, found that nine different types of commercially used dental composites released formaldehyde into the mouth. \"A continuous release of formaldehyde was evident during the first ten days [after placement],\" they stated. Then it declined, but was still detectable after 115 days.\n\nThese two examples of estrogenic dental factors are included here to give an introductory idea about biocompatibility. Obviously, estrogenic chemicals and formaldehyde from dental materials are not compatible with the human system and will contribute to body-wide toxicity and a heightening (and eventually a fatiguing) of the immune response.\n\nA test called the Clifford Materials Reactivity Testing (CMRT) is now available to monitor dental biocompatibility. The purpose of the test, according to its provider, Clifford Consulting and Research of Colorado Springs, Colorado, is to screen a patient for sensitivities to chemical groups and compounds, to identify dental materials that can be used with that patient with the greatest degree of safety and least amount of risk, and to help the dentist put together a biocompatible treatment plan.\n\nThe test looks for specific antibody formation in response to various dental materials, and interprets this response as a gauge for biocompatibility. (When the body encounters a foreign or suspect substance in the bloodstream, it produces antibodies, or immune defense proteins, to tag and neutralize the intruding substance.) The test looks for antigens to various chemical groups such as acrylates, urethanes, toluenes, nickel, aluminum, and mercury. The testers take a small amount of blood and give it a \"challenge\" of the dental material (the antigen) to see whether antibodies are formed in response.\n\nThe test is recommended for people with existing sensitivity problems such as environmental illness or multiple chemical sensitivity, those who already have a compromised physiology (an illness or chronic condition), or those who have a prior history of reacting negatively to dental materials. It's also helpful if you are planning major dental restorative work, including dental implants, crowns, bridges, or mercury amalgam replacement. The results are cross-referenced with a list of over 1,900 dental and medical trade-named products so that the dentist can select materials of minimal reactivity for individual patients.\n\nThe test looks at how the patient will react not only to newly placed dental materials, but the corrosion byproducts of dental materials already in the patient's mouth. In other words, say you had a gold crown put on a molar five years ago; whatever trace amounts of metals or byproducts of the corrosion process have been generated and released by this crown will be accounted for in the test. For example, according to Clifford Consulting and Research, testing of more than 12,800 patients shows that 1% do not handle gold in the mouth well, and that twenty-five percent have problems with silver. In other words, \"no single material is 100% suited to all patients.\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Clifford Materials Reactivity Testing:\n\nClifford Consulting and Research, Inc., P.O. Box 17597,\n\nColorado Springs, CO 80935;\n\ntel: 719-550-0008;\n\nfax: 719-550-0009;\n\ne-mail: wjclifford@ccrlab.com;\n\nwebsite: www.ccrlab.com.\n\nResolution of long-standing dental problems may turn around many chronic symptoms-that you had previously never suspected were associated with events in your mouth. You may find that certain emotional states that came upon you now and then start to lift; you may have thought depression or anxiety were \"just who you were,\" but now that your mercury amalgams are gone, so is your depression.\n\nNow that you have removed a fair amount of physical toxicity from your system, from the protocols in chapters 4\u2014 encompassing your liver, intestines, kidneys, lymphatic system, and teeth, you have sufficient momentum to move a layer deeper into yourself\u2014into the realm of emotions. Emotions, too, can be toxic; certain strongly held but unexpressed emotions can act like toxins if they are left to fester. So our next phase of detoxification in Chapter 8 moves into the emotional life and offers ways to start resolving buried, repressed, or neglected emotional issues.\n\n# CHAPTER 8\n\n# Guidelines for Emotional Detoxification: Unresolved Emotional Issues Can Become Toxic to the Body\n\nIn the course of following the detoxification suggestions in this book you have undoubtedly come up against some old, perhaps painful or intractable emotions. You may have been surprised by their existence, or by the energy with which they reminded you of their existence. It is highly common for people cleansing the colon to reexperience charged emotional material from many years back, as if they had unknowingly walked into some unsuspected personalized museum of hurts and trauma.\n\nIn many respects, physical detoxification makes it possible to get to the emotions, buried, as it were, under layers of physical toxins and dysfunctional physiological states. Internal organ cleansing often helps you identify this next level of toxicity.\n\nLike the intestines, the body is a remarkable storage device for unprocessed thoughts and feelings. Most people spend a lifetime depositing unresolved emotional issues throughout the body; in fact, it is not only painful experiences that are stored; all sorts of memories get lodged in the body's tissues. In her book, _Being-in-Dreaming_ , Florinda Donner, an anthropologist and compatriot of Carlos Castaneda in the mysteries of Toltec shamanism, explains how she uses a little wooden mallet to tap her body for memories. She systematically taps every inch of her body, noting that everyday events tend to get stored in the chest, back, and abdomen, while dream content tends to go into the legs and hips.\n\n\"All that concerns you now is that remembering dreams has to do with physical pressure on the specific spot where that vision is stored,\" she says, adding that events experienced in dreams are hard to remember \"because the body stores them in different places.\" I used to think this was a fascinating but unsubstantiated idea until in the course of giving a friend a backrub and probing the left shoulder blade, she suddenly remembered being chased by a large turtle in a dry riverbed when she was eight years old. She had stored the actual daytime experience\u2014like a home movie, vivid with the fright, surprise, adrenaline rush\u2014at the bottom of her left shoulder blade.\n\n**_In many respects, physical detoxification makes it possible to get to the emotions, buried, as it were, under layers of physical toxins and dysfunctional physiological states. Internal organ cleansing often helps you identify this next level of toxicity_**.\n\nWhile the body is almost endlessly accommodating in this coping strategy, eventually it runs out of space, or the load in one place is too big a burden. Then the price comes due for your longterm emotional storage: your unresolved, stored emotions become toxic to the body and will start to make you sick. In this chapter, then, we will explore the contribution of unresolved emotional issues to body-wide toxicity, and suggest a strategy for clarifying and dissolving this subtler level of toxicity. But first let's consider some more examples of the consequences of using the body as an emotional storage depot.\n\n## How Unresolved Emotions Can Contribute to Illness\n\nEmotional issues carry a powerful charge capable of affecting your biology in profound and often undesirable ways. For example, unresolved emotional issues are often associated with gynecological problems, such as uterine fibroids, ovarian cysts, and excessive menstrual bleeding, because the body must express itself, explains John Diamond, M.D., a homeopathic physician who practices in Reno, Nevada.\n\nDr. Diamond, who was educated in South Africa, cites a saying in that country that \"If you don't cry above, you'll cry below.\" The crying below refers to the uterus, whose \"tears\" are menstrual blood. Dr. Diamond explains that he frequently sees strong emotions, if left unaddressed, manifesting as body symptoms, such as excessive menstrual bleeding.\n\nThe body is always metaphorical, he says. \"It always _listens_ to the psyche of the individual and responds accordingly, in its own way, translating withheld tears into copious menstrual flow.\" The emotional issue is an energy that _has_ to go somewhere, _has_ to be expressed physiologically, says Dr. Diamond.\n\nA uterine fibroid (a noncancerous fibrous mass that grows on the inside wall of the uterus) often is about emotional suppression relating to either sexuality or relationships with men; the heavy bleeding it occasions is often associated with prolonged internalized sadness and withheld emotions; and an ovarian cyst (a noncancerous fluid-filled sac on an ovary) can be correlated with a history of sexual abuse, according to Dr. Diamond.\n\nHe cites the case of Flora, a thirty-eight-year-old woman with heavy, painful menstrual bleeding occasioned by multiple uterine fibroids. The woman, a mother of four, related that her heavy menstruation started two years earlier when her husband talked about divorcing her. Even though the marriage remained \"intact,\" Flora was anxious, unsettled, and fearful about the prospect of a divorce, and \"stuffed\" the emotions and sadness into her body, says Dr. Diamond. Why her uterus?\n\nHere we see the role of individual constitutional weakness and genetic predisposition. Her mother at her age had had the identical uterine problem, and both had started menstruating at a late age (fifteen), which can be indicative of later reproductive system problems, says Dr. Diamond. Flora's uterus was the designated weak point in her body, so if she had strong emotional content to store (suppress) somewhere, it would go there, and when the body could no longer cope with the continuous toxic presence of these emotions, it would discharge them through this same weak point: the uterus. Metaphorically, the fibroid was saying to Flora that her \"ugly secret\" was out: she could no longer hide her emotional pain and the way she tried to bury it in a hard shell.\n\nAfter a comprehensive treatment program involving homeopathic remedies and herbs, Flora regained her health and did not need a hysterectomy (surgical removal of the uterus). The homeopathic remedies helped bring her emotions to the surface for clarification and expression. Then she resolved her emotional problems with her husband and started having normal periods again.\n\nA hysterectomy, which conventional medicine routinely recommends for fibroids and related uterine problems, would not have corrected the deep-set cause of the imbalance, says Dr. Diamond. It would not have addressed \"the true pathology of the internalized emotional fear and anger she had harbored for so long.\" It could have driven these emotions even deeper into her body, producing a more serious health problem such as cancer or mental illness.\n\nUnexpressed, stuffed emotions can also contribute to the emergence of rheumatoid arthritis, according to board-certified rheumatologist Norman Levin, M.D., who practices in Aldie, Virginia. He cites the case of Tara, thirty-six, who came to him suffering from rheumatoid arthritis. In addition to addressing the physiological and biochemical aspects of her medical problem, Dr. Levin encouraged Tara to ask herself _why_ she was having arthritis now. Nobody gets a serious illness out of the blue, as if you accidentally walk through an illness ray and emerge sick, he says. There are always emotional and psychological factors involved in the development of an illness, he says.\n\n\"Patients need to be encouraged to find the meaning in their health crisis,\" Dr. Levin notes. Echoing Dr. Diamond's observation, he adds: \"The body speaks its own language, and often illnesses affecting different parts of the body are associated with certain emotional states.\" For example, Dr. Levin often finds that unexpressed anger or resentment underlie the emergence of arthritis; the arthritis is a physical manifestation or expression of those emotions, he says.\n\nIn Tara's case, she had been a successful realtor who was suddenly relocated to an unfamiliar territory, having to abandon the market area she knew very well. It _felt_ to her like she was being fired. At the same time she was uprooted from the safe and familiar routines of her job and environment, a romantic relationship was breaking up. So the job change and termination of a relationship gave Tara a double hit of emotional trauma, acting as _precipitating_ factors for her arthritis, which began soon afterwards, says Dr. Levin.\n\nIn the case of Michelle, thirty-nine, her fibromyalgia was emotionally linked to a feeling of carrying the weight of the world on her shoulders. Fibromyalgia is a highly painful and debilitating illness in which numerous muscles throughout the body are chronically inflamed and painful, severely limiting movement. Ironically, Michelle had been an aerobics instructor before her muscles refused to let her move without extreme pain, spasm, and fatigue. \"She was the kind of person who tried to do and be everything for everybody,\" comments Dr. Levin. \"As we discovered, this attitude was at the core of her muscle aches.\"\n\nAbout four weeks into her treatment program (a comprehensive alternative medicine approach), Michelle had enough freedom from pain to start to examine her psychoemotional state to see why she had fibromyalgia. She saw that she was a superachiever, a woman who always sought to be the best in every situation, be it motherhood, teaching aerobics, being a wife. She cared so much for everyone else she forgot to care for herself. She saw how everybody in her life _leaned_ on her for support, that she could never say no to _carrying_ their burdens.\n\nThe words say it all: the burden she was carrying for others was too much for her muscles. She was like not the mythic Atlas, capable of carrying the entire world on his Titan shoulders; she was a human woman with limits. Following its own innate intelligence, her body manifested fibromyalgia\u2014muscle gridlock, so to speak\u2014to force her to realize her approach to life was toxic. Michelle realized she could say no, turn down another plea for help, and ask for help herself. This understanding was the final turn of the key in the lock of her recovery.\n\nNot everyone with fibromyalgia has this exact state of mind, says Dr. Levin, but it is fairly representative of the broad strokes. \"A person feels burdened and overwhelmed, as if everybody is _leaning_ on their body, as if they are _carrying_ the world's weight on their shoulders.\" Eventually, the body translates this metaphorical self-assessment into biological reality, and the psychological weight gets shifted to the actual muscles, and they ache, spasm, and refuse to carry it.\n\nThe longer a condition like this persists without the person understanding\u2014and undoing\u2014the emotional etiology, the more entrenched the translation into biology becomes. In other words, the emotional state may be a prime precipitating factor, but because of its charge and power, it starts to rearrange the body's structure, physiology, and biochemistry\u2014only the tip of the illness iceberg, the part that conventional medicine tends to focus on\u2014in accordance with its view of reality If I feel like I'm carrying the weight of the world on my shoulders, my shoulders will start to believe it, and hurt accordingly.\n\nThink of it this way: the unexpressed or unresolved anger, fear, resentment, or whatever the emotionally charged issue might be, act as free radicals in the body, as toxic agents capable of destabilizing physiological structures and processes just as powerfully and effectively as mercury, an organochlorine, or any other toxic substance described in this book. This is no judgment on emotions; it is not saying some are good and some bad. Rather, the point here is that when they get _stuck_ , emotions become a physiological problem for the body. When stuck, their powerful energy charge starts to work negatively on the body.\n\nAll emotions are healthy because they are what tie the mind and body together, explains Candace Pert, Ph.D., in _Molecules of Emotion._ But they need to be expressed and let go of so they don't fester, build, or escalate uncontrollably. When they are not expressed, and instead are repressed, this sets up a \"dis-integrity\" within the body, causing it to act at cross-purposes with being a unified whole organism. This repression generates stress that leads to blockages and an insufficient flow of \"peptide signals to maintain function at the cellular level.\" This blocked flow creates the conditions of weakened function and immunity that can precipitate disease, says Dr. Pert.\n\nThe peptide signals are modes of communication between the biochemicals of emotion, Dr. Pert explains. Emotional biochemistry consists of neuropeptides (key brain chemicals that translate thoughts and feelings into biochemical and physiological responses) and their receptors. Dr. Pert's research has established that emotions are equal players in the health or illness of the body, as considerable in their influence as chemicals, toxic substances, or other material biological substances. The neuropeptides are the \"substrates of emotion,\" in continuous communication with the immune system and its cells. Emotions have a \"cellular signal\" that is involved in translating emotional and mental information into physical reality, \"literally transforming mind into matter.\"\n\nIn fact, Dr. Pert says our brains are not really just in our skulls; they are _mobile_ brains, with communication networks throughout the body. Intelligent information travels throughout the network, which includes the hormonal, gastrointestinal, immune, and central nervous systems. According to this model, as we think and feel, so do our molecules act. Emotions are part of our cellular consciousness, because every cell has receptors that are conduits for emotional energy. First the emotions happen, then the neuropeptides are released. The fact that information flows throughout the mobile brain network demonstrates that \"the body is the actual outward manifestation, in physical space, of the mind.\"\n\nBut what about the stuck emotions you are unaware of, the ones you have buried deep and cannot retrieve or even remember that they exist? Sometimes clinical hypnotherapy can be helpful in aiding your dive into the subconscious to retrieve them, according to Joseph Riccioli, M.D., N.D., a physician and naturopath practicing in Clifton, New Jersey, who uses clinical hypnosis to help patients get over depression, cancer, and other serious health problems.\n\nMedical science has established that our thoughts and emotions \"talk\" to our cells. Ever since the 1970s, scientists (including Dr. Pert) have been building a case for this and calling it psychoneuroimmunology, or PNI. PNI says the mind and emotions (psyche) communicate with the nervous (neuro) and immune (immunology) systems, both to their detriment or advantage, depending on whether it is consciously or unconsciously directed, and depending on the content of that communication.\n\nUnfortunately, most of the time this dialogue is unconscious, says Dr. Riccioli. We get the results (the health problem) without understanding the causes (the emotional content), or we may be aware of a part of the picture, but missing the other pieces to make it comprehensible. Below are a few vivid examples from Dr. Riccioli's clinical casebook that make the point.\n\nGrace, thirty-six, had an embarrassing problem: she chronically pulled her hair out the way other people bite their fingernails. Her scalp was a mess of patchy, straggly hair and great gaps where she had ripped the hair out. In the course of working with Grace, Dr. Riccioli learned that she was using the hair-pulling as a way to discharge stress and tension associated with two traumatic events earlier in her life, in the thrall of which emotions she was still gripped. As a child, she had been publicly humiliated in school, and in early adulthood, her husband became an active alcoholic and deserted their marriage.\n\n**_Psychoneuroimmunology says the mind and emotions (psyche) communicate with the nervous (neuro) and immune (immunology) systems, both to their detriment or advantage, depending on whether it is consciously or unconsciously directed, and depending on the content of that communication._**\n\nGrace thought these two events happened because of her and therefore she deserved to be punished for them, even though she couldn't figure out why. Her pent-up, conflicted anger was expressed through her hair-pulling; because she couldn't find rest or resolution with respect to these two intense experiences, she was transiting back and forth continuously from her present age and those earlier times. Once she understood the connection and remembered the earlier experiences clearly and in full, she was able to discharge the old emotions and stop pulling her hair, says Dr. Riccioli.\n\nVera, forty-two, was a woman with angry ovaries. She had a serious case of ovarian cancer and was about to undergo surgery. Dr. Riccioli learned Vera had endured a highly abusive marital relationship, such that she at times had to lock herself in a room to protect herself from her husband. She knew she had a lot of anger towards him, but had stuffed it away inside herself and never dealt with it. \"Vera never dealt with her anger and it eventually turned into guilt,\" and she assumed she must have done something wrong to provoke her husband, notes Dr. Riccioli.\n\nAccording to mind-body logic, if she had been wrong, guilty of bad behavior, she must be punished, and what better biological punishment than ovarian cancer, in one of the key organs of her femininity? \"Just as her husband literally beat her up, so she symbolically transferred the beating to her ovaries,\" says Dr. Riccioli. \"You might say Vera stored the energy of the abuse, and perhaps the unexpressed anger, too, in her ovaries.\" During the course of several hypnotherapy sessions, Vera understood the connections and interactions between her relationship with her husband, her emotions, and her bodily symptoms.\n\nThe preceding examples illustrate the meaning of a useful term coined by noted women's doctor and health educator, Christiane Northrup, M.D. She calls them \"toxic emotions.\" Toxic emotions are powerful, strongly held, and typically unconscious beliefs and emotions that act as precipitators, catalysts, or seeds for a variety of illnesses, from mild to mortal. Belief becomes biology when it remains unaddressed, undischarged, or even unidentified, says Dr. Northrup. When emotions are unexpressed, they stay _in_ the body \"like small ticking time bombs\u2014they are illnesses in incubation.\" She relates a vivid case to make her point.\n\nA forty-one-year-old business executive was suffering from hot flashes. This is early to have the first signs of menopause, but because she had had a hysterectomy with removal of both ovaries a few years earlier, the surgery probably speeded up the hormonal timetable a bit, edging her, if prematurely, into perimenopause. She was under stress at work, taking four times the normal dose of estrogen for women with hot flashes, and not getting any relief. Two years into her treatment with Dr. Northrup, the woman finally revealed she had been sexually molested at age six by an adult male. She had felt frozen, numb, unable to speak at the time. Later she felt ashamed and told nobody about the incident, thinking that somehow she must have done something wrong to warrant this kind of punishment.\n\nThe hysterectomy, which was done as a result of years of unremitting uterine pain, says Dr. Northrup, was actually a step in the right direction, not so much medically, as emotionally. It was her body's way of focusing the woman's attention on the \"scene of the crime,\" in this case, literally. Even deeper, the woman felt oppressed by \"the original sin of being female,\" and tried to compensate for this felt inferiority by overworking. She overworked to prove herself in the workplace and to avoid contact with the deep emotional pain she bore in her uterus and the shame that made her feel unworthy and bad, explains Dr. Northrup.\n\nShe adds that the \"seeds\" for the woman's physical problems were planted by the emotional trauma of sexual abuse. It did not necessarily create it in a linear chain of causality, but this early abuse \"set a pattern of discomfort in her body-mind,\" and her body-mind expressed the pain continually at the place of the original trauma. Undoing the lifelong effects of this toxic seed meant going back in herself to experience and \"expiate it, exorcise it\" from her cellular memory bank.\n\nDr. Northrup calls this process of feeling, expressing, and releasing old repressed emotions \"emotional incision and drainage.\" Buried emotions, festering and toxic, are like abscesses that a doctor needs to lance and drain. Similarly, emotions, walled off and generating pain, must be \"lanced\" and \"drained\" so healthy new tissues (metaphorically, self-attitudes and behaviors) can form.\n\nAs Carolyn Myss, Ph.D., the highly popular advocate of the seamless mind-body model, remarks, the first principle of illness is that biography becomes biology. All our thoughts and feelings first enter our biological system as energy. \"Every thought you have had has traveled through your biological system and activated a physiological response,\" she says. Some have dropped depth charges, causing physiological reactions throughout the body. \"In this way our biographies are woven into our biological systems, gradually, slowly, every day.\"\n\nMost of us carry destructive beliefs and feelings around with us, and these steadily undermine our health, gradually converting our biology to be in accordance with our definition of reality. For example, if you are continuously \"pissed off' at somebody or a situation, your body may translate this into chronic urinary tract infections in which your \"piss\" acidifies under the influence of this strongly held, undischarged emotion. Until you discharge the emotion, you may be subject to recurrent bouts of urinary tract infection.\n\nVarious emotions, such as low self-esteem or self-worth, have been associated with the development of cancer, and these toxic emotions can have a powerful influence on the course of the disease and the inability of well-indicated therapies to stop it. \"Many cancer patients cannot recover from their cancer until these memories are discovered and treated.\"\n\nIt sounds counter to common sense, but what you _don't_ know can hurt you, \"because the repressed memory of trauma is traumatic,\" explains Arthur Janov, M.D., in _Why You Get Sick and How You Get Well._ Dr. Janov is the well-known founder of the primal scream technique, which he introduced in the 1970s. It is our repressed memory of a painful or deprived or abusive childhood that is itself traumatic, exerting a continuous debilitating effect on our body, says Dr. Janov.\n\nNeurosis, the state of carrying unresolved, conflicted emotion and states of mind, is the foundation of illness, he says. There is nothing more healing than to actually _feel_ and fully experience the memories and emotions\u2014our \"internal reality\"\u2014from earlier in life that have been repressed or shunted out of our awareness. Childhood pain is like a psychological big bang that can set the course for the rest of our life, until we deal with it. We may spend a lifetime trying to repress the painful memory, but \"all that energy and activation has to find its way into some kind of disease,\" says Dr. Janov.\n\nDr. Janov relates the case of a woman, twenty-nine, who had been bulimic for eleven years. After working with Dr. Janov and his primal therapy approach, she realized her bulimia was a coping mechanism traceable to her difficult birth. During her birthing, she got stuck in the birth canal and went into a frenzy\u2014angry, afraid, and fighting for her life. She couldn't breathe and her body went into convulsions. In her adult life, during a crisis, she would stuff herself with food and then vomit. She realized her bulimia was mimicking the trauma of her birth. Throwing up was her way of disgorging the stress of not being able to breathe, and therefore of discharging the tension of the crisis.\n\n\"I seemed to have recreated the airlessness of my birth through the throwing up of my bulimia,\" she told Dr. Janov. When she had a crisis as an adult and felt her breathing start to constrict, she _had_ to vomit; this was how her body learned to deal with crises. Once she understood this energy equation, she was able to free herself from the bulimia. \"We have learned that what makes people sick is the key to making them well again,\" comments Dr. Janov. But to \"fully respond to traumatic aspects of their childhood for the first time,\" people have to bring the content from out of unconscious storage back into consciousness.\n\n### The Healthy Living Space Expert Interview: Patricia Kaminski, Flower Essence Practitioner\n\nThere are many ways to access, express, and release buried emotions, but the one we will focus on in this chapter is called flower essence therapy. It is a gentle but deep practice originally developed in the 1930s by a British bacteriologist, Edward Bach, M.D. He found a way to use the subtle energies of plant blossoms to highlight, clarify, and resolve human emotions.\n\nDr. Bach created a series of thirty-eight formulas, now called the Bach Flower Remedies, in which an infusion of flower blossoms into distilled water\u2014when taken orally as drops over a period of weeks\u2014could produce profound clarifications in specific emotions such as grief, sadness, fear, anxiety, and others. His idea was to use these flower essences to address emotional, psychological, even spiritual issues underlying physical and medical problems.\n\nIn the late 1970s, Patricia Kaminski joined with Richard Katz to extend Dr. Bach's principles of emotional healing through flower essences to develop remedies based on plants grown in North America. Their assumption was that, it being fifty years later and another continent, the remedies might be more effectively produced from indigenous plants and encompass a broader range of emotional nuance. Humans are emotionally more complex today than in Dr. Bach's time, and the American psyche may need homegrown flower energies, botanical essences more in resonance with the landscape in which the end users live.\n\nToday, their line of flower essences number more than 100 (based on the freshly harvested and infused blossoms of plants, bushes, and trees) and through their Flower Essence Society, based in Nevada City, California, they have an active network of colleagues, students, and practitioners throughout the world.\n\nImplicit in the model of flower essences, whether they come from Bach or Katz and Kaminski, is the startling idea that somehow the energy of plant blossoms has a relationship with the spectrum of human emotions. Flower essence practitioners suggest that the plant kingdom, through its blossoms, embodies in a pure, almost abstract form, the essence of human emotions. The purity of this expression enables the plant blossoms, when prepared in the right way, to have a clarifying effect on human emotions\u2014clarifying in the sense of the impact an archetype (a pure emotion) can have on its manifestation (a pure emotion expressed, distorted, or blocked by a human psyche).\n\nFlower essence practitioners also postulate that the human energy field, or aura (discussed in more detail in Chapter 9), has a basic affinity with the plant kingdom. It is even suggested that in some manner the human energy field embodies the full spectrum of the plant kingdom and, analogically speaking, the myriad blossoms occurring in the plant world are like little stars, or points of pure consciousness, in the human energy field. Whatever the theoretical explanation, flower essence practitioners operate on the assumption\u2014one backed by decades of empirical results and clinical observation of clients\u2014that a flower essence can help a person experience, clarify, and resolve distorted emotions.\n\nFlower essences _\"encourage_ rather than _compel_ change, working by vibrational resonance rather than by biochemical intervention,\" explains Kaminski. They evoke and stimulate \"an inner dialogue with hidden aspects of the Self, awakening profound psychological archetypes, and giving us access to their message.\" Through this kind of inner dialogue, Kaminski says, flower essences can catalyze \"deep emotional and mental changes\" powerful enough to in turn spark physiological alterations, reversals of symptoms, and often healings.\n\n**_How do we know if we're toxic, either physically or emotionally? \"I think we should assume that we all are,\" says Kaminski. \"The question is this: do we have ways of efficiently recycling that toxicity, processing it the way the liver handles toxins? What kind of soul apparatus do we have that helps us deal with toxicity?\"_**\n\nAs Kaminski sees it, flower essences \"unite the human soul with the soul of nature\" and \"rekindle a vital connection\" between these two expressions of soul. Flower essences \"massage\" the sensibilities of the soul, Kaminski explains. \"They reinstill our capacity to receive living forces from nature, qualities that allow our souls to be permeable rather than hardened,\" as well as vital instead of mechanical.\n\nIn flower essence thinking, the term \"soul\" has a vital, real role, and would still have this quality even if matters of the soul were not now a nationally validated topic in publishing and teaching, ever since Thomas More's best-selling _Care of the Soul_ was published in 1992. The soul is the substrate of our emotions and sense of self, the deep, pure feeling state within, the inner, permanent self. In her book _Flowers That Heal_ , Kaminski says using flower essences dredges unpleasant emotions to the surface of our awareness. \"As we witness these parts of ourselves, we have the opportunity to understand, to redeem and to cleanse these emotions.\" With flower essences, we heal from the inside out.\n\nHow do we know if we're toxic, either physically or emotionally? \"I think we should assume that we all are,\" says Kaminski. \"The question is this: do we have ways of efficiently recycling that toxicity, processing it the way the liver handles toxins? What kind of soul apparatus do we have that helps us deal with toxicity? You can't medicate away the pain of the soul with psychiatric drugs. We need to be involved in creating in the soul those structures and processes analogous to the liver that enable us to process emotional toxicity. With flower essences, you actually flood the body with a positive 'medicine,' with positive emotional archetypes that shift the cellular structure in a different direction.\"\n\nIn other words, says Kaminski, toxicity is a condition we are bound to encounter on the emotional level, so we need to develop ways to _process_ it. Even better, we need to process it in such a way that in the end we have gained in consciousness. We are more aware of ourselves, of our emotional totality. This way, Kaminski says, emotional detoxification through flower remedies can lead to more consciousness. She adds that flower essence therapy is \"a cutting edge, vanguard 'medicine' that is trying to build a bridge between psyche and soma, or the body.\"\n\nHere's an example from Kaminski's case files that illustrates these points. She consulted with a fifty-eight-year-old business executive who had sustained a heart attack. He had tremendous stress in his life and it seemed his primary way of relating to the world was through hostility. He was in advertising, a highly competitive industry, and his job description was virtually to walk all over everybody else, to triumph, win the client, do the best, make the most amount of money, and the usual ultra-Type A personality behaviors. Naturally, he had high blood pressure as well as a weakened heart after the \"attack.\"\n\nWhen he came to Kaminski, he had already started a nutritional therapy regimen and meditation. She started him on four flower essences specific to his condition of stress, hostility, and a rigidified heart, using Impatiens, Zinnia, Borage, and Holly.\n\nIn terms of its effect on human emotions, the flower Impatiens is aptly named: it dissolves impatience. This man was always snapping his fingers, tapping his fingers, expressing his impatience, says Kaminski. \"The impatience was everywhere in him: the need to drive fast, to be somewhere in a hurry, to get things done in a hurry.\"\n\nIndividuals who need Impatiens \"find it difficult to be within the flow of time; their tendency is to rush ahead of experience,\" comments Kaminski in her masterwork on flower essence prescribing, _Flower Essence Repertory_ (cowritten with husband Richard Katz). They have too much fiery force, and this flares up easily into irritation, impatience, intolerance, and anger. Impatiens as a flower essence actually imparts the opposite qualities: patience, acceptance, and the ability to flow with the pace of life and others, explains Kaminski.\n\nKaminski gave the man Zinnia flower essence because much of his joy of life was gone. He had lost his sense of humor and ability to easily laugh; he took himself too seriously. He had developed an overly somber sense of self, a dullness and over-seriousness, explains Kaminski. Zinnia would help reverse that by evoking \"childlike humor and playfulness,\" a lightheartedness, an inner joyfulness, and a sense of detached perspective on oneself. It would help him appreciate the possibility that playfulness and laughter could have a role in a responsible life, says Kaminski.\n\nThe third flower essence prescribed was Borage. The emotional issues to do with Borage were to a large degree at the \"heart\" of the man's heart condition, says Kaminski. The patterns of imbalance that indicate the need for Borage are heavy-heartedness and a lack of confidence in facing difficult circumstances, Kaminski explains. Borage instills a sense of courage and buoyancy that helps one rise above difficulties. \"Borage is often at the core of outer hostility because underneath this is grief and vulnerability that emotionally weigh down the heart and prevent the heart from expressing its own real emotion. Borage helps the heart to experience this ebullience and lightness, filling the soul with fresh forces of optimism and enthusiasm.\"\n\nFinally, Holly was indicated to address this man's sense of feeling cut off from love, and of being filled with jealousy, suspicion, anger, and envy. On the positive side, Holly would help evoke the feeling of love and the ability to extend it to others, an open heart and a sense of compassion, says Kaminski. This remedy helps restore a person's \"ability to feel unity and wholeness,\" she notes.\n\nAfter a few weeks of taking the four remedies\u2014he took them several times daily as drops\u2014the man began to see changes in his emotional nature. Previously, when his young daughter walked into a room, he would be impatient for her to leave, talking to her briefly, feeling increasingly irritated. But now he found himself \"fascinated\" by what she had to say; he felt inclined to play with her as opposed to having to order himself to do so out of fatherly duty. He started to see he didn't have to \"jump all the time into reality,\" but that he could sit back and allow it to unfold before him, says Kaminski. He didn't have to control everything any more.\n\n\"Learning to trust the world is the antidote for hostility,\" Kaminski notes. This man had not been trusting of the world, and eventually his body mimicked his core emotional stance and attacked him just as he was chronically attacking the world through his hostility. A key element in easing hostility is teaching a person to trust the world again, to let the heart relax its vigilance, she adds.\n\n\"Through the flower essences, he gained enough of an edge on himself that he could start to _witness_ life rather than _react_ to it all of the time. He has had remarkable changes in his heart condition since then. All of the physiological indicators for heart health are now very favorable for him.\" Through the flower essences and the way they worked to clarify his emotional knots and give him a position of detached awareness about them\u2014what Kaminski calls the \"witness\" stance\u2014he was able to dissolve much of his emotional toxicity and thereby relieve its steady, sickening pressure on his physical heart. He became able to have vulnerability and emotional space with himself in the context of his family, and this was intensely healing for him, says Kaminski. He no longer was \"incredibly actively hostile\" at work: he didn't _need_ to exhibit that kind of behavior any more.\n\n\"For me, the question with emotional toxicity is not that we get the 'poison' out of the body, but that we learn how to handle the daily flow of it, that we learn how emotional toxicity originates. So the goal is not so much a cleansing of emotional toxicity but an active change in how we _deal_ with it,\" Kaminski explains. Ideally, through flower essences, we can deal with it with more consciousness, more awareness of its true message.\n\n## HEALTHY LIVING SPACE DETOXIFIER #35 Remedying the Four Basic States of Emotional Toxicity with Flower Essences\n\nBased on more than twenty years of clinical consultation with hundreds of clients, Kaminski describes four basic states of emotional toxicity. Using these four categories, she says it is possible for a person to categorize oneself and thereby select appropriate remedies to start the process of emotional clarification. The four basic categories are hostility, depression, stress\/tension, and anxiety\/fear, all of which are presented in more detail below. The strategy is simple: a group of compatible remedies will undo the emotional knots that produce the overall style of reactivity, such as hostility or depression.\n\nThe idea is simple: examine your typical style of emotional reaction when you are pressed up against the wall. What is your usual response when \"push comes to shove\"? Do you get mad, depressed, stressed out, or afraid? From this small act of self-observation, you can see which of the four categories you fit into, says Kaminski. Then within the chosen category, decide again which of the nuances of that basic emotional state best describes your emotional style, and select one flower essence and use it for a month or two (see figure 8-1).\n\nThese are not absolute categories, by the way. They are meant only as provisional definitions of emotional experience, says Kaminski\u2014general, perhaps broad, guideposts to help orient you to your emotional reality. Further, the flower remedies are not first aid conveniences, like aspirin, producing a quick resolution of the problem. They work slowly, deeply, organically, from within you; generally you should allow a month or two for lasting results, she notes.\n\n**Figure 8-1. The Four Basic States of Emotional Toxicity and Their Appropriate Flower Essences**\n\nIt is useful to compare the action of flower remedies to intestinal cleansers. A kind of dredging work is started by taking either. In the case of intestinal cleansers, odd, uncomfortable, but transient mental and physical states are experienced as the intestines get cleansed. Similarly, while taking flower remedies you can expect an upwelling of strange, unfamiliar, or deeply familiar emotions; the process may make you feel unsettled, but then, it's supposed to. You can't expect to clarify and resolve emotions without some confusion and seeming emotional chaos during the process. But remember: it is a _process_ , and one with an eventual positive and beneficial conclusion. So it is worth it to stick out the ups and downs of the emotional process flower remedies can precipitate.\n\n### _Hostility_\n\nThis first categoiy includes feelings of hostility and lack of trust. As we saw in the case study above, it especially afflicts the heart, both physically and as the center of our affective life. This affliction then gets projected aggressively outwards onto the world in overly assertive, aggressive, even potentially violent, behavior. It is about a loss of trust by the heart and a lack of open space for the heart, for a sense of vulnerability and emotional exposure. Borrowing a term from traditional Chinese medicine, Kaminski calls this state _yang_ , meaning, fiery, intense, strong, hard, active, outward-tending. But it is a state of emotional imbalance.\n\nFlower essences recommended for this emotional state to help curb the overly _yang_ disposition of the heart:\n\n_Holly_ : Holly helps one develop an equanimity and sense of spaciousness in the soul, says Kaminski, freeing one up from the felt need to react hard against the world. It helps one be more inclusive and expansive, to feel less isolated from others, to experience love in connection with other people.\n\n_Oregon Grape_ : This is indicated if a person feels paranoid or self-protective or expects hostility from others even when this expectation is not warranted by the facts. The person tends to feel they are likely to be attacked, from anywhere. In contrast, taking the remedy will help generate a sense of loving inclusion of others, the expectation of good will coming from others, and the ability to trust in the goodness of other people.\n\n_Willow_ : This remedy is needed when one feels resentful, irritable, bitter, or that life is unfair and has cast one as a victim, or when one has too many hard edges to the personality. Willow will help produce a feeling of acceptance, forgiveness, a flowingness with life, the ability to yield and bend with circumstances, to stop rigidly holding on to negative emotions or some old, unnecessary part of oneself, and the willingness to take responsibility for one's life. Willow helps to decongest dammed up, internalized emotions that can lead to\u2014as it were, congeal into\u2014painful, stiff joints and muscles, even arthritis.\n\n_Beech_ : Beech is needed if you feel critical, judgmental, intolerant, and have perfectionist expectations of others; also if you have an oversensitivity to your social and physical environment. You project your inner sense of inferiority, imperfection, and insecurity onto others, and seek to feel safe by condemning others. Beech helps undo this by generating feelings of tolerance and acceptance of the differences and imperfections of others.\n\n_Zinnia_ : As described earlier, Zinnia helps evoke feelings of childlike playfulness, humor, amusement, and laughter as a counterbalance to the tendency to be heavy and hard-hearted.\n\nFlower essences recommended to help heal the underlying lack of emotional vulnerability that accompanies hostility:\n\n_Borage_ : Borage also helps undo the heaviness of the heart and opens up the hidden areas of grief in the heart. Borage instills courage, upliftment, a sense of light-filled buoyancy, and \"fresh forces of optimism and enthusiasm.\"\n\n_Bleeding Heart_ : This essence is indicated if you form relationships based on fear or possessiveness or emotional codependence, or if you have poured great amounts of love into a relationship that subsequently failed (or the person died or moved away), leaving you wounded and heartbroken. The remedy helps one rebuild heart forces from within, generating a capacity to love unconditionally with an open heart, and to forgive those who have hurt you.\n\n_Golden Ear Drops_ : You can benefit from this flower essence if there are suppressed toxic memories of childhood or feelings of pain and unexpiated trauma from earlier events that produce emotional instability. Look for evidence of emotional amnesia and the \"unconscious residue of traumatic memories,\" says Kaminski. This remedy will help cleanse the heart and may literally produce tears as a form of emotional discharge, she adds. It helps you reconnect with your childhood and see it as a source of emotional well-being, not toxicity.\n\n_Calendula_ : A person can benefit from this remedy if one tends to use sharp, cutting, argumentative words as a psychological defense in communication with others. Calendula brings warmth to the spoken word, so that it generates good will and understanding and not division and hurt. It also helps to calm down the adrenal glands, which can be revved up in a state of chronic argumentativeness and verbal aggressiveness.\n\n### _Depression_\n\nAs the second category of emotional toxicity, depression is in many respects the opposite of hostility It is a _yin_ condition, a turning inward, a cold, damp, passive, soft, yielding, sinking quality. There is no fire here, and the person retreats from the world and, in effect, from active expressions of self and will. Cancer is typical of this state of emotional toxicity, says Kaminski. It would be mistaken to think of depression purely in biochemical terms, as only a brain-bound condition, she says.\n\nIt is also about whether our will has fire in it, and enthusiasm for life and taking action. A person with this condition is liable to develop alcohol dependency, but alcohol is poison for depressives, says Kaminski. \"Alcohol has so much fire in it that it drowns out our own fire.\" Depression is often \"a loss of connection (especially in the will) with the spiritual world and a loss of hope in that connection; one becomes submerged in the narrowness of the personality (the opposite of the hostile person who inflames their sense of self).\"\n\nSometimes the person's sense of self is so uninflamed\u2014so passive and watery, so to speak, so lacking in inner will\u2014that one is easily overcome by foreign disease agents, such as cancerous cells, or even parasitic energy organisms, thoughts, and feelings from the outer world, Kaminski notes. \"At its worst, one can become overcome and possessed by entities because one's ego is not strong enough to ward off these negative disease-producing influences.\" (For more on this subject, see Chapter 9.)\n\nFlower essences recommended to help heal the deep underlying wounds of depression include:\n\n_St. John's Wort_ : A need for this remedy is indicated by physical and psychic vulnerability, the lack of rootedness in the world leading to an overexposure to the light. Such people have a tendency to get sunburned and develop allergies as part of their oversensitivity, and they have a very active psychic life, but an ungrounded one, leading them to loose their connection to the physical body, says Kaminski. \"This weak association to the body results in a propensity for invasion or attack from negative elemental forces or other entities, especially during sleep.\" This remedy helps a person _circulate_ the light throughout the physical body and undo psychic blockages that drain off one's vital energy.\n\n_California Wild Rose_ : Here the condition is apathy, resignation, and the inability to organize and spark the forces of will through the heart so that one can take hold of one's responsibilities and life tasks. People needing this remedy find it hard to take emotional risks, need to insulate themselves against wounding, and may even be alienated from the social world. The remedy instills a feeling of love for the Earth and human life, and an enthusiasm to serve others. This flower gives one the courage to love, to love life, and even to wear the \"crown of thorns\" life inevitably gives you.\n\n**_Sometimes the person's sense of self is so uninflamed\u2014so passive and watery, so to speak, so lacking in inner will\u2014that one is easily overcome by foreign disease agents, such as cancerous cells, or even parasitic energy organisms, thoughts, and feelings from the outer world._**\n\n_Scotch Broom_ : This remedy is indicated when a person feels weighed down and depressed, overcome with despair or pessimistic views, especially with regard to one's relationship to large social events. The soul feels burdened with negative scenarios of great social upheaval and travail, and this unrelieved weight leads to resignation, paralysis of the will, and depression. Scotch Broom sparks feelings of optimism about being able to affect the world positively; it imparts a sense of caring, encouragement, hope, and purposefulness.\n\n_Mustard_ : Feelings of melancholy, gloom, despair, and depression without obvious cause suggest this remedy. The soul feels darkness and is overwhelmed by it, especially as its onset seems inexplicable. Mustard helps one gain emotional equanimity and joy, and identify forgotten or suppressed events that may have triggered or sustained the depression.\n\n_Gentian_ : If you feel doubt or discouragement after a setback, Gentian can help. Gentian helps build resilience in the face of impediments and obstacles, and it provides encouragement to see these setbacks as opportunities to learn more about oneself by shifting one's perspective on life and identity. Skepticism and doubt get transformed into faith and confidence.\n\n_Sweet Chestnut_ : This remedy is helpful when you feel strong despair and anguish and something akin to what is traditionally called \"the Dark Night of the Soul.\" If you need this you are probably suffering acutely, seemingly being tested to the breaking point. You probably feel deeply alone, possibly suicidal. Sweet Chestnut helps instill a deep sense of courage and faith that comes from trusting the spiritual world to not let you down and to help you out of the \"rock bottom\" of your depression.\n\n_Love-Lies-Bleeding_ : In this case, a person feels an intensification of pain and suffering due to isolation. The suffering can be intense, either bodily or psychological in nature, and it pushes one's awareness deeply inward; such a person is \"truly deep-pressed,\" comments Kaminski. This remedy does not so much provide immediate relief from this state as it helps the person move consciousness outward from the over-personal identification and isolation into something more transpersonal. It is a shift in perspective and meaning from the inside to the outside, from the personal to something beyond it; it provides a context that thereby eases the suffering because suffering now has a role and purpose.\n\n_Yerba Santa_ : This remedy is recommended when you feel emotionally constricted, especially in the chest; when you have internalized grief and melancholy; or have deeply repressed emotions stored away, as it were, in the deepest recesses of the heart. Typically, your breathing is disturbed and congested, and you are liable to respiratory diseases as the soul wastes away within, consumed by the intense energies of unrelieved sadness or grief. Yerba Santa helps free up the flow of emotions so that feeling and breathing may be harmonized.\n\n### Stress\/Tension\n\nThese two conditions tend to go together. As a strong sense of pressure, stress\/tension is \"the younger brother of hostility and afflicts many,\" notes Kaminski. \"Such persons are not as overtly hostile or aggressive, but they still are predominantly _yang_ in their soul orientation. They need to develop more inner space, a quality of soul spaciousness. Such persons find it hard to be at rest; they always need to be doing something; they feel hurried and pressured; and they set up situations in their life in which they will be hurried and pressured.\"\n\nKaminski notes that in some respects, this condition is \"a strong imprint or overlay from Western culture itself and may not be part of a person's true temperament. Many people are profoundly influenced by the overall speeded-up quality of the urban environment in which everything rushes by intensely; people end up unconsciously copying this energy pattern in their own life. Such persons could be called adrenaline junkies, because they thrive on tension, pressure, deadlines, and performance expectations, all of which tend to over-stimulate the adrenal glands. The faster something is, the more impatient they are to make it go even faster. They will switch TV channels constantly or upgrade their computer every year.\"\n\nTwo essences specifically deal with the overall quality of pressure, tension, and stress imparted by Western culture to all of us:\n\n_Yarrow Special Formula_ : This formula focuses on enhancing the integrity of our energy body, the flow of physical and subtle energies through our physical body and aura. It also helps undo the toxic influence of noxious environmental energies such as radiation, pollution, geopathic stress (harmful energies from the Earth itself\u2014see Chapter 12 for more details), and even computer stress. This remedy combines yarrow and sea salt to strengthen the human energy field with pure formative forces from the plant and mineral kingdoms so that we can withstand and resist the tension\/pressure and noxious influences from the environment.\n\n_Five Flower Formula_ : This is an all-purpose first-aid remedy (also known as Rescue Remedy) combining five flower essences (Cherry Plum, Clematis, Impatiens, Rock Rose, and Star of Bethlehem). It is especially useful in cases of sudden, acute shock or trauma. \"It is also a good overall balancer for tension, stress, emergencies, life in the fast lane, some cases of insomnia, subliminal levels of panic, and general _dis_ -ease,\" comments Kaminski. It instills calmness and stability in the face of physical or emotional emergencies, high stress, sudden panic, disorientation, or loss of consciousness. It helps the soul stay connected with the body, and in cases of nonemergency, it can be used in the early stages of inner work as a way of stabilizing the self so that it can calmly contact the soul within.\n\nOther essences helpful for easing the pattern of stress\/tension:\n\n_Impatiens_ : As already mentioned, Impatiens can alleviate the sense of impatience, irritation, urgency, and tension, and allow instead a sense of patient acceptance of the flow of events and the movement of time.\n\n_Canyon Dudleya_ : This remedy is particularly indicated for people who have distorted psychic experiences, who are preoccupied with mediumism, or who need more order and less self-inflation in their soul lives. Canyon Dudleya will help catalyze a healthy, balanced, grounded spiritual opening.\n\n_Yarrow._ People who experience extreme vulnerability to others and to their environment, who are easily depleted energetically, or who too easily absorb negative influences and psychic toxicity from around them will benefit from Yarrow. This flower essence instills a sense of inner radiance and auric strength, a sensitivity that is not vulnerable to constant permeation by outside influences. Persons who have allergies, environmental sensitivities, and psychosomatic diseases will benefit from Yarrow.\n\n_Scleranthus_ : Scleranthus deals with hesitation, confusion, indecision, and conditions of unresolved duality in the quest for inner wholeness. People for whom this remedy is indicated tend to vacillate when they need to make decisions; this chronic uncertainty can get translated into bodily symptoms, especially ones that continually shift, come and go, increase and decrease in intensity. Scleranthus can help produce inner resolve and decisiveness.\n\n_Indian Pink_ : The person requiring this flower essence tends to have their psychic forces easily shattered by overactivity and find it hard to remain centered and grounded during this intense activity. Indian Pink people tend to try to do too many things at once, and they spin out of control and out of focus. They can be emotionally volatile and appear haggard or depleted. Indian Pink helps such a person stay centered and focused even during times of stress.\n\n_Star Tulip_ : The Star Tulip person tends to feel hardened, cut off, unable to feel attuned to things or any sense of quiet inner presence as is needed to pray or meditate. The remedy helps one listen within, to become aware of the subtle influences perpetually at work within and around them; it enhances one's ability to pray, dream, meditate, or feel connected to the spiritual worlds.\n\n### _Anxiety\/Fear_\n\nThis fourth basic state of emotional toxicity includes loss of confidence and low self-esteem in the context of low-grade chronic fear and anxiety. \"This is the lesser sister of depression,\" says Kaminski. \"It works predominantly in a _yin_ way in the soul. These are persons who under-perform, who do not push or pressure themselves with respect to externals though they may carry a huge amount of pressure or anxiety that churns them up from within. Such persons hold in too much psychic force especially in the abdomen or solar plexus area of the body. They tend to cave in, give way to fear and limiting notions of self. They need courage and strength, more _yang_ force.\"\n\nThese are the flower remedies recommended to help undo this condition of chronic anxiety and fear:\n\n_Mimulus_ : This essence is helpful for easing the effects of known, everyday fears, for people who are hypersensitive to small ordinary fears. The solar plexus churns with agitation and unease at all these unrelieved fears. Mimulus restores courage to the soul to face these fears and life challenges with confidence and inner light, and it helps highlight the deeper, more basic fear underlying the multitude of surface panics: \"the fear of the physical body, or of physical life itself,\" sometimes traceable to the moment of birth, says Kaminski.\n\n_Scarlet Monkeyflower_ : The person needing this flower essence tends to fear intense feelings and represses all strong emotions, leaving them unable to resolve issues of anger and powerlessness. The person is afraid of their shadow-self emotions, the ones they believe originate from the unconscious. Scarlet Monkeyflower helps instill emotional honesty and helps integrate the shadow feelings.\n\n_Sticky Monkeyflower._ This one deals with repressed sexual feelings or acting out in inappropriate sexual behavior. It is also concerned with a person's inability to experience human warmth during sexual relations; in a sense, it is about a deep fear of sex and intimacy, says Kaminski, and even more, it is about a fear of exposure of one's essential self during moments of physical intimacy. Sticky Monkeyflower thereby helps a person integrate sexual urges with human warmth and intimacy.\n\n_Buttercup_ : Buttercup is indicated when one feels low selfworth and high self-doubt, and when one is unable to acknowledge one's own inner light and uniqueness especially when measured against conventional external standards. Buttercup, in keeping with its bright, almost glistening yellowness, helps one radiate an inner light so as to become free of the need for outer recognition. Buttercup helps you honor your own value from within and bring this quality out into the world.\n\n**_\"This is exactly what flower essence therapy is intended to achieve: not to mask outer symptoms but to help the soul find the right balance with regard to these basic soul dispositions.\" The toxicity comes when these soul aspects aren't being attended to in a conscious way._**\n\n_Sunflower._ The Sunflower person has a distorted sense of self, either tending to self-inflation or self-diminishment, to arrogance or low self-esteem. There is often an imbalance in their relationship to the masculine (or Father archetype), even if the person is a male. Sunflower evokes a balanced sense of ego or self-identity, neither too much nor too little, marked by light, warmth, and compassion.\n\n_Penstemon_ : Here one feels sorry for oneself and self-pity is the order of the day. One is also unable to bear the difficulties of one's circumstances (such as a handicap or disability from birth or an accident) and succumbs to a sense of persecution. Hence, Penstemon instills a feeling of inner fortitude, an ability to persevere despite the heavy challenges facing one.\n\n_Mountain Pride_ : The Mountain Pride person often vacillates and even withdraws in the face of a hard challenge, lacking assertiveness and the inner nerve to stand up for one's convictions. This is for people who confuse peace with passivity. Mountain Pride evokes a sense of \"forthright masculine energy, a warrior-like spirituality that confronts and transforms,\" explains Kaminski.\n\n_California Wild Rose_ : See the Depression category for a description of this essence.\n\n_Blackberry._ Here a person has a hard time translating goals and ideals into actual action in the world. It is as if the will cannot link itself with the world; there is a significant gap between aims and achievements. Appropriately, the person's metabolism is sluggish, mirroring the sluggishness of the will. Blackberry helps one develop a sense of \"exuberant manifestation\" in which one can clearly, directly channel one's will forces into concrete action.\n\nAs mentioned earlier, these categories and their subdivisions are not cast in stone. \"As the emphasis here is emotional toxicity, I refer to the dysfunctional qualities we often see when these four temperaments are out of balance,\" comments Kaminski. \"There is usually one we gravitate towards more than others, but certainly these are not mutually exclusive categories. In fact, we can switch into different basic responses as we go through different life cycles. Even the seasons can affect our responses.\" For example, there is a normal disposition to \"depression\" in the winter months when there is the least amount of sunlight.\n\nWriters such as James Hillman, in _The Soul's Code_ , wisely counsel us to cultivate this depression in the winter, to take time to go into our depths. If we did, he argues, we would see more depth in our culture, more ability to deal with grief and loss, and less need to medicate. \"This is exactly what flower essence therapy is intended to achieve,\" notes Kaminski. \"Not to mask outer symptoms, but to help the soul find the right balance with regard to these basic soul dispositions.\"\n\nThe toxicity comes when these soul aspects aren't being attended to in a conscious way, she explains. \"I use the analogy of a river that flows harmoniously without being impeded, dammed, or otherwise regulated from without. This river (our emotional life) is able to cleanse itself of toxic waste effectively, even to regenerate itself from rather extreme states of toxicity if given the opportunity to come back into dynamic balance.\"\n\nAs Patricia Kaminski makes clear, flower essences can highlight, clarify, and help resolve emotional knots in the psyche. It can bring light into the dark regions of our emotional life and help us see where we have been tied up, repressed, or otherwise blocked by stuck or buried emotions. Flower essences, especially St. John's Wort, discussed above, can also make us aware of psychic blockages or even parasitic elements in our overall energy field. Sometimes these blockages or the presence of foreign energies in our aura can compromise us to a degree beyond which flower essence therapy can be therapeutic. That is because such blockages or presences operate beyond our emotional sphere, in the sphere of energy and psychic or spiritual influences, and require a different approach. This is the subject of the next chapter.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about flower\n\nessences mentioned in this chapter:\n\nPatricia Kaminski,\n\nFlower Essence Society,\n\nP.O. Box 459,\n\nNevada City, CA 95959;\n\ntel: 800-736-9222 or 530-265-9163;\n\nfax: 530-265-0584;\n\ne-mail: mail@flowersociety.org;\n\nwebsite: www.flowersociety.org.\n\n# CHAPTER 9\n\n# Guidelines for Spiritual Detoxification: Cleansing the Space around the Body\n\nCall to mind, if you will, the last time you walked into a room and felt accosted by the strange, unpleasant, possibly obnoxious energy in that space. It may not have been anything you heard, anyone in particular you observed, or even an odor you detected. But nonetheless you _felt_ something, an almost baleful atmosphere emanating from the room\u2014or movie theater, restaurant, store, basement, potentially any public or domestic space you find yourself in. Perhaps you felt a prickling at the back of your neck; maybe your skin went clammy, your breathing tightened, you felt panicked or anxious, or noticed anomalous aches or pains suddenly emerging in your body.\n\nWhat's happened? Using senses common to humans, though ones not generally acknowledged, you have _sensed_ the spiritual quality of a living space\u2014\"spiritual\" in the literal sense of spirit, meaning not physically embodied but immaterial, an energy essence or influence, a vibration. The part of you that did the sensing is often called the aura, and this is the subtle energy field\u2014a kind of sixth sense\u2014said to envelop the body of every human being and to resemble a luminous egg, according to the reports of psychics. The aura can see where we can't with our ordinary eyes, and it can register subtle influences, both healthy and unwholesome, that can affect our minds and bodies. It is another aspect of our overall _living space_ and like the other aspects, our physical body and our home, it can either support or hinder our well-being.\n\n### Poisons in Our Invisible Atmosphere\n\nWhile the direct perception of the aura (and thus confirmation of its existence to skeptics) depends on one having psychic abilities, the inference that there is an invisible energy envelope around the human body is much easier to support. After all, you have just inferred its existence when you felt the unpleasant vibrations upon entering that room.\n\nThe matter of auric influences has been a topic of discussion for many centuries in both Western and Eastern cultures. The sixteenth century German philosopher Henry Cornelius Agrippa advised his readers against associating with people who were evil-minded because \"their souls are filled with harmful radiations which will infect your surroundings in an unhealthy way.\" Agrippa recommended that his contemporaries interested in preserving their health associate with \"good and fortunate people, for these can help us greatly by being near to us.\"\n\nMore recently, in 1934, a German physician named Dr. Friedrich Markus Huebner noted that \"one can be poisoned\" not only by unwholesome foods, gases, and microbes, but by \"unwholesome people.\" Dr. Huebner explained that \"our invisible atmosphere\" is the part of us that first senses the subtle poison, and it becomes \"negatively charged by contact with harmful radiations.\" These radiations then exert unhealthy influences on the mind and body, eventually contributing to the emergence of symptoms of dysfunction and disease.\n\nDr. Heubner further noted that it was not uncommon for a person, upon coming in contact with a man or woman with a \"malignant aura,\" to get sudden giddiness, stomach pains, perspiration, or a fit of nervous laughter. \"Our organism is hard put to throw off an intoxication that menaces it or already seems to be active,\" he stated. According to these experts, it is the energy field or aura around the body that registers, and holds, these influences, and it is from the aura, as if it were a staging ground, that these external energies work their way into the physical body.\n\nSo what does the aura look like? One way to conceptualize it is to think of it in terms of heat radiations seen over a flat landscape or over an empty tarred parking lot on a hot summer day. You can actually see the heat rising from the surface, even though normally \"heat\" is not visible. Another way is to think of an egg. The consensus among psychics describing the human aura is that it resembles a translucent white egg extending perhaps one to three feet away from the physical body on all sides. Others have called it a \"radiant luminous cloud.\" Either way, every human has a \"personal radiation field through which we touch our environment.\" Clairvoyants report that in a state of vibrant health, a human standing in his aura resembles a figure inside a sun, as the brilliant aura radiates light in all directions.\n\nHowever, the aura is a fragile energy field, subject to injury and compromise, according to psychic healer Ruth Berger. In her handbook on aura cleansing, _The Secret is in the Rainbow: Aura Interrelationships_ , she notes that the aura can be ripped, torn, and penetrated. It can sustain cuts, slashes, gaps, slits, and openings, Berger reports. \"Picture a piece of cloth or other material dotted with cuts, slits, tears, cigarette burns, and frayed edges, and you will be able to imagine what it would take to repair such an aura.\"\n\nHow does the aura get damaged? Powerful emotional events, focused insults, psychic and verbal abuse, accidents, violence, bad thoughts directed at you from other people, drunkenness, and substance abuse\u2014there are numerous ways in which the delicate energy field around the human body can be polluted, soiled, or injured. All of us are subject to these auric influences, and the older we get, the more likely it is that our auric field matches Berger's description.\n\nThe cumulative effect of many energy insults to the aura is eventually tangible, physical symptoms, Berger explains. The aura develops splotches, dirty, murky, grayish areas like stains on blotting paper; it may also bear circles, checks, squares, lines, squares, or other strange unnatural-looking shapes, each of which is a kind of seed for an illness. In fact, psychic healers often report that observation of these abnormalities in the energy field is a valuable diagnostic tool, enabling them to see a health problem before it manifests physically.\n\nFor example, say somebody shouts at you and abuses you verbally and, in your view, quite unfairly. The abuse makes you feel bad, probably depressed, and it remains in your aura as a kind of repetitive thought, a harmful echo, a constant bruising influence, and thus a seed of illness. It weakens your aura and in turn makes your physical organism vulnerable, a little off balance. \"This could result in colds, headaches, and general ill health,\" says Berger. \"You must be aware of how negative thoughts can implant themselves in the body in order to take up the proper defense.\"\n\n**_According to psychic healers who work regularly with patients who have compromised auras, the energy field of the typical person bears hundreds of negative \"seeds\" of possible future illness left over from insults, abuses, traumas, strong emotional experiences, bad thoughts, and the rest._**\n\nAccording to psychic healers who work regularly with patients who have compromised auras, the energy field of the typical person bears hundreds of negative \"seeds\" of possible future illness left over from insults, abuses, traumas, strong emotional experiences, bad thoughts, and the rest. In fact, not only have \"seeds\" been observed, but numerous anomalous and frankly bizarre energy aberrations and \"devices\" have been noted. In fact, some have reported the presence of foreign sentient beings in the aura, even the spectral presence\u2014auric ghosts, if you will\u2014of deceased family members, relatives, or generational forebears.\n\nAs long as these \"seeds\" remain in the aura, they continue to exert an influence as \"spiritual toxins.\" Metaphorically, they are another kind of free radical, less tangible than the molecular kind but equally capable of producing illness. Your aura is part of your overall living space and thereby is subject to toxic influences. Over time these spiritual toxins can contribute to a general weakening of the immune system and the body's physical integrity and to the emergence of numerous mental and physical symptoms, even degenerative disease\u2014 _unless_ they are expunged. This chapter provides practical techniques to accomplish this.\n\n### The Outer and Inner Aura in Disease\n\nThe link between unseen influences in the energy field around the body and the emergence of mental and physical symptoms has occupied progressive researchers and physicians for the past one hundred years. Similarly, devising workable methods for auric detoxification has been the concern of psychic healers for a long time. Among scientific researchers, the effort of course has been to somehow quantify, even to make visible, the auric field and to use it as a diagnostic tool in assessing a patient's health and predisposition to illness.\n\nWhile the aura and its \"contents\" have been discussed for millennia by healers, spiritual leaders, and psychics, in 1908, the scientific world took a major step forward in demonstrating the existence of the aura. Dr. Walter Kilner, a physician at London's St. Thomas' Hospital, developed a photographic process called the Kilner Screen that made the human aura relatively visible, thereby offering doctors a remarkable diagnostic potential.\n\nThe aura is triple-layered, transiently disfigured in infections but permanently distorted in pathological conditions, Dr. Kilner reported. With the Kilner Screen a physician could now see aspects of the human auric field, a perception that formerly had been available only to medical clairvoyants. In fact, Dr. Kilner devoted many pages to medical descriptions of the aura in disease, drawing correlations between observed energy anomalies in the aura and diagnosed medical conditions or pathologies in the physical bodies of patients. In Dr. Kilner's view, the connection between auric status and physical health was unassailable. \"It is extremely difficult to imagine any departure from health that can occur without in some manner influencing one or more of the auric forces, and in consequence the aura itself.\"\n\nThe human raiment of light received further clarification through the more recent work of John C. Pierrakos, M.D., a physician and practicing clairvoyant. Dr. Pierrakos described the \"thrilling phenomenon\" of the cloudlike auric envelope and its precise chromatic pulsations relative to body metabolism, breathing rate, emotional state, humidity, and air ionization. According to his own observations, the aura, which appears nearly perfectly oval, pulsates in a continual expansion-contraction rhythm at the rate of fifteen to twenty-five times per minute in a person who is in a wakeful state.\n\n\"The auric envelope viewed as a whole is blue-gray to sky blue, and it illuminates the periphery of the body in the way the rays of the rising sun light up the profile of dark mountains.\" Dr. Pierrakos postulates that the aura springs from a \"longitudinal core of energy\" in the physical body (possibly connected with the central nervous system) that radiates outward through the skin to activate the immediate atmosphere and form the aura. For those who can see the aura, it provides information about \"the systemic implications of an illness, whether the primary symptoms surface in physical disabilities, emotional imbalances, or warped thinking.\"\n\nThe practical, diagnostic correlations of the subtle human anatomy (energy features that work in complement with organic ones) with medicine were advanced in 1967 by neuropsychiatrist Shafica Karagulla, M.D., when she investigated the applications of clairvoyance for aura diagnosis. Dr. Karagulla worked with a team of psychics, matching their psychic perceptions of internal organs, auric colors, and the functional integrity of the chakras with known physical pathologies. For Dr. Karagulla's sensitives, the healthy energy field is a matrix of light frequencies penetrating the body like a \"sparkling web of light beams.\"\n\nThis energy web can be tightly or loosely woven around the person; it can appear coarse or fine, dull or bright, she said. The auric web of unhealthy people, however, shows gaps, breaks in the pattern, holes, minute energy whirlpools that have separated from the normal energy stream, and a jumble of lines of force whose texture is like scar tissue. The essential point is that \"the energy web or body showed the condition clearly many months before it became apparent in the physical body,\" commented Dr. Karagulla.\n\nDr. Karagulla described the case of \"SA\" in whose thyroid gland (in the throat) a nodule had been found and excised. The energy field of SA's throat was dull, ranging from gray and blue to a darker shade of gray. The energy level of the thyroid gland was \"flabby, extremely low, as though dead,\" said Dr. Karagulla. Describing the overall impression made by SA's aura, she noted: \"There was an unusual granular, sandlike appearance of the etheric energy distributed over the surface of the whole etheric field [another name for the auric energy field], which dipped deep into the physical body as well.\" In other words, Dr. Karagulla saw confirmations of SA's poor health in the low energy state of and deformations in her aura.\n\nDr. Karagulla described the abdominal region of another patient. The solar plexus energy region of this person was \"wilted\" and \"broken into fragments,\" especially around the belly button, while the energy outside that area, but still in the abdomen, moved at a faster rate and was brighter, all of which indicated an energy imbalance in the abdomen. The medical diagnosis was that this person had a blocked colon in the part near the spleen (on the left side).\n\nDr. Karagulla further reported that within the auric envelope are a series of whirling wheels of color, known anciently in both the East and West as chakras. These energy centers are arranged vertically with reference to the spinal column and in association with the endocrine glands. The chakras are \"spiral cones of whirling energy,\" reported Dr. Karagulla in _Breakthrough to Creativity._ Breaks or disturbances in these spiral cones indicate an imbalance in the function of the physical body in that area, Dr. Karagulla explained. \"If any of these major vortices show a dullness or irregularity or 'leak' in this central point or core, we look for some serious pathology in the physical body in the area.\"\n\n**Figure 9-1. Locations of the Chakras.**\n\n## HEALTHY LIVING SPACE DETOXIFIER #36 Salting Your Energy Centers\n\nBecause the focus of this book is pragmatic rather than theoretical, it's time to try a technique that will both produce noticeable immediate benefits and demonstrate the reality of the aura and its influence on the mind and body.\n\nYou can use this technique as a once-weekly mind-body detoxifier, as an emergency auric detoxification method (for moments in which you feel \"polluted\"), as an occasional auric toner for improved well-being, or every morning (or evening) before your shower or bath. In this exercise, you use a heavy saltwater solution to anoint different, parts of your body to cleanse your energy centers, or chakras, thereby cleansing them of auric pollution. It is generally agreed that there are seven major centers along the spine from groin to head, but there are also numerous minor centers (some say as many as seventy-two) arrayed across the rest of the body; both major and minor energy centers are addressed in this detoxification technique (see figure 9-1).\n\nBefore you begin this exercise, notice your state of mind and body: How do you _feel?_ What is your emotional state? Are you irritable or calm? Does your body feel comfortable or tense? Do you feel sluggish or alert? Do you have a headache? Is your breathing loose or tight? What kinds of thoughts are occupying your attention?\n\n_Directions_ : Fill a plastic quart container (such as a yogurt container) with one-half pound of food-grade salt, then add warm water to the top. Stir to mix the salt with the water. Take your clothes off and step into a shower stall; pull the shower curtain to keep the salt water from splattering your bathroom. Put your hand into the salt water (make sure the salt is thoroughly mixed in) so that a small amount of the thick salty water remains on your palm when you remove your hand. Daub the top of your head in a circular motion with this salt water in your hand.\n\nYou may wish to make a statement as you apply the salt water; one such statement is: \"In the name of the Father, the Son, and the Holy Ghost, may these centers be clear and full of light.\" You can say anything you wish; the principle is that by making a statement it helps to focus the physical action. Make the statement each time you daub an energy center.\n\nThere are many energy centers on the head. After you daub the crown, daub your brow, then both sides of your nose, behind both ears, the temples, at the top back of the head, the back of the neck, the front of the neck. Next do your upper chest, then your solar plexus, pubic area, and groin. Be careful (especially men) not to get salt water directly on your genitals; it can irritate the urethra. With each energy center (chakra), take a fresh portion of salt water from the container. Finally, do the soles of your feet.\n\nNow return to the top portion of your torso and do the armpits, insides of the elbows, the wrists, and the palms. Next, anoint your liver (lower front right side), spleen (lower front left side), both kidneys (same place, but in the back), the two shoulder blades, the inguinal lymph node area at the tops of the thighs (a few inches down from the groin on each leg), the front and back of the knees, the ankles, and the soles of the feet again.\n\nIf there is any salt water remaining, daub it any place where you sense it is needed and empty the rest over your head. One clue to where else may need salting is that during the anointing you may find you shiver or shake as you apply the salt to certain areas. Commonly, the back of the neck and the solar plexus (upper abdomen) reflect a great deal of auric pollution; the contact of the salt water with these sites can produce temporary physical sensations. Often the occurrence of shakes, shivers, sighs, groans, or even shouts as you apply the salt can be interpreted as a kind of feedback, confirming the appropriateness of the salt-based detoxification technique. In other words, getting a reaction of this type tells you there was in fact some auric pollution there; however, not reacting in these ways does not mean your aura was clear. People react differently to energy clearings.\n\nRemain standing with the salt all over your body for about ten seconds, then shower off all the salt. Take another inventory of your state of mind and body. Do you notice any differences? I am certain you will feel better. The more you use this technique and the closer you observe your starting and finishing mind-body state, the more you will come to appreciate the cleansing effect of the salt water. Also, the more acute the sense of auric pollution\u2014recall that contaminated room we imagined entering at the start of this chapter\u2014the more immediate and unarguable the psychic and physical relief you will probably experience after salting your energy centers.\n\nThis technique is not curative, in the sense of permanently removing deeply embedded influences from your aura, but it is definitely palliative. You will feel better after doing it and can preempt the influence of unwholesome auric energies upon your thoughts, feelings, and actions. You will probably come to realize that sometimes that spike of irritation, that surge of anger, that flare-up of pique, is not _you;_ it does not come authentically from you but from a transiently resident energy influence, in a sense, something you picked up at the mall.\n\n_Variations_ : Should you be so fortunate as to live near the ocean, a daily swim, float, or simple immersion in the salt water will accomplish the same effect as the anointing of your energy centers in the shower stall. If you own or have access to a flotation tank, thirty to sixty minutes in this salt-rich enclosed water environment will produce excellent results. Not only will your energy centers have a cleansing, but your whole body will feel astonishingly relaxed, rested, and clean.\n\n### Charting the Subconscious Mind in the Aura\n\nThe medical applications of auric diagnosis have taken a great leap forward with the innovative work of Shakuntala Modi, a board-certified psychiatrist based in Wheeling, West Virginia, who has successfully used hypnotherapy to help relieve patients of numerous physical and psychological symptoms due to spiritual toxins. Dr. Modi's approach is to hypnotize patients, then interview them while they are in a light trance state to ascertain the subconscious factors underlying their health problems, both physiological and psychological in nature.\n\nBased on her research with hypnotized patients, conducted over more than fifteen years, Dr. Modi is confident in correlating numerous health conditions, such as stammering, panic disorder, headaches, weight gain, allergies, schizophrenia, uterine disorders, and others, with various types of unresolved energy field disturbances. What Dr. Modi refers to as a person's \"soul history\" is arrayed in the aura. She explains that this energy field can get \"polluted\" with unhealthy thoughts and influences just as the physical body gets overburdened with toxins.\n\nResearch conducted among her patient base showed that 77 out of 100 patients reported having foreign \"beings\" in their aura (they described them as \"demons\") who were responsible for the psychological and physical symptoms for which they were seeking help, reports Dr. Modi. These inimical energies were described as being black, red, or gray blobs in various shapes, \"ranging from tiny dots to giant-sized creatures,\" as \"black and gray pebbles all over the body,\" or a \"piece of coal,\" but they could also assume shapes more like animals, gargoyles, devils, or monsters; while in hypnotic trance, some people said their bodies seemed to be infested with \"freckles,\" which they understood to be the foreign energies; still others said they perceived these bizarre external energies to exist in many layers in the aura.\n\nThese toxic energies can make patients \"convulse, shake, jerk, froth at the mouth, spit, hiss, growl,\" and to feel other strong, negative emotions, says Dr. Modi. Her research demonstrated that these foreign energy beings are \"the most common cause for depression\" and \"the single leading cause for psychiatric problems in general, directly or indirectly.\" In one case, persistent shooting nerve pain in the heel was due to gray and dark foreign energies \"packed in multiple layers\" in the patient's heel; in another case, a man had chronic back pain due to a similar layering of toxic energies; while still other patients reported \"black blobs\" in their stomach, bladder, shoulders, jaw, eyes, hands, or head, producing chronic pain and discomfort in those areas (see figure 9-2).\n\n**Figure 9-2. Picture of Body Showing Energy Interferences.**\n\nOf equal strangeness is Dr. Modi's report that many patients, under hypnosis, discovered that various foreign \"devices\" were lodged in their auras. These devices act as spiritual toxins, draining the body of energy or inducing symptoms and the energy conditions for physical illness. For example, Dr. Modi found that some patients had \"energy absorbers,\" which interfered with nerve impulses. Their effect is to absorb a person's energy, \"making them chronically fatigued, sluggish, and sleepy,\" and they can interfere with memory and brain functions as well. One patient reported detecting a \"yellow-green slime\" in the digestive tract. This had the physical effect of interfering with digestion and nutrient absorption, and produced stomach pain, comments Dr. Modi.\n\n**_The crucial point is that once this realm of toxic influence is acknowledged\u2014once you accept that it is possible such things can exist\u2014the deep roots of a great deal of toxicity, illness, and disease have been identified._**\n\nThere are a multitude of cases showing the deleterious effects of spiritual toxins in the aura from the casebooks of Dr. Modi and other \"spirit releasement\" practitioners, whether psychics or psychiatrists. The crucial point is that once this realm of toxic influence is acknowledged\u2014once you accept that it is possible such things can exist\u2014the deep roots of a great deal of toxicity, illness, and disease have been identified. Not to remove these bad energy influences is to ensure their continuing negative impact on our health and well-being and to relegate many people to _unnecessary_ (because it is treatable) discomfort and suffering.\n\nWhen these foreign presences and devices were removed from the patients' auras through a process called spirit releasement, says Dr. Modi, the acute symptoms \"often cleared up immediately,\" relieving patients of their long-term discomfort.\n\nIt's not always spirit attachment per se that creates the problem, sometimes it is only a matter of proximity. According to the late British psychiatrist, Arthur Guirdham, M.D., many common illnesses can be attributed to the aura registering contact with evil entities or discarnate beings or with the onset of \"far memory,\" otherwise known as recall of past lives. In this model, the body's reaction is to channel the mind's suppression of this information into certain illness categories, including migraines, epilepsy, asthma, childhood tics, intestinal distress, Meniere's disease, shingles, and later in life, obsessional neurosis, heart problems, and in some instances, cancer.\n\nDr. Guirdham, a psychic himself, recognized the psychic side of many illness conditions, both physical and mental. The illness results from the body recoiling from the inimical contact or from the force of the psyche's repression of the awareness of the contact, Dr. Guirdham says. He reminds us that epilepsy was once called the \"sacred disease,\" and was associated with psychic abilities and contact with discarnate presences. It remains so today in most instances, even if it is not medically recognized as such. \"In such cases, the individual is escaping in _petit mal_ from what he is afraid to receive in full consciousness.\" Epilepsy is a defensive response, both crude and immediate, to the encroaching presence of \"invading entities,\" and as the child grows older it is often succeeded by migraines and asthma. Asthma in a child, says Dr. Guirdham, is often \"a straight response to night terrors,\" the semi-conscious perception of astral entities or discarnate beings, usually of an unwholesome valence.\n\nIncreasingly, practitioners from diverse fields, such as psychology and medicine, are beginning to acknowledge this energy factor in disease. In fact, the matter of infestation by entities has a distinguished past, being one of the eight sections of classical Ayurveda, the traditional medical-mystical science of India. In Ayurveda, _bhuta-vidya_ is the science of entities and considers the effect of bhutas, or entities, on human health and sanity, and the ways one can eliminate these noxious presences. One _bhuta-vidya_ practitioner is Parisian-trained physician Samuel Sagan, M.D., who directs the Clairvision School in Sydney, Australia. A significant portion of his work involves clearing entities and \"perverse energies\" from the auras of clients.\n\n\"Entity\" is an all-purpose term denoting a nonphysical being or presence that gets attached to a human being and there acts as an energy parasite. Entities drain life force energy; they inspire perverse or inappropriate behavior; and they contribute to various types of ill health and disease, says Dr. Sagan. What do they look like? Patients have described them as blobs, clouds, amorphous black shapes, amoeba, octopus-like, spider-like, and other \"monstrous\" shapes.\n\nIn Dr. Sagan's view, for the most part, entities derive from human beings, specifically, from shattered fragments of the astral body after one has died. These float around like miniature persons, but with a singular, if not fixated, focus, usually upon a single desire for revenge, pleasure, or other strongly charged emotional condition. The entities get attached to humans in moments of stress, shock, or trauma; during electroshock therapy; during serious illness (involving fluid loss, hemorrhaging, or extreme energy depletion); under anesthesia or recreational drug use; _in utero_ and during birth; and in other situations when the aura opens up or is vulnerable and when its natural defense system collapses.\n\nIn fact, Dr. Sagan advises having a \"systematic entity checkup\" with a qualified practitioner after undergoing high-risk situations, such as abortions, miscarriage, delivery, or the death of a close relative. Each of these is a condition in which one's natural defense system is vulnerable to entity penetration.\n\nEntities are experts at camouflage, and are intent on not being detected by their hosts. Their goal is to decrease the person's mental acuity, to make the awareness blurry and confused, so as to enable them to elude detection. The last thing an entity wants is for the host to wake up to the fact they are being \"tricked and manipulated by something foreign,\" says Dr. Sagan.\n\nEntities in a human's energy field act like toxic substances, and are capable of producing numerous health problems, such as, for example, depression. Dr. Sagan relates a case in which a man, age twenty-six, a former heroin addict, was heavily depressed. During treatment (in which the patient is put into a kind of light trance enabling one to see the intruders in their energy body), he saw a black cloud around his heart. The primary emotion associated with this black cloud was fear of death. He understood it had entered his auric space when he began using heroin at age twelve, and been reawakened by a recent car accident.\n\nThe entity's purpose was to make the man suffer, to harm him, to make him take drugs. \"It lives in people's minds like a headache,\" he told Dr. Sagan. \"If you need a headache, it will come to you and give you one.\" The man further understood that the entity was creating his depression. \"It secretes the depression like a dark cloud into my heart.\" Removal of the entity produced complete relief from depression in two days, says Dr. Sagan.\n\nHere is another eye-opening case. In this one, a dead step-father returned to haunt his stepdaughter's womb and produce a fibroid, a noncancerous but medically dangerous fibrous growth in the uterus. In this woman, age forty-two, her fibroid had grown very large\u2014uncomfortably so\u2014and was producing increased menstrual bleeding. The stepfather had raped the woman when she was ten; when he died by suicide, the woman (as a girl) was unable to process her emotions and went through a \"time of madness\"; his despair and rage revisited the woman at this time of vulnerability and lodged in her uterus like a fetus. The fibroid started to shrink once the stepfather's presence was removed from it, says Dr. Sagan.\n\nHere is a third case, this time involving entities and migraines. A woman, age forty-eight, had excruciating migraines that would take eight days to subside. She discovered that dug in around her skull was \"a big insect,\" like a crayfish, with legs dug in around her occiput and eyebrows. The insect had a tube inserted into the woman's head and it was feeding off her brain's energy and her anger like a tick, she reported. Its purpose was to confuse her, to keep her from thinking clearly. \"The migraine comes out of the tick like a poison,\" she said. It came out of its belly and was injected under her skin, producing terrible pain. Once Dr. Sagan removed the astral entity, the migraines stopped immediately.\n\nBased on his clinical experience, Dr. Sagan concludes that \"the work on entities has proved a remarkably quick and efficient way of improving the health and well-being of a number of clients.\" He adds that when you discover and clear the entity behind\u2014that is to say, responsible for\u2014a physical illness, the results can be \"spectacular.\" The earlier you can do this, the better, before the body has translated a functional disorder, produced by an entity, into an entrenched, crystallized disease.\n\nEntities do not necessarily produce all illnesses, and sometimes they stop short at energy drainage or psychological interference, such as producing emotional outbursts, sugar cravings, and alcohol or drug abuse. Their preferred \"food\" is strong human emotion, especially of a negative kind, such as anger, dismay, frustration, emotional pain, melancholy, sadness, or depression, as well as sensual enjoyment, says Dr. Sagan. The most common bodily locations for entities are the large intestine (especially the left iliac region), the vagina, uterus, and ovaries, mainly because, from their viewpoint, these are commodious, cavelike environments, suitable for staying resident and hidden.\n\nMany\u2014perhaps most\u2014people still regard the concept of entities in the aura as something close to preposterous, yet Dr. Sagan reports that \"hundreds\" of clients who had never even heard of the possibility, much less dismissed it as being impossible, had, during a heightened state of perception, become aware of a foreign, parasitic presence attached to their energy body.\n\nIn most cases, a person doesn't have just one entity attached, but usually many, states William J. Baldwin, D.D.S., Ph.D., one of the country's preeminent educators and practitioners in the field of what he calls spirit releasement therapy. Dr. Baldwin is the author of _Spirit Releasement Therapy: A Technique Manual_ , an accomplished remover of attached entities, and the director of the Center for Human Relations in Enterprise, Florida.\n\n\"A living person can have dozens, even hundreds, of attached spirits as they occupy no physical space,\" states Dr. Baldwin. Parts of the body that have a physical weakness tend to attract parasitic energy forms, especially discarnate but earth-bound human spirits; the physical weakness matches an ailment or disability they had while alive. \"The discarnate entity retains the psychic energy pattern of its own ailments following death and can produce in the host any mental aberration or emotional disturbance and any symptom of physical illness,\" explains Dr. Baldwin.\n\nSimilarly, any symptom, both mental and physical, or strong emotion (especially repressed strong feelings such as unexpressed grief or rage) can \"act like a magnet\" and attract a discarnate being or other energy form with similar emotions, conditions, needs, or feelings. He calls this situation in which a foreign energy is present in the aura and exerting a powerful but unsuspected influence on the host \"spirit possession syndrome.\"\n\nFor those initially skeptical of the possibility that essentially invisible and inimical energy entities reside in their auras, Dr. Baldwin suggests considering the following list of symptoms, odd sensations, and inexplicable feelings typically reported by people who later discovered such entities present around them: feelings of being blocked or sabotaged, especially when things seem favorable; hearing intrusive thoughts and voices; the sense that some of your behaviors or statements are inconsistent with your sense of yourself; recurrent dreams or nightmares; addictions; sexual or relationship problems; feelings of abandonment or of not being alone in your body; the sense of being followed or stalked; irrational fears, anger, sadness, guilt, or other negative emotional states; chronic unhappiness as if over eveiything and anything. Any, some, or all of these symptoms can indicate an attached negative\u2014that is to say, toxic\u2014entity in your energy space, says Dr. Baldwin.\n\n  **_The Healthy Living Space_ Info Tip**\n\nDr. Modi specializes in spirit releasement work using hypnotherapy in her clinical practice.\n\nTo contact:\n\nShakuntala Modi, M.D.,\n\n1025 Main Street,\n\n416 Holly Building,\n\nWheeling, WV 26003;\n\ntel: 304-233-7246.\n\nSamuel Sagan, M.D., offers courses in training sensitivity to perceive foreign energies in the aura and techniques for removing them.\n\nClairvision, P.O. Box 33,\n\nRoseville, 2069 Australia;\n\ntel: 61-2-9888-1999;\n\ne-mail: info@c1airvision.org;\n\nwebsite: www.clairvlslon.org.\n\nWilliam J. Baldwin, D.D.D., Ph.D.,\n\nCenter for Human Relations,\n\nP.O. Box 4061,\n\nEnterprise, FL 32725;\n\ne-mail: doctorbill@aol.com;\n\nwebsite: www.spiritreleasement.org.\n\nThere are also many empirical signs that are suggestive of entity attachment, says Dr. Baldwin. All it takes is a little extra measure of self-awareness and a comparison of present and earlier mental and emotional states, a kind of identity baseline, if you will. Dr. Baldwin explains that \"a newly formed spirit attachment\" can generate any of these symptoms: the sudden onset of drug dependency; speech or behavior that is out of character in terms of affect or content; unfamiliar, uncharacteristic reactions to otherwise familiar situations; bodily movements, especially repetitive ones, that seem out of your control; a sense of having lost one's normal sense of oneself; conspicuous but inexplicable personality changes; sudden change in beliefs, dietary preferences, or taste in clothing, or even subtle changes in facial features.\n\n## HEALTHY LIVING SPACE DETOXIFIER #37 Request a Cleansing from Above\n\nHow does Dr. Modi remove these harmful energies? As with physical detoxification, a deep cleansing is involved, but this one has a twist. Here again, most of us need to make a leap of faith, but if we accept one end of the subtle world of invisible energies, namely, the role of negative auric entities, we may as well accept the other, positive pole; cleansers from above, so to speak.\n\n\"I routinely ask the angels to cleanse and heal the person by scrubbing every part and every organ of the patient's body, removing all the negative entities, energies, and devices, and any leftover residue,\" states Dr. Modi unabashedly. Next, Dr. Modi asks the angelic helpers to fill the person with light and to shield the body with light and \"spiritual mirrors and rays of blinding hot white Light.\" Dr. Modi also advises clients to visualize a bright violet flame burning around their bodies, incinerating impurities and infusing the energy field with a strong protective quality.\n\nNo special ability is required to request this kind of otherworldly cleansing, only the intent to be rid of the foreign energies and a belief that seemingly invisible help can accomplish this important task. Merely _asking_ for assistance from above is a powerful technique in starting to clear the aura of pollutants. The power of asking is an open secret, and a paradoxical one. It works, but you have to believe in something you probably can't see. You will not necessarily be aware of anything during the subtle cleansing, but almost surely you will feel some easing of your symptoms shortly afterwards, probably within a day.\n\n_Directions_ : While sitting in a quiet place, request a cleansing from above. If you are comfortable with the concept of invisible benign helpers, then ask that beings of light, friendly spirits, angels, your higher self, inner self, guardian angel, a saint, or God\u2014whatever form the numinous takes in your belief system\u2014to remove any negative entities, energies, and devices from your aura. Ask that they then put a protective bubble or shield of spiritual light around your aura and reflective mirrors on the outside to keep away further contaminating energies. Visualize a column of brilliant pure white light coming from the highest divine source above you and entering your body through your head, filling all of your body down to your toes, cleansing every aspect of your physical body and surrounding aura. Finally, ask that you be protected from this point forward from any further inimical influence.\n\n### The Healthy Living Space Expert Interview: Rev. Leon S. LeGant, Psychic Healer\n\nTo an extent, once we grant the possibility that the energy field surrounding our physical body can contain injurious energies and presences, it is possible to take steps on our own to rid ourselves of these intruders. At the same time, it is a good idea to arrange a consultation with a qualified psychic healer to do a thorough job of cleansing your aura. The combination of self-help steps and professional treatment is generally advisable. After all, as we discussed in earlier chapters, there is much we can do to effect a cleansing of our digestive system, including liver, kidney, and intestinal detoxification, but a few acupuncture treatments, for example, can help a great deal in balancing the overall energy flow within the body and through these organs.\n\nSince 1994, Rev. Leon S. Legant, a self-described \"clairvoyant reader and spiritual healer\" practicing in San Rafael, California, has removed unsuspected spiritual toxins from the energy fields of some two thousand clients. Working with this world of subtle influence and effect every day for years has given LeGant an \"insider's\" view of the mechanics of spiritual toxicity.\n\nThe energy field is \"basically a storage unit,\" LeGant explains, containing information packed in layers on \"survival, our emotions, how the body runs its energy, how it drains it, our affinities, self-image, and much more.\" Foreign energy affects and tends to lodge in one of the layers of the aura, depending on its nature and the corresponding location it chooses in the aura. If it lodges in the auric storage area for survival issues, the foreign energy will stimulate thoughts, feelings, fears, even outer episodes to do with survival, explains LeGant. How does one know if there is something foreign in the aura? You will first notice the problem it is causing, says LeGant; you will not necessarily (unless you are trained in this kind of sensitivity) notice the energy itself.\n\n\"The first thing for people to realize as a step in their healing is to acknowledge that many of their problems are not produced by their own energy. What they are feeling isn't really who they are in many cases; there is something foreign causing these states of mind or feelings about themselves.\" Once the foreign energies are removed, you can then sense the difference in yourself made by their absence, LeGant adds. You will feel the change, and looking back to how you felt earlier, see the effect on you of the foreign energy and the degree to which you thought it was yourself.\n\nOur auras, like the skin on our physical bodies, are subject to bruises and penetrations every day, says LeGant. The difference is that we usually are unaware of when our energy field has been breached, leaving us vulnerable to harmful energy influences. \"When someone's aura is ripped or torn open, this is called a whack. A whack happens when someone or an experience contradicts a person and injures the aura.\" Harsh, insulting, demeaning words, or any kind of strong verbal abuse, can \"whack\" the aura, especially when they focus on the person's weak, sore, or painful spots. You may get over the words, but their effect still resonates in your aura; in a sense, the words are still lodged there.\n\nLeGant explains that \"sometimes the aura looks like someone fired buckshot at it, or there is a giant rip starting at the aura's outermost edge. Once there is a whack, the person's energy begins to drain out through that rip or opening in the aura, instead of flowing evenly through the aura.\" Once the aura's integrity has been violated, it increases our vulnerability to further pollution by inimical energies, says LeGant. The situation is similar to when your immune system is weakened by illness or an unbalanced lifestyle; this leaves your body open to the effect of opportunistic germs in your environment. In this case, the aura is your energetic immune system, and, once weakened by \"whacks,\" it can allow unwholesome energies to enter and attach themselves to our vulnerable spots.\n\n**_Those painful areas or points of vulnerability exist in the aura like pictures or condensed images of experiences, says LeGant. He calls these \"core pictures;\" and he says they are central to an understanding of how spiritual toxins work_**.\n\nThose painful areas or points of vulnerability exist in the aura like pictures or condensed images of experiences, says LeGant. He calls these \"core pictures,\" and he says they are central to an understanding of how spiritual toxins work. \"Because of the intense stimulation of an event, the body believes it is having an important experience, causing it to store all that information from the experience within the brain. From a clairvoyant perspective, that stored information looks like a mass of energy about the size of a softball, located in the limbic system but 'bleeding' into the brain's cortex.\"\n\nTypically, a person has thousands of core pictures, says LeGant, accumulated (and unprocessed) from this and other lifetimes. Each is a condensed, crystallized picture or residue of an emotional intensity, be it trauma, injury, insult, abuse, or humiliation. When something happens in our outer world that is similar to the content of a core picture, the core picture gets lit up, emphasized, and reactivated, and we see our reality through it. If it involved illness or dysfunction, these conditions too start to manifest. In essence, each core picture is an \"energetic block\" that acts like a toxin and detracts from our mental clarity, emotional well-being, and physical health.\n\n\"When a core picture becomes stuck in the brain, its influence is extreme. It wraps around our flame of consciousness. Our essence shines through the core picture and projects that past time experience into our mental body [our thoughts and mental processes construed as if they are a body on their own] and thus into our reality.\" For example, say a young girl has an abusive, alcoholic father. The girl stores her memories of her father's abusiveness in a \"solidified\" form as a core picture. This core picture then becomes a filter throughout her life for how she sees her reality and relationships with men.\n\n\"Now as your consciousness projects that core picture into your reality, you are in your fifth relationship with an alcoholic and abusive man. With each new relationship, you attracted someone like your father. It only takes one core picture to produce such devastating effects.\" But it gets still more complex for two reasons. First, most of our core pictures are not derived from our current life, which means it is unlikely we would ever suspect they are in play; and second, it is the core pictures that attract the spiritual toxins described above by Dr. Modi. \"Core pictures are like cheese on Ritz crackers for unwholesome energies and beings,\" says LeGant.\n\nCore pictures are \"anchor points\" allowing \"demons,\" discarnate earthbound entities, and other injurious energies to attach to the person's aura. LeGant uses the general term \"spirit guides\" to describe the range of beings that can occupy (or infest) our auric space\u2014\"beings without bodies, sources of influence.\" A spirit guide in this model is not necessarily a beneficial presence; in fact, it can be literally demonic.\n\nThe larger point is that spirit guides, whatever they are, claim \"seniority\" over the host's thoughts, feelings, actions, health\u2014life. They are parasites that have claimed a person's body as their own. \"Most people have many dozens, if not hundreds, of spirit guides in their auric space, and only a very small percentage of people have seniority to their spirit guides.\"\n\nThe combination of core pictures and attached entities often explains why people are not able to clear out the bad energy by themselves, notes LeGant. \"I first ask why the spiritual consciousness of the client has not or is not healing those blocks on their own. I believe everyone knows how to heal himself. But if they are not doing so, that usually means their consciousness is sitting in a core picture that is stuck in the system.\"\n\nOne of the central points in this book is that toxicity affects every aspect of our being, from the physical body to the energy field surrounding it. The combination of core picture and attached entity is similar to a situation observed on the molecular level and known as circulating immune complexes (CIC), a key issue in physiological toxicity (see Chapter 2, p. 74).\n\nTo review: foreign proteins \"leak\" into the blood system, usually from the intestines. The immune system tags these proteins as foreign invaders (antigens) and surrounds them with immune cells (antibodies); the antigen-antibody complex is known as a CIC.\n\nIn a healthy person, CICs are neutralized, but in someone with a compromised immune system, they tend to accumulate in the blood and burden the detoxification pathways or produce allergies. If too many CICs accumulate, the kidneys are usually unable to excrete them through the urine, and they get stored in the body's soft tissues where they produce inflammation and further stress on the immune system. The overload can lead to a variety of chronic health conditions.\n\nMeanwhile, the entity complexes, left unresolved in the auric space, produce numerous symptoms, conditions, and illnesses, just as a glut of free radicals and CICs do in the physical body. A \"spirit guide\" may lodge itself in an organ and start depriving the person of the full use of this organ, explains LeGant. Say it is the lungs, and the person is a smoker. The tobacco smoke is the entry point for a type of being. \"Once the beings get into the lungs, they start to lodge themselves in the cells and there becomes less room for the person's own energy in that organ and more of this foreign energy.\"\n\nLeGant explains that sometimes \"just the presence of that foreign energy in an organ for so long and not having your own healing energies running through the tissues can cause a health problem, even cancer.\" He further states that sometimes foreign energy beings can activate cancer genes and start a cancer process, or attract viruses to the host to produce serious illnesses, or provoke the host into taking self-destructive actions, such as neglecting their health or exposing themselves to infection.\n\n\"However, as a rule of thumb, the spiritual toxins come first and that attracts or produces the physical problems,\" says LeGant. \"You can clairvoyantly 'read' someone who has a cancer being in their body _before_ they have cancer, and you can remove that cancer being and enable the person to avoid developing cancer.\" This can be taken one step further, he adds. If you remove a fair amount of the spiritual toxins from a person's aura, that person tends to be less affected by or can better handle physical toxins they come across. \"You can reverse whatever physical manifestations are present if you remove enough of the foreign energy and are very thorough about this. A cancer can go into remission, a disease can go away.\"\n\nThe practical conclusion from this information is that it is advisable to undertake a physical detoxification program in conjunction with work that removes spiritual toxins as well, advises LeGant. \"It is prudent to deal with _both_ poles of toxicity, the physical and the spiritual sides.\" If you do a physical cleanse program and overlook possible auric toxicity, you have not removed the seeds of the physical toxicity. At the same time, the body always needs a helping hand when it comes to handling an extra load of detoxification, so if you remove spiritual toxins, be sure to undertake a complementary physical detoxification program because the auric shifts will provoke corresponding physiological ones.\n\nTypically, LeGant, using psychic methods, removes a layer of four or five core pictures and numerous attached entities, then watches his client's \"subconscious or spiritual self release the remaining anchor points throughout their body, aura, and energy centers. If they are not healing themselves at that point, I will remove another layer of core pictures and then another until they are successfully able to clear themselves.\"\n\nDuring a typical session, which may last from 30 to 120 minutes, LeGant \"erases\" core pictures and \"removes\" the attached beings according to what is most prominent in the client's aura. Current life experiences tend to \"light up\" specific core pictures, making them more noticeable and the client more uncomfortable under their influence, says LeGant, but it is the foreground core pictures to which he devotes his attention in a clearing session. If you schedule regular clearings at the rate of once monthly, the subconscious starts cooperating, shifting material towards the forefront of your awareness such that in the few days before a session, new core pictures and their attached beings become illuminated in the aura\u2014\"lit up,\" LeGant calls it. It is as if they start to queue for their forthcoming release.\n\nHow do you know if you have spiritual toxins in your aura? LeGant's estimation is that most people have them, but usually go about their lives oblivious of their presence, attributing their effects to other causes. One rule of thumb is that if you keep coming up against a brick wall in your life\u2014an attitude that won't shift, a destructive or self-limiting self-definition, an obstacle, opponent, or seeming enemy that won't go away\u2014then it is likely you have a foreign energy of some kind at work, invisible to you, in your space.\n\nAnother way to think of this is in a way parallel to how we view the intestines. Many people with regular eliminations would consider it ridiculous to suggest that nonetheless they might be chronically constipated and benefit from an intestinal cleansing. Yet many are, and many would. Similarly, it would surprise many Westerners to learn that they have intestinal dysbiosis and possibly some parasites, yet the trend in laboratory testing indicates that many do (see Chapter 3). Pollution in the intestines, pollution in the aura\u2014in essence they are the same.\n\n  **_The Healthy Living Space_ Info Tip**\n\nLeon S. LeGant does most of his spirit releasement and psychic healing work over the telephone, and works with clients across the United States. To contact:\n\nRev. Leon S. LeGant,\n\n27 Aquinas Drive,\n\nSan Rafael, CA 94901;\n\ntel: 888-459-4585.\n\nWe tend to become accustomed to our dysfunctions, and mask them or account for them in ways that enable us to keep living with a minimum of disruption. While this may seem pragmatic, it's not curative; we're still stuck with these invisible yet potent energies in our space that can often make our life a lot harder than it needs to be. The best confirmation that you need a psychic clearing of your aura comes afterwards, when you look back and wonder how you managed to function before all this negative energy was removed, just as, in the weeks after a colon cleanse, you wonder how your intestines ever functioned properly\u2014because now you can _feel_ the difference in yourself.\n\n## HEALTHY LIVING SPACE DETOXIFIER #38 Do a 30-Minute Spirit-Detoxifying Bath\n\nWhile you are likely to feel immediate relief from some mental or physical symptoms during or immediately after a session (LeGant calls it a \"reading\"), in the following one to two weeks, the physical body starts to deal with the ramifications of this major energy shift in its auric field. Also, if LeGant removes five beings, your auric space, now freed from the magnetizing core pictures, can release perhaps thirty more beings on its own, and possibly shatter a few pictures as well.\n\nThe way you experience this is similar to what we defined earlier in the book as the Herxheimer effect in terms of physical detoxification. In the physical case, the body temporarily feels more toxic as toxins are mobilized, neutralized, and released from the body. Similarly, in the auric case, your state of mind, emotions, and thoughts, and to an extent, your body as well, feel more toxic for a short while as the aura purges beings and energies and your physical body makes the corresponding adjustments.\n\nWhat experts in physical detoxification call the Herxheimer Effect, psychic healers like Leon LeGant call a \"growth period.\" Removing foreign energies, pictures, and beings from your auric space generates a profound shift in your mind-body constitution, says LeGant. In a sense, you have disturbed a long-term ecology, although it is an unbalanced ecology in which your body has learned to accommodate \"hostile demons, painful core pictures, and energy blocks\" in much the same way your liver and intestines learn to accommodate environmental toxins. \"As spirits, we can change our energy in the snap of a finger, but our bodies require a period of time to catch up and adjust to those changes.\"\n\nIn the weeks following a psychic clearing, you continue to heal yourself, bringing up and releasing physical, emotion, and mental pain, as well as more beings, says LeGant. \"As that energy releases, you may experience its effects, which include an edgy and uncomfortable sensation. You may feel that your life is falling apart, that chaos has broken loose in your mind, that you have lost all sense of your self, among other transient mind states.\" These pass, just as the Herxheimer reaction to detoxification subsides. Typically, the intense phase of the growth period lasts only one week. \"Once a person finishes releasing, they begin to recreate themselves and their environment. I have seen these kinds of transformations thousands of times; after removing demonic influences, a person has the freedom to change any and every element of their life they choose.\"\n\nIt is advisable to give your body some tangible support as it reorganizes itself now that it has been freed from the deleterious influence of negative energies. One practical way is to take specially formulated baths, and it is especially helpful to do this in the days _immediately_ following a psychic clearing, says LeGant. \"Epsom salts baths are very useful for clearing pain out of the body and for releasing beings, foreign energy, and stress,\" says LeGant.\n\n_Directions_ : You may have tried the Epsom salts bath following directions in Chapter 4 for the purposes of general stress relief. Now the focus is more refined\u2014to help your body release accumulated energetic toxins.\n\nFill a bathtub with very hot water, as hot as you are comfortable lying in. Add bubble bath and an aromatherapy essential oil to the water to provide bubbles, scent, and additional detoxifying agents. If you wish to relax deeply, use lavender or melissa essential oils; if you need to center, reintegrate, and reorganize yourself, use rosemary; if you need a cleansing, rejuvenating energy, use pine, juniper, or eucalyptus.\n\nHere is a tip about adding bubbles and oils: Measure the bubble bath into a plastic quart container; dribble in the aromatherapy oils, anywhere from five to thirty drops, understanding that more equals stronger. Then fill the container with hot water and dribble its contents into the stream of water coming out of the faucet. This improves the miscibility and dispersion of the bubble bath material and oils into the hot water.\n\nAdd two to four pounds of Epsom salts (magnesium sulfate) to the water, stirring well. As with the aromatherapy oils, more Epsom salts equals a deeper relaxation, but don't use more than four pounds per bath. Epsom salts typically come in a four-pound milk-carton-like package, costing between two and three dollars.\n\nAs additional support for your post-spirit-releasement detoxification bath, consider bringing a CD-player into the bathroom and playing your favorite music while you soak. It is advisable, however, to use a calming music, something that might evoke a meditative, relaxed state of mind, such as Chopin's _Etudes_ , or Steven Halpern's _Spectrum Suite_ (see Chapter 4 for more on music therapy for stress reduction). Also supportive of the detox mood is dim lighting, such as candlelight or indirect lighting from another room, while the bathroom itself is dark.\n\nSoak for at least twenty minutes, but thirty or longer is better. You may find yourself sighing, moaning, groaning, whimpering, shouting, or emitting other surprising vocal noises while you soak. You may also experience transient emotions, such as grief, anger, sadness, or despair\u2014any of the gamut of feelings. Do not be alarmed, but be amused if you wish: your body is doing its releasing work. In a sense, you are only becoming aware of what was already there in your emotional field.\n\nWhen you are finished soaking and climb out of the tub, do not towel dry. Lie on some dry towels on your bed and air dry over the next thirty to forty-five minutes. Your skin is still covered with a film of Epsom salts and aromatherapy oils, and these will continue working while you air dry. Often this phase of the detoxification bath is the most relaxing; don't be surprised if you fall asleep for a short time.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor a source of Dead Sea mineral salts (unprocessed, in their natural form):\n\nMasada Marketing Corp.,\n\nP.O. Box 4118,\n\nChatsworth, CA 91313-4118;\n\ntel: 800-368-8811 or 818-717-8300;\n\nfax: 818-717-8400;\n\nwebsite:\n\nwww.secondwave.net\/masadamarketing;\n\ne-mail: sam@masada-spa.com.\n\nAlso: Ashtar Natural Dead Sea Products,\n\nAmig Group,\n\nP.O. Box 53, A.l.E.\n\nAmman, Jordan 11512;\n\nfax: 962-6-5534562;\n\nwebsite: www.fjcc.com\/amig.\n\n_Variation_ : Try substituting Dead Sea mineral salts for the Epsom salts for a twenty- to thirty-minute growth-period soak. The mineral salts concentration of Dead Sea bath salts is 27-33% higher than the average ocean concentration of 3.4%. The specific salts include magnesium, potassium, sodium, and calcium chlorides. For presumably thousands of years, the Dead Sea has been the destination for millions seeking the therapeutic benefit of its mineral salts for muscle pain, muscle tension, arthritis, and other conditions. You can also mix Dead Sea bath salts with aromatherapy oils (such as citrus, eucalyptus, lavender, rose) for extra detoxifying benefits.\n\nHere is a general recommendation: add a full cup (250 g, or about 1\/2 pound) of Dead Sea bath salts to a tub of hot water; stay in the water fifteen to twenty minutes, but do not use any soap, then rinse off and rest in or on your bed for thirty minutes.\n\nAccording to one Dead Sea salts processor, the high mineral content of the salts \"stimulates blood and lymph circulation, an effect that helps drain out the trapped fluid in the joints, which causes arthritic pain, thus improving joint movement.\" Further, the salts \"exfoliate and revitalize the skin, eliminating toxins\" and they help relieve aches and pains, relax the muscles, and relieve neck and back tension and stiffness.\n\n## HEALTHY LIVING SPACE DETOXIFIER #39: Shake Yourself Free of Toxic Energies\n\nThink of a dog just out of the rain or from swimming in a lake: it shakes itself vigorously, sending water droplets every-where, perhaps on you. It's an effective drying technique, at least for the dog. In the same way, vigorously shaking your whole body is an effective way to scatter negative, heavy energies that have adhered to it, suggests psychic protection expert William Bloom. A leader of psychic protection workshops in Europe, author of several books on the subject, and a man especially \"sensitive to atmospheres\" since infancy, Bloom has developed a variety of practical techniques for clearing the energy field of obtrusive, foreign energies.\n\n\"Moving and shaking your body will free up your own frozen energy, as well as moving any that belongs to someone or something else,\" he explains. Energy, whether it comes from our own emotional reactions or the energy extrusions of others around us, can get stuck in the aura. It can get \"blocked or glued into you,\" Bloom says. When the energy doesn't move, it can start to stagnate in your energy field and eventually create symptoms, such as pains and aches, heaviness or weariness, distress, and illness.\n\n\"The quickest way to unload it is to move your body.\" Bloom suggests shaking, stretching, skipping, bouncing\u2014moving around vigorously, doing any kind of strong physical movement that gets your entire body moving and shaking. Make sure you move everything: elbows, hands, shoulders, head, torso, hips, legs, and feet. Wobble your body, even your cheeks and tongue. One simple way to produce the desired body-wide shaking is to use a rebounder or mini-trampoline, recommended in Chapter 6 with reference to stimulating the lymphatic system.\n\nIt might help to pretend you are that wet dog scattering water droplets in all directions. Think of the water droplets as \"molecules\" of toxic energy. This moving will loosen and release the stuck energy, Bloom states. Feel free to make noises, too, such as screams, gasps, and moans.\n\nOther activities of similar effectiveness include putting your hands under running water, taking a shower, standing in a windy place or before a fan, and changing your clothes and giving your worn ones a good shake and thumping to remove the energetic traces of the contaminants in question.\n\n## HEALTHY LIVING SPACE DETOXIFIER #40 Scrub Your Aura Clean with a Golden Sponge\n\nWhile it is hard to duplicate the masterful auric cleanse you get from a professional psychic healer, there are several practical steps you can take on your own to do a fair amount of auric scrubbing, says LeGant. The following is a self-help technique for removing energy imprints, influences, detritus, and residual after-effects of arguments, encounters, or excursions into unpleasant environments such as we pick up every day.\n\n_Directions_ : This is an exercise developed by LeGant, which you can do while soaking in an Epsom salts bath (Healthy Living Space Detoxifier #38), as you lie in bed waiting to fall asleep, or any time during the day when you feel aurically toxic. Visualize your aura as a big golden bubble extending a few feet out from your skin in all directions. Give it an electric blue fringe to help see where its edges are. Tell your aura, which might be extended further out, to retract to within two feet of your body.\n\nNext, visualize a golden sponge and instruct the sponge to move around inside your aura and energy centers (chakras) to collect (sponge up) all the \"pictures and energies that allow whacks in the aura to exist,\" says LeGant. Tell the golden sponge to absorb everything of this nature. If, for example, you have been obsessing about a topic\u2014an insult, abuse, argument, money, sex, an embarrassment, an unexpressed grievance\u2014specifically direct the sponge to absorb all the energy associated with it.\n\nWhen you feel the sponge is full and has absorbed all it can, put it in an imaginary container, seal the lid, throw the container away from you, and explode a bomb under it. Sometimes you may need to do several sponges in succession to collect all the negative energy. Be sure to blow up every sponge you use for this purpose. \"If you feel you want to keep clearing your aura, keep creating sponges, bringing them into your aura, sponging things up until you feel clear,\" says LeGant.\n\nIt's a good idea to consider the golden sponge technique a regular part of your detoxification routines, he adds. \"People get whacked aurically very easily and frequently during the course of a day or week, and the older whacks get reactivated, too, so you need to be constantly clearing out the effect of these whacks.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #41 Turn Your Body into Transparent Glass\n\nThis exercise, which I have modified from instructions provided by Leon LeGant, is appropriately performed while in the tub or a flotation tank. For some people, water acts as a psychic enhancer, making it easier to visualize the steps in the exercise. The goal of this exercise is to make your body neutral and transparent, thereby highlighting all the foreign, undesirable energy, enabling you to remove it.\n\n_Directions_ : Immerse yourself in a bathtub full of hot water with all the additions specified in Healthy Living Space Detoxifier #38. Wait ten to fifteen minutes in the water before starting this exercise. When you feel relaxed and focused, begin. Starting at your feet, visualize that your body is turning to transparent, colorless glass. In other words, first see your feet and ankles as clear glass. You will probably have some obstacles or opposition to this, but this is the point of the exercise\u2014to highlight the negative energies so you can remove them. As you try to see your body as transparent glass, all the foreign energies will become apparent; that's when you can remove them.\n\nYou may come up with a better way to do this, but try \"pushing\" the dark energies, when you see them, up through your body to drain out through the back of your neck. Why there? Because the back of the neck, a recognized energy center (chakra) is known as the psychic gate; it's like a swinging door for the entry and departure of psychic energies.\n\nPretend that you can open a valve in the back of your neck, and as you push the energies up from the feet, they start draining out from your neck. Also, visualize a large, highly absorbent sponge situated on the outside of your neck, capable of absorbing all the toxic energy that will be draining out of your neck. You might focus on the neck drain, seeing more and more \"liquid\" or dark energy flowing out, thereby enabling dark liquids lower down in your body to move towards the drain. Or you may need to more aggressively push the dark energies to the drain, somewhat like using a squeegee to move water across a surface. Sometimes, when I perceive there is a great deal of energy rubbish present in my body, I visualize using a shovel or even a tiny bulldozer to push it all to the neck drain.\n\nAs the energy rubbish moves toward the drain, see the parts of the body from which this energy has just departed as being clear as glass. By the time everything has drained out of your neck (and don't forget to drain your head), you should be able to see your whole body as glassine, if only for a moment. You may have to repeat this exercise two or three times before you can feel confident you have become transparent as glass. When you have completed one pass through the body, blow up the sponge; if you are repeating the exercise, install a new sponge at the neck drain, and destroy it once you have drained the body. Don't be alarmed if during the drainage, you find yourself twitching, shaking, moaning, groaning, or even shouting. It is your body registering the energy transaction, and is a transient, though sometimes to others in the house a somewhat amusing, occurrence.\n\n**_Your energy body, like your intestines once they get engaged in a detoxification program, has its own peristalsis, its own rhythm of pushing out foreign toxic energies. The more you can keep up with this natural energy rhythm, the better you will feel because you are always current with the latest batch of energy rubbish awaiting removal from your auric space._**\n\nIn the week following a psychic clearing, it is highly beneficial to do this exercise every day, even twice daily. Your energy body, like your intestines once they get engaged in a detoxification program, has its own peristalsis, its own rhythm of pushing out foreign toxic energies. The more you can keep up with this natural energy rhythm, the better you will feel because you are always current with the latest batch of energy rubbish awaiting removal from your auric space.\n\nYou may experience the energy rubbish in many different ways. For example, I have seen it as thick green sludge, as dark blackish tar, as caked, dried, sandy-colored mud, and as a sickly pale green vapor. Sometimes you might experience a particular accumulation in one body area, such as the groin, solar plexus, or neck, although it can be anywhere in the body. You may find that the flow of energy rubbish up from the feet to the neck gets slowed down, perhaps halted, at one spot. Don't fight it. Visualize another sponge and use this to absorb the toxic sludge in this particular area; then blow up the sponge and resume your full body drain. Once the localized energy obstacle has been removed, the overall drainage may proceed rapidly and without obstruction.\n\nThe experience of purging your body and space of the foreign energies by doing the glass body exercise may feel intense for a few moments; it certainly requires some concentration. When you have completed your glass body exercise, get out of the tub, lie down on a towel on your bed for about twenty to thirty minutes, and don't do anything else. This gives your mind and body time to relax and reorganize after the intensity of the toxic cleanup.\n\n## HEALTHY LIVING SPACE DETOXIFIER #42 Cutting the Cords and Ties That Bind\u2014Psychic Plumbing\n\nAccording to many psychic practitioners, one of the ways we receive a continuous stream of foreign and often toxic energy from other people, especially those emotionally close to us, is by way of psychic cords. These psychic cords, or energy connections, have also been described as \"hooks, nets, cables, tree trunks, and many other variations linking us inextricably to the other person,\" explains Judy Hall, an expert in psychic protection and energy work.\n\nEnergy cords are connections to people we have been close to (parents, siblings, relatives), are presently close to (spouses, partners, children, friends), or with whom we have some type of energetically charged relationship (employees, coworkers, neighbors, casual acquaintances, even strangers). Energy links are formed, connecting us to the person; perhaps we established the cord, perhaps they did\u2014either way, it functions as a kind of umbilical cord through which thoughts, feelings, attitudes, and other charged emotional states or conditions of negativity can continuously flow. The effect, potentially, is to keep us constantly toxic from this source, receiving energy input that is not always in our best interests and may in fact be psychically injurious.\n\nFor example, you can count on having energy cords connecting you to your parents, regardless of their status, dead or alive, close or estranged. It is not always this way, but it is often the case that energy cords to your parents can exist all over your body; you may in fact discover dozens, even hundreds of cables from your mother and father hooked into your body. They may appear to you as an inch wide or five or six inches in diameter. You may detect a large trunk line surrounded by a dozen smaller input cables. They tend to be organized around the body's major and minor chakras, arrayed along the spinal column and all over the head.\n\nPsychically attuned practitioners report that they can hear, sense, feel, or see emotional content flowing through these cables from their parents and other people. In some cases, these cables drain away our vital energy, as if a powerful suction device at the other end of the cable is steadily sucking our _qi_ out of us. Sometimes people unconsciously establish cords into high-energy individuals to siphon off\u2014some call it vampirize\u2014energy, leaving us unaccountably depleted until we figure out what has happened and remove the drainage cord.\n\nIn infancy and early childhood, cables from our parents were useful, even necessary for our survival; but as adults, they can be toxic artifacts, riveting us to old, inappropriate, or regressive modes of being derived from our parents and our earliest relationships with them. You may feel you cannot sever your dependency on your mother or father, or that you keep hearing or feeling their personalities or attitudes; in both cases, the presence is inappropriate, even oppressive at times.\n\nCutting a tie or energy cord to a loved one does not mean you are severing all links with that person; it just means you are unencumbering yourself of inappropriate old energy links. Your relationship may actually improve because it has been clarified and updated\u2014that is, brought into the present, out of the past where it had stagnated due to the energy cords. \"If you feel you owe someone something, you probably need more than most to cut the ties,\" advises Hall. \"Guilt is disempowering, and highly abusive on both sides.\"\n\nThe cords\u2014whether they carry guilt, parental \"shoulds\" and \"don'ts,\" or other forms of family invalidation\u2014provide the energy antecedents for unhealthy emotional states (as discussed in Chapter 8) and unbalanced physiological states (as discussed in chapter 5). Thus removing the cords can be a significant part of your detoxification program.\n\nTo cut energy cords, it first is necessary to see or at least sense them. Surprisingly, one doesn't have to be too psychic to accomplish this; the intent to discern these cords is often a sufficiently strong factor in making the existence of such cords apparent. One way that might be helpful is to look for the cords while you are having your detoxification bath. Name the person in question. You could say, for example: \"I wish to become aware of energy cords from my mother.\" Or whichever individual you suspect of being corded into. It is quite likely that your mind will project some form of image in response to this request; you may see cords hooked into a certain part of your body, or you may sense some discomfort, tingling, heaviness, or other sensation in a particular area.\n\n**_The cords\u2014whether they carry guilt, parental \"shoulds\" and \"don'ts,\" or other forms of family invalidation\u2014provide the energy antecedents for unhealthy emotional states and unbalanced physiological states. Thus removing the cords can be a significant part of your detoxification program._**\n\nOnce you see the cords, remove them from yourself and from the person to whom they are attached, Judy Hall says. How do you remove the cords? Hall recommends that you visualize cutting them with a pair of golden scissors, or slicing them with a sharp hoop of light. She also says it is effective to cut them with a visualized mental laser of light or to imagine yourself in a \"mental bonfire\" in which all energy ties named will be burned up, while leaving your own energies intact. You may wish to mentally tell the person to whom you are corded what you are doing and why. When you are finished with the exercise, pile the cords up like kindling for a bonfire and burn them so that no trace remains. You could also visualize tossing them into a swiftly flowing river. If you do not destroy the remains of the cords, Hall cautions, they can return to you or sprout and grow back again. She also advises sealing the part of the body from which the cords were removed with a healing light.\n\n\"In cutting the ties, the energy changes,\" Hall explains. Your life is likely to change for the better because cutting the ties that bind you to other people restores your autonomy and psychic energy. Hall recommends examining all your relationships to see if they are based on inappropriate cords. All relationships will benefit from cutting the energy cords, she notes. It makes good detoxification sense, she says, to \"regularly clear our ties.\"\n\nLeon LeGant also works with cord removals and offers a variant to Hall's approach. When you see the cords, unscrew them the way you open a tamper-proof bottle of pills, pushing down and turning; this frees the cords from your energy body. Then using the golden sponge image, instruct it to collect all the cords; then destroy the sponge by visualizing placing a bomb under it and blowing it up. Or put it in a large container and blow it up inside that. Somehow the psychic sponge handles the cord and collects its connections at the other end, says LeGant. It is advisable to work on one person at a time, allowing your body a few days to regroup to the new energy configuration, which is now free of a fair number of cords.\n\n## HEALTHY LIVING SPACE DETOXIFIER #43 Ground Yourself to the Center of the Earth\n\nPeople often talk about \"centering\" themselves, and although the concept is sound and helpful, often the practice used does not accomplish the task. One method that ensures real centering is a technique called the \"grounding cord,\" says LeGant. Not only does it provide deep centering\u2014rooting yourself to the imagined core of the planet itself\u2014but it acts as a kind of lightning rod, sending spiritual toxins straight through you (who remains unscathed) and into the Earth.\n\nYou can use this technique any time you find yourself in an unfriendly environment\u2014remember the toxic room we mentioned at the start of the chapter?\u2014or in a formal meeting, a charged situation among people, or every day as a regular part of your meditative or exercise routine.\n\n_Directions_ : Visualize a hollow energy tube extending from your waist to the center of the Earth. It might help to see yourself sitting on this tube as if it is a tree trunk (or a pillar) that comes out of the center of the planet. It is hollow like a chute, and you can throw things down it; it can be any color you choose, and it is about the width (plus a little) of your body. It starts around your waist and includes your groin (seat of the root chakra) and goes all the way down to the core of the Earth, in whatever way you can conceive this. Sometimes it helps to see the grounding cord anchored into a specific place, like a pylon in a rock face when climbing; you might for example see it as attached to a gold plate with your name on it on the subterranean rock.\n\nBy the way, don't worry about whether the center of the planet is molten lava or hollow, or that you don't feel you can even imagine what it might be. What is essential is that you purposefully connect yourself with the \"center\" of the planet and use this connection as a way of grounding foreign, hostile, or simply nonself energy, in the way a lightning rod functions on a house.\n\nForeign energy, which includes energy from those closest to you as well as from strangers and those who wish you (or everyone in sight) ill, now has a place to go upon hitting your aura. Without the grounding cord, hostile foreign energy lodges somewhere in your aura and works its way into your body; with the grounding cord, it can pass through you into the ground. You remain neutral to its influence. Generally, you will need to destroy your grounding cord every one to two weeks and generate a new one.\n\nOnce you become familiar with the workings of your grounding cord, you can extend its uses, says LeGant. \"You can allow your grounding cord to release energy out of your body and aura, along with all of the pain in your body or from a particular place that is experiencing pain.\" If you have a headache, or stomachache, or menstrual cramps, try establishing a minigrounding cord from that organ or body area to the center of the Earth and directing the cord to drain out the pain. Or perhaps you are feeling despair, apathy, or hopelessness; direct the cord to drain these negative states from your mind-body. Spend five to ten minutes focused on this draining.\n\nIf it helps your visualization of the grounding cord, imagine it has an on\/off switch, suggests LeGant. When you are ready to \"drain,\" flip the on switch; when you are finished, flip the off switch. Allow the natural gravity of the Earth to effortlessly suck out all the pain and negativity from your aura. You might view the pain as water circling a bathtub drain, flowing out of the tub (your body) down the pipe (grounding cord).\n\nOne final point on the grounding cord. Resistance is futile, says LeGant. If you come in contact with \"unfriendly\" energy or situations, resisting this undesired field of influence can be counterproductive, LeGant explains. It will actually draw the unwanted energy closer to you. \"If you start resisting it, it's going to stick right to you.\" Further, you have to learn to be aware of when you're in resistance. \"You have to learn to catch yourself doing it. The problem is if you go into resistance, you're actually going into fear. You're expecting it to attack or pollute you, and you are basically opening the door for that energy to slam into you and do exactly what you fear it will.\"\n\nThe best way to \"repel\" foreign, inimical energy is to go neutral to it; then you do not attract or repel it, but rather you act as if it didn't exist or that it had no ability to affect you. \"You can have your own conversation with it. 'I see you, but I'm neutral to you and I'm senior to you and you don't have to effect me.'\"\n\nThen you send the energy down the grounding cord, or if it is a situation, put a grounding cord under it\u2014the people, building, whatever the physical environment and its contents\u2014and allow the hostile energy to funnel down the cord to the center of the Earth. One of LeGant's colleagues once defused an armed bank robbery in progress by putting a grounding cord under the robbery principals; after a few minutes, they got flustered, confused, even frightened, and ran out of the bank, abandoning the robbery.\n\nAs a further precaution, LeGant recommends generating a protection symbol around your aura, such as an outward facing red rose, a six-pointed star, a cross, \"or whatever you're comfortable with.\" Place a clear mental image of this protection symbol on the electric blue edge of your aura, says LeGant. This is the energy equivalent of strengthening your immune system to protect your body against unwanted biological substances.\n\n## HEALTHY LIVING SPACE DETOXIFIER #44 Protect Your Space by Connecting to a Higher Source\n\nAt this level of detoxification and health maintenance, you need a kind of free-ranging spiritual antioxidant to protect your space against unwanted intruders and foreign energies, or in the language of Modi and LeGant, \"beings.\" LeGant recommends going to the highest source possible, what he recognizes as the Supreme Being. Call this highest source God, divinity, the Creator, Source, Goddess, Allah, Brahman, Ain Sof\u2014anything at all; the point is that you request spiritual assistance from the highest or most exalted or most evolved source you can conceive of and feel comfortable conceiving of, advises LeGant.\n\n_Directions_ : LeGant says to visualize a bright golden cord of light extending from the center of your head straight up to this highest source. You don't have to be a saint or mystic to do this; simply tell the golden cord to connect itself to this highest source, and it will do the right thing for you, says LeGant. \"If your _intention_ is to send the golden cord to the Supreme Being, that's where it will go. Hook yourself up and ask for healing, ask to have some of the foreign energy cleared out of your space.\"\n\nYou can also ask your higher source to guide you to information or experts to help you further detoxify your spiritual space. \"You can also talk directly, have an interactive relationship with the Supreme Being. Just say hello and listen for responses. You will hear them in your mind, and you may think they are your own thoughts.\" LeGant recommends that after saying \"hello\" and requesting assistance, try to still your thoughts, then wait and listen, even if it runs to a few days, even months.\n\n\"You might hear that 'hello' come back.\" Naturally, we all will feel cognitively wobbly at this point, for how can we be certain it is the Supreme Being responding? It may sound preposterous, but LeGant suggests you \"ask the Supreme Being to make itself real for you,\" however paradoxical, impossible, or wild that eventuality sounds. \"Keep asking the questions, keep noticing the answers, keep seeing that back-and-forth relationship\u2014then you will develop certainty about the dialog,\" says LeGant. This kind of connection surely is the ultimate immune defense.\n\nBear in mind that these approaches are palliative, not curative. In other words, they are effective short-term steps for spiritual detoxification and energy immune defense. Core pictures are deeply embedded in the auric space, however, and generally require a trained psychic practitioner to remove.\n\n## HEALTHY LIVING SPACE DETOXIFIER #45 Cleanse and Protect Your Aura with Pomanders and \"Air Conditioners\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor audio tapes, classes, and instruction including tips and techniques for psychic clearing and protection as well as psychic clearing practitioner referrals:\n\nBerkeley Psychic Institute,\n\n2018 Allston Way,\n\nBerkeley, CA;\n\ntel: 510-548-8020, 510-644-1600, or 800-433-5288;\n\nwebsite: www.berkeleypsychic.com;\n\ne-mail: dejavu@dnai.com.\n\nAnother approach that clears auric energy blockages and balances the aura is called Noetic Field Therapy, based on original research by Dr. Robert D. Waterman _(Through the Eyes of Soul: Theory and Practice of Noetic Field Therapy_ , 1999). For information and referrals:\n\nJohn M. Browning,\n\ntel: 800-251-3354 or 703-532-7796;\n\ne-mail:\n\nenergytherapy@starpower.net;\n\nwebsite: energytherapy.net.\n\nWouldn't it be remarkably convenient if there was something natural but powerful you could spray into your aura that would help cleanse and protect it? It turns out there is, and it is known as a pomander. The term \"pomander\" is an old English word denoting a fragrant herbal bouquet once worn close to the body as an air freshener in a time when people didn't bathe very often or use deodorants. It might be comprised of small flower blossoms or small fruits flavored with spices. The pomander's intent was for protection, disinfection, or for clearing the local atmosphere (a few feet in radius) of psychic pollutants.\n\nIn recent years, British master color therapist Mike Booth has reintroduced and expanded the concept of pomanders. The fifteen Pomanders produced by his Aura-Soma Products company in Tetford, England, are capable of precise cleansing and protection of the aura. The application is simple: dribble three drops of the selected Pomander onto the left palm, rub the hands together, then drape the aura from head to foot with the delicious herbal scents, and gently inhale them through your cupped hands.\n\nThe Pomanders come in tiny crush-proof plastic bottles and in mister-sprayer dispensers, making them suitable for carrying with you during the day or while traveling. Each Pomander contains the essences of forty-nine herbs, essential oils, flowers, minerals, and crystals (preserved in a light alcohol base), in a special ratio enabling it to affect different aspects of one's emotional and psychic dimensions, explains Booth. Each is a different color as well. This makes sense because psychics describe the aura as a multicolored field surrounding the body, and that its colors and intensities change constantly in accordance with our moods and thoughts; so introducing the essences of colors to your aura is a way of reestablishing balance through resonance\u2014like balancing like, color for color.\n\n\"The Pomanders are created mostly for the protection of persons who are in the process of [psychically] opening up,\" says Booth. In practical terms, that would mean after (or during the time) you have tried some of the aura detoxifying exercises in this chapter. According to Booth, the Pomanders help balance the electromagnetic field of the user, providing protection against negative influences while allowing positive ones to enter the aura. \"By placing these naturally fragranced colored essences into your hands and moving them through the aura, the electromagnetic field that surrounds your body, like a gentle breeze you create a delicate web around yourself,\" explains Booth. \"This colored 'web' protects you from energies you do not want and allows the positive energies to enter. Kirlian photography [which photographs energy emissions from living organisms] has shown [that the Pomanders] create little valves within the electromagnetic field that allow the positive energies in and filter out energies that are less helpful.\"\n\nThe Pomanders can also change negative energy in rooms or houses. Here you apply the specific Pomander to your hand, as above, but then \"wave\" its fumes around the room in question, distributing the aromatic vapors to all corners. While doing this, it helps to visualize the fumes as sweeping away all negative energies in the room, says Booth. Especially helpful here are the mister-sprayer Pomander dispensers (suitable for spraying around the body) and the Air Conditioners, four-fluid-ounce spray bottles suitable for misting an indoor environment. \"These enable you to clear and condition your space with the energies of color,\" says Booth.\n\nWith this in mind, we can move on to the actual Pomanders that will be helpful for detoxification.\n\n### White Pomander\n\nThe predominant essential oils in this blend include Kajeput, California laurel, and laurel. The crystal energies are morganite, quartz, and selenite. This Pomander is recommended for protecting and balancing all seven chakras of the body, for protecting the whole electromagnetic field, for mitigating the energetic effects of radiation, and which may help to alleviate allergies caused by outside toxic substances and energies. Of the fifteen Pomanders produced by Aura-Soma, the White Pomander, the original one of the series, contains all forty-nine herbs in equal balance. \"It is suitable for cleansing, purifying, and protecting, and may be used in any space to cleanse the atmosphere, bring the light in, and renew energies,\" says Booth. \"It is also helpful during detoxification.\"\n\n### Deep Red Pomander\n\nThis one contains a predominance of cedar and laurel essential oils, as well as crystal energies of garnet, ruby, car-nelite, strawberry-quartz, hematite, and neptunite. It is especially helpful in dealing with stress, and of the entire line of Pomanders \"this is the one that grounds most intensively, energizes, and provides the most effective protection,\" says Booth. It is recommended for use after meditation and rituals. Further, \"it protects against negative influences from Earth energies and on the other hand, it sensitizes for Earth energies.\" (See \"geopathic stress\" in Chapter 12.)\n\nFor healers, the Deep Red Pomander is recommended for times when you feel your client is draining your energy. \"This one provides the strongest protection for energy zapping, as it is energizing and may restore physical energies after tiredness and fatigue or depletion through drugs.\" It also supports the intuition and removes fears regarding survival issues surrounding, for example, money or health. \"The Deep Red Pomander may help those in their teens who have poltergeist activity, and it may be used in house-clearing to neutralize the effects of psychic activity or when visiting sacred sites and power places that have very often been used in an incorrect manner,\" or whose energy aspects or water have become toxic, adds Booth. (Again, see Chapter 12.)\n\n### Serapis Bey Master Quintessence\n\nSerapis Bey is one of fifteen similar products made by AuraSoma called Master Quintessences. In general, they deal with more refined energies and states of awareness than the Pomanders. As with the Pomanders, they are color-differentiated, and contain herbs, essential oils, and crystal energies. The Serapis Bey Quintessence is a clear formula suitable for cleansing, says Booth. It is recommended for healers and therapists to use to \"clear\" a treatment room of ambient energies and energy toxins following a session and to help the practitioner to cleanse and rebalance his aura.\n\nIt is believed that this spicy and flowery Quintessence \"stretches\" the electromagnetic field of the user, enabling one to integrate one's recent healing experience or insight. \"Serapis Bey is good for detoxifying at all levels,\" Booth notes. As with the Pomanders, the Quintessence is available in a small plastic bottle or as a sprayer\/mister dispenser.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Pomanders and Air Conditioners:\n\nAura-Soma USA Inc.,\n\nTrish and Will Hunter,\n\nP.O. Box 1688,\n\nCanyon Lake, TX 78130;\n\ntel: 830-935-2355;\n\nfax: 830-935-2508;\n\ne-mail: usa-info@aura-soma.net.\n\nFor information about color therapy, Aura-Soma training programs, lectures, and residential workshops, contact:\n\nAura-Soma International Academy of Colour Therapeutics,\n\nDev Aura, Little London, Tetford, Horncastle,\n\nLincolnshire LN9 6QL, England;\n\ntel: (44)-1507-533218;\n\nfax: (44) 1507-534025;\n\ne-mail: info@asiact.org.\n\nFor information concerning the products of Aura-Soma, contact: Aura-Soma Products, Ltd.\n\nSouth Road, Tetford Horncastle, Lincolnshire\n\nLN9 6QB England\n\ntel: (44) 1507-533581\n\nfax: (44) 1507-533412\n\ne-mail: info@aura-soma.co.uk\n\nwebsite: www.aura-soma.net.\n\nNote: \"Aura-Soma,\" the \"Prism and Rainbow\" device, and the \"Laurel\" device are registered trademarks of Aura-Soma, Ltd.\n\n### Spiritual Detoxification Is a Lifelong Activity\n\nOnce you begin to purge spiritual toxins from your system, you have started a process that can, if you wish it, transform your life and give you a focus for meditation and inner work for the rest of your life. Once you enter the field of the aura, and the larger, even subtler energy fields beyond it (yet still connected to you), you find auric space is remarkably _larger_ than you would have expected. In fact, some people, after doing spiritual detoxification work for a few years, are astonished at how vast is this auric space and how replete it is with beings and energies that can be purged.\n\nIt is not that we are phenomenally polluted, so contaminated that a life of constant detoxification is required to cleanse our space. Rather, the further you go with this level of detoxification, the more you enter the realm of basic spiritual growth and unfoldment. In so doing, you unavoidably enter the domain of the transpersonal\u2014that is, the realm of parts of yourself you didn't know you had, and that are not necessarily even from this time, this life, this body. Let me put this in perspective by an example from an old religious tradition.\n\nJainism is an ancient religion of India, older than Buddhism and possibly Hinduism as well. One of its tenets is spiritual cleansing, but Jainists take a very broad view of what this entails. Jainism says that the condition and color status of the aura is an index to the soul condition of the individual. Originally, and in our pure condition, our essential \"color\" is a noncolor of stainless, crystalline, transparent clarity, but through our life actions, thoughts, feelings, and the rest of everyday living, other colors can appear in the aura. How we live and what kind of cumulative effect our actions have determine (and attract) what colors appear in the aura. Jainism calls these colors _lesyas_ , and there are six of them: white, yellow (or rose), flaming red, dove-grey, dark blue, and black.\n\nOne of the chief goals of Jainist spiritual practice is to purify the aura and remove the colors ( _lesyas_ ) so that only the original crystalline purity remains. As the noted scholar of Indian religions, Heinrich Zimmer, stated, \"the advance of the individual toward perfection and emancipation is the result of an actual physical process of cleansing taking place in the sphere of subtle matter [our auric space]\u2014literally a cleansing of the crystallike life-monad.\"\n\nJainism proposes that when this life-monad (the original transparent clarity of our being) is purged of all the life-action colorings and contaminations, \"it literally shines with a transparent lucidity\" and that when it is completely cleansed, \"it is immediately capable of mirroring the highest truth of man and the universe, reflecting reality as it really is.\" Surely that's a benefit worth detoxifying for. But here is yet another way of framing the spiritual side of detoxification.\n\nAccording to British psychic Mona Rolfe, Ph.D., in _The Sacred Vessel_ , an essential element of the ancient tradition of initiation was preparation and purification. To understand more about one's deepest self and the cosmos, one had to purify the body and mind of all the aspects that weighted or darkened it. Candidates for initiation purified their bodies daily with fresh clean water and foods, what we would call \"kosher\" today.\n\nThe point of all this incessant detoxification was to enable the human form to contain more light, which is to say, more spiritual illumination. \"Buddha is an immense force of light, the greatest force of light in the world,\" explained Dr. Rolfe, but equally powerful was the light of the Christ. For anyone desiring to be touched by these sources of immense light, they must first be purified and prepared. Toxic substances actually impede the entry of light\u2014more consciousness, self-awareness, understanding, and insight\u2014into the mind and body.\n\n\"Your soul is guiding your personality through certain experiences on Earth in order that you may become purified, body, soul, and spirit, sufficiently so that the Buddhic force may touch you and hold you for all time,\" explains Dr. Rolfe. You needn't worry that spiritual detoxification will make a Buddhist of you; where Dr. Rolfe says \"Buddhic,\" substitute whatever description of spiritual energy you prefer, or simply psychological insight. The essential point is that spiritual detoxification can make it easier for you to embody more spiritual light, insight, compassion, even wisdom, if you wish. And it might make it easier for you to extend this understanding to your living space, the subject of the next several chapters.\n\n# Part Three\n\n# _Creating the Healthy_ Home _Living Space_\n\n# CHAPTER 10\n\n# Cleaning Up the Home's Indoor Environment: Overcoming the Effects of Sick Building Syndrome\n\nWhen we consider the home as a living space that has something to do with our health, we find the same factors at play as we did with the body. Toxins. Certain factors about the modem house\u2014specifically, its building materials, furnishings, and air quality and circulation\u2014can be toxic factors and thus harmful to our health.\n\nJust as we saw with toxicity at the physiological level of the body, it isn't always a question of a single poison negatively affecting our system. House-bome toxins often are multiple and together can produce a toxic, immune-system weakening effect. We may be able to tolerate one, or several, toxins in our home environment, but for each of us, there is a limit, a point at which our system can no longer accommodate the toxic load and we get sick. It may start as allergies, headaches, or irritability, but it may progress to a wide range of symptoms and baffling new health conditions including chronic fatigue syndrome, multiple chemical sensitivity, and environmental illness. Physicians are increasingly faced with a strange new kind of patient, the hypersensitive, one who is allergically reactive, seemingly to almost anything of modern manufacture.\n\nThe trouble is, with respect to toxins generated in our home living space, we are not yet sufficiently accustomed as a culture even to think of the home as a source of toxicity. Potentially, most of the structural elements of our inside living space, even many items we take for granted as part of everyday living, can be toxic: paints, lumber, carpets, sofas, furniture, household cleaning products, computers, overhead lights, appliances, the unfinished basement. It's a long list of potential suspects, potential toxic sources.\n\nIt's not that we should suddenly start to fear our living space as poisoned and out to hurt us, but the news that aspects of modem homes, schools, office buildings, and nearly all indoor living or work spaces may be hazardous to our health should counsel us to look carefully at the materials we surround ourselves with and to think about ways to neutralize their ill effects or replace them with nontoxic alternatives.\n\n**_Sick building syndrome refers to the negative health effect of a building on its occupants. More formally, according to the EPA, it refers to \"situations in which the building occupants experience acute health and comfort effects that appear to be linked to time spent in a building, but no specific illness or cause can be identified.\"_**\n\nSince the majority of Americans probably spend about ninety percent of their time in indoor environments, the importance of that inside living space being health-promoting becomes strikingly evident. Since the 1980s, medical attention has started to focus on this previously unsuspected field of toxicity such that the negative health relationship between an interior living space and the health of its occupants is now often referred to as \"sick building syndrome.\"\n\n### Sick Building Syndrome: The Disease of Modern Architecture\n\nOne of the dominant trends in late twentieth century Western architecture was to create increasingly self-sufficient internal environments, buildings whose inner space was almost completely sealed off from the outside world. To an extent, this trend gained emphasis in the wake of the 1973 oil embargo, which riveted attention on the need to conserve fuels and maximize their efficiency. The result was both office buildings and homes that were indeed more fuel efficient, but they achieved this savings at a hidden cost, one that is only now becoming clear. In effect, by saving heating fuel, we created a new illness, known as sick building syndrome (SBS).\n\nPut as simply as possible, SBS refers to the negative health effect of a building on its occupants. More formally, according to a statement by the United States Environmental Protection Agency (EPA), sick building syndrome refers to \"situations in which the building occupants experience acute health and comfort effects that appear to be linked to time spent in a building, but no specific illness or cause can be identified.\" EPA also describes a similar condition called \"building-related illness\" (BRI) for situations in which \"symptoms of diagnosable illness are identified and can be attributed directly to airborne building contaminants.\" SBS symptoms can be generated by almost any type of modern indoor environment, such as office building, school, hospital, public assembly hall, or home, in potentially any country.\n\nFrom a layperson's viewpoint, there is not a great deal of difference between SBS and BRI: both describe how a building can make its occupants sick. Whether the exact individual causes can be identified, as in BRI, or cannot be pinpointed, as in SBS, is almost a moot point, because as we are instructed by alternative medicine, toxicity is a cumulative, additive influence. The body has a capacity to handle a certain amount of toxicity\u2014the toxic load or total toxic burden\u2014but when that capacity is reached, and exceeded, any additional toxic exposure will make us sick. So one specific factor in an interior environment may not sicken us, but when added to the toxic influence of several others, the end result may be illness.\n\nSBS symptoms tend to fall into a familiar cluster that includes headache; eye, nose, or throat irritation; sinus infections; dry cough; dry or itchy skin; dizziness; nausea, concentration difficulties; reduced memory; fatigue; drowsiness; whole body weakness; hoarseness or changed voice; skin reddening; stinging or smarting of the skin; runny nose or eyes; and odor sensitivity, according to the EPA. BRI symptoms include cough, chest tightness, fever, chills, and muscle aches (see figure 10-1).\n\nHow widespread is the problem? According to the prestigious British medical journal, _The Lancet_ , SBS is \"an increasingly common problem.\" The symptoms of SBS, said _The Lancet_ , can be \"uncomfortable, even disabling, and whole workplaces can be rendered non-functional\" as a result.\n\n**Figure 10-1. Typical Health Symptoms of Sick Building Syndrome and Building-Related Illness**\n\nAccording to a 1984 study by the World Health Organization, thirty percent of new and remodeled buildings worldwide may generate symptoms of SBS or BRI, mainly due to indoor air quality. The air circulation mechanics and quality of sealed in living spaces tend to be the prime carrier of a great deal of the sick building pollutants. A more recent estimate suggests that twenty percent of American office workers are affected by SBS. A study by Honeywell Techanalysis of 600 of its own office workers found that 24% said they felt (based on symptoms experienced) there were air quality problems in their office environment. Also, between seven percent and eleven percent of these same employees reported having a tired feeling, stuffed up nose, eye irritations, headaches, or labored breathing.\n\nResearchers at Cornell University in Ithaca, New York, found that workers in poorly ventilated offices are twice as likely to complain of SBS symptoms as those in well-ventilated environments. The 1998 study looked at four multistory office buildings over a course of three consecutive days, measuring levels of thirty-six \"potential worker irritants,\" such as temperature, carbon monoxide and carbon dioxide levels, dust mass, carpet dust, light levels, airborne particulate counts, and nicotine and formaldehyde levels. The researchers collected over 1,500 questionnaires from workers in the four buildings concerning SBS-type symptoms.\n\nThe study results were instructive and clearly supported the SBS thesis. \"Both the workers with very few symptoms and those with more intense symptoms show a clear pattern of increased problems by the end of the day, suggesting that something is making the workers who are more sensitive feel sick,\" concluded head researcher Alan Hedge, director of the Human Factors Laboratory at Cornell's College of Human Ecology. The workers had a \"clear pattern\" of feeling worse at the end of each day.\n\nWhile no single \"culprit\" was identified as the \"cause\" of SBS symptoms, Hedge noted that small accumulations of carbon dioxide from human breathing (due to inadequate ventilation) had been noted in these environments and are related to SBS. He further explained that workers were much more likely to report SBS symptoms when indoor carbon dioxide levels were higher than 650 parts per million (ppm). A typical outdoor concentration of carbon dioxide is 325 ppm. \"This suggests that SBS symptoms may be associated with building ventilation performance.\" An earlier study (1993) by Hedge of 1,324 workers in nine buildings had linked SBS symptoms to \"the amount of man-made mineral fibers in settle office dust,\" again pointing to poor air circulation as the main problem in SBS.\n\nOther studies have established the connection between carbon dioxide concentration and the incidence of SBS symptoms. Finnish researchers at the Helsinki University of Technology reviewed the results of 41 studies involving 60,000 subjects and found that almost all the studies showed that poor ventilation rates were clearly linked with a worsening of SBS conditions. They also reported that about fifty percent of the studies showed that as indoor carbon dioxide concentrations dropped below 800 ppm, the rate of SBS symptoms also dropped \"significantly.\"\n\nOther studies, reported by Michael J. Hodgson, M.D., M.P.H., of the University of Pittsburgh School of Medicine, confirm and extend Hedge's conclusions. For example, Hodgson reported that in Germany, among 2,000 office workers with work-related symptoms, a fifty percent higher than average rate of upper respiratory tract infections was noted. These infections could be directly attributed to inadequate mechanically ventilated buildings\u2014in other words, buildings where you couldn't open a window to get fresh air, but had to breath continually recirculated indoor air.\n\nOne study reported by Hodgson showed that seventy percent of buildings already classified as sources of SBS symptoms had poor circulation of fresh outdoor air. This study also showed that fifty percent to seventy percent of these buildings had poor internal air distribution (meaning the air didn't move much at all); sixty percent had poor filtration of pollutants derived from outside the building; sixty percent had indoor standing water (capable of supporting unfavorable biological growths); and twenty percent had humidifiers that functioned poorly. The result of this array of toxic indoor air pollutants\u2014including toxic airborne compounds released from building materials, furniture, office machines, paints, glues, and other standard items found in the modern office or home\u2014is \"a complex mixture of very low levels of individual pollutants.\"\n\nA Danish study from 1987 examined the relationship of SBS-type symptoms and indoor air quality in 3,757 office workers and their environments in 14 different buildings. The researchers found that people working in the oldest buildings (and therefore not mechanically ventilated) had the least number of SBS symptoms; they also found the highest rates of certain symptoms (mucous membrane irritation, headache, fatigue, and lethargy) among workers who spent their time in buildings with mechanical ventilation, meaning, without the regular circulation of fresh outside air. The study also found that twenty-eight percent of the office workers said they had irritation in the mucous linings (nose, throat) and thirty-six percent had headache, unusual fatigue, or malaise (general poor feeling).\n\nA British study that same year examined the SBS incidence among 4,373 office workers in 42 different buildings and under 47 different ventilation conditions. The researchers found that eighty percent of the people had at least one work-related symptom and more than forty percent had SBS symptoms such as stuffy nose, dry throat, or headache. Women tended to have more SBS symptoms than men, and the typical range of symptoms per person in any of the forty-two different buildings was from one to five. In buildings in which the indoor air supply was cooled or humidified, the incidence of symptoms was much higher. The study further revealed that eighty percent had at least one work-related symptom, fifty-seven percent had lethargy, forty-seven percent a blocked nose, forty-six percent dry throat, forty-six percent headaches, nine percent chest tightness, and nine percent breathing difficulties.\n\nAnother study in Denmark had fifty-four \"air quality judges\" assess the indoor air quality in twenty randomly selected offices and public assembly halls. They found that air quality deteriorated more from indoor pollution sources (from the outgassing of building and furnishings materials, tobacco smoke, and the ventilation system) than from the effects of the occupants (such as carbon dioxide exhalation). Specifically, twenty percent of the assessed indoor air pollution was due to the building materials, forty-two percent to the ventilation systems, twenty-five percent to tobacco smoke, and only thirteen percent to the people in those spaces.\n\nGovernment researchers at the Institute of Environmental Epidemiology in Singapore found that 19% of 2,856 office workers from 56 randomly selected public and private sector buildings reported SBS symptoms. The same researchers, in studying a different group of 2,160 people in 67 offices in Singapore discovered that SBS symptoms are more prevalent among workers who report high levels of physical and mental stress; the study implied that stress makes one more susceptible to the toxic indoor influences that produce SBS symptoms.\n\nNot only can stress bring on susceptibility to SBS influences, but if you are already allergic, being in a \"sick building\" is likely to make your allergies worse, or produce SBS symptoms faster than in people who are not constantly allergic. Researchers at Kristianstad University in Sweden found that people with allergies (in this case, high school students) \"note discomfort earlier\" than people without allergies when exposed to sick building influences. A higher percentage of already allergic people suffered weekly symptoms than nonallergic people did, but even more interesting (or alarming) was the finding that only forty-five percent of the students in the Swedish high school studied were non-allergic, meaning fifty-five percent had regular allergies.\n\nElsewhere in Sweden, researchers at the Department of Lung Medicine and Asthma Research Center at Uppsala University found that 21% of a randomly selected group of 418 men and women, aged 20-45 years, reported experiencing one or more SBS symptoms on a weekly basis; these results led the researchers to conclude that \"sick building symptoms are common in the general population.\"\n\nA study from New Zealand came up with similar results. Researchers at Massey University in Palmerston North found that 80% of 360 office workers studied experienced some SBS symptoms, with forty percent of these reporting lethargy, stuffy noses, dry throat, and headaches. These researchers used a British questionnaire from a study of an English population, the statistical results of which were the same. \"Sick building syndrome was found to be sufficiently prevalent in both surveys to warrant concern.\"\n\nOften individuals suffering with SBS or BRI do not make the connection between their strange list of symptoms and where they spend the bulk of their time. Given the sealed-in nature of many modern homes and apartments, it is possible to be in a toxic living space twenty-four hours a day, on the job and at home. Similarly, most conventional physicians do not make the link between indoor environment and symptom picture, and often patients visit one doctor ten times or ten different doctors in search of a comprehensive diagnosis for their condition: not acutely sick, but not well either. You end up with what one SBS expert characterized as \"a basket of symptoms with no clear cause.\"\n\nFor some reason, women tend to have more SBS symptoms than men, according to a study at the University of Hamburg in Germany of 2,517 female employees. The women had a higher incidence than men in general bodily complaints and sensory irritations along with a \"more negative evaluation of the indoor climate\"\u2014which may be scientific jargon for saying they were both more aware of and sensitive to the quality of their workspace.\n\nIn especially sensitive individuals, typical SBS and BRI symptoms can progress to more long-term dysfunction and a constant state of allergic reaction, as in multiple chemical sensitivity, environmental illness, and chronic fatigue syndrome, which share overlapping symptoms. In these cases, one is hypersensitive to substances found in a typical indoor environment and is in effect unable to function in those environments.\n\n### Indoor Pollutants\u2014A Matter of \"Genuine Concern\"\n\nIf one's reaction to SBS is caught early on, generally the symptoms can be reversed quickly, by removing yourself from the toxic space and\/or neutralizing the toxic effect of various elements in that indoor living space. But if your exposure to SBS influences goes on unchecked for a longer period, it can start to produce deeper-set damage (to the liver and immune system, primarily), which though reversible, may take months to undo.\n\nBy way of illustration of the first situation\u2014quick reversal of symptoms by leaving the toxic environment\u2014here is a case history from Michael Hodgson, M.D., M.P.H. Dr. Hodgson describes a forty-year-old woman who started having headaches, dizziness, difficulty in concentrating, and runny nose soon after she started working in a \"hermetically sealed\" modem office building. Indoor carbon dioxide levels were 600-700 ppm, or about double, or more, than normal outdoor levels; there were also measurable amounts of formaldehyde and volatile organic compounds (outgasses of various equipment and building materials).\n\nVarious neuropsychological tests showed that this woman had symptoms of solvent neurotoxicity. This means the constant low-grade but poisonous gas emissions from various components of the office environment were affecting how her nervous system worked, to the extent that they were toxic to her neurons (nerve cells). When the woman stayed away from this unhealthy office for four weeks, working elsewhere, all of her test results returned to normal and the toxic effects went away. If she went back to the toxic office, her symptoms returned. According to Dr. Hodgson, when a formal study was made of the \"toxic\" building and its occupants, a definite relationship was shown between the level of \"respirable suspended particulates\" (airborne, breathable toxic substances) and the degree of symptoms in \"a large proportion\" of the people there.\n\nHere is a case that shows the long-term effects of not dealing with SBS symptoms when they first show up. Susan Lange, O.M.D., L.Ac., is a licensed acupuncturist, doctor of Oriental medicine, and cofounder of Meridian Center, an alternative medicine clinic in Santa Monica, California. It took her almost twelve years to figure out that the reason she had environmental illness was that she had endured exposure to numerous toxic substances much earlier in her life, and the effects were still active.\n\nSpecifically, Dr. Lange had sustained petrochemical toxicity, due to working in an office heated by kerosene gas stoves and with poor ventilation; later, she worked in a hospital whose building materials were contaminated with molds and fungus; then she worked in another \"health\" clinic in which the air conditioning filters were dirty such that they vented microbially contaminated air into the offices. Things got so bad for Dr. Lange that she even became allergic to her own house.\n\n**_The liver, as the body's prime detoxification organ, gets overloaded and becomes unable to do its normal detoxification, so the body gets overwhelmed by toxins and moves into a state of constant dysfunction and, paradoxically, hypersensitivity to new toxic influences._**\n\nThe combined effect of all the toxins she had been exposed to over the years made her \"incredibly ill,\" with numerous SBS symptoms including heart palpitations, concentration difficulties, and allergies to seventy percent of the foods and substances in her environment. She nearly fainted when she inhaled perfume or gas fumes, and \"when I walked into my own house, I felt like passing out.\" The story has a happy ending: Dr. Lange was able to undo the cumulative toxicity in her system through a carefully designed alternative medicine detoxification program.\n\nThat is the good news: despite the rising incidence of hypersensitive, highly reactive individuals, their condition is not permanent. With effective detoxification practices and alternative medicine-based restorative protocols, it can be reversed, enabling them to function once again in the modem world without getting sick every time they walk upon a new carpet or stand by a photocopier machine. As we learned in earlier chapters, in cases of hypersensitivity, often the liver is the key to both the problem and its solution. The liver, as the body's prime detoxification organ, gets overloaded and becomes unable to do its normal detoxification, so the body gets overwhelmed by toxins and moves into a state of constant dysfunction and, paradoxically, hypersensitivity to new toxic influences.\n\nEven though its unhealthy results can be reversed, that doesn't change the fact that the problem of SBS is clearly widespread and probably increasing as buildings become ever more \"modem\" and thus sealed off from the outside environment. The main toxic sources, according to the EPA, fall into four categories.\n\nFirst is inadequate ventilation. Before 1973, U.S. ventilation standards required that each building occupant have 15 cubic feet per minute (cfm) of outside air; after 1973, the amount was reduced by 66% to only 5 cfm.\n\nSecond are chemical contaminants from indoor sources, such as outgassing from carpets, adhesives, upholstery, manufactured wood products, photocopiers, pesticides, cleaning products, tobacco smoke, and unvented heating devices, all of which emit volatile organic compounds (VOCs).\n\nThird are chemical contaminants from outdoor sources that enter the building through the ventilation system. These can include motor vehicle exhaust, and airborne toxins from plumbing vents, and building exhausts.\n\nFourth are biological contaminants, such as bacteria, molds, pollen, and viruses that may, for example, breed and spread in stagnant water present in ducts, humidifiers, air conditioners, ceiling tiles, carpets, or insulation. Toxic microorganisms such as _Staphylococcus_ , _Streptococcus_ , and _Aspergillus_ can actually grow in a building's duct work and then get spread through the air through the building's ventilation system.\n\nManagers of office buildings and large indoor spaces are beginning to get the idea that these spaces might be harmful to the health of the people who work inside them. A trade publication called _Today's Facility Manager_ published a survey of an unspecified number of facility managers, which showed that 89% had heard of SBS and 100% considered it a \"genuine concern,\" especially given the high incidence of allergies; 69% said their facility had already been tested for SBS and poor indoor air quality, and of these, 55% said the testing occurred because of employee complaints, 31% because of insufficient air circulation or filtration in their building, and 17% as part of a preventive maintenance program.\n\nThe reader at this point might well ask: \"What have SBS-conditions reported in office buildings, schools, and other large indoor spaces to do with my home or apartment?\" The answer is that the data reported for SBS conditions in these larger modem indoor environments are highly relevant to the typical modem Western indoor living space. Many of the same toxic factors are at play in the home (and will be discussed later in this chapter, along with their practical remedies), and while perhaps many homes do not have photocopiers (that outgas toxic VOCs), many homes should be concerned with radon emissions (a toxic \"natural\" gas emission from the Earth) from their basements. The modem home can be just as much a source of SBS symptoms as the modem office building, school, or hospital (see figure 10-2).\n\n**Figure 10-2. 8 Major Indoor Pollutants and the Symptoms They Produce**\n\nOne of the ways the \"genuine concern\" of facility managers and the increasing research data on sick building syndrome is being translated into the domestic front is through a discipline called Bau-biologie. A German-based initiative, the term means the living or biological house. More specifically, Bau-biologie (pronounced BOW-bee-oh-low-gee) is \"the study of the impact of the building environment upon the health of people and the application of this knowledge to the construction of healthy homes and work places.\" It is also the science of the \"holistic interactions between life and the living environment,\" according to The Institute for Bau-biologie, the American branch of this innovative movement, based in Clearwater, Florida.\n\nThe Institute's prime purpose, it states, is to help people realize that health hazards may exist in their homes and workplaces even though they may not be aware of them, and that not knowing about these hazards can be harmful to one's health, even to the larger outdoor environment. Based on research developed since the 1970s, the Institute has trained and certified over seventy \"environmental inspectors\"\u2014architects, builders, electricians, medical practitioners, engineers, and others\u2014who can evaluate indoor living spaces for the numerous health hazards identified by Bau-biologie.\n\nOther individuals and groups are taking up the European cue as well. John and Lynn Marie Bower formed the Healthy House Institute in 1992 in Bloomington, Indiana, to spread practical knowledge about how to build nontoxic homes and how to detoxify already built ones. Through their books, workshops, and website, they provide information to homeowners, designers, architects, and contractors \"interested in making houses healthy places in which to live.\"\n\nBuilding a nonpolluting natural house requires considerable research and planning and, until recently, most people tended to think the issue involved whether to live in a city or the country, in an apartment or an A-frame in the woods. But it's not so much the structure of the new house that matters, as what it's built with, says Bower. Almost everything, formerly taken for granted as \"standard\" building materials and furnishings, must be rethought.\n\nParticle board and interior plywood walls, for example, can outgas formaldehyde from their glues; new synthetic carpets emit toxic fumes, can become havens for microorganisms, and generate synthetic dust in the indoor air; indoor paints, varnishes, and clear finishes can also release minute amounts of toxic fumes (VOCs). Tighter construction for increased energy efficiency (with less fresh air circulation) and the use of synthetic building materials (that outgas dozens of chemical pollutants or release particulates) both contribute to poor indoor air quality, which becomes a vehicle for distributing many other toxic elements in the indoor living space. Studies have shown that indoor air may contain concentrations of air pollutants 100 times higher than found in outside air. \"Your house is where you spend the most time, so it is important to build it so that it won't contribute to your ill health,\" comments John Bower.\n\nConsider the data on pesticide pollution inside the home, for example. According to a 1997 study conducted by the Special Education Department at the University of South Florida in Tampa, the majority of U.S. homes emit the pesticide chlordane into the living air space. How is this possible? For about thirty years, up until 1988 when it was banned, chlordane (a potent nerve toxin) was routinely used on building foundations to kill termites. Typically, 100-200 gallons of chlordane were applied underneath the home's concrete foundation. The pesticide may have been banned, but the regulation didn't ban the possibility of outgassing from the long-lived chemical residues.\n\nStudies now show that chlordane vapor is leaking upwards (through foundation cracks or around pipes that enter the home) and contaminating the indoor air. Researchers report that seventy-five percent of U.S. homes built before 1988 routinely contain air levels of chlordane; one study showed that thirty-four percent of homes built before 1982 had air chlordane levels beyond the safety limit. The University of South Florida research team estimated that 100-185 million U.S. residents \"are breathing questionable levels of chlordane in their homes daily,\" and that 10-20 million could be living in homes where the indoor air levels of chlordane are \"exceeding\" the recommended safe limits.\n\nWhat can chlordane do to you? It can have a negative impact on many parameters of mental function, as demonstrated in 1987 when 250 adults and children were exposed to a toxic dose of chlordane at an apartment complex. The building surfaces and grounds were sprayed with chlordane, and its fumes entered the apartments for years afterwards, outgassing from the wooden surfaces of the apartment complex. Seven years later, scientists at the University of Southern California School of Medicine at Los Angeles ran a battery of neurological tests on 216 of the former residents of this building to determine if they showed signs of neurotoxicity, which would indicate the pesticide was toxic to the nervous system. It was.\n\nSimple reaction time for the chlordane-exposed residents was thirty-four percent slower; memory recall was twenty percent diminished; digit symbol attention was nineteen percent slower; balance dysfunction was sixteen percent increased; vocabulary use was twelve percent slower; mood state\/tension was seventy percent elevated; mood state\/depression was seventy-four percent increased; mood state\/anger was sixty-nine percent more prevalent; vigor was twenty-seven percent reduced; fatigue was seventy-five percent elevated; and confusion increased by eleven percent. It wasn't just the brain that was affected. Other symptoms traced to chlordane toxicity included asthma, allergies, excess phlegm, chronic bronchitis, wheezing, shortness of breath, headaches, and indigestion.\n\nThe same researchers analyzed nine patients in depth in an attempt to profile \"chronic neurobehavioral impairment\" due to chlordane exposure. They administered various tests to the nine people and found abnormal balance with eyes closed in seven, abnormal color discrimination in six, verbal recall problems in five, and various other perceptual disorders in four patients. \"These observations suggest that chlordane causes protracted neurotoxicity,\" the scientists concluded.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more information, including study materials, books, workshops, practitioner referrals, monitoring equipment, contact: Bau-biologie USA,\n\n1401A Cleveland Street,\n\nP.O. Box 387,\n\nClearwater FL, 33755;\n\ntel: 727-461-4371;\n\nfax: 727-441-4373;\n\ne-mail: baubiologie@earthlink.net;\n\nwebsite:\n\nwww.bau-biologieusa.com.\n\nAlso:\n\nThe Healthy House Institute,\n\n430 N. Sewell Road,\n\nBloomington, IN 47408;\n\ntel\/fax: 812-332-5073;\n\ne-mail:\n\nhealthy@bloomington.in.us;\n\nwebsite: www.hhinst.com.\n\nChlordane is only one of many toxic chemicals that are potentially inside our living space polluting our indoor air. Even the way we clean our indoor living space may actually contribute to its toxicity, explains Lynn Marie Bower, author of a 700-page guide to nontoxic home cleaning, _Creating a Healthy Household._ \"Many of the cleaners, polishes, and waxes you've been using could be making you and your family sick.\" Bower, who states she has been hypersensitive for twenty years, explains that children, the elderly, pregnant women, and those who are already sick, asthmatic, or allergic \"are especially at risk.\"\n\n## The Healthy Living Space Expert Interview: Michael Riversong, Design Ecologist and Environmental Assessor\n\nAs the years progress, healthy living space experts like the Browers are gaining more colleagues across the spectrum of disciplines and applications. One is Michael Riversong of Cheyenne, Wyoming, a multifaceted expert in creating healthy living spaces who calls his discipline Design Ecology.\n\nRiversong combines the principles of Bau-biologie, feng shui (the ancient Chinese art of placement and energy relationships, discussed in Chapter 11), and geobiology (the effect of Earth energies on humans, discussed in chapter 12). \"Design Ecology is the process of addressing the effect of design on the physical, mental, and spiritual health of people,\" he comments in his book, _Design Ecology_ (1996). His practice encompasses \"the integration of music, environmental education, and advanced technology to create improvements in the living environment.\"\n\nOne of Riversong's specialties is an \"environmental assessment,\" a thorough review of all aspects of a home or small business environment for the presence and influence of toxic elements, electromagnetic fields, radon, VOCs, and other toxins, followed by practical recommendations for eliminating these toxic factors. \"I measure a lot of things with various instruments, all the while looking at the design of the place to see if there are communications being made by the design that are contrary to the goals of the people using the building.\"\n\nFor \"communications\" substitute the term \"toxic factors\" and you get an idea of what Riversong looks at when he's called in to evaluate an indoor living space. When Riversong refers to \"various instruments,\" he is in part thinking of his handy threesome: a multifrequency gauss meter, a Geiger counter, and an electrostress meter. The gauss meter shows him the level of the magnetic fields present in an indoor environment, especially in the home's bedroom(s). Magnetic fields, says Riversong, can exert strong influences on humans, either healthy or toxic, depending on their type.\n\nRelying on Swedish standards, Riversong says that magnetic fields higher than 2.5 milligauss (mG) are harmful to adults, and above 1.0 are \"risky\" for children. The combination of the human body's relatively high iron content, the possible iron content in the bed (metal spring mattress, metal bed frame), and an existing magnetic field can produce toxic effects in the bedroom. Let's say his gauss meter gives him a 4.0 milligauss reading in the bedroom. Sometimes the problem might be as simple as a miswired electrical junction box. One wire is misconnected, creating a \"very heavy\" electromagnetic field, Riversong notes. The remedy is simple: have an electrician fix the faulty wiring.\n\nIn other cases, an appliance, such as a television, can produce the problem. Say the head of the bed is up against a wall, on the other side of which sits the back of a television. The magnetic field emanated by that television goes through the wall and affects the sleeper in the bed. \"The bed gets some electromagnetic interference from that television, especially as, even if the television is turned off, there is always current flowing through it.\" (For more on the subject of electropollution, see below.) \"Home surveys should always start in the bedroom. This is vital because we depend on our nightly sleep for renewal of all biological systems,\" and disturbances in the electromagnetic fields can upset this crucial self-renewal mechanism of the body.\n\nRiversong relies on another device for measuring electromagnetic disturbances\u2014a surprisingly simple, low-cost approach developed by his mentor, the late Vince Wiberg, a healthy house wizard who practiced his trade for thirty years in Los Angeles. Get a cheap transistor A.M. radio and tune it to a high frequency range, such as 1500-1600, and in between stations, so you get only static. Walk around a building, or inside it, and listen to the sounds. \"You can actually hear the electromagnetic disturbances,\" says Riversong. \"You'll hear a buzzing type of sound, and different kinds of buzzes with different kinds of interference.\" This only works with an A.M. radio, he stresses.\n\n**_Riversong takes the idea of preventive house design a step further and regularly consults with practitioners of alternative medicine. He encourages them to make it a standard procedure that an environmental assessment should be performed on the home of every new patient to see if whatever is bothering them has an environmental cause._**\n\nRiversong emphasizes that the homeowner will get the best results by having an environmental assessment before the house is built. Then the various disciplines that comprise Design Ecology can have a _preventive_ aspect\u2014you can use health-promoting or at least nontoxic building materials to prevent illness in those who live within the indoor environment they create. \"I encourage people to have me in on the planning stages of a remodeling job or new building construction, because if they do this, they will prevent so many problems later on.\" This is preventive medicine for a house; this is the way you produce a health house, says Riversong.\n\nIn fact, Riversong takes the idea of preventive house design a step further and regularly consults with practitioners of alternative medicine, such as naturopaths, acupuncturists, and chiropractors. He encourages them to make it a standard procedure that an environmental assessment should be performed on the home of every new patient to see if whatever is bothering them has an environmental cause. \"The ideal scenario is to check the house to find out what's wrong with the patient.\"\n\nHere are a few case studies from Riversong's practice that illustrate the point:\n\n  * The owner of a house with an attached office under construction was concerned with microwave contamination since his future home was near a large telephone and research installation. Riversong found a \"microwave beam\" in the owner's office (already completed); this beam was draining the man's energy and causing other health problems. Riversong had him redesign his office so that he was not sitting in the microwave beam, after which his health improved \"and he found a whole new store of personal energy.\"\n\nWhen Riversong found another microwave beam going through what was to be the master bedroom, he was able to counteract this by installing microwave-blocking film on the bedroom windows. \"A follow-up measurement after completion of the house showed microwave levels in the bedroom were very low, but right outside the window, levels were extremely high due to bouncing of the waves off the covering.\" The health of the occupants has been excellent since they moved in, says Riversong.\n\n  * Children who lived in a beach house were constantly coming down with colds and coughs. Riversong took a scraping of material from the bedroom carpet and examined it under the microscope. He found numerous insect parts, mostly from ultratiny creatures too small to be seen without a microscope. He recommended a thorough housecleaning (especially of the carpets, which harbor the numerous small insects found at the seashore) and a new air filtration system for the bedroom. The children's health began to improve after these changes were made.\n  * Residents of a home in Montana reported constant fatigue. Riversong notes that large parts of this state are underlain by high underground water tables, sometimes dangerously close to the surface. In this case, water regularly seeped into his clients' basement. Riversong, calling on his knowledge of dowsing and geobiology (see Chapter 12 for details on these subjects), installed a length of thick copper wire in the ground near the house; this neutralized the energy-draining effect of the standing water. \"The experiment worked, and they reported an immediate surge of personal energy.\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about environmental assessments, music, books, devices, and other services, contact: Michael Riversong,\n\nP.O. Box 2775,\n\nCheyenne, WY 82003;\n\ntel: 307-635-0900 or 303-829-0774;\n\ne-mail: MRiversong@earthlink.net;\n\nwebsite: www.designecology.com.\n\n## HEALTHY LIVING SPACE DETOXIFIER #46 Get the Electropollution Out of Your Home\n\nOne of the pervasive features of modern technological society is the presence of electrical, magnetic, and electromagnetic fields produced by human-made instruments and devices. While these fields are natural in the sense that all of nature, including the human body, runs on or emits differing energy fields of these types, they can be harmful to our health when they are too strong, when there are too many, or when we live too close to them.\n\nThe negative health influence of these invisible fields is now referred to variously as electropollution, electrosmog, EMF stress, or electrical hypersensitivity, and it's one of the prime contributing factors in SBS-type symptoms. Electrical hypersensitivity (or its variant, electrical sensitivity) is used to indicate a form of environmental illness, \"specifically a chronic illness triggered by exposure to electromagnetic fields,\" according to Lucinda Grant, author of _The Electrical Sensitivity Handbook_.\n\nGrant says that being electrically sensitive means an illness you already have gets worse when you are near electrical appliances, power lines, or other significant sources of EMFs. Or you may experience symptoms _only_ when in the presence of EMFs. The electrically sensitive (ES) person reacts sooner and more strongly to otherwise \"normal\" EMF fields that do not produce negative reactions in other people. ES people experience a \"spreading\" effect with sensitivity, in that their vulnerability to weaker, shorter EMF inputs increases and they get the reactive symptoms faster.\n\n**Figure 10-3. Typical Field Strengths for Common Appliances and Household Devices 28**\n\nTypical ES symptoms include headache, nausea, fatigue, dizziness, tingling or prickling sensation on the skin, burning skin or eyes, memory loss, concentration difficulties, muscle or joint pain, fluctuations in heart rate, and sometimes paralysis, says Grant. \"Electrical sensitivity has the great potential for being a missing link that uncovers explanations for many noticeable, recurring health problems people currently have no medical answers for and no long-term relief,\" comments Grant.\n\nSources of ES\/EMF toxicity can include fields inside your house\u2014generated by appliances, lighting, wiring, power panels, heating devices\u2014or any electrical devices, or outside, such as the home's proximity to power lines.\n\nA given home may have strong or weak EMFs depending on the situation. For example, if you use an electric hair dryer, you are generating a short-term EMF; it dissipates when the device is turned off. But your electric power panel emits a steady EMF regardless of which household appliance or other electrical device is operating (see figure 10-3).\n\nNote that some household appliances, such as the electric can opener and hair dryer emit very high EMFs at close range. Unless one is already electrically hypersensitive, these shortterm exposures to relatively high EMFs will not produce much harm to the body because the time of exposure is of such short duration: with a can opener; perhaps twenty seconds; a hair dryer, less than five minutes. But the clothes washer and dryer, television, and fluorescent fixtures can be more damaging depending on their location. If any of these items is situated on the other side of a bedroom wall (specifically, against the wall that abuts the head of the bed), or in the case of the fluorescent lights, directly over the bed, then their effect can be cumulative, even though their actual EMF strength is not as strong as the can opener. These are often called electromagnetic hot spots.\n\nThe constancy of human exposure is what produces the damage. What kind of damage? While this is a subject still under intense discussion and research, evidence is mounting suggesting causal links between EMF exposure and immune dysfunction, depression, childhood leukemia, central nervous system cancer, melanoma, and breast cancer.\n\nFor example, researchers at Pacific Northwest Laboratory in Richland, Washington, concluded that exposure to extremely low-frequency electric or magnetic fields (generated by typical 60-Hertz alternating electrical current found in U.S. households) can contribute to the onset of depression and depression-related suicide. They postulated that these fields disrupt the body's normal day\/night rhythm (the circadian cycle) and the chemicals responsible for maintaining it, such as the neurotransmitter serotonin and the hormone melatonin. They further postulated that these low-frequency fields accomplished the disruption by negatively affecting the brain's pineal gland, which regulates the circadian rhythm and its chemicals.\n\nOther research since the mid-1980s has further substantiated the various negative health impacts of prolonged exposure to EMFs of varying strengths or duration. Documented effects include: alteration of brain waves from cell phone signals; increases in leukemia rates in adults and children; interference with hormone and intercell communication activity; a slowing of visual reaction time; impaired nervous system activity; changes in the brain's hippocampus; alterations in immune system function; significant reduction in REM sleep; decline in insulin levels; undesirable changes in the blood-brain barrier; changes in cell cycles and proliferation; decrease in appetite and ability to sleep; and tumor formation.\n\nWaveGuide, a public interest website that summarized medical data on EMFs from dozens of studies, noted that these studies \"indicate biological effects at exposure levels _far below_ what would be explained by 'thermal effects' and well within the range people are commonly exposed to every day.\" Most of the exposures studied, commented WaveGuide, \"lie _far below_ the current advisory exposure standards in the United States.\"\n\nHow does an EMF disturb human health? It may be harmful to human health in much the same way that free radicals upset physiology, suggests Michael Riversong. Electropollution consists of a high level of positive ions in an environment, and positive ions, despite their confusing name, are bad for your health. Riversong suggests thinking of positive ions and free radicals as essentially identical, certainly in regard their biological effects.\n\nA free radical, he explains, is \"simply a piece of a molecule that is missing electrons, so it wants to 'eat' any electrons in the vicinity, and it will take them from living cells.\" A positive ion is also \"hungry\" for more ions, and it will take them from negative ions (the healthy, uplifting ones), which are \"free\" or in excess as electrons. Both positive ions and free radicals have been documented to contribute to the formation of numerous illnesses and bodily dysfunctions.\n\n\"Positive ions in a house result from many things. Any formaldehyde in the air, for example, is a bit of a free radical. This is one of the reasons it is harmful; it outgasses from wood products as a type of positive ion. Another source of positive ions in the house is plain old dust and grit because in many cases it has degraded to the point where it has positive ions on its molecules. And when your house is sealed in, as many modem houses are, you get positive ions from the lack of fresh air circulation. People (and pets) take in all the electrons they can inside a house, so that few are left in the air. What is left are insect parts, dust, and chemicals.\" Riversong explains that houses situated near busy roads are subject to indoor air pollution from the nitrous oxides from automobile exhaust. These, too, he says, are \"essentially positive ions and will 'eat' electrons.\"\n\nHere are some real-life case studies that show the effects of chronic EMF (and thus heightened positive ion) exposure. The first is provided by Veronica Strong, a British health-care practitioner and director of Dowsing for Health in the Vale of Evesham, England. Strong consulted with a woman in her late forties who spent eighteen hours a day in bed, incapacitated. She had suffered from chronic fatigue for three years; she was unable to walk more than fifty yards without feeling exhausted; and she was getting worse every week. Strong determined that the woman's main problem was unrelieved electromagnetic stress. There was a television in her bedroom; its cable ran in the wall alongside her bed, and a TV booster was connected to it. Once these three factors were corrected and relocated, the woman started to recover rapidly, such that within only three weeks, she was able to go on long walks, and now puts in a twelve-hour workday.\n\nMichael Riversong relates a case involving the staff of a small newspaper office. Several staff members had intermittent health problems. They would have mysterious pains that came and went and followed no pattern; the doctors attending these people were unable to make a diagnosis. The newspaper editor said he didn't like being in his office, and was concerned about the high turnover rate in his business. Previous employees and editors had resigned for health problems, and one had died. Riversong measured high magnetic field levels of 10 mG at the editor's desk. Remember, the Swedish standard for \"acceptable\" EMF exposure is only 2.5 mG, so this was four times the safe level.\n\nThe probable reason for the high EMFs, Riversong found out, was that the newspaper office was next door to an appliance store; the EMFs from TVs and refrigerators on display (and thus always on) came through the wall. The editor relocated his office and tried to work as much in other locations as possible.\n\nIn a second case, Riversong consulted with a woman who had severe headaches when she came home from college. A gauss mater reading of the woman's bedroom showed normal EMFs except at the head of her bed where it registered between 5-11 mG. The baffling thing was that there was nothing in her room to account for the high EMF, but when Riversong examined the room immediately on the other side of the wall, he found an old motorized clock on a shelf directly in line with the woman's headboard. The remedy was simple: Riversong moved the clock and the woman's headaches stopped.\n\n\"Sometimes simply moving appliances will take care of problems. We're lucky when we can get off that easily,\" comments Riversong. In other situations, you may need to turn off electrical power at night, or have an electrician install a special relay so that you can turn off the power by a bedside remote switch. \"Many people have experienced sudden, dramatic improvements in health when taking control of their electrical environment in this way.\"\n\nHere are some practical, fairly low-cost devices a householder can employ to minimize or neutralize electromagnetic field exposure. This is not meant to be a definitive list (nor an endorsement of any products), but merely a quick look at some available options.\n\n### Measure Your EMF Exposure\n\nThe more you know about your indoor electromagnetic environment the better informed you are for making a smart choice about a neutralization technique. There are a number of handheld consumer-oriented EMF-measuring devices (in a price range of about $19 to $140) called gauss meters that can be used to get EMF readings in various rooms of your home.\n\n_E.L.F.-Zone Gauss Meter._ This unit has a minimum sensitivity of 0.25 mG, but reads EMFs from 20-20,000 Hz. Its three lights (red, yellow, green) are calibrated to come on when danger, caution, and safe thresholds, respectively, are reached in an environment.\n\n_Dr. Gauss_ : Called \"The EMF Detective,\" this battery-powered device can measure magnetic field radiation from power lines, computers, and kitchen appliances, and other hidden sources of EMF radiation. Its measurement range works on two scales, from 0-1 mG and 0-10 mG.\n\n_Trifield Meter_ : This unit can measure magnetic and electric fields, EMF, and radio\/microwave pollution across an \"extremely wide frequency response,\" according to the manufacturer. Specifically, its magnetic readings range from 0.2 mG to 3.0 mG and 1 mG to 100 mg; electric from 1 kV\/m-100 kV\/m; and RF\/microwave from 0.01 mVV\/cm2 to 1.0 mVV\/cm2.\n\n### Minimize or Neutralize Your Indoor EMF Exposure\n\nOnce you have identified the EMF \"hot spots\" in your home, you can take steps to reduce or eliminate their injurious effects on your health, using a variety of devices.\n\n_Magnetic Shielding Foil (\"MuMetal\")_ : This is an alloy of eighty percent nickel and fifteen percent iron (plus copper, molybdenum or chromium) used for years in industrial applications to shield delicate electronic components from EMFs. For consumer-household use, the material (in fifteen-inch-wide sheets) can be trimmed (in length) to fit the required size and form a magnetic barrier against emitters such as cell phones, microwave ovens, doorbell transformers, computer terminals, and buried wiring. According to the manufacturer, one thickness of the material (0.004 inches) draped over the offending object can shield its EMF emissions by seventy-five percent. Essentially, the MuMetal attracts and holds the magnetic field within its own molecular structure, preventing it from passing any further.\n\n_Mag-Stop Plates_ : These are thick magnetic alloy plates (made of MuMetal; twenty-four by thirty inches, weighing six pounds) used to shield EMF emissions from electric circuit boxes, side-by-side computer terminals (such as in an office), appliances on the other side of bedroom walls, or \"any situation where you need a flat shielding material on a wall, floor, or ceiling.\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about gauss meters, shielding materials, and other EMF measuring devices, contact:\n\nLessEMF Inc.,\n\n26 Valley View Lane,\n\nGhent, NY 12075;\n\ntel: 888-LESS-EMF;\n\nfax: 518-392-1946;\n\ne-mail: lessemf@lessemf.com;\n\nwebsite: lessemf.com.\n\nFor Total Shield, see:\n\nThe Cutting Edge Catalog,\n\nBefit Enterprises, Ltd.,\n\nP.O. Box 5034,\n\nSouthampton, NY 11969;\n\ntel: 800-497-9516 or 516-287-3813;\n\ne-mail: cutcat@cutcat.com;\n\nwebsite: cutcat.com.\n\nFor QLink, see:\n\nClarus Products International, L.L.C.,\n\n1330 Lincoln Avenue, #210,\n\nSan Rafael, CA 94901;\n\ntel: 800-425-2787 or 415-459-3829;\n\nfax: 415-459-2230;\n\ne-mail: info@darus.com;\n\nwebsite: www.clarusproducts.com.\n\nFor more about flower essences, contact:\n\nFlower Essence Society,\n\nP.O. Box 459,\n\nNevada City, CA 95959;\n\ntel: 800-736-9222 or 530-265-9163;\n\nfax: 530-265-0584;\n\ne-mail: mail@flowersociety.org;\n\nwebsite: www.flowersociety.org.\n\n_Total Shield_ : This unit is designed to neutralize EMFs by producing a 7.83 Hz field capable of blanketing a 20,000 square foot area, according to its manufacturer. The 7.83 Hz field is considered beneficial because it is the Schumann Resonance, the \"pulse rate\" of the planet itself. By \"blanketing\" an indoor area with this frequency, the device is believed to overcome the negative effects of EMFs present.\n\n### Shield Your Body from EMF Exposure\n\nIt is now possible to enjoy EMF shielding in your personal body and auric space by portable devices, so that as you move around in your home, into different rooms, each with their own EMF problems, you are still protected.\n\n_QLink_ : One of the more popular of personal EMF shielding devices is called the QLink, a pendant made by Clarus Products, a company founded in 1992 and specializing in subtle energy devices based on a principle they call \"Sympathetic Resonance Technology.\" According to Clarus, the QLink works like a tuning fork or permanent resonator, and can \"mimic the harmonic energy range of a person who is balanced, in a state of well-being, and 'in the zone.'\" Clarus further explains that the QLink requires no battery, but instead is powered by the wearer's own electromagnetic and other bioenergies when it is worn.\n\nWearing the pendant (made from lightweight, biocompatible acrylic) enables the wearer's energy systems to readjust to this example of physiological and energetic balance, and to \"relax into\" it. Its protective field is placed at about eighteen inches around the body. Clarus reports that user benefits include increased mental focus and concentration, heightened stamina and energy during the day, greater emotional balance, reduced emotional reactivity, less jet lag, and less of a \"fried\" feeling after extended computer use. According to Clarus, one-third of users who report noticeable benefits say they observe these positive changes within a week of using the QLink; the rest say it takes a month or so to register significant benefits.\n\nClarus also provides the same protective technology in a ClearWave digital clock, which shields EMFs for up to thirty feet in all directions (or 113,000 cubic feet), including up and down; other models of the ClearWave shield EMFs up to forty and fifty feet in all directions. Clarus' Portable Ally is a battery-driven tabletop device (4 by 2 by 0.75 inches high) that provides EMF shielding for up to 40 feet.\n\n### How Flower Essences Can Reduce Your EMF Stress\n\nThe next time overhead fluorescent lights irritate you, try taking a flower remedy to calm yourself. Research has documented that taking two flower essence formulas can reduce your system's reaction to the intense environmental stimulation produced by fluorescent lights and their accompanying electromagnetic fields. California researcher Jeffrey R. Cram, Ph.D., studied the benefits of using Yarrow Special Formula and Five-Flower Formula, two brand-name mixtures of different flower essences to reduce systemic EMF stress. A flower essence is a highly dilute infusion of flower blossoms taken as liquid drops, and available at most natural foods stores.\n\nDr. Cram monitored the brain waves and their changes in twenty-four subjects who took one or the other remedy in response to. fluorescent light exposure. \"The two flower essences were found to reduce physiological activation and stress on the human organism,\" concluded Dr. Cram. He theorized that these flower essences were able to \"strengthen emotional equilibrium and equanimity in the face of stresses and environmental impacts, thus reducing the typical 'fight or flight' stress response.\" In other words, the flower essence formulas were able to address the person's physiological and emotional reaction to the stress of the electromagnetic fields of the lights.\n\n## HEALTHY LIVING SPACE DETOXIFIER #47 Benefits of Nontoxic Carpets\u2014Don't Let Your Carpets Kill You Softly\n\nOne common assumption is that if you keep an old carpet clean with regular vacuuming and occasional shampooing, you have eliminated the potential health hazards that rugs and carpets may bear. Another common assumption holds that a new carpet is the height of purity and cleanliness and surely a guarantee against the presence of germs and other possibly harmful residents of old carpets.\n\nNeither assumption is correct. Let's start with the truth about new carpets. The alarming fact is that the newer the carpet the more toxic it might be. New carpets, because they are treated with a great number of chemicals and preservatives, can actually outgas toxic fumes and not only slowly but consistently pollute the indoor air quality of a house, but also generate allergies or seriously irritate those already overburdened with allergens. It's a distressing thought: here you are comfortably sitting in your apartment living room, enjoying the brand new plush rug underfoot, and now you learn this same rug might be continuously releasing into your indoor air molecules containing substances that might eventually make you sick.\n\nOf the modern machine-made carpets made from synthetic fibers, ninety percent of these are tufted, with the tufts held in place by a latex substance. Further, the yam libers of such carpets tend to be treated to prevent stains and wear. The carpet's backing material often includes a vinyl coating or a layer of foam-cushioning made from a synthetic material. All of these materials are potential allergens and potentially toxic when outgassed into a room. More specifically, synthetic carpets may contain pesticides (antimicrobial substances), neurotoxic solvents (such as toluene and xylene), and benzene, a carcinogen.\n\nCarpet toxicity is an increasingly serious problem, due to the high degree of carpeted interiors in the United States. In 1993, for example, it was estimated that seventy percent of the floors in the United States had a carpet, that 1.5 billion square yards of carpeting had been purchased in that one year (equivalent to 5.5 square yards per American), and that from 1960 to 1993, carpet purchases had grown by 654% in the United States. So if carpets are a source of indoor toxic emission, a great number of households are potentially at risk, and the greater the amount of indoor carpeting in a home, the greater the chance that indoor air quality is polluted in every room.\n\nAccording to John Bower of the Healthy House Institute, in 1988 the United States Environmental Protection Agency had to remove 27,000 yards of new carpeting from EPA offices after they found that dozens of their employees (about ten percent in all) had become sick from being in contact with the new carpets. EPA research revealed that a carpet substance called 4-phenylcyclohexene (found in the latex backing, abbreviated as 4-PC) was the main toxic substance sickening their employees. The chemical that gives carpets the \"new carpet\" smell, 4-PC has been found in concentrations of 20 ppb in new carpets, although the levels decline to 1-2 ppb by about two months after installation. This toxic substance has been linked to numerous respiratory and mucous membrane problems and allergic reactions, as well as nausea, fatigue, and memory loss.\n\n**_New carpets, because they are treated with a great number of chemicals and preservatives, can actually outgas toxic fumes and not only slowly but consistently pollute the indoor air quality of a house, but also generate allergies or seriously irritate those already overburdened with allergens._**\n\nAt least the EPA figured out it was their new carpets that were sickening their employees, and in fact this led to considerable scientific investigation of carpet toxicity. Unfortunately, in far too many cases, office managers and householders fail to make the connection and remain highly allergic, if not chronically sick or seriously incapacitated, from unsuspected toxic carpet gas emissions.\n\nIn 1995, Rosalind Anderson, Ph.D., a researcher at Anderson Laboratories in Dedham, Massachusetts, reported on her analysis of gas emissions (outgassing) from over 300 different carpet samples obtained from commercial outlets, carpet mills, or patients' homes. None of the carpet samples were older than 40 years and most had been in use from one week to twelve years. Dr. Anderson performed 500 different laboratory tests to measure the effect of carpet outgassing on laboratory mice.\n\nHer results showed that the gas emissions from carpets decreased the breathing rate of mice immediately on contact, from a normal rate of 280 breaths\/minute to 235\/minute, after only 8 minutes of exposure. When Dr. Anderson removed the mice from contact with the gas emissions, their respiration quickly normalized. On further inspection of the mice, Dr. Anderson discovered that one or more exposures to the outgassing produced distress symptoms in the mice, including swelling of the face, bleeding beneath the skin surface, altered posture, balance problems, hyperactivity, tremors, convulsions, limb paralysis, and, in some cases, death.\n\nDr. Anderson next studied 125 carpet samples for neurotoxicity, that is, negative effects on the activity of nerve and brain cells. She found that ninety percent of these samples produced at least one toxic effect and that sixty percent produced three or more \"severe neurotoxic effects\" in at least twenty-five percent of the mice. In other words, nearly all of the carpet samples were able to sicken one-quarter of the mice population in the study.\n\nWhy are new carpets often killer carpets? Because over 200 different chemicals are now found in the typical modem carpet, and many of these chemicals are toxic to humans (and mice), capable of producing \"diverse toxic effects,\" says Dr. Anderson, such as flu-like symptoms, muscle pain, fatigue, headaches (lasting for months after the initial exposure), memory loss, and concentration difficulties, among others.\n\nEven before Dr. Anderson's published research in 1995, evidence had been coming out, in part from her, linking carpet outgassing with illness. In 1991, the New York State attorney general released a consumer alert advising people who smoked, had allergies, or suffered from respiratory disorders that they would be \"more prone to experiencing symptoms when exposed to new carpeting.\" The alert stated that the carpet chemicals were a serious risk for children and the fetuses of pregnant women. Notable symptoms produced by toxic emissions included flu-like feelings, rashes, asthma, respiratory conditions that worsen, and multiple chemical sensitivities. The consumer alert was prompted by the fact that the United States Consumer Product Safety Commission had by 1991 received some 500 consumer complaints about suspected carpet toxicity. Individuals had reported a variety of symptoms, such as chills, fever, burning eyes, nausea, dizziness, vision problems, memory deficits, cough, numbness, depression, nervousness, concentration problems, and others.\n\nConsumer advocate and health educator Debra Lynn Dadd relates a remarkable case study showing the health-damaging effects of new carpets on householders. A man installed seventy-nine yards of new synthetic carpeting in his home-based business and noticed that on the first day, not only was the indoor space filled with a pungent odor from the carpeting, but dozens of spiders had become immobilized on its surface. Formerly, they had been observed to crawl freely through the house. A few months later, the man and his family became sick, reporting headaches, nausea, concentration difficulties, unusual thirst, and mucous membrane burning, including of the eyes.\n\nSoon the symptoms worsened, to include depression, skin rashes, and insomnia. The carpets were removed, but even after almost a year away from them, the family remained hypersensitive to many substances. A laboratory analysis of the carpet showed that it was outgassing at least six toxic substances including formaldehyde, and that another thirty-five toxic substances had been used in the carpet's manufacture.\n\nOf considerable toxicity are styrene (a carcinogen) and 4-PC, found in the styrene butadiene latex backing used on about ninety-five percent of carpets (applied in liquid form at the carpet mill). An estimated two-thirds of the 2.7 billion pounds of carpet yarns sold in the United States as of the mid-1990s were nylon, while only 0.5% were wool. The glue that holds the synthetic carpet fibers together contains 4-PC, the toxic substance discussed earlier.\n\nResearchers writing in the British scientific journal _Nature_ recently reported yet another toxic factor in modern rugs. Carpets can trap benzene from car exhaust and outgas this into the home's indoor air. Carpets (as well as wood and linoleum) are absorbent and can trap airborne benzene, a documented carcinogen, and eventually release it into the air.\n\nBut it's not only new rugs that can trap airborne particles; in fact, old carpets can be troublesome as well. The carpet acts as a kind of floor-based air filter, collecting everything that's in the air, including, for example, hydrocarbons from car exhaust in the street or pesticides from a yard, tracked in by feet onto the rug. Airborne chemical compounds from sources other than carpets (such as paint fumes if a room is repainted while the rug is uncovered) can end up trapped inside the carpets to be released into the air later. Carpets also house dust mites (which leave highly allergenic excrement) and mold and mildew, allergens that can breed in damp carpets.\n\nThere is an additional problem with carpets and house dust. Dust and tracked-in soil accumulate in the carpets, then get released into the indoor air when people walk on them. Researchers examined the house dust collected from homes in North Carolina and found traces of fourteen pesticides and ten polycyclic aromatic hydrocarbons. They further discovered that the smaller the particle of house dust, the greater the concentrations of any of these twenty-four indoor air pollutants.\n\nApparently it is possible to quietly sicken under the influence of toxic carpets without manifesting any overt symptoms for a long time, observes Cindy Duehring, research director for Environmental Access Research Network in Minot, North Dakota. \"Lack of acute reactions to toxic carpeting are common, but the chemicals may be causing serious damage nonetheless,\" Duehring comments. \"The lack of acute adverse reactions (e.g., headaches, breathing difficulties, seizures, etc.) does not mean that the chemicals coming off of the toxic carpet are not causing slow, silent damage.\" Duehring argues that carpeting can \"make a person much more susceptible to getting a chronic illness in the future\" and that sustained exposure to unhealthy carpets sets you up for future problems \"by dosing yourself with dangerous chemicals on a regular basis with carpeting.\"\n\nAs mentioned already, the main problem with synthetic carpets is their outgassing of toxic substances known as a group by the term volatile organic compounds, or VOCs. Technically, this group includes hundreds of organic compounds present in the Earth's atmosphere (containing hydrogen, carbon, and other elements); these compounds evaporate easily, hence the designation volatile. Outgassing refers to when VOCs evaporate or release gases from the substrate in which they are present (such as in the latex finish of carpet backings). Natural sources of VOCs include fossil fuel deposits, oil sands, volcanoes, trees, even bacteria. Human-made VOCs tend to be toxic, or more toxic than naturally emitted ones, and derive from motor vehicles, solvent use, industrial processes, and gasoline evaporation.\n\nMore specifically, toxic VOCs are found in\u2014or outgassed from\u2014carpets, photocopier machines, paints, solvents, varnishes, adhesives and glues, spray shoe polish, water repellents, dry-cleaning fluids and chemicals, underarm deodorants, and many others. Here are but a few examples of VOCs found in common household products: tetrachloroethylene (dry-cleaned clothes); chloroform (chlorinated water); benzene (tobacco smoke); formaldehyde (fabrics, pressed wood products, building insulation, even cosmetics); styrene (carpets, plastics); and para-dichlorobenzene (moth balls and deodorizers).\n\nGiven the nature of modern, tightly sealed homes and the lack of proper ventilation, it is not uncommon for a typical indoor air to contain VOC levels 100 to 1,000 times higher than outside air pollution levels. Put differently, the EPA reported that the average American home was likely to have VOC levels two to five times higher than outside, regardless of whether the homes were situated in a rural area or in cities. It is a shocking thought that the air outside the home may be less polluted than indoors. Further, in 1989, based on a study of indoor air quality in ten newly constructed public access buildings, the EPA reported that over 900 VOCs were identified in the ambient indoor air.\n\nAs discussed previously, one of the new, serious, and medically vexing conditions to emerge as a result of prolonged exposure and allergic reaction to VOCs is multiple chemical sensitivity syndrome (MCS). This condition has multiple symptoms, affects multiple body organs and senses, and has multiple causes, thereby making it impossible for conventional medicine to diagnose according to its standard single-causative agent approach.\n\nTypically, MCS produces many symptoms involving the neurological, immune, respiratory, skin, gastrointestinal, and\/or musculoskeletal systems. Medical studies show that about sixty-six percent of people suffering from chronic fatigue and fibromyalgia (chronic muscle pain) also have MCS. Studies report that typical symptoms include all of the ones reported above (in connection with carpets) as well as sleeping and breathing difficulties, memory loss, mood swings, mental confusion, sneezing, itching, drowsiness, wheezing, runny or stuffy nose, muscle and joint pain, vaginal burning, frequent urination, constipation or diarrhea, joint and limb swelling, migraines, stomach pain, impaired balance, and increased sensitivity to odors, loud noises, bright lights, touch, extremes of heat and cold, and EMFs.\n\nUsually, a person sustains an initial acute or chronic toxic exposure, which shocks the system and overloads the central nervous system and liver, then the initial sensitivity broadens or spreads to include reactivity to many more substances commonly found in the modern Western environment. It does not have to be a high dose once the system is overburdened with toxicity. Once the triggering sensitivity happens, low exposures to common chemicals can generate major allergic reactions, usually out of proportion to the dose of the exposure. Eventually, you become allergic (chemically sensitive) to seemingly everything chemical. Perhaps this is why some medical authorities think MCS would be better labeled Toxicant-Induced Loss of Tolerance to indicate a toxic exposure that leads to the loss of tolerance to widely encountered chemicals to which you would not expect a reaction.\n\nAccording to William J. Rea, M.D., director of the Environmental Health Center in Dallas, Texas, and for many years an expert in this field, chemical sensitivity is \"an adverse reaction to ambient doses of toxic chemicals in our air, food, and water at levels which are generally accepted as subtoxic.\" Another study (with mice) concluded: \"A detailed chemical and microbial evaluation of the carpets and carpet emissions showed volatile organic compounds, pesticide residues, and microbiological flora, but at insufficient quantities to result in acute toxicity.\"\n\nUnfortunately, what is nontoxic and tolerable, or of \"insufficient quantity\" to one individual, can be quite toxic, sufficient, and intolerable to another. The trouble here is you can't know until your system has already reacted; another problem is that toxicity is cumulative and may take a long time to produce \"acute\" effects; a third problem is that not every individual corresponds to whatever norm for toxicity toleration is used in scientific studies from which safety standards are determined.\n\nIn fact, yet another study confirmed this effect. Scientists at the Centers for Disease Control and Prevention in Atlanta, Georgia, measured VOCs in human blood, noting that they are found in the general population at the high parts-per-trillion range, \"but some people with much higher levels have apparently been exposed to VOC sources away from the workplace.\" Of especial significance here was the CDC's observation that while \"most of the internal dose of these compounds is quickly eliminated [once they're in the blood], there is a fraction that is only slowly removed, and these compounds may bioaccumulate.\"\n\n**_The MCS-afflicted person becomes in effect allergic and potentially reactive to almost anything synthetic. In fact, the growing body of statistics on MCS is sobering: 17% to 34% of Americans report symptoms of chemical sensitivity._**\n\nThe MCS-afflicted person becomes in effect allergic and potentially reactive to almost anything synthetic. In fact, the growing body of statistics on MCS is sobering: 17% to 34% of Americans report symptoms of chemical sensitivity; 6.3% of Californians have been given an MCS diagnosis; up to 80% of these also have chronic fatigue syndrome, 55% to 65% have fibromyalgia, and 50% have traditional allergies.\n\nIn fact, there is such a degree of symptom overlap among MCS, chronic fatigue, environmental illness, fibromyalgia, and acute allergic reactivity that one wonders if there are real distinguishing differences between these conditions or if they are simply a variation on the same theme of sustained chronic toxicity. A study of 100 new MCS cases showed that 88% also had the symptoms of chronic fatigue syndrome, 49% had fibromyalgia symptoms, and 47% had all three. \"These extensive overlaps highlight the need to screen patients for all three disorders whenever anyone is suspected,\" concluded Albert Donnay, executive director of MCS Referral & Resources in Baltimore, Maryland.\n\nWhatever the outcome of this question of overlapping disorders, the fact remains that, for the householder, uncorrected long-term exposure to toxic factors in the indoor living space can lead to any of these conditions in the extra-sensitive person or in one whose system is overburdened with toxicity from other sources. In fact, if you find yourself suffering from several of these conditions, it is worth considering that your house may be making you sick, that some element in your indoor living space is toxic.\n\nOne way to determine this possibility on a physiological basis is to have one of the many biochemical tests for toxicity detailed in Chapter 3. Then if you test positive for toxic contaminants in your system, it is prudent to consider a comprehensive physiological detoxification program (see chapters 4\u2014) to flush the poisons out of your body and to help your liver regain its natural ability to detoxify your body. In the meantime, here are some practical steps to reduce your exposure to potentially toxic carpets.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about product emissions testing, contact: Anderson Laboratories,\n\nP.O. Box 323,\n\n773 West Hartford Main Street,\n\nWest Hartford, VT 05084;\n\ntel: 802-295-7344;\n\nfax: 802-295-7648;\n\nwebsite:\n\nwww.andersonlaboratories.com.\n\nFor Nature's Carpet:\n\nColin Campbell & Sons Ltd.,\n\n1428 West 7th Avenue,\n\nVancouver, BC, V6H 1C1, Canada,\n\ntel: 800-667-5001 or 604-734-2758;\n\nfax: 604-734-1512;\n\nwebsite: www.colcam.com.\n\nFor Hendricksen Nat\u00fcrlich, contact:\n\nNatural Home,\n\nP.O. Box 1677,\n\nSebastopol, CA 95473;\n\ntel: 707-824-0914;\n\nfax: 800-329-9398;\n\ne-mail: nathome@monitor.net;\n\nwebsite:\n\nwww.naturalhomeproducts.com.\n\n### Test Your Carpet for Toxicity\n\nIf you suspect your carpets are making you sick, you can have a professional toxicology laboratory test a sample to confirm or refute your suspicions. The same laboratory that produced the startling research by Rosalind Anderson in 1995 now offers product emissions testing of a range of items including carpets, mattresses, diapers, wallpaper, paint, adhesives, mothballs, insulation, air fresheners, perfumes, boots, even telephones.\n\nA client sent in her telephone because she believed somehow it was making her sick. She was right: Anderson Laboratories found a cotton pad saturated with pesticide inside the mouth piece. The lab also reported that a survey of the first one hundred families who sent them carpets for emissions testing all had new symptoms arise after installing the carpets and these symptoms were consistent with those associated with sick building syndrome and multiple chemical sensitivity. Laboratory analyses of product emissions range from $750 to $3500 depending on the extensiveness of the data requested.\n\n### Do a \"Bake-Out\" on New Carpeting\n\nIf you are unable to remove new carpeting or must move into a new, renovated, or refurnished house or apartment with unremovable new carpets, one option is to do a carpet \"bake-out.\" If the carpets have not yet been installed, ask the carpet provider to air the carpets in a well-ventilated area for seventy-two hours before bringing them to your home. Once the carpets are installed, stay away from your home for a few days. Before you leave, open all the windows, run the ventilation system (such as exhaust fans) at full capacity, and set the indoor heat at 100\u00b0F (38\u00b0C). The idea is to set the heat and ventilation systems to rim at much higher than normal levels\u2014overheating and overventilating. Let them run at that level for about forty-eight hours.\n\nThe theory behind the home bake-out is a bit like the self-cleaning oven: heat the house up and force the VOCs out of the carpets and, through the open windows, out of the house. It is believed that the high temperature forces the VOCs to be released quickly, compared to normal or ambient indoor conditions under which the VOCs are released more slowly and over a longer period, even up to years. Some studies have shown that a bake-out can reduce VOC emissions by sixty-five percent. However, other experts suggest that the bake-out may not have an effect on certain compounds such as formaldehyde, 4-PC, and BHT, which do not decay rapidly.\n\nAfter the bake-out, use a balanced mechanical home ventilation system, such as a heat recovery ventilator, that will continuously exhaust the indoor air (containing VOCs from all sources, including carpets) and replace it with fresh outside air. The recommended replacement rate of indoor with outdoor air is that thirty-three percent of the home's air should be exchanged every hour. This approach will reduce your exposure to airborne VOCs. However, healthy building house expert John Bower comments: \"Bake-outs aren't effective with long-term outgassing sources such as particle board and medium-density fiberboard.\"\n\n### Do a Nontoxic Carpet Cleaning\n\nA thorough cleaning of your carpets can reduce the likelihood of their contaminating your indoor space. Obviously a rigorous vacuuming is in order, followed by a hot water extraction and\/or steam shampooing. Annie Berthold-Bond, an expert in environmentally safe options and author of _Better Basics for the Home_ , recommends renting a steam-cleaning water-extraction machine and using your own nontoxic detergents for cleaning your carpets.\n\nAlmost always, commercial carpet cleaners will use detergents that have some amount of toxic substances such as perfumes, additives, or antimicrobial pesticides. As a nontoxic alternative, Berthold-Bond suggests mixing one-fourth cup of perfume-free, all-purpose liquid detergent (she advises using a product called Infinity Heavenly Horsetail, available at natural foods stores) with about four gallons of water. She also uses antifungal essential oils in the rinse water, notably tea tree oil (a natural essential oil known in aromatherapy as a powerful disinfectant; available at most natural foods outlets). Tea tree oil can also be applied as an antifungal carpet spray (not to be rinsed off) in a ratio of two teaspoons to two cups of water.\n\n### Try Woven Wool Carpets\n\nIf your budget allows it, you might consider replacing your synthetic, allergenic carpets with a natural fiber alternative, such as woven wool carpets. Although woven wool carpets may still contain yarn that has been treated with pesticide mothproofing, it is possible that even if you react adversely to synthetic carpets you may be able to tolerate these. In addition, woven wool carpets use a great deal less latex than conventional synthetic latex-backed carpets; woven wool carpets also tend to have fewer outgassing VOCs than the average synthetic carpet with a glued backing.\n\nA number of companies now offer natural woven wool carpets for the environmentally sensitive. One such firm is Colin Campbell & Sons of Vancouver, British Columbia, which has made their MCS-friendly \"Nature's Carpet\" since 1992. The company reports that their \"clean scoured wool\" is \"totally biodegradable\" and hasn't been subject to harsh chemicals during the mothballing stage of regular wool products; they apply a nontoxic latex made from a natural rubber base to the undersides of the carpets; no chemical dyes are used to color the yams; and the dual backing of the carpet is made from natural jute (a hemp product).\n\n\"The complete absence of chemicals, at every stage of the carpet's manufacture and composition, means there is no toxic outgassing,\" explains Colin Campbell & Sons. An independent laboratory testing of Nature's Carpet in 1996 (by Anderson Laboratories) confirmed the company's claims. It found no evidence of sensory irritation or pulmonary iritation produced by a sample of the company's 100% wool carpet.\n\nAnother highly regarded source of natural carpets is Hendricksen Nat\u00fcrlich of Sebastopol, California. Since 1989, this company has been providing environmentally friendly carpets made from natural wool fibers. The carpets are free of formaldehyde, stain-repellents, flame retardants, and synthetic blends, and contain minimal outgassing glues.\n\nWool has many merits for the allergic, notes the company, for after all, it is the original that synthetics strive to copy. \"Wool releases soil more easily than any other fiber, has the greatest resilience, absorbs noise and noxious gases from the air, and resists burns.\" In the warm months, it acts as a natural air conditioner by absorbing and releasing humidity and by definition is produced without hazardous chemicals, states Hendricksen Nat\u00fcrlich. According to a list they publish, obtained from a carpet manufacturer, at least forty-two potentially toxic chemicals are \"commonly\" used in carpet manufacture.\n\n### The Trouble with Indoor Air\u2014Worse Pollution Indoors than Out\n\nEver since the outbreak of Legionnaire's disease in Philadelphia in 1976, attention has increasingly focused on the relationship between indoor air quality and human health. In that initially baffling outbreak in Philadelphia, a microbial contaminant was distributed throughout a hotel's interior by way of the ventilation system, sickening many of those guests who breathed the air. It's an extreme example that illustrates the need for fresh, pure, constantly ventilated indoor air, and the consequences of not having it.\n\nAs we discovered above, if you live in an apartment or a tightly sealed new home and have new or even recently installed wall-to-wall carpets, potentially you are breathing air contaminated with numerous VOCs outgassed from the carpets. However, carpets are not the only source of airborne pollutants found in the typical modern home, although they are certainly a vivid example of how the mechanism of indoor air pollution works. Unfortunately, there are a great number of things in the modern home that can poison you.\n\nIndoor air quality can also be negatively affected by other factors including combustion gases, carbon monoxide, VOCs from other sources, formaldehyde, radon, biological contaminants, dust mite fragments and feces, molds, pollen, minerals and metals (notably asbestos, found in building materials; and lead, found in some paints, dinnerware, food cans, furniture, toys, dust, and soil), cleaning products made from synthetic chemicals, synthetic fibers and fabrics, household furnishings, pesticides, and tobacco smoke, among others.\n\nFor example, combustion gases (invisible gases left over from burning substances) can come from gas and oil furnaces, gas dryers, water heaters (gas or oil), and boilers (gas or oil). Carbon monoxide (formed as a result of the incomplete burning of carbon), can be released inside a home from fireplaces (wood, natural gas), stoves (coal or wood-burning), heaters (natural gas, propane, or oil), and hot water tanks. Radon, a radioactive gas from inside the Earth, can enter a home through cracks in the foundation and through well water.\n\nFormaldehyde can outgas from urea-formaldehyde foam insulation, glues used in manufactured pressed-wood products (particleboard, plywood, fiberboard), clear finishes, resin treatments on fabrics, durable press drapes, environmental tobacco smoke, paint preservatives, veneered kitchen cabinets, and wet fingernail hardeners and polishes, among others. Symptoms of formaldehyde toxicity include watery eyes, a burning sensation in the eyes and throat, nausea, breathing difficulties, chest tightness, skin rashes, allergic reactions, and, for the extrasensitive, asthma attacks; it is also regarded as a carcinogen.\n\nA recent study of formaldehyde emissions from fifty-five domestic consumer and construction products measured over a twenty-four-hour period showed that among the high emitters are polyurethane floor finishings, wet fingernail hardeners and polishes, permanent press shirts and draperies, and pre-pasted wallpaper. They found that when wet, the polyurethane can emit one thousand times more formaldehyde than bare particleboard, usually considered a high emitter.\n\nTypically, formaldehyde is present in both indoor and outdoor air at an assumed \"safe\" level of less than 0.03 ppm, but it becomes problematic for humans when it reaches airborne levels greater than 0.1 ppm. The less circulation of indoor air there is, and the less an infusion of fresh outside air, the greater the chance that formaldehyde levels will build to toxic concentrations. Further, formaldehyde levels rise when the rooms are humid and hot, and drop when they are cool and dry, arguing for the desirability of indoor dehumidification.\n\nFor some people, a tiny amount of formaldehyde may be way too much. According to the EPA, ten percent of all individuals are hypersensitive to formaldehyde, reacting allergically at much lower concentrations than are generally considered safe. Hypersensitives can react severely to formaldehyde at a 0.1 ppm even though the \"permissible\" work place level established by OSHA (Occupational Safety and Hazard Administration) is much higher, at 0.75 ppm.\n\n**_Volatile organic compounds (VOCs) in indoor air can number from 20 to several hundred\u20141,000 different VOCs have been detected in indoor air altogether, drawn from many samples\u2014and their concentrations inside tend to be larger than you would find outdoors, sometimes 100 times higher._**\n\nThe above are only the most well-known offenders on the list of possible indoor pollutants. In fact, one researcher has tabulated seventy-nine different organic chemical compounds likely to be found in a typical indoor environment (home or office). You're likely to encounter these in: refrigerants, linoleum floor covering, epoxy paint, gypsum board, polyurethane wood finish, moth crystals\/deodorizers, floor wax, cement flagstone, wallpaper, and plastics, among many other sources. VOCs in indoor air can number from 20 to several hundred\u20141,000 different VOCs have been detected in indoor air altogether, drawn from many samples\u2014and their concentrations inside tend to be larger than you would find outdoors, sometimes 100 times higher.\n\nAs mentioned earlier, studies have shown that indoor air can often be more polluted than outside air, containing a higher percentage of a variety of airborne contaminants. Using air conditioning rather than natural ventilation (open windows, fans) can make the situation worse, contributing to the onset of numerous sick building syndrome symptoms. Here's a look at some of the scientific findings on indoor air quality (LAQ) and illness:\n\n  * Researchers in France found exposure to air conditioning in offices (versus natural ventilation) was associated with an increased rate of SBS symptoms and sickness absence.\n  * A study involving 2,761 office workers from five different buildings showed that SBS symptoms decreased by forty percent to fifty percent after the workers were relocated to a single new building with superior indoor ventilation. Their old building had relied on mechanical ventilation, air conditioning, humidification, and had sealed windows. Among the symptoms that improved with better indoor air quality were skin and eye irritations, respiratory problems, nose and throat irritations, fatigue, and headache. Even better, sixty percent of the workers who were originally symptomatic were free of all symptoms once they were in the new, properly ventilated building.\n  * Taiwanese researchers looked at indoor air quality and respiratory illness symptoms in 264 nursing workers at 28 day-care centers and found a strong connection between dampness (indoor humidity) and SBS symptoms. The dampness concentrated airborne fungi and molds (notably _Aspergillus_ strains), and those centers that had no air conditioning or air cleaning systems had higher concentrations.\n  * Among 1,144 office workers, those working under ventilation conditions involving mechanical heating and ventilating, air conditioning, or fan coil units had a higher risk of experiencing SBS symptoms than those people working under conditions of natural ventilation. Symptoms included throat irritation, nasal discharges, blocked nose on awakening, migraines, and coughing.\n  * German researchers studied the role of airborne dust and SBS symptoms in a group of 133 office workers who had complained of symptoms after sound-absorbing mineral fiber boards were installed as suspended ceilings in their work spaces. More than half the 133 employees (79 in all) had complained of itching, eye irritation (burning, reddening), and upper respiratory tract irritations, and 50% of the employees had consulted a doctor for treatment of their symptoms. It turned out the fiber content in the office air was quite high, sufficient for the researchers to link it with the health complaints.\n  * Another Taiwanese study showed that inside building dampness is directly correlated with eye irritations, coughs, lethargy, and fatigue, based on a study of 1,237 workers in 19 air-conditioned offices. Reports of eye irritation were higher when stuffy odor or mold was present, still higher when both were present, higher still with the presence of water damage in the building, and the highest when all \"dampness exposure\" factors were present.\n  * According to research at Texas Tech University Health Sciences Center in Lubbock, Texas, buildings with IAQ complaints often have high airborne concentrations of _Penicillium_ strains (representing 89% to 100% of the total fungi), while buildings with a low incidence of IAQ problems have an indoor air \"fungal ecology\" (the overall balance of biological contaminants) similar to what's found outdoors. Fungal populations may vary outdoors, but the indoor fungal populations \"tend to remain unchanged,\" researchers concluded.\n  * An earlier study by the same group looked at forty-eight schools in which there had been concerns about indoor air quality and SBS symptoms. While five families of fungi (including _Penicillium_ ) comprise about ninety-five percent of the typical outdoor air, in twenty schools _Penicillium_ was the dominant organism found in the indoor air and at rates higher than found outdoors. In eleven schools, significant amounts of _Stachybotrys atra_ were isolated in the classrooms (under wet carpets, wet walls, behind vinyl wall coverings), leading researchers to conclude that SBS symptoms were associated with the presence of these two fungi.\n  * On the positive side, researchers at McGill University in Montreal, Quebec, found that when office workers had a new ventilation system that allowed them to individually adjust the ventilation at their worksite, their productivity rose by eleven percent and SBS symptoms declined when measured sixteen months later. In other words, bad indoor air quality leads to lost productivity and illness, while proper indoor ventilation can lead to enhanced productivity and better health.\n\nAs the research cited above suggests, molds and fungi are a big problem in the indoor environment. Experts are paying more attention to household fungi and toxigenic molds as possible main contributing factors in sick building syndrome and its many allergy-like symptoms. One fungus that is now particularly under study is called _Stachybotrys chartarum_ , a toxic fungus that can cause health problems in humans and animals. For example, _S. chartarum_ can produce skin rashes, nervous system complaints, nausea, vomiting, diarrhea, gallbladder-like colic pain, nosebleeds, and numerous respiratory problems typical of mild to strong allergic reaction.\n\nHow does it get into the house? It enters the indoor living space when the house has sustained water damage from broken pipes, flooding, roof leaks, sewage backups, or excessive condensation. These are ideal growth conditions for the fungus, which is present in soil, dust, dirt, the paper covering of sheetrock, wallpaper, ceiling tiles, various paper goods, natural fiber carpets, insulation material, wet cellulose products, and other organic debris. The fungus can grow in the damp conditions yet not be noticed by occupants; when observed, it is black, shiny if wet and powdery if dry.\n\nAmong the mostly commonly occurring microbes that can infest homes, offices, and schools are _Cladosporium, Altemaria, Fusarium, Penicillium_ (4 species typically encountered), and _Aspergillus_ (fifteen different species commonly encountered in homes). Often you can smell their presence before you actually spot them, in the form of a foul, musty odor when you enter the room. According to researchers at Georgia Tech's Indoor Environmental Research Program, in many respects the air pollution from molds and fungi is not much different than that generated by hazardous chemicals. \"Many of the volatile compounds produced by the cultured fungi are identical to those originating from solvent-based building materials and cleaning supplies.\"\n\nIn fact, the volatile organic compounds from molds and fungi can include hexane, methylene chloride, benzene, and acetone, thereby adding \"heavily\" to a building's existent level of toxic VOCs. In one case, the researchers found high levels of hexane in the indoor air but could not find an actual emitting source until they factored in the possibility of the ambient hexane levels being the result of microbiological contamination.\n\nOne of the most likely places to find any of seventeen fungi growing indoors is in some part of the home's HVAC system (heating, ventilation, air conditioning). For example, _Acremonium spp._ grows in humidifier water and HVAC fiberglass insulation; _Altemaria spp._ in cooling systems, refrigerator coils, and on dust in ductwork; _Aspergillus spp._ in an evaporative air cooler, HVAC fiberglass insulation, cooling systems, coils, fans, filters, and dust in ductwork; _Fusarium spp._ in humidifier water; _Penicillium spp._ in air conditioners, HVAC ducts, filters, fans, and humidifier water. The same is true of at least eight harmful bacteria that grow indoors: _Micropolyspora faeni_ flourishes in the humidifier; _Pseudomonas aeruginosa_ in the evaporative air cooler, humidifier, and indoor dust; and _Thermoactinomycetes vulgaris_ in air conditioners and humidifier water.\n\n## HEALTHY LIVING SPACE DETOXIFIER #48 Filter Your Home's Indoor Air to Remove Toxins and Allergens\n\nThe scientific reports give a vivid and sometimes chilling picture of the degree to which elements of our indoor living space can make us sick when we live in a home or work in an office that lacks good air circulation. Not necessarily every home or interior work or living space is toxic, or toxic to the degree that it guarantees some degree of illness. On the other hand, most of the VOCs are invisible, and many are odorless, so how can you tell if you're living under their baleful influence? One way is home testing.\n\n### Be a Professional House Doctor\n\nA company in Iowa is now providing a set of seven inexpensive do-it-yourself Healthy Home Test Kits by which you can monitor indoor levels of molds and bacteria, lead, carbon monoxide, radon, asbestos, and formaldehyde. The kits range in price from about four dollars to forty dollars. No special knowledge of laboratory procedures or instrumentation is required to get reliable results, the company (Professional House Doctors, Inc., or PHD) explains.\n\nFor example, to test the levels of molds and bacteria, you expose a culture plate to the indoor air for fifteen minutes, store it in a dark place for seventy-two hours, then compare the results (that is, what has grown in the plate) against a chart to find out what contaminants you have. For formaldehyde, you expose the monitor to your indoor air for eight hours, then send it to the PHD laboratory for evaluation. For lead, you dip the lead test dauber in water and rub it on the item in question; if the item or dauber turns pink, the item contains lead. The carbon monoxide test is even easier: when the \"CO Alert Badge\" changes from tan to gray\/black, it means you have too much carbon monoxide in your indoor air; the badge returns to its normal color when exposed to fresh air.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about Healthy\n\nHome Test Kits, contact:\n\nProfessional House Doctors, Inc.,\n\n1406 E. l4th Street,\n\nDes Moines, IA 50316,\n\ne-mail: info@prohousedr.com;\n\nwebsite: www.prohousedr.com.\n\nFor formaldehyde monitors, contact:\n\nAir Quality Research,\n\n100 E. Main Suite, C,\n\nCarrborro, NC 27510;\n\ntel: 800-818-5894;\n\ne-mail: ppecevich@mindspring.com;\n\nwebsite: www.airqualityresearch.com.\n\nMonitors specifically for formaldehyde are available for home and office use as originally developed by Air Quality Research, an organization founded in 1982 at the School of Public Health at the University of California at Berkeley. One unit (PF-1) determines indoor levels of formaldehyde in work and home spaces; the second unit (PF-20) is a clip-on device to measure individual exposure levels for people in a work space.\n\n### Decrease Your Mold Exposure\n\nIf you suspect mold in your indoor living space, a key preventive step to take is keep the indoor humidity level below forty percent and ventilate (with exhaust fans) all showers and cooking areas. Use a humidifier or air conditioner during the humid months; add mold inhibitors to indoor paints before applying to the walls if you are repainting; scour the bathrooms with mold-killing products (nontoxic, nonallergenic: see below); refrain from carpeting basement and bathroom floors; remove any carpets and upholstery that have sustained water damage.\n\nIf your home has walls or floors that have sustained water damage, disinfect the surface with a chlorine solution of one cup of laundry bleach per gallon of water (provided you're not allergic to chlorine). It is prudent to do this with adequate ventilation, wearing goggles and rubber gloves.\n\nIt is essential to regularly clean the water reservoir in humidifiers because they can act like \"little mold factories . . . pools of standing stagnant water\" that during a season can allow mold to grow, infiltrate the ducts, and get blown through the house.\n\nThe most effective step to take to purify your indoor air of VOCs is air filtration, and here you have several choices. These include mechanical filtration by way of HEPA (high efficiency particulate air) filters, either whole house or individual room systems; electrostatic filtration, by electronically charged plastic panel filters inserted into a home's central heating system; ultraviolet radiation; and negative ion generators.\n\nBefore we review the different technological options for air filtering, let's get a sense for how fine the filtration mesh must be, which is to say, how small the toxic airborne particles are. The key measurement in this domain is the micron, which is equal to 1\/25,000 of an inch. The HEPA systems are so called because they can deal with particles as small as 0.1-0.3 microns in diameter, but is that fine enough? Yes, for most microorganisms.\n\nAmong the airborne viruses, adenovirus (associated with the common cold) is 0.08 microns in width; rhinovirus (also with colds) 0.023; influenza (flu) is 0.1; respiratory syncytial virus (pneumonia) is 0.22; paramyxovirus (mumps) is 0.23; varicellazoster (chickenpox) is 0.16; and mycobacterium tuberculosis (TB) is 0.86. Now for bacteria: _Acinetobacter_ (opportunistic infections) is 1.3; _Corynebacterium diphtheriae_ (diphtheria) is 1.0; _Klebsiella pneumoniae_ (opportunistic infections) is 0.4; _Staphylococcus aureus_ (opportunistic infections) is 1.0; and _Streptocococcus pneumoniae_ (pneumonia, otitis media [ear infection]) is 0.9. Among the fungi, _Cryptococcus neoformans_ (cryptococcosis) is 5.5 microns wide, considerably larger than the viruses and bacteria.\n\n### HEPA Mechanical Filtration\n\nThis approach is generally rated as capable of removing 99.99% of airborne particulates, such as dust, pollen, plant and mold spores, asbestos dust, animal hair, and tobacco smoke particles, if they are no smaller than 0.1 microns in diameter. There are _many_ HEPA filters available, so the ones mentioned here are presented only to give the reader an idea of how a HEPA unit works and where to start looking for a suitable model.\n\nThe Safe Zone 2500 (weight: fifteen pounds) can handle a medium-sized room of about 300-650 square feet and uses both a carbon-activated and a HEPA filter. The activated carbon canister (weight: 8 pounds) is especially good at removing indoor odors, such as in a kitchen. One user stated that the unit \"cut the dust buildup on our furnishings by probably 70%.\" The Safe Zone 3000 (weight: 22 pounds) air-cleans a 400-850 square foot room; one user employed it to ventilate her home nail salon business.\n\nThe Safe Zone 4000 is a whole-house, three-stage filtration unit that works with a home's HVAC system (heating, ventilation, and air conditioning) to clean and circulate indoor air. In addition to the HEPA and charcoal filters, the unit's third filter is a VOC cartridge capable of removing dangerous chemical gases outgassed from household products and fixtures. At 56 pounds, it is not as readily portable as the other Safe Zones, but is rated to air-filter 560-1100 square feet.\n\n### Activated Charcoal Air Filtration\n\nIn some instances, the hypersensitive person might require a non-HEPA indoor air filtering system. Chemical fumes and pollutants can contaminate the HEPA filter such that it begins to smell of them\u2014a perfume, a new wall paint, or whatever chemical particulate it has absorbed. The Aireox Activated Carbon Air Purifier is recommended when a person could become allergic (chemically reactive) to the substances collected in the HEPA filter or when the primary offending substance is chemical fumes.\n\nThe carbon filter\u2014the unit can be set unobtrusively on a desktop\u2014can remove 99.95% of dust, pollen, and particulates to 0.5 microns, as well as most indoor fumes and odors. According to the manufacturer, \"if your work place is loaded with VOCs, consider the activated carbon purifiers as the safest air purification.\"\n\n### Electrostatic Air Filters\n\nThe concept here is to remove particles from airstreams having large steady flow rates, such as you have with indoor heating or air conditioning. Air filters are placed within a home's central heating and\/or air conditioning system to catch tiny airborne allergenic particles before (or as) they are blown through the ducts. In a sense, the furnace or air handling system becomes a whole-house air-cleaning device. The device is inserted in the place normally occupied by the filter in the existing furnace or air conditioner; the normal filter is usually fiberglass and removes only five percent to ten percent of particles.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about Electrostatic Air Filter and OxyClean Air Filter, contact:\n\nCrystal Enterprises,\n\n770 Big Tree Dr., Building 114,\n\nLongwood, FL 32750;\n\ntel: 800-724-2220;\n\nfax: 407-260-9690;\n\ne-mail: webmaster@oxyfilters.com;\n\nwebsite: www.oxyfilters.com.\n\nFor Safe Zones 2500,3500, and 4000:\n\nAirFilters.net,\n\n11905 NW 35th Street,\n\nCoral Springs, FL 33065;\n\ntel: 800-757-1836 or 954-752-1836;\n\nfax: 954-752-0113;\n\ne-mail: info@airfilters.net;\n\nwebsite: www.airfilters.net.\n\nFor Dust Fighter 95 and Bio-Fighter Anti-Microbial UV Light System (Nomad, Nomad 2, Triad):\n\nDust Free,\n\nP.O. Box 519,\n\n1112 Industrial,\n\nRoyse City, TX 75189;\n\ntel: 800-441-1107;\n\nfax: 800-929-9712;\n\nwebsite: www.dustfree.com.\n\nThe Electrostatic Air Filter removes 99.97% of indoor air allergens of 1.3 microns in size and larger. Its surface has been treated with an antimicrobial substance to deactivate bacteria, molds, fungi, and other microbiological contaminants as they pass through it. The company also makes an OxyClean Air Filter, which they claim can remove foul odors and harmful gases present in indoor air. The filter can be used with activated charcoal pads to remove odors and gases.\n\nAccording to the manufacturer, the filters use the magnetic effect of static electricity and special filtering media to attract and trap airborne particulates; in other words, the static electricity on the fibers of the filter attract the contaminants and, as the air blows across these electrically charged fibers, the static charge actually increases. The filter does not require replacement, but maintenance every thirty to sixty days (washing it out with a garden hose) is recommended.\n\nAnother brand called Dust Fighter 95 uses five layers of filtration media to trap about ninety-five percent of particulates passing through it. According to Dust Free, the manufacturer, numerous clients with allergy problems have reported improvement after using the air filters. A nurse (with allergies) who works with allergy patients found the filters \"made a noticeable difference\" in the amount of dusting she had to do in her'home. A child with post-nasal drip and leg pain every morning found that these symptoms abated over the course of a few months once the unit was installed in his room.\n\nAn asthmatic child who started coughing the minute the home's central air conditioning blew dust through her room, had considerable reduction in her asthma attacks once a filter was in place. A woman said she no longer woke every morning with a sinus headache and dry sinuses; another user said the unit made it \"so much easier to breathe\" in her home; and still another said the filters reduced \"moderately severe\" allergies to house dust and animal fur.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor a source of numerous products dealing with all the topics mentioned in this chapter, as well as air filtration, see:\n\nThe Cutting Edge Catalog,\n\nBefit Enterprises,\n\nP.O. Box 5034,\n\nSouthampton, NY 11969;\n\ntel: 800-497-9516 or 516-287-3813;\n\nfax: 516-287-3112;\n\ne-mail: cutcat@cutcat.com;\n\nwebsite: www.cutcat.com.\n\nFor Aireox Activated Carbon Air Purifier, Care 2000 Air Purifier, and a general catalog of environmentally-friendly products and equipment: Nirvana Safe Haven,\n\n3441 Golden Rain Road, Suite 3,\n\nWalnut Creek, CA 94595;\n\ntel: 800-968-9355 or 925-472-8868;\n\nfax: 925-938-9019;\n\ne-mail: daliya@nontoxic.com;\n\nwebsite: www.nontoxic.com.\n\n### Ultraviolet Light Air Purification\n\nThe idea behind this approach is to kill airborne pathogens with ultraviolet (UV) radiation, a naturally occurring frequency of light that has proven antimicrobial capabilities, especially against molds, bacteria, yeasts, and viruses. The Bio-Fighter Anti-Microbial UV Light System, for example, kills harmful microbes in the ductwork and air space of the home's HVAC system; it comes in four sizes, according to the size of the indoor environment needing purification. The device, which is installed in the ductwork or on either side of an A-coil, is meant to work alongside a dust filter or electrostatic air filter system already in place in a home.\n\nThe UV light radiates a surface to prevent microbes from growing there, and it disinfects the airstream as it passes through the HVAC system. The \"kill rate\" for microbials passing through the UV light is fifteen percent to twenty percent with one UV lamp per single pass, according to the manufacturer, which adds that one lamp is normally enough to control microbial growth on the coils and drain pan of the HVAC system. Because indoor air is constantly recirculated, the kill rate is cumulative, and soon almost all the airborne pathogens are destroyed. UV light is basically safe for humans provided you do not look directly at the emitting source for more than five seconds; doing so can damage the eyes.\n\nThe Care 2000 Air Purifier is a device that combines HEPA air filtration and UV cleansing. It is billed as capable of removing mold, dander, odors, chemicals, dust mites, and virus and bacteria particulates with a system that changes the air six times an hour in a 1500-square-foot room. The UV features, says the manufacturer, make the unit especially advantageous for \"vulnerable, immune-compromised, newborn, chronically ill, emphy-semic, or bronchitic people.\"\n\nThe unit has five different filters, including a 10-micron particulate prefilter (to capture the big stuff, such as dust, pollen, dander, and mold particles); a fifteen-pound activated carbon filter (to absorb odors, gases, fumes, and VOCs); a 0.3-micron primary microfiltration 99.97% HEPA filter; a germicidal dual ultraviolet light, which constantly bathes the HEPA media, the other filters, and the air passing through them, killing any trapped microbes; and a carbon-impregnated post-filter, which reinforces the air cleansing performed by the first four filters.\n\n## HEALTHY LIVING SPACE DETOXIFIER #49 Install an \"Indoor Waterfall\" to Add Negative Ions to Your Air\n\nOne of the popular ways of improving indoor air quality is through a negative ion generator, or what colloquially is referred to as an \"indoor waterfall.\" Diffusing indoor ambient air with negative ions produces a refreshing, reinvigorating effect on the room's occupants similar to the excellent results you get from sitting before a plunging waterfall or breaking waves at the beach. Your body and its energy field are refreshed and healthfully stimulated with negative ions.\n\nThe term \"negative ions\" of course is confusing, for why is something with a positive value labelled negative? To answer that, we have to first understand what an ion is: a charged particle in the air produced when something forceful acts on a molecule (it could be carbon dioxide, nitrogen, water, or oxygen) such that it releases an electron. The molecule is now a positive ion, meaning, it's short an electron.\n\nThis now displaced electron attaches itself to the nearest available molecule, which becomes a negative ion\u2014paradoxically now more than complete with the extra ion attached and thus capable of imparting a health benefit. It is the negative ions that exert the revitalizing effect, leading one commentator to dub them \"vitamins of the air.\"\n\nA negative ion generator takes advantage of this natural activity by emitting negative ions of oxygen, which once in the air collide with particulate matter (typically positive ions, such as dust, pollen, mold spores, bacteria, chemicals). The negative ions surrender their extra ions to the positively charged particles or positive ions (which, technically are deficient in ions, having lost some). Other positively charged particles start to surround this newly negatively charged particle, building up around it like a snowball. These react with dust and pollutants to form larger particles. Eventually, the \"snowballs\" of positive ions get too heavy to stay up in the air and fall to the floor where they can be vacuumed up.\n\n**_The modern city and the tightly sealed office or apartment building act as \"ion prisons,\" concentrating unhealthy levels of positive ions. Consider these sources of air pollution: car exhaust, factory fumes, dust, tobacco smoke, fumes from cooking and heating. All of these grab negative ions and either neutralize or convert them to positive ions._**\n\nThe modem city and the tightly sealed office or apartment building act as \"ion prisons,\" concentrating unhealthy levels of positive ions. Consider these sources of air pollution: car exhaust, factory fumes, dust, tobacco smoke, fumes from cooking and heating. All of these grab negative ions and either neutralize or convert them to positive ions. For example, the typical ion count of fresh country air is 2,000-5,000 negative ions per cubic centimeter (ccm); the negative ion count at big waterfalls, such as Yosemite Falls, is estimated to be 100,000 ccm. In contrast, indoor environments can have negative ion counts as low as 100 ccm.\n\nInside the buildings, the steel and concrete building materials also absorb all the negative ions, as do synthetic building materials, furniture coverings, clothing, metal ducts covering the ventilation outlets, and plastics (which emit a positive static charge). Air conditioned buildings and homes (as well as trains and planes) become laden with positive ions because the air conditioning system's metal blowers, filters, and ducts strip negative ions from the air before it is vented into the rooms.\n\nYou end up with an intensely positive-ion-dominated internal environment whose negative ion level may be below 100 ccm. The minimum level considered supportive for top human functioning is about ten times that, or 1,000 ccm.\n\nPositive ions exert numerous ill health effects. In a sense, they are the delivery system of the sick building syndrome and its effects. Research shows, for example, that they can alter the functioning of the human nervous system; induce a higher secretion of serotonin (the brain chemical associated with sleep, mood, nerve impulses); and slow down by thirty-three percent the sweeping action of tiny throat hairs and cut mucous flow, thereby lowering our resistance to airborne allergens.\n\nPositive ions have been shown to raise blood pressure; decrease blood albumin; generate various allergy symptoms (dryness, burning, and itching of the nose, dry and scratchy throat, dizziness, breathing problems, eye irritations); decrease alertness; and increase depression, tension, irritability, and insomnia. Generally, the principal ill effects of positive ions are to produce tension and irritation, exhaustion, and an overstimulation of the thyroid gland (in the throat). The Santa Ana winds (the dry desert or sea winds, known by many different names worldwide) concentrate positive to negative ions at the rate of 33:1, producing numerous symptoms of physical and mental discomfort in people subjected to them.\n\nNegative ions, in contrast, produce a great number of health benefits, according to more than 700 scientific papers. As early as 1932, researchers discovered that workers became ebullient when situated next to an electrostatic generator when it generated negative ions, but morose when it emanated positive ions. Subsequent research has shown that negative ions can reduce anxiety and neurosis, increase appetite and thirst, even stimulate sexual activity; they can sharpen reaction time, voluntary movement dexterity, and mental performance while reducing error rates.\n\nHyperactive children get calmer, students concentrate better, and teachers feel less fatigued when under the influence of negative ions. People report relief from allergies, migraines, and postoperative pains; infections and bums heal faster\u2014apparently negative ions kill infection-propagating germs. Negative ions may also accelerate the blood's delivery of oxygen to the body's cells and enhance our ability to absorb and use oxygen.\n\nOther benefits of negative ions include reducing the severity of depression, counteracting the effects of tobacco smoke on mucous membranes, increasing work capacity, relieving allergy symptoms (such as sneezing, watery eyes, itchy nose, exhaustion from insomnia), providing an emotional lift, reducing germ counts (in one study, by fifty percent within six hours, by seventy percent in twenty-four hours), and disinfecting indoor air. Even better, houseplants growing under the influence of negative ions showed a marked increase in size and growth rate.\n\nThere are numerous inexpensive portable or desktop-size negative ion generators now available on the consumer market. How do they work? Essentially, an extremely low current at a high voltage is applied to a group of sharp points known as the ion emitter within the device; electrons build up on these sharp points, then are ejected into the air and attach themselves to oxygen atoms. The electrons ejected have a negative charge, which means the oxygen molecules they attach to become negative ions. Here are some details on merely a few of the units available.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about Electrocorp Zestron Ionair Ionizers, contact:\n\nElectrocorp, 595 Portal Street,\n\nCotati, CA 94931;\n\ntel: 800-525-0711 or\n\n707-665-9616, ext. 201;\n\nfax: 707-665-9620, ext. 201;\n\ne-mail: electrocor@aol.com;\n\nwebsite: www.electrocorp.nu.\n\nFor lonAir Wein VI-2000 High Density\n\nNegative Ion Generator:\n\nThe lonAir Company,\n\nDept. Purify Online,\n\n2574 North University Drive, Suite 201,\n\nFort Lauderdale, FL 33322;\n\ntel: 800-478-7324;\n\nfax: 954-742-7882;\n\nwebsite: www.purifyonline.com.\n\nFor Comtech Research units:\n\nComtech Research,\n\n4360 Walls Ford Road,\n\nMansfield, MO 65704;\n\ntel: 417-741-6934;\n\nfax: 417-741-6056;\n\ne-mail: info@comtech-pcs.com;\n\nwebsite: www.comtech-pcs.com\/ions.\n\n### Electrocorp Zestron Ionair Ionizers\n\nVarious Electrocorp units may be used on a tabletop, in an automobile, greenhouse, laundry room, RV, truck, boat, or hotel room. The IG700 is a tabletop ionizer, but it can also be hung from a ceiling (IG1000). The IG200 works in a car and can be plugged into the cigarette lighter outlet. The IG300 and IG350 are fan-driven and work in a car or room. The IG900 is a desk model for use in an office or reception area. In terms of ion density emitted, the IG700, IG1000, and IG900 emit 4.5 million negative ions per ccm\/second, measurable about three feet from the unit, while the IG300 and IG350 produce 500,000, and the IG200 produces 100,000. The units weigh between one and two pounds and can be purchased for around $150 each or less.\n\n### IonAir Wein VI-2000 High Density Negative Ion Generator\n\nThis unit is especially recommended (by its manufacturer) for relief of positive ion overload and radiation from working with computers and video displays. It is now scientifically recognized that long-term exposure to computer radiation and positive ion emission produces numerous symptoms, dubbed Video Operator Distress Syndrome. The symptoms include increased fatigue, eyestrain, blurred vision, rashes, headaches, back pain, irritability, depression, anxiety, and apathy. This unit covers a 400-square-foot area, plugs into any wall outlet, and can operate noiselessly on a desk next to the computer, emitting an estimated ion output of 70 trillion ions\/second and an ion density of 1.6 trillion ions\/cc at 2.5 cm from the device. The unit sells for about $110.\n\n### Comtech Research IG-033A\n\nThis unit, measuring 5 by 5 inches, can cover 400 square feet, or a room twenty feet by twenty feet, emitting an ion output estimated at 93 trillion ions\/second and an ion density of 1.2 trillion ions\/cc, and only 0.04 ppm of ozone close to the emitter. Comtech notes that it is safe to place the ionizer about four feet from your computer, safe in that it will not damage the computer yet still fill the air around it with negative ions.\n\n## HEALTHY LIVING SPACE DETOXIFIER #50 Use Houseplants to Filter Out Toxins from Your Indoor Air\n\nOne of the effective ways to improve indoor air quality may be something you already have in place in your home: houseplants. Joint research by the National Aeronautics and Space Administration (NASA) and the Associated Landscape Contractors of America beginning in the 1970s showed that common houseplants are capable of ridding indoor air of a variety of contaminants and VOCs. Somehow the plants are able to absorb the airborne toxins without ill effects to themselves and either deposit them in their leaves and roots, or in some cases, biodegrade the molecules into food.\n\nIn the NASA study, spider plants were placed in a closed chamber twelve cubic feet in size, the air of which contained either carbon monoxide at a concentration of 120 ppm or nitrogen oxide at a concentration of 50 ppm. NASA scientists observed that after twenty-four hours, the spider plants had removed ninety-six percent of the carbon monoxide and ninety-nine percent of the nitrogen oxide. They also found that the plant golden pothos was able to remove seventy-five percent of the carbon monoxide in the same time period.\n\nAmong the plants best suited for removing formaldehyde are aloe vera (ninety percent), elephant ear philodendron (eighty-six percent), and ficus (weeping fig, forty-seven percent). For removing benzene, there is English ivy (ninety percent), peace lily (eighty percent), Janet Craig (corn plant, seventy-nine percent), and golden pothos (sixty-seven percent). Also rated well for benzene removal are gerbera daisy and chrysanthemums. The peace lily can also remove fifty percent of the trichloroethylene from an indoor space in twenty-four hours, the study found.\n\n### General Air Filtration\n\nIn terms of overall effectiveness in detoxifying indoor air, NASA ranks the houseplants in this order: heart leaf philodendron, elephant ear philodendron, green spider plant, lacy tree philodendron, aloe vera, golden pothos, Chinese evergreen, mini-schefflera, peperomia, and peace lily.\n\n### Formaldehyde\n\nPlants with a large leaf surface area perform best in removing formaldehyde, the scientists found, and these include heart-leaf philodendron, elephant ear philodendron, green spider plant, lacy tree philodendron, golden pothos, Chinese evergreen, mini-schefflera, peperomia, peace lily, corn plant, snake plant, bromeliad, aloe vera, bamboo palm, pot mum, orchid, dumbcane, wameckii, Madagascar dragon tree, mass cane, poinsettia, weeping fig, variegated lily-turf, banana, azalea, Mother-in-Law's tongue, miniature umbrella plant, arrowhead plant, and oyster plant.\n\nSmall azalea plants can remove formaldehyde from a sealed chamber at the rate of about 1 ppm per hour, which is equivalent to 264 ug of formaldehyde. This means one azalea plant can remove 6,336 ug of formaldehyde per day from an office or home. To put this in perspective, the typical indoor air concentration of formaldehyde is about 5,436 ug, based on a contamination level of 0.2 ppm per 1,200 foot area.\n\n### Toluene\n\nUseful in removing this toxic chemical are Chinese evergreen and miniature umbrella plant.\n\n### Trichloroethylene\n\nThis toxic substance comes from dry cleaning, inks, paints, varnishes, lacquers, and adhesives, and can be effectively removed from indoor air by Gerbera daisy, chrysanthemum, peace lily, Warneckei, and marginata.\n\n### Benzene\n\nThis substance derives chiefly from tobacco smoke, gasoline fumes, synthetic fibers, plastics, inks, oils, and detergents. The following plants can help filter it out of your indoor air: English ivy, marginata, Janet Craig, chrysanthemum, Gerbera daisy, Warneckei, and peace lily.\n\n### Carbon Monoxide\n\nSpider plant and golden pothos can remove this toxic substance from indoor air.\n\n### Orchids\n\nThe Dendrobium orchid can remove airborne pollutants such as acetone, methyl alcohol, and ethyl acetate as exhaled by humans. They can also remove xylene and carbon dioxide from indoor air and seem to be more effective at doing this when it is dark; they not only remove the toxins, but they also release oxygen. Used in conjunction with plants that filter air better during daylight hours (such as _Dracaena marginata_ ), the result is a stabilized day and night balance of carbon dioxide and oxygen in tightly sealed indoor environments.\n\n### Hydroponically Grown Plants\n\nAs a variation on this approach, research developed at the University of Koln in Germany reports that plants grown by hydroculture (also called hydroponics, that is, in water, without any soil) are also effective in removing airborne toxins, notably formaldehyde, benzene, phenol, and nicotine.\n\nApparently, the plants are able to transform up to ninety percent of these pollutants into sugars, new plant material, and oxygen, rather than merely storing them in their own plant matter. Especially competent when used hydroponically are _Ficus benjamina_ and golden pothos. Ficus and golden pothos together are effective against airborne nicotine, because the former removes nicotine and breaks it down (biodegrades it), while the latter stores nicotine in its growing leaves.\n\nThe German researchers said that you can \"substantially\" improve the plants' air-cleaning abilities by installing a special ventilator in their pots. This device acts as an air filter and draws indoor air over the roots for enhanced removal of toxins. The researchers further reported that one to two hydroponic plants can cleanse a room with an air space of 100 square feet. An additional advantage to using hydroponic plants used as air filters is that because they lack soil, they do not add fungus spores to the indoor air; this is a desirable factor for people with allergies. The plants also humidify the air, reducing the buildup of static electricity (or positive ions).\n\nNot everyone thinks plants are good toxic air filters. As a qualifying note, research since the NASA studies in the mid-1980s has challenged their findings, downgrading the ability of houseplants to filter out airborne toxins. Researchers at Ball State University in Muncie, Indiana, put plants in a sealed chamber and subjected the air to a continuously outgassing source of formaldehyde (particleboard). At the end of the trial, the ambient levels of formaldehyde were slightly lower, but nothing on the scale of the original NASA findings.\n\n**_\"The ability of over forty interior plants to remove various indoor air polluting chemicals from sealed chambers has been extensively tested and documented over the past ten years,\" declared B.C. Wolverton, Ph.D. in a 1992 report._**\n\nAccording to Lynn Marie Bower in _The Healthy Household_ , if you have a great number of houseplants in a room or throughout your home, it raises the living space's relative humidity (from the regular watering), and insofar as formaldehyde outgasses faster in damp, high-humidity conditions, it may create suitable conditions for higher airborne concentrations of formaldehyde rather than lower. Further, it will probably take a lot of plants to duplicate NASA's botanical air filtering effect. In _Home Safe Home_ , author Debra Lynn Dadd estimates that to get the same results that NASA reported you would need seventy-two plants for a nine-by-twelve-foot room.\n\nContinuing research by B.C. Wolverton, Ph.D., director of Wolverton Environmental Services in Picayune, Mississippi, one of the NASA's principal researchers on this topic, tends to support the ability of plants to filter indoor air. He is the author of thirty-eight technical reports published by NASA and _How to Grow Fresh Air\u201450 Houseplants that Purify Your Home or Office._\n\n\"The ability of over forty interior plants to remove various indoor air polluting chemicals from sealed chambers has been extensively tested and documented over the past ten years,\" he declared in a 1992 report. In 1993, Dr. Wolverton concluded that a low-light-requiring interior plant can remove \"significant quantities\" of formaldehyde, xylene, and ammonia from a sealed room. In 1996, one of his research colleagues reported that the accumulation of particulate matter (tiny particles of various chemicals) on horizontal surfaces inside a living space can be reduced by up to twenty percent by foliage plants.\n\nEven if the true air-filtering ability of plants is less than the NASA research indicates, your indoor air quality is bound to benefit considerably by having a fair amount of plant life around; the oxygen-releasing activity of houseplants in itself improves ambient air quality.\n\n### Clean Air Plant System\n\nHere is a practical way to take advantage of the research linking indoor plants with air filtration. Sunshine Tropical Foliage in Homestead, Florida, has developed a combination houseplant, carbon charcoal filter, and small fan. The system uses one of five houseplants: White Bird of Paradise, Dracaena Marginata, Philodendron Xanadu, Spathiphyllum Supreme, or Golden Pothos Totem.\n\nA small fan is placed inside a pot containing one of the plants. The filtration system draws room air down through the soil in the pot where the plant's roots trap the dust and particulate pollutants and, with help from the microorganisms in the soil, break them down into harmless substances. The air then passes through the charcoal filter and is exhausted back into the room as clean air.\n\nAccording to Sunshine Tropical Foliage, if you run the unit twenty-four hours a day, it will change the indoor air in a typical 100-square-foot room about ten times. \"That is more than adequate to give you good indoor air quality.\" The company claims the charcoal filter never needs to be replaced because the microorganisms around the plant's roots are \"constantly eating the pollutants.\"\n\n### The Breathing Wall\n\nThis is a more ambitious, wall-sized solution to the problem of indoor air pollution and sick building syndrome, developed by Canadian scientist Wolfgang Amelung, director of Genetron Systems, Inc., of Downsview, Ontario. The Breathing Wall is an \"indoor ecosystem\" in the form of a sheet of moss and fern-covered lava rock, five feet high by fifteen feet long. It's kept constantly wet and is supported by an aquarium filled with fish and aquatic plants.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about the Clean Air Plant System, contact:\n\nSunshine Tropical Foliage,\n\n13805 SW 248 Street,\n\nHomestead, FL;\n\ntel: 305-251-6966;\n\nfax: 305-257-1421;\n\ne-mail: stf@netrus.net; website:\n\nwww.zonelO.com\/tech\/cleanair\/cleanair.htm.\n\nFor information about the Breathing Wall, contact:\n\nGenetron Systems, Inc.,\n\n4801 Keele Street, Unit 34,\n\nDownsview, Ontario M3J 3A4,\n\nCanada;\n\ntel: 416-665-8155;\n\nfax: 416-665-8779;\n\nwebsite: www.flux.com\/Genetron.\n\nAccording to Genetron, there are fans behind the structure that draw the room's air across the plants and water, and through the \"breathing wall,\" thereby absorbing the airborne pollutants. \"Air passing nearby the continuously flowing water is filtered to remove water soluble pollutants and improve its overall quality,\" states Genetron. The Breathing Wall can be connected to a ventilation or air conditioning unit to distribute this plant-purified air throughout an entire building or home. The Breathing Wall is able to prevent unpleasant smells, allergens, and other contaminants (fumes, dust, fibers) from being released into a room's air.\n\nAs a filtration system, it \"inhales sick air and exhales clean air.\" The Wall, says Dr. Amelung, is \"a self-contained ecosystem in an enclosed space . . . like putting life on a planet that doesn't have life.\" The planet in this case is the sealed indoor living or work space.\n\nThe Breathing Wall was originally devised as a pilot project for a large Toronto-based insurance company's building. Called the Canada Life Environmental Room, it was an 1,100-square-foot meeting room featuring a Breathing Wall containing 8,000 different life forms, including mosses, ferns, orchids, and bulrushes, as well as frogs, insects, mollusks, salamanders, and fish. The wall was prepunctured with thousands of little holes through which ambient air could be drawn and filtered by the living organisms. The biological air cleaning system was later installed in other commercial settings and was studied by NASA for use in their proposed space station. Genetron reports that \"the environmental quality in this partially sealed room is equal to that achieved by the most modem engineered system of air turnover.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #51 Get the Radon Out of Your Basement\n\nNot to be overlooked in our review of toxic elements in the home is radon, a gas derived from uranium and radium. Radon is a carcinogen linked with lung cancer, and it can enter a home through the basement or building foundation. Uranium, although it is found almost everywhere on the planet, is a radioactive element; it goes through a series of changes, changing into as many as a dozen other unstable elements until it settles into a stable form of lead. Radium is one of the in-between changes uranium makes, and radium, found in soil, emits a gas known as radon. While radon itself is a short-lived naturally occurring gas, sometimes existing for only a few days, it decays into various even more short-lived elements (or decay products) called radon progeny or \"radon daughters,\" such as polonium, lead, and bismuth. You don't want these in your basement, and you definitely don't want to inhale them.\n\nRadon is present in the outdoor atmosphere in highly dilute concentrations, so breathing it there causes no harm; but inside a building, its levels can concentrate to a toxic degree. Radon can be found in soil, rock under the home, well water, building materials, and loose underground pipe fittings; from any of these sources, it can enter the breathable air inside a living space. It seems that pressure differences between a home's interior (the inside of a tightly sealed construction) and the outside draws the radon gas into the home through existing cracks in the foundation. Then lack of adequate ventilation and fresh air circulation prevents the radon from getting diluted in the indoor air, and allows it to concentrate. Radon levels are usually higher below the ground than above.\n\nIt is helpful to think of radon exposure as somewhat akin to X-ray exposure. When the dangers of radon first came to national attention in the 1980s, it was through the unfortunate experience of a worker at a Pennsylvania nuclear plant who was triggering radiation alarms. What was puzzling was that his radiation exposure was not from the nuclear plant; rather, it came from a geologic formation called the Reading Prong, which had far higher than average radium concentrations, and the man's home sat over this radium factory. It turned out he and his wife were getting radon exposure at the equivalent of 455,000 chest X-rays per year\u2014just by living in their home and breathing the radon-enriched indoor air.\n\nThe cancer risk of radon exposure is well established. In fact, radon is now regarded as the second leading cause of lung cancer, the first being tobacco smoke. The National Cancer Institute (NCI) estimates that at least 15,000 people die of lung cancer due to residential radon each year. The NCI studied the morbidity data on 68,000 miners who were exposed to different levels of radon in their work and found they were dying of lung cancer at five times the expected rate for the general population. A 1998 mortality estimate by the National Academy of Sciences placed radon-caused cancer deaths at between 15,400 and 21,800 per year.\n\nAccording to a 1991 national residential radon survey conducted by the EPA, the average indoor radon level is 1.3 pic-ocuries\/liter (pCi\/L ) compared to 0.4 picocuries\/liter outdoors. \"Millions of homes and buildings contain high levels of radon gas,\" said the EPA in a 1996 fact sheet. How high is dangerous? The EPA says very high radon levels start at 10 pCi\/L, and reports it has tested some homes at 30 pCi\/L; levels between 210 pCi\/L are \"probably somewhat dangerous\"; while \"typical\" indoor levels are 0.5-2 pCi\/L, which might constitute a slightly increased risk. How dangerous is 4 pCi\/L? If you're exposed to this level of radon, it's about thirty-five times as much radiation as the Nuclear Regulatory Commission would permit as a \"safe\" level if you stood at the fence of a radioactive waste site.\n\nEPA advises that all homeowners should check their homes for radon, and homes in radon-intense regions of the country (with concentrations over 4 pCi\/L) should not only test, but also take measures to correct the problem.\n\n### Regional Radon Reports\n\nWant to find out if you're living in radon country? The Columbia University Department of Statistics and Lawrence Berkeley National Laboratory have prepared a highly detailed U.S. map of radon exposure levels, available on the following website: www.stat.columbia.edu\/radon. Click on the part of the map corresponding to where you live, and the full-color map shows you the radon concentration in that area.\n\nAnother radon information database is offered by Air Check, Inc., which provides data per state, including the number of radon tests performed, average results, and a concentration report based on county. (See this website: www.radon.com\/radon\/radon_map.html; e-mail: info@radon.com) In Virginia, for example, 113,856 tests have been performed, with an average statewide radon level of 4.2 pCi\/L, or \"probably somewhat dangerous,\" according to EPA; 74.6% of the tests showed radon at less than 4.0 pCi\/L and 19.5% of the results were 4.0-9.9 pCi\/L. However, 73 tests (0.1% of total tests) showed radon levels above 100 pCi\/L, indicating the state has some toxic radon neighborhoods.\n\n### Home Radon Testing\n\nThere are numerous local or regional radon testing services available for home radon monitoring, and simple home radon testing kits are available at hardware stores. Businesses that offer professional building inspections also may do radon tests. The tests may be based on short-term (a few days) or long-term (measured over many weeks) samples.\n\nFor an initial, short-term evaluation of your home's possible radon levels, AirChek offers a mail-in charcoal packet radon test, which they report has been used to determine radon levels in more than two million homes and 250,000 schoolrooms. The basic test takes ninety-six hours, but the company also has a long-term radon test kit (ninety days to one year). If your radon levels come in at the risk level, you might wish to proceed to a longer-term monitoring, or even to what is called a continuously operating radon gas monitor that provides you with \"radon alerts\" when ambient levels get too high.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about AirChek\n\nRadon Test kits, contact:\n\nAirChek,\n\n570 Butler Bridge Road,\n\nFletcher, NC 28732;\n\ntel: 800-AIR-CHEK or 828-684-0893;\n\nfax: 828-684-8498;\n\nwebsite: www.radon.com.\n\nIf your radon level tests out above the EPA safety level, then you will need to engage professional radon \"mitigation\" experts to develop a plan to rid your home of this gas; it may involve structural changes and\/or the installation of specialized equipment. For example, you might need to install a radon ventilation system (to ventilate crawl spaces or the soil under the building's foundations) or to seal any foundation cracks that permit radon to enter the home's interior.\n\n## HEALTHY LIVING SPACE DETOXIFIER #52 Take the SADness Out of Your Indoor Space with Full Spectrum Lighting\n\nIn thinking about the nontoxic, natural home and how to detoxify it, many people tend to overlook their lights. Light is a nutrient, and its absence can affect your health. The wrong kind of light can act as a toxin. Working indoors under standard fluorescent lights deprives you of the full spectrum light you need; living in an apartment in a big city also cuts down your exposure to natural sunlight. You get sunlight deprived as a result. You may know it as the winter blues, a season-long slump, a draining away of energy, vitality, and enthusiasm, and a corresponding desire for a quick\u2014perhaps permanent\u2014trip to the tropics.\n\nThe \"winter blues\" is now recognized as a legitimate medical condition known as seasonal affective disorder, or SAD for short (see figure 10-4). SAD refers to the measurable negative effect that a lack of natural sunlight has on the body and mind of the person deprived. Put simply, SAD is a light-mediated depression. The National Institute of Mental Health estimates that perhaps 10 million Americans suffer from SAD every year, while another 25 million are affected by a milder form of this dysfunction, which chiefly manifests as a seasonal depression and fatigue.\n\nFor example, an estimated one percent of people living in Florida experience SAD while ten percent of those living in Alaska have the problem. Other estimates place the incidence of SAD at ten percent of the general population of Northern New England, five percent of those living in the BaltimoreWashington, D.C., area and less than two percent of those living in Southern California.\n\nOther symptoms of SAD include a desire for extra sleep accompanied by a failure to feel more rested from the expenditure under the covers; increased appetite and irresistible food cravings, especially for carbohydrates (sugars, starches, alcohol), which lead to undesired, unneeded weight gain; diminished sex drive; body aches or pains; memory loss; concentration problems; indecisiveness; low self-esteem; suicidal thoughts; anxiety; and irritability, even to the extent of violent outbursts. Physical activity decreases, and one feels sluggish and sedentary. In women\u2014seventy-five percent of SAD sufferers are female\u2014SAD can produce a worsening of menstrual discomfort; in both men and women, it can create a desire to withdraw from the world and to avoid social contacts. SAD sufferers feel sad, sometimes to the point of grief.\n\n**Figure 10-4. Typical Symptoms of SAD\u2014Seasonal Affective Disorder**\n\nYou almost feel as if you have entered a period of winter hibernation\u2014or would like to. Most people feel a diluted form of some of the SAD symptoms as a natural response to winter, but people with SAD are hypersensitive to these changes and suffer the physiological impact of reduced sunlight in a more intense form.\n\nAnother way to look at SAD is that it is a season-long dose of jet lag. Your body's internal clock gets thrown off by the steadily diminishing amount of daily sunlight (the shorter days of winter) and gets disoriented. Another way to put it is that the decreasing length of the \"photoperiod,\" that part of the twenty-four-hour cycle in which natural sunlight is available, generates the symptoms of SAD.\n\nSAD appears to deliver its negative effects through the hormonal system, specifically melatonin, a key player in the body's biological clock. People with SAD are said to have a disturbance in their biological clock; it runs slow\u2014or too slow\u2014in the winter. A sunlight deficiency leads to a reduction in the secretion of melatonin by the brain's pineal gland; melatonin is crucial to the body's sleep\/wake cycle.\n\nThe chief recommended therapy for SAD is called bright light therapy, which means being exposed to full spectrum bright lights from one-half hour to three hours daily during the winter months. The light is conveyed by way of full spectrum fluorescent bulbs or special light boxes. The concept behind light therapy is that the bright light helps to reset the biological clock and thereby restore normal physiological (and pineal gland-melatonin) function. For those with mild SAD symptoms, a device called a dawn\/dusk simulator can be effective; this device gradually raises the light of a lamp over a preset time period, imitating the arrival of morning light.\n\nIt's important to realize that light is a nutrient, like food, air, and water, and that full spectrum light is the best photonutrient. \"Our eyes are not only organs of sight, but they are also photo receptors for the brain glands,\" comments Nicholas Harmon, the president of Verilux, a company based in Stamford, Connecticut, that makes various full spectrum lighting devices (see below). \"These glands produce hormones which affect our mood, health, and sense of well-being.\"\n\n### Light Boxes\n\nTypically, the therapeutic 'light exposure is at 2,500 lux (a technical measure of brightness), which is five times brighter than the average \"well lit\" office and twenty-five times brighter than the average living room, which can be as low as 100 lux (although normal indoor light is stated to be about 300-500 lux). Light boxes that emit 10,000 lux produce resolution of SAD symptoms even faster. To put this all in context, bright summer sunlight emits 100,000 lux.\n\nThe user sits or stands about two feet from the box; or you can place the light box in any part of the house where you are likely to perch for a time while awake, such as a home office, library, den, kitchen. \"The light box provides a measured amount of balanced spectrum light equivalent to standing outdoors on a clear spring day.\" The light enters the body through the eyes, which transmit the photonutrients to the brain where they help reset the body's internal clock. A typical selftreatment is thirty minutes at 10,000 lux or one hour at 5,000 lux once a day. Generally, a SAD person begins to notice a response\u2014many describe it as \"energizing\"\u2014within two to four days of beginning light therapy, and the \"cure\" is often complete within two weeks.\n\nThe North Star 10,000 Bright Light Box provides 10,000 lux at 26 inches, requiring the user to sit in front of it (the box measures 13\" high by 24\" long by 4\" deep) for about thirty minutes daily to get the optimal benefit. The Ott-Lite, developed by pioneering \"photobiologist\" John Nash Ott, Ph.D., is a full spectrum device in the form of a desktop lamp, a clamp-on portable light, or a light box of 2,500 or 10,000 lux strengths.\n\nAmjo Sunrise II and III are light boxes that have an adjustable lux strength; you can have 10,000 lux at 12 inches, 18 inches, or 24 inches, depending on how many bulbs are turned on inside the box. Amjo notes that they wrap a special tape around the base of the tubes to reduce or eliminate electrical, RF, or UV radiation from the cathode tubes, and that the tubes are hum and flicker-free. The Brite Lite IV from Apollo Light Systems provides 10,000 lux at 26 inches and has an electronic ballast on the tubes that eliminates the flicker and magnetic emissions typical of conventional tubes.\n\n### Full-Spectrum Fluorescent Tubes\n\nIf your home (or office) has conventional strip fluorescent lights, you might want to consider replacing them with full-spectrum lights. This way you eliminate a potentially toxic factor from your living space and introduce a health-promoting one at the same time.\n\nDr. Ott's early research on the relationship between light and human health revealed that standard partial-spectrum fluorescent lights were actually detrimental to health. He found that he could not successfully grow plants indoors under artificial lights and that the cathode radiation emitted by common fluorescent tubes made his plants mutate and form unnaturally.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about North Star 10,000 Bright Light box, contact:\n\nAlaska Northern Lights,\n\nP.O. Box 1801,\n\nHomer, AL 99603;\n\ntel: 907-235-6953 or 800-880-6953;\n\nfax: 907-235-7665;\n\ne-mail: nlights@xyz.net; website:\n\nwww.alaskanorthernlights.com.\n\nFor Brite Lite IV:\n\nApollo Light Systems,\n\n352 West 1060 South,\n\nOrem, UT 84058;\n\ntel: 800-545-9667 or 801-226-2370;\n\ne-mail: info@apollolight.com;\n\nwebsite: www.apollolight.com.\n\nFor Sunrise II, III, and Dawn Simulators: Amjo Corp.,\n\nP.O. Box 8304,\n\nWest Chester, OH 45069;\n\ntel: 513-942-2770 or 877-289-2656;\n\nfax: 513-942-2771;\n\ne-mail: support@sadlight.com;\n\nwebsite: www.sadlight.com.\n\nFor Vita-Lite: Natural Lighting.com,\n\n1939 Richvale, Houston, TX 77062;\n\ntel: 281-486-9583 or 888-900-6830;\n\nfax: 281-486-0352;\n\ne-mail: larry@naturallighting.com;\n\nwebsite: www.naturallighting.com.\n\nFor Ott-Lites:\n\nEnvironmental Lighting Concepts, Inc.,\n\n3923 Coconut Palm Drive, #101,\n\nTampa, FL 33619;\n\ntel: 800-842-8848 or 813-621-0058;\n\nfax: 813-626-8790;\n\nwebsite: www.ott-lite.com.\n\nDr. Ott further discovered that artificial partial-spectrum light is another form of potentially dangerous indoor pollution. It illuminates more intensely on the yellow part of the color spectrum and can cause eyestrain and eye fatigue. Further, said Dr. Ott, light from fluorescent tubes (as well as from televisions and computer monitors) can cause red blood cells to clump together after a long-term exposure. The clumping then contributes to reduced mental alertness and a sense of chronic tiredness.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor Natural Spectrum light bulbs:\n\nVerilux, 9 Viaduct Road,\n\nStamford, CT 06907;\n\ntel: 203-921-2430 or 800-786-6850;\n\nfax: 203-921-2427;\n\ne-mail: verilux@ergolight.com;\n\nwebsite: www.ergolight.com.\n\nNatural sunlight, or full-spectrum lighting, in contrast, stimulates the red blood cells to unclump and does not strain or fatigue the eyes. Research suggests that full-spectrum lighting can improve your mood. When he installed radiation-shielded, full-spectrum lights in schools, Dr. Ott observed a significant reduction in behavioral problems and learning disabilities and an improvement in academic performance. Other research has supported Dr. Ott's findings. Canadian psychologist Warren E. Hathaway, Ph.D., studied 327 school children in five different elementary schools and found that under full-spectrum lighting they had fewer dental cavities, better attendance, higher rates of academic achievement, and greater physical development than students in classrooms with standard fluorescent tube lighting.\n\n\"Full spectrum lighting, containing all wavelengths, sparks the delicate impulses which regulate brain and autonomic functions of the body, regulating these functions and maintaining health,\" comments John Downing, O.D., Ph.D., director of the Light Therapy Department at the Preventive Medical Center of Marin in Santa Rosa, California. \"In order to maintain health, it is important to be exposed to light containing the full wavelength spectrum found in natural sunlight.\"\n\nReplacing conventional fluorescent tubes with full-spectrum, radiation-shielded ones can make simulated light a nutrient for your system. Vita-Lite, for example, \"simulates the full color and ultraviolet spectrum of sunlight,\" according to the manufacturer who reports that these lights have been used in thousands of schools, medical care facilities, factories, banks, dental offices, retail establishments, grocery stores, and florist shops.\n\n### Healthy Light Bulbs\n\nVerilux has developed Natural Spectrum light bulbs that they describe as \"the closest replication of natural sunlight available.\" According to Verilux, their bulbs show the colors of the spectrum accurately in accordance with natural light, increase the contrast between black and white, and thereby reduce eye strain, glare, and ocular fatigue.\n\n### Dawn Simulators\n\nThese are programmable units (set to a specified wake-up time by the user) that can emit up to 400 watts of incandescent light. The Pi-Square SunUp and SunRizr by Amjo are examples of this approach, and are recommended for waking up refreshed on dark winter mornings.\n\n**_(Dr. Ott's research on the relationship between light and human health revealed that standard partial-spectrum fluorescent lights were actually detrimental to health. He further discovered that artificial partial-spectrum light is another form of potentially dangerous indoor pollution and can cause eyestrain and eye fatigue._**\n\nThe idea is to simulate the dawning and duration of natural sunlight so that your bedroom gradually brightens in the morning (presumably in the deep of SAD-flavored winter). The dawn-simulated light can either be directed at the user's pillow or indirectly elsewhere in the room; an ideal situation, says Amjo, is track lighting with three to eight 50-watt bulbs directly over, but about three to four feet above, the sleeper's head. The dawn simulator activates the lights according to its timer schedule, and you can set this timer so that the light reaches its full intensity in anywhere between one minute to three hours.\n\nAccording to Sheri Lundstrom, founder of Light Therapy Products of Plymouth, Minnesota, and an admitted SAD sufferer, the dawn simulator prompts the brain's pineal gland to start stimulating the feel-good hormone, serotonin. Light is crucial in the production of serotonin, and serotonin is essential to the body because it influences numerous aspects of our physiology, such as pain, digestion, body temperature, blood pressure and clotting, immunity, sleep, and daily body rhythms.\n\nBut the shorter days of northern latitude winters, buildings with no outside windows, frequent cloudy skies, or not spending much time outside in the winter leads to a serotonin depletion and its host of symptoms. This is why Lundstrom characterizes light therapy as a \"natural Prozac\" for winter depression. She comments: \"Research subjects have shown marked improvement in energy, mood, social interest, productivity, quality of sleep, and quality of awakening with dawn simulation.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #53 Go Green with Your Household Products\n\nFrom a certain perspective, it seems that most commercial products that are supposed to make our lives run smoother, more efficiently, and with more ease end up being potentially toxic for us. This is especially so if you are already allergic or chemically reactive to many of the fixtures of modern life, and it is acutely so if you have symptoms of sick building syndrome or multiple chemical sensitivities.\n\nMany of the items we take for granted actually contain allergenic or toxic elements. These items household products (air fresheners, cleaners, bleaches, dishwashing detergents, floor wax, drain cleaners, carpet shampoos, laundry fabric softeners, oven cleaners, toilet bowl cleansers), paints, stains, thinners, home and garden pesticides, pet supplies (flea collars, cat litter, flea and tick products), auto products, art and craft supplies, cosmetics and personal care products, perfumes and colognes, hair conditioners and shampoos, dental and oral hygiene products, feminine hygiene products, nail polish, and skin products (deodorants, shaving creams).\n\nThe simple, possibly shocking, fact is that most of these products contain known toxins\u2014some are poisons on purpose, such as disinfectants and pesticides\u2014and when you use them you are exposing yourself to a poison, either through skin contact or by breathing in the product's vapors. Your exposure may be minimal and not necessarily produce an immediate or noticeable reaction, but you need to think in terms of long-range effects. Toxic exposure is often cumulative; the toxins remain in your system, and your liver is unable to remove them. Eventually your body is overloaded by a number of toxins, and you get sick. Perhaps you get an allergy first, then a gastrointestinal problem, then a hormonal imbalance, and eventually worse, more debilitating and chronic conditions emerge.\n\nThe matter of toxicity in common household and personal products is relative to the individual. Various governmental authorities, such as the EPA, have standards for \"safe\" levels of presumed or proven toxic elements in commercial products, but these standards are based on averages, norms for human biochemical tolerance. Your tolerance may fall above or well below these averages.\n\nIf you are healthy, detoxified, and fortified with nutrients, there is a strong chance your system can tolerate more exposure to toxins than the average. But if you are already allergic, and your immune system and liver are stressed by chronic exposure to toxic substances, one more toxin\u2014from something as seemingly innocuous as your automatic dishwasher detergent\u2014could send you down into a new level of dysfunction and ill health. You tolerated the first six toxic exposures, but the seventh made you sick. Your body's total toxic burden was exceeded.\n\nUnfortunately, once you are chemically reactive or chronically allergic to human-made substances, it starts to extend to other items you wouldn't think could be harmful, yet, for you, they are. You need to concern yourself with furniture, upholstery, curtains, vacuum cleaners, bedding, clothes, even, in extreme cases, the ink used in books and magazines.\n\nGiven these physiological facts, it is prudent to carefully consider the products with which you customarily surround yourself in your home. Are they full of toxic chemicals? Do they contain known or suspected carcinogens? Is there evidence that they can produce allergies or symptoms that fall under the various medical categories of chronic fatigue, fibromyalgia, environmental illness, sick building syndrome, and multiple chemical sensitivity? If so, it's time to start thinking green\u2014that is, thinking in terms of environmentally friendly products. But remember, there are two environments here: the outside world, which includes your home, and your body, which is your inner environment. An environmentally friendly, green product does not pollute either; many conventional household and personal products do.\n\nThe problem is the synthetic organic chemicals they contain. Today, an estimated 75,000 chemicals are in use in commercial products. Of these, only 1.5% to 3.0% (or 1,200 to 1,500) have been tested for carcinogenicity, that is, to see if they can cause cancer. So far, about 200 have been identified as carcinogens. But in 1995, the National Toxicology Program stated that somewhere between 5% and 10% of the existing industrially used chemicals could be carcinogenic, based on the 1.5-3.0% of chemicals they had tested. In other words, 3,750 to 7,500 chemicals found in products we might encounter or use could be carcinogenic.\n\nSince nobody knows yet which of the remaining 75,000 chemicals are carcinogenic, how will you know if you're using one? Worrying about cancer isn't even the chief concern here. The minute a chemical produces an allergic reaction, such as a sneeze, a rash, a gastrointestinal upset, your body is telling you\u2014your liver actually\u2014that it doesn't like this substance and considers it inappropriate for further contact. Either it doesn't want to or can't detoxify the substance and throws out an allergic reaction as notification of the fact. Allergies are the first act in a health tragedy that can lead to dozens of different dysfunctional conditions, including but not limited to cancer.\n\nHow can a chemical that's labeled \"organic\" be harmful to my health? It all comes down to petroleum, which, technically, is a natural, organic product, being derived ultimately from plant substances. Many of the toxic products contain petroleum as a base, employing its hydrocarbon composition (combinations of hydrogen and carbon atoms); hence the terms petrochemical and organic chemical. Yet these organic chemicals are synthesized, invented, as it were; hence the term synthetic.\n\nIn nature, for each compound there is an enzyme that can break it down and neutralize it, but in the case of synthetic organic chemicals, no balancing natural enzymes exist to break them down because the chemicals themselves have been added to nature, not derived from it. This means petrochemicals persist in the environment for a long time; it also means they can persist in the human body, producing illness.\n\n\"Organic chemicals are some of the most hazardous substances ever made,\" states a fact sheet published by Seventh Generation, a supplier of green products based in Burlington, Vermont. \"The average household contains 63 different organic chemical products for a total of approximately ten gallons of hazardous petrochemicals . . . [and] most of them are found in an unlikely place\u2014common household cleaners.\"\n\nHere are a few examples of how common household products can be toxic. Window cleaners contain diethylene glycol, which depresses the nervous system. Toilet bowl cleaners contain chlorinated phenols, which are known to be toxic to the circulatory and respiratory systems. The phenols often found in disinfectants are harmful to these two systems as well. Laundry detergents and all-purpose cleaners often contain a chemical called nonyl-phenol-ethoxylate, which biodegrades slowly and into even more toxic compounds than itself.\n\nYou will find formaldehyde in some spray and wick deodorants; this chemical (discussed earlier) is a suspected carcinogen and a known irritant of the respiratory system. The petroleum solvents found in floor cleaners can damage mucous membranes. A substance commonly found in stain removers, called per-chloroethylene, has been proven to cause damage to the kidneys and liver. Butyl cellosolve, a synthetic chemical you'll find in various cleaners, damages bone marrow, the kidneys, liver, and nervous system. This is just the beginning of a long list of offenders.\n\nLet's consider fabric softeners in detail. Thanks to EPA research collated by Julia Kendall (a multiple chemical sensitivity sufferer), former co-chair of Citizens for a Toxic-Free Marin (California), we know that at least nine toxic chemicals are found in many brands of this product.\n\nA typical fabric softener can contain alpha-terpineol (causes central nervous system [CNS] disorders, irritates mucous membranes, can cause edema, loss of muscular coordination, headache); benzyl acetate (a pancreatic carcinogen that irritates eyes and breathing passages); benzyl alcohol (causes headache, vomiting, nausea, dizziness, blood pressure drops); camphor (causes CNS disorders, dizziness, confusion, nausea, twitching, convulsions); chloroform (a neurotoxin and carcinogen, vapors can cause headache, nausea, vomiting, drowsiness, kidney\/liver damage); ethyl acetate (causes headache, stupor, anemia, liver\/kidney damage); limonen (a carcinogen); linalool (a narcotic, causes CNS disorders, affects muscle and heart activity); and pentane (depresses the CNS, causes headache, nausea, vomiting, dizziness, eye irritation, skin rash). Incidentally, this is only a partial listing of the symptoms each of these nine substances can cause.\n\nNext consider the relationship between toxicity and the ingredients in various common fragrance-based (scented) household products. Examples of these are perfume, soap, shaving cream, deodorant, air freshener, hairspray, hand lotion, Vaseline lotion, after-shave, and many more similar products.\n\nAcetone, found in cologne, dishwashing liquid, and nail polish remover, is a CNS depressant and can cause dryness of the mouth and throat, dizziness, nausea, slurred speech, even coma in severe cases. Benzaldehyde, found in perfume, cologne, hairspray, laundry bleach, deodorants, shaving cream, shampoo, and dishwasher detergent, is a CNS depressant, is irritating to all mucous membranes and the gastrointestinal tract, and can damage the kidneys. Even though FDA banned the use of methylene chloride in 1988, it can still show up in products because the chemical fragrance industry is protected by ingredient secrecy laws. Methylene chloride is a carcinogen, producing carbon monoxide in the body and reducing the blood's ability to transport oxygen. Again, this is just a brief review of the long list of toxic household products.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more information about and sources of environmentally-friendly, non-toxic, green household products: Seventh Generation,\n\nOne Mill Street, Box A-26,\n\nBurlington, VT 05401;\n\ntel: 802-658-3773;\n\nfax: 802-658-1771;\n\ne-mail: recycle@seventhgen.com;\n\nwebsite: www.seventhgen.com.\n\nSee also:\n\nReal Goods,\n\n200 Clara Street,\n\nUkiah, CA 95482;\n\ntel: 800-762-7325,800-994-4243, or\n\n707-468-9292;\n\nfax: 800-508-2342;\n\nwebsite: www.realgoods.com.\n\nFor information about and sources of products for environmentally safe and low-toxic building and interiors: Shelter Ecology,\n\n43 Pine Ridge Road,\n\nAsheville, NC 28804;\n\ntel & fax: 828-251-5888;\n\ne-mail: sheltereco@aol.com; website:\n\nwww.ioa.com\/~shelterecology.\n\nFor information about household prod ucts from an advocacy organization: Environmental Health Network,\n\nP.O. Box 1155,\n\nLarkspur, CA 94977;\n\ntel: 415-541-5075;\n\ne-mail: wilworks@lanminds.com;\n\nwebsite: www.users.lmi.net\/wilworks.\n\nCare for perfume? Perhaps you won't after reviewing some of the toxic constituents that may be found in various perfumes and fragrances. In general, perfumes may contain upwards of 5,000 natural or synthetic sources of sweet smells, of which, as of 1991, 84% have not been tested for toxicity to humans, or tested only minimally. Scents, incidentally, show up not just in perfume, but cosmetic and hygienic products, drugs, detergents, cleaning products, plastics, industrial greases, oils, solvents, and even foods. The National Academy of Sciences, back in 1986, targeted fragrances as one of six categories of chemicals requiring testing for neurotoxicity, that is, for their capacity to damage the nervous system. The other five groups included food additives, heavy metals, air pollutants, insecticides, and solvents.\n\nSome of the chemicals found in fragrances have been identified as neurotoxic, carcinogenic, or capable of producing birth defects, while a few have even been designated \"hazardous waste disposal chemicals,\" according to EPA standards. Of a list of 2,983 chemicals (a partial list) used by the fragrance industry, 884 of them were identified as toxic substances. Toluene, a chemical we first encountered earlier in this chapter in toxic carpets, is a big player in the fragrance industry, found in every fragrance sample in a 1991 EPA study, as well as in auto parts stores and in the fragrance sections of department stores. Toluene, among other things, is linked with asthma attacks, and it can trigger asthma episodes even in people who never previously had respiratory symptoms. Insofar as a 1986 study showed that seventy-two percent of asthma patients have adverse reactions to perfume, one reasonably wonders if the seeming ubiquity of toluene is a major factor in new cases of asthma.\n\nIn fact, the statistics on perfume and human health are very surprising, even the basis of advocacy movements to ban fragrances in public places. The American Lung Association states that 14.6 million Americans every year sustain a dangerous and\/or painful reaction to fragrance chemicals in the form of an asthma attack; 5,000 of these people die from the effects of the asthma attack. Louisiana state medical officials reported that twenty percent of asthmatics in their state undergo an asthma attack as a result of exposure to perfumes or fragrances.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor more on how-to recommendations for \"less toxic living,\" see: www.betterbasics.com, edited by Annie Berthold-Bond.\n\nRosalind Anderson, research director of Anderson Laboratories (visited above under toxic carpets) reports that laboratory mice experienced neurotoxic effects, pulmonary irritation, and airflow limitation as a direct response to being exposed to fragrances. In her view, this evidence was sufficient to advise humans to stop using fragrances, suggesting that sensitivity may be cumulative; you may not be chemically reactive today, but keep using the various scented household products for five or ten years (or months), and you may be.\n\nDon't stop your review of household toxins with the perfumes and scented products. There may be problems with your bedding. If your mattress is made of polyurethane foam plastic\u2014your innerspring mattress can also be wrapped in this substance\u2014this has most likely been sprayed with chemical fire retardants and has a covering of polyester plastic fabric. Exposure to polyurethane foam outgassing has been linked with bronchitis, skin and eye irritations, and coughing. This is probably due in large part to the fact that it out-gasses toluene diisocyanate, a toxin known to produce respiratoiy problems. In effect, every time you lie down on your bed and, by your body weight, apply pressure on the mattress, you may be forcing the outgassing of this and other toxic chemicals into the bedroom's indoor air.\n\nFurther, if you have a polyester mattress pad, polyester pillow, polyester comforter, and if you use polyester sheets, you are exposing yourself to additional toxins. That's because the sheets maybe treated with a permanent press finish based on formaldehyde. \"Amid this cloud of plastic vapors, is it any wonder that millions of Americans take drugs to get to sleep at night?\" comments nontoxic solutions expert Debra Lynn Dadd. She might as well have added: Is it any wonder an estimated fifteen percent of Americans are hypersensitive to common household products?\n\n**_The key principle with respect to toxic household products is simple and twofold: avoid and substitute. Don't use ones that contain synthetic chemicals and find nontoxic green substitutes, of which there are many. When you have a choice between natural and nontoxic versus conventional and toxic, be prudent andgo green._**\n\nThe key principle with respect to toxic household products is simple and twofold: avoid and substitute. Don't use ones that contain synthetic chemicals (or at least reduce your use of them and thus your exposure to their contents) and find nontoxic green substitutes, of which there are many. When you have a choice between natural and nontoxic versus conventional and toxic, be prudent and go green. In many cases, you don't need to have a commercially prepared green substitute to go green; there are many homemade nontoxic solutions to standard household cleaning and personal care concerns.\n\nThis is by now a well-researched subject and a handful of excellent books detail your options. You might consider these as starters: _Home Safe Home_ , by Debra Lynn Dadd (1997); _Better Basics for the Home_ , by Annie Berthold-Bond, (1999), and _Clean House, Clean Planet_ , by Karen Logan (1997).\n\n# CHAPTER 11\n\n# The Feng Shui of a Healthy Home: Detoxifying the Energy Aspects of Your Living Space\n\nAs I write this, spring is imminent, and with it, the need to thoroughly air out my two-story home. As the weather warms up and the heater gets turned off for the season, all the doors and windows can be opened wide to encourage fresh spring air to rush through the interior space and blow out all the stagnant furnace-heated indoor air. My house has many windows, and they are situated in such a way that air can blow straight through the house, from the back porch to front garden. There are no troubling curves or strange comers or dead-end corridors where the air can get either trapped or slowed down. With all this fresh air cleansing my interior space, I am bound to feel invigorated every time I am inside now, and I will have more of a _feeling_ for the outside while I'm indoors\u2014I won't feel so shut in against the cold.\n\nAiring one's house in the springtime of course is not a new idea, but it is a useful image to get across an idea central to the practical guidelines in this chapter: the role\u2014and _circulation_ \u2014of energy in an indoor space. By energy, I don't mean gas, oil, electricity, or any other physically tangible form of energy; rather, I mean something more subtle, yet still one that can be sensed and adjusted.\n\nIn China, it's called _qi_ (pronounced _chee_ also spelled ch'i), and in traditional Chinese medicine (and its vast philosophical foundations) it has a pivotal role as the prime agent of healing and health. In terms of a healthy living space, the circulation of _qi_ in environments, both indoors and outdoors, is regulated by a practical discipline gaining increasing popularity and application in North America and known as feng shui (pronounced _fung schway_ ). This chapter will explore some of the many ways this vital energy can be used both to detoxify your indoor living space and to make it one that imparts health.\n\n### Charting the Movement of Life Force Energy\n\nBefore going any further, let's be sure we have a sense of the meaning of the two concepts of _qi_ and feng shui.\n\nThe broadest observation we can make here is that both come out of a profound cosmological model put forth by classical China. This model encompassed heaven and earth, energy and matter, thoughts and emotions, health and illness, and had many practical applications including music, landscape utilization and architecture (feng shui), medicine (acupuncture), diet, herbology, physiology, meditation and \"internal alchemy\" (Taoism), philosophy ( _I-Ching_ , or the _Book of Changes)_ , and energy management on all levels. At the heart of all these applications is energy, or _qi._\n\n_Qi_ is a Chinese word variously translated to mean \"vital energy,\" \"essence of life,\" and \"living force.\" (In other cultures, it is known variously as _prana, dynamis_ , od, orgone, and other names.) Generally, it's easiest to think of it as essential life force energy and to appreciate that it has a role in our body (managed through acupuncture) as well as our house and its surroundings (managed through feng shui).\n\nIn Chinese medicine, the proper (balanced) flow of _qi_ along energy channels (meridians) within the body is crucial to a person's health and vitality. There are many types of _qi_ , classified according to source, location, and function (such as activation, warming, defense, transformation, and containment). Within the body, _qi_ and blood are closely linked, because each is considered to flow through the body along with the other. _Qi_ may be stagnant (non-moving), deficient (partially absent), or excessive (inappropriately abundant) in a given organ system. _Qi_ has two essential qualities: _yang_ (active, fiery, moving, bright, energizing) and _yin_ (passive, watery, stationary, dark, calming). The manipulation and readjustment of _qi_ to treat disease and ensure maximum health benefit is the basic principle of acupuncture, although other remedies and therapies can be used to influence _qi_.\n\nIn a simplistic sense, feng shui is a form of acupuncture applied to environments, both interior and exterior. Feng shui means \"wind\" _(feng)_ and \"water\" _(shui)_ , and evokes a sense of the constant movement and interplay of these two elements in a natural environment. In practical terms today, people tend to take the term to mean \"the art of placement,\" in reference to designing interior spaces to maximize the health benefits. According to scholars, the term \"feng shui\" has been used only for the last one hundred years, and this complex science of energy and landscape was earlier known as _Kan Yu_ , which means time theory _(kan)_ and geographical theory _(yu)_ , and addresses the flow and change of _qi_ through a landscape over time. _Qi_ is a living, dynamic energy, constantly moving and changing.\n\n_Kan Yu_ , according to one expert, is about the study of the environment, both inside and out, both natural and constructed. The term evokes this state of mind: \"Raise the head and observe the sky above. Lower the head and observe the environment around us.\"\n\n_Kan Yu_ properly concerns itself with the location and orientation of a site, whether it is a room, home, garden, pagoda, city, or grave. The proper place for a site is dependent on \"a complex interaction of location and direction of topographical and artificial features\" as charted over time, explains another expert, Thomas Lee, a chartered land and hydrographical surveyor based in Hong Kong. \"The technique of _Kan Yu_ is mainly concentrated on the observation of surrounding features and understanding the flow of ' _Qi_ ' in terms of time. Energy is therefore believed to flow periodically, cycling hourly, monthly, or annually, etc., under a rigid mathematical formula that has been applied for over thousands of years,\" explains Lee.\n\nThe purpose of this energy flow is to balance and harmonize environments. \"The degree of harmony in an individual place is solely dependent on the balance of Yin and Yang (Ch'i) in the theory,\" says Lee. Even though _qi_ is invisible and hard to detect by conventional means, he adds, \"it is an energy which brings good influence to the living organisms within its influence.\"\n\nThe idea here is that _qi_ not only flows through an environment (a landscape or a home), but it _changes_ , with the seasons, the planetary influences, with the use given to that environment (e.g., a polluted landscape, a cluttered house or a clean, well-ordered one). The practical goal is to be aware of these flow changes and to utilize them for the health and well-being of those living in them. In the Chinese model, the benefits of maximizing the flow of _qi_ include not only health, but prosperity, longevity, family and community cohesiveness, even political stability. \"Utilizing these principles, people are creating comfortable, safe, and re-energizing environments for themselves and the people who visit their buildings or dwellings.\" When feng shui principles are applied, the results are \"happier attitudes, more prosperity, more sense of harmony and balance, an improvement in health and well-being.\"\n\n**_Regarding the living space, feng shui looks at numerous factors, including the site; the orientation of the house (or apartment); the layout of the rooms; the qualities of light, air, and heat circulation; the placement of the furniture; the layout of individual rooms; and architectural shapes and angles within the house. All of these factors have marked health effects on the people living under their influence._**\n\nAlthough this chapter focuses only on the movement of _qi_ through a living space, it is instructive to appreciate the _wide_ scope of Chinese thinking on energy flow, one that takes in the entire world, from bedroom to planet. Regarding the living space, feng shui looks at numerous factors, including the site; the orientation of the house (or apartment); the layout of the rooms; the qualities of light, air, and heat circulation; the placement of the furniture; the layout of individual rooms; and architectural shapes and angles within the house.\n\nAll of these factors, and many more not cited here, have marked health effects on the people living under their influence. These effects can be toxic or health enhancing. Recall, if you will, my opening example of airing the house in springtime. It is not only air that I invite to flow through my house when I fling open the windows and doors; _qi_ will flow into the house with the air. If my house is laid out properly, the _qi_ will reach every corner, flow through every room, and invigorate every aspect of my living space; but if my house is laid out poorly (as are many, if not most, modern North American homes), the _qi_ flow will be blocked, compromised, curtailed, or rendered stagnant, and my living space will become a health-deteriorating one.\n\nBear in mind, because this is only an image to get an idea across, _qi_ flows, or is blocked from flowing, in a house regardless of whether the windows and doors are open; everything in your house and the house itself either helps or hinders its flow on a moment-to-moment basis, and as the _qi_ flows, so goes your health.\n\n## HEALTHY LIVING SPACE DETOXIFIER #54: Evaluate the Energy Flow in Your Home with the Ba-Gua\n\nHow is one to know if one's house or apartment has good or bad feng shui, has _qi_ flowing or stagnating? Increasingly, North Americans are seeking the advice of qualified feng shui advisors to evaluate their living spaces to answer this precise question.\n\nOne of the evaluation tools commonly used by feng shui practitioners is a standardized energy template or blueprint called the _Ba-Gua_ (also spelled _Pa-Kua)_. It is a kind of generic map of the ideal flow of _qi_ through an interior environment, dividing this space into eight sections. Fortunately, you don't have to be a feng shui expert to compare the layout of your living space with the ideal _qi_ template of the _Ba-Gua_.\n\nThe _Ba-Gua_ is a complex symbol, based on complex ideas. In a simplistic sense, suitable for our purposes of home detoxification, it's useful to know that at its core is the yin\/yang symbol, representing the basic twofold polarized nature of energy as active and passive, hot and cold, dry and wet. The eight figures consisting of solid (yang) and broken (yin) thick lines are called trigrams; these are taken from the classic Chinese book of divination and philosophy, the _I-Ching_ (which contains sixty-four trigrams) and represent different combinations of the five elements and the outcome of those combinations (see figure 11-1).\n\nThe five elements are in dynamic relationship with one another, subject to cycles of influence and passivity (see figure 11-2). The feng shui expert uses the five elements to restore energetic balance to a place. As feng shui experts Sophia Tang Shaul and Chris Shaul, operators of the website _168 Feng Shui Advisors_ and a feng shui consultancy in Burbank, California, note: \"Used incorrectly, the elements can cause harm to relationships, health, or money prosperity. When used correctly, they can strengthen relationships, health, and money.\"\n\n**Figure 11-1. The _Ba-Gua_ Wheel of Energy Flow in a Home**\n\n**Figure 11-2. The Five Elements in Dynamic Relationship**\n\nAccording to the theory of feng shui, the way the elements are arranged (the variations on the trigrams) can be _predictive_ of life situations, including health, prosperity, and longevity in a living space. You might think of the _Ba-Gua_ template as a geometrical model of energy relationships with practical applications for how your house is laid out and how you are likely to fare living within this layout.\n\n_Directions_ : Draw a floor plan of your house or apartment, showing all the different rooms and their compass orientation, windows, doors, and the placement of furniture. If your home has more than one story, make a separate diagram for each floor.\n\nMark the center point in your house diagram by drawing an \"X\" from opposite comers. If your house or apartment is not square, rectangular, or symmetrical, square it off on the diagram, then draw the two diagonal lines from the opposing corners to find the midpoint.\n\nNext, align the _Ba-Gua_ with your floor plan. Set the center of the _Ba-Gua_ diagram on top of the midpoint on your floor plan, so that your front door corresponds to the bottom trigram of the _Ba-Gua_ , marked \"Career\/Kan.\" This is called the \"mouth of _qi,\"_ and marks where the vital life energy enters the house, like a guest, and how it is \"greeted\" by the home's interior layout. It does not have to be the front door; if the main traffic pattern in your home is through a side or back door, then, functionally, this is the mouth of _qi_ in your living space.\n\nNow that you have matched the two diagrams, label your home floor plan with the terminology from the _Ba-Gua_. See where your bedroom, kitchen, living room, and other functional rooms show up in the ideal template. But first a quick word of clarification on the terms and concepts of this ideal template, starting with the mouth of _qi_ then moving around the _Ba-Gua_ clockwise.\n\n  * Career: Think of this first section as your financial livelihood and career path, your spiritual path, and generally the way you approach life.\n  * Knowledge: Matters of contemplation, introspection, meditation, study, and seeking guidance from within occur here.\n  * Family\/Health: This area pertains to relationships with parents, authority figures, one's ancestors, and influences from the past.\n  * Wealth: Here we observe the flow of abundance and prosperity, but the blessings can take any form\u2014money, materials, good will, good fortune.\n  * Fame\/Rank: Your individuality and uniqueness of personal expression are grounded here.\n  * Marriage: This section of the house focuses on your primary relationships, such as with a partner, yourself, your friends, and colleagues.\n  * Children: Not only literally about children, this energy field deals with creativity, to anything that you can birth out of yourself.\n  * Helpful People: This points to the constellation of allies and supporters in your life, even helpful situations, such as finding answers from books; this section of the house connotes your relationship to philanthropy and generosity, and can be well used as an altar room.\n  * _Tai Ji_ : The _Ba-Gua_ 's central space is about health and vitality at the center of our living space.\n\nThis preliminary assessment of the layout of your living space contrasted with the ideal energy blueprint will give you ideas and insights as to how your house works, the degree to which it is in accord with the natural flow of energy, and the places where a feng shui adjustment would be advisable. Doing a _Ba-Gua_ analysis of your domestic living space is analogous to doing a biochemical assay of your blood and your body's nutrition, as we explored in Chapter 3. Both approaches give you a base line from which to evaluate your physiological and domestic status against a presumed ideal template, and to see to what degree your condition deviates from the optimal. As this chapter proceeds, the health significance of the deviation between your house and the _Ba-Gua_ will become clear.\n\nAccording to feng shui, a misalignment, such as having the bedroom in the career space, or the bathroom in the marriage space can be unfortunate; in fact, it can disturb or imbalance your life and possibly make you sick. In other words, it can be a toxic influence. Obviously, you cannot tear the house apart if the rooms are wrongly allocated, and in many cases this is not necessary, as feng shui offers many practical ways to alleviate the harmful influence of misaligned rooms. Nor is the _Ba-Gua_ meant as an ultimate architectural authority; some feng shui experts see it as an interpretive overlay, while others see is as a tool for calculating the attributes of a building.\n\nAccording to feng shui practitioner Terah Kathryn Collins, you can apply the _Ba-Gua_ to individual rooms, taking the principal doorway into the room as the orienting \"mouth of _qi_.\" While the _Ba-Gua_ alignment of the entire house is of greater importance and holds more \"structural _qi_ \" than that of individual rooms, it is still useful to chart the layout of individual rooms against the template. Once the evaluation and the practical feng shui adjustments are made, says Collins, \"a positive change occurs that directly relates to the person's desires and goals in less than thirty days.\"\n\n### The Healthy Living Space Expert Interview: Stanley Aaga Bartlett, Feng Shui Master\n\nThe _Ba-Gua_ represents \"an example of a perfect, balanced piece of the whole, a core template for understanding the flow of energy through a living space,\" observes Stanley Aaga Bartlett, a feng shui practitioner based in East Burke, Vermont. After twenty years in the real estate business, Bartlett shifted his professional interests to the use of feng shui for improving residential and commercial properties, and now consults nationally as a feng shui expert. \"The _Ba-Gua_ is a playful tool of discernment, a tool to better understand the clues as to what your environment is telling you about your life and a tool with which to change your reality. You can even carry the _Ba-Gua_ configuration around with you and check out various environments, looking for clues as to what is going on.\" The _Ba-Gua'_ s eight sides correspond to the various central aspects of our lives, and when each of the eight sides of the _Ba-Gua_ is in balance, then the whole is in balance, and our living space can fully support our well-being, Bartlett explains.\n\nForemost in Bartlett's approach to living space is the following principle gleaned over his years of feng shui practice: _As our environment goes, so go we_. In other words, our living space can either support our health and psychological well-being, or hinder them, Bartlett explains. \"There is a direct relationship between what happens in our environment and what happens in our lives, and the reverse, too. So if there is some imbalance in our living environment, whether it is physical or nonphysical, it affects our health, our mental and emotional attitudes.\"\n\nTo truly achieve domestic balance, you must deal with both the obvious and the subtle, with the \"practical\/tangible issues and the transcendental\/intangible aspects,\" Bartlett says. \"On the physical level, you have the obvious issues such as color, lighting, floor plans, electrical pollution, air quality, the placement of objects, furniture, and doorways. On the non-physical side, you have electromagnetic influence, Earth energy imbalance, predecessor and lingering energies from previous occupants of the space.\"\n\nIt is crucial to use a feng shui approach that integrates both aspects, Bartlett says. \"Considering only the practical aspects of feng shui might be 40% effective in creating harmonious space, but when the practical is combined with the transcendental, then it may have a 140% effectiveness. Knowledge of these transcendental aspects allows a feng shui practitioner to see through our environment what we are unable to see in our selves.\"\n\n_Everything_ in our living space has an effect on us in some way, even the angles of the furniture, the geometric shapes that surround us. Further, how our living space is situated with respect to its environment also has a pronounced effect on our well-being, says Bartlett. In other words, how our house or apartment building sits on the landscape, the direction it faces, the houses or other buildings that face it, even the angles of those neighboring houses and which part of our home they face\u2014all these factors and more come into play when a feng shui expert looks at a living space with a view to making health-supporting adjustments.\n\nHeightened sensitivity certainly comes in handy in this kind of work. \"When I walk into an environment, I see a panoramic view, a holographic pattern, if you will, of what is happening in that environment,\" Bartlett says. Somehow, his \"internal computer gathers vast information\" about the conditions of the immediate environment. Bartlett senses the subconscious of the living space, the influences that, perhaps not consciously known to the space's inhabitants, nonetheless are subtly shaping their lives, thoughts, feelings, and health.\n\nHe might hear an almost inaudible sound from an electrical transformer or power line nearby that is out of balance and creating a \"gnawing, annoying, or irritating vibration\" that affects the occupants of the house, even though they are not consciously aware of the origin of the toxic energy. \"Perhaps the environment is telling me that the front door entrance is constricted and dark, the colors inside are dark and not very balanced and the wall is too close to the entrance as I walk in and it is compressing my energy field. You might also notice that the Earth energies are depleting your _qi_ on one side of a room, but on the other side of the room you sense that the Earth energies are much more supportive.\" Bartlett says he often gets distracted by \"another irritant\" when he realizes the energy of the space suggests there must have been an argument in that room recently.\n\n\"I see the history of what has happened in a space. There are energies that linger\u2014they're called _ling_ energies in Chinese thinking, and mean the energies of the preceding occupants. For example, if there was an argument in this space yesterday or even a traumatic event one hundred years ago in this house, the energy _patterns_ of that experience will sometimes remain, or linger, in that environment.\n\n\"Our environments have an energy consciousness to them, and there is a lot we can do to create a different reality in our lives and in our environment by using the tools of feng shui wisely,\" Bartlett says. \"The goal of feng shui is to help us use this information to arrange our reality in more harmonious ways. Many wonderful stories exist of how people have used these teachings to bring health, prosperity, and good fortune to their lives.\" Even though feng shui often deals with the highly subtle energy influences in our environment, it is also deeply practical, never overlooking the obvious, such as the effect of chronic clutter in a living space.\n\n## HEALTHY LIVING SPACE DETOXIFIER #55: Tidy Up All the Cluttered Areas in Your Home\n\nUnattended clutter has repercussions in our life. \"I look for clues in the environment on a physical level to see what is going on in one's life and what one is (sometimes unconsciously) creating,\" says Bartlett. A cluttered living space is often at the root of many dysfunctional aspects of a person's life, and clearing out clutter can be \"one of the most powerful, yet unglamorous solutions to feng shui issues. When you are feeling stuck in life, start getting rid of clutter and watch how quickly new possibilities surface.\"\n\n**_A dense, cluttered environment most definitely affects us. Not only do cluttered areas attract dust, but they become sites of stagnant qi, places around and through which invigorating qi cannot flow. Areas of stagnant qi then attract other energies, thoughts, objects that are in agreement with that level of stasis and stagnation_**.\n\n_Directions_ : Examine your living space with a neutral, detached, objective eye. Is it cluttered? Are there piles of old stuff, debris, unused possessions, clothes, books, postcards that only gather dust, closets bursting with forgotten objects, stacks of old magazines, crates of glass bottles, or too much furniture in too small a space? Probably few people can truthfully say there are no clutter zones in their homes. So start throwing stuff out, winnowing out what you do not need any more, and recycling, giving away, or selling the rest. Keep only those things \"you truly love and need in your environment.\"\n\nA dense, cluttered environment most definitely affects us, says Bartlett. \"We say in feng shui, less is more, get rid of the old to make room for the new, and cluttered closets are like cluttered minds.\" Not only do cluttered areas attract dust, but they become sites of stagnant _qi_ , places around and through which invigorating _qi_ cannot flow. Areas of stagnant _qi_ then attract other energies, thoughts, objects that are in agreement with that level of stasis and stagnation and these merely strengthen this stagnant quality, heightening the attraction for what Bartlett calls the \"transcendental\/intangible aspects\" of a living space, in this case, negative, toxic aspects (discussed below).\n\nBartlett offers a vivid case study from his practice that illustrates the life-hindering effect of unattended clutter. Bartlett consulted with a woman named Joan who was having health, relationship, and career problems. \"She was experiencing health issues, relationship deprivation, a lack of money, and an inability to achieve success in her career as an actress and theater director. She was stuck in survival and forced to accept menial jobs and was even experiencing difficulty securing regular menial work.\"\n\nHer house was full of too many beautiful antiques, books, baskets, and other objects such that the overall visual impact was one of intense clutter. \"Joan thought she was simply going to have a feng shui expert come in and help her arrange all her things, move her furniture into the correct feng shui positions, and that this would resolve all her problems,\" says Bartlett. But that would have been fruitless unless she made a substantial reduction in her clutter. Bartlett agreed to return for more consultation after she had cleaned up some of the clutter. \"Joan began to understand that on some level she was surrounding herself with things out of a deep fear of lack. Somehow it felt safer if she kept lots of things around her. It also became clear to her that there were emotions hidden behind all these things that she had not been willing to look at for a long time.\"\n\nA short while later, Joan called him up, saying she had thrown out, sold, or given away a fair amount of her stuff. She said: \"I don't know what's going on but ever since you were here and I started clearing the clutter, I've been getting a lot of job offers.\"\n\nBartlett visited Joan again, and suggested another round of clearing, this time focusing on dense, old energies lingering in the house and how those energies were associated with Joan herself. \"We did a healing of her environment to fill her home with loving light.\"\n\nBartlett left Joan a second time to complete the deeper housecleaning. This time, she threw out more stuff, cleaned the windows and walls, dusted everything, gave the place a proper scrubbing. Joan called back again. \"I told you I hadn't been in a relationship for 19 years [since the death of her husband] and that I had some health issues. But all of a sudden, I'm getting inquiries from men.\" A few weeks later, Joan called Bartlett again. \"This is really the ultimate,\" she told him. \"I just had a call from a woman I knew well years ago; she had a very successful theater company. She made it big then lost everything, but now she's getting ready to make it big again and invited me to be her assistant director. So I'm selling my house and moving to the city.\" Joan's life continued to improve. \"Romance is in, career is great, and the body is much healthier,\" comments Bartlett.\n\nIn a sense, the ultimate (and a little amusing) result of Joan's rigorous clutter-removing work was to get rid of her house. This of course is not always required. The essential point in this case study is that Joan had a high level of clutter and thus stagnant energy in three areas of her living space: Career (\"mouth of _qi_ \"), Marriage, and Family\/Health. As a result, her career was stalled, she had no love relationships, and her health was deteriorating.\n\n  **_The Healthy Living_ Space Info Tip**\n\nStanley Aaga Bartlett can be\n\ncontacted at\n\nBartlett Designs,\n\nP.O. Box 119,\n\nEast Burke, VT 05832-0119;\n\ntel & fax: 802-626-9492;\n\ne-mail: stanlu@aol.com;\n\nwebsite: www.bartlettdesigns.com.\n\nWhen the clutter was removed and the stagnant energy exorcised from these domestic spaces, these three factors in her life reconfigured because she had reenergized their correlates in her living environment. \"For Joan, by clearing her attachments to physical things and clearing out her clutter, it allowed new possibilities to enter her life. This is such a powerful thing to do for one's health and well-being because when one is stuck emotionally, energetically, one tends to be stuck in one's environment, and that's when things do not happen very well in our lives.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #56: Repaint All the Interior Walls with Your Color of Happiness\n\nBefore addressing any of the specifics of individual room layouts and alignments, there is a simple but powerful way to impart fresh _qi_ to every inch of your living space. This is especially advised if you are moving into a new home or apartment because this detoxifying step is one you can\u2014and should\u2014do immediately upon taking possession of the place, even before you bring in your belongings.\n\nEssentially you want to expunge the physical residues of all previous occupants _and_ impart the unique stamp of your own energy and well-being. There are three interlinked cleaning tasks you can perform in your new space: repainting the walls, cleaning the carpets, and washing all the windows. While these are obvious hygienic, house-warming measures, they can become powerful feng shui tools by adding a special element: your _intention_.\n\nI will illustrate this point with an example from my own experience. My wife and I once rented a fairly new home for a year. The owner gave us a reduction in the first month's rent in exchange for our repainting the interior. This suited us fine, not only for the cost savings, but because we could impart our own energy to the walls that would surround us for the next twelve months.\n\nWith every slap and swish of the brush, we imparted a sense of peace, stillness, and spiritual presence to the fresh white paint, gradually filling our interior environment with a new vibration\u2014ours. We happened to be in pretty good spirits at the time\u2014dare I say happy?\u2014so this emotional quality flavored our intentions and whatever other qualities we sought to impart to the paint. Sometimes we played music to accompany the brush strokes, such as a Mozart sonata or vocal music of a gentle lilting quality, knowing that whatever vibrations we surrounded ourselves with during the \"thoughtful\" painting, along with the quality of our own internal state, the fresh paint (the \"inner skin\" of our living space) would \"remember\" and reflect back to us. We had a wonderful year within those newly painted walls.\n\n_Directions_ : \"Embrace your environment as if it were a part of you, which it is,\" says Bartlett. He suggests that even before starting your thoughtful painting that you hold \"the image of the perfect environment in your mind.\" This image, Bartlett explains, \"might elicit a feeling of love, protection, and perfect harmony from all around you.\" It should be so strong that \"when you near the entrance to your driveway, you begin to get this feeling that it is really good to be coming home.\" Then when you open the front door, \"your heart opens wide with this overwhelming feeling of love and harmony.\"\n\nThe remarkable property of energy is that once you have formed this intention, once you have evoked this sense of heart-opened harmony and balance, you can impart it to your living space, through the paint, the substance with which you wash the windows, and the substances you use to clean your carpets. As long as you hold this feeling quality of the image, you can imbue everything around you with it\u2014putting a little white magic spell on your walls, so to speak.\n\nBartlett recommends lightly sanding all the wall surfaces and cleaning them with soap and water before painting. This is another level to clearing out the old, dense energies of the space and bringing the environment up to a level of purity and neutrality upon which you can now impart your own intention and vibration. If your floors are carpeted, Bartlett advises vacuuming then cleaning them thoroughly with a nontoxic carpet-cleaning solution. You may need to use Clorox bleach to remove the residual mildew in and underneath the carpets. (Be careful, though: you don't want to bleach the color out of your rugs.) Similarly, scrub and sanitize all windows, both inside and outside, removing all traces of dirt and residue. \"Windows are the eyes of the _qi_ (life force energy) and affect one's clarity, so replace any broken panes and clean all the windows.\"\n\n\"Do this with clear intention, that you are clearing out older, denser energies, and filling your home with loving light,\" says Bartlett. \"Intention is very important. Your thought-energy is crucial in creating a new reality for you.\" Once you have evoked a sense of your ideal living environment and the feeling-quality associated with it, try to stay in this meditative space throughout your housecleaning work. The positive, life-affirming vibrations you impart to your walls, paint, windows, and floorings, will return to nourish and support you throughout your stay in that home.\n\n## HEALTHY LIVING SPACE DETOXIFIER #57: Clear Your Space of All Predecessor Energy\n\nThoughtful painting and imparting the color of happiness to your interior walls is the outer, physical aspect of clearing out old, dense energies of former occupants of your living space, says Bartlett. But you also need to address the subtler aspects of having had former residents in your living room and bedrooms. The central idea here is that the older, denser energies of former residents will interrupt or contribute to the stagnation of _qi_ in your house, unless you take steps to remove them.\n\n\"If you were to examine these dense, old, stuck energies with Kirlian photography, using a camera that takes pictures of energies in color, you would see these as a kind of clouded, murky fog,\" says Bartlett. \"On the other hand, a light-filled energy would be very clear and crystalline-like. That's what we want to create in our environment because the denser energies tend to keep us stuck.\"\n\nLet's ground this observation with some examples. Eira Hellberg, a Swedish collector of occult lore, remarked some years ago that often when examining an apartment with the intent to rent, she might discern \"something unpleasant in the walls ... an atmosphere that exhales something distinctly human.\" Even though the physical details of the room\u2014wallpaper, carpets, furniture, the colors and layout\u2014seem congenial and hygienic, there is still something off, exuding \"an uncomfortable sense of oppression.\" The person showing the apartment may be friendly, but your feeling of unease does not go away.\n\nHellberg comments: \"If our amiable hosts in their comfortable room cannot overcome our uneasiness, the reason is that the room and the objects in it are _saturated_ with the mental activity of the occupants, their will- and thought-waves, which are beamed back on any receptive individual who comes within range of them.\"\n\nIn other cases, intense feelings, such as of despair or violence, can remain in a room or \"in\" the former occupant's furniture (in the case of a furnished rented room) and exert a harmful, if unsuspected, influence on the new tenants of the living space. For example, French occult scholar Paul Sedir noted in 1923 that \"human beings exercise a real influence on the objects in their environment. A chair that has been flung across the room in a rage stores anger inside it; the things used by a niggardly housewife make subsequent owners stingy.\"\n\nA feng shui expert echoes this observation: \"If the last tenants' marriage ended in divorce, or someone was very sick, patterns of this energy stay with the house until it is cleared.\" Similarly, if a particular room had formerly been given over to young children with the intent of their having great fun in it, \"lingering childish energy might impede your plans for a tranquil retreat.\"\n\nStuck energy and \"psychic debris\" can accumulate in all rooms in a home, but especially in comers, nooks, and crannies, comments another expert. Many people, unknowingly, live in the \"psychic equivalent of a garbage dump.\" Everything that happens in a room is recorded in all of its structures; \"repetitive patterns get deeply imprinted, as do moods and atmospheres.\" Further, it is conceivable that if previous persons living in a house, or room, had health problems and were, for example, overweight, or had eating disorders, the energy supporting this, or associated with its expression, is still present in the house and may influence a new generation of weight disorders.\n\n_**Stuck energy and \"psychic debris\" can accumulate in all rooms in a home, but especially in corners, nooks, and crannies, comments another expert. Many people, unknowingly live in the \"psychic equiualent of a garbage dump.\" Everything that happens in a room is recorded in all of its structures.**_\n\nGerman occult researcher Willy Schr\u00f6dter relates how a Viennese family \"suffered inexplicable health\" for two years. It turns out an oil painting that hung in their living room was, according to a clairvoyant who discerned this, \"a hate radiator.\" Once the painting was removed and _burned_ , all occupants of that household regained their health. In other words, the painting radiated energy that acted as an unrelieved toxin to all those in its sphere. Schrodter cites the examples of others for whom hygiene encompassed the realm of subtle influences. One man surrounded himself only with new furniture and objects, so as \"not to have any old thoughts or feelings hanging about him.\"\n\nThe reality of \"mental atmospheres\" and their effects on consciousness must be accepted, Schr\u00f6dter and many other eso-tericists state, thereby confirming what feng shui has always said about _ling_ energies, or predecessor energies. Schr\u00f6dter comments that according to the Tibetans, \"the Earth's atmosphere is injured by unkind thoughts\" and that when people send out bad thoughts, they collect in a manner similar to water vapor condensing into rain, and fall down on people in the form of diseases. This phenomenon of \"raining toxicity\" can happen at any level of residence\u2014country, city, community, neighborhood, single house, or single room.\n\n_Directions_ : There are many practical steps for clearing an interior space of lingering predecessor energies, according to feng shui.\n\n### Remove Old Energies with an Energy Blanket\n\nThis is a practical step advised by Carol Bridges, an instructor at the Nine Harmonies School of Feng Shui in Nashville, Indiana. Bridges suggests visualizing that you wear \"energy gloves,\" and that if you stretch your arms before you, you can send \"strands of light\" across your property (including your home or apartment). Imagine that a \"team of spirit energies\" (or helpful forces) are cooperating with you in this space-clearing work (that is, cleaning the subtle \"airs\" around you), and that they occupy the basic compass directions.\n\nNext, visualize that you and the helpers hold an \"energy blanket\" underneath the entire property, so that it is twelve feet underground. Asking that the blanket collect all foreign, inimical, and predecessor energies from this site, raise the blanket up through the house until you and the helpers are holding it twelve feet above the top of the room, house, or apartment building, says Bridges. Coming up through the house, the blanket would have collected all these energies, which may or may not present themselves to you in a variety of displeasing forms. Bundle up and tie the blanket and ask the spirit team to take it away.\n\n### Use Incense, Candles, and Expelling Mudras\n\nOne way to remove older, denser energies is to burn a strongsmelling incense such as frankincense, sage, or other pungent incense while walking through each room and allowing the smoke to infiltrate the air and furnishings, recommends Bartlett. As you fill each room with the aromatic fumes, visualize the older, denser energies clearing out and \"loving light\" filling the room throughout. You can also bum candles in each room to \"burn off the old and symbolically fill your home with loving light,\" Bartlett adds.\n\nAmong the \"transcendental cures\" Bartlett uses in his feng shui practice is the expelling mudra. This is a special hand gesture meant to change the energy field in a room. Curl the middle and ring finger of each hand toward your palm while keeping the index and pinky fingers pointing straight out. Hold the backs (nails) of the middle fingers with the ball of your thumb. Flick the two middle fingers upward \"expelling all older, denser energies.\" According to Bartlett, this mudra can be used to clear the aura, the land, an automobile, an airplane, and any room, including a motel room or a doctor's office, among many other venues.\n\nOther simple but effective ways to change the energy of your interior space include singing, dancing, chiming bells, drumming, ringing gongs, striking cymbals, toning (emitting tones or other nonverbal sounds), or playing music in each room, says Bartlett. According to Celtic folklore, one of the principal reasons church bells were tolled frequently in villages, especially on Sundays, was to drive away all the lingering spirits, both good and bad. The spirits could not abide the sound of the bells and considered it \"poison.\"\n\nWhile performing these space-clearing exercises, it is important, says Bartlett, to visualize your living space \"filled with loving light and that only those energies that are harmonious to your being are present.\"\n\n### Keep the Lights On\n\nMost people think it is energy-efficient to keep only the minimum of lights on in a house in the evening. However, from a feng shui viewpoint, this is not advised. It is better for the circulation of _qi_ through an interior environment if a light is left on in each room, preferably twenty-four hours a day, says Bartlett. \"During the evening hours or on cloudy days, leave a light on for twenty-four hours in the basement to keep the _qi_ active.\n\n\"It is not advisable to keep all the lights turned off, other than the one area you are currently in, as this can cause the energy in the rest of the house to become tired. This is especially important in the winter, as it helps to avoid winter blues and depression. A small light left on twenty-four hours a day in the basement will also help keep the foundation of your home activated.\" If the _qi_ is moving and the foundation \"activated,\" there is no room for energy stagnation, which is a reservoir for predecessor energy. \"Carefully positioned lighting can cause stuck energy areas to flow. When we keep the _qi_ flowing, that tends to be an environment where lingering energies choose not to hang out.\"\n\nOn a practical level, Bartlett recommends conserving electricity (since he's advising to keep all the lights on) by using energy-efficient, full-spectrum fluorescent bulbs in each area of the house and by designating these as the lights that are kept on in the evenings. You can use 12-volt halogen track lighting in the kitchen instead of fluorescents, if desired, Bartlett adds.\n\n### Clapping in Every Room\n\nAccording to \"space-clearing\" expert Karen Kingston, you can disperse static energy in a room by clapping your hands loudly in every corner, crevice, and closet in that room. After clapping, be sure to wash your hands under running water to clear away \"any bits of psychic debris that may have clung to you\" while clapping.\n\n### Purify Secondhand Furnishings\n\nDespite the fantastic bargain you may have achieved on that antique coffee table or Spanish clothes dresser or sofa-bed, it is quite likely that these furnishings bear energy residues of their former owners and users and that these residues, if not expunged, will visit your energy field and produce discomfort. Let's reprise a technique introduced in Chapter 9 as Healthy Living Space Detoxifier #40, Scrub Your Aura Clean with a Golden Sponge.\n\nFirst, apply this golden energy sponge to the furnishing in question, mentally swabbing all its surfaces and interior, visualizing that this golden energy field is absorbing the toxic residues. Then imagine that there is a drain under the table, dresser, or sofa-bed, and that this drain goes all the way to the center of the Earth. Instruct the drain to draw to it all the negative, foreign, toxic, predecessor energy presently within the object. You may get a visual impression of sludge or mucus flowing slowly out of the furniture and down the drain. Continue with this draining until no further exudate is visible, or for at least five minutes if you don't have any visuals on the proceedings. Then visualize a bright golden Sun over the furniture and slowly bring it into the object so that sunlight permeates every molecule of its structure. These steps should make the table, dresser, or sofa-bed fit for your aura.\n\n## HEALTHY LIVING SPACE DETOXIFIER #58: The Health Benefits of Ghost-Busting\n\nPredecessor energies may involve more than the residual thought patterns and energy waves of previous occupants of your living space. They may involve the purely \"transcendental\" aspects of energy imprints, a term Bartlett uses to refer to \"the realm beyond our current intellectual understanding.\" But what we cannot understand with our minds, our bodies and auras can nevertheless register, even if it is possible ghosts in the house.\n\nAccording to the classical description of _ling_ energies from the feng shui canon, the life force energy, or _qi_ , of a deceased person may return to their familiar earthbound haunts, such as a former living space and inadvertently (or sometimes purposefully) annoy, irritate, or deeply trouble the current living occupants, says Bartlett.\n\nHe was recently asked by some householders to remove the influence of a former occupant who had died twelve years earlier and whose lingering presence was producing \"weird ghostlike events\" and other negative energies for the new owners. To some, the return of a deceased former resident\u2014perhaps he is a relative or parent\u2014can be perceived as a pleasant event, but for others it can be quite disruptive. In either case, it can be confusing and intrusive to one's own _qi_ to be visited by the _ling_ energy of a deceased being.\n\nIn ghost-busting cases such as these, Bartlett relies on his various space-clearing techniques, as described above. In some instances, occupants with no previous training or experience in metaphysical matters may be able to expel the foreign energies, or they may have to engage a consultant, says Bartlett. If you possess any physical objects belonging to the former and now deceased occupants, it is helpful to purify these of all energy residues of the previous owners, using the techniques described above in \"Purify Secondhand Furnishings.\" Otherwise, these residues act as magnets or homing signals, enabling the deceased to find a familiar landmark in an otherwise troubling (because no longer \"physical\" the way it used to be) environment.\n\nIf it is the case of a lingering ghost, in most cases, the intelligence inhabiting the ghostly form does not know it is dead and no longer lawfully associated with the physical plane. Or if they know they're dead, sometimes they don't know what to do next, or feel a strong gravitational pull towards both their former domicile and the Earth plane in general, almost out of habit.\n\n\"It is appropriate to lovingly invite these _ling_ particles [or spirits of the deceased] to return to the light for completion of their soul's journey,\" comments Bartlett. It may sound ridiculously simple, but often telling them these facts is sufficient to move them on their way. The hard part may be _believing_ that the ghosts are really there in your house and are capable of exerting toxic effects on you.\n\nHere are two examples from my experience as a householder that help to illustrate this possibly challenging, unsettling observation.\n\nOnce I stayed at a small ranch in New Mexico and found I couldn't sleep in the appointed guest room. I tossed and turned, and strange thoughts ran through my mind. I perspired, fretted, worried, got irritated, and got up. I usually fall asleep within five minutes of my head reaching the pillow, and if I don't, I figure it's because I went to bed too early. But this was different. My wife was unable to sleep, either. The next morning we learned from the owner that generally nobody was able to sleep well in the room.\n\nBetween the three of us, we mustered enough psychic power to investigate the problem. The impression we gathered was that underneath the room in question was a casket containing the remains of a teenage Native American girl who had been buried there some two hundred years earlier. She had lived in the space where the ranch house now stood and had died young and under considerable trauma, due to a severe sudden illness. Our impression was that \"she\" didn't approve of where she was buried and was still traumatized by the events, which to us were several centuries ago, but to her, were the eternal present, having happened only \"minutes\" ago.\n\nAs we were not about to excavate the ground under the room, not to mention tear up the floor to get to it, we called on higher authorities for help. To the extent we were capable, we counseled the girl and advised her to move on, and let the spiritual helpers do the rest. That night we slept wonderfully in the room, as if it had never been the site of insomnia for many.\n\nIn this example, had someone taken over this room as a bedroom and not been aware of the possibility of _ling_ energies, quite likely, physical, emotional, and mental symptoms would have developed in accordance with the \"fault lines\" of that person's constitution. The toxic emanations of the restless dead girl could have produced irritation, chronic sleeplessness or broken sleep, nightmares, foreign dreams, paranoia, despair, fear, panic, headaches, elevated blood pressure\u2014any of a long list of possible symptoms.\n\nIn the second example, I once rented a home in coastal California. There was a two-car garage, which I appropriated as my office and filled with bookshelves and a large worktable. However, I soon noticed that every time I spent more than ten minutes in the garage, I became irritated, even angry. Even though that is my \"fault line\" psychologically, I have learned to use it as a dowsing rod to detect foreign, inimical energies. Feeling irritation is one way I receive information from my aura that there is an energy imbalance in my immediate environment; of course other times it's just because I didn't get my way.\n\nMy wife and I isolated the toxic energy influence to an area of about three feet square in the central part of the garage. It looked like a vertical column of energy, like a smoky quartz crystal column\u2014nothing I'd want to have on my coffee table to be sure. Using a pendulum, we could chart where the field started and stopped; even so, being within ten to fifteen feet of it was enough to irritate me and drive me from the garage. We placed a very large clear quartz crystal on the edge of this toxic field to deflect the energy somewhat, but I still felt it anywhere I stood in the garage. Soon I stopped working in the garage and even avoided going into it unless I had to.\n\nThe day before we moved out of the house\u2014we lived there only one year and did not like it at all\u2014we discovered what the problem was in the garage. We saw two men lying on the garage floor, dead of knife wounds; police cars were parked in the driveway. The events took place approximately in the mid-1950s, which meant their toxic influence had lingered in the garage for more than forty years. Again, we asked for \"backup\" from the spiritual world, and watched as \"they\" escorted the lingering souls of the slain men off into the light. As soon as they were gone, the vertical column shattered into a thousand pieces and the vile energy spot was purified. The garage was now fit for human occupation, although I suspect the next residents used it for their car.\n\nThe lesson in this garage haunting is the same as with the dead girl under the floorboards. The toxic emanations were capable of producing no end of unfortunate effects on those within their field of influence. In matters like these, unless you can discern what is self and not-self as far as thoughts and inclinations go, you are likely, regrettably, to act out as your own, to some degree, the baleful influences your aura is registering. Sufficient immersion in that angry, literally murderous energy field in the garage could have provoked somebody to a significant outburst of rage, or worse. This is the kind of _ling_ you don't want lingering under any circumstances.\n\nEven _ling_ energies of a more benign nature are not to be encouraged. \"These energies disturb your _qi_ , producing interference with your energy, so that things get too crowded energetically,\" explains Bartlett. \"Unless these spirits are personal spirit guides, they are best not kept around your living space for long periods of time because then it becomes a case of too much interruption, too much interference.\" They become an additional toxic element in your living space, and can distract you from your focus on positive activities and your own creativity, says Bartlett. \"In my experience, when your environment is free and clear of entities and ghosts, then you are allowed to co-create in more ease-filled and perfect ways.\"\n\nGhost-busting of course is not something everyone will feel comfortable undertaking, nor is it particularly advisable to try to remove restless spirits if you are just beginning to acknowledge subtle energies. Often in such cases, it is prudent to engage the services of a qualified feng shui practitioner, a dowser, or a psychic trained in this particular type of space clearing. It is not unusual for a well-seasoned dowser to be skilled in all three of these practices.\n\n  **_The Healthy Living_ Space Info Tip**\n\nFor information about dowsing, including books, practitioner referrals, educational conferences, and publications, contact:\n\nThe American Society of Dowsers,\n\nP.O. Box 24,\n\nDanville, VT 05828;\n\ntel: 800-711-9530 or 802-684-3417;\n\ne-mail: ASD@Dowsers.org;\n\nwebsite: www.dowsers.org.\n\n## HEALTHY LIVING SPACE DETOXIFIER #59: Eliminate Poison Arrows and Cutting Qi\n\nIt sounds strange at first hearing, but according to feng shui practitioners, the quality of your life within your house can be affected, for better or worse, by the way other houses face yours. Obviously noisy neighbors can be irritating even when you are indoors, but this is different.\n\nThe way other houses _face_ your house can be an energy toxic element. Face means the orientation of the other houses, their geometry with respect to yours, the angles and comers and how they're situated in relation to your house and particular sections (as defined by the _Ba-Gua_ ) of your house. When these geometric features of neighboring houses are injurious, they fall into the category feng shui terms \"poison arrows\" and \"cutting _qi_ ,\" and this becomes something you need to address for your long-term well-being in your home.\n\n_Qi_ flows best and is most supportive of health when it can always move in gentle, meandering curves, but not in straight lines, and it should always be able to keep moving. When it enters a house, _qi_ must be able to depart from a different door; it can't flow backwards. But you don't want the _qi_ rushing into your house like an assault team, which can be the energy effect of an unchecked poison arrow.\n\nA poison arrow (also called a \"secret arrow\") is any straight line or sharp angle that heads directly for you, like an energy arrow shot from the bow of an opposing architectural structure. Here's how feng shui practitioner Angel Thompson explains it: \"When it is forced into straight lines or sharp angles, instead of gently falling and rising like the tummy of a sleeping baby, _chi_ compresses into a barb, like an arrow, as sharp as the tip of a knife. . . .\" She adds that no matter where you live, \"there are many secret arrows around you.\"\n\nHere is a way to gain a visceral sense of the problem of poison or secret arrows. Sit before the comer of a rectangular table so that your eyes are about six inches from and level with the right angle. Observe how this makes you feel. It is very likely that within a matter of minutes, or less, your eyes will feel uncomfortable, as if assaulted, or even penetrated; you may feel that spears are piercing your eyes and you want to pull your face away from the line of vision. But what has happened? The table hasn't attacked you\u2014or has it? In terms of feng shui, it has: it has thrown poison arrows at you by virtue of its geometrical relationship to you. On an energy level, the two lines that form the right angle continue through space, beyond the confines of the physical table, and pass through whatever is before it, in this case, your eyes.\n\nYou can expand the example to the sphere of an entire house. Say a neighboring house is situated at right angles to your front door, so that when you stand at your open door, you are confronted by the right-angle comer of that house. It doesn't matter if it's fifty or 100 feet away; you'll still get the poison arrow effect, just as you did when sitting before the table. Your energy field will register it as a hostile, even aggressive energy focus.\n\nOther examples of poison arrows can include T-intersections on roads, driveways, sidewalks, rooflines, corners, streetlights, walls, signs, porches, pillars, clock towers, statues, any very large imposing object, the sharp edge of any nearby building, sharply defined hills and landscape contours\u2014essentially any straight line, extended from an architectural or landscape feature, that comes right at you or at your front door. Other toxic situations include having your house at the center of a fork in two roads, at the outside of a bend in the road, or in the center of a circular traffic pattern, such as a rotary or roundabout. Remember, in and of themselves, each of these features is not necessarily harmful, but when placed in the wrong geometrical relationship to your house, they become toxic.\n\nPoison arrows are harmful for your health because they embody \"cutting _qi,\"_ which is also known as noxious _qi_ , \"the cutting breath.\" They interrupt, distort, divert, or pollute the natural, health-promoting flow of basic life force energy through an environment, in this case, your home through its front door, the \"mouth of _qi_.\" \"All these structures give out an energy that is too strong and will overwhelm your home, often with sad or tragic consequences,\" explains Lillian Too, a well-known feng shui expert and author. Poison arrows can block the good, healthy _qi_ from flowing into your home, leading to energy stagnation. Poison arrows generate _shar qi_ , says Too, \"the killing breath,\" capable of producing every type of misfortune and ill luck, especially disease and fatal illness.\n\n_**Poison arrows are harmful for your health because they embody \"cutting**_ **qi,\" _which is also knows as noxious_** **qi, _\"the cutting breath.\" They interrupt, distort, divert, or pollute the natural, health-promoting flow of basic life force energy through an environment, in this case, your home through its front door, the \"mouth of_** **qi.\"**\n\nSome feng shui experts say the reason poison arrows are harmful is that they project \"high-velocity _qi_ ,\" basic energy that's moving too fast (because it's moving in a straight line) to be health promoting. _Qi_ is supposed to flow in curves, but when it is forced to move in a straight line, it gathers in power and acts like a speeding projectile or flying arrow. Others note that one main reason they're called \"secret\" arrows is that most people are not immediately aware of their effects, which build over time, from minor irritations to major disturbances. But meanwhile your energy field, or aura, feels as if it is facing directly into a sharpened knife point, or as if it is the stationary target receiving arrows in its body from an expert bowman.\n\n\"A secret arrow is like having a stick stuck in your side: not only can it be annoying, but eventually it can also cause harm,\" notes Nancilee Wydra, founder of the Pyramid School of Feng Shui in Vero Beach, Florida. She adds that even a person with a normally positive, uplifted temperament can gradually be worn down by this \"insidious or subtle pressure.\" Secret arrows can distort your own _qi_ field, and make you feel unsettled, confused, disoriented, and as if you have lost your grounding and sense of life direction.\n\nHere's a practical way to visualize the probable effect of poison arrows. Imagine you stand within a large inflated balloon that extends perhaps three or four feet away from your physical body in all directions. Next, imagine all the poison arrows you documented on your list (see below) come at this balloon from the outside and cause indentations, punctures, and disfigurations. Now imagine that this balloon around you can transmit its energy to your physical body, and you ar _e feeling_ the effects of these dents, holes, and distortions in your physical organism.\n\nThis \"imagination\" is actually not too far off the mark from what actually happens. The balloon corresponds to your aura, or energy field, which surrounds you and transmits energies to your physical body. Sooner or later, depending on your inherent vitality, your body will register the effects of these distortions in its energy envelope and will manifest this registration in the form of symptoms and, if unchecked, eventual health problems. Just to be precise, even though your aura can sustain punctures and holes, it does not \"deflate,\" although its local integrity collapses inward to an extent at the site of the injury.\n\nThe feng shui answer to poison arrows is deflection. In nearly all cases, you won't be able to move the offending object or source of poison arrows, but you can effectively block their energy effects.\n\n_Directions_ : There are many ways to deflect or calm the incoming toxic _qi_ , according to long-established principles of feng shui.\n\n  * Before we get into the practical steps, here is a preliminary exercise. Open your front door, stand in the doorway, and survey your environment. Look for poison arrows. Examine the buildings nearby: do they have sharp angles projected at your front door? Repeat this exercise by standing before every window in your house, looking out at your neighbors from inside. Make a list of all possible secret and poison arrows, no matter how slight or inconsequential they may seem. Contemplate for a moment the combined effect of these poison arrows on your energy field, using the simple example of the effect of a table's comer on your eyes to get a visceral sense of the assault on your aura.\n  * Plant bushy trees, shrubs, or a dense hedge to block off the poison arrow. This is like placing a shield between yourself, your front door, or window, and the source of the poison arrows. The natural foliage will absorb the toxic energy stream of the poison arrow, so that it does not reach your house.\n  * Employ mirrors, shiny reflective surfaces, or plaques, even polished brass doorknobs, orienting these towards the source of the poison arrows. If a poison arrow from an abutting house is focused through a window, hang a reflecting mirror inside your window but facing outwards towards the arrow. (For more on mirrors, see Healthy Living Space Detoxifier #60, below.)\n  * Calm the incoming _qi_ according to Five Element theory. This approach is typical of feng shui's marvelous sense of using restored balance as a way of eliminating problems. Simon Brown, a design engineer and feng shui consultant, explains that insofar as the _Ba-Gua_ gives the orientation for the five elements (according to Chinese philosophy), you can use the interrelationships of these elements to deflect poison arrows and \"calm\" or \"drain\" their effect. For example, earth drains fire, metal drains earth, water drains metal.\n\nSay you have an incoming poison arrow from the southwest, which is the orientation for earth, which is \"drained\" by metal, explains Brown. To calm or drain the cutting _qi_ from the southwest, place an object associated with metal _qi_ , such as a gold or silver disc in the southwest corner of your house. \"Alternatively, place a large statue made of cast-iron, bronze or other metal, which includes circular shapes, in the garden between your house and the corner.\" The same applications can easily be worked out with the other elements and compass directions, according to the _Ba-Gua._\n\n  * If your home is disturbed by traffic patterns or is at the end of a T-intersection or other toxic traffic arrangement, hanging wind chimes in your hall, just outside the front door, or in your backyard can help cleanse and purify the _qi_ around your house so that it no longer transmits disturbed energy patterns into your living space.\n  * If you have a tree, post, or pole directly facing your front door, the ideal solution is to remove the tree, because it blocks the flow of _qi_ into your living space. If you can't remove the _qi_ , Angel Thompson recommends placing a small mirror in the middle of the tree (or post or pole) so that it faces your front door. This mirror acts as a \"symbolic hole,\" says Thompson, and absorbs the noxious _qi_.\n  * If your walkway leads directly in a straight line from the road or driveway to your front door (thereby generating a poison arrow), arrange bricks, stones, or potted plants on the edge of the path along its course to deflect and soften the _qi_ flow and to break up its straightness.\n\n## HEALTHY LIVING SPACE DETOXIFIER #60: Pacify Your Neighborhood with Feng Shui Mirrors\n\nProbably most people have experienced obnoxious, noisy, or irritating neighbors whose lifestyle activities we wish we could curtail or eliminate. Or maybe our neighbors don't like us, or don't like anybody, and transmit a steady field of ill will and negativity in our direction. Or their back porch faces our back porch and they are too close for comfort. Or your house abuts a funeral parlor or cemetery.\n\nThere are many scenarios, both mild and serious, in which the presence of neighboring houses (and their occupants) sometimes exceeds our ability to cope. Short of moving away to live somewhere else, there are some palliative steps you can take using feng shui principles.\n\n_Directions_ : One of the simplest of these methods is the mirror, which some practitioners refer to as \"the aspirin of feng shui.\" Specifically helpful here is the _Ba-Gua_ mirror, an octagonal piece of reflecting mirror set inside an octagonal frame with some of the traditional _Ba-Gua_ signs and traditional colors of red, green, and gold. If you have troubling neighbors, place a series of these mirrors (they're about four inches in diameter) around the outside of your house and facing the direction of the irritation.\n\nUse the _Ba-Gua_ mirror to redirect intrusive energies such as from an intrusive neighbor, says Stanley Bartlett. \"Simply point the mirror with loving intention towards the intrusive energy and it will lovingly redirect it back to the source of the intrusion. That will redirect the intrusive energies as if to say 'return to sender.'\" Using mirrors to deflect threatening _qi_ achieves two purposes at once, according to Sarah Rossbach, author of _Interior Design with Feng Shui._ \"The mirror both offensively reflects back malignant ch'i and defensively provides protection. Any size will do.\"\n\n  **_The Healthy Living_ Space Info Tip**\n\nFor a source of practical feng shui devices, objects, mirrors, and other feng shui tools, as well as books and practitioner listings, contact:\n\nFeng Shui Warehouse,\n\nP.O. Box 6689,\n\nSan Diego, CA 92166;\n\ntel: 800-399-1599 or 619-523-2158;\n\ne-mail: fengshuiWH@aol.com;\n\nwebsite: www.fengshuiwarehouse.com.\n\nUsing mirrored glass walls (such as you see on many urban skyscrapers) is also effective if you care to spend the money to refit your windows or redo one side of your house. Further, Rossbach suggests, as a way of removing \"the overbearing forces altogether,\" a convex mirror that reflects the images upside down. However, Bartlett cautions against over-using mirrors or using them randomly (especially inside the house), as \"they are powerful remedies and can cause chaotic energies when misused.\"\n\nIn terms of influencing your neighbors, there is more you can do than setting up effective defense perimeters against their noisy intrusions, explains Bartlett. You can send them nice vibrations. The simplest way to do that is by purifying your space; this change in the energy pattern of the neighborhood gets registered, at some level, by your neighbors. If they are sensitive, they may detect that somehow the change came from your house, and they may want to make changes themselves, says Bartlett.\n\n\"Because everything affects everything, if you focus on your environment, and make that as whole and healthy and clearly aligned as possible, then you immediately affect your neighbors,\" Bartlett says. \"Inevitably, after doing a house clearing, sometime later\u2014a week, a month or two\u2014I'll get a call from somebody a street or two away in that same neighborhood asking me to do feng shui on their house. As I understand it, on some level, they _felt_ the positive changes. They heard, felt, or sensed the energetic change in their neighborhood and it sparked them and woke them up, and now they want some of it, too. That's one way to affect your neighbors in a positive way.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #61: Optimizing the Mouth of Qi at Your Front Door\n\nOne of the most important elements of your living space to check and correct is your front door, or whichever entrance you use most frequently. This is your point of transition from outer to inner world, from the external environment into your private domestic space. The threshold into your home, and to a lesser extent, the threshold into any room in the house, is where the life force energy enters like an exhaled breath. Feng shui calls it \"the mouth of _qi_ ,\" and if your home's \"mouth\" is unclean or obstructed, your health will suffer.\n\n\"The way _qi_ flows throughout the house is important,\" explains Bartlett, and the front door can be the key to a healthy flow. Your home's mouth of _qi_ represents your relationship with the outer world, both natural and social: energy, money, opportunities, good health, well-being, friends, and good fortune flow in through the front door, but if the flow of this energy (and its numerous expressions) is blocked, the _qi_ cannot get to your marriage comer, or prosperity section, or any of the other six _Ba-Gua_ life divisions, and your life, correspondingly, will be negatively affected.\n\nYou need to think about the matter of _qi_ flow in both a symbolic and practical sense, because both realms will help you detoxify your home using feng shui principles. How the _qi_ flows, or doesn't, affects the parts of your life that correspond to the parts of your house where the _qi_ is obstructed. The front door is like a _qi_ valve, which can let in positive energy and block out negative energy. \"Inhibited, obstructed, or confused flow of chi is probably the number one problem to solve with feng shui,\" comments Angel Thompson. There are numerous ways in which the _qi_ flow can be blocked, compromised, or diverted, and any of these arrangements can \"frustrate, anger, even enrage the chi,\" with the result that the occupants feel the same way. \"Tension, poor health, and bad luck await those who occupy the space.\"\n\n\"What you see first upon entering a house is important, as this creates your first subconscious experience of its atmosphere,\" observes Bartlett, and this is especially true of your own living space. Preferable is a view of a foyer or vestibule or living room with its evocation of rest, leisure, and quiet entertainment; not preferable, for example, is a view of a bathroom or kitchen, says Bartlett. Particularly undesirable in a two-story home is a direct alignment of second-floor toilet, staircase, and front door; in the symbolic energy calculus of feng shui, your wealth (an expression of _qi)_ flows right out the front door and\/or down the toilet.\n\nA first view of a kitchen is undesirable, says Bartlett, because the first thought you get upon entering your home is food. The result may be eating or bingeing tendencies, which lead to weight problems. Feng shui consultant Terah Kathryn Collins relates a case study of a woman who was unable to overcome her bulimia. Collins, upon visiting the woman's house, found that her front door opened directly into the kitchen. The solution was to transform the kitchen (at least what could be seen from the home's threshold) into an atrium, with palms, bromeliads, and a flowing water and rock feature. The kitchen functions were moved down to one end of the room and redesigned as a kitchen nook. The woman thereafter found it much easier to correct her unbalanced eating habits since her home (and its mouth of _qi_ ) was not constantly stimulating her to overeat.\n\nYou can get a sense of the flow of energy through your home's mouth of _qi_ by standing at the threshold of your home, facing in. Imagine a gently forceful breeze flowing into your house through this open front door; it is a breeze that brings all good things, invigorates everything it contacts, clears out stagnation, and brings well-being _\u2014provided_ its flow is unobstructed. Follow this flow of energy through your house: what path does it take? What gets in its way? Where does it get stuck? Is there an exit point, a back door, out of which it can flow when it's finished inside your home?\n\nThe information you gather from this simple exercise can be the basis for applying feng shui \"remedies\" to improve the _qi_ flow in your living space. The key point to remember is this: obstructed _qi_ flow produces energy stagnation, and this negatively affects the health of the home's occupants.\n\n  * Keep hallways and access areas free of clutter and unnecessary objects, says Bartlett. If possible, avoid zigzagging flows (such as multiple L-layouts in hallways and attached rooms) from one section of the house to another. These \"twist up your energy field, causing confusion and an uncentered feeling, which can be quite exhausting.\" Further, it makes it hard for the _qi_ to flow smoothly, or at all, through the house, says Bartlett.\n  * Make sure your front door opens fully and is not obstructed by anything behind it. The hindrance in opening fully obstructs the _qi_ flow and it can chronically provoke a reaction of annoyance and irritation by those who open the door, expecting it to yield fully. \"When doors are allowed to open fully, the people walking through them can open fully to promising opportunities, resources, and circumstances in their lives.\"\n  * Another typical and undesirable feature at the entrance is constriction, such as if there is a wall close to the entrance on the inside. \"If upon entering you come to a wall close to the entrance, then this would be said to limit your opportunities in life,\" says Bartlett. It inhibits _qi_ and, according to traditional feng shui thinking, it guarantees you a life of struggle, if not failure within three years.\n\nIf this is the case in your home, place a mirror (not a feng shui mirror but a conventional larger reflecting mirror) on the wall to face the door, \"thereby expanding the reflected image so that the wall feels more expansive.\" Alternatively, you can hang a picture or photograph of an outdoor scene, \"featuring a stream flowing down through a valley, thereby allowing your sensitivities to go beyond the wall in front of you,\" says Bartlett.\n\n  * If you live near a place of excellent _qi_ , such as by the ocean, a park, a river, lake, mountain view, or other natural setting of high quality _qi_ , there is a simple way to invite this _qi_ to flow into your home. Place a feng shui mirror outside your front door (or the door or window facing the auspicious scene) in such a way that it reflects the scenic environment into your house. You should be able to see the scene reflected in the outside mirror as you stand at your front door (or window).\n  * If the immediate space inside your front door is dark or constrained, this oppresses the mouth of _qi_ and \"chokes\" the occupants' good luck. Or if the entrance is a narrow hallway, this can generate health problems such as respiratory problems and difficult births; further, the psychological impact of such a darkened entryway is depressing and can lead to melancholy moods. The feng shui \"cure\" is to install a very bright ceiling light (at least 100 watts) and a mirror on the opposite wall; both will invigorate the _qi_ and create a sense of depth.\n  * Feng shui offers other practical low-cost devices either to ward off harmful _qi_ or to maximize inflowing positive _qi._ Wind chimes, hung under eaves, can temper and redirect malignant energy or summon positive _qi_ into a living space. Rounded prisms or small-faceted crystal balls can convert overly strong _qi_ and disperse it throughout an interior space.\n\n## HEALTHY LIVING SPACE DETOXIFIER #62: Setting Up a Feng Shui-Friendly Bedroom\n\nPossibly of equal importance to your home's mouth of _qi_ is the disposition of your bedroom, where you spend up to one-third of your life. There are many key factors to consider in applying feng shui to your bedroom, including the following recommendations about bed placement:\n\n  * Do not situate your bed such that your feet, as you lie in bed, face the bedroom door. In classical Chinese thinking, this is the \"death position.\" In old China, corpses were laid out with their feet pointing towards the door to facilitate their passage into the next world. _Do_ situate your bed so that the bed faces the door but displaced to the side. In other words, your feet face the same wall that contains the door, yet none of the bed directly faces the door opening. Remember, that on a smaller scale, the door into your bedroom is the mouth of _qi_ for that room and its function, so the same principles apply here as they do for your home's main front door.\n  * Put the bed against a solid wall if possible, but not a window because your _qi_ can escape through the window while you sleep, leaving you depleted in the morning.\n  * Regarding bed placement, the important principle is to have the correct relationship with your bedroom's mouth of _qi_ , explains Bartlett. Ideally, you want to be in a \"commanding position,\" he says, so that \"when you're lying in bed on your back, you can clearly see the doorway to the bedroom. This gives you a feeling of being in control, of having power over your environment.\" If anything enters the bedroom, you will see it as it enters and not be taken by surprise by its unannounced entry. \"On a subconscious level, if you are not able to see the bedroom's doorway at night when you sleep, this might create a feeling of uncertainty and insecurity.\"\n\nOne long-term consequence of having your bed positioned wrongly with respect to the bedroom's mouth of _qi_ is that \"your opportunities in life might be limited,\" says Bartlett. \"You might find that you're not easily manifesting and creating things that you want and need.\" Bartlett cites an example from his case records.\n\n**_One long-term consequence of having your bed positioned wrongly with respect to the bedroom's mouth of qi is that \"your opportunities in life might be limited,\" says Bartlett, \"You might find that you're not easily manifesting and creating things that you want and need.\"_**\n\nHe consulted with a woman who lived in an apartment; she told Bartlett she felt stifled in her career and wanted to establish a relationship with someone. \"Her bed was placed in such a way that it was not in the commanding position, yet when I explained this to her, she was resistant to changing it.\" She told Bartlett she felt comfortable being \"tucked away\" into the room's corner; he countered by suggesting that it was perhaps \"part of her subconscious need to protect herself.\" He encouraged her to try moving the bed as an experiment, to see how she felt under the new conditions. \"She did this, then all those difficult aspects immediately changed in her life.\"\n\n  * Do not place your bed under an exposed beam or low slanted ceiling. \"These angular protrusions are carriers of sha [negative _qi_ ] and can cause headaches, confused thinking, and even financial and career problems.\" An exposed beam that runs the length of the room, bisecting the bed from above, can provoke marital discord, literally by placing an energy barrier between the partners. Beams that cross the bed horizontally can generate aches, pains, even diseases in the body regions under the crossing influence, such as a beam across the abdomen leading to digestive disorders.\n  * Do not place mirrors facing the bed. If you wake in the middle of the night and see your reflection, it can be very unsettling, even shocking.\n  * Generally, it is not advisable to place electronic equipment in the bedroom, such as computers, televisions, fax machines, or radios. These generate a constant electromagnetic field, which can negatively affect you while sleeping, and they can confuse the function of the bedroom (rest and regeneration) with profession and career (such as using the computer for writing, bookkeeping, or Internet research). They serve as continual reminders of work obligations, and this is contrary to the state of mind that induces relaxation and rest.\n  * There are two ways to minimize or eliminate the effect of electromagnetic fields upon you as you sleep at night, says Bartlett. First, if your head is near an electrical outlet (not desirable but often unavoidable), you can install a switch in the circuit box that shuts off all electrical power to the bedroom at night. Second, you can place a flat spiral dielectric resonator under your bed. \"This device balances the energetic field under a bed and will neutralize most noxious zones. Aside from improved sleep, many people report they suddenly feel more comfortable.\"\n  * Don't use the bedroom as a storage facility. Piles of materials, such as books, clothes, boxes, or other objects in corners of the room or under the bed lead to areas of stagnant, nonflowing _qi_ , and these conditions will eventually have a negative impact on the health of the room's occupants. These piles can also produce jumpiness and disturb one's sleep.\n  * Don't situate the head of your bed against a wall that abuts the bathroom. Feng shui consultant Angel Thompson relates a case in which a woman consulted her for relationship help, reporting that her affairs were constantly falling apart and \"going down the toilet.\" Thompson found that the head of the woman's bed was against the same wall that abutted the toilet. So symbolically and energetically, her relationship\/marriage section of the _Ba-Gua_ was in the toilet space, which flushed away all the marriage _qi_ generated or cultivated in the bedroom.\n\nThompson recommended moving the bed to another wall; this apparently worked as, soon after, the woman got married. If you can't move your bed, place a mirror in the bathroom on the wall opposite the toilet to deflect the toilet energy from entering the bedroom, Thompson suggests. Alternatively, place a mirror on the bedroom wall opposite the bed \"to symbolically move the bed to the opposite side of the room\" and away from the toilet wall.\n\n  * There are several ill-advised locations within the house for a bedroom, according to feng shui practitioners: under a toilet, a washing machine, or stove (on the next floor up); over a garage; at the end of a long corridor. The flow of _qi_ down a long corridor is too fast and strong and can attract health problems over time; it's even more harmful if your bed faces the door that opens into the corridor. The situation of a bed over a garage leads to an inability to develop foundations in your life, to the sense of rootlessness and a transitory residence in the home.\n\n## HEALTHY LIVING SPACE DETOXIFIER #63: Setting Up a Feng Shui-Friendly Bathroom\n\nFeng shui experts agree that a key principle with bathrooms is that the _qi_ must be able to flow in and out of this room without any obstruction or any place in which it can get stuck and produce stagnation. They also agree that feng shui imbalances in this section of the house can produce health problems, as this case study from the practice files of Terah Kathryn Collins vividly shows.\n\nCollins was called in to consult with a woman who had moved into a new house only nine months previously. During that short stay, she had sustained an inexplicable sudden bout of clumsiness and three falling injuries. She had tripped or fallen as many times in nine months as she had in her entire life of forty-five years, so she suspected something in her environment might be stimulating this. Collins determined that while most of the woman's house was in good feng shui balance, there was a significant problem with the bathroom. For one thing, it was in the wrong place\u2014that is, the home's sole toilet was situated between two bedrooms, both of which opened into it with their doors in direct opposition to each other.\n\nThis configuration speeded up the flow of _qi_ as it passed through the bathroom and into the bedrooms, especially as their doors were usually left open, says Collins. \"Living with the rushing Ch'i can cause you to move too quickly,\" she told the homeowner, who admitted that in the three falling incidents, in which she was injured, she was \"rushing to do something.\" In other words, her own actions were influenced by the imbalanced flow of _qi._ it rushed through, and so did she, except she got off balance each time and fell.\n\nOnce Collins had identified the source of the problem, she was able to apply effective feng shui solutions. Keep the bathroom doors closed whenever possible; keep the bathroom drains closed and the toilet lid down; hang two round faceted crystals in the bathroom, one by the window (to draw the _qi_ into the bathroom from outside), the other in the middle of the bathroom (to balance the flow from the two bedrooms when both doors were open).\n\nCollins further recommended that the woman hang in the bathroom a piece of art that to her symbolized good health, and also hang a large wardrobe mirror on the outside of the bathroom door leading into the bedroom currently being used as an office to increase productivity in that room. A few months after the changes were completed, the woman reported that ever since the changes, she had been healthy, calmer, and had not fallen or tripped.\n\nAs mentioned above, _qi_ needs to move easily and briskly in and out of the bathroom, but water is another element of crucial importance to this room. Sink, toilet, shower stall, bathtub\u2014from out of the bottom of each flows water. Feng shui experts counsel that it is vital to observe the energy symbolism here so that water does not get equated with money, and as your bathroom goes, so goes your prosperity. It is a truism\u2014yet true nonetheless\u2014that if your toilet faces your mouth of _qi_ (your home's main entrance), you run the real risk of having your good fortune (incoming _qi)_ flushing down the toilet every day. Here are some additional tips for the feng shui-friendly bathroom:\n\n  * A potted plant can encourage _qi_ to flow into the bathroom, and blue is an excellent color for the walls or fittings because this color relates to the water element, advises feng shui advisor Richard Webster.\n  * However, he counsels if your bathroom is located in the middle of your home, in the good luck area (the center of the _Ba-Gua_ octagon), this is most unfortunate because \"it will send negative ch'i throughout the house\" and your good luck down the toilet. Obviously it is not practical to rip out the toilet and move it, so Webster suggests placing _Ba-Gua_ mirrors on all four walls of the bathroom to produce the effect of making the room symbolically disappear. In a sense, its otherwise negative energy gets trapped and neutralized in an infinite hall of mirrors, and becomes incapable of escaping this energy matrix to exert an influence on the rest of your home.\n  * Webster also says it is unfortunate if your bathroom is located at the end of a long corridor, because this sends a _shar_ (poison arrow) heading directly into it, and beneficial _qi_ entering your house through its mouth of _qi_ goes down the toilet and is lost to the home's occupants. The solution is to keep the toilet lid down when not in use, to keep the bathroom door closed, and to place a mirror on the outside of the bathroom door\u2014again to make the room \"symbolically\" disappear and at least to disarm the poison arrow flying down the corridor.\n  * Another problem with the bathroom is if it is located in the wealth section of the _Ba-Gua_ , as this is likely to produce financial or abundance problems. Again, the issue is one of flushing your wealth down the toilet. If your toilet is located in this section, Australian architect, engineer, and feng shui consultant George Birdsall advises placing a crystal in the bathroom window to attract incoming _qi_ from outside, keeping the toilet lid down and the bathroom door closed, and placing a feng shui ( _Ba-Gua_ ) mirror on the outside of the bathroom door to \"seal off this room's energy from the rest of the home or business.\"\n\n### The Health Costs of Having an Energy-Imbalanced Home\n\nAt first glance, many of the feng shui observations and recommendations cited here and throughout the burgeoning literature of the ancient art of placement seem odd, perhaps ridiculous, and surely impossible according to our general sense of the way the world works. How can a toilet flush affect our well-being? But feng shui describes the world and its workings at a subtler level of operation than we are accustomed to credit it as having; it works in a sphere of influence about which we are normally oblivious. Feng shui deals with causes, and we are usually aware only of the effects, in the form of dysfunction, ill health, or misfortune.\n\nEven though it seems hard to grant that your prosperity can be symbolically flushed down the toilet if your bathroom is situated in the wrong part of your house, this eventuality has been demonstrated empirically in hundreds of documented case studies reported by feng shui practitioners. The _Ba-Gua_ is not an arbitrary or fancifully contrived scheme; it is, according to feng shui, a true-to-life model of energy relationships at any level of reality\u2014a hologram of the cosmos, if you will\u2014and especially at the level of a residence. Functions within each of its eight sections are influenced, if not directed, by that section's symbolic role within the overall energy configuration and the dynamic interplay of the five elements during the course of the year.\n\nPerhaps the best way to demonstrate this point is to relate the following remarkable case study of the terrible effect of bad feng shui arrangements in a home upon its occupants, as explained by Jenny Liu, an environmental designer and feng shui professional. Liu relates the extreme toxic effects a house had on its teenage resident and how she corrected the bad feng shui.\n\nA woman called Liu and said her teenage son was rebellious, having \"mental problems,\" and generally driving her crazy. Visiting the woman's house, Liu immediately saw that the orientation of the house itself was highly unfavorable. It was a \"backwards-sitting house,\" which means a hill rises in front of the house, and behind the home, the ground slopes downward. For well-being, the arrangement should be the opposite, Liu says. The hill before the house blocks the inflow of good energy into the house, while the incline out back allows energy to drain away from the house.\n\nThere were other more arcane aspects to the home's orientation\u2014its orientation and energy style were contrary to that of its occupants\u2014that ensured that the home itself would produce internal trouble for its occupants, especially any males living within it. It would weaken their health (producing head colds and respiratory problems) and generate unstable relationships. Inspecting the interior of the home, Liu found that a fireplace directly faced the main entrance (mouth of _qi)._ \"Having a major circulation path where people are constantly coming into conflict with fire energies is a potential cause of family arguments,\" Liu told the woman, who confirmed that her family members had many disagreements.\n\nFurther, a stove and an exposed overhead beam were located in the center or \"heart\" of the home. This spelled further trouble, because the stove produced temper, while the exposed beam \"suppresses heart energies that can cause family members to feel congested and stifled.\" Both features were triggers for constant quarrels among family members. The son's bedroom door was directly in line with that of his parent's master bedroom, a configuration that set the stage for conflicts between the occupants of both rooms. Further, the two bedrooms were separated by a bathroom, which faced the _Ba-Gua_ direction associated with the liver and eldest son, in classical Chinese thinking. Functionally, this meant that the bathroom's inherently \"degrading energies\" would conflict with the son's liver energies, and make him moody and easily tired, and hinder his body's ability to detoxify itself.\n\nLiu found the bedrooms to be dark, poorly ventilated, and intensely cluttered, and the teenage son's room had numerous wall posters depicting violent images. Just to complicate matters further, the home's previous owner had died in the house and the present owner's teenage son had tried to commit suicide three times already. Liu's overall assessment was that this house was a \"container of _sha qi,\"_ or harmful, negative, health-dampening energies. Anyone living in such a house, she added, \"is bound to get very sick, have abnormal behavior, mental problems, and suffer a great deal of trouble and money loss.\" Astonishingly, the woman's family had endured this \"sick house\" for ten years, although she had tried on occasions to move; failing this, she admitted her family was \"slowly falling apart.\"\n\nLiu's main recommendation was to move out of the home immediately. It was too full of negative, stagnant energies for her family to cope any longer; already the males had succumbed to its bad influence. Liu noted that the woman's husband's aura was extremely dark and that his face showed signs of kidney exhaustion; even so, he was unaware that the house was having a bad influence on him, and he denied that there were any problems afoot inside his home at all, so he refused to move.\n\nLiu set out to purify the _sha qi_ , or negative energies. She advised the family to clean up the clutter and get rid of everything they didn't need; to cover the fireplace with a screen to hide it; to hang crystal balls and place living green plants on either side of it to neutralize the fire energy; to place a mirror behind the stove to reflect negative energies away from the home's center; and to hang a wind chime between the son's room and that of his parent's to disperse some of the conflicted energy between them. Liu further instructed the son to take down all the violent posters and put in their place images of nature and uplifting themes related to his personal goals.\n\n\"These are temporary means of ameliorating some of the negative energies, so that they may live a bit more comfortably while they search for a new home,\" observes Liu, noting that the father finally consented to move. These measures would not, she added, change the bad effects of the land and home's structural configuration, but they would help the family enhance their own positive energies so as to resist the debilitating effect of their home. These measures would help them \"break the cycle of having the house's energies constantly degenerating their well-being.\" While the combined toxic elements of this _sha qi_ house did not entirely produce all the ill-health and poor results in this family, they _amplified_ the residents' latent vulnerabilities, tendencies, and unresolved conflicts to the extent that they became almost unbearable and lethal.\n\nWith the exception of the issue of the home's orientation, most of the health-destroying toxic elements in this \"sick house\" existed inside the house. In the next chapter, we take a look outside the house to see what effect subtle exterior environmental influences and toxic agents can exert on our health.\n\n# CHAPTER 12\n\n# _Cleansing the Home's Exterior Environment: Neutralizing the Effects of Geopathic Stress_\n\nIt comes as an alarming discovery to many to learn that after having removed all the toxic elements in the home to make it a bau-biologie-friendly residence, as discussed in Chapter 10, and having arranged the home according to feng shui principles to make it a _qi_ -friendly living space, as in chapter 11, there may still be another underlying toxic element, one of equal if not greater potency to produce ill health. This one literally _underlies_ the home.\n\nThe influence is called geopathic stress, or geopathogenic stress, and refers to the ability of the Earth to produce illness or pathology. It isn't that the planet itself is inherently toxic, but that in certain places, the combination of natural and\/or humanmade elements can produce toxic effects. If your home is situated over such a confluence of potentially toxic energies, you may develop any of a wide range of illnesses and physical dysfunctions, including chronic fatigue syndrome, cancer, and multiple sclerosis. It gets worse if your bedroom, or merely your bed, sits over such an underground field of noxious influence, principally because most people spend one-third of their lives here.\n\nTypically, the term geopathic stress encompasses the health-altering effects of underground water streams (50-450 feet below the surface and 10-20 inches wide, although they can be any width), mineral and ore deposits and concentrations, caves, geological faults, underground cavities, or specially \"disturbed\" places within various energy \"grids\" purported to exist on or below the Earth's surface. It can also include human-made features such as mining installations, foundations for tall buildings, underground transport systems such as subways, sewers and drains, and public utility installations.\n\nThese various geobiological influences, according to the researchers, emit what have variously been described as \"noxious fields,\" \"earthrays,\" \"Earth radiations,\" and \"black streams\" that penetrate everything above them, destabilize the electromagnetic fields of living organisms, compromise immune function, and produce illness. The key is that various factors can distort the Earth's natural, and usually beneficial electromagnetic field (EMF); in fact, the planet's EMF is subject to natural gradients, of variations of stronger or weaker strengths. As one expert puts it, \"Geopathic stress occurs when the Earth's magnetic field is disturbed, either naturally or artificially, and the background field we normally experience is changed.\"\n\nOnce the natural radiation is disturbed, it can become harmful to biological life, including ours as humans, if we spend a lot of time over it. If you spend enough time over a geopathic stress zone (and the amount of time can vary dramatically from hours to years), some form of illness or dysfunction is likely to develop, according to the Dulwich Health clinic in London, one of England's premier geopathic stress (GS) and treatment facilities. \"By diagnosing well over 40,000 people, we have found GS in people with most types of serious and long-term illnesses.\"\n\nThe effect is especially enhanced if we sleep over it, because while relaxed, the body is more susceptible to the harmful influences, and the brain and nervous system must spend some of their energy dealing with the noxious influence instead of resting. Normally during sleep the brain accomplishes the body's \"housekeeping,\" creating some eighty percent of required new cells and directing the body to absorb nutrients and adjust its hormone levels. \"GS will interfere with this process and leave your immune system weak. Geopathic stress does not cause any illness, but lowers your immune system and your ability to fight off viruses and bacteria.\" Dulwich Health emphasizes that usually all these body functions return to normal once GS is removed from the environment and its effects cleansed from the body.\n\nPut more technically, advocates of the discipline of geo-pathic stress explain that harmful Earth energies, in the form of ionizing and nonionizing electromagnetic radiation that emanates naturally from the planet's \"geophysiology,\" rise upwards to the surface of the planet and can negatively affect all life living over and in its influence. Fortunately, geopathic stress, once detected, can be minimized, reversed, or eliminated according to a variety of simple but effective techniques.\n\n### _Disease Can Be a Problem of Location_\n\nKnowledge of geopathic stress emerged in Germany in the early days of the twentieth century when a researcher named Baron Gustav Freiherr von Pohl discovered a link between cancer clusters and the patients' homes. In 1929, working in Vilsbiburg, Germany, a village of 3,300 people and 565 houses, von Pohl correlated the addresses and sleeping locations of 54 people who had died of cancer with the existence of noxious underground water streams that passed beneath the homes or bedrooms of the cancer victims.\n\nVon Pohl was acting on a hunch inspired by the earlier Winzer-Melzer survey of Stuttgart in the 1920s, which had determined that major geological faults in that city traversed the districts that had the highest cancer mortality rates. The tentative conclusion was that an unknown but noxious radiation emanating from the earth faults might be an important and overlooked contributory cause of the cancers.\n\nThrough dowsing, von Pohl located all the major subterranean water veins (lying at a depth of forty-four to fifty meters with a width of three to four meters) under Vilsbiburg, then mapped their courses on the city street plan. Next he crosschecked this with the residences of the fifty-four recent cancer fatalities and arrived at a startling conclusion. \"The completed check of my map confirmed all the beds of the 54 cancer deaths were where I had drawn the radiation currents,\" von Pohl wrote in 1932 in his now classic _Earth Currents: Causative Factor of Cancer and Other Diseases._\n\nVon Pohl concluded that those who died from cancer had \"slept in a strongly ray-infected place\" at the time of their diagnosis. In fact, he declared: \"All diseases of human beings, animals, trees and plants are actually caused by earth currents; the currents weaken the organism and make it more vulnerable to disease.\" Von Pohl was confident about his discoveries but tended to take a conservative stance regarding developing a hypothetical model. \"My observations set down in this book about negative electrical earth currents are in the main virgin territory for medical science. We not only hope but expect that more doctors will research and advance this new knowledge for the benefit of mankind.\"\n\nEighteen months later, von Pohl returned to Vilsbiburg and found that the beds of another ten cancer mortalities were situated directly over the intersection of underground streams. Following von Pohl's indications, another researcher named Dr. Hager investigated the residences of 5,348 people who had died of cancer over a 21-year span in the town of Stettin; he found that in all cases \"strong earth rays\" and a subterranean water vein had intersected underneath their homes. \"Medical science has now a preventative measure which did not exist previously,\" noted von Pohl. \"If one makes sure one's bed does not stand above a strong underground current and one tries not to work above these underground currents, one should not get cancer.\"\n\nThere are very few people whose constitution is robust enough to permit sleeping \"in a bed full of radiation\" without some form of adverse health effect, von Pohl added. In fact, von Pohl documented the direct relationship between sleeping in a bed situated over Earth currents and numerous health complaints: nervous itching, insomnia, abdominal pains, asthma, shivers, headaches, migraine, gallbladder dysfunction, rheumatism, heart spasms, tuberculosis, heavy menstruation, thrombosis, mental illness, and epilepsy, among others. Not only did von Pohl describe the correlations between bed location and noxious underground emissions, he demonstrated the curative benefit of simply moving the bed to avoid the geopathogenic zones. He thereby produced remarkable reversals in numerous immedicable health conditions.\n\nVon Pohl had determined the existence of the underground streams using the services of professional dowsers, which to most scientists was not a \"legitimate\" methodology. In 1960, another German researcher named Jacob Stangle replicated von Pohl's research using sophisticated geomagnetometers and confirmed von Pohl's 1929 findings. Since von Pohl's day, a host of researchers from Germany and other countries have confirmed and considerably extended his model of \"cancer houses\" and \"cancer beds.\"\n\nFor example, in the 1930s, Swiss researcher Dr. S. Jenny studied the effects of geobiological faults and subterranean water streams on the health of laboratory mice. He documented these geopathic effects on 25,000 mice over a period of twelve years of study. Dr. Jenny found that when the cage of the mice was placed over a disturbance in the geomagnetic field, the mice grew agitated, gnawed their own tails, and sometimes ate their offspring; they also often developed cancer. When they were moved to a neutral zone, free of geopathic stress, the mice settled down. Given the choice, the mice chose to stay over the neutral (healthy) zone rather than over the geopathically stressed one. Not long after, Professor K.E. Lotz of the School of Architecture of Biberach, Germany, analyzed 400 cancer deaths and found that 383 were related to beds situated over geological faults, underground water veins, or disturbances in the natural geomagnetic field.\n\n**_Austrian researchers found that when a person was exposed to the noxious energy emissions of the Earth there were measurable changes in a majority of the twenty-three biochemical factors considered, such as the blood values of serotonin and melatonin as welt as levels of important minerals such as zinc and calcium._**\n\nIn 1989, Austrian researchers finished a twenty-four-month study on the short-term effects of geopatho-genic sites on human health. Researchers led by Otto Bergsmann, M.D., a specialist in internal medicine at the University of Vienna in Austria, tested 985 sites with 462,421 different measurements included within 6,943 individual tests. They found that when a person was exposed to the noxious energy emissions of the Earth there were measurable changes in a majority of the twenty-three biochemical factors considered, such as the blood values of serotonin (an important brain chemical, or neurotransmitter) and melatonin (the sleep-regulating hormone) as well as levels of important minerals such as zinc and calcium. \"There were many other altered states to biological functions noted as a result of short-term exposure to these forces.\"\n\nHans Nieper, M.D., another noted German researcher and former director of the Paracelsus Silbersee Hospital in Hannover, Germany, concluded on the basis of his clinical observations that seventy-five percent of his multiple sclerosis patients spent too much time over a geopathogenic zone. He found similar correlations for cancer, reporting that ninety-three percent of people with malignancies acquired them as a result of residing over \"a crossing in a geopathogenic zone.\"\n\nDr. Nieper explained that the geopathogenic zone did not necessary produce the cancer all by itself, but the incidence of cancer cases is certainly higher in correlation with such zones than otherwise, and the effect of living or sleeping over these noxious zones \"increases the risk of gene lability,\" which becomes a biochemical foundation for the emergence of cancer. Dr. Nieper characterized the \"geopathically stressed zone\" as \"the ultimate push button\" that makes the cancer happen. Like von Pohl, Dr. Nieper strongly advised those suffering from the ill effects of geopathically stressed areas to remove themselves from the site of geopathogenic exposure.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #64: When Nothing Else Heals, Try Moving Your Bed_\n\nDespite the complexity of the geopathic stress theory and its dismaying illumination of another toxic environmental energy, often the solution is radically simple. Move your bed, even a few feet, and you may get well. While the subject of geopathic stress and geopathogenic zones is still practically an unknown medical category in North America, in Europe, especially Germany and England, it has been recognized to a far greater extent by physicians as a factor in illness and its reversal. In fact, some physicians now state that as a rule of thumb if a patient fails to respond to the indicated therapies (either conventional or alternative), it is possible that a geopathogenic influence exists in their living space.\n\nIn an influential article in the British _Journal of Alternative and Complementary Medicine_ , physician Anthony Scott-Morley of the Institute of Bioenergetic Medicine looked at how geopathic stress might block natural therapeutics. He proposed that habitual exposure to \"geopathic disturbance\" produced by \"localized variations in the geomagnetic flux of the earth\" might be a basic reason why natural therapies like acupuncture and homeopathy fail to provide a definitive cure in some patients. He reasoned there must be other mitigating factors that compromised what was otherwise \"excellent and appropriate therapy.\"\n\nScott-Morley reported that according to \"experienced practitioners who are aware of these stresses,\" thirty percent to fifty percent of chronically sick patients exhibit some degree of geo-pathic stress. \"A geopathic stress may be defined,\" wrote Scott-Morley, \"as a geomagnetic disturbance which is geographically localized and which disrupts the homeostatic mechanisms of the sensitive patient. It does seem that geopathic stresses energetically weaken the body so that the patient may be more prone to disease-forming processes.\" A more recent study concluded that geopathic stress can undermine \"both the body's subtle energy system (the etheric body, chakras, and meridians), and the body's electrical system (brain, heart, and muscles), thus delaying healing and recovery.\"\n\nVon Pohl produced stunning turnabouts in clinical conditions merely by moving the patient's bed out of a dowsed geopathogenic zone in the bedroom. Since his time, other practitioners have applied his insight and helped many suffering individuals reverse their otherwise intractable health conditions. Notable among these practitioners has been Kathe Bachler, an Austrian dowser who investigated over 3,000 homes and 12,000 cases of geopathic stress among individuals in fifteen countries. Bachler concluded that toxic underground energy emissions\u2014she called them \"noxious Earth energies\"\u2014represent a major contributing factor in the emergence of many health problems.\n\n\" _All_ serious illnesses I have found up to now occur above interference crossings: twitching, tinnitus, facial paralysis, angina, asthma, heart attacks, strokes, inflammation of the kidneys, cancer (in more than 700 cases without exception). Doctors say that our immune system is weakened by constant radiation so that, depending on one's constitution and the prevalent influences, different diseases can develop.\" Bachler specialized in correlating poor school performance in children and children's health problems with geopathogenic zones or what she refers to as \"interference zones.\" Below is a sampling of her empirical field results.\n\n  * A young mother had two boys with serious behavior problems. Every night, one of the boys, depending on which bed he was sleeping in, would toss and turn, and throw his arms about as if he were having a fit. Bachler found he was sleeping over an interference zone that got activated every night when the boiler on the other side of the bedroom wall was turned on. She moved both beds and the children slept soundly thereafter.\n  * A boy was a chronic bed-wetter and, after seven years, no physician could provide an answer to the embarrassing problem. Undersized for his age, nervous, and irritable in the morning, he was sleeping on a crossing of two interference zones. \"With other bed-wetters\u2014all of whom lie on interference zones\u2014a cure can often be immediate and often achieved simply by moving the bed at random,\" says Bachler.\n  * Poor school performance is often linked with \"geobiological influences.\" Bachler found that students who were always tired, were the slowest in doing school work, had the greatest difficulty in concentrating, were the most forgetful or the worst behaved, or were frequently ill \"were all victims of interference zone crossings.\"\n  * A twelve-year-old girl developed a severe backache during a gym class, and the pain did not go away afterwards. She was confined to her bed for eight weeks with cramp-like pain, but no physical cause could be identified for her discomfort. Bachler found she was sleeping over an interference zone. The girl's bed was moved immediately and in a few weeks the mother reported her daughter's back problem had healed.\n\nIn the cases above, and in the thousands of others treated by Bachler, moving the bed of the one who was ill reversed the health problem, usually in a dramatically brief period. In every house, even in every room in a house, \"there are also good places where we all have the chance of becoming healthier in body, mind, and spirit,\" she concluded.\n\nDuring the last three decades of the twentieth century, as the concept of geopathic stress gained currency in the progressive flanks of American research and alternative medicine, dowsers independently verified and extended the original findings. For example, Vermont dowser Herbert Douglas stated that based on ten years of research and investigation of sixty cases of arthritis, twenty of cancer, and nine of cataracts, in every case, intersecting lines of energy passed under the bedroom in direct relation to the affected part of the body.\n\nIn almost all the cancer cases, Douglas found \"a network of water veins creating anywhere from thirty to fifty crossings.\" In some cases, clefts or breaks in a subterranean rocky ledge underneath the ground (but directly underneath the bed) created the harmful energy emissions, he said. Douglas said the easiest way to visualize this hidden but measurable effect was to lay a series of wooden laths across the patient's bed, to correspond with the dowsed underground energy lines. With actual clients, Douglas laid the laths on the bed, according to his dowsing indications, then had the patient assume their customary sleeping position on top of them. \"Repeatedly, the crossing of the laths indicates precisely where the person is afflicted,\" states Douglas.\n\nFrench research in the 1940s demonstrated the connection between subterranean energy emissions (ionizing radiation) and the incidence of cancer. Pierre Cody made a seven-year study in the French city of Le Havre using an electrometer to measure air ion concentrations at various locations in the cellars underlying 7,000 \"cancer beds\" in the homes of people who had died of cancer. Cody discovered that the air ion concentrations were ten times higher when measured on the spot in the cellar directly under the bed than at spots even a few feet away; he further found that the typical width of the band of toxic radiation was about four and one-half feet. His study also showed that the radiation was capable of rising vertically, up from within the ground and through the patient's bed, without spreading out horizontally.\n\nBritish dowser and geopathic stress investigator Alf Riggs also built an impressive empirical base for his conclusions about geopathogenic zones. He studied the energy emissions in 14,000 homes in fifteen countries for correlations between noxious underground radiation and illness. Among the health problems, he studied 2,200 cases of chronic fatigue syndrome (myalgic encephalomyelitis, as the British call it) and found that one of the two major causes of chronic fatigue is exposure to specific types of Earth radiation, the other being energy depletion due to viruses.\n\n\"As a result of these observations, I contend that chronic fatigue syndrome is initially a bioelectric problem,\" says Riggs. This illness often results \"from an imbalance in the value of bioelectric fields at cell level caused by the invasion of specific electromagnetic fields\" from the outside edges of the underground water stream. Riggs further concludes that disease is, in fair measure, \"a problem of location, with the fields generated from Earth and manmade sources having the ability to invade the cell, altering the values of the bioelectric systems within.\"\n\nAs a subterranean water stream flows under a house (or simply under the Earth's surface), it interacts with the geological strata present and creates a positive electrical field, more specifically, a direct-current-generated magnetic field, Riggs explains. Small bands of energy appear on both outside edges of the flowing water and radiate upwards. According to Riggs, most of the biological damage caused by geopathogenic zones produced by underground water is due to these two outside bands. Riggs estimates it is the chief factor in seventy-two percent of cases.\n\nHe reports that after moving a patient's bed into a neutral zone in the bedroom, \"away from the harmful Earth rays and manmade radiation,\" chronic fatigue patients \"often enjoy a measurable improvement,\" although they do not always have a full recovery from this intervention alone. Usually what completes the recovery is some form of energy therapy or \"bioelectric treatment\" by a skilled practitioner. This combination of approaches has resulted in significant improvement in over eighty percent of the cases Riggs has overseen, including many long-term cases involving severe disabilities.\n\nRiggs further explains that these frequencies have deleterious effects on women by interfering with their estrogen cycle. Riggs investigated 1,600 cases of energy depletion, and found that young girls can sleep in beds positioned over the outside edge lines of underground streams for years without being noticeably affected until they reach puberty, when without exception they develop chronic fatigue. Boys may sleep on such noxious lines without ill effect, but adolescent girls always suffer far worse because of the negative interaction between the emitted electromagnetic fields and estrogen, says Riggs.\n\nNot only can the position of the bed be a negative health factor when it sits over a geopathogenic zone, but sometimes the composition of the bed itself can relay the noxious energies to the sleeper, says Riggs. In one case, he found that the geomagnetic field under and around a single bed was within normal range (that is, the bed lay over a zone, but the zone exerted only a minor influence), but the mattress springs and the metal in the headboard increased the field by ninety percent, rendering it toxic to the person in the bed.\n\nThis person had been ill for six years with a variety of problems including chronic fatigue, depression, mental confusion, and candidiasis (systemic yeast infection). The occupant's health problems had begun right after she had bought the bed six years before. Riggs recommended getting rid of this bed, and using instead one made entirely of wood with a nonspring mattress; the bed was also moved elsewhere in the bedroom to a neutral zone. Once she made these changes, the woman's health started to return, even though she had received no therapeutic benefit from years of consulting a list of qualified health practitioners.\n\n## _HEALTHY LIVING SPACE DETOXIFIER #65: Scan Your Home and Environment for Signs of Geopathic Stress_\n\nThe evidence continues to mount linking bed location and noxious underground emissions with health problems. But how can you tell if your bedroom, or the location of your bed, is making you sick because it sits over a geopathogenic zone? British natural health practitioner and geopathic stress educator Jane Thurnell-Read offers the following \"do-it-yourself' indicators:\n\n  * Look for places of persistent dampness in your house. Even though physical factors such as poor ventilation may seem to be the cause, the reason the damp flourishes may be due to an unsuspected geopathic zone.\n  * Often your pulse rate (measured at the wrist) rises significantly when you spend time in the influence of a geopathic zone.\n  * Bees, ants, and cats flourish over geopathic energy zones that are harmful to humans. If you have these animals in your environment, study their behavior; wherever they congregate, you should not. Thurnell-Read notes a curious correspondence between high levels of geopathic stress and homes that have cats that were once strays. She adds that in many cases the homeowners had discouraged the strays from staying, but they had remained nonetheless, probably because of a preference for the noxious (to humans) geopathic zones. Further, look for where your cat sleeps; some cats often choose drafty, unpleasant (to humans) areas, preferences that might indicate the presence of harmful Earth energies.\n  * Most trees will tend to lean away from negative Earth energies, with the exception of oaks, ash, willows, elder, and elm, which like geopathogenic zones. If you can discern this botanical behavior on your property, see if the tree leaning fits a generic pattern of distortion and if this pattern can be traceable or definable to a particular area of ground, such as the ground under which a water vein might pass. Also look for cancerous-like growths on apple trees, abnormal sap increase in cherry trees, and plum and pear trees that rot or wither. These phenomena indicate geopathic stress, says Thurnell-Read.\n  * Other techniques may be used, but they require a little advanced training or, in some cases, a little enhanced sensitivity. You can dowse your environment, using rods or a pendulum, asking the device to indicate a \"yes\" or \"no\" answer to designate the presence or absence of geopathic stress. You can use muscle testing or kinesiology to measure your body's response (or that of someone else) to different energies and locations in your environment; here, typically, a muscle goes weak when the body is in contact with something unbeneficial to it.\n\nAnother way to determine if your house sits over a geopathogenic zone is to fill out the following questionnaire (see figure 12-1), developed by professional dowsers and geopathic stress educators Tom Passey and Robert D. Egby. These are some indicators of probable geopathogenic zones in your bedroom.\n\nIf you answer yes to at least one third of these, it is worth considering calling in an expert to investigate your house for possible geopathogenic zones. It certainly can't hurt things to move your bed provisionally to a different spot in your bedroom and observe the results in your health and well-being over the next two weeks.\n\n**Figure 12-1. 13 Telltale Signs that You're Probably Living in a Geopathic Zone**\n\nAnother approach is to get a geopathic survey performed on your house. One organization that provides this service is the Geo Group, based in Medina, Washington. \"A geopathic ( _geo_ -Earth, _pathic_ -disease) survey determines if there are Earth energies . . . under a home or office and if they are neutral or negative (polluted or potentially harmful),\" the organization explains. All the Geo Group requires from a homeowner in order to prepare a geopathic survey is a map of the home, its interior layout and how it sits on its property; the location and types of furniture in each room; a listing of known problems or unusual conditions; and the names of the people living in the home.\n\nBased on twenty years experience, the Geo Group's founder Chuck Pettis recommends a geopathic survey if someone in your family is hyperactive, cranky, or moody for no discernible reason; a place in your home doesn't feel right, or feels scary or evil; you feel energy is being drained from you or pulled out of you; you feel someone or something is watching you; you feel unproductive; there is someone in the house with a history of illness or a disease that doesn't improve; there is a feeling of unease; strange things happen in your home.\n\nThe Geo Group notes that while geopathogenic zones can produce these effects, the effects can be exacerbated or complemented by the presence of \"human entities (ghosts) and nonhuman entities (invisible malevolent beings).\" (See Chapter 9.)\n\nPettis, an expert in creating \"sacred space,\" comments: \"Entities tend to take up residence around water lines or springs. They seem to be attracted to the field that the moving water gives off.\" Although he does not mention this, it is logical to presume that if you have, without knowing it, unwholesome energies or beings in your aura, by a kind of negative resonance this in itself can attract you to live in an environment that has corresponding geopathogenic energies.\n\nThat there is a kind of feedback loop or field of resonant attraction between your energy field, entities, and sites of geo-pathic stress is confirmed by Australian psychic practitioner, Samuel Sagan, M.D., whose views on auric pollution we visited in Chapter 9. Living in a house full of noxious lines, also called earth ray lines or black streams, \"often leads to catching multiple perverse energies and, possibly, entities,\" he says. (Perverse energies are not entities as such, but discordant nonsentient energy fields.) Dr. Sagan notes that if you meditate for long periods over noxious lines you may \"catch\" an entity; similarly, sleeping or meditating above a sewage pipe, a tank of stagnant water, or an underground creek can lead to entity attachment. Any mass of stagnant water, including a water bed, \"can attract several 'unattached' entities.\"\n\nFurther, it is prudent to be aware, to whatever degree possible, of the potential presence of \"spirits of the place\" or nature spirits at your living location, advises Dr. Sagan. He notes that the Australian city of Sydney has many sacred sites with heightened \"land energies.\" Some of these sites are uplifting and beneficial to humans, but others are not, and are \"completely incompatible\" with healthy human life. Some buildings have been unadvisedly constructed at such heightened land energy sites, Dr. Sagan comments. He says \"unadvisedly\" because they are sites where no human dwelling should ever be situated. Such places \"will attract fragments [perverse energies] like sticky paper attracting flies.\" Architects usually design homes on paper only and are \"completely insensitive\" to land energies; if they unknowingly pick the wrong place, it is a losing proposition. Remove one entity, and ten more will come back within the week, says Dr. Sagan.\n\nHe has witnessed homes where the residents and their animal pets were constantly attacked by entities and perverse energies, and where the \"spirits of the place were constantly waging war on them.\" The residents became depressed, mentally disturbed, even severely ill as a result. In such cases, it is better to move than to try to combat or remove the spirits of the land. Lest this seem too fantastic a scenario, one needs only remember the plot line of Steven Spielberg's film _Poltergeist_ (1982), in which the troubled spirits of Native Americans made the life of a California family unbearable because their home, while being constructed, had disturbed an ancient burial ground. Of course, nobody removed the spirits; rather they removed the house by imploding it.\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about a geopathic survey of your home, contact:\n\nThe Geo Group,\n\nP.O. Box 602,\n\nMedina, WA 98039;\n\nwebsite: www.geo.org\/dowsel.htm;\n\ne-mail: chuckp@geo.org.\n\nAlso:\n\nFor information about dowsing, including books, practitioner referrals, educational conferences and publications, contact: The American Society of Dowsers,\n\nP.O. Box 24,\n\nDanville, VT 05828;\n\ntel: 800-711-9530 or 802-684-3417;\n\ne-mail: ASD@Dowsers.org;\n\nwebsite: www.dowsers.org.\n\nThis is why Dr. Sagan, like other experts in geopathology, recommends that before moving into a house, you have the house and its surrounding land dowsed for negative energies. If there are traces of previous residents or lingering \"ghosts\" in the land, in many cases these can be removed, if done expertly. In ancient Indian culture, for example, recognition of this was commonplace. When a person moved into a new dwelling, the local priests would perform a _yajna_ , or fire ceremony, including a _bhuta-shuddhi_ , or entity-clearing and purification ritual, to cleanse the energy field of the home and its surroundings, making it fit for human habitation. Something in the spirit of this approach is still recommended today by geopathic practitioners.\n\n### _The Healthy Living Space Expert Interview: Patrick MacManaway, M.D., Ch.B., Geomancer_\n\nScottish-born physician Patrick MacManaway has made a career out of treating sick homes, a discipline he dubs \"whole Earth geomancy.\" With offices both in Shelburne, Vermont, and Strathmiglo, Scotland, Dr. MacManaway operates on the principle that ill effects in a building's residents may be (and often are) traceable to harmful energies emanating from the ground over which they live or work.\n\nDr. MacManaway goes to homes and figures out the sources of geopathic stress that hinder the residents' health. He does this through a process he calls \"therapeutic dowsing,\" which is dowsing with a medical focus; once he has identified the problem\u2014geopathic zone, underground water, radioactive fields, disturbed energy flows, \"lines of various Earth grids or traumatized psychic energies\"\u2014he uses a variety of techniques to neutralize the ill effects.\n\nFor him, geopathic stress denotes a place where the energy relationship between people and environment has become toxic. \"Often there is a great deal of associated emotional trauma and dissonance held in the energy field, with or without the presence of disturbed and disoriented spirits. Geopathic stress may be first experienced as a disturbed, heavy, cold or unfriendly atmosphere in a place, and it progressively gives rise to emotional fatigue and a sense of being overwhelmed, difficulty in human communication and relationship, and it leads to depleted physical health. This energetic soup of trauma and dissonance can be very disturbing to attune to, and I typically know how bad the problems are by how tightly I feel myself contract and tighten as I arrive on the site. The tension that develops in my body, most noticeable in and around my gut and abdomen, gives me a feel for how the people resident in that building feel most of the time.\"\n\nHe defines whole Earth geomancy as \"a holistic environmental health service that is committed to teaching and providing healthy solutions for homes, work environments, and landscapes.\" But what is geomancy? It is a science barely known in North America, but one that is widely employed by practitioners in the British Isles and Northern Europe. \"Geomancy,\" says Dr. MacManaway, \"is the study of the dynamic and interwoven relationship between human consciousness and its subtle energetic matrix with the consciousness and subtle energetic matrix of the Earth.\" Both humans and planet, Dr. MacManaway explains, exist in a perpetual \"dynamic interdependent relationship,\" and geomancy is the discipline that both explains and rebalances this relationship.\n\nOne's ties, both emotional and energetic, to one's house are often more profound than people think, says Dr. MacManaway. \"The human relationship with a place is the primary relationship a human being has. Stress questionnaires have shown that the most stressful thing you can do to yourself is to move; this is an even stronger influence than changing your job, which is second, and getting married, which is third. I think this speaks to the primacy of the relationship between a person and place. Almost in the same way as coming into mood with a piece of music played in the background, humans tend to be primarily reactive to the subtle energetic environment they find themselves in.\"\n\nThe trouble is that our system may be \"reactive\" to an environment, but we are not necessarily aware of it. Most of the time the effect is \"off our radar screens,\" says Dr. MacManaway. The body recognizes the influence\u2014geopathic stress\u2014but the mind doesn't know how to interpret it, so in a sense, it minimizes or even ignores it. \"How rare this is, and how little we notice\u2014so much of our time we are in landscapes of place and of people that are woven with pain and fear, anxiety and hurt. We may notice the initial contraction, but once pulled in to ourselves, we no longer notice how small we have become. We have simply adapted to our environment, and in our adaptation, defined limits to our energetic and emotional availability. We've been taught to disbelieve our imagination, so the place from which we get this level of geomantic information is locked away for many people.\"\n\n**_\"Geopathic stress may be first experienced as a disturbed, heavy, cold or unfriendly atmosphere in a place, and it progressively gives rise to emotional fatigue and a sense of being overwhelmed, difficulty in human communication and relationship, and it leads to depleted physical health,\" says (Patrick MacManaway, M.D)._**\n\nLocked for many, but fortunately not for Dr. MacManaway. In his process of therapeutic dowsing, he scans a residence or its surroundings for evidence of geopathic stress. \"The first layer I scan for is the basic nature and health of the Earth's energies themselves, including water veins and any other detectable lines or pockets of energies. The next layer I look for is the accumulated psychoemotional residue that 'floats' in that first energy layer.\" This residue can include what feng shui characterizes as _ling_ or \"predecessor energy\" (discussed in Chapter 11), the presence of discamate spirits, the residues of human trauma, or imbalances in the subtle energies of nature, what mystics and clairvoyants call the \"elemental and devic kingdoms\"\u2014nature spirits to the rest of us. All of these factors can have a bearing on human health, and when they are disturbed, the effect can be harmful.\n\nThe most direct physiological effects come from prolonged human exposure to underground water streams or veins, says Dr. MacManaway, to which he attributes eighty percent of geo-pathic stress. \"Underground water is a passive receptive energy form; it absorbs energy into itself. It will drain psychic and emotional energy out of a living space so that a room above it may feel cold or appear darker than it actually is, measured objectively.\" It has a kind of vampiric effect, producing a sensation not unlike having a hole in your energy field, he explains.\n\n\"People residing over such a water stream will tend to get very sleepy and lethargic.\" And if it drains energy, the immune system is likely to suffer, he adds, and when that happens, any of a long list of possible health problems may develop and persist until the Earth energy cause is resolved. The data from the Bergsmann study, cited earlier in the chapter, testifies to this observation. \"That study showed shifts in plasma immunoglobulins, changes in the EKG, EEG, and quite a variety of both blood-based and electrophysiological measurements, and both specific and nonspecific measurements of immune function.\"\n\nThe biggest shift noted was in serotonin, which affects mood and the biological measuring of time (as discussed previously). \"That's one of the key factors that one sees with geopathic stress\u2014a disorientation in time sense. A body will tend to sleep heavily, sleep long, but wake unrefreshed. Many times you find gynecological dysfunction, problems with the menstrual cycle and with conception, both in animals and humans. One of my observations from working with geopathic stress is that it is quite frequent in women of menstruating age who are exposed to underground water to have menstrual irregularities for a month or so _after_ I've rectified the geopathic stress. There seems to be a resetting of the pineal measurement of time when the geopathic imbalance is resolved.\"\n\nOf special interest here, says Dr. MacManaway, is the fact that all of the physiological parameters studied by Dr. Bergsmann in relation to geopathic stress would also shift under direct electromagnetic field exposure. But this fact does not take us away from geopathogenic zones at all, because, as Dr. MacManaway explains, \"geopathic stress is related to geomagnetic anomaly (disturbances or imbalances) and to the physiological pathway for naturally occurring electromagnetic disturbance, which affects the energy of geopathic stress. EMF problems of humanmade origin go through the _same_ physiological pathway.\"\n\nNot only do they share the same pathways, but they interact; further, geopathic zones with their associated EMFs also interact with atmospheric phenomena. \"Lightning tends to strike where underground water veins cross under the Earth, particularly where you have a deeper, faster flowing vein that is crossed at a shallower level by a weaker one. All of atmospheric radiation and pollution grounds itself in a more intensely focused way over underground water veins. So if there are local EMF sources and you're on a water vein, then you are probably getting a double dose of geopathic stress.\"\n\nLet's see how these models of geomancy, therapeutic dowsing, and neutralization of geopathic stress play out in actual case histories drawn from Dr. MacManaway's files. \"The most florid and exciting stories are those involving psychic disturbance. Psychic disturbance is very depleting of the energy field and leaves people drained, lethargic, irritable, and, if sensitive, psychically manipulated and physically quite unwell. Some degree of psychic involvement is classic in chronic conditions such as chronic fatigue, although whether the energy depletion or psychic attachment comes first can be hard to sort out.\"\n\nA woman named Patty contacted Dr. MacManaway after thirteen years of \"one disaster after another,\" as she told him. She lived in a 400-year-old house on the border of Wales and England. The house sat in the \"Y\" of a lane that came down from a church at the base of a hill, which meant it was \"almost certainly\" situated on what British dowsers call a \"ley line.\" Dr. MacManaway found that behind the church and up on the hill was a big boulder\u2014a \"glacial erratic\"\u2014which marked the border between Wales and England. \"It was known locally as a magic stone, and if you went around it three times and made a wish, your wish would come true. It had also been the site of incredible atrocities.\" In the old days, anyone found on the wrong side of the border was subject to having his hands chopped off at this stone, says Dr. MacManaway.\n\nFactoring the influence of the stone and its conduit, the ley line, Dr. MacManaway concluded that \"there was significant psychic and emotional trauma here as well as a very strong energy in the house.\" Patty did not have any physical health issues, \"but her whole life was miserable.\" Lots of bad things had happened in it\u2014the list of phenomena is startling:\n\nThree businesses that Patty opened in the house failed; she was a painter, yet shortly after moving into the house, her creative energy dried up and she wasn't able to sell any of her work either. Her neighbors treated her with hostility; several parts of the house felt cold, appeared dark, and felt fearful to her. There were certain rooms she would not enter at night. The plumbing in one of the bathrooms was always breaking down. Patty said she was aware of ghosts in the house; friends and family who visited the house hated it, and all had terrible nightmares when they slept over. Her mother came to visit, got ill, and eventually died in the house; two infants almost died while staying there. Patty's neighbors redug a drain and caused very expensive flooding and water damage in her house, such that she had to wage a six-year legal battle with them for reparations.\n\n  **_The Healthy Living Space_ Info Tip**\n\nPatrick MacManaway, M.D., may be contacted at:\n\nWestbank Natural Health Centre,\n\nStrathmiglo, Fife, KY14 7QP,\n\nScotland, UK;\n\ntel. & fax: 01337-868-945.\n\nAlso at: Whole Earth Geomancy,\n\n4076 Shelburne Road, Suite 6,\n\nShelburne, VT 05482;\n\ntel. & fax: 802-985-2266;\n\ne-mail: PatchMac@aol.com;\n\nwebsite:\n\nwww.geomancy.org\/patrick\/patrserv.html.\n\nEven though she had a large garden with fruit trees, in the thirteen years she lived there, no birds ever visited the property. Dr. MacManaway found this fact alone highly suggestive of major geopathic stress, as birds do not like geopathic zones. When Patty would try to sell the property, major problems would erupt and require up to two years to set right before the house was marketable again. In fact, nobody ever made a bid on the house.\n\nIn desperation, Patty went to the magic stone, circumambulated it three times, and fervently wished to be rid of her house. Three nights later, reality obliged and her house almost burned down, generating thousands of dollars of repairs and many months' delay before the house could go back on the real estate market. All told, there were two floods and two fires.\n\nThe \"magic stone\" seemed to be the focal point for the disturbances at Patty's house, Dr. MacManaway concluded. \"It was the focal point for a lot of the psychic trauma this house contained.\" Dr. MacManaway performed what he calls \"Earth acupuncture\" around the house (described in more detail below). He burned the aromatic herb sage in the house, and daubed the interior of the home with water from a holy spring. He assembled seven other people and together they did a \"collective blessing\" on the house, ringing bells, lighting candles, offering prayers, and focusing love and attention on the situation. Basically, he was attempting to detoxify the geopathogenic zone and neutralize its harmful effect on Patty's life and fortunes.\n\nJust as a structured physical detoxification program produces a healing crisis for the body and a temporary worsening of symptoms, eventually yielding to greatly improved health and well-being, so too does a geomantic housecleaning. Patty went through a few weeks of \"very intense and unpleasant withdrawal experiences, gastric upset, heavy flu-like symptoms and emotions that were all over the place,\" says Dr. MacManaway. Why did Patty get the brunt of the geopathic detoxification?\n\nFor years, her physiology and energy body had been living within the disturbed matrix of the geopathogenic zone and its harmful radiations. \"There is a constant interchange between us and our environment with every breath, every heartbeat\u2014it's a two-way flow of energy. So whatever dissonance is present in our environment, depending on how strong a personality and constitution we have, we will react to its influence like a thermometer reacting to heat.\" In a sense, Patty's body had grown accustomed to this toxic energy milieu and to an extent reset itself to the imbalanced biological parameters. But once the geo-biological parameters were corrected and restored to normal (or at least their effects were neutralized), Patty's system was allowed to go back to normal; getting there was what caused all the detoxification reaction she experienced.\n\nDoes everyone who wants a detoxified home have to go through what Patty did? Not necessarily. \"Not everybody is equally sensitive and not everybody has been chronically exposed,\" says Dr. MacManaway. \"In extreme cases, I observe in clients two or three nights of very vivid dreaming and restless and disturbed sleep, followed by seven to ten days of detoxification symptoms typical of withdrawal from the harmful energies.\" These can include flu-like symptoms, gastrointestinal upset, excess mucus and phlegm, and \"a fragile emotional state.\"\n\nOften Dr. MacManaway uses his body's own reaction to the change in environment to give him reliable feedback that the geomantic work has been successful. \"As my work proceeds on site and the healing process goes along\u2014Earth acupuncture easing _qi_ flow and restoring vitality, psychic intervention allowing the release and easing of long-stored hurts\u2014the pain, dissonance, and trauma are gently brought to a harmony of peace and balance. A state of grace often descends upon the place. The way I generally notice that my work is taking effect is that my body starts to relax and my mood to lighten. This is for me a most interesting phenomenon, and one that is so repeatable that it has become predictable. I never know which stage in the process of healing is going to bring the energy field into peace and balance, but I know that creeping feeling of ease that accompanies it, almost like the relaxing warmth of easing into a hot bath.\"\n\nOnce her body had detoxified and adjusted to the new energy parameters of the geobiologically balanced house, things started to clear and get better for Patty. About four weeks after the geomantic work, her neighbors started treating her more cordially; she resumed painting; she sold some paintings; she received a statement of interest in the house. \"The house felt lighter and brighter,\" says Dr. MacManaway. \"Since your visit, the birds are everywhere in the front garden and back. The house is now peaceful and no longer has an icy feel to it,\" Patty told Dr. MacManaway. The flocks of finches and a fair number of robins visiting Patty's property perhaps made it a little easier to sell, which she did a few months later.\n\nDr. MacManaway relates another case in which geopathic stress produced constant migraine headaches in a resident. The client had noticed that along with the onset of his migraines, a number of newly planted shrubs and trees on his property were not thriving, and some had died. Dr. MacManaway found that \"harmful Earth rays\" crossed the client's bed. As a remedy, he placed copper rods in the ground around the house to neutralize the rays. As the client testified afterwards, \"He told me I might have some withdrawal symptoms that first night in the form of a headache, which I did, but soon I was waking up with no more migraine headaches and with much more energy.\"\n\nAnother case shows how both humans and plants can be affected by geopathogenic zones. Dr. MacManaway was called in to consult with a woman who had been diagnosed with Hodgkin's disease, a form of cancer. He found a harmful ray at chest level going across the woman's bed. She remarked that her husband suffered a chest disease. His placement on the bed corresponded to where the harmful ray crossed the bed. In the couple's garden, an apple tree that was falling over and had to be supported and two azaleas that hadn't bloomed in seven years were both situated over a harmful energy line. \"The results of Dr. MacManaway's work is the azaleas look healthier and one has flowered,\" the woman commented. \"The apple tree is firmer in the ground. My husband's health is much better and his sleep is less restless. I, too, am feeling much healthier.\"\n\nVery often fruit trees affected by geopathic stress \"pick up tremendously and put out much greater blossom and fruit after the Earth energies are 'sweetened' for them,\" says Dr. MacManaway. \"I 'rescued' one plum tree which had never had any fruit on it and heard that next season it had a bumper crop of plums.\" He also rescued a warren of commercially grown rabbits, as his next case highlights.\n\nDr. MacManaway was called in to help a rabbit breeder whose Angora rabbits were not doing well. For months, all of the rabbits bom in one hutch at the end of the barn (out of seven hutches) died soon after birth. \"Three post-mortems had been performed by the veterinarian who was unable to explain the deaths. I found a toxic line running under the hutch and balanced it with simple Earth acupuncture techniques.\" The litter of rabbits being carried by the doe in that hutch at the time of Dr. MacManaway's visit died shortly after birth, but subsequently there were no further problems or unexplained mortalities.\n\n\"Interestingly for me, this would suggest that the problems had been intra-uterine, affecting the mother rather than primarily affecting the baby rabbits. Pineal function is greatly responsible for governing the fertility and childbearing cycle, and, as discussed by Dr. Bergsmann, it can be disordered by exposure to geopathic stress.\"\n\n## _HEALTHY LIVING SPACE DETOXIFIER #66: Take Your Energy Body Out of the Curry-Hartmann Grid_\n\nThe majority of research in the field of geopathic stress for the past one hundred years has come out of Germany. An additional element was added to the theoretical model of geopatho-genic zones by the complementary work of two German researchers, Ernest Hartmann, M.D., and Manfred Curry, M.D.\n\nIn the late 1940s, Dr. Hartmann described a tightly woven checkerboard energy grid comprised of naturally occurring charged and magnetically oriented lines running North-South and East-West across the entire surface of the planet. According to Dr. Hartmann, alternate lines are positively or negatively charged, and where N-S and E-W lines meet, there can be double positive charges, double negative charges, or a mix of both charges. It is the intersections that are problematic and that figure into the expanded model of geopathic stress. \"At the more powerful crossing points of this system, there is a fifty percent increase of radioactivity,\" and the grid of Hartmann lines are distorted during earthquakes and take thirty minutes to return to normal.\n\n**_Remaining for any length of time over a Hartmann knot, either in a bed or at a desk, is considered unhealthy. The new discipline of \"geocancerology\" has demonstrated that when a bed is situated over a Hartmann knot and a subterranean water vein, the deleterious effect is amplified. \"Cells located on a geopathogenic point become tired or, contrarily, defend themselves by multiplying.\"_**\n\nAccording to one commentator, the Hartmann network \"appears as a structure of radiations rising vertically from the ground like invisible, radioactive walls\" nine inches wide. The lines are spaced about six feet, six inches (north to south) by eight feet, two inches (east to west). A crossing of the rays in this \"terrestrial grille\" is called a Hartmann knot, and here \"there is very frequently found a geopathogenic point having great importance for the health of a human being, even to the point of chronic illness.\"\n\nRemaining for any length of time over a Hartmann knot, either in a bed or at a desk, is considered unhealthy. The new discipline of \"geocancerology\" has demonstrated that when a bed is situated over a Hartmann knot and a subterranean water vein, the deleterious effect is amplified. \"Cells located on a geopathogenic point become tired or, contrarily, defend themselves by multiplying.\" The Hartmann grid \"penetrates everywhere, whether over open ground or through dwellings.\"\n\nDr. Manfred Curry, in turn, described a global gridwork of electrically charged lines, also naturally occurring, that run diagonally to the north and south poles\u2014NE-SW and SE-NW. The lines are believed to be about nine feet (three meters) apart, although the width can vary; the medically significant \"Curry crossings\" occur as double lines about forty-seven yards apart. Again, the problematic area is where the lines crisscross.\n\nDr. Curry contended that positively charged spots on his \"Curry Net\" can cause cells to proliferate, and potentially lead to cancerous growths, while the negatively charged points at the intersections can produce inflammations. So confident was he of the etiological role of his gridwork, that Dr. Curry claimed he could forecast which part of a patient's body would be the most affected by knowing where the subterranean rays crossed the bed.\n\nThe geomancer who seeks to neutralize areas of geopathic stress at a residence must take into consideration the combined effects of these two energy grids, says Dr. MacManaway. Researchers in the field of geobiology generally contend that in some way the CurryHartmann grid (a meshwork of intersecting lines) represents an interface between incoming cosmic rays (whose existence is well-documented and confirmed by astronomers and astrophysicists) and the Earth's various layers and types of energy (see figure 12-2).\n\n**Figure 12-2. Pictures of Hartmann Net and Curry Grid**\n\nOne theory is that the combined webwork is an \"earthing grid for cosmic rays\" subject to distortion by geological fault lines and underground mining. Another interpretation is that the Curry-Hartmann grid describes the planet's geomagnetic field with \"its active and neutral zones, the active areas being lines 21 cm wide with the neutral zone lying between the lines.\"\n\nWhatever the Curry-Hartmann energy grid proves to be, for geomancer Dr. MacManaway it has a certain experiential validity. \"Certainly a lot of dowsers, and I myself, find energy grids in the ground. My observation is that as far as geopathic stress goes, these grids tend more to be carriers of energy than causative problems. They are usually never problematical unless there is underground water present, too.\" It's like a fish net, and unless there is a fish in the net, the net itself is not a problem.\n\n\"These carrier grids definitely hold any psychoemotional residue of previous occupants of that space, so if there has been emotional trauma expressed in the landscape, it will get held in this grid which is analogous to the human nervous system.\" Dr. MacManaway notes that when there is unresolved trauma in the land, the grid lines will tend to compact and draw themselves close together, \"almost like forming a scar in the lattice work where the land has been wounded.\" When the psychoemotional tension is released at a location, the lines tend to space themselves further apart again. He also says that when working with the grids, he often observes them clearing within twenty minutes, while energy residues in the home may take up to two weeks to fully be purged.\n\nBut how do human emotional residues get into this energy grid in the first place? Dr. MacManaway explains that there is a constant transfer of subtle information between the human energy body, or aura, and the equivalent energy \"body\" of the landscape. Human traumas might include domestic quarrels or violence, traffic accidents, sudden intense emotional experiences, former military battles, or forgotten burial grounds. \"Patterns will tend to repeat themselves over and over again as people just express the energies that are there.\" Let's take a moment to amplify the idea of toxic residues of psychological and emotional trauma in the landscape. \"The land has memory and therefore every act of violence is registered on Mother Earth,\" explains English spiritual teacher and dowser, Reshad Field in his book, _Here to Heal._\n\nHe notes, for example, that when he goes to a place where strip mining has been done, he gets a tingling in his solar plexus and a desire to leave the place immediately. There is a place in the Scottish Highlands called Glencoe where a massacre once occurred; few people ever stop there, Field says, because \"they can sense the smell of blood.\" Something nasty still remains from the terrible events that happened there many years ago.\n\nNear Sedona, Arizona, Field was asked to consult on the possible reasons for the strange atmosphere around a creek that flowed through a canyon. For the most part the land held great charm, he noted, except for one area. There the temperature of the air was colder, people didn't talk to one another, and something was clearly amiss. Finally, Field identified the source of the disturbance. According to dowsing vocabulary, it was a negative energy spiral, which means, the energy was flowing in the wrong way. But the reason it did was that there was \"the most terrible atmosphere of grief and violence present,\" as it were, trapped in the landscape. It was Field's impression that somebody had been either raped or murdered on this spot.\n\nIn a manner similar to acupuncture, the geomancer can \"aid in bringing back harmony to the land by redeeming the long-lasting effects of man's mistakes,\" Field says. In this case, he inserted three rods into the ground to release the stuck, traumatized energy. People accompanying Field said they felt \"a tremendous sense of relief,\" and Field noted that the air temperature rose an estimated ten degrees, leaving a warm, inviting warmth where there had been a chill.\n\nThe Curry-Hartmann grid lines are like \"capillaries in the Earth's subtle, energy body; they are its warp and woof,\" says Dr. MacManaway, as if the combined effect of the Curry-Hartmann grid lines were a globally encompassing energy carpet (see figure 12-3). Fortunately, any carpet can be cleaned. Here are two of the techniques Dr. MacManaway uses to neutralize geopathic stress and to remove psychoemotional residues from the CurryHartmann energy grid:\n\n**Figure 12-3. The Geopathic Room: Layers of Energy Disturbance**\n\n### _Eliminating Detrimental Energy from an Underground Stream_\n\nDowsers commonly employ a rebar, which is a steel or iron reinforcing rod used with concrete pouring for house foundations. Once you, or a dowser, geomancer, or psychic has identified the location of an underground stream and found the place and angle at which it passes underneath your home, hammer the rebar (it should be several feet in length) into the ground over the water vein upstream, says Dr. MacManaway.\n\nTypically, this will be ten to twenty feet away from the house, and depending on the size of the metal and the strength and volume of the water vein, \"you can clear a 150-300 yard section of vein by doing this.\" He calls this \"staking the heart of the vampire,\" referring to fact that the underground vein can literally suck energy out of people who live in a house situated over its \"mouth.\" On certain kinds of water veins, it is better to situate the rebar over the edges of the current, while in other cases, it works better if you pin the center of the vein with the rebar.\n\nMore often than not, Dr. MacManaway locates the rebar upstream, sometimes at the edge of the property line, sometimes leaving a small part of the bar above the ground's surface. \"I tend to sink most of mine under the surface although the occasional practitioner will leave part of the bar protruding. Doing this is like putting an antenna in the ground. You ground the atmospheric electromagnetic field and you allow an ion exchange from the Earth upward so that it is locally concentrated. That way you change the water vein's radiating effect, even though you are not actually moving its physical course.\" More traditionally, other dowsers tend to hammer the rebar all the way into the ground. In any event, be sure there are no electric cables, cable lines, sewer pipes, or other necessary human-placed pipes or conduits under the ground where you propose to place a rebar.\n\n### _Space-Clearing a Hartmann-Curry Knot_\n\nHere Dr. MacManaway uses what he calls \"an Earth acupuncture technique.\" He finds a nodal point in the energy web then inserts long copper needles (fifteen inches long by one-half-inch wide) into the ground, leaving them in place for twenty to forty minutes. He inserts these either into the \"knot\" in the Hartmann or Curry grid, as close to it as he can get, or at the appropriate distance to be able to affect it. Somehow the copper \"acupuncture\" needles \"release and disperse the energy held there.\"\n\n## _HEALTHY LIVING SPACE DETOXIFIER #67: Five Simple Techniques for Detoxifying a Room or Environment_\n\nHere are several simple, do-it-yourself techniques for cleansing an interior space or local outdoor environment, derived from the experiences of two seasoned geopathic stress experts:\n\n### _Salt_\n\nDr. MacManaway may also use table salt placed in an open dish for a few days in the room of the house where the problem is most acute. The salt absorbs the toxic energy during this time; in fact, it is well recognized as capable of absorbing heavy or depressed vibrations out of the atmosphere in a room. Leave the bowl of salt exposed in a targeted room; after a few days, dump the salt in a compost heap outside the house or dissolve it in water and sprinkle it around the edge of the property. Repeat as needed, or do this regularly if you desire. \"Salt is a wonderful cleanser of atmospheric influences,\" he notes.\n\n### _Burning Cauldron of Salt_\n\nAs a variant, try burning salt. This is a technique borrowed from magical rituals, where the salt (Epsom salt, not sodium chloride) is used to purify the local atmosphere of inimical influences. Use a fireproof container, such as a metal cauldron, and fill it with two cups of Epsom salt. Just before lighting, coat it with one to two cups of ninety-nine percent alcohol (found at drugstores). Light with a long fireplace match. The fire should burn yellow and produce almost no smoke; be sure to keep the cauldron away from any flammable materials. The more alcohol you use, the longer the salt will burn. You can employ this technique when you have a particularly disturbed room or if you are planning on dedicating a room to a pursuit, such as art or meditation that requires a clean energy field.\n\n### _Sound_\n\nDr. MacManaway also uses sound, delivered by a set of tuning forks, to help release trapped emotions from a knot in the energy grid of a property. Similarly, he might assemble a group of men and women and have them stand over the troubled area (or as close to it as they can get) and then have them \"just express themselves with a tone or sound.\" People express grunts or tones, but it tends to be very dissonant and chaotic as they release the stored energy through themselves. But gradually the noise becomes more harmonious and congruent. Once the group is in harmony, then that is generally a sign that the space has been cleared.\" He adds that prayer and \"focused loving attention to channel healing\" to a site may be what does the real work.\n\n### _Rock the Walls with Wagner_\n\nIn Chapter 9, British psychic protection expert William Bloom showed us how to use physical vibration, or shaking, to release stuck energy from our bodies. The same technique can be used to clear a room of baleful influences. Here vibration, from music or sound, dislodges the toxic traces from the room's atmosphere. \"Vibration is naturally created by sweeping, mopping, beating, wiping, and vacuuming, all of which work directly into the fabric of the floors and walls,\" Bloom comments.\n\nWith regard to sound, Bloom advises filling a troubled room with loud music that has a strong bass vibration, such as Johannes Brahms or rock music. (My preference is some of the stirring, high-decibel moments in Richard Wagner's operas.) Also useful, if available, are Scottish bagpipes, drums, cymbals, bells, vibrant chanting, or even trumpets. The Tibetans traditionally used ten-foot long trumpets to cleanse the atmosphere; Christian churches have relied on full-strength organ bass notes and clanging bells to shake the toxic energies out of a space.\n\nBloom advises playing your music system full blast for five minutes every few weeks to vibrate your house as a kind of interior energy cleansing, an approach bound to please most teenagers and the recidivistic Woodstockers in many of us. While the music is vibrating the room, air all blankets and coverings, and thump, shake, and thwack\u2014a tennis racket is helpful here\u2014all chairs, sofas, mattresses, and cushions, says Bloom.\n\n### _Flower Essences_\n\nOften Dr. MacManaway pours drops of flower essences (in liquid form) in the ground at the site where he would place copper needles, in other words, at the troubled zone. Or he might dilute them in water and sprinkle the mixture around the site, much as one applies plant food in water; or he may put the essences in an aerosol mister and spray the area this way. There are some 1,600 commercially available flower essences, so Dr. MacManaway typically dowses for the appropriate ones, finding \"they tend to be specific to the kind of trauma that occurred at the site.\" However, he finds that often he uses a preparation called Rescue Remedy or Five Flower Formula as a \"safe first aid catch-all essence.\"\n\nFurther, he often gives the same essences to the house's occupants, so they can make the same shift as their property and home. \"This minimizes their withdrawal symptoms\u2014that is, the transitory healing crisis\u2014and it minimizes their likelihood of recreating the same problem just because they have become habituated to it, as this can sometimes happen. People can recreate the original trauma that you've cleared because it has become an established pattern of their own behavior.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #68: Put in an Energy Drainage Pipe from Your House to the Center of the Earth\n\nDo you happen to remember the movie by Steven Spielberg called _Poltergeist_? A California family found their house had been built over native burial grounds and the spirits were intensely restless. In fact, they were so disturbed by the placement of the house (part of a hillside housing development) that after they had finished frightening and even abducting the occupants, they caused the house to implode and effectively cease to exist. It was their retribution for the obliviousness of the builders and residents and for the continuing desecration of their sanctified land that the presence of the house represented.\n\nThis is a dramatic and of course exaggerated scenario of what can happen when the house and its occupants are out of harmony with their immediate environment and when they are unaware of (initially) and ineffectual against (later) the existence of and incursion by nonphysical beings and energies into their domestic terrain. The energy field of the house or apartment is in many respects similar to that surrounding the human being. Foreign, inappropriate, and\/or generally unhealthy or unwholesome energies may reside in it, or pass through, or squat uninvited in it for a time, contributing their influences and \"flavors\" to the overall energy mix of the home. In most cases, the presence of such beings or nonphysical energies is not beneficial to the residents in terms of physical and mental well-being.\n\nIn Healthy Living Space Detoxifier #43: Ground Yourself to the Center of the Earth (Chapter 9), we reviewed the benefits of establishing a kind of master energy drainage pipe between your physical body and the center of the planet. The same can be done for a home. Why not let the planet itself absorb the negativity or toxic energy from a room or given locality?\n\nIt may seem counterintuitive, but energetically, the planet is well equipped to absorb negativity flushed down an energy drainage pipe; in many respects, it can handle it easier than physical toxicity. Even though it is toxic to the Earth when we dump physical chemicals and poisons into the landscape, it is different when we \"drain\" toxic energies into the Earth. It is capable of absorbing and transmuting much of our local accumulation, suggests William Bloom. We often forget \"what a huge and magnificent creature of energy she is,\" he says, referring to the planet in its energy aspect.\n\nBloom recommends visualizing a \"spinning plug-hole\" directly under the toxic place in question, and seeing the vortex spin itself down to the center of the Earth, carrying the toxins with it like grey water down a sink drain. In a sense, you can \"instruct\" the \"spinning plug-hole\" to collect and remove the toxic energies you have identified or sensed in a room or local environment, seeing the drain as a kind of vertical vacuum cleaner. This technique is especially useful if you need to drain the energy out of a room in which a group has discharged \"a load of emotion and pain,\" says Bloom.\n\nIn terms of specific recommendations, the idea is to visualize the drainage pipe as being set under a specific room or, better, under the entire house. Imagine there is a drainage pipe with an open diameter as wide as you house. Set your house, room, or apartment on it, or set the pipe under it, and make sure the connection is secure. Make sure the pipe connects firmly to the center of the planet as well. You could visualize a name plate \"down there\" that says \"my house\" or \"the house (or room, or apartment) of the so-and-so family.\"\n\nOnce you have the drainage pipe in place, you tell it to release and drain the toxicity from the room or house. You could visualize a lever at the top of the pipe marked \"release\"; set the lever on release. All this may seem a bit like magic, but because you're affecting energy with energy\u2014the toxic residues in a space with a mentally energized construct\u2014it works. The bad energies will drain out. It is advisable to renew this pipe every day. You can do this by refreshing the visualization: go through the steps again, making the image stronger and more vivid, and instruct it again to release toxic energies from the living space.\n\nThis drainage pipe will release a fair amount of unpleasant energy from your living space. It will work even better if you put into play many of the suggestions in Chapter 9, even, ideally, having at least one psychic reading or energy clearing session with a qualified psychic practitioner. Remember, you and your living environment are in a perpetual feedback loop. Events and toxic exposures in one will affect the other.\n\nIf it is physically toxic, and you are trying to detoxify your body, the toxins in the home will come into you. If your home is detoxified, chemically and energetically, but you are not, the toxic energies in your own energy field or aura may contribute to a retoxification of your living space. As I have stressed throughout the book, it is essential to detoxify both aspects of your living space\u2014body and home\u2014at the same time to get the best results.\n\nThere is another benefit to putting in an energy drainage pipe for your living space. It grounds the living space. It gives the space an energy center, a calm, focused midpoint. When you are in this space, you will feel more grounded yourself. You will feel calmer, more centered, relaxed, less reactive, less emotionally volatile, and less mentally agitated. You are sitting in a living space that is grounded to the center of the planet. That has a wonderful psychic and psychological effect on everyone in that space. It also acts as a subtle protection device for your home, discouraging robbers or others that might be otherwise motivated to damage, vandalize, or rob the home. As an energy equation, groundedness equals strength, much like having a strong, vital immune system is the best protection against infectious disease.\n\nEvery time you are able to drain off residual toxic energy, you are helping the energy situation of the entire planet or what might be called the planet's \"psychic ecology,\" Bloom suggests. A great deal of the planet's vibrational pollution comprises negative attitudes and stuck energy, he says. Energy is stuck in the body and the home\u2014in both aspects of our living space. There is in fact a \"huge amount of invisible psychic pollution floating around the whole planet,\" and it might be a major contributing factor, even a root cause, of the ubiquitous toxic pollution, creating the disturbed energy framework that makes it possible for toxic substances to accumulate.\n\n## HEALTHY LIVING SPACE DETOXIFIER #69: How to Take the Stress Out of a Geopathic Zone\n\nAs the role of geopathic stress in human health becomes gradually accepted and more widely studied, low-cost neutralizing devices are now appearing on the market to enable individuals to clear suspected geopathic stress out of their homes. The following is a review of a few of the units available and some of the health claims put forward by their manufacturers or distributors.\n\n### _RadiTech_\n\nThis a device that comes in four sizes, according to the size of the geopathically stressed area. It plugs into a wall socket yet draws almost no current. Physically, the device is an unassuming thin white canister varying in height (according to the model) from about six inches to a little less than two feet tall. It operates on the principle of \"stabilizing multiwave oscillators,\" by which it neutralizes the electromagnetic field (which causes the geopathic stress in the first place) of an entire building, apartment, house, hospital room, or office, and removes the effects of geopathic stress from the individuals within its radiating field. There is also a portable RadiTech unit that one can carry into an office, supermarket, or other environment for \"moving\" protection.\n\nFurther, the device apparently adjusts the balance between positive and negative ions in an environment, so that negative ions predominate, producing an effect similar to fresh mountain air, which is high in healthful negative ions. The manufacturer (Dulwich Health clinic of London) claims that the RadiTech works at maximum efficiency within a few moments of installation and that as the geopathic stress is cleared out of a person, one is likely to experience transitory withdrawal symptoms as the body releases toxins. These symptoms typically include brief headaches, irritability, flu-like feelings, upset stomach, and disturbed sleep.\n\nThe affidavits for the RadiTech are copious and impressive, illustrating its positive effects on numerous health conditions as well as further exemplifying the multiple ways in which geopathic stress can compromise your health, according to Dulwich Health. The following are examples of how RadiTech has helped relieve health problems:\n\n  * Prominent among the health conditions benefited has been chronic fatigue. A woman who suffered from \"tremendous limitations in energy levels\" and a \"mystery paralysis\" for thirty years found she could now clean her house again, go out shopping, and resume seeing clients. Another chronic fatigue sufferer\u2014ill for six years and diagnosed with chronic fatigue for the previous two\u2014was able to discontinue use of her wheelchair.\n  * A woman experienced her first pain-free menstruation in over a year.\n  * Old injuries tend to start to heal suddenly. A man's old leg sore started to heal within seven days. Another elderly man with long-term open leg sores found they started to heal and close up.\n  * A woman installed a RadiTech in her house after she tried to commit suicide. After six weeks with the unit, \"she . . . [was] a changed person, optimistic, and making plans for her future.\" She went out dancing (before she never wanted to leave the house); she ate well for the first time in years; and she went off her tranquilizers and antidepressants.\n  * People with chronic migraines have benefited. One man who had endured migraines for twenty years found that they virtually ceased, with only one minor attack in five months. Another man who for twenty-seven years had food-allergy-related migraines found he could now eat all his formerly allergenic foods with no resulting migraine. A woman used to wake up at night with an \"oppressive feeling and a splitting headache\"; in fact, she had suffered from migraines since childhood and was unresponsive to conventional pain-killing drugs. After a few days with the unit in her bedroom, she felt much \"brighter and ha[d] more energy . .. and no longer .. . [got] up five or six times a night.\"\n  * Other conditions that improved include asthma; eczema; hyperactivity; arthritis; neck, back, and leg pain; nervous stomach; neck gland swelling; and moodiness, among many others. Speaking of the device's effect on an entire home, one user commented: \"It feels as if my house has been spring-cleaned.\"\n\n### _Energeia_\n\nThis device, which bills itself as a geopathic stress neutralizer and is plugged into a wall socket, can clear a home of distorted energies, raise the occupants' energy levels, put a \"protective field\" around them, and enhance one's sense of harmony and well-being, according to its British manufacturer, Geomack Products. The Energeia device comes in three sizes, one of which is portable. It weighs from between one and eight pounds (depending on the model), and can reduce the harmful electromagnetic field emitted from household appliances, computer equipment, or electric utility poles.\n\nAccording to Geomack, \"Harmful energy fields are emitted from the Earth, electrical products, and transmitters. Energeia works in conjunction with a property's electrical wiring system to convert detrimental distorted energy fields into well-balanced beneficial energies.\"\n\nOne application of the Energeia that has drawn positive response is its use in British natural health clinics.\n\n  * Dr. Laurent Bannock of the Nutrafit Clinic in Wimbome, Dorset, reported noticing a \"distinctive difference in the 'feel'\" of his building's atmosphere after installing two units. Various unexplained headaches and migraines that had bothered staff members disappeared, and when Dr. Bannock used a portable Energeia with his laptop computer, he found it \"made a tremendous difference in my ability to concentrate at the screen for long periods of time, without feeling ill or suffering from a bad headache.\"\n  * Liz Morris of the Heavenly Bodies Therapy Clinic in South Wonston, Winchester, Hantshire, reported that prior to using the unit she had been unable to sleep well at night, and that for the previous thirty months, three to four hours a night was the best sleep she could expect. She also suffered from depression and was on antidepressants, but once she started taking these, she slept so well that she had trouble waking in the morning. She developed irritability, self-doubt, concentration problems, and lethargy, among many other symptoms. Three months after using the Energeia, Morris reported being \"a different person\"; her symptoms had abated and she was \"back at the top of the hill instead of at the bottom.\"\n\n  **_The Healthy Living Space_ Info Tip**\n\nFor information about RadiTech, contact:\n\nDulwich Health,\n\n130 Gipsy Hill,\n\nLondon SE19 1PL,\n\nEngland;\n\ntel: 0181-670-5883;\n\nfax: 0181-766-6616;\n\nwebsite: www.dulwichhealth.uk.com.\n\nFor Energeia, contact:\n\nGeomack Products,\n\nP.O. Box 519,\n\nSouthampton S016 7RW,\n\nEngland;\n\ntel: 23-8076-0100;\n\nfax: 23-8076-0200;\n\ne-mail: mail@geomack.com;\n\nwebsite: www.geomack.com.\n\nFor products available in the United States from:\n\nThe Healing Center,\n\n1924 Juan Tabo NE, Suite E,\n\nAlbuquerque, NM 87112;\n\ntel: 505-292-222, ext. 1;\n\nfax: 505-821-2325;\n\ne-mail: addleasy@aol.com.\n\n  * Mayne Sundewall-Hopkins, a therapist in Looe, Cornwall, with more than twenty years' practice, reported that ever since she moved into her house seven years ago, her health started to deteriorate. \"Something didn't feel right, but I couldn't put my finger on what or why,\" she said. She had problems upon problems: aching muscles, digestive disturbances, allergies, joint stiffness, mood swings, concentration difficulties, constant flu\/cold symptoms, fatigue, stress. \"It was as though something like a leech was continually emptying me of energy,\" she said, adding that even regular acupuncture and other natural healing modalities failed to improve her condition. This, as we saw above, is a classic indicator of geopathic stress.\n\nBy good fortune, Sundewall-Hopkins came across the Energeia device and installed one in her home and realized she had identified the source of all her problems: geopathic stress. \"The distorted energies in the house prevented me from getting well. My immune system suffered and thus everything gradually worsened,\" she explained. \"With the unit in the house, I felt better within hours.\"\n\n## HEALTHY LIVING SPACE DETOXIFIER #70: Selecting Energy Wells in a Geography of Enlightenment\n\nOne of the most practical steps you can take, suggests Samuel Sagan, M.D., is to become sensitive to Earth and ambient energies in local environments, even in as small a space as a single room in a house. Many living spaces are crisscrossed with toxic energy lines and geopathogenic zones, as discussed above, yet they often present positive Earth energies, too. Dr. Sagan calls these positive, uplifting sources energy wells. Both rooms and landscapes have energy wells, and if you can identify them, they can help restore your health by reestablishing energetic balance, he says.\n\nIf you are gifted with clairvoyance, an energy well resembles \"a little fountain of energy, a little 'geyser,' a column of light,\" says Dr. Sagan. Even if you cannot see one, you may be able to sense the presence of an energy well. You may find your thoughts uplifted, more positive, exalted, happy, or inspired, or you may experience some degree of healing of a discomfort as you reside, even for just a few moments, within an energy well. Dr. Sagan reports that typically an energy well is quite small, perhaps one or two feet in diameter.\n\nHow do you sense an energy well? Listen to the \"belly signal,\" advises Dr. Sagan. The solar plexus, and especially navel area, has a keen ability to act as a dowsing rod for sensing ambient energies, he says. This is the energy basis of the folk wisdom about having a \"gut feeling.\" Once you know what neutral feels like in your \"gut\" or solar plexus, you can sense energy changes in that region in response to sensing new environments. In fact, when sensing for energy lines, geopathic or otherwise, it is more helpful to think in terms of \"walls\" not \"lines,\" Dr. Sagan comments. He makes the interesting observation that what you will feel is more like a \"wall of vibration\" or \"a wall of etheric smoke\" than a two-dimensional line.\n\nDeveloping a sensitivity to energy wells within your home, property, or immediate landscape can have a definite practical value. It could lead you to completely reorganize how you use and allocate living space, suggests Dr. Sagan. \"All this means that you have to reconsider your habits and to start using the space in harmony with the energy of the house.\" You may wish, to the degree it is possible, to redesign the interior of your home so that you sleep, meditate, eat, work, think on energy wells to get their maximum benefit for consciousness and the functions of your mind and body.\n\nDoing this, suggests Dr. Sagan, provides \"immense\" benefits for your physical, mental, and spiritual health. \"By systematically choosing spots of high-quality energy, we can influence our state of health and consciousness positively.\" Dr. Sagan notes that people often select an environment, even which chair to sit on in a room, based on resonant vibrations. There is a correspondence between one's energy quality and that of a room or environment, and thus we are drawn (or repelled) accordingly.\n\nFor example, in his experience, many people unconsciously select toxic spots in which to sit in a house, movie theater, or other enclosed space; this selection is energetically appropriate (but not therapeutic) because it reinforces, or is in resonance with, the quality of their own energy imbalances. Modifying the principle of homeopathy: like draws like.\n\nBut once you have detoxified a certain amount of your mental, emotional, psychic, and physiological toxins, you will start being in resonance with the health-affirming energy wells rather than the geopathic spots. You will start moving within what Dr. Sagan elegantly calls \"a geography of enlightenment,\" selecting your every environment\u2014the seat in a movie theater, the table in a restaurant, the plush chair in a living room\u2014based on your sensing of energy qualities and high and low points in a given living space. You will avoid the toxic sites and gravitate toward the healthy ones. It is possible, Dr. Sagan assures us, to so fine-tune this sensing ability that standing in the doorway of a new interior living space, you can detect the toxic sites and the energy wells, and thereby avoid the first and move towards the second.\n\n# AFTERWORD\n\n# _Life in a Healthy Living Space_\n\nSo what happens next? What is life like after detoxification? How can we use detoxification as a foundation for further exploration in our lives and the world?\n\nIn a sense, the experience of detoxification can be like a reentry into the physical world, and even into our own bodies. This is especially so if we undertake our first detoxification intensive in our forties or fifties. Purging ourselves of half a lifetime of toxic accumulation can be acutely stimulating. It's as if we are free to think, feel, and sense anew, if for the first time. Our life works better in all its aspects.\n\nPerhaps as a result of following the seventy detoxifying steps in _The Healthy Living Space_ you are considering the matter of medicine and health care. The mental clarity produced by your detoxification gives you the \"space\" to consider some interesting questions. Perhaps alternative medicine and its nontoxic worldview is new to you; maybe you never critically examined the principles and practices of conventional medicine even though you have been a recipient of its \"products\" all your life.\n\nConversely, maybe you are an \"old hand\" at alternative medicine and in agreement with its principles. Either way, it is important to note that medicine is not neutral: it embodies a philosophy of matter, a way of looking at and understanding the physical world. It expresses an orientation toward the body and the environment. It invades and attacks; or it supports and nurtures. It has ramifications in the world, and affects our total environment. Your choice of medicine\u2014essentially it comes down to conventional and alternative systems\u2014affects how you relate to your body. It is quite likely that the detoxification experience has given you a new perspective on this.\n\nYou might ask yourself three questions: Does a given medical approach or an entire field contribute to or diminish my existing level of toxicity? Does it stimulate my waking up to my role and responsibility in the physical world, or does it keep me in the dark as a nonplayer? Does it make me more conscious or less? Many of these matters are discussed in my book _Physician: Medicine and the Unsuspected Battle for Human Freedom._\n\nAfter detoxifying ourselves, we have a new understanding of our relationship to the world, and our responsibility for that relationship. We as consumers participate in the pollution of the planet and the sickening of human bodies by all our choices in the marketplace. Now we have woken up to that responsibility and we are making new, more informed consumer choices. But there's more.\n\nWe have a new sense of how our physical well-being influences our psychological and emotional well-being, and vice versa. We no longer divide our life into categories, each separately treatable. Now we see they are interconnected. So, too, we see our home afresh. Our body and home form a living system, and this living system itself has a relationship to the larger environment.\n\nIt is quite likely that the detoxification experience has given you a new appreciation of how you relate to your living space, your home. You may see the need to adopt a new style of relationship consistent with the conclusions you reached about how you will treat your body from now on.\n\nHow we relate to either our body or our home represents a philosophy of matter, represents a definition we have formed of both. Are things living or inert? Is consciousness important? Does it have a role in the world? Do our decisions really affect the planet? The detoxification experience may open up some large questions for you and encourage you to consider big subjects that previously you had neglected or thought too remote from your interests. The matter of toxicity and detoxification is one way you can see your inter-relatedness with the entire planet and all of matter. It's a doorway, and once through it, you will have a vista of far more than free radicals, organochlorines, and the liver's two-phase detoxification pathway.\n\nPerhaps some of the ideas in Chapter 12 were provocative. The concept that subtle, technically \"invisible\" energies from the landscape can affect how we feel and how our bodies function is certainly a different perspective than that offered by conventional science and medicine. In this book, for the most part, we focused on those landscape energies that were harmful to health. But are there Earth energies that are good for us, that elevate consciousness and improve our health just by our being in proximity to them? Yes, there are.\n\nJust as many traditional societies routinely practiced detoxification, they also knew about beneficial Earth energies and used them in their rituals, placement of holy buildings, and other aspects of their lives. Such places are known today variously as sacred sites, holy places, power points, vortexes, and pilgrimage destinations, among other names. Such places have a definite \"energy\" to them, but it is subtle.\n\nDetoxification actually prepares you to be more sensitive and receptive to beneficial Earth energies in much the same way as ritual purification and detoxification were employed in traditional societies for the purposes of preparing people for spiritual experiences. Detoxification frees you up to consider the concept of Earth energies, and it fine-tunes your senses so you can make some discoveries on your own. This field is called geomancy, and many aspects of this subject are discussed in my forthcoming book, _The Galaxy on Earth: A Traveler's Guide to the Planet's Visionary Geography._\n\nTaking the large view, we can say that the planet itself is our total living space. Do we want it to be toxic or healthy? Detoxification of the planet truly starts at home with the choices we make. When we choose to detoxify, it has a global impact. One person detoxifying affects everything. It may seem like a minute influence, but big effects are compounded by many minute influences working together. The detoxification experience itself shows us convincingly that our seemingly inconspicuous efforts to purify ourselves of toxicity can change the world and help make it the healthy living space it was always meant to be.\n\n# _Endnotes_\n\n## _Chapter 1_\n\n Theo Colborn, Dianne Dumanoski, and John Peterson Myers, _Our Stolen Future. Are We Threatening Our Fertility, Intelligence and Survival\u2014A Scientific Detective Story_ (New York: Dutton\/Penguin Book, New York, 1996): 106.\n\n For more information about POPs, see: www.stoppops.org. This is a politically active organization working to eliminate POPs in the world environment and food supply. See also: www.ewg.org (Environmental Working Group, 1718 Connecticut Avenue, N.W., Suite 600. Washington, D.C., 20009; e-mail: info@ewg.org).\n\n Eric Dewailly et al., \"Concentration of Organochlorines in Human Brain, Liver, and Adipose Tissue Autopsy Samples from Greenland,\" _Environmental Health Perspectives_ 107:10 (October 1999): 823-828.\n\n \"Alpine Lake Traps 'Dirty Dozen' Poisons,\" Reuters Limited, 11 April 2000, EnviroLink News Service www.envirolink.org\/environews.\n\n Sandra Steingraber, _Living Downstream. An Ecologist Looks at Cancer and the Environment_ (Reading, Mass.: Addison-Wesley Publishing Company, 1997): 177.\n\n Some experts now use the term \"toxics\" to distinguish toxic substances from \"toxins,\" which, technically, are poisons from plants and animals, and from \"toxic substances,\" which tend to indicate only chemicals and not radiation. \"Toxics\" thereby refers to chemicals, nuclear radiation, and even electromagnetic fields and noise. See: John Harte et al., _Toxics A to Z. A Guide to Everyday Pollution Hazards_ (Berkeley: University of California Press, 1991): xiv.\n\n \"Enjoy Vibrant Health in a Toxic World,\" Health Designs International (1997) www.healthdesigns.com.\n\n Iris R. Bell, _Clinical Ecology: A New Medical Approach to Environmental Illness_ (Bolinas, Calif.: Common Knowledge Press, 1982).\n\n Sherry A. Rogers, _Tired or Toxic? A Blueprint for Health_ (Syracuse: Prestige Publishing, 1990): 5, 10, 11.\n\n This list of toxicity symptoms is adapted from several sources, including a detailed toxicity screening questionnaire published by Health Designs.com. To take the test and get numerical toxicity rankings of the results, see: www.healthdesigns.com\/DetoxificationTest.html.\n\n Catherine Hoffman, Dorothy Rice, Hai-Yen Sung, \"Persons with Chronic Conditions,\" _The Journal of the American Medical Association_ 276 (November 13, 1996): 1473-1479.\n\n Peter Bennett, Stephen Barrie, and Sara Faye, _7-Day Detox Miracle: Restore Your Mind and Body's Natural Vitality with This Safe and Effective Life-Enhancing Program_ (Rocklin, Calif.: Prima Health\/Prima Publishing, 1999): 30-33.\n\n W. John Diamon, W. Lee Cowden, with Burton Goldberg, _An Alternative Medicine Definitive Guide to Cancer_ (Tiburon, Calif.: Future Medicine Publishing, 1997): 545.\n\n The EHIS has been publishing its chemical profiles since 1978 by Congressional mandate. The full text of the _9th Report on Carcinogens, 2000_ , is available at <http:\/\/ntp-server.niehs.nih.gov\/> NewHome Roc\/9RocFacts\/html. You can also contact them at: National Toxicology Program-RoC, MD EC-14, P.O. Box 12233, Research Triangle Park, NC 27709; tel: 919-541-4096; email: jameson@niehs.nih.gov.\n\n Philip J. Landrigan, \"Commentary: Environmental Disease\u2014A Preventable Epidemic,\" _American Journal of Public Health_ , 82 (July 1992): 941-943.\n\n Philip J. Landrigan, _Environmental Neurotoxicology_ (Washington, D.C.: National Academy Press, 1992): 2.\n\n Robert W. Pinner et al., \"Trends in Infectious Diseases Mortality in the United States,\" _Journal of the American Medical Association_ , 275:3 (January 17, 1996): 189-193.\n\n David W. Talmage et al., _Biologic Markers in Immuno-toxicology_ (Washington, D.C.: National Academy Press, 1992): 1.\n\n Cynthia Wilson, \"Chemical Sensitivities: A Global Problem,\" _Earth Island Journal_ , Spring 1998 www.earthisland.org\/ejournal\/spring98.\n\n Richard Leviton, \"Environmental Illness\u2014A Special Report,\" _Yoga Journal_ , November\/December 1990.\n\n William J. Rea, \"Chemical Hypersensitivity and the Allergic Response,\" _Ear, Nose, and Throat Journal_ 67 (January 1988). William J. Rea and Monte J. Mitchell, \"Chemical Sensitivity and the Environment,\" _Immunology & Allergy Practice_ (September\/October 1982). William J. Rea and Alfred R. Johnson, \"20th Century Illness,\" _Total Health_ , December 1986. Yaqin Pan, Alfred R. Johnson, and William J. Rea, \"Alipathic Hydrocarbon Solvents in Chemically Sensitive Patients,\" _Clinical Ecology_ V, no. 3 (1987).\n\n John L. Laseter, Ildefonso R. DeLeon, William J. Rea, and Joel R. Butler, \"Chlorinated Hydrocarbon Pesticides in Environmentally Sensitive Patients,\" _Clinical Ecology_ II, no. 1 (Fall 1983).\n\n Janette D. Sherman, _Chemical Exposure and Disease. Diagnostic and Investigative Techniques_ (Princeton: Princeton Scientific Publishing, 1994): 88, 91, 97, 101-102, 118-119.\n\n Agrow Reports, \"World Non-Agricultural Pesticides Markets,\" March 2000; data summarized in \"Non-Ag Pesticides Market Growing,\" September 12, 2000, by Pesticide Action Network North America (49 Powell Street, Suite 500, San Francisco, CA 94102; tel: 415-981-1771; fax: 415-981-1991; e-mail: panna@panna.org; website: www.panna.org). The organization offers a free weekly e-mail newsletter called PANUPS, available on their website.\n\n Joel Grossman, \"Dangers of Household Pesticides,\" _Environmental Health Perspectives_ 103:6 (June 1995): 550-554.\n\n The National Library of Medicine offers a searchable online database for toxic substances and their effects, including toxicology, lists of hazardous substances, and carcinogens. See <http:\/\/toxnet.nlm.nih.gov>.\n\n Sherman, Chemical Exposure and Disease, 223, 230.\n\n EPA, Office of Toxic Substances, \"Broad Scan Analysis of the FY82 National Human Adipose Tissue Survey Specimens,\" EPA-560\/5-86-035 (Washington, D.C.: 1986).\n\n P.A. Stehr-Green, \"Demographic and Seasonal Influences on Human Serum Pesticide Residue Levels,\" _Journal of Toxicology and Environmental Health_ 27 (1989): 405-421.\n\n F. Adeshina, E.L. Todd, \"Organochlorine Compounds in Human Adipose Tissue from North Texas,\" _Journal of Toxicology and Environmental Health_ 29 (1990): 147-156.\n\n J.L. Jacobson et al., \"Determinants of Polychlorinated Biphenyls (PCBs), Polybrominated Biphenyls (PBBs), and Dichlorodiphenyl Trichchloroethane (DDT) Levels in the Sera of Young Children,\" _American Journal of Public Health_ 79 (1989): 1401-1404.\n\n Walter J. Crinnion, \"Environmental Medicine, Part 1: The Human Burden of Environmental Toxins and Their Common Health Effects,\" _Alternative Medicine Review_ 5 (February 2000): 53-54.\n\n Benjamin C. Blount, et al., \"Levels of Seven Urinary Phthalate Metabolites in a Human Reference Population,\" _Environmental Health Perspectives_ , 108 (October 2000): 979-982.\n\n Kenneth Bock and Nellie Sabin, _The Road to Immunity. How to Survive and Thrive in a Toxic World_ (New York: Pocket Books\/Simon & Schuster, 1997): 81.\n\n Al Meyerhoff, \"We Must Get Rid of Pesticides in the Food Supply,\" _USA Today Magazine_ , November 1993, 51-53.\n\n Peter Montague, \"Illegal Poisons in Our Food,\" _Rachel's Environment & Health Weekly_ 9 May 1996, www.rachel.org.\n\n J.E. Davies, \"Changing Profile of Pesticide Poisonings,\" _New England Journal of Medicine_ 316:13 (March 26, 1987): 807-808.\n\n James Liebman et al., \"Rising Toxic Tide: Pesticide Use in California, 1991-1995,\" 1997, Pesticide Action Network & Californians for Pesticide Reform, San Francisco, CA. For more information, see www.igc.org\/cpr; see also www.panna.org. Both organizations, based in San Francisco, are coalitions of numerous citizens groups around the world working to oppose the misuse of pesticides and to promote sustainable, ecologically-based agriculture.\n\n \"EWG Air Monitoring Finds Toxic Pesticides Drifting from California Farm Fields,\" Environmental Working Group, Washington, D.C., 13 January 1999 www.ewg.org.\n\n Robert Repetto et al., _Pesticides and the Immune System: The Public Health Risk_ (Washington, D.C.: World Resources Institute, 1996): 3, 9-16, 18, 21, 56, 46-48.\n\n \"Technical Report,\" Beyond Pesticides\/National Coalition Against the Misuse of Pesticides 15:7 (2000). 701 E. Street SE, Suite 200, Washington D.C., 20003; tel: 202-543-5450; fax: 202-543-4791; website: www.beyondpesticides.org. This organization is a good source of information about the hazards of pesticides and the means of taking advocacy action to limit their use.\n\n Jacqueline D. Savitz et al., _Factory Farming: Toxic Waste and Fertilizer in Virginia_ , 1990-1995, Environmental Working Group, Washington, D.C. (n.d.) www.ewg.org. EWG offers a free downloadable report on toxic wastes used in fertilizer for each of the fifty states.\n\n John Cary Stewart, _Drinking Water Hazards: How to Know If There are Toxic Chemicals in Your Water and What to Do If There Are_ (Hiram, Ore.: Envirographics, 1990): 158.\n\n \"Tap Water in Central Valley Tainted With Banned Pesticide,\" Environmental Working Group, Washington, D.C., 15 November 1999 www.ewg.org.\n\n \"Government Underestimates Infant Exposure to Toxic Weed Killer,\" Environmental Working Group, Washington, D.C., 28 July 1999 www.ewg.org.\n\n Although arsenic is a naturally occurring element (the 20th most common in the Earth's crust), it can be toxic to humans. High-dose exposure to arsenic can produce gastrointestinal irritations, problems in swallowing, abnormally low blood pressure, excessive thirst, even convulsions and cardiovascular collapse in cases of acute exposure. Arsenic has also been linked to cancer and diabetes. In addition to ingesting it in water, people consume arsenic through fish and seafood.\n\n MSNBC Staff and Wire Reports, \"How Much Arsenic in Our Water?\" MSNBC News\/Environment, 25 February 2000 www.msnbc.com\/news.\n\n Butch Kinerney, \"New Map Shows Arsenic in Nation's Ground Water,\" UniSci, Daily University Science News, 10 May 2000 www.unisci.com\/stories.\n\n Denise Riedel Lewis et al., \"Drinking Water Arsenic in Utah: A Cohort Mortality Study,\" _Environmental Health Perspectives_ 107 (May 1999): 359-365.\n\n P\u00e4ivi Kurttio et al., \"Arsenic Concentrations in Well Water and Risk of Bladder and Kidney Cancer in Finland,\" _Environmental Health Perspectives_ 107 (September 1999): 705-710.\n\n \"Update 1. U.S. Toxic Pollution 3 Times Worse Than Thought\u2014EPA,\" Reuters Limited, 12 May 2000, EnviroLink News Service www.envirolink.org\/environews.\n\n Steingraber, _Living Downstream_ , 197-202.\n\n Hans-Rudolf Buser and Markus D. Muller, \"Occurrence of the Pharmaceutical Drug Clofibric Acid and the Herbicide Mecoprop in Various Swiss Lakes and in the North Sea,\" _Environmental Science and Technology_ , 32:1 (1998): 188-192. Janet Raloff, \"Drugged Waters,\" _Science News_ , 153:12 (March 21, 1998): 187-189. Peter Montague, \"Drugs in the Water,\" _Rachel's Environment & Health Weekly_, September 3, 1998.\n\n Christian G. Daughton and Thomas A. Ternes, \"Pharmaceuticals and Personal Care Products in the Environment: Agents of Subtle Change?\" _Environmental Health Perspectives_ 107: supp. 6 (December 1999): 907-938. Also: Peter Montague, \"Pay Dirt from the Human Genome,\" _Rachel's Environment & Health Weekly_, #702, 6 July 2000 www.rachel.org.\n\n Other toxic substances are emitted from motor vehicles or formed as byproducts of incomplete combustion, contributing to air pollution. These substances include benzene, toluene, xylene, formaldehyde, acetaldehyde, and 1,3-butadiene. Some of these are known or suspected carcinogens.\n\n W.S. Linn and H. Gong, Jr., \"The 21st Century Environment and Air Quality Influences on Asthma,\" _Current Opinions in Pulmonary Medicine_ 5 (January 1999): 21-26.\n\n William S. Linn et al., \"Air Pollution and Daily Hospital Admissions in Metropolitan Los Angeles,\" _Environmental Health Perspectives_ 108 (May 2000): 427-434.\n\n South Coast Air Quality Management District, \"Smog and Health,\" July 1996 www.aqmd.gov. The South Coast AQMD is the smog control agency for Los Angeles and three neighboring counties.\n\n S. Takafuji and T. Nakagawa, \"Air Pollution and Allergy,\" _Journal of Investigative Allergology and Clinical Immunology_ 10 (January-February 2000): 5-10.\n\n G. D'Amato, \"Outdoor Air Pollution in Urban Areas and Allergic Respiratory Diseases,\" _Monaldi Archices of Chest Diseases_ 54:6 (December 1999): 470-474.\n\n Rob McConnell et al., \"Air Pollution and Bronchitic Symptoms in Southern California Children with Asthma,\" _Environmental Health Perspectives_ 107 (September 1999): 757-760.\n\n Joachim Heinrich et al., \"Respiratory Diseases and Allergies in Two Polluted Areas in East Germany,\" _Environmental Health Perspectives_ 107 (January 1999): 53-62.\n\n Bart D. Ostro et al., \"Air Pollution and Health Effects: A Study of Medical Visits Among Children in Santiago, Chile,\" _Environmental Health Perspectives_ 107 (January 1999): 69-73.\n\n South Coast Air Quality Management District, \"Smog and Health\" www.aqmd.gov.\n\n Annette Peters et al., \"Associations Between Mortality and Air Pollution in Central Europe,\" _Environmental Health Perspectives_ 108 (April 2000): 283-287.\n\n Joe Thornton, _Pandora's Poison: Chlorine, Health and a New Environmental Strategy_ (Cambridge, Mass.: MIT Press, 2000).\n\n Ibid., vii.\n\n The production of chlorine gas is the \"single root cause\" for all the pollutants known as organochlorines, says Thornton. A powerful electric current is passed through a saltwater solution (sodium chloride plus water). This changes sodium chloride, entirely natural and a part of nature and our body, into chlorine gas, \"a heavy, violently reactive, greenish gas.\" Most of the chlorine gas is then used as the \"feedstock\" or base material for making 11,000 organochlorines, also foreign to nature. The chlorine industry produces about 40 million tons of chlorine gas annually, most of which is used to make chlorinated organic chemicals.\n\n Some organochlorines can persist a very long time in the atmosphere. The half-life of chloropentafluoroethane is 381 years; of dichlorotetrafluoroethane, 152 years; of dichlorodifluoromethane, 69; and of trichlorotrifluoroethane, 58. A half-life means how long it takes for a substance to degrade to 37% of its original concentration\u2014in other words, for about two-thirds of it to degrade. The half-lives of 20 other organochlorines range from 0.4 years (chloroform) to 31 years (trichlorofluoormethane). The persistence of organochlorines in water involves unbelievable stretches of time. Scientists estimate that 1,2-dichloroethylene found in water will require 21 billion years to degrade; heaxachloroethane will take 1.8 billion years; perchloroethylene, 990 million years; 1,1-dichloroethylene, 120 million; and trichloroethylene, 1.3 million years. (Data from Thornton, _Pandora's Poison_ , 27, 33.)\n\n Ingrid Gerhard et al., \"Chlorinated Hydrocarbons in Women with Repeated Miscarriages,\" _Environmental Health Perspectives_ 106 (October 1998): 675-681.\n\n A. Mayani et al., \"Dioxin Concentrations in Women with Endometriosis,\" _Human Reproduction_ 12 (1997): 373-375.\n\n It is shocking to learn, as Thornton documents, that the global organochlorine problem and the entire chlorine industry were the result of trying to find a use for a chemical byproduct. In the late 19th century, German chemists found a faster way to make alkali, needed for the manufacture of paper, soap, glass, textiles, and other products. They introduced an electrical current into brine and got alkali, hydrogen, and chlorine gas. The trouble was they got a lot of chlorine gas: to make one ton of alkali, nine-tenths of a ton of chlorine gas was generated. It was dangerous to store and impossible to get rid of, so new markets and applications had to be found for it. These included, during the 20th century, bleaching powder; chemical weapons and war gas; degreasing solvents; and PCBs, DDT and other pesticides.\n\n Jay M. Gould, _The Enemy Within: The High Cost of Living Near Nuclear Reactors_ (New York: Four Walls\/Eight Windows, 1996): 15-56. See also Ralph Graeub, _The Petkau Effect. The Devastating Effect of Nuclear Radiation on Human Health and the Environment_ (New York: Four Walls\/Eight Windows, 1994).\n\n Interference with the functioning of the thyroid gland is a serious matter. The thyroid gland, one of the body's seven endocrine glands, is located just below the larynx in the throat, with interconnecting lobes on either side of the trachea. The thyroid is the body's metabolic thermostat, controlling body temperature, energy use, and, in children, the body's growth rate. The thyroid controls the rate at which organs function and the speed with which the body uses food; it affects the operation of all body processes and organs. Of the hormones the thyroid synthesizes and releases, T3 (triiodothyronine) represents seven percent and T4 (thyroxine) accounts for almost ninety-three percent of the thyroid's hormones active in all of the body's processes; T4 is converted into T3 outside the thyroid gland. Iodine is essential to forming normal amounts of thyroxine. The secretion of both these hormones is regulated by the thyroid-stimulating hormone, or TSH, secreted by the pituitary gland in the brain. The thyroid also secretes calcitonin, a hormone required for calcium metabolism.\n\n Jay M. Gould, \"The Current Cancer Epidemic and the Baby Boom Generation,\" _Nuclink. The Journal of Current Radiation and Public Health and Issues_ 17 February 1999 www.radiation.org.\n\n Joseph J. Mangano, \"Improvements in Local Infant Health After Nuclear Power Reactor Closing,\" _Environmental Epidemiology and Toxicology_ 2:1 (March 2000): 32-36.\n\n Jay M. Gould, \"Why Cancer Rates in the Hamptons Are So High,\" _Nuclink. The Journal of Current Radiation and Public Health and Issues_ , 14 September 1998 www.radiation.org.\n\n Russell L. Blaylock, _Excitotoxins\u2014The Taste That Kills_ (Santa Fe, N. Mex.: Health Press, 1994).\n\n For a comprehensive list and discussion of the toxicity of each food additive, see \"Chemical Cuisine: CSPI's Guide to Food Additives,\" Center for Science in the Public Interest www.cspinet.org. Address: CSPI, 1875 Connecticut Avenue, NW, Suite 300, Washington, D.C. 20009; tel: 202-332-9110; fax: 202-265-4954; e-mail: cspi@cspinet.org. CSPI also publishes _Nutrition Action Healthletter_ , a monthly nutrition newsletter.\n\n Doctor's Guide to the Internet (Medical & Other News), \"Food Additive Can Cause Severe Allergic Reactions,\" 5 November 1997 www.docguide.com.\n\n Carol Simontacchi, _The Crazy Makers. How the Food Industry is Destroying Our Brains and Harming Our Children_ (New York: Jeremy P. Tarcher\/Putnam, Penguin Putnam, Inc., 2000): 35, 93.\n\n Marc Kaufman, \"Antibiotics in Animal Feed\u2014A Growing Public Health Hazard, Worries Rise Over Effect of Antibiotics in Animal Feed,\" _The Washington Post_ , March 17, 2000.\n\n Kirk E. Smith et al., \"Quinolone-Resistant _Campylobacter Jejuni_ Infections in Minnesota,\" _New England Journal of Medicine_ , 340:20 (May 20, 1999): 1525-1532.\n\n M. Kathleen Glynn et al., \"Emergence of Multidrug Resistance _Salmonella Enterica_ Serotype Typhimurium DT104 Infections in the United States,\" _New England Journal of Medicine_ 338:19 (May 7, 1998) 1333-1338. Also Denise Grady, \"A Move to Limit Antibiotic Use in Animal Feed,\" _The New York Times_ , March 8, 1999.\n\n Food poisoning from bacterially-tainted food is a serious matter: about 76 million cases are reported annually in the United States, of which 5,000 result in fatalities.\n\n Shannon Brownlee, \"Agribusiness Threatens Public Health with Antibiotics in Animal Feed,\" _The Washington Post_ , May 21, 2000.\n\n Michelle Allsopp et al., \"Recipe for Disaster. A Review of Persistent Organic Pollutants in Food,\" Greenpeace Research Laboratories, March 2000 www.greenpeace.org.\n\n Environmental Working Group, \"Moms... and POPs\" www.ewg.org.\n\n Allsopp et al., \"Recipe for Disaster\" www.greenpeace.org.\n\n Environmental Working Group was founded in 1993 and is today, with its staff of eighteen researchers, a leading source of information for public interest groups campaigning to protect the environment. Its website (www.ewg.org) offers a searchable database on numerous environmental issues.\n\n Environmental Working Group, \"Moms . . . and POPs\" www.ewg.org. EWG compiled this information based on the FDA \"Total Diet Study, 1991-1997\" and USDA \"Continuing Survey of Food Intake by Individuals, 1994-1996.\"\n\n EPA, \"EPA's Agency-Wide Multimedia Persistent, Bioaccumulative, and Toxic Pollutants Initiative, 1999 Accomplishments Report,\" EPA 743-R-00-003, July 2000 www.epa.gov\/pbt\/accomp99.htm.\n\n Joe Thornton proposes that it will take six human generations, or about 150 years, before all the PCBs and the other organochlorines have been removed from human fat cells\u2014even if no further organochlorines were introduced into the global environment starting today. In other words, we may be contaminated for centuries, he says. Thornton further suggests that our natural detoxification mechanisms may be ineffective against many organochlorines, although they are able to degrade some of them into more easily excreted chemical forms. (Thornton, _Pandora's Poison_ , 43, 215).\n\n## _Chapter 2_\n\n Peter Bennett, Stephen Barrie, and Sara Faye, _7-Day Detox Miracle: Restore Your Mind and Body's Natural Vitality with This Safe and Effective Life-Enhancing Program_ (Rocklin, Calif.: Prima Health\/Prima Publishing, 1999): 113-114.\n\n Rudolph M. Ballentine, _Radical Healing. Integrating the World's Great Therapeutic Traditions to Create a New Transformative Medicine_ , (New York: Three Rivers Press, 1999): 294-297.\n\n Frank Edward Allen, \"One Man's Suffering Spurs Doctors to Probe Pesticide-Drug Link,\" _The Wall Street Journal_ , October 14, 1991.\n\n Kenneth Bock, and Nellie Sabin, _The Road to Immunity: How to Survive and Thrive in a Toxic World_ (New York: Pocket Books\/Simon & Schuster, 1997): 34-36.\n\n There are six types of free-oxygen radicals, including superoxide, hydroxy radical, lipid peroxy radical, singlet oxygen, hydrogen peroxide, and hypochlorous acid.\n\n Stephen B. Edelson, \"Free Radical Pathology: A Unified Cause of Chronic Illness,\" Environmental and Preventive Health Center of Atlanta, 1995 www.ephca.com.\n\n Genox Corporation, Baltimore, Md., providers of Oxidative Stress Profile www.genox.com.\n\n D. Harman, \"Free Radicals in Aging,\" _Molecular Cell Biochemistry_ 84:2 (December 1998): 155-61; \"Aging: Phenomena and Theories,\" _Annals of the New York Academy of Sciences_ 854 (November 20, 1998): 1-7; \"Extending Functional Life Span,\" _Experimental Gerontology_ 33:1-2 (January-March 1998): 95-112; \"Free Radical Involvement in Aging: Pathophysiology and Therapeutic Implications,\" _Drugs Aging_ 3 (January-February 1993): 60-80.\n\n J.E. Gallagher et al., \"Comparison of DNA Adduct Levels in Human Placenta from Polychlorinated Biphenyl Exposed Women and Smokers in which CYP 1A1 are Similarly Elevated,\" _Terato Carcino Mutagen_ 14 (1994): 183-92.\n\n Joseph Pizzorno, _Total Wellness. Improve Your Health by Understanding the Body's Healing Systems_ (Rocklin, CA: Prima Publishing, 1996): 99.\n\n Ibid., 101-102.\n\n Toluene, a hydrocarbon distilled from crude oil, is a common additive in gasoline. It is vented into the air every time a car, truck, bus, or airplane is driven. A catalytic converter can remove about ninety-five percent of toluene from automobile exhaust. Toluene is also an additive in model glues, paints, inks, resins, and adhesives; and is found in dyes, detergents, linoleum, perfumes, lacquers, prescription drugs, and saccharin, among other products. You will also encounter toluene in tobacco smoke from cigarettes. For more information about toluene, and other toxic substances, see John Harte et al., _Toxics A to Z: A Guide to Everyday Pollution Hazards_ (Berkeley: University of California Press, 1991).\n\n Bennett, Barrie, and Faye, _7-Day Detox Miracle_ , 99-101.\n\n Since the Center was founded in 1974, more than 30,000 patients have passed through its bio-detoxification program or have received treatment for toxicity-related illness. \"We analyze difficult cases from all angles using environmental, nutritional, allergy, medication, and surgical strategies tailored to the individual patient,\" states an information release by the Center. Reknowned for its carefully executed, chemically-reduced indoor environment for sensitive patients, the Center has also consulted on \"less-toxic building technology\" for 17,000 private and public buildings. Dr. Rea is the author of two books, the medical textbook _Chemical Sensitivity_ , Vols. 1-5 (CRC Press-Lewis Publishers, 1992-1997), and coauthor of _Your Home, Your Health and Well-Being_ (Ten Speed Press, 1988) as well as over 100 peer-reviewed medical articles.\n\n W.J. Rea and G.H. Ross, \"Food and Chemicals as Environmental Incitants,\" _Bol Asoc Med P R_ 83 (July 1991): 310-315.\n\n W.J. Rea et al., \"Confirmation of Chemical Sensitivity by Means of Double-Blind Inhalant Challenge of Toxic Volatile Chemicals,\" _Bol Asoc Med P R_ 83 (September 1991): 389-393.\n\n Sherry A. Rogers, _The E.I. Syndrome: An Rx for Environmental Illness_ (Syracuse: Prestige Publishing, 1986); _Tired or Toxic: A Blueprint for Health_ (Syracuse: Prestige Publishing, 1990); _You Are What You Ate: An Rx for the Resistant Diseases of the 21st Century_ (Syracuse: Prestige Publishing, 1988). Also Richard Leviton, \"Environmental Health\u2014Sherry Rogers, M.D.,\" _East West Journal_ , July\/August 1991.\n\n Walter J. Crinnion, \"Environmental Medicine, Part 1: The Human Burden of Environmental Toxins and Their Common Health Effects,\" _Alternative Medicine Review_ 5:1 (February 2000): 52-63.\n\n W. Crinnion, \"Results of a Decade of Naturopathic Treatment for Environmental Illnesses: A Review of Clinical Records,\" _Journal of Naturopathic Medicine_ 7:2 (1998): 21-27. See also W. Crinnion, \"Chronic Fatigue Syndrome and Environmental Overload,\" (n.d.) www.naturopathic.org\/Library\/articles.lay\/WC.CFS.html.\n\n M.S. Wolff et al., \"Blood Levels of Organochlorine Residues and Risk of Breast Cancer,\" _Journal of the National Cancer Institute_ 85 (1993): 648-652.\n\n J.R. Davis et al., \"Family Pesticide Use and Childhood Brain Cancer,\" _Archives of Environmental Contamination and Toxicology_ 24 (1993): 87-92.\n\n Crinnion, \"Environmental Medicine, Part 1\" 57.\n\n C.G. Ohlson and L. Hardell, \"Testicular Cancer and Occupational Exposures with a Focus on Xenoestrogens in Polyvinyl Chloride Plastics,\" _Chemosphere_ 40 (May-June 2000): 1277-82.\n\n Ted Schettler et al., _In Harm's Way: Toxic Threats to Child Development_ (Cambridge, Mass.: GBPSR, 2000): 1-8. This report is available in full online at www.igc.org\/psr\/ihw.htm. Or by mail: Greater Boston Physicians for Social Responsibility, 11 Garden Street, Cambridge, MA 02138; tel: 617-497-7440; fax: 617-876-4277; e-mail: psrmabo@igc.org.\n\n Peter Montague, \"Dumbing Down the Children-Part 1,\" _Rachel's Environment & Health Weekly_, #687, February 17, 2000 www.rachel.org.\n\n Peter Montague, \"Dumbing Down the Children-Part 1,\" _Rachel's Environment & Health Weekly_, #687, March 2, 2000.\n\n Ibid.\n\n Peter Montague, \"Toxics and Violent Crime,\" _Rachel's Environment & Health Weekly_, #551, June 19, 1997.\n\n Peter Montague, \"Toxics Affect Behavior,\" _Rachel's Environment & Health Weekly_, #529, January 16, 1997. See also Herbert L. Needleman et al., \"Bone Lead Levels and Delinquent Behavior,\" _Journal of the American Medical Association_ , 275 (February 7, 1996): 363-369.\n\n Roger D. Masters, \"Environmental Pollution, Toxic Chemicals, Crime and Disease,\" at www.dartmouth.edu\/~rmasters\/research.html.\n\n Crinnion, \"Environmental Medicine, Part 1,\" 60.\n\n The thyroid gland is located just below the larynx in the throat, and has interconnecting lobes on either side of the trachea, giving it the effect of a bowtie on the trachea. This gland is the body's metabolic thermostat, controlling body temperature, energy use, and, in children, the body's growth rate. The thyroid controls the rate at which organs function and the speed with which the body uses food; it affects the operation of all body processes and organs. Of the hormones the thyroid synthesizes and releases, T3 (triiodothyronine) represents seven percent and T4 (thyroxine) accounts for almost ninety-three percent of the thyroid's hormones active in all of the body's processes. The thyroid also secretes calcitonin, a hormone required for calcium metabolism and for satisfactory digestion. Conventional medicine tends to downplay, if not ignore, the importance of the thyroid in many health problems, and it tends to use inadequate laboratory tests to measure thyroid hormone levels, thereby giving, in many cases, a false picture of thyroid hormone status. The thyroid gland also is one of the first of the endocrine glands to be damaged by exposure to nuclear radiation, indicating its sensitivity to environmental poisons.\n\n M.H. Li and L. Hansen, \"Enzyme Induction and Acute Endocrinic Effects in Prepubertal Feamel Rats Receiving Environmental PCB\/PCDF, PCDD Mixtures,\" _Environmental Health Perspectives_ 104 (1996): 712-722.\n\n J.F. Leatherland, \"Changes in Thyroid Hormone Economy Following Consumption of Environmentally Contaminated Great Lakes Fish,\" _Toxicology and Industrial Health_ 14 (1998): 41-57.\n\n Susan Porterfield, \"Vulnerability of the Developing Brain to Thyroid Abnormalities: Environmental Insults to the Thyroid System,\" _Environmental Health Perspectives_ 102:2 (1994): 125-130.\n\n There are five main groups or categories of endocrine-disrupting chemicals: 1) phthalates (used as plasticizers; can imitate estradiol, a type of estrogen); 2) alkylphenols (used in industrial and domestic detergents); 3) bisphenol A (found in dental lacquers and used to coat the interior of metal food cans; estrogenic to breast cancer cells); 4) organochlorine pesticides (lindane, DDT, altrazine; may increase breast cancer risk; DDT can block the action of male hormones); 5) polychlorinated biphenyls (PCBs) and dioxins (dioxins are produced during incineration and industrial processes; PCBs were once used in electrical equipment, and still present in transformers and capacitors); and 6) others (the cosmetics preservative parabens; food antioxidant BHA; fungicide vinclozolin; and a group of natural plant compounds called phytoestrogens). From research provided by Dr. A. Michael Warhurst, \"Introduction to Hormone Disrupting Chemicals,\" Friends of the Earth, London, England, 1998 www.website.lineone.net\/~mwarhurst.\n\n Dieldrin is an organochlorine insecticide, used extensively in agriculture since the 1940s. \"It is one of the most persistent of all pesticides, remaining for years in soils, accumulating in the fatty tissues of living organisms, and consequently, concentrating in the food chain.\" Dieldrin is no longer used in the United States on food crops, yet dieldrin residues continue to be detected in foods, especially carrots, com, cucumbers, and sweet potatoes. It is highly persistent in water, more so than other organochlorines, and enters the human body through the skin, eyes, lungs, and gastrointestinal tract. (Harte et al., _Toxics A to Z_ , 207-208.)\n\n A.P. Hoyer et al., \"Organochlorine Compounds and Breast Cancer\u2014Is There a Connection Between Environmental Pollution and Breast Cancer?\" _Ugeskr Laeger_ 162 (February 14, 2000): 922-926.\n\n S.A. Ahmed et al., \"Gender and Risk of Autoimmune Diseases: Possible Role of Estrogenic Compounds,\" _Environmental Health Perspectives_ 107, supp. 5 (October 1999): 681-686.\n\n N. Olea et al., \"Inadvertent Exposure to Xenoestrogens in Children,\" _Toxicology and Industrial Health_ 15 (January-March 1999): 151-158.\n\n J.P. Sumpter, \"Xenoendocrine Disrupters\u2014Environmental Impacts,\" _Toxicology Letters_ , 102-103 (December 28, 1998): 337-42.\n\n Gina M. Solomon, \"Endocrine Disruptors. What Should We Do Now?\" Natural Resources Defense Council, 1997 www.nrdc.org.\n\n Theo Colborn, Frederick S. vom Saal, and Ana M. Soto, \"Developmental Effects of Endocrine-Disrupting Chemicals in Wildlife and Humans,\" _Environmental Health Perspectives_ 101 (October 1993): 378-384.\n\n OCCs are synthetic, organic, chlorinated chemicals\u2014in other words, industrially produced chemicals (synthetic) based around carbon (organic), to which chlorine has been added (chlorinated). Examples of OCCs are chloroform, DDT and PCBs, certain pesticides, and dioxin. Normally carbon and chlorine are not found bound together into a molecule in nature, other than as a byproduct of volcanic eruptions and forest fires, and as a natural exudate of marine algae.\n\n Peter Montague, \"Dangers of Chemical Combinations,\" _Rachel's Environment & Health Weekly_, #498, June 13, 1996. See also Steven F. Arnold et al., \"Synergistic Activation of Estrogen Receptor with Combinations of Environmental Chemicals,\" _Science_ 272 (June 7, 1996): 1489-1492.\n\n Montague, \"Dangers of Chemical Combinations.\"\n\n It is encouraging to note that in Massachusetts it is now a law that certain chemicals may not be used in schools and day-care centers, and any use of pesticides requires parental notification first. Although thirty other states have some form of chemical regulation, the Massachusetts law, passed in June 2000, is considered the most far-reaching. Other states, such as California, New York, and Wisconsin, are attempting to pass similar laws. For more information about pesticides and schools, contact: Beyond Pesticides\/NCAMP (National Coalition Against the Misues of Pesticides), 701 E. Street SE, Suite 200, Washington, D.C., 20003; tel: 202-543-5450; fax: 202-543-4791; e-mail: info@beyondpesticides.org; website:www.beyondpesticides.org.\n\n Kagan Owens and Jay Feldman, \"The Schooling of State Pesticide Laws: Review of State Pesticide Laws Regarding Schools,\" _Pesticides and You_ 18:3 (1998): 9-23. Published by Beyond Pesticides\/NCAMP (National Coalition Against the Misues of Pesticides), 701 E. Street SE, Suite 200, Washington, D.C., 20003; tel: 202-543-5450; fax: 202-543-4791; e-mail: info@beyondpesticides.org; website: www.beyondpesticides.org.\n\n World Resources Institute, \"Threats from Environmental Estrogens,\" 1995 www.wri.org.\n\n Theo Colborn, Dianne Dumanoski, and John Peterson Myers, _Our Stolen Future: Are We Threatening Our Fertility, Intelligence, and Survival?\u2014A Scientific Detective Story_ (New York: Dutton\/Penguin Group, 1996): 187-209.\n\n Clinical molecular medicine is so named, says Dr. Edelson, because it integrates environmental medicine, which assesses the effect of the environment on human illness, and applied immunology (how the immune system functions), toxicology (how toxins make us sick), and clinical nutritional biochemistry (the scientific way of using nutrients to improve health based on an understanding of cellular chemical processes that are not working optimally).\n\n Stephen B. Edelson, \"Requirements for Chemical Detoxification,\" 1998, The Edelson Center for Environmental and Preventive Medicine, 3833 Roswell Road, Suite 110, Atlanta, GA 30342; tel: 404-841-0088; fax: 404-841-64156; website: www.ephca.com; also: www.edelsoncenter.com.\n\n Elson M. Haas, \"Detoxification and the Detox Diet: An Important Healing Process,\" 1999 www.elsonhaas.com.\n\n Elson M. Haas, M.D., \"General Detoxification and Cleansing,\" (n.d.) HealthWorld Online www.healthy.net. See also _The Detox Diet: The How-to and When-to Guide for Cleansing the Body_ (Berkeley, Calif.: Celestial Arts, 1996): 21-32. Elson Haas, M.D., may be contacted at Preventive Medical Center of Marin, Inc., 25 Mitchell Blvd., Suite 8, San Rafael, CA 94903; tel: 415-472-2343; fax: 415-472-7636; e-mail: lora_pmc2000@hotmail.com; website: www.elsonhaas.com.\n\n Elson M. Haas, \"Spring Cleansing,\" 2000 www.elsonhaas.com.\n\n Strictly speaking, for some, especially Westerners raised on animal protein, soybeans are not easy to digest and can produce gastrointestinal discomfort. However, for the purposes of this example, the statements made about soybeans are accurate.\n\n## _Chapter 3_\n\n Natural killer (NK) cells are central to the immune system's ability to withstand toxins and potential disease. They are a type of nonspecific, free-ranging immune cell produced in the bone marrow and matured in the thymus gland, located behind the sternum. NK cells can recognize and quickly destroy virus and cancer cells on first contact. \"Armed\" with an estimated 100 different biochemical poisons for killing foreign proteins, they can kill target cells without having encountered them previously. As with antibodies, their role is surveillance, to rid the body of aberrant or foreign cells before they can grow and produce cancer or infection.\n\n Symptoms of mercury toxicity make a very long list: anorexia, depression, fatigue, insomnia, arthritis, multiple sclerosis, moodiness, irritability, memory loss, nausea, diarrhea, gum disease, swollen glands, headaches, and many more. Mercury toxicity has been shown to have a destructive effect on kidney function and to contribute to cardiovascular disease, neuropsychological dysfunction, reproductive disorders, and birth defects.\n\nMercury fillings, or amalgams, have been used in dentistry since the 1820s, but not until 1988 did the routine use of mercury raise serious enough questions for the Environmental Protection Agency (EPA) to declare scrap dental amalgam a hazardous waste. Even so, the majority of dentists routinely install mercury fillings in patients' mouths, despite the fact that it is a certified poison. Evidence now shows that mercury amalgams are the major source of mercury exposure for the general public\u2014at rates six times higher than those found in fish and seafood. Studies by the World Health Organization show that eight amalgams in a single mouth can release 3-17 mcg of mercury per day. A Danish study of a random sample of 100 men and 100 women showed that increased blood mercury levels were related to the presence of more than four amalgam fillings in the teeth.\n\n DMPS stands for 2,3-dimercaptopropane-l-sulfonate. It is a substance used by progressive physicians as an evaluation tool to measure suspected mercury levels in a patient. DMPS is the chelating (binding-up) agent of choice for the removal of elemental mercury from the human body. It can be given orally, intravenously, or intramuscularly, and is useful for people who have been exposed to mercury amalgam through their dental fillings, or for those who show evidence of or suspect heavy metal toxicity from other sources.\n\n The pivotal concept in EDS is that an energetic event transfers its signal through an acupuncture meridian to the nervous system, with the end result being a cellular pathology, or some kind of organic disturbance. Essentially, a trained practitioner conducts an \"interview\" with the patient's organs and tissues, gathering information about the basic functional status of those systems and their energy pathways. Not only can EDS show the degree of stress affecting an organ, it is able to monitor the progress of therapy. EDS practitioners assert that this enables them to avoid the trial and error and general guesswork that often accompany difficult diagnoses.\n\nAs a form of computerized information gathering, EDS places a blunt, noninvasive electric probe at specific points on a person's hands, face, or feet, corresponding to acupuncture points at the beginning or end of energy meridians. The theory behind EDS is that very small electrical discharges from these points serve as information signals about the condition of the body's organs and systems, useful for the physician in evaluation and developing a treatment plan. EDS uses a scale of 0 to 100, with 45-55 being \"normal\" or \"balanced.\" Readings above fifty-five are interpreted as indicating an inflammation of the organ associated with the meridian tested, while readings below forty-five to fifty suggest organ stagnation and degeneration.\n\n Fats and oils are made of building blocks called fatty acids. A fatty acid is made of a chain of carbon atoms with hydrogen atoms attached, and an acid group of atoms at the end able to combine with glycerol. When three fatty acids attach to one molecule of glycerol, this makes a simple fat called a triglyceride. The chain of carbon atoms can be short (two to six atoms), medium (eight to ten), or long (twelve to thirty). Butyric acid, found in milk fat, butter, and cream, is a short-chain fatty acid. A fatty acid that has its full quota of hydrogen atoms is a saturated fatty acid (animal fats and hardened fats, solid at room temperature); a fatty acid with less than its full allotment of hydrogen atoms is an unsaturated fatty acid (many plant and fish oils, liquid at room temperature). When a fatty acid lacks only two hydrogen atoms, it is a monounsaturated fatty acid (such as oleic acid); a fatty acid lacking four or more hydrogen atoms is a polyunsaturated fatty acid (linoleic acid). Unsaturated fats required in the diet are called essential fatty acids and include linoleic acid (an omega-6 oil), found in corn, beans, and some nuts and seeds; and alpha linolenic acid (an omega-3 oil), found in fish, flaxseed, and walnuts.\n\n Dr. Robert C. Greenberg, \"Biological Terrain,\" Biological Technologies International, 1998 www.bioterrain.com.\n\n King James Medical Laboratory, Inc., \"Frequently Asked Questions Regarding the Analysis of Metals in Hair Specimens,\" 1998 www.kingjamesomegatech-lab.com.\n\n Affinity Labeling Technologies specifies five proteins found in gingival crevicular fluid, each indicating a different aspect of possible (or probable) dental infections under way. 1) Human alkaline phosphatase: Normally found inside the body's cells, when this protein is present in the gingival fluid, it indicates inflammation or tissue destruction at the sampling site. It will not be found around a healthy tooth; also, the higher the level of this protein, the greater the degree of periodontal disease. 2) Bacterial alkaline phosphatase: This protein is produced by various species of anaerobic (not requiring oxygen) bacteria in the mouth and associated with dental disease. This substance, used by bacteria for their own growth, is not found around healthy teeth; thus its presence indicates infection. 3) Human serum albumin: This protein normally appears only in the blood; when it is encountered in the mouth, it means the blood vessels in the mouth have become permeable, allowing this protein to \"leak\" out. Its presence usually indicates inflammation. 4) Bacterial proteases: These are enzymes that break down other proteins and are secreted by various bacteria found in the mouth. They are responsible for a fair amount of the connective tissue damage that happens with periodontal disease. 5) Human antibodies: The immune system releases specific immune cells (antibodies) in response to the presence of pathogenic microorganisms present in the mouth in the case of dental infection. See \"GCF Components: Proteins Present in Gingival Crevicular Fluid (GCF) Samples (Not Added by ALT, Inc.) and the Possible Reason(s) for Their Presence,\" Affinity Labeling Technologies, 2000 www.altcorp.com.\n\n The term intestinal dysbiosis refers to an imbalance of intestinal flora, or the total microbial population of the large and small intestines. Specifically, these flora are friendly, beneficial bacteria, such as _Lactobacillus acidophilus_ and _Bifidobacterium bifidum_ , and unfriendly bacteria such as _Escherichia coli_ and _Clostridium per-fringens._ In dysbiosis, the unfriendly bacteria predominate; they begin fermentation, producing toxic byproducts, such as ammonia, amines, nitrosamines, phenols, cresols, indole, and skatole, which interfere with the normal elimination cycle. Dysbiosis is considered a primary cause or major cofactor in the development of many health problems, such as acne, yeast overgrowth (by _Candida albicans)_ , chronic fatigue, depression, digestive disorders, bloating, food allergies, PMS, rheumatoid arthritis, and cancer.\n\n Richard S. Lord, \"Dysbiosis Metabolic Markers,\" MetaMetrix Clinical Laboratory, 2000 www.metametrix.com.\n\n Gary Kaplan, \"Chronic Pain, Fatigue, and Sinus Headaches,\" _Alternative Medicine Digest_ , October\/November 1997, 56-63.\n\n James Braly, \"Detecting Hidden Food Allergies,\" _Alternative Medicine Digest_ , June\/July 1998, 30-34.\n\n Ibid.\n\n Richard Leviton, \"The Allergy-free Body,\" _Alternative Medicine Digest_ , April 1995, 8-13.\n\n## _Chapter 4_\n\n Peter Bennett, Stephen Barrie, and Sara Faye, _7-Day Detox Miracle: Restore Your Mind and Body's Natural Vitality with This Safe and Effective Life-Enhancing Program_ (Rockland, Calif.: Prima Health\/Prima Publishing, 1999): 164.\n\n Ibid.\n\n F. Batmanghelidj, _Your Body's Many Cries for Water_ , Global Health Solutions, Inc., P.O. Box 3189, Falls Church, VA 22043; tel: 703-848-2333; fax: 703-848-2334; website: www.watercure.com.\n\n Terry Grossman, _The Baby Boomers' Guide to Living Forever. An Introduction to Immortality Medicine_ (Golden, Colo.: The Hubristic Press, 2000): 53.\n\n You can be exposed to VOCs in your drinking water even if you rely on untreated (unchlorinated) well water. In 1999, the United States Geological Service (USGS) reported that about 42 million Americans were consuming water from an untreated groundwater aquifer containing at least one VOC. Out of 3,000 water sources tested, VOC levels in 6% of urban wells and in 1.5% of rural wells exceeded the established safety levels for VOCs, said USGS. In urban areas, about fifty percent of the wells had at least one VOC, and thirty percent had two or more, while in rural areas, fifteen percent of the wells had one VOC, and six percent had two or more. The four most commonly detected VOCs in wells were chloroform, methyl-tert-butylether, tetra-chloroethene, and trichloroethene. See \"About 42 Million Americans Drink Water Containing at Least One VOC,\" Earth Vision Reports, 12 October 1999 www.earthvision.net; this is a publication of Global Environment & Technology Foundation, at: www.getf.org\n\n M. Thomason et al, \"Study of Water Treatment Effects on Organic Volatiles in Drinking Water,\" _Journal of Chromatography_ 158 (October 1, 1978): 437-47.\n\n _Cryptosporidium parvum_ is a parasite capable of causing human disease. It is generally not removed or deactivated when water is chlorinated. In 1993, 400,000 people in Milwaukee, Wisconsin, developed diarrhea (the disease is called cryptosporidiosis) after drinking municipal water contaminated with this microorganism, a single-celled protozoa. The parasite is transmitted through the stool of an infected person or animal. The presence of chlorine in water actually makes the parasite enter a self-protective cystic form, in which it surrounds itself with a hard impermeable shell, like an egg, which is chlorine resistant and very hard to destroy. Thus, chlorination supports the toxicity of _Cryptosporidium_ in water. Chlorine is also ineffective against _Giardia lamblia_ , another hard-shelled cyst that produces human illness.\n\n G.A. Boorman, \"Drinking Water Disinfection Byproducts: Review and Approach to Toxicity Evaluation,\" _Environmental Health Perspectives_ 107, supp. 1 (February 1999): 207-17.\n\n T.F. Lin and S.W. Hoang, \"Inhalation Exposure to THMs from Drinking Water in South Taiwan,\" _Science of the Total Environment_ 246 (January 31, 2000): 41-49.\n\n Kirsten Waller et al., \"Trihalomethanes in Drinking Water and Spontaneous Abortion,\" _Epidemiology_ 9 (March 1998): 134-140. Shanna H. Swan et al., \"A Prospective Study of Spontaneous Abortion in Relation to Amount and Source of Drinking Water Consumed in Early Pregnancy,\" _Epidemiology_ 9 (March 1998): 126-133.\n\n R.D. Morris et al., \"Chlorination, Chlorination By-Products, and Cancer: A Meta-Analysis,\" _American Journal of Public Health_ 82 (July 1992): 1347-1348.\n\n M. Wrensch et al., \"Spontaneous Abortions and Birth Defects Related to Tap and Bottled Water Use in San Jose, California, 1980-1985,\" _Epidemiology_ 3 (March 1992): 98-103.\n\n L. Fenster et al., \"Tap or Bottled Water Consumption and Spontaneous Abortion in a Case-Control Study of Reporting Consistency,\" _Epidemiology_ 3 (March 1992): 120-4.\n\n M.D. Gallagher et al., \"Exposure to Trihalomethanes and Adverse Pregnancy Outcomes,\" _Epidemiology_ 9 (September 1998): 484-489.\n\n L. Dodds et al., \"Trihalomethanes in Public Water Supplies and Adverse Birth Outcomes,\" _Epidemiology_ 10 (May 1999): 233-7.\n\n Judith B. Klotz and Laurie A. Pyrch, \"A Case Control Study of Neural Tube Defects and Drinking Water Contaminants,\" Agency for Toxic Substances and Disease Registry, Atlanta, Ga., January 1998.\n\n Frank. L. Bove et al., \"Public Drinking Water Contamination and Birth Outcomes\" _American Journal of Epidemiology_ , 141 (May 1, 1995: 850-862.\n\n Andrew T.L. Chen et al., \"Re: 'Public Drinking Water Contamination and Birth Outcomes,'\" _American Journal of Epidemiology_ 143 (June 1, 1996: 1179-1180.\n\n P. Magnus et al., \"Water Chlorination and Birth Defects,\" _Epidemiology_ 10 (September 1999): 513-17.\n\n P. D. Lilly et al., \"Trihalomethane Comparative Toxicity: Acute Renal and Hepatic Toxicity of Chloroform and Bromodichloromethane Following Aqueous Gavage,\" _Fundamentals of Applied Toxicology_ 40 (November 1997): 101-110.\n\n Kenneth Cantor, \"Bladder Cancer, Drinking Water Source, and Tap Water Consumption: A Case-Control Study,\" _Journal of the National Cancer Institute_ , 79 (December 1987): 1269-1279. See also Peter Montague, \"Chlorine Chemicals in Our Water Linked to Human Bladder Cancer,\" _Rachel's Environment & Health Weekly_, #84, July 4, 1988 www.rachel.org.\n\n R.J. Bull et al., \"Use of Biological Assay Systems to Assess the Relative Carcinogenic Hazards of Disinfection By-products,\" _Environmental Health Perspectives_ 46 (December 1982): 215-27.\n\n M.S. Kanarek and T.B. Young, \"Drinking Water Treatment and Risk of Cancer Death in Wisconsin,\" _Environmental Health Perspectives_ 46 (December 1982): 179-86.\n\n J.K. Dunnick et al., \"Bromodichloromethane, a Trihalaomethane that Produces Neoplasms in Rodents,\" _Cancer Research_ 47 (October 1, 1987): 5189-93.\n\n J.K. Dunnick and R.L. Melnick, \"Assessment of the Carcinogenic Potential of Chlorinated Water: Experimental Studies of Chlorine, Chloramine, and Trihalomethanes,\" _Journal of the National Cancer Institute_ 85 (May 19, 1993): 817-22.\n\n R.D. Morris, \"Drinking Water and Cancer,\" _Environmental Health Perspectives_ 103, supp. 8 (November 1995): 225-31.\n\n W.D. King and L.D. Marrett, \"Case-Control Study of Bladder Cancer and Chlorination By-products in Treated Water (Ontario, Canada),\" _Cancer Causes Control_ 7 (November 1996): 596-604.\n\n T.J. Doyle et al., \"The Association of Drinking Water Source and Chlorination By-products with Cancer Incidence Among Postmenopausal Women in Iowa: A Prospective Cohort Study,\" _American Journal of Public Health_ 87 (July 1997): 1168-76.\n\n K.P. Cantor et al., \"Drinking Water Source and Chlorination By-products. I. Risk of Bladder Cancer,\" _Epidemiology_ 9 (January 1998): 21-28.\n\n M.E. Hildesheim et al., \"Drinking Water Source and Chlorination Byproducts. II. Risk of Colon and Rectal Cancers,\" _Epidemiology_ 9 (January 1998): 29-35.\n\n K.P. Cantor et al., \"Drinking Water Source and Chlorination Byproducts in Iowa. III. Risk of Brain Cancer,\" _American Journal of Epidemiology_ 150 (September 15, 1999): 552-60.\n\n KDF uses high-purity copper-zinc granules to remove heavy metals, microorganisms, and chlorine from water. KDF has been demonstrated to remove ninety-nine percent of free chlorine from water, ninety-five percent of chlorine in municipal water, and ninety-eight percent of water-soluble, positively charged ions of lead, mercury, copper, nickel, chromium, mercury, and other dissolved metals. For more information, contact KDF Fluid Treatment, Inc., 1500 KDF Drive, Three Rivers, MI 49093; tel: 616-273-3300; fax: 616-273-4400; website: www.kdfft.com.\n\n \"News You Can Use: Everybody Out of the Pool,\" _Non-Toxic Times_ , 1 (August 2000): 4-5, published by Seventh Generation, One Mill Street, Box A26, Burlington, VT 05401; tel: 802-658-3773; fax: 802-6581771; e-mail: recycle@seventhgen.com; website: www.seventhgen.com. Seventh Generation is also an online source of \"environmentally conscious products,\" including alternatives to chlorine-based products.\n\n A.R. Hinman et al., \"The U.S. Experience with Fluoridation,\" _Community Dental Health_ , 13, supp. 2 (September 1996): 5-9.\n\n Toxic waste remaining from the manufacture of phosphate fertilizer is the source of a large portion of the fluorides dumped in American water. The waste left over from making phosphate fertilizer contains about nineteen percent fluorine. Municipalities all over the United States purchase \"fresh pollution concentrate\" in the form of fluorosilicic acid from Florida companies. This acid is composed of tetrafluorosiliciate gas and other types of fluorine gases collected by pollution scrubbers in the manufacture of phosphate fertilizer. Fluorosilicic acid would otherwise be classified as a hazardous waste if governmental and industry \"health\" authorities hadn't set it up to be used to fluoridate public drinking water. Fluorosilicic acid contains at least two radionuclides (radiation products): polonium-210 and radon-222; both are decay products of uranium, which is found in phosphate rock used to make the fertilizer. When we talk about radionuclides, we're talking about carcinogens. See George C. Glasser, \"Fluoride and the Phosphate Connection,\" _Earth Island Journal_ , Summer 1998, 12.\n\n Fluoride has always been the aluminum industry's \"most devastating pollutant,\" and in 1938, when the aluminum industry had to step up production to meet wartime demands, it had to find new ways to deal with its toxic byproduct. Ironically, the industry sustained a number of damage suits claiming damage to human health from exposure to fluoride wastes. Conveniently, the aluminum industry found a way to dump its toxic waste product into America's public drinking water in the 1940s under the guise of dental benefits, even though very little hard evidence had been generated to support the contention. See Joel Griffith, \"Fluoride: Industry's Toxic Coup,\" _Earth Island Journal_ , Spring 1998 www.earthisland.org.\n\n Richard Leviton, \"The Fluoride Lie\u2014Santa Cruz Refuses to Comply with State Law,\" _Alternative Medicine_ , August-September 1998, 105-109.\n\n Gar Smith, \"Why Fluoride Is an Environmental Issue,\" _Earth Island Journal_ , Summer 2000 www.earthislandjournal.org.\n\n L. Seppa et al., \"Caries in the Primary Dentition, After Discontinuation of Water Fluoridation Among Children After Receiving Comprehensive Dental Care,\" _Community Dental and Oral Epidemiology_ 28 (August 2000): 281-8.\n\n B.A. Burt et al., \"The Effects of a Break in Water Fluoridation on the Development of Dental Caries and Fluorosis,\" _Journal of Dental Research_ 79 (February 2000): 761-9.\n\n W. Kunzel and T. Fischer, \"Caries Prevalence After Cessation of Water Fluoridation in La Salud, Cuba,\" _Caries Research_ 34 (January-February 2000): 20-5.\n\n J.J. Warren et al., \"Fluorosis of the Primary Dentition: What Does It Mean for Permanent Teeth?\" _Journal of the American Dental Association_ 130 (July 1999): 922.\n\n K.E. Heller et al., \"Dental Caries and Dental Fluorosis at Varying Fluoride Concentrations,\" _Journal of Public Health Dentistry_ 57 (Summer 1997): 136-43.\n\n R.H. Selwitz et al., \"Dental Caries and Dental Fluorosis Among Schoolchildren Who Were Lifelong Residents of Communities Having Either Low or Optimal Levels of Fluoride in Drinking Water,\" _Journal of Public Health Dentistry_ 58 (Winter 1998): 28-35.\n\n L. Morgan et al., \"Investigation of the Possible Associations Between Fluorosis, Fluoride Exposure, and Childhood Behavior Problems _,\" Pediatric Dentistry_ 20 (July-August 1998): 244-52.\n\n D.G. Pendrys et al., \"Risk Factors for Enamel Fluorosis in Optimally Fluoridated Children Born After the U.S. Manufacturers' Decision to Reduce the Fluoride Concentration of Infant Formulas,\" _American Journal of Epidemiology_ 148 (November 15, 1998): 967-74.\n\n A. Bardsen et al., \"Dental Fluorosis Among Persons Exposed to High and Low-Fluoride Drinking Water in Western Norway,\" _Community Dental and Oral Epidemiology_ 27 (August 1999): 259-67.\n\n A.E. Villa et al., \"Dental Fluorosis in Chilean Children: Evaluation of Rsik Factors,\" _Community Dental and Oral Epidemiology_ 26 (October 1998): 310-15.\n\n A. Bardsen and K. Bjorvatn, \"Risk Periods in the Development of Dental Fluorosis,\" _Clinical Oral Investigations_ 2 (December 1998): 155-60.\n\n Morton Walker, \"Fluoridation Brings Hazards to Human Health,\" _Townsend Letter for Doctors & Patients_, No. 202, May 2000, 30-36.\n\n G.J. Judd, \"Mass Fluoridation Causes Alarming Rise in Cancer Deaths,\" _Health Freedom News_ , May 1995, 10.\n\n E. Tohyama, \"Relationship Between Fluoride Concentration in Drinking Water and Mortality Rate from Uterine Cancer in Okinawa Prefecture, Japan,\" _Journal of Epidemiology_ 6 (December 1996): 184-91.\n\n W. John Diamond, M.D., and W. Lee Cowden, M.D., with Burton Goldberg, _An Alternative Medicine Definitive Guide to Cancer_ (Tiburon, Calif.: Future Medicine Publishing, 1997): 580-582.\n\n Collation of published researched provided by Health and Longevity Resource Center, 1998 www.all-natural.com.\n\n S. Hillier et al., \"Fluoride in Drinking Water and Risk of Hip Fracture in the UK: A Case-Control Study,\" _Lancet_ 355 (January 22, 2000): 265-9.\n\n John R. Lee, \"Fluoridation and Hip Fractures,\" _Earth Island Journal_ , Summer 2000 www.earthisland.org.\n\n M. Diesendorf et al., \"New Evidence on Fluoridation,\" _Australian New Zealand Journal of Public Health_ 21 (April 1997): 187-90.\n\n J.K. Maurer et al., \"Two-Year Carcinogenicity Study of Sodium Fluoride in Rats,\" _Journal of the National Cancer Institute_ 82 (July 4, 1990): 1118-26.\n\n Richard G. Foulkes, \"The Fluoride Connection,\" _Townsend Letter for Doctors & Patients_, April 1998, 11.\n\n Leviton, \"The Fluoride Lie,\" 107.\n\n Bob Woffinden, \"A Clear and Present Danger,\" _Earth Island Journal_ , Summer 2000 www.earthisland.org.\n\n P.J. Mullenix et al., \"Neurotoxicity of Sodium Fluoride in Rats,\" _Neurotoxicology and Teratology_ , 17 (March-April 1995): 169-177.\n\n Joel Griffith, \"Fluoride: Industry's Toxic Coup,\" _Earth Island Journal_ , Spring 1998 www.earthisland.org.\n\n Gar Smith, \"Why Fluoride Is an Environmental Issue,\" _Earth Island Journal_ , Summer 2000 www.earthislandjournal.org.\n\n Joseph Dispenza, _Live Better Longer. The Parcells Center Seven-Step Plan for Health and Longevity_ (San Francisco: Harper San Francisco, 1997): 38-45.\n\n Research summarized from C.H. Van Middelem, \"Fate and Persistence of Pesticides,\" presented at Pesticides in the Environment, a 1965 symposium, reprinted at www.veggiewash.com.\n\n For a good review of the subject and its ramifications, see Martin Teitel and Kimberley Wilson, _Genetically Engineered Food:Changing the Nature of Nature_ (Rochester, Vt.: Park Street Press, 1999). See also the author's website, the Council for Responsible Genetics: www.gene-watch.org.\n\n Richard Leviton, \"Bioengineered Foods\u2014Putting Consumer Health and Safety Last,\" _Alternative Medicine_ , November 1998, 104-110.\n\n Ibid., 108.\n\n Julie A. Nordlee et al., \"Identification of a Brazil-nut Allergen in Transgenic Soybeans,\" _New England Journal of Medicine_ 334 (March 14, 1996): 688-692.\n\n \"Frequently Asked Questions About Genetically Engineered Food,\" Summer 1999, The Council for Responsible Genetics, 5 Upland Road, Suite 3, Cambridge, MA 02140; tel: 617-868-0870; fax: 617-491-5344; website: crg@gene-watch.org.\n\n Francesca Lyman, \"'Transgenic' Pollution a New Concern,\" MSNBC News, 28 September 1999 www.msnbc.com.\n\n Rich Charnes, \"Genetically Altered Food: Myths and Realities,\" EarthSave, 2000 www.earthsave.org. Earth Save International, 1509 Seabright Avenue, Suite Bl, Santa Cruz, CA 96062; tel: 831-423-0293 or 800-362-3648; fax: 831-423-1313.\n\n Joe Cummins, \"Gene Tinkering Blues: Allergy,\" _Gene Tinkering Blues_ , February 1997 www.holisticmed.com\/ge\/allergy.html.\n\n Marion Nestle, \"Allergies to Transgenic Foods\u2014Questions of Policy,\" _New England Journal of Medicine_ , 334 (March 14, 1996): 726-728.\n\n Ronnie Cummins, \"Son of Frankenfoods: GE Trees, Fish, and Functional Foods,\" _BioDemocracy News_ #28, July 2000 www.purefood.org.\n\n \"Summary of U.S. Consumer Polls on GE Foods,\" Center for Food Safety, 666 Pennsylvania Avenue, SE, Suite 302, Washington D.C., 20003; tel: 202-547-9359; fax: 202-547-9429; website: www.centerforfoodsafety.org.\n\n Peter Montague, \"Genetically Altering the World's Food,\" _Rachel's Environment & Health Weekly_, #639, February 25, 1999.\n\n Richard Leviton, \"The Tainted Milk Mustache\u2014How Monsanto and the FDA Spoiled a Staple Food,\" _Alternative Medicine_ , January 1999, 94-108.\n\n Samuel S. Epstein, \"Unlabeled Milk from Cows Treated with Biosynthetic Growth Hormones: A Case of Regulatory Abdication,\" _International Journal of Health Sciences_ 26:1 (1996): 173-185.\n\n Steve Emmott, \"EU Scientific Committee Warns of Human Health & Cancer Hazards of Monsanto's Recombinant Bovine Growth Hormone (rGBH),\" March 1999 www.purefood.org.\n\n Susan E. Hankinson et al., \"Circulating Concentrations of Insulin-like Growth Factor 1 and Risk of Breast Cancer,\" _The Lancet_ 351 (May 9, 1998): 1393-1396.\n\n June M. Chan et al., \"Plasma Insulin-like Growth Factor 1 and Prostate Cancer Risk: A Prospective Study,\" _Science_ 279 (January 23, 1998): 563-566.\n\n \"Summary of U.S. Consumer Polls on GE Foods,\" Center for Food Safety www.centerforfoodssafety.org.\n\n For a complete list of labeling requirements and loopholes, see \"Background and Status of Labeling of Irradiated Foods,\" BioDemocracy and Organic Consumers Association, 28 June 2000 www.purefood.org.\n\n See John W. Gofman, \"Preventing Breast Cancer: The Story of a Major, Proven, Preventable Cause of this Disease\" (Committee for Nuclear Responsibility, San Francisco, Calif., 1996). \"Radiation from Medical Procedures in the Pathogenesis of Cancer and Ischemic Heart Disease\" (Committee for Nuclear Responsibility, San Francisco, Calif., 1999). \"Radiation-Induced Cancer from Low-Dose Exposure\" (Committee for Nuclear Responsibility, San Francisco, Calif., 1990).\n\n Richard Leviton, \"FDA's Mission Impossible\u2014Selling the American Public the Food Irradiation Safety Con,\" _Alternative Medicine_ , May 1998, 103-108.\n\n Jeffrey Reinhardt, \"Liver Cancer: Danger of Radiolytic Products in the Diet,\" BioDemocracy and Organic Consumers Association, 26 June 2000 www.purefood.org.\n\n Wenonah Hauter, \"Overview of Food Irradiation Reports,\" _Public Citizen_ , 28 December 1999 www.citizen.org.\n\n See Gary Gibbs, _The Food That Would Last Forever_ (Garden City Park, N.Y.: Avery Publishing Group, 1993).\n\n S.G. Srikantia, Professor of Food and Nutrition, University of Mysore, India, in testimony before U.S. Congressional hearings on Food Irradiation, June 19, 1987. Published at www.citizen.org.\n\n \"Memorandum, Irradiation and the New Proposed National Organic Program Area of Concern,\" Center for Food Safety, 5 April 2000 www.centerforfoodsafety.org.\n\n \"How It Hurts: Key Research on the Unwholesomeness of Irradiated Foods,\" Public Citizen, 2000, www.citizen.org.\n\n Wenonah Hauter, \"Food Irradiation: Do You Know Where Your Dinner Has Been?\" Public Citizen, 1999 www.citizen.org.\n\n Patricia Kane, \"Lifting Depression with Nutrition,\" _Alternative Medicine Digest_ , March 1998, 64-70.\n\n A. Ascherio et al., \"Trans-fatty Acids and Coronary Heart Disease,\" _New England Journal of Medicine_ 340 (June 24, 1999): 1994-8.\n\n Research compiled and summarized by Mary G. Enig at www.enig.com. Dr. Enig is a nutritionist, former faculty research associate with the Lipid Research Group at the University of Maryland, director of nutritional sciences at Enig Associates, Inc., and the author of two books on fats.\n\n For more information, see: Mary G. Enig, _Trans-Fatty Acids in the Food Supply: A Comprehensive Report Covering 60 Years of Research_ , 2d ed. (Bethesda, Md.: Bethesda Press, 1995): available from www.enig.com; e-mail: marye@enig.com. Also Mary G. Enig, _Know Your Fats: The Complete Primer for Understanding the Nutrition of Fats, Oils, and Cholesterol_ (Bethesda, Md.: Bethesda Press, 2000): Bethesda Press, Suite 340, Meadows Park Building, 12501 Prosperity Drive, Silver Spring, MD 20904; tel: 301-680-8800; fax: 301-680-8100; website: www.bethesdapress.com.\n\n Mary G. Enig, \"Published Estimates of Trans-Fatty Acid Availability and Consumption in Various Countries,\" (n.d.) www.enig.com.\n\n Bruce Fife, _The Detox Book. How to Detoxify Your Body to Improve Your Health, Stop Disease, and Reverse Aging_ (Colorado Springs, Colo.: Health Wise, 1997): 99.\n\n Elson M. Haas, _The Detox Diet. A How-To & When-To Guide for Cleansing the Body_ (Berkeley, Calif.: Celestial Arts, 1996): 119.\n\n Lewis Harrison, _30-Day Body Purification. How to Cleanse Your Inner Body & Experience the Joys of Toxin-Free Health_ (Paramus, N.J.: Reward Books\/Prentice Hall, 1995): 80-81, 83.\n\n Jacquelin Krohn, M.D., Frances Taylor, and Jinger Prosser, _The Whole Way to Natural Detoxification. Clearing Your Body of Toxins_ (Point Roberts, Wash.: Hartley & Marks, 1996): 132-33.\n\n Fife, _The Detox Book_ , 93.\n\n John Updike, _Rabbit Redux_ (New York: Alfred A. Knopf, 1971): 194.\n\n Charles Tart, quoted by Ralph Losey in \"The Problem of the Subtle Sybil Effect,\" www.sun-angel.com.\n\n Untitled, unsigned editorial], _Cogenesis Journal_ , Eos CoCreations, P.O. Box 483, Cave Creek, AZ 85327; tel: 602-502-0686; website: [www.cogenesis.com.\n\n Also known as bitter salts or heptahydrate, Epsom salt is a naturally occurring mineral called epsomite, with a high magnesium sulfate content. For commercial use, it is made through the action of sulphuric acid on magnesite (magnesium carbonate\/oxide). During the bath, the high magnesium concentration in the salt enables water to be pulled out through the body's skin pores and, with it, toxins capable of excretion through the skin. It is not uncommon for the bath water to turn a light grey color after one has been soaking for twenty minutes or so; this is an indication that toxins have been excreted through the skin. Epsom salt is also an effective, safe, nontoxic laxative, taken as a drink, mixed with water, per instructions on the package. As a topical agent, Epsom salt is recommended for providing some relief from stiffness, soreness, joint tightness, arthritic pain, discomfort from tender muscles, bruises, sprains, or strains, and to improve sleep and increase the body's energy levels through deep relaxation.\n\n Valerie Ann Worwood, _The Fragrant Mind. Aromatherapy for Personality, Mind, Mood, and Emotion_ (Novato, Calif.: New World Library, 1996): 131-132.\n\n Ibid., 193-194.\n\n Chrissie Wildwood, _The Encyclopedia of Aromatherapy_ (Rochester, Vt.: Healing Arts Press, 1996): 83.\n\n The approach was developed in 1954 by Dr. John C. Lilly, a pioneer in brain and behavioral research and later famous for his work with dolphins. Dr. Lilly experimented with the effects on the brain of restricting external sensory stimulation; he thought it would put the brain to sleep. Instead, it woke it up. Using himself as his first test subject, Dr. Lilly removed all visual, sound, tactile, and temperature stimuli, and found his brain operated independently of external sensory data, and functioned at an even higher level as a result. He found that the greatest benefit to floating in salt in a sensory deprivation chamber was the profound state of relaxation it induced. He stated that two hours in the flotation tank is equivalent to eight hours of restful sleep and that floating is an excellent way to destress after an overloaded work day.\n\n The autonomic nervous system (ANS) can be likened to your body's automatic pilot. It keeps you alive through breathing, heart rate, and digestion, without your being aware of it or participating in its activities. The ANS has two divisions: the sympathetic nervous sysyem, which expends body energy; and the parasympathetic nervous system, which conserves body energy. The sympathetic nervous system is associated with arousal and stress; it prepares us physically when we perceive a threat or challenge by increasing our heart rate, blood pressure, and muscle tension. The parasympathetic nervous system slows heart rate and increases intestinal and most gland activity.\n\n According to information provided by ThinkTank International Pte. Ltd. (Singapore) about the PathFinder floatation tank; website: www.thinktank.com.sg.\n\n Thomas H. Fine and Roderick Borrie, \"Flotation REST in Applied Psychophysiology\" (paper presented at the Sixth International Conference on REST, San Francisco, Calif., April 17, 1997) International REST Investigators Society (IRIS) www.concentric.net\/%7Etfine\/flotrest.shtml.\n\n According to information provided about Oasis Flotation Systems by Biofeedback Instrument Corporation, New York, N.Y.; website: www.biof.com\/oasis.\n\n According to information provided by Samadhi Tank Corporation, Nevada City, CA; website: www.samadhitank.com.\n\n Comments by Steven Halpern drawn from \"Push-Button Relaxation,\" 1997 www.stevenhalpern.com; from an interview by Richard Leviton, 1992; and from Richard Leviton, \"Healing Vibrations,\" _Yoga Journal_ , January-February 1994.\n\n## _Chapter 5_\n\n First developed by 19th-century European physicians, naturopathy is an umbrella term to describe the original basis of natural, holistic, and alternative medicine. A naturopath understands that Nature heals and that the body is inherently self-healing when given the appropriate support therapies, such as diet; herbs; nutritional supplementation; detoxification and internal cleansing regimens; exercise; heat and water therapy; massage; bodywork; counseling; and relaxation. Naturopathy emphasizes disease prevention and health promotion. It is currently licensed as a medical profession in at least eleven U.S. states, with the number of new states accepting naturopathy growing steadily. In a few states, naturopaths function as primary care physicians, and insurance companies cover their services. There are several accredited four-year naturopathic medical colleges now in operation.\n\n These cruciferous vegetables all contain a nutritionally valuable component called glucosinolates. Studies have shown that these compounds can increase the body's detoxification activities and help protect against the effects of carcinogens. According to nutritionist Jeffrey Bland, Ph.D., the beneficial activity of cruciferous vegetables may be due not only to glucosinolates, \"but to the synergistic interaction of many substances inherent in cruciferous vegetables.\" Dr. Bland further notes that it is important not to overcook this type of vegetable, because the heat destroys or denatures the enzymes, thereby diminishing their benefit. See Jeffrey S. Bland, _The 20-Day Rejuvenation Diet Program, With the Revolutionary Phytonutrient Diet_ (New Canaan, Conn.: Keats Publishing, 1997): 65-66.\n\n Bile is a thick, viscous, straw-colored fluid produced by the liver, from which it passes into the common bile duct and then into the small intestine as needed. Bile is also stored in the gallbladder and discharged as needed. It is typically produced at the rate of a pint to a quart per twenty-four hours, although during fasting, little bile is formed. Bile functions as a digestive juice and emulsifying agent, facilitating the digestion of oils and fats in the small intestine. Essentially, bile emulsifies the fats so that digestive enzymes from the pancreas can break them down. When consumed food (especially fats) reaches the first part of the small intestine (duodenum), it triggers the liver and gallbladder to release bile into the intestine. Bile also stimulates peristalsis, or the rhythmic contractions of the intestines that move fecal matter along to its final elimination. Normally bile is ejected from the liver or gallbladder storage only during small intestine digestion.\n\n Laurel Vukovic, _14-Day Herbal Cleansing. A Step-by-Step Guide to All Natural Inner Cleansing Techniques for Increased Energy, Vitality and Beauty_ (Paramus, N.J.: Prentice Hall, 1998), 73, 145-6.\n\n Carolyn Reuben, _Cleansing the Body, Mind, and Spirit_ (New York: Berkley Books, 1998): 178.\n\n RR. Bradley, ed., _British Herbal Compendium_ , Vol. 1 (Bournemouth, Dorset, U.K.: British Herbal Medicine Association, 1992): 112-114. J. Yamahara et al., \"Gastrointestinal Motility Enhancing Effect of Ginger and Its Active Constituents,\" _Chemical Pharmacology Bulletin_ 38 (1990): 430-1.\n\n Paul Pitchford, _Healing with Whole Foods. Oriental Traditions and Modern Nutrition_ (Berkeley, Calif.: North Atlantic Books, 1993): 497.\n\n Helene Silver. _Rejuvenate. A 21-Day Natural Detox Plan for Optimal Health_ (Freedom, Calif.: The Crossing Press, 1998), 42.\n\n Donald J. Brown, N.D., _Herbal Prescriptions for Better Health. Your Everyday Guide to Prevention, Treatment, and Care_ (Rocklin, Calif.: Prima Publishing, 1996): 151-158.\n\n Vukovic, _14-Day Herbal Cleansing_ , 75.\n\n Daniel B. Mowrey, _Herbal Tonic Therapies_ (New Canaan, Conn.: Keats Publishing, 1993): 239-241.\n\n Marie Nadine Antol, _Healing Teas. How to Prepare and Use Teas to Maximize Your Health_ (Garden City Park, N.Y.: Avery Publishing Group, 1996): 122-123.\n\n David Hoffmann, _The Holistic Herbal. A Herbal Celebrating the Wholeness of Life_ (Shaftesbury, Dorset, U.K.: Element Books, 1988): 62.\n\n Bennett, _7-Day Detox Miracle_ , 187-88.\n\n Ibid, 179-186.\n\n Carol A. Newall, Linda A. Anderson, and J. David Phillipson, _Herbal Medicines. A Guide for Health-Care Professionals_ (London: The Pharmaceutical Press, 1996): 60-61.\n\n Michael Tierra, \"How to Do a Liver Flush,\" 1999 www.planetherbs.com.\n\n Silver. _Rejuvenate_ , 51, 49.\n\n Christopher Hobbs, L.Ac, _Healing with Herbs & Foods_ (Loveland, Colo.: Botanica Press\/Interweave Press, 1994).\n\n Serafino Corsello, M.D, \"Reversing Lupus and Diabetes,\" _Alternative Medicine Digest_ , February-March 1998, 44-53.\n\n William Lee Cowden, M.D, \"Health Hazard in Your Intestines,\" _Alternative Medicine Digest_ , November 1996, 26-29.\n\n W.C. Willett et al, \"Relation of Meat, Fat, and Fiber Intake to the Risk of Colon Cancer in a Prospective Study Among Women,\" _New England Journal of Medicine_ 323 (December 13, 1990): 1664-72.\n\n B. Trock et al, \"Dietary Fiber, Vegetables, and Colon Cancer: Critical Review and Meta-Analyses of the Epidemiologic Evidence,\" _Journal of the National Cancer Institute_ 82 (April 1990): 650-61.\n\n E. Benito et al, \"A Population-Based Case-Control Study of Colorectal Cancer in Majorca,\" _International Journal of Cancer_ 45 (January 15, 1990): 69-76.\n\n C.F. Babbs, \"Free Radicals and the Etiology of Colon Cancer,\" _Free Radical Biological Medicine_ 8:2 (1990): 191-200.\n\n \"The Colon: Sewer or Cesspool?\" Angel Healing Center, 1999 www.cyberport.net\/angel.\n\n Dr. Bernard Jensen, _Dr. Jensen's Guide to Better Bowel Care_ (Garden City Park, N.Y.: Avery Publishing Group, 1999): 27, 37-38, 51-54.\n\n \"Medicinal Activated Charcoal Tablets and Biscuits,\" J.L. Bragg (Ipswich) Ltd., Suffolk, England www.users.globalnet.co.uk.\n\n Robert C. Atkins, M.D., _Dr. Atkins' Vita-Nutrient Solution. Nature's Answer to Drugs_ (New York: Simon & Schuster, 1998): 237-238.\n\n Richard C. Kaufman, \"The Universal Antidote and Detoxifier That Extends Life: Activated Charcoal,\" _Journal of the Megahealth Society_ , July 1989 www.healingtools.tripod.com.\n\n S.M. Lipson and G. Stotzky, \"Specificity of Virus Adsorption to Clay Minerals,\" _Canadian Journal of Microbiology_ 31 (January 1985): 50-53.\n\n M.D. Sobsey and T. Cromeans, \"Effects of Bentonite Clay Solids on Poliovirus Concentration from Water by Microporous Filter Methods,\" _Applied Environmental Microbiology_ 49 (April 1985): 795-8.\n\n V.M. Illrin et al., \"A New Sorbent for Concentrating Viruses from an Aqueous Medium,\" _Lik Sprava_ , May-June 1995, 5-6, 177-9.\n\n W.A. Hartman and D.B. Martin, \"Effect of Suspended Bentonite Clay on the Acute Toxicity of Glyphosphate to Daphnia pulex and Lemna minor,\" _Bulletin of Environmental Contamination and Toxicology_ 33 (September 1984): 355-361. And R.D. Fairshter and A.F. Wilson, \"Paraquat Poisoning: Manifestations and Therapy,\" _The American Journal of Medicine_ 59 (December 1975): 751-753. D. I. Blodgett, et al. \"Adsorption of Aflatoxin with Feed-Approved Clays,\" Virginia Com Board Project Progress Report (n.d.) www.yerbaprima.com\/bentproc.htm.\n\n J.M. Santurio et al., \"Effect of Sodium Bentonite on the Performance and Blood Variables of Broiler Chickens Intoxicated with Aflatoxins,\" _British Poultry Science_ 40 (March 1999): 115-9.\n\n M.A. Abdel-Wahhab et al., \"Effect of Aluminosilicates and Bentonite on Aflatoxin-Induced Developmental Toxicity in Rats,\" _Journal of Applied Toxicology_ 19 (May-June 1999): 199-204.\n\n T.C. Schell et al., \"Effects of Feeding Aflatoxin-Contaminated Diets with and without Clay to Weanling and Growing Pigs on Performance, Liver Function, and Mineral Metabolism,\" _Journal of Animal Science_ 71 (May 1993): 1209-18.\n\n R.A. McKenzie, \"Bentonite as Therapy for Lantana camara Poisoning of Cattle,\" _Australian Veterinary Journal_ 68 (April 1991): 146-8.\n\n Ran Knishinsky, _The Clay Cure_ (Rochester, Vt.: Healing Arts Press, 1998). See also \"Cleanse Yourself Internally with Liquid Clay\u2014The Bentonite Cure,\" _Alternative Medicine_ (December 1998-January 1999), 14-15.\n\n \"Chlorella Q & A\" www.earthrise.com.\n\n W. John Diamond, M.D, Lee W. Cowden, M.D, and Burton Goldberg, _An Alternative Medicine Definitive Guide to Cancer_ (Tiburon, Calif.: Future Medicine Publishing, 1997): 809-811.\n\n \"Greater Health and Longevity: Chlorella, the Green Algae Superfood, May Be the Answer,\" _Alternative Medicine Digest_ 12 (1996): 56.\n\n Information provided by Taiwan Chlorella Manufacturing Co, Ltd, Taipei, Taiwan; website: www.taiwanchlorella.com.\n\n Randal Merchant, M.D, \"1999: Research Update,\" Health & Happiness Publishing, The Press Room, 2435 E. North St, #116, Greenville, SC 29615; tel: 800-694-2774; e-mail: info@health-books.com; website: www.health-books.com\/PressRoom.\n\n Randal Merchant et al, \"Dietary _Chlorella Pyrenoidosa_ for Patients with Malignant Glioma: Effects of Immuno-Competence, Quality of Life, and Survival,\" _Phytotherapy Research_ 4:6 (1990): 220-231.\n\n R.S. Pore, \"Detoxification of Chlordecone Poisoned Rats with Chlorella and Chlorella Derived Sporopollenin,\" _Drug and Chemical Toxicology_ 7:1 (1984): 57-71.\n\n I. Hagino, \"Effect of Chlorella on Fecal and Urinary Cadmium Excretion in 'Itai-Itai,'\" _Japan Journal of Hygiene_ 30:1 (1975): 77.\n\n T. Nagano et al, \"Absorption and Excretion of Cadmium by the Rat Administered Cadmium-Containing Chlorella,\" _Eisei Kagaku_ 24:4 (1978): 7182-7186.\n\n Randal Merchant et al, \"Nutritional Supplementation with _Chlorella pyrenoidosa_ for Patients with Fibromyalgia Syndrome: A Pilot Study,\" _Phytotherapy Research_ 14 (May 2000): 167-73.\n\n Merchant, \"1999: Research Update.\"\n\n Ayurveda is the traditional medicine of India, based on many centuries of empirical use. Its name means \"end of the Vedas\" (which were India's sacred scripts), implying that a holistic medicine may be founded on spiritual principles. Ayurveda describes three metabolic, constitutional, and body types (doshas) in association with the basic elements of Nature. These are vata (air and ether, rooted in intestines), pitta (fire and water\/stomach), and kapha (water and earth\/lungs). Ayurvedic physicians use these categories (which also have psychological aspects) as the basis for prescribing individualized formulas of herbs, diet, massage, breathing, meditation, exercise and yoga postures, and detoxification techniques.\n\n Dr. Michael Tierra, L.Ac, O.M.D, \"The Wonders of Triphala: Ayurvedic Formula for Internal Purification,\" Online Articles, 1996 www.planetherbs.com.\n\n Rudolf M. Ballentine, M.D., _Radical Healing. Integrating the World's Great Therapeutic Traditions to Create a New Transformative Medicine_ (New York: Three Rivers Press, 1999): 306.\n\n Jacqueline Krohn, M.D., Frances Taylor, and Jinger Prosser, _The Whole Way to Natural Detoxification. Clearing Your Body of Toxins_ (Point Roberts, Wash.: Hartley & Marks, 1996): 17.\n\n L.J. Cheskin et al., \"Mechanisms of Constipation in Older Persons and Effects of Fiber Compared with Placebo,\" _Journal of the American Geriatric Society_ 43 (June 1995): 666-9.\n\n J. Stevens et al., \"Comparison of the Effects of Psyllium and Wheat Bran on Gastrointestinal Transit Time and Stool Characteristics,\" _Journal of the American Dietetics Association_ 88 (March 1988): 323-6.\n\n M. Tomas-Ridocci et al., \"The Efficacy of Plantago ovata as a Regulator of Intestinal Transit. A Double-Blind Study Compared to Placebo,\" _Rev Esp Enferm Dig_ 82 (July 1992): 17-22.\n\n W. Ashraf et al., \"Effects of Psyllium Therapy on Stool Characteristics, Colon Transit and Anorectal Function in Chronic Idiopathic Constipation,\" _Alimentary Pharmacological Therapy_ 9: (December 1995): 639-47.\n\n All cholesterol is not unhealthy; some is required for the body's metabolic and hormonal functions; but one form of cholesterol is believed to be injurious. To understand this, we need to discuss lipoproteins. These are found in two principal forms: low- and high-density. Low-density lipoproteins (LDLs), which are made from protein and fat molecules, circulate in the blood and act as the primary carriers of cholesterol to the cells of the body. An elevated level of LDL, often called \"bad\" cholesterol, contributes to atherosclerosis, a circulatory system disease that leads to a buildup of plaque deposits on the inner walls of the arteries. These plaque deposits can eventually cause heart attacks, stroke, and other illnesses. A diet high in saturated fats can increase levels of LDLs in the blood. In contrast, high-density lipoproteins (HDLs) contain a larger amount of protein and less fat than LDLs. HDLs readily absorb cholesterol and related compounds in the blood and transport them to the liver for elimination. A higher ratio of HDL to LDL cholesterol in the blood is associated with a reduced risk of cardiovascular disease.\n\n J.W. Anderson et al., \"Cholesterol-Lowering Effects of Psyllium-Enriched Cereal as an Adjunct to a Prudent Diet in the Treatment of Mild to Moderate Hypercholesterolemia.\" _American Journal of Clinical Nutrition_ 56 (July 1992): 93-8.\n\n W.L. Haskell et al., \"Role of Water-Soluble Dietary Fiber in the Management of Elevated Plasma Cholesterol in Healthy Subjects,\" _American Journal of Cardiology_ 70 (September 15, 1992): 840.\n\n Technically, this physiological experience, or sensation, is called the Jarisch-Herxheimer reaction or effect, or sometimes \"herx,\" for short. It is named after the Austrian dermatologist, Jarisch Adolf Herxheimer, who identified it in 1895; the effect was confirmed by his brother, Karl Herxheimer, M.D, also a dermatologist but working in Germany. Originally it was used in reference to syphilis, but doctors found it accurately described the phenomena of other illnesses, such as tuberculosis, arthritis, and candidiasis (yeast infestation by _Candida albicans_ ). The term refers to a temporary increase of symptoms under the influence of treatment; in the case of syphilis, it referred to the temporary worsening or intensification of symptoms as a result of administering anti-syphilitic drugs, or antibiotics. In that case, the syphilis organisms were literally dying off at a rate faster than the body could clean up and remove their toxic debris. The term is now being used in the context of Lyme disease, caused by the spirochete _Borrelia burgdorferi_ , related to _Treponema pallidum_ , the organism responsible for syphilis. But it is also being used even more generally for comparable experiences during detoxification. Although the experience is unpleasant, it is a sure sign that the detoxification protocols are working. It demonstrates the ironic truth of the following statement: You wouldn't feel this miserable if you weren't actually getting better.\n\n Thomas Fleming, ed, _PDR\u00ae for Herbal Medicines_ (Montvale, N.J.: Medical Economics Company, 1998): 1090.\n\n Mowrey, _Herbal Tonic Therapies_ , 215-218.\n\n Richard Anderson, N.D, N.M.D, \"The Key to Excellent Health: Cleansing the Colon,\" _Alternative Medicine_ , March 1999, 18-22. _Cleanse & Purify Thyself_, Book 1.5, Richard Anderson (publisher), 1998, 48-49, 66-67.\n\n Anderson, \"The Key to Excellent Health: Cleansing the Colon,\" 21-22.\n\n Ann Louise Gittleman, _Guess What Came to Dinner. Parasites and Your Health_ (Garden City Park, N.Y.: Avery Publishing Group, 1993): xiii.\n\n An immunoglobulin is one of a class of five specially designed antibody proteins produced in the spleen, bone marrow, or lymph tissue and involved in the immune system's defense response to foreign substances. The main types of immunoglobulins, grouped according to their concentration in the blood, are: IgG (80%), IgA (10-15%), IgM (510%), IgD (less than 0.1%), and IgE (less than 0.01%). Technically, all antibodies are immunoglobulins.\n\n Silena Heron, N.D, and Eric Yarnell, N.D, \"Treating Parasitic Infections with Botanical Medicines,\" _Alternative & Complementary Therapies_ 5 (August 1999): 214-224.\n\n Leo Galland, M.D, \"Persistent GI Upset a Signal of Hidden Giardiasis,\" _Cortlandt Forum_ (1990):120-21.\n\n Leo Galland, M.D, et al, \" _Giardia lamblia_ Infection as a Cause of Chronic Fatigue,\" _Journal of Nutritional Medicine_ 1 (1990): 27-31.\n\n \"Results of Testing for Intestinal Parasites by State Diagnostic Laboratories,\" United States, 1987, _Morbidity and Mortality Weekly_ 40 (SS-4) (1992): 25-30.\n\n J. Lee, G.M. Park et al., \"Intestinal Parasite Infections at an Institution for the Handicapped in Korea,\" _Korean Journal of Parasitology_ 38 (September 2000): 179-81.\n\n K.J. Lee, Y.K. Ahn, and T.S. Yong, \"A Small-Scale Survey of Intestinal Parasite Infections Among Children and Adolescents in Legaspi City, the Philippines,\" _Korean Journal of Parasitology_ 38 (September 2000): 183-5.\n\n M. Youssef et al., \"Bacterial, Viral, and Parasitic Enteric Pathogens Associated with Acute Diarrhea in Hospitalized Children from Northern Jordan,\" _FEMS Immunology and Medical Microbiology_ 28 (July 2000): 257-63.\n\n A.L. el Idrissi, M. Lyagoubi et al., \"Prevalence of Intestinal Parasitic Disease in Three Provinces in Morocco,\" _Eastern Mediterranean Health Journal_ 5 (January 1999): 86-102.\n\n Joseph Pizzorno, N.D., _Total Wellness. Improve Your Health by Understanding the Body's Healing Systems_ (Rocklin, Calif.: Prima Publishing, 1996): 215.\n\n J. Flegr, P. Kodym, and V. Tolarova, \"Correlation of Duration of Latent _Toxoplasma gondii_ Infection with Personality Changes in Women,\" _Biological Psychology_ 53 (May 2000): 57-68.\n\n Pizzorno, _Total Wellness_ , 79.\n\n Michael Murray, N.D., and Joseph Pizzorno, N.D., _Encyclopedia of Natural Medicine_ , rev. 2d ed. (Rocklin, Calif.: Prima Publishing, 1998): 436-437.\n\n R. Sharma, C.K. Joshiu, and R.K. Goyal, \"Berberine Tannate in Acute Diarrhea,\" _Indian Pediatrics_ 7 (1970): 496-501.\n\n V.P. Choudry, M. Sabir, and V.N. Bhide, \"Berberine in Giardiasis,\" _Indian Pediatrics_ 9 (1972): 143-6.\n\n S.A. Kamat, \"Clinical Trial with Berberine Hydrochloride for the Control of Diarrhea in Acute Gastroenteritis,\" _Journal of the Association of Physicians of India_ 129 (1967): 525-9.\n\n J. Lai, S. Chandra et al., \"In Vitro Antihelmintic Action of Some Indigenous Medicinal Plants on _Ascardia galli_ Worms,\" _Indian Journal of Physiological Pharmacology_ 20 (April-June 1976): 64-68.\n\n V. Munoz et al., \"The Search for Natural Bioactive Compounds through a Multidisciplinary Approach in Bolivia. Part II. Antimalarial Activity of Some Plants Used by Mosetene Indians,\" _Journal of Ethnopharmacology_ 69 (February 2000): 139-55.\n\n A.D. Frame et al., \"Plants from Puerto Rico with Anti-Mycobacterium Tuberculosis Properties,\" _Puerto Rican Health Science Journal_ 17 (September 1998): 243-52.\n\n A. Singh, S.P. Singh, and R. Bamezai, \"Postnatal Efficacy of _Momordica charantia_ Peel, Pulp, Seed, and Whole Fruit Extract in the Detoxification Pathway of Suckling Neonates and Lactating Mice,\" _Cancer Letters_ 122 (January 9, 1998): 121-126.\n\n P. Scartezzini and E. Speroni, \"Review on Some Plants of Indian Traditional Medicine with Antioxidant Activity,\" _Journal of Ethnopharmacology_ 71 (July 2000): 23-43.\n\n O.V. Rybaltovskii, \"On the Discovery of Cucurbitin\u2014a Component of Pumpkin Seed with Antihelmintic Action,\" _Med Parazitol_ (Mosk) 35 (July-August 1966): 487-8. R.J. Blagrove and G.G. Lilley, \"Characterisation of Cucurbitin from Various Species of the Cucurbitaceae,\" _European Journal of Biochemistry_ 103 (February 1980): 577-84.\n\n S.J. Bhatia et al, \" _Lactobacillus acidophilus_ Inhibits Growth of _Campylobacter pylori_ in Vitro,\" _Journal of Clinical Microbiology_ 27 (October 1989): 2328-30.\n\n M. Racccach, R. McGrath, and H. Daftarian, \"Antibiosis of Some Lactic Acid Bacteria Including _Lactobacillus acidophilus_ toward _Listeria monocytogenes,\" International Journal of Food Microbiology_ 9 (August 1989): 25-32.\n\n M.Y. Lin and F.J. Chang, \"Antioxidative Effect of Intestinal _Bifidobacterium longum_ ATCC 15708 and _Lactobacillus acidophilus_ ATCC 4356,\" _Dig Dis Sci_ 45 (August 2000): 1617-22.\n\n O. Sreekumar and A. Hosono, \"Immediate Effect of _Lactobacillus acidophilus_ on the Intestinal Flora and Fecal Enzymes of Rats and the in Vitro Inhibition of _Escherichia coli_ in Coculture,\" _Journal of Dairy Science_ 83 (May 2000): 931-9.\n\n N.M. de Roos and M.B. Katan, \"Effects of Probiotic Bacteria on Diarrhea, Lipid Metabolism, and Carcinogenesis: A Review of Papers Published Between 1988 and 1998,\" _American Journal of Clinical Nutrition_ 71 (February 2000): 405-11.\n\n G.H. Mclnbtosh, P.J. Royle, and M.J. Playne, \"A Probiotic Strain of _L. acidophilus_ Reduces DMH-Induced Large Intestinal Tumors in Male Sprague-Dawley Rats,\" _Nutr Cancer_ 35:2 (1999): 153-9.\n\n L.I. Chernyshova, \"Effects of Dysbacteriosis and Impairment of Immunity Formation in the Early Neonatal Period on Morbidity of Children During the 1st Year of Life and Ways of Its Reduction,\" _Pediatriia_ 6 (1989): 24-29.\n\n PS. Moshchich, L.I. et al, \"Prevention of Dysbactriosis in the Early Neonatal Period Using a Pure Culture of Acidophilic Bacteria,\" _Pediatriia_ 3 (1989): 25-30.\n\n H.S. Gill et al, \"Enhancement of Natural and Acquired Immunity by _Lactobacillus rhamnosus_ (HN001), _Lactobacillus acidophilus_ (HN017), and _Bifidobacterium lactis_ (HN019),\" _British Journal of Nutrition_ 83 (February 2000): 167-76.\n\n Natasha Trenev, _Probiotics. Nature's Internal Healers_ (Garden City Park, N.Y.: Avery Publishing Group, 1998): 74-75.\n\n John Anderson, \" _Acidophilus_ \u2014Why Your Intestines Need This Friendly Bacteria,\" _Alternative Medicine_ , November 1998, 72-76.\n\n Bennett, _7-Day Detox Miracle_ , 192.\n\n Diamond, _An Alternative Medicine Definitive Guide to Cancer_ , 963-964.\n\n T. May et al., \"Effect of Fiber Source on Short-Chain Fatty Acid Production and on the Growth and Toxin Production by _Clostridium difficile,\" Scandinavian Journal of Gastroenterology_ 29 (October 1994): 916-22.\n\n R.K. Buddington et al., \"Dietary Supplement of Neosugar Alters the Fecal Flora and Decreases Activities of Some Reductive Enzymes in Human Subjects,\" _American Journal of Clinical Nutrition_ 63 (May 1996): 709-16.\n\n Trenev, Probiotics. Nature's Internal Healers, 126-7.\n\n## _Chapter 6_\n\n W. John Diamond, M.D., W. Lee Cowden, M.D., with Burton Goldberg, _An Alternative Medicine Definitive Guide to Cancer_ (Tiburon, Calif.: Future Medicine Publishing, 1997): 968.\n\n Morton Walker, D.P.M., \"Jumping for Health,\" _Townsend Letter for Doctors_ (July 1995): 42-48.\n\n John Anderson, \"Rebounders\u2014Bounce Your Way to Better Health,\" _Alternative Medicine Digest_ (October-November 1997): 42-46.\n\n A. Bhattacharya et al., \"Body Acceleration Distribution and 02 Uptake in Humans During Running and Jumping,\" _Journal of Applied Physiology_ 49 (November 1980): 881-887.\n\n Walker, \"Jumping for Health,\" 42-48.\n\n H. Erichsen and H. Bottcher, \"Trampoline Therapy with Brain-Injured Children and Adolescents,\" _Rehabilitation_ 15 (May 1976): 100-2.\n\n J.K. Stanghelle et al., \"Effect of Daily Short Bouts of Trampoline Exercise During Eight Weeks on the Pulmonary Function and the Maximal Oxygen Uptake of Children with Cystic Fibrosis,\" _International Journal of Sports Medicine_ Supplement 1 (February 9, 1988): 32-36.\n\n Helene Silver, _Rejuvenate. A 21-Day Natural Detox Plan for Optimal Health_ (Freedom, Calif.: The Crossing Press, 1998): 128.\n\n Farida Sharan, M.D., _Iridology. A Complete Guide to Diagnosing Through the Iris and to Related Forms of Treatment_ (Wellingborough, U.K.: Thorsons Publishing Group, 1989): 141.\n\n W. J. Evans, \"Vitamin E, Vitamin C, and Exercise,\" _American Journal of Clinical Nutrition_ 72 (supp.) (August 2000): 647S-52S.\n\n D.A. Leaf et al, \"The Exercise-Induced Oxidative Stress Paradox: The Effects of Physical Exercise Training,\" _American Journal of Medicine and Science_ 317 (May 1999): 295-300.\n\n M.C. Polidori et al, \"Physical Activity and Oxidative Stress During Aging,\" _International Journal of Sports Medicine_ 21 (April 2000): 154-7.\n\n Bruce Fife, N.D, _The Detox Book. How to Detoxify Your Body to Improve Your Health, Stop Disease and Reverse Aging_ (Colorado Springs, Colo.: HealthWise, 1997): 124.\n\n Electrolytes are substances in the blood, tissue fluids, between the cells, or in the urine that conduct an electrical charge, either negative or positive. Specifically, potassium, magnesium, phosphate, sulfate, bicarbonate, sodium, chloride, and calcium are all electrolytes. These substances provide inorganic chemicals needed to run cellular reactions and to control mechanisms such as the conduction of electrochemical impulses to nerves and muscles. The body also requires adequate stores of electrolytes for crucial enzymatic reactions involved in metabolism.\n\n Saunas come in two modalities: dry, radiant heat and wet heat, or steam. Most of the scientific studies of the benefits of sauna therapy or heat stress have been based on dry heat saunas. Similarly, most detoxification experts who discuss sauna therapy advise dry over wet heat. In a dry sauna, no moisture is introduced into the chamber, while in a wet sauna, moisture is introduced. A wet heat sauna is akin to the traditional Native American sweat lodge, where water is splashed on super-hot rocks inside a tightly sealed chamber to produce steam.\n\n Blood pressure is the force of the blood against the walls of the arteries, veins, capillaries, and heart chambers as it is pumped through the body and as the heart beats. Two measurements are involved: systolic (the heart contracts and pumps blood) and diastolic (the heart rests and fills with blood). The ratio of systolic and diastolic equals blood pressure, as in an average or normative reading of 120\/80.\n\n J. Leppaluoto, M. Tuominen et al, \"Some Cardiovascular and Metabolic Effects of Repeated Sauna Bathing,\" _Acta Phsyiologica Scandinavia_ 128 (September 1986): 77-81.\n\n K. Kauppinen, \"Facts and Fables About Sauna,\" _Annals of the New York Academy of Science_ 813 (March 15, 1997): 654-62.\n\n A. Eisalo and O.J. Luurila, \"The Finnish Sauna and Cardiovascular Diseases,\" _Annals of Clinical Research_ , 20:4 (1988): 267-70.\n\n T.A. Parpalei, L.G. Prokof'eva, and V.G. Obertas, \"The Use of the Sauna for Disease Prevention in the Workers of Enterprises with Chemical and Physical Occupational Hazards,\" _Vrach Delo_ 5 (May 1991): 93-95.\n\n K.V. Sudakov, V.V. Sinitchkin, and A.A. Khasanov, \"Systemic Responses in Man Exposed to Different Heating and Cooling Treatments in a Sauna,\" _Pavlov Journal of Biological Science_ 23 (July-September 1988): 89-94.\n\n W. Dean, \"Effect of Sweating [letter],\" _Journal of the American Medical Association_ 246 (August 7, 1981): 623.\n\n \"Have a Sauna At Home: Healing Benefits from a Plug-In Radiant Heat Sauna,\" _Alternative Medicine Digest_ 17 (March 1997): 72-74.\n\n Zane R. Gard, M.D., and Erma J. Brown, \"Literature Review and Comparison Studies of Sauna\/Hyperthermia in Detoxification,\" _Townsend Letter for Doctors and Patients_ (August\/September 1999): 76-86.\n\n \"Aromatherapy Spa\u2014Detoxify, Relax Muscles, and Enhance Immunity,\" _Alternative Medicine Digest_ 15 (November 1996): 57-58.\n\n Laurel Vukovic, _14-Day Herbal Cleansing_ (Paramus, N.J.: Prentice Hall, 1998): 209.\n\n Joseph Dispenza, _Live Better Longer. The Parcells Center Seven-Step Plan for Health and Longevity_ (San Francisco: Harper San Francisco, 1997): 31-34.\n\n## _Chapter 7_\n\n Biological dentistry stresses the use of nontoxic restoration materials for dental work and focuses on the unrecognized impact that dental toxins and hidden dental infections can have on overall health. Typically, a biological dentist will emphasize the following: the safe removal of mercury amalgam; avoidance or removal of root canals; the investigation of possible jawbone infections (cavitations) as a \"dental focus\" or source of body-wide illness centered in the teeth; and the health-injuring role of misalignment of teeth and jaw structures.\n\n Morton Walker, D.P.M., _Elements of Danger. Protect Yourself Against the Hazards of Modem Dentistry_ (Charlottesville, Va.: Hampton Roads, 2000): 63-65.\n\n Joachim Thomsen, D.D.S., \"The Frequent Involvement of 'Vital' Teeth in Focal Disturbances,\" _American Journal of Acupuncture: Special EAV Issue_ (1989): 94-99.\n\n Richard Leviton, \"Cancer, Miscarriage, and Your Teeth,\" _Alternative Medicine_ (July 1998): 56-62.\n\n Hal Huggins, D.D.S., M.S., \"Dental Toxins. Your Teeth May Be Making You Sick,\" _Alternative Medicine_ (May 1998): 48-54.\n\n Burton Goldberg and the Editors of Alternative Medicine Digest, _Alternative Medicine Guide to Chronic Fatigue, Fibromyalgia, & Environmental Illness_ (Tiburon, Calif.: Future Medicine Publishing, 1998): 191-192.\n\n M.M. van Benschoten, \"Acupoint Energetics of Mercury Toxicity and Amalgam Removal with Case Studies,\" _American Journal of Acupuncture_ 22:3 (1994): 251-262.\n\n Christopher Hussar, D.D.S, D.O, \"No More Chronic Pain,\" _Alternative Medicine Digest_ (November 1996): 30-34.\n\n J.W. Osborne, \"Dental Amalgam and Mercury Vapor Release,\" _Advanced Dental Research_ 6 (September 1992): 135-8.\n\n L.J. Hahn et al, \"Dental 'Silver' Tooth Fillings: A Source of Mercury Exposure Revealed by Whole-Body Image Scan and Tissue Analysis,\" _FASEB Journal_ 3 (1989): 2641-2646. Also Fritz Lorscheider et al, \"Mercury Exposure from 'Silver' Tooth Fillings: Emerging Evidence Questions a Traditional Dental Paradigm,\" _FASEB Journal_ 9 (1995): 504-508.\n\n W. J. Crinnion, \"Environmental Medicine, Part Three: Long Term Effects of Chronic Low-Dose Mercury Exposure,\" _Alternative Medicine Review_ 5 (June 2000): 209-23.\n\n J.W. Reinhardt, \"Side-Effects: Mercury Contribution to Body Burden from Dental Amalgam,\" _Advanced Dental Research_ 6 (September 1992): 110-113.\n\n A. Jokstad, Y. Thomassen et al, \"Dental Amalgam and Mercury,\" _Pharmacological Toxicity_ 70 (April 1992): 308-13.\n\n W. Melillo, \"How Safe is Mercury in Dentistry?\" _The Washington Post Weekly Journal of Medicine, Science, and Society_ , September 1991, 4.\n\n A. Lussi and V. Schoenberg, \"The Mercury Release of Different Amalgams in Vitro,\" _Schweiz Monatsschr Zahnmed_ 101:11 (1991): 1405-8.\n\n H. Lichtenberg, \"Mercury Vapor in the Oral Cavity in Relation to the Number of Amalgam Surfaces and the Classic Symptoms of Chronic Mercury Poisoning,\" _Journal of Orthomolecular Medicine_ 11:2 Second Quarter (1996): 87-94.\n\n Hal Huggins, D.D.S, M.S., and Thomas E. Levy, M.D, J.D, _Uninformed Consent. The Hidden Dangers in Dental Care_ (Charlottesville, Va.: Hampton Roads, 1999): 170-171.\n\n N. Galic et al, \"Dental Amalgam Mercury Exposure in Rats,\" _Biometals_ 12 (September 1999): 227-31.\n\n E. Berdouses et al, \"Mercury Release from Dental Amalgams: An _In Vitro_ Study Under Controlled Chewing and Brushing in an Artificial Mouth,\" _Journal of Dental Research_ 74 (May 1995): 1185-93.\n\n Walker, Elements of Danger, 90.\n\n For an excellent bibliography of published scientific articles documenting the effects of mercury, see \"Our References\" at www.iaomt.org.\n\n R. L. Siblerud and E. Kienholz, \"Evidence that Mercury from Silver Dental Fillings May be an Etiological Factor in Multiple Sclerosis,\" _Science of Total Environment_ 142 (March 15, 1994): 191-205.\n\n R.L. Siblerud, \"A Comparison of Mental Health of Multiple Sclerosis Patients with Silver\/Mercury Dental Fillings and Those with Fillings Removed,\" _Psychol Rep_ 70 (June 1992): 1139-51.\n\n S. Ziff, \"Consolidated Symptom Analysis of 1,569 Patients,\" _BioProbe Newsletter_ 9 (March 1993): 7-8.\n\n Diana Echeverria et al., \"Behavioral Effects of Low-Level Exposure to Elemental Hg Among Dentists,\" _Neurotoxicology and Teratology_ 17 (March-April 1995): 161-168.\n\n R.L. Siblerud, J. Motl, and E. Kienholz, \"Psychometric Evidence that Mercury from Silver Fillings May be an Etiological Factor in Depression, Excessive Anger, and Anxiety,\" _Psychol Rep_ 74 (February 1994): 67-80.\n\n A. Kingman, T. Albertini, and L.J. Brown, \"Mercury Concentrations in Urine and Whole Blood Associated with Amalgam Exposure in a U.S. Military Population,\" _Journal of Dental Research_ 77 (March 1998): 461-71.\n\n Walker, _Elements of Danger_ , 138.\n\n M.J. Vimy et al, \"Maternal-Fetal Distribution of Mercury,\" G. Drasch, \"Public Annoucement 25,\" _Bio-Probe Newsletter_ , January 1994. M.J. Vimy and F.L. Lorscheider, \"Intra-Oral Mercury Released from Dental Amalgams,\" _Journal of Dental Research_ 64:8 (1985); 1069-71.\n\n Huggins and Levy, _Uninformed Consent_ , 172-175.\n\n A.O. Summers et al, \"Mercury Released from Dental 'Silver' Fillings Provoks an Increase in Mercury- and Antibiotic-resistant Bacteria in Oral and Intestinal Floras of Primates,\" _Antimicrobial Agents in Chemotherapy_ 37 (April 1993): 825-34.\n\n _Candida albicans_ is a yeast-like fungus found widely in nature, in the soil, on vegetables and fruits, and in the human body. It is frequently present in small quantities in the intestines and in the vagina. Provided its population is small, _Candida_ is generally not harmful to the human body. A _Candida_ overgrowth, a condition called candidiasis, can become pathogenic and cause allergic reactions throughout the body. These reactions can lead to a wide range of symptoms, including depression, fatigue, weight gain, anxiety, rashes, headaches, and muscle cramping.\n\n Richard Leviton, \"Fibroids and Male Impotence,\" _Alternative Medicine Digest_ , (August-September 1997): 46-53.\n\n The peripheral nervous system (PNS) is a part of the central nervous system and consists of a network of about 93,000 miles of nerves and nerve channels inside the body. The PNS works with the five senses and motor branch of the body to translate sensory information from the outside world into appropriate muscle movements by the body.\n\n L.J. Hahn et al, \"Whole-Body Imaging of the Distribution of Mercury Released from Dental Fillings into Monkey Tissues,\" _FASEB Journal_ 4 (1990): 3256-3260.\n\n The sympathetic nervous system is involved with responding to stress or physical threats to the body. It prepares the body's systems by increasing the heart rate, blood pressure, and muscle tension; it also controls the contraction and expansion of blood vessels; the activity of the connective tissue, a body-wide network; and the voltage (called membrane potential) that exists across the cell wall in every cell. So when a toxic substance starts affecting the sympathetic nervous system, it potentially affects every operation in the body.\n\n Richard Leviton, \"Migraines, Seizures, and Mercury Toxicity,\" _Alternative Medicine Digest_ (January 1998): 60-66.\n\n R.F. Kidd, \"Results of Dental Amalgam Removal and Mercury Detoxification using DMPS and Neural Therapy,\" _Alternative Therapies_ 6 (July 2000): 49-55.\n\n Y. Omura et al., \"Significant Mercury Deposits in Internal organs Following the Removal of Dental Amalgam,\" _Acupuncture Electrotherapy Research_ 21 (April-June 1996): 133-60.\n\n Y. Omura, S.L. Beckman, \"Role of Mercury (Hg) in Resistant Infections and Effective Treatment of _Chlamydia trachomatis_ and Herpes Family Viral Infections (and Potential Treatment for Cancer) by Removing Localized Hg Deposits with Chinese Parsley and Delivering Effective Antibiotics Using Various Drug Uptake Enhancement Methods,\" _Acupuncture Electrotherapy Research_ 20 (August-December 1995): 195-229.\n\n Maya Muir, \"Current Controversies in the Diagnosis and Treatment of Heavy Metal Toxicity,\" _Alternative & Complementary Therapies_ (June 1997), 170-178.\n\n A.L. Miller, \"Dimercaptosuccinic Acid (DMSA), a Non-Toxic, Water-Soluble Treatment for Heavy Metal Toxicity,\" _Alternative Medicine Review_ 3 (June 1998): 199-207.\n\n J. Forman, J. Moline, et al., \"A Cluster of Pediatric Metallic Mercury Exposure Cases Treated with Meso-2,3-Dimercaptosuciccinic Acid (DMSA),\" _Environmental Health Perspectives_ 106 (June 2000): 575-77.\n\n G. Englund Sandborgh et al., \"DMSA Administration to Patients with Alleged Mercury Poisoning from Dental Amalgams: A Placebo-Controlled Study,\" _Journal of Dental Research_ 73 (March 1994): 620-8.\n\n P. Grandjean et al., \"Placebo Response in Environmental Disease: Chelation Therapy of Patients with Symptoms Attributed to Amalgam Fillings,\" _Journal of Occupational Medicine_ 39 (August 1997): 707-14.\n\n H.V. Aposhian, \"Mobilization of Mercury and Arsenic in Humans by Sodium 2,3-Dimercapto-l-Propane Sulfonate (DMPS),\" _Environmental Health Perspectives_ 106, supp. 4 (August 1998): 1017-25.\n\n H.V. Aposhian et al., \"Urinary Mercury after Administration of 2,3-Dimercaptopropane-l-Sulfonic Acid: Correlation with Dental Amalgam Score,\" _FASEB Journal_ 6 (April 1992): 2472-6.\n\n D. Gonzalez-Ramirez et al, \"Sodium 2,3-Dimercaptopropane-l-Sulfonate Challenge Test for Mercury in Humans: II. Urinary Mercury, Porphyrins, and Neurobehavioral Changes of Dental Workers in Monterrey, Mexico,\" _Journal of Pharmacological Experimental Therapy_ 272 (January 1995): 264-74.\n\n R.K. Zalups, \"Influence of 2,3-Dimercaptopropane-l-Sulfonate (DMPS) and Meso-2,3-Dimercaptosuccinic Acid (DMSA) on the Renal Disposition of Mercury in Normal and Uninephrectomized Rats Exposed to Inorganic Mercury,\" _Journal of Pharmacology and Experimental Therapies_ 267 (November 1993): 791-800.\n\n J. Nerudova et al, \"Mobilization of Mercury by DMPS in Occupationally Exposed Workers and in Model Experiments on Rats: Evaluation of Body Burden,\" _International Journal of Occupational Medicine and Environmental Health_ , 13:2 (2000): 131-46.\n\n Hal Huggins, D.D.S, M.S., \"Appointment Scheduling,\" Hugnet FAQs, 1997 www.hugnet.com.\n\n Daniel F. Royal, D.O, \"Health Hazard in Your Teeth,\" _Alternative Medicine Digest_ (July 1996): 40-44.\n\n Huggins and Levy, _Uninformed Consent_ , 212.\n\n Mark A. Breiner, D.D.S, _Whole-Body Dentistry. Discover the Missing Piece to Better Health_ (Fairfield, Conn.: Quantum Health Press, LLC, 1999): 102-106.\n\n Dr. Karen Evans and Dr. Stephen R. Evans, \"Cavitations,\" 1999 by Affinity Labeling Technologies, Inc, 235 Bolivar Street, Lexington, KY 40508; tel: 606-388-9445; fax: 606-388-9645.\n\n P. Schupbach, V. Osterwalder, and B. Guggenheim, \"Human Root Caries: Microbiota of a Limited Number of Root Caries Lesions,\" _Caries Research_ 30:1 (1996): 52-64.\n\n K. Okuda, Y. Ebihara, \"Relationships Between Chronic Oral Infectious Diseases and Systemic Diseases,\" _Bulletin of the Tokyo Dental College_ 39 (August 1998): 165-74.\n\n F.A. Scannapieco, \"Position paper of the American Academy of Periodontology: Periodontal Disease as a Potential Risk Factor for Systemic Diseases,\" _Journal of Periodontology_ 69 (July 1998): 841-50.\n\n W.J. Loesche, \"Association of the Oral Flora with Important Medical Diseases,\" _Current Opinions in Periodontology_ 4 (1997): 21-8.\n\n G.J. Debelian, I. Olsen, and L. Tronstad, \"Systemic Diseases Caused by Oral Microorganisms,\" _Endodontics and Dental Traumatology_ 10 (April 1994): 57-65.\n\n V. Lucas and G.J. Roberts, \"Odontogenic Bacteremia Following Tooth Cleaning Procedures in Children,\" _Pediatric Dentistry_ 22 (March-April 2000): 96-100.\n\n R. Hayrinen-Immonen et al, \"Oral Health of Patients Scheduled for Elective Abdominal Aortic Correction with Prosthesis,\" _EuropeanJournal of Vascular and Endovascular Surgery_ 19 (March 2000): 294-8.\n\n I. Tar, K. Bagyi et al., \"Screening of Patients Referred to Our Clinic for Odontogenic Focal Diseases,\" _Fogorv Sz_ 92 (October 1999): 295-300.\n\n X. Li, L. Tronstad and I. Olsen, \"Brain Abscesses Caused by Oral Infection,\" _Endodontic Dental Traumatology_ 15 (June 1999): 95-101.\n\n B. Bertrand, P. Rombaux et al., \"Sinusitis of Dental Origin,\" _Acta Otorhinolaryngol Belg_ 51:4 (1997): 315-22.\n\n Hubert N. Newman, \"Focal Infection,\" _Journal of Dental Research_ 75:12 (1996): 1912-1919.\n\n J.H. Meurmann, \"Dental Infections and General Health,\" _Quintessence International_ 28 (December 1997): 807-11.\n\n George E. Meinig, D.D.S., F.A.C.D., _Root Canal Cover-up_ (Ojai, Calif.:, Bion Publishing, 1994): 7.\n\n W.A. Price, \"Dental Infections and Related Degenerative Diseases,\" _The Journal of the American Medical Association_ 84 (January 24, 1925): 254-59.\n\n Breiner, _Whole-Body Dentistry_ , 99-100.\n\n M. Georgopoulou, E. Kontakiotis, and M. Nakou, \"In Vitro Evaluation of the Effectiveness of Calcium Hydroxide and Paramonochlorophenol on Anaerobic Bacteria from the Root Canal,\" _Endodontic Dental Traumatology_ 9 (December 1993): 249-53.\n\n N. Olea et al., \"Estrogenicity of Resin-Based Composites and Sealants Used in Dentistry,\" _Environmental Health Perspectives_ 104 (March 1996): 298-305.\n\n H. Oysaed, I.E. Ruyter, T. J. and Sjovik Kleven, \"Release of Formaldehyde from Dental Composites,\" _Journal of Dental Research_ 67 (October 1988): 1289-94.\n\n## _Chapter 8_\n\n Florinda Dormer, _Being-in-Drearning. An Initiation into the Sorceror's World_ (San Francisco: Harper San Francisco, 1991): 126-127.\n\n John Diamond, M.D., \"Reversing Fibroids and Ovarian Cysts,\" _Alternative Medicine Digest_ (January 1998): 98-102.\n\n Norman Levin, M.D., \"Rheumatoid Arthritis and Fibromyalgia,\" _Alternative Medicine Digest_ (September 1997): 100-105.\n\n Candace B. Pert, _Molecules of Emotion. Why You Feel the Way You Feel_ (New York: Scribner, 1997): 187, 189, 192.\n\n Joseph Riccioli, M.D., N.D., \"Talking to Your Body,\" _Alternative Medicine_ (March 1999): 72-80.\n\n Christiane Northrup, M.D., _Women's Bodies, Women's Wisdom. Creating Physical and Emotional Health and Healing_ , rev. ed. (New York: Bantam Books, 1998): 19-21, 55-56.\n\n Carolynn Myss, _Anatomy of the Spirit. The Seven Stages of Power and Healing_ (New York: Harmony Books, 1996): 40.\n\n W. John Diamond, M.D., and W. Lee Cowden, M.D., with Burton Goldberg _An Alternative Medicine Definitive Guide to Cancer_ (Tiburon, Calif.: Future Medicine Publishing, 1997): 616.\n\n Dr. Arthur Janov, _Why You Get Sick and How You Get Well. The Healing Power of Feelings_ (West Hollywood, Calif.: Dove Books, 1996) 5, 6, 8, 16, 141-143.\n\n Others have followed suit to expand the original concept of flower essences. Today, an estimated 20 different brands of flower remedies, based on plants native to many landscapes, from Australia to India to Alaska, offer about 1,500 different blends for a diverse range of psychological conditions. Innovative remedies are now made from gems and crystals, certain types of sea creatures, even the essence of glaciers.\n\n Patricia Kaminski and Richard Katz, _Flower Essence Repertory. A Comprehensive Guide to North American and English Flower Essences for Emotional and Spiritual Well-Being_ , rev. ed.( Nevada City, Calif.: The Flower Essence Society, 1994): 26.\n\n Patricia Kaminski, _Flowers That Heal. How to Use Flower Essences_ (Dublin, Ireland: Newleaf\/Gill & Macmillan, Inc., 1998): 27, 30, 104.\n\n Kaminski and Katz, _Flower Essence Repertory_ , 329.\n\n## _Chapter 9_\n\n Willy Schr\u00f6dter, quoting Henry Cornelius Agrippa, in _History of Energy Transference: Exploring the Foundations of Modem Healing_ (York Beach, Maine: Samuel Weiser, 1999): 115.\n\n Willy Schr\u00f6dter, quoting Dr. Friedrich Markus Huebner, in _History of Energy Transference_ , 116.\n\n Willy Schrodter, quoting Hiro Hasegawa, in _History of Energy Transference_ , 107.\n\n Ruth Berger, _The Secret is in the Rainbow: Aura Interrelationships_ (York Beach, Maine: Samuel Weiser, 1979): 47-57.\n\n Walter J. Kilner, _The Human Aura_ (Secaucus, N.J.: The Citadel Press, 1965): 160.\n\n John C. Pierrakos, M.D. _Core Energetics. Developing the Capacity to Love and Heal_ (Mendocino, Calif.: Life Rhythm, 1990): 73-74, 91.\n\n Shafica Karagulla, M.D., and Dora van Gelder Kunz, _The Chakras and the Human Energy Fields_ (Wheaton, Ill.: Quest Books, 1989): 207-208.\n\n Shafica Karagulla, M.D., _Breakthrough to Creativity. Your Higher Sense Perception_ (Santa Monica, Calif.: DeVorss & Company, 1967): 123-146.\n\n Shakuntala Modi, M.D., _Remarkable Healings. A Psychiatrist Discovers Unsuspected Roots of Mental and Physical Rlness_ (Charlottesville, Va.: Hampton Roads, 1997): 295, 297, 334-336, 357-361.\n\n Arthur Guirdham, _The Psyche in Medicine_ (Jersey, U.K.: Neville Spearman, 1978): 44, 48, 73, 223.\n\n Samuel Sagan, M.D., _Entity Possession. Freeing the Energy Body of Negative Influences_ (Rochester, Vt.: Destiny Books, 1997).\n\n William J. Baldwin, D.D.S., _Spirit Releasement Therapy. A Technique Manual_ , 2d ed. (Terra Alta, W.Va.: Headline Books, 1992): 13, 15.\n\n Modi, _Remarkable Healings_ , 353.\n\n Adapted from suggestions by Shakuntala Modi, _Remarkable Healings_ , 290-291.\n\n The Dead Sea in Israel is the world's lowest point, at 293 meters (approximately 900 feet) below sea level. It is considered a unique source of concentrated natural minerals and salts, which have been used over the centuries for cosmetic and therapeutic purposes.\n\n Information provided by the manufacturer, Ashtar Natural Dead Sea Products; website: www.fjcc.com\/amig\/ashtar.\n\n William Bloom, _Psychic Protection: Creating Positive Energies for People and Places_ (London: Judy Piatkus Publishers, 1996): 70-72.\n\n According to occult scholar Willy Schrodter, the nape of the neck, where the medulla oblongata passes through the occiput (at the base of the skull and the top of the neck), is important in occultism. It is a place of strong and easily detected psychic radiation, and is known by different names: _Uls_ (ancient Egyptian magicians); _Lus_ (Jewish Qabalists); \"divine mouth\" (Indian Kriya yoga); \"and the exit-point of a hypothetical subtle meta-organism,\" says Schrodter. When you sense people behind you and it makes you uncomfortable, your psychic gate at the nape of the neck is registering an energy input from this other source. In Schrodter, _History of Energy Transference_ , 31. Another writer, Peter Dawkins, calls this site the eighth or alta major or Pan chakra, and says it is associated with universal consciousness. \"It acts as a control chakra and integrates the powers and qualities of all the other chakras. It is the primary gateway for divine inspiration . . . and is perhaps the most sensitive area of the whole body.\" In: Peter Dawkins, _Zoence\u2014The Science of Life. Rediscovering the Sacred Spaces of Your Life_ (York Beach, Maine: Samuel Weiser, 1998): 72.\n\n Judy Hall, _The Art of Psychic Protection_ (Forres, Scotland: Findhom Press, 1996): 97-105.\n\n Irene Dalichow and Mike Booth, _Aura-Soma. Healing Through Color, Plant, and Crystal Energy_ (Carlsbad, Calif.:, Hay House, 1996): 23-24.\n\n Heinrich Zimmer, _Philosophies of India_ , ed. Joseph Campbell (Princeton: Princeton University Press\/Bollingen Series XXVI, 1951): 231,251.\n\n Mona Rolfe, _The Sacred Vessel_ (Sudbury, England: Neville Spearman Limited, 1978): 59-62.\n\n## _Chapter 10_\n\n \"Indoor Air Facts No. 4 (Revised): Sick Building Syndrome (SBS),\" Office of Radiation and Indoor Air, Environmental Protection Agency, April 1991 www.epa.gov\/iaq\/pubs\/sbs.html.\n\n C.A. Redlich, J. Sparer, and M.R. Cullen, \"Sick-Building Syndrome,\" _The Lancet_ 349:9057 (1997): 1013-16.\n\n Adrian Bejan, \"A Cure for Sick Buildings,\" 1994 www.tc.cornell.edu\/er\/sci93\/dis04sickb\/dis04sickb.html.\n\n J.E. Woods, G.M. Drewry and PR. Morey, \"Office Worker Perceptions of Indoor Air Quality Effects on Discomfort and Performance.\" In B. Seifert et al (eds): _Indoor Air '87. Proceedings of the 4th International Conference on Indoor Air Quality and Climate._ Berlin, Institute for Water, Soil and Air Hygiene, 1987.\n\n \"Bad Office Air, Not Bad Vibes, May Cause Many Symptoms of Sick Building Syndrome, a New Cornell Study Finds,\" Cornell University News, 24 Feb. 1998 www.news.cornell.edu\/releases\/Feb98\/sick.building.ssl.html.\n\n O.A. Seppanen, W.J. Fisk, et. al., \"Association of Ventilation Rates and CO2 concentrations with Health and Other Responses in Commercial and Institutional Buildings,\" _Indoor Air_ 9 (December 1999): 226-52.\n\n Michael Hodgson, M.D., M.P.H., \"The Medical Evaluation\" and \"The Sick Building Syndrome: Effects of the Indoor Environment on Health.\" Cited in _Occupational Medicine: State of the Art Reviews_ 10 (January-March 1995): 167-194.\n\n P. Skov, O. Valbjorn: Danish Indoor Climate Study Group. \"The 'Sick' Building Syndrome in the Office Environment: The Danish Town HaH Study,\" _Environ Int_ 13 (1987): 339-349.\n\n S. Burge et al. \"Sick Building Syndrome: A Study of 4373 Office Workers,\" _Annals of Occupational Hygiene_ 31 (1987): 493-504.\n\n P.O. Fanger, et al., \"Air Pollution Sources in Offices and Assembly Halls, Quantified by the Olf Unit,\" _Energy Building_ , 12 (1988):7-19.\n\n PL. Ooi, K.T. et al., \"Epidemiology of Sick Building Syndrome and Its Associated Risk Factors in Singapore,\" _Occupational Environmental Medicine_ 55 (March 1998): 188-193.\n\n PL. Ooi and K.T. Goh, \"Sick Building Syndrome: An Emerging Stress-Related Disorder?\" _International Journal of Epidemiology_ 26 (December 1997): 1243-9.\n\n L. Lundin, \"Allergic and Non Allergic Students' Perceptions of the Same High School Environment,\" _Indoor Air_ 9 (June 1999): 92-102.\n\n E. Bjornsson et al., \"Symptoms Associated to the Sick Building Syndrome in a General Population Sample: Associations with Atopy, Bronchial Hyper-Responsiveness and Anxiety,\" _International Journal of Tubercular Lung Disease_ 2 (December 1998): 1023-28.\n\n R.A Phipps, W.E. Sisk, and G.L. Wall, \"A Comparison of Two Studies Reporting the Prevalence of the Sick Building Syndrome in New Zealand and England,\" _New Zealand Medical Journal_ 112 (June 25, 1999): 228-30.\n\n Arnold Mann, \"This Place Makes Me Sick. Modern, Airtight Offices are Causing More Cases of Sick-Building Syndrome, Just Ask Southwest Airlines.\" _Time_ , December 21, 1998, 152-25.\n\n M. Bullinger, M. Morfeld et al., \"The Sick-Building Syndrome\u2014Do Women Suffer More?\" _Zentralbl Hyg Umweltmed_ 202 (August 1999): 235-41.\n\n Michael J. Hodgson, M.D., M.P.H., \"Clinical Diagnosis and Management of Building-Related Illness and the Sick-Building Syndrome,\" _Occupational Medicine: State of the Art Reviews_ 4 (October-December 1989): 602-603.\n\n Burton Goldberg and the Editors of _Alternative Medicine Digest, Alternative Medicine Guide to Chronic Fatigue & Environmental Illness_ (Tiburon, Calif.: Future Medicine Publishing, 1998): 251-255.\n\n \"Is Your Building Sick?\" _TFM on-line, Today's Facility Manager_ 18 July 1999 www.tfmgr.com.\n\n An encouraging trend is found in the fact that the International Institute for Bau-biologie has established an affiliation with the University of Natural Medicine of Santa Fe, New Mexico. This will enable students of alternative medicine to include Bau-biological principles in their understanding of health and healing, and thus gain an additional therapeutic edge in treating clients. For more information, contact: The University of Natural Medicine, P.O. Box 4089, Santa Fe, NM 87502; e-mail:natmedu@aol.com; tel: 505-424-7800 or 800-8933367; fax: 505-424-7878; website: www.unaturalmedicine.edu.\n\n Richard W. Pressinger, M.Ed., \"Environmental Causes of Learning Disabilities and Child Neurological Disorders,\" University of South Florida, 1997 www.chem-tox.com.\n\n Kaye H. Kilburn, M.D., and John C. Thornton, \"Protracted Neurotoxicity from Chlordane Sprayed to Kill Termites,\" _Environmental Health Perspectives_ 103 (July-August 1995): 690-694.\n\n K.H. Kilburn, \"Chlordane as a Neurotoxin in Humans,\" _Southern Medical Journal_ 90 (March 1997): 299-304.\n\n Comments by John and Lynn Marie Brower derived from their website www.hhinst.com.\n\n An electromagnetic field (EMF) is created by the interaction of an electric field (surrounding an electric charge) and a magnetic field (surrounding a source of magnetism), when an electric current passes through a wire. Put slightly differently, an EMF is a biologically active but invisible energy field that accompanies electrical current flow. For ease of layperson understanding, you could think of an EMF as the energy aura emitted by the interaction of the electric and magnetic fields, and which surrounds the carrier of the electric current, such as a wire, appliance, electric panel, power line, etc. Electric fields are found around electric typewriters, lamps, digital clocks, heaters, and water pipes, among other objects.\n\n Lucinda Grant, in _The Electrical Sensitivity Handbook: How Electromagnetic Fields Are Making People Sick_ , Weldon Pub., 1995, reproduced in part at www.northlink.com\/~lgrant\/whatises.html. For further information (and a six-issue\/year newsletter called _Electrical Sensitivity News')_ , contact: Electrical Sensitivity Network, P.O. Box 4146, Prescott, AZ 86302.\n\n Lynn Marie Bower, quoting data provided by the Electric Power Research Institute study, in _The Healthy Household: A Complete Guide for Creating a Healthy Indoor Environment_ (Bloomington: The Healthy House Institute, 1995): 383-384.\n\n Ellen Sugarman, quoting J.R. Guager, \"Household Appliance Magnetic Field Survey. IEEE Transactions on Power Apparatus and Systems,\" PA-104 (September 1985), in _Warning: The Electricity Around You May Be Hazardous to Your Health_ (New York: Simon & Schuster, 1992) and reprinted at WaveGuide website www.wave-guide.org\/exposure.html.\n\n B.W. Wilson, \"Chronic Exposure to ELF Fields May Induce Depression,\" _Bioelectromagnetics_ 9:2 (1988): 195-205.\n\n Data review provided by WaveGuide's \"What Are the Studies Telling Us?\" www.wave-guide.org\/ccwti\/studies.html.\n\n A free radical is an unstable, toxic molecule of oxygen with an unpaired electron that steals an electron from another molecule and produces harmful effects. Free radicals are formed when molecules within cells react with oxygen (oxidize) as part of normal metabolic processes. Free radicals then begin to break down cells, especially the cell membranes, often in a matter of minutes to an hour. A single free radical can destroy a cell. Their work is enhanced if there are not enough free-radical quenching nutrients, such as vitamins C and E, in the cell. While free radicals are normal products of metabolism, uncontrolled free-radical production plays a major role in the development of degenerative disease, including cancer and heart disease. Free radicals harmfully alter important molecules, such as proteins, enzymes, fats, even DNA. Other sources of free radicals include pesticides, industrial pollutants, smoking, alcohol, viruses, most infections, allergies, stress, even certain foods, and excessive exercise.\n\n Veronica Strong, \"Biosensing to Counter Geopathic Stress,\" www.positivehealth.com\/permit\/Articles\/Environment\/strong33.html.\n\n Jeffrey R. Cram, \"Flower Essences Reduce Stress Reaction to Intense Environmental Stimulus,\" Flower Essence Society, 5 February 1999 www.flowersociety.org\/DrZ.htm.\n\n \"Carpeting, Indoor Air Quality, and the Environment,\" _Environmental Building News_ November\/December 1994 www.environmentalbuilding.com\/Archives\/Features\/Carpets\/Carpets.html.\n\n John Bower, \"Healthy Construction Recommendations for Health People\" (speech presented at the Energy Efficient Building Association Excellence in Housing Conference, Dallas, TX, February 1994) published at www.hhinst.com.\n\n David Pearson, _The Natural House Catalog: Everything You Need to Create an Environmentally Friendly Home_ (New York: Fireside\/Simon & Schuster, 1996): 141.\n\n Rosalind C. Anderson, \"Toxic Emissions from Carpets,\" _Journal of Nutritional and Environmental Medicine_ , 5:4 (1995): 375-386.\n\n Robert Abrams, \"Consumer Alert: Chemicals in New Carpets Pose Potential Health Hazard,\" April 1991, cited at www.holisticmed.com\/carpet\/tcl.txt.\n\n Debra Lynn Dadd, _Home Safe Home: Protecting Yourself and Your Family from Everyday Toxics and Harmful Household Products_ (New York: Jeremy P. Tarcher\/Putnam, 1997): 316-319.\n\n Merritt McKinney, \"Home Carpets Catch & Hold Benzene from Car Exhaust,\" 9 March 2000 www.sightings.com.\n\n Robert G. Lewis et al., \"Distribution of Pesticides and Polycyclic Aromatic Hydrocarbons in House Dust as a Function of Particle Size,\" _Environmental Health Perspectives_ 107 (September 1999): 721-726.\n\n Cindy Duehring, \"Carpet Concerns, Part Four: Physicians Speak Up as Medical Evidence Mounts,\" and \"Toxic Carpeting: Important Points to Remember,\" Environmental Access Research Network, 1995 www.holisticmed.com\/carpet\/tc4.txt.\n\n Mauiy M. Beecher, and Shirley Linde, _Healthy Homes in a Toxic World: Preventing, Identifying, and Eliminating Hidden Health Hazards in Your Home_ (New York: John Wiley & Sons, 1992): 160-166.\n\n For a comprehensive source of information about MCS and numerous sources of toxicity, as well as legal and political aspects, contact: Chemical Injury Information Network (CIIN), publishers of the monthly newsletter, _Our Toxic Times._ P.O. Box 301, White Sulphur Springs, MT 59645; tel: 406-547-2255; fax: 406-547-2455; website: ciin.org\/newsletter.htm. CIIN was founded in 1990 as an advocacy organization run by chemically injured persons primarily for the benefit of other chemically injured individuals. It is considered a major clearinghouse for information on the adverse effects of chemical exposures.\n\n Fibromyalgia is a multiple-symptom syndrome primarily involving widespread muscle pain (myalgia), which can be debilitating in its severity. The pain seems to be caused by the tightening and thickening of the myofascia, the thin film or tissue that holds the muscle together. Typical tender sites include the neck, upper back, rib cage, hips, and knees. Other symptoms include general fatigue and stiffness, insomnia and sleeping disorders, anxiety, depression, mood swings, allergies, carpal tunnel syndrome, headaches, the sense of \"hurting all over,\" tender skin, numbness, irritable bowel symptoms, dizziness, and exercise intolerance. Post-traumatic fibromyalgia is believed to develop after a fall, whiplash, or back strain, whereas primary fibromyalgia has an uncertain origin. The majority of fibromyalgia sufferers are women between the ages of thirty-four and fifty-six.\n\n Peter Montague, \"A New Mechanism of Disease,\" _Rachel's Environment & Health Weekly_, February 12, 1998.\n\n William R. Rea, M.D. et al, \"Considerations for the Diagnosis of Chemical Sensitivity,\" in D.W. Talmage et al.: _Biologic Markers in Immunotoxicology_ (Washington, D.C.: National Academy Press, 1992): 169. See also Thomas Orne and Paul Benedetti, \"Multiple Chemical Sensitivity,\" American Council on Science and Health. 1991 www.hcrc.org\/contrib\/acsh\/booklets\/mcsdoc.html.\n\n J.S. Tepper, V.C. Moser et al., \"Toxicological and Chemical Evaluation of Emissions from Carpet Samples,\" _American Industrial Hygiene Association Journal_ 56 (February 1995): 158-70.\n\n D.L. Ashley, M.A. Bonin et al., \"Measurement of Volatile Organic Compounds in Human Blood,\" _Environmental Health Perspectives_ 104, supp. 5 (October 1996): 871-77.\n\n Research from multiple medical sources collated by Albert Donnay, executive director, MCS Referral & Resources, July 1998 www.mcsrr.org\/factsheets\/mcsdisorders.html. For more information: MCS Referral & Resources, 508 Westgate Road, Baltimore, MD 21229; tel: 410-362-6400; fax: 410-362-6401.\n\n Albert Donnay, \"Overlapping Disorders: Chronic Fatigue Syndrome, Fibromyalgia Syndrome, Multiple Chemical Sensitivity & Gulf War Syndrome,\" MCS Referral & Resources, 29 May 1997 www.mcsrr.org\/factsheets\/mcsdisorders.html.\n\n Julius Anderson, \"Reactions to Carpet Emissions: A Case Series,\" _Journal of Nutritional and Environmental Medicine_ 7 (1997): 177-185.\n\n Bake-outs have been tested on commercial buildings. It has been found that the best results are obtained when the interior is heated to at least 90\u00b0F and kept there for several days (about three days); a one-day bake-out is usually not long enough to release the maximum VOCs possible through this technique. Even so, overall release of VOCs is not necessarily high. The average obtained over several tests was twenty percent to thirty percent (based on measuring fifteen to twenty of the most abundant indoor VOCs), but one building achieved a ninety-four percent reduction, and another had a sixty percent reduction of concentrations; another had an initial VOC decrease of sixty-five percent one day after the bake-out, but this had declined to only six percent after a month. See John R. Girman, \"Volatile Organic Compounds and Building Bake-Out,\" in _Occupational Medicine_ \u2014 _State of the Art Reviews 4_ (October-December 1989): 695-712.\n\n Information Technology Specialists, Inc. REED (Residential Energy Efficiency Database), Site 4, Box 16, RR1, Carvel, Alberta, Canada TOE OHO; tel: 780-829-3594; fax: 780-892-3598; e-mail: info@its-canada.com; website: www.its-canada.com\/reed.\n\n John Bower, _The Healthy House_ (Bloomington, Ind.: The Healthy House Institute, 1997): 536.\n\n Annie Berthold-Bond, _Better Basics for the Home_ (New York: Three Rivers Press, 1999): 52-54.\n\n T.J. Kelly et al., \"Emission Rates of Formaldehyde from Materials and Consumer Products Found in California Homes,\" _Environmental Science & Technology_ 33 (January 1, 1999): 81.\n\n Hal Levin, \"Building Materials and Indoor Air Quality,\" in _Occupational Medicine\u2014State of the Art Reviews 4_ (October-December 1989): 667-693.\n\n John R. Girman, \"Volatile Organic Compounds and Building Bake-Out,\" in _Occupational Medicine\u2014State of the Art Reviews 4_ (October-December 1989): 695-712.\n\n D. B. Teculescu et al., \"Sick-Building Symptoms in Office Workers in Northeastern France: A Pilot Study,\" _Int Arch Occup Environmental Health_ 71 (July 1998): 353-56.\n\n J. Bourbeau et al., \"Prevalence of the Sick Building Syndrome Symptoms in Office Workers Before and After Being Exposed to a Building with an Improved Ventilation System,\" _Occupational Environmental Medicine_ 53 (March 1996): 204-210.\n\n C.S. Li, C.W. Hsu, and M.L. Tai, \"Indoor Pollution and Sick Building Syndrome Symptoms Among Workers in Day-Care Centers,\" _Archives of Environmental Health_ 52 (May-June 1997): 200-207.\n\n D. Vincent et al, \"Ventilation System, Indoor Air Quality, and Health Outcomes in Parisian Modern Office Workers,\" _Environmental Research_ 75 (November 1997): 100-12.\n\n B. Thriene et al., \"Man-Made Mineral Fibre Boards in Buildings\u2014Health Risks Caused by Quality Deficiencies,\" _Toxicology Letters_ 88 (November 1996): 299-303.\n\n G.H. Wan and C.S. Li, \"Dampness and Airway Inflammation and Systemic Symptoms in Office Building Workers,\" _Archives of Environmental Health_ 54 (January-February 1999): 58-63.\n\n J.J. McGrath et al., \"Continually Measured Fungal Profiles in Sick Building Syndrome,\" _Current Microbiology_ 38 (January 1999): 33-36.\n\n J.D. Cooley, et al., \"Correlation Between the Prevalence of Certain Fungi and Sick Building Syndrome,\" _Occupational Environmental Medicine_ 55 (September 1998): 579-84.\n\n D. Menzies et al., \"Effect of a New Ventilation System on Health and Well-Being of Office Workers,\" _Archives of Environmental Health_ 52 (September\/October 1997): 360-367.\n\n \"Fungi Suspected 'Culprit' in 'Sick Building Syndrome,\" Georgia Tech Alumni Association www.alumni.gatech.edu\/news\/topics\/sum96\/RealFungus.html.\n\n W.J. Kowalski and William Bahnfleth, \"Airborne Respiratory Diseases and Mechanical Systems for Control of Microbes,\" Pennsylvania State University, Architectural Engineering Department, July 1988 www.engr.psu.edu\/www\/dept\/arc\/server\/wjk\/ardtie.html.\n\n \"Molds in the Environment,\" Centers for Disease Control and Prevention-National Center for Environmental Health Factsheet, 3 April 1997 www.cdc.gov\/nceh\/pubcatns\/facts\/molds\/molds.htm.\n\n Eckardt Johanning, M.D., \"Hazardous Molds in Homes and Offices: Stachybotrys atra and Others ...\" Enviro Village Library, Acumen Technologies, 1997 www.envirovillage.com\/Papers.\n\n \"Fungi Suspected 'Culprit' in 'Sick Building Syndrome,\"' Georgia Tech Alumni Association www.alumni.gatech.edu\/news\/topics\/sum96\/RealFungus.html.\n\n W.J. Kowalski, and William Bahnfleth, Ph.D., P.E., \"Airborne Respiratory Diseases and Mechanical Systems for Control of Microbes,\" Pennsylvania State University, Architectural Engineering Department July 1998 www.engr.psu.edu\/www\/dept\/arc\/server\/wjk\/ardtie.html.\n\n \"Negative Ions: Vitamins of the Air?\" Don Strachan and Jim Karnstedt, _New Realities_ , reprinted on Library-Electrocorp website: www.net-gain.com\/electrocorp.\n\n Research data compiled as \"Medical Studies\" by The IonAir Company; website: www.purifyonline.com\/studies_chart.html.\n\n Virginia Peart, \"Indoor Air Quality in Florida: Houseplants to Fight Pollution,\" Fact Sheet HE 3208, Florida Cooperative Extension Service, University of Florida, April 1993.\n\n B.C. Wolverton, \"Removal of Formaldehyde from Sealed Experimental Chambers by Azalea, Poinsettia, and Dieffenbachia,\" Research Report No. WES\/100\/01-91\/005 January 1991, published by Wolverton Environmental Products, 514 Pine Grove Road, Picayune, MS 39466; website: www.wolvertonenvironmental.com.\n\n Data on trichloroethylene and benzene air filtration from Plants for Clean Air Council, 3458 Godspeed Road, Davidsonville, MD 21035; fax: 410-956-9039; website: www.plants4cleanair.org.\n\n Data drawn from these sources: \"Clean Air Plants & Sick Building Syndrome,\" interiorlandscape.com, 1997. W. Prescod, \"More Indoor Plants as Air Purifiers,\" _Pappus_ 11:4 (1992). United States Environmental Protection Agency, \"Sick Building Syndrome,\" Air and Radiation, Indoor Air Facts, 4, 1991. B.C. Wolverton, _InteriorLandscape Plants and Their Role in Improving Indoor Air Quality_ (Picayune, Miss.:, Wolverton Environmental Services, 1990).\n\n B.C. Wolverton, Ph.D. and John Wolverton, \"Improving Indoor Air Quality Using Orchids and Bromeliads,\" Wolverton Environmental Services (514 Pine Grove Road, Picayune, MS 39466) December 1991 www.wolvertonenvironmental.com.\n\n \"Hydroculture: The Cure for 'Sick Building Syndrome,'\" and from information published by Inter Urban Water Farms Online (3638 University Avenue, Suite 225, Riverside, CA 92501; tel: 909-342-7947; fax: 909-342-7984. This is a good source for information about hydroponics.\n\n B.C. Wolverton and John Wolverton, \"Interior Plants and Their Role in Indoor Air Quality: An Overview,\" Research Report # WES\/100\/06-92-008, Wolverton Environmental Services. C. Wolverton and John Wolverton, \"Plants and Soil Microorganisms: Removal of Formaldehyde, Xylene, and Ammonia from the Indoor Environment,\" _Journal of the Mississippi Academy of Sciences_ 38 (AugustSeptember 1993). Virginia I. Lohr and Caroline H. Pearson-Mims, \"Particulate Matter Accumulation on Horizontal Surfaces in Interiors: Influence of Foliage Plants,\" _Atmospheric Environment_ 30:14 (1996).\n\n \"The Breathing Wall from Genetron,\" _TFM On-line (Today's Facility Manager)_ , 6 September 1999 www.tfmgr.com.\n\n Bower, _The Healthy House_ , 501.\n\n \"Biological Effects of Ionizing Radiation (BEIR) VT Report: 'The Health Effects of Exposure to Indoor Radon,' Executive Summary,\" The National Academy of Sciences, 19 February 1998 www.epa.gov\/iaq\/radon\/beirvil.html.\n\n Melatonin, a hormone produced by the pea-sized, light-sensitive pineal gland in the center of the brain, regulates the body's internal clock, or circadian rhythm, which determines the twenty-four-hour sleep-wake cycle. Low melatonin levels have been associated with sleeping disturbances and light-related conditions such as SAD. With aging, the peak in melatonin secretion is about one hour later than normal (normal peak secretion time is about 2 a.m.), and the maximum peak of melatonin is only one-half the level of young adults.\n\n Nicholas Harmon, \"Lose Those Winter Blues with Verilux's Newest Full Spectrum Lighting,\" press release, 4 November 198 www.ergolight.com.\n\n \"SAD Info,\" Light Therapy Products; website: www.lighttherapyproducts.com.\n\n Overuse of light therapy is possible and may produce side effects if overused for four to six weeks. The symptoms include irritability, agitation, eye strain, and fatigue. About one percent of users are prone to mania from consistent overuse.\n\n Alfred J. Lewy, M.D., \"Treating Chronobiologie Sleep and Mood Disorders with Bright Light,\" _Psychiatric Annals_ 17 (October 1987).\n\n \"Tests Prove Children Do Better in School with Full Spectrum, Visually-Efficient Lighting,\" press release, Verilux, 13 May 1998 www.ergolight.com.\n\n John Downing, O.D., Ph.D., F.C.S.O., \"Clinical EEG and Neurophysiological Case Studies in Ocular Light Therapy,\" in _Light Years Ahead: The Illustrated Guide to Full Spectrum and Colored Light in Mindbody Healing_ , ed. Brian J. Breiling (Berkeley: Celestial Arts,1996): 133-164.\n\n Sheri Lundstrom, \"Light Therapy a 'Natural prozac' for Winter Depression, _Twin City Wellness_ , November 1997, reprinted at www.lighttherapyproducts.com.\n\n _The Seventh Generation Guide to a Toxic-Free Home_ (Burlington, Vt. Seventh Generation, n.d.): 11.\n\n \"Just the FAQs: Answers to Frequently Asked Questions about Petroleum and Household Cleaners,\" Seventh Generation, 16 June 1998 www.seventhgen.com\/petrol.htm.\n\n Julia Kendall, \"Chemicals Found in Fabric Softeners by U.S. Environmental Protection Agency (EPA),\" 1995 www.immuneweb.org\/articles\/fabricsoftener.html.\n\n Julia Kendall, \"Twenty Most Common Chemicals in Thirty-One Fragrance Products Based on a ] 1991 EPA Study,\" Environmental Health Network (n.d.) [www.users.lmi.net\/wilworks.\n\n Julia Kendall, \"Making Sense of Scents,\" collated scientific research (n.d.) www.ourlittleplace.com\/scents.html.\n\n Damon Franz and Holly Prall, \"Smelling Good but Feeling Bad,\" _E Magazine_ , XI (January-February 2000).\n\n Debra Lynn Dadd, _Home Safe Home: Protecting Yourself and Your Family from Everyday Toxics and Harmful Household Products_ (New York: Jeremy P. Tarcher\/Putnam, 1997): 281-282.\n\n## _Chapter 11_\n\n Larry Sang, _The Principles of Feng Shui_ , 2d ed. (Monterey Park, Calif.: American Feng Shui Institute, 1996). For more information about feng shui training, class schedules, and books, contact: American Feng Shui Institute, 108 North Ynez, Suite 202, Monterey Park, CA 91754; tel: 626-571-2757; fax: 626-571-2065; e-mail: fsinfo@amfengshui.com; website: www.amfengshui.com.\n\n Thomas Lee, \"Temporal Location Theory, Kan Yu (Feng Shui)\u2014An Ancient Chinese Theory on Site Location\" (paper presented at Geoinformatics '95 conference, The Association of Chinese Professionals in Geographic Information System, Hong Kong, 1995), published at <http:\/\/home.hkstar.com\/~starvsn\/fsgeoinf.html>.\n\n Thomas Lee, \"Kan Yu\u2014The Book of Change Concept in Environmental and Architectural Planning\" (paper presented at \"Greening to the Blue\" conference, Yale University, New Haven, 1996), published at <http:\/\/home.hkstar.com\/~starvsn\/kyyale.html>.\n\n Sophia Tang Shaul and Chris Shaul, \"Why Feng Shui?\" 168 Feng Shui Advisors website, June 1998 www.168fengshui.com.\n\n Here are more examples of the yin\/yang polarity. Yin: passive, cold, death, winter, female, night, even numbers, moon, water; yang: active, hot, life, summer, male, day, odd numbers, sun, fire.\n\n In Chinese philosophy, the five elements are wood, fire, earth, metal, and water. In contrast, the Western philosophical tradition originally accorded five also, but differently: earth, water, fire, air, and ether (for space); later, the fifth element, ether, was relegated to the metaphysicians while the chemists tended to deal with only the first four. The Chinese elements have to be understood with a fair degree of \"poetic license,\" in the sense that they do not mean that, literally, trees and branches are the wood element, or that steel and iron are the sole expression of the metal element. The Chinese elements of wood and metal partake in some degree of the qualities of the Western elements of ether and air. The Chinese elements are assigned many qualities, including direction, time of year, yin or yang shadings, and their own trigram.\n\n Sophia Tang Shaul and Chris Shaul, \"The Five Elements, Part One,\" 168 Feng Shui Advisors website, 1999 www.168fengshui.com.\n\n Terah Kathryn Collins, _The Western Guide to Feng Shui_ (Carlsbad, Calif.: Hay House, 1996): 63.\n\n Stanley Aaga Bartlett, \"Feng Shui for Lightworkers: Feng Shui and the Spirit of Change,\" _Planet Lightworker_ , 1999 www.planetlight-worker.com.\n\n Ibid.\n\n Willy Schrodter, quoting Eira Hellberg in _History of Energy Transference_ (York Beach, Maine: Samuel Weiser, 1999): 118.\n\n Willy Schrodter, quoting Paul Sedir, in _History of Energy Transference_ , 120.\n\n Carol Bridges, \"The Bones of Your Home,\" in _The Feng Shui Anthology: Contemporary Earth Design_ , ed. Jami Lin (Miami: Earth Design, Inc., 1997): 400.\n\n Karen Kingston, _Creating Sacred Space with Feng Shui_ (New York: Broadway Books, 1997): 30.\n\n Schr\u00f6dter, History of Energy Transference, 124.\n\n Carol Bridges, \"The Bones of Your Home,\" 401.\n\n Stanley Bartlett, \"Feng Shui for Lightworkers: Part III, Getting Rid of Density and Creating Light Centers,\" _Planet Lightworker, 1999_ www.planetlightworker.com.\n\n Kingston, _Creating Sacred Space with Feng Shui_ , 95-98.\n\n Angel Thompson, _Feng Shui. How to Achieve the Most Harmonious Arrangement of Your Home and Office_ (New York: St. Martins Griffin, 1996): 34.\n\n Lillian Too, _Feng Shui Fundamentals: Health_ (Rockport, Mass.: Element Books, 1997): 47.\n\n Nancilee Wydra, _Feng Shui. The Book of Cures_ (Lincolnwood, III.: Contemporary Books, 1996): 134.\n\n Simon Brown, _Practical Feng Shui_ (London: Ward Lock, 1997): 62.\n\n Thompson, _Feng Shui_ , 83.\n\n Sarah Rossbach, _Interior Design with Feng Shui_ (New York: Arkana\/Penguin Books, 1991): 21-22.\n\n Thompson, _Feng Shui_ , 38-39.\n\n Collins, _The Western Guide to Feng Shui_ , 44.\n\n Stanley Bartlett, \"Feng Shui for Lightworkers: Part II, Practical\/Physical Aspects of Feng Shui,\" _Planet Lightworker_ , 1999 www.planetlightworker.com.\n\n Collins, _The Western Guide to Feng Shui_ , 41.\n\n Kirsten M. Lagatree, _Feng Shui. Arranging Your Home to Change Your Life_ (New York: Villard Books, 1996): 29.\n\n Rossbach, _Interior Design with Feng Shui_ , 84.\n\n Lagatree, _Feng Shui. Arranging Your Home to Change Your Life_ , 40.\n\n The flat spiral dielectric resonator is available through Stanley Bartlett.\n\n Thompson, _Feng Shui_ , 101.\n\n Collins, _The Western Guide to Feng Shui_ , 82-86.\n\n Richard Webster, _101 Feng Shui Tips for the Home_ (St. Paul, Minn.: Llewellyn Publications, 1998): 93-97.\n\n George Birdsall, _The Feng Shui Companion: A User-Friendly Guide to the Ancient Art of Placement_ (Rochester, Vt.: Destiny Books, 1997): 117-118.\n\n The reader should bear in mind that feng shui encompasses a great deal more than what I have surveyed in this chapter. It has a more complex model of energy influences, timing, and location, and a copious amount of practical solutions for all aspects of our environment. What I've presented in this chapter, is merely a pr\u00e9cis of what is possible, in terms both of analysis and remedy, as well as the range of thinking available from practitioners.\n\n Jenny Liu, \"Sick House, Sick People,\" 1998 www.spiritweb.org.\n\n## _Chapter 12_\n\n Jane Thumell-Read, _Geopathic Stress. How Earth Energies Affect Our Lives_ (Rockport, Mass.: Element Books, 1995): 2.\n\n Information provided by Dulwich Health at www.dulwichhealth.uk.com.\n\n Ionizing radiation consists of high-energy rays capable of taking electrons out of molecules and forming ions in the substances through which it passes, thereby producing genetic mutations that can lead to cancer. Examples include X-rays, mammography, and other nonnuclear medical sources such as radiation therapy, used in conventional cancer treatment, and food irradiation, used to sterilize foods. An ion is an atom, or group of atoms, that has a negative or positive charge _as a result_ of having lost or gained one or more electrons. Ionizing radiation can break atomic bonds and affect chromosomes, thereby making gene changes. Nonionizing radiation does not possess enough energy to remove electrons from atoms. This makes it unable to alter genes.\n\n Gustav Freiherr von Pohl, _Earth Currents. Causative Factor of Cancer and Other Diseases_ (Stuttgart, Germany: Frech-Verlag, 1983): 11, 12, 13, 22, 33.\n\n J.M. Gobet, \"Geobiology\u2014The Holistic House,\" (n.d.) www.earthtransitions.com\/articles\/geobiology.htm.\n\n Serotonin is a neurotransmitter involved in many key body functions (and can produce problems when its levels are off), such as hunger and appetite regulation, sleep induction, cardiovascular activity, motor activity, respiration, control of body temperature, perception and moods and mood disorders (depression). Deficiencies or imbalances in serotonin levels are associated with violence, aggressive behavior, suicide, schizophrenia, and Parkinson's, among other health problems.\n\n Melatonin, a hormone produced by the pea-sized, light-sensitive pineal (pronounced pie-NEEL) gland in the center of the brain, regulates the body's internal clock, or circadian rhythm, which determines the twenty-four-hour sleep-wake cycle. With aging, the peak in melatonin secretion is about one hour later than normal (normal peak secretion time is about 2 a.m.), and the maximum peak of melatonin is only one-half the level of young adults. Low melatonin levels have been associated with sleeping disturbances and light-related conditions such as seasonal affective disorder (SAD). Eating vitamin- and mineral-rich foods and increasing your exposure to bright light can improve the body's natural melatonin production.\n\n Alf Riggs, \"Harmful Effects from Earth Radiation & Electrical Fields,\" <http:\/\/whale.to\/Earth_Radiation\/Riggsl.html>.\n\n Hans Nieper, M.D., \"Modern Medical Cancer Therapy Following the Decline of Toxic Chemotherapy,\" _Townsend Letter for Doctors & Patients_, November 1996, 88-89. _Dr. Nieper's Revolution in Technology, Medicine and Society_ (MIT, Oldenburg, 1985): 206, 222. Lecture notes from Professional Medical Seminar, Los Angeles, California, July 4, 1986, 13-15, 22, 28 (From A. Keith Brewer International Science Library, Richland Center, WI; tel: 608-647-6513; fax: 608-647-6797; e-mail: drbrewer@mwt.net; website: www.mwt.net\/~drbrewer\/other.htm). See also Hans Alfred Nieper, et al., _The Curious Man: The Life and Works of Dr. Hans Nieper_ (Garden City Park, N.Y.: Avery Publishing Group, 1998).\n\n Anthony Scott-Morley, \"Geopathic Stress: The Reason Why Therapies Fail?\" _Journal of Alternative Medicine_ , May 1985.\n\n D. Freshwater, \"Geopathic Stress,\" _Complement Ther. Nurs. Midwifery_ 3 (December 1997): 160-2.\n\n Kathe Bachler, \"Noxious Earth Energies and Their Influence on Human Beings\" (talk given in 1987), published at <http:\/\/whale.to\/Earth_Radiation\/Bachler.htm>. See also Kathe Bachler, _Discoveries of a Dowser_ (n.p.: Veritas, 1981) and _Earth Radiation_ (Manchester, U.K.: Wordmasters, Ltd., 1989).\n\n In some senses, the presence of a geopathogenic zone and what it dictates for bedroom arrangement may have to take priority over what a feng shui analysis reveals about the mouth of _qi_ and the optimal flow of _qi_ through a bedroom. It is possible of course, that when you move the bed out of the toxic zone you also move it into the healthy _qi_ zone. It is also possible to consult a professional who is skilled in both disciplines.\n\n Christopher Bird, quoting Herbert Douglas, in _The Divining Hand. The 500-Year-Old Mystery of Dowsing_ (Black Mountain, N.C.: New Age Press, 1979): 268-269.\n\n Bird, _The Divining Hand_ , 273.\n\n Alf Riggs, \"Myalgic Encephalomyelitis\" <http:\/\/www.simnet.is\/vgv\/jardarur.htm>.\n\n Thurnell-Read, _Geopathic Stress_ , 63-64.\n\n Tom Passey and Robert D. Egby, \"Is the Place Where You Sleep Making You Sick?\" <http:\/\/members.spree.com\/achievers\/pagell.htm>. Egby is a professional \"hypnoanalyst\" and director of the Center for Inner Healing and Meditation Studies. P.O. Box 1494, Hightstown, NJ 08520; tel: 609-581-2415.\n\n Chuck Pettis, _Secrets of Sacred Space: Discover and Create Places of Power_ (St. Paul, Minn.: Llewellyn Publications, 2000): 87.\n\n Samuel Sagan, M.D., _Entity Possession. Freeing the Energy Body of Negative Influences_ (Rochester, Vt.: Destiny Books, 1997), 98,106-7. See his website at: www.clairvision.org. Dr. Sagan's Clairvision School in Sydney, Australia, offers courses in developing sensitivity to land energies and energy qualities of homes.\n\n Dr. Sagan recommends leaving a food offering for the bhutas you are removing. It is well known that entities have food cravings\u2014usually for sugar, chocolate, heavy fried foods, and meats\u2014and inspire or compel their hosts to overconsume these foods. In his practice of bhuta-suddhi (\"entity purification\" or cleansing of the elemental energy layer, the one just beyond the physical), he puts out a sugar cube or two, offering it to the entities; at the end of the ritual, he wraps the cube (or sweet, if he used something else sugary) in a leaf and buries it. A certain portion of the energy essence (or at least the debris remaining from the ritual clearing) of the entity adheres to the sugar and is thus removed from the scene.\n\n Patrick MacManaway, \"A Definition of Geomancy,\" MAG E-zine, Mid-Atlantic Geomancy, Winter Solstice 1996 www.geomancy.org\/ezines\/ezine_4\/ezine_4c.html.\n\n In Dr. MacManaway's usage of the term a ley line is a narrow track of dowsable Earth energy that usually runs in straight lines and upon which in Europe old churches are oriented. Ley lines are not the same thing as a geopathogenic zone, although they can have strong effects on one's state of mind and to a lesser extent, upon one's physiology. Within circles of dowsers, there is disagreement whether a ley line is the track of subtle telluric energy (more recent definition), or is merely a straight line of alignment between ancient structures, such as old churches, standing stones, and other megalithic structures, but one which has no discernible energy (old definition). \"Pre-Reformation English and European churches will always have an energy ley line running down their long axis, which is the same pattern as you find in temple space in much of the world,\" says Dr. MacManaway.\n\n Ernest Hartmann, M.D., (1915-1992) practiced medicine in Eberbach, Germany, for over forty years. In 1961, he created the Institute for Geobiology and served for almost thirty years as its chairman.\n\n Manfred Curry, M.D., lived from 1899-1953.\n\n David Cowan and Rodney Girdlestone, _Safe as Houses? Ill Health and Electro-Stress in the Home_ (Bath, England: Gateway Books, 1996): 124.\n\n Blanche Merz, _Points of Cosmic Energy_ (Saffron Walden, England: C.W. Daniel Company, 1987): 17-20.\n\n J.M. Gobet, \"Geobiology\u2014The Holistic House,\" (n.d.) www.earthtransitions.com\/articles\/geobiology.htm.\n\n Reshad Field, _Here to Heal_ (Shaftesbury, U.K.: Element Books, 1985): 81, 88-89.\n\n Joan McFarlane, \"Dowsing Geopathogenic, Electromagnetic, and Other Irritation Zones\" <http:\/\/members.tripod.com\/Reid_J\/Joan.htm>.\n\n Mary K. Kuhner, \"Useful Ritual Techniques,\" 22 August 1996, posted on soc.religion.paganism (internet newsgroup), University of Washington at Seattle, by mkkuhner@phylo.genetics.washington.edu.\n\n William Bloom, _Psychic Protection. Creating Positive Energies for People and Places_ (London, England: Judy Piatkus Publishers, 1996): 65-66.\n\n Rescue Remedy is a formula originally developed in the 1930s by English physician Edward Bach. It contains the essences of five flowers, including Star of Bethlehem (for trauma and shock), clematis (for a tendency to pass out into unconsciousness), cherry plum (for being on the verge of a breakdown), impatiens (for irritability and tension), and rock rose (for frozen terror and panic). The combination produces a first aid effect on an overly charged emotional body. Rescue Remedy is available in the United States from Belson Bach USA, 100 Research Drive, Wilmington, MA 01887; tel: 978-988-3833; fax: 508988-0223. For more about Bach flower remedies, see these websites: www.bachflower.com; www.nelsonbach.com; www.bachcentre.com. The same formula is also available as Five Flower Formula, made by Flower Essence Society, RO. Box 459, Nevada City, CA 95959; tel: 800736-9222 or 530-265-9163; fax: 530-265-0584; e-mail: mail@flowersociety.org; website: www.flowersociety.org.\n\n William Bloom, _Psychic Protection. Creating Positive Energies for People and Places_ (London: Judy Piatkus Publishers, 1996): 126-7.\n\n Bloom, _Psychic Protection_ , 16.\n\n From information provided by Geomack Products under \"Alternative Health Practitioners\" www.geomack.force9.co.uk\/geo0.7.html.\n\n Samuel Sagan, M.D., _Awakening the Third Eye_ (Roseville, Australia: Clairvision, 1997): 173-180.\n\n# _Index_\n\n4-phenylcyclohexene, ,\n\n_9th Report on Carcinogens, 2000_ ,\n\nAAL Reference Laboratories, Inc.,\n\nAce Pump Corp.,\n\nacetone, ,\n\nN-acetyl-cysteine,\n\n_Acremonium spp_.,\n\nactivated charcoal, \u2013\n\nactivated intermediates,\n\nAffinity Labelling Technologies,\n\naging, \u2013\n\nAgrippa, Henry Cornelius,\n\nAir Check, Inc.,\n\nair conditioning, \u2013\n\nair filters, \u2013\n\nair pollution\n\nillness correlated with, \u2013\n\nmajor pollutants, ,\n\nmortality rates and, \u2013\n\n_See also_ indoor air\n\nAir Quality Research,\n\nairborne pesticides,\n\nAirChek,\n\nAireox Activated Carbon Air Purifier,\n\nAirFilters.net,\n\nAlaska Northern Lights,\n\nalbumin, ,\n\naldehyde detoxification pathway, \u2013\n\nalginate, \u2013\n\naliphatic hydrocarbon solvents,\n\nallergies\n\nCandida albicans and,\n\nfood\n\neliminating, \u2013\n\nidentifying,\n\nsymptoms of, \u2013\n\ntests for, \u2013\n\nwheat,\n\nfrom air pollution,\n\nfrom genetically modified foods,\n\ntriggers of,\n\nalpha-lipoic acid,\n\nalpha-terpineol,\n\nALS (amyotrophic lateral sclerosis),\n\n_Alternaria_ , ,\n\nAlzheimer's disease, , ,\n\nAmelung, Wolfgang,\n\nAmerican Academy of Biological Dentistry,\n\nAmerican cheese,\n\nAmerican Institute of Reboundology,\n\nAmerican Society of Dowsers,\n\namino acid conjugation,\n\namino acids, , ,\n\nAmjo Corp.,\n\nAmjo Sunrise light boxes,\n\nammonia, ,\n\namyotrophic lateral scierosis (ALS),\n\nanaphylaxis,\n\nAnderson Laboratories,\n\nAnderson, Richard,\n\nAnderson, Rosalind, ,\n\nandrogens,\n\nanemia,\n\nangina,\n\nanti-idiotope allergen,\n\nantibiotic resistance, , \u2013\n\nantibiotics in livestock feed, \u2013\n\nantimony, test for, \u2013\n\nantioxidants\n\naction of,\n\nas cleansing support,\n\nblood test for,\n\ndefined, ,\n\nnutrient,\n\nplant,\n\nanxiety reduction, \u2013, \u2013\n\nApollo Light Systems,\n\nAqua-Stream shower water filter, ,\n\nArctic, pollution in, ,\n\nArise & Shine Cleanse Thyself program, \u2013\n\naromaSpa, \u2013\n\naromatherapy,\n\narsenic\n\neffects on body, , n46\n\nlevels considered safe,\n\nlevels in drinking water,\n\ntest for, \u2013\n\narthritis\n\ndental focal infection and,\n\nemotional problems and, \u2013\n\ngeopathic stress and,\n\ntrigger of,\n\nAshtar Natural Dead Sea Products,\n\naspartame (NutraSweet), ,\n\n_Aspergillus_ , , ,\n\naspirin,\n\nasthma, , , \u2013,\n\natrazine, \u2013\n\nAura-Soma International Academy of Colour Therapeutics,\n\nAura-Soma Products, Ltd.,\n\nAura-Soma USA Inc.,\n\nauras\n\ndamaged, \u2013, \u2013\n\ndetecting toxins in,\n\ndetoxification methods\n\nrequesting a cleansing, \u2013\n\nsalting energy centers, \u2013\n\nusing pomanders, \u2013\n\nusing psychic methods, \u2013\n\nentities and, \u2013\n\nill-health and, \u2013\n\nnature and characteristics of, \u2013, \u2013\n\nof the unhealthy, \u2013\n\nprofessional detoxification, \u2013\n\nprotecting, \u2013\n\nautoimmune disease.\n\nautomobile emissions, , n55\n\nautonomic nervous system, n113\n\nAyurveda, \u2013, On51\n\nazaleas,\n\nB-vitamins, test for, \u2013\n\n_Ba-Gua_ , \u2013\n\nbaby food, contaminated,\n\nBach, Edward,\n\nBach Flower Remedies,\n\nBachler, Kathe,\n\nbacteria\n\nantibiotic-resistant,\n\nsizes of,\n\nbagels,\n\nbake-outs, , n54\n\nBaldwin, William J., ,\n\nBallentine, Rudolf,\n\nBarrie, Stephen, , ,\n\nBartlett Designs,\n\nBartlett, Stanley Aaga, \u2013\n\nBassett Aromatherapy,\n\nbathrooms, \u2013\n\nbaths\n\nbody-detoxifying, \u2013\n\nchlorine from, \u2013\n\nrelaxation,\n\nspirit-detoxifying, \u2013, \u2013\n\nBatmanghelidj, F.,\n\nBau-biologie, \u2013,\n\nBDCM (bromodich1oromethane),\n\nbedding, \u2013\n\nbedrooms, \u2013\n\nbeds, \u2013, ,\n\nbeech remedy,\n\nbeets, \u2013\n\nBefit Enterprises,\n\nBell, Iris,\n\nBennett, Peter, , ,\n\nbentonite, \u2013,\n\nbenzaldehyde,\n\nbenzene\n\nfrom carpets,\n\nhealth problems associated with,\n\nincidence in humans,\n\nplants which absorb, ,\n\nsources of,\n\nsymptoms produced by,\n\nbenzoate(s), ,\n\nbenzopyrene,\n\nbenzyl acetate,\n\nbenzyl alcohol,\n\nbenzyl butyl phthalate,\n\nberberine, \u2013\n\nBerger, Ruth,\n\nBergsmann, Otto,\n\nBerkeley Psychic Institute,\n\nBernard, Claude,\n\nBerthold-Bond, Annie, ,\n\nBeta-BHC (benzene hexachloride),\n\nbeta-carotene, test for,\n\nBieler's broth,\n\n_Bifidobacterium bifidum_ , ,\n\nbile, n3\n\nbile lubricants,\n\nbilirubin, test for,\n\nBio-Fighter Anti-Microbial UV Light System,\n\nBioCalex, \u2013\n\nbiochemical defenses, test for,\n\nBioDemocracy and Organic Consumers Assoc., ,\n\nbiodetoxification,\n\nBiofeedback Instrument Corporation,\n\nbiological dentistry, n1\n\nBiological Technologies International, Inc.,\n\nBiological Terrain Assessment, , \u2013,\n\nBioProbe, Inc.,\n\nbiotransformation,\n\nbirds,\n\nBirdsall, George,\n\nbismuth, test for, \u2013\n\nbitter melon, \u2013\n\nblackberry remedy,\n\nbladder cancer, , ,\n\n_Blastocystis hominus_ ,\n\nBlaylock, Russell L.,\n\nbleeding heart remedy,\n\nblood levels of toxins, mothers and infants,\n\nblood pressure, n16\n\nblood tests\n\ncomparisons with known patterns\/standards,\n\nnutrient and toxic element, \u2013\n\nBloom, William, ,\n\nBock, Kenneth,\n\nborage remedy,\n\nboron, test for, \u2013\n\nbotulinum,\n\nbovine growth hormone (BGH), \u2013\n\nbowel movement frequency,\n\nBower, John,\n\nBower, Lynn Marie, , ,\n\nbrain cancer, ,\n\nbrain symptoms,\n\nBraly, James, M.D.,\n\nbread, cracked wheat,\n\nbreast cancer\n\nBaby Boomer mortality,\n\ncase of, \u2013\n\ndental problems and,\n\nnuclear counties mortality,\n\norganochlorine compounds and, ,\n\nXcray-caused,\n\nbreast-feeding,\n\nbreast milk,\n\nBreathing Wall, \u2013\n\nBreiner, Mark A.,\n\nBRI. _See_ building-related illness\n\nBridges, Carol,\n\nbright light therapy, , n91\n\nBrite Lite IV light box,\n\nbromodichloromethane (BDCM), ,\n\nbronchitis,\n\nBrown, Simon,\n\nBrowrung, John M.,\n\nbuilding-related illness (BRI)\n\ndefined,\n\nsymptoms of, I\n\n_See also_ indoor air; sick building syndrome\n\nbulimia case,\n\nburdock, \u2013\n\nbutter,\n\nbuttercup remedy,\n\nbutyl cellosolve,\n\ncadmium, test for, \u2013\n\ncalcium, test for, \u2013\n\ncalendula remedy,\n\nCalifornia wild rose remedy, ,\n\ncamphor,\n\ncancer\n\nchildhood,\n\ndaily development of,\n\ngeopathie stress and, \u2013,\n\nrates. and nuclear radiation, \u2013\n\ntriggers of,\n\n_See also_ specific types of cancer\n\n_Candida albicans_\n\nand intestinal dysbiosis,\n\nallergies and,\n\ncleansing toxins from, \u2013\n\ndefined, n32\n\nFOS and,\n\nCanyon Dudleya remedy,\n\nCarbon Based Corporation,\n\ncarbon dioxide, , ,\n\ncarbon monoxide, , , \u2013\n\ncarbon tetrachloride,\n\ncarcinogens, \u2013\n\nCare 2000 Air Purifier, \u2013\n\ncannine dye,\n\ncarpets\n\n'bake-outs' of new,\n\nbenzene from,\n\ncleaning,\n\ndangers of new, , \u2013\n\nEnvironmental Protection Agency,\n\ngas emissions from, \u2013\n\nhouse dust and,\n\nneurotoxicity of,\n\n'new carpet' smell,\n\n01d,\n\nsub-acute reactions to,\n\nsymptoms in humans from, \u2013\n\nsymptoms in mice from,\n\ntesting for toxicity,\n\nuse in U.S., \u2013\n\nwool, \u2013\n\ncarrots,\n\nCarson, Rachel L., _xxi_\n\ncascara sagrada, \u2013\n\ncataracts, ,\n\ncats,\n\ncauliflower mosaic virus, \u2013\n\ncavitations\n\ndangers of, \u2013, \u2013\n\ndebridement of, \u2013\n\ndefined, ,\n\nincidence of,\n\nCayce, Edgar,\n\nCDSA (Comprehensive Digestive Stool Analysis), ,\n\nCellMate Blood Test Report, ,\n\nCenter for Food Safety, ,\n\ncentral nervous system (CNS) disorders, , \u2013\n\nchakras,\n\ncharcoal water filtration,\n\ncheddar cheese,\n\ncheeseburgers,\n\nchemical sensitivity. _See_ multiple chemical sensitivities (MCS)\n\nchemicals\n\ncommercially used,\n\nhousehold, logging exposure to, \u2013\n\nindustrial, , ,\n\n'organic,' \u2013\n\nChernobyl, ,\n\nchickens,\n\nchildren\n\nair pollution and, \u2013\n\nbehavior problems,\n\nDMSA administration to,\n\ngeopathic stress and, \u2013\n\ninfants\n\natrazine in,\n\nDBCP in,\n\ndioxin in,\n\nhypothyroidism in,\n\n_L. acidophilus_ for,\n\nunderweight newborns,\n\nneurological disorders in, \u2013\n\nsexual maturation, \u2013\n\nsusceptibility of,\n\nchloral hydrate, \u2013\n\nchlordane\n\namounts in home building,\n\neffects of, , , \u2013\n\nfoods found in,\n\nincidence in foods,\n\nlevels in homes,\n\nchlordecone, \u2013\n\nchlorella, \u2013\n\nchlorinated hydrocarbon pesticides,\n\nchlorine, \u2013, , n68\n\nchlorine absorption by the body, \u2013\n\nchlorine chemistry,\n\nchlorine industry, n72\n\nchloroform, , , , ,\n\nchloropyrifos,\n\ncholesterol, , \u2013\n\ncholinesterase,\n\nchronic fatigue syndrome\n\ncase of,\n\nchemicals associated with,\n\ncoincident with fibromyaIgia, \u2013,\n\ncoincident with MCS, \u2013,\n\ngeopathic stress and, \u2013,\n\nMCS and, \u2013\n\nmercury and,\n\ntoxicity and,\n\ncilantro,\n\ncirculating immune complexes (CIC), ,\n\n_Cladosporium_ ,\n\nClarus Products International,\n\nClean Air Plant System, \u2013\n\ncleansing\n\nauras\n\nrequesting a cleansing, \u2013\n\nsalting energy centers, \u2013\n\nusing pomanders, \u2013\n\nusing psychic methods, \u2013\n\nintestinal, \u2013\n\nwhole body, \u2013\n\n_See also_ liver cleans ing; detoxification\n\nclear cell cancer,\n\nClearWave digital clock,\n\nClifford Consulting and Research,\n\nClifford Materials Reactivity Testing, \u2013\n\nclofibric acid, \u2013\n\nClorox, \u2013\n\nclutter, \u2013\n\ncochineal extract,\n\nCody, Pierre,\n\ncoffee,\n\nColburn, Theo, \u2013, , \u2013\n\nColin Campbell & Sons Ltd.,\n\nCollins, Terah Kathryn, ,\n\ncolon. _See_ intestinal tract\n\ncolon cancer\n\nchlorinated water and,\n\ndiet associated with, \u2013\n\nfecal free radicals and, \u2013\n\nfiber-rich diets preventing,\n\nslow transit times and, \u2013\n\nTHM levels and, l48\n\ncolon cleansers, \u2013\n\ncolonies,\n\nCommittee for Nuclear Responsibility,\n\nComprehensive Digestive Stool Analysis, ,\n\nComprehensive Parasitology Profile, \u2013,\n\nComtech Research,\n\nComtech Research IG-033A ionizer,\n\nconjugation,\n\nconsensus reality,\n\nconstipation, ,\n\nconsurner products\n\nadvice on,\n\nassessing,\n\nchemicals used in, \u2013\n\ncommon toxic, \u2013\n\npotentially toxic, ,\n\ncopper toxicity,\n\ncore pictures, \u2013,\n\nCorsello, Serafina,\n\nCram, Jeffrey R.,\n\ncramps,\n\nCrirmion, Walter J., N.D., , ,\n\ncruciferous vegetables, \u2013, \u2013\n\n_Cryptosporidia_ , ,\n\n_Cryptosporidium parvum_ , n7\n\nCrystal Enterprises,\n\nCulturelle, ,\n\nCurry-Hartmann Grid, \u2013\n\nCuny, Manfred,\n\ncutting _qi_ , \u2013\n\ncytochrome P450\n\ndefined, \u2013\n\nfunction of,\n\nnutrients needed by,\n\nvulnerability to mercury,\n\nCzech Republic,\n\nDadd, Debra Lynn, , ,\n\ndairy products, \u2013\n\nDAMS (Dental Amalgam Mercury Syndrome),\n\ndandelion root,\n\nDaughton, Christina G.,\n\ndawn simulators, \u2013\n\nDBCP (dibromochloropropane),\n\nDDE\n\n(dichlorodiphenyldichloroethylene), ,\n\nDDT (dichlorodiphenyltrichloroethane)\n\nimmune damage from, \u2013\n\nin children,\n\nin fish, ,\n\nin foods, \u2013\n\nincidence in chemically sensitive patients,\n\nspread of,\n\ndeath rates\n\nfrom chemical exposures,\n\ninfectious disease, \u2013\n\ndehydration, \u2013\n\ndental\n\ncomposites,\n\nfluorosis, \u2013\n\nfoci, \u2013\n\ninfections, n8\n\nDental Amalgam Mercury Syndrome (DAMS),\n\ndental material biocompatibility, \u2013\n\ndental mercury amalgam fi1lings\n\nalloy composition,\n\nannual use,\n\nantibiotic-resistant bacteria and, \u2013\n\ncontroversy over, \u2013,\n\nmercury levels per number of, \u2013\n\nmercury release rates from,\n\nmultiple scleros is patients with, \u2013\n\noutgassing from, \u2013\n\nremoval of\n\nassessing need for, \u2013\n\ncase study of, \u2013\n\ncautions during,\n\nincorrect,\n\nnutritional support for,\n\npatient satisfaction after,\n\nproblems prompting,\n\nsymptom reduction after, \u2013\n\ntiming of,\n\ntoxicity of, \u2013\n\nurinary mercury per number of,\n\ndentists, , , \u2013\n\ndepression\n\ncase of entities and, \u2013\n\nelectromagnetic fields and,\n\nflower essences for, \u2013\n\nnature of,\n\ntoxicity and,\n\ntrans-fatty acids and,\n\nDES (diethylstilbestrol),\n\ndetoxification\n\naura\n\nrequesting a cleansing, \u2013\n\nsalting energy centers, \u2013\n\nusing pomanders, \u2013\n\nusing psychic methods, \u2013\n\nbenefits of, \u2013, \u2013, \u2013\n\nde-fined, _xxii-xxiii_ , \u2013,\n\nglobal consequences of, \u2013\n\nimportance of, _xxv_ ,\n\nliver as primary organ for,\n\nnutrients supporting, \u2013\n\nphases of, \u2013\n\nshaking for, \u2013\n\nspiritual, \u2013, \u2013\n\nwhole body, \u2013\n\n_See also_ colon cleansers; liver cleansing\n\nDetoxification Profile,\n\ndetoxification program components, \u2013\n\ndetoxification system, rnaladaptation of, \u2013\n\nDevelopmental Natural Resources,\n\ndevelopmental neurotoxicants, \u2013\n\nDiamond, John,\n\ndiarrhea,\n\ndibutyl phthalate,\n\ndieldrin\n\nbasics of, n37\n\nbreast cancer and,\n\ndaily dietary intake of,\n\nresidues in foods,\n\n_Dientamoeba fragilis_ ,\n\ndiesel engine exhaust,\n\ndiet\n\nassessing,\n\nhigh-fiber, ,\n\nlow-fiber, \u2013\n\nmeat, \u2013\n\ndiethyl phthalate,\n\ndigestive tract symptoms,\n\ndill pickles,\n\ndioxin, , \u2013,\n\ndizziness,\n\nDMPS (2,3-dimercaptopropane-l-sulfonic acid), , \u2013, \u2013\n\nDMSA (dimercaptosuccinic acid), , ,\n\nDNA repair enzymes, test for,\n\ndoctors,\n\nDoctor's Data, ,\n\nDonner, Florinda,\n\nDouglas, Herbert,\n\nDowning, John,\n\ndowsing, therapeutic, \u2013\n\nDr. Gauss,\n\nDragon River Herbals, ,\n\ndrainage pipes, energy\n\nhomes to earth, \u2013\n\npeople to earth, \u2013\n\ndrinking water\n\nchlorinated\n\ncarcinogenic effects, \u2013\n\norganochlorine byproducts from \u2013, \u2013\n\nreproductive problems and, \u2013\n\nusage of,\n\nconsumption of,\n\nfluoridated\n\ncancer and, \u2013\n\ncavity reduction and, \u2013\n\n**The Healthy Living Space**\n\ndental fluorosis from, \u2013\n\nextent of, \u2013\n\ninception of,\n\nlead levels and,\n\ntypical fluoride concentrations,\n\npersonal care product residues in, \u2013\n\nphannaceutical drug residues in, \u2013\n\nuntreated, VOCs in, n5\n\n_See also_ water (tap)\n\ndry cleaning solvent case, \u2013\n\nDuehring, Cindy,\n\nDulwich Health,\n\nDursban,\n\ndust,\n\nDust Fighter  fi1ter, \u2013\n\nDust Free,\n\ndysbiosis\n\ndefined, \u2013, , n9\n\nproblems from, , ,\n\ndyspepsia, \u2013\n\near symptoms of toxicity,\n\nEarth acupuncture, \u2013\n\nEarth Essentials,\n\nEarth Partnership, Inc.,\n\nEarthrise Nutritionals, Inc.,\n\nEast Germany,\n\nEdelson, Stephen B., M.D., \u2013\n\nEgby, Robert D.,\n\nEI (environmental illness), \u2013, \u2013\n\nElectrocorp,\n\nelectrodermal screening (EDS), \u2013, n4\n\nelectrolytes, , n14\n\nelectromagnetic field, n26\n\nelectropollution\n\naction of, \u2013\n\nbody-shielding devices, \u2013\n\ncases involving, \u2013\n\ndelecting,\n\nexposure reduction devices, \u2013\n\nfield strengths per source,\n\nhot spots,\n\nlow-frequency,\n\nmeasuring devices,\n\nsymptoms due to, ,\n\nElectroStatic Air Filter,\n\nelectrostatic air filters, \u2013\n\nE.L.F.-Zone Gauss Meter,\n\nemotional symptoms of toxicity,\n\nemotions, toxic,\n\nendocrine system\n\ndisruptors, n36\n\nglands in,\n\nsymptoms of damage to, \u2013\n\nendometriosis,\n\nEndosulfan, ,\n\nendotoxins,\n\nEnergeia, \u2013\n\nenergy cords\n\ncutting, \u2013\n\ndefined, \u2013\n\nenergy level symptoms,\n\nenergy refreshing, \u2013\n\nenergy toxins, \u2013, \u2013\n\nenergy wells, \u2013\n\n_Entamoeba histolytica_ ,\n\nentities\n\ncases of, \u2013\n\ndefined, \u2013\n\nfood cravings of, n21\n\npoints of residence, ,\n\nsymptoms of, \u2013\n\nenvironment, toxins in, \u2013,\n\nenvironmental estrogens\n\nconcentrations in body,\n\ndefined,\n\neffects of, , \u2013\n\nfetal susceptibilities to, \u2013,\n\nillnesses linked with, \u2013\n\nmarine life affected by,\n\nmultiplicity of, \u2013\n\nsources of, ,\n\nEnvironmental Health Center, , n14\n\nEnvironmental Health Information Service, \u2013\n\nEnvironmental Health Network,\n\nenvironmental illness (EI), \u2013, \u2013\n\nEnvironmental Lighting Concepts,\n\nEnvironmental Management, Inc.,\n\nenvironmental medicine,\n\nenzymatic activity test, \u2013\n\nenzymes\n\ndefined,\n\nliver, action on toxins,\n\nmixed function oxidative, \u2013\n\nepilepsy,\n\nEpsom salt, , , n108\n\nEpste in, Samuel,\n\nestrogen,\n\nestrogenicity,\n\nethyl acetate, ,\n\nEverpure, Inc.,\n\nEverpure water filters, ,\n\nEWG (Environmental Working Group),\n\nexcitotoxins, \u2013\n\nexercise, \u2013\n\nexotoxins,\n\nexpelling mudra,\n\nextracellular connective tissue matrix,\n\neye symptoms,\n\nfacial paralysis,\n\nfacility managers, and SBS,\n\nFagan, John B.,\n\nFarin, Jacob, ,\n\nfasting\n\nbenefits of,\n\nfrequency of,\n\nmental, \u2013\n\nprecautions, \u2013\n\nsome drinks for, \u2013\n\nwater,\n\nfatty acids\n\nbasics of, n5\n\ndefined,\n\nfunctions of,\n\nsaturated, \u2013\n\ntest for, \u2013\n\nunsaturated,\n\nfear and flower essence therapy, \u2013\n\nfeces\n\nfree radical generation in, \u2013\n\ntests of, \u2013,\n\nFeldman, Jay,\n\nfeng shui\n\nbasics of, \u2013, , n37\n\nconf1icts with geopathogenic zones,\n\nevaluation tool for, \u2013\n\nFeng Shui Warehouse,\n\nfertilizers, toxic components of,\n\nfetuses, \u2013, ,\n\nFIA (Functional Intracellular Analysis), ,\n\nfibroids, uterine, ,\n\nfibromyalgia\n\nchronic fatigue and, \u2013,\n\ndefined, n46\n\nemotional problems and,\n\nMCS and, \u2013, \u2013,\n\npesticides and,\n\nFife, Bruce, ,\n\nFinlandia Sauna, \u2013,\n\nfish, ,\n\nFit vegetable wash, \u2013\n\nfive elements, n6\n\nFive Flower Formula remedy, \u2013, n33\n\nflatulence,\n\nFlotation REST, \u2013, n112\n\nFlower Essence Society, ,\n\nflower essence therapy\n\nbasics of, \u2013, \u2013, \u2013\n\ncase study, \u2013\n\nemotion-essences summary,\n\nfor anxiety\/fear, \u2013\n\nfor depression, \u2013\n\nfor EMF stress, \u2013\n\nfor hostility, \u2013\n\nfor stress\/tension, \u2013\n\nfluoridated water. _See_ drinking water, fluoridated\n\nfluoride\n\nadult daily intake of,\n\nill effects of, \u2013\n\nindustrial emissions, , n36\n\nremoving from tap water, \u2013\n\nsource of, n35\n\n_See also_ drinking water, fluoridated\n\nFluoride-X filter,\n\nfluorosilicic acid, n35\n\nfood additives,\n\nfoods\n\ncontaminants of,\n\ndairy products, \u2013\n\nDDT-laden,\n\ngenetically modified\n\navoiding, \u2013\n\ncommercially prepared (list), \u2013\n\ncrops containing GMOs,\n\ndangers of, \u2013\n\npublic's view of, \u2013\n\nheptachlor-laden,\n\nirradiated\n\ndangers of, \u2013\n\nenergy sources used for,\n\nexposures allowed on,\n\ninception of,\n\nlabelling of, \u2013\n\nnutritional deficits of,\n\nPOP-laden,\n\ntesting for GMOs in, \u2013\n\nformaldehyde\n\ncase history involving,\n\nhypersensitives and,\n\nmonitors,\n\nplants which absorb,\n\nproducts contained in,\n\n'safe' level of,\n\nsources of, , , ,\n\nsymptoms produced by,\n\ntoxicity symptoms of,\n\nFOS (fructo-oligosaccharides), \u2013\n\nfragrances, \u2013\n\nfree radicals\n\naction of,\n\ndefined, \u2013, , n32\n\neffects of,\n\nfecal, \u2013\n\nsources of, , \u2013,\n\ntest for, \u2013\n\nfront door area _qi_ , \u2013\n\nfructo-oligosaccharides (FOS), \u2013\n\nfull spectrum lights, \u2013\n\nFunctional Intracellular Analysis (FIA),\n\nfungi, , , \u2013\n\nfurnishings,\n\n_Fusarium_ , ,\n\nGeneral Ecology, Inc.,\n\nGenetic ID, Inc., \u2013,\n\nGenetically Engineered Food Alert,\n\ngenetically modified organisms\n\n(GMO)\n\ncrops containing (list),\n\ncross-species examples, \u2013\n\ndefined, \u2013\n\n_See also_ foods, genetically modified\n\nGenetron Systems, Inc.,\n\nGenox Corporation,\n\ngentian remedy,\n\nGeo Group,\n\ngeocancerology,\n\nGeomack Products,\n\ngeomancy\n\ncase studies employing, \u2013\n\ndefined,\n\ngeopathic stress\n\nbiochemical factors and,\n\ndefined, \u2013,\n\ndevices for clearing spaces of, \u2013\n\neffects of, ,\n\nentities and, \u2013\n\nhealth problems near, \u2013\n\nindicators of, \u2013\n\nneutralizing, \u2013, n13\n\nwomen's estrogen cycle and,\n\nghost-busting, \u2013\n\n_Giardia lambia_ , ,\n\nginger root,\n\ngingival crevicular fluid, n8\n\nGittleman, Ann Louise,\n\ngliadin,\n\nglobal distillation, \u20135\n\nGlobal Environment Technologies,\n\nglucuronidation,\n\nglutamate,\n\nglutathione, , ,\n\nglutathione conjugation,\n\nglutathione peroxidase, test for,\n\nglutathione S-transferase,\n\ngluten,\n\nglycine,\n\nGMOs. _See_ genetically modified organisms\n\nGoffman, John W.,\n\ngolden ear drops remedy,\n\ngolden sponge, ,\n\ngolf courses,\n\nGould, Jay, ,\n\nGrant, Lucinda,\n\ngrapes,\n\nGreat Smokies Diagnostic Laboratory, ,\n\ngreen drink,\n\nGreenberg, Robert C., \u2013\n\ngreens.\n\nGrossman. Terry,\n\ngrounding cords\n\nhomes to earth, \u2013\n\npeople to earth, \u2013\n\ngrowth periods, \u2013\n\nGTC Nutrition Company,\n\nGuirdham, Arthur, M.D.,\n\ngutta percha alternative, \u2013\n\nHaas, Elson M., \u2013,\n\nHailey, Boyd,\n\nhair analysis test, \u2013,\n\nhair-pulling case, \u2013\n\nHall, Judy,\n\nHalpern, Steven,\n\nhamburger,\n\nHarman, Denham,\n\nHarmon, Nicholas,\n\nHarrison, Lewis,\n\nHartmann, Ernest,\n\nHathaway, Warren E.,\n\nheadaches\n\ncase of entities and,\n\nchloropyrifos and,\n\nexcitotoxins and,\n\ngeopathic stress and,\n\ntartrazine and,\n\ntriggers of,\n\nHealing Center,\n\nHealth Mate Sauna, \u2013,\n\nHealthy Home Test Kit,\n\nHealthy House Institute, ,\n\nhealthy living resources,\n\nheart problems, ,\n\nheat depuration,\n\nheavy metals\n\nfound in tap water,\n\npoisoning from,\n\ntesting feces for, \u2013\n\ntesting hair for, \u2013\n\ntesting urine for, \u2013\n\nHellberg, Eira,\n\nHendricksen Nat\u00fcrlich, ,\n\nHEPA air filters, \u2013\n\nheptachlor, ,\n\nheptachlor epoxide,\n\nherbicide exposure case, \u2013\n\nHeron, Silena,\n\nHerxheimer reaction, , , \u2013\n\nhexachlorobenzene, , ,\n\nhives,\n\nHobbs, Christopher,\n\nHodgson, Michael,\n\nHoffman, David,\n\nHolistic Dental Association,\n\nholly remedy,\n\nHolyk, Peter, ,\n\nhomes\n\nbathroom considerations, \u2013\n\nbed location in, \u2013\n\nbedroom considerations, \u2013\n\nchlordane in,\n\nclutter in, \u2013\n\nconstruction, \u2013\n\ndesign and arrangement, \u2013\n\nenergy flow evaluation, \u2013\n\nenergy imbalance costs, \u2013\n\nenergy of, \u2013\n\nenergy refreshing of, \u2013\n\nenvironmental assessments of, \u2013,\n\nfront door area _qi_ , \u2013\n\nfurnishings,\n\ngeopathic zone testing for, \u2013,\n\nghost-busting, \u2013\n\nlights in,\n\nlocation of. _See_ geopathic stress\n\nneighboring homes' influences, \u2013\n\npainting interior, \u2013\n\nroom detoxification, \u2013\n\nunderground water near, \u2013, \u2013,\n\nhormones (fed to livestock),\n\nhormones (human)\n\ncentral,\n\ndefined,\n\nthyroid T4,\n\nhostility and flower essence therapy, \u2013\n\nhot flashes case, \u2013\n\nhouseplants as air filters, \u2013\n\nHuebner, Friedrich Markus,\n\nHuggins, Hal, , ,\n\nhuman milk,\n\nhumic water,\n\nhumidifiers, ,\n\nHussar, Christopher,\n\nHVP (hydrolyzed vegetable protein),\n\nhydrocarbons,\n\nhydroch1oric acid dumping,\n\nhydrogen fluoride dumping,\n\nhydrogenated fats,\n\nhydrolyzed vegetable protein (HVP),\n\nhydroponic plants, \u2013\n\nhyperactivity,\n\nhypertension,\n\nhypnotherapy, \u2013\n\nhypochondriacs, ,\n\n'hypochondriasis,'\n\nhypoglycemia,\n\nhypothyroidism,\n\nice cream,\n\nIGF-1 (insulin-like growth factor), \u2013\n\nillness\n\nchronic, \u2013\n\nenvironmental, \u2013\n\nrate from workpiace exposures,\n\nimmune system\n\nanti-cancer capability of,\n\nintestines as part of,\n\noverloaded, ,\n\npesticides and, \u2013\n\nresponse to toxins in blood,\n\nspiritual, \u2013\n\n1mmuno  B1oodprint test, \u2013,\n\nimmuno Laboratories,\n\nimmunoglobulins, , n68\n\nimpatiens remedy,\n\nindian pink remedy, \u2013\n\nIndividualized Optimal Nutrition (ION) test, \u2013,\n\nindoor air\n\ncarbon monoxide in,\n\nchemicals found in, ,\n\nchlordane in, \u2013\n\ndamp, , ,\n\ndust in, \u2013\n\nfactors affecting,\n\nfiltering\n\nwith houseplants, \u2013\n\nwith machines, \u2013\n\nformaldehyde in, \u2013\n\nfungi in, , , \u2013\n\nionizing, \u2013\n\nmolds in, , \u2013, \u2013\n\norganochlorines in,\n\npollutant sources,\n\nproductivity and,\n\npurifying, \u2013\n\nsick building syndrome and, \u2013,\n\ninfants. _See under_ children\n\nInfinity Heavenly Horsetail,\n\ninsulin-like growth factor (IGF-1), \u2013\n\nInternational Academy of Oral Medicine and Toxicology,\n\nInternational Dental Health Foundation,\n\nintestinal tract\n\nbacteria in\n\nbeneficial, \u2013,\n\nharmful,\n\nbasics of, \u2013, \u2013\n\ncolon\n\naverage American male,\n\ncleansers for, \u2013\n\ndiseased,\n\nnature and function of, \u2013\n\ntransit times in, ,\n\nleaky, , ,\n\nlength of,\n\nmucoid plaque in, \u2013\n\nparasites in, \u2013,\n\nsmall intestine, \u2013\n\nstagnation symptoms,\n\ntoxicity of, \u2013\n\n_See also_ parasites\n\nInuit,\n\nion generators, \u2013\n\nION (Individualized Optimal Nutrition) test, \u2013,\n\nion levels, \u2013,\n\nIonAir Company,\n\nIonAir Wein VI-2000 High Density Negative Ion Generator, \u2013\n\nionizing radiation, n3\n\nions\n\ndefined,\n\nnegative, , \u2013\n\npositive,\n\niron, test for, \u2013\n\nJainism, \u2013\n\nJanov, Arthur,\n\nJarro-Dophilus, \u2013,\n\nJarrow Formulas,\n\nJenny, S.,\n\nJensen, Barnard,\n\njoints, ,\n\nKaminski, Patricia, \u2013,\n\nKan Yu,\n\nKane, Patricia,\n\nKaragulla, Shafica, M.D.,\n\nKaufman, Richard C.,\n\nKDF shower water filter, , , 560n32\n\nKellogg, John Harvey,\n\nKendall, Julia,\n\nkidney inflammation,\n\nKilner, Walter,\n\nkinesiological muscle testing, ,\n\nKing James Medical Laboratory, ,\n\nKlinghardt, Dietrich,\n\nKnishinsky, Ran,\n\nKron, Jacquelin,\n\n_Lactobacilius acidophilus_ , , \u2013\n\nLake Baikal,\n\nlakes,\n\nLange, Susan,\n\nlasagna with meat,\n\nlaxative types,\n\nlead, , ,\n\nlead poisoning, \u2013, \u2013\n\nleaky gut syndrome,\n\nlearning disorders,\n\nLeGant, Leon S., \u2013\n\nlemon juice,\n\nLessEMF Inc.,\n\nlethargy, ,\n\nleukemia, , , \u2013\n\nLevin, Norman,\n\nley lines, n23\n\nLichtenberg, H.,\n\nlight therapy, , n91\n\nlightning,\n\nlights\n\ndawn simulators, \u2013\n\nfeng shui of,\n\nfull-spectrum\n\nfluorescent tubes, \u2013\n\nlight boxes, \u2013\n\nlight bulbs,\n\npartia1 spectrum, \u2013\n\nLilly, John C., n112\n\nlimonen,\n\nlinalool,\n\nlindane,\n\nling energies, , , \u2013,\n\nlipid peroxides, , ,\n\nliquid clay. _See_ bentonite\n\nLiquid Needie Rebalancer,\n\nLiu, Jenny,\n\nliver\n\nbacklogged, \u2013\n\ncleansing frequency,\n\ncommon toxicity of,\n\nfunction of, , ,\n\noverloaded, ,\n\ntest for health of,\n\nliver cleansers\n\ncommercial, \u2013,\n\nhomemade, ,\n\nliver cleansing\n\nbenefits of,\n\ncastor oil compresses during,\n\nflushing\n\noverview of,\n\nprotocols for, \u2013\n\nfoods for, \u2013\n\nfrequency recommended,\n\nherbs for \u2013\n\nprogram for \u2013\n\nresults to expect, \u2013\n\nliving space. _See_ homes\n\nLogan, Karen,\n\nlogging one's chemical exposures, \u2013\n\nLord, Richard S.,\n\nLos Angeles air pollution,\n\nLotz, K.E.,\n\nlove-lies-bleeding remedy,\n\nLundstrom, Sheri,\n\nlung cancer, ,\n\nlung symptoms, , \u2013\n\nlymph, ,\n\nlymphatic system\n\ndefined, , ,\n\ndysfunctional, \u2013\n\nfunction of, \u2013\n\nstimulating, \u2013\n\nlymphocytes, viability test for,\n\nlymphoma, ,\n\nMacManaway, Patrick, \u2013\n\nMag-Stop Plates,\n\nmagnesium\n\ndeficiency,\n\nmetabolic role,\n\ntests for, , \u2013, \u2013\n\nmagnetic fields, \u2013\n\nmagnetic shielding material, \u2013\n\nmanganese toxicity,\n\nmanufactured foods,\n\nmarine life,\n\nmarine pollution,\n\nMasada Marketing Corp.,\n\nMasters, Roger D.,\n\nMCS (multiple chemical sensitivities)\n\nCFS, fibromya1gia, and, \u2013\n\ndefined,\n\nincidence of, , ,\n\nVOCs and, \u2013\n\nmeditation,\n\nMeinig, George, D.D.S.,\n\nmelatonin, , n88, n7\n\nmelons,\n\nmenstrual bleeding,\n\nmental symptoms, ,\n\nMerchant, Randal,\n\nmercury\n\nabsorber of,\n\naction of, \u2013\n\nbinder of,\n\nchelators of, , \u2013\n\nconversion to methylmercury, \u2013\n\ndangers of, ,\n\ndisplacing magnesium,\n\nexposure standards, \u2013\n\nnervous system effects from, \u2013\n\npersistence in body,\n\ntests for, \u2013,\n\ntransmission to fetuses,\n\nmercury toxicity\n\ncases of, \u2013, \u2013\n\nhealth problems associated with, , \u2013, n2\n\nin dentists, \u2013\n\nmood altering effects of, \u2013\n\nsources of, \u2013,\n\nstudies on rats, \u2013\n\nMetagenics,\n\nMetaMetrix Clinical Laboratory,\n\nmethionine,\n\nmethyl alcohol,\n\nmethyl bromide,\n\nmethylmercury, , ,\n\nmethylene chloride,\n\nMeunnann, J.H.,\n\n'Mickey Finn:' \u2013\n\nmicrofiltration water filters, \u2013\n\nmicronutrients, testing for,\n\n_Micropolyspora faeni_ ,\n\nmicrowave pollution,\n\nmilk thistle, \u2013\n\nmimulus remedy,\n\nMineral Rich, ,\n\nmirrors, , \u2013,\n\nmiscarriage, , \u2013\n\nModi, Shakuntala, ,\n\nmolds, , \u2013, \u2013\n\nmolecular medicine, n51\n\nmolybdenum, test for, \u2013\n\nMom's Veggiewash, \u2013\n\nmonobutyl phtha1ate,\n\nmonosodium glutamate (MSG),\n\nMonsanto, ,\n\nmontmorillonite. _See_ bentonite\n\nMother & Others,\n\nmountain pride remedy, \u2013\n\nmoving between homes, \u2013\n\nMowrey, Daniel B.,\n\nMSG (monosodium glutamate),\n\nmucoid plaque, \u2013\n\nmultiple chemical sensitivities\n\n(MCS)\n\nchemicals in people with,\n\nproof of,\n\ntoxicity and, ,\n\nVOCs and, \u2013\n\nmultiple sclerosis, \u2013,\n\nMuMetal,\n\nmuscle symptoms,\n\nmuscle testing, ll,\n\nmusic therapy, \u2013\n\nmustard remedy, \u2013\n\nNambudripad Allergy Elimination Technique, , \u2013\n\nNambudripad, Devi S., ,\n\nnasal symptoms,\n\nNational Human Adipose Tissue Survey,\n\nNational Toxicology Program, \u2013\n\nnatural killer cells, , nl\n\nNatural Solutions Environmental, Inc.,\n\nNature's Carpet, , \u2013\n\nnaturopathy, nl\n\nnausea,\n\nneck, nape of, n18\n\nNeedak Manufacturing,\n\nNeedak Soft-Bounce rebounder,\n\nnegative ion generators, \u2013\n\nnegative ions, , \u2013\n\nneighbor pacification, \u2013\n\nneighboring homes' influences, \u2013\n\nnervous system\n\nautonomic, nll3\n\ncentral, disorders of, , \u2013\n\nmercury and, \u2013\n\nperipheral, n34\n\nsympathetic, n36\n\nNestle, Marion,\n\nneurological disorders, \u2013, ,\n\nneuropeptides, \u2013\n\nneurosis,\n\nNF Formulas,\n\nNickel, David J., \u2013\n\nnickel, test for, \u2013\n\nNieper, Hans,\n\nNirvana Safe Haven, ,\n\nnitrites,\n\nnitrogen dioxide, ,\n\nnitrogen oxide,\n\nnitrosamines,\n\nNoetic Field Therapy,\n\nnonionizing radistion, n3\n\nnonyl-phenol-ethoxylate,\n\nNordic Naturals,\n\nNorth Star ,000 Bright Light Box,\n\nNortheast Center for Environmental Medicine,\n\nNorthrup, Christiane,\n\nnuclear counties,\n\nnuclear power plants,\n\nnuclear weapons testing, \u2013\n\nNutraFiora,\n\nNutraSweet (aspartame), ,\n\nnutrients, intracellular,\n\nnutritional deficiencies,\n\nOCCs (organochlorine compounds). _See_ organochlorines\n\nOlestra,\n\noral\n\nbacterial populations,\n\ndiseases and heart problems, ,\n\nfocal infections, \u2013\n\ntoxicity symptoms,\n\ntoxicity tests, \u2013,\n\norchids,\n\nOregon grape remedy, \u2013\n\nOrganic Crop Improvement Assn.,\n\nOrganiclean E-Commerce Division,\n\nOrganiclean Fruit and Vegetable Wash, ,\n\norganochlorines\n\naverage dietary intake of, \u2013\n\nbodily damage from, \u2013\n\ndefined,\n\nformation of,\n\nhalf-lives of, n69\n\nin dolphins,\n\nin humans, \u2013\n\nmultiple, in blood,\n\npersistence of, , n69, n93\n\nsources of,\n\nOtt, John Nash,\n\nOtt-Lite,\n\noutgassing\n\nof carpets,\n\nfrom dental mercury amalgam fillings, \u2013\n\novarian cancer case, \u2013\n\nOwens, Kagan,\n\noxidation, \u2013,\n\nOxidative Protection Screen test, \u2013,\n\noxidative stress, , \u2013,\n\nOxidative Stress Profile, \u2013\n\nOxyClean Air Filter,\n\nOxystat, ,\n\nozone,\n\nPacific Northwest Laboratory,\n\nPacked Blood Cell Elements Profile, \u2013,\n\npacked red blood cell intracellular analysis, , \u2013\n\npaint exposure case,\n\npainting home interiors, \u2013\n\nPantox Laboratories,\n\nPantox Profile, ,\n\nParagon shower water filter, ,\n\nParagon Water Systems,\n\nParaquat,\n\nparasites\n\naction of,\n\narthropod,\n\nformula for purging, \u2013\n\nincidence of, \u2013\n\nsymptoms from, \u2013\n\ntest for, \u2013,\n\ntreatments for, \u2013\n\nworms,\n\nParkinson's disease\n\nexcitotoxins and, ,\n\npesticide exposure and,\n\ntriggers of,\n\nparticulate matter, as alr pollutant, ,\n\nPassey, Tom,\n\npathological detoxifiers,\n\npatient evaluation, \u2013\n\nPBBs (polybrominated biphenyls), ,\n\nPBTs (persistent, bioaccumulative, toxic substances), \u2013\n\n4-PC (4-phenylcyclohexene), ,\n\nPCBs (polychlorinated biphenyls)\n\namount produced worldwide,\n\nbanning of,\n\nin dolphins,\n\nin humans, , ,\n\nthyroid function and, \u2013\n\nuses of, \u2013\n\npeanut butter,\n\npeanuts,\n\n_Penicillium_ , ,\n\npenstemon remedy,\n\npentane,\n\nperchioroethylene,\n\nperfumes, \u2013\n\nperiodontal disease, \u2013\n\nperipheral nervous system, n34\n\npersistent organic pollutants. _See_ POPs\n\nPert, Candace,\n\npesticides\n\nalrborne,\n\nannual market for,\n\nCalifornia agricultural use of, \u2013\n\ncancer and,\n\n'drift' from, ,\n\nexposure case,\n\nhousehold, \u2013\n\nin house dust,\n\nlicensed agricultural,\n\nproportion not reaching target,\n\nproscription in schools, n47\n\nresidue levels in humans,\n\nusage rates, \u2013, \u2013\n\nwater contamination from, \u2013,\n\nPettis, Chuck,\n\npH, intracellular, test for, \u2013\n\nPhase I detoxification\n\naction of, ,\n\ndefined,\n\nensymes involved in, \u2013\n\nnutrients needed for,\n\nsubstances blocking,\n\ntriggering substances,\n\nPhase II detoxification\n\ndefined,\n\nnutrients needed for,\n\npathways in, \u2013\n\nphenols,\n\nphosphorus, test for, \u2013\n\nphthalates in urine,\n\nPhyto-Pro,\n\nPi-Square SunUp dawn simulator,\n\npickles,\n\nPierrakos, John C.,\n\npineal function,\n\npizza, pepperoni,\n\nPizzorno, Joseph, ,\n\nPlanetary Formulas,\n\nPLH Products,\n\n\"poison arrows,\" cutting _qi_ , \u2013\n\nPolari Tea,\n\npolychlorinated biphenyls. _See_ PCBs\n\npolycyclic aromatic hydrocarbons,\n\npolyurethane,\n\npolyurethane foam,\n\npolyvinyl chloride (PVC), \u2013\n\npomanders, \u2013\n\nPOPs (persistent organic pollutants), \u20134, ,\n\npositive ions,\n\npotassium, test for, \u2013\n\npotato chips,\n\npotatoes,\n\np, p-DDE,\n\nprebiotics, \u2013\n\nPrestige Publishing,\n\nPrice, Weston A.,\n\nProAlgen,\n\nprobiotics, \u2013\n\nproduce\n\ncleaning,\n\norganically grown,\n\nwas hes for, \u2013\n\nProfessional House Doctors,\n\nprostate cancer\n\ncase of,\n\ndental problems and,\n\nrates of,\n\nproteins, metal-binding, test for,\n\nPsenner, Roland,\n\n_Pseudomonas aeruginosa_ ,\n\npsyllium husks and seeds, \u2013\n\nPublic Citizen, ,\n\npumpkin pie,\n\npumpkin seeds, \u2013\n\nPur Recovery Engineering,\n\nPur water filters, ,\n\nPure-Pro USA Corp.,\n\nPure-Pro water filters, ,\n\nPVC (polyvinyl chloride), \u2013\n\npyrethroids, \u2013\n\n_qi_ , \u2013\n\ncutting, \u2013\n\nQLink, \u2013\n\nradiation\n\neffects of, \u2013\n\nionizing, n3\n\nradioactive fallout, \u2013\n\nradiolytic products, \u2013\n\nRadiTech, \u2013,\n\nradon\n\nassessing homes for,\n\ndangers of, \u2013\n\ndeaths attributed to,\n\nmode of entry, ,\n\nresidential levels of,\n\nsources of,\n\nU.S. concentration maps,\n\nraisins,\n\nRancho Seco nuclear plant,\n\nRandolph, Theron,\n\nRau, Thomas,\n\nRea, William J., , , ,\n\nReal Goods,\n\nReboundAIR rebounder,\n\nrebounding, \u2013\n\nrectal cancer, ,\n\nredox,\n\nreduction,\n\nrelaxation suggestions, \u2013\n\nreproductive problems, \u2013\n\nRescue Remedy, \u2013, n33\n\nresistivity, intracellular, test for, \u2013\n\nrestricted envirorunental stimulation therapy (REST), \u2013, \u2013\n\nreverse osmosis water filters, \u2013\n\nRiccioli, Joseph,\n\nRiggs, Alf,\n\nRiversong, Michael, \u2013\n\nRogers, Sherry, , , \u2013,\n\nroot canals, \u2013, \u2013\n\nRossbach, Sarah,\n\nRoundup,\n\nRoundup Ready,\n\nRoyal Center of Advanced Medicine,\n\nRoyal, Daniel F., ,\n\nSAD (seasonal affective disorder), \u2013\n\nSafe Zone filters,\n\nSagan, Samuel, , , ,\n\nsalmonella,\n\nSamadhi Tank Company,\n\nsauna units, \u2013\n\nsaunas, \u2013, \u2013, n15\n\nSBS. _See_ sick building syndrome\n\nscarlet monkeyflower remedy, \u2013\n\nSchr\u00f6dter, Willy,\n\nScientific Consulting Service,\n\nscleranthus remedy,\n\nscotch broom remedy,\n\nScott-Morley, Anthony, \u2013\n\nscrubs, \u2013, \u2013\n\nSeagull IV X-1 water filters, \u2013,\n\nseals, and POPs in body fat,\n\nseasonal affective disorder (SAD), \u2013\n\nSedir, Paul,\n\nseizures,\n\nserotonin, \u2013, , , n6\n\nSeventh Generation,\n\nsexual maturation, \u2013\n\nshaking, \u2013\n\nSharan, Farida, \u2013\n\nShaul, Chris,\n\nShaul, Sophia Tang,\n\nShelter Ecology,\n\nShelton, Bruce,\n\nSherman, Janette D., ,\n\nshower water filters, \u2013\n\nshowers, chlorine exposure in, \u2013\n\nsick building syndrome (SBS)\n\nair quality and, \u2013\n\ndefined, \u2013\n\ndust and, \u2013\n\nincidence of, \u2013\n\nstress and,\n\nsymptoms of, \u2013\n\ntoxin sources surrounding, ,\n\nwomen and,\n\n_See also_ building-related illness; indoor air\n\nsilicofluoride,\n\nSilver, Helene,\n\nSimontacchi, Carol, \u2013\n\nskin\n\nbrushing, \u2013\n\nscrubs for, \u2013\n\nsymptoms,\n\nsmectite. _See_ bentonite\n\nsmog, \u2013\n\nsolvents, , \u2013\n\nsour cream,\n\nsoy products, \u2013\n\nsoybeans, ,\n\nSpecial Provocative Mercury Test, ,\n\nSpectraCell Laboratories,\n\nspider plants,\n\nspiritual\n\nantioxidant, \u2013\n\ndetoxification, \u2013, \u2013\n\nguide, \u2013\n\ntoxins, \u2013\n\nSprite Slim-Line shower water filter, ,\n\nSt. John's wort remedy,\n\n_Stachybotrys atra_ ,\n\n_Stachybotrys chartarum_ ,\n\nStangle, Jacob, \u2013\n\nstar tulip remedy,\n\nsteam distillation, \u2013\n\nSteingraber, Sandra,\n\nSteven Halpern's Inner Peace Music,\n\nsticky monkeyflower remedy,\n\nstomach cancer,\n\nstool tests, \u2013,\n\nstrawberries,\n\nstress\n\nflower essence therapy for, \u2013\n\nrelieving, \u2013\n\ntypes of, \u2013\n\n_See also_ geopathic stress\n\nstrokes, ,\n\nstrontium-90,\n\nstyrene,\n\nsulfation,\n\nsulfur dioxide,\n\nsulfuric acid dumping,\n\nSun-Pure Water Purifier, ,\n\nsunflower remedy,\n\nSunRizr dawn simulator,\n\nSunshine Tropical Foliage,\n\nsuperoxide dismutase, test for,\n\nsweet chestnut remedy,\n\nsympathetic nervous system, n36\n\nsymptoms\n\nper organ or organ system, \u2013\n\nvague, \u2013\n\nSynercid,\n\nTagamet,\n\nTanalbit, ,\n\nTart, Charles,\n\ntartrazine,\n\nteeth-brushing,\n\ntension and flower essence therapy, \u2013\n\nTernes, Thomas A.,\n\nTerraFlo CBLX water filter, ,\n\ntesticular cancer, \u2013\n\ntests available, body nutrients and toxins, \u2013\n\ntetrachloroethylene, \u2013\n\n_Thermoactinomycetes vulgaris_ ,\n\nThinkTank International,\n\nTHMs. _See_ trihalomethanes\n\nThompson, Angel,\n\nThornton, Joe,\n\nThree Mile Island,\n\nthroat symptoms, ,\n\nThurnell-Read, Jane,\n\nthyroid gland\n\nfunction of, n74, n32\n\nincidence of malfunctioning,\n\nPCBs and, \u2013\n\nthyroid hormones\n\nT3,\n\nT4,\n\nTierra, Michael,\n\ntinnitus,\n\ntoluene\n\nasthma and,\n\nbasics of, n12\n\nbenzoates as block to cleansing of,\n\nhouseplants which absorb,\n\nincidence in fat tissue,\n\nincidence in fragrances,\n\ntoluene diisocyanate,\n\ntomatoes,\n\nToo, Lillian,\n\ntooth extractions,\n\nTOPAS (Toxicity Prescreening Assay),\n\nTotal Antioxidant Status test, ,\n\ntotal environmental load,\n\nTotal Shield,\n\ntoxaphene,\n\ntoxic bowel, ,\n\ntoxic load, tests for, \u2013\n\ntoxic substances\n\nannual dumping of,\n\ncarcinogenic,\n\n'toxic triggers,'\n\nToxicant-Induced Loss of Tolerance,\n\ntoxicity\n\nas disease,\n\nassessment criteria, \u2013\n\ndegrees of,\n\neffects of, _xxii_ ,\n\ngradual development of, ,\n\nlong-term, ,\n\nmistaken as fatigue,\n\nunrelieved,\n\ntoxins\n\nblood levels of, in mothers and infants,\n\nbuildup in environment,\n\ndefined,\n\nenergy, \u2013, \u2013\n\nexposures to, versus body levels of,\n\nfat-soluble, ,\n\nhuman milk,\n\nincidence in humans, , \u2013,\n\nindoor sources of, ,\n\nlevels in body, \u2013\n\nmigration of, , \u2013\n\nnon-chemical,\n\n'safe' levels of,\n\nspiritual, \u2013\n\ntoxics versus, n6\n\ntypes of,\n\nubiquity of, _xxi-xxii_ , \u2013\n\ntoxoplasmosis,\n\nTrace Mineral Systems,\n\ntrampolining, \u2013\n\ntrans-fatty acids\n\navoiding consumption of,\n\nconsumption statistics,\n\ndangers of, \u2013\n\ndefined,\n\nhistory of,\n\nnature of,\n\ntransgenic,\n\ntransit times, , ,\n\ntrauma, unresolved,\n\ntrees, ,\n\ntrichloroethane,\n\ntrichloroethylene, , ,\n\nTrifield Meter,\n\ntrigrams, \u2013\n\ntrihalomethanes (THMs)\n\ncarcinogenic effects of, \u2013\n\ndefined,\n\nlevels considered acceptable,\n\nmodes of ingestion,\n\nreproductive effects of, \u2013\n\nTriphala, \u2013\n\ntubulin,\n\ntwitching,\n\nUltra Clear, \u2013,\n\nultraviolet light air purification, \u2013\n\nultraviolet purification water filters, \u2013\n\nuranium, \u2013,\n\nuric acid, test for,\n\nurine\n\ncolor of,\n\nmercury in,\n\nphthalates in,\n\ntesting for heavy metals, \u2013\n\nUrine Elements Profiles, \u2013,\n\nvan Benschoten, M.M., O.M.D., \u2013\n\nvanadium, test for, \u2013\n\nVariel Health International,\n\nventilation systems, , \u2013,\n\nVerilux, ,\n\nVideo Operator Distress Syndrome,\n\nvilli, intestinal,\n\nVilsbiburg, Germany, \u2013\n\nViracin, ,\n\nVirginiamycin,\n\nvirus sizes,\n\nVita-Lite, \u2013\n\nVitamin C, test for,\n\nVitamin E, test for,\n\nVitamin Research Products,\n\nVOCs (volatile organic compounds), \u2013, \u2013\n\nexamples of,\n\nhouseplants as absorbers of, \u2013\n\nhuman-made,\n\nin blood,\n\nin tap water, \u2013\n\nindoor levels of,\n\nnatural sources of,\n\nsources of, \u2013, \u2013,\n\nvon Pohl, Gustav Freiherr, ,\n\nWalker, Morton, ,\n\nwater consumption\n\nbenefit of,\n\nbest times for,\n\ninadequate, \u2013\n\nindicator of adequate, \u2013\n\nminimum recommended,\n\nwater contamination, \u2013, , \u2013\n\nwater filter types, \u2013\n\nwater filtration,\n\nwater, stagnant,\n\nwater (tap)\n\nchlorination of,\n\ncontaminants in, \u2013\n\nreproductive problems from, \u2013\n\n_See also_ drinking water\n\nWaterman, Robert D.,\n\nWaterwise, Inc.,\n\nWaveGuide,\n\nWebster, Richard,\n\nweight symptoms,\n\nwhacks, auric,\n\nwhite blood cell counts,\n\nWhole Blood Elements Profile, \u2013,\n\nwillow remedy,\n\nWolverton, B.C.,\n\nwomen\n\ncolon cancer in, ,\n\nendocrine disruptors and,\n\nhealth problems and geopathic stress, ,\n\npregnant and nursing,\n\nsick building syndrome in,\n\n_See also_ breast cancer\n\nWorld Resources Institute,\n\nWydra, Nancilee,\n\nX-rays,\n\nxenobiotics,\n\nxenoestrogens. _See_ environmental estrogens\n\nxylene, , ,\n\nYarnell, Eric,\n\nyarrow remedy,\n\nYarrow Special Formula remedy, ,\n\nYerba Prima, Inc.,\n\nYerba Santa remedy,\n\nyin\/yang polarities, n5\n\nZestron Ionair ionizers,\n\nzinc, test for, \u2013\n\nzinnia remedy, \n**Hampton Roads Publishing Company**\n\n... _for the evolving human spirit_\n\nHampton Roads Publishing Company  \npublishes books on a variety of subjects, including metaphysics, health, integrative medicine, visionary fiction, and other related topics.\n\nFor a copy of our latest catalog, call toll-free\n\n(800) 766-8009, or send your name and address to:\n\nHampton Roads Publishing Company, Inc.\n\n1125 Stoney Ridge Road\n\nCharlottesville, VA 22902\n\ne-mail: hrpc@hrpub.com\n\nwww.hrpub.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"The Project Gutenberg Etext of Supplemental Nights, Volume 2\nby Richard F. Burton\n#13 in our series by Sir Richard Francis Burton\n\nCopyright laws are changing all over the world. Be sure to check\nthe copyright laws for your country before distributing this or\nany other Project Gutenberg file.\n\nWe encourage you to keep this file, exactly as it is, on your\nown disk, thereby keeping an electronic path open for future\nreaders.  Please do not remove this.\n\nThis header should be the first thing seen when anyone starts to\nview the etext. Do not change or edit it without written\npermission. 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FOR PUBLIC DOMAIN ETEXTS*Ver.10\/04\/01*END*\n\n\n\n\n\n\nThis etext was scanned by JC Byers\n(http:\/\/www.capitalnet.com\/~jcbyers\/index.htm) and proofread by\nLynn Bornath, JC Byers, Diane Doerfler, Peggy Klein, P.J.\nLaBrocca, Robert Sinton, and Mats Wernersson.\n\n\n\n\n\n\n\n\n                          SUPPLEMENTAL\n                             NIGHTS\n                  To The Book Of The Thousand\n                   And One Nights With Notes\n                      Anthropological And\n                          Explanatory\n\n                               By\n                       Richard F. Burton\n\n                           VOLUME TWO\n              Privately Printed By The Burton Club\n\n\n\n                     To Henry Irving, Esq.\n\nMy Dear Irving,\n\n     To a consummate artist like yourself I need hardly suggest\nthat The Nights still offers many a virgin mine to the\nPlaywright; and I inscribe this volume to you, not only in\nadmiration of your genius but in the hope that you will find\nmeans of exploiting the hidden wealth which awaits only your\n\"Open Sesame!\"\n\n                    Every yours sincerely,\n                       Richard F. Burton.\n\nLondon, August 1, 1886.\n\n\n\n\n\n\n                Contents of the Twelfth Volume.\n\n\n13.  Al-Malik Al-Zahir Rukn Al-Din Bibars Al-Bundukdari and the\n     Sixteen Captains of Police\n     a.   First Constable's History\n     b.   Second Constable's History\n     c.   Third Constable's History\n     d.   Fourth Constable's History\n     e.   Fifth Constable's History\n     f.   Sixth Constable's History\n     g.   Seventh Constable's History\n     h.   Eighth Constable's History\n          ha.  The Thief's Tale\n     i.   Ninth Constable's History\n     j.   Tenth Constable's History\n     k.   Eleventh Constable's History\n     l.   Twelfth Constable's History\n     m.   Thirteenth Constable's History\n     n.   Fourteenth Constable's History\n          na.  A Merry Jest of a Clever Thief\n          nb.  Tale of the Old Sharper\n     o.   Fifteenth Constable's History\n     p.   Sixteenth Constable's History\n14.  Tale of Harun Al-Rashid and Abdullah Bin Nafi'\n     a.   Tale of the Damsel Torfat Al-Kulub and the Caliph Harun\n          Al-Rashid\n15.  Women's Wiles\n16.  Nur Al-Din Ali of Damascus and the Damsel Sitt Al-Milah\n17.  Tale of King Ins Bin Kays and His Daughter with the Son of\n     King Al-'Abb\u00e1s\n18.  Tale of the Two kings and the Wazir's Daughters\n19.  The Concubine and the Caliph\n20.  The Concubine of Al-Maamun\n\nAppendix: Variants and Analogues of Some of the Tales in Vols. XI\n                            and XII.\n                       by W. A. Clouston\n\nThe Sleeper and the Waker\nThe Ten Wazirs; or the History of King Azadbakht and His Son\nKing Dadbin and His Wazirs\nKing Aylan Shah and Abu Tammam\nKing Sulayman Shah and His Niece\nFiruz and His Wife\nKing Shah Bakht and His Wazir Al-Rahwan\nOn the Art of Enlarging Pearls\nThe Singer and the Druggist\nThe King Who Kenned the Quintessence of Things\nThe Prince Who Fell in Love With the Picture\nThe Fuller, His Wife, and the Trooper\nThe Simpleton Husband\nThe Three Men and Our Lord Isa\nThe Melancholist and the Sharper\nThe Devout Woman Accused of Lewdness\nThe Weaver Who Became a Leach by Order of His Wife\nThe King Who Lost Kingdom, Wife, and Wealth\nAl-Malik Al-Zahir and the Sixteen Captains of Police\nThe Thief's Tale\nThe Ninth Constable's Story\nThe Fifteenth Constable's Story\nThe Damsel Tuhfat Al-Kulub\nWomen's Wiles\nNur Al-Din and the Damsel Sitt Al-Milah\nKing Ins Bin Kays and his Daughter\n\nAdditional Notes:\n     Firuz and His Wife\n     The Singer and the Druggist\n     The Fuller, His Wife, and the Trooper\n\n\n\n\n\n\n\n                      Supplemental Nights\n\n                       To The Book Of The\n\n                  Thousand Nights And A Night\n\n\n\n\n\n\n            AL-MALIK AL-ZAHIR RUKN AL-DIN BIBARS AL-\n             BUNDUKDARI AND THE SIXTEEN CAPTAINS OF\n                         POLICE.[FN#1]\n\n\n\n\nThere was once in the climes[FN#2] of Egypt and the city of\nCairo, under the Turks, a king of the valiant kings and the\nexceeding mighty Soldans, hight Al-Malik al-Z\u00e1hir Rukn al-Din\nBibars al-Bundukd\u00e1ri,[FN#3] who was used to storm the Islamite\nsconces and the strongholds of \"The Shore\"[FN#4] and the Nazarene\ncitadels. His Chief of Police in the capital of his kingdom{} was\njust to the folk, all of them; and Al-Malik al-Zahir delighted in\nstories of the common sort and of that which men purposed in\nthought; and he loved to see this with his own eyes and to hear\ntheir sayings with his own ears. Now it fortuned that he heard\none night from a certain of his nocturnal reciters[FN#5] that\namong women are those who are doughtier than the doughtiest men\nand prower of prowess, and that among them are some who will\nengage in fight singular with the sword and others who beguile\nthe quickest-witted of Walis and baffle them and bring down on\nthem all manner of miseries; wherefore said the Soldan, \"I would\nlief hear this of their legerdemain from one of those who have\nhad to do with it, so I may hearken unto him and cause him\ndiscourse.\" And one of the story-tellers said, \"O king, send for\nthe Chief of Police of this thy city.\" Now 'Alam al-Din[FN#6]\nSanjar was at that time Wali and he was a man of experience, in\naffairs well versed; so the king sent for him and when he came\nbefore him, he discovered to him that which was in his mind.\nQuoth Sanjar, \"I will do my endeavour for that which our lord\nseeketh.\" Then he arose and returning to his house, summoned the\nCaptains of the watch and the Lieutenants of the ward and said to\nthem, \"Know that I purpose to marry my son and make him a bridal\nbanquet, and I desire that ye assemble, all of you, in one place.\nI also will be present, I and my company, and do ye relate that\nwhich you have heard of rare occurrences and that which hath\nbetided you of experiences.\" And the Captains and Runners and\nAgents of Police answered him, \"'Tis well: Bismillah--in the name\nof Allah! We will make thee see all this with thine own eyes and\nhear it with thine own ears.\" Then the Chief of Police arose and\ngoing up to Al-Malik al-Zahir, informed him that the assembly\nwould meet on such a day at his house; and the Soldan said, \"'Tis\nwell,\" and gave him somewhat of coin for his spending-money. When\nthe appointed day came the Chief of Police set apart for his\nofficers and constables a saloon, which had latticed casements\nranged in order and giving upon the flower-garden, and Al-Malik\nal-Zahir came to him, and he seated himself and the Soldan, in\nthe alcove. Then the tables were spread for them with food and\nthey ate: and when the bowl went round amongst them and their\nsouls were gladdened by meat and drink, they mutually related\nthat which was with them and, revealed their secrets from\nconcealment. The first to discourse was a man, a Captain of the\nWatch, hight Mu'\u00edn al-Din[FN#7] whose heart was wholly occupied\nwith the love of fair women; and he said, \"Harkye, all ye people\nof high degree, I will acquaint you with an extraordinary affair\nwhich fortuned me aforetime.\" Then he began to tell\n\n\n\n\nThe First Constable's History.[FN#8]\n\n\n\nKnow ye that when I entered the service of this Emir,[FN#9] I had\na great repute and every low fellow and lewd feared me most of\nall mankind, and when I rode through the city, each and every of\nthe folk would point at me with their fingers and sign at me with\ntheir eyes. It happened one day, as I sat in the palace of the\nPrefecture, back-propped against a wall, considering in myself,\nsuddenly there fell somewhat in my lap, and behold, it was a\npurse sealed and tied. So I hent it in hand and lo! it had in it\nan hundred dirhams,[FN#10] but I found not who threw it and I\nsaid, \"Lauded be the Lord, the King of the Kingdoms!\"[FN#11]\nAnother day, as I sat in the same way, somewhat fell on me and\nstartled me, and lookye, 'twas a purse like the first: I took it\nand hiding the matter, made as though I slept, albeit sleep was\nnot with me. One day as I thus shammed sleep, I suddenly sensed\nin my lap a hand, and in it a purse of the finest; so I seized\nthe hand and behold, 'twas that of a fair woman. Quoth I to her,\n\"O my lady, who art thou?\" and quoth she, \"Rise and come away\nfrom here, that I may make myself known to thee.\" Presently I\nrose up and following her, walked on, without tarrying, till we\nstopped at the door of a high-builded house, whereupon I asked\nher, \"O my lady, who art thou? Indeed, thou hast done me\nkindness, and what is the reason of this?\" She answered, \"By\nAllah, O Captain[FN#12] Mu'in, I am a woman on whom love and\nlonging are sore for desire of the daughter of the Kazi Am\u00edn\nal-Hukm.[FN#13] Now there was between me and her what was and\nfondness for her fell upon my heart and I agreed upon an\nassignation with her, according to possibility and convenience;\nbut her father Amin al-Hukm took her and went away, and my heart\ncleaveth to her and yearning and distraction waxed sore upon me\nfor her sake.\" I said to her, marvelling the while at her words,\n\"What wouldst thou have me do?\" and said she, \"O Captain Mu'in, I\nwould have thee lend me a helping hand.\" Quoth I, \"Where am I and\nwhere is the daughter of the Kazi Amin al-Hukm?\"[FN#14] and quoth\nshe \"Be assured that I would not have thee intrude upon the\nKazi's daughter, but I would fain work for the winning of my\nwishes. This is my will and my want which may not be wroughten\nsave by thine aid.\" Then she added, \"I mean this night to go with\nheart enheartened and hire me bracelets and armlets and anklets\nof price; then will I hie me and sit in the street wherein is the\nhouse of Amin al-Hukm; and when 'tis the season of the round and\nfolk are asleep, do thou pass, thou and those who are with thee\nof the men, and thou wilt see me sitting and on me fine raiment\nand ornaments and wilt smell on me the odour of Ottars; whereupon\ndo thou question me of my case and I will say, 'I hail from the\nCitadel and am of the daughters of the deputies[FN#15] and I came\ndown into the town for a purpose; but night overtook me all\nunawares and the Zuwaylah Gate[FN#16] was shut against me and all\nthe other portals and I knew not whither I should wend this\nnight. Presently I saw this street and noting the goodly fashion\nof its ordinance and its cleanliness, I sheltered me therein\nagainst break of day.' When I speak these words to thee with\ncomplete self-possession,[FN#17] the Chief of the watch will have\nno ill suspicion of me, but will say, 'There's no help but that\nwe leave her with one who will take care of her till morning.'\nThereto do thou rejoin, ''Twere best that she night with Amin\nal-Hukm and lie with his wives[FN#18] and children until dawn of\nday.' Then straightway knock at the Kazi's door, and thus shall I\nhave secured admission into his house, without inconvenience, and\nwon my wish; and--the Peace!\" I said to her, \"By Allah, this is\nan easy matter.\" So, when the night was blackest, we rose to make\nour round, followed by men with girded swords, and went about the\nways and compassed the city, till we came to the street[FN#19]\nwhere was the woman, and it was the middle of the night. Here we\nsmelt mighty rich scents and heard the clink of rings: so I said\nto my comrades, \"Methinks I espy a spectre;\" and the Captain of\nthe watch cried, \"See what it is.\" Accordingly, I undertook the\nwork and entering the thoroughfare presently came out again and\nsaid, \"I have found a fair woman and she telleth me that she is\nfrom the Citadel and that dark night surprised her and she saw\nthis street and noting its cleanness and goodly fashion of\nordinance, knew that it belonged to a great man[FN#20] and that\nneeds must there be in it a guardian to keep watch over it, so\nshe sheltered her therein.\" Quoth the Captain of the watch to me,\n\"Take her and carry her to thy house;\" but quoth I, \"I seek\nrefuge with Allah![FN#21] My house is no strong box[FN#22] and on\nthis woman are trinkets and fine clothing. By Allah, we will not\ndeposit the lady save with Amin al-Hukm, in whose street she hath\nbeen since the first starkening of the darkness; therefore do\nthou leave her with him till the break of day.\" He rejoined, \"Do\nwhatso thou willest.\" So I rapped at the Kazi's gate and out came\na black slave of his slaves, to whom said I, \"O my lord, take\nthis woman and let her be with you till day shall dawn, for that\nthe lieutenant of the Emir Alam al-Din hath found her with\ntrinkets and fine apparel on her, sitting at the door of your\nhouse, and we feared lest her responsibility be upon you;[FN#23]\nwherefore I suggested 'twere meetest she night with you.\" So the\nchattel opened and took her in with him. Now when the morning\nmorrowed, the first who presented himself before the Emir was the\nKazi Amin al-Hukm, leaning on two of his <DW64> slaves; and he was\ncrying out and calling for aid and saying, \"O Emir, crafty and\nperfidious, yesternight thou depositedst with me a woman and\nbroughtest her into my house and home, and she arose in the dark\nand took from me the monies of the little orphans my\nwards,[FN#24] six great bags, each containing a thousand\ndinars,[FN#25] and made off; but as for me, I will say no\nsyllable to thee except in the Soldan's presence.\"[FN#26] When\nthe Wali heard these words, he was troubled and rose and sat down\nin his agitation; then he took the Judge and placing him by his\nside, soothed him and exhorted him to patience, till he had made\nan end of talk, when he turned to the officers and questioned\nthem of that. They fixed the affair on me and said, \"We know\nnothing of this matter but from Captain Mu'in al-Din.\" So the\nKazi turned to me and said, \"Thou wast of accord to practice upon\nme with this woman, for she said she came from the Citadel.\" As\nfor me, I stood, with my head bowed ground-wards, forgetting both\nSunnah and Farz,[FN#27] and remained sunk in thought, saying,\n\"How came I to be the dupe of that randy wench?\" Then cried the\nEmir to me, \"What aileth thee that thou answerest not?\" Thereupon\nI replied, \"O my lord, 'tis a custom among the folk that he who\nhath a payment to make at a certain date is allowed three days'\ngrace: do thou have patience with me so long, and if, at the end\nof that time, the culprit be not found, I will be responsible for\nthat which is lost.\" When the folk heard my speech they all\napproved it as reasonable and the Wali turned to the Kazi and\nsware to him that he would do his utmost to recover the stolen\nmonies adding, \"And they shall be restored to thee.\" Then he went\naway, whilst I mounted without stay or delay and began to-ing and\nfro-ing about the world without purpose, and indeed I was become\nthe underling of a woman without honesty or honour; and I went my\nrounds in this way all that my day and that my night, but\nhappened not upon tidings of her; and thus I did on the morrow.\nOn the third day I said to myself, \"Thou art mad or silly;\" for I\nwas wandering in quest of a woman who knew me[FN#28] and I knew\nher not, she being veiled when I met her. Then I went round about\nthe third day till the hour of mid-afternoon prayer, and sore\nwaxed my cark and my care for I kenned that there remained to me\nof my life but the morrow, when the Chief of Police would send\nfor me. However, as sundown-time came, I passed through one of\nthe main streets, and saw a woman at a window; her door was ajar\nand she was clapping her hands and casting sidelong glances at\nme, as who should say, \"Come up by the door.\" So I went up,\nwithout fear or suspicion, and when I entered, she rose and\nclasped me to her breast. I marvelled at the matter and quoth she\nto me, \"I am she whom thou depositedst with Amin al-Hukm.\" Quoth\nI to her, \"O my sister, I have been going round and round in\nrequest of thee, for indeed thou hast done a deed which will be\nchronicled and hast cast me into red death[FN#29] on thine\naccount.\" She asked me, \"Dost thou speak thus to me and thou a\ncaptain of men?\" and I answered, \"How should I not be troubled,\nseeing that I be in concern for an affair I turn over and over in\nmind, more by token that I continue my day long going about\nsearching for thee and in the night I watch its stars and\nplanets?\"[FN#30] Cried she, \"Naught shall betide save weal, and\nthou shalt get the better of him.\"[FN#31] So saying, she rose and\ngoing to a chest, drew out therefrom six bags full of gold and\nsaid to me, \"This is what I took from Amin al-Hukm's house. So an\nthou wilt, restore it; else the whole is lawfully[FN#32] thine;\nand if thou desire other than this, thou shalt obtain it; for I\nhave monies in plenty and I had no design herein save to marry\nthee.\" Then she arose and opening other chests, brought out\ntherefrom wealth galore and I said to her, \"O my sister, I have\nno wish for all this, nor do I want aught except to be quit of\nthat wherein I am.\" Quoth she, \"I came not forth of the Kazi's\nhouse without preparing for thine acquittance.\" Then said she to\nme, \"When the morrow shall morn and Amin al-Hukm shall come to\nthee bear with him till he have made an end of his speech, and\nwhen he is silent, return him no reply; and if the Wali ask,\n'What aileth thee that thou answerest me not?' do thou rejoin, 'O\nlord and master[FN#33] know that the two words are not alike, but\nthere is no helper for the conquered one[FN#34] save Allah\nAlmighty.' The Kazi will cry, 'What is the meaning of thy saying,\nThe two words are not alike?' And do thou retort, 'I deposited\nwith thee a damsel from the palace of the Sultan, and most likely\nsome enemy of hers in thy household hath transgressed against her\nor she hath been secretly murdered. Verily, there were on her\nraiment and ornaments worth a thousand ducats, and hadst thou put\nto the question those who are with thee of slaves and\nslave-girls, needs must thou have litten on some traces of the\ncrime.' When he heareth this from thee, his trouble will redouble\nand he will be amated and will make oath that thou hast no help\nfor it but to go with him to his house: however, do thou say,\n'That will I not do, for I am the party aggrieved, more\nespecially because I am under suspicion with thee.' If he\nredouble in calling on Allah's aid and conjure thee by the oath\nof divorce saying, 'Thou must assuredly come,' do thou reply, 'By\nAllah, I will not go, unless the Chief also go with me.' Then, as\nsoon as thou comest to the house, begin by searching the\nterrace-roofs; then rummage the closets and cabinets; and if thou\nfind naught, humble thyself before the Kazi and be abject and\nfeign thyself subjected, and after stand at the door and look as\nif thou soughtest a place wherein to make water,[FN#35] because\nthere is a dark corner there. Then come forward, with heart\nharder than syenite-stone, and lay hold upon a jar of the jars\nand raise it from its place. Thou wilt find there under it a\nmantilla-skirt; bring it out publicly and call the Wali in a loud\nvoice, before those who are present. Then open it and thou wilt\nfind it full of blood, exceeding for freshness, and therein a\nwoman's walking-boots and a pair of petticoat-trousers and\nsomewhat of linen.\" When I heard from her these words, I rose to\ngo out and she said to me, \"Take these hundred sequins, so they\nmay succour thee; and such is my guest-gift to thee.\" Accordingly\nI took them and leaving her door ajar returned to my lodging.\nNext morning, up came the Judge, with his face like the\nox-eye,[FN#36] and asked, \"In the name of Allah, where is my\ndebtor and where is my property?\" Then he wept and cried out and\nsaid to the Wali, \"Where is that ill-omened fellow, who aboundeth\nin robbery and villainy?\" Thereupon the Chief turned to me and\nsaid, \"Why dost thou not answer the Kazi?\" and I replied, \"O\nEmir, the two heads[FN#37] are not equal, and I, I have no\nhelper;[FN#38] but, an the right be on my side 'twill appear.\" At\nthis the Judge grew hotter of temper and cried out, \"Woe to thee,\nO ill-omened wight! How wilt thou make manifest that the right is\non thy side?\" I replied \"O our lord the Kazi, I deposited with\nthee and in thy charge a woman whom we found at thy door, and on\nher raiment and ornaments of price. Now she is gone, even as\nyesterday is gone;[FN#39] and after this thou turnest upon us and\nsuest me for six thousand gold pieces. By Allah, this is none\nother than a mighty great wrong, and assuredly some foe[FN#40] of\nhers in thy household hath transgressed against her!\" With this\nthe Judge's wrath redoubled and he swore by the most solemn of\noaths that I should go with him and search his house. I replied,\n\"By Allah I will not go, unless the Wali go with us; for, an he\nbe present, he and the officers, thou wilt not dare to work thy\nwicked will upon me.\" So the Kazi rose and swore an oath, saying,\n\"By the truth of Him who created mankind, we will not go but with\nthe Emir!\" Accordingly we repaired to the Judge's house,\naccompanied by the Chief, and going up, searched it through, but\nfound naught; whereat fear fell upon me and the Wali turned to me\nand said, \"Fie upon thee, O ill-omened fellow! thou hast put us\nto shame before the men.\" All this, and I wept and went round\nabout right and left, with the tears running down my face, till\nwe were about to go forth and drew near the door of the house. I\nlooked at the place which the woman had mentioned and asked,\n\"What is yonder dark place I see?\" Then said I to the men, \"Pull\nup[FN#41] this jar with me.\" They did my bidding and I saw\nsomewhat appearing under the jar and said, \"Rummage and look at\nwhat is under it.\" So they searched, and behold, they came upon a\nwoman's mantilla and petticoat-trousers full of blood, which when\nI espied, I fell down in a fainting-fit. Now when the Wali saw\nthis, he said, \"By Allah, the Captain is excused!\" Then my\ncomrades came round about me and sprinkled water on my face till\nI recovered, when I arose and accosting the Kazi (who was covered\nwith confusion), said to him, \"Thou seest that suspicion is\nfallen on thee, and indeed this affair is no light matter,\nbecause this woman's family will assuredly not sit down quietly\nunder her loss.\" Therewith the Kazi's heart quaked and fluttered\nfor that he knew the suspicion had reverted upon him, wherefore\nhis colour yellowed and his limbs smote together; and he paid of\nhis own money, after the measure of that he had lost, so we would\nquench that fire for him.[FN#42] Then we departed from him in\npeace, whilst I said within myself, \"Indeed, the woman falsed me\nnot.\" After that I tarried till three days had passed, when I\nwent to the Hammam and changing my clothes, betook myself to her\nhome, but found the door shut and covered with dust. So I asked\nthe neighbours of her and they answered, \"This house hath been\nempty of habitants these many days; but three days agone there\ncame a woman with an ass, and at supper-time last night she took\nher gear and went away.\" Hereat I turned back, bewildered in my\nwit, and for many a day after I inquired of the dwellers in that\nstreet concerning her, but could happen on no tidings of her. And\nindeed I wondered at the eloquence of her tongue and the\nreadiness of her talk; and this is the most admirable of all I\nhave seen and of whatso hath betided me. When Al-Malik al-Zahir\nheard the tale of Mu'in al-Din, he marvelled thereat. Then rose\nanother constable and said, \"O lord, hear what befel me in bygone\ndays.\"\n\n\n\n\nThe Second Constable's History.\n\n\n\nI was once an overseer in the household of the Emir Jam\u00e1l al-Din\nal-Atwash al-Mujhidi, who was made governor of the two provinces,\nShark\u00edyah and Gharb\u00edyah,[FN#43] and I was dear to his heart and\nhe hid from me naught of whatso he desired to do; and he was eke\nmaster of his reason.[FN#44] It came to pass one day of the days\nthat it was reported to him how the daughter of Such-an-one had a\nmint of monies and raiment and ornaments and at that present she\nloved a Jewish man, whom every day she invited to be private with\nher, and they passed the light hours eating and drinking in\ncompany and he lay the night with her. The Wali feigned not to\nbelieve a word of this story, but he summoned the watchmen of the\nquarter one night and questioned them of this tittle-tattle.\nQuoth one of them, \"As for me, O my lord, I saw none save a\nJew[FN#45] enter the street in question one night; but I have not\nmade certain to whom he went in;\" and quoth the Chief, \"Keep\nthine eye on him from this time forward and note what place he\nentereth.\" So the watchman went out and kept his eye on the\nJudaean. One day as the Prefect sat in his house, the watchman\ncame in to him and said, \"O my lord, in very sooth the Jew goeth\nto the house of Such-an-one.\" Whereupon Al-Atwash sprang to his\nfeet and went forth alone, taking with him none save\nmyself.\"[FN#46] As he went along, he said to me, \"Indeed, this\ngirl is a fat piece of meat.\"[FN#47] And we gave not over going\ntill we came to the door of the house and stood there until a\nhand-maid came out, as if to buy them something wanted. We waited\ntill she opened the door, whereupon, without question or answer,\nwe forced our way into the house and rushed in upon the girl,\nwhom we found seated with the Jew in a saloon with four da\u00efses,\nand cooking-pots and candles therein. When her eyes fell on the\nWali, she knew him and rising to her feet, said, \"Well come and\nwelcome and fair cheer! By Allah, great honour hath betided me by\nmy lord's visit and indeed thou dignifiest my dwelling.\" Hereat\nshe carried him up to the dais and seating him on the couch,\nbrought him meat and wine and gave him to drink; after which she\nput off all that was upon her of raiment and ornaments and tying\nthem up in a kerchief, said to him, \"O my lord, this is thy\nportion, all of it.\" Then she turned to the Jew and said to him,\n\"Rise, thou also, and do even as I:\" so he arose in haste and\nwent out very hardly crediting his deliverance.[FN#48] When the\ngirl was assured of his escape, she put out her hand to her\nclothes and jewels and taking them, said to the Chief, \"O Emir,\nis the requital of kindness other than kindness? Thou hast\ndeigned to visit me and eat of my bread and salt; so now arise\nand depart from us without ill-doing; or I will give a single\noutcry and all who are in the street will come forth.\" So the\nEmir went out from her, without having gotten a single dirham;\nand on this wise she delivered the Jew by the seemliness of her\nstratagem. The company admired this tale, and as for the Wali and\nAl-Malik al-Zahir, they said, \"Ever devised any the like of this\ndevice?\" and they marvelled with the utterest of marvel. Then\narose a third constable and said, \"Hear what betided me, for it\nis yet stranger and rarer.\"\n\n\n\n\nThe Third Constable's History.\n\n\n\nI was one day abroad on business with certain of my comrades;\nand, as we walked along behold, we fell in with a company of\nwomen, as they were moons, and among them one, the tallest of\nthem and the handsomest. When I saw her and she saw me, she\nlagged behind her companions and waited for me till I came up to\nher and bespake her. Quoth she, \"O my lord (Allah favour thee!) I\nsaw thee prolong thy looking on me and I fancied that thou\nknewest me. An it be thus, let me learn more of thee.\" Quoth I,\n\"By Allah, I know thee not, save that the Most High Lord hath\ncast the love of thee into my heart and the goodliness of thy\nqualities hath confounded me; and that wherewith the Almighty\nhath gifted thee of those eyes that shoot with shafts hath\ncaptivated me.\" And she rejoined, \"By Allah, indeed I feel the\nlike of that which thou feelest; ay, and even more; so that\nmeseemeth I have known thee from childhood.\" Then said I, \"A man\ncannot well effect all whereof he hath need in the\nmarket-places.\" She asked me, \"Hast thou a house?\" and I\nanswered, \"No, by Allah, nor is this city my dwelling-place.\"\nRejoined she, \"By Allah, nor have I a place; but I will contrive\nfor thee.\" Then she went on before me and I followed her till she\ncame to a lodging-house[FN#49] and said to the Housekeeper, \"Hast\nthou an empty room?\" The other replied, \"Yes:\"[FN#50] and my\nmistress said, \"Give us the key.\" So we took the key and going up\nto see the room, entered to inspect it; after which she went out\nto the Housekeeper and giving her a dirham, said to her \"Take the\ndouceur of the key[FN#51] for the chamber pleaseth us, and here\nis another dirham for thy trouble. Go, fetch us a gugglet of\nwater, so we may refresh ourselves and rest till siesta-time pass\nand the heat decline, when the man will depart and bring our bag\nand baggage.\" Therewith the Housekeeper rejoiced and brought us a\nmat, two gugglets of water on a tray, a fan and a leather rug. We\nabode thus till the setting-in of mid-afternoon, when she said,\n\"Needs must I make the Ghusl-ablution ere I fare.\"[FN#52] Said I,\n\"Get water wherewith we may both wash,\" and drew forth from my\npocket a score or so of dirhams, thinking to give them to her;\nbut she cried, \"Refuge with Allah!\" and brought out of her pocket\na handful of silver, saying, \"But for destiny and that the\nAlmighty hath caused the love of thee fall into my heart, there\nhad not happened that which hath happened.\" Quoth I, \"Accept this\nin requital of that which thou hast spent;\" and quoth she, \"O my\nlord, by and by, whenas mating is prolonged between us, thou wilt\nsee if the like of me looketh unto money and means or no.\" Then\nthe lady took a jar of water and going into the lavatory, made\nthe Ghusl-ablution[FN#53] and presently coming forth, prayed the\nmid-afternoon prayer and craved pardon of Allah Almighty for the\nsin into which she had fallen. Now I had asked her name and she\nanswered, \"Rayh\u00e1nah,\"[FN#54] and described to me her\ndwelling-place. When I saw her make the ablution, I said within\nmyself, \"This woman doth on this wise, and shall I not do the\nlike of her doing?\" Then quoth I to her, \"Peradventure[FN#55]\nthou wilt seek us another jar of water?\" Accordingly she went out\nto the Housekeeper and said to her, \"O my sister, take this Nusf\nand fetch us for it water wherewith we may wash the\nflags.\"[FN#56] So the Housekeeper brought two jars of water and I\ntook one of them and giving her my clothes, entered the lavatory\nand bathed. When I had made an end of bathing, I cried out,\nsaying, \"Harkye, my lady Rayhanah!\" However none answered me. So\nI went out and found her not; but I did find that she had taken\nmy clothes and all that was in them of silver, to wit, four\nhundred dirhams. She had also carried off my turband and my\nkerchief and I lacked the wherewithal to veil my shame; so I\nsuffered somewhat than which death is less grievous and abode\nlooking about the place, hoping that haply I might espy a rag\nwherewith to hide my nakedness. Then I sat a little and presently\ngoing up to the door, smote upon it; whereat up came the\nHousekeeper and I said to her, \"O my sister, what hath Allah done\nwith the woman who was here?\" She replied, \"The lady came down\njust now and said, 'I'm going to cover the boys with the\nclothes,' adding, 'and I have left him sleeping; an he awake,\ntell him not to stir till the clothes come to him.'\" Then cried\nI, \"O my sister, secrets are safe with the fair-dealing and the\nfreeborn. By Allah, this woman is not my wife, nor ever in my\nlife have I seen her before this day!\" And I recounted to her the\nwhole affair and begged of her to cover me, informing her that my\nprivate parts were clean unconcealed. She laughed and cried out\nto the women of the lodging-house, saying, \"Ho, F\u00e1timah! Ho,\nKhad\u00edjah! Ho, Har\u00edfah! Ho, San\u00ednah!\" Whereupon all those who were\nin the place of women and neighbours flocked to me and fell\na-mocking me and saying, \"O pimp,[FN#57] what hadst thou to do\nwith gallantry?\" Then one of them came and looked in my face and\nlaughed, and another said, \"By Allah, thou mightest have known\nthat she lied, from the time she said she liked thee and was in\nlove with thee! What is there in thee to love?\" A third said,\n\"This is an old man without wisdom;\" and all vied one with other\nin exercising their wits upon me, I suffering mighty sore\nchagrin. However, one of the women took compassion on me after a\nwhile, and brought me a rag of thin stuff and cast it on me. With\nthis I covered my shame, and no more, and abode awhile thus: then\nsaid I in myself, \"The husbands of these women will presently\ngather together upon me and I shall be disgraced.\" So I went out\nby another door of the lodging-house, and young and old crowded\nabout me, running after me and crying, \"A madman! A\nmadman![FN#58] till I came to my house and knocked at the door;\nwhereupon out came my wife and seeing me naked, tall, bare of\nhead, cried out and ran in again, saying, \"This is a maniac, a\nSatan!\" But, when my family and spouse knew me, they rejoiced and\nsaid to me, \"What aileth thee?\" I told them that thieves had\ntaken my clothes and stripped me and had been like to slay me;\nand when I assured them that the rogues would have slaughtered\nme, they praised Allah Almighty and gave me joy of my safety. So\nconsider the craft this woman practised upon me, and I pretending\nto cleverness and wiliness. Those present marvelled at this story\nand at the doings of women; then came forward a fourth constable\nand said, \"Now that which hath betided me of strange adventures\nis yet stranger than this, and 'twas after the following\nfashion.\"\n\n\n\n\nThe Fourth Constable's History.\n\n\n\nWe were sleeping one night on the terrace-roof, when a woman made\nher way through the darkness into the house and, gathering into a\nbundle all that was therein, took it up that she might go away\nwith it. Now she was big with child and nigh upon her time of\ndelivery; so, when she packed up the bundle and prepared to\nshoulder it and make off with it, she hastened the coming of the\nlabour-pangs and bare a child in the dark. Then she sought for\nthe fire-sticks and when they burned, kindled the lamp and went\nround about the house with the little one, and it was weeping.\nThe wail awoke us, as we lay on the roof, and we marvelled. So we\nrose to see what was to do, and looking down through the opening\nof the saloon,[FN#59] saw a woman, who had lit the lamp, and\nheard the little one crying. As we were peering, she heard our\nwords and raising her head to us, said, \"Are ye not ashamed to\ndeal thus with us and bare our shame? Wist ye not that the day\nbelongeth to you and the night to us? Begone from us! By Allah,\nwere it not that ye have been my neighbours these many years, I\nwould assuredly[FN#60] bring down the house upon you!\" We doubted\nnot but that she was of the Jinn and drew back our heads; but,\nwhen we rose on the morrow, we found that she had taken all that\nwas with us and made off with it;[FN#61] wherefore we knew that\nshe was a thief and had practised on us a device, such as was\nnever before practised; and we repented, whenas repentance\navailed us naught. The company, hearing this tale, marvelled\nthereat with the utmost marvelling. Then the fifth constable, who\nwas the lieutenant of the bench,[FN#62] came forward and said,\n\"This is no wonder and there befel me a story which is rarer and\nstranger than this.\"\n\n\n\n\nThe Fifth Constable's History.\n\n\n\nAs I sat one day at the door of the Prefecture, behold, a woman\nsuddenly entered and said as though consulting me. \"O my lord, I\nam the wife of Such-an-one the Leach, and with him is a company\nof the notables[FN#63] of the city, drinking fermented drinks in\nsuch a place.\" When I heard this, I misliked to make a scandal;\nso I bluffed her off and sent her away unsatisfied. Then I rose\nand walked alone to the place in question and sat without till\nthe door opened, when I rushed in and entering, found the company\neven as the woman aforesaid had set out, and she herself with\nthem. I saluted them and they returned my salam and rising,\ntreated me with honour and seated me and served me with meat.\nThen I informed them how one had denounced them to me, but I had\ndriven him away and had come to them by myself; so they thanked\nme and praising me for my kindness, brought out to me from among\nthem two thousand dirhams[FN#64] and I took them and went away.\nNow two months after this adventure, there came to me one of the\nKazi's officers, with a paper, wherein was the judge's writ,\nsummoning me to him. So I accompanied the officer and went in to\nthe Kazi, whereupon the plaintiff, he who had taken out the\nsummons, sued me for two thousand dirhams, declaring I had\nborrowed them of him as the agent or guardian of the woman. I\ndenied the debt, but he produced against me a bond for that sum,\nattested by four of those who were in company on the occasion;\nand they were present and bore witness to the loan. I reminded\nthem of my kindness and paid the amount, swearing that I would\nnever again follow a woman's counsel. Is not this marvellous? The\ncompany admired the goodliness of his tale and it pleased\nAl-Malik al-Zahir; and the Wali said, \"By Allah, this is a\nstrange story!\" Then came forward the sixth constable and said to\nthose present, \"Hear my adventure and that which befel me, to\nwit, that which befel Such-an-one the Assessor, for 'tis rarer\nthan this and finer.\"\n\n\n\n\nThe Sixth Constable's History.\n\n\n\nA certain Assessor one day of the days was taken with a woman and\nmuch people assembled before his house and the Lieutenant of\npolice and his posse came to him and rapped at the door. The\nAssessor peered from house-top and seeing the folk, said, \"What\ndo ye want?\" Replied they, \"Speak with the Lieutenant of police\nSuch-an-one.\" So he came down and as he opened the door they\ncried to him, \"Bring forth the woman who is with thee.\" \"Are ye\nnot ashamed? How shall I bring forth my wife?\" \"Is she thy wife\nby book[FN#65] or without marriage-lines?\" \"She is my wife\naccording to the Book of Allah and the Institutes of His\nApostle.\" \"Where is the contract?\" \"Her lines are in her mother's\nhouse.\" \"Arise thou and come down and show us the writ.\" \"Go from\nher way, so she may come forth.\" Now, as soon as he got wind of\nthe matter, he had written the bond and fashioned it after the\nfashion of his wife,[FN#66] to suit with the case, and he had\nwritten therein the names of certain of his friends to serve as\nwitnesses and forged the signatures of the drawer and the wife's\nnext friend and made it a contract of marriage with his wife and\na legal deed.[FN#67] Accordingly, when the woman was about to go\nout from him, he gave her the contract he had forged, and the\nEmir sent with her a servant of his, to carry her home to her\nfather. So the servant went with her and when she was inside she\nsaid to him, \"I will not return to the citation of the Emir: but\nlet the Assessors present themselves and take my contract.\"\nHereupon the servant carried this message to the Lieutenant of\npolice, who was standing at the Assessor's door, and he said,\n\"This is permissible.\" Then said the Assessor to the servant,\n\"Fare, O eunuch, and fetch us Such-an-one the Notary;\" for that\nhe was his friend and 'twas he whose name he had forged as the\ndrawer-up of the contract.[FN#68] So the Lieutenant sent after\nhim and fetched him to the Assessor, who, when he saw him, said\nto him, \"Get thee to Such-an-one, her with whom thou marriedst\nme, and cry out upon her, and when she cometh to thee,[FN#69]\ndemand of her the contract and take it from her and bring it to\nus.\" And he signed to him, as much as to say, \"Bear me out in the\nlie and screen me, for that she is a strange woman and I[FN#70]\nam in fear of the Lieutenant who standeth at the door; and we\nbeseech Allah Almighty to screen us and you from the woes of this\nworld. Amen.\" So the Notary went up to the Lieutenant, who was\namong the witnesses, and said, \" 'Tis well. Is she not Such-an-\none whose marriage-contract we drew up in such a place?\" Then he\nbetook himself to the woman's house and cried out upon her;\nwhereat she brought him the forged contract and he took it and\nreturned with it to the Lieutenant of police.[FN#71] When the\nofficer had taken cognizance of the document and professed\nhimself satisfied, the Assessor said to the Notary, \"Go to our\nlord and master, the Kazi of the Kazis, and acquaint him with\nthat which befalleth his Assessors.\" The Notary rose to go, but\nthe Lieutenant feared for himself and was urgent in beseeching\nthe Assessor and in kissing his hands till he forgave him;\nwhereupon the Lieutenant went away in the utmost concern and\naffright. On such wise the Assessor ordered the case and carried\nout the forgery and feigned marriage with the woman; and thus\nescaped calumny and calamity by the seemliness of his\nstratagem.[FN#72] The folk marvelled at this with the uttermost\nmarvel and the seventh constable said, \"There befel me in\nAlexandria the God-guarded a wondrous thing, and 'twas\nthis.\"[FN#73]\n\n\n\n\nThe Seventh Constable's History.\n\n\n\nThere came one day an old woman to the stuff-bazar, with a casket\nof mighty fine workmanship, containing trinkets, and she was\naccompanied by a young baggage big with child. The crone sat down\nat the shop of a draper and giving him to know that the girl was\npregnant by the Prefect[FN#74] of Police of the city, took of\nhim, on credit, stuffs to the value of a thousand dinars and\ndeposited with him the casket as security. She opened the casket\nand showed him that which was therein and he found it full of\ntrinkets of price; so he trusted her with the goods and she\nfarewelled him and carrying the stuffs to the girl who was with\nher, went her way. Then the old woman was absent from him a great\nwhile, and when her absence was prolonged, the draper despaired\nof her; so he went up to the Prefect's house and asked anent the\nwoman of his household who had taken his stuffs on credit; but\ncould obtain no tidings of her nor happen on any trace of her.\nThen he brought out the casket of jewellery and showed it to\nexperts, who told him that the trinkets were gilt and that their\nworth was but an hundred dirhams. When he heard this, he was sore\nconcerned thereat and presenting himself before the Deputy of the\nSultan made his complaint to him; whereupon the official knew\nthat a sleight had been served upon him and that the sons of\nAdam[FN#75] had cozened him and conquered him and cribbed his\nstuffs. Now the magistrate in question was a man of experience\nand judgment, well versed in affairs; so he said to the draper,\n\"Remove somewhat from thy shop, including the casket, and to-\nmorrow morning break the lock and cry out and come to me and\ncomplain that they have plundered all thy shop.[FN#76] Also mind\nthou call upon Allah for aid and wail aloud and acquaint the\npeople, so that a world of folk may flock to thee and sight the\nbreach of the lock and that which is missing from thy shop: and\non this wise display it to every one who presenteth himself that\nthe news may be noised abroad, and tell them that thy chief\nconcern is for a casket of great value, deposited with thee by a\ngreat man of the town and that thou standest in fear of him. But\nbe thou not afraid and still say ever and anon in thy saying, 'My\ncasket was the casket of Such-an-one, and I fear him and dare not\nbespeak him; but you, O company and all ye who are present, I\ncall you to witness of this for me.' And if there be with thee\nmore than this saying, say it; and the old woman will assuredly\ncome to thee.\" The draper answered with \"To hear is to obey\" and\ngoing forth from the Deputy's presence, betook himself to his\nshop and brought out thence the casket and a somewhat making a\ngreat display, which he removed to his house. At break of day he\narose and going to his shop, broke the lock and shouted and\nshrieked and called on Allah for aid, till each and every of the\nfolk assembled about him and all who were in the city were\npresent, whereupon he cried out to them, saying even as the\nPrefect had bidden him; and this was bruited abroad. Then he made\nfor the Prefecture and presenting himself before the Chief of\nPolice, cried out and complained and made a show of distraction.\nAfter three days, the old woman came to him and bringing him the\nthousand dinars, the price of the stuffs, demanded the\ncasket.[FN#77] When he saw her, he seized her and carried her to\nthe Prefect of the city; and when she came before the Kazi, he\nsaid to her, \"Woe to thee O Sataness; did not thy first deed\nsuffice thee, but thou must come a second time?\" She replied, \"I\nam of those who seek their salvation[FN#78] in the cities, and we\nforegather every month: and, yesterday we foregathered.\" He asked\nher, \"Canst thou cause me to catch them?\" and she answered, \"Yes;\nbut, an thou wait till to-morrow, they will have dispersed; so I\nwill deliver them to thee to-night.\" The Emir said to her, \"Go;\"\nand said she, \"Send with me one who shall go with me to them and\nobey me in whatso I shall say to him, and all that I bid him he\nshall not gainsay and therein conform to my way.\" Accordingly, he\ngave her a company of men and she took them and bringing them to\na certain door, said to them, \"Stand ye here, at this door, and\nwhoso cometh out to you seize him; and I will come out to you\nlast of all.\" \"Hearing and obeying,\" answered they and stood at\nthe door, whilst the crone went in. They waited a whole hour,\neven as the Sultan's deputy had bidden them, but none came out to\nthem and their standing waxed longsome, and when they were weary\nof waiting, they went up to the door and smote upon it a heavy\nblow and a violent, so that they came nigh to break the wooden\nbolt. Then one of them entered and was absent a long while, but\nfound naught; so he returned to his comrades and said to them,\n\"This is the door of a dark passage, leading to such a\nthoroughfare; and indeed she laughed at you and left you and went\naway.\"[FN#79] When they heard his words, they returned to the\nEmir and acquainted him with the case, whereby he knew that the\nold woman was a cunning craft-mistress and that she had mocked at\nthem and cozened them and put a cheat on them, to save herself.\nWitness, then, the wiles of this woman and that which she\ncontrived of guile, for all her lack of foresight in presenting\nherself a second time to the draper and not suspecting that his\nconduct was but a sleight; yet, when she found herself hard upon\ncalamity, she straightway devised a device for her deliverance.\nWhen the company heard the seventh constable's story, they were\nmoved to mirth galore, than which naught could be more; and\nAl-Malik al Zahir B\u00edbars rejoiced in that which he heard and\nsaid, \"Verily, there betide things in this world wherefrom kings\nare shut out, by reason of their exalted degree!\" Then came\nforward another person from amongst the company and said, \"There\nhath reached me through one of my friends a similar story bearing\non the malice of women and their wiles, and it is more wondrous\nand marvellous, more diverting and more delectable than all that\nhath been told to you.\" Quoth the company there present, \"Tell us\nthy tale and expound it unto us, so we may see that which it hath\nof extraordinary.\" And he began to relate\n\n\n\n\nThe Eighth Constable's History.\n\n\n\nYe must know that a company, amongst whom was a friend of mine,\nonce invited me to an entertainment; so I went with him, and when\nwe came into his house and sat down on his couch, he said to me,\n\"This is a blessed day and a day of gladness, and who is he that\nliveth to see the like of this day? I desire that thou practice\nwith us and disapprove not our proceedings, for that thou hast\nbeen accustomed to fall in with those who offer this.\"[FN#80] I\nconsented thereto and their talk happened upon the like of this\nsubject.[FN#81] Presently, my friend, who had invited me, arose\nfrom among them and said to them, \"Listen to me and I will\nacquaint you with an adventure which happened to me. There was a\ncertain person who used to visit me in my shop, and I knew him\nnot nor he knew me, nor ever in his life had he seen me; but he\nwas wont, whenever he wanted a dirham or two, by way of loan, to\ncome to me and ask me, without acquaintance or introduction\nbetween me and him, and I would give him what he required. I told\nnone of him, and matters abode thus between us a long while till\nhe began a-borrowing at a time ten or twenty dirhams, more or\nless. One day, as I stood in my shop, behold, a woman suddenly\ncame up to me and stopped before me; and she was a presence as\nshe were the full moon rising from among the constellations, and\nthe place was a-light by her light. When I saw her, I fixed my\neyes on her and stared in her face; and she fell to bespeaking me\nwith soft voice. When I heard her words and the sweetness of her\nspeech, I lusted after her; and as soon as she saw that I longed\nfor her, she did her errand and promising me an assignation, went\naway, leaving my thoughts occupied with her and fire a-flame in\nmy heart. Accordingly I abode, perplexed and pondering my affair,\nthe fire still burning in my heart, till the third day, when she\ncame again and I could hardly credit her coming. When I saw her,\nI talked with her and cajoled her and courted her and craved her\nfavour with speech and invited her to my house; but, hearing all\nthis, she only answered, \"I will not go up into any one's house.\"\nQuoth I, \"I will go with thee\" and quoth she, \"Arise and come\nwith me.\" So I rose and putting into my sleeve a kerchief,\nwherein was a fair sum of silver and a considerable, followed the\nwoman, who forwent me and ceased not walking till she brought me\nto a lane and to a door, which she bade me unlock. I refused and\nshe opened it and led me into the vestibule. As soon as I had\nentered, she bolted the entrance door from within and said to me,\n\"Sit here till I go in to the slave-girls and cause them enter a\nplace whence they shall not see me.\" \"'Tis well,\" answered I and\nsat down: whereupon she entered and was absent from me an\neye-twinkling, after which she returned to me, without a veil,\nand straightway said, \"Arise and enter in the name of Allah.\" So\nI arose and went in after her and we gave not over going till we\nreached a saloon. When I examined the place, I found it neither\nhandsome nor pleasant, but desolate and dreadful without symmetry\nor cleanliness; indeed, it was loathsome to look upon and there\nwas in it a foul smell. After this inspection I seated myself\namiddlemost the saloon, misdoubting; and lo and behold! as I sat,\nthere came down on me from the dais a body of seven naked men,\nwithout other clothing than leather belts about their waists. One\nof them walked up to me and took my turband, whilst another\nseized my kerchief that was in my sleeve, with my money, and a\nthird stripped me of my clothes; after which a forth came and\nbound my hands behind my back with his belt. Then they all took\nme up, pinioned as I was, and casting me down, fell a-haling me\ntowards a sink-hole that was there and were about to cut my\nthroat, when suddenly there came a violent knocking at the door.\nAs they heard the raps, they were afraid and their minds were\ndiverted from me by affright; so the woman went out and presently\nreturning, said to them, \"Fear not; no harm shall betide you this\nday. 'Tis only your comrade who hath brought you your dinner.\"\nWith this the new-comer entered, bringing with him a roasted\nlamb; and when he came in to them, he asked, \"What is to do with\nyou, that ye have tucked up sleeves and bag-trousers?\" Replied\nthey, \"This is a head of game we've caught.\" As he heard these\nwords, he came up to me and peering in my face, cried out and\nsaid, \"By Allah, this is my brother, the son of my mother and\nfather! Allah! Allah!\" Then he loosed me from my pinion-bonds and\nbussed my head, and behold it was my friend who used to borrow\nsilver of me. When I kissed his head, he kissed mine and said, \"O\nmy brother, be not affrighted;\" and he called for my clothes and\ncoin and restored all to me nor was aught missing. Also, he\nbrought me a porcelain bowl full of sherbet of sugar, with lemons\ntherein, and gave me to drink; and the company came and seated me\nat a table. So I ate with them and he said to me, \"O my lord and\nmy brother, now have bread and salt passed between us and thou\nhast discovered our secret and our case; but secrets with the\nnoble are safe.\" I replied, 'As I am a lawfully-begotten child\nand a well-born, I will not name aught of this nor denounce you!\"\nThey assured themselves of me by an oath; then they brought me\nout and I went my way, very hardly crediting but that I was of\nthe dead. I lay ill in my house a whole month; after which I went\nto the Hammam and coming out, opened my shop and sat selling and\nbuying as was my wont, but saw no more of that man or that woman\ntill, one day, there stopped before my shop a young\nTurkoman,[FN#82] as he were the full moon; and he was a\nsheep-merchant and had with him a leathern bag, wherein was\nmoney, the price of sheep he had sold. He was followed by the\nwoman, and when he stopped over against my shop, she stood by his\nside and cajoled him, and indeed he inclined to her with great\ninclination. As for me, I was dying of solicitude for him and\nbegan casting furtive glances at him and winked at him, till he\nchanced to look round and saw me signing to him; whereupon the\nwoman gazed at me and made a signal with her hand and went away.\nThe Turkoman followed her and I deemed him dead without a doubt;\nwherefore I feared with exceeding fear and shut my shop. Then I\njourneyed for a year's space and returning, opened my shop;\nwhereupon, behold, the woman as she walked by came up to me and\nsaid, \"This is none other than a great absence.\" I replied, \"I\nhave been on a journey;\" and she asked, \"Why didst thou wink at\nthe Turkoman?\" I answered, \"Allah forfend! I did not wink at\nhim.\" Quoth she, \"Beware lest thou thwart me;\" and went away.\nAwhile after this a familiar of mine invited me to his house and\nwhen I came to him, we ate and drank and chatted. Then he asked\nme, \"O my friend, hath there befallen thee aught of sore trouble\nin the length of thy life?\" Answered I, \"Tell me first, hath\nthere befallen thee aught?\" He rejoined, \"Know that one day I\nespied a fair woman; so I followed her and sued her to come home\nwith me. Quoth she, 'I will not enter any one's house but my own;\nso come thou to my home, an thou wilt, and be it on such a day.'\nAccordingly, on the appointed day, her messenger[FN#83] came to\nme, proposing to carry me to her; and when he announced his\npurpose I arose and went with him, till we arrived at a goodly\nhouse and a great door. He opened the door and I entered,\nwhereupon he bolted it behind me and would have gone in; but I\nfeared with exceeding fear and foregoing him to the second door,\nwhereby he would have had me enter, bolted it and cried out at\nhim, saying, 'By Allah, an thou open not to me, I will slay\nthee;[FN#84] for I am none of those whom thou canst readily\ncozen!' 'What deemest thou of cozening?' 'Verily, I am startled\nby the loneliness of the house and the lack of any keeper at its\ndoor; for I see none appear.' 'O my lord, this is a private\ndoor.' 'Private or public, open to me.' So he opened to me and I\nwent out and had gone but a little way from the door when I met a\nwoman, who said to me, 'A long life was fore-ordained to thee;\nelse hadst thou never come forth of yonder house.' I asked, 'How\nso?' and she answered, 'Enquire of thy friend Such-an-one,'\n(naming thee), 'and he will acquaint thee with strange things.'\nSo, Allah upon thee, O my friend, tell me what befel thee of\nwondrous and marvellous, for I have told thee what befel me.\" \"O\nmy brother, I am bound by a solemn oath.\" \"O my friend, false\nthine oath and tell me.\"[FN#85] \"Indeed, I dread the issue of\nthis.\" But he urged me till I told him all, whereat he marvelled.\nThen I went away from him and abode a long while, without further\nnews. One day, I met another of my friends who said to me, \"A\nneighbour of mine hath invited me to hear singers\" but I\nsaid:--\"I will not foregather with any one.\" However, he\nprevailed upon me; so we repaired to the place and found there a\nperson, who came to meet us and said, \"Bismillah!\"[FN#86] Then he\npulled out a key and opened the door, whereupon we entered and he\nlocked the door after us. Quoth I, \"We are the first of the folk;\nbut where be the singers' voices?\" He replied, \"They're within\nthe house: this is but a private door; so be not amazed at the\nabsence of the folk.\" My friend said to me, \"Behold, we are two,\nand what can they dare to do with us?\" Then he brought us into\nthe house, and when we entered the saloon, we found it desolate\nexceedingly and dreadful of aspect. Quoth my friend, \"We are\nfallen into a trap; but there is no Majesty and there is no Might\nsave in Allah, the Glorious, the Great!\" And quoth I, \"May God\nnever requite thee for me with good!\"[FN#87] Then we sat down on\nthe edge of the dais and suddenly I espied a closet beside me; so\nI peered into it and my friend asked me, \"What seest thou?\" I\nanswered, \"I see there wealth in store and corpses of murdered\nmen galore. Look.\" So he looked and cried, \"By Allah, we are down\namong the dead!\" and we fell a-weeping, I and he. As we were\nthus, behold, four men came in upon us, by the door at which we\nhad entered, and they were naked, wearing only leather belts\nabout their waists, and made for my friend. He ran at them and\ndealing one of them a blow with his swordpommel, knocked him\ndown, whereupon the other three rushed upon him. I seized the\nopportunity to escape while they were occupied with him, and\nespying a door by my side, slipped into it and found myself in an\nunderground room, without issue, even a window. So I made sure of\ndeath, and said, \"There is no Majesty and there is no Might save\nin Allah, the Glorious, the Great!\" Then I looked at the top of\nthe vault and saw in it a range of glazed and \nlunettes;[FN#88] so I clambered up for dear life, till I reached\nthe lunettes, and I out of my wits for fear. I made shift to\nremove the glass and scrambling out through the setting, found\nbehind them a wall which I bestrode. Thence I saw folk walking in\nthe street; so I cast myself down on the ground and Allah\nAlmighty preserved me, and when I reached the face of earth,\nunhurt, the folk flocked round me and I acquainted them with my\nadventure. Now as Destiny decreed, the Chief of Police was\npassing through the market-street; so the people told him what\nwas to do and he made for the door and bade raise it off its\nhinges. We entered with a rush and found the thieves, as they had\nthrown my friend down and cut his throat; for they occupied not\nthemselves with me, but said, \"Whither shall yonder fellow wend?\nVerily, he is in our grasp.\" So the Wali hent them with the\nhand[FN#89] and questioned them of their case, and they confessed\nagainst the woman and against their associates in Cairo. Then he\ntook them and went forth, after he had locked up the house and\nsealed it; and I accompanied him till he came without the first\nhouse. He found the door bolted from within; so he bade raise it\nand we entered and found another door. This also he caused pull\nup, enjoining his men to silence till the doors should be lifted,\nand we entered and found the band occupied with new game, whom\nthe woman had just brought in and whose throat they were about to\ncut. The Chief released the man and gave him back whatso the\nthieves had taken from him; and he laid hands on the woman and\nthe rest and took forth of the house a mint of money, with which\nthey found the purse of the Turkoman sheep-merchant. They at once\nnailed up the thieves against the house-wall, whilst, as for the\nwoman, they wrapped her in one of her mantillas and nailing her\nto a board, set her upon a camel and went round about the town\nwith her. Thus Allah razed their dwelling-places and did away\nfrom me that which I feared from them. All this befel, whilst I\nlooked on, and I saw not my friend who had saved me from them the\nfirst time, whereat I wondered to the utterest of wonderment.\nHowever, some days afterward, he came up to me, and indeed he had\nrenounced the world and donned a Fakir's dress; and he saluted me\nand went away.[FN#90] Then he again began to pay me frequent\nvisits and I entered into conversation with him and questioned\nhim of the band and how he came to escape, he alone of them all.\nHe replied, \"I left them from the day on which Allah the Most High\ndelivered thee from them, for that they would not obey my say; so\nI sware I would no longer consort with them.\" Quoth I, \"By Allah,\nI marvel at thee, for that assuredly thou wast the cause of my\npreservation!\" Quoth he, \"The world is full of this sort; and we\nbeseech the Almighty to send us safety, for that these wretches\npractice upon men with every kind of malpractice.\" Then I said to\nhim, \"Tell me the rarest adventure of all that befel thee in this\nvillainy thou wast wont to work.\" And he answered, \"O my brother,\nI was not present when they did such deeds, for that my part with\nthem was to concern myself with selling and buying and feeding\nthem; but it hath reached me that the rarest thing which befel\nthem was on this wise.\"\n\n\n\n\nThe Thief's Tale.\n\n\n\nThe woman who acted decoy for them and trapped their game and\nused to inveigle damsels from marriage-banquets, once caught them\na woman from a bride-feast, under pretence that she had a wedding\nin her own house, and fixed for her a day when she should come to\nher. As soon as the appointed time arrived, the woman presented\nherself and the other carried her into the house by a door,\ndeclaring that it was a private wicket. When she entered the\nsaloon, she saw men and braves[FN#91] and knew that she had\nfallen into a snare; so she looked at them and said, \"Harkye, my\nfine fellows![FN#92] I am a woman and in my slaughter there is no\nglory, nor have ye against me any feud of blood-wite wherefor ye\nshould pursue me; and that which is upon me of raiment and\nornaments ye are free to take as lawful loot.\" Quoth they, \"We\nfear thy denunciation;\" but quoth she, \"I will abide with you,\nneither coming in nor going out.\" So they said, \"We grant thee\nthy life.\" Then the Captain looked on her and she pleased him; so\nhe took her for himself, and she abode with him a whole year\ndoing her very best in their service, till they became familiar\nwith her and felt assured of her faith. One night of the nights\nshe plied them with drink and they drank till they became\ndrunken; whereupon she arose and took her clothes and five\nhundred dinars from the Captain; after which she fetched a razor\nand shaved off all their beards. Then she took soot from the\ncooking-pots and blackening their faces[FN#93] opened the doors\nand fared forth; and when the thieves recovered from their drink,\nthey abode confounded and knew that the woman had practiced upon\nthem. All present marvelled at this his story and the ninth\nconstable came forward and said, \"I will tell you a right\npleasant tale I heard at a wedding.\"\n\n\n\n\nThe Ninth Constable's History.\n\n\n\nA certain singing-girl was fair of favour and bruited of repute,\nand it happened one day that she fared forth to a garden\na-pleasuring. As she sat in the summer-house, behold, a man\nlopped of the hand stopped to beg of her, and suddenly entered in\nat the door. Then he touched her with his stump, saying, \"An\nalms, for the love of Allah!\"[FN#94] but she answered, \"Allah\nopen!\" and insulted him. Many days after this, there came to her\na messenger and gave her the hire of her going forth.[FN#95] So\nshe took with her a hand-maid and an accompanyist;[FN#96] and\nwhen she came to the place appointed, the messenger brought her\ninto a long passage, at the end whereof was a saloon. \"So\" (quoth\nshe) \"we entered therein and found nobody, but we saw the room\nmade ready for an entertainment with candles, dried fruits and\nwine, and in another place we saw food and in a third beds.\nThereupon we sat down and I looked at him who had opened the door\nto us, and behold he was lopped of the hand. I misliked this, and\nwhen I sat a little longer, there entered a man, who filled the\ncandelabra in the saloon and lit the waxen candles; and behold,\nhe also was handlopped. Then flocked the folk and there entered\nnone except he were lopped of the hand, and indeed the house was\nfull of these companions.[FN#97] When the session was complete,\nthe host came in and the company rose to him and seated him in\nthe place of honour. Now he was none other than the man who had\nfetched me, and he was clad in sumptuous clothes, but his hands\nwere in his sleeves, so that I knew not how it was with them.\nThey brought him food and he ate, he and the company; after which\nthey washed hands and the host began casting at me furtive\nglances. Then they drank till they were drunken, and when they\nhad taken leave of their wits, the host turned to me and said,\n'Thou dealtest not in friendly fashion with him who sought an\nalms of thee, and thou saidst to him, \"How loathsome art thou!\"'\u00b4\nI considered him and behold, he was the lophand who had accosted\nme in my pleasance.[FN#98] So I asked, 'O my lord, what is this\nthou sayest?' and he answered, 'Wait; thou shalt remember it.' So\nsaying, he shook his head and stroked his beard, whilst I sat\ndown for fear. Then he put out his hand to my mantilla and\nwalking-boots and laying them by his side, cried to me, 'Sing, O\naccursed!' Accordingly, I sang till I was tired out, what while\nthey occupied themselves with their case and drank themselves\ndrunk and the heat of their drink redoubled. Presently, the\ndoorkeeper came to me and said, 'O my lady, fear not; but when\nthou hast a mind to go, let me know.' Quoth I, 'Thinkest thou to\ndelude me?' and quoth he, 'Nay, by Allah! But I have ruth on thee\nfor that our Captain and chief purposeth thee no good and\nmethinketh he will kill thee this night.' Said I to him, 'An thou\nbe minded to do me a favour, now is its time;' and said he, 'When\nour Chief riseth to his need and goeth to the Chapel of Ease, I\nwill precede him with the light and leave the door open; and do\nthou wend whithersoever thou wiliest.' Then I sang and the\nCaptain cried, ''Tis good.' Replied I, 'Nay, but thou'rt\nloathsome.' He looked at me and rejoined, 'By Allah, thou shalt\nnever more scent the odour of the world!' But his comrades said\nto him, 'Do it not,' and gentled him, till he added, 'An it must\nbe so, and there be no help for it, she shall tarry here a whole\nyear and not fare forth.' My answer was, 'I am content to submit\nto whatso pleaseth thee: if I have failed in respect to thee,\nthou art of the clement.' He shook his head and drank, then arose\nand went out to do his need, whilst his comrades were occupied\nwith what they were about of merry-making and drunkenness and\nsport. So I winked to my friends and we all slipped out into the\ncorridor. We found the door open and fled forth, unveiled[FN#99]\nand unknowing whither we went; nor did we halt till we had fared\nafar from the house and happened on a Cook cooking, of whom I\nasked, 'Hast thou a mind to quicken the dead?' He said, 'Come\nup;' so we went up into the shop, and he whispered, 'Lie down.'\nAccordingly, we lay down and he covered us with the Halfah\ngrass,[FN#100] wherewith he was used to kindle the fire under the\nfood. Hardly had we settled ourselves in the place when we heard\na noise of kicking at the door and people running right and left\nand questioning the Cook and asking, 'Hath any one passed by\nthee?' Answered he, 'None hath passed by me.' But they ceased not\nto go round about the shop till the day broke, when they turned\nback, disappointed. Then the Cook removed the reeds and said to\nus, 'Rise, for ye are delivered from death.' So we arose, and we\nwere uncovered, sans veil or mantilla; but the Cook carried us up\ninto his house and we sent to our homes and fetched us veils; and\nwe repented to Allah Almighty and renounced singing, for indeed\nthis was a mighty narrow escape after stress.\"[FN#101] Those\npresent marvelled at this, and the tenth constable came forward\nand said, \"As for me, there befel me that which was yet rarer\nthan all ye have yet heard.\" Quoth Al-Malik al-Zahir, \"What was\nthat?\" And quoth he, \"Deign give ear to me.\"\n\n\n\n\nThe Tenth Constable's History.\n\n\n\nA robbery of stuffs had been committed in the city and as it was\na great matter I was cited,[FN#102] I and my fellows:\nthey[FN#103] pressed hard upon us: but we obtained of them some\ndays' grace and dispersed in search of the stolen goods. As for\nme, I sallied forth with five men and went round about the city\nthat day; and on the morrow we fared forth into the suburbs. When\nwe found ourselves a parasang or two parasangs away from the\ncity, we waxed athirst; and presently we came to a garden. There\nI went in alone and going up to the waterwheel,[FN#104] entered\nit and drank and made the Wuzu-ablution and prayed. Presently, up\ncame the keeper of the garden and said to me, \"Woe to thee! Who\nbrought thee to this water-wheel?\" and he smote me and squeezed my\nribs[FN#105] till I was like to die. Then he bound me with one of\nhis bulls and made me work the water-wheel, flogging me as I\nwalked round with a cattle-whip[FN#106] he had with him, till my\nheart was a-fire; after which he loosed me and I went out,\nknowing not the way. Now when I came forth, I fainted: so I sat\ndown till my trouble subsided; then I made for my comrades and\nsaid to them, \"'I have found money and malefactor, and I\naffrighted him not neither troubled him, lest he should flee; but\nnow, come, let us go to him, so we may contrive to lay hold upon\nhim.\" Then I took them and we repaired to the keeper of the\ngarden, who had tortured me with tunding, with the intent to make\nhim taste the like of that which he had done with me and lie\nagainst him and cause him eat many a stick. So we rushed to the\nwaterwheel and seized the keeper. Now there was with him a youth\nand, as we were pinioning the gardener, he said, \"By Allah, I was\nnot with him and indeed 'tis six months since I entered this\ncity, nor did I set eyes on the stuffs until they were brought\nhither.\" Quoth we, \"Show us the stuffs;\" upon which he carried us\nto a place wherein was a pit, beside the waterwheel, and digging\nthere, brought out the stolen goods with not a thread or a stitch\nof them missing. So we took them and carried the keeper to the\nPrefecture of Police where we stripped him and beat him with\npalm-rods till he confessed to thefts manifold. Now I did this by\nway of mockery against my comrades, and it succeeded. The company\nmarvelled at this story with the utmost marvelling, and the\neleventh constable rose and said, \"I know a story yet stranger\nthan this: but it happened not to myself.\"\n\n\n\n\nThe Eleventh Constable's History.\n\n\n\nThere was once in times of yore a Chief Officer of Police and\nthere passed by him one day of the days a Jew, hending in hand a\nbasket wherein were five thousand dinars; whereupon quoth that\nofficer to one of his slaves, \"Art able to take that money from\nyonder Jew's basket?\" \"Yes,\" quoth he, nor did he tarry beyond\nthe next day ere he came to his lord, bringing the basket. \"So\"\n(said the officer) \"I bade him 'Go, bury it in such a place;'\nwhereupon he went and buried it and returned and told me. Hardly\nhad he reported this when there arose a clamour like that of\nDoomsday and up came the Jew, with one of the King's officers,\ndeclaring that the gold pieces belonged to the Sultan and that he\nlooked to none but us for it. We demanded of him three days'\ndelay, according to custom and I said to him who had taken the\nmoney, 'Go and set in the Jew's house somewhat that shall occupy\nhim with himself.' Accordingly he went and played a mighty fine\ntrick, which was, he laid in a basket a dead woman's hand,\npainted with henna and having a gold seal-ring on one of the\nfingers, and buried that basket under a slab in the Jew's home.\nThen we came and searched and found the basket, whereupon without\na moment of delay we clapped the Jew in irons for the murder of a\nwoman. As soon as it was the appointed time, there entered to us\nthe man of the Sultan's guards, who had accompanied the Jew, when\nhe came to complain of the loss of the money,[FN#107] and said,\n'The Sultan sayeth to you, Nail up[FN#108] the Jew and bring the\nmoney, for there is no way by which five thousand gold pieces can\nbe lost.' Wherefore we knew that our device did not suffice. So I\nwent forth and finding a young man, a Haur\u00e1ni,[FN#109] passing\nalong the road, laid hands on him forthright and stripped him,\nand whipped him with palm-rods. Then I threw him in jail, ironed,\nand carrying him to the Prefecture, beat him again, saying to\nthem, 'This be the robber who stole the coin.' And we strove to\nmake him confess; but he would not. Accordingly, we beat him a\nthird and a fourth time, till we were aweary and exhausted and he\nbecame unable to return a reply; but, when we had made an end of\nbeating and tormenting him, he said, 'I will fetch the money this\nvery moment.' Presently we went with him till he came to the\nplace where my slave had buried the gold and he dug there and\nbrought it out; whereat I marvelled with the utmost marvel and we\ncarried it to the Prefect's house. When the Wali saw the money\nand made sure of it with his own eyes, he rejoiced with joy\nexceeding and bestowed on me a robe of honour. Then he restored\nthe coin straightway to the Sultan and we left the youth in\ndurance vile; whilst I said to my slave who had taken the money,\n'Say me, did yonder young man see thee, what time thou buriedst\nthe money?' and he replied, 'No, by Allah the Great!' So I went\nin to the young man, the prisoner, and plied him with\nwine[FN#110] till he recovered, when I said to him, 'Tell me how\nthou stolest the money?' Answered he, 'By Allah, I stole it not,\nnor did I ever set eyes on it till I brought it forth of the\nearth!' Quoth I, 'How so?' and quoth he, 'Know that the  cause of\nmy falling into your hands was my parent's imprecation against\nme; because I entreated her evilly yesternight and beat her and\nshe said to me, 'By Allah, O my son, the Lord shall assuredly gar\nthe oppressor prevail over thee!' Now she is a pious woman. So I\nwent out forthright and thou sawest me on my way and didst that\nwhich thou didst; and when beating was prolonged on me, my senses\nfailed me and I heard a voice saying to me, 'Fetch it.' So I said\nto you what I said and the Speaker[FN#111] guided me till I came\nto the place and there befel what befel of the bringing out of\nthe money.' I admired this with the utmost admiration and knew\nthat he was of the sons of the pious. So I bestirred myself for\nhis release and cured him and besought him of acquittance and\nabsolution of responsibility.\" All those who were present\nmarvelled at this story with the utmost marvel, and the twelfth\nconstable came forward and said, \"I will tell you a pleasant\ntrait that I heard from a certain person, concerning an adventure\nwhich befel him with one of the thieves.\n\n\n\n\nThe Twelfth Constable's History.\n\n\n\nI was passing one day in the market, when I found that a robber\nhad broken into the shop of a shroff, a changer of monies, and\nthence taken a casket, wherewith he had made off to the\nburial-ground. Accordingly I followed him thither and came up to\nhim, as he opened the casket and fell a-looking into it;\nwhereupon I accosted him, saying, \"Peace be on you!\"[FN#112] And\nhe was startled at me; so I left him and went away from him. Some\nmonths after this, I met him again under arrest, in the midst of\nthe guards and \"men of violence,\"[FN#113] and he said to them,\n\"Seize this man.\" So they laid hands on me and carried me to the\nChief of Police, who said, \"What hast thou to do with this\nwight?\" The robber turned to me and looking a long while in my\nface, asked, \"Who took this man?\" and the officer answered, \"Thou\nbadest us take him; so we took him.\" And he cried, \"I ask refuge\nof Allah! I know not this man, nor knoweth he me; and I said not\nthat to you but of a person other than this.\" So they released\nme, and a while after the thief met me in the street and saluted\nme with the salam, saying, \"O my lord, fright for fright! Hadst\nthou taken aught from me, thou hadst a part in the\ncalamity.\"[FN#114] I replied to him, \"Allah be the judge between\nthee and me!\"[FN#115] And this is what I have to recount. Then\ncame forward the thirteenth constable and said, \"'I will tell you\na tale which a man of my friends told me.\"\n\n\n\n\n\nThe Thirteenth Constable's History.\n\n\n\nI went out one night of the nights to the house of a friend and\nwhen it was the middle of the night, I sallied forth alone to hie\nme home. When I came into the road, I espied a sort of thieves\nand they espied me, whereupon my spittle dried up; but I feigned\nmyself drunken and staggered from side to side, crying out and\nsaying, \"I am drunken.\" And I went up to the walls right and left\nand made as if I saw not the thieves, who followed me afoot till\nI reached my home and knocked at the door, when they went away.\nSome few days after this, as I stood at the door of my house,\nbehold, there came up to me a young man, with a chain about his\nneck and with him a trooper, and he said to me, \"O my lord, an\nalms for the love of Allah!\" I replied, \"Allah open!\" and he\nlooked at me a long while and cried, \"That which thou shouldst\ngive me would not come to the worth of thy turband or thy\nwaistcloth or what not else of thy habit, to say nothing of the\ngold and the silver which were about thy person.\" I asked, \"And\nhow so?\" and he answered, \"On such a night, when thou fellest\ninto peril and the thieves would have stripped thee, I was with\nthem and said to them, Yonder man is my lord and my master who\nreared me. So was I and only I the cause of thy deliverance and\nthus I saved thee from them.\" When I heard this, I said to him,\n\"Stop ;\" and entering my house, brought him that which Allah\nAlmighty made easy to me.[FN#116] So he went his way; and this is\nall I have to say. Then came forward the fourteenth constable and\nsaid, \"Know that the tale I have to tell is rarer and pleasanter\nthan this; and 'tis as follows.\"\n\n\n\n\nThe Fourteenth Constable's History.\n\n\n\nI had a draper's shop before I entered this corporation,[FN#117]\nand there used to come to me a person whom I knew not, save by\nhis face, and I would give him whatso he sought and have patience\nwith him, till he could pay me. One night, I foregathered with\ncertain of my friends and we sat down to liquor: so we drank and\nwere merry and played at T\u00e1b;[FN#118] and we made one of us Wazir\nand another Sultan and a third Torchbearer or Headsman.[FN#119]\nPresently, there came in upon us a spunger, without bidding, and\nwe went on playing, whilst he played with us. Then quoth the\nSultan to the Wazir, \"Bring the Parasite who cometh in to the\nfolk, without leave or license, that we may enquire into his\ncase; after which I will cut off his head;\" so the headsmen arose\nand dragged the spunger before the Sultan who bade cut off his\nhead. Now there was with them a sword, that would not cut clotted\ncurd;[FN#120] so the headsmen smote him therewith and his head\nflew from his body. When we saw this, the wine fled from our\nbrains and we became in the foulest of plights. Then my friends\nlifted up the corpse and went out with it, that they might hide\nit, whilst I took the head and made for the river. Now I was\ndrunken and my clothes were drenched with the blood; and as I\npassed along the road, I met a robber. When he saw me, he knew me\nand cried to me, \"Such-an-one!\" \"Well?\" said I, and he rejoined,\n\"What is that thou hast with thee?\" So I acquainted him with the\ncase and he took the head from me. Then we fared on till we came\nto the river, where he washed the head and considering it\nstraitly, exclaimed, \"By Allah, verily this be my brother, the\nson of my sire, and he used to spunge upon the folk;\" after which\nhe threw that head into the river. As for me, I was like a dead\nman for dread; but he said to me, \"Fear not, neither do thou\ngrieve, for I acquit thee of my brother's blood.\" Presently, he\ntook my clothes and washed them and dried them and put them on\nme; after which he said to me, \"Get thee gone to thy house.\" So I\nreturned to my house and he accompanied me, till I came thither,\nwhen he said to me, \"Allah never desolate thee! I am thy friend\nSuch-an-one, who used to take of thee goods on credit, and I owe\nthee a kindness; but henceforward thou wilt never see me more.\"\nThen he went his ways. The company marvelled at the manliness of\nthis man and his clemency[FN#121] and courtesy, and the Sultan\nsaid, \"Tell us another of thy stories, O Shahrazad.\"[FN#122] She\nreplied, \" 'Tis well! They set forth[FN#123]\n\n\n\n\nA Merry Jest of a Clever Thief.\n\n\n\nA thief of the thieves of the Arabs went one night to a certain\nman's house, to steal from a heap of wheat there, and the people\nof the house surprised him. Now on the heap was a great copper\ntasse, and the thief buried himself in the corn and covered his\nhead with the tasse, so that the folk found him not and went\ntheir ways; but, as they were going, behold, there came a mighty\ngreat fart[FN#124] forth of the corn. So they went up to the\ntasse and raising it, discovered the thief and laid hands on him.\nQuoth he, \"I have saved you the trouble of seeking me: for I\npurposed, in breaking wind, to direct you to my hiding-place;\nwherefore do you be easy with me and have ruth on me, so may\nAllah have ruth on you!\" Accordingly they let him go and harmed\nhim not. \"And for another story of the same kind\" (she\ncontinued), \"hearken to\n\n\n\n\nThe Tale of the Old Sharper.\n\n\n\nThere was once an old man renowned for clever roguery, and he\nwent, he and his mates, to one of the markets and stole thence a\nquantity of stuffs: then they separated and returned each to his\nquarter. Awhile after this, the old man assembled a company of\nhis fellows and, as they sat at drink, one of them pulled out a\ncostly piece of cloth and said, \"Is there any one of you will\ndare sell this in its own market whence it was stolen, that we\nmay confess his superior subtlety?\" Quoth the old man, \"I will;\"\nand they said, \"Go, and Allah Almighty open to thee the door!\" So\nearly on the morrow, he took the stuff and carrying it to the\nmarket whence it had been stolen, sat down at the very shop out\nof which it had been purloined and gave it to the broker, who\nhent it in hand and cried it for sale. Its owner knew it and\nbidding for it, bought it and sent after the Chief of Police, who\nseized the Sharper and seeing him an old man of grave presence\nand handsomely clad said to him, \"Whence hadst thou this piece of\nstuff?\" Quoth he, \"I had it from this market and from yonder shop\nwhere I was sitting.\" Quoth the Wali, \"Did its owner sell it to\nthee?\" and quoth the robber, \"Not so; I stole it, this and other\nthan it.\" Then said the Chief, \"How camest thou to bring it for\nsale to the place whence thou stolest it?\" \"I will not tell my\ntale save to the Sultan, for that I have a profitable counsel\nwherewith I would lief bespeak him.\" \"Name it!\" \"Art thou the\nSultan?\" \"No!\" \"I'll not tell it save to himself.\" Accordingly\nthe Wali carried him up to the Sultan and he said, \"I have a\ncounsel for thee, O my lord.\" Asked the Sultan, \"What is thy\ncounsel?\" And the thief said, \"I repent and will deliver into thy\nhand all who are evildoers, and whomsoever I bring not, I will\nstand in his stead.\" Cried the Sultan, \"Give him a robe of honour\nand accept his profession of penitence.\" So he went down from the\npresence and returning to his comrades, related to them that\nwhich had passed, when they confessed his subtlety and gave him\nthat which they had promised him. Then he took the rest of the\nbooty and went up therewith to the Sultan, who, seeing him,\nrecognised him and he was magnified in the royal eyes and the\nking commanded that naught should be taken from him. After this,\nwhen he went down, the Sultan's attention was diverted from him,\nlittle by little, till the case was forgotten, and so he saved\nthe booty for himself. Those present marvelled at this and the\nfifteenth constable came forward and said, \"Know that among those\nwho make a trade of trickery are those whom Allah Almighty taketh\non their own testimony against themselves.\" It was asked him,\n\"How so?\" and he began to relate\n\n\n\n\nThe Fifteenth Constable's History.[FN#125]\n\n\n\nIt is told of a thieving person, one of the braves, that he used\nto rob and cut the way by himself upon caravans, and whenever the\nChief of Police and the Governors sought him, he would flee from\nthem and fortify himself in the mountains. Now it came to pass\nthat a certain man journeyed along the road wherein was that\nrobber, and this man was single-handed and knew not the sore\nperils besetting his way. So the highwayman came out upon him and\nsaid to him, \"Bring out that which is with thee, for I mean to\nkill thee and no mistake.\" Quoth the traveller, \"Kill me not,\nbut annex these saddle-bags and divide that which is in them and\ntake to thee the fourth part.\" And the thief answered, \"I will\nnot take aught but the whole.\"[FN#126] Rejoined the traveller,\n\"Take half, and let me go;\" but the robber replied, \"I will have\nnaught but the whole, and eke I will kill thee.\" So the wayfarer\nsaid, \"Take it.\" Accordingly the highwayman took the saddle-bags\nand offered to slay the traveller, who said, \"What is this? Thou\nhast against me no blood-feud that should make my slaughter\nincumbent.\" Quoth the other, \"Needs must I kill thee;\" whereupon\nthe traveller dismounted from his horse and grovelled before him,\nbeseeching the thief and bespeaking him fair. The man hearkened\nnot to his prayers, but cast him to the ground; whereupon the\ntraveller raised his eyes and seeing a francolin flying over him,\nsaid, in his agony, \"O Francolin,[FN#127] bear testimony that\nthis man slayeth me unjustly and wickedly; for indeed I have\ngiven him all that was with me and entreated him to let me go,\nfor my children's sake; yet would he not consent. But be thou\nwitness against him, for Allah is not unmindful of deeds which\nthe oppressors do.\" The highwayman paid no heed to what he heard,\nbut smote him and cut off his head. After this, the rulers\ncompounded with the highwayman for his submission, and when he\ncame before them, they enriched him and he became in such favour\nwith the lieutenant of the Sultan that he used to eat and drink\nwith him and there befel between them familiar converse which\nlasted a long while till in fine there chanced a curious chance.\nThe lieutenant of the Sultan one day of the days made a banquet,\nand therein was a roasted francolin, which when the robber saw,\nhe laughed a loud laugh. The lieutenant was angered against him\nand said to him, \"What is the meaning of thy laughter? Seest thou\nany fault or dost thou mock at us, of thy lack of good manners?\"\nAnswered the highwayman, \"Not so, by Allah, O my lord; but I saw\nyonder francolin, which brought to my mind an extraordinary\nthing; and 'twas on this wise. In the days of my youth, I used to\ncut the way, and one day I waylaid a man, who had with him a pair\nof saddle-bags and money therein. So I said to him, 'Leave these\nsaddle-bags, for I mean to slay thee.' Quoth he, 'Take the fourth\npart of that which is in them and leave me the rest;' and quoth\nI, 'Needs must I take the whole and kill thee without mistake.'\nThen said he, 'Take the saddle bags and let me wend my way;' but\nI answered, 'There is no help but that I slay thee.' As we were\nin this contention, behold, he saw a francolin and turning to it,\nsaid, 'Bear testimony against him, O Francolin, that he slayeth\nme unjustly and letteth me not go to my children, for all he hath\ntaken my money.' However, I had no pity on him neither hearkened\nto that which he said, but smote him and slew him and concerned\nnot myself with the evidence of the francolin.\" His story\ntroubled the lieutenant of the Sultan and he was enraged against\nhim with sore rage; so he drew his sword and smiting him, cut off\nhis head while he sat at table; whereupon a voice recited these\ncouplets--\n\n\"An wouldst not be injur\u00e8d, injure not; * But do good and from\n     Allah win goodly lot,\nFor what happeth by Allah is doomed to be * Yet thine acts are\n     the root I would have thee wot.\"[FN#128]\n\nNow this voice was the francolin which bore witness against him.\nThe company present marvelled at this tale and all cried, \"Woe to\nthe oppressor!\" Then came forward the sixteenth constable and\nsaid, \"And I for another will tell you a marvellous story which\nis on this wise.\"\n\n\n\n\nThe Sixteenth Constable's History.\n\n\n\nI went forth one day of the days, intending to travel, and\nsuddenly fell upon a man whose wont it was to cut the way. When\nhe came up with me he offered to slay me and I said to him, \"I\nhave naught with me whereby thou mayst profit.\" Quoth he, \"My\nprofit shall be the taking of thy life.\" I asked, \"What is the\ncause of this? Hath there been enmity between us aforetime?\" and\nhe answered, \"Nay; but needs must I slay thee.\" Thereupon I ran\naway from him to the river side; but he caught me up and casting\nme to the ground, sat down on my breast. So I sought help of the\nShaykh of the Pilgrims[FN#129] and cried to him, \"Protect me from\nthis oppressor!\" And indeed he had drawn a knife to cut my throat\nwhen, lo and behold! there came a mighty great crocodile forth of\nthe river and snatching him up from off my breast plunged into\nthe water, with him still hending knife in hand, even within the\njaws of the beast: whilst I abode extolling Almighty Allah, and\nrendering thanks for my preservation to him who had delivered me\nfrom the hand of that wrong-doer.[FN#130]\n\n\n\n\n            TALE OF HARUN AL-RASHID AND ABDULLAH BIN\n                         NAFI'.[FN#131]\n\n\n\nKnow thou, O King of the Age, that there was in days of yore and\nin ages and times long gone before, in the city of Baghdad, the\nAbode of Peace, a Caliph Harun al-Rashid hight, and he had cup-\ncompanions and tale-tellers to entertain him by night. Among his\nequerries was a man named Abdullah bin N\u00e1fi', who stood high in\nfavour with him and dear to him, so that he did not forget him a\nsingle hour. Now it came to pass, by the decree of Destiny, that\nit became manifest to Abdullah how he was grown of small account\nwith the Caliph, who paid no heed unto him nor, if he absented\nhimself, did he ask after him, as had been his habit. This was\ngrievous to Abdullah and he said within himself, \"Verily, the\nsoul of the Commander of the Faithful and his Wazir are changed\ntowards me and nevermore shall I see in him that cordiality and\naffection wherewith he was wont to treat me.\" And this was\nchagrin-full to him and concern grew upon him, so that he recited\nthese couplets:--\n\n\"Whoso's contemned in his home and land * Should, to better his\n     case, in self-exile hie:\nSo fly the house where contempt awaits, * Nor on fires of grief\n     for the parting fry;\nCrude Ambergris[FN#132] is but offal where * 'Tis born; but\n     abroad on our necks shall stye;\nAnd Kohl at home is a kind of stone, * Cast on face of earth and\n     on roads to lie;\nBut when borne abroad it wins highest worth * And thrones between\n     eyelid and ball of eye.\"\n\n(Quoth the sayer,) Then he could brook this matter no longer; so\nhe went forth from the dominions of the Prince of True Believers,\nunder presence of visiting certain of his kith and kin, and took\nwith him nor servant nor comrade, neither acquainted any with his\nintent, but betook himself to the road and fared deep into the\nwold and the sandwastes, unknowing whither he went. After awhile,\nhe unexpectedly fell in with travellers who were making the land\nof Hind and journeyed with them. When he came thither, he lighted\ndown in a city of that country and housed him in one of the\nlodging-houses; and there he abode a while of days, relishing not\nfood neither solacing himself with sleep; nor was this for lack\nof dirhams or dinars, but for that his mind was occupied with\nmusing upon the shifts of Destiny and bemoaning himself for that\nthe revolving sphere had turned against him in enmity, and the\ndays had decreed unto him the disfavour of our lord the\nImam.[FN#133] After such fashion he abode a space of days, and\npresently he homed him in the land and took to himself friends\nand got him many familiars, with whom he addressed himself to\ndiversion and good cheer. He used also to go a-pleasuring with\nhis companions and their hearts were solaced by his company and\nhe entertained them every evening with stories and displays of\nhis manifold accomplishments[FN#134] and diverted them with\ndelectable verses and told them abundance of stories and\nhistories. Presently, the report of him reached King Jamh\u00far, lord\nof Kashgar of Hind, who sent in quest of him, and great was his\ndesire to see him. So Abdullah repaired to his court and going in\nto him, kissed ground before him; and Jamhur welcomed him and\ntreated him with kindness and bade lodge him in the guest-house,\nwhere he abode three days, at the end of which the king sent to\nhim a chamberlain of his chamberlains and bade bring him to the\npresence. When he came before him, he greeted him, and the\ntruchman accosted him, saying, \"Verily, King Jamhur hath heard of\nthy report, that thou art a pleasant cup-companion and an\neloquent teller of night-tales, and he would have thee company\nwith him o' nights and entertain him with that which thou knowest\nof histories and pleasant stories and verses.\" And he made\nanswer, ' To hear is to obey!\" (Quoth Abdullah bin Nafi',) So I\nbecame his boon-companion and entertained him by night with tales\nand talk; and this pleased him with the utmost pleasure and he\ntook me into favour and bestowed on me robes of honour and set\napart for me a lodging; indeed he was bountiful exceedingly to me\nand could not brook to be parted from me a single hour. So I\nsojourned with him a while of time and every night I caroused and\nconversed with him till the most part of the dark hours was past;\nand when drowsiness overcame him, he would rise and betake\nhimself to his sleeping-place, saying to me, 'Forsake not my\nservice and forego not my presence.' And I made answer with\n'Hearing and obeying.' Now the king had a son, a nice child,\ncalled the Emir Mohammed, who was winsome of youth and sweet of\nspeech: he had read books and had perused histories and he loved\nabove all things in the world the telling and hearing of verses\nand tales and anecdotes. He was dear to his father King Jamhur,\nfor that he owned no other son than he on life, and indeed he had\nreared him in the lap of love and he was gifted with exceeding\nbeauty and loveliness, brilliancy and perfect grace: he had also\nlearnt to play upon the lute and upon all manner instruments and\nhe was used to converse and company with friends and brethren.\nNow it was his wont when the king arose seeking his\nsleeping-chamber, to sit in his place and require me to entertain\nhim with tales and verses and pleasant anecdotes; and on this\nwise I abode with them both a great while in all joyance and\ndelight, and the Prince still loved me with mighty great love and\ntreated me with the utmost tenderness. It fortuned one day that\nthe king's son came to me, after his sire had withdrawn, and\ncried, 'O Ibn Nafi'!' 'At thy service, O my lord;' 'I would have\nthee tell me a wondrous story and a marvellous matter, which thou\nhast never related either to me or to my father Jamhur.' 'O my\nlord, what story is this that thou desirest of me and of what kind\nshall it be of the kinds?' 'It mattereth little, so it be a\ngoodly story, whether it befel of olden tide or in these times.'\n'O my lord, I know by rote many stories of various kinds; so\nwhich of the kinds preferrest thou, and wilt thou have a story of\nmankind or of Jinn kind?' ' 'Tis well! An thou have espied aught\nwith thine eyes and heard it with thine ears, tell it me.' Then\nhe bethought himself and said to me, 'I conjure thee by my life,\ntell me a tale of the tales of the Jinn and that which thou hast\nheard of them and seen of them!' I replied, 'O my son, indeed\nthou conjurest me by a mighty conjuration; so lend an ear to the\ngoodliest of stories, ay, and the strangest of them and the\npleasantest and rarest.' Quoth the Prince, 'Say on, for I am\nattentive to thy speech;' and quoth I, 'Hear then, O my son,\n\n\n\n\nThe Tale of the Damsel Tohfat al-Kulub and the Caliph Harun al-\nRashid.\n\n\n\nThe Viceregent of the Lord of the three Worlds, Harun al-Rashid,\nhad a boon-companion of the number of his boon-companions, by\nname Ishak bin Ibrahim al-Nadim al-Mausili,[FN#135] who was the\nmost accomplished of the folk of his time in smiting upon the\nlute; and of the Commander of the Faithful's love for him, he set\napart for him a palace of the choicest of his palaces, wherein he\nwas wont to instruct hand-maidens in the arts of singing and of\nlute playing. If any slave-girl became, by his instruction,\nclever in the craft, he carried her before the Caliph, who bade\nher perform upon the lute; and if she pleased him, he would order\nher to the Harim; else would he restore her to Ishak's palace.\nOne day, the Commander of the Faithful's breast was straitened;\nso he sent after his Wazir Ja'afar the Barmecide and Ishak the\ncup-companion and Masrur the eunuch, the Sworder of his\nvengeance; and when they came, he changed his habit and disguised\nhimself, whilst Ja'afar and Ishak and Masrur and Al-Fazl[FN#136]\nand Y\u00fanus[FN#137] (who were also present) did the like. Then he\nwent out, he and they, by the postern, to the Tigris and taking\nboat fared on till they came to near Al-T\u00e1f,[FN#138] when they\nlanded and walked till they came to the gate of the high street.\nHere there met them an old man, handsome in his hoariness and of\na venerable bearing and a dignified, agreeable of aspect and\napparel. He kissed the earth before Ishak al-Mausili (for that he\nknew only him of the company, the Caliph being disguised, and\ndeemed the others certain of his friends), and said to him, \"O my\nlord, there is presently with me a hand-maid, a lutanist, never\nsaw eyes the like of her nor the like of her grace, and indeed I\nwas on my way to pay my respects to thee and give thee to know of\nher, but Allah, of His favour, hath spared me the trouble. So now\nI desire to show her to thee, and if she take thy fancy, well and\ngood; otherwise I will sell her.\" Quoth Ishak, \"Go before me to\nthy quarters,[FN#139] till I come to thee and see her.\" The old\nman kissed his hand and went away; whereupon quoth Al-Rashid to\nhim, \"O Ishak, who is yonder man and what is his want?\" The other\nreplied, \"O my lord, this is a man Sa'\u00edd the Slave-dealer hight,\nand 'tis he that buyeth us maidens and Mamelukes. He declareth\nthat with him is a fair slave, a lutanist, whom he hath withheld\nfrom sale, for that he could not fairly sell her till he had\npassed her before me in review.\" Quoth the Caliph, \"Let us go to\nhim so we may see her, by way of solace, and sight what is in the\nslave-dealer's quarters of slave-girls;\" and quoth Ishak,\n\"Command belongeth to Allah and to the Commander of the Faithful.\"\nThen he forewent them and they followed in his track till they\ncame to the slave-dealer's quarters and found a building tall of\nwall and large of lodgment, with sleeping-cells and chambers\ntherein, after the number of the slave-girls, and folk sitting\nupon the wooden benches. So Ishak entered, he and his company and\nseating themselves in the place of honour, amused themselves by\nlooking at the hand-maids and Mamelukes and watching how they\nwere bought and sold, till the vending came to an end, when some\nof the folk went away and some remained seated. Then cried the\nslave-dealer, \"Let none sit with us except whoso purchaseth by\nthe thousand dinars and upwards.\" Accordingly those present\nwithdrew and there remained none but Al-Rashid and his suite;\nwhereupon the slave-dealer called the damsel, after he had caused\nset her a chair of Faww\u00e1k,[FN#140] lined with Grecian brocade,\nand she was like the sun shining high in the shimmering sky. When\nshe entered, she saluted and sitting down, took the lute and\nsmote upon it, after she had touched its strings and tuned it, so\nthat all present were amazed. Then she sang thereto these\ncouplets:\n\n\"Breeze o' Morn, an thou breathe o'er the loved one's land, *\n     Deliver my greeting to all the dear band!\nAnd declare to them still I am pledged to their love * And my\n     longing excels all that lover unmanned:\nO ye who have blighted my heart, ears and eyes, * My passion and\n     ecstasy grow out of hand;\nAnd torn is my sprite every night with desire, * And nothing of\n     sleep can my eyelids command.\"\n\nIshak exclaimed, \"Brava, O damsel! By Allah, this is a fair\nhour!\" Whereupon she sprang up and kissed his hand, saying, 'O my\nlord, in very sooth the hands stand still before thy presence and\nthe tongues at thy sight, and the loquent when confronting thee\nwax dumb; but thou art the looser of the veil.\"[FN#141] Then she\nclung to him and cried, \"Stand;\" so he stood and said to her,\n\"Who art thou and what is thy need?\" She raised a corner of the\nveil, and behold she was a damsel as she were the full moon\nrising or the levee glancing, with two side-locks of hair which\nfell down to her anklets. She kissed his hand and said to him, \"O\nmy lord, know that I have been in these quarters some five\nmonths, during which I have withheld myself from sale till thou\nshouldst be present and see me; and yonder slave-dealer also made\nthy coming a pretext for not vending me, and forbade me for all I\nsought of him night and day that he should cause thee come hither\nand vouchsafe me thy company and gar me and thee forgather.\"\nQuoth Ishak, \"Tell me what thou wouldst have;\" and quoth she, \"I\nbeseech thee, by Allah Almighty, that thou buy me, so I may be\nwith thee by way of service.\" He asked, \"Is that thy desire?\" and\nshe answered, \"Yes.\" So Ishak returned to the slave-dealer and\nsaid to him, \"Ho thou, Shaykh Sa'\u00edd!\" Said the old man, \"At thy\nservice, O my lord,\" and Ishak continued, \"In the corridor is a\nchamber and therein wones a damsel pale and wan. What is her\nprice in dirhams and how much dost thou ask for her?\" Quoth the\nslave-dealer, \"She whom thou mentionest, O my lord, is called\nTohfat al-Humak\u00e1?\"[FN#142] Ishak asked, \"What is the meaning of\nAl-Humaka?\" and the old man answered, \"Her price hath been\nweighed and paid an hundred times and she still saith, Show me\nhim who would buy me; and when I show her to him she saith, This\none I mislike; he hath in him such and such a default. And in\nevery one who would fain buy her she noteth some defect or other,\nso that none careth now to purchase her and none seeketh her, for\nfear lest she find some fault in him.\" Quoth Ishak, \"She seeketh\nat this present to sell herself; so go thou to her and inquire of\nher and see her price and send her to the palace.\" Quoth Sa'id,\n\"O my lord, her price is an hundred dinars, though, were she free\nof this paleness that is upon her face, she would be worth a\nthousand gold pieces; but wanton folly and wanness have\ndiminished her value; and behold I will go to her and consult her\nof this.\" So he betook himself to her and enquired of her, \"Wilt\nthou be sold to Ishak bin Ibrahim al-Mausili?\" She replied,\n\"Yes,\" and he said, \"Leave folly, for to whom doth it happen to\nbe in the house of Ishak the cup-companion?\"[FN#143] Thereupon\nIshak went forth the slave-dealer's quarters and overtook Al-\nRashid who had preceded him; and they ceased not walking till\nthey came to their landing-place, where they embarked in the boat\nand fared on to Thaghr al-Kh\u00e1nakah.[FN#144] As for the slave-\ndealer, he sent the damsel to the house of Ishak al-Nadim, whose\nslave-girls took her and carried her to the Hammam. Then each\ndamsel gave her somewhat of her gear and they decked her with\nearrings and bracelets, so that she redoubled in beauty and\nbecame as she were the moon on the night of its full. When Ishak\nreturned home from the Caliph's palace, Tohfah rose to him and\nkissed his hand; and he saw that which the hand-maids had done\nwith her and thanked them for so doing and said to them, \"Let her\nhome in the house of instruction and bring her instruments of\nmusic, and if she be apt at song teach her; and may Allah\nAlmighty vouchsafe her health and weal!\" So there passed over her\nthree months, while she homed with him in the house of\ninstruction, and they brought her the instruments of music.\nFurthermore, as time went on she was vouchsafed health and\nsoundness and her beauty waxed many times brighter than before\nand her pallor was changed to white and red, so that she became a\nseduction to all who saw her. One day, Ishak bade summon all who\nwere with him of slave-girls from the house of instruction and\ncarried them up to Al-Rashid's palace, leaving none in his house\nsave Tohfah and a cookmaid; for that he thought not of Tohfah,\nnor did she come to his memory, and none of the damsels reminded\nhim of her. When she saw that the house was empty of the slave-\ngirls, she took the lute (now she was singular in her time for\nsmiting upon the lute, nor had she her like in the world, no, not\nIshak himself, nor any other) and sang thereto these couplets:--\n\n\"When soul desireth one that is its mate * It never winneth dear\n     desire of Fate:\nMy life for him whose tortures tare my frame, * And dealt me pine\n     he can alone abate!\nHe saith (that only he to heal mine ill, * Whose sight is\n     medicine to my doleful state),\n'O scoffer-wight, how long wilt mock my woe * As though did Allah\n     nothing else create?' \"\n\nNow Ishak had returned to his house on an occasion that called\nfor him; and when he entered the vestibule, he heard a sound of\nsinging, the like whereof he had never heard in the world, for\nthat it was soft as the breeze and more strengthening than\noil[FN#145] of almonds. So the pleasure of it get hold of him and\ndelight so seized him, that he fell down fainting in the\nvestibule. Tohfah heard the noise of footfalls and laying the\nlute from her hand, went out to see what was the matter. She\nfound her lord Ishak lying aswoon in the entrance; so she took\nhim up and strained him to her bosom, saying, \"I conjure thee in\nAllah's name, O my lord, tell me, hath aught of ill befallen\nthee?\" When he heard her voice, he recovered from his fainting\nand asked her, \"Who art thou?\" She answered, \"I am thy slave-\ngirl Tohfah;\" and he said to her, \"Art thou indeed Tohfah?\"\n\"Yes,\" replied she; and he, \"By Allah, I had indeed forgotten\nthee and remembered thee not till this moment!\" Then he looked at\nher and said, \"Verily, thy case is altered to other case and thy\nwanness is changed to rosiness and thou hast redoubled in beauty\nand loveliness. But was it thou who was singing just now?\" She\nwas troubled and affrighted and answered, \"Even I, O my lord;\"\nwhereupon Ishak seized upon her hand and carrying her into the\nhouse, said to her, \"Take the lute and sing; for never saw I nor\nheard thy like in smiting upon the lute; no, not even myself!\"\nQuoth she, \"O my lord, thou mockest me. Who am I that thou\nshouldst say all this to me? Indeed, this is but of thy\nkindness.\" Quoth he, \"Nay, by Allah, I said but the truth to thee\nand I am not of those on whom pretence imposeth. For these three\nmonths nature hath not moved thee to take the lute and sing\nthereto, and this is naught save a rare thing and a strange. But\nall this cometh of strength in the art and thy self-restraint.\"\nThen he bade her sing; and she said, \"Hearkening and obedience.\"\nSo she took the lute and tightening its strings to the sticking\npoint, smote thereon a number of airs, so that she confounded\nIshak's wit and for delight he was like to fly. Then she returned\nto the first mode and sang thereto these couplets:--\n\n\"By your ruined stead aye I stand and stay, * Nor shall change or\n     dwelling depart us tway!\nNo distance of homestead shall gar me forget * Your love, O\n     friends, but yearn alway:\nNe'er flies your phantom the babes of these eyne * You are moons\n     in Nighttide's murkest array:\nAnd with growing passion mine unrest grows * And each morn I find\n     union dissolved in woes.\"\n\nWhen she had made an end of her song and laid down the lute,\nIshak looked fixedly on her, then took her hand and offered to\nkiss it; but she snatched it from him and said to him, \"Allah, O\nmy lord, do not that!\"[FN#146] Cried he, \"Be silent. By Allah, I\nhad said that there was not in the world the like of me; but now\nI have found my din\u00e1r in the art but a d\u00e1nik,[FN#147] for thou\nart more excellent of skill than I, beyond comparison or\napproximation or calculation! This very day will I carry[FN#148]\nthee up to the Commander of the Faithful, Harun al-Rashid, and\nwhen his glance lighteth on thee, thou wilt become a Princess of\nwomankind. So Allah, Allah upon thee, O my lady, whenas thou\nbecomest of the household of the Prince of True Believers, do not\nthou forget me!\" She replied, \"Allah, O my lord, thou art the\nroot of my fortunes and in thee is my heart fortified.\" Thereat\nhe took her hand and made a covenant with her of this and she\nswore to him that she would not forget him. Then said he to her,\n\"By Allah, thou art the desire of the Commander of the Faithful!\nNow take the lute and sing a song which thou shalt sing to the\nCaliph, when thou goest in to him.\" So she took the lute and\ntuning it, improvised these couplets:--\n\n\"His lover hath ruth on his woeful mood * And o'erwept him as\n     still by his couch he[FN#149] stood:\nAnd garred him drink of his lip-dews and wine[FN#150] * Ere he\n     died and this food was his latest good.\"\n\nIshak stared at her and seizing her hand, said to her, \"Know that\nI am bound by an oath that, when the singing of a damsel pleaseth\nme, she shall not end her song but before the Prince of True\nBelievers. But now tell me, how came it that thou tarriedst with\nthe slave-dealer five months and wast not sold to any one, and\nthou of this skill, especially when the price set on thee was no\ngreat matter?\" Hereat she laughed and answered, \"O my lord, my\nstory is a wondrous and my case a marvellous. Know that I belonged\naforetime to a Maghribi merchant,who bought me when I was three\nyears old, and there were in his house many slave-girls and\neunuchs; but I was the dearest to him of them all. So he kept me\nwith him and used not to address me otherwise than, 'O\ndaughterling,' and indeed to this moment I am a clean maid. Now\nthere was with him a damsel, a lutanist, and she reared me and\ntaught me the art, even as thou seest. Then was my master removed\nto the mercy of Allah Almighty[FN#151] and his sons divided his\nmonies. I fell to the lot of one of them; but 'twas only a little\nwhile ere he had wasted all his wealth and there was left him\nnaught of coin. So I gave up the lute, fearing lest I should fall\ninto the hand of a man who knew not my worth, for well I wot that\nneeds must my master sell me; and indeed but a few days passed\nere he carried me forth to the quarters of the slave-merchant who\nbuyeth damsels and displayeth them to the Commander of the\nFaithful. Now I desired to learn the art and mystery; so I\nrefused to be sold to other than thou, until Allah (extolled and\nexalted be He!) vouchsafed me my desire of thy presence;\nwhereupon I came out to thee, as soon as I heard of thy coming,\nand besought thee to buy me. Thou heartenedst my heart and\nboughtest me; and since I entered thy house, O my lord, I have\nnot taken up the lute till now; but to-day, when I was left\nprivate by the slave-girls, I took it; and my purpose in this was\nthat I might see if my hand were changed[FN#152] or not. As I was\nsinging, I heard a footfall in the vestibule; so springing up, I\nlaid the lute from my hand and going forth to see what was to do,\nfound thee, O my lord, after this fashion.\" Quoth Ishak, \"Indeed,\nthis was of thy fair fortune. By Allah, I know not that which\nthou knowest in this art!\" Then he arose and opening a chest,\nbrought out therefrom striped clothes,[FN#153] netted with jewels\nand great pearls and other costly gems and said to her, \"In the\nname of Allah, don these, O my lady Tohfah.\" So she arose and\ndonned that dress and veiled herself and went up with Ishak to\nthe palace of the Caliphate, where he made her stand without,\nwhilst he himself went in to the Prince of True Believers (with\nwhom was Ja'afar the Barmaki) and kissing the ground before him,\nsaid to him, \"O Commander of the Faithful, I have brought thee a\ndamsel, never saw eyes of seer her like for excellence in singing\nand touching the lute; and her name is Tohfah.\" Al-Rashid asked,\n\"And where be this Tohfah[FN#154] who hath not her like in the\nworld?\" Answered Ishak, \"Yonder she standeth, O Commander of the\nFaithful ;\" and he acquainted the Caliph with her case from first\nto last. Then said Al-Rashid, \" 'Tis a marvel to hear thee praise\na slave-girl after this fashion. Admit her that we may look upon\nher, for verily the morning may not be hidden.\" Accordingly,\nIshak bade admit her; so she entered, and when her eyes fell upon\nthe Prince of True Believers, she kissed ground before him and\nsaid, \"The Peace be upon thee, O Commander of the faithful Fold\nand Asylum of all who the true Creed hold and Quickener of\njustice in the Worlds threefold! Allah make thy feet tread on\nsafest wise and give thee joy of what He gave thee in generous\nguise and make thy harbourage Paradise and Hell-fire that of\nthine enemies!\" Quoth Al-Rashid, \"And on thee be the Peace, O\ndamsel! Sit.\" So she sat down and he bade her sing; whereupon she\ntook the lute and tightening its strings, played thereon in many\nmodes, so that the Prince of True Believers and Ja'afar were\nconfounded in sprite and like to fly for delight. Then she\nreturned to the first mode and improvised these couplets:--\n\n\"O mine eyes! I swear by him I adore, * Whom pilgrims seek\n     thronging Araf\u00e1t;\nAn thou call my name on the grave of me, * I'll reply to thy call\n     tho' my bones go rot:\nI crave none for friend of my heart save thee; * So believe me,\n     for true are the well-begot.\"\n\nAl-Rashid considered her comeliness and the goodliness of her\nsinging and her eloquence and what other qualities she comprised\nand rejoiced with joy exceeding; and for the stress of that which\novercame him of delight, he descended from the couch and sitting\ndown with her upon the floor, said to her, \"Thou hast done well,\nO Tohfah. By Allah, thou art indeed a choice gift!\"[FN#155] Then\nhe turned to Ishak and said to him, \"Thou dealtest not justly, O\nIshak, in the description of this damsel, nor didst thou fairly\nset forth all that she comprised of charms and art; for that, by\nAllah, she is inconceivably more skilful than thou; and I know of\nthis craft that which none knoweth save I!\" Exclaimed the Wazir\nJa'afar, \"By Allah, thou sayst sooth, O my lord, O Commander of\nthe Faithful. Indeed, she hath done away my wit, hath this\ndamsel.\" Quoth Ishak, \"By Allah, O Prince of True Believers, I\nhad said that there was not on the face of the earth one who knew\nthe art of the lute like myself; but when I heard her, my skill\nbecame nothing worth in mine eyes.\" Then said the Caliph to her,\n\"Repeat thy playing, O Tohfah.\" So she repeated it and he cried\nto her, \"Well done!\" Moreover, he said to Ishak, \"Thou hast\nindeed brought me a marvellous thing, one which is worth in mine\neyes the empire of the world.\" Then he turned to Masrur the\neunuch and said to him, \"Carry Tohfah to the chamber of honour.\"\nAccordingly, she went away with the Castrato and the Caliph\nlooked at her raiment and ornaments and seeing her clad in\nclothing of choice, asked Ishak, \"O Ishak, whence hath she these\nrobes?\" Answered he, \"O my lord, these are somewhat of thy\nbounties and thy largesse, and they are a gift to her from me. By\nAllah, O Commander of the Faithful, the world, all of it, were\nlittle in comparison with her!\" Then the Caliph turned to the\nWazir Ja'afar and said to him, \"Give Ishak fifty thousand dirhams\nand a robe of honour of the choicest apparel.\" \"Hearing and\nobeying,\" replied Ja'afar and gifted him with that which the\nCaliph ordered him. As for Al-Rashid, he was private with Tohfah\nthat night and found her a pure virgin and rejoiced in her; and\nshe took high rank in his heart, so that he could not suffer her\nabsence a single hour and committed to her the keys of the\naffairs of the realm, for that which he saw in her of good\nbreeding and fine wit and leal will. He also gave her fifty\nslave-girls and two hundred thousand dinars and a quantity of\nraiment and ornaments, gems and jewels worth the kingdom of\nEgypt; and of the excess of his love for her, he would not\nentrust her to any of the hand-maids or eunuchs; but, whenever he\nwent out from her, he locked the door upon her and took the key\nwith him, against he should return to her, forbidding the damsels\nto go in to her, of his fear lest they should slay her or poison\nher or practice on her with the knife; and in this way he abode\nawhile. One day, as she sang before the Commander of the\nFaithful, he was delighted with exceeding delight, so that he\noffered to kiss her hand;[FN#156] but she drew it away from him\nand smote upon her lute and broke it and wept. Al-Rashid wiped\naway her tears and said, \"O desire of the heart, what is it\nmaketh thee weep? May Allah not cause an eye of thine to shed\ntears!\" Said she, \"O my lord, what am I that thou shouldst kiss\nmy hand? Wilt thou have Allah punish me for this and my term come\nto an end and my felicity pass away? For this is what none ever\nattained unto.\" He rejoined, \"Well said, O Tohfah. Know that thy\nrank in my esteem is high and for that which delighted me of what\nI saw in thee, I offered to do this, but I will not return unto\nthe like thereof; so be of good cheer, with eyes cool and clear,\nfor I have no desire to other than thyself and will not die but\nin the love of thee, and thou to me art queen this day, to the\nexclusion of al! humankind.\" Therewith she fell to kissing his\nfeet; and this her fashion pleased him, so that his love for her\nredoubled and he became unable to brook severance from her a\nsingle hour. Now Al-Rashid one day went forth to the chase and\nleft Tohfah in her pavilion. As she sat perusing a book, with a\ncandle-branch of gold before her, wherein was a perfumed candle,\nbehold, a musk-apple fell down before her from the top of the\nsaloon.[FN#157] So she looked up and beheld the Lady Zubaydah\nbint al-Kasim,[FN#158] who saluted her with a salam and\nacquainted her with herself, whereupon Tohfah sprang to her feet\nand said, \"O my lady, were I not of the number of the\nnew,[FN#159] I had daily sought thy service; so do not thou\nbereave me of those noble steps.\"[FN#160] The Lady Zubaydah\ncalled down blessings upon her and replied, \"I knew this of thee;\nand, by the life of the Commander of the Faithful, but that it is\nnot of my wont to go forth of my place, I had come out to do my\nservice to thee.\" Then quoth she to her, \"Know, O Tohfah, that\nthe Commander of the Faithful hath deserted all his concubines\nand favourites on thine account, even myself hath he abandoned on\nthis wise, and I am not content to be as one of the mistresses;\nyet hath he made me of them and forsaken me, and I have sought\nthee, so thou mayst beseech him to come to me, though it be but\nonce a month, in order that I may not be the like of the hand-\nmaids and concubines nor take rank with the slave-girls; and this\nis my need of thee.\" Answered Tohfah, \"Hearkening and obedience!\nBy Allah, O my lady, I would that he might be with thee a whole\nmonth and with me but one night, so thy heart might be heartened,\nfor that I am one of thy hand-maids and thou in every case art my\nlady.\" The Princess Zubaydah thanked her for this and taking\nleave of her, returned to her palace. When the Caliph came back\nfrom the chase and course, he betook himself to Tohfah's pavilion\nand bringing out the key, opened the lock and went in to her. She\nrose to receive him and kissed his hand, and he gathered her to\nhis breast and seated her on his knee.[FN#161] Then food was\nbrought to them and they ate and washed their hands; after which\nshe took the lute and sang, till Al-Rashid was moved to sleep.\nWhen aware of this, she ceased singing and told him her adventure\nwith the Lady Zubaydah, saying, \"O Prince of True Believers, I\nwould have thee favour me with a favour and hearten my heart and\naccept my intercession and reject not my supplication, but fare\nthee forthright to the Lady Zubaydah.\" Now this talk befel after\nhe had stripped himself naked and she also had doffed her dress;\nand he said, \"Thou shouldst have named this ere we stripped\nourselves naked, I and thou!\" But she answered, saying, \"O\nCommander of the Faithful, I did this not except in accordance\nwith the saying of the poet in these couplets,\n\n\"Of all intercessions can none succeed, * Save whatso Tohfah bint\n     Marj\u00e1n sue'd:\nNo intercessor who comes enveiled;[FN#162] * She sues the best\n     who sues mother-nude.\"\n\nWhen Al-Rashid heard this, her speech pleased him and he strained\nher to his bosom. Then he went forth from her and locked the door\nupon her, as before; whereupon she took the book and sat perusing\nit awhile. Presently, she set it aside and taking the lute,\ntightened its strings; and smote thereon, after a wondrous\nfashion, such as would have moved inanimate things to dance, and\nfell to singing marvellous melodies and chanting these couplets:--\n\n\"Cease for change to wail,* The world blames who rail;\nBear patient its shafts  * That for aye prevail.\nHow often a joy          * Grief-garbed thou shalt hail:\nHow oft gladding bliss   * Shall appear amid bale!\"\n\nThen she turned and saw within the chamber an old man, handsome\nin his hoariness and stately of semblance, who was dancing in\ngoodly and winning wise, a dance whose like none might dance. So\nshe sought refuge with Allah Almighty from Satan the Stoned and\nsaid, \"I will not give over what I am about, for whatso the Lord\nwilleth, He fulfilleth.\" Accordingly, she went on singing till\nthe Shaykh came up to her and kissed ground before her, saying,\n\"Well done, O Highmost of the East and the West! May the world be\nnot bereaved of thee! By Allah, indeed thou art perfect of\nmanners and morals, O Tohfat al-Sud\u00far![FN#163] Dost thou know\nme?\" Cried she, \"Nay, by Allah, but methinks thou art of the\nJann.\" Quoth he, \"Thou sayst sooth; I am Ab\u00fa al-Taw\u00e1if[FN#164]\nIblis, and I come to thee every night, and with me thy sister\nKamariyah, for that she loveth thee and sweareth not but by thy\nlife; and her pastime is not pleasant to her, except she come to\nthee and see thee whilst thou seest her not. As for me, I\napproach thee upon an affair, whereby thou shalt gain and rise to\nhigh rank with the kings of the Jann and rule them, even as thou\nrulest mankind; and to that end I would have thee come with me\nand be present at the festival of my daughter's wedding and the\ncircumcision of my son;[FN#165] for that the Jann are agreed upon\nthe manifestation of thy command. And she answered, \"Bismillah;\nin the name of the Lord.\"[FN#166] So she gave him the lute and he\nforewent her, till he came to the Chapel of Ease,[FN#167] and\nbehold, therein was a door and a stairway. When Tohfah saw this,\nher reason fled; but Iblis cheered her with chat. Then he\ndescended the steps and she followed him to the bottom of the\nstair, where she found a passage and they fared on therein, till\nthey came to a horse standing, ready saddled and bridled and\naccoutred. Quoth Iblis, \"Bismillah, O my lady Tohfah;\" and he\nheld the stirrup for her. So she mounted and the horse heaved\nlike a wave under her and putting forth wings soared upwards with\nher, while the Shaykh flew by her side; whereat she was\naffrighted and clung to the pommel of the saddle;[FN#168] nor was\nit but an hour ere they came to a fair green meadow, fresh-\nflowered as if the soil thereof were a fine robe, purfled with\nall manner bright hues. Amiddlemost that mead was a palace\ntowering high in air, with crenelles of red gold, set with pearls\nand gems, and a two-leaved door; and about the gateway were much\npeople of the chiefs of the Jann, clad in costliest clothing.\nWhen they saw the Shaykh, they all cried out, saying, \"The Lady\nTohfah is come!\" And as soon as she reached the palace-gate, they\npressed forward in a body, and dismounting her from the horse's\nback, carried her into the palace and fell to kissing her hands.\nWhen she entered, she beheld a palace whereof seers ne'er saw the\nlike; for therein were four halls, one facing other, and its\nwalls were of gold and its ceilings of silver. It was high-\nbuilded of base, wide of space, and those who descried it would\nbe posed to describe it. At the upper end of the hall stood a\nthrone of red gold set with pearls and jewels, up to which led\nfive steps of silver, and on its right and on its left were many\nchairs of gold and silver. Quoth Tohfah, \"The Shaykh led me to\nthe estrade and seated me on a chair of gold beside the throne,\nand over the dais was a curtain let down, gold and silver wrought\nand broidered with pearls and jewels.\" And she was amazed at that\nwhich she beheld in that place and magnified her Lord (extolled\nand exalted be He!) and hallowed Him. Then the kings of the Jann\ncame up to that throne and seated themselves thereon; and they\nwere in the semblance of Adam's sons, excepting two of them, who\nappeared in the form and aspect of the Jann, each with one eye\nslit endlong and jutting horns and projecting tusks.[FN#169]\nAfter this there came up a young lady, fair of favour and seemly\nof stature, the light of whose face outshone that of the waxen\nflambeaux; and about her were other three women, than whom none\nfairer abode on face of earth. They saluted Tohfah with the salam\nand she rose to them and kissed ground before them whereupon they\nembraced her after returning her greeting[FN#170] and sat down on\nthe chairs aforesaid. Now the four women who thus accosted Tohfah\nwere the Princess Kamariyah, daughter of King Al-Sh\u00edsb\u00e1n, and her\nsisters; and Kamariyah loved Tohfah with exceeding love. So, when\nshe came up to her, she fell to kissing and embracing her, and\nShaykh Iblis cried, \"Fair befal the accolade! Take me between\nyou.\" At this Tohfah laughed and Kamariyah said, \"O my sister, I\nlove thee, and doubtless hearts have their witnesses,[FN#171]\nfor, since I saw thee, I have loved thee.\" Replied Tohfah, \"By\nAllah, hearts have sea-like deeps, and thou, by Allah, art dear\nto me and I am thy hand-maid.\" Kamariyah thanked her for this and\nkissing her once more said, \"These be the wives of the kings of\nthe Jann: greet them with the salam! This is Queen\nJamrah,[FN#172] that is Queen Wakh\u00edmah and this other is Queen\nShar\u00e1rah, and they come not but for thee.\" So Tohfah rose to her\nfeet and bussed their hands, and the three queens kissed her and\nwelcomed her and honoured her with the utmost honour. Then they\nbrought trays and tables and amongst the rest a platter of red\ngold, inlaid with pearls and gems; its raised rims were of or and\nemerald, and thereon were graven[FN#173] these couplets:--\n\nTo bear provaunt assigned,    * By hands noble designed,\nFor the gen'rous I'm made     * Not for niggardly hind!\nSo eat safe all I hold        * And praise God of mankind.\n\nAfter reading the verses they ate and Tohfah looked at the two\nkings who had not changed shape and said to Kamariyah, \"O my\nlady, what be this feral and that other like unto him? By Allah,\nmine eye may not suffer the sight of them.\" Kamariyah laughed and\nanswered, \"O my sister, that is my sire Al-Shisban and the other\nis hight Maymun the Sworder; and of the arrogance of their souls\nand their insolence, they consented not to change their created\nshapes. Indeed, all whom thou seest here are nature-fashioned\nlike them; but on thine account they have changed favour, for\nfear lest thou be disquieted and for the comforting of thy mind,\nso thou mightest become familiar with them and be at thine ease.\"\nQuoth Tohfah, \"O my lady, verily I cannot look at them. How\nfrightful is this Maymun, with his monocular face! Mine eye\ncannot brook the sight of him, and indeed I am in affright of\nhim.\" Kamariyah laughed at her speech, and Tohfah continued, \"By\nAllah, O my lady, I cannot fill my eye with the twain!\"[FN#174]\nThen cried her father Al-Shisban to her, \"What be this laughing?\"\nSo she bespoke him in a tongue none understood but they two and\nacquainted him with that which Tohfah had said; whereat he\nlaughed a prodigious loud laugh, as it were the roaring thunder.\nPresently they ate and the tables were removed and they washed\ntheir hands; after which Iblis the Accursed came up to Tohfah and\nsaid to her, \"O my lady, thou gladdenest the place and\nenlightenest and embellishest it with thy presence; but now fain\nwould these kings hear somewhat of thy singing, for Night hath\ndispread her pinions for departure and there abideth of it but a\nlittle.\" Quoth she, \"Hearing and obeying.\" So she took the lute\nand touching its strings with rare touch, played thereon after\nwondrous wise, so that it seemed to those who were present as if\nthe palace surged like a wave with them for the music. Then she\nbegan singing and chanting these couplets,\n\n\"Folk of my faith and oath, Peace with you be! * Quoth ye not I\n     shall meet you you meet me?\nI'll chide you softerwise than breeze o' morn, * Sweeter than\n     spring of coolest clarity.\nI' faith mine eyelids are with tears chafed sore: * My vitals\n     plain to you some cure to see.\nMy friends! Our union to disunion changed * Was aye my fear for\n     'twas my certainty.\nI'll plain to Allah of all ills I bore; * For pine and yearning\n     misery still I dree.\"\n\nThe kings of the Jann were moved to delight by that sweet singing\nand seemly speech and thanked Tohfah therefore; and Queen\nKamariyah rose to her and threw her arms round her neck and\nkissed her between the eyes, saying, \"By Allah, 'tis good, O my\nsister and coolth of mine eyes and core of my heart!\" Then said\nshe, \"I conjure thee by Allah, give us more of this lovely\nsinging;\" and Tohfah answered with \"To hear is to obey.\" So she\ntook the lute and playing thereon in a mode different from the\nformer fashion, sang these couplets:--\n\n\"I, oft as ever grows the pine of me, * Console my soul with hope\n     thy sight to see.\nHaply shall Allah join our parted lives, * E'en as my fortunes\n     far from thee cast He!\nThen oh! who thrallest me by force of love-- * Seiz\u00e8d by fond\n     affection's mastery\nAll hardships easy wax when thou art nigh; * And all the far\n     draws near when near thou be.\nAh! be the Ruthful light to lover fond, * Love-lorn, frame-     wasted, ready Death to dree!\nWere hope of seeing thee cut off, my loved; * After thine absence\n     sleep mine eyes would flee!\nI mourn no worldly joyance, my delight * Is but to sight thee\n     while thou seest my sight.\"\n\nAt this the accursed Iblis was hugely pleased and thrust his\nfinger up his fundament,[FN#175] whilst Maymun danced and said,\n\"O Tohfat al-Sudur, soften the sound;[FN#176] for, as pleasure\nentereth into my heart, it arresteth my breath and blood.\" So she\ntook the lute and altering the tune, played a third air; then she\nreturned to the first and sang these couplets:--\n\n\"The waves of your[FN#177] love o'er my life have rolled; * I\n     sink while I see you all aid withhold:\nYou have drowned my vitals in deeps of your love, * Nor can heart\n     and sprite for your loss be consoled:\nDeem not I forget my troth after you: * How forget what Allah\n     decreed of old?[FN#178]\nLove clings to the lover who nights in grief, * And 'plains of\n     unrest and of woes ensouled.\n\nThe kings and all those who were present rejoiced in this with\njoy exceeding and the accursed Iblis came up to Tohfah and\nkissing her hand, said to her, \"Verily there abideth but little\nof the night; so tarry with us till the morrow, when we will\napply ourselves to the wedding[FN#179] and the\ncircumcision.\"[FN#180] Then all the Jann went away, whereupon\nTohfah rose to her feet and Iblis said, \"Go ye up with Tohfah to\nthe garden for the rest of the night.\" So Kamariyah took her and\nwent with her into the garden, which contained all manner birds,\nnightingale and mocking-bird and ringdove and curlew[FN#181] and\nother than these of all the kinds. Therein were all manner of\nfruits: its channels[FN#182] were of gold and silver and the\nwater thereof, as it broke forth of its conduits, was like the\nbellies of fleeing serpents, and indeed it was as it were the\nGarden of Eden.[FN#183] When Tohfah beheld this, she called to\nmind her lord and wept sore and said, \"I beseech Allah the Most\nHigh to vouchsafe me speedy deliverance and return to my palace\nand to my high estate and queendom and glory, and reunion with my\nlord and master Al-Rashid.\" Then she walked about that garden and\nsaw in its midst a dome of white marble, raised on columns of\nblack teak whereto hung curtains purfled with pearls and gems.\nAmiddlemost this pavilion was a founfain, inlaid with all kinds\nof jacinths, and thereon a golden statue of a man and beside it a\nlittle door. She opened the door and found herself in a long\ncorridor: so she followed it and entered a Hammam-bath walled\nwith all kinds of costly marbles and floored with a mosaic of\npearls and jewels. Therein were four cisterns of alabaster, one\nfacing other, and the ceiling of the bath was of glass \nwith all varieties of colours, such as confounded the\nunderstanding of those who have insight and amazed the wit of\nevery wight. Tohfah entered the bath, after she had doffed her\ndress, and behold the Hammam-basin was overlaid with gold set\nwith pearls and red balasses and green emeralds and other jewels:\nso she extolled Allah Almighty and hallowed Him for the\nmagnificence of that which she saw of the appointments of that\nbath. Then she made her Wuzu-ablution in that basin and\npronouncing the Prohibition,[FN#184] prayed the dawn-prayer and\nwhat else had escaped her of orisons;[FN#185] after which she\nwent out and walked in that garden among jessamine and lavender\nand roses and chamomile and gillyflowers and thyme and violets\nand basil royal, till she came to the door of the pavilion\naforesaid. There she sat down, pondering that which would betide\nAl-Rashid after her, when he should come to her apartment and\nfind her not; and she plunged into the sea of her solicitude,\ntill slumber overtook her and soon she slept. Presently she felt\na breath upon her face; whereupon she awoke and found Queen\nKamariyah kissing her, and with her her three sisters, Queen\nJamrah, Queen Wakh\u00edmah and Queen Shar\u00e1rah. So she arose and\nkissed their hands and rejoiced in them with the utmost joy and\nthey ceased not, she and they, to talk and converse, what while\nshe related to them her history, from the time of her purchase by\nthe Maghrabi to that of her coming to the quarters of the slave-\ndealer, where she besought Ishak al-Nadim to buy her,[FN#186] and\nhow she won union with Al-Rashid, till the moment when Iblis came\nto her and brought her to them. They gave not over talking till\nthe sun declined and yellowed and the hour of its setting drew\nnear and the day departed, whereupon Tohfah was urgent in\nsupplication[FN#187] to Allah Almighty, on the occasion of the\nsundown-prayer, that he would reunite her with her lord\nAl-Rashid. After this, she abode with the four queens, till they\narose and entered the palace, where she found the waxen tapers\nlit and ranged in candlesticks of gold and silver, and censing\nvessels of silver and gold filled with lign-aloes and ambergris,\nand there were the kings of the Jann sitting. So she saluted them\nwith the salam, kissing the earth before them and doing them\nservice; and they rejoiced in her and in her sight. Then she\nascended the estrade and sat down upon her chair, whilst King Al-\nShisban and King Al-Muzfir[FN#188] and Queen L\u00fal\u00faah and other\nkings of the Jann sat on chairs, and they brought choice tables,\nspread with all manner meats befitting royalties. They ate their\nfill; after which the tables were removed and they washed their\nhands and wiped them with napkins. Then they brought the wine-\nservice and set on tasses and cups and flagons and beakers of\ngold and silver and bowls of crystal and gold; and they poured\nout the wines and they filled the flagons. Then Iblis took the\nbowl and signed to Tohfah to sing: and she said, \"To hear is to\nobey!\" So she hent the lute in hand and tuning it, sang these\ncouplets,\n\n\"Drink wine, O ye lovers, I rede you alw\u00e0y, * And praise his\n     worth who loves night and day;\n'Mid the myrtle, narcissus and lavender, * And the scented herbs\n     that bedeck the tray.\"\n\nSo Iblis the Damned drank and said, \"Brava, O desire of hearts!\nBut thou owest me still another aria.\" Then he filled the cup and\nsigned to her to sing. Quoth she, \"Hearkening and obedience,\" and\nchanted these couplets,\n\n\"Ye wot, I am whelmed in despair and despight, * Ye dight me\n     blight that delights your sight:\nYour wone is between my unrest and my eyes; * Nor tears to melt\n     you, nor sighs have might.\nHow oft shall I sue you for justice, and you * With a pining\n     death my dear love requite?\nBut your harshness is duty, your farness near; * Your hate is\n     Union, your wrath is delight:\nTake your fill of reproach as you will: you claim * All my heart,\n     and I reck not of safety or blame.\"\n\n\nAll present were delighted and the sitting-chamber was moved like\na wave with mirth, and Iblis said, \"Brave, O Tohfat al-Sudur!\"\nThen they left not liquor-bibbing and rejoicing and making merry\nand tambourining and piping till the night waned and the dawn\nwaxed near; and indeed exceeding delight entered into them. The\nmost of them in mirth was the Shaykh Iblis, and for the stress of\nthat which befel him of joyance, he doffed all that was on him of\n clothes and cast them over Tohfah, and among the rest a\nrobe broidered with jewels and jacinths, worth ten thousand\ndinars. Then he kissed the earth and danced and he thrust his\nfinger up his fundament and hending his beard in hand, said to\nher, \"Sing about this beard and endeavour after mirth and\npleasance, and no blame shall betide thee for this.\" So she\nimprovised and sang these couplets:--\n\n\"Barbe of the olden, the one eyed goat! * What words shall thy\n     foulness o' deed denote?\nBe not of our praises so pompous-proud: * Thy worth for a dock-\n     tail dog's I wot.\nBy Allah, to-morrow shall see me drub * Thy nape with a\n     cow-hide[FN#189] and dust thy coat!\"\n\nAll those present laughed at her mockery of Iblis and wondered at\nthe wittiness of her visnomy[FN#190] and her readiness in\nversifying, whilst the Shaykh himself rejoiced and said to her,\n\"O Tohfat al-Sudur, verily, the night be gone; so arise and rest\nthyself ere the day; and to-morrow there shall be naught save\nweal.\" Then all the kings of the Jinn departed, together with\nthose who were present of guards; and Tohfah abode alone,\npondering the case of Al-Rashid and bethinking her of how it went\nwith him after her going, and of what had betided him for her\nloss, till the dawn lightened, when she arose and walked about\nthe palace. Suddenly she saw a handsome door; so she opened it\nand found herself in a flower-garden finer than the first--ne'er\nsaw eyes of seer a fairer than it. When she beheld this garth,\nshe was moved to delight and she called to mind her lord Al-\nRashid and wept with sore weeping and cried, \"I crave of the\nbounty of Allah Almighty that my return to him and to my palace\nand to my home may be nearhand!\" Then she walked about the\nparterres till she came to a pavilion, high-builded of base and\nwide of space, never espied mortal nor heard of a grander than\nit. So she entered and found herself in a long corridor, which\nled to a Hammam goodlier than that aforetime described, and its\ncisterns were full of rose-water mingled with musk. Quoth Tohfah,\n\"Extolled be Allah! Indeed, this[FN#191] is none other than a\nmighty great king.\" Then she pulled off her clothes and washed\nher body and made her Ghusl-ablution of the whole person[FN#192]\nand prayed that which was due from her of prayer from the evening\nof the previous day.[FN#193] When the sun rose upon the gate of\nthe garden and she saw the wonders thereof, with that which was\ntherein of all manner blooms and streams, and heard the voices of\nits birds, she marvelled at what she beheld of the rareness of\nits ordinance and the beauty of its disposition and sat musing\nover the case of Al-Rashid and pondering what was come of him\nafter her. Her tears coursed down her cheeks and the Zephyr blew\non her; so she slept and knew no more till she suddenly felt a\nbreath on her side-face, whereat she awoke in affright and found\nQueen Kamariyah kissing her, and she was accompanied by her\nsisters, who said, \"Rise, for the sun hath set.\" So Tohfah arose\nand making the Wuzu-ablution, prayed her due of prayers[FN#194]\nand accompanied the four queens to the palace, where she saw the\nwax candles lighted and the kings sitting. She saluted them with\nthe salam and seated herself upon her couch; and behold, King Al-\nShisban had shifted his semblance, for all the pride of his soul.\nThen came up Iblis (whom Allah damn!) and Tohfah rose to him and\nkissed his hands. He also kissed her hand and blessed her and\nasked, \"How deemest thou? Is not this place pleasant, for all its\ndesertedness and desolation?\" Answered she, \"None may be desolate\nin this place;\" and he cried, \"Know that this is a site whose\nsoil no mortal dare tread;\" but she rejoined, \"I have dared and\ntrodden it, and this is one of thy many favours.\" Then they\nbrought tables and dishes and viands and fruits and sweetmeats\nand other matters, whose description passeth powers of mortal\nman, and they ate their sufficiency; after which the tables were\nremoved and the dessert-trays and platters set on, and they\nranged the bottles and flagons and vessels and phials, together\nwith all manner fruits and sweet-scented flowers. The first to\nraise the bowl was Iblis the Accursed, who said, \"O Tohfat al-\nSudur, sing over my cup.\" So she took the lute and touching it,\ncarolled these couplets,\n\n\"Wake ye, Ho sleepers all! and take your joy * Of Time, and boons\n     he deign\u00e8d to bestow;\nThen hail the Wine-bride, drain the wine-ptisane * Which, poured\n     from flagon, flows with flaming glow:\nO Cup-boy, serve the wine, bring round the red[FN#195] * Whose\n     draught gives all we hope for here below:\nWhat's worldly pleasure save my lady's face, * Draughts of pure\n     wine and song of musico?\"\n\nSo Iblis drained his bowl and, when he had made an end of his\ndraught, waved his hand to Tohfah; then, throwing off that which\nwas upon him of clothes, delivered them to her. The suit would\nhave brought ten thousand dinars and with it was a tray full of\njewels worth a mint of money. Presently he filled again and gave\nthe cup to his son Al-Shisban, who took it from his hand and\nkissing it, stood up and sat down again. Now there was before him\na tray of roses; so he said to her, \"O Tohfah, sing thou somewhat\nupon these roses.\" She replied, \"Hearkening and obedience,\" and\nchanted these two couplets,\n\n\"It proves my price o' er all the flowers that I * Seek you each\n     year, yet stay but little stound:\nAnd high my vaunt I m dy\u00e8d by my lord * Whom Allah made the best\n     e'er trod on ground.[FN#196]\"\n\nSo Al-Shisban drank off the cup in his turn and said, \"Brava, O\ndesire of hearts!\" and he bestowed on her that was upon him, to\nwit, a dress of cloth-of-pearl, fringed with great unions and\nrubies and purfled with precious gems, and a tray wherein were\nfifty thousand dinars. Then Maymun the Sworder took the cup and\nbegan gazing intently upon Tohfah. Now there was in his hand a\npomegranate-flower and he said to her, \"Sing thou somewhat, O\nqueen of mankind and Jinn kind upon this pomegranate-flower; for\nindeed thou hast dominion over all hearts.\" Quoth she, \"To hear\nis to obey;\" and she improvised and sang these couplets,\n\n\"Breathes sweet the zephyr on fair part\u00e8rre; * Robing lute in the\n     flamings that fell from air:\nAnd moaned from the boughs with its cooing rhyme * Voice of\n     ring-doves plaining their love and care:\nThe branch dresses in suit of fine sendal green* And in wine-hues\n     borrowed from bloom Gulnare.\"[FN#197]\n\nMaymun the Sworder drained his bowl and said to her, \"Brava, O\nperfection of qualities!\" Then he signed to her and was absent\nawhile, after which he returned and with him a tray of jewels\nworth an hundred thousand ducats, which he gave to Tohfah.\nThereupon Kamariyah arose and bade her slave-girl open the closet\nbehind the Songstress, wherein she laid all that wealth; and\ncommitted the key to her, saying, \"Whatso of riches cometh to\nthee, lay thou in this closet that is by thy side, and after the\nfestivities, it shall be borne to thy palace on the heads of the\nJinn.\" Tohfah kissed her hand and another king, by name\nMun\u00edr,[FN#198] took the bowl and filling it, said to her, \"O\nferly Fair, sing to me over my bowl somewhat upon the jasmine.\"\nShe replied with, \"Hearkening and obedience,\" and improvised\nthese couplets,\n\n\"'Twere as though the Jasmine (when self she enrobes * On her\n     boughs) doth display to my wondering eyne;\nIn sky of green beryl, which Beauty enclothes, * Star-groups like\n     studs of the silvern mine.\"\n\nMunir drank off his cup and ordered her eight hundred thousand\ndinars, whereat Kamariyah rejoiced and rising to her feet, kissed\nTohfah on her face and said to her, \"Be the world never bereaved\nof thee, O thou who lordest it over the hearts of Jinn-kind and\nmankind!\" Then she returned to her place and the Shaykh Iblis\narose and danced, till all present were confounded; after which\nthe Songstress {sic!} said, \"Verily, thou embellishest my festivities, O\nthou who commandest men and Jinn and rejoicest their hearts with\nthy loveliness and the beauty[FN#199] of thy faithfulness to thy\nlord. All that thy hands possess shall be borne to thee in thy\npalace and placed at thy service; but now the dawn is nearhand;\nso do thou rise and rest thee according to thy custom.\" Tohfah\nturned and found with her none of the Jinn; so she laid her head\non the floor and slept till she had gotten her repose; after\nwhich she arose and betaking herself to the lakelet, made the\nWuzu-ablution and prayed. Then she sat beside the water awhile\nand meditated the matter of her lord Al-Rashid and that which had\nbetided him after her loss and wept with sore weeping. Presently,\nshe heard a blowing behind her;[FN#200] so she turned and behold,\na Head without a body and with eyes slit endlong: it was of the\nbigness of an elephant's skull and bigger and had a mouth as it\nwere an oven and projecting canines as they were grapnels, and\nhair which trailed upon the ground. So Tohfah cried, \"I take\nrefuge with Allah from Satan the Stoned!\" and recited the Two\nPreventives;[FN#201] what while the Head drew near her and said,\n\"Peace be with thee, O Princess of Jinn and men and union-pearl\nof her age and her time! Allah continue thee on life, for all the\nlapsing of the days, and reunite thee with thy lord the\nImam!\"[FN#202] She replied, \"And upon thee be Peace; O thou whose\nlike I have not seen among the Jann!\" Quoth the Head, \"We are a\nfolk who may not change their favours and we are hight Ghuls:\nmortals summon us to their presence, but we cannot present\nourselves before them without leave. As for me, I have gotten\nleave of the Shaykh Abu al-Tawaif to appear before thee and I\ndesire of thy favour that thou sing me a song, so I may go to thy\npalace and question its Haunters[FN#203] concerning the plight of\nthy lord after thee and return to thee; and know, O Tohfat al-\nSudur, that between thee and thy lord be a distance of fifty\nyears' journey for the bon\u00e2-fide traveller.\" She rejoined,\n\"Indeed, thou grievest me anent him between whom and me is fifty\nyears' journey;\" but the Head[FN#204] cried to her, \"Be of good\ncheer and of eyes cool and clear, for the sovrans of the Jann\nwill restore thee to him in less than the twinkling of an eye.\"\nQuoth she, \"I will sing thee an hundred songs, so thou wilt bring\nme news of my lord and that which betided him after me.\" And\nquoth the Head, \"Do thou favour me and sing me a song, so I may\ngo to thy lord and fetch thee tidings of him, for that I desire,\nbefore I go, to hear thy voice, so haply my thirst[FN#205] may be\nquenched.\" So she took the lute and tuning it, sang these\ncouplets:--\n\n\"They have marched, yet no empty stead left they: * They are\n     gone, nor heart grieves me that fled be they:\nMy heart forebode the bereaval of friends; * Allah ne'er bereave\n     steads wherefrom sped be they!\nThough they hid the stations where led were they, * I'll follow\n     till stars fall in disarray!\nYe slumber, but wake shall ne'er fly these lids; * 'Tis I bear\n     what ye never bore--well-away!\nIt had irked them not to farewell who fares * With the parting-\n     fires that my heart waylay.\nMy friends,[FN#206] your meeting to me is much * But more is the\n     parting befel us tway:\nYou're my heart's delight, or you present be * Or absent, with\n     you is my soul for aye!\"\n\nThereupon the Head wept exceeding sore and cried, \"O my lady,\nindeed thou hast solaced my heart, and I have naught but my life;\nso take it.\" She replied, \"Nay, an I but knew that thou wouldst\nbring me news of my lord Al-Rashid, 'twere fainer to me than the\nreign of the world;\" and the Head answered her, \"It shall be done\nas thou desirest.\" Then it disappeared and returning to her at\nthe last of the night, said, \"O my lady, know that I have been to\nthy palace and have questioned one of its Haunters of the case of\nthe Commander of the Faithful and that which befel him after\nthee; and he said, 'When the Prince of True Believers came to\nTohfah's apartment and found her not and saw no sign of her, he\nbuffeted his face and head and rent his raiment.' Now there was\nin thy chamber the Castrato, the chief of thy household, and the\nCaliph cried out at him, saying, 'Bring me Ja'afar the Barmaki\nand his father and brother at this very moment!' The Eunuch went\nout, bewildered in his wit for fear of the King, and when he\nstood in the presence of Ja'afar, he said to him, 'Come to the\nCommander of the Faithful, thou and thy father and thy brother.'\nSo they arose in haste and betaking themselves to the presence,\nsaid, 'O Prince of True Believers what may be the matter?' Quoth\nhe, 'There is a matter which passeth description. Know that I\nlocked the door and taking the key with me, betook myself to my\nuncle's daughter, with whom I lay the night; but, when I arose in\nthe morning and came and opened the door, I found no sign of\nTohfah.' Quoth Ja'afar, 'O Commander of the Faithful have\npatience, for that the damsel hath been snatched away, and needs\nmust she return, seeing that she took the lute with her, and 'tis\nher own lute. The Jinns have assuredly carried her off, and we\ntrust in Allah Almighty that she will return.' Cried the Caliph,\n'This[FN#207] is a thing which may nowise be!' And he abode in\nher apartment, nor eating nor drinking, while the Barmecides\nbesought him to fare forth to the folk; and he weepeth and\ntarrieth on such fashion till she shall return. This, then, is\nthat which hath betided him after thee.\" When Tohfah heard his\nwords, they were grievous to her and she wept with sore weeping;\nwhereupon quoth the Head to her, \"The relief of Allah the Most\nHigh is nearhand; but now let me hear somewhat of thy speech.\" So\nshe took the lute and sang three songs, weeping the while. The\nHead exclaimed, \"By Allah, thou hast been bountiful to me, the\nLord be with thee!\" Then it disappeared and the season of sundown\ncame: so she rose and betook herself to her place in the hall;\nwhereupon behold, the candles sprang up from under the earth and\nkindled themselves. Then the kings of the Jann appeared and\nsaluted her and kissed her hands and she greeted them with the\nsalam. Presently appeared Kamariyah and her three sisters and\nsaluted Tohfah and sat down; whereupon the tables were brought\nand they ate; and when the tables were removed there came the\nwine-tray and the drinking-service. So Tohfah took the lute and\none of the three queens filled the cup and signed to the\nSongstress. Now she had in her hand a violet, so Tohfah\nimprovised these couplets:--\n\n\"I'm clad in a leaf-cloak of green; * In an honour-robe\n     ultramarine:\nI'm a wee thing of loveliest mien * But all flowers as my vassals\n     are seen:\nAn Rose title her 'Morn-pride,' I ween * Nor before me nor after\n     she's Queen.\"\n\nThe queen drank off her cup and bestowed on Tohfah a dress of\ncloth-of-pearl, fringed with red rubies, worth twenty thousand\nducats, and a tray whereon were ten thousand sequins. All this\nwhile Maymun's eye was upon her and presently he said to her,\n\"Harkye, Tohfah! Sing to me.\" But Queen Zalzalah cried out at\nhim, and said \"Desist,[FN#208] O Maymun. Thou sufferest not\nTohfah to pay heed to us.\" Quoth he, \"I will have her sing to\nme:\" and many words passed between them and Queen Zalzalah cried\naloud at him. Then she shook and became like unto the Jinns and\ntaking in her hand a mace of stone, said to him, \"Fie upon thee!\nWhat art thou that thou shouldst bespeak us thus? By Allah, but\nfor the respect due to kings and my fear of troubling the session\nand the festival and the mind of the Shaykh Iblis, I would\nassuredly beat the folly out of thy head!\" When Maymun heard\nthese her words, he rose, with the fire shooting from his eyes,\nand said, \"O daughter of Iml\u00e1k, what art thou that thou shouldst\noutrage me with the like of this talk?\" Replied she, \"Woe to\nthee, O dog of the Jinn, knowest thou not thy place?\" So saying,\nshe ran at him, and offered to strike him with the mace, but the\nShaykh Iblis arose and casting his turband on the ground cried,\n\"Out on thee, O Maymun! Thou dost always with us on this wise.\nWheresoever thou art present, thou troublest our pleasure! Canst\nthou not hold thy peace until thou go forth of the festival and\nthis bride-feast be accomplished? When the circumcision is at an\nend and ye all return to your dwellings, then do as thou willest.\nFie upon thee, O Maymun! Wottest thou not that Imlak is of the\nchiefs of the Jinn? But for my good name, thou shouldst have seen\nwhat would have betided thee of humiliation and chastisement; yet\non account of the festival none may speak. Indeed thou exceedest;\ndost thou not ken that her sister Wakhimah is doughtier[FN#209]\nthan any of the Jann? Learn to know thyself: hast thou no regard\nfor thy life?\" So Maymun was silent and Iblis turned to Tohfah\nand said to her, \"Sing to the kings of the Jinns this day and to-\nnight until the morrow, when the boy will be circumcised and each\nshall return to his own place.\" Accordingly she took the lute and\nKamariyah said to her (now she had a citron in hand), \"O my\nsister, sing to me somewhat on this citron.\" Tohfah replied, \"To\nhear is to obey,\" and improvising, sang these couplets,\n\n\"I'm a dome of fine gold and right cunningly dight; * And my\n     sweetness of youth gladdeth every sight:\nMy wine is ever the drink of kings * And I'm fittest gift to the\n     friendliest sprite.\n\nAt this Queen Kamariyah rejoiced with joy exceeding and drained\nher cup, crying, \"Brava! O thou choice Gift of hearts!\"\nFurthermore, she took off a sleeved robe of blue brocade, fringed\nwith red rubies, and a necklace of white jewels worth an hundred\nthousand ducats, and gave them to Tohfah. Then she passed the cup\nto her sister Zalzalah, who hent in her hand herb basil, and she\nsaid to Tohfah, \"Sing to me somewhat on this basil.\" She replied,\n\"Hearing and obeying,\" and improvised and sang these couplets,\n\n\"I'm the Queen of herbs in the s\u00e9ance of wine * And in Heaven\n     Na'\u00edm are my name and sign:\nAnd the best are promised, in garth of Khuld, * Repose, sweet\n     scents and the peace divine:[FN#210]\nWhat prizes then with my price shall vie? * What rank even mine,\n     in all mortals' eyne?\"\n\nThereat Queen Zalzalah rejoiced with joy exceeding and bidding\nher treasuress bring a basket, wherein were fifty pairs of\nbracelets and the same number of earrings, all of gold, crusted\nwith jewels of price, whose like nor mankind nor Jinn-kind\npossessed, and an hundred robes of vari- brocades and an\nhundred thousand ducats, gave the whole to Tohfah. Then she\npassed the cup to her sister Shararah, who had in her hand a\nstalk of narcissus; so she took it from her and turning to the\nSongstress, said to her, \"O Tohfah, sing to me somewhat on this.\"\nShe replied, \"Hearkening and obedience,\" and improvised these\ncouplets,\n\n\"With the smaragd wand doth my form compare; * 'Mid the finest\n     flowers my worth's rarest rare:\nMy eyes are likened to Beauty's eyne, * And my gaze is still on\n     the bright part\u00e8rre.\"\n\nWhen she had made an end of her song, Shararah was moved to\ndelight exceeding, and drinking off her cup, said to her, \"Brava,\nO thou choice Gift of hearts!\" Then she ordered her an hundred\ndresses of brocade and an hundred thousand ducats and passed the\ncup to Queen Wakhimah. Now she had in her hand somewhat of\nNu'uman's bloom, the anemone; so she took the cup from her sister\nand turning to the Songstress, said to her, \"O Tohfah, sing to me\non this.\" Quoth she, \"I hear and I obey,\" and improvised these\ncouplets,\n\n\"I'm a dye was dyed by the Ruthful's might; * And all confess me\n     the goodliest sight:\nI began in the dust and the clay, but now * On the cheeks of fair\n     women I rank by right.\"\n\nTherewith Wakhimah rejoiced with joy exceeding and drinking off\nthe cup, ordered her twenty dresses of Roum\u00ed brocade and a tray,\nwherein were thirty thousand ducats. Then she gave the cup to\nQueen Shu'\u00e1'ah,[FN#211] Regent of the Fourth Sea, who took it and\nsaid, \"O my lady Tohfah, sing to me on the gillyflower.\" She\nreplied, \"Hearing and obeying,\" and improvised these couplets,\n\n\"The time of my presence ne'er draws to a close, * Amid all whose\n     joyance with mirth o'erflows;\nWhen topers gather to sit at wine * Or in nightly shade or when\n     morning shows,\nI filch from the flagon to fill the bowls * And the crystal cup\n     where the wine-beam glows.\"\n\nQueen Shu'a'ah rejoiced with joy exceeding and emptying her cup,\ngave Tohfah an hundred thousand ducats. Then up sprang Iblis\n(whom Allah curse!) and cried, \"Verily, the dawn lighteneth;\"\nwhereupon the folk arose and disappeared, all of them, and there\nabode not one of them save the Songstress, who went forth to the\ngarden and entering the Hamman made her Wuzu-ablutions and prayed\nwhatso lacked her of prayers. Then she sat down and when the sun\nrose, behold, there came up to her near an hundred thousand green\nbirds, which filled the branches of the trees with their\nmultitudes and they warbled in various voices, whilst Tohfah\nmarvelled at their fashion. Suddenly, appeared eunuchs, bearing a\nthrone of gold, studded with pearls and gems and jacinths, both\nwhite and red, and having four steps of gold, together with many\ncarpets of sendal and brocade and Coptic cloth of silk sprigged\nwith gold; and all these they spread in the centre of the garden\nand setting up the throne thereon, perfumed the place with virgin\nmusk, Nadd[FN#212] and ambergris. After that, there came a queen;\nnever saw eyes a fairer than she nor than her qualities; she was\nrobed in rich raiment, broidered with pearls and gems, and on her\nhead was a crown set with various kinds of unions and jewels.\nAbout her were five hundred slave-girls, high-bosomed maids, as\nthey were moons, screening her, right and left, and she among\nthem like the moon on the night of its full, for that she was the\nmost worthy of them in majesty and dignity. She ceased not\nwalking till she came to Tohfah, whom she found gazing on her in\namazement; and when the Songstress saw her turn to her, she rose\nto her, standing on her feet, and saluted her and kissed ground\nbetween her hands. The queen rejoiced in her and putting out her\nhand to her, drew her to herself and seated her by her side on\nthe couch; whereupon the Songstress kissed her hands and the\nqueen said to her, \"Know, O Tohfah, that all which thou treadest\nof these carpets belongeth not to any of the Jinn, who may never\ntread them without thy leave,[FN#213] for that I am the queen of\nthem all and the Shaykh Abu al-Tawaif Iblis sought my permission\nto hold festival[FN#214] and prayed me urgently to be present at\nthe circumcision of his son. So I despatched to him, in my stead,\na slave-girl of my slave-girls, namely, Shu'\u00e1'ah Queen of the\nFourth Sea, who is vice-reine of my reign. When she was present\nat the wedding and saw thee and heard thy singing, she sent to\nme, informing me of thee and setting forth to me thy grace and\namiability and the beauty of thy breeding and thy\ncourtesy.[FN#215] So I am come to thee, for that which I have\nheard of thy charms, and hereby I do thee a mighty great favour\nin the eyes of all the Jann.\"[FN#216] Thereupon Tohfah arose and\nkissed the earth and the queen thanked her for this and bade her\nsit. So she sat down and the queen called for food; when they\nbrought a table of gold, inlaid with pearls and jacinths and\njewels and bearing kinds manifold of birds and viands of various\nhues, and the queen said, \"O Tohfah, in the name of Allah! Let us\neat bread and salt together, I and thou.\" Accordingly the\nSongstress came forward and ate of those meats and found therein\nsomewhat the like whereof she had never eaten; no, nor aught more\ndelicious than it, while the slave-girls stood around the table,\nas the white compasseth the black of the eye, and she sat\nconversing and laughing with the queen. Then said the lady, \"O my\nsister, a slave-girl told me of thee that thou saidst, 'How\nloathly is what yonder Jinni Maymun eateth!\"[FN#217] Tohfah\nreplied, \"By Allah, O my lady, I have not any eye that can look\nat him,[FN#218] and indeed I am fearful of him.\" When the queen\nheard this, she laughed till she fell backwards and said \"O my\nsister, by the might of the graving upon the seal-ring of\nSolomon, prophet of Allah, I am queen over all the Jann, and none\ndare so much as cast on thee a glance of the eye;\" whereat Tohfah\nkissed her hand. Then the tables were removed and the twain sat\ntalking. Presently up came the kings of the Jinn from every side\nand kissed ground before the queen and stood in her service; and\nshe thanked them for this, but moved not for one of them.[FN#219]\nThen appeared the Shaykh Abu al-Taw\u00e1if Iblis (Allah curse him!)\nand kissed the earth before her, saying, \"O my lady, may I not be\nbereft of these steps!\"[FN#220] She replied, \"O Shaykh Abu\nal-Taw\u00e1if, it behoveth thee to thank the bounty of the Lady\nTohfah, who was the cause of my coming.\" Rejoined he, \"Thou\nsayest sooth,\" and kissed ground. Then the queen fared on towards\nthe palace and there arose and alighted upon the trees an hundred\nthousand birds of manifold hues. The Songstress asked, \"How many\nare these birds?\" and Queen Wakhimah answered her, \"Know, O my\nsister, that this queen is hight Queen al-Shahb\u00e1[FN#221] and that\nshe is queen over all the Jann from East to West. These birds\nthou seest are of her host, and unless they appeared in this\nshape, earth would not be wide enough for them. Indeed, they came\nforth with her and are present with her presence at this\ncircumcision. She will give thee after the measure of that which\nhath been given to thee from the first of the festival to the\nlast thereof;[FN#222] and indeed she honoureth us all with her\npresence.\" Then the queen entered the palace and sat down on the\ncouch of the circumcision[FN#223] at the upper end of the hall,\nwhereupon Tohfah took the lute and pressing it to her breast,\ntouched its strings suchwise that the wits of all present were\nbewildered and Shaykh Iblis cried to her, \"O my lady Tohfah, I\nconjure thee, by the life of this noble queen, sing for me and\npraise thyself, and cross me not.\" Quoth she, \"To hear is to\nobey; still, but for thine adjuration, I had not done this. Say\nme, doth any praise himself? What manner thing is this?\" Then she\nimprovised these couplets:\n\n\"In all f\u00eates I'm Choice Gift[FN#224] to the minstrel-race;\nFolk attest my worth, rank and my pride of place,\nWhile Fame, merit and praises with honour engrace.\"\n\nHer verses pleased the kings of the Jann and they cried, \"By\nAllah, thou sayst sooth!\" Then she rose to her feet, hending lute\nin hand, and played and sang, whilst the Jinns and the Shaykh Abu\nal-Taw\u00e1if danced. Presently the Father of the Tribes came up to\nher bussing her bosom, and gave her a Br\u00e1hmani[FN#225] carbuncle\nhe had taken from the hidden hoard of Y\u00e1fis bin N\u00fah[FN#226] (on\nwhom be the Peace), and which was worth the reign of the world;\nits light was as the sheen of the sun and he said to her, \"Take\nthis and be equitable therewith to the people of the world.\"\n[FN#227] She kissed his hand and rejoiced in the jewel and said,\n\"By Allah, this befitteth none save the Commander of the\nFaithful.\" Now Queen Al-Shahba laughed with delight at the\ndancing of Ibl\u00eds and she said to him, \"By Allah, this is a goodly\npavane!\" He thanked her for this and said to the Songstress, \"O\nTohfah, there is not on earth's face a skilfuller than Ishak\nal-Nadim;[FN#228] but thou art more skilful than he. Indeed, I\nhave been present with him many a time and have shown him\npositions[FN#229] on the lute, and there has betided me with him\nthat which betided. Indeed, the story of my dealings with him is\na long one but this is no time to repeat it; for now I would show\nthee a shift on the lute, whereby thou shalt be exalted over all\nfolk.\" Quoth she, \"Do what seemeth good to thee.\" So he took the\nlute and played thereon a wondrous playing, with rare divisions\nand marvellous modulations, and showed her a passage she knew\nnot; and this was goodlier to her than all that she had gotten.\nThen she took the lute from him and playing thereon, sang and\npresently returned to the passage which he had shown her; and he\nsaid, \"By Allah, thou singest better than I!\" As for Tohfah, it\nbecame manifest to her that her former practice was all of it\nwrong and that what she had learnt from the Shaykh Abu al-Taw\u00e1if\nIblis was the root and foundation of all perfection in the art\nand its modes. So she rejoiced in that which she had won of skill\nin touching the lute far more than in all that had fallen to her\nlot of wealth and honour-robes and kissed the Master's hand. Then\nsaid Queen Al-Shahba, \"By Allah, O Shaykh, my sister Tohfah is\nindeed singular among the folk of her time, and I hear that she\nsingeth upon all sweet-smelling blooms.\" Iblis replied, \"Yes, O my\nlady, and I am in extremest wonderment thereat. But there\nremaineth somewhat of sweet-scented flowers, which she hath not\nbesung, such as myrtle and tuberose and jessamine and the\nmoss-rose and the like.\" Then the Shaykh signed to her to sing\nsomewhat upon the rest of the flowers, that Queen Al-Shahba might\nhear, and she said, \"Hearing and obeying.\" So she took the lute\nand played thereon in many modes, then returned to the first and\nsang these couplets,\n\n\"I'm one of the lover-retinue * Whom long pine and patience have\n     doom\u00e8d rue:\nAnd sufferance of parting from kin and friends * Hath clothed me,\n     O folk, in this yellow hue:\nThen, after the joyance had passed away, * Heart-break, abasement\n     and cark I knew,\nThrough the long, long day when the lift is light, * Nor, when\n     night is murk, my pangs cease pursue:\nSo, 'twixt fairest hope and unfailing fear, * My bitter tears\n     ever flow anew.\"\n\nThereat Queen Al-Shahba rejoiced with joy exceeding and cried,\n\"Brava, O queen of delight! No one is able to describe thee. Sing\nto us on the Apple.\" Quoth Tohfah, \"Hearkening and obedience.\"\nThen she recited these couplets,\n\n\"I surpass all forms in my coquetry* For mine inner worth and\n     mine outer blee;\nTend me noble hands in the sight of all * And slake with pure\n     waters the thirst of me;\nMy robe is of sendal, and eke my veil * Is of sunlight the\n     Ruthful hath bidden be:\nWhen my fair companions are marched afar, * In sorrow fro' home\n     they are forced to flee:\nBut noble hands deign hearten my heart * With beds where I sit in\n     my high degree; [FN#230]\nAnd where, like full moon at its rise, my light * 'mid the\n     garden-fruits thou shalt ever see.\"\n\nQueen Al-Shahba rejoiced in this with exceeding joy and cried\n\"Brava! By Allah, there is none excelleth thee.\" Tohfah kissed\nthe ground, then returned to her place and versified on the\nTuberose, saying,\n\n\"I'm a marvel-bloom to be worn on head! * Though a stranger among\n     you fro' home I fled:\nMake use of wine in my company * And flout at Time who in\n     languish sped.\nE'en so doth camphor my hue attest, * O my lords, as I stand in\n     my present stead.\nSo gar me your gladness when dawneth day, * And to highmost seat\n     in your homes be I led:\nAnd quaff your cups in all jollity, * And cheer and ease shall\n     ne'er cease to be.\"\n\nAt this Queen Al-Shahba rejoiced with exceeding joy and cried,\n\"Brava, O queen of delight! By Allah, I know not how I shall do\nto give thee thy due! May the Most High grant us the grace of thy\nlong continuance!\" Then she strained her to her breast and bussed\nher on the cheek; whereupon quoth Iblis (on whom be a curse!),\n\"This is a mighty great honour!\" Quoth the queen, \"Know that this\nlady Tohfah is my sister and that her biddance is my biddance and\nher forbiddance my forbiddance. So all of you hearken to her word\nand render her worshipful obedience.\" Therewith the kings rose in\na body and kissed ground before Tohfah, who rejoiced in this.\nMoreover, Queen Al-Shahba doffed dress and habited her in a suit\nadorned with pearls, jewels and jacinths, worth an hundred\nthousand ducats, and wrote for her on a slip of paper[FN#231] a\npatent appointing her to be her deputy. So the Songstress rose\nand kissed ground before the Queen, who said to her, \"Of thy\nfavour, sing to us somewhat concerning the rest of the\nsweet-scented flowers and herbs, so I may hear thy chant and\nsolace myself with witnessing thy skill.\" She replied, \"To hear\nis to obey, O lady mine,\" and, taking the lute, improvised these\ncouplets,\n\n\"My hue excelleth all hues in light, * And I would all eyes\n     should enjoy my sight:\nMy site is the site of fillets and pearls * Where the fairest\n     brows are with jasmine dight:\nMy light's uprist (and what light it shows!) * Is a silvern zone\n     on the waist of Night.\"\n\nThen she changed the measure and improvised these couplets,\n\n\"I'm the gem of herbs, and in seasons twain * My tryst I keep\n     with my lovers-train:\nI stint not union for length of time * Nor visits, though some be\n     of severance fain;\nThe true one am I and my troth I keep, * And, easy of plucking,\n     no hand disdain.\"\n\nThen, changing measure and the mode, she played so that she\nbewildered the wits of those who were present, and Queen\nAl-Shahba, moved to mirth and merriment, cried, \"Brava, O queen\nof delight!\" Presently she returned to the first mode and\nimproved these couplets on Nenuphar,\n\n\"I fear me lest freke espy me, * In air when I fain deny me;\nSo I root me beneath the wave, * And my stalks to bow down apply\nme.\"\n\nHereat Queen Al-Shahba rejoiced with exceeding joy, and cried,\n\"Brava, O Tohfah! Let me hear more of thy chant.\" Accordingly,\nshe smote the lute and changing the mode, recited on the Moss-\nrose these couplets,\n\n\"Look on Nasr\u00edn[FN#232] those branchy shoots surround; * With\n     greenest leafery 'tis deckt and crowned:\nIts graceful bending stem draws every gaze * While beauteous\n     bearing makes their love abound.\"\n\nThen she changed measure and mode and sang these couplets on the\nWater-lily,\n\n\"O thou who askest S\u00fasan[FN#233] of her scent, * Hear thou my\n     words and beauty of my lay.\n'Emir am I whom all mankind desire' * (Quoth she) 'or present or\n     when ta'en away.'\"\n\nWhen Tohfah had made an end of her song, Queen Al-Shahba rose and\nsaid, \"I never heard from any the like of this;\" and she drew the\nSongstress to her and fell to kissing her. Then she took leave of\nher and flew away; and on like wise all the birds took flight\nwith her, so that they walled the horizon; whilst the rest of the\nkings tarried behind. Now as soon as it was the fourth night,\nthere came the boy who was to be circumcised, adorned with jewels\nsuch as never saw eye nor heard ear of, and amongst the rest a\ncrown of gold crusted with pearls and gems, the worth whereof was\nan hundred thousand sequins. He sat down upon the couch and\nTohfah sang to him, till the chirurgeon[FN#234] came and they\nsnipped his foreskin in the presence of all the kings, who\nshowered on him a mighty great store of jewels and jacinths and\ngold. Queen Kamariyah bade her Eunuchs gather up all this and lay\nit in Tohfah's closet and it was as much in value as all that had\nfallen to her, from the first of the festivities to the last\nthereof. Moreover, the Shaykh Iblis (whom Allah curse!) bestowed\nupon the Songstress the crown worn by the boy and gave the\ncircumcisee another, whereat Tohfah's reason took flight. Then\nthe Jinn departed, in order of rank, whilst Iblis farewelled\nthem, band after band. Seeing the Shaykh thus occupied with\ntaking leave of the kings, Maymun seized his opportunity, the\nplace being empty, and taking up Tohfah on his shoulders, soared\naloft with her to the confines of the lift, and flew away with\nher. Presently, Iblis came to look for the Songstress and see\nwhat she purposed, but found her not and sighted the slave-girls\nslapping their faces: so he said to them, \"Fie on you! What may\nbe the matter?\" They replied, \"O our lord, Maymun hath snatched\nup Tohfah and flown away with her.\" When Iblis heard this, he\ngave a cry whereto earth trembled and said, \"What is to be done?\"\nThen he buffetted his face and head, exclaiming, \"Woe to you!\nThis be none other than exceeding insolence. Shall he carry off\nTohfah from my very palace and attaint mine honour? Doubtless,\nthis Maymun hath lost his wits.\" Then he cried out a second time,\nso that the earth quaked, and rose on his wings high in air. The\nnews came to the rest of the kings; so they flew after him and\novertaking him, found him full of anxiety and affright, with fire\nissuing from his nostrils, and said to him, \"O Shaykh\nal-Tawaif,[FN#235] what is to do?\" He replied, \"Know ye that\nMaymun hath carried off Tohfah from my palace and attainted mine honour.\"\nWhen they heard this, they cried, \"There is no Majesty and there\nis no Might save in Allah the Glorious, the Great. By God he hath\nventured upon a grave matter and verily he destroyeth self and\nfolk!\" Then Shaykh Iblis ceased not flying till he fell in with\nthe tribes of the Jann, and they gathered together a world of\npeople, none may tell the tale of them save the Lord of\nAll-might. So they came to the Fortress of Copper and the Citadel\nof Lead,[FN#236] and the people of the sconces saw the tribes of\nthe Jann issuing from every deep mountain-pass[FN#237] and said,\n\"What be the news?\" Then Iblis went in to King Al-Shisban and\nacquainted him with that which had befallen; whereupon quoth he,\n\"Verily, Allah hath destroyed Maymun and his many! He pretendeth\nto possess Tohfah, and she is become queen of the Jann! But have\npatience till we devise that which befitteth in the matter of\nTohfah.\" Iblis asked, \"And what befitteth it to do?\" And\nAl-Shisban answered, \"We will fall upon him and kill him and his\nhost with cut of brand.\" Then quoth Shaykh Iblis, \"'Twere better\nto acquaint Queen Kamariyah and Queen Zalzalah and Queen\nShararah and Queen Wakhimah; and when they are assembled, Allah\nshall ordain whatso He deemeth good in the matter of her\nrelease.\" Quoth Al-Shisban, \"Right is thy rede\" and they\ndespatched to Queen Kamariyah an Ifrit hight Salhab who came to\nher palace and found her sleeping, so he roused her and she said,\n\"What is to do, O Salhab?\" Cried he, \"O my lady, come to the\nsuccour of thy sister the Songstress, for Maymun hath carried her\noff and attainted thine honour and that of Shaykh Iblis.\" Quoth\nshe, \"What sayst thou?\" and she sat up straight and cried out\nwith a great cry. And indeed she feared for Tohfah and said, \"By\nAllah, in very sooth she used to say that he gazed at her and\nprolonged the gaze; but ill is that whereto his soul hath\nprompted him.\" Then she rose in haste and mounting a Sataness of\nher Satans, said to her, \"Fly.\" So she flew off with her and\nalighted in the palace of her sister Shararah, whereupon she sent\nfor her sisters Zalzalah and Wakhimah and acquainted them with\nthe tidings, saying, \"Know that Maymun hath snatched up Tohfah\nand flown off with her swiftlier than the blinding leven.\" Then\nthey all flew off in haste and lighting down in the place where\nwere their father Al-Shisban and their grandfather the Shaykh Abu\nal-Taw\u00e1if, found the folk on the sorriest of situations. When\ntheir grandfather Iblis saw them, he rose to them and wept, and\nthey all wept for the Songstress. Then said Iblis to them,\n\"Yonder hound hath attainted mine honour and taken Tohfah, and I\nthink not other wise[FN#238] but that she is like to die of\ndistress for herself and her lord Al-Rashid and saying, 'The\nwhole that they said and did was false.'\"[FN#239] Quoth\nKamariyah, \"O grandfather mine, nothing is left for it but\nstratagem and device for her deliverance, for that she is dearer\nto me than everything; and know that yonder accursed when he\nwaxeth ware of your coming upon him, will ken that he hath no\npower to cope with you, he who is the least and meanest of the\nJann; but we dread that he, when assured of defeat, will slay\nTohfah; wherefore nothing will serve but that we contrive a\nsleight for saving her; else will she perish.\" He asked, \"And\nwhat hast thou in mind of device?\" and she answered, \"Let us take\nhim with fair means, and if he obey, all will be well;[FN#240]\nelse will we practice stratagem against him; and expect not her\ndeliverance from other than myself.\" Quoth Iblis, \"The affair is\nthine; contrive what thou wilt, for that Tohfah is thy sister and\nthy solicitude for her is more effectual than that of any other.\"\nSo Kamariyah cried out to an Ifrit of the Ifrits and a calamity\nof the calamities,[FN#241] by name Al-Asad al-Tayy\u00e1r, the Flying\nLion, and said to him, \"Hie with my message to the Crescent\nMountain,[FN#242] the wone of Maymun the Sworder, and enter and\nsay to him, My lady saluteth thee with the salam and asketh thee,\n'How canst thou be assured for thyself of safety, after what thou\nhast done, O Maymun? Couldst thou find none to maltreat in thy\ndrunken humour save Tohfah, she too being a queen? But thou art\nexcused, because thou didst not this deed, but 'twas thy drink,\nand the Shaykh Abu al-Taw\u00e1if pardoneth thee, because thou wast\ndrunken. Indeed, thou hast attainted his honour; but now restore\nher to her palace, for that she hath done well and favoured us\nand rendered us service, and thou wottest that she is this day\nour queen. Belike she may bespeak Queen Al-Shahba, whereupon the\nmatter will become grievous and that wherein there is no good\nshall betide thee; and thou wilt get no tittle of gain. Verily, I\ngive thee good counsel, and so the Peace!'\" Al-Asad answered\n\"Hearing and obeying,\" and flew till he came to the Crescent\nMountain, when he sought audience of Maymun, who bade admit him.\nSo he entered and kissing ground before him, gave him Queen\nKamariyah's message, which when he heard, he cried to the Ifrit,\n\"Return whence thou comest and say to thy mistress, 'Be silent\nand thou wilt show thy good sense.' Else will I come and seize\nupon her and make her serve Tohfah; and if the kings of the Jinn\nassemble together against me and I be overcome by them, I will\nnot leave her to scent the wind of this world and she shall be\nneither mine nor theirs, for that she is presently my sprite\n[FN#243] from between my ribs; and how shall any part with his\nsprite?\" When the Ifrit heard Maymun's words, he said to him, \"By\nAllah, O Maymun, art thou a changeling in thy wits, that thou\nspeakest these words of my lady, and thou one of her page-boys?\"\nWhereupon Maymun cried out and said to him, \"Woe to thee, O dog\nof the Jinns! Wilt thou bespeak the like of me with these words?\"\nThen he bade those who were about him bastinado Al-Asad, but he\ntook flight and soaring high in air, betook himself to his\nmistress and told her the tidings: when she said, \"Thou hast done\nwell, O good knight!\" Then she turned to her sire and said to\nhim, \"Hear that which I shall say to thee.\" Quoth he, \"Say on;\"\nand quoth she, \"I rede thee take thy troops and go to him, for\nwhen he heareth this, he will in turn levy his many and come\nforth to thee; whereupon do thou offer him battle and prolong the\nfight with him and make a show to him of weakness and giving way.\nMeantime, I will devise me a device for getting at Tohfah and\ndelivering her, what while he is busied with you in battle; and\nwhen my messenger cometh to thee and informeth thee that I have\ngotten possession of Tohfah and that she is with me, return thou\nupon Maymun forthwith and overthrow him and his hosts, and take\nhim prisoner. But, an my device succeed not with him and we fail\nto deliver Tohfah, he will assuredly practice to slay her,\nwithout recourse, and regret for her will remain in our hearts.\"\nQuoth Iblis, \"This is the right rede\" and bade call a march among\nthe troops, whereupon an hundred thousand knights, doughty wights\nof war, joined themselves to him and set out for the country of\nMaymun. As for Queen Kamariyah, she flew off to the palace of her\nsister Wakhimah, and told her what deed Maymun had done and how\nhe declared that, whenas he saw defeat nearhand, he would slay\nTohfah; adding, \"And indeed, he is resolved upon this; otherwise\nhad he not dared to work such sleight. So do thou contrive the\naffair as thou see fit, for in rede thou hast no superior.\" Then\nthey sent for Queen Zalzalah and Queen Shararah and sat down to\ntake counsel, one with other, concerning what they had best do in\nthe matter. Presently said Wakhimah, \" 'Twere advisable we fit\nout a ship in this our island home and embark therein, disguised\nas Adam's sons, and fare on till we come to anchor under a little\nisland that lieth over against Maymun's palace. There will we sit\ndrinking and smiting the lute and singing; for Tohfah will\nassuredly be seated there overlooking the sea, and needs must she\nsee us and come down to us, whereupon we will take her by force\nand she will be under our hands, so that none shall be able to\nmolest her any more. Or, an Maymun be gone forth to do battle\nwith the Jinns, we will storm his stronghold and take Tohfah and\nraze his palace and slay all therein. When he hears of this, his\nheart will be broken and we will send to let our father know,\nwhereat he will return upon him with his troops and he will be\ndestroyed and we shall have rest of him.\" They answered her,\nsaying, \"This is a good counsel.\" Then they bade fit out a ship\nfrom behind the mountain,[FN#244] and it was fitted out in less\nthan the twinkling of an eye; so they launched it on the sea and\nembarking therein, together with four thousand Ifrits, set out,\nintending for Maymun's palace. They also bade other five thousand\nIfrits betake themselves to the island under the Crescent\nMountain and there lie in wait for them ambushed well. Thus fared\nit with the kings of the Jann; but as regards Shaykh Abu\nal-Taw\u00e1if Iblis and his son Al-Shisban the twain set out, as we\nhave said, with their troops, who were of the doughtiest of the\nJinn and the prowest of them in wing-flying and horsemanship,\nand fared on till they drew near the Crescent Mountain. When the\nnews of their approach reached Maymun, he cried out with a mighty\ngreat cry to the troops, who were twenty thousand riders, and\nbade them make ready for departure. Then he went in to Tohfah and\nkissing her, said, \"Know that thou art this day my life of the\nworld, and indeed the Jinns are gathered together to wage war on\nme for thy sake. An I win the day from them and am preserved\nalive, I will set all the kings of the Jann under thy feet and\nthou shalt become queen of the world.\" But she shook her head and\nshed tears; and he said, \"Weep not, for I swear by the virtue of\nthe mighty inscription borne on the seal-ring of Solomon, thou\nshalt never again see the land of men; no, never! Say me, can any\none part with his life? Give ear, then, to my words; else will I\nslay thee.\" So she was silent. And forthright he sent for his\ndaughter, whose name was Jamrah,[FN#245] and when she came, he\nsaid to her, \"Harkye, Jamrah! Know that I am going to fight the\nclans of Al-Shisban and Queen Kamariyah and the Kings of the\nJann. An I be vouchsafed the victory over them, to Allah be the\nlaud and thou shalt have of me largesse;[FN#246] but, an thou see\nor hear that I am worsted and any come to thee with ill news of\nme, hasten to kill Tohfah, so she may fall neither to me nor to\nthem.\" Then he farewelled her and mounted, saying, \"When this\ncometh about, pass over to the Crescent Mountain and take up\nthine abode there, and await what shall befal me and what I shall\nsay to thee.\" And Jamrah answered \"Hearkening and obedience.\" Now\nwhen the Songstress heard these words, she fell to weeping and\nwailing and said, \"By Allah, naught irketh me but severance from\nmy lord Al-Rashid; however, when I am dead, let the world be\nruined after me!\"[FN#247] And she was certified in herself that\nshe was assuredly lost. Then Maymun set forth with his army and\ndeparted in quest of the hosts of the Jinn, leaving none in the\npalace save his daughter Jamrah and Tohfah and an Ifrit which was\ndear to him. They fared on till they met with the army of\nAl-Shisban; and when the two hosts came face to face, they fell\neach upon other and fought a fight, a passing sore than which\nnaught could be more. After a while, Al-Shisban's troops began to\ngive way, and when Maymun saw them do thus, he despised them and\nmade sure of victory over them. On this wise it befel them; but\nas regards Queen Kamariyah and her company they sailed on without\nceasing, till they came under the palace wherein was Tohfah, to\nwit, that of Maymun the Sworder; and by the decree of the Lord of\ndestiny, the Songstress herself was at that very time sitting on\nthe belvedere of the palace, pondering the affair of Harun\nal-Rashid and her own and that which had befallen her and weeping\nfor that she was doomed to death. She saw the vessel and what was\ntherein of those we have named, and they in mortal guise, and\nsaid, \"Alas, my sorrow for this ship and for the men that be\ntherein!\" As for Kamariyah and her many, when they drew near the\npalace, they strained their eyes and seeing the Songstress\nsitting, cried, \"Yonder sitteth Tohfah. May Allah not bereave us\nof her!\" Then they moored their craft and, making for the island\nwhich lay over against the palace, spread carpets and sat eating\nand drinking; whereupon quoth Tohfah, \"Well come and welcome to\nyonder faces! These be my kinswomen and I conjure thee by Allah,\nO Jamrah, that thou let me down to them, so I may sit with them\nawhile and enjoy kindly converse with them and return.\" Quoth\nJamrah, \"I may on no wise do that;\" and Tohfah wept. Then the\nfolk brought out wine and drank, while Kamariyah took the lute\nand sang these couplets,\n\n\"By Allah, had I never hoped to greet you * Your guide had failed\n     on camel to seat you!\nFar bore you parting from friend would greet you * Till meseems\n     mine eyes for your wone entreat you.\"\n\nWhen Tohfah heard this, she cried out so great a cry, that the\nfolk heard her and Kamariyah said, \"Relief is nearhand.\" Then the\nSongstress looked out to them and called to them, saying, \"O\ndaughters of mine uncle, I am a lonely maid, an exile from kin\nand country: so for the love of Allah Almighty, repeat that\nsong!\" Accordingly Kamariyah repeated it and Tohfah swooned away.\nWhen she came to herself, she said to Jamrah, \"By the rights of\nthe Apostle of Allah (whom may He save and assain!) unless thou\nsuffer me go down to them and look on them and sit with them for\na full hour, I will hurl myself headlong from this palace, for\nthat I am aweary of my life and know that I am slain to all\ncertainty; wherefore will I kill myself, ere you pass sentence\nupon me.\" And she was instant with her in asking. When Jamrah\nheard her words, she knew that, an she let her not down, she\nwould assuredly destroy herself. So she said to her, \"O Tohfah,\nbetween thee and them are a thousand cubits, but I will bring the\nwomen up to thee.\" The Songstress replied, \"Nay, there is no help\nbut that I go down to them and solace me in the island and look\nupon the sea anear; then will we return, I and thou; for that, an\nthou bring them up to us, they will be affrighted and there will\nbetide them neither joy nor gladness. As for me, I wish but to be\nwith them, that they may cheer me with their company neither give\nover their merrymaking, so peradventure I may broaden my breast\nwith them, and indeed I swear that needs must I go down to them;\nelse I will cast myself upon them.\" And she cajoled Jamrah and\nkissed her hands, till she said, \"Arise and I will set thee down\nbeside them.\" Then she took Tohfah under her armpit and flying up\nswiftlier than the blinding leven, set her down with Kamariyah\nand her company; whereupon she went up to them and accosted them,\nsaying, \"Fear ye not: no harm shall befal you; for I am a mortal,\nlike unto you, and I would fain look on you and talk with you and\nhear your singing.\" So they welcomed her and kept their places\nwhilst Jamrah sat down beside them and fell a-snuffing their\nodours and saying, \"I smell the scent of the Jinn![FN#248] Would\nI wot whence it cometh!\" Then said Wakhimah to her sister\nKamariyah, \"Yonder foul slut smelleth us and presently she will\ntake to flight; so what be this inaction concerning her?\"[FN#249]\nThereupon Kamariyah put out an arm long as a camel's neck, and\ndealt Jamrah a buffet on the head, that made it fly from her body\nand cast it into the sea. Then cried she, \"Allah is\nAll-great!\"[FN#250] And they uncovered their faces, whereupon\nTohfah knew them and said to them, \"Protection!\" Queen Kamariyah\nembraced her, as also did Queen Zalzalah and Queen Wakhimah and\nQueen Shararah, and the first-named said to her, \"Receive the\ngood tidings of assured safety, for there abideth no harm for\nthee; but this is no time for talk.\" Then they cried out,\nwhereupon up came the Ifrits ambushed in that island, hending\nswords and maces in hand, and taking up Tohfah, flew with her to the\npalace and made themselves masters of it, whilst the Ifrit\naforesaid, who was dear to Maymun and whose name was\nDukh\u00e1n,[FN#251] fled like an arrow and stinted not flying till he\ncame to Maymun and found him fighting a sore fight with the Jinn.\nWhen his lord saw him, he cried out at him, saying, \"Fie upon\nthee! Whom hast thou left in the palace?\" Dukhan answered,\nsaying, \"And who abideth in the palace? Thy beloved Tohfah they\nhave captured and Jamrah is slain and they have taken the palace,\nall of it.\" At these ill tidings Maymun buffeted his face and\nhead and said, \"Oh! Out on it for a calamity!\" Then he cried\naloud. Now Kamariyah had sent to her sire and reported to him the\nnews, whereat the raven of the wold[FN#252] croaked for the foe.\nSo, when Maymun saw that which had betided him (and indeed the\nJinn smote upon him and the wings of eternal severance overspread\nhis host), he planted the heel of his lance in the earth and\nturning its head to his heart, urged his charger thereat and\npressed upon it with his breast, till the point came forth\ngleaming from his back. Meanwhile the messenger had made the\nfriendly host with the news of Tohfah's deliverance, whereat the\nShaykh Abu al-Taw\u00e1if rejoiced and bestowed on the bringer of lief\ntidings a sumptuous robe of honour and made him commander over a\ncompany of the Jann. Then they charged home upon Maymun's host\nand wiped them out to the last man; and when they came to Maymun,\nthey found that he had slain himself and was even as we have\nsaid. Presently Kamariyah and her sister Wakhimah came up to\ntheir grandfather and told him what they had done; whereupon he\ncame to Tohfah and saluted her with the salam and congratulated\nher on deliverance. Then he made over Maymun's palace to Salhab;\nand, taking all the rebel's wealth gave it to the Songstress,\nwhile the troops encamped upon the Crescent Mountain.\nFurthermore, the Shaykh Abu al-Taw\u00e1if said to Tohfah, \"Blame me\nnot,\" and she kissed his hands, when behold, there appeared to\nthem the tribes of the Jinn, as they were clouds, and Queen\nAl-Shahba flying in their van, drawn sword in grip. As she came\nin sight of the folk, they kissed ground between her hands and\nshe said to them, \"Tell me what hath betided Queen Tohfah from\nyonder dog Maymun and why did ye not send to me and report to\nme?\" Quoth they, \"And who was this dog that we should send to\nthee on his account? Indeed he was the least and lowest of the\nJinn.\" Then they told her what Kamariyah and her sisters had done\nand how they had practiced upon Maymun and delivered the\nSongstress from his hand, fearing lest he should slay her when he\nfound himself defeated; and she said, \"By Allah, the accursed was\nwont to lengthen his looking upon her!\" And Tohfah fell to\nkissing Al-Shahba's hand, whilst the queen strained her to her\nbosom and kissed her, saying, \"Trouble is past; so rejoice in\nassurance of deliverance.\" Then they rose and went up to the\npalace whereupon the trays of food were brought and they ate and\ndrank; after which quoth Queen Al-Shahba, \"O Tohfah, sing to us,\nby way of sweetmeat[FN#253] for thine escape, and favour us with\nthat which shall solace our minds, for that indeed my thoughts\nhave been occupied with thee.\" And quoth Tohfah, \"Hearkening and\nobedience, O my lady.\" So she improvised and sang these couplets,\n\n\"Breeze of East[FN#254] an thou breathe o'er the dear ones' land\n     * Speed, I pray thee, my special salute and salam:\nAnd say them I'm pledged to love them and * In pine that passeth\n     all pine I am.\"\n\nThereat Queen Al-Shahba rejoiced and with her all who were\npresent; and they admired her speech and fell to kissing her; and\nwhen she had made an end of her song, Queen Kamariyah said to\nher, \"O my sister, ere thou go to thy palace, I would fain bring\nthee to look upon Al-'Ank\u00e1,[FN#255] daughter of Bahram J\u00far, whom\nAl-'Anka, daughter of the wind, carried off, and her beauty; for\nthat there is not her fellow on earth's face.\" And Queen\nAl-Shahba said, \"O Kamariyah, I also think it were well an I\nbeheld her.\" Quoth Kamariyah, \"I saw her three years ago; but my\nsister Wakhimah seeth her at all times, for she is near to her\npeople, and she saith that there is not in the world fairer than\nshe. Indeed, this Queen Al-Anka is become a byword for beauty and\ncomeliness.\" And Wakhimah said, \"By the mighty inscription on the\nseal-ring of Solomon, there is not her like for loveliness here\nbelow.\" Then said Queen Al-Shahba, \"An it needs must be and the\naffair is as ye say, I will take Tohfah and go with her to\nAl-Anka, so she may look upon her!\" So they all arose and\nrepaired to Al-Anka, who abode in the Mountain Kaf. When she saw\nthem, she drew near to them and saluted them, saying, \"O my\nladies, may I not be bereaved of you!\" Quoth Wakhimah to her,\n\"Who is like unto thee, O Anka? Behold, Queen Al-Shahba is come\nto thee.\" So Al-Anka kissed the Queen's feet and lodged them in\nher palace; whereupon Tohfah came up to her and fell to kissing\nher and saying, \"Never saw I a seemlier than this semblance.\" Then\nshe set before them somewhat of food and they ate and washed\ntheir hands; after which the Songstress took the lute and smote\nit well; and Al-Anka also played, and they fell to improvising\nverses in turns, whilst Tohfah embraced Al-Anka every moment.\nAl-Shahba cried, \"O my sister, each kiss is worth a thousand\ndinars;\" and Tohfah replied, \"And a thousand dinars were little\ntherefor;\" whereat Al-Anka laughed and after nighting in her\npavilion on the morrow they took leave of her and went away to\nMaymun's palace. Here Queen Al-Shahba farewelled them and taking\nher troops, returned to her capital, whilst the kings also went\naway to their abodes and the Shaykh Abu al-Taw\u00e1if applied himself\nto diverting Tohfah till nightfall, when he mounted her on the\nback of one of the Ifrits and bade other thirty gather together\nall that she had gotten of treasure and raiment, jewels and robes\nof honour. Then they flew off, whilst Iblis went with her, and in\nless than the twinkling of an eye he set her down in her sleeping\nroom, where he and those who were with him bade adieu to her and\nwent away. When Tohfah found herself in her own chamber[FN#256]\nand on her couch, her reason fled for joy and it seemed to her as\nif she had never stirred thence: then she took the lute and tuned\nit and touched it in wondrous fashion and improvised verses and\nsang. The Eunuch heard the smiting of the lute within the chamber\nand cried, \"By Allah, that is the touch of my lady Tohfah!\" So he\narose and went, as he were a madman, falling down and rising up,\ntill he came to the Castrato on guard at the gate of the\nCommander of the Faithful and found him sitting. When his fellow\nneutral saw him, and he like a madman, slipping down and\nstumbling up, he asked him, \"What aileth thee and what bringeth\nthee hither at this hour?\" The other answered, \"Wilt thou not\nmake haste and awaken the Prince of True Believers?\" And he fell\nto crying out at him; whereupon the Caliph awoke and heard them\nbandying words together and Tohfah's slave crying to the other,\n\"Woe to thee! Awaken the Commander of the Faithful in haste.\" So\nquoth he, \"O Sawab, what hast thou to say?\" and quoth the Chief\nEunuch, \"O our lord, the Eunuch of Tohfah's lodging hath lost his\nwits and crieth, 'Awaken the Commander of the Faithful in haste!'\n\" Then said Al-Rashid to one of the slave-girls, \"See what may be\nthe matter.\" Accordingly she hastened to admit the Castrato, who\nentered at her order; and when he saw the Commander of the\nFaithful, he salamed not neither kissed ground, but cried in his\nhurry, \"Quick: up with thee! My lady Tohfah sitteth in her\nchamber, singing a goodly ditty. Come to her in haste and see all\nthat I say to thee! Hasten! She sitteth awaiting thee.\" The\nCaliph was amazed at his speech and asked him, \"What sayst thou?\"\nHe answered, \"Didst thou not hear the first of the speech? Tohfah\nsitteth in the sleeping-chamber, singing and lute-playing. Come\nthy quickest! Hasten!\" Accordingly Al-Rashid sprang up and donned\nhis dress; but he believed not the Eunuch's words and said to\nhim, \"Fie upon thee! What is this thou sayst? Hast thou not seen\nthis in a dream?\" Quoth the Eunuch, \"By Allah, I wot not what\nthou sayest, and I was not asleep;\" and quoth Al-Rashid, \"An thy\nspeech be soothfast, it shall be for thy good luck, for I will\nfree thee and give thee a thousand gold pieces; but, an it be\nuntrue and thou have seen this in dream-land, I will crucify\nthee.\" The Eunuch said within himself, \"O Protector, let me not\nhave seen this in vision!\" then he left the Caliph and running to\nthe chamber-door, heard the sound of singing and lute-playing;\nwhereupon he returned to Al-Rashid and said to him, \"Go and\nhearken and see who is asleep.\" When the Prince of True Believers\ndrew near the door of the sleeping-chamber, he heard the sound of\nthe lute and Tohfah's voice singing; whereat he could not\nrestrain his reason and was like to faint for excess of delight.\nThen he pulled out the key, but his hand refused to draw the bolt:\nhowever, after a while, he took heart and applying himself,\nopened the door and entered, saying, \"Methinks this is none other\nthan a vision or an imbroglio of dreams.\" When Tohfah saw him,\nshe rose and coming to meet him, pressed him to her breast; and\nhe cried out a cry wherein his sprite was like to depart and fell\ndown in a fit. She again strained him to her bosom and sprinkled\non him rose-water mingled with musk, and washed his face, till he\ncame to himself, as he were a drunken man, and shed tears for the\nstress of his joy in Tohfah's return to him, after he had\ndespaired of her returning. Then she took the lute and smote\nthereon, after the fashion she had learnt from Shaykh Iblis, so\nthat Al-Rashid's wit was bewildered for excess of joy and his\nunderstanding was confounded for exultation; after which she\nimprovised and sang these couplets,\n\n\"That I left thee my heart to believe is unlief; * For the life\n     that's in it ne'er leaveth; brief,\nAn thou say 'I went,' saith my heart 'What a fib!' * And I bide\n     'twixt believing and unbelief.\"\n\nWhen she had made an end of her verses, Al-Rashid said to her, \"O\nTohfah, thine absence was wondrous, yet is thy presence still\nmore marvellous.\" She replied, \"By Allah, O my lord, thou sayst\nsooth;\" then, taking his hand, she said to him, \"O Commander of\nthe Faithful, see what I have brought with me.\" So he looked and\nspied treasures such as neither words could describe nor\nregisters could document, pearls and jewels and jacinths and\nprecious stones and unions and gorgeous robes of honour, adorned\nwith margarites and jewels and purfled with red gold. There he\nbeheld what he never had beheld all his life long, not even in\nidea; and she showed him that which Queen Al-Shahba had bestowed\non her of those carpets, which she had brought with her, and that\nthrone, the like whereof neither Kisr\u00e0 possessed nor C\u00e6sar, and\nthose tables inlaid with pearls and jewels and those vessels\nwhich amazed all who looked on them, and that crown which was on\nthe head of the circumcised boy, and those robes of honour, which\nQueen Al-Shahba and Shaykh Abu al-Taw\u00e1if had doffed and donned\nupon her, and the trays wherein were those treasures; brief, she\nshowed him wealth whose like he had never in his life espied and\nwhich the tongue availeth not to describe and whereat all who\nlooked thereon were bewildered, Al-Rashid was like to lose his\nwits for amazement at this spectacle and was confounded at that\nhe sighted and witnessed. Then said he to Tohfah, \"Come, tell me\nthy tale from beginning to end, and let me know all that hath\nbetided thee, as if I had been present.\" She answered,\n\"Hearkening and obedience,\" and acquainting him with all that had\nbetided her first and last, from the time when she first saw the\nShaykh Abu al-Taw\u00e1if, how he took her and descended with her\nthrough the side of the Chapel of Ease; and she told him of the\nhorse she had ridden, till she came to the meadow aforesaid and\ndescribed it to him, together with the palace and that was\ntherein of furniture, and related to him how the Jinn rejoiced in\nher, and whatso she had seen of their kings, masculine and\nfeminine, and of Queen Kamariyah and her sisters and Queen\nShu'a'ah, Regent of the Fourth Sea, and Queen Al-Shahba, Queen of\nQueens, and King Al-Shisban, and that which each one of them had\nbestowed upon her. Moreover, she recited to him the story of\nMaymun the Sworder and described to him his fulsome favour, which\nhe had not deigned to change, and related to him that which befel\nher from the kings of the Jinn, male and female, and the coming\nof the Queen of Queens, Al-Shahba, and how she had loved her and\nappointed her her vice-reine and how she was thus become ruler\nover all the kings of the Jann; and she showed him the writ of\ninvestiture which Queen Al-Shahba had written her and told him\nwhat had betided her with the Ghulish Head, when it appeared to\nher in the garden, and how she had despatched it to her palace,\nbeseeching it to bring her news of the Commander of the Faithful\nand of what had betided him after her. Then she described to him\nthe flower-gardens, wherein she had taken her pleasure, and the\nHammam-baths inlaid with pearls and jewels and told him that\nwhich had befallen Maymun the Sworder, when he bore her off, and\nhow he had slain himself; in fine, she related to him everything\nshe had seen of wonders and marvels and that which she had beheld\nof all kinds and colours among the Jinn. Then she told him the\nstory of Al-Anka, daughter of Bahram Jur, with Al-Anka, daughter\nof the wind, and described to him her dwelling-place and her\nisland, whereupon quoth Al-Rashid, \"O Tohfat al-Sadr,[FN#257]\ntell me of Al-Anka, daughter of Bahram Jur; is she of the\nJinn-kind or of mankind or of the bird-kind? For this long time\nhave I desired to find one who should tell me of her.\" Tohfah\nreplied, \"'Tis well, O Commander of the Faithful. I asked the\nqueen of this and she acquainted me with her case and told me who\nbuilt her the palace.\" Quoth Al-Rashid, \"Allah upon thee, tell it\nme;\" and quoth Tohfah, \"I will well,\" and proceeded to tell him.\nAnd he was amazed at that which he heard from her and what she\nreported to him and at that which she had brought back of jewels\nand jacinths of various hues and precious stones of many sorts,\nsuch as amazed the beholder and confounded thought and mind. As\nfor this, Tohfah was the means of the enrichment of the\nBarmecides and the Abbasides, and they had endurance in their\ndelight. Then the Caliph went forth and bade decorate the city:\nso they decorated it and the drums of glad tidings were beaten;\nand they made banquets to the people for whom the tables were\nspread seven days. And Tohfah and the Commander of the Faithful\nceased not to enjoy the most delightsome of life and the most\nprosperous till there came to them the Destroyer of delights and\nthe Severer of societies; and this is all that hath come down to\nus of their story.\n\n\n\n\n                     WOMEN'S WILES[FN#258]\n\n\n\nOn the following night Dunyazad said to her sister Shahrazad, \"O\nsister mine, an thou incline not unto sleep, prithee tell us a\ntale which shall beguile our watching through the dark hours.\"\nShe replied:--With love and gladness.[FN#259] It hath reached me,\nO magnificent King, that whilome there was in the city of\nBaghdad, a comely youth and a well-bred, fair of favour, tall of\nstature, and slender of shape. His name was Al\u00e1 al-D\u00edn and he was\nof the chiefs of the sons of the merchants and had a shop wherein\nhe sold and bought. One day, as he sat in his shop, there passed\nby him a merry girl[FN#260] who raised her head and casting a\nglance at the young merchant, saw written in a flowing hand on\nthe forehead[FN#261] of his shop door these words, \"THERE BE NO\nCRAFT SAVE MEN'S CRAFT, FORASMUCH AS IT OVERCOMETH WOMEN'S\nCRAFT.\" When she beheld this, she was wroth and took counsel with\nherself, saying, As my head liveth, there is no help but I show\nhim a marvel-trick of the wiles of women and put to naught this\nhis inscription!\" Thereupon she hied her home; and on the morrow\nshe made her ready and donning the finest of dress, adorned\nherself with the costliest of ornaments and the highest of price\nand stained her hands with Henna. Then she let down her tresses\nupon her shoulders and went forth, walking with coquettish gait\nand amorous grace, followed by her slave-girl carrying a parcel,\ntill she came to the young merchant's shop and sitting down under\npretext of seeking stuffs, saluted him with the salam and\ndemanded of him somewhat of cloths. So he brought out to her\nvarious kinds and she took them and turned them over, talking\nwith him the while. Then said she to him, \"Look at the\nshapeliness of my shape and my semblance! Seest thou in me aught\nof default?\" He replied, \"No, O my lady;\" and she continued, \"Is\nit lawful in any one that he should slander me and say that I am\nhumpbacked?\" Then she discovered to him a part of her bosom, and\nwhen he saw her breasts his reason took flight from his head and\nhis heart clave to her and he cried, \"Cover it up,[FN#262] so may\nAllah veil thee!\" Quoth she, \"Is it fair of any one to decry my\ncharms?\" and quoth he, \"How shall any decry thy charms, and thou\nthe sun of loveliness?\" Then said she, \"Hath any the right to say\nof me that I am lophanded?\" and tucking up her sleeves, she\nshowed him forearms as they were crystal; after which she\nunveiled to him a face, as it were a full moon breaking forth on\nits fourteenth night, and said to him, \"Is it lawful for any to\ndecry me and declare that my face is pitted with smallpox or that\nI am one-eyed or crop-eared?\" and said he, \"O my lady, what is it\nmoveth thee to discover unto me that lovely face and those fair\nlimbs, wont to be so jealously veiled and guarded? Tell me the\ntruth of the matter, may I be thy ransom!\" And he began to\nimprovise,[FN#263]\n\n\"White Fair now drawn from sheath of parted hair, * Then in the\n     blackest tresses hid from sight,\nFlasheth like day irradiating Earth * While round her glooms the\n     murk of nightliest night.\"\n\n--And Shahrazad perceived the dawn of day and ceased to say her\npermitted say. Whereupon cried Dunyazad her sister, \"O sister\nmine, how delectable is this tale and how desirable!\" She\nreplied, saying, \"And where is this compared with that which I\nwill recount to thee next night, Inshallah?\"\n\n             The Hundred and Ninety-seventh Night.\n\nNow when came the night, quoth Dunyazad to her sister Shahrazad,\n\"O sister mine, an thou incline not unto sleep, prithee finish\nthy tale which shall beguile our watching through the dark\nhours.\" She replied:--With love and gladness! It hath reached me,\nO auspicious King, that the girl said to the young merchant,\n\"Know, O my lord, that I am a maid oppressed of my sire, who\nspeaketh at me and saith to me, Thou art loathly of looks and\nsemblance and it besitteth not that thou wear rich raiment; for\nthou and the slave-girls are like in rank, there is no\ndistinguishing thee from them. Now he is a richard, having a\nmighty great store of money and saith not thus save because he is\na pinchpenny, and grudgeth the spending of a farthing; wherefore\nhe is loath to marry me, lest he be put to somewhat of expense in\nmy marriage, albeit Almighty Allah hath been bounteous to him and\nhe is a man puissant in his time and lacking naught of worldly\nweal.\" The youth asked, \"Who is thy father and what is his\ncondition?\" and she answered, \"He is the Chief Kazi of the well-\nknown Supreme Court, under whose hands are all the Kazis who\nadminister justice in this city.\" The merchant believed her and\nshe farewelled him and fared away, leaving in his heart a\nthousand regrets, for that the love of her had prevailed over him\nand he knew not how he should win to her; wherefore he woned\nenamoured, love-distracted, unknowing if he were alive or dead.\nAs soon as she was gone, he shut up shop and walked straightway\nto the Court, where he went in to the Chief Kazi and saluted him.\nThe magistrate returned his salam and treated him with\ndistinction and seated him by his side. Then said Ala al-Din to\nhim, \"I come to thee seeking thine alliance and desiring the hand\nof thy noble daughter.\" Quoth the Kazi, \"O my lord merchant,\nwelcome to thee and fair welcome; but indeed my daughter\nbefitteth not the like of thee, neither beseemeth she the\ngoodliness of thy youth and the pleasantness of thy compostition\nand the sweetness of thy speech;\" but Ala al-Din replied, \"This\ntalk becometh thee not, neither is it seemly in thee; if I be\ncontent with her, how should this vex thee?\" So the Kazi was\nsatisfied and they came to an accord and concluded the marriage\ncontract at a dower precedent of five purses[FN#264] ready money\nand a dower contingent of fifteen purses, so it might be hard for\nhim to put her away, her father having given him fair warning,\nbut he would not be warned. Then they wrote out the contract\ndocument and the merchant said, \"I desire to go in to her this\nnight.\" Accordingly they carried her to him in procession that\nvery evening, and he prayed the night prayer and entered the\nprivate chamber prepared for him; but, when he lifted the head\ngear from the bride's head and the veil from her face and looked,\nhe saw a foul face and a favour right fulsome; indeed he beheld\nsomewhat whereof may Allah never show thee the like! loathly,\ndispensing from description, inasmuch as there were reckoned in\nher all legal defects.[FN#265] So he repented, when repentance\navailed him naught, and knew that the girl had cheated him.--And\nShahrazad perceived the dawn of day and ceased to say her\npermitted say. Whereupon cried Dunyazad, her sister, \"O sister\nmine, how delectable is thy story and how sweet!\" She replied,\nsaying, \"And where is this compared with that which I will\nrecount to thee next night an I be spared and suffered to live by\nthe King, whom Almighty Allah preserve?\"\n\n              The Hundred and Ninety-eight Night.\n\nNow whenas came the night, quoth Dunyazad to her sister\nShahrazad, \"O sister mine, an thou incline not unto sleep,\nprithee finish thy story which shall beguile our watching through\nthe dark hours, for indeed 'tis a fine tale and a wondrous.\" She\nreplied:--With love and gladness! It hath reached me, O generous\nKing, that the unhappy merchant carnally knew the loathly bride,\nsore against the grain, and abode that night troubled in mind, as\nhe were in the prison of Al-Daylam.[FN#266] Hardly had the day\ndawned when he arose from her side and betaking himself to one of\nthe Hammams, dozed there awhile, after which he made the\nGhusl-ablution of ceremonial impurity[FN#267] and donned his\nevery day dress. Then he went out to the coffee house and drank a\ncup of coffee; after which he returned to his shop and opening\nthe door, sat down, with concern and chagrin manifest on his\ncountenance. After an hour or so, his friends and intimates among\nthe merchants and people of the market began to come up to him,\nby ones and twos; to give him joy, and said to him, laughing, \"A\nblessing! a blessing! Where be the sweetmeats? Where be the\ncoffee?[FN#268] 'Twould seem thou hast forgotten us; and nothing\nmade thee oblivious save that the charms of the bride have\ndisordered thy wit and taken thy reason, Allah help thee! We give\nthee joy, we give thee joy.\" And they mocked at him whilst he\nkept silence before them, being like to rend his raiment and shed\ntears for rage. Then they went away from him, and when it was the\nhour of noon, up came his mistress, the crafty girl, trailing her\nskirts and swaying to and fro in her gait, as she were a branch\nof Ban in a garden of bloom. She was yet more richly dressed and\nadorned and more striking and cutting[FN#269] in her symmetry and\ngrace than on the previous day, so that she made the passers stop\nand stand in espalier to gaze upon her. When she came to Ala\nal-Din's shop, she sat down thereon and said to him, \"Blessed be\nthe day to thee, O my lord Ala al-Din! Allah prosper thee and be\ngood to thee and perfect thy gladness and make it a wedding of\nweal and welfare!\" He knitted his brows and frowned in answer to\nher; then asked her, \"Wherein have I failed of thy due, or what\nhave I done to harm thee, that thou shouldst requite me after\nthis fashion?\" She answered, \"Thou hast been no wise in default;\nbut 'tis yonder inscription written on the door of thy shop that\nirketh me and vexeth my heart. An thou have the courage to change\nit and write up the contrary thereof, I will deliver thee from\nthine evil plight.\" And he answered, \"Thy requirement is right\neasy: on my head and eyes!\" So saying, he brought out a\nsequin[FN#270] and summoning one of his Mamelukes, said to him,\n\"Get thee to Such-an-one the Scribe and bid him write us an\nepigraph, adorned with gold and lapis lazuli, in these words,\n\"THERE BE NO CRAFT SAVE WOMEN'S CRAFT, FOR INDEED THEIR CRAFT IS\nA MIGHTY CRAFT[FN#271] AND OVERCOMETH AND HUMBLETH THE FALSES OF\nMEN.\" And she said to the white slave \"Fare thee forthright.\" So\nhe repaired to the Scribe, who wrote him the scroll, and he\nbrought it to his master, who set it on the door and asked the\ndamsel, \"Is thy heart satisfied?\" She answered, \"Yes! Arise\nforthwith and get thee to the place before the citadel, where do\nthou foregather with all the mountebanks and ape-dancers and\nbear-leaders and drummers and pipers and bid them come to thee\nto-morrow early, with their kettle-drums and flageolets, whilst\nthou art drinking coffee with thy father-in-law the Kazi, and\ncongratulate thee and wish thee joy, saying, 'A blessing, O son\nof our uncle! Indeed, thou art the vein[FN#272] of our eye! We\nrejoice for thee, and if thou be ashamed of us, verily we pride\nourselves upon thee; so, although thou banish us from thee, know\nthat we will not forsake thee, albeit thou forsake us.' And do\nthou fall to throwing dinars and dirhams amongst them; whereupon\nthe Kazi will question thee, and do thou answer him, saying, My\nfather was an ape-dancer and this is our original condition; but\nour Lord opened on us the gate of fortune and we have gotten us a\nname amongst the merchants and with their provost.' Upon this he\nwill say to thee, 'Then thou art an ape-leader of the tribe of\nthe mountebanks?' and do thou rejoin, 'I may in nowise deny my\norigin, for the sake of thy daughter and in her honour.' The Kazi\nwill say, 'It may not be that thou shalt be given the daughter of\na Shaykh who sitteth upon the carpet of the Law and whose descent\nis traceable by genealogy to the loins of the Apostle of\nAllah,[FN#273] nor is it meet that his daughter be in the power\nof a man who is an ape-dancer, a minstrel.' Then do thou reply,\n'Nay, O Efendi, she is my lawful wife, and every hair of her is\nworth a thousand lives, and I will not put her away though I be\ngiven the kingship of the world.' At last be thou persuaded to\nspeak the word of divorce and so shall the marriage be voided and\nye be saved each from other.\" Quoth Ala al-Din, \"Right is thy\nrede,\" and locking up his shop, betook himself to the place --And\nShahrazad perceived the dawn of day and ceased saying her\npermitted say. Whereupon cried Dunyazad, her sister, \"O sister\nmine, how goodly is thy story and how sweet!\" She replied,\nsaying, \"And where is this compared with that which I will\nrecount to thee next night, Inshallah!\"\n\n              The Hundred and Ninety-ninth Night.\n\nAnd whenas came the night, quoth Dunyazad to her sister, \"O\nsister mine, an thou incline not unto sleep, pray finish thy tale\nwhich shall beguile our watching through the dark hours.\" She\nreplied:--With love and gladness! It hath reached me, O generous\nKing, that the young merchant betook himself to the place before\nthe citadel, where he foregathered with the dancers, the drummers\nand pipers and instructed them how they should do, promising them\na mighty fine reward. They received his word with \"Hearing and\nobeying;\" and he betook himself on the morrow, after the morning\nprayer, to the presence of the Judge, who received him with\nhumble courtesy and seated him by his side. Then he addressed him\nand began questioning him of matters of selling and buying and of\nthe price current of the various commodities which were carried\nto Baghdad from all quarters, whilst his son-in-law replied to\nall whereof he was questioned. As they were thus conversing,\nbehold, up came the dancers and drummers with their drums and\npipers with their pipes, whilst one of their number preceded\nthem, with a long pennon-like banner in his hand, and played all\nmanner antics with voice and limbs. When they came to the Court-house, the Kazi cried, \"I seek refuge with Allah from yonder\nSatans!\" and the young merchant laughed but said naught. Then\nthey entered and saluting his worship the Kazi, kissed Ala\nal-Din's hands and said, \"A blessing on thee, O son of our uncle!\nIndeed, thou coolest our eyes in whatso thou doest, and we\nbeseech Allah for the enduring greatness of our lord the Kazi,\nwho hath honoured us by admitting thee to his connection and hath\nallotted to us a portion in his high rank and degree.\" When the\nJudge heard this talk, it bewildered his wit and he was dazed and\nhis face flushed with rage, and quoth he to his son-in-law, \"What\nwords are these?\" Quoth the merchant, \"Knowest thou not, O my\nlord, that I am of this tribe? Indeed this man is the son of my\nmaternal uncle and that other the son of my paternal uncle, and\nif I be reckoned of the merchants, 'tis but by courtesy!\" When\nthe Kazi heard these words his colour changed--And Shahrazad\nperceived the dawn of day, whereupon cried Dunyazad her sister,\n\"O sister mine, how delectable is thy story and how desirable!\"\nShe replied, saying, \"And where is its first compared with its\nlast? But I will forthwith relate it to you an I be spared and\nsuffered to live by the King, whom may Allah the Most High keep!\"\nQuoth the King within himself, \"By the Almighty, I will not slay\nher until I hear the end of her tale!\"\n\n                    The Two Hundredth Night.\n\nNow whenas came the night, quoth Dunyazad to her sister, \"O\nsister mine, an thou incline not unto sleep, prithee finish thy\ntale which shall beguile our watching through the dark hours.\"\nShe replied:--With love and gladness! It hath reached me, O\nauspicious king, that the Kazi's colour changed and he was\ntroubled and waxed wroth with exceeding wrath and was like to\nburst for stress of rage. Then said he to the young merchant,\n\"Allah forfend that this should last! How shall it be permitted\nthat the daughter of the Kazi of the Moslems cohabit with a man\nof the dancers and vile of origin? By Allah, unless thou\nrepudiate her forthright, I will bid beat thee and cast thee into\nprison and there confine thee till thou die. Had I foreknown that\nthou wast of them, I had not suffered thee near me, but had spat\nin thy face, for that thou art more ill-omened than a dog or a\nhog.\"[FN#274] Then he kicked him down from his place and\ncommanded him to divorce; but he said, \"Be ruthful to me, O\nEfendi, for that Allah is ruthful, and hasten not: I will not\ndivorce my wife, though thou give me the kingdom of Al-Irak.\" The\njudge was perplexed and knew that compulsion was not permitted of\nHoly Law;[FN#275] so he bespake the young merchant fair and said\nto him, \"Veil me,[FN#276] so may Allah veil thee. An thou divorce\nher not, this dishonour shall cleave to me till the end of time.\"\nThen his fury gat the better of his wit and he cried, \"An thou\ndivorce her not of thine own will, I will forthright bid strike\noff thy head and slay myself; Hell-flame but not shame.\"[FN#277]\nThe merchant bethought himself awhile, then divorced her with a\nmanifest divorce and a public[FN#278] and on this wise he won\nfree from that unwelcome worry. Then he returned to his shop and\npresently sought in marriage of her father her who had done with\nhim what she did[FN#279] and who was the daughter of the Shaykh\nof the guild of the blacksmiths. So he took her to wife and they\nabode each with other and lived the pleasantest of lives and the\nmost delightsome, till the day of death: and praise be to Allah\nthe Lord of the Three Worlds.\n\n\n\n\n           NUR AL-DIN ALI OF DAMASCUS AND THE DAMSEL\n                     SITT AL-MILAH.[FN#280]\n\n\n\nThere was once, in days of yore and in ages and times long gone\nbefore, a merchant of the merchants of Damascus, by name Abu\nal-Hasan, who had money and means, slave-blacks and slave-girls,\nlands and gardens, houses and Hammams in that city; but he was\nnot blessed with boon of child and indeed his age waxed great. So\nhe addressed himself to supplicate[FN#281] Allah Almighty in\nprivate and in public and in his bows and his prostrations and at\nthe season of prayer-call, beseeching Him to vouchsafe him,\nbefore his decease, a son who should inherit his wealth and\npossessions. The Lord answered his prayer; his wife conceived and\nthe days of her pregnancy were accomplished and her months and\nher nights; and the travail-pangs came upon her and she gave\nbirth to a boy, as he were a slice of Luna. He had not his match\nfor beauty and he put to shame the sun and the resplendent moon;\nfor he had a beaming face and black eyes of B\u00e1bil\u00ed\nwitchery[FN#282] and aquiline nose and carnelian lips; in fine,\nhe was perfect of attributes, the loveliest of folk of his time,\nsans dubitation or gainsaying. His father joyed in him with\nexceeding joy and his heart was solaced and he was at last happy:\nhe made banquets to the folk and he clad the poor and the widows.\nPresently he named the boy S\u00edd\u00ed Nur al-Din Ali and reared him in\nfondness and delight among the hand-maids and thralls. When he\nhad passed his seventh year, his father put him to school, where\nhe learned the sublime Koran and the arts of writing and\nreckoning; and when he reached his tenth year, he was taught\nhorsemanship and archery and to occupy himself with arts and\nsciences of all kinds, part and parts.[FN#283] He grew up\npleasant and polite, winsome and lovesome; a ravishment to all\nwho saw him, and he inclined to companying with brethren and\ncomrades and mixing with merchants and travelled men. From these\nhe heard tell of that which they had witnessed of the wonders of\nthe cities in their wayfare and heard them say, \"Whoso journeyeth\nnot enjoyeth naught;[FN#284] especially of the city of Baghdad.\"\nSo he was concerned with exceeding concern for his lack of travel\nand disclosed this to his sire, who said to him, \"O my son, why\ndo I see thee chagrined?\" Quoth he, \"I would fain travel;\" and\nquoth Abu al-Hasan, \"O my son, none travelleth save those whose\nneed is urgent and those who are compelled thereto by want. As\nfor thee, O my son, thou enjoyest ample means; so do thou content\nthyself with that which Allah hath given thee and be bounteous to\nothers, even as He hath been bountiful to thee; and afflict not\nthyself with the toil and tribulation of travel, for indeed it is\nsaid that travel is a piece of Hell-torment.\"[FN#285] But the\nyouth said, \"Needs must I journey to Baghdad, the House of\nPeace.\" When his father saw the strength of his resolve to travel\nhe fell in with his wishes and fitted him out with five thousand\ndinars in cash and the like in merchandise and sent with him two\nserving-men. So the youth fared forth, on the blessing of Allah\nAlmighty;[FN#286] and his parent went out with him, to take leave\nof him, and returned to Damascus. As for Nur al-Din Ali, he\nceased not travelling days and nights till he entered Baghdad\ncity, and laying up his loads in the Wak\u00e1lah,[FN#287] made for\nthe Hammam-bath, where he did away that which was upon him of the\nsoil of the road and doffing his travelling clothes, donned a\ncostly suit of Yaman\u00ed stuff, worth an hundred dinars. Then he\nloaded his sleeve with a thousand miskals of gold and sallied\nforth a-walking and swaying gracefully as he paced along. His\ngait confounded all those who gazed upon him, as he shamed the\nbranches with his shape and belittled the rose with the redness\nof his cheeks and his black eyes of Babil\u00ed witchcraft: thou\nwouldst deem that whoso looked on him would surely be preserved\nfrom bane and bale;[FN#288] for he was even as saith of him one\nof his describers in these couplets:--\n\n\"Thy haters and enviers say for jeer * A true say that profits\n     what ears will hear;\n'No boast is his whom the gear adorns; * The boast be his who\n     adorns the gear!'\"\n\nSo Sidi Nur al-Din went walking in the highways of the city and\nviewing its edifices and its bazars and thoroughfares and gazing\non its folk. Presently, Ab\u00fa Now\u00e1s met him. (Now he was of those\nof whom it is said, \"They love fair lads,\" and indeed there is\nsaid what is said concerning him.)[FN#289] When he saw Nur al-Din\nAli, he stared at him in amazement and exclaimed, \"Say, I take\nrefuge with the Lord of the Daybreak!\" Then he accosted the youth\nand saluting him, asked him, \"Why do I see my lord lone and lorn?\nMeseemeth thou art a stranger and knowest not this country; so,\nwith leave of my lord, I will put myself at his service and\nacquaint him with the streets, for that I know this city.\" Nur\nal-Din answered, \"This will be of thy favour, O nuncle.\" Abu\nNowas rejoiced at this and fared on with him, showing him the\nstreets and bazars, till they came to the house of a\nslave-dealer, where he stopped and said to the youth, \"From what\ncity art thou?\" \"From Damascus,\" replied Nur al-Din; and Abu\nNowas said, \"By Allah, thou art from a blessed city, even as\nsaith of it the poet in these couplets,\n\n'Now is Damascus a garth adorned * For her seekers, the Houris\n     and Paradise-boys.'\"\n\nSidi Nur al-Din thanked him and the twain entered the mansion of\nthe slave-merchant. When the people of the house saw Abu Nowas,\nthey rose to do him reverence, for that which they knew of his\nrank with the Commander of the Faithful; and the slave-dealer\nhimself came up to them with two chairs whereon they seated\nthemselves. Then the slave-merchant went inside and returning\nwith a slave-girl, as she were a branch of Ban or a rattan-cane,\nclad in a vest of damask silk and tired with a black and white\nheaddress whose ends fell down over her face, seated her on a\nchair of ebony; after which he cried to those who were present,\n\"I will discover to you a favour as it were a full moon breaking\nforth from under a cloud-bank.\" They replied, \"Do so;\" whereupon\nhe unveiled the damsel's face and behold, she was like the\nshining sun, with shapely shape and dawn-bright cheeks and\nthready waist and heavy hips; brief, she was endowed with an\nelegance, whose description is unfound, and was even as saith of\nher the poet,[FN#290]\n\n\"A fair one, to idolaters if she herself should show, They'd\n     leave their idols and her face for only Lord would know;\nAnd if into the briny sea one day she chanced to spit, Assuredly\n     the salt sea's floods straight fresh and sweet would grow.\"\n\nThe dealer stood at the hand-maid's head and one of the merchants\nsaid, \"I bid a thousand dinars for her.\" Quoth another, \"I bid\none thousand one hundred dinars;\" and a third, \"I bid twelve\nhundred.\" Then said a fourth merchant, \"Be she mine for fourteen\nhundred ducats.\" And the biddings standing still at that sum, her\nowner said, \"I will not sell her save with her consent: an if she\ndesire to be sold, I will sell her to whom she willeth.\" The\nslave-dealer asked him, \"What is her name?\" Answered the other,\n\"Her name is Sitt al-Mil\u00e1h;\"[FN#291] whereupon the dealer said to\nher, \"With thy leave, I will sell thee to yonder merchant for\nthis price of fourteen hundred dinars.\" Quoth she, \"Come hither\nto me.\" So the man-vendor came up to her and when he drew near, she\ngave him a kick with her foot and cast him to the ground, saying,\n\"I will not have that oldster.\" The slave-dealer arose, shaking\nthe dust from his dress and head, and cried, \"Who biddeth more of\nus? Who is desirous?\"[FN#292] Said one of the merchants, \"I,\" and\nthe dealer said to her, \"O Sitt al-Milah, shall I sell thee to\nthis merchant?\" She replied, \"Come hither to me;\" but he\nrejoined, \"Nay; speak and I will hear thee from my place, for I\nwill not trust myself to thee nor hold myself safe when near\nthee.\" So she cried, \"Indeed I will not have him.\" Then the\nslave-dealer looked at her and seeing her fix eyes on the young\nDamascene, for that in very deed he had fascinated her with his\nbeauty and loveliness, went up to him and said to him, \"O my\nlord, art thou a looker-on or a buyer? Tell me.\" Quoth Nur\nal-Din, \"I am both looker-on and buyer. Wilt thou sell me yonder\nslave-girl for sixteen hundred ducats?\" And he pulled out the\npurse of gold. Hereupon the dealer returned, dancing and clapping\nhis hands and saying, \"So be it, so be it, or not at all!\" Then\nhe came to the damsel and said to her, \"O Sitt al-Milah, shall I\nsell thee to yonder young Damascene for sixteen hundred dinars?\"\nBut she answered, \"No,\" of bashfulness before her master and the\nbystanders; whereupon the people of the bazar and the\nslave-merchant departed, and Abu Nowas and Ali Nur al-Din arose\nand went each his own way, whilst the damsel returned to her\nowner's house, full of love for the young Damascene. When the\nnight darkened on her, she called him to mind and her heart hung\nto him and sleep visited her not; and on this wise she abode days\nand nights, till she sickened and abstained from food. So her\nlord went in to her and asked her, \"O Sitt al-Milah, how findest\nthou thyself?\" Answered she, \"O my lord, dead without chance of\ndeliverance and I beseech thee to bring me my shroud, so I may\nlook upon it ere I die.\" Therewith he went out from her, sore\nconcerned for her, and betaking himself to the bazar, found a\nfriend of his, a draper, who had been present on the day when the\ndamsel was cried for sale. Quoth his friend to him, \"Why do I see\nthee troubled?\" and quoth he, \"Sitt al-Milah is at the point of\ndeath and for three days she hath neither eaten nor drunken. I\nquestioned her to-day of her case and she said, 'O my lord, buy\nme a shroud so I may look upon it ere I die.'\" The draper\nreplied, \"Methinks naught aileth her but that she is in love with\nthe young Damascene, and I counsel thee to mention his name to\nher and declare to her that he hath foregathered with thee on her\naccount and is desirous of coming to thy quarters, so he may hear\nsomewhat of her singing. An she say, 'I reck not of him, for\nthere is that to do with me which distracteth me from the\nDamascene and from other than he,' know that she saith sooth\nconcerning her sickness; but, an she say thee other than this,\nacquaint me therewith.\" So the man returned to his lodging and\ngoing in to his slave-girl said to her, \"O Sitt al-Milah, I went\nout for thy need and there met me the young man of Damascus, and\nhe saluted me with the salam and saluteth thee; he seeketh to win\nthy favour and prayed me to admit him as a guest in our dwelling,\nso thou mayst let him hear somewhat of thy singing.\" When she\nheard speak of the young Damascene, she gave a sob, that her soul\nwas like to leave her body, and answered, \"He knoweth my plight\nand how these three days past I have not eaten nor drunken, and I\nbeseech thee, O my lord, by Allah of All-Might, to do thy duty by\nthe stranger and bring him to my lodging and make excuse to him\nfor me.\" When her master heard this, his reason fled for joy, and\nhe went to his familiar the draper and said to him, \"Thou wast\nright in the matter of the damsel, for that she is in love with\nthe young Damascene; so how shall I manage?\" Said the other, \"Go\nto the bazar and when thou seest him, salute him, and say to him,\n'Thy departure the other day, without winning thy wish, was\ngrievous to me; so, an thou be still minded to buy the maid, I\nwill abate thee of that which thou badest for her an hundred\nsequins by way of gaining thy favour; seeing thou be a stranger\nin our land.' If he say to thee, 'I have no desire for her,' and\nhold off from thee, be assured that he will not buy; in which\ncase, let me know, so I may devise thee another device; and if he\nsay to thee other than this, conceal not from me aught.\" So the\ngirl's owner betook himself to the bazar, where he found the\nyouth seated at the upper end of the place where the merchants\nmostly do meet, selling and buying and taking and giving, as he\nwere the moon on the night of its full, and saluted him. The\nyoung man returned his salam and he said to him, \"O my lord, be\nnot offended at the damsel's speech the other day, for her price\nshall be lowered to the intent that I may secure thy favour. An\nthou desire her for naught, I will send her to thee or an thou\nwouldst have me abate to thee her price, I will well, for I\ndesire nothing save what shall content thee; seeing thou art a\nstranger in our land and it behoveth us to treat thee hospitably\nand have consideration for thee.\" The youth replied, \"By Allah, I\nwill not take her from thee but at an advance on that which I\nbade thee for her afore; so wilt thou now sell her to me for one\nthousand and seven hundred dinars?\" And the other rejoined, \"O my\nlord, I sell her to thee, may Allah bless thee in her!\" Thereupon\nthe young man went to his quarters and fetching a purse, sent for\nthe girl's owner and weighed out to him the price aforesaid,\nwhilst the draper was between the twain. Then said he, \"Bring her\nforth;\" but the other replied, \"She cannot come forth at this\npresent; but be thou my guest the rest of this day and night, and\non the morrow thou shalt take thy slave-girl and go in the ward\nof Allah.\" The youth agreed with him on this and he carried him\nto his house, where, after a little, he bade meat and wine be\nbrought, and they ate and drank. Then said Nur al-Din to the\ngirl's owner, \"I would have thee bring me the damsel, because I\nbought her not but for the like of this time.\" So he arose and\ngoing in to the girl, said to her, \"O Sitt al-Milah, the young\nman hath paid down thy price and we have bidden him hither; so he\nhath come to our quarters and we have entertained him, and he\nwould fain have thee be present with him.\" Therewith the damsel\nrose deftly and doffing her dress, bathed and donned sumptuous\napparel and perfumed herself and went out to him, as she were a\nbranch of Ban or a cane of rattan, followed by a black\nslave-girl, bearing the lute. When she came to the young man, she\nsaluted him and sat down by his side. Then she took the lute from\nthe slave-girl and screwing up its pegs,[FN#293] smote thereon in\nfour-and-twenty modes, after which she returned to the first and\nsang these couplets,\n\n\"My joy in this world is to see and sit near thee. * Thy love's\n     my religion; thy Union my pleasure.\nAttest it these tears when in memory I speer thee, * And\n     unchecked down my cheeks pours the flood without measure.\nBy Allah, no rival in love hast to fear thee; * I'm thy slave as\n     I sware, and this troth is my treasure.\nBe not this our last meeting: by Allah I swear thee * Thy\n     severance to me were most bitter displeasure!\"\n\nThe young man was moved to delight and cried, \"By Allah, thou\nsayest well, O Sitt al-Milah! Let me hear more.\" Then he\nlargessed her with fifty gold pieces and they drank and the cups\nmade circuit among them; and her seller said to her, \"O Sitt\nal-Milah, this is the season of farewelling; so let us hear\nsomewhat thereon.\" Accordingly she struck the lute and touching\nupon that which was in her heart, improvised these couplets,\n\n\"I thole longing, remembrance and sad repine, * Nor my heart can\n     brook woes in so lengthened line.\nO my lords think not I forget your love; * My case is sure case\n     and cure shows no sign.\nIf creature could swim in the flood of his tears, * I were first\n     to swim in these floods of brine:\nO Cup-boy withhold cup and bowl from a wretch * Who ne'er ceaseth\n     to drink of her tears for wine!\nHad I known that parting would do me die, * I had shirked to\n     part, but--'twas Fate's design.\"\n\nNow whilst they were thus enjoying whatso is most delicious of\nease and delight, and indeed the wine was to them sweet and the\ntalk a treat, behold, there came a knocking at the door. So the\nhouse-master went out, that he might see what might be the\nmatter, and found ten head of the Caliph's eunuchs at the\nentrance. When he saw this, he was startled and said, \"What is to\ndo?\" \"The Commander of the Faithful saluteth thee and requireth\nof thee the slave-girl whom thou hast exposed for sale and whose\nname is Sitt al-Milah.\" \"By Allah, I have sold her.\" \"Swear by\nthe head of the Commander of the Faithful that she is not in thy\nquarters.\" The slaver made oath that he had sold her and that she\nwas no longer at his disposal: yet they paid no heed to his word\nand forcing their way into the house, found the damsel and the\nyoung Damascene in the sitting-chamber. So they laid hands upon\nher, and the youth said, \"This is my slave-girl, whom I have\nbought with my money;\" but they hearkened not to his speech and\ntaking her, carried her off to the Prince of True Believers.\nTherewith Nur al-Din's pleasure was troubled: he arose and donned\nhis dress, and his host said, \"Whither away this night, O my\nlord?\" Said he, \"I purpose going to my quarters, and to-morrow I\nwill betake myself to the palace of the Commander of the Faithful\nand demand my slave-girl.\" The other replied, \"Sleep till the\nmorning, and fare not forth at the like of this hour.\" But he\nrejoined, \"Needs must I go;\" and the host said to him, \"Go in\nAllah his safeguard.\" So the youth went forth and, drunkenness\nhaving got the mastery of his wits, he threw himself down on a\nbench before one of the shops. Now the watchmen were at that hour\nmaking their rounds and they smelt the sweet scent of essences\nand wine that reeked from him; so they made for it and suddenly\nbeheld the youth lying on the bench, without sign of recovering.\nThey poured water upon him, and he awoke, whereupon they carried\nhim off to the office of the Chief of Police and he questioned\nhim of his case. He replied \"O my lord, I am an alien in this\ntown and have been with one of my friends: I came forth from his\nhouse and drunkenness overcame me.\" The Wali bade carry him to\nhis lodging; but one of those in attendance upon him, Al-Mur\u00e1di\nhight, said to him, \"What wilt thou do? This man is robed in rich\nraiment and on his finger is a golden ring, whose bezel is a ruby\nof great price; so we will carry him away and slay him and take\nthat which is upon him of clothes and bring to thee all we get;\nfor that thou wilt not often see profit the like thereof,\nespecially as this fellow is a foreigner and there is none to ask\nafter him.\"[FN#294] Quoth the Chief, \"This wight is a thief and\nthat which he saith is leasing.\" Nur al-Din said, \"Allah forfend\nthat I should be a thief!\" but the Wali answered, \"Thou liest.\"\nSo they stripped him of his clothes and taking the seal-ring from\nhis finger, beat him with a grievous beating, what while he cried\nout for succour, but none succoured him, and besought protection,\nbut none protected him. Then said he to them, \"O folk, ye are\nquit[FN#295] of that which ye have taken from me; but now restore\nme to my lodging.\" They replied, \"Leave this knavery, O rascal!\nthine intent is to sue us for thy clothes on the morrow.\" The\nyouth cried, \"By the truth of the One, the Eternal One, I will\nnot sue any for them!\" but they said, \"We find no way to this.\"\nAnd the Prefect bade them bear him to the Tigris and there slay\nhim and cast him into the stream. So they dragged him away, while\nhe wept and said the words which shall nowise shame the sayer:\n\"There is no Majesty and there is no Might save in Allah, the\nGlorious, the Great!\" When they came to the Tigris, one of them\ndrew the sword upon him and Al-Muradi said to the sworder, \"Smite\noff his head;\" but one of them, hight Ahmad, cried, \"O folk, deal\nsoftly with this poor wretch and slay him not unjustly and\nwickedly, for I stand in fear of Allah Almighty, lest He burn me\nwith his fire.\" Quoth Al-Muradi, \"A truce to this talk!\" and\nquoth the Ahmad aforesaid, \"An ye do with him aught, I will\nacquaint the Commander of the Faithful.\" They asked, \"How, then,\nshall we do with him?\" and he answered, \"Let us deposit him in\nprison and I will be answerable to you for his provision; so\nshall we be quit of his blood, for indeed he is a wronged man.\"\nAccordingly they agreed to this and taking him up cast him into\nthe Prison of Blood,[FN#296] and then went their ways. So far as\nregards them; but returning to the damsel, they carried her to\nthe Commander of the Faithful and she pleased him; so he assigned\nher a chamber of the chambers of choice. She tarried in the\npalace, neither eating nor drinking, and weeping sans surcease\nnight and day, till, one night, the Caliph sent for her to his\nsitting-hall and said to her, \"O Sitt al-Milah, be of good cheer\nand keep thine eyes cool of tear, for I will make thy rank higher\nthan any of the concubines and thou shalt see that which shall\nrejoice thee.\" She kissed ground and wept; whereupon the Prince\nof True Believers called for her lute and bade her sing: so in\naccordance with that which was in her heart, she sang these\nimprovised couplets,\n\n\"By the sheen of thy soul and the sheen of thy smile,[FN#297] *\n     Say, moan'st thou for doubt or is't ring-dove's moan?\nHow many have died who by love were slain! * Fails my patience\n     but blaming my blamers wone.\"\n\nNow when she had made an end of her song, she threw the lute from\nher hand and wept till she fainted away, whereupon the Caliph\nbade carry her to her chamber. But he was fascinated by her and\nloved her with exceeding love; so, after a while, he again\ncommanded to bring her in to the presence, and when she came, he\nordered her to sing. Accordingly, she took the lute and chanted\nto it that which was in her heart and improvised these couplets,\n\n\"Have I patience and strength to support this despair? * Ah, how\n     couldst thou purpose afar to fare?\nThou art swayed by the spy to my cark and care: * No marvel an\n     branchlet sway here and there![FN#298]\nWith unbearable load thou wouldst load me, still * Thou loadest\n     with love which I theewards bear.\"\n\nThen she cast the lute from her hand and fainted away; so she was\ncarried to her sleeping-chamber and indeed passion grew upon her.\nAfter a long while the Prince of True Believers sent for her a\nthird time and commanded her to sing. So she took the lute and\nchanted these couplets,\n\n\"O of piebald wild ye dunes sandy and drear, * Shall the teenful\n     lover 'scape teen and tear?\nShall ye see me joined with a lover, who * Still flies or shall\n     meet we in joyful cheer?\nO hail to the fawn with the Houri eye, * Like sun or moon on\n     horizon clear!\nHe saith to lovers, 'What look ye on?' * And to stony hearts,\n     'Say, what love ye dear?'[FN#299]\nI pray to Him who departed us * With severance-doom, 'Be our\n     union near!'\"\n\nWhen she had made an end of her verse, the Commander of the\nFaithful said to her, \"O damsel, thou art in love.\" She replied,\n\"Yes;\" and he asked, \"With whom?\" Answered she, \"With my lord and\nsovran of my tenderness, for whom my love is as the love of the\nearth for rain, or as the desire of the female for the male; and\nindeed the love of him is mingled with my flesh and my blood and\nhath entered into the channels of my bones. O Prince of True\nBelievers, whenever I call him to mind my vitals are consumed,\nfor that I have not yet won my wish of him, and but that I fear\nto die, without seeing him, I had assuredly slain myself.\"\nThereupon quoth he, \"Art thou in my presence and durst bespeak me\nwith the like of these words? Forsure I will gar thee forget thy\nlord.\" Then he bade take her away; so she was carried to her\npavilion and he sent her a concubine, with a casket wherein were\nthree thousand ducats and a collar of gold set with seed-pearls\nand great unions, and jewels, worth other three thousand, saying\nto her, \"The slave-girl and that which is with her are a gift\nfrom me to thee.\" When she heard this, she cried, \"Allah forfend\nthat I be consoled for the love of my lord and my master, though\nwith an earth-full of gold!\" And she improvised and recited these\ncouplets,\n\n\"By his life I swear, by his life I pray; * For him fire I'd\n     enter unful dismay!\n'Console thee (cry they) with another fere * Thou lovest!' and I,\n     'By 's life, nay, NAY!'\nHe's moon whom beauty and grace array; * From whose cheeks and\n     brow shineth light of day.\"\n\nThen the Commander of the Faithful summoned her to his presence a\nfourth time and said, \"O Sitt al-Milah, sing.\" So she recited and\nsang these couplets,\n\n\"The lover's heart by his beloved is oft dishearten\u00e8d * And by\n     the hand of sickness eke his sprite dispirit\u00e8d,\nOne asked, 'What is the taste of love?\"[FN#300] and I to him\n     replied, * 'Love is a sweet at first but oft in fine\n     unsweetened.'\nI am the thrall of Love who keeps the troth of love to\n     them[FN#301] * But oft they proved themselves 'Urk\u00fab[FN#302]\n     in pact with me they made.\nWhat in their camp remains? They bound their loads and fared\n     away; * To other feres the veil\u00e8d Fairs in curtained litters\n     sped;\nAt every station the beloved showed all of Joseph's charms: * The\n     lover wone with Jacob's woe in every shift of stead.\"\n\nWhen she had made an end of her song, she threw the lute from her\nhand and wept herself a-swoon. So they sprinkled on her\nmusk-mingled rose-water and willow-flower water; and when she\ncame to her senses, Al-Rashid said to her, \"O Sitt al-Milah, this\nis not just dealing in thee. We love thee and thou lovest\nanother.\" She replied, \"O Commander of the Faithful, there is no\nhelp for it.\" Thereupon he was wroth with her and cried, \"By the\nvirtue of Hamzah[FN#303] and 'Ak\u00edl[FN#304] and Mohammed, Prince\nof the Apostles, an thou name in my presence one other than I, I\nwill assuredly order strike off thy head!\" Then he bade return\nher to her chamber, whilst she wept and recited these couplets,\n\n'Oh brave!' I'd cry an I my death could view; * My death were\n     better than these griefs to rue,\nDid sabre hew me limb by limb; this were * Naught to affright a\n     lover leal-true.\"\n\nThen the Caliph went in to the Lady Zubaydah, complexion-altered\nwith anger, and she noted this in him and said to him, \"How\ncometh it that I see the Commander of the Faithful changed of\ncolour?\" He replied, \"O daughter of my uncle, I have a beautiful\nslave-girl, who reciteth verses by rote and telleth various\ntales, and she hath taken my whole heart; but she loveth other\nthan myself and declareth that she affecteth her former lord; so\nI have sworn a great oath that, if she come again to my sitting-\nhall and sing for other than for me, I will assuredly shorten her\nhighest part by a span.\"[FN#305] Quoth Zubaydah, \"Let the\nCommander of the Faithful favour me by presenting her, so I may\nlook on her and hear her singing.\" Accordingly he bade fetch her\nand she came, upon which the Lady Zubaydah withdrew behind the\ncurtain,[FN#306] where the damsel saw her not, and Al-Rashid said\nto her, \"Sing to us.\" So she took the lute and tuning it, recited\nthese couplets,\n\n\"O my lord! since the day when I lost your sight, * My life was\n     ungladdened, my heart full of teen;\nThe memory of you kills me every night; * And by all the worlds\n     is my trace unseen;\nAll for love of a Fawn who hath snared my sprite * By his love\n     and his brow as the morning sheen.\nLike a left hand parted from brother right * I became by parting\n     thro' Fortune's spleen.\nOn the brow of him Beauty deigned indite * 'Blest be Allah, whom\n     best of Creators I ween!'\nAnd Him I pray, who could disunite * To re-unite us. Then cry\n     'Ameen!'\"[FN#307]\n\nWhen Al-Rashid heard the end of this, he waxed exceeding wroth\nand said, \"May Allah not reunite you twain in gladness!\" Then he\nsummoned the headsman, and when he presented himself, he said to\nhim, \"Strike off the head of this accursed slave-girl.\" So Masrur\ntook her by the hand and led her away; but, when she came to the\ndoor, she turned and said to the Caliph, \"O Commander of the\nFaithful, I conjure thee, by thy fathers and forefathers, behead\nme not until thou give ear to that I shall say!\" Then she\nimprovised and recited these couplets,\n\n\"Emir of Justice, be to lieges kind * For Justice ever guides thy\n     generous mind;\nAnd, oh, who blamest love to him inclining! * Are lovers blamed\n     for l\u00e2ches undesigned?\nBy Him who gave thee rule, deign spare my life * For rule on\n     earth He hath to thee assigned.\"\n\nThen Masrur carried her to the other end of the sitting-hall and\nbound her eyes and making her sit, stood awaiting a second order:\nwhereupon quoth the Lady Zubaydah, \"O Prince of True Believers,\nwith thy permission, wilt thou not vouchsafe this damsel a\nportion of thy clemency? An thou slay her, 'twere injustice.\"\nQuoth he, \"What is to be done with her?\" and quoth she, \"Forbear\nto slay her and send for her lord. If he be as she describeth him\nin beauty and loveliness, she is excused, and if he be not on\nthis wise then kill her, and this shall be thy plea against\nher.\"[FN#308] Al-Rashid replied, \"No harm in this rede;\" and\ncaused return the damsel to her chamber, saying to her, \"The Lady\nZubaydah saith thus and thus.\" She rejoined, \"God requite her for\nme with good! Indeed, thou dealest equitably, O Commander of the\nFaithful, in this judgment.\" And he retorted, \"Go now to thy\nplace, and tomorrow we will bid them bring thy lord.\" So she\nkissed ground and recited these couplets,\n\n\"I indeed will well for whom love I will: * Let chider chide and\n     let blamer blame:\nAll lives must die at fixt tide and term * But I must die ere my\n     life-term came:\nThen Oh whose love hath afflicted me * Well I will but thy\n     presence in haste I claim.\"\n\nThen she arose and returned to her chamber. Now on the morrow,\nthe Commander of the Faithful sat in his hall of audience and his\nWazir Ja'afar bin Yahya the Barmecide came in to him; whereupon\nhe called to him, saying, \"I would have thee bring me a youth who\nis lately come to Baghdad, hight Sidi Nur al-Din Ali the\nDamascene.\" Quoth Ja'afar, \"Hearing and obeying,\" and going forth\nin quest of the youth, sent to the bazars and Wakalahs and Khans\nfor three successive days, but discovered no trace of him,\nneither happened upon the place of him. So on the fourth day he\npresented himself before the Caliph and said to him, \"O our lord,\nI have sought him these three days, but have not found him.\" Said\nAl-Rashid, \"Make ready letters to Damascus. Peradventure he hath\nreturned to his own land.\" Accordingly Ja'afar wrote a letter and\ndespatched it by a dromedary-courier to the Damascus-city; and\nthey sought him there and found him not. Meanwhile, news was\nbrought that Khorasan had been conquered;[FN#309] whereupon\nAl-Rashid rejoiced and bade decorate Baghdad and release all in\nthe gaol, giving each of them a ducat and a dress. So Ja'afar\napplied himself to the adornment of the city and bade his brother\nAl-Fazl ride to the prison and robe and set free the prisoners.\nAl-Fazl did as his brother commanded and released all save the\nyoung Damascene, who abode still in the Prison of Blood, saying,\n\"There is no Majesty, and there is no Might save in Allah, the\nGlorious, the Great! Verily, we are God's and to Him are we\nreturning.\" Then quoth Al-Fazl to the gaoler, \"Is there any left\nin the prison?\" Quoth he, \"No,\" and Al-Fazl was about to depart,\nwhen Nur al-Din called out to him from within the prison, saying,\n\"O our lord, tarry awhile, for there remaineth none in the prison\nother than I and indeed I am wronged. This is a day of pardon and\nthere is no disputing concerning it.\" Al-Fazl bade release him;\nso they set him free and he gave him a dress and a ducat.\nThereupon the young man went out, bewildered and unknowing\nwhither he should wend, for that he had sojourned in the gaol a\nyear or so and indeed his condition was changed and his favour\nfouled, and he abode walking and turning round, lest Al-Muradi\ncome upon him and cast him into another calamity. When Al-Muradi\nlearnt his release, he betook himself to the Wali and said, \"O\nour lord, we are not assured of our lives from that youth,\nbecause he hath been freed from prison and we fear lest he\ncomplain of us.\" Quoth the Chief, \"How shall we do?\" and quoth\nAl-Muradi, \"I will cast him into a calamity for thee.\" Then he\nceased not to follow the Damascene from place to place till he\ncame up with him in a narrow stead and cul-de-sac; whereupon he\naccosted him and casting a cord about his neck, cried out, \"A\nthief!\" The folk flocked to him from all sides and fell to\nbeating and abusing Nur al-Din,[FN#310] whilst he cried out for\naidance but none aided him, and Al-Muradi kept saying to him,\n\"But yesterday the Commander of the Faithful released thee and\nto-day thou robbest!\" So the hearts of the mob were hardened\nagainst him and again Al-Muradi carried him to the Chief of\nPolice, who bade hew off his hand. Accordingly, the hangman took\nhim and bringing out the knife, proceeded to cut off his hand,\nwhile Al-Muradi said to him, \"Cut and sever the bone and\nfry[FN#311] not in oil the stump for him, so he may lose all his\nblood and we be at rest from him.\" But Ahmad, he who had before\nbeen the cause of his deliverance, sprang up to him and cried, \"O\nfolk, fear Allah in your action with this youth, for that I know\nhis affair, first and last, and he is clear of offence and\nguiltless: he is of the lords of houses,[FN#312] and unless ye\ndesist from him, I will go up to the Commander of the Faithful\nand acquaint him with the case from beginning to end and that the\nyouth is innocent of sin or crime.\" Quoth Al-Muradi, \"Indeed, we\nare not assured from his mischief;\" and quoth Ahmad, \"Set him\nfree and commit him to me and I will warrant you against his\ndoings, for ye shall never see him again after this.\" So they\ndelivered Nur al-Din to him and he took him from their hands and\nsaid to him, \"O youth, have ruth on thyself, for indeed thou hast\nfallen into the hands of these folk twice and if they prevail\nover thee a third time, they will make an end of thee; and I in\ndoing thus with thee, aim at reward for thee and recompense in\nHeaven and answer of prayer.\"[FN#313] So Nur al-Din fell to\nkissing his hand and blessing him said, \"Know that I am a\nstranger in this your city and the completion of kindness is\nbetter than its commencement; wherefore I pray thee of thy favour\nthat thou make perfect to me thy good offices and generosity and\nbring me to the city-gate. So will thy beneficence be\naccomplished unto me and may God Almighty requite thee for me\nwith good!\" Ahmad replied, \"No harm shall betide thee: go; I will\nbear thee company till thou come to thy place of safety.\" And he\nleft him not till he brought him to the city-gate and said to\nhim, \"O youth, go in Allah's guard and return not to the city,\nfor, an they fall in with thee again, they will make an end of\nthee.\" Nur al-Din kissed his hand and going forth the city, gave\nnot over walking, till he came to a mosque that stood in one of\nthe suburbs of Baghdad and entered therein with the night. Now he\nhad with him naught wherewith he might cover himself; so he\nwrapped himself up in one of the mats of the mosque and thus\nabode till dawn, when the Muezzins came and finding him seated in\nsuch case, said to him, \"O youth, what is this plight?\" Said he,\n\"I cast myself on your protection, imploring your defence from a\ncompany of folk who seek to slay me unjustly and wrongously,\nwithout cause.\" And one of the Muezzins said, \"I will protect\nthee; so be of good cheer and keep thine eyes cool of tear.\" Then\nhe brought him old clothes and covered him therewith; he also set\nbefore him somewhat of victual and seeing upon him signs of fine\nbreeding, said to him, \"O my son, I grow old and desiring help\nfrom thee, I will do away thy necessity.\" Nur al-Din replied, \"To\nhear is to obey;\" and abode with the old man, who rested and took\nhis ease, while the youth did his service in the mosque,\ncelebrating the praises of Allah and calling the Faithful to\nprayer and lighting the lamps and filling the spout-pots[FN#314]\nand sweeping and cleaning out the place of worship. On this-wise\nit befel the young Damascene; but as regards Sitt al-Milah, the\nLady Zubaydah, the wife of the Commander of the Faithful, made a\nbanquet in her palace and assembled her slave-girls. And the\ndamsel came, weeping-eyed and heavy-hearted, and those present\nblamed her for this, whereupon she recited these couplets,\n\n\"Ye blame the mourner who weeps his woe; * Needs must the mourner\n     sing, weeping sore;\nAn I see not some happy day I'll weep * Brine-tears till followed\n     by gouts of gore.\"\n\nWhen she had made an end of her verses, the Lady Zubaydah bade\neach damsel sing a song, till the turn came round to Sitt\nal-Milah, whereupon she took the lute and tuning it, carolled\nthereto four-and-twenty carols in four-and-twenty modes; then she\nreturned to the first and sang these couplets,\n\n\"The World hath shot me with all her shafts * Departing friends\n     parting-grief t' aby:\nSo in heart the burn of all hearts I bear * And in eyes the\n     tear-drops of every eye.\"\n\nWhen she had made an end of her song, she wept till she garred\nthe bystanders weep and the Lady Zubaydah condoled with her and\nsaid to her, \"Allah upon thee, O Sitt al-Milah, sing us somewhat, so we may hearken to thee.\" The damsel replied, \"Hearing\nand obeying,\" and sang these couplets,\n\n\"People of passion, assemble ye! * This day be the day of our\n     agony:\nThe Raven o' severance croaks at our doors; * Our raven which\n     nigh to us aye see we.\nThe friends we love have appointed us * The grievousest\n     parting-dule to dree.\nRise, by your lives, and let all at once * Fare to seek our\n     friends where their sight we see.\"\n\nThen she threw the lute from her hand and shed tears till she\ndrew tears from the Lady Zubaydah who said to her, \"O Sitt\nal-Milah, he whom thou lovest methinks is not in this world, for\nthe Commander of the Faithful hath sought him in every place, but\nhath not found him.\" Whereupon the damsel arose and kissing the\nPrincess's hands, said to her, \"O my lady, an thou wouldst have\nhim found, I have this night a request to make whereby thou mayst\nwin my need with the Caliph.\" Quoth the Lady, \"And what is it?\"\nand quoth Sitt al-Milah, \"'Tis that thou get me leave to fare\nforth by myself and go round about in quest of him three days,\nfor the adage saith, Whoso keeneth for herself is not like whoso\nis hired to keen![FN#315] An if I find him, I will bring him\nbefore the Commander of the Faithful, so he may do with us what\nhe will, and if I find him not, I shall be cut off from hope of\nhim and the heat of that which is with me will be cooled.\" Quoth\nthe Lady Zubaydah, \"I will not get thee leave from him but for a\nwhole month; so be of good cheer and eyes cool and clear.\"\nWhereat Sitt al-Milah rejoiced and rising, kissed ground before\nher once more and went away to her own place, and right glad was\nshe. As for Zubaydah, she went in to the Caliph and talked with\nhim awhile; then she fell to kissing him between the eyes and on\nhis hand and asked him for that which she had promised to Sitt\nal-Milah, saying, \"O Commander of the Faithful, I doubt me her\nlord is not found in this world; but, an she go about seeking him\nand find him not, her hopes will be cut off and her mind will be\nset at rest and she will sport and laugh; and indeed while she\nnourisheth hope, she will never take the right direction.\" And\nshe ceased not cajoling him till he gave Sitt al-Milah leave to\nfare forth and make search for her lord a month's space and\nordered a riding-mule and an eunuch to attend her and bade the\nprivy purse give her all she needed, were it a thousand dirhams a\nday or even more. So the Lady Zubaydah arose and returning to her\npalace bade summon Sitt al-Milah and, as soon as she came,\nacquainted her with that which had passed; whereupon she kissed\nher hand and thanked her and called down blessings on her. Then\nshe took leave of the Princess and veiling her face with a\nmask,[FN#316] disguised herself;[FN#317] after which she mounted\nthe she-mule and sallying forth, went round about seeking her\nlord in the highways of Baghdad three days' space, but happed on\nno tidings of him; and on the fourth day, she rode forth without\nthe city. Now it was the noon-hour and fierce was the heat, and\nshe was aweary and thirst came upon her. Presently, she reached\nthe mosque of the Shaykh who had lodged the young Damascene, and\ndismounting at the door, said to the old Muezzin, \"O Shaykh, hast\nthou a draught of cold water? Verily, I am overcome with heat and\nthirst.\" Said he, \"'Tis with me in my house.\" So he carried her\nup into his lodging and spreading her a carpet, seated her; after\nwhich he brought her cold water and she drank and said to the\neunuch, \"Go thy ways with the mule and to-morrow come back to me\nhere.\" Accordingly he went away and she slept and rested herself.\nWhen she awoke, she asked the old man, \"O Shaykh, hast thou aught\nof food?\" and he answered, \"O my lady, I have bread and olives.\"\nQuoth she, \"That be food which befitteth only the like of thee.\nAs for me, I will have naught save roast lamb and soups and\nreddened fowls right fat and ducks farcis with all manner\nstuffing of pistachio-nuts and sugar.\" Quoth the Muezzin, \"O my\nlady, I have never heard of this chapter[FN#318] in the Koran,\nnor was it revealed to our lord Mohammed, whom Allah save and\nassain!\"[FN#319] She laughed and said, \"O Shaykh, the matter is\neven as thou sayest; but bring me pen-case and paper.\" So he\nbrought her what she sought and she wrote a note and gave it to\nhim, together with a seal-ring from her finger, saying, \"Go into\nthe city and enquire for Such-an-one the Shroff and give him this\nmy note.\" Accordingly the oldster betook himself to the city, as\nshe bade him, and asked for the money-changer, to whom they\ndirected him. So he gave him ring and writ, seeing which, he\nkissed the letter and breaking it open, read it and apprehended\nits contents. Then he repaired to the bazar and buying all that\nshe bade him, laid it in a porter's crate and made him go with\nthe Shaykh. The old man took the Hamm\u00e1l and went with him to the\nmosque, where he relieved him of his burden and carried the rich\nviands in to Sitt al-Milah. She seated him by her side and they\nate, he and she, of those dainty cates, till they were satisfied,\nwhen the Shaykh rose and removed the food from before her. She\npassed that night in his lodging and when she got up in the\nmorning, she said to him, \"O elder, may I not lack thy kind\noffices for the breakfast! Go to the Shroff and fetch me from him\nthe like of yesterday's food.\" So he arose and betaking himself\nto the money-changer, acquainted him with that which she had\nbidden him. The Shroff brought him all she required and set it on\nthe heads of Hammals; and the Shaykh took them and returned with\nthem to the damsel, when she sat down with him and they ate their\nsufficiency, after which he removed the rest of the meats. Then\nshe took the fruits and the flowers and setting them over against\nherself, wrought them into rings and knots and writs, whilst the\nShaykh looked on at a thing whose like he had never in his life\nseen and rejoiced in the sight. Presently said she to him \"O\nelder, I would fain drink.\" So he arose and brought her a gugglet\nof water; but she cried to him, \"Who said to thee, Fetch that?\"\nQuoth he, \"Saidst thou not to me, I would fain drink?\" and quoth\nshe, \"'I want not this; nay, I want wine, the solace of the soul,\nso haply, O Shaykh, I may refresh myself therewith.\" Exclaimed\nthe old man, \"Allah forfend that strong drink be drunk in my\nhouse, and I a stranger in the land and a Muezzin and an Imam,\nwho leadeth the True Believers in prayer, and a servant of the\nHouse of the Lord of the three Worlds!\" \"Why wilt thou forbid me\nto drink thereof in thy house?\" \"Because 'tis unlawful.\" \"O\nelder, Allah hath forbidden only the eating of blood and\ncarrion[FN#320] and hog's flesh: tell me, are grapes and honey\nlawful or unlawful?\" \"They are lawful.\" \"This is the juice of\ngrapes and the water of honey.\" \"Leave this thy talk, for thou\nshalt never drink wine in my house.\" \"O Shaykh, people eat and\ndrink and enjoy themselves and we are of the number of the folk\nand Allah is indulgent and merciful.\"[FN#321] \"This is a thing\nthat may not be.\" \"Hast thou not heard what the poet saith?\" And\nshe recited these couplets,\n\n\"Cease thou to hear, O Sim'\u00e1n-son,[FN#322] aught save the say of\n     me; * How bitter 'twas to quit the monks and fly the\n     monast'ry!\nWhen, on the F\u00eate of Palms there stood, amid the hallowed\n     fane,[FN#323] * A pretty Fawn whose lovely pride garred me\n     sore wrong to dree.\nMay Allah bless the night we spent when he to us was third, *\n     While Moslem, Jew, and Nazarene all sported fain and free.\nQuoth he, from out whose locks appeared the gleaming of the morn,\n     * 'Sweet is the wine and sweet the flowers that joy us\n     comrades three.\nThe garden of the garths of Khuld where roll and rail amain, *\n     Rivulets 'neath the myrtle shade and B\u00e1n's fair branchery;\nAnd birds make carol on the boughs and sing in blithest lay, *\n     Yea, this indeed is life, but, ah! how soon it fades away.'\"\n\nShe then asked him, \"O Shaykh, an Moslems and Jews and Nazarenes\ndrink wine, who are we that we should reject it?\" Answered he,\n\"By Allah, O my lady, spare thy pains, for this be a thing\nwhereto I will not hearken.\" When she knew that he would not\nconsent to her desire, she said to him, \"O Shaykh, I am of the\nslave-girls of the Commander of the Faithful and the food waxeth\nheavy on me and if I drink not, I shall die of indigestion, nor\nwilt thou be assured against the issue of my case.[FN#324] As for\nme, I acquit myself of blame towards thee, for that I have bidden\nthee beware of the wrath of the Commander of the Faithful, after\nmaking myself known to thee.\" When the Shaykh heard her words and\nthat wherewith she threatened him, he sprang up and went out,\nperplexed and unknowing what he should do, and there met him a\nJewish man, which was his neighbour, and said to him, \"How cometh\nit that I see thee, O Shaykh, strait of breast? Eke, I hear in\nthy house a noise of talk, such as I am unwont to hear with\nthee.\" Quoth the Muezzin, \"'Tis of a damsel who declareth that\nshe is of the slave-girls of the Commander of the Faithful, Harun\nal-Rashid; and she hath eaten meat and now would drink wine in my\nhouse, but I forbade her. However she asserteth that unless she\ndrink thereof, she will die, and indeed I am bewildered\nconcerning my case.\" Answered the Jew, \"Know, O my neighbour,\nthat the slave-girls of the Commander of the Faithful are used\nto drink wine, and when they eat and drink not, they die; and I\nfear lest happen some mishap to her, when thou wouldst not be\nsafe from the Caliph's fury.\" The Shaykh asked, \"What is to be\ndone?\" and the Jew answered, \"I have old wine that will suit\nher.\" Quoth the Shaykh, \"By the right of neighbourship, deliver\nme from this descent[FN#325] of calamity and let me have that\nwhich is with thee!\" Quoth the Jew, \"Bismillah, in the name of\nAllah,\" and passing to his quarters, brought out a glass flask of\nwine, wherewith the Shaykh returned to Sitt al-Milah. This\npleased her and she cried to him, \"Whence hadst thou this?\" He\nreplied, \"I got it from the Jew, my neighbour: I set forth to him\nmy case with thee and he gave me this.\" Thereupon Sitt al-Milah\nfilled a cup and emptied it; after which she drank a second and a\nthird. Then she crowned the cup a fourth time and handed it to\nthe Shaykh, but he would not accept it from her. However, she\nconjured him, by her own head and that of the Prince of True\nBelievers, that he take the cup from her, till he received it\nfrom her hand and kissed it and would have set it down; but she\nsware him by her life to smell it. Accordingly he smelt it and\nshe said to him, \"How deemest thou?\" Said he, \"I find its smell\nis sweet;\" and she conjured him by the Caliph's life to taste\nthereof. So he put it to his mouth and she rose to him and made\nhim drink; whereupon quoth be, \"O Princess of the Fair,[FN#326]\nthis is none other than good.\" Quoth she, \"So deem I: hath not\nour Lord promised us wine in Paradise?\" He answered, \"Yes! The\nMost High saith, 'And rivers of wine, delicious to the\ndrinkers.'[FN#327] And we will drink it in this world and in the\nnext world.\" She laughed and emptying the cup, gave him to drink,\nand he said, \"O Princess of the Fair, indeed thou art excusable\nin thy love for this.\" Then he hent in hand from her another and\nanother, till he became drunken and his talk waxed great and his\nprattle. The folk of the quarter heard him and assembled under\nthe window; and when the Shaykh was ware of them, he opened the\nwindow and said to them, \"Are ye not ashamed, O pimps? Every one\nin his own house doth whatso he willeth and none hindereth him;\nbut we drink one single day and ye assemble and come, panders\nthat ye are! To-day, wine, and to-morrow business;[FN#328] and\nfrom hour to hour cometh relief.\" So they laughed together and\ndispersed. Then the girl drank till she was drunken, when she\ncalled to mind her lord and wept, and the Shaykh said to her,\n\"What maketh thee weep, O my lady?\" Said she, \"O elder, I am a\nlover and a separated.\" He cried, \"O my lady, what is this love?\"\nCried she, \"And thou, hast thou never been in love?\" He replied,\n\"By Allah, O my lady, never in all my life heard I of this thing,\nnor have I ever known it! Is it of the sons of Adam or of the\nJinn?\" She laughed and said, \"Verily, thou art even as those of\nwhom the poet speaketh, in these couplets,\n\n\"How oft shall they admonish and ye shun this nourishment; * When\n     e'en the shepherd's bidding is obey\u00e8d by his flocks?\nI see you like in shape and form to creatures whom we term *\n     Mankind, but in your acts and deeds you are a sort of\n     ox\"[FN#329]\n\nThe Shaykh laughed at her speech and her verses pleased him. Then\ncried she to him, \"I desire of thee a lute.\" So he arose and\nbrought her a bit of fuel.[FN#330] Quoth she, \"What is that?\" and\nquoth he \"Didst thou not say: Bring me fuel?\" Said she, \"I do not\nwant this,\" and said he, \"What then is it that is hight fuel,\nother than this?\" She laughed and replied, \"The lute is an\ninstrument of music, whereunto I sing.\" Asked he, \"Where is this\nthing found and of whom shall I get it for thee?\" and answered\nshe, \"Of him who gave thee the wine.\" So he arose and betaking\nhimself to his neighbour the Jew, said to him, \"Thou favouredst\nus before with the wine; so now complete thy favours and look me\nout a thing hight lute, which be an instrument for singing; for\nshe seeketh this of me and I know it not.\" Replied the Jew,\n\"Hearkening and obedience,\" and going into his house, brought him\na lute. The old man carried it to Sitt al-Milah, whilst the Jew\ntook his drink and sat by a window adjoining the Shaykh's house,\nso he might hear the singing. The damsel rejoiced, when the old\nman returned to her with the lute, and taking it from him, tuned\nits strings and sang these couplets,\n\n\"Remains not, after you are gone, or trace of you or sign, * But\n     hope to see this parting end and break its lengthy line:\nYou went and by your wending made the whole world desolate; * And\n     none may stand this day in stead to fill the yearning eyne.\nIndeed, you've burdened weakling me, by strength and force of you\n     * With load no hill hath power t'upheave nor yet the plain\n     low li'en:\nAnd I, whenever fain I scent the breeze your land o'erbreathes, *\n     Lose all my wits as though they were bemused with heady\n     wine.\nO folk no light affair is Love for lover woe to dree * Nor easy\n     'tis to satisfy its sorrow and repine.\nI've wandered East and West to hap upon your trace, and when *\n     Spring-camps I find the dwellers cry, 'They've marched,\n     those friends o' thine!'\nNever accustomed me to part these intimates I love; * Nay, when I\n     left them all were wont new meetings to design.\"\n\nNow when she had ended her song, she wept with sore weeping, till\npresently sleep overcame her and she slept. On the morrow, she\nsaid to the Shaykh, \"Get thee to the Shroff and fetch me the\nordinary;\" so he repaired to the money-changer and delivered him\nthe message, whereupon he made ready meat and drink, according to\nhis custom, with which the old man returned to the damsel and\nthey ate their sufficiency. When she had eaten, she sought of him\nwine and he went to the Jew and fetched it. Then the twain sat\ndown and drank; and when she waxed drunken, she took the lute and\nsmiting it, fell a-singing and chanted these couplets,\n\n\"How long ask I the heart, the heart drowned, and eke * Refrain\n     my complaint while I my tear-floods speak?\nThey forbid e'en the phantom to visit me, * (O marvel!) her\n     phantom my couch to seek.\"[FN#331]\n\nAnd when she had made an end of her song, she wept with sore\nweeping. All this time, the young Damascene was listening, and\nnow he likened her voice to the voice of his slave-girl and then\nhe put away from him this thought, and the damsel had no\nknowledge whatever of his presence. Then she broke out again into\nsong and chanted these couplets,\n\n\"Quoth they, 'Forget him! What is he?' To them I cried, * 'Allah\n     forget me when forget I mine adored!'\nNow in this world shall I forget the love o' you? * Heaven grant\n     the thrall may ne'er forget to love his lord!\nI pray that Allah pardon all except thy love * Which, when I meet\n     Him may my bestest plea afford.\"\n\nAfter ending this song she drank three cups and filling the old\nman other three, improvised these couplets,\n\n\"His love he hid which tell-tale tears betrayed; * For burn of\n     coal that 'neath his ribs was laid:\nGiv'n that he seek his joy in spring and flowers * Some day, his\n     spring's the face of dear-loved maid.\nO ye who blame me for who baulks my love! * What sweeter thing\n     than boon to man denayed?\nA sun, yet scorcheth he my very heart! * A moon, but riseth he\n     from breasts a-shade!\"\n\nWhen she had made an end of her song, she threw the lute from her\nhand and wept, whilst the Shaykh wept for her weeping. Then she\nfell down in a fainting fit and presently recovering, crowned the\ncup and drinking it off, gave the elder to drink, after which she\ntook the lute and breaking out into song, chanted these couplets,\n\n\"Thy parting is bestest of woes to my heart, * And changed my\n     case till all sleep it eschewed:\nThe world to my being is desolate; * Then Oh grief! and O\n     lingering solitude!\nMaybe The Ruthful incline thee to me * And join us despite what\n     our foes have sued!\"\n\nThen she wept till her voice rose high and her wailing was\ndiscovered to those without; after which she again began to drink\nand plying the Shaykh with wine, sang these couplets,\n\n\"An they hid thy person from eyen-sight, * They hid not thy name\n     fro' my mindful sprite:\nOr meet me; thy ransom for meeting I'll be[FN#332] * Or fly me;\n     and ransom I'll be for thy flight!\nMine outer speaks for mine inner case, * And mine inner speaks\n     for mine outer plight.\"\n\nWhen she had made an end of her verses, she threw the lute from\nher hand and wept and wailed. Then she slept awhile and presently\nawaking, said, \"O Shaykh, say me, hast thou what we may eat?\" He\nreplied, \"O my lady, I have the rest of the food;\" but she cried,\n\"I will not eat of the orts I have left. Go down to the bazar and\nfetch us what we may eat.\" He rejoined, \"Excuse me, O my lady, I\ncannot rise to my feet, because I am bemused with wine; but with\nme is the servant of the mosque, who is a sharp youth and an\nintelligent. I will call him, so he may buy thee whatso thou\nwantest.\" Asked she, \"Whence hast thou this servant?\" and he\nanswered, \"He is of the people of Damascus.\" When she heard him\nsay \"of the people of Damascus,\" she sobbed such a sob that she\nswooned away; and when she came to herself, she said, \"Woe is me\nfor the people of Damascus and for those who are therein! Call\nhim, O Shaykh, that he may do our need.\" Accordingly, the old man\nput his head forth of the window and called the youth, who came\nto him from the mosque and sought leave to enter. The Muezzin\nbade him come in, and when he appeared before the damsel, he knew\nher and she knew him; whereupon he turned back in bewilderment\nand would have fled at hap-hazard; but she sprang up to him and\nheld him fast, and they embraced and wept together, till they\nfell to the floor in a fainting fit. When the Shaykh saw them in\nthis condition, he feared for himself and fared forth in fright,\nseeing not the way for drunkenness. His neighbour the Jew met him\nand asked him, \"How is it that I behold thee astounded?\" Answered\nthe old man, \"How should I not be astounded, seeing that the\ndamsel who is with me is fallen in love with the mosque servant\nand they have embraced and slipped down in a swoon? Indeed, I\nfear lest the Caliph come to know of this and be wroth with me;\nso tell me thou what is thy device for that wherewith I am\nafflicted in the matter of this damsel.\" Quoth the Jew, \"For the\npresent, take this casting-bottle of rose-water and go forthright\nand sprinkle them therewith: an they be aswoon for this their\nunion and embrace, they will recover, and if otherwise, then take\nto flight.\" The Shaykh snatched the casting-bottle from the Jew\nand going up to the twain, sprinkled their faces, whereupon they\ncame to themselves and fell to relating each to other that which\nthey had suffered, since both had been parted, for the pangs of\nseverance. Nur al-Din also acquainted Sitt al-Milah with that\nwhich he had endured from the folk who would have killed[FN#333]\nhim and utterly annihilated him; and she said to him, \"O my lord,\nlet us for the nonce leave this talk and praise Allah for reunion\nof loves, and all this shall cease from us.\" Then she gave him\nthe cup and he said, \"By Allah, I will on no wise drink it,\nwhilst I am in this case!\" So she drank it off before him and\ntaking the lute, swept the strings and sang these couplets,\n\n\"O absent fro' me and yet present in place, * Thou art far from\n     mine eyes and yet ever nigh!\nThy farness bequeathed me all sorrow and care * And my troublous\n     life can no joy espy:\nLone, forlorn, weeping-eyelidded, miserablest, * I abide for thy\n     sake as though banisht I:\nThen (ah grief o' me!) far thou hast fared from sight * Yet canst\n     no more depart me than apple of eye!\"\n\nWhen she had made an end of her verse, she wept and the young man\nof Damascus, Nur al-Din, wept also. Then she took the lute and\nimprovised these couplets,\n\n\"Well Allah wots I never nam\u00e8d you * But tears o'erbrimming eyes\n     in floods outburst;\nAnd passion raged and pine would do me die, * Yet my heart rested\n     wi' the thought it nurst;\nO eye-light mine, O wish and O my hope! * Your face can never\n     quench mine eyes' hot thirst.\"\n\nWhen Nur al-Din heard these his slave-girl's verses, he fell\na-weeping, while she strained him to her bosom and wiped away his\ntears with her sleeve and questioned him and comforted his mind.\nThen she took the lute and sweeping its strings, played thereon\nwith such performing as would move the staidest to delight and\nsang these couplets,\n\n\"Indeed, what day brings not your sight to me, * That day I\n     rem'mber not as dight to me!\nAnd, when I vainly long on you to look, * My life is lost, Oh\n     life and light o' me!\"\n\nAfter this fashion they fared till the morning, tasting not the\nnourishment of sleep;[FN#334] and when the day lightened, behold\nthe eunuch came with the she-mule and said to Sitt al-Milah, \"The\nCommander of the Faithful calleth for thee.\" So she arose and\ntaking by the hand her lord, committed him to the Shaykh, saying,\n\"This is the deposit of Allah, then thy deposit,[FN#335] till\nthis eunuch cometh to thee; and indeed, O elder, my due to thee\nis the white hand of favour such as filleth the interval betwixt\nheaven and earth.\" Then she mounted the mule and repairing to the\npalace of the Commander of the Faithful, went in to him and\nkissed ground before him. Quoth he to her, as who should make\nmock of her, \"I doubt not but thou hast found thy lord;\" and\nquoth she, \"By thy felicity and the length of thy continuance on\nlife, I have indeed found him!\" Now Al-Rashid was leaning back;\nbut, when he heard this, he sat upright and said to her \"By my\nlife, true?\" She replied, \"Ay, by thy life!\" He said, \"Bring him\ninto my presence, so I may see him;\" but she said, \"O my lord,\nthere have happened to him many hardships and his charms are\nchanged and his favour faded; and indeed the Prince of True\nBelievers vouchsafed me a month; wherefore I will tend him the\nrest of the month and then bring him to do his service to the\nCommander of the Faithful.\" Quoth Al-Rashid, \"Sooth thou sayest:\nthe condition certainly was for a month; but tell me what hath\nbetided him.\" Quoth she, \"O my lord (Allah prolong thy\ncontinuance and make Paradise thy place of returning and thine\nasylum and the fire the abiding-place of thy foes!), when he\npresenteth himself to serve thee, he will assuredly expound to\nthee his case and will name to thee his wrongdoers; and indeed\nthis is an arrear that is due to the Prince of True Believers, by\nwhom may Allah fortify the Faith and vouchsafe him the victory\nover rebel and froward wretch!\" Thereupon he ordered her a fine\nhouse and bade furnish it with carpets and vessels of choice and\ncommanded them to give all she needed. This was done during the\nrest of the day, and when the night came, she sent the eunuch\nwith a suit of clothes and the mule, to fetch Nur al-Din from the\nMuezzin's lodging. So the young man donned the dress and\nmounting, rode to the house, where he abode in comfort and luxury\na full-told month, while she solaced him with four things, the\neating of fowls and the drinking of wine and the sleeping upon\nbrocade and the entering the bath after horizontal\nrefreshment.[FN#336] Furthermore, she brought him six suits of\nlinen stuffs and took to changing his clothes day by day; nor was\nthe appointed time of delay accomplished ere his beauty and\nloveliness returned to him; nay, his favour waxed tenfold fairer\nand he became a seduction to all who looked upon him. One day of\nthe days Al-Rashid bade bring him to the presence; so his\nslave-girl changed his clothes and robing him in sumptuous\nraiment, mounted him on the she-mule. Then he rode to the palace\nand presenting himself before the Caliph, saluted him with the\ngoodliest of salutations and bespake him with Truchman's[FN#337]\nspeech eloquent and deep-thoughted. When Al-Rashid saw him, he\nmarvelled at the seemliness of his semblance and his loquence and\neloquence and asking of him, was told that he was Sitt al-Milah's\nlord; whereupon quoth he, \"Indeed, she is excusable in her love\nfor him, and if we had put her to death wrongfully, as we were\nminded to do, her blood would have been upon our heads.\" Then he\naccosted the young man and entering into discourse with him,\nfound him well-bred, intelligent, clever, quick-witted, generous,\npleasant, elegant, excellent. So he loved him with exceeding love\nand questioned him of his native city and of his sire and of the\ncause of his journey to Baghdad. Nur al-Din acquainted him with\nthat which he would know in the goodliest words and concisest\nphrases; and the Caliph asked him, \"And where hast thou been\nabsent all this while? Verily, we sent after thee to Damascus and\nMosul and all other cities, but happened on no tidings of thee.\"\nAnswered the young man, \"O my lord, there betided thy slave in\nthy capital that which never yet betided any.\" Then he acquainted\nhim with his case, first and last, and told him that which had\nbefallen him of evil from Al-Muradi and the Chief of Police. Now\nwhen Al-Rashid heard this, he was chagrined with sore chagrin and\nwaxed wroth with exceeding wrath and cried, \"Shall this thing\nhappen in a city wherein I am?\" And the H\u00e1shim\u00ed vein [FN#338]\nstarted out between his eyes. Then he bade fetch Ja'afar, and\nwhen he came between his hands, he acquainted him with the\nadventure and said to him, \"Shall this thing come to pass in my\ncity and I have no news of it?\" Thereupon he bade Ja'afar fetch\nall whom the young Damascene had named,and when they came, he\nbade smite their necks: he also summoned him whom they called\nAhmad and who had been the means of the young man's deliverance a\nfirst time and a second, and thanked him and showed him favour\nand bestowed on him a costly robe of honour and made him Chief of\nPolice in his city.[FN#339] Then he sent for the Shaykh, the\nMuezzin, and when the messenger came to him and told him that the\nCommander of the Faithfull summoned him, he feared the\ndenunciation of the damsel and walked with him to the palace,\nfarting for fear as he went, whilst all who passed him by laughed\nat him. When he came into the presence of the Commander of the\nFaithful, he fell a-trembling and his tongue was tied,[FN#340] so\nthat he could not speak. The Caliph smiled at him and said, \"O\nShaykh, thou hast done no offence; so why fearest thou?\" Answered\nthe old man (and indeed he was in the sorest of that which may be\nof fear), \"O my lord, by the virtue of thy pure forefathers,\nindeed I have done naught, and do thou enquire of my manners and\nmorals.\" The Caliph laughed at him and ordering him a thousand\ndinars, bestowed on him a costly robe of honour and made him\nheadman of the Muezzins in his mosque. Then he called Sitt\nal-Milah and said to her, \"The house wherein thou lodgest with\nall it containeth is a largesse to thy lord: so do thou take him\nand depart with him in the safeguard of Allah Almighty; but\nabsent not yourselves from our presence.\" Accordingly she went\nforth with the young Damascene and when she came to the house,\nshe found that the Prince of True Believers had sent them gifts\ngalore and good things in store. As for Nur al-Din, he sent for\nhis father and mother and appointed for himself agents in the\ncity of Damascus, to receive the rent of the houses and gardens\nand Wakalahs and Hammams; and they occupied themseves with\ncollecting that which accrued to him and sending it to him every\nyear. Meanwhile, his father and mother came to him, with that\nwhich they had of monies and merchandise of price and,\nforegathering with their son, found that he was become of the\nchief officers and familiars of the Commander of the Faithful and\nof the number of his sitting-companions and nightly entertainers,\nwherefore they rejoiced in reunion with him and he also rejoiced\nin them. The Caliph assigned them solde and allowances; and as\nfor Nur al-Din, his father brought him those riches and his\nwealth waxed and his estate was stablished, till he became the\nrichest of the folk of his time in Baghdad and left not the\npresence of the Commander of the Faithful or by night or by day.\nHe was vouchsafed issue by Sitt al-Milah, and he ceased not to\nlive the goodliest of lives, he and she and his father and his\nmother, a while of time, till Abu al-Hasan sickened of a sore\nsickness and departed to the mercy of Allah Almighty. Presently,\nhis mother also died and he carried them forth and shrouded them\nand buried and made them expiations and funeral\nceremonies.[FN#341] In due course his children grew up and became\nlike moons, and he reared them in splendour and affection, while\nhis wealth waxed and his case never waned. He ceased not to pay\nfrequent visits to the Commander of the Faithful, he and his\nchildren and his slave-girl Sitt al-Milah, and they abode in all\nsolace of life and prosperity till there came to them the\nDestroyer of delights and the Sunderer of societies; and laud to\nthe Abiding, the Eternal! This is all that hath come down to us\nof their story.\n\n\n\n\n           TALE OF KING INS BIN KAYS AND HIS DAUGHTER\n            WITH THE SON OF KING AL-'ABBAS.[FN#342]\n\n\n\nThere was once, in days of yore and in ages and times long gone\nbefore, in the city of Baghdad, the House of Peace, a king mighty\nof estate, lord of understanding and beneficence and generosity\nand munificence, and he was strong of sultanate and endowed with\nmight and majesty and magnificence. His name was Ins bin Kays bin\nRab\u00ed' al-Shayb\u00e1n\u00ed,[FN#343] and when he took horse, there rode\nabout him riders from the farthest parts of the two\nIraks.[FN#344] Almighty Allah decreed that he should take to wife\na woman hight 'Af\u00edfah, daughter of Asad al-Sund\u00fasi, who was\nendowed with beauty and loveliness and brightness and perfect\ngrace and symmetry of shape and stature; her face was like the\ncrescent moon and she had eyes as they were gazelle's eyes and an\naquiline nose like Luna's cymb. She had learned cavalarice and\nthe use of arms and had mastered the sciences of the Arabs; eke\nshe had gotten by heart all the dragomanish[FN#345] tongues and\nindeed she was a ravishment to mankind. She abode with Ins bin\nKays twelve years, during which time he was not blessed with\nchildren by her; so his breast was straitened by reason of the\nfailure of lineage, and he besought his Lord to vouchsafe him a\nson. Accordingly the queen conceived, by permission of Allah\nAlmighty; and when the days of her pregnancy were accomplished,\nshe gave birth to a maid-child, than whom never saw eyes fairer,\nfor that her face was as it were a pearl pure-bright or a lamp\nraying light or a candle gilt with gold or a full moon breaking\ncloudy fold, extolled be He who her from vile water dight and\nmade her to the beholders a delight! When her father saw her in\nthis fashion of loveliness, his reason fled for joy, and when she\ngrew up, he taught her writing and belles-lettres and philosophy\nand all manner of tongues. So she excelled the folk of her time\nand surpassed her peers; and the sons of the kings heard of her\nand all of them longed to look upon her. The first who sought her\nto wife was King Nabh\u00e1n[FN#346] of Mosul, who came to her with a\ngreat company, bringing an hundred she-camels, laden with musk\nand lign-aloes and ambergris and five score loaded with camphor\nand jewels and other hundred laden with silver monies and yet\nother hundred loaded with raiment of silken stuffs, sendal and\nbrocade, besides an hundred slave-girls and a century of choice\nsteeds of swift and generous breeds, completely housed and\naccoutred, as they were brides; and all this he had laid before\nher father, demanding her of him in wedlock. Now King Ins bin\nKays had bound himself by an oath that he would not marry his\ndaughter save to him whom she should choose; so, when King Nabhan\nsought her in marriage, her father went in to her and consulted\nher concerning his affair. She consented not and he repeated to\nNabhan that which she said, whereupon he departed from him. After\nthis came King Bahr\u00e1m, lord of the White Island, with treasures\nricher than the first; but she accepted not of him and he\nreturned disappointed; nor did the kings cease coming to her\nsire, on her account, one after other, from the farthest of the\nlands and the climes, each glorying in bringing more than those\nwho forewent him; but she heeded not any one of them. Presently,\nAl-'Abb\u00e1s, son of King Al-'Az\u00edz, lord of the land of Al-Yaman and\nZab\u00edd\u00fan[FN#347] and Meccah (which Allah increase in honour and\nbrightness and beauty!) heard of her; and he was of the great\nones of Meccah and Al-Hij\u00e1z,[FN#348] and was a youth without hair\non his side-face. So he presented himself one day in his sire's\nassembly, whereupon the folk made way for him and the king seated\nhim on a chair of red gold, crusted with pearls and gems. The\nPrince sat, with his head bowed ground-wards, and spake not to\nany: whereby his father knew that his breast was straitened and\nbade the cup-companions and men of wit relate marvellous\nhistories, such as beseem the sessions of kings; nor was there\none of them but spoke forth the goodliest of that which was with\nhim; but Al-'Abb\u00e1s still abode with his head bowed down. Then the\nking bade his sitting-companions withdraw, and when the chamber\nwas private, he looked at his son and said to him, \"By Allah,\nthou cheerest me with thy coming in to me and chagrinest me for\nthat thou payest no heed to any of the familiars nor of the\ncup-companions. What is the cause of this?\" Answered the Prince,\n\"O my papa, I have heard tell that in the land of Al-Ir\u00e1k is a\nwoman of the daughters of the kings, and her father is called\nKing Ins bin Kays, lord of Baghdad; she is famed for beauty and\nloveliness and brightness and perfect grace, and indeed many of\nthe kings have sought her in marriage; but her soul consented not\nunto any one of them. Wherefore my thought prompteth me to travel\nherwards, for that my heart cleaveth to her, and I beseech thee\nsuffer me to go to her.\" His sire replied, \"O my son, thou\nknowest that I have none other than thyself of children and thou\nart the coolth of mine eyes and the fruit of my vitals; nay, I\ncannot brook to be parted from thee a single hour and I purpose\nto seat thee on the throne of the kingship and espouse thee to\none of the daughters of the kings, who shall be fairer than she.\"\nAl-Abbas gave ear to his father's word and dared not gainsay him;\nwherefore he abode with him awhile, whilst the love-fire raged in\nhis vitals. Then the king took rede with himself to build his son\na Hammam and adorn it with various paintings, so he might display\nit to him and divert him with the sight thereof, to the intent\nthat his body might be solaced thereby and that the accident of\ntravel might cease from him and he be turned from his purpose of\nremoval from his parents. Presently he addressed himself to the\nbuilding of the bath and assembling architects and artisans from\nall his cities and citadels and islands, assigned them a\nfoundation-site and marked out its boundaries. Then the workmen\noccupied themselves with the building of the Hammam and the\nordinance and adornment of its cabinets and roofs. They used\npaints and precious minerals of all kinds, according to the\ncontrast of their colours, red and green and blue and yellow and\nwhat not else of all manner tincts; and each artisan wrought at\nhis craft and each painter at his art, whilst the rest of the\nfolk busied themselves with transporting thither vari-\nstones. One day, as the Master-painter wrought at his work, there\ncame in to him a poor man, who looked long upon him and observed\nhis mystery; whereupon quoth the artist to him, \"Knowest thou\naught of painting?\" Quoth the stranger, \"Yes;\" so he gave him\ntools and paints and said to him, \"Limn for us a rare semblance.\"\nAccordingly the pauper stranger entered one of the bath-chambers\nand drew on its walls a double border, which he adorned on both\nsides, after a fashion than which eyes never saw a fairer.\nMoreover, amiddlemost the chamber he limned a picture to which\nthere lacked but the breath,[FN#349] and it was the portraiture\nof Mariyah, daughter to the king of Baghdad. Then, when he had\nfinished the portrait, he went his way and told none of what he\nhad done, nor knew any wight the chambers and doors of the bath\nand the adornment and ordinance thereof. Presently the chief\nartisan came to the palace and sought audience of the king who\nbade admit him. So he entered and kissing the earth, saluted him\nwith a salam beseeming Sultans and said, \"O king of the time and\nlord of the age and the tide, may prosperity endure to thee and\nacceptance and eke thy degree over all the kings both morning and\nevening[FN#350] exalted be! The work of the bath is accomplished,\nby the king's fair fortune and the purity of his purpose, and\nindeed, we have done all that behoved us and there remaineth but\nthat which behoveth the king.\" Al-Aziz ordered him a costly robe\nof honour and expended monies galore, giving unto each who had\nwroughten after the measure of his work. Then he assembled in the\nHammam all the Lords of his realm, Emirs and Wazirs and\nChamberlains and Nabobs, and the chief officers of his kingdom\nand household, and sending for his son Al-Abbas, said to him, \"O\nmy son, I have builded thee a bath, wherein thou mayst take thy\npleasance; so enter that thou mayst see it and divert thyself by\ngazing upon it and viewing the beauty of its ordinance and\ndecoration.\" \"With love and gladness,\" replied the Prince and\nentered the bath, he and the king and the folk about them, so\nthey might divert themselves with viewing that which the\nworkmen's hands had worked. Al-Abbas went in and passed from\nplace to place and chamber to chamber, till he came to the room\naforesaid and espied the portrait of Mariyah, whereupon he fell\ndown in a fainting-fit and the workmen went to his father and\nsaid to him, \"Thy son Al-Abbas hath swooned away.\" So the king\ncame and finding his son cast down, seated himself at his head\nand bathed his face with rose-water. After awhile he revived and\nthe king said to him, \"I seek refuge with Allah for thee, O my\nson! What accident hath befallen thee?\" The Prince replied, \"O my\nfather, I did but look on yonder picture and it bequeathed me a\nthousand qualms and there befel me that which thou beholdest.\"\nTherewith the king bade fetch the Master-painter, and when he\nstood before him, he said to him, \"Tell me of yonder portrait and\nwhat girl is this of the daughters of the kings; else I will take\nthy head.\" Said the painter, \"By Allah, O king, I limned it not,\nneither know I who she is; but there came to me a poor man and\nlooked hard at me. So I asked him, Knowest thou the art of\npainting? and he answered, Yes. Whereupon I gave him the gear and\nsaid to him, Limn for us a rare semblance. Accordingly he painted\nyonder portrait and went away and I wot him not neither have I\never set eyes on him save that day.\" Hearing this, the king\nordered all his officers to go round about in the thoroughfares\nand colleges and to bring before him all strangers they found\nthere. So they went forth and brought him much people, amongst\nwhom was the pauper who had painted the portrait. When they came\ninto the presence, the Sultan bade the crier make public\nproclamation that whoso wrought the portrait should discover\nhimself and have whatso he wished. Thereupon the poor man came\nforward and kissing the ground before the king, said to him, \"O\nking of the age, I am he who limned yonder likeness.\" Quoth\nAl-Aziz, \"And knowest thou who she is?\" and quoth the other,\n\"Yes, this is the portrait of Mariyah, daughter of the king of\nBaghdad.\" The king ordered him a robe of honour and a slave-girl\nand he went his way. Then said Al-Abbas, \"O my papa, give me\nleave to seek her, so I may look upon her: else shall I farewell\nthe world, withouten fail.\" The king his father wept and\nanswered, \"O my son, I builded thee a Hammam, that it might turn\nthee from leaving me, and behold, it hath been the cause of thy\ngoing forth; but the behest of Allah is a determinate\ndecree.\"[FN#351] Then he wept again and Al-Abbas said to him,\n\"Fear not for me, for thou knowest my prowess and puissance in\nreturning answers in the assemblies of the land and my good\nbreeding and accomplishments together with my skill in rhetoric;\nand indeed for him whose father thou art and whom thou hast\nreared and bred and in whom thou hast united praiseworthy\nqualities, the repute whereof hath traversed the East and the\nWest, thou needest not fear aught, more especially as I purpose\nbut to seek pleasuring and return to thee, an it be the will of\nAllah Almighty.\" Quoth the king, \"Whom wilt thou take with thee\nof attendants and what of monies?\" Replied Al-Abbas, \"O my papa,\nI have no need of horses or camels or weapons, for I purpose not\nwarfare, and I will have none go forth with me save my page 'Amir\nand no more.\" Now as he and his father were thus engaged in talk,\nin came his mother and caught hold of him; and he said to her,\n\"Allah upon thee, let me gang my gait and strive not to divert me\nfrom what purpose I have purposed, for needs must I go.\" She\nreplied, \"O my son, if it must be so and there be no help for it,\nswear to me that thou wilt not be absent from me more than a\nyear.\" And he sware to her. Then he entered his father's\ntreasuries and took therefrom what he would of jewels and\njacinths and everything weighty of worth and light of load: he\nalso bade his servant Amir saddle him two steeds and the like for\nhimself, and whenas the night beset his back,[FN#352] he rose\nfrom his couch and mounting his horse, set out for Baghdad, he\nand Amir, whilst the page knew not whither he intended.[FN#353]\nHe gave not over going and the journey was joyous to him, till\nthey came to a goodly land, abounding in birds and wild beasts,\nwhereupon Al-Abbas started a gazelle and shot it with a shaft.\nThen he dismounted and cutting its throat, said to his servant,\n\"Alight thou and skin it and carry it to the water.\" Amir\nanswered him with \"Hearkening and obedience\" and going down to\nthe water, built a fire and broiled the gazelle's flesh. Then\nthey ate their fill and drank of the water, after which they\nmounted again and fared on with diligent faring, and Amir still\nunknowing whither Al-Abbas was minded to wend. So he said to him,\n\"O my lord, I conjure thee by Allah of All-might, wilt thou not\ntell me whither thou intendest?\" Al-Abbas looked at him and in\nreply improvised these couplets,\n\n\"In my vitals are fires of desire and repine; * And naught I\n     reply when they flare on high:\nBaghdad-wards I hie me on life-and-death work, * Loving one who\n     distorts my right judgment awry:\nA swift camel under me shortcuts the wold * And deem it a cloud\n     all who nearhand espy:\nO '\u00c1mir make haste after model of her * Who would heal mine ill\n     and Love's cup drain dry:\nFor the leven of love burns the vitals of me; * So with me seek\n     my tribe and stint all reply.\"\n\nWhen Amir heard his lord's verses, he knew that he was a slave of\nlove and that she whom he loved abode in Baghdad. Then they fared\non night and day, traversing plain and stony way, till they\nsighted Baghdad and lighted down in its environs[FN#354] and\nthere lay their night. When they arose in the morning, they\nremoved to the bank of the Tigris where they encamped and\nsojourned a second day and a third. As they abode thus on the\nfourth day, behold, a company of folk giving their beasts the\nrein and crying aloud and saying, \"Quick! Quick! Haste to our\nrescue, Ho thou the King!\" Therewith the King's chamberlains and\nofficers accosted them and said, \"What is behind you and what\nhath betided you?\" Quoth they, \"Bring us before the King.\" So\nthey carried them to Ins bin Kays; and when they saw him, they\nsaid to him, \"O king, unless thou succour us, we are dead men;\nfor that we are a folk of the Ban\u00fa Shayb\u00e1n,[FN#355] who have\ntaken up our abode in the parts of Bassorah, and Hodhayfah the\nwild Arab hath come down on us with his steeds and his men and\nhath slain our horsemen and carried off our women and children;\nnor was one saved of the tribe but he who fled; wherefore we\ncrave help first by Allah Almighty, then by thy life.\" When the\nking heard their speech, he bade the crier proclaim in the\nhighways of the city that the troops should busk them to march\nand that the horsemen should mount and the footmen fare forth;\nnor was it but the twinkling of the eye ere the kettle-drums beat\nand the trumpets blared; and scarce was the forenoon of the day\npassed when the city was blocked with horse and foot. Presently,\nthe king reviewed them and behold, they were four-and-twenty\nthousand in number, cavalry and infantry. He bade them go forth\nto the enemy and gave the command of them to Sa'ad ibn al-W\u00e1kid\u00ed,\na doughty cavalier and a dauntless champion; so the horsemen set\nout and fared on along the Tigris-bank. Al-Abbas, son of King\nAl-Aziz, looked at them and saw the flags flaunting and the\nstandards stirring and heard the kettle-drums beating; so he bade\nhis page saddle him a blood-steed and look to the surcingles and\nbring him his harness of war, for indeed horsemanship[FN#356] was\nrooted in his heart. Quoth Amir, \"And indeed I saw Al-Abbas his\neyes waxed red and the hair of his hands on end.\" So he mounted\nhis charger, whilst Amir also bestrode a destrier, and they went\nforth with the commando and fared on two days. On the third day,\nafter the hour of the mid-afternoon prayer, they came in sight of\nthe foe and the two armies met and the two ranks joined in fight.\nThe strife raged amain and sore was the strain, whilst the dust\nrose in clouds and hung in vaulted shrouds, so that all eyes were\nblinded; and they ceased not from the battle till the night\novertook them,[FN#357] when the two hosts drew off from the\nmellay and passed the night, perplexed concerning themselves.\nWhen Allah caused the morning to morrow, the two hosts were\naligned in line and their thousands fixed their eyne and the\ntroops stood looking one at other. Then sallied forth Al-H\u00e1ris\nibn Sa'ad between the two lines and played with his lance and\ncried out and improvised these couplets,\n\n\"You are in every way this day our prey; * And ever we pray\u00e8d\n     your sight to see:\nThe Ruthful drave you Hodhayfah-wards * To the Brave, the Lion\n     who sways the free:\nSay, amid you's a man who would heal his ills, * With whose lust\n     of battle shrewd blows agree?\nThen by Allah meet me who come to you * And whoso is wronged\n     shall the wronger be.\"[FN#358]\n\nThereupon there sallied forth to him Zuhayr bin Hab\u00edb, and they\nwheeled about and wiled a while, then they exchanged strokes.\nAl-Haris forewent his foe in smiting and stretched him weltering\nin his gore; whereupon Hodhayfah cried out to him, \"Gifted of\nAllah[FN#359] art thou, O Haris! Call out another of them.\" So he\ncried aloud, \"I say, who be a champion?\" But they of Baghdad held\nback from him; and when it appeared to Al-Haris that\nconsternation was amongst them, he charged down upon them and\noverrolled the first of them upon the last of them and slew of\nthem twelve men. Then the evening caught him and the Baghdadis\nbegan addressing themselves to flight. No sooner had the morning\nmorrowed than they found themselves reduced to a fourth part of\ntheir number and there was not one of them had dismounted from\nhis horse. Wherefore they made sure of destruction and Hodhayfah\nrushed out between the two lines (now he was reckoned good for a\nthousand knights) and cried out, \"Harkye, my masters of Baghdad!\nLet none come forth to me but your Emir, so I may talk with him\nand he with me; and he shall meet me in combat singular and I\nwill meet him, and may he who is clear of offence come off safe.\"\nThen he repeated his words and said, \"How is it I see your Emir\nrefuse me a reply?\" But Sa'ad, the Emir of the army of Baghdad,\nanswered him not, and indeed his teeth chattered in his mouth,\nwhen he heard him summon him to the duello. Now when Al-Abbas\nheard Hodhayfah's challenge and saw Sa'ad in this case, he came\nup to the Emir and asked him, \"Wilt thou suffer me to answer him\nand I will be thy substitute in replying him and in monomachy\nwith him and will make my life thy sacrifice?\" Sa'ad looked at\nhim and seeing knighthood shining from between his eyes, said to\nhim, \"O youth, by the virtue of Mustaf\u00e0 the Chosen Prophet (whom\nAllah save and assain), tell me who thou art and whence thou\ncomest to bring us victory.\"[FN#360] Quoth the Prince, \"This is\nno place for questioning;\" and quoth Sa'ad to him, \"O Knight, up\nand at Hodhayfah! Yet, if his Satan prove too strong for thee,\nafflict not thyself on thy youth.\"[FN#361] Al-Abbas cried, \"Allah\nis He of whom help is to be sought;\"[FN#362] and, taking his\narms, fortified his purpose and went down into the field, as he\nwere a fort of the forts or a mountain's contrefort. Thereupon\nHodhayfah cried out to him, saying, \"Haste thee not, O youth! Who\nart thou of the folk?\" He replied, \"I am Sa'ad ibn al-Wakidi,\ncommander of the host of King Ins, and but for thy pride in\nchallenging me, I had not come forth to thee; for thou art no\npeer for me to front nor as mine equal dost thou count nor canst\nthou bear my brunt. Wherefore get thee ready for the last\nmarch[FN#363] seeing that there abideth but a little of thy\nlife.\" When Hodhayfah heard this speech, he threw himself\nbackwards,[FN#364] as if in mockery of him, whereat Al-Abbas was\nwroth and called out to him, saying, \"O Hodhayfah, guard thyself\nagainst me.\" Then he rushed upon him, as he were a swooper of the\nJinn,[FN#365] and Hodhayfah met him and they wheeled about a long\nwhile. Presently, Al-Abbas cried out at Hodhayfah a cry which\nastounded him and struck him a stroke, saying, \"Take this from\nthe hand of a brave who feareth not the like of thee.\" Hodhayfah\nmet the sabre-sway with his shield, thinking to ward it off from\nhim; but the blade shore the target in sunder and descending upon\nhis shoulder, came forth gleaming from the tendons of his throat\nand severed his arm at the armpit; whereupon he fell down,\nwallowing in his blood, and Al-Abbas turned upon his host; not\nhad the sun departed the dome of the welkin ere Hodhayfah's army\nwas in full flight before Al-Abbas and the saddles were empty of\nmen. Quoth Sa'ad, \"By the virtue of Mustafa the Chosen Prophet,\nwhom Allah save and assain, I saw Al-Abbas with the blood upon\nhis saddle-pads, in clots like camels' livers, smiting with the\nsword right and left, till he scattered them abroad in every\ngorge and wold; and when he hied him back to the camp, the men of\nBaghdad were fearful of him.\" But as soon as they saw this\nvictory which had betided them over their foes, they turned back\nand gathering together the weapons and treasures and horses of\nthose they had slain, returned to Baghdad, victorious, and all by\nthe knightly valour of Al-Abbas. As for Sa'ad, he foregathered\nwith his lord, and they fared on in company till they came to the\nplace where Al-Abbas had taken horse, whereupon the Prince\ndismounted from his charger and Sa'ad said to him, \"O youth,\nwherefore alightest thou in other than thy place? Indeed, thy\nrights be incumbent upon us and upon our Sultan; so go thou with\nus to the dwellings, that we may ransom thee with our souls.\"\nReplied Al-Abbas, \"O Emir Sa'ad, from this place I took horse\nwith thee and herein is my lodging. So, Allah upon thee, mention\nnot me to the king, but make as if thou hadst never seen me\nbecause I am a stranger in the land.\" So saying, he turned away\nfrom him and Sa'ad fared on to his palace, where he found all the\ncourtiers in attendance on the king and recounting to him that\nwhich had betided them with Al-Abbas. Quoth the king, \"Where is\nhe?\" and quoth they, \"He is with the Emir Sa'ad.\" So, when the\nEmir entered, the king looked, but found none with him; and\nSa'ad, seeing at a glance that he longed to look upon the youth,\ncried out to him, saying, \"Allah prolong the king's days! Indeed,\nhe refuseth to present himself before thee, without order or\nleave.\" Asked the king, \"O Sa'ad, whence cometh this man?\" and\nthe Emir answered, \"O my lord, I know not; but he is a youth fair\nof favour, amiable of aspect, accomplished in address, ready of\nrepartee, and valour shineth from between his eyes.\" Quoth the\nking, \"O Sa'ad, fetch him to me, for indeed thou describest to me\nat full length a mighty matter.\"[FN#366] And he answered, saying,\n\"By Allah, O my lord, hadst thou but seen our case with\nHodhayfah, when he challenged me to the field of fight and the\nstead of cut-and-thrust and I held back from doing battle with\nhim! Then, as I thought to go forth to him, behold, a knight gave\nloose to his bridle-rein and called out to me, saying, 'O Sa'ad,\nwilt thou suffer me to be thy substitute in waging war with him\nand I will ransom thee with myself?' and quoth I, 'By Allah, O\nyouth, whence comest thou?' and quoth he, 'This be no time for\nthy questions, while Hodhayfah standeth awaiting thee.'\"\nThereupon he repeated to the king all that had passed between\nhimself and Al-Abbas from first to last; whereat cried Ins bin\nKays, \"Bring him to me in haste, so we may learn his tidings and\nquestion him of his case.\" \"'Tis well,\" replied Sa'ad, and going\nforth of the king's presence, repaired to his own house, where he\ndoffed his war-harness and took rest for himself. On this wise\nfared it with the Emir Sa'ad, but as regards Al-Abbas, when he\ndismounted from his destrier, he doffed his war-gear and reposed\nhimself awhile; after which he brought out a body-dress of\nVenetian[FN#367] silk and a gown of green damask and donning\nthem, bound about his head a turband of Damietta stuff and zoned\nhis waist with a kerchief. Then he went out a-walking in the\nhighways of Baghdad and fared on till he came to the bazar of the\ntraders. There he found a merchant, with chess before him; so the\nPrince stood watching him, and presently the other looked up at\nhim and asked him, \"O youth, what wilt thou bet upon the game?\"\nHe answered, \"Be it thine to decide.\" Said the merchant, \"Then be\nit an hundred dinars,\" and Al-Abbas consented to him; whereupon\nquoth he, \"Produce the money, O youth, so the game may be fairly\nstablished.\" Accordingly Al-Abbas brought out a satin purse,\nwherein were a thousand dinars, and laid down an hundred dinars\ntherefrom on the edge of the carpet, whilst the merchant produced\nthe like, and indeed his reason fled for joy when he saw the gold\nin possession of Al-Abbas. The folk flocked about them, to divert\nthemselves with watching the play, and they called the bystanders\nto witness the wager and after the stakes were duly staked, the\ntwain fell a-playing. Al-Abbas forebore the merchant, so he might\nlead him on, and dallied with him a full hour; and the merchant\nwon and took of him the hundred dinars. Then said the Prince,\n\"Wilt thou play another partie?\" and the other said, \"O youth, I\nwill not play again, save for a thousand dinars.\" Quoth the\nyouth, 'Whatsoever thou stakest, I will match thy stake with its\nlike.\" So the merchant brought out a thousand dinars and the\nPrince covered them with other thousand. Then the game began, but\nAl-Abbas was not long with him ere he beat him in the house of\nthe elephant[FN#368] nor did he cease to do thus till he had\nbeaten him four times and won of him four thousand dinars. This\nwas all the merchant had of money; so he said, \"O youth, I will\nplay thee another game for the shop.\" Now the value of the shop\nwas four thousand dinars; so they played and Al-Ahbas beat him\nand won his shop, with whatso was therein; upon which the other\narose, shaking his clothes,[FN#369] and said to him, \"Up, O\nyouth, and take thy shop.\" Accordingly Al-Abbas arose and\nrepairing to the shop, took possession thereof, after which he\nreturned to the place where he had left his servant 'Amir, and\nfound there the Emir Sa'ad, who was come to bid him to the\npresence of the king. The Prince consented to this and\naccompanied him till they came before King Ins bin Kays,\nwhereupon he kissed the ground and saluted him and\nexaggerated[FN#370] the salutation. So the king asked him,\n\"Whence comest thou, O youth, and whither goest thou?\" and he\nanswered, \"I come from Al-Yaman.\" Then said the king, \"Hast thou\na need we may fulfil to thee; for indeed thou hast strong claims\nto our favour after that which thou didst in the matter of\nHodhayfah and his folk.\" And he commanded to cast over him a\nmantle of Egyptian satin, worth an hundred dinars. He also bade\nhis treasurer give him a thousand dinars and said to him, \"O\nyouth, take this in part of that which thou deservest of us; and\nif thou prolong thy sojourn with us, we will give thee slaves and\nservants.\" Al-Abbas kissed ground and said, \"O king, Allah grant\nthee abiding weal, I deserve not all this.\" Then he put his hand\nto his pouch and pulling out two caskets of gold, in each of\nwhich were rubies whose value none could estimate, gave them to\nthe king, saying, \"O king, Allah cause thy welfare to endure, I\nconjure thee by that which the Almighty hath vouchsafed thee,\nheal my heart by accepting these two caskets, even as I have\naccepted thy present.\" So the king accepted the two caskets and\nAl-Abbas took his leave and went away to the bazar. Now when the\nmerchants saw him, they accosted him and said, \"O youth, wilt\nthou not open thy shop?\" As they were addressing him, up came a\nwoman, having with her a boy bare of head, and stood looking at\nAl-Abbas, till he turned to her, when she said to him, \"O youth,\nI conjure thee by Allah, look at this boy and have ruth on him,\nfor that his father hath forgotten his skull-cap in the shop he\nlost to thee; so, an thou see fit to give it him, thy reward be\nwith Allah! For indeed the child maketh our hearts ache with his\nexcessive weeping, and the Lord be witness for us that, had they\nleft us aught wherewith to buy him a cap in its stead, we had not\nsought it of thee.\" Replied Al-Abbas, \"O adornment of\nwomankind,[FN#371] indeed, thou bespeakest me with thy fair\nspeech and supplicatest me with thy goodly words! But bring me\nthy husband.\" So she went and fetched the merchant, whilst a\ncrowd collected to see what Al-Abbas would do. When the man came,\nhe returned him the gold he had won of him, art and part, and\ndelivered him the keys of the shop, saying, \"Requite us with thy\npious prayers.\" Therewith the woman came up to him and kissed his\nfeet, and in like fashion did the merchant her husband: and all\nwho were present blessed him, and there was no talk but of\nAl-Abbas. Thus fared it with him; but as for the merchant, he\nbought him a head of sheep[FN#372] and slaughtering it, roasted\nit and dressed birds and other meats of various kinds and colours\nand purchased dessert and sweetmeats and fresh fruits; then he\nrepaired to Al-Abbas and conjured him to accept of his\nhospitality and visit his home and eat of his provaunt. The\nPrince consented to his wishes and went with him till they came\nto his house, when the merchant bade him enter: so Al-Abbas went\nin and saw a goodly house, wherein was a handsome saloon, with a\nvaulted ceiling. When he entered the saloon, he found that the\nmerchant had made ready food and dessert and perfumes, such as\nmay not be described; and indeed he had adorned the table with\nsweet-scented flowers and sprinkled musk and rose-water upon the\nfood; and he had smeared the saloon walls with ambergris and had\nburned aloes-wood therein and Nadd. Presently, Al-Abbas looked\nout of the window of the saloon and saw by its side a house of\ngoodly ordinance, tall of base and wide of space, with rooms\nmanifold and two upper stories crowning the whole; but therein\nwas no sign of inhabitants. So he said to the merchant, \"Verily,\nthou exaggeratest in doing us honour; but, by Allah, I will not\neat of thy meat until thou tell me what hath caused the voidance\nof yonder house.\" Said he, \"O my lord, that was Al-Ghitrif's\nhouse and he passed away to the mercy of the Almighty and left no\nheir save myself; whereupon the mansion became mine, and by\nAllah, an thou have a mind to sojourn in Baghdad, take up thine\nabode in this house, whereby thou mayst be in my neighbourhood;\nfor that verily my heart inclineth unto thee with affection and I\nwould have thee never absent from mine eyes, so I may still have\nmy fill of thee and hearken to thy speech.\" Al-Abbas thanked him\nand said to him, \"By Allah, thou art indeed friendly in thy\nconverse and thou exaggeratest in thy discourse, and needs must I\nsojourn in Baghdad. As for the house, if it please thee to lodge\nme, I will abide therein; so accept of me its price.\" Therewith\nhe put hand to his pouch and bringing out from it three hundred\ndinars, gave them to the merchant, who said in himself, \"Unless I\ntake his dirhams, he will not darken my doors.\" So he pocketed\nthe monies and sold him the mansion, taking witnesses against\nhimself of the sale. Then he arose and set food before Al-Abbas\nand they sat down to his good things; after which he brought him\ndessert and sweetmeats whereof they ate their sufficiency, and\nwhen the tables were removed they washed their hands with musked\nrose-water and willow-water. Then the merchant brought Al-Abbas a\nnapkin scented with the smoke of aloes-wood, on which he wiped\nhis right hand, and said to him, \"O my lord, the house is become\nthy house; so bid thy page transport thither the horses and arms\nand stuffs.\" The Prince did this and the merchant rejoiced in his\nneighbourhood and left him not night nor day,[FN#373] so that\nAl-Abbas said to him, \"By the Lord, we distract thee from thy\nlivelihood.\" He replied, \"Allah upon thee, O my lord, name not to\nme aught of this, or thou wilt break my heart, for the best of\ntraffic art thou and the best of livelihood.\" So there befel\nstraight friendship between them and all ceremony was laid aside.\nMeanwhile[FN#374] the king said to his Wazir, \"How shall we do in\nthe matter of yonder youth, the Yam\u00e1ni, on whom we thought to\nconfer gifts, but he hath gifted us with tenfold our largesse and\nmore, and we know not an he be a sojourner with us or not?\" Then\nhe went into the Harim and gave the rubies to his wife Afifah,\nwho asked him, \"What is the worth of these with thee and with\nother of the kings?\" Quoth he, \"They are not to be found save\nwith the greatest of sovrans nor can any price them with monies.\"\nQuoth she, \"Whence gottest thou them?\" So he recounted to her the\nstory of Al-Abbas from beginning to end, and she said, \"By Allah,\nthe claims of honour are imperative on us and the King hath\nfallen short of his devoir; for that we have not seen him bid the\nyouth to his assembly, nor hath he seated him on his left hand.\"\nWhen the king heard his wife's words, it was as if he had been\nasleep and awoke; so he went forth the Harim and bade kill\npoultry and dress meats of every kind and colour. Moreover, he\nassembled all his courtiers and let bring sweetmeats and dessert\nand all that beseemeth the tables of kings. Then he adorned his\npalace and despatched after Al-Abbas a man of the chief officers\nof his household, who found him coming forth of the Hammam, clad\nin a jerkin[FN#375] of fine goats' hair and over it a Baghd\u00e1di\nscarf; his waist was girt with a Rustaki[FN#376] kerchief and on\nhis head he wore a light turband of Damietta[FN#377] stuff. The\nmessenger wished him joy of the bath and exaggerated in doing him\nhonour: then he said to him, \"The king biddeth thee in\nweal.\"[FN#378] \"To hear is to obey,\" quoth Al-Abbas and\naccompanied the officer to the king's palace. Now Afifah and her\ndaughter Mariyah were behind the curtain, both looking at him;\nand when he came before the sovran he saluted him and greeted him\nwith the greeting of kings, whilst all present gazed at him and\nat his beauty and loveliness and perfect grace. The king seated\nhim at the head of the table; and when Afifah saw him and\nconsidered him straitly, she said, \"By the virtue of Mohammed,\nprince of the Apostles, this youth is of the sons of the kings\nand cometh not to these parts save for some noble purpose!\" Then\nshe looked at Mariyah and saw that her favour was changed, and\nindeed her eye-balls were as dead in her face and she turned not\nher gaze from Al-Abbas a twinkling of the eyes, for that the love\nof him had sunk deep into her heart. When the queen saw what had\nbefallen her daughter, she feared for her from reproach\nconcerning Al-Abbas; so she shut the casement-wicket that the\nPrincess might not look upon him any more. Now there was a\npavilion set apart for Mariyah, and therein were boudoirs and\nbowers, balconies and lattices, and she had with her a nurse, who\nserved her as is the fashion with the daughters of the Kings.\nWhen the banquet was ended and the folk had dispersed, the King\nsaid to Al-Abbas, \"I would fain have thee abide with me and I\nwill buy thee a mansion, so haply we may requite thee for thy\nhigh services; and indeed imperative upon us is thy due and\nmagnified in our eyes is thy work; and soothly we have fallen\nshort of thy deserts in the matter of distance.\"[FN#379] When the\nyouth heard the king's speech, he rose and sat down[FN#380] and\nkissing ground, returned thanks for his bounty and said, \"I am\nthe King's thrall, wheresoever I may be, and under his eye.\" Then\nhe told him the tale of the merchant and the manner of the buying\nof the house, and the king said, \"In very truth I would fain have\nhad thee in my neighbourhood and by side of me.\" Presently\nAl-Abbas took leave of the king and went away to his own house.\nNow it chanced that he passed under the palace of Mariyah, the\nking's daughter, and she was sitting at a casement. He happened\nto look round and his eyes met those of the Princess, whereupon\nhis wit departed and he was ready to swoon away, whilst his\ncolour changed, and he said, \"Verily, we are Allah's and unto Him\nare we returning!\" But he feared for himself lest severance\nbetide him; so he concealed his secret and discovered not his\ncase to any of the creatures of Allah Almighty. When he reached\nhis quarters, his page Amir said to him, \"I seek refuge for thee\nwith Allah, O my lord, from change of colour! Hath there betided\nthee a pain from the Lord of All-might or aught of vexation? In\ngood sooth, sickness hath an end and patience doeth away\ntrouble.\" But the Prince returned him no answer. Then he brought\nout ink-case[FN#381] and paper and wrote these couplets:\n\nI cry (and mine's a frame that pines alw\u00e0y), * A mind which fires\n     of passion e'er waylay;\nAnd eyeballs never tasting sweets of sleep; * Yet Fortune spare\n     its cause I ever pray!\nWhile from world-perfidy and parting I * Like Bishram with\n     Hind,[FN#382] that well-loved may;--\nYea, grown a bye-word 'mid the folk but aye * Spend life\n     unwinning wish or night or day.\n\"Ah say, wots she my love when her I spied * At the high lattice\n     shedding sunlike ray?\"\nHer glances, keener than the brand when bared * Cleave soul of\n     man nor ever 'scapes her prey:\nI looked on her in lattice pierced aloft * When bare her cheat of\n     veil that slipped away;\nAnd shot me thence a shaft my liver pierced * When thrall to care\n     and dire despair I lay\nKnowst thou, O Fawn o' the palace, how for thee * I fared from\n     farness o'er the lands astray?\nThen read my writ, dear friends, and show some ruth * To wight\n     who wones black-faced, distraught, sans stay!\n\nAnd when he ended inditing, he folded up the letter. Now the\nmerchant's wife aforesaid, who was the nurse of the king's\ndaughter, was watching him from a window, unknown of him, and\nwhen she saw him writing and reciting, she knew that some rare\ntale attached to him; so she went in to him and said, \"Peace be\nwith thee, O afflicted wight, who acquaintest not leach with thy\nplight! Verily, thou exposest thy life to grievous blight. I\nconjure thee by the virtue of Him who hath afflicted thee and\nwith the constraint of love-liking hath stricken thee, that thou\nacquaint me with thine affair and disclose to me the truth of thy\nsecret; for that indeed I have heard from thee verses which\ntrouble the mind and melt the body.\" Accordingly he acquainted\nher with his case and enjoined her to secrecy, whereof she\nconsented, saying, \"What shall be the recompense of whoso goeth\nwith thy letter and bringeth thee its reply?\" He bowed his head\nfor shame before her and was silent; and she said to him, \"Raise\nthy head and give me thy writ\": so he gave her the letter and she\nhent it and carrying it to the Princess, said to her, \"Take this\nepistle and give me its answer.\" Now the dearest of all things to\nMariyah was the recitation of poesy and verses and linked rhymes\nand the twanging of lute-strings, and she was versed in all\ntongues; wherefore she took the writ and opening it, read that\nwhich was therein and understood its purport. Then she threw it\nto the ground and cried, \"O nurse, I have no answer to make to\nthis letter.\" Quoth the nurse, \"Indeed, this is weakness in thee\nand a reproach to thee, for that the people of the world have\nheard of thee and commend thee for keenness of wit and\nunderstanding; so do thou return him an answer, such as shall\ntrick his heart and tire his soul.\" Quoth she, \"O nurse, who may\nbe the man who presumeth upon me with this correspondence? Haply\n'tis the stranger youth who gave my father the rubies.\" The woman\nsaid, \"It is himself,\" and Mariyah said, \"I will answer his\nletter in such fashion that thou shalt not bring me other than\nit.\" Cried the nurse, \"So be it.\"[FN#383] Thereupon the Princess\ncalled for ink-case and paper and wrote these couplets:--\n\nThou art bold in the copy thou sentest! May be * 'Twill increase\n     the dule foreign wight must dree!\nThou hast spied me with glance that bequeaths thee woe * Ah! far\n     is thy hope, a mere foreigner's plea!\nWho art thou, poor freke, that wouldst win my love * Wi' thy\n     verse? What seeks thine insanity?\nAn thou hope for my favours and greed therefor; * Where find thee\n     a leach for such foolish gree?\nThen rhyme-linking leave and fool-like be not * Hanged to Cross\n     at the doorway of ignomy!\nDeem not that to thee I incline, O youth! * 'Mid the Sons of the\n     Path[FN#384] is no place for me.\nThou art homeless waif in the wide wide world; * So return thee\n     home where they keen for thee:[FN#385]\nLeave verse-spouting, O thou who a-wold dost wone, * Or minstrel\n     shall name thee in lay and glee:\nHow many a friend who would meet his love * Is baulked when the\n     goal is right clear to see!\nSo begone and ne'er grieve for what canst not win * Albe time be\n     near, yet thy grasp 'twill flee.\nNow such is my say and the tale I'd tell; * So master my meaning\n     and--fare thee well!\n\n\nWhen Mariyah had made an end of her verses, she folded the letter\nand delivered it to the nurse, who hent it and went with it to\nAl-Abbas. When she gave it to him, he took it and breaking it\nopen, read it and comprehended its contents; and when he reached\nthe end of it, he swooned away. After awhile, he came to himself\nand cried, \"Praise be to Allah who hath caused her return a reply\nto my writ! Canst thou carry her another missive, and with Allah\nAlmighty be thy requital?\" Said she, \"And what shall letters\nprofit thee, seeing that such is her reply;\" but he said,\n\"Peradventure, she may yet be softened.\" Then he took ink-case\nand paper and wrote these couplets:--\n\nReached me the writ and what therein didst write, * Whence grew\n     my pain and bane and blight:\nI read the marvel-lines made wax my love * And wore my body out\n     till slightest slight.[FN#386]\nWould Heaven ye wot the whole I bear for love * Of you, with\n     vitals clean for you undight!\nAnd all I do t' outdrive you from my thought * 'Vails naught and\n     'gainst th' obsession loses might:\nCouldst for thy lover feel 'twould ease his soul; * E'en thy dear\n     Phantom would his sprite delight!\nThen on my weakness lay not coyness-load * Nor in such breach of\n     troth be traitor-wight:\nAnd, weet ye well, for this your land I fared * Hoping to 'joy\n     the union-boon forthright:\nHow many a stony wold for this I spanned; * How oft I waked when\n     men kept watch o'night!\nTo fare fro' another land for sight of you * Love bade, while\n     length of way forbade my sprite:\nSo by His name[FN#387] who molt my frame, have ruth, * And quench\n     the flames thy love in me did light:\nThou fillest, arrayed with glory's robes and rays, * Heaven's\n     stars with joy and Luna with despight.\nThen who dare chide or blame me for my love * Of one that can all\n     Beauty's boons unite?\n\nWhen Al-Abbas had made an end of his verses, he folded the letter\nand delivering it to the nurse, charged her keep the secret. So\nshe took it and carrying it to Mariyah, gave it to her. The\nPrincess broke it open and read it and apprehended its purport;\nthen cried she, \"By Allah, O nurse, my heart is chagrined with\nexceeding chagrin, never knew I a sorer, because of this\ncorrespondence and of these verses.\" And the nurse made answer to\nher, \"O my lady, thou art in thy dwelling and thy palace and thy\nheart is void of care; so return to him a reply and reck not.\"\nAccordingly, the Princess called for ink-case and paper and wrote\nthese couplets:--\n\nHo thou who wouldst vaunt thee of cark and care; * How many\n     love-molten, tryst-craving be there?\nAn hast wandered the wold in the murks of night * Bound afar and\n     anear on the tracks to fare,\nAnd to eyne hast forbidden the sweets of sleep, * Borne by Devils\n     and Marids to dangerous lair;\nAnd beggest my boons, O in tribe-land[FN#388] homed * And to urge\n     thy wish and desire wouldst dare;\nNow, woo Patience fair, an thou bear in mind * What The Ruthful\n     promised to patient prayer![FN#389]\nHow many a king for my sake hath vied, * Craving love and in\n     marriage with me to pair.\nAl-Nabhan sent, when a-wooing me, * Camels baled with musk and\n     Nadd scenting air.\nThey brought camphor in boxes and like thereof * Of pearls and\n     rubies that countless were;\nBrought pregnant lasses and <DW64>-lads, * Blood steeds and arms\n     and gear rich and rare;\nBrought us raiment of silk and of sendal sheen, * And came\n     courting us but no bride he bare:\nNor could win his wish, for I 'bode content * To part with far\n     parting and love forswear;\nSo for me greed not, O thou stranger wight * Lest thou come to\n     ruin and dire despair!\n\nWhen she had made an end of her verses, she folded the letter and\ndelivered it to the nurse, who took it and carried it to\nAl-Abbas. He broke it open and read it and comprehended its\ncontents; then took ink-case and paper and wrote these improvised\ncouplets:--\n\nThou hast told me the tale of the Kings, and of them * Each was\n     rending lion, a furious foe:\nAnd thou stolest the wits of me, all of them * And shotst me with\n     shaft of thy magic bow:\nThou hast boasted of slaves and of steeds and wealth; * And of\n     beauteous lasses ne'er man did know;\nHow presents in mighty store didst spurn, * And disdainedst\n     lovers both high and low:\nThen I followed their tracks in desire for thee, * With naught\n     save my scymitar keen of blow;\nNor slaves nor camels that run have I; * Nor slave-girls the\n     litters enveil, ah, no!\nBut grant me union and soon shalt sight * My trenchant blade with\n     the foeman's woe;\nShalt see the horsemen engird Baghdad * Like clouds that wall the\n     whole world below,\nObeying behests which to them I deal * And hearing the words to\n     the foes I throw.\nAn of <DW64> chattels ten thousand head * Wouldst have, or Kings\n     who be proud and prow\nOr chargers led for thee day by day * And virgin girls high of\n     bosom, lo!\nAl-Yaman land my command doth bear * And my biting blade to my\n     foes I show.\nI have left this all for the sake of thee, * Left Aziz and my\n     kinsmen for ever-mo'e;\nAnd made Al-Ir\u00e1k making way to thee * Under nightly murks over\n     rocks arow;\nWhen the couriers brought me account of thee * Thy beauty,\n     perfection, and sunny glow,\nThen I sent thee verses whose very sound * Burns the heart of\n     shame with a fiery throe;\nYet the world with falsehood hath fals\u00e8d me, * Though Fortune was\n     never so false as thou,\nWho dubbest me stranger and homeless one * A witless fool and a\n     slave-girl's son!\n\nThen he folded the letter and committed it to the nurse and gave\nher five hundred dinars, saying, \"Accept this from me, for by\nAllah thou hast indeed wearied thyself between us.\" She replied,\n\"By Allah, O my lord, my aim is to bring about forgathering\nbetween you, though I lose that which my right hand possesseth.\"\nAnd he said, \"May the Lord of All-might requite thee with good!\"\nThen she carried the letter to Mariyah and said to her, \"Take\nthis letter; haply it may be the end of the correspondence.\" So\nshe took it and breaking it open, read it, and when she had made\nan end of it, she turned to the nurse and said to her, \"This one\nfoisteth lies upon me and asserteth unto me that he hath cities\nand horsemen and footmen at his command and submitting to his\nallegiance; and he wisheth of me that which he shall not win; for\nthou knowest, O nurse, that kings' sons have sought me in\nmarriage, with presents and rarities; but I have paid no heed\nunto aught of this; how, then, shall I accept of this fellow, who\nis the ignoramus of his time and possesseth naught save two\ncaskets of rubies, which he gave to my sire, and indeed he hath\ntaken up his abode in the house of Al-Ghitrif and abideth without\nsilver or gold? Wherefore, Allah upon thee, O nurse, return to\nhim and cut off his hope of me.\" Accordingly the nurse rejoined\nAl-Abbas, without letter or answer; and when she came in to him,\nhe looked at her and saw that she was troubled, and he noted the\nmarks of anger on her face; so he said to her, \"What is this\nplight?\" Quoth she, \"I cannot set forth to thee that which\nMariyah said; for indeed she charged me return to thee without\nwrit or reply.\" Quoth he, \"O nurse of kings, I would have thee\ncarry her this letter and return not to her without it.\" Then he\ntook ink-case and paper and wrote these couplets:--\n\nMy secret now to men is known though hidden well and true * By\n     me: enough is that I have of love and love of you:\nI left familiars, friends, and kin to weep the loss of me * With\n     floods of tears which like the tide aye flowed and flowed\n     anew:\nThen, left my home myself I bore to Baghdad-town one day, * When\n     parting drave me there his pride and cruelty to rue:\nI have indeed drained all the bowl whose draught\n     repression[FN#390] was * Handed by friend who bitter\n     gourd[FN#391] therein for drinking threw.\nAnd, oft as strove I to enjoin the ways of troth and faith, * So\n     often on refusal's path he left my soul to sue.\nIndeed my body molten is with care I'm doom\u00e8d dree; * And yet I\n     hoped relenting and to win some grace, my due.\nBut wrong and rigour waxed on me and changed to worse my case; *\n     And love hath left me weeping-eyed for woes that aye pursue.\nHow long must I keep watch for you throughout the nightly gloom?\n     * How many a path of pining pace and garb of grief endue?\nAnd you, what while you joy your sleep, your restful pleasant\n     sleep, * Reck naught of sorrow and of shame that to your\n     friend accrue:\nFor wakefulness I watched the stars before the peep o' day, *\n     Praying that union with my dear in bliss my soul imbrue;\nIndeed the throes of long desire laid waste my frame and I * Rise\n     every morn in weaker plight with hopes e'er fewer few:\n\"Be not\" (I say) \"so hard of heart!\" for did you only deign * In\n     phantom guise to visit me 'twere joy enough to view.\nBut when ye saw my writ ye grudged to me the smallest boon * And\n     cast adown the flag of faith though well my troth ye knew;\nNor aught of answer you vouchsafe, albe you wot full well * The\n     words therein address the heart and pierce the spirit\n     through.\nYou deemed yourself all too secure for changes of the days * And\n     of the far and near alike you ever careless grew.\nHadst thou (dear maid) been doomed like me to woes, forsure hadst\n     felt * The lowe of love and Laza-hell which parting doth\n     enmew;\nYet soon shalt suffer torments such as those from thee I bear *\n     And storm of palpitation-pangs in vitals thine shall brew:\nYea, thou shalt taste the bitter smack of charges false and foul,\n     * And public make the privacy best hid from meddling crew;\nAnd he thou lovest shall approve him hard of heart and soul * And\n     heedless of the shifts of Time thy very life undo.\nThen hear the fond Salam I send and wish thee every day * While\n     swayeth spray and sparkleth star all good thy life ensue!\n\nWhen Al-Abbas had made an end of his verses, he folded the scroll\nand gave it to the nurse, who took it and carried it to Mariyah.\nWhen she came into the Princess's presence, she saluted her; but\nMariyah returned not her salutation and she said, \"O my lady, how\nhard is thy heart that thou grudgest to return the salam! Accept\nthis letter, because 'tis the last that shall come to thee from\nhim.\" Quoth Mariyah, \"Take my warning and never again enter my\npalace, or 'twill be the cause of thy destruction; for I am\ncertified that thou purposest my disgrace. So get thee gone from\nme.\" And she bade beat the nurse who went forth fleeing from her\npresence, changed of colour and 'wildered of wits, and gave not\nover going till she came to the house of Al-Abbas. When the\nPrince saw her in this plight, he became like a sleeper awakened\nand cried to her, \"What hath befallen thee? Acquaint me with thy\ncase.\" She replied, \"Allah upon thee, nevermore send me to\nMariyah, and do thou protect me, so the Lord protect thee from\nthe fires of Gehenna!\" Then she related to him that which had\nbetided her with Mariyah which when Al-Abbas heard, there took\nhim the pride and high spirit of the generous and this was\ngrievous to him. The love of Mariyah fled forth of his heart and\nhe said to the nurse, \"How much hadst thou of Mariyah every\nmonth?\" Quoth she, \"Ten dinars\" and quoth he, \"Be not concerned.\"\nThen he put hand to pouch and bringing out two hundred ducats,\ngave them to her and said,\"Take this wage for a whole year and\nturn not again to serve anyone of the folk. When the twelvemonth\nshall have passed away, I will give thee a two years' wage, for\nthat thou hast wearied thyself with us and on account of the\ncutting off the tie which bound thee to Mariyah.\" Also he gifted\nher with a complete suit of clothes and raising his head to her,\nsaid, \"When thou toldest me that which Mariyah had done with\nthee, Allah uprooted the love of her from out my heart, and never\nagain will she occur to my thought; so extolled be He who turneth\nhearts and eyes! 'Twas she who was the cause of my coming out\nfrom Al-Yaman, and now the time is past for which I engaged with\nmy folk and I fear lest my father levy his forces and ride forth\nin quest of me, for that he hath no child other than myself nor\ncan he brook to be parted from me; and in like way 'tis with my\nmother.\" When the nurse heard his words, she asked him, \"O my\nlord, and which of the kings is thy sire?\" He answered, saying,\n\"My father is Al-Aziz, lord of Al-Yaman, and Nubia and the\nIslands[FN#392] of the Banu Kaht\u00e1n, and the Two\nSanctuaries[FN#393] (Allah of All-might have them in His\nkeeping!), and whenever he taketh horse, there ride with him an\nhundred and twenty and four thousand horsemen, each and every\nsmiters with the sword, besides attendants and servants and\nfollowers, all of whom give ear to my word and obey my bidding.\"\nAsked the nurse, \"Why, then, O my lord, didst thou conceal the\nsecret of thy rank and lineage and passedst thyself off for a\nforeigner and a wayfarer? Alas for our disgrace before thee by\nreason of our shortcoming in rendering thee thy due! What shall\nbe our excuse with thee, and thou of the sons of the kings?\" But\nhe rejoined, \"By Allah, thou hast not fallen short! Indeed, 'tis\nincumbent on me to requite thee, what while I live, though from\nthee I be far distant.\" Then he called his man Amir and said to\nhim, \"Saddle the steeds.\" When the nurse heard his words and\nindeed she saw that Amir brought him the horses and they were\nresolved upon departure, the tears ran down upon her cheeks and\nshe said to him, \"By Allah, thy separation is saddening to me, O\ncoolth of the eye!\" Then quoth she, \"Where is the goal of thine\nintent, so we may know thy news and solace ourselves with thy\nreport?\" Quoth he, \"I go hence to visit 'Ak\u00edl, the son of my\npaternal uncle, for that he hath his sojourn in the camp of\nKundah bin Hish\u00e1m, and these twenty years have I not seen him nor\nhath he seen me; so I purpose to repair to him and discover his\nnews and return. Then will I go hence to Al-Yaman, Inshallah!\" So\nsaying, he took leave of the nurse and her husband and set out,\nintending for 'Akil, the son of his father's brother. Now there\nwas between Baghdad and 'Ak\u00edl's abiding-place forty days'\njourney; so Al-Abbas settled himself on the back of his steed and\nhis servant Amir mounted also and they fared forth on their way.\nPresently, Al-Abbas turned right and left and recited these\ncouplets,\n\n\"I'm the singular knight and my peers I slay! * I lay low the foe\n     and his whole array:\nI fare me to visit my friend Al-Ak\u00edl, * And in safety and\n     Allah-lauds,[FN#394] shorten the way;\nAnd roll up the width of the wold while still * Hears 'Amir my\n     word or in earnest or play.[FN#395]\nI spring with the spring of a lynx or a pard * Upon whoso dareth\n     our course to stay;\nO'erthrow him in ruin and abject shame, * Make him drain the\n     death-cup in fatal fray.\nMy lance is long with its steely blade; * A brand keen-grided,\n     thin-edged I sway:\nWith a stroke an it fell on a towering hill * Of the hardest\n     stone, this would cleave in tway:\nI lead no troops, nor seek aid save God's, * The creating Lord\n     (to whom laud alw\u00e0y!)\nOn Whom I rely in adventures all * And Who pardoneth l\u00e2ches of\n     freeman and thrall.\"\n\nThen they fell a-faring night and day, and as they went, behold,\nthey sighted a camp of the camps of the Arabs. So Al-Abbas\nenquired thereof and was told that it was the camp of the Banu\nZohrah. Now there were around them herds and flocks, such as\nfilled the earth, and they were enemies to Al-Akil, the cousin of\nAl-Abbas, upon whom they made daily raids and took his cattle,\nwherefore he used to pay them tribute every year because he\nlacked power to cope wth them. When Al-Abbas came to the skirts\nof the camp, he dismounted from his destrier and his servant Amir\nalso dismounted; and they set down the provaunt and ate their\nsufficiency and rested an hour of the day. Then said the Prince\nto his page, \"Fetch water from the well and give the horses to\ndrink and draw up a supply for us in thy bag,[FN#396] by way of\nprovision for the road.\" So Amir took the water-skin and made for\nthe well; but, when he came there, behold, two young men slaves\nwere leading gazelles, and when they saw him, they said to him,\n\"Whither wendest thou, O youth, and of which of the Arabs art\nthou?\" Quoth he, \"Harkye, lads, fill me my water-skin, for that I\nam a stranger astray and a farer of the way, and I have a comrade\nwho awaiteth me.\" Quoth the thralls, \"Thou art no wayfarer, but a\nspy from Al-Ak\u00edl's camp.\" Then they took him and carried him to\ntheir king Zuhayr bin Shabib; and when he came before him, he\nsaid to him, \"Of which of the Arabs art thou?\" Quoth Amir, \"I am\na wayfarer.\" So Zuhayr said, \"Whence comest thou and whither\nwendest thou?\" and Amir replied, \"I am on my way to Al-Ak\u00edl.\"\nWhen he named Al-Ak\u00edl, those who were present were excited; but\nZuhayr signed to them with his eyes and asked him, \"What is thine\nerrand with Al-Ak\u00edl?\" and he answered, \"We would fain see him, my\nfriend and I.\" As soon as Zuhayr heard his words, he bade smite\nhis neck;[FN#397] but his Wazir said to him, \"Slay him not, till his\nfriend be present.\" So he commanded the two slaves to fetch his\nfriend; whereupon they repaired to Al-Abbas and called to him,\nsaying, \"O youth, answer the summons of King Zuhayr.\" He\nenquired, \"What would the king with me?\" and they replied, \"We\nknow not.\" Quoth he, \"Who gave the king news of me?\" and quoth\nthey, \"We went to draw water, and found a man by the well. So we\nquestioned him of his case, but he would not acquaint us\ntherewith, wherefore we carried him willy-nilly to King Zuhayr,\nwho asked him of his adventure and he told him that he was going\nto Al-Ak\u00edl. Now Al-Ak\u00edl is the king's enemy and he intendeth to\nbetake himself to his camp and make prize of his offspring, and\ncut off his traces.\" Said Al-Abbas, \"And what hath Al-Ak\u00edl done\nwith King Zuhayr?\" They replied. \"He engaged for himself that he\nwould bring the King every year a thousand dinars and a thousand\nshe-camels, besides a thousand head of thoroughbred steeds and\ntwo hundred black slaves and fifty hand-maids; but it hath\nreached the king that Al-Ak\u00edl purposeth to give naught of this;\nwherefore he is minded to go to him. So hasten thou with us, ere\nthe King be wroth with thee and with us.\" Then said Al-Abbas to\nthem, \"O youths, sit by my weapons and my stallion till I\nreturn.\" But they said, \"By Allah, thou prolongest discourse with\nthat which beseemeth not of words! Make haste, or we will go with\nthy head, for indeed the King purposeth to slay thee and to slay\nthy comrade and take that which is with you.\" When the Prince\nheard this, his skin bristled with rage and he cried out at them\nwith a cry which made them tremble. Then he sprang upon his horse\nand settling himself in the saddle, galloped till he came to the\nKing's assembly, when he shouted at the top of his voice, saying,\n\"To horse, O horsemen!\" and couched his spear at the pavilion\nwherein was Zuhayr. Now there were about the King a thousand\nsmiters with the sword; but Al-Abbas charged home upon them and\ndispersed them from around him; and there abode none in the tent\nsave Zuhayr and his Wazir. Then Al-Abbas came up to the door of\nthe tent wherein were four-and-twenty golden doves; so he took\nthem, after he had tumbled them down with the end of his lance.\nThen he called out saying, \"Ho, Zuhayr! Doth it not suffice thee\nthat thou hast abated Al-Akil's repute, but thou art minded to\nabate that of those who sojourn round about him? Knowest thou not\nthat he is of the lieutenants of Kundah bin Hisham of the Banu\nShayban, a man renowned for prowess? Indeed, greed of his gain\nhath entered into thee and envy of him hath gotten the mastery of\nthee. Doth it not suffice thee that thou hast orphaned his\nchildren[FN#398] and slain his men? By the virtue of Mustafa, the\nChosen Prophet, I will make thee drain the cup of death!\" So\nsaying. he bared his brand and smiting Zuhayr on his\nshoulder-blade caused the steel issue gleaming from his throat\ntendons; then he smote the Wazir and clove his crown asunder. As\nhe was thus, behold, Amir called out to him and said, \"O my lord,\ncome help me, or I be a dead man!\" So Al-Abbas went up to him\nguided by his voice, and found him cast down on his back and\nchained with four chains to four pickets of iron.[FN#399] He\nloosed his bonds and said to him, \"Go in front of me, O Amir.\" So\nhe fared on before him a little, and presently they looked, and,\nbehold, horsemen were making to Zuhayr's succour, and they\nnumbered twelve thousand riders led by Sahl bin Ka'ab bestriding\na coal-black steed. He charged upon Amir, who fled from him, then\nupon Al-Abbas, who said, \"O Amir, hold fast to my horse and guard\nmy back.\" The page did as he bade him, whereupon Al-Abbas cried\nout at the folk and falling upon them, overthrew their braves and\nslew of them some two thousand riders, whilst not one of them\nknew what was to do nor with whom he fought. Then said one of\nthem to other, \"Verily, the King is slain; so with whom do we\nwage war? Indeed ye flee from him; but 'twere better ye enter\nunder his banners, or not one of you will be saved.\" Thereupon\nall dismounted and doffing that which was upon them of war-gear,\ncame before Al-Abbas and proffered him allegiance and sued for\nhis protection. So he withheld his brand from them and bade them\ngather together the spoils. Then he took the riches and the\nslaves and the camels, and they all became his lieges and his\nretainers, to the number (according to that which is reported) of\nfifty thousand horses. Furthermore, the folk heard of him and\nflocked to him from all sides; whereupon he divided the loot\namongst them and gave largesse and dwelt thus three days, and\nthere came gifts to him. After this he bade march for Al-Akil's\nabiding place; so they fared on six days and on the seventh they\nsighted the camp. Al-Abbas bade his man Amir precede him and give\nAl-Akil the good news of his cousin's coming; so he rode on to\nthe camp and, going in to Al-Akil, acquainted him with the glad\ntidings of Zuhayr's slaughter and the conquest of his\nclan.[FN#400] Al-Akil rejoiced in the coming of Al-Abbas and the\nslaughter of his enemy and all in his camp rejoiced also and cast\nrobes of honour upon Amir; while Al-Akil bade go forth to meet\nAl-Abbas, and commanded that none, great or small, freeman or\nslave, should tarry behind. So they did his bidding and going\nforth all, met Al-Abbas at three parasangs' distance from\nthe camp; and when they met him, they dismounted from their\nhorses and Al-Akil and he embraced and clapped palm to\npalm.[FN#401] Then rejoicing in the coming of Al-Abbas and the\nkilling of their foeman, they returned to the camp, where tents\nwere pitched for the new-comers and skin-rugs spread and game\nslain and beasts slaughtered and royal guest-meals spread; and\nafter this fashion they abode twenty days in the enjoyment of all\ndelight of life. On this wise fared it with Al-Abbas and his\ncousin Al-Akil; but as regards King Al-Aziz, when his son left\nhim, he was desolated for him with exceeding desolation, both he\nand his mother; and when tidings of him tarried long and the\ntryst-time passed without his returning, the king caused public\nproclamation to be made, commanding all his troops to get ready\nto mount and ride forth in quest of his son Al-Abbas, at the end\nof three days, after which no cause of hindrance or excuse would\nbe admitted to any. So on the fourth day, the king bade muster\nthe troops who numbered four-and-twenty thousand horse, besides\nservants and followers. Accordingly, they reared the standards\nand the kettle-drums beat the general and the king set out with\nhis power intending for Baghdad; nor did he cease to press\nforward with all diligence, till he came within half a day's\njourney of the city, when he bade his army encamp on the Green\nMeadow. There they pitched the tents, till the lowland was\nstraitened with them, and set up for the king a pavilion of green\nbrocade, purfled with pearls and precious stones. When Al-Aziz\nhad sat awhile, he summoned the Mamelukes of his son Al-Abbas,\nand they were five-and-twenty in number besides ten slave-girls,\nas they were moons, five of whom the king had brought with him\nand other five he had left with the prince's mother. When the\nMamelukes came before him, he cast over each and every of them a\nmantle of green brocade and bade them mount similar horses of one\nand the same fashion and enter Baghdad and ask after their lord\nAl-Abbas. So they rode into the city and passed through the\nmarket-streets and there remained in Baghdad nor old man nor boy\nbut came forth to gaze on them and divert himself with the sight\nof their beauty and loveliness and the seemliness of their\nsemblance and the goodliness of their garments and horses, for\nall were even as moons. They gave not over going till they came\nto the palace,[FN#402] where they halted, and the king looked at\nthem and seeing their beauty and the brilliancy of their apparel\nand the brightness of their faces, said, \"Would Heaven I knew of\nwhich of the tribes these are!\" And he bade the Eunuch bring him\nnews of them. The castrato went out to them and questioned them\nof their case, whereto they replied, \"Return to thy lord and\nenquire of him concerning Prince Al-Abbas, an he have come unto\nhim, for that he left his sire King Al-Aziz a full-told year ago,\nand indeed longing for him troubleth the King and he hath levied\na division of his army and his guards and is come forth in quest\nof his son, so haply he may light upon tidings of him.\" Quoth the\nEunuch, \"Is there amongst you a brother of his or a son?\" and\nquoth they, \"Nay, by Allah, but we are all his Mamelukes and the\npurchased of his money, and his sire Al-Aziz hath sent us to make\nenquiry of him. Do thou go to thy lord and question him of the\nPrince and return to us with that which he shall answer thee.\"\nAsked the Eunuch, \"And where is King Al-Aziz?\" and they answered,\n\"He is encamped in the Green Meadow.\"[FN#403] The Eunuch returned\nand told the king, who said, \"Indeed we have been unduly\nnegligent with regard to Al-Abbas. What shall be our excuse with\nthe King? By Allah, my soul suggested to me that the youth was of\nthe sons of the kings!\" His wife, the Lady Afifah saw him\nlamenting for his neglect of Al-Abbas, and said to him, \"O King,\nwhat is it thou regrettest with this mighty regret?\" Quoth he,\n\"Thou knowest the stranger youth, who gifted us with the rubies?\"\nQuoth she, \"Assuredly;\" and he, \"Yonder youths, who have halted\nin the palace court, are his Mamelukes, and his father, King\nAl-Aziz, lord of Al-Yaman, hath pitched his camp on the Green\nMeadow; for he is come with his army to seek him, and the number\nof his troops is four-and-twenty thousand horsemen.\" Then he went\nout from her, and when she heard his words, she wept sore for him\nand had compassion on his case and sent after him, counselling\nhim to summon the Mamelukes and lodge them in the palace and\nentertain them. The king hearkened to her rede and despatching\nthe Eunuch for the Mamelukes, assigned unto them a lodging and\nsaid to them, \"Have patience, till the King give you tidings of\nyour lord Al-Abbas.\" When they heard his words, their eyes ran\nover with a rush of tears, of their mighty longing for the sight\nof their lord. Then the King bade the Queen enter the private\nchamber opening upon the throne-room and let down the curtain\nbefore the door, so she might see and not be seen. She did this\nand he summoned them to his presence; and, when they stood before\nhim, they kissed ground to do him honour, and showed forth their\ncourtly breeding and magnified his dignity. He ordered them to\nsit, but they refused, till he conjured them by their lord\nAl-Abbas: accordingly they sat down and he bade set before them\nfood of various kinds and fruits and sweetmeats. Now within the\nLady Afifah's palace was a souterrain communicating with the\npavilion of the Princess Mariyah: so the Queen sent after her and\nshe came to her, whereupon she made her stand behind the curtain\nand gave her to know that Al-Abbas was son to the King of\nAl-Yaman and that these were his Mamelukes: she also told her\nthat the Prince's father had levied his troops and was come with\nhis army in quest of him and that he had pitched his camp on the\nGreen Meadow and had despatched these Mamelukes to make enquiry\nof their lord. Then Mariyah abode looking upon them and upon\ntheir beauty and loveliness and the goodliness of their raiment,\ntill they had eaten their fill of food and the tables were\nremoved; whereupon the King recounted to them the story of\nAl-Abbas and they took leave of him and went their ways. So\nfortuned it with the Mamelukes; but as for the Princess Mariyah,\nwhen she returned to her palace, she bethought herself concerning\nthe affair of Al-Abbas, repenting her of what she had done; and\nthe love of him took root in her heart. And, when the night\ndarkened upon her, she dismissed all her women and bringing out\nthe letters, to wit, those which Al-Abbas had written her, fell\nto reading them and weeping. She left not weeping her night long,\nand when she arose in the morning, she called a damsel of her\nslave-girls, Shaf\u00edkah by name, and said to her, \"O damsel, I\npurpose to discover to thee mine affair and I charge thee keep my\nsecret, which is that thou betake thyself to the house of the\nnurse, who used to serve me, and fetch her to me, for that I have\ngrave need of her.\" Accordingly, Shafikah went out and repairing\nto the nurse's house, entered and found her clad in clothing\nother and richer than what she had whilome been wont to wear. So\nshe saluted her and asked her, \"Whence hadst thou this dress,\nthan which there is no goodlier?\" Answered the nurse, \"O\nShafikah, thou deemest that I have seen no good save of thy\nmistress; but, by Allah, had I endeavoured for her destruction, I\nhad acted righteously, seeing that she did with me what she did\nand bade the Eunuch beat me, without offence by me offered: so\ntell her that he, on whose behalf I bestirred myself with her,\nhath made me independent of her and her humours, for he hath\nhabited me in this habit and given me two hundred and fifty\ndinars and promised me the like every year and charged me to\nserve none of the folk.\" Quoth Shafikah, \"My mistress hath a need\nfor thee; so come thou with me and I will engage to restore thee\nto thy dwelling in safety and satisfaction.\" But quoth the nurse,\n\"Indeed her palace is become unlawful and forbidden to me[FN#404]\nand never again will I enter therein, for that Allah (extolled\nand exalted be He!) of His favour and bounty hath rendered me\nindependent of her.\" Presently Shafikah returned to her mistress\nand acquainted her with the nurse's words and that wherein she\nwas of prosperity; whereupon Mariyah confessed her unmannerly\ndealing with her and repented when repentance profited her not;\nand she abode in that her case days and nights, whilst the fire\nof longing flamed in her heart. On this wise happened it to her;\nbut as regards Al-Abbas, he tarried with his cousin Al-Akil\ntwenty days, after which he made ready for the journey to Baghdad\nand bidding bring the booty he had taken from King Zuhayr,\ndivided it between himself and his cousin. Then he sent out\na-marching Baghdad-wards and when he came within two days'\njourney of the city, he summoned his servant Amir and said to\nhim, \"Mount thy charger and forego me with the caravan and the\ncattle.\" So Amir took horse and fared on till he came to Baghdad,\nand the season of his entering was the first of the day; nor was\nthere in the city little child or old greybeard but came forth to\ndivert himself with gazing on those flocks and herds and upon the\nbeauty of those slave-girls; and their wits were wildered at what\nthey saw. Soon afterwards the news reached the king that the\nyoung man Al-Abbas, who had gone forth from him, was come back\nwith booty and rarities and black slaves and a conquering host\nand had taken up his sojourn without the city, whilst his servant\nAmir was presently come to Baghdad, so he might get ready for his\nlord dwelling-places wherein he should take up his abode. When\nthe King heard these tidings of Amir, he sent for him and caused\nbring him before him; and when he entered his presence, he kissed\nthe ground and saluted with the salam and showed his fine\nbreeding and greeted him with the goodliest of greetings. The\nKing bade him raise his head and, this done, questioned him of\nhis lord Al-Abbas; whereupon he acquainted him with his\nadventures and told him that which had betided him with King\nZuhayr and of the army that was become at his command and of the\nspoil he had secured. He also gave him to know that Al-Abbas was\nto arrive on the morrow, and with him more than fifty thousand\ncavatiers, obedient to his orders. When the king heard his words,\nhe bade decorate Baghdad and commanded the citizens to equip\nthemselves with the richest of their apparel, in honour of the\ncoming of Al-Abbas. Furthermore, he sent to give King Al-Aziz the\nglad tidings of his son's return and informed him of all which he\nhad heard from the Prince's servant. When the news reached King\nAl-Aziz, he joyed with exceeding joy in the approach of his son\nand straightway took horse, he and all his host, while the\ntrumpets blared and the musicians played, so that the earth\nquaked and Baghdad also trembled, and it was a notable day. When\nMariyah beheld all this, she repented in all possible penitence\nof that which she had done against Al-Abbas and the fires of\ndesire raged in her vitals. Meanwhile, the troops[FN#405] sallied\nforth of Baghdad and went out to meet those of Al-Abbas, who had\nhalted in a garth called the Green Island. When he espied the\napproaching host, he strained his sight and, seeing horsemen\ncoming and troops and footmen he knew not, said to those about\nhim, \"Among yonder troops are flags and banners of various kinds;\nbut, as for the great green standard that ye see, 'tis the\nstandard of my sire, the which is reserved to him and never\ndisplayed save over his head, and thus I know that he himself is\ncome out in quest of me.\" And he was certified of this, he and\nhis troops. So he fared on towards them and when he drew near\nthem, he knew them and they knew him; whereupon they lighted down\nfrom their horses and saluting him, gave him joy of his safety\nand the folk flocked to him. When he came to his father, they\nembraced and each greeted other a long time, whilst neither of\nthem could utter a word, for the greatness of that which betided\nthem of joy in reunion. Then Al-Abbas bade the folk take horse;\nso they mounted and his Mamelukes surrounded him and they entered\nBaghdad on the most splendid wise and in the highest honour and\nglory. Now the wife of the shopkeeper, that is, the nurse, came\nout, with the rest of those who flocked forth, to divert herself\nwith gazing upon the show, and when she saw Al-Abbas and beheld\nhis beauty and the beauty of his host and that which he had\nbrought back with him of herds and slave-girls, Mamelukes and\n<DW64>s, she improvised and recited these couplets,\n\n\"Al-Abb\u00e1s from the side of Ak\u00edl is come; * Caravans and steeds he\n     hath plunder\u00e8d:\nYea; horses he brought of pure blood, whose necks * Ring with\n     collars like anklets wher'er they are led.\nWith dom\u00e8d hoofs they pour torrent-like, * As they prance through\n     dust on the level stead:\nAnd bestriding their saddles come men of war, * Whose fingers\n     play on the kettle-drum's head:\nAnd couched are their lances that bear the points * Keen grided,\n     which fill every soul with dread:\nWho wi' them would fence draweth down his death * For one deadly\n     lunge soon shall do him dead:\nCharge, comrades, charge ye and give me joy, * Saying, 'Welcome\n     to thee, O our dear comr\u00e0de!'\nAnd who joys at his meeting shall 'joy delight * Of large gifts\n     when he from his steed shall 'light.\"\n\nWhen the troops entered Baghdad, each of them alighted in his\ntent, whilst Al-Abbas encamped apart on a place near the Tigris\nand issued orders to slaughter for the soldiers, each day, that\nwhich should suffice them of oxen and sheep and to bake them\nbread and spread the tables: so the folk ceased not to come to\nhim and eat of his banquet. Furthermore, all the country-people\nflocked to him with presents and rarities and he requited them\nmany times the like of their gifts, so that the lands were filled\nwith his renown and the fame of him was bruited abroad among the\nhabitants of wold and town. Then, as soon as he rode to the house\nhe had bought, the shopkeeper and his wife came to him and gave\nhim joy of his safety; whereupon he ordered them three head of\nswift steeds and thoroughbred and ten dromedaries and an hundred\nhead of sheep and clad them both in costly robes of honour.\nPresently he chose out ten slave-girls and ten <DW64> slaves and\nfifty mares and the like number of she-camels and three hundred\nof sheep, together with twenty ounces of musk and as many of\ncamphor, and sent all this to the King of Baghdad. When the\npresent came to Ins bin Kays, his wit fled for joy and he was\nperplexed wherewith to requite him. Al-Abbas also gave gifts and\nlargesse and bestowed robes of honour upon noble and simple, each\nafter the measure of his degree, save only Mariyah; for to her\nindeed he sent nothing. This was grievous to the Princess and it\nirked her sore that he should not remember her; so she called her\nslave-girl Shafikah and said to her, \"Hie thee to Al-Abbas and\nsalute him and say to him, 'What hindereth thee from sending my\nlady Mariyah her part of thy booty?'\" So Shafikah betook herself\nto him and when she came to his door, the chamberlains refused\nher admission, until they should have got for her leave and\npermission. When she entered, Al-Abbas knew her and knew that she\nhad somewhat of speech with him; so he dismissed his Mamelukes\nand asked her, \"What is thine errand, O hand-maid of good?\"\nAnswered she, \"O my lord, I am a slave-girl of the Princess\nMariyah, who kisseth thy hands and offereth her salutation to\nthee. Indeed, she rejoiceth in thy safety and blameth thee for\nthat thou breakest her heart, alone of all the folk, because thy\nlargesse embraceth great and small, yet hast thou not remembered\nher with anything of thy plunder, as if thou hadst hardened thy\nheart against her.\" Quoth he, \"Extolled be He who turneth hearts!\nBy Allah, my vitals were consumed with the love of her; and, of\nmy longing after her I came forth to her from my mother-land and\nleft my people and my home and my wealth, and it was with her\nthat began the hardheartedness and the cruelty. Natheless, for\nall this, I bear her no malice and there is no help but that I\nsend her somewhat whereby she may remember me; for that I sojourn\nin her country but a few days, after which I set out for the land\nof Al-Yaman.\" Then he called for a chest and thence bringing out\na necklace of Greek workmanship, worth a thousand dinars, wrapped\nit in a mantle of Greek silk, set with pearls and gems and\npurfled with red gold, and joined thereto a couple of caskets\ncontaining musk and amber-gris. He also put off upon the girl a\nmantle of Greek silk, striped with gold, wherein were divers\nfigures and portraitures depictured, never saw eyes its like.\nTherewithal the girl's wit fled for joy and she went forth from\nhis presence and returned to her mistress. When she came in to\nher, she acquainted her with that which she had seen of Al-Abbas\nand that which was with him of servants and attendants and set\nout to her the loftiness of his station and gave her that which\nwas with her. Mariyah opened the mantle, and when she saw that\nnecklace (and indeed the place was illumined with the lustre\nthereof), she looked at her slave-girl and said to her, \"By\nAllah, O Shafikah, one look at him were dearer to me than all\nthat my hand possesseth! Oh, would Heaven I knew what I shall do,\nwhen Baghdad is empty of him and I hear of him no news!\" Then she\nwept and calling for ink-case and paper and pen of brass, wrote\nthese couplets:\n\nLongsome my sorrows are; my liver's fired with ecstasy; * And\n     severance-shaft hath shot me through whence sorest pangs I\n     dree:\nAnd howso could my soul forget the love I bear to you? *\n     You-wards my will perforce returns nor passion sets me free:\nI 'prison all desires I feel for fear of spies thereon * Yet\n     tears that streak my cheek betray for every eye to see.\nNo place of rest or joy I find to bring me life-delight; * No\n     wine tastes well, nor viands please however savoury:\nAh me! to whom shall I complain of case and seek its cure * Save\n     unto thee whose Phantom deigns to show me sight of thee?\nThen name me not or chide for aught I did in passion-stress, *\n     With vitals gone and frame consumed by yearning-malady!\nSecret I keep the fire of love which aye for severance burns; *\n     Sworn slave[FN#406] to Love who robs my rest and wakes me\n     cruelly:\nAnd ceaseth not my thought to gaze upon your ghost by night, *\n     Which falsing comes and he I love still, still unloveth me.\nWould Heaven ye wist the blight that I for you are doomed to bear\n     * For love of you, which tortures me with parting agony!\nThen read between the lines I wrote, and mark and learn their\n     sense * For such my tale, and Destiny made me an outcast be:\nLearn eke the circumstance of Love and lover's woe nor deign *\n     Divulge its mysteries to men nor grudge its secrecy.\n\nThen she folded the scroll and givng it to her slave-girl, bade\nher bear it to Al-Abbas and bring back his reply. So Shafikah\ntook the letter and carried it to the Prince, after the\ndoorkeeper had sought leave of him to admit her. When she came in\nto him, she found with him five damsels, as they were moons, clad\nin rich raiment and ornaments; and when he saw her, he said to\nher, \"What is thy need, O hand-maid of good?\" Presently she put\nout her hand to him with the writ, after she had kissed it, and\nhe bade one of his slave-girls receive it from her.[FN#407] Then\nhe took it from the girl and breaking the seal, read it and\ncomprehended its contents; whereupon he cried, \"Verily, we be\nAllah's and unto Him we shall return!\" and calling for ink-case\nand paper, wrote these improvised couplets:--\n\nI wonder seeing how thy love to me * Inclined, while I in heart\n     from love declined:\nEke wast thou wont to say in verseful writ, * \"Son of the\n     Road[FN#408] no road to me shall find!\nHow oft kings flocked to me with mighty men * And bales on back\n     of Bukhti[FN#409] beast they bind:\nAnd noble steeds of purest blood and all * They bore of choicest\n     boons to me consigned;\nYet won no favour!\" Then came I to woo * And the long tale o'\n     love I had designed,\nI fain set forth in writ of mine, with words * Like strings of\n     pearls in goodly line aligned:--\nSet forth my sev'rance, griefs, tyrannic wrongs, * And ill device\n     ill-suiting lover-kind.\nHow oft love-claimant, craving secrecy, * How oft have lovers\n     'plained as sore they pined,\nHow many a brimming bitter cup I've quaffed, * And wept my woes\n     when speech was vain as wind!\nAnd thou:--\"Be patient, 'tis thy bestest course * And choicest\n     medicine for mortal mind!\"\nThen unto patience worthy praise cleave thou; * Easy of issue and\n     be lief resigned:\nNor hope thou aught of me lest ill alloy * Or aught of dross\n     affect my blood refined:\nSuch is my speech. Read, mark, and learn my say! * To what thou\n     deemest ne'er I'll tread the way.\n\nThen he folded the scroll and sealing it, entrusted it to the\ndamsel, who took it and bore it to her mistress. When the\nPrincess read the letter and mastered its meaning, she said,\n\"Meseemeth he recalleth bygones to me.\" Then she called for pens,\nink, and paper, and wrote these couplets:\n\nLove thou didst show me till I learnt its woe * Then to the\n     growth of grief didst severance show:\nI banisht joys of slumber after you * And e'en my pillow garred\n     my wake to grow.\nHow long in parting shall I pine with pain * While\n     severance-spies[FN#410] through night watch every throe?\nI've left my kingly couch and self withdrew * Therefrom, and\n     taught mine eyelids sleep t'unknow:\n'Twas thou didst teach me what I ne'er can bear: * Then didst\n     thou waste my frame with parting-blow.\nBy oath I swear thee, blame and chide me not: * Be kind to\n     mourner Love hath stricken low!\nFor parting-rigours drive him nearer still * To narrow home, ere\n     clad in shroud for clo':\nHave ruth on me, since Love laid waste my frame, * 'Mid thralls\n     enrolled me and lit fires that flame.\n\nMariyah rolled up the letter and gave it to Shafikah, bidding her\nbear it to Al-Abbas. Accordingly she took it and going with it to\nhis door, proceeded to enter; but the chamberlains and\nserving-men forbade her, till they had obtained her leave from\nthe Prince. When she went into him, she found him sitting in the\nmidst of the five damsels before mentioned, whom his father had\nbrought for him; so she gave him the letter and he tare it open\nand read it. Then he bade one of the damsels, whose name was\nKhafifah and who came from the land of China, tune her lute and\nsing anent separation. Thereupon she came forward and tuning her\nlute, played thereon in four-and-twenty modes: after which she\nreturned to the first and sang these couplets,\n\n\"Our friends, when leaving us on parting-day, * Drave us in wolds\n     of severance-grief to stray:\nWhen bound the camels' litters bearing them, * And cries of\n     drivers urged them on the way,\nOutrusht my tears, despair gat hold of me * And sleep betrayed\n     mine eyes to wake a prey.\nThe day they went I wept, but showed no ruth * The severance-spy\n     and flared the flames alw\u00e0y:\nAlas for lowe o' Love that fires me still! * Alack for pine that\n     melts my heart away!\nTo whom shall I complain of care, when thou * Art gone, nor fain\n     a-pillow head I lay?\nAnd day by day Love's ardours grow on me, * And far's the tent\n     that holds my fondest may:\nO Breeze o' Heaven, bear for me a charge * (Nor traitor-like my\n     troth in love betray!),\nWhene'er thou breathest o'er the loved one's land * Greet him\n     with choice salam fro' me, I pray:\nDust him with musk and powdered ambergris * While time endures!\n     Such is my wish for aye.\"\n\nWhen the damsel had made an end of her song, Al-Abbas swooned\naway and they sprinkled on him musked rose-water, till he\nrecovered from his fainting-fit, when he called another damsel\n(now there was on her of linen and raiment and ornaments that\nwhich undoeth description, and she was a model of beauty and\nbrightness and loveliness and symmetry and perfect grace, such as\nshamed the crescent moon, and she was a Turkish girl from the\nland of the Roum and her name was H\u00e1fizah) and said to her, \"O\nHafizah, close thine eyes and tune thy lute and sing to us upon\nthe days of severance.\" She answered him, \"To hear is to obey\"\nand taking the lute, tightened its strings and cried out from her\nhead,[FN#411] in a plaintive voice, and sang these couplets,\n\n\"My friends! tears flow in painful mockery, * And sick my heart\n     from parting agony:\nMy frame is wasted and my vitals wrung * And love-fires grow and\n     eyes set tear-floods free:\nAnd when the fire burns high beneath my ribs * With tears I\n     quench it as sad day I see.\nLove left me wasted, baffled, pain-begone, * Sore frighted, butt\n     to spying enemy:\nWhen I recal sweet union wi' their loves * I chase dear sleep\n     from the sick frame o' me.\nLong as our parting lasts the rival joys * And spies with fearful\n     prudence gain their gree.\nI fear me for my sickly, langourous frame * Lest dread of parting\n     slay me incontinently.\"\n\nWhen Hafizah had ended her song, Al-Abbas cried to her, \"Brava!\nVerily, thou quickenest hearts from griefs.\" Then he called\nanother maiden of the daughters of Daylam by name Marj\u00e1nah, and\nsaid to her, \"O Marjanah, sing to me upon the days of parting.\"\nShe said, \"Hearing and obeying,\" and recited these couplets,\n\n\"'Cleave to fair Patience! Patience 'gendereth weal': * Such is\n     the rede to us all sages deal:\nHow oft I plained the lowe of grief and love * Mid passions cast\n     my soul in sore unheal.\nHow oft I waked and drained the bitter cup * And watched the\n     stars, nor sleep mine eyes would seal!\nEnough it were an deal you grace to me * In writ a-morn and\n     garred no hope to feel.\nBut Thoughts which probed its depths would sear my heart * And\n     start from eye-brows streams that ever steal:\nNor cease I suffering baleful doom and nights * Wakeful, and\n     heart by sorrows rent piece-meal:\nBut Allah purged my soul from love of you * When all knew secrets\n     cared I not reveal.\nI march to-morrow from your country and * Haply you'll speed me\n     nor fear aught unweal;\nAnd, when in person you be far from us, * Would heaven we knew\n     who shall your news reveal.\nWho kens if home will e'er us two contain * In dearest life with\n     union naught can stain!\"\n\nWhen Marjanah had made an end of her song, the Prince said to\nher, \"Brava, O damsel! Indeed, thou sayest a thing which had\noccurred to my mind and my tongue was near to speaking it.\" Then\nhe signed to the fourth damsel, who was a Cairene, by name Sitt\nal-Husn, and bade her tune her lute and sing to him upon the same\ntheme. So the Lady of Beauty tuned her lute and sang these\ncouplets,\n\n\"Patience is blest for weal comes after woe * And all things\n     stated time and ordinance show;\nHaps the Sultan, hight Fortune, prove unjust * Shifting the\n     times, and man excuse shall know:\nBitter ensueth sweet in law of change * And after crookedness\n     things straightest grow.\nThen guard thine honour, nor to any save * The noble knowledge of\n     the hid bestow:\nThese be vicissitudes the Lord commands * Poor men endure, the\n     sinner and the low.\"\n\nWhen Al-Abbas heard her make an end of her verses, they pleased\nhim and he said to her, \"Brava, O Sitt al-Husn! Indeed, thou hast\ndone away with anxiety from my heart and hast banished the things\nwhich had occurred to my thought.\" Then he sighed and signing to\nthe fifth damsel, who was from the land of the Persians and whose\nname was Marz\u00edyah (now she was the fairest of them all and the\nsweetest of speech and she was like unto a lustrous star, a model\nof beauty and loveliness and perfection and brightness and\njustness of shape and symmetric grace and had a face like the new\nmoon and eyes as they were gazelle's eyes) and said to her, \"O\nMarziyah, come forward and tune thy lute and sing to us on the\nsame theme, for indeed we are resolved upon faring to the land of\nAl-Yaman.\" Now this maiden had met many of the monarchs and had\nforegathered with the great; so she tuned her lute and sang these\ncouplets,\n\n\"Friend of my heart why leave thou lone and desolate these eyne?\n     * Fair union of our lots ne'er failed this sitting-stead of\n     mine!\nAnd ah! who dwellest singly in the heart and sprite of me, * (Be\n     I thy ransom!) desolate for loss of friend I pine!\nBy Allah! O thou richest form in charms and loveliness, * Give\n     alms to lover who can show of patience ne'er a sign!\nAlms of what past between us tway (which ne'er will I divulge) *\n     Of privacy between us tway that man shall ne'er divine:\nGrant me approval of my lord whereby t' o'erwhelm the foe * And\n     let my straitness pass away and doubtful thoughts malign:\nApproof of thee (an gained the meed) for me high rank shall gain\n     * And show me robed in richest weed to eyes of envy fain.\"\n\nWhen she had ended her song, all who were in the assembly wept\nfor the daintiness of her delivery and the sweetness of her\nspeech and Al-Abbas said to her, \"Brava, O Marz\u00edyah! Indeed, thou\nbewilderest the wits with the beauty of thy verse and the polish\nof thy speech.\"[FN#412] All this while Shafikah abode gazing\nupon her, and when she beheld the slave-girls of Al-Abbas and\nconsidered the charms of their clothing and the subtlety of their\nsenses and the delicacy of their delivery her reason flew from\nher head. Then she sought leave of Al-Abbas and returning to her\nmistress Mariyah, sans letter or reply, acquainted her with what\nshe had espied of the damsels and described to her the condition\nwherein he was of honour and delight, majesty, venerance and\nloftiness of rank. Lastly, she enlarged upon what she had seen of\nthe slave-girls and their case and that which they had said and\nhow they had incited Al-Abbas anent returning to his own country\nby the recitation of songs to the sound of the strings. When the\nPrincess heard this her slave-girl's report, she wept and wailed\nand was like to leave the world. Then she took to her pillow and\nsaid, \"O Shafikah, I will inform thee of a something which is not\nhidden from Allah the Most High, and 'tis that thou watch over me\ntill the Almighty decree the accomplishment of His destiny, and\nwhen my days are ended, take thou the necklace and the mantle\nwith which Al-Abbas gifted me and return them to him. I deem not\nhe will survive me, and if the Lord of All-might determine\nagainst him and his days come to an end, do thou give one charge\nto shroud us and entomb us both in one tomb.\" Then her case\nchanged and her colour waxed wan; and when Shafikah saw her\nmistress in this plight, she repaired to her mother and told her\nthat the lady Mariyah refused meat and drink. Asked the Queen,\n\"Since when hath this befallen her?\" and Shafikah answered,\n\"Since yesterday's date;\" whereat the mother was confounded and\nbetaking herself to her daughter, that she might inquire into her\ncase, lo and behold! found her as one dying. So she sat down at\nher head and Mariyah opened her eyes and seeing her mother\nsitting by her, sat up for shame before her. The Queen questioned\nher of her case and she said, \"I entered the Hammam and it\nstupefied me and prostrated me and left in my head an exceeding\npain; but I trust in Allah Al-mighty that it will cease.\" When\nher mother went out from her, Mariyah took to chiding the damsel\nfor that which she had done and said to her, \"Verily, death were\ndearer to me than this; so discover thou not my affair to any and\nI charge thee return not to the like of this fashion.\" Then she\nfainted and lay swooning for a whole hour, and when she came to\nherself, she saw Shafikah weeping over her; whereupon she pluckt\nthe necklace from her neck and the mantle from her body and said\nto the damsel, \"Lay them in a damask napkin and bear them to\nAl-Abbas and acquaint him with that wherein I am for the stress\nof severance and the strain of forbiddance.\" So Shafikah took\nthem and carried them to Al-Abbas, whom she found in readiness to\ndepart, being about to take horse for Al-Yaman. She went in to\nhim and gave him the napkin and that which was therein, and when\nhe opened it and saw what it contained, namely, the mantle and\nthe necklace, his chagrin was excessive and his eyes turned in\nhis head[FN#413] and his rage shot out of them. When Shafikah saw\nthat which betided him, she came forward and said to him, \"O\nbountiful lord, verily my mistress returneth not the mantle and\nthe necklace for despite; but she is about to quit the world and\nthou hast the best right to them.\" Asked he, \"And what is the\ncause of this?\" and Shafikah answered, \"Thou knowest. By Allah,\nnever among the Arabs nor the Ajams nor among the sons of the\nkings saw I a harder of heart than thou! Can it be a slight\nmatter to thee that thou troublest Mariyah's life and causest her\nto mourn for herself and quit the world for the sake of thy\nyouth?[FN#414] Thou wast the cause of her acquaintance with thee\nand now she departeth this life on thine account, she whose like\nAllah Almighty hath not created among the daughters of the\nkings.\" When Al-Abbas heard from the damsel these words, his\nheart burned for Mariyah and her case was not light to him, so he\nsaid to Shafikah, \"Canst thou bring me in company with her; so\nhaply I may discover her concern and allay whatso aileth her?\"\nSaid she, \"Yes, I can do that, and thine will be the bounty and\nthe favour.\" So he arose and followed her, and she preceded him,\ntill they came to the palace. Then she opened and locked behind\nthem four-and-twenty doors and made them fast with padlocks; and\nwhen he came to Mariyah, he found her as she were the downing\nsun, strown upon a T\u00e1if rug of perfumed leather,[FN#415]\nsurrounded by cushions stuffed with ostrich down, and not a limb\nof her quivered. When her maid saw her in this state, she offered\nto cry out; but Al-Abbas said to her, \"Do it not, but have\npatience till we discover her affair; and if Allah (be He\nextolled and exalted!) have decreed her death, wait till thou\nhave opened the doors to me and I have gone forth. Then do what\nseemeth good to thee.\" So saying, he went up to the Princess and\nlaying his hand upon her bosom, found her heart fluttering like a\ndoveling and the life yet hanging to her breast.[FN#416] So he\nplaced his hand on her cheek, whereat she opened her eyes and\nbeckoning to her maid, said to her by signs, \"Who is this that\ntreadeth my carpet and transgresseth against me?\"[FN#417] \"O my\nlady,\" cried Shafikah, \"this is Prince Al-Abbas, for whose sake\nthou forsakest the world.\" When Mariyah heard speak of Al-Abbas,\nshe raised her hand from under the coverlet and laying it upon\nhis neck, inhaled awhile his scent. Then she sat up and her\ncomplexion returned to her and they abode talking till a third\npart of the night was past. Presently, the Princess turned to her\nhandmaid and bade her fetch them somewhat of food, sweetmeats,\nand fruits, fresh and dry. So Shafikah brought what she desired\nand they ate and drank and abode on this wise without lewdness,\ntill night went and light came. Then said Al-Abbas, \"Indeed, the\nmorn breaketh. Shall I hie to my sire and bid him go to thy\nfather and seek thee of him in wedlock for me, in accordance with\nthe book of Allah Almighty and the practice of His Apostle (whom\nmay He save and assain!) so we may not enter into transgression?\"\nAnd Mariyah answered, saying, \"By Allah, 'tis well counselled of\nthee!\" So he went away to his lodging and naught befel between\nthem; and when the day lightened, she recited these couplets,\n\n\"O friends, morn-breeze with Morn draws on amain: * A\n     Voice[FN#418] bespeaks us, gladding us with 'plain.\nUp to the convent where our friend we'll sight * And wine more\n     subtile than the dust[FN#419] we'll drain;\nWhereon our friend spent all the coin he owned * And made the\n     nursling in his cloak contain;[FN#420]\nAnd, when we oped the jar, light opalline * Struck down the\n     singers in its search waylain.\nFrom all sides flocking came the convent-monks * Crying at top o'\n     voices, 'Welcome fain!'\nAnd we carousing sat, and cups went round, * Till rose the\n     Venus-star o'er Eastern plain.\nNo shame in drinking wine, which means good cheer * And love and\n     promise of prophetic strain![FN#421]\nHo thou, the Morn, our union sundering, * These joyous hours to\n     fine thou dost constrain.\nShow grace to us until our pleasures end, * And latest drop of\n     joy fro' friends we gain:\nYou have affection candid and sincere * And Love and Joy are best\n     of Faiths for men.\"\n\nSuch was the case with Mariyah; but as regards Al-Abbas, he\nbetook himself to his father's camp, which was pitched on the\nGreen Meadow, by the Tigris-side, and none might thread his way\nbetween the tents, for the dense network of the tent ropes. When\nthe Prince reached the first of the pavilions, the guards and\nservants came out to meet him from all sides and walked in his\nservice till he drew near the sitting-place of his sire, who knew\nof his approach. So he issued forth his marquee and coming to\nmeet his son, kissed him and made much of him. Then they returned\ntogether to the royal pavilion and when they had seated\nthemselves therein and the guards had taken up their station in\nattendance on them, the King said to Al-Abbas, \"O my son, get\nready thine affair, so we may go to our own land, for that the\nlieges in our absence are become as they were sheep lacking\nshepherd.\" Al-Abbas looked at his father and wept till he\nfainted, and when he recovered from his fit, he improvised and\nrecited these couplets,\n\n\"I embraced him,[FN#422] and straight I waxt drunk wi' the smell\n     * Of a fresh young branch wont in wealth to dwell.\nYea, drunken, but not by the wine; nay, 'twas * By draughts from\n     his lips that like wine-cups well:\nFor Beauty wrote on his cheek's fair page * 'Oh, his charms! take\n     refuge fro' danger fell!'[FN#423]\nMine eyes, be easy, since him ye saw; * Nor mote nor blearness\n     with you shall mell:\nIn him Beauty showeth fro' first to fine * And bindeth on hearts\n     bonds unfrangible:\nAn thou kohl thyself with his cheek of light * Thou'll find but\n     jasper and or in stelle:[FN#424]\nThe chiders came to reproach me when * For him longing and pining\n     my heart befel:\nBut I fear not, I end not, I turn me not * From his life, let\n     tell-tale his tale e'en tell:\nBy Allah, forgetting ne'er crossed my thought * While by life-tie\n     bound, or when ends my spell:\nAn I live I will live in his love, an I die * Of love and\n     longing, I'll cry, ''Tis well!'\"\n\nNow when Al-Abbas had ended his verses, his father said to him,\n\"I seek refuge for thee with Allah, O my son! Hast thou any want\nthou art powerless to win, so I may endeavour for thee therein\nand lavish my treasures in its quest.\" Cried Al-Abbas, \"O my\npapa, I have, indeed, an urgent need, on whose account I came\nforth of my mother-land and left my people and my home and\naffronted perils and horrors and became an exile, and I trust in\nAllah that it may be accomplished by thy magnanimous endeavour.\"\nQuoth the King, \"And what is thy want?\" and quoth Al-Abbas, \"I\nwould have thee go and ask for me to wife Mariyah, daughter of\nthe King of Baghdad, for that my heart is distracted with love of\nher.\" Then he recounted to his father his adventure from first to\nlast. When the King heard this from his son, he rose to his feet\nand calling for his charger of parade, took horse with\nfour-and-twenty Emirs of the chief officers of his empire. Then\nhe betook himself to the palace of the King of Baghdad who, when\nhe saw him coming, bade his chamberlains open the doors to them\nand going down himself to meet them, received him with all honour\nand hospitality and carried him and his into the palace; then\ncausing make ready for them carpets and cushions, sat down upon\nhis golden throne and seated the guest by his side upon a chair\nof gold, framed in juniper-wood set with pearls and jewels.\nPresently he bade bring sweetmeats and confections and scents and\ncommanded to slaughter four and-twenty head of sheep and the like\nof oxen and make ready geese and chickens and pigeons stuffed and\nboiled, and spread the tables; nor was it long before the meats\nwere served up in vessels of gold and silver. So they ate their\nsufficiency and when they had eaten their fill, the tables were\nremoved and the wine-service set on and the cups and flagons\nranged in ranks, whilst the Mamelukes and the fair slave-girls\nsat down, with zones of gold about their waists, studded with all\nmanner pearls, diamonds, emeralds, rubies and other jewels.\nMoreover, the king bade fetch the musicians; so there presented\nthemselves before him twenty damsels with lutes and\npsalteries[FN#425] and viols, and smote upon instruments of music\nplaying and performing on such wise that they moved the assembly\nto delight. Then said Al-Aziz to the King of Baghdad, \"I would\nfain speak a word to thee; but do thou not exclude from us those\nwho are present. An thou consent unto my wish thine is ours and\non thee shall be whatso is on us;[FN#426] and we will be to thee\na mighty forearm against all unfriends and foes.\" Quoth Ins bin\nKays, \"Say what thou wilt, O King, for indeed thou excellest in\nspeech and in whatso thou sayest dost hit the mark.\" So Al-Aziz\nsaid to him, \"I desire that thou marry thy daughter Mariyah to my\nson Al-Abbas, for thou knowest what he hath of beauty and\nloveliness, brightness and perfect grace and his frequentation of\nthe valiant and his constancy in the stead of cut-and-thrust.\"\nSaid Ins bin Kays, \"By Allah, O King, of my love for Mariyah, I\nhave appointed her mistress of her own hand; accordingly,\nwhomsoever she chooseth of the folk, to him will I wed her.\" Then\nhe arose to his feet and going in to his daughter, found her\nmother with her; so he set out to them the case and Mariyah said,\n\"O my papa, my wish followeth thy word and my will ensueth thy\nwill; so whatsoever thou chooseth, I am obedient to thee and\nunder thy dominion.\" Therewith the King knew that Mariyah\ninclined to Al-Abbas; he therefore returned forthright to King\nAl-Aziz and said to him, \"May Allah amend the King! Verily, the\nwish is won and there is no opposition to that thou commandest.\"\nQuoth Al-Aziz, \"By Allah's leave are wishes won. How deemest\nthou, O King, of fetching Al-Abbas and documenting the\nmarriage-contract between Mariyah and him?\" and quoth Ins bin\nKays, \"Thine be the rede.\" So Al-Aziz sent after his son and\nacquainted him with that which had passed; whereupon Al-Abbas\ncalled for four-and-twenty mules and ten horses and as many\ncamels and loaded the mules with fathom-long pieces of silk and\nrugs of leather and boxes of camphor and musk and the camels and\nhorses with chests of gold and silver. Eke, he took the richest\nof the stuffs and wrapping them in wrappers of gold-purfled\nsilk, laid them on the heads of porters,[FN#427] and they fared\non with the treasures till they reached the King of Baghdad's\npalace, whereupon all who were present dismounted in honour of\nAl-Abbas and escorting him in a body to the presence of Ins bin\nKays, displayed to the King all that they had with them of things\nof price. The King bade carry all this into the store rooms of\nthe Harim and sent for the Kazis and the witnesses, who wrote out\nthe contract and married Mariyah to Al-Abbas, whereupon the\nPrince commanded slaughter one thousand head of sheep and five\nhundred buffaloes. So they spread the bride-feast and bade\nthereto all the tribes of the Arabs, men of tents and men of\ntowns, and the banquet continued for the space of ten days. Then\nAl-Abbas went into Mariyah in a commendable and auspicious hour\nand lay with her and found her a pearl unthridden and a goodly\nfilly no rider had ridden;[FN#428] wherefore he rejoiced and was\nglad and made merry, and care and sorrow ceased from him and his\nlife was pleasant and trouble departed and he ceased not abiding\nwith her in most joyful case and in the most easeful of life,\ntill seven days were past, when King Al-Aziz resolved to set out\nand return to his realm and bade his son seek leave of his\nfather-in-law to depart with his wife to his own country. So\nAl-Abbas spoke of this to King Ins, who granted him the\npermission he sought; whereupon he chose out a red\ncamel,[FN#429] taller and more valuable than the rest of the\ncamels, and loading it with apparel and ornaments, mounted\nMariyah in a litter thereon. Then they spread the ensigns and the\nstandards, whilst kettle-drums beat and the trumpets blared, and\nset out upon the homewards way. The King of Baghdad rode forth\nwith them and companied them three days' journey on their route,\nafter which he farewelled them and returned with his troops to\nBaghdad. As for King Al-Aziz and his son, they fared on night and\nday and gave not over going till there remained but three days'\njourney between them and Al-Yaman, when they despatched three men\nof the couriers to the Prince's mother to report that they were\nbringing with them Mariyah, the King's daughter of Baghdad, and\nreturning safe and laden with spoil. When the Oueen-mother heard\nthis, her wit took wings for joy and she adorned the slave-girls\nof Al-Abbas after the finest fashion. Now he had ten hand-maids,\nas they were moons, whereof his father had carried five with him\nto Baghdad, as hath erst been set forth, and the remaining five\nabode with his mother. When the dromedary-posts[FN#430] came,\nthey were certified of the approach of Al-Abbas, and when the sun\neasted and their flags were seen flaunting, the Prince's mother\ncame out to meet her son; nor on that day was there great or\nsmall, boy or grey-beard, but went forth to greet the king. Then\nthe kettle-drums of glad tidings beat and they entered in the\nutmost of pomp and the extreme of magnificence; so that the\ntribes and the townspeople heard of them and brought them the\nrichest of gifts and the rarest of presents and the Prince's\nmother rejoiced with joy exceeding. They butchered beasts and\nspread mighty bride-feasts for the people and kindled\nfires,[FN#431] that it might be visible afar to townsman and\ntribesman that this was the house of hospitality and the stead of\nthe wedding-festival, to the intent that, if any passed them by,\nit should be of his own sin against himself. So the folk came to\nthem from all districts and quarters and in this way they abode\ndays and months. Presently the Prince's mother bade fetch the\nfive slave-girls to that assembly; whereupon they came and the\nten damsels met. The queen seated five of them on her son's right\nhand and the other five on his left and the folk gathered about\nthem. Then she bade the five who had remained with her speak\nforth somewhat of poesy, so they might entertain therewith the\nseance and that Al-Abbas might rejoice thereat. Now she had clad\nthem in the costliest of clothes and adorned them with trinkets\nand ornaments and moulded work of gold and silver and collars of\ngold, wrought with pearls and gems. So they paced forward, with\nharps and lutes and zithers and recorders and other instruments\nof music before them, and one of them, a damsel who came from the\nland of China and whose name was B\u00e1'\u00fathah, advanced and screwed\nup the strings of her lute. Then she cried out from the top of\nher head and recited these couplets,\n\n\"Indeed your land returned, when you returned, * To whilom light\n     which overgrew its gloom:\nGreen grew the land that was afore dust-brown. * And fruits that\n     failed again showed riping bloom:\nAnd clouds rained treasures after rain had lacked, * And plenty\n     poured from earth's re-opening womb.\nThen ceased the woes, my lords, that garred us weep, * With tears\n     like dragons' blood, our severance-doom,\nWhose length, by Allah, made me yeam and pine, * Would Heaven, O\n     lady mine, I were thy groom!\"\n\nWhen she had ended her song, all who were present were delighted\nand Al-Abbas rejoiced in this. Then he bade the second damsel\nsing somewhat on the same theme. So she came forward and\ntightening the strings of her harp, which was of balass\nruby,[FN#432] raised her voice in a plaintive air and improvised\nthese couplets,\n\n\"Brought the Courier glad news of our absentees,[FN#433] * To\n     please us through those who had wrought us unease:\nCried I, 'My life ransom thee, messenger man, * Thou hast kept\n     thy faith and thy boons are these.'\nAn the nightlets of union in you we joyed * When fared you naught\n     would our grief appease;\nYou sware that folk would to folk be true, * And you kept your\n     oaths as good faith decrees.\nTo you made I oath true lover am I * Heaven guard me when sworn\n     from all perjuries:\nI fared to meet you and loud I cried, * 'Aha, fair welcome when\n     come you please!\"\nAnd I joyed to meet you and when you came, * Deckt all the\n     dwelling with tapestries,\nAnd death in your absence to us was dight, * But your presence\n     bringeth us life and light.\"\n\nWhen she had made an end of her verse, Al-Abbas bade the third\ndamsel (who came from Samarkand of Ajam-land and whose name was\nRumm\u00e1nah) sing, and she answered, \"To hear is to obey.\" Then she\ntook the zither and crying out from the midst of her head,\nrecited and sang these couplets,[FN#434]\n\n\"My watering mouth declares thy myrtle-cheek my food to be * And\n     cull my lips thy side-face rose, who lily art to me!\nAnd twixt the dune and down there shows the fairest flower that\n     blooms * Whose fruitage is granado's fruit with all\n     granado's blee.[FN#435]\nForget my lids of eyne their sleep for magic eyes of him; *\n     Naught since he fared but drowsy charms and languorous air I\n     see.[FN#436]\nHe shot me down with shaft of glance from bow of eyebrow sped: *\n     What Chamberlain[FN#437] betwixt his eyes garred all my\n     pleasure flee?\nHaply shall heart of me seduce his heart by weakness' force *\n     E'en as his own seductive grace garred me love-ailment dree.\nFor an by him forgotten be our pact and covenant * I have a King\n     who never will forget my memory.\nHis sides bemock the bending charms of waving Tamarisk,[FN#438] *\n     And in his beauty-pride he walks as drunk with coquetry:\nHis feet and legs be feather-light whene'er he deigns to run *\n     And say, did any ride the wind except 'twere\n     Solomon?\"[FN#439]\n\nTherewith Al-Abbas smiled and her verses pleased him. Then he\nbade the fourth damsel come forward and sing (now she was from\nthe Sundown-land[FN#440] and her name was Balakhsh\u00e1); so she came\nforward and taking the lute and the zither, tuned the strings and\nsmote them in many modes; then she returned to the first and\nimprovising, sang these couplets,\n\n\"When to the s\u00e9ance all for pleasure hied * Thy lamping eyes\n     illumined its every side;\n While playing round us o'er the wine-full bowl * Those\n     necklace-pearls old wine with pleasure plied,[FN#441]\nTill wits the wisest drunken by her grace * Betrayed for joyance\n     secrets sages hide;\nAnd, seen the cup, we bade it circle round * While sun and moon\n     spread radiance side and wide.\nWe raised for lover veil of love perforce * And came glad tidings\n     which new joys applied:\nLoud sang the camel-guide; won was our wish * Nor was the secret\n     by the spy espied:\nAnd, when my days were blest by union-bliss * And to all-parting\n     Time was aid denied,\nEach 'bode with other, clear of meddling spy * Nor feared we hate\n     of foe or neighbour-pride.\nThe sky was bright, friends came and severance fared * And\n     Love-in-union rained boons multiplied:\nSaying 'Fulfil fair union, all are gone * Rivals and fears lest\n     shaming foe deride:'\nFriends now conjoin\u00e8d are: wrong passed away * And meeting-cup\n     goes round and joys abide:\nOn you be Allah's Peace with every boon * Till end the dooming\n     years and time and tide.\"\n\nWhen Balakhsha had ended her verse, all present were moved to\ndelight and Al-Abbas said to her, \"Brava, O damsel!\" Then he bade\nthe fifth damsel come forward and sing (now she was from the land\nof Syria and her name was Rayh\u00e1nah; she was passing of voice and\nwhen she appeared in an assembly, all eyes were fixed upon her),\nso she came forward and taking the viol (for she was used to play\nupon all instruments) recited and sang these couplets,\n\n\"Your me-wards coming I hail to sight; * Your look is a joy\n     driving woe from sprite:\nWith you love is blest, pure and white of soul; * Life's sweet\n     and my planet grows green and bright:\nBy Allah, you-wards my pine ne'er ceased * And your like is rare\n     and right worthy hight.\nAsk my eyes an e'er since the day ye went * They tasted sleep,\n     looked on lover-wight:\nMy heart by the parting-day was broke * And my wasted body\n     betrays my plight:\nCould my blamers see in what grief am I, * They had wept in\n     wonder my loss, my blight!\nThey had joined me in shedding torrential tears * And like me\n     a-morn had shown thin and slight:\nHow long for your love shall your lover bear * This weight o'er\n     much for the hill's strong height?\nBy Allah what then for your sake was doomed * To my heart, a\n     heart by its woes turned white!\nAn showed I the fires that aye flare in me, * They had 'flamed\n     Eastern world and earth's Western site.\nBut after this is my love fulfilled * With joy and gladness and\n     mere delight;\nAnd the Lord who scattered hath brought us back * For who doeth\n     good shall of good ne'er lack.\"\n\nWhen King Al-Aziz heard the damsel's song, both words and verses\npleased him and he said to Al-Abbas, \"O my son, verily long\nversifying hath tired these damsels, and indeed they make us\nyearn after the houses and the homesteads with the beauty of\ntheir songs. These five have adorned our meeting with the charm\nof their melodies and have done well in that which they have said\nbefore those who are present; so we counsel thee to free them for\nthe love of Allah Almighty.\" Quoth Al-Abbas, \"There is no command\nbut thy command;\" and he enfranchised the ten damsels in the\nassembly; whereupon they kissed the hands of the King and his son\nand prostrated themselves in thanksgiving to the Lord of\nAll-might. Then they put off that which was upon them of\nornaments and laying aside the lutes and other instruments of\nmusic, kept to their houses like modest women and veiled, and\nfared not forth.[FN#442] As for King Al-Aziz, he lived after this\nseven years and was removed to the mercy of Almighty Allah; when\nhis son Al-Abbas bore him forth to burial as beseemeth kings and\nlet make for him perlections and professional recitations of the\nKoran. He kept up the mourning for his father during four\nsuccessive weeks, and when a full-told month had elapsed he sat\ndown on the throne of the kingship and judged and did justice and\ndistributed silver and gold. He also loosed all who were in the\njails and abolished grievances and customs dues and righted the\noppressed of the oppressor; so the lieges prayed for him and\nloved him and invoked on him endurance of glory and continuance\nof kingship and length of life and eternity of prosperity and\nhappiness. The troops submitted to him, and the hosts from all\nparts of the kingdom, and there came to him presents from each\nand every land: the kings obeyed him and many were his warriors\nand his grandees, and his subjects lived with him the most\neaseful of lives and the most delightsome. Meanwhile, he ceased\nnot, he and his beloved, Queen Mariyah, in the most enjoyable of\nlife and the pleasantest, and he was vouchsafed by her children;\nand indeed there befel friendship and affection between them and\nthe longer their companionship was prolonged, the more their love\nwaxed, so that they became unable to endure each from other a\nsingle hour, save the time of his going forth to the Divan, when\nhe would return to her in the liveliest that might be of longing.\nAnd after this fashion they abode in all solace of life and\nsatisfaction till there came to them the Destroyer of delights\nand the Severer of societies. So extolled be the Eternal whose\nsway endureth for ever and aye, who never unheedeth neither dieth\nnor sleepeth! This is all that hath come down to us of their\ntale, and so the Peace!\n\n\n\n\n                SHAHRAZAD AND SHAHRYAR.[FN#443]\n\n\n\nKing Shahryar marveled at this history[FN#444] and said, \"By\nAllah, verily, injustice slayeth its folk!\"[FN#445] And he was\nedified by that, wherewith Shahrazad bespoke him and sought help\nof Allah the Most High.  Then said he to her, \"Tell me another of\nthy tales, O Shahrazad; supply me with a pleasant story and this\nshall be the completion of the story-telling.\"Shahrazad replied,\n\"With love and gladness! It hath reached me, O\nauspicious King, that a man once declared to his mates, 'I will\nset forth to you a means of security against annoy.'  A friend of\nmine once related to me and said, \"We attained to security\nagainst annoy, and the origin of it was other than this; that is,\nit was the following'\"[FN#446]\n\n\n\n\n             TALE OF THE TWO KINGS AND THE WAZIR'S\n                       DAUGHTERS.[FN#447]\n\n\n\nI overtravelled whilome lands and climes and towns and visited\nthe cities of high renown and traversed the ways of dangers and\nhardships.  Towards the last of my life, I entered a city of the\ncities of China,[FN#448] wherein was a king of the Chosro\u00ebs and\nthe Tobbas[FN#449] and the C\u00e6sars.[FN#450]  Now that city had\nbeen peopled with its inhabitants by means of justice and equity;\nbut its then king was a tyrant dire who despoiled lives and souls\nat his desire; in fine, there was no warming oneself at his fire,\n[FN#451] for that indeed he oppressed the believing band and\nwasted the land.  Now he had a younger brother, who was king in\nSamarkand of the Persians, and the two kings sojourned a while\nof time, each in his own city and stead, till they yearned unto\neach other and the elder king despatched his Wazir to fetch his\nyounger brother.  When the Minister came to the King of Samarkand\nand acquainted him with his errand, he submitted himself to the\nbidding of his brother and answered, \"To hear is to obey.\"  Then\nhe equipped himself and made ready for wayfare and brought forth\nhis tents and pavilions.  A while after midnight, he went in to\nhis wife, that he might farewell her, and found her with a\nstrange man, lying by her in one bed.  So he slew them both and\ndragging them out by the feet, cast them away and set forth on\nhis march.  When he came to his brother's court, the elder king\nrejoiced in him with joy exceeding and lodged him in the pavilion\nof hospitality beside his own palace. Now this pavilion\noverlooked a flower-garden belonging to the elder brother and\nthere the younger abode with him some days.  Then he called to\nmind that which his wife had done with him and remembered her\nslaughter and bethought him how he was a king, yet was not exempt\nfrom the shifts of Time; and this affected him with exceeding affect,\nso that it drave him to abstain from meat and drink, or, if he\nate anything, it profited him naught.  When his brother saw him\non such wise, he deemed that this had betided him by reason of\nseverance from his folk and family, and said to him, \"Come, let\nus fare forth a-coursing and a-hunting.\"  But he refused to go\nwith him; so the elder brother went to the chase, while the\nyounger abode in the pavilion aforesaid.  Now, as he was\ndiverting himself by looking out upon the flower-garden from the\nlatticed window of the palace, behold, he saw his brother's wife\nand with her ten black slaves and ten slave-girls.  Each slave\nlaid hold of a damsel and another slave came forth and did the\nlike with the queen; and when they had their wills one of other\nthey all returned whence they came.  Hereat there betided the\nKing of Samarkand exceeding surprise and solace and he was made\nwhole of his malady, little by little.  After a few days, his\nbrother returned, and finding him cured of his complaint, said to\nhim, \"Tell me, O my brother, what was the cause of thy sickness\nand thy pallor, and what is the reason of the return of health to\nthee and of rosiness to thy face after this?\"  So he acquainted\nhim with the whole case and this was grievous to him; but they\nhid their affair and agreed to leave the kingship and fare forth\na-pilgrimaging and adventuring at hap-hazard, for they deemed\nthat there had befallen none the like of what had befallen them.\nAccordingly, they went forth and as they journeyed, they saw by\nthe way a woman imprisoned in seven chests, whereon were five\npadlocks, and sunken deep in the midst of the salt sea, under the\nguardianship of an Ifrit; yet for all this that woman issued out\nof the ocean and opened those padlocks and coming forth of those\nchests, did what she would with the two brothers, after she had\npractised upon the Ifrit.  When the two kings saw that woman's\nfashion and how she circumvented the Ifrit, who had lodged her in\nthe abyss of the main, they turned back to their kingdoms and the\nyounger betook himself to Samarkand, whilst the elder returned to\nChina and contrived for himself a custom in the slaughter of\ndamsels, which was, his Wazir used to bring him every night a\ngirl, with whom he lay that night, and when he arose in the\nmorning, he gave her to the Minister and bade him do her die.\nAfter this fashion he abode a long time, whilst the folk murmured and God's creatures were destroyed and the commons cried\nout by reason of that grievous affair into which they were fallen\nand feared the wrath of Allah Almighty, dreading lest He destroy\nthem by means of this.  Still the king persisted in that practice\nand in his blameworthy intent of the killing of damsels and the\ndespoilment of maidens concealed by veils,[FN#452] wherefore the\ngirls sought succour of the Lord of All-might, and complained to\nHim of the tyranny of the king and of his oppression.  Now the\nking's Wazir had two daughters, sisters german, the elder of whom\nhad read the books and made herself mistress of the sciences and\nstudied the writings of the sages and the stories of the cup-\ncompanions,[FN#453] and she was a maiden of abundant lore and\nknowledge galore and wit than which naught can be more.  She\nheard that which the folk suffered from that king in his misuage\nof their children; whereupon ruth for them gat hold of her and\njealousy and she besought Allah Almighty that He would bring the\nking to renounce that his new and accursed custom,[FN#454] and the\nLord answered her prayer.  Then she consulted her younger sister\nand said to her, \"I mean to devise a device for freeing the\nchildren of folk; to wit, I will go up to the king and offer\nmyself to marry him, and when I come to his presence, I will send\nto fetch thee.  When thou comest in to me and the king had his\ncarnal will of me, do thou say to me, 'O my sister, let me hear a\nstory of thy goodly stories, wherewith we may beguile the waking\nhours of our night, till the dawn, when we take leave of each\nother; and let the king hear it likewise!'\"  The other replied,\n\"'Tis well; forsure this contrivance will deter the king from\nthis innovation he practiseth and thou shalt be requited with\nfavour exceeding and recompense abounding in the world to come,\nfor that indeed thou perilest thy life and wilt either perish or\nwin to thy wish.\"  So she did this and Fortune favoured her and\nthe Divine direction was vouchsafed to her and she discovered her\ndesign to her sire, the Wazir, who thereupon forbade her, fearing\nher slaughter.  However, she repeated her words to him a second\ntime and a third, but he consented not.  Then he cited to her a\nparable, which should deter her, and she cited to him a parable\nof import contrary to his, and the debate was prolonged between\nthem and the adducing of instances, till her father saw that he\nwas powerless to turn her from her purpose and she said to him,\n\"There is no help but that I marry the King, so haply I may be a\nsacrifice for the children of the Moslems: either I shall turn\nhim from this his heresy or I shall die.\"  When the Minister\ndespaired of dissuading her, he went up to the king and\nacquainted him with the case, saying, \"I have a maiden daughter\nand she desireth to give herself in free gift to the King.\"\nQuoth the King, \"How can thy soul consent to this, seeing that\nthou knowest I abide but a single night with a girl and when I\narise on the morrow, I do her dead, and 'tis thou who slayest\nher, and again and again thou hast done this?\" Quoth the Wazir,\n\"Know, O king, that I have set forth all this to her, yet\nconsented she not to aught, but needs must she have thy company\nand she chooseth to come to thee and present herself before thee,\nalbeit I have cited to her the sayings of the sages; but she hath\nanswered me with more than that which I said to her and\ncontrariwise.\"  Then quoth the king, \"Suffer her visit me this\nnight and to-morrow morning come thou and take her and kill her;\nand by Allah, an thou slay her not, I will slay thee and her\nalso!\"  The Minister obeyed the king's bidding and going out from\nthe presence returned home.  When it was night, he took his elder\ndaughter and carried her up to the king; and when she came before\nhim she wept;[FN#455] whereupon he asked her, \"What causeth thee\nto weep? Indeed, 'twas thou who willedst this.\"  She answered, \"I\nweep not but of longing after my little sister; for that, since\nwe grew up, I and she, I have never been parted from her till\nthis day; so, an it please the King to send for her, that I may\nlook on her, and listen to her speech and take my fill of her\ntill the morning, this were a boon and an act of kindness of the\nKing.\"  So he bade fetch the damsel and she came.  Then there\nbefel that which befel of his union with the elder\nsister,[FN#456] and when he went up to his couch, that he might\nsleep, the younger sister said to her elder, \"Allah upon thee, O\nmy sister, an thou be not asleep, tell us a tale of thy goodly\ntales, wherewith me may beguile the watches of our night, ere day\ndawn and parting.\"  Said she, \"With love and gladness;\" and fell\nto relating to her, whilst the king listened.  Her story was\ngoodly and delectable, and whilst she was in the midst of\ntelling it, the dawn brake.  Now the king's heart clave to the\nhearing of the rest of the story; so he respited her till the\nmorrow; and, when it was the next night, she told him a tale\nconcerning the marvels of the land and the wonders of Allah's\ncreatures which was yet stranger and rarer than the first.  In\nthe midst of the recital, appeared the day and she was silent\nfrom the permitted say.  So he let her live till the following\nnight, that he might hear the end of the history and after that\nslay her.  On this wise it fortuned with her; but as regards the\npeople of the city, they rejoiced and were glad and blessed the\nWazir's daughters, marvelling for that three days had passed and\nthat the king had not put his bride to death and exulting in that\nhe had returned to the ways of righteousness and would never\nagain burthen himself with blood-guilt against any of the maidens\nof the city.  Then, on the fourth night, she related to him a\nstill more extraordinary adventure, and on the fifth night she\ntold him anecdotes of Kings and Wazirs and Notables.  Brief, she\nceased not to entertain him many days and nights, while the king still\nsaid to himself, \"Whenas I shall have heard the end of the tale,\nI will do her die,\" and the people redoubled their marvel and\nadmiration.  Also, the folk of the circuits and cities heard of\nthis thing, to wit, that the king had turned from his custom and\nfrom that which he had imposed upon himself and had renounced his\nheresy, wherefor they rejoiced and the lieges returned to the\ncapital and took up there abode therein, after they had departed\nthence; and they were constant in prayer to Allah Almighty that\nHe would stablish the king in his present stead.\" \"And this,\" said\nShahrazad, \"is the end of that which my friend related to me.\"\nQuoth Shahryar,[FN#457] \"O Shahrazad, finish for us the tale thy\nfriend told thee, inasmuch as it resembleth the story of a King\nwhom I knew; but fain would I hear that which betided the people\nof this city and what they said of the affair of the King, so I\nmay return from the case wherein I was.\"  She replied, \"With love\nand gladness!\"  Know, O auspicious king and lord of right rede\nand praiseworthy meed and prowest of deed, that, when the folk\nheard how the king had put away from him his malpractice and\nreturned from his unrighteous wont, they rejoiced in this with\njoy exceeding and offered up prayers for him.  Then they talked\none with other of the cause of the slaughter of the maidens, and\nthe wise said, \"Women are not all alike, nor are the fingers of\nthe hand alike.\"  Now when King Shahryar heard this story he came\nto himself and awakening from his drunkenness,[FN#458] said, \"By\nAllah, this story is my story and this case is my case, for that\nindeed I was in reprobation and danger of judgment till thou\nturnedst me back from this into the right way, extolled be the\nCauser of causes and the Liberator of necks!\" presently adding,\n\"Indeed, O Shahrazad, thou hast awakened me to many things and\nhast aroused me from mine ignorance of the right.\" Then said she\nto him, \"O chief of the kings, the wise say, 'The kingship is a\nbuilding, whereof the troops are the base, and when the\nfoundation is strong, the building endureth;' wherefore it\nbehoveth the king to strengthen the foundation, for that they\nsay, 'Whenas the base is weak, the building falleth.'  In like\nfashion it befitteth the king to care for his troops and do\njustice among his lieges, even as the owner of the garden careth\nfor his trees and cutteth away the weeds that have no profit in\nthem; and so it befitteth the king to look into the affairs of\nhis Ryots and fend off oppression from them.  As for thee, O\nking, it behoveth thee that thy Wazir be virtuous and experienced\nin the requirements of the people and the peasantry; and indeed\nAllah the Most High hath named his name[FN#459] in the history of\nMus\u00e0 (on whom be the Peace!) when he saith, 'And make me a Wazir\nof my people, Aaron.'  Now could a Wazir have been dispensed\nwithal, Moses son of Imr\u00e1n had been worthier than any to do\nwithout a Minister.  As for the Wazir, the Sultan discovereth\nunto him his affairs, private and public; and know, O king, that\nthe likeness of thee with the people is that of the leach with\nthe sick man; and the essential condition of the Minister is that\nhe be soothfast in his sayings, reliable in all his relations,\nrich in ruth for the folk and in tenderness of transacting with\nthem.  Verily, it is said, O king, that good troops be like the\ndruggist; if his perfumes reach thee not, thou still smellest the\nfragrance of them; and bad entourage be like the blacksmith; if\nhis sparks burn thee not, thou smellest his evil smell.  So it\nbefitteth thee to take to thyself a virtuous Wazir, a veracious\ncounsellor, even as thou takest unto thee a wife displayed before\nthy face, because thou needest the man's righteousness for thine\nown right directing, seeing that, if thou do righteously, the\ncommons will do right, and if thou do wrongously, they will also\ndo wrong.\"  When the King heard this, drowsiness overcame him and\nhe slept and presently awaking, called for the candles; so they\nwere lighted and he sat down on his couch and seating Shahrazad\nby him, smiled in her face.  She kissed the ground before him and\nsaid, \"O king of the age and lord of the time and the years,\nextolled be the Forgiving, the Bountiful, who hath sent me to\nthee, of His grace and good favour, so I have incited thee to\nlonging after Paradise; for verily this which thou wast wont do\nwas never done of any of the kings before thee.  Then laud be to\nthe Lord who hath directed thee into the right way, and who from\nthe paths of frowardness hath diverted thee!  As for women, Allah\nAlmighty maketh mention of them also when He saith in His Holy\nBook, 'Truly, the men who resign themselves to Allah[FN#460] and\nthe women who resign themselves, and the true-believing men and\nthe true-believing women and the devout men and the devout women\nand truthful men and truthful women, and long-suffering men and\nlong-suffering women, and the humble men and the humble women,\nand charitable men and charitable women, and the men who fast and\nthe women who fast, and men who guard their privities and women\nwho guard their privities, and men who are constantly mindful of\nAllah and women who are constantly mindful, for them Allah hath\nprepared forgiveness and a rich reward.'[FN#461]  As for that\nwhich hath befallen thee, verily, it hath befallen many kings\nbefore thee and their women have falsed them, for all they were\nmore majestical of puissance than thou, and mightier of kingship\nand had troops more manifold.  If I would, I could relate unto\nthee, O king, concerning the wiles of women, that whereof I\nshould not make an end all my life long; and indeed, in all these\nmy nights that I have passed before thee, I have told thee many\ntales of the wheedling of women and of their craft; but soothly\nthe things abound on me;[FN#462] so, an thou please, O king, I\nwill relate to thee somewhat of that which befel olden kings of\nperfidy from their women and of the calamities which overtook\nthem by reason of these deceivers.\"\" Asked the king, \"How so?\nTell on;\" and she answered, \"Hearkening and obedience.  It hath\nbeen told me, O king, that a man once related to a company the\nfollowing tale of\n\n\n\n\n             THE CONCUBINE AND THE CALIPH.\"[FN#463]\n\n\n\nOne day of the days, as I stood at the door of my house, and the\nheat was excessive, behold, I saw a fair woman approaching, and\nwith her a slave-girl carrying a parcel.  They gave not over\ngoing till they came up to me, when the woman stopped and asked\nme, \"Hast thou a draught of water?\"  answered I, \"Yes, enter the\nvestibule, O my lady, so thou mayst drink.\"  Accordingly she\ncame in and I went up into the house and fetched two gugglets of\nearthenware, smoked with musk[FN#464] and full of cold water.\nShe took one of them and discovered her face, the better to\ndrink; whereupon I saw that she was as the rising moon or the\nresplendent sun and said to her, \"O my lady, wilt thou not come\nup into the house, so thou mayst rest thyself till the air cool\nand afterwards fare thee to thine own place?\"  quoth she, \"Is\nthere none with thee?\" and quoth I, \"Indeed I am a bachelor and\nhave none belonging to me, nor is there a wight in the\nsite;[FN#465] whereupon she said, \"An thou be a stranger, thou\nart he in quest of whom I was going about.\"  So she went up into\nthe house and doffed her walking dress and I found her as she\nwere the full moon. I brought her what I had by me of food and\ndrink and said to her, \"O my lady, excuse me: this is all that is\nready;\" and said she, \"This is right good[FN#466] and indeed 'tis\nwhat I sought.\" Then she ate and gave the slave-girl that which\nwas left; after which I brought her a casting-bottle of musked\nrose-water, and she washed her hands and abode with me till the\nseason of mid-afternoon prayer, when she brought out of the\nparcel she had with her a shirt and trousers and an upper\ngarment[FN#467] and a gold-worked kerchief and gave them to me;\nsaying, \"Know that I am one of the concubines of the Caliph, and\nwe be forty concubines, each of whom hath a cicisbeo who cometh\nto her as often as she would have him; and none is without a\nlover save myself, wherefore I came forth this day to get me a\ngallant and now I have found thee.  Thou must know that the\nCaliph lieth each night with one of us, whilst the other nine-\nand-thirty concubines take their ease with the nine-and-thirty\nmasculines, and I would have thee company with me on such a day, when do\nthou come up to the palace of the Caliph and sit awaiting me in\nsuch a place, till a little eunuch come out to thee and say to\nthee a certain watch-word which is, 'Art thou Sandal?'  Answer\n'Yes,' and wend thee with him.\"  Then she took leave of me and I\nof her, after I had strained her to my bosom and thrown my arms\nround her neck and we had exchanged kisses awhile.  So she fared\nforth and I abode patiently expecting the appointed day, till it\ncame, when I arose and went out, intending for the trysting-place; but a friend of mine met me by the way and made me go home\nwith him.  I accompanied him and when I came up into his sitting-\nchamber he locked the door on me and walked out to fetch what we\nmight eat and drink.  He was absent until midday, then till the\nhour of mid-afternoon prayer, whereat I was chagrined with sore\nconcern.  Then he was missing until sundown, and I was like to\ndie of vexation and impatience; and indeed he returned not and I\npassed my night on wake, nigh upon death, for the door was locked\non me, and my soul was like to depart my body on account of the\nassignation.  At daybreak, my friend returned and opening the\ndoor, came in, bringing with him meat-pudding[FN#468] and\nfritters and bees' honey, and said to me, \"By Allah, thou must\nneeds excuse me, for that I was with a company and they locked\nthe door on me and have let me go but this very moment.\"  I\nreturned him no reply; however, he set before me that which was\nwith him and I ate a single mouthful and went out running at\nspeed so haply I might overtake the rendezvous which had escaped\nme.  When I came to the palace, I saw over against it eight-and-\nthirty gibbets set up, whereon were eight-and-thirty men\ncrucified, and under them eight-and-thirty[FN#469] concubines as\nthey were moons.  So I asked the cause of the crucifixion of the\nmen and concerning the women in question, and it was said unto\nme, \"The men thou seest crucified the Caliph found with yonder\ndamsels, who be his bed-fellows.\"  When I heard this, I\nprostrated myself in thanksgiving to Allah and said, \"The\nAlmighty require thee with all good, O my friend!\"  For had he\nnot invited me and locked me up in his house that night, I had\nbeen crucified with these men, wherefore Alhamdolillah--laud to\nthe Lord!  \"On this wise\" (continued Shahrazad), \"none is safe\nfrom the calamities of the world and the vicissitudes of Time,\nand in proof of this, I will relate unto thee yet another story\nstill rarer and stranger than this.  Know, O king, that one said\nto me: A friend of mine, a merchant, told me the following tale of\n\n\n\n\n               THE CONCUBINE OF AL-MAAMUN[FN#470]\n\n\n\nAs I sat one day in my shop, there came up to me a fair woman, as\nshe were the moon at its rising, and with her a hand-maid.  Now I\nwas a handsome man in my time; so that lady sat down on my\nshop[FN#471] and buying stuffs of me, paid the price and went her\nways.  I asked the girl anent her and she answered, \"I know not\nher name.\"  Quoth I, \"Where is her abode?\"  Quoth she, \"In\nheaven;\" and I, \"She is presently on the earth; so when doth she\nascend to heaven and where is the ladder by which she goeth\nup?\"[FN#472]  The girl retorted, \"She hath her lodging in a\npalace between two rivers,[FN#473] that is, in the palace of Al-\nMaam\u00fan al-H\u00e1kim bi-Amri 'llah.\"[FN#474]  Then said I, \"I am a\ndead man, without a doubt;\" but she replied, \"Have patience, for\nneeds must she return to thee and buy other stuffs of thee.\"  I\nasked, \"And how cometh it that the Commander of the Faithful\ntrusteth her to go out?\" and she answered, \"He loveth her with\nexceeding love and is wrapped up in her and crosseth her not.\"\nThen the slave-girl went away, running after her mistress;\nwhereupon I left the shop and followed them, so I might see her\nabiding-place. I kept them in view all the way, till she\ndisappeared from mine eyes, when I returned to my place, with\nheart a-fire.  Some days after, she came to me again and bought\nstuffs of me:  I refused to take the price and she cried, \"We\nhave no need of thy goods.\"  Quoth I, \"O my lady, accept them\nfrom me as a gift;\" but quoth she, \"Wait till I try thee and make\nproof of thee.\"  Then she brought out of her pocket a purse and\ngave me therefrom a thousand dinars, saying, \"Trade with this\ntill I return to thee.\"  So I took the purse and she went away\nand returned not till six months had passed.  Meanwhile, I traded\nwith the money and sold and bought and made other thousand dinars\nprofit on it.  At last she came to me again and I said to her,\n\"Here is thy money and I have gained with it other thousand\nducats;\" and she, \"Let it lie by thee and take these other\nthousand dinars.  As soon as I have departed from thee, go thou\nto Al-Rauzah, the Garden-holm, and build there a goodly pavilion,\nand when the edifice is accomplished, give me to know thereof. So saying, she left me and went away.\nAs soon as she was gone, I betook myself to Al-Rauzah and fell to\nbuilding the pavilion, and when it was finished, I furnished it\nwith the finest of furniture and sent to tell her that I had made\nan end to the edifice; whereupon she sent back to me, saying,\n\"Let him meet me to-morrow about day-break at the Zuwaylah gate\nand bring with him a strong ass.\"  I did as she bade and,\nbetaking myself to the Zuwaylah gate, at the appointed time,\nfound there a young man on horseback, awaiting her, even as I\nawaited her.  As we stood, behold, up she came, and with her a\nslave-girl.  When she saw that young man, she asked him, \"Art\nthou here?\" and he answered, \"Yes, O my lady.\"  Quoth she, \"To-\nday I am invited by this man: wilt thou wend with us?\" and quoth\nhe, \"Yes.\"  Then said she, \"Thou hast brought me hither against\nmy will and parforce.  Wilt thou go with us in any case?\"[FN#475]\nHe cried, \"Yes, yes,\" and we fared on, all three, till we came\nto Al-Rauzah and entered the pavilion.  The dame diverted herself\nawhile with viewing its ordinance and furniture, after which she\ndoffed her walking-dress and sat down with the young man in the\ngoodliest and chiefest place.  Then I fared forth and brought\nthem what they should eat at the first of the day; presently I\nagain went out and fetched them what they should eat at the end\nof the day and brought for the twain wine and dessert and fruits\nand flowers.  After this fashion I abode in their service,\nstanding on my feet, and she said not unto me, \"Sit,\" nor \"Take,\neat\" nor \"Take, drink,\" while she and the young man sat toying\nand laughing, and he feel to kissing her and pinching her and\nhopping over the ground[FN#476] and laughing.  They remained thus\nawhile and presently she said, \"Hitherto we have not become\ndrunken; let me pour out.\"  So she took the cup, and crowning it,\ngave him to drink and plied him with wine, till he lost his wits,\nwhen she took him up and carried him into a closet.  Then she\ncame out, with the head of that youth in her hand, while I stood\nsilent, fixing not mine eyes on her eyes neither questioning her\nof the case; and she asked me, \"What be this?\" \"I wot not,\" answered I; and she said,  \"Take it and throw it in the\nriver.\"  I accepted her commandment and she arose and stripping\nherself of her clothes, took a knife and cut the dead man's body\nin pieces, which she laid in three baskets, and said to me,\n\"Throw them into the river.\" I did her bidding and when I\nreturned, she said to me, \"Sit, so I may relate to thee yonder\nfellow's case, lest thou be affrighted at what accident hath\nbefallen him.  Thou must know that I am the Caliph's favourite\nconcubine, nor is there any higher in honour with him than I; and\nI am allowed six nights in each month, wherein I go down into the\ncity and tarry with my whilome mistress who reared me; and when I\ngo down thus, I dispose of myself as I will.  Now this young man\nwas the son of certain neighbours of my mistress, when I was a\nvirgin girl. One day, my mistress was sitting with the chief\nofficers of the palace and I was alone in the house, and as the\nnight came on, I went up to the terrace-roof in order to sleep\nthere, but ere I was ware, this youth came up from the street and\nfalling upon me knelt on my breast.  He was armed with a dagger\nand I could not get free of him till he had taken my maidenhead\nby force; and this sufficed him not, but he must needs disgrace\nme with all the folk for, as often as I came down from the\npalace, he would stand in wait for me by the way and futtered me\nagainst my will and follow me whithersoever I went.  This, then, is\nmy story, and as for thee, thou pleasest me and thy patience\npleaseth me and thy good faith and loyal service, and there\nabideth with me none dearer than thou.\"  Then I lay with her that\nnight and there befel what befel between us till the morning,\nwhen she gave me abundant wealth and took to meeting me at the\npavilion six days in every month.  After this wise we passed a\nwhole year, at the end of which she cut herself off from me a\nmonth's space, wherefore fire raged in my heart on her account.\nWhen it was the next month, behold , a little eunuch presented\nhimself to me and said, \"I am a messenger to thee from Such-an-\none, who giveth thee to know that the Commander of the Faithful\nhath ordered her to be drowned, her and those who are with her, six-\nand-twenty slave-girls, on such a day at Dayr al-Tin,[FN#477] for\nthat they have confessed of lewdness, one against other and she\nsayeth to thee, 'Look how thou mayst do with me and how thou\nmayst contrive to deliver me, even an thou gather together all\nmy money and spend it upon me, for that this be the time of\nmanhood.'\"[FN#478]  Quoth I, \"I know not this woman; belike it is\nother than I to whom this message is sent; so beware, O Eunuch,\nlest thou cast me into a cleft.\"  Quoth he, \"Behold, I have told\nthee that I had to say,\" and went away, leaving me in sore\nconcern on her account.  Now when the appointed day came, I\narose and changing my clothes and favour, donned sailor's\napparel; then I took with me a purse full of gold and buying a\nright good breakfast, accosted a boatman at Dayr al-Tin and sat\ndown and ate with him; after which I asked him, \"Wilt thou hire\nme thy boat?\"  Answered he, \"The Commander of the Faithful hath\ncommanded me to be here;\" and he told me the tale of the\nconcubines and how the Caliph purposed to drown them that day.\nWhen I heard this from him, I brought out to him ten gold pieces\nand discovered to him my case, whereupon he said to me, \"O my\nbrother, get thee empty gourds, and when thy mistress cometh,\ngive me to know of her and I will contrive the trick.\"  So I\nkissed his hand and thanked him and, as I was walking about,\nwaiting, up came the guards and eunuchs escorting the women, who\nwere weeping and shrieking and farewelling one another.  The\nCastratos cried out to us, whereupon we came with the boat, and\nthey said to the sailor, \"Who be this?\" Said he, \"This be my mate\nwhom I have brought to help me, so one of us may keep the boat,\nwhilst another doth your service.\"  Then they brought out to us\nthe women, one by one, saying \"Throw them in by the Island;\" and\nwe replied, \"'Tis well.\"  Now each of them was shackled and they\nhad made fast about her neck a jar of sand.  We did as the\nneutrals bade us and ceased not to take the women, one after\nother, and cast them in, till they gave us my mistress and I\nwinked to my mate.  So we took her and carried her out into mid-stream, where I threw to her the empty\ngourds[FN#479] and said to her, \"Wait for me at the mouth of the\nCanal.\"[FN#480] Then we cast her in alongside the boat, after we had loosed the jar of sand from her legs[FN#480a] and done off her shackles, and returned.\nNow there remained one woman after her: so we\ntook her and drowned her and the eunuchs went away, whilst we\ndropped down the river with the craft till we came to the mouth of Khalij, where I saw my mistress\nawaiting me.  We haled her into the canoe and returned to our\npavilion on Al-Rauzah.  Then I rewarded the sailor and he took his boat and\nwent away; whereupon quoth she to me, \"Thou art indeed the friend\never faithful found for the shifts of Fortune.\"[FN#481]  and I\nsojourned with her some days; but the shock wrought upon her so\nthat she sickened and fell to wasting away and redoubled in langour and\nweakness till she died.  I mourned for her with exceeding mourning and buried her; after\nwhich I removed all that was in the pavilion and abandoned the\nbuilding.   Now she had brought to that pavilion a little coffer\nof copper and laid it in a place whereof I knew not; so, when the\nInspector of Inheritances[FN#482] came, he rummaged the house and\nfound the coffer, with the key in the lock.  Presently he opened it and seeing it full of jewels and jacinths and earrings and seal-rings and precious stones (and 'twas a matter such as is not found save with kings and sultans), took it, and me with it, and he and his men ceased not to\nput me to the question with beating and torment till I confessed to them the whole affair, from beginning to end.  Thereupon they carried me to the Caliph and I told him all that had passed between me and her; and he said to\nme, \"O man, depart this city, for I release thee on account of\nthy courage and because of thy constancy in keeping thy secret\nand thy daring in exposing thyself to death.\"  So I arose\nforthwith and fared from his city; and this is what befel me.\n\n\n\n\n                     Variants and Analogues\n                               of\n                       Some of the Tales\n                               in\n                      Volumes XI. and XII.\n\n                      By. W. A. Clouston.\n\n    Author of \"Popular Tales and Fictions: Their Migrations\n                   and Transformations,\" Etc.\n\n\n\n                            Appendix\n\n\n\n          Variants and Analogues of Some of the Tales\n                     in Volumes XI and XII\n\n                       By W. A. Clouston.\n\n\n\n\n\n           THE SLEEPER AND THE WAKER--Vol. XI. p. 1.\n\n\n\nFew if the stories in the \"Arabian Nights\" which charmed our\nmarvelling boyhood were greater favourites than this one, under\nthe title of \"Abou Hassan; or, the Sleeper Awakened.\"  What\nrecked we in those days whence it was derived?--the story--the\nstory was the thing!  As Sir R. F. Burton observes in his first\nnote, this is \"the only one of the eleven added by Galland, whose\noriginal has been discovered in Arabic;\"[FN#483] and it is\nprobable that Galland heard it recited in a coffee-house during\nhis residence in Constantinople.  The plot of the Induction to\nShakspeare's comedy of \"The Taming of the Shrew\" is similar to\nthe adventure of Ab\u00fa al-Hasan the Wag, and is generally believed\nto have been adapted from a story entitled \"The Waking Man's\nFortune\" in Edward's collection of comic tales, 1570, which were\nretold somewhat differently in Goulart's \"Admirable and Memorable\nHistories,\" 1607; both versions are reprinted in Mr. Hazlitt's\n\"Shakspeare Library,\" vol. iv., part I, pp. 403-414.  In Percy's\n\"Reliques of Ancient English Poetry\" we find the adventure told\nin a ballad entitled \"The Frolicksome Duke; or, the Tinker's Good\nFortune,\" from the Pepys collection: \"whether it may be thought\nto have suggested the hint to Shakspeare or is not rather of\nlatter date,\" says Percy, \"the reader must determine:\"\n\nNow as fame does report, a young duke keeps a court,\nOne that pleases his fancy with frolicksome sport:\nBut amongst all the rest, here is one, I protest,\nWhich will make you to smile when you hear the true jest:\nA poor tinker he found lying drunk on the ground,\nAs secure in a sleep as if laid in a swownd.\n\nThe duke said to his men, William, Richard, and Ben,\nTake him home to my palace, we'll sport with him then.\nO'er a horse he was laid, and with care soon convey'd\nTo the palace, altho' he was poorly arrai'd;\nThen they stript off his cloaths, both his shirt, shoes, and\nhose,\nAnd they put him in bed for to take his repose.\n\nHaving pull'd off his shirt, which was all over durt,\nThey did give him clean holland, this was no great hurt:\nOn a bed of soft down, like a lord of renown,\nThey did lay him to sleep the drink out of his crown.\nIn the morning when day, then admiring[FN#484] he lay,\nFor to see the rich chamber both gaudy and gay.\n\nNow he lay something late, in his rich bed of state,\nTill at last knights and squires they on him did wait;\nAnd the chamberling bare, then did likewise declare,\nHe desired to know what apparel he'd ware:\nThe poor tinker amaz'd, on the gentleman gaz'd,\nAnd admired how he to this honour was rais'd.\n\nTho' he seem'd something mute, yet he chose a rich suit,\nWhich he straitways put on without longer dispute;\nWith a star on his side, which the tinker offt ey'd,\nAnd it seem'd for to swell him no little with pride;\nFor he said to himself, Where is Joan my sweet wife?\nSure she never did see me so fine in her life.\n\nFrom a convenient place, the right duke his good grace\nDid observe his behavior in every case.\nTo a garden of state, on the tinker they wait,\nTrumpets sounding before him: thought he this is great:\nWhere an hour or two, pleasant walks he did view,\nWith commanders and squires in scarlet and blew.\n\nA find dinner was drest, both for him and his guests,\nHe was placed at the table above all the rest,\nIn a rich chair, or bed, lin'd with fine crimson red,\nWith a rich golden canopy over his head:\nAs he sat at his meat, the musick play'd sweet,\nWith the choicest of singing his joys to compleat.\n\nWhile the tinker did dine, he had plenty of wine.\nRich canary with sherry and tent superfine,\nLike a right honest soul, faith, he took off his bowl,\nTill at last he began for to tumble and roul\nFrom his chair to the floor, where he sleeping did snore,\nBeing seven times drunker than ever before.\n\nThen the duke did ordain, they should strip him amain,\nAnd restore him his old leather garments again:\n'Twas a point next the worst, yet perform it they must,\nAnd they carry'd him strait, where they found him at first;\nThen he slept all the night, as indeed well he might,\nBut when he did waken, his joys took their flight.\n\nFor his glory to him so pleasant did seem,\nThat he thought it to be but a meer golden dream;\nTill at length he was brought to the duke, where he sought\nFor a pardon as fearing he had set him at nought;\nBut his highness he said, Thou'rt a jolly bold blade,\nSuch a frolick before I think never was plaid.\n\nThen his highness bespoke him a new suit and cloak,\nWhich he gave for the sake of this frolicksome joak;\nNay, and five hundred pound, with ten acres of ground\nThou shalt never, said he, range the counteries round,\nCrying old brass to mend, for I'll be thy good friend,\nNay, and Joan thy sweet wife shall my duchess attend.\n\nThen the tinker reply'd, What! must Joan my sweet bride\nBe a lady in chariots of pleasure to ride?\nMust we have gold and land ev'ry day at command?\nThen I shall be a squire I well understand:\nWell I thank your good grace, and your love I embrace,\nI was never before in so happy a case.\n\nThe same story is also cited in the \"Anatomy of Melancholy,\" part\n2, sec. 2, memb. 4, from Ludovicus Vives in Epist.[FN#485] and Pont.\nHeuter in Rerum Burgund., as follows:\n\n\"It is reported of Philippus Bonus, that good Duke of Burgundy,\nthat the said duke, at the marriage of Eleonora, sister to the\nKing of Portugal, at Bruges in Flanders, which was solemnized in\nthe deep of winter, when as by reason of the unseasonable (!)\nweather he could neither hawk nor hunt, and was now tyred with\ncards, dice, &c., and such other domestical sports, or to see\nladies dance, with some of his courtiers, he would in the evening\nwalk disguised all about the town.  It so fortuned as he was\nwalking late one night, he found a country fellow dead drunk,\nsnorting on a bulk; he caused his followers to bring him to his\npalace, and there stripping him of his old clothes, and attiring\nhim after the court fashion, when he waked, he and they were all\nready to attend upon his excellency, persuading him that he was\nsome great duke.  The poor fellow, admiring how he came there,\nwas served in state all the day long; after supper he saw them\ndance, heard musick, and the rest of those court-like pleasures;\nbut late at night, when he was well tipled, and again fast asleep\nthey put on his old robes, and so conveyed him to the place where\nthey first found him.  Now the fellow had not made them so good\nsport the day before, as he did when he returned to himself; all\nthe jest was to see how he looked upon it.  In conclusion, after\nsome little admiration, the poor man told his friends he had seen\na vision, constantly beleeved it, would not otherwise be\nperswaded; and so the jest ended.\"\n\nI do not think that this is a story imported from the East: the\nadventure is just as likely to have happened in Bruges as in\nBaghd\u00e1d; but the exquisite humor of the Arabian tale is wanting-\n-even Shakspeare's Christopher Sly is not to be compared with\nhonest Ab\u00fa al-Hasan the Wag.\n\nThis story of the Sleeper and the Waker recalls the similar\ndevice practised by the Chief of the Assassins--that formidable,\nmurderous association, the terror of the Crusades--on promising\nnovices.  Von Hammer, in his \"History of the Assassins,\" end of\nBook iv., gives a graphic description of the charming gardens\ninto which the novices were carried while insensible from\nhashish:\n\nIn the center of the Persian as well as the Assyrian territory of\nthe Assassins, that is to say, both at Alamut and Massiat, were\nsituated, in a space surrounded by walls, splendid gardens--true\nEastern paradises.  There were flower-beds and thickets of\nfruit-trees, intersected by canals, shady walks, and verdant\nglades, where the sparkling stream bubbled at every step; bowers\nof roses and vineyards; luxurious halls and porcelain kiosks,\nadorned with Persian carpets and Grecian stuffs, where\ndrinking-vessels of gold, silver, and crystal glittered on trays\nof the same costly materials; charming maidens and handsome boys\nof Muhammed's Paradise, soft as the cushions on which they\nreposed, and intoxicating as the wine which they presented.  The\nmusic of the harp was mingled with the songs of birds, and the\nmelodious tones of the songstress harmonized with the murmur of\nthe brooks.  Everything breathed pleasure, rapture, and\nsensuality.  A youth, who was deemed worthy by his strength and\nresolution to be initiated into the Assassin service, was invited\nto the table and conversation of the grand master, or grand\nprior, he was then intoxicated with hashish and carried into the\ngarden, which on awaking he believed to be Paradise; everything\naround him, the houris in particular, contributing to confirm the\ndelusion.  After he had experienced as much of the pleasures of\nParadise, which the Prophet has promised to the faithful, as his\nstrength would admit; after quaffing enervating delight from the\neyes of the houris and intoxicating wine from the glittering\ngoblets; he sank into the lethargy produced by debility and the\nopiate, on awakening from which, after a few hours, he again\nfound himself by the side of his superior.  The latter endeavoured\nto convince him that corporeally he had not left his side, but\nthat spiritually he had been wrapped into Paradise and had there\nenjoyed a foretaste of the bliss which awaits the faithful who\ndevote their lives to the service of the faith and the obedience\nof their chiefs.\n\n\n\n\n THE TEN WAZIRS; OR, THE HISTORY OF KING \u00c1Z\u00c1DBAKHT AND HIS SON\n                        Vol. XI. p. 55.\n\n\n\nThe precise date of the Persian original of this romance\n(\"Bakhty\u00e1r N\u00e1ma\") has not been ascertained, but it was probably\ncomposed before the beginning of the fifteenth century, since\nthere exists in the Bodleian Library a unique Turk\u00ed version, in\nthe Uygur language and characters, which was written in 1434.\nOnly three of the tales have hitherto been found in other Asiatic\nstorybooks.  The Turk\u00ed version, according to M. Jaubert, who\ngives an account of the MS. and a translation of one of the tales\nin the Journal Asiatique, tome x. 1827, is characterised by\n\"great sobriety of ornament and extreme simplicity of style, and\nthe evident intention on the part of the translator to suppress\nall that may not have appeared to him sufficiently probable, and\nall that might justly be taxed with exaggeration;\" and he adds\nthat \"apart from the interest which the writing and phraseology\nof the work may possess for those who study the history of\nlanguages, it is rather curious to see how a T\u00e1t\u00e1r translator\nsets to work to bring within the range of his readers stories\nembellished in the original with descriptions and images\nfamiliar, doubtless, to a learned and refined nation like the\nPersians, but foreign to shepherds.\"\n\nAt least three different versions are known to the Malays-\n-different in the frame, or leading story, if not in the\nsubordinate tales.  One of those is described in the second\nvolume of Newbold's work on Malacca, the frame of which is\nsimilar to the Persian original and its Arabian derivative,\nexcepting that the name of the king is Z\u00e1dbokhtin and that of the\nminister's daughter (who is nameless in the Persian) is Mahrwat.\nTwo others are described in Van den Berg's account of Malay,\nArabic, Javanese and other MSS. published at Batavia, 1877: p.\n21, No. 132 is entitled \"The History of Ghul\u00e1m, son of\nZ\u00e1dbukht\u00e1n, King of Ad\u00e1n, in Persia,\" and the frame also\ncorresponds with our version, with the important difference that\nthe robber-chief who had brought up Ghul\u00e1m, \"learning that he had\nbecome a person of consequence, came to his residence to visit\nhim, but finding him imprisoned, he was much concerned, and asked\nthe king's pardon on his behalf, telling him at the same time how\nhe had formerly found Ghul\u00e1m in the jungle; from which the king\nknew that Ghul\u00e1m was his son.\"  The second version noticed by Van\nden Berg (p. 32, No. 179), though similar in title to the Persian\noriginal, \"History of Prince Bakhty\u00e1r,\" differs very materially\nin the leading story, the outline of which is as follows: This\nprince, when his father was put to flight by a younger brother,\nwho wished to dethrone him, was born in a jungle, and abandoned\nby his parents. A merchant named Idr\u00eds took charge of him and\nbrought him up. Later on he became one of the officers of state\nwith his own father, who had in the meanwhile found another\nkingdom, and decided with fairness, the cases brought before him.\nHe was, however, put in prison on account of a supposed attempt\non the king's life, and would have been put to death had he not\nstayed the execution by telling various beautiful stories.  Even\nthe king came repeatedly to listen to him.  At one of these\nvisits Bakhty\u00e1r's foster-father Idr\u00eds was present, and related to\nhis adopted son how he had found him in the jungle.  The king, on\nhearing this, perceived that it was his son who had been brought\nup by Idr\u00eds, recognised Bakhty\u00e1r as such, and made over to him the\nkingdom.\"--I have little doubt that this romance is of Indian\nextraction.\n\n\n\n\n     STORY OF KING DADBIN AND HIS WAZIRS.--Vol. XI. p. 94.\n\n\n\nThis agrees pretty closely with the Turk\u00ed version of the same\nstory (rendered into French by M. Jaubert), though in the latter\nthe names of the characters are the same as in the Persian, King\nD\u00e1d\u00edn and the Waz\u00edrs K\u00e1mg\u00e1r and K\u00e1rd\u00e1r.  In the Persian story,\nthe damsel is tied hands and feet and placed upon a camel, which\nis then turned into a dreary wilderness.  \"Here she suffered from\nthe intense heat and from thirst; but she resigned herself to the\nwill of Providence, conscious of her own innocence.  Just then\nthe camel lay down, and on the spot a fountain of delicious water\nsuddenly sprang forth; the cords which bound her hands and feet\ndropped off; she refreshed herself by a draught of the water, and\nfervently returned thanks to Heaven for this blessing and her\nwonderful preservation.\"  This two-fold miracle does not appear\nin the Turk\u00ed and Arabian versions.  It is not the cameleer of the\nKing of Persia, but of King D\u00e1d\u00edn, who meets with the pious\ndamsel in the wilderness.  He takes her to his own house and one\nday relates his adventure to King D\u00e1d\u00edn, who expresses a wish to\nsee such a prodigy of sanctity.  The conclusion of the Persian\nstory is quite dramatic: The cameleer, having consented, returned\nat once to his house, accompanied by the king, who waited at the\ndoor of the apartment where the daughter of K\u00e1mg\u00e1r was engaged in\nprayer.  When she had concluded he approached, and with\nastonishment recognised her.  Having tenderly embraced her, he\nwept, and entreated her forgiveness.  This she readily granted,\nbut begged that he would conceal himself in the apartment while\nshe should converse with K\u00e1rd\u00e1r, whom she sent for.  When he\narrived, and beheld her with a thousand expressions of fondness,\nhe inquired how she had escaped, and told her that on the day the\nking had banished her into the wilderness, he had sent people to\nseek her and bring her to him.  \"How much better would it have\nbeen,\" he added, \"had you followed my advice, and agreed to my\nproposal of poisoning the king, who, I said, would one day\ndestroy you as he had done your father!  But you rejected my\nadvice, and declared yourself ready to submit to whatever\nProvidence should decree.  Hereafter you will pay more attention\nto my words.  But now let us not think of what is past.  I am\nyour slave, and you are dearer to me than my own eyes.\"  So\nsaying, he attempted to clasp the daughter of K\u00e1mg\u00e1r in his arms,\nwhen the king, who was concealed behind the hangings, rushed\nfuriously on him and put him to death.  After this he conducted\nthe damsel to his palace, and constantly lamented his\nprecipitancy in having killed her father.--This tale seems to\nhave been taken from the Persian \"T\u00fat\u00ed N\u00e1ma,\" or Parrot-book,\ncomposed by Nahkshab\u00ed about the year 1306;[FN#486] it occurs in\nthe 51st Night of the India Office MS. 2573, under the title of\n\"Story of the Daughter of the Vaz\u00edr Kh\u00e1ssa, and how she found\nsafety through the blessing of her piety:\"  the name of the king\nis Bahram, and the Waz\u00edrs are called Kh\u00e1ssa and Khal\u00e1ssa.\n\n\n\n\n       STORY OF AYLAN SHAH AND AB\u00da TAMM\u00c1M--Vol. XI p. 112.\n\n\n\nThe catastrophe of this story forms the subject of the Lady's\n37th tale in the text of the Turkish \"Forty Vez\u00edrs,\" translated\nby Mr. E. J. W. Gibb.  This is how it goes:\n\nIn the palace of the world there was a king, and that king had\nthree vez\u00edrs, but there was rivalry between them.  Two of them\nday and night incited the king against the third, saying, \"He is\na traitor.\"  But the king believed them not.  At length they\npromised two pages much gold, and instructed them thus: \"When the\nking has lain down, ere he yet fall asleep, do ye feign to think\nhim asleep, and while talking with each other, say at a fitting\ntime, 'I have heard from such a one that yon vez\u00edr says this and\nthat concerning the king, and that he hates him; many people say\nthat vez\u00edr is an enemy to our king.'\"  So they did this, and when\nthe king heard this, he said in his heart, \"What those vez\u00edrs\nsaid is then true; when the very pages have heard somewhat it\nmust indeed have some foundation.  Till now, I believed not those\nvez\u00edrs, but it is then true.\"  And the king executed that vez\u00edr.\nThe other vez\u00edrs were glad and gave the pages the gold they had\npromised.  So they took it and went to a private place, and while\nthey were dividing it one of them said, \"I spake first; I want\nmore.\"  The other said, \"If I had not said he was an enemy to our\nking, the king would not have killed him; I shall take more.\"\nAnd while they were quarrelling with one another the king passed\nby there, and he listened attentively to their words, and when he\nlearned of the matter, he said, \"Dost thou see, they have by a\ntrick made us kill that hapless vez\u00edr.\"  And he was repentant.\n\n\n\n\n  STORY OF KING SULAYMAN SHAH AND HIS NIECE.--Vol. XI. p. 131.\n\n\n\nThe Persian original has been very considerably amplified by the\nArabian translator.  In the \"Bakhty\u00e1r N\u00e1ma\" there is not a word\nabout the two brothers and their fair cousin, the attempted\nmurder of the infant, and the adventures of the fugitive young\nprince.  This story has also been taken from the \"T\u00fat\u00ed N\u00e1ma\" of\nNakhshab\u00ed, Night the 50th of the India Office MS. 2573, where,\nunder the title of \"Story of the Daughter of the Kaysar of Roum,\nand her trouble by reason of her son,\" it is told somewhat as\nfollows:\n\nIn former times there was a great king, whose army was numerous\nand whose treasury was full to overflowing; but, having no enemy\nto contend with, he neglected to pay his soldiers, in consequence\nof which they were in a state of destitution and discontent.  At\nlength one day the soldiers went to the prime minister and made\ntheir condition known to him.  The vaz\u00edr promised that he would\nspeedily devise a plan by which they should have employment and\nmoney.  Next morning he presented himself before the king, and\nsaid that it was widely reported the Kaysar of Roum had a\ndaughter unsurpassed for beauty--one who was fit only for such a\ngreat monarch as his Majesty; and suggested that it would be\nadvantageous if an alliance were formed between two such great\npotentates.  The notion pleased the king well, and he forthwith\ndespatched to Roum an ambassador with rich gifts, and requested\nthe Kaysar to grant him his daughter in marriage.  But the Kaysar\nwaxed wroth at this, and refused to give his daughter to the\nking.  When the ambassador returned thus unsuccessful, the king,\nenraged at being made of no account, resolved to make war upon\nthe Kaysar; so, opening the doors of his treasury, he distributed\nmuch money among his troops, and then, \"with a woe-bringing host,\nand a blood-drinking army, he trampled Roum and the folk of Roum\nin the dust.\"  And when the Kaysar was become powerless, he sent\nhis daughter to the king, who married her according to the law of\nIslam.\n\nNow that princess had a son by a former husband, and the Kaysar\nhad said to her before she departed, \"Beware that thou mention\nnot thy son, for my love for his society is great, and I cannot\npart with him.\"[FN#487]  But the princess was sick at heart for\nthe absence of her son, and she was ever pondering how she should\nspeak to the king about him, and in what manner she might\ncontrive to bring him to her.  It happened one day the king gave\nher a string of pearls and a casket of jewels.  She said, \"With\nmy father is a slave who is well skilled in the science of\njewels.\"  The king replied, \"If I should ask that slave of thy\nfather, would he give him to me?\"  \"Nay,\" said she, \"for he holds\nhim in the place of a son.  But if the king desire him, I will\nsend a merchant to Roum, and I myself will give him a token, and\nwith pleasant wiles and fair speeches will bring him hither.\"\nThen the king sent for a clever merchant who knew Arabic\neloquently and the language of Roum, and gave him goods for\ntrading and sent him to Roum with the object of procuring the\nslave.  But the daughter of the Kaysar said privily to the\nmerchant, \"That slave is my son; I have, for a good reason, said\nto the king that he is a slave; so thou must bring him as a\nslave, and let it be thy duty to take care of him.\"  In due\ncourse the merchant brought the youth to the king's service; and\nwhen the king saw his fair face, and discovered in him many\npleasing and varied accomplishments, he treated him with\ndistinction and favour, and conferred on the merchant a robe of\nhonour and gifts.  His mother saw him from afar, and was pleased\nwith receiving a secret salutation from him.\n\nOne day the king had gone to the chase, and the palace remained\nvoid of rivals; so the mother called in her son, kissed his fair\nface, and told him the tale of her great sorrow.  A chamberlain\nbecame aware of the secret, and another suspicion fell upon him,\nand he said to himself, \"The harem of the king is the sanctuary\nof security and the palace of protection.  If I speak not of\nthis, I shall be guilty of treachery and shall have wrought\nunfaithfulness.\"  When the king returned from the chase, the\nchamberlain related to him what he had seen, and the eking was\nangry and said, \"This woman hath deceived me with words and\ndeeds, and has brought hither her desire by craft and cunning.\nThis conjecture must be true, else why did she play such a trick?\nAnd why did she hatch such a plot?  And why did she send the\nmerchant?\"  Then the king, enraged, went into the harem, and the\nqueen saw from his countenance that the occurrence of the night\nbefore had become known to him, and she said, \"Be it not that I\nsee the king angry?\"  He said, \"How should I not be angry?  Thou,\nby craft and trickery, and intrigue, and plotting, hast brought\nthy desire from Roum--what wantonness is this that thou hast\ndone?\"  And then he thought to slay her, but he forbore, because\nof his great love for her.  But he ordered the chamberlain to\ncarry the youth to some obscure place, and straightway sever his\nhead from his body.  When the poor mother saw this, she well-nigh\nfell on her face, and her soul was near leaving her body.  But\nshe knew that sorrow would not avail, and so she restrained\nherself.\n\nAnd when the chamberlain took the youth into his own house, he\nsaid to him, \"O youth, knowest thou not that the harem of the\nking is the sanctuary of security?  What great treachery is this\nthat thou hast perpetrated?\"  The youth replied, \"That queen is\nmy mother, and I am her true son.  Because of her natural\ndelicacy, she said not to the king that she had a son by another\nhusband.  And when yearning came over her, she contrived to bring\nme here from Roum; and while the king was engaged in the chase,\nmaternal love stirred in her, and she called me to her and\nembraced me.\"  On hearing this, the chamberlain said to himself,\n\"What is passing in his mother's breast?  What I have not done I\ncan yet do, and it were better that I preserve this youth some\ndays, for such a rose may not be wounded through idle words, and\nsuch a bough may not be broken by a breath.  For some day the\ntruth of this matter will be disclosed, and it will become known\nto the king when repentance may be of no avail.\"  So he went\nbefore the king and said, \"That which was commanded have I\nfulfilled.\"  On hearing this the king's wrath was to some extent\nremoved, but his trust in the Kaysar's daughter was departed;\nwhile she, poor creature, was grieved and dazed at the loss of\nher son.\n\nNow in the palace-harem there was an old woman, who said to the\nqueen, \"How is it that I find thee sorrowful?\"  And the queen\ntold the whole story, concealing nothing.  This old woman was a\nheroine in the field of craft, and she answered, \"Keep thy mind\nat ease; I will devise a stratagem by which the heart of the king\nwill be pleased with thee, and every grief he has will vanish\nfrom his heart.\"  The queen said that, if she did so, she should\nbe amply rewarded.  One day the old woman, seeing the king alone,\nsaid to him, \"Why is thy former aspect altered?  And why are\nthere traces of care and anxiety visible on thy countenance?\"\nThe king then told her all.  Then said the old woman, \"I have an\namulet of the charms of Sulayman, in the Syriac language, and in\nthe writing of the jinn (genii).  When the queen is asleep, do\nthou place it on her breast, and whatever it may be, she will\ntell the truth of it.  But take care, fall not asleep, but listen\nwell to what she says.\"  The king wondered at this and said,\n\"Give me that amulet, that the truth of this matter may be\nlearned.\"  So the old woman gave him the amulet, and then went to\nthe queen and explained what she had done, and said, \"Do thou\nfeign to be asleep, and relate the whole of thy story\nfaithfully.\"\n\nWhen a watch of the night was past, the king laid the amulet upon\nhis wife's breast, and she thus began: \"By a former husband I had\na son, and when my father gave me to this king, I was ashamed to\nsay I had a tall son.  When my yearning passed all bounds, I\nbrought him here by an artifice.  One day that the king was gone\nto the chase I called him into the house, when, after the way of\nmothers, I took him in my arms and kissed him.  This reached the\nking's ears; he unwittingly gave it another construction, and cut\noff the head of that innocent boy, and withdrew from me his own\nheart.  Alike is my son lost to me and the king angry.\"  When the\nking heard these words he kissed her and exclaimed, \"O my life,\nwhat an error is this thou hast committed!  Thou hast brought\ncalumny upon thyself, and hast given such a son to the winds, and\nhast made me ashamed!\"  Straightway he called the chamberlain,\nand said, \"That boy whom thou hast killed is the son of my\nbeloved and the darling of my beauty! Where is his grave, that we\nmay make there a guest-house?\"  The chamberlain said, \"That youth\nis yet alive.  When the king commanded his death, I was about to\nkill him, but he said, 'That queen is my mother.  Through modesty\nbefore the king, she revealed not the secret that she has a tall\nson.  Kill me not; it may be that some day the truth will become\nknown, and repentance profiteth not, and regret is useless.\"  The\nking commanded them to bring the youth; so they brought him\nforthwith.  And when the mother saw the face of her son, she\nthanked God and praised the Most High, and became one of the\nMuslims, and from the sect of unbelievers came into the faith of\nIslam.  And the king favoured the chamberlain in the highest\ndegree, and they passed the rest of their lives in comfort and\nease.\n\n\n\n\n             FIRUZ AND HIS WIFE.--Vol. XI. p. 185.\n\n\n\nThis tale, as Sir R. F. Burton remarks, is a rechauff\u00e9 of that of\nthe King and the Wazir's Wife in the \"Malice of Women,\" or the\nSeven Waz\u00edrs (vol. vi. 129); and at p. 308 we have yet another\nvariant.[FN#488]  It occurs in all the Eastern texts of the Book\nof Sindib\u00e1d, and it is commonly termed by students of that cycle\nof stories \"The Lion's Track,\" from the parabolical manner in\nwhich the husband justifies his conduct before the king.  I have\ncited some versions in the Appendix to my edition of the Book of\nSindib\u00e1d (p. 256 ff.), and to these may be added the following\nVenetian variant, from Crane's \"Italian Popular Tales,\" as an\nexample of how a story becomes garbled in passing orally from one\ngeneration unto another generation:\n\nA king, averse from marriage, commanded his steward to remain\nsingle.  The latter, however, one day saw a beautiful girl named\nVigna and married her secretly.  Although he kept her closely\nconfined in her chamber, the king became suspicious, and sent the\nsteward on an embassy.  After his departure the king entered the\napartment occupied by him, and saw his wife asleep.  He did not\ndisturb her, but in leaving the room accidentally dropped one of\nhis gloves on the bed.  When the husband returned he found the\nglove, but kept a discreet silence, ceasing, however, all\ndemonstration of affection, believing his wife had been\nunfaithful.  The king, desirous to see again the beautiful woman,\nmade a feast and ordered the steward to bring his wife.  He\ndenied that he had one, but brought her at last, and while every\none else was talking gaily at the feast she was silent.  The king\nobserved it and asked the cause of her silence, and she answered\nwith a pun on her own name, \"Vineyard I was, and Vineyard I am.\nI was loved and no longer am. I know not for what reason the\nVineyard has lost its season.\" Her husband, who heard this,\nreplied, \"Vineyard thou wast, and Vineyard thou art: the Vineyard\nlost its season, for the lion's claw.\"  The king, who understood\nwhat he meant, answered, \"I entered the Vineyard; I touched the\nleaves; but I swear by my crown that I have not tasted the\nfruit.\"  Then the steward understood that his wife was innocent,\nand the two made peace, and always after lived happy and\ncontented.\n\nSo far as I am aware, this tale of \"The Lion's Track\" is not\npopularly known in any European country besides Italy; and it is\nnot found in any of the Western versions of the Book of Sindib\u00e1d,\ngenerally known under the title of the \"History of the Seven Wise\nMasters,\" how, then, did it reach Venice, and become among the\npeople \"familiar in their mouths as household words?\"  I answer,\nthat the intimate commercial relations which long existed between\nthe Venetian Republic and Egypt and Syria are amply sufficient to\naccount for the currency of this and scores of other Eastern\ntales in Italy.  This is not one of those fictions introduced\ninto the south of Europe through the Ottomans, since Boccaccio\nhas made use of the first part of it in his \"Decameron,\" Day I.\nnov. 5; and it is curious to observe that the garbled Venetian\npopular version has preserved the chief characteristic of the\nEastern story--the allegorical reference to the king as a lion\nand his assuring the husband that the lion had done no injury to\nhis \"Vineyard.\"\n\n\n\n\n   KING SHAH BAKHT AND HIS WAZIR AL-RAHWAN.--Vol. XI. p. 127.\n\n\n\nWhile the frame-story of this interesting group is similar to\nthat of the Ten Waz\u00edrs (vol. i. p. 55), insomuch as in both a\nking's favourite is sentenced to death in consequence of the\nfalse accusations of his enemies, and obtains a respite from day\nto day by relating stories to the king, there is yet a very\nimportant difference: Like those of the renowned Shahrazad, the\nstories which Al-Rahwan tells have no particular, at least no\nuniform, \"purpose,\" his sole object being to prolong his life by\ntelling the king an entertaining story, promising, when he has\nended his recital, to relate one still \"stranger\" the next night,\nif the king will spare his life another day.  On the other hand,\nBakhty\u00e1r, while actuated by the same motive, appeals to the\nking's reason, by relating stories distinctly designed to exhibit\nthe evils of hasty judgements and precipitate conduct--in fact,\nto illustrate the maxim,\n\n     Each order given by a reigning king,\n          Should after long reflection be expressed;\n     For it may be that endless woe will spring\n          From a command he paused not to digest.\n\nAnd in this respect they are consistent with the circumstances of\nthe case, like the tales of the Book of Sindib\u00e1d, from which the\nframe of the Ten Waz\u00edrs was imitated, and in which the Waz\u00edrs\nrelate stories showing the depravity and profligacy of women and\nthat no reliance should be placed on their unsupported\nassertions, and to these the lady opposes equally cogent stories\nsetting forth the wickedness and perfidy of men.  Closely\nresembling the frame-story of the Ten Waz\u00edrs, however, is that of\na Tamil romance entitled, \"Alakeswara Kath\u00e1,\" a copy of which,\nwritten on palm leaves, was in the celebrated Mackenzie\ncollection, of which Dr. H. H. Wilson published a descriptive\ncatalogue; it is \"a story of the R\u00e1j\u00e1 of Alakepura and his four\nministers, who, being falsely accused of violating the sanctity\nof the inner apartments, vindicate their innocence and disarm the\nking's wrath by relating a number of stories.\"  Judging by the\nspecimen given by Wilson, the well-known tale of the Lost Camel,\nit seems probable that the ministers' stories, like those of\nBakhty\u00e1r, are suited to their own case and illustrate the truth\nof the adage that \"appearances are often deceptive.\"  Whether in\nthe Siamese collection \"Nonthuk Pakkaranam\" (referred to in vol.\ni. p. 191) the stories related by the Princess Kankras to the\nKing of Pataliput (Palibothra), to save her father's life, are\nsimilarly designed, does not appear from Benfey's notice of the\nwork in his paper in \"Orient and Occident,\" iii. 171 ff.  He says\nthat the title of the book, \"Nonthuk Pakkaranam,\" is taken from\nthe name of a wise ox, Nonthuk, that plays the principal part in\nthe longest of the tales, which are all apparently translated\nfrom the Sanskrit, in which language the title would be Nandaka\nPrakaranam, the History of Nandaka.\n\nMost of the tales related by the wazir Al-Rahwan are not only in\nthemselves entertaining, but are of very considerable importance\nfrom the story-comparer's point of view, since in this group\noccur Eastern forms of tales which were known in Italy in the\n14th century, and some had spread over Europe even earlier.  The\nreader will have seen from Sir R. F. Burton's notes that not a\nfew of the stories have their parallels or analogues in countries\nfar apart, and it is interesting to find four of them which\nproperly belong to the Eastern texts of the Book of Sindibad,\nwith the frame-story of which that of this group has so close an\naffinity.\n\n\n\n\n         THE ART OF ENLARGING PEARLS.--Vol. XI. p.197.\n\n\n\n\"Quoth she, I have a bangle; sell it and buy seed pearls with the\n                     price; then round them\n              and fashion them into great pearls.\"\n\nFor want of a more suitable place, I shall here reproduce an\naccount of the \"Method of making false pearls\" (nothing else\nbeing meant in the above passage), cited, from Postl. Com. Dict.,\nin vol. xxvi. of Rees' Cyclop\u00e6dia,\" London, 1819:\n\n\"Take of thrice distilled vinegar two pounds, Venice turpentine\none pound, mix them together into a mass and put them into a\ncucurbit, fit a head and receiver to it, and after you have luted\nthe joints set it when dry on a sand furnace, to distil the\nvinegar from it; do not give it too much heat, lest the stuff\nswell up.  After this put the vinegar into another glass cucurbit\nin which there is a quantity of seed pearls wrapped in a piece of\nthin silk, but so as not to touch the vinegar; put a cover or\nhead upon the cucurbit, lute it well and put it in bal. Mari\u00e6,\nwhere you may let it remain a fortnight.  The heat of the balneum\nwill raise the fumes of the vinegar, and they will soften the\npearls in the silk and bring them to the consistence of a paste,\nwhich being done, take them out and mould them to what bigness,\nform, and shape you please.  Your mould must be of fine silver,\nthe inside gilt; you must also refrain from touching the paste\nwith your fingers, but use silver-gilt utensils, with which fill\nyour moulds.  When they are moulded, bore them through with a\nhog's bristle or gold wire, and then tread them again on gold\nwire, and put them into a glass, close it up, and set them in the\nsun to dry.  After they are thoroughly dry, put them in a glass\nmatrass into a stream of running water and leave them there\ntwenty days; by that time they will contract the natural hardness\nand solidity of pearls.  Then take them out of the matrass and\nhang them in mercurial water, where they will moisten, swell, and\nassume their Oriental beauty; after which shift them into a\nmatrass hermitically closed to prevent any water coming to them,\nand let it down into a well, to continue there about eight days.\nThen draw the matrass up, and in opening it you will find pearls\nexactly resembling Oriental ones.\"  (Here follows a recipe for\nmaking the mercurial water used in the process, with which I need\nnot occupy more space.)\n\nA similar formula, \"To make of small pearls a necklace of large\nones,\" is given in the \"Lady's Magazine\" for 1831, vol. iv., p.\n119, which is said to be extracted from a scarce old book.  Thus,\nwhatever mystery may surround the art is Asiatic countries there\nis evidently none about it in Europe.  The process appears to be\nsomewhat tedious and complicated, but is doubtless profitable.\n\nIn Philostratus' Life of Appolonius there is a curious passage\nabout pearl-making which has been generally considered as a mere\n\"traveller's tale\":  Apollonious relates that the inhabitants of\nthe shores of the Red Sea, after having calmed the water by means\nof oil, dived after the shell-fish, enticed them with some bait\nto open their shells, and having pricked the animals with a\nsharp-pointed instrument, received the liquor that flowed from\nthem in small holes made in an iron vessel, in which it hardened\ninto real pearls.--It is stated by several reputable writers that\nthe Chinese do likewise at the present day.  And Sir R. F. Burton\ninforms me that when he was on the coast of Midian he found the\nArabs were in the habit of \"growing\" pearls by inserting a grain\nof sand into the shells.\n\n\n\n\n         THE SINGER AND THE DRUGGIST.--Vol. XI. p. 203.\n\n\n\nThe diverting adventures related in the first part of this tale\nshould be of peculiar interest to the student of Shakspeare as\nwell as to those engaged in tracing the genealogy of popular\nfiction.  Jonathan Scott has given--for reasons of his own--a\nmeagre abstract of a similar tale which occurs in the \"Bah\u00e1r-i-\nD\u00e1nish\" (vol. iii. App., p. 291), as follows:\n\n\n                        PERSIAN VERSION\n\nA young man, being upon business in a certain city, goes on a\nhunting excursion, and, fatigued with the chase, stops at a\ncountry house to ask refreshment.  The lady of the mansion\nreceives him kindly, and admits him as her lover.  In the midst\nof their dalliance the husband comes home, and the young man had\nno recourse to escape discovery but to jump into a basin which\nwas in the court of the house, and stand with head in a hollow\ngourd that happened to be in the water.  The husband, surprised\nto see the gourd stationary in the water, which was itself\nagitated by the wind, throws a stone at it, when the lover slips\nfrom beneath it and holds his breath till almost suffocated.\nFortunately, the husband presently retires with his wife into an\ninner room of the house, and thus the young man was enabled to\nmake good his escape.\n\nThe next day he relates his adventure before a large company at a\ncoffee-house.  The husband happens to be one of the audience,\nand, meditating revenge, pretends to admire the gallantry of the\nyoung man and invites him to his house.  The lover accompanies\nhim, and on seeing his residence is overwhelmed with confusion;\nbut, recovering himself, resolves to abide all hazards, in hopes\nof escaping by some lucky stratagem.  His host introduces him to\nhis wife, and begs him to relate his merry adventure before her,\nhaving resolved, when he should finish, to put them both to\ndeath.  The young man complies, but with an artful presence of\nmind exclaims at the conclusion, \"Glad was I when I awoke from so\nalarming a dream.\"  The husband upon this, after some questions,\nis satisfied that he had only told his dream, and, having\nentertained him nobly, dismisses him kindly.\n\nThe story is told in an elaborate form by Ser Giovanni\nFiorentino, in \"Il Pecorone\" (The Big Sheep, or, as Dunlop has\nit, The Dunce), which was begun in 1378 but not published till\n1554 (at Milan).  It is the second novel of the First Day and has\nbeen thus translated by Roscoe:\n\n\n                     SER GIOVANNI'S VERSION\n\nThere were once two very intimate friends, both of the family of\nSaveli, in Rome; the name of one of whom was Bucciolo; that of\nthe other Pietro Paolo, both of good birth and easy\ncircumstances.  Expressing a mutual wish to study for a while\ntogether at Bologna they took leave of their relatives and set\nout.  One of them attached himself to the study of the civil law,\nthe other to that of the canon law, and thus they continued to\napply themselves for some length of time.  But the subject of\nDecretals takes a much narrower range than is embraced by the\ncommon law, so Bucciolo, who pursued the former, made greater\nprogress than did Pietro Paolo, and, having taken a licentiate's\ndegree, he began to think of returning to Rome.  \"You see, my\ndear fellow student,\" he observed to his friend Paolo, \"I am now\na licentiate, and it is time for me to think of moving\nhomewards.\"  \"Nay, not so,\" replied his companion; \"I have to\nentreat you will not think of leaving me here this winter.  Stay\nfor me till spring, and we can return together.  In the meantime\nyou may pursue some other study, so that you need not lose any\ntime;\" and to this Bucciolo at length consented, promising to\nawait his relative's own good time.\n\nHaving thus resolved, he had immediate recourse to his former\ntutor, informing him of his determination to bear his friend\ncompany a little longer, and entreating to be employed in some\npleasant study to beguile the period during which he had to\nremain.  The professor begged him to suggest something he should\nlike, as he should be very happy to assist him in its attainment.\n\"My worthy tutor,\" replied Bucciolo, \"I think I should like to\nlearn the way in which one falls in love, and the best manner to\nbegin.\"  \"O very good!\" cried the tutor, laughing.  \"You could\nnot have hit upon anything better, for you must know that, if\nsuch be your object I am a complete adept in the art.  To lose\nno time, in the first place go next Sunday to the church of the\nFrati Minori (Friars Minor of St. Francis), where all the ladies\nwill be clustered together, and pay proper attention during\nservice in order to discover if any one of them in particular\nhappens to please you.  When you have done this, keep your eye\nupon her after service, to see the way she takes to her\nresidence, and then come back to me.  And let this be the first\nlesson--the first part--of that in which it is my intention to\ninstruct you.\"  Bucciolo went accordingly, and taking his station\nthe next Sunday in the church, as he had been directed, his eyes,\nwandering in every direction, were fixed upon all the pretty\nwomen in the place, and upon one in particular, who pleased him\nabove all the rest.  She was by far the most beautiful and\nattractive lady he could discover, and on leaving church he took\ncare to obey his master and follow her until he had made himself\nacquainted with her residence.  Nor was it long before the young\nlady began to perceive that the student was smitten with her;\nupon which Bucciolo returned to his master and informed him of\nwhat he had done.  \"I have,\" said he, \"learned as much as you\nordered me, and have found somebody I like very well.\"  \"So far,\ngood,\" cried the professor, not a little amused at the sort of\nscience to which his pupil had thus seriously devoted himself--\n\"so far, good!  And now observe what I have next to say to you:\nTake care to walk two or three times a day very respectfully\nbefore her house, casting your eyes about you in such a way that\nno one may catch you staring in her face; look in a modest and\nbecoming manner, so that she cannot fail to notice and be struck\nwith it.  And then return to me; and this, sir, will be the\nsecond lesson in this gay science.\"\n\nSo the scholar went and promenaded with great discretion before\nthe lady's door, who observed that he appeared to be passing to\nand fro out of respect to one of the inhabitants.  This attracted\nher attention, for which Bucciolo very discreetly expressed his\ngratitude by looks and bows, which being as often returned, the\nscholar began to be aware that the lady liked him.  He\nimmediately went and told the professor all that had passed, who\nreplied, \"Come, you have done very well.  I am hitherto quite\nsatisfied.  It is now time for you to find some way of speaking\nto her, which you may easily do by means of those gipsies who\nhaunt the streets of Bologna, crying ladies' veils, purses and\nother articles for sale.  Send word by her that you are the\nlady's most faithful, devoted servant, and that there is no one\nin the world you so much wish to please.  In short, let her urge\nyour suit, and take care to bring the answer to me as soon as you\nhave received it.  I will then tell you how you are to proceed.\"\n\nDeparting in all haste, he soon found a little old pedlar woman,\nquite perfect in the trade, to whom he said he should take it as\na particular favour if she would do one thing, for which he would\nreward her handsomely.  Upon this she declared her readiness to\nserve him in anything he pleased.  \"For you know,\" she added, \"it\nis my business to get money in every way I can.\"  Bucciolo gave\nher two florins, saying, \"I wish you to go for me to-day as far\nas the Via Maccarella, where resides a young lady of the name of\nGiovanna, for whom I have the very highest regard.  Pray tell her\nso, and recommend me to her most affectionately, so as to obtain\nfor me her good graces by every means in your power.  I entreat\nyou to have my interest at heart, and to say such pretty things\nas she cannot refuse to hear.\"  \"O leave that to me, sir,\" said\nthe little old woman, \"I will not fail to say a good word for you\nat the proper time.\"  \"Delay not,\" said Bucciolo, \"but go now,\nand I will wait for you here;\" and she set off at once, taking\nher basket of trinkets under her arm.  On approaching the place,\nshe saw the lady before the door, enjoying the air, and curtseying\nto her very low, \"Do I happen to have anything here you would\nfancy?\" she said, displaying her wares.  \"Pray, take something,\nmadam--whatever pleases you best.\"  Veils, stays, purses, and\nmirrors were now spread in the most tempting way before the\nlady's eyes.  Out of all these things her attention seemed to be\nmost attracted by a beautiful purse, which, she observed, if she\ncould afford, she should like to purchase. \"Nay, madam,\"\nexclaimed the crone, \"do not think anything about the price--take\nanything yo please, since they are all paid for already, I assure\nyou.\"  Surprised at hearing this, and perceiving the very\nrespectful manner of the speaker, the lady rejoined, \"Do you know\nwhat you are saying?  What do you mean by that?\"  The old woman,\npretending now to be much affected, said, \"Well, madam, if it\nmust be so, I shall tell you.  It is very true that a young\ngentleman of the name of Bucciolo sent me hither; one who loves\nyou better than all the world besides.  There is nothing he would\nnot do to please you, and indeed he appears to very wretched\nbecause he cannot speak to you, and he is so very good, that it\nis quite a pity.  I think it will be the death of him, and then\nhe is such a fine--such an elegant--young man, the more is the\npity!\"  on hearing this, the lady, blushing deeply, turned\nsharply round upon the little old woman, exclaiming, \"O you\nwicked creature! were it not for the sake of my own reputation, I\nwould give you such a lesson that you should remember it to the\nlatest day of your life!  A pretty story to come before decent\npeople with! Are you not ashamed of yourself to let such words\ncome out of your mouth?\"  Then seizing an iron bar that lay\nacross the doorway, \"Ill betide you, little wretch!\" she cried,\nas she brandished it.  \"If you ever come this way again, depend\non it, you will never go back alive!\"  The trembling old trot,\nquickly bundling up her wares, scampered off, in dread of feeling\nthat cruel weapon on her shoulders, nor did she think of stopping\ntill she had reached the place where Bucciolo stood waiting her\nreturn.  Eagerly inquiring the news and how she succeeded, \"O\nvery badly--very badly,\" answered the crone.  \"I was never in\nsuch a fright in all my life.  Why, she will neither see nor\nlisten to you, and if I had not run away, I should have felt the\nweight of a great iron bar upon my shoulders.  For my own part, I\nshall go there no more; and I advise you, signor, to look to\nyourself how you proceed in such affairs in future.\"\n\nPoor Bucciolo became quite disconsolate, and returned in all\nhaste to acquaint the professor with this unlucky result.  But\nthe professor, not a whit cast down, consoled him, saying, \"Do\nnot despair; a tree is not levelled at a single stroke, you know.\nI think you must have a repetition of your lesson to-night.  So\ngo and walk before her door as usual; notice how she eyes you,\nand whether she appears angry or not, and then come back again to\nme.\"  Bucciolo accordingly proceeded without delay to the lady's\nhouse.  The moment she perceived him she called her maid and said\nto her, \"Quick, quick--hasten after the young man--that is he,\nand tell him from me that he must come and speak with me this\nevening without fail--without fail.\"  The girl soon came up with\nBucciolo, and thus addressed him: \"My lady, signor, my lady,\nGiovanna, would be glad of your company this evening, she would\nbe very glad to speak with you.\"  Greatly surprised at this,\nBucciolo replied, \"Tell your lady I shall be most happy to wait\nupon her,\" so saying, he set off once more to the professor, and\nreported the progress of the affair.  But this time the master\nlooked a little more serious; for, from some trivial\ncircumstances put together, he began to entertain suspicions that\nthe lady was (as it really turned out) no other than his own\nwife.  So he rather anxiously inquired of Bucciolo whether he\nintended to accept the invitation.  \"To be sure I do,\" replied\nhis pupil.  \"Then,\" said the professor, \"promise that you will\ncome here before you set off.\"  \"Certainly I will,\" answered\nBucciolo readily, and took his leave.\n\nNow Bucciolo was far from suspecting that the lady bore so near a\nrelationship to his respected tutor, although the latter began to\nbe rather uneasy as to the result, feeling some twinges of\njealousy which were by no means pleasant.  For he passed most of\nhis winter evenings at the college where he gave lectures, and\nnot unfrequently remained there for the night.  \"I should be\nsorry,\" said he to himself, \"if this young gentleman were\nlearning these things at my expense, and I must therefore know\nthe real state of the case.\"  In the evening his pupil called\naccording to promise, saying, \"Worthy master, I am now ready to\ngo.\"  \"Well, go,\" replied the professor; \"but be wise, Signor\nBucciolo--be wise and think more than once what you are about.\"\n\"Trust me for that,\" said the scholar, a little piqued: \"I shall\ngo well provided, and not walk into the mouth of danger unarmed.\"\nAnd away he went, furnished with a good cuirass, a rapier, and a\nstiletto in his belt.  He was no sooner on his way than the\nprofessor slipped out quietly after him, dogging his steps\nclosely, until, trembling with rage, he saw him stop at his own\nhouse-door, which, on a smart tap being given, was quickly opened\nby the lady herself and the pupil admitted.  When the professor\nsaw that it was indeed his own wife, he was quite overwhelmed and\nthought, \"Alas, I fear this young fellow has learned more than he\nconfesses at my expense;\" and vowing to be revenged, he ran back\nto the college, where arming himself with sword and dagger, he\nthen hastened to his house in a terrible passion. Arriving at his\nown door, he knocked loudly, and the lady, sitting before the\nfire with Bucciolo, instantly knew it was her husband, so taking\nhold of Bucciolo, she concealed him hurriedly under a heap of\ndamp clothes lying on a table near the window for ironing, which\ndone, she ran to the door and inquired who was there.  \"Open\nquickly,\" exclaimed the professor.  \"You vile woman, you shall\nsoon know who is here!\"  On opening the door, she beheld him with\na drawn sword, and cried in well-affected alarm, \"O my dearest\nlife, what means this?\"  \"You know very well what it means,\" said\nhe.  \"The villain is now in the house.\"  \"Good Heaven! what is\nthat you say?\" exclaimed the lady.  \"Are you gone out of your\nwits? Come and search the house, and if you find anybody, I will\ngive you leave to kill me on the spot.  What! do you think I\nshould now begin to misconduct myself as I never before did \u2013 as\nnone of my family ever did before? Beware lest the Evil One\nshould be tempting you, and, suddenly depriving you of your\nsenses, draw you to perdition!\"  But the professor, calling for\ncandles, began to search the house from the cellar upwards--among\nthe tubs and casks--in every place but the right place--running\nhis sword through the beds and under the beds, and into every\ninch of the bedding--leaving no corner or crevice of the whole\nhouse untouched.  The lady accompanied him with a candle in her\nhand, frequently interrupting him with, \"Say your beads--say your\nbeads, good signor; it is certain that the Evil One is dealing\nwith you, for were I half so bad as you esteem me, I would kill\nmyself with my own hands.  But I entreat you not to give way to\nthis evil suggestion: oppose the adversary while you can.\"\nHearing these virtuous observations of his wife, and not being\nable to discover any one after the strictest search, the\nprofessor began to think that he must, after all, be possessed,\nand presently extinguished the lights and returned to the\ncollege.  The lady, on shutting the door after him, called out to\nBucciolo to come from his hiding place, and then, stirring the\nfire, began to prepare a fine capon for supper, with some\ndelicious wines and fruits.  And thus they regaled themselves,\nhighly entertained with each other, nor was it their least\nsatisfaction that the professor had just left them, apparently\nconvinced that they had learned nothing at his expense.\n\nProceeding to the college the next morning, Bucciolo, without the\nleast suspicion of the truth, informed his master that he had\nsomething for his ear which he was sure would make him laugh.\n\"How so?\" demanded the professor. \"Why,\" said his pupil, \"you\nmust know that last night, just as I had entered the lady's\nhouse, who should come in but her husband, and in such a rage!\nHe searched the whole house from top to bottom, without being\nable to find me.  I lay under a heap of newly-washed clothes,\nwhich were not half dry.  In short, the lady played her part so\nwell that the poor gentleman forthwith took his leave, and we\nafterwards ate a fine capon for supper and drank such wines--and\nwith such zest!  It was really one of the pleasantest evenings I\never spent in my life.  But I think I'll go and take a nap, for I\npromised to return this evening about the same hour.\"  \"Then be\nsure before you go,\" said the professor, trembling with\nsuppressed rage, \"be sure to come and tell me when you set out.\"\n\"O certainly,\" responded Bucciolo, and away he went.  Such was\nnow the unhappy tutor's condition as to render him incapable of\ndelivering a single lecture during the whole day, and such was\nhis extreme vexation and eagerness for evening, that he spent his\ntime in arming himself with sword and dagger and cuirass,\nmeditating only upon deeds of blood.  At the appointed time came\nBucciolo, with the utmost innocence, saying, \"My dear master, I\nam going now.\"  \"Yes, go,\" replied the professor, \"and come back\nto-morrow morning, if you can, and tell me how you have fared.\"\n\"I intend doing so,\" said Bucciolo, and departed at a brisk pace\nfor the house of the lady.\n\nArmed cap-\u00e0-pie, the professor ran out after him, keeping pretty\nclose to his heels, with the intention of catching him just as he\nentered.  But the lady, being on the watch, opened the door\nsuddenly for the pupil and shut it in her husband's face.  The\nprofessor began to knock and to call out with a furious noise.\nExtinguishing the light in a moment, the lady placed Bucciolo\nbehind the door, and throwing her arms round her husband's neck\nas he entered, motioned to her lover while thus she held his\nenemy to make his escape, and he, upon the husband's rushing\nforward, slipped out from behind the door unperceived.  She then\nbegan to scream as loud as she could, \"Help! Help! The professor\nhas gone mad! Will nobody help me?\" for he was in an ungovernable\nrage, and she clung faster to him than before.  The neighbours\nrunning to her assistance and seeing the peaceable professor\narmed with deadly weapons, and his wife crying out, \"Help, for\nthe love of Heaven!--too much study hath driven him mad!\"{ they\nreadily believed such to be the fact.  \"Come, good signor,\" they\nsaid, \"what is all this about?  Try to compose yourself--nay, do\nnot struggle so hard, but let us help you to your couch.\"  \"How\ncan I rest, think you,\" he replied, \"while this wicked woman\nharbours paramours in my house?  I saw him come in with my own\neyes.\"  \"Wretch that I am!\" cried his wife.  \"inquire of all my\nfriends and neighbours whether any one of them ever saw anything\nthe least unbecoming in my conduct.\"  The whole party with one voice entreated\nthe professor to lay such thoughts aside, for there was not a\nbetter lady breathing, or one who set a higher value upon her\nreputation.  \"But how can that be,\" said he, \"when I saw him\nenter the house, and he is in it now?\"  In the meanwhile the\nlady's two brothers arrived, when she began to weep bitterly,\nexclaiming, \"O my dear brothers, my poor husband has gone mad,\nquite mad--and he even says there is a man in the house.  I\nbelieve he would kill me if he could; but you know me too well to\nlisten for a moment to such a story,\" and she continued to weep.\n\nThe brothers then accosted the professor in no gentle terms:  \"We\nare surprised, signor--we are shocked to find that you dare\nbestow such epithets on our sister.  What can have led you, after\nliving so amicably together, to bring these charges against her\nnow?\" \"I can only tell you,\" answered the professor, \"that there\nis a man in the house.  I saw him enter.\"  \"Then come, and let us\nfind him.  Show him to us,\" retorted the incensed brothers, \"for\nwe will sift this matter to the bottom.  Show us the man, and we\nwill then punish her in such a way as will satisfy you.\"  One of\nthe brothers, taking his sister aside, said, \"First tell me, have\nyou really got any one hidden in the house? Tell the truth.\"\n\"Heavens!\" cried his sister, \"I tell you, I would rather suffer\ndeath.  Should I be the first to bring a scandal on our house?  I\nwonder you are not ashamed to mention such a thing.\"  Rejoiced to\nhear this, the brothers, directed by the professor, at once\ncommenced a search.  Half frantic, he led them at once to the\ngreat bundle of linen, which he pierced through and through with\nhis sword, firmly believing that he was killing Bucciolo, all the\nwhile taunting him at every blow.  \"There! I told you,\" cried his\nwife, \"that he was mad.  To think of destroying your own property\nthus!  It is plain he did not help to get them up,\" she\ncontinued, whimpering--\"all my best clothes!\"\n\nHaving now sought everywhere in vain, one of the brothers\nobserved, \"He is indeed mad,\" to which the other agreed, while he\nagain attacked the professor in the bitterest terms:  \"You have\ncarried matters too far, signor; your conduct to our sister is\nshameful, and nothing but insanity can excuse it.\"  Vexed enough\nbefore, the professor upon this flew into a violent passion, and\nbrandished his naked sword in such a way that the others were\nobliged to use their sticks, which they did so very effectively\nthat, after breaking them over his head, they chained him down\nlike a maniac upon the floor, declaring he had lost his wits by\nexcessive study, and taking possession of his house, they\nremained with their sister all night.  Next morning they sent for\na physician, who ordered a couch to be placed as near as possible\nto the fire, that no one should be allowed to speak or reply to\nthe patient, and that he should be strictly dieted until he\nrecovered his wits; and this regimen was diligently\nenforced.[FN#489]\n\nA report immediately spread through Bologna that the good\nprofessor had become insane, which caused very general regret,\nhis friends observing to each other, \"It is indeed a bad\nbusiness; but I suspected yesterday how it was--he could scarcely\nget a word out as he was delivering his lecture, did you not\nperceive?\"  \"Yes,\" said another, \"I saw him change colour, poor\nfellow.\"  And by everybody, everywhere, it was decided that the\nprofessor was mad.  In this situation numbers of his scholars\nwent to see him, and among the rest Bucciolo, knowing nothing of\nwhat had happened, agreed to accompany them to the college,\ndesirous of acquainting his master with last night's adventure.\nWhat was his surprise to learn that he had actually taken leave\nof his senses, and being directed on leaving the college to the\nprofessor's house, he was almost panic-struck on approaching the\nplace, beginning to comprehend the whole affair.  Yet, in order\nthat no one might be led to suspect the truth, he walked into the\nhouse along with the rest, and on reaching a certain apartment\nwhich he knew, he beheld his poor tutor almost beaten to a mummy,\nand chained down upon his bed, close to the fire.  His pupils\nwere standing round condoling with him and lamenting his piteous\ncase.  At length it came to Bucciolo's turn to say something to\nhim, which he did as follows:  \"My dear master, I am as truly\nconcerned for you as if you were my own father, and if there is\nanything in which I can be of service to you, command me as your\nown son.\"  To this the poor professor only replied, \"No,\nBucciolo, depart in peace, my pupil; depart, for you have learned\nmuch, very much, at my expense.\"  Here his wife interrupted him:\n\"You see how he wanders--heed not what he says--pay no attention\nto him, signor.\"  Bucciolo, however, prepared to depart, and\ntaking a hasty leave of the professor, he proceeded to the\nlodging of his friend Pietro Paolo, and said to him, \"Fare you\nwell.  God bless you, my friend.  I must away; and I have lately\nlearned so much at other people's expense that I am going home.\"\nSo saying, he hurried away, and in due course arrived in safety\nin Rome.\n\nThe affliction of the professor of Giovanni's sprightly tale will\nprobably be considered by most readers as well-merited\npunishment; the young gallant proved an apt scholar in the art of\nlove, and here was the inciter to evil repaid with the same coin!\n\nStraparola also tells the story, but in a different form, in his\n\"Pleasant Nights\" (Piacevoli Notti), First Day, second novella;\nand his version is taken into a small collection entitled\n\"Tarlton's Newes out of Purgatorie,\" first published in or before\n1590--a catchpenny tract in which, of course, Dick Tarlton had\nnever a hand, any more than he had in the collection of jests\nwhich goes under his name.\n\n\n                  STRAPAROLA'S VERSION[FN#490]\n\nIn Pisa, a famous cittie of Italye, there lived a gentleman of good\nlinage and landes, feared as well for his wealth, as honoured\nfor his vertue, but indeed well thought on for both; yet the\nbetter for his riches.  This gentleman had one onelye daughter,\ncalled Margaret, who for her beauty was liked of all, and desired\nof many.  But neither might their sutes nor her owne prevaile\nabout her father's resolution, who was determyned not to marrye\nher, but to such a man as should be able in abundance to maintain\nthe excellency of her beauty.  Divers yong gentlemen proffered\nlarge feoffments, but in vaine, a maide shee must bee still: till\nat last an olde doctor in the towne, that professed phisicke,\nbecame a sutor to her, who was a welcome man to her father, in\nthat he was one of the welthiest men in all Pisa; a tall\nstripling he was and a proper youth, his age about foure score,\nhis heade as white as milke, wherein for offence sake there was\nleft never a tooth.  But it is no matter, what he wanted in\nperson he had in the purse, which the poore gentlewoman little\nregarded, wishing rather to tie herself to one that might fit her\ncontent, though they lived meanely, then to him with all the\nwealth in Italye.  But shee was yong, and forcst to follow her\nfather's direction, who, upon large covenants, was content his\ndaughter should marry with the doctor, and whether she likte him\nor no, the match was made up, and in short time she was married.\nThe poore wench was bound to the stake, and had not onely an olde\nimpotent man, but one that was so jealous, as none might enter\ninto his house without suspition, nor shee doo any thing without\nblame; the least glance, the smallest countenance, any smile was\na manifest instance to him that she thought of others better than\nhimselfe.  Thus he himselfe lived in a hell, and tormented his\nwife in as ill perplexitie.\n\nAt last it chaunced that a young gentleman of the citie, comming\nby her house, and seeing her looke out at her window, noting her\nrare and excellent proportion, fell in love with her, and that so\nextreamelye, as his passions had no meanes till her favour might\nmittigate his heart sicke discontent.  The yong man that was\nignorant in amorous matters, and had never beene used to courte\nanye gentlewoman, thought to reveale his passions to some one\nfreend that might given him counsaile for the winning of her\nlove, and thinking experience was the surest maister, on a daye\nseeing the olde doctor walkinge in the churche that was\nMargaret's husband, little knowing who he was, he thought this\nthe fittest man to whom he might discover his passions, for that\nhee was olde and knew much, and was a phisition that with his\ndrugges might helpe him forward in his purposes, so that seeing\nthe olde man walke solitary, he joinde unto him, and after a\ncurteous salute, tolde him that he was to impart a matter of\ngreat import to him, wherein, if hee would not onely be secrete,\nbut indevour to pleasure him, his pains should bee every way to\nthe full considered.  You must imagine, gentleman, quoth Mutio,\nfor so was the doctor's name, that men of our profession are no\nblabs, but hold their secrets in their hearts bottome, and\ntherefore reveale what you please, it shall not onely be\nconcealed, but cured, if either my art or counsaile may doo it.\nUpon this, Lyonell, so was the young gentleman called, told and\ndiscourst unto him from point to point, how he was falne in love\nwith a gentlewoman that was married to one of his profession,\ndiscovered her dwelling and the house, and for that he was\nunacquainted with the woman, and a man little experienced in love\nmatters, he required his favour to further him with his advice.\nMutio at this motion was stung to the hart, knowing it was his\nwife hee was fallen in love withall, yet to conceale the matter,\nand to experience his wive's chastity, and that if she plaide\nfalse, he might be revenged on them both, he dissembled the\nmatter, and answered that he knewe the woman very well, and\ncommended her highly: but said she had a churle to her husband,\nand therefore he thought shee would bee the more tractable: Trye\nher, man, quoth hee, fainte harte never wonne faire lady, and if\nshee will not be brought to the bent of your bowe, I will provide\nsuch a potion as shall dispatch all to your owne content: and to\ngive you further instructions for oportunitie, knowe that her\nhusband is foorth every after-noone from three till sixe.  Thus\nfarre I have advised you, because I pitty your passions, as my\nselfe being once a lover, but now I charge thee reveale it to\nnone whomsoever, least it doo disparage my credit to meddle in\namorous matters.\n\nThe yong gentleman not onely promised all carefull secrecy, but\ngave him harty thanks for his good counsell, promising to meete\nhim there the next day, and tell him what newes.  Then hee left\nthe old man, who was almost mad for feare his wife any way should\nplay false; he saw by experience brave men came to besiege the\ncastle, and seeing it was in a woman's custodie, and had so weeke a\ngovernor as himselfe, he doubted it would in time be delivered\nup: which feare made him almost franticke, yet he drivde of the\ntime great torment, till he might heare from his rival.  Lionello\nhe hastes him home and sutes him in his braverye, and goes downe\ntoward the house of Mutio, where he sees her at the windowe, whome\nhe courted with a passionate looke, with such humble salute as\nshee might perceive how the gentleman was affectionate.\nMargaretta, looking earnestlye upon him, and noting the\nperfection of his proportion, accounted him in her eye the flower\nof all Pisa, thinkte herselfe fortunate if shee might have him\nfor her freend, to supply the defaultes that she found in Mutio.\nSundry times that afternoone he past by her window, and he cast\nnot up more loving lookes, then he received gratious favours,\nwhich did so incourage him that the next daye betweene three and\nsixe hee went to her house, and knocking at the doore, desired to\nspeake with the mistris of the house, who hearing by her maid's\ndescription what he was, commaunded him to come in, where she\nintertained him with all courtesie.\n\nThe youth that never before had given the attempt to court a\nladye, began his exordium with a blushe; and yet went forward so\nwell, that hee discourst unto her howe hee loved her, and that if\nit might please her to accept of his service, as of a freende\never vowde in all dutye to bee at her commaunde, the care of her\nhonour should bee deerer to him than his life, and hee would be\nready to prise her discontent with his bloud at all times.  The\ngentlewoman was a little coye, but, before they part, they\nconcluded that the next daye at foure of the clock hee should\ncome thither and eate a pound of cherries, which was resolved on\nwith a succado des labras, and so with a loath to depart they\ntooke their leaves.  Lionello as joyfull a man as might be, hyed\nhim to the church to meete his olde doctor, where he found him in\nhis olde walke: What newes, syr, quoth Mutio, how have you sped?\nEven as I can wishe, quoth Lionello, for I have been with my\nmistrisse, and have found her so tractable, that I hope to make\nthe olde peasant, her husband, looke broadheaded by a paire of\nbrowantlers.  How deepe this strooke into Mutio's hart, let them\nimagine that can conjecture what jelousie is; insomuch that the\nolde doctor askte when should be the time.  Marry, quoth\nLionello, tomorrow, at foure of the clocke in the afternoone, and then\nMaister Doctor, quoth hee, will I dub the old squire knight of\nthe forked order.\n\nThus they past on in that, till it grew late, and then Lyonello\nwent home to his lodging and Mutio to his house, covering all his\nsorrows with a merrye countenance, with full resolution to\nrevenge them both the next daye with extremitie.  He past the\nnight as patiently as he could, and the next daye, after dinner,\nawaye hee went, watching when it should bee foure of the clocke.\nAt the hour justly came Lyonello and was intertained with all\ncurtesie; but scarce had they kist, ere the maide cryed out to her\nmistresse that her maister was at the doore; for he hasted,\nknowing that a horne was but a litle while in grafting.\nMargaret, at this alarum, was amazed, and yet for a shift chopt\nLionello into a great driefatte[FN#491] full of feathers,[FN#492]\nand sat her downe close to her woorke.  By that came Mutio in\nblowing, and as though hee came to looke somewhat in haste,\ncalled for the keyes of his chamber, and looked in everye place,\nsearching so narrowlye in everye corner of the house, that he\nleft not the very privie unsearcht.  Seeing he could not finde\nhim, hee said nothing, but fayning himselfe not well at ease,\nstaide at home; so that poor Lionello was faine to staye in the\ndrifatte till the olde churle was in bed with his wife; and then\nthe maide let him out at a backe doore, who went home with a flea\nin his eare to his lodging.\n\nWell, the next day he went againe to meete his doctor, whome he\nfounde in his wonted  walke.  What newes? Quoth Mutio, how have\nyou sped?  A poxe of the olde slave, quoth Lyonello; I was no\nsooner in and had given my mistresse one kisse, but the jelous\nasse was at the doore; the maide spied him, and cryed her\nmaister; so that the poore gentlewoman, for very shifte, was\nfaine to put me in a driefatte of feathers that stoode in an olde\nchamber, and there I was faine to tarry while[FN#493] he was in\nbed and a-sleepe, and then the maide let me out, and I departed.\nBut it is no matter; 'twas but a chaunce, and I hope to crye\nquittance with him ere it be long.  As how? Quoth Mutio.  Marry,\nthus, quoth Lionello: shee sent me woord by her maide this daye\nthat upon Thursday next the olde churle suppeth with a patient of\nhis a mile out of Pisa, and then I feare not but to\nquitte[FN#494] him for all.  It is well, quoth Mutio; fortune bee\nyour frende.  I thanke you, quoth Lionello: and so, after a\nlittle more prattle, they departed.\n\nTo bee shorte, Thursdaye came, and about sixe of the clocke,\nfoorth goes Mutio no further than a freendes house of his, from\nwhence he might descrye who went into his house; straight hee saw\nLionello enter in, and after goes hee, insomuche that hee was\nscarcelye sitten downe, before the mayde cryed out againe, my\nmaister comes.  The goodwife, that before had provided for after-\nclaps,[FN#495] had found out a privie place between two seelings\nof a plauncher,[FN#496] and there she thrust Lionello, and her\nhusband came sweting.  What news, quoth shee, drives you home\nagaine so soone, husband?  Marry, sweete wife, quoth he, a\nfearfull dream that I had this night, which came to my\nremembrance, and that was this: me thought there was a villaine\nthat came secretlye into my house, with a naked poinard in his\nhand, and hid himselfe, but I could not finde the place; with\nthat mine nose bled, and I came backe; and, by the grace of God, I\nwill seeke every corner in the house for the quiet of my minde.\nMarry, I pray you doo, husband, quoth she.  With that he lockt in\nall the doors, and began to search every chamber, every hole,\nevery chest, every tub, the very well; he stabd every feather bed\nthrough, and made havocke like a mad man, which made him thinke\nall was in vaine; and hee began to blame his eies that thought\nthey saw that which they did not.  Upon this he rest halfe\nlunaticke, and all night he was very wakefull, that towards the\nmorning he fell into a dead sleepe, and then was Lionello\nconveighed away.\n\nIn the morning when Mutio wakened, hee thought how by no meanes\nhee should be able to take Lionello tardy: yet he laid in his\nhead a most dangerous plot; and that was this:  Wife, quoth he, I\nmust the next Monday ride to Vycensa, to visit an olde patient of\nmine; till my returne, which will be some ten dayes, I will have\nthee staye at our little graunge house in the countrey.  Marry,\nvery well content, quoth she.  With that he kist her, and was\nverye pleasant, as though he had suspected nothing, and away hee\nflings to the church, where he meetes Lionello.  What, sir, quoth\nhe, what news? is your mistresse yours in possession?  No, a\nplague of the olde slave, quoth hee.  I think he is either a\nwitch or els woorkes by magick; for I can no sooner enter into\nthe doores, but he is at my backe, and so he was againe\nyesternight; for I was not warm in my seate before the maide\ncryed, my maister comes; and then was the poore soule faine to\nconveigh me betweene two seelings of a chamber, in a fit place\nfor the purpose, wher I laught hartely to myself too see how he\nsought every corner, ransakt every tub, and stabd every feather\nbed, but in vaine; I was safe enough until the morning, and then,\nwhen he was fast asleepe, I lept out.  Fortune frownes on you,\nquoth Mutio.  I,[FN#497] but I hope, quoth Lionello, this is the\nlast time, and now shee will begin to smile; for on Monday next\nhe rides to Vicensa, and his wife lyes at the grange house a\nlittle (out) of the towne, and there in his absence I will\nrevenge all forepast misfortunes.  God sent it be so, quoth\nMutio; and so took his leave.\n\nThese two lovers longd for Monday, and at last it came.  Early in\nthe morning Mutio horst himselfe and his wife, his maide and a\nman, and no more, and away he rides to his grange house, wher,\nafter he had brok his fast, he took his leave, and away towards\nVicensa.  He rode not far ere, by a false way, he returned into\na thicket, and there, with a company of cuntry peasants, lay in\nan ambuscade to take the young gentleman.  In the afternoon comes\nLionello galloping, and as soon as he came within sight of the\nhouse, he sent back his horse by his boy, and went easily afoot,\nand there, at the very entry, was entertained by Margaret, who\nled him up the staires, and convaid him into her bedchamber,\nsaying he was welcome into so mean a cottage.  But, quoth she,\nnow I hope fortun shall not envy the purity of our loves.  Alas!\nalas! mistris, cried the maid, heer is my maister, and 100 men\nwith him, with bils and staves.  We are betraid, quoth Lionel,\nand I am but a dead man.  Feare not, quoth she, but follow me:\nand straight she carried him downe into a low parlor, where\nstoode an olde rotten chest full of writinges; she put him into\nthat, and covered him with olde papers and evidences, and went to\nthe gate to meet her husband.\n\nWhy, Signor Mutio, what meanes this hurly burly? quoth she.  Vile\nand shameless strumpet as thou art, thou shalt know by and by,\nquoth he.  Where is thy love?  All we have watcht him and seen\nhim enter in.  Now, quoth he, shall neither thy tub of feathers\nor thy seeling serve, for perish he shall with fire, or els fall\ninto my handes.  Doo thy worst, jealous foole, quoth she, I ask\nthee no favour.  With that, in a rage, he beset the house round,\nand then set fire on it.  Oh, in what perplexitie was poore\nLionello in that he was shut in a chest, and the fire about his\neares! and how was Margaret passionat, that knew her lover was in\nsuch danger!  Yet she made light of the matter, and, as one in a\nrage, called her maid to her and said: Come on, wench, seeing thy\nmaister, mad with jelousie, hath set the house and al my living\non fire, I will be revengd on him: help me heer to lift this old\nchest where all his writings and deeds are; let that burne first,\nand as soon as I see that on fire I will walke towards my\nfreends, for the olde foole will be beggard, and I will refuse\nhim.  Mutio, that knew al his obligations and statutes lay there,\npuld her back and bad two of his men carry the chest into the\nfield, and see it were safe, himselfe standing by and seeing his\nhouse burned downe sticke and stone.  Then, quieted in his mind,\nhe went home with his wife and began to flatter her, thinking\nassuredly that he had burnt her paramour, causing his chest to be\ncarried in a cart to his house in Pisa.  Margaret, impatient,\nwent to her mother's and complained to her and her brethren of\nthe jealousie of her husband, who maintaned her it to be true,\nand desired but a daies respite to proove it.\n\nWel, hee was bidden to supper the next night at her mother's, she\nthinking to make her daughter and him freends againe.  In the\nmeane time he to his woonted walk in the church, and there,\npr\u00e6ter expectationem, he found Lionello walking.  Wondring at\nthis, he straight enquires what newes.  What newes, Maister\nDoctor, quoth he, and he fell in a great laughing; in faith\nyesterday, I scapt a scouring, for syrrha, I went to the grange-\nhouse, where I was appointed to come, and I was no sooner gotten\nup the chamber, but the magicall villeine, her husband, beset\nthe house with bils and staves, and that he might be sure no\nseeling nor corner should shrowde me, he set the house on fire,\nand so burnt it downe to the ground.  Why, quoth Mutio, and how\ndid you escape?  Alas, quoth he, wel fare a woman's wit; she\nconveighed me into an old chest full of writings, which she knew\nher husband durst not burne, and so I was saved and brought to\nPisa, and yesternight, by her maide, let home to my lodging.\nThis, quoth he, is the pleasantest jest that ever I heard; and\nupon this I have a sute to you: I am this night bidden foorth to\nsupper, you shall be my guest, onely I will crave so much favour,\nas after supper for a pleasant sporte, to make relation what\nsuccesse you have had in your loves.  For that I will not sticke,\nquoth he, and so he conveyed Lionello to his mother-in-law's\nhouse with him, and discovered to his wive's brethren who he was,\nand how at supper he would disclose the whole matter; For, quoth\nhe, he knowes not that I am Margaret's husband.  At this all the\nbrethren bad him welcome, and so did the mother to, and Margaret,\nshe was kept out of sight.  Supper time being come they fell to\ntheir victals, and Lionello was carrowst unto by Mutio, who was\nvery pleasant, to drawe him into a merry humour, that he might to\nthe ful discourse the effect and fortunes of his love.  Supper\nbeing ended, Mutio requested him to tel to the gentlemen what had\nhapned between him and his mistresse.  Lionello, with a smiling\ncountenance, began to describe his mistresse, the house and\nstreet where she dwelt, how he fell in love with her, and how he\nused the councell of this doctor, who in all his affaires was his\nsecretarye.  Margaret heard all this with a great feare, and when\nhe came to the last point, she caused a cup of wine to be given\nhim by one of her sisters, wherein was a ring that he had given\nMargaret.  As he had told how he had escapt burning, and was\nready to confirme all for a troth, the gentlewoman drunke to him,\nwho taking the cup and seeing the ring, having a quick wit and a\nreaching head, spide the fetch, and perceived that all this while\nthis was his lover's husband to whome hee had revealed these\nescapes; at this drinking the wine and swallowing the ring into\nhis mouth he went forward.  Gentlemen, quoth he, how like you of\nmy loves and my fortunes?  Wel, quoth the gentlemen; I pray you\nis it true?  As true, quoth he, as if I would be so simple as to\nreveal what I did to Margaret's husband; for, know you,\ngentlemen, that I knew this Mutio to be her husband whom I\nnotified to be my lover; and for that he was generally known\nthroughout Pisa to be a jealous fool, therefore, with these tales\nI brought him into paradice, which are follies of mine owne\nbraine; for, trust me, by the faith of a gentleman, I never spake\nto the woman, was never in her companye, neyther doo I know her\nif I see her.  At this they all fell in a laughing at Mutio, who\nwas ashamde that Lionello had so scoft him.  But all was well;\nthey were made friends, but the jest went so to his hart that he\nshortly after died, and Lionello enjoyed the ladye.\n\nSer Giovanni's story, Roscoe observes, is \"curious as having\nthrough the medium of translation suggested the idea of those\namusing scenes in which the renowned Falstaff acquaints Master\nFord, disguised under the name of Brooke, with his progress in\nthe good graces of Mrs. Ford.  The contrivances likewise by which\nhe eludes the vengeance of the jealous husband are similar to\nthose recounted in the novel, with the addition of throwing the\nunweildy knight into the river.  Dunlop says that the same story\nhas been translated is a collection entitled 'The Fortunate,\nDeceived, and Unfortunate Lovers,' and that Shakspeare may\nprobably also have seen it in 'Tarlton's Newes out of\nPurgatorie,' where the incidents related in the Lovers of Pisa\nare given according to Straparola's story.  Moli\u00e8re made a happy\nuse of it in his 'Ecole des Femmes,' where the humour of the\npiece turns upon a young gentleman confiding his progress in the\naffections of a lady to the ear of her guardian, who believed he\nwas on the point of espousing her himself.\"  Two other French\nplays were based upon the story, one of which was written by La\nFontaine under the title of \"Le Ma\u00eetre en Droit.\"  Readers of\n\"Gil Blas\" will also recollect how Don Raphael confides to\nBalthazar the progress of his amour with his wife, and expresses\nhis vexation at the husband's unexpected return.\n\nIt is much to be regretted that nothing is known as to the date\nand place of the composition of the Breslau edition of The\nNights, which alone contains this and several other tales found\nin the collections of the early Italian novelists.\n\n\n\n\nTHE KING WHO KENNED THE QUINTESSENCE OF THINGS.--Vol. XI. p. 212.\n\n\n\nAlthough we may find, as already stated, the direct source of\nthis tale in the forty-sixth chapter of Al-Mas'\u00fadi's \"Meadows of\nGold and Mines of Gems,\" which was written about A.D. 943, yet\nthere exists a much older version--if not the original form--in a\nSanskrit collection entitled, \"Vet\u00e1lapanchavinsat\u00ed,\" or Twenty-\nfive Tales of a Vampyre.  This ancient work is incorporated with\nthe \"Kath\u00e1 Sarit S\u00e1gara,\" or Ocean of the Streams of Story,\ncomposed in Sanskrit verse by Somadeva in the 11th century, after\na similar work, now apparently lost, entitled \"Vrihat Kath\u00e1,\" or\nGreat Story, written by Gunadhya, in the 6th century.[FN#498]  In\nthe opinion of Benfey all the Vampyre Tales are of Buddhist\nextraction (some are unquestionably so), and they probably date\nfrom before our era.  As a separate work they exist, more or less\nmodified, in many of the Indian vernaculars; in Hind\u00ed, under the\ntitle of \"Baital Pach\u00eds\u00ed\"; in Tamil, \"Vedala Kadai\"; and there\nare also versions in Telegu, Mahratta, and Canarese.  The\nfollowing is from Professor C. H. Tawney's complete translation\nof the \"Kath\u00e1 Sarit S\u00e1gara\" (it is the 8th recital of the\nVet\u00e1la):\n\n\n                         INDIAN VERSION\n\nThere is a great tract of land assigned to Br\u00e1hmans in the\ncountry of Anga, called Vrikshaghata.  In it there lived a rich\nsacrificing Br\u00e1hman named Vishnusv\u00e1min.  And he had a wife equal\nto himself in birth.  And by her he had three sons born to him,\nwho were distinguished for preternatural acuteness.  In course of\ntime they grew up to be young men.  One day, when he had begun a\nsacrifice, he sent those three brothers to the sea to fetch a\nturtle.  So off they went, and when they had found a turtle, the\neldest said to his two brothers, \"Let one of you take the turtle\nfor our father's sacrifice; I cannot take it, as it is all\nslippery with slime.\"  When the eldest said this, the two younger\nones answered him, \"If you hesitate about taking it, why should\nnot we?\"  When the eldest heard that, he said, \"You two must take\nthe turtle; if you do not, you will have obstructed your father's\nsacrifice, and then you will certainly sink down to hell.\" When\nhe told the younger brothers this, they laughed and said to him,\n\"If you see our duty so clearly, why do you not see that your own\nis the same?\"  Then the eldest said, \"What, do you not know how\nfastidious I am?  I am very fastidious about eating, and I cannot\nbe expected to touch what is repulsive.\"  The middle brother,\nwhen he heard this speech of his, said to his brother, \"Then I am\na more fastidious person than you, for I am a most fastidious\nconnoisseur of the fair sex.\"  When the middle one said this, the\neldest went on to say, \"Then let the younger of you two take the\nturtle.\"  Then the youngest brother frowned, and in his turn said\nto the two elder, \"You fools, I am very fastidious about beds; so\nI am the most fastidious of the lot.\"\n\nSo the three brothers fell to quarrelling with one another, and\nbeing completely under the dominion of conceit, they left that\nturtle and went off immediately to the court of the king of that\ncountry, whose name was Prasenajit, and who lived in a city named\nVitankapura, in order to have the dispute decided.  There they\nhad themselves announced by the warder, and went in, and gave the\nking a circumstantial account of their case.  The king said,\n\"Wait here, and I will put you all in turn to the proof;\" so they\nagreed and remained there.  And at the time that the king took\nhis meal, he had them conducted to a seat of honour, and given\ndelicious food fit for a king, possessing all the six flavours.\nAnd while all were feasting around him, the Br\u00e1hman who was\nfastidious about eating along of the company did not eat, but sat\nthere with his face puckered up with disgust.  The king himself\nasked the Br\u00e1hman why he did not eat his food, though it was\nsweet and fragrant, and he slowly answered him, \"I perceive in\nthis food an evil smell of the reek from corpses, so I cannot\nbring myself to eat it, however delicious it may be.\"  When he\nsaid this before the assembled multitude, they all smelled it by\nthe king's orders, and said, \"This food is prepared from white\nrice and is good and fragrant.\"  But the Br\u00e1hman who was so\nfastidious about eating would not touch it, but stopped his nose.\nThen the king reflected, and proceeded to inquire into the\nmatter, and found out from his officers that the food had been\nmade from rice which had been grown in a field near the burning\ngh\u00e1t of a certain village.  Then the king was much astonished,\nand, being pleased, he said to him, \"In truth you are very\nparticular as to what you eat; so eat of some other dish.\"\n\nAnd after they had finished their dinner, the king dismissed the\nBr\u00e1hmans to their apartments and sent for the loveliest lady of\nhis court.  And in the evening he sent that fair one, all whose\nlimbs were of faultless beauty, splendidly adorned, to the second\nBr\u00e1hman, who was so squeamish about the fair sex.  And that\nmatchless kindler of Cupid's flame, with a face like the full\nmoon of midnight, went, escorted by the king's servants, to the\nchamber of the Br\u00e1hman.  But when she entered, lighting up the\nchamber with her brightness, that gentleman who was so fastidious\nabout the fair sex felt quite faint, and stopping his nose with\nhis left hand, said to the king's servants, \"Take her away; if\nyou do not, I am a dead man: a smell comes from her like that of\na goat.\"  When the king's servants heard this, they took the\nbewildered fair one to their sovereign, and told him what had\ntaken place.  And the king immediately had the squeamish\ngentleman sent for, and said to him, \"How can this lovely woman,\nwho has perfumed herself with sandal-wood, camphor, black aloes,\nand other splendid scents, so that she diffuses exquisite\nfragrance through the world, smell like a goat?\"  But though the\nking used this argument to the squeamish gentleman he stuck to\nhis point; and then the king began to have his doubts on the\nsubject, and at last, by artfully framed questions, he elicited\nfrom the lady herself that, having been separated in her\nchildhood from her mother and nurse, she had been brought up on\ngoat's milk.\n\nThen the king was much astonished, and praised highly the\ndiscernment of the man who was fastidious about the fair sex, and\nimmediately had given to the third Br\u00e1hman, who was fastidious\nabout beds, in accordance with his taste, a bed composed of seven\nmattresses placed upon a bedstead.  White smooth sheets and\ncoverlets were laid upon the bed, and the fastidious man slept\nupon it in a splendid room.  But, before half a watch of the\nnight had passed, he rose up from that bed, with his hand pressed\nto his side, screaming in an agony of pain.  And the king's\nofficers, who were there, saw a red crooked mark on his side, as\nif a hair had been pressed deep into it. And they went and told\nthe king, and the king said to them, \"Look and see if there is\nnot something under the mattress.\"  So they went and examined the\nbottom of the mattresses one by one, and they found a hair in the\nmiddle of the bedstead underneath them all.  And they took it and\nshowed it to the king, and they also brought the man who was\nfastidious about beds, and when the king saw the state of his\nbody, he was astonished.  And he spent the whole night in\nwondering how a hair could make so deep an impression on his skin\nthrough seven mattresses.[FN#499]\n\nAnd the next morning the king gave three hundred thousand gold\npieces to those fastidious men, because they were persons of\nwonderful discernment and refinement.  And they remained in great\ncomfort in the king's court, forgetting all about the turtle, and\nlittle did they reck of the fact that they had incurred sin by\nobstructing their father's sacrifice.[FN#500]\n\nThe story of the brothers who were so very \"knowing\" is common to\nmost countries, with occasional local modifications.  It is not\noften we find the knowledge of the \"quintessence of things\"\nconcentrated in a single individual, as in the case of the ex-\nking of our tale, but we have his exact counterpart--and the\ncircumstance is significant--in No. 2 of the \"Cento Novelle\nAntiche,\" the first Italian collection of short stories, made in\nthe 13th century, where a prisoner informs the king of Greece\nthat a certain horse has been suckled by a she-ass, that a jewel\ncontains a flaw, and that the king himself is a baker.  Mr.\nTawney, in a note on the Vet\u00e1la story, as above, refers also to\nthe decisions of Hamlet in Saxo Grammaticus, 1839, p. 138, in\nSimrock's \"Quellen des Shakespeare,\" I, 81-85; 5, 170; he lays\ndown that some bread tastes of blood (the corn was grown on a\nbattlefield); that some liquor tastes of iron (the malt was mixed\nwith water taken from a well in which some rusty swords had\nlain); that some bacon tastes of corpses (the pig had eaten a\ncorpse); lastly, that the king is a servant and his wife a\nserving-maid.  But in most versions of the story three brothers\nare the gifted heroes.\n\nIn \"M\u00e9lusine\"[FN#501] for 5 Nov. 1885, M. Ren\u00e9 Basset cites an\ninteresting variant (in which, as is often the case, the \"Lost\nCamel\" plays a part, but we are not concerned about it at present)\nfrom Radloff's \"Proben der Volksliteratur der t\u00fcrkischen St\u00e4mme\nS\u00fcd-Sibiriens,\" as follows:\n\n\n                        SIBERIAN VERSION\n\nMeat and bread were set before the three brothers, and the prince\nwent out.  The eldest said, \"The prince is a slave;\" the second,\n\"This is dog's flesh;\" the youngest, \"This bread has grown over\nthe legs of a dead body.\"  The prince heard them.  He took a\nknife and ran to find his mother.  \"Tell me the truth,\" cried he-\n-\"were you unfaithful to my father during his absence?  A man who\nis here has called me a slave.\"  \"My son,\" replied she, \"If I\ndon't tell the truth, I shall die; if I tell it, I shall die.\nWhen thy father was absent, I gave myself up to a slave.\"  The\nprince left his mother and ran to the house of the shepherd:\n\"The meat which you have cooked to-day--what is it?  Tell the\ntruth, otherwise I'll cut your head off.\"  \"Master, if I tell it,\nI shall die; if I don't, I shall die.  I will be truthful.  It was\na lamb whose mother had no milk; on the day of its birth, it was\nsuckled by a bitch: that is to-day's ewe.\"  The prince left the\nshepherd and ran to the house of the husbandman: \"Tell the truth,\nor else I'll cut off your head.  Three young men have come to my\nhouse. I have placed bread before them, and they say that the\ngrain has grown over the limbs of a dead man.\"  \"I will be frank\nwith you.  I ploughed with my plough in a place where were\n(buried) the limbs of a man; without knowing it, I sowed some\nwheat, which grew up.\"  The prince quitted his slave and returned\nto his house, where were seated the strangers.  He said to the\nfirst, \"Young man, how do you know that I am a slave?\"  \"Because\nyou went out as soon as the repast was brought in.\"  He asked the\nsecond, \"How do you know that the meat which was served was that\nof a dog?\" \"Because it has a disagreeable taste like the flesh of\na dog.\"  Then to the third: \"How come you to know that this bread\nwas grown over the limbs of a dead person?\"  \"What shall I say?\nIt smells of the limbs of a dead body; that is why I recognised\nit.  If you do not believe me, ask your slave; he will tell you\nthat what I say is true.\"\n\nIn the same paper (col. 516) M. Ren\u00e9 Basset cites a somewhat\nelaborate variant, from Stier's \"Ungarische Sagen und M\u00e4rchen,\"\nin which, once more, the knowledge of the \"quintessence of\nthings\" is concentrated in a single individual.\n\n\n                       HUNGARIAN VERSION\n\nA clever Magyar is introduced with his companions in disguise\ninto the camp of the king of the T\u00e1t\u00e1rs, who is menacing his\ncountry.  The prince, suspicious, causes him to be carefully\nwatched by his mother, a skilful sorceress.  They brought in the\nevening's repast.  \"What good wine the prince has!\" said she.\n\"Yes,\" replied one, \"but it contains human blood.\"  The sorceress\ntook note of the bed from whence these words proceeded, and when\nall were asleep she deftly cut a lock of hair from him who had\nspoken, crept stealthily out of the room, and brought this mark\nto her son.  The strangers started up, and when our hero\ndiscovered what had been done to him, he cut a lock from all, to\nrender his detection impossible.  When they came to dinner, the\nking knew not from whom the lock had been taken.  The following\nnight the mother of the prince again slipped into the room, and\nsaid, \"What good bread has the prince of the T\u00e1t\u00e1rs!\"  \"Very\ngood,\" replied one, \"it is made with the milk of a woman.\"  When\nall were asleep, she cut a little off the moustache of him who\nwas lying in the bed from which the voice proceeded.  This time\nthe Magyars were still more on the alert, and when they were\napprised of the matter, they all cut a little from their\nmoustaches, so that next morning the prince found himself again\nfoiled.  The third night the old lady hid herself, and said in a\nloud voice, \"What a handsome man is the prince of the T\u00e1t\u00e1rs!\"\n\"Yes,\" replied one, \"but he is a bastard.\"  When all were asleep,\nthe old lady made a mark on the visor of the helmet of the one\nfrom whence had come the words, and then acquainted her son of\nwhat she had done.  In the morning the prince perceived that all\nthe helmets were similarly marked.[FN#502] At length he\nrefrained, and said, \"I see that there is among you a master\ngreater than myself; that is why I desire very earnestly to know\nhim.  He may make himself known; I should like to see and know\nthis extraordinary man, who is more clever and powerful than\nmyself.\"  The young man started up from his seat and said, \"I\nhave not wished to be stronger or wiser than yourself.  I have\nonly wished to find out what you had preconcerted for us.  I am\nthe person who has been marked three nights.\"  \"It is well, young\nman.  But prove now your words:  How is there human blood in the\nwine?\"  \"Call your butler and he will tell you.\"  The butler came\nin trembling all over, and confessed that when he corked the wine\nhe had cut his finger with the knife, and a drop of blood had\nfallen into the cask.  \"But how is there woman's milk in the\nbread?\" asked the king.  \"Call the bakeress,\" he replied, \"and\nshe will tell it you.\"  When they questioned her, she confessed\nthat she was kneading the bread and at the same time suckling her\nbaby, and that on pressing it to her breast some milk flowed and\nwas mixed with the bread.  The sorceress, the mother of the king,\nwhen they came to the third revelation of the young man,\nconfessed in her turn that the king was illegitimate.\n\nMr. Tawney refers to the Chevalier de Mailly's version of the\nThree Princes of Serendip (Ceylon):  The three are sitting at\ntable, and eating a leg of lamb, sent with some splendid wine\nfrom the table of the emperor Bahr\u00e1m.  The eldest maintains that\nthe wine was made of grapes that grew in a cemetery; the second,\nthat the lamb was brought up on dog's milk; while the third\nasserts that the emperor had put to death the son of the waz\u00edr.\nAnd that the latter is bent on vengeance.  All these statements\nturn out to be well-grounded.  Mr. Tawney also refers to parallel\nstories in the Breslau edition of The Nights; namely, in Night\n458, it is similarly conjectured that the bread was baked by a\nsick woman; that the kid was suckled by a bitch, and that the\nsultan is illegitimate; and in Night 459, a gem-cutter guesses\nthat a jewel has an internal flaw, a man skilled in the pedigrees\nof horses divines that a horse is the offspring of a female\nbuffalo, and a man skilled in human pedigrees that the mother of\nthe favourite queen was a rope-dancer.  Similar incidents occur\nin \"The Sultan of Yemen and his Three Sons,\" one of the\nAdditional Tales translated by Scott, from the Wortley-Montague\nMS., now in the Bodleian Library, and comprised in vol. vi. of\nhis edition of \"The Arabian Nights Entertainments,\" published at\nLondon in 1811.\n\nAn analogous tale occurs in Mr. E. J. W. Gibb's recently-\npublished translation of the \"History of the Forty Vezirs (the\nLady's Fourth Story, p. 69 ff.), the motif of which is that \"all\nthings return to their origin:\"\n\n\n                        TURKISH ANALOGUE.\n\nThere was in the palace of the world a king who was very desirous\nof seeing Khizr[FN#503] (peace on him!), and he would even say,\n\"If there be any one who will show me Khizr, I will give him\nwhatsoever he may wish.\"  Now there was at that time a man poor\nof estate, and from the stress of his poverty he said to himself,\n\"Let me go and speak to the king, that if he provide for me\nduring three years, either I shall be dead, or the king will be\ndead, or he will forgive me my fault, or I shall on somewise win\nto escape, and in this way shall I make merry for a time.\"  So he\nwent to the king and spake these words to him.[FN#504]  The king\nsaid, \"An thou show him not, then I will kill thee,\" and that\npoor man consented.  Then the king let give him much wealth and\nmoney, and the poor man took that wealth and money and went to\nhis house.  Three years he spent in merriment and delight, and he\nrested at ease till the term was accomplished.  At the end of\nthat time he fled and hid himself in a trackless place and he\nbegan to quake for fear.  Of a sudden he saw a personage with\nwhite raiment and shining face, who saluted him.  The poor man\nreturned the salutation, and the radiant being asked, \"Why art\nthou thus sad?\"  but he gave no answer.  Again the radiant being\nasked him and sware to him, saying, \"Do indeed tell to me thy\nplight, that I may find thee some remedy.\"  So that hapless one\nnarrated his story from its beginning to its end, and the radiant\nbeing said, \"Come, I will go with thee to the king, and I will\nanswer for thee.\"  So they arose.\n\nNow the king wanted that hapless one, and while they were going\nsome of the king's officers who were seeking met them, and they\nstraightway seized the poor man and brought him to the king.\nQuoth the king, \"Lo, the three years are accomplished; come now,\nand show me Khizr.\"  The poor man said, \"My king, grace and\nbounty are the work of kings--forgive my sin.\"  Quoth the king,\n\"I made a pact; till I have killed thee, I shall not have\nfulfilled it.\"  And he looked to his chief vez\u00edr and said, \"How\nshould this be done?\"  Quoth the vez\u00edr, \"This man should be hewn\nin many pieces and then hung up on butchers' hooks, that others\nmay see and lie not before the king.\"  Said that radiant being,\n\"True spake the vez\u00edr;--all things return to their origin.\"  Then\nthe king looked to the second vez\u00edr and said, \"What sayest thou?\"\nHe replied, \"This man should be boiled in a cauldron.\"  Said that\nradiant being, \"True spake the vez\u00edr;--all things return to their\norigin.\"  The king looked to the third vezir and said, \"What\nsayest though?\"  The vez\u00edr replied, \"This man should be hewn in\nsmall pieces and baked in an oven.\"  Again said that elder, \"True\nspake the vez\u00edr;--all things return to their origin.\"  Then quoth\nthe king to the fourth vez\u00edr, \"Let us see what sayest thou?\"  The\nvez\u00edr replied, \"O king, the wealth thou gavest this poor creature\nwas for the love of Khizr (peace on him!).  he, thinking to find\nhim, accepted it; now that he has not found him he seeks pardon.\nThis were befitting, that thou set free this poor creature for\nthe love of Khizr.\"  Said that elder, \"True spake the vez\u00edr;--all\nthings return to their origin.\"  Then the king said to the elder,\n\"O elder, my vez\u00edrs have said different things contrary the one\nto the other, and thou hast said concerning each of them, 'True\nspake the vez\u00edr; - all things return to their origin.'  What is\nthe reason thereof?\"  That elder replied, \"O king, thy first\nvez\u00edr is a butcher's son; therefore did he draw to his origin.\nThy second vez\u00edr is a cook's son, and he likewise proposed a\npunishment as became his origin.  Thy third vez\u00edr is a baker's\nson; he likewise proposed a punishment as became his origin.\nBut thy fourth vez\u00edr is of gentle birth; compassion therefore\nbecomes his origin, so he had compassion on that hapless one, and\nsought to do good and counselled liberation.  O king, all things\nreturn to their origin.\"[FN#505]  And he gave the king much\ncounsel, and at last said, \"Lo, I am Khizr,\" and\nvanished.[FN#506]\n\nThe discovery of the king's illegitimate birth, which occurs in\nso many versions, has its parallels in the story of the Nephew of\nHippocrates in the \"Seven Wise Masters,\" and the Lady's 2nd Story\nin Mr. Gibb's translation of the \"Forty Vez\u00edrs.\"  The\nextraordinary sensitiveness of the third young Br\u00e1hman, in the\nVet\u00e1la story, whose side was scratched by a hair that was under\nthe seventh of the mattresses on which he lay, Rohde (says\nTawney), in his \"Griechische Novellistik,\" p. 62, compares with a\nstory told by Aelian of the Sybarite Smindyrides, who slept on a\nbed of rose-leaves and got up in the morning covered with\nblisters.  He also quotes from the Chronicle of Tabari a story of\na princess who was made to bleed by a rose-leaf lying in her\nbed.[FN#507]\n\nThe eleventh recital of the Vet\u00e1la is about a king's three\nsensitive wives:  As one of the queens was playfully pulling the\nhair of the king, a blue lotus leaped from her ear and fell on\nher lap; immediately a wound was produced on the front of her\nthigh by the blow, and the delicate princess exclaimed, \"Oh! oh!\"\nand fainted. At night, the second retired with the king to an\napartment on the roof of the palace exposed to the rays of the\nmoon, which fell on the body of the queen, who was sleeping by\nthe king's side, where it was exposed by her garment blowing\naside; immediately she woke up, exclaiming, \"Alas! I am burnt,\"\nand rose up from the bed rubbing her limbs.  The king woke up in\na state of alarm, crying out, \"What is the meaning of this?\"\nThen he got up and saw that blisters had been produced on the\nqueen's body.  In the meanwhile the king's third wife heard of it\nand left her palace to come to him.  And when she got into the\nopen air, she heard distinctly, as the night was still, the sound\nof a pestle pounding in a distant house.  The moment the gazelle-\neyed one heard it, she said, \"Alas! I am killed,\" and she sat\ndown on the path, shaking her hands in an agony of pain.  Then\nthe girl turned back, and was conducted by her attendants to her\nown chamber, where she fell on her bed and groaned.  And when her\nweeping attendants examined her, they saw that her hands were\ncovered with bruises, and looked like lotuses upon which black\nbeetles had settled.\n\nTo this piteous tale of the three very sensitive queens Tawney\nappends the following note:  Rohde, in his \"Griechische\nNovellistik,\" p. 62, compares with this a story told by Tim\u00e6us,\nof a Sybarite who saw a husbandman hoeing a field, and contracted\nrupture from it.  Another Sybarite, to whom he told the tale of\nhis sad mishap, got ear-ache from hearing it.  Oesterley, in his\nGerman translation of the Bait\u00e1l Pach\u00eds\u00ed, points out that Grimm,\nin his \"Kinderm\u00e4rchen,\" iii. p. 238, quotes a similar incident\nfrom the travels of the Three sons of Giaffar:  out of four\nprincesses, one faints because a rose-twig is thrown into her\nface among some roses; a second shuts her eyes in order not to\nsee the statue of a man; a third says, \"Go away; the hairs in\nyour fur cloak run into me;\" and the fourth covers her face,\nfearing that some of the fish in a tank may belong to the male\nsex.  He also quotes a striking parallel from the \"Elites des\ncontes du Sieur d'Onville:\" Four ladies dispute as to which of\nthem is the most delicate.  One has been lame for three months\nowing to a rose-leaf having fallen on her foot; another has had\nthree ribs broken by a sheet in her bed having been crumpled; a\nthird has held her head on one side for six weeks owing to one\nhalf of her head having three more hairs on it than the other; a\nfourth has broken a blood-vessel by a slight movement, and the\nrupture cannot be healed without breaking the whole limb.[Poor\nthings!]\n\n\n\n\nTHE PRINCE WHO FELL IN LOVE WITH THE PICTURE.--Vol. XI. p. 226.\n\n\n\nIn the Persian tales of \"The Thousand and One Days,\" a young\nprince entered his father's treasury one day, and saw there a\nlittle cedar chest \"set with pearls, diamonds, emeralds, and\ntopazes;\" on opening it (for the key was in the lock) he beheld\nthe picture of an exceedingly beautiful woman, with whom he\nimmediately fell in love.  Ascertaining the name of the lady from\nan inscription on the back of the portrait, he set off with a\ncompanion to discover her, and having been told by an old man at\nBaghdad that her father at one time reigned in Ceylon, he continued\nhis journey thither, encountering many unheard-of adventures by\nthe way.  Ultimately he is informed that the lady with whose\nportrait he had become enamoured was one of the favourites of\nKing Solomon.  One should suppose that his would have effectually\ncured the love-sick prince; but no:  he \"could never banish her\nsweet image from his heart.\"[FN#508]\n\nTwo instances of falling in love with the picture of a pretty\nwoman occur in the \"Kath\u00e1 Sar\u00edt S\u00e1gara.\"  In Book ix., chap. 51,\na painter shows King Prithvir\u00fapa the \"counterfeit presentment\" of\nthe beauteous Princess Rapalat\u00e1, and \"as the king gazed on it his\neye was drowned in that sea of beauty her person, so that he\ncould not draw it out again.  For the king, whose longing was\nexcessive, could not be satisfied with devouring her form, which\npoured forth a stream of the nectar of beauty, as the partridge\ncannot be satisfied with devouring the moonlight.\"  In Book xii.,\nchap. 100, a female ascetic shows a wandering prince the portrait\nof the Princess Mand\u00e1ravat\u00ed, \"and Sundarasena when he beheld that\nmaiden, who, though she was present there only in a picture,\nseemed to be of romantic beauty and like a flowing forth of joy,\nimmediately felt as if he had been pierced with the arrows of the\ngod of the flowery bow [i.e. K\u00e1ma].\"  In chapter 35 of Scott's\ntranslation of the \"Bah\u00e1r-i-D\u00e1nish,\" Prince Ferokh-Faul opens a\nvolume, \"which he had scarcely done when the fatal portrait of\nthe fair princess who, the astrologers had foretold, was to\noccasion him so many perils, presented itself to his view.  He\ninstantly fainted, when the slave, alarmed, conveyed intelligence\nof his condition to the sultan, and related the unhappy cause of\nthe disorder.\"  In Gomberville's romance of Polexandre, the\nAfrican prince, Abd-el-Malik, falls in love with the portrait of\nAlcidiana, and similar incidents occur in the romance of\nAgesilaus of Colchos and in the Story of the Seven Waz\u00edrs (vol.\nvi.); but why multiply instances?  Nothing is more common in\nAsiatic fictions.\n\n\n\n\n    THE FULLER, HIS WIFE, AND THE TROOPER.--Vol. XI. p. 231.\n\n\n\nIn addition to the versions of this amusing story referred to on\np. 231--all of which will be found in the second volume of my\nwork on \"Popular Tales and Fictions,\" pp. 212-228--there is yet\nanother in a Persian story-book, of unknown date, entitled,\n\"Shamsa \u00fa Kuhkuha,\" written by Mirza Berkhorder Turkman, of which\nan account, together with specimens, is given in a recently-\npublished little book (Quaritch), \"Persian Portraits: a sketch of\nPersian History, Literature, and Politics,\" by Mr. F. F.\nArbuthnot, author of \"Early Ideas: a Group of Hindoo Stories.\"\n\nThis version occurs in a tale of three artful wives--or, to\nemploy the story-teller's own graphic terms, \"three whales of the\nsea of fraud and deceit: three dragons of the nature of thunder\nand the quickness of lightning; three defamers of honour and\nreputation; namely, three men-deceiving, lascivious women, each\nof whom had from the chicanery of her cunning issued the diploma\nof turmoil to a hundred cities and countries, and in the arts of\nfraud they accounted Satan as an admiring spectator in the\ntheatre of their stratagems.[FN#509]  One of them was sitting in\nthe court of justice of the kazi's embrace; the second was the\nprecious gem of the bazaar-master's diadem of compliance; and the\nthird was the beazle and ornament of the signet-ring of the life\nand soul of the superintendent of police.  They were constantly\nentrapping the fawns of the prairie of deceit within the grasp of\ncunning, and plundered the wares of the caravans of tranquillity\nof hearts of strangers and acquaintances, by means of the edge of\nthe scimitar of fraud.  One day this trefoil of roguery met at\nthe public bath, and, according to their homogeneous nature they\nintermingled as intimately as the comb with the hair; they tucked\nup their garment of amity to the waist of union, entered the tank\nof agreement, seated themselves in the hot-house of love, and\npoured from the dish of folly, by means of the key of hypocrisy,\nthe water of profusion upon the head of intercourse; they rubbed\nwith the brush of familiarity and the soap of affection the\nstains of jealousies from each other's limbs.  After a while,\nwhen they had brought the pot of concord to boil by the fire of\nmutual laudation, they warmed the bath of association with the\nbreeze of kindness, and came out.  In the dressing-room all three\nof them happened simultaneously to find a ring, the gem of which\nsurpassed the imagination of the jeweler of destiny, and the like\nof which he had never beheld in the storehouse of possibility.\nIn short, these worthy ladies contended with each other for\npossession of the ring, until at length the mother of the bathman\ncame forward and proposed that they should entrust the ring to\nher in the meanwhile, and it should be the prize of the one who\nmost cleverly deceived and befooled her husband, to which they\nall agreed, and then departed for their respective\ndomiciles.[FN#510]\n\nMr. Arbuthnot's limits pertained only of abstracts of the tricks\nplayed upon their husbands by the three ladies--which the story-\nteller gives at great length--and that of the kazi's wife is as\nfollows:\n\nThe kazi's wife knows that a certain carpenter, who lived close\nto her, was very much in love with her.  She sends her maid to\nhim with a message to say that the flame of his love had taken\neffect upon her heart, and that he must make an underground\npassage between his house and her dwelling, so that they might\ncommunicate with each other freely by means of the mine.  The\ncarpenter digs the passage, and the lady pays him a visit, and\nsays to him, \"To-morrow I shall come here, and you must bring the\nkazi to marry me to you.\"  The next day the kazi goes to his\noffice; the lady goes to the carpenter's house, and sends him to\nbring her husband, the kazi, to marry them.  The carpenter\nfetches him, and, as the kazi hopes for a good present, he comes\nwillingly enough, but is much surprised at the extreme likeness\nbetween the bride and his own wife.  The more he looks at her,\nthe more he is in doubt; and at last, offering an excuse to fetch\nsomething, he rushes off to his own house, but is forestalled by\nhis spouse, who had gone thither by the passage, and on his\narrival is lying on her bed.  The kazi makes some excuses for his\nsudden entry into her room, and, after some words, goes back to\nthe carpenter's house; but his wife had preceded him, and is\nsitting in her place.  Again he begins the ceremony, but is\nattracted by a black mole on the corner of the bride's lip, which\nhe could have sworn was the same as that possessed by his wife.\nMaking more excuses, and in spite of the remonstrances of the\ncarpenter, he hurries back to his house once more; but his wife\nhad again got there before him, and he finds her reading a book,\nand much astonished at his second visit.  She suggests that he is\nmad, and he admits that his conduct is curious, and returns to\nthe carpenter's house to complete the ceremony.  This is again\nfrequently interrupted, but finally he marries his own wife to\nthe carpenter, and, having behaved in such an extraordinary\nmanner throughout, is sent off to a lunatic asylum.\n\nFor the tricks of the two other ladies, and for many other\nequally diverting tales, I refer the reader to Mr. Arbuthnot's\npleasing and instructive little book, which is indeed an\nadmirable epitome of the history and literature of Persia, and\none which was greatly wanted in these days, when most men, \"like\nthe dogs in Egypt for fear of the crocodiles, must drink of the\nwaters of information as they run, in dread of the old enemy\nTime.\"\n\nI have discussed the question of the genealogy of this tale\nelsewhere, but, after a somewhat more minute comparative analysis\nof the several versions, am disposed to modify the opinion which\nI then entertained.  I think we must consider as the direct or\nindirect source of the versions and variants the \"Miles\nGloriosus\" of Plautus, the plot of which, it is stated in the\nprologue to the second act, was taken from a Greek play.  It is,\nhowever, not very clear whether Berni adapted his story from\nPlautus or the \"Seven Wise Masters\"; probably from the former,\nsince in both the lady is represented, to the captain and the\ncuckold, as a twin sister, while in the S. W. M. the crafty\nknight pretends that she is his leman, come from Hungary with\ntidings that he may now with safety return home.  On the other\nhand, in the S. W. M., as in Plautus, the lovers make their\nescape by sea, an incident which Berni has altered to a journey\nby land--no doubt, in order to introduce further adventures for\nthe development of his main plot.  But then we find a point of\nresemblance between Berni and the S. W. M., in the incident of\nthe cuckold accompanying the lovers part of their way--in the\nlatter to the sea-shore; while in Plautus the deceived captain\nremains at home to prosecute an amour and get a thrashing for his\nreward (in Plautus, instead of a wife, it is the captain's slave-\ngirl).  It is curious that amidst all the masquerade of the\nArabian story the cuckold's wife also personates her\nsupposititious twin-sister, as in Plautus and Berni.  In Plautus\nthe houses of the lover and the captain adjoin, as is also the\ncase in the modern Italian and Sicilian versions; while in Berni,\nthe S. W. M., the Arabian, and the Persian story cited in this\nnote they are at some distance.  With these resemblances and\nvariations it is not easy to say which version was derived from\nanother.  Evidently the Arabian story has been deliberately\nmodified by the compiler, and he has, I think, considerably\nimproved upon the original: the ludicrous perplexity of the poor\nfuller when he awakes, to find himself apparently transformed\ninto a Turkish trooper, recalls the nursery rhyme of the little\nwoman \"who went to market her eggs for to sell,\" and falling\nasleep on the king's highway a pedlar cut off her petticoats up\nto the knees, and when she awoke and saw her condition she\nexclaimed, \"Lawk-a-mercy me, this is none of I!\" and so on.  And not less diverting is the pelting the blockhead receives from his brother fullers--altogether, a capital story.\n\n\n\n\n        TALE OF THE SIMPLETON HUSBAND.--Vol. XI. p. 239.\n\n\n\nThe \"curious\" reader will find European and Asiatic versions of this amusing story in \"Originals and Analogues of some of\nChaucer's Canterbury Tales,\" published for the Chaucer Society, pp. 177-188 and (in a paper contributed by me: \"The Enchanted Tree\") p. 341-364.\n\n\n\n\n    TALE OF THE THREE MEN AND OUR LORD ISA.--Vol XI. p. 250.\n\n\n\nUnder the title of \"The Robbers and the Treasure-Trove\" I have\nbrought together many European and Asiatic versions of this wide-\nspread tale in \"Chaucer Analogues,\" pp. 415-436.\n\n\n\n\n     THE MELANCHOLIST AND THE SHARPER. -- Vol. XI. p. 264.\n\n\n\nA similar but much shorter story is found in Gladwin's \"Persian\nMoonshee,\" and story-books in several of the Indian vernaculars\nwhich have been rendered into English:\n\nA miser said to a friend, \"I have now a thousand rupees, which I\nwill bury out of the city, and I will not tell the secret to any\none besides yourself.\"  They went out of the city together, and\nburied the money under a tree.  Some days after the miser went\nalone to the tree and found no signs of his money.  He said to\nhimself, \"Excepting that friend, no other has taken it away; but\nif I question him he will never confess.\"  He therefore went to\nhis (the friend's) house and said, \"A great deal of money is come\ninto my hands, which I want to put in the same place; if you will\ncome to-morrow, we will go together.\"  The friend, by coveting\nthis large sum, replaced the former money, and the miser next day\nwent there alone and found it.  He was delighted with his own\ncontrivance, and never again placed any confidence in friends.\n\nOne should suppose a miser the last person to confide the secret\nof his wealth to any one; but the Italian versions bear a closer\nresemblance to the Arabian story.  From No. 74 of the \"Cento\nNovelle Antiche\" Sacchetti, who was born in 1335 and is ranked by\nCrescimbini as next to Boccaccio, adapted his 198th novella,\nwhich is a most pleasing version of the Asiatic story:\n\n\n                        ITALIAN VERSION.\n\nA blind man of Orvieto, of the name of Cola, hit upon a device to\nrecover a hundred florins he had been cheated of, which showed he\nwas possessed of all the eyes of Argus, though he had unluckily\nlost his own.  And this he did without wasting a farthing either\nupon law or arbitration, by sheer dexterity; for he had formerly\nbeen a barber, and accustomed to shave very close, having then\nall his eyes about him, which had been now closed for about\nthirty years.  Alms seemed then the only resource to which he\ncould betake himself, and such was the surprising progress he in\na short time made in his new trade that he counted a hundred\nflorins in his purse, which he secretly carried about him until\nhe could find a safer place.  His gains far surpassed anything he\nhad realised with his razor and scissors; indeed, they increased\nso fast that he no longer knew where to bestow them; until one\nmorning happening to remain the last, as he believed, in the\nchurch, he thought of depositing his purse of a hundred florins\nunder a loose tile in the floor behind the door, knowing the\nsituation of the place perfectly well.  After listening some time\nwithout hearing a foot stirring, he very cautiously laid it in\nthe spot; but unluckily there remained a certain Juccio\nPezzichernolo, offering his adoration before an image of San\nGiovanni Boccadoro, who happened to see Cola busily engaged\nbehind the door.  He continued his adorations until he saw the\nblind man depart, when, not in the least suspecting the truth, he\napproached and searched the place.  He soon found the identical\ntile, and on removing it with the help of his knife, he found the\npurse, which he very quietly put into his pocket, replacing the\ntiles just as they were, and, resolving to say nothing about it,\nhe went home.\n\nAt the end of three days the blind mendicant, desirous of\ninspecting his treasure, took a quiet time for visiting the\nplace, and removing the tile searched a long while in great\nperturbation, but all in vain, to find his beloved purse.  At last,\nreplacing things just as they were, he was compelled to return in\nno very enviable state of mind to his dwelling; and there\nmeditating on his loss, the harvest of the toil of so many days,\nby dint of intense thinking a bright thought struck him (as\nfrequently happens by cogitating in the dark), how he had yet a\nkind of chance of redeeming his lost spoils.  Accordingly in the\nmorning he called his young guide, a lad about nine years old,\nsaying, \"My son, lead me to church,\" and before setting out he\ntutored him how he was to behave, seating himself at his side\nbefore the entrance, and particularly remarking every person who\nshould enter into the church.  \"Now, if you happen to see any one\nwho takes particular notice of me, and who either laughs or makes\nany sign, be sure you observe it and tell me.\"  The boy promised\nhe would; and they proceeded accordingly and took their station\nbefore the church\n\nWhen the dinner-hour arrived, the father and son prepared to\nleave the place, the former inquiring by the way whether his son\nhad observed any one looking hard at him as he passed along.\n\"That I did,\" answered the lad, \"but only one, and he laughed as\nhe went past us.  I do not know his name, but he is strongly\nmarked with the small-pox and lives somewhere near the Frati\nMinori.\"  \"Do you think, my dear lad,\" said his father, \"that you\ncould take me to his shop, and tell me when you see him there?\"\n\"To be sure I could,\" said the lad.  \"Then come, let us lose no\ntime,\" replied the father, \"and when we are there tell me, and\nwhile I speak to him you can step on one side and wait for me.\"\nSo the sharp little fellow led him along the way until he reached\na cheesemonger's stall, when he acquainted his father, and\nbrought him close to it.  No sooner did the blind man hear him\nspeaking with his customers than he recognised him for the same\nJuccio with whom he had formerly been acquainted during his days\nof light.  When the coast was a little clear, our blind hero\nentreated some moments' conversation, and Juccio, half suspecting\nthe occasion took him on one side into a little room, saying,\n\"Cola, friend, what good news?\"  \"Why,\" said Cola, \"I am come to\nconsult you, in great hopes you will be of use to me.  You know\nit is a long time since I lost my sight, and being in a destitute\ncondition, I was compelled to earn my subsistence by begging\nalms.  Now, by the grace of God, and with the help of you and of\nother good people of Orvieto, I have saved a sum of two hundred\nflorins, one hundred of which I have deposited in a safe place,\nand the other is in the hands of my relations, which I expect to\nreceive with interest in the course of a week.  Now if you would\nconsent to receive, and to employ for me to the best advantage,\nthe whole sum of two hundred florins it would be doing me a great\nkindness, for there is no one besides in all Orvieto in whom I\ndare to confide; nor do I like to be at the expense of paying a\nnotary for doing business which we can as well transact\nourselves.  Only I wish you would say nothing about it, but\nreceive the two hundred florins from me to employ as you think\nbest.  Say not a word about it, for there would be an end of my\ncalling were it known I had received so large a sum in alms.\"\nHere the blind mendicant stopped; and the sly Juccio, imagining\nhe might thus become master of the entire sum, said he should be\nvery happy to serve him in every way he could and would return an\nanswer the next morning as to the best way of laying out the\nmoney.  Cola then took his leave, while Juccio, going directly\nfor the purse, deposited it in its old place being in full\nexpectation of soon receiving it again with the addition of the\nother hundred, as it was clear that Cola had not yet missed the\nmoney.  The cunning old mendicant on his part expected that he\nwould do no less, and trusting that his plot might have\nsucceeded, he set out the very same day to the church, and had\nthe delight, on removing the tile, to find his purse really\nthere.  Seizing upon it with the utmost eagerness, he concealed\nit under his clothes, and placing the tiles exactly in the same\nposition, he hastened home whistling, troubling himself very\nlittle about his appointment of the next day.\n\nThe sly thief Juccio set out accordingly the next morning to see\nhis friend Cola, and actually met him on the road.  \"Whither are\nyou going?\" inquired Juccio.  \"I was going,\" said Cola, \"to your\nhouse.\"  The former, then taking the blind man aside, said, \"I am\nresolved to do what you ask; and since you are pleased to confide\nin me, I will tell you of a plan that I have in hand for laying\nout your money to advantage.  If you will put the two hundred\nflorins into my possession, I will make a purchase in cheese and\nsalt meat, a speculation which cannot fail to turn to good\naccount.\"  \"Thank you,\" quoth Cola, \"I am going to-day for the\nother hundred, which I mean to bring, and when you have got them\nboth, you can do with them what you think proper.\"  Juccio said,\n\"Then let me have them soon, for I think I can secure this\nbargain; and as the soldiers are come into the town, who are fond\nof these articles, I think it cannot fail to answer; so go, and\nHeaven speed you.\"  And Cola went; but with very different\nintentions from those imagined by his friend--Cola being now\nclear-sighted, and Juccio truly blind.  The next day Cola called\non his friend with very downcast and melancholy looks, and when\nJuccio bade him good day, he said, \"I wish from my soul it were a\ngood, or even a middling, day for me.\"  \"Why, what is the\nmatter?\"  \"The matter?\" echoed Cola; \"why, it is all over with\nme: some rascal has stolen a hundred florins from the place where\nthey were hidden, and I cannot recover a penny from my relations,\nso that I may eat my fingers off or anything I have to expect.\"\nJuccio replied, \"This is like all the rest of my speculations.  I\nhave invariably lost where I expected to make a good hit.  What I\nshall do I know not; for if the person should choose to keep me\nto the agreement I made for you, I shall be in a pretty dilemma\nindeed.\"  \"Yet,\" said Cola, \"I think my condition is still worse\nthan yours.  I shall be sadly distressed and shall have to amass\na fresh capital, which will take me ever so long. And when I have\ngot it, I will take care not to conceal it in a hole in the\nfloor, or trust it, Juccio, into any friend's hands.\"  \"But,\"\nsaid Juccio, \"if we could contrive to recover what is owing by\nyour relations, we might still make some pretty profit of it, I\ndoubt not.\"  For he thought, if he could only get hold of the\nhundred he had returned it would still be something in his way.\n\"Why,\" said Cola, \"to tell the truth, if I were to proceed\nagainst my relations I believe I might get it; but such a thing\nwould ruin my business, my dear Juccio, for ever:  the world\nwould know I was worth money, and I should get no more money from\nthe world; so I fear I shall hardly be able to profit by your\nkindness, though I shall always consider myself as much obliged\nas if I had actually cleared a large sum.  Moreover, I am going\nto teach another blind man my profession, and if we have luck you\nshall see me again, and we can venture a speculation together.\"\nSo far the wily mendicant, to whom Juccio said, \"Well, go and try\nto get money soon, and bring it; you know where to find me, but\nlook sharp about you and the Lord speed you; farewell.\"\n\"Farewell,\" said Cola; \"and I am well rid of thee,\" he whispered\nto himself, and going upon his way, in a short time he doubled\nhis capital; but he no longer went near his friend Juccio to know\nhow he should invest it.  He had great diversion in telling the\nstory to his companions during their feasts, always concluding,\n\"By St. Lucia! Juccio is the blinder man of the two:  he thought\nit was a bold stroke to risk his hundred to double the amount.\"\n\nFor my own part, I think the blind must possess a more acute\nintellect than other people, inasmuch as the light, exhibiting\nsuch a variety of objects to view, is apt to distract the\nattention, of which many examples might be adduced.  For\ninstance, two gentlemen may be conversing together on some matter\nof business, and in the middle of a sentence a fine woman happens\nto pass by, and they will suddenly stop, gazing after her; or a\nfine equipage or any other object is enough to turn the current\nof their thoughts.  And then we are obliged to recollect\nourselves, saying, \"Where was I?\" \"What was it that I was\nobserving?\"--a thing which never occurs to a blind man. The\nphilosopher Democritus very properly on this account knocked his\neyes out in order to catch objects in a juster light with his\nmind's eye.\n\nIt is impossible to describe Juccio's vexation on going to church\nand finding the florins were gone.  His regret was far greater\nthan if he had actually lost a hundred of his own; as is known to\nbe the case with all inveterate rogues, half of whose pleasure\nconsists in depriving others of their lawful property.\n\nThere are many analogous stories, one of which is the well-known\ntale of the merchant who, before going on a journey, deposited\nwith a dervish 1,000 sequins, which he thought it prudent to\nreserve in case of accidents. When he returned and requested his\ndeposit, the dervish flatly denied that he ever had any of his\nmoney.  Upon this the merchant went and laid his case before the\nkazi, who advised him to return to the dervish and speak\npleasantly to him, which he does, but receives nothing but abuse.\nHe informed the kazi of this, and was told not to go near the\ndervish for the present, but to be at ease for he should have his\nmoney next day.  The kazi then sent for the dervish, and after\nentertaining him sumptuously, told him that, for certain reasons,\nhe was desirous of removing a considerable sum of money from his\nhouse; that he knew of no person in whom he could confide so much\nas himself; and that if he would come the following evening at a\nlate hour, he should have the precious deposit.  On hearing this,\nthe dervish expressed his gratification that so much confidence\nshould be placed in his integrity, and agreed to take charge of\nthe treasure.  Next day the merchant returned to the kazi, who\nbade him go back to the dervish and demand his money once more,\nand should he refuse, threaten to complain to the kazi.  The\nresult may be readily guessed: no sooner did the merchant mention\nthe kazi than the rascally dervish said, \"My good friend, what\nneed is there to complain to the kazi?  Here is your money; it\nwas only a little joke on my part.\"  But in the evening, when he\nwent to receive the kazi's pretended deposit, he experienced the\ntruth of the saw, that \"covetousness sews up the eyes of\ncunning.\"\n\nA variant of this is found in the continental \"Gesta Romanorum\" (ch.\ncxviii. of Swan's translation), in which a knight deposits ten\ntalents with a respectable old man, who when called upon to\nrefund the money denies all knowledge of it.  By the advice of an\nold woman the knight has ten chests made, and employs a person to\ntake them to the old man and represent them as containing\ntreasure; and while one of them is being carried into his house\nthe knight enters and in the stranger's presence demands his\nmoney, which is at once delivered to him.\n\nIn Mr. Edward Rehatsek's translated selections from the Persian\nstory-book \"Shamsa \u00fa Kuhkuha\" (see ante, p. 329), printed at\nBombay in 1871, under the title of \"Amusing Stories,\" there is a\ntale (No. xviii.) which also bears some resemblance to that of\nthe Melancholist and the Sharper; and as Mr. Rehatsek's little\nwork is exceedingly scarce, I give it in extenso as follows:\n\nThere was in Damascus a man of the name of Zayn el-Arab, with the\nhoney of whose life the poison of hardships was always mixed.\nDay and night he hastened like the breeze from north to south in\nthe world of exertion, and he was burning brightly like straw,\nfrom his endeavours in the oven of acquisition in order to gain a\nloaf of bread and feed his family.  In course of time, however,\nhe succeeded in accumulating a considerable sum of money, but as\nhe had tasted the bitter poison of destitution, and had for a\nvery long time carried the heavy load of poverty upon his back,\nand fearing to lose his property by the chameleon-like changes of\nfortune, he took up his money on a certain night, carried it out\nof the city, and buried it under a tree.  After some time had\npassed be began sorely to miss the presence of his treasure, and\nbetook himself to the tree to refresh his eyes with the sight of\nit.  But when he dug up the ground at the foot of the tree he\ndiscovered that his soul-exhilarating deposit was refreshing the\npalate of some one else.  The morning of his prosperity was\nsuddenly changed into the evening of bitterness and\ndisappointment.  He was perplexed to what friend to confide his\nsecret, and to what remedy to fly for the recovery of his\ntreasure.  The lancet of grief had pierced the liver of his\npeace, and the huntsman of distress had tied up the wings and\nfeet of the bird of his serenity.  One day he went on some\nbusiness to a learned and wise man of the city with whom he was\non a footing of intimacy.  This man said to him, \"It is some time\nsince I perceived the glade of your circumstances to have been\ndestroyed by the burning coals of restlessness, and a sad change\nto have taken place in your health.  I do not know the reason,\nnor what thorn of misfortune has pierced the foot of your heart,\nnor what hardship has dawned from the east of your mind.\"  Zayn\nel-Arab wept tears of sadness and said, \"O thou standard coin\nfrom the mint of love! the treachery of misfortune has brought a\nstrange accident upon me, and the bow of destiny has let fly an\nunpropitious arrow upon my feeble target.  I have a heavy heart\nand great sorrow, and were I to reveal it to you perhaps it would\nbe of no use and would plunge you also into grief.\"  The learned\nman said, \"Since the hearts of intimate friends are like looking-\nglasses and are receiving the figures of mutual secrets, it is at\nall times necessary that they should communicate to each other\nany difficulties which they have fallen into, that they may\nremove them by taking in common those steps which prudence and\nforesight should recommend.\"  Zayn el-Arab replied, \"Dear friend,\nI had some gold, and fearing lest it should be stolen, I carried\nit to such and such a place and buried it under a tree, and when\nI again visited the place, I perceived the garment of my beloved\nJoseph to be sprinkled with the blood of the wolf of deception.\"\nThe learned man said, \"This is a grave accident, and it will be\ndifficult to get on the track of your gold.  Perhaps some one saw\nyou bury it:  he who has taken it will have to give an account of\nit in the next world, for God is omniscient.  Give me ten days'\ndelay, that I may study the book of expedients and stratagems,\nwhen mayhap somewhat will occur to me.\"\n\nThat knowing man sat down for ten days in the school of\nmeditation, and how much so ever he turned over the leaves of the\nvolume of his mind from the preface to the epilogue, he could hit\nupon no plan.  On the tenth day they again met in the street, and\nhe said to Zayn el-Arab, \"Although the diver of my mind has\nplunged deeply and searched diligently in this deep sea, he has\nbeen unable to seize the precious pearl of a wise plan of\noperation:  may God recompense you from the stores of His hidden\ntreasury!\"  They were conversing in this way when a lunatic met\nthem and said, \"Well, my boys, what secret- mongering have you\ngot between you?\"  The learned man said to Zayn el-Arab, \"Come,\nlet us relate our case to this crazy fellow, to see the flower of\nthe plant that may bloom from his mind.\"  Zayn el-Arab replied,\n\"Dear friend, you, with all your knowledge, cannot devise\nanything during ten days:  what information are we likely to gain\nfrom a poor lunatic who does not know whether it is now day or\nnight?\"  The learned man said, \"There is no telling what he may\nsay to us. But you know that the most foolish as well as the most\nwise have ideas, and a sentence uttered at random has sometimes\nfurnished a clue by which the desired object may be attained.\"\nMeanwhile a little boy also came up, and perceiving the lunatic\nstopped to see his tricks.  The two friends explained their case\nto the lunatic, who then seemed immersed in thought for some\ntime, after which he said, \"He who took the root of that tree for\na medicine also took the gold,\" and having thus spoken, he turned\nhis back upon them and went his way.  They consulted with each\nother what indication this remark might furnish, when the little\nboy who had overheard the conversation, asked what kind of tree\nit was.  Zayn el-Arab replied that it was a jujube tree.  The boy\nsaid, \"This is an easy matter: you ought to inquire of all the\ndoctors of this town for whom a medicine has been prescribed of\nthe roots of this tree.\"  They greatly admired the boy's\nacuteness and also of the lunatic's lucky thought.[FN#511]  The\nlearned man was well acquainted with all the physicians of the\ncity and made his enquiries, till he met with one who informed\nhim that about twenty days ago he had prescribed for a merchant\nof the name of Khoja Semender, who suffered from asthma, and that\none of the remedies was the root of that jujube tree.  The\nlearned man soon discovered the merchant's house, found him\nenjoying excellent health, and said to him, \"Ah, Khoja, all the\ngoods of this world ought to be surrendered to procure health.\nBy the blessing of God, you have recovered your health, and you\nought to give up what you found at the root of that tree, because\nthe owner of it is a worthy man and possesses nothing else.\"  The\nhonest merchant answered, \"It is true, I have found it, and it is\nwith me.  If you will describe it I will deliver it into your\nhands.\"  The exact sum being stated, the merchant at once\ndelivered up the gold.\n\nIn the \"Kath\u00e1 Sarit S\u00e1gara,\" Book vi. ch. 33, we have probably\nthe original of this last story:  A wealthy merchant provided a\nBr\u00e1hman with a lodging near his own house, and every day gave him\na large quantity of unhusked rice and other presents and in\ncourse of time he received like gifts from other great merchants.\nIn this way the miserly fellow gradually accumulated a thousand\nd\u00edn\u00e1rs, and going into the forest he dug a hole and buried it in\nthe ground, and he went daily to carefully examine the spot.  One\nday, however, he discovered that his hoard had been stolen, and\nhe went to his friend the merchant near whose house he lived,\nand, weeping bitterly, told him of his loss, and that he had\nresolved to go to a holy bathing-place and there starve himself\nto death.  The merchant tried to console him and dissuade him\nfrom his resolution, saying, \"Br\u00e1hman, why do you long to die for\nthe loss of your wealth?  Wealth, like an unseasonable cloud,\nsuddenly comes and goes.\"  But the Br\u00e1hman would not abandon his\nfixed determination to commit suicide, for wealth is dearer to\nthe miser than life itself.  When he was about to depart for the\nholy place, the king, having heard of it, came and asked him,\n\"Br\u00e1hman, do you know of any mark by which you can distinguish\nthe place where you buried your d\u00edn\u00e1rs?\"  He replied, \"There is a\nsmall tree in the wood, at the foot of which I buried that\nmoney.\"  Then said the king, \"I will find the money and give it\nback to you, or I will give it you from my own treasury;--do not\ncommit suicide, Br\u00e1hman.\"\n\nWhen the king returned to his palace, he pretended to have a\nheadache, and summoned all the physicians in the city by\nproclamation with beat of drum.  And he took aside every one of\nthem singly and questioned them privately, saying, \"What patients\nhave you, and what medicines have you prescribed for each?\"  And\nthey thereupon, one by one, answered the king's questions.  At\nlength a physician said, \"The merchant M\u00e1tridatta has been out of\nsorts, O king, and this is the second day I have prescribed for\nhim n\u00e1gabal\u00e1 (the plant Uraria Lagopodioides).\"  Then the king\nsent for the merchant, and said to him, \"Tell me, who fetched you\nthe n\u00e1gabal\u00e1?\" The merchant replied, \"My servant, your highness.\"\nOn hearing this, the king at once summoned the servant and said\nto him, \"Give up that treasure belonging to a Br\u00e1hman, consisting\nof a store of d\u00edn\u00e1rs, which you found when you were digging at\nthe foot of the tree for n\u00e1gabal\u00e1.\"  When the king said this to\nhim the servant was frightened, and confessed immediately; and\nbringing the money left it there.  Then the king summoned the\nBr\u00e1hman and gave him, who had been fasting meanwhile, the d\u00edn\u00e1rs,\nlost and found again, like a second soul external to his body.\nThus did the king by his wisdom recover to the Br\u00e1hman his wealth\nwhich had been taken away from the root of the tree, knowing that\nthat simple grew in such spots.\n\n\n\n\nTALE OF THE DEVOUT WOMAN ACCUSED OF LEWDNESS.--Vol. XI. p. 270.\n\n\n\nThis is one of three Arabian variants of Chaucer's Man of Law's Tale (the Story of Constance), of which there are numerous versions--see my paper entitled \"The Innocent Persecuted Wife,\" pp. 365-414 of \"Originals and Analogues of some of Chaucer's Canterbury Tales.\"\n\n\n\n\nTHE WEAVER WHO BECAME A LEACH BY ORDER OF HIS WIFE.--Vol. XI. p. 282.\n\n\n\nSomewhat resembling this, but much more elaborate, is the amusing\nstory of Ahmed the Cobbler, in Sir John Malcolm's \"Sketches of\nPersia,\" ch. xx., the original of which is probably found in the\ntale of Harisarman, book vi. ch. 30, of the \"Kath\u00e1 Sarit S\u00e1gara,\"\nand it has many European variants, such as the German story of\nDoctor Allwissend, in Grimm's collection, and that of the\nCharcoal Burner in Sir George Dasent's \"Tales from the Fjeld.--\nAccording to the Persian story, Ahmed the Cobbler had a young and\npretty wife, of whom he was very fond. She was ever forming grand\nschemes of riches and splendour, and was firmly persuaded that\nshe was destined to great fortune. It happened one evening, while\nin this frame of mind, that she went to the public baths, where\nshe saw a lady retiring dressed in a magnificent robe, covered\nwith jewels, and surrounded by slaves. This was the very\ncondition she had always longed for, and she eagerly inquired the\nname of the happy person who had so many attendants and such fine\njewels. She learned it was the wife of the chief astrologer to\nthe king. With this information she returned home. Ahmed met her\nat the door, but was received with a frown, nor could all his\ncaresses obtain a smile or a word; for several hours she\ncontinued silent, and in apparent misery; at length she said,\n\"Cease your caresses, unless you are ready to give me a proof\nthat you do really and sincerely love me.\" \"What proof of love,\"\nexclaimed poor Ahmed, \"can you desire that I will not give?\"\n\"Give over cobbling, it is a vile, low trade, and never yields\nmore than ten or twelve d\u00edn\u00e1rs a day. Turn astrologer; your\nfortune will be made, and I shall have all I wish and be happy.\"\n\"Astrologer!\" cried Ahmed--\"astrologer! Have you forgotten who I\nam--a cobbler, without any learning--that you want me to engage\nin a profession which requires so much skill and knowledge?\" \"I\nneither think nor care about your qualifications,\" said the\nenraged wife; \"all I know is that if you do not turn astrologer\nimmediately, I will be divorced from you to-morrow.\" The cobbler\nremonstrated, but in vain. The figure of the astrologer's wife,\nwith her jewels and her slaves, took complete possession of her\nimagination. All night it haunted her: she dreamt of nothing\nelse, and on awakening declared that she would leave the house if\nher husband did not comply with her wishes. What could poor Ahmed\ndo? He was no astrologer; but he was dotingly fond of his wife,\nand he could not bear the idea of losing her. He promised to\nobey, and having sold his little stock, bought an astrolabe, an\nastronomical almanac, and a table of the twelve signs of the\nzodiac. Furnished with these, he went to the market-place, crying,\n\"I am an astrologer! I know the sun, and the moon, and the stars,\nand the twelve signs of the zodiac; I can calculate nativities; I\ncan foretell everything that is to happen.\" No man was better\nknown than Ahmed the Cobbler. A crowd soon gathered round him.\n\"What, friend Ahmed,\" said one, \"have you worked till your head\nis turned?\" \"Are you tired of looking down at your last,\" cried\nanother, \"that you are now looking up at the stars?\" These and a\nthousand other jokes assailed the ears of the poor cobbler, who\nnotwithstanding continued to exclaim that he was an astrologer,\nhaving resolved on doing what he could to please his beautiful\nwife.\n\nIt so happened that the king's jeweller was passing by. He was in\ngreat distress, having lost the richest ruby belonging to the\nking. Every search had been made to recover this inestimable\njewel, but to no purpose; and as the jeweller knew he could no\nlonger conceal its loss from the king, he looked forward to death\nas inevitable. In this hopeless state, while wandering about the\ntown, he reached the crowd around Ahmed, and asked what was the\nmatter. \"Don't you know Ahmed the Cobbler?\" said one of the\nbystanders, laughing. \"He has been inspired and is become an\nastrologer.\" A drowning man will catch at a broken reed: the\njeweller no sooner heard the sound of the word astrologer than he\nwent up to Ahmed, told him what had happened, and said, \"If you\nunderstand your art, you must be able to discover the king's\nruby. Do so, and I will give you two hundred pieces of gold. But\nif you do not succeed within six hours, I will use my influence\nat court to have you put to death as an impostor.\" Poor Ahmed was\nthunderstruck. He stood long without being able to speak,\nreflecting on his misfortunes, and grieving, above all, that his\nwife, whom he so loved, had, by her envy and selfishness, brought\nhim to such a fearful alternative. Full of these sad thoughts, he\nexclaimed aloud, \"O woman! woman! thou art more baneful to the\nhappiness of man than the poisonous dragon of the desert!\" Now\nthe lost ruby had been secreted by the jeweller's wife, who,\ndisquieted by those alarms which ever attend guilt, sent one of\nher female slaves to watch her husband. This slave, on seeing her\nmaster speak to the astrologer, drew near; and when she heard\nAhmed, after some moments of abstraction, compare a woman to a\npoisonous dragon, she was satisfied that he must know everything.\nShe ran to her mistress, and, breathless with fear, cried, \"You\nare discovered by a vile astrologer! Before six hours are past\nthe whole story will be known, and you will become infamous, if\nyou are even so fortunate as to escape with life, unless you can\nfind some way of prevailing on him to be merciful.\" She then\nrelated what she had seen and heard; and Ahmed's exclamation\ncarried as complete conviction to the mind of the terrified lady\nas it had done to that of her slave. The jeweller's wife hastily\nthrowing on her veil, went in search of the dreaded astrologer.\nWhen she found him, she cried, \"Spare my honour and my life, and\nI will confess everything.\" \"What can you have to confess to me?\"\nsaid Ahmed, in amazement. \"O nothing--nothing with which you are\nnot already acquainted. You know too well that I stole the king's\nruby. I did so to punish my husband, who uses me most cruelly;\nand I thought by this means to obtain riches for myself and have\nhim put to death. But you, most wonderful man, from whom nothing\nis hidden, have discovered and defeated my wicked plan. I beg\nonly for mercy, and will do whatever you command me.\" An angel\nfrom heaven could not have brought more consolation to Ahmed than\ndid the jeweller's wife. He assumed all the dignified solemnity\nthat became his new character, and said, \"Woman! I know all thou\nhast done, and it is fortunate for thee that thou hast come to\nconfess thy sin and beg for mercy before it was too late. Return\nto thy house; put the ruby under the pillow of the couch on which\nthy husband sleeps; let it be laid on the side farthest from the\ndoor; and be satisfied thy guilt shall never be even suspected.\"\nThe jeweller's wife went home and did as she was instructed. In\nan hour Ahmed followed her, and told the jeweller he had made his\ncalculations, and found by the aspect of the sun and moon, and by\nthe configuration of the stars, that the ruby was at that moment\nlying under the pillow of his couch on the side farthest from the\ndoor. The jeweller thought Ahmed must be crazy; but as a ray of\nhope is like a ray from heaven to the wretched, he ran to his\ncouch, and there, to his joy and wonder, found the ruby in the\nvery place described. He came back to Ahmed, embraced him, called\nhim his dearest friend and the preserver of his life, gave him\ntwo hundred pieces of gold, declaring that he was the first\nastrologer of the age.\n\nAhmed returned home with his lucky gains, and would gladly have\nresumed his cobbling but his wife insisting on his continuing to\npractice his new profession, there was no help but to go out\nagain next day and proclaim his astrological accomplishments. By\nmere chance he is the means of a lady recovering a valuable\nnecklace which she had lost at the bath, and forty chests of gold\nstolen from the king's treasury, and is finally rewarded with the\nhand of the king's daughter in marriage.\n\n\n\n\nSTORY OF THE KING WHO LOST KINGDOM, WIFE AND WEALTH.--Vol. XI. p. 319.\n\n\n\nIn the \"Indian Antiquary\" for June 1886 the Rev. J. Hinton\nKnowles gives a translation of what he terms a Kashm\u00edr\u00ed Tale,\nunder the title of \"Pride Abased,\" which, he says, was told him\nby \"a Brahman named Mukund B\u00e1y\u00fa, who resides at Suth\u00fa, Sr\u00ednagar,\"\nand which is an interesting variant of the Waz\u00edr Er-Rahwan's\nsecond story of the King who lost his Realm and Wealth:\n\n\n                   KASHMIRI VERSION.[FN#512]\n\nThere was once a king who was noted throughout his dominions for\ndaily boasting of his power and riches. His ministers at length\nbecame weary of this self-glorification, and one day when he\ndemanded of them, as usual, whether there existed in the whole\nworld another king as powerful as he, they plainly told him that\nthere was such another potentate; upon which he assembled his\ntroops and rode forth at their head, challenging the neighbouring\nkings to fight with him. Ere long he met with more than his\nmatch, for another king came with a great army and utterly\ndefeated him, and took possession of his kingdom. Disguising\nhimself, the humbled king escaped with his wife and two boys, and\narriving at the sea shore, found a ship about to sail. The master\nagreed to take him and his family and land them at the port for\nwhich he was bound. But when he beheld the beauty of the queen,\nhe became enamoured of her, and determined to make her his own.\nThe queen was the first to go on board the ship, and the king and\nhis two sons were about to follow, when they were seized by a\nparty of ruffians, hired by the shipmaster, and held back until\nthe vessel had got fairly under way. The queen was distracted on\nseeing her husband and children left behind, and refused to\nlisten to the master's suit, who, after having tried to win her\nlove for several days without success, resolved to sell her as a\nslave. Among the passengers was a merchant, who, seeing that the\nlady would not accept the shipmaster for her husband, thought\nthat if he bought her, he might in time gain her affection.\nAccordingly he purchased her of the master for a large sum of\nmoney, and then told her that he had done so with a view of\nmaking her his wife. The lady replied that, although the shipman\nhad no right thus to dispose of her, yet she would consent to\nmarry him at the end of two years, if she did not during that\nperiod meet with her husband and their two sons; and to this\ncondition the merchant agreed. In the meanwhile the king, having\nsorrowfully watched the vessel till it was out of sight, turned\nback with his two boys, who wept and lamented as they ran beside\nhim. After walking a great distance, he came to a shallow but\nrapid river, which he wished to cross, and, as there was no boat\nor bridge, he was obliged to wade through the water. Taking up\none of his sons he contrived to reach the other side in safety,\nand was returning for the other when the force of the current\novercame him and he was drowned.\n\nWhen the two boys noticed that their father had perished, they\nwept bitterly. Their separation, too, was a further cause for\ngrief. There they stood, one on either side of the river, with no\nmeans of reaching each other. They shouted, and ran about hither\nand thither in their grief, till they had almost wearied\nthemselves into sleep, when a fisherman came past, who, seeing\nthe great distress of the boys, took them into his boat, and\nasked them who they were, and who were their parents; and they\ntold him all that had happened. When he had heard their story, he\nsaid, \"You have not a father or mother, and I have not a child.\nEvidently God has sent you to me. Will you be my own children and\nlearn to fish, and live in my house?\" Of course, the poor boys\nwere only too glad to find a friend and shelter. \"Come,\" said the\nfisherman kindly, leading them out of the boat to a house close\nby, \"I will look after you.\" The boys followed most happily, and\nwent into the fisherman's house, and when they saw his wife they\nwere still better pleased, for she was very kind to them, and\ntreated them as if they had been her own children. The two boys\nwent to school, and when they had learned all that the master\ncould teach them, they began to help their adoptive father, and\nin a little while became most expert and diligent young\nfishermen.\n\nThus time was passing with them, when it happened that a great\nfish threw itself on to the bank of the river and could not get\nback again into the water. Everybody in the village went to see\nthe monstrous fish, and cut a slice of its flesh and took it\nhome. A few people also went from the neighbouring villages, and\namongst them was a maker of earthernware. His wife had heard of\nthe great fish and urged him to go and get some of the flesh. So\nhe went, although the hour was late. On his arrival he found that\nall the people had returned to their homes. The potter had taken\nan axe with him, thinking that the bones would be so great and\nstrong as to require its use in breaking them. When he struck the\nfirst blow a voice came out of the fish, like that of some one in\npain, at which the potter was greatly surprised. \"Perhaps,\"\nthought he, \"the fish is possessed by a bh\u00fat.[FN#513] I'll try\nagain,\" whereupon he struck another blow with his axe. Again the\nvoice came forth from the fish, saying, \"Woe is me! woe is me!\"\nOn hearing this, the potter thought, \"Well, this is evidently not\na bh\u00fat, but the voice of an ordinary man. I'll cut the flesh\ncarefully.  May be that I shall find some poor distressed\nperson.\" So he began to cut away the flesh carefully, and\npresently he perceived a man's foot, then the legs appeared, and\nthen the entire body. \"Praise be to God,\" he cried, \"the soul is\nyet in him.\" He carried the man to his house as fast as he could,\nand on arriving there did everything in his power to recover him.\nA large fire was soon got ready, and tea and soup given the man,\nand great was the joy of the potter and his wife when they saw\nhim reviving.[FN#514] For some months the stranger lived with\nthose good people, and learnt how to make pots and pans and other\narticles and thereby helped them considerably. Now it happened\nthat the king of that country died and it was the custom of the\npeople to take for their sovereign whomsoever the late king's\nelephant and hawk should select. And so on the death of the king\nthe royal elephant was driven all over the country, and the hawk\nwas made to fly about, in search of a successor and it came to\npass that the person before whom the elephant saluted and on whom\nthe hawk alighted was considered as the divinely-chosen one.\nAccordingly the elephant and the hawk went about the country, and\nin the course of their wanderings came by the house of the potter\nwho had so kindly succoured the poor man whom he found in the\nbelly of the monstrous fish; and it chanced that as they passed\nthe place the stranger was standing by the door, and behold, no\nsooner did the elephant and hawk see him than the one bowed down\nbefore him and the other perched on his hand. \"Let him be king!\nlet him be king!\" shouted the people who were in attendance on\nthe elephant, and they prostrated themselves before the stranger\nand begged him to accompany them to the palace.[FN#515]\n\nThe ministers were glad when they heard the news, and most\nrespectfully welcomed their new king. As soon as the rites and\nceremonies necessary for the installation of a king had been\nobserved, his majesty entered on his duties. The first thing he\ndid was to send for the potter and his wife and grant them some\nland and money. In this and other ways, such as just judgments,\nproper laws, and kindly notices of all who were clever and good,\nhe won for himself the good opinion and affection of his subjects\nand prospered in consequence thereof. After a few months,\nhowever, his health was impaired, and his physicians advised him\nto take out-door exercise. Accordingly, he alternately rode,\nhunted and fished. He was especially fond of fishing, and\nwhenever he indulged in this amusement, he was attended by two\nsons of a fisherman, who were clever and handsome youths.\n\nAbout this time the merchant who bought the wife of the poor king\nthat was carried away by the rapid river visited that country for\npurposes of trade. He obtained an interview with the king, and\ndisplayed before him all his precious stones and stuffs. The king\nwas much pleased to see such treasures, and asked many questions\nabout them and the countries whence they had been brought. The\nmerchant satisfied the king's curiosity, and then begged\npermission to trade in that country, under his majesty's\nprotection, which the king readily granted, and ordered that some\nsoldiers should be placed on guard in the merchant's courtyard,\nand sent the fisherman's two sons to sleep in the premises.\n\nOne night those two youths not being able to sleep, the younger\nasked his brother to tell him a story to pass the time, so he\nreplied, \"I will tell you one out of our own experience: Once\nupon a time there lived a great and wealthy king, who was very\nproud, and his pride led him to utter ruin and caused him the\nsorest afflictions.. One day when going about with his army,\nchallenging other kings to fight with him, a great and powerful\nking appeared and conquered him. He escaped with his wife and two\nsons to the sea, hoping to find a vessel, by which he and his\nfamily might reach a foreign land. After walking several miles\nthey reached the sea-shore and found a ship ready to sail. The\nmaster of the vessel took the queen, but the king and his two\nsons were held back by some men, who had been hired by the master\nfor this purpose, until the ship was under way. The poor king\nafter this walked long and far till he came to a rapid river. As\nthere was no bridge or boat near, he was obliged to wade across.\nHe took one of his boys and got over safely, and was returning\nfor the other when he stumbled over a stone, lost his footing,\nand was carried down the stream; and he has not been heard of\nsince. A fisherman came along, and, seeing the two boys crying,\ntook them into his boat, and afterwards to his house, and became\nvery fond of them, as did also his wife, and they were like\nfather and mother to them. All this happened a few years ago, and\nthe two boys are generally believed to be the fisherman's own\nsons. O brother, we are these two boys! And there you have my\nstory.\"\n\nThe tale was so interesting and its conclusion so wonderful that\nthe younger brother was more awake than before. It had also\nattracted the attention of another. The merchant's promised wife,\nwho happened to be lying awake at the time, and whose room was\nseparated from the warehouse by a very thin partition, overheard\nall that had been said, and she thought within herself, \"Surely\nthese two boys must be my own sons.\" Presently she was sitting\nbeside them and asking them many questions. Two years or more had\nmade great difference in the persons of both the boys, but there\nwere certain signs which a hundred years could not efface from a\nmother's memory. These, together with the answers which she\nelicited from them, assured her that she had found her own sons\nagain. Tears streamed down her face as she embraced them, and\nrevealed to them that she was the queen, their mother, about whom\nthey had just been speaking. She then told them all that had\nhappened to her since she had been parted from them and their\npoor father, the king; after which she explained that although\nthe merchant was a good man and very wealthy yet she did not like\nhim well enough to become his wife, and proposed a plan for her\ngetting rid of him. \"My device,\" said she, \"is to pretend to the\nmerchant that you attempted my honour. I shall affect to be very\nangry and not give him any peace until he goes to the king and\ncomplains against you. Then will the king send for you in great\nwrath and inquire into this matter. In reply you may say it is\nall a mistake, for you regard me as your own mother, and in proof\nof this you will beg the king to summon me into his presence,\nthat I may corroborate what you say. Then I will declare that you\nare really my own sons, and beseech the king to free me from the\nmerchant and allow me to live with you in any place I may choose\nfor the rest of my days.\"\n\nThe sons agreed to this proposal, and next night, when the\nmerchant was also sleeping in the house, the woman raised a great\ncry, so that everybody was awakened by the noise. The merchant\ncame and asked the cause of the outcry, and she answered, \"The\ntwo youths who look after your warehouse have attempted to\nviolate me, so I screamed in order to make them desist.\" On\nhearing this the merchant was enraged. He immediately bound the\ntwo youths, and, as soon as there was any chance of seeing the\nking, took them before him preferred his complaint. \"What have\nyou to say in your defence?\" said the king, addressing the\nyouths; \"because, if what this merchant charges against you be\ntrue, I will have you at once put to death. Is this the gratitude\nyou manifest for all my kindness and condescension towards you?\nSay quickly what you have to say.\" \"O king, our benefactor,\"\nreplied the elder brother, \"we are not affrighted by your words\nand looks, for we are true servants. We have not betrayed your\ntrust in us, but have always tried to fulfil your wishes to the\nutmost of our power. The charges brought against us by this\nmerchant are unfounded. We have not attempted to dishonour his\nwife; we have rather always regarded her as our own mother. May\nit please your majesty to send for the woman and inquire further\ninto this matter.\"\n\nThe king consented, and the woman was brought before him. \"Is it\ntrue,\" he asked her \"what the merchant, your affianced husband,\nwitnesses against these two youths?\" \"O king,\" she replied, \"the\nyouths whom you gave to help the merchant have most carefully\ntried to carry out your wishes. But the night before last I heard\ntheir conversation. The elder was telling the younger a tale,\nfrom his own experience, he said. It was a story of a conceited\nking who had been defeated by another more powerful than he, and\nobliged to fly with his wife and two children to the sea. There,\nthrough the vile trickery of the master of a vessel, the wife was\nstolen and taken away to far distant lands, where she became\nengaged to a wealthy trader; while the exiled king and his two\nsons wandered in another direction, till they came to a river, in\nwhich the king was drowned. The two boys were found by a\nfisherman and brought up as his own sons. These two boys, O king,\nare before you, and I am their mother, who was taken away and\nsold to the trader, and who after two days must be married to\nhim. For I promised that if within a certain period I should not\nmeet with my husband and two sons I would be his wife. But I\nentreat your majesty to free me from this man. I do not wish to\nmarry again, now that I have found my two sons. In order to\nobtain an audience of your majesty, this trick was arranged with\nthe two youths.\"\n\nBy the time the woman had finished her story the king's face was\nsuffused with tears and he was trembling visibly. When he had\nsomewhat recovered he rose from the throne and going up to the\nwoman and the two youths embraced them long and fervently. \"You\nare my own dear wife and children,\" he cried. \"God has sent you\nback to me. I, the king, your husband, your father, was not\ndrowned as you supposed; but was swallowed by a great fish and\nnourished by it for some time, and then the monster threw itself\nupon the river's bank and I was extricated. A potter and his wife\nhad pity on me and taught me their trade, and I was just\nbeginning to earn my living by making earthen vessels when the\nlate king of this country died, and I was chosen king by the\nroyal elephant and hawk--I who am now standing here.\" Then his\nmajesty ordered the queen and her two sons to be taken into the\ninner apartments of the palace, and explained his conduct to the\npeople assembled. The merchant was politely dismissed from the\ncountry. And as soon as the two princes were old enough to govern\nthe kingdom, the king committed to them the charge of all\naffairs, while he retired with his wife to a sequestered spot and\npassed the rest of his days in peace.\n\nThe tale of Sarwar and N\u00edr, \"as told by a celebrated Bard from\nBaraut, in the Merath district,\" in vol. iii. of Captain R. C.\nTemple's \"Legends of the Panj\u00e1b\" (pp. 97-125), though differing in\nform somewhat from the Kashm\u00edr\u00ed version, yet possesses the\nleading incidents in common with it, as will be seen from the\nfollowing abstract:\n\n                        PANJ\u00c1B\u00cd VERSION.\n\nAmb\u00e1 the r\u00e1j\u00e1 of P\u00fan\u00e1 had a beautiful wife named Aml\u00ed and two\nyoung sons, Sarwar and N\u00edr. There came to his court one day a\nfak\u00edr. The r\u00e1j\u00e1 promised to give him whatsoever he should desire.\nThe fakir required Amb\u00e1 to give up to him all he possessed, or\nlose his virtue, and the r\u00e1j\u00e1 gave him all, save his wife and two\nchildren, receiving in return the blessings of the fak\u00edr, Then\nthe r\u00e1j\u00e1 and the r\u00e1n\u00ed went away; he carrying Sarwar in his bosom,\nand she with N\u00edr in her lap. For a time they lived on the fruits\nand roots of the forest. At length the ran\u00ed gave her husband her\n(jewelled) bodice to sell in the bazar, in order to procure food.\nHe offered it to Kundan the merchant, who made him sit down, and\nasked him where he had left the ran\u00ed, and why he did not bring her\nwith him. Amb\u00e1 told him that he had left her with their two boys\nunder the banyan-tree. Then Kundan, leaving Amb\u00e1 in the shop,\nwent and got a litter, and proceeding to the banyan-tree showed\nthe r\u00e1n\u00ed the bodice, and said, \"Thy husband wishes thee to come\nto him.\" Nothing doubting, the r\u00e1n\u00ed entered the litter, and the\nmerchant sent it off to his own house. Leaving the boys in the\nforest, he returned to Amb\u00e1, and said to him that he had not\nenough money to pay the price of the bodice, so the r\u00e1j\u00e1 must\ntake it back. Amb\u00e1 took the bodice, and coming to the boys,\nlearned from Sarwar how their mother had been carried away in a\nlitter, and he was sorely grieved in his heart, but consoled the\nchildren, saying that their mother had gone to her brother's\nhouse, and that he would take them to her at once. Placing the\ntwo boys on his shoulders he walked along till he came to a\nriver. He set down N\u00edr and carried Sarwar safely across, but as\nhe was going back for the other, behold, an alligator seized him.\nIt was the will of God: what remedy is there against the writing\nof Fate? The two boys, separated by the river, sat down and wept\nin their sorrow. In the early morning a washerman was up and\nspreading his clothes. He heard the two boys weeping and came to\nsee. He had pity on them and brought them together. Then he took\nthem to his house, and washed their faces and gave them food. He\nput them into a separate house and a Brahman cooked for them and\ngave them water.[FN#516] He caused the brothers to be taught all\nkinds of learning, and at the end of twelve years they both set\nout together to seek their living. They went to the city of\nUjjain, and told the r\u00e1j\u00e1 their history--how they had left their\nhome and kingdom. The r\u00e1j\u00e1 gave them arms and suitable clothing,\nand appointed them guards over the female apartments.[FN#517] One\nday a fisherman caught an alligator in his net. When he cut open\nits body, he found in it R\u00e1j\u00e1 Amb\u00e1, alive.[FN#518] So he took him\nto the r\u00e1j\u00e1 of Ujjain, and told how he had found him in the\nstomach of an alligator. Amb\u00e1 related his whole history to the\nr\u00e1j\u00e1; how he gave up all his wealth and his kingdom to a fak\u00edr;\nhow his wife had been stolen from him; and how after safely\ncarrying one of his young sons over the river in returning for\nthe other he had been swallowed by an alligator. On hearing of\nall these misfortunes the r\u00e1j\u00e1 of Ujjain pitied him and loved him\nin his heart: he adopted Amb\u00e1 as his son; and they lived together\ntwenty years, when the r\u00e1j\u00e1 died and Amb\u00e1 obtained the throne.\n\nMeanwhile the beautiful R\u00e1n\u00ed Aml\u00ed, the wife of Amb\u00e1, had\ncontinued to refuse the merchant Kundan's reiterated proffers of\nlove. At length he said to her, \"Many days have passed over thee,\nlive now in my house as my wife.\" And she replied, \"Let me bathe\nin the Ganges, and then I will dwell in thy house.\" So he took\nelephants and horses and lakhs of coin, and set the r\u00e1n\u00ed in a\nlitter and started on the journey. When he reached the city of\nUjjain, he made a halt and pitched his tents. Then he went before\nR\u00e1j\u00e1 Amb\u00e1 and said, \"Give me a guard, for the nights are dark.\nHitherto I have had much trouble and no ease at nights. I am\ngoing to bathe in the Ganges, to give alms and much food to\nBrahmans. I am come, r\u00e1j\u00e1, to salute thee, bringing many things\nfrom my house.\"\n\nThe r\u00e1j\u00e1 sent Sarwar and N\u00edr as guards. They watched the tents,\nand while the rain was falling the two brothers began talking\nover their sorrows, saying \"What can our mother be doing? Whither\nhath our father gone?\" Their mother overheard them talking, and\nby the will of God she recognised the princes; then she tore open\nthe tent, and cried aloud, \"All my property is gone! Who brought\nthis thief to my tent?\" The r\u00e1n\u00ed had both Sarwar and N\u00edr seized,\nand brought before R\u00e1j\u00e1 Amb\u00e1 on the charge of having stolen her\nproperty. The r\u00e1j\u00e1 held a court, and began to ask questions,\nsaying, \"Tell me what hath passed during the night. How much of\nthy property hath gone, my friend? I will do thee justice,\naccording to thy desire: my heart is grieved that thy goods are\ngone.\" Then said the ran\u00ed, \"Be careful of the young elephant! The\nlightning flashes and the heavy rain is falling. Said N\u00edr, 'Hear,\nbrother Sarwar, who knows whither our mother hath gone?' And I\nrecognised my son; so I made all this disturbance, raja [in order\nto get access to thee]\". [FN#519] Hearing this, R\u00e1j\u00e1 Amb\u00e1 rose up\nand took her to his breast--Aml\u00ed and Amb\u00e1 met again through the\nmercy of God. The r\u00e1j\u00e1 gave orders to have Kundan hanged, saying,\n\"Do it at once; he is a scoundrel; undo him that he may not\nlive.\" They quickly fetched the executioners and put on the\nnoose; and then was Kundan strangled. The r\u00e1n\u00ed dwelt in the\npalace and all her troubles passed far away. She fulfilled all\nher obligations, and obtained great happiness through her virtue.\n\n\n                        TIBETAN VERSION.\n\nUnder the title of \"Krisa Gautami\" in the collection of \"Tibetan\nTales from Indian Sources,\" translated by Mr. Ralston from the\nGerman of Von Schiefner, we have what appears to be a very much\ngarbled form of an old Buddhist version of our story. The heroine\nis married to a young merchant, whose father gives him some\narable land in a hill district, where he resides with Krisa\nGuatami his wife.\n\nWhen the time came for her to expect her confinement, she\nobtained leave of her husband to go to her parents' house in\norder that she might have the attendance of her mother. After her\nconfinement and the naming of the boy, she returned home. When\nthe time of her second confinement drew near, she again expressed\nto her husband a desire to go to her parents. Her husband set out\nwith her and the boy in a waggon; but by the time they had gone\nhalf way she gave birth to a boy. When the husband saw that this\nwas to take place he got out of the waggon, sat under a tree, and\nfell asleep. While he was completely overcome by slumber a snake\nbit him and he died. When his wife in her turn alighted from the\nwaggon, and went up to the tree in order to bring him the joyful\ntidings that a son was born unto him, he, as he had given up the\nghost, made no reply. She seized him by the hand and found that\nhe was dead. Then she began to weep. Meantime a thief carried off\nthe oxen. After weeping for a long time, and becoming very\nmournful, she looked around on every side, pressed the new-born\nbabe to her bosom, took the elder child by the hand, and set out\non her way. As a heavy rain had unexpectedly fallen, all the\nlakes, ponds, and springs were full of water, and the road was\nflooded by the river. She reflected that if she were to cross the\nwater with both the children at once, she and they might meet\nwith a disaster, and therefore the children had better be taken\nover separately. So she seated the elder boy on the bank of the\nriver, and took the younger one in her arms, walked across to the\nother side and laid him down upon the bank. Then she went back\nfor the elder boy. But while she was in the middle of the river,\nthe younger boy was carried off by a jackal. The elder boy\nthought that his mother was calling him, and sprang into the\nwater. The bank was very steep, so he fell down and was killed.\nThe mother hastened after the jackal, who let the child drop and\nran off. When she looked at it, she found that it was dead. So\nafter she had wept over it, she threw it into the water. When she\nsaw that the elder was being carried along by the stream, she\nbecame still more distressed. She hastened after him, and found\nthat he was dead. Bereft of both husband and children, she gave\nway to despair, and sat down alone on the bank, with only the\nlower part of her body covered. There she listened to the howling\nof the wind, the roaring of the forest and of the waves, as well\nas the singing of various kinds of birds. Then wandering to and\nfro, with sobs and tears of woe, she lamented the loss of her\nhusband and her two children.\n\nShe meets with one of her father's domestics, who informs her\nthat her parents and their servants had all been destroyed by a\nhurricane, and that \"he only had escaped\" to tell her the sad\ntidings. After this she is married to a weaver, who ill-uses her,\nand she escapes from him one night. She attaches herself to some\ntravellers returning from a trading expedition in the north, and\nthe leader of the caravan takes her for his wife. The party are\nattacked by robbers and the leader is killed. She then becomes\nthe wife of the chief of the robbers, who in his turn finds death\nat the hands of the king of that country, and she is placed in\nhis zenana.\n\nThe king died, and she was buried alive in his tomb, after having\nhad great honour shown to her by the women, the princes, the\nministers, and a vast concourse of people. Some men from the\nnorth who were wont to rob graves broke into this one also. The\ndust they raised entered into Krisa Gautami's nostrils, and made\nher sneeze. The grave-robbers were terrified, thinking that she\nwas a demon (vet\u00e1la), and they fled; but Krisa Gautami escaped\nfrom the grave through the opening which they had made. Conscious\nof all her troubles, and affected by the want of food, just as a\nviolent storm arose, she went out of her mind. Covered with\nmerely her underclothing, her hands and feet foul and rough, with\nlong locks and pallid complexion, she wandered about until she\nreached Sravast\u00ed. There, at the sight of Bhagavant, she recovered\nher intellect. Bhagavant ordered Ananda to give her an overrobe,\nand he taught her the doctrine, and admitted her into the\necclesiastical body, and he appointed her the chief of the\nBhikshun\u00eds who had embraced discipline.[FN#520]\n\nThis remarkable story is one of those which reached Europe long\nanterior to the Crusades. It is found in the Greek martyr acts,\nwhich were probably composed in the eighth century, where it is\ntold of Saint Eustache, who was before his baptism a captain of\nTrajan, named Placidus, and the same legend reappears, with\nmodifications of the details, in many medi\u00e6val collections and\nforms the subject of several romances. In most versions the motif\nis similar to that of the story of Job. The following is the\noutline of the original legend, according to the Greek martyr\nacts:\n\n\n                    LEGEND OF ST. EUSTACHE.\n\nAs Placidus one day hunted in the forest, the Saviour appeared to\nhim between the antlers of a hart, and converted him. Placidus\nchanged his name into Eustache, when he was baptised with his\nwife and sons. God announced to him by an angel his future\nmartyrdom. Eustache was afflicted by dreadful calamities, lost\nall his estate, and was compelled to go abroad as a beggar with\nhis wife and his children. As he went on board a ship bound for\nEgypt, his wife was seized by the shipmaster and carried off.\nSoon after, when Eustache was travelling along the shore, his two\nchildren were borne off by a lion and a leopard. Eustache then\nworked for a long time as journeyman, till he was discovered by\nthe emperor Trajan, who had sent out messengers for him, and\ncalled him to court. Reappointed captain, Eustache undertook an\nexpedition against the Dacians. During this war he found his wife\nin a cottage as a gardener--the shipmaster had fallen dead to the\nground as he ventured to touch her--and in the same cottage he\nfound again his two sons as soldiers: herdsmen had rescued them\nfrom the wild beasts and brought them up. Glad was their meeting\nagain! But as they returned to Rome they were all burnt in a\nglowing bull of brass by the emperor's order, because they\nrefused to sacrifice to the heathen gods.[FN#521]\n\nThe story of Placidus, which forms chapter 110 of the continental\n\"Gesta Romanorum,\" presents few and unimportant variations from\nthe foregoing: Eustatius came to a river the water of which ran\nso high that it seemed hazardous to attempt to cross it with both\nthe children at the same time; one therefore he placed upon the\nbank, and then passed over with the other in his arms, and having\nlaid it on the ground, he returned for the other child. But in\nthe midst of the river, looking back, he beheld a wolf snatch up\nthe child he had just carried over and run with it into the\nadjoining wood. He turned to rescue it, but at that instant a\nhuge lion approached the other child and disappeared with it.\nAfter the loss of his two boys Eustatius journeyed on till he\ncame to a village, where he remained for fifteen years, tending\nsheep as a hired servant, when he was discovered by Trajan's\nmessengers, and so on.\n\nThe story is so differently told in one of the Early English\ntranslations of the \"Gesta Romanorum\" in the Harleian MSS. 7333\n(re-edited by Herrtage for the E.E.T. Soc., pp. 87-91) that it is\nworth while, for purposes of comparison, reproducing it here in\nfull:\n\n\n                  OLD ENGLISH \"GESTA\" VERSION.\n\nAverios was a wise emperour regnyng in the cite of Rome; and he\nlet crye a grete feste, and who so ever wold come to that feste,\nand gete victory in the tournement, he shuld have his doughter to\nwyf, after his decesse. So there was a doughti knyght, and hardy\nin armys, and specially in tournement, the which hadde a wyf, and\ntwo yong children, of age of thre yere; and when this knyght had\nherd this crye, in a clere morowenyng[FN#522] he entred in to a\nforest, and there he herd a nyghtingale syng upon a tre so\nswetly, that he herd never so swete a melody afore that tyme. The\nknyght sette him doun undre the tre, and seid to him self, \"Now,\nLord, if I myght knowe what this brid[FN#523] shold\nbemene!\"[FN#524] There come an old man, and seid to him, \"That\nthou shalt go within thes thre daies to the emperours feste, and\nthou shalt suffre grete persecution or thou come there; and if\nthou be constant, and pacient in all thi tribulacion, thy sorowe\nshal turne the[FN#525] to grete joy; and, ser, this is the\ninterpretacion of his song.\" When this was seid, the old man\nvanysshed, and the brid fly away. Tho[FN#526] the knyght had\ngrete merveill; he yede[FN#527] to his wif, and told her the\ncas.[FN#528] \"Ser.\" quod she, \"the will of God be fulfilled, but\nI counsell that we go to the feste of the emperour and that ye\nthynk on the victory in the tournement, by the which we may be\navaunced[FN#529] and holpen.\"[FN#530] When the knyght had made\nall thing redy, there come a grete fire in the nyght; and\nbrent[FN#531] up all his hous and all his goodis, for which he\nhad grete sorowe in hert; nevertheles, notwithstondyng all this,\nhe yede forthe toward the see, with his wife, and with his two\nchildryn; and there he hired a ship, to passe over. When thei\ncome to londe, the maister of the shippe asked of the knyght his\nhire for his passage, for him, and for his wif and for his two\nchildryn. \"Dere frend,\" said the knyght to him, \"dere freed,\nsuffre me, and thou shalt have all thyn, for I go now to the\nfeste of the emperour, where I trust to have the victory in\nturnement, and then thou shalt be wele ypaied.\" \"Nay, by the\nfeith that I owe to the emperour,\" quod that other, \"hit shal not\nbe so, for but if [FN#532] you pay now, I shal holde thi wif to\nwed,[FN#533] tyll tyme that I be paied fully my salary.\" And he\nseid that, for he desired the love of the lady. Tho the knyght\nprofren his two childryn to wed, so that he myght have his wif;\nand the shipman seid, \"Nay, such wordis beth[FN#534] vayn, for,\"\nquod he, \"or[FN#535] I wol have my mede, or els I wolle holde thi\nwif.\" So the knyght lefte his wif with him, and kyst her with\nbitter teris; and toke the two childryn, scil. oon on his arme,\nand that othir in his nek, and so he yede forth to the turnement.\nAftir, the maister of the shippe wolde have layn by the lady, but\nshe denyed hit, and seid, that she had lever dey[FN#536] than\nconsente therto. So within short tyme, the maister drew to a\nfer[FN#537] lond, and there he deied; and the lady beggid her\nbrede fro dore to dore, and knew not in what lond her husbond was\nduellinge. The knyght was gon toward the paleis, and at the last\nhe come by a depe water, that was impossible to be passid,\nbut[FN#538] hit were in certein tyme, when hit was at the lowist.\nThe knyght sette doun oo[FN#539] child, and bare the othir over\nthe water; and aftir that he come ayen[FN#540] to fecche over the\nothir, but or[FN#541] he myght come to him, there come a lion,\nand bare him awey to the forest. The knyght pursued aftir, but he\nmyght not come to the lion; and then he wept bitterly, and yede\nayen over the water to the othir child; and or he were ycome, a\nbere had take the child, and ran therwith to the forest. When the\nknyght saw that, sore he wepte, and seid, \"Alias! that ever I was\nbore, for now have I lost wif and childryn. O thou brid! thi song\nthat was so swete is yturned in to grete sorowe, and hath ytake\naway myrth fro my hert.\" Aftir this he turned toward the feste,\nand made him redy toward the turnement; and there he bare him so\nmanly, and so doutely in the turnement and that twies or thries,\nthat he wan the victory, and worship, and wynnyng of that day.\nFor the emperour hily avauncid him, and made him maister of his\noste,[FN#542] and commaundid that all shuld obey to him; and he\nencresid, and aros from day to day in honure and richesse. And he\nwent aftirward in a certain day in the cite, [and] he found a\nprecious stone, colourid with thre maner of colours, as in oo\npartie[FN#542] white, in an othir partie red, and in the thrid\npartie blak. Anon he went to a lapidary, that was expert in the\nvertue of stonys; and he seid, that the vertue of thilke[FN#544]\nstone was this, who so ever berith the stone upon him, his\nhevynesse[FN#545] shall turne in to joy; and if he be\npovere,[FN#546] he shal be made riche; and if he hath lost\nanything, he shall fynde hit ayen with grete joy. And when the\nknyght herd this, he was glad and blith, and thought in him self,\n\"I am in grete hevynesse and poverte, for I have lost all that I\nhad, and by this stone I shal recovere all ayen, whether hit be\nso or no, God wote!\" Aftir, when he must go to bataile of the\nemperour he gadrid togidre[FN#547] all the oste, and among them\nhe found two yong knyghtis, semely in harneis,[FN#548] and wele\ni-shape, the which he hired for to go with him yn bataill of the\nemperour. And when thei were in the bataill, there was not oon in\nall the batail that did so doutely,[FN#549] as did tho[FN#550]\ntwo knyghtis that he hired; and therof this knyght, maister of\nthe ost, was hily gladid. When the bataill was y-do,[FN#551]\nthes two yong knyghtes yede to her oste[FN#552] in the cite; and\nas they sat to-gidir, the elder seid to the yonger, \"Dere frend,\nhit is long sithen[FN#553] that we were felawys,[FN#554] and we\nhave grete grace of God, for in every batail we have the victory;\nand therfore I pray you, telle me of what contre ye were ybore,\nand in what nacion? For I askid never this of the or now; and if\nthou wilt telle me soth,[FN#555] I shall telle my kynrede and\nwhere I was borne.\" And when oo felawe spak thus to the othir, a\nfaire lady was loggid[FN#556] in the same ostry;[FN#557] and when\nshe herd the elder knyght speke, she herkened to him; but she\nknew neither of hem,[FN#558] and yit she was modir of both, and\nwyf of the maister of the oste,[FN#559] the which also the\nmaister of the shippe withheld for ship-hire, but ever God kept\nher fro synne. Then spake the yonger knyght, \"Forsoth, good man,\nI note[FN#560] who was my fader or who was my modir, ne[FN#561]\nin what stede[FN#562] I was borne; but I have this wele in mynde\nthat my fader was a knyght, and that he bare me over the water,\nand left my eldir brothir in the lond; and as he passid over ayen\nto fecche him, there come a lion, and toke me up, but a man of the\ncite come with houndis, and when he saw him, he made him to leve\nme with his houndis.\"[FN#563] \"Now sothly,\" quod that othir, \"and\nin the same maner hit happid vith me. For I was the sone of a\nknyght, and had only a brothir; and my fader brought me and my\nbrothir, and my modir, over the see toward the emperour; and for\nmy fader had not to pay to the maister of the ship for the\nfraught, he left my modir to wed; and then my fader toke me with\nmy yong brothir, and brought us on his bak, and in his armys,\ntyll that we come unto a water, and there left me in a side of\nthe water, and bare over my yong brothir; and or my fader myght\ncome to me ayene, to bare me over, ther come a bere, and bore me\nto wode;[FN#564] and the people that saw him, make grete cry, and\nfor fere the bere let me falle, and so with thelke[FN#565] poeple\nI duellid x. yere, and ther I was y-norisshed.\" When the modir\nherd thes wordis, she seid, \"Withoute doute thes ben my\nsonys,\" and ran to hem anon, and fil upon her[FN#566] nekkes, and\nwepte sore for joy, and seid, \"A! dere sonys, I am your modir,\nthat your fader left with the maister of the shippe; and I know\nwele by your wordis and signes that ye beth true brethern. But\nhow it is with your fader, that I know not, but God, that all\nseth,[FN#567] yeve[FN#568] me grace to fynd my husbond.\" And alle\nthat nyght thes thre were in gladnes. On the morow the modir\nrose up, and the childryn, scil. the knyghtis, folowid; and as\nthei yede, the maister of the oste mette with hem in the strete,\nand though he were her fader, he knew hem not, but[FN#569] as\nthei had manli fought the day afore; and therfor he salued hem\nhonurably, and askid of hem what feir lady that was, that come\nwith hem? Anon as his lady herd his voys, and perceyved a certeyn\nsigne in his frount,[FN#570] she knew fully therby that it was\nher husbond; and therfore she ran to him, and clypt him, and kyst\nhim, and for joy fille doun to the erth, as she had be ded. So\naftir this passion, she was reised up; and then the maister seid\nto her, \"Telle me, feir woman, whi thou clippest me, and kyssist\nme so?\" She seid, \"I am thi wif, that thou leftist with the\nmaister of the ship; and thes two knyghtis bene your sonys. Loke\nwele on my front, and see.\" Then the knyght byheld her wele, with\na good avisement,[FN#571] and knew wele by diverse tokyns that\nshe was his wif; and anon kyst her, and the sonys eke; and\nblessid hiely God, that so had visited hem. Tho went he ayen to\nhis fond, with his wif, and with his children, and endid faire\nhis lif.\n\nFrom the legend of St. Eustache the romances of Sir Isumbras,\nOctavian, Sir Eglamour of Artois, and Sir Torrent of Portugal are\nderived. In the last, while the hero is absent, aiding the king of\nNorway with his sword, his wife Desonelle is delivered of twins,\nand her father, King Calamond, out of his hatred of her, causes\nher and the babes to be put to sea in a boat; but a favourable\nwind saves them from destruction, and drives the boat upon the\ncoast of Palestine. As she is wandering aimlessly along the\nshore, a huge griffin appears, and seizes one of her children, and\nimmediately after a leopard drags away the other. With submission\nshe suffers her miserable fate, relying on the help of the Holy\nVirgin. The king of Jerusalem, just returning from a voyage,\nhappened to find the leopard with the child, which he ordered to\nbe saved and delivered to him. Seeing from the foundling's golden\nring that the child was of noble descent, and pitying its\nhelpless state, he took it into his palace, and brought him up as\nif he were his own son, at his court. The dragon with the other\nchild was seen by a pious hermit, St. Antony, who, though son of\nthe king of Greece, had in his youth forsaken the world. Through\nhis prayer St. Mary made the dragon put down the infant. Antony\ncarried him to his father, who adopted him and ordered him to be\nbaptised. Desonelle wandered up and down, after the loss of her\nchildren, till she happened to meet the king of Nazareth hunting.\nHe, recognising her as the king of Portugal's daughter, gave her\na kind welcome and assistance, and at his court she lived several\nyears in happy retirement. Ultimately she is re-united to her\nhusband and her two sons, when they have become famous knights.\n\nThe following is an epitome of \"Sir Isumbras,\" from Ellis's\n\"Specimens of Early English Metrical Romances\" (Bohr's ed. p. 479\nff.):\n\n\n                    ROMANCE OF SIR ISUMBRAS.\n\nThere was once a knight, who, from his earliest infancy, appeared\nto be the peculiar favourite of Fortune. His birth was noble; his\nperson equally remarkable for strength and beauty; his\npossessions so extensive as to furnish the amusements of hawking\nand hunting in the highest perfection. Though he had found no\nopportunity of signalising his courage in war, he had borne away\nthe prize at numberless tournaments; his courtesy was the theme\nof general praise; his hall was the seat of unceasing plenty; it\nwas crowded with minstrels, whom he entertained with princely\nliberality, and the possession of a beautiful wife and three\nlovely children completed the sum of earthly happiness.\n\nSir Isumbras had many virtues, but he had one vice. In the pride\nof his heart he forgot the Giver of all good things, and\nconsidered the blessings so abundantly showered upon him as the\nproper and just reward of his distinguished merit. Instances of\nthis overweening presumption might perhaps be found in all ages\namong the possessors of wealth and power; but few sinners have\nthe good fortune to be recalled, like Sir Isumbras, by a severe\nbut salutary punishment, to the pious sentiments of Christian\nhumility.\n\nIt was usual with knights to amuse themselves with hawking or\nhunting whenever they were not occupied by some more serious\nbusiness; and, as business seldom intervened, they thus amused\nthemselves every day in the year. One morning, being mounted on\nhis favourite steed, surrounded by his dogs, and with a hawk on\nhis wrist, Sir Isumbras cast his eyes on the sky, and discovered\nan angel, who, hovering over him, reproached him with his pride,\nand announced the punishment of instant and complete degradation.\nThe terrified knight immediately fell on his knees; acknowledged\nthe justice of his sentence; returned thanks to Heaven for\ndeigning to visit him with adversity while the possession of\nyouth and health enabled him to endure it; and, filled with\ncontrition, prepared to return from the forest. But scarcely had\nthe angel disappeared when his good steed suddenly fell dead\nunder him, the hawk dropped from his wrist; his hounds wasted and\nexpired; and, being thus left alone, he hastened on foot towards\nhis palace, filled with melancholy forebodings, but impatient to\nlearn the whole extent of his misfortune.\n\nHe was presently met by a part of his household, who, with many\ntears, informed him that his horses and oxen had been suddenly\nstruck dead with lightning, and that his capons were all stung to\ndeath with adders. He received the tidings with humble\nresignation, commanded his servants to abstain from murmurs\nagainst Providence, and passed on. He was next met by a page, who\nrelated that his castle was burned to the ground, that many of\nhis servants had lost their lives, and that his wife and children\nhad with great difficulty escaped from the flames. Sir Isumbras,\nrejoiced that Heaven had yet spared those who were most dear to\nhim, bestowed upon the astonished page his purse of gold as a\nreward for the intelligence.\n\n          A doleful sight then gan he see;\n          His wife and children three\n             Out of the fire were fled:\n          There they sat, under a thorn,\n          Bare and naked as they were born,\n             Brought out of their bed.\n          A woful man then was he,\n          When he saw them all naked be,\n             The lady said, all so blive,\n          \"For nothing, sir, be ye adrad.\"\n          He did off his surcoat of pallade,[FN#572]\n             And with it clad his wife.\n          His scarlet mantle then shore[FN#573] he;\n          Therein he closed his children three\n             That naked before him stood.\n\nHe then proposed to his wife that, as an expiation of their sins,\nthey should at once under take a pilgrimage to Jerusalem; so,\ncutting with his knife a sign of the cross on his bare shoulder,\nhe set off with the four companions of his misery, resolving to\nbeg his bread till they should arrive at the Holy Sepulchre.\nAfter passing through \"seven lands,\" supported by the scanty alms\nof the charitable, they arrived at length at a forest, where they\nwandered during three days without meeting a single habitation.\nTheir food was reduced to the few berries which they were able to\ncollect; and the children, unaccustomed to such hard fare, began\nto sink under the accumulated difficulties of their journey. In\nthis situation they were stopped by a wide and rapid though\nshallow river. Sir Isumbras, taking his eldest son in his arms,\ncarried him over to the opposite bank, and placing him under a\nbush of broom, directed him to dry his tears, and amuse himself\nby playing with the blossoms till his return with his brothers.\nBut scarcely had he left the place when a lion, starting from a\nneighbouring thicket, seized the child and bore him away into the\nrecesses of the forest. The second son became, in like manner,\nthe prey of an enormous leopard; and the disconsolate mother,\nwhen carried over with her infant to the fatal spot, was with\ndifficulty persuaded to survive the loss of her two elder\nchildren. Sir Isumbras, though he could not repress the tears\nextorted by this cruel calamity, exerted himself to console his\nwife and humbly confessing his sins, contented himself with\npraying that his present misery might be accepted by Heaven as a\npartial expiation.\n\n          Through forest they went days three,\n          Till they came to the Greekish sea;\n             They grette,[FN#574] and were full wo!\n          As they stood upon the land,\n          They saw a fleet sailand,[FN#575]\n             Three hundred ships and mo.[FN#576]\n          With top-castels set on-loft,\n          Richly then were they wrought,\n             With joy and mickle[FN#577] pride:\n          A heathen king was therein,\n          That Christendom came to win;\n             His power was full wide.\n\nIt was now seven days since the pilgrims had tasted bread or\nmeat, the soudan's[FN#578] galley, therefore, was no sooner moored\nto the beach than the hastened on board to beg for food. The\nsoudan, under the apprehension that they were spies, ordered them\nto be driven back on shore; but his attendants observed to him\nthat these could not be common beggars; that the robust limbs and\ntall stature of the husband proved him to be a knight in\ndisguise, and that the delicate complexion of the wife, who was\n\"bright as blossom on tree,\" formed a striking contrast to the\nragged apparel with which she was very imperfectly covered. They\nwere now brought into the royal presence; and the soudan,\naddressing Sir Isumbras, immediately offered him as much treasure\nas he should require, on condition that he should renounce\nChristianity and consent to fight under the Saracen banners. The\nanswer was a respectful but peremptory refusal, concluded by an\nearnest petition for a little food; but the soudan, having by\nthis time turned his eyes from Sir Isumbras to the beautiful\ncompanion of his pilgrimage, paid no attention to his request.\n\n          The soudan beheld that lady there,\n          Him thought an angel that she were,\n             Comen a-down from heaven;\n          \"Man! I will give thee gold and fee,\n          An thou that woman will sellen me,\n             More than thou can neven.[FN#579]\n          I will give thee an hundred pound\n          Of pennies that been whole and round,\n             And rich robes seven:\n          She shall be queen of my land,\n          And all men bow unto her hand,\n             And none withstand her steven.\"[FN#580]\n          Sir Isumbras said, \"Nay!\n          My wife I will nought sell away,\n             Though ye me for her sloo![FN#581]\n          I weddid her in Goddislay,\n          To hold her to mine ending day,\n             Both for weal and wo.\"\n\nIt evidently would require no small share of casuistry to\nconstrue this declaration into an acceptance of the bargain, but\nthe Saracens, having heard the offer of their sovereign,\ndeliberately counted out the stipulated sum on the mantle of Sir\nIsumbras; took possession of the lady, carried the knight with\nhis infant son on shore; beat him till he was scarcely able to\nmove, and then returned for further orders. During this\noperation, the soudan, with his own hand, placed the regal crown\non the head of his intended bride; but recollecting that the\noriginal project of the voyage to Europe was to conquer it, which\nmight possibly occasion a loss of some time, he delayed his\nintended nuptial, and ordered a fast-sailing vessel to convey her\nto his dominions, providing her at the same time with a charter\naddressed to his subjects, in which he enjoined them to obey her,\nfrom the moment of her landing, as their legitimate sovereign.\n\nThe lady, emboldened by these tokens of deference on the part of\nher new lord, now fell on her knees and entreated his permission\nto pass a few moments in private with her former husband, and the\nrequest was instantly granted by the complaisant Saracen. Sir\nIsumbras, still smarting from his bruises, was conducted with\ngreat respect and ceremony to his wife, who, embracing him with\ntears, earnestly conjured him to seek her out as soon as possible\nin her new dominions, to slay his infidel rival, and to take\npossession of a throne which was probably reserved to him by\nHeaven as an indemnification for his past losses. She then\nsupplied him with provisions for a fortnight; kissed him and her\ninfant son; swooned three times, and then set sail for Africa.\n\nSir Isumbras, who had been set on shore quite confounded by this\nquick succession of strange adventures, followed the vessel with\nhis eyes till it vanished from his sight, and then taking his son\nby the hand led him up to some rocky woodlands in the\nneighbourhood. Here they sat down under a tree, and after a short\nrepast, which was moistened with their tears, resumed their\njourney. But they were again bewildered in the forest, and, after\ngaining the summit of the mountain without being able to descry a\nsingle habitation, lay down on the bare ground and resigned\nthemselves to sleep. The next morning Sir Isumbras found that his\nmisfortunes were not yet terminated. He had carried his stock of\nprovisions, together with his gold, the fatal present of the\nsoudan, enveloped in a scarlet mantle; and scarcely had the sun\ndarted its first rays on the earth when an eagle, attracted by\nthe red cloth, swooped down upon the treasure and bore it off in\nhis talons. Sir Isumbras, waking at the moment, perceived the\ntheft, and for some time hastily pursued the flight of the bird,\nwho, he expected, would speedily drop the heavy and useless\nburthen; but he was disappointed; for the eagle, constantly\ntowering as he approached the sea, at length directed his flight\ntowards the opposite shore of Africa. Sir Isumbras slowly\nreturned to his child, whom he had no longer the means of\nfeeding; but the wretched father only arrived in time to behold\nthe boy snatched from him by a unicorn. The knight was now quite\ndisheartened. But his last calamity was so evidently miraculous\nthat even the grief of the father was nearly absorbed by the\ncontrition of the sinner. He fell on his knees and uttered a most\nfervent prayer to Jesus and the Virgin, and then proceeded on his\njourney.\n\nHis attention was soon attracted by the sound of a smith's\nbellows: he quickly repaired to the forge and requested the\ncharitable donation of a little food; but was told by the\nlabourers that he seemed as well able to work as they did, and\nthey had nothing to throw away in charity.\n\n          Then answered the knight again,\n          \"For meat would I swink[FN#582] fain.\"\n             Fast he bare and drow,[FN#583]\n          They given him meat and drink anon.\n          And taughten him to bear stone:\n             Then had he shame enow.\n\nThis servitude lasted a twelvemonth, and seven years expired\nbefore he had fully attained all the mysteries of his new\nprofession. He employed his few leisure hours in fabricating a\ncomplete suit of armour: every year had brought him an account of\nthe progress of the Saracens; and he could not help entertaining\na hope that his arm, though so ignobly employed, was destined at\nsome future day to revenge the wrongs of the Christians, as well\nas the injury which he had personally received from the\nunbelievers.\n\nAt length he heard that the Christian army had again taken the\nfield; that the day was fixed for a great and final effort; and\nthat a plain at an inconsiderable distance from his shop was\nappointed for the scene of action. Sir Isumbras rose before day,\nbuckled on his armour, and mounting a horse which had hitherto\nbeen employed in carrying coals, proceeded to the field and took\na careful view of the disposition of both armies. When the\ntrumpets gave the signal to charge, he dismounted, fell on his\nknees, and after a short but fervent prayer to Heaven, again\nsprang into his saddle and rode into the thickest ranks of the\nenemy. His uncouth war-horse and awkward armour had scarcely less\neffect than his wonderful address and courage in attracting the\nattention of both parties; and when after three desperate\ncharges, his sorry steed was slain under him, one of the\nChristian chiefs make a powerful effort for his rescue, bore him\nto a neighbouring eminence, and presented to him a more suitable\ncoat of armour, and a horse more worthy of the heroic rider.\n\n          When he was armed on that stead,\n          It is seen where his horse yede,[FN#584]\n             And shall be evermore.\n          As sparkle glides off the glede,[FN#585]\n          In that stour he made many bleed,\n             And wrought hem wonder sore.\n          He rode up into the mountain,\n          The soudan soon hath he slain,\n             And many that with him were.\n          All that day lasted the fight;\n          Sir Isumbras, that noble knight,\n             Wan the battle there.\n          Knights and squires have him sought,\n          And before the king him brought;\n             Full sore wounded was he.\n          They asked what was his name;\n          He said, \"Sire, a smith's man;\n             What will ye do with me?\"\n          The Christian king said, than,\n          \"I trow never smith's man\n             In war was half so wight.\"\n          \"I bid[FN#586] you, give me meat and drink\n          And what that I will after think,\n             Till I have kevered[FN#587] my might.\"\n          The king a great oath sware,\n          As soon as he whole were,\n             That he would dub him knight.\n          In a nunnery they him leaved,\n          To heal the wound in his heved,[FN#588]\n             That he took in that fight.\n          The nuns of him were full fain,\n          For he had the soudan slain,\n             And many heathen hounds;\n          For his sorrow they gan sore rue;\n          Every day they salved him new,\n             And stopp\u00e8d well his wounds.\n\nWe may fairly presume, without derogating from the merit of the\nholy sisters or from the virtue of their salves and bandages,\nthat the knight's recovery was no less accelerated by the\npleasure of having chastised the insolent possessor of his wife\nand the author of his contumelious beating. In a few days his\nhealth was restored; and having provided himself with a \"scrip\nand pike\" and the other accoutrements of a palmer, he took his\nleave of the nuns, directed his steps once more to the \"Greekish\nSea,\" and, embarking on board of a vessel which he found ready to\nsail, speedily arrived at the port of Acre.\n\nDuring seven years, which were employed in visiting every part of\nthe Holy Land, the penitent Sir Isumbras led a life of continued\nlabour and mortification: fed during the day by the precarious\ncontributions of the charitable, and sleeping at night in the\nopen air, without any addition to the scanty covering which his\npilgrim's weeds, after seven years service, were able to afford.\nAt length his patience and contrition were rewarded. After a day\nspent in fruitless applications for a little food,\n\n          Beside the burgh of Jerusalem\n          He set him down by a well-stream,\n             Sore wepand[FN#589] for his sin.\n          And as he sat, about midnight,\n          There came an angel fair and bright,\n             And brought him bread and wine;\n          He said, \"Palmer, well thou be!\n          The King of Heaven greeteth well thee;\n             Forgiven is sin thine.\"\n\nSir Isumbras accepted with pious gratitude the donation of food,\nby which his strength was instantly restored, and again set out\non his travels; but he was still a widower, still deprived of his\nchildren, and as poor as ever; nor had his heavenly monitor\nafforded him any hint for his future guidance. He wandered\ntherefore through the country, without any settled purpose, till\nhe arrived at a \"rich burgh,\" built round a \"fair castle,\" the\npossessor of which, he was told, was a charitable queen, who\ndaily distributed a florin of gold to every poor man who\napproached her gates, and even condescended to provide food and\nlodging within her palace for such as were distinguished by\nsuperior misery. Sir Isumbras presented himself with the rest;\nand his emaciated form and squalid garments procured him instant\nadmittance.\n\n          The rich queen in hall was set;\n          Knights her served, at hand and feet,\n             In rich robes of pall:\n          In the floor a cloth was laid;\n          \"The poor palmer,\" the steward said,\n             \"Shall sit above you all.\"\n          Meat and drink forth they brought;\n          He sat still, and ate right nought,\n             But looked about the hall.\n          So mickle he saw of game and glee\n          (Swiche mirthis he was wont to see)\n             The tears he let down fall.\n\nConduct so unusual attracted the attention of the whole company,\nand even of the queen, who, ordering \"a chair with a cushion\" to\nbe placed near the palmer, took her seat in it, entered into\nconversation with him on the subject of his long and painful\npilgrimage, and was much edified by the moral lessons which he\ninterspersed in his narrative. But no importunity could induce\nhim to taste food: he was sick at heart, and required the aid of\nsolitary meditation to overcome the painful recollections which\ncontinually assailed him The queen was more and more astonished,\nbut at length left him to his reflections, after declaring that,\n\"for her lord's soul, or for his love, if he were still alive,\"\nshe was determined to retain the holy palmer in her palace, and\nto assign him a convenient apartment, together with a servant to\nattend him.\n\nAn interval of fifteen years, passed in the laborious occupations\nof blacksmith and pilgrim, may be supposed to have produced a\nvery considerable alteration in the appearance of Sir Isumbras;\nand even his voice, subdued by disease and penance, may have\nfailed to discover the gallant knight under the disguise which he\nhad so long assumed. But that his wife (for such she was) should\nhave been equally altered by the sole operation of time; that the\nair and gestures and action of a person once so dear and so\nfamiliar to him should have awakened no trace of recollection in\nthe mind of a husband, though in the midst of scenes which\npainfully recalled the memory of his former splendour, is more\nextraordinary. Be this as it may, the knight and the queen,\nthough lodged under the same roof and passing much of their time\ntogether, continued to bewail the miseries of their protracted\nwidowhood. Sir Isumbras, however, speedily recovered, in the\nplentiful court of the rich queen, his health and strength, and\nwith these the desire of returning to his former exercises. A\ntournament was proclaimed; and the lists, which were formed\nimmediately under the windows of the castle, were quickly occupied\nby a number of Saracen knights, all of whom Sir Isumbras\nsuccessively overthrew. So dreadful was the stroke of his spear,\nthat many were killed at the first encounter; some escaped with a\nfew broken bones; others were thrown headlong into the castle\nditch; but the greater number consulted their safety by a timely\nflight; while the queen contemplated with pleasure and\nastonishment the unparalleled exploits of her favourite palmer.\n\n          Then fell it, upon a day,\n          The Knight went him for to play,\n             As it was ere his kind;\n          A fowl's nest he found on high;\n          A red cloth therein he seygh[FN#590]\n             Wavand[FN#591] in the wind.\n          To the nest he gan win;[FN#592]\n          His own mantle he found therein;\n             The gold there gan he find.\n\nThe painful recollection awakened by this discovery weighed\nheavily on the soul of Sir Isumbras. He bore the fatal treasure\nto his chamber, concealed it under his bed, and spent the\nremainder of the day in tears and lamentations. The images of his\nlost wife and children now began to haunt him continually; and\nhis altered demeanour attracted the attention and excited the\ncuriosity of the whole court, and even of the queen, who could\nonly learn from the palmer's attendant that his melancholy seemed\nto originate in the discovery of something in a bird's nest. With\nthis strange report she was compelled to be satisfied, till Sir\nIsumbras, with the hope of dissipating his grief, began to resume\nhis usual exercises in the field; but no sooner had he quitted\nhis chamber than the \"squires\" by her command broke open the\ndoor, discovered the treasure, and hastened with it to the royal\napartment. The sight of the gold and the scarlet mantle\nimmediately explained to the queen the whole mystery of the\npalmer's behaviour. She burst into tears; kissed with fervent\ndevotion the memorial of her lost husband; fell into a swoon; and\non her recovery told the story to her attendants, and enjoined\nthem to go in quest of the palmer, and to bring him at once\nbefore her. A short explanation removed her few remaining doubts;\nshe threw herself into the arms of her husband, and the reunion\nof this long separated couple was immediately followed by the\ncoronation of Sir Isumbras and by a protracted series of\nfestivities.\n\nThe Saracen subjects of the Christian sovereign continued, with\nunshaken loyalty, to partake of the plentiful entertainments\nprovided for all ranks of people on this solemn occasion;\t but no\nsooner had the pious Sir Isumbras signified to them the necessity\nof their immediate conversion, than his whole \"parliament\"\nadopted the resolution of deposing and committing to the flames\ntheir newly-acquired sovereign, as soon as they should have\nobtained the concurrence of the neighbouring princes. Two of\nthese readily joined their forces for the accomplishment of this\nsalutary purpose, and invading the territories of Sir Isumbras\nwith an army of thirty thousand men, sent him, according to usual\ncustom, a solemn defiance. Sir Isumbras boldly answered the\ndefiance, issued the necessary orders, called for his arms,\nsprang upon his horse, and prepared to march out against the\nenemy; when he discovered that his subjects had, to a man,\nabandoned him, and that he must encounter singly the whole host\nof the invaders.\n\n          Sir Isumbras was bold and keen,\n          And took his leave at the queen,\n             And sighed wonder sore:\n          He said, \"Madam, have good day!\n          Sickerly, as you I say,\n             For now and evermore!\"\n          \"Help me, sir, that I were dight\n          In arms, as it were a knight;\n             I will with you fare:\n          Gif God would us grace send,\n          That we may together end,\n             Then done were all my care.\"\n          Soon was the lady dight\n          In arms, as it were a knight;\n             He gave her spear and shield:\n          Again[FN#593] thirty thousand Saracens and mo.[FN#594]\n          There came no more but they two,\n             When they met in field.\n\nNever, probably, did a contest take place between such\ndisproportioned forces. Sir Isumbras was rather encumbered than\nassisted by the presence of his beautiful but feeble helpmate;\nand the faithful couple were upon the point of being crushed by\nthe charge of the enemy, when three unknown knights suddenly made\ntheir appearance, and as suddenly turned the fortune of the day.\nThe first of these was mounted on a lion, the second on a\nleopard, and the third on a unicorn. The Saracen cavalry, at the\nfirst sight of these unexpected antagonists, dispersed in all\ndirections. But flight and resistance were equally hopeless:\nthree and twenty thousand unbelievers were soon laid lifeless on\nthe plain by the talons of the lion and leopard and by the\nresistless horn of the unicorn, or by the swords of their young\nand intrepid riders; and the small remnant of the Saracen army\nwho escaped from the general carnage quickly spread, through\nevery corner of the Mohammedan world, the news of this signal and\ntruly miraculous victory.\n\nSir Isumbras, who does not seem to have possessed the talent for\nunravelling mysteries, had never suspected that his three\nwonderful auxiliaries were his own children, whom Providence had\nsent to his assistance at the moment of his greatest distress;\nbut he was not the less thankful when informed of the happy\ntermination of all his calamities. The royal family were received\nin the city with every demonstration of joy by his penitent\nsubjects; whose loyalty had been completely revived by the recent\nmiracle. Magnificent entertainments were provided; after which\nSir Isumbras, having easily overrun the territories of his two\npagan neighbours, who had been slain in the last battle,\nproceeded to conquer a third kingdom for his youngest son; and\nthe four monarchs, uniting their efforts for the propagation of\nthe true faith, enjoyed the happiness of witnessing the baptism\nof all the inhabitants of their respective dominions.\n\n          They lived and died in good intent;\n          Unto heaven their souls went,\n             When that they dead were.\n          Jesu Christ, heaven's king,\n          Give us, aye, his blessing,\n             And shield us from care!\n\nOn comparing these several versions it will be seen that, while\nthey differ one from another m some of the details, yet the\nfundamental outline is identical, with the single exception of\nthe Tibetan story, which, in common with Tibetan tales generally,\nhas departed very considerably from the original. A king, or\nknight, is suddenly deprived of all his possessions, and with his\nwife and two children becomes a wanderer on the face of the\nearth; his wife is forcibly taken from him; he afterwards loses\nhis two sons; he is once more raised to affluence; his sons,\nhaving been adopted and educated by a charitable person, enter\nhis service; their mother recognises them through overhearing\ntheir conversation; finally husband and wife and children are\nhappily re-united. Such is the general outline of the story,\nthough modifications have been made in the details of the\ndifferent versions-- probably through its being transmitted\norally in some instances. Thus in the Arabian story, the king is\nruined apparently in consequence of no fault of his own; in the\nPanj\u00e1b\u00ed version, he relinquishes his wealth to a fak\u00edr as a pious\naction; in the Kashm\u00edr\u00ed and in the romance of Sir Isumbras, the\nhero loses his wealth as a punishment for his overweening pride;\nin the legend of St. Eustache, as in the story of Job, the\ncalamities which overtake the Christian convert are designed by\nHeaven as a trial of his patience and fortitude; while even in\nthe corrupted Tibetan story the ruin of the monarch is reflected\nin the destruction of the parents of the heroine by a hurricane.\nIn both the Kashm\u00edr\u00ed and the Panj\u00e1b\u00ed versions, the father is\nswallowed by a fish (or an alligator) in re-crossing the river to\nfetch his second child; in the Tibetan story the wife loses her\nhusband, who is killed by a snake, and having taken one of her\nchildren over the river, she is returning for the other when,\nlooking back, she discovers her babe in the jaws of a wolf: both\nher children perish: in the European versions they are carried\noff by wild beasts and rescued by strangers--the romance of Sir\nIsumbras is singular in representing the number of children to be\nthree. Only in the Arabian story do we find the father carrying\nhis wife and children in safety across the stream, and the latter\nafterwards lost in the forest. The Kashm\u00edr\u00ed and \"Gesta\" versions\ncorrespond exactly in representing the shipman as seizing the\nlady because her husband could not pay the passage-money: in the\nArabian she is entrapped in the ship, owned by a Magian, on the\npretext that there is on board a woman in labour; in Sir Isumbras\nshe is forcibly \"bought\" by the Soudan. She is locked up in a\nchest by the Magian; sent to rule his country by the Soudan;\nrespectfully treated by the merchant in the Kashm\u00edr\u00ed story, and,\napparently, also by Kandan in the Panj\u00e1b\u00ed legend; in the story of\nSt. Eustache her persecutor dies and she is living in humble\ncircumstances when discovered by her husband.--I think there is\ninternal evidence, apart from the existence of the Tibetan\nversion, to lead to the conclusion that the story is of Buddhist\nextraction, and if such be the fact, it furnishes a further\nexample of the indebtedness of Christian hagiology to Buddhist\ntales and legends.\n\n\n\n\nAL-MALIK AL-ZAHIR AND THE SIXTEEN CAPTAINS OF POLICE.--Vol. XII.\n                             p. 3.\n\n\n\nWe must, I think, regard this group of tales as being genuine\nnarratives of the exploits of Egyptian sharpers. From the days of\nHerodotus to the present time, Egypt has bred the most expert\nthieves in the world. The policemen don't generally exhibit much\nability for coping with the sharpers whose tricks they so well\nrecount; but indeed our home-grown \"bobbies\" are not particularly\nquick-witted.\n\n\n\n\n              THE THIEF'S TALE.--Vol. XII. p. 42.\n\n\n\nA parallel to the woman's trick of shaving off the beards and\nblackening the faces of the robbers is found in the well-known\nlegend, as told by Herodotus (Euterpe, 121), of the robbery of\nthe treasure-house of Rhampsinitus king of Egypt, where the\nclever thief, having made the soldiers dead drunk, shaves off the\nright side of their beards and then decamps with his brother's\nheadless body.\n\n\n\n\n         THE NINTH CONSTABLE'S STORY.--Vol. XII. p. 46.\n\n\n\nThe narrow escape of the singing-girl hidden under a pile of\nhalfah grass may be compared with an adventure of a fugitive\nMexican prince whose history, as related by Prescott, is as full\nof romantic daring and hair's-breadth 'scapes as that of\nScanderbeg or the \"Young Chevalier.\" This prince had just time to\nturn the crest of a hill as his enemies were climbing it on the\nother side, when he fell in with a girl who was reaping chian, a\nMexican plant, the seed of which is much used in the drinks of\nthe country. He persuaded her to cover him with the stalks she\nhad been cutting. When his pursuers came up and inquired if she\nhad seen the fugitive, the girl coolly answered that she had, and\npointed out a path as the one he had taken.\n\n\n\n\n       THE FIFTEENTH CONSTABLE'S STORY.--Vol. XII. p. 59.\n\n\n\nThe concluding part of this story differs very materially from\nthat of the Greek legend of Ibycus (fl. B.C. 540), which is thus\nrelated in a small MS. collection of Arabian and Persian\nanecdotes in my possession, done into English from the French:\n\nIt is written in the history of the first kings that in the reign\nof a Grecian king there lived a philosopher named Ibycus, who\nsurpassed in sagacity all other sages of Greece. Ibycus was once\nsent by the king to a neighbouring court. On the way he was\nattacked by robbers who, suspecting him to have much money,\nformed the design of killing him. \"Your object in taking my life,\"\nsaid Ibycus, \"is to obtain my money; I give it up to you, but\nallow me to live.\" The robbers paid no attention to his words,\nand persisted in their purpose. The wretched Ibycus, in his\ndespair, looked about him to see if any one was coming to his\nassistance, but no person was in sight. At that very moment a\nflock of cranes flew overhead. \"O cranes!\" cried Ibycus, \"know\nthat I have been seized in this desert by these wicked men, and I\ndie from their blows. Avenge me, and demand from them my blood.\"\nAt these words the robbers burst into laughter: \"To take away\nlife from those who have lost their reason,\" they observed, \"is\nto add nothing to their hurt.\" So saying, they killed Ibycus and\ndivided his money. On receipt of the news that Ibycus had been\nmurdered, the inhabitants of the town were exasperated and felt\ngreat sorrow. They caused strict inquiries to be made for the\nmurderers, but they could not be found. After some time the\nGreeks were celebrating a feast. The inhabitants of the adjoining\ndistricts came in crowds to the temples. The murderers of Ibycus\nalso came, and everywhere showed themselves. Meanwhile a flock of\ncranes appeared in the air and hovered above the people, uttering\ncries so loud and prolonged that the prayers and ceremonies were\ninterrupted. One of the robbers looked with a smile at his\ncomrades, saying, by way of joke, \"These cranes come without\ndoubt to avenge the blood of Ibycus.\" Some one of the town, who\nwas near them, heard these words, repeated them to his neighbour,\nand they together reported them to the king. The robbers were\ntaken, strictly cross-examined, confessed their crime, and\nsuffered for it a just punishment. In this way the cranes\ninflicted vengeance on the murderers of Ibycus. But we ought to\nsee in this incident a matter which is concealed in it: This\nphilosopher, although apparently addressing his words to the\ncranes, was really imploring help from their Creator; he hoped,\nin asking their aid, that He would not suffer his blood to flow\nunavenged. So God accomplished his hopes, and willed that cranes\nshould be the cause that his death was avenged in order that the\nsages of the world should learn from it the power and wisdom of\nthe Creator.\n\nThis ancient legend was probably introduced into Arabian\nliterature in the 9th century, when translations of so many of the\nbest Greek works were made; and, no doubt, it was adapted in the\nfollowing Indian (Muslim) story:[FN#595]\n\nThere was a certain pir, or saint, of great wisdom, learning, and\nsanctity, who sat by the wayside expounding the Kur\u00e1n to all who\nwould listen to him. He dwelt in the out-buildings of a ruined\nmosque close by, his only companion being a maina, or\nhill-starling, which he had taught to proclaim the excellence of\nthe formula of his religion, saying, \"The Prophet is just!\" It\nchanced that two travellers passing that way beheld the holy man\nat his devotions, and though far from being religious persons yet\ntarried a while to hear the words of truth. Evening now drawing\non, the saint invited his apparently pious auditors to his\ndwelling, and set before them such coarse food as he had to\noffer. Having eaten and refreshed themselves, they were\nastonished at the wisdom displayed by the bird, who continued to\nrepeat holy texts from the Kur\u00e1n. The meal ended, they all lay\ndown to sleep, and while the good man reposed, his treacherous\nguests, who envied him the possession a bird that in their hands\nmight be the means of enriching them, determined to steal the\ntreasure and murder its master. So they stabbed the sleeping\ndevotee to the heart and then seized hold of the bird's cage.\nBut, unperceived by them, the door of it had been left open and\nthe bird was not to be found. After searching for the bird in\nvain, they considered it necessary to dispose of the body, since,\nif discovered, suspicion would assuredly fall upon them; and\ncarrying it away to what they deemed a safe distance they buried\nit. Vexed to be obliged to leave the place without obtaining the\nreward of their evil deeds, they again looked carefully for the\nbird, but without success; it was nowhere to be seen, and so they\nwere compelled to go forward without the object of their search.\nThe maina had witnessed the atrocious deed, and unseen had\nfollowed the murderers to the place were they had buried the\nbody; it then perched upon the tree beneath which the saint had\nbeen wont to enlighten the minds of his followers, and when they\nassembled flew into their midst, exclaiming, \"The Prophet is\njust!\" making short flights and then returning. These unusual\nmotions, together with the absence of their preceptor, induced\nthe people to follow it, and directing its flight to the grave of\nits master, it uttered a mournful cry over the newly-covered\ngrave. The villagers, astonished, began to remove the earth, and\nsoon discovered the bloody corse.  Surprised and horror-stricken,\nthey looked about for some traces of the murderers, and\nperceiving that the bird had resumed the movements which had\nfirst induced them to follow it, they suffered it to lead them\nforward. Before evening fell, the avengers came up with two men,\nwho no sooner heard the maina exclaim, \"The Prophet is just'\" and\nsaw the crowd that accompanied it, than they fell upon their\nknees, confessing that the Prophet had indeed brought their evil\ndeeds to light; so, their crime being thus made manifest, summary\njustice was inflicted upon them.\n\n\n\n\n     TALE OF THE DAMSEL TUHFAT AL-KULUB.--Vol. XII. p. 70.\n\n\n\nAn entertaining story, but very inconsistent in the character of\nIblis, who is constantly termed, in good Muslim fashion, \"the\naccursed,\" yet seems to be somewhat of a follower of the Prophet,\nand on the whole a good-natured sort of fellow. His mode of\nexpressing his approval of the damsel's musical \"talent\" is, to\nsay the least, original.\n\n\n\n\n                WOMEN'S WILES.--Vol. XII. p. 99.\n\n\n\nA variant--perhaps an older form--of this story occurs in the\ntale of Prince Fadlallah, which is interwoven with the History of\nPrince Calaf and the Princess of China, in the Persian tales of\n\"The Thousand and One Days\":\n\nThe prince, on his way to Baghdad, is attacked by robbers, his\nfollowers are all slain, and himself made prisoner, but he is set\nat liberty by the compassionate wife of the robber-chief during\nhis absence on a plundering expedition. When he reaches Baghdad\nhe has no resource but to beg his bread, and having stationed\nhimself in front of a large mansion, an old female slave\npresently comes out and gives him a loaf. At this moment a gust\nof wind blew aside the curtain of a window and discovered to his\nadmiring eyes a most beautiful damsel, of whom he became\nimmediately enamoured. He inquired of a passerby the name of the\nowner of the mansion, and was informed that it belonged to a man\ncalled Mouaffac, who had been lately governor of the city, but\nhaving quarrelled with the k\u00e1z\u00ed, who was of a revengeful\ndisposition, the latter had found means to disgrace him with the\nkhal\u00edf and to have him deprived of his office. After lingering\nnear the house in vain till nightfall, in hopes of once more\nobtaining a glimpse of this beauty, he retired for the night to a\nburying-ground, where he was soon joined by two thieves, who\npressed upon him a share of the good cheer with which they had\nprovided themselves, but while the thieves were feasting and\ntalking over a robbery which they had just accomplished, the\npolice suddenly pounced upon them, and took all three and cast\nthem into prison.\n\nIn the morning they were examined by the k\u00e1z\u00ed, and the thieves,\nseeing it was useless to deny it, confessed their crime.  The\nprince then told the k\u00e1z\u00ed how he chanced to fall into company of\nthe thieves, who confirmed all he said, and he was set at\nliberty. Then the k\u00e1z\u00ed began to question him as to how he had\nemployed his time since he came to Baghd\u00e1d, to which he answered\nvery frankly but concealed his rank. On his mentioning the brief\nglance he had of the beautiful lady at the window of the\nex-governor's house, the k\u00e1z\u00ed's eyes sparkled with apparent\nsatifaction, and he assured the prince that he should have the\nlady for his bride; for, believing the prince to be a mere\nbeggarly adventurer, he resolved to foist him on Mouaffac as the\nson of a great monarch. So, having sent the prince to the bath and\nprovided him with rich garments, the k\u00e1z\u00ed despatched a messenger\nto request Mouaffac to come to him on important business. When\nthe ex-governor arrived, the k\u00e1z\u00ed told him blandly that there was\nnow an excellent opportunity for doing away the ill will that had\nso long existed between them. \"It is this,\" continued he: \"the\nprince of Basra, having fallen in love with your daughter from\nreport of her great beauty, has just come to Baghd\u00e1d, unknown to\nhis father, and intends to demand her of you in marriage. He is\nlodged in my house, and is most anxious that this affair should\nbe arranged by my interposition, which is the more agreeable to\nme, since it will, I trust, be the means of reconciling our\ndifferences.\" Mouaffac expressed his surprise that the prince of\nBasra should think of marrying his daughter, and especially that\nthe proposal should come through the k\u00e1z\u00ed, of all men. But the k\u00e1z\u00ed\nbegged him to forget their former animosity and consent to the\nimmediate celebration of the nuptials. While they were thus\ntalking, the prince entered, in a magnificent dress, and was not\na little astonished to be presented to Mouaffac by the\ntreacherous k\u00e1z\u00ed as the prince of Basra, who had come as a suitor\nfor his daughter in marriage. The ex-governor saluted him with\nevery token of profound respect, and expressed his sense of the\nhonour of such an alliance: his daughter was unworthy to wait\nupon the meanest of the prince's slaves. In brief, the marriage\nis at once celebrated, and the prince duly retires to the bridal\nchamber with the beauteous daughter of Mouaffac. But in the\nmorning, at an early hour, a servant of the k\u00e1z\u00ed knocks at his\ndoor, and, on the prince opening it, says that he brings him his\nrags of clothes and is required to take back the dress which the\nk\u00e1z\u00ed had lent him yesterday to personate the prince of Basra. The\nprince, having donned his tattered garments, said to his wife,\n\"The k\u00e1z\u00ed thinks he has married you to a wretched beggar, but I\nam no whit inferior in rank to the prince of Basra--I am also a\nprince, being the only son of the king of Mosel,\" and then\nproceeded to recount all his adventures. When he had concluded\nhis recital, the lady despatched a servant to procure a suitable\ndress for the prince, which when he had put on, she said, \"I see\nit all: the k\u00e1z\u00ed, no doubt, believes that by this time we are all\noverwhelmed with shame and grief. But what must be his feelings\nwhen he learns that he has been a benefactor to his enemies!\nBefore you disclose to him your real rank, however, we must\ncontrive to punish him for his malicious intentions. There is a\ndyer in this town who has a frightfully ugly daughter-- but leave\nthis affair in my hands.\"\n\nThe lady then dressed herself in plain but becoming apparel, and\nwent out of the house alone. She proceeded to the court of the\nk\u00e1z\u00ed, who no sooner cast his eyes upon her than he was struck\nwith her elegant form. He sent an officer to inquire of her who\nshe was and what she had come about. She made answer that she was\nthe daughter of an artisan in the city. and that she desired to\nhave some private conversation with the k\u00e1z\u00ed. When the officer\nreported the lady's reply, the k\u00e1z\u00ed directed her to be conducted\ninto a private chamber, where he presently joined her, and\ngallantly placed his services at her disposal. The lady now\nremoved her veil, and asked him whether he saw anything ugly or\nrepulsive in her features. The k\u00e1z\u00ed on seeing her beautiful face\nwas suddenly plunged in the sea of love, and declared that her\nforehead was of polished silver, her eyes were sparkling\ndiamonds, her mouth a ruby casket containing a bracelet of\npearls. Then she displayed her arms, so white and plump, the\nsight of which threw the k\u00e1z\u00ed into ecstasies and almost caused\nhim to faint. Quoth the lady, \"I must tell you, my lord, that\nwith all the beauty I possess, my father, a dyer in the city,\nkeeps me secluded, and declares to all who come to ask me in\nmarriage that I am an ugly, deformed monster, a mere skeleton,\nlame, and full of diseases.\" On this the k\u00e1z\u00ed burst into a tirade\nagainst the brutal father who could thus traduce so much beauty,\nand vowed that he would make her his wife that same day. The\nlady, after expressing her fears that he would not find it easy\nto gain her father's consent, took her leave and returned home.\n\nThe k\u00e1z\u00ed lost no time in sending for the dyer, and, after\ncomplimenting him upon his reputation for piety, said to him, \"I\nam informed that behind the curtain of chastity you have a\ndaughter ripe for marriage. Is not this true?\" Replied the dyer,\n\"My lord, you have been rightly informed. I have a daughter who\nis indeed fully ripe for marriage, for she is more than thirty\nyears of age; but the poor creature is not fit to be a wife to\nany man. She is very ugly, lame, leprous, and foolish. In short,\nshe is such a monster that I am obliged to keep her out of all\npeople's sight.\" \"Ha!\" exclaimed the k\u00e1z\u00ed, \"you can't impose on\nme with such a tale. I was prepared for it. But let me tell you\nthat I myself am ready and willing to marry that same ugly and\nleprous daughter of yours, with all her defects.\" When the dyer\nheard this, he looked the k\u00e1z\u00ed full in the face and said, \"My\nlord, you are welcome to divert yourself by making a jest of my\ndaughter.\" No,\" replied the k\u00e1z\u00ed \"I am quite in earnest. I demand\nyour daughter in marriage.\" The dyer broke into laughter, saying,\n'By Allah, some one has meant to play you a trick, my lord. I\nforewarn you that she is ugly, lame, and leprous.\"  \"True,\"\nresponded the k\u00e1z\u00ed, with a knowing smile; \"I know her by these\ntokens. I shall take her notwithstanding.\" The dyer, seeing him\ndetermined to marry his daughter, and being now convinced that he\nhad been imposed upon by some ill-wisher, thought to himself, \"I\nmust demand of him a round sum of money which may cause him to\ncease troubling me any further about my poor daughter.\" So he\nsaid to the k\u00e1z\u00ed, \"My lord, I am ready to obey your command; but\nI will not part with my daughter unless you pay me beforehand a\ndowry of a thousand sequins.\" Replied the k\u00e1z\u00ed, \"Although,\nmethinks, your demand is somewhat exorbitant, yet I will pay you\nthe money at once.\" which having done, he ordered the contract to\nbe drawn up. But when it came to be signed the dyer declared that\nhe would not sign save in the presence of a hundred men of the\nlaw.  \"Thou art very distrustful,\" said the k\u00e1z\u00ed, \"but I will\ncomply in everything, for I am resolved to make sure of thy\ndaughter.\" So he sent for all the men of law in the city, and\nwhen they were assembled at the house of the k\u00e1z\u00ed, the dyer said\nthat he was now willing to sign the contract; \"But I declare,\" he\nadded, \"in the presence of these honourable witnesses, that I do\nso on the condition that if my daughter should not prove to your\nliking when you have seen her, and you should determine to\ndivorce her, you shall oblige yourself to give her a thousand\nsequins of gold in addition to the same amount which I have\nalready received from you. \"Agreed,\" said the k\u00e1z\u00ed, \"I oblige\nmyself to it, and call this whole assembly to be witnesses. Art\nthou now satisfied?\" \"I am,\" replied the dyer, who then went his\nway, saying that he would at once send him his bride.\n\nAs soon as the dyer was gone, the assembly broke up, and the k\u00e1z\u00ed\nwas left alone in his house. He had been two years married to the daughter\nof a merchant of Baghd\u00e1d, with whom he had hitherto lived on very\namicable terms. When she heard that he was arranging for a second\nmarriage, she came to him in a great rage. \"How now,\" said she,\n\"two hands in one glove! two swords in one scabbard! two wives in\none house! Go, fickle man! Since the caresses of a young and\nfaithful wife cannot secure your constancy, I am ready to yield\nmy place to my rival and retire to my own family. Repudiate me--\nreturn my dowry--and you shall never see me more.\" \"I am glad you\nhave thus anticipated me,\" answered the k\u00e1z\u00ed, \"for I was somewhat\nperplexed how to acquaint you of my new marriage.\" So saying, he\nopened a coffer and took out a purse of five hundred sequins of\ngold, and putting it into her hands, \"There, woman,\" said he,\n\"thy dowry is in that purse: begone, and take with you what\nbelongs to you. I divorce thee once; I divorce thee twice, three\ntimes I divorce thee. And that thy parents may be satisfied thou\nart divorced from me, I shall give thee a certificate signed by\nmyself and my nayb.\" This he did accordingly, and his wife went\nto her father's house, with her bill of divorce and her dowry.\n\nThe k\u00e1z\u00ed then gave orders to furnish an apartment sumptuously for\nthe reception of his bride. The floor was spread with velvet\ncarpets, the walls were hung with rich tapestry, and couches of\ngold and silver brocade were placed around the room. The bridal\nchamber was decked with caskets filled with the most exquisite\nperfumes. When everything was in readiness, the k\u00e1z\u00ed impatiently\nexpected the arrival of his bride, and at last was about to\ndespatch a messenger to the dyer's when a porter entered,\ncarrying a wooden chest covered with a piece of green taffeta.\n\"What hast thou brought me there, friend?\" asked the k\u00e1z\u00ed. \"My\nlord,\" replied the porter, setting the chest on the floor, \"I\nbring your bride.\" The k\u00e1z\u00ed opened the chest, and discovered a\nwoman of three feet and a half, defective in every limb and\nfeature. He was horrified at the sight of this object, and\nthrowing the covering hastily over it, demanded of the porter,\n\"What wouldst thou have me do with this frightful creature?\" \"My\nlord,\" said the porter, \"this is the daughter of Omar the dyer,\nwho told me that you had espoused her out of pure inclination.\"\n\"O Allah!\" exclaimed the k\u00e1z\u00ed, \"is it possible to marry such a\nmonster as this?\" Just then, the dyer, well knowing that the k\u00e1z\u00ed\nmust be surprised, came in. \"Thou wretch,\" cried the k\u00e1z\u00ed, \"how\ndost thou dare to trifle with me? In place of this hideous\nobject, send hither your other daughter, whose beauty is beyond\ncomparison; otherwise thou shalt soon know what it is to insult\nme.\" Quoth the dyer, \"My lord, I swear, by Him who out of\ndarkness produced light, that I have no other daughter but this.\nI told you repeatedly that she was not for your purpose, but you\nwould not believe my words. Who, then, is to blame?\" Upon this\nthe k\u00e1z\u00ed began to cool, and said so the dyer, \"I must tell you,\nfriend Omar, that this morning there came to me a most beautiful\ndamsel, who pretended that you were her father, and that you\nrepresented her to everybody as a monster, on purpose to deter\nall suitors that came to ask her in marriage.\" \"My lord,\"\nanswered the dyer, \"this beautiful damsel must be an impostor;\nsome one, undoubtedly, owes you a grudge.\" Then the k\u00e1z\u00ed, having\nreflected for a few minutes, said to the dyer, \"Bid the porter\ncarry thy daughter home again. Keep the thousand sequins of gold\nwhich I gave thee, but ask no more of me, if thou desirest that\nwe should continue friends.\" The dyer, knowing the implacable\ndisposition of the k\u00e1z\u00ed, thought it advisable to content himself\nwith what he had already gained, and the k\u00e1z\u00ed, having formally\ndivorced his hideous bride, sent her away with her father. The\naffair soon got wind in the city and everybody was highly\ndiverted with the trick practiced on the k\u00e1z\u00ed.\n\nIt will be observed that in the Arabian story there are two\nclever devices: that of the lady who tricks the boastful\nmerchant, whose motto was that men's craft is superior to women's\ncraft, into marrying the ugly daughter of the k\u00e1z\u00ed; and that of\nthe merchant to get rid of his bad bargain by disgusting the k\u00e1z\u00ed\nwith the alliance. The scene at the house of the worthy judge--\nthe crowd of low rascals piping, drumming, and capering, and\nfelicitating themselves on their pretended kinsman the merchant's\nmarriage--is highly humorous. This does not occur in the Persian\nstory, because it is the k\u00e1z\u00ed, who has been duped into marrying\nthe dyer's deformed daughter, and she is therefore simply packed\noff again to her father's house.\n\nThat the tales of the \"Thousand and One Days\" are not (as is\nsupposed by the writer of an article on the several English\nversions of The Nights in the \"Edinburgh Review\" for July 1886,\np. 167) mere imitations of Galland[FN#596] is most certain, apart\nfrom the statement in the preface to Petis' French translation,\nwhich there is no reason to doubt--see vol. x. of The Nights, p.\n166, note 1. Sir William Ouseley, in his Travels, vol. ii., p.\n21, note, states that he brought from Persia a manuscript which\ncomprised, inter alia, a portion of the \"Haz\u00e1r \u00fa Yek R\u00faz,\" or the\nThousand and One Days, which agreed with Petis' translation of\nthe same stories. In the Persian collection entitled \"Shamsa \u00fa\nKuhkuha\" occur several of the tales and incidents, for example,\nthe Story of Nasiraddoli King of Mousel, the Merchant of Baghd\u00e1d,\nand the Fair Zeinib, while the Story of the King of Thibet and\nthe Princess of the Naimans has its parallel in the Turkish \"Kirk\nVaz\u00edr,\" or Forty Vaz\u00edrs. Again, the Story of Couloufe and the\nBeautiful Dilara reminds us of that of Haji the Cross-grained in\nMalcolm's \"Sketches of Persia.\" But of the French translation not\na single good word can be said--the Oriental \"costume\" and\nphraseology have almost entirely disappeared, and between Petis\nde la Croix and the author of \"Gil Blas\"--who is said to have had\na hand in the work--the tales have become ludicrously\nFrenchified. The English translation made from the French is, if\npossible, still worse. We there meet with \"persons of quality,\"\n\"persons of fashion,\" with \"seigneurs,\" and a thousand and one\nother inconsistencies and absurdities. A new translation is much\nto be desired. The copy of the Persian text made by Petis is\nprobably in the Paris Library and Ouseley's fragment is doubtless\namong his other Oriental MSS. in the Bodleian. But one should\nsuppose that copies of the \"Haz\u00e1r \u00fa Yek R\u00faz\" may be readily\nprocured at Ispah\u00e1n or Tehr\u00e1n, and at a very moderate cost, since\nthe Persians now-a-days are so poor in general that they are\neager to exchange any books they possess for the \"circulating\nmedium.\"\n\n\n\n\n  NUR AL-DIN AND THE DAMSEL SITT AL-MILAH.--Vol. XII. p. 151.\n\n\n\nThis is an excellent tale; the incidents occur naturally and the\nreader's interest in the fortunes of the hero and heroine never\nflags. The damsel's sojourn with the old Muezzin--her dispatching\nhim daily to the shroff--bears some analogy to part of the tale\nof Ghanim the Slave of Love (vol. ii. of The Nights), which, by\nthe way, finds close parallels in the Turkish \"Forty Vaz\u00edrs\" (the\nLady's 18th story in Mr. Gibb's translation), the Persian\n\"Thousand and One Days\" (story of Aboulcasem of Basra), and the\n\"Bagh o Bah\u00e1r\" (story of the First Dervish). This tale is, in\nfact, a compound of incidents occurring in a number of different\nArabian fictions.\n\n\n\n\n TALE OF KING INS BIN KAYS AND HIS DAUGHTER.--Vol. XII. p. 191.\n\n\n\nHere we have another instance of a youth falling in love with the\nportrait of a pretty girl (see ante, p. 328). The doughty deeds\nperformed by the young prince against thousands of his foes throw\ninto the shade the exploits of the Bedouin hero Antar, and those\nof our own famous champions Sir Guy of Warwick and Sir Bevis of\nHampton.\n\n\n\n\n                       ADDITIONAL NOTES.\n\n\n\n                  FIRUZ AND HIS WIFE, p. 301.\n\n\n\nI find yet another variant of this story in my small MS.\ncollection of Arabian and Persian anecdotes, translated from the\nFrench (I have not ascertained its source):\n\nThey relate that a lord of Basra, while walking one day in his\ngarden, saw the wife of his gardener, who was very beautiful and\nvirtuous. He gave a commission to his gardener which required him\nto leave his home. He then said to his wife, \"Go and shut all the\ndoors.\" She went out and soon returned, saying, \"I have shut all\nthe doors except one, which I am unable to shut.\" The lord asked,\n\"And where is that door?\" She replied \"That which is between you\nand the respect due to your Maker: there is no way of closing\nit.\" When the lord heard these words, he asked the woman's\npardon, and became a better and a wiser man.\n\nWe have here a unique form of the wide-spread tale of \"The Lion's\nTrack,\" which, while it omits the husband's part, yet reflects\nthe virtuous wife's rebuke of the enamoured sultan.\n\n\n\n\n              THE SINGER AND THE DRUGGIST, p. 305.\n\n\n\nIf Straparola's version is to be considered as an adaptation of\nSer Giovanni's novella-- which I do not think very probable--it\nmust be allowed to be an improvement on his model. In the Arabian\nstory the singer is first concealed in a mat, next in the oven,\nand again in the mat, after which he escapes by clambering over\nthe parapet of the druggist's roof to that of an adjoining house,\nand his subsequent adventures seem to be added from a different\nstory. In Ser Giovanni's version the lover is first hid beneath a\nheap of half-dried clothes, and next behind the street door, from\nwhich he escapes the instant the husband enters, and the latter\nis treated as a madman by the wife's relatives and the\nneighbours--an incident which has parallels in other tales of\nwomen's craft and its prototype, perhaps, in the story of the man\nwho compiled a book of the Wiles of Woman, as told in \"Syntipas,\"\nthe Greek version of the Book of Sindibad. In Straparola the\nlover--as in the Arabian story--is concealed three times, first\nin a basket, then between two boardings, and lastly in a chest\ncontaining law papers; and the husband induces him to recount his\nadventures in presence of the lady's friends, which having\nconcluded, the lover declares the story to be wholly fictitious:\nthis is a much more agreeable ending than that of Giovanni's\nstory, and, moreover, it bears a close analogy to the latter part\nof the Persian tale, where the lover exclaims he is right glad to\nfind it all a dream. Straparola's version has another point of\nresemblance in the Persian story--so far as can be judged from\nScott's abstract--and also in the Arabian story: the lover\ndiscovers the lady by chance, and is not advised to seek out some\nobject of love, as in Giovanni; in the Arabian the singer is\ncounselled by the druggist to go about and entertain wine\nparties. Story-comparers have too much cause to be dissatisfied\nwith Jonathan Scott's translation of the \"Bah\u00e1r-i- D\u00e1nish\"--a\nwork avowedly derived from Indian sources--although it is far\nsuperior to Dow's garbled version. The abstracts of a number of\nthe tales which Scott gives in an appendix, while of some use,\nare generally tantalising: some stories he has altogether omitted\n\"because they are similar to tales already well known\"\n(unfortunately the comparative study of popular fictions was\nhardly begun in his time); while of others bare outlines are\nfurnished, because he considered them \"unfit for general\nperusal.\" But his work, even as it is, has probably never been\n\"generally\" read, and he seems to have had somewhat vague notions\nof \"propriety,\" to judge by his translations from the Arabic and\nPersian. A complete English rendering of the \"Bah\u00e1r-i-D\u00e1nish\"\nwould be welcomed by all interested in the history of fiction.\n\n\n\n\n         THE FULLER, HIS WIFE AND THE TROOPER, p. 329.\n\n\n\nThe trick played on the silly fuller of dressing him up as a\nTurkish soldier resembles that of the Three Deceitful Women who\nfound a gold ring in the public bath, as related in the Persian\nstory-book, \"Shamsa \u00fa Kuhkuha:\"\n\nWhen the wife of the superintendent of police was apprised that\nher turn had come, she revolved and meditated for some time what\ntrick she was to play off on her lord, and after having come to a\nconclusion she said one evening to him, \"To-morrow I wish that we\nshould both enjoy ourselves at home without interruptions, and I\nmean to prepare some cakes.\" He replied, \"Very well, my dear; I\nhave also longed for such an occasion.\" The lady had a servant\nwho was very obedient and always covered with the mantle of\nattachment to her. The next morning she called this youth and\nsaid to him, \"I have long contemplated the hyacinth grove of thy\nsymmetrical stature; and I know that thou travellest constancy\nand faithfully on the road of compliance with all my wishes, and\nthat thou seekest to serve me. I have a little business which I\nwish thee to do for me.\" The servant answered, \"I shall be happy\nto comply. Then the lady gave him a thousand dinars and said, \"Go\nto the convent which is in our vicinity; give this money to one\nof the kalandars there and say to him, 'A prisoner whom the Am\u00edr\nhad surrendered to the police has escaped last night. He closely\nresembles thee, and as the superintendent of the police is unable\nto account to the Am\u00edr, he has sent a man to take thee instead of\nthe escaped criminal. I have compassion for thee and mean to\nrescue thee. Take this sum of money; give me thy dress; and flee\nfrom the town; for if thou remainest in it till the morning thou\nwilt be subjected to torture and wilt lose thy life.'\" The\nservant acted as he was bid, and brought the garments to his\nmistress. When it was morning she said to her husband, \"I know\nyou have long wished to eat sweetmeats, and I shall make some\nto-day.\" He answered, \"Very well.\" His wife made all her\npreparations and commenced to bake the sweetmeats. He said to\nher, \"Last night a theft was committed in a certain place, and I\nsat up late to extort confessions; and as I have spent a\nsleepless night, I feel tired and wish to repose a little.\" The\nlady replied, \"Very well.\"\n\nAccordingly the superintendent of the police reclined on the\npillow of rest; and when the sweetmeat was ready his wife took a\nlittle and putting an opiate into it she handed it to him,\nsaying, \"How long will you sleep? To-day is a day of feasting and\npleasure, not of sleep and laziness. Lift up your head and see\nwhether I have made the sweets according to your taste.\" He\nraised his head, swallowed a piece of the hot cake and lay down\nagain. The morsel was still in his throat when consciousness left\nand a deep sleep overwhelmed him. His wife immediately undressed\nhim and put on him the garments of the kalandar. The servant\nshaved his head and made some tattoo marks on his body. When the\nnight set in the lady called her servant and said, \"Hyacinth, be\nkind enough to take the superintendent on thy back, and carry him\nto the convent instead of that kalandar, and if he wishes to\nreturn to the house in the morning, do not let him.\" The servant\nobeyed. Towards dawn the superintendent recovered his senses a\nlittle; but as the opiate had made his palate very bitter, he\nbecame extremely thirsty. He fancied that he was in his own\nhouse, and so he exclaimed, \"Narcissus, bring water.\" The\nkalandars awoke from sleep, and after hearing several shouts of\nthis kind, they concluded that he was under the influence of\nbang, and said, \"Poor fellow! the narcissus is in the garden;\nthis is the convent of sufferers, and there are green garments\nenough here. Arise and sober thyself, for the morning and\nharbinger of benefits as well as of the acquisition of the\nvictuals for subsistence is approaching.\" When the superintendent\nheard these words he thought they were a dream, for he had not\nyet fully recovered his senses. He sat quietly, but was amazed on\nbeholding the walls and ceiling of the convent: he got up, looked\nat the clothes in which he was dressed and at the marks tattooed\non his body, and began to doubt whether he was awake or asleep.\nHe washed his face, and perceived that the caravan of his\nmustachios had likewise departed from the plain of his\ncountenance.\n\nIn this state of perplexity he went out of the convent and\nproceeded to his house. There his wife, with her male and female\nservants, was expecting his arrival. He approached the house and\nplaced his hand on the knocker of the door, but was received by\nHyacinth, who said, \"Kalandar, whom seekest thou?\" The\nsuperintendent rejoined, \"I want to enter the house.\" Hyacinth\ncontinued, \"Thou hast to-day evidently taken thy morning draught\nof bang earlier and more copiously than usual, since thou hast\nfoolishly mistaken the road to thy convent. Depart! This is not a\nplace in which vagabond kalandars are harboured. This is the\npalace of the superintendent of the police. and if the symurgh\nlooks with incivility from the fastness of the west of Mount K\u00e1f\nat this place, the wings of its impertinence will at once become\nsinged.\" The superintendent said, \"What nonsense art thou\nspeaking? Go out of my way, for I do not relish thy imbecile\nprattle.\" But when he wanted to enter, Hyacinth struck him with a\nbludgeon on the shoulder, which the superintendent returned with\na box on the ear, and both began to wrestle together. At that\nmoment the lady and her maid-servants rushed forth from the rear\nand assailed him with sticks and stones, shouting, \"This kalandar\nwishes in plain daylight to force his way into the house of the\nsuperintendent. What a pity that the superintendent is sick, or\nelse this crime would have to be expiated on the gallows!\" In the\nmeantime all the neighbours assembled, and on seeing the\nshameless kalandar's proceedings they cried, \"Look at that\nimpudent kalandar who wants forcibly to enter the house of the\nsuperintendent.\" Ultimately the crowd amounted to more than five\nhundred persons, and the gentleman was put to flight and pursued\nby all the little boys, who pelted him with stones till they\nexpelled him from the town.\n\nAt the distance of three farsangs from the town there was a\nvillage where the superintendent concealed himself in the corner\nof a mosque. During the evenings he went from house to house and\nbegged for food to sustain life, until his mustachios again grew\nand the tattooed scars gradually began to disappear. Whenever\nanyone inquired for the superintendent at his house, he was\ninformed by the servants that the gentleman was sick. After one\nmonth had expired, the grief of separation and the misery of his\ncondition had again driven him back to the city. He went to the\nconvent because fear hindered him from going to the house. His\nwife happened one day to catch a glimpse of him from her window,\nand perceived him sitting in the same dress with a company of\nkalandars. She felt compassion for him, called the servant and\nsaid, \"The superintendent has had enough of this!\" She made a\nloaf of bread and put some opiate into it, and said, \"When the\nkalandars are asleep, you must go and place this loaf under the\npillow of the superintendent.\" The servant obeyed, and when the\ngentleman awoke in the middle of the night he was surprised to\nfind the loaf. He fancied that when his companions had during the\nnight returned from begging, they had placed it there, and so he\nate some of it. During the same night the servant went there by\nthe command of the lady, took his master on his back and carried\nhim home. When it was morning, the lady took off the kalandar's\nclothes from her husband and dressed him in his own garments, and\nbegan to make sweetmeats as on the former occasion. After some\ntime he began to move, and his wife exclaimed, \"O superintendent,\ndo not sleep so much. I have told you that we shall spend this\nday in joy and pleasure, and it was not fair of you to pass the\ntime in this lazy way. Lift up your head and see what beautiful\nsweetmeats I have baked for you.\" When he opened his eyes, and\nsaw himself dressed in his own clothes and at home, the rosebush\nof his amazement again brought forth the flowers of astonishment,\nand he said, \"God be praised! What has happened to me?\" He sat\nup, and exclaimed, \"Wife, things have happened to me which I can\nscarcely describe.\" She replied, \"From the uneasy motions which\nyou have made in your sleep, it appears you must have had\nextraordinary dreams.\" \"Dreams, forsooth,\" said he, \"since the\nmoment I lay down I have experienced the most strange\nadventures.\" \"Certainly,\" rejoined the lady, \"last night you have\nbeen eating food disagreeing with your constitution, and to-day\nthe vapours of it have ascended into your brains, and have caused\nyou all this distress.\" The superintendent said, \"Yes last night\nwe went to a party in the house of Serjeant Bahman, and there was\nroasted pillau, of which I ate somewhat more than usual, and the\nvapour of it has occasioned me all this trouble.\"[FN#597]\n\nStrikingly similar to this story is the trick of the first lady\non her husband in the \"Fabliau des Trois Dames qui trouverent un\nAnel.\" Having made him drunk, she causes his head to be shaved,\ndresses him in the habit of a monk, and carries him, assisted by\nher lover, to the entrance of a convent. When he awakes and sees\nhimself thus transformed he imagines that God by a miraculous\nexercise of His grace had called him to the monastic life. He\npresents himself before the abbot and requests to be received\namong the brethren. The lady hastens to the convent in\nwell-feigned despair, and is exhorted to be resigned and to\ncongratulate her husband on the saintly vow he has taken. \"Many a\ngood man, ' says the poet, \"has been betrayed by woman and by her\nharlotry. This one became a monk in the abbey, where he abode a\nvery long time. Wherefore, I counsel all people who hear this\nstory told, that they ought not to trust in their wives, nor in\ntheir households, if they have not first proved that they are\nfull of virtues. Many a man has been deceived by women and by\ntheir treachery. This one became monk against right, who would\nnever have been such in his life, if his wife had not deceived\nhim.''[FN#598]\n\nThe second lady's trick in the fabliau is a very close parallel\nto the story in The Nights, vol. v. p. 96.[FN#599] She had for\ndinner on a Friday some salted and smoked eels, which her husband\nbade her cook, but there was no fire in the house. Under the\npretext of going to have them cooked at a neighbour's fire she\ngoes out and finds her lover, at whose house she remains a whole\nweek. On the following Friday, at the hour of dinner, she enters\na neighbour's house and asks leave to cook the eels, saying that\nher husband is angry with her for having no fire, and that she\ndid not dare to go back, lest he should take off her head. As\nsoon as the eels are cooked she carries them piping hot to her\nown house. The husband asks her where she has been for eight\ndays, and commences to beat her. She cries for help and the\nneighbours come in, and amongst them the one at whose fire the\neels had been cooked, who swears that the wife had only just left\nher house and ridicules the husband for his assertion that she\nhad been away a whole week. The husband gets into a great rage\nand is locked up for a madman.\n\nThe device of the third lady seems a reflection of the\n\"Elopement,\" but without the underground tunnel between the\nhouses of the wife and the lover. The lady proposes to her lover\nto marry him, and he believes that she is only jesting, seeing\nthat she is already married, but she assures him that she is\nquite in earnest, and even undertakes that her husband will\nconsent. The lover is to come for her husband and take him to the\nhouse of Dan Eustace, where he has a fair niece, whom the lover\nis to pretend he wishes to espouse, if he will give her to him.\nThe wife will go thither, and she will have done her business\nwith Eustace before they arrive. Her husband cannot but believe\nthat he has left her at home, and she will be so apparelled that\nhe cannot recognise her. This plan is accordingly carried out.\nThe lover asks the husband for the hand of his niece in marriage,\nto which he joyously consents, and without knowing it makes a\npresent of his own wife. \"All his life long the lover possessed\nher, because the husband gave and did not lend her; nor could he\never get her back.\"\n\nLe Grand mentions that this fabliau is told at great length in\nthe tales of the Sieur d'Ouville, tome iv. p. 255. In the\n\"Faceti\u00e6 Bebelian\u00e6,\" p. 86, three women wager which of them will\nplay the best trick on her husband. One causes him to believe he\nis a monk, and he goes and sings mass, the second husband\nbelieved himself to be dead, and allows himself to be carried to\nthat mass on a bier; and the third sings in it quite naked.\n(There is a very similar story in Campbell's \"Popular Tales of\nthe West Highlands.\") It is also found, says Le Grand, in the\n\"Convivales Sermones,\" tome i. p. 200, in the \"Delices de\nVerboquet,\" p. 166; and in the Faceti\u00e6 of Lod. Dom\u00e9nichi, p. 172.\nIn the \"Contes pour Rire,\" p. 197, three women find a diamond, and\nthe arbiter whom they select promises it, as in the fabliau, to\nher who concocts the best device for deceiving her husband, but\ntheir ruses are different.\n\n\n\nEnd of Supplemental Nights Volume 2.\n\n\n\n\n\n\n                   Arabian Nights, Volume 12\n                           Footnotes\n\n\n\n[FN#1] Bresi. Edit., vol. xi. pp. 321-99, Nights dccccxxx-xl.\n\n[FN#2] Arab. \"Ikl\u00edm\" from the Gr. {kl\u00edma}, often used as amongst\nus (e.g. \"other climes\") for land.\n\n[FN#3] Bibars whose name is still famous and mostly pronounced\n\"Baybars,\" the fourth of the Baharite Mamelukes whom I would call\nthe \"Soldans.\" Originally a slave of Al-S\u00e1lih, seventh of the\nAyyubites, he rose to power by the normal process, murdering his\npredecessor, in A. D. 1260; and he pushed his conquests from\nSyria to Armenia. In his day \"Saint\" Louis died before Tunis (A.\nD. 1270).\n\n[FN#4] There are sundry S\u00e1hils or shore-lands. \"Sahil Misr\" is\nthe River-side of Cairo often extended to the whole of Lower\nEgypt (vol. i. 290): here it means the lowlands of Palestine once\nthe abode of the noble Philistines; and lastly the term extends\nto the sea-board of Zanzibar, where, however, it is mostly used\nin the plur. \"Saw\u00e1hil\"=the Shores.\n\n[FN#5] Arab. \"Samm\u00e1r\" (from Samar, = conversatio nocturna), = the\nstory-teller who in camp or house whiles away the evening hours.\n\n[FN#6] \"Flag of the Faith:\" Sanjar in old Persian=a Prince, a\nKing.\n\n[FN#7] \"Aider of the Faith.\"\n\n[FN#8] These policemen's tales present a curious contrast with\nthe detective stories of M. Gaboriau and his host of imitators.\nIn the East the police, like the old Bow Street runners, were and\nare still recruited principally amongst the criminal classes on\nthe principle of \"Set a thief,\" &c. We have seen that the\nBarmecide Wazirs of Baghdad \"anticipated Fourier's doctrine of\nthe passionel treatment of lawless inclinations,\" and employed as\nsubordinate officers, under the Wali or Prefect of Police,\naccomplished villains like Ahmad al-Danaf (vol. iv. 75), Hasan\nShuuman and Mercury Ali (ibid.) and even women (Dalilah the\nCrafty) to coerce and checkmate their former comrades. Moreover a\ngird at the police is always acceptable, not only to a\ncoffee-house audience, but even to a more educated crowd; witness\nthe treatment of the \"Charley\" and the \"Bobby\" in our truly\nEnglish pantomimes.\n\n[FN#9] i.e. the Chief of Police, as the sequel shows.\n\n[FN#10] About \u00a34.\n\n[FN#11] i.e. of the worlds visible and invisible.\n\n[FN#12] Arab. \"Mukaddam:\" see vol. iv, 42.\n\n[FN#13] \"Faithful of Command;\" it may be a title as well as a P.\nN. For \"Al-Am\u00edn,\" see vol. iv. 261.\n\n[FN#14] i. e. \"What have I to do with, etc.?\" or \"How great is\nthe difference between me and her.\" The phrase is still popular\nin Egypt and Syria; and the interrogative form only intensifies\nit. The student of Egyptian should always try to answer a\nquestion by a question. His labours have been greatly facilitated\nby the conscientious work of my late friend Spitta Bey. I tried\nhard to persuade the late Rogers Bey, whose knowledge of Egyptian\nand Syrian (as opposed to Arabic) was considerable, that a simple\ngrammar of Egyptian was much wanted; he promised to undertake it,\nbut death cut short the design.\n\n[FN#15] Arab. \"Naww\u00e1b,\" plur. of N\u00e1ib (lit. deputies,\nlieutenants)=a Nabob. Till the unhappy English occupation of\nEgypt, the grand old Kil'ah (Citadel) contained the palace of the\nPasha and the lodgings and offices of the various officials.\nForeign rulers, if they are wise, should convert it into a fort\nwith batteries commanding the town, like that of Hyderabad, in\nSind.\n\n[FN#16] For this famous and time-honoured building, see vol. i.\n269.\n\n[FN#17] Arab. \"Tamk\u00edn,\" gravity, assurance.\n\n[FN#18] Arab. \" Iy\u00e1l-hu\" lit. his family, a decorous\ncircumlocution for his wives and concubines.\n\n[FN#19] Arab. \"Darb,\" lit. a road; here a large thoroughfare.\n\n[FN#20] When Mohammed Ali Pasha (the \"Great\") began to rule, he\nfound Cairo \"stifled\" with filth, and gave orders that each\nhouseholder, under pain of confiscation, should keep the street\nbefore his house perfectly clean. This was done after some\nexamples had been made and the result was that since that time\nCairo never knew the plague. I am writing at Tangier where a\nMohammed Ali is much wanted.\n\n[FN#21] i.e. Allah forfend!\n\n[FN#22] Arab. \"Mustauda'\" = a strong place where goods are\ndeposited and left in charge.\n\n[FN#23] Because, if she came to grief, the people of the street,\nand especially those of the adjoining houses would get into\ntrouble. Hence in Moslem cities, like Damascus and Fez, the H\u00e1r\u00e1t\nor quarters are closed at night with strong wooden doors, and the\nguards will not open them except by means of a silver key.\nMohammed Ali abolished this inconvenience, but fined and\nimprisoned all night-walkers who carried no lanterns. See\nPilgrimage, vol. i. 173,\n\n[FN#24] As Kazi of the quarter he was ex-officio guardian of the\norphans and their property, and liable to severe punishment\n(unless he could pay for the luxury) in case of fraud or neglect.\n\n[FN#25] Altogether six thousand dinars = \u00a33000. This sentence is\nborrowed from the sequel and necessary to make the sense clear.\n\n[FN#26] i.e. \"I am going at once to complain of thee before the\nking unless thou give me due satisfaction by restoring the money\nand finding the thief.\"\n\n[FN#27] The Practice (of the Prophet) and the Holy Law (Koranic):\nsee vols. v. 36, 167 and i. 169.\n\n[FN#28] In the corrupt text \"Who knew me not;\" thus spoiling the\npoint.\n\n[FN#29] Arab. \"Maut Ahmar\" = violent or bloody death. For the\nvarious  deaths, see vol. vi. 250.\n\n[FN#30] i.e. for lack of sleep.\n\n[FN#31] i.e. of the Kazi.\n\n[FN#32] Arab. \"Mub\u00e1h,\" in the theologic sense, an action which is\nnot sinful (har\u00e1m) or quasi-sinful (makr\u00fah); vulgarly \"permitted,\nallowed\"; so Shahrazad \"ceased to say her say permitted\" (by\nShahryar).\n\n[FN#33] Arab. \"Y\u00e1 Khawand\"; see vol. vii. 315.\n\n[FN#34] i.e. we both make different statements equally credible,\nbut without proof, and the case will go against me, because thou\nart the greater man.\n\n[FN#35] Arab. \"Irtiy\u00e1d\" = seeking a place where to stale, soft and\nsloping, so that the urine spray may not defile the dress. All\nthis in one word!\n\n[FN#36] Arab. \"Bah\u00e1r,\" the red buphthalmus sylvester often used\nfor such comparisons. In Algeria it is called 'Ar\u00e1wah: see the\nJardin Parfum\u00e9, p. 245, note 144.\n\n[FN#37] i.e. parties.\n\n[FN#38] i.e. amongst men.\n\n[FN#39] Almost as neat as \"o\u00f9 sont les neiges d'autan?\"\n\n[FN#40] Arab. \" \u00c1d\u00ed,\" one transgressing, an enemy, a scoundrel.\n\n[FN#41] It was probably stuck in the ground like an amphora.\n\n[FN#42] i.e. hush up the matter.\n\n[FN#43] In Egypt; the former being the Eastern of the Seven\nProvinces extending to the Pelusium branch, and the latter to the\nCanobic. The \"Bar\u00e1ri\" or deserts, i.e. grounds not watered by the\nNile, lie scattered between the two and both are bounded South by\nthe Kal\u00fab\u00edyah Province and Middle Egypt.\n\n [FN#44] i.e. a man ready of wit and immediate of action, as\nopposed to his name Al-Atwash = one notable for levity of mind.\n\n[FN#45] The negative is emphatic, \"I certainly saw a Jew,\" etc.\n\n[FN#46] The \"Irish bull\" is in the text; justified by--\n\n          They hand-in-hand, with wand'ring steps and slow\n          Through Eden took their solitary way,\n\n[FN#47] As we should say, \"There are good pickings to be had out\nof this job.\" Even in the last generation a Jew or a Christian\nintriguing with an Egyptian or Syrian Moslemah would be offered\nthe choice of death or Al-Islam. The Wali dared not break open\nthe door because he was not sure of his game.\n\n[FN#48] The Jew rose seemingly to fetch his valuables and ran\naway, thus leaving the Wali no proof that he had been there in\nMoslem law which demands ocular testimony, rejects circumstantial\nevidence and ignores such partial witnesses as the policeman who\naccompanied his Chief. This I have before explained.\n\n[FN#49] Arab. \"Raba',\" lit. = spring-quarters. See Marba', iii. 79.\n\n[FN#50] Arab. \"Ni'am,\" an exception to the Abb\u00e9 Sicard's rule.\n\"La consonne N est l'expression naturelle du doute chez toutes\nles nations, par ce que le son que rend la touche nasale, quand\nl'homme incertain examine s'il fera ce qu'on lui demande; ainsi\nNE ON, NE OT, NE EC, NE IL, d'o\u00f9 l'on a fait non, not, nec, nil.\n\n[FN#51] For this \"Hal\u00e1wat al-Mift\u00e1h,\" or sweetmeat of the\nkey-money, the French denier a Dieu, Old English \"God's penny,\"\nsee vol. vii. 212, and Pilgrimage i. 62.\n\n[FN#52] Showing that car. cop. had taken place. Here we find the\nirregular use of the inn, perpetuated in not a few of the monster\nhotels throughout Europe.\n\n[FN#53] For its rules and right performance see vol. vi. 199.\n\n[FN#54] i.e. the \"Basil(issa),\" mostly a servile name, see vol.\ni. 19.\n\n[FN#55] Arab. \"La'alla,\" used to express the hope or expectation\nof some event of possible occurrence; thus distinguished from\n\"Layta\" --Would heaven! utinam! O si! etc.-- expressing desire or\nvolition.\n\n[FN#56] Arab. \"Bal\u00e1t,\" in Cairo the flat slabs of limestone and\nsandstone brought from the Turah quarries, which supplied stone\nfor the J\u00edzah Pyramids.\n\n[FN#57] Arab. \"Y\u00e1 Mu'arras!\" here = O fool and disreputable; see\nvol. i. 338.\n\n[FN#58] These unfortunates in hot climates enjoy nothing so much\nas throwing off the clothes which burn their feverish skins: see\nPilgrimage iii. 385. Hence the boys of Eastern cities, who are\nperfect imps and flibbertigibbets, always raise the cry \"Majn\u00fan\"\nwhen they see a man naked whose sanctity does not account for his\nnudity.\n\n[FN#59] Arab. \"Daur al-K\u00e1'ah\" = the round opening made in the\nceiling for light and ventilation.\n\n[FN#60] Arab. \"La-nakhsifanna\" with the emphatic termination\ncalled by grammarians \"N\u00fan al-taakid\"--the N of injunction. Here\nit is the reduplicated form, the Nun al-Sak\u00edlah or heavy N. The\naddition of L\u00e1 (not) e.g. \"L\u00e1 yazrabanna\" = let him certainly not\nstrike answers to the intensive or corroborative negative of the\nGreek effected by two negations or even more. In Arabic as in\nLatin and English two negatives make an affirmative.\n\n[FN#61] Parturition and death in warm climates, especially the\ndamp-hot like Egypt are easy compared with both processes in the\ntemperates of Europe. This is noticed by every traveller. Hence\nprobably Easterns have never studied the artificial Euthanasia\nwhich is now appearing in literature. See p. 143 \"My Path to\nAtheism,\" by Annie Besant, London: Freethought Publishing\nCompany, 28, Stonecutter Street, E. C., 1877, based upon the\nUtopia of the highly religious Thomas Moore. Also \"Essay on\nEuthanasia,\" by P. D. Williams, Jun., and Mr. Tollemache in the\n\"Nineteenth Century.\"\n\n[FN#62] i.e. he whose turn it is to sit on the bench outside the\npolice-office in readiness for emergencies.\n\n[FN#63] Arab. \"'Ud\u00fal\" (plur. of '\u00c1dil), gen. men of good repute,\nqualified as witnesses in the law-court, see vol. iv. 271. It is\nalso used (as below) for the Kazi's Assessors.\n\n[FN#64] About \u00a380.\n\n[FN#65] Arab. \"Kit\u00e1b\" = book, written bond. This officiousness of\nthe neighbours is thoroughly justified by Moslem custom; and the\nsame scene would take place in this our day. Like the Hind\u00fa's,\nbut in a minor degree, the Moslem's neighbours form a volunteer\npolice which oversees his every action. In the case of the Hind\u00fa\nthis is required by the exigencies of caste, an admirable\ninstitution much bedevilled by ignorant Mlenchhas, and if\n\"dynamiting\" become the fashion in England, as it threatens to\nbecome, we shall be obliged to establish \"Vigilance Committees\"\nwhich will be as inquisitorial as caste\n\n[FN#66] e.g. writing The contract of A. with B., daughter of\nSuch-an-one, etc.\n\n[FN#67] Arab. \"Hujjat,\" which may also mean an excuse.\n\n[FN#68] The last clause is supplied by Mr. Payne to stop a gap in\nthe broken text.\n\n[FN#69] The text idiotically says \"To the King.\"\n\n[FN#70] In the text \"Nahnu\"=we, for I, a common vulgarism in\nEgypt and Syria.\n\n[FN#71] This clause has required extensive trimming; the text\nmaking the Notary write out the contract (which was already\nwritten) in the woman's house.\n\n[FN#72] Arab. \"Husn tadb\u00edr\" = lit. \"beauty of his contrivance.\"\nHusn, like {kal\u00f2s}, pulcher, beau and bello, is applied to moral and\nintellectual qualities as well as to physical and material. Hence\nthe {kal\u00f2 g\u00e9roon} or old gentleman which in Romaic becomes Calogero, a\nmonk.\n\n[FN#73] i.e. that some one told me the following tale.\n\n[FN#74] Arab. \"Mutawall\u00ed\": see vol. i. 259.\n\n[FN#75] i.e. his Moslem neighbours.\n\n[FN#76] In the text is a fearful confusion of genders.\n\n[FN#77] Her object was to sue him for the loss of the pledge and\nto demand fabulous damages.\n\n[FN#78] Arab. \"Ya'tamid\u00fana hud\u00e0-hum\" = purpose the right direction,\na skit at the devotees of her age and sex; and an impudent\ncomment upon the Prefect's address \"O she-devil!\"\n\n[FN#79] The trick has often been played in modern times at fairs,\nshows, etc. Witness the old Joe Miller of the \"Moving Multitude.\"\n\n[FN#80] Apparently meaning the forbidden pleasures of wine and\nwassail, loose talk and tales of women's wiles, a favourite\nsubject with the lewder sort of Moslem.\n\n[FN#81] i.e. women's tricks.\n\n[FN#82] The \"Turkoman\" in the text first comes in afterwards.\n\n[FN#83] Arab. \"K\u00e1sid,\" the old Anglo-lndian \"Cossid\"; see vol.\nvii. 340.\n\n[FN#84] Being a merchant he wore dagger and sword, a safe\npractice as it deters attack and far better than carrying hidden\nweapons, derringers and revolvers which, originating in the\nUnited States, have now been adopted by the most civilised\nnations in Europe.\n\n[FN#85] I have noted (vol. ii. 186, iv. 175) the easy expiation\nof perjury amongst Moslems, an ugly blot in their moral code.\n\n[FN#86] i.e. Enter in the name of Allah.\n\n[FN#87] i.e. Damn your soul for leading me into this danger!\n\n[FN#88] Arab. \"Saff Kamar\u00edy\u00e1t min al-Zuj\u00e1j.\" The Kamar\u00edyah is\nderived by Lane (Introd. M.E.) from Kamar=moon; by Baron Von\nHammer from Khum\u00e1rawayh, second of the Banu-Tul\u00fan dynasty, at the\nend of the ixth century A.D., when stained glass was introduced\ninto Egypt. N.B.--It must date from many centuries before. The\nKamariyah are  glass windows about 2 feet high by 18\ninches wide, placed in a row along the upper part of the\nMashrab\u00edyah or projecting lattice-window, and are formed of small\npanes of brightly-stained glass set in rims of gypsum-plaster,\nthe whole framed in wood. Here the allusion is to the \"Mamrak\" or\ndome-shaped skylight crowning the room. See vol. viii. 156.\n\n[FN#89] i.e. easily arrested them.\n\n[FN#90] The reader will not forget the half-penitent Captain of\nBandits in Gil Blas.\n\n[FN#91] Arab. \"Abt\u00e1l\" = champions, athletes, etc., plur. of Batal,\na brave: so Batalat = a virago. As the root Batala = it was vain, the\nform \"Batt\u00e1l\" may mean either a hero or a bad lot: see vol. viii.\n335; x. 74,75.\n\n[FN#92] Arab. \"Fity\u00e1n;\" plur. of Fat\u00e0; see vol. i, 67.\n\n[FN#93] This was in popular parlance \"adding insult to injury:\"\nthe blackening their faces was a promise of Hell-fire.\n\n[FN#94] Arab. \"Shayyan li 'll\u00e1h!\" lit.=(Give me some) Thing for\n(the love of) Allah. The answer in Egypt. is \"Allah\nya't\u00edk:\"=Allah will give it thee (not I), or, \"Yaftah 'Allah,\"=\nAllah open (to thee the door of subsistence): in Marocco \"Sir f\u00ed\nh\u00e1lik\" (pron. Sirf h\u00e1k)= Go about thy business. In all cities\nthere is a formula which suffices the asker; but the Ghash\u00edm\n(Johny Raw) who ignores it, is pestered only the more by his\nprotestations that \"he left his purse at home,\" etc.\n\n[FN#95] i.e. engaged her for a revel and paid her in advance.\n\n[FN#96] Arab. \"Ras\u00edlah\"=a (she) partner, to accompany her on the\nlute.\n\n[FN#97] Suggesting that they are all thieves who had undergone\nlegal mutilation.\n\n[FN#98] Arab. \"Nuzhat-\u00ed:\" see vol. ii. 81.\n\n[FN#99] Arab. \"Muhattak\u00e1t;\" usually \"with torn veils\" (fem.\nplur.) here \"without veils,\" metaphor. meaning in disgrace, in\ndishonour.\n\n[FN#100] For this reedy Poa, see vol. ii. 18.\n\n[FN#101] I have repeatedly noticed that singing and all music\nare, in religious parlance, \"Makr\u00fah,\" blameable though not\nactually damnable; and that the first step after \"getting\nreligion\" is to forswear them.\n\n[FN#102] i.e. to find the thief or make good the loss.\n\n[FN#103] i.e. the claimants.\n\n[FN#104] Arab. \"Sakiyah:\" see vol. i. 123.\n\n[FN#105] The lower orders of Egypt and Syria are addicted to this\nbear-like attack; so the <DW64>s imitate fighting-rams by butting\nwith their stony heads. Let me remark that when Herodotus (iii.\n12), after Psammenitus' battle of Pelusium in B.C. 524, made the\nremark that the Egyptian crania were hardened by shaving and\ninsolation and the Persians were softened by wearing head-cloths,\nhe tripped in his anthropology. The Iranian skull is naturally\nthin compared with that of the negroid Egyptian and the <DW64>.\n\n[FN#106] Arab. \"Farkalah,\" {phrag\u00e9llion} from flagellum; cattle-whip\nwith leathern thongs. Lane, M.E.; Fleischer Glos. 83-84; Dozy\ns.v.\n\n[FN#107] This clause is supplied to make sense.\n\n[FN#108] i.e. to crucify him by nailing him to an upright board.\n\n[FN#109] i.e. a native of the Hauran, Job's country east of\nDamascus, now a luxuriant waste, haunted only by the plundering\nBadawin and the Druzes of the hills, who are no better; but its\nstretches of ruins and league-long swathes of stone over which\nthe vine was trained, show what it has been and what it will be\nagain when the incubus of Turkish mis-rule shall be removed from\nit. Herr Schuhmacher has lately noted in the Hauran sundry Arab\ntraditions of Job; the village Naw\u00e1, where he lived; the Hammam\n'Ayy\u00fab, where he washed his leprous skin; the Dayr Ayy\u00fab, a\nmonastery said to date from the third century; and the Makan\nAyyub at Al-Mark\u00e1z, where the semi-mythical patriarch and his\nwife are buried. The \"Rock of Job\", covered by a mosque, is a\nbasaltic monolith 7 feet high by 4, and is probably connected\nwith the solar worship of the old Ph\u0153nicians.\n\n[FN#110] This habit \"torquere mero,\" was a favourite with the\nmedi\u00e6val Arabs. Its effect varies greatly with men's characters,\nmaking some open-hearted and communicative, and others more\ncunning and secretive than in the normal state. So far it is an\nexcellent detection of disposition, and many a man who passes off\nwell when sober has shown himself in liquor a rank snob.\nAmong the lower orders it provokes what the Persians call\nBad-mast\u00ed (le vin m\u00e9chant) see Pilgrimage iii. 385.\n\n[FN#111] This mystery is not unfamiliar to the modern\n\"spiritualist;\" and all Eastern tongues have a special term for\nthe mysterious Voice. See vol. i. 142.\n\n[FN#112] Arab. \"Alaykum:\" addressed to a single person. This is\ngenerally explained by the \"Salam\" reaching the ears of Invisible\nControls, and even the Apostle. We find the words cruelly\ndistorted in the Pentamerone of Giambattista Basile (partly\ntranslated by John E. Taylor, London: Bogue, 1848), \"The Prince,\ncoming up to the old woman heard an hundred Licasalemme,\" p. 383.\n\n[FN#113] Arab. \"Al-Zalamah\"; the policeman; see vol. vi. 214.\n\n[FN#114] i.e. in my punishment.\n\n[FN#115] i.e. on Doomsday thou shalt get thy deserts.\n\n[FN#116] i.e. what I could well afford.\n\n[FN#117] Arab. Hirfah=a trade, a guild, a corporation: here the\nofficers of police.\n\n[FN#118] Gen. \"tip-cat\" (vol. ii. 314.) Here it would mean a rude\nform of tables or backgammon, in which the players who throw\ncertain numbers are dubbed Sultan and Wazir, and demean\nthemselves accordingly. A favourite bit of fun with Cairene boys\nof a past generation was to \"make a Pasha;\" and for this\nproceeding, see Pilgrimage, vol. i. 119.\n\n[FN#119] In Marocco there is great difficulty about finding an\nexecutioner who becomes obnoxious to the Th\u00e1r, vendetta or\nblood-revenge. For salting the criminal's head, however, the\nsoldiers seize upon the nearest Jew and compel him to clean out\nthe brain and to prepare it for what is often a long journey.\nHence, according to some, the local name of the Ghetto,\nAl-Mall\u00e1h,=the salting-ground.\n\n[FN#120] Mr. Payne suspects that \"laban,\" milk, esp. artificially\nsoured (see vol. vi, 201), is a clerical error for \"jubn\"=cheese.\nThis may be; but I follow the text as the exaggeration is greater\n\n[FN#121] i.e. in relinquishing his blood-wite for his brother.\n\n[FN#122] The Story-teller, probably to relieve the monotony of\nthe Constables' histories, here returns to the original cadre. We\nmust not forget that in the Bresl. Edit. the Nights are running\non, and that the charming queen is relating the adventure of\nAl-Malik al-Zahir.\n\n[FN#123] Arab. \"Za'amu\"=they opine, they declare, a favourite\nterm with the Bresl. Edit.\n\n[FN#124] Arab. \"Zirtah\" the coarsest of terms for what the French\nnuns prettily termed un sonnet; I find ung sonnet also in Nov.\nii. of the Cent nouvelles Nouvelles. Captain Lockett (p. 32)\nquotes Strepsiades in The Clouds {bront\u00e2 komid\u00ea papp\u00e1x} \"because he cannot\nexpress the bathos of the original (in the Tale of Ja'afar and\nthe old Badawi) without descending to the oracular language of\nGiacoma Rodogina, the engastrymythian prophetess.\" But Sterne was\nby no means so squeamish. The literature of this subject is\nextensive, beginning with \"Peteriana, ou l'art de peter,\" which\ndistinguishes 62 different tones. After dining with a late friend\nen gar\u00e7on we went into his sitting-room and found on the table 13\nbooks and booklets upon the Crepitus Ventris, and there was some\nastonishment as not a few of the party had never seen one.\n\n[FN#125] This tale is a replica of the Cranes of Ibycus. This was\na Rhegium man who when returning to Corinth, his home, was set\nupon by robbers and slain. He cast his dying eyes heavenwards and\nseeing a flight of cranes called upon them to avenge him and this\nthey did by flying over the theatre of Corinth on a day when the\nmurderers were present and one cried out, \"Behold the avengers of\nIbycus!\" Whereupon they were taken and put to death. So says\nPaulus Hieronymus, and the affecting old tale has newly been sung\nin charming verse by Mr. Justin H. McCarthy (\"Serapion.\" London:\nChatto and Windus).\n\n[FN#126] This scene is perfectly true to Badawi life; see my\nPilgrimage iii. 68.\n\n[FN#127] Arab. \"Durr\u00e1j\": so it is rendered in the French\ntranslation of Al-Masudi, vii. 347.\n\n[FN#128] A fair friend found the idea of Destiny in The Nights\nbecome almost a night-mare. Yet here we suddenly alight upon the\ntrue Johnsonian idea that conduct makes fate. Both extremes are\nas usual false. When one man fights a dozen battles unwounded and\nanother falls at the first shot we cannot but acknowledge the\npresence of that mysterious \"luck\" whose laws, now utterly\nunknown to us, may become familiar with the ages. I may note that\nthe idea of an appointed hour beyond which life may not be\nprolonged, is as old as Homer (Il. vi. 487).\n\nThe reader has been told (vol. vii. 135) that \"Kaz\u00e1\" is Fate in a\ngeneral sense, the universal and eternal Decree of Allah, while\n\"Kadar\" is its special and particular application to man's lot,\nthat is Allah's will in bringing forth events at a certain time\nand place. But the former is popularly held to be of two\ncategories, one Kaz\u00e1 al-Muham which admits of modification and\nKaz\u00e1 al-Muhkam, absolute and unchangeable, the doctrine of\nirresistible predestination preached with so much energy by St.\nPaul (Romans ix. 15-24); and all the world over men act upon the\nformer while theoretically holding to the latter. Hence \"Chinese\nGordon\" whose loss to England is greater than even his friends\nsuppose, wrote \"It is a delightful thing to be a fatalist,\"\nmeaning that the Divine direction and pre-ordination of all\nthings saved him so much trouble of forethought and afterthought.\nIn this tenet he was not only a Calvinist but also a Moslem whose\ncontradictory ideas of Fate and Freewill (with responsibility)\nare not only beyond Reason but are contrary to Reason; and\nalthough we may admit the argumentum ad verecundiam, suggesting\nthat there are things above (or below) human intelligence, we are\nnot bound so to do in the case of things which are opposed to the\ncommon sense of mankind. Practically, however, the Moslem\nattitude is to be loud in confessing belief of \"Fate and Fortune\"\nbefore an event happens and after it wisely to console himself\nwith the conviction that in no way could he have escaped the\noccurrence. And the belief that this destiny was in the hands of\nAllah gives him a certain dignity especially in the presence of\ndisease and death which is wanting in his rival religionist the\nChristian. At the same time the fanciful picture of the Turk\nsitting stolidly under a shower of bullets because Fate will not\nfind him out unless it be so written is a freak of fancy rarely\nfound in real life.\n\nThere are four great points of dispute amongst the schoolmen in\nAl-Islam; (1) the Unity and Attributes of Allah, (2) His promises\nand threats, (3) historical as the office of Im\u00e1m; and (4)\nPredestination and the justice thereof. On the latter subject\nopinions range over the whole cycle of possibilities. For\ninstance, the Mu'tazilites, whom the learned Weil makes the\nProtestants and Rationalists of Al-Islam, contend that the word\nof Allah was created in subjecto, ergo, an accident and liable to\nperish, and one of their school, the K\u00e1diriyah ( = having power)\ndenies the existence of Fate and contends that Allah did not\ncreate evil but left man an absolutely free agent. On the other\nhand, the Jabariyah (or Mujabbar = the compelled) is an absolute\nFatalist who believes in the omnipotence of Destiny and deems\nthat all wisdom consists in conforming with its decrees.\nAl-Mas'udi (chapt. cxxvii.) illustrates this by the saying of a\nMoslem philosopher that chess was the invention of a Mu'tazil,\nwhile Nard (backgammon with dice) was that of a Mujabbar proving\nthat play can do nothing against Destiny. Between the two are the\nAshariyah; trimmers whose stand-point is hard to define; they\nwould say, \"Allah creates the power by which man acts, but man\nwills the action,\" and care not to answer the query, \"Who created\nthe will ?\" (See Pocock, Sale and the Dabistan ii. 352). Thus\nSa'adi says in the Gulistan (iii. 2), \"The wise have pronounced\nthat though daily bread be allotted, yet it is so conditionally\nupon using means to acquire it, and although calamity be\npredestined, yet it is right to secure oneself against the\nportals by which it may have access.\" Lastly, not a few doctors\nof Law and Religion hold that Kaza al-Muhkam, however absolute,\nregards only man's after or final state; and upon this subject\nthey are of course as wise as other people, and--no wiser. Lane\nhas treated the Moslem faith in Destiny very ably and fully\n(Arabian Nights, vol. i. pp. 58-61), and he being a man of\nmoderate and orthodox views gives valuable testimony.\n\n[FN#129] Arab. \"Shaykh al-Hujj\u00e1j.\" Some Santon like Hasan al-\nMar\u00e1bit, then invoked by the Meccan pilgrims: see Pilgrimage, i.\n321. It can hardly refer to the famous Hajj\u00e1j bin Y\u00fasuf al-Sakaf\u00ed\n(vol. iv. 3).\n\n[FN#130] Here the Stories of the Sixteen Constables abruptly end,\nafter the fashion of the Bresl. Edit. They are summarily\ndismissed even without the normal \"Bakhsh\u00edsh.\"\n\n[FN#131] Bresl. Edit. vol xi. pp. 400-473 and vol. xii. pp. 4-50,\nNights dccccxli.-dcccclvii. For Kashghar, see vol. i. 255.\n\n[FN#132] Mr. Payne proposes to translate \"'Anbar\" by amber, the\nsemi-fossilised resin much used in modern days, especially in\nTurkey and Somaliland, for bead necklaces. But, as he says, the\nsecond line distinctly alludes to the perfume which is sewn in\nleather and hung about the neck, after the fashion of our ancient\npomanders (pomme d' ambre).\n\n[FN#133] i.e. The Caliph: see vol. i. p. 50.\n\n[FN#134] Arab. \"Adab :\" see vol. i. 132, etc. In Moslem dialects\nwhich borrow more or less from Arabic, \"B\u00ed-adab\u00ed\" = being without Adab, means rudeness, disrespect, \"impertinence\" (in its modern\nsense).\n\n[FN#135] i.e. Isaac of Mosul, the greatest of Arab musicians: see\nvol. iv. 119.\n\n[FN#136] The elder brother of Ja'afar, by no means so genial or\nfitted for a royal frolic. See Terminal Essay.\n\n[FN#137] Ibn Hab\u00edb, a friend of Isaac, and a learned grammarian\nwho lectured at Basrah.\n\n[FN#138] A suburb of Baghdad, mentioned by Al Mas'\u00fadi.\n\n[FN#139] Containing the rooms in which the girl or girls were\nsold. See Pilgrimage i. 87.\n\n[FN#140] Dozy quotes this passage but cannot explain the word\nFaww\u00e1k.\n\n[FN#141] \"A passage has apparently dropped out here. The Khalif\nseems to have gone away without buying, leaving Ishac behind,\nwhereupon the latter was accosted by another slave-girl, who came\nout of a cell in the corridor.\" So says Mr. Payne. vol. ii. 207.\nThe \"raiser of the veil\" means a fitting purchaser.\n\n[FN#142] i.e. \"Choice gift of the Fools,\" a skit upon the girl's\nname \"Tohfat al-Kul\u00fab\" = Choice gift of the Hearts. Her folly\nconsisted in refusing to be sold at a high price, and this is\noften seen in real life. It is a Pundonor amongst good Moslems\nnot to buy a girl and not to sleep with her, even when bought,\nagainst her will.\n\n[FN#143] \"Every one cannot go to Corinth.\" The question makes the\nassertion emphatic.\n\n[FN#144] i.e. The Narrows of the (Dervishes') convent.\n\n[FN#145] Arab. \"Akw\u00e0 min dahni 'l-lauz.\" These unguents have been\nused in the East from time immemorial whilst the last generation\nin England knew nothing of anointing with oil for incipient\nconsumption. A late friend of mine, Dr. Stocks of the Bombay\nEstablishment, and I proposed it as long back as 1845; but in\nthose days it was a far cry from Sind to London.\n\n[FN#146] The sequel will explain why she acted in this way.\n\n[FN#147] i.e. Thou hast made my gold piece (10 shill.) worth only\na doit by thy superiority in the art and mystery of music.\n\n[FN#148] Arab. \"Uadd\u00edki,\" Taadiyah (iid. of Ad\u00e1, he assisted)\nmeans sending, forwarding. In Egypt and Syria we often find the\nform \"Waddi\" for Addi, imperative.\n\n[FN#149] Again \"he\" for \"she\".\n\n[FN#150] i.e. Honey and wine.\n\n[FN#151] i.e. he died.\n\n[FN#152] i.e. if my hand had lost its cunning.\n\n[FN#153] Arab. \"Thiy\u00e1b 'Am\u00fadiyah\": 'Amud = tent prop or column, and\nKhatt 'Am\u00fad = a perpendicular line.\n\n[FN#154] i.e. a choice gift. The Caliph speaks half ironically.\n\"Where's this wonderful present etc?\" So further on when he\ncompares her with the morning.\n\n[FN#155] Again the usual pun upon the name.\n\n[FN#156] Throughout the East this is the action of a servant or a\nslave, practised by freemen only when in danger of life or\nextreme need and therefore humiliating.\n\n[FN#157] It had been thrown down from the Mamrak or small dome\nbuilt over such pavilions for the purpose of light by day and\nventilation by night. See vol. i. 257, where it is called by the\nPersian term \"B\u00e1dhanj.\"\n\n[FN#158] The Nights have more than once applied this patronymic\nto Zubaydah. See vol. viii. 56, 158.\n\n[FN#159] Arab. \"Mutahaddis\u00edn\" = novi homines, upstarts.\n\n[FN#160] i.e. thine auspicious visits.\n\n[FN#161] He being seated on the carpet at the time.\n\n[FN#162] A quotation from Al-Farazdat who had quarrelled with his\nwife Al-How\u00e1r (see the tale in Ibn Khallikan, i. 521), hence \"the\nnaked intercessor\" became proverbial for one who cannot be\nwithstood.\n\n[FN#163] i.e. Choice Gift of the Breasts, that is of hearts, the\ncontinens for the contentum.\n\n[FN#164] Pron. \"Abuttaw\u00e1if,\" the Father of the (Jinn-)tribes. It\nis one of the Moslem Satan's manifold names, alluding to the\nnumber of his servants and worshippers, so far agreeing with that\namiable Christian doctrine, \"Few shall be saved.\"\n\n[FN#165] Mr. Payne supplies this last clause from the sequence.\n\n[FN#166] i.e. \"Let us go,\" with a euphemistic formula to defend\nher from evil influences. Iblis uses the same word to prevent her\nbeing frightened.\n\n[FN#167] Arab. \"Al-Mustar\u00e1h,\" a favourite haunting place of the\nJinn, like the Hamm\u00e1m and other offices for human impurity. For\nits six names Al-Khal\u00e1, Al-Hushsh, Al-Mutawazz\u00e1, Al-Kan\u00edf,\nAl-Mustar\u00e1h, and Mirh\u00e1z, see Al-Mas'udi, chap. cxxvii., and\nShir\u00edshi's commentary to Hariri's 47, Assembly.\n\n[FN#168] Which, in the East, is high and prominent whilst the\ncantle forms a back to the seat and the rider sits as in a baby's\nchair. The object is a firm seat when fighting: \"across country\"\nit is exceedingly dangerous.\n\n[FN#169] In Swedenborg's \"Arcana C\u0153lestia\" we read, \"When man's\ninner sight is opened, which is that of his spirit; then there\nappear the things of another life which cannot be made visible to\nthe bodily sight.\" Also \"Evil spirits, when seen by eyes other\nthan those of their infernal associates, present themselves by\ncorrespondence in the beast (fera) which represents their\nparticular lust and life, in aspect direful and atrocious.\" These\nare the Jinns of Northern Europe.\n\n[FN#170] This exchange of salams was a sign of her being in\nsafety.\n\n[FN#171] Arab. \"Shaw\u00e1hid,\" meaning that heart testifies to heart.\n\n[FN#172] i.e. A live coal, afterwards called Zalzalah, an\nearthquake; see post p. 105. \"Wakh\u00edmah\" = an unhealthy land, and\n\"Shar\u00e1rah\" = a spark.\n\n[FN#173] I need hardly note the inscriptions upon the metal trays\nsold to Europeans. They are usually imitation words so that\ninfidel eyes may not look upon the formul\u00e6 of prayer; and the\nsame is the case with table-cloths, etc., showing a fancy Tohgra\nor Sultanic sign-manual.\n\n[FN#174] i.e. I cannot look at them long.\n\n[FN#175] Evidently a diabolical way of clapping his hands in\napplause. This description of the Foul Fiend has an element of\ngrotesqueness which is rather Christian than Moslem.\n\n[FN#176] Arab. \"Rikk\u00ed al-Saut,\" which may also mean either \"lower\nthy voice,\" or \"change the air to one less touching.\"\n\n[FN#177] \"Your\" for \"thy.\"\n\n[FN#178] i.e. written on the \"Guarded Tablet\" from all eternity.\n\n[FN#179] Arab. \"Al-'Urs w'al-Tuh\u00far\" which can only mean, \"the\nwedding (which does not drop out of the tale) and the\ncircumcision.\"\n\n[FN#180] I here propose to consider at some length this curious\ncustom which has prevailed amongst so many widely separated\nraces. Its object has been noted (vol. v. 209), viz. to diminish\nthe sensibility of the glans, no longer lubricated with prostatic\nlymph; thus the part is hardened against injury and disease and\nits work in coition is prolonged. On the other hand, \"pr\u00e6putium\nin coitu voluptatem (of the woman) auget, unde femina pr\u00e6putiatis\nconcubitum malunt quam cum Turcis ac Jud\u00e6is \" says Dimerbroeck\n(Anatomie). I vehemently doubt the fact. Circumcision was\ndoubtless practised from ages immemorial by the peoples of\nCentral Africa, and Welcker found traces of it in a mummy of the\nxvith century B.C. The Jews borrowed it from the Egyptian\npriesthood and made it a manner of sacrament, \"uncircumcised\"\nbeing = \"unbaptised,\" that is, barbarian, heretic; it was a seal of\nreconciliation, a sign of alliance between the Creator and the\nChosen People, a token of nationality imposed upon the body\npolitic. Thus it became a cruel and odious protestation against\nthe brotherhood of man, and the cosmopolitan Romans derided the\nverp\u00e6 ac verpi. The Jews also used the term figuratively as the\n\"circumcision of fruits\" (Lev. xix. 23), and of the heart (Deut.\nx. 16); and the old law gives copious historical details of its\norigin and continuance. Abraham first amputated his horny\n\"calotte\" at \u00e6t. 99, and did the same for his son and household\n(Gen. xvii. 24-27). The rite caused a separation between Moses\nand his wife (Exod. iv. 25). It was suspended during the Desert\nWanderings and was resumed by Joshua (v. 3-7), who cut off two\ntons weight of prepuces. The latter became, like the scalps of\nthe Scythians and the North-American \"Indians,\" trophies of\nvictory; Saul promised his daughter Michol to David for a dowry\nof one hundred, and the son-in-law brought double tale.\n\nAmongst the early Christians opinions concerning the rite\ndiffered. Although the Founder of Christianity was circumcised,\nSt. Paul, who aimed at a cosmopolitan faith discouraged it in the\nphysical phase. St. Augustine still sustained that the rite\nremoved original sin despite the Fathers who preceded and\nfollowed him, Justus, Tertullian, Ambrose and others. But it\ngradually lapsed into desuetude and was preserved only in the\noutlying regions. Paulus Jovius and Munster found it practised in\nAbyssinia, but as a mark of nobility confined to the descendants\nof \"Nicaules, queen of Sheba.\" The Abyssinians still follow the\nJews in performing the rite within eight days after the birth and\nbaptise boys after forty and girls after eighty days. When a\ncircumcised man became a Jew he was bled before three witnesses\nat the place where the prepuce had been cut off and this was\ncalled the \"Blood of alliance.\" Apostate Jews effaced the sign of\ncircumcision: so in 1 Matt. i. 16, fecerunt sibi pr\u00e6putia et\nrecesserunt a Testamento Sancto. Thus making prepuces was called\nby the Hebrews Meshookim = recutiti{s}, and there is an allusion to\nit in 1 Cor. vii. 18, 19, {m\u00e8 episp\u00e1sthai} (Farrar, Paul ii. 70). St.\nJerome and others deny the possibility; but Mirabeau (Akropodie)\nrelates how Father Conning by liniments of oil, suspending\nweights, and wearing the virga in a box gained in 43 days 7\u00bc\nlines. The process is still practiced by Armenians and other\nChristians who, compelled to Islamise, wish to return to\nChristianity. I cannot however find a similar artifice applied to\na circumcised clitoris. The simplest form of circumcision is mere\namputation of the prepuce and I have noted (vol. v. 209) the\ndifference between the Moslem and the Jewish rite, the latter\naccording to some being supposed to heal in kindlier way. But the\nvarieties of circumcision are immense. Probably none is more\nterrible than that practiced in the Province Al-As\u00edr, the old\nOphir, lying south of Al-Hij\u00e1z, where it is called Salkh,\nlit. = scarification. The patient, usually from ten to twelve years\nold, is placed upon raised ground holding m right hand a spear,\nwhose heel rests upon his foot and whose point shows every\ntremour of the nerves. The tribe stands about him to pass\njudgment on his fortitude, and the barber performs the operation\nwith the Jumbiyah-dagger, sharp as a razor. First he makes a\nshallow cut, severing only the skin across the belly immediately\nbelow the navel, and similar incisions down each groin; then he\ntears off the epidermis from the cuts downwards and flays the\ntesticles and the penis, ending with amputation of the foreskin.\nMeanwhile the spear must not tremble and in some clans the lad\nholds a dagger over the back of the stooping barber, crying, \"Cut\nand fear not!\" When the ordeal is over, he exclaims, \"Allaho\nAkbar!\" and attempts to walk towards the tents soon falling for\npain and nervous exhaustion, but the more steps he takes the more\napplause he gains. He is dieted with camel's milk, the wound is\ntreated with salt and turmeric, and the chances in his favour are\nabout ten to one. No body-pile or pecten ever grows upon the\nexcoriated part which preserves through life a livid ashen hue.\nWhilst Mohammed Ali Pasha occupied the province he forbade\n\"scarification\" under pain of impalement, but it was resumed the\nmoment he left Al-Asir. In Africa not only is circumcision\nindigenous, the operation varies more or less in the different\ntribes. In Dahome it is termed Addagwibi, and is performed\nbetween the twelfth and twentieth year. The rough operation is\nmade peculiar by a double cut above and below; the prepuce being\ntreated in the Moslem, not the Jewish fashion (loc. cit.). Heated\nsand is applied as a styptic and the patient is dieted with\nginger-soup and warm drinks of ginger-water, pork being\nespecially forbidden. The Fantis of the Gold Coast circumcise in\nsacred places, e.g., at Accra on a Fetish rock rising from the\nsea. The peoples of Sennaar, Taka, Masawwah and the adjacent\nregions follow the Abyssinian custom. The barbarous Bissagos and\nFellups of North Western Guinea make cuts on the prepuce without\namputating it; while the Baquens and Papels circumcise like\nMoslems. The blacks of Loango are all \"verp\u00e6,\" otherwise they\nwould be rejected by the women. The Bantu or Caffre tribes are\ncircumcised between the ages of fifteen and eighteen; the \"Fetish\nboys,\" as we call them, are chalked white and wear only grass\nbelts; they live outside the villages in special houses under an\nold \"medicine-man,\" who teaches them not only virile arts but\nalso to rob and fight. The \"man-making\" may last five months and\nends in f\u00eates and dances: the patients are washed in the river,\nthey burn down their quarters, take new names, and become adults,\ndonning a kind of straw thimble over the prepuce. In Madagascar\nthree several cuts are made causing much suffering to the\nchildren; and the nearest male relative swallows the prepuce. The\nPolynesians circumcise when childhood ends and thus consecrate\nthe fecundating organ to the Deity. In Tahiti the operation is\nperformed by the priest, and in Tonga only the priest is exempt.\nThe Maories on the other hand fasten the prepuce over the glans,\nand the women of the Marquesas Islands have shown great cruelty\nto shipwrecked sailors who expose the glans. Almost all the known\nAustralian tribes circumcise after some fashion: Bennett supposes\nthe rite to have been borrowed from the Malays, while Gason\nenumerates the \"Kurrawellie wonkauna\" among the five mutilations\nof puberty. Leichhardt found circumcision about the Gulf of\nCarpentaria and in the river-valleys of the Robinson and\nMacarthur: others observed it on the Southern Coast and among\nthe savages of Perth, where it is noticed by Salvado. James\nDawson tells us \"Circumciduntur pueri,\" etc., in Western\nVictoria. Brough Smyth, who supposes the object is to limit\npopulation (?), describes on the Western Coast and in Central\nAustralia the \"Corrobery\"-dance and the operation performed with\na quartz-flake. Teichelmann details the rite in Southern\nAustralia where the assistants--all men, women, and children\nbeing driven away--form a \"manner of human altar\" upon which the\nyouth is laid for circumcision. He then receives the normal two\nnames, public and secret, and is initiated into the mysteries\nproper for men. The Australians also for Malthusian reasons\nproduce an artificial hypospadias, while the Karens of New Guinea\nonly split the prepuce longitudinally (Cosmos p. 369, Oct. 1876);\nthe indigens of Port Lincoln on the West Coast split the virga:--\nFenditur usque ad urethram a parte infera penis between the ages\nof twelve and fourteen, says E. J. Eyre in 1845. Missionary\nSch\u00fcrmann declares that they open the urethra. Gason describes in\nthe Dieyerie tribe the operation 'Kulpi\" which is performed when\nthe beard is long enough for tying. The member is placed upon a\nslab of tree-bark, the urethra is incised with a quartz-flake\nmounted in a gum handle and a splinter of bark is inserted to\nkeep the cut open. These men may appear naked before women who\nexpect others to clothe themselves. Miklucho Maclay calls it\n\"Mika\" in Central Australia: he was told by a squatter that of\nthree hundred men only three or four had the member intact in\norder to get children, and that in one tribe the female births\ngreatly outnumbered the male. Those mutilated also marry: when\nmaking water they sit like women slightly raising the penis, this\nin coition becomes flat and broad and the semen does not enter\nthe matrix. The explorer believes that the deed of kind is more\nquickly done (?). Circumcision was also known to the New World.\nHerrera relates that certain Mexicans cut off the ears and\nprepuce of the newly born child, causing many to die. The Jews\ndid not adopt the female circumcision of Egypt described by Huet\non Origen--\"Circumcisio feminarum fit resectione {t\u00eas nymph\u00eas}           (sive\nclitoridis) qu\u00e6 pars in Australium mulieribus ita crescit ut\nferro est co\u00ebrcenda.\" Here we have the normal confusion between\nexcision of the nymph\u00e6 (usually for fibulation) and circumcision\nof the clitoris. Bruce notices this clitoridectomy among the\nAbyssinians. Werne describes the excision on the Upper White Nile\nand I have noted the complicated operation among the Somali\ntribes. Girls in Dahome are circumcised by ancient sages femmes,\nand a woman in the natural state would be derided by every one\n(See my Mission to Dahome, ii. 159) The Australians cut out the\nclitoris, and as I have noted elsewhere extirpate the ovary for\nMalthusian purposes (Journ Anthrop. Inst., vol. viii. of 1884).\n\n[FN#181] Arab. \"Kayraw\u00e1n\" which is still the common name for\ncurlew; the peewit and plover being called (onomatopoetically)\n\"B\u00edbat\" and in Marocco Yah\u00fadi, certain impious Jews having been turned into the Vanellus Cristatus which still wears the black skullcap of the Hebrews.\n\n[FN#182] Arab. \"Saw\u00e1ki,\" the leats which irrigate the ground and are opened and closed with the foot.\n\n[FN#183] The eighth (in altitude) of the many-storied Heavens.\n\n[FN#184] Arab. \"Ihramat li al-Sal\u00e1t,\" i.e. she pronounced the\nformula of Intention (Niyat) without which prayer is not valid, ending with Allaho Akbar = Allah is All-great. Thus she had clothed herself, as it were, in prayer and had retired from the world pro temp.\n\n[FN#185] i.e.. the prayers of the last day and night which she\nhad neglected while in company with the Jinns. The Hammam is not a pure place to pray in; but the Farz or Koranic orisons should be recited there if the legal term be hard upon its end.\n\n[FN#186] Slaves, male as well as female, are as fond of talking over their sale as European dames enjoy looking back upon the details of courtship and marriage.\n\n[FN#187] Arab. \"Du'\u00e1,\" = supplication, prayer, as opposed to\n\"Sal\u00e1t\" = divine worship, \"prayers.\" For the technical meaning of the latter see vol. iv. 65. I have objected to Mr. Redhouse's distinction without a difference between Moslems' worship and prayer: voluntary prayers are not prohibited to them and their praises of the Lord are mingled, as amongst all worshippers, with petitions.\n\n[FN#188] Al-Muzfir = the Twister; Zaf\u00e1ir al-Jinn = Adiantum capillus veneris. L\u00fal\u00faah = The Pearl, or Wild Heifer; see vol. ix. 218.\n\n[FN#189] Arab. \"Bi jildi 'l-baker.\" I hope that captious critics will not find fault with my rendering, as they did in the case of Fals ahmar = a red cent, vol. i. 321.\n\n[FN#190] Arab. \"Far\u00e1sah\" = lit. knowing a horse. Arabia abounds in tales illustrating abnormal powers of observation. I have noted this in vol. viii. 326.\n\n[FN#191] i.e. the owner of this palace.\n\n[FN#192] She made the Ghusl not because she had slept with a man, but because the impurity of Satan's presence called for the major ablution before prayer.\n\n[FN#193] i.e. she conjoined the prayers of nightfall with those of dawn.\n\n[FN#194] i.e. those of midday, mid-afternoon and sunset.\n\n[FN#195] Arab. \"Sahb\u00e1\" red wine preferred for the morning\ndraught.\n\n[FN#196] The Apostle who delighted in women and perfumes. Persian poetry often alludes to the rose which, before white, was dyed red by his sweat.\n\n[FN#197] For the etymology of Juln\u00e1r--Byron's \"Gulnare\"--see vol. vii. 268. Here the rhymer seems to refer to its origin; Gul (Arab. Jul) in Persian a rose; and An\u00e1r, a pomegranate, which in Arabic becomes N\u00e1r = fire.\n\n[FN#198] i.e. \"The brilliant,\" the enlightened.\n\n[FN#199] i.e. the moral beauty.\n\n[FN#200] A phenomenon well known to spiritualists and to \"The\nHouse and the Haunter.\" An old Dutch factory near Hungarian Fiume is famed for this mode of \"obsession\": the inmates hear the sound of footfalls, etc., behind them, especially upon the stairs; and see nothing.\n\n[FN#201] The two short Koranic chapters, The Daybreak (cxiii.)\nand The Men (cxiv. and last) evidently so called from the   words which occur in both (versets i., \"I take refuge with\"). These \"Ma'\u00fazat\u00e1ni,\" as they are called, are recited as talismans or preventives against evil, and are worn as amulets inscribed on parchment; they are also often used in the five canonical prayers. I have translated them in vol. iii. 222.\n\n[FN#202] The antistes or fugleman at prayer who leads off the\norisons of the congregation; and applied to the Caliph as the\nhead of the faith. See vol. ii. 203 and iv. 111.\n\n[FN#203] Arab. \" 'Umm\u00e1r\" i.e. the Jinn, the \"spiritual creatures\" which walk this earth, and other non-humans who occupy it.\n\n[FN#204] A parallel to this bodiless Head is the Giant Face,\nwhich appears to travellers (who expect it) in the Lower Valley of the Indus. See Sind Re-visited, ii. 155.\n\n[FN#205] Arab. \"Ghal\u00edl\u00ed\" = my yearning.\n\n[FN#206] Arab. \"Ahb\u00e1bu-n\u00e1\" plur. for singular = my beloved.\n\n[FN#207] i.e. her return.\n\n[FN#208] Arab. \"Arja'\" lit. return! but here meaning to stop. It is much used by donkey-boys from Cairo to Fez in the sense of \"Get out of the way.\" Hence the Spanish arre! which gave rise to arriero = a carrier, a muleteer.\n\n[FN#209] Arab. \"Afras\" lit.=a better horseman.\n\n[FN#210] A somewhat crippled quotation from Koran lvi. 87-88, \"As for him who is of those brought near unto Allah, there shall be for him easance and basil and a Garden of Delights (Na'\u00edm).\"\n\n[FN#211] i.e. Queen Sunbeam.\n\n[FN#212] See vol. i. 310 for this compound perfume which contains musk, ambergris and other essences.\n\n[FN#213] I can hardly see the sequence of this or what the\ncarpets have to do here.\n\n[FN#214] Here, as before, some insertion has been found\nnecessary.\n\n[FN#215] Arab. \"Dukh\u00falak\" lit.=thy entering, entrance, becoming familiar.\n\n[FN#216] Or \"And in this there shall be to thee great honour over all the Jinn.\"\n\n[FN#217] Mr. Payne thus amends the text, \"How loathly is yonder Genie Meimoun! There is no eating (in his presence);\" referring back to p. 88.\n\n[FN#218] i.e. \"I cannot bear to see him!\"\n\n[FN#219] This assertion of dignity, which is permissible in\nroyalty, has been absurdly affected by certain \"dames\" in\nAnglo-Egypt who are quite the reverse of queenly; and who degrade \"dignity\" to the vulgarest affectation.\n\n[FN#220] i.e. \"May thy visits never fail me!\"\n\n[FN#221] i.e. Ash-, verging upon white.\n\n[FN#222] i.e. \"She will double thy store of presents.\"\n\n[FN#223] The Arab boy who, unlike the Jew, is circumcised long\nafter infancy and often in his teens, thus making the ceremony\nconform after a fashion with our \"Confirmation,\" is displayed\nbefore being operated upon, to family and friends; and the seat is a couch covered with the richest tapestry. So far it resembles the bride-throne.\n\n[FN#224] Tohfah.\n\n[FN#225] i.e. Hindu, Indian.\n\n[FN#226] Japhet, son of Noah.\n\n[FN#227] Mr. Payne translates \"Take this and glorify thyself\nwithal over the people of the world.\" His reading certainly makes\nbetter sense, but I do not see how the text can carry the\nmeaning. He also omits the bussing of the bosom, probably for\nartistic reasons.\n\n[FN#228] A skit at Ish\u00e1k, making the Devil praise him. See vol.\nvii. 113.\n\n[FN#229] Arab. \"Maw\u00e1zi\" (plur. of Mauza')=lit. places, shifts,\npassages.\n\n[FN#230] The bed (farsh), is I presume, the straw-spread (?)\nstore-room where the apples are preserved.\n\n[FN#231] Arab. \"Farkh warak\", which sounds like an atrocious\nvulgarism.\n\n[FN#232] The Moss-rose; also the eglantine, or dog-rose, and the\nsweet-briar, whose leaf, unlike other roses, is so odorous.\n\n[FN#233] The lily in Heb., derived by some from its six (shash)\nleaves, and by others from its vivid cheerful brightness. \"His\nlips are lilies\" (Cant. v. 13), not in colour, but in odoriferous\nsweetness.\n\n[FN#234] The barber is now the usual operator; but all operations\nbegan in Europe with the \"barber-surgeon.\"\n\n[FN#235] Sic in text xii. 20. It may be a misprint for Ab\u00fa\nal-Tawaif, but it can also mean \"O Shaykh of the Tribes (of\nJinns)!\"\n\n[FN#236] The capital of King Al-Shisban.\n\n[FN#237] Arab \"Fajj\", the Spanish \"Vega\" which, however, means a mountain-plain, a plain.\n\n[FN#238] i.e. I am quite sure: emphatically.\n\n[FN#239] i.e. all the Jinn's professions of affection and\npromises of protection were mere lies.\n\n[FN#240] In the original this apodosis is wanting: see vol. vi. 203, 239.\n\n[FN#241] Arab. \"D\u00e1hiyat al-Daw\u00e1h\u00ed;\" see vol. ii. 87.\n\n[FN#242] Arab. \"Al-Jabal al-Mukawwar\"= Cha\u00eene de montagnes de\nforme demi circulaire, from Kaur, a park, an enceinte.\n\n[FN#243] Arab. \"R\u00fah\u00ed\" lit. my breath, the outward sign of life.\n\n[FN#244] i.e. K\u00e1f.\n\n[FN#245] i.e. A bit of burning charcoal.\n\n[FN#246] Arab. \"Al-yad al-bayz\u00e1,\"=lit. The white hand: see vol. iv. 185.\n\n[FN#247] Showing the antiquity of \"Apr\u00e8s moi le d\u00e9luge,\" the fame of all old politicians and aged statesmen who can expect but a few years of life. These \"burning questions\" (e.g. the Bulgarian) may be smothered for a time, but the result is that they blaze forth with increased violence. We have to thank Lord Palmerston (an Irish landlord) for ignoring the growth of Fenianism and another aged statesman for a sturdy attempt to disunite the United Kingdom. An old notion wants young blood at its head.\n\n[FN#248] Suggesting the nursery rhyme:\n\n          Fee, fo, fum\n          I smell the blood of an Englishman.\n\n[FN#249] i.e. why not at once make an end of her.\n\n[FN#250] The well-known war-cry.\n\n[FN#251] Lit. \"Smoke\" pop. applied, like our word, to tobacco.\nThe latter, however, is not here meant.\n\n[FN#252] Arab. \"Ghur\u00e1b al-bayn,\" of the wold or of parting. See\nvol. vii. 226.\n\n[FN#253] Arab. \"Hal\u00e1wah\"; see vol. iv. 60.\n\n[FN#254] Here the vocative particle \"Y\u00e1\" is omitted.\n\n[FN#255] Lit. \"The long-necked (bird)\" before noticed with the\nRukh (Roc) in vol. v. 122. Here it becomes a Princess, daughter of Bahr\u00e1m-i-G\u00far (Bahram of the Onager, his favourite game), the famous Persian king in the fifth century, a contemporary of Theodosius the younger and Honorius. The \"Ank\u00e1\" is evidently the Iranian S\u00edmurgh.\n\n[FN#256] \"Chamber\" is becoming a dangerous word in English. Roars of laughter from the gods greeted the great actor's declamation, \"The bed has not been slept in! Her little chamber is empty!\"\n\n[FN#257] Choice Gift of the breast (or heart).\n\n[FN#258] From the Calc. Edit. (1814\u201318), Nights cxcvi.\u2013cc., vol. ii., pp. 367\u2013378. The translation has been compared and collated with that of Langl\u00e8s (Paris, 1814), appended to his Edition of the Voyages of Sindbad. The story is exceedingly clever and well deserves translation.\n\n[FN#259] It is regretable that this formula has not been\npreserved throughout The Nights: it affords, I have noticed, a\npleasing break to the long course of narrative.\n\n[FN#260] Arab. \"Ban\u00e1t-al-haw\u00e1\" lit. daughters of love, usually\nmeaning an Anonyma, a fille de joie; but here the girl is of good repute, and the offensive term must be modified to a gay,\nfrolicsome lass.\n\n[FN#261] Arab. \"Jabhat,\" the lintel opposed to the threshold.\n\n[FN#262] Arab. \"Ghatt\u00ed,\" still the popular term said to a child showing its nakedness, or a lady of pleasure who insults a man by displaying any part of her person.\n\n[FN#263] She is compared with a flashing blade (her face) now\ndrawn from its sheath (her hair) then hidden by it.\n\n[FN#264] The \"Muajjalah\" or money paid down before consummation was about \u00a325; and the \"Mu'ajjalah\" or coin to be paid contingent on divorce was about \u00a375. In the Calc. Edit ii. 371, both dowers are \u00a335.\n\n[FN#265] i.e. All the blemishes which justify returning a slave to the slave-dealer.\n\n[FN#266] Media: see vol. ii. 94. The \"Daylamite prison\" was one of many in Baghdad.\n\n[FN#267] See vol. v. 199. I may remark that the practice of\nbathing after copulation was kept up by both sexes in ancient\nRome. The custom may have originated in days when human senses\nwere more acute. I have seen an Arab horse object to be mounted by the master when the latter had not washed after sleeping with a woman.\n\n[FN#268] On the morning after a happy night the bridegroom still\noffers coffee and Halw\u00e1 to friends.\n\n[FN#269] i.e. More bewitching.\n\n[FN#270] Arab. \"Shar\u00edf\u00ed\" more usually Ashrafi, the Port. Xerafim,\na gold coin = 6s.\u20137s.\n\n[FN#271] The oft-repeated Koranic quotation.\n\n[FN#272] Arab. \"'Irk\": our phrase is \"the apple of the eye.\"\n\n[FN#273] Meaning that he was a Sayyid or a Shar\u00edf.\n\n[FN#274] i.e. than a Jew or a Christian. So the Sultan, when\nappealed to by these religionists, who were as usual squabbling\nand fighting, answered, \"What matter if the dog tear the hog or\nthe hog tear the dog\"?\n\n[FN#275] The \"Shar\u00ed'at\" forbidding divorce by force.\n\n[FN#276] i.e. protect my honour.\n\n[FN#277] For this proverb see vol. v. 138. 1 have remarked that\n\"Shame\" is not a passion in Europe as in the East; the Western\nequivalent to the Arab. \"Hay\u00e1' 'would be the Latin \"Pudor.\"\n\n[FN#278] Arab. \"Tal\u00e1kan b\u00e1inan,\" here meaning a triple divorce\nbefore witnesses, making it irrevocable.\n\n[FN#279] i.e. who had played him that trick.\n\n[FN#280] The Bresl. Edit. (vol. xii. pp. 50-116, Nights\ndcccclviii- dcccclxv.) entitles it \"Tale of Abu al-Hasan the\nDamascene and his son S\u00edd\u00ed Nur al-D\u00edn ' Al\u00ed.\" S\u00edd\u00ed means simply,\n\"my lord,\" but here becomes part of the name, a practice\nperpetuated in Zanzibar. See vol. v.283.\n\n[FN#281] i.e. at the hours of canonical prayers and other\nsuitable times he made an especial orison (du'\u00e1) for issue.\n\n[FN#282] See vol. i.85, for the traditional witchcraft of\nBabylonia.\n\n[FN#283] i.e. More or less thoroughly.\n\n[FN#284] i.e. \"He who quitteth not his native country diverteth not himself with a sight of the wonders of the world.\"\n\n[FN#285] For similar sayings, see vol. ix. 257, and my Pilgrimage\ni. 127.\n\n[FN#286] i.e. relying upon, etc.\n\n[FN#287] The Egyptian term for a khan, called in Persia\ncaravanserai (karw\u00e1n-ser\u00e1i); and in Marocco funduk, from the\nGreek; whence the Spanish \"fonda.\" See vol. i. 92.\n\n[FN#288] Arab. \"Baliyah,\" to jingle with \"B\u00e1biliyah.\"\n\n[FN#289] As a rule whenever this old villain appears in The\nNights, it is a signal for an outburst of obscenity. Here,\nhowever, we are quittes pour la peur. See vol. v. 65 for some of his abominations.\n\n[FN#290] The lines are in vols. viii. 279 and ix. 197. I quote Mr.\nPayne.\n\n[FN#291] Lady or princess of the Fair (ones).\n\n[FN#292] i.e. of buying.\n\n[FN#293] Arab. \"\u00c1z\u00e1n-h\u00fa\" = lit. its ears.\n\n[FN#294] Here again the policeman is made a villain of the\ndeepest dye; bad enough to gratify the intelligence of his\ndeadliest enemy, a lodging-keeper in London.\n\n[FN#295] i.e. You are welcome to it and so it becomes lawful\n(hal\u00e1l) to you.\n\n[FN#296] Arab. \"Sijn al-Dam,\" the Carcere duro inasprito (to\nspeak Triestine), where men convicted or even accused of\nbloodshed were confined.\n\n[FN#297] Arab. \"Mab\u00e1sim\"; plur. of Mabsim, a smiling mouth which shows the foreteeth.\n\n[FN#298] The branchlet, as usual, is the youth's slender form.\n\n[FN#299] Subaudi, \"An ye disdain my love.\"\n\n[FN#300] In the text \"sleep.\"\n\n[FN#301] \"Them\" and \"him\" for \"her.\"\n\n[FN#302] 'Urk\u00fab, a Jew of Yathrib or Khaybar, immortalised in the A.P. (i. 454) as \"more promise-breaking than 'Urk\u00fab.\"\n\n[FN#303] Uncle of Mohammed. See vol. viii. 172.\n\n[FN#304] First cousin of Mohammed. See ib.\n\n[FN#305] This threat of \"'Orf with her 'ead\" shows the Caliph's lordliness.\n\n[FN#306] Arab. \"Al-Bashkh\u00e1nah.\"\n\n[FN#307] i.e. Amen. See vol. ix. 131.\n\n[FN#308] When asked, on Doomsday, his justification for having\nslain her.\n\n[FN#309] Khorasan which included our Afghanistan, turbulent then as now, was in a chronic state of rebellion during the latter part of Al-Rashid's reign.\n\n[FN#310] The brutality of a Moslem mob on such occasions is\nphenomenal: no fellow-feeling makes them decently kind. And so at executions even women will take an active part in insulting and tormenting the criminal, tearing his hair, spitting in his face and so forth. It is the instinctive brutality with which wild beasts and birds tear to pieces a wounded companion.\n\n[FN#311] The popular way of stopping hemorrhage by plunging the stump into burning oil which continued even in Europe till\nAmbrose Par\u00e9 taught men to take up the arteries.\n\n[FN#312] i.e. folk of good family.\n\n[FN#313] i.e. the result of thy fervent prayers to Allah for me.\n\n[FN#314] Arab. \"Al-Ab\u00e1r\u00edk\" plur. of lbrik, an ewer containing\nwater for the Wuzu-ablution. I have already explained that a\nMoslem wishing to be ceremonially pure, cannot wash as Europeans do, in a basin whose contents are fouled by the first touch.\n\n[FN#315] Arab. \"N\u00e1ihah\", the pr\u00e6fica or myriologist. See vol. i. 311. The proverb means, \"If you want a thing done, do it\nyourself.\"\n\n[FN#316] Arab. \"Burka',\" the face veil of Egypt, Syria, and\nArabia with two holes for the eyes, and the end hanging to the\nwaist, a great contrast with the \"Lith\u00e1m\" or coquettish fold of transparent muslin affected by modest women in Stambul.\n\n[FN#317] i.e. donned petticoat-trousers and walking boots other than those she was wont to wear.\n\n[FN#318] \"Surah\" (Koranic chapter) may be a clerical error for\n\"S\u00farah\" (with a S\u00e1d) = sort, fashion (of food).\n\n[FN#319] This is solemn religious chaff; the Shaykh had doubtless often dipped his hand abroad in such dishes; but like a good Moslem, he contented himself at home with wheaten scones and olives, a kind of sacramental food like bread and wine in\nsouthern Europe. But his retort would be acceptable to the True\nBeliever who, the strictest of conservatives, prides himself on\nimitating in all points, the sayings and doings of the Apostle.\n\n[FN#320] i.e. animals that died without being ceremonially\nkilled.\n\n[FN#321] Koran ii. 168. This is from the Chapter of the Cow where \"that which dieth of itself (carrion), blood, pork, and that over which other name but that of Allah (i.e. idols) hath been invoked\" are forbidden. But the verset humanely concludes:\n\"Whoso, however, shall eat them by constraint, without desire, or as a transgressor, then no sin shall be upon him.\"\n\n[FN#322] i.e. son of Simeon = a Christian.\n\n[FN#323] Arab. and Heb. \"Haykal,\" suggesting the idea of large\nspace, a temple, a sanctuary, a palace which bear a suspicious\nlikeness to the Accadian \u00ca-kal or Great House = the old Egyptian Perao (Pharaoh?), and the Japanese \"Mikado.\"\n\n[FN#324] Wine, carrion and pork being lawful to the Moslem if\nused to save life. The former is also the sovereignest thing for\ninward troubles, flatulence, indigestion, etc. See vol. v. 2, 24.\n\n[FN#325] Arab. \"N\u00e1zilah,\" i.e., a curse coming down from Heaven.\n\n[FN#326] Here and below, a translation of her name.\n\n[FN#327] \"A picture of Paradise which is promised to the\nGod-fearing! Therein are rivers of water which taint not; and\nrivers of milk whose taste changeth not; and rivers of wine,\netc.\"--Koran xlvii. 16.\n\n[FN#328]  Let us have wine and women, mirth and laughter,\n          Sermons and soda-water the day after.\n                                   Don Juan ii. 178.\n\n[FN#329] The ox (Bakar) and the bull (Taur, vol. i. 16) are the\nMoslem emblems of stupidity, as with us are the highly\nintelligent ass and the most sagacious goose.\n\n[FN#330] In Arab. \"'Ud\" means primarily wood; then a lute. See\nvol. ii. 100. The Muezzin, like the schoolmaster, is popularly\nsupposed to be a fool.\n\n[FN#331] I have noticed that among Arab lovers it was the fashion to be jealous of the mistress's nightly phantom which, as amongst mesmerists, is the lover's embodied will.\n\n[FN#332] i.e. I will lay down my life to save thee from sorrow--a common-place hyperbole of love.\n\n[FN#333] Arab. \"Katl.\" I have noticed the Hibernian \"kilt\" which\nis not a bull but, like most provincialisms and Americanisms, a\nsurvival, an archaism. In the old Frisian dialect, which agrees\nwith English in more words than \"bread, butter and cheese,\" we\nfind the primary meaning of terms which with us have survived\nonly in their secondary senses, e.g. killen = to beat and slagen\n= to strike. Here is its great value to the English philologist.\nWhen the Irishman complains that he is \"kilt\" we know through the Frisian what he really means.\n\n[FN#334] The decency of this description is highly commendable\nand I may note that the Bresl. Edit. is comparatively free from\nerotic pictures.\n\n[FN#335] i.e. \"I commit him to thy charge under God.\"\n\n[FN#336] This is an Americanism, but it translates passing well\n\"Al-il\u00e1j\" = insertion.\n\n[FN#337] Arab. (and Heb.) \"Tarjum\u00e1n\" = a dragoman, for which see\nvol. i. 100. In the next tale it will occur with the sense of\npolyglottic.\n\n[FN#338] See vol. i. p. 35.\n\n[FN#339] After putting to death the unjust Prefect.\n\n[FN#340] Arab. \"Lajlaj.\" See vol. ix. 322.\n\n[FN#341] Arab. \"Maw\u00e1lid\" lit. = nativity festivals (plur. of\nMaulid). See vol. ix. 289.\n\n\n[FN#342] Bresl. Edit., vol. xii. pp. 116-237, Nights\ndcccclxvi-dcccclxxix. Mr. Payne entitles it \"El Abbas and the\nKing's Daughter of Baghdad.\"\n\n[FN#343] \"Of the Shayban tribe.\" I have noticed (vol. ii. 1) how\nloosely the title Malik (King) is applied in Arabic and in\nmedi\u00e6val Europe. But it is ultra-Shakespearean to place a Badawi\nKing in Baghdad, the capital founded by the Abbasides and ruled\nby those Caliphs till their downfall.\n\n[FN#344] i.e. Ir\u00e1k Arab\u00ed (Chald\u00e6a) and 'Ajami (Western Persia).\nFor the meaning of Al-Ir\u00e1k, which always, except in verse, takes\nthe article, see vol. ii. 132.\n\n[FN#345] See supra, p. 185. Mr. Payne suspects a clerical error\nfor \"Turkum\u00e1niyah\" = Turcomanish; but this is hardly acceptable.\n\n[FN#346] As fabulous a personage as \"King Kays.\"\n\n[FN#347] Possibly a clerical error for Zab\u00edd, the famous capital\nof the Tah\u00e1mah or lowlands of Al-Yaman.\n\n[FN#348] The Moslem's Holy Land whose capital is Meccah.\n\n[FN#349] A hinted protest against making a picture or a statue\nwhich the artist cannot quicken; as this process will be demanded of him on Doomsday. Hence also the Princess is called M\u00e1riyah (Maria, Mary), a non-Moslem name.\n\n[FN#350] i.e. day and night, for ever.\n\n[FN#351] Koran xxxiii. 38; this concludes a \"revelation\"\nconcerning the divorce and marriage to Mohammed of the wife of\nhis adopted son Zayd. Such union, superstitiously held incestuous\nby all Arabs, was a terrible scandal to the rising Faith, and\ncould be abated only by the \"Commandment of Allah.\" It is hard to\nbelieve that a man could act honestly after such fashion; but we\nhave seen in our day a statesman famed for sincerity and\nuprightness honestly doing things the most dishonest possible.\nZayd and Abu Lahab (chap. cxi. i.) are the only contemporaries of\nMohammed named in the Koran.\n\n[FN#352] i.e. darkened behind him.\n\n[FN#353] Here we have again, as so common in Arab romances, the\nexpedition of a modified Don Quixote and Sancho Panza.\n\n[FN#354] Arab. \"Arzi-h\u00e1\" = in its earth, its outlying suburbs.\n\n[FN#355] The king's own tribe.\n\n[FN#356] i.e. he was always \"spoiling for a fight.\"\n\n[FN#357] In the text the two last sentences are spoken by Amir\nand the story-teller suddenly resumes the third person.\n\n[FN#358] Mr. Payne translates this \"And God defend the right\" (of\nplunder according to the Arabs).\n\n[FN#359] Arab. \"Lill\u00e1hi darruk\"; see vol. iv. 20. Captain Lockett (p.28) justly remarks that \"it is a sort of encomiastic exclamation of frequent occurrence in Arabic and much easier to comprehend than translate.\" Darra signifies flowing freely (as milk from the udder) and was metaphorically transferred to bounty and to indoles or natural capacity. Thus the phrase means \"your flow of milk is by or through Allah.\" i.e., of unusual abundance.\n\n[FN#360] The words are euphemistic: we should say \"comest thou to our succour.\"\n\n[FN#361] i.e. If his friend the Devil be overstrong for thee,\nflee him rather than be slain; as\n\n          He who fights and runs away\n          Shall live to fight another day.\n\n[FN#362] i.e. I look to Allah for aid (and keep my powder dry).\n\n[FN#363] i.e. to the next world.\n\n[FN#364] This falling backwards in laughter commonly occurs\nduring the earlier tales; it is, however, very rare amongst the\nBadawin.\n\n[FN#365] i.e. as he were a flying Jinni, swooping down and\npouncing falcon-like upon a mortal from the upper air.\n\n[FN#366] This may be (reading Imraan = man, for Amran = matter) \"a masterful man\"; but I can hardly accept it.\n\n[FN#367] Arab. \"Bunduk\u00ed,\" the adj. of Bunduk, which the Moslems evidently learned from Slav sources; Venedik being the Dalmatian corruption of Venezia. See Dubrovenedik in vol. ii. 219.\n\n[FN#368] i.e. the castle's square.\n\n[FN#369] In sign of quitting possession. Chess in Europe is\nrarely played for money, with the exception of public matches:\nthis, however, is not the case amongst Easterns, who are also for the most part as tricky as an old lady at cribbage rightly named.\n\n[FN#370] i.e. he was as eloquent and courtly as he could be.\n\n[FN#371] Arab. \"Ya Z\u00ednat al-Nis\u00e1,\" which may either be a P.N. or\na polite address as Bella f\u00e9 (Handsome woman) is to any feminine\nin Southern Italy.\n\n[FN#372] Arab. \"Raas Ghanam\": this form of expressing singularity\nis common to Arabic and the Eastern languages, which it has\ninfluenced.\n\n[FN#373] This most wearisome form of politeness is common in the\nMoslem world, where men fondly think that the more you see of\nthem the more you like of them. Yet their Proverbial Philosophy\n(\"the wisdom of many and the wit of one\") strongly protests\nagainst the practice: I have already quoted Mohammed's saying,\n\"Zur ghibban, tazid Hibban\"--visits rare keep friendship fair.\n\n[FN#374] This clause in the text is evidently misplaced (vol.\nxii.144).\n\n[FN#375] Arab. Dara' or Dira'=armour, whether of leather or\nmetal; here the coat worn under the mail.\n\n[FN#376] Called from Rustak, a quarter of Baghdad. For Rust\u00e1k\ntown see vol. vi. 289.\n\n[FN#377] From Damietta comes our \"dimity.\" The classical name was Tami\u00e1this apparently Coptic gr\u00e6cised: the old town on the shore famed in Crusading times was destroyed in A.H. 648 = 1251.\n\n[FN#378] Easterns are always startled by a sudden summons to the presence either of King or Kazi: here the messenger gives the youth to understand that it is in kindness, not in anger.\n\n[FN#379] i.e. in not sending for thee to court instead of\nallowing thee to live in the city without guest-rite.\n\n[FN#380] In sign of agitation: the phrase has often been used in\nthis sense and we find it also in Al-Mas'udi.\n\n[FN#381] I would remind the reader that the \"Daw\u00e1t\" (ink-case)\ncontains the reed-pens.\n\n[FN#382] Two well-known lovers.\n\n[FN#383] On such occasions the old woman (and Easterns are hard\nde dolo vetularum) always assents to the sayings of her prey,\nwell knowing what the doings will inevitably be.\n\n[FN#384] Travellers, Nomads, Wild Arabs.\n\n[FN#385] Whither they bear thee back dead with the women crying\nand keening.\n\n[FN#386] Arab. Azn\u00e1n\u00ed = emaciated me.\n\n[FN#387] Either the Deity or the Love-god.\n\n[FN#388] Arab. \"Him\u00e0\" = the tribal domain, a word which has often\noccurred.\n\n[FN#389] \"O ye who believe! seek help through patience and\nprayer: verily, Allah is with the patient.\" Koran ii. 148. The\npassage refers to one of the battles, Bedr or Ohod.\n\n[FN#390] Arab. \"Sirr\" (a secret) and afterwards \"Kitm\u00e1n\"\n(concealment) i.e. Keeping a lover down-hearted.\n\n[FN#391] Arab. \"'Alkam\" = the bitter gourd, colocynth; more\nusually \"Hanzal.\"\n\n[FN#392] For \"Jaz\u00edrah\" = insula, island, used in the sense of\n\"peninsula,\" see vol. i. 2.\n\n[FN#393] Meccah and Al-Medinah. Pilgrimage i. 338 and ii. 57,\nused in the proverb \"Sharr fi al-Haramayn\" = wickedness in the\ntwo Holy Places.\n\n[FN#394] Arab. Al-hamd (o li'llah).\n\n[FN#395] i.e. play, such as the chase, or an earnest matter, such as war, etc.\n\n[FN#396] Arab. \"Mizwad,\" or Mizw\u00e1d = lit. provision-bag, from Z\u00e1d = viaticum; afterwards called Kirbah (pron. Girbah, the popular term), and Sakl. The latter is given in the Dictionaries as Ask\u00e1lah = scala, \u00e9chelle, stage, plank.\n\n[FN#397] Those blood-feuds are most troublesome to the traveller, who may be delayed by them for months: and, until a peace be patched up, he will never be allowed to pass from one tribe to their enemies. A quarrel of the kind prevented my crossing Arabia from Al-Medinah to Maskat (Pilgrimage, ii. 297), and another in Africa from visiting the head of the Tanganyika Lake. In all such journeys the traveller who has to fight against Time is almost sure to lose.\n\n[FN#398] i.e. his fighting-men.\n\n[FN#399] The popular treatment of a detected horse-thief, for\nwhich see Burckhardt, Travels in Arabia (1829), and Notes on the\nBedouins and Wahabys (1830).\n\n[FN#400] Arab \"Ash\u00edrah\": see vol. vii. 121.\n\n[FN#401] Arab. \"Mus\u00e1fahah\" -. see vol. vi. 287.\n\n[FN#402] In the text, \"To the palace of the king's daughter.\"\n\n[FN#403] Arab. \"Marj Sal\u00ed'\" = cleft meadow (here and below). Mr.\nPayne suggests that this may be a mistranscription for Marj Sal\u00ed'\n(with a S\u00e1d) = a treeless champaign. It appears to me a careless\nblunder for the Marj akhzar (green meadow) before mentioned.\n\n[FN#404] The palace, even without especial and personal reasons,\nnot being the place for a religious and scrupulous woman.\n\n[FN#405] \"i.e. those of El Aziz, who had apparently entered the\ncity or passed through it on their way to the camp of El Abbas.\"\nThis is Mr. Payne's suggestion.\n\n[FN#406] Arab \"Hatif\"; gen. = an ally.\n\n[FN#407] Not wishing to touch the hand of a strange woman.\n\n[FN#408] i.e. a mere passer-by, a stranger; alluding to her\ntaunt.\n\n[FN#409] The Bactrian or double-humped dromedary. See vol. iii.\n67. Al-Mas'udi (vii. 169) calls it \"Jamal f\u00e1lij,\" lit. = the\npalsy-camel.\n\n[FN#410] i.e. Stars and planets.\n\n[FN#411] i.e. Sang in tenor tones which are always in falsetto.\n\n[FN#412] Arab. Tahz\u00edb = reforming morals, amending conduct,\nchastening style.\n\n[FN#413] i.e. so as to show only the whites, as happens to the\n\"mesmerised.\"\n\n[FN#414] i.e. for love of and longing for thy youth.\n\n[FN#415] i.e. leather from Al-T\u00e1if: see vol. viii. 303. The text\nhas by mistake T\u00e1lif\u00ed.\n\n[FN#416] i.e. she was at her last breath, when cured by the magic\nof love.\n\n[FN#417] i.e. violateth my private apartment.\n\n[FN#418] The voice (Sh\u00e1zz) is left doubtful: it may be girl's,\nnightingale's, or dove's.\n\n[FN#419] Arab. \"Hib\u00e1\" partly induced by the rhyme. In desert\ncountries the comparison will be appreciated: in Sind the fine\ndust penetrates into a closed book.\n\n[FN#420] i.e. he smuggled it in under his 'Ab\u00e1-cloak: perhaps it was a better brand than that made in the monastery.\n\n[FN#421] i.e. the delights of Paradise promised by the Prophet.\n\n[FN#422] Again, \"he\" for \"she,\" making the lover's address more courtly and delicate.\n\n[FN#423] i.e. take refuge with Allah from the evil eye of her\ncharms.\n\n[FN#424] i.e. an thou prank or adorn thyself: I have translated literally, but the couplet strongly suggests \"nonsense verses.\"\n\n[FN#425] Arab. \"Sant\u00edr:\" Lane (M.E., chapt. xviii.) describes it as resembling the Kan\u00fan (dulcimer or zither) but with two oblique peg-pieces instead of one and double chords of wire (not treble strings of lamb's gut) and played upon with two sticks instead of the little plectra. Dozy also gives Santir from {psalt\u00e9rion}, the Fsaltr\u00fan of Daniel.\n\n[FN#426] i.e. That which is ours shall be thine, and that which\nis incumbent on thee shall be incumbent on us = we will assume\nthy debts and responsibilities.\n\n[FN#427] This passage is sadly disjointed in the text: I have\nfollowed Mr. Payne's ordering.\n\n[FN#428] The Arab of noble tribe is always the first to mount his\nown mare: he also greatly fears her being put out to full speed\nby a stranger, holding that this should be reserved for occasions\nof life and death; and that it can be done to perfection only\nonce during the animal's life.\n\n[FN#429] The red (Ahmar) dromedary like the white-red (Sabah)\nwere most valued because they are supposed best to bear the heats\nof noon; and thus \"red camels\" is proverbially used for wealth.\nWhen the head of Abu Jahl was brought in after the Battle of\nBedr, Mahommed exclaimed, \"'Tis more acceptable to me than a red\ncamel!\"\n\n[FN#430] i.e. Couriers on dromedaries, the only animals used for\nsending messages over long distances.\n\n[FN#431] These guest-fires are famous in Arab poetry. So\nAl-Har\u00edr\u00ed (Ass. of Banu Haram) sings:--\n\n          A beacon fire I ever kindled high;\n\ni.e. on the hill-tops near the camp, to guide benighted\ntravellers. Also the Lam\u00edyat al-Ajam says:--\n\n          The fire of hospitality is ever lit on the high\nstations.\n\nThis natural telegraph was used in a host of ways by the Arabs of\nThe Ignorance; for instance, when a hated guest left the camp\nthey lighted the \"Fire of Rejection,\" and cried, \"Allah, bear him\nfar from us!\" Nothing was more ignoble than to quench such fire:\nhence in obloquy of the Faz\u00e1r tribe it was said:--\n\n          Ne'er trust Faz\u00e1r with an ass, for they\n               Once roasted ass-pizzle, the rabble rout:\n          And, when sight they guest, to their dams they say,\n               \"Piss quick on the guest-fire and put it out!\"\n     (Al-Mas\"udi vi. 140.)\n\n[FN#432] i.e. of rare wood, set with rubies.\n\n[FN#433] i.e. whose absence pained us.\n\n[FN#434] Mr. Payne and I have long puzzled over these enigmatical\nand possibly corrupt lines: he wrote to me in 1884, \"This is the\nfirst piece that has beaten me.\" In the couplet above (vol. xii.\n230) \"Rayh\u00e1n\u00ed\" may mean \"my basil-plant\" or \"my food\" (the latter\nKoranic), \"my compassion,\" etc.; and S\u00fas\u00e1n\u00ed is equally ancipitous\n\"My lilies\" or \"my sleep\": see Bard al-Susan = les douceurs du\nsommeil in Al-Mas'\u00fadi vii. 168.\n\n[FN#435] The \"Nik\u00e1\" or sand hill is the swell of the throat: the\nGhaur or lowland is the fall of the waist: the flower is the\nbreast anent which Mr. Payne appropriately quotes the well-known\nlines of Fletcher:\n\n          \"Hide, O hide those hills of snow,\n          That thy frozen bosom bears,\n          On whose tops the pinks that grow\n          Are of those that April wears.\"\n\n[FN#436] Easterns are right in regarding a sleepy languorous look\nas one of the charms of women, and an incitement to love because\nsuggestive only of bed. Some men also find the same pleasure in a\nlacrymose expression of countenance, seeming always to call for\nconsolation: one of the most successful women I know owes her\nexceptional good fortune to this charm.\n\n[FN#437] Arab. \"H\u00e1jib,\"eyebrow or chamberlain; see vol. iii. 233.\nThe pun is classical used by a host of poets including Al-Har\u00edr\u00ed.\n\n[FN#438] Arab. \"Tarfah.\" There is a Tarfia Island in the\nGuadalquivir and in Gibraltar a \"Tarfah Alto\" opposed to \"Tarfah\nbajo.\" But it must not be confounded with Tarf = a side, found in\nthe Maroccan term for \"The Rock\" Jabal al-Tarf = Mountain of the\nPoint (of Europe).\n\n[FN#439] For Solomon and his flying carpet see vol. iii. 267.\n\n[FN#440] Arab. \"Bil\u00e1d al-Maghrib (al-Aksa,\" in full) = the\nFarthest Land of the setting Sun, shortly called Al-Maghrib and\nthe people \"Maghribi.\" The earliest occurrence of our name\nMorocco or Marocco I find in the \"Mar\u00e1kiyah\" of Al-Mas'udi (iii.\n241), who apparently applies it to a district whither the Berbers\nmigrated.\n\n[FN#441] The necklace-pearls are the cup-bearer's teeth.\n\n[FN#442] In these unregenerate days they would often be summoned to the houses of the royal family; but now they had \"got religion\" and, becoming freed women, were resolved to be\n\"respectable.\" In not a few Moslem countries men of wealth and\nrank marry professional singers who, however loose may have been their artistic lives, mostly distinguish themselves by decency of behaviour often pushed to the extreme of rigour. Also jeune coquette, vieille d\u00e9vote is a rule of the world, Eastern and Western.\n\n[FN#443] Bresl. Edit., vol. xii p. 383 (Night mi).  The king is called as usual \"Shahrb\u00e1n,\" which is nearly synonymous with\nShahry\u00e1r.\n\n[FN#444] i.e. the old Sindibad-N\u00e1meh (see vol. vi. 122), or \"The Malice of Women\" which the Bresl. Edit. entitles, \"Tale of the King and his Son and his Wife and the Seven Wazirs.\"  Here it immediately follows the Tale of Al-Abbas and Mariyah and occupies pp. 237-383 of vol. xii. (Nights dcccclxxix-m).\n\n[FN#445] i.e. Those who commit it.\n\n[FN#446] The connection between this pompous introduction and the story which follows is not apparent.  The \"Tale of the Two Kings and the Wazir's Daughters\" is that of Shahrazad told in the third person, in fact a rechauff\u00e9 of the Introduction.  But as some three years have passed since the marriage, and the d\u00e9no\u00fbement of the plot is at hand, the Princess is made, with some art I think, to lay the whole affair before her husband in her own words, the better to bring him to a \"sense of his duty.\"\n\n[FN#447] Bresl. Edit., vol. xii. pp. 384-412.\n\n[FN#448] This clause is taken from the sequence, where the elder brother's kingdom is placed in China.\n\n[FN#449] For the Tobbas = \"Successors\" or the Himyaritic kings, see vol. i. 216.\n\n[FN#450] Kay\u00e1sirah, opp. to Ak\u00e1sirah, here and in many other\nplaces.\n\n[FN#451] See vol. ii. 77. King Kulayb (\"little dog\") al-W\u00e1'il, a powerful chief of the Banu Ma'ad in the Kas\u00edn district of Najd, who was connected with the war of Al-Bas\u00fas.  He is so called because he lamed a pup (kulayb) and tied it up in the midst of his Him\u00e0 (domain, place of pasture and water), forbidding men to camp within sound of its bark or sight of his fire.  Hence \"more masterful than Kulayb,\" A.P. ii. 145, and Al-Hariri Ass. xxvi. (Chenery, p. 448).  This angry person came by his death for wounding in the udder a trespassing camel (Sorab) whose owner was a woman named Bas\u00fas.  Her friend (Jas\u00fas) slew him; and thus arose the famous long war between the tribes W\u00e1'il Bakr and Taghlib. It gave origin to the saying, \"Die thou and be an expiation for the shoe-latchet of Kulayb.\"\n\n[FN#452] Arab. \"Mukhaddar\u00e1t,\" maidens concealed behind curtains and veiled in the Harem.\n\n[FN#453] i.e. the professional R\u00e1wis or tale-reciters who learned stories by heart from books like \"The Arabian Nights.\"  See my Terminal Essay, vol. x. 163.\n\n[FN#454] Arab. \"Bid'ah,\" lit. = an innovation, a new thing, an\ninvention, any change from the custom of the Prophet and the\nuniversal practice of the Faith, whether it be in the cut of the beard or a question of state policy.  Popularly the word =\nheterodoxy, heresy; but theologically it is not necessarily used in a bad sense.  See vol. v. 167.\n\n[FN#455] About three parts of this sentence have been supplied by\nMr. Payne, the careless scribe having evidently omitted it.\n\n[FN#456] Here, as in the Introduction (vol. i. 24), the king\nconsummates his marriage in presence of his virgin sister-in-law,\na process which decency forbids amongst Moslems.\n\n[FN#457] Al-Mas'udi (vol. iv. 213) uses this term to signify\nviceroy in \"Shahry\u00e1r Sajast\u00e1n.\"\n\n[FN#458] i.e. his indifference to the principles of right and\nwrong, which is a manner of moral intoxication.\n\n[FN#459] i.e. hath mentioned the office of Wazir (in Koran xx.\n30).\n\n[FN#460] i.e. Moslems, who practice the Religion of Resignation.\n\n[FN#461] Koran xxxiii. 35.  This is a proemium to the\n\"revelation\" concerning Zayd and Zaynab.\n\n[FN#462] i.e. I have an embarras de richesse in my repertory.\n\n[FN#463] The title is from the Bresl. Edit. (vol. xii. pp. 398-402).  Mr. Payne calls it \"The Favourite and her Lover.\"\n\n[FN#464] The practice of fumigating gugglets is universal in\nEgypt (Lane, M. E., chapt. v.); but I never heard of musk being so used.\n\n[FN#465] Arab.  \"Laysa fi 'l-diy\u00e1ri dayy\u00e1r\"--a favourite jingle.\n\n[FN#466] Arab.  \"Khayr Kathir\" (pron. Kat\u00edr) which also means\n\"abundant kindness.\"\n\n[FN#467] Dozy says of \"Hunayn\u00ed\" (Ha\u00edn\u00ed), Il semble \u00eatre le nom\nd'un v\u00eatement.  On which we may remark, Connu!\n\n[FN#468] Arab.  Har\u00edsah:  see vol. i. 131.  Westerns make a sad mess of this dish when they describe it as une sorte d'olla podrida (the hotch-pot), une p\u00e2t\u00e9e de viandes, de froment et de l\u00e9gumes secs (Al-Mas'udi viii. 438).  Whenever I have eaten it, it was always a meat-pudding, for which see vol. i. 131.\n\n[FN#469] Evidently one escaped because she was sleeping with the Caliph, and a second because she had kept her assignation.\n\n[FN#470] Mr. Payne entitles it, \"The Merchant of Cairo and the\nFavourite of the Khalif el Mamoun el Hakim bi Amrillah.\"\n\n[FN#471] See my Pilgrimage (i. 100): the seat would be on the\nsame bit of boarding where the master sits or on a stool or bench in the street.\n\n[FN#472] This is true Cairene chaff, give and take; and the\nstranger must accustom himself to it before he can be at home\nwith the people.\n\n[FN#473] i.e. In Rauzah-Island: see vol. v. 169.\n\n[FN#474] There is no historical person who answers to these name, \"The Secure, the Ruler by Commandment of Allah.\"  The cognomen applies to two soldans of Egypt, of whom the later Abu al-Abbas Ahmad the Abbaside (A.D. 1261-1301) has already been mentioned in The Nights (vol. v. 86).  The tale suggests the earlier Al-Hakim (Abu Ali al-Mans\u00far, the Fatimite, A.D. 995-1021), the God of the Druze \"persuasion;\" and the tale-teller may have purposely blundered in changing Mans\u00far to Maam\u00fan for fear of offending a sect which has been most dangerous in the matter of assassination and which is capable of becoming so again.\n\n[FN#475] Arab.  \"'Al\u00e0 kulli h\u00e1l\" = \"whatever may betide,\" or\n\"willy-nilly.\"  The phrase is still popular.\n\n[FN#476] The dulce desipere of young lovers, he making a buffoon of himself to amuse her.\n\n[FN#477] \"The convent of Clay,\" a Coptic monastery near Cairo.\n\n[FN#478] i.e. this is the time to show thyself a man.\n\n[FN#479] The Eastern succedaneum for swimming corks and other\n\"life-preservers.\"  The practice is very ancient; we find these gourds upon the monuments of Egypt and Babylonia.\n\n[FN#480] Arab.  \"Al-Khal\u00edj,\" the name, still popular, of the\nGrand Canal of Cairo, whose banks, by-the-by, are quaint and\npicturesque as anything of the kind in Holland.\n\n[FN#480a] A few lines higher up it was \"her neck\"; but the jar may have slipped down.\n\n[FN#481] We say more laconically \"A friend in need.\"\n\n[FN#482] Arab.  \"N\u00e1zir al-Maw\u00e1r\u00eds,\" the employ\u00e9 charged with the disposal of legacies and seizing escheats to the Crown when Moslems die intestate.  He is usually a prodigious rascal as in the text.  The office was long kept up in Southern Europe, and Camoens was sent to Macao as \"Provedor dos defuntos e  ausentes.\"\n\n[FN#483] Sir R. F. Burton has since found two more of \"Galland's\" tales in an Arabic text of The Nights, namely, Aladdin and Zeyn al-Asnam.\n\n[FN#484] i.e. wondering; thus Lady Macbeth says:\n\n          \"You have displaced the mirth, broke the good meeting,\n          With most admired disorder.\"---Macbeth, iii. 4\n\n[FN#485] Ludovicus Vives, one of the most learned of Spanish\nauthors, was born at Valentia in 1492 and died in 1540.\n\n[FN#486] There was an older \"T\u00fat\u00ed N\u00e1ma,\" which Nakhshab\u00ed\nmodernised, made from a Sanskrit story-book, now lost, but its\nmodern representative is the \"Suka Saptat\u00ed,\" or Seventy (Tales) of a Parrot in which most of Nakhshabi's tales are found.\n\n[FN#487] According to Lescallier's French translation of the\n\"Bakhty\u00e1r N\u00e1ma,\" made from two MSS. = \"She had previously had a lover, with whom, unknown to her father, she had intimate\nrelations, and had given birth to a beautiful boy, whose\neducation she secretly confided to some trusty servants.\"\n\n[FN#488] There is a slight mistake in the passage in p. 313\nsupplied from the story in vol. vi.  It is not King Shah Bakht, but the other king, who assures his chamberlain that \"the lion\" has done him no injury.\n\n[FN#489] Such was formerly the barbarous manner of treating the insane.\n\n[FN#490] From \"Tarlton's Newes out of Purgatorie.\"\n\n[FN#491] A basket\n\n[FN#492] In the fabliau \"De la Dame qui atrappa un Pr\u00eatre, un\nPr\u00e9v\u00f4t, et un Forestier\" (or Constant du Hamel), the lady, on the pretext that her husband is at the door, stuffs her lovers, as they arrive successively, unknown to each other, into a large tub full of feathers and afterwards exposes them to public ridicule.\n\n[FN#493] Until.\n\n[FN#494] Requite.\n\n[FN#495] Accidents.\n\n[FN#496] A boarding.\n\n[FN#497] The letter I is very commonly substituted for \"ay\" in\n16th century English books.\n\n[FN#498] Oesterley mentions a Sanskrit redaction of the Vampyre Tales attributed to Sivad\u00e1sa, and another comprised in the \"Kath\u00e1rnava.\"\n\n[FN#499] And well might his sapient majesty \"wonder\"!  The humour of this passage is exquisite.\n\n[FN#500] In the Tamil version (Babington's translation of the\n\"Ved\u00e1la Kadai\") there are but two brothers, one of whom is\nfastidious in his food, the other in beds: the latter lies on a bed stuffed with flowers, deprived of their stalks.  In the\nmorning he complains of pains all over his body, and on examining the bed one hair is found amongst the flowers.  In the Hind\u00ed version, the king asks him in the morning whether he had slept comfortably.  \"O great King,\" he replied; \"I did not sleep all night.\"  \"How so?\" quoth he.  \"O great King, in the seventh fold of the bedding there is a hair, which pricked me in the back, therefore I could not sleep.\"  The youth who was fastidious about the fair sex had a lovely damsel laid beside him, and he was on the point of kissing her, but on smelling her breath he turned away his face, and went to sleep.  Early in the morning the king (who had observed through a lattice what passed) asked him, \"Did you pass the night pleasantly?\"  He replied that he did not, because the smell of a goat proceeded from the girl's mouth, which made him very uneasy.  The king then sent for the procuress and ascertained that the girl had been brought up on goat's milk.\n\n[FN#501] M\u00e9lusine: Revue de Mythologie, Litt\u00e9rature Populaire,\nTraditions, et Usages.  Dirig\u00e9e par H. Gaidoz et E. Rolland.--\nParis.\n\n[FN#502] The trick of the clever Magyar in marking all the other sleepers as the king's mother had marked himself occurs in the folk-tales of most countries, especially in the numerous versions of the Robbery of the King's Treasury, which are brought together in my work on the Migrations of Popular Tales and Fictions (Blackwood), vol. ii., pp. 113-165.\n\n[FN#503] A mythical saint, or prophet, who, according to the\nMuslim legend, was despatched by one of the ancient kings of\nPersia to procure him some of the Water of Life.  After a tedious journey, Khizr reached the Fountain of Immortality, but having drank of its waters, it suddenly vanished.  Muslims believe that Khizr still lives, and sometimes appears to favoured individuals, always clothed in green, and acts as their guide in difficult enterprises.\n\n[FN#504] \"Spake these words to the king\"--certainly not those\nimmediately preceding! but that, if the king would provide for\nhim during three years, at the end of that period he would show Khizr to the king.\n\n[FN#505] Mr. Gibb compares with this the following passage from Boethius, \"De Consolatione Philosophi\u00e6,\" as translated by\nChaucer:  \"All thynges seken ayen to hir propre course, and all thynges rejoysen on hir retourninge agayne to hir nature.\"\n\n[FN#506] In this tale, we see, Khizr appears to the distressed in white raiment.\n\n[FN#507] In an old English metrical version of the \"Seven Sages,\" the tutors of the prince, in order to test his progress in general science, secretly place an ivy leaf under each of the four posts of his bed, and when he awakes in the morning--\n\n          \"Par fay!\" he said, \"a ferli cas!\n          Other ich am of wine y-drunk,\n          Other the firmament is sunk,\n          Other wexen is the ground,\n          The thickness of four leav\u00e8s round!\n          So much to-night higher I lay,\n          Certes, than yesterday.\"\n\n[FN#508] See also the same story in The Nights, vols. vii. and\nviii., which Mr. Kirby considers as probably a later version.\n(App. vol. x. of The Nights, p. 500).\n\n[FN#509] So, too, in the \"Bah\u00e1r-i-D\u00e1nish\" a woman is described as being so able a professor in the school of deceit, that she could have instructed the devil in the science of stratagem: of another it is said that by her wiles she could have drawn the devil's claws; and of a third the author declares, that the devil himself would own there was no escaping from her cunning!\n\n[FN#510] There is a similar tale by the Spanish novelist Isidro de Robles (circa 1660), in which three ladies find a diamond ring in a fountain; each claims it; at length they agree to refer the dispute to a count of their acquaintance who happened to be close by.  He takes charge of the ring and says to the ladies, \"Whoever in the space of six weeks shall succeed in playing off on her husband the most clever and ingenious trick (always having due regard to his honour) shall possess the ring; in the meantime it shall remain in my hands.\"  (See Roscoe's \"Specimens of the Spanish Novelists,\" Chandos edition, p. 438 ff.)  This story was probably brought by the Moors to Spain, whence it may have passed into France, since it is the subject of a faliau, by Haisiau the trouv\u00e8re, entitled \"Des Trois Dames qui trouverent un Anel,\" which is found in M\u00e9on's edition of Barbazan, 1808, tome iii. pp. 220-229, and in Le Grand, ed. 1781, tome iv. pp. 163-165.\n\n[FN#511] Idiots and little boys often figure thus in popular\ntales:  readers of Rabelais will remember his story of the Fool and the Cook; and there is a familiar example of a boy's\nprecocity in the story of the Stolen Purse--\"Craft and Malice ofWomen,\" or the Seven Wazirs, vol. vi. of The Nights.\n\n[FN#512] I have considerably abridged Mr. Knowles' story in\nseveral places.\n\n[FN#513] A species of demon.\n\n[FN#514] This is one of the innumerable parallels to the story of Jonah in the \"whale's\" belly which occur m Asiatic fictions. See, for some instances, Tawney's translation of the \"Kath\u00e1 Sarit S\u00e1gara,\" ch. xxxv. and lxxiv.; \"Indian Antiquary,\" Sept. 1885, Legend of Ahl\u00e1; Miss Stokes' \"Indian Fairy Tales,\" pp. 75, 76; and Steel and Temple's \"Wide-Awake Stories from the Panj\u00e1b and Kashm\u00edr,\" p. 411. In Lucian's \"Vera Historia,\" a monster fish swallows a ship and her crew, who live a long time in the extensive regions comprised in its internal economy. See also Herrtage's \"Gesta Romanorum\" (Early English Text Society), p. 297.\n\n[FN#515] In the Arabian version the people resolve to leave the choice of a new king to the royal elephant because they could not agree among themselves (vol. i., p. 323), but in Indian fictions such an incident frequently occurs as a regular custom. In the \"Sivandhi Sthala Purana,\" a legendary account of the famous temple at Trichinopoli, as supposed to be told by Gautama to Matanga and other sages, it is related that a certain king having mortally offended a holy devotee, his capital and all its inhabitants were, in consequence of a curse pronounced by the enraged saint, buried beneath a shower of dust. \"Only the queen escaped, and in her flight she was delivered of a male-child. After some time, the chiefs of the Chola kingdom, proceeding to elect a king, determined, by the advice of the saint, to crown whomsoever the late monarch's elephant should pitch upon. Being turned loose for this purpose, the elephant discovered and brought to Trisira-m\u00e1l\u00ed the child of his former master, who accordingly became the Chola king.\" (Wilson's Desc. Catal. of Mackenzie MSS., i. 17.) In a Manipur\u00ed story of two brothers, Turi and Basanta--\"Indian Antiquary,\" vol. iii.--the elder is chosen king in like manner by an elephant who meets him in the forest, and takes him on his back to the palace, where he is immediately placed on the throne. See also \"Wide-Awake Stories from the Panj\u00e1b and Kashm\u00edr,\" by Mrs. Steel and Captain Temple, p. 141; and Rev.\nLal Behari Day's \"Folk-Tales of Bengal,\" p. 100 for similar\ninstances. The hawk taking part, in this story, with the elephant in the selection of a king does not occur in any other tale known to me.\n\n[FN#516] So that their caste might not be injured. A dhob\u00ed, or\nwasherman, is of much lower caste than a Br\u00e1hman or a Khshatriya.\n\n[FN#517] A responsible position in a r\u00e1j\u00e1's palace.\n\n[FN#518] \"And Jonah was in the belly of the fish three days and three nights.\" R\u00e1j\u00e1 Amb\u00e1 must have been fully twelve years in the stomach of the alligator.\n\n[FN#519] This device of the mother to obtain speech of the king is much more natural than that adopted in the Kashmiri version.\n\n[FN#520] The story of Ab\u00fa S\u00e1bir (see vol. i. p. 81 ff.) may also be regarded as an analogue. He is unjustly deprived of all his possessions, and, with his wife and two young boys, driven forth of his village. The children are borne off by thieves, and their mother forcibly carried away by a horseman. Ab\u00fa S\u00e1bir, after many sufferings, is raised from a dungeon to a throne. He regains his two children and his wife, who had steadfastly refused to cohabit with her captor.\n\n[FN#521] Introduction to the romance of \"Torrent of Portingale,\" re-edited (for the Early English Text Society, 1886) by E. Adam, Ph.D., pp. xxi. xxii.\n\n[FN#522] Morning.\n\n[FN#523] Bird.\n\n[FN#524] Mean; betoken.\n\n[FN#525] Thee.\n\n[FN#526] Tho: then.\n\n[FN#527] Yede: went.\n\n[FN#528] Case.\n\n[FN#529] Avaunced: advanced; promoted.\n\n[FN#530] Holpen: helped.\n\n[FN#531] Brent: burnt.\n\n[FN#532] But if: unless.\n\n[FN#533] To wed: in pledge, in security.\n\n[FN#534] Beth: are.\n\n[FN#535] Or: either.\n\n[FN#536] Lever dey: rather die.\n\n[FN#537] Far, distant.\n\n[FN#538] Unless.\n\n[FN#539] Oo: one.\n\n[FN#540] Ayen: again.\n\n[FN#541] Or: ere, before.\n\n[FN#542] Army; host.\n\n[FN#543] Part.\n\n[FN#544] That.\n\n[FN#545] Grief, sorrow.\n\n[FN#546] Poor.\n\n[FN#547] Gathered, or collected, together.\n\n[FN#548] Arms; accoutrements; dress.\n\n[FN#549] Bravely.\n\n[FN#550] Those.\n\n[FN#551] Done, ended.\n\n[FN#552] Their lodgings, inn.\n\n[FN#553] Since.\n\n[FN#554] Comrades.\n\n[FN#555] Truly.\n\n[FN#556] Lodged.\n\n[FN#557] Inn.\n\n[FN#558] Hem: them.\n\n[FN#559] Chief of the army.\n\n[FN#560] I note: I know not.\n\n[FN#561] Nor.\n\n[FN#562] Place.\n\n[FN#563] That is by means of his hounds.\n\n[FN#564] A wood.\n\n[FN#565] Those.\n\n[FN#566] Her: their.\n\n[FN#567] Looks towards; attends to.\n\n[FN#568] Give.\n\n[FN#569] Excepting; unless.\n\n[FN#570] Face; countenance.\n\n[FN#571] Care; close examination.\n\n[FN#572] Palata, Lat. (Paletot, O. Fr. ), sometimes signifying a particular stuff, and sometimes a particular dress.  See Du\nCange.\n\n[FN#573] Cut; divided\n\n[FN#574] Wept.\n\n[FN#575] Sailing.\n\n[FN#576] More.\n\n[FN#577] Much.\n\n[FN#578] Sultan.\n\n[FN#579] Name.\n\n[FN#580] Voice, i.e., command.\n\n[FN#581] Slew.\n\n[FN#582] Labour.\n\n[FN#583] Drew.\n\n[FN#584] Went.\n\n[FN#585] Burning coal.\n\n[FN#586] Pray; beg.\n\n[FN#587] Recovered.\n\n[FN#588] Head.\n\n[FN#589] Weeping.\n\n[FN#590] Saw.\n\n[FN#591] Waving.\n\n[FN#592] Began to climb.\n\n[FN#593] Against.\n\n[FN#594] More.\n\n[FN#595] From an early volume of the \"Asiatic Journal,\" the\nnumber of which I did not \"make a note of--thus, for once at\nleast, disregarding the advice of the immortal Captain Cuttle.\n\n[FN#596] \"It was no wonder,\" says this writer, \"that his (i.e.\nGalland's) version of the 'Arabian Nights' achieved a universal popularity, and was translated into many languages, and that it provoked a crowd of imitations, from 'Les Mille et Un Jours' to the 'Tales of the Genii.'\"\n\n[FN#597] This is a version of The Sleeper and the Waker--with a vengeance! Ab\u00fa Hasan the Wag, the Tinker, and the Rustic, and others thus practiced upon by frolic-loving princes and dukes, had each, at least, a most delightful \"dream.\" But when a man is similarly handled by the \"wife of his bosom\"--in stories, only, of course--the case is very different as the poor chief of police experienced. Such a \"dream\" as his wife induced upon him we may be sure he would remember \"until that day that he did creep into his sepulchre!\"\n\n[FN#598] I call this \"strikingly similar\" to the preceding\nPersian story, although it has fewer incidents and the lady's\nhusband remains a monk; she could not have got him back even had she wished; for, having taken the vows, he was debarred from returning to \"the world \" which a kalandar or dervish may do as often as he pleases.\n\n[FN#599] \"The Woman's trick against her Husband.\"\n\n\nEnd of the Project Gutenberg Etext of Supplemental Nights, Volume 2\nby Richard F. Burton\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nUTOPIA\n\n# UTOPIA\n\nSECOND EDITION\n\nThomas More\n\nTRANSLATED AND INTRODUCED BY  \nCLARENCE H. MILLER  \nWITH A NEW AFTERWORD BY  \nJERRY HARP\n\nPublished with assistance from the foundation established in memory of Oliver Baty Cunningham of the Class of 1917, Yale College.\n\nAfterword and Suggestions for Further Reading copyright \u00a9 2014 by Yale University. Translation copyright \u00a9 2001 by Clarence H. Miller.\n\nAll rights reserved.  \nThis book may not be reproduced, in whole or in part, including illustrations, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press), without written permission from the publishers.\n\nYale University Press books may be purchased in quantity for educational, business, or promotional use. For information, please e-mail sales.press@yale.edu (U.S. office) or sales@yaleup.co.uk (U.K. office).\n\nDesigned by Rebecca Gibb. Set in Adobe Garamond type by Keystone  \nTypesetting, Inc.  \nPrinted in the United States of America.\n\nLibrary of Congress Control Number: 2012954304  \nISBN 978-0-300-18610-9 (pbk.)\n\nA catalogue record for this book is available from the Library of Congress and from the British Library.\n\n10 9 8 7 6 5 4 3 2 1\n\n## CONTENTS\n\nIntroduction\n\nA Chronology of More's Life\n\nUTOPIA\n\nThomas More to Peter Giles, Greetings\n\nBook 1\n\nBook 2\n\nThomas More to His Friend Peter Giles, Warmest Greetings\n\nAfterword by Jerry Harp\n\nNotes\n\nSuggestions for Further Reading\n\nIndex\n\n## INTRODUCTION\n\nThe circumstances under which More composed _Utopia,_ as he recounts them in the opening of the book, give us some clues about one of its central issues: public service versus contemplative withdrawal. More was a busy London lawyer in the service of Henry VIII on a trade commission negotiating in the spring and early summer of 1515 in Bruges. In the midst of this activity came three months of leisure from late July to late October; the negotiations were interrupted because the Flemish ambassadors had to return to consult with their prince. Released from business and public service, More had time for thought and contemplation: he wrote what would become the second book of _Utopia,_ a description of the island and its people, customs, and form of government\u2014a sort of fantastic, Lucianic travelogue. After he returned home he wrote the first book, a semi-Platonic dialogue framework concerning the question of whether it is useful to serve as the counsellor of a prince. The first book, in other words, argues about the alternatives of engagement or retreat.\n\nUnlike Platonic dialogues, which reach solutions, or even Ciceronian dialogues, which are more open-ended but clearly suggest the superiority of one outlook, the first book of _Utopia_ comes to no conclusion. It does not take long for the reader to see that Hythloday (whose name means \"peddler of nonsense\") contradicts his position by telling the story of his sojourn in the court of Cardinal Morton, who listens to Hythloday's advice about punishing thieves and suggests that it might be tried in a modified form. Nor does he defeat More in the argument about the necessity of communism; he simply evades it by insisting that More would agree with him if he had seen how communism has transformed the \"good place,\" Utopia. But unfortunately the \"good place\" is also \"no place.\" And not everything about the \"good place\" is good: apart from its policies on euthanasia and divorce, which might be tolerated in a non-Christian society, it practices capital punishment with a harshness not far from what Hythloday condemns in Book 1; its military and especially its colonial policies also leave much to be desired. At the end of Book 2 we are left with Hythloday's passionate condemnation of the outrageous social injustices of European society, but we are not really any closer to believing that they could be cured by communism, partly because we know that it has never been introduced in society as a whole and never could be, and we are never told how it was introduced or sustained in Utopia. Even if we thought it might work in Western nations, would we want to live in such a faceless and regimented society? The citizens sometimes seem like robots; the houses and even the cities seem almost interchangeable. Apart from Utopus, we never learn the name of a single Utopian. Did Hythloday never have any special friends there? One could say that he is not interested in autobiography but only in the Utopians' economic, social, and political institutions, but should children simply be taken from their parents and moved to a different family? Should whole populations be shifted back and forth to the continental colonies to provide demographic stability?\n\nSuch facts and questions make us realize that More's Utopia does not fit the ordinary meaning of the word as it came down in modern languages, where it signifies an unreservedly \"good place\" (though it still includes the notion that it is \"no place,\" that it can never be actualized). And More's _Utopia_ should not be read (as it often has been and sometimes still is read) as presenting More's notion of a purely positive and desirable society. What the character More says was believed by the real More, though the character's range of opinion is circumscribed by the context in which he appears; the real More had opinions and ideas about issues the character does not address. The character argues at some length that it is reasonable and salutary to become the counsellor to a king\u2014a problem the real More resolved for himself when he became a member of Henry's privy council two years after _Utopia_ was first published. He also disagrees that communism would be a social panacea. He concludes by saying there are both good and bad features in Utopia: he says he disagrees with the Utopians' religious practices, their methods of warfare, and especially their communism, but, apart from communism, he does not tell us what specific features he disagrees with or why he does so. The real More does not have his character spell out these disagreements because the experience of the book is not supposed to give the reader a view of a perfect society or analyze what is good or bad about Utopia. Rather the work encourages taking a new view of social and political problems by seeing alleged (and strange) solutions to them and challenges readers to try to find out what they approve or disapprove of and why. To quote from Edward Surtz's acute and comprehensive introduction to an earlier Yale translation of _Utopia:_\n\nIs the success of _Utopia_ due to dialogue? After all, dialogue is symbolic of open-mindedness, humility, and inquiry. Somehow or other, More succeeds in involving readers in the dialogue. It is no accident that _Utopia_ ends with challenges. Is the Utopian view of war, religion, and communism really absurd? Is the Utopian vision really hopeless and unachievable? _Utopia_ therefore is an open-ended work\u2014or, better, a dialogue with an indeterminate close. More asks the right questions\u2014 which can never be answered fully.\n\nThe central character of the book, and the real More's most original character, is of course Hythloday, who practically identifies himself with Utopia, to which he is unreservedly committed. His names suggest the bipolarity of his character. He is Raphael (God's healer), and Hythloday (the peddler of nonsense): on the one hand a passionate analyst of social injustice, an intense and outraged defender of the oppressed, the poor, the sick, and the weak, a proponent of freedom from crushing toil, an enthusiastic promoter of intellectual pursuits, a supporter of equal rights for women; on the other hand, he is an advocate of inhuman social engineering, colonial exploitation, assassination, bloody warfare (however brief), \"ethnic cleansing\" of the Zapoletes, and capital punishment for someone who commits adultery twice. He is narrow-minded, unrealistic, humorless, puritanical, stubborn, tactless, and\u2014according to some of his critics\u2014even self-indulgent and narcissistic. And yet he is an energetic, credible character, whom we do not find entirely admirable or entirely repellent, like Utopia itself. But also like Utopia, he is always intense and intriguing. He asks and answers important questions, and however much or little we may like his answers, he makes us aware of the urgency of the questions.\n\nThe way Hythloday speaks reflects his character and ideas, and some of the range and tensions of his style can be perceived even in a translation. Hythloday's sentences range from turbulent and often strained complexity, when he is contrasting Europe with Utopia, to simple, straightforward ease when he is describing Utopia.\n\nWhen Hythloday imagines a session of the French king's council and projects the advice he would give, he launches into a 464-word sentence\u2014suspended, unrealistically intricate, almost interminable\u2014and ends by asking More, \"How do you imagine, my dear More, my listeners would react to this speech?\" With wry understatement More replies in four words: \"Certainly not very favorably.\" Well satisfied, Hythloday takes a deep breath and soars off into another imaginary council session about raising revenues, this time in a sentence of 926 words, a syntactical extravaganza so convoluted that he himself almost loses track of it. To Hythloday's concluding inquiry, More again replies with good-humored litotes and goes on to point out, in two- or three-line sentences, that the manner of advice is as important as the matter. Among editors and commentators, so far as I know, only J. H. Lupton has pointed out these strained, overburdened sentences, and until now among English translators only Ralph Robinson attempted to reproduce them. Nowhere else did More write Latin sentences that so deliberately go beyond what ordinary Latin syntax can bear. And Hythloday does this precisely when he brings the ideal kingdoms of Achoria and Macaria, his anticipations of Utopia, into jarring and irreconcilable conflict with the military and economic corruption of Europe. The two worlds, ideal and real, collide and the ordinary syntax accepted by speakers of Latin cannot contain them.\n\nSurely More expected his readers to be disconcerted, if not totally flummoxed, by these marathon sentences. It is not so much that Hythloday has lost his grip on reality; he understands French militarism and fiscal chicanery only too well, as the extensive commentary by Fr. Surtz will testify. Rather, his only reaction to real corruptions is to grip them in one hand and smash them into ideal solutions in the other. And the syntactic explosion leads us to the \"ideality\" of Utopia. As Richard Sylvester pointed out, \"Hythlodaeus' argument . . . moves from a firm grasp on a past historical situation [the punishment of thieves discussed at Cardinal Morton's court], to a hypothetical revision of contemporary history [the French council set over against the Achorians], and, finally, to a totally aloof fabrication [the purely imaginary council on raising money and the Macarians, near neighbors of the Utopians].\"\n\nWhen Hythloday describes his ideal commonwealth, his sentences undergo a remarkable change: they are predominantly brief, factual, straightforward, syntactically simple. Usually he is simply describing Utopian things as they are, and they are mostly simple, whether it be the doors of the houses,\n\nThere is no house which does not have a door opening on the street and a back door into the garden;6\n\nor the selection of candidates for ruler,\n\nFor each of the four quarters of the city names one person and proposes him to the senate;\n\nor the universal work at farming,\n\nFarming is the one occupation in which all of them are skilled, men and women alike;\n\nor the color of their cloaks,\n\nthroughout the island they are all of the same color, that of the natural wool;\n\nor the shifting of people to maintain uniform populations,\n\nThis limit is easily maintained by transferring persons from households with too many people to those with too few;\n\nor the distribution of goods,\n\nAnd when it is distributed equitably to everyone, it follows that no one can be reduced to poverty or forced to beg;\n\nor the lack of seeking for offices,\n\nAnyone who campaigns for public office becomes disqualified for holding any office at all;\n\nor their exclusion of lawyers,\n\nthey ban absolutely all lawyers as clever practitioners and sly interpreters of the law;\n\nor their recruitment of soldiers,\n\nIn each city they choose troops from a list of volunteers;\n\nor their strict keeping of a truce,\n\nWhen they make a truce with their enemies, they keep it so religiously that they do not violate it even under provocation;\n\nor their withholding of honor and office from those who do not believe in the immortality of the soul, divine providence, and future rewards and punishment,\n\nthey bestow no honors on such a person, they assign him to no office, they put him in charge of no public responsibility.\n\nIn such simple sentences, which I have rather randomly sampled from the second book, everything seems so balanced and rectilinear, so simple and straightforward, so effortless and obviously desirable. And such simple sentences may tend to lull us into simple unquestioning acceptance of what they say as simple fact. They tell us what the Utopians do but leave many unanswered questions about how they manage to do it. How are the four candidates for ruler chosen in each quarter of the city? What happens if someone is no good at farming or refuses to do it? Or if someone dyes his cloak? Or objects to being separated from his family and friends in a population shift? How do you know whether someone is seeking an office? The only sure sign is absolute refusal to accept it. In the absence of lawyers the judge is supposed to protect the interests of the accused. But what if the judge dislikes the defendant and admires the prosecutor? What if he is stupid? How was he chosen? How was the prosecutor chosen? Are there no rules of evidence? What if too few soldiers volunteer to fight? What happens if, during a truce, the enemy ambushes a patrol? What if someone has ambitions to be a magistrate but conceals them? What if he does not believe in the immortality of the soul but conceals his disbelief? How are the priests chosen? By whom?\n\nHythloday does not answer these questions. He considers them simply irrelevant because the difficulties they embody spring from pride, which has no place in Utopia. The institutions of the Utopians clearly cannot work unless pride is eliminated. And how is pride eliminated? By the institutions, especially the abolition of private property. The institutions cannot be introduced unless they have already been introduced. But the ease and lucidity of Hythloday's sentences tend to mask such difficulties. The thing is simply there. No need to ask how it got there or can manage to stay there.\n\nNaturally, Hythloday's syntax is not always so curt and pat, even in his description of Utopia, but when his sentences swell and become somewhat involuted and turbulent it is usually when he is contrasting the ideal life of Utopia with the corruptions of Europe, as when he condemns Europe's distribution of labor, or attitude toward gold or hunting, or futilely complex laws, or abuse of treaties, or the Zapoletan (that is, Swiss) mercenaries. But these are merely occasional aftershocks of the great quake of his marathon sentences in Book 1, and he soon reverts to the simplicity of Utopian syntax.\n\nIn Book 1 the length and complexity of Hythloday's sentences lead us to believe that More himself could agree with most of what he says until the approach to Utopia (by way of Achoria and Macaria) dissociates him from More, interrupts the debates about counsel and private property (leaving them unresolved), and frees Hythloday to present the simplicities of Utopia in simple sentences, which are so unlike More's way of thinking and writing elsewhere as to suggest that he meant us to probe them with questions\u2014not only about obvious difficulties such as Utopian warfare, divorce, euthanasia, or colonialism, but throughout. So much of what the Utopians do is admirable, but how in heaven's name do they manage to do it?\n\nThere is, for example, a glaring omission in Utopia, which Hythloday's limpid and easy sentences may cause us to overlook: the political structure of Utopia has no central executive authority for its fifty-four independent city-states. Although the whole island has no single governor, it does have an annual senate composed of three wisemen from each city. If there were any real self-centered rivalry among these cities, even any normal conflict of interest, such a senate, without any executive machinery whatever, would be quite ineffectual. We, like More, have only to look to the city-states of ancient Greece or Renaissance Italy to see what would happen. Then, too, we are told about the punishments for various crimes in Utopia (among which the absence of theft may not be surprising, but what about assault or murder?), but, without a police force, who catches the criminal? Who checks whether he has the proper papers to be out of his own city-state? What prevents him from stowing away on a ship to the mainland? Questions proliferate endlessly. We are always being brought back to the basic paradox: the institutions cannot be introduced unless they have already been introduced.\n\nAnother remarkable feature of Hythloday's style, which is related to the deceptive simplicity of Utopia, is his diction. We are not surprised that he is fond of words like \"equal\" (which he uses 26 times) or \"easy\" (24 times). After all, his main thesis is that equality of goods makes just government easy in Utopia. But another group of frequent words suggests his inability to deal with specific problems in concrete circumstances and reflects the universalist, absolute, all-or-nothing cast of his mind: \"all\" (200 times), \"nothing\" (76), \"none\" (68), \"whole\" (62), \"one\" (35), \"the same\" (33), \"any\" (33), \"no one\" (29), \"entirely\" (24), \"each\" (19), \"never\" (19), \"anything\" (17), \"everywhere\" (14), \"anywhere\" (13), \"only\" (11), \"universal\" (9), \"ever\" (8), \"never\" (7). In the samples of short sentences given above, I made no attempt to include any of these words, and yet I noticed that they had inevitably appeared (\"There is NO house\"; \"Farming is the ONE occupation in which ALL of them are skilled\"; \"Whoever seeks ANY office becomes ineligible for ALL offices\"). Moreover, as I checked the instances, I found that they often tended to occur together with other words from this absolutist cluster.\n\nA few examples must suffice to suggest the effect of such diction.\n\nThe island has fifty-four cities, ALL of them large and splendid and having EXACTLY THE SAME language, customs, institutions, and laws. They have the SAME layout and they look the SAME, insofar as the terrain allows.\n\nFrom them [the storehouses] EACH head of household goes to get whatever he and his household need and takes away WHATEVER he wants, paying no money and giving ABSOLUTELY NOTHING in exchange for it. For why should he be denied ANYTHING, since there is plenty of EVERYTHING and NO ONE need fear that ANYONE would want to ask for more than he needs? For why should ANYONE be suspected of asking for too much if he is certain he will NEVER lack for ANYTHING?\n\nOn the other hand, here, where EVERYTHING belongs to EVERYONE, NO ONE doubts that (as long as care is taken that the public storehouses are full) NOTHING WHATEVER will be lacking to ANYONE for his own use. For the distribution of goods is not niggardly; NO ONE is a pauper or a beggar there, and though NO ONE has anything, ALL are rich.\n\nSuch reiterated universalist diction is the source of what most readers feel to be objectionable about the Utopians: their faceless anonymity and homogeneity. But such diction is not characteristic of More himself, either as a character in _Utopia_ or in his other Latin writings. And it should cause us to ask questions similar to those raised by Hythloday's simple sentences: how do you get everyone always to do the same one thing everywhere, wholly and completely, without anyone anywhere at all deviating significantly in anything, with no exceptions, with no one ever wishing to contravene the universal system, with all in equal conformity, with never a dissenting voice, with nowhere a refusal to comply? Hythloday's answer is one and the same, always: introduce Utopian institutions, based on the sharing of everything. Only then will everyone be totally and completely committed to the common good (respublica). But only a people raised, educated, and trained under Utopian institutions can make the institutions work. As with Hythloday's simple sentences, we are brought back to the paradox, the dilemma, the \"double-bind\": nowhere can such institutions be introduced except where they have already been introduced\u2014nowhere.\n\nBut Hythloday should not be viewed as merely narrowminded, solipsistic, and naive. When he condemns the injustices of Europe, his voice and his sentences are not incompatible with those of More himself. His passionate denunciations of the greedy oppression of the poor and his compassionate indignation at the lazy, corrupt self-indulgence of the rich are intense and memorable; everyone remembers the sheep who were once gentle but now devour people (p. 22), or his thunderous peroration against economic and social injustice, culminating in the condemnation of the conspiracy of the rich who look out for themselves under the pretext of serving the commonwealth (pp. 132). Only when he flees such corruption and breaks through to the simplicities of Utopia do his sentences fracture Latin syntax and soar beyond what More's Latin, even at its most muscular, would attempt. And the simple sentences and universalist diction of his description of Utopia do not make him seem merely simple-minded. They also help him to make us think that this has happened, that it could happen (in spite of all our nagging doubts about how it could have happened or how it could happen in the world we know); and, even more, he makes us think that some of it should happen (in spite of the thought-provoking anomalies in Utopian behavior) because the Utopians really believe in the common good; and Hythloday makes us almost believe in their belief, and so we believe him even while we disbelieve him, just as (because of her virtuoso and contrasting styles) we believe and disbelieve his great compeer, the Folly of Desiderius Erasmus' _Moriae Encomium_.\n\n### THE EARLY EDITIONS AND THE LATIN TEXT\n\nErasmus and Peter Giles supervised the printing of the first edition (Louvain, 1516), adding commendatory and sometimes analytic letters from well-known humanists of the time; these letters, though they provide a useful context for reading _Utopia,_ have not been included in this translation because they are highly specialized and sometimes inflated. I have included More's prefatory letter to Giles and his second letter to Giles, which appeared only in the second edition, in 1517. Erasmus and Giles were also probably responsible for the marginal notes that appeared in the first edition, and they may have had a hand in changing some Latin proper and place names from Latin to Greek forms. In the early editions many of the marginal notes are merely labels to mark off sections equivalent to paragraphs (of which the early editions have none). I have omitted such paragraph markers and included only the marginal notes that make some independent comment on the text. I have supplied the paragraphing in this translation.\n\nThomas Lupset supervised the second edition (Paris, 1517), which was corrected by More. Erasmus transmitted copy for the third edition (Basel, March 1518), which was also corrected by More (sometimes differently from 1517). In fact, it is possible that More corrected 1517 later than 1518m (as the March 1518 edition is known). The Basel edition of November 1518 was simply a reprint of 1518m, as was the Florence edition of 1520. It is significant that More, Erasmus, and Johann Froben, the publisher of the Basel editions, originally intended to include the translations of Lucian by More and Erasmus, though they were never included because the volume had grown too large; Lucian's fantastic and satirical flair was one of the ingredients in the savory stew of _Utopia_. Moreover, 1518m and 1518n (the November 1518 edition) also included the Latin epigrams of More and Erasmus; several of More's concern issues of good and bad government.\n\nThe Latin text on which this translation is based is derived from all the texts and variants of the first three editions, the only ones in which More actually had a hand; these materials are presented in full in the comprehensive and definitive editions of Fr. Surtz and Andr\u00e9 Pr\u00e9vost. My Latin text corresponds very closely with the Cambridge edition of George Logan and others, which is the most accurate and usable modern text. For these editions see the list of books for further reading at the end of this volume. I have also consulted the remarks and emendations in the Latin editions by V. Michels and T. Ziegler (1895), J. H. Lupton (1895), and Marie Delcourt (1936).\n\n### A NOTE ON THE TRANSLATION\n\nSince _Utopia_ has been translated by Ralph Robinson, Gilbert Burnet, and several translators in the twentieth century, it may be asked why another translation should be undertaken. But the truth is that in spite of many translations, some of them frequently reprinted, _Utopia_ has not fared as well as it deserves in English. Robinson's pioneering translation, published in 1551, is quite accurate (with only a few exceptions) and was usually consulted by subsequent translators, but it was made at a time when English did not yet have the strength, either in diction or style, to reproduce the elaborate Latin of _Utopia_. Hence his translation, though lively and vivid, often seems wordy and awkward (as it probably also did to his contemporaries who read Latin). But at least Robinson did not omit words or phrases when they were inconvenient, and he tried to match the varied style of the Latin. The same cannot be said for those who walked in his footsteps. By the time of Gilbert Burnet's translation in 1684, English prose was a varied and powerful instrument, fully capable of matching More's Latin; but fashionable prose in the Restoration was simple, lucid, \"easy\" (in reaction against the florid, baroque \"excesses\" of the earlier part of the century). Hence Burnet streamlined and simplified More's Latin, ignoring its extreme stylistic variations, occasionally clipping and pruning it as well.\n\nG. C. Richards, the first twentieth-century translator (1923), tried like Robinson to be fully faithful to the Latin, in details and style, but he kept too close to the Latin so that his English often turned out to be awkward and unidiomatic. H. V. S. Ogden (1949) explicitly mentions the translations of Robinson and Burnet, admitting that he has borrowed wording from both, especially Burnet. In fact his translation is swift and readable (like Burnet's) but only at the cost of simplification in detail and the suppression of elaborations in More's Latin. The same false tendency is carried even further in Paul Turner's translation (1965) and further yet in the translation by John Sheehan and John P. Donnelly (1989), who professedly simplify and prune the text for American college students. The translation by Robert M. Adams (1975) is also uniformly swift and breezy, and even its corrected form (1995) does not match the stylistic variations of the Latin.\n\nIn fact, More's Latin is often anything but colloquial and easy (pace Turner and Surtz). In his first letter to Giles, More describes Hythloday's language as unpolished, informal, and extemporaneous, giving his style the label \"casual simplicity.\" But hardly anything in this letter can be taken at face value. When Hythloday speaks about Utopia itself his sentences are indeed generally simple and easy (Utopia is, after all, a simple and easy solution to social and economic problems), but when he denounces the corruptions of Europe in Book 1 and in the peroration to Book 2, his sentences are complex, lengthy, elaborate, muscular, even muscle-bound. Some translators seem to think that such complications are natural in Latin but not in English. But in fact, English can follow the normal complications of Latin well enough, and the unnatural complications in the Latin ought not to be merely smoothed out in the English. They are an important part of the meaning.\n\nHence I have tried to translate all the details of the Latin in idiomatic English that matches the simplicity, complexity, or even unnatural strain of the Latin.\n\n## A CHRONOLOGY OF MORE'S LIFE\n\n1477  \nBorn in London, February 7\n\nc. 1482\u201390  \nSchool at St. Anthony's\n\nc. 1490\u201392  \nPage in the household of John Morton, Archbishop of  \nCanterbury and chancellor of England\n\nc. 1492\u201394  \nStudent at Oxford University\n\n1494\u20131501  \nStudied law in London\n\nc. 1500\u20131504  \nTo test his vocation to the priesthood resided in Carthusian  \nmonastery next to his law school, Lincoln's Inn\n\n1504\u20135  \nMarried Joan Colt, late 1504 or early 1505\n\n1509  \nNegotiated with Antwerp merchants on behalf of London companies\n\n1511  \nJoan, who bore him four children, died, and More  \nmarried Alice Middleton\n\n1510\u201318  \nUndersheriff of London\n\n1513\u201318  \nWrote _The History of Richard III_ in both Latin and English\n\n1515  \nMember of a delegation sent to Bruges to revise  \na commercial treaty\n\n1516  \n _Utopia_ published at Louvain\n\n1517  \nHelped quell a riot by a mob of London apprentices, May 1\n\n1517  \nNegotiated with the French at Calais and Boulogne about  \nsuits arising from the recent war, September\u2013December\n\n1518  \nBecame a member of the king's council and  \nMaster of Requests\n\n1520  \nAccompanied Henry VIII to a meeting with Francis I at the  \nField of Cloth of Gold near Calais\n\n1520\u201321  \nNegotiated with Emperor Charles V and the  \nHansa merchants at Calais and Bruges\n\n1521  \nKnighted\n\n1523  \nIn _Responsio ad Lutherum_ defended Henry VIII against Luther's attack\n\n1523  \nSpeaker of the House of Commons\n\n1524  \nMoved into the large mansion he built upriver in Chelsea\n\n1525  \nChancellor of the Duchy of Lancaster\n\n1524\u201325  \nHigh Steward of the Universities of Oxford (1524) and Cambridge (1525)\n\n1528  \nCommissioned by Bishop Tunstall of London to read and refute Lutheran books in English\n\n1529\u201333  \nWrote seven polemical books in English against Lutheranism\n\n1529  \nAttended the congress at Cambrai at which peace was negotiated between France and the Empire\n\n1529\u201332  \nLord Chancellor of England\n\n1534  \nRefused to swear to the Act of Succession because it  \nrepudiated papal supremacy, April 13\n\n1534  \nImprisoned in the Tower, April 17\n\n1534\u201335  \nWrote _A Dialogue of Comfort Against Tribulation_ in the Tower\n\n1535  \nConvicted of treason on the perjured evidence of  \nRichard Rich, July 1\n\n1535  \nBeheaded in the Tower, July 6\nUTOPIA\nOn the Best Form of a Commonwealth  \nand  \non the New Island of Utopia\n\na Truly Precious Book  \nNo Less Profitable than Delightful  \nby  \nthe Most Distinguished and Learned Gentleman  \nThomas More  \nCitizen and Undersheriff  \nof the Illustrious City of London\n\n_A Six-line Stanza on the Island of Utopia  \nby the Poet Laureate  \nAnemolius  \nThe Son of Hythloday's Sister_\n\nCalled once \"No-place\" because I stood apart.  \nNow I compete with Plato's state, perhaps  \nSurpass it; what he only wrote about  \nI have alone in fact become: the best  \nIn people, wealth, in laws by far the best.  \n\"Good-place\" by rights I should be called.\n\n### THE UTOPIAN ALPHABET A QUATRAIN IN THE UTOPIAN LANGUAGE\n\nThe literal meaning of these lines:  \nWhen I was not an island, the commander Utopus made me into an island. I alone of all the nations on earth, without philosophy, have presented to mortals a philosophical state. Freely I share what I have; not unwillingly I accept what is better.\n\n## Thomas More to Peter Giles, Greetings\n\nI am almost ashamed, my dear Peter Giles, to have delayed for almost a year in sending you this little book about the Utopian commonwealth, which I'm sure you expected within six weeks. You knew, after all, that I was spared the labor of finding my matter, and did not have to give any thought to its arrangement; all I had to do was repeat what you and I heard Raphael say. For that reason there was no need to strive for eloquence, since his language could hardly be polished, first because it was informal and extemporaneous, and also because he is a person, as you know, not as well versed in Latin as in Greek; the closer my language came to his casual simplicity, the more accurate it would be, and in this matter accuracy is all that I ought to, and in fact do, aim for.\n\nI grant you, Peter, that with all this already taken care of, I was relieved of so much effort that there was almost nothing left for me to do. If this had not been so, thinking up the subject matter and arranging it might have required not a little time and study, even from someone of not inconsiderable intelligence and not totally without learning. But if I had been required to write not only accurately but also elegantly, no amount of time or study would have enabled me to do it. As it is, all these concerns, which would have cost me so much labor, are removed and all that remained to do was to write what I heard\u2014not a difficult task.\n\nBut nevertheless, even to perform this trifling task, other chores left me almost no time at all. I am constantly pleading one case, hearing another, acting as arbitrator, handing down decisions as a judge, visiting one person or another on business or because it is my duty to do so; I am out practically all day dealing with others, and the rest of my time is devoted to my family, and so I leave nothing for myself, that is for writing.\n\nWhen I get home, I have to talk with my wife, chat with my children, confer with the servants. All this I count as part of my obligations, since it needs to be done (and it does if you do not wish to be a stranger in your own home); and you must do everything you can to make yourself as agreeable as possible to the persons you live with, whether they were provided by nature, chance, or your own choice, as long as you do not spoil them by your familiarity or turn servants into masters through over-indulgence. As I am doing such things, as I said, a day, a month, a year slips by.\n\nWhen do I write then? And as yet I have said nothing about sleep and nothing at all about eating, and for many that takes up no less time than sleep itself, which consumes almost half our lives. The only time I get for myself is what I steal from sleep and eating. Because that is so little, I progressed slowly, but because it was at least something, I did make progress, and I sent _Utopia_ to you, my dear Peter, so that you can read it and let me know if I have missed anything. For, though on that score I do not lack all confidence in myself (and I only wish that my intelligence and learning were a match for my not inconsiderable memory), still I am not confident enough to think that nothing has escaped me.\n\nAs you know, John Clement, my young assistant, was there with us, for I do not allow him to miss out on any conversation which could be profitable to him because from this sprout which is beginning to grow green with proficiency in Latin and Greek I expect someday a marvelous harvest. He has made me feel very doubtful about one point: as far as I remember Hythloday told us that the bridge which spans the river Anyder at Amaurot is five hundred yards long, but my boy John says that is two hundred yards too many and that the river is no more than three hundred wide. Please try to remember that point. For if you agree with him, I will go along with you both and believe I am mistaken. But if you do not recall, I will stand by what I think I remember myself, for just as I have taken great pains to prevent any inaccuracy in the book, so too, when I am in doubt, I would rather say something inaccurate than tell a lie, because I would rather be honest than clever.\n\nIn fact, it would be easy to remedy this defect if you would find out from Raphael himself about it, in person or by letter. And you need to do the same concerning another difficulty which has arisen\u2014who is more to blame for it, I or you or Raphael himself, I do not know. For it did not occur to us to ask, or him to mention, in what part of that new world Utopia is located. Indeed, to remedy this oversight I would be willing to give a sizeable sum, partly because I am ashamed not to know in which ocean the island lies about which I have recounted so much, partly because there are one or two people here, but especially one person, a devout man and a theologian by profession, who is amazingly eager to go to Utopia, not out of idle curiosity or any hankering after novelties but in order to nourish and spread our religion, which has made such a good beginning there. To do this properly he has decided to see to it beforehand that he is sent by the pope and made the bishop of the Utopians. He has no scruples whatever about begging for this bishopric, since he considers such ambition to be holy if it is not based on honor or gain but rather springs from piety.\n\nTherefore, my dear Peter, I beg you to contact Hythloday, either in person if that is convenient or by letter if you are separated, and see to it that this work of mine contains nothing false and lacks nothing true. And perhaps it would be best to show him the book. For there is no one else capable of correcting any errors and even he cannot do so unless he reads through what I have written. Then too, this will let you see whether he is pleased or annoyed at me for writing this work. For if he himself has decided to commit his labors to writing, he may not want me to do so. And I certainly would not want to deprive his narrative of the bloom and charm of novelty by making the commonwealth of Utopia public.\n\nBut in fact, to tell you the truth, I myself have not yet made up my mind whether or not to publish it at all. For the tastes of mortals are so various, the temperaments of some are so bitter, their minds so ungrateful, their judgments so preposterous that a person would do far better to follow his own bent and lead a merry life than to wear himself out trying to publish something useful or entertaining for an audience so finicky and ungrateful. Most people know nothing about learning; many despise it. Dummies reject as too hard whatever is not dumb. The literati look down their noses at anything not swarming with obsolete words. Some like only ancient authors; many like only their own writing. One person is so dour that he cannot abide jokes; another is so witless that he cannot stand anything witty. Some have so little nose for satire that they dread it the way someone bitten by a rabid dog fears water. Others are so changeable that their approval depends on whether they are sitting down or standing up.\n\nThey sit around in taverns and over their cups they pontificate about the talents of writers, condemning each author just as they please, pulling him down through his writings as if they had grabbed him by the hair, while they themselves are safe and out of harm's way, as the saying goes, because these good men have their whole heads smooth-shaven so that there is not a single hair to grab on to.\n\nFurthermore, some are so ungrateful that, even though a work has given them great pleasure, they still do not like the author any better because of it. They are not unlike illmannered guests who, after they have been lavishly entertained at a splendid banquet, finally go home stuffed without saying a word of thanks to the host who invited them. Go on, now, and at your own expense provide a banquet for persons of such delicate palates and various tastes, who will remember and repay you with such gratitude!\n\nNevertheless, my dear Peter, raise with Hythloday the points I mentioned. Afterwards I will be free to consider the matter once more. But in fact, if he himself gives his consent\u2014 since it is late to be wise now that I have finished all the work\u2014in all other considerations about publishing I will follow the advice of my friends, and especially yours. Farewell, my dearest Peter Giles, with regards to your excellent wife, and be as fond of me as ever, since I am fonder of you than ever.\nA Discourse  \non the Best Form of a Commonwealth  \nSpoken by the Remarkable Raphael Hythloday\n\nas Reported by  \nthe Illustrious Thomas More  \na Citizen and the Undersheriff  \nof the Famous British City of London\n\n## BOOK 1\n\nRecently the invincible king of England, Henry the eighth of that name, who is lavishly endowed with all skills necessary for an outstanding ruler, had some matters of no small moment which had to be worked out with Charles, the most serene prince of Castile. To discuss and resolve these differences he sent me to Flanders as his ambassador; I was the companion and colleague of the incomparable Cuthbert Tunstall, whom he recently appointed to be Master of the Rolls, to the enormous satisfaction of everyone. I will say nothing in his praise, not because I am afraid that my friendship might seem to make me an unreliable witness, but because his virtue and learning are beyond my power to proclaim them and because they are everywhere so renowned and well known that there is no need for me to do so, unless I intend to display the sun by the light of a lantern, as they say.\n\nAs had been agreed, we were met at Bruges by those to whom the prince had entrusted the negotiations, all of them outstanding men. Their leader and chief was the Mayor of Bruges, a splendid man, but their spokesman and mastermind was George de Themsecke, the Provost of Cassel, who is not only a trained orator but also a naturally eloquent speaker; he is very skilled in the law as well, and also an extraordinarily deft negotiator because he is both intelligent and very experienced. After one or two meetings we could not reach agreement on some points, and so they bade us farewell for some days and set out for Brussels to ask for the pronouncement of their prince.\n\nMeanwhile, as my business required, I made my way to Antwerp. While I was staying there, I was often visited by Peter Giles, among others, though no other visitor was more delightful to me. A native of Antwerp, he holds a post of great responsibility and prestige (and he is worthy of the most prestigious), since for this young man it would be hard to say which is greater, his learning or his virtue. For he is most virtuous and very widely read, and also good-natured toward everyone, but toward his friends he is so responsive, warmhearted, loyal, and unfeignedly affectionate that it would be hard to find even one or two anywhere that you would think comparable to him in every aspect of friendship. He has a modesty rarely to be found; no one is further from false poses; no one combines more prudence with simplicity. Then, too, his elegant speech and his innocent wit are so attractive that his delightful companionship and his charming conversation alleviated my longing for my country, household, wife, and children, though I was tormented by my desire to see them again, for at that time I had been away from home for more than four months.\n\nOne day, after I had heard mass at the church of St. Mary, which is remarkable for its beautiful architecture and its large congregation, when the service was over and I was getting ready to return to my lodgings, I happened to see Giles conversing with a stranger who was getting up in years. His face was sunburned, his beard untrimmed, his cloak hanging carelessly from his shoulder; from his face and bearing I thought he looked like a sea captain. But then, when Peter saw me, he came up and greeted me. When I tried to answer, he took me a little aside and said, \"Do you see this man?\" (At the same time he indicated the person I had seen him talking to.) \"He is the one,\" he said, \"I was just getting ready to bring straight to you.\"\n\n\"He would have been all the more welcome to me on your account.\"\n\n\"Actually on his own,\" he said, \"if you knew him. For there is no mortal alive today who can give more information about unknown peoples and lands, and I know that you are very eager to hear about them.\"\n\n\"My guess was not far off, then,\" I said, \"for when I first set eyes on him, I immediately thought he was a sea captain.\"\n\n\"But in fact,\" he said, \"you were far off the mark. Certainly he has sailed, not like Palinurus, but rather like Ulysses, or even better like Plato. This man, who is named Raphael\u2014his family name is Hythloday\u2014has no mean knowledge of the Latin language but is especially proficient in Greek; he has devoted himself to Greek more than to Latin because he has totally committed himself to philosophy and he knew that in that field there is nothing of any importance in Latin except some works of Seneca and Cicero. Out of a desire to see the world he left to his brothers his heritage in his homeland (he is from Portugal), joined Amerigo Vespucci, and was his constant companion in the first three of the four voyages which everyone is now reading about; but on the last voyage he did not come back with him. He sought and practically wrested from Amerigo permission to be one of the twenty-four who were left behind in a fort at the farthest point of the last voyage. And so he was left behind in accordance with his outlook, since he was more concerned about his travels than his tomb. Indeed he often used to say, 'Whoever does not have an urn has the sky to cover him,' and 'from everywhere it is the same distance to heaven.' This attitude of his would have cost him dearly if God had not been merciful to him. However, after the departure of Vespucci, he traveled through many lands with five companions from the fort, and finally, by an extraordinary stroke of luck, he was transported to Ceylon and from there he reached Calicut, where he opportunely found some Portuguese ships and at last, beyond all expectation, he got home again.\"\n\nWhen Peter had told me this I thanked him for his kindness in taking so much trouble to introduce me to someone whose conversation he hoped I would enjoy, and then I turned to Raphael. After we had greeted each other and spoken the usual amenities that are exchanged when strangers meet for the first time, we went off to my house, where we conversed sitting in the garden on a bench covered with grassy turf.\n\nAnd so he told us how, after the departure of Vespucci, he and his companions who had remained in the fort gradually began to win the good graces of the people of that land by encountering and speaking well of them, and then they started to interact with them not only with no danger but even on friendly terms, and finally they gained the affection and favor of some ruler, whose name and country escape me. He told how, through the generosity of the ruler, he and five of his companions were liberally supplied with provisions and ships on the sea and wagons on the land\u2014together with a trustworthy guide who took them to other rulers to whom he heartily recommended them. After many days' journey, he said, he discovered towns and cities and commonwealths that were very populous and not badly governed.\n\nOn both sides of the equator, it is true, extending almost as far as the space covered by the orbit of the sun there lie vast empty wastelands, scorched with perpetual heat. The whole region is barren and ugly, rugged and uncultivated, inhabited by wild beasts and serpents and by people who are no less wild than the beasts and no less dangerous. But when you have traveled further, everything gradually becomes milder. The heavens are less fierce, the ground is green and pleasant, the creatures are more gentle, and finally one sees peoples, cities, towns, which not only trade continually among themselves and with near neighbors but also carry on commerce with distant nations by land and sea. From that point on they were able to visit many countries in all directions since there was no ship traveling anywhere in which he and his comrades were not eagerly welcomed.\n\nHe told us that in the first regions they traveled they saw flat-bottomed vessels, spreading sails made of wickerwork or of stitched papyrus, and in other places of leather. But afterwards they found ships with curved keels, canvas sails, and in fact all the features of our own vessels. The sailors were not unskilled in seamanship and celestial navigation, but he told us that they were extremely grateful to him for introducing them to the magnetic compass, with which they had been totally unfamiliar. For that reason they usually were afraid to commit themselves to the open sea and they did not venture to do so except during the summer. But now they have such confidence in the compass that they scorn the winter weather and are careless rather than secure; thus there is a danger that the device which they thought would do them so much good will do them great harm because of their imprudence.\n\nTo present what he told us about the things he saw in each and every place would take a long time and would be beyond the scope of this work. And perhaps I will speak of it elsewhere, especially those points of which it would be useful not to be ignorant, above all whatever correct and prudent provisions he observed among civilized nations. We asked him very eagerly about such matters, and he was quite willing to explain them, but we paid no attention to monsters, for nothing is less novel than they are. Indeed, there is almost no place where you will not find Scyllas and rapacious Celaenos and man-eating Laestrigonians and such prodigious monsters, but it is not everywhere that you will find soundly and wisely trained citizens. But just as he noted many ill-considered practices among those newly discovered nations, so too he recounted not a few features that could serve as patterns to correct the errors of our own cities, nations, peoples, and kingdoms. These, as I said, will have to be presented elsewhere. At present I intend to relate only what he told us about the customs and institutions of the Utopians, but first I will present the conversation which led him on, as it were, to mention that commonwealth. For after Raphael had very judiciously analyzed some of our errors and some of theirs (and certainly there are plenty in both places) and had presented some wiser provisions both here and there\u2014and he had such a mastery of the customs and institutions of every nation he visited that you would imagine he had spent his whole life there\u2014Peter was amazed by him and said, \"My dear Raphael, why do you not enter into the service of some king, for I am convinced that there is none who would not be extremely glad to have you, because this learning of yours and your knowledge of peoples and places would not only serve to delight him but would also make you fit to inform him of precedents and aid him with advice. In this manner you could at one and the same time promote your own interests enormously and be of great assistance to your relatives and friends.\"\n\n\"As for my relatives and friends, I am not much concerned about them because I have done my duty by them well enough: others do not give up their possessions until they are old and sick, and even then they do so reluctantly, when they can no longer retain them; but I divided my possessions up among my relatives and friends when I was not only healthy and vigorous but also young. I think they ought to be satisfied with my generosity, and beyond that they should not demand and expect me to hand myself over into servitude to kings for their sake.\"\n\n\"A fine thing to say,\" said Peter. \"I want you to go into the service of kings, not be in servitude to them.\"\n\n\"There is,\" he said, \"only one syllable's difference between them.\"\n\n\"But I am of the opinion,\" said Peter, \"that, whatever name you give it, it is still the course by which you can not only profit others, both privately and publicly, but also make your own position a happier one.\"\n\n\"Would I make it happier by following a course which is abhorrent to me? But as it is, I live as I please, and I certainly suspect that is very seldom the case with the grandees of court. Surely there are plenty of people who strive to gain the favor of powerful men, so that you need not consider it any great loss if I and one or two like me are not among them.\"\n\nThen I said, \"It is clear, my dear Raphael, that you are not greedy for wealth or power; I respect and revere a person with your attitude no less than I do any of the high and mighty. But it seems obvious to me that you would be acting in a fashion worthy of yourself and of your noble and truly philosophical nature if you could bring yourself to apply your intelligence and industry to public affairs, even at the cost of some private inconvenience. You will never be able to do this to such good effect as you could if you became a counsellor to some great prince and urged upon him what is right and honorable, as I am sure you would. For the stream of good and evil, as if from a never-failing spring, flows from the prince down upon the whole people. And your learning is so complete, even if you had no great experience, and your experience is so full, even if you had no learning at all, that you would be an outstanding counsellor to any king whatever.\"\n\n\"You are wrong on two counts, my dear More,\" he said. \"First about me, and then about the way things are. For I do not have the ability you attribute to me, and even if I had it in full measure, I would sacrifice my contemplative leisure to active endeavor without contributing anything to the common good. First of all, the princes themselves, almost all of them, are more devoted to military pursuits (in which I neither have nor desire any skill) than they are to the beneficent pursuits of peacetime; and they are far more interested in how to acquire new kingdoms by hook or crook than in how to govern well those they have already acquired. Moreover, among the counsellors to kings, there is none who is not so truly wise as not to need\u2014or at least thinks he is so wise as not to tolerate\u2014the advice of any other counsellor, except that they support and fawn on any and all absurdities propounded by the prince's favorites, whose favor they strive to win by flattery. Certainly nature seems to have arranged it so that everyone is delighted with his own insights. So the crow dotes on its chick, and the monkey on its whelp.\n\n\"But in a conclave made up of those who envy the insights of others or exalt their own, if anyone should propose something which he has read was done in other eras or which he has seen done in other places, his listeners there immediately act as if their whole reputation for wisdom were at risk, as if they would thereafter be considered totally stupid if they cannot propose something to undermine the proposals of others. If all else fails, then this is their last resort: these things pleased our ancestors, they say, and would that we were as wise as they! And with this remark they take their seat thinking they have said the last word on the subject, as if it were a very dangerous matter if anyone were detected to be wiser than his ancestors on any point. In fact if those ancestors have instituted some truly excellent policy, we are quite content to dismiss it. But if they might have taken a wiser course on some point, we immediately and eagerly seize the pretext of tradition to maintain it. And I have encountered such arrogant, absurd, and captious judgments often enough in other places, but once even in England.\"\n\n\"What,\" I said, \"you were in our country?\"\n\n\"I was,\" he said, \"and I spent some months there, not long after the revolt of the Englishmen from the west against the king was put down with such a miserable slaughter of the rebels. While there I was much obliged to the most Reverend Father John Morton, Cardinal Archbishop of Canterbury, and at that time also Lord Chancellor of England. He was a man, my dear Peter (for More already knows what I am about to say) no more venerable for his authority than for his prudence and character. He was of medium height, not stooped over though he was of an advanced age. His looks inspired reverence, not fear. In company he was not standoffish, but grave and serious. Sometimes he enjoyed handling suitors roughly, but harmlessly, so as to gauge the intelligence and presence of mind each would display. He was delighted with such qualities, provided they were devoid of all impudence, since they were related to his own character, and he embraced them as valuable in getting things done. His speech was polished and pointed; he was very skilled in the law; his intelligence was incomparable; his memory was so excellent as to be prodigious. These extraordinary natural gifts he had improved by study and practice. The king seemed to rely very much on his advice and while I was there he seemed to be the mainstay of the commonwealth. This was not surprising: thrust immediately from school into the court at a very young age, active in important affairs throughout his life, continually whirled about by violent changes of fortune, he had learned practical wisdom in the midst of many and serious perils, and wisdom so won is not easily forgotten.\n\n\"One day when I happened to be dining at his table, a layman who was skilled in the laws of your country was there. Following up some remark or other, he launched on a elaborate encomium of the rigorous justice which was at that time applied to thieves in England. They were executed everywhere, he said, sometimes as many as twenty at a time hanging on one gallows, and he remarked that he was all the more amazed that the country was cursed to have so many of them prowling about everywhere, since so few escaped punishment. Then I said (and I dared to speak my mind freely in the presence of the Cardinal): 'You should not be at all surprised. For this punishment of thieves is both beyond the limits of justice and not in the public interest. As a punishment for theft it is too harsh, and even so it is not a sufficient deterrent: simple theft is not so serious a crime as to deserve capital punishment, and no penalty is great enough to keep people from stealing if they have no other way to make a living. Thus, in this matter, not only you but most of the world seem to imitate bad teachers who are more eager to beat their pupils than to instruct them. For heavy and horrible punishments are imposed on thieves when it would be much better to make some provision for their livelihood, so that no one should labor under the cruel necessity first of stealing and then of dying for it.'\n\n\" 'We have made sufficient provision for that,' he said. 'There are trades; there is farming. From them they can make a living, as long as they do not willingly prefer to be criminals.'\n\n\" 'You will not get out of it that way,' I said. 'First of all, we will overlook the many soldiers who come home crippled from foreign or domestic wars, as they recently did from the battle against the Cornishmen and not long before that from the French wars. They have sacrificed their limbs for the commonwealth or the king; their disability does not allow them to practice their former trades and they are too old to learn a new one. These,' I said, 'let us overlook, since wars happen only now and then. Let us consider what is never not happening. Now there is a multitude of noblemen who not only live like drones on the labor of others\u2014namely the tenants of their estates whom they bleed white by raising their rents (for this is the only kind of frugality they recognize, and otherwise they are so prodigal as to reduce themselves to beggary)\u2014but they also travel with a huge crowd of retainers, none of whom has ever learned how to make a living. As soon as their master dies or they get sick, they are immediately thrown out. For lords would rather support idle men than invalids, and often the heir of a dying master cannot support a household as large as his father's, at least at first. Meanwhile the outcasts vigorously starve unless they vigorously steal. For what are they to do? After tramping around a bit they will have ruined their clothes and their health. Disfigured as they are by disease and clad in rags, no nobleman will deign to take them in and no farmer dares to do so. For the farmers are not unaware that a person who has been brought up in idle ease and pleasure and who has been used to swaggering about like a bully, girt with sword and buckler, looking down his nose at the whole neighborhood and despising everyone but himself, is hardly likely to be a reliable and faithful servant for a poor farmer, working with hoe and mattock for miserable wages and scanty keep.'\n\n\"To this the lawyer replied, 'But this is precisely the sort of person we should cherish the most. For since they are more high-spirited and lofty-minded than artisans and farmers, they provide the strength and power of an army if we ever have to fight a war.'\n\n\" 'Indeed,' I said, 'you might as well say that we should cherish thieves for the sake of warfare, for you will never lack for thieves as long as you have the retainers. In fact robbers are no slouches as soldiers and soldiers are not the most lethargic of thieves\u2014so finely matched are the two callings. But this problem, though it is widespread among you, is not peculiar to you; it is shared by almost all nations. But France is infected with another pestilence besides, one that is even more virulent: the whole country is occupied and filled with mercenaries, even during peacetime (if it can be called that). Their justification is the same as yours for maintaining idle retainers here: those foolosophers think that the public welfare consists in having strong and stout armed forces in a state of readiness, especially veterans, for they have no confidence in untried troops, just as if they should seek out a war precisely to avoid having inexperienced soldiers, and people should be gratuitously slaughtered (as Sallust nicely puts it) lest hand and spirit should grow sluggish through inactivity. Just how deadly it is to maintain such beasts France has learned to her cost, and the same is made clear by the examples of the Romans, the Carthaginians, and the Syrians, and of many other nations as well: standing armies of mercenaries, on one occasion after another, destroyed not only their government but also their fields and even their cities. How little this was necessary is made clear by the fact that not even French soldiers, thoroughly trained in warfare to their very fingertips, can very often boast that they came off better than your draftees\u2014not to put it more strongly lest I seem to be flattering present company. But your troops, whether urban artisans or rough and untrained farmers, are not thought to be very much afraid of the idle retainers of noblemen, except for some whose physique does not lend itself to strength and boldness or whose brave spirit has been broken by the poverty of their families. There is little enough danger that those retainers whose vigorous and strong bodies (for noblemen do not deign to ruin any but choice physiques) are now either grown flabby with idleness or soft with almost ladylike activities, no danger, I say, that such retainers would be unmanned if they were taught a good craft to earn a living and were exercised in manly labors. However that may be, I certainly do not see that it can ever contribute to the common good to prepare for war (which you never have unless you wish to) by maintaining such a huge crowd of people who undermine the peace, to which we ought to pay so much more attention than to war. But this is not the only problem which makes it necessary to steal. There is another, more peculiar (so far as I know) to you Englishmen.'\n\n\" 'What is that?' said the Cardinal.\n\n\" 'Your sheep,' I said, 'which are ordinarily so meek and require so little to maintain them, now begin (so they say) to be so voracious and fierce that they devour even the people themselves; they destroy and despoil fields, houses, towns. I mean that wherever in the realm finer and therefore more expensive wool is produced, noblemen, gentlemen, and even some abbots (holy men are they), not content with the annual rents and produce which their ancestors were accustomed to derive from their estates, not thinking it sufficient to live idly and comfortably, contributing nothing to the common good, unless they also undermine it, these drones leave nothing for cultivation; they enclose everything as pasture; they destroy homes, level towns, leaving only the church as a stable for the sheep; and as if too little ground among you were lost as game preserves or hunting forests, these good men turn all habitations and cultivated lands into a wilderness. And so that one glutton, a dire and insatiable plague to his native country, may join the fields together and enclose thousands of acres within one hedge, the farmers are thrown out: some are stripped of their possessions, circumvented by fraud or overcome by force; or worn out by injustices, they are forced to sell. One way or another, the poor wretches depart, men, women, husbands, wives, orphans, widows, parents with little children and a household which is numerous rather than rich, since agriculture requires many hands, they depart, I say, from hearth and home, all that was known and familiar to them, and they cannot find any place to go to. All their household furnishings, which could not be sold for much even if they could wait for a buyer, are sold for a song now that they must be removed. They soon spend that pittance in their wanderings, and then finally what else is left but to steal and to hang\u2014justly, to be sure\u2014or else to bum around and beg? For that matter, even as vagrants they are thrown into jail because they are wandering around idly, though no one will hire them, even when they offer their services most eagerly. For since no seed is sown, there is no farm labor, and that is all they are accustomed to. One herdsman or shepherd is sufficient to graze livestock on ground that would require many hands to cultivate and grow crops.\n\n\" 'And for this reason the price of grain has risen sharply in many places. Even the price of wool has gone up so high that poorer people who ordinarily make cloth out of it in this country cannot buy it, and for that reason many of them are out of a job and reduced to idleness. For after pastureland was expanded, huge herds of sheep were carried off by a murrain, as if God were punishing the owners' greed by visiting on the sheep a pestilence which might more justly have been hurled at the heads of their owners. But even if the number of sheep should increase enormously, the price still does not go down, because, though the sellers cannot be said to have a monopoly since more than one is selling, still it is certainly an oligopoly. For the sheep have almost all come into the hands of a few, and these men are so rich that they are under no necessity to sell until they want to, and they do not want to until they get the price they want.\n\n\" 'For the same reason other kinds of livestock are similarly high-priced, and all the more so because, once the farmhouses have been torn down and agriculture neglected, there is no one to see to the breeding of animals. For even those rich landholders do not rear other animals as they do sheep. Rather they buy them lean and cheap in some distant market and then sell them dear after fattening them up in their pastures. And for that reason, I think, the full disadvantage of this system has not yet been felt. I mean that up to now they have raised the prices only in the places where the animals are sold. But when the time comes that they are taken from the breeders faster than they can be bred, then finally the numbers will also gradually decrease where they are bought, so that here also there must needs be a severe shortage. Thus the very feature that seemed to make your island extremely fortunate has been turned into an instrument of its destruction by the wicked greed of a few men. For these high food prices are the reason why everyone dismisses as many as he can from his household\u2014to go where, I ask you, except to go begging or else, as a noble spirit can more easily be persuaded to do, to turn to robbery.\n\n\" 'What shall we say when this miserable poverty and want is coupled with wanton luxury? For the retainers of noblemen, artisans, and one might say even some peasants and, in sum, all classes of society indulge in extravagant sartorial display and excessive, luxurious cuisine. And then the cookshops, the brothels, the bawdy houses, and those other sorts of bawdy houses, the wine bars and alehouses, and then so many crooked games of chance, dice, cards, backgammon, tennis, bowling, quoits, don't all these quickly empty pockets and send their votaries off to rob someone? Get rid of these pernicious plagues, make laws requiring that villages and towns be rebuilt by those who have torn them down or be handed over to those who are willing to restore and rebuild them. Keep the rich from cornering the market and from having a licensed monopoly, as it were. Let fewer people be supported in idleness, let agriculture be restored, let cloth working be reinstated as an honest trade which will give useful employment to this idle mob, whether those whom poverty has already turned into thieves or those who are now vagabonds or idle servants\u2014in either case they will turn out to be thieves.\n\n\" 'Certainly unless you remedy these evils, it is pointless for you to boast of the justice administered in the punishment of thieves, a justice which is specious rather than either just or expedient. In fact when you bring people up with the worst sort of education and allow their morals to be corrupted little by little from their earliest years, and then punish them at last as grown men when they commit the crimes which from childhood they have given every prospect of committing, what else are you doing, I ask you, but making them into thieves and then punishing them for it?'\n\n\"As I was saying this, the lawyer was already getting ready to speak and had decided to employ that common method of disputants who are more diligent in repeating than in replying\u2014so high is their opinion of memory. 'A very fine speech indeed,' he said, 'especially for a stranger who has only had more opportunity to hear about these matters than to get any precise knowledge of them, as I shall make clear in a few words. For first I shall recount in an orderly way what you have said; then I shall show on what points your ignorance of our affairs has misled you; finally, I shall rebut and refute all your arguments. Therefore, to begin with the first task I promised to undertake, you seem to have made four\u2014'\n\n\" 'Be quiet,' said the Cardinal, 'for it seems hardly likely that you will reply in a few words after such a beginning. Hence, for the present we will relieve you of the trouble of replying, but we reserve that whole task for you when you two meet again, which I wish to be tomorrow, if nothing prevents you or Raphael here from meeting then. But meanwhile, my dear Raphael, I would very much like to hear why you think theft should not be punished with execution, or what other punishment you would enact that would contribute more to the common good. For even you do not think we should put up with it. But if people rush into thievery now when it is punishable by death, and then if they could once be sure of their lives, what force, what fear could possibly restrain criminals? They would interpret the mitigation of the punishment almost as an incentive or reward for wrongdoing.'\n\n\" 'Most gracious Father,' I said, 'it seems to me to be entirely and absolutely unjust to take a person's life because he has taken some money. For a human life cannot be equated with the goods of fortune, not even the whole sum of them. But if they say that this punishment is redress not for money but for the transgression of justice or the violation of laws, would it not be right to call this extreme justice extreme injury? For we ought not to approve of legal decrees so Manlian that the slightest infraction causes the sword to be unsheathed nor should we accept the Stoic maxim that all sins are equal, making no distinction between killing a person or stealing a coin from him, for between these two crimes (if fairness means anything at all) there is no similarity or relationship. God forbade us to kill anyone, and are we so ready to kill someone because he has taken a bit of money? But if someone should interpret that command to mean that the power to kill anyone is taken away except when human law declares a person should be killed, what is to prevent human beings from using the same principle to decide to what degree rape, adultery, or perjury are permissible? In fact, God has deprived us of the right to kill not only others but also ourselves, but if the mutual consent of human beings to specific laws allowing them to kill one another has enough force to release their agents from the bonds of God's commandment and enable them, with no precedent from God, to execute anyone condemned to death by human law, will that not mean that God's commandment has only as much force as is granted to it by human law? And indeed on this principle human beings may decide to what degree God's commands are to be observed in all fields. Finally, the law of Moses, though it was harsh and severe because it was made for slaves, and stubborn ones at that, still punished theft with a fine, not death. Let us not think that in his new law of mercy, by which he commands us as a father does his children, God has granted us greater license to be cruel to one another.\n\n\" 'These are the reasons why I think this punishment is wrong. And I think there is no one who does not understand how absurd and even dangerous it is to society to punish theft and murder in the same way. For when a thief sees that he is in no less danger if he is convicted of theft than if he had also been condemned for murder, that consideration alone will drive him to kill someone whom otherwise he would only have robbed. For apart from the fact that there is no more danger if he is caught, murder makes him more safe and gives him a greater hope of concealing his crime, since the witness to it has been eliminated. Thus by using excessively harsh measures to terrify thieves we encourage them to kill the innocent.\n\n\" 'As for the usual question of what punishment would be more advantageous, in my judgment it would be quite a bit easier to find a better one than to find one that is worse. For why should we doubt the utility of that way of punishing criminals which we know was once preferred for so long by the Romans, who were quite expert in the art of governing? Those convicted of serious crimes were condemned to quarry stone or dig out ore, constantly shackled and guarded. But as for me, on this point I reserve my highest approval for the system practiced by a people generally called the Polylerites whom I encountered in the course of my travels in Persia. Their population is not small and their institutions are not lacking in prudence; except that they pay an annual tribute to the king of Persia, they are otherwise free and allowed to make their own laws. But because they are a long way from the ocean and almost entirely surrounded by mountains, and because they are content with the produce of their land, which is by no means infertile, they neither visit others nor are visited very often themselves. In accord with the ancient policy of their country, they do not seek to extend their territory, and what they already have is easily protected by the mountains and the tribute paid to their overlord. They have absolutely no armed forces, their lifestyle is hardly splendid but it is comfortable, and they are happy rather than renowned or illustrious. Indeed, even their name, I think, is not very well known except to their immediate neighbors.\n\n\" 'And so, among them whoever is convicted of theft restores what was taken to its owner, not (as elsewhere) to the prince, for they consider he has no more right to stolen goods than the thief himself. If the goods have been lost, their equivalent is paid from the possessions of the thief, and whatever is left is handed over intact to his wife and children. He himself is condemned to hard labor.\n\n\" 'Moreover, unless the theft was committed with violence, they are not shackled or imprisoned but left free and unconstrained as they work on public projects. Shirkers and slackers are not restrained with shackles but egged on with the lash. If they work energetically, they are subjected to no humiliation; they are locked up in their cells only at nighttime after roll call. Except for constant labor their lives are not uncomfortable. Since they are doing public works they are fed at public expense, and not badly, but in different ways in different places. In some places what is spent on them is collected as alms; and this method, though it is unpredictable, has nevertheless been found to be the most productive because the people there are compassionate. In other places public revenues are set aside for that purpose. There are places where they levy a tax on private individuals to support the prisoners. Actually, in other places they do not do public works, but when a private person needs workmen he goes to the city square and hires some of them for that day at a fixed wage, which is a little less than what a freeman would cost. Moreover, if a slave is lazy it is permissible to whip him. Thus no one ever lacks work. And over and above his keep, each of them brings something into the public treasury every day.\n\n\" 'They are all dressed in one color and they are the only ones who wear it. Their hair is not shaved off but it is clipped a bit short above the ears. A little piece of one ear is cut off. Their friends are allowed to give any of them food, drink, and clothing of the right color. But it is death to give them money, both for the donor and the recipient, nor is it any less dangerous for a freeman to take money from them for any reason whatsoever or for a slave (for that is what they call the convicts) to lay a hand on a weapon. Each district has its own distinguishing badge, which it is a capital crime to throw away, just as it is to be seen outside the district boundaries or to say anything to a slave from another district. To plan an escape is no safer than to attempt it. In fact, to be an accessory to such a plan is death for a slave and enslavement for a freeman. On the other hand, rewards are allotted to informers: for a freeman money, for a slave freedom, and for either one pardon and amnesty for their complicity, to keep it from seeming safer to carry out a criminal plan than to repent of it.\n\n\" 'This law and the system I have described constitute their policy in this matter. It is perfectly obvious how humane and advantageous it is since vengeance is managed in such a way as to eliminate the vice and preserve the person, and to handle him in such a way that he has to be good and will spend the rest of his life making up for the harm he has done. Furthermore, there is so little fear that they will revert to their former ways that travelers who intend to make a journey consider no guides to be safer than these slaves, whom they exchange from district to district. For there is no opportunity whatever to commit robbery: they are unarmed; money is of no use except as evidence of a crime; punishment is in store for them if they are caught; and there is absolutely no hope of escaping anywhere. For how could a person whose clothes are totally different from anyone else's cover up his escape and disguise himself unless he ran away naked? And even then his ear would give him away. But couldn't they at least conspire to overthrow the republic?\u2014that surely is the real danger. As if any district could hope to do so without sounding out and enlisting the slave gangs of many other districts! They are so far from being able to conspire that they cannot even meet or converse or greet one another. And then how can we believe anyone would dare to trust his companions with such a plot, since it is dangerous for them to remain silent and most advantageous to reveal it? On the other hand, if they are patient and obedient, if they give good reason to believe that they will lead reformed lives in the future, none need despair of regaining his freedom; indeed not a year goes by in which some slaves who have recommended themselves by their patience are not reinstated.'\n\n\"When I had said this, and had added that I saw no reason why this system could not be set up also in England, and with much more benefit than the justice which the lawyer had praised so highly, then he (namely the lawyer) said: 'This system could never be established in England without enormous danger to the commonwealth.' As he said this he shook his head, puckered his mouth, and fell silent. And everyone there jumped on his bandwagon.\n\n\"Then the Cardinal said, 'It is not easy to predict whether the outcome would be favorable or not without at least trying it out. But when the death sentence has been pronounced, if the prince were to grant a reprieve without any right of asylum in order to see how the system would work, and then if in fact it turned out to be useful, it would be right to establish it. If not, then thieves who had been condemned earlier could be executed at that time; on the part of the government this would be neither less nor more unjust than immediate execution, and during the trial period it would pose no danger. In fact, it seems clear to me that it would not be a bad idea to treat vagabonds also in the same way, for in spite of the many laws made against them, we have still made no progress.'\n\n\"When the Cardinal had said this, there was no one there who did not vie with the others in praising what they had scorned when I proposed it, but especially the part about the vagabonds because the Cardinal himself had added it on.\n\n\"I do not know whether or not it would be better to say nothing about what happened then, for it was quite silly. But I will tell it anyway, for it was not malicious and it has some bearing on our subject. A certain hanger-on was standing around. It seems he wanted to play the fool, but he did it so well that he seemed to be one, raising a laugh with such witless jokes that the laughter was directed more often at him than at the jokes. But every now and then he came up with something not entirely absurd, so as to confirm the proverb 'Throw the dice often enough and you will sooner or later get a lucky combination.' One of the guests said that in my discourse I had made good provision for thieves and the Cardinal had also taken care of the vagabonds, and now all that remained was to make public provision for those whom disease or old age had rendered destitute and who were incapable of returning to the jobs by which they had earned their living. 'Leave that to me,' said the hanger-on. 'I will see to it that this is also properly taken care of. In fact, I am desperately anxious to ship off this sort of person somewhere out of my sight; they annoy me so much with their wailing and whining and pleas for money, though they can never sing so pretty a tune as to extract a penny from me. Actually, one of two things happens: either I don't want to give them anything or I don't have anything to give. And so they have now begun to get wise. To keep from wasting their effort, they keep silent when they see me passing by. Good lord, they no more hope for anything from me than if I were a priest. I would decree by law that all such beggars be divided up and parceled out among the Benedictine monasteries where they would become lay brothers (as they are called); and the women I would order to become nuns.'\n\n\"The Cardinal smiled and took it as a joke, but the others took it seriously. A certain friar, however, a theologian, was so delighted by a joke aimed at priests and monks that he himself also began to make merry, though he was otherwise so serious as to be almost sour. 'But even this,' he said, 'will not free you from beggars unless you also look out for us friars.'\n\n\" 'But that is already taken care of,' said the hanger-on. 'For the Cardinal looked out for you marvelously well when he proposed that vagabonds should be confined and put to work, for you are the greatest vagabonds of all.'\n\n\"After they all had looked at the Cardinal and saw that he did not reject this joke either, they were not at all loath to enjoy it, all except the friar. Needled in this fashion, he was indignant and furious (nor am I surprised that he was), so much so that he couldn't even refrain from hurling insults. He called the fellow a scoundrel, a backbiter, a sneak, and a son of perdition, all the while citing terrible threats from Holy Scripture. Now the buffoon began to do some serious buffoonery, for he was clearly on his own ground.\n\n\"'Do not grow angry, my good friar,' he said, 'for it is written, \"In your patience you shall possess your souls.\"'\n\n\"The friar replied (and I will give his very own words), 'I am not angry, you jailbird, or at least I do not sin. For the psalmist says, \"Be angry and do not sin.\"'\n\n\"Then the friar was gently advised by the Cardinal to control his emotions, but he said, 'No, my lord, my language springs from nothing but good zeal, as it should, for holy men have had good zeal, whence it is said, \"Zeal for your house has consumed me,\" and we sing in church, \"Those who mock Elisha as he goes up to the house of God feel the zeal of the bald man,\" just as perhaps this mocking and ribald rascal will feel it.'\n\n\" 'Perhaps you are acting out of a laudable feeling,' said the Cardinal, 'but it seems to me that you would act, if not in a holier, then certainly in a wiser way, if you would not put yourself on the level of a fool and set out to cap his absurdities with your own.'\n\n\" 'No, my lord,' he said, 'I would not act more wisely. For Solomon, the wisest of men, says, \"Reply to a fool in accord with his folly,\" as I am now doing, and I am showing him the pit into which he will fall if he does not watch out. For if the multitude which mocked Elisha, who was just one bald man, felt the zeal of the bald man, how much more will be heaped on a single person who mocks a multitude of friars, for many of them are bald. And also we have a papal bull which excommunicates anyone who makes fun of us.'\n\n\"When the Cardinal saw there would be no end to it, he sent the hanger-on away with a motion of his head and opportunely turned the conversation to another subject. A little later he arose from the table and, dismissing us, devoted himself to hearing the petitions of suitors.\n\n\"See, my dear More, how I have burdened you with a long discourse, and I would be quite ashamed of myself for doing so if you had not eagerly importuned me and seemed to listen as if you wanted no detail of this conversation omitted. Though I should have been more brief, still I did at least feel obliged to tell it to show how judiciously they scorned the plan when I proposed it and how the very same persons immediately reversed themselves and approved it when the Cardinal did not disapprove of it. Their flattery of him went so far that they seriously favored and almost accepted the ideas of his hangeron because his master took them as a joke and hence did not scorn them. From this you can judge how high an estimation courtiers would have of me and my advice.\"\n\n\"Indeed, my dear Raphael,\" I said, \"you have given me much pleasure, you told the whole story so judiciously and so deftly. Moreover, while you spoke I seemed not only to have returned to my homeland but also to have grown young again because of fond memories of the Cardinal, in whose household I was educated as a boy. When you honored his memory so highly, you cannot imagine how much dearer you became to me on that account, though you were already most dear. But I am by no means ready to change my mind yet. No, I am convinced if you could bring yourself not to shrink from the courts of princes, you could contribute a great deal to the common good through your advice. No duty of a good man (and you are one, of course) is more important than that. Then too, since your friend Plato thinks that commonwealths will be happy only when philosophers become kings or kings become philosophers, how far will we be from happiness if philosophers will not even deign to impart their advice to kings.\"\n\n\"They are not so disagreeable as that; they would do so gladly. Indeed they have already done so by publishing many books, if those in power were prepared to accept their good advice. But undoubtedly Plato clearly foresaw that unless kings became philosophers, they would never give their approval to the advice of philosophers, because since childhood they have been thoroughly imbued and infected with misguided notions. He also found this out for himself when he was with Dionysius. But don't you think that, if I proposed sound measures to some king and tried to eradicate from his mind the seeds of corruption, I would be banished or held up as a laughingstock!\n\n\"Come now, imagine that I serve the French king and sit in his council chamber, as the king himself presides in a secret session, surrounded by a most judicious circle of advisers who are very eagerly seeking out wiles and stratagems to keep Milan and win back Naples (which is always slipping from his fingers), and then to overthrow Venice and make all of Italy subject to him, and then to bring Flanders, Brabant, and finally all of Burgundy into his control, and other peoples as well, whose realms he has long had it in mind to invade. At this meeting, while one urges that a treaty be struck with Venice, to last only as long as it suits the French, and that the French share their plans with them and even give them some share of the spoils, which they can reclaim when matters have been satisfactorily settled; while another advises them to hire German mercenaries, another to soothe the Swiss with payments of money; someone else, on the other hand, thinks that his divine majesty the emperor ought to be propitiated with a votive offering, as it were, of gold; while another thinks it best to strike a bargain with the king of Aragon, granting him the kingdom of Navarre (which belongs to someone else) as the price of peace; and on the same occasion another suggests that the prince of Castile should be snared by the prospect of a marriage alliance and that some nobles of his court should be brought over to the French side by giving them reliable pensions; when the greatest difficulty of all is encountered, namely what to do in the meantime about England; but they agree that a peace treaty should be negotiated with them, for a weak bond should always be tightened by the strictest terms; let them be called friends but be suspected as enemies; and that therefore the Scots should be stationed in readiness, poised on all occasions to attack immediately if the English make any moves; moreover, that some exiled noblemen be supported secretly (for treaties forbid that it be done openly) who can claim that the kingdom is rightfully his so that the French king will have a rein to check an English king he does not trust\u2014at this council, I say, amidst such a mass of suggestions, surrounded by such distinguished men, all vying to give advice about going to war, if such a nobody as I were to stand up and give an order to tack in a different direction, expressing the opinion that Italy should be ignored and that the king should stay at home, that France is a kingdom so large that it is not easy for one man to rule it (much less should the king imagine he should consider adding others to it); and then if I should put before them the measures adopted by the Achorians, whose country faces the island of Utopia on the southeast side; if I should tell them that they had once fought a war to gain for their king a realm which he claimed to inherit because of some ancient marriage tie, and that, when they finally won it, they saw that they would endure no less suffering in keeping it than they did in gaining it, but rather that the seeds of war were always sprouting up, either rebellion within or incursions from without against the subjected people, so that they were always having to fight either for them or against them; that they never had an opportunity to disband their army, and that at the same time they were being stripped of their resources, their money was being carried out of the country, their blood was being spilled to provide someone else a smidgeon of glory, that they were no safer during peacetime; that at home the war had corrupted morals, imbued the citizens with a lust for robbery, that slaughter in warfare made them completely reckless, that they scorned the laws because the king was so distracted by trying to take care of two kingdoms that he couldn't concentrate on either one. When they saw that otherwise there would be no end to these great troubles, they finally took counsel together and very courteously gave their king the choice of retaining whichever of the kingdoms he wished; but they said he could not have power over both because they were too numerous to be governed by half a king (indeed no one is willing to share even a muledriver with someone else). And so the good prince left his new kingdom to one of his friends (who was soon afterwards banished) and was forced to be content with his old one. Furthermore, if I showed that all these abortive wars, which had thrown so many countries into turmoil for his sake would exhaust his treasury, destroy his people, and in the end still come to nothing through some mishap or other; and that therefore he should care for the kingdom of his ancestors, improve it as much as he could, make it as flourishing as possible; he should love his own and be loved by them; he should live with them, govern them kindly and leave other kingdoms alone, since the kingdom which has fallen to his lot is enough, and more than enough, for him\u2014how do you imagine, my dear More, my listeners would react to this speech?\"\n\n\"Certainly not very favorably,\" I said.\n\n\"Let us proceed, then,\" he said. \"If counsellors were in a discussion with some king or other and were thinking up schemes to fill up his treasury, while one person suggests increasing the value of the currency when the king pays out money and decreasing it exorbitantly when he collects it so that he can discharge a large debt with a little money and collect a great deal when he is owed only a little; while another urges him to pretend he is going to war and to use that pretext to raise money and then, when it suits him, to make peace with religious ceremonies, pulling the wool over the people's eyes and making them think that he is a conscientious, merciful prince who wishes to spare them bloodshed; while another reminds him of certain antiquated, moth-eaten laws, long since fallen into disuse, laws which everyone ignores since no one even remembers that they were passed, and advises that he should therefore enforce them with fines, noting that no source of revenue could be more productive, none more honorable, since it has the appearance of a concern for justice; while another advises him to prohibit many practices with heavy fines, especially those that are contrary to the public interest, noting that later he can make a monetary arrangement with those whose interests are hurt by the laws and that thus he can win the gratitude of the people and make a double profit, first from fining those whom greed has led into his trap and then by selling dispensations to others (the higher the price the better the prince, since he is reluctant to grant a private person the right to obstruct the common good, and therefore does it only for a high price); while someone else persuades him to put pressure on judges to rule in his favor in all cases and advises him to summon them to his palace where they are to discuss his affairs in his own presence, saying that thus no case will seem so flimsy that his judges (whether out of love of contradiction, or a desire to seem original, or a wish to curry favor) cannot, in his presence, find some loophole for a false verdict, noting that when the judges give differing opinions and argue about a case that is as clear as day, the truth can be called into question and the king will have a convenient handle to interpret the law in his own favor, pointing out that the others will acquiesce out of shame or fear and thus the judgment can be fearlessly rendered in court, nor can there be any lack of pretexts for someone ruling in the prince's favor, since he has on his side either equity or the letter of the law or a twisted interpretation of the language, or something that outweighs all laws in the minds of conscientious judges, the indisputable royal prerogative; while everyone agrees completely with that saying of Crassus that no amount of gold is sufficient for a king, since he has to maintain an army, and moreover that a king can do no wrong, no matter how much he wants to, since all the possessions of all his subjects, and even their own persons, belong to him, and since nothing belongs to anyone unless the king graciously refrains from taking it away from him, and that he should leave as little as possible to his subjects since his safety consists in keeping the people from enjoying too much wealth or freedom, which render them less willing to put up with harsh and unjust commands, whereas on the other hand poverty and privation break their spirits and make them patient, depriving the oppressed of the lofty aspirations needed for rebellion; at this point, if I should stand up and contend that all this advice is both dishonorable and harmful to the king, for not only his honor but also his safety depends more on the people's wealth than on his own; if I were to show that the people choose a king for their own sake, not his, since his labor and effort enable them to live in comfort and safety; and that therefore a prince should be more concerned with the welfare of his people than with his own, just as it is the duty of a shepherd, insofar as he is a real one, to feed his sheep and not himself; that experience itself shows how wrong they are in thinking that the poverty of the people is the safeguard of peace, for where can you find more quarrels than among beggars? who is more intent on changing things than someone who is most dissatisfied with his present state in life? or, finally, who is more driven to create a general disturbance in the hope of gaining something than someone who has nothing to lose? But if a king is so scorned and hated by his subjects that he cannot make them do their duty unless he harasses them with maltreatment, plundering, and confiscation and reduces them to poverty, it would certainly be better for him to abdicate his throne than to retain it by methods which may keep the name of authority but have certainly lost all of its majesty, for it does not befit the dignity of a king to rule over beggars but rather over wealthy and happy subjects; that was certainly what was meant by that upright and lofty spirit Fabricius, when he replied that he would rather rule over the rich than be rich himself. Indeed, for one person to wallow in pleasure and luxury while he is surrounded on all sides by grieving and groaning, that is to be the guardian not of a kingdom but of a prison; finally, just as a physician is totally incompetent if he cannot cure a disease except by means of another disease, so too someone who does not know how to improve the lives of citizens except by depriving them of the comforts of life is admitting that he does not know how to rule over a free people; instead he should cure either his sloth or his pride, for these are usually the vices that make his people despise and hate him; he should live harmlessly on his own income, adapt his expenses to his income; he should curb crime and, by educating his people properly, prevent it rather than allow it to increase and then punish it; he should not be hasty to revive laws which are customarily ignored, especially those which are long disused because they were never desirable; he should never take something as a fine which a private person would not be allowed to accept because to do so would be criminal and deceitful. At this point, what if I told them that the Macarians, who are also not very far from the Utopians, have a law requiring their king to swear formally and solemnly on the very first day of his reign that he will never have in his treasury at one time more than a thousand pounds in gold or the equivalent amount of silver? They say that a king who was more concerned about the welfare of his land than about his own wealth made this law to prevent the heaping up of so much treasure as to impoverish his people; for he saw that this amount would be enough either for the king to fight against rebels or for the kingdom to repel a hostile invasion but would be too little to encourage him to invade other countries\u2014and that was the primary reason for making the law. A secondary reason was that he thought it would make enough money available for the ordinary business transactions of the citizens; and since any money which accrues over that limit has to be paid back, he reckoned that a king would not seek out methods of extortion. A king such as this would be feared by malefactors and loved by his law-abiding subjects. If I should obtrude such notions and others like them on persons who are violently opposed to them, don't you suppose they would turn deaf ears as I told my tale?\"\n\n\"Deaf as a post, undoubtedly,\" I said. \"And, by heaven I am not surprised, and, to tell you the truth, I don't think you should obtrude such speeches or give advice which you are certain they will never accept. For how can it do any good or how can such an odd discourse influence the thinking of those whose minds are prejudiced and dead set against such notions? In private conversation with good friends this academic philosophy is not unpleasant. But there is no room for it in the council chambers of kings, where great matters are handled with great authority.\"\n\n\"That is what I said,\" he replied. \"Among princes there is no room for philosophy.\"\n\n\"Yes indeed, there is,\" I said, \"but not for this academic philosophy which considers anything appropriate anywhere. But there is another sort of philosophy better suited to public affairs. It knows its role and adapts to it, keeping to its part in the play at hand with harmony and decorum. This is the sort you should use. Otherwise, during a performance of a comedy by Plautus, when the slaves are joking around together, if you should come out onto the stage dressed like a philosopher and recite the passage from _Octavia_ where Seneca argues with Nero, wouldn't it have been better for you to have a non-speaking part than to jumble together tragedy and comedy by reciting something inappropriate? By hauling in something quite diverse, you would spoil and distort the play then being presented, even if what you add were better in itself. Whatever play is being presented, play your part as best you can and do not disturb the whole performance just because a more elegant play by someone else comes to mind.\n\n\"That's how it is in the commonwealth; that's how it is in the councils of princes. If you cannot thoroughly eradicate corrupt opinions or cure long-standing evils to your own satisfaction, that is still no reason to abandon the commonwealth, deserting the ship in a storm because you cannot control the winds. You should not din into people's ears odd and peculiar language which you know will have no effect on those who believe otherwise, but rather by indirection you should strive and struggle as hard as you can to handle everything deftly, and if you cannot turn something to good at least make it as little bad as you can. For everything will not be done well until all men are good, and I do not expect to see that for quite a few years yet.\"\n\n\"In that way,\" he said, \"I would be doing no more than trying to remedy the madness of others by succumbing to their madness myself. For if I want to tell the truth, then I have to say such things. I do not know whether it is proper for a philosopher to say what is false, but it certainly isn't for me. Though that discourse of mine might perhaps have been irksome and repugnant to them, I do not see why it should seem odd to the point of absurdity. If I were to describe everything Plato imagines in his _Republic_ or what the Utopians do in theirs, these things might be better (as they surely are), but they might still seem strange, because here we have private property and there all things are held in common.\n\n\"As for my speech (except that those who have decided to run headlong down a different path cannot be pleased by someone who calls them back and points out the dangers), but otherwise what was there in it that it is not fitting and even obligatory to say anywhere? Indeed if we are to avoid as odd or absurd everything that has been made to seem alien by the corrupt morals of mankind, we Christians will have to ignore almost all Christ's teachings, and he forbade us to ignore them, so much so that the teachings which he himself whispered in the ears of his disciples, he commanded them to preach openly from the rooftops. And most of his teachings are far more alien to our common customs than that speech of mine was, except that preachers (following your advice, I imagine), whenever mankind refuses to make their behavior conform to the rule of Christ, adapt Christ's teaching to the behavior as if it were a ruler made of lead, so as to make the two match in some way or other. I don't see what good that does except to allow people to be wicked with a better conscience.\n\n\"And that, indeed, is all the good I would do in the councils of princes. For my opinion would either be different, and that would amount to having no opinion at all, or it would be the same, and I would be the abettor, as Terence's Mitio says, of their madness. For I do not see what you mean by that indirect approach of yours which you think enables you to manage things deftly even if you cannot make everything good, and at least make them as little bad as you can. For there is no room there to dissemble or to look the other way: you must approve of advice that is clearly quite bad and subscribe to measures that are utterly pestilential. Anyone who gave faint praise to wicked advice would be taken for a spy or perhaps a traitor. There will be no occasions on which you can do any good, since you have fallen among colleagues who will corrupt the best of men before they themselves will be reformed; either you will be depraved by their evil way of life or, if you remain honest and innocent, you will be made a screen for the wickedness and folly of others. That is how far you are from being able to improve anything by that indirect approach.\n\n\"That is why Plato, in a very elegant simile, explains why wise men are right to refrain from taking on governmental tasks: when they see people rushing out on the streets only to be soaked by never-ending rain and they cannot persuade them to get under a roof and out of the rain, they get under shelter themselves, knowing that they will accomplish nothing by going out except to get drenched together with the rest and considering it sufficient, when they cannot cure the folly of others, at least to remain in safety themselves.\n\n\"But actually, my dear More (to tell you truly what I really think), it seems to me that wherever there is private property, where everything is measured in terms of money, it is hardly ever possible for the common good to be served with justice and prosperity, unless you think justice is served when all the best things go to the worst people or that happiness is possible when everything is shared among very few, who themselves are not entirely happy, while the rest are plunged into misery.\n\n\"Therefore, when I turn over in my mind the most prudent and holy institutions of the Utopians, who have very few laws and yet manage so well that virtue is rewarded and yet, since everything is equalized, everyone has plenty of everything, and then when I contrast their customs with those of other nations, always issuing ordinances but none of them all ever achieving order, where whatever a person can get he calls his own private property, where a mass of laws, enacted day after day, are never enough to ensure that anyone can protect what each calls his own private property or even adequately distinguish it from what belongs to someone else (as can easily be seen from the infinite lawsuits which are always being filed and are never finished), when I consider these things, I say, I have a higher opinion of Plato and I am not surprised that he would not deign to make any laws for people who would not accept laws requiring that all goods be shared equally by all. In his great wisdom he easily foresaw that the one and only path to the welfare of the public is the equal allocation of goods; and I doubt whether such equality can be maintained where every individual has his own property. For where everyone tries to get clear title to whatever he can scrape together, then however abundant things are, a few men divide up everything among themselves, leaving everyone else in poverty. And it usually happens that each sort deserves the lot of the other, since the one is rapacious, wicked, and worthless, and the other is made up of simple, modest men who by their daily labor contribute more to the common good than to themselves.\n\n\"Thus I am firmly persuaded that there is no way property can be equitably and justly distributed or the affairs of mortal men managed so as to make them happy unless private property is utterly abolished. But if it remains, there will also always remain a distressing and unavoidable burden of poverty and anxiety on the backs of the largest and best part of the human race. I grant their misery may be somewhat alleviated but I contend that it cannot be fully eliminated. I mean, if you decreed that no one could own more than a certain amount of land and that there be a legal limit to the money anyone can possess, if some laws were enacted that could keep the prince from being too powerful or the people too headstrong, that would keep offices from being solicited or put up for sale, or keep them from entailing many expenses (for otherwise they provide opportunities to rake in money by fraud and spoliation or it becomes necessary to put rich men in offices which ought to be held by wise men), such laws, I say, could mitigate and alleviate these ills, just as applying continual poultices can relieve the symptoms of sick bodies that are beyond healing. But as long as everyone has his own property, there is no hope whatever of curing them and putting society back into good condition. In fact, while you are trying to cure one part you aggravate the malady in other parts; curing one disease causes another to break out in its place, since you cannot give something to one person without taking it away from someone else.\"\n\n\"Quite the contrary,\" I said, \"it seems to me that no one can live comfortably where everything is held in common. For how can there be any abundance of goods when everyone stops working because he is no longer motivated by making a profit, and grows lazy because he relies on the labors of others. And then, when people are driven by want and there is no law which enables them to keep their acquisitions for their own use, wouldn't everyone necessarily suffer from continual bloodshed and turmoil? Especially when the magistrates no longer have any respect or authority, for I cannot conceive how they could have any among people who are all placed on one level.\"\n\n\"I am not surprised that you think so,\" he said, \"since you have no conception of the matter, or only a false one. But if you had been with me in Utopia and had seen their customs and institutions in person as I did (for I lived there more than five years, and I would never have wanted to leave except to reveal that new world to others) you would quite agree that you had never seen a people well governed anywhere but there.\"\n\n\"But you would surely have a hard time persuading me,\" said Peter Giles, \"that a better governed people can be found in that new world than in the one we know, since our intellects are no worse than theirs and our governments are older, I imagine, than theirs, so that long experience has brought to light many features which make our lives more comfortable, to say nothing of some things we have discovered by chance which no amount of ingenuity would have sufficed to invent.\"\n\n\"As for the antiquity of governments,\" said Raphael, \"you could give a more accurate judgment if you had read through the histories of that world: if they are trustworthy, there were cities there before there were people here. As for what ingenuity has invented or chance revealed up till now, that could have happened in either place. But certainly I think that even though we may surpass them in intelligence, they still leave us far behind in diligence and zeal to learn.\n\n\"According to their chronicles before we landed there they had never heard anything about us Ultra-equatorials (for that is what they call us) except that some twelve hundred years ago a ship was driven to Utopia by a storm and shipwrecked there. Some Romans and Egyptians were cast upon the shore and never left there again. Notice how their diligence turned this single occasion to their advantage. There was no useful skill in the whole Roman empire which they did not learn from the explanations of the strangers or did not manage to discover from the hints and clues they were given. Such was the enormous gain they made on this one occasion when some men from here were driven to their shores. But if a similar accident ever brought one of them from there to here, the incident has been completely forgotten, just as posterity perhaps will also forget that I was once there. One meeting alone was enough for them to appropriate all of our useful inventions, but I think it will be a long time before we will accept any institution of theirs which is better than ours. And I think that is the only reason why they manage their affairs more prudently and live more happily than we do, though we are not inferior to them in intelligence or resources.\"\n\n\"Therefore, my dear Raphael,\" I said, \"I beg and implore you, describe the island to us. And do not try to be brief but explain in order their fields, rivers, cities, population, customs, institutions, laws, and, in short, whatever you think we would want to know. And you should think we want to know whatever we don't know yet.\"\n\n\"There is nothing I would rather do,\" he said, \"for I have all this at my fingertips. But it will take some free time.\"\n\n\"Then let us go inside to eat lunch,\" I said. \"Afterwards we will take as much time as we want.\"\n\n\"Agreed,\" he said. And so we went in to eat lunch. After lunch we came back to the same place and sat down on the same bench, and having instructed the servants that we were not to be interrupted, Peter Giles and I urged Raphael to keep his promise. When he saw that we were attentive and eager to hear, he sat there quiet and thoughtful for a little while, and then began as follows.\n\nTHE END OF THE FIRST BOOK\nThe Discourse of Raphael Hythloday  \non the Best Form of a Commonwealth\n\nas Reported by  \nThomas More, Undersheriff of London\n\n## BOOK 2\n\nThe island of the Utopians is two hundred miles across in the middle, where it is widest, and throughout most of the island it is not much narrower, but toward both ends it narrows a bit. These ends, curling around into a circle with a circumference of five hundred miles, make the whole island look like a new moon. The sea flows in between the horns through a strait about eleven miles wide and then spreads out into a huge empty space protected from the wind on all sides, like an enormous, smooth, unruffled lake; thus almost the whole inner coast serves as a harbor and allows ships to go from shore to shore in all directions, much to the advantage of the people. The jaws of the strait are dangerous, on one side because of shallows, on the other because of rocks. In just about the middle of the channel, one rock stands out, visible and hence harmless; they have built and garrisoned a tower on it. The other rocks are hidden and treacherous. The channels are known only to the Utopians themselves, and hence it hardly ever happens that a foreigner enters the bay without a Utopian pilot. Indeed they themselves find it hard to enter it safely, except that they set their course by means of some signals on the shore. By moving these to different locations, they can easily lure an enemy fleet to shipwreck, no matter how large it is.\n\nMAP OF UTOPIA,  \nWOODCUT FROM THE NOVEMBER 1518 EDITION  \n(Beinecke Rare Book and Manuscript Library, Yale University)\n\nOn the outside coast there are not a few ports. But everywhere the landing places are so well defended, either naturally or artificially, that a few troops can keep a huge army from coming ashore. According to report, however (and the appearance of the place bears it out), their land was once not surrounded by the ocean. But Utopus, who conquered the island and named it after himself (for before that time it had been called Abraxa) and who brought its crude and rustic mob to a level of culture and humanity beyond almost all other mortals, after he won the victory at his first assault, had a channel cut fifteen miles wide at the point where the land adjoined the continent, and thus caused the sea to flow all around the land. And since he set not only the inhabitants to this task but also employed his own soldiers (to keep the inhabitants from thinking the work was imposed on them as a humiliation), the labor was shared by a great multitude of workers and was finished in an incredibly short time, so that the neighboring peoples (who at first ridiculed the project as silly) were overwhelmed with wonder and fear.\n\nThe island has fifty-four cities, all of them large and splendid and having exactly the same language, customs, institutions, and laws. They have the same layout and they look the same, insofar as the terrain allows. Those which are closest to each other are separated by twenty-four miles. None is so isolated that it is more than a day's journey on foot from another city. Every year each city sends three old and experienced citizens to Amaurot to discuss problems common to the whole island. For that city, which is located at the navel of the land, so to speak, and hence is most convenient as a meeting place for the delegates from everywhere, is the capital and chief city.\n\nThe land is so well distributed that no city has less than twelve miles of ground on all sides, though it may have much more in some directions, namely where the cities are furthest apart from one another. None of them is driven by any desire to extend its boundaries. Indeed, whatever land they have, they consider themselves its tenant-farmers, not its landlords. In the countryside, throughout the fields, they have conveniently located houses, each provided with farming tools. They are inhabited by the citizens, who take turns going out to live there. No country household has fewer than forty men and women, besides the two slaves bound to the land; it is presided over by a master and mistress who are sober and mature. Every thirty households are ruled by one phylarch. Every year twenty from each household return to the city, having fulfilled their two-year stint in the country. They are replaced by twenty substitutes from the city, who are to be trained by those who have already been there a year and hence are more skilled in farmwork; the substitutes themselves will train another group the following year, for if everyone were new and equally ignorant of farming, the crops would suffer from lack of skill. Although this system of exchanging farmers is customary, to keep anyone from being forced to live this hard life for a long time, nevertheless many who have a natural bent for agricultural pursuits apply for and are allowed additional years.\n\nThey farm the land, raise cattle, cut wood, and convey it to the cities by the most convenient route, whether by sea or by land. They raise a huge number of chickens, and they have a marvelous method of doing it. The hens do not sit on the eggs. For the Utopians themselves tend a great number of eggs, keeping them alive and hatching them them in constant warmth. As soon as the chicks emerge from the shell, they recognize and follow human beings around as if they were their mothers.\n\nThey raise very few horses and none but high-spirited ones, which serve no other purpose than the training of young people in horsemanship. For ploughing and hauling they use oxen; they grant that they are inferior to horses in short sprints, but they consider them superior over the long haul and less subject to diseases; moreover, they require less effort and expense to maintain, and when they have served out their term, they can be used for food.\n\nGrain they use only for bread. For they drink either wine made from grapes or cider made from apples or pears or else plain water, which they often boil with honey or licorice, of which they have plenty. Although they know (and they know it very well) how much produce is needed by a city and its surrounding population, they plant far more grain and raise far more cattle than they need for their own use, giving the surplus to their neighbors. All the supplies that are necessary but not available in the country they get from the city, giving nothing in exchange; the city magistrates provide them the goods with no bargaining. For every month many of them gather there on the feast day. On the day of harvesting, the phylarchs of the farmers inform the city magistrates how many citizens should be sent out; since they arrive at precisely the right time, such a large crowd of workers gets the harvest almost completely done in one day if they have good weather.\n\n### THEIR CITIES, ESPECIALLY AMAUROT\n\nIf you know one of their cities, you know them all, so similar are they in all respects (so far as the terrain allows). And so I will describe one of them (it doesn't much matter which one). But why choose any one except Amaurot? For it is the most notable and takes precedence over the others because the senate meets there; and no other is better known to me, since I lived there for five whole years.\n\nAmaurot, then, is situated on the gentle slope of a mountain; its shape is almost square. Beginning almost at the crest of the hill, it stretches two miles down to the river Anyder; its width is slightly greater along the river than it is at the hilltop. The source of the Anyder is eighty miles above Amaurot, a small spring which is amplified by tributaries, two of them sizeable, until, when it reaches the city itself, it is five hundred yards wide. Then for sixty miles it flows on, getting wider and finally flowing into the ocean. In the space between the city and the coast, and also for some miles above the city, the tide flows and ebbs for six whole hours in a swift current. Seawater flows in to a point thirty miles upstream, filling the whole channel of the Anyder and driving the river water upstream. It also makes the water salty somewhat higher up; from there the river gradually grows fresh and it is pure when it flows by the city. And at ebb tide it flows pure and fresh nearly all the way to the mouth of the river.\n\nThe city is connected to the opposite bank of the river by a bridge made not of pilings and planks but of beautifully arched stonework; it is placed at a point furthest from the sea so that ships can sail unobstructed along that whole side of the city. They also have another stream, not large but very gentle and pleasant, which gushes from a spring on the same mountain where the city is located; it flows down through the middle of the city into the Anyder. The Amaurotians have fortified the head and spring of this stream, which is located a little outside the city, surrounding it with walls that link it to the city, so that if an enemy ever attacks them, the water cannot be diverted or contaminated. From this stream the water is channeled in tile conduits to the various districts in the lower parts of the city. Where the terrain makes this impossible, rainwater collected in large cisterns serves the same purpose.\n\nThe city is surrounded by a high, thick wall with many towers and bastions. On three sides the wall is surrounded by a moat that is dry but wide and deep and blocked by thorn hedges; on the fourth side the river itself serves as a moat. The streets are laid out to facilitate traffic and to offer protection from the wind. The buildings are by no means ugly; the houses extend in a continuous row along the whole block, facing the row on the other side of the street; the housefronts along each block are separated by a street twenty feet wide. Behind the houses, a large garden, as long on each side as the block itself, is hemmed in on all sides by the backs of the rowhouses.\n\nThere is no house which does not have a door opening on the street and a backdoor into the garden. The double doors, which open easily with a push of the hand and close again automatically, allow anyone to come in\u2014so there is nothing private anywhere. For every ten years they exchange the houses themselves by drawing lots. The Utopians place great stock by these gardens; in them they grow vines, fruit trees, herbs, and flowers, all so bright and well tended that I have never seen anything more flourishing and elegant. In gardening they are motivated not only by their own pleasure but also by competition among the various blocks to see which has the best garden. And certainly you will not easily find any feature of the whole city that is of greater use to the citizens or gives them more pleasure. For that reason the founder of the city seems to have devoted more attention to these gardens than he did to anything else.\n\nFor they say that in the very beginning Utopus himself laid out the whole plan of the city. But he left it to succeeding ages to complete the adornment and landscaping that could not be completed during one lifetime. Thus in their annals, which have been diligently and scrupulously kept up since the island was captured 1,760 years ago, it is recorded that at first their dwellings were humble, mere huts and shacks, built of wood gathered at random, the walls plastered with mud. The roofs came to a point and were thatched with straw. But now all houses have a handsome appearance and are built three stories high. The outer sections of the walls are made of fieldstone, quarried rock, or brick, and the space between is filled up with gravel and cement. The roofs are flat and are coated with a sort of plaster which is not expensive but is formulated so as to be fireproof and more weather-resistant than lead. They commonly use glass (which is very plentiful there) to keep out the wind; sometimes they also use thin linen, soaked in clear oil or treated with resin\u2014a method which has two advantages: it lets in more light and keeps out more drafts.\n\n### THEIR MAGISTRATES\n\nEvery year each group of thirty families elects its magistrate, who in their ancient language was called a syphogrant but is known as a phylarch in the modern tongue. Ten syphogrants with their households are presided over by an official once called a tranibor, now known as a protophylarch. Finally, all the syphogrants, who number two hundred, having sworn to choose the person they consider the most capable, elect the ruler by secret ballot, choosing him from the four candidates named by the people. For each of the four quarters of the city names one person and proposes him to the senate. The ruler remains in office for life, unless his tenure is interrupted because he is suspected of trying to become a tyrant. They elect the tranibors every year, but they do not lightly change them. All the other magistrates hold office for one year.\n\nEvery third day, and sometimes oftener if circumstances require it, the tranibors gather to advise the ruler. They make decisions about public affairs; if there are any disputes among private persons (and there are very few) they settle them in a timely fashion. They always invite two syphogrants into the senate, different ones on every occasion; and they have provided that no measures concerning public affairs be adopted unless they have been discussed in the senate three days before a decision is reached. To enter into schemes concerning affairs of state outside the senate or public assemblies is a capital crime. These measures were taken, they say, to make it hard for the ruler and the tranibors to conspire to change the form of government and set up a tyranny over the people. And for the same reason matters of great moment are presented at the assemblies of the syphogrants, who report the matter to the households, take counsel among themselves, and report their recommendations to the senate. Sometimes a matter will be referred to the council of the whole island.\n\nThen, too, the senate has a rule that no point is discussed on the same day it is brought up, but rather it is put off till the next meeting; they do this so that someone who blurts out the first thing that occurs to him will not proceed to think up arguments to defend his position instead of looking for what is of use to the commonwealth, being willing to damage the public welfare rather than his own reputation, ashamed, as it were, in a perverse and wrong-headed way, to admit that his first view was short-sighted. From the start such a person should have taken care to speak with deliberation rather than haste.\n\n### OCCUPATIONS\n\nFarming is the one occupation in which all of them are skilled, men and women alike. They are all trained in it from childhood on, partly by instruction in the classroom, partly by being taken out to play at it, as it were, in the fields near the city, not merely looking on but doing the work themselves for bodily exercise.\n\nBesides farming (which, as I said, is common to all of them) everyone is taught some trade of his own. The ordinary ones are working with wool or linen or laboring as a stone mason, blacksmith, or carpenter. No other trade there employs any number worth mentioning. As for their clothing\u2014which is uniform throughout the island for all age groups and varies only to indicate sex or marital status, and which is not unappealing to the eye, allows freedom of movement, and is adapted to either heat or cold\u2014as for their clothing, I say, each household makes its own.\n\nEverybody learns one or the other of these trades, including women as well as men. But women, as the weaker sex, engage in lighter crafts, mostly working with wool or linen. The other trades, which require more strength, are relegated to the men. Generally children take up their father's trade, for most are naturally inclined to it. But if anyone is drawn to another occupation, he is transferred by adoption into another household where he can work at the trade he wants to pursue. The move is supervised not only by his father but also by the magistrates, to make sure the master of his adoptive household is respectable and responsible. Actually, if someone has mastered one trade and wants to learn another besides, he gets permission to do so by the same procedure. When he has mastered both, he practices whichever he wants to, unless the city has a greater need for the other.\n\nThe chief and practically the only function of the syphogrants is to take care and see to it that no one lounges around in idleness but rather that everyone practices his trade diligently, but not working from early morning till late at night, exhausted by constant labor like a beast of burden. For such grievous labor is fit only for slaves, and yet almost everywhere it is the way workmen live, except in Utopia. Dividing the day and night into twenty-four equal hours, they devote only six to work, three before noon, when they go to lunch. After lunch they take two hours of rest in the afternoon, then three more given over to work, after which they have dinner. Counting the first hour after noon as ending at one o'clock, it is eight o'clock when they go to bed. Sleep takes up eight hours.\n\nThe intervals between work, meals, and sleep they are allowed to spend however they like, provided that the time they have free from work is not wasted in debauchery and idleness but spent well in some other pursuit, according to their preference. Many devote these intervals to intellectual activities. For every day they have regular lectures in the hours before dawn; attendance is required only from those who have been specially chosen to devote themselves to learning. But a great number of men, and also women, from all orders of society flock to hear these lectures, some one sort, some another, as each is naturally inclined. But if someone wishes to spend this same time practicing his trade (as do many whose temperaments are not suited to any abstract discipline), they are quite free to do so; indeed they are also praised for doing so, since their labor contributes to the common good.\n\nAfter dinner they devote one hour to recreation, during the summer in the gardens, during the winter in the common rooms where they have their meals. There they either play music or entertain themselves with conversation. They do not so much as know about dice and other such pointless and pernicious games, but they do play two games not unlike chess. In one of them numbers fight against each other, one taking over the other; in the other game virtues are lined up in a battlefront against the vices. This game shows very cleverly both how the vices fight among themselves but join forces against the virtues, and also which vices are opposed to which virtues, what forces they bring to bear openly, what instruments they use to attack indirectly, what defenses the virtues use to fend off the forces of the vices, how they evade their assaults, and finally by what methods one side or the other wins the victory.\n\nBut at this point, it is necessary to examine the matter in more detail to avoid making a mistake. If only six hours are devoted to work, you might think that there would necessarily be some shortage of supplies. But that is so far from being true that six hours is not only enough to produce abundantly all the necessities and comforts of life but is even more than enough. This you, too, will understand if you consider what a large part of the population in other countries live their lives in idleness. First, almost all the women do, and they make up almost half the population. Or in places where the women work, the men take their place and lie around snoring. Add to that the huge idle crowd of priests and religious, as they are called. Throw in all the rich, especially the landlords of estates who are commonly called gentlemen and nobles. Include with them their retainers, that rank cesspool of worthless swashbucklers. Add, finally, the strong and sturdy beggars who feign some disease as a pretext for their idleness. You will certainly find that it takes far fewer than you thought to produce everything that mortals use.\n\nNow consider how few of these workers are occupied in necessary trades, since, where money is the measure of everything, many completely futile and superfluous crafts must be practiced just to support over-indulgence and wanton luxury. Now if that same crowd who are presently working were divided up among the few trades needed to produce the few commodities that nature requires, the resulting abundance of goods would drive prices down so low that craftsmen could not make a living. But if all those who work away at pointless tasks and, together with them, that whole crowd of lazy, languid idlers (any single one of whom consumes twice as much as any of the workers who produce the goods), if they all were put to work\u2014and useful work at that\u2014you can easily see how little time would be enough and more than enough time to produce all the goods required for human needs and conveniences\u2014and pleasures, too, as long as they are true and natural ones.\n\nAnd this very point is confirmed by the experience of the Utopians. For there, in the whole city and the surrounding territory, out of all the men and women who are old enough and strong enough to work, barely five hundred are exempted from work. Among them the syphogrants, who are legally relieved from work, nevertheless do not exempt themselves; they work so as to motivate others to work by giving a good example. The same immunity is enjoyed by those to whom the people give total leisure to pursue various branches of learning, but only after the priests have recommended them and the syphogrants have chosen them by a secret ballot. If any of them disappoints the hopes they had in him, he is put back to work; and on the other hand, it happens, not infrequently, that an artisan, devoting his free time to intellectual pursuits, works so diligently and makes such progress that he is exempted from working at his trade and promoted to the scholarly class. From this order of scholars are chosen ambassadors, priests, tranibors, and finally the ruler himself, who was called Barzanes in their ancient language, but is named Ademus in the modern tongue. The remaining group, which is neither idle nor devoted to useless trades, is so large that it is easy to imagine how many goods they produce in so few hours.\n\nApart from what I have just said, they have it easier because in most of the necessary trades they do not need to expend as much labor as in other nations. First of all, building or repairing structures everywhere else requires the continuous effort of so many workers for the simple reason that what a father has built his worthless heir allows to fall gradually into disrepair. Thus what could have been maintained with a minimum of effort has to be totally rebuilt, at great expense, by the next heir. Moreover, it often happens that a house that cost someone enormous sums to build seems contemptible to someone of more fastidious taste; after a short time it falls into ruin through neglect and the owner builds another house somewhere else, at no less expense. But among the Utopians, from the time when everything was settled and the commonwealth was established, it very rarely happens that a new site is chosen on which to build houses; and they not only repair damage quickly when it happens but they take preventive measures against it. The result is that their buildings last a very long time and require very little work, and sometimes construction workers have so little to do that they are set to shaping timbers or squaring and fitting stones at home, so that if they ever need to build anything, it can be constructed more quickly.\n\nNow as for their clothing, notice how little labor it requires. First of all, at work they wear informal garments made of leather or skins which last for seven years. When they go out in public they put on cloaks which cover these rough clothes; throughout the island they are all of the same color, that of the natural wool. Thus they not only get along with much less woolen cloth than anywhere else, but it also costs much less. But linen is easier to work and hence they use more of it; they are concerned only about the whiteness of linen and the neatness of wool, for they place no value on fineness of weave. The result is that in other places four or five woolen cloaks and the same number of silk shirts are not enough for one person, and if he is a bit fastidious, not even ten will do, but there everybody is content with one, which generally lasts for two years. Naturally there is no reason why he should want any more, for if he got them he would have no more protection against the cold, and his clothing would not look the least bit more fashionable.\n\nTherefore, since everyone is employed in a useful trade and the trades themselves require less labor, the result is a great abundance of everything, so that sometimes they bring out an enormous number of people to repair the public roads, if any have deteriorated. It happens very often, when there is no occasion even for that kind of work, that they publicly decree a shorter workday. For the magistrates do not compel anyone to engage in superfluous labor against his will, since the structure of the commonwealth is primarily designed to relieve all the citizens from as much bodily labor as possible, so that they can devote their time to the freedom and cultivation of the mind. For that, they think, constitutes a happy life.\n\n### SOCIAL RELATIONS\n\nNow is the time, I think, to explain how they treat each other, how they interact with one another, and what system they have for distributing goods.\n\nAnd so, while the city is made up of households, the households themselves consist mostly of blood relatives. Girls, when they grow up and marry, move into the dwellings of their husbands. But sons and, after them, grandsons remain in the household and are subject to the oldest parent, unless his mind is failing because of old age; in that case he is replaced by the next oldest. But to keep the city from being either over- or underpopulated, they see to it that no household (and each city, apart from its territory, has six thousand of them) has fewer than ten or more than sixteen adults. For it is not possible to set a limit for children. This limit is easily maintained by transferring persons from households with too many people to those with too few. But if it should happen that the whole city grows too large, they use the excess to supply underpopulated cities. But if it should happen that throughout the island the whole mass of the population should swell inordinately, they sign up citizens from each city and send them as colonists to live under their own laws on the nearest part of the continent, wherever the natives have a lot of land left over and uncultivated; they adopt any natives who choose to live with them. Assenting willingly to the same style of life and the same customs, the natives are easily assimilated, and that to the advantage of both groups. For by means of their institutions the Utopians make the land easily support both peoples, whereas before it provided a meager and skimpy living for only one. The natives who refuse to live under their laws are driven out of the territory the Utopians have marked off for their use; if they resist, the Utopians make war against them. For they think it is quite just to wage war against someone who has land which he himself does not use, leaving it fallow and unproductive, but denying its possession and use to someone else who has a right, by the law of nature, to be maintained by it. If any of their cities is ever accidentally so reduced in population that they cannot replenish it from other parts of the island and still keep the full quota in those cities (which they say has only happened twice in their whole history because of a virulent plague), then they resupply it with citizens immigrating from a colony. For they would rather allow the colonies to disappear than let any of the cities on the island shrink in size.\n\nBut, to return to the citizens' way of life, the oldest man, as I said, presides over a household. Wives serve their husbands and children their parents, and generally the younger serve the older. Each city is divided into four equal districts. In the middle of each district is a marketplace for all sorts of commodities. The products of each household are taken to designated houses there and each kind of goods is separately stored in a warehouse. From them each head of household goes to get whatever he and his household need, and he takes away whatever he wants, paying no money and giving absolutely nothing in exchange for it. For why should he be denied anything, since there is plenty of everything and no one need fear that anyone would want to ask for more than he needs? For why should anyone be suspected of asking for too much if he is certain he will never lack for anything? Certainly fear of want makes all kinds of animals greedy and rapacious, but only mankind is made so by pride, which makes them consider their own glory enhanced if they excel others in displaying superfluous possessions; in the Utopian scheme of things there is no place at all for such a vice.\n\nAdjoining the marketplaces I mentioned are food markets, to which vegetables, fruit, and bread are brought, and also fish and edible birds and beasts are conveyed from designated places outside the city where there is a stream to wash away refuse and offal. From here they bring the cattle which have been slaughtered and cleaned by the hands of bondsmen. For they do not allow their own citizens to become accustomed to butchering animals; they think that to do so gradually eliminates compassion, the finest feeling of human nature. They do not allow anything filthy or foul to be brought into the city, for air tainted by such rottenness might engender disease.\n\nFurthermore, each block has spacious halls located at equal intervals, each known by its own name. The syphogrants look after them, and to each of them are assigned thirty families (namely fifteen on either side) who eat their meals there. Stewards from each hall gather in the market at a designated hour and get food according to the number of mouths they have to feed.\n\nBut their first priority is the sick, who are cared for in public hospitals. They have four of them on the outskirts of the city, a little outside the walls; they are as capacious as four little towns so that no matter how many people are sick they do not need to be crowded uncomfortably together, and so that those who have contagious diseases that can be transferred from one person to another can be kept at a distance from the main body of the patients. These hospitals are so equipped and provided with everything that promotes health, the care provided in them is so gentle and solicitous, the doctors who are in constant attendance are so skilled that, although no one is sent there against his will, there is still almost no one in the whole city who would not rather be lodged there than at home when he is in failing health.\n\nAfter the stewards of the hospitals have received the food prescribed by the physicians, the best of what is left is divided equitably among the halls, according to the number fed by each one, except that they pay special attention to the ruler, the high priest, and the tranibors, and also to ambassadors and all foreigners (if there are any, for they are few and far between); but when there are any, designated residences are furnished and prepared for them. At the times fixed for lunch and dinner, the whole syphograncy, alerted by the blast of a bronze trumpet, convenes in these halls, except for those who are bedridden in the hospitals or at home. Nevertheless, no one is forbidden to take home food from the marketplace once the halls have been supplied with their quotas, for they know that no one would lightly choose to do so; though no one is prohibited from eating at home, still no one does it willingly, for it is not considered proper and it would be foolish to go to the trouble of preparing an inferior meal at home when a splendid and sumptuous one is ready and waiting in a hall nearby.\n\nIn this hall slaves perform all the chores which are somewhat heavy or dirty. But the women are solely responsible for preparing and cooking the food and making arrangements for the whole meal, each household taking its turn. They sit at three tables or more, according to the number of diners. The men sit with their backs to the wall, the women on the outside, so that if they should suddenly feel ill, as happens, sometimes, when they are pregnant, they can get up and go out to the nurses without disturbing the seating arrangement.\n\nThe nurses are seated separately with the nursing infants in a little room assigned to them; it never lacks a fire and clean water and also cradles so that when they want they can either lay them down or take off their swaddling clothes and let them refresh themselves by playing freely. Every mother nurses her own child unless death or disease prevents it. When that happens, the wives of the syphogrants immediately find a nurse, and that is not hard to do. For those who can are more than willing because everyone praises their compassion and the infant who is brought up this way takes the nurse as its natural mother.\n\nChildren who are under five sit in the nurses' den. Other minors, among whom they include members of both sexes who are not yet old enough to marry, either serve the diners, or, if they are too young and not strong enough for that, stand by\u2014and that in absolute silence. Both groups eat what is handed to them by those seated at table, nor is any other time set aside for them to eat.\n\nThe syphogrant and his wife sit at the head table, which is the place of honor and overlooks the whole assembly, since it is placed crosswise in the highest part of the chamber. Next to them sit two of the oldest persons, for they sit in groups of four at all the tables. But if a church is located in that syphograncy, the priest and his wife sit with the syphogrant so as to preside. On both sides of them sit younger people, and then older people again, and so on throughout the whole hall. And so people sit with their coevals, and yet they are mixed in with a different age group. They say that this arrangement was adopted so that the dignity of the elders and the respect due them would keep the young people from indulging in improper language or behavior, since nothing can be done or said at table which would escape the notice of the persons sitting nearby on all sides.\n\nThe dishes of food are not served to the highest places and then downward to the others, but rather the choicest pieces are served first to the old people (whose places are marked) and then equitable shares are served to the rest. But some of the delicacies which are not in sufficient supply to be distributed to the whole hall are given by the old people, as they see fit, to those sitting near them. Thus respect for the elders is maintained and yet everyone has the same advantage from it.\n\nLunch and dinner always begin with some reading that concerns morals, but it is brief lest it be tedious. Taking off from this, the elders begin the discussion, but not in a gloomy and sour fashion. And they do not take up the whole meal with long disquisitions. No, they would much rather listen to the young people, and they even deliberately challenge them so as to learn about the temperament and intelligence of each of them as revealed in the free give and take of tabletalk.\n\nLunches are quite brief, dinners more ample because the one is followed by work and the other by rest and sleep during the night, which they think contribute more to good digestion. They never dine without music and after dinner they never lack for tasty desserts. They light incense and sprinkle perfumes and spare no effort to cheer up the diners. For they tend to incline to the position that no kind of pleasure ought to be forbidden as long as no harm comes of it.\n\nThis is the way they live in the city. But in the country, since they live far apart, they all eat in their own homes. No household has any shortage of food, since, after all, everything eaten by the city-dwellers comes from the farmers.\n\n### HOW THE UTOPIANS TRAVEL\n\nIf someone wants to visit friends who live in another city or is simply taken with a desire to see the place, he easily gets the permission of his syphogrant and tranibor unless a necessary job keeps him from going. He is sent out as part of a group, with a letter from the ruler which grants them permission and sets the day they must be back. They are provided with a carriage and a public slave to drive the oxen and take care of them. But unless there are women in the group, they leave the carriage behind as more of a hindrance than a help. Throughout the whole journey they carry nothing with them; yet they lack for nothing and are at home everywhere. If they stay anywhere longer than one day, each of them works at his trade and is treated very kindly by his fellow craftsmen.\n\nIf someone takes it upon himself to wander outside his territory, when he is caught without the ruler's passport, he is treated with contempt, brought back as a runaway, and severely punished. If he dares to repeat the offense, he is punished with slavery. But if someone is taken with a longing to wander through the fields belonging to his own city, he is not prohibited from doing so, as long as he gets his father's permission and his wife's consent. But wherever he goes in the countryside, he is not given any food until he has done the work allotted to the morning or however much work is usually done there before dinner. Under this regulation he is allowed to go anywhere within the boundaries of his city's territory, for he will be no less useful to the city than if he were in it.\n\nSo you see that nowhere is there any chance to be idle; there is no excuse for laziness, no wine taverns, no alehouses, no brothels, no occasion to be corrupted, no hideouts, no hangouts. With the eyes of everyone upon them, they have no choice but to do their customary work or to enjoy pastimes which are not dishonorable. Such behavior on the part of the people is bound to produce an abundance of everything. And when it is distributed equitably to everyone, it follows that no one can be reduced to poverty or forced to beg.\n\nIn the senate at Amaurot (to which, as I said before, three representatives come every year from each city), once they have determined what surpluses are at hand in each place and what places have shortages, they immediately make up the deficiencies of the one with the excess supplies of the other, and they provide them as a free gift, receiving nothing in return from those to whom they gave them. But if they gave something to a city and received nothing in return, they also get what they need from some other city and pay nothing for it. Thus the whole island is like one household.\n\nWhen they have enough provisions for themselves (which they do not think they do unless they have provided for two years, since the next year's outcome is uncertain), they export to other countries vast quantities of grain, honey, wool, linen, timber, red and purple dye, fleece, wax, tallow, leather, and also livestock. They give one-seventh of all this to the poor in that country and sell the rest at a moderate price. In exchange they not only acquire goods they do not have at home (they lack almost nothing except iron) but also they bring back to their homeland enormous quantities of silver and gold. They have continued this practice for such a long time that they now have everywhere a greater supply of those metals than you would think possible. Hence they do not much care whether they are paid in cash or credit, and they accept promissory notes for most of what is owed them, but never from private persons; instead they make the usual legal documents binding on the city government. When the loan comes due, the city requires it to be paid by the private debtors and puts it in the public treasury; then the city enjoys the use of it until the Utopians call it in. For the most part they never do, since they think it is hardly right to claim what is of no use to them from those who have a use for it. But if circumstances require that they lend part of it to another nation they call it in, or when they are obliged to go to war; that is the only reason they keep all of the treasure which they have at home, as protection against extreme danger or sudden emergencies. They use it especially to pay enormous wages to foreign mercenaries, whom they would much rather expose to danger than their own citizens. They are also aware that with large sums of money even the enemies themselves can be bought and set against one another, either through treason or open hostilities.\n\nThis is the reason they reserve such an incalculable treasure, although they do not keep it as treasure but in a form I am really ashamed to tell you. I am afraid you will not believe what I say, and all the more rightly so since I am aware that if I had not seen it in person I would have been reluctant to believe it if someone else told it to me. For in general the more foreign something is to the habits of the listeners, the harder it must be for them to believe it. But actually, a prudent judge of the matter will perhaps be less surprised that they handle silver and gold in their own way rather than ours, since all their arrangements are so different from ours. In fact, since they themselves have no use for money but rather keep it as protection against events which might or might not happen, in the meantime they keep gold and silver (from which money is made) in a form that lets no one place more value on it than it deserves by its nature. And obviously it deserves far less than iron, without which mortals could no more live, by heaven, than they could without fire or water, whereas nature gave to gold and silver no use which we could not easily do without; the folly of mankind gives them value because they are rare, but nature, on the other hand, like a kind and gracious mother, made the most useful elements openly available, like air, water, and earth, but she hid away what is vain and unprofitable in the most remote recesses.\n\nNow if in their society these metals were put away in some tower, the ruler and the senate might be suspected of deceiving the people by some trick and getting some good from it for themselves\u2014such is the foolish anxiety of the mob. And then if they made platters out of them or other vessels made by goldsmiths, if ever the occasion arose to melt them down and use them to pay mercenaries, they realize that once people had begun to delight in them they would be reluctant to give them up. To obviate these difficulties they have thought up a method quite compatible with the rest of their arrangements but very far removed from ours (for we value gold very highly and hide it away quite carefully), a method which is therefore hard to believe unless you have experienced it. Whereas they eat and drink from vessels of earthenware and glass, beautifully crafted but inexpensive, they use gold and silver, not only in the common halls but also in private houses, to make all the chamberpots and lowliest containers. Moreover, the chains and heavy shackles used to restrain the slaves are made of the same metals. Finally, the most notorious criminals wear gold rings in their ears, gold rings on their fingers, a gold collar around their necks, and even a gold band around their heads. By these means they see to it that the same metals which other nations give up with almost as much grief as if their guts were being pulled out have so little value that if circumstances required the Utopians to part with all such metals none of them would think they had lost as much as a single farthing.\n\nFurthermore, they gather pearls on the seashore and even diamonds and rubies on some cliffs; they do not look for them, however, but when they have found some by chance, they polish them. They use them to deck out their infants, who are boastful and proud of such gems in their earliest childhood; but, as they get a little older and notice that such trinkets are worn only by children, they become ashamed of them of their own accord and, with no urging from their parents, they give them up just as our children discard their baubles, necklaces, and dolls when they grow up.\n\nThese arrangements, so different from those of other peoples, have produced quite different feelings and attitudes. That never became clearer to me than in the incident of the Anemolian ambassadors. They came to Amaurot while I was there and since they had come to discuss important matters, the three citizens chosen by every city had come before they arrived. All the ambassadors from neighboring countries, who had landed there before and were familiar with the customs of the Utopians, knew that they did not revere sumptuous clothing, considered silk contemptible, and even associated gold with disgrace; and so they used to come clothed as modestly as possible. But the Anemolians lived further away and had less contact with them. Hence, when they saw that all the Utopians wore one and the same rough garment, they thought they did so because they had nothing better to wear and, with more pride than wisdom, they decided to set themselves up as gods by the elegance of their trappings and to dazzle the eyes of the poor Utopians by the splendor of their garb.\n\nAnd so when the three ambassadors made their entry, their retinue of a hundred retainers was dressed in particolored garments, mostly made of silk, but the ambassadors themselves, who were noblemen in their own country, were garbed in cloth of gold, with large chains and earrings of gold, and also golden rings on their fingers, and on top of that strings of pearls and gems hanging from their hats, and in sum, decked out in everything that the Utopians use to punish slaves, to mark off someone in disgrace, or to make toys for children. And so it was a sight to see how they ruffled their feathers when they compared their finery with the clothing of the Utopians (for the people had poured out onto the streets). On the other hand, it was no less delightful to observe how totally mistaken their hopes and expectations were and how far they were from the consideration they thought they would receive. For in the eyes of all the Utopians, except for the very few who had had some good reason to travel to foreign countries, all their splendid trappings seemed shameful. They greeted all the retainers of the lowest rank reverently as if they were lords. But they considered the ambassadors to be slaves because they wore golden chains, and so they passed over them with no respect whatever. In fact, you could have also seen children there who had thrown away their gems and pearls. When they saw such gems affixed to the hats of the ambassadors, they nudged their mothers and said: \"Look, mother, that big lout is still wearing little pearls and gems, as if he were a little boy!\" But the mother would reply in all seriousness: \"Hush, my son, I think he is one of the ambassadors' fools.\" Others criticized those golden chains as useless because they were so fine that a slave could easily break them and so loosely fastened that a slave could shake them off whenever he wanted and run off anywhere he wanted, footloose and fancy-free.\n\nBut after the ambassadors had lived there for a day or two and seen such an enormous amount of gold treated as if it were worthless and contemned there as much as it was honored in their countries, and when they also noticed that the chains and shackles of only one runaway slave contained more gold and silver than the trappings of all three of them, they were crestfallen and sheepishly put away all the finery which they had so haughtily displayed, especially after they had talked more informally with the Utopians and learned their customs and opinions.\n\nIndeed they are amazed that any mortal can take delight in the dubious sparkle of a tiny gem or precious stone when he can look at a star or even at the sun, or how anyone could be so insane as to imagine that he is nobler because of fine-spun woolen thread, since that wool (however fine-spun) was once worn by a sheep, which was at the same time nothing more than a sheep. They are likewise amazed that gold, which in itself is useless, is now prized so highly everywhere that mankind itself, which gave it value and for whose use it got that value, is valued much less than the gold itself, so much so that some beef-witted blockhead, who has morals to match his folly, nevertheless has many wise and good men in his service, for no better reason than that he has a heap of gold coins. And if some turn of Fortune or trick of the law (which turns things topsy-turvy no less than Fortune herself) should transfer this heap from the heir to the lowest lout in the whole household, the master would shortly enter the service of his servant as if he were a mere adjunct and appendage of the coins. But what they find most amazing and despicable is the insanity of those who all but worship the rich, to whom they owe nothing and who can do them no harm; they do so for no other reason except that they are rich, knowing full well that they are so mean and tightfisted that they will certainly never give them one red cent during their whole lives.\n\nThese opinions and others like them they have formed partly from their upbringing, since they were brought up in a commonwealth whose institutions are farthest removed from those kinds of folly, and partly from instruction and books. For though not many in each city are dispensed from physical labor and assigned to do nothing but study (namely those in whom they have perceived from their childhood remarkable talent, extraordinary intelligence, and devotion to learning), nevertheless all children are introduced to good books, and throughout their lives a good many people, both men and women, devote to learning the hours I have mentioned as free from labor.\n\nThey learn the various branches of knowledge in their own language, which has no lack of vocabulary, is not unpleasant to the ear, and is not surpassed by any other in the expression of thought. It has spread throughout most of that part of the world, though everywhere else it is corrupted in various ways.\n\nOf all the philosophers whose names are so famous in this known part of the world, they had not so much as heard of any before our arrival, and yet in music, dialectic, arithmetic, and geometry, they have made almost the same discoveries as our own ancient writers did. But though they measure up to our ancient writers in almost all respects, they are not up to the discoveries of modern dialecticians. In fact, they have not discovered a single one of those rules about restrictions, amplifications, and suppositions which have been so subtly excogitated in the _Parva logicalia_ and which are taught to young men everywhere in our world. And then, as for second intentions, they are so far from being able to understand them that none of the Utopians could see man in general, as they say, even when we pointed him out with our finger, though, as you know, he is plainly colossal and bigger than any giant. But they are very expert in the orbits of stars and the movement of heavenly bodies. In fact, they have devised instruments of various designs which enable them to understand very accurately the movements and positions of the sun and moon and also the other stars which are visible in their hemisphere. But as for the conjunctions and oppositions of the planets and the whole fraud of divination by the stars, they have never so much as dreamed of it. By means of signs that they have perceived from long observation they predict rainstorms, winds, and other changes in the weather. But concerning the causes of those phenomena, and concerning tides and the saltiness of the ocean, and in general concerning the origin and nature of the heavens and the world, they agree on some points with our own ancient philosophers, and on others, just as the ancients disagreed with one another, they also differ from all the ancients and propose new theories, and yet they do not entirely agree among themselves.\n\nIn that area of philosophy which deals with ethics, they discuss the same issues as we do. They inquire about the goods of the mind and body and external goods, and whether the designation \"good\" applies to all of these or only to the gifts of the mind. They discuss virtue and pleasure, but the primary and principal controversy is about what they think human happiness consists in, whether one thing or many. On this point they seem over-inclined to the position which claims that all or the most important part of human happiness consists of pleasure. And what is even more surprising, they claim support for this self-indulgent view even from religion, which is sober and strict and, indeed, almost gloomy and stern. For they never analyze happiness unless they combine some religious principles with the rational analysis of philosophy, since they think that without such principles reason by itself is too weak and deficient to investigate true happiness.\n\nThese principles are of this sort: that the soul is immortal, and by the beneficence of God is born for happiness; that our virtues and good deeds will be rewarded after this life, and our crimes have punishments prepared for them. Though these are religious principles, the Utopians still think that reason leads them to believe and grant them; if they are eliminated, the Utopians have no hesitation in affirming that no one could be so stupid as not to feel that he ought to pursue his own pleasure by hook or crook. He would only be concerned not to sacrifice a greater pleasure for a lesser one and not to pursue one that would be requited by pain. For they think it would be truly insane to pursue virtue, which is harsh and difficult, and not only to banish the pleasures of life but even to seek out pain of your own accord, and to expect to get nothing out of it (for how can you get anything out of it if you get nothing after death, since you have spent your whole life here without pleasure, that is, wretchedly?). But as it is, they think happiness consists not in every sort of pleasure but in pleasure that is good and honorable, for they believe that our nature is drawn to pleasure as the highest good by virtue itself, whereas the opposite faction attributes happiness to virtue alone.\n\nAnd then they define virtue as living according to nature; to that end, they say, we were created by God. We follow the guidance of nature when we obey reason in choosing and avoiding things. Furthermore, reason above all inspires mortals to love and revere the majesty of God, to whom we owe our very existence and our capacity to be happy. Secondly, reason admonishes and encourages us to lead lives with as little anxiety and as much joy as possible and, beyond that, to exert ourselves in helping all others achieve the same end because of our natural fellowship. For not even the gloomiest and sternest advocate of virtue, who despises pleasure so much that he would impose toil, vigils, and mortifications on you, would refrain from enjoining you to do as much as you can to alleviate the poverty and distress of others, and he would think it praiseworthy and humane for one human being to rescue and comfort another, since the very essence of humanity (and no virtue is more proper to human beings) is to relieve the distress of others, eliminate sadness from their lives, and restore them to a joyful life, that is, to pleasure. Why should nature not impel us to do the same for ourselves? For either a joyful life, that is, a life of pleasure, is wrong and in that case we should not only not help anyone to achieve it but rather we should do all we can to make everyone avoid it as harmful and deadly, or if you are not only allowed but even required to obtain it for others, why not do so first of all for yourself? You should be no less well-disposed to yourself than to others. For when nature prompts you to be good to others, she does not require you to turn around and be cruel and merciless to yourself. Nature herself, they say, prescribes as the aim of all our actions a joyful life, that is, pleasure, and they define virtue as following the prescriptions of nature. But when nature invites mortals to help each other to lead cheerful lives (and she is certainly right to do so, since no one is so far above the rank of human beings that nature should care for him alone, whereas in fact she is equally concerned about all those whom she groups together as belonging to the same species), she also, of course, forbids you time after time to seek your own advantages in ways that create disadvantages for others.\n\nTherefore they think that not only private agreements must be kept but also public laws which have either been promulgated by a good ruler or which a people not oppressed by a tyrant or deceived by some trick have laid down by common consent to govern the distribution of vital commodities, that is, the means to pleasure. As long as these laws are not broken, to look out for your own good is prudent; to promote the public good is pious. But to deprive someone else of pleasure to promote your own is wrong; on the other hand, to deprive yourself of something to give it to someone else is a work of humanity and kindness and it always brings you more good than it takes away. For it is counterbalanced by gifts given in return, and also your consciousness of having done a good deed and the thought of the love and good will of those you have benefited will give you mental pleasure that outweighs any loss of bodily comfort. Finally, as religion makes clear to true believers, God will repay the loss of brief and paltry pleasures with enormous and never-ending joy. Following this line of reasoning and having considered the matter long and hard, they think that all our actions, including also our virtuous deeds, are directed toward pleasure as our happiness and final end.\n\nThey define pleasure as any motion or state of the mind or body which produces delight in accord with the guidance of nature. Not without reason do they add that the impulse must be in accord with nature. For just as not only our senses but also our reason pursues whatever is pleasurable by nature, that is, pleasures not achieved through wrongdoing, or acquired with the loss of a greater pleasure, or followed by hardship, so too they hold that all those unnatural pleasures which mortals agree to call delightful by the emptiest of fictions (as if it were in their power to change the thing by changing the name) are so far from contributing to happiness that they actually hinder it because, once they have taken over the mind, they occupy it totally and leave no room for true and genuine pleasures. For a great many things are not pleasurable by their very nature and are, in fact, for the most part bitter, but through the perverse enticement of evil desires they are not only thought to be the greatest pleasures but are even included among the primary reasons for living.\n\nAmong those who pursue false pleasures they include those whom I mentioned before who think that the finer the gown they wear the better they are. On this one point they are wrong twice over. They are no less deceived in thinking the gown is better than in imagining they themselves are. For if you consider the usefulness of a garment, why is wool woven with fine thread better than wool woven with coarser thread? But they think they excel in fact, not merely in their illusions. They ruffle their feathers; they believe that they are more valuable because of their clothes. And on that basis, honors they would not have dared hope for in cheaper clothes they demand as rightly due to their elegant gown, and they are outraged if someone passes them by without due deference.\n\nAnd then isn't it equally stupid to be much taken with empty and worthless honors? For what natural pleasure is there in someone's baring his head to you or bending his knee? Will that relieve the pain in your own knee or cure the delirium in your head? It is amazing how some are caught up in this imaginary, specious pleasure: delightfully insane, they flatter themselves and take pride in their imagined nobility simply because they happen to be descended from a long series of ancestors who are considered to be rich, above all rich landlords (for nowadays there is no other source of nobility except wealth), and yet they think they are not a whit the less noble even if their ancestors have left them no wealth or they themselves have squandered it.\n\nWith these they group the persons I mentioned before who are enthralled by gems and precious stones and almost think they have been deified if they ever get a fine specimen, especially if it is the sort most highly valued in their own times; for not all sorts are highly regarded by all persons and at all times. But they do not buy such a stone unless it is removed from its gold setting and exposed, and even then not unless the seller swears and guarantees that it is a genuine jewel and a true gemstone; so afraid are they that their eyes may be deceived by a counterfeit substituted for a real stone. For why should your eyes be any less delighted by a counterfeit since they cannot distinguish it from a real one? To you each of them should have equal value, no less so, by heaven, than they would to a blind man.\n\nWhat about people who keep superfluous wealth under lock and key, taking delight not in using the amassed treasure but merely in contemplating it? Do they feel any real delight or rather are they not deluded by a false pleasure? How about those who are subject to a different vice and hide away their gold, intending not only never to use it but perhaps never even to see it any more; in their anxiety not to lose it, they lose it. For surely it is lost if it is buried in the ground so as to be of no use to you and perhaps not to any other mortal. But still, when the treasure is hidden away, you feel carefree and happy. If a thief took it away and you died ten years later without knowing of the theft, in all those years that you lived after the money was stolen, what difference did it make to you whether it was removed or remained safe? In either case its usefulness to you was the same.\n\nTo these categories of absurd enjoyment they add gambling (a sort of madness they know of only through hearsay, not experience) and also hunting and falconry. For what pleasure can there be, they say, in throwing dice on a gaming table? Even if there were any pleasure in it, you have done it so often that mere repetition should have made you sick of it. How can it be delightful to hear the barking and howling of dogs?\u2014isn't that a disgusting noise? Why do hunters feel more pleasure when a dog chases a hare than when a dog chases a dog? For in either case the action is the same, that is, running, if that is what pleases you. Or if you are attracted by the hope of carnage and the expectation of seeing the slaughter with your own eyes, you ought instead to be moved to compassion when you see a little hare torn to pieces by a dog, a weak creature tormented by a stronger one, a timid creature fleeing from a ferocious beast, a harmless creature from a cruel hound. And so the Utopians have assigned the whole business of hunting to the butchers, whose trade (as I said before) is conducted entirely by slaves, considering it beneath the dignity of free men. They consider it the lowest function of the trade. The other activities of butchers are more useful and honorable, since they contribute much more and destroy animals only out of necessity, whereas the hunter seeks nothing but pleasure from the slaughter and butchering of some poor little creature. Even in beasts themselves, according to the Utopians, such an eagerness to view carnage springs from a cruel disposition, or else the continual indulgence in such brutal pleasure finally degenerates into cruelty.\n\nThough the herd of mortals consider such pursuits as these and others like them (for there is no end to them) to be pleasures, the Utopians firmly hold that they have nothing to do with pleasure, since there is no natural sweetness in them. Though they ordinarily produce sensual joy (which seems to be the function of pleasure), the Utopians are unwilling to change their minds. The reason they seem pleasant is not the nature of the things themselves but the perverse habits of their devotees, whose vicious attitudes cause them to embrace what is bitter as sweet, just as the defective tastebuds of pregnant women make them think that pitch and tallow are sweeter than honey. And yet no one's judgment, if it is vitiated by disease or habit, can change the nature of pleasure, or of anything else for that matter.\n\nTrue pleasures they divide into various classes, assigning some to the mind, others to the body. To the mind they attribute understanding and the sweetness which springs from the contemplation of the truth. To these they add the pleasure of looking back on a lifetime of good deeds and the sure hope of happiness to come.\n\nThey divide bodily pleasure into two kinds: one is the sweetness which pervades the senses, either when the supplies our natural heat has used up are replenished (as they are by food and drink) or else when the excessive elements overburdening our bodies are discharged. This happens when we purge our intestines of excrement, or go about generating children or when the itching in some part of the body is alleviated by rubbing or scratching. But sometimes pleasure results not from the replenishment sought by our bodily members nor from relieving them of excess but from some secret but remarkable power which tickles, excites, and attracts our senses to itself, such as the pleasure arising from music.\n\nThey claim that there is another kind of bodily pleasure which consists in the balanced and quiet condition of the body, that is, when a person's health is not disturbed by any disease. Such health, as long as it is not interrupted by any pain, is delightful in itself, even though it is not affected by any external pleasure. Though it is less obvious and affects the senses less grossly than the insistent desire for food and drink, nevertheless many Utopians hold it to be the greatest pleasure of all. Almost all of them believe that it is a great pleasure and the foundation and basis, as it were, of all the others, since it is the only one which keeps our lives peaceful and desirable; and, if you take it away, there is no room left for any pleasure at all. For the mere absence of pain without health they regard as insensibility, certainly not as pleasure.\n\nThey have long since rejected the position of those who think that stable and undisturbed health should not be considered to be a pleasure because, they say, its presence can be felt only through some external stimulus (for they, too, have debated this question intensely). But now they are in almost complete agreement with the opposite position, that health is actually essential to pleasure. For according to them, disease brings pain, which is unalterably opposed to pleasure, in the same way as disease is opposed to health. Why not conclude, in turn, that there is pleasure in undisturbed health? On this point they do not think it makes any difference whether the disease is a pain or the pain comes from the disease; in either case the effect is the same. Thus, if health itself is a pleasure or if it necessarily brings pleasure with it as fire brings heat, the result in either case is that, wherever health is, stable pleasure cannot be lacking.\n\nMoreover, when we eat, they say, what happens is that health, which has begun to fail, now has food as its ally in the battle against hunger. As it gradually becomes stronger, the very progress toward its ordinary vigor brings with it the pleasure of being reinvigorated. And so if health finds joy in the struggle, will it not rejoice when the victory is won? But when it has at last happily recovered its former strength, which was the sole object of the whole struggle, will it immediately become insensible and fail to recognize and embrace its own good? The idea that health is not perceived they consider to be very far from the truth. For when we are awake, who does not perceive that he is healthy\u2014except someone who is not? Who can be so constricted by dullness and lethargy that he does not admit that health is delightful and enjoyable? And what is enjoyment but another name for pleasure?\n\nAbove all they embrace the pleasures of the mind, which they consider the first and foremost of all pleasures. They think that mental pleasure springs primarily from the practice of the virtues and the consciousness of a good life. Of the pleasures supplied by the body they give the first place to health. As for the pleasure of eating and drink and whatever else falls under a similar category of delight, they think they should be sought, but only for the sake of health, for such activities are not enjoyable in themselves but only insofar as they counter the unnoticed encroachments of ill health. And therefore a wise man, they say, should ward off disease rather than seek medicine for it and avoid pain rather than seek relief from it; just so it would be better not to have any need for such pleasure than to be relieved by it.\n\nIf anyone thinks that this kind of pleasure makes him happy, he must also confess that his life would be the happiest of all if it could be spent in perpetual hunger, thirst, and itching, followed by eating, drinking, scratching, and rubbing\u2014 and who can fail to see that such a life would be not only foul but also miserable? Certainly these are the lowliest of all pleasures, since they are the least unadulterated and never occur except in conjunction with the pain contrary to them. Thus the pleasure of eating is coupled with hunger, and not in equal proportions, for the pain is both longer and more intense. For it begins before the pleasure and never departs until the pleasure also ceases. Therefore they do not place much stock in such pleasures, except insofar as necessity demands them. But they also rejoice in them and gratefully acknowledge the kindness of Mother Nature, who uses the sweetest pleasures to entice her offspring to do what they must always be doing out of necessity. How irksome our lives would be if the daily ailments of hunger and thirst had to be warded off by drugs and bitter medications like the other diseases which afflict us less often?\n\nThey gladly cherish beauty, strength, agility as special and enjoyable gifts of nature. Certainly the pleasures which are mediated by our ears, eyes, and noses and which nature assigned as proper and peculiar to the human race (for no other kind of creature admires the design and beauty of the world, or is moved by the beauty of fragrances except to distinguish kinds of food, or recognizes the harmonious or discordant intervals in sounds), these pleasures, I say, they cultivate as adding a certain enjoyable spice to their lives. In all of them, however, they impose the limitation that a lesser should not impede a greater pleasure or that a pleasure should not cause pain at some later time\u2014and they think this will necessarily happen if the pleasure is dishonorable.\n\nThey think it is certainly quite mad for someone to despise a beautiful figure, to deplete his strength, to turn agility into torpor, to wear out his body with fasting, to ruin his health, and to scorn the other favors bestowed by nature, unless he neglects his own good so as to work more avidly for the the good of others or the public welfare, and in return for his effort he expects greater pleasure from God. Otherwise to inflict pain on oneself without doing anyone any good\u2014simply to gain the empty shadow of virtue or to be able to bear with less distress adversities that may never come\u2014this they consider to be insane and the mark of a mind that is both cruel to itself and ungrateful to nature, rejecting her benefits and not deigning to be beholden to her.\n\nThis is their view of virtue and pleasure; and in the absence of religious inspiration from heaven revealing something holier, they think human reason can discover no truer doctrine. I do not have time now to examine whether or not their teaching is correct, nor is it necessary, since I undertook to present their principles, not to defend them. But whatever validity their precepts may have, I am fully persuaded that nowhere will you find a more extraordinary people or a happier commonwealth.\n\nPhysically they are agile and vigorous, stronger than you would expect from their height, though they are not undersized. Though their soil is not uniformly fertile and their weather is not particularly favorable, they protect themselves from the climate by moderation in their diet and they work hard to remedy the defects of the soil, so that nowhere in the world will you find a more abundant supply of crops and cattle or bodies more vigorous and subject to fewer diseases. You can see them there diligently employing the usual agricultural methods of improving infertile soil by skill and effort, but you could also see a forest that they uprooted with their own hands and planted in another place. The reason for doing this was not greater production but transportation: they wanted the timber closer to the sea or rivers or the cities themselves, since it takes less labor to move crops by land over long distances than it does to transport timber.\n\nThey are an easy-going people, cheerful and clever. They enjoy their leisure but they endure physical labor well enough as long as it is useful (but otherwise they are hardly fond of it); in intellectual pursuits they are indefatigable. When we told them about the literature and learning of the Greeks (for in Latin there is nothing except the poets and historians that would be likely to interest them very much) it was amazing how eagerly they pressed us to help them master Greek by giving them instruction. And so we began to read, at first more out of a desire not to seem lazy than from any hope that much good would come of it. But when we had made a little progress, their diligence immediately made us anticipate that ours would not be wasted. They began to imitate the shape of the letters so easily, to pronounce the words so readily, to memorize so quickly, and to recite so accurately that we would have thought it miraculous except that the majority of them had undertaken this study not only on their own initiative but also at the explicit command of the senate, and hence they were selected from the most talented and mature scholars. And so in less than three years there was nothing in that language which they had not mastered; they read good authors with no hesitation, unless they encountered some textual crux. I tend to think they mastered Greek all the more easily because it is somewhat related to their own language. I suspect that the Utopian people originally sprang from the Greeks because their language, which is otherwise closest to Persian, preserves some vestiges of Greek in the names of cities and magistrates.\n\nOn the fourth voyage, instead of trade goods I took on board a fair-sized packet of books because I was fully determined to return only after a long time, if ever. From me they got most of Plato's works, more of Aristotle's, and also Theophrastus _On Plants,_ which was mutilated in several places, I'm sorry to say. During the voyage the book had not been put away properly and a playful monkey came upon it; he mischievously ripped out some pages here and there and tore them up. Of the grammarians they have only Lascaris, for I did not take Theodore with me nor any dictionary except Hesychius and Dioscorides. They are very fond of Plutarch's books and they are also much taken with the wit and elegance of Lucian. Of the poets they have Aristophanes, Homer, and Euripides, and also Sophocles in the small typeface of Aldus. Of the historians they have Thucydides and Herodotus, as well as Herodian.\n\nFurthermore, as for medical books, my companion Tricius Apinatus had brought with him some shorter works of Hippocrates and the _Microtechne_ of Galen; for these books they have a high regard. Even though there is hardly a country in the world that has less need of medicine, still it is nowhere more honored, precisely because they consider a knowledge of it as one of the finest and most useful branches of science. When they investigate the secrets of nature using the resources of science, they not only experience wonderful pleasure from doing so but they also think they win the highest approbation from the creator and maker of the world. For they suppose that he, like other workmen, set up the marvelous mechanism of this world for mankind to view and contemplate (and men are the only creatures he made capable of doing so) and that therefore he is fonder of a careful observer and meticulous admirer than he is of some lazy blockhead who ignores such a marvelous spectacle as if he were a mindless brute.\n\nAnd so the natural talent of the Utopians, trained by study, is marvelously effective in inventing techniques which make some contribution to a comfortable life. Two of these they owe to us, printing and papermaking, but even these they owe not only to us but in large part to themselves. For when we had shown them some books printed by Aldus on paper and had spoken a bit about the material for making paper and the technique of printing letters, though we did not really explain it (since none of us was expert in either process), they immediately and most ingeniously figured it out. And whereas before they had written only on vellum, bark, and papyrus, they immediately tried to make paper and to print with type. Though at first they did not get it quite right, by frequent attempts they soon mastered both techniques, and they became so proficient that if they had copies of Greek texts, there could have been no lack of printed editions. But as it is, they have no more than what I have mentioned, but what they have they have disseminated in many thousands of printed copies.\n\nAny sightseers who visit them are especially welcome if they are recommended by unusual intellectual gifts or knowledge of many lands gained by traveling widely (and for that reason they welcomed us warmly when we landed), for they are eager to learn what is happening everywhere in the world. But not very many come there to trade. For what can they bring except iron or gold and silver, which they would prefer to take home than to export? As for their own exports, they think it more advantageous to deliver them themselves than to have others pick them up, for in that way they learn more about foreign countries everywhere and they keep their seamanship and nautical skills from getting rusty.\n\n### SLAVES\n\nPrisoners of war they do not consider to be slaves except those captured in wars they themselves have fought. The children of slaves and the slaves of foreign countries whom they have obtained are not kept in slavery. Their slaves are those who have committed a serious crime in Utopia or foreigners who have been condemned to death for committing some crime (and these are by far the larger number), for the Utopians acquire many of them, sometimes cheaply, more often gratis, and take them away. These kinds of slaves they not only keep constantly at work but also in chains. Utopian slaves, however, they treat more harshly since they consider them baser and deserving of more severe punishment because they had an extraordinary education and the best of moral training, yet still could not be restrained from wrongdoing. Another class of slaves is made up of poor, overworked drudges from other nations who choose of their own accord to be slaves among the Utopians. These they treat decently and, except that they make them work a bit harder (since they are used to it), they are treated not much less kindly than the citizens. If they wish to depart (and that does not happen very often), they are not kept against their will nor are they sent away empty-handed.\n\nThey care for the sick, as I said, with great concern, omitting nothing whatever in the way of medicine or diet that might restore them to health. They sit with those who are suffering from an incurable disease, talk with them, console them, and do what they can to alleviate their pain. But if someone suffers from a disease which is not only incurable but also constantly and excruciatingly painful, then the priests and the magistrates point out that he can no longer live a useful life, that he is a heavy burden to himself and to others, and that he has outlived his own death; they encourage him to make a decision not to maintain the sickness and disease any longer and urge him not to hesitate to die, but rather to rely on hope for a better life; since he lives in a prison where he is cruelly tormented on the rack, he should escape from this miserable life on his own or willingly allow others to rescue him from it. This would be a wise act, they say, since death would deprive him of no advantages but would save him from suffering; and since in doing so he would be following the advice of the priests, the interpreters of God's will, it would also be a pious and holy deed.\n\nThose who agree with these arguments voluntarily starve themselves to death or are put to sleep and dispatched with no sensation of dying. But they do not do away with anyone who is unwilling, and they do not in any way diminish their attendance on him. Those who are persuaded and die in this way are treated with honor; but otherwise anyone who commits suicide for reasons not approved by the priests and senate is deemed unworthy of either burial or cremation and is ignominously thrown into a swamp without a proper funeral.\n\nA woman does not marry until she is eighteen, a man not until he is four years older than that. If a man and a woman are convicted of engaging in secret intercourse before marriage, they are both severely reprimanded and they are forbidden ever to marry anyone unless the ruler remits the sentence. But both the master and the mistress of the household where the offense was committed fall into utter disgrace for not doing their duty with sufficient diligence. They punish this offense so severely because they foresee that few would join together in married love, living their whole lives with one person and enduring besides the troubles that come with marriage, if they were not carefully restrained from promiscuous intercourse.\n\nMoreover, in choosing spouses they have a custom which seemed to us absolutely absurd and thoroughly ridiculous, but they observe it strictly and seriously. The bride, whether virgin or widow, is presented naked to the groom by a sober and respected matron, and the groom in turn is shown naked to the bride by some honorable man. When we laughed at this custom and criticized it as ridiculous, they in turn were amazed at the extraordinary folly of all other nations: when they are buying a colt\u2014a matter of no great expense\u2014they are so cautious that even if the animal is almost completely exposed they refuse to buy it unless the saddle and saddlecloth are removed so as to reveal any sores that might be hidden beneath them; yet in choosing a spouse\u2014a matter which will make them either happy or miserable for the rest of their lives\u2014they are so careless that they judge her whole person by a mere handsbreadth, that is, by her face only, since the rest of her is wrapped up in her clothes, and according to that judgment they join themselves to her, not without great danger of not getting along with her if they later find something offensive. For not everyone is so wise as to pay attention only to character, and even in the marriages of the wise the gifts of the body add something to the virtues of the mind. Certainly some ugly deformity concealed beneath clothing can completely alienate a man's mind from his wife when his body can no longer be separated from her. If such a deformity should occur after the wedding, then everyone must put up with his lot; but before the wedding the laws should see to it that no one is duped or deceived.\n\nAll the more care needs to be taken because, of all the countries in that part of the world, they are the only one that is monogamous, and their marriages are almost never dissolved except by death, though adultery or unbearably offensive conduct can be grounds for divorce. The offended party gets permission from the senate to remarry; the offender is disgraced and can never remarry. Otherwise, it is absolutely forbidden to put away a wife against her will and without any blame on her part because of some bodily disfigurement. They consider it cruel to desert someone at the very time she is in most need of comfort and they think it would make her uncertain and insecure about her old age, which brings diseases with it and is itself a disease.\n\nBut sometimes it happens that two people are temperamentally incompatible, and if they have each found someone else with whom they hope they can live more agreeably, they separate by mutual consent and remarry, but not, however, without the permission of the senate, which does not permit divorce unless the senators and their wives have examined the case very carefully. Even then they do not do it readily because they know that the expectation of easily remarrying is hardly a means of strengthening the love of married couples.\n\nAdulterers are punished with the harshest servitude, and if both were married the injured parties may divorce their spouses and marry each other if they wish to; otherwise they may marry whomever they like. But if one of the injured parties continues to love such an undeserving spouse, the marriage can remain intact, as long as the innocent party is willing to accompany the criminal condemned to hard labor; and it happens sometimes that the affectionate concern of the one and the repentance of the other move the ruler to mercy so that he sets them free again. But if the crime is repeated, it is punished with death.\n\nTheir laws do not prescribe punishments for other crimes, but rather the senate determines penalties according to how heinous or venial each particular offense seems to be. Husbands chastise their wives and parents their children, unless an offense is so serious that open punishment is advisable in order to maintain public morality. But generally the most serious crimes are punished with servitude, which they consider no less grievous to the criminal and much more advantageous to the commonwealth than to execute wrongdoers and immediately get rid of them altogether. They do more good by their labor than by their death, and they offer a long-standing example to deter others from similar crimes. If slaves are rebellious and unruly, then they are finally slaughtered like wild beasts that cannot be restrained by bars or chains. But if they are patient, they are not left entirely without hope. If they are tamed by long suffering and show that they regret the sin more than the punishment, their servitude may be either mitigated or revoked, sometimes by the ruler's prerogative, sometimes by popular vote.\n\nAttempted seduction is no less dangerous than seduction itself. In fact, in all sorts of crimes, they equate the clear and deliberate attempt with the completed deed, for they do not think that the mere incompletion of the deed should benefit someone who did everything he could to complete it.\n\nThey are very fond of fools: they consider it quite shameful to treat them with contempt, and they have nothing against finding enjoyment in their foolery, since they think that will do the most good for the fools themselves. If anyone is so strict and gloomy that he never laughs at any word or deed, they do not entrust fools to him, out of fear that he would not treat them kindly enough, since to him they would be not only useless but not even entertaining\u2014and that is the only talent they have.\n\nTo mock someone for being disfigured or crippled is considered shameful and disfiguring, not to the person mocked but to the mocker, since it is stupid for him to blame someone for a defect which it is not in his power to avoid.\n\nThey consider it lazy and negligent not to keep up natural beauty by grooming, but they consider seeking help from cosmetics a disgraceful affectation. They know from experience itself that no physical beauty recommends wives to their husbands as much as respect and an upright character. Some men may be snared by beauty alone, but none can be held except by virtue and compliance.\n\nThey not only deter from crime by punishments, but they also foster virtue by rewarding it with honors. And so in the marketplace they set up statues of outstanding men who have done extraordinary service to the commonwealth, thus preserving the memory of their good deeds so that posterity may have the glory of their ancestors as a spur and incentive to virtuous deeds.\n\nAnyone who campaigns for public office becomes disqualified for holding any office at all. The Utopians live together amiably, since no magistrate is arrogant or terrifying; they are called fathers and they live up to the name. Honor is willingly paid to them (as is proper); it is not exacted from those unwilling to give it. The ruler is not singled out by his clothes or a crown but rather by the sheaf of grain he carries: the sign of the high priest is a wax candle borne before him.\n\nThey have very few laws, for very few suffice for persons trained as they are. Indeed, one of their primary charges against other nations is that endless volumes of laws and interpretations are not sufficient. But they consider it quite unjust to bind people by laws which are so numerous no one can read through all of them or so obscure that no one can understand them. Moreover, they ban absolutely all lawyers as clever practitioners and sly interpreters of the law. For they think it is practical that everyone should handle his own case and present the facts to the judge as he would to a lawyer; in this way there will be less confusion and the truth will be easier to determine, since he tells his story without having learned any evasion from a lawyer, while the judge weighs all the details carefully and protects simple souls from the false accusations of crafty litigants. In other countries, such straightforwardness is difficult to obtain because there is a mass of incredibly intricate laws. But among them everyone is knowledgeable about the laws. For, as I said, there are very few laws, and as for interpretations, they consider the most obvious the most correct. For though all laws (they say) are promulgated to inform everyone of his duty, a subtle interpretation will inform very few (for few can understand it); on the other hand, the simpler and more obvious meaning of the laws is clear to everyone. Otherwise, as far as ordinary people are concerned (and they constitute the largest group that needs to be informed), it would make no difference if you formulated no laws at all or if, after you have formulated them, you interpret them in such a way that no one can understand them without great intelligence and long analysis. The dull judgment of ordinary people is not adequate to that task, and they do not have enough time, occupied as they are in making a living.\n\nInspired by the virtues of the Utopians, those of their neighbors who are free and can choose as they please (for the Utopians themselves have long since liberated many of them from tyranny) ask for and obtain Utopians to act as their magistrates, some for a year, some for five years; when they have served their term, they bring them back to Utopia with great honor and praise, and take replacements with them back to their own country. And certainly these countries are providing very well and very effectively for the public welfare, which depends, for good or bad, on the character of the magistrates. What persons could they choose more wisely than those whose honesty cannot be undermined by bribes (since they will soon return to a place where money is useless) and who cannot be swayed by some person or faction, since they have no connections among that people? Wherever these two vices, favoritism and greed, get a hold on judicial decisions, all justice, which is the mainstay of the commonwealth, is immediately undermined. The peoples who recruit magistrates from them are called allies by the Utopians; the others on whom they have bestowed benefits are called friends.\n\nThey do not make treaties with any nation\u2014such treaties as other nations so often make, break, and remake. What good is a treaty, they say, as if nature did not sufficiently bind one human being to another? And if someone scorns nature, do you think he will be concerned with mere words? They are especially drawn to that view because in that part of the world treaties and agreements between princes are not usually observed with very much good faith.\n\nIn Europe, of course, and especially in those parts which follow the faith and religion of Christ, the authority of treaties is everywhere holy and inviolable, partly because of the goodness and justice of the princes themselves, partly out of reverence and respect for the popes, who themselves undertake nothing which they do not carry out most scrupulously and likewise command all princes to keep their promises to the letter; if any prince reneges, the pope makes him comply by pastoral censure and sharp reproof. Certainly they are right in thinking that it is quite shameful for those who are specifically called the faithful not to be faithful to their treaties.\n\nBut in that new world, which is as far from us in customs and way of life as it is removed from us by the distance the equator puts between us, no one has confidence in treaties: the more ceremoniously and solemnly the knot of a treaty is tied, the more quickly it is untied; it is easy to find some defect in the wording, which they often intentionally devise with some clever loophole, so that the language can never bind them so tightly that they cannot somehow escape, breaking both the treaty and their word. If such craftiness, or rather downright fraud and deceit, occurred in a private transaction it would be contemptuously decried as sacrilegious and deserving of the gallows\u2014and that by the very same persons who are proud of having advised the prince to do the same. Thus it happens that justice seems either to be nothing more than a plebeian and humble virtue, far beneath the exalted dignity of a king, or at least there seem to be two kinds of justice: one is fit for ordinary people, lowly and creeping along the ground, fenced in on all sides, totally encumbered with chains and unable to escape; the other kind is a virtue proper to princes, which is more august than the ordinary virtue and hence much freer\u2014 forbidden, in fact, to do only what it does not wish to do.\n\nSuch behavior on the part of the princes there, who have so little respect for treaties, is the reason, I think, that the Utopians make no treaties; perhaps they would change their minds if they lived here. But even if treaties were strictly observed, they still think the practice of making them at all is a bad custom because it implies that nations think they are natural-born enemies to each other (just as if there were no natural ties between two peoples separated only by a little distance, a hill or a creek) and that they would rightly try to destroy one another if they were not bound by treaties; and that even if they have entered into a treaty, they are not united in friendship but rather have permission to prey upon each other, insofar as nothing which the treaty forbids is couched with sufficient care because of some oversight in the language. On the other hand, the Utopians think that no one should be considered an enemy if he has done no harm, and that the natural bond which unites us should replace treaties, and that men are more adequately bound to one another by good will than by agreements, more strongly joined by their hearts than by their words.\n\n### MILITARY PRACTICES\n\nThey loathe war as positively bestial (though no sort of beast engages in it as constantly as mankind), and unlike almost all nations they consider nothing more inglorious than glory won in warfare. Therefore, though they regularly devote themselves to military training on certain appointed days so that they will not be incapable of fighting when circumstances require it\u2014and not only the men do so but also the women\u2014 they are reluctant to go to war and do so only to defend their own territory, or to drive an invading enemy from the territory of their friends, or else, out of compassion and humanity, they use their forces to liberate a oppressed people from tyranny and servitude. When they come to the aid of their friends, it is not always to defend them but sometimes also to requite and avenge injuries inflicted on them. But they do this only if they have been consulted before any steps are taken and if, after they have verified the facts, demanded restitution, and been refused, they themselves declare war. They decide to do this not only when an enemy has invaded and plundered one of their friends, but also, and even more fiercely, when their friends' merchants in any part of the world have been unjustly accused under some pretext of justice, either by using unjust laws speciously or by interpreting good laws perversely.\n\nThis was the only reason for the war which the Utopians fought a little before our time on behalf of the Nephelogetes against the Alaopolitans: some Nephelogete merchants among the Alaopolitans had been treated unjustly under some pretext of justice (or so the merchants thought). Certainly, whether the cause was just or unjust, it was avenged by a hideous war, in which the surrounding nations also added their energy and resources to the hostile forces of the major opponents so that some prosperous peoples were ravaged, others were badly shaken. One disaster followed upon another until finally the surrender and enslavement of the Alaopolitans put an end to the war. The Utopians, who sought nothing for themselves, subjected the vanquished to the Nephelogetes\u2014a people hardly to be compared with the Alaopolitans in their heyday.\n\nSo fierce are the Utopians even when they are punishing only monetary injuries against their friends; but they are not so when the injury is against themselves. If they should be cheated out of their property, as long as they are subjected to no physical force, they set limits to their anger: they merely refrain from trade with that nation until restitution is made, not because they care less for their own citizens than for their allies but rather they are more offended by their friends' loss of money than by their own because their friends' merchants are severely injured by such a loss, since it comes from their own private possessions. But their own citizens lose nothing but public property, goods which were abundant at home, even superfluous, for otherwise they would not have been exported. So the loss is hardly perceived by anyone. Hence they feel that it would be cruel to punish an injury by killing many people when it causes no inconvenience to any of the Utopians in their lives or livelihood. But if any of their citizens is unjustly disabled or killed, wherever it may be, whether it be done by a public decision or by a private citizen, they send ambassadors to ascertain the facts, and if the malefactors are not handed over to them they cannot be put off but declare war immediately. If the guilty persons are handed over for punishment, they are sentenced to death or servitude.\n\nThey are not only grieved by a bloody victory but also ashamed of it, thinking that it is stupid to pay too much for merchandise, however valuable it may be. But if they conquer and crush an enemy by skill and cunning, they glory mightily in the victory, holding public parades to celebrate it and putting up a monument as if for a hard-won victory. For they boast that they have acted with courage and fortitude only when they have won the victory as no other creature but man is able to win it, that is, by the power of his wits. For bears, lions, boars, wolves, dogs, and other animals (they say) fight with the power of their bodies; and though most of them surpass us in strength and ferocity, we outdo them all in intelligence and reasoning.\n\nTheir one and only aim in warfare is to gain the objective which, if they had obtained it beforehand, would have kept them from going to war at all. Or, if circumstances make that impossible, they seek to punish those they consider culpable so severely that fear will keep them from daring to do such a thing in the future. These are the goals they set for their undertaking, and they try to achieve them quickly, but yet in such a way that a concern for avoiding danger takes precedence over winning praise and glory.\n\nAnd so, immediately after declaring war, they see to it that many notices certified by their official seal are put up secretly and simultaneously in the most conspicuous places in the enemy's territory, promising a huge reward to anyone who does away with the enemy's prince; they also assign lesser, but still very substantial, sums for the deaths of those individuals they list in the same notices. These are the persons who, apart from the prince himself, were responsible for plotting against the Utopians. They double the reward assigned to the assassin if he brings them any of the proscribed persons alive; in fact, they offer the same rewards to the proscribed persons themselves, and throw immunity into the bargain, if they turn against their comrades. Thus their enemies quickly suspect all outsiders and even among themselves they are neither trusting nor trustworthy so that they live in a state of utter panic and no less peril. For it has very often turned out (as is well known) that a good number of them, and among them the prince himself, have often been betrayed by those they trusted the most. So easy is it to get someone to commit any crime whatsoever by means of bribes, and for that reason the Utopians set no limits to their bribes. Keeping in mind the great risks they are urging people to take, they take care to balance the magnitude of the danger with the lavishness of the reward; hence they promise not only enormous quantities of gold but also personal and perpetual title to rich estates in the safe and secure territory of their friends, and they faithfully keep their promises.\n\nOther nations condemn this practice of bidding for and buying off an enemy as a barbarous, degenerate crime, but the Utopians think it does them great credit: it shows them to be wise, since in this way they win great wars without fighting at all, and also humane and compassionate, since by killing a few malefactors they spare the lives of many innocent persons who would have fallen in battle, both their own soldiers and those of the enemy; for they pity the rank-and-file of the enemy's soldiers almost as much as their own citizens because they know they do not go to war of their own accord but are driven to it by the madness of princes.\n\nIf this procedure is not successful, they sow and cultivate the seeds of dissension by encouraging the brother of the prince or some nobleman to have hopes of gaining the throne. If such internal factions languish, they stir up neighboring peoples and set them against their enemy by digging up some ancient claim such as is never lacking to kings.\n\nWhen they have promised resources for war, they supply money lavishly, but their citizens very sparingly. They hold their own people so very dear and value each other so highly that they would not be willing to exchange a single one of their own citizens for the enemy's prince. But they are not at all reluctant to pay out gold and silver, since they keep it only for this purpose and would live no less comfortably if they spent all of it. Then too, apart from the wealth they have at home, they also have a limitless treasure abroad, since many nations, as I said before, owe them money. And so they hire mercenaries from everywhere and send them to war, especially the Zapoletes.\n\nThese people live five hundred miles to the east of Utopia. Rough, rude, and fierce, they prefer to live in the forests and rugged mountains where they were brought up. They are a hardy people, able to endure heat, cold, and hard labor. They have no interest in agriculture, no acquaintance with refinements, no concern about their houses or clothes; they care only about their flocks. They live mostly from hunting and plundering. They are born only for warfare; they zealously seek opportunities to fight and when they find one they embrace it eagerly. They set out in great numbers and offer themselves cheaply to whoever needs soldiers. The only skill they have to live on is one that aims at death.\n\nThey fight fiercely and with complete loyalty for whoever pays them. But they bind themselves for no fixed period. They sign on with the stipulation that if an enemy offers them higher wages tomorrow they will take his side, and if they are lured with slightly higher pay they will return to the side they abandoned. There are very few wars in which a great many of them are not fighting in both armies. And so it happens every day that blood relatives who were hired by the same side and lived together amicably are separated a little later in opposing armies and fight each other as enemies. Forgetting both kinship and friendship, they run each other through with violent hostility, trying to kill each other for no other reason than that they were hired for a pittance by opposing princes. They reckon their wages so strictly that adding one penny to their daily pay can easily cause them to change sides. They have quickly become greedy through and through, and yet it does them no good for what they gain with their blood they immediately squander on debauchery, and wretched debauchery at that.\n\nThese people fight for the Utopians against any mortals whatsoever because they hire their services for more than they can get anywhere else. And just as the Utopians seek good men in order to use them, so too they also enlist these wicked men in order to use them up. When they need to use them, they urge them on with great promises and expose them to the greatest dangers so that most of them do not return to claim what they were promised. To the survivors they faithfully keep their promises so as to make them eager to undertake similar exploits. Nor do they have any qualms about doing away with so many of them, since they believe the human race would owe them a great debt of gratitude if they could purge the whole world of such loathsome and wicked scum.\n\nApart from the Zapoletes, they use the forces of those for whom they have taken up arms, and after that the auxiliary troops of other friendly nations. As a last resort they add their own citizens, from whom they choose a man of proven valour to command the whole army. Under him they appoint two men who remain private citizens as long as he is safe, but if he is captured or killed, one of the two succeeds him, and in case of a mishap he himself is succeeded by the third, so that if the commander is in danger (and the fortunes of war are quite various) the whole army does not panic.\n\nIn each city they choose troops from a list of volunteers. No one is sent out to foreign wars against his will, for they are convinced that if someone is by nature fearful he will not only not fight vigorously himself but he will also inspire fear in his comrades. But if their country is invaded during a war, cowards of this sort, as long as they are physically fit, are dispersed among better troops in the ships or they are spread out here and there on the walls so that they have no place to run away to. Thus shame in the presence of their friends, the confrontation with the enemy, and the absence of any hope of escaping overcome fear, and often they make a virtue out of extreme necessity.\n\nThough no one is sent to a foreign war unwillingly, if women are willing to accompany their husbands to battle the Utopians are so far from preventing them that they exhort them to do so and encourage them with praise. Each accompanies her husband to the front and is stationed shoulder to shoulder with him in the battle line. Moreover, each soldier is surrounded by his children and relatives by blood or marriage so that they all have help close by from the persons who are by nature most highly motivated to help one another. It is a great disgrace for one spouse to return without the other or for a son to come back after the loss of a parent. The result is that once it comes to hand-to-hand combat, if the enemy stands his ground, the battle is so long and grim that it ends in a general slaughter.\n\nCertainly they take every precaution to avoid having to fight themselves, as long as they can wage war using mercenaries to take their place. But when they can no longer avoid entering the fray, the courage with which they fight matches the prudence with which they avoided fighting as long as they could. They do not give their all in a first furious attack but rather they grow stronger gradually and over a period of time, and they are so resolute that they would rather die than retreat. For one thing, they are certain that everyone at home is provided for, and they do not need to worry about their children (such concern generally breaks the spirits of lofty souls); so their courage is proud and contemptuous of defeat. Moreover, their skill in the arts of war gives them confidence. Finally, sound ideas, instilled in them from childhood on, both by instruction and through the institutions of the commonwealth, give them courage: they hold life neither so cheap as to throw it away recklessly nor so perversely dear as to cling to it greedily and shamefully when honor requires them to give it up.\n\nWhen the battle is at its fiercest everywhere, a picked group of sworn and dedicated young men seek out the enemy commander. Sometimes they attack him openly; sometimes they try to ambush him. They assail him from close by and from a distance and they attack him in a wide, unbroken phalanx, continuously replacing the exhausted men with fresh troops. And unless he saves himself by running away, it rarely happens that he is not killed or captured alive by his enemies.\n\nIf they win a victory, they do not slaughter the defeated; they would rather capture than kill those they have put to flight. And they never pursue retreating troops without keeping in reserve at least one battalion drawn up under its colors. They do this so regularly that if the rest of their own forces have been defeated and they win the victory with their last battalion, they would rather let the whole enemy army escape than get into the habit of pursuing the fugitives with their own forces in disarray. They remember something that happened to them more than once: when the main body of the whole Utopian army had been overwhelmed and put to flight, while the enemy was exulting in the victory and pursuing them as they ran away in all directions, a few of their own troops held in reserve and on the lookout for opportunities suddenly attacked the enemy troops, who were scattered and straggling and careless from overconfidence, and thus changed the whole outcome of the battle; snatching certain and undoubted victory from their enemies' hands, the conquered turned the tables and conquered the conquerors.\n\nIt is not easy to say whether they are more clever in laying ambushes or more cautious in avoiding them. You would think they are preparing to flee when that is the last thing they intend; on the other hand, when they do intend to flee, you would imagine that is the last thing they have in mind. For if they feel they are at a disadvantage either in numbers or location, then they either move their camp silently at night, or escape by some stratagem, or withdraw gradually by day, keeping their ranks in such good order that they are no less dangerous in retreat than when they attack. They fortify their camp very carefully with a wide and very deep moat; the earth they dig up is piled up on the inside. In such work they do not use the services of common laborers. It is done by the hands of the soldiers themselves, and the whole army joins in the work except for the armed soldiers outside the rampart who keep watch against sudden attacks. With so many soldiers pitching in, they build massive fortifications around a large area with incredible speed.\n\nThey wear armor which is strong enough to ward off blows but does not hinder movement and gestures\u2014so much so that they feel no inconvenience even in swimming. For swimming in armor is one of the ordinary rudiments of their military training. At long range their weapon is the arrow which they shoot with great force and accuracy, not only on foot but also from horseback. At close quarters they strike not with swords but with battle-axes, which are deadly because of their sharp blade and their weight, whether used to hack or thrust. They are very skilled in devising siege engines. Once they are made, they conceal them very carefully, lest they become known before it is time to use them and turn out to be more ridiculous than useful. In designing them their primary concern is to make them easy to move and aim.\n\nWhen they make a truce with their enemies, they keep it so religiously that they do not violate it even under provocation. They do not lay enemy territory waste or burn their crops; they even do what they can to keep the grain from being trampled by men and horses, for they think it may be of some use to them. They injure no unarmed civilians except for spies. They offer amnesty to cities that surrender and even those taken by siege they do not sack; instead they execute those who prevented the surrender; they enslave the rest of the defenders, but the civilian populace they leave unharmed. If they find persons who urged the town to surrender, they grant them a share in the property of the condemned; they divide up the rest and give it to their auxiliaries, for none of the Utopians takes any of the booty.\n\nWhen the war is over, they assess the costs not against the friends for whom they incurred them but against the losers; they demand part of it in money, which they reserve for similar use in warfare, and part in estates within enemy territory, from which they forever enjoy a not inconsiderable income. They now have revenues of this sort in many nations; it accumulated gradually in various ways and now amounts to 700,000 ducats a year. To take care of it they send out collectors of revenue, who live there in grand style and play the part of great lords. But there is plenty left over to put into the treasury, unless they choose to give credit to the nation that owes it, which they often do until they need it, and even then it rarely happens that they demand all of it. They also bestow some of these estates on those whom they have persuaded to place themselves in great danger, as I mentioned before.\n\nIf some prince takes up arms against them and is preparing to invade their domain, they immediately confront him with a huge force outside their own boundaries, for they are reluctant to wage war within their own territory and no exigency could ever induce them to allow foreign auxiliaries on their island.\n\n### THE RELIGIONS OF THE UTOPIANS\n\nThere are various religions not only throughout the island but also within individual cities: some worship the sun as god, others the moon, others a different planet. Others worship some ancient paragon of either virtue or glory, venerating such a person not only as a god but as the supreme god. But the vast majority, and those by far the wiser ones, accept none of those gods and believe there is a certain single deity, unknown, eternal, infinite, inexplicable, diffused throughout this whole universe not physically but by his power, in a manner that is beyond human comprehension; him they call their parent. To him alone they attribute the origin, increase, progress, changes, and goals of all things; him and no other they honor as divine.\n\nActually, though all the others hold different beliefs on some points, they agree with the monotheists in thinking that there is some one supreme being who made and rules the universe, and in their native language they all agree in calling him Mythras, but they differ in that they identify the supreme power variously, each asserting that whatever he considers to be supreme is in fact that single nature to whose divine majesty, by the consensus of all nations, the whole creation is attributed. But gradually they are all abandoning these superstitious variations and joining together in that one religion which seems more reasonable than the others. And there is no doubt that the other beliefs would have vanished long ago if it were not that, whenever something untoward happened to someone who was considering changing his religion, fear made him think that it was not accidental but was sent from heaven, as if the divinity whose cult he was forsaking were avenging a wicked affront to himself.\n\nBut after they had heard from us the name, the teaching, the behavior, and the miracles of Christ, and the no less miraculous constancy of so many martyrs who freely shed their blood and thus brought many peoples, from far and wide, over to their religion, you would not believe how eagerly they also were converted, whether through the secret inspiration of God or because Christianity seemed closest to the sect which is predominant among them, although I think it was a matter of no small moment with them to hear that Christ approved of life in common for his disciples and that it is still practiced among the most genuine Christian communities. But certainly, whatever the reason, no small number of them were converted to our religion and were washed clean in the sacred waters of baptism.\n\nBut because there was, I am sorry to say, no priest among the four of us (for only that number remained after two of us had given up the ghost), they received the other sacraments but still lacked those which among us are conferred only by priests. But they know about them and long for them most intensely. In fact, they also earnestly discuss among themselves whether someone chosen from among their number could receive the sacerdotal character without the dispatch of a Christian bishop. And in fact it seemed they were about to choose someone, but when I left they had not yet done so.\n\nEven those who do not agree with the Christian religion still do not frighten anyone away from it; they do not oppose anyone who has embraced it, except that one of our community was repressed while I was there. Shortly after he was baptized, over our objections, he harangued publicly about Christianity with more zeal than prudence, and he began to get so carried away that he not only ranked our religion above all the rest but condemned all the others outright. He cried out against them as profane; he denounced their worshipers as wicked, sacrilegious, and worthy to be punished in eternal fire. When he had preached like this for a long time, they arrested him and tried him, not for despising their religion but for exciting riots among the people. They convicted him and sentenced him to exile, for it is one of their oldest policies that no one should come to any harm because of his religion.\n\nFor Utopus had learned that before his arrival the inhabitants squabbled incessantly about religion and he had noticed that the sects, which generally disagreed with each other and fought for their country in separate groups, provided the opportunity for him to conquer all of them. Hence, from the very beginning, after he had obtained the victory, he decreed first of all that everyone could practice the religion of his choice and could also strive to convert others to it, but only so long as he advocated it calmly and moderately with rational arguments. And if he could not win others over by persuasion, he was not to assail their religions bitterly nor use force against them, and he was to refrain from insults. Anyone who quarrels insolently about religion is punished with exile or enslavement.\n\nUtopus laid down these rules not only for the sake of peace, which he saw was completely undermined by constant strife and implacable hatred, but also because he thought such a decree would benefit religion itself. In religious matters he did not venture to dogmatize rashly because he was uncertain whether or not God wishes to have varied and manifold kinds of worship and hence inspires different people with different views. Certainly he thought that to use force and threats to make everyone accept what you believe to be true is both arrogant and absurd. Then too, if one religion should be actually true and the rest false, still he easily foresaw that in the long run the the truth would sooner or later emerge and prevail by its own force as long as the matter was handled reasonably and moderately. But if the struggle is conducted with arms and uprisings, since the worst people are always the most headstrong, the best and holiest religion, embroiled among empty superstitions, will be choked like grain among thorns and briars. And so he left the whole matter open and left everyone free to believe whatever he wanted, except that he solemnly and strictly forbade that anyone should sink so far below the dignity of human nature as to think that the soul dies with the body or that the world is ruled by mere chance and not by providence.\n\nAnd for this reason they believe that after this life punishments are ordained for vices and rewards for virtues. Anyone who thinks otherwise they do not even include in the category of human beings since he has degraded the lofty nature of his soul to the base level of a beast's wretched body. Still less will they count him as one of their citizens, since he would set no store whatever by all their laws and morality if it were not for fear. For who can doubt that someone who has nothing to fear but the law and no hope of anything beyond bodily existence would strive to evade the public laws of his country by secret chicanery or to break them by force in order to satisfy his own personal greed? For that reason they bestow no honors on such a person, they assign him to no office, they put him in charge of no public responsibility. He is universally looked down on as a lazy and spineless character. But he is not subjected to any punishment because they are convinced that it is not within a person's power to believe whatever he wishes; they neither compel him by any threats to mask his opinion nor accept any pretexts or lies, which they utterly despise as next door to deliberate malice. Still they do forbid him to argue for his opinion, but only among the common people. Otherwise, in private, among priests and prudent men, they not only permit him to argue but also encourage it, confident that in the end his madness will yield to reason.\n\nThere are also others, and they are by no means few (since their position is not forbidden as completely unreasonable or wicked) who go to the opposite extreme and believe that the souls of brute beasts are also immortal, although not comparable to ours in dignity nor destined for the same happiness.\n\nAlmost all of them are certain and fully persuaded that human happiness will be so boundless that they mourn for everyone who is sick but not for anyone who dies, unless they see that he is torn from life anxiously and unwillingly. For they take this to be a very bad sign, as if such a soul, despairing and conscious of guilt, fears to leave life because of some secret presentiment of future punishment. Moreover, they think God will hardly be well pleased when someone who is summoned does not come running eagerly but is dragged off reluctant and unwilling. Therefore when they see such a death they are dismayed and they carry out the dead persons with grief and in silence; after praying that God in his mercy will kindly forgive the infirmities of such souls, they cover the body with earth. On the other hand, when someone dies joyfully and full of good hope, they do not mourn him, but rather they conduct his funeral with song; commending his soul to God with great affection, they finally cremate his body with reverence, not grief, and erect on that spot a column inscribed with the virtues of the dead person. After they have returned home, they tell of his character and deeds, and no part of his life is rehearsed more often or more eagerly than his cheerful death.\n\nThey think this commemoration of his uprightness is a very strong inducement to virtue for the living and the most acceptable form of veneration to the dead, whom they also believe to be present when they are talked about, though invisible to us because the eyesight of mortals is too dull to see them. For it would not be suitable to the condition of the blessed to lack the liberty of going wherever they want, and it would be ungrateful of them to have no desire whatever to visit their friends, to whom they were united in mutual love and charity while they were alive; such charity they suppose, like other good qualities, is increased, not diminished, in good men after their death. Thus they believe that the dead are present among the living, observing what they say and do, and for that reason they go about their business more confidently because of their trust in such protectors; their belief in the presence of their ancestors also deters them from secret wrongdoing.\n\nThey have nothing to do with fortune-telling and other vain, superstitious divinations, which other people take quite seriously but which they consider ridiculous. But miracles which happen apart from any natural cause they revere as works and witnesses which manifest the presence of a deity. They say such miracles often happen there, and sometimes, during great crises, they pray publicly for a miracle with great confidence and they do obtain it.\n\nThey think the worship which pleases God is the contemplation of nature and the praise which springs from it. But there are others, and they are by no means few, who neglect learning in the name of religion, who do not strive to attain any knowledge, and who allow themselves no leisure at all. They are determined to earn happiness after death solely by keeping busy in the service of others. And so some tend the sick, others repair the roads, clear out ditches, rebuild bridges, dig turf, sand, or stones, fell and cut up trees, cart lumber, crops, and other provisions into the cities. They perform their services not only for the public but also for private citizens, and they work even harder than slaves. They willingly and cheerfully undertake any tasks which are rough, difficult, dirty, and shunned by most people because of the toil, disgust, and hopelessness they entail. They see to it that others have leisure, while they themselves are continually engaged in labor and toil, but nevertheless they take no credit for it. They neither censure the lives of others nor extol their own. The more they conduct themselves like slaves the more everyone honors them.\n\nThey are divided into two sects. The one is celibate and not only abstains from any sexual activity but also eats no meat (and some of them no animal products at all), totally rejecting the pleasures of this life as harmful, longing only for those of the world to come, which they strive to obtain by toil and vigils. Meanwhile, confident that they will soon obtain them, they are cheerful and energetic. The other group, no less devoted to labor, prefers to marry: they do not spurn the consolations of marriage, and they think that just as they owe such activity to nature, they owe children to their country. They do not refuse any pleasure which does not interfere with their work. They like to eat the flesh of animals precisely because they think such food gives them the strength to do all kinds of work. The Utopians consider this group more prudent; the other they regard as holier. If they claimed on rational grounds to prefer celibacy to marriage and a hard life to a comfortable one, the Utopians would laugh at them; but since they profess to be motivated by religion, the Utopians respect and revere them. On no other subject are they more cautious about making any rash pronouncements than on matters concerning religion. In their language these persons are given the special title \"Buthrescae,\" which could be translated into Latin as \"religiosi.\"\n\nTheir priests are extremely holy and therefore very few. For each city has no more than thirteen, one for each church, except during wartime, when seven of them set out with the army and are replaced by substitutes for the time being. But when the priests return, each assumes his former position. Until the time when the substitutes, in an orderly succession, replace priests who have died, they become attendants of the high priest (for one priest has authority over the others). They are elected by the people in the same way as other magistrates, that is, by secret ballot, in order to avoid partisan strife. Once elected, they are consecrated by their own college of priests.\n\nThey preside over divine worship, attend to religious matters, and act as guardians of morality. To be summoned by them and rebuked for dishonorable conduct is considered to be a great disgrace. But their role is to exhort and admonish; to repress and punish wrongdoers is the function of the ruler and other magistrates. The priests, however, do excommunicate those they find to be thoroughly vicious. There is almost no other punishment which they fear more, for such persons are both dejected by their infamy and tormented by a bad conscience. They may not even be physically safe for very long. For unless they quickly convince the priests that they are repentant, they will be seized by the senate and punished for their impiety.\n\nChildren and young people are educated by the priests, and they devote no more attention to learning than to character and virtue. They take the greatest pains from the very first to instill in the tender and impressionable minds of children sound opinions conducive to preserving the common good. When such ideas are thoroughly absorbed in childhood, they persist throughout all of manhood and they are extremely useful in protecting the status of the commonwealth, which decays only because of vices which spring from perverse attitudes.\n\nThe wives of the priests are the very finest women in the country, unless the priests themselves are women, for that sex is not excluded; but they are rarely elected and must be widows of advanced years.\n\nNo magistrates are held in greater honor among the Utopians, so much so that even if they commit a crime they are not subject to a public tribunal but are left to God and their own consciences. For they do not think it is right to lay human hands on anyone, however vicious, who has been dedicated to God in such a special way as a holy offering, so to speak. It is easier for them to observe this custom because priests are so few and are chosen so carefully. For it is very unlikely that someone who is the cream of the crop and is elevated to a position of such dignity only because of his virtue should degenerate into corruption and vice. And even if that very thing should happen\u2014for human nature is changeable\u2014nevertheless there would certainly be no reason to fear that the public would be in any great danger, because the priests are so few and have no power beyond what derives from the honor paid them. In fact the very reason they have so few and scattered priests is to keep the dignity of the order, now held in such high esteem, from being cheapened by bestowing the honor on many, especially since they think it is hard to find very many who are equal to the dignity of the office, for which merely mediocre virtues are insufficient.\n\nTheir reputation at home is no greater than the esteem in which they are held by foreign nations. This becomes quite clear, I think, if we note the reason for it. When troops are engaged in battle, the priests kneel at a distance but not very far away, dressed in their sacred vestments; lifting up their hands to heaven, they pray first of all for peace, and then for victory for their own forces, but without bloodshed on either side. When their soldiers win they rush into the battle line and restrain the fury of their forces against the routed troops. Merely to see them and make oneself known to them by calling out is enough to save anyone's life; to touch their flowing garments also protects the remaining goods of fortune from any damage due to the war. Hence they are venerated by the countries all around them, who attribute to them such genuine majesty that oftentimes they provide as much protection for their own citizens as they do for their enemies. For sometimes it has happened that, when their battle line was thrown back in despair and had turned to flee, as the enemy was rushing in to kill and plunder, the intervention of the priests has stopped the slaughter and separated the two armies so that a peace was devised and established on equitable terms. For nowhere is there a nation so savage, cruel, and barbarous that they do not hold their persons to be sacrosanct and inviolable.\n\nThe first and last days of each month and likewise of each year are celebrated as feastdays; the months are marked off by the orbit of the moon, just as the year is established by the course of the sun. In their language they call all of the first days \"cynemerni,\" the last days \"trapemerni,\" names that are equivalent to \"first-feastday\" and \"last-feastday.\" Their churches are remarkable not only for their workmanship but also for their capacity to hold immense crowds\u2014which is necessary because there are so few of them. They are all dimly lit, and they say this resulted not from lack of skill but from the deliberate policy of the priests, who believe that too much light distracts our thoughts, whereas dim and doubtful lighting concentrates the mind and intensifies religious devotion.\n\nSince religion is not the same for everyone there, yet all the forms of it, however varied and different, converge from various directions on one goal, the worship of the divine nature, nothing is seen or heard in the churches which is not held in common by all the religions. If any denomination has a rite peculiar to it, they provide for it in their own homes. Public worship is conducted according to a ritual which does not at all detract from any of the private devotions. Therefore no images of the gods are seen in churches so that everyone can be free to imagine the form of God as he wishes according to his own religion. They invoke God by no other name than Mythras, a name they all apply to the one divine nature, whatever it may be. No prayers are devised which everyone cannot say without offending his own denomination.\n\nAnd so on the last-feastdays they gather in church in the evening, still fasting and ready to give thanks to God for the success they enjoyed during the year or month just coming to an end. On the next day, which is the first-feastday, they flock to church in the morning to pray for success and happiness in the following year or month which begins on that feastday. But on the last-feastdays, at home, before they go to church, wives throw themselves at the feet of their husbands, and children do the same before their parents; they confess that they have sinned either through commission or negligence, and they beg forgiveness for their offenses. In this way if some little cloud of strife has arisen in the household, it is dispelled by such atonement so that they can attend the sacrifices with clear and untroubled minds, for they are too conscientious to worship with a disturbed conscience. Therefore those who feel anger or hatred toward someone do not intrude on the sacrifices unless they are reconciled and purged of such feelings, for fear of some swift and severe punishment.\n\nWhen they get there, the men sit on the right side of the church, the women separately on the left. Then too, they position themselves so that the male members of each household sit in front of the master of that household, and the matron of each household sits in the last row of the women. Thus they see to it that all the actions of everyone are observed in public by the persons whose authority maintains discipline at home. Moreover they are also very careful to intermingle everywhere young persons with their elders; otherwise, if children were entrusted to children, they might spend in childish tomfoolery the time that they should devote to cultivating a religious fear of the heavenly beings, the greatest and practically the only incitement to virtue.\n\nIn their sacrifices they do not kill any animals; they do not think that a merciful God, who bestowed life on animals precisely that they might live, takes any pleasure in bloodshed and slaughter. They burn incense and other fragrant substances. They also display many candles, not because they do not know that such things add nothing to God's nature, no more than human prayers do, but they like this harmless mode of worship and people feel that somehow such perfumes, lights, and other ceremonies lift up the human heart and make it rise more eagerly in divine worship.\n\nIn church the people wear white garments; the priests are clothed in vestments of various colors, marvelous in both workmanship and design, though the materials are not especially expensive, and they are not woven with gold threads or encrusted with rare gems; rather they are fashioned out of the feathers of various birds, so elegantly and skillfully that the costliest material would not match the value of the workmanship. Moreover, these feathers and plumes of birds and the set patterns in which they are arranged on the priests' garments are said to contain certain secret mysteries which, if rightly understood (and the interpretation is carefully handed down by the priests), remind them of the benefits bestowed on them by God and of the devotion they owe him in return, as well as their duty to each other.\n\nWhen the priest, dressed in this way, comes out of the sacristy, everyone immediately prostrates himself on the ground out of reverence; on all sides the silence is so profound that the spectacle itself inspires a certain fear, as if in the presence of some divinity. They remain on the ground for a while and then arise at a signal from the priest. Then they sing the praises of God, accompanied by musical instruments, which are mostly shaped differently from those seen in our part of the world. Most of them surpass ours in sweetness of tone, but some of them are incomparably superior to ours. But in one respect their music is undoubtedly far ahead of ours: whether instrumental or vocal, it imitates and expresses natural feelings so well, the sound matches the sense of the words so closely (whether they express supplication or joy, peace or turmoil, sadness or anger), and the shape of the melody matches the meaning so well that it quite wonderfully stirs up, pierces, and inflames the hearts of the hearers. Finally the priest and the people recite together certain customary and fixed forms of prayer, composed in such a way that everyone can apply to himself what they all recite together.\n\nIn these prayers each one recognizes God as the creator and ruler of the universe and also the source of all good things. He thanks God for bestowing so many benefits on him, but especially because through God's kindness he was placed in the happiest form of commonwealth and has been allotted the religion which he hopes is the truest. If he is mistaken in this matter or if there is some form of commonwealth or religion which is better and more approved by God, he prays that God in his goodness will cause him to recognize it, for he is prepared to follow wherever God leads him. But if this form of commonwealth is the best and this religion is truest, he asks that God will both make him steadfast and lead other mortals to the same way of life and the same idea of God\u2014unless there is in fact something in this variety of religions which pleases his inscrutable will.\n\nFinally he prays that by an easy death God may take him to himself, how soon or late he certainly does not dare to determine. But, provided that God's majesty is not offended by it, he would much rather go to him by a very difficult death than be kept away from him any longer, even by a prosperous way of life. After saying this prayer they once more prostrate themselves on the ground and after a little while they get up again, go to eat lunch, and spend the rest of the day playing games or doing military exercises.\n\nI have described to you as accurately as I can the plan of their commonwealth, which I certainly consider to be not only the best but also the only kind worthy of the name. For elsewhere they always talk about the public good but they are concerned with their own private welfare; here, where there is no private property, everyone works seriously for the public good. And for good reason in both places, for elsewhere is there anyone who does not know that unless he looks out for his own personal interest he will die of hunger, no matter how flourishing the commonwealth may be; therefore necessity causes him to think he should watch out for his own good, not that of others, that is, of the people. On the other hand, here, where everything belongs to everyone, no one doubts that (as long as care is taken that the public storehouses are full) nothing whatever will be lacking to anyone for his own use. For the distribution of goods is not niggardly; no one is a pauper or a beggar there, and though no one has anything, all are rich.\n\nFor what greater wealth can there be than to be completely spared any anxiety and to live with a joyful and tranquil frame of mind, with no worries about making a living, not vexed by a wife's complaints and demands, not fearing a son will end up in poverty, not concerned about a daughter's dowry, but secure about the livelihood and happiness of himself and his own, his wife, children, grandchildren, great-grandchildren, great-great-grandchildren, and however long a line of descendants noblemen presume they will have. Indeed those who worked before but are now disabled are no less provided for than those who are still working.\n\nAt this point I wish that someone would venture to compare with this equity the justice to be found in other nations, where I'll be damned if I can find any trace whatever of justice or equity. For what sort of justice is it for some nobleman or goldsmith or moneylender or, in short, any of the others who either do nothing at all or something that is not very necessary for the commonwealth, to live luxuriously and splendidly in complete idleness or doing some superfluous task? And at the same time a laborer, a teamster, a blacksmith or farmer works so long and so hard that a beast of burden could hardly sustain it, performing tasks so necessary that without them no commonwealth could survive at all for even a single year, and yet they earn such a meager living and lead such miserable lives that beasts of burden seem to be better off, since they do not have to work so incessantly, their fodder is not much worse (and to them it tastes better), and in the meantime they are not afraid of what will happen to them. These workers are driven to toil without profit or gain in the present; they are crushed by the thought that they will be poverty-stricken in their old age, for their daily wages are not enough for that very day, much less can they accumulate any surplus which might be put aside every day to provide for their old age.\n\nIs a commonwealth not unjust and ungrateful if it lavishes so many benefits on noblemen, as they are called, and goldsmiths, and the rest of that crew who are either idle or else merely flatterers and providers of empty pleasures, but makes no proper provision for farmers, colliers, laborers, teamsters, and blacksmiths, without whom there would be no commonwealth at all; unmindful of their sleepless labors and forgetting their many and great contributions, it first uses up the labors of their flourishing years, and then, when they are worn down by old age and diseases, it is totally ungrateful and rewards them with a miserable death. And how about this: every day the rich scrape away something from the wages of the poor, not only by private chicanery but also by public laws. Before, it seemed unjust that those who deserve the most from the commonwealth should receive the least, but now, by promulgating a law, they have transmuted this perversion into justice. From my observation and experience of all the flourishing nations everywhere, what is taking place, so help me God, is nothing but a conspiracy of the rich, as it were, who look out for themselves under the pretext of serving the commonwealth. They think up and devise all ways and means, first of keeping (and having no fear of losing) what they have heaped up through underhanded deals, and then of taking advantage of the poor by buying their labor and toil as cheaply as possible. Once the rich have decreed in the name of the public (including the poor) that these schemes must be observed, then they become laws.\n\nBut after these depraved creatures, in their insatiable greed, have divided among themselves all the goods which would have sufficed for everyone, they are still very far from the happiness of the Utopian commonwealth; there, once the use of money was abolished, and together with it all greed for it, what a mass of troubles was cut away, what a crop of crimes was pulled up by the roots! Is there anyone who does not know that fraud, theft, plunder, strife, turmoil, contention, rebellion, murder, treason, poisoning, crimes which are constantly punished but never held in check, would die away if money were eliminated? And also that at the very instant when money disappeared, so would fear, anxiety, worries, toil, and sleepless nights? Indeed, poverty itself, which seems to be merely the lack of money, would itself immediately fade away if money were everywhere totally abolished.\n\nTo make this clearer, imagine some barren year of bad harvests when many thousands of people die of hunger. I maintain it is clear that at the end of this famine, if you examined the barns of the rich, you would find so much grain that if it had been divided among those swept away by starvation and disease, no one would have noticed any effect at all of the failure of weather and soil. It would have been easy to provide food if that blessed money, that invention very clearly designed to open the way to what we need to live, were not the only barrier to keep us from it. I have no doubt that the rich also understand this and are not unaware how much better it would be to lack no necessities than to abound in so many superfluities, to be relieved of so many troubles than to be hemmed in by such great wealth. And in fact I have no doubt that everyone's concern for his own well-being or the authority of our savior Christ (who is so wise that he cannot be unaware of what is best and so good that he would never advise what he knew was not the best) would long since have easily drawn the whole world to adopt the laws of this commonwealth, if it were not held back by one and only one monster, the prince and parent of all plagues, pride.\n\nPride measures prosperity not by her own advantages but by the disadvantages of others. She would not even wish to be a goddess unless there were some wretches left whom she could order about and lord it over, whose misery would make her happiness seem all the more extraordinary, whose poverty can be tormented and exacerbated by a display of her wealth. This infernal serpent, pervading the human heart, keeps men from reforming their lives, holding them back like a suckfish.\n\nSince pride is too firmly fixed in the minds of men to be easily plucked out, I am glad that this form of commonwealth, which I would gladly see adopted by everyone, is at least enjoyed by the Utopians; they have followed ethical principles which enabled them to lay the foundations of a commonwealth that is not only most happy but also, so far as human prescience can foresee, likely to last forever. For now that they have eradicated factional strife and ambition at home, along with the other vices, there is no danger that they can be disturbed by domestic discord, which has been the sole reason for the downfall of many prosperous and splendidly fortified cities. But as long as their domestic tranquility and wholesome social structure is preserved, the envy of all the surrounding princes cannot shock or unsettle their dominion, though in the past they have often unsuccessfully tried to do so.\n\nWhen Raphael had ended his tale, there occurred to me quite a few institutions established by the customs and laws of that nation which seemed to me quite absurd, not only in their way of waging war, their religious beliefs and practices, and other institutions as well, but also (and above all) in the very point which is the principal foundation of their whole social structure, namely their common life and subsistence with no exchange of money. That one fact entirely undermines all nobility, magnificence, splendor, and majesty, which are (in the popular view) the true adornments and ornaments of a commonwealth. Nevertheless, I knew that his talk had worn him out, and I was not sure whether he could endure to listen to an opinion contrary to his own\u2014especially since I remembered that he had reproached some persons precisely because they thought they would not be considered wise unless they could find some way of picking apart the ideas of others\u2014and so, having praised their regimen and his own exposition, I took his hand and led him in to dinner, though first I said we would have another time to consider these matters more thoroughly and to confer more fully. I only wish this would happen someday!\n\nMeanwhile, just as I can hardly agree with all the points he made (even though he is a person of unquestionable learning and wide experience of human affairs), so too I readily confess that in the Utopian commonwealth are very many features which in our societies I would wish rather than expect to see.\n\nTHE END OF THE SECOND BOOK\n\nThe End of the Afternoon Discourse  \nof Raphael Hythloday  \nabout the Laws and Institutions  \nof the Little-known Island of Utopia  \nRecorded by the Most Illustrious  \nand Learned Gentleman  \nMaster Thomas More  \nCitizen and Undersheriff of London\n\n## Thomas More to His Friend Peter Giles,  \nWarmest Greetings\n\nMy dear Peter, I was thoroughly delighted with the judgment you know about, delivered by that very sharp fellow in the form of a dilemma directed against my _Utopia:_ if the story is being presented as true, I find some things in it rather absurd; if it is a fiction, then I think that More's usual good judgment is lacking on some points. I am very grateful to this man, my dear Peter, whoever he may be, who I suspect is learned and whom I see as a friendly critic. I do not know whether any other critique since the book came out has pleased me as much as this one. For, first of all, motivated either by his regard for me or for the work itself, it seems that he did not begrudge the effort of reading it all the way through, and that not cursorily and hastily the way priests read the divine office (if they do so at all) but deliberately and carefully so as to weigh the details thoughtfully. And then, after criticizing some points, and not very many at that, he declares that he approves of the rest, not thoughtlessly but judiciously. Finally, even in the language with which he castigates me he praises me more highly than those who deliberately set out to praise me. For he gives a clear indication what a splendid opinion he has of me when he complains that he is disappointed when he reads a passage that is not as precise as it should be, whereas I myself exceed my own hopes if I happen to be able to publish something in the whole lot that is at least not absolutely absurd.\n\nBut in fact, to deal with him no less frankly in turn, I do not see why he should consider himself so eagle-eyed and, as the Greeks say, sharp-sighted, if he discovers some things rather absurd in the institutions of the Utopians or finds that in setting up a commonwealth I have not thought through some matters in a sufficiently practical way, as if there were no absurdities elsewhere in the world, or as if any of all the philosophers everywhere had so devised a commonwealth, a ruler, or a household so perfectly as to propound nothing that could not be improved. On that point, if it were not that I consider as sacred the memory of the most extraordinary men who have been hallowed from ancient times, I could certainly point out features from each of them which everyone would undoubtedly agree in condemning.\n\nBut when he is in doubt whether the work is true or fictitious, on this point I think his own usual good judgment is lacking. Nevertheless, I do not deny that if I had decided to write about the commonwealth and a story such as this had occurred to me, I would not have shrunk from a fictional presentation which would make the truth slip more pleasantly into the mind like medicine smeared with honey. But certainly I would have managed it so that, even though I might have wanted to deceive the ignorant mob, I would at least have inserted some pointed hints which would have let the more learned discover what I was about. Thus even if I had done nothing more than assign to the ruler, river, city, and island such names as would have informed learned readers that the island is nowhere, the city is a phantom, the river has no water, the ruler no people\u2014which would not have been hard to do and would have been much more elegant than what I actually did, for if I had not been forced by historical accuracy, I am not so stupid as to use those barbarous and meaningless names Utopia, Anyder, Amaurot, and Ademus.\n\nBut my dear Giles, since I see that some people are so cautious, wary, and sagacious that they can hardly be induced to believe what we simple and credulous souls wrote down at Hythloday's dictation, lest such persons should mistrust not only the accuracy of the story but also my own credibility, I am glad that I can say for my brainchild what Mysis in Terence says to keep Glycerius' boy from being considered a changeling: \"By heaven, I thank goodness that there were some freeborn matrons present at the birth.\" For luckily for me it so happens that Raphael told his tale not only to you and me but also to many very respectable and upstanding men. I do not know whether he related more numerous or notable details but I am sure he told them no fewer and no less remarkable matters than he did to us.\n\nBut if these incredulous persons will not take even their word for it, they can visit Hythloday himself, for he has not yet died. I just heard from some persons who recently returned from Portugal that on the first day of last March he was healthy and vigorous as ever. Therefore let them ask him for the truth or question him to ferret it out, as long as they understand that I am responsible only for my own work, not for the trustworthiness of others. Farewell, dearest Peter, to you and your charming wife and pretty little daughter, to whom my wife wishes long life and good health.\n\n## AFTER WORD\n\nJERRY HARP\n\nPoet, translator, lawyer, statesman, social philosopher, martyr, and (as of 1935) canonized saint, Thomas More remains\u2014in his friend Erasmus's phrase\u2014a \"man for all seasons,\" one who in his integrity is suited to all occasions. He was formed to no small degree by the cultural movement known as Renaissance humanism, with its emphases on the study of ancient texts, the deepening of a historical sense, the cultivation of the art of rhetoric, and devotion to active service in the world. The terms \"Renaissance\" and \"humanism\" come trailing clouds of ambiguity, so some sorting of their meaning is in order.\n\nThe idea that the Renaissance\u2014roughly 1400 to 1650, give or take (depending on where one stood in the world)\u2014was a time of great cultural renewal immediately following the \"Dark Ages\" owes a lot to the work of Jacob Burkhardt. Although his writing has been immensely influential, many generations of scholars have challenged certain of his ideas. More recent work has stressed, for example, the continuities that carry from the ancient, through the medieval, and into the early modern world. One sign of the continuity is the occurrence of various smaller-scale renaissances leading up to the major period known as _the_ Renaissance. There was the Carolingian Renaissance of the late eighth and ninth centuries, which brought into greater prominence study of the Bible, the church fathers, and the Latin classics, along with a reform of handwriting that made the copying of manuscripts more efficient. Later came the Ottonian Renaissance (tenth century), with its emphasis on historical writing, revitalization of monastic and cathedral schools, and increased circulation of classical learning. Perhaps most widely known is the Renaissance of the \"long twelfth century\" (roughly 1050 to 1250), which saw a surge of cultural energies in a variety of spheres: further revival of the classics of ancient Latinity, the rise of scholasticism, and the emergence of theology as an academic discipline, as well as developments in art, architecture, vernacular literature, and music. These \"Dark Ages\" were not so dark as some reports might lead us to believe.\n\nThat there was greater continuity between the fall of Rome and the beginning of the Renaissance is one reason for the use, in the past few decades, of the term \"early modern\" in preference to \"Renaissance.\" To put the matter simply, \"early modern\" stresses the period's relationship to what follows (modernity and even postmodernity), whereas \"Renaissance\" emphasizes the period's relationship to the past, and also implies that culture somehow died out in the intervening period (returning us again to the idea of the Dark Ages). While the term \"early modern\" avoids the problem of connoting a rebirth of something that never really died out, it also introduces its own distinct difficulties into the discussion. For example, it tends to gloss over important differences between early and late modernity, implying a smoother trajectory of cultural development than is fitting. Nevertheless, it can be salutary to choose a new set of problems to negotiate; the new questions can call forth insights and work that otherwise might not occur. Besides, in placing greater emphasis on the Renaissance as harbinger of the new, the term \"early modern\" actually extends the work of Burkhardt, who ends his great study by proclaiming the Italian Renaissance the \"leader of modern ages.\"\n\nThe evidence of continuity does not, however, negate that something distinctive and new was happening in the period commonly referred to as the Renaissance, merely that the Burkhardtian view of disjunction overstates the case. Taking shape in the run-up to the Renaissance is what might be termed the requisite intellectual infrastructure in the form, for example, of the many manuscripts that medieval monks had been busy copying for centuries, and then in the form of printed books in the middle of the fifteenth century. As Jack Goody has pointed out, the Renaissance of early modern Europe is one instance of an identifiable pattern in which a critical mass of material culture enables a recovery and circulation of much older texts, which in some cases then lead to an outpouring of further work and experimentation. Even if many of the ancient texts had continued to be known, at least by a learned cohort, their further circulation was required to open the floodgates of Renaissance work.\n\nA distinctive movement within the early modern era was what scholars in the nineteenth century termed humanism. Douglas Bush said of this movement that it is a \"medieval fusion of classical wisdom with Christian faith, and the only real change in later times was that the classical element, philosophically and aesthetically, became a less inferior partner.\" With regard to the aesthetic inheritance, the humanists' work brought the importance of style in human discourse into greater prominence, not as mere ornamentation but as part and parcel of signification. In other words, they emphasized rhetoric over dialectic (logic); it's not that they were against logic, but rather that they were deeply aware that far more than logic is needed to make discourse meaningful. When Aristotle defined rhetoric as the \"faculty of observing in any given case the available means of persuasion,\" he was writing about oratory, but rhetoric also has to do with the art of structuring discourse. In other words, rhetoric is about the ways that human experience, insight, and wisdom are encoded in language.\n\nThis concern with the style and structuring of discourse was at the heart of the humanists' educational and cultural reform of the medieval dispensation they inherited, and _Utopia_ exemplifies these ambitions with great force. In the Dialogue of Counsel, when Hythloday maintains that princes would be impervious to his advice, the character More takes him to task for his idea that the language of truth is singular; Hythloday's preference is for the language of \"academic philosophy\" (p. 43), the scholasticism that the humanists criticized for its obsessive concern with hyper-subtle logic. As in his letter to Martin Dorp, More derided a trifling, pointless, and at times captious concern with logical quibbles. Generally, the humanists wanted more Cicero and less of the Aristotle of the logical works in their educational program.\n\nWith their focus on rhetoric and style, the humanists developed the discipline of textual criticism, seeking as they did to establish reliable texts and accurate translations. A signal example of this work with texts, one that connects with the humanists' historical-mindedness, is Lorenzo Valla's demonstration that the Donation of Constantine was a forgery. Relying largely on historical details, many of them philological, Valla showed that the document could not have come from the Emperor Constantine's hand.\n\nIn relation to their other labors, the humanists also undertook a variety of literary experiments, such as the classic and quirky texts by Erasmus ( _Praise of Folly_ ), More ( _Utopia_ ), and Fran\u00e7ois Rabelais ( _Gargantua and Pantagruel_ ). It was as if all of the scholarly work and public service issued in outbreaks of sheer creativity and serious play. For all of the lasting value of the humanists' scholarship, it is these texts at play that most of us now read most assiduously. It is not without reason that, with regard to More's text, one of the great theologians of our day discerns a strong link between utopia and festivity.\n\nThe humanist emphasis on rhetoric and style fostered skills fitting for administrative workers in a world of increasingly centralized states and expanding bureaucratic church structures. These skills were fostered by such precursors of the humanists as the lawyers and notaries who mediated commercial activity in thirteenth-century Italy, workers who embellished documents and letters with allusions to the Latin classics. Further, their training in Roman law led them to study the literature of ancient Rome as well. As this world of centralized bureaucracy grew, so did the need for well-trained and fluent language workers. Whatever its other merits, humanism trained people to work in this increasingly bureaucratized world, the one for which Thomas More was shaped.\n\nHis early formation took place in the grammar school (where young boys were drilled in the niceties of Latin grammar) at St. Anthony's in London. Having finished his course of study there, he embarked on an apprenticeship at the home of Archbishop (later Cardinal and then Lord Chancellor) Morton, who shows up as a character in _Utopia._ As a page in Morton's household at Lambeth, More would have not only learned how to play a formal role on public occasions, but also extended his rhetorical skills by engaging in debates and taking on various personae to perform fluent and convincing discourses. Having entered his fourteenth year, he left Lambeth to enter Canterbury College, Oxford, where he studied for two years before leaving to take up legal studies at New Inn, London. We do not know why he left Oxford without taking a degree, though it is worth noting that it was not uncommon to study at a university for a year or two before moving on to legal training. After two years at New Inn, he moved on to Lincoln's Inn to continue his training in the law. It was during his time at Lincoln's Inn that he was also part of a circle of scholars\u2014such as William Grocyn, John Colet, and Thomas Linacre\u2014interested in the new humanist learning. These figures came from a variety of walks of life including medicine, law, theater, education, and publishing. But the most influential person More met at this time was Erasmus, who first visited England in 1499. Because neither was proficient in the other's native tongue, they spoke Latin, the lingua franca of European intellectual life, a language in which they could exchange ideas, interests, and witticisms. Early in their friendship, they undertook a friendly contest of translating work of the ancient Greek satirist Lucian from Greek into Latin. Erasmus saw _Utopia_ through the press of Thierry Martens at Louvain. It appeared near the end of 1516.\n\n_Utopia_ keeps appearing as if out of nowhere, showing up in the here and now to take us elsewhere. According to Thomas More's Greek pun, \"Utopia\" is the good place ( _eu-topos_ ) that is no place ( _ou-topos_ ). Early on he referred to his book by the Latin _Nusquama_ (Nowhere), but it was his Greek coinage that entered the language, and it shows up everywhere. Although the term has come to mean an imaginary and ideal place, an impractical social scheme, More's text works in more complex ways than popular usage allows. _Utopia_ is a nowhere that opens into new discursive spaces. Were the realm of the present and pragmatic concern to dominate entirely, we would be led into stagnation. The nowhere of _Utopia_ \u2014the work as well as the genre and mode of thinking\u2014provides one way to keep consciousness on the move even though it is an impossible place (even the mathematical dimensions of the island cannot work out).\n\nMore's great text indeed uncannily recurs. Many editions have been noted. Back in the early 1980s, I spent the better part of a month tracking them down\u2014this for one of those marvelous, old-fashioned graduate school exercises that I hope students still undertake, at least on occasion. I stopped counting somewhere upward of 260. Had neither limitation of time nor lack of initiative intervened, I suspect I could have found many more, and the ensuing decades have produced editions no doubt by the score, such as the one you are reading now. They tend to proliferate during times of conspicuous social stress, such as the world wars. Does this pattern of publication disclose a desire to escape to nowhere during tumultuous times? Or perhaps to reflect on what a good place might be? Does it show an interest, felt if not explicitly contemplated, in the complexities of how to speak of social change?\n\nWe do well to read the text in more complex terms than as a blueprint to an ideal state. Its longer title\u2014 _On the Best Form of a Commonwealth and on the New Island of Utopia_ \u2014cues us into as much. The conjunction separates as well as joins the two parts. The \"best form of a commonwealth,\" along with what it could mean to talk about and work toward it (dominant in Book 1), is not the same as the \"new island of Utopia\" (which dominates Book 2). After all, as Clarence Miller states in the introduction to this edition, few would want to live in such a regimented world as Hythloday describes. He even seems to forget a principle of justice that he urges. In his dialogue with Cardinal Morton, he argues convincingly that execution is an unjust and finally ineffectual punishment for thievery, but then in the description of his beloved isle, we learn that the Utopians practice capital punishment for a repeated offense of adultery (p. 99). Both offenses violate a biblical commandment (one the sixth, the other the seventh), so we might take this slip\u2014criticism of one death sentence and implied praise of the other\u2014as something of a wink from the author.\n\nSomething other is at stake here than a picture of a perfect world, which must exist elsewhere for More anyway. For this student of St. Augustine, we work to better this world as best we can even when we know we'll fail. In _The City of God_ (Bk. 15, chs. 1\u20136), Augustine locates the beginnings of the human city in the fratricide in which Cain kills Abel, and he makes the rather astute point that of the two, it is only Cain who builds a literal city. For a citizen of the City of God, life in this world is constant pilgrimage. It is thus fitting that, as several have pointed out, the life of Utopia is that of a monastery writ large. Although the life of a monk traditionally involves staying home, the monastic way of renunciation reminds us that our lives are pilgrimage in a much deeper sense than locomotion can account for; the monk is one who uses the monastic rule to \"follow the path to God.\"\n\nMore wrote about the complications of forming an ideal commonwealth in this world, in a Latin poem titled _Quis Optimus Reipubae Status_ (\"What Is the Best Form of Government\"), in which the speaker raises the question of which rules better, a senate or a king. He wrote the poem around the time he was composing _Utopia,_ with which it appeared in 1518. Here the poem is rendered into English heroic couplets:\n\nWhich one excels, a senate or a king?  \nLikely the senate\u2014it's a common thing:  \nThe best in greater numbers of the good.  \nBut how to find the numbers that you would?  \nIt's easier to hold one bad in view.  \nA senate often lives between the two,  \nBut kings don't walk on any middle ground.  \nFlawed senates may get counsel from the sound,  \nBut kings subdue the wisest of the earth.  \nOne is elected by the people. By birth  \nThe other reigns while holding in derision  \nThe whole of his subjected population.  \nWhile greedy kings will chew their people up,  \nAn evil senate still leaves room for hope.  \nThe old tale says, Endure the sated pest;  \nA hungry one invades, worse than the last.  \nBut greedy kings are never satisfied;  \nA leech hangs on until the body's dried.  \nDissent will throw a senate into shambles.  \nNot so the king, but that is where one gambles.  \nWhen disagreement rules in weighty things . . .  \nBut hey, what started all this anyway?  \nAre there people whom you hold in sway  \nTo shape their mode of rule? If so, you're king.  \nDon't worry whether there would be abuse.  \nAsk rather, Would it be of any use?\n\nSure enough, Utopia is governed by a senate, but it was founded by a king (Utopus) who held sway to establish the system of rule and then legislated his position out of existence\u2014an unlikely scenario, as things go in this world, and one that raises the question of whether democracy can be imposed on a people. It may be said of Utopus that, in taking on the role of king to create a system of democratic rule, the \"latter end of his commonwealth forgets the beginning,\" as Antonio says of Gonzalo's ideal state. The voice at the end of More's poem makes a similar point with the question \"Would it be of any use?\"\n\nGiven the ways of worldly power, could I escape\u2014if I were king\u2014the corrupting influence of my kingly position? How might one answer this question from More's point of view, given life in a fallen world of fallen people whose institutions are compromised by the effects of social sin, where social justice falls by the wayside, and where the dangerous folly of the deluded rich and powerful holds sway more than the wise folly of the City of God (foolish in the eyes of this world) and what passes for wisdom and common sense is far too often the \"common nonsense\" that arises from the distorting influence of our desires and fears, which are misshapen by life in a fallen world? Besides the corruptions of power, our struggles merely to survive compromise our desires to live well; as Lewis Mumford wrote, Utopia addresses the complication that \"our attempts to live the good life are constantly perverted by our efforts to gain a living.\" From Hythloday's point of view, the only solution is to establish Utopian institutions. But how do we get those institutions up and running? As Miller points out, they can be introduced only if they already exist. You can't get there from here.\n\n_Utopia_ ushers its readers into a style of thought that confronts such complications as these. It is therefore advisable to attend to the style of this masterpiece of Neo-Latin prose. As Elizabeth McCutcheon points out, litotes\u2014an affirmation formed by the negation of its opposite\u2014plays no small part in More's text; she counted more than 140 instances in the Latin of the Yale edition. Because litotes does not do quite the same work as straightforward affirmation, it fosters a mindset that allows nuances and ambiguities otherwise easily glossed over, and thus encourages a habit of understanding in more complex terms than simple affirmation or negation accomplishes. The figure of litotes is one of many that Stephen Greenblatt points to as forming the incongruous relationship between the two realms of the text\u2014the world where the dialogue takes place and the island that Hythloday describes. Clearly, they are related, but their relationship is unstable and distorted. One can even map out certain significant shifts in the style of Hythloday's sentences. When he speaks of the injustices of his contemporary Europe, his sentences are of moderate length, similar to More's Latin prose in his other works. However, when Hythloday contrasts the real with his ideal, or describes the Utopians' simple way of life, he employs styles that represent \"extremes that cannot be found anywhere else in More's Latin prose.\" When contrasting the real with his ideal, he employs \"marathon sentences\" that go \"beyond what ordinary Latin syntax can bear.\" When describing his ideal, his sentences are simple and brief. Much of the meaning of the text depends on experience of its style, as of its shifts of style. Happily, Miller's translation preserves these stylistic features, and more.\n\nRecognition of the need for a variety of styles and discourses was central to the humanist program. In _Utopia,_ the character More emphasizes that there are many languages of truth. The language that a given speaker chooses should respond to the situation at hand and the persons addressed. Here he makes use of the metaphor of the world as a stage, arguably the most common of Renaissance commonplaces. As he says to Hythloday, \"But there is another sort of philosophy better suited to public affairs. It knows its role and adapts to it, keeping to its part in the play at hand with harmony and decorum\" (p. 43). The Latin of this passage even refers literally to the boards of the theater, the stage ( _scaena_ ). We may be reminded of the anecdote, related by More's son-in-law, about the future Lord Chancellor: during his apprenticeship in the home of Archbishop Morton, where \"though [More] was young of years, yet would he at Christmas-tide suddenly sometimes step in among the players, and never studying for the matter, make a part of his own there presently among them, which made the lookers-on more sport than all the players beside.\" The humanist spirit recognized that the stage of the world demands improvisation with many language styles.\n\nIt may seem curious that More speaks of this theatrical metaphor as a philosophy. But there is a way of understanding ourselves and our world implicit in the metaphor and its relationship to the arts of rhetoric, one that includes a variety of roles and language worlds by means of which humans negotiate experience. One term for this way of understanding is what has been styled relationism. Because everything is related to everything else, there is no singular statement to make about anything. Further, because every statement (such as the one I am making now) conceals as much as it reveals, any statement, no matter how true, must be supplemented by other statements. There is no final word, as there is no final interpretation. What makes a given utterance relevant has everything to do with the situation within which it is made, who is speaking, and who is addressed. None of this is to say that humans cannot know something true\u2014were this the case, we could not know it\u2014but rather that there is no absolute human perspective or singular language of truth, as Hythloday insists that there is.\n\nGiven this relationist style of thinking, we may discern in the humanists' work early stirrings toward what Bernard Lonergan identified in his aptly titled \"The Transition from a Classicist World-View to Historical-Mindedness.\" This transition moves from a monolithic understanding of what it means to be human to the insight that there is a great variety of valid ways to be human, as there is a great variety of valid languages of knowing. The _Utopia_ suggests that we are permitted to reflect on how we might change our institutions, revise our social structures, conduct our cultural and civic lives differently. For all of its monolithic structure and faceless anonymity, even Utopia is permeable to outside influence\u2014Utopians eagerly learn Greek, printing, and papermaking (pp. 92\u201395). Reading _Utopia_ means entering into a dialogue, with oneself and others, that continues to this day.\n\nAnother influence on the emerging historical-mindedness of More's era was the ongoing exploration of the globe, as learning about the varieties of culture beyond Europe enabled a greater appreciation of the forms that human societies, and everyday human life, can take. Hythloday himself accompanied Amerigo Vespucci on three of his voyages. Accounts of these journeys provide descriptions of peoples with striking similarities to the Utopians. Thus, we learn of a society in which, as in Utopia, women accompany men into battle, an Epicurean sensibility reigns, and people \"hold their habitations in common.\" Perhaps most striking is the attitude toward gold and other objects that pass for wealth in Europe: \"They do not value gold, nor pearls, nor gems, nor such other things as we consider precious here in Europe.\" The description calls immediately to mind the Utopians' use of gold for chamber pots and shackles, as well as the marvelous anecdote of the Anemolian ambassadors (pp. 76\u201378).\n\nOf all the questions that surround _Utopia,_ the most vexing has been that of More's attitude toward common ownership of property. Character More objects to the Utopians' community of goods. Even so, it is difficult to ignore the forceful language that he assigns to Hythloday in defense of common property:\n\n. . . it seems to me that wherever there is private property, where everything is measured in terms of money, it is hardly ever possible for the common good to be served with justice and prosperity, unless you think justice is served when all the best things go to the worst people or that happiness is possible when everything is shared among very few, who themselves are not entirely happy, while the rest are plunged into misery. (p. 46)\n\nBut character More responds that private ownership is necessary as a goad, for the promise of profit motivates people to work, and too many are lazy louts without the promise of one day owning their own demesne, no matter how small. After the account of Utopian life, he intervenes again, this time with the contention that community of goods \"entirely undermines all nobility, magnificence, splendor, and majesty, which are (in the popular view) the true adornments and ornaments of a commonwealth\" (p. 134). The appeal to the \"popular view\" might raise an eyebrow or two, for More remained throughout his life critical of general opinion as a source of wisdom, as did his friend Erasmus, whose _Praise of Folly_ tells the tale. We do well to bear in mind that it was More's critical stance with regard to the popular position that got him imprisoned and killed in the end.\n\nA further complication in considering the Utopians' community of goods is its prominence in the traditions that More revered. Thus, in Plato's _Republic_ \u2014which casts a long shadow over More's text\u2014common ownership is a way of life for the Guardian class, part of the training in virtue for those who are to lead and protect. Then all we need do is shift from Athens to Jerusalem to read the second chapter of the great chronicle of the early Christian communities, Acts of the Apostles, to discover that the believers \"held everything in common, and they sold their belongings and possessions and divided them to all according as anyone had need\" (2:44\u201345). This is similar to what Hythloday has done\u2014divided up his possessions among relatives and friends\u2014thus freeing himself for a philosophic life (p. 15). More would have known some echo of the early believers' community of possessions during the four years he lived as a guest of the Carthusians at their Charterhouse in London. This sharing of goods was a well-established part of monastic tradition; the \"vice of private ownership must be uprooted from the monastery,\" St. Benedict wrote. Closer to the composition of _Utopia,_ Erasmus included in his Adagia \" _Amicorum communia omnia_ \" (\"Among friends all is held in common\"). In the 1515 edition, Erasmus gave this adage pride of place at the opening of the collection, stating, \"Since there is nothing more wholesome or more generally accepted than this proverb, it seemed good to place it as a favourable omen at the head of this collection of adages.\" Kathy Eden takes this three-word Latin adage as her point of departure in a marvelous book-length study of Erasmus and the humanists.\n\nOne of the more nuanced treatments of common ownership occurs in St. Thomas Aquinas's _Summa Theologiae._ Although Aquinas is sometimes associated with the scholastic philosophy against which the humanists inveighed, they generally regarded him not as a scholastic philosopher, but rather as a theologian and Doctor (that is, teacher) of the church. On the one hand, Aquinas set out three reasons why private ownership is helpful to human life: (1) it provides motivation to work; (2) it allows for orderliness in human affairs; and (3) it enables peaceful social relations (ST 2\u20132ae, Q. 66, A. 2). All three of these accord with the views that character More espouses. On the other hand, in describing humans' fundamental relationship to the world, Aquinas articulates a vision of communal possession:\n\nCommunity of goods is ascribed to the natural law, not that the natural law dictates that all things should be possessed in common and that nothing should be possessed as one's own: but because the division of possessions is not according to the natural law, but rather arose from human agreement which belongs to positive law, as stated above. (Q. 57, AA. 2\u20133)\n\nHence, private ownership is not incompatible with natural law, which also does not require it; rather, private ownership is an \"addition thereto devised by human reason\" (2\u20132ae, Q. 66, A. 2, Reply to Objection 1). The natural law recognizes that humans hold all the world in common. Private ownership is merely a provisional, pragmatic, and contingent means whereby humans make creative use of what is fundamentally communal. One valid reading of the _Utopia_ sees the proposition concerning community of property as a way of loosening the metaphysical grip on ownership, reminding readers that we own things in this world merely by convention, not by nature.\n\nIf Aquinas had lived in the twentieth century\u2014a speculation I may be allowed given the fictive space of utopian writing\u2014he might have joined with Paul Ricoeur in considering utopian thought alongside ideology. A working definition of ideology, cast in Thomistic terms, is the taking of the provisional and pragmatic for the metaphysical. Thus, stating a right to private ownership as a metaphysical given is an example of the false consciousness wherein Ricoeur finds ideology to function. In his reading, the best function of utopian thinking is as an antidote to ideology, for such thinking provides an opportunity to play one's identity out and away from the prison house of the here and now. As he put it, \"This function of utopia is finally the function of the nowhere. To be here, _Da-sein,_ I must also be able to be nowhere.\" Utopian thought relates to identity because part of identity is prospective, who and what we desire and strive to be\u2014\"What we call ourselves is also what we expect and yet what we are not.\" But ideology and utopia will not remain separate; they tend to interweave, and one issue worth further reflection is how the two function together as well as tend to tear apart, in _Utopia_ and elsewhere.\n\nUp for further consideration too is the place of this text in the trajectories of human consciousness in the tumultuous times of the early modern era. As already adverted to, part of the humanist movement related to shifts in awareness emerging from exploration of the planet. But other forces figured into these alterations also. One of these was the introduction of print (an innovation that the Utopians took to, as Hythloday witnesses). Working just after the incunabulum\u2014or cradle\u2014of print, the humanists were perhaps the first generation of European writers fully to embrace this technology, an important part of the era's seismic shifts in thinking and sensibility. As one scholar put it, \"As an institution the printing press represented an autonomous and cosmopolitan site for the production of knowledge free of lay and ecclesiastical control.\" Print, in other words, helped to create a new kind of imaginative space, and a new feel for how intellection happens. The printed book or pamphlet allows an impression of a free-floating island of discourse, and even though no discourse is ever really broken off from the dialogues taking place in the human lifeworld, print technology was related (though not reducible) to new ways, including more cosmopolitan and mobile ways, of imagining and conducting intellectual life. Like Cardinal Morton breaking into the speech of the ponderous lawyer (p. 26), the discourse emergent with print disrupted older styles. These changes also created tensions. Hanan Yoran has gone perhaps as far as anyone in showing the discursive tensions of _Utopia._ As he points out, nearly all symbolic action has been eliminated from this island, where even law\u2014that most contentiously symbol-laden of realms\u2014is fantasized into a commonsense reign of the obvious (p. 102). For a community of scholars to whom all of human life is interwoven with symbolic action, such a space as this is a no-place indeed. The ways in which print technology figured into these tensions, along with the creative possibilities it allowed, might be further scanned.\n\nAnother area for further study is the family resemblance between _Utopia,_ along with humanist discourse generally, and certain strains of postmodern thought. With their emphases on the performativity of human identity, the slipperiness of language, and the provisionality of all human discourses\u2014 along with their commitment to literary experimentation and serious engagement in play\u2014the humanists may be taken in some ways as precursors to Derrida and company. While the early humanists were far from deconstructionists avant la lettre, they could nevertheless have appreciated Derrida's assertion that when it comes to philosophical statement, political discourse, or ethical judgment, negotiation is always necessary, and there is always something about negotiation that \"gets one's hands dirty\" even when one is negotiating \"in the name of purity.\" More was in Flanders busy with negotiations when the opportunity to write _Utopia_ fell into his soiled hands.\n\nAfter Hythloday's description of Utopia, character More says to the reader, but not to Hythloday, that he harbors some objections to the Utopian way of life. He would prefer at this point, however, to avoid contention, so instead of arguing, he extends a hand to Hythloday: \"I took his hand and led him in to dinner, though first I said we would have another time to consider these matters more thoroughly and to confer more fully. I only wish this would happen someday!\" (p. 134). The narrative comes to its close with this gesture of friendship and a prospective note, the desire to talk more fully at some unspecified time in the future. Even with his objections, his final sentence\u2014prospective as well\u2014ends the text with wistful agreement, as he states, \"I readily confess that in the Utopian commonwealth are very many features which in our societies I would wish rather than expect to see\" (p. 135). Let us confer, then, so that we may discover what event may yet arrive out of that nowhere that is the future.\n\n## NOTES\n\n### _Introduction_\n\n. J. H. Hexter argues in _More's Utopia: The Biography of an Idea_ (Princeton: Princeton University Press, 1952) and in Edward Surtz, S.J., and J. H. Hexter, eds., _Utopia,_ vol. 4 of _The Complete Works of St. Thomas More_ (New Haven: Yale University Press, 1965), pp. xv\u2013xxiii, that because there are biographical details about the speaker of Book 2 that fit the Hythloday of Book 1, the opening of Book 1 up to the point where Giles asks Hythloday why he does not serve a prince must have also been written in Flanders. He also argues that the passionate peroration of Book 2, in which Hythloday excoriates Europe and defends communism, was written in London. However probable or plausible Hexter's arguments are, More could have written all of Book 1 in London, adding biographical details in Book 2 and possibly its peroration after he finished Book 1.\n\n. The term \"dystopia\" had to be invented to accommodate the negative features of More's fantasy.\n\n. _Utopia,_ with an introduction and notes by Edward Surtz, S.J. (New Haven: Yale University Press, 1964), p. xxvi.\n\n. In my translation I have tried to reproduce stylistic elements of _Utopia_ as well as its meaning. I have borrowed and modified some passages from my article \"Style and Meaning in _Utopia:_ Hythloday's Sentences and Diction.\"\n\n. \" 'Si Hythlodaeo Credimus': Vision and Revision in Thomas More's _Utopia,_ \" _Soundings_ (formerly _The Christian Scholar_ ) 51 (1968): 271\u201389; reprinted in _Essential Articles for the Study of Thomas More,_ ed. Richard S. Sylvester and Germain Marc'hadour (Hamden, Conn.: Archon Books, 1977), pp. 290\u2013301 (p. 298).\n\n. How the other capital sins survive in the absence of pride and just which of the other capital sins cause the crimes punished by the Utopians are also difficult questions.\n\n. Valerian Paget's _More's Millennium_ (1909) was simply a modernization of Robinson's translation and has no independent value.\n\n. The translation in the Yale edition (Surtz and Hexter, eds., 1965) was a reworking of Richards' translation by Edward Surtz, but the result was usually not an improvement.\n\n### _Utopia_\n\n. More had been undersheriff since September 3, 1510. As such he presided over the court at one of the sheriff's jails, where he heard various (mostly minor) cases.\n\n. The poem, \"by the Poet Laureate Anemolius,\" was probably written by More himself. On the name Anemolius, see note 186, below.\n\n. In the first edition (Louvain, 1516) and in the running heads of the two Basel editions of 1518, this letter is described as a \"preface.\"\n\n. Peter Giles (c. 1486\u20131533) was a humanist friend of More and Erasmus. He was a corrector at the press of Dirk Martens in his native city of Antwerp and was a clerk of that city from 1512 on.\n\n. The name \"Utopia\" derives from Greek _ou_ (\"not\") and _topos_ (\"place\"), meaning \"no place\" (More also called it by the equivalent Latin name \"nusquama\"). \"Utopia\" includes a pun because the initial \"u\" may also be derived from Greek _eu_ (\"good\"). Hence Utopia is a good place which is no place.\n\n. More visited Giles in Antwerp in September 1515; together with a letter dated 3 September 1516, he sent the manuscript of _Utopia_ to Erasmus for publication.\n\n. The angel Raphael is a saving guide and healer in the biblical book of Tobias. Raphael's surname, Hythloday, is derived from Greek words meaning \"peddler of nonsense.\"\n\n. More refers to the principal divisions of rhetoric according to the classical tradition: invention (finding matter), disposition (arranging it), and eloquence (stylistic elaboration).\n\n. This description applies to most of Hythloday's description of Utopia itself, but hardly to the elaborate and often passionate eloquence of Hythloday's language in much of Book 1 and in his peroration at the end of Book 2. Almost nothing in this letter (or in _Utopia_ itself, for that matter) can be taken at face value.\n\n. More describes himself (accurately) as devoted to the active life which Hythloday rejects.\n\n. John Clement (c. 1500\u20131572), one of the first students at Colet's humanist school, St. Paul's, became a page and pupil in More's household about 1514; later he became a distinguished physician.\n\n. \"Anyder\" is coined from the Greek for \"waterless\"; \"Amaurot\" from the Greek for \"made dark or dim.\"\n\n. Sidenote: _Note the theological distinction between lying and speaking a falsehood_. Though it is apparently not found among the theologians, the distinction between _mentiri_ (tell a lie) and _mendacium dicere_ (speak a falsehood, with no intention of deceiving) derives from Aulus Gellius (11.11.1\u20134) and was well known. Erasmus, and perhaps Peter Giles, probably added the sidenotes that appeared in the margins of the original edition.\n\n. Sidenote: _A holy ambition!_\n\n. Sidenote: _Human judgments are ungrateful_.\n\n. Sidenote: _Persons with no \"nose\"_ [appreciation of wit] _he calls \"flat-nosed.\"_\n\n. Sidenote: _A saying_. See Erasmus, _Adages_ 293 in _The Collected Works of Erasmus_ (Toronto: University of Toronto Press, 1974\u2013), vol. 31, p. 311. _The Collected Works of Erasmus_ is hereinafter referred to as _CWE,_ followed by the volume number in italic and the page number in roman type: _CWE 31,_ 311.\n\n. The metaphor seems to be drawn from wrestling.\n\n. Sidenote: _A remarkable comparison_.\n\n. Erasmus, _Adages_ 28 ( _CWE 31,_ 76\u201377).\n\n. In August 1513 Henry's army had been victorious at the Battle of the Spurs and briefly occupied Th\u00e9rouanne and Tournai; but his French campaigns, then and later, were as futile and destructive as those of the French kings in Italy.\n\n. The difficulties were mainly connected with the wool trade between England and Flanders. They were serious enough for Wolsey to be worried early in 1515 that Charles would seize the English fleet for back taxes.\n\n. By 1515 Charles V, later Holy Roman Emperor (1519), was Duke of Burgundy and Prince of Castile.\n\n. Sidenote: _Cuthbert Tunstall_. Tunstall (1474\u20131559), bishop of London (1522) and later Durham (1530), was a close friend whom More admired throughout his lifetime. On 12 May 1516 Tunstall became Master of the Rolls and Vice-chancellor; as such he was chief of the twelve assistants to the Lord Chancellor.\n\n. Sidenote: _An adage_. See Erasmus, _Adages_ 1406\u20137 ( _CWE 33,_ 245).\n\n. Jean (or perhaps Jacques) de Halewyn, Seigneur de Maldeghem.\n\n. De Themsecke (d. ca. 1536), a doctor of the law and a member of Charles V's council at Mechlin, was employed on many diplomatic missions. (Cassel is now in northern France.)\n\n. On or before 25 July 1515.\n\n. Sidenote: _Peter Giles_. In Flemish his name is \"Gillis\" or \"Gilles,\" but the usual English translation of his Latin name (\"Aegidius\") is \"Giles.\"\n\n. Giles (1486\u20131533) was learned in the law and edited classical and humanist works. Since 1512 he had been chief clerk of the court of justice at Antwerp.\n\n. Cf. Matt. 10:16. The same combination was part of the printer's mark of Johann Froben, who printed the two 1518 editions of _Utopia_.\n\n. More left England 12 May 1515.\n\n. Palinurus, Aeneas' steersman, dozed at the helm, fell overboard, and drowned ( _Aeneid_ 5.833\u201361), unlike the alert Odysseus and observant Plato who learned much from their travels ( _Odyssey_ 1.1\u20134; Diogenes Laertius 3.6\u20137.18\u201322).\n\n. More expressed the same opinion in his _Letter to Oxford,_ in _The Complete Works of St. Thomas More,_ vol. 15, ed. Daniel Kinney (New Haven: Yale University Press, 1986), p. 143. (Works in this series are hereinafter cited as _CWM,_ followed by the volume number in italic and the page number in roman type: _CWM 15,_ 143).\n\n. In 1515 the Portuguese excelled in exploration, especially in the far east.\n\n. The voyages (1503\u20134) of the Florentine explorer Amerigo Vespucci (1451\u20131512), who was in the employ of the King of Portugal, were described in two Latin narratives (of disputed authenticity) published about 1507; one of the versions mentions the twenty-four mariners left behind in a fort at the farthest point of the voyage (Cape Frio in southeast Brazil).\n\n. Lucan, _Pharsalia_ 7.818\u201319; cf. Augustine, _City of God_ 1.12.\n\n. Sidenote: _Apophthem;_ cf. Erasmus, _Apophthegmata_ 7, Anaxagoras Clazomenus 4, and Cicero, _Tusculan Disputations_ 1.43.104.\n\n. The Portuguese had visited Calicut (a city on the west coast of India, not Calcutta) by 1487 and established a station there in 1511.\n\n. This is not an ordinary bench covered with sod. The small woodcut of the scene in the two editions of 1518 shows that it was a long wooden box filled with earth and covered on top with growing grass.\n\n. The torrid zone between the Tropic of Cancer and the Tropic of Capricorn, the northern and southern limits between which the sun's orbit was thought to move.\n\n. Scylla was a six-headed sea monster ( _Odyssey_ 12.73\u2013100, 234\u201359; _Aeneid_ 3.424\u201332); Celaeno was one of the harpies, disgusting birds with women's faces ( _Aeneid_ 3.209\u201358); the Laestrigonians were giant cannibals ( _Odyssey_ 10.17\u2013133).\n\n. It seems likely that at this point More inserted the bulk of Book 1, the dialogue about counseling kings, which was written after Book 2, when More had returned to London. (See the Introduction, p. vii, and p. 141n.1.) In this addition More does not limit himself to describing Utopian institutions but gives Raphael's narration about the Polylerites, Achorians, and Macarians.\n\n. Hythloday paraphrases a definition of liberty given by Cicero in a context similar to this one ( _De officiis_ 1.20.69\u201370).\n\n. Cf. Erasmus, _Adages_ 115, 121, 3064 ( _CWE 31,_ 158\u201360, 167\u201368).\n\n. A Cornish rebellion was crushed at the Battle of Blackheath on 22 June 1497.\n\n. More had admired Morton (1420\u20131500) since the time he was a page in his household (c. 1490\u201392). He is portrayed as skilled and shrewd in More's _Richard III (CWM 2,_ 90\u201392).\n\n. In his _Description of England_ (1587), ed. Georges Edelen (Ithaca, N.Y.: Cornell University Press, 1968), p. 87, William Harrison reported that in the reign of Henry VIII alone 72,000 thieves and vagabonds were hanged.\n\n. Since Hythloday was in England in late 1497 and early 1498 he may be referring to English skirmishes in France in the early 1490s. But as he speaks in 1515, he may also be thinking of the much heavier casualties in Henry VIII's futile French campaigns of 1512\u201313.\n\n. Plato uses the figure of the drones to describe an oligarchy ruled by rich men who exploit the poor and contribute nothing to society ( _Republic_ 8.552B\u2013C).\n\n. The parallel between soldiers and robbers is a frequent theme among humanists; see, for example, Erasmus, _Complaint of Peace (CWE 27,_ 317).\n\n. In the time of Francis I the French relied mostly on Swiss and German mercenaries.\n\n. The Latin \"morosophi\" (transliterated from Greek) means literally \"foolish wisemen\" (the reverse of the modern \"sophomore\"). See Lucian, _Alexander_ 40. Erasmus uses it in _The Praise of Folly,_ tr. Clarence H. Miller (New Haven: Yale University Press, 1979), p. 13; in _De copia, Opera omnia_ 1.12C; and in _Adagia_ (prol., _CWE 31,_ 23).\n\n. _Bellum Catalinae_ 16.3.\n\n. Foreign mercenaries often wreaked havoc in France during the Hundred Years' War (1337\u20131453).\n\n. The Greek historian Herodian (mentioned in Book 2 as one of the authors Hythloday brought to Utopia) describes how several emperors were murdered by the barbarian mercenaries of the Praetorian guard. After the first Punic War, foreign mercenaries revolted against their Carthaginian employers. From the thirteenth to the sixteenth century, the Mamelukes (originally mercenaries from Turkey and Circassia) ruled despotically a large empire consisting of Egypt, Syria, and other parts of the Middle East.\n\n. Especially Italy, which was often devastated by foreign mercenaries; Machiavelli, who firmly opposed the use of mercenaries, gives many examples of the harm they caused.\n\n. The English defeated the French decisively at Cr\u00e9cy (1346), Poitiers (1356), and Agincourt (1415).\n\n. Between the thirteenth and eighteenth centuries, much arable land was enclosed by hedges or ditches and used to pasture sheep. Hythloday's arguments against enclosure were widespread, and though it had its supporters (mostly because of the profitability of the wool trade), it undoubtedly caused much suffering to farm laborers and destroyed many villages.\n\n. Long before and after 1515 many sumptuary laws were passed against extravagant display, especially in clothing, but they were honored more in the breach than the observance.\n\n. During the reigns of Henry VII and Henry VIII laws were passed forbidding gaming and alehouses, limiting enclosure, restoring land from pasture to tillage, and restricting monopolies, but with little effect.\n\n. Sidenote: _This shows the Cardinal's usual way of interrupting anyone who talks too much_.\n\n. A proverbial saying (Erasmus, _Adages_ 924), derived primarily from Cicero, _De officiis_ 1.10.33.\n\n. Sidenote: _Manlian edicts from Livy_. Proverbial for \"harshly unjust\" (Erasmus, _Adages_ 987). The Roman consul Manlius executed his son for winning a victory without having permission to do so (Livy 8.7.1\u201322).\n\n. Stoics such as Zeno, Seneca, and Epictetus believed that virtue consisted in ignoring exterior forces and remaining faithful to the interior dictates of reason about what is right; such faithfulness has no degrees but is either kept or not. Cicero presents and refutes the paradox in _De finibus_ 4.10.21\u201323; Horace ridicules it in _Satires_ 1.3.113\u201324.\n\n. Exod. 20:13, Deut. 5:17. (All scriptural references are to the Vulgate text and numbering.)\n\n. Exod. 22:1\u20134.\n\n. The Mosaic law does, of course, prescribe death as a punishment for various crimes. And even under the more merciful Christian dispensation, Hythloday does not always condemn capital punishment; as the remedy of last resort, it is employed by the Polylerites and the Utopians (pp. 30, 99).\n\n. A name formed from Greek _polus_ (\"much\") and _leros_ (\"nonsense\").\n\n. Sidenote: _We should note this, since we do otherwise_. Erasmus expresses the same opinion in _The Education of a Christian Prince (CWE 27,_ 270).\n\n. Sidenote: _But nowadays the servants of noblemen find such a haircut attractive_.\n\n. Erasmus, _Adages_ 1612.\n\n. If a criminal could reach a place of asylum or sanctuary (usually a church) he could not be arrested, though during the reign of Henry VII the privilege was discussed and somewhat curtailed. It is debated in More's _Richard III (CWM 2,_ 27\u201333).\n\n. Sidenote: _An entertaining exchange between a friar and a fool_.\n\n. Cf. Erasmus, _Adages_ 113 ( _CWE 31,_ 154\u201355).\n\n. Sidenote: _A proverb frequently bandied about among beggars_. There seems to be no recorded or recognized proverb here (though it might still have been a frequent saying among beggars); there may be some allusion to the priest who passed by the wounded Samaritan (Luke 10:31).\n\n. Unordained members of religious orders were called \"lay brothers.\"\n\n. Sidenote: _He alludes to the Horatian phrase \"doused with Italian vinegar.\"_ See _Satires_ 1.7.32 and Erasmus, _Adages_ 1252 ( _CWE 33,_ 164). The phrase is here translated as \"needled.\"\n\n. John 17:12, 2 Thess. 2:3.\n\n. Luke 21:19.\n\n. Ps. 4:5. Sidenote: _How well the people in the story speak in character!_\n\n. Ps. 68:10.\n\n. Sidenote: _Apparently the friar, in his ignorance, misuses \"zelus\" as if it were neuter like \"scelus.\"_ In 4 Kings 2:23\u201325 some children mocked Elisha because of his baldness; when he cursed them two bears came out of the woods and tore forty-two of them to pieces. The friar quotes a hymn attributed to Adam of St. Victor, sung within the octave of Easter. In the ordinary pronunciation of Erasmus' time _zelus_ (\"zeal\") could sound like _scelus_ (\"crime\"). The confusion produces the following result: those who mocked Elisha . . . feel the crime of the bald man.\n\n. Prov. 26:5. But the preceding verse says: \"Do not answer the fool according to his folly lest you become like him.\"\n\n. Perhaps alluding to Ps. 7:16.\n\n. _Republic_ 5.473C\u2013D, _Epistles_ 7.326A\u2013B.\n\n. During his three sojourns at Syracuse, Plato failed in his attempt to reform the tyrant Dionysius or his son (also Dionysius); see his _Epistles_ 7 and Plutarch, _Dion_. 4.1\u20135.3, 10.1\u201320.2.\n\n. Here Hythloday launches into a 464-word sentence, suspended, unrealistically intricate, interminable (as Lupton called it), which ends with \"react to this speech.\" Though translators (with the exception of Robinson) have generally broken up this sentence to make it easier, such manipulation is unjustified: the sentence is no easier in Latin than in English. Its difficulty springs from Hythloday's difficult outlook.\n\n. In 1515, the time of More's imagined interview with Hythloday, the king of France was Francis I, who continued the policy of his predecessors Charles VII and Louis XII. All three invented claims to Milan and Naples, but their military adventures in Italy foundered in confusion and intrigue.\n\n. The French won Milan in 1499, lost it in 1512, regained it in 1515. They won Naples in 1495, lost it in 1496, regained it in 1501, and lost it in 1503.\n\n. Sidenote: _Indirectly he is discouraging the French from acquiring Italy_. At the battle of Agnadello (1509), France defeated Venice and deprived it of its territory on the mainland. By 1515, when the Venetians helped Francis I in his campaign again Milan, the French king restored Verona to his Venetian ally. Hythloday wonders if the French king is ready to turn on his recent ally once more.\n\n. After the death of Charles the Rash, Duke of Burgundy (1477), Louis XI of France tried to seize all the vast Burgundian holdings, though many parts clearly did not belong to France.\n\n. The German mercenary footsoldiers were surpassed only by the Swiss; both were despised and excoriated by Erasmus and many humanists.\n\n. Emperor Maximilian of Hapsburg, grandfather of Charles V, was usually impecunious and totally unreliable. A votive offering was normally an expensive gift left in a church or shrine in thanksgiving for a favor from God or a saint.\n\n. With the help of troops sent by a duped Henry VIII, Ferdinand II, King of Aragon and regent of Castile, occupied southern Navarre in 1512 and annexed it to Castile in 1515.\n\n. Charles V, prince of Castile and the future emperor (1519) was often affianced for dynastic reasons, especially to French brides.\n\n. Francis I did make a treaty with England in April 1515.\n\n. The Scots were traditionally allies of France against England.\n\n. The French had supported several pretenders to the English throne during the reigns of Henry VII and Henry VIII: Lambert Simnel, Perkin Warbeck, Edmund de la Pole, and his brother Richard.\n\n. Erasmus, _Adages_ 860 ( _CWE 32,_ 215).\n\n. Cf. More's epigram \"On Lust for Power\": \"Among many kings there will be scarcely one, if there is really one, who is satisfied to have one kingdom. And yet among many kings there will be scarcely one, if there is really one, who rules a single kingdom well\" ( _CWM 3\/2,_ 257).\n\n. Sidenote: _A notable example_. From Greek _a-_ (\"without\") and _choros_ (\"place, country\").\n\n. More here echoes Erasmus' _Adages_ 1401 ( _CWE 33,_ 237\u201343): \"Sparta is your portion; make it flourish.\"\n\n. Hythloday presents his second imaginary council in an even longer marathon sentence (926 words); it is just as extravagant in Latin as in this English translation.\n\n. Fraudulent manipulation of the currency was practiced by Edward IV, Henry VII, and (later) Henry VIII.\n\n. In 1492 Henry VII not only levied taxes for a pretended war against France but accepted a bribe from Charles VIII of France for not fighting it.\n\n. Henry VII's ministers Empson and Dudley were notorious for such chicanery.\n\n. The royal prerogative, the special, inherited claims of the king apart from common law, was a subject of considerable dispute even in More's time, though it became more heated in the following century.\n\n. Hythloday adapts Cicero's statement in _De officiis_ 1.8.25: \"Recently Marcus Crassus said that no amount of money is enough for one who wishes to be head of state unless it produces enough income to maintain an army.\"\n\n. Among the techniques mentioned by Aristotle by which tyrants maintain their power are keeping subjects poor and humble-spirited and pretending to rule for the advantage of the citizens ( _Politics_ 5.9.4, 8, 11, 1313b, 1314a\u2013b).\n\n. The biblical and Homeric figure of kings as shepherds was widespread; in his speech at the opening of Parliament in 1529 More compared kings to shepherds. See also his Latin epigrams against tyranny ( _CWM 3\/2,_ 162\u201365, 168\u201369).\n\n. The saying derives from Manlius Curius Dentatus (Plutarch, _Moralia_ 194F) but it was also attributed to Gaius Fabricius Luscinus by classical and medieval authors.\n\n. From the Greek _makarios_ (\"happy\"); the Greek word introduces each of the beatitudes (Matt. 5:3\u201311).\n\n. More may be thinking of Henry VII, who had an enormous sum in his treasury when he died.\n\n. Sidenote: _A proverb_. See Erasmus, _Adages_ 1387 ( _CWE 31,_ 376).\n\n. The following argument centers on the moral and rhetorical notion of decorum (Cicero, _De officiis_ 1.27.93\u201339.141, _Orator_ 21.69\u201322.74, and _De oratore_ 3.55.109\u201312). It is also based on the conflict between rhetorical persuasion, which deals with probable truths, and philosophical logic, which produces demonstrable truths.\n\n. Sidenote: _The philosophy of the schools_. In the text and sidenote this philosophy is designated \"scholastica.\" The only academic philosophy in More's time was that of the universities, which we nowadays call scholasticism, so that in this case \"academic\" and \"scholastic\" are practically synonymous. The humanists generally attacked the hairsplitting excesses of scholastic philosophy and favored a more rhetorical approach to literature and life. See, for example, More's _Letter to Dorp (CWM 15,_ 29\u201339, 49\u201370).\n\n. Sidenote: _A marvelous comparison_. _Octavia_ is a tragedy once attributed to Seneca in which Seneca discusses the abuse of power with Nero. Cf. Erasmus, _Adages_ 91, \"to be subservient to your role\" ( _CWE 31,_ 131\u201332).\n\n. Sidenote [in Greek]: _A mute role_. John Clement plays such a part in _Utopia_.\n\n. Plato allows rulers (even presumably philosopher-kings) to lie to their subjects for a useful purpose ( _Republic_ 3.21.414B\u2013415D, 5.8.459C\u2013D). Quintilian says that \"everyone must allow, what even the sternest of the Stoics admit, that the good man will sometimes tell a lie\" ( _Institutes_ 12.1.38).\n\n. Matt. 10:27, Luke 12:3.\n\n. The so-called Lesbian ruler was made of lead so as to accommodate itself to measuring curved surfaces; see Erasmus, _Adages_ 493 ( _CWE 31,_ 465).\n\n. _Adelphoe_ 1.2.145\u201347.\n\n. _Republic_ 6.10.496D\u2013E.\n\n. See p. 101.\n\n. According to Diogenes Laertius (3.23), \"the Arcadians and Thebans, when they were founding Megalopolis, invited Plato to be their legislator; but . . . when he discovered that they were opposed to equality of possessions, he refused to go.\" In the the _Republic_ Plato prescribes community of property (and of wives and children) only for the guardians (5.12.464B\u2013E), but in the _Laws_ he says that in the best state it would be observed by the whole populace (5.739B\u2013D).\n\n. More summarizes Aristotle's arguments in the _Politics_ (2.1.2.1260b\u20134.13.1267b) against Plato's advocacy of communism. Aristotle's arguments had been adopted by the medieval scholastics such as Thomas Aquinas in his commentary on Aristotle's _Politics_ (2.1\u20137).\n\n. The numerical equivalents of the Greek letters in \"Abraxas\" (the usual form, rather than \"Abraxa\") add up to 365. The name was given to the highest of the 365 heavens invented by the heretic Basilides.\n\n. Sidenote: _A greater task than cutting through the Isthmus_. Several attempts to dig a canal across the Isthmus of Corinth failed so that the attempt became proverbial for failure (Erasmus, _Adages_ 3326).\n\n. Sidenote: _Common effort lightens a burden_.\n\n. According to Erasmus, in Utopia More \"represented the English commonwealth in particular\" ( _CWE 7,_ 23.281). In 1587, according to William Harrison's _Description of England_ (1587), ed. Georges Edelen (Cornell University Press, 1968, pp. 86\u201387), England had fifty-three counties, which, together with London, make it match the citystates of Utopia. The city-states are mostly independent but loosely federated, each having its own governor; they are united only by codes and customs, as well as a triennial meeting of a senate.\n\n. Sidenote: _Likeness breeds concord_.\n\n. Sidenote: _But such a desire is the curse of modern commonwealths_.\n\n. From a Greek compound meaning \"ruler of a tribe.\"\n\n. Pliny mentions artificial incubation ( _Natural History_ 10.76.154\u201355) but it seems not to have been practiced in More's time.\n\n. That is, they do not use it to make beer or ale, as the English do.\n\n. Sidenote: _The advantage of communal labor_.\n\n. From a Greek adjective meaning \"without water.\" Amaurot resembles London in its tidal river (the Thames) and smaller stream (Fleet Ditch, except that London's stream was foul and unpleasant).\n\n. Sidenote: _The same thing happens to the Thames in England_.\n\n. Sidenote: _In this feature London is also like Amaurot_. But Amaurot has the advantage of having its bridge above the city, not below it.\n\n. Sidenote: _This is reminiscent of Plato_. See _Republic_ 3.22.416D.\n\n. Sidenote: _Virgil also praised the usefulness of gardens_. See _Georgics_ 4.116\u201348.\n\n. That is, 244 B.C., when Aegis IV became king of Sparta; he was killed because of the egalitarian reforms he wished to introduce. See Richard Schoeck, \"More, Plutarch, and King Aegis: Spartan History and the Meaning of History,\" _Philological Quarterly_ 35 (1956): 366\u201375; reprinted in _Essential Articles for the Study of Thomas More,_ ed. Richard Sylvester and Germain Marc'hadour (Hamden, Conn.: Archon Books, 1977), pp. 275\u201380.\n\n. In More's time lead was commonly used to roof important buildings. William Harrison, in his _Description of England_ (1587), ed. Georges Edelen (Ithaca, N.Y.: Cornell University Press, 1968), speaks of \"fine alabaster burned, which they call plaster of Paris, whereof in some places we have great plenty and that very profitable against the rage of fire,\" but he is describing the plastering of interior walls, not roofs, which he says are covered with shingles, straw, sedge, reeds, or slate (p. 196).\n\n. Glass windows were uncommon in homes during More's time; oiled linen, sheets of horn, or lattices of wicker or wood were used instead. Hythloday means that oiled linen is brighter and more impervious than linen alone, not that it is superior to glass.\n\n. Sidenote: _In the Utopian language \"tranibor\" means \"chief director.\"_ \"Syphogrant\" seems to be derived from the Greek compound meaning \"wise old man\" (or perhaps \"old man of the sty\" = steward). \"Tranibor\" seem to come from a Greek compound meaning \"plain eater.\" But other meanings have also been suggested. In fact, Hythloday continues to use the older terms \"syphogrant\" and \"tranibor,\" not \"phylarch\" (ruler of a tribe) or \"protophylarch\" (chief phylarch).\n\n. Thus there are six thousand families in Utopia, excluding the countryside (see p. 54).\n\n. Sidenote: _A remarkable way of electing officials_.\n\n. Sidenote: _Tyranny is hateful to the well-ordered commonwealth_.\n\n. Utopia is a federation of democratic republics: the households elect the syphogrants, who elect the tranibors and governor (whom they can also remove from office). The syphogrants also select the class of scholars, from which all high officials are chosen.\n\n. Sidenote: _Disputes should be settled quickly, but nowadays they are deliberately and lengthily prolonged_.\n\n. Sidenote: _Nothing should be decided hastily_.\n\n. But for the whole island of Utopia there is no single executive branch to carry out or enforce the deliberations or decisions of this council.\n\n. Sidenote: _Would that the same thing were done in our councils_.\n\n. Sidenote: _This is the meaning of the proverb \"take counsel at night.\"_ See Erasmus, _Adages 1143 (CWE 33,_ 96).\n\n. Sidenote: _Farming is an occupation common to everyone, though here it is fobbed off on a few despised workers_.\n\n. Plato ( _Republic_ 7.797A\u2013B) and Aristotle ( _Politics_ 7.15.5.1336a). Plato specifically advises that \"to make a good farmer [a man] must play [in childhood] at tilling land\" ( _Republic_ 1.643B\u2013C).\n\n. Sidenote: _Trades should be learned to satisfy needs, not luxury_.\n\n. Sidenote: _Let everyone learn the trade for which he has a natural aptitude_.\n\n. Unlike the Utopians, Plato insists that each craftsman must have only one trade ( _Republic_ 2.11.370A\u2013C, 2.13.474B\u2013C; _Laws_ 8.846D\u2013E).\n\n. Sidenote: _The idle are to be expelled from the commonwealth_.\n\n. Statutes during the reign of Henry VII required laborers to work from daybreak to nightfall in spring and summer and from before 5 a.m. to between 7 and 8 p.m. in fall and winter.\n\n. Sidenote: _The work of laborers should be kept within bounds_.\n\n. Sidenote: _But nowadays playing at dice is the sport of princes_.\n\n. More surely knew how inaccurate Hythloday is here, since women in his time had duties at least as heavy as they have now.\n\n. That is, members of the religious orders.\n\n. Sidenote: _A very perceptive observation_.\n\n. This number would be made up of the governor, the two hundred syphogrants, the twenty tranibors, the thirteen priests, the scholars, and the ambassadors.\n\n. \"Barzanes\" derives from the Hebrew for \"son of\" and the Greek Doric form for \"of Zeus.\" A Chaldean named \"Mithrobarzanes\" appears in Lucian's _Menippus_ , which More translated. \"Ademus\" derives from the Greek for \"without a people.\"\n\n. An average of twelve adults in each household would produce a population of seventy-two thousand in each city. Adding children and slaves would probably bring it to more than one hundred thousand (of whom only five hundred are exempt from work).\n\n. Hythloday's and the Utopians' rather facile justification of colonialism offers many difficulties. For example, if no one is using or occupying the land, why does anyone have to be driven from it by force? Is farming the only satisfactory use of land?\n\n. Sidenote: _Thus they avoid having a crowd of idle servants_.\n\n. That is, each of the four sides of a block has thirty houses, with a hall in the middle of each side.\n\n. Sidenote: _They always take freedom into account lest anyone act under compulsion_.\n\n. That is, without the help of servants.\n\n. Sidenote: _Praise and a sense of duty are the best way to encourage citizens to act properly_.\n\n. Sidenote: _The education of the young_.\n\n. Sidenote: _Priests above the prince, though nowadays even bishops are the lackies of princes_.\n\n. Sidenote: _Nowadays even monks rarely observe this custom_.\n\n. Sidenote: _Nowadays physicians condemn this practice_.\n\n. Cf. 2 Thess. 3:10.\n\n. Sidenote: _O holy commonwealth, worthy to be imitated even by Christians_.\n\n. Sidenote: _See how they never forget their sense of community_.\n\n. Sidenote: _What a clever fellow!_\n\n. Sidenote: _What a magnificent contempt for gold!_\n\n. Sidenote: _A very fine story_. \"Anemolian\" is from the Greek word for \"windy.\"\n\n. Sidenote: _O what a craftsman!_\n\n. Sidenote: _He calls it dubious because the gems are fake, or at least because their glitter is scanty and dim_.\n\n. Cf. Lucian, _Demonax_ 41; see also _CWM 13,_ 8.\n\n. Sidenote: _How true and well put!_\n\n. Sidenote: _How much wiser are the Utopians than the general run of Christians!_\n\n. More uses \"philosophy\" in the older, broader sense of the investigation of all the arts and sciences, including mathematics and the natural sciences (which was often called \"natural philosophy\").\n\n. They have mastered the quadrivium, the second tier of university studies (music, arithmetic, geometry, and astronomy); of the first tier, the trivium (grammar, logic or dialectic, and rhetoric), dialectic is mentioned here. Grammar and rhetoric they would learn in their literary studies.\n\n. Sidenote: _There seems to be some underlying satire in this passage_.\n\n. Peter of Spain's thirteenth-century _Little Logicbook,_ with its finespun categories and distinctions, was dissected and mocked by More in his _Letter to Dorp,_ which he wrote in 1515, near the time he wrote the second book of _Utopia_. On complicated \"rules about restrictions, amplifications, and suppositions\" see More's text and Daniel Kinney's introduction to _Letter to Dorp_ in _CWM 15,_ liv\u2013lvii, 29\u201339.\n\n. \"First intention\" refers to the intellect's direct perception of an object; a \"second intention\" is the intellect's perception of or reflection on a first intention. It has no objective existence outside the mind.\n\n. That is, the universal concept of man that applies to each man in particular. From the fourteenth century through More's time, scholastic philosophers from the camps of the Realists and the Nominalists quarreled elaborately about whether and how universals had any real existence.\n\n. Sidenote: _But nowadays these practitioners rule the roost among Christians_. More wrote a number of Latin epigrams ridiculing judicial astrology ( _CWM 3\/2,_ 158, 166, 208, 214\u201316, 348\u201349).\n\n. Sidenote: _Natural science the most uncertain study of all_.\n\n. These three categories of goods (external goods and goods of the mind and of the body) derive primarily from the Aristotelian tradition. Generally the Aristotelians applied \"good\" to all three categories; the Stoics, only to the goods of the mind.\n\n. Sidenote: _The Utopians measure happiness by honorable pleasure_. That is, they are inclined to the Epicurean position that pleasure is the highest good. Beginning with Lorenzo Valla's _The True and False Good_ (1444\u201349) and with the help of such thinkers as Ficino, Pico, and Erasmus, Epicurean philosophy had been rehabilitated and shown to consist not in mere hedonism but rather in the calm pleasures of the mind. But the Utopians differ sharply from the Epicureans, who did not believe in immortality and thought the gods were unconcerned about mankind.\n\n. Sidenote: _First principles of philosophy should be derived from religion_.\n\n. Sidenote: _The theology of the Utopians_.\n\n. Sidenote: _The immortality of the soul, about which not a few Christians nowadays have doubts_. The fifth Lateran Council (1513) affirmed as dogma the immortality of the soul. The philosopher most closely associated with the dispute concerning the immortality of the soul was Pietro Pomponazzi, whose treatise _On the Immortality of the Soul_ (1516) argued that the doctrine could not be proved by reason but has to be derived from revealed religion.\n\n. Sidenote: _Just as not just any pleasure should be sought after, so too pain should not be pursued except for the sake of virtue_. This is in keeping with the teachings of Epicurus; see Diogenes Laertius 10.130\u201332. The opposite faction is the Stoics.\n\n. Sidenote: _This is a teaching of the Stoics_.\n\n. Sidenote: _But nowadays some seek out pain, as if religion consisted in it, whereas pain is only to be borne if it occurs by natural necessity or to someone performing the duties of piety_. Seneca, whose Stoicism is often severe and uncompromising, agrees with the Utopians: \"Our motto . . . is 'Live according to Nature'; but it is quite contrary to nature to torture the body, to hate unlabored elegance, to be dirty on purpose, to eat food that is not only plain, but disgusting and forbidding\" ( _Epistulae morales,_ 5.4). Unlike some Stoics, Seneca is not entirely unsympathetic with Epicurus: \"the teachings of Epicurus are upright and holy and, if you consider them closely, austere; for his famous doctrine of pleasure is reduced to small and narrow proportions, and the rule that we Stoics lay down for virtue, his same rule he lays down for pleasure\u2014he bids that it obey Nature\" ( _De vita beata_ 13.1). The Utopians combine elements of Stoicism and Epicureanism, and add to the blend belief in divine providence, the immortality of the soul, and rewards and punishments in the afterlife\u2014doctrine not specifically Christian but not uniformly held until the advent of Christianity.\n\n. An Epicurean (not a Stoic) teaching; see Diogenes Laertius 10.138.\n\n. Sidenote: _A remarkable hypothesis, and a very apt one_.\n\n. More wrote a Latin epigram against the cruelty of hunters ( _CWM 3\/2,_ p. 123).\n\n. Sidenote: _But nowadays this is the craft practiced by godlike courtiers_.\n\n. More is probably thinking of hunting dogs.\n\n. Sidenote: _This point should be noted with special care_. In _The Confutation of Tyndale's Answer_ (1532\u201333), More argued on religious grounds that \"besides the taming of the body, fasting and our pain taken therein pleaseth god done with devotion, and serveth us for obtaining many and great gifts of grace ( _CWM 8\/1,_ p. 72).\n\n. Sidenote: _But nowadays blockheads and dolts are chosen to be educated; the most talented minds are corrupted by pleasures_.\n\n. The pupil and successor of Aristotle.\n\n. Constantine Lascaris (d. 1501) and Theodore of Gaza (d. 1475) wrote grammars of Greek. The dictionary of Hesychius (fl. ca. a.d. 400) was first published in 1514; Dioscorides (fl. ca. a.d. 50) wrote a handbook of medical and botanical terms.\n\n. Plutarch (ca. A.D. 50\u2013120) was a favorite Greek writer among Renaissance humanists, both for his _Moralia_ and for his _Parallel Lives_ of eminent Greeks and Romans. Several pieces by the satirist Lucian (b. ca. A.D. 120) were translated by More and Erasmus and first published in 1506; they were reprinted ten times in the sixteenth century.\n\n. In the early sixteenth century the Venetian printer Aldus Manutius was famous for his compact, elegantly printed editions of classical authors in both Latin and Greek. In 1508 he printed the first enlarged edition of Erasmus' huge and elaborate collection of proverbs, _Adagia,_ which brought Erasmus almost instant fame.\n\n. Thucydides and Herodotus are the leading historians of ancient Greece. Herodian (ca. A.D. 170\u2013240) wrote a Greek history of the Roman emperors who reigned from A.D. 180 to 238.\n\n. A name in keeping with that of Hythloday himself: \"tricae apinaeque\" became proverbial meaning \"stuff and nonsense\" (Erasmus, _Adages_ 143, _CWE 31,_ 184).\n\n. Hippocrates (fifth century B.C.) and Galen (second century A.D.) were the leading Greek writers on medicine. _Microtechne_ was a medieval summary of Galen.\n\n. Sidenote: _The remarkable fairness of this people_.\n\n. The non-hereditary character of Utopian slavery distinguishes it from both ancient slavery and feudal serfdom. In More's time it was generally agreed that Christians should not be enslaved, but the same was not true of African negroes and American Indians.\n\n. Such euthanasia, naturally, is contrary to Catholic teaching; at Morton's court Hythloday himself had said that God has forbidden us to kill ourselves (p. 27, above), but he also told More and Giles earlier that he did not intend to discuss whether or not Utopian moral principles are correct (pp. 91\u201392). More has a long psychological analysis of suicide in _A Dialogue of Comfort (CWM 12,_ 129\u201356).\n\n. According to canon law in More's time, girls could not marry before the age of twelve and boys not before fourteen. Plato ( _Republic_ 5.9.460E, _Laws_ 4.721A\u2013B) and Aristotle ( _Politics_ 7.14.6.1335a) set the age of marriage for women at at least twenty and for men over thirty.\n\n. Sidenote: _This practice is somewhat immodest, but it is far from imprudent_. Plato requires similar premarital inspections in _Laws_ 6.771E\u2013772A, 11.925A.\n\n. In More's time the Church permitted separation in the case of adultery but did not allow remarriage. In his commentaries on 1 Cor. 7:10\u201311 and 39 Erasmus favored relaxing the prohibition of remarriage.\n\n. See Erasmus, _Adages_ 1537 ( _CWE 33,_ 309\u201310).\n\n. The Latin for \"fool\" here is _morio,_ wordplay on More's name; Erasmus had exploited the same pun in the prefatory letter of his _Encomium Moriae (The Praise of Folly),_ which is dedicated to More. Thomas More kept a fool, Henry Patenson, in his household; Patenson appears in Holbein's sketch of More's family and is mentioned by More in his _Confutation of Tyndale's Answer (CWM 8\/2,_ 900\u2013901).\n\n. An error in the 1516 edition is corrected differently in the editions of 1517 and 1518, in both cases probably by More himself. One correction could mean that only crafty lawyers are excluded; the other must mean that all lawyers are excluded because all lawyers are crafty. The latter interpretation seems preferable.\n\n. The rulers and popes of More's time were notorious for breaking treaties or making them with the deliberate intention of breaking them. This was especially true of the popes Alexander VI and Julius II. Machiavelli said Alexander VI \"never did anything, never thought of anything other than to deceive men. . . . And never was there a man who had greater success in asserting, and with greater oaths in affirming a thing, who observed it less\" ( _The Prince_ 18).\n\n. A common false etymology derived \"bellum\" (war) from \"belua\" (beast)\u2014or the other way around. For a full account of pacificism in More and his humanist contemporaries see R. P. Adams, _The Better Part of Valor: More, Erasmus, Colet, and Vives on Humanism, War, and Peace, 1496\u20131535_ (Seattle: University of Washington Press, 1962).\n\n. One of the key texts giving the rules for fighting a \"just\" war was Cicero, _De officiis_ 1.11.34\u20131.13\u201340.\n\n. Greek compounds meaning \"cloud-born\" and \"citizens of a country without people.\"\n\n. From a Greek compound meaning \"busy sellers\"\u2014that is sellers and resellers of their military services.\n\n. Sidenote: _A people not unlike the Swiss_.\n\n. Sidenote: _Above all the commander should be assailed so as to end the war sooner_.\n\n. A ducat was a gold coin minted primarily by Venice and worth about a quarter of a pound sterling at that time. The 700,000 ducats mentioned here would be worth many hundred times that much today.\n\n. The Utopians do not have an ordinary treasury; perhaps deposits owed the Utopians and placed in the treasuries of other countries are what is meant here.\n\n. Among the ancient Persians, Mithras was the supreme deity, identified with light.\n\n. Sidenote: _Monasteries_. Communism was practiced by religious orders in More's time, as it still is; on communism among the early Christians see Acts 2:44\u201345 and 4:32\u201337.\n\n. Of the seven sacraments, only baptism and matrimony can be administered by laymen.\n\n. In sacramental theology \"character\" is a technical term meaning the indelible quality bestowed on a soul by sacraments that cannot be repeated: baptism, confirmation, and holy orders.\n\n. Sidenote: _People must be drawn to religion by hearing it praised_.\n\n. In Christian England, More approved of punishing religious dissent or heresy, but that was because the true religion had been revealed there, as it had not in Utopia; as More said in _A Dialogue Concerning Heresies (CWM 6,_ 345\u201346), \"if it were now doubtful and ambiguous whether the church of Christ were in the right rule of doctrine or not, then were it very necessary to give them all good audience that could and would anything dispute on either party for it or against it, to the end that if we were now in a wrong way, we might leave it and walk in some better.\"\n\n. Sidenote: _A remarkable opinion about the souls of animals_.\n\n. In _A Dialogue Concerning Heresies (CWM 6,_ 211, 213) More wrote concerning saints: \"For if their holy souls live, there will no wise man ween them worse and of less love and charity to men that need their help, when they be now in heaven, than they had when they were here in earth. . . . When saints were in this world at liberty and might walk the world about, ween we that in heaven they stand tied to a post?\"\n\n. Sidenote: _The active life_.\n\n. \"Buthrescae\" is a Greek word meaning \"extraordinarily religious.\" In More's Europe the adjective \"religious\" was applied to members of religious orders, who differed, however, from the Buthrescae in that they combined labor with prayer, study, and contemplation.\n\n. Hythloday must mean that the priests supervise the education of children, for in each city there are many thousands of children and only thirteen priests.\n\n. In _The Confutation of Tyndale's Answer (CWM 8\/1,_ 260\u201361) More accepts the traditional view that women may not be ordained as priests.\n\n. Sidenote: _But what a flock of them we have!_\n\n. Sidenote: _O these priests are far holier than ours!_\n\n. The first Greek compound means \"dog days\" (or perhaps \"starting days); the second means \"turning days.\"\n\n. There may be more than one service on every feastday, but even so the churches would have to be very large indeed: only thirteen of them serve about one hundred thousand inhabitants of each city.\n\n. Cf. Matt. 5:23\u201324.\n\n. Sidenote: _But among us the most defiled strive to get closest to the altar_.\n\n. Latin _superos,_ which includes the one God, the other gods believed in by some of the Utopians, and their ancestors who are in heaven.\n\n. A startling idea, but perhaps Hythloday (or the Utopians) mean that for children this tends to be true.\n\n. In his _Four Voyages_ Vespucci mentions that the American Indians made vestments of feathers.\n\n. Fr. Surtz notes that many of More contemporaries, especially Erasmus, objected to the elaborateness of church music and urged that it be composed so as to emphasize the meaning of the words ( _CWM 4,_ 555\u201356).\n\n. Hythloday is so carried away that he speaks as if he is still in Utopia.\n\n. Goldsmiths often functioned as bankers.\n\n. Sidenote: _Note this, reader!_\n\n. Sidenote: _A striking phrase_\n\n. The remora has a suck-disk on top of its head, by which it attaches itself to larger fish or ships; impressed by its tenacity, the ancients thought it could impede the progress of a ship.\n\n. What \"More\" says here is in keeping with his earlier Aristotelian arguments against community of property. Aristotle continually associates nobility and the highest virtue with wealth; he defines magnificence as \"suitable expenditure on a grand scale\" ( _Nicomachean Ethics_ 4.2.1.1122a). But many readers get the impression that More lets the mask of the character \"More\" slip to reveal a hint of irony. For a discussion of the critical disputes about this passage see Thomas I. White, _\"Festivitas, utilitas, et opes:_ The Concluding Irony and Philosophical Purpose of Thomas More's _Utopia,\" Albion_ 10 (1978): 135\u201350.\n\n. This sentence is incomplete in the Latin and has been left so in the translation.\n\n. _Andria_ 4.4.770\u201371.\n\n### _Afterword_\n\n. Clarence H. Miller, _Humanism and Style: Essays on Erasmus and More_ (Bethlehem, Penn.: Lehigh University Press, 2011), pp. 105\u20137.\n\n. Jacob Burkhardt, _The Civilization of the Renaissance in Italy,_ trans. S. G. C. Middlemore (1860; repr., Vienna: Phaidon Press, 1937).\n\n. See, for example, the following: Douglas Bush, _The English Renaissance and Humanism_ (Toronto: University of Toronto Press, 1939); Paul Oskar Kristeller, _Renaissance Thought: The Classic, Scholastic, and Humanist Strains_ (New York: Harper and Row, 1955); Charles Trinkaus, _The Scope of Renaissance Humanism_ (Ann Arbor: University of Michigan Press, 1983); and Charles G. Nauert, _Humanism and the Culture of Renaissance Europe_ (Cambridge: Cambridge University Press, 1995).\n\n. See Bush, _The English Renaissance,_ pp. 13\u201338; Kristeller, _Renaissance Thought,_ pp. 3\u20137; Trinkaus, _Scope of Renaissance Humanism,_ pp. 3\u201331; and Nauert, _Humanism,_ pp. 1\u20137. Nevertheless, the old idea of the medieval period as a \"Dark Age\" is not without its defenders. See Jack Goody, _Renaissances: The One or the Many?_ (Cambridge: Cambridge University Press, 2010), p. 11.\n\n. See R. N. Swanson, _The Twelfth-Century Renaissance_ (New York: Manchester University Press, 1999).\n\n. Burkhardt, _Civilization of the Renaissance,_ p. 292.\n\n. Goody, _Renaissances,_ pp. 5, 7\u201342. For an interesting study of the cultural energies released by print, see Alexandra Halasz, _The Marketplace of Print: Pamphlets and the Public Sphere in Early Modern England_ (Cambridge: Cambridge University Press, 1997).\n\n. Bush, _The English Renaissance,_ p. 82. For a helpful introduction to humanism in England, see Clare Carroll, \"Humanism and English Literature in the Fifteenth and Sixteenth Centuries,\" in _The CambridgeCompanion to Renaissance Humanism,_ ed. Jill Kraye (Cambridge: Cambridge University Press, 1996), pp. 246\u201368. For a circumstantial and critical history of humanism, see Anthony Grafton and Lisa Jardine, _From Humanism to the Humanities: Education and the Liberal Arts in Fifteenth- and Sixteenth-Century Europe_ (Cambridge: Harvard University Press, 1986).\n\n. _The Basic Works of Aristotle,_ ed. Richard McKeon (1947; repr., New York: Modern Library, 2001), p. 1329.\n\n. See John M. Perlette, \"Irresolution as Solution: Rhetoric and the Unresolved Debate in Book 1 of More's _Utopia,_ \" _Texas Studies in Literature and Language_ 29:1 (Spring 1987): 28\u201353. Perlette reads the debate in _Utopia_ about the worth of advising a prince on public affairs as an extension of the ancient debate between Plato and the Sophists.\n\n. This is J. H. Hexter's coinage for the dialogue in Book 1 concerning the usefulness of advising a prince. See his _More's \"Utopia\": The Biography of an Idea_ (New York: Harper and Row, 1952), p. 102.\n\n. Thomas More, \"Letter to Martin Dorp,\" in _In Defense of Humanism: Letter to Martin Dorp; Letter to the University of Oxford; Letter to Edward Lee; Letter to a Monk, with a New Text and Translation of Historia Richardi Tertii,_ ed. Daniel Kinney, vol. 15 of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1986), pp. 26\u201339.\n\n. Lorenzo Valla, _On the Donation of Constantine,_ trans. G. W. Bowersock (Cambridge: Harvard University Press, 2007), p. 73.\n\n. Harvey Cox, _The Feast of Fools: A Theological Essay on Festivity and Fantasy_ (Cambridge: Harvard University Press, 1969), pp. 82\u201397. For helpful studies of the centrality of play to human life, see Johan Huizinga, _Homo Ludens: A Study of the Play Element in Culture_ (Boston: Beacon Press, 1955), and Hugo Rahner, _Man at Play: Or, Did You Ever Practice Eutrapelia?_ trans. Brian Battershaw and Edward Quinn (London: Burns and Oates, 1963).\n\n. Nauert, _Humanism,_ pp. 5\u20136.\n\n. Peter Ackroyd, _The Life of Thomas More_ (New York: Doubleday, 1998), p. 32.\n\n. Ibid., pp. 71\u201380. See also Richard Marius, _Thomas More: A Biography_ (Cambridge: Harvard University Press, 1984), pp. 71\u201378.\n\n. Ackroyd, _Life of Thomas More,_ pp. 82\u201395; Marius, _Thomas More,_ pp. 83\u201387, 153.\n\n. See Hexter, _More's \"Utopia,\"_ p. 99.\n\n. Alan F. Nagel, \"Lies and the Limitable Inane: Contradiction in More's _Utopia,\" Renaissance Quarterly_ 26:2 (Summer 1973), pp. 175\u201377).\n\n. See the essay \"Editions of _Utopia_ \" by Edward Surtz, S.J., in _Utopia,_ ed. Edward Surtz, S.J., and J. H. Hexter, vol. 4 of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1965), pp. clxxxiii\u2013cxciii; and R. W. Gibson and J. Max Patrick, _Thomas More: A Preliminary Bibliography of His Works and of Moreana to the Year 1750 with a Bibliography of Utopiana_ (New Haven: Yale University Press, 1961), pp. 3\u201357.\n\n. It is striking that in his early twenties, More delivered a series of lectures on St. Augustine's _City of God._ See Ackroyd, _Life of Thomas More,_ pp. 104\u20136.\n\n. See Hexter, _More's \"Utopia,\"_ pp. 85\u201396; Marius, _Thomas More,_ pp. 67\u201369; and Ackroyd, _Life of Thomas More,_ p. 99. Hexter clarifies that there are also groups within Utopia who live a professed life that more closely resembles monasticism.\n\n. _The Rule of St. Benedict,_ trans. Anthony C. Meisel and M. L. del Mastro (New York: Image Books, 1975), p. 106.\n\n. More's Latin, along with a prose translation, appears in Clarence H. Miller et al., eds., _Latin Poems,_ vol. 3, part 2, of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1984), pp. 238\u201341.\n\n. The quotation comes from the Arden edition of Shakespeare's _The Tempest,_ ed. Frank Kermode (London: Methuen, 1954), 2.1.153\u201354.\n\n. In his _More's \"Utopia\"_ (Toronto: University of Toronto Press, 2000), Dominic Baker-Smith emphasizes More's concern with what modern theology has identified by the term \"social sin,\" the distortions of human values woven into the social fabric. See pp. 115\u201316.\n\n. The phrase comes from Bernard Lonergan, _Insight: A Study of Human Understanding_ (1957; repr., Toronto: University of Toronto Press, 1992).\n\n. Lewis Mumford, _The Story of Utopias_ (New York: Peter Smith, 1941), p. 78.\n\n. Besides his introduction to this edition, see also Clarence H. Miller, _Humanism and Style,_ pp. 71\u201379.\n\n. Elizabeth McCutcheon, \"Denying the Contrary: More's Use of Litotes in the _Utopia,_ \" in _Essential Articles for the Study of Thomas More,_ ed. R. S. Sylvester and G. P. Marc'hadour (Hamden, Conn.: Archon Books, 1977), p. 263.\n\n. Stephen Greenblatt, _Renaissance Self-Fashioning: From More to Shakespeare_ (Chicago: University of Chicago Press, 1980), pp. 22\u201326.\n\n. See Miller's _Humanism and Style,_ pp. 71\u201379.\n\n. Ibid., 76.\n\n. Ibid., 72.\n\n. William Roper, _Sir Thomas More,_ in _Two Early Tudor Lives,_ ed. Richard S. Sylvester and Davis P. Harding (New Haven: Yale University Press, 1962), p. 198. With regard to More's devotion to the theatrical metaphor, see Greenblatt, _Renaissance Self-Fashioning,_ pp. 26\u201337. See also Thomas More, _The History of Richard III,_ ed. George Logan (Bloomington: Indiana University Press, 2005), pp. 94\u201395, 126\u201328.\n\n. See Dominic Baker-Smith, \"' _Civitas philosophica_ ': Ideas and Community in Thomas More,\" in _A Companion to Thomas More,_ ed. A. D. Cousins and Damian Grace (Madison, N.J.: Fairleigh Dickinson University Press, 2009), pp. 165\u201377.\n\n. The use of the term \"relationism\" as a way of understanding the sociology of knowledge emerged in the work of Karl Mannheim; see his _Ideology and Utopia: An Introduction to the Sociology of Knowledge_ (New York: Harcourt, Brace and Company, 1936), pp. 71\u201378 and passim. Among more recent scholars, the one who has done the most, as far as I know, to develop the implications of the term is Walter J. Ong; see his _Fighting for Life: Contest, Sexuality, and Consciousness_ (1981; repr., Amherst: University of Massachusetts Press, 1989), pp. 29\u201334; and \"Hermeneutic Forever: Voice, Text, Digitization, and the 'I,'\" _Oral Tradition_ 10 (1995): 3\u201326. See also my \"Ong, Evolution, and the Method of Dialogue,\" _Explorations in Media Ecology_ 9:1\u20134 (2010): 103\u201318.\n\n. For further considerations, see Charles Trinkaus, \"The Question of Truth in Renaissance Rhetoric and Anthropology,\" in _Scope of Renaissance Humanism,_ pp. 437\u2013449.\n\n. Bernard Lonergan, _A Second Collection,_ ed. William F. J. Ryan, S.J., and Bernard J. Tyrrell, S.J. (Toronto: University of Toronto Press, 1974), pp. 1\u20139. See also Anthony Grafton, _What Was History? The Art of History in Early Modern Europe_ (Cambridge: Cambridge University Press, 2007).\n\n. For helpful insights into such dialogue, see David M. Bevington, \"The Dialogue in 'Utopia': Two Sides to the Question,\" _Studies in Philology_ 58:3 (July 1961): 496\u2013509.\n\n. Martin Waldseem\u00fcller, _Cosmographiae Introductio,_ trans. Joseph Fischer and Franz von Wieser (University Microfilms, 1966), pp. 91\u201397.\n\n. Ibid., p. 98.\n\n. Roper, _Sir Thomas More,_ p. 198. See also Ackroyd, _Life of Thomas More,_ pp. 96\u2013101.\n\n. _Rule of St. Benedict,_ ch. 33.\n\n. _Collected Works of Erasmus: Adages Ii1 to Iv100,_ trans. Margaret Mann Phillips (Toronto: University of Toronto Press, 1982), p. 29.\n\n. Kathy Eden, _Friends Hold All Things in Common: Tradition, Intellectual Property, and the \"Adages\" of Erasmus_ (New Haven: Yale University Press, 2001).\n\n. All references to Aquinas are to _Summa Theologica,_ 3 vols., trans. Fathers of the English Dominican Province (New York: Benziger Brothers, 1947).\n\n. Paul Ricoeur, _Lectures on Ideology and Utopia,_ ed. George H. Taylor (New York: Columbia University Press, 1988), p. 310.\n\n. Ibid., p. 311.\n\n. Ibid., p. 312.\n\n. See Walter J. Ong, S.J., _Interfaces of the Word: Studies in the Evolution of Consciousness and Culture_ (Ithaca: Cornell University Press, 1977), pp. 166\u201381; and Halasz, _Marketplace of Print._\n\n. Hanan Yoran, _Between Utopia and Dystopia: Erasmus, Thomas More, and the Humanist Republic of Letters_ (New York: Rowman and Littlefield, 2010), p. 63.\n\n. Ibid., pp. 177\u201386.\n\n. See Walter J. Ong, _Ramus, Method, and the Decay of Dialogue: From the Art of Discourse to the Art of Reason_ (1958; repr., Chicago: University of Chicago Press, 2004), pp. 307\u201318.\n\n. For a fine introduction to these and related themes in the work of Jacques Derrida, see his \"Structure, Sign, and Play in the Discourse of the Human Sciences,\" in _Writing and Difference,_ trans. Alan Bass (Chicago: University of Chicago Press, 1978), pp. 278\u201393.\n\n. Jacques Derrida, _Negotiations: Interventions and Interviews, 1971\u20132001,_ trans. Elizabeth Rottenberg (Stanford: Stanford University Press, 2002), pp. 13\u201314.\n\n. I would like to thank Amanda Joyce, Mike Smolinsky, and Mary Szybist for their helpful comments, as well as my students Charlotte Markle and Justine Minette for their hours of helpful conversation about More's _Utopia._ The errors and folly are mine.\n\n## SUGGESTIONS FOR FURTHER READING\n\nUPDATED BY JERRY HARP\n\n### BIOGRAPHIES\n\nAckroyd, Peter. _The Life of Thomas More_ (New York: Doubleday, 1998). The most reliable full biography.\n\nChambers, R. W. _Thomas More_ (London: Jonathan Cape, 1935; repr., Ann Arbor: University of Michigan Press, 1958). A classic biography, well informed and beautifully written, but it neglects the polemical works.\n\nGuy, John. _Thomas More_ (New York: Oxford University Press, 2000). Emphasizes especially the political struggles. Guy acknowledges that some issues that have exercised More biographers cannot be resolved based on the available evidence; thus, as he points out, each reader tends to make More over into his or her desired image.\n\nMarius, Richard. _Thomas More: A Biography_ (Cambridge: Harvard University Press, 1984). The biography that makes the most extensive use of the full range of More's writings. Marius performed a helpful service by writing against the grain of the hagiographic tendencies of past More biographies, though his conclusions occasionally overreach the evidence; he rides his hobbyhorse a bit vigorously at times.\n\nRoper, William. _Sir Thomas More,_ in _Two Early Tudor Lives,_ ed. Richard S. Sylvester and Davis P. Harding (New Haven: Yale University Press, 1962). A handy edition of the first biography of More, written by his son-in-law\u2014personal and poignant.\n\n### BIBLIOGRAPHIES\n\nGeritz, Albert. _Thomas More: An Annotated Bibliography of Criticism, 1935\u20131997_ (Westport, Conn.: Greenwood Press, 1998). Excellent, very full. Pages 215\u2013309 are devoted to _Utopia_ alone.\n\nGibson, R. W., and J. Max Patrick. _Thomas More: A Preliminary Bibliography of His Works and of Moreana to the Year 1750 with a Bibliography of Utopiana_ (New Haven: Yale University Press, 1961). Early editions and translations of _Utopia_ up to 1750.\n\nLakowski, Romauld Ian. \"A Bibliography of Thomas More's _Utopia,_ \" <http:\/\/extra.shu.ac.uk\/emls\/01-2\/lakoutop.html>. This excellent Internet bibliography, devoted exclusively to _Utopia,_ also exists in a printed form: _Early Modern Literary Studies_ 1.2 (August 1995).\n\nWentworth, Michael D. _The Essential Sir Thomas More: An Annotated Bibliography of Major Modern Studies_ (New York: G. K. Hall, 1995). Entries 380\u2013640 are devoted to _Utopia_ alone.\n\nThe indices of the journal _Moreana_ will provide a plethora of articles on _Utopia._\n\n### EDITIONS\n\n_Utopia: Latin Text and English Translation,_ ed. George M. Logan, Robert M. Adams, and Clarence H. Miller (Cambridge: Cambridge University Press, 1995). A reliable and usable Latin text, with a compact introduction and notes.\n\n_Utopia,_ ed. Edward Surtz, S.J., and J. H. Hexter, vol. 4 of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1965). Full-fledged edition with elaborate introduction, textual variants, and a very full commentary.\n\n_L'Utopie de Thomas More,_ ed. Andr\u00e9 Pr\u00e9vost (Paris: Mame, 1978). The French equivalent of Surtz's edition, with a facsimile of the edition of November 1518, an elaborate introduction, a French translation, and a very full commentary.\n\n_Utopia,_ trans. Ralph Robinson, Everyman's Library (New York: Alfred A. Knopf, 1992). The first English translation. This edition includes a helpful introduction by Jenny Mezciems.\n\n### STUDIES OF _UTOPIA_\n\nBaker-Smith, Dominic. _More's Utopia._ (Toronto: University of Toronto Press, 2000). A multidimensional introduction. Baker-Smith discusses the historical background and literary precedents in detail. He shows the continuing relevance of the work, especially with regard to what he terms social sin, the injustices woven into the social fabric.\n\nBevington, David M. \"The Dialogue in 'Utopia': Two Sides to the Question,\" _Studies in Philology_ 58:3 (July 1961): 496\u2013509. Emphasizes the open-endedness of the dialogue that _Utopia_ invites the reader to join.\n\nCave, Terence (Ed.). _Thomas More's \"Utopia\" in Early Modern Europe: Paratexts and Contexts_ (Manchester: Manchester University Press, 2008). Provides studies of the material\u2014e.g., prefaces, poems, essays, letters\u2014that accompanied early editions of _Utopia_ in a variety of languages.\n\nCousins, A. D., and Damian Grace (Eds.). _A Companion to Thomas More_ (Madison, N.J.: Fairleigh Dickinson University Press, 2009). Brings together studies ranging from the early epigrams to the Tower works.\n\nGreenblatt, Stephen. _Renaissance Self-Fashioning: From More to Shakespeare_ (Chicago: University of Chicago Press, 1980). Explores the unstable and distorted relationship between the two books of _Utopia,_ which Greenblatt also connects to More's role-playing.\n\nHexter, J. H. _More's \"Utopia\": The Biography of an Idea_ (New York: Harper and Row, 1952). A detailed study of the evolution of _Utopia._\n\nLogan, George. _The Meaning of More's \"Utopia\"_ (Princeton: Princeton University Press, 1983). Provides a close reading of _Utopia_ as a product of Northern humanism, with constant reference to the classical and medieval antecedents, as well as contemporary influences.\n\nLogan, George M. (Ed.). _The Cambridge Companion to Thomas More_ (Cambridge: Cambridge University Press, 2011). A collection of essays on More's life and writings, by some leading More scholars.\n\nMcCutcheon, Elizabeth. \"Denying the Contrary: More's Use of Litotes in the _Utopia,_ \" in _Essential Articles for the Study of Thomas More,_ ed. R. S. Sylvester and G. P. Marc'hadour (Hamden, Conn.: Archon Books, 1977), 263\u201374. Demonstrates the significance of More's frequent use of litotes in _Utopia._ As a figure of discourse, litotes allows a more complex understanding than simple affirmation or negation can accomplish.\n\nMcCutcheon, Elizabeth. _My Dear Peter: The \"Ars Poetica\" and Hermeneutics for More's \"Utopia\"_ (Angers: Moreanum, 1983). An excellent study of litotes and the oblique style of _Utopia._\n\nMiller, Clarence H. \"Style and Meaning in More's _Utopia:_ Hythloday's Sentences and Diction,\" in _Humanism and Style: Essays on Erasmus and More_ (Bethlehem, Penn.: Lehigh University Press, 2011). Demonstrates the significance of the shifts of style in Hythloday's Latin.\n\nNagel, Alan F. \"Lies and the Limitable Inane: Contradiction in More's _Utopia,_ \" _Renaissance Quarterly_ 26:2 (Summer 1973), pp. 173\u2013180. A consideration of the contradictions, tensions, and impossibilities woven into More's text.\n\nNelson, William (Ed.). _Twentieth Century Interpretations of Utopia: A Collection of Critical Essays_ (Englewood Cliffs, N.J.: Prentice-Hall, 1968). Provides a broad range of commentaries and perspectives.\n\nPerlette, John M. \"Irresolution as Solution: Rhetoric and the Unresolved Debate in Book 1 of More's _Utopia,_ \" _Texas Studies in Literature and Language_ 29:1 (Spring 1987): 28\u201353. A reading of the Dialogue of Counsel as an extension of the ancient debate between philosophy and sophistry.\n\nSylvester, Richard S. \"'Si Hythlodaeo Credimus': Vision and Revision in Thomas More's _Utopia,_ \" _Soundings_ 51 (1968): 271\u201389. Reprinted in _Essential Articles for the Study of Thomas More,_ ed. R. S. Sylvester and G. P. Marc'hadour (Hamden, Conn.: Archon Books, 1977), pp. 290\u2013301. The best brief introduction to the ironies of _Utopia._\n\nSylvester, R. S., and G. P. Marc'hadour (Ed.). _Essential Articles for the Study of Thomas More._ (Hamden, Conn: Archon Books, 1977). A collection that lives up to its title.\n\nYoran, Hanan. _Between Utopia and Dystopia: Erasmus, Thomas More, and the Humanist Republic of Letters_ (New York: Rowman and Littlefield, 2010). A study of the early humanists' desire to create a virtual community transcending national boundaries and institutional affiliations, and the impossibility of doing so. Yoran provides a reading of _Utopia_ in light of this impossibility.\n\n## INDEX\n\nAbraxa. _See_ Utopia\n\nAchoria (ideal kingdom), xii, xv,\n\nAdams, Robert M., xxii\n\nAdemus (ruler of Utopia), , , n169\n\nadultery, , , n227\n\nage, , ,\n\nagriculture, \u201324, , \u201355, ,\n\nAlaopolitans (enemies of Utopia),\n\nAmaurot (Utopian city), , , , n12; description of, \u201357; resemblance to London, nn138\u201340\n\nambassadors, \u201378\n\nAnemolians, \u201378\n\nAnemolius,\n\nanimals, , , ,\n\nAntwerp, xxv,\n\nAnyder (Utopian river), , , , n12\n\nApinatus, Tricius,\n\nAristophanes,\n\nAristotle, , n110, n157, n267\n\nartisans, ,\n\nastronomy,\n\nBarzanes. _See_ Ademus\n\nbeggary, , , \u201333, ,\n\nbribes, ,\n\nbrothels, ,\n\nBruges, xxvi,\n\nBrussels,\n\nBurnet, Gilbert, xxi, xxii\n\nButhrescae, , \u201381n249\n\ncapital punishment, viii, , \u201332, , ; for adultery, x, ; for foreign wrongdoers, ; religion and, , n68; for theft,\n\nCarthage,\n\nCelaeno (monster), , n42\n\ncelibacy,\n\nCharles V (Holy Roman Emperor), xxvi, , n23, n96\n\nchildren, viii\u2013ix, \u201361, , ; and books, ; and gemstones, , \u201378; obligation to reproduce, ; and religion, \u201324, , n250; of slaves, ; warfare and,\n\nChrist, , ,\n\nChristianity, , . _See also_ religion\n\nchurches, , \u201328\n\nCicero, vii, , n65, n109\n\ncities, , , ; city-states, xvi, xvii; description of, , \u201358; enemy, \u201315; social relations in, \u201367\n\ncivilization,\n\nClement, John, , n11\n\nclothing, , , , \u201385\n\ncolonialism, ix, xvi, , n171\n\nColt, Joan (More's first wife), xxv, xxvi\n\ncommunism, viii, ix, x,\n\nconformity, xviii\n\ncontemplation, vii\n\ncraftsmen,\n\ncrime: bribes and, ; causes of, \u201320, ; money and, ; priests and, ; punishment of, xvi, \u201334, , \u2013101\n\ndeath, \u201321\n\ndeath penalty. _See_ capital punishment\n\nDelcourt, Marie, xxi\n\ndestitution. _See_ poverty\n\n_Dialogue of Comfort Against Tribulation, A_ (More), xxviii\n\nDionysius, , n87\n\nDioscorides,\n\ndisabled persons,\n\ndisease, , , , ; old age as, ; pain and, ; rarity of, ; treatment of the sick, ; warding off,\n\ndisputes, settlement of,\n\ndivorce, viii, xvi, ,\n\nDonnelly, John P., xxii\n\neducation, , \u201380, , \u201324\n\nElisha (biblical), , n83\n\nenclosures, , n59, n61\n\nEngland, , , n131; crime and punishment in, \u201334; Hythloday in, \u201319; peace treaties and, ; relations with France, ; trade with Flanders, n22\n\nErasmus, Desiderius, xix, xx, n13, n93, n218\n\nethics, \u201383\n\nethnic cleansing, x\n\nEuripides,\n\nEurope, xi, xv, xxii\n\neuthanasia, viii, xvi, \u201397, n224\n\nexcommunication,\n\nexile, as punishment,\n\nexports, ,\n\nfalconry,\n\nfamilies, xiv, \u201368\n\nfarming, xiii, xiv, xvii, ; enclosures and, ; Plato on, n157; tenant-farmers, ; war veterans and, ,\n\nfeastdays, , , n255\n\nfood, \u201372, ; health and, , , ; money and, \u201333\n\nfools, , n229\n\nFrance, , , , n49,\n\nFrancis I (king of France), xxvi, n89, n91\n\nfreedom,\n\nFroben, Johann, xx\n\nGalen, , n221\n\ngambling, , , n61\n\ngames, , ,\n\ngardens, \u201358,\n\ngemstones, , \u201378,\n\nGiles, Peter, , , nn29\u201330; on government, ; meeting with Hythloday, \u201312; More's letters to, xx, xxii, \u20137, \u201339; and printing of _Utopia,_ xix; qualities of,\n\nGod, , , . _See also_ religion\n\ngold, , , ; as bribe, ; Utopian use of, \u201376, \u201379\n\ngoods, equality of, xvi, ,\n\ngovernment, xvi, xx; in ancient Rome, ; antiquity of, ; wise men and,\n\nGreek (ancient), , , , \u201395\n\nhappiness, \u201382, , ,\n\nhealth (physical), \u201389,\n\nHenry VIII (king of England): Charles V and, ; French wars of, n21, n49; laws under, n61; More in service of, vii, ix, xxvi, xxvii\n\nHerodian, , n56, n219\n\nHerodotus, , n219\n\nHesychius,\n\nHippocrates, n221\n\n_History of Richard III, The_ (More), xxvi\n\nHoly Roman Empire, xxvii\n\nHomer,\n\nhonors, bestowing of, xiv\n\nhospitals, \u201369\n\nhouseholds, , \u201368\n\nhumanists, xix, n117\n\nhunting, \u201387,\n\nHythloday, Raphael (advocate of Utopia), , , ; Anemolius and, ; bipolarity of, x; characteristics of, x\u2013xi; diction of, xvi\u2013xviii; language of, xxii\u2013xxiii; learning of, , ; qualities of, \u201312; significance of name, viii, x, ; syntax of, xi\u2013xii, xv; travels of, \u201314\n\nidleness, , , ; injustice and, \u201331; travel and,\n\ninjustice, viii, , ; class differences and, \u201332; Hythloday's stand against, x, xix\n\ninstitutions, xvii, , , ; absurdity of, ; paradox of, xvi, xviii\n\nItaly, , , n21, n57\n\njustice, , , \u201331\n\nkings, , , , n111; philosopher-kings, ; power of, \u201341; warfare and, \u201340. _See also_ princes\n\nlabor, , \u201366; common good and, ; dispensation from, ; injustice and, \u201332; as punishment, , \u201330, ; religion and, \u201322; transportation and, ; travel and, \u201373; workday in Utopia, ,\n\nLaestrigonians (monsters), , n42\n\nlanguage, xvii\n\nLascaris,\n\nLatin, xi\u2013xii, xviii, , , ; early editions of _Utopia_ in, xx\u2013xxi; philosophy and, \u201312; translation of, xxi\u2013xxiii; works by More in, xxvi\n\nlaws, xvii, , , \u20132; antiquated (disused), , ; injustice and, 131, ; private property and, , ; punishment of thieves, \u201332; religion and, ; scorn for,\n\nlawyers, xiii, , , \u20132\n\nlivestock, , \u201324, ,\n\nLogan, George, xxi\n\nLucian, xx, , n217\n\nLupset, Thomas, xx\n\nLupton, J. H., xi, xxi\n\nLuther, Martin, xxvii\n\nLutheranism, xxvii\n\nluxury, ,\n\nMacaria (ideal kingdom), xii, xv,\n\nmagistrates, \u201360, , , ; dining arrangements and, \u201371; election of, ; and euthanasia, ; honor of, ; Utopians as magistrates for other peoples, \u20133\n\nmarriage, \u201399,\n\nmedicine, ,\n\nmen, , , ; in church, ; husbands, , , ; idleness and, ; military training and,\n\nmercenaries: employed by Utopia, , \u201311, ; in European wars, , , n93. _See also_ soldiers\n\nMichels, V., xxi\n\n_Microtechne_ (Galen),\n\nMiddleton, Alice (More's second wife), xxvi\n\nmoney, , , ; abolition of, , ; buying off enemies with, , ; given to prisoners, ; precious metals and, , ; private property and, ; royal treasury and, \u201343; theft of, \u201327; value of currency, ; war reparations,\n\nmonks, \u201334\n\nmonsters,\n\nmorals\/morality, , ,\n\nMore, Thomas, vii, xviii, , ; business activities of, ; chronology of life, xxv\u2013xxviii; death of, xxviii; as lawyer, vii; letters to Peter Giles, \u20137, \u201339; marathon sentences of, xi\u2013xii; on religious heresy, n245\n\n_Moriae Encomium_ (Erasmus), xix\n\nMorton, Cardinal John: on crime and punishment, viii, xii, \u201334, , \u201332; More and, xxv, , n47\n\nMoses (biblical),\n\nmurder, xvi, \u201328\n\nmusic, , , n261\n\nMythras (god), ,\n\nnature, \u201383, , ; gifts of, ; religion and, ; science and,\n\nNephelogetes (allies of Utopia),\n\nnoblemen, \u201322; injustice and, , ; as landlords, ; peace treaties and, ; retainers of,\n\nnuns,\n\noffice seeking, xiii, xiv, xvii,\n\nOgden, H. V. S., xxii\n\n_On Plants_ (Theophrastus),\n\npain, ,\n\npapacy, xvii, , , n231\n\n_Parva logicalia,_\n\npassports,\n\nPersia,\n\nPersian language,\n\nphilosophers\/philosophy, , \u201380, n117, n192; ethics, \u201383; philosopher-kings, ; on pleasure, n201\n\nphylarchs, , ,\n\nPlato, , , \u201336, , n120; dialogues of, vii; Dionysius and, , n87; on farming, n157; on government, ; on laws, ; on oligarchy, n50; _Republic,_ 44, n126\n\nPlautus,\n\npleasure, , \u201384, n201; false, \u201387; true, \u201391\n\nPlutarch, , n217\n\npolice, absence of, xvi\n\nPolylerites, \u201331, n68\n\nPortugal, ,\n\npoverty, , , \u201333, , ; power of kings and, ; pride and,\n\nprayers, , , \u201329\n\nPr\u00e9vost, Andr\u00e9, xxi\n\nprices: abundance of goods and, ; of food, , ; of livestock, \u201324\n\npride, xv, , ,\n\npriests, xv, , , , ; dining arrangements and, ; election of, ; and euthanasia, , ; and idleness, ; insignia of, ; religious education and, \u201324, n250; reputation of, \u201325; vestments of, \u201328\n\nprinces, \u201317, ; dispensations and, ; enemy, , ; and philosophy, ; and stolen goods, ; treaties of, ; and warfare, , . _See also_ kings\n\nprinting, \u201395\n\nprivate property, xv, ; abolition of, ; absence of, ; equality of goods and, \u201348; money and,\n\npublic service, vii\n\npunishment, of crime, \u201334, , \u2013101,\n\nreason, , , , ,\n\nreligion, x, , , \u201384, ; capital punishment and, , n68; children and, \u201324, , n250; labor and, \u201322; laws and, ; nature and, ; of Utopia, \u201329. _See also_ Christianity\n\n_Republic_ (Plato), , n126\n\n_Responsio ad Lutherum_ (More), xxvii\n\nRestoration, xxii\n\nretainers, idle, , \u201322\n\nRich, Richard, xxviii\n\nRichards, G. C., xxii\n\nRobinson, Ralph, xi, xii, xxi\n\nRome, ancient, , ,\n\nsacrifices,\n\nSallust,\n\nsatire,\n\nscience,\n\nScylla (monster), , n42\n\nseamanship, \u201314,\n\nseduction,\n\nSeneca, , , n118, n207\n\nsexual activity, ,\n\nSheehan, John, xxii\n\nsheep, , \u201324, , n59\n\nshortages,\n\nsilver, , \u201376,\n\nsin, ,\n\nslaves, , \u201331, \u201370, \u201396; character of Utopian slavery, n223; in country households, ; execution of, ; golden chains of, , \u201378; as hunters and butchers, ; prisoners of war as, ; slavery as punishment, , \u201396,\n\nsocial classes, xix, \u201334\n\nsocial relations, \u201372\n\nsoldiers, xiii, xiv, \u201321; priests and, ; as thieves, \u201321, n51; Utopian military practices, , \u201315. _See also_ mercenaries\n\nSolomon (biblical),\n\nSophocles,\n\nSpain, king of,\n\nStoics, , n65, n207\n\nsuicide, \u201397\n\nsuperstition,\n\nsurpluses,\n\nSurtz, Edward, x, xii, xxi, xxii,\n\nSylvester, Richard, xii\n\nsyphogrants. _See_ magistrates\n\nSyria,\n\ntaverns,\n\ntaxes, , n106\n\nTerence, ,\n\nThemsecke, George de,\n\nTheophrastus,\n\nthieves: causes of theft, , , \u201325; punishment of, xii, , , \u201332, n48; soldiers as, \u201321, n51\n\nThucydides, , n219\n\ntradition (custom),\n\ntranibors (chief magistrates), , , , n146\n\ntranslations, xi, xxi\u2013xxiii, n88\n\ntravel, \u201395\n\ntreaties, xv, \u201337, \u20135, n231\n\nTunstall, Bishop Cuthbert, xxvii, , n24\n\nTurner, Paul, xxii\n\nuniversalism, xvii, xviii\n\nUtopia, , ; absence of central authority in, xvi, n153; agriculture in, \u201355; allies and friends of, , \u20136; alphabet of, ; cities of, \u201354, \u201358; comparison with Europe, xv; Hythloday in, ; laws in, ; meaning of name, ix, , ; military practices in, \u201315; occupations in, \u201366; physical description of, , , \u201354, \u201358; religions of, \u201329; slavery in, \u201396, n223; social relations in, \u201372; travel in, \u201395; visitors to,\n\n_Utopia_ (More): criticism of, ; early editions of, xix\u2013xxi; Latin text of, xx\u2013xxi; publication of, xxvi; translation of, xxi\u2013xxiii\n\nUtopus (conqueror-founder of Utopia), viii, , , ,\n\nvagabonds, , , n48\n\nVespucci, Amerigo, , n36, n260\n\nvices, , , ; eradication of, ; punishments for,\n\nvirtues, , , , , ; rewards for, ; Stoics on, n65; treaties and,\n\nwarfare, ix, x, xvi, , ; colonialism and, ; corruption of morals and, ; crippled veterans of, \u201320; priests and, , ; princes and, ; slavery and, ; thieves as soldiers, \u201321; Utopian military practices, \u201315; women and, \u201312\n\nwealth, , \u201386,\n\nwomen, x, , , ; age at marriage, , n225; in church, ; idleness and, ; and meal preparation, ; and military training, ; occupations of, ; as priests, , n251; warfare and, \u201312; wives, , , ,\n\nZapoletes (Utopian mercenaries), x, xv, \u201311\n\nZiegler, T., xxi\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n### HOT MAMA\n\nby\n\n### Jennifer Estep\n\nBook Two in the Bigtime series\n\nHOT MAMA\n\nCopyright 2007, 2011, 2012, 2013 by Jennifer Estep\n\nExcerpt from **Jinx**\n\nCopyright 2008, 2011, 2012, 2013 Jennifer Estep\n\nThis is a work of fiction. Names, characters, places, brands, media, and incidents are either the product of the author's imagination or are used fictitiously. Any resemblance to actual or fictional characters or actual or fictional events, locales, business establishments, or persons, living or dead, is entirely coincidental. The fictional characters in this story have no relation to any other fictional characters, except those in works by this author.\n\nAll rights reserved by the author.\n\nSmashwords Edition\n\nISBN: 9781465797452\n\nCover Art by The Killion Group\n\nDigital Formatting by Author E.M.S.\n\n# Other books by Jennifer Estep\n\nThe Bigtime series\n\nKarma Girl\n\nHot Mama\n\nJinx\n\nA Karma Girl Christmas (holiday story)\n\nNightingale\n\n~\n\nThe Elemental Assassin series\n\nBooks\n\nSpider's Bite\n\nWeb of Lies\n\nVenom\n\nTangled Threads\n\nSpider's Revenge\n\nBy a Thread\n\nWidow's Web\n\nDeadly Sting\n\nHeart of Venom\n\nThe Spider\n\nPoison Promise\n\nE-novellas\n\nThread of Death\n\nKiss of Venom\n\n~\n\nThe Mythos Academy series\n\nBooks\n\nTouch of Frost\n\nKiss of Frost\n\nDark Frost\n\nCrimson Frost\n\nMidnight Frost\n\nKiller Frost\n\nE-novellas\n\nFirst Frost\n\nSpartan Frost\n\n# Table of Contents\n\nCopyright\n\nOther books by Jennifer Estep\n\nHot Mama\n\nDedication\n\nPART ONE\u2014WEDDING BELLS\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\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nPART TWO\u2014THE HONEYMOON'S OVER\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nPART THREE\u2014BREAKUP BLUES\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nEpilogue\n\nExcerpt from JINX\n\nAbout the Author\n\n# HOT MAMA\n\nFiona Fine is the hottest fashion designer in Bigtime, N.Y.\u2014literally. That's because she moonlights as Fiera, a superhero with superstrength and volatile, fire-based powers. As Fiera, she's also a member of the Fearless Five, the city's most powerful and popular superhero team.\n\nHowever, Fiona's been through a lot lately, including the death of her fianc\u00e9, who was murdered by an ubervillain. But Fiona is ready to move on with her life, so it seems like good karma when she meets sexy businessman Johnny Bulluci at a friend's wedding.\n\nBut Fiona has little time to think of love thanks to Siren and Intelligal\u2014the city's newest ubervillains who crash the wedding and then go on a city-wide crime spree. Fiona doesn't know exactly what the ubervillains are up to, but if she doesn't figure it out, she's the one who just might go up in flames this time....\n\n# Dedication\n\nTo my mom, the best person I know.\n\nAnd to Andre, who coined the phrase \"So, are you going to eat that?\"\n\n# PART ONE\n\nWEDDING BELLS\n\n# Chapter One\n\nMy wedding day.\n\nIt was supposed to be the happiest day of my life. A time of joy and celebration and new beginnings. The day every girl dreams of from the time she's old enough to play dress-up in her mother's clothes.\n\nIt was exactly that sort of day.\n\nJoy. Hope. New beginnings.\n\nBut it wasn't _mine_.\n\nCarmen Cole twirled in front of the full-length mirror. Her white satin wedding dress swung out in an arc then gathered back in on itself. Thousands of crystals dotted the fitted bodice and full skirt, giving the dress a shimmering, ethereal air. A matching crystal necklace sparkled like a ring of stars around her neck.\n\n\"How do I look, Fiona?\" Carmen turned her blue eyes to mine.\n\nI hated to admit it, but Carmen looked fantastic. Absolutely fantastic. A rosy flush tinted her cheeks. Excitement brightened her eyes. Even her auburn hair glistened underneath her simple lace veil.\n\n\"You look fabulous. After all, you're wearing a Fiona Fine original.\"\n\nCarmen frowned at her reflection. \"I know it's one of your more subdued designs, but I still think it's a little much.\"\n\nI crossed my arms over my chest. An errant spark flew from my thumb and landed on the beige carpet. I squashed it with my stiletto. A little much? Please. If Carmen had gotten her way, she would have worn holey jeans, worn-out sneakers, and a ratty T-shirt with some cutesy saying on it to the wedding.\n\nLuckily, hotter heads had prevailed. Mine. Then again, it was easy to get your way when you had the ability to shoot fire out of your fingertips. Getting my way was one of the prime benefits of being a superhero. My favorite benefit.\n\nJust because I moonlight as a superhero doesn't mean that I can't be a little selfish **\u2014** or enjoy the perks of having superpowers. Usually, I'm perfectly happy just being Fiera, one of many superheroes in Bigtime, New York, fighting evil, cracking skulls, and making life miserable for all those pesky ubervillains who want to take over the city, then the world. But every once in a while, I enjoy showing off my fiery skills, especially when it's for the greater good, such as making sure Carmen didn't look like a bag lady at her own wedding.\n\nA knock sounded on the door, the knob turned, and Lulu Lo zipped her motorized wheelchair into the room. A royal-blue dress covered her slender form, bringing out the smoothness of her porcelain skin and the cobalt streaks in her spiky, black hair. Since we were both bridesmaids, I wore a matching gown, but with a few modifications\u2014a lower bodice, a tighter fit, and a higher slit up the side.\n\n\"Nice dress, Sister Carmen.\" Lulu whistled. \"That'll make Sam sit up and take notice.\"\n\nCarmen grinned. Another spark shot out from my thumb. Sam had already taken plenty of _notice_ of Carmen, despite my efforts to the contrary. The two of them were always sneaking off to have wild sex in some corner of the manor house.\n\n\"Of course Sam will notice,\" I snapped. \"I designed the dress. Ours too, if you'll remember. They're all fabulous.\"\n\n\"Well, you do look very _hot_ , Fiona.\" Lulu laughed.\n\nI glowered at Lulu. Just because I was a member of the Fearless Five, one of the most esteemed superhero teams in the world, didn't mean that I didn't get snarly from time to time\u2014or that civilians like Lulu had the right to poke fun at me.\n\nOf course, none of this would be happening if Carmen, aka Karma Girl, hadn't insisted that we tell Lulu our secret superhero identities. Carmen had argued that Lulu deserved to know the truth since she'd helped save us from the Terrible Triad, a group of ubervillains. Lulu was also the main squeeze of Henry Harris, aka Hermit of the Fearless Five, and he'd wanted to tell her the truth as well. The other two members of the Fearless Five, Sam \"Striker\" Sloane and Sean \"Mr. Sage\" Newman, had agreed with Carmen.\n\nSo the four of them told Lulu everything, despite my protests. Once the shock wore off, Lulu ingratiated herself with the rest of the Fearless Five. Now, everybody else treated her like one of the gang. She even had her own room in the top-secret, underground compound with the rest of us.\n\nI ignored Lulu whenever possible. It was bad enough that she knew our real identities. I didn't want to invite her any more into our lives. Lulu was a computer hacker. She did all sorts of highly illegal things, like breaking into the FBI mainframe and swapping corporate secrets, but nobody cared except me. Not even my father, the esteemed police chief of Bigtime, as well as a member of the Fearless Five.\n\nIn return for my blatant hostility, Lulu zinged me with heat-related puns whenever we crossed paths. _Fiona's hot. Fiona's smokin'. Fiona's on fire._ Like I hadn't heard them all a hundred thousand times before. Ha, ha, ha, ha. Lulu could have at least come up with something original, if she was going to mock me on a daily basis.\n\nMy eyes fixed on Lulu's hair. I could turn those blue streaks red in a heartbeat. Heat pulsed through my body. My fingers twitched. Just one little spark...\n\n\"Fiona,\" Carmen warned. \"There will be no flare-ups today. You promised Sam.\"\n\nI had promised Sam. And my father. And Henry. And even Carmen. Three times each. I let go of the fire coursing through my veins and banked it deep inside me. It didn't matter anyway. Carmen would have just done her empathy thing and used the ambient energy in the room to buffer Lulu and herself from my heat. Carmen had the ability to tap into other people and use their own energy against them. I hated her power, mainly because I hadn't figured out a way to counteract it yet. Most of the time, I either punched or flamb\u00e9ed my way through danger. But I couldn't do that with Carmen, because she gave just as good as she got.\n\nLulu smirked at me and motored away. She'd probably max out my credit cards or do some other devious, identity-theft thing as soon as the wedding ended. I didn't know what Henry saw in her. Maybe he was just glad that he'd finally found someone who understood all the techno-babble he spouted on a daily basis.\n\nLulu left the door open, and classical music drifted in, along with the murmur of distant conversations. I eyed the clock on the wall. Five minutes to go. Good. The sooner this spectacle was over with, the better. I wasn't in the mood for a wedding today. Not any day. Not anymore.\n\nCarmen picked up on my dark thoughts and stared at me in the mirror. \"I know this has been hard for you, Fiona. The engagement, the wedding, everything. I'm sorry. I wish things were different. I wish Tornado were still here...\"\n\nHer soft, Southern twang trailed off under my hot gaze. Hard for me? She had no idea.\n\nIt'd been more than a year since my fianc\u00e9, Tornado, had been murdered. Carmen had exposed the superhero's secret identity as Travis Teague to the world, including our archenemies, the Terrible Triad. The ubervillains had killed Travis and used Carmen to get to the rest of us. We'd been captured, stuffed in glass tubes, and almost sucked dry of our superpowers before Carmen had saved us by getting dumped into a vat of radioactive goo and developing superpowers herself.\n\nSometimes, I couldn't believe the irony of it. Carmen exposing superheroes, becoming one herself, and now marrying one. Things never seemed to turn out the way you thought they would, especially in Bigtime.\n\nMostly though, I still couldn't believe that Travis was gone. Forever. My heart twisted, and the burning fire inside me flickered and dimmed. My eyes dropped to the square, diamond engagement ring on my finger. Travis had given it to me a week before he'd died. I hadn't taken it off since.\n\n\"Fiona? Are you okay?\" Carmen asked.\n\nI wasn't. Not even close. But this was Carmen's big day, and I didn't want to ruin it for her.\n\n\"I'm fine,\" I lied. \"In fact, I was thinking that it's time for me to get out and start dating again. I've done the men of Bigtime a cruel, heartless injustice, depriving them of my fabulous company all this time.\" I tossed my long, blond hair over my shoulder for effect.\n\nCarmen's face lit up like I'd just hit her with a fireball between the eyes. \"That's wonderful, Fiona! Just wonderful!\"\n\nHer blue eyes grew cloudy and distant, the way they always did when she was listening to the strange whispers in her head. Carmen called them her inner voice, her instincts. I thought she had more than a few loose rocks rattling around in all that empty space.\n\n\"Maybe you'll meet somebody at the reception,\" she murmured.\n\nI huffed. Please. I'd been active on the social scene ever since I'd moved to Bigtime some fifteen years ago, and I knew everybody invited to the wedding. There wasn't a man among them that I'd date, let alone sleep with.\n\nI twisted the ring on my finger. The silver solidium band heated up on my hot hand, and the diamond glowed like a tiny moon. Still, I would like to find somebody. It'd be nice to be part of a couple again. To laugh and talk and have dinner with someone who wasn't a relative or an employee or a fellow superhero. To find somebody who looked at me the way that Sam looked at Carmen.\n\nPlus, I liked sex. A lot. It sucked to go without.\n\nMy hand stilled. Maybe that's what I should do. Get drunk at the reception, have a one-night stand with some anonymous guy to take the edge off, and then start looking for someone suitable. Someone more long-term. The only problem with my plan was that it would take an ocean of champagne to get me drunk, given my fast-burning metabolism. Well, it was a good thing Sam was richer than almost everyone else on the planet put together. He could afford a couple hundred thousand dollars' worth of bubbly if it meant me getting lucky.\n\nThe music quickened and swelled, and the conversations faded away. The air hummed with energy and anticipation.\n\n\"Time to go.\" Carmen smoothed down her billowing skirt. Her hand trembled just a bit.\n\nI picked up her long train, careful not to singe the fabric with my fingers. I'd spent too much time sewing the damn thing to ruin it now. Carmen turned and grabbed my arm.\n\n\"Do you think this is the right thing to do? Do you think we should go through with it? Do you think we're ready? You know how badly my last wedding turned out.\" Panic filled her blue eyes.\n\n_Badly_ was the understatement of the century. Right before the wedding, Carmen had found her fianc\u00e9 boinking her best friend and discovered that the two were her town's resident superhero and ubervillain. That, of course, had set Carmen off on her little mission to expose the identity of every superhero and ubervillain who crossed her path. Which, of course, was how Carmen had met Sam and the rest of us. Karma, she called it. Destiny, kismet, fate. I just thought of it as bad luck on our part.\n\nBut I bit back the sarcastic retort I'd been ready to let loose. The nosy reporter had grown on me, despite my best efforts. And she had saved my life and everyone else's. I owed her for that. Plus, it was my solemn duty as a bridesmaid to support the bride **\u2014** even if Carmen occasionally made me want to put my fist through a wall.\n\n\"Do you love Sam?\"\n\nCarmen nodded. Some of the tension left her body. \"With all my heart.\"\n\n\"Then, it'll be fine,\" I said. \"Sam loves you, and you love him. You're going to have a fabulous wedding, a fantastic honeymoon, and a wonderful life together. Plus, you're wearing a Fiona Fine original couture gown. And what could possibly be better than that?\"\n\n# Chapter Two\n\nAfter Carmen calmed down, we made our way through Sublime, Sam's mansion on the outskirts of Bigtime. Roughly the size of a small country, the manor house contained just about every antique and art object known to man and superhero. Polished suits of armor, colorful paintings, detailed sculptures, exquisite tapestries. Even though I'd been prowling the halls for years now, the rich furnishings still impressed me. And it took a lot to impress me.\n\nCarmen tiptoed her way through the manor, struggling to stay upright in her towering heels. I stalked along behind, holding up the train so it wouldn't get snagged on a piece of furniture, and wishing the bride-to-be would move a little faster. I could always zing Carmen with a hot flash. That would get her moving. But I couldn't risk ruining the gown. Not now. I'd already had to redo it twice because of some temperamental flare-ups on my part.\n\nCarmen stumbled on the edge of an Oriental rug and almost fell on her face.\n\n\"Stupid shoes,\" she muttered, glaring at me.\n\nHer three-inch strappy sandals had been another hotly contested point between us. Carmen had wanted to wear these ugly, flat, ballet slippers that had gone out of style twenty years ago. I'd told her point-blank that she wasn't wearing those monstrosities with a Fiona Fine wedding gown. She wasn't ruining my hard work with her fashion faux pas. Of course, I had to roast most of Carmen's worn-out tennis shoes before she agreed to wear the sandals, but the important thing was that I, and designer fashion, had triumphed yet again.\n\n\"I don't see how you walk in these things,\" Carmen said, tugging on one of the white straps wrapped around her ankle.\n\n\"It's easy,\" I snapped. \"Millions of women do it every day. Now quit whining and walk. You've got a superhero to marry.\"\n\nCarmen gave me another annoyed glance, but she shuffled forward. Despite Carmen's time covering the Bigtime society beat, fashion wasn't her forte. That was my domain. And I knew, like all good designers know, that no outfit is complete without a pair of killer shoes, preferably stilettos. The higher, the better.\n\nAfter a couple more stumbles and a string of curses, we stopped in front of a pair of doors that led outside. Lulu sat there waiting, along with Chief Sean Newman. Sunlight streamed in through the glass, warming the alcove where we stood. The ceremony was taking place in the luscious gardens that surrounded the enormous estate, as befitting a traditional May wedding.\n\nI peeked through the doors. Henry Harris, the best man, and Sam Sloane, the anxious groom, had already taken their places in front of the minister at the far end of the long aisle.\n\n\"Carmen, Fiona,\" Chief Newman rumbled in his deep Irish brogue. \"You both look beautiful.\"\n\n\"Chief.\" I kissed my father on the cheek.\n\nI'd always called my father _chief_ , ever since I was a little girl and had first seen him in his police uniform. It was a habit I continued out of necessity. Nobody knew about our family connection, except the other members of the Fearless Five and now Lulu. It was safer that way. Since people, and more importantly ubervillains, didn't know about our relationship, they couldn't kidnap and use us against each other, either in our real lives or as our superhero alter egos, Fiera and Mr. Sage.\n\nThe first notes of the classic wedding march sounded, and the hundreds of guests outside rose to their feet.\n\n\"Here we go,\" Carmen whispered, her face pale and slightly sweaty.\n\n\"Heaven help us all,\" I muttered.\n\nTwo ushers stepped forward, and the double doors creaked open. The chief took Carmen's arm, while Lulu motored out. I waited for her to get halfway to the end before stepping outside and strolling along the rose-covered aisle.\n\nA thick, white carpet stretched three hundred feet toward a raised dais, which stood beneath an enormous trellis strung with silver roses and purple pansies. Oak and elm trees hovered in the background, providing splashes of green to the scene. Men and women dressed in dark tuxedos, sparkling gowns, and flashing jewels stood on either side of the aisle, adding even more color to the gardens. The sweet, thick scent of the flowers mixed with the men's and women's spicy colognes and heady perfumes. A full orchestra sat on one side of the dais, but their instruments couldn't quite drown out the low, steady drone of bumblebees in the distance. The sun hung like a ball of orange sherbet in the evening sky, and a faint breeze ruffled my long hair. Late spring was a perfect time for a wedding.\n\nMy blue eyes traced over the area, drinking in the sights and sounds. Everything that I'd wanted for my own wedding was right here. Everything except Travis. I clutched my bouquet of roses and turned my engagement ring around.\n\nI took my place next to Lulu and winked at Sam, trying not to look as sad as I felt. I was happy for him and Carmen, truly I was. But I couldn't stop the pang of longing and jealousy that stabbed my heart with every beat of the soaring music.\n\nThe businessman-superhero gave me a nervous grin and tugged on his sleeves. With his dark hair and light, grayish eyes, Sam looked like he'd just stepped out of a men's magazine.\n\nSo did Henry, for once. His glasses gleamed in the late-afternoon sunlight, along with his mocha skin. He flashed me a shy smile and yanked on his tie. Like Carmen, Henry would have worn his usual attire to the wedding\u2014a rumpled sweater vest, plaid pants, and a polka-dot bow tie\u2014if I hadn't intervened. I'd had to melt a couple of the wires attached to his precious computers, but Henry had eventually seen things my way. Most people did, sooner or later. Especially when I turned up the heat.\n\nThe music swelled to a thundering crescendo, and Carmen stepped into view. Sam's face grew soft and dreamy and dopey at the sight of her. I smiled. Too much? Please. Score another one for Fiona Fine Fashions.\n\nMy father escorted Carmen down the aisle. She beamed like a neon light. If she got any happier, she'd blow a bulb. A few people tried to catch her gaze, but the only one Carmen acknowledged was her friend Jasper, Bigtime's resident mad bomber. Then, Carmen turned her attention to Sam, and the rest of the world fell away. She only had eyes for him, and he for her. The two of them couldn't stop staring at each other. Cold envy frosted over my aching heart.\n\nThe music faded away, and the minister stepped forward.\n\n\"Dearly beloved, we are gathered here today...\"\n\n* * *\n\nThirty minutes later, Carmen and Sam professed their true and undying love for each other. They kissed, everyone cheered, and the wedding ended. I half expected to see white doves and fluffy bunnies stampede down the long aisle, but thankfully, people just blew soap bubbles at the happy couple.\n\nThe blushing bride, handsome groom, and other members of the wedding party stayed in the garden to pose for the all-important, post-ceremony pictures. Meanwhile, the rich, distinguished guests drifted out onto the mile-wide lawn where hundreds of tables, gallons of champagne, and thousands of pounds of chicken cordon bleu and other choice foodstuffs had been set out. Waiters and caterers zipped through the crowd, dishing up food and drinks as fast as they could.\n\nI spotted Kyle Quicke among the caterers. I'd gone to Bigtime University with Kyle, and his family owned Quicke's, one of my favorite restaurants in Bigtime. I watched Kyle cut a slice of chocolate cake and hand it to a waiting guest. My stomach rumbled. I could use some food. A lot of food.\n\n\"Come on, Fiona. It's time for the pictures,\" the chief murmured in my ear, dashing my hopes of a stealthy getaway.\n\nI rolled my eyes and turned back toward the dais.\n\n* * *\n\nForty-five minutes later, I tapped my fingers on my arm. \"How many pictures are you going to take?\" I snapped at the photographer. Patience was not one of my virtues. In fact, I didn't think it was any sort of virtue at all.\n\nThe short, fat man wilted under my hot gaze. Most everybody did. My eyes went to his digital camera. One little flare-up, and it would look like something that came out of a kiln. There'd definitely be no more pictures then....\n\n\"Fiona.\" The chief's blue eyes narrowed in warning.\n\nI shot him a sour look and dampened my temper. That was the problem with your father being a psychic superhero. You could never get away with anything.\n\n\"Actually, I'm done with the wedding party. I just want to get a few more shots of the bride and groom,\" the photographer said.\n\n\"It's about bloody time,\" I muttered.\n\nI stomped out of the garden and onto the lawn. My stilettos sank into the damp earth, but I yanked up my feet and kept going. Another benefit of having superstrength. Henry, Lulu, and my father followed at a more sedate pace.\n\nMost of the male guests had settled down at the tables and were busy stuffing themselves with chicken, spring vegetables, and fancy sourdough rolls baked in the shape of curling vines and flowers. Not so for the women. Every woman under sixty gathered around the wide, long bar, downing glass after glass of champagne. Some stared at nothing in particular. A few snuffled into their crystal flutes, while others dabbed at their runny mascara. They weren't tears of joy. Quite the contrary. Three hundred women's dreams of becoming Mrs. Sam Sloane had just been dashed by a lowly society reporter. It was more than enough to make the stoutest society matron weepy, depressed, and drunk as a skunk.\n\nI shoved my way through the crowd, zapped a few people with hot flashes to make them move, and grabbed the biggest champagne glass I could find. Bubbles fizzed up in the golden liquid. I drained the glass and twitched my nose to ease the sudden tickling sensation. I didn't need to sneeze flames in front of a thousand people.\n\n\"Fiona! Fiona Fine!\"\n\nI turned at the sound of my name and spotted Joanne James fluttering her hand at me. Joanne was a tall, skinny woman with a rather large chest and too-smooth features. Hair blacker than shoe polish brushed her slender shoulders, contrasting with the ropes of pearls that encircled her gaunt neck.\n\nOh boy. I was going to need something a lot stronger than champagne. I ordered a gin and tonic from one of the bartenders and downed two doubles before Joanne sashayed her way through the crowd to my side.\n\n\"Fiona, darling!\"\n\n\"Joanne, darling!\"\n\nWe air-kissed the way that society women are supposed to. My eyes raked over Joanne's outfit, a gunmetal-gray halter dress with a slinky, sequined skirt. Not bad, but it wasn't one of mine. Joanne also sized me up, her gaze critiquing everything from my shoes to my chandelier earrings. Standard operating procedure among the women in Bigtime. Once that was out of the way, we made polite chitchat about how fabulous and thin we both looked before Joanne got down to business.\n\n\"I'm getting married again next month. I was wondering if I could come in and talk to you about another gown.\"\n\nI almost choked on my drink. \"What will this one be? Number five?\"\n\nJoanne James went through husbands like they were tissues\u2014she used one up and then tossed him aside for another.\n\n\"Six actually.\"\n\n\"Who's the lucky fellow?\" My lips only twitched a little. It was amazing the things you could say with a straight face.\n\n\"Berkley Brighton, the whiskey billionaire.\"\n\n\"Of course.\"\n\nJoanne James didn't waste her time on small-time fish. She only went after the big, big catches. She was Bigtime's resident black widow, but she didn't kill her husbands. Instead, Joanne bled her hubbies dry, added their money to her own considerable fortune, and somehow managed to wiggle out of paying for anything\u2014even her own divorce attorneys. In a way, it was her own personal superpower. Joanne was a legend in Bigtime, and more than a few society mamas urged their daughters to emulate her marriage merry-go-round.\n\n\"I thought you would have asked Bella Bulluci to design your gown. She did your last two, didn't she?\" I couldn't resist the dig. Joanne had gone to Bella after complaining that my services were too expensive. As if. Fashion genius like mine was priceless.\n\nJoanne swallowed some champagne. \"To be honest, I did ask Bella first, but she turned me down.\"\n\n\"Really? Why?\"\n\nJoanne waved her hand, and I squinted at the sudden glare. The diamond boulder on her finger could easily feed the people of a third world country for years. \"Oh, she said she was taking some time off to concentrate on family affairs because of her father's death. I think she just didn't want to bother with it.\"\n\nBella Bulluci was one of Bigtime's most popular designers, next to me, of course. Bella had plenty of talent, but I'd always thought her creations were a bit conservative, tame even. Bella was very fond of solid colors and subtle pinstripes. I was more of a polka-dot, plaid, and leopard-print girl. All rolled into one. With neon sequins and a feather boa to match.\n\nI signaled the bartender and ordered another drink. I didn't like being anyone's second choice, but my eyes strayed back to Joanne's ring. That thing had to be at least ten flawless carats.\n\nBeing a superhero had plenty of perks, but there was one major drawback\u2014it was a pricey occupation. We all had to pitch in to keep the Fearless Five out and about fighting crime. Carmen and Henry didn't contribute much to our annual budget. They couldn't with the pitiful paychecks they earned as newspaper reporters at _The Expos_ _\u00e9_. My father wasn't much better off. Even though he was the chief of police, his salary wasn't what it should be, mainly because most of the city's budget went to repairing the municipal buildings, bridges, and overpasses that us superheroes and ubervillains obliterated during our epic battles.\n\nThat left Sam and me to shoulder the monetary load. With his various business interests and billion-dollar bank balance, Sam gave the most for the greater good. But I chipped in at least five million every year. Sometimes more. Outfitting Joanne James with wedding gown number six would keep us all in black leather and orange-red spandex for the foreseeable future.\n\n\"Have your assistant call the store, and we'll set something up for later this week,\" I promised and downed my third gin and tonic.\n\nJoanne smiled, her lips lavender against her pale face. \"Fiona, darling, it's always a pleasure doing business with you.\"\n\nWe air-kissed again and exchanged more meaningless pleasantries. Then, Joanne strutted back through the crowd to Berkley Brighton, a short, square man who'd made his fortune selling Brighton's Best whiskey. Joanne latched on to his arm, and the pretty young things who'd been clustered around the boisterous billionaire scattered like minnows fleeing a hungry barracuda. Joanne wasn't someone you wanted to mess with\u2014especially when she was husband hunting. Berkley actually beamed at Joanne, happy to see his honey.\n\nI snorted. Poor guy. He might as well just sign over his family's secret whiskey recipe to Joanne right now. It would save him a lot of trouble and hefty lawyers' fees down the road.\n\nWhile I'd been talking to Joanne, Carmen and Sam had joined the festivities. They walked from table to table, greeting the wedding guests and basking in the afterglow of the ceremony. After paying their respects to the bride and groom, people finished their dinners and drifted out onto the tile dance floor that had been planted on the lawn for the grand occasion. A twelve-piece band to one side of the floor played a loud, brassy version of \"The Right Thing To Do\" by Carly Simon, Carmen's favorite singer.\n\nMy eyes scanned the glittering crowd. Joanne and Berkley. Carmen and Sam. Henry and Lulu. Even my father was dancing with one of Bigtime's rich, lonely widows. Couples, couples, everywhere. But no Travis.\n\nNo Travis.\n\nThe happy society scene and all the couples burned me out. I needed some peace and quiet. Now.\n\nI shoved through the crowd, wrenched open a side door, and stomped inside the manor. The usual rich, shiny trappings greeted me, but for once, I didn't pay attention to them. Sam wouldn't like it if I accidentally melted some ancient knight's suit of armor or fried another one of his Monets. The mood I was in, they'd go up like dry newspaper.\n\nThe music and laughter and happy sounds faded away, replaced by the _thwack_ of my heels on the hardwood floors. I walked into one of the many game rooms that populated the manor and sank down onto the smooth leather couch. A big-screen TV took up one wall, while a pool table crouched in the middle of the floor. Dart boards and various other sports-like contraptions filled the rest of the area, but I didn't really see them. I didn't see any of it.\n\nI twisted the ring on my finger. It wasn't nearly as big as Joanne James's was, but it meant the world to me, even now. Travis. My heart squeezed like a dishrag being wrung out.\n\n\"A beautiful bridesmaid alone by herself. What a sad, sad clich\u00e9,\" a low, cultured voice called out.\n\nI looked up. A man stood in the doorway. He topped out at just over six feet, with a mane of tawny blond hair that curled around the collar of his impeccable tuxedo. Flashing green eyes contrasted with his golden skin, making him look like a sleek lion in the gathering shadows. He strode into the room, his black suit flowing with easy grace around his perfect figure. It fit him well. Then again, just about anything would have looked good on him.\n\nMy eyes widened. If Sam resembled a male model, then this guy was the Goliath of male models. Yummy.\n\nThe man stared at me, and his eyes crinkled in amusement. The merriment dancing in his sharp gaze made him look that much better, even if he seemed to be making fun of me. I didn't like people making fun of me, and I especially didn't like being looked down on. I got to my feet and tossed my long hair back. With my stilettos, he only had half an inch on me.\n\n\"I'm not a clich\u00e9,\" I snapped.\n\n\"Really? You were one of the bridesmaids, right?\"\n\n\"Yes.\"\n\n\"And you're sitting here all alone.\"\n\n\"Yes.\"\n\n\"And you certainly are beautiful.\"\n\n\"Oh, yes.\"\n\nModesty is another one of my nonexistent virtues. On a scale of one to ten, I'm a solid eight and a half. With my blond hair, blue eyes, and up-to-there legs, I've got the Barbie look men love down pat. The only problem is they think I'm as dumb as one of the plastic dolls. The same thing goes for my alter ego, Fiera. But more than one ubervillain had gotten badly burned by underestimating me.\n\nStill, the compliment pleased me. Every woman likes to be told she's beautiful, but coming from Mr. Model, it sounded...better. Truer. Sexier.\n\n\"If all that's not a clich\u00e9, then I don't know what is.\" His voice was deep with a hint of an accent I couldn't quite place. White teeth gleamed in his tan face, adding to his already staggering sex appeal.\n\nI crossed my arms over my chest and flipped through my mental Rolodex of Bigtime society players. No match. He must be new in town. I certainly would have remembered him. My eyes drifted over his suit, which draped perfectly over his broad shoulders and chest. Oh yeah. I would have remembered him.\n\nI suddenly realized that I was twisting the ring on my finger. Bloody hell. I'd gone from pining over Travis to ogling a complete stranger in the space of a minute. I really did need to get lucky before my hormones made me have a total meltdown. Literally.\n\nThe man continued. \"You certainly looked sad and lonely sitting there, staring into space.\"\n\n\"I was doing nothing of the sort.\"\n\nI couldn't tell him that I'd been looking at the ring my murdered fianc\u00e9 had given me before he'd died. My pain was my own. I didn't go blabbing about it to strangers. Besides, no one except the Fearless Five had even known Travis and I were engaged. It was another little secret we'd decided to keep to ourselves.\n\n\"I was just taking a break from the festivities,\" I replied in my best cool, bored society voice. \"All that happiness can be a bit grating after a while.\"\n\n\"Really? You know we could create our own festivities, you and me.\"\n\nI stifled a laugh. That was one of the lamest lines I'd ever heard. \"Really? And how could we do that?\"\n\n\"Let me show you.\"\n\nHe flashed me a devilish grin, pulled me into his arms, and planted his lips on mine.\n\n# Chapter Three\n\nFor a moment, I couldn't believe it. Who the bloody hell did this jerk think he was, kissing me? I was Fiera, for crying out loud. Superhero du jour. Protector of the innocent. Defender of democracy. I could snap his neck like a pretzel stick. I could light his ass up like a firecracker with a mere thought.\n\nI thought of doing both\u2014at the same time. Then, something strange happened.\n\nI realized that I liked kissing him.\n\nA lot.\n\nA whole lot.\n\nHe had fantastic lips. Soft, firm, smooth. He tasted like fizzy champagne and smelled of some subtle, spicy soap. The combination made my head spin more than the three drinks I'd just had.\n\nAt five foot nine, I'm no small, petite thing, but I felt dwarfed by him. His sculpted chest felt like sun-warmed stone under my hands, and his heart thumped under my clenched fingers. His arms held me securely in place. I opened my mouth to tell him something, I wasn't quite sure what, and he dipped his tongue in. The taste of him, the feel of his mouth, his tongue on mine overwhelmed my senses. I felt like I'd been zapped with a couple dozen stun guns\u2014all weak and twitchy.\n\nHe plundered my mouth like a pirate seeking buried treasure. Nibbling my lips. Skimming my teeth. Probing with his tongue. I couldn't resist him, and I didn't really want to. I'd been thinking about having a one-night stand. Let's see how Mr. Model measured up.\n\nSo I opened my mouth wider, and my tongue met his. Then, I went on the offensive. Nibbling on _his_ lips. Skimming _his_ teeth. Probing with _my_ tongue. He pulled me closer until I thought we would melt into each other. I certainly felt like I was on fire in more ways than one.\n\nHis fingers skimmed my neck and traced down to the tops of my breasts. He slid his hand inside the scooped neckline and stroked my chest. My nipples sprang to attention. His other hand went through the slit in the side of my dress and moved up my leg with quick, sure purpose. Damn, he didn't waste any time. Smooth, sexy, and bold. I loved it. Absolutely loved it.\n\nA warm, pleasant tingling started between my thighs and spread throughout my body. My stomach quivered the tiniest bit, and my breath came in soft gasps. My hormones had already kicked into overdrive. If he kept this up much longer, I'd have to throw him onto the couch instead of through the wall. Or on the floor. Or maybe on the pool table. It looked sturdy enough\u2014\n\n\"Ahem.\"\n\nA cool, feminine voice dampened the liquid fire burning inside me.\n\n\" _Ahem_.\"\n\nThe man sighed in disappointment against my mouth. He pressed his lips to mine once more, withdrew his hands, and stepped back. I tried not to stagger. I felt like I was drunk. And I never got drunk.\n\n\"Hello, Bella,\" the man said. Regret tinged his deep voice, and his hot green gaze burned into me.\n\nThe intensity of his stare made me shiver, a rare thing for me.\n\nI turned my head. Bella Bulluci hovered just inside the door. A forest-green dress of her own design clung to her curvy body, while a simple silver necklace accentuated her graceful neck. A small pair of angel wings dangled from the end of the chain. Bella's foot tapped out a rapid pattern, smacking onto the wooden floor.\n\n\"Fiona.\"\n\n\"Bella.\"\n\nI not-so-discreetly yanked the bodice of my dress back into its proper place. It didn't have far to go, really.\n\nBella's hazel eyes flicked to the man beside me. \"Well, I see you've met Johnny.\"\n\n\"Johnny?\" So that's what his name was.\n\n\"Johnny Bulluci. My older brother.\"\n\n\"Your brother?\"\n\n\"Guilty as charged, I'm afraid.\" Johnny strolled over to his sister and planted a chaste kiss on her cheek.\n\nMy eyes zipped back and forth between them. Side by side, the resemblance was obvious. Same tawny hair, same golden skin, same killer cheekbones. The only difference was their eyes. Bella's were a soft hazel while Johnny's were as green as polished jade. Johnny also had almost a foot on his sister. Bella was a bit on the short side.\n\nBella looked at me, then her brother. She shook her head and looped an arm around his waist. \"Seducing another unsuspecting bridesmaid, brother dear?\" she asked in a teasing tone.\n\n\"Make a habit of it, do you, Johnny?\" I asked, smoothing down my skirt.\n\n\"Only with the exceptionally beautiful ones.\" He winked.\n\nI crossed my arms over my still-tingling chest. His sister had just caught us making out like a couple of sex-starved teenagers, and the man still had the nerve to wink at me. Johnny Bulluci had no shame. I rather liked that about him.\n\n\"Johnny's moved back home from Greece,\" Bella explained. \"He's been looking after our business interests overseas.\"\n\nSo that's why I hadn't seen him around Bigtime before. It explained the accent too.\n\n\"He arrived in town several weeks ago, and I thought the wedding would be a perfect opportunity for him to meet and mingle. I didn't know he was going to disappear. I should warn you, Fiona, my brother is a notorious playboy. Always has been.\" Bella's voice was light, but there was a hint of disapproval to it. Her foot kept up its annoyed, tapping pattern.\n\n\"No harm done,\" I said in a cool voice and flipped my hair over my shoulder. \"Johnny and I were just getting better acquainted.\"\n\n\"Indeed,\" Johnny said, his eyes catching mine again. \"Indeed\u2014\"\n\nFrantic screams slashed through the air. A bright white light flashed outside, followed by a crack of thunder. The ground trembled as though it were going to split open beneath our feet.\n\nWe stood there, stunned.\n\n\"Grandfather!\" Bella shouted.\n\nShe ran out the door. Johnny gave me another quick, regretful look and rushed after her.\n\nI stuck my head out into the hall to make sure that they weren't going to double back and drag me with them, but the pair had already disappeared from sight. Damn, Bella could scoot in those heels. Carmen could definitely learn a thing or two from her.\n\nI locked the door to the game room, ran to the pool table, and twisted a knob hidden underneath one of the corners. The green felt top slid back with a whisper, revealing a row of neatly folded costumes, masks, and shoes. The pool table was one of many objects throughout the manor that held more than just a good time.\n\nIt took me about thirty seconds to ditch the bridesmaid dress, stilettos, and earrings and shimmy into my costume. The orange-red spandex molded to my body like plastic wrap, familiar and comfortable as always. I shoved my feet into my chunky, steel-toed, kick-ass boots and slapped on my flame-shaped mask. Then, I stuffed my dress and heels into the pool table and twisted the knob. The hidden compartment disappeared from sight.\n\nThe screams had quieted down, but cracks of thunder ripped through the air like gunfire. My hands curled into fists. Sparks flared to life in my hair. Flames licked at my fingers.\n\nUnless I missed my guess, it was time for Fiera to go kick some ubervillain ass.\n\nGoody.\n\n* * *\n\nI ran through the halls until I reached the doors that led outside, zoomed down the white carpet that had served as the aisle, and sprinted to the edge of the garden. A row of thick holly bushes separated the lush, flowering plants from the smooth lawn. I slid to a halt and peeked through one of the small gaps in the glossy greenery.\n\nBigtime's finest stood on the lawn, blank looks on their faces. They seemed not to notice the overturned tables, broken dishes, spilled food, and general chaos that surrounded them. Acrid smoke snaked up from an oak tree that had been split in two. Smoldering, snapped branches and shredded leaves littered the singed grass.\n\nA woman dressed in a neon blue suit stood in the middle of the dance floor. Siren. She had a tangle of black curls and an hourglass figure that would make even a blind monk look twice at her. The silver zipper on her suit was open almost to her navel, showing off enough buoyant cleavage to raise the _Titanic_. A mask in the shape of a zigzagged lightning bolt covered her bright blue eyes.\n\nBut she wasn't alone. A woman in a shiny silver suit hovered next to Siren in a floating chair. Intelligal. Thick, black goggles highlighted her dark eyes, making them seem as big and round as quarters. A silver cowl clung to her egg-shaped head, hiding her hair from sight, and her suit was zipped up to her neck. It was a good thing too. Intelligal didn't have the body to pull off the evil sexpot look. She had about as many curves as a board.\n\nSo, our new neighborhood ubervillains had decided to crash the wedding. I wasn't terribly surprised. Siren and Intelligal had come to town about six months ago, right after the members of the Terrible Triad had disappeared. The two had been raising hell ever since. Breaking into computer companies, stealing electronic and media equipment, robbing anyone and everyone they could get their hands on.\n\nBut even worse than the general mayhem the pair caused were their powers. It was bad enough that Siren could create powerful lightning bolts and energy balls with her bare hands. But her real weapon was her voice. One throaty whisper from Siren was all it took to turn most people into mindless, willing slaves. Her power especially seemed to work on men. Naturally. Of course, her looks didn't hurt either when it came to turning men into helpless puppets.\n\nIntelligal didn't seem to have any superpowers of her own, but she really didn't need them. Her Intellichair did the work for her. The device was a flying, open-topped tank that wrapped around the ubervillain's legs and torso. Equipped with heat-seeking missiles, machine-gun turrets, and a force field, the chair was like something out of a James Bond movie. Only better. The Intellichair was also scratch-proof, fire-proof, and virtually indestructible. Most things made out of solidium were. I knew. I'd been trying to melt and smash the chair to pieces for months with no success. I hadn't even been able to put so much as a dent in the dull gray metal.\n\n\"Ladies and gentlemen, I'm terribly sorry if I frightened you, but I'm afraid my invitation to the wedding got lost in the mail,\" Siren cooed into the microphone she'd taken from the band leader. \"And I just couldn't miss _the_ social event of the season.\"\n\nHer voice was low and husky and throaty. To the men, it whispered of sexual fantasies come to life. To the women, it was the conspiratorial tone of a trusted friend and confidant. Just about everyone in the crowd nodded their heads, accepting Siren's explanation for her little lightning storm.\n\nSiren's dulcet tones curled through the air, seeking out everyone within hearing distance. I felt the humming melody in her voice coiling around me like a boa constrictor, trying to squeeze my will into hers. Trying to make me one of her mindless lackeys. But I was too strong-willed and too temperamental to be manipulated by the likes of Siren. I grabbed on to the roaring fire inside me and concentrated on the heat. The coils burned away, and my head cleared.\n\nSo was Carmen. The blushing bride eyed the busty ubervillain with open hatred. Soot streaked her white dress, and her auburn hair had tumbled out of its elegant updo. Carmen's fingers twitched, and I knew she wanted to tap into Siren's powers and give the ubervillain a taste of her own medicine. But she couldn't. Not without blowing her cover as Karma Girl and ruining what was left of her wedding.\n\nI looked for the other members of the Fearless Five. Sam and Henry both wore slightly vacant, dazed expressions. Sam's tie had come undone, and Henry's glasses dangled on the end of his nose. They'd been getting better at resisting Siren's call, but her hypnotic voice still affected them. I snorted. Men. Please. They really were the weaker sex, in every sense of the word. The first time we'd gone up against her, Siren had convinced Sam aka Striker to attack me. It had taken one of my fireballs to the chest to bring him back to his senses. But since Striker regenerated, the only thing my fireball had really hurt was his pride.\n\nMy father looked bemused and a bit pained by the whole thing. Siren purred into the microphone again, and the chief massaged his temples. With his psychic powers, he was well beyond Siren's call, although her voice always left him with one hell of a headache.\n\nLulu sat next to the chief and gazed at Intelligal's chair. A calculating expression filled her smooth face, and her fingers tapped against the arm of her wheelchair like it was a keyboard. The computer hacker was probably wondering how she could get Henry aka Hermit to make her a similar contraption. I didn't blame her. Although I had little use for gadgets, given my natural superpowers, even I had to admit the Intellichair was a cool, deadly gizmo. Especially since it had been kicking our asses for the last six months.\n\n\"I do hope you all will forgive my rude outburst,\" Siren continued. \"I really want to make a good impression on everyone, being so new in town. I've heard such lovely, lovely things about Bigtime and all of you. I just want to fit in.\"\n\nI rolled my eyes. Good grief, she was simpering now. Did the woman have no pride? She was giving ubervillains everywhere a bad name. As much as I hated evildoers like Malefica, the Wallflower, and Hydra, at least I had some respect for them. They would never have lowered themselves to play to a crowd of civilians.\n\n\"Oh, get on with it, Siren. Quit showing off,\" Intelligal snapped and crossed her arms over her flat chest. The geeky ubervillain had little use for her partner's sexy antics. That was two of us.\n\n\"You always spoil my fun.\" Siren stuck her lip out and pouted.\n\n\"I'll kill her for you, Siren!\" one of the wealthy businessmen volunteered.\n\n\"Me too!\"\n\n\"And me!\"\n\n\"No, let me do it!\"\n\nSiren smiled, and several of the older men in the crowd clutched their chests, ready to have coronary episodes if she so much as crooked her pinky. \"Thank you, darlings, but that won't be necessary.\"\n\nI thought she muttered the word _yet_ , but I couldn't be sure. The two ubervillains might work together, but they definitely were not the best of friends. Too bad they wouldn't take each other out and save me the trouble.\n\n\"Intelligal is going to come over to you now. If you'll be so kind as to put your necklaces, watches, rings, and other valuables into her bag, we would greatly appreciate it.\"\n\nSiren's husky voice wrapped around me again, and I burned away the tight, grasping tendrils. Intelligal zoomed into the glassy-eyed crowd. She hit a button, and the side of her chair opened up. A mechanical arm holding a soft, cloth bag stretched out.\n\nThe ubervillain floated through the crowd, robbing Bigtime's richest, while Siren cooed into the microphone and kept them in line. Although I wanted to lob a fireball or three her way, I couldn't risk it. There were too many people around. Frying civilians was not good for my image. Or for sales of my action figures, calendars, and other officially licensed merchandise.\n\nIntelligal motored round and round, collecting millions in jewels, watches, and other baubles. Of course, there were a few willful holdouts who weren't under Siren's spell and didn't want to give up their precious shinies. Joanne James was among them. Intelligal had to threaten to shoot Berkley with the machine guns mounted on her chair before Joanne parted with her pearls and enormous engagement ring. Even then, it took her a while to decide between imminent, bloody death and going without her status symbols.\n\nTo my surprise, Johnny Bulluci was also among the people who resisted Siren's throaty call. Unlike the others, he didn't have a blank look on his face. He and Bella hovered protectively over an old man in a navy tuxedo. That must be the grandfather they'd rushed out to save. What was his name? Something with an R. Reed, Reynolds, Richards, Roger, Rob, Roberto! That was it. Roberto Bulluci. Bobby for short.\n\nIntelligal drifted their way, and Johnny's eyes narrowed to slits. He shifted his stance, and his jacket opened up, exposing his white shirt. I remembered the solid feel of his chest next to mine. His lips on mine. His hands on my body. I could still smell his spicy scent. My temperature roared up a few hundred degrees. The man was sexy, there was no doubt about that. None whatsoever. Even now, I was thinking about how we could pick up where we'd left off earlier. But I pushed that thought aside. My duty as a superhero came first\u2014pummeling ubervillains whenever possible.\n\nIntelligal stopped in front of the Bullucis. Johnny stepped forward to shield his family from the ubervillain. His hands clenched and unclenched into fists, and Johnny looked like he wanted to leap onto Intelligal's chair and punch her over and over again. The old man put a wrinkled, restraining hand on Johnny's arm and shook his head. Johnny shot his grandfather an annoyed look, but he relaxed just a bit. To my surprise, Bella's lips tightened into a thin line, and her hazel eyes glittered with rage. Bella? Angry? The mild-mannered designer never got angry, not even when she lost out on the Bigtime Fashion Designer of the Year Award to me. Despite his age, Bobby was just as worked up as his grandchildren. Even he shot the ubervillain a look of disgust and loathing.\n\nJohnny handed over a thick watch and his engraved cufflinks. Bella took off her silver chain and charm and threw them into the bag, while Bobby passed over a diamond pinky ring. Intelligal moved on to her next victims, but all three of the Bullucis kept their eyes on her, wishing that looks could kill.\n\nAfter about twenty minutes, Intelligal finished her mission. The mechanical arm clamped down on the bag, securing it and the goodies inside. She floated back to the dance floor, where Siren waited.\n\n\"Well, ladies and gentlemen, it's been a real pleasure. But I'm afraid it's time for us to leave.\"\n\n\"Siren, don't go!\"\n\n\"Please stay!\"\n\n\"We'll do anything you want!\"\n\nMen and a few women called out to the ubervillain, pleading with her to stay.\n\nSiren smiled. \"I know you will, darlings, but I really must go before the Fearless Five show up. Now, why don't you all do me a favor and talk quietly among yourselves for, say, ten minutes? That would make me _so_ happy. And you want to make me happy, don't you?\"\n\nJust about everyone in the crowd immediately turned to each other and began chatting about the weather, the wedding, and how fabulous Siren looked. Siren tossed the microphone back to the befuddled bandleader and hopped onto the arm of Intelligal's chair. The two of them zoomed up off the dance floor.\n\nAh, my cue at last.\n\nI grabbed the fire pulsing through me, and my body exploded into flames.\n\n# Chapter Four\n\nI plowed through the prickly holly bushes and pointed my finger. Flames arced outward, and a line of fire roared up between the ubervillains and the crowd. A few people snapped out of their trances, but most just stared at the flames. I pointed my other finger. Another line of fire sprang up, cutting off the ubervillains' escape route. I sprinted forward through the flames toward the evildoers.\n\nSiren and Intelligal froze. The chair hovered in midair, and their heads snapped around to me.\n\n\"You!\" Siren screamed.\n\nHer voice morphed from sultry and simpering to sharp and jagged in an instant. It felt like cold razors slicing into my brain. I clapped my hands to my head, trying to block out the horrible sound.\n\nA bluish energy ball popped into her hand, and Siren reared back to throw it at me. But a streak of lightning cut through the air and slammed into Siren's back. Shocked, the ubervillain flew off the arm of Intelligal's chair and hit the ground. Behind her, Carmen waved at me and smiled.\n\n\"Get her, Intelligal!\" Siren shrieked through a mouthful of dirt. The harsh tone in her voice cut through me again.\n\nIntelligal zoomed up over the dancing flames and punched some buttons on her floating chair. Two flaps opened on the front of the metal contraption, and missiles rocketed out\u2014straight at me. I waited until they were in range, lobbed two fireballs at them, and rolled out of the way. My fireballs slammed into the missiles, and they exploded in midair. The thunderous roar and resulting shockwave jolted the rest of the crowd out of its sheep-like state. People screamed and stampeded and sprinted down the lawn toward the metal gates a mile away. Smoke, soot, and ash darkened the spring sky.\n\nSiren scrambled to her feet and sent a lightning bolt my way. I ducked it and retaliated with a fireball. Intelligal zipped back and forth over us, trying to get a clear shot at me with some more of her pesky missiles.\n\nOut of the corner of my eye, I spotted Johnny Bulluci. He ran through the wall of flames and launched himself onto Intelligal's chair like it was something he did every day. What the hell did he think he was doing? I was the hero here, not him. The fool was going to get himself killed, and me along with him.\n\nBut I was too busy trying to dodge Siren's energy balls and fry her alive to pay much attention to Johnny. I tried to get close enough to touch Siren, but she kept flinging lightning bolts at me. She was too smart to let me get within arm's reach. Siren knew I could crush her to goo with my bare hands. I would, given half a chance. I'd wrap my hands around her neck and squeeze her like an orange.\n\nAfter a few minutes of back-and-forth action, I started to wear Siren down. Her lightning bolts grew weaker, shorter, and slower, and she started backing away from me. I crouched down, ready to pounce on her, when something smacked into me.\n\n\"Oof!\" I hit the ground so hard my flame-shaped mask left an impression in the smoldering grass.\n\nIt took me a second to realize Johnny Bulluci had fallen on top of me. Intelligal must have shaken him off her chair. Johnny didn't look like it, but the guy weighed a ton. I knew. That's what I bench-pressed.\n\n\"Move, move, move!\" I roared.\n\nJohnny rolled off me, onto his feet, and pulled me up in one smooth motion. I shoved him aside and raced toward the ubervillains. Too little, too late.\n\nSiren hopped back onto Intelligal's chair. A helicopter rotor sprang up from somewhere inside the device and started to whine and whir. Good grief, how many gadgets did the woman have on that thing? The two of them motored up, up, and away. I tossed a couple fireballs in the air, but Intelligal easily steered clear of them. Seconds later, the ubervillains disappeared from sight.\n\nDamn. Just when I was getting warmed up.\n\n\"Fiera! Fiera!\" someone shouted.\n\nI turned. Chief Newman pointed at the lines of fire still burning on the lawn and threatening to scorch the gardens. I concentrated, pulling the heat back into my own body, and the flames snuffed out. The chief and the others started forward to see if I was injured. I waved them off and marched over to Johnny, who was staring into the sky as if he could see where the ubervillains had gone. Bella and Bobby Bulluci stood next to him, alternating their gazes between Johnny and the wild blue yonder.\n\n\"What the hell did you think you were doing? You almost got yourself killed,\" I snapped.\n\n\"I was trying to take back what was mine and everybody else's,\" Johnny growled, not the least bit intimidated by the orange-red flames surrounding my body.\n\nI leaned forward and focused my anger\u2014and heat\u2014on him. \"That's not your job, your responsibility. That's what I'm here for. And I didn't just go a couple of rounds with some ubervillains so you could try to play hero and die a horrible, static-charged death.\"\n\n\"As you can see, I'm fine.\" His green eyes narrowed. \"And if I want to _play hero_ as you put it, it's no business of yours. Just because you're a superhero doesn't mean you can tell the rest of us what to do.\"\n\nThat's exactly what it meant, at least to me. Putting life and limb on the line for the fine citizens of Bigtime entitled me to a few perks. Bossing people around was one of them.\n\nJohnny continued to glare at me. My hair sparked and crackled with fire. Arrogant man. First, he tried to seduce me, and now he thought he knew how to do my job better than I did. Johnny Bulluci didn't know how close he was to getting strangled. I also had the oddest desire to singe his clothes off, yank him toward me, and kiss him until his eyeballs melted. Well, I'd stop before the eyeball-melting part. That could get a little icky.\n\nI opened my mouth to berate Johnny some more when a white blur zipped by.\n\nSwifte stopped in the middle of the lawn and struck a heroic pose. Hands on his hips, chest out, chin up. He was a tall, thin man who reminded me of a string bean. As his name suggested, Swifte was built for speed, not strength. The afternoon sun hit his back, and all the colors of the rainbow shimmered in the superhero's opalescent spandex suit. Silver sparkles glittered on the edges of the wing-shaped mask that covered his face. I eyed him with distaste. Swifte's getup was rather over the top, even for me. Superheroes, especially men, should not wear head-to-toe white. Ever. He so needed a makeover.\n\nThe superhero was so quick that he didn't even stir up any of the sooty dust or hot embers that coated the grass. A black bag dangled from his gloved hand.\n\n\"You're a little late, aren't you, Swifte?\" I sniped. \"The party's over.\"\n\nThe superhero relaxed his media pose. He shrugged. \"I was around. I heard the commotion and thought you might need a hand, especially since none of your friends showed up to help. Where's the rest of the F5 gang?\"\n\n\"Oh, they're around somewhere,\" I replied in a vague voice.\n\n\"Of course they are,\" Swifte murmured, looking at Carmen.\n\nI thought Carmen blushed. But it was probably just the heat lingering in the air that painted her cheeks a bright pink.\n\nSwifte gazed out at the smoke and debris. \"Well, you acquitted yourself nicely, Fiera. Even without your friends' help.\"\n\nI flipped my hair over my shoulder. \"I always do.\"\n\n\"Well, I'll just drop these off and be on my way.\" Swifte shoved the bag at me and tipped his head to Carmen and Sam. \"My congrats to the happy bride and groom.\"\n\nThe superhero put his hands back on his hips and struck another pose. Then, he disappeared. One second he was here, the next he was halfway to Cypress Mountain.\n\nI hefted the bag. The telltale clink and rattle of jewels and chains sounded.\n\n\"Is that what I think it is?\" Henry asked, pushing his soot-streaked glasses up his nose.\n\nI opened the bag and peered inside at the gleaming gems. \"Yep. Somehow, Swifte lifted the stolen jewels off Intelligal's chair.\"\n\n\"That's pretty impressive,\" Lulu chimed in. \"I didn't even see him.\"\n\n\"Swifte is quick, I'll give him that,\" I said.\n\nHe was also rather annoying, popping up when you least expected him to, sort of like a zit. And he always zoomed off before you had a chance to talk to him. The only people who ever got to say more than twenty words to Swifte were the reporters from the Superhero News Network and other media outlets. He always had plenty of time to stop and talk to them. Showboat.\n\n\"Well, now that everything's settled, can I have my watch back?\" Johnny said.\n\nI couldn't believe his nerve. He'd gotten in my way, let the ubervillains escape, and now he wanted his shinies back? How shallow. Even for a rich, spoiled playboy.\n\n\"I don't think so,\" I snapped. \"Not after the way you behaved. You can wait. The Fearless Five will get your items back to you in a few days.\"\n\nIt was standard operating procedure in cases like this. We'd sort through the items, determine what belonged to whom, and mail them back to the rightful owners. It kept people like Joanne James from claiming that certain pieces of jewelry belonged to them when they really didn't.\n\nJohnny's face hardened, his lips forming a tight line. I tore my gaze away from them and tried not to think about just how firm and warm they'd been earlier.\n\n\"Let's go, Johnny. Now.\" Bella's voice was sharp as a stiletto and twice as pointed.\n\n\"Yes, come along, Johnny,\" Bobby Bulluci added. \"I hardly think Fiera's going to keep anything for herself. We can get the watch back later.\"\n\nI eyed the grandfather. Even though he was over seventy, Bobby Bulluci was still a handsome man. He stood straight and tall, with a trim body. His hair was a burnished silver, but his eyes were the same remarkable green that Johnny's were. If Bobby was what Johnny would look like in another forty years, then the younger Bulluci was one lucky devil.\n\nJohnny stared at his sister and grandfather. He turned back to me and took a deep breath. \"The watch was my father's. I'd like it back. Please. It has a lot of sentimental value to me, to my whole family.\"\n\nI started to say no, but something in his eyes stopped me. A bright flash of pain I knew all too well. The pain of losing someone you loved. I remembered what Joanne had told me about Bella's father dying.\n\nI held out the bag. \"Take it.\"\n\nJohnny reached for the bag. Our fingers brushed, and I felt the need to blow off some steam. A few sparks flew out of my fingertips and landed on Johnny's torn, dirty tux. He paused and patted them out.\n\n\"Sorry about that,\" I said. \"Job hazard.\"\n\n\"No problem.\"\n\nJohnny dug through the fancy baubles until he found what he was looking for\u2014a simple silver watch. All that decorated the timepiece's black face were a pair of wings, inlaid in mother-of-pearl. No diamonds, no jewels, nothing fancy. It wasn't even a Rolex. Odd. Given how much he wanted it back, you'd think the watch would have been made out of solid platinum.\n\nJohnny fastened the timepiece around his left wrist. \"Thank you.\"\n\n\"No problem.\"\n\nJohnny took his grandfather's arm, and the Bullucis headed toward the gates, where the rest of the wedding guests milled about, waiting for their stretch limos to come take them home. That left just the six of us.\n\nThe chief put a concerned hand on my shoulder. \"Are you all right, Fiona? That was a pretty nasty fight.\"\n\nI stretched my arms over my head and did a mental inventory of my body. I felt a little sore in places, but that was it. With my superstrength, it took a lot to hurt me.\n\n\"Oh, I'm fine. It's nothing a little liquor won't cure. I need a drink. Fighting ubervillains is thirsty work.\"\n\nI strolled over to the bar, which had somehow survived the fires and explosions and lightning bolts, and grabbed a bottle of champagne. I yanked the cork out, took a long swig of the warm, bubbly liquid, and raised the bottle to the others. \"Cheers.\"\n\n# Chapter Five\n\n\"Are you sure we don't need to stay here?\" Carmen asked.\n\nI rolled my eyes. \"If you ask me that again, I'm going to burn your tongue.\"\n\nThe five of us, plus Lulu, were down in the secret library underneath Sublime. Superhero Central, Carmen called it. To me, it was our War Room, despite the books and magazines and encyclopedias that clustered in the floor-to-ceiling shelves.\n\nAfter calming down the frantic wedding guests and escorting them off the grounds, the Fearless Five had gone into full-fledged superhero mode. We'd spent the last couple hours in the library trying to track down Siren and Intelligal. With no luck.\n\n\"Intelligal must have some sort of cloaking device on that chair,\" Henry muttered.\n\nHe was using his power, mind-melding with his supercomputer, to try and locate the ubervillains. A soft, bluish-white glow connected his fingers to the keyboard. The monitor reflected numbers and letters onto his round glasses, making them gleam. The symbols flashed by too fast for me to follow, but Henry had no trouble deciphering them with his photographic memory.\n\n\"There's no trace of them. Not even a heat signature on the radar.\"\n\n\"I'm sure you'll find her, babe,\" Lulu said, pounding away on her own computer nearby. \"You always do.\"\n\n\"Maybe we should stay here,\" Carmen repeated.\n\n\"And abandon your honeymoon plans?\" Chief Newman said. \"There's no need for that.\"\n\nCarmen paced around the room. She picked up one of her Rubik's Cubes and turned it round and round in her hands as she slid the plastic rows of colors back and forth. \"I feel like Siren and Intelligal are up to something. Something big.\"\n\n\"Aren't they always? It's sort of what ubervillains do.\" I leaned back in my chair and put my feet up on the round table that dominated the middle of the room. My black stilettos rested on top of the giant _F5_ insignia carved into the heavy wood.\n\nCarmen shook her head. \"This feels bigger than that. Like world-domination big.\"\n\n\"Chief?\" I asked my father.\n\nHe laced his fingers together. \"I think Carmen's right. They're definitely up to something. They took a big risk crashing the wedding, and I've got the headache to prove it.\"\n\n\"See? We should stay. Besides, those bitches ruined my wedding. They should pay.\"\n\nCarmen's eyes flashed neon blue with rage. They only did that when she was very, very angry. Even I didn't like to go up against Carmen when she was angry.\n\nSam put an arm around her shoulder and hugged her to him. Carmen's eyes brightened, taking on a different sort of glow.\n\n\"Don't worry, they will,\" Sam said. \"The chief and the others will take care of them. You don't want to cancel our trip, do you?\"\n\nAn extravagant, no-holds-barred, month-long honeymoon around the world was Sam's wedding present to Carmen. She'd be a fool to cancel on him just to track down a couple of measly ubervillains. But if she did, I'd be happy to take her place, and eat and shop my way around the globe. I could use a vacation, especially after today. As much as I loved being a superhero, there were times when the epic battles got a bit old. Unfortunately, ubervillains were a dime a dozen in this town, and somebody was always up to something. Which meant we superheroes always had to be on the lookout for trouble with a capital T.\n\n\"No, of course not,\" Carmen said, leaning into Sam that much more. \"But you know how I worry.\"\n\n\"There's nothing to worry about, Sister Carmen,\" Lulu said. \"Henry will track them down, and then Fiona will fry their asses.\"\n\n\"Sounds like a plan to me,\" I said. \"I'm always up for a good barbecue.\"\n\nIn the end, Carmen gave in. She and Sam got into a limo and headed for the Bigtime Airport, where Sam's private jet was waiting to take them to London. They didn't even make it out of the driveway before they tinted the windows and started canoodling in the backseat.\n\nAfter seeing them off, in more ways than one, the rest of us talked for a few more minutes about the ubervillains and what they might be up to. But nobody had any bright, earth-shattering ideas, so we sorted through the stolen jewelry and packed it up to mail back to its rightful owners. Then, we closed down shop for the night. Too tired to make the drive back to my apartment in the city, I went to my suite down the hall.\n\nOur underground lair beneath Sublime took up almost as much space as the manor house did aboveground. Wing after wing contained training rooms, gyms, kitchens, sick bays, and everything else a superhero could ever want or need. My suite was an enormous area, bigger than most apartments, and featured a bedroom, bathroom, living room, and kitchen all rolled into one, along with a pitifully small closet. The others' suites all had a similar setup except for one thing\u2014mine boasted three oversized refrigerators.\n\nI rustled around in one of the metal behemoths and came up with a platter of cold cuts, three kinds of cheese, and a loaf of crusty French bread. I made myself seven sandwiches and downed them with a six-pack of lemon-lime soda and a box of chocolate snack cakes. Due to my fiery metabolism, I could eat whatever I wanted to, whenever I wanted to, and never gain an ounce. A whole pizza for breakfast, ten burgers and a bucket of fries for lunch, a couple of steaks and a pound of pasta for dinner. The only downside was that I constantly had to eat to keep up my strength. It got a little tiring sometimes. Not to mention grossed people out. My father and Travis were the only ones who didn't seem disturbed by my never-ending hunger and enormous appetite.\n\nTravis. An icy hand squeezed my heart. I plopped down on the king-sized bed and picked up the picture of him on my nightstand. Blond hair, brown eyes, terrific smile. Travis Teague beamed at me. I'd missed him today. A lot. Carmen and Sam's wedding had only reminded me of what I'd lost when Travis had been killed over a year ago. Of what I should have right now. I turned my ring around on my finger.\n\nBut as I looked at Travis's picture, my thoughts drifted back to Johnny Bulluci and our encounter in the game room. The man could kiss, that was for sure. A couple of sparks flew from my fingertips and landed on the tile floor. If Johnny could work with his hands as well as he did with his lips and tongue, well, he could seduce me anytime.\n\nJohnny Bulluci could be my rebound guy, I decided. Someone to have a little fun and a lot of steamy sex with before I started looking for the next Mr. Right. Travis had been my Mr. Right, but he was gone now. I would always miss him, but he wouldn't want me to be a nun the rest of my life. Travis had been generous to a fault. He would understand me moving on. He'd encourage me to. I put Travis's picture on the nightstand and carefully positioned it so he was smiling at me.\n\nYes, it was time to get out and live a little.\n\nIt was time for Fiona Fine to get back into the dating game.\n\n* * *\n\nThe next morning, I drove my convertible into the city. I put the top down, stepped on the gas, and let the warm wind whip my blond locks into a tangled mess. The sporty car had been my present to myself for my thirtieth birthday a couple months ago. I loved the sleek design, the smooth ride, the fire-engine-red color. The four hundred horses under the hood weren't too shabby either.\n\nBefore work, I swung by my apartment, located in the penthouse in Tip-Top Tower, one of the most expensive and exclusive buildings in Bigtime. I opened the door, flicked on the lights, and tossed my keys onto a table. My apartment featured ivory-colored granite floors and walls, fireproof up to three thousand degrees. No thick, shag carpet for me. I'd ruined too much of it in my life to risk having it in my home. No chintzy curtains either. Instead, metal screens covered the tall windows like blinds, giving me privacy from prying eyes, without being a fire hazard.\n\nI walked into the living room. A long sofa the color of Bing cherries sat against one wall, flanked by four black armchairs. The rest of the furniture was done in various shades of red, black, and white. Sometimes all three. Paintings behind shatterproof and fire-resistant glass added even more color to the pale walls.\n\nI took a quick shower, dried off in about two seconds, and moved to my favorite part of my apartment\u2014my closet. I flicked on the lights in the walk-in closet, which took up more than two thousand square feet. Rack after rack of shirts, skirts, slacks, shoes, and more, along with drawers full of jewelry and handbags, crouched inside the massive space. The bright fabrics, outrageous patterns, and bold colors always brought a smile to my face. Life was too short for drab beiges and tame taupes and basic blacks. Give me electric blue and lime green and shocking purple any day.\n\nNow, on to the most important question of the day. What to wear? So many color combinations, so little time. I flipped through the racks of clothes, choosing a short, hot-pink skirt and a sleeveless, black, V-necked top with huge pink polka dots. A pair of matching kitten heels, a heart-shaped bag, and plastic, hoop earrings finished off the look.\n\nI left the convertible in the building's parking garage and walked the few blocks to my store. It was just after nine, and most folks were grabbing a last cup of coffee and a doughnut before heading off to work. I stopped in front of Bryn's Bakery, eyeing the delectable-looking bear claws through the windows. The bag of blueberry bagels, tub of cream cheese, and quart of apple juice I'd had for breakfast were already long gone.\n\nA siren screamed out, and a brigade of fire trucks roared by. I squinted into the morning sun. Black smoke boiled up from a building a couple of blocks away. I stood on the sidewalk, torn between the bear claws and whipping my spare Fiera costume and boots out of the special, air-compressed compartment in my purse.\n\n\"Out of my way! Move it! Now!\"\n\nA woman dressed in a formfitting, ebony jumpsuit strutted by me. Green and gold snakes curled around her arms like bracelets, hissing and snapping at people to get them to move out of the way. More snakes writhed on top of her sky-high headdress. Ah, Black Samba was coming to the rescue today.\n\nBlack Samba marched into the street and held out her hand. She chanted in an odd, sibilant language. Five seconds later, a city bus rounded a corner and stopped in front of her. Along with the snakes, Black Samba had some sort of magic voodoo powers I didn't really understand. I preferred a simpler style, punching and smashing and frying to get things done. But the mumbo-jumbo worked for her.\n\nBlack Samba leapt up onto the hood, told the driver to head toward the fire, then scampered up onto the roof. Bus tops were her preferred mode of transportation around the city. For some reason, her snakes didn't like the subway. At least, that's what I'd heard through the superhero grapevine.\n\n\"Wait for me!\" an older, feminine voice called out.\n\nA diamond-topped walking stick snapped against the sidewalk, and the crowd parted to let Granny Cane through. She looked like your average grandmother, except for the flowered purple mask that covered her face and the cape-like flow to her matching angora sweater. Granny Cane prowled the streets of Bigtime, getting thieves to try to mug her before wailing on them with her enormous handbag and walking stick and dragging them to the police station. Granny was a lot stronger and sturdier than she looked.\n\n\"Hurry up, old woman!\" Black Samba snapped, stalking up and down the top of the bus. Her snakes hissed in time to her footsteps.\n\n\"Don't make me wash your mouth out with soap, missy.\" Granny pointed her cane at the other superhero. \"There's no need to be disrespectful to your elders.\"\n\nGranny hopped onto the bus, and it pulled away into traffic. Well, with Black Samba and Granny Cane on the way, there was no need for me to get involved. They could handle a simple fire-and-rescue operation. Besides, too many superheroes on the scene wasn't really a good thing. Everybody always wanted to get in on the drama. When there wasn't enough danger to go around, the stronger superheroes tended to assert themselves, which often led to ego-bruising and spats. We all wanted to be the best\u2014and most popular. It was sort of like being back in high school. With superpowers, a good image, and action-figure sales to maintain **\u2014** instead of a cool quotient, high grades, and clear skin.\n\nI did, however, still need to get to work. So, I forced myself away from the bakery and walked on.\n\nFiona Fine Fashions sat in the middle of Retail Row in the heart of Bigtime's downtown shopping district. The front of the multistory building housed the runway and store, while workers sewed garments and more in the back. A huge, marble F towered three stories above the sidewalk and announced the store's presence to shoppers.\n\nI breezed through the front door. A bell chimed, letting everyone know that I had arrived. The inside of the store was rather like my apartment, in that it featured white marble walls and floors. Everything was simple and clean and colorless, the better to focus attention on my bright, bold, daring clothes. Racks of the Fiona Fine originals sat throughout the store, while a long runway cut through the middle of the open space. Models in various shapes and sizes trotted up and down the catwalk, showing off the latest, greatest designs to customers, who sat on plush leather chairs munching on champagne and gourmet chocolates. Techno music pulsed out a snappy, happy beat through the sound system. Clerks bustled back and forth, fetching food and taking orders. I waved to them and to the society women I recognized. They fluttered their hands at me and kept on eating and drinking. I'd learned a long time ago that society ladies felt they deserved to be pampered, so I catered to their whims.\n\nMy eyes traced over the store with pride. The gleaming marble and pricey clothes were a long way from the simple home where I'd grown up in Ireland. Like so many others, my father had moved the family here in search of the American dream. I was one of the lucky ones who'd gotten mine. I sewed until my fingers bled to get to the top of the fashion world, and I did everything I could to stay there.\n\nI strolled through the store, punched in the security code on the back door, and headed for the factory floor. The whine and whir of sewing machines, ringing phones, and other equipment replaced the pulsing music. Men and women sat in colorful, ergonomically designed cubicles and sewed dresses, tops, skirts, and more. Some placed dainty crystals on supple leather handbags, while a few worked with sapphires, rubies, and emeralds. The gems glistened and sparkled under the white lights. My jewelry line was launching this fall, and it was going to be just as fabulous as everything else I designed.\n\nMore than a few folks called out and waved to me, and I answered them in kind. My workers earned top dollar and excellent benefits for their time, diligence, and quality craftsmanship. I could afford to be generous. Most of my creations retailed for at least five thousand dollars, and everything was backordered for six weeks. The only other designer who commanded similar prices was Bella Bulluci.\n\nJohnny Bulluci's handsome face flashed through my mind. I'd have to ask around and get some more dirt on him. I remembered the suave way he'd kissed me at the wedding. A spark sizzled on the tip of my thumb. Oh yes. I definitely wanted my rebound relationship to ramp up as soon as possible.\n\nI put that thought away and stepped into Piper Perez's office, which was next to mine. Piper typed away on a keyboard and murmured into a headset attached to one side of her cats-eye glasses. She was an average-sized woman with glossy black hair, tan skin, and warm, brown eyes.\n\nPiper also was my chief financial officer and anal to a fault. Everything on her desk was arranged from tallest (her computer monitor) to shortest (a pink sticky pad) and sorted by color, shape, _and_ times used during the day. Piper was insanely proud of her straitlaced, obsessive tendencies. She should be. Her organizational and accounting skills earned her more than most CEOs.\n\n\"Morning, Piper.\"\n\nPiper said her goodbyes to her caller, punched a button on her headset, and ended the conversation. \"Morning, Fiona.\"\n\nShe stopped typing and handed me a stack of papers. \"The front desk gave me your latest messages by mistake. I went ahead and placed the oldest ones on top so you can return those calls first. I also put the cost-comparison reports about the fabric and gemstone suppliers on your desk. And Joanne James called me personally, demanding to know when you would be available. Her highness wants to come in for a fitting.\"\n\nI groaned, tempted to burn the message and drop the charred remains in Piper's spotless steel trash can. Then, I thought about that flashing rock on Joanne's finger. I really could use a couple of new Fiera suits, and the flame-proof fabric I had to make them out of wasn't cheap. I might be rich, but I wasn't going to turn down a hefty payday, even if I had to deal with the black widow of Bigtime. \"All right. Have somebody set it up. But be sure we have plenty of champagne on hand that day.\"\n\n\"For Joanne?\"\n\n\"No, for me. I'm going to need it.\"\n\n\"If you think you're going to need it then, just wait a minute,\" Piper said.\n\n\"What? What did you say?\"\n\n\"Oh nothing,\" Piper chirped in a cheery voice that didn't fool me for a minute.\n\nMy eyes narrowed. \"What are you up to?\"\n\nPiper blinked. \"Nothing. Nothing at all, Fiona.\"\n\nAt times like this, the chief's mind-reading powers would so come in handy. I thought about sweating the truth out of Piper but decided against it. She'd just spend the rest of the morning in the bathroom touching up her makeup until it was perfect again. Piper was as obsessive about her appearance as she was about her desk. I gave her a last suspicious look, opened the door to my office, and stepped inside.\n\nWords escaped me.\n\nFlowers, flowers, and more flowers covered every inch of my desk, surrounding shelves, and windowsills. Roses, tulips, orchids, and more painted the room in a wild, vivid rainbow of color. My nose twitched at the heady, rich scents all mingled and mashed together.\n\nPiper leaned against the doorframe. \"They've been arriving every half hour since eight this morning.\"\n\n\"Where and who did all these come from?\" I fingered the petals on a blue orchid. I loved flowers, especially orchids, but I never bought them for myself. Unfortunately, they had a tendency to go up like smoke around me.\n\nPiper pulled out a card embossed with silver filigree. _Fiona. So sorry we got interrupted yesterday at the wedding. Dinner tonight? I'll pick you up at seven at your apartment. Johnny Bulluci_.\n\nI managed not to squeal. Instead, I took the card from her and read it myself. Three times.\n\n\"Who's the guy?\" Piper asked. \"I haven't heard you mention him before. Then again, I haven't heard you mention anyone in a long time.\"\n\n\"Bella's brother. He's new in town,\" I said in an absent tone, my eyes still fixed on the card, which was made out of thick, creamy paper. I traced over the silver letters with my finger, careful not to scorch the stationary.\n\n\"Bella? As in Bella Bulluci? Our archrival in the design world?\"\n\n\"The one and the same.\"\n\nI read the card again. Tingles spread through my heated body. Johnny Bulluci liked me. Enough to want to see me again. Either that or he just thought I was extremely easy. I remembered his spicy smell, his firm hands on my feverish body, the way I'd offered no resistance to his skillful advances. Yeah, he probably thought I was easy. Didn't bother me any.\n\n\"You know, I don't remember seeing you with this guy at the wedding. So what exactly were the two of you doing yesterday? And where?\" Piper's dark eyes sparkled. \"Or should I even ask? It must have been something to get this kind of enthusiastic response.\"\n\nI flipped my long, blond hair over my shoulder. \"Why, Piper, don't you know? It's always something with me.\"\n\nPiper just snickered.\n\n# Chapter Six\n\nI spent the rest of the day working on my new fall lines. Fiona Fine Fashions was an empire unto itself, and I had my fingers in everything from evening wear for society matrons to couture wedding gowns to sportswear to baby clothes. Yes, baby clothes. It was never too early to start teaching kids about fabulous fashion.\n\nToday, though, my main focus was on Fine Finds by Fiona, an affordable businesswear collection for professional women debuting in stores across America later this year. I was taking the basic, black work suit and dressing it up with stripes, plaids, paisleys, and most importantly, color. Your job might be boring, but your clothes should never be.\n\nAt least, I tried to work. I kept staring at the flowers clustered around me. It was like sitting in my own personal jungle. My eyes traced over the kaleidoscope sprays with their soft, velvety petals. Inspired, I fished a large pad out of my messy desk and started sketching. Two hours later, I had over a dozen drawings, each one bigger and bolder and better than the last. I'd been looking for a theme for next year's spring line. _Flower power_ would be perfect. After all, flowers featured everything that I loved\u2014wild patterns, brilliant colors, sleek designs.\n\nA knock on the door interrupted the flow of my creative juices.\n\nPiper held up three white paper bags that looked like they were ready to pop. \"I went to Quicke's for a meeting and thought you might be hungry. I got you a couple of grilled chicken sandwiches, a Caesar salad, large fries, a dozen cookies, and the biggest raspberry tea I could find.\"\n\nI glanced at the clock. Three in the afternoon already. My stomach rumbled. \"Thanks, Piper. You're a lifesaver.\"\n\nPiper deposited the bags on my crowded desk. She'd long ago grown used to my enormous appetite, although she thought I had some sort of eating disorder, like binging and purging. Every so often, I'd come into the office to find pamphlets on anorexia and other eating-disorder clinics and programs on my desk, courtesy of Piper. I reduced them to ash, ignored Piper's prying looks, and continued to eat like there was no tomorrow. I couldn't exactly tell her the truth\u2014that I was a superhero with a superhigh metabolism. Piper was my friend, but I had a secret identity to protect. The Terrible Triad had taught me the importance of anonymity.\n\nI polished off the food in about fifteen minutes under Piper's watchful eyes. She stuck her head into my office several times that afternoon, probably hoping to catch me hurling into one of the empty paper bags, but I disappointed her by slaving away at my desk.\n\nIn addition to my work as head of Fiona Fine Fashions, I also had to attend to my duties and business as Fiera. Being a superhero was just as much about public relations as it was about saving kittens from trees and babies from burning buildings. Most of us had our own websites, merchandise, fan clubs, and more so we could control who did what with our image. The same went for the ubervillains. Unfortunately, evil appealed to just as many folks as good did.\n\nSome superheroes, namely Swifte, really hammed it up. They blogged and did webcasts and wrote tell-all biographies, regaling the masses with their latest daring adventures and narrow escapes. I and the other members of the Fearless Five were much more restrained. We had a simple website where folks could e-mail us and sign up for our newsletter, as well as buy our merchandise, namely copies of my annual calendar and latest action figures. The others didn't have action figures. At least not officially sanctioned, royalty-paying ones. They weren't so into the marketing end of things, especially Sam. The man would have been a shadow if he could.\n\nBut I didn't keep any money I made as Fiera. Neither did the rest of the Fearless Five. Not a penny. We split the profits evenly and donated them all to worthy causes. My cut went to a charity that helped burn victims get reconstructive surgery and to the Bigtime Municipal Building Restoration Fund. After all, if you tear it down, you really should help rebuild it. And I'd destroyed more than one building in Bigtime over the years.\n\nOf course, some folks like Gentleman George and the Baseballer spurned the spotlight, preferring to keep to themselves and avoid the hoopla. Sometimes, I thought about going that route. Taking down the website, discontinuing my annual calendar, canceling my few public appearances. But part of me liked the attention. Modesty wasn't one of my virtues, and I enjoyed being recognized for saving the city every couple of weeks. My father had taught me long ago that I had special gifts that I must use to help others. He called it my duty, my purpose in life. I just thought it was fun. At least, it was most of the time.\n\nI logged into my Fiera accounts and checked on sales of the calendar and a few other things that needed my attention. All of the money flowed from the website through untraceable, numbered overseas accounts to our favorite charities. Sam was a real whiz at setting up things like that. Plus, Henry had put tons of security gizmos on my computer, ensuring that only I could access the Fiera files hidden on my hard drive.\n\nOnce that was done, I switched over to my e-mails. I got dozens of e-mails and instant messages every day, most of them from fanboys who wanted me to autograph their inflatable Fiera dolls or their tighty-whities\u2014sometimes both. I usually ignored those. I also got a lot of letters from kids asking me to find their lost puppies or to please, please, _please_ bring their pet hamster back to life. Those I answered as best I could. Despite my superpowers, I couldn't help everybody, couldn't save everybody. I did what I could, when I could. That was another thing my father had taught me. You could be a hero, do your duty, and still have a life outside the masks and capes and spandex costumes. If you didn't, you'd drive yourself crazy.\n\nBy the time I finished my sketches, returned messages, and caught up on my paperwork, superhero-related and otherwise, it was after five. Time to go home and get ready for my night out on the town with the delectable Johnny Bulluci.\n\nGoody.\n\n* * *\n\nI walked home in record time. Thankfully, there were no more fires or other emergencies to distract me. After grabbing a couple of apples from the fruit bowl on the kitchen counter, I strolled into the bathroom, peeled off my clothes, and ran some water into the claw-foot tub. While I chomped on the tart fruit, I fluttered my fingers, heating the water until it was nice and toasty. Then, I did the whole _I-have-a-hot-date-and-might-even-have-sex-tonight_ routine. Lathered up with some vanilla-scented soap. Washed my hair. Shaved everything.\n\nI dried off in about three seconds, went into the bedroom, and flicked on the light in my closet to begin my search for the perfect first-date outfit. I wanted something cool, something sophisticated, something that said, _We'll probably have sex if you behave yourself and buy me a nice, big dinner first_. After flipping through three racks, I picked a tight scarlet dress covered in matching sequins that set off my blond hair and rosy skin. Heavy black eye shadow, red lipstick, a ruby solitaire pendant, and red stilettos polished off the outfit. The dress didn't have too many buttons down the back, and the short skirt would provide easy access, if desired. I closed my eyes and thought of Johnny's lips on mine. There would probably be some desire tonight. Maybe even some satisfaction, if Johnny was a very good boy. Or a very bad one, depending on your point of view.\n\nI twirled around in front of the mirrors that lined one wall of the bedroom. I looked hot. Smokin' even, as Lulu would say. I put my hands on my hips, admiring my reflection. My engagement ring caught the light and flashed it back to me. I twisted it around on my finger. My good mood vanished.\n\nShould I really be doing this? Going out with somebody else? Thinking about sleeping with someone else? Travis was gone, but I still loved him. I always would. I didn't know if there was room in my heart for someone else. If I could even love somebody again. If I really even wanted to.\n\nIf you loved someone, you gave him power over you. Power to hurt you, whether he did it on purpose or not. I didn't ever want to feel the way I had when Travis died. Alone. Grief-stricken. Helpless. Numb. Maybe I should wait awhile longer to get back into the social scene. Maybe I should cancel\u2014\n\nThe phone rang, interrupting my troubled thoughts.\n\n\"Fiona Fine.\"\n\n\"It's me,\" the chief's voice rumbled through the receiver. \"I just wanted to check in with you. I'm on call tonight.\"\n\nEach member of the Fearless Five took turns sitting in the underground library and monitoring SNN and the local police scanners in case an ubervillain decided to wreak havoc or some natural disaster tore through Bigtime. The rest of us were equipped with cell phones, ready to respond in case bad stuff went down. With Carmen and Sam gone on their honeymoon, the chief, Henry, and I would be pulling double duty for the next month.\n\n\"How are things at the manor? Any sign of Siren and Intelligal yet?\"\n\n\"Not a trace of them so far. Everything's quiet. Henry and Lulu are working on tracking them down. Carmen and Sam called earlier. They're having a wonderful time in London, although Carmen's still worried about the ubervillains.\"\n\nI was surprised the happy couple had found time to call or to actually see any of London. You'd think they would have spent all day in bed, given their propensity for doing so at Sublime. Their happiness was so annoying sometimes.\n\nMy father paused. \"Have fun on your date tonight. Make sure he treats you right.\"\n\nI rolled my eyes. \"How many times have I asked you not to peer into my mind?\"\n\n\"I'm your father, Fiona. I'll be doing it until the end of time.\"\n\n\"Just because you're psychic doesn't give you the right to use your powers on others whenever you want. Remember that lecture? You used to give it to me whenever I'd singe one of the bullies in my class with a fireball or pick them up and spin them around until they screamed for me to stop.\" I'd gotten more than a few of those talks.\n\nThe chief chuckled. \"I remember you spent more time in the principal's office than a classroom full of kids.\" His voice grew serious. \"You should go out, Fiona. Travis would want you to get on with your life.\"\n\nI looked at the ring on my finger. \"I know he would. But it's easier said than done.\"\n\n\"Go out. Have fun. Be your usual charming self.\"\n\nI sniffed. \"I'm always that.\"\n\n\"I know. Now, I need to go. Henry wanted me to look over some new computer program he's invented to track Siren and Intelligal's crimes.\"\n\nHenry and his computer programs. Sometimes, I wondered if the man was even human or just a robot in disguise. If he were a robot, though, you'd think his designer would have given him some fashion sense. Polka-dot bow ties and sweater vests were not the stuff women's dreams were made of. Except maybe Lulu's.\n\n\"Well, you two be careful tonight. Call me if you need me. I love you.\" I always ended our conversations like this now. I'd learned the hard way that just because we were superheroes didn't mean we were invincible. Quite the opposite, unfortunately.\n\n\"I love you, too,\" the chief rumbled.\n\nI hung up. For a moment, I stared at my engagement ring. Then, my eyes went to the mirror. I smoothed my dress down and turned sideways. I did look good tonight. Damn good. Self-esteem was another area where I didn't have a lot of problems.\n\nI let out a long, hot breath. I might as well go out. The chief was right. It was time for me to move on, whether I thought I was ready to or not.\n\nBesides, it'd be a shame to get all dressed up for nothing.\n\n* * *\n\nThe doorbell rang at exactly seven o'clock, just as I was powdering my nose. My date had arrived right on time. I loved punctual men.\n\nI opened the door to find another bouquet of orchids outside. These were picture-perfect and more beautiful than all the rest combined. They made the ones in my office look like cheap, plastic imitations. I could just make out Johnny's green eyes through the exquisite purple-and-white petals.\n\n\"Do you know where I might find a beautiful bridesmaid?\"\n\nHis voice was just as rich and cultured as I remembered. His faint accent sounded sophisticated, sultry, and utterly sexy. I'd always been a sucker for accents.\n\n\"I'm not a bridesmaid anymore,\" I said, stepping back to let him inside.\n\n\"But you're still beautiful.\"\n\n\"Aren't you the charmer?\"\n\nHe winked. \"Always, Fiona. Always.\" Johnny tossed the flowers onto a nearby table and swept me into his arms. The man didn't waste any time, that was for sure.\n\nHis lips covered mine, and all thought vanished. I wrapped my arms around his neck and pulled him closer. He opened his mouth, and I slid my tongue inside the hot depths. He tasted like mouthwash, tart and tangy. His spicy soap wafted up my nose, and I breathed in, enjoying the aroma. Whatever that scent was, it complemented him perfectly. Johnny's hands traced circles up and down my back as I explored his mouth with my tongue. It was a long, slow, lingering kiss. The sort that whispered of even better, slower, longer, harder things to come.\n\nAfter a good two minutes, we came up for air.\n\n\"You know, we could just order in,\" Johnny murmured, pressing a kiss to the hollow of my throat. My pulse roared under his lips.\n\n\"Dinner first,\" I said, stepping back.\n\nI had an image to maintain, and I didn't want him to think I was too eager. Or know that single kiss had me ready to toss him over my shoulder, carry him back to the bedroom, and ravish him from head to toe. Three times over.\n\nThe truth was the intense attraction unnerved me. One look, one kiss, one minute alone with him, and I was all hot and bothered. At least, more so than usual. I'd never believed in love at first sight, but lust was another matter. Or was it? Did Johnny Bulluci really have that much of an effect on me? Or were my hormones just hot to trot because of the long drought?\n\nI didn't know the answers to my questions, so I concentrated on him. His tawny mane of hair glistened under the dim lights, and his eyes were as green as polished turquoise. Johnny wore a navy-blue suit with a faint pinstripe. It looked fantastic on him. Or he looked fantastic wearing it. I couldn't decide which one it really was. Maybe I should get him to model some designs for me, provided I could lure him away from Bella. With him in the store, sales would go through the roof. If I could stop the society types from tearing off his clothes, something I was dangerously close to doing myself.\n\n\"Well, I suppose you're right,\" Johnny said. \"We really should eat dinner first, before getting to dessert.\"\n\nJohnny smiled. A sly, impish, wicked sort of grin spread across his face, while his eyes devoured me from head to toe. I had no doubt what he meant by dessert. I shivered, something I never did, not even when Frost hit me with his freezoray gun.\n\nJohnny offered me his arm, and away we went. A limo waiting downstairs took us to Quicke's, a restaurant famous for its speedy service, reasonable prices, terrific food, and most important, generous portions. A blue neon light flashed on and off above the revolving door, announcing the restaurant's name. Quicke's was located downtown, a couple of blocks from my store, and it was one of my favorite places to get a fast burger or ten for lunch. I practically kept the place in business with my massive orders.\n\nJohnny opened the door for me, and when I stepped through, he put his hand on the small of my back to guide me inside. Some women would have been insulted by this, but not me. Maybe it's because I moonlight as a superhero, but I like strong, take-charge men. They're the only ones who can keep up with me.\n\nKyle Quicke was working as the host tonight, the latest in a long line of Quickes to run the restaurant. With his chestnut hair, light eyes, and thin physique, Kyle was cute in a boy-next-door sort of way. We'd gone to college together, back when we'd both been penniless nobodies. I, of course, had made my fortune in fashion, while Kyle had gotten his business degree and taken over the family restaurant. During his tenure, Quicke's had increased its already healthy profits so much that the restaurant had opened a number of smaller caf\u00e9s and eateries throughout Bigtime.\n\n\"Hello, Kyle.\"\n\n\"Fiona Fine. Always a pleasure.\"\n\nI threw my long hair over my shoulder. \"Of course it is. And you as well. Loved the food you did for the wedding.\" What little of it I'd gotten to eat had been delicious as always.\n\n\"Thanks. I aim to please.\" Kyle's eyes flicked over us. \"Table for two?\"\n\n\"Obviously.\"\n\n\"This way.\"\n\nKyle grabbed a couple of leather-bound menus and zipped through the crowded restaurant, darting around tables and bustling waiters with a grace I didn't know he possessed. I followed, and Johnny put his hand on my back again. Another shiver swept through me. A girl could get used to that strong, firm touch. In all sorts of places.\n\nThe inside of Quicke's looked like your typical bar and grill, only a bit nicer. Round tables covered with white linen tablecloths took up most of the first floor and part of the second. Roses perched in tall vases on every table, along with skinny votive candles. A bar with a gleaming brass rail and mirror behind it ran the whole length of the building. Shouts and curses drifted out from behind the swinging doors that led to the kitchen, along with the sizzle of frying meat and the yeasty aroma of baking bread. My stomach quivered in anticipation at the mouthwatering scents.\n\nThe restaurant's patrons wore everything from torn jeans and sneakers to designer ball gowns and tuxedos. Quicke's catered to all segments of Bigtime's population, including superheroes and ubervillains. Framed posters, newspaper clippings, and autographed pictures of heroes and villains covered the brick walls. Action figures crouched in mock-fighting positions on the shelves behind the bar, along with car replicas and plush toys. I was pleased to note there were more than a few Fiera figures among the mix, including one scene where a six-inch-tall version of me pummeled Scorpion. I smiled.\n\nI scanned the tables, but I didn't see any neon-colored costumes among the crowd. Quicke's was one of the few places considered to be neutral territory in Bigtime. Both superheroes and ubervillains could eat here without worrying about being hit with heat rays or thrown through the plate-glass windows. In fact, with its fast turnaround time, more than one superhero and ubervillain grabbed a sandwich or salad at Quicke's before heading out to do battle for the night. Or even in the middle of a fight. Debonair, a sometimes hero, sometimes thief with teleportation powers, was particularly fond of doing that.\n\nKyle seated us at a cozy table in back next to one of the windows that fronted part of Paradise Park. The colorful lights of the carousel spun round and round in the distance like a never-ending comet. Shrieks and shouts of glee penetrated the window, along with the jingle and jangle of the various other rides and attractions at the park.\n\nKyle gave us our menus and went back to his post at the front of the restaurant. A fat waiter named Ray came over to fill our water glasses and take our orders. Johnny decided on a black-pepper-seasoned steak with the usual trimmings, and the waiter turned to me.\n\n\"Your usual, Miss Fine?\" Ray asked. Most of the staff at the restaurant knew me by name.\n\nI started to say you betcha, but caught myself just in time. My usual consisted of half a dozen various sandwiches, five jumbo orders of fries, onion rings, hot 'n' spicy buffalo wings, two bowls of soup, a couple liters of soda, and the cake of the day. Three of them.\n\n\"I'll have the grilled chicken special.\" It came with only a couple of measly sides. It wasn't going to be enough food. I tried not to cringe.\n\nThe waiter looked at me, waiting for the rest of my order.\n\n\"That's all.\"\n\n\"That's all? Are you sure, Miss Fine?\"\n\n\"Positive.\" I snapped the menu shut and gave it to him.\n\nThe waiter gave me an odd look, but he bowed and walked away.\n\nJohnny grinned. \"They seem to know you pretty well here, Fiona.\"\n\nI shrugged. \"My store's a couple blocks over. I eat here a lot. I was rather surprised you decided to bring me here for dinner. I expected Chezanne's or one of the other fancy French places.\"\n\n\"Why's that?\"\n\nI stared at his expensive attire, then at the rough brick walls and simple bar that made up the restaurant. \"This place seems a little tame for you.\"\n\nIt was his turn to shrug. \"I like good food. And Quicke's has the best in town from what I remember.\"\n\n\"From what you remember?\"\n\nJohnny nodded. \"I've spent the past few years mostly overseas looking after our European interests, but I grew up in Bigtime and went to college here.\"\n\nFrom what I'd read and heard on the society circuit, Bulluci Industries was a massive corporation. The Bullucis had immigrated from the Mediterranean region a couple of generations ago, and they'd achieved the American dream of owning their own business and becoming fabulously wealthy. Besides Bella's fashion empire, the Bullucis had several oil wells, mining interests, olive oil plants, and even a couple of automotive and motorcycle factories.\n\n\"You went to Bigtime U? I don't remember you.\" And I certainly would have. My eyes traced over his firm body. Oh yeah. I would have remembered. And done everything in my power to make sure he remembered me too.\n\n\"I think it was a little before your time. I'm seven years older than Bella. You were probably still in high school when I was at Bigtime U.\"\n\nActually, I was probably still in Ireland at the time. We hadn't come to America until my mother died after a long battle with breast cancer. Seven years older than Bella? Let's see. That would make him about thirty-six, thirty-seven. Not too old. That was another mark in his favor, along with his surprisingly good choice of restaurant.\n\nI opened my mouth to ask Johnny another question, when a soft, breathy voice cut in.\n\n\"Well, if it isn't Fiona Fine, designer to the rich and famous.\"\n\nMy eyes narrowed. I knew that voice, and I didn't like the person it belonged to one bit. I looked up to find Erica Songe hovering over us. Erica was a news reporter for SNN, the Superhero News Network. She was one of SNN's rising stars, having gone from being the substitute late-night weather girl on the weekends to an evening news on-air personality in a matter of months.\n\nNow, I don't like reporters in general. They're always getting in the way when I'm trying to roast some ubervillains, shoving a camera in my face afterward, or attempting to uncover my real identity, just like Carmen had done. Why did they have to be so nosy and demanding all the time? I couldn't understand why reporters just didn't stay away and let me do my kick-ass thing. It would work out better for everyone. At least the photographers and videographers knew enough to steer clear of me. Or at least stay a couple hundred feet away. Ruin a few five-thousand-dollar cameras and people weren't so eager to take your picture anymore.\n\nBut Erica Songe was the worst of the worst. Just because she was on television, she thought that made her special. Erica had come into my store when she'd first gotten into town and ordered several pricey suits for her on-air appearances. The clothes she wanted were on backorder, along with everything else. The reporter had hit the roof when I told her that she'd have to wait just like everyone else. She thought she deserved special treatment just because she was a member of the media. Please.\n\nErica Songe had acted more spoiled than Joanne James ever dreamed of being. Screaming. Yelling. Threatening the staff. Demanding that we treat her like the princess she was. Or else. It had taken every bit of patience I had not to drop-kick Erica out onto the sidewalk. I'd settled for ripping up her orders and barring her from the store. For life. She tried to act cool about it, always coming up and speaking to me and being as sweet as honey whenever we crossed paths. But I knew she hated me. The feeling was mutual.\n\n\"Nice to see you too, Erica.\"\n\n\"Who's your handsome friend?\" she asked, her eyes zeroing in on Johnny.\n\nErica had long, black hair, blue eyes, and lips that had so much collagen in them they looked like they were about to explode. Her size-two figure was poured into a black dress that showed off as much tanned flesh as it concealed. Erica had a sultry, sex-kitten vibe to her that turned more than one head in the room, not all of them male.\n\n\"My handsome date is Johnny Bulluci.\" I reached over and put my hand on top of Johnny's, staking my claim for the evening.\n\nHis warm thumb drew lazy circles on the back of my hand. I looked up. Johnny's eyes were on me, not Erica. Score another one for Johnny Bulluci. I loved attentive men.\n\n\"Care if I join you?\" Erica asked.\n\n\"Actually, we were right in the middle of a very intense, very private discussion. We don't really feel like company tonight. Sorry, Erica. I'm sure you understand.\"\n\nA tight, annoyed smile curved Erica's oversized lips, and I knew she wanted to wrap her purse strap around my throat and strangle me with it. But her face smoothed over after only a few seconds. Nothing rattled her for long.\n\n\"Of course, Fiona.\" Erica drew a small piece of paper out of her handbag. \"Just let me leave my card with Johnny. In case he ever gets tired of...well, whatever.\" She licked her lips and gave him her best I'm-so-slutty-I'll-do-you-under-the-table stare.\n\nI snatched Erica's card out of her hands before Johnny could even reach for it and tore it into pieces. \"So sorry, Erica. The only number Johnny needs to remember tonight and for the foreseeable future is mine.\"\n\nErica's mouth opened and closed. She seemed shocked that someone would spoil her fun when she was trying to be cute and coy and seductive. Erica turned on her heel and stalked off. I watched her sashay over to the bar, where Kelly Caleb, another SNN reporter, perched on a round stool.\n\nThe two of them couldn't have been more different. While Erica's barely there clothes screamed trashy whore, Kelly wore a navy dress with a modest neckline that brushed the tops of her ankles. Kelly's blond hair was pulled back into a ponytail, and her face was free of the heavy makeup she wore on television. Erica had enough lipstick and eyeliner on to paint a clown's face.\n\nErica snapped her fingers, and the bartender rushed to get her a drink. Erica turned to Kelly, gesturing wildly and no doubt saying all sorts of horrid things about me. Kelly's gaze met mine, and her eye twitched, almost in a wink. The blond woman's wide lips quivered, as though she was trying not to laugh at Erica's tirade.\n\nI didn't particularly care for Kelly either. She was, after all, a reporter and thus the enemy, but I waved at her and smiled. Erica might think that she was the baddest bitch in the room, but she had nothing on me. I was Fiona Fine, for crying out loud. I'd been playing the society game in Bigtime before she'd even thought about having fat stuffed in her lips or saline in her breasts.\n\n\"Well that was interesting.\" Johnny leaned back in his chair and crossed his arms over his chest. \"I take it she's not a friend of yours.\"\n\n\"I can't stand that woman. Something about her just rubs me the wrong way.\" Actually, it was everything about her.\n\n\"Well, I'm glad you got rid of her. You're the one I asked to dinner, not her.\"\n\nJohnny's eyes met mine. My heart fluttered.\n\nOur food arrived, and we spent the next half hour eating, talking, and laughing. Johnny Bulluci had a teasing, devilish sense of humor I found refreshing after all the posing, pretentious playboys that populated the Bigtime society scene. He was frank, honest, and not above telling humiliating stories about himself. I particularly enjoyed the one about him stealing his father's car when he was thirteen and taking Bella to ride on the carousel in Paradise Park. And he had the most fantastic laugh, warm and throaty and deep. It made me smile, something I hadn't done much of in a very long time.\n\nThe only problem was that I polished off my grilled chicken, vegetables, a side salad, and half a bottle of wine before Johnny was even halfway done with his steak, baked potato, and cheddar cheese soup. And he didn't seem too interested in his food. He'd put his fork down ten minutes ago and was listening intently to everything I said. Now, I didn't really mind that, as it was always nice to be the center of attention, especially when that attention was coming from a handsome man. But ignoring a black-pepper-seasoned steak from Quicke's? That was just criminal.\n\n\"So, are you going to eat that?\" I asked, eyeing his half-finished steak.\n\nJohnny stared at the meat, then at me. My stomach chose that moment to let out an ominous rumble. Mount St. Fiera needed some fuel. I cringed. My enormous appetite and related bodily functions were the only things that ever really embarrassed me.\n\n\"Sorry,\" I said. \"I didn't have any lunch today.\" Lame, but it was the only excuse I could think of.\n\nJohnny laughed and pushed his plate over to me. \"I don't mind. I like a woman with a healthy appetite.\"\n\nHealthy appetite? Please. If he only knew.\n\nI polished off Johnny's steak, a basket of sourdough rolls, and an enormous chocolate souffl\u00e9.\n\n\"Didn't have lunch, eh?\" Johnny asked, amused by the stacks of plates, glasses, and silverware littering the table.\n\n\"Or breakfast either,\" I lied.\n\nWe finished with our meal. Johnny insisted on paying, which I found to be very old-fashioned and charming. I might be a modern, liberated woman, but it was nice to be treated to an evening out every now and again, especially since my big hunger pangs led to so many big bills. It was another point in his favor.\n\nOnce the check was squared away, we went out through the restaurant's side exit. We stood there in the warm twilight, listening to the cheery carousel music from the park. Now came the tricky part of the evening. Did I really want to invite Johnny back to my apartment? Sure, we'd made out at the wedding and shared a steamy kiss earlier this evening, but did I want to go the rest of the way? Was I ready to do that? Especially with someone I'd known only two days? It had seemed like a terrific idea at the wedding, but now, I wasn't so sure.\n\nOh, I liked sex, and I wasn't ashamed of it. I'd had my share of ill-advised, torrid affairs in college, including a couple of one-night stands. But with Travis, sex had come to mean something else entirely. It had been making love, sharing my body with someone who cared about me as much as I cared about him. Not to mention the fact that we'd been together five years. I wasn't sure I knew how to be with anybody else. My hand crept to my engagement ring. Or if I even wanted to, sex or no sex. There was always self-fulfillment, so to speak.\n\n\"How about a stroll through the park?\" Johnny suggested.\n\n\"That'd be nice,\" I said, relieved I wouldn't have to make a decision just yet.\n\nWe crossed the street and entered Paradise Park, one of Bigtime's biggest tourist attractions. The park featured just about every carnival ride and game in the known universe. A carousel, Ferris wheel, ring tosses, water guns, strength tests. Even now, after nine in the evening, thousands of people crowded into the park to eat funnel cakes, get dizzy and sick on the spinning rides, and try to win overpriced stuffed animals for their honeys.\n\nThe air smelled of popcorn and grease. Still hungry, I got the biggest cone of cherry-flavored cotton candy the vendor had. The pure, spun sugar melted on my tongue and quieted my stomach. There was nothing better than sugar for a quick boost of energy. It was why I kept an emergency stash of candy bars in my desk at work.\n\nJohnny and I wandered arm-in-arm through the park, just another couple taking in the sights and sounds. To my surprise, he seemed content to stroll along the gum-littered pavement alongside the harried parents, shrieking children, and bawdy vendors. All the other men on the society circuit would have blown up like a balloon if they so much as scuffed their polished wingtips. They would have sneered in disdain at the noisy families and demanded that the park be closed down just for their visit. Not Johnny.\n\n\"You know, this doesn't exactly strike me as your kind of scene,\" I said.\n\n\"Why not?\"\n\n\"From all appearances, you seem to be just another rich, spoiled playboy.\" Subtlety was another area where I was lacking. \"Yet here you are walking through a park instead of some posh art gallery.\"\n\n\"Oh, I'm all of those things,\" Johnny said. \"Rich, spoiled, a playboy. But I happen to like parks. Posh art galleries are so boring. Besides, you're one to talk.\"\n\n\"Oh?\"\n\n\"You seem...different from the other women I've met in Bigtime,\" he said.\n\n\"Different? Different how?\"\n\nHe shrugged. \"Like you've got more on your mind than what you're going to wear to the latest society bash. Sometimes, you look almost...sad.\"\n\nSad? My cool, haughty mask must be slipping. \"Sad?\" I forced myself to laugh. \"I'm a fabulously wealthy woman who designs fabulous clothes. What do I have to be sad about?\" Just because my fianc\u00e9 had been murdered by ubervillains was no reason to feel blue. Yeah right.\n\nJohnny stared at me with his gorgeous green eyes. \"I don't know. But I can see it.\"\n\nI didn't respond.\n\nJohnny bought two tickets for the Ferris wheel. We climbed on board and sailed up into the night sky. We went round and round and up and down. Finally, the ride stopped to let the lovers on board have a little private time with each other. Our cart was near the top of the ride, giving us a spectacular view of the park and the city lights. Everything looked fresh and clean from this height, and the lights resembled twinkling, colored stars that had fallen from the sky.\n\n\"It's just as beautiful as I remember,\" Johnny said. \"My father, James, used to bring Bella and me here at least once a week to ride the carousel and visit the animals in the zoo.\" A sad, wistful tone crept into his voice.\n\nI remembered what Joanne James had said about Bella's father dying. \"He passed away recently, didn't he? Your father.\"\n\nJohnny's eyes hardened. \"He was murdered. A couple months ago.\"\n\n\"I'm so sorry.\" I put a hand on his arm. \"I know what it's like to lose someone you love so brutally.\"\n\n\"Is that why you wear an engagement ring, even though there's no fianc\u00e9 around?\"\n\nSurprised, my thumb sparked. I clenched my fist to keep it from igniting. \"How do you know that?\"\n\n\"I asked around. There's no fianc\u00e9 or boyfriend in the picture. In fact, no one even knows who you were engaged to, according to Bella.\"\n\nI dropped my hand from his arm and twisted the ring around on my finger. Nobody had known about my engagement to Travis because we'd thought it would be safer that way. But he'd still died. Travis deserved better than to remain some anonymous, faceless figure from my past.\n\n\"I was engaged to a wonderful man. His name was Travis. He was murdered too.\"\n\n\"I'm sorry,\" Johnny said in a quiet voice. \"I didn't mean to pry.\"\n\n\"No, it's good to talk about him. My friends are afraid to ask me about him, to even mention his name. They don't want to stir up bad memories.\"\n\nJohnny put his arm around me, and I leaned into him. His shoulder felt warm and solid under my cheek. \"Tell me about him, Fiona. Tell me about Travis.\"\n\nSo I did. I told him how kind and caring Travis was, how much I loved him, our plans and hopes and dreams. The only thing I left out was our nighttime occupation as Bigtime's resident superheroes and how it had led to his death.\n\n\"What happened to the person who killed him?\" Johnny asked when I wound down.\n\nThe image of Malefica plunging into a vat of radioactive goo flashed through my mind. \"Oh, she got what she deserved in the end. What about your father? What was he like?\"\n\n\"He was a lot like your Travis. Kind, caring, considerate. He was a wonderful father. The only thing we ever argued about was\u2014\" Johnny cut off his words.\n\n\"Was what?\"\n\nHe hesitated. \"My taking over part of...the family business.\"\n\n\"I thought you'd done that already.\"\n\n\"Most of it, yes. But there was one area I wasn't particularly interested in. That's where my father and I disagreed. In fact, I hung up on him the last time we talked about it. It was the last time I ever spoke to him.\" Sadness and guilt tightened his handsome face.\n\nI squeezed his hand. \"Have the police caught the person responsible for his death?\"\n\n\"The police are useless. They can't do anything about it. But I can. And I will. The people responsible are going to pay. More than they ever dreamed of.\" The vehemence in his voice startled me. His eyes glittered with anger and a deadly promise.\n\nJohnny turned to me, and some of the anger melted away. \"But let's think about happier things. Here we are stuck on top of a Ferris wheel, the whole city at our feet. It would be a shame to let the view go to waste, wouldn't it?\"\n\nJohnny pulled me toward him. His eyes glowed like a lion's in the twilight, illuminating the chiseled planes of his face. My heart quickened. My lips parted. I leaned forward. He did the same. And then\u2014\n\nMy cell phone rang.\n\n# Chapter Seven\n\nWe stared at each other, frozen in an almost-kiss. My phone kept ringing out the old song \"Light My Fire,\" by The Doors. The loud, blaring rock tune could mean only one thing. Fiera was needed. My desire to kiss Johnny warred with my duty and responsibilities as a superhero. It was a short, tough battle, but duty won out. Drat. Why did my father have to raise me to always do the right thing?\n\n\"Please excuse me.\" I dug through my purse, grabbed my phone, looked at the caller ID, and tapped the screen. \"This better be good, Henry.\"\n\n\"Sorry to interrupt, Fiona, but we've got a tip on a possible location for Siren and Intelligal. The chief wants you here ASAP. We're going to go after them and see if we can shut them down.\"\n\nI was about to respond when Johnny's phone started to ring. I recognized the tune as \"I Won't Back Down,\" by Tom Petty. He gave me a sheepish grin and answered the call. Johnny listened for a moment, then turned away from me and started speaking in a low, hushed voice.\n\nI focused on Henry. \"All right, I'll be there as soon as I can. But tell the chief that he's going to pull one of my shifts for interrupting my date.\" Sometimes, I wondered whether my father used his psychic radar to monitor me a little too closely, especially now that Travis was gone.\n\n\"You're out on a date? That's wonderful, Fiona!\" I could almost see Henry beaming at me through the phone. \"It's about time you started dating again. Are you having a good time? Did he take you somewhere nice? He should try to make a good impression on the first date. Did he bring you flowers? Or chocolates?\"\n\nI rolled my eyes. Ever since he'd hooked up with Lulu, Henry thought he knew a thing or two about love. Like most men, the computer geek would be forever clueless.\n\n\"He made a terrific impression. See you soon.\" I hung up.\n\nJohnny finished his call and turned to me.\n\n\"That was Henry. He...keeps an eye on the store for me at night. There's a problem with one of my suppliers. I'm afraid I'm going to have to cut the evening short and go deal with it.\"\n\nJohnny shook his head. \"This seems to be the night for emergencies. Bella called. One of my business managers is concerned about a potential investor. Seems he wants to back out of a previously negotiated deal. I'm afraid I have to leave as well.\" Regret tinged his voice, as though he didn't want the evening to end.\n\nI didn't either. To my surprise, I'd been having a wonderful time, memories of Travis and all. There was more to Johnny Bulluci than just a handsome face and hard body. A lot more. I wanted to see him again. Soon. So, I told him.\n\n\"I'd like to see you again too, Fiona. Why don't we meet up Monday night?\" Johnny asked. \"Unfortunately, I have to work this weekend, or I would suggest we get together sooner.\"\n\nTonight was Thursday. Monday, Monday. I flipped through my mental calendar. Work in the morning, followed by an afternoon of PR as Fiera, then a benefit that night.\n\nI shook my head. \"I can't. I have to attend the annual fundraiser for the Bigtime Observatory. It'll be so boring, all those scientific, astronomer types talking about star charts and aliens and whatnot, but I already sent in my RSVP.\"\n\nIn addition to always doing my duty as a superhero, my father had also drilled it into my head how important it was to keep the promises I made, no matter how small or trivial they might seem. But that didn't mean I still couldn't have some fun with Johnny.\n\n\"You could meet me there, and we could sneak out afterwards,\" I suggested.\n\n\"It's a date.\"\n\nHe grinned, and my heart fluttered. The man had one hell of a smile. And great eyes. And a terrific body. And... The list went on for quite a while.\n\n\"So I guess this is good night,\" Johnny said.\n\n\"I guess so.\"\n\nI stared into Johnny's eyes. Then, I leaned in and pressed my lips to his. He put his arms around me and pulled me closer. This kiss wasn't like the others. Oh, it was still hot and passionate. But there was a gentle sweetness to it, a sense of letting down our cool, guarded, society fa\u00e7ades. The rich, charming playboy and the bitchy fashion designer were getting to know each other, faults and all. The Ferris wheel jerked and began to drift downward. We kissed until the ride stopped.\n\n* * *\n\nI had Johnny drop me off outside my store to keep up appearances. I waited until the limo's taillights disappeared, hailed a cab, and went back to my apartment to get my convertible.\n\nTwenty minutes later, I got off the interstate and turned onto a pothole-filled dirt road that seemed to end in the middle of the woods. I kept my foot on the gas, plunging the car into a dark stand of pine trees. My tires fell into a well-worn track, and my headlights bounced off a rock wall up ahead. I flicked them on and off five times in rapid succession. The wall slid aside, and I drove into a lighted tunnel, which stretched out for another mile before expanding into an underground garage.\n\nThe garage, like the rest of the underground Fearless Five compound, was hollowed out of the extensive network of caverns beneath Sublime. Wrenches, screwdrivers, and other tools lined the garage walls, along with cans of oil, wires, and other automotive parts. Five black vans huddled together inside the open space. They were our preferred mode of transportation when we went out into the city at night to fight crime and help with natural disasters. The vans resembled the one from the old _A-Team_ television show, only the F5 vehicles had more computer equipment and electronic gizmos stuffed inside them than collagen in Erica Songe's pouty lips. Thanks to Sam's money and Henry's technological genius, the vans were also bullet-, bomb-, and ubervillain-proof.\n\nHenry's small, white scooter was also parked inside the garage, along with my father's city-issued sedan. So was Lulu's van. I frowned. What was the computer guru doing down here this late at night? She should be at home trying to hack into the CIA mainframe or hijack the SNN nightly newscast or something equally geeky.\n\nI parked my convertible and strolled over to the metal door set into the far wall. The lights flickered, reminding me that I had only thirty seconds left before the garage would be filled with a potent sleeping gas. I punched in 555, the security code, on the blinking keypad and stepped inside. The door whooshed shut and locked behind me, another layer of security. If someone ever stumbled into the garage from the outside, she wouldn't be able to go any farther into the compound without the code. And once the gas took effect, we could transport her out of the garage without her ever being the wiser. At least, that was the theory. No one had ever breached our defenses before, except Carmen, and now Lulu.\n\nI strode down the halls until I reached the double doors that led into the library. The others were gathered inside, costumes on. My father wore a green-and-white outfit with a flowing cape that transformed him into Mr. Sage, while Henry's black-and-white uniform and matching goggles morphed him into Hermit. Lulu wore her usual getup of designer jeans and a Bulluci fleece pullover. The cobalt streaks in her hair gleamed under the flickering screen mounted on one wall. Blueprints winked on and off like a pair of bloodshot eyes blinking at me.\n\n\"Finally. We've been waiting almost an hour,\" Hermit said.\n\nI flipped my hair over my shoulder. \"I came as fast as I could. I had to get rid of my date first, you know. I couldn't just fire up and come running.\"\n\n\"Was he a little too hot for you to handle, Fiona?\" Lulu joked.\n\nI glared at the computer hacker. Another heat-related pun. Ha, ha, ha, ha. One of these days I was going to show her just how hot under the collar I could get. My fingers curled into loose fists. Red-hot sparks landed on the Persian rug under my feet.\n\nMy father stomped them out and put a restraining hand on my arm. \"Go get changed and meet us back in the garage. There's no time to waste.\"\n\nLulu zoomed out of the room, followed by Henry and my father. I jogged down to the equipment room, punched in the code, and went inside. I grabbed one of my many superhero suits hanging behind a pair of glass doors along one side of the room and the rest of my gear from another compartment. While I shimmied into the spandex, I glanced at a rack of swords sitting in the middle of the room. I thought about taking them along but decided against it. Weapons were Sam's\u2014Striker's\u2014thing. Not mine. My flames and fists were enough.\n\nFiera costume on, I retraced my route to the garage, stopping when I got to the kitchen. I glanced down the hall. Another minute wouldn't hurt. Besides, I needed to keep up my strength in case I got to toast Siren and Intelligal tonight. It was hard to do my sacred superhero duty on an empty stomach. I grabbed three bags of chocolate chip cookies from one of the kitchen counters and continued on to the garage.\n\nHermit was already in the driver's seat in one of the vans. The side door was open, and the engine idled. I started to get into the van when I realized Lulu was strapped inside. In my seat.\n\n\"She's coming?\"\n\n\"Of course,\" Hermit said, turning around. \"She helped me track down Siren and Intelligal. Since Karma Girl and Striker are on hiatus, I asked her to come along and man the van and comm links, while I help you guys in the field.\"\n\n\"Oh, goody.\"\n\nI slid into Karma Girl's usual spot in the very back and crossed my arms over my chest. Lulu ignored my hot stare, pulled out her laptop, and started banging away. My father turned around in his seat and gave me a look, clearly wanting me to play nice. That was something else I'd never quite learned to do. At least, not very well. Being an only child, I was used to getting my way. All the time. And I liked it.\n\nI rolled my eyes. But parental guilt could still make me do funny things sometimes, like be polite to a criminal mastermind. \"So where are our two ubervillains at?\" I asked, attempting to be marginally friendly.\n\n\"Right now, they're holed up in an abandoned warehouse on the edge of town, a couple of miles away from the old Snowdom Ice Cream Factory. Or what's left of it.\"\n\nIn other words, the ubervillains were squatting next to a big pile of rubble. The Fearless Five had spent some time at the factory last year during our run-in with the Terrible Triad. The factory had paid the price for being the site of a superhero-ubervillain battle. Not to mention the fact that Carmen had stuffed enough explodium bombs in the building to decimate the whole city.\n\n\"How did you find them?\" I asked. \"I thought they were untraceable or some such nonsense.\"\n\nHermit glanced at me in the rearview mirror. \"I created a new computer program to track the exhaust fumes from Intelligal's chair. It's powered with a combination of electricity, gas, and some sort of radioactive isotope I haven't identified yet. The point is that it lets off a particular combination of fumes and radiation.\"\n\n\"Clever as always, Hermit,\" Mr. Sage said.\n\n\"I couldn't have done it without Lulu's help,\" Hermit said.\n\nLulu smiled at her honey's praise, a goofy grin spreading across her heart-shaped face. She stared at his back, her eyes soft and dreamy. Her obvious affection for him almost made her part of the team. Almost.\n\n\"So what do you think they're doing at that factory? Braiding each other's hair?\"\n\nI snickered. More than likely, the two ubervillains were trying to pull it out. For partners in crime, they didn't seem to get along too well.\n\n\"I have no idea. Their crime wave doesn't make any sense,\" Hermit said. \"One day, they're stealing electronic equipment. The next, they're loading up on computer chips. The day after that, they're taking jewels from Bigtime's finest. I can't figure out what they're up to.\"\n\nI shrugged. \"Does it really matter? They're bad, we're good, and we're going to stop them.\" Being a superhero had always been rather black-and-white to me. No gray allowed. Nobody looked good in gray, especially superheroes.\n\nHermit drove through the streets of Bigtime, stopping at a red light. Suddenly, Swifte was there. The speedy, iridescent-clad superhero skidded to a stop in front of the van. He waved at us and dashed away to do whatever it was he did all by himself late at night. Before the light changed, three police cars hauled ass in the same direction. A white van bearing the SNN logo zoomed by a couple seconds after that.\n\n\"What's the problem?\" I asked.\n\n\"Big traffic accident on the southbound side of the interstate. Looks like fifty cars involved. Injuries, gasoline fires, people trapped in their vehicles, the usual,\" Lulu explained, staring at her computer screen.\n\n\"Should we divert?\" Mr. Sage asked.\n\nLulu pounded a few more keys. \"Nah. Swifte's already there, along with the Invisible Ing\u00e9nues. It looks like they've got things under control.\"\n\n\"How can you tell the Ing\u00e9nues are on the scene?\" I asked.\n\n\"Because they're not quite so invisible when they're covered with soot and ash. Look.\"\n\nLulu turned the screen around. Sure enough, I spotted two shapely outlines pulling people out of burning, crumpled cars.\n\n\"Good for them. Who knows? Maybe they'll finally get the recognition they deserve, and not be so _invisible_ anymore.\" I laughed. Lulu wasn't the only one who could do bad puns.\n\nThe others looked at me like I'd just sprouted a third eye. Lulu shook her head and went back to her typing. Mr. Sage stared out the window, while Hermit concentrated on his driving.\n\nI ignored them. It was funny. Johnny would have laughed at my joke. Johnny. My hands crept up to my lips. I could still feel his lips on mine, still smell his rich, exotic scent. Maybe I'd call him when I got home. Rich, spoiled playboys kept late hours. It might not be too late for a booty call\u2014\n\n\"We're here,\" Hermit said, stopping the van and killing the headlights.\n\nI peered out the front. We sat at the end of a long, deserted street. Buildings squatted all around us. Even though it was dark, I could see the graffiti on the walls and the broken doors and windows. Trash and debris cluttered the streets and alleyways, spilling out of the tops of metal cans and Dumpsters. A few streetlights flickered on and off, as though in pain.\n\n\"Where are we?\" I asked.\n\n\"About two miles east of Good Intentions Lane,\" Lulu said.\n\nThat explained it. Good Intentions Lane made a mockery of its own name. It was one of the worst streets in Bigtime and the major hangout of drug-running gangs like the Westsiders. Every couple of weeks, we had to come down here and clean out the riffraff. The gangbangers were like cockroaches, though. You could never get them all, and they always scuttled back out from their hiding places when the lights were turned off **\u2014** or when there were no superheroes or cops around.\n\n\"The factory's a block ahead on the right,\" Hermit said. \"I figured you could go through the front, Fiera, while Mr. Sage and I went in through the back.\"\n\n\"Works for me.\" We always sent the muscle in first. Me.\n\nWe piled out of the van. Hermit, Mr. Sage, and I activated the cameras in the _F5_ insignias on our costumes and shoved miniature two-way transmitters into our ears.\n\n\"Check, check,\" Lulu said, her voice crackling in my ear.\n\nWe checked back to her.\n\nLulu's slender fingers danced over the keyboard, and her laptop whirred and sputtered. \"All right. You guys are online. I've got visual and audio on all of you.\"\n\n\"Then let's rock 'n' roll,\" I said.\n\n\"Be careful,\" Lulu said.\n\nHermit squeezed her hand. \"Always.\"\n\nWe locked Lulu in the van and crept down the block. Discarded candy wrappers and empty soda cans littered the street, crackling under our boots. Enormous rats scurried back and forth in the debris, their eyes red pinpricks of light in the semidarkness. I glared at the odious little monsters. Maybe if I barbecued a couple, the others would get the message and run away. Probably not. Rats were nasty, tenacious creatures. Rather like ubervillains.\n\nThe wind shifted, whipping trash into the air and gusting down the street toward us. The smell of rotten garbage, wet fur, and fresh dirt filled the night. I also caught a whiff of something sickeningly sweet, almost like perfume. My nose twitched. For a second, I felt slow and limp and languid, like my body wasn't my own to command. Then, the scent was gone, overpowered by the filth around us. The weak sensation faded. My eyes scanned the street, but I didn't spot anything moving among the piles of trash, except for the foot-long rodents.\n\nWe rounded the corner of a building. A cracked parking lot stretched out in front of us, leading to a low, square building set back from the dilapidated street a couple hundred yards.\n\n\"This is it. This is where I tracked Intelligal's chair to. It used to be a television factory way back when.\" Hermit hit a few buttons on his handheld supercomputer. \"Here are the entrances and exits. You should go in through the front _here_ , Fiera. Mr. Sage and I will come in back _here_ , and we'll meet in the middle _there_.\" He pointed to various squiggles on the screen.\n\n\"Front door. Got it.\"\n\nWe huddled together in the shadows. Our eyes went to each other's faces. Despite our powers, we knew that one night all of us might not come back. It had happened before when we'd least expected it. And it could happen again. I put my fist over my heart. Mr. Sage and Hermit followed suit. Courage, all.\n\nThen, one by one, we eased toward the factory and the ubervillains inside.\n\n* * *\n\nI waited for Mr. Sage and Hermit to slip into the dark alley that ran along the side of the building before I scurried across the parking lot. I made it to the entrance without incident and glanced over my shoulder. Nothing.\n\nA set of uneven steps led up to the front door, which was boarded over with rotten-looking two-by-fours. I paused. Looking. Listening. Nothing. I started up the steps, which shrieked and groaned like a woman in labor under my weight. I winced. Good grief. I bet they heard that in the next time zone. I might have a variety of superpowers and skills, but moving quietly wasn't one of them. So, I did what I do best **\u2014** I put a little muscle into the situation.\n\nI yanked the boards off the building and tossed them over my shoulder. The door followed three seconds later. Darkness spilled out from the interior, and I stepped inside. I made my way through a series of abandoned offices that lined the front of the building. Moonlight shimmered in through the broken windows, painting the square, empty rooms with an eerie gray glow. The usual assortment of crushed cigarettes, empty beer bottles, and used condoms covered the floor. How romantic.\n\nI found a set of metal doors that led into the factory and peered through the grimy windows on them. The doors opened up into one enormous room. Assembly lines, pipes, and more snaked through the open space. I couldn't see the back wall for all the metal zigzagging everywhere. I couldn't see very far into the factory at all.\n\n\"Anything, guys?\" I whispered.\n\n\"Nothing so far, Fiera. We're at the back door ready to come inside,\" Hermit said in my ear.\n\n\"Lulu?\" I asked.\n\n\"Everything's fine, as far as I can tell.\"\n\n\"All right. Here I go.\"\n\nI tried the metal knob. Locked. Not for long. I gave it a good tug, pulled the knob off, yanked the door open, and stepped through. I paused. Waiting. Watching. Listening. Everything was still and quiet and hushed inside. I moved slowly, staying in the shadows and trying to keep my chunky boots from clacking too loudly on the hard floor. I went a hundred feet, then two hundred, then three. Pipes, pipes, and more pipes. The place seemed endless.\n\nI stumbled and almost cracked my head on a metal support beam. I muttered a curse and looked down. A power cable curled around my feet. Unlike everything else in the building, it wasn't coated with a layer of gray dust and cobwebs. It looked brand-spanking-new.\n\nAt last. A clue.\n\nKeeping one eye on the black cable, I moved farther into the factory. I realized it was getting brighter. There was a light up ahead.\n\n\"I think I've got something,\" I whispered. \"Lights in the factory.\"\n\n\"Then go with your hot self,\" Lulu replied.\n\nI rolled my eyes and kept moving. I rounded the edge of one of the assembly lines and stopped. I'd finally gotten to the belly of the beast, as it were. Halogen bulbs illuminated the scene. Computers, power cables, and more clustered together. Bits and pieces of metal littered the floor and a couple of steel tables. Soldering irons, wire cutters, and other tools crouched on wooden stools and plastic milk crates. Blueprints and schematics and weird diagrams hung on a corkboard. It was your usual ubervillain, mad-scientist lab. Intelligal was hard at work on something.\n\nSome sort of giant, radio-karaoke-like thing stood in the middle of the makeshift workshop. I spotted a cordless microphone, a bunch of wires, and what looked like a flat-screen plasma monitor. I squinted. Where those diamonds lining the thin screen?\n\n\"Are you getting this, Lulu?\" I whispered.\n\n\"Every last pixel,\" the computer hacker muttered. \"Although I don't have a clue as to what that thing is or does.\"\n\nI didn't really care what it did. When in doubt, smash. That was my philosophy. I raised my fist and focused, ready to punch through that metal like it was a piece of paper. Suddenly, I realized the hairs on the back of my neck were standing straight out. Not good.\n\nI hit the ground rolling. A blue lightning bolt zigzagged over my head and rattled off into the darkness. Now, I don't have my father's psychic abilities or even Carmen's inner voice, but after some fifteen years as a superhero, I know when an ubervillain's trying to get the drop on me. Siren had been quiet as death, but the electrical charge in the air from her powers had given her away.\n\nI popped up on my feet and turned around, a fireball in my hand. \"Hello, Siren. So nice to see you again.\"\n\nThe ubervillain stood behind me, looking as voluptuous as ever in her electric-blue suit. For once, the zipper was at a reasonable level, showing just a hint of cleavage. Evidently, Siren knew her charms would be utterly wasted on me. That or she was just cold.\n\n\"I wish I could say the same, Fiera. You and yours just keep showing up wherever Intelligal and I go,\" Siren purred. \"Wouldn't you much rather put those nasty fireballs away and have a nice talk with me? I'm sure we have a lot in common, being such strong, powerful women.\"\n\nHer husky tone wrapped around my body, slowly tightening its grip. I concentrated on the fire inside me, burned away the coils, and ignored the hypnotic pull of her voice. That trick was really wearing thin. I hated women who relied solely on their bodies to get them through life. I wasn't above using my feminine wiles to work things to my advantage every once in a while, but I also used my brains, brawn, and general bitchiness in equal parts as well. Besides lightning bolts, all Siren seemed to have was her body. What would the ubervillain do when her voice went? Or when her bazooka boobs started to sag? She wouldn't be nearly so alluring then.\n\n\"Honey,\" I said. \"You're going to have to do a lot better than that. Your two-bit persuasive thing might work on weak-willed, sex-starved men, but not me. Where'd you learn that trick anyway? Hookers High?\" When in doubt, taunt the enemy. It always worked on the society circuit.\n\nSiren's eyes glittered with rage. A crackling energy ball popped into her hand. Her white, French-tipped nails curled around it like she was caressing her lover. \"I could just electrocute you.\"\n\nMy body burst into flames. \"You could try.\"\n\nA loud roar filled the air, cutting off Siren's clich\u00e9d retort. Good grief. It sounded like a couple of tanks were rolling my way. The floor trembled beneath my feet.\n\n\"Lulu, what the hell is that?\" I shouted.\n\nLulu squawked something in my ear, but I couldn't hear her over the rumbling. Ten seconds later, I got the answer. A sleek, silver-and-black motorcycle grumbled to a stop in between Siren and me. A motorcycle? In the middle of a factory?\n\nA figure clad in black leather pants sat astride the huge vehicle. Angel wings shimmered on the back of his tough-guy jacket. Johnny Angel. He was a minor player in the superhero-ubervillain scene in Bigtime, not really on one side or the other. Angel was more intent on riding his motorcycle around the city late at night and raising hell than anything else. The local motorcycle gangs loved him and often tagged along, whooping and hollering and making pests of themselves.\n\nBut Angel wasn't all bad. Sometimes, he helped out people who needed it. Damsels in distress mostly. Women about to be mugged. Women who were carjacked. Women fleeing abusive boyfriends. To them, he was an angel. To the Fearless Five, he was a minor blip on the radar screen.\n\n\"Well, if it isn't Johnny Angel. Back from the dead so soon?\" Siren cooed. \"And here I thought Intelligal had finished you off for good with her heat-seeking missiles.\"\n\nI frowned. What the hell was Siren talking about? Johnny Angel wasn't dead. He was sitting right there between us\u2014\n\nAngel swung one leg over his bike, and I caught a glimpse of his face. His smooth, unlined face. With a start, I realized that he was around my age. Usually, this wouldn't be of any importance, as most superheroes and ubervillains fell in the twenty-to-forty age range, with a few exceptions like my father, who was creeping up on fifty-five, or the Tween Terrors, who hadn't hit puberty yet. But the last time I'd seen Johnny Angel, he'd been limping down an alley, struggling to keep up with the thug he'd been chasing. Lines and sweat had painted his strained, red face. Gray had glistened in his brownish hair. He'd moved slowly, as though every step hurt.\n\nHe'd been an older man, just like my father.\n\nNot this guy. He moved with a loose, easy confidence. Blond hair. Tan face. Light eyes. Black-and-silver, wing-shaped mask. That was all I saw before Angel turned to face Siren. I frowned. Johnny Angel might not be an ubervillain, but he should know better than to turn his back on me. I was Fiera, for crying out loud. I could melt his ass like a candle, if I wanted to.\n\n\"There's a new Angel in town,\" he snarled in a low, throaty voice. \"And you're going to pay for what you did to my predecessor.\"\n\nSiren put her hands on her hips and laughed. The dulcet tones rang like church bells, all sweetness and light. \"Now why would you want to hurt little ole me?\" Siren lowered the zipper on her suit, exposing more of her buoyant chest. She focused her eyes on Angel like a snake trying to hypnotize a helpless bunny. \"I'm just a simple girl who's trying to make a name for herself in the big, bad city.\"\n\nYeah right. And I was the freaking Tooth Fairy.\n\nAngel stilled, struggling against the pull of Siren's voice. Since the ubervillain's attention was focused on Angel, her tone didn't bother me as much as usual. I circled to one side, easing to the left of them.\n\n\"Are you guys hearing this?\" I muttered.\n\n\"Every word,\" Mr. Sage whispered in my ear. \"We're almost there to back you up. Let them keep talking.\"\n\nSiren and Angel stared at each other, ignoring me. I took the chance to study the man. He was taller than his predecessor, his skin darker, his hair longer. My gaze swept lower, tracing over his leather jacket and the pants he'd poured himself into. And he was definitely in much better shape than the previous Angel. The old guy had a bit of a potbelly.\n\nNow, I'd run into my fair share of superheroes and ubervillains over the years. I'd seen more buff bodies, rippling abs, and bulging biceps in skintight spandex than I could remember. But Angel gave buns of steel a whole new meaning. I frowned. But there was something strange about his skin. It almost looked hard\u2014\n\n\"Am I interrupting something?\" a tight, cold voice cut in.\n\nMy head snapped up. Intelligal hovered over us in her chair. Damn. I'd forgotten about her during my perusal of Johnny Angel's new, improved body.\n\n\"It's about time you got here,\" Siren snapped. \"Take care of them. Now.\"\n\nIntelligal hit a button, and two missiles spewed out of the depths of her chair.\n\nRight at Angel and me.\n\n# Chapter Eight\n\nI immediately hurled a fireball at my incoming missile. The two met in midair and exploded. Smoke, ash, and sharp bits of flying silver metal filled the factory. I ignored the smoldering debris and turned to help Angel. But it was too late. The missile was bearing down on him. Angel was going to get blown to hell and back in two seconds.\n\nBut he didn't run or duck or even try to leap out of the way. Instead, Angel curled his hands into fists, held them down by his legs, and flexed. Please. All the chiseled muscles in the world weren't going to save him from a rocket. The missile hit him in the chest and exploded.\n\nI threw up my hands to ward off the bloody man-chunks and brain matter that were coming my way. To my surprise, nothing hit me, except a few bits of metal shrapnel that stung my hands and arms. Dust and soot hung like thick clouds in the air. I waved the smoke away from my face and squinted through the soupy fog.\n\nMy mouth gaped open. Johnny Angel stood in the exact spot where the missile had hit. He was still alive, although his bike was a smoldering piece of melted junk now. Incredible.\n\n\"Is that all you've got?\" Angel mocked.\n\nThe explosion had ripped into his white T-shirt, black leather jacket, pants, and boots, but the man himself didn't seem to have a scratch on him. His hair wasn't even mussed. Now _that_ was a neat trick.\n\n\"Do you care to explain this, Intelligal?\" Siren hissed, taking a few steps back.\n\n\"He...he should be dead!\" Intelligal sputtered. \"Those are hundred-load explodium missiles! He should be nothing but a stain on the floor!\"\n\nAngel kicked aside his melted motorcycle and stepped forward. \"Well, I guess it's a good thing I have a superstrong exoskeleton then, isn't it?\"\n\nMy eyes narrowed. Wait a minute. The old Angel hadn't had any superpowers that I'd known of, just a souped-up motorcycle. It looked like this guy was the new and improved model. My eyes strayed back to the hard body. In more ways than one.\n\n\"Don't just hover up there!\" Siren screeched. \"Gas them, you fool!\"\n\nI started forward to body slam Siren into next week when a puff of light blue smoke exploded in my face. I coughed and gagged and tried not to retch. It was the same odd smell that I'd inhaled before outside the factory. The sweet, noxious stench worked its way down my throat and into my lungs. My arms and legs felt like gelatin, all loose and wobbly. Beside me, Angel choked and sputtered. We both slid to the ground, caught in a cloud of blue gas. I tried to focus. Where were Hermit and Mr. Sage?\n\n\"Hurry up! Get it loaded!\" Intelligal's voice drifted through the fog.\n\nMetal screeched. Something hit the floor. Siren cursed.\n\nSomehow, I rolled over onto my hands and knees. Those bitches didn't know who they were dealing with. I was Fiera. Member of the Fearless Five. Protector of the innocent. Superhero du jour. A little knockout gas wasn't going to keep me from kicking their asses. Slowly, very, _very_ slowly, I crawled forward. Metal dug into my hands and scraped my knees, but I ignored the pain and kept going. After about twenty feet, I broke free of the cloud. I drew in a deep breath of soot-filled air. I still felt weak and disjointed, but some of the feeling returned to my arms and legs.\n\nToo late. Siren stuffed something in the side of Intelligal's chair and hopped on the arm. The helicopter rotor popped up out of the back, and the chair whirred and flew right over me. I pointed my finger at them, but I couldn't find the strength to muster up so much as a single spark. I couldn't even roast a marshmallow, the shape I was in right now.\n\nA pair of familiar green boots flashed in front of my face. \"Fiera! Are you all right?\" Mr. Sage asked, his eyes bright with concern.\n\nI coughed some more. \"Johnny Angel...still...in...there,\" I wheezed. My tongue felt like I'd eaten a whole tube of sugar-flavored toothpaste, sticky and gooey sweet. Yuck.\n\n\"I'll get him,\" Hermit said.\n\nHe took a breath, held it, and plunged into the blue gas.\n\n\"A little help, please!\" Hermit called out. \"He's too heavy for me!\"\n\nMr. Sage reached into the cloud and helped drag Angel free of the fumes. I got to my feet, wobbling back and forth like a newborn fawn. The others weren't in much better shape.\n\nSomething smacked against the front of the building. The windows that weren't already busted out shattered with a collective roar. The factory quivered and trembled like a feather floating on the breeze.\n\n\"Intelligal's firing explodium missiles at the factory! The whole place is going to come down on top of you! Get out! Get out now!\" Lulu screamed in my ear.\n\nI took another breath. My head cleared, and most of my strength returned. I put my arm under Johnny Angel's shoulder, and my knees threatened to buckle. Hermit was right. He was heavier than he looked. The other superhero and I half carried, half dragged Angel toward the back of the factory.\n\n\"Back there!\" Hermit shouted, pointing toward a door a couple hundred feet away.\n\nMr. Sage stopped long enough to grab some of Intelligal's weird schematics and drawings, then followed us.\n\nA hundred feet to go. More explosions ripped through the building. Pipes creaked and snapped like dry twigs, while the floor ripped and bucked like water.\n\nFifty feet. Fire shot through the air above our heads. Smoke seared my lungs.\n\nTwenty, ten, five...\n\nWe stumbled out the door and down some steps. The night air, still reeking of garbage, revived me the rest of the way, and the fire inside me flared back to life. Behind us, the building moaned and creaked. The front section collapsed with a low, long groan, and the resulting shockwave threw us fifteen feet forward. We scrambled up and kept hobbling along, trying to get clear of the dust and debris.\n\nAbout a quarter mile away, we lunged out into the street and stopped. Angel slumped against the side of a nearby building. Hermit put one knee on the ground, while Mr. Sage rested a hand on his shoulder. I put my hands on my hips, trying to get my breath back. Superstrong exoskeleton indeed. Superheavy was more like it. The four of us stood there, gulping down air.\n\nAfter a few minutes, Johnny Angel straightened. His eyes swept over us. Then, he turned and started to walk away.\n\n\"Hey! Come back here!\" I grabbed his arm.\n\n\"Let go of me!\" he snapped.\n\nHe tried to shake me off, but I tightened my grip. It was like trying to squeeze a cement block, but I held on. Superstrength came in handy sometimes.\n\n\"Who are you? What happened to the old Johnny Angel?\"\n\n\"None of your damn business.\"\n\nI pointed a smoking finger at his chest. His skin felt like solid steel through the thin fabric of his ripped, blackened T-shirt. \"When you waltz in when I'm trying to apprehend two ubervillains and almost get my friends and me blown to smithereens, then you make it my business. What the hell did you think you were doing in there?\"\n\nHis face hardened. Angry gold flecks sparked in the depths of his eyes, which glowed a rich green. \"None of your damn business.\"\n\n\"We're not going to hurt you, if that's what you're worried about,\" Mr. Sage said in a calm, soothing voice. \"We only want to talk.\"\n\nHermit nodded. Smoke smudged his thick goggles.\n\nJohnny Angel stared at the three of us. Outnumbered. He nodded. \"All right. We'll talk.\"\n\nHe stared down at my hand on his arm. I dropped it, even though I didn't want to let go.\n\nAngel leaned back against the side of the pockmarked building. From his pants pocket, he fished out a pack of cigarettes and a black lighter that had somehow survived Intelligal's missile. Silver angel wings gleamed on the dark surface. He lit his cigarette, drew in a long drag, and blew smoke out his nostrils, showing us what a badass he was. Please. One snap of my fingers, and that cancer stick would blow up in his face like a grenade. I should light up the cigarette anyway. Didn't he know those things would kill you? Or perhaps his lungs were as hard as his chest. And his head.\n\nAfter another couple of puffs of his cigarette, Johnny spoke. \"I'm the new Angel. I have been for a few months now, ever since Siren and Intelligal killed my predecessor.\"\n\n\"We hadn't heard. How did it happen?\" Mr. Sage asked, his voice soft and kind. My father. Always the diplomat.\n\nAngel stared at the ruined factory, which smoldered behind us. \"The two muscled in on some of the High Riders' territory. The motorcycle gang asked the old Angel to get rid of them. He owed the gang a favor, so he took the job. It was going to be his last gig, since he'd been training me to take over as Angel. One night, he tracked down the ubervillains and confronted them. He told them to get out of town or else. They just laughed at him, and Intelligal fired off some of her heat-seeking explodium missiles. He tried to outrun the missiles on his motorcycle, but he never had a chance.\" Sad, bitter anger colored his low, throaty voice.\n\n\"Because he didn't have superpowers,\" Hermit said.\n\nAngel nodded. \"All he had was his bike. It wasn't fast enough.\"\n\nI stood to one side and listened to the question-and-answer session. Generational superheroes and ubervillains were rare, but not unheard of. Fathers and mothers who moonlighted for the greater good or evil often passed down their powers or some variation thereof to their children. When the old folks got tired of fighting or creating crime, the youngsters donned the cape and spandex and took up the family mantle. It didn't even have to be a family member. More than one aging hero and villain had plucked a young orphan off the streets, trained her, and introduced her to Bigtime and the world as the new whomever. Pistol Pete and Hangglider were classic examples. If I remembered correctly, Johnny Angel had been around in three incarnations now, counting the new guy.\n\n\"But you have powers, don't you?\" Mr. Sage asked. \"I believe you mentioned something about an exoskeleton?\"\n\nAngel rolled up the sleeve of his tattered T-shirt and flexed his bicep. At first, I couldn't see the difference between his bicep and anybody else's. Well, it might have been a bit more defined than the average superhero's. But when I looked closer, I realized there was a hard look to his skin, almost like it was stretched over steel. Plus, I could just make out a faint, weblike pattern in it.\n\n\"When I concentrate and focus on what I want to do, I'm impervious to pretty much everything. Missiles, guns, grenades, lasers.\" Angel's eyes flicked to me. \"Fire.\"\n\n\"Fascinating,\" Hermit murmured, tapping a few buttons on his computer and angling the camera embedded in his suit for a closer look.\n\n\"We're very sorry for your loss,\" Mr. Sage said. \"Believe me, we know how hard it is to lose a member of your team. But killing Siren and Intelligal won't bring back your friend and mentor. And it won't ease your pain. We need to capture them and turn them over to the police before they hurt someone else. That's what superheroes do. It's our duty. Dishing out your own brand of vigilante justice isn't the way.\"\n\n\"Turn them over to the police? Why? So they can spend a couple of weeks in prison before they break out and come back to Bigtime? The police in this town are useless. They always have been, and they always will be,\" Angel scoffed.\n\nOuch. I glanced at Mr. Sage. His face was smooth and unreadable, but his eyes glittered. Being the chief of police, he didn't think his men were useless. Quite the opposite. They provided valuable support to the superheroes in town, as well as taking care of the more ordinary, mundane crimes.\n\n\"I'm going after Siren and Intelligal, and there's nothing you can do to stop me.\" Angel's green eyes shimmered in his tight face. \"Get in my way, and you'll be sorry.\"\n\n\"What are you going to do?\" I asked. \"Kill us too?\"\n\n\"If you get in my way\u2014yes. I will kill you, superheroes or not. I won't hesitate. Not for a second. Remember that. Or suffer the consequences.\"\n\nWith those words, Angel turned and stalked into the darkening night.\n\n# Chapter Nine\n\n\"Can you believe the nerve of that guy? Telling us, the Fearless Five, to stay out of his way or he'll kill us? He must have a pair of steel balls on him to go along with that exoskeleton,\" I snapped.\n\n\"Fiona,\" my father chided. \"There is no need for such language.\"\n\n\"Well, he must,\" I muttered.\n\nAfter Johnny Angel huffed and puffed and stormed away, Lulu had brought the van around, and we'd gone back to Sublime. Now, we were gathered in the underground library, reviewing the mission and planning our next move. The others clustered in front of the film screen, staring at the footage from inside the factory. I sat in the corner, munching on the cookies I hadn't had a chance to eat in the van, along with a couple of cheese burritos, a tray of tortilla chips and hot salsa, Spanish rice, refried beans, and a gallon of raspberry iced tea. The fight with the ubervillains and dragging Mr. Two-Ton Ego around had zapped a lot of my energy, so I'd made Henry stop at Ol\u00e9, an all-night Mexican joint, before we headed home.\n\n\"Right now, I'm more worried about that doomsday-looking contraption the ubervillains were building. Not to mention whatever it was they gassed you with, Fiona,\" Lulu said.\n\n\"Gas, missiles, flying chairs, weird inventions. It comes with the territory.\" I took another bite of my cheese burrito. Spicy Monterey Jack cheese melted in my mouth. Ah, so good.\n\n\"Speaking of the gas, I collected a sample for the chief to analyze,\" Henry said, holding up a glass tube full of light blue fumes. \"I got it while I was trying to drag Angel out of the cloud. It looks like a fascinating chemical compound.\"\n\nOnly Henry would stop in the middle of a battle to take a scientific sample of the enemies' weapon of choice. I rolled my eyes. I loved him like a brother, but he was such a complete geek.\n\n\"Judging from the effect the gas had on Fiona, I'd say it contained some sort of superpotent muscle relaxer,\" Chief Newman said, lacing his long fingers together. \"Perhaps even a power inhibitor.\"\n\n\"Smelled like cheap perfume to me,\" I said. \"As befits Siren.\"\n\nMy nose twitched. I'd taken a steamy shower and washed my hair twice, but I could still smell the gas. It clung to my body like Joanne James on a widowed billionaire's arm. Even the extra onions and hot sauce on my burritos couldn't quite overpower the sweet, jaw-locking taste in my mouth.\n\n\"Whatever it is, I need to make some sort of antidote for it,\" the chief continued. \"Especially since it didn't seem to affect Siren and Intelligal in the slightest.\"\n\nI looked at my father. \"Whatever you do, just make sure it doesn't taste like toothpaste.\"\n\n* * *\n\nI spent the next few days working. During regular business hours, I finished up the business-wear line for fall and took care of a thousand other fashion-related details. Then, at night, it was time to suit up as Fiera, answer my fan mail, and squash the evildoers who crossed my path.\n\nThe rest of the Fearless Five worked just as hard as I did at their day jobs, and at night, we all focused on Siren and Intelligal. The chief started cooking up some antidote to neutralize the stuff they'd gassed us with, as well as putting the final touches on some earplugs to block Siren's hypnotic voice. But Henry couldn't figure out what their radio-like device did, and Lulu couldn't find whatever abandoned building they were holed up in.\n\nStill, we searched, prowling the streets in the F5 van. A few times, Johnny Angel zoomed by on a new, shiny, silver bike. We tried to flag him down, but he always ignored us, the hardheaded, egotistical snot. We even gave chase in the van a couple of times, but he was always too quick for us to catch or darted into some narrow space where we couldn't follow. Show-off.\n\nAs for the other Johnny, Johnny Bulluci, I didn't see him, since we were both so busy with work. That didn't stop me from thinking about him, though. I replayed our date over and over again in my head. He was a rich, spoiled playboy. Smooth, suave, used to getting his own way. But he'd also been kind, considerate, and caring. How many guys would have sat there listening to a woman gush about her dead fianc\u00e9? Not many. We'd clicked, connected that night in the park. I was beginning to think maybe Johnny Bulluci had potential beyond being Rebound Guy.\n\nJohnny hadn't forgotten about me, either. The day after our date, he had a black-pepper-seasoned steak from Quicke's delivered to my office. I'd squealed so loudly when I opened the package that Piper thought I'd gotten a box full of diamonds or something equally expensive. I'd returned the favor by sending two tickets to the Ferris wheel and a funnel cake to Bulluci Enterprises. Two hours later, I'd received a stick of cotton candy and a stuffed purple rabbit. We exchanged gag gifts the rest of the day. A sense of humor was another thing I liked in a man. Life was too short to be taken seriously. Especially when you went around town wearing body-hugging, orange-red spandex and shooting fire out of your fingertips.\n\nMonday rolled around. I left the final details of the fall collection in Piper's capable, anal-retentive, obsessive-compulsive hands and went home early to get ready for my latest public appearance as Fiera.\n\nEvery couple of months, SNN, _The Expos_ _\u00e9_ , _The Chronicle_ , and the other media outlets in the city sponsored _Meet Bigtime's Superheroes_ in Paradise Park. The events were a chance for regular folks to mix and mingle with their favorite superheroes and learn about our powers, as well as raise money for some worthy causes. The members of the Fearless Five took turns attending the functions, and I was the lucky superhero today. I didn't mind too much, though. It was fun to show off in front of people. Kids, in particular, always got a kick out of my fireball-juggling skills.\n\nOnce I was suited up, I flipped a switch just inside the door. It looked like any other light switch, but it was a special feature Henry had installed for me when I'd moved into the apartment. The switch sent out a signal that scrambled the building's security cameras and locked down all the elevators except for the one I used for five minutes. Since I had the whole floor to myself, I didn't have to worry about nosy neighbors\u2014just about getting out of the building without being seen.\n\nI rode the elevator down to the fifth subbasement, the very bottom of Tip-Top Tower. The doors pinged open, revealing a dark space filled with metal trash chutes and overflowing dumpsters. Black mold covered the damp walls, and the air smelled of rotten pizza, greasy French fries, and overripe fruit. Tip-Top Tower might be one of the most exclusive buildings in Bigtime, but it still had to have a place to collect tenants' trash, just like every other apartment complex. Nobody, not even the maintenance men, came down here unless they absolutely had to. Lucky for me.\n\nI walked to the very back corner of the subbasement, where a metal door stood. The lock and knob were long gone, and the door had a smushed look to it, like a bag of chocolate that had been out in the sun too long. I pointed my finger at the door, letting out a steady stream of concentrated flame, and traced along its borders. The metal heated up in seconds, casting a red-hot glow onto the rotting trash. I pulled the door open, stepped through, and repeated my welding process, sealing it behind me. The building managers thought the door had been melted shut by a long-ago gas explosion. They didn't know it was my secret way out of the building. Slipping out of my apartment in my costume unseen by the other tenants was a necessary skill that I'd perfected over the years, especially since I couldn't just go up to the roof and fly away like Hangman and Rocket Ron and the other winged types.\n\nI strolled through the dark passageway, using my flaming hand as a torch. The passageway twisted and turned underneath a couple of city blocks. It was part of a network of old tunnels that ran under most of the downtown area and was one of the main reasons I'd chosen to live in Tip-Top Tower.\n\nA pinprick of light flashed up ahead, and I dimmed my hand. The tunnel opened up into an alley close to the Bigtime Public Library. A few weeds and some loose bricks cluttered the entrance, along with the backside of a metal Dumpster. I shimmied through the space between the Dumpster and the alley wall. I peeked around the corner of the metal container, but I was the only person in the alley. With its Dumpsters, bits of smelly garbage, and cracked pavement, the alley wasn't the sort of place most people would willingly walk into, even during the middle of the afternoon.\n\nThere were other ways out of my apartment building, but I'd found this to be the quickest and safest. Even if someone moved the Dumpster and stumbled into the passageway, they'd never get through the door at the far end without some sort of superpower. Even then, there was nothing in the tunnel that could be traced back to me. I wasn't dumb enough to leave costumes lying around or stupid enough to write FIERA WAS HERE on the walls.\n\nOnce I was satisfied that no one was watching, I slipped out into the main street with the city's other citizens. Most folks didn't give me a second look. There were so many superheroes in Bigtime these days that people didn't get too excited unless we were going fist-to-fist with ubervillains on top of a skyscraper somewhere. Still, a few people stopped and asked me for autographs, which I graciously signed.\n\nI made my way to Paradise Park. An enormous banner across the front entrance proclaimed that it was _A_ _Super Day for Superheroes!_ Thousands of people, parents with kids mostly, along with the requisite fanboys, crowded into the park anxious to see their heroes in the flesh and fur. And of course, some folks from _Slaves for Superhero Sex_ had shown up. _SSS_ was a cult group whose members put themselves in danger in order to get close to heroes and villains. You could spot them a mile away. They always thought they had to dress up in cheerleading, French maid, and other costumes to attract the attention of their favorite hero. I thought they were all fruitcakes.\n\nI headed for the check-in station, where a twentysomething woman carrying an enormous clipboard barked orders at her teenage underlings. A headset clung to the side of her square sunglasses, while pens, paperclips, a stun gun, and more hung in the mesh khaki vest that covered her chest. Abby Appleby. She stabbed a pen at a boy who didn't look old enough to shave yet.\n\n\"You! Go take the Caffeinator a fresh supply of chocolate bars and coffee. Now!\"\n\nThe poor kid grabbed a box of candy and a bag of beans and scurried off to do her bidding like he had jets attached to his sneakers.\n\nAbby zeroed in on me and plastered a smile on her face. \"Fiera. Glad you could make it. If you'll follow me, I'll show you to your booth. Everybody else is here already.\"\n\nAbby Appleby was one of the premiere event planners in Bigtime. She did everything from weddings to birthday parties to funerals. Abby had a reputation for throwing fantastic events, as well as bringing everything in on time and under budget\u2014no matter how many toes and legs and other things she had to break to get it done. She was also one of Piper's best friends. It was no wonder, given how tightly the two of them were wound.\n\n\"Here we are,\" Abby said, stopping in front of a wooden booth decorated with orange-and-red cardboard flames and the _F5_ insignia.\n\nNext door, a tall, Nordic-looking woman created snow cones with her bare hands, covered them with sugary, blue goo, and dished them out to a line of anxious kids. A giant snowflake flashed like a strobe light on her ice-blue costume.\n\n\"You're putting me next to Wynter again?\"\n\nWynter looked up from her snow cone making. Her blue eyes frosted over at the sight of me. Fire and ice never mixed, and Wynter and I had never gotten along. She was a bit cold and distant for my liking. I'd never once seen her smile or laugh, not even when she was taking out Hot Stuff, her archenemy.\n\n\"Sorry,\" Abby apologized, pointing to another booth. \"The only other spot I have is down at the end with Halitosis Hal, and you know I can't put the two of you together.\"\n\nI followed her finger and spotted the other superhero. Halitosis Hal was a short guy with a wide but solid frame. His costume was a putrid green color, but it worked with his dark skin and hair. He was busy handing out gas masks to the folks lined up in front of his booth, which was decorated with cardboard cutouts of garlic, anchovies, and other smelly things.\n\nNo, Abby couldn't put me down there. As his name suggested, Halitosis Hal's superpower was his superbad breath. One whiff of it could stop the strongest ubervillain in his tracks. In addition to smelling like something that died two months ago, Hal's breath also contained a mixture of nitrous and other flammable gases. Whenever we got too close to each other, things blew up. And not in a good way.\n\n\"I understand,\" I said, slipping into the booth.\n\nFor the next three hours, I juggled fireballs, made my fingers flash like sparklers, bench-pressed parents, and scampered around in my skintight spandex costume, much to the delight of shrieking kids and fawning fanboys. I, of course, was among the more popular superheroes in attendance. Folks lined up three-deep around my booth to see me light myself on fire, get autographs, and snap up official Fiera merchandise.\n\nEverything was going fine until Kelly Caleb arrived. She had her game face on today\u2014a smart red suit, perfect makeup, flawless hair. She was like a blond version of Erica Songe, except her suit wasn't slit down to her bellybutton and up to her hips. Kelly wandered through the rows of booths with her SNN cameraman, chatting and shaking hands with the superheroes like they were her close, personal friends. She even hugged Halitosis Hal and Pistol Pete. Hugging. Please. Superheroes did not _hug_ people. At least, not this superhero.\n\nThe news reporter stopped in front of my booth and gestured for her cameraman to shoot some video of me juggling fireballs. I thought about using one of them to melt the television camera but decided against it. I'd promised my father that I wouldn't set anything on fire at the park today. I supposed that included nosy news reporters.\n\n\"Fiera.\"\n\n\"Kelly.\"\n\n\"I was wondering if I could have a moment of your time.\" The news reporter gave me her trademark toothy grin.\n\n\"I'm sort of busy right now.\"\n\nMy pointed tone didn't faze her. \"It will only take a few minutes. You and the other members of the Fearless Five haven't given me an interview in months, ever since you introduced Karma Girl as your new member. I'm beginning to think you're ignoring me on purpose. And I'd hate for that to be the case. Heroes like Swifte give me all the interviews I want. Since you're so busy, maybe I should go see if he's here. I'm sure he'd love to be the featured lead on the evening news, especially since he's got that new video game to promote. From what I hear, it's selling extremely well. Even better than your action figures.\"\n\nKelly's voice was mild, but her tone was just as sharp as mine had been. The message was clear\u2014play nice or else. If it had just been me, I would have ignored her. But I had a duty to the rest of the F5 team to make us look good **\u2014** whether I wanted to or not. Unfortunately, duty was about doing a lot of things you didn't want to.\n\nSo, I snuffed out the fireballs and plastered something that resembled a smile on my face. \"Well, I suppose I can give you a few minutes.\"\n\nKelly shoved a microphone at me and peppered me with questions. What I and the other team members had been up to. How Karma Girl was fitting in with the rest of the Fearless Five. If we'd heard anything from Malefica and the other members of the Terrible Triad. She skillfully moved from one topic to the next and crammed in more questions in five minutes than most reporters could get to in half an hour. Even I couldn't deny that Kelly was good at her job. She was still a reporter, though. One that I didn't like, no matter how fabulous she looked in the suit that I'd designed.\n\nAfter another round of nosy questions, Kelly wrapped up the interview and told her cameraman to stop shooting.\n\n\"Now, that wasn't so bad, was it?\" Kelly asked.\n\n\"No,\" I muttered. \"I suppose not. It could have been worse. Erica Songe could have been here.\"\n\nI hadn't meant for Kelly to hear me, but she did. The reporter's face tightened at the mention of the other woman.\n\n\"Not a fan either?\" I asked.\n\n\"You know we're all just one big, happy family at SNN.\" Kelly smiled, although it looked more like a grimace. \"Erica has her uses.\"\n\nI couldn't resist twisting the knife a little. \"Like covering Black Samba and Granny Cane rescuing those trapped firefighters a couple days ago? I saw the news and was surprised to see Erica on the scene. I thought superheroes and ubervillains were your exclusive beat at SNN. Or is that one of Erica's uses?\"\n\nIce filled Kelly's eyes, turning them the same cold blue color as Wynter's snow cones. \"I know what you're implying, but nobody muscles in on my territory. I'm the number one superhero reporter in this city, and I plan on keeping it that way\u2014whatever it takes. Now, if you'll excuse me, Fiera, I need to get this on the air for the six o'clock news. Thanks for the interview.\"\n\nKelly jerked her head at the cameraman, and the two of them left the park. Sometimes, I thought corporate politics were just as dangerous as superhero-ubervillain battles, although without the building-leveling explosions. I shook my head and returned to my fans.\n\nAfter Kelly left, the event wound down, and the other superheroes started packing up, ready to go out and prowl the streets for the night. I gave my regards to Abby, who was now barking orders at the cleanup crew, and headed for home.\n\nI started to walk back to the alley next to the library. On an impulse, I cut through a row of pine trees that marked the park's borders and headed for Bigtime Cemetery. A wrought-iron gate surrounded the sloping, green expanse, and marble tombstones, pinnacles, and angel statues dotted the manicured lawn. The wild carousel music and shrieks of glee from the park faded away to mere whispers.\n\nMy steps grew slower, heavier as I headed for my destination, but I pushed on until I reached a white marble tombstone. The words _Travis Templeton Teague. Beloved by all_ flowed across the marker. A few wilted flowers and cracked, faded, weather-worn action figures surrounded the gravesite. People didn't leave as many flowers and cards as they used to. Superheroes and ubervillains came and went in Bigtime, and most folks were slowly forgetting about Travis aka Tornado.\n\nNot me. I would never forget him. Never. I crouched down, straightening the action figures and arranging the flowers into a tidy pile. I did that for a long time, thinking about him and how much love there had been between us. My heart ached, and hot tears steamed off my flushed cheeks.\n\nTravis had been taken from me before his time. It was cruel, unfair, and there was nothing I could do about it. But he was in a better place now, watching over me. I knew he was. I twisted my engagement ring around my finger. I would give anything for Travis to be here with me now. But that would never happen.\n\nIt was time to move on. To date again. To laugh again. To fall in love again. My father was right. I couldn't live in the past forever, and Travis wouldn't want me to. He would want me to be happy, to be with someone who made me happy, whether it was Johnny Bulluci or someone else further down the line. I quit twisting my ring and stood.\n\n\"I love you, Travis. I always will.\" I pressed a kiss to my hand and put it on top of the tombstone. The sun-warmed marble felt smooth as glass under my fingers.\n\nThen, I turned and walked away.\n\n# Chapter Ten\n\nAfter leaving the cemetery, I slipped back into my apartment, stripped off my superhero suit, and got ready for the observatory benefit.\n\nI took extra care with my _I-might-have-sex-tonight_ beauty rituals. Daydreaming about Johnny all weekend had fired up my hormones even more than usual. I liked him. He liked me. Why shouldn't we have a little fun after the party tonight?\n\nThen, it was time to pick my dress for the evening. I hadn't seen Johnny in a few days, and I wanted to remind him just how fabulous I was. I walked up and down the rows of clothes in my massive closet, pulling things out, tossing them aside, grabbing even more outfits. After about thirty minutes of contemplation, I decided on a jade-green ball gown with a high neck, long sleeves, and a skirt that reached to my ankles.\n\nFrom a distance, the dress looked very prim and proper, but the back was completely exposed, showing off my shoulders and muscles, while the sequined fabric clung to my body, hinting at what lay beneath. I pulled my hair back into a smooth, coiled bun. Strappy silver stilettos and long, dangly emerald earrings completed the sophisticated look. Perfect.\n\nI eyeballed my purse. Something was missing from the mess inside. Lipstick, compact, cell phone, credit cards. I remembered what I'd forgotten. I rummaged around in my nightstand drawer and drew out a couple of dust-covered condoms. I stared at the foil packages, and the old, nagging doubt flared up inside me.\n\nDespite my visit to the cemetery, I couldn't quite shake the feeling that I was betraying Travis by moving on. But Travis would want me to be happy. He would. And Johnny made me happy. For the moment. He might turn out to be a loathsome little toad tomorrow, but tonight, he was Mr. Right.\n\nI pushed aside my guilt and stuffed the condoms in my purse. If you were going to play around, you should be safe doing it. I was already on the pill, but a girl couldn't be too careful, even if she was a superhero. Better to have the condoms and chicken out than push ahead without them.\n\nI drove my convertible to the benefit, top up to preserve my hair, handed the keys to a tuxedo-clad valet, and walked up the curving steps to the observatory. Situated on a towering hill on the outskirts of town, the Bigtime Observatory was the highest point in the city, farther up than even the gleaming skyscrapers downtown. A museum to all things star-related sat on top of the hill, along with the round, white dome that housed the observatory's powerful telescope and other sensitive scientific equipment. The observatory was also connected to a nature center and park, where people could come and get up close and personal with smaller woodsy animals like owls, otters, and foxes. A manmade river flowed down the steep hill, through the woods and animal habitats that surrounded the observatory, forming a waterfall and small lake, before continuing its journey toward the city and out into Bigtime Bay.\n\nEvery spring, the scientists who ran the observatory and nature center put away their pocket protectors, telescopes, and black glasses, and threw a party to raise enough money to keep operating for another year. In addition to scientific research, the observatory was a favorite with Bigtime teachers, who brought thousands of students to the facility every year to stargaze, visit the animals, and swim in the lake. Science wasn't my favorite thing, not by a long shot, but my pockets were deep enough to get me invited to the party every year.\n\nI gave my engraved invitation to the guy working the door and stepped inside. The observatory featured square, white rooms, sharp angles, and high ceilings. Scientific instruments with names too long to pronounce, detailed star charts, and planetary images adorned the walls, along with interactive displays about physics and astronauts and trips to the moon. The displays had already captivated some of Bigtime's finest, who were pushing buttons and staring at the flashing lights like kids high on sugar.\n\nI walked to the museum's main auditorium, where most of the school programs were held. The semicircle of hard, wooden chairs had been replaced with circular tables, and big models of planets dangled from the ceiling. A band played swing tunes on the stage at the far end of the room, while waiters dispensed food to the hungry crowd. Through it all, the observatory's scientists tried to blend in with the suave, sophisticated businessmen and women. But it was easy to tell who was who. The scientists kept tugging on their too-tight ties and ill-fitting dresses.\n\nI grabbed a glass of champagne and roamed through the crowd looking for Johnny Bulluci. All the usual partygoers were in attendance, with Joanne James and Berkley Brighton holding court in the middle of the auditorium. The sparkle from Joanne's engagement ring would blind a bat at thirty paces. My father stood near them, schmoozing with a couple of lonely widows. I caught his eye and waved.\n\nTo my surprise and displeasure, news reporter Erica Songe was also on the scene. She stood at the far end of the room next to a short, slender woman. The contrast between the two was striking. Erica's voluptuous form was poured into a pink dress that looked like transparent pieces of tissue sewn together. The flimsy garment exposed far more than it covered up, and more than a few men had their eyes firmly fixed on Erica's cleavage. How trashy.\n\nThe other woman wore a black, kimono-style garment that covered her from neck to feet. The bulky fabric swallowed her up, hiding any hints about her figure. Her mousy, brown hair was pulled back into a tight bun, and a scowl painted her pasty face. Tortoiseshell glasses perched on her nose, which was set rather high in the air. She might as well have had NERDY SCIENTIST tattooed on her chest. A tall, fat cameraman hovered nearby, downing champagne like he was dying of thirst. I didn't blame him. I'd drink too if I had to work with the likes of Erica Songe.\n\nA familiar motorized whir caught my ear, and I zeroed in on Henry and Lulu. Henry had traded in his usual polka-dot bow ties for one of my classy tuxedoes, while Lulu wore a scarlet dress that showed off her pale skin and blue-and-black hair. The two huddled in a corner by themselves, engaged in an intense conversation. Henry held on to Lulu's hand and said something, almost pleading with her. Lulu shook her head. Henry dropped her hand, turned on his heel, and stalked away. Lulu stared at his retreating back. What was that about?\n\nDetermined to find out, I marched over to the other woman. \"Trouble in paradise?\"\n\n\"You'd like that, wouldn't you, Fiona?\" Lulu looked away, but not before I caught the gleam of tears in her dark eyes. \"For Henry to dump me.\"\n\nThe hurt tone in Lulu's voice made me hesitate. I didn't like the computer hacker. I'd made no secret of that. We'd come too close to getting killed last year to take any more chances than necessary. Letting Lulu Lo, one of Bigtime's computer geniuses and not-quite-aboveboard citizens, in on our secret identities wasn't the smartest thing we'd ever done.\n\nBut Carmen trusted her, and Henry loved her. And Lulu loved Henry, from what I'd seen. I loved Henry too, and I wanted him to be happy. If Lulu made him happy, well, I supposed I could live with her. After all, I'd come to live with Carmen. So, I decided to put my personal feelings aside and do my duty to support Henry\u2014and his hacker honey.\n\n\"No, I don't want Henry to dump you.\"\n\nLulu snorted. \"Oh, come off it, Fiona. You can't stand me.\"\n\nI ignored her comment. \"Why would Henry dump you? Are the two of you fighting? The two of you never fight.\" In addition to having superstrength, I was also supertenacious, especially when it came to the people I cared about.\n\n\"It's nothing. Absolutely nothing,\" Lulu muttered. She hit a button on her wheelchair and zoomed away.\n\nI started to melt her tires to get her to stop, when a low voice sounded behind me.\n\n\"Hello, beautiful.\"\n\nMy heart fluttered. I put my most dazzling smile on my face and turned around. Johnny Bulluci stood behind me, wearing a perfectly fitted tuxedo that showed off his golden hair and bronze skin. My eyes zipped up and down him. The man looked yummy enough to eat. And I was always hungry. For all sorts of things.\n\nJohnny leaned in and planted a kiss on my cheek. His lips felt warm even against my flushed skin. I closed my eyes, drinking in his spicy aroma.\n\n\"Hello yourself, handsome.\"\n\nJohnny's eyes raked over me in a slow fashion. He let out a low whistle. \"Nice dress.\"\n\nI smoothed down the slinky fabric. \"I thought you might like it.\"\n\n\"I do. A lot.\" He gave me a wolfish grin and took my hand. \"Come on, I want you to meet my grandfather.\"\n\nJohnny led me through the chattering crowd. More than a few eyebrows rose when we passed, and hushed whispers broke out in our wake. Gossiping about everyone and everything was one of the main activities in Bigtime society. Actually, in Bigtime in general. Tomorrow, we'd be the talk among the city's matrons, more than a few who'd probably like to foist their daughters off on Johnny. Too bad. The man was mine. At least for tonight. I tightened my grip on his hand. And I was planning on keeping him all to myself.\n\nJohnny strolled over to a table set against the wall, where Bella sat with their grandfather. Tonight, Bella wore a simple, peach-colored, satin sheath dress that brought out her tan skin. Pearl combs glinted in her amber-colored hair, while her usual silver angel charm hung around her throat. The two of them rose at our approach.\n\n\"Fiona.\"\n\n\"Bella.\"\n\n\"Nice dress,\" we said in unison.\n\nWe looked at each other, not sure if we were being catty or not. A slow smile spread across Bella's face. I grinned in return, and we laughed.\n\nJohnny drew me forward. \"Grandfather, this is Fiona Fine. Fiona, this is my grandfather, Roberto.\"\n\n\"I've long been a fan of your work, Miss Fine.\" Roberto Bulluci's voice was just as rich and cultured as Johnny's, except his accent was far more pronounced. He bowed low, pressed a kiss to my hand, and gave me a sly wink that would have been right at home on a much younger man. \"And please call me Bobby. All my friends do.\"\n\nI laughed. \"Well, I see where Johnny gets his charm from.\"\n\n\"Johnny has told us quite a bit about you, including your fondness for steaks. You must come have dinner with us one evening, Miss Fine,\" Bobby said. \"Perhaps Bella will let me have some red meat and wine if you do.\"\n\nBella's lips pursed as though she'd bitten into a lemon. \"Grandfather, you know the doctor said you shouldn't eat\u2014\"\n\nBobby Bulluci waved his hand. A silver ring set with diamonds sparkled on his pinkie. \"Bah! I'm seventy-two years old, Bella. If I want to eat steak and drink fine wine in my golden years, then I should be able to **\u2014** no matter what any doctor says. They're all fools anyway.\"\n\nBella put her hands on her hips. \"Not if you want to live to see seventy-three.\"\n\nBobby eyed his no-nonsense granddaughter. \"Just like your father, you are. Everything must always be by the book.\" His tone was light and teasing.\n\n\"Well, perhaps if he were here, I'd have an easier time keeping you in line and out of trouble,\" Bella said.\n\nHer words hit a little too close to home. A cold shadow fell over the three of them at the mention of James Bulluci. Johnny's hand tightened around mine.\n\n\"Let's get some champagne,\" I said, trying to lighten the darkening mood.\n\nIt worked. Just like always. A waiter came around, and we grabbed tall flutes filled with bubbling, golden liquid. I kept up a steady stream of chatter, and slowly but surely, the darkness faded away and the ice broke. Then again, it usually did when I was around.\n\nBella and I talked about our fall lines. I was going for a hip yet preppy look. Lots of bright, multicolored plaid. Lots of blazers. Lots of chunky jewelry. Bella, meanwhile, had chosen to focus on every Bigtime woman's favorite color\u2014black. How boring. I was sure her clothes would be exquisitely made and very beautiful, but the girl really needed to mix it up a bit. She needed some color in her life\u2014in more ways than one.\n\nAfter we exhausted the world of fashion, Bella drifted off to speak to one of the society types about her latest dress, leaving me alone with the men. After a couple of false starts, the boys and I found a topic we could talk about\u2014soccer.\n\n\"Soccer is the true football,\" I said. \"Americans really don't know what they're missing. Anybody can kick a ball with his foot. Bouncing it off his head is what takes _real_ talent.\"\n\nBobby's green eyes lit up. \"Ah! A woman after my own heart.\"\n\nWe chatted about various European leagues and teams and the latest scandals, and Johnny jumped in with his thoughts. After about half an hour of sports talk, Johnny took my arm again.\n\n\"I'm sorry to leave you alone, but we really need to mingle for a while, Grandfather,\" Johnny said. \"I'm sure Fiona has some friends that she'd like to say hello to tonight.\"\n\n\"Of course, of course. Just as I have some lovely ladies that I would like to speak to as well.\"\n\nBobby winked and pointed at a group of older women standing by the bar. The spry septuagenarian strode off, his walk tall and strong. He strolled right into the midst of the women, and more than a few of the elderly ladies perked up at his presence.\n\n\"Your grandfather is a real character,\" I said. \"Very charming, very lively.\"\n\nJohnny smiled. \"He's always up to something. That's why I love him.\"\n\nWe moved off into the crowd and made a circuit of the auditorium, speaking to people we knew, including Chief Newman. My father's eyes grew dark and curious when I introduced him to Johnny.\n\n\"I've heard a lot about you, Mr. Bulluci,\" my father said, shaking Johnny's hand.\n\nJohnny looked at me. \"I hope it's been good.\"\n\n\"So far. So far, my boy.\"\n\nThe two of them started talking about manly things, like cars and sports and fine cigars. Johnny turned out to be quite an expert on the last subject, giving my father tips about what cigars to buy and where to get them from.\n\n\"I didn't know you smoked,\" I said.\n\nJohnny shrugged. \"Occasionally. Not so much since I came back to Bigtime. Americans are very uptight about it, far more so than Europeans. It takes the fun out of it. Plus, Bella won't let me smoke in the house. She's a real stickler for things like that.\"\n\nAs the conversation progressed, my father kept lacing his fingers together and peering into Johnny's eyes, as though deep in thought. His own eyes seemed to be glowing ever so faintly. I frowned. Johnny moved off to speak to one of Bella's friends, and I grabbed my father's arm.\n\n\"Stop that!\" I hissed.\n\n\"Stop what?\"\n\n\"Trying to read his mind.\"\n\n\"Would I do something like that?\"\n\nMy father smiled and tried to look innocent. Please. I didn't have to be a psychic to know he was faking.\n\n\"You've always done it, ever since I started dating boys in high school. Remember the guy who took me to the prom? I thought he was going to have a heart attack when you started asking him about the hotel room he'd booked for the two of us for after the dance.\"\n\nMy father chuckled at the memory. I rolled my eyes.\n\n\"So what's the verdict?\" I asked.\n\nMy father frowned. \"I'll get back to you on that.\"\n\nJohnny returned before I could interrogate Chief Newman further about his cryptic comment. A widow dripping with jewels wiggled in between us and pulled my father onto the dance floor. Johnny and I made another lap around the room. It was close to midnight, and I was ready to move on to the next part of the evening\u2014whatever it might entail.\n\n\"Are you ready to blow this joint?\" I asked.\n\nJohnny opened his mouth to reply, when a low, sultry voice cut in.\n\n\"Mr. Bulluci. Hello. So nice to see you again.\" Erica Songe batted her long, black lashes at Johnny. They were fake, just like her lips and boobs. \"Oh, Fiona. I didn't see you there.\"\n\n\"Erica,\" I said through gritted teeth. Up close, the reporter's filmy dress was practically transparent, giving everyone a view of all her charms, including Johnny. \"That's a nice little dress you're wearing.\"\n\n\"Isn't it just fabulous?\"\n\nErica put her hands on her curvy hips and preened. The woman actually thought I was being serious and giving her a genuine compliment. Please.\n\n\"Actually, I prefer a more understated look on a woman.\" Johnny smiled and put his arm around my waist. \"It leaves more to the imagination.\"\n\n\"But sometimes the real thing's so much better than your imagination,\" she purred.\n\nI rolled my eyes. The woman had no shame and zero class. Slut.\n\nJohnny's cool green eyes flicked over her. \"Not in this case, I think.\"\n\nErica's mouth opened and closed as though she couldn't believe there was a man alive who didn't want her. Who could resist her. She pouted, sticking her lower lip out as far as it would go. If she kept that up much longer, her whole face would explode.\n\nBut I had to give Erica credit, she recovered quickly. She brushed off Johnny's comment like it was nothing more than a piece of lint sticking to her see-through dress. She gestured to her cameraman, who was still guzzling champagne, and whipped out a microphone. The guy wobbled over and pointed his video camera in our general direction.\n\n\"Well, while I have you cornered, what do you think about the benefit? I'm covering this tonight for SNN.\"\n\n\"Really? Why?\" I asked. \"Shouldn't you be out chasing down superheroes and ubervillains like Kelly Caleb?\"\n\n\"Kelly is a bit limited in what she can do, poor girl. The network wants to broaden its horizons beyond just superheroes. I'm the go-to girl for the more high-profile assignments these days,\" Erica replied in a smooth tone.\n\nIn other words, Erica had somehow wrangled the assignment away from the other reporter. I didn't like reporters much, but Kelly Caleb was good at her job. If there was any scoop to be had, she was the one who knew about it. I wondered who Erica had to sleep with to beat out the other woman. That was the only way I could imagine that she'd one-upped Kelly.\n\n\"Care to comment about the benefit, Johnny?\" Erica shoved the microphone and her chest in Johnny's face.\n\n\"Well,\" he said, keeping his eyes fixed on Erica's face. \"I think the observatory is an important asset to the community...\"\n\nWhile Johnny expounded on the virtues of the Bigtime Observatory, nature center, and sundry related programs, I wandered away and grabbed a couple of crab cake appetizers from a passing waiter. I would have taken the whole tray away from the guy, but there were too many witnesses.\n\nInstead, I polished off the batter-dipped bites, licked the crumbs off my fingers, and watched Erica interrogate Johnny. A scuffle sounded, and the kimono-clad scientist I'd spotted with the reporter earlier stopped next to me. The woman stared at the curvaceous news reporter. A disapproving frown covered her white, makeup-free face.\n\n\"Not enjoying the show?\" I asked.\n\n\"Oh, preening for the camera and throwing herself at men is what Erica does best.\" The woman sniffed. \"After a while, the spectacle gets old.\"\n\n\"You sound like you know her pretty well.\"\n\n\"Unfortunately,\" the woman replied, \"we share the same gene pool.\"\n\nI eyed her. With her headache-inducing bun and buttoned-up-to-there dress, the woman didn't look a thing like Erica _I'm-so-slutty_ Songe. \"Cousins?\"\n\n\"Sisters. Unfortunately.\"\n\n\"I'm Fiona Fine, and you are...\"\n\n\"Irene Songe.\"\n\nWe shook hands. Irene's grip felt as cold and passionless as an ice cube. It matched her pale face. We stood there drinking champagne and watching Erica do her best to let Johnny Bulluci know she was up for anything, anytime. After about five minutes, Johnny managed to extricate himself from Erica's clutches and head back to me. Erica trailed along behind him, not quite ready to admit defeat. The cameraman grabbed some more champagne. It was a wonder the guy could still stand **\u2014** or that his liver didn't burst.\n\nErica stopped short when she spotted her sister talking to me. \"Enjoying yourself, Irene?\"\n\n\"Of course,\" Irene replied.\n\nThe two women looked at each other with barely disguised hostility. Definitely no love lost there. One sister an exhibitionist, the other a repressed, wound-up scientist. I couldn't imagine why they didn't get along.\n\nErica turned to Johnny and flashed him another smile. She opened her mouth to proposition Johnny yet again, but he beat her to the punch.\n\n\"It was so nice talking to you, Erica. But if you'll excuse us, Fiona and I have somewhere we need to be. Now.\"\n\nJohnny offered me his arm, and we walked away. I looked back over my shoulder and smiled at Erica. Her face reddened, and her hands clenched into fists, as though she wanted to stab me with her microphone. Let her try. I could kick her ass blindfolded. Irene whispered something to her, and Erica turned and snapped at her sister.\n\n\"Forget about her,\" Johnny said. \"You're the only woman I have eyes for tonight.\"\n\nI smiled at him. \"Smooth. Very, very smooth.\"\n\n\"Let's get out of here,\" Johnny said, his eyes burning into mine. \"And go somewhere a bit more private.\"\n\nMy stomach quivered. \"Let's go,\" I said.\n\nHe pulled me toward the door.\n\n# Chapter Eleven\n\nWe strolled out the back of the auditorium and onto a stone balcony that overlooked the gardens and nature preserve that surrounded the observatory. Acres of flowers, grassy knolls, and low bushes stretched out before us into the dark night before melting into the black woods. A few other couples stood on the balcony, whispering, drinking, and enjoying the scenic view.\n\n\"Care to go for a walk in the moonlight?\" Johnny asked.\n\n\"I'd love to.\"\n\nWe picked our way down the stone steps and onto one of the crushed-shell paths that wound through the tame jungle. Antique streetlamps lit the way, along with the moon and stars above. A steady breeze blew from the north, pushing a few wisps of clouds across the sky and bringing with it the heady scent of roses, orchids, and more. But all I was aware of was Johnny walking next to me. His spicy cologne tickled my nose, and his body felt warm and solid next to mine.\n\nWe strolled in silence, leaving the noise and gaiety and lights of the benefit behind. Crickets chirped in the dew-covered grass. A bullfrog let out a loud bellow. Doves cooed and fluttered in the trees above. We crossed a stone bridge that arched over the manmade river. Water gurgled below our feet.\n\n\"Come on,\" Johnny said. \"Let's take a detour.\"\n\nHe took my hand and led me down a path that curved back under the bridge. After going under the bridge, the river tumbled down a hill, forming a small, fifteen-foot-high waterfall. The water pooled into a shallow lake a little less than two miles across before rushing on down the hillside. Fiberglass picnic tables and stone benches dotted the lake's edge. Brilliant moonlight bounced off the water's surface, making it shimmer like liquid silver. Cattails and other vegetation surrounded the water's edge, and a few water lilies floated in the pool, bouncing up and down on the rippling waves.\n\n\"It's beautiful,\" I whispered.\n\n\"Want to go for a swim?\"\n\n\"Now? In the middle of the night?\"\n\nI loved to swim, loved to lose myself in cool, soothing water and rhythmic strokes. But coming up to the lake for some late-night swimming, skinny-dipping, and a little necking was a popular pastime for local teens and college students. When Johnny had said he wanted to go somewhere more private, I'd been thinking about my apartment, specifically my bed.\n\nJohnny grinned, his teeth flashing in the darkness. \"Why not? There's nobody around but the two of us. I won't tell if you won't tell.\"\n\nHe slipped off his jacket and shoes and put them on a nearby bench. His shirt and pants soon followed. All I could do was stare at the man before me. I'd thought there was a body to die for under those clothes, but I had no idea how deadly it was. Sculpted biceps. Rippling abs. Solid muscles. My eyes dipped lower. Everywhere. Yummy. Johnny Bulluci was one fine-looking man. I closed my hands to keep unwanted sparks from shooting out of my fingertips.\n\nI ogled him a minute longer, then laughed. \"Cherubs?\"\n\nJohnny put his hands on his hips, proud of his boxers. Fat, happy-looking cherubs frolicking on puffy clouds decorated the slick, silk fabric. \"Just because I'm no angel doesn't mean I can't wear them. Now, are you coming or not?\"\n\nI hesitated, twisting the ring around my finger. The time for being wishy-washy was long gone. I stopped and let out a long, hot breath. \"I'm coming.\"\n\nI stripped off my dress and high heels, folding the fabric into a neat pile. It would have a million wrinkles in it, but I could always make another one just like it. In a moment, I was standing there in a lacy green bra and matching panties. I always wore the good underwear when there was a chance of someone else taking it off.\n\nJohnny's eyes traced over my body. \"You know, I think Erica was right. The real thing is so much better than my imagination.\"\n\nI smiled and lifted my chin up. \"Of course it is, when I'm involved.\"\n\nWe strolled down to the water's edge and plunged into the frothy spray. We treaded water a few minutes, letting our bodies adjust to the cool temperature. Even though it was the middle of May, the water hovered around seventy degrees, a bit chilly for most people. Not Johnny, though. He swam and dove like a duck. If anything, the water invigorated him. Or perhaps it was just the sight of me in my wet skivvies. Yeah, that's what I was going with.\n\n\"Is the water too cold for you?\" Johnny asked.\n\n\"No.\" I was never cold. But the hunger in Johnny's eyes made me shiver.\n\nWe swam back and forth in the pool, shrieking and splashing water and pulling each other under like a couple of teenagers. After about half an hour, we headed for shore. We flopped onto the grass and stared at the smattering of stars high above.\n\n\"It's so peaceful here,\" I murmured. \"So beautiful.\"\n\n\"Not as beautiful as you,\" Johnny whispered in a husky voice.\n\nI stared into his eyes. This was it. Decision time. The old guilt flared to life inside me. But Travis was gone, and Johnny was here. Now. With me.\n\nSo, I kissed him. His tongue met mine, and I was lost.\n\nMy bra and panties disappeared before I knew what had happened. So did Johnny's boxers. Everything else faded away until there was nothing but the soft grass below us, the stars above, and Johnny's hands on me.\n\nWe lay there for a long time. Kissing. Stroking. Caressing.\n\nJohnny pulled back. He pushed a wisp of hair off my face and stared into my eyes. His own glistened like jewels in the moonlight.\n\n\"I want to do something for you, Fiona,\" Johnny said. \"Do you trust me?\"\n\nI went quiet and still. I wasn't good at trusting people. I'd gone through too many battles with too many ubervillains to give my trust lightly to anyone. But there was something about Johnny that made me want to trust him. Made me want to lower my defenses. Made me want to believe in him. \"Yes, I do.\"\n\n\"Then roll over onto your stomach.\"\n\nI did, and to my surprise, he started to massage me. There was nothing overtly sexual about it. Just Johnny's firm hands kneading my body with the detachment of a large Swedish woman. He started at my feet, moving up my calves to my thighs. His fingers worked on my ass, molding, sculpting, before going up to my lower back and shoulders. He even worked on my head, loosening my hair from its wet bun and massaging my scalp.\n\nHeaven. Sheer heaven.\n\n\"Now, the other side,\" Johnny whispered.\n\nI turned over onto my back, eager for more. Much, much more. Johnny repeated the process, this time starting with my neck and working his way down. His fingers pressed gently into me, soothing away my doubts and fears. His sure hands moved to my breasts, massaging each one in turn. My nipples hardened under his touch, but Johnny ignored them and went on with the massage, focusing on my stomach, before sliding lower.\n\nI trembled when he reached my pelvic area. I thought he might slip his fingers inside me, but Johnny moved on as though he wasn't the least bit interested in that part of my anatomy. He worked on my thighs, then calves, then feet. Slowly, the fireball of guilt and tension in my stomach faded away, replaced by a different, though no less dangerous, sort of liquid heat.\n\n\"That was wonderful,\" I sighed when he finished. \"Absolutely wonderful.\"\n\nJohnny grinned. A devilish light sparked in his eyes. \"Baby, if you thought that was wonderful, how about this?\"\n\nHe eased my thighs apart and lowered his mouth to me. I gasped at the sudden, unexpected sensation.\n\n\"Do you like that, baby?\" he asked.\n\nI buried my fingers in his hair and whimpered.\n\n\"I'll take that as a yes.\"\n\nJohnny delved into me like I was a delectable dessert just waiting to be sampled, and he was a sugar addict. His tongue. His fingers. Stroking. Caressing. Driving me crazy.\n\nSweet, painful pressure built up in my body, and I thought my hair would catch fire from the sensation. But I'd learned long ago to dampen down the physical manifestation of my power during sex. To let all that fire melt into liquid desire coursing through my veins.\n\n\"Johnny...I...can't...wait...much...longer,\" I panted, fingers clenching the grass.\n\n\"You don't have to, baby,\" he murmured. \"I'm ready for you.\"\n\nWe kissed, then I rolled Johnny onto his back. I reached for my purse and pulled out one of the condoms. Johnny tried to take it from me, but I pushed him back against the ground.\n\n\"Let me,\" I whispered, tearing the packet open with my teeth.\n\nI trailed my fingers up and down his long length before slipping the condom over him. Johnny's eyes widened at the sensation, and he hissed.\n\nThen, I got on my knees and straddled him, taking his stiff shaft deep inside my fiery body. Johnny was so much more than just ready for me. _Hard_ didn't do the man justice. Even _steely_ couldn't quite describe it.\n\n\"Oh, baby, you're so tight,\" Johnny murmured. \"So tight and hot and wet.\"\n\nI rocked back and forth, thrusting against him. Johnny's hands stroked my breasts as I rode him, squeezing my nipples until they ached and throbbed along with the rest of me. It only made me want him that much more. All that sweet liquid fire bubbled up in my veins like a volcano about to erupt. A moment later, it did. An orgasm tore through my body, and I cried out. So did he, and we rode up into the stars together.\n\nSometime later, I slumped over Johnny with him still inside me.\n\n\"That was fantastic,\" I said, nuzzling his neck. Sweat covered my body, and my heartbeat slowly returned to normal. Smoke puffed away from the ends of my fingers, but you could barely see it in the darkness.\n\n\"That was fantastic, but we're not done yet,\" Johnny said.\n\nHe eased me off him and got rid of the condom. Then, he pulled me to my feet and led me down into the lake. The water washed away the slick sweat and cooled my feverish body, as much as it could. Johnny kissed the back of my neck while his hands slid around and covered my breasts, rubbing my nipples until they were hard and aching once more.\n\n\"Come on,\" he whispered, withdrawing his hands.\n\nI followed him. Johnny walked up the bank and tugged me over to one of the picnic tables that flanked the lake.\n\n\"This looks about right,\" he said.\n\n\"Yes, it does.\"\n\nJohnny reached for me, but I eluded his grasp, grabbing another condom from my purse. Then, I went down on my knees and took his shaft in my mouth. Johnny gasped in surprise. He staggered, and his back went against the table.\n\n\"You're not the only one who likes to be shocking,\" I said.\n\nI ran my tongue up and down his penis while my fingers stroked him. Johnny twitched and trembled with every flick of my hot tongue. Now, I was the one in control. Just the way I liked it.\n\nNot for long, though. Johnny reached down and plucked the condom from me, unrolling it in record time. Then, he picked me up and placed me on top of the picnic table. My legs locked around his waist. Johnny leaned over me, bracing his hands on the cool fiberglass. He stared into my eyes for a heartbeat. Then, he thrust into me. His hungry mouth covered mine, hushing my cries and moans of pleasure.\n\nJohnny plunged into me over and over again. My hands were everywhere. His hair. Neck. Chest. Abs. Our tongues dueled back and forth even as we pushed together. I couldn't get enough of him. I urged him on, wanting him to go deeper and deeper.\n\nSo, he did.\n\nAnd I loved it.\n\nEvery fantastic, pleasurable, white-hot second.\n\n# Chapter Twelve\n\nAfterward, we lay there on the cool grass for a long time, looking at the stars and lying in each other's arms.\n\n\"This is perfect, absolutely perfect,\" I said.\n\n\"Yes, yes it is,\" Johnny replied in a soft tone.\n\n\"So where do we go from here?\"\n\nJohnny propped his elbow up and stared into my eyes. \"I want to keep seeing you, Fiona, if that's what you're asking. I might be a rich, spoiled playboy, but I have been known to engage in actual relationships, every now and then. Especially with such an incredible woman like you.\"\n\nIt was exactly what I was asking\u2014and exactly the response I wanted.\n\n\"Good. I might be a bitchy fashion designer, but I've also been known to indulge in relationships\u2014with certain equally fabulous people.\"\n\n\"And do I fit that bill?\" Johnny asked.\n\n\"Absolutely. I want to keep seeing you too.\" I trailed my fingers down his abdomen. \"All of you.\"\n\nJohnny grinned and reached for me again.\n\n* * *\n\nEventually, we put on our clothes and headed back to the observatory. It was closer to morning than midnight now, and the birds and bugs and bullfrogs had quieted down to sleep. Only a few drunken stragglers remained at the benefit, sitting at tables and guzzling down what was left of the champagne, along with the science types who had organized the event. But they were too busy counting money and tallying up checks and contributions to notice Johnny and me and our disheveled appearance.\n\nJohnny walked me outside, and a sleepy-looking valet retrieved my convertible. I threw my purse in the car and turned to face Johnny.\n\n\"Dinner tomorrow night?\" he asked, cupping my cheek in his hand.\n\nI hesitated. I liked Johnny, I really did, but the sex hadn't been as casual as I thought it would be. Quite the opposite. I'd been looking for a rebound guy, not somebody long-term. But this felt like the beginning of something big, something serious. Despite our earlier promises to keep seeing each other, I didn't know exactly how I felt about the sexy businessman. Other than hot and bothered.\n\n\"Come on. You promised grandfather you'd come. You wouldn't want him to miss out on his steak and wine, would you? He was really looking forward to it,\" Johnny wheedled.\n\nI laughed. \"Far be it from me to let your grandfather down. Dinner tomorrow night. Or rather, tonight. Eight o'clock?\"\n\n\"It's a date.\" Johnny flashed me another sexy grin, leaned in, and kissed me.\n\nIt was another fifteen minutes before I was able to get in my car and drive away.\n\n* * *\n\nI entered my apartment at six in the morning. I tossed my purse on a table and sank onto the scarlet-upholstered sofa. A smile spread across my face. Tonight had been fabulous. Completely, wonderfully, absolutely, perfectly fabulous. I stretched my arms over my head. The dry spell was over. I was totally satisfied. Johnny Bulluci should have been a sculptor instead of a businessman. The things that man could do with his hands. And tongue. And lips...\n\nMy thoughts strayed for a little while, replaying the evening over and over again in my mind. But my stomach rumbled, and I realized that I was starving. Sex really was great for burning calories, not that I had any problem in that department anyway. And it was definitely a lot more fun than exercise.\n\nI was halfway through a gallon of strawberry cheesecake ice cream when the phone rang. I frowned. Who could be calling me at this hour? Johnny? My heart quickened.\n\n\"Fiona Fine.\"\n\n\"Well, it's about time you got home,\" Carmen said. \"I left you a message hours ago.\"\n\nI looked at the answering machine. Sure enough, the red light blinked on and off. I'd been so busy thinking about Johnny that I hadn't even noticed it. \"Sorry. I was out.\"\n\n\"Don't worry. I didn't want anything important. But tell me, how was your date?\"\n\n\"What date?\" I mumbled through a mouthful of melted ice cream.\n\n\"The hot date that ended in you having fantastic sex.\"\n\nThe spoon slipped from my fingers. \"How do you know that?\"\n\n\"I can hear it in your voice. You're practically purring, Fiona. And the chief called and said you disappeared with some hot new guy you've been seeing. That the two of you were looking at each other a certain way. I just put two and two together.\"\n\nI cursed and picked up my spoon. Carmen's inherent nosiness could be so annoying sometimes, especially when coupled with her new empathic powers. Didn't I have any secrets from anybody anymore?\n\n\"I'm happy for you, Fiona. That you found somebody you like. I really am. It's time you got back out there again. Now, if only I could get Lulu to\u2014\" Carmen cut off her sentence.\n\n\"Get Lulu to what?\"\n\n\"Nothing. Never mind.\"\n\n\"What's going on with Henry and Lulu?\" I asked. \"It looked like they were fighting at the benefit.\"\n\nCarmen was silent. \"You'll have to ask Lulu about that.\"\n\n\"I did. She told me to buzz off. She almost ran over my feet with her wheelchair.\" I scraped the bottom of the ice cream carton.\n\n\"Well, I wouldn't worry about it. The two of them will work things out. Eventually.\"\n\nThe last luscious bite of ice cream melted in my mouth. Gone already. Damn. I should have gotten the grocery service to deliver more than just one gallon. \"Well, if they do break up, can I finally set Lulu's hair on fire?\"\n\n\"Fiona! That's not very nice.\"\n\n\"What?\" I asked. \"She's always telling me how _hot_ I am. I'd like to show her how right she is. Just once.\"\n\n\"Henry and Lulu are not going to break up. They're just having a little problem right now.\"\n\n\"But you're not going to tell me what it is.\"\n\n\"No. It's not my place to say anything.\" Carmen switched gears. \"Now, back to you and the hot guy. How was it? I want all the juicy details.\"\n\nI sat back against the cushy sofa. Carmen was the closest thing to a girlfriend that I had, and I couldn't exactly talk to my father about this. Plus, I felt like bragging. Just a little bit. All right. A lot. Who wanted to be modest when Johnny Bulluci was in the picture?\n\n\"Wonderful. Absolutely wonderful. The man knows how to use his hands very well, if you know what I mean. As well as other parts of his anatomy.\"\n\n\"So, who is the mystery guy?\"\n\n\"Johnny Bulluci. Bella Bulluci's brother.\"\n\n\"I didn't know Bella had a brother.\"\n\n\"Well, she does. I met him at the wedding. He was the guy who thought he could take out Siren and Intelligal by himself.\"\n\n\"You met him at the wedding? Really?\"\n\nI rolled my eyes at the note of triumph in Carmen's voice. \"Yes, really.\"\n\n\"Tell me about him,\" she said.\n\nSo I did. I told Carmen how funny and charming and wickedly sweet Johnny was. How he made me laugh. And how he had reduced me to a puddle of oozing mush.\n\n\"He sounds terrific,\" Carmen said.\n\n\"He is,\" I admitted. \"I just don't know what to do about it.\"\n\n\"What do you mean?\"\n\n\"Well,\" I said. \"At first, I was just going to have a fling with him to get my feet wet again, so to speak. But I really like him. A lot more than I thought I would. That's complicated enough by itself. Then, there's the whole secret-identity thing. I've never figured out how you tell a guy, O _h, by the way, I moonlight as one of the most famous superheroes in the world. I can create fireballs with my hands and bench-press a couple thousand pounds._ It's not exactly pillow talk. Most guys are intimidated by things like that, especially the superhuman strength part.\"\n\n\"You're not thinking about telling him that you're Fiera, are you? You've only known the guy a few days!\"\n\nI wanted to point out that Carmen hadn't even known Sam's real identity before she'd slept with him, but I was too relaxed and mellow to quibble over that fact. Right now, anyway.\n\n\"Of course I'm not going to tell him that I'm Fiera. How stupid do you think I am? I've been a superhero a long time now. It's not something you just blurt out to people.\"\n\nIt'd be a long, long, _long_ time before I told Johnny what I did in my spare time. If ever. I'd had more than one guy dump me as a result of my nighttime escapades. Guys had one of two reactions when they learned about my secret identity\u2014they freaked out and broke up with me, or they got totally kinky and wanted me to dress up in the Fiera costume for them. It was rather embarrassing when you had to get your father to mind-wipe your boyfriend because he couldn't handle the fact that you could break his arm like a peppermint stick or barbecue him with a thought. Or when he wanted you to play dress-up in the bedroom all the time.\n\n\"Well, whenever you tell him, if Johnny can't understand your other job, he's not worth having. Just take it one day at a time. That's what Sam and I did in the beginning.\" Carmen hesitated. \"But you need to be careful, Fiona.\"\n\n\"Why's that?\" I got off the couch and opened the refrigerator door, still hungry.\n\n\"The chief told me about your run-in with Siren and Intelligal the other night. Those two are up to something, I'm sure of it. Something big. Something dangerous.\"\n\n\"Is that what the voices in your head are telling you?\" I snickered.\n\n\"Yes,\" Carmen snapped. She tended to get a little touchy about her powers sometimes, just like newbie heroes always did. \"That's what the voices in my head are telling me.\"\n\n\"All right, Mom, I'll be careful.\" I pulled a frozen pizza out of the refrigerator, tore off the wrapper, and put it on a baking sheet.\n\n\"Good.\"\n\nI wiggled my fingers, spreading flames over the ham, pineapple, and cheese concoction. \"So how's the honeymoon going? Where's Sam at?\"\n\n\"Right here beside me.\"\n\nMy eyes narrowed. \"Are you guys still in bed? Isn't it like noon over there by now?\"\n\n\"Eleven o'clock actually,\" Carmen said in a cheery voice. \"You didn't think you were the only one who got lucky last night, did you?\"\n\n* * *\n\nAfter eating my pizza, a bag of chips, three cinnamon buns, and washing it all down with a six-pack of soda, I went to bed. Due to my late night out, I didn't get to work until almost noon. Good thing I owned the place, or I would have been so fired. I strolled into Piper's dust-free office, and she gave me a sly smile.\n\n\"Someone must have had a good time last night,\" Piper said, her eyes sparkling. \"A _very_ good time.\"\n\n\"Why do you say that?\"\n\nShe jerked her head at my office. I opened the door to find more flowers and chocolates crammed into the room than I'd seen in my entire life. There were three times as many as before, and they covered every single surface, even the floor. It looked like the Petal Pusher had set up her evil, flower-filled lair inside. I picked up one of the boxes scrunched next to a towering bouquet of white roses. Chocolate truffles filled with raspberry. Johnny Bulluci definitely had good taste.\n\nPiper leaned in the doorway. \"Like I said, you must have had a _really_ good time last night.\"\n\nI just grinned.\n\nI spent the rest of the day working. At least I tried to. It took almost an hour to clear out some of the flowers so I could actually get to my desk. More often than not, I ended up mentally replaying last night with Johnny. I'd catch myself staring at the flowers and chocolates and smiling. Well, at the flowers anyway. The chocolates didn't last past twelve-thirty.\n\nEventually, I buckled down and got busy. I okayed Piper's final pricing suggestions on the fall line and contacted the necessary fabric suppliers. I called a few of Bigtime's rich society types to let them know their orders would be available for pickup next week and signed a variety of forms and initialed them in triplicate. I even answered all my outstanding Fiera e-mails.\n\nPiper came in my office around three, bearing bags full of burgers, soups, salads, and desserts from Quicke's.\n\n\"You're a lifesaver,\" I said, unwrapping a double cheeseburger as fast as I could without burning off the cellophane. I bit into the warm, cheese-covered bun. Mayonnaise, chargrilled meat, tomatoes, onions, and more exploded onto my tastebuds. Ah, heaven.\n\n\"I know. You should give me a raise. A big one.\" Piper snitched a French fry from one of the bags.\n\nI nodded, too caught up in my burger to protest or smack her hand away from the fries. I didn't like people messing with my food, especially eating what was mine. That was how nasty, fire-filled accidents happened.\n\n\"And don't forget. Joanne James is coming in at four to talk about her next wedding dress.\"\n\nI groaned. \"Is that today?\"\n\nPiper nodded. \"Yep. You told me to set it up, remember?\"\n\nI groaned again and stared at the white paper bags. \"So that's why you brought me all this food. You were softening me up. Trying to bribe me with burgers so I wouldn't weasel out of the appointment.\"\n\n\"Guilty as charged,\" Piper said, breezing out. \"But it worked, didn't it?\"\n\nI just took another bite of my burger.\n\n* * *\n\nJoanne James arrived at exactly four. She glided through the front door like she was the queen of Bigtime, even though she wouldn't officially get that title until she married Berkley. She wore a sleek, lavender Bulluci suit and matching heels that set off her blacker-than-black hair. Amethysts bigger than small potatoes ringed her thin throat. They looked like something out of a Cracker Jack box. Except I knew the gems were real. The necklace had to weigh a ton. I didn't see how Joanne held her neck up with that thing strung on it, much less walked around. When you factored in her golf-ball-size engagement ring, the jewelry she wore probably weighed more than her scrawny body. Like most women on the society circuit, Joanne was painfully thin.\n\n\"Fiona, darling!\" she cried out, removing a pair of oversized black sunglasses studded with diamonds.\n\n\"Joanne, darling!\"\n\nWe air-kissed and told each other how fabulous we both looked. Once the fake pleasantries were out of the way, I led Joanne to one of the many spacious consultation rooms that branched off the storefront. The area featured plush chairs, a low settee, flowers, and sample books of all the garments I'd designed, along with trays of fabric swatches. A round, raised dais and three-sided mirror stood in the middle of the area. It was everything I needed to outfit the fabulously wealthy\u2014and unbelievably picky.\n\nA knock sounded on the door, and one of the clerks came by with a tray of vanilla-cr\u00e8me-filled gourmet chocolates and champagne. Joanne nibbled on one of the sweet confections, while I gulped down two glasses of the bubbly.\n\n\"Leave the bottle, and keep the booze coming,\" I muttered to the clerk. \"Slip me something stronger, if you've got it.\"\n\nShe nodded with sympathy. She'd had a few run-ins with Joanne. The clerk left, and Joanne settled herself on the settee, sitting up smartly as to not wrinkle her five-thousand-dollar suit.\n\n\"First of all, let's talk price,\" I said.\n\nIt was always better to lock down the money up front. That way, I wouldn't spend my time and energy coming up with something fabulous only to have some penny-pinching rich bitch try to stiff me on the bill. Joanne was particularly notorious for doing that. Sometimes, I thought she should have gone into business instead of trolling for husbands. She was a tough negotiator.\n\nJoanne waved her hand. The sparkles from her ring made my eyes hurt. \"Money is no problem. Charge whatever you want for the dress. Berkley is footing the bill for everything. It's his wedding gift to me. I told him that I wanted the wedding of my dreams, and he is more than happy to give it to me. He's very sweet that way.\"\n\n_Sweet_ wasn't the word I would use. _Complete, utter, blind fool_ would be more appropriate. But if Berkley Brighton wanted to blow his whiskey millions on the wedding and throw some of that money my way, who was I to argue? Still, I couldn't stop myself from asking the painfully obvious question.\n\n\"Haven't you already had the wedding of your dreams? You have been married five times now.\"\n\n\"Of course. The first one left a lot to be desired, but the last four were absolutely wonderful. But there's always room for improvement, Fiona.\" Joanne polished off her chocolate. \"Now, on to the color. I was thinking of something in a rich red. Burgundy maybe, or perhaps scarlet.\"\n\n\"Scarlet?\" I said. \"Are you sure? What about white? Or maybe a nice ivory or pale pink?\"\n\n\"I look like a ghost in white and utterly washed out in ivory. Pale pink? Disgusting. Besides, it's not like I'm some quaking virgin who's never been with a man. I've been around the block, several times, and we all know it.\"\n\nDespite her other faults, Joanne didn't mince words or pull punches, not even at herself. I admired that small part of her personality.\n\n\"All right, red it is,\" I said, pulling out a thick book.\n\nFor the next half hour, Joanne looked at fabric swatches. She settled on a crushed velvet in a deep, wine-red color that contrasted nicely with her blacker-than-black hair and pale skin. Next, it was on to the dress itself. I showed her some preliminary sketches I'd done for various lines.\n\n\"No, no, no, no.\" She flipped through the pages. \"Too poofy. Too saccharine. Too much tulle. Too many sequins. Wait a minute. This. Now this, I like.\"\n\nJoanne pointed to a sketch, and I snorted. I should have known she'd pick that one. The dress featured a long, flowing, almost transparent silk skirt and a lace-up, leather bustier. The dress was a little too slutty for your traditional brides, looking more hooker than virgin. I'd actually been thinking of adding it to my lingerie line for next year.\n\n\"Are you sure?\" I asked. \"I was going to use that garment for a more intimate line of apparel.\"\n\n\"Absolutely. That will look marvelous on me.\" Joanne's voice rang with certainty.\n\nLingerie for a wedding dress? Well, it'd certainly look good in red.\n\nJoanne stripped down to her undies so I could take her measurements. I couldn't really tell how old Joanne was, giving her ageless-looking face and rumored propensity for plastic surgery. But she still had a fabulous body, trim and toned, if a bit on the bony side. Tiny amethysts sparkled on her lilac-colored underwear, matching the jewels around her neck. Talk about flaunting your wealth.\n\nOnce that was done, I draped a piece of white muslin over her and started shaping it into the dress she'd picked out so she could see how it would fit and look.\n\n\"The wedding's in two months. When will you have the dress done?\" Joanne asked.\n\n\"We'll do the final fitting in a couple of weeks.\"\n\n\"Good.\" Joanne's eyes met mine in the mirror. \"I want this dress to be especially fabulous, Fiona. The best thing you've ever done for me. This is going to be my last wedding, and I want to look my very best.\"\n\nI stifled a giggle. \"Your last wedding? Are you sure about that, Joanne?\"\n\nA dreamy, faraway look crept over Joanne's unnaturally smooth face. \"Oh yes. Berkley is everything I've ever wanted in a man. Kind, caring, sweet, richer than a sultan, fantastic in bed. He's a very generous lover, the best I've ever had.\"\n\nI almost swallowed a pin. The woman looked as besotted as a teenager with a movie star. Joanne James truly in love? A scary, scary thought.\n\n\"He always makes sure that I'm satisfied, and he does the most marvelous thing with his...\"\n\nSomehow, I managed to focus on the dress and block out most of the details of Joanne's sex life with Berkley. Those were mental images I definitely did not need.\n\n\"...then again, you'd know how important the bedroom is in a relationship, wouldn't you, Fiona?\"\n\n\"What? What did you say?\" I mumbled through a mouthful of pins.\n\n\"I'm talking about you and Johnny Bulluci.\" Joanne gave me a sly look in the mirror. \"The two of you looked quite cozy at the observatory benefit last night. Until you disappeared, that is. Tell me, are the waterfall and lake still as beautiful as I remember?\"\n\n\"I wouldn't know,\" I replied in an even tone, resisting the urge to stick a sharp pin in Joanne's bony ass.\n\nI'd known we'd be the talk of town after slipping out of the benefit, but Joanne's catty tone still irked me. What I did with Johnny was nobody's business but mine. Especially since I wasn't quite sure what I was doing with Johnny yet. Other than having amazing sex.\n\n\"Now, now, Fiona. We're all friends here. Like father, like son.\"\n\nI shook my head. \"I really have no idea what you're talking about.\"\n\n\"I'm talking about the fact that James Bulluci and I used to meet at the observatory, go down to the waterfall, and become otherwise engaged for several hours. I hadn't realized Johnny had taken up his father's bad habits until I saw you two slip away from the party last night.\"\n\n\"You knew Johnny's father?\"\n\n\"Oh yes. James Bulluci and I were quite the item at one time, although he was several years older than me.\" Joanne preened. \"I was in high school, and he was finishing up his second business degree at Bigtime U when we started dating. He was such a sweet boy. He even gave me his college football ring. I wore it until when we broke up shortly before he graduated. I hated to give it back, though.\"\n\n\"Why? Was it covered with diamonds?\" I couldn't resist the dig.\n\n\"Oh no. It was made out of silver and onyx, if you can believe that. Completely worthless. But it had the most beautiful design of wings on it.\"\n\nI frowned. Wings? On a college class ring? I flashed back to Johnny's cherub boxers and his silver watch, and Bella's necklace. The Bullucis seemed to have a strange fascination with winged, angel-like spirits. One that bordered on obsession. Who knew? Maybe they were the ones who supplied Angel with all his embossed gear.\n\nA vague unease filled me at that odd idea. I couldn't quite put my finger on why connecting Angel to the Bullucis upset me. But it did. For a moment, I felt like Carmen listening to the voices in her head. That couldn't be good.\n\n\"James was very, very good with his hands, from what I remember. We spent many hours down by that waterfall skinny-dipping and doing other things...\"\n\nJoanne chattered on about James Bulluci and his magnificent hands, chasing my stray thought away. I finished pinning up the dress, and Joanne twirled around and looked at the rough copy.\n\n\"You've outdone yourself again, Fiona. This will be perfect, absolutely perfect.\"\n\nIt'd come with a perfect price tag too. I'd make sure of that. Joanne slipped out of the dress and back into her suit. She fluffed out her hair and put on a fresh coat of lavender lipstick.\n\nJoanne eyed me in the mirror. \"Come on, Fiona. Everybody saw you with Johnny at the benefit. Give me something, some little detail to share with the girls at the spa.\"\n\nI stared at Joanne. Then, I smiled. \"Like father, like son.\"\n\n# Chapter Thirteen\n\nI finished up with Joanne James and sped home. I was due at the Bulluci manor at eight, leaving me less than two hours to get my fabulous self ready. Of course, I always looked fabulous, but I really wanted to knock Johnny's socks off. Along with the rest of his clothes.\n\nI took a shower and dried my hair in a nanosecond. Then, I paced back and forth through my closet, trying to find the perfect dress to wear, while I scarfed down a couple bags of key-lime cookies. The gown couldn't be too casual\u2014or too revealing. I wouldn't want to give Bobby Bulluci a heart attack by shoving my cleavage at him \u00e1 la Erica Songe. Although from the way the old man had worked the crowd at the benefit the other night, maybe I was the one who should watch out. Charm seemed to run in the family, except for Bella, who was far too serious and uptight for her own good.\n\nI decided on a short, sleeveless, silver lam\u00e9 number with lots of cowgirl-like fringe in strategic places. The requisite stilettos and bag polished off the look, along with large diamond studs and a matching tennis bracelet. Joanne James wasn't the only one who could sparkle like an electrified disco ball when she wanted to.\n\nBy the time I was dressed, it was creeping up on seven-thirty. I put the gas pedal to the floor and roared across town in my convertible to the Bulluci manor in record time, traveling fifteen miles through heavy traffic in about ten minutes. Swifte couldn't have done much better himself.\n\nTires squealing, I took an exit off the interstate and threaded my way through more traffic before pulling onto Lucky Way. Old, twisted cypress trees and Spanish moss hung over Lucky Way, bathing it in long, soft shadows. Lawns as smooth as carpet led up from the street to some of the priciest homes in Bigtime. We were talking castles here, turrets, moats, and all. Joanne James had a mansion in the area. So did Berkley Brighton, Nate Norris, Wesley Weston, and all the other power players.\n\nI double-checked the address Johnny had given me and turned into a driveway that curled up a steep hill. The wrought-iron gates swung open at my approach. The black pavement curved into a circle that went by the front of the house before looping back around on itself and flowing down the hill. I pulled up to the front steps, got out of the car, and paused to stare at the massive building. I'd been to the mansion a couple of times before during some of Bella's fashion shows, but it still wowed me. It wasn't as large as Sublime, but the house had an old-world feel to it that I admired. The red-tile roof, wide stone arches, and numerous balconies and patios reminded me of a villa on the coast of Italy or Greece. Perhaps Bobby Bulluci had built it to remind himself of his homeland. Either way, the manor had a bright, cheery feel. In the light of day, it would be even more impressive.\n\nA tuxedo-clad butler greeted me at the front door, crooked his finger, and beckoned me to follow him. We moved deeper into the house, and my eyes roamed over the rich furnishings. In a way, it was exactly like Sublime. Tapestries covered the walls, and statues and other pricey knickknacks clustered in curio cabinets.\n\nBut as I looked around, I noticed there was a definite, odd pattern to the finery. Almost every single item had some sort of angel-like figure on it. Fat cherubs smiled from frescoes on the walls. Wings adorned the backs of tall chairs. Miniature halos dangled from a pair of crystal wind chimes. Even the light fixtures looked like marshmallow clouds drifting by. I'd have to ask Johnny about his family's strange obsession.\n\nWe reached a massive wooden door. The butler knocked once before pulling it open. My gaze went to the knob, which was shaped like angel's head, halo and all. It was kind of creepy.\n\n\"Miss Fiona Fine.\" The butler bowed and left the room.\n\nI strode inside. Johnny, Bella, and Bobby sat on a long sofa staring at an enormous plasma-screen television. Soccer players shouted and screamed and slammed into each other on the flickering monitor.\n\n\"Johnny. Bella. Bobby.\"\n\nThey echoed my greeting, but Johnny and Bobby's eyes remained glued to the television. I snorted. Men. Bella rose and walked over to me. She wore a shimmering dress of the palest pink imaginable. Despite Joanne James's hatred of it, the color looked wonderful on Bella, and the embroidery on the bodice was exquisite and dainty. I peered closer at the pattern. Tiny angels ran along the high-fitting neckline, and the usual silver angel charm ringed Bella's throat. What was it with these people and angels?\n\n\"Sorry about them,\" Bella said, rolling her eyes at her male relatives. \"I told them that you'd be here at eight sharp, but they just had to check out the highlights from today's games. Didn't you, Johnny?\" Bella leaned over the couch and mussed her brother's tawny locks.\n\nJohnny swatted her hand away. \"Fiona likes soccer too. She completely understands. Don't you, Fiona?\"\n\nHe looked and sounded like such a little boy that I had to laugh. \"Believe me, I understand. My friend Sam never leaves the house without seeing what the latest football and hockey scores are.\"\n\nAfter several more loud, jovial, clich\u00e9d comments, the sportscaster bid his captivated audience good night, and Johnny clicked off the television. \"Now it's over.\"\n\nBobby came around the couch and bowed to me. \"And now we can properly entertain our lovely, lovely guest.\" The old man pressed a dry kiss to my hand.\n\n\"All right, that's enough of that,\" Johnny growled in a playful tone. He took my hand away from his grandfather.\n\n\"Afraid the old man might be too smooth for you, eh, Johnny boy?\" Bobby elbowed his grandson in the ribs and shot me a playful look.\n\n\"Of course not,\" Johnny replied in an even tone. \"I not only have my father's charm, I have my mother's good looks as well. No woman can resist the combination.\"\n\nBella crossed her arms over her chest. \"I don't think Fiona will be as easily swayed as that.\"\n\n\"Oh, I don't know. I think I've made considerable progress so far. We had a wonderful time at the observatory benefit, didn't we, Fiona?\" Johnny grinned.\n\nThe feel of Johnny's hands, his lips, his tongue on me, in me flashed through my head. For the first time in a very long time, I blushed. \"Yes, we did,\" I murmured, catching his dancing eyes. \"Yes, we did.\"\n\nBella looked back and forth between me and her brother, frowning. She really needed to lighten up. Bobby clapped his hands together.\n\n\"Come! Let's eat! I want my steak and wine and chocolate.\" Bobby beamed at me. \"And not necessarily in that order.\"\n\n\"Grandfather...\" Bella warned.\n\nBobby pulled himself up to his full height. \"We have a guest tonight, Bella. I will not insult her by eating that tasteless gruel the doctor calls food in front of her. The chef has prepared an excellent meal, and we will all dine well tonight.\"\n\nBella sighed, giving in to the old man. I didn't envy her. Bobby Bulluci was quite a handful, even if he was on the downhill side of seventy. I put my hand on Johnny's arm, and he escorted me to the dining room. It was a long, narrow room with a long, narrow table that looked like it could seat five hundred people and then some. More tapestries lined the walls, including one that featured more fat, happy cherubs shining their gleaming halos on puffy clouds.\n\nThe more I stared at the cherubs, the more I wondered why the Bullucis had so many of them. So, I asked.\n\n\"You guys seem to have a thing for angels. Everywhere I turn, there they are. Is your family crest a halo or something?\"\n\nBella's mouth dropped open. Bobby's step faltered. Johnny's arm tensed under my fingertips.\n\n\"I'm sorry, did I say something wrong?\"\n\nMy eyes flicked back and forth between the three of them. What was _that_ about? You'd think I'd just uncovered a skeleton in the hall closet or their secret identities as superheroes or something.\n\nSecret identities as superheroes... I frowned. I'd seen a lot of angels lately, mostly adorning the clothes and sundry gear of Johnny Angel. Could the Bullucis be connected to him? If so, how? Bobby and Bella certainly weren't the motorcycle rider. And it wasn't like Johnny could actually be Angel...could he?\n\nNo, I decided. Johnny didn't have Angel's attitude problems. Besides, Johnny hadn't even been in the country until recently, and Angel had been around Bigtime for years. At least, his old incarnation had been. I didn't know how long the new guy had been prowling the streets.\n\nJohnny forced himself to relax. \"Of course not. Angels aren't our...family crest.\" He looked at his grandfather. \"But we all like them. It's sort of a family...tradition to collect them.\"\n\n\"Oh.\"\n\nI didn't quite believe his explanation, although there was no real reason for me not to. But I decided to let the matter drop. I wasn't as nosy and paranoid as Carmen. I didn't think everyone was harboring a secret identity or wearing a spandex suit underneath his tuxedo. In fact, I didn't really care who was who, as long as they stayed out of my way.\n\nWe sat down at one end of the table. Servants hustled in, bringing with them tempting dishes that smelled like they'd come down from heaven itself. My stomach rumbled, and I forgot about my unease and suspicions. The bags of cookies were long gone.\n\n\"Grilled filet mignon, roasted potatoes, fried zucchini, Italian bread, butter, a Caesar salad, and enough olive oil to grease a car.\" Bella looked at her grandfather. \"Going all out, aren't we?\"\n\nBobby smiled. \"Well, I had to make sure we had enough food for our guest. Johnny told me that she has quite the healthy appetite.\"\n\nBella's eyes flicked to me. For the second time that evening, I found myself blushing. My stomach gurgled again, this time loud enough for everyone to hear. Bella looked shocked by the grumbling sound, while the men seemed amused.\n\n\"Guilty as charged,\" I said, reaching for the nearest dish.\n\n* * *\n\nWe spent the next hour dining and making small talk. The Bullucis were a delightful bunch, merry and funny with no hints of the self-importance and pretentiousness that plagued so many on the society circuit. They didn't take themselves or anyone else too seriously. Even Bella loosened up a bit and let me see that she had a dry, biting sense of humor under her by-the-book fa\u00e7ade. She was the straight woman who kept the men in line. As much as she could anyway.\n\nBut I was most impressed by the food. Everything was just as scrumptious as Bobby had promised it would be. The vegetables were fresh, crisp, and perfectly seasoned. The bread was warm and chewy. And the steak was so tender I could have cut it with a toothpick.\n\nI took big portions of everything and wolfed them down in about fifteen minutes. Then, I went back for seconds. I finished those up about the time the others had washed down their first helpings with a superb bottle of wine. I eyed the platters, wondering if I dared to take any more food. I didn't want to come across as a total glutton. Johnny saw me eyeing the potatoes with barely restrained desire.\n\n\"Let me guess. You didn't have breakfast or lunch again today,\" Johnny said, sliding his half-eaten steak over to me.\n\nI looked down at the juicy, medium-well steak. A man who gave up his food for me? I could get used to this. Quite easily. The thought startled me, but not enough to make me turn down the meat. I stabbed my fork at him. \"You guessed it.\"\n\nSo, I ate the rest of his steak.\n\nAnd the bread.\n\nAnd the potatoes.\n\nAnd the wine.\n\nIn the end, I gobbled down more than the three of them put together. Bella looked back and forth at me and the now-empty platters. Then, she turned her gaze to her brother. Johnny shrugged. Bella probably thought I had an eating disorder, just like Piper did. I'd probably find more pamphlets on my desk in the morning. Ah, well. The steak alone had been worth it.\n\nMore servants appeared to clear the dishes away and present us with a monstrous three-tiered cake. The flavor came as a bit of a surprise to me, although it really shouldn't have.\n\nIt was angel food, of course.\n\nThe cake, topped with cherries, chocolate sauce, and whipped cream, was just as wonderful as everything else had been. I ate four big pieces and put my fork down in contentment. Steak and potatoes were wonderful, high-calorie, filling, stick-to-your ribs food. I'd be all right until I got my midnight snack.\n\n\"Well, I don't know about you young folks, but this old man is ready for bed,\" Bobby announced, pushing his chair away from the table. \"Too much wine and food make me sleepy these days.\"\n\nBella also rose to her feet. \"I'll help you, Grandfather.\" She looked at Johnny and me. \"I'm sure Johnny and Fiona would like some time to themselves.\"\n\nJohnny turned to me. \"I'll show you around the house.\" The devilish glint in his eyes told me that the tour would probably end up in his bedroom. I wasn't sure quite how I felt about that. Other than hot and panting.\n\nBella and Bobby disappeared up a flight of stairs, while Johnny led me down a series of wide hallways. Strolling through the house was a little like walking through a church. More angels decorated everything from paintings to furniture to even the carpet under our feet.\n\nWe reached a hall filled with large oil paintings. Smiling, similar-looking people sat in the portraits, their eyes filled with light and joy. I didn't need Johnny to tell me that I was staring at pictures of the Bullucis through the centuries.\n\nJohnny tugged me down the hall, stopping at a portrait of a middle-aged man on a motorcycle. \"This is my father, James. I wasn't sure if you remembered him or not. He wasn't much for the society scene, preferring to concentrate on looking after our business interests. He rarely attended events, preferring to leave that to Bella and Grandfather.\"\n\nI stared at the picture. Blue eyes. Tan skin. A mane of chestnut hair. James Bulluci had been a handsome man. But the more I looked at the picture, the more it bothered me. There was something else very familiar about James Bulluci. Something in his eyes and the set of his jaw reminded me of\u2014\n\n\"And this is my mother, Lucia,\" Johnny continued, pointing to the next portrait.\n\nMy tremulous thought fled. Johnny hadn't been bragging earlier when he'd said he'd inherited his mother's good looks. Blond hair. Green eyes. Fabulous figure. Tawny skin. Lucia Bulluci was a goddess in mortal form.\n\n\"Funny, I don't remember her either,\" I said, staring at the picture.\n\n\"She died when I was a teenager,\" Johnny said. Sadness colored his voice, a sadness I recognized.\n\n\"My mother died when I was young too,\" I said, squeezing his hand.\n\n\"What about your father? You don't talk about him much.\"\n\nI hesitated. I couldn't tell Johnny who my father really was, but I didn't want to deceive him either. \"Oh, he's...around. His work takes up a lot of his time, and we don't see as much of each other as we'd like.\"\n\nIt wasn't a complete lie. The only times I got to see the chief these days were when we went out as Fiera and Mr. Sage to do battle with Bigtime's ubervillains or braved the society crowd at the latest benefit.\n\n\"What does he do for a living?\"\n\n\"He's in, ah, security.\" Another half-truth.\n\n\"Like a bodyguard?\"\n\n\"Sort of.\" Did protecting the citizens of Bigtime from ubervillains qualify you to be a bodyguard? I supposed it did.\n\n\"What's his name?\"\n\nFinally, an easy one. \"Sean.\"\n\n\"Sean? That's very Irish.\"\n\nI laughed. \"We're a very Irish sort of family. My mother's name was Finola.\"\n\n\"Irish, huh? Does that mean if I kiss you, I'll have good luck for the rest of the year?\"\n\nI tossed my hair over my shoulder. \"I don't know about having good luck in the future, but you just might get lucky tonight.\"\n\n\"Well, then, let's test your theory out right now,\" he murmured.\n\nJohnny dipped his head forward and kissed me. His lips felt like liquid fire on my own. I opened my mouth, and Johnny darted his tongue inside. He tasted like chocolate and wine, and his spicy scent filled my nostrils. Yummy.\n\nJohnny eased me up against the wall. We kept kissing, our tongues moving back and forth. I raised one leg around his waist. Johnny's sure, confident hands moved up and down my body. Kneading. Caressing. Sculpting. He moved from my breasts to my stomach down to the junction between my thighs. I hissed as he stroked me through my silken panties, and a warm, heavy wetness pooled between my legs.\n\n\"Would you care to continue the tour? Say, upstairs?\" Johnny nibbled on the side of my neck.\n\nTingles of pleasure shot through me. My neck was _very_ sensitive. I dug my fingers into his hair and pulled him closer. \"Is that where your bedroom is?\"\n\n\"Mmm-hmm.\" Johnny continued to kiss my neck as one hand cupped my breast. \"So what do you say?\"\n\nBreathless, I couldn't speak. I raised his head and stared in his shimmering green eyes. I opened my mouth to respond, but Johnny swooped in and cut me off with another hot kiss.\n\nThen, the worst thing in the world happened.\n\nMy cell phone rang.\n\n# Chapter Fourteen\n\n\"That's becoming a rather annoying habit of yours,\" Johnny said, his lips still touching mine.\n\n\"Sorry,\" I muttered. \"It's one I'm trying to break.\"\n\nJohnny stepped back and dropped his hands. I bit my lip. All worked up and no relief in sight. The things I gave up for my city. Stupid superhero duty.\n\nI dug through my purse, grabbed my phone, and answered the call. \"What _is_ it, Henry? I'm right in the middle of something.\"\n\n\"We've got a lead on Siren and Intelligal,\" Henry replied, not the least bit put off by my fierce growl. He'd long ago grown used to it. \"We're heading out soon to see if we can track them down and stop them. The chief wants you here ten minutes ago.\"\n\nI stared at Johnny, who leaned forward and pressed another kiss to my neck. I closed my eyes. Mmm. \"Do I have to?\" Whining just a bit.\n\n\"Well, you don't _have_ to, of course,\" Henry replied. \"We could go after the ubervillains by ourselves. But since you're the strongest and since we're down two members, it might be a good idea for you to come in. You do have the most firepower, after all.\"\n\nI snorted. \"Firepower? You've been hanging around Lulu too long.\"\n\n\"I have not,\" Henry snapped back. \"And you should be nicer to her, Fiona. If I get my way, Lulu will be part of the team permanently.\"\n\nI narrowed my eyes. Permanently? I didn't like the sound of that. What was up with Henry and the computer hacker? \"What do you mean by that?\"\n\n\"None of your business at the moment. Now, are you coming in or not?\" he asked.\n\nJohnny's hand slipped under my short skirt and started creeping upward. He stroked my thigh. For a moment, I couldn't speak.\n\n\"I'll be there as soon as I can.\" I ended the call.\n\n\"Let me guess. Henry has more problems at the store.\" Johnny arched an eyebrow.\n\n\"Unfortunately.\"\n\n\"Perhaps you should think about getting another night manager. He always seems to call you at the most inappropriate moments,\" Johnny said.\n\n\"You're telling me. I'm sorry I have to go, Johnny. I was having a fabulous time.\" Regret tinged my voice. It had been wonderful spending the evening with Johnny and his family. And then being alone afterward with the man himself.\n\n\"I know, and I understand. But since you have to leave, we should make the most of this brief time we have together, shouldn't we?\"\n\nJohnny flashed me a wicked, wanton grin and sank to his knees in front of me. Johnny stroked my thighs with his fingers. I gasped at the sensation.\n\n\"Tell me to stop, and I will,\" he said in a quiet voice.\n\nI couldn't say a word, and I didn't want to. For a moment, Travis's face flashed through my mind. Then, Johnny moved his hand, and the image disappeared.\n\n\"Go ahead.\" My voice was thick and husky with passion, desire, need.\n\nBefore I knew what he was doing, Johnny had pushed my skirt up and my panties down. I stepped out of the filmy lace and kicked them down the hall. Johnny hooked one of my long legs over his shoulder. Then, he leaned forward and put his mouth on me.\n\nLicking. Tasting. Teasing. Driving me insane.\n\nI dug my hands into his shoulders. \"Johnny!\" Begging him to stop. Wanting him to go on.\n\nJohnny chose to think of my strangled cry as one of encouragement. He quickened his pace, his tongue probing deeper and deeper with every sure stroke, as though he wanted to lick his way to the very core of me. Liquid desire flowed like thick lava through my veins. Pressure built deep in my stomach. My whole body trembled.\n\nAnd then\u2014\n\nAn eruption of epic proportions. But in a good way.\n\nI just stood there, wrapped in the soft blanket of sexual bliss. Afterglow, people called it. I felt as warm and fuzzy as a fleece blanket that had just tumbled out of the dryer.\n\n\"Wow!\" That was all I could say.\n\n\"I aim to please,\" Johnny said, rising to his feet. \"Again...and again...and again...\" He planted more kisses on my throat. His tongue flicked over my rapid, throbbing pulse.\n\n\"I have to go,\" I said in a weak tone, even as I drew him closer.\n\nWe kissed long and slow and deep. Johnny's hands moved up and down my body. I kneaded his back, his chest, his abdomen with my grasping fingers. I wanted to touch him. All of him. Over and over again until we were both sweaty, satisfied, and utterly spent. And then some. But Johnny had other ideas.\n\n\"Oh, I think we have a little more time left,\" he said, grinning.\n\nJohnny went down on his knees again.\n\n* * *\n\nIt was close to midnight by the time I reached the underground garage at Sublime and parked my convertible next to one of the Fearless Five vans. It'd been almost half an hour since I'd left Johnny, and I couldn't stop smiling. I'd even been humming on the drive across town. Humming! I never hummed. Superheroes did _not_ hum.\n\nI made my way to the equipment room and changed into my orange-red Fiera outfit. I felt good tonight, charged, pumped up. Like I could take on Siren and Intelligal with one hand tied behind my back. And it was all thanks to Johnny. Oh, the things that man could do with his tongue...\n\nA loud squawk sounded, cutting through my sexual daydreams.\n\n\"Fiona, are you almost ready? You've been in there almost twenty minutes.\" The chief's voice cracked over the underground intercom system.\n\n\"Coming.\"\n\nSince my dinner had been burned away by my time with Johnny, I stopped by the kitchen, made myself ten tomato-and-cheese sandwiches, and grabbed a two-liter soda before heading to the library. The others waited inside.\n\n\"Hello, all,\" I said in a cheery tone. \"Ready to go kick some ubervillain ass?\"\n\nThey just stared at me.\n\n\"What?\" I asked. \"What's wrong? Why are you all looking at me like that?\"\n\nThe chief cleared his throat. \"You're glowing, Fiona.\"\n\nI looked down. Sure enough, a soft, orangey glow enveloped my entire body. I hadn't even noticed it, and I was pretty good about keeping my flare-ups in check. Even when I let myself go, most of the time I emanated more of an angry red. I glowed orange only when I was extremely relaxed or very, very happy. I was both right now.\n\n\"Well, well, someone must have gotten lucky tonight,\" Lulu drawled. \"Did we have a _hot_ date, Fiona?\"\n\nI narrowed my eyes at the computer hacker. A spark flew from my fingertip. My orange glow took on a reddish tint. One day I was going to get her for those stupid heat-related puns. One day soon...\n\nMy father cleared his throat again.\n\nBut not today. Too many witnesses around. So instead of frying Lulu from the inside out, I tossed my hair back.\n\n\"As I matter of fact, I did have a _hot date_ tonight. Unlike you two. Let me guess. You've been sitting here all night pounding away on your keyboards trying to track down ubervillains. Is that your idea of a good time, or are the two of you still not over your tiff from the benefit?\"\n\nLulu's lips tightened into a thin line. Henry stopped typing. The two of them looked at me, then each other, then back at their computers. I stared at my father. He shook his head, his way of telling me not to butt in. I glared at him. Lulu had just shoved her way into my business, but I couldn't get a little get-back? _So_ not fair.\n\nI learned long ago there was only one thing to do when life wasn't fair. So, I marched over to the table with its _F5_ insignia, put my feet up, cracked open my soda, and started eating my melted, cheese-and-tomato sandwiches.\n\n# Chapter Fifteen\n\nWhile I polished off my midnight snack, Henry and Lulu used one of their umpteen computer programs to pinpoint Siren and Intelligal's exact location. Surprise, surprise, the ubervillains were holed up in another dilapidated warehouse. This time, though, they were near the downtown area. So, we loaded up the van, exited the garage, and headed toward Bigtime.\n\nWe'd just gotten on the interstate when Lulu frowned.\n\n\"Uh-oh,\" she said.\n\n\"What?\" I asked. \"What's wrong now?\"\n\n\"SNN just cut into their charity benefit. Halitosis Hal and Pistol Pete were engaging in a mock fight to raise money for the children's hospital. Pete would fire his pistol at Hal, but his breath would stop the bullets in midair. Funny stuff.\"\n\n\"But now...\" I prompted.\n\n\"They're live at the construction site where the city's building the new sports complex. The whole thing's collapsed.\" Lulu stared at her laptop. \"Kelly Caleb's on the scene. She's saying that Swifte got trapped inside while he was trying to rescue some of the construction crew.\"\n\n\"Why is a construction crew working on a building after midnight?\" I asked.\n\n\"It's a city contract, and the sports complex is two months behind schedule due to the snow we had this winter. They've been working round-the-clock shifts for the last six weeks to try to get back on track,\" Mr. Sage said. \"Is anyone else on the scene besides Swifte? Anyone providing security or backup for him?\"\n\nLulu shook her head. \"Not yet. The police, fire, and rescue squads are there, but no other superheroes.\" She hesitated. \"It looks pretty bad.\"\n\nShe turned the laptop around to us. The skeleton dome of the sports complex had folded in on itself, like a deck of cards knocked over by a gust of wind. Hoarse screams and shouts could be heard over Kelly Caleb's clipped voice. Sirens flashed blue and red in the distance while dust drifted through the air, obscuring the camera's view. Lulu was right. It looked bad.\n\nI stared at my father. His was face smooth, but his eyes glowed neon green. He was reaching out with his psychic powers, trying to tap into people's vibrations, emotions, fears. \"It is bad. The worst accident we've had in some time. Turn around, Hermit. We need to go help.\"\n\n\"What about Siren and Intelligal?\" I asked.\n\n\"They'll have to wait,\" Mr. Sage said. \"This is more important. Agreed?\"\n\nI'd been looking forward to getting another chance to wrap my hands around Siren's scrawny neck and squeeze until her head popped off her overinflated body. But I'd just have to settle for moving rocks around and saving sweaty construction guys instead. It wasn't a bad trade-off. \"Agreed.\"\n\nHermit whipped a U-turn, and we headed back toward the collapsed building. You could see the smoke and dust miles away, billowing into the night sky like a mushroom cloud.\n\n\"Hermit?\" Mr. Sage asked.\n\n\"I know,\" he answered. \"Step on it.\"\n\nHermit's hands tightened around the wheel, and we made it to the construction site in less than seven minutes. He might be unnaturally fond of computers, but Hermit could drive when he put his mind to it. I thought it was one of his most useful skills.\n\nWe stopped the van in a dark alley about two blocks from the construction site. Lulu outfitted us with two-way earpieces and turned on the cameras in our suits. Once we were set up, Hermit, Mr. Sage, and I jogged over to the scene, while Lulu stayed in the van to monitor us and the news channels.\n\nPeople were everywhere, like ants running around a pool of hot honey. They crowded into the streets, climbed onto parked cars, and hung out apartment windows high above to get a better view of the action. They chattered like magpies, their voices building to a cacophony of high-pitched, excited sound.\n\nI plowed into the crush, trying to clear a path. \"Excuse me, excuse me, coming through. Now. Move, please. Thank you.\" I kept up a steady stream of patter as I zinged people with hot flashes to get them out of the way.\n\nFinally, though, someone took notice of us.\n\n\"Look! It's Fiera!\"\n\n\"And Mr. Sage and Hermit!\"\n\n\"The Fearless Five are here!\"\n\nClaps, cheers, and whistles sounded. People got the message and stepped aside to let us by. I smiled and shot a few sparks off my fingertips, my signature move. Folks roared in response. It was always nice to be recognized. That was another perk of being a superhero. People knew you wherever you went, and you never had to wait in line for anything.\n\nA woman darted through an opening in the crowd and clutched at my arm. \"Please! Please! You've got to help them! My husband is trapped in there!\"\n\nI'm used to such tearful pleas. I've heard thousands of them during my time as a superhero. But the raw panic in her eyes touched my heart. I'd felt the same way about Travis aka Tornado every time he'd gone out on a mission. Panicked and worried and slightly crazed.\n\n\"I'll do my best, ma'am.\" I patted her arm, careful not to burn her, and gently shook off her hand.\n\nThe three of us made our way to the police barricade, where the esteemed members of the press waited. My eyes scanned over the crowd, and I spotted reporters from _The Chronicle_ , _The Expos_ _\u00e9_ , and of course, SNN. Word of our arrival had already spread, because Kelly Caleb was ready and waiting for us. She shoved her microphone in my face the second we approached the cops. The white lights of her television camera cranked up like headlights in my face. I squinted against the blinding glare.\n\n\"Fiera! Fiera! Have you come to help? What's the status of the trapped workers? Do you think you're too late already?\"\n\n\"Later, Kelly,\" I snapped and kept moving. Unlike Swifte, I didn't have time to give interviews _before_ I saved people.\n\nBesides, reporters were always so difficult. Why couldn't they just leave us alone to do our jobs? Public's right to know, my ass. All the public needed to know was that I would do my superhero duty\u2014to protect and serve\u2014no matter what.\n\nWe went around the police barricade and entered the construction site, which was cordoned off from the street by a metal chain-link fence topped with razor wire. Cranes and bulldozers and backhoes sat behind the fence, along with tin trailers, cement mixers, trucks, and portable toilets.\n\nPolice, fire, and other emergency rescue officials hovered around the back of a pickup truck, where a beefy guy with a dirt-covered face sat. His arm was in a makeshift sling, and cuts and bruises dotted his bulging body. He looked like he'd just barely escaped the collapse himself, but he was busy looking at several papers.\n\nThe police let us through, just like they always did, and Mr. Sage headed straight for Beefcake.\n\n\"What's the situation, sir?\" Mr. Sage asked in a calm but strong voice.\n\nSome of the tension dribbled out of the cluster of people around us. That was my father for you. He could calm down a hurricane if he wanted to. He'd done it many times before.\n\n\"I'm Jim, the foreman. I was overseeing the night shift when she came down. Scariest thing I've ever lived through.\" His eyes flicked to the destroyed building.\n\n\"Any idea what caused the collapse?\" Hermit asked, punching some buttons on his handheld computer. No doubt he was accessing the building's blueprints, looking for structural weak spots and the like.\n\n\"Not a clue.\" Jim hesitated. \"But right before she came down, I heard this...sound. Sort of like purring or something. It was the oddest thing.\"\n\nSome of the policemen chuckled. A few meowed. The foreman shot them angry stares.\n\n\"Well, that's what it sounded like to me,\" he growled. \"Instead of just standing around, why don't you boys in blue get in gear and get my folks out?\"\n\nThat shut the cops up real quick. I gave Jim an encouraging nod.\n\n\"Please. Let us offer our assistance,\" Mr. Sage cut in.\n\nJim nodded. \"I've been going over the blueprints. We've got some folks in sector two.\" The foreman pointed to the area in question. \"Swifte arrived about two minutes after she went down, but more of the building gave way and sealed off the entrance. He went in there, but he hasn't come back out. Neither has anyone else.\"\n\n\"All right,\" Mr. Sage said. \"The first thing we need to do is shore up that section so it won't cave in on us when we go in. Fiera, you move the big pieces out of the way. I'll use my telekinesis to help and make sure that everything doesn't come falling down on you.\"\n\nI nodded. Standard operating procedure. This wasn't the first collapsed building we'd seen in Bigtime. We'd run into and dug our way out of more than a few ourselves.\n\n\"Let me help too,\" a cool voice cut through the air.\n\nI snapped my head around. Johnny Angel stood behind us, looking like his usual badass self in his shiny black leather.\n\n\"What are you doing here?\" I snapped. \"I thought you were busy with your vengeance mission. Shouldn't you be plotting ways to run down ubervillains with your bike?\"\n\n\"I heard about the collapse on television. I just want to help,\" he said in a quiet, defensive tone.\n\n\"Us too,\" a pair of feminine voices chimed in.\n\nMy eyes darted over the crowd. I couldn't see them, but I knew they were there. Finally, I spotted two pairs of footprints in the dirt about ten feet away. I squinted and could just make out the faint outlines of two curvy figures. It was like I was looking at something through cloudy water.\n\n\"The Invisible Ing\u00e9nues. It's been a while since we've seen you girls.\" I'd been hanging around Lulu too long. I just couldn't seem to stop with the bad puns.\n\n\"Oh, we've been around...\"\n\n\"Just like always.\"\n\nThe Ing\u00e9nues were sisters who had a strange habit of finishing each other's sentences. I'd wondered more than once if they were joined at the hip. I couldn't tell, though, since they were invisible.\n\nI looked at Hermit, who shrugged. I turned to Mr. Sage. His eyes glowed for half a second, and he nodded.\n\n\"All right, Angel, Ing\u00e9nues,\" Mr. Sage said. \"We can use all the help we can get. Let's move, people.\"\n\nJim told us a few more pertinent details. Some of his folks had called in sick, and he'd been working with a skeleton crew. \"There were thirteen of us inside when she went down. Me and five others made it out. That leaves seven, plus Swifte.\"\n\nOnce we got all the information we could from Jim, we stepped through the police barricade and approached the building. The sports center\u2014what was left of it\u2014looked like a doll's house that had been upset by a child's angry tantrum. Steel beams as thick as my body stuck up at weird angles. Bricks, concrete blocks, wires, cables, and more spilled down from the towering heap of metal. Every once in a while, something deep inside the structure would moan and creak and groan.\n\n\"Right here,\" Hermit said. \"That's where the entrance was.\"\n\nSteel beams and pieces of building the size of small cars covered the exit. Nothing I couldn't handle. I cracked my knuckles and did a few stretches to limber up. It was always important to stretch before undertaking big tasks. I might be a superhero, but I could still blow out my knee or tear my biceps.\n\n\"Which side do you want, Ing\u00e9nues?\"\n\nLike most superheroes, the girls had some muscle on them, especially when they worked together.\n\n\"We'll take...\"\n\n\"The left side.\"\n\n\"Good. I've got the right then.\"\n\nI looked over my shoulder, making sure no one was behind me so they wouldn't get hit by the boulders I was about to toss back. Then, I went to work.\n\nSo did Johnny Angel. In addition to having a superhard exoskeleton, Angel was stronger than average. After twenty minutes, the four of us had cleared a space big enough for a couple of people to walk through. I peered into the darkness. More beams lay inside crisscrossed over each other like the teeth of a zipper. Whoever went in would have to be very, very careful. It would be worse than walking through a maze of razor wire **\u2014** one that could come down on your head and crush you at any moment.\n\n\"It might work better if Fiera and I went inside together. She can hold up the roof and move things as needed, while I drag the people out,\" Angel suggested. \"As long as I keep concentrating on my exoskeleton, the debris won't hurt me.\"\n\nMr. Sage peered into the dark hole. \"You're right. Hermit and I will stay here with the Ing\u00e9nues and monitor the building from the outside. I'll use my telekinesis to try to steady some of the structural points. Be careful. Both of you.\"\n\nI pressed my fist to my heart. Mr. Sage and Hermit did the same. The Ing\u00e9nues had seen the salute before, but Angel stared at us. Curiosity glimmered in his eyes.\n\nI turned to him. \"All right, Angel. Follow me.\"\n\nI took a deep breath and stepped inside the collapsed building.\n\n* * *\n\nThe inside was even worse than the outside. The main support beams had crumpled in on each other like cheap tinfoil. Dust hung in the air like a wet blanket, and debris littered the uneven ground. Rocks slid and shifted under our feet with every step we took. It would have been safer to walk through a minefield, but we plunged inside anyway. We had to. People were counting on us. I wrapped myself in flames to light the way, careful not to burn too bright, so I wouldn't ignite whatever gas or chemicals might be in the air.\n\nI didn't have many fears, but being in a small, unstable space was one of them. I could feel the weight of the building hovering over my head. All that metal. All that steel. All that concrete. Just waiting to come crashing down and bury me forever. It reminded me of the unbreakable glass tube I'd been stuck in last year when the others and I had been captured by the Terrible Triad. I'd felt the tube pressing in on me in the exact same manner. I panted for air.\n\n\"Just take deep breaths,\" Angel said, touching my arm. \"It'll help.\"\n\nI stiffened, hating myself for letting my nervousness show. Still, the feel of his hand reminded me that I wasn't alone in this. And that there were eight people somewhere in here that were in much worse shape than me. I had a duty to help them. I couldn't afford to chicken out now.\n\nI drew in a breath and let it out slowly. My nerves steadied.\n\n\"Fiera, turn left. According to the heat-sensing camera in your suit, there's a thermal image about a hundred yards ahead of you,\" Hermit's voice crackled in my ear.\n\n\"Copy that.\" I jerked my head at Angel. \"This way.\"\n\nWe eased through the metal maze. Ten yards in, twenty, fifty, a hundred. I upped the dimmer on my body and peered into the darkness, trying to see something, anything that resembled a human form. A flash of shimmering white caught my eye. \"Swifte!\"\n\nI eased over to the superhero. Now I knew why he hadn't made it back out of the building. Two metal beams had crashed down on him. One lay on his leg, the other on his back. His costume was torn in several places, and blood blackened the dirt under his body. For a moment, I thought he was unconscious or maybe even dead, but Swifte turned his head to stare at my boots.\n\n\"Hey, Swifte. What's shaking?\" I asked.\n\nThe superfast superhero grinned into the dirt. \"Not much at the moment.\" His voice was raspy and strained. The beam must be pressing down on his lungs, making it hard for him to breathe.\n\n\"You know, I've never seen you like this before. Usually, you just zip in and zip out of these situations,\" I said, being my usual bitchy self to lighten the mood.\n\nHe shrugged. Or at least, he tried to. He got about halfway there before stopping in pain. \"Normally I do. But it's a little more difficult when there's a thousand beams raining down on you at once.\"\n\n\"Well, hold still. We're going to get you out of here.\"\n\nSwifte shook his head an inch. \"Get the others out first. I'm all right.\" He grimaced.\n\n\"Are you sure?\" Angel asked. \"Because you look pretty messed up to me, man.\"\n\n\"Get the others out first,\" Swifte wheezed. \"They're only a couple hundred yards up ahead.\"\n\n\"Noble to the end, huh, Swifte?\" I said.\n\n\"Just doing my job, Fiera. Just doing my job.\"\n\nI could respect that. While we talked, I heard the sound of buttons being punched in my ear.\n\n\"Swifte's right. I've got more images about two hundred yards farther in,\" Hermit said.\n\n\"All right. We're going to get the others out. But we'll be back for you in a few minutes. Okay, Swifte?\"\n\nThe superhero chuckled. \"Of course. Unfortunately, I'm not going anywhere at the moment.\"\n\nI nodded to Angel. \"Let's go.\"\n\nHermit guided us through the remains of the ruined building. It was slow going, but ten minutes later, we stumbled onto the rest of the construction crew. They huddled together in a space about the size of a large desk that was amazingly free of debris. Other than cuts and scrapes, a few broken bones, and having a decade or so scared out of them, the men and women were in remarkably good shape. Somebody upstairs had been smiling on them tonight.\n\nAngel surprised me again. One by one, he pulled the men and women to their feet. He spoke to them in soft, soothing tones and guided them back through the metal maze. If I hadn't known better, I would have thought that Johnny Angel was one of Bigtime's finest superheroes. I stayed behind to help calm the others while they waited for their turn. Every other minute, the building creaked and shrieked like a tree caught in a hurricane, but it stayed up, thanks to my father and his mind of steel.\n\nAngel put his arm around the last construction crew member, and the three of us headed back to Swifte. I picked up the beams pinning down the speedy superhero, and Angel pulled Swifte out from under them. The superhero's suit was in tatters, and blood stained the opalescent fabric a dark scarlet. A long, deep gash decorated his right thigh where shrapnel had cut into him. I winced. Blood didn't bother me, but Swifte's wound looked nasty and painful.\n\n\"I don't think you'll be running any marathons anytime soon,\" Angel said, staring at the superhero's leg.\n\n\"That's what you think,\" Swifte said.\n\nHe took a step as though to dart away. I caught him before he smacked into the ground.\n\n\"Easy, big boy,\" I said. \"There'll be plenty of time to disappear on us later.\"\n\nSwifte mumbled something under his breath.\n\n\"You're welcome,\" I replied.\n\nI picked up Swifte and carried him outside, while Angel took care of the construction guy. The four of us emerged from the building, and the crowd erupted into wild, happy cheers. Cameras flashed. Sirens sounded. People whistled and clapped and yelled until they were hoarse. The roar was deafening.\n\nI put Swifte down in the back of an ambulance and struck my best, classic superhero pose. Hands on hips, stomach in, breasts out. The crowd cheered even louder. I smiled and shot more sparks off my fingertips. A little good press never hurt anyone. There were so many heroes in Bigtime these days it was hard to stand out from the crowd.\n\nA couple of paramedics came over to treat Swifte. Mr. Sage and Hermit moved to stand beside me.\n\nMr. Sage put a hand on my shoulder. \"Good work, Fiera.\"\n\n\"Thanks, Dad,\" I said in a soft voice only he could hear.\n\nMy gaze went to the people we'd pulled from the rubble. I stared at the worried husbands and wives who hovered over their injured spouses as the paramedics checked them out. I thought of Travis. He would have loved to have been part of such a successful rescue. But he wasn't here. And he would never be again.\n\nBut the thought didn't make me feel so empty, so hollow, as it usually did. I could remember the good times now, instead of just the aching pain of Travis's loss. I knew Johnny Bulluci had a great deal to do with that.\n\nSpeaking of Johnnies, I drifted over to Angel, who was standing next to his motorcycle smoking a cigarette. I didn't really want to say _thank you_ , especially to someone who'd threatened to kill us if we got in the way of his pursuit of Siren and Intelligal. But I had to. My silly superhero honor wouldn't let me not do it. \"Thanks for helping me out in there. I don't do so well in tiny spaces.\"\n\n\"No problem. I'm glad you let me help.\" Angel's eyes wandered over to the construction-crew members, who were busy hugging and kissing their weeping spouses and children. \"We did a good thing tonight.\"\n\n\"Yes, we did.\"\n\nWe exchanged a smile, in sync for once. Angel turned and got on his motorcycle.\n\nSomething clinked, and a spot of silver on the black pavement caught my eye.\n\n\"Hey, wait! You lost your watch.\" I leaned down, picked up the silver watch, and turned it over so I could see what time it was. It had to be almost three in the morning by now\u2014\n\nI froze.\n\nA pair of silver angel wings decorated the watch's black face. I recognized it immediately. It was the same watch Siren and Intelligal had stolen at Carmen's wedding. The same watch I'd handed back to Johnny Bulluci.\n\nJohnny Bulluci.\n\nJohnny Angel.\n\nMy eyes flew to his face. And I knew they were one and the same.\n\n\"Thanks,\" he said, taking the watch from my sparking fingers. \"I'd hate to lose this.\"\n\nAngel snapped the timepiece around his wrist and gave me a mock salute. Then, he fired up his motorcycle and rumbled away into the night.\n\nAll I could do was stare at his retreating form.\n\n# PART TWO\n\nTHE HONEYMOON'S OVER\n\n# Chapter Sixteen\n\nAfter shaking hands with everyone, working the crowd, and reluctantly giving interviews to reporters from SNN and the two newspapers in town, _The Chronicle_ and _The Expos_ _\u00e9_ , we piled into the van and headed back to Sublime.\n\nThe others chattered about the rescue and wondered who or what could have brought down the massive sports complex. I sat in the back and stared out the tinted windows. Brooding.\n\nJohnny Bulluci was Johnny Angel. I was still trying to figure out what it meant. How I felt about it. The guy I'd been having a hot fling with was actually a...a what? He wasn't a hard-core ubervillain like Malefica, but Angel was no, well, angel. At least, his last incarnation hadn't been.\n\nJohnny's anger over his father's death. Angel's vow of revenge on Siren and Intelligal for killing his predecessor. All those freaking angels in the Bulluci mansion. I hated to admit it, but Carmen was right. We superheroes weren't the brightest bunch when it came to disguising our real identities. I could have firebombed myself for not seeing Angel's real identity sooner. Say, before I'd gone and slept with him.\n\n\"You're awfully quiet, Fiona,\" the chief said in his thick brogue. \"Is something wrong?\"\n\n\"I'm just tired.\" I was tired. And cranky. And famished. Not to mention the shock I'd just received.\n\n\"Well, I don't want to pry, but if you want to talk about it, I'm here for you.\"\n\nGrateful that he wasn't going to read my mind tonight, I nodded and returned to my brooding.\n\n* * *\n\nI crashed in my underground suite at Sublime and didn't get up until almost four the next afternoon. I hadn't slept well. In my dreams, Johnny Bulluci had kept morphing into Johnny Angel and back again until I couldn't tell where one man started and the other ended. Not the most pleasant dream to have about your lover. Especially when you moonlighted as a superhero who was supposed to fight crime, and your honey was out to kill a couple of ubervillains, no matter what. And it was your responsibility to stop bad things like murders from taking place, no matter how much someone might deserve it.\n\nGrumbling, I got out of bed, showered, and went to the kitchen. I wolfed down five pizzas, three hamburgers, a twelve-pack of soda, two chocolate cheesecakes, an apple pie, and three gallons of vanilla ice cream. I always ate more when I was angry or upset. Johnny's secret identity was more than enough to make me both.\n\nAfter polishing off my late lunch, I went to the library looking for Henry. I wanted him to get me all the information on Johnny Bulluci that he could find. I opened the double doors and stopped short. Instead of Henry, Lulu sat inside the spacious room. The fan in her laptop whirred to life at my arrival.\n\n\"Oh, you're here\u2014\" I started to say _again_ , but I caught myself.\n\n\"Nice to see you too, Fiona,\" Lulu replied.\n\n\"Where are the others?\"\n\nLulu hit a few buttons on her computer. \"The chief had some big city meeting he had to go to about the collapse of the sports complex. Evidently, the powers that be want to know why it happened and how much it's going to cost to clean it up and get the project back on track.\"\n\nNo big shock there. Assessing the damage was the first thing the Bigtime city government did the morning after accidents like last night's incident. They set aside one morning a week to devote to all the buildings, cars, and streetlights the city's superheroes and ubervillains destroyed. Epic battles might have been bad for the historic structures in the city, but they kept the local construction companies in business. There was always a building boom of one kind or another in Bigtime.\n\n\"And Henry?\" He was the one I was really interested in at the moment.\n\n\"He's down at the Complete Computer Company trying out some new microprocessor for his technology column in the newspaper. He won't be back until late tonight.\"\n\nDamn. Where was the computer geek when I needed him? There was nothing I could do about it, though, short of calling Henry and demanding that he torpedo his job to help me. I knew from past experience that wouldn't work. For some strange reason, Henry actually liked his crummy day job at _The Expos_ _\u00e9_. He would play hooky from work only if one of the Fearless Five was in mortal danger\u2014or his precious computer was about to explode.\n\nI sat down in my usual seat at the F5 table and drummed my fingers on the top. Sparks flew everywhere, adding more scars to the scorched wood. My stomach rumbled. Lunch was gone already.\n\n\"Do you need something, Fiona?\" Lulu asked.\n\n\"Why do you ask?\"\n\n\"Because you look just like Carmen did when she was hot on the trail of a superhero she was about to expose. All tense and twitchy and bothered.\"\n\nBeing compared to Carmen aka Karma Girl Cole didn't help my sour mood. But I stopped drumming my fingers. Lulu would find out about Johnny when I told Henry and the chief. I might as well tell her what I wanted and get the information now. I didn't want to wait a second longer than necessary to get the goods on Johnny Bulluci. I wasn't good at waiting. Patience was something else I'd never seen the virtue in.\n\n\"I need everything you can dig up on Johnny Bulluci.\"\n\nLulu arched an eyebrow. \"Why is that? Is he your new honey?\"\n\n\"Yes, he's my new honey,\" I snarled.\n\n\"Mmm-hmm.\" Lulu shot me a coy look. \"So that's who had you all _hot and bothered_ last night.\"\n\nI rolled my eyes at her obvious pun. \"Yes, that's who had me all _hot and bothered_ last night. But there's more to it than that.\"\n\n\"Sure there is,\" Lulu smirked.\n\nShe was getting on my last nerve, so I did my best to shut her up. \"Oh, there's a lot more to it. I'm pretty sure he's really Johnny Angel. You know, the guy who rides around on the motorcycle? The one who threatened to kill us?\"\n\nLulu stopped typing. Her head snapped up. \"No way!\"\n\n\"Way.\"\n\nI told her everything. About Johnny's anger over his father's death, the family obsession with angels, the watch. All of it.\n\nLulu let out a low whistle. \"So you've been getting hot 'n' heavy with the guy who's vowed to destroy Siren and Intelligal, and the Fearless Five if you get in his way.\"\n\nI shot my finger at her. \"You've got it.\"\n\n\"And I thought I had problems,\" Lulu muttered.\n\nMy eyes narrowed. \"What do you mean by that? What's going on with you and Henry?\"\n\nLulu looked at her computer screen and refused to meet my hot, searching gaze.\n\n\"Oh, come on,\" I snapped. \"I spilled to you, now you spill to me. I won't laugh, if that's what you're worried about.\" At least, I'd try not to.\n\n\"Henry asked me to marry him,\" Lulu said in a soft tone.\n\nStraight arrow, geek-to-the-max Henry aka Hermit Harris had proposed marriage to one of the most notorious computer hackers in Bigtime. A woman who could expose us with a click of her mouse. Fabulous. Just fabulous.\n\n\"When?\"\n\n\"Three weeks ago.\"\n\nI thought back. \"Wasn't that the weekend you guys went to that big ubergeek conference in Ashland?\"\n\nLulu glared at me. \"It was not an ubergeek conference. It was a computer symposium about Internet security.\"\n\nI snorted. \"Like there's a difference.\"\n\n\"Anyway,\" Lulu continued with her story. \"We stayed at this quaint little bed-and-breakfast just outside the city and went to the conference during the day. One night when we came back to the room, everything was covered in rose petals. There were flowers and champagne. Henry had hidden the ring in a chocolate cake. He had it made in the shape of a laptop. It was so romantic.\" Lulu's eyes went all soft and dreamy.\n\nRose petals? Champagne? A ring hidden in a computer-shaped chocolate cake? Henry was more of a romantic than I'd given him credit for. In a completely geeky sort of way.\n\n\"Well, that's wonderful, isn't it?\"\n\nMy voice didn't come out too strangled. I, of course, thought Henry was making a terrible mistake, just like I'd thought Sam had been making a terrible mistake when he'd started boinking Carmen when the Terrible Triad was after us. But I forced myself to be polite. For once. For Henry's sake.\n\nLulu stared at me like I'd just said the dumbest thing in the world. Maybe I had. A superhero and a computer hacker? Not a good combo.\n\n\"Look at me, Fiona. I'm in a wheelchair, in case it's escaped your notice. It's not all hearts and flowers, you know.\"\n\n\"So? Lots of people are in wheelchairs. It doesn't seem to bother Henry any, so why should it bother you?\"\n\nLulu sighed and pushed a wisp of black-and-blue hair out of her face. \"You wouldn't understand.\"\n\n\"Is it because of sex?\" In relationships, just about everything came back to sex in the end. Sex or money. Or both. \"The two of you do have sex, don't you?\"\n\n\"Of course we have sex,\" she snapped. \"Just because I'm in a wheelchair doesn't mean I don't have needs. That Henry doesn't have needs. In fact, Henry and I have sex quite frequently\u2014\"\n\nI held my hand up. \"Don't tell me. I don't need the mental image of you two supernerds going at it. It's bad enough Carmen and Sam do the nasty in every corner of the manor imaginable. I don't want to hear about anyone else or I'll never be able to sit on the furniture again. So if sex isn't the problem, what is?\"\n\nLulu stared at her still legs. \"I'm not just paralyzed. I can't have kids either,\" she mumbled.\n\n\"What?\" I asked, straining to hear her. \"What did you say?\"\n\n\"I can't have kids. I can't marry Henry because I can't have kids. There. I've told you what's wrong. Are you happy now?\" Tears shimmered in Lulu's dark eyes.\n\n\"So what?\"\n\nHer mouth gaped open. \"So what?\"\n\nI shrugged. \"So what? So you can't have kids. Lots of women and men can't have kids. Besides, do the two of you even want kids right now? Aren't you a little young for that? You're not even thirty yet.\"\n\n\"No, we don't want kids _right now_. But someday we would, and I can't give them to Henry.\" More tears puddled in her eyes, threatening to spill down her cheeks.\n\n\"You could always adopt,\" I pointed out. \"There are lots of great kids out there who need a good home.\"\n\nLulu shook her head. \"It wouldn't be the same.\"\n\n\"Why the hell not?\"\n\nLulu stared at me as if the answer should be obvious. I was getting rather tired of that look. I wasn't a mind reader like my father was. I couldn't discern someone's innermost thoughts with a single, soul-searing gaze. Beat it out of them, yes. Fry them alive? Always. But glean pertinent information with a quick look? No, not so much.\n\n\"Because of Henry's power.\"\n\n\"What does Henry's power have to do with you not being able\u2014Ah.\" The lightbulb switched on inside my head. \"If you can't have kids, then Henry couldn't pass his mind-melding power on to the next generation of Lo-Harrises.\"\n\n\"Bingo. If we adopted a kid one day, we'd love her to death, but she wouldn't have Henry's power.\"\n\n\"But you don't even know that Henry's kid would get his power anyway. Or that it would manifest in the same way. Sometimes, these things skip a couple of generations.\"\n\nMy father and I were a prime example. It had been my mother and her fiery temper that had influenced my power, not my father's calm sensibility. Johnny and James Bulluci were another pair that proved my point. Johnny had a power his father had probably never dreamed of. But that's what happened when you battled ubervillains on a regular basis. The villains always seemed to live in the nastiest places, surrounded by acres of radioactive waste. The goo wasn't good for your skin or hair\u2014or for your genes. More than one hero had had her powers altered by being exposed to radioactive waste over the years. And if it didn't get you, then it would more than likely get your kids and change them in some way\u2014either good or bad.\n\n\"I don't think Henry would care about whether or not his kid had powers. He'd love him or her just the same.\"\n\nLulu shook her head. \"I've run the numbers. There's a good chance the kid would either have Henry's power or some other manifestation of it. I don't want to take that chance, that opportunity, away from Henry.\"\n\nRun the numbers? Lulu was making decisions about her love life based on some statistics a computer program had spit out. How romantic.\n\nThe other woman eyed me. \"Haven't you ever thought about having kids? About passing your powers on? Isn't that what you would want?\"\n\nI thought about it. I'd been a fire-starting hellion when I was a kid. Everyone on the street where we'd lived in Ireland had thought that I was an arsonist and hopped up on steroids. Only the fact that my father was a policeman had kept me out of juvenile detention. Even as a kid, I had a tendency to beat up bullies.\n\nTravis and I had talked about having children, about the fact that we might pass our powers on to them. Travis had been thrilled with the idea, but I was more ambivalent about it. Don't get me wrong. I loved having superstrength and the ability to zap a pizza with my eyeballs. But powers weren't the beat-all, end-all of the world, as Carmen was so fond of reminding me. Sure, superheroes got plenty of perks, but being one was a lot of hassle too. The long hours and late nights. The constant beatings and narrow escapes. The continual drain on my finances. Having to make nice with the likes of Kelly Caleb, Erica Songe, and other members of the press. My constant need to eat everything in sight. It got old sometimes.\n\nBut the most important thing I'd learned over my years of being a superhero was this\u2014having powers couldn't keep you safe from all the big, bad things out there in the world. Travis's death was proof of that.\n\nI answered Lulu. \"It'd be nice to pass my power on, but it wouldn't determine if I loved my kid or not. And it sure as hell wouldn't keep me from being with the man I loved.\"\n\nI opened my mouth to further argue my point, but Lulu snapped her hand up.\n\n\"I don't want to talk about it anymore,\" she said through gritted teeth. \"Talking won't fix anything, especially not my shriveled-up ovaries and useless legs. Let's get back to your problem.\"\n\nShe pounded away on her laptop, signaling the end of our conversation. For now.\n\nI started drumming my fingers on the table again while Lulu typed and clicked and muttered under her breath.\n\n\"Oh, go hoover down a pizza or something,\" Lulu snapped about five minutes later. \"I can't concentrate with you giving me the laser gaze. All those heat waves make my computer freak out.\"\n\n\"Fine,\" I sniffed, threw my hair over my shoulder, and flounced out of the library.\n\nI stalked to the underground kitchen, but I didn't _hoover_ down a pizza as Lulu had so indelicately suggested. Instead, I ate three boxes of Oreos and drank two gallons of milk. I was just finishing up when Lulu's voice bellowed out of the intercom.\n\n\"I've got the information. You can come back to the library now.\"\n\n\"Gee, thanks,\" I said, draining the last of my milk.\n\nBy the time I returned to the library, Lulu had compiled several of inches of paper on Johnny Bulluci. A printer whirred and chugged in the background, spitting out more reams.\n\n\"How did you get so much information so fast?\" I asked.\n\n\"Carmen showed me how she does it,\" Lulu said. \"She's still working on Frost and Scorpion's real identities, although they didn't leave a trail for her to follow like Malefica did. Carmen doesn't think any of them died at the ice cream factory, not even Malefica.\"\n\n\"I doubt any of them are dead myself,\" I replied. \"Unfortunately, ubervillains are very resilient. One or all the members of the Terrible Triad will come back to Bigtime someday, and we'll be ready for them, real identities or not.\"\n\n\"Anyway, I set up a couple of computer programs to facilitate the process. I'm just printing the last of it now,\" Lulu said, tidying up some pages. \"This is all pretty normal stuff. School, college, business honors. Lots of friends, female and otherwise. Until three months ago.\"\n\n\"When his father died, and he took over as Johnny Angel.\"\n\n\"You betcha.\" Lulu grabbed the last of the pages from the printer and shoved them into a thick blue binder. \"After that, the life of Johnny Bulluci gets a bit more murky.\"\n\n\"Naturally.\"\n\nLife was always murky in the world of superheroes and ubervillains. For someone like Johnny, who wasn't on one side or the other, it would be positively gray. And I hated gray.\n\nLulu tossed me the binder. \"Knock yourself out.\" She paused. \"And try not to set those on fire, okay? I don't want to have to print them out again.\"\n\n\"I won't, Mom,\" I muttered, settling myself at the round, wooden table.\n\nLulu steered her chair toward the door.\n\n\"Hey, Lulu?\"\n\nShe stopped. \"What?\"\n\n\"Thanks. I really do appreciate it.\"\n\nI'd said _thank you_ twice now in less than twenty-four hours. I really was going to have to quit freaking out and getting people to do me favors.\n\nShe nodded. \"Back at you.\"\n\n\"What did I do?\"\n\nLulu stared at me. \"You listened.\"\n\nThen, she opened the door and zoomed away.\n\n# Chapter Seventeen\n\nI waited until the sound of Lulu's wheelchair faded away. Then, I cracked open the binder and started reading.\n\nLulu had compiled quite a bit of information on James John aka Johnny Bulluci. Age thirty-six. Hair blond. Eyes green. Blah, blah, blah. I knew the boring facts already. I wanted to get to the good stuff.\n\nI skimmed through pages detailing Johnny's progress in high school and college, as well as the business accolades he'd received over the years. To my surprise, there were more than a few of those. I flipped through pages of earnings and stock reports. Since Bobby had retired and Johnny had taken over the majority of Bulluci Industries, the company had almost doubled its profits. Johnny was definitely more than just a sexy guy. He was a shrewd businessman who wasn't afraid to take risks. Sam would have approved.\n\nFinally, I found what I was looking for\u2014James Bulluci's obituary. It had appeared in both _The Chronicle_ and _The Expos_ _\u00e9_. Lulu had even downloaded the transcripts off the SNN archive service for me.\n\n_James Michael Bulluci, 58, died in a fiery car accident on Feb. 7. According to Bigtime police, an unknown driver apparently hit the rear of Bulluci's silver Mercedes, causing the gas tank to explode. Bulluci's body was badly burned and partially disintegrated. He was pronounced dead at the scene. The family will receive guests at 6 p.m. Feb. 10 at Bigtime Funeral Home. The burial will take place at 10 a.m. Feb. 11 at Bigtime Cemetery. In lieu of flowers, the family asks that donations be made to the Cure Cancer Research Facility_ _..._\n\nI frowned. Died in a fiery car crash? I wondered how the Bullucis had pulled off that lie. They must have paid someone in the coroner's office to look the other way. Or perhaps one of them had some sort of psychic power or mind-control gizmo. It didn't really matter how they had done it. Only that they had.\n\nI kept reading. The obit went on to detail James's life and his work at the helm of his family's company, as well as his many contributions to Bigtime charities. The local media had covered the funeral, of course, since the Bulluci family was so prominent in Bigtime society. I'd been out of town on a business trip and hadn't gone to the funeral, but I remembered Sam talking about attending and what a sad day it had been.\n\nI stared at a picture of Johnny with his arm around Bella, comforting his weeping sister. Johnny's mouth was set in a hard, tight line. Even though the picture was in black and white, his eyes practically glowed with fury. The casket stood in the foreground. It was closed. Not surprising. If Intelligal had killed James Bulluci with explodium missiles like Angel claimed, there wouldn't have been enough of him left to put in a spoon, much less a casket. Died in a car crash, my ass.\n\nAfter the funeral, there were no more mentions of James Bulluci or Johnny Angel. For a while. Then, about a month after the funeral, SNN reported an Angel sighting at the Everything Electronics Store in downtown Bigtime.\n\nWait a minute. That name sounded familiar. I closed my eyes and thought back. That was one of the places Siren and Intelligal had hit during their crime spree. I remembered because we'd gone tearing after them when they'd robbed the store, but we'd lost them in traffic. All Siren had to do was crook her finger, and twenty cars had slammed into each other. I snorted. Men. And people thought women drivers were hazardous. Please.\n\nI kept reading. According to the television transcripts, Angel had shown up just as the reporter was leaving. That's why we hadn't spotted him that time, but he'd been tracking them even then.\n\nI wondered how he did it. And how he'd known the ubervillains were in the factory a few days ago. Did he sit by the police scanner at night like the newspaper reporters did? Or did he prowl the streets like the roving crews for SNN? Maybe his father's old motorcycle gang friends had given him the heads-up. Perhaps his grandfather helped him. Or even Bella. She had to know the family secret. She might even have some sort of power herself, since Johnny did. She was probably too uptight to use it, though.\n\nAs the months went by, more and more sightings of Johnny Angel were reported. He always popped up where Siren and Intelligal had been, sometimes missing them by minutes. Occasionally, he'd save somebody from a burning building or chase off some would-be rapists, but he spent most of his time hunting the two ubervillains. Trying to get his revenge.\n\nRevenge. Johnny wanted revenge on the ubervillains for killing his father. I couldn't blame him for that. When I'd thought Travis had committed suicide because Carmen had exposed him, I'd wanted to tear her into little pieces and feed her to the fish in the marina. My father had to slip me sleeping pills for a week before I'd calmed down enough to even think about letting Carmen live.\n\nWhen I learned that Malefica had actually murdered the man I loved, I'd gone ballistic. If I could have gotten my hands on the ubervillain, I would have ripped the skin from her body an inch at a time, sewn it back on, and started all over again. And again. And again. So, I didn't begrudge Johnny his revenge. I understood the need for it all too well.\n\nBut in the end, revenge wasn't as satisfying as it seemed. There had been plenty of people and ubervillains who had done me wrong over the years. Caveman Stan, the Undertaker, Frost, Carmen, Malefica. Some of them had gotten their comeuppance and then some. Caveman Stan had been buried alive in a cave-in. Frost got attacked by his own monstrous creations. Malefica had disappeared into a vat of radioactive goo.\n\nBut it had all been so hollow, so anticlimactic. Oh, the idea of revenge tempted you with its sweet, deadly song, whispered sly promises in your ear. But it didn't bring back the person you loved. It didn't change the past. It didn't heal your hurt. Revenge only made you feel that much more empty inside. At least it had me.\n\nNo, Fiona Fine didn't begrudge Johnny Bulluci his revenge. But as Fiera, the superhero, I was honor-bound to stop him. I was in the business of saving lives, not taking them. Not even the lives of ubervillains the world would be better off without. It wasn't for me to decide who lived and who died. That had been one of the first lessons my father had drilled into my head when I'd decided to become a superhero like him. I might be powerful, but I wasn't God. And according to Carmen, karma took care of everybody in the end, good and bad. From what I'd seen so far, she was right. Bad things had a way of happening to bad people. It might take a while, longer than it should, but in the end, you got what you deserved.\n\nBut if I stopped Johnny from taking his revenge, things would end between us. I knew his secret, what he did when he thought no one was watching. The knowledge would only fester and rankle between us until the connection we had turned into something sour and rotten.\n\nI didn't want that to happen. But I couldn't ignore my calling, my duty either. And I didn't know if I could risk telling him the truth about me, about what I did when no one was watching.\n\nSo what was I going to do?\n\n* * *\n\nI sat in the library staring into space, brooding, and eating candy bars until the others came in around eleven that night.\n\n\"Fiona, is something wrong?\" the chief asked, his green eyes bright with concern.\n\nI licked a bit of caramel off my finger and shoved the binder over to him. \"Oh no. Nothing. Nothing at all. Everything's just dandy. In fact, I've been reading up on a good friend of mine. Johnny Bulluci. Aka Johnny Angel.\"\n\nMy father froze, his fingers hovering over the binder. \"Johnny Bulluci is really Angel? Are you sure?\"\n\n\"You've got to be kidding,\" Henry said, his glasses gleaming in the dim light.\n\nI shook my head. \"Unfortunately, I'm not. It's all there. Read it for yourselves.\"\n\nHenry and the chief pored over the pages, while I told them about Johnny's watch and all the angels floating around the Bulluci household.\n\n\"You found this information yourself?\" the chief asked Lulu.\n\nShe shrugged. \"I've been getting pointers from Carmen. She's right, you know. It really is easy to figure out who you guys are.\"\n\n\"It makes sense,\" Henry said, pushing his glasses up his nose. He leaned over his computer and started to type. \"Each Bulluci generation has had at least one son in it, and we know Johnny Angel is a generational superhero\/ubervillain. Not to mention the fact that one of Bulluci Industries' specialties is the production of custom motorcycles. It all fits together. The secret identity, the business, everything.\"\n\nSomething Johnny had said before echoed in my mind. \"He told me once that the only thing he and his father ever argued about was his taking over part of the family business. I didn't think anything of it at the time, but he must have been talking about becoming Johnny Angel.\"\n\n\"But now he is Angel,\" Lulu pointed out. \"And we know what he wants.\"\n\n\"To kill Siren and Intelligal for murdering his father,\" I replied.\n\nChief Newman put his hand on mine. \"Fiona, I know Johnny is your...special friend, but we can't let him kill Siren and Intelligal. That's not how we do things around here. We put ubervillains behind bars. We don't execute them. Just because we have superpowers doesn't mean that we're above the law. And neither is Angel.\"\n\n\"I know,\" I said in a low tone. \"We've got to stop him. No matter what.\"\n\n\"No matter how much you might like him,\" my father said in a kind tone. \"No matter how much you think you might come to care for him.\"\n\n\"I do.\" I rubbed my aching head. \"I do care for him. And that makes this whole thing suck even more.\"\n\n# Chapter Eighteen\n\nWe sat there in silence, each of us digesting this latest secret-identity expos\u00e9.\n\nAnd a loud bell blared out.\n\nHenry jumped like Siren had just electrified him with a couple thousand volts. I thought his polka-dot bow tie would pop right off his neck, along with his plaid sweater vest. Henry punched buttons on one of his many computers. His fingers flashed over the keyboard faster than Swifte zipping down the freeway.\n\n\"I've got them!\" he said. \"I've got Siren and Intelligal!\"\n\nWe clustered around his computer.\n\n\"Where? Where are they?\" I asked.\n\nHenry punched some more keys. \"They're downtown. Right next to Oodles o' Stuff.\"\n\nOodles o' Stuff was the shopping center of shopping centers in Bigtime. The mall of malls. The store of stores. The multistoried building had more levels than a wedding cake and featured everything from clothes to jewelry to makeup to consumer electronics. The only thing I didn't like about the store were its subbasements, where last season's Fiona Fine originals could be had for up to seventy-five percent off. Oodles' extreme sales cut into my profit margins far more than I liked.\n\n\"What would they be doing there?\" I asked. \"They wouldn't be dumb enough to try to rob the place, would they?\"\n\nLike Quicke's, Oodles was considered neutral territory for everyone in Bigtime. After all, even villains had to shop for essentials from time to time\u2014namely, boots, masks, stilettos, and neon, sequined spandex. Oodles carried all of the above in large quantities.\n\nBut the owners of Oodles weren't stupid. They knew better than to rely on the kindness of ubervillains\u2014or your average shoplifter. The store had one of the best security systems in Bigtime. Guards, dogs, lasers, cameras, steel doors, ink-filled security tags. Not to mention the superheroes who shopped there during regular business hours. Oodles had more security than some of the banks in town.\n\nThen again, Siren could charm just about anybody out of just about anything she wanted to in record time. Oodles' security wouldn't faze her or Intelligal a bit.\n\n\"Well, Swifte did stop them from taking everyone's jewels at the wedding, and Oodles has more jewelry than anyone else in town,\" Lulu pointed out. \"Maybe Siren and Intelligal need to stock up on diamonds or something. There were gems all along the border of their radio device.\"\n\n\"I don't know why they're there,\" Henry said, still typing. \"But they've been popping up in and around the area for a couple of days now. Maybe they've set up shop somewhere downtown. Intelligal must have figured out how I found them last time, because she masked the exhaust system on her chair. But I locked in on that strange gas she gave you guys. It contained a small amount of a very rare radioactive isotope that I was able to track through the atmosphere\u2014\"\n\nI held up my hand. \"Enough of the geek talk. It doesn't matter what they're doing at Oodles, only that we've tracked them down. Let's go.\" I took a deep breath. \"Before Johnny Angel beats us to them.\"\n\n* * *\n\nTen minutes later, we were suited up and in the van. Hermit zoomed out of the underground garage and raced through the empty streets.\n\nWorries and fears rattled around inside my head like electrified dice. Siren and Intelligal were deadly enough on their own, but now we had Johnny Bulluci aka Johnny Angel to worry about. What would I do if Angel showed up? What would I do if he tried to kill the ubervillains? Or worse yet, tried to hurt one of us for getting in his way? I didn't know the answers to my troubling questions.\n\nMy father leaned over and took my hand. \"Don't worry, Fiera. It will work out all right in the end. You'll see.\"\n\nI squeezed his hand, comforted as always by his cheerful, optimistic calm.\n\n\"Mr. Sage is right. We might not even see Angel tonight,\" Hermit called out from the driver's seat. \"I was only able to trace the ubervillains by using my radioactive isotope tracker. Angel's not exactly known for his technological devices.\"\n\nI brightened. \"So he probably can't even track them on his own. Excellent.\"\n\n\"Unless he's watching for us,\" Lulu piped up. \"The van's pretty easy to spot. If I were Angel, I wouldn't even try to find Siren and Intelligal on my own. I'd just be looking out for the Fearless Five. Because where you guys are, ubervillains are sure to follow.\"\n\nMy good mood vanished, and I glared at her. \"You're not helping.\"\n\n\"Well, forgive me for thinking.\"\n\nWe rode in silence the rest of the way. Since it was so late, downtown was largely deserted, and Hermit was able to park right in front of Oodles. Normally, at this time of night, steel bars and shutters would have covered the store's plate-glass windows, along with the revolving doors that led inside. But the bars were up, and light spilled out from the building. Siren and Intelligal were here all right. Ubervillains were always so brazen when it came to strong-arm robberies.\n\n\"Same procedure as before?\" I asked. \"You boys in the back, me in the front?\"\n\nMr. Sage nodded. \"You've got it. And Hermit and I have whipped up something special for tonight, just in case Intelligal decides to spread some more of her gas around.\" He held up a small, white pill.\n\nI took the pill and gulped it down. \"So this will keep me from having gas?\" I snickered.\n\nLulu groaned. Hermit rolled his eyes. Mr. Sage just smiled.\n\n\"Oh, come on, Lulu,\" I said. \"It's not any worse than the corny puns you come up with.\"\n\n\"That was terrible, Fiera,\" Lulu said. \"Absolutely terrible.\"\n\n\"Yes, this will keep you from being affected by the gas,\" Hermit said. \"It absorbs the gas before it gets into your bloodstream, sort of like an _RID_ pill. But it only works for about twenty minutes. Mr. Sage and I haven't come up with a permanent solution yet.\"\n\n\"And it will probably only work half as long for you, Fiera,\" my father added. \"Maybe even less, given your rapid metabolism.\"\n\n\"You know you could make millions from flatulent Americans if you could perfect it.\"\n\nNo one responded. Ah well. Some people just didn't have the vision I did. While Mr. Sage and Hermit ate their pills, Lulu gave us all earpieces and activated the cameras in the _F5_ insignias on our suits. Then, it was time to get out and about and see if we could catch us a couple of ubervillains hell-bent on destruction, world domination, and the like.\n\nMr. Sage and Hermit ran around to the back of the building. I marched over to the first door I saw and shoved through it, not even bothering to be quiet or sneaky. I wanted to get this over and done with before Angel showed up. I didn't want to face Johnny just yet. Not until I figured out how I really felt about him. And how big an obstacle our secret identities were going to be to our blossoming relationship. With Travis, it had been so easy. We'd both known who we were when the lights went out, and we'd both been on the same side. But things wouldn't be that simple with Johnny and me. Not by a long shot.\n\nA couple of large, vicious-looking Dobermans lay inside the door. At least, they would have been vicious-looking if they weren't snoring like lumberjacks and piled on top of each other like puppies. Ten feet behind the dogs, four guards slumped against counters full of designer handbags. They, too, were sleeping, and slightly goofy smiles curved their lips.\n\nI sniffed. The sweet, sickening stench of the ubervillains' gas lingered in the air, although there was a slightly more floral aroma to it this time, almost like jasmine. It didn't take a rocket scientist to figure out that Siren and Intelligal had used the blue gas or some variation of it to knock out the guards and dogs. At least they hadn't killed them.\n\n\"Hey, guys, I found the dogs and the guards. At least, the ones on this side of the building.\" I pointed the camera in my suit at the snoring piles so Lulu could get a look at them. \"Everyone seems to be having sweet dreams.\"\n\n\"They'll be all right,\" Lulu squawked in my ear. \"They have good, steady vital signs, and according to our analysis, the gas gets flushed out of people's bodies pretty quickly. They'll probably just sleep for a couple of hours. Keep going, Fiera.\"\n\nThe first floor contained women's clothes and accessories, from lacy lingerie to simple socks to stern business suits to slick panty hose. Even though the store had closed only a few minutes ago at midnight, all the clothes hung neatly from the racks, and the sweaters, shirts, and other items were folded and stacked on wooden tables in tidy piles. The staff at Oodles prided itself on making sure everything was meticulously arranged and easy to find.\n\nI was happy to note that some of my goods, including my latest collection of affordable cocktail dresses, were prominently featured in a display near the middle of the first floor. I stopped to straighten a mannequin's wayward spaghetti strap and continued on. I walked up to one of the many maps located throughout the store. My eyes scanned over the diagrams and lists of products and services available on each floor. If I were Siren and Intelligal, where would I go? What would I need to complete my scheme for world domination? I thought back to the device I'd seen in the abandoned warehouse and the spare bits of metal that had been lying around. Parts. That's what I'd need.\n\n\"I'm heading up to the third floor, where the consumer electronics are,\" I said to the others.\n\n\"Roger that,\" Mr. Sage replied in a tinny voice. \"Hermit and I will go up to the fifth floor, where the jewelry's located.\"\n\nI headed for the nearest escalator and walked up a flight of frozen steps. The second floor was devoted to home goods, like kitchen appliances, bedding, and shower curtains. I resisted the urge to see how many of my new linens were left on the shelves and climbed the next flight of stopped stairs.\n\nThe staff at Oodles was also known for its marketing savvy, which was why the third floor was split between menswear and consumer electronics, with electronics taking up the majority of the space. Someone had realized long ago that men were far more likely to drop a couple thousand dollars on plasma-screen TVs instead of business suits.\n\n\"I'm on three. I'm heading in,\" I whispered to the others.\n\n\"Be careful,\" Hermit whispered back to me.\n\nI slid behind a row of suits and headed for the electronics side of the floor. I stopped every few feet, looking and listening, but all I could hear was the hum of the air-conditioning system, and the others breathing in my ear. I wandered through the rows of televisions, computers, and digital cameras, searching for the ubervillains and exchanging hushed updates with the rest of the gang. After a few minutes of searching, I found the ubervillains' spur-of-the-moment lair.\n\nIt was the same setup as before. Someone had shoved a cash register and displays of chocolate bars off a long counter to make way for the strange, radio-like device. Lots of tools and blueprints and papers were also strewn about the area, and more electronic junk sat in a shopping cart. I picked up something that looked like a subwoofer for a speaker. Maybe the villains were getting into the car stereo installation business. I tossed it aside.\n\n\"You getting this, Lulu?\" I asked.\n\n\"Yep, but focus on the blueprints this time,\" Lulu chirped in my ear. \"I've got plenty of pictures of the actual device from the other night.\"\n\nI did as she asked, pointing my chest and the camera hidden there at the writings and scribblings and schematics on the counter.\n\n\"Interesting. It seems to be some sort of sound device,\" Lulu murmured.\n\nI snorted. \"I could have told you that. The thing looks like a giant radio.\"\n\nI wandered among the debris, getting pictures of everything for Lulu. Siren and Intelligal were nowhere to be found. They must have been supremely confident to leave their doomsday device sitting here by itself. Or supremely stupid. It was all the same when it came to ubervillains. Besides me, the only thing that was talking was a television set tuned to SNN, where Erica Songe preened for the camera.\n\nWhen I was done taking images for Lulu, I went back to the radio-looking gizmo and reared back my fist.\n\n\"Shall I smash it to bits?\" I asked the others.\n\n\"Why don't you just stand guard over it?\" Hermit suggested. \"I'd love to take a look at the device and see some of Intelligal's handiwork. The ubervillains aren't up here on the fifth floor, although there seems to be some jewelry missing. We'll be down in a minute.\"\n\nI rolled my eyes. Hermit was such a technology nerd. But I did as he asked, leaning against a display filled with the newest laser-jet printers. I opened my mouth to tell him to double-time it when a mechanical whir sounded above Erica Songe's voice. Intelligal floated into view.\n\nThe two of us stared at each other.\n\n\"You again! Can't you go set yourself on fire or something?\" Intelligal muttered.\n\nI didn't bother responding. That was another comment I'd heard about a thousand times. Instead, I grabbed the biggest printer I saw off one of the tables and hurled it at the ubervillain. The action surprised her, and she didn't have time to get her force field up before the printer smacked into the bottom of the chair.\n\nThe plastic case shattered, and parts pinged everywhere, along with a good amount of colored ink. Intelligal fumbled for her controls. The chair spun round and round like a child's top before slamming into a thick, metal support beam and sliding to the floor. A bit of smoke spewed out from the back of the trunk, like a car with a coughing fit.\n\nThe impact threw Intelligal out of her seat, and she hit a counter hard before sliding off. She sprawled on the floor, and I stood over her. The ubervillain looked small and sad and lost without her massive, gadget-filled chair. She stared up at me, her eyes dark and unreadable behind her thick, black glasses.\n\nSomeone started to clap.\n\nI turned. Johnny Angel stood behind me, looking dangerous in his head-to-toe black leather. My heart sank. Where had he come from?\n\n\"Very nicely done,\" he said.\n\n\"What are you doing here?\" I snapped. \"Are you following us?\"\n\nAngel tipped his head. \"Guilty as charged, I'm afraid. I saw your van outside and couldn't resist following you. Now, if you'll just step aside, I'll finish the job you so elegantly started. I owe the old Angel that much.\" His voice twisted with guilt, and I knew that Angel, that Johnny, was thinking of his father.\n\nI looked at Intelligal. The ubervillain blanched and scooted backward on her heels, like a crab scrabbling along a sandy beach. She bumped into a glass counter and stopped, trapped. She didn't have any powers without her ruined chair; otherwise she would have used them on me. Right now, Intelligal was as helpless as any other Bigtime citizen. My heart heavy, I turned back to Angel.\n\n\"I'm afraid I can't let you do that,\" I said. \"It's my duty to turn her over to the police.\"\n\n\"And it's my duty to avenge my predecessor, no matter what. Now, get out of my way.\" Angel took a step forward. His hands curled into fists. \"Don't make me hurt you to get to her.\"\n\n\"I can't let you kill her so you can take your revenge. I'm in the business of stopping ubervillains, not murdering them.\"\n\n\"Then it's a good thing I don't have such qualms.\"\n\nAngel marched straight at me. I knew he wasn't going to stop. Not when I was standing between him and Intelligal. Angel wouldn't stop. Neither would Johnny Bulluci. It was my bad fortune they were one and the same.\n\nAnd that I was Fiera, member of the Fearless Five. Protector of the innocent. Defender of democracy. Superhero du jour. I couldn't just let Angel murder Intelligal, no matter how much he wanted to, no matter how much she might deserve it. My duty, my code of honor as a superhero, wouldn't let me.\n\nSo, I slow-pitched a fireball at him, making sure to give him enough time to defend himself. Angel stopped and flexed, his skin taking on a hard, chiseled look. The liquid ball of heat hissed through the air and exploded on his chest. Angel's jacket and T-shirt disintegrated into ash. On the bright side, that gave me an eyeful of nicely toned man-flesh. On the dark side, it didn't slow him down a bit.\n\nSmoke boiled in the store, setting off the fire alarms. Sprinklers dropped down from the ceiling and spewed out a puffy, white, heat- and flame-retardant foam. Oodles had contingency plans for just about everything, including a superhero-ubervillain battle in the middle of the store. It had happened before when Gentleman George and the Dapper Duke both wanted the same silk ascot. George had won, but by the time the fire was out, the ascot had been reduced to ash.\n\nAngel laughed and tapped his chest. It sounded like metal ringing. \"Superstrong exoskeleton, remember? Your fireballs won't hurt me a bit, but I'll forgive you for that one, Fiera.\"\n\nI looked back at the ubervillain. Intelligal had flopped over on her hands and knees and started crawling around the counter. Sweat poured down her face, and a bit of blood darkened the side of her silver suit. She must have injured herself when the chair hit the support beam.\n\nAngel kept coming, his green eyes as hard as jade. My eyes flicked around, trying to figure out a way to stop him without hurting him too much. He was on me in a second.\n\n\"Move out of my way, Fiera,\" Angel growled, his eyes focused on Intelligal. \"Or else.\"\n\nI was out of options and out of time.\n\nSo, I punched him.\n\n# Chapter Nineteen\n\nI really should have restrained myself. It was like trying to punch through granite or slamming my hand into a wall of concrete. Pain pulsed through my fingers, up my wrist, and into my arm and shoulder.\n\n\"Son of a bitch, that hurt!\" I cursed, shaking my hand. I'd have bruises tomorrow for sure.\n\n\"I told you, Fiera. You...can't...hurt...me,\" Angel emphasized. \"Now stand aside. Or else.\"\n\n\"Or else what?\" I taunted, stalling for time.\n\nTime to do what, I hadn't a clue. My options were severely limited. Johnny Angel was strong, and with his heavy exoskeleton, I'd have a hard time wrestling him or pinning him to the ground until the others came to help. I couldn't keep him at bay with my fireballs without destroying the entire floor. The only thing I could do would be to beat on him a little, but my blows wouldn't have any effect as long as he was concentrating on his exoskeleton.\n\n\"Or else, I'll move you. And I don't think you'll like it.\"\n\nMy eyes darted back and forth. I didn't want to hurt Angel aka Johnny Bulluci, but I couldn't let him kill Intelligal either. I just couldn't. Not and still call myself a superhero. I spotted a long, thick pipe among the junk in the shopping cart. I came up with a hasty plan. The only kind of plan I did, really.\n\nI sighed. A great, big, heaving _I'm-such-a-wimpy-girl-I-just-have-to-give-in-before-I-break-a-nail_ sigh. \"All right, you can have her. Just let me tie my shoes first.\"\n\nMy excuse was totally lame, but it worked. The girly sigh, combined with my Barbie doll looks and baby blues, gets 'em every time. Angel's eyes dropped to my shoes, and the hard, chiseled look left his skin.\n\n\"Tie your shoes? But you're wearing boots\u2014\"\n\nI reached over, grabbed the pipe, and smacked Angel with it. That got his attention. He staggered back and fell to one knee.\n\n\"Kinky. I like it,\" he mumbled. Blood welled up out of the cut I'd sliced across his left cheek.\n\nHe got back up and charged at me, but he forgot to concentrate on forming his exoskeleton\u2014exactly what I'd been counting on. I sidestepped him and whacked him across the back with the pipe. But Angel limped to his feet again. And again. And again.\n\nEvery time Angel got back up, I hit him. Every time, the pipe got a bit heavier in my hands. Every cut, every bruise made me feel sick inside. Every blow I gave him was like a knife in my own heart. Despite his faults and desperate need for vengeance, I really liked Angel aka Johnny Bulluci. I hated to hurt him. I hated to hurt all innocents. But he'd left me with no choice. Sometimes, being a superhero really sucked. Damn duty.\n\nAngel's skeleton might have been superstrong, but so was I. With the pipe, I was able to put a few dents in him. Drops of blood slid down his chest like a trail of rubies. The sight made me want to retch. But he kept coming at me. Stupid, stubborn fool.\n\nDidn't the man realize that he was beaten? Evidently not, because Angel stumbled to his feet after a long moment of kissing the floor. He seesawed back and forth like a kid's teeter-totter. Then, he took a step forward. He wasn't going to stop until I killed him. Well, I wasn't going to do that. Not to Angel. Not to Johnny. I'd had enough of this. I marched over to him.\n\n\"You really should have stayed down that last time,\" I snarled.\n\nI reared back and gave him the ole Fist-o-Might. I put everything I had into the punch. All my strength, all my anger, all my guilt, all my shame. My fist cracked against his jaw, and Angel crumpled to the ground like a paper doll. I let him. Maybe the fall would knock some sense into that hard head of his, although I doubted it.\n\nThis time, Angel didn't get up. He lay there on the floor. I drew in a deep breath and threw the pipe away. I didn't want to look at it. Or the blood trickling out of the many wounds I'd inflicted on Angel. On Johnny. I closed my eyes. What had I done?\n\nA gloved hand settled on my shoulder. I shrieked, reached back for the hand, and flipped the person it belonged to over my shoulder. A blur of green slammed into the floor in front of me.\n\nMr. Sage's masked face stared up at me. His mouth opened and closed in pain, and his eyes rolled around in his head like marbles. I cringed. First my lover, now my father. I was batting a thousand today. Who was next? Some orphan out on the street? How about a senior citizen on his way home from the bingo parlor? Maybe I could knock Lulu around a little. That would make my day complete.\n\n\"Oh, sorry about that. You startled me.\"\n\nI reached down and helped Mr. Sage to his feet. His green eyes were dark and dazed. I kept a hand on his arm to keep him from plummeting back to the floor.\n\nHermit moved over to the unconscious Angel. \"Fiera, what, uh, happened here?\"\n\n\"I had Intelligal cornered, and Angel showed up. He wanted me to step aside so he could finish her off\u2014Intelligal!\" During my sickening fight with Angel, I'd forgotten about the supersmart ubervillain.\n\nI whipped around. She was gone, along with all the blueprints, the shopping cart full of stuff, and the curious machine she'd been working on. Her chair sat where it had fallen, slowly getting covered with sticky, white foam, just like everything else in the store.\n\nDamn. I put my fist through a computer monitor, not even caring about the mess I made. I couldn't believe I'd been so stupid as to let her get away. She'd been injured, for crying out loud. What kind of superhero was I? I beat up on people who helped me and let ubervillains escape. Maybe it was time to hang up the skintight spandex and mask. I punched through another monitor.\n\n\"Easy, Fiera. Easy,\" Mr. Sage said, stepping back to avoid the flying debris. \"It's not your fault she got away, so please don't destroy any more of Oodles' stock than necessary.\"\n\n\"Was there any sign of Siren?\" I asked.\n\nHermit shook his head. \"Not a trace of her, although there was a smashed jewelry case on the fifth floor. It looks like Intelligal broke in alone.\" He spotted the ruined Intellichair and brightened. \"But at least she left her chair behind. I've been itching to get my hands on it and see the technology she's got hard-wired inside it.\"\n\nHermit pulled out his computer and started to walk toward the chair.\n\nMr. Sage's eyes glowed. \"No, Hermit!\"\n\nA tall, skinny cylinder popped up on top of the chair, and a red light flashed on and off. Red lights were never, ever good. I didn't have to be psychic to know what was going to happen next. I grabbed Hermit by the back of his black-and-white suit and sprinted toward the escalator, dragging him along behind me.\n\n\"Get Johnny!\" I screamed at my father.\n\nMr. Sage used his telekinesis to pick up Johnny's unconscious form, which zipped ahead of him. Johnny sailed down the frozen escalator with Mr. Sage right behind. Hermit pounded down after them.\n\n\"Go, go, go!\" I shouted.\n\nBeing the strongest and fire-resistant, I stayed in the back this time, so I could shield the others from the force of the explosion\u2014\n\nIntelligal's chair self-destructed with a roar. I don't know what sort of explosive she had in that thing, but it packed one hell of a punch. Even better than my trusty Fist-o-Might. The others had just reached the bottom of the escalator and I was halfway down when the shockwave from the explosion knocked me into them. I felt like I was a human bowling ball, and they were a set of pins. We went down in a tangle of limbs, boots, and spandex. The building shook, and fire raced down the escalator, threatening to roast the others. I scrambled on top, trying to shield them from the searing heat. But it wasn't going to be enough.\n\nAnother set of sprinklers dropped down from the ceiling and spit out a pale, blue foam, adding to the mounds of white. The color scheme reminded me of some sort of winter wonderland. But the blue foam did its job, and the fire snuffed out just as it hit the bottom of the escalator\u2014inches away from our feet. A second later, fans rumbled to life, pulling the smoke and soot from the explosion up out of the building. Bless Oodles and its efficient safety system.\n\nThe foam covered everything. By the time we got to our feet, we were ankle-deep in the stuff. Everybody was a little shaken up, but nobody had any serious injuries. Nobody except Johnny. Hermit managed to prop him up against a table full of blenders. Blood dripped from the cuts on his face and chest, a bright, scarlet contrast to the white and blue around us.\n\n\"Let's go,\" Mr. Sage said. \"The ubervillains are gone, and the fire department will be on their way to assess the damage. There's nothing more we can do here tonight. We need to get Angel back to Sublime and treat his injuries.\"\n\nI sighed. I knew I'd done the right thing by stopping him, but seeing him there on the floor beaten, bloody, and bruised made something twist deep inside me. I knelt down beside him and brushed his hair back from his face.\n\n\"I'm so sorry, Johnny. So sorry,\" I whispered. \"But you wouldn't stop. You wouldn't listen to me. And I couldn't let you do it. I just couldn't.\"\n\nHe didn't respond. So, I picked up Angel, heaved him over my shoulder, and followed the others out of the building.\n\n* * *\n\nTwo hours later, I stood outside one of the underground sick bays in Sublime. I stared through the tinted glass window at the unconscious Johnny Angel. After waking up the unconscious guards and filling them in on what had happened, we'd ridden back to the manor in silence.\n\nI'd put Johnny in one of the sick bays we used when we had unexpected guests. The chief and Henry had treated Angel's cuts and bruises with their magic mojo pills, salves, and ointments. I hadn't beaten him as badly as I feared, and the chief promised me that he would make a full recovery. It didn't ease my guilt any, though. Now, the four of us waited outside the room trying to decide what we were going to do when Angel woke up.\n\n\"Beating up the guy you're sleeping with. That's twisted, Fiona, even for a superhero,\" Lulu said, attempting a joke.\n\n\"He didn't exactly give me a choice,\" I growled. \"I couldn't let him just kill Intelligal, especially when she was completely out of it. They call that _murder_ , you know. \"\n\nMy father placed a comforting hand on my shoulder. \"You did what had to be done. We know you'd never intentionally harm someone, Fiona. Don't we, Lulu?\"\n\nThe computer hacker didn't quite meet his gaze. Or mine. \"Sure, we know that, Chief Newman.\"\n\n\"What you think I might do or not do doesn't matter. The fact is that I beat Angel to a bloody pulp.\"\n\nIf it had been Scorpion or Frost or some other ubervillain in that bed, I would have been beaming like a comet, proud of whipping an evildoer. Now, I just felt sick and cold and tired.\n\nMy eyes traced over Johnny's face. We'd left his mask on, of course, but bruises had already started to form on his golden skin. A white T-shirt stretched over his chest, while a cotton sheet covered the rest of his body. An IV dripped antibiotics into his arm, while more machines monitored his heart and blood pressure. Johnny looked so sad, so battered, so broken lying in that hospital bed. And it was all because of me.\n\nEven worse, it could happen again if Johnny kept on with his mission of vengeance. I couldn't bear to stop him. Not Johnny. Not again. My heart couldn't take it.\n\nIn that instant, I made a decision. Probably a bad decision, probably the wrong move to make, but it was the only plan I could come up with. I looked at the others.\n\n\"I have to tell him that I know who he is, and I have to tell him who I am too.\"\n\nEveryone stilled. Then, Lulu whistled. Henry pushed his glasses up his nose. The chief laced his fingers together.\n\nFinally, Chief Newman spoke. \"Do you think that's a good idea, Fiona?\"\n\nI shrugged. \"I haven't got a clue. But it's the only way I can think of to get him to stop this vendetta. We got lucky tonight. If Intelligal hadn't been injured, she could have killed us both. She could have gassed us and shot us and not thought twice about it. Next time, we might not be so lucky. Maybe if Johnny knows who I really am and knows I'll see that justice is served, he'll think twice about killing the ubervillains.\"\n\n\"You think he cares for you?\" the chief asked. His eyes glowed a second, like the quick flare of a cigarette.\n\nI thought back to the time I'd spent with Johnny. Our dinner at Quicke's. Riding the Ferris wheel. Sneaking out of the observatory. Swimming in the lake. Dinner at his house. The way he kissed me. The way he held me. The way he made my heart quicken just by thinking about him.\n\n\"I don't know exactly how he feels about me,\" I replied. \"But I care about him. I have to try to stop him before he hurts himself **\u2014** or somebody else.\"\n\nFrom inside the sick bay, Angel let out a low moan. The sound tugged at my heart. He was beginning to come around. I put my mask back, opened the door, and stepped inside the room. His head turned in my direction.\n\n\"Well, if it isn't my favorite sadist,\" Angel said. His voice was thick and slurred like he'd been drinking instead of unconscious.\n\nI settled myself in a chair next to the bed. My eyes flicked to the many monitors that surrounded Angel. Heart, lungs, brain. Everything was fine, despite the smackdown I'd given him. Relief filled my body, along with guilt and shame.\n\n\"I'm sorry about that,\" I said in a low voice, not quite looking into his bleary eyes. \"But you didn't give me a choice.\"\n\n\"No, I suppose I didn't.\" He rubbed his bruised jaw. \"You've got quite the right hook going there. Especially when you're swinging something heavy to go along with it.\"\n\nI didn't respond. There was nothing I could say to change the fact that I'd just kicked his ass. He was taking it better than I had expected, though. Most men would have been curled up in a fetal position and whimpering by now if they'd been knocked out by a woman, even one with superstrength. No matter whether they were ubervillains or not, getting beaten up by a girl was enough to make the strongest man just fall to pieces.\n\n\"Where am I?\" Angel asked, looking around at the gleaming medical equipment.\n\n\"At Fearless Five headquarters. In one of our sick bays. It's where we bring team members to rest and recuperate when we get injured **\u2014** or other people who need medical attention.\"\n\nAngel struggled to sit up. I put my hand under his shoulder to help him, but he flinched and jerked away. I dropped my hand and stepped back, my heart heavy and aching. I twisted my engagement ring around my finger. His eyes went to the ring, and my hand stilled.\n\nAngel managed to raise himself up. He fell back against the bed, panting a bit from the effort. When he got his breath back, his eyes went around the room again, more focused now. Taking in every little detail. Then, he peeked under the covers. In addition to being the chief of police, my father was also a first-rate doctor. He'd cut Angel's skin-tight pants off so he could examine his whole body. I'd been torn between watching and leering and staying away. In the end, I'd stayed away, not wanting to see any more of the damage I'd done than necessary.\n\n\"Nice jammies,\" Angel said. \"Do they come with the room?\"\n\nHow he could crack a joke at a time like this, I didn't know. But some of the anguish dripped out of my heart. \"You might say that.\"\n\nAngel examined the rest of his body, flexing his arms and legs and making sure everything was in working order, more or less. His hands went up to his face and the mask that was still there. His head snapped around to me.\n\n\"Did you...?\"\n\nI shook my head. \"No, I didn't peek and see who you really are under there. Neither did the others.\"\n\nTension ebbed out of his body.\n\n\"I didn't need to look under your mask,\" I continued. \"Because I knew who you were already.\"\n\n\"Really?\" he said in a fake, cheery, carefree voice. \"And who do you think I am?\"\n\n\"Johnny Bulluci.\"\n\nAngel looked at me. His shoulders sagged. Then, he did the last thing I expected him to do. He reached up and yanked his mask off. Johnny Bulluci stared back at me. Green eyes. Tawny hair. Golden skin. My gaze traced over his features. He was still as handsome as handsome could be, even with the blue and purple bruises that colored his face.\n\n\"How did you figure it out?\" he asked in a low voice.\n\n\"You gave yourself away with all those angels,\" I replied in a soft tone. \"All angels, all the time.\"\n\nJohnny's green eyes narrowed, and he looked at me. Wondering. Thinking. Remembering. And he knew. Just as I had known. The knowledge blazed in his eyes.\n\nThere was only one thing left to do. I reached up and took off my own mask.\n\n\"Hello, Johnny,\" I said.\n\n# Chapter Twenty\n\nThe act shocked him as much as his unmasking had startled me.\n\nJohnny's eyes widened. \"Fiona? You're really...Fiera?\"\n\n\"In the flaming flesh,\" I said, tossing my mask on the foot of the bed.\n\nWe stared at each other. Then, Johnny threw his head back and laughed.\n\nHe laughed...\n\nAnd laughed...\n\nAnd laughed...\n\n\"What's so funny?\" I growled.\n\n\"This! Us!\" he said, wiping away tears of mirth from his battered face. \"Look at us. You're a superhero, and I'm...well, I'm not exactly as pure as vanilla. We just got into a knock-down, drag-out brawl over the life of an ubervillain who doesn't deserve to live. You beat the snot out of me, and now I find out the superhero who kicked my ass is, in fact, the woman I've been seeing. It's absurd. It's ridiculous. It's just...funny!\"\n\nJohnny started laughing again. Then, he winced and grabbed his taped ribs. His laughter died on his lips. My heart sank down into my rumbling stomach. There was nothing funny about what I'd done to Johnny Bulluci, nothing humorous or amusing about the beating I'd given him. All to save Intelligal's worthless ass.\n\n\"It's not funny,\" I said in a cold tone. \"It's\u2014\" I bit my words off, shocked and more than a little disturbed by what I'd been about to say.\n\n\"It's what?\" Johnny asked. \"What were you going to say?\"\n\nI rolled my eyes. \"It's karma.\"\n\n\"Karma? Where are you getting that from?\"\n\nI sighed. \"I have this _friend_ , for lack of a better word. She thinks that everything we do affects everybody else in the world and that our actions determine events in our lives. That there are no coincidences, basically. That everything is destiny. Fate. Kismet. Karma.\"\n\nI was glad Carmen wasn't here to hear me use her catchphrase and superhero name. I'd scoffed at her view of the world more than once. If she knew I was bandying around words like _karma_ , I'd never hear the end of it. Ever. Lulu would probably tell her all about it, though.\n\nJohnny's green eyes narrowed. \"Karma? As in Karma Girl? Wait a minute. Isn't that the name of one of the Fearless Five now? The new member who took over Tornado's slot after Malefica killed him\u2014\" Johnny stopped, guessing the rest of my secret.\n\n\"The man you were engaged to, the guy you're so in love with, it was Tornado, wasn't it? The superhero?\"\n\nI closed my eyes and nodded. \"Yes, he was the man I was engaged to. And yes, Karma Girl is the name of the woman who took his place on the Fearless Five team.\"\n\nJohnny sat still, digesting everything. Me being Fiera. Me being engaged to a superhero who'd been murdered. Him being beaten down by the woman he was sleeping with. I couldn't quite read the emotions swirling in his beautiful eyes, but I had to try to explain things to him. Johnny Bulluci had become very important to me these last few days. I wanted him to understand me, accept me. I needed him to.\n\n\"Johnny, I\u2014\"\n\nA whoosh sounded at the far end of the room, and the door opened. To my surprise, the others trooped in. Costumes on. Masks off.\n\n\"What...what are you doing?!\" I squealed.\n\nThe chief and Henry exchanged unapologetic shrugs.\n\n\"He knows who you are, he'd figure out the rest of us eventually. We thought we'd just speed along the process,\" Henry said.\n\nI sat back in my chair, dazed. Good grief, if we kept this up, everybody in the greater Bigtime metropolitan area would know our secret identities. We really needed to stop telling people what we did in our spare time.\n\nJohnny's mouth dropped open. \"Chief Newman? The chief of police in Bigtime?\" He stared at the chief, who was still wearing his green-and-white costume. \"You're Mr. Sage, the psychic superhero?\"\n\n\"Guilty as charged,\" the chief said, looking at me. \"I'm also Fiona's father.\"\n\nJohnny's head snapped back to me. \"Hold on. You said your father was a bodyguard. That his name was Sean and that you didn't get to see him as often as you'd like because of his work.\"\n\nI winced and shrugged. \"The chief's first name is Sean. And he is a bodyguard, sort of. I mean, he _is_ a superhero. That's like being a bodyguard, except of the whole city instead of just one person. And we don't _really_ spend a lot of time together\u2014\"\n\n\"Except when you're out chasing ubervillains,\" Johnny finished.\n\nI winced again.\n\nHis eyes moved over to Henry and his black-and-white outfit. \"And who are you?\"\n\nHenry stepped forward and offered Johnny his hand to shake. \"I'm Henry Harris, aka Hermit.\"\n\n\"Hermit, the technical whiz, right?\" Johnny gave Henry's hand a hard squeeze.\n\nThe superhero stepped back, wincing and cradling his hand. \"Guilty as charged.\"\n\n\"You're the Henry who calls Fiona at odd hours with those weird emergencies?\"\n\n\"That's me,\" Henry said. \"Sorry if I've been interrupting your dates lately, but we've had some situations going on these past couple days, as you're well aware of.\"\n\nJohnny's eyes moved over to Lulu. They lingered on the blue streaks in her black hair. \"And who are you? Karma Girl? You're not dressed in silver. In fact, you don't really look like her. And, not to be rude, but she's not in a wheelchair. Unless that's part of your disguise or something.\"\n\nLulu laughed. \"Oh no. I'm not Karma Girl. Not even close. I'm Lulu Lo.\"\n\nJohnny frowned. \"I've never heard of a superhero named Lulu.\"\n\n\"Oh, I'm not a superhero. I just help the Five out from time to time. I'm sort of a sidekick. A groupie, if you will.\"\n\nI glared at Lulu. Superheroes did not have _groupies_. At least, not the real pros like us. Debonair might have legions of them, but not the Fearless Five. Lulu made us sound like we were some sort of strange rock band. Or encouraged those weird people who belonged to _Slaves for Superhero Sex_.\n\nHenry put his arm around Lulu. \"She's also my fianc\u00e9e.\"\n\nLulu scowled. \"Actually, that has yet to be decided,\" she said, shrugging off his touch.\n\nHenry pushed his glasses up his nose. \"Well, when exactly are you going to get around to deciding that?\"\n\nHe sounded more than a little annoyed. I stared at the mild-mannered computer guru. Henry never got really cross, not even when I melted his computer wires. But now, he sounded rather like me when I hadn't eaten for a couple of hours **\u2014** ready to explode.\n\n\"I told you that I needed more time to think about your proposal, and I told you why,\" Lulu snapped.\n\n\"It's been almost a month already,\" Henry snapped back. \"Either you love me and want to marry me, or you don't.\"\n\n\"Of course I love you,\" Lulu replied. \"With all my heart. Even if I want to throttle you right now.\"\n\nHenry's eyes narrowed. Lulu crossed her arms over her chest and glared at him. Johnny looked back and forth between the couple, apparently amused by the squabbling supernerds.\n\nMy father stepped in between the feuding lovers and cleared his throat. \"Why don't we take this discussion to the library and leave Fiona and Johnny alone? I'm sure they have plenty of things to discuss.\"\n\nHe squeezed my hand once, then walked out of the room, followed by Henry.\n\nLulu's dark eyes flicked back and forth between the two of us. \"Yes, lots of things to...discuss.\" She snickered, then motored away.\n\nThe door whooshed shut behind them, and I let out a long breath.\n\n\"Sorry about that,\" I said. \"Usually, we're not so boisterous. At least not Lulu and Henry. They're the quiet ones of the gang. They hardly ever fight and never in front of other people.\"\n\n\"It's okay,\" Johnny said. \"It took my mind off my ribs for a few minutes.\"\n\nI grabbed his hand and stared into his eyes. \"Johnny, I want you to know how sorry I am. I truly, truly am. But I couldn't let you kill Intelligal. I just couldn't.\"\n\nHe dropped his eyes. \"You know she's evil. She and Siren have tried to kill you before. And when you figured out who I was, you knew the two of them killed my father. Why couldn't you have just let me deal with her the way I wanted to?\"\n\n\"Because I'm a superhero. I have a code of honor that I follow. I don't kill ubervillains, no matter how heinous their crimes are. I only hurt others to protect myself or innocent people. If I went around killing ubervillains or punishing everyone who broke the law in whatever manner I saw fit, I wouldn't be a superhero anymore. I'd just be a vigilante, dispensing what I saw as justice. Like Violet Crush and Sahara and all the others.\"\n\n\"A vigilante? Is that what you think I am?\" Johnny asked, his eyes bright and hard in his bruised face.\n\n\"No.\" I shook my head. \"I don't think you're a vigilante. I think you're a son who misses his father very, very much.\"\n\n\"Then, why didn't you let me have her? Why?\"\n\n\"Because killing her and even Siren too won't take away your pain. It won't bring your father back, Johnny. Nothing will do that. Believe me, I know.\"\n\nHis eyes burned into mine. \"What do you know about pain? You still have your father by your side. Fighting with you. Just like always. Mine is dead, Fiona. Dead. And he's never coming back.\"\n\nTravis's face flashed through my mind. His kind eyes. His sweet, happy smile. I turned my ring around my finger.\n\n\"I know your pain, your need for vengeance, a lot better than you think. Malefica murdered Tornado, the man I loved. The man I was going to marry. Don't you think I wanted to kill her for that? Don't you think I wanted to rip her black heart out of her chest and squeeze it until it burst like a pi\u00f1ata?\"\n\n\"You said that the person who killed your fianc\u00e9 got what she deserved. What happened to her? What happened to Malefica?\"\n\nI flashed back to that fateful night in the Snowdom Ice Cream Factory. I was trapped in a glass tube. And cold. So cold. And angry. And fearful. \"Carmen Cole, Karma Girl, dropped her into a vat of radioactive something-or-other.\"\n\n\"So, she's dead.\"\n\nI shook my head. \"We don't know that. No one's seen her since. Ubervillains are sort of like movie monsters. They always come back, even if they've been beaten, stabbed, shot, burned alive, and beheaded. I won't believe Malefica or the other members of the Terrible Triad are dead until I see their bodies. \"\n\n\"But Malefica is probably dead. Or at least horrible mutated.\" Johnny's voice was flat, cold, icy.\n\n\"Yes.\"\n\n\"You got your revenge, Fiona. Malefica paid for what she did to you and yours. So why are you denying me my vengeance?\"\n\nI let out a long breath. \"Inadvertently, yes, I did get my revenge. But it didn't bring Travis back. It didn't change the pain I felt when he was murdered. Can't you see? If you pursue your vendetta against Siren and Intelligal, it won't help you in the end. You could even die in the process, and where would that leave Bobby and Bella? Grieving for you _and_ your father. Would you cause them more pain just to kill a couple of worthless ubervillains?\"\n\nJohnny didn't respond. He looked at me, then turned over and rolled onto his side toward the wall. Away from me.\n\nSuddenly tired, I let out a long sigh. Our conversation was clearly over. For now.\n\nSo I did the only thing I could. I opened the door and left the room.\n\n* * *\n\nI left Johnny to his brooding, stalked to my suite, and crashed onto the king-size bed. It was approaching dawn, and I was worn out. For once, I was even too tired to eat.\n\nI fell into a dark, dreamless sleep and woke up around noon. To my surprise, I felt a little better. I knew Johnny's secret, and now he knew mine. I didn't really like keeping secrets, especially from a man I was sleeping with. One that I had come to care about. It was so much easier just to say what you wanted to, when you wanted to. Consequences be damned.\n\nI took a hot shower and pondered my wardrobe for the day. I planned to give Johnny the grand tour of the Fearless Five headquarters, and I wanted to look damn good when he saw me. I needed to look good. I had a feeling I'd have to amp up the old Fiona Fine charm in order to get Johnny to forget about our boxing match and his kissing the ground multiple times last night. If he ever really could.\n\nI riffled through the small, dark area. Unfortunately, I didn't have quite the selection or closet space as I did in my apartment back in Bigtime. There were no lights in the closet, no stacks of shoes, no rows of purses, no chests filled with jewelry. Just a couple of pitiful metal racks that weren't nearly long enough. In fact, Sam had limited me to one measly, ordinary-sized closet. He said the underground space was too valuable to use housing my enormous clothing collection. The man was so misguided sometimes. Just because the love of his life preferred ratty T-shirts and ripped jeans to real clothes didn't mean the rest of us should have to suffer.\n\nI settled on an available-but-casual outfit so I wouldn't seem too eager. I'd done that already the other night, when Johnny had me moaning and begging for more against the wall. I put on a pair of tight, low-cut jeans that hugged my curves and a sleeveless, black top that laced up the front. It looked like something a working girl might have worn in an Old West saloon. Combined with the jeans, a pair of stilettos, and a black velvet choker, I had the whole _hooker-with-a-heart-of-gold_ look going on. I turned, admiring myself in the mirror over the dresser. I liked it. I hoped Johnny would too.\n\nI closed my eyes, strategizing. I'd go check on Johnny and, if he felt like it, give him the grand tour of the place. Then, maybe we could have lunch. My stomach rumbled. Well, maybe Johnny could have lunch, and I could have whatever he didn't eat. And everything else in the refrigerators. All the refrigerators. Upstairs and downstairs.\n\nI put on my makeup, brushed my golden hair until it gleamed, and headed for the sick bay. I peeked in through the window, not wanting to disturb Johnny if he was still asleep. But he was gone. The covers on the bed were thrown back, and the machines stood still and silent. My eyes flicked up and down the corridor. No broken glass, no busted door. Johnny hadn't bashed his way out of the sick bay like Carmen had once. So where was he?\n\nI spent the next twenty minutes stalking up and down the carpeted corridors of our underground lair. I checked the kitchen, the game room, the gym, the entertainment room, the garage. I reached the library. If he wasn't in here, I'd send out a red alert. I cracked the door open and stuck my head in. Johnny was inside. Relief flooded my body, followed by a far more potent feeling.\n\nI stared at him. Green eyes. Tawny skin. Golden hair. A hard body sent down from heaven itself. It was no wonder the Bullucis were obsessed with angels. With their golden good looks, they could have passed for the winged guardians themselves.\n\nJohnny prowled around the room with a predator's deadly grace. The bruises on his face and arms had faded to a soft purple, and most of the cuts had already sewn themselves shut. Johnny, Angel, must be a quick healer too. Most superheroes and ubervillains were. You sort of had to be in this business.\n\n\"Hey there, handsome,\" I drawled in a light tone and stepped inside. I wanted things to go better today than they had last night. I needed them to. \"What are you doing in here?\"\n\n\"I'm looking for a phone,\" Johnny said, eyeing Henry's many computers, wires, and other gadgets. \"That's just about the only thing you people don't seem to have around here.\"\n\n\"Why do you want a phone?\" My heart sank. I didn't want him to leave just yet. Not until things were right between us, and I'd convinced him to let the Fearless Five handle Siren and Intelligal.\n\n\"When I'm out riding around as Johnny Angel, I check in with Bella and grandfather a couple of times a night. It's been over twelve hours since I last contacted them.\"\n\nI slapped my head against my hand. \"I'm so sorry! I should have thought of that last night. They're probably worried sick.\" I would have been. I'd always been worried when Travis had gone out on missions by himself. \"Come on. You can use the phone in my room.\"\n\nI led Johnny down the twisting corridors to the one that housed our underground suites. His eyes flicked over the signs on the doors that designated whose room was where. They lingered on the one that read _Tornado_.\n\nI opened the door to my suite, and we stepped inside. Johnny strode ahead, taking in the furniture and knickknacks. Johnny's gaze focused on Travis's picture before settling on the enormous bed. A wave of hot, fiery anticipation shot through my body.\n\n\"So this is where you stay when you're fighting crime. I like it. It looks like you.\"\n\nI glanced around. Red sofa, black chairs, tile floors, white walls, fireproof paintings. The room was almost an exact copy of my apartment, except without the extra, much-needed closet space. \"Thanks. The phone's over there by the bed.\"\n\nJohnny's lips twitched up into a seductive smile. \"Really? How interesting.\" He gave me a long, steady stare that made my hormones flare to life. \"Perhaps we'll explore that area of the suite later. Unfortunately, right now I really do need to make my phone call.\"\n\nA few sparks shot off the tips of my fingers and landed on the slick tile floor.\n\n\"Dial 5 to get out of the manor,\" I said in a helpful, slightly flustered tone. Like Joanne James, I was no innocent, young thing, but something about Johnny Bulluci made me quiver deep down inside.\n\nJohnny punched in a series of numbers on the black phone. \"Hey, it's me.\"\n\n\"Where the hell have you _been_?\"\n\nI winced. I could hear Bella's high-pitched screech across the room.\n\n\"Grandfather and I have been up all night!\"\n\n\"I'm sorry,\" Johnny said. \"I was unavoidably detained, but I'm fine now. In fact, I'm with Fiona.\"\n\n\"Fiona?\" Bella squawked. \"Did the two of you have a sleepover? Is that why you didn't call and check in last night? You should have told me where you were going, Johnny. If I'd known that you and your new girlfriend were having fun, then I would have done something nice for myself. Like gone to bed at a decent hour instead of sitting up, waiting and worrying about you.\"\n\n\"We did something like that,\" he replied. \"I'm sorry I didn't call sooner.\"\n\nBella spent the next five minutes tearing into her brother for his lack of manners and total disregard for her feelings. \"The next time you stay out all night with your girlfriend, you'd better call me, Johnny Bulluci. Or else it will be the last time you're ever with a woman. And I do mean _ever_.\"\n\n\"I'm sorry, Bella. I really am. It won't happen again.\" Johnny glanced at me. \"Although I suppose I should tell you that I probably won't be home tonight either. Fiona and I are right in the middle of something important. So, don't wait up for me.\"\n\n\"You\u2014you\u2014you!\"\n\nBella sputtered incoherently, then slammed down the phone. The crack sounded like a concrete block being splintered. Johnny winced and rubbed his ear.\n\n\"She sounded really angry. I'm sorry I got you into trouble.\"\n\nJohnny shrugged and put the phone back in its cradle. \"This isn't the first time I've forgotten to check in. Bella will get over it. Eventually.\"\n\n\"I hope so. For your sake. I always thought Bella was so cool, so calm, so collected, but she didn't sound like someone I'd want to cross. Not even as Fiera.\"\n\n\"You have no idea. Bella can get a little worked up at times. And when she does, well, strange things happen.\" Johnny ran his fingers through his thick hair. \"But I'm yours for another day, at least. So, how do you want to spend it?\"\n\nI knew exactly how I'd like to spend the day. In bed. With Johnny. I stared at him, wondering exactly how injured he still was\u2014\n\n\"But before we do anything, do you think I could get some real clothes to wear?\"\n\nMy eyes traced over his body, and I realized that Johnny was only wearing the white T-shirt and pajama bottoms we kept in the sick bay. The clothes were so thin they were practically transparent. More sparks fluttered from my fingertips, and I curled my hands into fists to keep them from igniting. Now was not the time to ogle Johnny. Not after I'd beaten him senseless only a few hours ago.\n\n\"Come on,\" I said. \"We'll find you some clothes, then I'll show you around.\"\n\n* * *\n\n\"Are you sure about this, Fiona?\" Johnny asked half an hour later. \"I know this might be upsetting for you.\"\n\nI closed my eyes. \"It's fine. Really, it is.\"\n\nAfter half an hour of rifling through the suites and pitifully small closets, we'd come up with nada. Nothing came close to fitting Johnny. My father's and Sam's shirts and pants were too small across the chest and too long. Henry's clothes were too short and too mismatched. Besides, I wasn't about to make Johnny wear polka-dot bow ties with plaid sweater vests. That was just cruel. Which meant the only other place to look for men's clothes in the manor was in Travis's room.\n\nI opened the door to Travis's suite, and we stepped inside. My heart twisted, the way it always did when I came in here. A fine layer of dust covered the coffee table and entertainment center. The bed was bare, stripped of its sheets, and the magazines and knickknacks had been removed from the bookshelves. A couple of months ago, I'd packed up Travis's things and stacked them in some cardboard boxes in a corner of the room. I hadn't had the heart to get rid of them or donate them to charity. Not yet. I turned the ring around on my finger. The diamond glowed under my hot hands.\n\nJohnny put a hand on my shoulder. \"Are you sure this is okay? I really don't mind wearing the pajamas.\"\n\nMy eyes slid over his perfect biceps. I didn't mind him wearing the pajamas either. Or even nothing at all. That was the problem. \"No, this is fine. Someone should get some use out of Travis's clothes. I've been meaning to donate them to charity. I just haven't gotten around to it yet.\"\n\nI stepped away from Johnny and tore into one of the boxes marked _Clothes_. I handed him a pair of jeans and a T-shirt that featured several white, cartoon clouds. \"Here. Try these. They look like they're about your size.\"\n\nJohnny nodded and stepped into the bathroom. He came back out wearing the clothes I'd given him. \"They fit well enough,\" he said, turning around. \"How do I look?\"\n\nHe looked perfect, of course, just like always. But my eyes misted over just a bit. That had been one of Travis's favorite T-shirts. I'd given it to him as a gag gift for his birthday last year.\n\n\"Fine,\" I said, dropping my eyes. \"Let's get out of here.\"\n\nWe closed the door, and I took a deep breath. Remembering Travis was always a little difficult for me. So was going into his old room and realizing that he'd never come striding into it again. Still, it wasn't quite as hard as it had been in the past. Johnny had made it a little easier for me. I only hoped I could do the same for him and his pain over his father's death.\n\nI took a step forward. My head spun, and I braced myself against the wall.\n\n\"Hey, are you all right?\" Johnny asked, his eyes bright with concern. \"If me wearing Travis's clothes bothers you that much\u2014\"\n\n\"It's not that. Sorry. I'm just a little light-headed. I get that way when I don't eat enough.\"\n\nJohnny stared at me. \"Really?\"\n\nMy stomach chose that moment to let out what amounted to a plaintive wail. _Feed me_... _feed me_... _feed me now_ , it whined.\n\nJohnny's eyes widened. \"I didn't know a person's stomach could make that particular sound. Especially for that long.\"\n\n\"Fiery metabolism,\" I explained. \"I have to eat quite a bit every couple hours or I get light-headed and weak. As you can tell from my complaining stomach.\"\n\n\"Don't worry about it. I could use a little something to eat myself. Food first. Then, we'll explore. Deal?\"\n\nMy stomach rumbled again. \"Deal.\"\n\nI led Johnny to the underground kitchen and cracked open the refrigerators. I pulled out almost everything that was in there. Johnny started making some turkey-and-Swiss sandwiches. I couldn't wait any longer, so I grabbed a frozen pizza and tore off the cellophane covering. I wiggled my fingers, and flames covered the dish. Thirty seconds later, the pizza was done. I put it on a plate, folded the pizza over, and prepared to sink my teeth into the cheesy, greasy concoction.\n\n\"Can I have some of that?\" Johnny stared at me, a knife in one hand and a fork in the other.\n\n\"Sorry,\" I said, putting down the pizza. \"Everybody around here's pretty used to my, ah, habits. Sometimes, when I'm hungry, I forget my manners around other people, including how to share my food. Here. Have some. Please.\"\n\nJohnny cut himself off a large slice, while I inhaled the rest of it. We sat there in silence, munching on pizza and sandwiches. After about twenty minutes, Johnny pushed his plate away. I kept eating.\n\nMy eyes zeroed in on the half-eaten sandwich on his plate. \"So, are you going to eat that?\"\n\nJohnny shook his head and laughed. He slid his plate over to me. \"I seem to be doing this a lot around you.\"\n\n\"Oh? What's that?\"\n\n\"Missing meals.\"\n\nI rolled my eyes at his bad joke. Johnny chuckled, and the tension between us lightened. Soon, we were laughing and talking and joking just like we had before we'd discovered each other's secret identities\u2014and our confrontation at Oodles o' Stuff. We were having such a good time I almost forgot that less than twenty-four hours earlier, we'd been at each other's throats. Almost.\n\n\"I know this must have come as a shock to you,\" I said, polishing off a bag of chocolate chip cookies. \"Our secret lair. Me having superpowers. Especially the superstrength. That freaks a lot of people out, especially guys.\"\n\n\"Actually, I think I always sort of knew,\" Johnny said.\n\n\"Really? How?\"\n\n\"I'm not quite sure. When I was Angel and I ran into you guys, I thought you looked very familiar. That you reminded me of somebody I knew. I just couldn't quite put my finger on who it was.\"\n\nI narrowed my eyes. \"You don't happen to have an inner voice that whispers to you, do you?\"\n\nJohnny shook his head. \"Sorry. Psychic powers aren't my forte.\"\n\nGood. We had enough mind readers around here. It was getting to where a person couldn't even _think_ anything in private. Especially anything naughty.\n\n\"And there was the food, of course. You ate so much and were so thin I thought maybe you were anorexic or something.\"\n\n\"You're not the only one.\"\n\nJohnny continued with his story. \"Plus, that night we had dinner with Bella and Grandfather, when we were looking at the portraits, your body was...warm, hot even. Unnaturally so.\" A rueful grin crossed Johnny's face. \"I thought it was me at first, but I guess I can't take credit for that now, can I?\"\n\n\"Oh, it was mostly you,\" I said, blushing just a bit as I remembered my outrageous behavior. I'd begged Johnny for more...and more...and more.\n\n\"Really? Well, then, I'll have to remember that.\"\n\nAnd there he was, standing in front of me. \"And exactly how I went about doing it.\"\n\nJohnny lowered his lips to mine. I closed my eyes, reveling in the sensation. His tongue licked the corner of my mouth, and I opened it. Johnny plunged his tongue inside even as he pulled me toward him. His hands ran up and down my back. I tangled my fingers in his hair. I breathed in, letting his spicy scent make my head swim.\n\nBut I couldn't fully enjoy the kiss. I kept seeing myself hitting Angel with that metal pipe, his blood zipping through the air. With a sigh, I broke off the kiss and stepped back.\n\n\"Johnny, about last night\u2014\"\n\nHe put his finger to my lips. Another fiery flare shot through me. \"Let's not talk about business today.\" He smiled that crooked grin I knew and loved so well. \"Didn't you say something about a tour?\"\n\n* * *\n\n\"This is incredible. Absolutely incredible.\" Wonderment filled Johnny's green eyes. Which were pretty wondrous in and of themselves.\n\nDuring the last hour, we'd done a complete sweep of the underground lair. The library, the kitchen, the suites, game room, equipment room, everything. Now, we stood in the training room watching a computer simulation of our previous battle with Siren and Intelligal at the factory near Good Intentions Lane.\n\n\"The equipment, the computers, the holograms. Where do you guys get the money for all this stuff?\" Johnny asked, his eyes moving from the lifelike holograms to the rows of computers and blinking lights. I could almost see him trying to calculate the cost of everything.\n\nI shrugged. \"Being a superhero isn't cheap, so it helps that Sam's a billionaire.\"\n\n\"Who's Sam?\" Johnny asked.\n\n\"Sam Sloane, the businessman,\" I replied. \"He's Striker, another member of the Fearless Five, and one of the wealthiest guys in the city. He has plenty of money to burn on our superhero gear. And I don't do so badly myself either. Fiona Fine Fashions has a very healthy profit margin.\"\n\nJohnny's eyes slid over my shirt and jeans. \"I'm sure it does, with such a beautiful woman running it.\"\n\nI smiled, pleased by the compliment. A lot of things about Johnny Bulluci pleased me.\n\nHe turned back to the holograms that flickered in the training room below us. \"Do you have more of these? I'd love to watch them.\"\n\n\"Of course. But they're so boring. Why would you want to look at\u2014\" I knew why Johnny would want to watch our training simulations. He wanted to know more about Siren and Intelligal. Every strength. Every weakness. Every habit. He had to, if he was going to kill them.\n\n\"Oh, Johnny.\"\n\n\"He was my father, and he was blown away. Can't you understand that?\" His voice was hard with emotion. Anger. Rage. Grief.\n\n\"Of course I can understand it.\" I put a hand on his shoulder. \"No matter what else has happened, you have to know that I'm sorry for your loss. Losing a loved one is never easy, no matter what the circumstances are. Losing your father the way you did is especially painful. I know you think that I'm against you, that the Fearless Five are against you, but we're on your side. We want Siren and Intelligal brought to justice just as much as you do. They're planning something\u2014something that could hurt a lot of innocent people. We've got to stop them before they get a chance to do it.\"\n\nHe didn't respond, but his eyes went to my hand on his shoulder. They lingered on my diamond ring.\n\n\"Is there anything I can say to get you to change your mind? To work with us?\" I said in a soft voice.\n\nJohnny stared at me. \"You have your duty, your code. And I have mine. I'm sorry, Fiona.\"\n\nThen, he turned and walked away.\n\n# Chapter Twenty-One\n\nJohnny returned to the sick bay and spent the rest of the afternoon in bed recovering his strength. I paced around the library and brooded for an hour, before heading to the kitchen and polishing off everything I could get my hands on. Cookies, chips, pies, pizzas, cold cuts, cheese, bread. If it wasn't nailed down, I ate it. If I'd had enough salt left, I would have gnawed on a couple of table legs. But I'd used it all on my French fries. Damn.\n\nThe chief came in late that afternoon after his shift and checked Johnny's vitals. I stood in a corner and watched.\n\n\"Well, it looks like you're healing up nicely, Johnny,\" the chief said, listening to Johnny's heart through a stethoscope. \"In addition to your exoskeleton, I also believe you have a bit of regeneration. Then again, most superheroes do.\"\n\n\"I'm not a superhero,\" Johnny said.\n\nHe didn't speak to me. He didn't even look in my direction.\n\n\"Well, everything appears to be in order. I think you can return home tomorrow, if you wish. Try not to do anything too strenuous for a couple of days, though. Okay?\" the chief asked.\n\nJohnny nodded his head. \"Okay. I'll get out of your hair first thing in the morning.\"\n\n\"I'll drive you back to Bigtime,\" I volunteered.\n\nHe stared at the tiled ceiling and shook his head. \"That's all right. I'll just call a cab to take me back into town.\"\n\n\"You can't exactly do that,\" I pointed out. \"We're a couple hundred feet underground here, you know. Plus, there's our whole secret-identity thing. Calling a cab would officially blow our cover.\"\n\n\"Then, I'll go upstairs and call a cab from there. I'll be fine. I always am.\"\n\nOur gazes locked. Johnny pulled the covers up to his chin and rolled over on his side.\n\nShutting me out again.\n\n* * *\n\nTired, grouchy, and frustrated, I went to bed early. And I dreamed...\n\nI stood between Johnny and Siren. We were all in costume, standing in the middle of an open field. Spring-green grass stretched for miles in every direction, contrasting with the gaudy colors of our suits. And the gunmetal-gray of the pistol in Johnny's hand. It was a large weapon, more like a cannon than an actual gun.\n\nSiren stared at me with her hypnotic blue eyes. \"Are you just going to stand there and let him kill me? Or are you going to be a real superhero and save me?\"\n\nHer voice sounded harsh and demanding. I could feel her will wrapping around me, squeezing my thoughts, my actions into her wishes and desires. My feet moved of their own accord until I stood in front of Johnny.\n\n\"Please, Johnny,\" I begged. \"Please don't do this. Don't make me choose between the two of you. She's not worth it. She's nothing compared to you. Please, please don't do this.\"\n\nJohnny raised his gun and pointed it right at me. I opened my mouth to beg, to plead with him one more time. But he didn't listen. He pulled the trigger. The bullet punched through the air, slamming into my heart. Breaking it\u2014\n\nI woke up with a gasp and bolted upright. My heart pounded. My hair crackled and sizzled and hissed with fire. My fingertips glowed. Sparks flew everywhere, smoldering on my silk sheets. My eyes zipped around the room.\n\nMy room. I was in my room. Slowly, I relaxed. It was a dream. Just a dream. Just a nightmare. I flopped back against the bed and rubbed my blazing eyes.\n\nFor now.\n\nI stripped the sheets off the bed so they wouldn't catch fire, then lay back down. I tried to go back to sleep, but for once, it eluded me. Hot and sweaty, I got out of bed and grabbed a couple of thick towels and an itty-bitty bikini from my stash of swimsuits.\n\nIt was time for a midnight swim, something I'd started doing after Travis had died. I hadn't slept well for months after his murder, and swimming was the only thing that calmed me down. The gentle splash of the water, the weightless feel of it against my skin, the rhythmic, repeated kick of my arms and legs soothed something primal deep inside me. For a little while, I could forget about my troubles, my pain, my anger. I could lose myself in the water. Plus, I didn't have to worry about getting emotional and setting the pool on fire. I'd melted more than one treadmill that way.\n\nI walked through the deserted halls of the underground lair. We were taking the night off, given the disastrous events of the previous evening. My father had gone back into the city to finish up some paperwork regarding the mess at Oodles. Henry and Lulu were around somewhere, probably huddled in the library _type-type-typing_ away on their computers, but no one was officially on call.\n\nI made my way to the exercise room. Unlike my pitiful closet space, Sam had spared no expense in outfitting the gym with the best treadmills, elliptical trainers, and strength-training devices money could buy. Row after row of gleaming machines sat in the long room, along with shelves full of exercise DVDs. Free weights, exercise balls, and yoga mats hung in racks in a corner of the room, which smelled like sweat, rubber, and old gym socks.\n\nWe couldn't afford to let ourselves go or ubervillains would wipe the floor with us. And then there was the PR side of things. Nobody\u2014not the public and certainly not SNN\u2014wanted to see out-of-shape superheroes. Even the older heroes like Granny Cane kept themselves in tip-top condition. They needed to in order to sell their toys and video games. Given my fiery metabolism, I didn't have a problem keeping the weight off, but the others had to work at it.\n\nI strolled past the machines, opened a door, and stepped into the pool room. Sam had spared no expense here either. Cushioned lounge chairs and umbrella-topped tables crouched on either side of the Olympic-sized pool, along with metal lockers full of scuba, snorkeling, and other diving equipment. In addition to saving people on the mean streets of Bigtime, we'd also done more than a few underwater rescues. At least once a month, some civilian ran his car off the Skyline Bridge and into the marina. Or a cruise ship tipped over and had to be righted before it sank to the bottom of Bigtime Bay.\n\nBut perhaps the most unusual feature of the pool room was the ceiling. Instead of the usual tiled ceiling that hovered above so many of the hallways in the underground lair, this one featured embedded 3-D holograms similar to those in the training room. A knob on one wall let you set the ceiling to whatever scene you wanted to look at, from rolling clouds, to lightning, to several of the better-known constellations. I chose the _night sky_ setting, remembering my time with Johnny at the observatory. The ceiling dimmed to a dusky gray, and a smattering of stars twinkled to life, along with a full moon. I hit another knob, and the soft sounds of crickets and other forest creatures filtered into the room. Ah, perfect.\n\nThe faintest hint of a spring breeze brushed my face, and I felt some of the tension ease out of my body. I spread my towels out next to the shallow end of the pool and did a few stretches to loosen up my tight muscles. Then, I climbed up the twenty-foot diving board, bounced once, and plunged into the deep, blue water.\n\nThe wetness closed over me, blocking out everything else. Johnny's stubbornness, his vendetta, my fruitless attempts to stop him, these new feelings I was developing for him. I swam and swam, going from one end of the pool and back again in a thoughtless rhythm.\n\nI popped up for air and found myself staring at a pair of bare feet. Slightly hairy, very male feet. My eyes snapped up.\n\nJohnny towered above me, clad only in his pajama bottoms. \"Hey there.\"\n\n\"Hey,\" I said, not quite sure what he was doing here. I swam over and sprawled across the steps that led down into the water.\n\nJohnny followed me. He sat down on the edge a few feet away and dangled his legs in the warm water. I tried not to notice how the fabric clung to his toned calves. Or how see-through it became when wet.\n\n\"I wanted to talk to you about earlier today,\" Johnny said. \"I acted like a complete ass. I'm sorry.\"\n\n\"There's no need to apologize.\"\n\n\"Yes, there is. You were trying to help me, and I shut you down.\" He drew in a deep breath. \"It's just, I don't know, I can't think straight where Siren and Intelligal are concerned. It just hurts too much. All I can focus on is my father and what they did to him. But I took my pain, my frustration, out on you today. I'm sorry for that.\"\n\n\"Forget about it. I understand. Really, I do.\"\n\nJohnny nodded. \"That's what makes it so hard sometimes.\"\n\nHe stared out at the shimmering water, then back at me. \"Care for some company?\"\n\n\"I'd love some. Doing laps gets pretty boring after a while,\" I joked, trying to lighten the mood.\n\nI swam back out into the middle of the pool, so Johnny would have room to ease himself down into the water. To my surprise, he followed my progress alongside the edge of the pool, grinning.\n\n\"Cannonball!\" Johnny shouted, throwing himself into the water.\n\nAn enormous wave washed over me. Water stung my face, and I laughed. Johnny came up for air, and we treaded water. We swam back and forth in the pool for a while.\n\nAfter about half an hour, we raced back to the shallow end and got out. We flopped onto the thick beach towels and stared at the fake stars high above. If I hadn't known better, I would have sworn I was lying on a deserted beach somewhere instead of underneath one of the finest homes in Bigtime.\n\n\"It all looks so real,\" Johnny said. \"Like you could almost reach up and pluck the stars out of the sky.\"\n\n\"Sam, Henry, and the chief do a good job with the holograms.\"\n\nWe lay there, still and silent and staring at the stars. I thought about asking Johnny one more time to forget his vendetta, to leave Siren and Intelligal to us. But I didn't want to ruin the moment, so I kept quiet **\u2014** for a change.\n\nMy stomach, though, had other ideas. It let out a long, gurgling roar that would have put a movie monster to shame.\n\nJohnny turned over on his side and looked at me. A grin spread across his face. \"Let me guess. No lunch or dinner, right?\"\n\nMy stomach rumbled like thunder. \"Oh no, I ate enough lunch and dinner for thirteen people. But unfortunately, my metabolism never, ever stops. Not even when I'm asleep.\"\n\n\"Well, we better get you fed before you waste away to nothing. We can't have you fighting crime on an empty stomach, now can we?\"\n\nI laughed, and Johnny pulled me to my feet. I grabbed my towels, and we headed for the underground kitchen.\n\nI plopped down on one of the metal swivel stools while Johnny yanked open the door on one of the refrigerators. He stared inside, then glanced over his shoulder at me.\n\n\"I know.\" I cringed just a bit. \"It's probably a little empty in there.\"\n\n\"I'm sure I can find something for us to snack on.\"\n\nJohnny turned back to the refrigerator, and I used the opportunity to study him. His pants were still slightly damp, accentuating every muscle in his perfect body. Exoskeleton or not, the man looked like a marble statue come to life. One that was perfectly proportioned in all the right places. My eyes dipped lower. Everywhere.\n\n\"So, what do you want?\" Johnny asked, rummaging around in the frosty depths. \"I think there's still enough roast beef left for a couple of sandwiches.\"\n\nI was hungry, but not for food for a change. I hungered for Johnny. And I was going to do something about it. Right now.\n\n\"Here, let me see what I can find.\"\n\nI strolled over to the refrigerator and scanned the contents. Johnny was right. There wasn't much left, except a couple of boxes of baking soda and a few condiments. Even I couldn't eat ketchup, mustard, or mayonnaise straight out of the bottle, no matter how famished I was. My eyes roamed over the metal racks. But it wasn't a complete loss. I plucked out a skinny can.\n\n\"Whipped cream?\" Johnny asked. \"You're going to eat chocolate whipped cream and nothing else?\"\n\nI grinned. \"Absolutely.\"\n\nI shook the can. Then, I leaned over and squirted a little on the corner of Johnny's mouth. I flicked my tongue over the chocolate frost and licked it off, lightning-quick. I repeated the process on the other corner of his mouth. Then, on the side of his neck. Then, in the hollow of his throat. Whipped cream had never tasted so good.\n\n\"Fiona...\" Johnny said in a husky tone.\n\n\"What's the matter? Don't you like whipped cream?\" I asked in a coy voice.\n\nJohnny put his hands on my hips and rested his forehead on mine. His light touch made me burn that much more. \"Are you sure you want to do this with everything that's happened? With everything that's between us? Because Siren and Intelligal are still out there, and I'm still\u2014\"\n\nI put a bit of whipped cream on my finger and pressed it to his lips. \"Enough about them. We're here now, and that's all that matters. Besides, I'm still hungry.\" I licked the chocolate confection away. \"Aren't you?\"\n\nJohnny stared at me. Then, a slow, wicked grin spread across his face. \"Baby, I'm always hungry.\"\n\nHe took the can from me and squirted some of the cream into his mouth. I reached up and pulled his lips down to mine. The frothy chocolate melted as our tongues stroked back and forth. Even as the whipped cream disappeared, the liquid heat inside me grew and grew until it was a bonfire raging out of control, burning me alive. And there was only one thing that would douse the flames, only one thing that would cool my desire. Johnny. Now.\n\nRight now.\n\nI sank to my knees and pulled Johnny down with me. He lowered me the rest of the way to the floor. The tile was cool and slick beneath us, but I didn't feel it. All I could feel, see, taste, hear, smell was Johnny.\n\nOur kisses grew longer, harder, deeper, until we were both frantic and dizzy with need. We clawed at each other's clothes, ripping aside the thin barriers. My bikini was shredded away. So were his pajama pants.\n\nJohnny squeezed my breasts. I raked my nails down his back. He kissed my neck. I nibbled on his ear. Our legs tangled together, and fire throbbed and pulsed through my body, rocking me to my molten core. With one thought, we moved into each other. The white-hot feel of Johnny sinking into me made me cry out with pleasure.\n\nIt was quick and hard and good. We climaxed and lay there panting on the tile.\n\nJohnny drew me into his arms and nuzzled my neck. I closed my eyes and stroked his damp hair, careful not to singe it with my sparking fingertips. A soft, warm glow wrapped around my body.\n\n\"Don't you go to sleep on me just yet,\" Johnny said.\n\nI looked at him through slitted eyes. \"Why not? Good sex always makes me sleepy. I think I held up my end of the bargain just now.\"\n\nJohnny reached over my head and grabbed the stray can of chocolate whipped cream. \"Because, baby, we still have half a can of whipped cream left.\"\n\nJohnny shook the metal can. He leaned over and spread the cream on my nipples, my stomach, then farther down to more interesting places. The creamy frost felt like dew drops on my flushed skin.\n\n\"Johnny, what are you up to?\"\n\nHe grinned and lowered his mouth to me.\n\n# Chapter Twenty-Two\n\nI woke up a couple hours later in Johnny's arms. We were still in the kitchen, stretched out in front of the refrigerator. A couple of beach towels covered our bodies. I stretched my arms above my head. Every part of me felt warm and languid and relaxed. A sigh of contentment escaped my lips. I felt so satisfied, so peaceful, so loved.\n\nA hand stroked my cheek. I turned, and Johnny was there. He rose up on one elbow and stared into my eyes.\n\n\"Hey there.\"\n\n\"Hey there yourself.\" I pressed my lips to his for a quick kiss. \"That was pretty incredible.\"\n\n\"Yes, it was. Although I'd use a better adjective than merely _incredible_. Lots of things are _incredible_. Give me something stronger.\"\n\nI arched an eyebrow. \"Fantastic?\"\n\nJohnny shook his head. \"Something stronger.\"\n\n\"Unbelievable?\"\n\n\"Nope.\"\n\n\"Tip-top?\"\n\nJohnny shuddered.\n\n\"Super-duper?\"\n\nJohnny snapped his fingers. \"Super-duper! That's it! Fiona Fine, we have just made _super-duper_ love together. Wouldn't you agree?\"\n\n\"And then some.\"\n\nWe kissed again, and Johnny flopped onto his back. He stared up at the ceiling, and his smile faded away. I knew he was thinking of his father. And how he would never experience such pleasure again.\n\n\"Tell me about your father,\" I asked, snuggling into the crook of his arm to distract him. \"I told you about Travis. Tell me about your father, James.\"\n\n\"Are you sure you want to hear about him now?\" Johnny asked. \"That's sort of an odd conversation for pillow talk.\"\n\nI gestured at the tile floor beneath us. \"Well, we're not on pillows, we're on the kitchen floor. And I'd like to know more about your father, more about Johnny Angel.\"\n\n\"All right.\"\n\nJohnny told me everything about his father. How he'd been heartbroken when his wife Lucia had died. How he'd struggled to raise Johnny and Bella alone. How he'd given them everything he had. His love, his time, his business. Johnny's love for his father radiated from him, along with some emotion I couldn't quite identify.\n\n\"When was the last time you spoke to him?\"\n\nJohnny's eyes darkened. \"The night before he died. I was still in Greece then, and he called me to talk about a business deal I was working on. Then, we started talking about the other family business\u2014Johnny Angel. He wanted me to come home to Bigtime and take over as Angel. He said he was getting too old, that he'd almost let a mugger get away from him a couple of weeks ago. I refused. I told him that life wasn't for me. The fighting, the patrolling, the gang wars. It wasn't what I wanted.\"\n\n\"And then he died,\" I prodded in a gentle tone.\n\nJohnny sucked in a deep breath. \"And then he died. Two days later, Bella called me, frantic. She hadn't heard from Dad. I flew home at once. We went out to where he'd last been seen. We found some of the pieces of his motorcycle. They were charred and burned almost beyond recognition. And we found his watch. The glass case was busted, but it was still in one piece. That watch was my grandfather's, the original Johnny Angel. My dad never went anywhere without it. We knew then that he was dead.\"\n\nMy heart ached for Johnny. I touched his face with my fingertips, and Johnny pressed a kiss to my hand.\n\n\"I turned over our Greek operations to one of the other business managers and moved back home. I vowed to find the people responsible and make them pay.\"\n\n\"Siren and Intelligal.\"\n\nJohnny nodded. \"A contact in one of the local biker gangs overheard the two of them talking about how they'd killed Angel. They were bragging about it at Quicke's one night. When I found out it was them, I swore a blood oath to my grandfather and Bella to kill them and avenge my father. And I will honor my vow.\"\n\nI opened my mouth to protest. To tell him once again that the Fearless Five would bring the ubervillains to justice. Johnny wasn't having any of it, though. His fingers slid down my breasts, past my stomach. They penetrated me, exploring. I closed my eyes. Heat waves of pleasure sizzled through my body. I was so hot you could have fried bacon on my forehead.\n\n\"That's...not...fair,\" I rasped, digging my hands into his smooth, broad shoulders. \"I...can't think...straight...much less...argue...when you...do that.\"\n\nJohnny grinned. \"Baby, nothing's fair in love and war. Isn't that how the old saying goes?\"\n\nHe didn't have to reach for me. I threw myself into his waiting arms.\n\n* * *\n\nWe made love four more times. On the floor. On the countertop. Against the refrigerator. Sitting on top of a stool. The last one was particularly interesting. I'd never look at a chair the same way again.\n\nWhen we were done, Johnny went back to the sick bay, and I slipped back into my suite. We both would have been much more comfortable in my bed, but I didn't want the others to know everything we'd done just yet. I wasn't worried about their disapproval. Despite Johnny's desperate need for revenge, the others knew he was a good person deep down inside. Besides, I'd never needed other people's approval anyway.\n\nNo, I didn't want the others to know everything just yet because if they knew, then I'd have to admit to myself that Johnny and I were having more than a casual fling. That we were officially a couple. An item. Together. And that I was slowly but surely starting to forget about Travis.\n\nI sat down on my bed. With a start, I realized that I hadn't thought about Travis in hours now. Days, really. Oh, he crossed my mind every now and then, but he didn't fill my thoughts anymore. My heart no longer ached like a rotten tooth, pulsing and throbbing with each strained beat. My eyes flickered to the beaming picture on my nightstand. Travis's blond hair and brown eyes looked strange to me, almost as if I were seeing him for the first time. The pain of his loss wasn't the only thing fading from my mind. My memories were getting fuzzed over as well. And that's what really scared me. I didn't want to forget him. Didn't want to forget anything about him.\n\nBut I was.\n\nMy fingers traced over his wide smile. The diamond on my finger sparkled like a small sun on my hand. \"Well, I suppose you know about me and Johnny by now. You'd like him, I think. He's a lot like you. Fun, energetic, full of life,\" I whispered. \"He'll never take your place, but I like him. And he likes me. He makes me laugh, just the way that you used to. I've missed that. Laughing.\"\n\nTalking to a picture of a dead man was rather silly, of course. He wasn't going to talk back to me or suddenly appear beside my bed as a shimmering ghost. Whispering to Travis was something Carmen would do while listening to those pesky voices in her head. But it made me feel a little better, even if Travis didn't respond. Even though he never would again.\n\nI snapped out the lights and went to sleep.\n\n* * *\n\nThe next morning, I stood in the sick bay with Johnny and the chief, who was giving him one more exam before discharging him. Johnny grinned at me while Chief Newman checked his temperature, pulse, and blood pressure. The last of his cuts and bruises had healed up, and Johnny looked no worse for wear, despite the fact I'd gone to town on him with a lead pipe. I didn't feel quite so guilty about it as I had before. I figured last night had made up for it a little bit. Pleasure. Pain. There was a thin line between the two.\n\nThe chief tapped his knees with a metal hammer, making a hollow, ringing sound. Johnny slipped his hand in mine, while the chief continued his ministrations. I liked the solid feel of Johnny's skin next to mine. All of his bare, naked, smooth, supple skin next to mine. His hard, muscled body on top of me. Under me. Beside me. In me\u2014\n\n\"Fiona? Are you listening?\" Chief Newman rumbled.\n\nI snapped back to reality. \"Of course. What did you say?\"\n\n\"I said that Johnny's fine, except he seems a little tired this morning. You wouldn't know anything about that, would you?\" The chief turned his blue eyes to mine.\n\nI flashed back to our time together in the kitchen. I could still taste the chocolate whipped cream on my lips. \"Of course not.\"\n\n\"I did tell him to avoid strenuous activities,\" the chief continued. \"But I don't think he heeded my warning.\"\n\n_Strenuous_ wouldn't be the word I would use to describe last night. Perhaps _vigorous_ or _long-lasting_ or the previously agreed upon _super-duper_.\n\nMy father kept staring at me. His eyes glowed, like a match burning. Johnny looked back and forth between the two of us, still grinning.\n\n\"Johnny doesn't seem to be one to heed warnings,\" I said.\n\n\"That sounds like someone else I know,\" the chief replied.\n\nI tossed my hair over my shoulder, but I couldn't help the blush that crept up my cheeks. I'd just been busted for having a little nookie-time fun by my own father. Way to go, Fiona.\n\nThe chief finished up his examination, pronouncing Johnny fit to take on the world once more. Johnny put on some more of Travis's old clothes, and we headed back to the city in my convertible. It was a beautiful May day, with plenty of sunshine and a steady breeze. I put the top down, tilted my face up, and drank in the vibrant rays. Johnny leaned his head against the back of his seat and did the same.\n\nWe didn't speak on the way back to Bigtime. I didn't know what to say to Johnny, other than to ask him once again to abandon his vendetta against Siren and Intelligal. I wasn't sure how he would react to that.\n\nWe pulled into the long, curving driveway of the Bulluci manor about ten that morning. I stopped the car outside the front door and killed the engine.\n\nJohnny turned to me. \"Come in. Stay with me for a while.\"\n\nI stared up at the house. A white lace curtain in one of the upstairs windows twitched and fell back into place. \"I'm not sure that's such a good idea. Unless I miss my guess, Bella's waiting somewhere inside ready to pounce on you and demand an explanation. Given how your last conversation went, I imagine there'll be some yelling involved too.\"\n\n\"That's precisely why I need you to come with me. To protect me from my big, bad, nasty sister.\" Johnny grinned.\n\nI found myself grinning back. \"I do so admire manly men, especially those who quake at the thought of facing their little sister.\"\n\nJohnny shuddered. \"You don't know Bella. She might seem mild-mannered and sweet, but once she gets going, she's hard to stop.\"\n\n\"Well, if you insist...\"\n\nThe truth was that I wanted to spend more time with Johnny. Every waking minute if I could. Anything to keep him close to me, to keep him safe. Anything to keep him from trying to fulfill his vendetta.\n\n\"I do insist with you, Fiona. Always.\"\n\nJohnny pulled me inside, and we wandered through the angel-filled rooms, searching for the rest of the Bullicis.\n\n\"Bella? Grandfather? I'm home,\" Johnny announced. His voice boomed through the sprawling mansion.\n\nNo response.\n\n\"Come on,\" Johnny said. \"I know where they are.\"\n\nHe led me to the den. Bella sat on the long sofa, watching some art restoration program on the enormous television. A sketch pad lay on her lap, but she snapped off the screen at the sight of us.\n\n\"Well, it's about time you got home,\" Bella said in a tight, cold voice. Her hazel eyes shimmered like liquid gold with anger. \"I hope the two of you enjoyed yourselves these last few days. Some of us were working.\"\n\n\"I didn't mean to be gone so long, but I ran into a bit of trouble. Fiona helped me out of it.\" Johnny strolled to Bella's side and planted a kiss on her cheek like nothing was wrong.\n\nBella narrowed her eyes. \"What sort of trouble? Did the two of you run out of condoms or something?\"\n\nI arched an eyebrow. I wouldn't have thought quiet, shy Bella would have the gumption to say the word _condoms_ out loud, much less in front of other people. Perhaps still waters really did run deep.\n\n\"Not exactly,\" Johnny said. \"Fiona beat the stuffing out of me.\"\n\nConfusion and worry spread across Bella's face. \"How would she be able to do that? Your exo\u2014\" She cut off her words and looked at me.\n\nI rolled my eyes and tossed my hair over my shoulder. All these innuendos and shortened sentences and half truths were really starting to annoy me. So I did what I did best. I snapped my fingers, and a fireball popped into my hand. With my other hand, I picked up the sofa\u2014with Bella still sitting on it. She let out a squeak of alarm and clung to the side of the furniture.\n\n\"This is how,\" I replied.\n\nBella's eyes zipped back and forth from me to the fireball to the sofa to Johnny and back to me. Comprehension filled her face. \"You're Fiera?\" she screeched. \"The superhero? A member of the Fearless Five?\"\n\nI winced. Bella could almost match Siren when it came to her piercing voice. \"That's right.\"\n\nBella's golden gaze flicked to her brother. \"Why are you telling us your secret identity?\" She tried to be cool and casual, but panic sparked in her eyes.\n\n\"You don't have to pretend, Bella. Fiona knows that I'm Johnny Angel. She knows everything. About me, about Father, all of it.\"\n\nBella's mouth dropped open. She almost tumbled off the sofa. \"Johnny! You didn't!\"\n\n\"He didn't really have a choice,\" I said. \"I did beat the stuffing out of him. But I knew who he was before then.\"\n\nBella turned on her brother. \"Johnny, how could you be so reckless? Letting someone guess your secret identity. The family's secret identity.\"\n\nI gestured at all the cherubs and angel wings and halos in the room. \"It wasn't that hard to figure out. In fact, it's pretty obvious. You know, you guys might want to think about redecorating just a little bit. The angels are a dead giveaway.\"\n\n\"That's what their grandmother always thought, but I never listened to her.\"\n\nWe turned at the sound of Bobby Bulluci's voice. The old man stood in the doorway, staring at the three of us. His eyes went to the fireball on my hand and the sofa I had hoisted in midair. Busted again. If I kept this up, I might as well just take an ad out in _The Chronicle_ or _The Expos_ _\u00e9_ announcing my secret identity to the entire world. Hell, maybe I'd just rip my mask off at the next superhero gathering at Paradise Park in front of Kelly Caleb. The SNN reporter would get the news out in no time flat. On the upside, I'd save a fortune in masks.\n\nBella rubbed her temples. \"Let me get this straight. You knew that my brother, the man you've been dating, was Johnny Angel, yet you still beat up on him?\"\n\n\"That's right.\"\n\n\"Johnny, do you care to explain this?\" Bella crossed her arms over her chest, looking more like an uptight schoolteacher than a cutting-edge fashion designer.\n\n\"It's a long story,\" Johnny said. \"Let's sit down.\"\n\nI put the furniture and Bella back on the floor and snuffed the fireball out of my hand. We sat down on the lowered sofa, and Johnny spent the next ten minutes filling Bella and his grandfather in on everything that had happened since we'd had our knock-down, drag-out brawl at Oodles o' Stuff.\n\n\"Incredible,\" Bobby said. \"All these years I've wondered about the Fearless Five, who they really were, and here you are, Fiera, sitting in my living room. Incredible.\"\n\nI shrugged in a modest sort of way. It was always nice to be thought of as incredible. My eyes went to Johnny. Or super-duper. I cleared my throat. Since all of our cards were on the table, I might as well appeal to Bella and Bobby to help me with my mission. My very personal mission.\n\n\"My friends and I have spent the last two days trying to convince Johnny to leave Siren and Intelligal to us. The two of them are planning something, and we need to figure out what it is and stop them before they hurt anyone else.\"\n\n\"It is not simply a matter of stopping them,\" Bobby said. Tears gathered in his eyes, and he suddenly seemed old and small and frail. \"I believe in the old ways. Eye for an eye, blood for blood. Those two killed my James, my son. They should be killed in turn to set things right.\"\n\n\"I understand your pain, sir, truly I do,\" I said. \"But killing them won't bring your son back. It won't take away your sadness.\"\n\nBella nodded. \"That's what I've been trying to tell them for months now. We've already lost Father. We don't need to lose you too, Johnny.\"\n\n\"He has a duty to his family, to carry on the legacy\u2014\" Bobby started.\n\n\"Oh, screw the stupid family legacy,\" Bella snapped. \"I'd rather have Johnny home safe and sound any day than lose him to your silly legacy.\"\n\nThe two of them glared at each other. Yikes. And I thought I could get hot under the collar when I was angry. I had nothing on Bella Bulluci.\n\nDisgusted with the men in her family, Bella grabbed her sketch pad off the sofa and threw it onto the glass-topped coffee table.\n\nCrack!\n\nThe pad hit the table with a loud smack, and the glass split down the middle. The two pieces fell on top of each other, along with an assortment of coasters, magazines, and a mug that hit the floor and shattered.\n\nWhen the noise faded and the glass and dust settled, I looked at Bella. Wondering.\n\n\"It's nothing,\" Bella muttered, avoiding my eyes. \"Just another part of the stupid family legacy.\"\n\nOne that was getting more interesting and unusual by the minute.\n\n\"Maybe you should go and let us talk about this, Fiona,\" Johnny said in a soft voice. He squeezed my hand. \"Please?\"\n\nI looked at the three of them. This was a family matter, and I wasn't part of the family. The thought made me sadder than I would have imagined.\n\n\"All right. I'll go. For now. But think about what I said. Let the Fearless Five handle the ubervillains. It's what we do.\"\n\n\"It's what Johnny Angel does too,\" Bobby replied. \"It's what we've done for three generations now. We take care of our own. We always have, and we always will. That's our family's real legacy.\"\n\nI didn't have a response. Johnny's eyes begged me to go before things got any more heated. So, I left.\n\n# Chapter Twenty-Three\n\nI drove to my office. Paperwork cluttered my desk, along with the remains of the wilted flowers Johnny had sent me. I stared at the dried-up brown petals and hoped they weren't an omen. That my relationship with Johnny wouldn't soon be as dead and decayed as they were.\n\nI rolled my eyes. Sheesh. I was getting as bad as Carmen and my father looking for doom-and-gloom portents of the future. I shoved the flowers into the trash can and stared at the piles of messages on my desk, the notes from suppliers, all the thousand small details that needed seeing to. I really needed to get to work.\n\nBut I couldn't concentrate on any of it, not even my Fiera fan mail. All I could think about was Johnny and his family. I paced around my office. Bella had agreed with me, that the Fearless Five should be the ones to take care of Siren and Intelligal. But Johnny and his grandfather hadn't. And I knew they wouldn't. Their honor, their code, their legacy was just as important to them as my superhero duty was to me. Why did men have to be such fools sometimes?\n\n\"If you don't cut that out, Fiona, you're going to poke holes in the floor with your stilettos,\" Piper called out from the doorway. \"You don't want a repeat of last time, do you?\"\n\nI grimaced. The last time Piper was referring to was when I'd put my foot through the floor after a particularly grueling fitting session with Joanne James. It was either that or put my foot through her bony ass. Joanne had gotten off lucky that day.\n\n\"Why don't you use the balls I got you?\" Piper suggested.\n\nIn addition to leaving eating disorder and other self-help pamphlets on my desk, Piper was also fond of giving me gifts like those rubber balls you squeeze in your hands. She thought I needed to relax.\n\nPiper kept staring at me, so I yanked open my desk drawer and rummaged through the mess inside until I came up with one of the rubber gizmos, which was a little smaller than a tennis ball. I rhythmically flexed my fist around the puny ball.\n\nPiper smiled, happy that she'd gotten me to do her bidding, and disappeared back into her office.\n\nI waited to make sure she wasn't coming back. Then, I squeezed the ball with all my might. It only took me a second to turn the rubber into a handful of goo. I tossed it in the trash and watched it smolder.\n\nPiper was right. It did make me feel better. I always enjoyed melting things.\n\n* * *\n\nI worked the rest of the day, stopping only to eat a quick but massive lunch from Quicke's. Every so often, I'd look at the phone, hoping Johnny would call. Hoping he'd tell me he'd changed his mind about going after Siren and Intelligal. That he was giving up his quest, his mission, his vendetta. But he never did. Men never called when you wanted them to.\n\nI left work around six and headed back out to Sublime. Tonight was my night to be on duty in the library. I rolled my neck around, trying to ease some of the tension that had built up there. I'd zapped all of the rubber balls I could find in my desk, but they hadn't helped much. Maybe I'd be able to do a few laps in the pool after my shift ended. Maybe not, the way things were going. It hadn't exactly been a banner week so far. Except for my time with Johnny in the kitchen.\n\nI stopped by the kitchen and made myself a couple of ham-and-cheese sandwiches, grabbed two bags of chips, and put a bottle of soda under my arm before heading to the library. One of the doors was cracked open, and loud, angry voices bounced down the hall. I frowned and quickened my pace. What the hell was going on?\n\n\"I don't see why you have to be so stubborn,\" Henry's voice floated through the door. \"I love you, and I want to marry you. What could be simpler than that?\"\n\nI peeked inside. Ah, the two lovenerds were squared off above their flickering computer monitors, each one glaring at the other.\n\n\"There's nothing simple about it, and you know it,\" Lulu snapped back. \"Your power is a precious gift. You deserve to have kids who will follow in your footsteps\u2014who'll have your mind-melding power and help others with it.\"\n\n\"You're acting like there's no hope,\" Henry said. \"But there are things we could try. Medical advances like in vitro fertilization, or a surrogate mother. And researchers are developing new methods all the time.\"\n\nLulu shook her head. \"But you should be able to have both\u2014a woman who loves you and who can give you kids. It shouldn't be this hard. Your power is too important to waste\u2014\"\n\n\"Oh, stuff my power.\" Henry shoved his glasses up his nose. \"I wouldn't care if my power went away tomorrow, as long as I had you.\"\n\nLulu's face softened. Even I melted a little. Wow. Giving up your power to be with the person you loved? That was the ultimate sacrifice for superheroes and ubervillains alike.\n\nLulu drew in a deep breath. \"We both know I can't have kids. And unless they come up with some sort of new science in the next couple of years, I won't ever be able to have children. I don't want you to give that up for me, Henry.\"\n\n\"But we can adopt,\" Henry persisted. \"It wouldn't matter to me. You know that.\"\n\nLulu shook her head. \"But they wouldn't have your powers. You're far too strong and your power is far too important to let it end with you.\"\n\nHenry opened his mouth to protest again.\n\nI cleared my throat, not wanting to hide outside any longer. \"I hate to interrupt, but it's my turn to be on call.\"\n\nThe computer gurus stared at me, shocked that I'd overheard their conversation. I strolled inside, sat down at the table, put my feet up, and started eating sandwiches like I hadn't heard a single word of their heated argument.\n\nHenry and Lulu glared at me, then each other. Both of them looked at their monitors and began to pound away on their computers. I supposed that's what geeks in love did after they had a fight. Scary.\n\nAfter I finished my sandwiches and assorted munchies, I flipped through the stack of fashion magazines Sam kept in the library for me. But my heart wasn't really into dissecting the latest looks from Paris and Milan. All I could think about was Johnny. I wondered where he was right now. What he was doing.\n\nWas he getting ready to suit up and prowl the streets as Angel? Or had he taken my advice to heart and was staying home where he'd be safe? I didn't know.\n\nAfter about an hour of silence, Lulu let out a loud yell. \"I've got them!\" she cried. \"I've got them! I've got them!\"\n\n\"Who? What? Where?\" Henry asked just like a good journalist would.\n\nThe two of us darted over to Lulu and peered at her computer monitor. The hacker's thin fingers pounded the keyboard so hard I thought she was going to punch through the plastic keys.\n\n\"They're holed up in one of the buildings down by the marina,\" she said, pointing to a city map on her wide screen. \"Right there next to the fish-packing plant.\"\n\nI put my hands on my hips. \"The marina? What would they be doing at the marina? There's nothing down there but boats and water and rotten fish. And, of course, the occasional dead body.\"\n\nLulu stared at me. \"How am I supposed to know what goes through the minds of ubervillains? But that's where they're at, according to my calculations.\"\n\n\"I'll call the chief,\" Henry said. \"If we're lucky, we can trap them in the building and take them down. There's nowhere for them to go but out into the water. It shouldn't be so easy for them to slip through our fingers this time.\"\n\nHe moved to his computer, pressed a few buttons, and spoke into a microphone. My father's voice boomed into the room, and Henry told him the situation.\n\nWhile they talked, I thought about calling Johnny. Perhaps if I asked him to fight with us instead of against us, it might satisfy him. Then, I remembered the pain in his eyes when he'd told me about his father's death. The cold, hard rage in his voice. No, Johnny Bulluci aka Johnny Angel wouldn't be satisfied until Siren and Intelligal were dead. But perhaps I could save him from himself. And my heart in the process.\n\nHenry cut the connection to the chief. \"He's in the area checking on a burglary. He'll be here in fifteen minutes.\"\n\n\"Good. Then let's go end this thing. Once and for all,\" I said.\n\n# Chapter Twenty-Four\n\nThe chief arrived, everyone suited up, and we piled into the van. Thirty minutes later, we skidded to a stop in front of the entrance to the Bigtime Marina.\n\nBigtime was situated on the eastern edge of New York, right on the Atlantic. The ocean cut into the middle of the city, almost like a jagged shark bite. It and the manmade river that flowed down the hill from the observatory formed Bigtime Bay. The bay's waters were calm and shallow for the most part, making it the perfect place to come for a swim or day of sailing. Some of the society folks like Berkley Brighton and Devlin Dash even had their own private islands out in the middle of the bay, offering them impressive views of the city skyline.\n\nHermit eased the van over a couple of speed bumps. During the day, the cobblestone marina was a pedestrian area closed to vehicles, so the boating types had to wait until after sunset to put their ships into the water. In addition to fleets of sailboats, the marina featured a maritime museum where kids could pet stingrays and the like, and stores that carried all things nautical, from clothes to scuba gear to bait. A wooden pier stretched out like a finger into the bay. It was popular with people from all walks of life, many who came to fish and feed the flocks of gulls.\n\nWe left the sailboats behind and headed for the less glamorous side of the bay, where the loading docks, shipping yards, and industrial plants crouched against the water's edge like barnacles.\n\n\"Before we go in, everyone be sure to take their gas pill.\" Hermit passed out the medication to each of us. \"Remember, it only lasts about twenty minutes, so we need to get in and take down Siren and Intelligal as quickly as possible.\"\n\n\"And we're sticking together this time,\" Mr. Sage said. \"I don't want anybody being ambushed by the ubervillains or Angel, if he decides to make an appearance. We all go in together. We all come out together. Agreed?\"\n\nHermit and I nodded.\n\n\"Then, let's go.\"\n\nLulu outfitted us with the usual cameras and transmitters, and the three of us left the van.\n\n\"Be safe,\" she called after us, looking at Hermit.\n\n\"Always,\" I replied, pressing my fist to my heart.\n\nWe tiptoed through the dark shadows, the bay a pool of black ink on our right. A few lights bobbed up and down on the water, and a foghorn sounded in the distance. A steady breeze blew the smell of salt to us. It made me want some pretzels.\n\n\"According to Lulu's calculations, Siren and Intelligal are holed up in that building over there.\" Hermit pointed to one of the frozen-fish-stick-processing plants.\n\nI wrinkled my nose. Fish sticks. I hated fish sticks. They were just about the only food I wouldn't eat. The stench alone was enough to make me gag. Why did ubervillains always have to pick the dingiest, dirtiest, most disgusting places imaginable for their supersecret, diabolically evil lairs? You'd think that, every once in a while, they'd spring for a nice room at the Bigtime Plaza or something. Most of them stole enough jewelry and other pricey baubles to stay anywhere they wanted to, anytime they wanted to. But no. Ubervillains skulked about in the shadows, and we always ended up saving the world in some abandoned, out-of-the-way dive.\n\nA loud rumble cut through the air, and my heart sank like a cement block tossed in the bay. I knew that sound. A pair of halogen headlights popped into view down the street, and Johnny Bulluci aka Johnny Angel slid his motorcycle to a halt in front of us. His eyes warmed at the sight of me, and he shot me a crooked grin. My heart sped up, and I found myself smiling back. Then, I remembered why Angel was here\u2014to kill the ubervillains who had murdered his father. My smile faded.\n\nStill, I had to give him a chance, no matter how farfetched it might be. \"What are you doing here, Angel?\"\n\n\"Just looking after things, including my girl. Is that a crime?\" he asked, his green eyes bright.\n\n\"Not as long as that's all you're doing. Is it?\"\n\nAngel shrugged. He drew his lighter out of his pocket and fired up a cigarette. \"I spoke with Grandfather and Bella. The three of us decided that I should work with you, instead of against you. For now.\"\n\nMeaning he'd play nice until we had Siren and Intelligal right where he wanted them. I opened my mouth to protest, but my father cut me off.\n\n\"All right, Angel,\" Mr. Sage said. \"We're a couple of hands short, so you can join us if you wish. But you follow our lead.\"\n\nAngel nodded his head. \"Of course.\"\n\nI stared at my father. What was he thinking? He was psychic; he had to know that Johnny Angel had no intention of letting us turn Siren and Intelligal over to the Bigtime police. Hell, I wasn't a psychic, and I could see it.\n\nJohnny parked his bike, and the four of us headed for the fish-stick plant. It was a short, squat building that jutted up against the side of the bay. We faced the back part of the building, where the docks were that the fisherman used to drop off their daily hauls of tuna, flounder, and shrimp. The front of the building faced one of the downtown streets a couple of blocks over.\n\n\"Don't you guys think this is sort of odd?\" Lulu said in my ear.\n\n\"What's that?\" my father asked.\n\n\"Well, in my somewhat limited experience, ubervillains usually choose abandoned buildings to set up their headquarters. According to my information, Fred's Fried Flounder Fish Sticks is the main supplier of fish sticks in Bigtime.\"\n\n\"Who cares?\" Angel asked. \"All that matters is that they're in there, and we're going to get them. One way or another. I'm tired of waiting. Let's go.\"\n\nAngel headed for the plant. I looked at my father, then Hermit. They shrugged. I shook my head and followed Angel.\n\nI not-so-gently wrenched open one of the loading-dock doors, and we stepped inside. The four of us formed a line and searched the area. Lulu was right. Fred's Fried Flounder Fish Sticks was very much a working operation. Everything was neat and orderly and clean, from the rows of forklifts to the stacks of cardboard boxes to the few workstations and desks that we passed.\n\nWe did a complete sweep of the facility. Other than a couple dozen industrial-size freezers full of frozen fish sticks and your usual assembly-line setup, there was nothing inside. No power cords. No radio-like device. No wires. No tools. No blueprints. No ubervillains. Nothing.\n\nWe backtracked to the center of the factory, hoping to find something we'd overlooked. Still nothing. Angel cursed and lit another cigarette. Hermit typed on his handheld computer and murmured to Lulu. My father laced his fingers together, deep in thought. I put my hands on my hips.\n\nA chill swept over me that had nothing to do with the freezers. My nose twitched. I smelled something rotten, and it wasn't fish sticks.\n\n\"This doesn't feel right,\" my father murmured.\n\n\"Oh, I wouldn't say that. I think it's about time you got here,\" a sultry voice drawled out.\n\nThe four of us turned. Siren stood behind us, holding a curious-looking microphone in her manicured hands. Diamonds gleamed on the black metal surface. Intelligal floated high above. She flipped a couple of switches on a device that looked like an oversized boom box. It, too, was studded with diamonds.\n\nAngel's hands clenched into tight fists. He started forward, but I caught his arm. He took a half a step forward. I tightened my grip, stopping him.\n\n\"Not yet,\" I whispered out of the corner of my mouth.\n\nFor a second, I thought Johnny would shake me off and launch himself at Siren. But he paused. I nodded my head at him. Maybe I had finally gotten through to him.\n\nBecause no matter what happened, Johnny wasn't going to kill the ubervillains. I wasn't going to let him. It wasn't the Fearless Five way. It wasn't my way, and I didn't want it to be his way either.\n\nAnd I hated to admit it, but Lulu was right. This was a little too pat, a little too rehearsed for my liking. The ubervillains were up to something. Well, more so than usual.\n\n\"You're probably wondering why we're here tonight,\" Siren said.\n\nHer soft, breathy voice curled around me like a rope. I reached for my inner fire and burned the coils away.\n\n\"Intelligal told me about Angel and Fiera's tussle the other night, and I had an idea. Instead of trying to fight you off, I should bend you to my will, make you all my little puppets. Intelligal and I are going to need some help to fully implement our scheme.\"\n\n\"And what scheme would that be?\" I said.\n\n\"Oh, the usual,\" Siren said. \"Take over Bigtime, then the world.\"\n\nThat was our cue to move. I loosened my grip on Johnny's arm and tensed my muscles, ready to strike. He did the same. My father's fingers fluttered, and Hermit fixed his computer on Intelligal's chair. But before we could move, Siren held the strange microphone up to her lips.\n\n\"Now, now, I don't want to fight. Why do the four of you?\" she purred.\n\nThe microphone amplified the sultry, hypnotic pull of Siren's voice a hundred times. A thousand times. I stopped cold. Maybe even a hundred thousand times. My brain screamed at my muscles to move, to attack, to lash out at Siren, but I couldn't quite make myself do it. The others stood frozen beside me.\n\n\"I don't want to fight, do you? Why don't you clasp your hands behind your backs and stand there like good superheroes?\" Siren lowered the zipper on her neon blue costume, exposing cleavage that would have put the Great Wall of China to shame.\n\nSince Siren's voice had more effect on men than women, the others did as she asked without question. Even my father, the great psychic, couldn't resist Siren's command with her new toy firmly in hand. The men had their hands behind their backs before she'd even finished speaking.\n\nNot me.\n\nMaybe I was just too much of a hothead to be easily controlled. Maybe my will was just a tad stronger than the others'. Or maybe I just hated the ubervillains more for what they'd done to the Bullucis, to James, to Johnny.\n\nSo, I didn't succumb to Siren's sultry song. Instead, I reached for my inner fire, concentrating on the searing, pulsing flames deep within me. I grabbed them and held on tight, focusing my energy on _my_ fire, _my_ anger, _my_ will. The sweet, gauzy haze of Siren's voice melted away like snow in a firestorm. My arms twitched.\n\nThe microphone. I had to destroy that damned microphone. Then the others would be free, and they could help me.\n\n\"What's going on?\" Lulu squawked in my ear. \"Hermit, Mr. Sage, what are you doing? Why are you listening to her? Hermit, can you hear me?\"\n\nI blocked the computer hacker's voice out of my mind. I couldn't answer her and fight off Siren at the same time.\n\nSiren noticed that I hadn't done as she'd asked. She frowned and raised the microphone to her pouty lips again. \"Siren says, _p_ _ut your hands behind your back, Fiera_.\"\n\nI took a step forward. Sparks flew from my fingertips. Then another step. My hair hissed with fire. Another step. My body started to glow like a liquefied ruby. One more step.\n\nEvery step got a little easier, a little faster. The bitch wasn't going to control my mind. I didn't care what kind of souped-up karaoke machine she had. No way, no how. I was Fiera, for crying out loud. Protector of the innocent. Superhero du jour.\n\n\"Go to hell, Siren,\" I muttered through gritted teeth.\n\n\"Gas her! Now!\" Siren roared.\n\nIntelligal hit a button on her chair, and that sickly sweet blue gas floated over me. I must have burned away Hermit's antigas pill, because the feeling immediately went out of my arms and legs. The fire inside me snuffed out. I fell to the cold, slick floor, my arms and legs flopping around like a fish trying to breathe in the bottom of a boat.\n\n\"Now, Siren says, _lie still_.\"\n\nI growled at the silky, hypnotic purr. I couldn't move my arms or legs, and I didn't have my fire to sustain me. So I did the only other thing I could think of. I bit down on my tongue. Hard. Blood filled my mouth. The coppery taste washed away some of the sugary sweet gas and grounded me.\n\n\"Never.\"\n\nSiren stared up at her sister. \"What's wrong with her? Why isn't she a puppet like the others? You told me this thing was foolproof.\" She tapped her long nail against the top of the microphone.\n\nIntelligal shrugged. \"I never said it was foolproof. Only that ninety-seven percent of the population could be put under your control by using it. As for Fiera, she seems to have a stronger will than the others. And she's female. You know you've never been able to get along with other women, even when you use your power to its fullest extent.\"\n\n\"You're right, of course. Oh well. Three superheroes will be more than enough to help us carry out our plan.\"\n\nThe ubervillain leaned over me, giving me a closeup view of her overinflated breasts. They matched her ego perfectly.\n\n\"Sorry, Fiera, but Intelligal's right. I'm just not into chicks. Since you won't play nicely, you won't play at all.\" Siren laughed. \"Siren says, _throw her in one of the freezers, Angel_. _Now_.\"\n\n\"What? Don't be a fool, Siren. Let me blast her with my explodium missiles,\" Intelligal said, flipping switches on her chair. \"Let's kill her. Immediately.\"\n\nLulu let out a loud shriek of dismay in my ear. I winced.\n\n\"Hold on, Fiera. I'm leaving the van! I'm getting out right now!\" the computer hacker screamed.\n\nThrough my earpiece, I heard the hydraulic lift hiss to life on the van. But it was too little, too late. Lulu wouldn't get here in time to save me. There wasn't anything she could do against two ubervillains anyway. And there wasn't any radioactive goo around that could turn her into Super Lulu.\n\n\"Come on, Siren,\" Intelligal said. \"Enough of this nonsense. One missile and Fiera will be history. Forever.\"\n\n\"Calm down,\" Siren snapped. \"Look at her. She's as helpless as a baby. She's no threat to us now.\"\n\nSiren dug her pointed boot into my ribs. I couldn't even feel it.\n\n\"Besides, you just finished calibrating the machine. I don't want the shockwaves from the explosion throwing it off. Otherwise, we'll have three pissed-off superheroes to deal with. I don't think they'd take too kindly to the brutal demise of their comrade. That just might be enough to snap them out of my trance. And your missiles make such a mess of everything.\" Siren fluffed out her black curls. \"You know how I hate to have bad hair.\"\n\nIntelligal scowled, but her hands dropped from the switches. I let out a quiet sigh. Thank heavens for ubervillains and their vanity.\n\nBut my relief was short-lived. Angel jerked forward like a robot. He put his hands under my arms and hauled me to my feet. I dangled against him like a wet noodle. Limp and completely lifeless.\n\n\"Please, Angel. Please don't do this. Fight her. I know you can.\" My tongue felt thick and heavy in my bloody mouth.\n\nAngel's eyes remained blank and expressionless as a piece of paper.\n\n\"Mr. Sage? Hermit? Snap out of it, guys! She's an ubervillain! Fight it! Fight her commands!\"\n\nMy two fellow superheroes didn't respond. They didn't even seem to hear me. Instead, they stared at their new mistress, eager for another task.\n\nSiren laughed. \"Tearful pleas won't do you any good, Fiera. Your friends are now under my control. If you haven't figured it out by now, this device that Intelligal rigged up amplifies my power incredibly. With it, I'm going to enslave the people of Bigtime, then the rest of the world...\"\n\nI tuned out her long-winded, self-congratulatory explanation. Ubervillains always thought they had to explain every single, tiny, minute detail of their schemes. When you've heard one plan to take over the world, you've heard them all. It doesn't really matter what the brilliant plan is **\u2014** as long as you figure out a way to stop it.\n\nBut one thing Siren said did catch my attention.\n\n\"...And if city officials don't believe I'm serious, well, I'll just have to remind them about our demonstration at the sports complex a few days ago.\"\n\n\"What? You used your karaoke thingy to bring down the sports complex?\" I asked. \"That's why it collapsed?\"\n\n\"Of course,\" Intelligal answered. \"That was our dry run, so to speak, to test the maximum power of the device, as well as its effect on solid matter. It was very effective, exceeding all of my calculations.\"\n\nSiren preened. \"All it took was a couple of throaty whispers set to the right frequency to make the whole thing come tumbling down.\"\n\nI opened my mouth to ask another question, but the ubervillains were ready to wrap up their gloating. Siren snapped her fingers. Angel, Mr. Sage, and Hermit stiffened to attention at the sound.\n\n\"Siren says, _take her to the freezer, Angel. Now_.\"\n\nThe superhero dragged me back toward one of the many freezers that lay inside the plant and threw me down. Siren, being ever-so-helpful, opened the door. Cold air blasted out. If my arms and legs hadn't felt like soggy tissue paper, the frigid chill would have made even me wince. I focused my eyes on Angel.\n\n\"Come on, Angel. Fight her. Fight Siren. Do it. For me. Remember that night by the lake? When we made love? Our time together in the kitchen?\" I pleaded, trying to spark some sort of memory. A super-duper memory.\n\n\"You two are an item? Well, isn't that sweet? Superheroes in love. Or at least lust.\" Siren slithered up to Angel, and her eyes roamed over his hard body. She leaned in and slid her manicured nails down Angel's chest to his crotch.\n\n\"If you can inspire such devotion from Fiera, I just might have to give you a ride once this is over with.\" She cupped him, and her face lit up. \"Oh my, what a nice package.\"\n\nI wanted to rip the bitch limb from limb for touching my man like that. My hands jerked and spasmed. If only the gas would wear off, I would fireball her trashy ass. Hell, I'd boil the whole bay with her in it.\n\nSiren snapped her fingers again, and Angel dragged me inside the freezer. It was one of those large, walk-in freezers favored by restaurants and, evidently, fish-stick factories. Boxes and boxes of fish sticks lined the walls. Blowers set into the ceiling churned out a steady stream of air, and a thick layer of ice covered everything. Frost gathered in Angel's tawny hair, making it gleam like pure silver.\n\nAs a superhero, I'd been in plenty of tight spots before. I'd been thrown through walls, slammed through floors, dropped off high-rise buildings. So I wasn't ready to panic just yet.\n\n\"Angel, please. I know you care about me. Don't you know how much I care about you? Don't you know how much I love you?\"\n\nI hadn't planned on saying the words. Hadn't really thought about them before. But as soon as they came out, I knew they were true. I did love Johnny Angel aka Johnny Bulluci. Somewhere, in the middle of this craziness, I'd fallen for the rich biker playboy. I loved him for his crooked grin, his devotion to his family, the way he could always make me laugh.\n\nSo I said it again. \"I love you, Angel.\"\n\nNothing.\n\nAngel didn't smile. Didn't look at me. Didn't even blink.\n\nI poured my heart out to him, and he didn't even care.\n\nHe didn't respond to my desperate, heartfelt plea, and that hurt me worse than anything Siren had in mind.\n\nI'd appealed to Johnny's feelings for me. Evidently, he didn't have any because it hadn't worked. It always worked in the movies. I'd seen it work for Carmen and Sam. What was wrong with Johnny and me?\n\nI shoved that painful thought aside and focused. I knew Johnny pretty well. What else did he care about? What else was important to him? What would snap him out of his trance? Finally, the answer hit me.\n\n\"Johnny Angel,\" I said in a hard, sharp voice. \"Your family needs you. I _need_ you. You're not responsible for your father's death. Your father made his choice a long time ago. He knew the dangers. He knew the risks. It's not your job to avenge him and uphold the family honor. But if you don't fight Siren, if you don't _try_ , we're all going to die. Your friends. Your family. Me. All of us are...going...to...die.\"\n\nFor a moment, Angel's face cleared. His hand slowly went to his jacket, and he fumbled with something in his pocket. Angel leaned over me. Concentrating. Trying. Something slipped from his fingers onto my stomach. I stretched my numb hand out and managed to cover up the lump of cold metal.\n\n\"Siren says, _leave her there_ ,\" the ubervillain cooed.\n\nAngel's eyes widened. Sweat beaded and froze on his forehead, but he didn't move away from me.\n\n\"Stop stalling!\" Siren roared into her microphone. \"Get out of there now!\"\n\nWith a jerk, Angel straightened. He turned and walked out of the freezer. The last thing I saw before he shut the door was his eyes. They were ice-green.\n\nFrozen.\n\nJust like his heart.\n\n# PART THREE\n\nBREAKUP BLUES?\n\n# Chapter Twenty-Five\n\nThe door slammed shut, and the metal bar on the other side clanged into place. Trapped. I was trapped in an industrial-strength freezer. With fish sticks. Things were definitely not going as planned. Then again, they rarely did in my line of work.\n\nBut what hurt more than my present situation was my heart, which felt as if the Ringer had used it as a punching bag. Johnny had succumbed to Siren's song. He hadn't cared enough about me to fight her off. Neither had Hermit or Mr. Sage. I wasn't thrilled with my team members right now, but Johnny's betrayal was the one that wounded me the most.\n\nOh sure, he was under the influence of a hypnotic ubervillain with more cleavage than a lingerie model. Oh sure, she had a fancy device that could enslave ninety-seven percent of the population. Pitiful excuses, at best. I'd told Johnny that I loved him, and he'd still abandoned me.\n\nMy fingers twitched. But maybe not entirely. Hands shaking, I uncurled my fingers. Johnny's, Angel's, lighter lay inside. A spark of hope flared to life inside me. In the end, Johnny had tried to help me. To give me a way out. Maybe I could forgive him, if I got out of this alive.\n\nMy body felt limp and tingly from Intelligal's power-diluting gas, but I didn't have time for such weakness. If I didn't get out of here soon, I'd freeze to death. Even with my fiery superpowers, it would only be a matter of minutes before I was one big icicle. I hated to be cold, and I hated fish. But here I was, trapped with both. Ah, the glamorous life of a superhero.\n\n\"Lulu?\" I asked, my voice weak and small. \"Lulu, can you hear me?\"\n\nThe computer hacker didn't answer. Not even static crackled in my ear. The thick door must be blocking the signal, meaning I was on my own. Fabulous.\n\nSomehow, I managed to roll over onto my knees. I could still taste the sickeningly sweet gas in my mouth, mixing and mingling with my blood. With my free hand, I scraped up a mound of frost off the floor and shoved it into my mouth. The ice crystals melted, washing away some of the gas. My head cleared, and I felt a little stronger.\n\nI kept repeating the process until I had cleansed my mouth. My arms and legs twitched and jerked and spasmed, recovering from the effects of Intelligal's gas. I crawled to the middle of the freezer and slung my numb limbs around until I faced the door. It was a thick metal door, designed to keep the cold in. Well, not for long.\n\nMy fingers trembled. I grasped the lighter and slowly clicked it. Once, twice, three times. Four, five. Nothing happened. I couldn't quite grip the cold, slick metal with my weak, tingling fingers. _Focus, Fiera, focus!_ I'd been in tighter spots than this, most notably when Prince Horrid had captured me with plans to add me to his harem as one of his pliant dancing girls. I'd gotten out of that mess. I'd get out of this one too. I was Fiera, for crying out loud. Member of the Fearless Five. Protector of the innocent. Superhero du jour. It was what I did.\n\nJust when I thought I couldn't hold it another second, the lighter sparked on. I cupped the tiny, weak flame like it was the most precious thing in the world. To me, it was. Slowly, my hands warmed. The lighter's small flame fed my own inner power. My fingertips started to glow as the fire inside me rekindled. I concentrated on burning the rest of the limp, languid feeling from my body. High metabolism, help me now.\n\nMy emotions had always fueled my powers, and I grabbed hold of them. I remembered how Siren had tricked us, how she'd turned my friends against me, how the bitch had put her hands all over Johnny. I focused my anger, let it rage through me with the heat of a thousand suns. The fire inside me grew and grew and grew.\n\nI formed a fireball with my hands. I took careful aim and threw it at the door. It exploded onto the cold metal, making it shriek and groan. Steam filled the freezer. I formed another fireball.\n\nThen another one...\n\nThen another one...\n\nThen another one...\n\nTen minutes later, the last remains of the door melted away. I got to my feet, still a little shaky, and stumbled through the melted edges of the white-hot metal.\n\n\"Fiera! Fiera! Where are you?\" Lulu's voice squawked in my ear.\n\n\"Over by one of the freezers,\" I said, sliding to the ground.\n\nA motor whirred, and Lulu stopped in front of me, tires smoking.\n\n\"Where the hell have you been?\" I muttered, trying to rub the rest of the feeling back into my arms and legs.\n\n\"I got here as quick as I could. I'm not Swifte, you know,\" Lulu said in a defensive tone. Her eyes dropped to her wheelchair and legs. \"Not by a long shot.\"\n\nI bit back my angry retort. It wasn't Lulu's fault. She'd done the best she'd could. And I had other things to think about right now. Like how to rescue the others and stop Siren's evil plan to take over the city. \"Did you hear Siren? Did you see where they went?\"\n\n\"I heard everything, but by the time I got out of the van, it was too late.\" Lulu shook her head. \"Siren and Intelligal shepherded the others into a car on the far side of the building. They sped away before I could fix a tracker to it.\"\n\nI cursed. This was no time to be sitting around. I grabbed on to the remains of the melted door and pulled myself up. At least, I tried to. My arms buckled, and I wobbled back and forth before falling and smacking my ass against the cold concrete floor. I cursed again, loud and long, hating my sudden weakness.\n\n\"Whoa there, tiger!\" Lulu put a hand on my shoulder. \"You're not in any position to be walking around right now. It's a miracle you got out of that freezer alive.\"\n\n\"Well, I can't sit still. And it's not like you can carry me out of here. Do you have another suggestion?\"\n\nLulu patted her lap. \"Hop on board the Lulu Lo Express. I can drive us both out of here.\"\n\n\"You can't be serious.\"\n\n\"Oh, I am. Henry likes to ride around on my lap all the time.\"\n\nI groaned and put my head in my hands. \"Too much information, Lulu. Too much information.\"\n\n* * *\n\nIn the end, I didn't have a choice. After a good five minutes of cursing and snarling and trying to heft myself into an upright position, I crawled up onto Lulu's lap, and she motored us back to the van.\n\n\"Damn, this thing can scoot,\" I said, trying to distract myself from the fact I was hanging on to Lulu's neck like we were lovers.\n\nI was extremely glad it was after midnight and pitch-black, and that there was no one around to witness my humiliation. I'd never live down the shame. Fiera, member of the Fearless Five, protector of the innocent, reduced to clinging to a wheelchair to get around. Some superhero I was. I couldn't even stand upright at the moment.\n\nLulu beamed and patted the side of the chair. \"Of course it can. I've got almost two hundred horses in the motor. It tops out at about fifty miles an hour.\"\n\nWe zoomed to a stop in front of the black van. I slid off Lulu's lap and into the carpeted interior. The computer hacker strapped herself in the motorized lift and joined me.\n\nLulu rustled through one of the first-aid boxes we kept inside and handed me a small, foil packet. \"Here. Take one of these. It should flush the rest of the radioactive gas out of your system.\"\n\nI took the packet, which contained a _Radioactive Isotope Diminisher_ , or _RID_ for short. The pills were the invention of some mad scientist who found himself constantly bombarded by radioactivity while he was researching something or other. They were like vitamins to superheroes, and the Fearless Five used them to keep from getting more mutated than we already were. They'd saved our asses on more than one occasion, especially Carmen's last year when she'd gone up against the Terrible Triad by herself.\n\nI swallowed the pill and felt the effects almost immediately. My limbs grew heavy and substantial once again, my superstrength returned, and my inner fire flared up to its usual slow, steady burn. I sat up. \"All right. That did the trick.\"\n\n\"Are you sure you're okay?\" Lulu asked. \"We can sit here and rest a few more minutes if you need to.\"\n\n\"I'm fine. And we don't have any time to rest. We need to get our boys back.\"\n\n\"How are we supposed to do that?\" Lulu asked. \"Carmen and Sam are halfway around the world, and Henry and the chief are under Siren's spell, along with Johnny. We're a little short of superheroes right now.\"\n\n\"Oh, don't worry about it. I know a couple of people who should be more than willing to help us,\" I said. \"Let's go.\"\n\n* * *\n\n\"Are you sure this is a good idea, Fiona?\" Lulu said.\n\nI put my hands on my hips. \"Of course I'm sure it's a good idea. All my ideas are good ones.\"\n\nI tossed my hair over my shoulder, trying to look more confident than I felt. The truth was I wasn't sure it was a good idea, but it was the only one I'd been able to come up with. Lulu and I needed help, and this was the only place we were likely to get it.\n\nWe stood outside the closed iron gates that led up to the Bulluci manor. Lulu pressed the button on the call box again. No response. We'd been standing out here for ten minutes, trying to get somebody to wake up and answer us.\n\n\"Oh, screw this.\" I took hold of the iron gates and looked over my shoulder. \"You might want to move back.\"\n\nLulu eyed my glowing hands, which were clenched around the metal. \"I think I'll do that very thing.\" She hit a button on her chair and zoomed out of the way.\n\nI ripped open the gate. The iron wasn't nearly as sturdy as it looked, and the gates cracked off their hinges. I didn't even have to put any real muscle into it. Part of the surrounding wall crumbled in on itself.\n\nSince Lulu's chair could move a lot faster than I could walk, I hopped back on Lulu's lap, and we scooted up the long driveway. I was getting almost used to sitting on top of the computer hacker. Almost.\n\nThe chair stopped. I climbed off and pounded on the front door, which was embossed with a giant B and another freaking angel wing. \"Bella! Bobby! Open up! Now!\"\n\nA light flared to life in one of the upstairs windows. A curtain on the front door twitched, and Bella cracked it open. She wore a pair of short pajamas with white clouds on them. Naturally.\n\nHer mouth dropped open at the sight of me. \"Fiera? I mean Fiona? I mean...oh, you. What's wrong?\" Her eyes flicked to Lulu. \"And who is this person with you?\"\n\nI took a deep breath. \"Johnny's in trouble. We need your help.\"\n\n# Chapter Twenty-Six\n\nLulu and I explained the situation to Bella. She woke up Bobby, they threw on some clothes, and we piled in the van and headed back to Sublime. Normally, I would have knocked them out or blindfolded them to keep them from seeing exactly where we were going, but I didn't have time. Besides, sneakiness wasn't my strong suit. That was more Sam and my father's thing.\n\nWe led Bella and Bobby through the underground garage to the library. It was a good thing I hadn't used blindfolds, because it was painfully slow going with the two Bullucis stopping every three feet to stare at something else.\n\n\"Come on, come on,\" I snapped. \"You guys can _ooh_ and _aah_ over the super-duper, supersecret superhero lair later.\"\n\nThe two of them picked up their pace, and we reached the library. Lulu wheeled around and plugged her laptop into Henry's network of computers. The Bullucis stood at the doorway, mouths hanging open.\n\n\"Oh, come in. It's not that sacred.\" I marched over, pulled them inside, and closed the double doors.\n\nBella and Bobby cautiously crept farther into the massive library.\n\n\"Please forgive my surprise. It's just that I never thought I'd be invited in _here_ ,\" Bobby said, running his hands over the _F5_ insignia carved into the table.\n\n\"Me either,\" Bella whispered. \"This is incredible. Look at all the books you have!\" Her eyes flicked over to Lulu and her computers. \"And the equipment. It's so amazing!\"\n\nAfter a few more minutes of staring and sputtering, Bella and Bobby sat down with Lulu and me at the big, round table.\n\n\"Explain it to me again,\" Bella said, rubbing her head. \"I'm still a little confused. Johnny joined up with you guys to take out the ubervillains, but Siren hypnotized the men.\"\n\n\"Right.\" I nodded.\n\n\"How was she able to do that?\" Bobby asked. \"Surely Mr. Sage would have been able to resist her. He's a powerful psychic in his own right.\"\n\n\"It's that damn microphone she had. It upped her power tremendously.\" I drummed my fingers on the table. Sparks flew everywhere. \"Nobody could resist it but me, and that's only because I was a woman. And totally pissed off.\"\n\n\"It's not just the microphone, Fiona. Weren't you paying attention to Siren? Didn't you listen to what she said?\" Lulu asked. \"She told you exactly what she and Intelligal are up to.\"\n\nI shrugged. \"Not really. I was more concerned with not being able to move at the time. And in the end, all ubervillains want the same thing\u2014to rule the world. The only difference is the crackpot scheme they come up with to try to help them do it.\"\n\n\"Well, let me tell you, Siren's plan is a doozy.\" Lulu typed on her computer. Images began to pop up on the film screen hanging on the wall.\n\n\"Hey, it's that radio thingy the two of them were protecting so fiercely,\" I said, staring at the box with its odd wires and diamond-studded frame.\n\n\"It's not just a radio thingy,\" Lulu corrected. \"It's a voice amplifier and projector. Or VAMP for short.\"\n\nI snorted. \"How fitting.\"\n\n\"Indeed. Anyway, the VAMP machine is designed to take a sound, like Siren's voice, and distribute it over a wide area.\"\n\n\"Like, say, the whole city?\"\n\nLulu shot her finger at me. \"Precisely. Siren mentioned something about a broadcast while she was rambling. They must be ready to use it.\"\n\n\"What happens if they do that?\" Bella asked.\n\n\"Siren's voice will spread out over Bigtime. Not only does the VAMP machine boost the range of her voice, it also ups the power, which is how the others were brainwashed into doing her bidding. It's really quite an impressive machine. Luckily, I thought to put a filter on everyone's comm links to screen Siren's voice, or I probably would have been under her spell too. But if we don't stop them, Siren will enslave everyone.\"\n\n\"Why would she want to do that?\" Bobby said.\n\n\"Who knows why ubervillains do what they do? I quit trying to figure out their motivations a long time ago,\" I said. \"She'll probably do something totally lame and clich\u00e9 and hackneyed like get people to bring her all their money.\"\n\n\"But wouldn't she have to keep talking into the machine the whole time to keep people under her control?\" Bella asked.\n\nWe looked at Lulu.\n\n\"Not necessarily,\" Lulu said. \"She could record her voice and just loop the recording so that it plays back over and over. The only problem with the VAMP machine is that it's not quite powerful enough to broadcast Siren's voice over the entire city. So she's going to need another power source or a way to piggyback the signal onto one of the radio or television stations.\"\n\n\"Then that's what we need to figure out,\" I said. \"Where the ubervillains are going to go to unleash their doomsday device.\"\n\n\"I'm on it,\" Lulu said, hands flashing across her keyboard.\n\nWe sat there in silence while the computer hacker went to work. Bella and Bobby kept shooting furtive looks around at the maps and globes and books. I got up and paced back and forth behind Lulu.\n\nLulu glared over her shoulder at me. \"You know I can't concentrate when you flounce around like that.\"\n\n\"Well, what do you want me to do?\" I snarled. \"I can't sit still, and I can't crack any skulls until we get a positive location for Siren and Intelligal.\"\n\n\"Oh, go eat what's left in the refrigerator or something. That's what you usually do.\"\n\nLulu turned her back to me and started typing again. I resisted the urge to light her hair on fire.\n\n\"Come on, Fiona. I'll go with you,\" Bella said. \"Do you want anything, Grandfather?\"\n\nBobby shook his head. \"No. I think I'm just going to sit here and look at everything. If that's all right with you, Fiona.\"\n\nI waved my hand. \"Fine. Just as long as you don't disturb her computer highness over there.\"\n\nLulu shot me another dirty look. I just tossed my hair over my shoulder.\n\n* * *\n\nBella and I walked down the deserted halls until we reached the underground kitchen. I opened the doors on the restocked refrigerators, desperately in need of ten thousand calories or so. It'd been a busy night so far, and tomorrow, rather, today would only be worse. I needed to keep my strength up.\n\n\"How about some cheesecake?\" I said, pulling a large pie out of one of the refrigerators. \"It's triple chocolate, one of Quicke's specialties.\" The restaurant delivered a couple dozen of the delectable desserts to Sublime every week.\n\n\"Well, I really shouldn't. I'm on a no-sugar diet...\" Bella's voice trailed off as she stared at the luscious cheesecake.\n\nNo-sugar diet? No wonder the poor thing was so uptight. No sugar, no fun, in my book.\n\n\"Well, you're going to have a piece tonight,\" I said, cutting her a slice. \"I think we've all earned it. I certainly have.\"\n\nBella took the cheesecake from me and poured herself a glass of milk. I thought she'd just pick at the yummy cake, but Bella downed it and came back for seconds. She was a quick eater, just like me. Maybe we'd get along better than I thought.\n\n\"I still can't quite believe you're a member of the Fearless Five. And that I'm sitting here with you in the supersecret Fearless Five headquarters.\" Bella took another big bite of her cheesecake. \"It's all a bit surreal.\"\n\n\"Tell me about it. I can't believe you come from a family of generational superheroes. I also can't believe your brother is actually Johnny Angel, and that I've been dating him.\" I eyed Bella. \"Do you moonlight as somebody too?\"\n\n\"Of course not. Johnny Angel is the only one in our family.\" She sounded offended, as though being a superhero was some vile occupation.\n\n\"Do you have an exoskeleton like Johnny does? Or some other sort of power?\"\n\nShe hesitated. \"I don't know if you'd call it a power, exactly. It's a little strange.\"\n\n\"Oh really? Strange how? Strange like you can manipulate the weather with your bare hands? Strange like you can create unbreakable force fields around yourself? Strange like you can create earthquakes just by thinking about them?\"\n\nI stared at the fashion designer. Perhaps if Bella had some hidden superpower, she could suit up and go out into the field with me. I could use all the help I could get when I went after Siren and Intelligal. My eyes flicked over her body. She was about Carmen's size, although quite a bit curvier. Surely, I could find her some sort of costume to wear\u2014\n\nBella laughed. \"Not that strange. I'm just lucky.\"\n\nSo much for that thought. \"Lucky? That's not really a power, is it?\"\n\nBella shrugged. \"Not really. Not like your power.\"\n\n\"So how does it even work?\"\n\n\"I just think about things, and stuff...happens. Especially when I'm stressed out.\"\n\nI thought back. \"Is that why the coffee table cracked when I was at your house?\"\n\nBella nodded. \"That's one of the problems with it. You can have good luck...\"\n\n\"Or bad luck,\" I finished. \"Show me. Show me how it works. Or doesn't work.\"\n\n\"I can't control it all the time, but I'll try.\"\n\nBella stared at me. I looked back at her. Her eyes didn't glow. Her hair didn't snap and crackle. Sparks didn't fly from her fingertips. She didn't seem to be doing anything at all other than eyeing me. Some power.\n\nBut I kept that thought to myself and raised my fork to take another bite of cheesecake. To my surprise, my hand wobbled, and the cheesecake fell onto the table. What a waste. It was a good thing I still had plenty left. I stabbed another bite. Again, the cheesecake slid off my fork and splattered onto the table. I frowned at the chocolate stains. I wasn't that clumsy, especially when it came to food.\n\nI eyed Bella with suspicion. \"Did you make me do that?\"\n\nShe smiled and cut another piece of cheesecake off with her fork. \"What do you think?\"\n\nBella raised the fork to her lips. She was just getting ready to pop it into her mouth when the dessert slipped off the silver tines and joined the rest of mine on the table.\n\nShe stared at the chocolatey mess and let out a long sigh. \"Unfortunately, it always seems to boomerang around back to me\u2014in a bad way.\"\n\nMaybe there was something to this luck thing. Too bad Bella didn't know how to control her power. Or at least make Siren and Intelligal have a string of bad, debilitating luck.\n\nWe sat there in silence. After I finished off the cheesecake, I made myself a dozen cucumber-and-tomato sandwiches, which I ate with three bags of chips, five liters of soda, a box of oatmeal-raisin cookies, another box of crackers, a pound of grapes, and a wheel of Gouda cheese.\n\nBella fixed her amber eyes on me. \"So what are your intentions regarding my brother?\"\n\nI almost choked on my sandwich. \"Excuse me?\"\n\n\"What are your intentions regarding Johnny? Do you care about him? Or is he just a fling to you? Some random guy you can have great sex with?\"\n\nI wiped the mayonnaise off my mouth, stalling for time. I'd fallen in love with Johnny Bulluci, but I wasn't sure quite how I felt about it. Much less what I should say to his sister about him.\n\n\"I ask because Johnny really likes you. I even think he's starting to fall in love with you,\" Bella said in a soft voice.\n\nMy mouth fell open. I couldn't speak.\n\n\"Do you care about him, Fiona? Because if you don't, you should walk away from him when this is over. I know Johnny appears as if he's cheerful and carefree and that nothing can hurt him, literally, but you could. He told us about your fianc\u00e9. How he was murdered. How much you loved him. How you still wear his engagement ring. If you're not over your fianc\u00e9 yet, you need to tell Johnny now, before he gets any more involved with you.\" Bella's eyes bored into mine. \"I won't let my brother get hurt, especially not by a superhero. Do you understand me?\"\n\nFor once, I chose my words carefully. \"I care about your brother a great deal. I'm not leading him on. That would never be my intention. As for Travis, I'll always love him. I'll always miss him. But he wouldn't want me to live my life in the past. He wasn't that sort of man. As for me and Johnny, I'm just trying to take it one day at a time. Things are a little...complicated between us right now, in case you haven't noticed.\"\n\nSome of the bright, angry glow drained out of Bella's golden eyes. My words seemed to satisfy her. At least she hadn't cracked the table this time. I'd hate to lose the rest of my food.\n\n\"I noticed,\" Bella said. \"Things are always complicated when it comes to my family.\"\n\nThe odd tone in her voice struck me. \"What do you mean?\"\n\nBella sighed again. \"It's a long story.\"\n\n\"I've got nothing but time until we find the ubervillains.\"\n\nBella stared at the refrigerator, but the vacant look in her eyes told me that she wasn't really seeing it. \"I don't know much about your childhood being the daughter of a superhero, but I grew up in a home where it was all Angel, all the time. If we weren't talking about Angel or motorcycles or ubervillains, my mother and brother and I were patching up my father and grandfather when they'd come home late at night. Can you imagine being a kid and going through that?\"\n\nI flashed back to my high school years, when I'd go out and prowl the streets with my father, fighting crime. They were some of my fondest memories. \"Oh, I can imagine.\"\n\n\"When I was a kid, I thought it was so fascinating that my family were the ones behind Johnny Angel. I used to ride on the motorcycle with my father and dream of the day when I'd get to be Johnny Angel.\" Bella's mouth twisted into a wry grin. \"Of course, I didn't realize then that Angel was more of a man's name. And a man's tradition.\"\n\n\"And when you grew up?\" I asked, sensing this story wasn't going to have a happy ending.\n\n\"I realized how silly it was. Dressing up in a costume, riding around town on a motorcycle, raising hell with other bikers. And I remembered the strain on my mother. How she'd sit up late at night worrying whether my father was going to come home or not. It's the same thing I did with my father before he died. And it's the same thing I do now with Johnny every night when he's gone.\" Bella rubbed her head. \"I don't want to have anything to do with superheroes and ubervillains and weekly battles anymore. I'm so tired of it all. Johnny Angel, the worrying, the constant fear. Johnny was too, until our father died.\"\n\nI squeezed her hand. \"It's not disrespectful, and it's not wrong of you to want to have a superfree life. Some people can't handle the lifestyle. Like you, I grew up around it. But to me, my powers have always been a part of who I am. I couldn't imagine not being a superhero, not trying to help people, but I can understand how you feel.\"\n\nI took a deep breath. \"When Travis was alive, he'd go out on missions by himself. I'd do the same thing you did\u2014sit up and worry. I wore out more carpets pacing back and forth than you can imagine.\"\n\nBella smiled. \"You do seem like a bit of a pacer.\"\n\n\"You have no idea.\"\n\n\"What are you going to do if Johnny wants to quit being Angel? Or if he doesn't?\"\n\n\"I don't know,\" I admitted. \"Let's worry about rescuing him first and dealing with Siren and Intelligal. Then, we'll talk about the complicated, relationship stuff.\"\n\n* * *\n\nAn hour and another cheesecake later, Bella and I went back to the library to check on Lulu and Bobby. The elderly Bulluci sat in front of one of the computer monitors, flipping through various television channels. He looked extremely bored, until he stumbled across a soccer game. That perked him up a bit.\n\nLulu sat nearby, _type-type-typing_ away on her computer like usual. Sometimes, I wondered if she and Henry did anything but stare at their monitors when they went out on dates. The two of them were never far from an electronic device of some kind or another.\n\n\"Anything yet?\" I asked.\n\nLulu shook her head. \"I'm still trying to narrow down the list of places Siren and Intelligal might go to turn on their boom box. It has to be somewhere fairly high up so they can hook into a radio or television signal. I've been focusing on SNN and the other local television stations downtown, but I don't know how the ubervillains would get past their security. Most of it is state-of-the-art.\"\n\nI snorted. \"Get past security? Please. Intelligal can just use those cursed missiles of hers to blast their way in.\"\n\nLulu shook her head. \"I don't think so. Remember in the fish-stick factory, Siren said the radio is very finely calibrated. That's the reason they didn't explode you right then and there. Siren didn't want to risk the shockwave disrupting the VAMP machine.\"\n\n\"So they'll have to do it real quiet-like. Sneak in and set everything up before Siren goes live.\"\n\n\"Bingo,\" Lulu said.\n\n\"Well, let's get cracking,\" I said.\n\nThe four of us started naming the tallest buildings in Bigtime. Bobby not-so-humbly pointed out that Bulluci Industries was housed in one of the highest skyscrapers in the city.\n\n\"What about the observatory?\" Bella asked. \"Isn't it officially the highest point in the city?\"\n\nI flashed back to the benefit and the scientific models I'd seen. More than a few of them remarked on the observatory's height. \"It sure is. Wouldn't that be the logical spot for the ubervillains to turn the volume up on their radio? Wouldn't they get the strongest, clearest signal from there?\"\n\n\"They would. Let me check on something.\" Lulu pounded away. After a couple of minutes, she stopped. \"That's funny.\"\n\n\"What's funny?\" I asked.\n\n\"I hacked into the observatory's database so I could see what programs were scheduled and how many people might be on the scene in case Siren and Intelligal tried to sneak in with the regular folks. Guess who's doing a live morning show there for SNN in a couple of hours?\"\n\n\"Who?\" Bobby asked.\n\n\"Erica Songe,\" Lulu replied. \"And the weird thing is, she's going when there are no school groups scheduled. No tours, no benefits, no press conferences, nothing.\"\n\n\"Erica Songe? What would that little twit be doing at the observatory\u2014\"\n\nAn odd thought struck me. Erica Songe. I thought back to my run-ins with the news reporter. Her hissy fit at the shop a couple months ago. Her relentless flirtation with Johnny. Her pushiness at the observatory benefit. All the pieces slowly formed a picture. For a moment, I felt just like Carmen.\n\n\"You know, Erica Songe is a trashy little thing, just like Siren is.\" I snapped my fingers together, remembering something else. \"And she has a sister. Irene something. A total geek, just like you.\"\n\nLulu stared at me. \"You don't think...\"\n\nI nodded. \"I do think. And so do you.\"\n\n\"Let me pull up some pictures of them off the Internet.\" Lulu's fingers smacked against her keyboard.\n\nBella looked back and forth between the two of us. \"What are the two of you talking about?\"\n\n\"We're talking about Siren actually being Erica Songe, a news reporter for SNN,\" I replied.\n\nBella, Bobby, and I hunched over Lulu's shoulder as she pulled up photos of the two women. I studied the pictures. Same black hair. Same blue eyes. Same collagen-injected lips. Same supersized boobs. There was no mistake about it. Erica Songe was Siren.\n\n\"Hold on a minute, let's see if I can get the sister too.\" Lulu hit more buttons on her computer.\n\nA photo of Irene popped up on the monitor. Lulu compared it to one of Intelligal. Same black glasses. Same sour expression. Same disdain for Siren and everyone else. I wanted to laugh. Carmen was right. Spandex costumes and bright masks really were very thin disguises. We all might as well go around with our real names tattooed on our foreheads.\n\n\"That's her! She's the one who killed my son!\" Bobby pounded his fist into his hand and let out a long string of curses. His face turned red, then purple with rage and fury.\n\nBella put her arm around her grandfather's shoulder, trying to comfort and calm him down before he had a heart attack. I turned to Lulu.\n\n\"It's them. We agree that it is. Now, let's figure out how we're going to stop them,\" I said.\n\n# Chapter Twenty-Seven\n\n\"The first thing we have to do is figure out how to resist Siren's voice, especially now that she's going to be plugged into that TRAMP machine,\" I said. \"I can't very well smash it to bits and throttle her if I'm under her spell.\"\n\n\"It's actually called a VAMP machine,\" Lulu corrected.\n\nI waved my hand. \"TRAMP, VAMP, whatever. They both describe her.\"\n\n\"I might be able to help with that,\" Lulu said. \"Henry and the chief were working on some earplugs to block Siren's voice. They're not perfected yet, but they might work well enough for us, since we'll be in the van.\"\n\n\"In the van? I don't want to stay in the van,\" Bobby protested. \"I want to go inside and save my grandson with you.\"\n\n\"You'll need all the help you can get, Fiona,\" Bella added. \"Why don't you let us come with you?\"\n\n\"Because it's too dangerous. I can't take a chance on one of you getting hurt. I can handle Siren and Intelligal by myself.\"\n\nProbably.\n\n\"But you can't handle Angel, Hermit, and Mr. Sage too,\" Bella pointed out. \"You're going to need some backup.\"\n\nI paced back and forth. \"I need Carmen and Sam, or rather Karma Girl and Striker. Where are they now, Lulu?\"\n\n\"Carmen called me yesterday before all of this went down. They'd just flown into Rome on Sam's private jet. There's no way they can get back here in time.\" Lulu looked at the Bullucis. \"I think they're right, Fiona. You're going to have to take us with you, whether you like it or not.\"\n\nI stared at the three of them, eyes shining, faces tight with hope. They were all so ready, willing, and eager to wade into battle and probably get themselves electrocuted or worse. Sending them up against Siren and Intelligal would be like throwing tender steaks to wolves\u2014or to me. The ubervillains would gobble them down without a second thought. I saved innocents. That was my calling, my duty. I didn't put them in danger.\n\nBut I didn't have a choice. Not if I had any hope of rescuing the others. Bella was right. I couldn't fight Siren and Intelligal and save the others at the same time. Even Swifte would have been hard-pressed to do it, no matter how speedy he was.\n\nSo, I thought about my three new teammates and how to minimize the danger to them. There wouldn't be any radioactive goo around, so they wouldn't become horribly mutated by going with me. At least, I didn't think they would. You could never tell what sort of effect something like that stupid VAMP machine might have on regular folks. Carmen was proof of that.\n\nBesides, Bella had a power that might come in handy. If luck was really a power, instead of just some weird, wild mojo, and if she could get it to work in our favor.\n\nNo radioactive goo and a little luck. Maybe things wouldn't go too badly. I snorted. Yeah, right.\n\n\"Fine, you can come with me, but you have to stay out of the way. Agreed?\"\n\nThe three of them nodded their heads.\n\n\"I mean it,\" I snapped. \"I'm not bringing you along so you can do something stupid, like die a noble, bloody death in order to save the others and the city. That's _my_ job. You'll do what I say, when I say it. Or else I'll lock you in the van and take the ubervillains on by myself. Got it?\"\n\nThey nodded again, a bit more reluctantly this time. Bobby, in particular, shot me a sour look.\n\nI resumed my pacing. \"Even with you guys tagging along, I'm still going to need some extra help. Siren I can handle no problem. It's Intelligal's chair and her stupid missiles that always have me ducking for cover. I need to get her out of that chair, or at least neutralize it, if we have any hope of freeing the others and destroying that VAMP thing.\"\n\n\"You know who you should talk to if you want more firepower,\" Lulu said.\n\n\"Who?\"\n\n\"Why, Jasper, of course.\"\n\nJasper was one of Lulu's more notorious friends and Bigtime's underground bomb expert. If you needed something that went _boom_ in a big way, you went to Jasper. What he did wasn't exactly legal, of course. The chief had thought about busting him many times, but since Jasper had helped Carmen rescue us last year, he'd given the demolitions expert a free walk. For now.\n\n\"Jasper?\" Bobby asked. \"The Jasper who lives downtown in that big brownstone?\"\n\nLulu nodded. \"The one and only.\"\n\n\"You know him?\" I asked.\n\nBobby shrugged. \"I've had use for a few of his items over the years.\"\n\n\"Oh really? Like what?\"\n\nBobby winked. \"Oh, this and that.\"\n\nI opened my mouth to question him further, when Lulu interrupted.\n\n\"We should get going,\" she said. \"It's after four now. Erica, I mean Siren, is set to go on the air at seven-ten sharp.\"\n\n\"Then, off to Jasper's we go,\" I said.\n\n* * *\n\nLulu and I went to the equipment room. I changed into a fresh superhero suit, one that didn't smell like frozen fish sticks, while she gathered up a couple of things we might need, including the earplugs my father and Henry had been working on and some more of the gas-blocking and _RID_ pills. Then, we went back to the library, grabbed Bella and Bobby, and loaded up the van.\n\nJust after five in the morning, Lulu and I stood in one of the nicer neighborhoods in Bigtime. A massive brownstone that even finicky Joanne James wouldn't have minded owning towered above us. Bella waited in the van, keeping the engine warm. Bobby sat in the passenger seat and peered out the window. The streets were still and quiet, and the air was cool and smelled of impending rain. The streetlights flickered on and off, confused by the grayish dawn.\n\nWide, shallow steps led up to the brownstone. Lulu hit a button on her wheelchair, and four jack-like devices sprang out from the sides. The jacks hissed as they planted themselves on the ground. They lifted the wheelchair several inches off the ground\u2014high enough so Lulu could roll herself up onto the first step. She started to repeat the process, but I was too impatient to wait. I picked up Lulu, chair and all, and carried her to the front door.\n\n\"I can make it up and down stairs by myself,\" Lulu protested. \"I've been doing it for years.\"\n\n\"We don't have time for you to be stubborn and independent,\" I said. \"Now wake up Jasper.\"\n\nLulu punched a box on the wall, and a security camera swiveled over to see who was calling at this early hour. I could almost see the lens in the camera widen at the sight of me. I wore my costume, and my hair crackled and sparked with fire. I didn't think Jasper got many visits from the members of the Fearless Five. Well, except from Carmen and Lulu.\n\nThe box clicked on.\n\n\"What's the word?\" a male voice asked over the static.\n\n\"The word is _silent night_ ,\" Lulu replied.\n\nThe camera lingered on us. Jasper wasn't going to let us in. I clenched my hand into a fist. Sparks and smoke hissed out from between my fingers. We'd get in one way or another. I'd make sure of that.\n\nThe door buzzed open.\n\n\"I thought the word was _boom-boom_ ,\" I whispered.\n\n\"He changed it right before Christmas.\" Lulu shrugged. \"Jasper has a strange sense of humor sometimes.\"\n\nNo kidding. We entered the brownstone. A tall, thin man appeared at the far end of the hallway. A bathrobe hung on his bony figure, and his hair stuck out at those weird angles that were only made possible by a good night's sleep. We made our way toward him. I moved slowly, keeping my hands in sight and the fire to a minimum. I'd been in the vicinity of one of Jasper's bomb blasts before, and I'd been sore for a week as a result. So, I didn't think it would be a good idea to startle the bomb guru. He might do something stupid, like try to blow us up.\n\n\"Lulu, how it's going?\" Jasper asked in a cautious voice, his eyes flicking over me.\n\nLulu extended her hand, and she and Jasper engaged in a series of slaps, high-fives, and other strange finger signals. \"Fine, Jasper. Just fine. I know what you're thinking, but don't worry about ole Fiera here. We didn't come here to bust you. In fact, quite the opposite. We need some supplies.\"\n\nJasper looked at me. \"You need supplies?\" he asked in a rather disbelieving voice.\n\nI shrugged. \"I'm going up against a couple of ubervillains bent on taking over Bigtime, and I'm four team members down. I can use all the help I can get. Or rather, all the bombs you can give me. And anything else you think might come in handy.\"\n\nJasper stared at me. Suddenly, a smile creased his face. \"Well, then, who am I to turn away a potential customer? Plus, any friend of Lulu's is a friend of mine. Come in, come in.\"\n\nHe turned and led us farther into the dark brownstone.\n\n\"You know I've long been a fan of your work, Fiera,\" Jasper said in a conversational voice. \"Your fireballs are most impressive. The intensity of the heat, the explosion on impact, the massive damage they cause. I'd trade my explodium bombs for them in a second.\"\n\nThe odd praise pleased me, even if it was coming from a somewhat mad bomber. \"Thanks. Lulu's told me a lot about your work too.\"\n\nJasper gave the computer hacker a pointed look that would have cut glass. \"Not too much, I hope.\"\n\nLulu patted him on the arm. \"Don't worry, Jasper. Your trade secrets are safe with me.\"\n\n\"Good to know,\" he said. \"Good to know.\"\n\nJasper punched in a security code and went down a flight of stairs. I picked up Lulu and followed him. It looked like Intelligal's lair, except it was much tidier. Wires and bits of metal were stacked in neat piles on top of several worktables. Soldering irons, pliers, and other tools hung from slots in a tall rack attached to one of the thick walls.\n\n\"You've done some redecorating, Jasper,\" Lulu said. \"The last time I came here, this place was a mess.\"\n\nJasper shrugged. \"I decided to get a little more organized. It was either that or blow myself up tripping over things.\"\n\nLulu and I froze. We stared at each other, then Jasper. The bomb guru paid no attention to our sudden nervousness.\n\nJasper pulled out a legal pad and a pencil from a desk in the corner. \"So tell me what you need.\"\n\nLulu outlined Siren and Intelligal's scheme to take over Bigtime and, subsequently, the world. \"So as you can see, Fiera here needs some help.\"\n\n\"What I need, specifically, is a way to disable Intelligal's chair. Of course, I've tried many times myself. Fireballs, superstrength, collisions. Nothing seems to put much of a dent in it, except whatever Intelligal used to make it self-destruct.\"\n\n\"It's probably made out of solidium,\" Jasper said. \"It's very rare and very expensive, but it's the strongest, toughest metal known to man. Explodium won't even scratch it. At least, not by itself. You have to mix some other explosives with it.\"\n\n\"I don't want to scratch it,\" I snapped. \"I want to completely destroy it. Think big fireball. Think catastrophic damage. Think crumpled lump of charred, twisted metal. That's what I'm aiming for. First the chair, then the VAMP device. Maybe even the ubervillains if they get in my way.\"\n\nJasper looked at me as though I were the love of his life. His eyes went all soft and warm. \"Ah, a woman after my own heart.\" He took my hand and pressed a kiss to it. \"For you, my lady, only the best.\"\n\nI stared at Lulu. Her lips twitched, and I could tell she was trying to keep from laughing. I rolled my eyes.\n\nJasper spent the next few minutes puttering around, grabbing strange-looking objects out of various safes and lead boxes hidden throughout the room. Finally, he pulled out a small, square box. He bowed to me and cracked open the lid. Nestled inside were ten round, brown-colored objects. They reminded me of chocolate-covered bon-bons, although I doubted they tasted as good. My stomach rumbled. I'd have to eat some more food before I went toe-to-toe with Siren and Intelligal.\n\n\"For you, Fiera, only the best. This is something new I've been working on. Explodium has become rather pass\u00e9 this last year. Everybody's working with it these days, and you can practically buy it on the street corner,\" Jasper said. \"But this, this is something special. I call it obliteron.\"\n\n\"Obliteron?\" I asked.\n\n\"Obliteron, because it not only destroys matter, it pretty much obliterates it.\"\n\nI nodded as if I knew exactly what he was talking about. As a superhero, I'm used to the creative, colorful, and sometimes ridiculous names my fellow heroes and ubervillains call themselves. Halitosis Hal was a prime example. But _obliteron_? For a bomb? Oh my.\n\nJasper picked up one of the bon-bons and rolled it around in his hand. \"Obliteron is a special form of explodium, a mixture of it and a few other key radioactive isotopes. I've managed to put my own stamp on it with these little beauties. Inside each of these thick, plastic shells is a liquid ball of obliteron.\"\n\nI eyed the bon-bon. \"If it's as dangerous as you say, do you think you should be tossing it around like that? I don't want to get blown up before I get to the ubervillains.\" That wouldn't help anybody, especially not Johnny, my father, and Henry.\n\nJasper waved his hand. \"Oh, in this form, it's perfectly harmless. You see, I've encased it in a special, highly protective plastic shell. You could even play baseball with it, and it wouldn't detonate.\"\n\n\"So how do I arm it?\" I asked, reaching for the small ball.\n\nJasper held it out of my reach. \"All it needs is a little heat to fire it up. The plastic disintegrates, and the obliteron becomes active. There's a twenty-second fuse, but I don't think you'll have to worry about that. A flare-up from you will more than do the trick. Along with a stabilizing agent, the outer shell also contains a form of heat-activated, superstrong glue. Once the shell starts to melt, it will stick to any surface you throw it at. So once it's hot, let it fly. When it blows, well, let's just say that you don't want to be anywhere near it.\"\n\nI thought about the observatory. Some of the rooms were cramped, and I had no idea where Siren and Intelligal would set up their VAMP device. I didn't want the others to get caught in the crossfire. \"What's the blast radius?\"\n\n\"Like explodium, obliteron produces a limited blast range, about thirty feet or so. But it packs a hell of a punch and it burns superhot. A couple of these balls would be more than enough to bring down any building, any structure in the greater Bigtime area.\"\n\nI eyed him. \"Like, say, the new paper mill they were going to build out next to the ice cream factory? The one that was going to infringe on part of the observatory's wildlife sanctuary? There was some sort of explosion out there a couple of weeks ago late at night when nobody was around. The damage was so severe that construction was halted. Permanently, I believe.\"\n\n\"I didn't have anything to do with that,\" Jasper said in a hurt, offended tone.\n\nI kept staring at him.\n\n\"I might not be as heroic as you are, Fiera, but I'm not a bad person. I might blow things up, but I never sell my goods to anyone who's going to use them to hurt other people.\"\n\n\"So who do you sell your wares to then?\"\n\nHe shrugged. \"People who want to collect on insurance, mostly. I also get a lot of business from the construction crews in town. A few of the more radical environmental groups. Sometimes, the bomb squad even calls me in to consult on cases. Now, do you want the obliteron or not?\"\n\n\"I do.\" I didn't really have a choice. Not if I wanted to save the others and the rest of Bigtime. Besides, I could always come back and bust Jasper later.\n\nJasper nodded. \"Good. Let me package them for you. Don't fire them up until you're absolutely sure you want to blow something up. Unlike some of my other work, these can't be diffused.\"\n\nHe rustled around the underground laboratory, fishing a heavy, lead briefcase out of one of the safes. Jasper carefully put ten of the obliteron bon-bons in the dark, foam-lined depths. Then, he turned and handed it to me.\n\n\"Let me carry them. Better not to take any more chances than necessary. I'm not as hot under the collar as other folks around here.\" Lulu shot me an amused look out of the corner of her eye.\n\nI gritted my teeth at the bad pun and tried to remain calm. Now was not the time to set Lulu's hair on fire. I would do that later. After I'd rescued everyone and reduced Siren and Intelligal to weeping, wailing heaps on the floor.\n\nLulu took the case from Jasper and set it on her lap. I eyed a clock on the wall. Just before six. Time to go.\n\n\"Ah, before you leave, there is the matter of my fee,\" Jasper reminded us in a soft voice.\n\nLulu and I looked at each other.\n\n\"How much?\" she asked.\n\nJasper tapped his finger on his lips. \"For ten of my obliteron delights? I'd say an even two million would cover it.\"\n\nI winced. There went my fat fee for designing Joanne James's latest wedding dress. Unless... The bomber wore a tattered bathrobe that had definitely seen better days. The slippers on his feet had holes in them, and his socks were threadbare. Plus, everything was the same, drab, gray color. Rather like Jasper's pale skin. I really, really hated gray.\n\n\"How would you like some clothes instead?\" I said. \"I happen to know a designer who does incredible work. She makes men's suits that are to die for. Very bold. Very colorful. Just the thing for you to entertain prospective clients in. Or to wear for a night out on the town with your lady. I'm sure she'd be happy to outfit you with a whole new wardrobe.\"\n\nJasper just blinked.\n\n* * *\n\nTen minutes later, Lulu and I left Jasper's brownstone.\n\n\"I can't believe Jasper agreed to waive his two million dollars for a closet full of _clothes_ ,\" Lulu said, maneuvering her wheelchair down the sidewalk. \"I think Jasper's taken one too many blows to the head recently. Or all the radioactivity is making him go soft.\"\n\n\"It's much simpler than that, Lulu. The truth is that everybody wants to look good,\" I said in a smug tone. \"Even mad bombers. And they're not just _clothes_. I plan on outfitting Jasper with the finest menswear Fiona Fine Fashions has to offer.\"\n\nBobby saw us coming. He hopped out of the front seat and opened the side door of the van. Too impatient to bother with the chair lift, I picked up Lulu and deposited her in the van, along with the precious case of Jasper's bombs.\n\n\"Did you get everything you needed?\" Bella asked from the driver's seat.\n\n\"And then some,\" I said, slamming the door shut. \"Now, let's go take those bitches out.\"\n\n# Chapter Twenty-Eight\n\nWe drove through the streets of Bigtime in silence. It was early, and most folks hadn't yet started their commute into the city from the outlying suburbs. Most of them probably hadn't even had their first cup of coffee yet. The sun was just creeping above the tops of the skyscrapers that housed _The Expos_ _\u00e9_ and _The Chronicle_.\n\nI took the opportunity to eat the emergency bucket of food I had stashed in the van. It was mostly junk food, stuff like candy bars and peanut butter crackers that would keep forever. But all that sweet, sweet sugar was more than enough to give me a boost of energy for the big battle ahead. I wasn't psychic like my father or Carmen, but I had a feeling I'd need it.\n\nWe pulled into the long drive that led up to the observatory a little after six thirty. We didn't see anyone. No cars climbing up the hill. No buses pulling in with sleepy, cranky students. Even the guardhouse at the bottom of the steep hill was empty, although the door looked like it had been ripped off its hinges, along with the gate that blocked the road up to the observatory. I wondered if that's what Siren had needed the others for **\u2014** a little extra muscle.\n\nBut other than the mangled gate, everything else seemed normal, and the place was deserted. Except for two maniacal ubervillains lurking around somewhere. Plus, my hoodwinked teammates and the man I loved.\n\nBella drove the van to one of the garages attached to the side of the observatory. She parked on the lowest level, out of sight from anyone who might be watching from inside. The only other vehicle in the garage was a small SUV bearing the colorful SNN logo. I rattled the door. Locked. I put a little muscle into it, yanked the whole thing off, and set the crumpled metal aside. I rustled through the interior of the vehicle, but there was nothing inside besides your usual assortment of bad CDs, gum wrappers, and empty fast-food cartons.\n\n\"Anything?\" Lulu asked, smacking her computer, which sat on top of Jasper's case of bombs on her lap.\n\n\"Nothing. But at least we know they're here,\" I replied. \"Something wrong over there?\"\n\n\"Stupid case,\" Lulu muttered. \"I'm trying to pull up the blueprints that I downloaded of the observatory, but the radioactivity from Jasper's bombs is interfering with my laptop. The case he gave us isn't quite as secure as it looks.\"\n\n\"Why don't you let me hold on to those?\" Bella asked, sliding the case out from under Lulu's computer. \"It might be better for us all.\"\n\n\"A little luck certainly couldn't hurt,\" I quipped. \"As long as it's good.\"\n\nBobby laughed. Bella glared at her grandfather.\n\n\"All right, folks, gather round,\" I said.\n\nMy three troops clustered near me.\n\n\"Okay, here's the game plan. I go in first. Every single time. If we run into Siren and Intelligal, you guys stay back out of sight. Let me take care of the ubervillains. If the situation gets really desperate, Bella can toss me a couple of the bombs so I can even things out. I said it before, and I'll say it again\u2014I don't want any of you trying to play hero. That's my job. Under no circumstances are you to engage Siren or Intelligal by yourselves. Agreed?\"\n\nI looked at each one of them in turn, giving them my _I'm-a-powerful-superhero-so-don't-even-think-about-messing-with-me_ look. It seemed to work, because they nodded.\n\n\"Okay then. Let's go get our boys back,\" I said.\n\n* * *\n\nWe took a few minutes to get our gear together. Lulu passed out the earplugs Henry and my father had been working on and hooked them into her laptop. Like our other equipment, the earplugs were equipped with microphones so we could all talk to each other. Then, Lulu piggybacked her computer onto one of the electrical boxes that lined the empty parking structure. She started to disable the alarm, but someone had already done it.\n\n\"Hermit,\" I said. \"That's what Siren needed him for. She probably got Mr. Sage and Johnny to move the gate, then let him do the rest.\"\n\nSince Hermit had done the hard part already, I opened one of the metal doors that led from the garage to the observatory, and we scuttled inside.\n\n\"Where do we go from here?\" I asked Lulu in a hushed voice.\n\nNormally, I would have gone in fireballs a'blazing, but I couldn't risk alerting the ubervillains to our presence. Not with the others following behind me like ducklings waddling after their mother.\n\n\"According to my calculations, they're probably in the main auditorium,\" Lulu whispered. \"That would be the most logical place to set up the VAMP device. And if I remember correctly, that's where SNN usually does their live feeds from.\"\n\n\"Then that's where we go.\"\n\nThe four of us made our way through the silent, empty halls. A few of the interactive displays flickered to life as we passed, spewing out facts about the observatory and other scientific babble about stars and planets and black holes. A couple of heated jerks from me, and the machines died a painful, fiery death. We stopped at the edge of one of the observatory's main crosswalks to catch our breath and get a little more direction.\n\n\"Time?\" I asked.\n\nBella checked her watch, which looked like a smaller, female version of Johnny's, complete with wings. \"Ten minutes until seven.\"\n\n\"We need to keep moving,\" I said, pushing away from the slick, marble wall.\n\nAfter some more slow going, we reached the auditorium. I moved the others back down the hall. Then I dropped down on my stomach, slid forward, and stuck my head around the doorjamb.\n\nWe'd guessed right. They were all inside. Siren, Intelligal, Johnny Angel, Mr. Sage, Hermit, and the same weary cameraman I'd seen the night of the observatory benefit. My eyes flicked over my fellow superheroes. They looked no worse for wear, even though their eyes and faces were as blank and smooth as sheets of glass.\n\nMy gaze lingered on Johnny, and my heart squeezed in on itself. He looked fine, physically. I just wondered what the toll would be when Johnny realized that Siren had roped him into doing her bidding. That he'd tried to kill me and take over the city on orders from one of the women he hated most in the world.\n\nAs for the ubervillains, they stood in the middle of the auditorium in all their foul glory. They must have felt supremely confident in their dastardly plan, because they didn't even have their masks on. I looked at Erica Songe and her geeky scientist sister, Irene. I'd been right on that count too. Stupid, stupid, stupid. Didn't they know something always went wrong with the best-laid plans of ubervillains? And here they were without their masks on, running around for the whole world to see.\n\nI shook my head. Sometimes, ubervillains just weren't as classy\u2014or as smart\u2014as they used to be. Frost certainly would never have tried to take over the world without wearing his mask. Neither would Scorpion or Mad Maria or any of the other colorful, certifiable characters I'd battled over the years. If things went wrong, as they so often did, then they wouldn't have their real identities to protect and hide them. Erica's, Siren's, ego was definitely as big as her overinflated chest.\n\nIntelligal fiddled with switches on the VAMP device, which sat on the auditorium stage. Her face was set in its usual dour expression. You'd think she would have been a little happier about her Frankenstein machine finally coming to life. Every once in a while, a loud squawk would fill the air. A series of wires snaked across the floor from the radio-like object and plugged into the SNN camera standing in the middle of the empty auditorium, along with some lights. I eyed the machines and wires. I'd definitely have to do something about those.\n\nAs for Siren, she used the fancy VAMP microphone to berate and bark orders to her hypnotized cameraman about the hot lighting ruining her makeup. At least, that's what she did when she wasn't cuddling up to Johnny and rubbing herself against him like a cat in heat. She purred something into his ear. The tramp. Johnny stared blankly ahead, as if he couldn't even feel Siren's slutty hands roaming all over him.\n\nI hoped he couldn't. More than that, I hoped he wasn't enjoying it, dazed though he was. My own hands clenched into fists. A couple of sparks shot up in the air. I was so going to enjoy reducing Siren to a crimson stain on the floor and throwing her ass in jail.\n\nI watched them for a few more seconds before sliding back out of the auditorium into the corridor.\n\n\"Well?\" Bobby asked in a low voice, his eyes bright. \"Are they in there? Is Johnny with them? How does he look?\"\n\n\"Easy, easy. They're all in there. Siren and Intelligal are fiddling with the VAMP machine. Johnny and the others are fine, if a little spaced out.\"\n\n\"So what's our next move?\" Bella asked, clutching the case full of bombs.\n\nI thought. Planning and strategy weren't my strengths, but improvising was. \"We'll do what we usually do. Or rather, what the Fearless Five usually do. I'll go in through the front door, and you guys circle around the back. Is there another way to get in there?\"\n\n\"There's another entrance on the far side,\" Lulu said, pounding keys. \"The one that leads out to the gardens.\"\n\nOf course. I flashed back to that wonderful night I'd spent with Johnny out next to the waterfall. I allowed myself to remember. All the hot kisses, the soft caresses, the whispered endearments. Then, I pushed those thoughts, those feelings aside. It was time to put on my game face. My angry, _I'm-going-to-kick-your-evil-ubervillain-ass_ game face.\n\n\"Here's what we're going to do. I'm going in through the front door and toss a couple of fireballs at them. While I've got Siren and Intelligal occupied, you guys go in the back and get the others out. Don't forget the cameraman. Stuff the extra earplugs in their ears. Maybe that will bring them around. If the ubervillains try to stop you, somebody throw the bombs at them and retreat with the others. Any questions?\"\n\nEveryone shook their heads no.\n\nI took a deep breath. \"Then, let's go.\"\n\n* * *\n\nLulu, Bella, and Bobby snuck off to skirt around the building and come in through the back of the auditorium. I gave them three minutes to get into position, then eased into the room, crouching behind a row of chairs. Intelligal was still fiddling with the VAMP machine. Siren stood nearby, checking her pale, flawless reflection in a small, compact mirror. I narrowed my eyes. Too bad the Pimpler wasn't here right now. He'd turn Siren's smooth face as red and ragged as the top of a pizza in a matter of seconds.\n\n\"How much longer?\" Siren asked.\n\nIntelligal checked a watch on her wrist. \"Five minutes until the SNN studio cuts to you.\"\n\n\"Excellent.\" Siren snapped her compact shut and fluffed out her black hair. \"Just think, in a few minutes, we'll own this city and everyone and everything in it. No more doing stupid live shots of superheroes flaunting their latest triumph. No more skulking around abandoned factories. No more taking insults from Bigtime's high and mighty. We'll be the ones in control. Permanently.\"\n\n\"Don't get ahead of yourself,\" Intelligal cautioned. \"Things could still go wrong.\"\n\nSiren waved her hand. \"You worry too much. Things won't go wrong. Karma Girl and Striker are nowhere to be found. We've taken care of Fiera, and Mr. Sage, Hermit, and Johnny Angel are under our control. \"\n\n\"Not completely,\" Intelligal snapped. \"They all balked when you wanted them to take off their masks.\"\n\nSiren's thick lips turned down in a poor imitation of a pout. Her lips had so much collagen in them that it didn't quite work. \"Yes, well, keeping your real identity secret is the very first thing you learn to do as a superhero or ubervillain. It's ingrained so deep in their subconscious even I couldn't break through that particular barrier with my hypnosis. At least, not yet. Don't worry, sister dear. We'll uncover their identities soon enough. And have the rest of the city's superheroes and ubervillains eating out of our hands. Let's talk about something more interesting. What's the first thing you're going to do when we take over the city?\"\n\nIntelligal spliced two wires together. \"Quit my miserable job here at the observatory. Perhaps use some of the other scientists as test subjects. I'm curious as to what the effect of explodium is on a normal human body. Angel's remains didn't give me much to work with.\"\n\nMy eyes went to Johnny. I thought I saw a bit of anger spark to life in his green eyes, but it was quickly swallowed up by the emptiness.\n\n\"And you, sister?\" Intelligal asked.\n\nSiren tapped a finger on her lips. \"I'm not sure. Perhaps I'll have a little fun with Mr. Johnny Angel here. He certainly has the equipment for it. And I plan on taking certain individuals down a few pegs from their lofty heights. Speaking of Johnnies, perhaps I'll start with Mr. Johnny Bulluci and his trashy cohort, Fiona Fine. What the world sees in that woman's fashion designs is beyond me. Yes, I think those two will be the first ones to feel the new city order under Siren.\"\n\nI smiled. That was my cue. Siren wanted a dose of Fiona Fine? Well, she was sure as hell going to get it. And then some. I straightened up and stepped into view.\n\n\"Oh, why wait another five minutes?\" I called out. \"Let's get the fun started now.\"\n\n# Chapter Twenty-Nine\n\nSiren and Intelligal's mouths dropped open so far that their teeth almost hit the floor.\n\n\"You...you...you're supposed to be dead!\" Siren shrieked. \"Frozen solid like the rest of those fish sticks!\"\n\nI winced at the high-pitched sound, grateful the earplugs muffled the worst of it. \"If I had a dollar for every time I'd heard that, I'd be even richer than I am right now. But enough chitchat.\" I took aim and lobbed a fireball at Siren. \"I've got some ubervillains to get rid of.\"\n\nSiren ducked down, and Intelligal soared up into the air. The fireball sailed right between the two ubervillains.\n\n\"You missed!\" Siren crowed in a loud voice.\n\nThe red-hot ball of fire streaked through the air like a meteor and slammed into the television camera in the middle of the auditorium. The wires and camera exploded like firecrackers. Metal and glass zipped everywhere. When the smoke cleared, all that was left of the camera and the large majority of wires attached to the VAMP machine was a puddle of melted black metal.\n\n\"No, I didn't.\" It was my turn to crow. \"Oops. Did I break your television camera? So sorry. I think your feed's been interrupted. Permanently.\"\n\n\"You're going to pay for that,\" Siren hissed.\n\n\"Bring it on, bitch.\"\n\nAn energy ball popped into Siren's hand, and she threw it at me. I ducked down, and the ball smacked against the back wall of the auditorium. Sparks flew everywhere, and static electricity gathered in my hair and on my fingertips.\n\nUnnoticed by the ubervillains, the door that led to the gardens behind them opened. Lulu motored into the room, followed by Bella and Bobby. Lulu clutched the open case full of bombs on her lap. The Bullucis ran over to the three frozen superheroes and the cameraman and stuffed the Siren-proof plugs into their ears.\n\nI threw another fireball, this time at Intelligal. She zipped out of the way, and the ball burst into flames on the ceiling. A display of planets hanging there went up like kindling. Somewhere in the distance, a fire alarm blared to life.\n\n\"Turn on the machine!\" Siren screamed. \"Now!\"\n\nIntelligal started to whirl her chair around, but I sent another fireball her way. She turned back to me. I kept lobbing my fireballs at the two ubervillains, trying to give the Bullucis enough time to get the others out the back door. Bella stuffed the earplugs in her brother's ears and shot me a thumbs-up. She half pushed, half dragged Johnny toward the door, where the others were waiting.\n\nIntelligal saw the sudden movement out of the corner of her eye. Her mouth dropped open for the second time in as many minutes. \"Get them, Siren! Don't let them get away!\"\n\nSiren spun, formed an energy ball in her hands, and hurled it through the air with a furious shriek. It seemed to grow in size as it sped through the room, as though all the ambient energy were attracted to it. The ball zoomed through the air toward the others. Right at Johnny's retreating back.\n\nMy heart froze in my chest. Even as I started running, I knew I wouldn't get there in time to save him. I wasn't going to be able to save Johnny. The man I loved was going to die.\n\nAgain.\n\nSuddenly, Lulu did the one thing I'd asked her not to\u2014she played the part of the self-sacrificing hero and zoomed in front of Bella and Johnny. Siren's energy ball hit her in the chest. Lulu's dark eyes lit up with an inner fire, and her arms and legs twitched wildly. Then, the computer hacker slumped over in her wheelchair. The metal case slipped from her lifeless fingers, and the bombs rolled over the floor like shiny, chocolate marbles.\n\n\"Lulu!\" Hermit screamed. He ran back inside the auditorium.\n\nThe sight of Lulu getting electrocuted also snapped the others out of their Siren-induced reverie. Mr. Sage and Angel stormed back into the room. I raced forward. Siren and Intelligal swiveled back and forth between us. They didn't like being in the middle of a superhero sandwich.\n\n\"Get the machine and let's get out of here!\" Siren screamed.\n\n\"I don't think so, bitch,\" I muttered.\n\nI put on an extra burst of speed and reached the VAMP machine the same time Intelligal did. A mechanical arm shot out of her chair and grabbed the device, ready to lift it into the trunk on the back of the chair. I grabbed the other side of the machine, latching on to the frame. The VAMP machine seesawed back and forth between us. In the background, I heard Siren battling the others. Energy balls zipped through the air. People screamed and shouted and cursed. Ash fluttered around like confetti.\n\n\"Oh, to hell with this,\" I said.\n\nWith a mighty roar, I yanked the VAMP machine toward me and drove my free hand through the metal casing at the same time. Glass tubes and bits of metal snapped deep inside the device, which let out something that sounded like the screech of a wounded animal.\n\n\"No, no, no!\" Intelligal screamed. \"Not my beautiful machine!\"\n\n\"Hell yeah, your beautiful machine.\" I grabbed some of the wires inside and yanked them out, along with my hand. Then, I reared back and punched another hole in the side of it. Intelligal's eyes bulged so far out of her head I thought they'd bust through her thick goggles.\n\n\"No!\" she wailed, sounding just like the broken machine.\n\nI let go of the device. Intelligal's chair floated down, and she hovered over the ruined machine. I knew from experience that she'd be out of commission for a few minutes. All the geeky, science types hated it when you destroyed their pet projects.\n\nI turned to the others. Bobby and Bella were pushing the unconscious Lulu out the door, while Hermit and Mr. Sage guarded their backs.\n\nBut Johnny wasn't retreating with the others. Instead, he stalked toward Siren, who shot energy ball after energy ball at him. Angel flexed and laughed as the bolts bounced harmlessly off his chest.\n\n\"Don't you know, you can't electrocute a rock?\" he snarled.\n\n\"Johnny!\" I shouted. \"Don't do it!\"\n\nHe didn't even looked at me. \"Stay out of this!\"\n\nI raced toward him, desperate to get to him before he got to Siren. My foot slipped on something, and I almost fell. Johnny shot ahead of me, reaching for Siren. I looked down. One of Jasper's bombs rolled past my feet. It kept going and going and going across the floor. The bomb stopped\u2014in front of the still-smoldering camera.\n\nFlames licked at it, and the plastic shell began to melt before my horrified eyes.\n\n\"Get out! Get out now! The whole place is going to blow!\" I screamed.\n\nWith a burst of speed, I grabbed Johnny's leather jacket and yanked him away from Siren. He struggled against me, but I was stronger. Thank heavens for superstrength. I picked him up, panting from the effort, spun around, and tossed him toward the door.\n\nMr. Sage did the rest. His eyes glowed, and he used his telekinesis to float a still-struggling Johnny outside. I looked over my shoulder. Siren and Intelligal hovered over the machine, trying to salvage it, oblivious to everything else. Including the bon-bon bombs rolling around on the floor.\n\n\"Get out now!\" I yelled. \"There are bombs everywhere!\"\n\n\"Go to hell, bitch!\" For once, Siren and Intelligal were in agreement. They screamed the epithet at me.\n\nWell, I'd tried. It was all I could do. My conscience was clear. I started running for the back door.\n\nBut I wasn't quite quick enough this time. A loud roar sounded behind me, followed by a rush of heat and fire. The shockwave from the bombs threw me forward. I slammed into the back wall of the auditorium and out into the gardens below. Darkness overcame me just before I hit the ground.\n\n# Chapter Thirty\n\nMy eyes fluttered open, and my dazed mind wondered what had happened. Where I was. Why the sun was searing my eyeballs. Then, I remembered. Ubervillains. Battle. Explosion. The usual.\n\nI was lying on the garden overlook on the back of the destroyed auditorium. The stone platform felt rough beneath my cheek. The rest of the fog faded from my mind. I wiggled my toes. Then, my fingers. Everything seemed to be in working order. Good to know. I focused on my arms and legs, moving them just a bit. Pain rippled through my body, but it wasn't anything I couldn't handle, being a superstrong superhero and all.\n\nI pushed myself up on my hands and knees. Debris fell off my back, and glass tinkled out of my hair and crunched under my fingers. I winced at the stinging sensation in my palms. Somehow, I made myself stand up and dug the plugs out of my ears. I groaned and staggered back as more pain ripped through my body. I'd be sore for a week because of this escapade. But that was the price of saving the city and the world yet again.\n\nI turned to look at the auditorium. Or what was left of it. The whole back of the building had been blown out, and nothing remained except the stone platform I was standing on and some smoldering bits of debris and rubble. Soot blackened the air, making it hard to breathe. Jasper had been right. His new bombs made Intelligal's explodium missiles look like toy sparklers. I'd have to come up with some extra-special designs for the bomb guru.\n\nMy eyes flicked around the empty, ruined shell of a room. Not a trace of the VAMP machine remained. There was a large, charred lump I thought might be Intelligal's almost indestructible chair. I didn't see any blood or body parts, though. I crept inside, careful of the rubble. Something glinted, and I walked over to it. A silver zipper, halfway undone, lay in one of the smoldering piles. I kicked it with my boot, and it disintegrated.\n\n\"Fiona! Fiera! Fiona!\"\n\nI winced at the loud voices. They didn't mesh so well with the harsh buzzing in my ears. I stepped outside to find the Bullucis scrambling up the broken stone steps.\n\n\"Fiona! Are you all right?\" Bella asked, putting her hand on my shoulder.\n\n\"Fine, I'm fine. Just a little shaken up.\"\n\nI looked down the steps, expecting to see Hermit, Mr. Sage, and Lulu at the bottom. But they weren't there, and they didn't appear.\n\n\"Lulu?\" I asked, feeling more concerned than I'd thought possible. The computer hacker wasn't so bad if she'd only stop peppering me with her cheesy, heat-related puns.\n\nBobby shook his head. \"She's in pretty bad shape. Mr. Sage sent us to come get you so he and Hermit could work on her. She needs to be taken back to the manor as soon as possible.\"\n\n\"I'm just glad you're all right, Fiona,\" Johnny said, pulling me into a rough, tight hug.\n\nI closed my eyes, savoring the feel of his warm body against mine. I pulled back and just looked at him. Johnny. He was alive, and the ubervillains were gone. Everything was going to be all right. I grabbed Johnny's face and pulled it down to mine. Then, I gave him the hottest kiss I could manage. I lost myself in the feel of him, the smell, the taste.\n\n\"Ahem.\"\n\nI ignored Bella.\n\n\"Ahem!\"\n\nIf she did that any louder, she was going to bring the rest of the observatory down around us. Reluctantly, I broke off the kiss. Johnny brushed my hair back from my face. I touched his lips with my fingers. His green eyes softened. Suddenly, he grinned.\n\n\"Now that's how I like to be greeted,\" Johnny said, reaching for me again.\n\nI stiff-armed him and stepped back. \"Hold it, buster. It's going to take a lot more than one kiss to get back in my good graces.\" I tossed my hair over my shoulder. Dust flapped off the limp, dirty locks.\n\nJohnny stuck out his lip in a mock pout. Bella and Bobby just laughed.\n\n* * *\n\nMy happiness at saving Johnny and the day was muted by the seriousness of Lulu's condition. Siren's energy ball had done quite a number on her. In addition to the burns on her body, the electricity had also interrupted her heartbeat and brain activities. The chief had to shock her twice with the defibrillator in the back of the van when she flat-lined. We raced back to Sublime in silence, and Henry and the chief put her in one of the sick bays moments after we parked in the underground garage.\n\nThe Bullucis and I stood vigil outside the room, watching the chief work on Lulu. An hour later, he came out. The chief looked exhausted, and worry lines tightened his long face.\n\n\"She's stabilized, but it's still touch-and-go. If she makes it through the night, then I think she'll be all right. Henry's in there with her. I suggest you all go home and get some rest. There's nothing you can do at the moment.\"\n\nOne by one, we drifted away. Johnny took his sister and grandfather home to change clothes and freshen up. The Bullucis promised to return as soon as they could. I made my way to my suite, stripped off my tattered, ruined costume, and took a long shower to wash away the grime of the fight and explosion. I changed into jeans, stilettos, and a fitted white shirt.\n\nI lay down on my bed, but I couldn't sleep. Even though I'd been up more than twenty-four hours, I was still too wired from everything that had happened. So, I got up and went to the kitchen, where I made a dozen roast-beef-and-cheddar-cheese sandwiches. I put four aside for Henry and the chief and downed the rest of them, along with a gallon of milk and a deep-dish apple pie topped with vanilla bean ice cream. My father found me inside just as I was scraping the last bite of melted ice cream out of the bowl.\n\n\"Sorry,\" I said, pushing him the plate full of sandwiches. \"If you wanted pie, you should have gotten here three minutes ago.\"\n\n\"Don't worry about it,\" he said, taking one of the sandwiches. \"I'm used to it.\"\n\nAfter the chief finished his first sandwich, he spoke again.\n\n\"You've been incredibly strong these last few hours. Escaping the freezer, figuring out the ubervillains' scheme. You saved us, Fiona. You saved all of us today. I don't think I've ever been prouder of you,\" the chief said, his eyes bright.\n\nI nodded. \"I only wish Lulu hadn't gotten hurt in the process. Is there any change?\"\n\nThe chief shook his head. \"We'll just have to wait and see. It's out of my hands now.\"\n\nHot tears gathered in my eyes. My father opened his arms, and we hugged for a long time.\n\n* * *\n\nThe chief left to go to his own suite and shower before he checked on Lulu again. I went down to the sick bay to give Henry a break. The doors swooshed open, and I stepped inside. The harsh, chemical smell of disinfectant and ointment filled the room. I walked over to the bed where Lulu lay.\n\nThe computer hacker's face was paler than usual, and her hair stood out like black-and-blue ink against the white of her pillow. Thick gauze bandages covered her chest, and she seemed about an inch away from death. Henry sat by her side, still in costume, holding her hand and whispering words of encouragement.\n\nI tapped him on the shoulder. Henry yelped and almost jumped out of his chair.\n\n\"Oh, it's you, Fiona. You scared me.\"\n\nI pulled up a chair and sat next to him. \"How is she?\"\n\nHenry sighed, his eyes dark and sad. \"The same. No worse, no better.\"\n\n\"Why don't you go get some rest? I'll sit with her until you get back,\" I said in a gentle tone.\n\n\"No, I'm not going anywhere. Not until she wakes up.\"\n\n\"You're dirty and exhausted. You're not any good to Lulu right now.\"\n\nHenry looked down at his ripped, torn costume. He shifted in his chair, and bits of ash flaked off the spandex. \"So what?\"\n\n\"So, you look almost as bad as she does. You certainly smell worse. Now go.\" I grabbed the back of his suit and pushed him toward the door. \"I'll watch her while you get cleaned up. I promise.\"\n\n\"Come get me if anything, anything at all, changes,\" Henry said, clutching the doorframe.\n\nI pulled his fingers off, but he grabbed on to the other side. \"I will. Now go.\"\n\nAfter a few more false starts and forceful shoves, I convinced Henry to leave Lulu to me for a little while. I plopped down in the chair beside her bed. After about a minute, I drummed my fingers on my knee. Shifted back and forth in my seat. Fiddled with my hair. If I'd had some gum, I would have blown big bubbles. I'd never been good at sitting still, and I absolutely hated waiting.\n\nBut I'd promised Henry. Plus, it was my fault that Lulu was hurt. I was the one who'd let her come along. I should have locked her and the Bullucis in the van where they would have been safe. Instead, Lulu had gone all noble on me and kept Siren from zapping Johnny. I should have been able to take out Siren before she got that shot off. I should have been quicker, smarter, stronger. Now, the computer hacker was paying the price for my mistakes\u2014and her own bravery.\n\n_Coulda, woulda, shoulda_. Unfortunately, after-the-fact clarity was another thing that went along with being a superhero.\n\n\"You know, you need to snap out of this coma thing you've got going on,\" I said in a conversational tone. \"It's not doing you any good, and you've got everybody else worried sick about you. Especially Henry. If you go and die on him, well, I'll have to kill you all over again.\"\n\nI rolled my eyes. I sounded dumb even to myself. Perhaps dying wasn't the best thing to talk to Lulu about. A good bedside manner was not my specialty. That was Chief Newman's department.\n\nSo, I sat there and talked to Lulu about all sorts of things. The battle, how Siren and Intelligal had been toasted by the bombs, how brave but stupid I thought she'd been saving Johnny, how I hoped that Johnny and I could get past today.\n\n\"I really, really like him,\" I said. \"And not just because the sex is incredible and he's one of the most gorgeous men I've ever had the privilege to sleep with. I like his sense of humor, his quick wit, his silly grins.\" I leaned forward and dropped my voice to a whisper. \"The truth is that I love him. But let's just keep that between you and me for right now.\"\n\nAfter going on for a good while about my feelings for Johnny, I lapsed into silence. I didn't have any more words left in me. So, I clutched Lulu's hand and tried to feel soothing and calm.\n\nI sat beside her, my head bowed, for almost an hour. A small rustle sounded. My head snapped up. Lulu's eyes fluttered open. My breath caught in my throat. This was a good sign, right? I mean, she wasn't going to die on us if she was awake, was she?\n\n\"Henry?\" Lulu croaked, her voice weak and raspy.\n\n\"Hey,\" I said, leaning over so she could see me. \"Henry's not here right now. It's me, Fiona. You're back at the manor in one of the sick bays.\"\n\n\"Henry?\"\n\n\"Henry's fine, and so is everybody else. The ubervillains are dead.\"\n\nAfter a moment of thinking and staring at me, Lulu's eyes cleared. \"Fiona. Of course. That's why it's so hot in here.\"\n\nI rolled my eyes. \"You must be getting better if you can crack bad jokes like that.\"\n\nLulu chuckled. At least she tried to. She gave up, gasping for air.\n\n\"Easy, easy. Don't overdo it.\"\n\nLulu nodded. \"I'll try. What happened? The last thing I remember is this big blue ball of energy slamming into me.\"\n\nI filled her in again on the battle at the observatory.\n\n\"I'm glad those bitches are dead,\" Lulu said. \"My chest feels like it's on fire.\"\n\n\"Do you want me to get the chief? I'm sure he can give you some more painkillers.\"\n\nLulu shook her head. \"No, I'm fine. Really sleepy, actually. Where's Henry?\"\n\n\"Get some sleep. He'll be back in a few minutes.\"\n\nLulu relaxed. \"You were right about one thing, Fiona.\"\n\n\"Really? What was that?\"\n\n\"Me being in a wheelchair and having kids or not having kids isn't that important. All that matters is Henry and me and how we feel about each other.\"\n\n\"So, does this mean there'll be another wedding around here in the near future?\" I asked. \"Because I know this designer who does really fabulous gowns.\"\n\n\"You can count on it,\" Lulu said, drifting off. \"Tell Henry I said _yes_... _yes_... _yes_...\"\n\nI smiled. \"I'll be sure to relay the message.\"\n\nLulu's eyes slid shut before I finished speaking.\n\n* * *\n\nLater that afternoon, the chief announced that Lulu had stabilized and that she should make a slow, but full, recovery. The computer hacker drifted in and out of consciousness the rest of the day. She came awake long enough to tell Henry that she wanted to marry him, then promptly fell asleep when he leaned over to kiss her. Ah, love among nerds. It was a beautiful thing.\n\nThe Bullucis returned that night bearing pasta, wine, and more. After a brief, somewhat subdued celebration in Lulu's room, which the guest of honor slept through, we moved the party to the library. Henry went to check on Lulu every few minutes, while my father entertained the Bullucis with superhero stories.\n\nBut I had a different sort of party in mind. One that involved only two people\u2014Johnny and me. I walked over and drew him away from the others.\n\n\"Let's get out of here,\" I said in a low voice. \"We need to talk.\"\n\nJohnny nodded. He put down his champagne, and we slipped out of the library.\n\n\"Where are we going?\" he asked.\n\n\"You'll see.\"\n\nI dragged Johnny down the halls until we reached my suite. I threw open the door, pulled him inside, and shut it behind us. Then, I wrapped my arms around his neck and lifted my lips to his.\n\nI loved him. I really, really loved him. I'd admitted it to myself and Lulu, and now I was going to tell Johnny. Somehow, some way in the last few days, I'd fallen in love with him. Travis would always be a part of me, always have a piece of my heart. But I was ready to get on with my life. And I wanted to do it with Johnny Bulluci. Mr. Right Now had turned into Mr. Forever.\n\nThe kiss went on for a long time. I tried to take a few steps to the side, where the bed was oh-so-conveniently waiting for us to make wild, crazy love in it. But Johnny wasn't cooperating. The kiss ended, and he pulled back.\n\n\"Is something wrong?\" I asked, eyeing the bed. If I could just maneuver him a few more feet to the right\u2014\n\n\"I've been thinking a lot today. About you, me, my father, Angel, everything. I...we...we can't be together, Fiona,\" he said in a low voice.\n\nAll thoughts of the bed and what we could do in it fled. \"What? Why?\" My voice came out as more of a shriek than a wail.\n\nJohnny ran his hand through his tawny hair. \"Because of Siren and Intelligal. Because...of everything.\"\n\n\"But it's over now,\" I protested. \"We stopped them. The ubervillains are dead. They'll never bother us again. Justice has been served.\"\n\n\"Thanks to you. You saved us all, Fiona. Your father, Henry, me, Lulu, my family. All I did was almost get myself and you killed.\" Johnny's eyes were dark and troubled. His gaze wandered around the room, settling on something over my shoulder.\n\n\"But\u2014\"\n\n\"No buts. You were right. This whole time, you were right. I let my need for vengeance blind me. I acted like a reckless, selfish fool, and I almost killed you in the process. I let an ubervillain take over my mind and make me her puppet. I put you in a fish freezer and left you to die.\"\n\n\"But you helped me escape,\" I pointed out. \"You gave me your lighter.\"\n\nJohnny's mouth twisted. \"Too little, almost too late. How can I expect you to forgive me for that? I can't even forgive myself for hurting you. How can we be together after everything that's happened? I'm sorry, Fiona. I don't deserve you. I never have, and I never will.\"\n\nJohnny stared at me, as if memorizing the curves of my face. Then, he opened the door and left the room.\n\nLeaving me alone.\n\n# Chapter Thirty-One\n\nTo say that I spent the next few days in a bad mood would be the understatement of the year. The century even. After Johnny left me, I stayed in a perpetual pissy state. I alternated between crying, swimming, and growling at everyone who crossed my path.\n\nAnd eating. I ate everything I could sink my teeth into. I always ate more when I was heartbroken.\n\nAlmost three weeks after the battle at the observatory, Carmen and Sam returned from their honeymoon. They looked rested, tanned, and more in love than ever. Their happiness and Lulu and Henry's engagement only made me more painfully aware of what I could have had with Johnny if he hadn't been such a hard-headed, stubborn ass.\n\nI tried to get through to him, of course. I called him and sent gag gifts and even showed up at the Bulluci mansion with dinner from Quicke's. But Johnny didn't take my calls. Didn't return my gifts. Didn't even acknowledge my visit.\n\nIt was a problem I couldn't fix by using my fists or fireballs or general fabulousness. I didn't know what to do.\n\nCarmen, being the unbearable, cheerful newlywed, made light of my disastrous love life.\n\n\"He'll come around eventually, Fiona,\" she murmured, her eyes vacant the way they always were when she was listening to the voices in her head. \"I just know he will.\"\n\nStupid voices in Carmen's head. They gave me a shred of hope. But as the days passed and Johnny didn't respond to my reconciliation attempts, my depression only grew. Four weeks after the incident in my suite, I paced around my office. Brooding as usual.\n\nA knock sounded, and Piper entered the room. She set a bag of doughnuts down on my desk. I tore into them like a dog snapping at a bone. Sugar was always good for a broken heart.\n\n\"Do you want to tell me what's bothering you?\" Piper asked.\n\n\"It's nothing a couple dozen of these babies won't fix,\" I said, shoving a chocolate-glazed pastry into my mouth.\n\n\"If you say so.\" Piper leaned against the doorway. \"Well, now that the fall line has finally been shipped out to our suppliers, I had a chance to look over your sketches for spring.\"\n\nI stared at her, another sticky doughnut halfway to my lips. \"What sketches?\"\n\nPiper held out a stack of papers. \"These sketches. Don't you remember?\"\n\nI took the papers from her and flipped through them. My heart sank. They were the drawings I'd done the first time Johnny had sent me flowers. Now, they just reminded me of what I had lost and would never have again. I tossed them aside and ate another doughnut, a cream-cheese-filled one this time.\n\nPiper flipped through the pages of discarded drawings. \"I love it, Fiona. The color, the patterns, everything. I think it will be one of your best lines ever.\"\n\nI snorted. \"It's crap, Piper. Garbage. Those designs don't have any edge, any real style. I'm going to totally redo the spring line for next year. I'll do something bold, something daring, something...in black, I think.\"\n\n\"Black? You only use black when you're depressed about something.\" Piper's eyes narrowed. \"Or when you have man troubles. What's his name and what's he done to you, Fiona? It's Johnny Bulluci, isn't it?\"\n\nJust hearing his name was painful. \"What makes you think that?\"\n\nPiper sighed. \"Because you've eaten half a dozen doughnuts in the space of about two minutes.\"\n\n\"So?\" I asked, popping another one of the sugary treats into my mouth. \"What does that prove?\"\n\n\"It doesn't _prove_ anything. But it's a bit of a record, even for you. You can talk to me, Fiona. I'm more than your business partner. I'm your friend too.\"\n\nPiper had such a sweet, earnest look on her face that it made me sigh.\n\n\"I know,\" I said, licking a bit of glaze off my finger. \"And I appreciate you wanting to help me through this. But honestly, the only thing that eases the pain is food.\"\n\nI reached into the bag for another doughnut. My hand clutched at air. I peered into the bag. Empty already.\n\nDamn.\n\n* * *\n\nThat night, I headed out to the manor, since it was my turn to be on call. Superhero duties went on, broken heart or no broken heart.\n\nI wandered into the library to find Lulu sitting inside. The computer hacker had recovered from Siren's energy blast, except for the burns on her chest and arms. Even those would heal with time, which meant Lulu was back to compiling information and weaving her web of wickedness on the Internet.\n\n\"Where are the others?\" I asked.\n\n\"Carmen and Sam had some society benefit to go to, and Henry and the chief had to work late.\" Lulu didn't even glance up from her monitor.\n\n\"Oh,\" I said, feeling deflated. I'd hoped somebody would be around. I'd wanted to go a few rounds in the training room with Sam to burn off some of my pent-up anger and frustration.\n\nLulu heard the sadness in my tone. \"Any word from Johnny?\"\n\nI shook my head and started pacing. \"Of course not.\"\n\nBored, I grabbed a Rubik's Cube off Carmen's desk in the corner and tossed it back and forth in my hands. I put it down when the plastic started to melt. Carmen got a little touchy about people messing with her stuff. But the cube was already too far gone. It slid off the desk and hit Lulu's knee before falling to the floor and soaking into the Persian rug.\n\n\"Watch it!\" Lulu said, rubbing her knee. \"Some of us aren't made out of steel, you know.\"\n\nI looked at the melted plastic, then at Lulu's knees. What the hell? My eyes widened.\n\nI leaned over Lulu and hit her leg. I didn't give her the ole Fist-o-Might, but I smacked her hard enough so that it would hurt.\n\n\"Ouch! What did you do that for?\" Lulu glared at me.\n\nI hit her again.\n\n\"Fiona! What the hell are you doing?\"\n\n\"I'm hitting your leg. Can you feel it?\"\n\n\"Hell yeah, I feel it, and it bloody well hurts\u2014\"\n\nLulu's eyes got as big and round as balloons. \"I can _feel_ it. I can feel you hitting _my leg_. Do it again, Fiona! Do it again!\"\n\nI was happy to oblige her.\n\n\"I can feel it! I can feel it! I can feel it!\" Lulu screamed.\n\nWe started laughing. I kept hitting Lulu, and we both kept laughing and screaming and crying until the chief and Henry walked in an hour later.\n\nThey thought we were insane. At first.\n\nOnce I explained why I was playing pin the fist on Lulu, the chief whisked her away to do some tests. A few hours later, we gathered in the sick bay to get the results.\n\nThe chief pointed to some X-rays he'd taken of Lulu's back and said a bunch of scientific mumbo-jumbo about neurons and electricity that I mostly tuned out. I loved my father, but he could be such a bore sometimes, especially when he was in doctor mode.\n\n\"Oh, get to the point, Dad,\" I snapped.\n\n\"Please, chief,\" Lulu begged. \"What does it all mean?\"\n\n\"Well, I can't be sure without running an extensive battery of tests, but it seems that Siren's energy ball has kick-started the dead nerves in your spine.\"\n\n\"And what does that mean?\" Lulu asked, clutching Henry's hand so hard I thought it would pop off.\n\n\"It means, my dear, that one day, I think you'll be able to walk again,\" Chief Newman said.\n\nThere was complete silence.\n\nThen, we all started screaming.\n\n* * *\n\nAs it turned out, the chief was right. Siren's energy bolt had done a number on Lulu. The huge amount of electricity she'd been hit with had fired up the synapses and nerves and other things that make up a person's spine. In short, Siren had jumpstarted Lulu's body into healing the damage done when she'd broken her back a few years ago. With massive amounts of physical therapy, she should be able to walk again.\n\nThe next night, the members of the Fearless Five gathered in the library to celebrate the good news with champagne and chocolate. I stood a little apart from the others, watching them laugh and talk and celebrate. I was happy for Lulu, truly I was, but I still felt sorry for myself. I wanted Johnny here to celebrate with us. I wanted...I just wanted _him_. Always. Forever.\n\nCarmen detached herself from Sam and strolled in my direction. I stifled a groan and downed the rest of my champagne. I didn't have to be a mind reader to know that another unwanted probe of my psyche was coming up. Carmen stood beside me, sipping from her own glass of champagne.\n\n\"It's wonderful, isn't it?\" she murmured.\n\n\"It's just ducky.\"\n\nShe sighed. \"I'm sorry you're hurting, Fiona. I really, really am. But you don't have to take it, you know. You can always do something about your situation.\"\n\n\"What do you want me to do? Beg Johnny to love me?\" I snapped. \"Johnny has made it perfectly clear he never wants to see me ever again.\"\n\n\"You're a fighter. You're Fiera, for crying out loud. Protector of the innocent and all that. So do what you do best. Fight for your man, Fiona. Fight for Johnny.\"\n\nCarmen moved back to Sam. He wrapped his arm around her waist, and she leaned her head on his shoulder.\n\nSuddenly tired, I said my good-nights to everybody and went to my suite. I flopped down on my bed, grabbed the picture of Travis, and stared into his dark, smiling eyes.\n\n\"It was always so easy with you and me,\" I said. \"Why can't it be that way with Johnny?\"\n\nI stroked his face with my fingers, but Travis didn't have any answers for me.\n\nMy fingers stilled. Or maybe he did.\n\nThe truth was things hadn't always been easy for Travis and me. Sure, we'd loved each other, but we'd also had our share of fights and arguments and problems just like everybody else did. But we'd worked through them all. Together.\n\nI thought about Carmen's words, about how I should fight for what I wanted. I thought about Johnny and how he made me feel. The way he listened to me, laughed with me, loved me.\n\n\"She's right,\" I whispered to Travis. \"I hate to admit it, but she's absolutely right.\"\n\nIt was time to fight. Time for a new beginning.\n\nStarting right now.\n\nI went over to the dresser, shoved aside some of my fashion magazines, and gently put Travis's picture down in the clear spot. He looked a little strange, a little out of place sitting there, but I knew I would get used to it.\n\nI'd gotten used to the pain of his loss, and I'd found new love in the process. Johnny would never take Travis's place, but he had an equal share of my heart. He was here now, and we could have a life together. A fabulous life.\n\nAll we had to do was fight for it.\n\nI'd never been very good at waiting, and I'd never been a quitter. I wasn't going to give up on Johnny now. Not now, not ever. Miracles really did happen. Lulu was proof of that.\n\nNow, it was time to make a miracle of my own.\n\n# Chapter Thirty-Two\n\nI placed a few strategic phone calls to recruit some spies and put my plan into action the very next night at Paradise Park.\n\nAfter a leisurely dinner at Quicke's, Bella, Bobby, and Johnny Bulluci strolled into the park at exactly eight o'clock. While Bobby yammered into Johnny's ear about something, Bella flashed me a discreet thumbs-up as they walked by. Everything was in place and right on schedule.\n\nI swallowed the rest of my raspberry-flavored cotton candy and followed the Bullucis as the three of them wandered around the park, keeping a good distance between us. I was also wearing a floppy hat, oversized sunglasses, and a tight-fitting, black trench coat so Johnny wouldn't spot me. Bella and Bobby chatted and laughed and even played a few of the carnival contests. Bella won every game she played, even the rigged ones. Bobby bought a large funnel cake from one of the vendors and ate all of it, despite his granddaughter's dire warnings and hot glares.\n\nJohnny just looked pained the whole time. I wondered if he was remembering our first date here. I hoped so. It was why I'd chosen the spot.\n\nBella pulled Johnny over to the enormous Ferris wheel. The ride stopped, and people filed off. At first, Johnny balked, not wanting to ride, but she whined and pleaded and begged until she got him to the front of the line. I elbowed people aside until I was right behind them. Meanwhile, Bobby went over and talked to the operator, whispering in his ear and slipping him some money.\n\nJohnny sat down in one of the swinging carts, scooting over to the far side. Bella started to get in next to him, but stopped.\n\n\"You know what? I just remembered there's somewhere else I need to be,\" Bella said.\n\nShe stepped aside. I tossed my hat and sunglasses to Bobby and took her place.\n\n\"Hi, Johnny. What's up?\" I slid in next to him and yanked the bar down over us.\n\n\"Sorry, folks. This is a private ride,\" the operator said to the other people in line.\n\nA groan went up through the crowd. The ride started, and we sailed into the air before Johnny could protest, much less get off. Bella and Bobby waved to us once and then disappeared to check out the rest of the park. Johnny didn't say anything, but his eyes looked frantic and confused. I smiled. It was always good to keep a man guessing.\n\nWe went round one time as the calliope music played. Then, just as we crested the top the second time, the wheel jerked to a halt. We dangled in the air, high above the shrieking carnival goers.\n\n\"Why are we stopped?\" Johnny asked. \"Did you have something to do with this?\"\n\n\"Of course I did. So did Bella and Bobby. It was really a team effort.\"\n\nJohnny sighed. \"So how long are we stuck up here?\"\n\n\"Oh, we're not coming down until you tell me why you dumped me,\" I said in a cheerful tone.\n\nJohnny stared at the city lights around us. \"We've been through this before, Fiona.\"\n\n\"Johnny, I don't care about the past. I only care about the future and you.\"\n\nHe seemed surprised. \"You don't care that I attacked you? You don't care that I was a complete ass to you and the rest of the Fearless Five? You don't care that I tried to kill you? You don't care that I let my need for vengeance blind me to everything else? You don't care\u2014\"\n\nI put a finger on his lips. \"Of course I care. But it's nothing that we can't work through. Together. I'm willing to try. Why aren't you?\"\n\n\"Because I'm not a superhero. I never have been, and I don't know that I can ever be one. You're Fiera, a member of the Fearless Five. You do good things, important things. I'm just Johnny Angel. A guy who rides around the city on a motorcycle and looks cool because it's what his father used to do, and his grandfather before him.\"\n\n\"I don't care whether or not you're a superhero. As for being Johnny Angel, it's part of your family legacy, and that's part of who you are. I understand that, and I don't want to change it.\"\n\n\"What about Travis?\" he asked in a soft voice.\n\n\"What about him?\"\n\n\"You're still in love with your dead fianc\u00e9, Fiona.\" Johnny looked away. \"You still have his things. You still have his picture by your bed. You still wear his ring.\"\n\nI thought back to my conversation with Bella in the kitchen when she'd accused me of the same thing. The constant looks Johnny shot at my ring and Travis's picture. It all became clear to me. Johnny thought I didn't love him. That's what he was really afraid of. He'd told Bella that, which is why she'd warned me to be careful with him, with his feelings.\n\nI'd made my peace with Travis and his death. Now, it was time for Johnny to do the same.\n\nSo, I did something I'd never done before\u2014I took the engagement ring off my finger. A white band marked its place on my hand.\n\n\"I love Travis. I always will. But there's plenty of room in my heart, and it's time to move on. That's what I want to do. With _you_. The man I love.\"\n\nJohnny's head snapped up. \"The man you love?\"\n\nI nodded and stuffed the ring in my pocket. \"The man I love. That would be you, stubborn, rock-headed ass that you are. I love you, Johnny Bulluci. Do you believe me? Or am I going to have to beat it into you?\"\n\nJohnny's eyes lit up. \"No, you don't have to beat it into me. At least, not anymore. I love you too, Fiona.\"\n\nWe kissed, and everything was fabulous. Just the way it should be.\n\nJust the way I was going to make sure it would always be.\n\n\"Have you ever made love on a Ferris wheel?\" Johnny murmured, pressing a kiss to the hollow of my throat.\n\nI cupped his head in my hands and stared into his magnificent eyes. Passion, love, and more than a little wickedness glinted in the green depths.\n\nI laughed. \"A Ferris wheel? Aren't you the adventurous one.\"\n\nJohnny grinned that sly, crooked grin I loved so well. \"Always, baby.\"\n\nHe leaned over and looked down at the people below. \"Exactly how long did you tell the operator to leave us dangling up here?\"\n\nI smiled. \"Thirty minutes.\"\n\n\"Well then, we've got plenty of time, haven't we?\" Johnny said and reached for me again.\n\nWe didn't come down for a very, very long time.\n\n# Epilogue\n\nThree months later\n\nJohnny and I lay sprawled on the floor in the underground kitchen at Sublime. I put my arm over the beach towel covering our bodies and snuggled closer to him.\n\n\"Super-duper once again, Mr. Bulluci,\" I purred, nibbling on his ear.\n\n\"I agree. Super-duper once again, Mrs. Bulluci.\"\n\nI flashed the square diamond on my hand at him. It was almost as big as Joanne James' was and more precious to me than anything\u2014except Johnny himself. \"Not Mrs. Bulluci just yet. We only got engaged last week. And I'm keeping my name. Or at least hyphenating it.\"\n\nJohnny grinned. \"Fiona Fine-Bulluci. I like the sound of that. Especially the last part.\"\n\nJohnny put his arm around me and pulled me closer. I rested my head on his shoulder, happy, content, and suitably satiated. After taking a dip in the pool, we'd come to the kitchen for a midnight snack. But, of course, things had gotten a little heated between the two of us, especially since I made a point to keep a couple of cans of chocolate whipped cream in the refrigerator at all times now.\n\nThe door to the kitchen creaked open, and a walker appeared, followed by Lulu. She shoved the metal device forward and dragged her feet toward it. Lulu couldn't walk on her own yet, but she was getting a little stronger every day. Her hands clenched around the frame, and her face was set and determined. Henry followed along behind her.\n\n\"Why don't you just use your wheelchair? Just for tonight?\" Henry asked.\n\n\"No,\" Lulu snapped. \"Using the walker is part of my physical therapy. You know that. Besides, I made a promise to myself that I was going to walk down the aisle at our wedding, and I intend to keep it. No matter how much it might hurt right now.\"\n\n\"But Lulu\u2014\"\n\nThe words died on Henry's lips as he caught sight of Johnny and me lying on the floor. Lulu spotted us a second later.\n\nLulu put a hand over her eyes. \"Don't mind us. We just came in to get some coffee. We don't see a thing.\" She peeked between her fingers. \"Well, not much. Nice abs, Johnny.\"\n\n\"Thanks, Lulu.\"\n\nI not-so-gently punched him in his perfect abs. Johnny just grinned. Henry stared at us while Lulu rustled around in one of the refrigerators.\n\nHenry straightened his ever-present, polka-dot bow tie. Then, he turned to Lulu and picked her up.\n\n\"Henry!\" she squealed. \"What are you doing?\"\n\nHe pressed a quick kiss to her lips. \"I don't think I'm in the mood for coffee anymore. Are you?\"\n\nLulu looked at Henry, then down at us.\n\n\"Ohhh,\" Lulu drawled. She wrapped her arms around his neck. \"Well, when you put it like that, I think I've had enough coffee tonight too.\"\n\nHenry headed for the door. Lulu waved at us.\n\n\"Hey, Fiona,\" she called out just before the door swung shut behind them. \"Try not to melt the floor tiles again, okay?\"\n\nI sat up, fingers sparking, ready to light Lulu's hair up like a firecracker. But before I could focus my power, Johnny pulled me back down on top of him. His hands moved down my body, and I started thinking about a different sort of fire.\n\n\"Let them have their fun, baby,\" he whispered, stroking me. \"So we can have ours.\"\n\nI threw my head back and enjoyed the liquid fire coursing through me. Then, I reached for the almost-empty can of whipped cream we'd dropped in our haste.\n\n\"Well, let the good times roll,\" I said.\n\nExcerpt from\n\n# JINX\n\nby Jennifer Estep\n\nBook Three in the Bigtime series\n\nPART ONE\n\nI HATE SUPERHEROES\n\nChapter One\n\nDinner with superheroes.\n\nIt's an interesting experience\u2014and one that I rather hate.\n\nThe empty wineglass floated past me, sailing along as though carried by a steady, invisible hand. I tried to pretend it wasn't there. That I didn't see it. That the glass was as invisible as the force propelling it forward. But that was hard to do since it landed on the table next to me.\n\nI further tried to pretend I didn't see the crystal carafe beside my elbow rise up, tip itself over, and pour ruby-red sangria into the waiting glass. I even tried to convince myself I didn't _really_ see the glass float back across the table.\n\nI failed miserably at all three.\n\nThe other people gathered round didn't pay any attention to the floating glass. Didn't slow their conversation or ignore their food for an instant.\n\nUnfortunately, floating glasses had become a normal sight around the Bulluci household these days\u2014no matter how I wished otherwise.\n\n\"Is that really necessary?\" I asked, my voice a little snappish. \"I would have been happy to pour you some more wine.\"\n\nChief Sean Newman held out his hand, and the glass drifted over to him. \"There was no need to bother you, Bella, when I could do it for myself.\"\n\n\"But you could have just asked,\" I persisted. \"You didn't have to use your powers like that.\"\n\n\"Please,\" Fiona Fine cut in, turning her blue eyes to me. \"What's the point of having superpowers if you don't use them?\"\n\nFiona grabbed the bread basket and waved her hand over the top. A few red-hot sparks shot off the ends of her fingertips, and the delicious smell of warm cheese bread filled the air.\n\n\"Lighten up, Bella,\" Fiona continued, putting the entire loaf on one of the dozen plates in front of her. \"We all know each other here\u2014alter egos and otherwise. It's not like there are other people around to catch us in the act.\"\n\nNo, there weren't any other people around. No _normal_ people anyway. Just me, Fiona, Chief Newman, my brother, Johnny, and my grandfather, Bobby.\n\nI'd barely touched my whole wheat ravioli, but I put my fork down. I wasn't hungry anymore. I never was when there were superheroes around.\n\nBut Fiona and Chief Newman weren't just _any_ superheroes. There were plenty of those in Bigtime, New York. No, they were Fiera and Mr. Sage, members of the Fearless Five\u2014the most powerful, elite team of heroes in the city. In addition to being stronger than five people put together, Fiera could also form fireballs with her bare hands, while Mr. Sage had all sorts of psychic powers, including telekinesis, or the ability to move objects with his mind.\n\nAnd now, they were part of my family.\n\nFiona had gotten engaged to my brother, Johnny, a couple of months ago after she'd saved him from two ubervillains who were trying to enslave the city. During all the commotion, Fiona had revealed her secret identity as Fiera to my grandfather and me and gotten us to help her rescue Johnny. And Chief Newman was Fiona's father, as well as her teammate.\n\nBut they weren't the only superheroes in the family these days.\n\nThe Fearless Five were a package deal. In addition to Fiera and Mr. Sage, we also got Karma Girl, Striker, and Hermit. Or Carmen Cole, Sam Sloane, and Henry Harris. That's how I thought of them. As nice, regular people who were mostly normal. Never as their alter egos. I tried to pretend those other people didn't exist.\n\nI tried to pretend a lot of things didn't exist.\n\nEspecially my own supposed superpower.\n\nMy grandfather, Bobby Bulluci, clapped his hands together. \"Come! Let's talk of other things.\" He turned to Fiona and Johnny. \"Are the two of you packed for your trip?\"\n\nJohnny had some business to take care of in the overseas divisions of Bulluci Industries, so he and Fiona had decided to make a working vacation out of it. The two were leaving tomorrow on a month-long trip to explore the Mediterranean.\n\n\"Of course,\" Johnny answered, flashing Fiona a grin. \"Although I don't know how we're going to get all of Fiona's clothes onto the plane.\"\n\nFiona reached over and punched my brother. Johnny flexed his bicep, which took on a hard look\u2014like his skin had suddenly morphed into metal. Fiona's fist smacked into his arm, and she frowned and shook her hand. Even with her great strength, it hurt to punch my brother when he formed his superhard, supertough exoskeleton. It made Johnny immune to just about everything. Kicks, punches, explosions, Fiona's flare-ups. That was good, since my brother had an annoying tendency to dress up in tacky, formfitting, black leather, zoom around town on his motorcycle, and fight ubervillains.\n\nInstead of an exoskeleton, I'd gotten something far less useful from the mutated family gene pool\u2014luck. As if that was any kind of superpower. Superannoying was more like it.\n\nFiona sniffed and tossed her blond hair over her shoulder. \"I've told you a million times you can never have too many clothes, especially when you're going on vacation. Besides, we're taking Sam's private jet. There'll be more than enough room for my things.\"\n\nJohnny gave Fiona another wicked smile. \"But, baby, you know I think you look fine in whatever you wear\u2014especially when it's nothing at all.\"\n\nFiona rolled her eyes. \"Please. There's nothing sexier than a well-dressed woman. Right, Bella?\"\n\n\"Of course,\" I murmured.\n\nFiona and I knew a few things about well-dressed women, since we both worked as fashion designers. Fiona fronted Fiona Fine Fashions, while I ran the design portion of Bulluci Industries. Fiona and I had completely different styles, and we'd been friendly rivals for years. She created garments that screamed _Here I am! Look at me! I'm fabulous!_ with their bright colors, wild patterns, and mounds of sequins and feathers. I preferred simpler styles, with muted hues, clean lines, and absolutely, positively no sequins. Ever.\n\nDon't get me wrong. I liked Fiona just fine. Her father too. And I was glad Johnny had found someone he wanted to spend the rest of his life with.\n\nBut there was nothing I hated more than superheroes and ubervillains. Dressing up in those silly costumes. Calling themselves absurd names. Plotting and scheming and planning elaborate ways to take over the city and rule the world. It was all so dramatically ridiculous.\n\nC'mon. Who would want to rule the world, really? It'd be nothing but a giant headache, with everyone constantly whining and crying at you. Not to mention all the paperwork and demands on your free time. But the ubervillains always tried to reign supreme, and the superheroes always stopped them. The cycle was endless.\n\nUnfortunately, I had lots of experience with superheroes. Or rather, pseudo heroes. All the men in my family masqueraded as Johnny Angel in their youth, riding around Bigtime on a tricked-out motorcycle, getting into trouble, and taking on ubervillains when the mood hit them. Masquerading as Johnny Angel was how my brother had first met Fiona a few months ago.\n\nAnd how my father, James, had died.\n\nI was happy for Johnny, but I couldn't help shuddering at the fact he'd added another superhero to the family tree. Five of them. Six, actually, if you counted Lulu Lo, the computer hacker who was engaged to Henry Harris.\n\nOh, I liked Fiona, Carmen, Sam, Henry, and Chief Newman just fine when they were themselves. It was their nightly habit of turning into Fiera, Karma Girl, Striker, Hermit, and Mr. Sage that had me concerned.\n\nAnd knowing the Fearless Five's secret identities was sort of like being in a mob family\u2014once you were in, you were _all_ the way in whether you wanted to be or not. And you couldn't get out, no matter how hard you tried. Whenever we had any of the heroes over for dinner, all they talked about were their latest epic battles and daring escapes. Or the new equipment Henry Hermit Harris had purchased for their underground lair. Or the current ubervillains populating Bigtime. Or a dozen other superhero-related things that made me grind my teeth. Last week, Fiona had even asked me if I thought her costume needed a redesign. Sheesh.\n\nMy power flared up at my dark thoughts. I didn't know how the other superheroes felt their power, but mine was sort of like standing in a ball of static electricity. My skin hummed. My fingertips itched. And worst of all, my caramel-colored hair frizzed out to alarming proportions. There wasn't a conditioner on the market that could tame it. Believe me, I'd tried them all. Together. At the same time.\n\nThe overall sensation wasn't uncomfortable so much as it was aggravating. Because the static, the power, the energy built and built until it had to be discharged. And when it did, well, watch out. More often than not, whatever was around me either exploded, shattered, fell from the sky, or spontaneously combusted. Sometimes all at once. My luck was like some sort of supercharged telekinesis I couldn't control. Stuff just happened, whether I wanted it to or not. And here's the really annoying thing about having luck as a superpower\u2014it can be good or bad.\n\nSometimes, if I thought about something, wanted it to happen, willed it to be, I'd get my heart's desire. I'd catch the subway a second before the doors closed. Snag the last seat in a crowded movie theater. Find the only dress in my size. I even won five hundred dollars in a sweepstakes as a kid just by staring at my entry form before I sent it in and wishing I could win.\n\nBut just as often, my luck turned on me. I'd catch the subway but rip my jacket on the doors. Get the last seat but sit down in a puddle of sticky soda. Find the perfect dress but forget my credit cards. Win the lottery but lose my ticket.\n\nLuck, the most capricious thing in the world. That was my supposed power. My curse was more like it.\n\nMy jinx.\n\nI always felt the static energy around me and did my best to keep it clamped down and under control. But the sudden surge told me that it was time for it to let loose\u2014and for something to happen. I could never tell whether that something would be good or bad, but I wasn't taking any chances.\n\nI slowly, carefully, calmly pushed my chair back from the table, making sure I was clear of the tablecloth, candles, bread basket, wineglasses, plates, silverware, and anything else I could drag down or knock off or upset in any way. Then, I stood.\n\nWith small, thoughtful steps, I backed around the chair until I stood five feet away from the table\u2014and out of range of everyone and everything. Now, nobody else would get caught in the crossfire if something crazy happened, like the chandelier above my head plummeting from the ceiling, despite the ten or so bolts that held it in place.\n\n\"Bella? Are you all right?\" Chief Newman asked, his eyes flashing a brilliant green. \"Is your power bothering you again?\"\n\nChief Newman had offered to work with me, to try to find some way to help me control my power. I'd refused. You couldn't control luck. I'd long ago given up hope of ever taming it, along with my hair.\n\nThe doorbell rang, saving me from an explanation.\n\n\"I'll get it,\" I said. \"It's probably more trick-or-treaters.\"\n\nIt was late October and still several days before Halloween, but little ghosts and ghouls and goblins had already started showing up asking for candy. Or else. Halloween was a two-week-long event in Bigtime that wouldn't wrap up until the night of the thirty-first. The extended holiday gave everybody, kids and adults alike, a chance to go around town all dressed up, instead of just the heroes and villains.\n\n\"What are you giving them?\" Fiona asked, her eyes gleaming at the thought of Halloween candy. \"Snickers? M&Ms? Chocolate Twinkies?\"\n\nThe only thing Fiona loved as much as Johnny was food. With her fire-based superpowers and high metabolism, Fiona could eat whatever she wanted to, whenever she wanted to, and never gain a pound. Besides her nighttime gig as a superhero, that was the only other thing I really hated about her. Well, that and her sky-high legs. I was just a couple inches over five feet. And her perfectly smooth blond hair and gorgeous baby blues. My tawny locks resembled a bush more often than not, while my hazel eyes just sort of faded into my bronze face. All right, so I really hated a lot of things about Fiona.\n\n\"Hardly. I'm giving them apples, fat-free trail mix, boxes of raisins, and bags of unpopped, butter-free microwave popcorn.\" I pointed to the far end of the long table, where I'd put the plastic bowls of goodies.\n\n\"What's the fun in that?\" Fiona said.\n\n\"Not contributing to the American epidemic of childhood obesity, for one,\" I snapped.\n\nFiona rolled her eyes. \"Your house is _so_ going to be covered in toilet paper in the morning.\"\n\nBobby cleared his throat. \"Actually, Bella, I took the liberty of buying some candy bars on my way home today. Just in case you ran out of apples.\"\n\n\"Chocolate? Where?\" Fiona demanded.\n\nI put my hands on my hips and glared at my grandfather. There was a devilish twinkle in his green eyes I knew all too well.\n\n\"And how many did you eat before you put them away?\"\n\nHis lips twitched. \"Bella, you've told me many times I shouldn't eat candy. I didn't have a single one.\"\n\nRight. And I looked good in a thong.\n\n\"Grandfather,\" I warned.\n\nBobby's heart, cholesterol, and blood pressure weren't the best in the world, something I was trying to change. With little success. My grandfather still ate like he was twenty-three, instead of seventy-three, despite doctor's orders and my constant nagging. And don't even get me started on his other bad habit\u2014motorcycle riding. Bobby had broken his leg two years ago gallivanting around town, and I'd moved back home to take care of and keep an eye on him.\n\nBobby ignored me. \"They're in the kitchen, Fiona, if you want to hand them out.\"\n\nFiona snapped to her feet. \"Count me in.\"\n\nBobby's eyes sparkled. \"Try to leave some for the kids.\"\n\nFiona sniffed and tossed her hair over her shoulder again before disappearing into the kitchen.\n\nI grabbed the bowls of apples, raisins, and popcorn and carried them to the front door. The static crackled around me like an invisible force field, but it seemed to be holding steady. For the moment. Fiona came out of the kitchen and fell in step beside me, candy bars in hand. She opened the door, and I smiled, ready to greet our visitors.\n\n\"Trick or treat!\" the kids shouted, holding out plastic orange pumpkins.\n\nThere were five of them, of course. Each one dressed like a member of the Fearless Five. A girl clad in orange-red spandex was supposed to be Fiera, and one in silver represented Karma Girl. One of the little boys sported an Irish green cape as Mr. Sage, while the other had on black leather and two long swords made out of aluminum foil for Striker. The man with them wore black-and-white goggles, representing Hermit.\n\nSuperheroes. More stupid superheroes. What happened to the good old days when kids dressed up as princesses and cowboys and monsters?\n\nMy smile faltered, but I held out the bowls. \"Who wants some apples?\"\n\nSilence. Dead silence. I didn't even hear crickets chirping in the front yard.\n\nThe kids looked at me, then each other, then at the man. No one said anything.\n\nMy power surged again. The static discharged.\n\nAnd the plastic bowls in my hands shattered.\n\nYou would have thought I had some explodium in the containers instead of healthy snacks. Raisins and popcorn showered us all, while bits of pulverized apple pelted my thick, curly hair and face. The few apples that survived the explosion intact bounced down the long driveway and out of sight. The pieces of the splintered bowls zipped through the air, embedding themselves in the stone steps like daggers around my feet. In a perfect circle, no less.\n\nI sighed and wiped a bit of apple juice off my nose. I'd long ago grown used to my power\u2014and the embarrassment that went along with it.\n\n\"I'm so sorry,\" I said, scooping raisins and popcorn into my hands. \"I have more inside. Let me get that.\"\n\nI'd been prepared for such a disaster. In fact, I always bought five of everything, whether it was candy or jewelry or clothes. Years of bad luck had taught me that my jinxed power would find a way to trash even the safest, sturdiest object. In the last six months, I'd gone through seven purses, dozens of shirts, and more shoes than I cared to admit. And two cars.\n\n\"Um, I think we'll just try the next house,\" the man replied, drawing the kids close to him.\n\nFiona not-so-gently shouldered past me. \"Don't worry. I've got some chocolate bars right here. They're a little melted, but they're still good.\"\n\n\"Yeah!\"\n\nThe kids stepped forward, and Fiona gave them each a chocolate bar. The girl in the Fiera costume got two. Naturally.\n\nSatisfied, the kids headed back down the driveway in search of more Halloween goodies to rot their teeth and drive their sugar levels through the roof.\n\nFiona smirked. \"See? I told you the kids would want candy.\"\n\nI sighed again. I should have known better. After all, it was almost Halloween.\n\nAnd the perfect time of year for my power to play tricks on me.\n\n# About the Author\n\nJennifer Estep is a _New York Times_ bestselling author, prowling the streets of her imagination in search of her next fantasy idea.\n\nJennifer is the author of the Elemental Assassin urban fantasy series. The books focus on Gin Blanco, an assassin codenamed the Spider who can control the elements of Ice and Stone. When she's not busy killing people and righting wrongs, Gin runs a barbecue restaurant called the Pork Pit in the fictional Southern metropolis of Ashland. The city is also home to giants, dwarves, vampires, and elementals\u2014Air, Fire, Ice, and Stone.\n\nJennifer is also the author of the Mythos Academy young adult urban fantasy series. The books focus on Gwen Frost, a 17-year-old Gypsy girl who has the gift of psychometry, or the ability to know an object's history just by touching it. After a serious freak-out with her magic, Gwen is shipped off to Mythos Academy, a school for the descendants of ancient warriors like Spartans, Valkyries, Amazons, and more.\n\nJennifer is also the author of the Bigtime paranormal romance series. The books feature sexy superheroes, evil ubervillains, and smart, sassy gals looking for love.\n\nFor more information on Jennifer and her books, visit her at www.JenniferEstep.com. You can also follow her on Facebook, Goodreads, and Twitter.\n\nHappy reading, everyone!\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## About the Book\n\n**From the seedy backstreets of London's Soho in the 60s to the tough, sexy world of international rock-stardom in the 70s, Georgia sees it all...**\n\nWhen nine-year-old orphan Georgia James is unexpectedly fostered by the kindly Celia and her bank manager husband she can hardly believe her luck. But then \u2013 on her fifteenth birthday \u2013 she suffers the cruellest betrayal of all at the hands of her foster father and is forced to run away, leaving everything she loves behind her.\n\nPenniless, sleeping rough, Georgia is soon introduced to the sleazy Soho world of brassy strippers, sweat shops, camaraderie and hardship. Fired by a fierce ambition, blessed with an extraordinary voice, her long struggle for fame and fortune begins. But even when she reaches the top she finds that the scars of the past can open up to ruin her...\n\n#####\n\nLesley Pearse was born in Rochester, Kent, but has lived in Bristol for over twenty-five years. She has three daughters and a grandson. She is the bestselling author of nineteen novels, including _Ellie, Georgia, Tara, Camellia_ and _Charity_ , all five of which are published by Arrow. She is one of the UK's best loved novelists with fans across the globe and sales of over three million copies of her books to date.\n\n##### _Also by Lesley Pearse_\n\n##### Tara*\n\n##### Charity*\n\n##### Ellie*\n\n##### Camellia*\n\n##### Rosie\n\n##### Charlie\n\n##### Never Look Back\n\n##### Trust Me\n\n##### Father Unknown\n\n##### Till We Meet Again\n\n##### Remember Me\n\n##### Secrets\n\n##### A Lesser Evil\n\n##### Hope\n\n##### Faith\n\n##### Gypsy\n\n##### Stolen\n\n##### Belle\n\n##### * Also available in Arrow Books\n\n#\n\n#####  \n\n## Contents\n\nCover\n\nAbout the Book\n\nAbout the Author\n\nAlso by Lesley Pearse\n\nTitle Page\n\nCopyright\n\nDedication\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\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\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\n##### Version 1.0\n\n##### Epub ISBN 9781409043942\n\n##### www.randomhouse.co.uk\n\n##### Published by Arrow Books 2011\n\n##### 2 4 6 8 10 9 7 5 3 1\n\n##### Copyright \u00a9 Lesley Pearse 1993\n\n##### Lesley Pearse has asserted her right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work\n\n##### Quotation in chapter 25 from _The Prophet_ by Kahlil Gibran\n\n##### This novel is a work of fiction. Names and characters are the product of the author's imagination and any resemblance to actual persons, living or dead, is entirely coincidental\n\n##### This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser\n\n##### First published in Great Britain in 1993 by William Heinemann  \nFirst published in Great Britain in paperback in 1993 by Mandarin Paperbacks  \nFirst published in paperback by Arrow Books in 1998\n\n##### Arrow Books  \nRandom House, 20 Vauxhall Bridge Road  \nLondon SW1V 2SA\n\n##### www.randomhouse.co.uk\n\n##### Addresses for companies within The Random House Group Limited can be found at: www.randomhouse.co.uk\/offices.htm\n\n##### The Random House Group Limited Reg. No. 954009\n\n##### A CIP catalogue record for this book is available from the British Library\n\n##### ISBN 9780099557456\n\n##### To my girls, Lucy, Sammy and Jo; without your love and support I couldn't have written it.\n\n##### A big thank you too for the real Georgia who was friend, confidante and inspiration.\n\n## Chapter 1\n\n_Grove Park, South London, February 1954_\n\nClanking keys and a ponderous step woke Georgia. Her ear was so finely tuned she knew which nun was coming, even her exact position.\n\nIt was Sister Agnes. Some of the nuns moved up the stairs in one fluid movement, some panted and huffed, pausing to rest halfway, but Sister Agnes despite her bulk and age ploughed on steadily to the top, her breath wheezing faintly.\n\nShe had reached the top now, passing the long, narrow, barred window, on her way to ring the early morning bell.\n\nGeorgia sat up in bed, rubbing her eyes. A murky grey light showed up twelve iron beds, six to each side of the large room. Small mounds in each, still fast asleep.\n\nThe heavy footsteps moved away from her dormitory, down towards where the bell hung on the wall just outside the big girls' room. Another pair of feet were coming down the stairs from the floor above, this time light and bouncy, almost running as they went on down further. That would be Sister Theresa on her way to make Mother Superior's early morning tea.\n\nA whimper made Georgia's head turn to the bed on her left. As the child stirred, so an unmistakable acrid smell of urine wafted across to her nostrils.\n\n'Pamela!' she hissed. 'Aggie will be in here any minute, run for the bathroom. I'll try and cover for you.'\n\nThe bell rang out in the uncarpeted corridor, drowning Pamela's reply and as the last echo reverberated round the convent, so heavy feet thudded towards them.\n\nPamela's first cry had been one of dismay to find she was wet, but her second was one of terror. Instead of shooting out of her bed, and running like a hare out of harm's way, she just cowered, small arms over her head, waiting for the beating she knew would soon come.\n\nGeorgia knew to protect Pamela she had to create a diversion. Tossing back her covers she leapt into the air.\n\nSister Agnes paused momentarily in the doorway in time to see Georgia's trial bounce, landing feet apart, hands clutching her pyjama trousers.\n\n'Get down this minute!' she shouted. The child looked like a chimney-sweep. As thin as a stick in oversized striped pyjamas, her crop of black curls standing out like a wire brush.\n\nOne hand flew up to hold down the starched wimple, the other lifted her habit clear of the floor.\n\n'How dare you?' her voice rasped as she swept down the room indignantly.\n\nGeorgia merely grinned at her, a yellow-brown face cut in two with the flash of white teeth. Another small bounce quickly followed by a stronger one, and she had flipped herself over and landed on her feet again, just yards from the exasperated nun. She had perfected this somersault only days earlier in the playroom, where she had launched herself from an old couch on to cushions in front of an enthusiastic audience. But landing on cold, hard lino had jarred her legs and back and she toppled back against the bed rail.\n\n'Morning, Sister Agnes,' she panted, hauling the baggy trousers back to her waist. 'Did you see how good it was?'\n\nSister Agnes was the oldest nun in the convent. Humourless, mean-spirited and cruel. Black hairs sprouted from her white flabby chin, a hooked nose with a jiggling wart next to it vying for attention, and sharp piggy eyes that could spot a misdemeanour almost through a wooden door.\n\n'This is a dormitory, not a gymnasium,' she sniffed. 'You are nine, it's high time you set a good example to the younger girls.'\n\nInstinctively the old nun knew Georgia was trying to distract her, and insolent interference was something she wouldn't tolerate. Georgia infuriated her. Not only was she scrawny with huge eyes that dominated her yellowy face, but also endless punishments and beatings couldn't wipe her ear-to-ear grin away. Despite her skinniness and her mixed blood she had managed to become the leader of the younger girls and worse still she was encouraging them all in acts of disobedience.\n\n'I'll deal with you later,' Sister Agnes swept the dormitory with her sharp eyes. Small girls jumping into their navy blue knickers, eyes avoiding her. 'What's been going on in here?'\n\n'There was a noise,' Georgia sidled away from the Sister, rolling her eyes round the room in pretended alarm. 'I think a bird's got in again.'\n\nIt was all she could think of on the spur of the moment. Only last summer a pigeon had found its way in and to the children's amusement Sister nearly had hysterics. The way she had sped from the room as the bird flapped around her veil was something they still giggled about.\n\n'We heard it too,' a chorus of agreement came from three of Georgia's closest allies. As they struggled into grey skirts and jumpers, they nodded at one another, waving their hands as if to indicate the flight path.\n\nSister spun round, her hands reaching up to her veil, eyes scanning, ears straining for the sound of wings or cooing. Jennifer, the youngest child in the dormitory, stood with her thumb in her mouth, her pyjama jacket almost reaching her thin, scabby knees.\n\nEvery girl was poised expectantly, breath like smoke in the cold air, eyes alternating between the hesitant nun and Georgia. Bravery vanished as the big woman turned slowly. Each girl blanched under her inspection, fingers hastily fumbling for buttons, eyes downcast. At best she was as sour as a crab apple, angry, she was dangerous.\n\n'Come here, girl.' Sister's voice echoed round the bare room. Her chins were quivering ominously, her face turning puce.\n\nGeorgia cast one frantic look at Pamela, hoping she had the sense to move now, then sauntered over to Sister.\n\nSister caught her shoulder with one hand, her other swung out and hit Georgia with her full strength across the face.\n\nGeorgia stumbled back against a bed rail catching her side with a crack. A rustle came from Pamela's bed on the other side of the room. Georgia gritted her teeth, willing Sister not to turn and catch sight of the girl. But Sister's sharp ears had picked up the sound too. She wheeled round and at the same time her nose twitched furiously. The hasty dressing was halted. Ten mouths dropped open in horror, Jennifer sucked vigorously on her thumb. Pamela just stood by her bed. Pyjamas steaming, fists covering her eyes, whimpering and shaking with fear.\n\nShe was a quiet, nervous child, still in the throes of grief from losing her mother. Straggly brown hair, a slight squint and a tendency towards fatness hadn't endeared her to anyone other than Georgia.\n\n'Seven years old and you still wet the bed,' Sister's bellow caused yet another trickle to splash on to the floor. 'You are worse than an animal, even they don't lie in their own filth!'\n\nOne claw shot out, grabbing the terrified child who didn't have the sense to run, and with the other she boxed her ears so hard that Pamela fell to the floor.\n\nThe sheer force of Sister's attack made Georgia spring forward. 'Don't you dare!' she yelled, lungeing at the black habit. She saw one heavy black shoe swing forward to kick the helpless child and she pummelled her fists against the nun's wide posterior. 'She can't help it. You only make her more frightened. Leave her alone you bully!'\n\nThe other children hopped from foot to foot on the icy lino. One of the older girls caught hold of Jennifer and began helping her to dress, anxious to get her out of the way.\n\nSister turned and caught Georgia by the wrists. Her face was purple now, her thin lips curling back.\n\n'Get downstairs and fill the coal scuttles,' she roared, spittle spraying the child's face. 'You won't get away with this insolence.'\n\nGeorgia backed away to her pile of clothes. If she said another word it was quite likely Sister would lock her in the cupboard they used as a punishment cell. Bread and water only, crouching in that black hole until bedtime, without even a blanket to wrap round her. She couldn't help Pamela any further and she wanted her breakfast.\n\nLater, as Georgia knelt in the outhouse shovelling coal, she could hear Pamela crying in the bathroom. It wasn't even screams of anger, just a wail of distress.\n\nShe could picture the scene. Sister Agnes would have her standing in a bath of cold water, scrubbing at her with a brush. Pinching, slapping and all the time lashing her with jibes about her bedwetting.\n\nThere'd be no breakfast for her. While the other girls ate their porridge, Pamela would be alone in the laundry, crying as she struggled to wash the sheets. Why did Aggie think punishment would make her stop doing it? Even Georgia knew Pamela couldn't help it.\n\n'Aggie's evil,' she chanted to herself as she wielded the shovel, banging it down hard on the coal, pretending Sister Agnes was under it. 'Why doesn't someone stop her?'\n\nGeorgia was always being punished, if she dawdled coming home from school, if she talked during meals or giggled in the chapel, so much so that it hardly concerned her any longer. She learned to accept that Sister Agnes would never like her, along with accepting she was a different colour from the other girls. It even amused her when Sister called her 'Devil's spawn'; it reminded her of tadpoles in the tank at school.\n\nShe had mentioned it to Sister Mary once and her laughter had banished any sinister thoughts.\n\n'You are like a little tadpole,' her blue eyes twinkled. 'But you'll change into a beautiful woman, just you wait.'\n\nUntil she was five or six there had always been the possibility she might be adopted one day. Most Sundays couples came to St Joseph's looking for a child to love. Some old, some young, some rich with cars and fur coats, some ordinary like the other girls' mothers at school. But they all had one thing in common, they wanted pretty blonde girls with blue eyes, the younger and sweeter the better.\n\nThere had been times when Georgia tried the ploys the other girls used. Climbing on to laps, tugging at clothes, beguiling smiles, letting her eyes fill with tears, but all she ever heard was the same remark.\n\n'She's a nice little thing, but we couldn't cope with mixed race I'm afraid.'\n\nGeorgia sighed deeply as she hauled the two heavy coal buckets across the yard and down the stone steps into the kitchen. She was resigned to staying here until she was fifteen and found a job. At least she had school.\n\nMost of the other girls hated school more than the convent. They were singled out as different from other children, not only by the way they were shepherded across the busy main road by one of the nuns, but by their badly fitting clothes, heavy shoes and lack-lustre hair. But to Georgia every day at school was an adventure, a chance to see the outside world, to learn about things and places, to feel normal.\n\nShe liked the pictures on the walls and growing beans in blotting paper, mixing powder paint and making puppets, the percussion band and stories. But most of all she liked Miss Powell and her music.\n\nMiss Powell was the headmistress. She had a kind of glamour in her dark suits and white frilly blouses, her blonde wavy hair swept up at the back. But when she sat at the piano and played, that was the very best.\n\nHymns, sea shanties, folk songs, beautiful haunting melodies that made pictures in Georgia's head. Without Miss Powell perhaps Georgia would never have found she could sing!\n\nSinging made her feel good. She could forget the convent and Sister Agnes, her dark skin and the people who didn't want a mixed-race child. When she sang people looked at her and listened, even her own teacher who grumbled because she didn't learn her multiplication tables looked proud of her.\n\n'You've been given a very special gift Georgia.' Miss Powell had smiled down at her the day she picked her to be Archangel Gabriel in the school nativity play. 'I've chosen you because your voice can do justice to the beauty of Christmas. I want everyone to be as proud of you as I am.'\n\nThat afternoon in December when she had stood on the stage wrapped in a white sheet with a tinsel halo, hearing applause ringing out round the assembly hall, had been the best moment in her life.\n\n'In the Bleak Mid-Winter' seemed so appropriate now as she rinsed the coal dust from her hands before joining the other children for breakfast. Her cheeks were icy, her hands and thighs chapped with the cold, and right now Sister Agnes was plotting her punishment.\n\nWhen Sister Agnes didn't retaliate immediately after the usual Saturday breakfast of porridge and boiled eggs, Georgia put punishment out of her mind. Keeping warm outside in the playground was more important than worrying what might happen later.\n\nSt Joseph's gave the impression of being a large country house. The gravelled drive, the sweeping lawn, the walled kitchen garden and the old knarled trees were all from a more elegant period.\n\nIn fact the large house was only a stone's throw from Grove Park station in South London. Minutes away were rows of shops and a street busy with cars and buses.\n\nThree floors, with basement and attics, it was too large to heat adequately. The once gracious drawing and dining rooms were now draughty dormitories. Only Mother Superior's sitting room held any comfort. Even the small chapel on the first floor was gradually becoming dingy through lack of maintenance.\n\nThe garden was beautiful in summer. The children ran on the grass, chasing each other around the trees. There was the smell of the flowers, the big bushes they could hide behind, and long days with little supervision.\n\nBut now in February it was torture. The wind whistled through thin gaberdine raincoats, catching sore places on bare legs, nipping at ears and fingers. If they played with the snow brushed up round the playground it soon made them colder. All they could do was huddle closer to the walls. Twenty-four girls from four to twelve waiting for the bell to ring for dinner. Pale, pinched faces, gazing longingly at the steamy laundry where the older girls were privileged enough to be up to their elbows in soapy suds or sweating over hot irons.\n\n'She'll call you in soon.' Susan Mullins a carroty-haired eleven-year-old with freckles moved closer to Georgia. 'Are you scared?'\n\nThe word had even spread to the bigger girls about Georgia's run-in with Aggie. It was almost worth being punished to see their approval. But however big and tough she felt here surrounded by admiring friends it didn't stop the need to keep going to the lavatory, or the moments of panic when she saw a nun's face at the window.\n\n'No,' Georgia gave a wobbly grin. 'I'll get a knife and cut off her wart, then she'll bleed to death.'\n\nThe door of the playroom opened just before tea-time. Georgia was curled up on one of the old settees reading an ancient comic, younger girls were racing around the big empty room, while older girls huddled in a corner by the hot pipes.\n\n'Georgia,' Sister Mary's voice made her jump. 'Mother Superior wants you.'\n\nSister Mary was the youngest of the nuns. Perhaps in her mid-thirties, but it was difficult to put an age to her. She was tall and slender, with a smooth, unlined face. She had the appearance of a china doll, dainty fair eyebrows set above eyes like summer skies, and rosy lips over small white teeth.\n\nYet despite Sister Mary's youth, she was tough enough to act as a mediator between them and Sister Agnes. Her rippling laughter, her understanding of children, her gentleness and soft voice gave each child a feeling of security. She had trained as a nurse. During the war she had been close to enemy lines and the older girls speculated why anyone so pretty had chosen to enter a convent instead of marrying and having children of her own.\n\nThe other girls from the middle dormitory were looking at Georgia in horror. Pamela's eyes filled up with tears, she clutched Georgia with her small podgy hands.\n\n'It's all my fault,' she whimpered. 'You'll get a beating now, just for sticking up for me.'\n\n'Don't worry,' Georgia said reassuringly, slipping an arm round the smaller child. 'I'm not afraid of her. Besides, I might be able to tell her how cruel Sister Agnes is to you.'\n\n'You're so brave,' Pamela sighed, her good eye on Georgia, the other one on the window. 'I wish I could be like you.'\n\nA statue of the Virgin Mary stood at the turn of the stairs, with a small night light in front of it. Georgia genuflected, screwing her eyes up tightly as she made a quick plea for mercy.\n\nThe wide hallway was very dark. It was oak panelled, the only natural light came from the window on the staircase, and a lone candle under a picture of the Sacred Heart. It was no use looking at the front door and considering escape. Even if she could reach the big bolt at the top she couldn't get far in the snow with only plimsoles on her feet and no coat. Instead she screwed her hands into fists, wiped her nose on her jumper sleeve and knocked at Mother Superior's door.\n\n'Come in!' Mother Superior's faint old voice crackled from within, like ancient parchment.\n\nGeorgia turned the brass knob with two hands, opened it just a crack, and tentatively put her head inside.\n\nMother Superior sat by a blazing log fire, her back to the window, a small, bowed figure in an oversized winged armchair.\n\n'Come on in, no one's going to bite you.'\n\nTo Georgia's surprise the tone was almost jovial, but then Sister Agnes was probably lurking behind the door.\n\nGeorgia slunk in, eyes down on the carpet, hands still holding the door.\n\n'Close that door,' Mother Superior snapped. 'We don't want to freeze.'\n\nIt was the 'we' that made Georgia glance up. A lady was sitting on the settee further back from the fire, looking at her. Mother Superior was wearing the smile she usually only reserved for Christmas and visitors.\n\nGeorgia closed the door carefully, arranging the heavy wool curtain over it to keep the draughts out. She had seen this lady before once or twice at school, yet she wasn't a teacher. Had Georgia been so bad they needed outside help now, to punish her?\n\nMother Superior reached out one tiny, bony hand, in a gesture that said Georgia was to come closer. She was rumoured to be eighty. Whether this was true or not Georgia had no idea, but she certainly was very wrinkled; not just around her eyes, but all over her face, as if she had shrunk a foot or two and all the spare skin remained.\n\n'Mrs Anderson is a children's officer. She's come here to talk to you.'\n\nGeorgia stood uneasily on the hearth rug, her stomach churning with fear. She knew what children's officers did, they were the ones who came and took girls away when they wouldn't behave. Yet for all that, Mrs Anderson didn't look fierce. She had that same look of authority Miss Powell had, and she sat as serenely as if she were in her own home. Her face was round and her hair cut almost like a man's, but her smile and pink cheeks were distinctly feminine.\n\n'Hallo Georgia,' the woman got up, taking Georgia by surprise as her strong, clear voice filled the room. 'I don't suppose you remember me, but I saw you at the Christmas play.'\n\n'You're going to take me away?' Georgia stuck out her small pointed chin defiantly. 'I didn't do anything but try and help Pamela. Sister Agnes is cruel and mean.'\n\nThe lady looked from both Georgia to Mother Superior in surprise.\n\nGeorgia was baffled now. Her entire childhood had been spent studying adults' secret looks. Whatever this lady had come for it wasn't to chastise her further.\n\n'Now then, Georgia,' Mother Superior's tone was honeyed, the warning of punishment hidden except to the two of them. She got up unsteadily and put one hand on Georgia's shoulder, bony fingers digging in her flesh just hard enough to remind her she hadn't been brought in to reveal secrets about anyone. 'Mrs Anderson has come here today to offer you a wonderful opportunity. Don't try to be difficult.'\n\n'Perhaps I should talk to Georgia on her own for a while?' Mrs Anderson's suggestion sounded more like a statement.\n\nGeorgia looked from one adult to the other, puzzled, but no longer frightened.\n\n'If you think that is necessary,' the older woman replied starchly. She straightened up her small, bent frame, her bloodless lips pursed with irritation. 'I have got some important jobs to do.' She bustled towards the door, every inch of her showing disapproval.\n\nMrs Anderson got up, took Georgia's hand and led her back to the settee.\n\n'She wasn't keen to go,' she said, lifting Georgia's face up with one finger to study it. 'So I'll have to be quick.'\n\nGeorgia liked her touch. It was like her manner, confident, kindly, maybe even motherly. Her eyes were grey, with tiny specks of green, bright and unwavering, a few tiny lines around them, maybe more from laughter than old age.\n\nThere was a lovely fresh smell about her. Like sheets when they had hung outside all day in the sunshine. She was a big woman, with ample hips and a bosom that pushed out the front of her jacket, but not exactly fat. Not as elegant as Miss Powell, but she looked more friendly.\n\n'I saw you at the school concert,' she said softly. 'I loved your voice and I couldn't forget you. When I discovered you had been here for years, I tried to find out if I could adopt you.'\n\nGeorgia's mouth dropped open in surprise.\n\n'Apparently that isn't possible. But I still want you to be my little girl. I want you to come and live with me if you'd like to.'\n\nIt was like a dream, yet the plump, warm hand holding hers was real enough.\n\n'You want me?' Georgia's wide mouth split into a grin which spread from ear to ear.\n\nTo her surprise Mrs Anderson's eyes seemed to be filling with tears.\n\n'Don't cry,' Georgia leaned closer, tentatively touching the lady's face. 'I can be ready in ten minutes.'\n\nMrs Anderson laughed then, the sort of laugh Georgia never heard from the nuns. It was the sound of freedom, a wonderful sound that somehow embodied life outside the convent. Georgia joined in, her nose wrinkling up with merriment.\n\n'Oh, Georgia, I knew you were my little girl when I first saw you,' she laughed, squeezing Georgia's hand still tighter. 'My goodness, you are a tonic.'\n\n'What's a tonic?' Georgia's face was suddenly more serious.\n\n'It's a kind of medicine you take, to make you feel better,' Mrs Anderson explained, her eyes still dancing with laughter. 'You've just banished every doubt in my mind.'\n\n'Do you really want to take me with you?' Georgia's eyes were wary. Sister Mary and Miss Powell could be relied on but she'd never met any other adults who didn't change their minds.\n\n'Yes, but I can't take you now. It will be tomorrow.'\n\nGeorgia thought quickly. She was sure she could trust Mrs Anderson. This wasn't one of those empty-headed ladies who came here looking for a small, cuddly plaything. She wasn't afraid of anything or anyone.\n\n'Can you do something for me then?' Georgia asked.\n\n'I'll try.'\n\n'Well get someone to stop Sister Agnes. She beats Pamela for wetting the bed and she can't help it.'\n\n'I'll do my best,' Mrs Anderson looked shocked. 'Has she ever beaten you?'\n\n'Loads of times,' Georgia said nonchalantly. 'But I'm bigger and tougher. I can stand up for myself. Pamela can't. She's only seven and her mummy and daddy are dead.'\n\n'But you haven't any parents either?' Mrs Anderson's voice dropped, she smoothed Georgia's cheek, then kissed her hair.\n\n'Yes,' Georgia looked up at her proudly. 'But I've been on my own since I was born. I've learnt to cope with things, and anyway I don't wet the bed.'\n\nMrs Anderson seemed to find that amusing.\n\n'Mr Anderson and myself live in a nice big house in Blackheath,' she explained. 'I'm very glad you don't wet the bed as I've bought a nice new one for you. You'll go to school nearby and we have the heath and Greenwich Park just across the road. But once you have settled in with us, I'll see what I can do for your friend.'\n\n'Have you got lots of children?' Georgia asked.\n\n'No, I haven't any,' Mrs Anderson's mouth was twitching with merriment at Georgia's rapt expression. 'But you'll soon make new friends at school.'\n\n'Will there be music there?' There had to be some hidden catch, but maybe Miss Powell and her piano was a small price to pay.\n\n'There certainly will, I play the piano myself and if you like we can arrange music and singing lessons.'\n\nGeorgia's eyes lit up, her mouth fell open and if it hadn't been for the door opening again, she would have whooped with delight. But Mother Superior shuffled into the room, her wrinkled face full of suspicion.\n\n'Have we had enough time?' her sarcasm was not wasted even on Georgia.\n\n'We'll have all the time in the world soon,' Mrs Anderson said sweetly. She bent over to kiss Georgia, and whispered in her ear. 'When you're my little girl.'\n\n'Run along now Georgia.' Mother Superior once more put on the expression for visitors, a smarmy smile, a patronizing tone and all the time her bony fingers fiddling with her Rosary. 'Mrs Anderson will be coming in the morning for you.'\n\nThe white tiled bathroom was full of steam. The floor was awash where less than an hour ago twenty other children had been bathed in the four large baths. Despite the steam the room was freezing, the windows rattling as a gale-force wind howled around the old convent.\n\nGeorgia wanted to dance and sing. She wanted to tell the world this was her last night. Tomorrow she would have her own room. A mother who would tuck her into bed. Someone who liked her singing and could play the piano.\n\nSince meeting Mrs Anderson earlier on, she had been kept apart from the other children. Mother Superior had even said she was to spend the night in the isolation room at the top of the house. But no one could silence Georgia's high spirits tonight. Alone in the bathroom she stripped off the matted grey jumper, the long, ugly skirt, her flannel petticoat, liberty bodice and her navy blue baggy knickers. Forgetting the propriety of never standing naked in sight of the Lord, even the shabby old vest was tossed away.\n\nShe picked up a small towel, wrapped it round her middle like a dress, and made believe she was a grown-up lady in front of a big audience.\n\n'In Dublin's fair city, where the maids are so pretty,' she sang at the top of her voice, dancing nimbly around the room. 'That's where I first set eyes on sweet Molly Malone.'\n\nThe door opened silently. Georgia was so engrossed in her performance, she didn't see Sister Agnes's approach, or hear the sharp intake of breath.\n\nCrack!\n\nGeorgia jumped in the air as if she'd been stung by a wasp, dropping her towel to the floor.\n\nSister Agnes had one of her favourite weapons in her hand. It was merely a thin, damp towel, but in her hands it was deadly. She was poised for mischief, flicking it accurately across Georgia's naked buttocks like a whip.\n\n'Admiring ourselves were we?' her bloated ugly face was contorted with suspicion. Already she was preparing the small towel for another blow.\n\n'I wasn't,' the small girl retorted indignantly, jumping to one side, hands raised to ward off more blows. 'I was just singing.'\n\n'Don't lie to me,' Sister roared, flicking the towel expertly to catch the child yet again. 'You are a wicked sinful girl with unclean thoughts. How dare you expose yourself?'\n\nIn her excitement Georgia had forgotten the incident in the dormitory, but it was clear Sister Agnes hadn't. Yet surely she wouldn't dare hurt her now, not when Mrs Anderson was coming back so soon?\n\n'Don't you touch me,' she yelled with all the volume she could muster. 'I've got a mother now!'\n\n'How dare you?' Sister Agnes dropped the towel and stalked towards her, her several chins quivering round her wimple with rage, beady eyes full of malice.\n\nGeorgia backed into the tiled wall, her bare toes scrabbling to get a grip on the wet floor. She was prepared now to stand her ground, not to let the old woman get the better of her.\n\n'Don't you hit me,' she yelled defiantly, her dark eyes blazing with new-found courage. 'I'll tell her!'\n\n'Tell her what you like. Do you think anyone will believe some half-witted nigger instead of me?'\n\nGeorgia braced herself. Time and time again Sister Agnes had thrown that word at her.\n\n'I'm not a nigger,' her eyes filled with tears. 'That's an evil word and so are you!'\n\nSister stared at her for a moment, clearly surprised at any child answering her back. Georgia's darkness showed up more clearly in here, against the white walls. Naked, she looked thin to the point of malnutrition, her limbs like sticks, her head seeming too big for her body.\n\nTo Sister Agnes, the child before her was a product of the Devil. A child born out of wedlock, abandoned at a few months, proof in her eyes that the mother was a whore.\n\nShe resented the way Georgia got attention both from adults and the other children by singing and play acting. No other child at St Joseph's ever had the nerve to answer back as she did and now she had been singled out for a new home with that insolent woman who dared suggest Georgia was undernourished. Mrs Anderson wasn't even a Catholic. What right did she have to criticise the care in St Joseph's?\n\nGeorgia hadn't reckoned with Sister coming armed with her small cane. Like a snake it appeared out of the folds of Sister's habit. Some fourteen inches of thin, bendy wood, polished and smooth with years of handling.\n\nSister Agnes was old, fat and out of breath. But Georgia was no match for her, not now Sister was filled with righteous indignation.\n\nMoving back, Georgia found herself trapped in the corner and she watched in horrified fascination as the old woman stooped over the bath and turned the taps on full to drown any noise. Still stooping, cane in one hand, the other on the tap, she turned slightly to look at Georgia, her lips curled into a sneer.\n\nGeorgia tried to slide along the wall. Her heart thumped and she felt as if her legs were embedded in cement.\n\nOne claw-like hand reached out and clamped on to Georgia's bony shoulder and the other hand lifted the thin cane up high.\n\nThere was a whistling noise and the cane flashed through the air, catching the child's arm, searing through the skin.\n\n'Please don't!' Georgia yelled, dancing in pain.\n\n'Bend over,' Sister bellowed. 'You've had this coming to you for a long time.'\n\n'Please, Sister,' Georgia whimpered. 'I'm sorry, I didn't mean what I said.'\n\n'Oh yes you did. You think you are special. You always have. It's about time someone took you to task, beat that proud look out of you.'\n\nGeorgia cowered further into the corner, slumping down onto her haunches, arms raised to protect her head.\n\nShe saw one black shoe shoot out from under the habit, kicking out her legs from under her, and her bottom crashed to the floor.\n\nThe next blow caught her on the thigh. She scrabbled to get away, but made the mistake of presenting her bottom as she did so.\n\nAgain and again the cane cut into her bottom, legs and back. She screamed in terror, but it was drowned by the rush of bath water.\n\n'Get in that bath!' Sister Agnes yelled.\n\nSkirting round Sister, Georgia moved quickly to the other side of the bath and jumped in. The water was scalding hot, but she didn't dare cry out. It came up to her armpits and burnt into the weals left by the cane.\n\nGeorgia had no fight left. She submitted to being dragged up and scrubbed.\n\n'Now, dry yourself and get up to bed!' Sister hissed. 'And don't take long about it.'\n\nThe door slammed behind her and Georgia groped blindly for the towel. She was shaking with cold. Her eyes stung and her body was on fire. Slowly she hauled herself out of the bath, and sunk on to a small stool. Her earlier happiness glugged down the drain with the bath water, and was replaced by tears of despair.\n\n'Georgia?'\n\nShe blinked at the sound of Sister Mary's voice at the door.\n\n'What is it?' Sister moved across the wet floor, arms outstretched, her face a picture of concern.\n\n'Sis \u2013, Sister Agnes,' Georgia stuttered.\n\nA dry, softer towel was wrapped round her, the smaller one deftly removed and wound round her hair like a turban.\n\n'What happened?' Sister asked, her tone gentle as always, in sharp contrast to Agnes's.\n\nGeorgia tried to explain. Another coughing fit engulfed her, this time coming in great whoops, bringing with it large quantities of fluid she had swallowed.\n\nSister Mary turned the child deftly onto her stomach across her own lap, patting her back until the attack stopped. Georgia could feel her soothing her wounds gently with the towel.\n\n'What did you do?' Sister's voice was soft, yet with a touch of steel.\n\n'I was singing and dancing, she said I was admiring myself. She called me a nigger.' Georgia sobbed.\n\nSister made no comment. Just lifted the child up into her arms and held her tightly against her chest, soothing her with endearments.\n\n'Let me get you dry and into bed,' her voice shook a little. 'You've got a big day ahead of you tomorrow. Sister Agnes won't touch you again.'\n\nScooping up Georgia in her arms, still only wrapped in the towel, she walked swiftly up the stairs with her in the direction of the isolation room.\n\n'Wait a moment,' she said as she dropped the child on the bed. 'I'll just go and find some pyjamas.'\n\nThe room was cosy at night. A small bedside lamp and a lighted gas fire gave the sparsely furnished room warmth that every other room in the convent lacked.\n\nAlthough she hurt all over, Georgia noticed that clean clothes had been placed on the chair for the morning. A tartan kilt and a much nicer jumper than she normally got to wear. Her sobs faded to hiccups.\n\n'Here we are,' Sister Mary bustled back into the room, a pair of pyjamas and vest over her arm.\n\nIn one hand she held a pot of ointment.\n\n'Lay down on your tummy,' she said gently. 'This will help the soreness.'\n\nAt first Georgia winced at each soft touch, but gradually under Sister Mary's healing hands, the pain lessened. Firmly, Sister turned her and more ointment was applied to her stomach, chest and arms.\n\n'That's better,' Sister said, picking up the vest and popping it over her head, quickly followed by the warm pyjamas. 'Now into bed with you and I'll dry your hair a bit more.'\n\n'Why is Sister Agnes so mean?' Georgia plucked up courage to ask, as her hair was rubbed vigorously.\n\n'I can't say anything about another Sister,' Mary said reprovingly with a twinkle in her eye. 'But you will find the world is full of all kinds of people, some nice, some plain nasty. Let's just say that maybe Sister Agnes isn't as happy inside as me.'\n\n'Why are you happy?' Georgia twisted her head round to look Mary full in the face.\n\n'Because God saw fit to send me here,' Sister smiled, her blue eyes twinkling. 'How else would I have met you?'\n\n'Why is this lady taking me to her home?'\n\nSister laughed, showing small even white teeth in the half light. 'So many questions! I expect she liked your courage and enthusiasm, just like I do.'\n\n'So does that mean I will be her little girl for ever?' Georgia's eyes were shining now, her sore body forgotten.\n\n'I think so,' Sister Mary wound a curl round her finger. 'She is a strong, caring woman Georgia, you'll have a good home with her and her husband. All you have to do is be a good girl and she'll take care of everything else.'\n\n'If I'm bad will she send me back here?' Georgia's eyes widened with fright.\n\n'I doubt that somehow,' Sister laughed soft and low. 'I don't think she's the type to give up on anything or anyone. But don't you get any ideas about testing her will you? Even the nicest people have their breaking point.'\n\nShe pulled a comb out of her pocket and ran it through Georgia's damp hair. Georgia glanced up and saw a tear trickling down the nun's cheek.\n\n'Why are you crying?' she whispered.\n\n'I'm just sad to know this is the last night I'll spend with you,' Sister replied, wiping her cheek with the back of her hand. 'We've been friends a long time. I was the one who undressed you the first night you came here. You clung to me like a little monkey.'\n\nShe smiled as she remembered.\n\nIt was a wild November night when Georgia arrived with a social worker. Just twenty-one months old, plump, with a halo of jet black curls, her thumb firmly planted in her mouth, her eyes as black as night.\n\nWhether she had been abandoned or orphaned wasn't known, just the name 'Georgia' passed on, her birth date of January 6th 1945 just an approximation.\n\nSister Mary had only been at St Joseph's a few weeks and she was appalled by the conditions. No toys, precious little warm clothing or bedding, children with running sores, threadworms and lice. She had been sent here because of her nursing training and youth, yet so far she had been unable to make a dent in the mountain of things wrong with the place.\n\nShe took Georgia into her arms, rocking her against her breast and watched her dark eyes beginning to droop. She knew she should insist the child was taken somewhere with proper facilities for babies, but she heard the exasperation in the social worker's voice, the complaints that every home was full, and her heart went out to the child.\n\nIt was love from that first night. Bathing, dressing, teaching and feeding, this was no longer duty, but joy. Small brown arms wrapped around her neck, damp sweet kisses, a constant reminder of everything she had given up by taking her vows.\n\nBut as the years passed, joy was tinged with fear. She saw Georgia's character forming, a bold clown, leader and entertainer, a child that rushed to the defence of anyone weaker and she knew Sister Agnes had the power and hate to crush it.\n\nMary had managed to change many things for the better in St Joseph's. Diet, hygiene and health were all improved, but still Mother Superior turned a blind eye to the sadistic cruelty of Sister Agnes, refusing to admit that women of her character had no place with children.\n\nWhen she heard Mrs Anderson wished to foster Georgia, Mary felt as if her heart was being torn out. Yet at the same time she wasn't prepared to sit by and watch while Georgia's proud spirit was broken, hear her voice silenced and see her turn into a cringing, empty shell.\n\n'Goodnight, my darling,' Sister Mary bent down over Georgia and kissed her cheek. 'Remember me in your prayers sometimes, maybe write to me when you have the time.'\n\n'I'll come back and visit you,' Georgia said sleepily, her eyelashes dropping over her cheeks.\n\n'Just sing for me once in a while,' Sister wiped back a tear from her cheek. 'I'll hear you wherever I am. God bless you.'\n\nGeorgia was asleep by the time she got to the door. Her dark tight curls forming a black halo on the pillow, one arm curled round her head. In that instant Sister Mary saw a glimpse of the beauty which was to come. Coffee skin with pink undertones, perfect bone structure. Features too angular for a mere child of nine, but the basic materials for a real beauty.\n\nSilently she closed the door, pausing for one moment to compose herself.\n\nWhen Mrs Anderson saw the weals on the child's body tomorrow, she knew with utter certainty that the caring woman would act fast and without mercy. Perhaps out of one child's misery, many others would be spared.\n\n'Protect and keep her Lord,' she whispered. 'And give me the strength to deal with Sister Agnes.'\n\n## Chapter 2\n\n_September 1956_\n\n'Drop me off here Daddy!' Georgia's voice had a tremor of apprehension as they turned into Kidbrooke Lane and the playing fields of the comprehensive school loomed in front of them.\n\nIt was a hot sunny morning, vivid splashes of colour in the suburban gardens, dahlias at their best as if trying to outdo one another in their brilliance.\n\n'Don't you want me to come in with you?' Brian Anderson pulled up, turning towards Georgia in his seat.\n\n'I'll look like a baby if you do.'\n\n'You are our baby,' Brian chuckled. 'But I know what you mean. Some things are better tackled alone.'\n\n'Were you scared on your first day at a big school?' Georgia leaned against his shoulder for a moment, drawing strength from the smell of starched shirt and aftershave.\n\n'Terrified,' he admitted, patting her small hand with his big one. 'But it wasn't as bad as I expected, nothing ever is.'\n\n'I'd better go now,' she straightened up, then leaned closer to kiss his smooth cheek. 'Do I really look all right?'\n\n'All right! You look perfect,' he smiled, wishing he could cuddle her one more time and banish that worried frown. 'Off you go now, and don't worry about anything, there will be hundreds of other new girls, just like you.'\n\nBrian Anderson watched as she crossed the road and walked along the railings to the gate. Scores of other girls were filling the tree-lined avenue, peace halted now the new term had started. But Brian Anderson hardly noticed the other girls, his eyes were just on Georgia.\n\nIn two years she had changed almost beyond recognition. She was taller, her stick-like limbs had filled out with good food, the once cropped hair allowed to curl on her shoulders and her skin had lost that yellowy tinge.\n\nThe navy-blue pleated skirt swung beneath a smart new blazer and she wore her beret at a jaunty angle. Yet the sight of her childish brown legs in long grey socks and the stiff, shiny satchel on her shoulder brought an unexpected lump to his throat.\n\n'Make them accept you Georgia,' he said softly as he put his car into gear and pulled away. 'Just the way you did me.'\n\nBrian Anderson knew better than anyone how it felt to be different. Brought up alone with his widowed mother in the big house on Blackheath where he still lived, he understood a child's need to be just like everyone else.\n\nHis mother had meant well keeping him away from other children. She wanted to protect him from harm, wrap him in a cocoon of devotion. A small, select private school where rough games were frowned on, evenings spent reading with her by the fire, or long walks in the summer. He had allowed himself to be nudged into banking as a career. Girls, dancing, drinking or sport were things that men did who weren't gentlemen. Brian didn't consider himself weak at bowing to his mother's wishes. He was merely a loner who didn't need change, new experiences or even challenge. But sometimes he would have preferred to have had a more outgoing life.\n\nAs Brian drove down towards Lewisham across the heath he caught a glimpse of himself in the driving mirror. Sandy thinning hair, neatly combed to one side, a round, plump fresh face which had barely changed from his teens. Pale blue eyes with gingery lashes and eyebrows. A straight small nose and the kind of even white teeth which owed much to his mother's care and attention. Not a handsome man, but as his mother had always pointed out, 'Clothes maketh a man.' His suits were all hand-tailored, navy blue with a faint pin stripe for the bank, light grey for social occasions and a navy blazer for weekends and holidays.\n\nHis shirts always went to the laundry, he liked his collars stiff and starchy, his ties subdued. He had four pairs of identical black leather lace-up shoes which he rotated daily.\n\nHe looked what he was, a fifty-year-old, respectable, dependable bank manager, neat and industrious.\n\nThe traffic was heavy as Brian approached Lewisham High Street, he tutted with irritation, realizing that for the first time ever he was going to be late.\n\nHe parked his Humber in the side road close to the bank, took his briefcase from the back seat and hurriedly locked the car door.\n\n'Good morning, Mr Anderson!'\n\nBrian looked up at the sound of his secretary's voice.\n\n'Good morning, Miss Bowden,' he smiled. 'I'm afraid I'm a little late. I took Georgia to her new school this morning.'\n\n'Don't worry,' Miss Bowden didn't miss the frown lines on his forehead. 'I purposely didn't make you any appointments this morning until after ten thirty. I anticipated you might get held up.'\n\nMiss Bowden had been his secretary for five years now. A sensible spinster in her mid-thirties, she was as dedicated to her job as Anderson himself. Her dark suit and white blouse, the sturdy flat shoes and neat brown hair were a constant reminder to the other, younger clerks that this was how a woman in banking should look.\n\n'I just hope Coulson was on time,' Brian took up his position on the outside of the pavement, irritated still more by the amount of early shoppers pushing their way along to the market. 'It's so long since he was expected to unlock the bank, I doubt he remembers how to.'\n\n'Of course he does,' Miss Bowden reassured her employer. 'Look, you can see yourself the lights are on.'\n\nAnderson had no need to be at the bank before nine thirty, but old habits died hard for him, and often he was behind his desk soon after eight thirty, well before the rest of the staff arrived. It had been this sort of reliability which got him promoted to manager, and although Celia kept telling him it was time he sat back and took things easier, he still liked to be there to unlock.\n\n'How was Georgia this morning?' Miss Bowden asked. 'Was she nervous? It's a big step going to such a huge school.'\n\n'A little nervous, but she'll be fine once she's settled in.' Anderson's expression softened a little. 'Remind me to telephone my wife later, will you?'\n\n'What a lovely girl she is!' Miss Bowden smiled warmly as they approached the bank door and rang the bell to be admitted. 'She's a credit to you both.'\n\n'Well, thank you Miss Bowden,' Brian's plump face beamed at the compliment'. Sometimes he felt a little overshadowed by Georgia and it was nice to know his staff at least felt he was responsible for the way she had shaped up, 'It hasn't all been easy you know, but she's been worth the disruption.'\n\nNo one knew how much he'd dreaded having a child of unknown background in his home, Celia least of all. He hid it away, just the same way he did so many things. Celia was like his mother, it was easier to go along with her wishes than argue.\n\nNow it made him blush when he remembered the way he reported to friends and colleagues about Georgia's lacerated back on her arrival. He took all the credit for caring for her, implied he intended to move heaven and earth to get St Joseph's shut down.\n\nHe had been horrified by her injuries, but it was Celia who coped with it, not him. Why had he been so afraid that one small child would ruin their lovely peaceful home? Why had he sulked silently while Celia threw herself into her new mother role wholeheartedly?\n\nOf course, he hadn't known then what benefits one child could bring with her. Perhaps if he'd realized he would lose his tag of 'Boring Old Anderson' overnight, he might have been less truculent. It had been like joining an exclusive club. Suddenly he was no longer exempt from conversations centred on family life. His staff took more interest in him and for the first time in his life he felt fully accepted.\n\nMaybe it had taken a little longer to learn to be a real father than he allowed his colleagues to see, but it had its moments of wonder. Taking Georgia for walks, teaching her to ride a bike and do her sums, gave him a kick he hadn't expected. Women looked at him in a different way, stopping to speak to him. He felt powerful, a man of action, not just a sandy-haired, middle-aged man clutching a briefcase.\n\nSo maybe the magic didn't reach as far as Celia responding with any real passion to him. Neither did a half-caste child make up for one of their own. But at least Celia and himself had a common interest. She looked younger, prettier, she laughed more, cuddled up to him at nights. Maybe in time that new warmth would turn to desire.\n\nHis office smelled of fresh polish. A clean sheet of paper was in his blotter and his pens were arranged neatly on a desk tray. Soon one of the junior clerks would bring him in fresh coffee and due to Miss Bowden's thoughtfulness he had time to collect his thoughts and stop dwelling on Georgia.\n\nReminders were all around him. The little pen-wipe she had sewn for him last Christmas with Daddy embroidered on it. A painting of him, carefully framed by Celia. Once he would never have considered hanging a picture of a man with flame red hair on his office wall, but he secretly loved Georgia's image of him. She'd caught his hidden self, a strong-looking man playing cricket. Almost handsome in his white slacks and sweater. It was a talking point with customers, it loosened them up and made them realize he was more than just a stuffed shirt.\n\nFinally, there was the photograph of the three of them, taken on holiday in Bournemouth. Celia in a low-cut cocktail dress, he in a dinner jacket and Georgia between them, laughing up at them, all dark curls, big eyes and dimples.\n\n'Your coffee, Mr Anderson,' he hadn't heard Miss Bowden come in. She put his cup in front of him and placed his diary beside it. 'Don't worry about her,' she patted him gently on the shoulder. 'Georgia's a match for anyone, you know that, and don't forget to telephone your wife.'\n\nGeorgia looked up at the school as she approached the main doors, her stomach churning with fear. It was the biggest school in South London, all glass and concrete, and although she had assured both her parents some of her old friends from Junior school would be there too, the truth was that most of them had found places elsewhere.\n\nIt was easy to identify the other first-years. Like her their uniforms were brand new, they stood white-faced and anxious, biting back tears, far smaller than the girls who sauntered by shouting to their friends, throwing each other's berets into the air.\n\nGirls, who looked like grown-up women, wearing prefect badges on their blazers directed the new girls to the main assembly hall. Georgia looked round with trepidation as teachers called out names and ordered the girls to stand in line.\n\nThe top class of the Junior school had only twenty-five children. In this hall alone there were nearer three hundred and she couldn't see one person she knew.\n\n'Georgia Anderson.'\n\nShe put her hand up and was ushered over to a line.\n\nThe teacher who had called her name came forward smiling warmly. She was younger than Georgia had expected, probably no more than thirty, and she was very elegant. Her blonde, sleek hair was cut short and swept up at the back, and she wore a black suit with a straight skirt and a white lacy shirt. Her light-brown eyes seemed to miss nothing. She reminded Georgia of Miss Powell, the headmistress who had played the piano, and that seemed a good omen.\n\n'My name is Miss Underwood,' she said in a crisp, well-modulated voice. 'I'll be your form teacher and I'm taking you now to your form room where I'll explain everything to you. You are in form 1B, remember that if nothing else, someone will guide you back to your class if you get lost. Follow me.'\n\nGeorgia followed the other girls in silence. As they started up the stairs she turned to the girl behind her.\n\n'Do you know anyone here?'\n\n'No one.' The small girl was near to tears. She hardly looked old enough to be going to a senior school, her baby blue eyes, pink cheeks and blonde pigtails looking out of place amongst the hard-faced bigger girls they'd seen strutting by.\n\n'Neither do I, I'm Georgia Anderson. What's your name?'\n\n'Christine Fellows,' the blonde girl whispered back. 'Do you think we'll be able to sit together?'\n\nBy morning break Georgia had tried to memorize every face. Christine had been given the desk next to her and although they hadn't been able to talk yet, at least she seemed friendly.\n\n'Do you think we'll ever find our way round this place?' Christine sighed as they filed out of the form room for break. 'Every lesson's in a different room. What if we get lost?'\n\n'We'd better stick together then,' Georgia giggled. 'I shouldn't think they'd punish us for getting lost in the first week!'\n\nAs they came down the last flight of stairs the number of girls converging into a large hallway had reached hundreds. Everyone was talking at once, a heaving mass of navy-blue striving to reach the doors leading out to the playground.\n\nChristine clung on to Georgia's blazer as they reached the hall. Surrounded by taller girls pushing and shoving, they inched their way forward blindly.\n\nThe crowd cleared suddenly as they stepped outside into bright sunlight. Both girls paused, looking around for the milk.\n\n'Another nigger in the first year!'\n\nThe remark was said loudly, with malice. Georgia's head swivelled round to see a group of girls, all around fourteen, standing by the milk crates.\n\nThinking the insult was intended for her she blushed scarlet, stopping in her tracks. Christine didn't appear to have heard as she walked towards the girls and lifted two bottles out of the crate.\n\n'What's up?' she asked as she came back, giving Georgia hers.\n\nGeorgia barely heard her as she watched a small West Indian girl being pushed away from the crates by a sullen-faced big girl.\n\n'Niggers get theirs round the corner,' she snarled at the frightened first-year. 'This is for whites only.'\n\nThe girl was brassy looking, with untidy bleached-blonde hair and her tie pulled down. Although she was actually wearing the uniform, she had done as much as was humanly possible to disguise it. Her skirt was short and tight, a wide 'waspy' belt holding it up. The sleeves of her shirt were rolled up, a heavy bust stretching the material to its limits. She wore nylons and casual shoes instead of the strong lace-ups and grey socks Georgia wore. A lovebite on her neck and a spotty, pasty face all added to her slovenly appearance.\n\n'Do you think that's true?' Georgia whispered to Christine.\n\n'What?'\n\n'That coloured girls have their milk somewhere else?' She was torn between moving away into the crowd before someone noticed her colour, or joining the black girl in her defence.\n\n'I shouldn't think so,' Christine looked puzzled. 'My sister came here. She never said.'\n\n'They'll have to put a separate crate for me if it is,' Georgia tried to smile. 'Halfway between the two, a greyish colour.'\n\nBy the time Georgia and Christine had gone round the corner to investigate, the West Indian girl had vanished amongst hundreds of other girls, but there were no more crates and it was obvious the big girl was playing a cruel joke.\n\n'Don't get upset about it,' Christine said. 'My sister told me all sorts of things they do to first-years. She said they held one girl's head down the toilet and flushed it.'\n\nThe day passed in a blur of new experiences. Books handed out, timetables which seemed formidable, so many new faces and names to be put to them, that Georgia forgot the incident at morning break.\n\nGeorgia parted from Christine at the school gates after arranging to meet her there the next day. Then joining a minority of girls going towards Blackheath, she turned left and crossed the road.\n\nThe sun was hot on her head and shoulders so Georgia took off her blazer and carried it. Up ahead on the corner of the road she was to turn into, a crowd of girls was gathering. Georgia quickened her pace to see what was going on.\n\nShe knew it was a fight, she could sense the tension in the air before she got there and she recognised the voice without even seeing over the other girls' heads.\n\n'You sneaky little bastard. I'll teach you not to go telling tales!'\n\nIt was the same big girl who had stopped the West Indian from getting any milk.\n\nGeorgie sidled round the crowd, intending to go on home, but the sight before her made her stop instantly.\n\nThe big girl had the small black girl by the hair and was slapping her face backwards and forwards like someone beating a carpet.\n\nGeorgia dropped her satchel and blazer and without thinking she ran the last few feet.\n\n'Stop it,' she grabbed hold of the bigger girl's shirt. 'She's smaller than you and she's new!'\n\nOnly as the girl paused and let go of her victim did Georgia feel a stab of fear.\n\n'And who the fuck d'you think you are, bloody Joan of Arc?'\n\nA roar of laughter went up from the crowd. They were all white girls, mostly third and fourth years, three or four of them Georgia had seen with the bully at break, the rest merely going home and enjoying a little diversion.\n\nSuddenly the tree-lined suburban street with its neat gardens seemed sinister and a long way from home. Georgia knew she'd got herself into something beyond her depth.\n\n'I know it's none of my business,' Georgia said, more calmly than she felt. 'But it isn't right to hit someone smaller than yourself.'\n\nThe West Indian girl was backing away, her eyes rolling with fear, her face swollen from the smacking. But like Pamela that day at St Joseph's, she hadn't the sense to run.\n\n'Don't she talk posh,' the bully smirked round at her audience. She looked back at Georgia and her mean mouth curved into a sneer. 'Oh, I get it,' she said, looking Georgia up and down. 'You've got a bit of nigger blood too!'\n\n'Yes, I'm half black,' Georgia held her head up proudly. 'That's a darn sight better than being all white and a bully.'\n\n'Bin taken out of the jungle by a priest and educated, have we?' The girl caught hold of Georgia's wrist before she could move away. She twisted it round and forced it behind Georgia's back, holding her in a tight grip. 'Well here we've got our own jungle, and we don't want no black bastards in it.'\n\n'Let me go,' Georgia yelled, kicking out at the girl's shins.\n\nTaken by surprise the girl let go. Georgia used the opportunity to run but a greasy-haired girl stood in front of her, grinning stupidly. She stood a foot taller than Georgia, her loose, sloppy mouth full of gum.\n\n'I've got her now Bev,' she called out. 'Come and give her a pasting.'\n\nThree of them were on her at once. One girl held her arms, another one caught her by the hair, and the girl they called Bev, slapped her round the face again and again.\n\nGeorgia tried to kick them, but together they were too strong for her. All she saw before Bev lifted her leg and kneed her in the stomach was the West Indian girl running up the road like a startled hare.\n\nWinded, Georgia staggered back against a tree.\n\n'I ain't finished with you yet,' Bev shouted at her. 'That's just a taster to show you who's boss round 'ere. Got the message?'\n\nDoubled over with pain, Georgia heard them run off down the road, laughing loudly.\n\nThe rest of the crowd dispersed as if by magic. One moment they were all gawping inanely, the next gone.\n\nHer parents had been a little tense about her coming to this school, now she knew why. Her face stung, the blow in her stomach had winded her and she felt sick with humiliation. By the time she collected her things and walked to the bus stop, the streets had cleared of school girls. She wanted to cry, ring up Celia to collect her, make her promise she'd find another school. Yet even as she thought these things, she knew she couldn't.\n\nCelia was talking on the telephone as she walked in the door. She waved, then went back to her conversation. Everything was just as it always was. The sun shining in the back of the house, lighting up a vase of flowers on the kitchen table. A smell of polish, a casserole in the oven. The prints on the walls, the thick, patterned carpet, chintz covers on the chairs. A spacious, middle-class home, a thousand times nicer than the one that girl must come from.\n\nCelia was at the telephone in a crisp blue summer dress with a white collar, carefully cut to minimize her wide hips. The kitchen table was laid with dainty china tea cups and a homemade cake. If she told her mother what had happened it would bring a cloud into this lovely home, a slur on all they had taught her.\n\nGeorgia slipped upstairs and washed her face. She was flushed, but as yet there was no bruising. Ten minutes later she came back downstairs wearing an old pink dress she'd almost grown out of.\n\n'Hullo darling,' Celia was in the kitchen making a pot of tea. 'How did it go?'\n\nShe loved her mother so much. She couldn't bear to see hurt take the smile off her face or worry spoil even one evening together.\n\n'It's a bit scary because it's big,' Georgia said, keeping her voice even and taking a seat with her back to the window so her mother couldn't see her clearly. 'I met a nice girl called Christine and I'm in form 1B.'\n\n'That's good.' Celia sat down at the kitchen table, stirred the tea and poured a little milk into the two cups. She cut the large cherry cake and placed a slice on a plate for Georgia. 'That means you are actually in the Grammar stream then. What's your teacher like?'\n\n'Nice,' Georgia said, looking down at the cake. 'Miss Underwood. She's youngish. Real elegant and sophisticated. But we'll only have her for registration and English, the rest of the time we go to other classrooms.'\n\n'What's the matter with your face, it looks flushed?' Celia's eagle eye missed nothing.\n\n'I ran down the road,' Georgia lied. 'I'm just hot.'\n\nAfter eating the cake she went back to her room under the pretence of doing homework. Her bedroom was the prettiest she had ever been in. Decorated in pink and white, it had lots of shelves and cupboards, and Celia had even bought her a desk to do her homework on. It wasn't a large room compared with her parents' room next door, but it was on the front of the house and the window overlooked the heath across the road. Across the landing was her playroom. Even in her wildest dreams back at St Joseph's she had never envisaged a room where she would be allowed to paint, dance, dress up and do whatever she liked.\n\nHer mother and father had taught her everything. How to speak properly and talk to people, how to dress, everything she had came from them, how could she burden them with worry about a bully?\n\nSt Joseph's had only taught her one thing that she clearly remembered. You had to stand up for yourself, or end up being bullied forever.\n\nThe next morning she braided her hair tightly.\n\n'Why on earth are you doing that?' Celia asked in surprise. 'It looks so much better down.'\n\n'It's a bit hot for school,' Georgia replied. 'There's so many big windows, and I sit right by one.'\n\nThere was no one guarding the milk crates at break and all day Georgia didn't even get a glimpse of Bev and her friends. She hoped that was the end of it, but she would be cautious just in case.\n\nOnce she'd said goodbye to Christine outside the school gates, she took off her blazer and beret and put them in her satchel. She rolled up her shirt sleeves and crossed the road.\n\nShe saw the four third-years cutting across the playing field. The way they scurried along, heads bent close together, made her sure they were planning to ambush her. Flinging her satchel over her shoulder she ran round the corner past the spot of yesterday's incident and paused, weighing up the area.\n\nThe wide grass verge between the pavement and the road was ideal for a soft landing, the trees good for cover. Her heart was thumping with fright, but she hid her satchel behind one tree, herself behind another, and waited.\n\n'She must have run like the bleedin' wind.'\n\nGeorgia suppressed a nervous giggle as one of the girls' surprised voice reached her. She didn't dare look out until they were right by her.\n\n'Either that or she's hiding in some gateway till we've passed.' She recognized the coarse voice as Bev's, the blonde bully. 'I'll give her an extra pasting for that, the scheming black bastard,' she added maliciously.\n\nGeorgia managed a peep from behind the trunk. Three of the girls checked each garden. The girl with the greasy hair hung back slightly, as if her heart wasn't really in it. The other two were either side of Bev, copying her actions as though they had no minds of their own. One was quite attractive in a sulky, voluptuous way, with dark, long hair. The fourth girl in the group was mousey blonde, smaller than her friends with sharp features and a brace on her teeth.\n\nGeorgia waited until they were less than five feet from her, took a deep breath, screwed up her fists and stepped out in front of them.\n\n'Looking for me?' she said, balancing on her toes.\n\n'Yes, you little shit,' Bev was clearly taken by surprise, she blinked furiously, the late afternoon sun in her eyes.\n\n'Planning to hit me again?' Georgia tried to keep her voice low and seemingly unruffled.\n\nThe four girls looked at one another in surprise.\n\n'That's fine with me as long as it's fair,' Georgia said. 'One of you against me, the others mustn't join in.'\n\n'You cheeky little bitch,' Bev spat at her. 'Who the hell do you think you are?'\n\n'I don't think I'm anyone,' Georgia said stoutly, moving lightly on her toes like a boxer. 'I know perfectly well who I am. Georgia Anderson.'\n\nThe girl with greasy hair laughed nervously and was already moving back, away from the others.\n\nIn groups of twos and threes, other girls were turning the corner. A buzz of excitement went up, calling for others to hurry and in moments a large circle was forming round them.\n\nGeorgia was afraid now. There was nothing to stop all these girls ganging up on her. She would have to act tough to get their approval.\n\n'Well, which one of you is going to give me the pasting?' Georgia said, looking directly at each of the ringleaders. This time the dark, attractive girl fell back. Her face growing a little pale.\n\nGeorgia focused her attention on the mousey blonde.\n\n'You is it?' she said arrogantly, 'Or do you want to back out too?'\n\nThe girl had a look of panic in her pale eyes. She glanced round at the other two girls, who merely shrugged their shoulders, and fell back with them.\n\nBev stood alone now, a slight tic in her cheek as though nervous. She was big and heavy, but she didn't look quite so confident now she knew she had no backup.\n\nThe crowd was tense, everyone waiting to see if this new kid was really foolish enough to fight.\n\n'Looks like you and me then,' Georgia smiled charmingly, fixing her big, dark eyes on Bev's small, mean ones.\n\n'You cocky little cow,' Bev sprang heavily at her.\n\nGeorgia waited until she was almost upon her then jumped lightly to one side. Bev stubbed her toe on the edging to the grass verge and nearly fell on her face. A roar of approval went up from the crowd and Georgia turned slightly to grin at them.\n\nBev recovered quickly and turned to catch Georgia again. Once again Georgia dodged. Bev's hand came out to grab her hair, but her fingers couldn't grasp the tight braids. Quick as a flash, Georgia pulled up her knee and jammed it up into Bev's stomach, kicking out at her shins as she moved back.\n\n'Get her, Bev,' the small blonde shouted. She was poised on her toes, clenching and unclenching her fists with blood lust in her eyes.\n\nBev was winded. Her face flushed like a tomato. She lunged heavily at Georgia again. This time Georgia put her foot out, caught her legs and tripped her up. Bev crashed down on to the pavement, flat on her face. Georgia stepped forward to screams of delight from the other girls.\n\nLeaning over and catching the girl by her hair, she twisted Bev's head up and round to look at her.\n\n'Had enough yet?' she said almost casually, as she saw a trace of blood coming from an angry red mark on the girl's forehead.\n\nThis was the danger point. If the other girls leapt in now they could beat her to a pulp.\n\nBev struggled to get up. Georgia waited until she was on all fours, then swiftly kicked her up the backside, sending her crashing to the pavement.\n\nThe other three girls had backed right away now, clearly terrified of being involved.\n\nThe circle of spectators moved in closer.\n\n'Wack her, little'un,' someone shouted. 'It's about time someone stood up to her.'\n\nGeorgia sat astride Bev's back, holding the girl firmly by the hair.\n\n'Hurts doesn't it?' she asked, teeth gritted. 'It hurts black girls too, or did you think we feel nothing?'\n\n'Get off,' Bev called out, her voice shaky as if on the point of tears. 'We was only teasing you.'\n\n'And I'm only teasing you,' Georgia pushed her head back down to touch the pavement. 'But I'll stop if you apologize and tell me you'll leave me alone in future.'\n\n'Fuck off,' Bev shouted, wriggling and trying to turn under Georgia. An unpleasant smell of sweat wafted up to Georgia.\n\n'That is very rude,' Georgia said, grinning round at her audience. 'You smell of B.O. too. Looks like I'll have to give you another taste.' She pulled sharply on the girl's hair, then crashed her head down again to the pavement.\n\nThis time Bev was sobbing.\n\n'You'll apologize?' Georgia looked around at the crowd. 'In front of witnesses?'\n\n'Yes,' the word came out like a groan.\n\n'Right. Repeat after me. Bev is a bully. She is also a fat, smelly slut.'\n\n'Bev is a bully,' the girl whimpered.\n\n'Louder,' Georgia tightened her grip again.\n\n'Bev is a bully,' the girl said.\n\n'Go on!'\n\n'She is also a fat, smelly slut.'\n\n'Very good. I will never, or allow my friends...'\n\n'I will never, or allow my friends,' Bev was crying freely now.\n\n'To bully, or frighten anyone, especially black girls.'\n\nGeorgia waited until Bev had finished. Still sitting on Bev's back she looked across at the other three girls who cowered against the wall.\n\n'That goes for you three too,' she said, lowering her voice to one of menace as she'd been taught in drama classes. 'I'll be watching.'\n\nCalmly she got up, crossed over to the tree to collect her satchel, slipped her blazer out, and swaggered off towards the bus stop.\n\nShe allowed herself only one glance back.\n\nBev stood alone, crying and dabbing at her forehead. Her friends had vanished, the rest of the girls were standing talking in small groups.\n\nOnce on the bus she could not stop shaking. She had been lucky, if Bev hadn't been like a charging rhinoceros she would have noticed that it wasn't physical strength that beat her, but preparation and speed. If Bev or another bully caught her unawares the next time she might be the loser.\n\n##### *\n\n'Is everything all right Georgia?' Celia came up to her bedroom after tea as Georgia was doing her homework. 'You didn't seem yourself yesterday or today. Is there something you want to tell me?'\n\n'No, Mummy,' Georgia looked up and smiled. 'I was just worried about all this homework. I don't know whether I'll have time for dancing and singing now.'\n\nCelia sat down on the bed.\n\n'You'll make time.' She picked up the teddy bear she'd given Georgia on her first day in the house and looked at him thoughtfully. 'Come on, the truth. I know something happened at school. Has someone bullied you?'\n\nGeorgia hadn't expected much the day she left St Joseph's in Celia's car. There was no picture in her mind of a house, or the kind of life she would lead with Mr and Mrs Anderson. She remembered the moment when the car stopped, the huge expanse of snow-covered heath on one side of the road, and the grey stone houses on the other.\n\n'This is ours,' Celia had taken her hand again and led her up to a red front door. It seemed tiny after the convent door, little panes of coloured glass and the porch with old blue and white tiles. She had hardly noticed Mr Anderson, all she had seen and felt was warmth and comfort. Soft carpet under her feet, a big fire in the grate and the piano standing by the window.\n\nThose first few weeks had been so exciting. New kinds of wonderful food, clothes that were brand new and toys that were all for her. Later there had been the dancing and singing lessons to give her new heights of happiness. But above all else it had been having a mother, someone who cared about her, listened and talked to her as if she was someone special.\n\n'There was just a little trouble yesterday,' she admitted. She knew her mother too well, she wouldn't give up until she got to the truth. 'But everything's okay now.'\n\n'Someone slapped you! I knew it,' Celia stiffened, dropping the teddy bear in her hands. 'Why didn't you tell me?'\n\n'Mum, I'm a big girl now,' Georgia laughed. 'I can stand up for myself. I talked to the girl today, it's over.'\n\n'What was it about?'\n\n'My posh voice, if you must know.' Georgia wasn't exactly lying, but she thought her mother could take that better than the issue of colour. She grinned cheekily. 'Maybe I'd best go back to talking like what I used to.'\n\n'Don't you dare!' Celia smiled. 'After all the coaching I've given you!'\n\n## Chapter 3\n\n_December 1959_\n\nGeorgia hurried to the church. The grass on the heath was thick with frost and the moon hung over the church spire as if endeavouring to impale itself. It was the last practice for the anthem the choir was going to sing at midnight mass on Christmas Eve.\n\nShe wore a grey duffel coat over a polo-necked white sweater and jeans, hair tied up in a pony-tail with a white ribbon, a long red scarf knotted round her neck.\n\nPeter was waiting on the church steps. Just the sight of him made her heart beat a little faster. He was so beautiful, gold blond hair gleaming under the porch lamp, his peachy skin as clear as her own. She could hardly wait to get up close and see those forget-me-not blue eyes and his wide, soft lips.\n\n'I thought you weren't coming.' His face broke into a relieved smile as she turned on to the church path.\n\n'I got held up,' she said breathlessly.\n\nFour months had passed since they'd met at a youth club debate, and since then there hadn't been one day when she hadn't thought about him. Was it possible to want someone so badly and not have the longing returned?\n\n'Mr Grey's having kittens,' he grinned, his soft lips parting to show perfect white teeth. 'We'd better go in.'\n\nAs Georgia stepped into the church she closed her eyes for a second and inhaled deeply. She loved churches. The incense, the candles, all the rich embroidery on the altar cloths, the smell of polish and flowers. Religion didn't come into it. To her it was a wonderful theatre, the choir part of a show they put on each weekend.\n\nFlinging her coat on a pew she slipped into the choir stalls, grinning sheepishly at the others. Eight women, six men and eight scruffy little boys. On Christmas Eve they would be transformed with starched ruffles and red cassocks, but for now they were just ordinary people who liked to sing just like her.\n\nThe choir master tapped his stick on a pew.\n\n'I'm glad you could make it Georgia,' Mr Grey's deep baritone was at odds with his stooped elderly body. His sarcasm unusual for such a gentle man. He wore a new Fair Isle cardigan in heathery shades, his pipe hanging out of one of the pockets. 'Now take it slowly. It's not a pop song, but a beautiful piece of music. I want the people in the back rows to hear you. Head up, chest out.'\n\nIt was the first time anyone in the choir had been chosen to sing a solo. She knew it was a great honour and she wanted it to be perfect.\n\nShe took a deep breath as the organ wheezed into life. The introduction filled the church with sound and Peter winked at her.\n\nHer voice reached each corner. Pure and clear, every word annunciated in the way Mr Grey had taught her.\n\nThe choir joined her. Sopranos soaring above her contralto, the tenors and bass giving it richness and warmth.\n\n'Very good,' Mr Grey shuffled forward up the step. He held his back as if it hurt, but his old face was alight with pleasure. 'If you sing it like that on Christmas Eve I should think Father O'Brady will get enough in the collection for his new roof. We'll do it once more, then a quick run through the carols, then you can all go early.'\n\n'You were very good tonight,' Peter walked out through the church door with her. 'I love to hear you sing.'\n\n'Thank you.' She smiled up at him, wondering if tonight she could find the words to ask him to her party.\n\nHe was always waiting for her. He walked home with her from choir practice, talked about anything and everything, yet he had never attempted to take it further.\n\n'Do you have to go straight home?'\n\nHis question took her by surprise. Peter was looking at his feet, he sounded as unsure of himself as she felt. 'I mean, could we go for a walk?'\n\n'Where?' she asked, not caring where it was as long as he was with her. She felt a flush creeping up her neck. Her teeth began to chatter more from anxiety than cold.\n\n'Over to the boating pond?'\n\nThe heath yawned in front of them. A big, empty dark space that was all theirs. A huge Christmas tree at the church steps lit up the darkness with tiny green, red, yellow and blue sparks of colour. The frosty grass scrunched beneath their feet and as they moved away from the light, so their shadows disappeared.\n\n'You're cold?' Peter paused and looked round at her.\n\nHer scarf was tied tightly round her neck, her breath like steam from a kettle.\n\n'My hands are,' she said, not wanting to admit she was freezing. 'I forgot my gloves.'\n\nHe took one of her hands and felt it.\n\n'Like ice,' he smiled. 'Put it in my pocket with mine.'\n\nHe held her hand in his pocket, running his thumb across her palm. A tiny shiver went down her spine, but this time it had nothing to do with the cold. She moved closer to him, huddling against his shoulder.\n\n'Better now?'\n\n'Much,' she smiled up at him. His ripe wide mouth made her feel weak inside. 'I've been meaning to ask you Peter. Would you like to come to my birthday party on January 6th?'\n\nHe didn't reply for a second, he looked straight ahead of him and she wondered if she'd asked too soon.\n\n'I thought you'd left me out. One of the boys at school mentioned it.'\n\nNow she felt foolish. Did he think she was only inviting him now out of politeness?\n\n'I didn't actually invite any boys,' she blushed. 'I just asked the girls to bring a partner.'\n\n'Does that mean I'd be your partner?'\n\n'Yes. If you want to be.' It was too late now for flirting and pretending disinterest as Christine suggested. 'I didn't ask you before because I was afraid you'd refuse.'\n\nShe hung her head, afraid to meet his eyes.\n\nHis fingers brushed her cheek as he lifted her face up to his.\n\n'Does that mean I can say you are my girl?'\n\nNo words came, just a nod of her head. His eyes almost closed and his hand cupped her head drawing her to him.\n\nHis lips touched hers tentatively, so light it could have been the touch of a moth's wing.\n\nClosing her eyes she just stood there, her heart pounding, her legs shaking. One moment his other hand was still in his pocket with hers, the next he withdrew it and crushed her to him, lips covering hers.\n\nThe deserted heath, the church behind them and her home in the distance fell away. All she could feel, see and smell was Peter. A soft, warm mouth on hers, the touch of stubble against her chin, the ecstasy of being in his arms at last.\n\nFour months of dreaming and hoping and at last the moment was here.\n\n'Let's run?' he whispered to her, his nose rubbing against hers. 'Maybe we won't notice the cold.'\n\nThe wind caught her hair and scarf as they ran hand in hand. They were laughing like small children, racing over the crisp grass.\n\n'I knew there was a shelter here,' he said breathlessly as they approached the silver pond. He pointed to a dark shape at one end, near a bus stop. 'It might not be so cold and at least we can sit down.'\n\nWithin seconds she was in his arms again. The soft inexperienced kisses soon becoming more adventurous and bolder.\n\nThey weren't aware of a man walking his dog, or the lone streetlamp casting a pool of golden light over a litter bin. The shelter smelled of mould and someone's abandoned chips in newspaper, but all they felt was one another's warm breath and the sweet agony of needing to get closer.\n\nHis tongue flickered over her lips, and she parted them, slipping her hands under his jacket for warmth.\n\nShe pressed closer to him, a warm, shaky feeling creeping all over her. Her breasts throbbed, she ached for him to touch them, yet was frightened that he would. Each kiss was longer than the last, tongues bolder, gaining experience with each one. Her body fitted to his, her fingers stroking, loving him. The hard boniness of his chest, the smell of soap and toothpaste. His fingers caressing her neck and the rough texture of his sweater.\n\n'We ought to get back,' he whispered, his lips buried in her neck. 'It's nearly half past nine.'\n\nReality came back with a jolt. Georgia jumped up, holding her watch towards the dim yellow light. Her eyes widened with fear as she saw he was right.\n\n'Dad will go mad,' she gasped. 'It feels as if we've only been here for minutes.'\n\nPeter stood in front of her, buttoning up her coat and winding the scarf back round her neck.\n\n'It's only just after our usual time,' he sounded calm and protective. 'Tell them we were talking.'\n\nThey ran then, hand in hand back across the heath, not stopping till they reached her house.\n\n'Ask them if you can come to the pictures tomorrow,' he said, smiling down at her, both panting from the run. 'I'll come and pick you up at seven.'\n\n'What if they say no?' she was torn between staying out with him and rushing in to make apologies.\n\n'I'll come anyway,' he laughed, bending to kiss her once more. 'Now go on in before you catch cold.'\n\n'Why didn't you ask me out before?' she whispered, poised to run in.\n\n'I was afraid you'd turn me down,' he whispered back.\n\n'You're late!' Celia said reprovingly.\n\nHer parents were watching television by the fire. The Christmas tree lights twinkled against the dark red curtains. Celia was already in her dressing-gown, pale blue wool, with a snippet of long winceyette showing beneath, her feet in slippers. She was knitting a pair of grey socks. Brian wore the brown cardigan he always put on when he took off his office suit, yet his tie was knotted as neatly as when he left for the office earlier that day. He had a glass of brandy on the small table by his side and he looked sleepy, glasses sliding down his nose.\n\nBy day the room was almost an extension of the garden, light streaming in the French windows, bushes just outside blending with plants inside. But by night it took on a different character, shrinking in size as the heavy curtains were drawn. A snug room that somehow embodied her parents' joint personalities. Celia in the baby grand piano, the Chinese vase lamps and the warmth of the roaring fire. Brian the plump, chintz-covered armchairs and settee, the delicate water colours on the walls, the leather-bound books close to his elbow.\n\nGeorgia looked from Celia to Brian as she unwound her scarf and unbuttoned her coat.\n\nHer father's love of order ran not only to arranging books in size, the fringe on the hearth rug brushed out flat, but also to timekeeping. Yet for once he didn't seem aware she was late.\n\n'I'm sorry,' Georgia panted. 'I was talking to Peter.'\n\n'Did you pluck up courage to invite him to the party?' Celia raised one eyebrow, letting her knitting droop to her lap. Georgia had spoken of this boy so often she felt she knew him almost as well as her daughter.\n\n'Yes,' Georgia wanted to sit on her mother's knee, wrap her arms round her and tell her everything. But the child in her was gone now, left back at the church steps when Peter took her hand. 'And he asked if he could take me to the pictures tomorrow.'\n\n'Did he now,' Celia's eyes were more green than grey in the light of the fire, twinkling like the Christmas tree lights.\n\n'Can I go then?'\n\n'I don't see why not,' Celia's soft pale lips curved into a smile. 'As long as he brings you straight home afterwards.'\n\n'Just a moment,' Brian sat up sharply and took off his glasses. 'Don't I have any say in this?'\n\nGeorgia gulped. She and Celia rarely asked Brian's opinion about anything and just lately he seemed to have noticed. His face had a polished look in the soft light, faded blue eyes puckered with irritation.\n\n'I'm sorry, Daddy,' Georgia went over to him and perched on the arm of his chair. His sandy hair was getting very thin and from her position slightly above him, she could see a bald patch as big as a half crown. She slid her arm around his neck, fondling his ears. 'Please don't be a grouch. I didn't ask you first because I was embarrassed.'\n\n'Who is this boy?' Brian's eyes softened just enough for her to know he was at least receptive.\n\n'Peter Radcliffe, he sings in the choir. You'd like him Daddy! He plays cricket.'\n\n'I'll reserve my judgement until I've met him,' Brian half smiled. 'If he isn't one of those ton-up boys and he can be trusted to behave himself with you, I can't really think of any reason to say no.'\n\nAs Georgia got into bed that night she could scarcely contain her excitement. If she closed her eyes she could taste Peter's lips again, feel that strange tugging sensation inside her.\n\nWas she in love? She had all the symptoms they mentioned in magazines. She could hardly wait till tomorrow to phone Christine and tell her Peter had finally kissed her.\n\nShe lay on her back looking at the new party dress hanging on the wardrobe. Celia had bought it just two days earlier and she couldn't wait to wear it.\n\nIt was beautiful. Red satin with a billowing underskirt of net. The bodice was tight and low cut, with just tiny cap sleeves. No one else would have a dress quite like it.\n\nCelia smiled as she peeped into Georgia's room an hour later.\n\nBrown arms hugging a teddy bear, face buried in the pillow. She had noticed the way Georgia looked at this boy. Guessed the big sighs and long periods of idle dreaming were because of him. Tomorrow she would want endless reassurance that he really did want to take her to the pictures!\n\nCelia closed the door softly and went into her own bedroom. Brian was already in bed, covers up to his chin.\n\n'What's this boy like?' he said unexpectedly.\n\n'Very nice,' Celia said sitting down on the edge of the bed. 'Polite, rather handsome, about to do his \"A\" levels.'\n\n'She's a bit young for a boyfriend,' he said.\n\n'I don't know,' Celia climbed into bed beside him. 'Most of her friends have one. Better someone like him than some hooligan hanging around on street corners. Don't get all edgy about it Brian, teenage romances are usually shortlived.'\n\nAs she switched out the light Brian turned over and put his arm round her.\n\n'Don't,' she said brusquely, moving away.\n\n'I only wanted to cuddle you,' his voice sounded peeved.\n\n'You always say that,' she brushed his arm away. 'And I'm not in the mood.'\n\n'Are you ever?' his tone was heavy with sarcasm, and with a sigh he rolled over.\n\nCelia lay there in the dark feeling just a little guilty. So much of their marriage was good, but she found it difficult to respond to him.\n\nMaybe that was why she wanted Georgia to have lots of boyfriends because her own experience was so limited.\n\nCelia Tutthill had always been a 'sensible' girl. Clothes chosen for their hard-wearing qualities rather than style, her hair cut short to save bother, swotting for her exams while others danced the night away and fell in love.\n\nShe came to this house as a lodger already set into spinsterhood at twenty-seven.\n\nMartha Anderson and her bachelor son Brian were wrapped up in each other and for over a year the only contact she had with either of them was when they passed on the stairs.\n\nIt was only when old Mrs Anderson became ill that Celia got involved. Getting shopping for them, helping Brian in the garden, occasionally giving the old lady her medicine and helping her out to the bathroom when Brian was away on business. Martha could be a tyrant, she had kept her son on a tight rein all his life, but Celia was touched by his devotion.\n\nMartha died suddenly one evening. One minute Brian was reading her the paper, the next she was dead, lying back on her lace-trimmed pillows, her wrinkled face suddenly younger.\n\nIt was fortunate Celia was off duty. She heard Brian cry out from across the landing and by the time she reached Martha's bedroom she found him sobbing, his head on his mother's breast.\n\nShe told herself she would stay only until he had got over his grief, then find another home. But without old Martha Anderson bullying him, Celia began to see another side of the lonely bachelor. He was capable yet sensitive, his gentleness was like a soothing balm after a day on the busy ward. She found herself looking forward to her weekends off, accepting his offers to share a meal, to go to the cinema, or even just listening to music together. When he asked her to marry him it seemed a perfect match. They both had their careers, and she could stay on in the house she'd come to love.\n\nIn her na\u00efvety Celia hadn't fully considered what marriage meant. It came as a shock to discover that the sensitive gentleman was also sensuous and demanding.\n\nA glimpse of stocking tops. A hint of nudity. A picture of something titillating in the paper, coupled with a drink or two would arouse him. Before she knew what was happening he was grappling with her, his mouth slobbering over her and suggesting things that made her flesh crawl.\n\nIn the new year of 1946 things came to a head. He was angry at being passed over for promotion, perhaps ashamed he'd spent the war behind a desk. But when he began to taunt her with her frigidity, blamed her for not producing a child, she felt leaving him was the answer.\n\nWhile Brian was away on a course in Brighton, Celia saw it as the perfect opportunity to make the break.\n\n'I can't go on the way things are,' she wrote. 'I blame myself because I can't respond to you the way you'd like. Perhaps I was never cut out for marriage. I care for you deeply, but I know that isn't enough. If I leave you now, maybe you will find happiness with someone else.'\n\nBut for once Brian surprised her by being unpredictable. He came home the moment he got the letter, catching her packing.\n\nTo this day she could see his face. Weak mouth quivering, eyes full of unshed tears. For once his appearance less than impeccable. He begged her to stay, insisting he wanted her on any terms.\n\nLater that year, Brian was finally promoted to manager of the bank in Lewisham High Street and Celia gave up nursing and became a children's officer in South London. They learned to compromise. She tried harder to please Brian, he didn't press her so often. When Georgia arrived, new happiness and purpose made her more loving, at times their lovemaking was tender, if not passionate. But Brian didn't seem to understand that she found it impossible now Georgia was older. Suppose she heard them? Somehow it seemed indecent at their age!\n\n'I want Georgia to marry for love,' Celia said to herself in the darkness. 'Just liking someone and sharing a home isn't enough for anyone.'\n\n##### *\n\nStrains of the Everly Brothers wafted down the stairs.\n\nThe door bell had been ringing constantly since eight that evening. Trudging feet, peals of laughter, shouts and giggles made them feel the playroom must now be packed to capacity.\n\n'Do you think I ought to go up there?' Brian looked up from his book. His round, plump face seemed more irritated than anxious, but his eyes held suspicion.\n\n'No,' Celia frowned. She had put on a new turquoise wool dress, her hair had been set that afternoon, just in case Georgia asked them to come up later. 'She won't get up to anything bad, and she knows where we are if anyone else does. This is her first grown-up party, don't spoil it for her.'\n\n'Who said anything about spoiling it?' His look was surly. He too had put on a clean shirt and his best suit and it felt rather silly being dressed up with nowhere to go. 'But they could be necking up there. We hardly know any of them! As for that dress!'\n\n'There's nothing much wrong with a bit of kissing,' Celia snapped. 'And she looks lovely in the dress.'\n\n'It's too adult,' he snapped at her. 'What are you trying to turn her into?'\n\nGeorgia had come into the sitting room to show it off, her hair put up in an elaborate French plait, tiny kiss-curls on her forehead. Toffee-coloured shoulders, her waist no bigger than a handspan and small breasts peeping out of the tight bodice.\n\n'She looks beautiful,' Celia retorted. 'I know it's sad to see her leave behind little white socks and pigtails. But she's a young woman now. We can't hold her back.'\n\n'I don't like her seeing so much of Peter either,' Brian snapped. 'After tonight I'm clamping down. He's been here almost every day since that time he took her to the pictures.'\n\n'Term starts again on Monday,' Celia said gently. 'He'll have homework and exams to think about. Georgia will be back to her dancing and singing lessons. What is it Brian,' she got up and went over to his chair, perching on the arm. 'You've been grumpy for days. Is it just the party? Or is there something else?'\n\n'Nothing you'd understand.'\n\n'Let's have a drink?' Celia ignored his cryptic reply. She got up and went over to the drinks cabinet.\n\n'Is that a good idea?' Brian raised one eyebrow.\n\n'Of course we can!' Celia looked over her shoulder, amused by his worried expression. 'It's our home, our daughter's birthday. I wasn't suggesting we got plastered.'\n\nShe poured herself a small gin and tonic, but gave Brian a larger one, hoping it would make him relax.\n\n'I don't like Georgia wearing so much eye makeup,' he snapped. 'It makes her look cheap.'\n\n'Don't be such a wet blanket,' Celia returned to her seat by the fire. Any other time she would have found his attitude rather touching, a father afraid that his little girl was turning into a woman. But now it merely irritated her.\n\n'Put a record on,' Celia said. The overhead light was swinging with the dancing going on overhead. They couldn't actually hear the music, just the thumping vibration of the bass notes.\n\nBrian got up slowly. He had put on some weight recently, although his good dark suit covered it well. As he bent down by the radiogram she could see he was getting a paunch which hung over his trousers. Yet even the extra pounds added something. He was actually improving with age. A few lines gave his youthful round face more character. Even the streaks of grey in his sandy hair gave him a note of distinction. Shame he had no remarkable features, his faded blue eyes were small, his nose just a fraction too wide, even his chin and mouth were weak. But he had good skin and neat straight teeth. A perfect face for a bank manager, not handsome enough for anyone to consider he might be indiscreet. Reassuring, neatly average.\n\n'I wonder if any of the boys can \"jive\" as well as Georgia and Christine?' Celia wanted to lighten the mood but she was running out of ideas.\n\nHe didn't comment, just took a record out of its sleeve and dusted it carefully.\n\nCelia was just gritting her teeth at his finicky manner when the phone rang out in the hall.\n\n'I'll get it.' She got up, moving to the door. 'I expect it's someone's mother checking what time the party ends.'\n\nBrian was back in his chair, drink in hand and the opening bars of 'Swan Lake' filled the room as Celia came in.\n\n'That was the police.' The relaxed wife and mother was replaced by a stern, committed social worker. Even her voice was crisper. 'I'll have to go. A child in Stepney has been hurt by his father. The rest of the family are at risk.' She reached behind an armchair for her briefcase and looked down at her clothes as if wondering if she was suitably dressed.\n\n'Isn't there anyone else in the world but you?'\n\n'Not tonight it seems,' she didn't notice his sarcasm, more concerned with a five-year-old with a fractured skull. 'I don't know how long this will take. I might even have to bring the kids back here till tomorrow. I hope Georgia won't mind.'\n\n'What about me?' Brian asked. 'Why don't you ask if I mind?'\n\nThis was the last straw. Everything revolved around Georgia, what she wanted to eat, where she wanted to go. Not once had Celia thought to consult him on anything. Now she was walking out with a hundred feet stamping on the floor above him, more concerned with some damn slum kid than her husband left on his own.\n\n'Don't be silly,' she bent to kiss him lightly on the cheek. 'You know perfectly well I'd rather be here with you. But it's my job, just like you'd have to go out if they robbed the bank tonight.'\n\nShe was gone before he could even think of a reply.\n\nTwo boys lounged on the stairs, below them sat a couple of girls talking with their heads close together. Celia hadn't met any of them before.\n\n'Everything all right?' she asked as they moved to let her past. 'Don't drop that on the carpet,' she looked disapprovingly at one of the boys' cigarettes dangling from his fingers.\n\nGeorgia was jiving with Christine, the red dress flaring out showing the layers of net and slim, shapely thighs. Her golden brown face, neck and shoulders glistened with perspiration, her dark eyes full of excitement. The French plait, which she'd spent hours arranging was already the worse for wear. Stray wispy curls were coming loose giving her a tempestuous look, like a gypsy dancer.\n\nChristine had changed since the early days at Kidbrooke. Still shorter than Georgia and a little plumper, but the baby face had gone. Her blonde 'beehive' quivered as she danced. She had tossed off her high heels and her shoulders were parting company with her low-necked turquoise dress, even the elaborate Eygptian make-up was smeared.\n\nGeorgia grinned as she saw Celia peering in the door, she beckoned for her to come in.\n\n'I can't stop,' Celia was shocked to see the room full of smoke and she was sure there were more than the twenty people her daughter had invited. Gangly lads propped up the walls, more sprawled on the floor. One girl was sitting on a boy's lap, their mouths glued together. A bunch of girls were giggling around the food table and still more were dancing. 'I've got to go out on an emergency.'\n\n'Oh, Mum!' Georgia's mouth turned down at the corners. 'I hoped you were going to come up and meet all my friends.'\n\nGeorgia's plaintive voice cut through her professional concern as Brian's never could.\n\n'I'm sorry, darling,' Celia tweaked Georgia's cheek. 'You know how these things are. I wanted to join you too. But I'm sure all your friends will enjoy themselves far better without me cramping their style.' She turned and smiled at Christine who offered her a sandwich.\n\n'No thank you dear, I really must go. I may even have to bring some children with me if there's no alternative. So make sure everyone leaves at twelve and don't let them disturb the neighbours. Daddy's downstairs if you want anything.'\n\n'Hallo, Mrs Anderson,' Peter was at her elbow. He was wearing smart grey trousers and a white shirt, the first time she had seen him in anything other than jeans. He had the same flushed, happy expression as Georgia, but mingled with concern. 'Is there anything I can do?'\n\nCelia liked Peter the first moment she met him. It wasn't the handsome face, the clear eyes or even the obvious intelligence. There was a kind of openness about him she found refreshing. The odd remark about his parents suggested they were more interested in his earning ability than scholastic achievements, his appreciation of her home was a pointer to his own being far more humble. Yet he didn't ingratiate himself, retaining his own character and belief in himself, while he soaked up information, from food he wasn't accustomed to, to Celia's work in Stepney.\n\n'Help Georgia to keep things under control,' she gave him a stern look, just to remind him of his place. 'Mr Anderson's downstairs. I'll be back just as soon as possible.'\n\n'Don't worry about us,' he glanced around the room as if already checking for trouble. 'Drive carefully won't you? It's frosty out there.'\n\nBy the time Celia had collected her coat from the bedroom, Georgia was dancing with Peter. She paused on the landing for just a second. Peter was no dancer, he shuffled awkwardly, barely in time to the music, but then her daughter's eyes were on his face, not his feet and the way Peter smiled at Georgia brought a lump to Celia's throat.\n\nBrian felt restless. There was nothing he wanted to watch on television and the noise upstairs was getting to him. He opened the drinks cabinet again and poured himself another drink.\n\nHe was almost glad Celia had to go out, at least now he had something tangible to base his anger on.\n\nIt wasn't the party upstairs, nor even the fact he hadn't been consulted about anything that bugged him. Neither had his mood just come on him.\n\nHe was well past fifty, hair thinning, body fatter. His staff called him 'Old Anderson' as if he already had one foot in the grave and sometimes he felt so lonely he wanted to scream.\n\nCelia was to blame. If she could arrange parties for Georgia, why couldn't she throw one or two for them? Her life was full, she had friends at her office, her clients from all walks of life, and she had Georgia.\n\nOnce she'd invited people to dinner, bought tickets for the theatre. They got invited out, they took walks together, they shared things. But lately the phone never rang except for Georgia or Celia, muffled conversations that made him feel shut out. How long was it since Celia cooked him a special meal, asked him what he'd like to do at the weekend? She only played the piano for Georgia and she cringed away from him as if he had leprosy.\n\nWhat had he got for being easy-going? A job he'd been pushed into. His youth lost in caring for his mother, a house that cost a fortune to run. A child who was someone else's reject and a frigid, domineering wife.\n\nHe poured himself another drink, curling his fingers round the glass and wincing at the fire.\n\n'I should have sold up when Mother died,' he said aloud. 'Travelled, changed my job, had some fun.' He glanced up at his parents' wedding picture on the mantelpiece as if half expecting her to reach out and box his ears for even thinking such thoughts. She looked pretty and guileless, gazing up at her uniformed husband in a classic pose of adoration. Was it becoming a war widow and bringing up her son alone that turned her into such a tyrant? She had approved of Celia, the only woman he ever heard her praise. That alone should have warned him off!\n\nHe could remember turning at the altar rail as Celia walked up the aisle on her uncle's arm, his heart almost bursting with pride.\n\nStout-shoed, tweed-skirted Nurse Tutthill, the plain, sensible girl who'd been there for him when he needed a friend was gone. In her place was a new Celia, curvy and feminine in a dark green, peplum-waisted costume and matching veiled hat. A soft voluptuous mouth accentuated with lipstick, shapely legs in sheer nylons, high-heeled shoes and a permanent wave.\n\nIn that moment he thought he'd got everything. Her husky voice promising to 'have and to hold', a waft of jasmine scent, and that small hand quivering as he slipped on the ring.\n\nBut just hours later in the hotel in Brighton that glow of pleasure turned to shame.\n\nHe could remember every detail of that night. The pale green satin eiderdown, the bedside lamps with silky tassels on the shades. The shiny walnut headboard, even the smell of the starched sheets.\n\nMaybe he was guilty of rushing things, but what man wouldn't when his fingers touched big firm breasts under silk?\n\n'Let me put my nightie on?' she whimpered.\n\n'You don't need clothes,' he said burying his face in her neck as his fingers fumbled to unwrap her.\n\nOn the train going to Brighton he had imagined taking her clothes off piece by piece, kissing every inch of soft flesh. Maybe he had only paid for sex before, but he thought he knew how to please women.\n\nHe turned out the light because she was embarrassed, sure that in darkness she would respond as passionately as him. But as his chest covered her naked breasts he couldn't hold back. Her skin was so silky he forgot caution, within moments he was pushing into her, squeezing her plump buttocks, whispering things he said to prostitutes.\n\nIt was only after that he realized she was icy, her face turned from him, every muscle and nerve-ending rigid with disgust. He touched her cheeks and found tears and in that moment he felt a complete failure.\n\n'I tried to please her,' he said aloud as he reached out for the bottle of gin.\n\nIt very nearly ended after the war. No more opportunities to get away and find a more amenable partner for the night. So much pent up excitement in the air as the troops came home, rebuilding all around them, yet for Brian everything stayed the same. Celia wanted a baby, he wanted promotion. A world war was fought and won against all odds, yet still he had a wife who stared blankly at the ceiling while he made love to her. Never actually refusing, but somehow that dutiful subservience made him feel dirty.\n\nPerhaps he should have let her leave back then, divorce wasn't such a big deal anymore.\n\nHe felt that uncomfortable feeling of frustration now. Yesterday at the bank it had been so strong he almost went up to the West End after work. Eight hours of working alongside ten women, watching breasts jiggle as they typed. Miss Baldwin the new clerk with her tight skirts and long slender legs curling round her stool as she served customers. At the Christmas party she'd wanted to kiss him, but always he had to be aware of his position. Clerks, assistant managers, they could have affairs, but not the manager, especially one with a social worker for a wife.\n\n'It isn't fair!' he looked up at the ceiling. The light was still quivering, but the music was softer now. All those kids up there, kissing, cuddling, their whole lives ahead of them. Was it wrong for a man of fifty to want a woman who liked lovemaking? Celia understood every wife-beater, every petty criminal, cared for the sick, the lame and the mentally unstable, why then couldn't she see what she was doing to her own husband?\n\nLaughter on the stairs made him sit up sharply. The record had long since finished, yet he hadn't noticed. He peered at his watch. It was twenty to twelve and he'd drunk half the bottle of gin.\n\n'She won't be back for hours,' he sighed, getting up unsteadily. 'I'd better go and see what they're doing upstairs.'\n\nThe furniture seemed to be in the wrong places. He banged against the settee, almost falling and he had trouble opening the door. It was cooler in the hall and only then did he notice he was still carrying the gin.\n\nChristine was smooching with John on the stairs. One rounded shoulder was free of her dress, an inch of white dimpled flesh above her stocking tops showing as Brian reached the first step.\n\n''Allo, 'allo,' he grinned. 'What 'ave we got 'ere?'\n\nChristine jumped a few inches away from the boy, pulled down her skirt and blushed charmingly. The earlier artful hairstyle was more like a tousled bird's nest now. Blonde tendrils surrounding her mascara-smeared face. She looked like a fallen angel, her blue eyes wide with surprise.\n\n'Hallo, Mr Anderson,' she said. 'Come to join the party?'\n\nThe boy stood up and Brian saw he had an erection.\n\n'I thought I might liven things up,' Brian smirked at the pimply-faced boy. 'But don't let me interrupt you!'\n\n'Can I have some of that?' Christine giggled, fluttering her eyelashes and pointing to the bottle.\n\n'I'll give you a drop in return for a kiss!'\n\nChristine picked up a glass just behind her and held it out to him.\n\nBrian lunged forward at her, almost tripping. The boy caught his arm.\n\n'Steady on!'\n\n'I'm just a little squiffy,' Brian said, plonking a wet kiss on Christine's cheek. 'But it's my daughter's birthday so why shouldn't I be?'\n\nHe glugged some of the gin in Christine's glass, then moved on past her, leaving the pair of them whispering behind him.\n\nThey were playing that record again, the same one Georgia had played almost non-stop since Christmas Day.\n\n'Till I kissed ya'. He'd found himself humming it in the car, singing it in the bath, but it had never sounded so good before.\n\nHe paused on the landing by the open playroom door. A small table lamp over in one corner was the only light. Four couples were shuffling round the floor, girls' heads buried in their partners' shoulders. The decorations were drooping down low over their heads, cigarette smoke swirled lazily up to the ceiling and the other twenty or so kids sat around the edges of the room locked in one another's arms.\n\nBut Brian barely noticed anyone. All he could see was Georgia in the middle of the room, Peter's mouth coming down to hers.\n\nHer upturned caramel face glowing in the soft light, full red lips open slightly, her breasts straining against the tight red bodice.\n\nNo woman had ever looked like that for him. He felt a quiver run down his spine, a stirring inside him.\n\n'Come on in then!' Christine was at his elbow, drawing him into the room.\n\nGeorgia barely moved. She turned, smiled at him, then looked back at Peter.\n\nBrian felt a rush of jealousy. Next to Peter he felt dwarfed, fat and old. The golden hair, the firm resolute chin, those bright blue candid eyes all irritated him.\n\nThis boy had it all, looks, brains and now Georgia. He would never get to fifty and wonder what had happened to his life.\n\n'Hallo Daddy,' she took Peter's hand and walked towards Brian. 'It's not twelve yet is it?' She kissed him just as she always did, one small hand touching his cheek lightly.\n\n'Thought I'd see what you are up to,' he lurched to one side, holding the wall for support. 'Are there any spare girls for me?'\n\n'Daddy!' Georgia frowned reprovingly. 'You've been drinking!'\n\n'Only one or two,' he said. 'I brought some up for you too.'\n\nShe half smiled, but Peter frowned.\n\n'Gin, Mr Anderson! Are you sure?'\n\nPeter's sanctimonious expression irritated Brian still further.\n\n'A man can drink in his own house,' he said sharply.\n\n'I'd like some,' Christine was already holding her now empty glass out to him, batting her eyelashes flirtatiously. Brian poured her a little, then himself one, and went round putting a splash in anyone's glass that was held out.\n\n'Let's have a dance Dad,' Georgia was worried now, he had been odd and moody all over Christmas and if her mother came home and found him like this, it might turn into a row.\n\n'That's nice, a dance with Daddy,' Brian held out his arms.\n\nGeorgia glanced around the room. She could sense everyone thought it was funny to see her straitlaced father drunk, but she was embarrassed.\n\nHe was a good dancer normally, with a fine sense of rhythm. Another time she would have been glad he wanted to join in, but she didn't like the way he clutched her to him, the fumes of alcohol on his breath, or being prevented from dancing with Peter.\n\n'Why don't you go back downstairs,' she wheedled with him. 'The party's nearly over now and I want to be with my friends for a little longer.'\n\n'Don't be like that,' he pouted, holding her still tighter. 'I don't get any fun.'\n\nChristine came up beside them.\n\n'Can I have some more gin please?' she asked. Her dress had split on her hip and creased across the stomach. She looked like a girl he picked up once in Church Street, Kensington.\n\n'Certainly,' Brian spun round, releasing Georgia, picking up his bottle and pouring a big measure into Christine's glass. 'You dance with me Chris? Georgia doesn't like her Daddy up here.'\n\nGeorgia took the opportunity to slip away, leading Peter out to the landing.\n\n'I'm sorry,' she said hanging her head. 'He isn't usually like this. I don't know what's got into him.'\n\n'Just booze,' Peter smiled and took her in his arms. 'My dad's like that half the time, you don't have to apologize.'\n\n'But he's spoilt the party,' Georgia sighed.\n\n'Not for me he hasn't,' Peter bent his head a little and rubbed her nose with his.\n\nGeorgia lifted her lips to his and wound her arms round his neck.\n\nShe longed to be entirely alone with him, somewhere warm and comfortable where they could relax knowing no one would come in.\n\nHis kisses drove her wild. She wanted more, each touch of his hand made her tremble.\n\nAt night she lay thinking about him, imagining his hands creeping under her clothes. She would wake suddenly, hot and sticky from dreaming he was in bed with her, his naked body pressed into hers.\n\nAgain and again they kissed, each kiss more passionate than the one before. Peter leant against the wall on the landing and she pressed herself into him, savouring the male hardness taunting her through her party dress.\n\nIt was Peter who finally broke away.\n\n'It's after twelve,' he said breathlessly. 'I don't want to go but I promised your mother. I'll have to keep in her good books if I'm ever to be allowed to be alone with you.'\n\n'Is that what you want then?' she smiled shyly.\n\nHis face was flushed, hair falling into half-closed eyes, even his lips were swollen with kissing.\n\n'You know I do,' he whispered, lifting her hand to his lips and nibbling at the tips of her fingers. 'I've dreamed of nothing else since that night on the heath.'\n\n'What do you imagine?' She covered his face with tiny kisses.\n\n'Making love to you,' he said dreamily. 'Peeling your clothes off. Exploring you.'\n\n'We can't do that,' she looked into his deep blue eyes, tracing one finger round his lips. 'I might have a baby!'\n\n'I'd marry you,' he said softly, burying his lips in her neck and stroking her breasts.\n\nShe loved it when he did that, she could feel her nipples growing hard and a delicious dizzy feeling creeping all over her.\n\n'Just because I was pregnant?' she was fishing for more.\n\n'No, because I love you,' he put one finger under her chin and lifted her face up to his. 'And so I could sleep with you and hold you for always.'\n\n'I love you too,' she whispered, burying her face in his neck.\n\nAs they rejoined the party, her father was trying to jive with Christine.\n\nGeorgia laughed. It wasn't often she saw her father like this. A lock of hair had fallen over his face, he was biting his lips trying to concentrate and each time Christine tried to twirl round, Brian moved the wrong way. His legs were rubbery, his arms were flaying about, his jacket hanging off one shoulder.\n\nThere was a good atmosphere in the room, laughter, chatter, relaxed, almost loving, it seemed a shame they had to break it up.\n\n'Time to go,' Peter moved first to the boys from his school.\n\n'Not yet,' Brian said, face flushed more with drink than the dance. 'It's only just warming up.'\n\nPeter smiled politely, but still went round getting everyone to leave. Georgia started to stack glasses on a tray.\n\nJohn, Christine's boyfriend was looking anxiously at his watch, then back to Christine as she twirled round with Brian.\n\n'I'm supposed to get her home by half twelve,' he shrugged his shoulders at Georgia. 'Her dad will blame me for getting her like that.'\n\n'Leave her to me,' Georgia said.\n\nCrossing the room she caught hold of Christine's arm, her father danced on alone, oblivious.\n\n'Please go now,' she said. 'Dad's being very silly and Mum will be furious if she comes in and finds him like that. Don't encourage him any more. John's waiting for you.'\n\n'Okay,' Christine smiled stupidly. Her eyes were like slits now, her mouth drooping and trails of mascara running down her cheeks. The split in her dress was bigger now, a bubble of white flesh peeping out and her stockings were laddered. 'Do I look drunk?'\n\n'The walk home will sober you up,' Georgia said, wiping away the mascara with a serviette. 'I'll phone you tomorrow.'\n\n'Lovely party,' someone shouted from the landing as Georgia bundled Christine into her coat. 'See you back at school!'\n\nCouple after couple left till finally there was only Peter, standing just a few inches from her in the hall, his navy blue duffel coat in his hand.\n\n'Happy birthday,' he said. His eyes were heavy with longing. His arms reached out for her, crushing her into his arms, lips hot with desire.\n\nFrom upstairs music suddenly blared out, breaking the moment.\n\nThey both started, looking up at the stairs.\n\n'I'd better stay with you,' Peter took a step towards the stairs.\n\n'I'll be all right,' she replied, tugging on his arm. 'Go on home. Mum will be back soon. She might think we're taking advantage.'\n\n'But what if he turns nasty?' Peter argued. 'My dad does sometimes.'\n\nHis concerned expression made her feel special. The hall light shining down on his blond hair turned him into a Greek god and in that second love overwhelmed her.\n\n'My Dad, the boring old bank manager!' she giggled. 'Don't be daft. He'll be like a little lamb.'\n\n'Well pack him off to bed before your mum sees him.'\n\n'I love you,' she said softly, reaching out for him one last time.\n\nHe pulled her to him fiercely. 'How can I go away to University now?' he said softly, burying his head in her neck, nibbling and kissing. 'I want to get a job and stay here with you.'\n\n'You may be fed up with me by then,' she said, holding his face between her two hands.\n\n'I'll never get fed up with you,' he said.\n\n'We'll see,' she sighed. 'Maybe a year from now you'll have forgotten my name.'\n\n'Not in twenty years,' he shook his head, kissing the tip of her nose, his fingers in her hair. 'I love you Georgia Anderson. For ever.'\n\nShe watched as he walked across the heath. Hands in pockets, shoulders hunched in the frosty darkness. Every now and then he turned and waved again, blowing one last kiss. As he blended into the darkness Georgia closed the door.\n\nThe taste of his lips still lingered, she could smell that soapy smell, hear him saying he'd love her for ever.\n\nGeorgia went up the stairs singing.\n\n'Never knew what I missed, until I kissed you. How did I exist until I kissed you. Oh you've got a way about you, now I can't live without you.'\n\nHer father was still dancing on his own in the playroom. He looked ridiculous, several buttons undone on his shirt, his tie hanging off and his trousers needed hitching up.\n\nThe playroom was a mess. Many of the Christmas garlands were hanging down. Full ashtrays lay everywhere, records and sleeves strewn across the floor, along with balloons and dirty plates.\n\nMost of the food was gone, a few sandwiches remained, covered with cigarette ash, one speared with cocktail sticks.\n\n'That's right, sing to me,' he said, staggering towards her, arms outstretched.\n\n'Go to bed, Daddy,' she laughed, too happy to be really cross. 'Mum will go ape when she sees you, especially if she brings kids home with her.'\n\n'I don't give a damn about your mother,' he said, his mouth hanging open wetly.\n\n'You don't mean that, and anyway I'm tired.' She picked up the gin bottle and held it to the light. 'Daddy! This is nearly empty!'\n\n'Christine had quite a lot,' he said. 'She's good fun. You're getting like your mother.'\n\nGeorgia looked back at him over her shoulder. He had a peeved, sour expression on his face, like a spoilt child.\n\n'Oh, Dad,' Georgia sighed, 'Go to bed, you're pathetic when you're drunk.'\n\n'Pathetic am I?' Suddenly his face turned from benign and very drunk, to angry and dangerous. 'You, of course, are perfect? I saw you necking with that Peter. How far did you go when you went out of the room?'\n\n'Don't be disgusting,' she said quietly, bending over to pick up the plates from the floor. It was tempting to go to bed and leave everything, including her father, but if her mother did bring children home they just might have to sleep in the room.\n\n'Did he put his hand up your frock?' he lunged towards her unexpectedly and his hand went under her skirt and clamped on to her bottom.\n\n'Stop it!' she shrieked, wheeling round and slapping his hand away. He had never said such things before and suddenly she was frightened of him.\n\n'Kiss me like you kissed him,' he said, putting both hands on her shoulders.\n\nAlarm bells were ringing in her head. Fathers didn't say or do such things. She must get away, lock herself in her bedroom, wait for her mother to come home.\n\n'Don't be stupid,' she tried to get his arms off, but he pushed her up against the wall.\n\n'The last thing I am is stupid.'\n\nThis wasn't her father but a stranger. His face was lungeing at her, bloated and flushed. His lips wet and sloppy, pinning her against the wall with all his fourteen stone. His mouth came down on hers and he pushed his tongue into her mouth.\n\nIt felt like a huge serpent, she gagged and pulled herself back from him, moving her head to one side.\n\n'Daddy no!' she shouted as she struggled to get away, but he was too strong, his arms went round her, pinning hers to her side. She tried to bite his cheek but one hand came up and slapped her full across the face.\n\nHis breath came in rasps, stinking of drink as he slobbered at her neck.\n\n'I'm going to have you,' he said thrusting his hand up under her skirt, pulling at her panties. 'You little black bitch.'\n\n'NO, Daddy. NO!' she fought with him, nails clawing his face, legs kicking out, pummelling at him with her fists. But the more she struggled the stronger he became.\n\n'Daddy it's me. Your daughter!' she shouted, willing her mother to walk in the door.\n\n'You're not my daughter,' he replied huskily, his eyes gleaming with lust.\n\nFor a second he loosened his hold, fumbling at his trousers. Like a shot Georgia was off running towards the door.\n\nHe caught her in a flying tackle, knocking her down in the open doorway, flat on her face, then leapt on top of her.\n\n'Don't, Daddy,' she screamed, still in her heart thinking he was playing some sick, drunken joke which had got out of hand. 'Let me go!'\n\nBut she was trapped now. She couldn't reach him with her nails or fists, and she could feel his erect penis against her leg, his fingers tearing at her panties.\n\nShe was trapped, so terrified she felt paralysed, yet she waited a second, assuming he must turn her on to her back, ready to lash out at him the moment he did.\n\nInstead he yanked her hips up towards him and thrust himself into her like a dog.\n\nThe pain was so bad her scream turned from one of terror to anguish. He caught hold of her arms by the elbow, digging his fingers into the soft flesh. Her forehead hit the floor and the force of his thrusting movements grated it on the cord carpet.\n\nAgain and again she tried to get away, but each time he got a firmer grip, first her arms, then holding her pubic hair with one hand, the other holding her shoulder. His breath burning into her back.\n\n'Daddy, no. _Pleease_ don't.' She couldn't even scream any longer. Just a pleading, tormented cry and all the while she could hear him grunting out dirty filthy things that hurt her even more than the pain of being torn in two.\n\nHe went slack suddenly.\n\nOne moment the hideous grunting noise filled her ears, the next all she could hear was her own sobbing. A draft of cold air on her legs and buttocks, alerted her that he had rolled off.\n\nFor a second she just lay there, face embedded in the carpet, too stunned to move. A groan made her lift her head.\n\nHe was lying next to her on his back, eyes closed, his mouth gaping, a trickle of saliva running down his chin.\n\nGeorgia moved then, recoiling in horror as if waking to find a snake beside her.\n\nFirst crawling on her knees, then as her hand came in contact with the wall, she pulled herself up, backing away, clapping her hand over her mouth as the full enormity of what had happened hit her.\n\nHe was just lying there. A crumpled, blank face the colour of raw meat. Shirt open to the waist, white flaccid chest with a sprinkling of gingery hairs. Trousers unzipped, flabby stomach oozing out, his penis like some spent reptile, brown and limp, nestling amongst the wiry foliage.\n\nThe house was silent, just the ticking of the grandfather clock in the hall below and the flapping of the 'Happy Birthday' banner Peter had hung across the chimney-breast.\n\nA faint smell made her gag. A hot, fishy smell which seemed to come from her. As she looked down she saw a trickle of white fluid, tinged with her own blood roll down the inside of her leg and splash silently to her ankle.\n\nSix years of happiness in this house, wiped out.\n\nHis legs were inside the darker playroom, the rest of him on the landing, bathed in bright clear light. The crisp white-painted doors, the thick carpet, all so familiar and comfortable. But nothing would ever be the same again.\n\nHate welled up in her like vomit.\n\nOne moment she was staring at him, the next she found herself in the kitchen, her hand reaching out for a knife.\n\nEight different kitchen knives. Without hesitating she took the biggest triangular one for chopping meat and holding it to her breast went back up the stairs.\n\nNo more tears now, just dry-eyed desolation. He groaned as she stood by his side looking down at him. His eyes were still closed, one hand resting on his chest, her own black hairs stuck to it.\n\nHolding the knife tightly with both hands she drew it up first to chest height, then closing her eyes, plunged it, right down to the part of him she hated the most.\n\nShe saw blood spurt up, heard him cry out as she withdrew the knife, ready to repeat it, but nausea washed over her and instead she turned, running back down the stairs.\n\nCelia parked her car and looked up at the house. She was exhausted, mentally and physically. The porch light was still on, as was the hall and staircase light, but she was relieved to hear the party was over.\n\nIt had been a harrowing evening. At times she wished she'd stayed with nursing, at least there she had the satisfaction of knowing her patients would often recover.\n\nThis family were born victims. A drunken father, a half-witted mother. Five dirty, neglected children living in unspeakable filth. How many years of child guidance would they need to wipe out the horrors they'd seen? There would be no happy endings for them. The father might get a prison sentence, the children a peaceful break in a children's home for a few weeks. But before long they would be back together again. The boys turning into clones of their father, the girls merely training to become the next generation of inadequate mothers.\n\nCelia sensed something was wrong the moment she opened the door.\n\nOutwardly everything looked normal, but there was an atmosphere of tension, which made the hairs on her neck stand on end.\n\n'Brian,' she called tentatively.\n\nThe door to the sitting room was open. She walked towards it, dropping her briefcase in the hall.\n\nIt was too silent. If they'd gone to bed the lights would be turned off.\n\nHer foot touched something hard. Looking down she saw it was her kitchen knife, its grey, steel blade almost concealed in the patterned carpet.\n\nAs she bent to pick it up she saw blood-stains.\n\n'Georgia,' she shouted, pushing the door back, almost afraid to look in.\n\nGeorgia was lying crumpled up on the settee, the fire nearly out in front of her, the Christmas tree lights highlighting the red party dress. The net of the skirt billowed round her, for one brief second she looked like a dying swan in a ballet.\n\nCelia sped the last few feet to her child, dropping to her knees in front of her, her hands automatically feeling for a pulse, eyes scanning for injury.\n\nHer pulse was slow, but not dangerously so. A rough, red patch on her forehead and a swelling across her cheek like a slap mark. But there was no obvious injury.\n\n'Darling, it's Mummy.' Celia caught Georgia up in her arms. 'Tell me what's happened?'\n\nNo reply. Not a flicker of anything in those coal black eyes. No emotion, no tears, not even a trace of recognition in the blank, vacant face.\n\nThe calm nurse vanished.\n\n'Answer me? Who did this? Where's Daddy?' She shook Georgia sharply.\n\nThere was no reply, but she sensed the child's eyes move towards the door fearfully and in the same instant saw dried blood on her hands and a splatter on her dress.\n\nSwiftly taking off her own coat she tucked it round Georgia.\n\n'I'll be back in two seconds,' she said breathlessly.\n\nShe saw Brian's head through the bannisters, well before she reached the top of the stairs.\n\n'Oh my God,' she gasped, racing the last few feet to his side.\n\nBrian was lying in a pool of blood, coming from his stomach, his hands were over it, blood trickling through his fingers.\n\nFor just one second she thought he was dead, but a faint pulse told her otherwise.\n\n'Hold on,' she said jumping up and running to the bathroom to fetch a towel to staunch the blood.\n\n'Which service do you require?' a disembodied voice replied to her emergency call.\n\n'Ambulance,' she shouted. 'Quickly!'\n\n'Someone's stabbed my husband,' she said hurriedly as she was connected, barking out the address. 'Hurry please, it's serious.'\n\nBack again at Brian's side, she held the towel tightly against his wound. He was unconscious still, but now that help was on its way she was able to think rather than act automatically.\n\nHer first reaction had been that someone had come into the party uninvited and the stabbing was a result of Brian asking them to leave.\n\nBut that didn't make sense! The other kids would have called for help. And why was the knife downstairs by the sitting room?\n\nThere was something very familiar about the situation. A child in shock refusing to speak. If her father had been hurt defending her she would have run for help.\n\nCelia moved the towel away from the wound and looked closely.\n\nHis trousers were unzipped. No sign of stab marks in the material and his penis was uncovered!\n\nShe heard herself cry out. Even through the smell of blood there were others. Body aromas, mingling with a stronger one of alcohol.\n\nLike a shot of insulin to a diabetic, the truth came to her.\n\n'You bastard!' she exclaimed, pushing the towel down hard again to cover him.\n\nShe got up, moving back from him in horror and disgust, an overwhelming nausea washed over her, and she clamped both hands over her mouth.\n\nFor the last ten years she had worked with the aftermath of rape and incest. She had learned to control her rage and disgust. But this wasn't a stranger, this was her husband, the man she had promised to love and cherish.\n\n'No,' she shouted. 'You couldn't do that!'\n\nBut he had, she knew it as certainly as if she'd seen the act take place. And now she wanted to finish what Georgia had started. Her hands moved towards his throat, her thumbs were on his windpipe, fingers digging into his neck.\n\nHis eyes flickered momentarily, his lips moved as if to speak.\n\n'No,' she stopped herself. 'It's too good for you. You'll pay for this you maggot. You'll pay!'\n\nShe forced herself to attend to him, just a nurse, going through the motions of emergency first aid. Not out of sympathy, love or any other finer motive. Just keeping him alive so he could pay the price for ruining her child's life.\n\nShe could hear the sirens coming closer. Her heart thumped painfully, her head reeling as she thought of the implications.\n\nGeorgia downstairs, still in her party dress, shocked and witless. Six years of building up her trust, over, through one man's lust.\n\nAnd it was all her fault. While she was out sorting out other families' problems, one had been festering here. She was the expert, the one who should have read the signs. Celia Anderson, the woman who knew all the answers had failed the child she loved.\n\n## Chapter 4\n\n'Georgia! Speak to me. You can't just shut me out.'\n\nCelia sat on the edge of the bed, leaning over to cup Georgia's small face in her hands.\n\n'You don't fool me!' her voice rose in exasperation. 'Dr Towle might think you have some kind of hysterical amnesia, but I know you better.'\n\nThe pretty pink and white bedroom with its rose sprigged curtains and bedspread, the books, games, teddy bears, even the posters of Elvis Presley and the Everly Brothers all seemed so very safe and cosy. Yet Brian had made a mockery of this childhood sanctuary.\n\nGeorgia had said nothing even when the doctor examined her. No tears, no screaming, not even angry accusation. Just a cold, emotionless silence that terrified Celia. Experience told her that rape cases often reacted like this. But this wasn't a case, this was personal.\n\nDr Towle confirmed that Georgia had been subjected to violent sexual intercourse. Brian's injuries and the entire scene Celia had walked in on just a few days earlier was surely proof enough of who was responsible, yet still the police were prevaricating.\n\nThere had been precious little sympathy for Georgia from the police when they arrived a few moments after the ambulance.\n\nInspector Forbes was a bigot. A big bully of a man, red-faced with hair to match. A lifetime immersed in the underworld of South London made him incapable of seeing beyond Georgia's colour. A man who'd peered down so many sewers, he'd begun to believe the whole world was one.\n\n'But everything you say is supposition,' he retorted almost angrily when Celia had once again told him the details of what she had discovered on her return to the house. 'Why should a man in his position, knowing full well you could come home at any time, force himself onto his daughter?'\n\n'Lust brought on by drink,' Celia burned with anger at his arrogance. She had come up against this man before and knew he loathed professional women almost as much as the blacks in his territory. She turned to the young policewoman with him, sure another woman would be sensitive enough to see the truth. 'You saw him lying there. You saw the way Georgia looked, don't tell me you agree with the inspector?'\n\n'Why wouldn't she tell me then?' the policewoman looked nervously at her superior officer, a sure sign she didn't dare oppose him. 'I think it's more likely someone else was involved, these teenage parties can get out of hand.'\n\nCelia heard the soft Sussex accent, noted the farmgirl complexion and knew immediately this girl had no personal experience to fall back on and even less intuition.\n\n'You don't believe my daughter was raped by her father?' Celia wanted to slap both of them to make them see sense.\n\n'But he isn't her father is he?' Inspector Forbes had a look of cunning in his bloodshot eyes. 'Mixed-race girls are volatile. She'd been drinking.'\n\n'He'd been drinking,' Celia corrected the man. 'Not Georgia.'\n\n'Then why won't she tell us?'\n\n'She's in shock.' Celia's eyes rolled with impatience. 'Do you really expect her to sit here and tell you the whole traumatic story?'\n\n'But the knife?' the inspector said, looking down at the weapon in his hand, dried blood sticking to its covering plastic bag. 'She could have killed him. I can't see a girl capable of wielding this, being incapable of defending herself against sex.'\n\n'He weighs fourteen stone,' Celia said through clenched teeth. 'Georgia about eight. If she'd had the knife handy before he attacked her she might have had a chance.'\n\n'Perhaps we'd better try again in the morning,' Forbes sighed. 'This isn't getting us anywhere. I don't see a man like Anderson being a rapist. Let's wait till I've interviewed the boyfriend. To be honest Mrs Anderson, I'm surprised at you not taking your husband's part.'\n\n'I've never heard of anyone attempting to cut off a man's penis unless that same organ was used against them,' Celia lost control entirely, almost screaming at him.\n\n'I do believe you are getting hysterical,' he said disdainfully. 'He has stomach wounds and his loose clothing was consistent with someone who tried to examine himself before losing consciousness.'\n\nIt was after four in the morning when they finally left the house. Georgia sedated by the doctor upstairs. Brian in hospital and there was no one she could turn to for help.\n\n'How can they be so stupid and blind?' she cried helplessly in the kitchen, her head on the table. 'I wish I'd let him die now.'\n\n'I know you can't bear to talk about what happened,' Celia stroked her daughter's face. 'But if you don't talk soon you'll be taken from me. You might even be charged with attempted murder. Think about it Georgia, unless Brian is charged with rape he'll walk out of that hospital a free man!'\n\nBut Georgia just lay in her bed, her eyes blank as if she were deaf and dumb.\n\n'What are we going to do?' Celia sat in the kitchen with Peter later the same morning, her face lined and drawn. She reached out to touch his hand, the social worker at odds with the distraught mother. 'She knows exactly what's going on. She's fully conscious, she goes out to use the toilet. Yet she's eaten nothing, only drinking the water I left by her bed.'\n\n'You're frightened they'll take her away from you?' Even after all the questioning Peter had been subjected to, he was still perceptive enough to understand Celia's fears.\n\nHe looked ill. His face was white, eyes ringed with dark circles. His chin had a rash of stubble, even his shoulders were stooped. Now he was risking more trouble by refusing to go back to school.\n\n'I don't know how much longer I can stall the children's department,' Celia said wearily, resting her head on her hands. She looked old. Her brown hair uncombed, wearing the same old tweed skirt and jumper she'd put on the morning after the event. 'Unless she starts talking they'll almost certainly take her. I keep trying to distance myself from it, work out what I'd do if Georgia and her foster mother were clients. And I don't like the answer I keep coming up with.'\n\n'What about him?' Peter winced as if even using Anderson's name hurt him.\n\n'He's out of danger now,' Celia snorted with anger. 'I never thought I'd admit to such a thing, but there've been times in the past few days I hoped he'd die. She only cut his abdomen, he's got a lot of stitches, but he'll survive.'\n\n'What's he been saying?' Peter asked.\n\nCelia's mouth trembled.\n\n'He claims he fell asleep in a chair, woke to see Georgia smuggling you out of her bedroom after the others had gone and that he had a row with her about it. He insists the last thing he remembers is Georgia coming at him with the knife and when he came round he was in hospital.'\n\n'But even my parents can vouch that I got home just after half past twelve.' Peter flushed an angry shade of red. 'I left only ten minutes after the others.'\n\n'A good lawyer would wipe him out,' Celia reassured him. 'But without Georgia's statement he can't even be charged.'\n\n'You mean he could come home here?' Peter's face blanched in horror.\n\n'It's his house.' Celia's greeny-grey eyes were blank with misery, her face contorted by dark thoughts.\n\nShe couldn't bring herself to tell Peter the full strength of it. Brian claimed to have caught them naked in bed together, that Peter had grabbed his clothes and made a run for it and Georgia screamed abuse at him and even threatened she would say Brian raped her if he told her mother.\n\nAt the hospital they all believed his story, but then they'd never met Georgia or Peter and Brian Anderson was charming and persuasive when he wanted to be.\n\n'Let me try speaking to her?' Peter leaned forward earnestly. 'I might be able to get through to her.'\n\n'I don't think she'll respond to you or any man,' Celia shook her head. 'But you can try. Just don't try to touch her that's all.'\n\n'Georgia!' Peter whispered in the darkened room. 'Are you awake?'\n\nThere was no reply. In the gloom all Peter could make out was a small lump in the bed, her dark hair sticking out of the covers like a chimney-sweep's brush.\n\nPeter walked across the room and drew back the curtains.\n\nIt was bitterly cold outside, a dark, grey day as if all the world was suddenly monochromatic. The heath across the road was deserted, bare branches of trees looked menacingly like skeletons.\n\n'I think it's going to snow,' he said, taking a chair just close enough to see her face.\n\nIt was crumpled, as if she'd aged ten years. Her eyes were open but they showed no signs of recognition.\n\n'You've got to talk,' he said, keeping his voice as normal as possible. 'Maybe not to me, but to your mother. She's tearing herself apart, and she's the one who is trained for things like this.'\n\nNo movement, not so much as a flicker of an eyebrow.\n\n'Would you like some music on?' he asked. 'I could get your record player and plug it in?'\n\nStill nothing.\n\nFor five days he'd waited patiently for this opportunity, convinced Georgia would open up to him. But she lay there like a doll, dark eyes staring into space and now he understood why Celia was so frightened.\n\n'School started again,' he went on. 'I haven't been though,' one hand reached out, but paused in mid air. He could feel tears pricking the back of his eyelids. 'I went to bed dreaming of you that night. I intended to come round to help you clear up in the morning. Then the police came.'\n\nHe had trusted the police until that morning. One moment he was lying in bed thinking of Georgia, the next he was bundled down the stairs into a squad car, accused of having sex with a minor.\n\nFor three hours they interrogated him. First in an almost jocular 'all-boys-together' manner suggesting he had been caught by Mr Anderson with his pants down. But later it turned vicious, with insinuations about Georgia's character that left Peter bewildered.\n\nIt was only when he discovered Mr Anderson had been stabbed that the hideous truth filtered through. He remembered the way Anderson had been as he left, Georgia's insistence she would get him to bed before her mother got home. Why had he let her persuade him to leave?\n\nMaybe it was foolhardy to take a swing at Inspector Forbes, to scream out his anger and frustration at their callous indifference to Georgia's pain, and stupidity at blaming him. But at that moment he didn't care if they locked him up forever.\n\nThere was no understanding or sympathy when he got home eight hours later. No concern that his girl had been raped, or even anger that he was being blamed. His father made lewd suggestions. His mother could see no further than Georgia's colour.\n\n'I'll never get over the shame,' she raved. 'My son mixed up with some wog. I won't have it Peter. Don't you dare go near her again.'\n\nCelia was the only one who shared his outrage. It was her arms he turned to for comfort instead of his own mother's, and now he hoped together they could bring Georgia out of her silence.\n\nPeter wriggled in the small cane chair. There was still no movement from Georgia in the bed. She didn't even look round to watch him.\n\n'I'm not going to give up,' he said petulantly. 'I shall just keep on coming until I bore you into telling me to shut my mouth. Day after day, year after year. I'll tell you what I ate for tea, what I did all day. You are my girl and I'll keep coming till you tell me to go.'\n\nHe peered at her to see if there was even a flicker of amusement.\n\n'Right, I'll try the music,' he said, getting up and making for the door.\n\nThe playroom was much as they'd left it the night of the party. The garlands hanging down, stray balloons and records still littering the floor. Only the food, glasses and plates had been removed.\n\nHe unplugged the record player and sorted through the records, selecting only one.\n\nThen carrying it back across the landing, he noticed the blood-stain.\n\nCelia had obviously tried to get it out of the carpet, but still it stayed, dark and menacing against the pale, flowery design.\n\nThat animal had taken her here, only feet away from the place where Peter had held her in his arms earlier. Rage washed over him, his hands shook and he could understand only too well why Georgia had got the knife.\n\nTaking a deep breath he pushed his way back into Georgia's room.\n\nThe record crashed down from the spigot, the arm moved across and for a second there was only a scratching noise.\n\n'How did I exist until I kissed ya. Oh, you've got a way about ya, now I can't live without ya. Never knew what I missed until I kissed ya.'\n\nIt sounded trite and silly under the circumstances. But they'd played it over and over the night of the party.\n\nHe hummed along with it, wondering what to do next.\n\nLeaning forward he saw a tear trickling down her cheek.\n\n'I didn't mean to upset you,' his hand reached out to touch her, but recoiled almost immediately as he remembered her mother's warning. 'It's just it's our song isn't it?' He was sure she was going to speak now, but still she was silent.\n\n'My feelings haven't changed,' he whispered. 'I love you.'\n\nIt was all he could do to prevent himself from sweeping her up in his arms. He wanted to hold that small crumpled body, breathe life back into that blank face, somehow he was sure the power of touch would succeed where words never could.\n\n'Now I can't live without you.' He sang the words softly, then turned away, stumbling to the door.\n\nGeorgia lay staring at the ceiling. The soreness had faded now, though when she turned over she could still feel the bruises on her buttocks. Nothing felt right anymore. Not her own body, or even her bed. She had felt like this before, in the early morning before Celia had collected her from St Joseph's. Sister Mary's kind words had softened some of the wounds Sister Agnes had inflicted on her, just as Celia's and Peter's tenderness had, but still inside was this core of terror that nothing could make go away.\n\n'Devil's spawn'! That was what Sister Agnes had said about her. Maybe it was her fault. Something bad within her that caused destruction and pain.\n\nShe couldn't stay here, not now. Celia and Peter might want to help, but they couldn't. What good would it do telling the police the truth? So Brian might get locked up, but what sort of hell would she have to drag Celia, Peter and herself through? And for what? It wouldn't make her forget what happened. She wouldn't even be able to stay with Celia.\n\nCelia came in with a tray of food around lunchtime. Once again Georgia lay silently, staring into space.\n\n'Look, dear,' Celia sounded as if she was losing patience. 'You must be hungry, even if you don't want to speak to me. Sit up and eat this, just for me.'\n\nThe smell of the casserole made her feel nauseous. Why did Celia keep bringing food, when all she really wanted was to be left alone?\n\n'I'll put it down here.' Celia's voice was firm and business-like as she placed the tray on the floor by Georgia's bed. 'I've got to go into the office this afternoon. You know how it is Georgia. When all's said and done the children's department will have a final say. By speaking to them now I might just get us a few weeks grace. Even the doctor has suggested you need to be taken into hospital. I can be back in a couple of hours. Will you be all right?'\n\nGeorgia nodded. The first indication she'd given her mother that she even understood what was going on around her.\n\nCelia sat down on the bed and took one of Georgia's hands in hers.\n\nIt was that same look Georgia had seen all those years ago when Celia had taken off her clothes and found the wounds on her back. So much pity and understanding, so willing to give her anything to make things right.\n\n'I love you darling. This evening I want you to come downstairs with me. I know why you don't want to speak,' her voice was breaking with emotion. 'But the sooner you open up to me, the sooner we can put this horrible affair behind us. Together we can find a solution.'\n\n''Bye darling,' she said at the door. 'Eat that dinner just for me?'\n\nIn that second Georgia's resolve nearly crumbled. Celia looked so careworn, the small lines round her eyes deeper, etched with unbearable sadness. She had made an effort to look smart again. Her navy pin-striped suit with its long jacket hid the plumpness, a small brooch at the neck of her white blouse, her hair washed and brushed back from the soft cheeks. Powder and lipstick had done their best to cover the putty-coloured skin. But the apple cheeks Georgia knew so well looked sunken.\n\nGeorgia waited until she heard the front door slam and the car start up. She sat up gingerly, looking down at the tray of food. She didn't want it, but it might make Celia feel better knowing she'd eaten something.\n\nShe forced herself to eat half the meat and some of the vegetables but however hard she tried she couldn't manage the apple pie and custard.\n\nHer legs were stiff and unsteady as she got out of bed, her face in the mirror was pale and drawn but however bad she felt she had to get dressed. She found a holdall in the cupboard on the landing and hastily packed a few warm clothes. She put on the new navy suit Celia had bought her before Christmas, which made her look older, and put her hair up in a French plait.\n\nFinally her thick, grey overcoat, a woolly hat and scarf and a pair of sheepskin-lined boots.\n\nThen stuffing her make-up, hairbrush and washing things in a handbag, she made her way downstairs.\n\nThe house gleamed. Celia always cleaned frantically when she was angry or anxious.\n\nAll the Christmas decorations were gone now, pine needles carefully swept away. Everything looked just the way it always did, except it could never be her home again.\n\nTaking a sheet of paper from Celia's desk in the dining room, she sat down to write a note, her eyes filling with tears.\n\n'Dear Mummy,' she paused, unable to see clearly through the tears, her mind suddenly blank of the right words.\n\n'I'm sorry. I had to go. We both know it's only a matter of days before someone takes me away and if I can't stay with you, I'd rather be alone. I'm old enough to get a job and somewhere to live. I will always love you and be grateful to you for taking me from that convent, no one could have had a better home than me.\n\n'Please don't blame yourself for what happened and try to forget about it. Don't try to find me, it will only make things worse.'\n\nShe paused again, sobs rising up within her. There was so much she wanted to say, so many thank yous. How did you say goodbye to someone who meant so much?\n\n'Explain to Peter for me. He must forget too and go on to university like he planned. Maybe one day we can meet again. You will never be out of my thoughts.\n\nI love you,\n\nGeorgia'\n\nLeaving the letter on the table, she lifted a ginger jar off the mantelpiece where Celia kept housekeeping money. She took twenty pounds and then returned to the desk.\n\n'PS, I took twenty pounds as a loan. I'll send it back just as soon as I can.' She added this on the bottom of the letter, then took it into the kitchen to leave it on the table.\n\nPicking up her bag she walked to the front door. Pausing for a moment to look back one more time.\n\nThere was snow her first day here. She remembered Celia sitting her on the stairs, taking off her shoes and rubbing her toes to warm them. Maybe the little girl who'd stared in wonder at the paintings, the thick, patterned carpets, the polished furniture and the grandfather clock had grown up, but that first impression would stay with her for ever.\n\nShe gulped back tears, opened the door resolutely and walked out, slamming it behind her.\n\nOutside, the cold air made her shrink back into her coat. The heath had a thin coating of frost, fog concealing the walls of Greenwich Park in the distance. It would be dark in a couple of hours. She must hurry now before anyone saw her.\n\nPeter would soon forget, whatever he said now. Brian had taught her how shallow men's love was.\n\nStill, she paused by the church, Peter's face dancing in front of her. She could see those blue eyes fill with emotion as she sang the Christmas Anthem at midnight mass. Remember the kiss he stole as they hung up their surplices in the vestry. She hated her father for a great many reasons now, but most of all for ending something so beautiful.\n\nStanding in Piccadilly with crowds of people milling around her, she felt numb. This was the centre of everything. The big city with its bright lights, smart shops and continuous noise. Neon lights flashing, the never-ending stream of traffic, strange smells. Men in bowler hats carrying furled umbrellas, office girls, shop assistants and shoppers.\n\nSwan and Edgar's windows were piled high with sale goods. Around her were the shouts of newspaper men, music from a one-man-band, wafts of fried onions from a hamburger stall, and Eros in the middle directing the circus. It was the ideal place to hide in, every day of the year girls swarmed to this area to begin life away from home and most probably started with less money than she had.\n\nAt six her legs and feet were aching. She'd seen two rooms with a To Let notice, but both times she climbed the stairs to enquire she'd been told the rooms were ten pounds a day.\n\nShe was baffled. Why would a grubby little room in Soho have such a high rent?\n\nThe answer came to her on her third attempt.\n\nA big-busted blonde girl of about thirty came to the door wearing a black negligee, a cigarette dangling from vermilion lips.\n\n'I've come about the room,' Georgia said wearily. 'Is it still vacant?'\n\nThe girl wore false eyelashes, one was peeling off and she had traces of mascara rubbed onto her cheeks. She looked Georgia up and down, taking in the wool coat, sheepskin boots and holdall.\n\n'You aren't a working girl,' her greyish face puckered into a frown.\n\n'Not yet,' Georgia tried to sound bright. 'I'm going to look for a job in a caf\u00e9 or something tomorrow.'\n\nThe blonde girl studied her for a moment. A quizzical look as if she thought someone was pulling her leg.\n\n'Come off it love!' she laughed, but it came out like a dry cough.\n\n'Sorry?' Georgia frowned. 'I don't understand.'\n\nThe girl just looked at her for a moment, a mixture of amusement and pity.\n\n'You must've heard what goes on in Soho?' She took another puff on her cigarette. 'These rooms are for the girls. Know what I mean?'\n\nGeorgia's mouth fell open. 'You mean?' she couldn't bring herself to say the word.\n\nThe blonde nodded, a warmer look spreading across her face. 'Try the evening paper, or go home love. There's nothing around here for a little thing like you.'\n\nGeorgia backed away feeling foolish and just a little tainted.\n\n'Good luck,' the girl said cheerfully. 'And watch what you are doing love!'\n\nEarlier Georgia had welcomed the darkness. Now it seemed like a threat. She was tired, cold and her holdall was getting heavier by the minute. She bought a paper and went into a caf\u00e9 to read it, wrapping her hands round a cup of tea to warm them.\n\nThe advertisements offered very little. Most of them were too expensive and almost all of them said 'references required'. She ringed eight which sounded possibilities, drank her tea then found a phone box.\n\nFive of them had already been let. One was right out near Wembley and at the other two there was no reply.\n\nTwo policemen walked past as she came out of the phone box. One looked at her bag and then directly at her.\n\nTrembling with fright, she moved off quickly, running down the narrow street, back towards Piccadilly.\n\nThe West End might have seemed exciting with the security of a friend beside her, or looking at Christmas lights from the safety of a car. But alone, in the dark it was menacing.\n\nSo many people pushing and shoving. Tramps mingling with couples out for the evening. Young office girls off to a dance, a gang of rough-looking Teddy boys shouting remarks at passing girls. Taxis, motorbikes, cars and buses, a madhouse of noise and bustle, a sinister undertone to everything. Her bag was heavy and each time someone turned to look at her Georgia sank into the shadows.\n\nTurning again back towards Soho, she walked deliberately up one street, looking at all the notices on doors, then down the next.\n\nThere were plenty of signs.\n\n'French Lessons, apply second floor.' 'Model first floor,' and even one saying 'Strict Instruction, ask for Mitzi.'\n\nGirls and women stood brazenly in doorways. Tight shirts, cigarettes dangling from lips like scarlet gashes, 'beehive' hair do's and heavy eyeliner, clinking keys in their hands. They stared openly at Georgia as she rushed by, her heavy bag banging against her legs.\n\n'Are you doing business?' A swarthy-looking man sidled up to her.\n\nShame overwhelmed her, her eyes filled with tears, all she could do was pretend she hadn't heard and carry on walking as if she had some place to go.\n\nIt was after nine and she'd been up and down each street. Almost every coffee bar she'd passed she enquired in.\n\nAlways the same answer.\n\n'Sorry. Try the paper.'\n\nShe saw one advertisement for a room in a shop window and went round to the house immediately.\n\nAn elderly lady came to the door, opened it just a crack and peered out.\n\nGeorgia put on her most beguiling smile.\n\n'I'm sorry to call so late. I believe you have a room to let?'\n\nShe reminded Georgia of story book grannies, white haired, wrinkled with a crocheted shawl round her shoulders. She opened the door a little wider and peered at Georgia standing on the pavement.\n\n'No blacks!'\n\nThe door slammed shut in her face, leaving Georgia standing there mouth agape, cheeks burning at the insult.\n\nThe cold seemed to have got right into her bones, even the sheepskin boots no longer kept out the cold.\n\nDown she went into the tube. At least there was a little warmth in the ladies' toilets. Locking herself in a cubicle she unpacked another jumper and put it on under her coat and suit.\n\nShe looked odd, shapeless under the many clothes, the woolly red hat seemed to drain even more colour from her face.\n\n'Have you run away from home?' The toilet attendant shuffled out of her small room, wearing a flowery crossover apron, as Georgia appeared for a second visit in an hour.\n\n'No,' Georgia said quickly. 'I'm waiting for someone.'\n\n'Well wait somewhere else,' she snapped. 'This is a public convenience. There's caf\u00e9s up there for meeting people.'\n\nIt was too late to find anywhere now. She was too scared to go to a proper hotel and if she hung around much longer the police might pick her up.\n\nInstead she went back into a caf\u00e9 and bought herself a meal, taking as long as possible to eat it while she thought what she should do next.\n\nThe caf\u00e9 overlooked a strip club. One of many she'd noticed while walking up and down. As she watched she saw men going over to the brightly-lit pictures of naked girls and some went on, down the stairs. All at once she realized she'd made a mistake coming to Soho. It was another myth, like happy families, fathers you could trust and the streets paved with gold.\n\nTomorrow morning she could try the flat-letting agency she'd spotted in Berwick Street. That just left tonight to get through.\n\nJust before midnight she left the warmth of the caf\u00e9. She was so tired now she could hardly put one foot in front of the other.\n\nThere was nothing for it but to look for somewhere to hide, and just wait till the morning.\n\nEarlier she had noticed some large boxes, piled outside a dress manufacturer's workshop. It was a small cul-de-sac tucked away behind the busier streets. She made her way there now, walking quickly to keep warm.\n\nShe passed an old tramp going through a box of abandoned fruit by the market; shuddering she moved even more quickly.\n\nLooking over her shoulder to check no one was watching, she looked in the boxes piled high against a doorway.\n\nOne of them was full of scraps of material.\n\nSelecting the largest of the boxes, she turned it on its side, placing the opening against the wall, then moving boxes either side of it, including the one with material, she crawled in.\n\nIt wasn't large enough to lie stretched out, but curled up it was adequate. Then she reached out, taking handfuls of material until she had covered the floor with a thick layer.\n\nOut of the wind it was much warmer, and the material soft to lie on. She wriggled out of her coat in the confined space, then using it as a blanket, her bag as a pillow, curled up to go to sleep.\n\nA hissing sound close to her head made her wake suddenly. She lifted her head and listened.\n\nShe heard feet walking quite close. Obviously it was a man who had just relieved himself against the wall.\n\nShe cringed in disgust. In the distance she could hear music, the sound of someone dropping a bottle nearby, and shouting coming from the end of the cul-de-sac.\n\nIt was too dark to see the time, she was stiff with cold and she ached to change position and stretch out.\n\nA cat mewed softly nearby and she heard rustlings which might possibly be a rat.\n\nIf she made any noise someone might investigate. The sort of men round here could be as bad as Brian. She turned on her back and bent her knees up, putting her hands under her head, trying hard to stay calm and not cry.\n\n'It's only for tonight,' she said to herself. 'You'll be laughing about this in a day or two.'\n\nTo take her mind off the cold she tried to imagine a bedsitter, small and cosy with a big fire.\n\n'I'll change my name,' she whispered to herself. 'Something glamorous.'\n\n'Come on love, ten quid's too much,' a booming male voice made her almost jump out of her skin.\n\n'Ten quid or nothing.' A woman with a cockney accent replied.\n\n'What! A tenner for here in an alley?' The man's voice was slurred with drink and the couple's footsteps were coming closer.\n\n'Take it or leave it,' the woman said defiantly. 'I can't take you anywhere now.' She sniffed loudly, she was so close Georgia could even hear her pull a handkerchief out of her pocket.\n\nGeorgia hardly dared breath. She could hear the man going through money in his pockets. She shook with fright, expecting any moment that he would sense her presence and root her out.\n\n'I'll give you a fiver,' he said reluctantly. 'That's enough for a bleedin' kneetrembler.'\n\n'You blokes are all the same,' the woman grumbled. 'Well, don't expect me to give you the full treatment.'\n\nA rustle of notes, the click of a handbag clasp and the rasp of fabric.\n\n'Give me a kiss then?' the man said. 'Don't expect me to get it up without some help.'\n\nGeorgia could hear her own heart beating. She had cramp in her leg, she was so cold she felt she might just die and the man's intentions brought her father right into the box with her.\n\nShe tried not to listen but the man's grunts penetrated even her gloved hands over her ears. Stiff joints creaked and the swishing sound of clothes being pushed aside.\n\nThey seemed to be at it forever, disgusting her so much she felt sick and faint. Sucking sounds, heavy breathing, words that made her blush with shame. Then a prolonged thumping sound, like a piece of meat being slapped against a plate.\n\nGeorgia wanted to cry. All those silly dreams she'd had of love. What was going on out there was the real world. Men weren't ruled by their hearts at all, they were just animals needing sex like food and it didn't matter to them who they shoved themselves into.\n\n'Was it good?' she heard the man ask.\n\n'Sure,' the woman replied, the sound of pinging elastic like gun shots in the quiet street.\n\nShe left first. Her high heels like castanets on the road, gradually fading away into the distance.\n\nA loud belch signalled the man was still close by. She heard the flick of a lighter, the smell of a cigarette, then unsteady movements and deep breathing. He stopped, the unmistakable sound of a zipper. A rumbling fart, and the hiss of urine, followed by a deep sigh.\n\nIt was too cold to sleep again. She had pulled everything out of her bag to lay over her, and handfuls more of the material from the other box, but still she shivered.\n\nThere was no more noise from people now. But cats continually prowled, frequently jumping on her box. There were other rustlings she couldn't identify, sometimes a gnawing sound which made her eyes open with terror.\n\nShe must have slept eventually, as she came to with a start when the sound of an engine roared nearby.\n\nWaiting a few moments, as the noise came closer and closer, she was finally brave enough to peep out round the edge of the box.\n\nIt was barely light, her breath like smoke in the cold grey dawn. At the crossroads less than a hundred yards away a dustcart had paused, four men throwing all the boxes and refuse into the back of the van.\n\nHastily she pushed all her belongings into her holdall, flung her coat over one arm and ran, her heart thumping like a steam engine.\n\nShe stopped for breath on the next corner and reached for her handbag to find a hanky. It was gone. In her haste she must have left it in the box.\n\nDropping her holdall, she flew back. All that was left of her bed for the night was a wet patch where one of the men had urinated, and a few scraps of fabric.\n\nA dull rumble alerted her the cart was now in the next street. She tore round the corner, catching it up and running along side.\n\n'Please stop,' she waved her arms frantically. 'Stop!'\n\nOne of the men jumped off the back of the cart.\n\nHe was very small and stocky, a woolly brown hat perched on fair greasy hair, dressed in dark green overalls and a donkey jacket.\n\n'What's up love?' He had a strange speech impediment, and his mouth twisted alarmingly. He was about fifty and very dirty.\n\n'My bag,' she said breathlessly. 'I left it in a box that you've just picked up.' She pointed in the direction they'd just left.\n\n'What sort of bag?' he frowned, looking at her dishevelled appearance, pieces of cloth in her hair.\n\n'A handbag, it's got my money and everything.'\n\n'Why was it in the box then?' He scratched his head through his hat.\n\n'Because I was in there too. But I heard you coming and ran away. I didn't realize I hadn't got it until after you'd gone.'\n\nThe man looked from Georgia to the dustcart. She could see it was fitted with a crusher device, already chomping away the last of the boxes from the road they were now in.\n\n'I can't get in there,' he said sharply. 'You should look after your stuff.'\n\n'But all my money's in there,' she said, tears springing to her eyes.\n\n'Can't help that,' he turned away from her.\n\n'What shall I do?' she ran after him clutching at his sleeve.\n\n'Go to the police,' he said wearily. 'It ain't my problem. What'cha sleeping in a box for anyway?'\n\nHe was gone before she could say anything more and Georgia felt as crushed as if she'd been in the machine herself.\n\nIt was a few moments before the full implications hit her. She went back to where she'd left her holdall and bent down in a doorway to repack it. She was so cold she was no longer shivering, and she knew she must find somewhere to get a hot drink.\n\nA sick feeling wound its way into her stomach. She couldn't get a drink or anything now. Every penny she had in the world was in that handbag.\n\nShe saw her reflection in the glass door. She was dirty, her hair full of bits of cloth and she had nothing at all other than a few clothes.\n\nSinking down onto her haunches she wept. Great heaving sobs that shook her shoulders and engulfed her whole body.\n\nMoney meant everything. She couldn't get a drink, or a meal, much less a room for the night. She couldn't use a phone or get on a bus or train.\n\nThere was no difference between her and the tramp she'd seen going through the rubbish last night!\n\nCold and hunger had brought on a new sort of apathy by five in the evening. A day spent walking up and down Oxford Street, eyes glued to the pavement, hoping to find a shilling or so to buy a drink. She'd been in the toilets of every shop from Marshall and Snelgrove to Selfridges, lingering in a cubicle as long as she dared, trying to summon the courage to swipe a handbag or purse from one of those wealthy ladies wreathed in shopping bags.\n\nBut each time she got close something stopped her. Sometimes an attendant watching, sometimes just a faint stirring of conscience. Eleven hours of walking and all she had to show for it was a yellow plastic comb and two halfpennies she'd found in a gutter.\n\nFor just a brief moment it seemed her luck had changed. Just off Tottenham Court Road she saw an advertisement being stuck up in a sweet shop window.\n\n'Bedsitter to let for business person. \u00a32 per week. 14 York Street.'\n\nShe had run there, memorising the address, so sure it was a sign from heaven that she forgot the cold and her aching feet.\n\n'I believe you have a room to rent,' she blurted out to the man who opened the door.\n\nHe was perhaps sixty, thin and bent with yellowish skin and a big, hooked nose.\n\n'How old are you?' He looked her up and down with sharp eyes. She felt sure he could see the creases in her clothes and the traces of fabric still clinging to her hat and coat.\n\n'Eighteen.' She could see a steep uncarpeted staircase behind him, it looked dark and dirty and she was more than a little scared. 'I've just started a job in Oxford Street.'\n\n'Your name?'\n\n'Georgia.' She paused suddenly remembering she intended to change it. She looked round and noticed a small shoe shop. The sign above it was James's.\n\n'Georgia James,' she said quickly.\n\nThe house was even worse inside than it looked from the street. Four narrow floors with uneven boards that creaked ominously. He led her further and further up, past so many brown-painted doors she lost count and finally right at the top he opened one.\n\nGeorgia gulped. Like the house itself it was long and narrow. A small window opposite the door so dirt-smeared it could have been curtained. The single bed sagged, the floor was bare linoleum, it looked as wretched as she felt.\n\n'I'll take it,' she said, taking no more than a cursory glance at the single gas ring and chipped sink.\n\n'A week in advance,' his hand shot out reminding her strangely of Sister Agnes. His fingernails were black and she noticed he had no teeth.\n\n'Could I give it you on Friday?' she forced herself to smile. 'I lost my bag this morning, it had all my money in it.'\n\n'I don't take charity cases,' he turned immediately, dismissing her. 'No advance rent, no room.'\n\n'Please,' she caught hold of his arm. 'Please, I promise I'll pay you.'\n\n'I wish I had a quid for everyone that's promised that,' his lips curled back. 'Go on push off. Go down the assistance if you've got no money.'\n\n'Look, take this,' she unfastened her watch. 'Please don't throw me out.'\n\nBut there was no softening in his dark eyes.\n\n'What do I want a cheap watch for?' his tone was contemptuous. 'Go on, push off. I can see you're trouble.'\n\nThere was nothing for it but to go on back down the stairs. He followed her in silence, Georgia pleaded silently with him to change his mind.\n\nShe turned at the door.\n\n'If I manage to get some money now, can I come back?' She could feel prickling tears and she hated herself for pleading with him.\n\n'Going to sell yourself?' he retorted, about to close the door.\n\n'How dare you?' She took a step closer to him, drawing herself up and glowering at him. But before she could even get the satisfaction of giving him her opinion of his disgusting house, the door slammed and he was gone.\n\nAt half past five she found herself in Berwick Street, the sixth or seventh time she'd been through the market that day. The men were packing up now, piling boxes of apples, flinging rail after rail of dresses into vans and handcarts.\n\nA weariness had replaced the earlier joviality. Stalls stripped down in hurried silence, the last-minute shoppers almost an intrusion. Hurricane lights softened the rubbish-strewn streets, yet drew attention to the gaunt, cold faces, highlighting abnormalities, turning every last man and woman into a Hogarth caricature.\n\nShe leaned against the lamp-post, her holdall at her feet, too weary to even go through the motions of pretending to wait for someone. Women hurried by, baskets full of shopping, rushing home to their families. Girls little older than her came tip-tapping down the street in high-heeled shoes, heads bent against the wind, perhaps planning the night ahead of them.\n\nThe notice on the club opposite beckoned her. 'Dancing girls wanted, apply within.' She'd looked furtively at the photographs of girls in nothing more than a G-string several times during the day, and now she was desperate enough to join them. Maybe it wasn't the kind of stage she'd seen in her dreams, or the kind of dancing she'd been trained for, but another night on the streets might kill her.\n\nA crippled girl was coming up a narrow street towards the crossroads where Georgia waited, catching her attention by her strange gait, bright red hair and unusual clothes. Caught for a moment under a street lamp she had the look of a Victorian maid in her long, waisted, russet coat with a fur collar, hopping, rather than walking, one foot dragging behind her.\n\nDespite her obvious affliction she was arresting. Glorious, shiny red hair tumbled around a small white face and her slim shoulders, somehow lighting up the whole street.\n\nShe made Georgia think of red squirrels. Not merely her colouring, or the fast hopping motion, but the nervous head movements, as if she anticipated danger. She stepped out into the road, hitching two bags of shopping up into her arms.\n\nA car roared round a corner just a few yards behind her. The girl's head jerked round sharply, she hesitated, then tried to run. To Georgia's horror, she stumbled, falling right into the path of the oncoming car.\n\nInstinctively Georgia waved her arms and shouted. The car swerved, going up on to the pavement, missing the girl by only inches. The driver made a rude gesture with his fist, and drove off round the corner.\n\nThe girl lay spread-eagled face down. Around her the dropped shopping bags spilled out their contents, oranges, potatoes, a bag of sugar, mingling with knitting wool and needles.\n\n'Are you hurt?' Georgia ran over, putting her hands under the girl's shoulders and lifting her up.\n\n'I don't think so.' The girl's voice was shaky, her face a ghastly white.\n\n'He only missed you by inches,' Georgia said, supporting the girl with her arm. She saw a trickle of blood run down the girl's forehead, as bright as berries against her white skin. 'You'll have a big lump there tomorrow.'\n\nShe led the girl over to a wall.\n\n'Stay there, I'll just pick up your stuff,' she said.\n\nAs she retrieved the vegetables and the knitting from the spilt milk she glanced up and noticed the girl was wearing a special built-up boot. She was leaning back against the wall panting, holding her head.\n\nShe carried the bags over to the girl.\n\n'Do you live round here?'\n\nThe girl dabbed at her forehead with a hanky, wincing as she saw the blood.\n\n'Over there,' she pointed to the corner where seconds earlier Georgia had stood. 'I feel funny.' Her accent wasn't a London one, the way she rolled the 'R' sounds smacked of the West Country.\n\n'I'll help you home,' Georgia put both bags in one hand and her spare arm round the girl.\n\nIt was difficult to know if the girl was pale because of her near miss, or naturally so. She was painfully thin, her wrists and hands smaller than a child's. But as Georgia helped her along it was the limp she noticed most. She barely put any weight on her bad leg, merely dragging it along behind her, hence the hopping gait she had observed earlier.\n\nGeorgia hesitated as they came to her own bag under the lamp-post.\n\n'Well that's something,' she said as she bent to pick it up too. 'I left a bag somewhere this morning and when I went back it was gone. Now where to?'\n\n'Just here.' The girl pointed to a door right next to the side of the caf\u00e9 where Georgia had smelt meat pies. She now looked faint, her colour turning from white to pale green and clearly she hardly knew what Georgia was saying. 'Can you open the door?'\n\nGeorgia took the keys from the girl's shaking hand. As she opened the door she could make out a narrow hallway with steep stairs ahead, the dank smell which wafted out suggested dirt and neglect.\n\n'There's no light bulb,' the girl said feebly. Georgia had a feeling she was about to be sick.\n\n'How far up is it?' she asked.\n\n'The top,' the girl sighed.\n\nGeorgia dropped the bags on the floor, bent down and picked the girl up, hoisting her over her shoulder.\n\n'Just hang on, I'll get you up there,' she said, feeling her way in the darkness.\n\nOnce past the bend of the stairs Georgia saw a glimmer of light ahead. The acrid smell indicated a bathroom. She slid her hand round the door and found a light switch.\n\nIt worked and gave Georgia enough light to continue up to the top.\n\nThe girl was so light it was like holding a small child, even through her thick coat Georgia could feel her bones.\n\nThere was only one door at the top. Still holding the girl Georgia unlocked the door and walked in with her.\n\nOne naked lightbulb lit up a pitiful attic room.\n\nIt was big with sloping ceilings that went from seven feet down to three on the window side. Untidy, a little dirty but the overall feeling was one of poverty.\n\nGeorgia put the girl in one of the two old armchairs, unsure whether she should go or stay. It was icy cold, yet when she glanced down at the girl she noticed beads of sweat on her forehead, mingling with the blood and the greeny tinge to her face was even more noticeable.\n\n'You look sick,' she said, looking round frantically for something to give her. Under the sink in the corner was a plastic bucket. She picked it up and handed it to her.\n\nThe girl retched violently. Georgia averted her eyes, trying hard not to gag, and knelt down to light the gas fire.\n\nLooking closer, the room was almost as comfortless as the one she'd seen earlier in the day. Two single old-fashioned beds, both with blankets just dumped on them like two heaps of rubbish. Armchairs with stuffing coming out of them, a table strewn with cups, plates and a sauce bottle. An old white sink was sandwiched in one corner with an ancient cooker, and a battered wardrobe with the door hanging open stood against the wall. Even the square of carpet was so old and dirty no real pattern was visible and the faded and peeling paper on the walls had patches of black mould.\n\nAgain and again the girl was sick. Georgia's empty stomach turned over as streams of vomit gushed into the bucket. She wanted to run out, away from the rancid smell, the miserable room and this odd, crippled girl.\n\nEverything had happened so quickly, a turn of events so unexpected that for a moment she had forgotten her own predicament. But it could be an important turn, at least she might be able to have a cup of tea if she stayed.\n\n'Let me bathe your forehead,' she said, going over to the sink and dampening a face flannel.\n\n'I don't know how I would have managed without you.' The girl looked up at Georgia. 'You've been so kind.'\n\nShe had the biggest, greenest eyes Georgia had ever seen. As intense as traffic lights, fringed with red-brown lashes, her skin so white it was almost transparent.\n\n'Are you feeling better now?' Georgia tried hard not to look at the brown mess in the bucket, still in the girl's hands. She bathed the wound gently, holding her breath so she wouldn't smell the vomit.\n\nA flush of embarrassment spread across the girl's face.\n\n'Yes,' she said weakly, trying to get up. 'But I need to empty this.'\n\n'Let me,' Georgia said, 'I'll get the bags we left downstairs. Stay where you are.'\n\nAs Georgia made her way back down the stairs with the evil-smelling bucket, she wanted to run. There was food in the bags and probably money in the handbag. She didn't owe this girl anything, she didn't even know her name.\n\nNothing had prepared her for the bathroom. It was filthy. The bath was ingrained with dirt, even a spider had climbed in and died. The walls above grubby white tiles were flaking and festering with mould. The toilet was corroded with limescale, the smell catching her in the back of her throat. Cracked lino, encrusted with dirt. Even the ancient geyser hanging over the bath was daubed with more filth.\n\nShe gagged as she tipped the bucket into the toilet. Holding it at arm's length she filled it with water and swilled it round. How could anyone live in such a place? It made her think of slums her mother spoke of over in Hackney.\n\nLeaving the bucket by the bathroom door, she went on downstairs, feeling her way in the darkness.\n\nHer coat was still on, her belongings down here, she could be far away by the time the girl upstairs had the energy to come and look.\n\nLooking over her shoulder back up the stairs, she rifled quickly through the girl's shopping, smelling rather than seeing. A loaf, some corned beef in greaseproof paper, eggs, cheese and bananas. She was so hungry it was all she could do not to rip the bread apart now and wolf it down.\n\nThe other bag contained some tins, potatoes and other vegetables along with the knitting.\n\nThen there was the handbag.\n\nIt was just a cheap plastic one, with a handle so worn it could break any minute, so different from the good leather one she'd lost in the dustcart.\n\n'Have you found the bathroom?' The girl's voice wafted down the stairs.\n\nGeorgia hesitated, her fingers on the bag clasp.\n\n'Yes. I'm just coming,' she called back up through the darkness.\n\nShe couldn't do it. Not to a crippled girl. There had to be another way.\n\n'It's awful down there isn't it?' The girl looked up as Georgia came in with the shopping bags and her own holdall. 'Whatever must you think?'\n\n'In my position I can't think anything,' Georgia managed a weak smile. 'Let me make you some tea?'\n\n'I'm Helen,' the girl turned in her chair and smiled. The greeny tinge had gone, her pixie-like face arrestingly white against the vivid red hair, yet a faint pink tinge had come back into her cheeks. 'What's yours?'\n\n'Georgia.' She moved over to the sink and filled the kettle, speaking over her shoulder. The sink was full of unwashed crockery and saucepans. 'Is it all right if I wash these things up?'\n\n'What did you mean, \"You couldn't afford to feel anything?\"' Helen's voice had a note of anxiety. Georgia turned on the hot tap wondering how she could explain.\n\n'Blow the geyser!' the girl said sharply. 'It'll light up then.'\n\nGeorgia blew and a sheet of flame leapt up, the water turned warm under her fingers, the first time she had been able to feel them all day.\n\n'Come on, what did you mean?'\n\nGeorgia gulped.\n\n'I haven't any money. I was just waiting for the club over the road to open when I saw you. I was going to ask for a job as a dancer.'\n\nThe girl got up, Georgia saw she was still shaking.\n\n'Stay where you are.' She pushed her back into the chair. 'Have some tea before anything else.'\n\n'But it's a strip club,' Helen said faintly. 'Are you serious?'\n\n'Don't I look like a stripper?' Georgia tried to laugh, but she was blushing. She pulled off her woolly hat and shook her hair free.\n\n'No you don't,' Helen retorted. 'You look about fifteen, cold and hungry. Have you run away?'\n\n'Sort of.' She avoided further questions by noisily washing the dishes. All she could think of was drinking hot, sweet tea, maybe a slice of that bread in the bag.\n\nEven the room didn't look so bad now. It was getting warmer, it was even growing cosy. Should she tell the girl everything? What if she turned her over to the police?\n\n'Tell me.' Helen's voice sounded sweet and warm as she handed her a mug of tea. 'I know you don't belong around here.'\n\nAs the warm sweet liquid went down, and the heat from the fire thawed out her icy toes so she felt able to tell a version of the truth.\n\n'I had a row with my foster father,' she said. 'I can't go back. But I lost all my money today and I don't know what to do.'\n\nHelen reached out and touched Georgia's knee. 'Stay and have some tea with me for now. But don't go to that club, Georgia. That's a last resort.'\n\nShe got up to remove her coat and held out a hand for Georgia's too.\n\nHelen prepared a meal then. Fried potatoes, corned beef and egg. She didn't pry further, merely chatted about general things, the market, her landlord who owned the caf\u00e9 on the corner and the two jobs she had.\n\n'I work on a stall in the afternoons and Saturdays,' she said. 'Then four nights a week I'm a cloakroom attendant in a nightclub.'\n\nThey sat at the table close to the fire to eat. Georgia wolfed hers down, hardly tasting it, pouring cup after cup of tea from the big brown pot.\n\nOnly when they were finished and sitting back in the armchairs did Helen bring the subject back to Georgia.\n\n'You haven't told me why you ran away,' her green eyes watchful, but sympathetic. 'Would you like to know about my first days in London.'\n\nHelen was certainly the strangest person Georgia had ever met. She looked so frail, yet a bright vitality shone out of the big green eyes. She claimed to have no real friends but she was as easy and comfortable to be with as Christine had been. The untidy, grubby room didn't go with her neat but old-fashioned appearance. Even the fiery, shiny hair waving over her small shoulders seemed to belong to another century where ladies had alabaster skin and kept their limbs covered at all times. She sat like a prim governess, knees and feet closely together, back straight, her boots concealed under the long brown dress, the only decoration a skimpy, cream lace collar. No one would guess that just outside her window was London's red light district.\n\n'Did you run away too?'\n\n'Hobbled,' Helen half smiled. 'I was seventeen, in a home for handicapped kids down in Plymouth. I knew if I didn't get out under my own steam they'd find me something they considered suitable for a cripple.'\n\nShe used the word 'cripple' like Georgia used the word 'black', getting it in first as if to protect herself.\n\n'What's the matter with your leg?' Georgia asked.\n\n'An injury when I was little,' Helen said lightly. 'I'm going to have an operation soon to put it right.'\n\n'Why London?' Georgia sensed the girl wasn't keen to talk about her leg, or her background.\n\n'Same reason as you I expect. A place to disappear, start again. When you're a kid it seems the place where dreams come true.'\n\n'But yours didn't?' Georgia glanced around her, noting how few possessions the girl had, a feeling that it was only intended as a temporary home, yet poverty had trapped her.\n\n'Well my dreams were a bit far-fetched,' Helen laughed. 'I thought I'd meet a millionaire, he'd whisk me off to the Ritz, you know, all that silly stuff.'\n\n'But you found a job and a home!'\n\n'Yes,' Helen frowned. 'Eventually, and it's not much to brag about is it? But believe me I've seen worse, especially the first few nights.'\n\n'I slept in a cardboard box last night,' Georgia sighed. 'I thought I'd die of cold.'\n\n'I can imagine,' Helen smiled in sympathy. 'I spent mine in St James's Park. It was summer so it wasn't so cold but I was scared witless. I didn't have a clue about anything.'\n\n'How did you get this place then?'\n\nHelen shook her head, blushing a little.\n\n'That came later,' she said uneasily, as if wondering whether she should divulge everything. She looked up at Georgia defiantly. 'I let a man take me in.'\n\nGeorgia was reminded of the man who approached her the night before.\n\n'You mean?' she couldn't actually say it.\n\n'Not for money,' Helen shot back. 'I just accepted a bed for the night. I suppose that amounts to the same thing, but then I was desperate.'\n\n'Was it awful?' Georgia whispered.\n\n'Yes,' Helen winced. 'He didn't hurt me or anything, but he was weird. He made me feel so dirty. After about three days I started to get frightened. I heard him talking about me to a friend on the telephone. I think he was intending to let this chap,' she broke off blushing. 'Well, you know!'\n\n'So what did you do?'\n\n'I nicked his money,' Helen giggled, blushing again. 'He left his wallet behind when he went out. By the time he got back I was gone.'\n\n'That was brave.' Georgia leaned forward in her chair, liking Helen far more for revealing she didn't intend to become a victim. 'What then?'\n\n'Bert in the caf\u00e9 downstairs had an advert on his window for this room. I took it and I've been here ever since.'\n\nGeorgia stared into the fire.\n\nAlready the standards she'd been taught by Celia were eroding. All day she'd contemplated theft, she was even prepared to become a dancer in a strip show. Helen's story made her feel a little better about herself, at least she wasn't the only one to fall prey to temptation.\n\nHelen got up again, hobbled across the floor and put the kettle on again.\n\n'Are you going to tell me about yourself?' she said gently from the sink.\n\nBeing crippled had made Helen perceptive. At twenty-one she had spent most of her life sitting quietly at the side lines of life watching others. But Georgia baffled her. It was obvious the girl had come from a good home. Her clothes were expensive, she spoke well and had beautiful manners. Yet there was deep sadness in the girl's eyes. Something more than a rich kid looking for adventure. Runaways were ten a penny in Soho. They bleated out their misfortunes, lied and conned. Never before had she met one where she felt compelled to tell her own shameful story to try and get them to open up.\n\n'I can't tell you everything,' Georgia said in a whisper, hanging her head. 'I want to, but I can't.'\n\nHelen just stood there, the kettle in her hand. The kid was so young, she might be a liability. For all Helen knew the police might be searching for her right now; if she had any sense she'd make some excuse to go out. Fetch her landlord and let him make the decision.\n\n'Stay here tonight,' Helen said softly. 'Maybe tomorrow you'll find it easier.'\n\n'Are you sure?' Georgia's head spun round to look at Helen. She could feel tears pricking her eyes at the unexpected compassion.\n\n'Quite sure.' Helen moved nearer, passing her hand lightly over Georgia's head in just the same motherly gesture Celia used. 'You didn't hesitate to help me.'\n\nThey had more tea and biscuits. Georgia told Helen about the way she lost her money and how she'd spent the day.\n\n'Would it really be so bad to get a job as a dancer?' she said, yawning widely.\n\n'It would in that club,' Helen smiled. 'But you're tired and I'm afraid I'll have to ask you to make up the bed. I'm sorry the sheets aren't clean. I was going to the laundrette tomorrow.'\n\nGeorgia leapt up, grabbing the blankets and smoothing out the sheets before Helen could change her mind.\n\n'I hope you won't mind if I use the bucket in the night?' Helen said without a trace of embarrassment after she had made a last trip to the bathroom. 'I can't go clonking downstairs, waking everyone up.'\n\nGeorgia winced, as she was reminded she too would have to go down there.\n\n'Doesn't anyone ever clean it?' she asked on her return.\n\n'I used to try,' Helen pulled a wry face. 'But I've given up. I go down the public baths once a week, and the other tenants are hardly ever here. My landlord keeps saying he'll do something to it, but he never does.'\n\nGeorgia hurriedly slipped out of her clothes and into her pyjamas. The bed felt damp and it was lumpy, the sheets and blankets smelled, but she was too tired to care. She put her arm under her head, nose buried in her pyjamas with the faint whiff of home and waited for Helen to put out the light.\n\nHelen wore thick stockings under her long skirt. As she peeled them off, sitting on the side of the bed, Georgia could see how wasted her leg was. She could have put her thumb and forefinger round the ankle, the calf only slightly bigger.\n\nHer other leg was normal, the muscles made bigger by taking the strain from the bad leg.\n\nEarlier, out in the street Helen had reminded her of a squirrel, later that image had changed to one of a prim governess, but now as she undressed so the image changed again.\n\nShe wore a white cotton petticoat, her gleaming hair cascading down her back, like a lady from a Pre-Raphaelite painting. Although she was too thin and her skin so pale, she was lovely in a mystical way. Her features were all small and childlike. Perfect bowed lips, green eyes sparkling under reddish-gold eyebrows and lashes. Not even a freckle as she would have expected with red hair, just pure white skin like a baby.\n\nGeorgia woke with a start, for a moment wondering where she was. It was still dark, but every now and then bright flashes of light splashed on to the walls accompanied by rumbling, banging and shouting from outside in the street.\n\nPulling a blanket round her, Georgia went over to the window and peered out through the dirty glass, rubbing a patch with her blanket to see better.\n\nThe stallholders were setting up. Skeleton-like stalls being hastily assembled by men in donkey jackets and woolly hats, brandishing mallets. The sound of tarpaulins being shaken, the trundle of trolley wheels and the slap of cardboard boxes hurled from vans.\n\nYesterday this same scene had seemed violent and brutal. The coarse jokes, the swearing and the raucous laughter had seemed like a glimpse into a madhouse as she shivered on the corner. Now it seemed jolly, almost a street party. An Indian in a red turban wheeled a rail of dresses. A woman, mummified with a scarf wound round her head and neck arranged flowers in pots. A tall thin man dancing on the spot to keep warm. Teasing banter as they pushed and shoved through the disorder, the cold air making each one of them move faster. Gleaming apples, oranges, tomatoes and bananas lay in boxes, like a feast about to be prepared.\n\nGeorgia looked at the clock. It was only seven. Although she wanted to get up she didn't think she should disturb Helen just yet. She climbed back into bed, buried her head in the pillow and went back to sleep.\n\nShe awoke again to the sound of splashing water. Helen stood at the sink, naked, washing herself as if she were alone. Sideways on she had the figure of a small boy, flat-chested, with a tiny, bony bottom and concave stomach, her thighs so slender it was a miracle she could walk at all.\n\nGeorgia closed her eyes again, afraid she might embarrass Helen, and waited for her to dress.\n\n'Cup of tea?' Helen was now wearing an old faded dressing-gown the colour of mushy peas.\n\nGeorgia sat up and looked at the clock.\n\nIt was only eight.\n\n'I always go out early,' Helen smiled. 'I go down to the library as soon as they open and read the papers.'\n\n'What for?'\n\n'To find out what's going on in the world. Besides it's warm in there, saves on heating. But I suggest you don't go far today as they may be looking for you.'\n\nGeorgia looked sharply at Helen. Could she be intending to go to the police to turn her in? Or did she mean she could stay?\n\nHelen frowned as though she had something on her mind.\n\n'I can't promise anything,' she said. 'Last night I wanted to invite you to stay permanently. But I'm not sure now. You're so young. We'd have to find you a job. I need time to think it over. I don't even know if I could live with another girl. But stay till tomorrow. By then we should both know if it's going to work.'\n\nHelen's honesty touched Georgia deeply, it lit up the drab, cold room and she felt ashamed that she had doubted her intentions.\n\n'But I've got no money,' Georgia reminded her. 'I can't help out until I get some.'\n\n'Another day won't break the bank.' Helen limped over and sat on the bed. 'Take the washing over to the laundrette across the street for me and tidy up. That's enough help for one day.'\n\nShe passed over a mug of tea, put some coins on the table, then under her dressing-gown she started to dress.\n\nHer knickers looked ancient, the petticoat was worn and thin. She sat down on the bed pulling on thick brown stockings held up with a garter of thick elastic. Next a thin sweater full of holes. The brown dress she'd been wearing the day before, and over the top a thick, gold-coloured cardigan which looked hand knitted.\n\nGeorgia gulped. She thought of her drawers full of clothes back in Blackheath, dainty underwear, soft sweaters, dresses hanging on padded, scented hangers.\n\n'You have to wrap up warm.' Helen seemed to sense Georgia's shock, turning to grin at her. 'This place is like a morgue and it's even worse on the stall.'\n\nPulling on her coat and a woolly hat over her ears, she left, shutting the door behind her.\n\nGeorgia sat for a moment, listening to the painful sound of the built-up boot clonking down the stairs.\n\nCelia came sharply into focus. She could almost smell bacon frying, hear the news on the radio. Celia would be sitting at the kitchen table, drinking tea from a bone china cup as she scanned through the post.\n\n'She'll be crying,' Georgia murmured, suddenly aware that until this moment she hadn't really considered how her mother would be feeling. 'She'll be frightened for me.'\n\nAn acute pain made her curl up into a ball. She didn't want to stay here in this dark, cold room. She wanted to be back home with Celia, fifteen was too young to be trying to fend for herself, how could she find a job alone? How was she going to manage?\n\nBut as she dug deeper into the bed, Brian's face came back to her. She could feel his breath on her face as he lunged at her in the playroom.\n\n'You're not my daughter.' Those were his words. Everything she remembered and loved belonged to him. The house, her clothes, the piano, even Celia belonged to him. She couldn't go back, not ever. It was over.\n\nSlowly she lifted her head from the covers. This room wasn't hers either, for now she was dependent on a crippled girl who'd been big-hearted enough to share what little she had.\n\n'You've got to make it work for you,' she whispered. 'Stop feeling sorry for yourself, the worst is over. Get up and get moving.'\n\nThe room was icy, but she must save the money in the gas meter. Jumping out of bed she moved the table to one side, taking the back of one of the upright chairs.\n\nDance exercises were the answer. She must get the blood flowing again, stretch those lazy muscles, warm herself up.\n\nIn the playroom in Blackheath, she had the record player to help her concentrate. Her own barre Brian had put up for her. She would watch each movement in the big mirror to see it was right. Back straight, eyes ahead, moving in time to the music.\n\nShe had to pretend the chair was a barre, imagine the music, forget she was wearing pyjamas and her feet were bare. Maybe she had no dancing teacher or parents now, but no one was going to stop her getting on the stage. All it took was willpower.\n\nFirst one leg, up, down, up, down, repeating it till she hurt. Then the other. Knee bends, high kicks, and toe touching until she was hot with exertion.\n\nVisions of dancing class came to her. Sixteen girls, each identical in black tights and leotards, pink ballet shoes. Hair scraped back into a tight bun, Miss Askell pounding the piano as they pli\u00e9-d and jet\u00e9-d at her instruction.\n\nToday Celia wasn't going to come for her, no flask of hot coffee while she got dressed, no hour of shopping before driving her over to her singing lesson in Greenwich. But Celia had believed in her talent enough to sacrifice each Saturday willingly. She must keep it up for Celia now.\n\nShe washed herself later in the sink, trying hard not to dwell on the sparkling bathroom she'd never see again. Then before she could get cold, she jumped into jeans and a sweater.\n\nShe filled a pillowcase with all the dirty clothes and bed linen, then made her way down the stairs, across the street.\n\nThe laundrette hadn't been open long. A woman in a blue overall was mopping the floor, she turned to look at Georgia.\n\n'Service wash, love?' she shouted above the noise of a radio.\n\nGeorgia stared blankly at the row of machines, round glass doors standing open.\n\n'I, I,' she stammered, blushing with embarrassment. 'What do I have to do?'\n\n'Well are you going to work, or have you got time to watch it yourself?' the woman asked, stepping nearer.\n\n'It's Helen's stuff,' Georgia said weakly. 'Do you know her?'\n\nThe woman's face broke into a smile, showing more gaps than teeth.\n\n'Oh, she gets special treatment.' She reached out and took the bag from Georgia. 'I see's to it for 'er. She comes back later. She ain't ill is she?'\n\n'No, she's fine. I'm just staying with her for a day or two,' Georgia reassured her. Already the woman had the pillowcase open, tossing the washing into two separate machines, whites in one, coloureds in another.\n\n'That's all right then,' the woman nodded at Georgia. 'Come back around twelve, it'll be ready then.'\n\nBack upstairs Georgia stared around her, wondering where to start.\n\nLast night the room had looked squalid, but by day it looked far worse. The small, dirty window cut a shaft of light through the middle, but under the eaves it was still in shadow. Yet looking objectively at the room, part of the reason it looked so wretched was the way the furniture was arranged. The wardrobe so close to the window blocked the light. The beds sticking out from the wall made it seem like a dormitory. So the wallpaper was stained, and the carpet would never curl comfortingly round anyone's feet, but there was room for improvement.\n\nStarting by the fireplace she cleared the furniture to the other side of the room by the sink. Then taking a small stiff brush she began to sweep the carpet on all fours. Again and again she went over it, until at last no more dust flew into the pan. Filling the bucket with hot soapy water she washed down the fire surround, the mantelpiece, the skirting boards and the floorboards around the carpet. Then changing the water again, she went over the surface of the carpet, being careful not to make it too wet.\n\nBy the time she had finished the whole room it was twelve. The furniture stacked by the fire, ready to rearrange. She couldn't count the number of buckets of black water that went down the drain. Her hair felt full of dust, she had tidemarks up to her elbows, but it was so satisfying.\n\nThe carpet was a red-brown. She could even see a swirly pattern now. The windows gleamed, letting in twice as much light, but most of all it smelled clean.\n\n##### *\n\n'Been up a chimney?' The woman in the laundrette grinned at her as she handed her the warm bag of washing.\n\n'Doing a bit of spring cleaning,' Georgia said shyly. She could feel the dirt on her face and her hands were bright red from all the cleaning fluid. 'How much do I owe you?'\n\n'Call it five bob,' the woman said. 'I dried everything real well and folded it. It's good 'Elen's got a bit of company, spends too much time on 'er own does that one.'\n\nThe market was packed with shoppers. Women from offices buying vegetables, young girls looking at clothes. Georgia kept her eyes down as she scuttled back across the street, aware of a policeman standing by the door of the caf\u00e9.\n\nWas he looking for her, or merely taking a break in his beat? Her heart thumped with anxiety as she slipped in the front door. What if he asked the lady in the laundrette about her?\n\nUpstairs again she felt safer and she had a great deal more to do before Helen got back.\n\nHelen smelled bleach and disinfectant as she stepped into the dark hallway. Instinctively her hand went out to the light switch and to her surprise a light on the stairs came on.\n\nShe blinked. Not a sweet wrapper or bus ticket in sight. Balls of fluff and a coating of grey grit on each stair had gone. The carpet was so worn in places she could see the stair treads showing through, but it was clean!\n\nSlowly she hauled herself up by the bannister. She was so cold she couldn't feel her toes, but for the first time ever she felt a rush of pleasure to be going home.\n\nThe open door to the bathroom beckoned her. She paused, leaning heavily against the wall, staring in amazement.\n\nThe smell of bleach was so strong it almost choked her, but in front of her was a clean bath, basin and toilet.\n\nGranted there was still a stain of limescale where the cold tap dripped in the bath, and nothing could be done about the chips and scratches, but the tiles were white! The window-sill was shiny, even the cracked lino was clean. The smell of stale urine was gone. They could even use the bath if they wanted to!\n\nBut there was another smell too as she went on up the stairs in a daze. An aroma of meat pie. Was she dreaming all this, or was it merely a faint memory of something from years ago?\n\nThe door opened as she turned on to the last flight of stairs. Georgia stood there, a smile of welcome on her face.\n\n'I thought you'd never come home,' she said, taking Helen's basket from her arm. 'I've got the tea ready.'\n\nAs Helen limped into the room she stopped suddenly.\n\nThe fire was lit. Curtains drawn. The table was now under the window laid for two, saucepans and kettle rattling on the cooker.\n\nGeorgia had moved everything and it looked almost like a real home. The beds flanked the walls either side of the fire place, made up properly, each covered with the bedspreads she had never bothered to get out of the wardrobe.\n\n'What have you done?' Helen asked, tears for no apparent reason pricking her eyelids.\n\nShe couldn't hold back the tears, she groped for the armchair and sat down heavily.\n\n'I'm sorry,' Georgia said, as if froms a long way off. 'I didn't mean to be bossy. I'll put it all back how it was tomorrow.'\n\n'It's,' Helen couldn't speak for a moment. 'It's just such a shock.'\n\nOnly then did she really see Georgia. She wore a red sweater and jeans, kneeling down in front of her, her heart-shaped face a picture of concern, her eyes bewildered and afraid.\n\n'I didn't mean to upset you. Please don't cry.'\n\nHelen's mouth shook, laughter bubbling up inside her. Georgia's face was so lovely, she hadn't fully taken it in the night before.\n\nThe big, dark doleful eyes, the sooty lashes, the curly hair tied up in a pom-pom on top of her small, beautifully-shaped head. The grace in her movements, the slender hands and the soft voice. Who would have thought a girl who looked like that could be capable of cleaning a stinking toilet?\n\n'I'm not cross,' she said weakly. 'I just didn't expect it.'\n\n'Why are you crying then?' Georgia's voice was tentative, like a child who isn't convinced she'd really won approval.\n\n'I've lived here for nearly four years,' Helen sniffed. 'In all that time no one has ever been up here except me. I can't remember the last time someone had a meal ready for me, or cared enough to tidy up for me. You don't know how good it feels.'\n\nGeorgia's eyes seemed to grow bigger, her wide, curvy mouth quivering.\n\n'I wanted to show you how much I appreciated your kindness,' she said softly, reaching out and touching Helen's cheek. 'If you'll let me stay and help me find a job I'll cook and clean for you every day.'\n\n'I thought you were such a child this morning.' Helen found herself crying again, remembering all the second thoughts she'd had during the day. 'I panicked because I thought you would want me to look after you. That's why I went out early. I even thought of going to the police and telling them you were here.'\n\n'Why didn't you?'\n\n'It seemed cowardly,' Helen sniffed. 'Besides, I half expected you to take the money and go. I couldn't see you wanting to stay with a cripple in a slum.'\n\n'That isn't how I see you,' Georgia looked shocked. 'Maybe I did at first, but then you became just another girl with a bad leg, someone all alone like me with nothing but bad memories behind her. But who knows? Maybe we can have a future together.'\n\n'Oh, Georgia,' Helen reached out for the younger girl, drawing her to her breast, an instinctive action she had never done to anyone before. 'I've got a feeling this was meant to be.'\n\nHelen felt Georgia's tears even through her coat and in that moment she knew something shocking had happened to her, something far worse than a family row.\n\n'What's the smell?' she said, lifting Georgia's chin up.\n\n'Meat pies from the market.' Georgia wiped her eyes with the back of her hand. 'I bought them with the change from the laundrette. I hope that was all right?'\n\n'I love meat pies,' Helen smiled. 'Especially when I've got company to eat them.'\n\n## Chapter 5\n\n'Sometimes I wonder if I'll ever be warm again.' Georgia held her toes up to the fire and massaged them with her fingers.\n\n'It's March next week, spring's nearly here,' Helen sighed in sympathy. She was stirring a pan of soup over the cooker, still wearing her coat. 'Why don't you do some exercises if it's that bad?'\n\n'I ache too much,' Georgia growled at Helen. 'Besides you always laugh at me.'\n\n'Sounds like a feeble excuse. What happened to your fighting spirit?'\n\n'Died of cold.' Georgia looked round at her friend and tried to smile. She didn't want to admit she'd been sick again that morning, or about her fears.\n\n'Did Janet get her letter from the sailor?' Helen plonked half a loaf down on the table and turned to get the teapot.\n\nGeorgia stood up, pushing her feet back into her slippers and went over to the cooker, peering into the pan.\n\n'No, I think she's given up on him now. She made some joke about how she could always pull men, but she hadn't figured out how to keep them yet.'\n\nGeorgia prodded the soup. Her insides were bashing together with hunger, yet just the thought of vegetable soup again made her feel queasy.\n\n'It's ready, pour it out will you?' Helen didn't notice Georgia's grimace as she got cups out of the cupboard.\n\n'I can't eat this,' Georgia dropped her spoon with a clatter after a few tentative mouthfuls. She picked up a hunk of dry bread and wolfed it down.\n\nHelen said nothing, just a slight raise of one gold eyebrow as she lifted the spoon to her lips.\n\n'Go on, say it,' Georgia challenged, her mouth full of bread. 'I'm a spoiled brat and it's my fault there's no money left this week.'\n\n'Did I say a word?' Helen retorted.\n\n'You don't have to,' Georgia sniffed. She picked up her spoon and tried again.\n\nHelen had asked her to buy belly of pork at the weekend, but she'd seen a piece of beef in the butcher's shop and bought that instead. Now she was discovering what a tight budget really meant.\n\n'It's a good job we can get vegetables for nothing,' Helen grinned. 'I bet you won't be so daft again.'\n\n'It's all right for you. You get a meal at the club later,' Georgia said. She had had to buy that beef, her stomach was screaming out for the sort of food Celia cooked. How was she to know how much beef cost? 'I bet you've been eating fruit all day too!'\n\n'What is it Georgia?' Helen put down her spoon and reached across the table to touch her hand. 'There's something wrong isn't there, something more than being hungry and cold. Do you want to go home?'\n\n'This is my home,' Georgia shovelled the soup in her mouth, barely tasting it.\n\nOf course she wanted to go home! Day after day she dreamed of Celia's hot meals waiting for her, the clean sweet-smelling house, her warm bedroom and all the other things that home meant. But there was no way back now, this room, her job as a machinist and Helen was all she had now.\n\n'Is it something at work then?' Helen could be so persistent. 'Is it too much for you?'\n\n'I'm just tired and cold,' Georgia tried to smile, even though she felt a lump coming up in her throat. 'I'll go to bed early tonight.'\n\nHelen said nothing more, just watched anxiously as Georgia finished her soup.\n\nShe hadn't believed Georgia would last a week in Soho, let alone seven. When she sent her up to Pop's workrooms to ask for a job, she hadn't expected her to even go in, let alone get the job and stick at it.\n\n'Pop' as everyone called him in the market was a fiery Greek, his sweatshop a place you had to be desperate to work in. Four tiny rooms over his material shop, stinking of paraffin, engine oil and damp cloth. Five industrial machines and a huge steam press assaulted the ears and if you were capable of dealing with that, there was still his other employees to cope with.\n\nJanet and Sally, two vicious-tongued women, ruled the roost, with Irene, Iris and Myrtle as their dim-witted sycophants.\n\nHelen had spent a week in the workshop herself, before moving on to what she considered a far easier life in the market, but somehow Georgia had not only managed to master the sewing machine, but she'd also made friends with the other women.\n\nHelen pulled two oranges out of her bag and tossed one to Georgia.\n\n'You see I didn't stuff myself with fruit all day,' she chuckled. 'Now, in return I want all the gossip.'\n\nGeorgia was such a child. Helen saw the way her big eyes lit up with glee, grubby fingernails digging into the thick peel, tearing it off and biting into the juicy flesh like a savage.\n\n'Iris isn't Pop's mistress,' Georgia said slurping at the fruit, juice running down her chin. 'She just kind of hints he's in love with her. I reckon she's got delusions of grandeur. She told me her boyfriend is a count!'\n\nIris, the cutter, was in her forties, still attractive in an overblown rose style. Flame-red hair copied slavishly from Ava Gardner, given to fox furs and the kind of glamour that related more to the war years than now.\n\n'What did Janet have to say about that?' Helen giggled.\n\nGeorgia wiped her mouth on her sleeve, she got up quickly and turned her back on Helen, returning moments later with two pairs of socks shoved up under her jumper in a parody of the busty Janet.\n\n'The Count of Monte Cristo?' she put her hands on her hips, imitating the tarty stance, wiggling across the room. 'Famous for his disappearing acts!'\n\nHelen's laughter pealed round the shabby, cold room. Georgia with her slim hips in tight jeans couldn't possibly look like the woman with her blonde bird's nest hair, tight low-cut dresses and ample curves, but she'd got the essence of her as she always did with people she imitated.\n\n'I suppose Iris went into one of her sulks?'\n\n'Not half,' Georgia reported gleefully. 'She cut out the dresses so fast we couldn't keep up. She kept banging the shears down on the cutting table so often I thought she might stab Janet.'\n\n'Are you keeping up with the others?' Helen fished, still convinced Georgia was hiding something. 'It must be hard when you aren't experienced like them.'\n\n'I'm not doing too bad,' Georgia stacked up the plates and took them over to the sink. 'Pop makes allowances for me, besides Myrtle always unpicks the bits I've done wrong. You know what she's like.'\n\nMyrtle was the sweet, uncomplaining one in the workroom. She perched in front of her machine all day like a drab little sparrow, offering very little in way of conversation. Her clothes were carefully pressed and mended but old and shabby. She rarely volunteered any information about herself, preferring to sit on the outskirts, looking in.\n\n'Is she still with that man?'\n\n'She must be,' Georgia looked round from the sink. 'She's got a huge bruise on her arm. I saw it when she took off her cardigan to wash before going home. I wonder why she doesn't leave him?'\n\n'Two kids and a council flat in Hackney, that's why,' Helen said, slowly nibbling the last segment of her orange. 'I don't suppose she thinks life has anything else to offer.'\n\n'But Janet and Sally left their husbands,' Georgia frowned. 'I can't see why anyone would stay with a man who beats you.'\n\n'That's easy for you to say,' Helen smiled. 'You are young and pretty. What chance has Myrtle got of finding a new man?'\n\n'I don't see why women need men. As far as I can see, they are nothing but trouble.'\n\nHelen picked up her knitting. She had noticed how often Georgia made disparaging remarks about men. Was it the influence of Sally and Janet? Or was it part of the chain of events which led Georgia to Soho? She still wouldn't open up fully, in seven weeks all she had was a sketchy vision of a comfortable middle-class home, dancing and singing lessons, then an unexplained row which led to her leaving it. Yet she had asked Helen to get someone to post a letter in Manchester to her foster mother. Why would she even bother if she didn't care, and why did she cry out in the night so often?\n\n'You're getting to be a right little cynic!' Helen said gently. 'To think I was hoping you'd introduce me to Mr Right!'\n\nHelen dreamed of men constantly. At night in her cloakroom job at 'Squires' she would smile at the smart men who came in, hoping against hope one day someone would overlook her limp. Only the thought that the operation she was waiting for would be successful kept her going. It was all very well to be liked for yourself, but she wanted romance and love, a husband and children. To dance in a man's arms, to walk without a limp down the aisle, to be desired.\n\nGeorgia knew Helen guessed there was more to her than she had revealed. She yearned to open up but she was afraid. It would be like opening a door, forcing herself to look at everything all over again. It was almost over now. She no longer jumped and ran when she saw a policeman, she had even adjusted to living in this place without television, music, or dancing and singing lessons. Maybe in time she could forget Peter, stop wanting his kisses. But if she told Helen now because of her fears it would all come back, and if her period was just late, she would have burdened Helen with it all for nothing.\n\n'We'll go out dancing when you've had the op,' Georgia smiled as if there was nothing on her mind. 'We'll make ourselves beautiful dresses and take London by storm.'\n\n'You've got to teach me to dance first,' Helen laughed. 'Anyway, I'd better get ready to go to work. Will you do my hair for me?'\n\nHelen couldn't imagine life now without Georgia. She filled the lonely hours before her night-time job with chatter and laughter. Sharing meals, shopping and cleaning the room was a pleasure where once there had only been a lonely void. When she came home tired from the club it felt secure to see her friend tucked up in bed and Sundays flew by with her company.\n\nBut Helen was a realist if nothing else, maybe Georgia was content to stay in now, but what would happen if she made new friends? Had it already happened? Was Georgia's troubled look because she was tired of living with a cripple who worked so many hours?\n\nYet if Georgia was growing weary of this place and her, it didn't show in the way she did her hair. Brushing it till it shone, coaxing curls round her fingers with an almost loving touch.\n\nGeorgia got up slowly the next morning, waiting for the expected feeling of nausea. Helen had gone out early as she did every morning, regardless of how cold it was, or how late she finished work. The gas meter had run out last night, she couldn't put the fire on, have a cup of tea or even wash in hot water. But she didn't feel sick and it was payday.\n\nThe window was frosted over on the inside. She scraped a small hole and peered out. Fridays were good. At lunchtime she could go into Sid's and have steak and kidney pie and tonight she could turn the gas fire on full and bask in front of it with a library book, knowing she could lie in tomorrow. Janet had promised to take her to a jumble sale up in Primrose Hill in the afternoon, she might find some dresses they could alter on Sunday, or if it was nice they could go for a walk in St James's Park.\n\nShe was still feeling cheerful when she arrived at work, bouncing up the stairs the way she did when she first started there.\n\n'Well that's a good start to the day,' Pop turned round from dumping some bales of cheap tweed on the floor. 'Let's hope the good mood lasts beyond tea-break!'\n\nGeorgia grinned impudently at him, a manner she'd learned from Janet and Sally. She'd been so frightened of him, and his machines on her first day that she almost wet herself, but now she knew his gruff manner hid a kind heart she often took advantage.\n\nPop still had a strong Greek accent despite living in England since he was eighteen. Portly, with thinning heavily-oiled hair it was hard to see him as the slim, handsome youth he was rumoured to have been. His dark eyes had faded a little, a melancholy olive face, thick fleshy lips and a large rubbery nose made Georgia think of an old clown. Yet perhaps it was true he'd been a hit with the ladies in his prime. He did have a comfortable, easy manner with women.\n\n'Can I make some tea now?' she fluttered her eyelashes at him. 'Our meter had run out so I couldn't have one at home.'\n\n'You girls!' Pop shrugged his shoulders. 'One of these days I'm going to make you work like they do down at Switalski's. On to that seat at half eight, standing over you till one. Maybe then I'd be able to get myself a decent car.'\n\nGeorgia took that as agreement, sliding into the staffroom before he changed his mind.\n\nThe staffroom was a joke. It was no bigger than a cupboard, the toilet adjoining it. A shelf for the kettle, three rickety chairs and the cracked window stuffed up with old rags.\n\nFrom behind her in the main workroom she could hear Pop and Iris discussing the cloth and which patterns should be used. She had been excited when she first came here, imagining she would make good clothes, but instead to her disappointment Pop specialised in making frumpy, cheap things for old ladies. Sometimes Janet and Sally would dress up in them during the lunch hour. Drab, shapeless dresses with white collars and cuffs, always in browns, dark blues and greens. Then the pair of them would do a striptease, peeling them off, more like pantomime dames than the show girls they pretended to be.\n\nWhile the kettle boiled Georgia watched through the open door. Janet was threading her machine, a cigarette hanging out the corner of her mouth, her blonde hair still in curlers with a shocking-pink chiffon scarf tied round them. Next to her was Sally her close friend, leaning forward whispering something.\n\nThey were both thirty, without husbands, and three children each. They even lived in the same block of tenement flats down near Charing Cross Road. Noisy, vulgar and aggressive, the pair had seemed like dragons on Georgia's first day, yet now she viewed them almost with admiration.\n\nSally's raven black 'beehive' stood up an alarming six inches from her head, a slick of greasy black fringe across her forehead, with lacquered kiss-curls fixed like cement on her ruddy cheeks. Her make-up was as startling as her hair. Heavy eyeliner and several coatings of thick mascara. Lips dark red and lustrous, a beauty spot painted on her cheek. Voluptuous and wanton, she scrutinised every man who had the misfortune to come into the workshop, dark, lust-filled eyes sparkling at their embarrassment.\n\nSally might be the one with the startling appearance, but it was Janet who had the real character and personality. By night she worked as a stripper, something she made no secret of. She could turn the most mundane of stories into comedy, and her observations about other people were bitingly astute.\n\nHer daytime appearance, the headscarf, shapeless sweater, crumpled skirt and no make-up, was at odds with the glamour snaps they'd all seen of her. Once the curlers were out, the warpaint and false eyelashes on, the metamorphosis from plain Janet Willoughby to exotic dancer Nicole was complete. If Pop and Sally were to be believed she bore more than a glancing likeness to Marilyn Monroe when she wiggled seductively onto the stage.\n\nIf it wasn't for the humour of these two women, Georgia might never have made it through her first week. They teased her, shouted, even swore at her, but an underlying sense of fairplay made them help and encourage her too, and when she was close to tears they had a knack of turning it to laughter.\n\n'You dun'alf talk posh!' Janet had remarked on her first tea-break. 'Why don'cha learn us to speak proper and I'll show you how to strip?'\n\nGeorgia blushed scarlet, twisting her hands in her lap as minutes later Janet came out of the toilet wearing only a length of fabric and proceeded to do a peek-a-boo dance routine with it. She was convinced Janet was naked under the material, as first one shoulder was bared, then the other. The other women sang for her, clapping their hands and stamping their feet and as Janet dropped the fabric as a climax, Georgia covered her face with her hands.\n\nSally grabbed her hands away, and to Georgia's astonishment Janet was standing there wearing a pair of pink, old ladies' bloomers and two paper roses pinned to a large brassi\u00e8re. The sight was so unexpected and hilarious Georgia almost fell off the seat with laughter, and she'd known then she could stand working at Pop's.\n\n'Isn't that kettle boiling yet?' Pop glanced up from the cutting table at her. Iris at his side sniffed loudly in disapproval. She was supposed to be the forewoman, but her instructions were never carried out by anyone other than Myrtle. Iris was wearing a flame-red two-piece which clashed with her red hair, a silk rose was pinned to her lapel as if she were going to a wedding.\n\n'That's not how it was done in my day,' was her favourite whine, covering everything from Georgia making tea so early in the morning, to the way Irene swept the floor.\n\n'When was that? Domesday?' Janet always retorted, sending Iris's heavily pan-sticked face into a vivid flush of frustration. She spoke vaguely about having a man 'in high places', alluded mysteriously to 'cocktails' after work and sometimes to 'our nest' in Brighton.\n\nGeorgia still had no clear picture about where the woman really came from. It was all snippets with no substance, even her carefully cultivated accent was a fake, as sometimes in anger she dropped it, and sounded more of a cockney than Janet or Sally.\n\nThe kettle boiled behind her and Georgia turned to make a big pot for everyone. She heard Myrtle turn on the steam press and at the same moment Irene came through the door late.\n\n'Do you know what the time is?' Iris's high voice rose above a belch of steam. 'This isn't how it was done in my day. We thought ourselves lucky to have a job, you could be dismissed at a moment's notice for unpunctuality.'\n\nIrene didn't answer, but by her shuffling gait coming into the staffroom to hang up her coat, Georgia knew it was one of her bad days.\n\nIrene was not quite right in the head, as Janet put it, 'A penny short of a full quid.' No one seemed to know what exactly was wrong. She could turn up on time for a week at a stretch, neatly dressed, and chat about her elderly mother in the Oval, books she'd read, and television programmes, as normal as everyone else. But then suddenly she'd change for a few days, like today, coming in late wearing men's trousers with a huge shapeless sweater thrown over the top, her dark hair all tousled as if she hadn't brushed it, top teeth missing, lipstick up to her ears, her eyes blank. She would say the oddest things at these times, about men who followed her. Strange spirits in her house, and odder still she would profess to live in Kensington with a man called James.\n\nBut whatever she was like, she worked harder than anyone, sweeping up, pressing, sewing on buttons at twice the speed of everyone else. Sally said she was over forty, but to Georgia, the smooth, unlined face was that of a girl, only the missing teeth suggested Sally was right.\n\nGeorgia gave everyone a cup of tea and sat down at her machine with her own. In front of her was a pile of grey wool skirts, her job was to do merely the seams, then pass them over for pressing. Later Sally would do the waistbands and zips.\n\n'You coming to the jumble tomorrow?' Janet shouted at her over the noise of her machine. 'Our Lyndsey's gonna take the other kids to the park for a bit so we can have some peace.'\n\n'Peace at a jumble?' Sally roared back. 'Have you warned her about the scraps you get into?'\n\nGeorgia felt suddenly dizzy as the hot tea went down in one long gulp. She sat back in her chair, wiping her brow with one hand. The paraffin stove seemed to smell much worse than usual and the hiss of the press sounded as if it was right in her ears.\n\nThe workroom was spinning. One moment Pop was standing on her right, the next on her left and the sickly smell of Iris's perfume caught her in the throat.\n\nWeakly she got up, groping almost blindly across the room, and as her stomach churned she put her hand over her mouth and ran the rest of the way to the toilet.\n\n'What's up with 'er?' she heard Sally shout, but her head was over the pan, vomiting as if her entire insides were coming up.\n\nOn and on it went until there was nothing left but green bile. She stood up and leaned against the toilet wall, so weak she felt she could slide to the floor.\n\n'Ow long's this bin goin' on?' Janet's voice behind her startled Georgia.\n\nFor a moment Georgia just stared at the older woman. There was no laughter now in those dark almond eyes, no hint of malice or sneering. Just sympathy and understanding.\n\n'About a week.'\n\n'Does 'Elen know?'\n\nGeorgia shook her head.\n\n'When did you last get the curse?'\n\n'Just before Christmas.'\n\n'When did you go with 'im?'\n\nTears came then. The sickness was going now but Janet had voiced her own fears and made it reality.\n\n'It was my birthday, January sixth.'\n\n'Is that why you left 'ome? Did yer ma find out?'\n\nOne moment Georgia was just hanging her head in shame, the next she was caught in Janet's soft arms. Her head on her shoulder, crying out all the fear and pain.\n\n'It's all right little 'un,' Janet whispered, kissing her hair and stroking her back. 'We can't talk now, but I'll 'elp you, don't you fret. At lunchtime I'll come back to your place and you can tell me about it. Now dry your eyes and try to smile. We don't want that nosy Iris getting wind of it, do we?'\n\n'Do you feel better now you've told me?' Janet said softly, as she came back into the attic room carrying two bags of fish and chips from the shop across the street.\n\n'Sort of,' Georgia whispered.\n\nAll morning she had thought of lies to tell. She even wanted to deny she could be pregnant, but once Janet sat down beside her in the other armchair, she seemed to know the right buttons to press to make her tell the truth.\n\nUntil now, Georgia had thought it was only women like her mother who could be relied on to be this sensitive. Janet with her curlers, plucked eyebrows and hourglass figure and bawdy jokes belonged to another world, yet she'd listened carefully, then went out to buy food.\n\n'I can take you to a doctor I know tomorrow.' Janet handed her a newspaper wrapped parcel. 'We 'ave to get it confirmed before we do anything else. He's a proper doctor, but 'e's bin struck off. If you go to an ordinary one he might just split on you. We'll make out you're sixteen anyway.'\n\n'But,'\n\n'I know, I know. You wants me to tell you we can wave a magic wand and make it right. I can't do that love. Let's just wait until we know for certain. It might just be the upset that's stopped your period.'\n\nPop sent Georgia out on an errand later in the afternoon and used the opportunity to call Janet into his office.\n\n'What's the matter with Georgia?' he asked bluntly. 'She's been looking pasty for days. Is the job too much for her?'\n\nHe knew Janet was capable of covering up for another girl she liked, but he wasn't in the charity business.\n\n'Just a tummy upset,' Janet distracted him with one of her sultry looks. 'She'll be fine in a day or two.'\n\n'Is there something I ought to know about her?' Pop was sure Janet knew something, she had that sly look in her dark almond eyes.\n\n'She needs a bit of tenderness,' Janet said, perching unasked on his cluttered desk. 'She ain't got no one but 'Elen.'\n\nPop sighed. His material shop downstairs was the legitimate part of his business. He ran the workshop and his market stall without declaring either to the Inland Revenue. All his employees until Georgia had been ones like Janet who could be trusted to keep their mouths shut. He didn't want any further headaches.\n\n'Don't you worry,' Janet picked up on his fears. 'She ain't some nark, or ever likely to be. Trust 'er Pop, she's a good kid.'\n\nAs Janet went back into the workroom she smiled to herself at Pop's na\u00efvety. He'd been married for donkey's years and had five children, the youngest Georgia's age, yet he hadn't suspected pregnancy. He might fiddle the taxman, but as an otherwise honourable man he was almost unaware of the evil some men were capable of.\n\nJanet knew. She knew all right.\n\nShe was the same age as Georgia on VE night, just another silly little girl out dancing in the streets. The big American looked so handsome in his blue uniform, it seemed so right to go and have a drink with him.\n\nShe knew what it felt like to scream your lungs out, and she knew too, even as she was doing it, no one would come to help. The whole of England was out celebrating the end of the war and how many other simple girls lost their virginity that night?\n\n'I 'ope she don't go the way I did,' Janet thought as she made her way back to her machine. She was lucky she didn't get pregnant, but she still hated what that man had turned her into. Off with any rich old man, taking what she could and using her body to trap them. At sixteen she saw Paris with one of them, trading her youth for nice clothes and the good life. Never mind who she hurt, as long as it wasn't her.\n\nYet she wasn't as tough as she thought, she still fell for Pete! Another fast-talking hustler just like herself. Just one year of wild good times, then everything turned sour. He sapped everything from her, the jewellery, the few bob she'd stashed away, he even took her looks. Why she stayed so long she never knew. Three kids, with each one she was pulled further and further down. Finally she ended up where she started, in Soho, a dirty, stinking rat hole of a flat without even a bath. He only came home in the end when he was broke, trying to push her out on the streets, anything for just one more stake. But she'd never done that. No man would make her sell herself in a doorway. Stripping was clean, taking the piss out of old wankers who couldn't get it up any other way, while she fed her kids and tried to build a new life.\n\n'But I won't let you go that way, baby,' she whispered under her breath as she picked up the cheap tweed. 'It ain't a life, it's just survival.'\n\nA raw wind caught Georgia's cheeks as she ran through the narrow back streets to Peabody Court.\n\nOnce away from the market it was quiet, offices closed for the weekend, too early in the morning for the night-time people to surface. Unpleasant odours seeped out of each dark alley, the many piles of vomit an indication of the previous night's revelry. Cellar trapdoors stood open, belching out a stench of beer, while the caf\u00e9s competed with the delicious aroma of fresh coffee and fried bacon.\n\nShe could never be sure whether she liked or feared Soho. For all the dirt, smells and danger that seemed to lurk round each dark corner, it had a warmer side that surprised her. A friendly wave from the old man in the corner sweet shop. A wolf whistle from the boy who worked in the Bastille coffee bar and a smile from the woman on her knees scrubbing her doorstep. Helen had no fear when she walked home late at night, the big bouncers in the clubs watched out for her, even the prostitutes and strippers knew her by name and Georgia was respected as being her friend.\n\n'Coo-ey!' The call made her look up. Janet was leaning over a small balcony on the third floor making a signal she was on her way down.\n\nThe buildings were dismal. Four storeys of soot-blackened dwellings housing sixty families. Small spiral stone staircases behind rusting prison-like bars gave more than a hint of the dark ages this place belonged to. Yet many of the windows sparkled defiantly, sporting brilliant white net curtains as if the owners wanted to prove they hadn't given up hope entirely.\n\nJanet's high heels clattered down the last few steps, bringing a touch of unexpected colour and glamour to the grey surroundings.\n\nWhite-blonde bouffant hair, a leopardskin swagger-jacket, red lipstick and tight red skirt matching perfectly. Now Georgia saw why she held her own against younger girls in the strip clubs, and why she'd earned the title of Soho's Marilyn Monroe.\n\n'You look lovely,' Georgia was touched that Janet had considered taking her to a doctor enough reason to dress up. 'Your hair's so pretty!'\n\n'I got it done last night,' Janet patted the masterpiece, and fluttered her spiky eyelashes. 'Course it's the first time you've seen me done up. Could I con anyone I was yer sister?'\n\n'Your skin's a bit pale,' Georgia giggled despite the turmoil inside her.\n\n'It's not far,' Janet tucked Georgia's hand under her arm and marched her quickly down to Charing Cross Road. They stopped at a door sandwiched between two record shops.\n\n'I'm scared,' Georgia hung back. 'I don't know what to say!'\n\nJanet took her cold face between both her gloved hands and kissed the end of her nose.\n\n'I'll be with you. Just agree with everything I say to him. It ain't so bad.'\n\nA bearded, tall, thin man answered the door, just as Janet was wiping her lipstick off Georgia's nose.\n\n'Good to see you Janet,' he smiled as if he really meant it. 'Come on up, it's bitterly cold isn't it?'\n\nHis voice brought back Blackheath into sharp focus. Resonant, educated. If she closed her eyes she could almost pretend it was Doctor Towle in his spacious antiseptic surgery in the village.\n\nJust one flight of shabby but clean stairs and they passed through a glass-panelled door.\n\n'How's things, Roger?' Janet swaggered into the flat as if she was no stranger to it. 'Never see you down the club anymore. Got a new lady?'\n\n'I spend my time with good books these days,' he laughed, implying that once he had been a regular visitor, waving one hand at a huge oak bookcase full of leather bound volumes. 'And you must be Georgia,' he smiled down at her, holding out his hand.\n\nGeorgia gulped. He had such nice eyes, pale blue, the colour of baby ribbon. When he smiled he looked younger than his fifty years, an unlined, almost boyish face.\n\n'Don't look so frightened,' he led her over to the couch and pulled out a cloth screen to put round it. 'Pop in there and take off your undies, then up on the couch. Janet will be right here with me.'\n\nGeorgia took off her coat, then hastily pulled her knickers off under her skirt. Behind the screen she could hear Janet talking softly. She gave him the date of her last period, mentioned the bouts of sickness as if she were an aunt.\n\nShe had expected someone seedier, maybe foreign, anything other than this tall, bearded man with his gentle voice and kind face.\n\n'If she is pregnant you know I cannot condone an abortion,' she heard him say. 'I hope you didn't think I would help with that?'\n\n'Of course not, Roger,' Janet's voice lost its cockney edge. 'She can stay with me, one more won't break the bank. We just wanted to be certain before I take her to the hospital.'\n\nGeorgia tried hard not to blush when he came back to her pulling on rubber gloves.\n\n'Put your feet up, and let your knees fall apart,' he smiled reassurance. 'Relax. It won't hurt.'\n\nBut it did hurt, not just physically, but mentally. It brought back that other examination after the rape. One more indignity, the shame of exposing herself to a man. Every muscle was tense as his fingers probed her. She screwed up her eyes, her toes and her fingers and wished she could just faint rather than submit to another minute of it.\n\nHis face was thoughtful as he removed his hand. He stood back and peeled the rubber gloves off as she hastily pulled her skirt down over her knees.\n\n'Sit up now and just let me see your breasts. Are they tender or enlarged?'\n\n'A bit,' she said, as she struggled to undo her bra.\n\n'Hm,' he said as he peered closely at them. 'No doubt about it my dear. You are pregnant, around eight to nine weeks I'd say.'\n\nGeorgia could contain herself no longer. Tears rolled down her cheeks and she covered her face with her hands.\n\n'I'm sorry,' he said gently, patting her shoulder. 'Ideally, every baby should be planned. But believe me, few are. In a few weeks you'll get used to the idea, in a couple of months you'll be waiting eagerly for it.'\n\nOnce out on the street, Janet steered her across the busy road to a coffee bar.\n\n'Better now?' Janet was surprised Georgia controlled herself so quickly, remembering her manners and thanking Roger for his verdict. If she didn't know better she would assume Georgia had already come to terms with it. 'I know it wasn't the result you hoped for. But at least we know for sure.'\n\nJanet took a booth at the far end of the coffee bar and ordered drinks for them.\n\nSoho was full of coffee bars, by day office workers bought sandwiches in them, by night they became meeting places for the young. This one had gay red and white checked curtains, with white formica tables and a big bulbous juke box, the only customers two taxi drivers eating breakfast.\n\n'I'd rather leave this till you've had time to think,' Janet said once the coffee was in front of them and she'd lit up a cigarette. 'But time is the one thing we're short on, so I won't go round the houses.'\n\nShe drew deeply on her cigarette.\n\n'We could go right now to the police and tell them the whole story. They'll arrest yer dad, and with a bit of luck they might offer help with an abortion in hospital.'\n\n'But what if they don't?' Georgia's eyes filled with fright. 'I don't know if I could bear to go through all that questioning. Besides, when Mum hears about the baby it will make her even more miserable. I can't do that!'\n\nJanet sighed deeply. The bastard who'd done this to Georgia had filled her dreams last night. She wanted him crucified as an example to any other man who might get the idea of raping a child in his care. But however much she wanted it she was aware of the problems. Social workers would step in, the kid'd be back in care and they'd probably make her go through with the baby too.\n\n'The other alternative is to have the baby, maybe get you into a home for unmarried mothers and have it adopted when it's born.'\n\nGeorgia stiffened. Eyes rolled in alarm, her lovely mouth tightening with hate.\n\n'I can't have it. I loathe it already. No one could expect me to keep his child inside me, could they?'\n\nJanet's almond eyes closed for a moment as she thought what she would have done if those Americans had left her pregnant.\n\n'Then there's only abortion,' she sighed. 'But that's risky.'\n\nWeek after week Janet met women who had illegal abortions. Some women like her who'd already had children and couldn't afford another, sometimes they were prostitutes who'd merely slipped up. But Georgia was a child, how could she help in something which could kill her?\n\n'How bad is risky?'\n\nJanet looked into the dark, determined eyes and saw the same kind of stubborn pluck that had kept her going through countless hardships.\n\n'Infection, blood poisoning, even death. I won't lie to you love, it's heavy.'\n\n'I don't care,' Georgia brushed tears from her eyes angrily. 'Anything's better than having it.'\n\nShe put one hand on her stomach tentatively, hardly able to believe there was a tiny baby growing in there.\n\n'It ain't a picnic,' Janet warned. 'It hurts so bad when the contractions come you'll want to die. I don't want to frighten you love, but it wouldn't be right for me not to spell it out.'\n\n'I can stand it,' Georgia stuck out her little pointed chin defiantly.\n\n'I hope so,' Janet said softly. 'I just hope so.'\n\n'Tonight's the night then,' Janet whispered to Georgia at work. Three, painfully slow weeks had passed since the visit to Roger. 'Did you weaken and tell Helen?'\n\n'No. I said I was minding your kids for you.' Georgia's face was pale but resolute. 'If I could get back home on Sunday she need never know. You know what a worry-guts she is!'\n\nA whole week spent waiting for Janet to contact the man. Another five days while he considered whether he would do it, then another nine of being so terrified she couldn't sleep at night.\n\nOn top of that was the worry about getting the ten pounds needed, along with hiding it from Helen. No lunch, sweets, new stockings or magazines. It was lucky Pop had asked them to work overtime on several occasions for a rush job, otherwise she would have had to borrow some of it from Janet. The longest three weeks she had ever known, but now the moment was close.\n\nWhat would happen if she got rushed to hospital? Would she be brave enough not to implicate Janet? And what if she did die? Would Janet go to prison for helping her?\n\nIt was raining hard as Georgia and Janet left work, for once the streets were almost empty. Neon lights from the clubs and bars were twinkling in puddles, the old yellow street lights giving St Anne's Court a Dickensian quaintness that belied the sordid activities which it was famous for.\n\nSally was the only other person who knew what was going to happen. She had already taken Janet's children down to her flat on the ground floor and she'd promised to look in the next morning to see how things were.\n\n'Well, this is it,' Janet said as she opened her front door. 'Sorry about the mess. I didn't have time this morning.'\n\nTwo months earlier Georgia would have considered Janet's home a slum, but after her attic room with Helen, it looked homely. Small boxy rooms, congested with furniture. A garish, orange patterned carpet vying with a red overstuffed three piece suite. Yet despite the clothes and toys strewn about, it was clean, bright and cosy. Photographs of her children, glass ornaments and seaside souvenirs jockeyed for position on the mantelpiece, window-sill and two shelves on the walls and gave a feeling of security.\n\n'We didn't even have a bath till last year,' Janet yelled at her as she made a cup of tea. 'I used to stand the kids in the sink. Posh ain't it?'\n\nGeorgia looked enviously at the white bathroom, almost as if she'd never seen one before. She may have improved her bathroom enough to use it, but it still made her shudder. Would there ever come a time when she and Helen could arrange their talcum powder, shampoo and face flannels like Janet had done? Or be able to invite friends round and not be embarrassed?\n\nGeorgia was in the bedroom getting herself prepared, when the abortionist arrived.\n\n'It's time,' Janet said softly from the doorway. 'Remember what I've told you. It's embarrassing, but not painful. I'll stay with you so don't panic!'\n\nGeorgia's stomach churned as she saw the man in the bathroom. He was short and very dark with hair that grew right down the back of his hands and thick eyebrows which met in the middle.\n\n'Call me Eric,' he smirked, not meeting her eyes. Seedy was the description that fitted him best. Clean enough, but with frayed cuffs to his shirt, and the trousers which didn't match his suit jacket were shiny with age. As he took off his jacket and rolled up his shirt sleeves, stale sweat wafted out.\n\nGeorgia could think of nothing to say. She stood awkwardly in her dressing-gown, her bare toes curling up on the cold lino as he turned to the bath and bent over.\n\nA tubular chair was already in place, Eric was furiously whisking a bowl of bright pink soapy water. The smell of carbolic took Georgia right back to her ordeal in the bathroom at the convent. She gagged involuntarily.\n\nThe closer she looked, the less she liked him. He had a paunch that hung over his trousers and through his thin shirt she could see bumps of a string vest. How could she let this man touch her?\n\n'I know this isn't the best way to meet people,' he said still whisking the soap. 'Try to think of me as a doctor.'\n\nGeorgia had an overwhelming desire to run away. He had a length of rubber tubing in his hands, testing it by submerging one end in the water and squeezing a round bulb in his hand. The other end of the tube had a firm nozzle, the bright pink water was splashing into the white bath.\n\n'Hop up on the chair,' he said, glancing round and taking her firmly by the hand. 'I expect it's been explained to you but I'll tell you once again.'\n\n'I can't,' Georgia said suddenly as panic washed over her. She had looked at those hairy hands, knew where they were going and the thought disgusted her.\n\n'Yes you can,' Janet said firmly behind her. 'Don't even look at Eric, sit on the edge of that chair and try to relax.'\n\n'I don't see or feel anything,' he said unconvincingly, shrugging his shoulders. 'I'm just here to help girls in trouble.'\n\nBetween the pair of them Georgia knew she could do nothing but agree.\n\nClutching the dressing-gown round her she stepped into the bath and perched on the chair. Janet came forward and lifted the gown, revealing the lower part of her body. Then positioning herself by Georgia's side, she took one hand in both of hers.\n\n'Now first I have to dilate the neck of the womb,' Eric said, already sliding two soap-covered fingers between her legs. 'When you got pregnant, that end sealed over with a thin membrane, first we have to break it, then pump the water in.'\n\nWhen Roger had examined her she was embarrassed, but secure in the knowledge he was a real doctor. This man with his sweaty smell and hairy hands made her feel tainted. As he felt round inside her he kept huffing and wheezing, if it hadn't been for Janet holding her hand so tight and the desperate need for his skill, then she would have pushed him away and run for it.\n\nGeorgia turned her eyes up to the ceiling, and tried to blank out from her mind what was happening.\n\nIt didn't exactly hurt, it was more humiliating than painful. She felt a dull ache at one point, very similar to a period pain.\n\n'That's done it,' Eric said cheerfully, withdrawing his hand and picking up the length of tubing. 'Now this end,' he lightly touched the rigid end and squirted the bulb again to see that it was working. A jet of pink soapsuds trickled across Georgia's thigh. 'This has to go right in there, then I pump the water. Janet please hold the other end under the water all the time, we mustn't risk an air bubble getting in.'\n\nOnce again his fingers felt their way in, this time with the rigid end firmly between them. He groped around for a moment or two then smiled.\n\n'That's it, it's going in now, you'll feel an odd sensation soon.'\n\nIt seemed as if he was pushing the tube forever and the sensation he mentioned came on, making her feel slightly nauseous.\n\n'Ready for the soap now,' he said, checking to see Janet was holding the other end down correctly. 'Here goes.'\n\nHe pumped for what seemed like minutes. The level in the bowl was going down, yet none was coming splashing out as Georgia had expected.\n\n'That's it, job done,' he said, withdrawing the tube quickly. 'It will start working within twelve hours, if it doesn't we'll have to try again. But I'll stake my reputation on you losing the baby before morning.'\n\nGeorgia quickly covered herself as Eric left the room. She was surprised how little it had hurt and she now felt a little foolish. As she climbed out of the bath she noticed her stomach was bloated, and she was shivering, but that was the only ill effect.\n\nShe went back into the living room and found her purse.\n\nEric was pulling on his raincoat, refusing a cup of tea from Janet. It was obvious he wanted to get out as quickly as possible, his tools already packed in a small leather briefcase.\n\n'Can't stop,' he said, his eyes darting about the room. 'Call it a tenner love, and remember not to spill any beans if you have to go into hospital, otherwise you might be very sorry.'\n\n'There's no need for any threats,' Janet moved towards him, her eyes flashing. 'She's one of the girls, she knows how to behave. I'll be contacting you if it doesn't work.'\n\nJanet closed the door behind him and thumped the wall beside it.\n\n'What a piece of human shit!' she exploded. 'He'd sell his own granny for a shilling. He couldn't care less about his victims. I bet he gets \u2013' she broke off suddenly as if remembering how young Georgia was. 'Never mind that! How yer feeling?'\n\n'Not so scared now,' Georgia smiled, a little colour coming back into her cheeks. 'I mean, I can't stop anything now can I? I just have to wait and see what happens.'\n\n'Good for you,' Janet smiled and tickled her under the chin. 'Now let's see what I've got in the cupboard to eat, you don't look as if you've eaten for days.'\n\nGeorgia was seeing another side of Janet now as she cooked chops for their tea, folded clothes and put away toys. She was very much a mother, not the good-time girl she liked to portray at work. The lounge was very warm and it was good to watch television again, almost the same snug feeling she'd had at home in Blackheath.\n\n'How did you come to be a stripper?' Georgia asked after they'd eaten the meal and Janet sat down with her coffee and cigarette, her slippered feet up on a pouffe.\n\n'It started out as a bit of a lark one night,' Janet grinned wickedly. 'My old man had left me with the three kids, I was a bit down and up to my eyes in debt. A mate asked me to go down the club for a few drinks. Well, I was so cheesed off I dolled myself up and out we went.'\n\nShe paused to light another cigarette, offering one to Georgia.\n\n'Sorry you don't do you. Anyway I got a bit pissed and when the strip act came on I fell about laughing it was so bad. The manager challenged me. \"Do it better than her and I'll give you twenty quid.\"'\n\n'So you did it?'\n\nJanet grimaced, as if remembering conflicting emotions.\n\n'Well that twenty quid meant food in the cupboard and new shoes for the kids. So I turns to him and says, \"All right mate, yer on.\" Next thing I knew he was wheeling me up on the stage.'\n\n'Were you any good?' Georgia was torn between shock and admiration.\n\n'Better than the other girl!' she laughed loudly, throwing back her head. 'I felt a bit silly, me undies was a bit rough and that. But I could dance and me bum and tits hadn't sagged that much. By the time I was waving my bra round me 'ead, the whole place was in an uproar. I got called for an encore.'\n\n'Did you get the twenty pounds?'\n\n'Yeah, and he asked me to appear the next night. A fiver for two ten minute slots. Can you think of a quicker way to make that kind of dough?'\n\nGeorgia shook her head.\n\n'I was planning to go to that club in Berwick Street just before I met Helen,' she admitted. 'Do you think I'd have been any good?'\n\n'Don't you even think of it,' Janet snapped. 'I knew all the people round 'ere. I knew what to watch out for. I make it sound glamorous, but believe me it ain't. You know what 'appens to old strippers?'\n\nGeorgia blushed.\n\n'They end up as winos, or brass,' she said. 'I had no alternative back then when the kids was small, but I keep it in mind. I do me turn and I come 'ome. I don't mess with the guys, and I saves some of the dough for a rainy day.'\n\n'Wouldn't you like to get married again?'\n\nJanet snorted with laughter. 'Chance'd be a fine thing. Who wants a woman with three kids for more than a night or two? They come round, want a free meal, a quick screw and they're on their way again. It's better to have a good mate like Sal than a man who just takes.'\n\nIt was like looking in a peepshow as Janet told her things about her former life.\n\n'I don't blame anyone but me,' Janet explained her philosophy of life. 'All the troubles I've had were my own doing. If I'd chosen a different path instead of what I thought was a short cut I'd be in clover now. I got no time for people who grizzle and whine. You make your own luck.'\n\nIt was after twelve when Janet took her into her bedroom. Next to the big double bed, a narrow single one was made up.\n\n'You sleep in there love, and wake me up if it starts. Anything yet?'\n\n'Nothing,' Georgia laughed. 'I don't know what to expect anyway.'\n\n'Well the soap acts as an irritant. It starts up contractions. They feel some'at like period pains. They go on for a bit, getting stronger and stronger, then if all goes well you'll lose it. It can be messy, so put a pad on now. But don't worry about the bed, I've put a rubber sheet on it and the sheets are old ones.'\n\nGeorgia lay awake long after Janet was snoring next to her.\n\nTo help her to sleep she lapsed into her favourite game, pretending to be a famous actress in a West End musical. She imagined the crowd shouting for an encore, and herself sweeping out of the stage door into a waiting limousine. As she raised one foot to get into the car, a voice behind her made her turn.\n\n'I knew I'd find you one day.'\n\nShe turned to see Peter standing there, his blond hair gleaming like gold under a street light, his arms open wide, waiting for her to run to him.\n\nWhen she woke, for a minute she didn't know where she was, until a dull ache in her stomach reminded her. It was still dark, and Janet snored softly in the bed next to her.\n\nCreeping out of bed silently, she fumbled for her dressing-gown and slippers and went into the lounge.\n\nIt was four o'clock, and still raining hard when she looked out of the window. Across the street four men were staggering up the steps of the Black Cat club. Two of them supported the third between them, and a fourth lurched unsteadily behind them.\n\nGeorgia perched on the arm of a chair and watched them. They paused under the club's canopy as if hoping to see a taxi. An illuminated sign was flashing on and off beside them, their faces turning red to green alternately. There was no one else about. Not so much as a light in an upstairs window or a passing car.\n\nShe turned away from the window to take some painkillers Janet had left on the table, and make some hot milk.\n\nShe turned to examine a photograph of Janet's children. It had been taken in a studio. What would happen to them if Janet didn't get rehoused in the next couple of years as she hoped? Would her daughter end up working in one of the clubs? Would the boys spend their nights in smoky billiard halls, waiting to be sucked into crime?\n\nWhen all this was over she would have to think hard about her own future. Soho might be a convenient place to hide, but as Janet had pointed out so clearly, it was full of life's losers.\n\nBy six the pain was so strong she went back to bed. Every now and then the soap oozed out in spurts, the smell of carbolic making her nauseous. She lay on one side, then the other, trying to find a way to get comfortable. Between the pains she counted. At first she could get up to forty, but gradually they got closer and closer together, until they were only seconds apart.\n\nIt wasn't just in her stomach now, but in her back too. She didn't want to wake Janet yet, but it was scary lying there in the darkness.\n\nThe curtains didn't quite meet in the middle. She focused on a triangle of sky, watching as it slowly turned from black to grey. In the distance she could hear a milk float, the crates rattling as it went over bumps, and slowly the hum of traffic increased from the direction of Charing Cross Road.\n\nIt was light enough to see Janet now. Her red nightdress bunched up around her neck, one pale arm curled round her head, wheezing as she slept.\n\nThere were no gaps between the pain now, just great waves of agony that grew gradually stronger. She tried to control herself, gripping the edge of the mattress, but even biting her lips together she couldn't stop a moan rumbling out.\n\nJanet sat up with a start.\n\n'Oh my God,' she said, rubbing her eyes, smearing the night before's mascara all over her cheeks. 'I'd forgotten about you. How is it?'\n\n'Bad,' Georgia said through gritted teeth. 'I've been trying not to wake you.'\n\n'You silly mare,' Janet jumped out of bed and wrapped herself in a shabby blue dressing-gown, her hair like a ball of tangled yellow wool. 'Are you losing any blood?'\n\nGeorgia gritted her teeth as a white hot pain gripped her. This time she felt blood flowing, warm and sticky on her legs.\n\nJanet moved over to her and pulled back the covers. She gasped as she saw the blood and hastily got two more towels.\n\nTime and place ceased to have any meaning as pain consumed her. She was aware of Janet sitting beside her, the cooling touch of a damp flannel on her face, the soothing words of encouragement, but pain drove out thought or hope.\n\nJanet was frightened now.\n\nShe could almost see the powerful contractions tearing at the child in front of her. Her face was contorted with the effort of not crying out. Bathed in sweat, rope-like veins on her forehead, yet still she had the iron-will not to beg for an ambulance.\n\nIt wasn't the first abortion she had helped with, but all those other girls were experienced, most had already got a child. Could she trust Georgia to tell her when she could no longer endure it?\n\nA tapping on the door sent Janet scurrying to open it.\n\n'Thank heavens you've come,' Janet blurted out. 'I'm scared Sal, I think she might die!'\n\nSally slipped in, closing the door behind her. She had left the children still in bed, intending to just check on Janet and Georgia before she went back to get them breakfast.\n\n'Why, what's happened?' She had only a thin coat over her nightdress, hair still in curlers. Her face without her usual thick make-up was as pale as an iced bun.\n\n'She ain't screaming or nothing, she's just taking it. But I don't think she can stand much more.'\n\nSally went whiter still when she saw Georgia. She was struggling now, tossing from side to side, her hair sticking to her head, her eyes rolling back.\n\n'I'll get an ambulance,' she gasped. 'What shall I say?'\n\n'Anything,' Janet almost screamed. 'Just get them here.'\n\nAs they looked back to the bed Georgia tried to sit up. Both women ran to her.\n\n'Stay where you are,' Janet pushed her back, her hands trembling with fright. 'You can't get up!'\n\n'Get me to the toilet,' Georgia croaked.\n\nThe two older women's eyes met. Sally nodded.\n\nTaking one arm each they half carried Georgia across the room, out across the tiny hall into the bathroom. Janet lifted the blood stained nightdress and sat her down, holding on to her shoulders.\n\nThey could see her knuckles turn white as she gripped the seat, her head lolling to one side like a broken doll.\n\n'I'm going to be sick,' she whispered.\n\nJanet grabbed the waste bin, pushing it in front of her just as Georgia retched. Once again her face went purple, eyes rolled back, she bared her teeth and her neck seemed to swell before their eyes.\n\n'Oh Gawd! She's havin' a fit!' Sally cried out. 'What do we do?'\n\nThe swollen veins in her forehead and only the whites of her eyes showing were enough to terrify the two older women but when a low roar rumbled in her throat, Sally crossed herself involuntarily.\n\n'Push it out!' Janet cried, wringing her hands with fear and impotence. 'Push!'\n\nEach second seemed like an hour as they waited in the icy bathroom. Janet held Georgia against her chest, sobbing now in fear. Sally knelt down in front of Georgia, eyes wide with panic.\n\nOnce again her face contorted, neck swelling, but this time her eyes opened wide and her hands gripped at the toilet seat. Her nostrils flared, teeth clenched together, and they both heard a slithering sound, quickly followed by a splash.\n\nShe slumped forward, only prevented from falling to the floor by Janet's arms.\n\n'Was that it?' Sally jumped up, wild-eyed, black hair tossed back from her white face.\n\n'I hope so,' Janet scooped Georgia up into her arms, lifting her as if she weighed nothing. 'You look. She's fainted. I'll get her back to bed.'\n\nSally came into the bedroom minutes later, she stood by the door, her face a pale green. Janet barely turned. She had Georgia back in bed now, tucking the covers round her.\n\n'Blood always looks like pints, even when you lose a teaspoonful,' she said, more calmly than she felt. 'At least she don't seem to be in pain.'\n\nGeorgia's eyelids fluttered, then opened.\n\n'It's over now darlin',' Janet whispered, smoothing back the girl's damp hair. 'You was a brave little soldier! How d'you feel?'\n\n'Just, kind of odd,' she said weakly.\n\n'Let's get you washed and into a clean bed,' Janet said. 'And Sal will make a nice cup of tea before you go back to sleep.'\n\nJanet came back into the bedroom with a clean nightdress over her arm, but stopped by the bed, tears pricking her eyelids.\n\nSally had Georgia lying on a white towel, just drying her legs. Georgia's skin looked much darker now, skin shining like a light brown pebble just tossed out of the sea. Such perfection, long, slender legs, tiny waist and small jutting breasts with dark brown nipples.\n\nBut it was her face that effected Janet most. It was like a small child's when it has just woken, smooth, unlined, dark pink lips in a moist bow. Teeth and the whites of her eyes almost dazzling. Delicate bones that transported her heart-shaped face from being just pretty, to magnificence.\n\nJanet had seen little beauty in her life. Her children's faces, flowers in the market, and a sunset once in Paris were about the only things she remembered as being truly beautiful. But as she looked down at Georgia, tears trickled down her cheeks.\n\n'What's the matter, Jan?' Georgia's voice was croaky.\n\n'You,' Janet retorted quickly. 'I was just thinking that you'd better not take up stripping otherwise there'll be no room for old bags like me anymore.'\n\nSally looked round at her friend, a smirk of understanding and affection playing on her lips.\n\n'Come on, you silly mare,' she said. 'Get the nightie on her before the poor kid freezes. I've got to get back to those little buggers downstairs!'\n\nGeorgia slept all day till nearly nine that night. Janet checked on her constantly, and gradually relief took the place of fear. There was no sign of fever. The flow of blood had turned to a trickle and she was sleeping as quietly and soundly as a baby.\n\nJanet was watching television around nine when she heard the door open behind her.\n\n'What on earth!' she exclaimed.\n\nGeorgia stood there fully dressed in her jeans and the red sweater she'd arrived in. She had her shoes on and her coat in her hand.\n\n'Where do you think you're going?' Janet demanded.\n\n'Home,' Georgia said simply.\n\n'Get back into bed this minute. You ain't going anywhere.' Janet jumped to her feet. 'Never heard nothing so daft.'\n\n'No, Jan,' Georgia said softly. 'It's time you had your children back, you've done enough. I'm all right now thanks to you. But I'm not going to impose any longer.'\n\n'Impose?' Janet roared. 'What's that supposed to mean?'\n\nGeorgia said nothing, just came over to Janet and put her arms round her, leaning her head on her shoulder.\n\n'I didn't realize till just now what you'd been risking,' she whispered. 'I'm so selfish sometimes. I don't think beyond me. You've been so wonderful, I'll never forget it. But your children belong here, not me.'\n\n'But \u2013'\n\nGeorgia lifted her head and put one finger on Janet's lips.\n\n'I'll tell Helen I've got a tummy ache and go to bed. If anything does happen then I'll pretend it's just a bad period. It's safer that way. Our little secret.'\n\n'You can't walk home!'\n\n'Of course I can,' Georgia smiled. 'I'll be back at work on Monday good as new. And I've got you to thank for that.'\n\nGeorgia's colour was almost normal now, her eyelids had a faint purple tinge, but the peachy tone was back in her cheeks.\n\nJanet shook her head. 'You're a funny kid,' she smiled reluctantly. 'Just promise me you'll put all this behind you now and try to forget.'\n\n'I shan't ever forget you,' Georgia said hugging Janet one more time. 'Not ever.'\n\n## Chapter 6\n\nThe smell of damp wool and paraffin on top of the roar of her machine was making Georgia's head ache. Three more identical dark green dresses to be finished before six and the light was so bad she could hardly keep the seam straight in front of her.\n\nShe wriggled, trying to ease her aching back into a more comfortable position, when suddenly the electricity was cut off, plunging the workroom into silence and darkness.\n\n'What's happened?' Georgia spun round in her seat.\n\nThere was enough light from the street lamps outside the window to see the other girls were no longer behind their machines. Irene was missing from the big steam press. Iris's scissors lay gleaming on the cutting table.\n\n'What's going on?' Georgia called out, but the only reply was a muffled giggle from the staffroom.\n\nShe stood up, leaving the half-finished garment still in the silent machine and took one step towards them.\n\nA faint, flickering yellow light, another giggle.\n\n'Happy birthday to you! Happy birthday to you!' Five voices burst out from the gloom.\n\nNow it was Georgia's turn to giggle as Pop appeared holding a cake. Sixteen candles burning and all the girls singing at the top of their lungs.\n\n'Happy birthday, dear Georgia, happy birthday to you!'\n\n'I thought you'd all forgotten,' Georgia's face broke into a wide smile.\n\nThe day had begun badly when Helen rushed out without remembering. Memories of all those other birthdays came flooding back. Celia waking her with a special breakfast, parcels and cards. A specially magical day where the birthday girl was treated like a princess.\n\nWhen she got into work and found they too hadn't remembered, she'd resigned herself to believing that in an adult world perhaps birthdays weren't important, and she'd been too embarrassed to mention it. Not once had she suspected they were planning something.\n\n'Forgotten?' Janet laughed. 'You reminded us enough times in the last week.'\n\nThey all stood there grinning like idiots, Iris with a bottle of sparkling wine in her hands, Sally with a plate of sausage rolls and Pop flushed with excitement.\n\n'But how could you keep me in suspense all day?' Georgia asked. She'd had to fight back tears of disappointment when she thought they'd forgotten what day it was.\n\n'I told them they had to,' Pop's sallow face was grotesque, lit only by the candles beneath it. He put the cake down on a machine and motioned for the lights to be put back on. 'I threatened them with the sack if they let on before five.'\n\n'Well, you mean old thing,' Georgia thumped him playfully in the chest. 'It's okay to have birthdays as long as they don't interfere with production?'\n\n'Come on,' Janet said, hands on hips, eyes gleaming. She had made an unusual effort with her appearance today, and now Georgia understood why. Her blonde hair was smooth for once, tied back at the nape of her neck with a ribbon, and she was wearing a black sheath dress with a white lace collar. 'One big blow, and don't forget to wish.'\n\nIt was this rag-bag group of people who had helped her to forget the trauma of last year. Pop with his funny Greek accent, Iris's tall tales, Irene's strange moods, Myrtle's quiet shyness, and the continual vulgar banter between Janet and Sally. But as they grouped around her, their faces full of affection, Georgia wanted to cry with happiness.\n\nTaking a deep breath, she blew the candles out. The same old wish that came to her night after night. But now she was sixteen surely it would come true?\n\n'There's more too,' Sally grinned, her mouth red and pouting. 'But we promised Helen we'd wait for her.'\n\nAs she spoke Georgia heard the familiar clonking sound of Helen's boots on the bare wood of the stairs.\n\n'She's coming!'\n\nHelen paused for breath in the doorway, one hand on her side as if severely winded. Two bright pink spots of excitement on her cheeks, wearing the same old russet coat she'd worn the first time Georgia met her.\n\n'Happy birthday,' she said, crossing the room to kiss Georgia and putting a large parcel in her lap. 'I wanted to give it to you this morning, but Janet and Sal said it would ruin their surprise.'\n\nGeorgia tore the wrapping off like a child.\n\nIt was a big, red baggy sweater, just like one Georgia had seen in Oxford Street weeks earlier. She looked up gleefully at Helen.\n\n'You copied it, you clever darling!'\n\nHelen blushed a becoming rose pink, her green eyes downcast with embarrassment.\n\n'They didn't like me asking to get it out of the window,' she said in a small voice. She was never at ease with large groups of people, if Janet hadn't insisted she came to the workroom and joined in she would have waited for Georgia in their room. 'They weren't pleased when I didn't want to buy it.'\n\nThe other girls waited expectantly. Pop with an amused smirk on his lips leant on one of the sewing machines watching silently.\n\nGeorgia pulled it over her shirt. It was the latest style, so huge it looked big enough for two, hanging right down over her bottom, the shawl neckline framing her face, the warm red complimenting her dark skin.\n\n'It's _wooonderful_!' she whooped, jumping around the room, lifting her dark curls and admiring herself in an old cracked mirror. 'I'm going to live and die in it.' She leapt back to Helen and hugged her. 'You are the best friend in the world. How did you knit it without me seeing you?'\n\n'Down in the library. At work in the evenings. Sometimes I even did a bit when you were asleep. It wasn't easy.'\n\nGeorgia could only hug the collar round her face and smile. It was the planned secrecy, the long hours of selfless work that made it such a special gift. A touch of the magic Celia used to weave.\n\n'Come on, open mine,' Sally cried, almost drowned by the others who offered their presents too.\n\nPop stood back and smiled.\n\nHe liked to see his girls happy like this together. He didn't mind one bit that work was halted, or that soon the already untidy workshop would be strewn with wrapping paper, empty glasses and cake crumbs. He had a feeling that quite soon he would be losing Georgia. His workshop wouldn't be the same without her, but a bright young thing like her wouldn't want to sweat over a sewing machine for ever.\n\nShe had changed so much in the year she'd been with him. She'd filled out a little, grown another inch and he'd seen her change from a fearful child to a delightful woman. Georgia was never still. She filled his workroom with chatter and movement. Again and again he had to ask her to get on with her work, but it was like asking the sun not to shine. And shine she did. The ready smile, the clowning, the eagerness to know anything and everything.\n\nHe hadn't known she could sing until one day in early April when he walked up the stairs to hear her in full flight.\n\n'Summertime' that was the song, and she sang it in a way that brought a lump to his throat.\n\nIt was lunchtime. She twirled an umbrella as though it were a parasol, a length of fabric wrapped round her. Another piece around her hair like a turban.\n\nSally, darkly seductive, lounged on a bale of cloth, a cigarette hanging from her scarlet lips. Janet perched on a machine, eyes shining with delight, her messy blonde head nodding in time to the beat. Iris painting her nails, glanced up now and then, not anxious to really be part of it. Irene grinned foolishly, her front teeth missing, standing hands on hips, wearing ridiculous men's trousers. And finally, quiet little Myrtle, eyes downcast, hands in her lap at her machine, drinking in every last word.\n\nNone of them were aware of him until she finished.\n\n'Sorry, Pop,' Georgia giggled, slapping her hand over her mouth.\n\n'I hope you intend to press that length after you take it off,' he said dourly.\n\nLater he wondered why he hadn't told her how much he enjoyed it.\n\nBut she didn't need encouragement. She sang again, day after day, like a little canary in a cage.\n\nThe market men ribbed him all the time as spring turned to summer and the windows were wide open.\n\n'Training nightingales up there are we? When do you become her agent and get your ten per cent?'\n\nPop watched as the girls gave Georgia little presents. A lamp for her room, a picture, a brightly-coloured scarf, earrings and a bracelet. He listened to her shrieks of glee, that gave equal pleasure to the giver.\n\nHe too had agonized for weeks over what he should give her. It couldn't be too expensive as the other girls would see it as favouritism. But he wanted her to know how much he appreciated her help.\n\nWithout this funny little girl his wife wouldn't be wearing a new fur coat now, and it had all started from Georgia's bossiness!\n\n'These dresses are horrible,' she had said one day back in May, eyeing up the row of finished garments ready for the market stall. They were blue, grey, brown and beige with neat white collars, ordinary dresses for women who wanted plain everyday clothes at rock bottom prices. 'Why on earth don't you make things for younger women? Pretty ones?'\n\n'Design me one then,' he challenged her, fully expecting her to back down.\n\n'All right, I will,' she said, drawing a rough sketch of a scoop-necked, full-skirted dress. 'Use that gingham,' she insisted pointing to a bale of cheap cotton meant for kitchen curtains. 'Put broderie anglaise round the hem and they'll be like Brigitte Bardot dresses.'\n\nHe had humoured her by making six. But it wasn't until she tried a pink one on with a wide belt that pulled her waist into a handspan and added a can-can petticoat that he became convinced. Perhaps it was only her brown skin and tiny waist that made it look chic and expensive, but it was worth a try.\n\n'Now get down there.' He shoved the rest of them into her arms, pointing down towards his stall in the market. 'Sell those for me today and I'll believe you.'\n\nHer face and figure were enough to get her noticed, yet she had that bouncy enthusiasm for life that set her apart from other young girls. In less than an hour the dresses were gone, sold to little office girls who hoped they'd look as good as Georgia did in hers. Thanks to Georgia's idea he'd had the best summer season ever and he had money put by for his expansion programme.\n\nIt was silly to think a young girl could change his life, but she had. She'd made him think big, want to reach out and grab things, just the way she did.\n\nHe was going to get a proper workshop. Employ a designer and sell his clothes everywhere. Georgia was right, the future lay with youngsters, not the staid old ladies he'd once catered for.\n\nIt was just before Christmas when the idea of Georgia's birthday present came to him. The workshop was festooned with paperchains. Cotton wool snow stuck to every window, doors decorated with large Santa Claus's, all made by Georgia out of old fabric.\n\nJanet ushered Andreous, an old friend, into his office, bringing three glasses with her.\n\n'Thought you'd want a seasonal drink,' she said, hands on hips, a sprig of mistletoe in her untidy hair. She winked at Andreous.\n\nAndreous owned the Acropolis club in Greek Street and like most women, Janet fancied him.\n\n'Who's the third glass for?' Pop asked, amused by her direct approach.\n\n'For me,' she said, plonking a kiss on his forehead.\n\nPop and Andreous were like brothers, the same olive skin and dark sad eyes. But women failed to notice Andreous's thinning hair and paunch. His charm and a certain mischievous, sensuous look had ladies doting on him.\n\nPop rolled his eyes at his friend who laughed uproariously, patting Janet on her ample backside.\n\n'You can have a drink if you strip for us,' Andreous said, leering at her breasts which almost popped out of her low-cut black blouse.\n\n'I don't want to get you too excited,' Janet put on a motherly expression and patted his cheeks. 'One glimpse of my luscious body sends old men's blood pressure sky 'igh.'\n\n'Off with you,' Pop said, his dark eyes twinkling. 'Andreous is here on business.'\n\nHe poured her a little brandy however and pushed it across his desk at her.\n\nThe idea popped into his head out of nowhere. Andreous had a club, he employed musicians and singers. Why not give it a try?\n\n'Get Georgia to sing for us!'\n\nShe downed the drink in one gulp, and bent to kiss Andreous lightly on the lips.\n\n'Your wish is my command,' she said in a deep throaty whisper as she wiggled out of the door.\n\n'Who's Georgia?' Andreous asked, dark eyes alight with the prospect of a new girl to ogle.\n\n'She's young, beautiful and I want you to just listen,' Pop said severely.\n\nThe machines all stopped seconds later. For a brief second it was silent, then a buzz of conversation started from the next room.\n\nPop waited, resting his head on both hands, his elbows on the desk.\n\nGeorgia started to sing, softly at first, but as she got into it so it became louder.\n\nIt was 'White Christmas', so corny and old hat Pop thought Andreous would walk out laughing.\n\n'She usually goes in for more spirited stuff,' Pop said.\n\n'Shush,' His friend silenced him and opened the office door so he could hear better.\n\nHer voice rang round the old building, filling each corner with sweetness. Perfectly in tune without any accompaniment, each word crystal clear. Andreous sat looking at the floor, his ears pricked up.\n\n'Well I'll be damned,' Andreous looked stunned as the song ended.\n\nPop felt a surge of excitement as Georgia burst into the 'Christmas Alphabet'. He stood up, pushed open the small hatch on the wall that allowed him to watch the girls while they worked and beckoned to his friend.\n\nAndreous peered in, Pop looking over his shoulder. Georgia was dancing round the workroom as she sang, a crown of tinsel on her dark curls, Christmas baubles hung on her ears. She wore a skimpy white blouse tucked into her jeans, slender brown arms waving in time to the song.\n\nAndreous turned and grinned at his friend.\n\n'She's gorgeous, now suppose you come clean.'\n\nIt took a little persuasion to overcome Andreous's conviction he wasn't being an old fool falling for a young pretty girl, still more to convince him Georgia could sing in front of a real audience. But all the time they talked, Georgia sang next door, gently nudging the club owner into seeing his idea was practical.\n\nPop wasn't an excitable man, yet his heart thumped as he waited while Georgia opened her other presents.\n\nThe envelope was in his hand, in it one of the hand-embossed invitation cards telling her that she was appearing at the Acropolis club on Sunday 15th April.\n\n'Happy Birthday, Georgia.' He stepped forward as she sat surrounded by bits of wrapping paper, envelopes and cards. 'It's the sort of present which isn't for just today, but maybe forever.'\n\n'Mysterious,' she laughed, taking the envelope and opening it. 'Is it a treasure hunt and this is the first clue?'\n\n'Sort of.'\n\nShe read the card then looked up at him, her smooth brow wrinkled into a frown. 'I'm sorry I'm not with it. Am I invited to this do?'\n\n'Yes,' he nodded gravely.\n\n'To go with you as your guest?'\n\n'Kind of.'\n\n'And are the other girls coming too?'\n\n'I hope they will.'\n\n'Well, thank you,' she said, clearly puzzled. 'That will be lovely.'\n\n'Did you see who is appearing that night?' He held his breath as he waited for her response.\n\n'Georgia James,' she smiled, but it was clear she hadn't cottoned on. 'Someone with the same name as me?'\n\n'No,' he could hardly contain himself. 'It's you who is appearing there. It's your chance to show an invited audience what you can do.'\n\nShe sat quite still. The other girls looked at one another in astonishment, not sure they had heard it right either.\n\n'You mean I can sing in this club, with a piano and all?'\n\n'A quartet,' he laughed. 'And you have to go along and practise a few times with them first, so they know what songs you do best.'\n\n'So who will be there?'\n\n'All Andreous's best customers.'\n\n'How did you arrange this,' she said, her face pale but with the beginning of a volcano in her eyes.\n\n'Andreous heard you sing. That's all you need to know.'\n\n'And he didn't think I was a joke?'\n\n'I didn't hear him laugh,' Pop said, a lump coming in to his throat as he saw her mouth curl into a wide, joyful smile. 'It's my hope this will change everything for you.'\n\n'Oh, Pop,' Georgia hurtled into his arms. 'That's the most wonderful, exciting birthday present. I can't believe it.'\n\nSomehow this sweet man had picked up on her dream, without her even realizing she had one. She'd talked of being an actress, a dancer, almost forgetting singing came as naturally to her as breathing.\n\nAll around her the other girls were laughing and joking, eating cake, drinking the wine. Each and every present had been carefully planned. But this one was so very special. As Pop said, 'Not just for now, but maybe forever.'\n\n'I can't say what I feel,' she whispered, winding her arms around his neck.\n\n'You don't have to, sweetie,' his voice was gruff with emotion. 'Just make it work for you on the night.'\n\n'Mum's left the office,' Georgia blurted out to Janet the next morning. 'I mean she's gone for good and no one knows where.'\n\nEven last night, with the party going on around her, Georgia had been dying to phone Celia. She had imagined the scenario, the shock, the surprise, even tears. Never once had she considered Celia might leave her job.\n\n'Now calm down,' Janet got up and put her arms round Georgia. 'Is it that surprising she left?'\n\n'But what do I do now?' Georgia's mouth drooped petulantly like a small child's.\n\n'You know where Peter lives,' Janet raised one eyebrow. 'Go on over there. He's bound to know where she is!'\n\nEltham High Street looked different, cleaner, brighter than she remembered. No drunks or tramps, spivs or tarts like Soho. Just middle-aged ladies with shopping baskets, younger women with prams and pushchairs and men driving Fords. Even the gang of youths who stood outside Olive's coffee bar watching the girls go by looked harmless. Suit jackets over carefully pressed jeans, hair cut in the same college-boy style favoured by office workers.\n\nGeorgia checked her appearance in a shop window as one of the boys whistled at her. Her red double-breasted coat came from a jumble sale but Helen had taken in the waist and shortened it for her. Although it was a little old-fashioned she knew it suited her. Her hair hung loose over her shoulders and she had new, pointed patent-leather shoes.\n\nAs she approached Haig Road she wanted to run. It was beginning to rain and she had no umbrella.\n\nPeter's house looked exactly the way it had when she rode passed it on her bike that first summer soon after she'd met him. A bit frowsy, litter in the bare front garden. The gate was hanging off its hinges, the front door still unpainted.\n\nShe knocked on the door, holding her breath with excitement, screwing up her eyes as she willed Peter to answer.\n\nBut instead it was a tall thin woman, a blue nylon overall over her clothes, a duster in her hand.\n\n'Is Peter in?' Georgia smiled at the woman. She had expected Peter's mother to be shorter and fat.\n\n'No,' she said sharply, her face tightening, her mouth a thin suspicious line.\n\n'Oh,' Georgia felt her heart lurch, instinctively knowing she should have written rather than called. 'I'm an old friend. Georgia.'\n\n'He's gone away,' Mrs Radcliffe stiffened visibly, already trying to close the door as if that was an end to their conversation.\n\n'Please,' Georgia moved closer, her bright smile wiped out by the hostility, her eyes pleading now. 'Can I come in and talk?'\n\n'What is there to talk about,' the woman looked at her with dead, cold eyes. 'He's at university. He's forgotten about you and he doesn't need any reminders.'\n\nThe woman's sharp words cut through her like a knife. There was no resemblance to Peter. Pale washed-out brown eyes, white skin with tiny, red broken veins, thin lips and a sharp, pointed nose. Even her hair was colourless. Never golden blonde like Peter's, just dreary light brown, fading to grey at her temples.\n\n'But let me explain,' Georgia pleaded. 'I can't find out where my mother is. You do know what happened don't you?'\n\n'I know only one thing,' the woman's thin face loomed up close to Georgia's, small eyes full of spite. 'You ran off leaving my boy so upset he could barely take his exams. He's got over that now. He's happy with a nice girlfriend. Go away and leave him in peace.'\n\n'Is he staying here for the holidays?' Georgia was getting desperate now. Not for one moment had she expected this.\n\n'No, he's with his girl.' Once again the door started to close.\n\nGeorgia moved forward and held the door. 'I don't want to bother him if he's got a girlfriend,' she blurted out, tears coming to her eyes. 'But he might know where my mother is.'\n\n'He doesn't,' she snapped, pushing Georgia's hand off the door. 'He hasn't seen her since you left. Why should he? You left him high and dry without a word. Got him into all that trouble with the police. Mud sticks you know, and you did nothing to clear his name.'\n\n'But \u2013' Georgia felt sick. She hadn't even considered Peter might be blamed in some way.\n\n'There's no \"buts\",' Mrs Radcliffe folded her arms across her bony chest, her narrow lips set in a straight uncompromising line. 'He's forgotten you. Now push off.'\n\nIt was that same kind of frosty prejudice she'd met when she first left home.\n\n'Please give him my address,' Georgia pleaded. 'Tell him I don't want anything but to find Mum.'\n\nShe thought then she had brought the woman round, she hesitated for a second, letting her arms drop to her sides.\n\n'All right,' she said grudgingly. 'But don't expect miracles. He's got enough on his plate.'\n\nGeorgia took out a notepad from her bag, quickly scribbled her address and passed it over.\n\n'Berwick Street,' Mrs Radcliffe looked at it, then at Georgia suspiciously. 'Soho?'\n\n'Yes. I share a flat with a friend.'\n\nShe sniffed, and put it in her pocket.\n\n'Thank you,' Georgia could feel her face burning with shame. She knew what the woman was thinking and it hurt so badly she wanted to die.\n\nThe door was closed before she even got to the gate. Georgia stood for a moment staring up at the house, her eyes filling with tears.\n\nShe could see a few books on an upstairs window-sill. Was that his room where she'd imagined him sleeping? Why couldn't his mother have been the fat, comfortable, jolly person she expected, opening up her arms in welcome?\n\nSo he was staying with his girlfriend! He had a new life and it didn't include her. Peter's world was closed to her, as firmly as his mother had shut that door.\n\nThe rain grew heavier, but Georgia hardly noticed it as she trailed down Shooters Hill Road towards Blackheath.\n\nThis morning she had been so full of joyful expectation and now it was all gone. Celia had left her job. Peter had been blamed. Instead of solving all their problems by running away, it looked as though she added to them. Celia had told her many times about hysterical teenagers who claimed rape to cover up for staying out late, or going willingly with a man. Was that what they believed of her?\n\nThe moment she saw the old house she knew Celia had long since left. The lawn in front of the house had been paved over and the railing removed. Apart from two forsythia bushes, there was nothing left of the garden. Now there was merely space for two cars, and every window sported net curtains.\n\nCelia had never liked nets. She believed they were unnecessary, spoiling natural light, fussy and old-fashioned.\n\nGeorgia stood just a little way away from the house wondering what to do.\n\nIt was possible that Brian still lived there and the last thing she wanted to do was see him.\n\nInstead she passed the house and called two doors away where Mrs Owen lived. She hadn't been a friend of the Andersons, just a gossipy neighbour, but if anyone knew where Celia had gone it would be her.\n\nShe rang the bell nervously.\n\nA short, plump, middle-aged lady answered the door, wiping her hands on a tea-towel.\n\n'Does Mrs Owen still live here?' Georgia's heart plummeted to even greater depths. She was like Mrs Owen, but smarter and several years younger.\n\n'She's gone to visit her daughter in Australia,' the woman smiled. 'I'm her sister.' Despite her smile she looked flustered, as if caught in the middle of something.\n\n'Oh,' Georgia frowned with disappointment.\n\n'Can I help?' the question was one of indifferent politeness, already she was looking past Georgia out towards the heath as if wondering how long this was going to take.\n\n'I was looking for someone who lived at number nine. Mrs Owen was friendly with her. I thought she might know where she went?'\n\nThe woman stepped forward out of her porch, glancing down the road as if to try and remember.\n\n'Oh yes, you must mean the Andersons,' she said. 'I never met them, but Nancy used to talk about them. They sold the house, dear. There's students living there now.'\n\nGeorgia sensed the brush off. The woman didn't want to talk, already she was retreating back into her doorway.\n\n'Thank you,' Georgia took a step back. 'I'm sorry to have bothered you.'\n\n'Sorry I couldn't help more,' the woman was already closing the door. 'Try the estate agents. It's Goodman and Smith, the first one you come to. They're bound to know.'\n\nBy the time Georgia got off the train at Charing Cross she was wet through and so cold her teeth were chattering.\n\nThe estate agents had given her an address, but it was Brian's.\n\n'Terrible mess the house was in,' the pink-faced, snobby estate agent told her. 'Seems his wife left him and he went to pieces afterwards. He got us to sell off everything. He said he was going abroad.'\n\n'But did he tell you where his wife went?' Georgia was tempted to tell him the whole story if only to get his full attention, but he was so cold and businesslike she merely pretended to be a niece.\n\n'Rumour had it she ran off months before,' the man said haughtily. 'I believe Anderson drank, we certainly found a great deal of evidence to bear that out. We had no reason to contact Mrs Anderson, the house was his sole property.'\n\nShe took Brian's address in New Cross out of politeness, but once outside she tore it up and threw it away.\n\nAs she opened the door to her room, Helen hobbled towards her, green eyes blazing like fireworks, her pale face flushed and hot-looking. Clothes were strewn all over the place and the air thick with the smell of burnt toast.\n\n'I've got a hospital bed at last,' she flung herself at Georgia. 'Next week. Isn't it wonderful?'\n\nGeorgia took a deep breath and tried hard to smile.\n\n'I'm so pleased for you,' she bit back tears and held Helen tightly. 'You've waited so long.'\n\n'I've got so much to do I don't know where to start.' Helen wriggled out of her arms, picking up things and throwing them down while all the time she trembled with excitement.\n\n'What have you got to do?' Georgia had to laugh despite her own misery. Helen was normally so placid and quiet, it was a diversion at least from her own troubles.\n\n'I'll have to give my notice at the club. Buy some new nighties, tell Bert I won't be here. So much.'\n\n'Now calm down,' Georgia said, taking Helen by the shoulders firmly. 'All that will take less than half an hour.'\n\nHelen tried to dance, hopping around on her one good leg, her smile stretching across her whole face. 'Oh, Georgia in a week or two we might be able to go out dancing. By the time you sing at the Acropolis I might be a normal girl.'\n\nAll the previous year Helen had been waiting for this bed. Twice before she had been accepted as a patient and then the operation had been cancelled just days before.\n\n'Don't build your hopes up too high,' Georgia said slowly. 'Think the worst, just in case.'\n\n'That's an odd thing for you to say,' Helen spun round and looked at Georgia sharply, colour draining from her face, as she remembered where Georgia had been going. 'Don't say you didn't find Peter?'\n\n'Worse,' Georgia slumped down into a chair. 'He doesn't care about me anymore. I think his mother hates me. Mum's vanished too.'\n\nIt was only after her abortion that Georgia had finally told Helen the whole story. It had been the opening up, the sharing of pain which had helped her to gain her old confidence. Once again Helen listened, her green eyes filling with tears and she rested her small red head on Georgia's dark one.\n\n'What can I say?' she whispered. 'I don't really believe they don't want to see you. How could anyone turn their back on you?'\n\n'But why didn't Mum leave an address?' Georgia sniffed. 'Surely she knew I'd want to contact her?'\n\n'People do funny things when they're hurt,' Helen said thoughtfully. 'But even though Peter's mother sounds like a real old witch, I'm sure she will pass on your note to Peter. She probably got a shock seeing you on her doorstep.'\n\n'I handled it all wrong,' Georgia sighed deeply. She was beyond crying now and she didn't want to spoil Helen's joy by dwelling on her own problems. 'I should have trusted Peter a year ago and written. You can't keep people in the dark and expect them to just know how you feel.'\n\n'You haven't had much luck have you,' Helen wound a strand of Georgia's hair round her finger, her small bony arms holding Georgia tightly.\n\nGeorgia looked at Helen. The built-up brown boot was peeping out from her long skirt, her green cardigan had tiny darns where moths had eaten it and she was about to face a serious operation which might leave her lamer than before. Yet never once had Georgia heard her sniffle about having no family.\n\n'No luck?' she forced herself to smile. 'I found you. I've got a job and a home. I've even got a chance at the Acropolis. How much more luck does anyone need?'\n\n##### *\n\nIt was after twelve that night when Mrs Radcliffe pulled the address out of her overall pocket. Peter had gone to bed early, his face white and strained.\n\nIt was lucky he hadn't got home from the library ten minutes earlier, otherwise he might have caught her sending that girl packing.\n\nWhy couldn't he be like her neighbours' sons, out with the lads on a motorbike instead of mooning around waiting for her? He had a fine career ahead of him, no mother would gladly see her only son going off with some wild black girl.\n\n'I'm doing this for your own good, son,' she muttered to herself, poking the fire up into a blaze. 'She's probably been on the game all this time. No good for you, my boy.'\n\nShe hesitated for a moment, then plunged the note into the flames before she could change her mind.\n\n'That's it over now.' She wiped her hands on her overall and straightened up. 'You'll thank me for it one day Peter.'\n\n## Chapter 7\n\n'You look exhausted,' Peter frowned with concern as Celia sank into a chair without even taking her coat off, an unopened letter in her hand.\n\nIt was after ten, a cold March night, yet another evening spent fruitlessly in pubs and coffee bars searching for Georgia.\n\n'I'll make some tea, then I'd better get home,' Peter bent down to light the gas fire. 'Are you just going to stare at that?'\n\n'It's from _him_ ,' Celia shuddered at the familiar neat script.\n\n'A letter can't hurt you,' Peter came closer and put one hand on her shoulder. 'Do you want me to open it?'\n\nShe shook her head and slid one finger under the flap.\n\n'The telephone bill,' she pursed her lips with annoyance. 'He's got a cheek, I've been gone three months!'\n\n'No letter?' Peter asked.\n\n'Just a curt note saying the long distance calls \u2013' she stopped suddenly in mid-sentence, making Peter turn his head.\n\n'What is it?'\n\n'A postcard too, from Georgia.'\n\n'What!' Peter came back to her side with one bound. 'Let me see.'\n\nCelia's hands were trembling, her eyes filling with tears. Peter snatched it from her, just the sight of her rounded, childish writing filling him with renewed hope.\n\n'Read it to me,' Celia whispered.\n\n'\"Dear Mum, I'm safe and well. I've got a nice room, a job and new friends.\"' Peter gulped, glanced at Celia's radiant face, then continued. '\"Don't worry about me please because everything's fine. Soon I'll be sixteen and then I can get in touch again. Give Peter my love, tell him I miss him. I love you, Georgia.\" '\n\nFor a moment they could only stare at one another, then Peter dropped down on to his knees beside Celia, running one finger over the few sentences as if committing them to memory.\n\n'Manchester,' he held the card closer to the light, examining the postmark. 'But it's dated January 29th, it's almost two months old.'\n\n'The evil swine,' Celia's face flushed with anger. 'He's sat on it for two months. How could he do that?'\n\n'Revenge?' Peter shrugged his shoulders. 'And all the time we've been wasting our time looking in London.'\n\nIf it hadn't been for Peter's obstinate strength, Celia might have buckled under the strain weeks ago. No one else seemed concerned that an underage rape victim was out there somewhere alone. The police had given up looking for her. Even the agencies who advertised their concern had come up with nothing. Wild goose chases to places where someone had reported a girl fitting her description. A call from a hospital in North London where a girl lay in a coma, another to view a body in the Deptford morgue. Each time Celia rushed there full of hope, or dread, only to discover the only similarity was dark hair and the right age. Even the children's department had lost interest, suggesting it was high time she concentrated on other children in her care. She couldn't count the cost of phone calls, stamps or petrol, that was all incidental. What frightened her most was running out of hope. It was Peter who kept her going night after night. Meeting her to check out yet a few more clubs, pubs or bedsitter houses, never daunted by the size of the task, never flagging in enthusiasm.\n\nKnowing Brian had raped Georgia was the worst thing that she'd ever been faced with. In the early days when Georgia lay in her bed refusing to speak, she kept that thought with her. Whatever came next had to come down on the scale of shock. Yet when she went into the kitchen and found Georgia's note saying she'd left, Celia went to pieces.\n\nThe empty bed, half-eaten meal, a holdall gone from the cupboard. When she threw herself down on the little bed and smelled her daughter on the sheets she thought her heart would break.\n\nEven now, months later the pain was still acute. All the things she once held so dear, gone for ever.\n\nTwo grubby little rooms were her home now, sharing a bathroom with four strangers. Belmont Road in Lewisham wasn't that far from Blackheath in miles, yet it felt as if she were on another planet.\n\nThe rooms were at the top of a large house, divided into a rabbit warren of bedsitters. Cold, draughty, threadbare carpets, dirty bathrooms and continual noise from the other tenants.\n\nIt had been less than a week after Georgia left that Celia arrived home to find Brian back. He was hunched up in a chair, a blanket round him. The expression on his pale face one of a cringing dog who fully expects to be whipped.\n\n'What on earth!' she exclaimed in horror.\n\n'You didn't know I was coming home then?' he said, his eyes cast down, hugging the blanket tighter around him.\n\nIf she'd had some warning of his discharge she would have made plans, at least prepared a speech to make her feelings quite plain. She wasn't prepared for the feeling of nausea that washed over her, or the terror at being alone with him.\n\n'Look, Celia,' he said, mistaking her silence for weakness. 'I know you believe the worst of me, but it wasn't like that at all.'\n\n'You louse,' she spat at him, pulling her coat tightly round her, wanting to walk right out again if it meant sharing the same air as him. 'Don't try to wriggle out of what you've done. Nothing will persuade me to forgive you, but don't insult my intelligence by lying!'\n\n'I knew you'd be like this,' he said in a petulant tone, a crocodile tear dripping down his cheek. 'That girl's the liar and you are a fool if you believe her.'\n\n'She said nothing to anyone,' Celia shook with rage. 'I saw the evidence myself remember. I know what happened as if I'd seen it on a film. You've ruined her life and I can't even bear to be in the same room as you.'\n\nIt would have been far better if she'd packed her bags and left that night. Only the certainty that Georgia would telephone kept her there a further two weeks. But two weeks was long enough for Brian to see it as compliance and she brought on herself that last ugly scene.\n\nLooking back, that period seemed like months. Sleeping in Georgia's old room, going through old diaries and address books, hoping for a lead. She didn't speak to him, not one word. She went out daily, knocking on doors, meeting old friends of her daughter, walking the streets till exhaustion and the prospect of a telephone call drove her home.\n\nBrian shuffled between the sitting room and the kitchen. He cooked food when she was out, leaving the dishes.\n\nCelia did nothing for him. She let the dishes pile up, his clothes stay where he dropped them. She bought no food, she didn't even pick up his mail.\n\nShe knew she was not behaving rationally. It would be better to scream abuse at him, hurt him as he'd hurt her. It was like an abscess that needed to be lanced. Ignoring it just prolonged the healing process, letting the poison slowly spread through both of them. But Celia remained trapped in a silent world, just as Georgia had been.\n\nShe barely noticed the house getting dirty. The dead flowers still sitting in vases and the kitchen bin full to overflowing. Brian was drinking, she couldn't avoid seeing the endless empty bottles, or miss the smell of whiskey gradually permeating round the house. She guessed that many nights he passed out in the chair downstairs, and she hoped he would drink himself to death.\n\nGeorgia was on her mind from the moment she opened her eyes in the morning till sleep finally closed them.\n\nShe could be in some awful tenement, hungry, cold, and in shock. Worse still she could have turned to someone who would betray her trust even more than Brian had.\n\n'Let's talk?' Brian came out into the hall when he heard her key in the door one evening.\n\nHe looked old, dishevelled and desperate, with several days' growth of stubble on his chin.\n\nCelia looked no better. She had lost weight, her once rounded rosy cheeks, sunken and grey. Hair in need of a trim, her tweed coat suddenly too large.\n\n'We can't go on like this,' he said, one hand on his stomach as though in pain. 'Tell me what you want of me and I'll do it.'\n\n'Kill yourself!' All the venom she ought to have released on him earlier came gushing out. 'But you aren't even man enough to do that!'\n\n'Don't Celia!' he begged. 'Come in the sitting room, we can't talk out here.'\n\nShe knew now it was foolish to hope he was going to make a full confession of his guilt, yet in that moment she believed he was.\n\nA stale smell of food and body odour filled the room. She sank onto a chair as far from Brian as possible and averted her eyes from a photograph of Georgia on the mantelpiece.\n\nBrian sat down by the fire and picked up a glass of whiskey.\n\n'Come on then, talk,' she said. 'Start with why you had to rape a fifteen year old, then go on to what you are planning to do about it.'\n\n'I've already told you I did no such thing,' his faded blue eyes looked hurt. 'Think about it again Celia. Why would she run away if she was innocent?'\n\n'You liar,' Celia hissed. 'How can you sit there and continue to make yourself believe that? Do you think you're talking to some half-baked office junior?'\n\n'If you think so little of my integrity,' his tongue flickering across his lips with nervousness, 'perhaps you'd better leave this house now.'\n\n'I will,' she snapped back at him. 'As for your integrity, you'd better stop all that drinking before you forget yourself and go and rape some other young girl on her way home.'\n\n'How dare you!' His face flushed with anger she'd never seen before. 'You are so self-righteous. If I'm drinking now it's because I can't bear you to think these things of me. We've been together for twenty years, yet you'd let the word of some nigger brat come between us.'\n\n'Don't Brian,' Celia got up to leave. She was sickened more by the insult to Georgia than his lies. 'I remember other pointers to your strange sexual tastes. If I have been self-righteous, it's about those. I actually believed I'd cured you!'\n\nHe leapt up, barring her way.\n\n'Cured me?' He caught hold of her arms, whiskey-soaked breath making her wince. 'You stupid, frigid cow! All men look at dirty books, there's nothing strange in that. It's you who's weird, not me. You cheated me, made out you married me for love, but all you wanted was this house. I found that out on our so-called honeymoon!'\n\n'Maybe that was your fault?' she said weakly.\n\n'My fault, eh?' he squeezed her arms tightly. 'I let you have everything your own way. I even came rushing back here when you said you were leaving to keep you on any terms. No wonder we never had a child of our own, you never gave it a chance. But even then I said nothing, I even let you steam-roller me into having that kid here.'\n\n'I did no such thing,' she retorted. 'You agreed to have Georgia quite readily.'\n\n'I went along with it to make you happy,' he shouted, his face growing flushed. 'You call yourself a social worker, dig into other people's problems, yet you can't see ones under your own nose.'\n\n'So you bottled up your lust to save it for her?'\n\nShe tried to push him away.\n\nAt that moment she knew he had got to a danger level. His pale eyes turned dark and a vein was ticking in his forehead.\n\n'I didn't bottle anything up,' he snarled. 'I've had plenty of women. Pretty, tarty girls, ones that like to be fucked and they were all more fun than you.'\n\nCelia was stunned, yet somehow she knew this much was true.\n\n'You sicken me,' she said, attempting to shrug off his hand. 'Let me go this instant and I'll pack and get out.'\n\n'Not so fast,' his voice deepened with menace and his fingers caught her hair, pulling her head back. 'Don't think you can walk out of here and then try to get my money or this house.'\n\n'I don't give a damn about this house,' she screamed at him. 'I don't want anything that reminds me I brought a helpless child into a house with a pervert!'\n\nHis fist hit her cheekbone like a sledgehammer, knocking her backwards over an armchair. He reached out and pulled her back up as if she weighed nothing.\n\n'I'll help you pack,' he hissed. 'Let's go through your clothes together, look at the thick knickers you wear, the great big ugly bras and corsets. You aren't a woman at all Celia, you are a man without a cock. I must have been mad to marry you!'\n\nIf she had ever had the slightest doubt Brian had raped Georgia it vanished that night. He was so strong and brutal, dragging her upstairs, pulling things out of her wardrobe and taunting her with everything from her sensible flat shoes to her choice of underwear.\n\nTime and time again she tried to make a run for it. But each time he blocked her way, slapping her again and again.\n\nIn the back of her mind she could remember telling women at work how to cope with violent men, yet now it was happening to her she was unable to defend herself.\n\nJust after eleven he eventually left her to return to the sitting room. Quickly she stuffed her clothes and personal belongings in a couple of suitcases.\n\n'I'm leaving now,' she returned nervously to the sitting room for her handbag and car keys, poised to make a dash for it if he attempted to hit her again.\n\n'Don't think I'll let you back in,' he said more calmly than she expected.\n\n'There is nothing here to bring me back,' she said proudly. 'I want you to spend the rest of your life reflecting on what you've done. I hope you never have one moment of peace or happiness again.'\n\nThe rooms in Belmont Park were the cheapest she could find. The rickety old furniture didn't bother her, she could wash the soiled blankets and buy new sheets. It was only the little things she missed, photographs, her sewing basket and dainty china teacups.\n\nIt was pure spite that made her write to the head office of the bank in Lombard Street. She laid out coldly and clearly the case against Brian and allowed his employers to consider whether they could trust him amongst young girls.\n\nIt didn't stop the pain of losing Georgia, but revenge had its own kind of sweetness.\n\nPeter made the tea and handed a cup to Celia. She had aged dramatically since the night of the party, not just more lines and grey hair, but that confident bossiness seemed to have vanished.\n\nThere had been times when he felt angry at Georgia. Not for running out on him, but for what her actions were doing to Celia. Surely she knew her mother better than to suppose she'd turn her over to the authorities?\n\nYet even in anger he ached for her. She was under his skin, in his head and heart. Just those few lines on a postcard made him tremble with longing. Nothing would make him give up on her, even if he had to continue his double life indefinitely.\n\nThe ugly scene with his parents soon after Georgia ran away had slammed home their prejudice and inadequacy. If it wasn't for Celia's deep understanding, he too might have been tempted to run off.\n\n'Damn little nigger slut,' his mother shouted at him when he tried to explain how frightened and miserable he was. 'That girl's made a laughing stock of you and now you tell me you're worried about her!'\n\nUntil he met Celia Anderson he'd never thought about his own mother's shortcomings, but that night he saw them all in close-up. Not one ounce of compassion in her thin, stringy body. Jealous of anyone who had more than her, suspicious of everything. The only time she was loving was when she had a few drinks inside her, a slut and an evil gossip.\n\nHe'd tried to reason with her, standing there in the living room, the table still strewn with the leftovers of breakfast even though it was tea-time, the fire concealed by steaming washing hanging to dry.\n\n'What makes you so bitter you have to take it out on Georgia?'\n\n'I've worked hard all my life,' she raved, her mouth wet with spittle, eyes screwed up with hatred. 'Cleaning offices, scrubbing floors when you were little, just to pay the rent. Your father pisses away every penny he earns down the pub. I was banking on you helping us later on.'\n\nIt was the last sentence that cut him to the quick. She was afraid that somehow Georgia would prevent him getting a degree. A degree to her meant nothing more than a well-paid job at the end of it.\n\n'You're best out of it, son,' his father said when his mother tried to drag him into it. He sat by the fire, still in mud-caked boots from the building site raising weary blue eyes towards his wife. 'My dad told me not to marry her. I wish I'd heeded his words, sometimes older people know best.'\n\nIt all came out that night like slugs leaving a trail of slime where they'd passed over a floor. His father's family were well-to-do, Josie's from a slum in Deptford. She faked pregnancy to force Geoffrey to marry her, but the plan backfired when his parents cut him off.\n\nPeter saw both sides of it as his parents shouted at one another. His weak but intelligent father, matched with an avaricious, cold woman. Disappointment and greed had killed any love. Geoffrey turned to the pub for friendship, Josie kept a tight rein on her son hoping he'd fulfil the expectations her husband hadn't lived up to.\n\nIt was safer to keep quiet about Georgia. He did his homework, ate his tea then made an excuse to get out to see Celia. It was ironic that his parents preferred to think he was down the snooker hall in Lewisham, rather than singing in the choir, or seeing a girl. They didn't guess he was knocking on doors, going in pubs and clubs brandishing a photo of the girl they hated.\n\nPeter put his coat on and wrapped a scarf round his neck.\n\n'It's Easter next week,' he said. 'Let's go to Manchester?'\n\nCelia got up from her chair and shuffled wearily across the floor to him.\n\n'Don't pin your hopes on finding her,' she reached up and kissed his cheek to soften her words. His eyes were shining with excitement, the way they had when Georgia first brought him home. 'It's almost as big as London!'\n\nCelia sighed deeply as they approached Manchester.\n\n'Why the big sigh?' Peter said softly.\n\n'I thought you were asleep,' she smiled round at him. 'I guess I'm thinking how it looks for a forty-plus woman to be running off with a handsome young lad like you.' She had to make jokes about their predicament, she had seen the looks people gave them, wondering if he was her son, or lover. Once they got out the pictures of Georgia the looks became even odder. His parents would hit the roof if they found out he wasn't really youth hostelling. How would she feel if the situation was reversed?\n\nPeter sat up and took the map out of the glove compartment.\n\n'I'm happy for them to think you're my mother,' he smiled. 'But it isn't really that is it? You feel guilty.'\n\nOne of the things she liked most about Peter was his perception. He understood Georgia's reasons for running, his own parents, the emptiness inside her and no doubt if he'd talked to Brian he would almost begin to understand him too. But it was this deep understanding of other people which could be his downfall. He had to make a life of his own, he was too young and clever to give up his own needs and education.\n\n'Will you promise me something?' She turned her eyes away from the road, reaching out to touch his hand.\n\nHe stiffened. Every muscle in his face and neck was strained, his eyes scanning the streets as they entered Manchester.\n\n'Depends,' he smiled faintly, returning the squeeze of her hand.\n\nHe was wearing jeans, a pale blue sweater under the denim jacket and his hair was creeping over his collar.\n\n'On what?'\n\n'Whether you want me to abandon the search,' he said, fixing those bright blue eyes on the side of her face as if trying to read her mind.\n\n'Not exactly,' she said choosing her words carefully. 'But I want you to promise me you will go on to university. I really do believe that Georgia will get in touch again as soon as she's sixteen. You'll only add another burden of guilt to her shoulders if you've found a job in London and not used your ability because of her.'\n\nHe frowned.\n\n'Maybe,' he stared out the window for a moment, thinking about Celia's words. 'All right, I promise,' he said. 'But you've got to promise me something too.'\n\n'Go on,' she half smiled.\n\n'That you'll get yourself sorted out too,' he said earnestly. 'Georgia won't write to the old house again, she'd ring me. So there isn't a great deal of point in you hanging around waiting. You look ill, you must look after yourself for her sake. Just as you said it wouldn't do for me to put another burden on her shoulders, neither must you.'\n\n'Fair enough,' she nodded. 'You've got a deal. After this weekend I'll rethink my life too.'\n\nCelia doubted that Georgia was in Manchester. Runaways went to places with a connection with their past. It was a false trail to put them off the scent. But that was a good sign. It meant wherever she was, she was happy to stay there and the message on the card about her sixteenth birthday an assurance she would reappear. Could it be that she had used Manchester, knowing it had one of the universities Peter was keen to go to?\n\nAs they passed the university she saw his eyes light up, for the first time his mind on something else other than Georgia.\n\nIt had been a gruelling weekend, calling in shops, pubs and clubs showing the photographs, knocking on doors, even asking children in the street. But at least the people were friendly here, no slammed doors or rude remarks and that at least was better than London.\n\n'You should apply here,' she said gently. 'If Georgia's around you may run into her, and anyway it's a fine place.'\n\n'Maybe,' he said, his face breaking into that enthusiastic smile she remembered so well. 'But that won't stop me searching other places in the holidays.'\n\n##### *\n\nAs the spring turned to summer Celia saw less of Peter. He was working hard for his exams and for the moment Georgia was taking second place.\n\nCelia pretended it was the same for her. She had taken on a temporary office job which gave her more time to continue her search.\n\nNight after night she went up to the West End of London, Chelsea and Earl's Court, wandering about, looking.\n\nJust a glimpse of a black curly mop of hair was enough to send her heart pounding. Each brown-skinned girl was studied closely, questions asked in coffee bars until she began to think she'd spoken to half the young girls in London.\n\nWhen Peter was accepted at Manchester University she was thrilled for him.\n\n'A new start,' she said hugging him. 'You must work hard now. I'll be here when you come home for Christmas. She's bound to write to you then.'\n\n'But you should find a better job,' he said, looking at her disapprovingly. 'It's no good telling me not to waste my talents when you are doing it too.'\n\n'After Christmas I will,' she promised.\n\nIn October Peter left for Manchester and Celia pressed twenty pounds into his hand.\n\n'Just a little nest egg to help you when you get there,' she said, trying not to dwell on how much she would miss him. 'Come and see me the moment you get back.'\n\nChristmas came and went without hearing anything. She knew Peter was hanging around by the phone and watching for the postman continually and she found it almost impossible to concentrate on anything.\n\nThey both bought birthday presents and cards. Celia made a cake and iced it in readiness.\n\nOn January 6th Celia had to go to work, but all day she jumped when the phone rang in the office, fully expecting good news from Peter.\n\nThere was nothing. Not that Thursday or the Friday.\n\n'What are we going to do now?' Peter telephoned her just before she left for home on Friday night.\n\n'Maybe she'd been working?' Celia said. 'She could be planning it tomorrow as it's Saturday.'\n\n'I haven't left the house,' he said wearily. 'Mum's getting really cheesed off with me. I'll have to go down to the library tomorrow morning, I've got loads of work to do.'\n\n'Don't worry,' Celia said. 'You can't stay in for ever. If she phones and you're out, she'll ring again.'\n\nThe weekend passed without a word, and finally on the Sunday night Peter came down to Celia's flat in Lewisham around nine.\n\n'Nothing,' he sighed, looking dejectedly at the brightly-coloured parcels on the table. 'She can't care about me anymore,' he said, flinging himself down into one of her chairs. 'It was all a dream after all.'\n\nCelia knelt by his side and cupped his face in her hands. 'Go back to Manchester. Get involved with things up there. Who knows, a letter might come any day. Your mother will send it on.'\n\n'But what about you?' he said, tears glistening in his eyes and his sensitive mouth dropping at the corners. 'You can't just wait here. There's no point in it.'\n\n'I think I need a new start too,' she said softly. 'I've applied for a job with the World Health Organization. I was going to back out if Georgia came home, but perhaps I should go.'\n\n'You should,' he said sadly. 'There's nothing to keep you here now. Where is it?'\n\n'In Africa. I don't know whether it will be a big hospital, or a small mission somewhere out in the wilds.'\n\n'You're very brave,' he said in a small voice. 'That's a huge step for you. I'm going to miss you so much. You will keep in touch, won't you?'\n\n'Of course, you silly boy,' she smiled and patted his face. 'I haven't another soul I want to write to and I'm sure once I'm there I'll be clinging to you like a life raft. Now if she does get in contact make sure you write immediately.'\n\n'Who else would I share it with?' he smiled.\n\nCelia smiled bravely as she kissed Peter goodbye. She knew more about rape victims than Peter could ever know. It was common for them to shy away from their old life. They wanted no reminders of their pain. Georgia was a strong person, but not strong enough to come back and re-live her ordeal. Perhaps one day she would be, but as Peter so rightly pointed out Celia couldn't just brood until then.\n\n'You must go,' she said to herself. 'Maybe helping other children will fill the void. Let Peter forget her, pack away the past and start afresh.'\n\nShe sat down at the table and opened a writing pad.\n\n'Dear Sirs, I am writing to confirm that I wish to take up the nursing position you recently offered me. I can be ready to leave at any time.\n\nYours sincerely,\n\nCelia Tutthill.'\n\n## Chapter 8\n\nThe Middlesex hospital was terrifying. From the smoke-blackened stone outside, to wards full of pain-filled faces.\n\nGeorgia looked anxiously at Helen as she climbed into her bed in the new white cotton nightdress Janet had made her. Until that morning excitement had brought a flush to her cheeks, but now she had a greeny tinge, her eyes troubled and afraid.\n\n'I'll be round every night,' Georgia squeezed Helen's small fragile hand. 'Just think of all the fun we'll have when your leg's fixed.'\n\n'It reminds me of the Home,' Helen whispered, glancing along the row of beds. Twenty in all, most of the patients lying still, no talking or laughter, a sense of distress in the air. 'Look at all those crutches, walking frames and wheelchairs,' she pointed over to the middle of the ward where a collection of aids was kept. 'Tell me I'll be able to walk like you when I get out?'\n\n'Of course you will,' Georgia gulped hard. 'Just rest, and think what you are going to wear to the Acropolis!'\n\nGeorgia felt scared without Helen that evening. From down in the street were all the usual noises of car doors banging, people shouting and bursts of loud music as a club opened its doors, but up in her room it was too silent.\n\nShe hadn't fully realized till now how much they needed one another. They had ups and downs, times when they argued and bickered. Yet their lives were intertwined. They were more than just friends, and alone she felt abandoned.\n\nJust outside her window were other young girls meeting friends, going dancing and to parties. At only sixteen life seemed to be passing her by.\n\nWould she ever meet a boy she wouldn't be frightened of? Why was it impossible to think of any male without holding him up in comparison to Peter?\n\nThat night as she lay staring at the ceiling, it seemed dirtier than ever. The paper was peeling off, the brown paintwork on the door scratched and puckered with age. Even the posters they'd put up didn't hide the fact the room was little more than a slum.\n\nThey had made some improvements. Newer bedspreads found at a jumble sale, a bright curtain round the hideous old sink, cushions, and a lampshade hid the naked light. But what it really needed was redecoration.\n\nShe woke early the next morning, an idea stopping her from sleeping any later.\n\nDressing quickly she ran downstairs to the caf\u00e9 where Bert was in the middle of frying several breakfasts.\n\nBert made her think of a bloodhound. Deep lines ran down his cheeks, baggy skin round his eyes, even his eyebrows and mouth drooped. One look at his sallow face was enough to think he'd spent his entire life in misery. But Georgia knew better now, she'd heard him laughing constantly, seen him walking through the market whistling, listened to him telling his customers jokes too many times to take notice of his face. He even made jokes about his sad expression, claiming it stopped anyone asking for protection money, one look at him was enough to buy a three course meal just to cheer him up.\n\n'Want a bit of help?' she called out gaily, squeezing behind the counter and pouring tea for the waiting customers before he could refuse her.\n\nBert and Babs were an odd couple. Bert had been brought up in the caf\u00e9 and had run it single-handedly until ten years ago when he married Babs. Now as they moved into middle-age, it had become their entire world.\n\nA steamy, warm place in which comfort, cleanliness and style meant less than cheap, plentiful food. The customers who came in for Babs' famous steak and kidney pies didn't mind the spindly chairs or the oilcloth-covered tables. Tea came in giant china mugs, bread was cut in doorsteps and when you ordered a full breakfast you got enough to last you all day.\n\nBert and Babs lived in a few chaotic rooms above the caf\u00e9. No one would guess that this frowsy pair had the additional income of rents from their property next door.\n\n'What's up with you?' Bert asked, amused by Georgia's bouncy appearance. 'In love, or after a free fry-up?'\n\nHelen looked upon Bert and Babs almost as parents and both girls gave them a hand in the caf\u00e9 when they needed it. Georgia had often thought privately that Bert could be a better landlord, but Helen never had a bad word to say about him.\n\nRight now Georgia had plans to butter him up.\n\n'Not love, and it's not food I'm after. I wanted to know if I could paint the room while Helen's in hospital?'\n\n'Gor blimey, Georgie,' He wiped one hand across his sweaty brow, flipping some fried bread over with the other. Bert didn't like to spend money. 'That room's big, it'll take more than one coat.'\n\n'I know,' she tried to look as if she really did. 'I used to help at home. Can I?'\n\n'What colour?' He looked at her sideways, his eyes narrowed.\n\n'White,' she said. 'With yellow paintwork.'\n\n'Sit down love and eat this,' he said, tipping a greasy egg on to the bread and picking up a couple of crispy rashers of bacon with his fingers. 'Let me get me thinking cap on!'\n\nGeorgia wolfed down the food as if she hadn't eaten for weeks.\n\nShe watched him as she ate.\n\n'Okay,' he said eventually, his sad, sallow face breaking into a grin. 'I'll get you the paint and brushes. But mind you do it proper like!'\n\n'Can you get it today?'\n\n'Blimey gal,' he groaned. Georgia had the appeal of a new puppy sometimes and he found it hard to resist. 'Ain't you gotta heart? 'Ow am I supposed to find time today?'\n\nAs Georgia came down the street at six, picking her way through the market refuse, Bert was smoking a cigarette at the door of the caf\u00e9. His face broke into a grin and he waved for her to come in.\n\n''Ere you are,' he put his hand round the door and pulled out a huge tin of white paint. 'The yella's coming tomorrow. Mind you do it proper!'\n\nIt was neither as easy or as much fun as Georgia imagined. She soon discovered the paint wouldn't stick to the ceiling unless she washed it first and it was soon clear it would take far longer than she imagined.\n\nShe could only do one bit at a time, moving the furniture one way, then back another. By the time she got back from seeing Helen it was already dark and hard to see bits she had missed.\n\n'You've got white flecks in yer 'air,' Sally laughed during the week. 'What on earth are you doing?'\n\n'I'll tell you when it's finished,' Georgia said.\n\nHelen was due to have the operation on Friday, and Georgia had been advised not to visit until Sunday afternoon.\n\nOn Saturday morning she got up early. The ceiling was finished now and the brilliant new whiteness was enough to spur her on. All day she worked like a slave and at twelve that night with blisters on her hands, she crawled into bed exhausted.\n\nShe woke early on Sunday. A ray of early sunshine bounced off the white walls. For a moment she just lay there looking, smiling to herself.\n\nIt looked wonderful, even with everything piled in the middle of the room. It could be an artist's studio or a nursery, so much more light and so clean she wanted to bounce up and down in the bed.\n\nGeorgia had a bath and washed her hair before going to the hospital. Her hands were red and sore but it had been worth it. Everything was back in place but it bore no resemblance to the way she'd seen the room that first night.\n\nShe wanted to attack the old furniture now. Paint the wardrobe, the table and chairs, make new curtains. When Helen came home it would be perfect.\n\n'Hallo dear', Sister smiled a greeting as Georgia peeped nervously round the door of the ward armed with a bunch of daffodils. 'Can I have a word before you see Helen?'\n\nGeorgia had met Sister Hall when she first brought Helen in. She was very tall and thin, a sharp face, softened only by gentle brown eyes. Yet she had a gracious, kindly manner, almost as if it hurt her personally to see her patients suffering.\n\n'How is she?' Georgia's smile faded as Sister ushered her into her office.\n\n'I'm afraid she's feeling very sorry for herself. She's in a lot of pain and she's not convinced her leg is any better.'\n\n'But it is, isn't it?' Georgia felt a prickle of fear, wiping out her earlier jubilation.\n\n'Yes, it's been more than successful, it's the nearest thing I've ever seen to a miracle. But of course she's going to need an awful lot of physiotherapy before she can walk a step.'\n\n'You mean she can't walk at all now?' Georgia's mouth fell open in horror. Somehow she had imagined that Helen would be jumping around in a few days.\n\n'No, of course not dear.' Sister Hall looked at Georgia as if she were simple. 'Besides she mustn't use the other leg otherwise the weak one will always be carried by the good one. Now go on in and cheer her up. But don't stay more than half an hour.'\n\nHelen had never looked so pale or ill. If her face hadn't been surrounded by the mass of red hair she would have disappeared into the pillow.\n\nHer leg was in traction, she looked terribly uncomfortable and her eyes were red-rimmed.\n\nShe tried to smile when she saw Georgia.\n\n'How are you feeling?' Georgia bent to kiss her.\n\n'Like hell,' Helen whispered, wincing at the pain. 'If I'd known it was as bad as this I would have stayed the way I was.'\n\n'Sister says it's a miracle,' Georgia tried to sound bright.\n\n'I don't believe that,' Helen turned her head to one side away from Georgia. 'I think they are afraid to tell me the truth.'\n\nPop had warned her people were often depressed after an operation, he had said it had something to do with the anaesthetic. He had advised her to disregard anything morbid Helen might say.\n\n'Don't be silly,' Georgia forced a laugh. 'You'll feel better in a day or two. The doctors know better than you do. Trust them Helen.'\n\n'Easy to say,' Helen muttered.\n\n'Sister said they'll be sending you off to a convalescent home after here. That will be like a holiday. I wish I could have a few weeks lying around.'\n\n'You'd make a worse patient than me,' Helen said tartly. 'And don't try to patronize me Georgia, I've lived with pain most of my life and when I say this is terrible, you can believe it.'\n\nEven though Helen never complained, Georgia was sure she was feeling sorry for herself.\n\n'If you want to be like that I'll go home,' Georgia said, convinced she could shame her friend out of it. 'I came to see how you were and spend time with you. Not for you to jump on every word I say.'\n\n'Please go,' Helen said, tears spilling down her cheeks. 'I expect I'll feel better in the next day or two, but right now I'm not fit company for a dog.'\n\n'I can't go,' Georgia felt tears pricking her eyelids. 'Not when you're like this.'\n\nHelen sighed deeply, forcing a watery smile.\n\n'Come back tomorrow, Georgia,' she turned her head into the pillow.\n\nGeorgia bent over the bed and kissed Helen on the cheek.\n\n'I love you,' she said softly. 'You're like a sister to me. Don't think you can get rid of me so you can feel all alone, because I'll be back tomorrow and the next day.'\n\nShe turned and walked quickly away, tears coursing down her cheeks.\n\nIt was only when she was downstairs in the street that she remembered Peter's words to her after the rape. 'I'll come day after day, till you're sick of me.' He hadn't got the chance to carry out his threat, she'd gone almost before he got home, and when she'd looked him up, he had someone else.\n\nOn both Monday and Tuesday evening Georgia went straight to the hospital from work. Helen was still very low and in a great deal of pain. Although she was pleased to see Georgia she was apathetic and touchy.\n\n'Don't think you've got to come here every night,' she said, trying hard to smile. 'I know you must be tired and hungry.'\n\n'But I like coming.'\n\nThat wasn't true, she hated it. The smell of the hospital made her queasy. Drips, oxygen masks, bedpans and syringes all hinted at things she didn't want to understand. She couldn't think of anything to say and even though she loved Helen, she couldn't bear to see her in pain.\n\n'Go on home,' Helen said, turning her face away. 'I want to sleep anyway.'\n\nBert was just about to close up the caf\u00e9 as Georgia got back, on an impulse Georgia put her head round the door.\n\n'Come and see the room,' she begged him. 'Helen's been a real grouch. I'll go mad if I don't talk to someone.'\n\n'Okay,' he smiled, guessing her request had more to do with worry about Helen than wanting to show her room off. 'Put the kettle on, we'll be up in a jiffy!'\n\n'Blimey ducks, what a difference!' Babs gasped as she came in, quickly followed by Bert.\n\nBig was the only way to describe her, wide hips, sagging bosom, hands and feet. Even her features were big, from her sharp eyes, her nose and a sloppy, shapeless, humorous mouth. If Georgia's features had been carved with a scalpel, Babs's had been shaped by a trowel. Yet it was an interesting, mobile face for all that, and her clothes enhanced her slovenly, yet colourful image.\n\nToday, she wore a red jumper and a bright blue skirt, topped with a washed-out yellow pinny. Thick stockings with a hole hastily botched together, and a fraying wisp of pink petticoat trailed behind her.\n\n'What a little palace ducks. Don't seem like the same room do it? Helen's going to be knocked out.'\n\nBabs stood still, hands on her ample hips, her sharp eyes taking in every last detail.\n\n'She's done this for Helen,' she thought. 'And I was the one who thought she'd be trouble when she turned up out of nowhere.'\n\n'You've done great,' Bert cast his eyes round the room as if hoping to find something to criticise. 'Not bad for a little 'un.'\n\n'If the singing don't work you could always take up decorating,' Babs chuckled. She put one big hand on Georgia's shoulder. 'Now tell us. 'Ow is she?'\n\n'Still very poorly,' Georgia's face fell, a doleful look back in the big dark eyes. 'I just wish she'd look on the bright side of things. They say everything is healing well, but I don't think she believes them.'\n\n'We was going up there Friday,' Bert said. 'I thought she was a fighter. She was always chirpy before, no matter what. If only she 'ad some family.'\n\n'She 'as,' Babs said rather sharply. 'Everyone up the market cares, and she's got Georgia. I'll 'ave to tell her a few 'ome truths.'\n\n'Don't be sharp with her,' Georgia turned to Babs, surprised by her tone. Babs was a mother figure to everyone. 'We just have to love her out of it. The thing I'm most worried about is that she won't be able to come to the Acropolis to see me. Not for me,' she added quickly. 'But she was dead set on having a lovely new dress and everything.'\n\n'Well we'll just 'ave to jog her memory,' Bert said, gazing appreciatively round the room. 'Maybe if she's got some goal in her mind she'll buck up.'\n\nGeorgia made them both some tea.\n\n'Do you mind if I paint the wardrobe and stuff?' she asked. She wanted something more to fill up the empty hours till Helen came home. 'They look a bit scruffy now.'\n\n'Course you can love,' Bert sank into one of the armchairs and winced as a spring shot up into his behind. 'I think we can find a better couple of chairs an' all.'\n\n'Really?' Georgia leant over him and kissed him on the cheek impulsively.\n\nHe smiled across at his wife. 'Those green 'uns would look a treat in 'ere wouldn't they?'\n\nBabs laughed, her wide mouth showing blackened teeth. 'I should take lessons from you in 'ow to get round my old man.'\n\nHelen was propped up in bed on Thursday night when Georgia went in as usual. A book open on the sheet in front of her and a huge basket of flowers by her bed from the stallholders in the market. Dozens of cards were propped up everywhere.\n\nOnce again she reminded Georgia of those ladies in old paintings. Her hair showering over the shoulders of the white, almost Victorian nightdress. Her eyes were still listless, but there was a faint hint of pink in her cheeks.\n\n'I'm never going to walk again,' she said gloomily.\n\n'Who said so?' Georgia gasped.\n\n'No one. I just know. They all feel sorry for me but it doesn't help.'\n\nGeorgia was torn two ways. Although Helen looked small and vulnerable in the big bed she knew her friend was tough. Should she sympathize and continue to let her wallow in self pity? Or should she be brutal to make her snap out of it?\n\n'There's only one person who can make you walk again and that's you.'\n\nThe moment the sharp words were out she felt a deep shame, but it was too late to retract them.\n\n'You think you are so bloody clever, don't you?' Helen sniffed. 'Everything you want comes to you. I bet you're glad I'm in here, I expect you have friends round every night, glad I'm not around to interfere.'\n\n'Oh yes, I'm having a ball,' Georgia shot back. 'I come here straight from work, tired and hungry and go back to an empty room alone.'\n\n'Don't give me that,' Helen pursed up her small mouth. 'I'm not stupid, even if I'm crippled, every night you've been here you can't wait to get out!'\n\nGeorgia just stared at Helen in shock.\n\n'That's not true,' she said weakly. 'But if you're going to be like that, I will go.'\n\n'Go on then,' Helen's cheeks were flushed. 'Go down and see Janet, she makes you laugh. Ask her to take you down the strip club.'\n\n'All right, I will,' Georgia turned away from the bed, colour draining from her cheeks. 'If you think that little of me, then I won't come back until you ask me to. Goodbye!'\n\nShe had made some curtains during the day at work when Pop wasn't watching. They were only cheap dress cotton but colourful and bright. She had been looking forward to hanging them, but now she felt resentful and bitter.\n\n'I don't know why I'm bothering,' she said, her voice echoing around the room. 'Heaven knows I'm doing all I can for her.'\n\nShe hung them anyway, tidied up and then crawled into bed feeling depressed and guilty.\n\n'I shouldn't have walked out like that,' she said to herself. 'I should have told her what I had been doing.'\n\nAs Bert and Babs were going to see Helen the next evening she went home instead with Janet. She'd had enough of her own company and a bit of laughter seemed the perfect antidote.\n\n'You did the right thing,' Janet reassured her. 'Leave her be till tomorrow, she'll have come round then. Poor kid, I expect she's feeling as bad about it as you.'\n\nIt was after eleven when she walked home, still chuckling to herself about Janet's family, when she spotted a policeman knocking on the door where she lived.\n\nHer first reaction was to run. For the last year police had been her biggest fear and she still hadn't quite got over it.\n\nBut reason got the better of her and she crossed over to where he stood, one hand on the bell.\n\nHe was young, male, surely she could think of some way to wriggle out of any trouble.\n\n'Can I help?' she asked, smiling brightly up into his lean face, batting her eyelashes when she realized he was nice-looking in a rather severe way. 'I live here.'\n\n'I'm trying to find Georgia James,' he said, blushing a little under her scrutiny.\n\n'That's me,' she said taking out her key.\n\n'I've been asked to take you to the Middlesex Hospital,' he said. 'You have a friend \u2013'\n\n'Helen?' she interrupted him, no longer caring if he looked like a film star. She turned pale under the street light. 'Has she?' she paused unable to say the word.\n\n'She's very ill,' he said gently, his brown eyes grave as he removed his helmet. 'She's been asking for you. Can you come now?'\n\nHe bundled her into a car and drove off at speed.\n\nGeorgia prayed silently as the car sped along the near empty streets.\n\n'What time were you called?' she asked him.\n\n'Around eight thirty,' he said, glancing across at her, seeing the big tears squeezing out from under her long lashes. 'It seems two other people had visited her earlier. She was poorly then but got worse after they left. I've been all round the neighbourhood trying to find you.'\n\n'It's the first night I've been out other than to visit her.' Georgia was crying now. 'I was nasty to her yesterday too. Oh God, please don't let her die!'\n\n'Now calm down,' he said soothingly. 'People often have ups and downs after ops, it doesn't necessarily mean she'll die. Besides you must go in there with a confident face, be strong for her. Is it the little redhead with the bad leg?'\n\n'Yes, do you know her?' Somehow it was comforting to think this young policeman knew about her.\n\n'Only by sight,' he said softly. 'Plucky little thing, always smiling even when she was frozen solid on that stall. Give her my regards won't you?'\n\nGeorgia ran like the wind once inside the hospital. Up the stairs two at a time, her hair streaming out behind her like a black flag.\n\nIt was quiet, the corridors deserted. The lights turned down for the night, bathed in a soft glow. There was none of the hustle and bustle of the day, just the odd clang of a bedpan on the sluice, and the tip-tapping sound of nurses' shoes on the polished floor.\n\n'Sister!' she called as she turned the corner and saw a familiar back view.\n\nSister Hall turned at her name, and came quickly towards Georgia, her hands outstretched. The thin tall woman had concern in every line of her body.\n\n'Oh Georgia, I'm so glad they could find you,' she said breathlessly. 'Helen's very sick I'm afraid.'\n\n'Why?' Georgia's eyes grew huge with fright, filling up with tears. 'She was doing so well.'\n\nThe sister smoothed down her apron, her lips quivering as if unable to find the right words.\n\n'We discovered an infection had set in earlier today, she was a bit feverish and we gave her more antibiotics. But her heart is weak too. We knew this before we operated and she fully understood the risk. Now it's her heart which is giving up.'\n\n'Is she going to die?'\n\nSister put one hand on Georgia's shoulder. Her expression saying it all.\n\n'I'm afraid so,' she whispered, he eyes glinting with tears.\n\nGeorgia just stared at the Sister.\n\n'But she can't,' she said. 'I've painted our room and everything.'\n\nSister half smiled.\n\n'I wish everyone could be cured with something so simple,' she said softly. 'Helen's a very brave girl. She has moaned to you because you were the only one she had. Go on in there now and try to comfort her. Tell her how much you care.'\n\nGeorgia had no experience with death. She stood for a moment trying to collect herself. She had faced the fact that Helen could possibly be left more crippled than she was before, but never had the possibility of her dying entered her head.\n\nEverything about the hospital seemed strange and dreamlike. The silent, empty corridors, the yellowish night lights, the smell of antiseptic. She blinked, hoping she was merely dreaming it, and that any moment she would find herself back in bed.\n\nHelen had been moved to a small room at the end of the ward, partitioned off with glass, curtains all round.\n\nShe was very still, ghostly pale. Her hair cascaded over the pillow like molten lava, her eyes closed, golden lashes lying on her cheeks. Arms as thin as sticks on the sheet, the long slender fingers which normally moved constantly, still and white.\n\nHer mouth looked like a young child's, so small and innocent, perfectly shaped, lips slightly parted.\n\nGeorgia crept to the side of the bed and leaned over her friend.\n\n'Helen,' she said softly.\n\nHer eyes opened.\n\n'Georgia,' she said weakly, struggling to move.\n\n'Stay where you are,' Georgia put one restraining hand on Helen's shoulder, feeling only bone under the white nightdress. She touched Helen's face lightly with one gentle caress. 'What sort of time is this to want to see a friend?'\n\nHelen's lips moved faintly in a flicker of amusement.\n\n'Did you get the curtains up?'\n\n'Yes,' Georgia took a deep breath, glad that Helen had led her into something other than her illness. 'You wait till you see our room. It's like a palace now. White walls, yellow door. I've even painted all the furniture yellow. It's like being in permanent sunshine.'\n\nFor a second Helen didn't answer. Her green eyes studied Georgia's face, as if she were trying hard to memorize it.\n\n'You've been like sunshine to me ever since I met you,' she said at length, her voice faint and breathless.\n\n'And you're my best and dearest friend,' Georgia said, taking Helen's small hand and holding it to her lips. 'I love you. I didn't mean what I said yesterday. I only didn't come because Bert and Babs were.'\n\n'I know that,' Helen turned her head slightly and looked hard at Georgia. 'They told me about the room. How you did it for me. I felt so ashamed at what I said about you having friends round there.'\n\n'It didn't matter,' Georgia felt tears threatening to run down her cheeks. 'You'll see it soon, I'll teach you to dance and we'll find ourselves a couple of rich men to take us out.'\n\n'You do it all for me,' Helen's voice was almost a whisper. 'Become a big star in the West End, wear lovely clothes and have hundreds of admirers. I'll be watching you.'\n\n'You'll do it with me!' Georgia tried to sound bossy and hard but it came out like a plea.\n\n'I'm dying, Georgia,' Helen spoke softly, one thin white hand reaching up to touch her friend's face, her expression one of tenderness. 'I knew I would after the op, it was all a dream, a lovely dream of dancing and being like you. But you must carry on, make my dream a reality.'\n\n'I can't without you,' Georgia's tears couldn't stop now. 'You are my family, everything.'\n\n'Don't cry for me,' Helen's eyes brimmed over. 'I'm not scared or anything now, I feel peaceful and content. I'm too tired of struggling, I'm happy to be going to a place where there's no pain, no striving for anything.'\n\n'But how will I manage without you? You are my only friend.'\n\n'You'll make new ones. Girls who won't hold you back like I would.'\n\n'You wouldn't hold me back,' Georgia pleaded with her, clutching Helen's hand and kissing the palm.\n\n'It's the way it has to be,' Helen's eyes seemed like emeralds, set on white velvet. 'Just think of me going somewhere good. Everything I have is yours. In the bottom of the wardrobe is a box. There's something special for you in there.'\n\nShe made a choking sound in her throat, a flicker of pain passed across her face, her eyes closed.\n\nGeorgia put her finger on the bell. She could hear her own heart pounding with terror, but she was afraid to leave Helen even to summon help.\n\nSister came running in.\n\nShe moved round to the other side of the bed and felt Helen's pulse, her eyes meeting Georgia's tear-filled ones across the tiny redhead.\n\nHelen opened her eyes again slowly.\n\n'Sing to me?'\n\nHelen looked at Sister with wide and troubled eyes.\n\nSister nodded.\n\n'I can't, not in here,' she whispered.\n\n'Please,' Helen's eyes pleaded. 'Just let me hear you one more time?'\n\nIt was so quiet in the small room. Georgia was aware of other patients sleeping just the other side of the partition. It seemed all wrong, to sing while her dearest friend slipped away.\n\n'Sing, Georgia,' Sister Hall whispered. 'Forget where you are.'\n\nGeorgia took a deep breath to calm herself.\n\n'Summertime. When the living is easy.' Georgia's rich contralto voice rang out around the small room.\n\n'The fish are jumping and the cotton is high. Your pa is rich and mama's good lookin'.' She looked down at Helen, her eyes were still glowing, yet they appeared to be getting dimmer.\n\n'So hush little baby don't ya cry.'\n\nHelen's face was at peace again, her eyes on Georgia, drinking in the words and the music.\n\n'One of these days, you're gonna rise up singin', you're gonna spread your wings and fly to the sky. One of these days you gonna rise up singing. So hush little baby, don't ya cry.'\n\nGeorgia felt something in the hand in hers. Not a movement or even a flicker, but she knew without being told Helen had gone.\n\n'No,' she cried out, leaning over to kiss her friend.\n\n'She can't,' she looked up in anguish at the Sister.\n\nSister silently closed Helen's eyes and put her two hands together on the sheet. Then she moved round the bed to embrace Georgia.\n\n'I never saw a more beautiful and peaceful death,' she said softly, holding Georgia against her shoulder. 'Your voice and song took her where she wanted to be. Now you must be strong too. She told me about your singing, she was so very proud of you. You must let her be your inspiration. Achieve everything she wanted for you.'\n\nGeorgia refused a lift home. London's streets held no terrors for her. It wasn't strangers she had to fear. But people she knew and trusted.\n\nWell she had no one now. Her mother, Peter, and now Helen, all gone.\n\nThe dark streets and alleys reflected her feelings. Dark, desolate, empty. Shafts of light here and there from street lamps splayed out an arc of gold light over a small area.\n\nHer life was just like that. A patch of light and happiness, only to go a little further and she was back in the darkness, alone again.\n\nIf she had known the risk Helen was taking she would have prevented it somehow. How could Helen have been so foolish if she knew her heart was weak too?\n\nWearily she climbed the stairs and opened the door of her room.\n\nAs she switched on the light, the first thing she saw was the woolly, emerald green shawl Helen put round her shoulders on cold nights.\n\nShe picked it up and held it to her face, drinking in the smell of Helen it carried in it, remembering the colour of her eyes.\n\nThere was no one to see her now, here alone with all the memories of Helen she could mourn her privately.\n\nNever again would she hear that familiar clonking sound on the stairs. Or wake to see Helen making tea in her long, white old-fashioned nightdress, her red hair flowing over her shoulders.\n\nSo many pictures trapped in her mind like a photograph album.\n\nHelen lying on the bed reading fashion magazines.\n\n'Do you think I could wear my hair like this?' she'd say, holding the glorious mane on top of her head. 'When my leg is normal again I won't need to hide under all this!'\n\nThere was the picture too of Helen on the market stall. Fur collar turned up on her russet coat, blowing on her fingers to warm them. Pale cheeks turned pink with the cold wind, her hair escaping in tendrils from her woolly hat as she animatedly teased the customers.\n\nThen there were the nights she didn't work at the club. Those were special nights when they would giggle and chat until the small hours, drinking tea and eating biscuits in bed and Helen would talk about the man she intended to marry.\n\n'He'll have to be big and strong. Dark hair, blue eyes, a sultry look like Elvis. But he'll adore me shamelessly and buy me beautiful clothes and expensive perfume.\n\n'We'll have two children. A boy with dark hair like him and a little girl with red. We'll have a house on Hampstead Heath with a garden full of roses.'\n\nShe sank onto Helen's bed, shawl in hands, rocking to and fro with grief, tears cascading down her cheeks. A bellow of rage started within her, filling the room with a terrible sound.\n\nHelen would never find that perfect man, run, dance or make love. She would never know the bliss of a man's arms around her who loved her, or have those children she'd longed for.\n\nIt was after three when Georgia finally crawled into bed, her eyes swollen, her heart numb with grief.\n\n##### *\n\n'Georgia!' she could hear Babs knocking on her door and calling. 'Georgia. Pop's been on the phone and wants to know if you're all right.'\n\n'Come in,' Georgia called out weakly. 'It's not locked.'\n\n'What is it ducks?' Babs came bustling in bringing a smell of fried bacon with her. 'Are you ill?'\n\n'Helen died last night.'\n\nBabs stopped short for a moment, her big mouth dropping open. 'But we only left her at seven!' Her work-reddened hands twitched at the stained apron, her mouth sagged as if suddenly all her teeth were gone.\n\nGeorgia had forgotten how long Helen had lived in Bert and Babs's room, and that they thought of her as a daughter. Until now she had been able to feel only her own grief, forgetting she didn't have the monopoly in loving Helen.\n\nBabs's face crumpled, tears welled up in her eyes, hands moving up to conceal them, shoulders heaving.\n\nSo often Helen and Georgia had poked fun at Babs. They laughed about the rag-bag collection of clothes she wore, the way her hair never looked clean, and the missing tooth which gave her a curious wobbly smile.\n\nMaybe she wasn't one of the world's great beauties, but now as Georgia saw her grief, she felt ashamed.\n\n'Her heart was weak, it just gave out. She had an infection in her leg too. She died just after midnight.'\n\nBabs's lips shook. She rolled her eyes up to the ceiling.\n\n'Why take 'er?' she said angrily as if addressing God personally.\n\n''Ow did you find out?' Babs whispered, creeping nearer to Georgia across the dim room, reaching out for her hand like a life raft.\n\n'I was with her, a policeman came here for me about eleven.'\n\n'Oh, you poor love,' now she enfolded Georgia in her arms, rocking her to and fro. 'I know 'ow close you were. We told 'er what you was doing. She was so ashamed she'd thought the worst of you.'\n\n'She was so brave,' Georgia buried her face in Babs's big, soft chest, once again starting to cry. 'She said she didn't mind dying and that I had to do everything we'd talked about for her.'\n\n'Then you must my darlin'.' Babs used that persuasive tone like Celia had, lifting her face up and dabbing at it with the corner of her apron. 'But you've gotta get up and go to work. Sooner or later you'll 'ave to.'\n\n'I know,' Georgia buried her head in Babs's shoulder. 'But I loved her, she was my friend, sister and mother all rolled into one. I don't even know what I should do about her funeral, or anything.'\n\n'Bert and me'll take care of that,' Babs said gently. Her voice, usually so loud, was hardly above a whisper. 'She was our Helen an' all.'\n\n'She said I was to have everything of hers.' Georgia raised a tear-stained face. 'She said there was a box in the wardrobe.'\n\n''Ave you looked in it?' Babs wiped at her own face with a corner of her apron.\n\n'No,' Georgia sniffed. 'I'm kind of afraid to.'\n\n'Well, let's do it together,' Babs said. 'Come on!'\n\nGeorgia got out of bed reluctantly. The mirror on the wardrobe door reflected back a sad waif of a girl. Tangled dark hair, red-rimmed eyes, wearing pyjamas that she'd outgrown several months before.\n\nShe found the chocolate box decorated with faded purple velvet flowers, tucked away under a pile of old jumpers.\n\nGeorgia put it on the table and drew back the curtains.\n\nBabs was lighting the fire.\n\n'Jesus it's cold in 'ere,' Babs said coming back to the table, slapping her raw hands together, eyes glistening with tears. 'Come on, get the lid off!'\n\nThere was a letter on the top addressed to Georgia.\n\nGeorgia picked it up and looked questioningly at Babs.\n\n'She went prepared, I'd say,' Babs smiled affectionately. 'But she was always one to think everything out.'\n\nUnder the letter were three bundles of notes with rubber bands round them.\n\n'There's about sixty quid 'ere,' Babs said in surprise, flicking through the pound and ten shilling notes.\n\n'There's more at the bottom, just change,' Georgia said pulling out handfuls of halfcrowns.\n\n'Read the letter,' Babs urged.\n\nGeorgia opened it, her hands shaking.\n\n'Dearest Georgia,\n\n'I feel a bit silly writing this, I keep hoping that when I get out of hospital I can get this out and we both can have a good laugh about it.\n\n'But I've had a feeling for quite some time that I might die. I've got a weak heart and I've been told about the risks. Anyway, I wanted to tell you, just in case I never got a chance to say it to your face, how much I loved you and how happy you made my last year.'\n\nGeorgia's eyes misted over, for a moment she couldn't see to read further.\n\n'I never had a family, or a close friend. I'd got so used to being on my own that I didn't even try to make friends anymore. But then you came along, filled up my life with your presence and suddenly I felt wanted and needed.\n\n'If I am dead when you read this I hope you'll be strong and not brood about me or feel guilty in any way.\n\n'Without this last year with you I would have died a lonely person. You enriched my life, you gave me laughter, the joy of sharing and most of all you gave me hope.\n\n'Carry on with your singing, fill the world with your beautiful voice. I'll be watching over you forever now. Watching to see you don't get tempted into bad things, or mix with evil people. I saw a lot of bad things while I worked at the club, gangsters, thieves, drugs and all sorts. Please be careful, don't be too trusting and watch out for men who will try to use you.\n\n'All the money in here is for you. Buy a beautiful dress for your special night, something red, sparkly and flashy, the kind of dress we planned to wear when we were rich. You are probably amazed that I had so much. I had to be really careful when I first came to London and somehow I never got out of the habit.\n\n'Give Bert and Babs my love, thank them for all they did for me. I thought of them as my parents. I wish them a long and happy life.\n\n'Don't cry for me Georgia. I'm happy now and I hope you get everything you've dreamed of.\n\nMy love always,\n\nHelen.'\n\nGeorgia read the letter and handed it to Babs.\n\nUnder the loose change were several photographs. Most were of Helen when she was a child, small, crumpled pictures of a painfully thin child with a mass of hair. There was one professional studio picture, taken before she left the home in Plymouth, her hair up, wearing a print dress with a large white lace collar. With this was a snap taken at Christmas of Helen and Georgia in the market, their arms around each other, laughing.\n\nGeorgia remembered the stallholder taking it, but she had never seen the snap until now. She held it out to Babs silently.\n\n'Fancy her sitting and writing all this before she went.' Babs was sobbing now, great tears rolling down her cheeks. She took the picture and looked at it, lifting it to her lips.\n\n'I'm glad you've got this, it's something to really remember her by, you both look so 'appy,' her voice shook and her lips trembled.\n\n'I'll never forget her anyway,' Georgia said brokenly. She scooped up the pictures and the money and put them carefully back. 'Now what shall I do with all this?'\n\n'Exactly what she said,' Babs said firmly. 'She wanted you all dolled-up for that night. You can't disappoint her.'\n\n## Chapter 9\n\nA rush of blood to his head, a tingling down his spine made Maxwell Menzies sit up sharply as the girl's voice soared out across the Acropolis.\n\nHe hadn't wanted to come tonight. Greek family parties were almost as boring as Jewish ones. But Andreous was his brother-in-law, and Miriam had insisted.\n\n'Summertime!'\n\nThe last time he'd heard that song was in New Orleans back in '58 and he defied anyone to surpass the fourteen-stone Negress who had turned his legs to jelly. This girl came close though, she might be young and slender, but her voice had the same depth and passion, and she was better to look at!\n\nMax forgot his drink and Miriam's tiresome family and friends all around him, all he could see and hear was the girl in front of him.\n\nBlack curls piled upon her head, fixed with glittery combs to match the long, spangled red dress. A hint of small brown breasts nestling below the plunging neckline, and when she turned, a deep 'V' of naked brown skin made his heart thump. But it was her eyes which held him, so sad and huge, at times glistening with tears.\n\n'This is it, Maxy,' he said to himself. 'You've found the crock of gold.'\n\n##### *\n\nFew men stood out in a crowd as Max did. It was not merely his rugged tanned face, height, wide shoulders and expensive clothes, but the sheer force of his personality.\n\nAs a young man he had flirted with boxing, but he was shrewd enough to know it would get him nothing other than cauliflower ears and a broken nose. He may not have won any titles, but it had left enough of a legend to intimidate his adversaries.\n\nA burst of applause broke round Max as the first number ended.\n\n'That was lovely,' Miriam put one plump, ring-laden hand on his. 'Fancy her just working in Pop's workshop!'\n\nMax glanced across at Pop. The man's eyes were glued to the stage, a smug look of satisfaction on his usually lugubrious features. Max had barely acknowledged this invitation when it arrived several weeks earlier, much less listened to Miriam rattling on about how excited Pop and Andreous were about this girl's voice, but now he wished he'd been attentive.\n\nShe was now singing the Everly Brothers' hit, 'Till I kissed you'. It was slower than their version, plucking at his emotions in a way the original never had and as her body swayed with the music, Max found himself slipping into a dream.\n\nThe London Palladium, then Vegas and Hollywood. He could see himself in a box looking down at her, hear the applause, see the sparkle of diamonds, smell money. The time was right. American men had dominated the charts for too long. This stunning girl with her powerful voice could be the one to change everything.\n\nAcross the same table, Pop too was struggling with his emotions, but unlike Max he had no thoughts of money or power.\n\nHelen's death sent shockwaves throughout the market. He'd seen Georgia bent to the point of breaking with grief. He hadn't heard her laugh in weeks, her face grey with pain, every line in her body showed the depths of her feeling. Yet somehow she'd found the guts to go on and rehearse.\n\nRight up to the moment she walked on to the stage, Pop had expected her to falter. Yet she'd picked up the microphone as if she was born to it, nodded to the band and straight into 'Summertime' as if she was merely in the workroom. Georgia had more sides than a threepenny bit. But until tonight he hadn't seen this adult and desirable woman.\n\nHer slender body moved sensuously to the beat, eyes flashing, hips undulating in her clingy dress. He couldn't help but wonder if his wife would be quite so maternal to Georgia in future.\n\nGlancing sideways, he saw Christina was as engrossed as the entire audience, one foot tapping, forgetting even her drink in front of her.\n\nIt was the last number of the first set. A Peggy Lee number she loved. 'Fever'.\n\nGeorgia hadn't copied the original version. Her voice was sweeter, more melodic and she sang it with just enough humour and pace to carry it off magnificently.\n\nThe audience applauded wildly.\n\nShe was just Georgia again, grinning as confidently as she did in the market. Eyes shining, beads of sweat glistening on that small brown forehead.\n\nThen as if remembering just where she was, she bowed deeply, and ran off stage.\n\nThe girls from the workshop were practically jumping out of their seats, yelling and stamping their feet, quite forgetting where they were.\n\nJanet had pulled out all the stops tonight. A long black dress, glittery earrings, like an actress at her premi\u00e8re.\n\nShe wiped a tear of pride from her cheeks and grinned at the other girls.\n\n'Our little Georgia,' she sniffed. 'I'm so bleedin' proud of her you'd think I trained her!'\n\n'Now that girl's got class. Don't you think so darling?' Max turned to his wife.\n\n'Oh yes,' Miriam gushed, encouraged by his dark eyes studying her so attentively. 'Couldn't you manage her?'\n\n'That's jumping the gun a bit,' he stroked her plump arm softly. 'She might be able to sing amongst her friends, but she's still a bit raw and we don't know anything about her.'\n\nWhen he married Miriam she had been as slim as Georgia, shiny dark hair with eyes to match. But like most Greek women she had turned to fat, almost as soon as the honeymoon was over. Her place was in the home, a mother and housewife, but tonight she had another purpose.\n\n'Talk to her darling. You know how I trust your judgement about these things!'\n\n'Oh, Maxy,' she cooed. 'What a sweet thing to say!'\n\nMax picked up his glass of brandy and downed it in one.\n\nMiriam could be so simple. Like a trusty dog, wagging its tail after a few kind words, forgetting how often her master stayed away from home and the women his name was linked with.\n\nShe knew how to make the best of herself though, despite her weight. Once her dark hair became grey she dyed it a dark auburn, an elaborate 'beehive' style which lengthened her round face and showed off her jewellery to advantage. The black dress she wore tonight was cleverly cut like all her expensive clothes, drawing the eye to her good points and camouflaging the bad. A low neck revealing her smooth olive shoulders and cleavage, sheer chiffon sleeves and draped empire line, hid away the damage a life of ease and plenty had caused.\n\n'She could be the one to change our lives,' Max whispered, running one finger down her arm sensuously. 'But we don't want to get into anything blind.'\n\nMiriam glowed at his words. He had sulked all day about coming. The best she had expected was for him to be pleasant to her family, then insist they leave before the party even got going.\n\nShe was only thirteen when her family left Greece and opened the restaurant in Greek Street. Her parents had led her to believe London would be wonderful, but all she saw in her teenage years was drudgery. Up to her elbows in greasy dishes, dark dreary rooms and a school where the other girls laughed at her accent. Pop and Andreous both worked as waiters then, and she knew her father had singled out Andreous for her.\n\nBack in Greece she would have welcomed the handsome young man with his soft eyes and ready laughter, but in London she saw them as a trap.\n\nAt seventeen she met Max and suddenly her mind was made up. Here was a man who wanted more out of life than waiting on tables. Little Ruth her younger sister could have Andreous.\n\nMax was a theatrical agent then. Even in his early twenties he showed signs of what was to come. Shrewd, manipulative, with an eye for the main chance. Exciting, not just the way he made her disobey her parents and slip out to meet him, but the way things happened around him.\n\nFaced with the risk of his oldest daughter bringing shame to their family, her father finally agreed to the wedding. Andreous married Ruth and when her father died, he left them the restaurant, and Miriam a few hundred pounds.\n\nAndreous turned the restaurant into a club and barely scraped a living out of it. But Max used Miriam's money to launch himself into the music world. He went to America and found talent, bringing them back to England and putting them on the road.\n\nAt twenty-three Max had been lean and hungry looking. But twenty years on, money, another stone in weight, he was in his prime. Sensuous, hooded eyes and fleshy lips promised passion. Thick black hair streaked with grey, detracted the eye from his Roman nose. A man who knew the effect he had on women and used it shamelessly. The handmade silk shirts, Italian shoes and his ostentatious jewellery were unnecessary adornments. No one ever forgot Max Menzies.\n\nBy the second set Georgia was getting into her stride. Her voice had a new maturity and range. Notes of caressing sweetness, mingled with shots of raw emotion that kept her audience spellbound.\n\n'When I fall in love', had them wiping a tear from their eyes. 'The Locomotion' made their feet itch to dance, and finally when she burst into 'Wonderful World' she wrenched the last dregs of emotion from everyone.\n\nAndreous was more than happy. His sophisticated and cool customers who rarely even acknowledged the entertainment were standing up to clap and cheer this young girl.\n\nTo think he had expected the evening to do little more than break even! They were all ordering drinks like there was no tomorrow, all because this young girl's voice had touched a sensitive spot.\n\nGeorgia mopped the sweat off her face and shoulders in the dressing room and turned to the men in the quartet.\n\n'Thank you all so much,' she said simply. They were packing up their instruments, obviously in a hurry to leave. 'I couldn't have got through it without you.'\n\nShe saw the exchanged glances. Four middle-aged men who'd seen everything and got credit for nothing.\n\n'You've got what it takes,' Jack the pianist smiled. 'But a word of warning. There's a lot of sharks out there. Watch what you're doing.'\n\n'Yes sir!' she grinned. 'But tell me how do I know the sharks from the nice fish?'\n\n'Extravagant promises,' he turned away to pack up his music. 'Now go on, be off with you and enjoy yourself.'\n\n'I hope we get to be together again,' Georgia couldn't find the right words to explain how much she owed her success to them.\n\n'So do we,' Jack touched her shoulder. 'Tonight was a real pleasure.'\n\nGreek music wafted round the club. One wall was covered in a mural of a Greek town, white houses like sugar cubes underneath a turquoise sky. Candles dripping on to bottles on every table, the vine-covered pergola round the bar all gave an impression they were far away from London.\n\nJanet and Sally were already up on the dance floor joining two snake-hipped waiters in Greek dancing. Georgia waved to them, then went over to join Pop, Andreous and their families.\n\n'This is my sister-in-law Miriam,' Andreous said after he'd hugged her and introduced his teenage sons, and Pop's five daughters.\n\n'I thought you were marvellous,' Miriam said, her double chin quivering with emotion. 'I had to find a hanky at the last song. You must tell me all about yourself.'\n\nGeorgia liked Miriam. Warmth and understanding flowed out of her, reminding her just a little of her own mother. Her dark eyes twinkled, her jewellery was dazzling and before Georgia knew it she was telling her all about her room, Helen and her hopes to be a professional singer.\n\n'You'll get there,' Miriam nodded. 'You've got all the right qualities. Your mother must be so proud of you Georgia, is she here tonight?'\n\n'My parents are dead.'\n\nShe didn't know why she said that. Was it pique because she'd had no word from Peter or Celia? Or protecting herself from the past?\n\n'I'm sorry darling,' Miriam leaned closer to her drowning her in expensive perfume. 'You are very young to be alone.'\n\nBut even reminders about being alone couldn't hurt her tonight. A photographer took her picture, men in evening dress kept coming up to kiss her. Women in beautiful dresses stopped by to compliment her. Finally the grey miserable feeling she'd had since Helen's death was actually fading.\n\n'You were very good.' A deep voice behind her made her turn and look up.\n\nFor a moment Georgia couldn't reply.\n\nHe was the most handsome man she'd ever seen. Making her think of thirties filmstars. His dinner-jacket fitted perfectly, his pin-tucked shirt like the kind she'd seen in glossy magazines. Pop had a clip-on bow tie, but Max's was the real thing. He was a human black panther, sleek, just a little flashy and perhaps dangerous.\n\n'Thank you,' she blushed.\n\n'This is my husband, Max.' She heard Miriam speak but could barely tear her eyes from his face.\n\n'May I?' He pulled up a chair, forcing his wife to move along. 'I expect someone will grab you any moment for a dance, but I just wanted to say how much I enjoyed your performance.'\n\nHe talked about everything and yet nothing, his hooded eyes never leaving her face. The only clear thing she remembered later was when he spoke of managing bands and artists.\n\nIt was obvious he made money out of it, just one look at the gold watch on his wrist, the shirt and suit said it all.\n\n'Come and dance?' he said later, holding out a hand.\n\n'I hope Miriam doesn't mind,' Georgia said as he scooped her into his arms. She felt uncomfortably aware of his big hard body against hers and she could see his wife watching them.\n\n'She's delighted to have a chat with Andreous and Ruth', he smiled down at her, his big hand lightly touching the naked skin on her back, sending shivers down her spine. 'They all gabble away in Greek the moment I'm out of earshot. Anyway you're the star tonight.'\n\n'I like Miriam,' she said weakly.\n\n'And so do I,' he smiled, showing flashing white teeth. 'We've been married for twenty years. She doesn't have to worry when I dance with a beautiful girl.'\n\n'Do you think the songs I picked were right for me?' she asked. She was very aware of the touch of his hand, his chin brushing her hair. She didn't want to be so aware of him, she wanted him to talk about her future.\n\n'I'd like to hear you sing some good soul music,' he said, smiling down at her. 'More up-to-date stuff. But only you know the sort of music that inspires you. You are the artist, not me.'\n\nIt was after four when Georgia finally got home.\n\nAs she unzipped the red dress and hung it up she felt a sudden pang of fear. Supposing nothing did come of tonight? How could she spend another year making dresses?\n\nThen there was Max. He'd been watching her with more than just passing interest. What she couldn't gauge was if that interest was professional, or personal.\n\nShe knew now with utter certainty that she was born to sing. If only her feelings about Max were as clean cut!\n\n'It's Miriam Menzies on the phone,' Pop shouted out into the workroom. 'Come and take it in my office.'\n\nTwo weeks had passed since the night at the Acropolis. A small photograph of her in a music paper and then nothing. Night after night spent alone in her room staring at the red spangly dress, watching her dream fade.\n\nGeorgia came back into the workshop looking stunned.\n\n'What did she want?' Janet switched off her machine and the others all followed suit.\n\n'She's invited me to go shopping with her on Saturday and lunch afterwards.'\n\n'Beats the casting couch routine,' Sally sniggered, leaning forward to listen.\n\n'But why?' Georgia turned puzzled eyes on Janet. 'Does this mean something?'\n\nJanet shrugged.\n\n'She's softening you up,' Sally smirked.\n\n'For Max?' Janet looked round sharply at her friend. 'I'd think 'e was capable of doing that 'imself. I'd be his for a gin and tonic.'\n\n'Did she mention him?' Sally asked.\n\nGeorgia shook her head.\n\n'Well girl,' Janet smiled. 'At least you'll get a free lunch. I bet that Miriam don't eat in Wimpy bars!'\n\n## Chapter 10\n\n'Almost Millionaires Row,' Georgia whispered to herself as she hurried along the tree-lined Hampstead avenue. Big detached houses set in neat gardens, gleaming cars on gravel drives. Images of au pairs, tennis clubs and holidays abroad. 'Max must be stinking rich.'\n\nThe Menzies's house was mock Georgian. Double-fronted with a glossy, navy blue door and brass coach lamps.\n\nTaking a deep breath, she walked up to the front door, the gravel scrunching under her feet, intensely aware of her shabby navy suit and run-down shoes.\n\nA girl opened the door, smiling welcomingly.\n\n'You must be Georgia?' she said. 'Come on in, Mrs Menzies won't be a moment.'\n\nShe was only a little older than Georgia, fair, with polished skin and an accent that suggested she could be Dutch or German. The plain black dress wasn't a uniform, but it had the same effect.\n\n'Take a seat,' she said, ushering Georgia into the sitting room, then disappeared down the hall to the back of the house.\n\nThe room was huge. The morning sun bathed the baby grand piano in the front window, glass doors led on to the garden at the back. The sort of room her mother would have described as 'More money than taste', but Georgia was impressed.\n\nA shaggy carpet curled round her feet. She looked around at luxurious fat couches with matching footstools, and huge exotic flower arrangements. Paintings with heavy gilt frames. A mahogany wall unit displaying family photographs encased in silver, along with a collection of porcelain figurines.\n\nThe house was silent, just the tick of a clock from the hall and birdsong from the garden. Georgia perched on the edge of a chair, taking careful note of everything, from the heavy curtains with tasselled pelmets to the imitation twin chandeliers.\n\nMiriam swept into the room wearing a blue silk dress, she looked pretty despite her bulk, and she smelled as exotic as her choice of flowers.\n\n'Hallo, dear,' she said, sitting down for a minute as she checked her bag, and patted her elaborate hairstyle. Georgia was pretty sure it wasn't her real colour, for the dark, burnished auburn didn't go with her dark eyes or eyebrows. It was swept up on to the top of her head, a mass of carefully arranged curls.\n\n'I thought we'd go to Kensington,' she said as she applied a touch more lipstick. 'How do you feel now after your debut?'\n\nHer plump hands were covered in rings, so many that Georgia wondered how she bent her fingers. Her nails were very long, painted a vivid pink to match her lips.\n\n'I'm not sure,' Georgia felt a little awkward. 'I want something more to happen I guess.'\n\n'It will,' Miriam stood up again, smoothing down her skirt, arranging a large, gold twisted chain round her plump neck. 'Max's meeting us for lunch, but let's forget that now. I love shopping, nothing better for giving one a boost.'\n\nSomehow Georgia had assumed they would go by bus or tube. She hadn't considered Miriam could drive, much less whisk her into a sleek black Rover, with seats like armchairs.\n\nMiriam attacked the shops like someone seeing food for the first time in weeks. She bought a cashmere sweater, stockings and a black cocktail dress without even checking the prices and all the time she chatted.\n\nGreece, holidays abroad, her two teenage boys. Between bursts she fired questions at Georgia and pointed out her favourite shops.\n\n'I don't want you to be offended,' Miriam said briskly as she whisked Georgia into the younger woman's department in Barkers, 'but I'm going to buy you something nice as a present.'\n\n'That's very kind of you,' Georgia almost stammered. 'But I hardly know you.'\n\n'Nonsense,' Miriam's eyes were already scanning the rails. 'I have my motives. I want Max to offer to be your manager. Now we can hardly impress him with you wearing that suit can we?'\n\nGeorgia blushed scarlet.\n\n'Don't look like that,' Miriam wheedled, putting one plump bejewelled hand under Georgia's chin and lifting it. 'You look fine for an interview for an office job, but entertainers have to sparkle. You wouldn't think twice if I was your mother.'\n\n'No, but \u2013'\n\nMiriam's nose flared.\n\n'No buts. I'm good with clothes and I know exactly what you need.' Her tone implied there was to be no further argument. 'Now a nice dress with a jacket. Something snazzy.'\n\nGeorgia watched helplessly as Miriam pulled out garment after garment. Each one of these outfits cost more than she earned in six weeks.\n\n'We want something that won't date,' Miriam pushed aside all the dresses with billowing net underskirts. 'And a jacket you can wear with other things.'\n\nThe assistant's arms were piled high. Black, red, blue and pink, stripes, spots and plain colours.\n\n'We'll try all these,' she pushed Georgia into the changing room. 'I think that black and white one is right for you, but we'll see.'\n\nTo Georgia everything was perfect. The size, the colours, but she hated Miriam watching her as she stood in her greying white bra and pants.\n\n'Put that one on again,' Miriam lifted down the one she'd mentioned earlier.\n\nIt was more sophisticated than the others. A silky low-necked dress in swirly black and white patterns, hugging Georgia's slim figure as if it was made for her. A short white jacket went over it, turning a dress that was made for dancing, into an outfit suitable for anything from church to an office.\n\nMirian dug one pink-nailed finger into her plump cheek and looked thoughtful.\n\n'That's it,' she said turning to the assistant. 'Would you mind cutting off the tags? She'll wear it now.'\n\nThe embarrassment grew as Miriam whisked her down into the shoe department and insisted she had a pair of Charles Jourdan black court shoes with two inch heels.\n\n'They're an investment,' she waved away Georgia's protest. 'If I have my way you'll be dancing all night soon. You can't do that if your feet hurt. Besides, if Max takes you on you'll soon be buying clothes much nicer than these things.'\n\nGeorgia's eyes strayed to her image as they passed a huge mirror on the way back to the ground floor.\n\nShe could easily pass for a model now. Elegant and poised. Her brown skin set off the white jacket, her hair tumbling over her shoulders stopped the outfit looking too matronly.\n\n'What do you see?' Miriam said softly at her elbow.\n\n'I look kind of glamorous,' Georgia giggled.\n\n'No, you look beautiful,' Miriam smiled. 'The glamour bit we have to work on. Your nails need shaping properly, you need lessons in make-up and grooming, but if I'm right about you, one day there'll be a million young girls copying you.'\n\nMiriam stopped by the perfume counter and sprayed her with Chanel No 5.\n\n'The finishing touch,' she explained. 'Now let's go and meet Max. Don't tell him about the dress and shoes, this is our little secret.'\n\n'Why?'\n\n'You know what men are like.' Miriam wrinkled her nose. 'He'll think I'm spoiling you.'\n\n'Is Max interested in me?' Georgia felt bold enough to ask as they crossed the busy High Street and made their way towards the parked car. Miriam puzzled her, it felt like a genuinely motherly act of kindness, yet she couldn't help suspecting a hidden motive.\n\n'I'm not quite sure,' Miriam frowned a little. 'He was impressed by your voice, but that isn't always enough.' She paused to unlock the car boot, took Georgia's bag and put it inside with her dress bag.\n\n'Let me tell you something about my Max,' she looked up at Georgia as she relocked it, pausing with one plump hand on the shiny paintwork. 'He's got a reputation as a hard man. You'll meet people that will tell you all sorts of bad things about him. They say he's cruel, unscrupulous, that he'd kill his granny for ten shillings.'\n\n'Would he?' Georgia's eyes were as big as saucers.\n\n'Maybe for a hundred pounds,' Miriam chuckled. 'But what few people will admit is, that he is the best. Maybe not the fairest, or the gentlest, but he's the best because he's tough. And if he decides he is going to make you a star, you will be.'\n\nMiriam stepped back on the pavement and linked her arm through Georgia's.\n\n'You won't be alone though,' she squeezed Georgia's arm. 'I'll be around too. Just remember I'm your friend. You can come to me if you have any problems.'\n\n'I like you,' Georgia said impulsively. She could remember her mother conspiring with her about her father just like this. Somehow it made it all seem safe.\n\n'And I like you too,' Miriam smiled.\n\nMiriam was seething with contradictory emotions. She really did like Georgia, yet still she set her up. Soothing her with maternal advice, stimulating her greed for the 'good life'. Now she was leading the girl into the final trap, and by tea-time she would be just another of her husband's assets. Max would be pleased with her, but just this once she wished she hadn't complied with his instructions.\n\nHarvey's in Church Street oozed quiet, expensive charm. Dark green paintwork with cream lace curtains on brass rails and windows as shiny as mirrors.\n\nMiriam swept in leaving Georgia to follow nervously.\n\n'Good afternoon, Mrs Menzies,' the headwaiter, a thin, whey-faced man glided towards them, a napkin over one arm. 'Mr Menzies is waiting for you.'\n\nGeorgia felt rather than saw the dark oak panelling, the cosy booths for intimate meals. All she could focus on was Max getting up from his table by the window, the width of his shoulders and a flash of white teeth.\n\nShe was out of her depth now. Foreign waiters, French menus. Even her new clothes couldn't hide the fact she had never been anywhere so grand before.\n\n'Hullo,' he said taking Georgia's hand firmly and leading her to the table as if understanding her fears. 'Has Miriam spent all my money?'\n\n'Only one little dress,' Miriam simpered, kissing him on the cheek.\n\n'Do you like chicken?' Max asked once the waiter had tucked in her chair.\n\n'Yes,' Georgia's voice was little more than a squeak. He looked even more handsome in his light grey suit and she was sure she couldn't eat a thing, much less choose.\n\n'Shall I order for you?' His big hand touched her briefly, like a secret message that he understood her fear. 'When I was your age I'd never eaten anywhere except at home.'\n\nOnce the meal arrived Georgia forgot her nervousness. Max told her a funny story about their son David getting lost in Barker's when he was small and how Miriam had wailed like a banshee until he was found, then amazed everyone by pulling down his trousers and smacking his bottom publicly.\n\n'David won't ever go in there now,' Miriam added. 'He's sure someone will remember him.'\n\nThere was something very reassuring in hearing such stories, soon both Max and Miriam were just another set of loving parents and Georgia found herself reciprocating with tales about Pop's workroom.\n\n'Miriam tells me you've no parents.' Max probed so gently she barely noticed it. 'Do you mind telling me what happened to them?'\n\n'I don't know anything about either of them,' she said. 'The records were lost, all I had was a name. Later on I was fostered.'\n\n'I wondered how you got to speak so nicely,' Max's thick lips spread into a wide flashing smile. 'Do you keep in touch?'\n\n'No. They split up after I left, it's not the same anymore.'\n\n'That's sad,' Max sighed. 'A girl of your age should have a family.'\n\n'At least it leaves me free to do as I like,' she said quickly.\n\n'And what is that?' Max enquired.\n\n'Sing again as soon as possible,' Georgia dug into an ice-cream sundae with childish relish.\n\nFor a moment or two there was silence. Georgia sensed Miriam was imploring Max with her eyes, but she hardly dared look up.\n\nDespite the couple's caring interest, Georgia knew this lunch had a real purpose. Was Max going to help her? Or had she blown it all by being too flippant?\n\nPower seeped out of him like a strange perfume. A tiny bell was ringing at the back of her head, warning her to get advice before she made, or let anyone else make any decisions. But who could she go to? Miriam had already told her he was the best!\n\n'I've been giving you a lot of thought,' Max spoke slowly, twisting his big gold ring around his carefully manicured finger. 'But one thing bothers me.'\n\n'What's that?' Georgia said eagerly, wiping her mouth on a serviette.\n\n'Your lack of experience,' he said, his thick dark eyebrows almost meeting as he frowned. 'I mean, your voice is superb, you move well and you have stage presence. But to make it in the music business you have to tour with a band putting on a complete entertainment.'\n\n'How do I find a band?' Georgia's heart sank. She didn't know anyone other than Andreous's quartet. Was Max trying to put her off?\n\n'Well I do have a band on my books which could be the one.' He pulled a notebook out of his pocket and studied it. 'Samson have a singer already, but he's the weak link,' he looked at Georgia again, his eyes narrowing. 'Unfortunately they are all very loyal to him. I've tried to make them part company with him before.'\n\n'But Georgia needn't be a threat to him,' Miriam leaned forward at the table, lightly touching Georgia's hand as if to reassure her. Her dark eyes bore into her husband as if willing him to find a way round the problem. 'Why couldn't Georgia sing and Ian do the harmonies and backing vocals?'\n\n'It might work,' Max beamed at his wife. 'A girl as pretty as Georgia would enhance the whole band.'\n\n'How much work would this involve?' Georgia asked, excitement bubbling up inside her. 'I mean would I have to give up my job?'\n\n'Certainly,' Max smiled at her na\u00efvety. 'Once you'd met them there would be rehearsals. Then off on the road. They play six nights a week now.'\n\n'Where?' she asked.\n\n'All over.'\n\n'But where would I sleep?'\n\n'In digs while you're away,' he said with a slight touch of amusement. 'But living where you do would be handy for the London gigs.'\n\n'Gigs?'\n\n'That's the musician's word for a job,' he said. 'But really the first step Georgia is for you to decide whether you want me to be your manager or not.'\n\nThere was an expectant silence.\n\n'I don't really understand what that means,' she said carefully.\n\n'It means I arrange all your jobs, pay you, organize transportation, publicity and countless other things.'\n\n'It means he virtually becomes your father,' Miriam's maternal, wobbly-chinned smile was reassuring. 'He looks out for you and guides you through all the problems.'\n\n'What do you get out of it then?' Even at the risk of appearing rude and ungrateful something told her she mustn't agree too readily.\n\n'A percentage,' he smiled, but his hooded eyes seemed a little colder now. 'Very little now, but one day when you go out for big money, I'll get my reward.'\n\n'But how much will I get now?' Her voice dropped. 'I mean I have to pay my rent and everything.'\n\nMax laughed then, disarming her.\n\n'This is all hypothetical remember. I haven't got the band to agree yet. But how does fifteen a week sound?'\n\n'Pounds?' Georgia's eyes flew open in amazement.\n\n'Well, of course,' he replied.\n\nGeorgia stared at him. That was more than twice the amount Pop paid her!\n\n'But what happens if the public don't like me. I don't get to make a record or anything?'\n\nMax shrugged his shoulders.\n\n'That's my problem,' he said. 'Do you think I got where I am today by backing losers?'\n\nMiriam smiled, reaching forward and taking Georgia's hands.\n\n'Grab it with both hands,' she said gently, her dark eyes full of excitement. 'Chances like this come just once. I'm sure you don't want to spend your life running up dresses?'\n\nGeorgia sat for a moment looking from one to the other.\n\nMiriam was motherly, a woman like her wouldn't lead her into anything bad. Max might be sharp, but just to look at him was to stare success in the face.\n\n'All right,' she smiled. 'I'd like you as a manager.'\n\n'Welcome to our family Georgia,' Max beamed, taking her hand and shaking it firmly. 'Once I've talked to the boys in the band we'll get an agreement drawn up. Meanwhile I think champagne is called for.'\n\nMax's big hands gripped the steering wheel to control his mounting excitement. She could really sing, she was beautiful, but best of all she was alone!\n\nStage-door Mamas were his b\u00eate noire. Hanging around, poking their noses in his business, demanding this and that and producing lawyers if he didn't comply.\n\nHe wouldn't even need hype this time. Georgia had the face, the body and the voice. She wasn't going to be a one hit wonder, the darling of the kids in youth clubs for two months, then dropped like a stone. No, Georgia was going to line his walls with gold records.\n\n'A \"Roller\", a mansion in the country,' he said aloud, turning his driving mirror so he could see himself. 'Well Maxy, the sky's the limit this time. Straight up, no messing!'\n\nWhen Max was onto something he wasted no time. After saying goodbye to his wife and Georgia he was straight off to Birmingham to see 'Samson'. They liked playing 'The Coconut Club', it was Saturday night with a day off tomorrow, all excellent reasons for them to be in a receptive mood.\n\nSeven unambitious lads from the East End and every one of them a first class musician. They'd paid their dues in London pubs and youth clubs, all they wanted out of life was a regular wage, a bit more crumpet and someone to organize them. Too loyal to split up and join bands with a name. Too thick to realize their own potential.\n\nIan McShane was their leader and singer. Max had plans for him at first, his pretty-boy blond looks could have given singers like Ricky Nelson a run for their money. But despite his gentle charm, Ian was an obstinate young cuss. He wouldn't play bubble-gum music, not even for the lures that Max put out. He had an obsession about soul music and that was exactly why Max wanted Georgia to join them.\n\nThe beaten-up old van was parked outside the club. The equipment all ready inside. Max brushed past the doorman with a few words and went into the club.\n\nWhy exactly it was called The Coconut Club, eluded him, as apart from a cut-out wooden palm tree outside it had no exotic aspirations. Inside it was like a catacomb, with small cave-like seating areas leading off the dance floor. The stage was at the far end, with a long bar which reached all down one wall.\n\nMax took a seat tucked away behind an archway. In a well-positioned mirror he could see Ian crawling around the stage checking wires, but to Max's surprise they were almost ready.\n\nNow with the main lights on, the club looked shabby. White walls gone yellow with smoke, velvet seats with no pile left, drink-ringed tables and a sticky, dirty carpet. Yet in an hour or two it would be transformed with subdued lighting, and five or six hundred kids looking for the Saturday night dream.\n\n'One, two, three, testing,' came the inevitable cry over the PA.\n\n'It seems all right,' Ian shouted to the boys. 'Let's get the balance now, then we can go for a few jars.'\n\nThe usual high pitched whine. Les and Patrick tuning up and John flexing his fingers on the trumpet valves.\n\nThe organ came in, a crashing from the drums and then they started to play.\n\nIt never ceased to surprise Max that these boys could actually unpack a van full of heavy equipment, with miles of wiring, cart it into a club or pub and within half an hour create a decent sound. The drums alone looked like a Chinese jigsaw puzzle to unravel. There were endless extension leads, lights and jack plugs. Yet, at the end of the evening they would pack it all away carefully, no matter how much they'd had to drink, only to repeat the performance the next night somewhere else.\n\nThey were all in their early twenties, all too thin and unhealthy looking, not a muscle between the lot of them.\n\nA bunch of underfed dogs, that's what they reminded him of. Mongrels with more courage and persistence than brain. All too keen to lick his hands for a titbit. On a long leash they were happy. A kick up the backside now and then to show them who was boss, then brush them up, show them off and they gave him undying loyalty.\n\nIn a year Max had learned little about them individually. At times he even had difficulty remembering their names. But then it was always Ian who spoke for them.\n\nHe was leaning lazily on the microphone. Butter-yellow hair falling into his eyes, pale skin and soft girlish mouth. It was a shame his voice wasn't that hot. He could have been a real matin\u00e9e idol. He wasn't a wanker like the rest of the bunch. A bit weak, a gentleman, but a likeable lad for all that.\n\nRod the drummer should have been their leader. He had the height, arrogance and charisma Ian lacked. His angular, almost Red Indian features drew the eye, his glossy black hair, his lean frame were a trap for any woman under fifty. Sometimes Max had an uneasy feeling this lad was biding his time, waiting for his moment.\n\nThe others he lumped together. Anaemic looking, sharp East End faces. In their scruffy jeans and plimsolls they could be the boys he grew up with.\n\nSometimes when he watched the lads performing, Max wished he was in their shoes. Girls looking up adoringly, already damp with excitement. By the interval they were ripe. One drink, a little snogging and before the equipment was packed those fresh-faced girls were waiting with their knickers in their handbags. He might have the money and the flash car, he might even promise a night in a swish hotel, but he still had to take knickers off manually.\n\n'That's it lads,' Ian called out. 'Down the pub.'\n\nMax got up from his seat.\n\n'Ye gods, it's the boss,' Norman leapt down from his organ and landed on the dance floor. 'Hallo Max, what brings you up here?'\n\nNorman was an irritating little know-all, with red hair and freckles, the boy he liked least.\n\n'To watch you,' Max said easily. 'Find out what I'm wasting my money on.'\n\n'We're going over to the pub,' Les said. 'Are you coming?'\n\nLes was the most easy going, possibly the dimmest of the bunch, with rounded shoulders, sallow skin and a hooked nose. Right now he had a spot on his greasy chin that flashed as vividly as a Christmas tree light.\n\n'Why not?' Max grinned, 'I'll even do the buying!'\n\nOnce at the pub Max humoured them by listening to the same old complaints. The van that overheated, no new sheet music for Norman. Why couldn't they have new band suits, and some time off?\n\n'Have you finished?' he said, looking from one to another, fixing them with his bright eyes. 'I didn't come all this way to have my ears bashed!'\n\n'Go on,' Norman said. 'What is it this time? Have we used too much petrol?'\n\n'No,' Max waited until he had their full attention. 'Have you ever wondered why you've never been approached by a record company?'\n\nThere was a nodding of heads all round. Speedy, the bass player stuck his head over Rod's shoulder and grinned inanely.\n\n'It's not because you aren't good enough musically, but because the voice isn't there,' Max said firmly.\n\nMax noted how Ian's face blanched, his blue eyes grew darker, his lip curled.\n\n'Are you saying you want me replaced?'\n\n'Not replaced, added to,' Max had to be careful not to antagonize Ian, after all he only intended them to train Georgia. 'If I could show you a girl singer who could blow the socks off everyone in the room, what would you say?'\n\n'Do we all get to shag her in the van too?' John said dryly.\n\n'She's not that type,' Max replied. John was better known for his irreverent humour than his classy trumpet solos. His brooding dark eyes were watching Max closely, missing nothing.\n\n'Then she's definitely out,' Norman broke into a high-pitched shriek of laughter.\n\n'I'm serious,' Max looked round at each of the boys, that 'Trust me because I know best' look he had perfected. 'I don't want to replace Ian. He has a big following with your girl fans, and he keeps you lot in order. But this girl will enhance everything you do. She's young, pretty and a fantastic singer, and if you've got half a brain, you'll let her rehearse with you and see what you think.'\n\n'Can we just have a moment to talk about it?' Ian said, he looked at each of his friends in silent appeal.\n\n'Fair enough,' Max got up from his stool. 'I'll give you ten minutes.'\n\nHe would have given anything to stay and hear what they were saying. Instead he took his drink to the other end of the bar and watched them from a distance.\n\n'I don't like it,' Rod said immediately Max was out of earshot. His dark eyes flashed with anger. 'He's got something up his sleeve.'\n\n'He's an arsehole,' John scowled. 'This is the first step to kicking Ian out.'\n\n'He can't get rid of Ian,' Speedy smirked. His nickname came from his slow, lethargic attitude. But he was the only one of them that had left school with any qualifications. 'We've got a contract with him as a complete band.'\n\nIan flashed a look of gratitude at Speedy. He looked and acted thick because it suited him to do so, but despite appearances, he was as sharp as a razor.\n\n'Thank, lads,' Ian said softly. 'Max is no fool. He must have found someone special. If she's as good as he thinks she is, she may be the one to help us, too.'\n\n'I don't want no fucking girl hanging out with us,' Norman stuck out his pointed chin defiantly. 'It'll be a drag and you know it. It's probably some tart he fancies and she'll be running back to him with stories.'\n\n'Possibly,' Ian said. 'But on the other hand she could be what we need. Max is right. I ain't got a brilliant voice. There's nothing to separate us from a thousand other bands. Why don't we agree to an audition? Nothing more. If we hate her, if she's his bit on the side, we'll refuse. He can't make us take her.'\n\n'But if she is good,' John surprised them by speaking up. 'Maybe we can swing it to make a record.'\n\n'And you Rod?' Ian knew his old friend only too well, if he didn't agree at this stage, there'd be trouble later.\n\n'It's your funeral,' Rod's slanty eyes flashed a warning message to each of them. 'Girls are trouble. We'll be fighting over who's going to screw her. She might even be Max's tart. But I'll agree to an audition. On our own, without Max sitting in. We'll discuss it further when we've heard her.'\n\nIan raised his hand, Max sauntered back.\n\n'Well?' He turned back to the bar, to order another round of drinks.\n\n'An audition, just us and her,' Ian said. 'We're not going to agree to anything till then.'\n\n'Fair enough,' Max smiled. 'Monday week at the usual place.'\n\nGeorgia was filled with self doubt.\n\nOn Saturday everything seemed perfect. But once she got home, doubts began to crowd her mind.\n\nHow could she even think of getting up on a stage and singing with seven men she didn't know. Living with them in digs, travelling hundreds of miles. She knew nothing about men and Brian kept popping back into her head.\n\nShe thought it was over, but now alone in her room he came back. His mouth slobbering on hers, his fingers digging into her arms and thighs. If a man she trusted could do that, how much more could a strange man do?\n\nOn Monday morning Max rang her before she'd even had time to sit down and start work.\n\n'I've talked to the boys, and they are happy for you to join them,' he said, in that clipped decisive way he had. 'I'm booking a rehearsal room for next Monday, can you make it?'\n\n'Yes,' she said looking across the room at Pop, wondering how she could just tell him she was leaving.\n\n'Good. I want you to come to my office today, about five thirty. I've got some records here I want you to listen to. I'll talk to you then about everything.'\n\nPop didn't know whether to laugh or cry when she told him her news.\n\n'I'm happy for you, my sweet,' he said chucking her under her chin. 'This place won't be the same without you.'\n\n'I'll be popping in to see you,' she said leaning her face against his shoulder and holding him.\n\n'That's what I was afraid you'd say,' he teased her. 'Just don't make it too often?'\n\nHe had his reservations about Max, even if Miriam was an old friend. It was no secret he screwed every performer he got his hands on. But in the same way London had beckoned to him as a boy, the music world was calling to Georgia.\n\nJanet listened carefully as Georgia recounted every detail of her day with the Menzies. She could see under Georgia's excitement there was something troubling her.\n\n'Out with it!' she said as she found Georgia alone at lunchtime in the tiny room adjoining the toilets. 'Did Max make a pass?'\n\n'No,' Georgia giggled.\n\nJanet sat on the broken chair and lit up a cigarette.\n\nThey were all reluctant to see Georgia leave. In Janet's case it was a protective instinct. Georgia was only a child, still limping mentally from what that man had done to her. 'Come off it, love!' she raised one eyebrow. 'I know you've got the screaming hab-dabs about something. What is it?'\n\n'I'm just a bit worried about being alone with seven men,' Georgia giggled and looked at her hands.\n\nJanet studied the younger girl as she sat on a bale of cloth, one leg tucked beneath her. She looked so pretty and fresh, the excitement of the phone call had put a pink glow in her cheeks, matching her gingham dress. Her hair curling over her shoulders like a doll in a toy shop. Somehow she had to give Georgia confidence, yet warn her gently too.\n\n'You must make it clear from the start that you aren't available,' Janet said carefully. 'Men aren't all rapists. But most of them will seize any opportunity going!'\n\n'How do I do that?' Georgia's eyes were full of fright.\n\n'Keep your distance,' Janet puffed thoughtfully. 'Get to know them individually. Men as friends are more truthful than women I've found. Don't have a dabble with one of them unless you're sure he's the right one.'\n\n'How do I know that?' She leaned forward to Janet, taking her hand. 'I'm scared.'\n\n'When it's right you'll know,' Janet smiled, stroking Georgia's face. 'Pick someone gentle and caring. Mother nature will do everything else.'\n\n'I wonder if I'll ever be as wise as you,' Georgia said wistfully.\n\n'Wisdom comes through suffering or old age,' Janet smiled. 'Don't wish either of them on yourself.'\n\n##### *\n\nMax's office was in an elegant town house in Berkeley Square. Smart iron railings, white steps led up to gleaming mahogany double doors. Up a graceful thickly-carpeted staircase with polished wood banisters to a door marked 'Menzies Enterprises'.\n\nA receptionist sat just inside. She barely looked up and continued to paint her nails a vivid pink.\n\n'Yes?'\n\n'I've an appointment with Mr Menzies,' Georgia said in a small voice.\n\n'He's with someone,' the girl said rudely. 'Sit down and wait.'\n\nIt seemed to Georgia that she was there for hours, but at least it gave her an opportunity to look around.\n\nBeyond the sulky, dark receptionist and her switchboard, she could see several rooms going off the corridor. In one a girl was busy typing, another girl beside her was getting the post ready.\n\nThe whole place was decorated in mossy green with white doors. Autographed photographs hung on every wall. Brenda Lee, Gene Vincent, Shirley Bassey and Jerry Lee Lewis.\n\nA door opened further along the passage and she heard Max's voice boom out, breaking the silence.\n\n'An audition won't be necessary I assure you,' he was saying to someone she couldn't see. 'I didn't get an office in Mayfair by selling crappy bands.'\n\nA small man in a grey suit came scurrying back along the corridor, he looked at Georgia and nodded.\n\n'You can go in now,' the receptionist still didn't look at Georgia. 'The far end of the corridor.'\n\nMax was sitting behind a huge desk as Georgia looked round the door tentatively. The window behind him overlooked the square.\n\n'Sit down,' he said, waving a cigar towards a chair and opening a desk diary.\n\nThick carpets, solid wood furniture and a huge cocktail bar in one comer, were even more evidence of his success.\n\nGeorgia looked up, a strange creepy sensation tickling the back of her neck.\n\nIn one corner of the room was a gold spider's web, complete with large gold spider, advancing on a gold fly.\n\nMax put the diary down, glancing up to see what she was looking at.\n\n'Do you like it?' he asked. 'It's real gold.'\n\n'I think it's awful,' she said and immediately blushed scarlet at her rudeness.\n\n'It cost a fortune,' he said in its defence, his thick lips curling a little. 'I designed it myself.'\n\n'I'm sure it's very clever and beautifully made,' she was almost trying to apologize. 'I just don't like spiders. They give me the creeps.'\n\nMax got up and went over to the bar.\n\n'Like a drink?' he said, over his shoulder.\n\nShe knew he was insulted. Just the stiffness in those wide shoulders warned her.\n\n'Just an orange juice,' she said. 'I can't stay long.' She hadn't anywhere to go. It was just something to say, but the moment the words were out of her mouth she knew that was wrong too.\n\nHe wheeled round, an angry flush across his cheeks. He reached her in two giant strides and put one great paw on her small shoulder.\n\n'I think we have to get one thing straight,' he said gruffly. 'If I'm to spend time, trouble and money on you, I expect total commitment from you. You sing when I make bookings for you, even when your grandmother has invited you out for tea. You don't tell me you have other plans!'\n\nUntil now Georgia had thought her days of being answerable to anyone were over. But one look at Max's stern face told her this wasn't so.\n\nHis jacket was off, draped over a chair, she saw the double 'M' monogram on his silk shirt, glanced up at the gold spider, and felt a tremor of fear.\n\nEven his face wasn't so inviting. He had dark stubble on the strong chin, his mouth looked bad-tempered and tough, every line in his big frame told her this man would be nothing like Pop.\n\n'I'm sorry,' she said. 'I didn't mean to hurry you or anything. And I won't ever let you down.'\n\n'Have you got a boyfriend?' his dark eyes narrowed, looking right into hers.\n\n'No,' she said, feeling very uncomfortable.\n\n'Well that's something. I don't want you preoccupied with any man at this stage.' He sat down again at his desk, tilting back his seat, playing with a pencil.\n\n'I'm only serious about singing,' she said, wishing she dared to tell him it was none of his business.\n\n'That's good,' he said, a faint smile playing at his lips. 'You see Georgia, joining a band isn't like any other job. These boys will soon be like your family. If you can't get on with them, really like them, you won't bring out the best in one another.'\n\n'I understand that.'\n\n'You may think you do now,' he smirked. 'Just wait until you've put up with smelling their socks. Listened to them farting in the van. Watched them snogging with girls when you are anxious to get home. That's the stuff that takes the fun out of it.'\n\nGeorgia tried hard to look serene. She was sure he was exaggerating.\n\n'I'll cope,' she said, more confidently than she felt.\n\n'Now for the music,' he picked up a small pile of records. 'I want you to play these until you know every word. We haven't got the music for these songs, the boys play them by ear. Make sure you really know them by next week.'\n\n'I haven't got a record player,' she whispered.\n\nMax looked up, surprise on his face. 'You're joking? All kids have record players!'\n\n'I can't afford one,' she replied, wringing her hands together, wishing she was anywhere but here alone with Max.\n\nShe felt him move, coming round to sit on his desk in front of her. For a moment he said nothing, just looking down at her.\n\nMax had that same feeling he'd had in the Acropolis. A tightening in his gut, a prickle in his heart. She looked so pretty, like a little Sunday school teacher in her pink, checked dress, her hair tied up with a ribbon. He remembered a day when he was fourteen being forced to admit he hadn't any boxing gloves for the same reason, and that prickle grew stronger.\n\n'I've got one you can have,' he said. 'I'll go and get it and give you a lift home.'\n\nAs Max climbed the stairs to Georgia's room carrying the Dansette record player he kept in the office, he felt as if he were going back some twenty-five years.\n\nThe smell of damp, a glimpse of the hideous bathroom, the worn, dusty carpet, the poverty. It was just like the place he lived in as a child. This place was silent, as if they were alone, yet he could almost hear the sounds that had filled his childhood. Babies crying, children shouting and adults screaming at each other, mingled with a stench of boiled cabbage and toilets.\n\n'How long have you lived here?' he asked, trying hard not to reveal his thoughts as she unlocked the door at the top of the stairs.\n\n'Over a year. It's not so bad inside and at least it's near everything.'\n\nShe ran in ahead of him, turning on a small lamp. He understood why, she wanted to soften the bareness.\n\nMax put the record player down. He felt huge against the low, sloping ceiling. 'You've made it nice,' was all he could say. He guessed she had painted it, wondered how she had come here, and above all whether the activities so close to this room had touched her.\n\n'Would you like some tea?' Georgia was flushed with embarrassment, she was moving her weight from one leg to the other, hoping he'd refuse and go.\n\n'I must get home,' he wanted to take her out to dinner, but in the mood he was in he might do or say the wrong thing. 'Shall I plug this in for you first?'\n\n'I can do it,' she said, watching as he placed it on a spindly coffee table. 'As soon as I earn some money I'll buy another and give you this back.'\n\n'Keep it,' he took her two hands in his, unable to control himself. 'It's a present.'\n\nShe just stood there looking at him. Her lips slightly parted, eyes like two dark pools.\n\n'Don't be embarrassed by having nothing darling,' his voice was husky. 'I started out like this too. There's no shame in it.'\n\nNo one had ever touched him like this. He'd had dozens of young girls with less than her, and never once wanted to give them anything. Max took what he wanted. Whether it was their youth, their talent or just their virginity. A meal or a night out in a hotel was the extent of his generosity.\n\nHe wanted to kiss her so bad it hurt. Yet somehow he knew if he touched her she'd back away and maybe she'd be lost to him forever.\n\n'It's not that,' she dropped her eyes from his. 'I've never brought a man up here before. It feels strange.'\n\nShe broke all the rules. Max understood girls who flirted and pretended to know everything, or even ones who ran a mile from being alone with him. She just stood there, still with her little soft hands in his, half child, half woman, too innocent to realize that this older man's interest in her was far from professional.\n\n'You've no need to feel strange with me Georgia,' he squeezed her hands then let them drop. 'I'll be going now, learn all the words, and I'll meet you at ten next Monday, outside Peter Robinson's in Oxford Street. Don't be late!'\n\n'Thank you Max,' she smiled took a step nearer him and standing on tip-toes kissed his cheek. 'You can't imagine how lovely it will be to hear music again.'\n\nHe heard the opening chords of 'Soul Train', even before he reached the street. Max put one hand up to his cheek where she'd kissed it and paused for a moment.\n\nThis was going to be tough. She wasn't going to fall into his arms like an over-ripe peach and just this once perhaps it would be him who got hurt.\n\n'I can't bear to leave you!' Georgia sobbed.\n\nAll week she had been in a state of hysterical excitement, but now as her last day at Pop's was almost over, she realized just how much they all meant to her.\n\nJanet's lips quivered, Sally was chain-smoking, the other women had fussed around her all day, giving her little treats. Myrtle had even run her up a pair of warm pyjamas.\n\n'You can always come back,' Pop said gently. 'The door will always be open for you. Good luck!'\n\nHe hadn't rebuked her all week, just long-suffering sighs at her high spirits. His sad, clown-like face was suddenly very dear to her.\n\n'Thank you for everything,' she said, running to hug him one last time. 'I'll never forget any of you.'\n\nPop held her tightly, his lips quivering.\n\n'You'll never be far from our thoughts. We'll be watching to see your name in lights.'\n\n'None of this would have happened without you.' She lay her head on his shoulder, fresh tears filling her eyes. 'I want to sing, but I'll miss you all.'\n\n'You'll make new friends,' he said softly against her hair. 'Not broken reeds like us lot.'\n\n'Drop us a line when you are singing in London,' Janet's voice was husky with emotion. 'And remember all the things I've taught you about men!'\n\nGeorgia sat hunched up in Max's Jaguar, shaking with fear. In jeans and a sweater, her hair tied up in a pony tail she looked about fourteen.\n\n'Now don't let any of them make passes at you,' Max said gruffly. 'Don't start making them tea and stuff otherwise you'll end up becoming mother to them all, and if you have any problems phone me, either at the office or home.'\n\n'What will I wear to the gigs?' she asked.\n\n'Miriam has all that in hand. I'll be popping in on Wednesday during the day to see you.'\n\nThe church hall was near Aldgate. Dilapidated and sad with ferns growing out of the roof, the wire-covered windows mostly broken.\n\nAs they got out of the car Georgia could hear music blasting out, surprising passers-by.\n\n'That's good,' Max said, grinning broadly. 'They've set up. Now don't be nervous, just sing and forget about everything else.'\n\nAs Max swept her into the dingy hall she almost laughed with relief. She had imagined strong, fierce men, but all she saw were boys, weeds in jeans and sweaters, cigarettes hanging out of their lips.\n\n'Hallo Georgia!' One of them jumped down off the stage, his pale face brightened by a wide smile. 'I'm Ian. Has Max filled you in on the music?'\n\n'I've played the records,' she blushed. Her voice seemed to echo round the hall too loudly now the rest of them had stopped playing and just stared silently at her. 'I know all the words, I think.'\n\n'Well, that's more than I do. I often ad lib.'\n\n'I'll shoot off now,' Max said, backing towards the door. 'Ring me tonight at seven, Ian. We'll talk then.'\n\nHe was gone in a flash, the doors shuddering behind him.\n\nFor a moment Georgia just stood there, eyes downcast. She knew this feeling so well, just the way it had been the first day Celia left her at the new school. Play interrupted as the other kids stared at her, then moving on, forgetting her.\n\n'Nervous?' Ian touched her shoulder lightly. 'Don't be, love, we all know what it's like. Come and meet the others.'\n\nShe knew it would take forever to remember their names. Ian with his gentle ways and angelic face stood out, and Rod the drummer for his dark brooding looks, but the others seemed so alike.\n\n'Speedy's really called Patrick,' Ian waved his hand at the auburn-haired, fresh-faced one nursing his bass guitar. 'You'll find out why we gave him that nickname. Norman on organ, an arrogant little shit-stirrer. Les on lead guitar, thick as two short planks and finally John and Alan the brass section.'\n\nWhy did they stare so hard? Was it because Max hadn't told them about her colour?\n\n'Let's get on with it!' Rod shouted irritably, banging on his side drum.\n\n'We'll start with \"Soul Train\". Norman sat down at the organ. 'I'll play it through to refresh your memory. It might be a good idea to sing by me so you don't get thrown by the backing.'\n\nClammy cold hands, butterflies in her stomach and as the introduction started her face broke out in a sweat.\n\n'I'm on the Soul Train, don't know where I'm going,' the first line came out as no more than a croak. She kept her eyes on Norman's hands dancing over the keys. The second line was easier, by the third she had forgotten her fears, throwing back her head for the chorus.\n\n'Soul train take me with you,' she sang moving back away from Norman, microphone in hand.\n\nThe trumpet and sax were playing thrilling little riffs, Rod's drums sent out a heavy beat and she could hear Speedy and Les joining in with vocals. Tingles went down her spine, she turned to sing to them, forgetting that she'd never seen or heard of them till a few moments ago.\n\nAs the music died away she looked round.\n\nAll seven of them were staring at her.\n\n'Did I do it wrong?' she asked turning pink with embarrassment.\n\n'You did it as good as perfect,' Ian grinned. 'Now let's do it again and I'll do some harmonies.'\n\nAs the day wore on she had a glimpse into each personality. Rod the drummer was the most dangerous, his slanty eyes seemed to be watching her closely. He oozed raw sexuality which made her uncomfortable, when he spoke it was a mere growl. Norman knew his music. Speedy was the calming member of the band and John, Les, and Alan were the quiet ones. But the biggest surprise of the day was Ian. When Max described him as the weak link, he couldn't have been further from the truth. His sweet face, the baby-soft, yellow hair and big blue eyes, hid a committed musician. He made the decisions, knowing each member's strengths and weaknesses and he balanced them like a juggler.\n\n'Sing from the groin,' he said at one point in the morning, making a thrusting gesture. 'This is rock not choir practice. You've got to sell yourself, not just your voice.'\n\n'Let's go down the pub for a few jars,' John said at two, packing away his trumpet before anyone could argue. 'This is the nearest thing we'll get to a holiday for months, so let's make the most of it.'\n\n##### *\n\n'We pool our money,' Ian smirked as Georgia watched him take a brown envelope out of his pocket to pay for the drinks while the others stood watching. 'We keep a fiver each, and the rest goes in here for our rent, food, fags and beer. If there's any left at the end of the week we divide it up.'\n\n'So what do I do?' she asked wondering if this meant she had to hand hers over too.\n\nA wall of hostility seemed to spring up round her.\n\n'Max didn't tell you it was only an audition?' Rod's lip curled back aggressively.\n\n'I'm sorry.' She could have died of shame. Max had put her in an impossible position. Why couldn't he have told her the truth?\n\nShe turned and ran to the toilet, tears springing into her eyes.\n\n'Fuck Max,' Rod said as the door banged behind her. 'He's so bloody \u2013' he paused, unable to find the right word.\n\n'Right?' Speedy grinned. 'Let's face it Rod, for once he's got his head screwed on.'\n\n'She's perfect,' John said quietly, staring at the door Georgia had rushed into. 'She's got the voice, the looks, everything.'\n\n'She'll be trouble,' Norman's face was sharp, as if searching for an alternative argument. 'What happens when she has to share a room with us? How can we change with her around?'\n\n'What do you think Rod?' Ian turned his lazy blue eyes on to his friend.\n\n'I think she's brilliant.'\n\nIan's mouth fell open. He had expected fierce opposition, at best a bet that he could get her into bed first.\n\n'I think she could be trouble,' Rod added quickly. 'I agree with Norman there. But more from Max than anyone. He fancies her, I saw it in his face. What we need to get straight is how she thinks of Max.'\n\n'Let's have a vote on her anyway?' Ian scanned the boys grouped around him. He could see Rod wanted her in, John too. Alan and Les would go along with the majority.\n\n'Hold on,' Speedy's slow voice halted them. 'I like her too, but what we've all got to think of is Ian's position. We've been together too long to split up now. If she stays we've got to integrate her right into the band, never allow her to think she's the important one. You've got to put more beef into it Ian, don't let her take over. Anyway you haven't said a word yourself.'\n\n'I think she's our saviour,' Ian blushed. 'She's just what we need. Did you hear all of you come alive in there? She's got that magic touch to make us all reach new heights. I don't care if it's going to cause trouble. I say we take her on.'\n\nWhen Georgia returned from the toilets she knew immediately they had reached a decision.\n\n'Give us two quid,' Rod said, holding out the brown envelope. 'That'll be your contribution until pay day. We'll work out your permanent share next week.'\n\n'You mean?' Georgia's eyes opened wide, her lips curving into a dimpled smile.\n\n'It means you're in now baby,' Speedy drawled. 'Welcome to Samson.'\n\n## Chapter 11\n\n'Is this it?' Georgia turned surprised eyes on Ian as he banged on a door between a Wimpy bar and a betting shop in Tottenham High Street.\n\n'Wait till you get inside,' he grinned, eyes full of mischief. 'Welcome to de shit-house!'\n\nThe door creaked open and a small, scrawny man in a dirty collarless shirt peered out, behind him was a flight of steep stairs, up which came a warm, fetid smell of dustbins.\n\n'You're early,' the man said. 'I haven't finished cleaning up yet.'\n\nGeorgia clutched the dress Miriam had given her to her chest, her legs turning to jelly.\n\n'You know we like to set up early to get a balance,' Ian prodded the old man in the chest playfully. 'Now come on Sid, don't be awkward.'\n\nGrumbling under his breath the old man tottered off down the stairs on bowed legs. Ian turned to the boys sitting in the van and waved for them to follow.\n\n'You'll soon get used to clubs by daylight,' Ian grinned at Georgia's horrified face and putting one hand in the small of her back, pushed her gently towards the stairs. 'By the time we get back here at half nine tonight it will look different. By two thirty when we leave it will seem wonderful. That's a promise.'\n\nAn hour later Georgia sat huddled on the edge of the stage watching the boys still putting the final touches to their equipment. All they had allowed her to carry down were the small items while they sweated profusely carting the heavy organ, speakers and amplifiers.\n\nThe basement club was a huge clammy cellar, the only seating consisted of rows of school-type benches lining the whitewashed walls. Two long strip lights lit the place, making it as inviting as a morgue. Sid was still washing the floor, or rather pushing the dirt round further with an ancient mop.\n\n'Which one do you belong to?'\n\nGeorgia assumed Sid was talking to her, although his eyes stayed glued to the mop in his hand.\n\n'I don't belong to anyone,' she said. 'I'm the new singer.'\n\n'Bit young for that aren't you?' He shuffled nearer her, pushing the full bucket with one foot. 'Did they tell you what a rough-house it is?'\n\n'Yes.' In fact it had only been mentioned on the way here when Ian gave her instructions to make for the dressing-room and stay there if trouble broke out. Georgia was more concerned with changing in a filthy, tiny hole with one forty watt bulb overhead and a mirror barely big enough to see her mouth in, than it being her sanctuary.\n\nNow that the boys' band suits hung in there too, there was barely room for any of them. How was she going to change into that awful dress Miriam had given her, without the entire band seeing her almost naked?\n\n'Some nights they pelt the band with glasses,' Sid said cheerfully. 'Course, they've never done it to this lot, because they like them. But you get on your toes if anything happens.'\n\nGeorgia's heart sank. She was frightened enough by just singing here, without the threat of violence too.\n\n'Nearly there now,' Rod called down from the stage. 'We'll run through a couple of numbers, then go and find something to eat.'\n\nGeorgia couldn't imagine anyone wanting to come and spend an evening here, even though the boys insisted it was always packed to capacity. They said it was one of their favourite venues, so what must the worst ones be like? The toilets were old and grubby with no hot water. Not one glass on the bar sparkled. The seating was almost non-existent and if fire broke out they'd all be trampled to death trying to get out that one narrow staircase.\n\nNorman began to play on the organ. A haunting little tune she remembered hearing on Radio Luxembourg while still at school.\n\n'I'm Mr Blue, when you say you're sorry,' she forgot for a moment her anxiety and joined in, turning round to face him. 'Then show it by going out on the sly, proving your love isn't true.'\n\nLes and Speedy immediately picked it up, leaning to the microphone and doing exaggerated backing vocals.\n\n'Do Wah doo, Call me Mr Blue.'\n\nIan stood back against a speaker, his mouth twitching with mirth, as Georgia, and the three boys aped the fifties song.\n\n'The balance seems fine,' Ian laughed aloud as they finished. 'Don't call us, we'll call you.'\n\nThe pub on the corner had been empty when they went in at seven. Now at half past eight it was getting crowded, just another reminder soon they would have to leave the comfortable bar and get back to the club.\n\nIan shot a glance at Georgia as though reading her mind.\n\n'I'll go back with Georgia, you lot stay on for a bit,' he said to the others, standing up and beckoning for her to come too. Georgia flashed a look of gratitude at him. She had been dwelling on the best way to get into her dress in that dressing-room without them seeing her underwear and she still hadn't found a way of removing her bra and zipping up the hated dress without revealing something.\n\nTo Georgia with only the scantiest real knowledge of men, it seemed the boys were obsessed about sex. They described breasts in detail, the size of a girl's nipples, the weight and feel of them. Rod claimed they all masturbated at least twice a day and insisted she would get used to it. They spoke of a mysterious thing called 'muff diving' which she knew was something rude but hadn't a clue what it was. No woman under fifty was safe from their bawdy banter and not once had she heard a tender or romantic suggestion.\n\nOnly Ian was different. He too laughed along with them, even pointed out women they passed in the van, but in Georgia's presence at least he didn't swear or say anything to make her blush, and now he was intuitive enough to know she was frightened of changing with them.\n\n'They aren't as bad as they seem,' he said as they crossed the High Street. Flynn's had bright lights over the door now and a big flashing neon sign had been switched on. It was dusk, and the street was full of groups of young people, giving the road a feeling of excitement that had been lacking earlier in the day. 'Believe it or not Georgia, they are merely trying to get you to open up. Until you do they'll keep it up.'\n\n'What do they want me to say?' she asked as they reached the club door. A small desk had been set up at the top of the stairs, but as yet there was no one manning it. 'Do they want me to say I had five men last night?'\n\n'No.' Ian laughed. He turned to her, brushing a strand of hair out of her eyes. 'They just want to know if there's a man in your life. Has Max come on to you. Where you come from and why you say so little.'\n\nSilence hadn't been intentional. She had been so busy listening to them she hadn't considered they might want to know about her. But even if she wanted to open up, how could she? One bit would lead to other bits she wanted left buried.\n\n'There isn't a man,' she hung her head. 'I don't know what you mean about Max. I'll tell you about my past when I feel safe, and I don't talk much because you lot all do it for me!'\n\nIan's hand came up, with one finger he lifted her chin, leaned forward and kissed her on the nose.\n\n'That's enough for me.'\n\nHe had the prettiest eyes she'd ever seen on a man. Saxe blue with tiny flecks of grey. Despite his blond hair, his eyelashes were thick and dark. But then there was a great deal to like about Ian. He was gentle, sensitive and his face held all the purity of a child. He was too thin, too pale. In a teeshirt and jeans he was like a rasher of bacon. Yet he had an adult quality which the other boys didn't share.\n\nTogether they went downstairs. The club looked completely different now. The strip lighting replaced by coloured spots on the walls, and the bar lit up. Only now did she appreciate that the rough wood on the bar was intentional, intended to look rustic. Even the floor didn't look dirty any longer, the smell of dustbins replaced by some kind of lavender smell.\n\n'Sid's been spraying the place for cockroaches,' Ian sniggered.\n\nGeorgia stood up on tip-toe, peering into the gloom suspiciously.\n\n'Not really, chump,' Ian laughed. 'It's just some kind of air freshener. Now get changed before the others get back.'\n\nThe dress was gold lurex. The top, boned and strapless, the skirt a full circle with layers of net underneath which scratched her legs unmercifully.\n\nGeorgia tried to see herself in the cracked mirror, holding the dress closed with one hand. She had been struggling to zip it up for five minutes, and now music was playing out in the club she could hardly go out and search for Ian to help her.\n\nThe shape of the dress was quite nice, but any yellow or brown tones didn't bring out the right colour in her face. She looked brassy. More suited to ballroom dancing than a soul band.\n\n'Can we come in now?' Rod's voice came through the door. 'Are you decent?'\n\n'Yes,' she backed up against the wall holding the dress closed behind her.\n\n'Blimey,' Rod whistled. 'Where on earth did Miriam dig that one up?'\n\nGeorgia could feel her lips quivering. She had hoped they would insist it was nice. Speedy and Alan pushed by Rod, pulling off their shirts as they went. Ian came in next.\n\n'I can't do up the zip,' she whispered to him, turning her back to him while clutching at the top.\n\nIan smiled. Her small narrow back reminded him of his younger sister's, little shoulder blades sticking out like tiny wings, skin so smooth and silky he was tempted to stroke it.\n\n'Zip it, don't kiss it,' John said behind him. 'You're supposed to be the gentleman.'\n\n'It's horrible isn't it?' Georgia turned back to Ian once he had fastened the hook and eye on the top. 'Do I really have to wear it?'\n\n'I can only suppose she thought it would contrast with our red suits,' Rod's deep voice chimed in. Georgia looked round and found him wearing only the scantiest of underpants barely covering a terrifying bulge. Undressed he looked like a man, broad shoulders and a sprinkling of dark hair on his chest. She blushed furiously.\n\n'It will have to do for tonight,' Ian realized her discomfort and distracted her. 'You'll look fine on stage.'\n\n'You can't wear stockings,' Speedy said in his slow, almost dreamy way, gazing at her legs with a slight leer as he too removed his jeans. 'They'll all be peering up your dress all night. Take them off.'\n\nGeorgia wanted to die. Nothing had prepared her for being cooped up in this tiny airless room with seven men only half dressed. Now they expected her to remove her stockings in front of them.\n\n'Speedy's right,' Ian said quietly. 'He could have been more tactful, but that's Speedy for you. Do your hair and stop worrying.'\n\n'You'd have been better in red,' John had already dressed, he was putting on a black shoe-string tie. 'That colour makes you sallow.'\n\nGeorgia was on the point of tears.\n\n'What shall I do?' she whispered.\n\n'Put plenty of rouge on your cheeks,' Ian said. 'And finish your hair, then we'll see.'\n\nSurreptitiously she tried to remove her stockings and suspender belt without anyone seeing.\n\nIt was a horrible feeling. If she bent forward her breasts popped out, her bare shoulders felt chilly. Suppose when she moved on stage the audience could see her knickers?\n\nSomehow she managed to wriggle into the corner, bent over and scooped her hair up into a top knot. Miriam had suggested she wore a 'beehive', but she'd tried that already at home and all she succeeded in doing was making herself look like a tart.\n\n'To hell with everyone,' she muttered to herself. 'I'll be me and if they don't like it, too bad.'\n\nShe stood up, pulling up the front of her dress, then turned.\n\nIan's face broke into a wide smile.\n\nHe could see the defiant look in her eyes. Her hair showered over her head in a mass of bubbly little curls. He stepped forward, and with one finger released a few tiny strands by her ears, curling them round his finger.\n\nHe was already in his suit. He looked bigger, the red contrasting well with his blond hair. His black shoes gleamed with polish, shirt dazzling white and a faint whiff of woody aftershave. Georgia could see now why he had such a big following of girls.\n\n'You need something in your hair,' he said thoughtfully. He was standing so close to her she was sure he could see right down her dress.\n\nGeorgia shamefacedly got out a gold feather plume that was with the dress and held it out.\n\n'Who did she think she was dressing?' He shook his head in disgust. 'That makes you look like something out of the Moulin Rouge!'\n\n'What about a flower?' Rod said, his arrogant face for once alight with interest. 'There's a vase full on the bar.'\n\n'Go and whip a couple, red ones,' Ian said.\n\nRod returned in minutes with two roses.\n\nIan came close to Georgia and pushed them into the band holding her hair.\n\n'That's better,' he said, standing back and smiling. 'Now put a bit more colour on your face, you might look like a clown in here, but up on the stage you'll look fine. Do you want me to do it for you?'\n\n'Yes,' she whispered, by now so nervous she couldn't do anything.\n\nIan stood in front of her carefully applying the rouge. He picked up a small eyeliner brush and added a little more.\n\n'That's better,' he said. 'Now use a lipstick brush and outline your lips in a darker colour.'\n\n'Are you sure?' she asked. She was surprised by his knowledge, and the way he handled her like a sister was very comforting.\n\n'Quite sure,' he said gently.\n\n'I'm in a bit of a state,' she whispered, not wanting the others to overhear.\n\n'Let me give you a cuddle?' he smiled. 'The best remedy for the collywobbles.' He put his arms round her and held her tightly, his lips close to her ear. 'I'm still nervous and I've been doing it for years. But it goes as soon as we get up there.'\n\n'Break it up,' Rod shouted. 'I thought Max said no passes were to be made?'\n\n'Not a pass,' Ian laughed, looking round at the others without letting go of her. 'Just a cuddle to banish nerves.'\n\n'You nervous?' Rod sounded amazed. 'If you can come along to rehearsals when you don't know anyone, you'll sail through this.'\n\n'Will I?' she said, still clinging to Ian.\n\n'Of course you will,' the boys all chimed in together.\n\n'Besides, we can get John to blast out on his horn if you can't reach the high notes,' added Norman.\n\nTheir faces touched something inside Georgia.\n\nThey were all seasoned professionals and she had been thrust on them whether they liked it or not. She was young, green as grass and she was holding them all up. Yet here they were being brotherly and kind, grouping around her offering her their support.\n\n'Thank you,' she said simply. 'I hope I don't let you down.'\n\n'You've got five minutes to have a pee,' Ian said with a lop-sided grin. 'Don't drop that net skirt in the bog.'\n\nFlynn's was filling up.\n\nGeorgia couldn't bear to peer through the door as the boys were doing. She could hear raucous laughter, shouting and stamping feet, it was frightening enough just to listen, without gawping at them too. Hard, pale-faced girls with beehives and heavily made-up eyes in tight sheath dresses and stilettos. The men tough and broad-shouldered in Italian boxy jackets and winkle-pickers. They seemed an unlikely bunch to appreciate a band who didn't play top twenty hits.\n\nBobby Vee's song 'Rubber Ball' was playing so loudly the speakers crackled, a smell of cigarettes, beer and cheap scent taking away all the oxygen she needed.\n\n'Never seen such a big crowd on a Thursday,' a raw-faced bouncer in an evening suit poked his head round the door. 'Looks like they all want to hear your new girl.'\n\n'And now, the moment you've all be waiting for,' the record was halted and the club manager's voice boomed out over the microphone. 'Your favourite band, _Samson_!'\n\nThe boys pushed past Georgia, Rod blowing a kiss. Ian took her hand and squeezed it.\n\n'We go next,' his eyes were full of understanding. 'Don't panic, let me lead you till you get into it. If you find you can't sing, just move with the backing.'\n\nA burst of applause and Rod played a roll on the drums.\n\nGeorgia felt a cold sweat breaking out all over her and she wanted to go to the toilet again.\n\n'You all know Ian McShane, but tonight you're in for an extra treat. Meet the lovely Georgia James, the band's new singer!'\n\nHer legs refused to move, yet Ian was dragging her up the three steps on to the stage.\n\n'Smile,' he said. 'Head up!'\n\nNorman played the opening bars. Georgia saw the expectant, upturned faces lining the stage only feet from her.\n\nSomehow she made it to the microphone. Her mouth was smiling, but her stomach churned.\n\nShe reached for the microphone, and found her mouth and throat as dry as a desert.\n\n'Turn towards the band,' Ian whispered as he adjusted his mike. 'I'll start.'\n\nShe forced herself to turn. Rod was smiling encouragement, John and Alan moved closer to cover any mistakes with their horns.\n\n'I'm on a soul train, don't know where I'm going,' she was mouthing the words but she could only hear Ian.\n\n'You can do it baby,' Rod said, his eyes sympathetic. 'Just keep singing till the voice comes back.'\n\nShe managed the second line a bit better, but by the time she got to the third, her voice came out despite her terror, and the fourth followed without even thinking about it.\n\nShe twirled down, a flash of gold net around slim brown legs. Mouth wide and red, teeth gleaming white. Her voice soared over the crowd.\n\n'I'm on the soul train and I'm coming baby to you.'\n\nTowards the end of the first set something else had taken over. She was dancing, smiling, bending down to young men in the front row and blowing kisses. She forgot the scratchy underskirt, kicking off her shoes as she immersed herself even more into the music. Whenever she turned towards the rest of the band they were grinning like Cheshire Cats, egging her on.\n\nHer voice found new heights, one moment deep and husky, raunchy and sexy, then sounds so pure she could hardly believe it was her.\n\nIan was better than he'd ever been at rehearsals. His singing was more punchy. As his head bent close to share the mike with her on some numbers, she knew he was putting his all into it.\n\nThe applause was deafening as the first set ended.\n\nThe D.J. ran on. 'Well what can I say?' he yelled, waving a hand at the departing band. 'Can that little girl sing, or what?'\n\nAs Georgia reached the dressing-room door she saw Max.\n\n'You were great,' his dark eyes shone like jet. 'I'm knocked out!'\n\n'I didn't know you were out there,' she was breathless, panting like a dog after a ten mile run.\n\n'Would I miss seeing you get started?' he said. 'I even brought a photographer.' He moved closer, taking her two arms in his big hands and squeezed them.\n\n'You were magic,' he said, his eyes burning into her. 'I've never seen the band so good.' He looked round at the boys grouped behind her and smiled broadly. 'Well done all of you. There's a reporter from _Melody Maker_ out there. If I'm not much mistaken you'll get the best review of your careers this week.'\n\n'Do we get a drink boss?' Norman as always was asking for something.\n\n'They're set up in the back bar,' Max grinned. 'And you've earned them.\n\n'We're privileged tonight,' Speedy drawled sardonically. 'This room's usually closed off. They only use it for card games.'\n\nA billiard table stood in one corner, covered with a cloth, in the middle of the room a large table had nine pints of beer standing on it.\n\n'Get a proper drink for Georgia,' Max barked out at John tossing him a note. 'And pay for that round while you're at it.'\n\n'What'll it be?' John smirked. 'Double champagne? Gin, brandy?'\n\n'Coke,' she grinned. 'And lots of ice.'\n\nGeorgia sat next to Ian.\n\n'Thank you for telling me to turn round,' she said softly. 'I thought my voice had gone for good.'\n\n'That's all right,' Ian smiled and leaned back in his seat. 'It's an old trick, works everytime. If you can't see the punters you can forget they are there.'\n\nIt didn't seem possible she'd only known these boys for four days. Just an hour ago she had wanted to run away and forget any dreams she'd had of being a singer. Now as she looked around her she knew their futures were interlinked. She would remember their faces forever, love them for all that support and understanding.\n\n'Her dress isn't right.' Speedy's voice rose over the others as he made sure Max listened. 'It's like something off \"Come dancing\", she needs something tight and sleazy.'\n\n'I think it looks good,' Max argued, he studied Georgia with hooded eyes. 'What do you think Ian?'\n\n'Well, Georgia would look good in a sack.' Ian leant back in his chair, his fair hair flopping over his eyes. 'But that dress is dated. It went out with Glenn Miller. I don't agree with Speedy though, she's not the sleazy type.' He paused, seeming to wake up a little, his soft lips twitching as if in silent amusement. 'I'd like to see her in something that shows her legs, so she can dance too.' He smiled and turned to Georgia. 'You held out on us there too. We had no idea you were such a good dancer.'\n\n'I'll look into it,' Max said his eyes running over Georgia, then moving back to Ian. He was wondering if something had occurred between them. Ian wasn't usually so forceful and there was a light in the boy's eyes he hadn't seen before. 'Anyway lads, is there anyone who doesn't think Georgia did well?'\n\n'She was brilliant,' Rod said unexpectedly. 'You've done us proud, Max.'\n\nGeorgia looked around with bright eyes. She felt fabulous, right now she could hardly wait to get back on the stage and do the rest of the performance.\n\nThe second half was even better. The audience were clapping, stamping their feet and smiling up at her.\n\nShe felt all-powerful, every nerve-ending twitching. She was no longer a frightened little sixteen-year-old. Up here she was a queen, and the audience her loyal subjects.\n\nThunderous applause rained down on them as they finished and the boys' faces told her it was all for her.\n\nBack in the dressing-room the smell of sweat and socks was even worse as they changed back into their ordinary clothes.\n\n'How are you feeling now?' Ian said softly as he unzipped her dress for her.\n\n'On top of the world,' she laughed, clutching the top of her dress to hide her breasts.\n\n'It's like a drug,' Ian smiled. 'Once tasted, you need it for evermore. But don't let it go to your head. Everything went well today, and the audience were great. Wait till you've done a really bad gig. When the crowd stand with their backs to you drinking, the lights fuse and the guitar strings break, and you know you've got a hundred-mile ride home in the van.'\n\nMax looked in the dressing-room, his big, square face looking unexpectedly healthy surrounded by the boys' pallor.\n\n'Okay boys, see you at the office at twelve tomorrow. Georgia, would you like a lift back with me?'\n\n'But the equipment?' she said, trying to get a shirt over her top half while still holding her dress round her.\n\n'Go on home,' Ian grinned at her. 'We can manage without you, besides, there's a bit more room in the van for us.'\n\nShe didn't want to go with Max. They would all be chatting about the gig and she wanted to be part of it. But without making herself look silly she could hardly refuse.\n\n##### *\n\n'How's it going with the boys?' Max asked as his Jaguar glided away down towards the West End. 'Any problems?'\n\n'Everything's just great,' Georgia looked out the window, it had begun to rain and the road looked like black tar studded with diamonds of reflected light.\n\n'Has Ian come on to you?'\n\nGeorgia's head turned sharply towards Max. Ian had said the same thing about Max. She wasn't sure what he meant at the time, but now it became clear.\n\n'Of course not,' she retorted.\n\n'Just asking, he seemed different tonight, that's all. I don't want you getting involved with any of them.'\n\n'Look Max,' something snapped inside her. 'I don't mind you telling me what to do where work is concerned. But I don't see what my private life has to do with you.'\n\nMax pulled the car over to the kerb and stopped abruptly.\n\n'Now look here,' he said, turning in his seat and grabbing her face in one big hand. 'If I turn you into a star you won't have a private life. One whiff of scandal, one bit of gossip can ruin a career. You've got to live at close quarters with that lot and if you hop into bed with one of them they'll be trouble. I've seen it all. I know.'\n\nShe could see something in his eyes. Something deep and disturbing. He was too old for her, she didn't quite trust him, yet she wanted him to kiss her.\n\nThere was a feeling in her stomach, something primitive. Warning bells were jangling in her head, but she could only see those full red lips and feel that hand on her face.\n\n'I feel something for you,' his voice was husky, those lips coming closer. 'I want to protect you, love you.'\n\nHer eyes were closing. She could feel herself drowning and his lips when they touched hers were soft and teasing.\n\n'No,' she moved back quickly, suddenly aware of his maleness.\n\nMax straightened up. She couldn't look at him, her eyes were firmly fixed on her hands in her lap. She knew he was studying her, his elbow resting on the steering wheel.\n\nThe silence was unbearable. She wanted to make a joke, anything to get him to start the car and take her home.\n\n'I can wait,' he said softly. 'There's something there, I know it. You're a little mystery girl Georgia. But I'll work it out in time.'\n\n'There's no mystery,' she said, perhaps too quickly. 'You're a married man. You're too old for me. You said yourself there mustn't be even a whiff of scandal.'\n\nHe turned on the ignition and the engine purred softly.\n\n'I'll take you home,' he said flatly. 'Forget everything, Georgia. I've never forced anyone in my life. I'm not going to start now.'\n\nIt was after nine when Georgia woke. Sunshine poured through a crack in the curtain, the sounds of the market as new and fresh as her first morning in the room.\n\nShe stretched like a cat, smiling to herself as she remembered the gig the night before.\n\nIt had been so wonderful and it was going to be repeated again and again.\n\nShe wanted to go out, stand in the sunshine and sing for joy. Get armfuls of flowers to fill the room. Buy something pretty to wear and to hell with the expense.\n\nThe boys were all in the office when she arrived.\n\nHer hair was hanging loose over her shoulders, the pink gingham dress that Pop had given her, freshly ironed with a wide white belt and sandals.\n\n'Well,' Max turned his attention to her, looked her up and down, noting the pink cheeks and bolder, confident stance. 'Something agrees with you!'\n\nThey were all looking at her, eight pairs of eyes, scrutinizing her.\n\n'It's such a lovely day,' she said lamely. She had put aside Max's behaviour the night before. If she was to be a star many more men would want to kiss her. Why blame him for being the first?\n\nMax cleared his throat. 'Sit down all of you,' he said, lighting up a fat cigar and leaning back in his chair. 'I've had _Melody Maker_ on the phone this morning. They are writing a review about you for the next edition, and they want to do a special in the next week or two.'\n\nHe looked intently at each of them.\n\n'It will be \"A day in the Life of\", type thing. The reporter will meet you in the morning, go with you to the gig, unload with you, set up, the whole bit. So I don't want any cock-ups, this could be the big break.'\n\n'How will he get in the van?' Norman said. 'I'm not sitting on a reporter's lap all night.'\n\n'He'll have his own car,' Max said impatiently. 'And a photographer with him. Ian can travel with them, he can be trusted not to tell them anything too damaging.'\n\nThe boys fell silent for a moment.\n\nBoth Ian and Rod were staring at Georgia as if they'd hardly heard Max. She wondered if Max had spoken about her before she came in.\n\n'I've got a list of gigs for next week,' Max said, handing round some copies. 'As you will see it's mostly in the North, leaving here on Monday for Birmingham, on to Preston Tuesday, Leeds Wednesday, Hull Thursday. Travel back to London after Hull. Friday at the Marquee, Saturday at the Bag of Nails.'\n\n'What about digs?' Ian asked, dragging his eyes away from Georgia.\n\n'I've got those here,' Max said, handing Ian another sheet. 'The rooms are booked, go to them before the gig please, so they know you are definitely coming. Georgia, there's a possibility you'll have to share with someone,' he paused, looking at her directly, through hooded eyelids, his tongue just flickering over his lips. 'I apologize, but it was late in the day to make the bookings.'\n\nHe got up and went over to a safe. 'Your money,' he said, handing out small brown envelopes. 'And the kitty for digs and petrol.' This last envelope he handed to Ian. 'Right, Blue Boar tomorrow, have a good rest today and Sunday, you won't be getting much next week.'\n\nHe looked across at Georgia.\n\n'I'd like a word with you please, in private. Will the rest of you wait in the office outside.'\n\nThe boys trooped out, each of them looking at Georgia, questioning looks in their eyes.\n\n'What is it?' she asked, suddenly afraid.\n\n'Well, dear,' he sat down and fiddled with a paper clip, his cigar burning away in a cut-glass ashtray. 'I'm sorry about the rooms. If you have to share make sure it's with Ian. He's the nearest thing to a gentleman.'\n\n'That's all right,' she smiled.\n\n'About last night, I was out of order and I'm sorry.'\n\n'There's nothing to be sorry for,' she hung her head. 'Perhaps it's my fault.'\n\n'The fault was all mine. But please let's put it aside. I don't want any bad feeling between us. You are a fine singer,' he went on, getting up from his desk and moving nearer her. He put one big hand on her hair, caressing it lightly. 'Perhaps the best I've handled, but you are very young, innocent and wet behind the ears. Now if there's anything in your past, you should tell me now, because believe me, one day it will come out.' He paused, looking closely at her, the hooded eyelids looked almost snakelike. 'I can handle anything as long as I know the truth in advance. But if you hide something from me I can't help. Do you understand?'\n\nGeorgia nodded.\n\n'Right, now the only other thing is about your dress. Miriam is taking that gold one back to the shop to change it. But she dug this one out this morning. It's old but we kind of thought it might suit you.'\n\nMax opened a cupboard in the wall and pulled out something white.\n\nGeorgia stood up and took it from him.\n\nIt was an exquisite dress. White silk with handkerchief points from the hip, each one decorated with tiny seed pearls.\n\n'Go and try it on,' he smiled at her awed face. 'The last time I saw that dress was on the evening of our wedding. Miriam was as slim as you then,' he pointed to the bathroom adjoining the office. 'In there.'\n\nIt fitted as if it were made for her.\n\nThe neckline was low both back and front, skimming over her hips, then flaring out gently to the knee. As Georgia tried a twirl in front of the mirror, the skirt moved, showing glimpses of brown thigh. Although it was a style from the Twenties, somehow it was timeless.\n\nShe bounced back into the office, twirling round in front of Max. Her eyes were shining with unsuppressed glee.\n\n'Perfect,' he smiled, putting a large hand on her shoulder to twirl her again. 'Takes me back a few years!'\n\nFor just one second she thought he was going to kiss her again. His eyes looked soft, the tip of his tongue just showing through his teeth.\n\n'Can I show the boys?' she grinned, tossing back her hair, backing away from him.\n\n'Of course,' he said, almost fondly. 'And wear your hair like that with it tomorrow.'\n\nAn hour later they were all in the pub around the corner. The boys were all sinking pints, but Georgia stayed with orange juice.\n\n'So what was the secret chat for?' Norman asked, his small eyes suspicious.\n\n'About behaving myself, not a breath of scandal,' she laughed. 'Is he serious? Will they really try to dredge up stuff about me?'\n\nIan took her to one side later. 'I got the same sort of chat from Max too, about you,' he said softly. 'I've got to be big brother it seems.' He leaned closer to her. 'Did something happen last night?'\n\n'Not really,' she blushed.\n\n'That means something did,' Ian took her hand in his, stroking it, yet hiding his action so the other boys wouldn't see. 'Look, I know he's got that fantastic charisma. He's rich, handsome and he can pull any bird he wants. But he's dangerous Georgia. You can't play games with him.'\n\n'I didn't,' she leaned closer to Ian, needing to unburden herself. 'For a moment I was tempted. But I'm scared of men.'\n\nShe expected him to laugh. To her surprise he didn't, only squeezed her hand tighter.\n\n'I know. I saw it the first day you were with us. I want us to be friends. Perhaps one day you'll tell me about it. When you want to, I'll be there.'\n\n'You all stared at me when I came in the office. Why?'\n\n'The stars in your eyes,' he laughed. 'We thought you had been in the sack with someone. But it was just the magic from last night's gig wasn't it?'\n\n'I felt so wonderful when I woke up,' Georgia confided. 'It was like the first warm day of spring after the long winter. I'm so happy I could burst.'\n\n'You aren't alone,' Ian looked across to the rest of the band. 'We all felt it too. We're going to make the big time together!'\n\n'She's something else,' Norman said dreamily over his pint after Georgia had left to go shopping.\n\n'Sings like an angel,' John said, his dark eyes smouldering. 'And moves like a whore, what more could we ask for?'\n\n'We'll have to be careful,' Ian looked around at his friends. 'I think Max has big ideas for her, and they don't necessarily include us.'\n\n'You mean he'll get her a recording contract and ditch us?' Rod's face clouded over.\n\n'He's only using us to train her. But if I'm right about her, she won't want to go it alone later,' Ian had been awake most of the night, weighing up every word Max had said about Georgia. 'She's the loyal type and I don't think she's got any family. We must stick with her, be that family. I don't think she'll allow Max to ditch us then.' He looked at his hands, the soft, pale face quietly determined. 'Max fancies her like mad too, so make sure she's never alone with him.'\n\nRod watched Ian carefully. They had been friends from school, shared everything from footballs, bikes and later girls. He had heard his friend tossing and turning last night in bed and he suspected it was more than just concern about the band.\n\n'Don't you feel bitter?' Rod wanted to goad Ian into admitting something. 'I mean until last night you were the front man, the one that got noticed?'\n\n'Not at all,' Ian smiled, his dark lashes fanning over his blue eyes, a hint of pink on his round face. 'She's got everything our band needs and deserves. I'm not a brilliant singer, we all know that. But last night she made me better, took me back to the fun we used to have back in the youth club, before we got all serious.'\n\n'You fancy her?' Speedy's auburn eyebrows rose questioningly. 'I mean we all do, but it's not like you to go ape over a girl.'\n\n'She's special,' Ian blushed a faint pink, picking up his pint to hide in. 'But something tells me there's someone, or something else. Until I get to the bottom of that there's no chance.'\n\n## Chapter 12\n\n_1963_\n\n'I'm sick and tired of this,' Georgia flung her stage dress on the floor of the dressing-room, stamping on it in rage. 'Maybe you lot are prepared to let Max walk all over you, but I've had it!'\n\nNo breeze came in the small open window. Eight people trying to change their clothes in a room less than ten feet square. Cigarette smoke, socks and sweat and a sink with only cold water.\n\nHammersmith Odeon. The name was synonymous with success, a big concert venue where fans queued for hours to get tickets to see their favourite stars. But Samson weren't the stars, just a bottom of the bill support group and they would remain that way until Georgia complied with Max's wishes.\n\nA year earlier Alex Rhodes, a scout from Decca, heard her sing and offered her a solo recording contract. If the man hadn't been such a creep, dismissing the band as if they were worthless, along with trying to seduce her, she might have won him round. But instead she'd lost her temper and insulted him. Now she looked around at the boys and saw what her defiance had done. Seven exhausted, pale, drawn faces, skin that reflected their bad diet and lack of fresh air.\n\n'We can't fight back,' Ian's blue eyes were cloudy with apathy. 'We haven't a leg to stand on.'\n\nMax tried everything to tempt the boys away from her. A new van, long contracts on cruise ships, more pay, new suits. At Georgia's insistence all these carrots were refused. Then he resorted to straightforward punishment.\n\nThe roughest digs. Booking them in at venues so far apart the travelling time was doubled. Seven gigs a week and no extra pay.\n\nThey were trapped. If they didn't do the gigs he booked them for, he could sack them for breach of contract.\n\n'Oh, Georgia,' Ian sighed. 'Haven't you learned yet the way it is? The guys in the record companies are all in it with Max. They've got lawyers, heavies and just about everything else on their side. They can afford to wait until we are desperate enough. They can squeeze each one of us dry, until we crack.'\n\n'But I don't understand their motives,' Georgia wanted to scream at the boys' resignation. 'We're good together. I wouldn't be the same without you.'\n\nThe first year with the band had been pure wonder. Each gig had been a dress rehearsal for the big moment when they would cut the first record. They could laugh at the thousands of miles of motorway burned up as they huddled together dreaming of a bright future. Joke about the poor food, wages, and seedy boarding houses. It had all been preparation for the time when Max swung them a contract.\n\nAs punishment made them draw closer to one another, Max resorted to humiliation.\n\nBottom of the bill. The band that opened concerts with big names like Gene Vincent, Adam Faith, Ricky Nelson, Billy Fury and any other name that was flavour of the month. A warm-up band that no one took seriously.\n\nMax knew that playing alongside big names would weaken Georgia's resolve far quicker than playing in dance halls. The support group was the one the fans chatted through, or missed by queuing for the toilet. They were so hyped up at the prospect of seeing stars, they rarely even listened to Samson. Worse still, Georgia could see at close quarters the beautiful clothes, the comfortable coaches and the money these star performers were paid, and each night of these big tours she and her band were reminded of their Cinderella role.\n\nBelow, in the theatre they could hear thousands of fans screaming, clapping, stamping their feet. Adam Faith was out there on the stage. They had prepared the audience for him, wound them up into near hysteria but already they were forgotten.\n\nEach night in major towns they would walk out the stage door unnoticed, past the screaming girl fans waving autograph books. Back to small boarding house, fish and chips and another night in lumpy beds, while the stars drank champagne in sumptuous hotel rooms.\n\nDisappointment and the endless travel was wearing down their loyalty to one another. Everyone of them had received offers to join another, more successful band. It was only a matter of time before the temptation of recognition, money and comfort broke down their bonds of friendship.\n\n'He's got us by the short and curlies,' John's dark eyes were dull now, his dry humour had turned to mere sarcasm. 'Leave us Georgia. You don't have to put up with this shit.'\n\n'John speaks for all of us,' Ian sighed wearily, leaning back against the wall. He was wearing his best dark suit, ready for the flashy end-of-tour party at the Hilton. But looking closely Georgia could see shiny marks from endless pressing, his best shoes paper thin on the soles. 'We love your loyalty Georgia. Most of us would have sold out a year ago for what Rhodes promised you.'\n\n'I won't compromise,' Georgia gave her dress another kick. It had been mended so often she winced if anyone looked closely at her in the wings and Max wouldn't foot the bill for a new one. 'Everything I know came from you. If I do as he asks you'll never earn any more. How are you going to buy houses, get married and have families?'\n\nShe was too tired to argue any longer. If she walked out on them tonight they'd be content to go back to the old club circuit where at least they got all the adulation. She wasn't even sure why she was fighting Max now. Was it for them, for her own ego, or just to keep Ian safe?\n\nShe had learned so much in those first few heady months with the band. Schooled by them her voice grew stronger, mature and power packed. They taught her how to whip an audience into a frenzy of excitement, hold them spellbound and hungry for more. Teasing, playing with the crowd came naturally, yet even as she moved them to tears with the emotion in her voice, her own life was as empty as a dry river.\n\nOutside of the band she had nothing. On nights when she went home alone to her room in Berwick Street she felt cut off, as if part of her was still travelling in the van, tucked in beside all those men. Without them she was like an electric guitar with no power source, and she missed both Celia and Helen to the point of distraction.\n\nThe boys taught her to live without embarrassment. She could strip off in front of them if necessary. Share a bed when it was cold. Change in the van when they were late. They zipped her into stage clothes, got her to the clubs on time. Critical yet supportive. They had rounded her out, educated her, and in return she listened to their problems, mended their clothes as they drove along and cuddled them when they were sad. They were brothers now, but they could never replace the maternal qualities of Celia or Helen.\n\nThere were fierce rows between them. Hours of frosty silence until the problem was resolved. Fights over girls nearly every week and arguments over places in the van.\n\nBut there was lots of laughter too.\n\nSo many times Georgia had walked into the dressing room and found one of the boys with a half-naked young groupie.\n\nHardly a night passed without one of the boys delaying their leaving by slipping out to make love to someone.\n\nGeorgia had seen it all in two years.\n\nSquirming with embarrassment in the back seat, as a girl gave Rod a blow job while they were driving home. Averting her eyes from the girl's bobbing head, trying to pretend she didn't see it.\n\nThere was the time when John had severe diarrhoea and had an accident in the van. Les, so hopelessly drunk he kept his head out of the window from Manchester to Leeds retching violently. Norman caught in the act of making love by the woman's husband, escaping from the small council house wearing only his underpants, clutching his clothes under his arm, with a sixteen-stone bricklayer in fast pursuit.\n\nLandladies who had threatened to call the police when the boys smuggled girls into their rooms. A Dutch cap left in the van that no one knew the owner of. Disasters on stage when Les's trousers split from waist to crutch and he didn't dare move. The driving lesson Rod gave her in the van where she'd swerved across a grassy island almost into the path of an articulated lorry. And the time Norman had jumped from the van into a field for a quick pee, and landed in a freshly-laid cowpat.\n\nLaughter, tears, fights and the promise of fame. An addictive, heady potion that bound them together.\n\nYet the very first night in digs with them was almost her last.\n\n'Is this it?' Georgia said as Norman pulled up at one of the worst hovels she'd ever seen.\n\nIt was a large crumbling house in central Birmingham. Once a substantial Victorian family home, now it was little more than a slum. The front garden was strewn with cardboard boxes, empty bottles and old carpets. Curtains hung from what looked suspiciously like string, the windows hadn't been cleaned in years.\n\n''Fraid so,' Ian smirked. 'What did you imagine? A cottage with roses round the door? A liveried doorman?'\n\n'No,' she said slowly, trying to quell the rising panic inside her. 'But I expected somewhere clean.'\n\nThe front door was opened by a fat, dirty looking man. He was wearing just a stained vest under his braces, his vast belly quivering like a bowl of jelly. A cigarette dangled from his lips, thinning hair hung in greasy strands down his forehead.\n\nHe didn't even speak, but turned and ambled back down the passageway that smelt of boiled cabbage and drains.\n\n'If it's any consolation,' Ian whispered as they made their way up the grubby staircase. 'This is the \"pits\", nowhere else we stay is as bad as this.'\n\n'Well let's hope we don't get to Birmingham that often,' she replied, wincing with distaste as she got a glimpse of a bathroom as bad as the one in Berwick Street when she first met Helen.\n\nThere were only two rooms, with four single beds apiece, she, Ian, John and Speedy were to share one of them. It was like a bad dream, bare lino cracked and peeling, mere hooks on the wall to hang clothes on, and a dressing table held up with a brick under one leg.\n\nThe sheets were supposedly clean, but the dirt was so ingrained she shuddered as she looked at them.\n\n'Watch out for the bed bugs,' Speedy grinned impishly, standing in front of the smeared mirror and combing back his quiff nonchalantly, amused by her horror.\n\n'I hope you are joking,' she shrieked.\n\n'He is,' Ian put an arm round her shoulder, giving Speedy a sharp look. 'We've never been bitten.'\n\nIt got steadily worse. No hot water in the bathroom and the meal they were served at tea-time was disgusting.\n\n'What is it?' she whispered to Speedy as she shuffled the greasy meat around her plate.\n\n'Probably gorilla,' he quipped, eating his regardless.\n\nNow she could laugh about that dining room. The violent orange roses on the wallpaper, the plastic gold Eiffel Tower and the clock shaped like a guitar. No landlord had ever been as comic as the man who plonked huge cracked mugs of strong tea down by each of them silently, stirred them with a knife, still with a cigarette dangling from the corner of his mouth.\n\nHis wife wandered in from time to time, her hair in curlers and the dirtiest apron Georgia had ever seen. She had a grey face, Georgia remembered, as if she hadn't been outdoors for twenty years, or even washed it often.\n\n'We wasn't told about no girl coming,' she said in sullen tones. The front of her hair was a yellowy colour, as if stained with nicotine to match her fingers. 'All that Mr Menzies said was that there was an extra person.'\n\n'I don't mind sharing with the boys,' Georgia tried to appease her.\n\n'Just as well,' the woman snapped. 'But I don't want no funny business in this house. Girls are always trouble.'\n\n'She probably hates another woman seeing the filth they live in,' she whispered to Rod as the woman swept out.\n\n'I heard that!' The woman stuck her head back through the doorway, her grey face contorted with rage. 'Some black tart telling me my house isn't clean. I never heard nothing like it!'\n\nHumiliation took away even the pleasure of her success that night, and later when she got into the lumpy, damp bed she felt dirty and demoralized.\n\nLooking back it seemed ridiculous that she cried the next morning. Floods of tears just because there was no hot water or a lock on the bathroom door.\n\n'I want to go home,' she sobbed to Ian, still wearing her thick pyjamas Myrtle had made her. 'If this is what it's going to be like, I don't want to be a singer.'\n\nBut Ian didn't laugh, or even get cross. One moment she was up there in the dreadful room, the next they were all in the van together and Rod drove them to the local public baths.\n\nThey could have made fun of her, embarrassed her still further, but that day she discovered their sensitivity and their loyalty to one another.\n\n'Feeling better now?' Rod asked when she came out of the baths with a pink scrubbed face and still damp hair. 'We won't let Max book us in a place like that again.'\n\nPerhaps it was inevitable she and Ian should become lovers, but at first it was just friendship borne out of mutual need. The other boys' insatiable need for sex didn't appear to be shared by Ian. While the others paired off to cruise each new town, Georgia and Ian explored together, went to the cinema, swimming or for walks.\n\nHe fitted into the hole vacated by Peter. His blond hair, blue eyes, sensitivity, warmth and compassion were all so similar. He was so easy to be with. He didn't probe at things, or make any demands. He could make her laugh, chat like another girl, yet take her side whenever the others closed ranks on her. Without her even realizing it, he was gradually taking the place of not only Peter but Helen and Celia too.\n\nIn November that first year the mild weather changed suddenly. They were on their way to Plymouth when the wind got up, buffeting the old van so hard, Norman got tired of driving and asked Rod to take over.\n\nAt Exeter it began to rain so hard the wipers couldn't remove the water fast enough from the windscreen.\n\n'We're going to be late,' John said dourly from the back seat. 'Can't you go faster?'\n\n'I can't fucking well see further than a yard ahead,' Rod snapped. 'If you can do better get up here.'\n\nStuck in the back with Ian, John and Speedy, Georgia dozed off to sleep, her head on Ian's shoulder.\n\n'Where are we?' she asked when she woke later. It was dark now but she could see street lighting ahead of them.\n\n'In Plymouth,' Ian said. 'We'll have to go straight to the club and set up. We haven't even got time for a cup of tea.'\n\nShe knew she had missed something while she was asleep. There was a tension in the air, as if they had been arguing.\n\nSomehow nothing went right that night. The bad weather kept the usually large audience away. Speedy broke a string in the first number. John had a coughing fit and Norman threw a tantrum because his Hammond organ had developed a fault.\n\nDuring the second set, just as they were finally getting themselves into the mood, the lights fused and the whole club was thrown into darkness.\n\nGeorgia just stood helplessly, still holding the microphone. She couldn't see anything, but the audience were shouting and screaming.\n\n'Calm down!' she tried to shout above their noise. 'It's only the fuses.' She had visions of a mass stampede in the darkness, bodies being trampled underfoot.\n\nThe manager came on to the stage holding a torch.\n\n'We're trying to fix it,' he shouted above the din. 'Just give us a minute.'\n\nStaff arrived with more torches. Georgia could see them dotted around the club, trying to get everyone to sit down and wait.\n\n'I knew it was going to be a disaster,' Norman said gloomily behind her. 'Any minute now he'll be sending them all off home and he'll blame us for overloading the circuit.'\n\nIt was after one when they finally left the club. The fuses were mended after twenty minutes of pandemonium and the show went on, but they had lost their edge.\n\nThe boarding house was 'The Gainsborough' a tall, sombre looking house of grey stone set on a corner, just behind the Lee's on Plymouth's front.\n\n'I can't be doing with staying up to let my guests in,' Mrs Pengelly the landlady snapped at them, standing in the doorway wearing a brown plaid dressing-gown wrapped round her formidable large body.\n\n'We're sorry,' Ian tried to charm her, explaining as best he could about the storm.\n\nThe rain was still lashing down and it was so cold Georgia's teeth were chattering. She hoped Mrs Pengelly would offer tea and perhaps even a hot water bottle, but instead she just slapped the bolts across the door behind them.\n\n'Rooms three, four and six,' she snapped, taking three keys from a hook on the wall. 'And don't make any noise.'\n\nGeorgia and Ian got the largest and coldest room, set on a corner of the house with two large windows which rattled ominously.\n\n'Christ it's cold in here,' Ian shivered as he pulled off his shirt. 'There's hardly any blankets on the beds either.'\n\nShe was glad to be sharing a room. The house was creepy with the wind howling outside and she was so cold and tired she felt like crying.\n\nWhen she got back from the bathroom wearing her striped pyjamas, Ian was already in the double bed, the covers pulled right up to his chin, curled up as though frozen. She prodded her narrow single bed and banged the hard pillow. It was like so many of their digs. Scratchy nylon sheets, a bedside light with no bulb and a bathroom full of notices about charges for a bath.\n\n'What was going on in the van on the way down here?' she asked once she'd turned the light out.\n\n'The usual,' Ian's voice sounded a long way off, almost drowned by the howling wind. 'Rod wanted to go and look up a couple of birds we used to see down here. I refused and he got a bit nasty.'\n\nRod had girls everywhere. He rarely stayed in digs with the rest of them, even if he didn't have a girl lined up before they arrived, he soon found one.\n\n'Why didn't you want to?' Ian had once been as keen as the others to look up girls, but in the past few months he had become almost a hermit. 'Didn't you like her?'\n\n'She was okay,' he sighed. 'I don't feel the need to prove myself like Rod does. He said I was turning queer.'\n\nRod was very cruel sometimes. He had taunted her too, suggesting there was something odd about a girl who showed no inclination to meet other men. There had been times when he made her so angry she had almost spat out her reasons for not trusting anyone enough. Yet something inside her told her it was better to keep her past firmly to herself.\n\n'Are you cold?' Ian asked, his voice sounded sweet and warm from across the room.\n\n'Freezing,' she said.\n\n'Come in here with me then, this bed's quite comfy.'\n\nGeorgia got out. Her feet were like ice, her breath was like smoke in the darkness. She pulled a blanket off her bed, laid it over Ian's and hopped in beside him.\n\nIt wasn't the first time they'd shared a bed. But it was the first when there wasn't someone else in the room with them. He was good to be close to. He always smelled nice and he didn't try anything on.\n\nGeorgia curled her back up against his stomach, his arm round her middle.\n\n'Better?' he whispered.\n\n'Much,' she curled her feet round so they lay on his warm legs.\n\n'I didn't want to go and see those girls because I prefer to be with you,' he said softly.\n\nShe understood everything with that one simple statement.\n\nIt was she who made the first move. She turned over and nestled her head in the crook of his arm. The warmth of his slim body was comforting, and the faint smell of aftershave stimulated long forgotten feelings.\n\nThe wind howled outside. A tree creaked somewhere down below in the garden. In the distance they could hear the sea hurtling onto the rocks. But the wild night was forgotten as he kissed her.\n\nThat gentle tugging inside her she remembered so well with Peter, warm lips tempting and teasing her, his arms round her so tightly, fear banished.\n\nHis kisses were more practised than Peter's had ever been. She felt herself moulding her body to his, aware her breathing was becoming fiercer.\n\nOne hand crept under her pyjamas jacket, stroking and soothing, slowly upwards till it reached her breast, fingers straying across her nipple.\n\nHalf of her mind was telling her to move away, to stop this now before it went any further, but the other half was responding and wanting more.\n\nHer hand crept down towards his waist, sliding under his T-shirt. His skin was silky, she could feel his spine under her fingers, knobbly and vulnerable.\n\n'Don't Georgia,' he whispered, trembling at her touch. 'Not if you don't mean it to go any further.'\n\nPerhaps it was his willingness to back away which made it safe. All she could do was pull him fiercely to her and offer her lips in silent agreement.\n\nA delicious warmth was spreading over her as his fingers became bolder and unbuttoned her pyjamas, then he moved down the bed to kiss her breasts. She held his head to her, tears filling her eyes with the beauty of it, breathing suspended as waves of sensual delight washed over her.\n\nSlowly his hand moved down to her waist, pushing down her pyjamas, smoothing her belly in circling movements, growing lower and lower with each one. As it moved down to her legs, caressing, teasing, she could feel a strange wetness between her legs and she longed for his fingers to make their way there.\n\nBut each time she thought he was going to touch her, so he moved his hand away, to her breast or to her bottom, his lips covering hers, and teasing her with his tongue.\n\nHer heart was pounding, breath coming loud and harsh. She grabbed his hand shamelessly and guided it to where she wanted it, groaning with delight as his fingers touched her parted lips.\n\nShe hadn't expected it to be so wonderful. Her own fingers had never felt like this. She was on fire, driven to some wonderful secret place, oblivious to anything other than those probing, gentle fingers and the heat of strange sensations washing over her.\n\nAgain and again her hands reached down to touch the bulge she could feel so close to her belly. His deep breathing would pause, she sensed he was willing her to touch it, yet she was afraid.\n\nHe moved away just slightly, his lips still on her breast, sucking at her nipples, taking his pants off with one hand. She stiffened involuntarily.\n\n'I'm not going to hurt you,' he whispered, burying his face in her neck. 'I love you Georgia, trust me.'\n\nInstinctively she knew this wasn't a trite remark made in the heat of passion to persuade her. Their close friendship had been building up to this moment and she was just too na\u00efve to see it before.\n\nHe took her hand, held it for a moment to his chest, then slowly moved it down onto his belly.\n\n'There,' he whispered as he closed her fingers round his penis. 'Is it so bad?'\n\nThere was laughter in his voice, nothing intense or scary. It wasn't even slimy as she'd expected, just hard and warm. He was kissing her again, his fingers probing deep within her and all at once she really wanted him.\n\n'My love,' he whispered. 'You can't imagine how often I've dreamed about this!'\n\nIt was she who rushed things, pulling him on to her, digging her fingers into his back and although she braced herself for pain, there was none.\n\nThe icy room, the wind outside had all faded. Waves of pleasure, coming hard and frantic, getting faster and faster, in time to his movements. Lips reaching for the other's. His hands on her buttocks, hers caressing his back. She was beginning to drift away onto some strange fiery plane, where all sense of time and place had ceased, when he moved back from her, dropping on to her, panting furiously.\n\nFor a moment she didn't understand, her hips were still undulating under him and their bellies were hot and sticky.\n\n'I'm sorry,' he whispered against her neck as his breathing slowly returned to normal. 'I only remembered at the last moment.'\n\n'Remembered what?' she whispered back, feeling strangely let down, but unsure why.\n\n'About Durex,' he said. 'I don't want to get you pregnant.'\n\nHe lay down next to her then, pulling her into his arms and kissing her deeply.\n\n'Did it hurt?' he whispered.\n\n'No,' she wound her fingers into his hair, her heart filled with tenderness for him. 'It was beautiful.'\n\n'Tell me about you,' his voice was husky with emotion. 'I know something happened before. Was it rape?'\n\nTheir act of love was an anaesthetic. She found she could tell him the whole story without pain. For Helen and Janet it had been abridged, the poison left inside the wound to flare up again unexpectedly. But now as she told Ian safe in his arms, she felt the healing process begin.\n\n'I'll make you forget,' he whispered, squeezing her to him. 'One day it will be so perfect he'll be wiped out forever.'\n\nGeorgia stroked his face, feeling the dampness round his eyes.\n\n'I love you, Ian!' she whispered. 'You've made me whole again.'\n\nShe did love Ian. How could she not love someone as special, sensitive and caring as him? He filled a place in her life so fully she didn't need anyone else. They had the same dreams and ambitions for their music. He'd driven the demons out of her head, shown her the beauty of sexual love.\n\nSecrecy was necessary. If Max found out they both knew he would move swiftly to put an end to it. Ian crept into her room after the others had gone to sleep, hidden hand-holding in the van, secret excursions together during afternoons in strange towns.\n\nShe saw new maturity in his angelic face. He put on a little more weight, his delicate features grew stronger.\n\nDisappointment and hardships bound them closer together. He didn't ask for commitment, he was content just to share his days and nights with her. Never asking about his part in her future, urging her to take Max's offer. Yet why with all this perfection were there times when she despised him! What sort of person was she that she was so irritated by his sheer goodness? Why did she want to scream at him, shake him, anything to make him wake up and see things as they really were.\n\nHer feelings for Max baffled her still further. She hated the way he was manipulating her and the boys, yet admired his single-mindedness at the same time. He was cruel, greedy, ruthless. As far apart from Ian as a predatory eagle to a chirpy robin. Sitting back in his plush office watching her struggle helplessly just like that fly in his gold spider's web. What weakness on her part made her feel that eventually he would get exactly just what he wanted? Not just a gold record or two, but maybe her body and soul!\n\nBut one thing was clear, which ever way she turned Ian would lose. Samson would flounder without her; if she stayed the pressure Max created would do the same. Had Ian really thought what would happen if she became a big star and his role was little more than a hanger-on?\n\n'Let's go,' Speedy picked up the red stage dress and brushed it off. He shot a look of understanding at Georgia. 'There's a party to go to, remember?'\n\nThis party was the one thing the boys had been looking forward to for weeks. It would be a glitzy, star-studded event that would hit the newspapers the next morning. Girls in plenty, free food and drink. Enough to make them forget the humiliations Max had thrown at them.\n\n'I don't fancy it,' Georgia said wearily. She was sick and tired of people for now. She wanted to be alone in her room, not getting dressed up and posing as the rising star.\n\n'Don't be daft,' Rod said brightly. 'You can't not turn up. We're all ready to go.' Rod looked like a star himself, in black leather trousers and a ruffled shirt, his hair cut in the new 'Beatles' style.\n\n'You all go,' she said firmly. 'Have a great time, get drunk and screw as many birds as you can find. I'm going home.'\n\nIan was looking at her, one eyebrow raised. A 'does that mean you want me to come home with you' face.\n\n'You go too Ian,' she said turning away from him so she couldn't see the sad look she knew would come into those beautiful blue eyes.\n\nHe followed her out the door, catching hold of her bare arm and squeezing it.\n\n'Have I done something?' he said, his eyes soft like a puppy's.\n\nFor a moment she weakened. His long dark lashes framing his eyes, a perfect straight nose and that soft vulnerable mouth. He was perfection, not just his angelic beauty but the depth of his love and understanding for her.\n\n'No, of course not,' she lifted a hand to his face, stroking it tenderly. 'I just can't cope tonight. I'd only spoil it for all of you. Get drunk and be silly and you can tell me all about it tomorrow night. We've got a week off remember? Maybe we can go somewhere together.'\n\n'I love you,' he took her face in both his hands, regardless of people barging along the corridor.\n\nHis kiss was sweet and lingering. She felt the dressing-room door open behind them and someone look out, but she no longer cared.\n\n'I love you too,' she whispered. 'Now go on and have fun.'\n\nHe was still standing by the door as she reached the end of the narrow corridor. Downstairs there were shrieks of girls' laughter, mingling with male voices and the popping of champagne corks. 'Love, love me do', the Beatles' song was playing at full volume.\n\nShe lifted her hand and blew him a kiss. She could see his soft lips curved into a smile, brushing back his floppy fair hair impatiently from his eyes. Tomorrow she had to come to a firm decision about him, it wasn't fair to take his love yet give him no real commitment in return.\n\nOutside in Hammersmith Broadway she slipped unnoticed through the huge crowd of fans waiting for the big stars to make their appearance. It was a hot night, only just getting dark, the traffic as heavy as if it were six in the evening instead of ten thirty. No one noticed the slim dark girl in a pink dress pushing her way through the crowd.\n\nShe would catch a tube to Piccadilly, wander about the way she used to do with Helen. Tomorrow she would go in and see Pop and the girls. By then she'd be feeling her old self again. It was just exhaustion that made her feel so prickly.\n\nIt was almost twelve as she approached Berwick Street. Nothing had changed here. It was as dirty, smelly and full of noise as always, but just seeing her front door made her more cheerful.\n\nBert had surprised her in the last year. He'd spent money on the house, turning the lower rooms into a suite of offices, and she was the only tenant left. What had once been the small landing outside her room was now a tiny bathroom just for her, only a shower, basin and toilet, but it was bright and new. A fitted carpet had been laid. Not only in her room but up the stairs too. The old cooker and sink had been replaced with a smart fitted sink, a baby belling cooker and a fridge. She knew Bert had made these improvements not for her, but as an investment. When she moved out he would treble the rent, or even sell the entire place at a huge profit. But at least she had no need to feel ashamed of where she lived now.\n\n'Georgia! What are you playing at?'\n\nShe had been so deep in her thoughts she hadn't spotted the new maroon MK 10 Jaguar parked outside. Max was sitting behind the wheel, his arm resting on the open window, wearing a white dinner jacket.\n\nHe could have passed for a film star as he leapt out of his car. His tanned, rugged face, the white jacket gleaming under the street light, his wide shoulders, animal grace and his sensuous features were enough to make any woman stop and stare.\n\n'Why aren't you at the party?' he asked.\n\n'I couldn't face it,' she sighed. 'Don't get at me Max. I'm not in the mood.'\n\n'I didn't come here to get at you,' he snapped. 'I was concerned about you. I was in the audience tonight and I thought you lacked your usual sparkle. Then the boys said you'd gone off alone. What's wrong?'\n\n'I'm just pissed off,' she snapped back at him, getting out her key and putting it in the door. 'Don't say I'm not even allowed an off day or I'll spit at you.'\n\nHis big hand covered hers on the lock.\n\n'Come and have something to eat with me?' His voice was softer, almost understanding. She could feel his body close to hers and for some reason it felt comforting. 'I'll take you somewhere nice and quiet, feed you up and let you relax. I didn't come to fight with you.'\n\n'I want to go to bed,' she said weakly. In fact she was very hungry and she knew the only food she had was a tin of baked beans.\n\n'You don't,' he insisted. 'I know perfectly well you didn't get up till two o'clock today. You may be tired and fed up, but you aren't sleepy. Now hop in the car and we'll go somewhere.'\n\nShe hadn't the will to argue further. Perhaps it was time she talked to him instead of ducking the issue. Maybe if she told him about how bad things were he might stop persecuting the band.\n\nHe drove silently up the narrow road to Oxford Street, then turned up towards Marble Arch.\n\n'Don't even think of taking me to the party,' she said quickly.\n\n'Not all roads lead to Park Lane,' he grinned. 'Though what on earth you've got against a bash like that I'll never know.'\n\n'Girls getting goosed in corners. Loud-mouthed louts making fools of themselves and all the PR birds falling over themselves to get one of the stars into bed,' she said bitterly. 'I can live without that.'\n\n'You sound like an old lady,' Max smiled. 'Could it be you wouldn't be the centre of attention?'\n\n'I'm just tired!'\n\nHe was right of course, when had she ever wanted to pass up a party before? Adam Faith was fun, he wasn't such a big shot, and if the PR girls were a pain, at least all the other performers agreed with her. 'But we can't go on like this Max, it isn't fair and you know it. Our band is far better than any of the others. We deserve more than bottom of the bill.'\n\nMax didn't answer. They were passing the Hilton's glass frontage, awash with golden light. A Rolls Royce drew up outside and the doorman rushed forward to welcome the occupant.\n\n'Petula Clark,' Max waved one big hand, his gold watch gleaming on a thick wrist. 'One day you'll have a car like that Georgia. Just stop being so stubborn and listen to reason.\n\nHe took her to a small restaurant in Chelsea, ushering her through the main area to a floodlit garden beyond.\n\n'It's too nice to be indoors tonight,' he smiled at Georgia's rapt face. She looked like a little girl tonight, in that pink cotton dress and her hair in a pony-tail. Surrounded as he was by predatory secretaries who hid behind masks of make-up, her innocence and straight talking was a tonic. 'The food's good here too.'\n\nIt reminded Georgia of places she'd seen on films. Honeysuckle covered walls, urns of bright petunias and pansies with little stone statues half hidden beneath the foliage.\n\nThey were the only people eating outside, soft music wafted out as they ate French onion soup and Max kept filling up her wine glass.\n\n'It's like being on holiday,' she grinned. Max could be so charming when he wanted to be. She hadn't eaten anything more than fry-ups and hamburgers for weeks and from inside the restaurant she could smell sizzling steaks being grilled.\n\n'Come away for a few days with me,' he said unexpectedly. 'We could catch a flight tomorrow to Spain. You could have this every night.'\n\nFor a moment she could just see a golden beach, turquoise sea and palms waving in a soft breeze. She could feel the sun on her shoulders, the sand between her toes.\n\n'No strings,' he smiled, sensing her temptation. 'Separate rooms. Just time to talk and relax.'\n\n'I couldn't,' she reluctantly pulled herself back to reality, Ian's face full of hurt. 'Besides I haven't any clothes.'\n\n'We could buy anything you need,' he leaned closer across the table, putting one big hand over hers. 'Of course you can come. What's stopping you?'\n\n'The boys,' she merely whispered it. 'They'd think I'd sold out.'\n\n'Would you come if I said I'd allow you to make a record with them?'\n\nGeorgia felt her stomach turn over. Was this blackmail, teasing, or merely trying to find out her price?\n\n'Are you serious?'\n\nHe was such a good-looking man. His face in the soft floodlights had a golden glow, his dark, hooded eyes so sexy she felt she wanted to reach out and touch him even if he was the enemy.\n\nThe waiter interrupted the moment by clearing away their soup bowls. It gave her time to collect herself.\n\nWould the thought of making a record absolve her from being alone in Spain with Max? Would Ian believe nothing happened?\n\n'Of course I'm serious,' he looked at her through half closed eyes, a faint smile twitching his lips.\n\n'What would Miriam say about this?' By turning the tables on him she was giving herself time to think.\n\n'I'd tell her and the boys we were going to see a promoter,' he said too glibly. He hadn't really planned anything tonight. It had all tumbled into his head when he saw her sad, troubled expression. He wanted her. He had from the moment he clapped eyes on her, her combination of girlish sweetness and the tough rock singer was enough to give any man a hard on. But for now he was content to woo her.\n\n'I can't make up my mind just like that,' she said weakly, tempted now to the point where she was almost agreeing.\n\n'There's plenty of time,' he smiled, sensing she was almost his. 'I'm serious Georgia. I can't fight you any longer this way. I can't swing a recording contract for the band, but if they came up with the right song, written by themselves, we could insist they got equal billing. After the first hit, Decca might review the situation.'\n\nThe steaks arrived with a huge bowl of salad. Georgia's mind was still churning over his idea. Was this another ploy to weaken her resolve? Could she hold him to it, get a real promise before they even talked seriously about getting on a plane?\n\nThe lack of contract for the boys wasn't important. If her first record was a hit, there would be enough exposure to launch them in their own right. They would have the shared royalties and her debt to them would be paid in full.\n\nThe steak was perfect, succulent and tender. The wine was giving her a rosy glow and the smell of honeysuckle filled her nostrils.\n\nMax was talking about one of his new bands. Georgia smiled as if she was listening carefully, but all the time her mind was on Ian.\n\nHe was the stumbling block. Samson for all their talent couldn't survive without a strong singer. They would have to replace him or lose all credibility. How could she even think of putting Ian in such a precarious position?\n\nThe steaks were followed by strawberries and cream in huge glass goblets. Max filled up her glass yet again, then sat back and lit up a cigar.\n\n'You're worried about Ian?' he said, looking at her with half-closed eyes. 'How long has it been going on?'\n\nHe hadn't been certain before. Secret looks between the pair of them, the lack of complaints about sharing rooms. Jokes from the other boys. Nothing definite to point to involvement with the lad, but he could see that look of concern in her eyes and he knew it wasn't for the rest of the band.\n\n'Eighteen months,' she sighed. It was too late for lies. Why should she cover it up anyway?\n\nMax crossed his legs, tilting back his chair. The floodlights reflecting on the foliage around him had turned his face green. He looked sinister now. Calculating and mean.\n\n'What made you get involved with him?' he snapped. 'He's a nice enough lad, but by God Georgia, couldn't you look ahead and see the problems?'\n\n'I could say the same about you,' she retorted. 'You've got a wife you don't seem to consider. Did you look ahead and see that coming?'\n\n'Leave Miriam out of this,' his mouth turned mean and bad tempered. 'We're talking about your future. I told you that lad was the weak link. He's useless as a singer and if you stay together he'll be a millstone round your neck. If it had been Rod I could have understood it, or Speedy. They've got a future. Ian's got nothing.'\n\nHe painted the scene she'd seen so often at the back of her mind. Ian waiting patiently at home while she got all the glory, how long would it be before his role was nothing more than a lap dog?\n\n'Don't you dare speak about Ian like that,' Georgia stood up, her chair tumbling over behind her. 'You may be my manager. But I won't stand by and let you belittle him.'\n\n'Sit down darling,' he reached out for her hand. 'People are looking!'\n\nOut of the corner of her eye she could see heads turning, middle-class socialites looking down their noses at her cheap cotton dress and assuming she was a pick-up.\n\n'I don't give a shit,' she snarled. 'You've got no idea how much of Samson is Ian have you? Without his ideas and drive they would never have got off the ground. If you had any sense you'd be looking for another slot to fit him into, on the management side, instead of trying to get rid of him.'\n\nA wave of red-hot anger was washing over her. This was the man who was responsible for all the problems, yet for a moment she'd been tempted to conspire with him!\n\n'It won't be me who gets rid of him,' Max arched one eyebrow. 'The lure of fame and wealth does many things to people.'\n\n'You should know,' she snarled. 'You wrote the dirty tricks book didn't you?'\n\nShe turned and ran then, straight through the restaurant, pushing aside a waiter, tears streaming down her cheeks, out into the street.\n\n'Is the young lady coming back, Mr Menzies?' The waiter approached tentatively, his thin body bending in supplication.\n\n'I doubt it,' Max snapped. 'Just get me a large brandy.'\n\n## Chapter 13\n\nAs Georgia rushed headlong towards Sloane Square, angry with herself for being foolish enough to believe she and Max had anything to discuss, the boys were just leaving the Hilton.\n\n'We'd better get a cab,' John wobbled unsteadily along the pavement.\n\nThey were all drunk. Ian so bad he could barely walk, lost in a silent world of his own.\n\n'We can't all get in one,' Rod said, slurring his words. 'Perhaps we ought to walk anyway.'\n\nSpeedy turned at a high-pitched shriek from behind them. A public relations girl from Decca was tottering towards them. Earlier in the evening her blonde hair had been piled up on her head in elaborate curls, now it was dishevelled, falling over her face.\n\n'Take me home Speedy,' she called out.\n\n'One down,' Rod murmured, pausing against a car showroom window and pressing his face against the glass. A gleaming red Mercedes was turning round slowly on a revolving platform, it made him feel even drunker.\n\nIan swayed, white-faced, eyes half closed a few feet from Rod. John and Alan were sitting on the kerb. Les was throwing up noisily behind a parked car.\n\nSpeedy and the girl crossed the road towards Knightsbridge. The tall blonde girl's white dress was so tight she could only hobble, she trailed a pink feather boa over her shoulder, unaware it was touching the pavement behind her. The pair of them had linked arms and it was hard to see which one was holding the other up.\n\n'He won't be much good to her when they get back,' Rod said to nobody in particular. 'But then I think he's already given her one in the toilets.'\n\nIt had been a wonderful party. Scores of girls milling around, convinced Rod was every bit as much of a star as Adam Faith, Billy Fury and all the others. One tart had given him a wank under the table and he could have pulled anyone he wanted if he'd put his mind to it.\n\nBut Ian was bugging him. He hadn't said or done anything other than drink himself stupid, and he knew it was all to do with Georgia.\n\n'Where's Norman?' he asked, lurching forward towards where John and Alan sat hunched mindlessly on the kerb.\n\n'Gone off with some girl,' John turned bloodshot eyes on Rod. 'Can you get it together to flag down a cab?'\n\nIan was usually the one that rounded them all up after nights like these, but one glance at his blank face was enough to know he was the one who needed looking after.\n\n'All right,' Rod lurched into the road and put up his hand.\n\nA taxi came seconds later.\n\n'Where to mate?' A raw-boned face under a flat cap looked out suspiciously.\n\n'Ladbroke Square,' Rod said. 'Just a minute while I get my mates.'\n\n'Don't any of you throw up in the back otherwise I'll rub your noses in it,' the taxi driver said sharply. 'And open the bloody windows, the fumes are enough to make me sick.'\n\nRod paid the driver as the others almost fell out the taxi. Les rushed back to the gutter, and once again vomited.\n\nThe house was in darkness. Timed switches on the wall turned themselves off even before they climbed slowly to the first landing. A smell of curry came from behind the door of a Jamaican family. Their flat was right at the top of the house. Three large rooms on the fourth floor and another up a flight of steep narrow steps to the attic.\n\nRod put his arm round Ian, supporting him as they made their way up. Alan and John had already reached the top now. Les was hauling himself up by the banisters like an old man.\n\n'What's been eating you tonight?' Rod asked Ian as they finally got in the front door.\n\n'Georgia,' Ian said stupidly, his mouth drooping. 'She's only staying with the band because of me and I can't bear it any longer.'\n\n'Come on now mate,' Rod eased Ian back into an armchair. The whole flat was filthy. He would have to find some bird to come and sort it out. Clothes in heaps, records all over the floor, piled up ashtrays, half eaten food going mouldy on plates left there weeks ago, and at least two dozen empty beer bottles.\n\nThe flat was a mixture of tastes. Speedy's collection of old books. Ian's posters and records of blues and soul artists. Rod himself was responsible for the red warning road lamps, picked up on another drunken binge. There were nude girl posters. A Salvador Dali print and a bull-fighting poster personalized with Norman's name. A black and red pair of knickers had been on the table lamp so long no one could even remember who they belonged to. A messy, tasteless place, but it was a storehouse of good memories.\n\nAlan was lurching up the steep stairs that led to the attic. His new grey leather jacket was stained with drink and he'd burned a hole in his one good pair of black trousers. His small, boyish face grinned inanely, his blond hair brushed down on his forehead made him look like a medieval page boy. He didn't even realize he was making for the wrong room. The attic belonged to Rod and Ian, but Rod couldn't be bothered to turf him out. Les was in the bathroom being sick yet again and John was attempting to make some coffee.\n\n'Get some in here sharpish.' Rod called out. He turned back to Ian and crouched down beside him. Even drunk, Ian was as immaculate as when he arrived at the party. Tie neatly knotted, shirt crisp and fresh. Even his hair was perfect, baby soft, shining pale yellow in the murky light.\n\nRod knew Ian loved Georgia. He'd even been jealous at first, hurt that anyone could come between him and his best friend. But Ian was level-headed. He had maintained all his old enthusiasm in the band, perhaps even increasing it. What had happened to make him react like this now?\n\n'Don't worry about it,' Rod felt a pang of sympathy for his friend. If a girl made him miserable, he just went out and found another, but Ian was different. 'Why don't we give her the elbow? Insist we want to go it alone that way she won't have to feel bad about us?'\n\n'Don't be such a fucking idiot,' Ian seemed to sober up suddenly, his blue eyes flashing ice cold. 'Do you really think I could fill Georgia's shoes as singer?'\n\n'We did all right before she joined us!' Rod said evenly.\n\n'We didn't know what a good singer could do then, did we?' Ian slumped back in his seat, eyes closing. 'You'd be looking to me to make it right. The audience would wonder why you had a wanker like me up front. I'm stuck in the bloody middle holding you both back. That's the reason she won't go. She's afraid for me.'\n\n'Aren't you forgetting we all used to sing before she joined us?' Rod had never seen Ian like this before. 'Mick Jagger's voice ain't so hot, and what about the Beatles? All we have to do is use a bit of strategy. Stop feeling sorry for yourself mate. Georgia loves you, anyone would think she wanted to dump you!'\n\n'Maybe she does!' Ian's head slumped down towards his chest. He looked defeated and old. 'Aren't I more like a brother to her? You can love someone, and be \"in love\" with them, the two aren't necessarily the same. She loves all of us. That's the bloody trouble.'\n\n'Of course she loves you,' Rod said firmly. He sat down on the arm of the chair and hugged his friend awkwardly. 'After all it's you she sleeps with!'\n\nRod wasn't one for soul searching. A girl to him was a diversion, nothing more. They all had their attractions, once he'd played them out, he moved on. He couldn't really understand why other men agonized over feelings, it was such a waste of energy.\n\n'You don't know anything about her,' Ian sniffed. 'Her father raped her when she was only fifteen. That's why there was never any blokes in her life until me. If any of you had been the one to press the right buttons you might be the one sharing her bed now.'\n\nRod was too drunk to think that one out, or even be shocked.\n\n'It's you she loves, you prat,' Rod hauled him up by the shoulders. 'You was always mates, right from the start and she ain't the type to use anyone. All you gotta do is to push her out on her own, 'cos she's too damned stubborn to do it herself.'\n\n'Then why didn't she come with me to the party tonight?'\n\n'I expect she knew we'd all end up rat-arsed like this,' Rod grinned. John was swaying behind him with three cups of coffee. 'Now drink that and piss off to bed. And stop seeing problems where there aren't any.'\n\nIan seemed to pull himself together a little after his coffee.\n\n'I shouldn't have told you about Georgia,' he said as he went on up the steep stairs to the attic. 'I only told you because I know you're a real human being under that flashy exterior. You won't let on to anyone?'\n\n'You're a wanker when you're pissed,' Rod called after him. 'Any more crap from you and I'll tell her what a feeble little prat you can be.'\n\nRod woke up, a faint smell of something in his nostrils.\n\nFor a moment he couldn't quite place where he was, he had cramp in his legs from lying curled up on the couch and as he moved he fell on to the floor knocking over the half drunk coffee.\n\n'Shit,' he exclaimed. It had splattered his new shirt and it would stain if he didn't see to it.\n\nStanding up, he looked around.\n\nThere was a smell, and a strange noise, a kind of faint roaring above him. His head felt as if it were full of cotton wool, his stomach was churning. Yet he was sure he wasn't imagining that smell.\n\n'Fire!' he yelled, running out the door and leaping up the stairs to the attic, three at a time. The door was closed, smoke billowing under it and as he opened it another cloud of it hit him in the face. Beyond the smoke were flames. Licking up by the window, fanned by the draught from the door.\n\nHe closed the door quickly, leaped down the stairs again and rushed into the other bedroom.\n\n'John, Les,' he shouted, shaking their arms. 'Fire! Ring the fucking fire brigade. Alan and Ian are in there burning alive!'\n\nJohn was up immediately. Dark eyes gleaming in the dark. He grabbed his trousers and ran for the door.\n\n'Les, for fuck's sake wake up,' Rod screamed now. 'Fire. Get out now!'\n\nHe stopped only long enough to grap a heap of blankets, his heart thumping like a steam hammer.\n\nRunning into the bathroom he turned the taps on full, then dunked the blankets in, hauled one out and carrying it dripping in his arms he made his way up the stairs again.\n\n'Get the other tenants out, Les,' he shouted back down the stairs as Les lumbered out. His sallow, long face stared stupidly, dark hair hanging over one eye. He wore only a pair of jeans, feet bare, his sunken chest and spindly arms almost pitiful. 'Get going!'\n\nThe smoke made him choke as he opened the door and the heat of the flames made him recoil momentarily. Putting the blanket over his head he got down on all fours and began to crawl across the floor.\n\nThe bed closest to the window was well and truly alight, flames licking over it, but the black smoke prevented him from seeing anything. The curtains had caught, taking the fire to the other bed, and he could just make out the outline of a body, flames just flicking out like evil fingers to consume it.\n\nHe threw the blanket over the body, standing just for a moment choking with the fumes. Then grabbing the body by the feet he hauled it away from the bed.\n\nThere was a thud as it hit the floor, but it was too late to be cautious. Blinded by the black smoke he hauled it back across the floor, towards the door.\n\nIt was Alan. His face blackened by smoke, burns on his arms and legs, his fair hair almost completely gone on one side of his head, sharp little features so familiar, but different seen black.\n\nJohn was coming back in the front door.\n\n'Bring me more blankets!' Rod shouted. 'In the bath.'\n\nRod stood coughing on the landing. His hands were burned but that was unimportant. Somehow he had to get Ian out of there too.\n\n'Don't go in again,' John flung him blankets, rushing up to look at Alan lying inert on the landing bending over him to feel for a pulse. 'The floor might go. Wait for the fire brigade.'\n\n'Try giving him the kiss of life,' Rod hissed. 'I'm going back.' He dropped down to his knees, one blanket over him, dragging the other behind him, as at last the sound of sirens wailed down the street.\n\n'Up the top,' Les directed the firemen at the front door. He had sobered up the moment he knew what had happened, but now he wished he'd had the presence of mind to grab shoes. He had cut his foot on a piece of broken glass and blood was pouring out.\n\n'You stay here lad,' the big fireman pushed him aside. 'Who else is in there?'\n\n'Three of my mates,' Les tried to follow but was pushed back. 'No, four, John went back into help.'\n\nThe two first fire engines were quickly joined by another two making a 'V' shape in the road. Firemen rushed in one after another, hauling hoses behind them.\n\nOutside a crowd was gathering, more stood behind the railings of the communal gardens in the centre of the square. Old women in nightdresses. Children in pyjamas, men wearing nothing but a pair of trousers like himself.\n\n'How did it start?' A fireman tugged at Les's arm as he stared up at the house. Flames were flickering on the roof now, turning the dark sky purple. Thick black choking smoke wafted down, almost concealing the ladder from the fire engine and the man on it wielding a high-pressure hose. Another two fire engines came roaring into the square, the men leaping out even before the engine stopped.\n\n'I don't know,' Les was crying now, tears splashing down over his cheeks, a dew drop gathering on his hooked nose. 'We'd been to a party. We were all drunk. Perhaps it was a cigarette.'\n\nJohn came out first, quickly followed by two firemen carrying a body on a stretcher.\n\nLes broke through the cordoned-off area and ran to John.\n\n'Who is it?' he asked. 'Where's Rod?'\n\n'That's Alan,' John's dark eyes were filled with tears. He reached out for Les like a child, burying his head on his shoulder. 'Rod went back to try and get Ian. I think he's copped it too.'\n\n'Is Alan dead?' He could hardly bear to ask.\n\n'Yes, overcome by the fumes before the fire reached him. I tried to revive him, but I didn't know how. Rod risked everything, for nothing.'\n\nIt was like some hideous nightmare. John and Les stood bare-chested, huddled together. Windows opened, more people drifted onto the square in their nightclothes, all around them were whispers, pointing them out and giving other neighbours their opinion as to who was still inside.\n\nTwo firemen came out again with a stretcher. Their faces black, the whites of their eyes showing clearly in contrast, under their helmets.\n\n'Who is it?' John cried out.\n\n'The one who went in to rescue them. He's in a bad way.' The firemen handed over the stretcher to the ambulance. 'But he's alive.'\n\n'And Ian?' They both knew the answer, but they had to ask.\n\n'He didn't stand a chance,' Another fireman came out of the doorway, he moved closer to them putting one big gloved hand on John's shoulder. 'He was dead before your friend even got in there. I'm sorry.'\n\nGeorgia jumped violently as the doorbell rang. For a moment she ignored it, thinking it was for one of the offices downstairs. Again and again it rang, and slowly she got out of bed to look out the window.\n\nSunday mornings were the only time the street was quiet. Empty of stalls, the only litter from drunken revellers the night before.\n\n'Who is it?' she called down, peering out over the window-sill.\n\n'It's John,' an ashen face turned up to her. He was wearing nothing over his bare chest but an old cardigan. 'There's been a fire!'\n\nShe threw a key down to him, then hastily pulled on jeans and a shirt. She was just pushing her feet into a pair of plimsolls when John came through the door.\n\n'Ian and Alan are both dead,' he said sinking into her arms like a child. 'Our flat's gone, and Rod's seriously hurt.'\n\nFor a moment Georgia just held John staring vacantly at a poster of poppies on her wall.\n\n'Ian, dead?' she whispered.\n\nShe made him coffee, putting in lots of sugar, and found him a shirt of Ian's left in her wardrobe. She was shaking, yet she couldn't cry. She knew she was behaving like a mere neighbour, not like someone who'd just been told her lover was dead. All the time she kept thinking it was a dream, that any minute she'd wake up and find she was alone.\n\nBut John was very real. He was icy cold to the touch. Bristles of beard showed through on his chin. His feet were bare and his eyes were full of tears, why couldn't she share them?\n\n'Let me find you something to put on your feet,' she said. 'Ian left some shoes here.'\n\n'Georgia, have you heard a word?' John shouted at her. 'Ian and Alan are dead, Rod maybe too by now and you worry about my feet?'\n\nHe grabbed her in his arms, holding her so tightly she could barely breathe, sobbing on her shoulder like a two-year-old. She could see Ian standing in the corridor at the Odeon last night, his hand lifted to return her kiss. All at once the reality hit her. She saw flames licking around that slender body, reaching out for his blond silky hair.\n\nJohn held her against his shoulder, patting her back as they sobbed together. 'My two best mates just wiped out and I never even said goodbye to either of them.'\n\nIt was just half past seven in the morning when John woke her and as the day slowly passed there was no end to the misery. Rod had been taken to Hammersmith hospital. Alan's body was already in the morgue and Ian's charred remains followed later. Les's foot was so badly cut it needed stitching and John, after rushing across London to find Georgia, had sunk into shock, barely speaking. Alan and he had been like brothers from the age of six, their whole adult life had been spent together, even most of their possessions were shared. Now he was gone, and the next closest friend, Ian, with him.\n\n'Can you tell me about Rod?' Georgia asked the nurses who scurried by, averting their eyes from the two boys still with smoke smuts on their faces sitting there shoeless and silent.\n\n'He's holding his own,' was all they would say. 'It's too soon to know yet.'\n\nLater, Speedy arrived at the hospital. He had gone home around ten in the morning to find the house still faintly smoking, the whole top floor caved in. He at least was calm enough to find John and Les other clothes and shoes from a nearby friend and to insist they washed and combed their hair.\n\nNorman's whereabouts were still unknown. Soon the parents of Ian, Alan and Rod would be wanting to talk to John and Les, but neither was capable of anything more than nodding and shaking their heads.\n\nMax came striding down the corridor around noon. He had been out when John had phoned him early that morning and Georgia suspected he had probably stayed out all night. He had stubble on his chin, his eyes bloodshot.\n\n'How did it happen?' he demanded of John, even though he could see the boy was hardly able to speak.\n\n'They don't know,' Georgia spoke up. 'They think Ian must have been smoking in bed and dropped it on the floor. What are we going to do?'\n\nShe waited for Max to say something harsh, but to her surprise his eyes filled with tears.\n\n'It's not fair,' he said, turning away so they couldn't see his face. 'So much to look forward to, snatched away in their prime.'\n\nGeorgia stood up, putting one hand on his arm, the other wiped away his tears with her hanky. His shoulders were slumped, his full lips quivering.\n\n'We've got to pray Rod will make it,' she said.\n\n'Speedy,' Max straightened up, getting back his self control. He sniffed, wiping one big hand across his eyes. 'Take John and Les to my house, get Miriam to put them to bed and call a doctor to give them something. Will you stay here with me Georgia?'\n\nShe nodded.\n\nThe clock hands moved so slowly. Each doctor who passed the waiting room made them straighten up and hope it was news. The police called to say that Rod's parents were away on holiday in Devon, but they were passing the information over to the police there. Alan's widowed mother had been informed as had Ian's family.\n\n'None of them were close to their families,' Max said softly. 'Alan's mother rejected him when her husband was killed in an accident when he was thirteen. Ian put up a show of caring about his, but I never knew any of them come to gigs or anything. Yet I still don't know how I'm going to face them.'\n\nIf only I'd taken Ian home with me last night,' Georgia blurted out. 'He wanted to come Max, but I insisted he went to the party. It's all my fault.'\n\n'Of course it isn't,' Max put his arm round her and cradled her to him. This was a side she'd never seen of him before and it touched her deeply. 'It could have happened at any time. If someone's number comes up, that's it. Can you imagine how low I feel now after what we were talking about last night?'\n\nAll at once Georgia felt tears welling up inside her. The same terrible loneliness she'd felt that night when Helen died. She put her head on Max's chest and sobbed and sobbed.\n\n'I wish I could find the right words,' he said softly stroking her hair. 'You don't deserve any of this, baby. I'll find a way to make it up to you.'\n\nThe door of the waiting room opened. A young woman doctor with pale brown hair and kind eyes was smiling.\n\n'He's going to be all right,' she said. 'We had some trouble with his breathing. But it's stabilized now.'\n\n'You mean he's not going to die?' Georgia sat up straight, wiping her eyes with the back of her hand.\n\n'Far from it,' the doctor laughed softly. 'He's got a few burns, but nothing too serious. He was a brave lad by all accounts. He won't be running for any buses for some time. I doubt he'll ever want a cigarette in his life again and of course the shock of seeing his two friends burned to death will take some getting over. But go along and see him for yourself.'\n\n'I'll wait here,' Max touched her shoulder. 'It's you he'll want to see. I'll wait till he's stronger.'\n\nGeorgia peeped round the door of the small side ward. There were three beds, but the others were empty. Rod was propped up on pillows, a drip in his arm, his face an angry shade of red. Both arms and his head were swathed in bandages.\n\n'How are you feeling?' she asked, standing tentatively at the end of the bed.\n\n'Sore,' he whispered, his voice unexpectedly croaky. 'I tried to save them,' a big tear rolled down his cheek.\n\n'You were wonderfully brave,' Georgia came closer, wanting to hug him but afraid of hurting him. 'You couldn't have done any more.'\n\n'I should have gone up and got Alan out of my bed,' Rod said. 'I might have seen the cigarette then.'\n\n'I should have taken Ian home with me,' she wiped a tear from her cheek.\n\nShe pulled a chair up close to his bed, laying her head on the covers. Without another word the burden of shared guilt passed from one to another.\n\n'He told me he loved you,' Rod's eyes opened wider, wincing as if they hurt. 'Make a deal with Decca now, Georgia. It's what he wanted.'\n\n'I don't want to sing again ever,' she whispered, unable to hold back her tears.\n\n'You must,' he croaked. 'Wherever Ian and Alan are they'll be waiting for it. Right now I don't believe I'll ever be able to play again either, or even want to, without them. But if I can manage it, so can you.'\n\nShe looked up and saw tears streaming down his face.\n\n'Don't Rod,' she got up and reached out tentatively to wipe them away.\n\n'Kiss me,' Rod whispered.\n\nGeorgia moved forward and put her lips on his.\n\n'I feel better already,' he tried hard to smile. 'One of these days I'll give you a real one and see what that does for you.'\n\nShe smiled at his brave attempt at flirtation, and patted his bandaged hand gently.\n\n'You can have all the kisses you want if you just get better. I think you must be the bravest man I ever met.'\n\n## Chapter 14\n\n'The time's right to cut a record Georgia,' Max sat in his office looking at her under hooded eyelids. 'You've had all that publicity. If we don't get it out now we'll lose the benefit.'\n\n'Do you have to be so ghoulish?' She leapt to her feet, eyes blazing at his callousness. 'Rod still can't hold drumsticks. John can hardly eat for grief. We've lost our saxophone player. Your timing stinks.'\n\nMax ignored her outburst. A fortnight had passed and it was time to crack the whip again.\n\n'I'm talking about you,' he said calmly. 'You cutting a record, not the boys.'\n\n'Don't start that again,' she stared at him coldly. 'Surely they can come in on it with me after what's happened?'\n\nMax studied Georgia. She had lost weight, her hair was dull and lank, she even had a couple of spots on her chin, her skin the colour of porridge. He didn't want to be cruel, but he knew if he gave her too much time she might never get back to work.\n\n'They can work as your backing group,' he said sharply. 'But that's all. I've got plans to find them a new male singer. Someone raunchy who'll get the girls screaming. I've also put the word out I want a new song for you.'\n\n'Why can't we record something we've written?' Georgia stuck out her lip. She was actually glad Max was going to find Samson a new singer. New blood was needed in the band and she knew Speedy, Rod and Norman hungered for stardom in their own right.\n\n'Because I want a big dramatic number with strings and a full orchestra,' he smirked. 'I don't want some drippy bit of bubblegum music.'\n\n'But what if we came up with the right song?'\n\n'Don't make me laugh,' he snapped. 'I'll listen to anything you come up with of course, but don't waste my time Georgia.'\n\nGeorgia wasn't surprised by Max's apparent heartlessness. He alone was behaving characteristically. In a strange way it was more comforting than fan letters and the endless sympathy poured on the band by people who scarcely knew Ian and Alan.\n\nShe had written the words of a song already, sitting alone in her room after the double funeral. Now with Max pushing her she might just find the strength to show the boys.\n\nIt had been such a beautiful funeral. A ray of sunshine shone through the high window at the crematorium, and rested on the two pine coffins side by side, each with a wreath of red roses. The chapel was packed to the doors with fans, old school friends, relatives and more than a sprinkling of musicians.\n\nAll the papers had covered the story, perhaps only because Adam Faith, Billy Fury, Helen Shapiro and Marty Wilde were there. But it meant a great deal to all of the band to see these stars take time off in their busy lives to pay their last respects.\n\nSo many bouquets of flowers and wreaths lined the entrance to the chapel and spilled out on to the forecourt. Some had sheets of music wound into them, some were just simple bunches brought along by unknown fans. The bright colours and the perfume belied the seriousness of the occasion. Few of the mourners wore dark clothes either, as if knowing the boys would prefer them to come the way they remembered them.\n\nGeorgia wore a simple black dress with a white collar. She knew Ian would have told her she looked like a missionary and urged her to wear red, but she was afraid his mother would misinterpret her actions.\n\nSpeedy and Les accompanied Norman with their guitars while Georgia sang 'Morning has Broken.'\n\nUp till that morning she had thought she couldn't sing, but it was Rod who convinced her. The bandages were off his head now, but his hands were still covered. His face was peeling like a bad case of sunburn, but he said he was well enough to attend the funeral.\n\n'You will sing,' he said, raising one singed eyebrow, at her protest. 'They would have wanted you to. It isn't your feelings that count, but theirs.'\n\nPut like that she could do nothing but agree, but when she stood up by the pulpit, she nearly lost her nerve.\n\n'Turn your back away from the audience,' she could almost hear Ian's whisper on stage at the first gig. 'If you can't see them, it's okay.'\n\nShe had turned to the altar to sing, looking up at the window high above. She concentrated on the patch of blue sky so much like Ian's eyes, and as Norman played the introduction and she filled her lungs with air, so her voice came back, filling the small chapel.\n\nDuring the prayers her thoughts were all with Ian and Alan. She could see them squabbling in the changing rooms over a bar of chocolate. Jumping fully clothed in the river at Oxford, sleepy in the back of the van. Laughing over a book of corny jokes. Alan was the one she knew least about. He hadn't been flashy like Rod, or even famous for his dry humour like John. He had none of Norman's know-all tendencies, Les's dimness, or Speedy's sloth to make him exceptional. Just a bright little cockney, happy to be privileged enough to play in a band, and his saxophone spoke for him.\n\nBut Ian. There were a million memories there. The way he wrapped his arms around her when he went to sleep, the kisses first thing in the morning. Day after day of love-filled hours. He'd taught her how to trust again. How to project her voice and wring the last emotion out of a song. One day she'd replace the lover, but her friend would always be in her head and heart.\n\nWhen Rod got up to speak on behalf of all of them, Georgia found it hard to hold back the tears. She wasn't used to seeing him in a formal dark suit, his hair slicked back from his face, and neither was she accustomed to seeing pain in his eyes.\n\nAs he left the altar steps he put his hands over his eyes, his shoulders bowed with the weight of his sorrow. His red peeling face, his bandaged hands were a testimony of his courage. At that moment Georgia understood why Ian had stood by Rod even when he was arrogant, selfish and sarcastic. Had he always known that Rod would be prepared to give his own life to try and save a friend?\n\nIt was late that evening, alone in her room that Georgia's grief and guilt finally surfaced and spilled out. She had been trying to write down her thoughts in a poem, but nothing came to her.\n\nAll she could see was the boys, standing in a group by the gates of the crematorium. They were lost without Ian to tell them what to do. Home, belongings and two of their members, gone. Broken by grief, tormented by the guilty feeling that they could have prevented the accident. Lost like a ship without a rudder.\n\nNorman's carroty hair caught the sunshine as he bowed it towards John to comfort him. Speedy had been in control earlier, talking to both Ian and Alan's mothers, but even he now seemed to have lost several inches in height, his shoulders hunched up in his borrowed suit. Les was deathly pale, his dark hair had lost its shine, his hooked nose almost like a beak, eyes glued to the path beneath him. Rod towered above them, straight-backed, but red-eyed, trying in vain to take Ian's place.\n\n'Why didn't you insist I came to the party? If you'd just stood up to me for once I might not have been so angry.' She put her hands over her eyes and wept. 'I almost went away with Max and now we've run out of time.'\n\nShe didn't know how long she sat by her window crying, but once it was all spent it was dark and Berwick Street was coming to life again as the neon lights were switched on.\n\nAs she undressed, words started to come. 'There's no time baby. It's all slipped away.' She could almost feel Ian's presence in the room, forcing her to pick up her pencil and write it down.\n\n'Do you know what sort of first record Max wants me to make?' she asked the boys a few days after her talk with Max. She had asked them all round for a meal, hoping that by bringing them together again, a little of the old spirit might return.\n\nShe had cooked enough spaghetti for an army and a saucepan full of sauce, there were four bottles of wine and a bowl of trifle in the fridge, but none of them seemed interested in either food or drink.\n\nJohn was staying with his parents in Dagenham. Les with his girlfriend in Wapping. Norman with his granny and Rod and Speedy had found one room together in Fulham. Their equipment was in the van and unless someone pulled them together it would stay there.\n\n'Some big, powerful ballad,' Rod said gloomily, he sat on the floor his back against her bed. His jeans had a hole in the knee and his dark hair needed washing.\n\n'That's right,' Georgia frowned. She wanted to provoke an argument, even at the risk of seeming uncaring. Anything was worth a try rather than this apathy. 'Of course, that's one of Max's objections to you. He thinks you are incapable of writing that type of song.'\n\n'He's never given us a chance,' Norman, sprawled on her bed, lifted his head, eyes sharp with indignation.\n\n'Well get to it,' Georgia snapped. 'Prove him wrong.'\n\n'What's the point?' Speedy sighed. He at least had managed to buy a new pair of trousers and a black polo neck sweater, but his auburn hair needed washing too. 'He won't let us get near the studio with you.'\n\n'He will you know,' Georgia smiled round at them. 'He said I could use you as backing. He even said he would listen to anything we wrote. If we could come up with the right song, written by us. You would all get royalties and publicity.'\n\nSpeedy's eyes lit up, his mouth curled up at the corners. 'Not a bad idea, but it'd have to be good.'\n\nGeorgia looked round at each of them. Five sad faces. Boys who once thought only of the next gig, or the girl in the next town. What would it take to get them all back in one flat again, and bring back their sparkle?\n\n'I've written some words,' she spoke casually, careful not to be too forceful. 'Shall I read them to you?'\n\nRod nodded. He was sitting up again now. His face had turned from red to brown at last. He still had one bandage on his right hand, but the rest of his burns were healing well.\n\n'Go on then.'\n\nShe reached up to a shelf and pulled down a notebook.\n\n'It's called \"No time\",' she said softly.\n\nThere's no time baby.\n\nIt's all slipped away,\n\nI thought we had forever,\n\nNow there's only a day.\n\nI dream of your kisses,\n\nI see only your eyes,\n\nWe have no tomorrows,\n\nOnly goodbyes.\n\nThere's no time baby.\n\nAll we have is now.\n\nThere's no time baby,\n\nThis is the last bow.\n\n'It's a bit morbid.' John spoke up, the first remark he'd made all evening. 'I mean I could just see Ian and you together.'\n\n'We can't forget what's happened,' she said gently. John troubled her the most, his dark eyes had lost their lustre, he was thinner than ever, his skin a putty colour. For years he had shared everything with Alan and now memories haunted him. 'All dramatic songs are sad. Don't you think we need that memory for us to write something worthwhile?'\n\n'It's not bad Georgia,' Norman surprised her. 'I can sort of make out a melody.'\n\n'Let's thrash it out together,' Rod's deep voice seemed to fill the room. 'If nothing else it's a challenge.'\n\nGeorgia hardly dared breathe while they thought it over. The notebook was passed from one to the other, already they looked brighter. She could somehow imagine her words padded out. Speedy and Norman had the ability to find a strong, beautiful melody. John could write the arrangements for brass. Les could be trusted to play a fantastic lead guitar, it only needed Rod to step forward and take command.\n\n'Let's open the wine,' Rod said, getting up and stretching. 'We'll have to find somewhere to rehearse and while we're at it, we'd better look for a new pad.'\n\n##### *\n\n'Where have you been?' Max was red with anger when Georgia walked into his office a fortnight later.\n\n'Resting, rehearsing, helping the boys get themselves organized again,' she sat down without being asked. 'You should have helped them with a flat.'\n\n'Don't come all that holier than thou crap with me,' Max stood up and took a threatening step towards her. 'I paid for the funeral. They've been on full wages. I even gave them extra money for new clothes.'\n\n'Where's the money from the \"Benefit\" gone?' she asked.\n\nSome of the musicians at the funeral had played a gig to raise money for the boys. A cheque for over two hundred pounds had been handed over to Max, but he hadn't passed it on.\n\n'I've got it here,' Max's lip was curling back, but she knew his anger was because he had been caught out. 'I was intending to give it to them today.'\n\n'Don't forget,' her pointed chin stuck out defiantly. 'Because I know you got an insurance payout to cover everything else you've spent.'\n\nShe had been into the office during the week while Max was out. Deirdre on reception had slipped out for a sandwich and Georgia used the time to snoop. Max had been doodling on his blotter. An insurance policy lay next to it and it hadn't taken her more than two minutes to read it, then slip back out before Deirdre even knew she'd been there. Max wasn't out of pocket at all. He was making claims for broken engagements, equipment and possibly claims on the boys' lives.\n\n'Now look here,' he blustered.\n\n'No, you look here,' she said more bravely than she felt. 'You owe the boys. I want you to come and listen to something we've written. Can you arrange a gig at the Marquee so we can try it out on an appreciative audience too?'\n\n'I don't know,' he shook his head. 'I've got a song lined up myself, you don't need the boys. I'll get them a singer and sort them out.'\n\n'Do as I say or I'll start talking too loudly,' she spoke softly, yet with enough menace to make him understand she meant it. 'The boys need to feel wanted right now. I need them too. Or have you forgotten my feelings?'\n\nHe hesitated.\n\nShe could see a rope-like vein quivering on his forehead. She really didn't know why she liked him. He was slippery and heartless. Yet there was something strong about him that was attractive and compelling, neither could she forget his gentleness on that awful day at the hospital.\n\n'Okay,' he sighed deeply. 'I'll arrange it. But don't think you can hold a gun to my head, young lady. If the song's no good, that's it.'\n\nGeorgia got up and walked round his desk. She dropped a kiss on his surprised face.\n\n'That's for saying yes,' she smiled down at him. 'And this is to make sure you listen when you come.' She bent over and kissed his lips gently, just lingering long enough so his hands came up to hold her, then jumped nimbly away out of reach.\n\nAs she backed towards the door, she glanced up at the gold spider's web. More and more she felt like that poor fly. Another year and she would probably be in the spider's jaws.\n\nMax folded his arms on his desk and lay his head down on them. He could never quite analyse his feelings about Georgia. If he wasn't such a cynic he could call it love. Any other girl singer who had worked for him had been in his bed before her first week was up. Yet somehow he'd never fathomed out a way to get Georgia. She couldn't be persuaded by flattery. Presents would merely be laughed at. She certainly wouldn't respond to just grabbing her. Yet she'd nearly agreed to come away with him.\n\nHe couldn't admit, even to himself that his first reaction to hearing about Ian's death had been almost pleasure. With Ian out of the running Georgia could be his, he could run her life the way he had always intended to.\n\nBut the pleasure had been hollow. Even he, tough as he was, cared for Ian, and Georgia's stricken face those first few days had cut him to the quick.\n\n'I should never have put her with that band,' he murmured, sitting up again, tipping back his chair and putting his feet up on his desk. 'I should have dangled her the carrot of a recording contract. Bought her new clothes and set her up in a flat.'\n\nYet Ian and the other boys had produced something in her he couldn't have managed alone. She kept her girlish breathless charm, while learning poise and timing. Her voice had gone from good to incredible and maybe Ian was responsible for adding that extra emotion. Everything was right for launching her. The tragedy of the boys' deaths was still in the public's mind.\n\nLike the phoenix she could soar out of the flames into stardom, and all he had to do was sit back and watch the money come rolling in.\n\nReminded of something, Max frowned and opened his desk drawer. The letter had come for her this morning, yet something had stopped him from giving it to her.\n\nHe turned it around thoughtfully in his hands. A blue envelope. Postmarked Manchester. The writing bold and masculine. It was probably only a fan letter, but there was no harm in checking.\n\nTaking a small silver dagger from his desk tray, he slid it under the flap carefully, in case he had to stick it down again later. He drew out the single sheet of paper and spread it out on his desk.\n\n'Dearest Georgia,\n\nI had almost given up hope of finding you. When you didn't contact me on your birthday three years ago I drew the conclusion that you didn't want me to. I understand after what happened that you wanted to sever all connections with your past. But just when I'd finally reconciled myself to a life without you, I read the story about the fire and the two boys burned to death.\n\nI might never have given it a second thought, but one of my friends up here had seen the band, and went on to tell me about the girl singer. He didn't remember her name, just how she looked, and of course I began to wonder. Then in the story about the funeral I saw the name Georgia. I've never heard of anyone else called that. Could there really be another Georgia who sings like an angel, dark-skinned with long curly hair and eyes like giant pansies? I doubt it somehow! It didn't require much detective work to find your manager's address and here I am writing with my heart thumping wondering how you'll receive this letter.\n\nEven if you have no further interest in me, please write back and tell me if Celia managed to find you? I kept in touch with her up till last Christmas, then gradually she stopped writing. She went to Africa to nurse, moving on several times. I guessed she stopped because she felt it was unhealthy for us both, but it has occurred to me she may now be reunited with you and you stopped her.\n\nPlease let me know how things are with you. I won't pester you, or bring back unwelcome reminders. I just need to know you are happy. Of course I'd like to think that you still hold a small torch for me, but I'm realistic if nothing else. I'm so sorry about your friends, it was a terrible tragedy. God bless you.\n\nMy love Peter'\n\nThere was a depth of passion in this letter that disturbed Max. Whoever this man was he came from a time before Georgia joined the band.\n\n'A childhood sweetheart?' he mused. 'And who is this Celia?'\n\nGeorgia had always been evasive about her past. The way she spoke, her education all hinted at a good home. Yet girls who came from good homes didn't normally turn their backs on them.\n\n'After what happened!' Max skimmed through the letter again. 'What could have happened?'\n\nCould she have been in trouble for seeing this boy? But if so who was Celia? An older sister? An aunt?\n\nMax pondered for some time. If he showed this to Georgia now, in her already disturbed state she might do anything. If she'd hidden from this young man once, she might have good reason to bolt again. Just a glance at the content of the letter and the bold handwriting was enough to know this wasn't some dim, uneducated lout. She was getting ideas above her station already; aided and abetted by intelligent friends she could break away from Max altogether.\n\nHe got up and went over to a typewriter on a small table. Taking a plain sheet of paper out of the drawer he inserted it, sat down and began to type.\n\n'Dear Mr Radcliffe,\n\nMiss James thanks you for your letter and has asked me to reply for her.\n\nAlthough she appreciates your concern, she feels she has nothing further to say to you. Her career as a singer is all important to her and leaves no time for socializing. She wishes me to assure you she is well and happy in her chosen career, and sincerely hopes you are too.\n\nYours sincerely,\n\nDeirdre Richards.\n\nP.P. Georgia James.\n\nMax pulled the letter out of the machine, signed it with a flourish, folded it and put it in an envelope.\n\n'Maybe that will dent his pride enough to leave her alone,' he said to himself. He picked up Peter's letter, screwed it up and tossed it into the bin.\n\nThe Marquee club seen by daylight had little to recommend it. A tiny stage, a plain wooden floor, the only seating further back in a cavern-like room by the bar. But then the people who flocked to the Marquee came to hear music, and they knew this club could be relied on to have the best.\n\nNorman played 'No time' through alone.\n\nGeorgia stood back in the shadows by the bar listening. She and Norman were alone now. The equipment was ready on the stage for tonight's gig, the rest of the boys were out getting sandwiches. A tingle ran down her spine, a rush of affection for Norman as she watched him crouched over his keyboard. This wasn't something he was playing under duress. He was putting his heart and soul into it, and it was good.\n\nIt cried out for strings, a full orchestra, but already it was a powerful melody, the kind that lingered in the mind long after it was finished. It could be a classic in the making.\n\nNorman just sat on his stool as he finished, he looked like a small elf with his red hair and sharp features, chin stuck forward, deep in thought.\n\n'It's brilliant,' Georgia clapped and ran over to him.\n\n'I'm pleased with it,' he blushed, for once less cocky. 'But wait till we get Speedy on bass and Les's throbbing lead. Up till now I've only been able to play it through on piano, but I know they are going to amaze you too.'\n\n'Oh Norman.' Impulsively she jumped up on the stage and threw her arms round him. 'It's beautiful.'\n\n'Ian helped me,' his head drooped as if embarrassed at saying such a thing. 'I felt his presence, almost as if he were humming it to me. You remember the way he used to?'\n\nGeorgia nodded. It had been odd that Ian who could play nothing more than 'chopsticks' on a piano had been able to invent melodies in his head.\n\n'God, I miss him,' Norman's eyes filled up with tears. 'I wish I could take back all the snidy things I said to him over the years.'\n\n'He wouldn't have had you any other way.' Georgia leaned over Norman and kissed his cheek. 'Now play it again and I'll sing.'\n\nAfter several false starts they got it together. Georgia's voice soared out across the empty club, lost in the beauty of Norman's melody.\n\nA loud clapping came from the front door as they finished. Georgia spun round to see Jack Fellows, the club owner, leaning against the wall.\n\nTall, stringy and untidy, Jack looked more like a struggling artist than the successful businessman he really was. His hair hung well past his ears, thinning on top. He had a long, pointed nose and a wide, smiling mouth that gave an indication to his inner nature. The Marquee was more than just a money spinner to him. He would rather have talented unknown bands playing than compromise an inch. But his high ideals had paid off, for his customers knew that any night in his club would be memorable, and the bands knew if Jack booked them, they were worth something.\n\n'That's a beautiful song,' he said, his thin face alight with enthusiasm. 'Did you write it?'\n\n'A joint effort,' Norman grinned. 'We're hoping Georgia might be able to record it.'\n\n'If Max approves?' He raised one bushy eyebrow. 'The man's a complete Philistine if he doesn't.'\n\nGeorgia bounced down Wardour Street just before nine. For the first time since Ian's death she felt the cloud hanging over her was moving back. In tight white jeans, a red T-shirt and boots she felt right. The Marquee was home ground, she didn't have to squeeze into the changing room and pour herself into something swish. She could just turn up, sing her heart out, then go home. No one dressed up for the Marquee, students, beatniks, office people and manual workers flocked there for music, and tonight she was going to give them something special.\n\n'Remember, we're not warming up for another band tonight,' she reminded the boys as they waited off stage for Jack to turn off the records and introduce them. She could see John standing alone, twiddling the valves of his trumpet, the first time he had ever played publicly without Alan beside him. 'You can do it John,' she reached out her arms to hug him. 'I feel the same about walking on without Ian, but we'll get through it.'\n\nShe could hear Jack out there on the stage. A joke with the audience about the price of beer, a word or two to remind them of Samson's recent loss of two members. She didn't have to look behind the worn curtains to know the club was packed to capacity, every face upturned, waiting for them.\n\nRod leapt on first, going straight to his stool and performing a dramatic drum roll. Norman was next, quickly followed by Les and Speedy and they launched into the opening number 'Soul Train'.\n\nGeorgia took John's hand in hers, leading him just as Ian had once led her, out into the spotlights.\n\nShe had to be better than her best tonight. She missed Ian's close harmonies, and Alan's sax, but putting that aside she sang first for the band. She put a new wildness in her dancing, strutting, teasing, bending to the audience till she knew the boys were on form again.\n\nJohn surprised her. He and Alan had tended to fall back on one another, staying together to play, never taking a lead. Now he moved forward, legs apart, blowing like she'd never heard him before, eyes closed, chest fully expanded, bringing out notes of such passion and sweetness, it was as if Alan's spirit had entered him.\n\nWhen the first set ended to wild applause Georgia was drenched in sweat.\n\n'You were something else tonight,' Rod grinned as she stripped off her T-shirt in the changing room and mopped at herself with a towel. 'When are we going to do the new number?'\n\n'Last,' Speedy said pulling open a can of coke and resting it for one moment on his sweat-covered forehead. 'If we do it too early it might kill the mood. Keep to our usual routine, then \"He's no good\" followed by \"No time\", and let's hope we all remember our parts.'\n\nMax came in just after the second set started. He stood by the side of the stage speaking to Jack Fellows giving the band no more than a cursory glance.\n\nGeorgia had relaxed sufficiently to notice there were fans in the audience from back when they played in London roadhouses and clubs a year before. She rewarded their loyalty by singing for them rather than Max, bending to touch outstretched hands, blowing kisses and finding the strength she thought she had lost in their smiling faces.\n\n'He's no good, he's no good, baby he's no good,' she sang cheekily to Max. Pulling off her hair ribbon and throwing it into the audience and tossing her mane of hair round her shoulders.\n\n'Finally,' Georgia mopped her brow to thunderous applause. 'We're going to do a totally new number we wrote ourselves. It's a breakaway from our usual stuff, but we hope you like it.'\n\nThe introduction started. She saw Max turn to look in surprise as Norman played the haunting first few bars. Tingles went down her spine, she tapped her feet to the beat and filled her lungs.\n\nAs the song went on, so Georgia drowned in it. She was singing to Ian all the things she wished she'd said while he was with her. And to Max too, to remind him she was her own person.\n\nShe knew without a shadow of a doubt it was the finest singing she'd ever done. Even if the audience walked out, her own ears had told her the truth.\n\nThe applause was simply deafening. On and on it went with calls for more. They left the stage once but had to go back and do another number.\n\nWhen it was finally over Max came forward.\n\n'So that's it?' He had the oddest expression, surprise, delight, mixed with a tiny amount of pique.\n\n'Yes,' Georgia smiled up at him. 'What do you think?'\n\nHer heart was in her mouth. If he turned her down now she had nothing more to offer.\n\n'A gold record,' he said, a smile stretching from ear to ear. 'You wrote it?'\n\n'I did the words, the boys did the rest,' she said simply, surprised that for once he wasn't hiding his enthusiasm behind criticism.\n\n'I'll get the studio booked for next week,' he said, putting one big hand on her shoulder and gripping it. 'This is it baby. I feel it in my water.'\n\n## Chapter 15\n\n'That's it then!' Max's voice crackled abrasively in their ear phones. Through the glass screen they could see him gesticulating wildly, as if he didn't believe his voice could really reach them. 'As they say on the movies, \"that's a wrap\".'\n\nGeorgia took off her head phones and wiped the perspiration from her forehead.\n\nShe was too tired to even think of celebrating. Nine hours of being stuck in a soundproof room, technicians staring at her through the glass as if she were a goldfish. This was an entirely new ball game to playing live.\n\nMixing, loop tapes and umpteen different tracks. Session men who'd filed in, played their parts then left. She had imagined they would just perform together over and over until it was perfect. She hadn't expected the separate instruments to be added, or harmonies put on afterwards. It was confusing, frustrating, and the constant stopping and starting irritating. But then Max had insisted they produced master tapes perfect enough for the disc to be cut from, a half-hearted demo tape just wasn't good enough.\n\nThe boys had come to the studio that morning dressed as if it were another gig. Rod in velvet trousers and a flowered shirt. Norman in a smart new green jacket. But now they looked like wilted flowers, hot and sweaty, hair sticking damply to their heads.\n\nSteven Albright, the producer was waiting for her, the boys grouped round him in the ante-room, waiting for his opinion.\n\nSteven had the look of an overgrown schoolboy. Not what they expected from a man in his thirties with four gold records already under his belt. Six foot tall, painfully thin, with greasy hair dangling over his thick specs. Even his clothes had a charity shop look about them. A city shirt with stiff collar, an old, stained school tie and suit jacket, then in contrast a faded pair of jeans and desert boots.\n\nIt was difficult to have confidence in someone who blinked owlishly behind his glasses and silently chewed a pencil. But he surprised them, not only was he alert to every last note, he had imagination, flair and a complete knowledge of many instruments.\n\n'Time to play the finished article,' he smiled warmly, his plummy, Old Etonian accent somehow reassuring. 'You look tired Georgia, but you did very well.'\n\nHe sat down at the controls and the introduction started.\n\nThe finished result was perfect. It was the sort of song that would be played last at every dance up and down the country. Bodies entwined, arms round each other's necks. A song for lovers everywhere.\n\n'It's good,' Steven turned off the tape as the last notes faded away. Like Max he was sparing with his praise. All day he had pushed them. They had seen him angry, frustrated, disinterested, even bored on occasions, but now at last his dark eyes shone with excitement and exhilaration. 'You can all be very proud of it. I'd say it will make it.'\n\nSomehow that simple statement meant more than gushing praise and for the first time all day, not one of them came back with a flippant remark.\n\n'You can all clear off now.' To Max it was business as usual. He had come in and out several times during the day, listening half-heartedly, flashing his gold watch and leaving again just as quickly.\n\n'You've got gigs in the Midlands for the next week. Put all this out of your minds and get on with playing.'\n\nMax had seemed preoccupied since the funeral. Georgia couldn't help wondering if he'd found another band who excited him more than Samson. She'd spotted brochures for new vans on his desk and receipts for band suits which she was sure weren't for them. Also there were three new girls in the office who barely acknowledged her. Was she being paranoid? Or was Max about to pull another stroke?\n\nThe next week or two was as if they'd gone back in time. If it hadn't been for the pressure of wondering what was happening in Decca's offices, and Ian and Alan's absence, they could have been back to the carefree days before Max put them on the cinema tours. Their re-appearance at dancehalls was enthusiastically received, old fans coming forward to show their pleasure at seeing them again.\n\nBut Georgia hadn't reckoned with all the old memories. She could handle it by day, wandering around town with John or Rod, even performing in places she associated with Ian. But by night when she crept into a cold, often damp bed, Ian's face came back to her.\n\nShe missed his jokes and chatter, all the little things he helped her with. The other boys were clumsy at zipping up dresses and putting make-up on her when there was no mirror handy. She missed him singing with her, complimenting her, spurring each other on. But it was making love that dominated her thoughts at night. She would torment herself remembering the way he stroked her. The thought of his kisses made her hot and damp. She longed for the blissful glow that followed making love, and waking early to find him aroused and holding her.\n\nWhen she lost Peter there had always been the hope he would return. Even when Helen died she had been able to comfort herself with the thought that she had left pain and poverty behind. But there was no sense to Ian's death. It wasn't right that he missed by just a few weeks the one thing he had aimed at all his life. And neither was it fair that everyone she loved was snatched from her so cruelly.\n\n'You can always count on me,' Rod said one evening as they made their way up the staircase to their rooms. 'My body's free anytime.'\n\nGeorgia stopped and turned to look at him. Two years ago she would have found his arrogance insulting. But now she saw it as a gesture of comfort and perception.\n\nThe boarding house was just like all the countless others they stayed in. Shabby, flowered wallpaper, worn at shoulder height with the hundreds of people that had gone up and down rubbing against it. Candlewick bedspreads, nylon sheets and plastic flowers.\n\nShe saw Rod then as other women saw him. His strange dark slanty eyes, high cheek-bones and thin, almost cruel lips. Raw sexuality seeped out of him, his height, coupled with wide shoulders, narrow hips and his shiny blue black straight hair gave the picture of a primitive savage.\n\n'Things aren't that bad,' she grinned.\n\n'No?' his eyes laughed at her.\n\n'I miss him so much it hurts,' she said softly. 'But it isn't just sex I miss.'\n\n'I didn't think it was,' he said, his hand reaching out and stroking her cheek. 'But sometimes another body can be very comforting.'\n\nWith one finger he traced round her lips. She felt goosebumps come up all over her and she couldn't move away.\n\n'I could make you forget at least for tonight,' he whispered. 'It doesn't have to be forever.'\n\nShe felt a tug in her stomach, a tingle of desire.\n\n'I'm not brave enough to chance it,' she took a step back from him and hesitated, looking down at him two steps beneath her. His eyes were half closed, narrow lips apart showing white teeth. For a moment she almost went back to him.\n\n'One day,' he smiled. 'Just for laughs!'\n\nThat night she thought of Rod's hands on her breasts. His tanned chest above her, skilful fingers playing with her and it was all she could do not to cry out.\n\nAs she let herself into her flat on their return to London, she found a note from Max summoning her to his office the next morning.\n\nThere was little hope of falling asleep after the note. Could this be the stroke she suspected? Or could it be that her dreams were finally about to be realized?\n\n'Go on in,' Deirdre on reception smiled a welcome as Georgia leapt up the stairs on the stroke of ten. 'He's waiting for you.'\n\nThis in itself was a good sign. Max frequently kept her waiting for hours.\n\nMax looked relaxed as she walked in the door. A pale lemon shirt open at the neck, sleeves rolled up revealing thick, brown arms. His chair was tilted back and he puffed on a cigar as if day-dreaming.\n\n'Hallo darling,' he stubbed out the cigar, leaping to his feet, reaching her in two giant strides and pecking her cheek. 'Sorry I couldn't give you more warning. I've been rushed off my feet.'\n\n'How are things going?' she asked, taking a seat by his desk, noticing many changes in the room.\n\nTwo years earlier Max had perhaps ten or so bands on his books. Now it looked as if he had expanded overnight. A new filing cabinet stood with drawers open. Stacks of contracts lay on his desk, glossy photographs of groups unknown to her scattered everywhere.\n\n'I've got some excellent news for you,' he returned to his desk, picking up a gold fountain pen from his blotter, shaking it, then signing a letter in front of him with a flourish.\n\n'They want it?' Georgia felt a rush of adrenalin to her head.\n\n'Yup. They are anxious to get the disc cut and released for the first week in September.'\n\n'I can't wait to tell the boys,' Georgia felt a bubble of glee rise up inside her.\n\n'The contract is just for you.'\n\nShe stared at him, mouth agape. He had that cold look in his eyes she knew so well. Had he actually managed to outmanoeuvre her despite everything?\n\n'Don't look like that,' he snapped at her. 'Their names will be on the recording as backing and co-writers, they'll get their royalties.'\n\n'You've got something up your sleeve.' She stood up, leaning towards him over the desk, dark eyes blazing. 'Are you trying to tell me this is the end of the line for us together?'\n\n'Georgia, darling,' he shrugged his shoulders, spreading his hands wide. 'I'm only thinking of you. The Palladium, the Albert Hall, that's where you're heading. You'll need an orchestra, not a bunch of dance hall musicians.'\n\n'But before we get there we'll still be playing in clubs and stuff,' she said desperately. 'I need them Max!'\n\n'Of course you do, for now.' He moved round the desk and caught hold of her arms. 'As from tomorrow when you sign with Decca you pay them just as you would a session musician.'\n\n'Whaaat!' Georgia stared at Max in horror. 'You mean I've got to tell them I'm their boss now?'\n\n'Just a re-arrangement of finances. I'll still be the one organizing everything. Financially this is far better for you. When you give interviews, television game shows etc, it means that money is all yours, and rightly so, you'll be the one working your butt off.'\n\n'But \u2013'\n\nMax interrupted her. 'Look here Georgia, I'm getting a little tired of this game. Hasn't it ever occurred to you the boys might want a change of direction? I'll get them a new singer. If they deserve it they'll get their own recording contract. Stop bloody well harping on about them.'\n\n'What time is it tomorrow?' she asked weakly. 'We've got to leave for the gig at twelve.'\n\n'Ten thirty. You can catch the train afterwards,' he grinned as if he was giving her a treat. 'The boys left this morning to pick up some new speakers. Soon you won't be travelling with them anyway. It will be limousines for you. Gold stars on the door of the changing room. Now clear off and buy yourself a smart outfit for tomorrow.'\n\nHe had sent them away from town on purpose to avoid any last minute rebellion, now he was opening his wallet and pulling out fifty pounds.\n\n'Something outstanding,' he said. 'I don't want them to think I kept you short.'\n\nGeorgia knew how Judas felt as he pocketed his thirty pieces of silver.\n\nDecca's offices were only a stone's throw from her room in Berwick Street.\n\nFor the last two years Georgia had looked through the big glass doors every time she passed by, dreaming of this very day.\n\nThe dream had come true, but why did she feel so empty?\n\nMax had been brusque with her when they met earlier. More interested in getting into the boardroom than speaking to her.\n\nShe had been left to sit outside, he hadn't even remarked on her outfit.\n\nWas this how it would be from now on? Alone all the time, watching out for people ready to stab her in the back? Once she would have been thrilled to be given fifty pounds to spend on clothes. All the boys would have come with her to choose them. Yet the boys were off buying speakers while she was signing away her future without them.\n\nWas the outfit she'd chosen right? A white knee length dress, with a diamond cutout showing her brown abdomen, and new white shoes. She could see herself mirrored in a chrome plant stand. Her hair in soft ringlets, a white ribbon nestling amongst the curls. Was an impression of innocence the right look? Ian would have insisted on something red and dramatic!\n\nFor a place that employed hundreds of people, it was eerily quiet. The faint tapping of a typewriter in the distance, the occasional ringing of telephones, and a buzz of conversation from the boardroom where Max had gone.\n\nA thick green carpet curled over her shoes. The seats were brown leather, big and comfortable. Huge plants stood in tubs and a brass-topped coffee table held a selection of quality magazines. It was more like a posh dental surgery than a place that dealt with music.\n\n'Would you like to come in now Georgia?' The blonde iceberg of a secretary was holding the door open, a false, rather cynical smile on her china doll face.\n\nBeyond the secretary's notebook, Georgia could see at least ten business men wreathed in cigarette smoke around a boardroom table. All at once she was more nervous than when she went on stage. Her palms were sticky and her stomach turned over.\n\n'This is Georgia,' Max stood up and pulled out a chair for her, his smile a contrived attempt at a fatherly one.\n\nShe sat down, thrown by the lack of interest in the men's faces.\n\n'Hallo Georgia.' A short fat man with small dark eyes held out his plump hand. 'I'm Jack Levy. Might I say on behalf of all of us, how much we have enjoyed hearing you sing. You're a girl with an exciting future.'\n\nGeorgia knew he was the top man at Decca. She had expected someone larger, not a nearly bald man with a wrinkled sallow face, a nose so huge it looked like a beak and gold-rimmed spectacles. He looked more like a banker than a maker of stars.\n\nShe glanced around the table.\n\nAlex Rhodes was there, avoiding her eyes. She had heard he had an administrative position now, perhaps that's why he had abandoned the tweed jacket and corduroys. His sandy head was bent over some papers, as if trying to forget this was the girl he couldn't lure away from her band all those months ago. Had he put Max up to the cinema tours? Had he hoped one day she'd come crawling to him?\n\nMaybe she wasn't exactly crawling, but it didn't feel like triumph either.\n\nMax's lawyer John Cohen she knew slightly. But here he blended in with all the others. They were all the same. Dark men, all of them greying at the temples. Not one of them less than forty, papers spread out in front of them. All with the same dark suits, gold watches, rings and cufflinks glinting, as if showing their allegiance to the same club. Not one look of interest or admiration. Tense faces as they approached yet another business deal.\n\n'We are offering you a three year contract,' Jack Levy went on. 'Under this contract we have the right to choose and oppose any songs which are offered to you. Though of course your opinion will be sought in this matter.\n\n'You may not work for any other record company during your contract with us. Also, work that is outside our field, a film for instance, would be vetted carefully before we agreed to it.'\n\nHe looked at her carefully over the top of his glasses, a shaft of sunlight played on his balding head and his nose seemed to grow larger.\n\n'Do you understand?'\n\nGeorgia nodded. These men were planning to take over her life. Once she'd signed with them she was just another pawn to be pushed anyway that suited them.\n\n'I want you to read through the contract,' he said more gently. 'It may seem difficult in parts, legal jargon does sound odd to someone who isn't used to it. But if there's anything you don't understand, please ask.'\n\nHe handed her a document.\n\nGeorgia had no idea what she was supposed to look out for. It seemed fairly straightforward, mainly revolving around her inability to do anything without their permission.\n\nThen her eyes caught an interesting section about immorality.\n\n'What does this mean?' she asked, pointing to the section.\n\n'Well, my dear,' he smiled condescendingly, his eyes so small behind his thick glasses they looked like currants. 'We understand that young people sometimes get led astray when they find themselves in your position. However, an established company like ourselves do not like scandal.\n\n'Should one of our clients get involved in something which could have a damaging effect on ourselves, we have the right to cancel their contract.'\n\n'What sort of scandal?' All the men were looking at her now, perhaps wondering if she had anything to hide.\n\nHe shrugged. 'Something criminal maybe. Drugs. Not turning up for performances continually. Loose behaviour.' He gave her a half smile as if implying she would never do any of these things.\n\n'I see, and is there a clause in here which protects me if, say, you didn't act in my best interests?' She wasn't going to let them think she was a pushover.\n\n'Why should that be necessary?' Jack Levy looked shocked.\n\nGeorgia could see he was another Max, he had the same expression on his face Max wore when confronted with his deviousness.\n\nHe looked round the room and laughed. 'Do you think all these people would be gathered here today if they didn't intend to look out for you?'\n\nGeorgia had a feeling they were rather more interested in the money aspect than her soul, but she thought it prudent to keep that to herself.\n\nShe continued to read the document, although most of it went over her head.\n\n'It seems okay,' she said at length.\n\n'Of course it is,' Max laughed, sucking on his huge cigar. 'We know what we're doing.'\n\nGeorgia had always imagined the signing of a record contract would be in a party atmosphere, with champagne, streamers and smiling faces.\n\nInstead she signed her name in chilly silence. It was witnessed by the secretary who ushered her into the room and then signed by Jack Levy and one of his colleagues. The men all shovelled their papers into their briefcases and got up to leave, barely glancing at her.\n\n'Is that it, Mr Levy?' she asked as they all filed out.\n\n'That's the boring part,' he said, smiling in a more pleasant manner. 'Now come with me and I'll show you a little more of our operation.'\n\n##### *\n\nIt wasn't until she was on the train that what was really happening finally sank in.\n\nShe was going to be a star. Making a lot of money for Decca, Max and the lawyers. Soon she would be able to move to a smarter home. Employ a cleaning lady, even a cook if she wanted one. She could have proper driving lessons and buy a car. She would be invited to smart parties and buy all the clothes she wanted. People would want her autograph.\n\nShe ought to have been dancing with happiness, yet somehow she felt cheated.\n\n'I've let you down,' she blurted out to the boys as she made her way into the changing room of The Purple Pussy Cat. 'I signed a contract with Decca this morning.'\n\nAll the way on the train she had wondered how to tell them. Should she pretend she was happy with everything? Insist their rights were protected? Assure them that everything would remain the same?\n\n'So that's why he let us get the speakers at last,' Rod sighed deeply. 'I wondered why he'd suddenly become so generous. He just wanted us out of the way.'\n\nSpeedy was watching Georgia. He saw how pale she looked, her hands shaking.\n\n'It's all right Georgia,' he slipped his arm round her shoulder. 'We're happy for you. We didn't expect a contract too.'\n\n'I did,' Norman's voice chipped in, his small face alight with pique. 'I worked my balls off with that song. Now she'll get all the credit.'\n\nGeorgia burst into tears and ran from the room.\n\n'You arsehole,' Speedy exploded with rage. 'What did you have to say that for? The poor kid is already eaten up with anxiety about us. Haven't you learned anything about her in two years?'\n\n'Speedy's right,' Rod sprawled on a bench, a faintly amused expression on his handsome face. 'Georgia's incapable of being selfish, which is more than I can say for myself. Max has beaten her down, the same as he's done to us countless times. But when it comes right down to it she deserves all the success coming to her.'\n\n'I don't begrudge her it,' Norman was sullen now, his lower lip stuck out like a sulky child. 'I just wonder where it leaves us.'\n\n'With a friend in important places,' Speedy snapped. 'Now for God's sake make it up with her. Show some pleasure that she's got what she worked so hard for. That's what real friends do.'\n\nOn September 1st, 'No Time' was released. Two days later it was on Juke Box Jury.\n\nThey were all together in the boys' new flat in Paddington. Later that night they would be playing at the Bag O' Nails in Soho, but Max had rung them while they were setting up the equipment to let them know he had managed to pull strings to get it on.\n\nGeorgia had no television, so they had leapt in the van and rushed back to Paddington. Now the programme was starting Georgia was biting her nails.\n\n'Stop that,' Rod smacked at her hand. 'If you nibble each time it gets on the air you'll be up to your elbows in a few weeks.'\n\nTheir old flat had been grubby and untidy, but it had been a home. This new flat might be tidy, with almost new furniture but it was soulless, like the boarding houses they spent so much time in. No clutter, no personal touches, just another reminder how much they had all lost.\n\n'Memphis Tennessee' was the first record played, a catchy number, guaranteed to hit the top twenty immediately. They listened impatiently while the jury deliberated and finally voted it a good song but one they doubted would make it.\n\n'They haven't a clue,' Speedy said in amazement. 'Does our future really lie with people like them?'\n\nThe next record was a ballad called 'Blue Nights', so dull and banal Rod pretended he had fallen asleep.\n\n'That's what I call a good song,' one of the jury enthused. 'Great backing, a little slow,' said another, 'But it will be a hit.'\n\n'They won't like ours, I know they won't,' Georgia said, sitting on the edge of her seat.\n\nAs David Jacobs introduced 'No Time', Georgia thought she saw him wince. It made no difference that the introduction still gave her goosebumps of pleasure, his impassive shiny face seemed to hold more than a trace of irritation.\n\nThe jury however leaned forward in their seats to listen. The youngest man had his eyes shut, resting his face in his hands.\n\n'Look they love it, they're all spellbound,' John shrieked.\n\nTo their surprise, the dreamy-eyed man said he loved it yet doubted it would be a hit. But the other three gave it a resounding thumbs up.\n\n'There you are,' Rod said triumphantly. 'Aren't I always right about everything?'\n\nMax left no stone unturned. He invited reporters to drop in on their London gigs. Copies of the record found their way on to every important desk in the music world. Bribes were passed out to give it plays on the radio.\n\nThe first week after its release was the worst. Rod reported it played first late at night, then again the next morning, but then nothing.\n\nAnxiety that it might not even make it into the top fifty was quickly dispelled by Max who was suddenly booking them into London clubs only and appearing nightly with photographers, behaving as if Georgia was the only person in his life.\n\n'I'm taking you to a party tonight,' he announced during the break at The Scene in Windmill Street. 'Nip off home and get changed the minute you've finished here. I want you all glammed up.'\n\nIt was on the tip of her tongue to protest she was too tired, but a look of excitement in Max's eyes hinted this was one party she couldn't miss.\n\n'Where is it?' she asked instead, remembering both Norman and Rod had dates later that night and all she had to look forward to was a new book.\n\n'Kensington,' Max had that kind of smug grin which meant it would be very swish. 'It's important, so don't let me down.'\n\n'Is that it?' Georgia saw big wrought-iron gates with the huge house beyond ablaze with lights. Loud music was wafting out, people dancing in front of one of the windows and scores of others sitting on the steps leading up to the front door.\n\nThey had turned off from Kensington High Street just minutes before, but with Holland Park lying in darkness next to this house, they could have been right out in the countryside.\n\n'Don't be scared.' Max patted her knee. 'It's the home of Louise Wainwright the heiress. She collects stars like others collect stamps or matchboxes, but she's a good sort.'\n\nEven with Max's arm firmly round her shoulders, Georgia felt tongue-tied with nerves as they walked into the hall.\n\nMax's cream suit looked just a little ostentatious under a crystal chandelier. Shouldn't she have worn a proper cocktail dress instead of her new white outfit?\n\nShe had seen glimpses of houses like this back in Blackheath. Antique furniture, Oriental rugs, art that had been collected over centuries, not snapped up from some shop in Chelsea.\n\n'Maxy, darling!' A bony woman with bulbous frogeyes and a flame-red, poker-straight bob, rushed to them. 'You've managed to bring Georgia, you absolute angel.'\n\nShe inclined one angular cheek to Max, drowning them both in 'Joy' perfume, then reached out for Georgia, enveloping her in a bony hug. Behind her on a wide sweeping staircase Georgia saw Vogue's top model, Bonnie Jackson in a long chiffon pink dress, her blonde hair caught up with a single rose.\n\n'This is Louise Wainwright,' Max said, winking at Georgia, suggesting silently she made out she had squeezed in the time for parties between other pressing engagements.\n\n'Thank you so much for inviting me,' Georgia stammered, feeling distinctly underdressed next to this formidable woman in her Dior emerald green evening dress. Diamonds glistened on her ears, more on her fingers.\n\n'My darling,' Louise gushed. 'It's my pleasure to meet you at last. All London's talking about you and your record.' She touched Georgia's hair lightly, her frog eyes nearly leaping out of her face. 'I didn't expect such devastating beauty. But come on in and meet everyone.'\n\nWith a glass of champagne in her hand, Georgia found her nervousness leaving her. The huge drawing room with doors open on to a floodlit garden beyond was too crowded to feel conspicuous. Some of the guests might have pedigrees going back to the Armada, others had forged their way out of far humbler homes than her own, but they all seemed to be linked by a common theme, they were people who were on the way up.\n\nA disc jockey modelling his patter on Alan Freeman kept the sounds coming and with Louise Wainwright holding her elbow she was whisked round to meet everyone.\n\n'Bill Johnson, the photographer,' Louise said breathlessly, as she introduced her to a gaunt-faced man in black leather. 'I expect you know his pictures, he did that wonderful spread in _Queen_ a few weeks ago.'\n\nGeorgia didn't know, but she smiled, shook his hand and Louise whisked her on to her next prot\u00e9g\u00e9.\n\nActors, models, debutantes and professional men. People from the advertising world, still more entrepreneurs doing everything from running art galleries to boutiques in King's Road.\n\nBy her third glass of champagne Georgia was doing the 'twist' with a red haired architect called Ivan. Louise had told her he won an award for an office block in the City, but he was more interested in talking about Georgia.\n\n'Get into films,' he said as if he knew all about it. 'That's where the real money lies. Demand to go to Hollywood, you'll be bigger than Doris Day.'\n\nThe entire party was like a film set. The backdrop was the gracious, high-ceilinged room, with its paintings and exquisite furniture. Jewellery glittered and jingled, perfume assaulted the nose. Outrageous dresses on perfect bodies. Dinner jackets and bow ties. Braying 'Sloane' accents mingled with cockney and Liverpool. One minute she was just another singer hoping for a break, now suddenly she found herself amongst kindred spirits.\n\nShe spotted Michael Caine's blond hair and glasses across the crowded room, dancing with an equally tall languorous brunette. His friend Terence Stamp was said to be there too but she didn't see him.\n\nDance after dance. Friendly chatter as if she had always been part of this exclusive club, shared jokes and invitations to everything from dinner to a country house weekend.\n\n'No Time' was played while she was in the dining room helping herself to food. She giggled with embarrassment as heads turned towards her, and Max caught her eye.\n\n'This is what it's all about.' He came up behind her and whispered in her ear. 'By tomorrow you'll be the name on everyone's lips.'\n\nAs Georgia sipped yet another drink she contemplated whether she would ever be rich enough to have even a small house of her own.\n\nApart from the drawing room, and the dining room she'd only had glimpses of the other rooms in this huge place. A library lined with old books, a sumptuous sitting room with silk-lined walls and matching peachy-coloured armchairs. Upstairs there were at least six bedrooms and there was another floor above that and a basement too. What would you do with all that space? Would you end up like Louise, throwing parties all the time just to fill it?\n\n'I think this had better be the last one,' Max's deep voice came from just behind Georgia as he handed her yet another glass of champagne, an hour later.\n\n'Who says?' Georgia giggled. They hadn't been together for more than a few minutes at one time all night, yet she'd been aware of his presence. Even amongst all these dynamic people he was still a man to catch the eye. His shoulders broader, his tan deeper than most. The expensive cut of his cream suit, the animal grace. 'Have you been watching me?'\n\nShe was on a real high, not just drunk. Everything really was going to happen for her at last. It was as if she were standing at a doorway, looking in on a whole new world and all it took was one more step to be part of it.\n\n'Of course I've been watching you,' Max said, taking her elbow and steering her towards the drawing room to dance. 'Everyone has, and I don't want you to get so drunk you make yourself look silly.'\n\nGeorgia leaned against his big chest. She was already a little wobbly but it felt good.\n\n'I'm proud of you,' Max said putting his arms round her. 'I don't think I've ever felt so proud of anything. Come and dance with me. You've danced with everyone else.'\n\nShe looked up at Max's rugged face, felt his arms around her and his words of praise ringing in her ears, gratitude overwhelmed her. The sharp practice, the bullying were all forgotten. This was the man who had believed in her. Without him she would still be back at Pop's on the sewing machine.\n\nThe wild dancing had slowed to smooching. Small groups of people sat on the stairs, in corners, even in the garden to talk.\n\nMax held her close, his lips against her hair. The heat of his body, the sweet music and the slow dancing made her feel almost unbearably happy.\n\n'The summer's nearly over,' Max said softly in her ear. 'How about those few days in Spain now while we wait for the record to reach number one?'\n\nShe giggled softly.\n\n'Why the giggle?' he asked putting one hand under her chin and lifting her face up to his. 'I just thought you needed a rest.'\n\nHis dark eyes weren't probing into her as they often did, they were soft and loving, his lips wide and smiling showing his white even teeth.\n\nIt was two years since that night when he'd kissed her, but it was a memory she hadn't forgotten. A strange and wonderful feeling was creeping over her, like she was being sucked into something she couldn't and didn't have to fight any longer. His hands were holding her so firmly, fingers just running down her spine, making her knees turn to jelly.\n\nHis eyes closed as his lips came down to meet hers, one hand moving up to caress her cheek. This time there was no fear, just a sense of something stronger than both of them and that this was the moment they'd been waiting for.\n\nShe felt his tongue first, flickering round her lips, sending out messages to each nerve-ending. Sensual, light touches that set her body on fire.\n\nThe music seemed to be engulfing her, as if written specially for them and slowly his mouth covered hers.\n\nNever before had a mere kiss been so intimate. It was everything, blocking out the other guests. It wasn't just a kiss but making love. She could feel her nipples growing hard, dampness seeping from her and at the same time she could feel his hardness pressing into her belly.\n\nInside she was trembling, her arms were reaching out to draw him closer, her tongue thrusting against his. Her fingers caressing his ears, neck and cheek, wanting him, now.\n\nShe had no idea how long they stayed wrapped in each other's arms. It was no longer teasing or tempting, but a need to possess so strong there was no question of backing away.\n\n'I've wanted you for so long baby,' he whispered against her neck, licking and stroking until her legs almost buckled under her. 'I want to kiss every inch of that beautiful body, I want it to sing for me.'\n\nGeorgia didn't remember Max suggesting they left, or even walking out the door. It was as if they floated out, one moment in a music-filled room, the next in the front garden, the smell of roses and damp earth mingling with the smell of their bodies.\n\nIt was hard to break away even to get in the car. Before Max turned on the ignition he had to pause to kiss her again.\n\nGolden light from a street lamp bathed them as his big hands cupped her breasts. She found her hands reaching for buttons on his shirt, wanting to touch flesh so badly it hurt.\n\nShe didn't ask where he was taking her, it didn't matter. Her pulse was racing, her limbs ached for him, if he had pulled into the side of the road that would have been enough.\n\nHe drove only a few streets away and stopped, turning to her and taking her in his arms again, his mouth devouring hers.\n\nWhen he released her and opened his car door, only then did she notice they were outside a block of Victorian flats with railings and marble steps up to plate glass doors.\n\nLike a child she took his hand trustingly. She leaned closer to him as he unlocked the door and his arm encircled her and drew her inside.\n\nUp a wide staircase with olive green carpet and delicate silk wall coverings. On past polished wood doors with brass numbers, the sort of place she imagined snooty ladies and gentlemen living.\n\nOn the second floor Max pulled out another key, he drew her closer and opened the door.\n\nA glimpse of honey coloured carpet stretching up a long narrow hall, gilt wall lights and hessian-covered walls.\n\nHe kissed her again, kicking the door shut behind them, then scooping her up into his arms, he carried her into the dark bedroom.\n\n'Watch this,' he whispered. She couldn't see much, just his hand reaching out for something in front of them. She heard him flick a switch and the room was filled with soft light.\n\nIt was a huge room dominated by a giant bed with a suede headboard. As he touched another switch, so gold velvet curtains slowly closed over the big window and sweet music filled the air.\n\n'What do you think of that?' he looked down at her in his arms. His expression was almost boyish, showing off his latest toy, but rather than spoiling the moment she loved him for it.\n\n'Amazing,' she smiled, unable to look at anything other than him. His big, rugged face looked softer now, his lips so red, eyes full of fire. 'But just kiss me again?'\n\nHe lowered her to the floor as he kissed her, his fingers already unzipping her dress, slowly peeling it from her shoulders, letting it drop to her waist.\n\n'Even more beautiful than I imagined,' he whispered, touching her breasts reverently. 'I've always loved you Georgia.'\n\nThe satin bedspread caressed her back. He kissed her feet as he took off her shoes, slowly licked her thighs as he moved slowly upwards to peel off her panties.\n\nOnly when Max stood up to tear off his own clothes did she realize something wasn't quite right. Her eyes didn't focus as they should, her breathing felt odd. The music seemed to be right inside her head and her heart was pounding with more than just desire. She was no stranger to drink, it was something else entirely. But she was helpless, a fire seemed to be raging within her, she wanted him so badly it burned.\n\nNaked, Max seemed even bigger. Wide muscular shoulders, tanned dark brown with a matt of black hair on his chest. His stomach was flat and taut with muscle and as he pulled off silk shorts, his penis rose up alarmingly large.\n\n'You're so perfect,' he said moving beside her and bending to kiss her small pink-tipped breasts, his tongue flickering over her nipples. 'I'm going to love you tonight like you've never been before.'\n\nLove-making with Ian had been sweet and tender, his hands gentle but unsurprising. With Max it was like being on a whirlwind of pleasure, one moment rough and demanding, the next so sensitive she wanted to cry. One moment his arms were crushing her, the next he moved away using only his tongue to tease her into rapture.\n\nAgain and again he brought her almost to the point of orgasm, then he'd pause, almost laughing at the way she tossed her head on the pillows and clawed at him for more. One moment he was on top of her, his hands holding her buttocks, driving himself into her, then suddenly he would roll over, taking her with him, lifting her gently, his lips back on her breasts.\n\nIt was naughty and delicious, rough and passionate, beautiful and gentle all at once. Georgia saw his red pointed tongue slithering down her body and she felt as if a pit of fire was about to consume her.\n\n'Make me come,' she heard herself cry out, catching his head between her hands and pushing him down onto her.\n\nBliss so exquisite she could only claw at the bedspread, writhing under his tongue. 'Harder, harder,' she shouted. 'It's wonderful!'\n\nShe was out of control now, carried away to a plane where only sensation mattered, all sense of reality gone. A million fireworks were going off inside her, she could hear brass bands.\n\nHer hands reached down to Max to draw him into her arms, but instead he knelt up beside her as she lay trembling.\n\nShe opened her eyes to find the room was spinning. Max was grinning down at her, tiny curls had crept round his face, lascivious red mouth and hairy chest made her think of a satyr.\n\n'Don't,' she said involuntarily. She couldn't say what it was that disturbed her, but she wanted a more tender look, more words of love.\n\n'Don't what?' He reached over to a drawer beside the bed, pulling out a packet of Durex. The ease he bit off the packet with his teeth and the speed with which he slid the rubber on his erect penis was further proof of how often he seduced girls.\n\n'I'm going to fuck you senseless now,' he whispered hoarsely as he scooped her up and turned her on to her stomach. Before she could protest he was pushing his way into her from behind, gripping her like a dog with a bitch.\n\n'Please don't, not like that,' she called out. Memories she'd thought she'd buried for ever were rushing back. But Max was oblivious.\n\nShe tried to wriggle away but he grabbed her round the middle so tight she felt nauseous, his breath rasping on her back.\n\nShe was no longer on a satin-covered bed with a man who'd taken her on a brief trip to heaven.\n\nIt was her father. Forcing his way into her, humiliating, degrading and hurting her. She could smell the smoky room, the drink on his breath. Feel the carpet beneath her, the prickle of her net petticoats against her thighs, even hear the filthy things he was saying.\n\n'No,' she whimpered. 'Please don't.'\n\nShe heard his breathing getting louder, grunts of animal joy, and finally he shuddered to a halt, slithering onto the bed beside her like a deflated balloon.\n\nFor a moment Georgia just lay on her stomach, face buried in the pillow. She wanted to cry, but no tears came.\n\n'Come here for a cuddle.' His voice wounded her further, she could hear the tenderness in it. He didn't even know he'd hurt her.\n\nThe room looked hideously vulgar now, all gold and ivory. This was his lair. Satin, mirrors, even the electronic devices to operate the curtains and music. Every last thing designed to impress and seduce silly drunken girls. She could see his silk shirt lying on the floor, the double 'M' monogram in gold thread glinting in the light.\n\n'What's up?' he asked, turning heavily towards her, one hand on her arm.\n\nWhat could she say? Was it possible to explain to any man that one moment she'd been almost fainting with desire, the next he made her remember a rapist?\n\n'Georgia,' his voice was soft as he realized something was very wrong. 'Tell me what it is? What did I do?'\n\n'I don't like it like that,' she whispered. 'Didn't you hear me telling you to stop?'\n\nHe sat up then, moving round to look at her. The face looking down at her was stricken with remorse.\n\n'I didn't hear you. I'm so sorry.'\n\nThe anger at him faded, replaced by a sick feeling about herself. He wasn't to blame, she was. What on earth had possessed her to even kiss him, let alone come here with him?\n\n'It's all right,' she whispered back. 'Get into bed, I'm just going to the bathroom.'\n\nHe moved then pulling back the gold satin covers to reveal black satin sheets.\n\n'It's just down the passage,' he said softly, crawling into bed. 'Come back quickly and we'll talk.'\n\nShe could see the sleepiness in his eyes. His face looked lined now, revealing his age. She pulled the covers up round him and dropped a chaste kiss on his forehead. As she reached the door she looked back.\n\nHis eyes were closed already, his hand snaking up to turn down the lights and music. Bending down she picked up her clothes and boots and backed out of the door.\n\nThe bathroom gleamed with opulent bad taste. Mirrored walls, soft lighting. A bath big enough for four people sunk into the floor. The basin set into white marble with gold taps, a purple carpet with matching sets of towels hanging on thick gold rails. Erotic pen sketches were framed in gilt, a naked male statue in one corner.\n\nWhen she opened a drawer under the washbasin there was even more evidence of who Max entertained in this place. A variety of cosmetics, even a vaginal douche.\n\nFor a moment she wanted to be sick. She was one of his groupies now. For two years she'd laughed at these vacuous, wide-eyed dolly birds. Not one memorable, designed like paper tissues to be used and thrown out. Yet mindless as they were, surely none of them had been foolish enough to think it was love that motivated Max?\n\nTen minutes later, Georgia stole back up the corridor towards the front door fully-dressed, her shoes in her hand.\n\nGlancing to her left, she could see Max lying on his back, mouth open, snoring softly. She opened the door and let herself out silently.\n\nAs the cool night touched her flaming face she knew with certainty she had been drugged. Her legs were unsteady, her heart beating too fast, throat dry. Some of that could be explained by the champagne, but not the strange way things looked, the brightness of colours, or her losing her inhibitions so suddenly.\n\nShe could blame Max for that perhaps. He was a practised seducer who would use anything at his disposal. But the rest of it was all her fault, and somehow she would have to live with that, and the repercussions later.\n\n## Chapter 16\n\nGeorgia heard the doorbell ring but ignored it. It was nearly four in the afternoon, and she was certain it was Max.\n\n'I can't face him,' she whispered to herself.\n\nThe door bell rang again.\n\nHer window was open, a soft warm breeze wafting in, the street below was silent, the way it always was on Sundays.\n\n'Georgia!' She heard a voice. 'Come on, open up.'\n\nNot Max but Rod. She didn't want to talk to him either, but he spotted her peering over the window-sill.\n\nSlowly she went downstairs. It had been nearly five in the morning when she got home. The drug she'd been given let her walk all the way home without noticing the distance, but it hadn't numbed the shame of what had happened, or allowed her to sleep. How was she going to face Max again?\n\n'Being a recluse, eh?' Rod grinned at her. He looked like an Apache warrior as the afternoon sun caught his high cheek bones and made his blue-black hair gleam. Even his faded denim shirt and jeans fitted the image. 'So how was the party?'\n\nRod's dark eyes skimmed round Georgia's room. It was unnaturally clean and tidy, as if she'd been cleaning all day, the smell of polish and disinfectant worse than a hospital. Her eyelids had a faint mauve tinge, the way she always looked when she'd been up all night worrying. Even her hairstyle, two childish fat plaits either side of her face was a sure sign she was insecure.\n\n'Someone's upset you,' he said with sharp perception, noting the blue-checked shirt she wore over her jeans was Ian's. 'Come on, tell me everything.'\n\n'The party was wonderful,' she said in a small voice, going over to the window and staring out rather than face his probing eyes. 'I'm just kind of hungover.'\n\nShe couldn't tell a man who shared Max's attitude to women how things were.\n\n'You're hurting,' he said softly, coming up behind her, sliding his arms round her waist and putting his chin on her shoulder. 'Is it missing Ian, or something more?'\n\nHis brotherly hug brought tears to her eyes. She wanted to tell him the truth, but she couldn't.\n\n'I'm fine, really,' she sniffed. 'I just did something last night I regret. Don't try and get me to tell you. I couldn't explain.'\n\nIt was odd how Ian's death had changed Rod. Two years ago she wouldn't have dared to be alone with him. But the tragedy had revealed more than his courage. Underneath the arrogance and sardonic wit, he had a deep insight into human behaviour.\n\nFor a moment he said nothing, just increased the pressure of his hug.\n\n'Don't waste time with regrets,' he said softly. 'I can imagine what happened, you met a man who sweet-talked you, now you feel like a tart.'\n\nGeorgia turned her head, nuzzling her cheek against his.\n\n'That's about the strength of it,' she sighed. Perhaps he wouldn't be so understanding if she told him it was Max, but at least in principle he knew the full story.\n\n'I do it all the time,' Rod gave a hollow little chuckle. 'I don't learn. But you will, you aren't as thick as me. The only advice I can give you is to tell the man the truth. Don't turn one mistake into a series of them as I usually do.'\n\n'You mean you regret things sometimes?' Georgia's dark eyes grew wide with surprise.\n\nShe could remember all the times the band had picked him up from girls flats, the cocky way he swaggered out, his ribald comments, never a look back at the pale sad face at the window.\n\n'Yeah,' he moved away from her and filled up the kettle. 'The ones that wanted more than I could give them.'\n\n'Have you ever thought it was love?' she said. 'You know, when everything seems so perfect, then \u2013' she stopped short, unsure she should say more.\n\n'Lots of times,' he smiled and took two mugs out of the cupboard. 'But you mean when you realize right in the middle of it, that it's a mistake?'\n\nShe nodded.\n\n'That's a bummer,' he laughed cheerfully, almost dispelling her gloom. 'I usually tell them I was attracted to them because they reminded me of a girl I lost, but I can't go through with it now I know what a fine person they are.'\n\nGeorgia giggled. 'Does it work?'\n\n'Rarely, they usually go so chilly on me I have to leave,' he pulled a woebegone face.\n\n'Oh Rod,' she was really laughing now. 'I'm glad you came round, you've made me feel much happier.'\n\n'Well, take Ian's old shirt off,' he said as he made coffee. 'Put something snazzy on and we'll go and see a film. Regret's a mug's game.'\n\nDelivery vans woke her early on Monday morning. She got out of bed and put the kettle on, but only then did she realize she had no milk left.\n\nSlipping on a pair of jeans, and the first sweater that came to hand, she ran down the stairs.\n\nThere was a cold, crisp autumn feeling in the air. Even though the sun was shining it was too low in the sky to reach her side of the street as it did in the summer.\n\n'Good morning Georgia.'\n\nShe stopped and looked around. The market men were always calling out to her. But this voice didn't belong to a cockney barrow boy.\n\nTwo men were pushing their way through the stalls towards her. Both had cameras slung round their necks, but she didn't make the connection immediately.\n\n'Good morning,' she replied, aware several stallholders had stopped work to watch. 'How do you know my name?'\n\nOne was in his thirties, neatly-combed brown hair, wearing a tweed jacket, a pleasant open face that invited trust. The other was older, fat-faced and balding, wearing a beige three-quarter-length raincoat.\n\n'Don't tell me you didn't know your record reached number forty on Saturday?' the younger one said, smiling as though amused by her surprise. 'Surely we aren't the first reporters to track you down?' As he spoke he lifted his camera and took a picture of her.\n\n'Yes. No,' she was flustered, aware her hair was uncombed and her sweater had a hole in the elbow. 'I mean, yes you are the first, and no I didn't know it had reached the charts.'\n\n'They reckon it could be number one in a couple of weeks,' the older man said. 'It could topple the Beatles. How do you feel about that?'\n\n'Thrilled.' She didn't actually feel anything, in fact she was sure it was some kind of hoax. If she'd known they were hanging around her door she wouldn't have come out at all. 'What paper are you from?'\n\n'The _Mirror_ ,' the older man's hand shot forward to grasp hers. 'Dave Barnet's the name, could I buy you some breakfast and have a little chat?'\n\nThe younger man lifted his camera again, she turned to him waving her hand to stop him, suddenly anxious.\n\n'Couldn't you have phoned or something? Are you from the _Mirror_ too?'\n\n'No, I'm from the _Evening News_ ,' the younger put down his camera and advanced on her. 'Giles Went-worth. I tried to telephone yesterday, but you aren't on the phone here. This must be just a temporary place?'\n\nHe was gazing around the market, noting the pile of empty boxes and refuse by her door. His tone was warm, yet she bristled at the implication behind his words.\n\n'No,' she said, running her hand through her hair, wondering if she should be charming, or send them away quickly. 'I like it here. It's where I belong.'\n\n'You were born here then?' the older man chimed in.\n\n'No, in the East End.' She saw the pit yawning before her. She would have to get a grip on herself, make a sensible background for herself before she said something she might regret later. 'Now look, I'm surprised and flattered by your interest,' she batted her eyelashes at the younger reporter. 'But it's too early for questions or pictures. I haven't even had a cup of tea yet this morning. Why don't you phone my manager Max Menzies and he'll arrange a proper interview?'\n\nThe market men were getting really interested now. They pressed in closer and from out of their depths came ribald comments and encouragement.\n\n'Give 'em a song Georgia!'\n\n'Tell 'em 'ow hard you've worked love!'\n\n'They seem to be your friends,' Giles Wentworth looked almost nervous at the crowd gathering around him. 'Could you pose with them for me?'\n\nShe never knew whose arms came up and lifted her off her feet, one moment she was vertical, the next horizontal lying in the men's arms supported by four of them.\n\nA click of the shutter, someone tickled her and another picture caught her laughing.\n\n'Just tell me Georgia,' the older man moved closer as the men released her. 'Is \"No Time\" in memory of Ian and Alan?'\n\nShe was flustered. She should have thought all this out before. One wrong statement now and she could be damned forever.\n\n'Not exactly, though the words came to me after their funeral.' Was that the right thing to say? She hadn't wanted anyone to dwell on Ian and Alan's death, and after the evening with Max she felt she'd betrayed Ian.\n\n'You were in love with Ian McShane,' the younger man said. It wasn't a question, more a statement he wanted confirmed. 'It must have been a terrible blow?'\n\n'The worst kind,' Georgia felt tears rise in her eyes. 'But you must excuse me now. I've got an appointment later on this morning.'\n\nAn hour later as she showered and washed her hair, the significance of the two reporters sank in. This was what she could expect now, people popping up at all times, trying to catch her unawares. Even though she didn't want to see Max, she had to. Like Rod said, regrets were a mug's game!\n\n'Why did you run out on me?' Max sat at his desk twiddling a paperclip. Thick dark stubble covered his lower face. If he wasn't wearing a different suit and tie she would have thought he hadn't been home since the party.\n\nRod had made her feel better yesterday. The news of the record making the charts suggested at last things were about to blast off. She had to find the right words, make Max realize that Saturday night was something behind them both.\n\n'It was all a mistake,' Georgia blushed, it sounded so cruel. 'Someone slipped something in my drink.'\n\nTo her surprise Max grinned, showing his white even teeth.\n\n'Well, that's a flattering thing to say. I take it you mean letting me make love to you was the mistake, not walking out the door?'\n\nFor a second she thought Max was relieved at her attitude, that he'd thought it over and decided it was a non-starter.\n\n'I shouldn't have let it happen,' she hung her head. 'You're married and my manager, you're even old enough to be my father. It would be best if we buried and forgot it.'\n\n'You don't claim it was rape then?'\n\nGeorgia's eyes flew wide open, she stiffened visibly.\n\n'What did I say?' Max got up and came round the desk towards her.\n\n'N \u2013 nothing,' she stammered.\n\nMax shook his head, a faint glimpse of amusement in his eyes. He perched on the desk just inches from her, staring at her hard.\n\n'I can't forget,' he said softly. 'You can play hard to get if you like. The terrified virgin if that's how you feel. But I know you enjoyed it, whether or not someone slipped something in your drink. I know I'm too old for you, but there's always been a spark of something between us, even you can't deny that.'\n\n'All right, maybe there was,' Georgia looked up defiantly. 'But it's something I don't want to repeat. If it had been right I would have been happy about it. All I feel now is misery and shame.'\n\n'You certainly know how to flatter a guy,' Max folded his arms, his mouth turning down at the corners. 'I can't say I won't try again because I want you Georgia, any way, any how. But if you want breathing space, you've got it. Can I be fairer than that?'\n\n'I want Saturday night to be wiped from your memory and mine,' she said in a whisper. 'I want us back where we were, business only. If you can't handle that then I'm sorry.'\n\nShe couldn't look up at him. She knew he was hurt, yet if he felt so badly why didn't he come round to her flat the next day, or even ask how she got home?\n\n'Is this all you came here for?' his tone was icy now, and she remembered all those secretaries and office girls who'd been sacked once he'd lost interest in them.\n\n'Not exactly.' She had to remember he needed her. She must be adult even if she felt like weeping inside. He wasn't going to grind her down like he did everyone else who crossed him. 'There were two reporters outside my door this morning. They said the record's at number forty. I thought we ought to talk about both things.'\n\n'Did you now,' Max sneered, not even a flicker of affection in his eyes. 'I suppose you realized the old man has some uses?'\n\n'Fuck off Max,' she snapped, getting to her feet. 'I've done my best to be straight with you. If you don't want to be my manager then I'll find someone else. Don't try and make me crawl to you, because I'll start to hate you.'\n\nHe caught hold of her wrist, digging his fingernails into her flesh.\n\n'Getting a bit above ourselves?' he hissed. His breath stank of cigars and uncleaned teeth. Georgia recoiled in disgust.\n\n'I learned that from you,' she snapped. 'Now either we talk about what we say to the press or I go out and say whatever I feel like. It's up to you!'\n\nHe dropped her wrist immediately, stood up and backed away, returning to his seat behind his desk. She could see a vein throbbing in his forehead and she guessed he was trying to control himself.\n\n'Let's pretend Saturday night ended with the gig at the Scene,' he said slowly, looking down at his big hands clenched on his blotter in front of him. 'I won't bring it up again, I'm just your manager.'\n\nIt was a hollow victory to see him like this. Part of her wanted to rush round the desk and hug him, to reassure him he wasn't old and ugly, that he would always have a place in her life. But she didn't dare. Max would take it as a sign of weakness.\n\n'So is it true? The record is number forty?' she said, sitting down again.\n\n'Yes.' He sighed deeply, as if struggling to control conflicting emotions. 'I heard last night, and came over to your place to celebrate with you. But you were out.'\n\n'I see.' She wasn't going to apologize, or explain where she was. 'I take it you found someone else to celebrate with?'\n\nHe half smiled then, rubbing one hand round his bristly chin.\n\n'You could say that.'\n\n'Well I suggest you get in that bathroom and shave.' Her tone was brisk and unsympathetic. 'I'll sit here and write a profile of myself while you're gone.'\n\nMax looked at her for a moment, then threw back his head and laughed. It echoed round the big office, shaking the framed photographs of Bill Hayley and Jerry Lee Lewis.\n\nGeorgia was stunned, but at least his laughter was better than morbid self-pity.\n\n'You can be a right little bitch sometimes,' he said, still spluttering with laughter. 'But you've certainly got a head on your shoulders.' He pushed a sheet of paper towards her and his gold pen. 'When I've made myself presentable we'll talk photographs and press releases. You'll need new clothes and somewhere better to live. Berwick Street might have been handy when your face was just another pretty unknown one. But by the time the _Evening News_ has printed something tonight you'll have fans and weirdos camping on your doorstep. With you alone at night in that house, anything could happen.'\n\nMax stayed in the bathroom for over half an hour. When he came back into the office he found Georgia resting her head on folded arms on his desk. The litter bin beside her was piled up high with screwed up pieces of paper.\n\n'Have you done it?' He came round behind the desk and looked down at what she'd written.\n\n'You've said nothing about yourself!' he exclaimed. 'Orphaned during the war. Left school without any qualifications, spent some time as a machinist before getting a first chance to sing at the Acropolis club in Greek Street.' He flung the paper down angrily. 'You'll have to come up with a bit more meat than that!'\n\n'There is nothing else,' she grinned up at him. 'We'll just have to be mysterious.'\n\nMax sat down heavily on his chair, loosening his tie and undoing his top button, a waft of Moustache aftershave reached Georgia's nose.\n\n'I wish I could believe you,' his eyes narrowed with suspicion. 'What is it you're hiding?'\n\n'Nothing.' Georgia stood up and stalked across the office to the window, keeping her back to him. 'I just don't think my childhood can possibly interest anyone. Who cares anyway what school I went to, or whether I was happy. It's now that counts. Let them ask me what I'm interested in, what books I've read, or who influenced me. That's enough for any fan surely?'\n\n'Just think, Rod,' Georgia said later that evening as together they pored over a copy of the _Evening News_ , in the dressing-room at the Marquee. 'In a couple of weeks we might be eating at the Ritz!'\n\nMax's phone had rung constantly all day with press wanting to know more about her. The paper in front of them proved the public was hungry for information.\n\nShe liked the picture of her in the arms of the stallholders in Berwick Street, but she wasn't exactly sure about the rest.\n\n'Berwick Street beauty tipped for the top,' was the headline.\n\n'Georgia James races up the charts with her record \"No Time\". Two years ago Georgia was selling frocks in Berwick Street on a market stall, now she's on her way to stardom with a heart-stopping ballad written by herself and her band Samson. Two of the boys in the band died in a fire earlier this year and the song is dedicated to their memory.'\n\n'Shit,' Rod exploded. 'Why did you say that?'\n\n'I didn't exactly,' Georgia said. 'Neither did I say I sold dresses.'\n\nMaybe the article was corny. Yet she couldn't help raise a smile about her love for Soho and the market.\n\n'They've made me sound like a cockney sparrow,' she giggled. 'Look at what Bert in the caf\u00e9 said!'\n\n'\"She's got a knockout voice. Of course she'll go all the way to the top, we never doubted it round here. She's a lovely girl and we're all very proud of her,\"' she read, wiping a tear from her eye. 'Bless him.'\n\nThat night people were turned away from the Marquee. Inside it was packed so tight with fans the temperature rose to the nineties. No dancing for the fans now, just pressed up together, hundreds of people clamouring to see her, prepared to face the discomfort just to hear her before she moved on to bigger venues.\n\nThe next day it was the _Mirror_ 's turn, taking it a stage further with pictures of her against piles of fruit, her door in the background. Another one of her on stage in Miriam's old white dress, belting out a song, head thrown back, hair like a storm.\n\nThey re-capped on the tragedy of Ian and Alan's death and the star-studded benefit concert that followed it.\n\n'No time', seemed almost prophetic now. There was no time to visit her old friends, no time to catch her breath. She had caught the public's imagination and as the week passed, so the record flew up the charts.\n\nMenzies Enterprises was besieged with callers. Club owners wanted to book the band while they could still afford them. Magazines wanted profiles. Radio and television producers offered guest appearances and fans stood outside hoping for a glimpse of her.\n\nBut to Georgia it became real when people turned to stare at her in the street and bolder ones came up to her and asked outright for her autograph.\n\nTwo days after the reporters found her home, Georgia was whisked off by Max to The Sunderland, a small hotel close to Sloane Square, leaving most of her belongings behind in Berwick Street.\n\n'We'll carry on paying the rent,' Max said as he closed the door behind them. 'Let the public think you're still here for the time being.'\n\nNo more long hours of lying in bed catching up from the gig the night before. No afternoons spent wandering around new towns or listening to records with the boys. Max would ring first thing in the morning to arrange her day, every minute planned to give her the maximum publicity.\n\nWhether she was buying clothes, or eating a meal, somehow the press were always there. Cameras zooming in on her, microphones poised to catch even the most trivial of remarks. She had to learn to think before speaking, to keep her guard up at all times.\n\nIt was exciting, yet frightening. Like that drug, it distorted her feelings, one moment she was high as a kite, the next strangely alone.\n\nMax was in his element as 'No Time' approached number one. He didn't miss a trick. Sending men ahead to gigs to erect crash barriers for crowd control, creating mass hysteria himself. Press releases were timed to keep her image high, and almost every day there was a picture of her in one of the papers.\n\nAt each performance the crowds got steadily larger. Queues stretched down the roads outside for hours before, and as she arrived in a limousine with blacked out windows the fans would surge forward, hands reaching out to touch her.\n\nGeorgia was amused yet frightened by Max's ploys. She understood that he had to promote her with everything within his power to make sure he got a return on his investment, but there was something vaguely dishonest about the flashy way he approached it.\n\nOnce it had been hard to get him to watch an entire set, but now he never missed one. He strutted around in his expensive clothes and gold jewellery, opening magnums of champagne. He ordered people around as if he were the star, with a retinue of vacant-looking girls posing as personal assistants clinging to him.\n\nSuddenly everything had to be the best. A luxury coach, to get the band to gigs. A couple of tough-looking men to act as 'minders.' A permanent photographer. Hairdresser and make-up artist. Public relations girls, a woman to look after their stage clothes, and roadies to unload and set up the equipment. Hotels were booked for them, dinners, parties and press conferences. It was a circus, with Max as the ringmaster. He didn't consult the band about anything. But who was paying for it all?\n\nThe big black Daimler cruised almost silently down the Strand. Georgia wriggled forward in her seat, her heart thumping with excitement.\n\nJust three hours ago she had heard 'No Time' was finally at number one, pushing the Beatles off their perch. Tonight she was going to a party at London's Savoy Hotel, where she was to be the guest of honour.\n\n'I used to come down here late at night with my friend,' she confided to the chauffeur. 'We used to watch all the rich people and pretend we were rich too.'\n\nIt was New Year's Eve, she remembered most distinctly. Helen with the fur collar of her coat turned up against the cold wind, her hair like gold under the street lights, shivering as they stood in a doorway to watch a stream of cars like this one, disgorging women in furs and silk dresses at the hotel.\n\n'You never guessed one day it would be you?' The man chuckled, surprised by the star's childish confession.\n\n'Not even in my wildest dreams,' Georgia laughed softly, breath hot on his neck, her perfume filling his nostrils. 'I wish Helen was with me now. She died before I got my first singing job.'\n\n'Are you nervous?'\n\nRobert Wells was over fifty. He'd been working for this company for almost ten years and in that time he'd driven more famous people than he could count. Actors, opera singers, lords, members of Parliament and film stars, but never once had he driven anyone so pent up with excitement.\n\n'Terrified,' she admitted. 'Do you think I look all right?'\n\nHe glanced into the mirror. All he could see clearly were her sparkling eyes, but he could remember the way she looked as she walked out of the hotel.\n\nA long red clingy dress, dark curls tumbling over her bare shoulders, and a face so lovely he could hardly drag his eyes from it.\n\n'All right?' he laughed softly. 'You look fabulous. You'll knock 'em all dead!'\n\nRobert slowed the car, ready to turn into the forecourt. The Savoy had never looked more beautiful. Floodlights had turned it into a golden temple framed by a black velvet sky. Gleaming plate glass doors, beyond, white marble, rich carpets and chandeliers. A perfect setting for this enchantress.\n\nGeorgia smoothed down her dress, spreading her fingers out to check she hadn't chipped the matching nail varnish. She could see Rod waiting for her by the door and she wondered if he was as nervous as her.\n\nHe looked so handsome. A white suit straight out of a Hollywood film, his black hair sleek and shiny, restyled with a middle parting, accentuating his Red Indian looks.\n\nThe car cruised slowly to a halt. A liveried doorman leapt forward to open her door. A group of fans pushed against the security men who tried to contain them.\n\n'Have a great time.' Robert turned round from his driving seat to look at Georgia one more time. 'I'll be back to pick you up later.'\n\n'Well, you look the business,' Rod said softly, taking her arm and leading her towards the open door.\n\n'I could say the same about you,' Georgia touched his bow tie lightly. 'Thank you for waiting out here for me, I'm scared stiff.'\n\nAhead of them as they walked up the few heavily-carpeted stairs, Georgia could see the ballroom. It was already very crowded, the soft music almost unheard under the barrage of chatter and clink of glasses.\n\n'I never thought we'd end up anywhere as posh as this,' Georgia giggled to Rod, pointing up to a chandelier above them. 'That's the real thing, not like the kind Max has in his hall. Don't any of you get too drunk and show us up!'\n\nThere was a hush as they walked in, people turned and stared at her and in that instant, Georgia felt a charge of something strange.\n\nIt lasted only a second. A glass of champagne was put in her hand, and all at once there were people clamouring to speak to her.\n\n'Congratulations on reaching number one. I'm so thrilled to meet you at last. I just love the song.' The flattery wrapped her in a warm blanket. These sophisticated people in evening clothes, dripping with jewels seemed to know so much about her. Every one of them looked important.\n\nAcross the crowded room she could see John and Norman with two leggy, blonde girls. Les looked almost handsome in a grey suit, as a red-haired woman talked to him earnestly. Speedy's auburn hair caught under the lights, complemented his grey velvet jacket and his dancing partner could have been a model.\n\nYet for all the glamour, there were no friends in the crowd. Where were all the other stars she'd met on tours? People who she could really talk to. Charming as most of the guests were, Georgia felt a little out of her depth. Lawyers, promoters, club owners, business men and their wives, surely if this party was thrown for her, real friends should have been invited too.\n\n'Georgia,' Max pushed his way through the crowd, took her hands and kissed both of them. He wore a dark dinner jacket with a plum-coloured cummerbund. 'You look gorgeous!'\n\nWith him was a tubby smaller man, reptilian eyes flickered behind gold-rimmed spectacles, his large forehead glistened with perspiration.\n\nWith one arm round her Max introduced them.\n\n'This is Al Green from Memphis, he flew over this afternoon to meet you.'\n\nThe name 'Al Green' was one Max often brought up in conversation. Georgia understood he was responsible for the glut of American pop stars that dominated the charts. He arranged tours for everyone from Elvis Presley downwards.\n\n'Hi there.' The man put a podgy hand into hers, his thin lips barely moved and she could see no pupils in his dark eyes. 'You're quite a girl Georgia. We've been hearing your name even back in Memphis. This is one helluva party honey.'\n\n'I've heard a lot about you, too,' she smiled politely. 'How nice of you to come all this way just to meet me.'\n\nShe didn't like him. It was ridiculous to feel something so strong when she'd only spoken a few words to him. Maybe it was just those eyes, how could anyone feel anything but repulsion for a reptile?\n\n'I don't know whether I'd have come if you hadn't been footing the bill,' he laughed, double chin wobbling, taking out a flamboyant red handkerchief and wiping his shiny brow. 'But now I'm here, I'm just loving it.'\n\nGeorgia looked round for Max, only to see his back view retreating into the crowd.\n\nFor a moment she just stared at the man. His dinner jacket was midnight blue, as he moved she caught a glimpse of silver lining. It was vulgar, even for someone in show business, the mark of a man who had no taste.\n\n'Me footing the bill?' She could feel her heart thumping just that little bit harder. Something smelled fishy, and she was going to get to the bottom of it.\n\n'Well, Max is your manager,' he licked his thin lips focusing on her cleavage. 'Don't that mean the same thing honey?'\n\nWhat was it Max had said right at the outset? 'I lay out all the money, and when you start earning that's when I'll get it back.'\n\nAll those chauffeur driven cars, champagne, new clothes, photographs, hairdressing. Everything was being logged down against money she was earning. But why should she pick up the tab for a party that was supposed to be for her? Or pay to fly this jerk over to meet her?\n\n'You and Max are lining up an American tour?' The man must have got his wires crossed. Maybe Max just implied the expenses would be met by him to try and impress Al.\n\n'Sure thing, honey,' he drawled. 'I'm gonna take a look see round this little island, get myself a piece of the action.'\n\n'How do you think I'll go down in the States?' she smiled sweetly. 'Has the record reached the charts there yet?'\n\n'It's been played honey,' he shot her a scathing look. 'But what our kids want is good old rock and roll.'\n\nAll at once Georgia understood. Max was offering free seats on the gravy train. This man wasn't interested in her. It was an excuse to get his podgy hands on some British rock and roll bands, kids with stars in their eyes and no experience. The pair of them were intending to expand their interests, using her earnings to finance it.\n\n'Where are you staying?' she asked through clenched teeth.\n\n'Here, honey,' he drawled. 'Max booked me into a suite overlooking the Thames.'\n\nHe couldn't even pronounce Thames correctly, making an awful th sound.\n\n'It was nice to meet you,' she lied. 'I must go and talk to the band now. Goodbye.'\n\nHer earlier euphoria vanished. Already the vultures were gathering and if she didn't keep one jump ahead, she might end up with nothing.\n\nSlipping out unnoticed to the reception desk, using the excuse she wanted to know who to thank personally for the evening, she discovered the whole event had been booked by Menzies Enterprises.\n\nChampagne by the truck load. Smoked salmon, caviar, breast of chicken, roast beef, fresh cream gateaux, mountains of salad, all paid for by her. She was the guest of honour and the mug who'd paid for it.\n\nNow she understood why none of her friends had been invited. It wasn't to celebrate her success. Just another way for Max to climb further up the ladder. She was just another trophy he'd won, and tonight he was displaying her publicly.\n\nAs the prattle of high-pitched snobby voices washed over Georgia, she felt murderous.\n\nThat feeling she'd had when she first walked in! She knew what it was now. Everyone here had the same motive as Al Green. They weren't interested in her talent, just how they could get a slice of the action.\n\nSoon these people would be involving Max in further deals. Films, tours, advertisements, public appearances. Money flowing backwards and forwards, but somehow never reaching her bank account.\n\nShe hadn't had a penny yet. Everything spent had been charged to Max's office. She didn't begrudge the boys new clothes, they deserved them. Norman's green mohair suit must have cost a hundred pounds, John's leather jacket another fifty, that was mere chicken feed to the amount of food and drink being consumed. Deirdre the receptionist from Max's office in a sparkly dress costing probably twice as much as her own. Miriam across the room with a pair of diamond earrings she kept touching protectively. Max no doubt had a new car in the garage.\n\nWas the man who owned the limousine company dancing with the Sloaney woman from the King's Road boutique? Could the fat man with bow legs be the printer? The tall man with the beard a director of the musical equipment store? They were leeches who would suck her blood until she was dry, then spit her out and look for another victim.\n\n'Wonderful party darling.'\n\nGeorgia smiled at the elegant redhead who sailed past her on Jack Levy's arm. She owned a string of secretarial agencies in Oxford Street and by Monday morning she would have a list of new contacts.\n\n'Don't let anyone know you've cottoned on,' she whispered to herself as she slipped into a toilet to compose herself. 'Just stay cool and observe. They'll soon find out you aren't quite as dumb as you look!'\n\nIt was easy to play the role of the little innocent. The glass in her hand was just water with ice and lemon, no one guarded their tongues when faced with a girl they thought was tipsy. Listening, watching, observing, remembering names and filing them away for another time.\n\nIt was late, almost one o'clock when she overheard something interesting.\n\nShe had stopped by the buffet, helping herself to some chicken and salad. Standing just a foot or two beside her were two men. One of them she knew slightly. He was the lawyer from Decca who had been there when she signed her contract. Slim, dark haired, an accent like cut glass which belied his swarthy Mediterranean looks. The other man was smaller, sandy haired and stout. But by the way he was speaking she suspected he was a lawyer too.\n\n'I knew he'd be trouble when he said he had Riox acting for him,' the man from Decca said. 'Damn me if the little guttersnipe didn't start asking all kinds of questions.'\n\nShe guessed the men were talking about a singer called Ricky Delaney. A tough Liverpool rock singer that Max had attempted to handle and then abandoned because of his wild behaviour. Or at least that was what Max claimed!\n\nWas this Riox a new manager on the scene?\n\n'I've heard the name,' the stout man replied frowning as if trying to put a face to the name. 'Is his office in Chancery Lane?'\n\nGeorgia sidled closer, pretending to be engrossed in the food.\n\n'The Strand,' the Decca man was swaying slightly as if he'd drunk too much. 'Old established law firm. Riox's got a bee in his bonnet about protecting the interests of young entertainers. Sharp as a razor.'\n\n'Eton man?' the stout man asked.\n\n'Rugby I believe. Strange fellow. French father, mother, one of the Asprey family. One has to admire his integrity, but there's such a thing as loyalty to one's peers.'\n\n'Did you have a good time?' Robert the chauffeur was waiting for her, the car gliding silently towards the steps at just a wave from one of the porters.\n\n'It was interesting,' she said slowly as she sank back into the comfortable leather seat.\n\nRobert glanced over his shoulder. The spark had gone from her, she wasn't tired, or drunk, just kind of sad.\n\n'Not your sort of people?' He had a desire to ask her if she wanted to ride in the front with him, but that would be too impertinent.\n\n'No,' her voice was faint, like a child about to cry. 'And I've got a feeling I'll need to watch my back from now on.'\n\nGeorgia sat before Simon Riox's desk and wondered if somehow she'd got it wrong.\n\nHe was only a junior partner in the law firm of Hollins, Burke and Gibson, too young, too lanky to be the formidable lawyer she'd imagined. His brown hair threatened to stand up in spikes, dark-rimmed glasses reminded her absurdly of Buddy Holly. Behind the thick glass his dark eyes were as soft as a spaniel's.\n\n'Of course I know who you are,' he laughed softly at her suggestion he wouldn't.\n\nHis voice at least was the kind she expected, deep, resonant, entirely at odds with his almost feminine small features. She could see now he was older than he looked, at least thirty-five. 'What I'd like to know is how you heard about me. I'm not a visible attraction like yourself.'\n\n'Eavesdropping,' she admitted. 'I heard a lawyer complaining you had too much integrity.'\n\nShe had tried four different lawyers' offices in the Strand before she found him and now as she sat in the small oak-panelled office she was beginning to lose her nerve.\n\nShe had put on a red trouser suit because it made her feel strong and decisive, but in this formal place it looked too loud.\n\n'I find that very flattering,' his eyes never left hers and all at once she knew she could open up to him. 'I expect you feel nervous Georgia, let me order some tea, then you can tell me what's troubling you. If I can help, I will. I'm sure I hardly need tell you anything you tell me is in the strictest confidence.'\n\n'I haven't had any money yet,' she said simply. 'I found out last night I was footing the bill for a party at the Savoy, and flying over men from the States who aim to fill their pockets at my expense.'\n\nHe listened carefully to the whole story, looking at her over his thick glasses from time to time, writing copious notes in a thin, spidery hand.\n\n'Don't be put off by my notetaking,' he said at length. 'I actually have an excellent memory, but writing it down helps cement it in here,' he tapped his forehead with a pencil.\n\n'I felt used last night,' she wrung her hands in her lap. 'Humiliated even. I mean can you imagine thinking you are the one they've all come to see, and then discovering almost every guest is on the make?'\n\n'I understand your feelings,' Riox smiled sympathetically. 'This kind of thing happens all the time, it's all show, glitz and promotion. Max was probably astute in throwing this party. Perhaps he even had good reason to oil the wheels in America. But he should have discussed it with you. Made it clear it was ultimately your expense.'\n\n'But that Al Green wasn't even interested in me. I just paid for a free holiday for him!'\n\n'I wish you'd come to me before you signed the contract with Decca. I could have acted for you independently, protected you from sharp business practice,' he sighed deeply, as if afraid he was too late to help. 'I despise people who prey on young talent and grow rich while they wear out their prot\u00e9g\u00e9.'\n\n'Is it too late to change things?'\n\n'The contract with Decca is rock hard.' He flicked through the copy she'd brought with her. 'But it's a fair one. However, your contract with Max Menzies is another story.'\n\n'Really?'\n\nRiox smirked, almost as if he relished a challenge.\n\n'Less than three months to go. He slipped up there, but maybe he was too busy thinking about Decca's contract. This will be a good lever to make changes. To make sure you get what you've earned. For goodness sake when he suggests you sign another, consult me first.'\n\n'Do you think it would be better to find a new manager then?'\n\n'Not necessarily.' Riox had a look of cunning in his eyes. 'Max Menzies may be too sharp, but he is the undisputed king of promotion. If you got yourself a gentler character, the chances are Max would outwit him, just for spite. There is only one way to beat him, and that is by being just as sharp yourself.'\n\n'What should I do?'\n\nSimon Riox weighed up the girl in front of him and he liked what he saw. He had heard her record soon after it was released, liked it so much he actually bought a copy, and since then he had watched her shoot to overnight success.\n\nPerhaps it was merely the blanket coverage in the press that had put him off her a little. Somehow he had formed the opinion she was just another vacuous pop star who would burn herself out with the high life. Now he could see beyond the beautiful face and passionate voice. She was highly intelligent, brave and resourceful. How many girls of her age would balk at throwing a Savoy party surrounded by all those socialites fawning at her? Indeed how many would stay sober and work out what was happening?\n\n'I was going to say I'll handle it all for you,' he smiled at her, suddenly boyish and mischievous. 'But I can see you are a woman of action. Go to him now, tell him you discovered last night's bun fight was at your expense. Demand to see his books and tell him I'm your lawyer. That is, if you want me to be?'\n\n'Yes, please,' she said softly. There was strength in him, she felt it now, flowing across his desk to her.\n\n'I'll write to him today. I'll outline how we expect your affairs to be managed in future. If you need advice on handling your money I can do that for you too.'\n\n'I'm so glad I came,' she breathed a deep sigh of relief, as she got to her feet. 'I felt murderous last night, but now I feel calm.'\n\n'I'm glad to hear it. A murdering pop star makes too many ripples,' he laughed, standing up and pushing back his chair. 'I think you'll find he will play reasonably fair from now on,' he said as he shook her hand. 'He'll probably be hiving off a bit, but a man like that is so practised at deceit even an auditor would have a job to catch him out. I'll do a little double checking from time to time. Just so he knows he can't be greedy. Ring me if you have any further problems.'\n\nGeorgia was fired up now. She caught a taxi straight to Berkeley Square and ran up the stairs two at a time.\n\nWithout waiting for Deirdre to invent some reason she couldn't see Max, she barged down the corridor and straight into his office.\n\n'What is it?' Max looked up from his desk, surprise and some irritation showing in his face. 'I'm busy.'\n\n'So am I,' she said sharply, slamming the door behind her. 'But neither of us is so busy we can ignore what's going on.'\n\nMax sighed deeply and pushed his papers away.\n\n'Five minutes,' he said. 'That's all I can spare.'\n\nShe sat down opposite him and crossed her legs.\n\nThere was never a time when she wasn't struck by his looks. The broad shoulders, the dark handsome face, gold jewellery, handmade suits, silk shirts and Italian shoes all added to the image of a man to be obeyed and feared. But she wasn't going to let him get the better of her this time.\n\n'Has it ever occurred to you that you work for me? Not the other way around?'\n\nMax's hooded eyes shot open, they were as dark as her own, for a moment stunned by her statement.\n\n'What?'\n\n'You heard.' She held herself in check. Not for one moment must she falter, or the chance might never come again. 'You're treating me like an idiot Max, and I won't stand for it any longer. I'm your best client. The golden goose if you like.'\n\n'Georgia darling,' he gave her his widest, most disarming smile. The one he always used while frantically thinking how he could outwit his opponent. 'What is this? What on earth do you think I'm guilty of?'\n\n'Using my earnings to finance other deals,' she said quietly. 'Last night for example. Not only did you omit to tell me I was paying for it, but Mr Green seems to believe I'm in the English holiday business.'\n\n'Now come on Georgia,' Max stood up threateningly. 'Have I ever kept you short? Don't you trust my judgement to make a shrewd move?'\n\n'A sharp move,' she corrected him. 'I wouldn't have paid to entertain that load of wankers. I'm the one with a number one hit. I don't need all that. You did it for your own ego, at my expense. I want to see your books. I want every penny accounted for. You can have your percentage, and I want the rest.'\n\n'But Georgia, you already owe me. I've been keeping you on full wages for over two years even when you didn't bring in a penny.'\n\n'Bullshit,' she exclaimed. 'I've done my sums. You've been on to a nice little earner from the day I started with Samson.'\n\n'But expenses have to come out of that,' he said smarmily. 'What's this all about Georgia? Do you want money for a new frock?'\n\n'No. I want what's mine,' she put her hands on her hips, glaring at him defiantly. 'The books Max, the ones you keep each week. I'm entitled to see them.'\n\n'Who's put you up to this?' He turned his back on her, moving over to the window.\n\n'No one. Are you surprised I have a brain as well as a voice?'\n\n'I never doubted you had a brain.' She knew Max was struggling to find a good excuse for his behaviour. 'You don't know the business like I do. Even a singer as good as you needs promotion.'\n\n'Thousands of pounds are being made out of me each day,' she snapped, striding across to him and grabbing his arm to turn him. 'Every gig, every personal appearance brings in a fortune. I hired you to manage my career. Not my money. Like I said. You work for me.'\n\n'How dare you?' His dark eyes flashed. She saw his hand clench into a fist.\n\n'Don't even think of hitting me,' she hissed. 'You've done some shabby things in your life I'm sure. But don't add hitting women to them. All I want out of you is straightness. Itemized accounts showing every penny I've earned, and every penny you claim you've spent on me. Then we'll come to the point where you hand over the rest.'\n\n'But you must understand that the money coming in now is paying back all I spent on you earlier,' he said with a snake-like smile. 'New equipment, the van, the band's suits.'\n\n'Bollocks,' she retorted. 'You might have spent money on Samson, but not on me.'\n\nShe waited just a moment or two for the implications of this statement to filter into his mind.\n\n'I'm on my own! Right?' she snapped. 'You diddled the boys out of a shared contract with me. When did I ever sign anything to take over their expenses?'\n\n'Come on, play fair,' his mouth opened and closed like a goldfish. 'You were part of that band!'\n\n'You didn't think so when you wanted me away from them. The only equipment I need is in here,' she touched her throat lightly with one red talon.\n\n'Who put you up to this?' his eyes narrowed.\n\n'No one. But I have found myself a lawyer just in case you're tempted to try and wriggle out of it. Simon Riox.'\n\nShe knew as she saw the colour drain from his face that Riox really was the best.\n\n'I understand my contract with you has only another three months to run,' she added airily.\n\nMax stared at her. For the first time ever she saw indecision in his eyes. He looked hunted.\n\n'Don't be like this Georgia. Aren't you forgetting that I was the man who believed in you two years ago, gave you a chance and supported you?'\n\n'I haven't forgotten anything,' she said softly. 'But let's clear the air Max. You knew you were on to a good thing, the moment you met me. You didn't do it for love, or charity. Just money. That's okay. It's a good pure motive. But during that time you've thrown a great deal of shit at me and I'm just calling time if you like.'\n\n'You wouldn't find another manager!' It wasn't a question, more a statement of belief in himself.\n\n'Not if you play straight. I like your strength and singlemindedness. But from now on there are new ground rules. I will not pay for this office, the legions of girls who work for you, yachts, gold toilet seats or villas in the South of France. I'll foot the bill for last night, but not for that cocksucking Yank's holiday and suite at the Savoy.'\n\n'But \u2013' he spluttered.\n\n'But nothing. I'm telling you that I don't need or want any of the hangers on. As from today I select who stays and who goes. I pay the boys, a couple of roadies and perhaps I'll find a wardrobe mistress I like.'\n\n'I only took people on to make life easier for you,' he was almost whining now. 'Some thanks I get for it.'\n\n'I can appreciate that,' she snapped. 'But from now on I want to have people around me I have chosen. I didn't wear myself out for two years, just to watch other people take over my life!'\n\n'And if I agree?' He looked deflated, for the first time since she had met him, unsure of himself.\n\n'You can continue to be my manager,' she smiled charmingly. 'I'm sure you don't want to kill the golden goose, not when it may lay a golden egg each month or so for years. But if not I'll start legal proceedings.'\n\n'Look Georgia,' he shrugged his shoulders, 'I'm not a crook. Most of the money hasn't even come in yet. I'm not a fool either. But you are young, and inexperienced with large sums of money.'\n\n'True,' she cut in on him. 'But unless you hand it over I'll never learn. By next Friday I want accounts and a cheque. If not you know what to expect. Have I made myself clear?'\n\n## Chapter 17\n\n_1965_\n\nMax pulled the girl closer to him, thrusting one big hand up her skirt and pulling at her lace panties.\n\n'Oh Max,' she sighed against his chest. 'You're so big. I don't know if I'm ready for this.'\n\n'Of course you are baby,' he buried his face in her blonde hair. 'Come and sit on my knee and show me those lovely titties.'\n\nA room that overlooked the Thames, pale green brocade settees, a drinks table laden with bottles, fruit piled up like a feast in a cut glass bowl. Through open double doors he could see a huge bed beyond, turned down in readiness for them, soft lamps illuminating it invitingly.\n\nBut Max's heart wasn't in it. Georgia had even spoiled this for him. Once he would have taken this dumb little dollybird on the office floor. Screwed her senseless in ten minutes and been off to a meeting within half an hour. Now for some reason he'd spent over fifty quid on a meal and champagne and even booked the pair of them into the same suite Al Green had stayed in.\n\nJenny pulled up her skirt and sat straddled across his lap. Once a glimpse of smooth white thigh above stocking tops was enough to give him a hard on in two seconds flat, but now the sight of white panties with dark blonde hair curling round the lace wasn't even giving him a twinge.\n\n'Look what Jenny's got for you,' she said in that idiotic baby voice. She unbuttoned her prim secretary's blouse to reveal breasts like two melons. 'Does Maxy want to hold them?'\n\nMax took hold of her breasts roughly, pulling them free of the restraining push-up bra. He tweaked her big pink nipples and waited for the expected sigh.\n\n'Oh Maxy, kiss them,' she said, pushing herself up against his face.\n\nOnce this was his idea of heaven. A willing sexy girl, pretty as paint, with knockers that made every man in the room turn round to look. Why did he still dream of small breasts, ones with tiny brown nipples that reminded him of the wild strawberries he used to find hop picking in Kent?\n\nHe was merely going through the motions, sucking at her tits, sticking his fingers up round her pants and hoping he'd get hard enough not to disgrace himself.\n\nBut he felt nothing. No passion, not even disgust. She could be an inflatable doll, bland, obedient and perfectly formed.\n\n'Suck me!' He pushed her down on to the floor between his legs, unzipping his fly and pulling her head to him. He would be all right in a moment, just stick it in that big red mouth, feel that drooling tongue and he'd forget Georgia.\n\n'I will if you lick me too,' she said already pushing his silk shorts aside. With one hand she released her skirt and wriggled out of her pants, leaving only her blouse, the suspender belt and stockings.\n\n'Let me see how good you are at it first,' he said drawing her head towards him. 'If you're a good girl I'll suck you all night.'\n\nThis girl wanted what they all wanted. Only an hour ago she had informed him she had a good voice. She really believed that if she went to bed with him then he'd be getting her a recording contract within a week. Didn't the silly bitch know London was full of girls who thought they could sing?\n\n'Is that good?' she asked, running her tongue down his cock. It was lucky he was so well hung. She hardly noticed it was still flaccid. He'd have to turn his mind on to something else. Make up some fantasy with three negro girls pandering to his every whim.\n\n'Wonderful,' he groaned, looking down at that triangle of damp hair in front of him. She was fingering herself and that usually turned him on a treat. 'Make yourself come for Maxy, darling,' he sighed. 'Be a rude little girl. I like that.'\n\nIt was beginning to work. The combination of her mouth and watching those fingers sliding in and out of that damp fanny made his cock rise up. He'd wait until it was right up then pause to take one of those pills in his pocket. Then he'd give her something to think about.\n\nIt was after two now. Max glanced at the clock by the bed wishing he could come and get it over and done with. That guy in the bar downstairs knew what he was doing giving him these pills, they really did keep you going. But he'd fucked little Jenny in every position he could think of and now she was almost asleep.\n\n'Sit on my face,' he ordered her. 'I'm going to wank myself off while I suck you.'\n\nThat was better. He could close his eyes, taste that damp hot pussy and make believe it was Georgia. Her waist was as tiny, even if her skin wasn't as smooth, as long as she didn't try to drag his hands up to those great melons again.\n\n'I love you Maxy.'\n\nWhy did they always have to say that? They didn't mean it anymore than he did. There was only one woman he loved and she didn't want him.\n\n'Think of those long, brown legs,' he told himself. 'That juicy pussy. She wanted you more than any other girl that night. You'll get her yet.'\n\nHis strokes were getting frantic. He willed Jenny not to speak and spoil the illusion. He was tired of being told what a wonderful lover he was. He wanted a girl who hadn't experienced any of this before.\n\nHe came then, quivering, shaking, his mouth biting into that hot, furry mound.\n\n'I came again,' she sighed, rubbing herself all down his chest. 'That's twelve times.'\n\n'No, you didn't,' he said rolling over to get away from her. 'Why do women make such preposterous claims? Twice maybe, that's enough for anyone.'\n\nShe was hurt now, sitting up in bed with her blonde hair falling over those huge breasts, arms clamped around her knees, ready to cry.\n\n'You don't really like me,' she said, full lips turning down at the corners. One of her false eyelashes was coming loose and she had mascara streaked down her cheek.\n\n'Of course I do.' Max reached up from his position on his side, half-heartedly fondling her leg. 'Now cuddle down and go to sleep. It's late.'\n\nShe fell asleep almost at once. Max heaved himself up on to the pillows and looked down at her. She looked about sixteen now. She said she was twenty-two but the truth was probably eighteen. Once a girl like her could have kept him interested for weeks. Silky blonde hair, eyes like a couple of cornflowers all wrapped up in a cool presentation which didn't give a hint of how sexy she was.\n\nHe was sorry now he had treated her badly, he liked his women to fall asleep smiling.\n\nWhat was happening to him? Once he wouldn't have given a shit whether he'd been a brute, or the world's greatest lover. Women liked a combination of the two, Miriam always told him that.\n\n'Poor Miriam,' he murmured softly.\n\nMax loved his wife. Perhaps he had never been 'in' love with her, but he cared deeply. She was a good woman, warm, comfortable and caring. Did she notice he stopped making love to her around the same time he met Georgia?\n\nHe had thought he had it all that night Georgia let him take her to his flat. A number one hit in the bag, money pouring in and the girl of his dreams at his side.\n\nSex didn't really come into it. He would have held her all night without making any demands and he still didn't understand why she'd suddenly frozen on him.\n\nBut even after she'd said she wanted it forgotten, Max hadn't given up hope. Georgia wasn't the type to give herself to a man without really caring.\n\nThen there was that damn party at the Savoy. Of course he was using her money, that's what the game was all about. But the way she attacked him was ridiculous.\n\nNow she had Simon Riox in her court and between them they'd sewn him up so tight he could barely manage to buy a new car.\n\nIf she just let him direct her to the places he'd negotiated deals, that would be something. But no, she didn't let him get a look in.\n\n'A thousand pounds that bird in Sloane Street offered me for getting Georgia to wear her clothes,' he muttered. 'The fucking bitch wouldn't even walk through the door of the boutique. She's even made some private deal with a chauffeur to ferry her about, that's another five hundred down the Swanney.'\n\nThe contract with the stage management team was torn up before it was even signed. Now she had two great hulks who were so protective to her Max couldn't get near enough to even smell her. She even had the men at Decca eating out of her hand.\n\nHer could overlook all that. After all he was making a fortune from her straight. It was the ideas she might give other entertainers that bothered him. If everyone ran their affairs themselves there would be no more room for men like Max.\n\n'First Love' had been even bigger than 'No Time' and that bouncy, snappy 'Dancing with you,' would still be played in clubs when he was getting his pension. Now he heard she'd written another winner, rumour had it Jack Levy had been heard singing it in his office, yet Georgia hadn't even mentioned it to her manager.\n\nThat was what hurt the most. Somewhere along the line Georgia had stopped needing him. She really believed he knew nothing about music, she didn't want to confide in him. She didn't even care if he was in the audience.\n\nTake his idea of booking the Albert Hall with a full orchestra just after she got her first hit. She laughed at that!\n\n'Don't be silly Max,' she said, standing there with her hands on her hips in a pair of jeans so tight it took his breath away. 'It would alienate my fans. It's too soon. I don't want them to think I've sold out already. Book me into places the kids can dance in. Let's carry on with soul music and rock. When I've made my first long player that's the time to lure the listening people in.'\n\nDidn't she know he wanted her to be more than just a pop star? He wanted her in cabaret in Las Vegas, chatting on the Johnny Carson show, then moving on to making a big musical. She could be Carmen!\n\nWhy did she have to make such a big thing about her band too? How could he separate them now without causing ripples? She pushed each one of them forward, made sure everyone knew what a stud Rod was, or that Speedy was the thinker in the band. They were supposed to be faceless men in the background. She'd managed to make each one of them a star in his own right. Les even claimed she was going to pay for him to have his nose fixed!\n\nHow the hell could Max push them back into line after she'd spoilt them all?\n\nWas there anything left up his sleeve to pull out?\n\n'More long tours,' he murmured, banging the pillow and lying down beside Jenny. 'That should knock the stuffing out of her, piss the boys off. I can get a few backhanders again and fiddle the expenses. In a year or two she'll be dying to sing at the Albert Hall.'\n\nMax smiled to himself as he slipped off his suit jacket and opened his office window. In his briefcase he had enough cash to buy himself the new 'Roller'. A white one this time with red upholstery and his double 'M' on the doors in gold.\n\nThe second LP was already the top selling album for '65. He had five gold records on his wall, and any day he would be getting the sixth.\n\nWhere did she get the inspiration from to write such brilliant songs? Every one fresh and new, the latest 'Devil Man' he liked to think was about him, but she only laughed when he suggested it.\n\nAugust in London was usually impossible, but right now it looked wonderful.\n\n'Strange how things work out,' he said aloud, posing in front of the mirror. That masseur in King's private health club in Pall Mall was doing him a power of good, he'd tightened up his muscles, made him look ten years younger. 'I send them off to Europe thinking they'll soon be urging me to bring them back home, and what do they do? Make it fucking well work for all of us!'\n\nSell-out concerts in every major city, promoters ringing Max offering him anything to book a return date. That was where this new wedge had come from. Five thousand pounds, slipped under the table. Riox, Georgia and the taxman couldn't trace it.\n\n'They've been working their arses off,' Max chuckled, opening his silver cigar box and lifting one out. He paused to smell it, savouring the moment before lighting it. 'Yet somehow they found time to write enough material for an LP and a string of new singles.'\n\nA tap on the door surprised him. He wasn't expecting any visitors today and Deirdre always used the intercom.\n\n'Come in!'\n\n'I'm sorry to disturb you.' It was Deirdre, her dark hair dyed a deep auburn and cut into a ridiculous copy of Cilla Black's. 'There's a man in reception and he won't leave.'\n\n'Who? What does he want?' Max's earlier feeling of well-being fizzled away. 'You know I don't see anyone without an appointment.'\n\nThere were times he wanted to get rid of this girl. He never had fancied her and she was far too thick with Georgia. But at least she wasn't hysterical like most of them.\n\n'I told him that. He say's he'll just sit there and wait until you are free. He says it's personal.'\n\n'Do I know him?'\n\n'I don't think so.' Deirdre was hopping from leg to leg, her mouth twitching with nervousness. 'He's different from the usual boys we get in here. He's kind of stern.'\n\n'I'll give him stern,' Max snapped. 'What's this creep's name?'\n\n'Peter Radcliffe,' she said softly. 'He isn't a creep Max, he's got a kind of,' she paused.\n\n'Peter Radcliffe?' The name came back to Max, as sharp and clear as if he'd heard it earlier today. 'All right, I'll see him.'\n\n'You will?' Her brown eyes opened wide in surprise, mouth dropping open.\n\n'Why not?' Max regained his composure quickly. 'I don't know him from Adam, but I can't have people cluttering the reception area all day. If he's selling something you'll be for it.'\n\nThe cigar was still unlit in his hand. Max grabbed his jacket from the back of the chair, put it back on, returning to his desk just as the knock came at his door.\n\n'Come in,' he lowered his voice to a growl, opened a file and bent his head over it as the door opened. 'I don't know who the hell you are, but you've got five minutes.'\n\n'I'm Peter Radcliffe, sir. An old friend of Georgia's.'\n\nMax looked up. The voice was deep and well rounded, just a trace of a London accent, but the face in front of him gave him a jolt.\n\nThat day as he wrote the fake letter back to Manchester, he had imagined some weedy academic with acne and greasy hair. This lad was like a Greek god, blond hair, blue eyes with the deepest golden tan Max had seen away from St Tropez. Big shoulders almost bursting out of a washed-out denim shirt, jeans that fitted so snugly round his narrow hips he could be a male model.\n\n'Georgia is away on tour,' Max said picking up the cigar and lighting it. 'But even if she were in London I doubt she would see you. Every day we get people claiming to be an old friend of hers.'\n\n'I understand that sir,' the boy had no trace of belligerence in his voice or face. 'But I do believe she would like to see me. I can't believe she's changed that much.'\n\n'Sit down. What did you say your name was again?' Max waved his cigar at a spare chair by his desk, tilted his chair back and narrowed his eyes.\n\n'Radcliffe, Peter.' He sat down, his arms bent slightly, resting his fists on his knees and leaning forward.\n\n'Tell me where you know Georgia from?' Max said. The lad was looking unswervingly into his eyes, he wasn't sure he liked that much honesty.\n\n'We were sweethearts,' Peter said. 'That sounds a bit trite I know, but it's the best explanation. She ran away from home.'\n\n'When was this?' Max wanted to know what happened, and where, but he'd have to take it one step at a time.'\n\n'January 1960,' Peter said. 'She was fifteen then, I was seventeen. She sent her mother a postcard saying she'd be in touch when she was sixteen. But we heard nothing more.'\n\n'Well, there we are,' Max shrugged his shoulders. 'It's an open and shut case. She didn't want to know.'\n\n'But I think she did,' Peter insisted. 'Her mother moved and the only address she knew was mine.'\n\n'Mother?' She's an orphan!' Max said quickly.\n\n'That's right,' Peter half smiled. 'I should have said foster mother. Anyway Celia moved on. I suspect Georgia called at my house and my parents never told me.'\n\n'Now why would they do that?'\n\n'Because she has mixed blood.' Peter raised one perfect golden eyebrow. 'They thought she would mess up my career.'\n\n'Which is?' Max had a sinking feeling this lad was going to be more difficult than he imagined.\n\n'Teaching,' Peter replied.\n\nMax felt a bubble of pleasure. Thank goodness he wasn't a lawyer. Any man with looks like his who wanted to spend his life with kids couldn't be dangerous.\n\n'But if all this happened five years ago why are you concerning yourself with her now? Aren't you just jumping on the band wagon?'\n\n'Her mother and I looked everywhere for her.' Peter didn't turn a hair at Max's suggestion. Pride and truth shone out of his handsome face. 'Until I read about the fire and a friend described Georgia James to me I was in the dark, assuming she had forgotten me. I wrote to this office then and I received a letter back making it quite clear she had no time for me.'\n\n'But why persist then?' Max puffed on his cigar, blowing the smoke up to the ceiling.\n\n'One of the tracks on her LP.'\n\nMax gulped. He knew immediately what the boy meant. He had listened to the track himself so many times, wondering where the inspiration came from. Yet until now it hadn't clicked.\n\n'We were so young, we thought we had it all,' Peter said slowly. 'Nights of crying for you, are you out there crying too?'\n\n'Don't be daft man,' Max chuckled. 'It's just a love song. They all have the same theme.'\n\n'Do they all mention kisses in the hall, a glimpse of blonde hair, waving goodnight in frosty air?'\n\n'You know the lyrics better than I do,' Max raised one eyebrow. 'But even I could work those words around to any number of old loves. Anyone could.'\n\nPeter reached behind him and pulled a cutting out of his back pocket.\n\n'This is from _Honey_ magazine,' he held it steady and Max knew he intended to read it to him. 'An interview. \"Do your lyrics have special significance, or do you write about general feelings?\" That was the journalist's question. Georgia replies, \"Mostly it's general, but now and then it's like a secret message to someone. 'The girl with red hair' is an old friend of mine who died. 'No Time', was for Ian and Alan. 'Crying' for someone very special.\"' Peter folded the paper and put it back in the pocket of his denim shirt buttoning it deliberately as he calmly stared at Max.\n\n'That doesn't mean it's you!' Max retorted, feeling just a little hot under the collar. How come he hadn't read this interview?\n\n'No?' Peter looked up at Max. 'Why does she go on to say \"It was someone who coloured my idea of love, when I was very young. Someone I lost and still hope I'll find again.\" Does that sound general?'\n\nMax gulped. He could see now why Georgia had been attracted to Ian. He was a watered down, punier version of this lad. Everything made sense now. The way she turned down dates, kept herself at arm's length. If only the women in his life had been so constant!\n\n'Let me tell you something in confidence,' Max leaned forward in his chair, trying hard to be chummy and pleasant, yet he knew he was going to assassinate Georgia's character.\n\n'We keep it well under wraps, but Georgia has had many love affairs. Every one to her is special for a week or two, then she moves on. Ian McShane is the only man she ever moped about, and if he'd lived it would have ended the same way as all the rest.' He wanted to add more, imply the girl was a bitch and a whore, but somehow he knew Peter Radcliffe might just get angry. 'Forget her Peter, she's not the girl she was any longer. Singing and making money is what drives Georgia these days. Stay with the beautiful memories you've got, don't try to see her and find yourself disillusioned.'\n\n'I'm not a kid,' Peter leaned forward, eyes flashing dangerously. 'I haven't sat on the sidelines of life pining for her. While she's been setting the world alight, I've done my share too. Of course there've been other men in her life. There've been women in mine too. I'm not some jerk with a broken heart.'\n\nMax noted the muscles straining the denim shirt. He hadn't got those just lifting books!\n\n'I didn't think you were a jerk,' Max said carefully. 'But I've been very close to Georgia for a long time. I know her as if she was my daughter. To be honest she can be a pretty cold, calculating girl.'\n\n'Has she talked about her childhood?'\n\nMax had a feeling he was being tested and he felt trapped.\n\n'Which incident?'\n\n'Why a fifteen-year-old felt running away from home was the only option?'\n\nMax tried to bluff it out. 'Oh, the row! She regrets that now.'\n\nPeter sat back in his chair. 'Just as I thought,' a cynical smile played on his lips. 'If Georgia was really close to you, she would have told you.'\n\n'Don't be a smart arse with me boy!' Max half stood up, clenching the edge of his desk. 'Come on, out with it!'\n\n'No,' Peter's lips moved into a straight determined line. 'I didn't come here to blab her secrets. Just tell her I called.'\n\nMax's heart thumped. This boy had the missing bit of the jigsaw and he wanted it.\n\n'Why don't we go down to the pub?' Max said. 'It's hot in here and I could do with a pint.'\n\n'It won't work Mr Menzies,' Peter looked right into his eyes. 'I'm not about to spill the beans, even with eight pints inside me. All I ask is that you give her my address.'\n\n'Sure,' Max pushed a pad and pencil across his desk. What was it about her that inspired such integrity? Riox couldn't be bought, or the boys in Samson, and now this guy?\n\nPeter stood up, flicking back his golden hair in a weary gesture and picked up the pencil.\n\n'Don't hold your breath,' Max said flippantly. 'The way things are going with Rod, she might just be married by then.'\n\nHe saw the lad's neck swell, and a tinge of pink colour rose to his tanned cheeks.\n\n'Just give it to her sir.' The 'sir' sounded more of a threat than respect. 'I'll be away till October, but if I haven't heard anything by Christmas I'll be back to check up on you.'\n\nIt was as if the sun had suddenly been blotted out. In the past few months Max had regained Georgia's affection. He'd helped her buy her first car, even gone with her to choose furniture for her flat. Granted he hadn't got beyond giving her the odd cuddle, but at least she looked on him as a father if nothing else.\n\nPerhaps it was because he played it straight with her about that flat. He'd wanted her to buy something ritzy, not the top floor in a block full of old Brigadiers and snooty blue rinses. But when she took him to see it he could see she had her heart set on it.\n\nIt was huge if nothing else, with a nice view of the Thames from the balcony. But the old lady who'd died hadn't had any repairs done for years. The kitchen was something out of the dark ages, the Regency striped wallpaper was almost peeling off the walls.\n\n'But can't you just see it Max?' she wheedled. 'The lounge all painted yellow, white carpets and huge comfy settees. I can get someone in to do all the work.'\n\nShe had magic eyes that day, she'd even managed to make him enthusiastic. It took him back to the thrill of buying his own first home.\n\nOf course she didn't let him get anyone in he knew. A poncy woman from Sloane Street drew up all the plans. Swedish pine in the kitchen, a bathroom designed to look like a tropical jungle. All Max got to do was oversee the workmen and make sure it was all done to Georgia's plans while she was away.\n\nYet she knew immediately she arrived home that he had put in some work for her. She didn't miss a thing. The row of little yellow duck soaps in the bathroom. The tubs of flowers on the balcony, food in the fridge, even a hot water bottle to air the new bed.\n\nThe woman from Sloane Street had laughed at his idea of hundreds of red roses, or a silver champagne bucket. But he had to admit the freesias and the Nottingham lace bedspread she did approve of him buying, were more Georgia's style.\n\n'Let me cook you and Miriam a meal,' she said, bouncing around the place like a puppy when she saw the finished work. 'You've been an absolute angel Max, without you I bet it wouldn't be so perfect.'\n\nHe had to settle for the meal. He would rather have sat in her tub amongst the palm trees with that ridiculous monkey grinning down at him, seeing those small breasts again and carrying her later into that pink and ivory bedroom. But instead he had to listen to Miriam advising her on the best places to buy bedlinen and china, and eat over-cooked roast beef.\n\nMiriam leaving him should have made everything right, yet for some strange reason he felt gutted. She went home to Greece for a holiday and the next thing a letter came saying she was staying there permanently.\n\n'I think I'll always love you,' she said in her letter. 'But I want more than being your hostess and housekeeper. I want to sit in the sun with a man who needs me, grow old with a man who desires me. I didn't come out here looking for that, but I found him, here in the village which has always been my real home. Be happy for me Maxy. Find someone for yourself that makes you feel like this too. I hope we can always be friends.'\n\nWhy wasn't he rushing around like a man with two cocks? He could have it all now. Georgia was so concerned she was even inviting him over for drinks and meals. He was top unattached male on every hostess's list. So why did he get a lump in his throat thinking about Miriam with her fat Greek? What made him keep recalling the way she looked on their wedding night?\n\nGuilty conscience maybe? Afraid that if Peter Radcliffe did find a way to Georgia he'd be snookered? She might not care about the guy any longer, but she sure as hell wouldn't like Max playing God. If she wrote him out of her life too, he'd be finished.\n\n## Chapter 18\n\n'Something wrong, Georgia?' Rod shook his head like a dog, spraying her with water. 'Don't you want a swim? You've been quiet all day.'\n\nGeorgia pushed her sunglasses up on to her head, dropped her book to the ground and smiled up at him.\n\n'Just thinking,' she reached out for her suntan oil and rubbed a little into her legs, stomach and arms. She could see now how tanned she was getting, her skin had turned from coffee coloured to a rich dark brown, her white bikini standing out in vivid contrast. 'I was just thinking about how lucky we've been. Even if we don't get time to appreciate it.'\n\nThree days earlier they were playing in Paris and with over a week before their next gig they had flown down to Barcelona to catch a few days of sunshine and sea.\n\nRod dropped his towel on to the ground beside her, picked up a bottle of water and guzzled it down before replying.\n\n'Thousands of miles of travelling, more success than we ever dreamed of, fame, money, the works. And now this place.'\n\nThe villa they were staying at belonged to the family of a saxophone player who had joined them for many of the European gigs. Set amongst pine trees, above a deserted stretch of beach, it was the closest Georgia had ever seen to paradise.\n\nSeen from the road it looked like a dilapidated fortress, peeling shutters firmly shut against the strong sun, a neglected, abandoned home. A huge old door creaked open as if protesting against visitors, but once inside, it was obvious it had been cared for with love, its outer neglect protecting it from unwanted intruders.\n\nDesigned in a traditional Spanish style, the villa was built round a central courtyard, complete with fountain, palm trees and a vine-covered pergola. A Moorish influence was strong, with vivid blue and yellow tiled steps to the upper floor, white stone walls and wrought-iron balconies. Purple bougainvillaea and scarlet hibiscus, scrambled up the walls. Urns full of bright orange lilies, geraniums and daisies filled each nook and cranny. Everything was simple, cool marble floors, heavy dark furniture, brightly coloured, scattered rugs and huge cushions, walls painted dazzling white. Yet for all its simplicity it was comfortable and inviting. Tiled bathrooms adjoined each of the spacious, airy bedrooms, a kitchen with every modern device to make their stay a happy one.\n\nTwo local women came in daily to clean, bringing with them fresh bread, salad, fruit and eggs. They scuttled about their work, heads down, clearly intimidated by five half naked young men with only one woman. Georgia's attempts at making herself understood were met by laughter and a torrent of fast Spanish.\n\nThe other boys had left yesterday to drive into Lloret further along the coast. Two days of swimming alone in clear turquoise sea, sunbathing without admirers and drinking the local wine had been enough for them. They wanted excitement which really meant girls and right now Georgia was hoping they wouldn't bring them back here.\n\n'Perhaps you should have gone with the boys?' She turned over on to her stomach, resting her chin on her hands. 'I wouldn't have minded being on my own.'\n\n'I can live without the fleshpots for a while,' Rod laughed softly, dropping down on to his towel and leaning back with the sun on his face. 'I don't mind you being silent, it's nice. Just unexpected that's all, after last night.'\n\nThey had sat out on the terrace overlooking the sea. Watching the sun like a huge fiery orange slowly dip into the sea. Talking about their childhood, revealing things to each other that once were taboo.\n\nRod was different without the other boys around. He dropped his sarcasm, his cynical approach to life, and when he began to admit what a failure he felt as a child, Georgia understood what had made him the way he was.\n\n'I thought it was my fault because my Dad left. Somehow I was responsible for the lack of money, the slum we lived in and Mum's bad temper. When she started to go out all the time drinking, I kept quiet, never admitting how much I wanted her home with me. She caught me once cuddling and sniffing her nightdress, she accused me of being weird. She never understood it was because I missed her, that her perfume kind of made me feel safe.'\n\nHe told her that he worried because he was too skinny. That he saved up to buy a bullworker and hid it in the wardrobe. He spoke of men that came late at night, his mother laughing one minute, crying the next. The insecurity with the lovers who didn't leave.\n\n'Some were nice,' Rod grinned. 'Took me to Southend or to the football, but I was afraid to like them in case they left too. But mostly they were mean types. They hit me when Mum wasn't around, resented me as much as I loathed them.'\n\n'Did you ever tell her how you felt?'\n\n'You can't tell Mum anything. She's either on a high when she laughs at everything, or so low she's practically suicidal,' Rod grimaced. 'If I went out she said I didn't care about her, if I stayed in she said I was spoiling her chances. I couldn't win with her.'\n\n'How do you feel about her now?' Georgia asked.\n\n'Embarrassed more than anything,' his dark eyes looked thoughtful. 'She's so flashy and empty-headed. I haven't got anything in common with her, but because I'm making money she pretends I mean everything to her.'\n\nGeorgia remembered her from the boys' funeral. Bleached blonde hair piled up in elaborate curls, wearing a black dress so low cut and tight it had embarrassed everyone. Not for one moment had she looked distressed at the boys' deaths. Her dark flirtatious eyes had merely wandered off to the more famous people present. She hadn't even shown any real pride in Rod's bravery at trying to save Ian and Alan.\n\n'It's because of her you don't like women much,' Georgia said softly. 'I mean, you pull so many girls, but you don't choose them for company do you?'\n\n'I guess that's right,' he laughed. 'Except for you I can't ever remember telling a girl the truth. I put on an act about everything. I don't feel anything inside for them. It's like there's a basic function missing.'\n\n'Emotion,' she said. 'But you do feel that Rod. You felt it when Ian and Alan died, it comes out in music. It's there all right, you just haven't had anyone tap the right keys.'\n\n'We're a lot alike,' he said ruefully as they moved back towards the house once the wind turned chilly. 'Tell me why you don't trust anyone enough to give love another stab. You can't still be mourning for Ian?'\n\nThey lit a log fire in the huge old hearth, sitting on cushions close to the blaze, opening yet another bottle of wine.\n\n'I don't think Ian has anything to do with it,' she said at length after staring into the blazing logs, thinking about Rod's question. 'I'll always miss him, just the same way I miss Helen, my old friend. But I was never convinced we were made for each other.'\n\n'The other guy?' Rod looked round at her and lightly touched her hand. 'Ian told me there was someone before.'\n\n'Maybe,' she said. 'I keep telling myself it's stupid to hold a torch for someone who doesn't give a toss for me. But it isn't only him either, there's my mother too.'\n\nRod listened carefully as she told him the whole story about running away from home.\n\n'Why hasn't she come forward?' she asked him. 'I mean my records have sold in just about every corner of the globe. How can she have missed seeing my picture or read things in the papers? It must be that she changed her mind about me, decided I was to blame. What other reason could there be for her silence?'\n\n'You're in a position now to do something about it,' he said thoughtfully, gazing into the fire, poking it viciously. 'You've got enough money and connections to get her found. Why don't you? At least that way you'd know for sure.'\n\n'I'm scared too,' she whispered. 'Everyone I've ever cared about deeply has gone. Sometimes I think I'm some sort of jinx. Peter, Celia, Helen and Ian. Even my real parents abandoned me. On top of all that I'm afraid if I dig too deeply I might just meet up with Brian, my foster father again.'\n\n'Maybe that's the real hurdle you need to overcome,' he said. 'That man can't hurt you again. By facing him you might be better for it.'\n\nRod's high cheek-bones stood out in sharp relief lit only by the fire, his black hair shining like a raven. A primitive face, like so many of the Spanish men, the whites of his eyes and teeth contrasting brilliantly against his dark skin.\n\n'I know it sounds daft,' she said reluctantly. 'But recently I've had a feeling he's watching me.'\n\n'That is daft,' Rod laughed, throwing back his hair. 'You've hardly stayed still long enough for us to watch you, let alone him.'\n\n'I don't mean he's actually watching me in the flesh,' she explained. 'More a feeling he's observing me from a distance, noting everything about me, biding his time, if you like.'\n\nThe reason she'd been silent all day was not from dwelling on the past, their lack of time to enjoy things, or even the thought of Brian watching her. It was Rod. She had a feeling something was starting up, something she couldn't fight, avoid, or protect herself from. Since Ian's death they had grown close. But last night there had been a brief fiery spark of something more. Should she quash it now? Save any further heartache, or take the initiative and worry about the results later?\n\nThis morning she'd woken without any solution, and silence had seemed the only way out, just as it had the morning after Brian raped her. By doing nothing and saying nothing she was preventing either steps back, or forward.\n\nThey had raked out the fire. Turned out the lights and walked together up the tiled stairs to the rooms above.\n\nThe moon shone down into the courtyard, reflecting clearly in the pool around the fountain and picking up the white of a towel left lying on a chair. Enough light to see the gallery on which they stood, wrought-iron railings going right round into a rectangle, dark wood doors invisible against the white stone walls, like yawning holes in a piece of cheese.\n\n'Back in Stepney I never thought I'd ever see a palm tree,' Rod said softly. 'The moon looks as if it's hanging on it like a Christmas tree bauble.'\n\nIt was beautiful. A star-sprinkled sky and the moon lighting up the feathery fronds of the tree.\n\nRod stood with her at the balcony, two pairs of hands so close on the railing, their sides just touching.\n\n'Whatever happens I'll remember this forever,' he said, so softly it was almost a whisper.\n\nShe could feel a current flowing between them, like two magnets so close they wanted to jump across the void. But instinctively she moved a step away to prevent it.\n\nHad Rod's words meant that he knew something was going to happen? That this was the moment when it started? Or did he mean that seeing the moon and the palm tree was the first time he'd been aware of the beauty of his surroundings?\n\n'Come with me for a swim?' Rod jumped up, dipping his hand in the pool round the fountain and splashing some on Georgia's hot back.\n\n'You swine,' she laughed, rolling over on to her back and rubbing the cold water off on the sunbed.\n\n'Please?' he said softly, a soft boyish plead in his eyes.\n\nHe was tanned almost as dark as Georgia now. His black brief trunks almost disappearing against his skin. His body had filled out remarkably in the last year, good food and a little exercise had given him broader shoulders and muscles in his arms. Slim hips, straight firm legs and a narrow waist coupled with his six foot height was enough to make any girl turn her head and look. But for Georgia it was his smooth olive glistening skin that attracted her and a face which reminded her of Red Indians. Such high cheek-bones, dark narrow eyes which could turn from laughter to anger in a second. A straight, proud nose and thin, stern lips. Even his hair completed the picture. It was much longer now, almost touching his shoulders, straight, sleek with blue black lights. Sometimes she wanted to put a band round his forehead and turn him from just Rod to a warrior.\n\n'All right,' she sighed, pulling the band from her hair and tossing it down. 'I'm just about frying anyway.'\n\nThey walked through the arched wrought-iron gates that led on to the patio where they'd sat last night, the tiles warm under their bare feet. There were steps hewn in rocks down to the beach, smooth, as if polished daily by someone they never saw. A small green lizard ran across in front of them, disappearing into a crack in the ground silently.\n\nIt was five in the afternoon, the desperate, sweaty heat at noon replaced by gentle soft warmth. A breeze was getting up, whipping little white horses on to the beach.\n\nThe nearest house was nearly a mile away, but here they were able to feel as if they were the only two people left in the world.\n\n'It's like heaven,' Rod said, pausing at the bottom of the steps. The last one had a big jump to the beach and he held out his hand to her.\n\nShe saw her hand go out to him. Only seconds in reality but it seemed like a lifetime before his fingers closed over hers. That spark again as skin touched skin, and she jumped, right into his arms.\n\nFor a moment they just stood together, his arms were round her, chests just touching. Her eyes were on line with his naked shoulders, the sea just visible beyond. Her hands rested on his hips. She dared not look up, not yet.\n\nHis hand moved so slowly up from her waist, over her shoulder blades, pausing on her shoulder lightly.\n\nFingers creeping up her neck, his thumb cupping round her chin, lifting her face to his.\n\nRod's eyes were closed, his mouth curved into a smile, just the tiniest glint of white teeth before he bent his head to hers.\n\nCoarse sand between her toes, more blowing up her legs, the breeze taking his hair and hers, bonding them together.\n\nA kiss that made her think of drowning. Sweet, peaceful on the surface, yet already an undercurrent rising to pull her down. His hand was holding her head, the other arm so firmly round her there was no escape, even if she'd wanted it.\n\nShe didn't know how long that kiss lasted. It was frantic, deep, world shattering. Strings inside her only vaguely remembered were being tugged and all the time her hands were caressing his smooth skin, pressing herself closer.\n\n'Oh Georgia,' he whispered at length, taking her face in both his hands and holding her away from him a little. 'Is this for real?'\n\nThey held each other as they walked to the sea, hands touching waists; the unexpected nakedness of their skin so available made it seem harder to touch.\n\nThe sea was warm. They entered it as one person, walking until it came up to their waists. Only then did Rod turn to her to hold her and kiss her again.\n\nA swim apart, then back together again, hands searching for each other, lips drawn back like an addict for a drug.\n\nSalty kisses, hair flapping like seaweed against their faces, skin that had lost the smooth warmth from the sun.\n\n'Let's go back now,' Rod said, taking her hand and walking up the beach.\n\nA shower stood just inside the courtyard. They stood under it together, blasted by the icy water and Rod unfastened the top of her bikini, letting it drop to their feet. Next the pants, one tie on each hip until she stood naked under the jet of water, reaching out for his trunks to push them down his legs.\n\nThe towels lying in the sun were hot to the touch. Rod wrapped one round her, pressing himself against her, rubbing her back, flicking back the strands of wet hair from her face, kissing her again and again until she could see nothing but him.\n\nA mattress still lay under the pergola where Georgia had retreated at midday to cool off. Rod led her there now, half carrying her, his lips on her neck and shoulders, breath hot and sweet.\n\nDappled sunshine filtered through the grapevine above them, the wind rustling the palm trees. Rod's lips were on her breasts, her whole body arching towards him.\n\nIan had been gentle and sensitive. Max rough, experienced and compelling, but Rod was so sensuous she wanted to scream out how much he was pleasing her.\n\nStroking, biting, kissing and probing. He made her feel like a woman and she could barely wait for him to enter her. Surprise at the hardness and length of him. Shock that he could immerse himself so far into the act of love he barely knew it was her. His mouth devoured hers, his tongue fierce and yet loving. His hands on her buttocks squeezing, kneading, a steady rhythm that sent shock waves pulsating through every nerve-ending.\n\nGeorgia could feel her heart thumping, it was joyous and wonderful, mixed with a fear that it would be over too soon.\n\nPerhaps Rod sensed it. He moved back from her, playing with her with his fingers, bringing her to the edge of an abyss again and again. She could stand it no longer. Grabbing his buttocks in her hands she pulled him to her, forcing him back inside her, head tossing from side to side, then reaching back to his mouth to kiss and bite him.\n\nWhen she came it was like a roller-coaster in pitch darkness, stars shooting past her, falling, then rising each time higher and higher. A hot feeling, burning, sucking her into oblivion.\n\nGeorgia lay with her eyes closed, holding his head against her shoulder, fingers wound into his hair. Words of love were on her lips, but she was afraid he would laugh at her.\n\nHow many times had she come upon him, during or just after lovemaking? In dressing rooms, hotels, even in the van. The girls would look like she felt now, but Rod's eyes were always cold and calculating. He was just a tom-cat with no finer feelings, he rarely even tried to protect his girls, often he made a sarcastic remark about them.\n\nShe felt him move away from her, taking his weight on his arms. Slowly she opened her eyes, fully expecting him to be looking around for a cigarette or a drink, already bored.\n\nBut instead he was looking down at her, narrow almost slanted eyes brimming with tears.\n\n'You are the most perfect woman in the world,' he whispered.\n\nNo words were necessary. She lifted her arms and reached for him pulling his face down to her again.\n\n'I'd like to stay here forever,' he said later that evening as he cooked her an omelette.\n\nGeorgia sat on a high stool at the breakfast bar. She had put on a red cotton housecoat with nothing underneath. He wore only a pair of faded denim shorts, his bare chest gleaming under the light above the cooker.\n\n'You wouldn't,' she said. 'We've got no sounds, no car, there's not even a bar for miles. What would we do all day?'\n\n'Play house,' he grinned. 'Make love all the time, go for walks, swim. I never thought I'd see the day when I dreaded the lads coming home.'\n\n'We've got tonight,' she said softly. 'They won't come back now.'\n\n'Let's pray they get arrested,' he said turning the omelette onto a plate. 'We won't answer the phone, we'll just let them languish in gaol.'\n\n'We'd better be careful,' Georgia blushed at the need to discuss something like contraception in such a tender moment. 'I don't want to get pregnant.'\n\nRod pulled a packet out of his pocket.\n\n'I'm a regular boy scout,' he grinned. 'Sorry I didn't have them with me this afternoon, but I didn't expect to be seduced. Let's just hope it isn't too late!'\n\nThey ate by candlelight close to the fire. The wind had got up, shaking the palm tree and rattling the iron gates that led to the beach. The empty plates were left on the table, while Rod fed her black grapes picked from the vine in the courtyard. They drank more wine, made love again, and dozed by the fire.\n\nGeorgia woke first the next morning. Sunshine was creeping through the slats in the blinds, making stripes on Rod's face and chest.\n\nIan had always slept curled up, one hand around his face like a small child, but Rod looked like a man. Flat on his back, arms and legs sprawled out, his angular face could have been carved from a piece of mahogany.\n\nHer cheeks, chin and lips felt tender from the dark bristles which had sprouted up overnight on his chin. Her arms, thighs and breasts felt bruised and pummelled, but her heart was singing.\n\nNo remorse, no fear, guilt or worry. Every moment of last night was one to savour. She was happy.\n\nWhen the phone rang later that morning Georgia stiffened. Jokes about the boys getting arrested were one thing, but she knew when Speedy and Norman went on the rampage it could happen.\n\n'How's it going?' Rod said quite casually, leaning nonchalantly against the wall blowing kisses at her. 'What are we doing? Nothing much, just going down to the beach. When are you coming back?'\n\nSilence while he listened.\n\n'Do we want to go to a party tonight?' Rod shouted out to Georgia. She could see him shaking his head, not quite tough enough to admit to his friends he didn't want to go. 'The usual stuff, loads of crumpet, booze and loose living.' Georgia shook her head. Rod grinned cheerfully.\n\n'No I don't think we'll bother. It's nice doing nothing. No, we don't care about the car. We'll get a taxi if we want to go anywhere.'\n\nAgain a silence as Rod listened.\n\n'Okay we'll meet you at the airport. We can pack your stuff for you and call a cab. Just don't miss the plane, blockhead or Max will castrate you.'\n\nHe put the phone down and leapt out to where Georgia sat in the courtyard.\n\n'Our prayers were answered. They don't want to come back here. They were feeling guilty because I was missing all the fun.'\n\nShe had never seen this side of Rod before. The boy in him that had been stamped out perhaps by his own mother. His eyes sparkled, his thin lips fuller. Once Rod would have refused to even make tea he was so full of arrogant chauvinistic ideas. Yet since they'd been alone he had cooked, washed up, even washed her hair and dried it. Now he was choosing to be alone with her, instead of rushing off to join more hedonistic pursuits.\n\n'So what's going on?' Georgia smiled. 'Let me guess. They've pulled some birds?'\n\n'Partly. Speedy got chatted up by an older woman with pots of money. She's a French film star, living in a fabulous Hollywood-style place along the coast from Lloret. Anyway they all went back there with her for drinks and she invited a bunch of English people she knows round, and it's turned into a mini orgy of sex, drugs and rock and roll.'\n\n'Drugs!' Georgia's eyes flew open.\n\n'Only cannabis I think,' Rod said airily. 'Anyway they want to stay there. Speedy said this place was too isolated and the road so awful he didn't fancy driving back to pick us up. I think he was relieved when I said we didn't want to join them, perhaps he was scared I'd nick his tart.'\n\n'Would you?' she laughed.\n\n'I'd be too frightened someone might nick you.'\n\nThree days more of loving and being loved. An unspoken knowledge that maybe once they got back on the plane everything could change. Long hours walking along the beach, swimming, lazing and talking. No dressing up, nor fans interrupting the peace. In a capsule where gigs, money and other people couldn't touch them.\n\nAgain and again Georgia looked for signs of his boredom. She had never known him able to even stay in one room for longer than an hour, he liked noise and confusion.\n\nBut there were no signs. When he flopped down on the beach he was totally relaxed, reaching out for her hand as if to reassure himself she was still with him. He told her more stories about his early youth, of meeting Ian and the others. Yet he never spoke of the future.\n\nHours and hours of lovemaking, made sweeter with the ending being so uncertain. But finally the last morning came and Georgia woke to find Rod standing by the bed with a cup of tea in his hand.\n\n'Time to go,' he said. 'I packed the boys' things and booked a cab. We've got just an hour to say goodbye.'\n\n'Is that it then?' she asked. His face was full of something she couldn't quite define. Sadness certainly, but something more.\n\n'You know how it will be when we get back,' he sat down beside her and put the tea into her hand. 'The fans will be there, the press. We'll revert to our usual ways.'\n\n'My usual ways would include you,' she whispered. 'Do you think I'll change with the click of a camera?'\n\n'No girl, you won't, but I will. The old flash Harry bit will come back. I'll be chatting up birds because it's the way I am. You'll get pissy and before we know it will turn ugly.'\n\nSomehow she knew he wasn't telling her he didn't care, rather that he cared too much to hurt her.\n\n'Are you always going to run from everything?' she asked. 'Because that's what it is Rod, running.'\n\n'No my sweet,' he turned and took her by the shoulders. 'I could try and trap you now, get a commitment from you and you'd stick to it whatever happened. But I know I'm not the one for you, not deep down where it counts.'\n\n'Was all this for nothing then?' She reached up and traced his tanned cheek-bones, ran one finger round his thin lips.\n\n'I wouldn't call it nothing,' he said looking right into her eyes. 'We've given each other something special in these few days, not just our bodies. You've made me realize I can feel love. I've freed you from that bogey man of a father. Perhaps you don't know it yet, but I have.'\n\nHe took her down to the beach later, kissed her again in the same spot where it all started and together they looked back up at the villa.\n\nIt was just nine in the morning, the sun barely rising above the palm tree. Through the iron gates they could see the cool courtyard with its splashes of purple, red and orange flowers. The white walls of the house almost a symbol of the peace they felt. The bedroom where they'd done so much of their lovemaking had the shutters wide open. The sheets white against the dark wood of the heavy oak bed.\n\n'We're too much alike,' he said softly. 'We could get lost in a power struggle. We both need gentler people to complement us.'\n\nShe knew he was right, but it hurt so much. Here alone they could be everything to each other, but once the real world stepped in between them, with jealousy, greed and vanity it couldn't be the same.\n\n'So we leave it here?' she whispered.\n\n'Yes baby,' his lips were against her hair, she felt his shoulders quiver and knew he was crying.\n\n## Chapter 19\n\n_January 1966_\n\n'Fasten your seat belt sir,' the red-haired stewardess leaned over to touch the black man's hand lightly. He had slept for almost the entire flight from New York and now they were landing.\n\n'Sure was fast,' he opened his eyes and sat up sleepily. 'Don't seem mor'an hour since I got on. I hope that's England down there?'\n\n'It is,' she smiled, more at his delightful Southern drawl than his little joke. 'Good old wet, cold London. Is it your first visit?'\n\n'First time in such comfort, ma'am,' he smiled showing gleaming white teeth. 'I was here last during the War.'\n\n'Well you're gonna find a few changes,' she said. 'The only thing the same is the weather.'\n\n'Sleeping Beauty's awake at last.' Sonia buckled herself into her own seat and smirked at Muriel her dark-haired friend on the seat next to her. 'If only they all slept like that our job would be easier.'\n\n'Did you find out about him?' Muriel enquired. It had been an uneventful flight and they had spent much of the time playing guessing games with each other about the passengers. The sleeping black man was the most intriguing because he'd given nothing away about himself.\n\n'I reckon he's going to see a woman. But I could hardly ask him, could I?'\n\nMost of the passengers were business men. Smart, seasoned travellers who either drank themselves into a stupor to relieve the boredom of the long flight, or put on their glasses and studied files and papers as if they were still in their office. No mystery with any of them, some of them flew backwards and forwards across the Atlantic like commuters.\n\nThere was a young couple near the rear of the plane who they guessed were newly married. The way they held hands, dozed on each other's shoulders and wrapped themselves up in each other was a sure sign. There were three younger men, travelling alone. Students, judging by their desert boots, jeans and thick sweaters, presumably running short of money as they only accepted free drinks.\n\nTwo blue-rinsed American ladies visiting their offspring in England. Five middle-aged couples who'd spent Christmas in New York. But it was the odd balls on the flight who gained the girls' real interest.\n\nCould that nasty little weed from Dallas be a dealer in pornography? Was the woman in a shabby coat with the huge frightened eyes a runaway nun? And was the group of couples from Philadelphia really on a church mission? Wasn't it more likely they had heard about swinging London and they just wanted a slice of the action?\n\nThe black man had a magnetic presence. Something that invited curious glances. Although he had spoken to no one, they sensed this was from being caught up in his own thoughts rather than unfriendliness. His clothes were old, but they had style. A leather officer's flying jacket with a fur collar, dark green cord trousers and a checked, warm-looking shirt, faded around the collar. Just the confident way he moved, looked and spoke gave a feeling that he was someone special, even if he had no money.\n\nThey knew from his passport that he was from New Orleans, that he was forty-three, a musician and his name was Samuel Cameron. If he hadn't fallen asleep they would have got to the bottom of it.\n\n'Such a good-looking man,' Muriel leaned closer to Sonia, dark eyes full of mischief. 'Just look at that face!'\n\nHe was sitting in an aisle seat, his head lolling over to one side as if he was about to go back to sleep. Rich goldeny-brown skin with high cheek-bones and long curling eyelashes. His nose was straight, almost Roman, lips wide and fleshy as might be expected with his dark skin, yet shapely and well defined. Even his ears were perfect, like two small shells flat against his beautifully shaped head.\n\n'I could go for him,' Sonia said. 'That black treacle voice gives me goose pimples.'\n\n'I'm more interested in his body,' Muriel giggled. She loved the way his hair was cut in a close crop, his broad shoulders and his muscular thighs. 'You know what they say about black men!'\n\n'Well it's too late now,' Sonia sighed. 'We should have kept him awake right from the start.'\n\nSamuel Cameron wasn't just tired. He was exhausted. Greyhound bus from New Orleans to New York, then two days of partying with old friends before catching the plane. It had seemed a good idea to avoid the expense of a New York hotel. But now he wasn't so sure. Men he hadn't seen since his de-mob, musicians he'd promised he'd look up when he was in town. Twenty years of catching up, packed into forty-eight hours.\n\nThey all believed landing a six week residence in Ronnie Scott's club in London was the big break he hoped for. But now in the early morning, tired and needing a bath and a shave, Sam had misgivings.\n\nThe money was little better than back home. By the time he'd sent something back for the kids and found himself somewhere to live there would be precious little left.\n\nYet he had to take a gamble. The war had changed nothing in the South. Still the same old prejudices. The whites got the good jobs, the decent homes. Which was worse, to leave his kids with his sister to chase a dream? Or to let those kids see him stay for their sakes and grow old and embittered?\n\n'Damn you Ellie for taking off,' he thought as through a gap in the clouds he caught a glimpse of London. 'If you'd been a proper mother we could have worked something out between us. But then you never cared for the kids.'\n\nHe ought to have known a barfly like Eleanor wouldn't settle for diapers and strollers instead of dancing all night and pretty clothes. Yet he managed to turn a blind eye to the way she neglected Jasmine and Junior, put up with her sulking, and worked longer hours to bring in more money. He didn't guess she was running off to meet this white guy. If it hadn't been for that freak storm and a cancelled gig, he wouldn't have found out his kids were alone at home, night after night.\n\n'I have to get out,' that was her explanation. 'The kids drive me crazy and you don't love me enough.'\n\nThat was the last night he spent with her. He could see her now as she threw herself into his arms begging forgiveness. The red satin dress she said her sister had lent her, damp from the rain, clinging to her curvy little body, her black hair a shower of tiny tight curls glistening with water. He'd believed her that night, loved her hard and long, promising to try harder to make her happy.\n\nBut the next evening when he got back from work, she was gone, taking everything worth selling and the savings from the box under the bed. She didn't even leave a note for her children.\n\nIt was left to the neighbours to fill him in about the man from Vegas who promised her a job as a dancer. And he had to tell his children Ellie wouldn't be coming back.\n\nFive years of trying to be mother and father. Finding babysitters so he could work. But Jasmine was ten now, Junior twelve, old enough to understand he wasn't deserting them for ever as Ellie had done. Just trying something new which might make a whole new life for all of them.\n\nThere were far more people in the arrivals lounge than he expected so early in the morning. Women rushing forward to hug their husbands. Children's faces alight with expectation. A feeling of pent up excitement in the air, shrieks of laughter, questions, perhaps the inevitable row brewing.\n\nSam stood still for a moment, his kitbag on one shoulder, his tenor saxophone in his right hand. Everyone had someone, except him.\n\n'Sam, over here.'\n\nHe could hear Clive's voice yet he couldn't see him. His eyes prickled with affection for the man who had not only arranged his contract, but also made the time to meet his flight.\n\n'Sam! Good to see you.'\n\nClive was pushing through the crowd, a wide grin spread over his small face. 'I thought you'd missed the flight, everyone else got off ages ago. What kept you?'\n\nSam's friend was smaller, paler than he remembered. His dark hair thinner, the moustache tinged with grey. But the grin was the same, pale brown eyes dancing with excitement, his mouth stretching from ear to ear.\n\n'Customs, what else!' Sam shrugged his shoulders. 'Only black guy on the flight. So it goes without saying they'd pull me.'\n\nFor a moment Sam felt unsure of himself. He had remembered Clive in old jeans and a sweat-soaked T-shirt. He hadn't expected the dapper grey pin-striped suit and neat polished shoes. Was this really the man who had dared to penetrate Harlem just to see and hear his jazz idols play?\n\n'Did they find anything interesting?' Clive took his sax from him, and squeezed his arm, the nearest thing he could get to hugging the man.\n\n'Just a pair of smelly socks I'd forgotten about under the sax,' Sam laughed. 'It sure is good to see you Clive. I've been havin' more than a touch of the collywobbles.'\n\nIt was strange to find their roles reversed. Three years earlier in the Ghetto club in Harlem, Sam had been the big man who saved the drunken Englishman from being rolled if not murdered. He had seen the raised eyebrows as the white fool flashed a wad of notes, and he found himself suddenly protective, just because the man had an accent that plucked at some forgotten chord.\n\nPulling a knife and bundling Clive out the club could have backfired on him badly, but fortunately by the time Clive sobered up, with his money still in his pocket, he was astute enough to understand Sam's motives.\n\nNow it was Clive's turn to be protective, leading Sam to his world.\n\nThat night in Harlem he thought Clive was a jerk. A white man who wished he was black, hung up on all the old jazz legends. But that was before he found out the guy was sincere, knew about music even if he did play the shittiest trombone he'd ever heard. And the guy had a big heart.\n\n'You got collywobbles?' Clive laughed heartily. 'You've got a contract, your air fare paid. You'll be knockin' em dead in Soho tomorrow.'\n\nIt hadn't seemed much to take Clive back to New Orleans with him. What else could he do with such a likeable jerk, hell bent on his own destruction? The man had a wife and kids at home in England. Someone had to straighten him out, let him hear some good music and get him on the plane home.\n\nHe didn't expect the guy to keep in touch. Clive was from the white middle classes with a high-flying job in the motor trade. He had money, good connections and education. Why would a man like that want to befriend an itinerant musician who lived in a cold water, one bedroom apartment with two kids to slow him down?\n\nPerhaps it was music that sowed the seeds of real friendship, but an understanding of each other made it grow. Clive had his problems. Sam had his and somehow as they shared them, they began to care.\n\n'What the heck!' Sam stopped and stared. Just outside the doors of the arrivals lounge was a vast crowd of kids, shouting and hollering, pushing and shoving. For a moment it reminded him of a riot scene he'd witnessed in Alabama, except most of these kids were white.\n\nPolicemen stood by, forty or fifty of them. The helmets and blue serge uniforms just the same as he remembered all those years before. Around twenty of them had linked arms to form a human fence, but the kids were still pushing to get through.\n\n'Georgia! Georgia!' they chanted. Girls, some no older than thirteen or fourteen, in white socks and school scarves, faces contorted in their screaming.\n\n'What's goin' on Clive?'\n\n'Shit.' Clive caught hold of Sam's arm, blushing furiously. 'I'd forgotten.'\n\n'Forgotten what? Is it a riot?'\n\n'Georgia, the singer. She's leaving with her band for the States. We may as well go back and get a drink at the bar, we won't get a taxi now.'\n\nIt was impossible to talk over the babble of noise from outside. Each time they started to speak, the doors would burst open and fans would rush in looking for another vantage point to get closer to their idol.\n\nThere was something hysterical about so many youngsters waiting for a celebrity Sam had never heard of. A sad reminder that he was too old and cynical now for such foolishness. A whole generation born and raised since he was last in England.\n\n'So who is this Georgia? I thought it was only the Beatles who got this kind of scene?' Sam laughed as he saw one girl trip another and she landed on her face, her short skirt flying up, revealing a pert bottom in white panties. 'I'd like to thank the person who's giving me this free show.'\n\n'Georgia James. Don't tell me you haven't heard of her?' Clive's face registered surprise. 'She's had five or six top twenty hits.'\n\n'I don't listen to that pop shit,' Sam grinned. 'Is she any good?'\n\n'Brilliant,' Clive's pale face took on that look of worship he'd once kept for giants like Duke Ellington. 'Beautiful girl with a voice that makes your toes curl up. She's one of the greats, buddy. Pop star or not.'\n\nThey sensed the arrival of the limousine even before they caught a glimpse of the gleaming black car. It started with a roar, a surging forward that made Sam leap to his feet despite his natural inclination to stay where he was. With long strides he made it to the window, forgetting even his kitbag and instrument left lying by the bar.\n\nHe was tall enough at six feet three to see over the heads of the fans, but even so he jumped up on a chair to see better.\n\nThe police had made a chain along each side of the car. More police were holding open the doors, and still more waited inside the departure lounge.\n\nBy now there were thousands of fans, elbowing each other, fighting to get to the front. Screaming at the top of their lungs, waving autograph books, handkerchiefs, scarves, even hats.\n\nA policeman rushed forward to open the car door. Sam saw first a long, slim, brown leg in white boots snake out of the car, then the other, and his eyes travelled upwards.\n\nShe was beautiful! Coffee coloured like the mulattos in New Orleans. Long black curly hair with a white fur pill box hat perched at a rakish angle that matched her short white fur coat.\n\nShe said something to the crowd. Dark eyes dancing with pleasure at seeing them, but a hefty man with shoulders like a barn door was grabbing her arm and urging her to get inside the airport.\n\nGeorgia looked up and Sam caught her eye, just for the briefest moment. Her hand flickered at him, and she was gone, dragged inside to the comparative safety of the airport.\n\n'Wow,' Sam jumped down, Clive had joined him, but his short stature had stopped him from seeing anything. 'She was something else, man.'\n\n'If you'd been here two weeks earlier you might have got to play with her,' Clive laughed at the animated expression on Sam's dark face. 'She often pops into Ronnie Scott's. She used to live in Soho and from what I hear there isn't one musician who wouldn't cut off his balls to play for her.'\n\n'I sure wouldn't go that far,' Sam smiled. 'But I'll have to check her out.'\n\n'I've found you a pad,' Clive said later in the taxi. 'I'd have liked you to stay with me and the wife, but Surbiton's too far out of London.'\n\nWas Anne, Clive's wife as enthusiastic about this Yank arriving in England? Clive had come to the States that time because his marriage was on the rocks. Maybe Sam was the man who sent her husband back, but as he remembered, sometimes women's minds worked differently to men's.\n\n'That's swell of you,' Sam sighed with relief. He didn't want to impose on Clive anymore than he had to. 'How much does it cost?'\n\n'Just a fiver a week will do,' Clive said. 'It belongs to a mate of mine. He's away on business right now. It's just a place he takes his chicks to. But don't have any wild parties there and upset anyone.'\n\nSam was too immersed in looking at London to reply.\n\nThere was nothing he could pinpoint to reassure himself this really was England. Somehow he hadn't expected streams of traffic, or tower blocks of apartments and offices. The only real point of reference was the biting cold.\n\nFunny little houses in rows. Millions of them as far as he could see. The big red double-decker buses and all those small chunky cars. The England Sam had imprinted on his mind was full of small green fields, thatched cottages and gardens full of flowers. Bars where old men played darts and dominoes. Women with scarves tied round their heads queueing up for rations. Little boys who asked the Americans for chewing gum and chocolate.\n\nLondon had meant Katy. Begging a lift in a jeep, or jumping on a train, barely noticing the bombed houses, or the narrow streets. Leave, to most of his friends had meant drinking, dancing and parties. To Sam it had been his girl in his arms.\n\n'Has it changed much?' Clive asked. It was strange to be with Sam again. Perhaps he'd built up the friendship to more than it really was. Maybe it was a mistake encouraging him to come over. Was that coldness on the other man's face, or just memories of something he'd never opened up about?\n\n'I guess so,' Sam looked round at Clive, his mouth straight and severe. 'But then I didn't know this part of London. Bayswater was one bit I knew. Soho and Whitechapel. But most of the time I was out at Lakenheath. Ask me again when I've seen them.'\n\n'I expect you went to the service men's club in Bayswater?' Clive needed to find some common ground. 'The Douglas House?'\n\n'That's right!' Sam grinned suddenly, happier memories chasing away the blues. 'Is it still there?'\n\n'Still there,' Clive chuckled, pulling out some cigarettes and offering Sam one. 'Packed with G.I.s on paydays. Maybe we can get up there one night. Georgie Fame and the Blue Flames play there.'\n\n'Mockingbird Hill!' Sam's eyes twinkled. 'I thought they were black guys when I first heard that song. Couldn't believe my eyes when I saw he was a little white guy.'\n\n'I think Georgie Fame would take that as a compliment,' Clive laughed.\n\nClive had been on the verge of cracking up when he went to the States. He was successful, he had a lovely wife, home and three kids, but he wasn't happy. It was like he had a self-destruct button attached to his chest. He wanted some excuse to push it, and blast away all the shit in his life. Sales managers, targets, social climbing, he'd had enough of all that. He wanted something more, something meaningful. He wanted to listen to music, fall in love with a wild woman, break out and be different.\n\nIf he hadn't met Sam he probably would never have got home again. His idealistic dream was already tarnished. He heard all the great jazz he hungered for. Yet the free spirits he'd expected were as trapped as himself. They had problems too, drugs, drink and poverty.\n\nSam put it all into perspective for him. Showed him the real America you didn't get on package holidays. The hospitality, the hypocrisy. Scenery he'd only ever dreamed of, poverty he wished he could forget. A country that had so much richness and beauty, but with an underlying core of greed and corruption.\n\n'You've already got the things that are important man,' Sam said. 'Go home to your wife and kids, learn to be happy with what you've got. Don't go crying for the moon.'\n\nClive knew when he said goodbye to Sam at New Orleans airport that if he couldn't break away from his own life, then he would make sure Sam got his just reward. No one in England could play the sax like him. Somehow he was going to make the breaks for this big-hearted black man. He had the connections in London, and he was going to use them.\n\n'Say, man,' Sam turned to Clive as the taxi came near to Piccadilly Circus. 'Hell, I don't know how to say this,' he paused suddenly embarrassed. 'I mean, I appreciate you setting this up. Finding me a place to stay an' all. But I don't want you to think you've gotta wet-nurse me. Okay?'\n\nColour had never been an issue in their friendship. But now looking out of the taxi window and seeing people who were predominantly white, Sam felt a pang of the old fear.\n\nClive was a true friend, but he had a white wife and kids, an important job. He didn't want to stretch that friendship so far it broke. In the States things were clearly defined. There was some sort of rough honesty in the prejudice. Here he didn't know how things stood, and he wouldn't find out for certain leaning on a white jazz buff.\n\n'Sure.' Clive grinned. 'I know what you're trying to say Sam. You need to find your feet on your own. Call me when you want company. I'd like to think we can get out to hear some bands together now and then. But I won't be on your back.'\n\n'Thanks man,' Sam really smiled at last. 'Say, can you imagine how great it is to see all this again?'\n\nHe knew this bit of London. Eros had been boarded up then. The shops hadn't looked so gawdy. But there was the White Bear he used sometimes and down this road was surely the theatre he'd taken little Katy to?\n\nThe taxi turned outside the Dominion, swung round to the right and stopped.\n\n'I remember this bit.' Sam got out, looking around like a kid on a Sunday school treat. 'Used to be a musical instrument shop here.'\n\n'Over there.' Clive paid the driver and slammed the door. 'Some things don't change. They call this bit Tin Pan Alley still. 'Cept now it's all electric guitars and organs. Back in the 'fifties there were men playing pianos in some shops to sell sheet music. Now it's record shops and coffee bars.'\n\nIt was still only nine in the morning. Clive had gone to work and Sam's body clock was so mixed up he didn't know whether he wanted to eat, sleep or get drunk.\n\nHis kitbag sat on the double bed. His saxophone on a settee and this tiny place was his for the next few weeks.\n\nOne room, a tiny kitchen and bathroom off the small hall. It had that quality he remembered about England, a sturdy, comfortable feel. Smart enough with its plain dove-grey walls and dark red curtains. The oak table and chairs scratched enough for it to feel homely. Clean but not antiseptic. A place where a loner like himself could be inconspicuous. Was it Clive who stocked up that icebox with eggs, bacon and beer? Enough coffee for a month and a cupboard full of tins?\n\nIt was on the third floor above a coffee bar. The window looked out over Charing Cross Road and there was that theatre almost winking at him.\n\nWhat was the show he'd taken her to see? Darned if he could remember now. All he could see was her face shining up at him.\n\nSuch a pretty face. Small and heart-shaped, with big brown eyes under those shiny bangs. He could still feel that shiny straight hair as he ran his fingers through it, thick, sweet-smelling and so silky it drew his fingers back to it again and again.\n\nIt had been her hair he noticed first that night in Lakenheath. He was playing with the band when he saw her walk through the crowd with her friend.\n\nAll the other girls had theirs curled. Weird styles rolled up round scarves and God knows what. She was just natural in a green and white print dress, her waist so small she looked about fourteen and that thick dark hair just touching her shoulders.\n\nShe didn't exactly dance. She just stood there jigging up and down and clapping her hands. As she moved her hair swayed like a heavy silk curtain, gleaming under the mirrored ball.\n\nHe shouldn't have even looked at a white girl. He wouldn't have dared back home. But it was different then, a feeling in the air that nothing was permanent. You had to take chances, because tomorrow you might just be dead.\n\n'Well Katy,' he said aloud as he watched the endless stream of traffic forcing itself down the narrow streets. 'What happened baby? Did you lose your nerve?'\n\nSuddenly he couldn't stay in. London was beckoning him. He had only his tuxedo and shirt to hang up, everything else could wait.\n\nOnce out on the street he stopped to get his bearings. It was all so familiar, yet so strange. It was usually night when he came here before. Yet surely that was the tube station he'd run to once when the sirens went off?\n\nLeicester Square. He knew where he was now. They went dancing here sometimes and she wore that little red costume he bought her with some sort of frill round the waist.\n\n'Wouldn't navy blue be more sensible?' she asked, giggling as she stood there in the dress shop. 'What about the points Sam? I haven't got enough.'\n\nThe Brits were funny about their ration coupons. He'd done a deal with the woman who owned the shop. Katy wasn't so worried about the money it cost, more about not handing over the damn points.\n\nHe was into Piccadilly without even being aware he was still walking. It was smaller than he remembered. What was that song they used to sing? Something about lights going on again in Piccadilly. They didn't look much, he'd expected it more like Vegas. Perhaps it was more spectacular at night?\n\nRegent Street. This was where the rich folks bought their clothes, wasn't it? Kind of like Fifth Avenue in New York. The girls were different to ones in the States. Kind of bolder, all that long flowing hair and big dark-ringed eyes. The short skirts were kinda neat though, women wouldn't go for those back home.\n\nHe didn't know why he got on the tube at Oxford Circus. One moment he was standing by the subway, the next walking down the stairs.\n\nThat dry, warm wind coming up from the platforms took him back. Strange dank smells that made him feel safe. No one felt safe on the New York subways. But he could remember everyone in London running to them when there was an air raid.\n\nPlatforms crowded with people. Women with picnics and knitting in baskets, small boys with a puppy or a kitten under their coat. A man with only one leg who played an accordian. Old ladies wrapped in blankets, babies crying, young girls giggling together and men who had to start work early trying hard to catch some sleep. Someone would start to sing and before long it was a party.\n\nPerhaps that was why Sam still liked Brits so much. That ability to hold on to what was important, no matter what life chucked at them. He was sure they would have licked the Germans even without the Yanks to help them.\n\nSam stood holding the strap, his body swaying with the motion of the train. The stations were all so familiar as he looked at the map. Lancaster Gate, that was the station he used from the Douglas House. Marble Arch, Bond Street, then Oxford Circus. He must have taken that ride forty or fifty times. Back then, the train had been packed with servicemen. Americans and Canadians in their blue uniforms. British soldiers in khaki, Royal marines in navy blue and a smattering of sailors in bellbottoms. There was always an atmosphere of comradeship, mixed with rivalry. Showing off for the shy girls with fresh faces, passing round cigarettes and telling jokes.\n\nNot a uniform in sight now. No conversation or jokes. Men in dark suits hugged their briefcases. Middle-aged ladies with bags bulging with shopping. Boys with their hair cut like the Beatles. Office girls with their knees neatly tucked together, eyes glued to magazines. What had happened in London that people avoided any human contact?\n\nIt was then he knew where he was going. He knew nobody else in London apart from Clive. He couldn't call into Ronnie Scott's until late tonight. What else could he do but make the trip to Whitechapel and get it out of his system?\n\nOutside Whitechapel Station he stood helplessly. It had changed, yet he couldn't say how.\n\nThere was the old hospital across the street, blackened by smoke, as forbidding as he remembered. Surely the main road was never that wide? Where were all the tiny shops he remembered and that Yiddish theatre on the corner?\n\nIt ought to smell the same. Smoke was still coming out of chimneys, the little market was still there. But now there was spice in the air instead of fish, the roar of traffic instead of the shouts of costermongers.\n\nSam stood for a moment watching the scene at the small market.\n\nDark faces outnumbered white. Indians in turbans behind the stalls, women in saris queueing next to plump West Indians. Girls in their late teens pushed strollers, pale, harassed, with untidy hair. A heavily pregnant young woman bent over to spank a small boy, his screams unnoticed by anyone but Sam. A big white Ford drew up at the kerb, disgorging four young black men in leather jackets. An old woman in a red woolly hat was sorting through some fruit left out for the garbage men, and a burly man hopped along on crutches, one trouser leg pinned up with a safety pin.\n\nIt was good to see there were no ragged urchins standing listlessly outside the pub on the corner. None of these people looked hungry or pinched. Rosy-faced children in bright, warm clothes hung on to their mothers' hands. There was food aplenty and money to spare.\n\nBut what had happened to the cheery greetings Sam remembered? There was isolation, just the way he noticed on the tube. Men in cloth caps, their jacket shoulders worn shiny with unloading ships down at the docks, were replaced by menacing looking youths, hair cut short like convicts. Car horns instead of the jangle of trolley bus bells.\n\nSam sighed. It had lost something more than its slums. Despondency had taken the place of character. Maybe there was less dirt and disease, but Whitechapel as he knew it was gone.\n\nAcross the street he saw a young black man holding a white girl's arm. In the Southern States that would be asking for trouble, yet here they passed unnoticed.\n\nThe Black Bull on the corner looked the same. White painted with black beams, the lattice windows twinkling invitingly. He had thought it was really old until Katy told him differently.\n\n'It's just mock Tudor,' she explained as he gawped at it. When Henry the Eighth was on the throne this was all fields.'\n\nWell fake or not, Sam liked it. Besides it stood on the corner of her street and back in '44 it had meant he was nearly there.\n\nIf it hadn't been for the Black Bull he might have thought he was mistaken. Valance Road had once been filled with tiny houses. A few gaps like broken teeth where bombs had dropped. Windows boarded up, shrapnel holes in some of the doors but it had been alive with people.\n\nNow it was quiet. New apartment blocks in the place of all the rows of old tiny houses he remembered. Scrubby grass stretched round them. Swings for the children, seats for the elderly. Those old houses were dark, damp places, infested with rats, overcrowded as his own childhood home, yet they had a quality of welcome which was lacking now.\n\nTo his right was another empty site, this one still not cleared. Stumpy grass grew over uneven ground, but behind it he could see that creepy railway arch.\n\n'Jack the Ripper got one of his victims here.' He could see Katy's eyes full of horror as they ran through that road one night soon after she moved here. It had been foggy, that evil yellow fog London was famous for, the kind that made every face look sinister.\n\nHe hadn't wanted her to move here. She should have stayed in Suffolk with her folks until they came round. A girl like her didn't belong in this place with its dark little alleys, the smells and the dirt. But he'd wanted to be with her so badly he didn't say any of that. London was the place where they didn't mind a girl having a black Yank. There was too much going on to dwell on tomorrow.\n\nHe remembered a hospital here. Not a big place like the one in the main street. A saint's name. Was it Peter? When did that get pulled down? It was an ugly, dark place, like a workhouse he'd seen pictures of before he came to England.\n\nIt was very cold. Sam zipped up his jacket and pulled up the fur collar. Hughes Mansions was up ahead of him. It looked different now too. Kind of forgotten. Old-fashioned compared with these new places.\n\nWhen Katy lived there it was a good modern place. She took pride in telling him they were all business people who lived there and for Sam who had been brought up in a shingle shack, it was the nearest thing he'd seen to luxury.\n\nA bathroom with a geyser to heat the hot water. Electric light, an indoor john and a proper kitchen with a boiler for the washing. Katy had a little room of her own, with roses on the walls and a satin-covered eiderdown on her bed.\n\nSam stopped at the gateway and stared up. It looked the same but it wasn't. Three blocks set on three sides of a square. Katy had lived in the one at the back. Five floors of red brick, stone balconies running from a central staircase. She used to wait on that landing in the summer, three floors up, just her head and shoulders visible above the stone wall.\n\nBut now there were four blocks, they seemed to press in on each other, shutting out light from the middle square.\n\n'You looking for someone?'\n\nSam was startled to find a middle-aged woman standing by him. She was pushing a basket on wheels piled high with laundry. Her hair in curlers with a scarf over them, a plaid, shapeless coat with a fur collar.\n\n'I used to know someone who lived here,' he said almost reluctantly. The expression on her tight face was one he knew meant resentment. 'It looks different now, but I can't make out why.'\n\n'Those three are new blocks,' she said pointing to the three buildings furthest from the road. 'They may be new but they ain't as warm and dry as the old one.'\n\nSam could see now. They were copies, the brickwork and doorways different. Only the one on the road was an original.\n\n'But why leave that one?' He pointed to the old one and frowned. 'It don't make sense.'\n\n'A bomb hit two of them.'\n\n'Bombed?' Sam felt his head reel. 'When?'\n\nShe looked at him as if he were simple-minded.\n\n'In the war of course!'\n\n'I meant what year. Which month,' he said quickly. His heart was pounding, a dry feeling in his throat.\n\n'I dunno,' she turned to walk away. 'I've only lived here five years.'\n\nShe was gone before Sam could question her further. He saw her pulling her basket over the central tarmac square and disappear into a flat across the other side.\n\nThere was a sick feeling in his belly. Was that why Katy didn't reply? They moved her on somewhere and his letters never got delivered?\n\nSam sat down on a low wall. He was cold, colder than he'd ever been before. The sky was black and threatening above him, the four apartment blocks were dark and gloomy, yet twenty-two years ago this had been the place where dreams came true.\n\nAll those years of thinking she'd just stopped caring. Never once had it crossed his mind she might be homeless, alone or frightened. Love had turned to bitterness. He had been so engrossed in self pity it had blinded him to considering she might have had her own problems.\n\nSam saw an old man turn into the yard. The man paused for a second, leaning heavily on his walking stick, his shoulders bent, breath like smoke in the cold air. He looked at Sam suspiciously, old rheumy eyes watering in the cold wind.\n\n'You waiting for someone?' His tone suggested Sam should wait somewhere else. The kind of arrogance Sam understood.\n\n'I came to look for an old friend,' he replied, standing up and taking a step towards the man. 'One of your neighbours just told me two of the blocks were bombed.'\n\n'You took yer time coming,' the old man said, taking a handkerchief out of his coat pocket and blowing his nose noisily. 'Twenty years ago that was.'\n\n'Well I'm American,' Sam said. 'I got posted to Germany, then I went back to the States. Did you live here when it happened?'\n\nThe old man wobbled on his stick. The suspicious look replaced by one of interest.\n\n'Yes son, thought I'd lost my wife and kids when I heard the news. But they was all right, it was the people in that block that copped it.' He lifted his stick and pointed to the block where Katy had lived. 'Was your friend in there?'\n\n'Yes,' Sam felt his eyes welling up. 'When did it happen? Did many die?'\n\n'March '45. It were one of those V.2S. No warning, nothing. Seven in the morning. Over a hundred and thirty killed.'\n\nThe sick feeling in Sam's stomach grew stronger. He swayed on his feet, closing his eyes momentarily.\n\n'What happened to the survivors?' he whispered.\n\n'Mostly men that had gone out to work.' The old man's tone softened, seeing the pain in the black man's eyes. 'Who was it you knew?'\n\n'She was called Katy. Small, dark-haired, shared a place with two other girls. I think it belonged to one of the girls' aunts. Number twenty-four it was.'\n\n'Don't remember her.' It wasn't often someone wanted to talk about the old days and he was anxious to prolong the conversation. 'But my daughter might. She's in there now, getting me some dinner. Come on up and ask her.'\n\nThe old man lived on the first floor of the old block. He took the stairs slowly, clinging on to the banisters.\n\nIt was dirty now, the way these blocks never looked back when Katy lived here. Someone had written on the walls, the stairs hadn't been swept for years and there was a smell of urine.\n\n'They was supposed to be moving us old 'uns out,' the old man held his chest and wheezed. 'But it never came about. I reckon I'll be here till the last. Eighty next month. Been here since they was built.'\n\n'Come on Dad. What kept you?' A rosy-faced woman came out on to the landing, a flowery apron over her dark jumper and skirt. Not fat exactly, but big hips and breasts. A round, unlined face as if life had been kind to her. Calm brown eyes and a soft mouth.\n\n'My fault, ma'am.' Sam flashed a brilliant smile at her, hoping charm would make it easier for her to talk to him. 'I kept him talking about the bombing here. He thought you might know my friend who lived in the back block.'\n\nShe smiled then, dimples showing in her pink cheeks. She had a girlish quality, even though she was in her late thirties.\n\n'You're American? You were here during the war?'\n\n'That's right ma'am,' he held out his hand. 'Sam Cameron. I've come over here to play at Ronnie Scott's. I just wandered down here to look at the old place, and it's gone.'\n\n'Well Sam,' she fluttered her eyelids a little at him, her hand in his was soft and warm. 'May I call you that? I'm Hilda Croft. Come in and have some coffee while Dad has his dinner. I'll see what I can remember.'\n\n'This is real kind of you,' Sam said once he was sitting by the fire with a cup of steaming coffee in his hands. 'It's my first day back in England and here I am in a room just like the one I remember from last time.'\n\nIt was almost identical to the living room in Katy's place. A heavy old polished table covered with a cloth and a sideboard of the same wood. A small couch covered in dark red material, worn thin on the arms and two matching armchairs, placed round the fire. If it hadn't been for the big television under the window and a gas fire replacing the coal one, he could have gone back in time.\n\nAt number twenty-four there'd been a wireless in the corner. A big shiny wood one with cut out sun-ray effect. He could almost see Katy perched on the arm of the chair, twiddling with the knobs to tune it. There was a wind-up gramophone too, pictures of horses on the walls and a plaster plaque of a cottage with roses round the door. It was always warm there, even though the girls had to carry bags of coal up the stairs.\n\n'I was only sixteen when the rocket hit.' Hilda sat down in the chair opposite him. She was wearing red fluffy slippers and her ankles were surprisingly slim. She made an impatient gesture to her father sitting at the table just behind them, half rising as if expecting him to ask her to cut up his dinner. 'I got married soon after and moved on out to Chigwell. But I know some of the people who lived in that block. What was your friend's name?'\n\n'Katy Collins,' Sam sipped the hot coffee, curling his fingers round the cup to warm them. 'Small, dark and pretty. She lived with Doris Lessing in number twenty-four.'\n\n'I remember Doris,' Hilda frowned. 'Big strapping girl. She was in the Fire Service.'\n\n'That's right,' Sam's face broke into a smile. 'Can you remember Katy?'\n\nHe could see she was struggling to put a face to the name. He willed her to remember.\n\n'She had a baby?'\n\n'No,' Sam chuckled. 'Not Katy.'\n\n'One of those girls did,' Hilda insisted. She frowned as if she knew she was right. 'I remember walking home with Doris once just after Christmas. She showed me the pattern for a coat she was knitting. She said her friend was about to have a baby.'\n\n'Maybe that was Ruth?' Sam said easily. 'The other girl who lived there.'\n\nHilda shook her head.\n\n'No, she was a glamour puss. She had long red hair and was always out dancing. The one that had the baby was a dressmaker. Doris told me she'd made lots of little nighties.'\n\nA strange feeling ran down Sam's spine.\n\n'Was she a dressmaker?' Hilda dropped her voice, aware she had hit a sensitive note.\n\nSam nodded. 'What happened to Doris?'\n\n'She copped it, and the other two. Same as most of the folk did in that block,' she stopped short, covering her mouth with her hand. 'Oh Sam, I'm sorry.'\n\nSam's face blanched. 'Are you sure?'\n\n'Of course I'm sure,' she lowered her eyes and twisted the ring on her finger round and round. 'I had stayed the night with my girlfriend across Mile End Road. We heard the bang, saw the smoke and dust. Someone said our flats had been hit and I ran all the way here. I knew Dad would be at work but Mum and the kids were here.'\n\n'Were they safe?'\n\n'Oh yes,' she closed her eyes as if reliving it. 'The rocket hit the back block, it left a crater thirty foot deep in the courtyard. The side block was almost entirely blown away too. All the windows here were broken. Some of the people were blown out of them. But Mum fell on the floor and the kids were still in bed.' She looked up at Sam again, a look of pain in those dark eyes, all flirtation gone.\n\n'There's lots of things I've forgotten, but not about that day. They laid the dead out there as they found them. It was the worst thing I ever saw. Old ladies and men, blown out their beds. Kids blown to pieces, arms legs,' she shuddered, pausing and wiping her hand over her eyes. 'Doris, Ruth and the other girl were all together, still in their nightclothes. I can remember seeing Ruth's red hair as they put her body on a stretcher, it was hanging down over the end, almost touching the ground. So pretty.'\n\nTears trickled down her cheeks. Sam reached out and touched her hand.\n\n'Don't mind me,' she sniffed. 'I wish I could say your girl wasn't there. But the three of them were all together. Doris in striped pyjamas. Ruth in a silky nightie and the other girl in between them, she had a blue dress. I even remember someone saying they must have been eating breakfast together.'\n\n'I'm sorry I brought it back,' he said.\n\n'It's okay,' she tried to smile. 'We got hardened somehow by it. Days and days of funerals, living in the hall across the road till it was safe to come back. We learned so much about one another, kind of shared the grief.'\n\n'But the baby. Was that with them?' He was sure Hilda was mixing Katy up with someone else. The only evidence was Doris making a coat. That could have been for almost anyone.\n\n'I don't remember,' she sighed deeply, frowning with concentration.\n\n'I remember about the baby.' The old man's voice surprised them both. They turned sharply. His open mouth revealed few teeth left and some half-chewed cabbage.\n\n'You don't remember anything,' Hilda said sternly. 'Get on with that dinner before it gets cold.'\n\n'I do,' he retorted, putting down his knife and fork. 'They never found it until the evening of the second day. I was out there helping clear the site. A young fireman found her, he heard crying under the ground. I was there when they lifted it out.'\n\nSam looked at Hilda, waiting for her to snap a denial at her father. Instead she looked thoughtful.\n\n'Dad's right,' she said slowly. 'I remember now. One of the women came tearing across the road to the church hall to tell us. There was women who'd lost their children, nearly out of their minds with grief. They all ran back to the site to see it.'\n\n'It was a black baby,' the old man piped up again.\n\nHilda looked sternly at her father, then back again to Sam.\n\n'Was it?' Sam said softly.\n\nShe didn't speak for a moment, as if trying to sort out real memories from rumours.\n\n'Mrs Kirkpatrick, she was the old girl who acted as midwife around here, said it was, but I never saw the baby. They said she was around ten weeks old. We were all so amazed the baby wasn't dead or even hurt. She was buried alive, still in her pram, trapped under rubble.'\n\n'Did anyone claim her?' Sam's voice was little more then a whisper. 'I mean, did this baby belong to a survivor?'\n\nHilda shook her head, eyes filling up with tears.\n\n'You think she was yours?'\n\nSam couldn't reply.\n\nA flashback took him back to that bedroom with the satin eiderdown. White curtains with red roses moving in the early morning breeze.\n\n'I should have been careful.' He was damp with perspiration, his face pressed into her small pink-tipped breasts. His hand against Katy's face looked so dark in contrast to her white skin. 'What if I've put you in the family way?'\n\n'I wouldn't care,' her laugh was like a caress. 'Besides, that only happens to other people.'\n\nHe remembered watching her as he dressed that last time, fastening his blue uniform slowly, anything to prolong the moment of leaving. She lay with her dark hair tangled on the pillow, dark eyes full of unshed tears.\n\n'You will come back?' her voice wobbled. 'I'm counting on you.'\n\n'I don't know,' Sam's voice shook as he answered Hilda. 'In her last letter she said she had a surprise for me. But I just thought she had made something for me. Why would a girl hide something like that?'\n\n'Maybe she was frightened you'd run out on her?' Hilda sighed. 'I mean men do, don't they?'\n\n'But we were going to get married,' Sam insisted. 'I left for Germany in April of '44. She wrote to me nearly every day. If I'd known I'd have found some way to get back. The last letter I got was dated March twenty-fifth. Then nothing.'\n\n'It was the morning of the twenty-seventh when it happened,' Hilda said softly.\n\nSam just stared at Hilda. She was counting on her fingers.\n\n'They said the baby was ten weeks old. That works right back to happening just before you left for Germany.'\n\nThe room was spinning. It was too hot. He wanted to run out denying what this woman was telling him.\n\nWas this some sort of bad dream? Why should he believe this baby belonged to Katy anyway? Twenty years distorted everything.\n\n'What happened to this baby?' he asked.\n\nHilda shook her head.\n\n'I don't know,' she said softly. 'Mrs Kirkpatrick would have known, and my mother too, but they both died years ago. I almost wish Dad hadn't brought you up here. Perhaps it would have been better if you'd never known about any of this.'\n\nA spotlight glinted on his saxophone as he raised it to his lips. He filled his lungs with air, fingers poised on the keys.\n\n'Watermelon man', a fitting number for a poor black man from New Orleans. He hadn't been prepared to sit in with the band tonight, but they'd insisted.\n\nScores of people sitting out there in the smoky darkness waiting to see if he could truly make his horn sing and all Sam could think of was Katy.\n\nHe tried to cut out the ugly scene Hilda had described and remember Katy how she was.\n\nThe notes came out as he pictured her running to him down the lane at Lakenheath. The fields were yellow with buttercups, she was wearing a pink dress that clung to her slim legs, her arms held out to him, hair flowing back from the small heart-shaped face. Cherry lips, cheeks pink with excitement.\n\n'I've found a place we can be together,' she shouted even before she reached him. 'I love you Sam.'\n\nHe didn't know he was putting all that into his music. It was all so real he could smell the fields, feel the sunshine on his neck and face, hear her voice.\n\nA roar washed over him, bringing him back to the present.\n\nOut beyond the stage he could see nothing but a blur of white faces and clapping hands.\n\nSo she was dead. Yet, in a way it was kinder than thinking she stopped loving him. He was back in the country that made him a man, and by that applause he knew he'd played a great solo.\n\nHe had turned a corner in his life. So Ellie had run out on him, but how could he blame her? Hadn't he ever only loved her with half a heart? He knew the truth at last about Katy and he had his children.\n\nMaybe the baby found in the rubble wasn't his, but he couldn't rest until he found that out for certain.\n\n'I'm counting on you.' Those were Katy's last words to him, and this time he wouldn't let her down.\n\n## Chapter 20\n\nThe headline mocked him even though he had moved away from the table where the newspaper lay. Even if he closed his eyes he could still see it.\n\n'Our Georgia, home again in triumph.'\n\n'How dare she smile like that after what she's put me through?' he muttered. Yet he still couldn't bring himself to screw up the paper and throw it to one side without reading further.\n\nMoving slowly back to the table, he bent over and tried to bring the small print into focus. His glasses were broken, and he had no money left to replace them. Dirt-encrusted fingers twitched with a kind of palsy. He was cold, his chest hurt, and now this.\n\n'Are you in there Mr Anderson?'\n\nHe heard the Irish voice at the door but ignored it. He could imagine Mrs Dooley's fat face squashed against his door as she listened, vast breasts straining her overall. Swollen purple feet incongruous in furry pink slippers.\n\nShe kept in with the landlords by acting as unpaid rent collector and stool-pigeon, while taking bribes from the black tenants upstairs to keep quiet when another friend had swelled their numbers even further.\n\n'I know you're in there!' she called out. 'You know what will happen if you don't pay me, don't you?'\n\nHe knew all right. Those bully boys from the landlord would come round and threaten him, maybe even throw him out on the street.\n\nIt wasn't as if the room was worth anything. A filthy hovel in Ladbroke Grove sharing the toilet with blacks and Irish. They hadn't mended the window since he fell against it, the gas meter was fixed to make money out of him and the sink was blocked up.\n\nA sagging single bed. A sink and old black cooker. Hooks on the door to hold his only coat. Two fireside chairs, one with a broken arm. A table and two chairs. The proportions of the room were all wrong. A giant black marble fireplace. An eight foot ceiling and a long narrow sash window covered in grey net curtain, in a room twelve foot long by six foot wide. Once the room next door had been part of it, until the landlords hastily erected a thin plasterboard partition. To Brian Anderson it seemed almost coffin shaped.\n\nGeorgia's face in the paper tormented him. She was wearing a ridiculous cowboy outfit, with a Stetson, fringed jacket, indecently short skirt and even cowboy boots. They said she'd taken America by storm and London airport was besieged with fans waiting for her arrival home.\n\n'They wouldn't like you so much if they knew what you'd done to your father!' he muttered, forcing the scissors between swollen, twisted fingers.\n\nHis room was full of her pictures. Glossy, glamour shots of her running along a beach in a red and white sarong, the wind catching her hair and holding it out behind her like a black flag. Another one and she was curled up sensuously in a white armchair, a hint of brown cleavage and those long, slim legs tucked round her. In sequinned evening dress, regal and beautiful. On stage, her hair sticking to her head with perspiration, that wide mouth open, head thrown back. Walking through fallen leaves in a park wearing jeans and a man's jacket over her shoulder, laughing as though interrupted in some private moment of fun.\n\nHe had them all. Each mention in the press. Every picture no matter how tiny. A private collection which gave him no pleasure. She was responsible for his plight and one day he would ruin her.\n\nSometimes he had dreams when he saw her coming for him with a knife. He would wake up sweating and shaking. But it soothed him to look at the pictures. She was the evil one, not him. He must never let himself forget that.\n\nHe picked up a thin, worn donkey-jacket and put it on. He had to go and collect his money from the post office, he couldn't risk Mrs Dooley calling the landlord again. Perhaps just one drink to warm him and stop the shakes.\n\nThe wind outside was icy. The soles of his shoes had holes in them and he had no socks left.\n\nWas he going mad? He knew there had been a better life before. He could see himself mowing a lawn, or sitting in a sweet-smelling room listening to a piano. He imagined opening a drawer and taking out a shirt fresh from the laundry, putting gold links into the cuffs and slapping cologne on his smooth shaven face. But if he had that life once, why was it he ended up here?\n\nMost of the time his mind was stubbornly fixed in Ladbroke Grove as if the startling memories that came to him were nothing more than something he'd seen on a film. Except for Georgia. Her face remained constant in his mind, only the details of how and why she made him suffer hazy.\n\nDown the grey street he shuffled, eyes down in the gutter. Past dustbins spewing out on to the path. Black faces everywhere. Standing on steps gossiping, lounging on corners with malice in their dark eyes.\n\nWhere had they all come from? Surely there was a time when everyone was white?\n\nHe crossed the road at the lights, turned left and into the post office.\n\nFor once it was quiet. Only two people before him. Sometimes the queue stretched right to the door. Black women with prams, out-of-work youngsters, all the old, sick and poor people of Ladbroke Grove waiting for their state benefits.\n\n'Have you got your book Mr Anderson?'\n\nHe liked the post office. They treated him properly here. Remembered his name and gave him respect.\n\nHe felt inside his coat and pulled out the crumpled yellow book.\n\n'You haven't signed it.' The big woman with red hair smiled at him. She was tapping her nails on the counter with impatience, but then she was a busy woman. 'Have you got a pen?'\n\n'Of course,' he said, feeling in his pocket again.\n\nThe snotty-nosed brats in his street called him 'The Banker'. He couldn't always remember why this was, but the pen was a clue.\n\nHe signed the book and pushed it under the grille. She counted out the notes and pushed them back to him.\n\n'Good morning,' he said, and shuffled off to the door, his money still in his hands.\n\nHis mind was cloudy again. It was like that most days now. Like the grey net at his window had got inside his head. It stopped him from thinking or planning and each day it grew thicker. He knew he needed to buy something but what was it?\n\nA bus stopped in front of him, then another right behind it. He was caught in the middle of a human whirlpool, trapped by women with pushchairs, old ladies with shopping baskets, young men in leather jackets and a group of school children.\n\nHe didn't see who pushed him, just a sharp thump in his chest, and the next thing he was on his back.\n\n'Are you okay?' A male voice was speaking to him, but he couldn't see anything but a blur of white above him.\n\nFor a moment he thought he'd just tripped, as he often did these days, yet why was he on his back instead of on his knees?\n\n'My money,' he clawed at the air, suddenly aware his hand was empty. 'Who's taken my money?'\n\nThe man was leaning over him, touching his shoulder, but he was talking to someone else. Something about a black man who pushed him and stole his money. He was urging another man to telephone the police.\n\nHe didn't know he was crying, just a damp feeling trickling down his face.\n\n'Poor old chap,' the man said. 'How could anyone be low enough to rob him?'\n\nThrough the grey mist he knew he must regain his dignity.\n\n'Help me up please,' he asked. 'I can't see very well, but I'd like to get in somewhere warm.'\n\nA hand went under his elbow and hoisted him up, he could see the man's face now. It was young, fresh and even kindly.\n\n'Would it be too much to trouble you for a cup of tea?' Brian asked. 'I need a moment to get over the shock.'\n\n'Of course,' the man said. He turned to someone behind him. 'I'll just take him into the caf\u00e9. Tell the police to come in there.' He took Brian's arm and led him into warmth. 'Sit down and I'll get you some tea.'\n\nIt was odd that the grey mist floated away suddenly. One moment confused, the next aware of everything. Just as if someone had opened curtains on a darkened room. Aware how shabby his clothes were, his dirty hands and the stubble on his chin. His tongue flickered across dry, cracked lips and he averted his eyes from a mirror on the wall.\n\nHe looked at least seventy. So thin, the skin on his face just hung in bags. Little hair left on the top of his head, but round his ears it sprouted out, grey\/brown and greasy. His mouth appalled him most. It was sunken, his lips drooping at the corners and when he opened it his teeth were stained brown.\n\nIt wasn't tea he wanted, but whiskey. He looked across the caf\u00e9 at the man who brought him in and wondered if he'd be good for a handout.\n\nThe man was young. Arty looking in his leather-patched tweed jacket. A long, serious face and his hair hanging over his collar. He could be a social worker or a teacher. Not someone with money.\n\nThe caf\u00e9 was one he often had lunch in when he first came to Ladbroke Grove. They greeted him like a friend then, telling him about specials on the menu and chatting about the news. He liked the red and white tablecloths, the smell of ground coffee and baking. It reminded him of his old home.\n\nFor some reason that eluded him, they asked him to leave. Was he drunk, or perhaps they didn't like his shabby clothes? Whatever the reason was they seemed to have forgotten it now, for the two plump women behind the counter were looking at him as they spoke to his rescuer, nodding in sympathy.\n\nThe man came back, putting down two teas on the table and sat opposite Brian. He was frowning, a kind of impatient look, as if he wanted to leave hurriedly, but couldn't bring himself to.\n\n'Are you feeling better now?' His brown eyes were gentle. A soft, generous mouth with a cleft chin. His light brown hair had a shine to it, flopping down to his eyes. 'The police will be here soon. They'll help you.'\n\n'I didn't see anything.' Brian felt confused now, the present mingling with the past. 'I felt someone push me, then I was on the floor. Did you see who did it?'\n\n'Two young black men,' the man frowned again. 'I saw you just standing there. I was just going to make you put the money away and they pounced. They moved so quick I couldn't do anything.'\n\n'Blacks!' Brian almost spat out the word. 'The place is overrun with them.'\n\n'I picked these up for you.' The young man dug into his pocket, and pulled out Brian's payment book and a couple of photographs.\n\n'Thanks.' Brian snatched up the book, but left the photographs on the table. Just the sight of them made his shakes come back. 'I'm sorry. I'm not well. You must excuse me.' This clarity of mind was far more painful than the grey mist and it was unreliable. He could start a conversation, think things out, then suddenly he could be thrown back to the confused man the children laughed at.\n\nThe photographs had been in his good overcoat. It wasn't until he tried to sell it that he found them. He must have had them all this time tucked into the back of his payment book.\n\n'I saw your name is Anderson,' the young man smiled, glancing down at the pictures. 'I'm John Adams. Who's this?' Adams picked up one of the photographs. It was one of Georgia in a swimsuit, taken on the beach at Hastings the summer before she ran away. 'What a pretty girl. She looks like Georgia, the singer.'\n\nBrian felt a hot flush creeping up his neck.\n\n'It is Georgia. She's my daughter.'\n\nHe had often wondered what it would be like to tell someone. He expected ridicule, laughter and questions, but he didn't expect the shock he saw in this man's eyes.\n\nHis jaw dropped. He blinked hard, picked up the picture and looked again. Lowering the picture he looked right into Brian's eyes, the sort of look that meant he wanted to believe it, but couldn't.\n\n'It's true,' Brian said. 'Look at the other one too. She's there again with my wife and I at a bank dinner and dance.'\n\nIt felt good to admit it. The hazy grey mist was lifting so quickly he could remember other things too.\n\nGeorgia sitting between them, in a light-coloured dress with a kind of ruffle round the neck, smiling at her father in dinner-jacket and bow tie. Celia, on her other side was in a chiffon, low-necked dress, a glass of wine in her hand. That night all his staff had been there, formality forgotten with Christmas coming. He'd given them little presents, shared jokes and pulled crackers.\n\nHe saw a red flush creeping up Adam's face, his eyes narrowed as he looked at the snap, going back to Brian's face, then back to the picture.\n\n'She changed her name of course,' Brian picked up his tea and tried to control his shaking hands. 'There's things I could tell you about her.' He paused, sensing something more in the man's expression.\n\nNot disbelief or scorn. But excitement.\n\n'How can she be your daughter?' the man asked. 'She's black.'\n\n'Half-caste.' Brian was surprised how easy he found to explain. 'We fostered her, gave her everything, singing and dancing lessons, nice clothes. We loved her like she was our own.'\n\n'But \u2013' The man's mouth was hanging open again.\n\n'But why am I like this if I've got a famous rich daughter?' Brian's laugh was hollow. 'Because she's a little bitch. That's why.'\n\nJohn Adams wasn't a man who was shocked easily. He had grown up in Ladbroke Grove and knew many of the odd characters who lived there. Pimps, prostitutes, thieves, faded actresses, doctors who'd been struck off, he'd even met a few titled people who'd fallen on hard times. It was an area where people ended up. A dustbin of human life. Yet if he believed everything he'd been told in pubs, listened to every old wino who bleated out a sob story then it would be him next for the funny farm.\n\nHe'd seen Anderson before. He knew he drank heavily and sometimes he shouted and talked to himself. He had guessed the man wasn't as old as he looked, and there were the rumours.\n\nDidn't Jock down at the Bell claim he was a retired bank manager? That a couple of years ago before he got this far out of it, he advised him on a couple of investments? Then there was Martha at the Black Horse, who felt sorry for him one night and put him to bed in her spare room. Later he'd written her a letter thanking her and that letter had been passed round the bar. Beautifully written in a stylish copperplate handwriting, the type of letter that could only be written by a man with a first-class upbringing and education.\n\nThe man lived in squalor now. His clothes were little better than a tramp's, he was dirty, unshaven, neglected and definitely half way round the bend. But his accent was impeccable. He had good manners. Despite his present appearance he could see Anderson was the man in the photograph. Suppose it was true? The story could make a fortune.\n\n'Look,' John leaned closer across the table. 'I'll be straight with you. If what you are telling me is true I can get you enough money to make up for what those guys took. I know you're ill. You may have even had a bang on the head. But if you are making it up for God's sake level with me now.'\n\n'Can I have made that up?' Brian pushed the picture of the family group towards him again. 'You can see plainly that's her, even if she was only a kid then. You can see it's me too, if you take away the fine clothes. How else would I have that picture if it wasn't true?'\n\nA picture five or six years old. How could it be a fake? Only a practised confidence trickster could engineer something like that, and this sad old man wasn't that.\n\n'Would you tell me everything about her?' Adams asked gently. 'I'm a writer you see.'\n\nCalling himself a writer was stretching the truth a bit. He'd been paid for a couple of articles on local history, written a few letters to _The Times_. But he could string a few words together, and he had got a couple of mates in Fleet Street.\n\n'All right. I will.'\n\nAdams could only stare at the old man. He'd expected haggling, even straight, sober people asked for money up front. He remembered one of his journalist friends words. 'They either spill the beans for money, or revenge.' Revenge would pay the most!\n\nBy the time a policeman had been and taken a statement and John Adams had bought him a big breakfast, Brian was feeling better.\n\nAdams had suggested he go home with him for a talk, he'd even suggested there might be some money in it for him. Maybe his luck was turning at last?\n\nIt was close to six o'clock when Adams showed Anderson the door. He had been tempted to ask the man to stay the night and finish what he'd started, but he was so overwhelmed by what he had heard he needed to be alone.\n\nHis feelings about Anderson had swung violently from pity to suspicion during the day, and now he wasn't sure whose side he was on.\n\nPity had been the major feeling when the man came out of the bathroom. He could see shame on his face, now it was scrubbed clean, further evidence that he hadn't always lived the way he did now.\n\n'I'll get these cleaned for you and return them,' he almost whispered, touching the clean shirt and old grey flannel trousers as if they'd come straight out of the window at Burton's.\n\n'They're yours.' John hid his own discomfort by digging out a pair of black shoes which had been left behind by a friend. 'Try these for size.'\n\nHe had felt bad taking a photograph of Anderson before the transformation. Even worse putting his old clothes into a bag and tying up the top to keep as evidence to back up his story. His mind was whirling with disgust at Georgia allowing her father to get to this stage, loathing at himself for cashing in on it, and an even greater desire to get to the hard truth.\n\nHe felt more pity too when he realized just how confused the man was. He had a story all right but it came out backwards, sideways, upside down, but rarely running straight.\n\nAdams' old tape recorder went round and round as he led Anderson through Georgia's first days at his home. He heard about her being beaten by the nuns, her injuries and the way she brought sunshine into their home. Sometimes it sounded as if she was just a toddler on arrival, sometimes far older. One moment his eyes filled with tears, the next they flashed with hate.\n\nBut when Anderson got to the bit about the party, nothing sounded right. The emotion had gone from his voice, it sounded like a story he'd rehearsed. No unnecessary detail as there had been earlier. Like a journalist's story in fact.\n\n'My wife had to go out you see,' he kept saying. 'I stayed downstairs because I didn't want to spoil their fun. But I sensed something was going on.'\n\nIn one huge gulp he told the tale of how he saw her come out of the bedroom with her boyfriend.\n\n'I packed the kids off sharpish,' he said. 'Then I asked her what she thought she was doing. She told me to mind my own business and I slapped her. Next thing I knew she was coming up the stairs with a knife. She lunged at me, sticking it in my stomach. When I came round I was in hospital and I'd nearly died.'\n\nThere was the scar to back it up. A vivid red slash against his white belly, the skin puckered and wrinkled around it. But even as he looked at it, he wondered why she had hit her father so low. A frenzied attack was usually in the direction of the heart.\n\nJohn wanted to go back over details. There was more, he knew it, but he was afraid to stop the man in case he dried up.\n\nHe could understand Georgia running away to escape punishment. But why did his wife leave? What had been left out?\n\nIf he was to believe everything Anderson told him, the man was a victim of not only his daughter's cruelty, but his wife and employers too. Why should the bank sack him? It didn't make sense.\n\nAnderson was an alcoholic, that much was plain. His hands shook, his eyes constantly strayed around the room as if searching for drink. But was he already one when Georgia ran away, or was she the trigger that had started his downward spiral?\n\nJohn Adams understood how bitterness could warp a person. A bright kid from the wrong side of the tracks who won a scholarship to public school, only to find he was a social outcast. Later at university he worked while everyone else enjoyed themselves. He got his 'First', but they got the girls, and later the good jobs.\n\nWhy did he end up in a laboratory testing paint while others less able set the world alight? All he had to show for his hard work was a poky one-bedroomed flat, a beaten-up Ford and no savings. Maybe he too could have turned to drink to cope with disappointment.\n\nBrian Anderson lay on his bed shivering. He'd managed to get home without going in the pub and he'd paid Mrs Dooley what he owed, but it was a mistake to be sober in this room. The damp patches seemed to press in on him, and the sink full of weeks-old dirty dishes appalled him. He had to remain sober though; unless he did, John Adams wouldn't help him any more.\n\nThis time with Adams' help he was going to pull his life together. There was no way she could spoil things now, not like she had before.\n\nIt was November 1960 when he went up to the West End. The house in Blackheath was sold at last. The money safely in a bank until he decided what to do with it. A nice little flat in New Cross. A fresh start in a place where no one knew him.\n\nHe told his landlord he had retired early because of an old war wound. He liked the sound of that, it made him sound romantic, a man of action. He wasn't going to drink again, not the way he had in those last dreadful months when everyone, Celia, the bank, friends and neighbours had turned against him. Maybe he would leave the country if he didn't find a job that suited him, but that night as he went to the West End he was just looking for some company.\n\nIt had been some years since he last visited Soho. As he stood in Piccadilly looking at the neon lights flashing out messages and advertisements he felt charged with new life. He cut a smart figure in his new blazer, military tie and grey slacks. Somewhere in that square mile was a woman who'd share drinks and supper with him, someone to make him laugh and forget the past.\n\nHe found her sitting in the White Bear bar. Red hair curling to her shoulders and a vivid green dress that echoed her eyes.\n\n'Is there anyone sitting here?' he said, pointing to the empty bench seat beside her.\n\n'Feel free,' she said, moving up just slightly and crossing her long, slim legs.\n\nHe knew she was a prostitute, but that made it all the easier. She might try to con him out of a few bob but it was better than picking up a girl on the streets.\n\n'Hasn't it changed around here?' he said brightly. 'Last time I came in here it was packed. Don't people drink here anymore?'\n\n'It's early yet,' she said languidly, looking at a cheap imitation of a diamond-studded bracelet watch. 'Are you from out of town?'\n\n'No,' he laughed lightly, letting her know straightaway he understood the West End. 'South London, just came up to have a little fun for a change.'\n\nShe was probably around thirty-five, though from a distance she looked nearer twenty-five. She smelled of apple blossom perfume that took him instantly back to a girl in Birmingham once. She wasn't pretty, her face was too long and thin, her lips rather thin, but she had good breasts, pushed up to reveal deep cleavage.\n\nThe best thing about her was that she wasn't obviously a tart. She could be a secretary or shop girl waiting for her boyfriend.\n\nHer red hair was maybe a little startling, the dress a flashy cheap one, but then that made it more exciting.\n\n'I suppose you're waiting for someone,' he said as she picked up her glass and drank the last drop. 'Can I buy you a drink while you wait?'\n\n'Okay,' she flashed a brilliant smile at him, which sent shivers of delight down to his toes. 'Gin and orange.'\n\nHer name was Paula, she said she'd been a dancer and she had a flat nearby, and she was open enough to name her price immediately.\n\nTen months had passed since Georgia's birthday and for the first time since that day he felt like his old self. Instead of gulping down drinks in an effort to forget, he found himself slowing down, listening to Paula's chatter, enjoying the pressure of her thigh on his, his mind calm, his body relaxed.\n\nHe told her he was a retired bank manager, hinted at wealth and encouraged her natural sympathy by telling her he was a widower and their only daughter lived miles away and never visited.\n\n'Well you aren't alone tonight, love,' she said warmly. 'Come on, drink up, let's find somewhere to dance.'\n\nEven when she softly asked for the money up front she did it gently, winding her soft arms round him and kissing him outside the club.\n\n'Better to get it over first,' she smiled, pressing herself against him. 'I want all the guys in there to think you are a date.'\n\nBrian had been in the Mandrake club before. A damp basement that smelled of mould and beer. Hot, stale air wafted up as they went down the dimly lit stairs. The music came from a juke box in one corner of the room, jangling and distorted. The seats were little more than wooden benches, the floor solid concrete and apart from a few candles spluttering in Chianti bottles, the only light came from the small bar. But with a pretty woman on his arm and the promise of a night of love, it could have been the Caf\u00e9 de Paris.\n\nIt didn't matter that each drink cost nearly a pound. Tomorrow would be soon enough to worry about money, right now he had a girl who cared about him. He was pleasantly tight, the club was warm and friendly. It had been so long since he held a woman in his arms, the smell of her perfume, the softness of her skin was like a soothing drug.\n\nWhen they walked up the stairs after one, the fresh air caught them by surprise. Paula was staggering in her high heels and Brian put his arm round her.\n\nAt the end of St Anne's Court they stopped for a moment in the shadows to kiss. It was quiet now, just the distant sound of music in another bar and fainter still the traffic from Piccadilly and Shaftesbury Avenue.\n\nShe kissed beautifully, slow, deliberate and sensuous, her tongue flickering across his, sending shudders of delight down Brian's spine. In her arms he could forget the mean streets, the glaring neon signs, the overflowing dustbins and the smell of rotting rubbish. It looked almost pretty, an old street lamp sending a golden arc of light across the road, Dickensian and quaint.\n\nFurther down the street the sound of high heels tapped out a staccato rhythm. As the footsteps came closer Brian heard a peal of laughter that made him stiffen involuntarily.\n\n'What is it?' she whispered, her lips against his neck.\n\nBrian didn't answer but concentrated on listening as the feet came closer.\n\nTwo pairs, one with a bouncy young step, the other older, more plodding.\n\n'Did you see that old bloke? He must have been at least fifty.'\n\nThe woman speaking wasn't the one with the laugh that jolted him, her voice common and rough. Brian shook himself and pulled Paula closer.\n\n'He was quite sweet though.'\n\nThis voice sounded exactly like Georgia's. He'd heard it night after night as his eyes closed with weariness and now he was hearing it again only a few yards from him.\n\n'Shall we go now?' Paula was saying, but Brian merely held her tighter, burying his lips in her neck as he watched over her shoulder.\n\nThe two women came under the yellow arc of light, their heads close together, blond against dark.\n\nBrian saw only the black pom-pom of hair, the big eyes that looked right into the alley where he and Paula stood and his blood ran cold.\n\n'Don't bite me!' Paula's squeal of pain made him loosen his grip on her. 'What's your game?'\n\nThe two women had passed the end of the alley. Brian ran forward, forgetting Paula.\n\nThe women stopped outside a door, standing close together whispering. Hearing his footsteps they looked round.\n\n'What's up mate?' the blonde one called out. 'Isn't one girl enough for you?'\n\nThe dark girl laughed, her hand poised to put a key in the door.\n\nIn profile Brian could see it wasn't Georgia. She was white, at least twenty-five and she had thick legs.\n\nHe backed away, feeling shaken and foolish.\n\nPaula was standing at the end of St Anne's Court with a puzzled look on her face.\n\nThe short fur jacket she wore over her green dress hanging off her shoulders.\n\n'What is it? You look as if you've seen a ghost.'\n\n'I thought I had. I could have sworn that was my daughter,' he said weakly.\n\n'That was Shirl and Denise,' she linked arms with him, urging him along. 'Nice girls don't roam around here at night. Come on, let's go back to my place.'\n\nThe small flat was just a block away. A door half hidden between two shops. They climbed up some narrow, grubby stairs and Paula opened a door on the first floor.\n\n'You did get a shock,' she smiled and drew him in. 'I'll just light the fire and make you a drink. You've gone all pale!'\n\n'You're very understanding,' he said, looking all around him furtively as if expecting someone to jump out on him.\n\nIt was one room, the walls and floor uneven as if the house was subsiding. A cooker and sink were curtained off in one corner, through the other door he could see a bath and toilet.\n\nShe was a hoarder. Every shelf, every surface was full of ornaments. Even the wardrobe wouldn't close because so many clothes were stuffed in there. The double bed was covered in red satin, a profusion of frilly pillows and soft toys arranged on it. Cheap prints hung over the bed, chosen for their garish colours rather than their artistic appeal. Behind the door hung a green silky dressing-gown with black lace and over one chair hung a red and black basque.\n\n'This is cosy,' his spirits rose again as she switched on twin red lamps either side of the bed.\n\n'Relax dear,' she said crossing the room to him. 'Take off your jacket and make yourself at home. Would you like some whiskey?'\n\nNever before had any prostitute offered him a drink. They took their money, did the business and then expected him to go. Maybe he was already special to her? Perhaps this could be the start of something good?\n\nShe poured him a large drink, dropped a kiss on his cheek and disappeared into the bathroom.\n\n'As they say at the movies,' she shouted through the door. 'Just slipping into something more comfortable.'\n\nBrian took off his jacket and shoes and sat on the bed nursing his drink.\n\n'There, I wasn't long was I?'\n\nPaula was standing in the doorway, a black negligee open to reveal a bra, knickers, stockings and suspenders.\n\nHer skin was very white, her thighs bulged at her stocking tops. Brian could say nothing, instead he reached out for her, putting his arms round her waist and buried his head in her breasts.\n\nShe smelt perfect, the perfume just a little too sweet and heady.\n\n'Let me undress you?' she said, bending down to him and lifting his face up with one finger. 'You're shy aren't you?'\n\nAs her fingers reached out to unbutton his shirt he felt restored. He pulled her down onto the bed beside him and covered her face with kisses. He could feel an erection starting and he knew she wouldn't insist on turning off the light.\n\n'You're lovely,' she whispered, her tongue flickering over his, 'Let me get your things off?'\n\nAs she slid his trousers down his legs she touched him lightly on the front of his white 'Y' fronts.\n\n'That looks very healthy,' she smiled up at him impishly, her auburn hair tumbling round her face.\n\nBrian could hear his heart hammering. She knelt up on the bed beside him slowly unbuttoning his shirt. As she undid the cuffs she kissed both his wrists and let her lips travel up the soft insides of his arms.\n\n'Now the pants,' she said, gripping the waistband firmly and lowering them, moving her lips down towards his penis as she pulled the pants right off his feet.\n\nHe was holding his breath now, wearing nothing but his socks and his vest. Her lips were only an inch away from his penis and her hand was poised to grasp it.\n\n'What a lovely big one,' she whispered, looking up at him and smiling. 'I think I've just got to kiss it!'\n\nHer tongue flicked over the end. He drew in his breath and watched her, leaning back on his elbows.\n\nIt was his favourite fantasy. A near-naked woman, about to take him in her mouth. Her breasts were full, spilling over the low cut bra, her skin very white and clear. The negligee had fallen off one shoulder and he could see a tiny sprinkling of freckles on her small shoulder.\n\nHer tongue darted out, long and pointed, she ran it along the length of his penis one hand reaching out to cup his balls.\n\nHe could feel her breasts touching his leg and he was desperately afraid he would come before they even got started.\n\n'Not yet,' she looked up and smiled seductively. 'First we have to get that vest off.' She gripped it by the bottom and quickly pulled it over his head.\n\n'Oh my God!' she gasped, moving back from him. 'What ever's happened to you?'\n\nHer remark was like a cold shower. His penis shrank back like a tortoise into its shell.\n\nBrian had forgotten the scar. In the past months it had become just another part of him, but seeing it through her eyes he saw how fearsome it looked.\n\nThe original gash was only an inch and a half. But during surgery they had opened it wider. It was diagonal across the fat part of his belly and as it had healed it had puckered so it looked like a pair of pursed lips.\n\n'It's nothing,' he said too quickly. 'I fell on a knife.' He cursed himself for not remembering the old war wound story. Women had enough imagination to understand the thrust of a bayonet. She would have shuddered delicately and changed the subject.\n\n'When?' her face was pale with fright now, the seductive look gone, replaced by morbid curiosity. 'Does it hurt?'\n\nAs if it wasn't bad enough her even remarking about it, she now reached out gingerly to touch it.\n\n'Don't,' he slapped her hand away.\n\n'Why not,' her eyes opened wide. 'Scars are interesting.'\n\nBrian closed his eyes for a second.\n\nIt was Georgia again. Somehow she'd even managed to spoil this night for him.\n\n'Don't be like that,' Paula wriggled up to lie beside him and leaned over his face to kiss him.\n\nBrian grabbed her fiercely, thrusting his tongue into her mouth.\n\nBut he felt nothing. No reaction. She smelled too sweet, it made him feel nauseous and underneath that perfume he could smell sweat.\n\n'Let me lick your prick again,' she said. 'You liked that.'\n\nShe moved back down the bed and once again Brian watched, holding his breath as her tongue slid out, red and pointy.\n\nBut all he could see now was the wound. Even from the angle he lay at it looked evil, like the mouth of an old crone.\n\nWhen she'd done this before it was sweet and exciting, but now he felt a sense of duty in her manner.\n\nShe took his penis in her mouth, sucking at it vigorously, but still it refused to grow. Her long red nails dug into his inner thighs and then she yawned.\n\n'Don't bother,' he said, pushing her away. 'You ruined everything anyway.'\n\nShe moved back from him, her eyes startled.\n\n'I'm sorry,' she said perching on the bed beside him. 'I didn't mean to hurt your feelings.'\n\n'Yes you did,' he said, getting up and reaching for his clothes. 'You women are all the same. Always got to spoil things.'\n\n'Just a minute!' she leapt off the bed and pulled her negligee tightly around her. She stood in front of him as he pulled on his shirt. 'Who the fuck do you think you're talking to? I spent all evening with you because I liked you. I brought you back here for the same reason. I may be a tart but it doesn't stop me having feelings.'\n\n'I paid you for a night out and sex,' he spat at her. 'I didn't expect you to pry into my private life.'\n\n'You call asking about a scar prying?' she sneered at him. 'If you were like that to your daughter it's no wonder she ran off.'\n\nRage welled up inside him. He saw only the thin pale face and the expression of aversion in those cold green eyes. One moment his hands were buttoning his shirt, the next his clenched fist shot out and punched straight into her face.\n\n'Filthy whore,' he shouted.\n\nWhen he saw her backing away on to the bed, blood gushing out of her nose, he was sorry. The anger wasn't directed at her, just at Georgia. He pulled on his pants and trousers, searching for the right words to make her forget.\n\nHe moved towards her, hands outstretched. She cowered back amongst the pillows, white-faced and frightened.\n\n'Don't you dare touch me!' The force with which she spat out the words surprised him, her eyes blazed and she scrabbled amongst the pillows behind her. 'I can defend myself you bastard!'\n\nA glint of silver under the red light. A long curved blade, her red nails curled round an ivory handle.\n\n'I didn't mean to hit you,' his words spilt out. 'It was just \u2013'\n\nThe knife drove all explanations out of his head, his legs turned to jelly.\n\n'I knew there was something weird about you,' she interrupted, getting up on to her knees, slashing out at the air between them. Her black negligee fell back, revealing her underwear and white skin. 'This could cut off your prick like a ripe banana. I'd finish what some other woman started.'\n\n'You don't understand,' Brian shoved his feet into his shoes. He was frightened now. She looked fiendish, white-faced, bloodied nose, red hair hanging over eyes that burned almost black with rage.\n\n'Oh, I do,' she snapped back. 'I've met dozens of perverts like you. Do you think I'm stupid just because I'm a prostitute? I know a woman made that scar and I can guess why.'\n\nBrian backed away towards the door as she jumped down off the bed moving steadily towards him, the knife held in front of her.\n\n'Get out you bastard,' she said through clenched teeth. 'And if I ever see you up this way again I'll swing for you.'\n\nThe bottle was the only way of blanking out that night. Alone in his new flat Paula and Georgia's faces haunted him. He had to go out, down to the crowded pub until he was drunk enough to blot them out.\n\nSome days he made an effort to forget. He washed and dressed smartly and presented himself at the labour exchange, but even there it seemed as if something was working against him. Manual work, that's all they offered. They ignored his education and qualifications, almost as if someone had been there first, whispering about him.\n\nHe wanted to make his flat a home. But daily he saw things sliding. Clothes dropped on the floor stayed there. Plates filled the sink and empty beer bottles multiplied overnight. His shirts looked grey, underwear and socks disintegrated and he never replaced them.\n\nBongo drums from a house behind his kept him awake. The landlord pushed notes under his door complaining of noise and smells coming from his flat. Was it before, or after he noticed how low his money was getting when he got arrested for being drunk?\n\nThere was another woman, he couldn't remember now who she was, or even where he met her, but for a time he got back on his feet. Did she help him get the night security job at Jarson's in Catford?\n\nA period of peace. Not exactly happy, but reconciled. That hot evening in August he walked to work thinking about moving to somewhere with a garden. He even went over to the town hall to look and see what concerts were coming up. He wasn't drinking then.\n\nThere were bands he'd never heard of. Silly names that didn't give a clue to what sort of music they played.\n\nFunny that he remembered so little about a period of over a year, yet he could see Catford Town Hall as clearly now as if a picture was in front of him.\n\nA rounded front, with two sets of doors going into the theatre and between them a sandwich board advertising the events. There were two teenage boys standing beside him.\n\n'See, I told you they are coming here,' one of them pointed to the programme. 'It's next Monday.'\n\nBrian looked to where the boy pointed.\n\n'Who are Samson?' he asked. 'Is it jazz or a dance band?'\n\nThey looked at him and laughed. He remembered they were both smartly dressed, Italian suits with short jackets and winklepicker shoes.\n\n'A soul band,' the smaller one replied. 'The best in England.'\n\n'That's them, there.' The other one pointed to a poster on the wall.\n\nThe picture was of a group of young men. Sitting in front of them was Georgia.\n\nHe blinked, afraid that his eyes were deceiving him.\n\nShe had that impudent grin that belonged to no one else. Head thrown back, hair longer than he remembered, tumbling over bare shoulders. A tight black dress barely covered her breasts, eyes that held his with all that old magnetism.\n\nIt was like the knife again. But this time he could feel it turning in the wound.\n\n'Who is she?' he whispered, turning to the boys, pleading silently with his eyes for them to say she was an American, Jamaican, anything but his daughter.\n\n'Georgia James!' The boy looked pityingly at him, as if unable to believe he didn't already know.\n\nThe evening sun seemed to fade instantly, a cold wind whipped around him. He felt sick, giddy and threatened.\n\nBlindly he made his way back across the road. He heard a car horn blast out at him and he stumbled on the kerb.\n\nThat poster drew him every morning as he walked home. He had to go and look at it, even though he knew her face would prevent him from sleeping when he got back to his flat. He would buy a bottle, drink it just to fall asleep, then he'd buy another on his way to work.\n\nHe risked getting the sack on Monday evening, slipping out leaving the warehouse unattended, just to see her.\n\nThe show finished at eleven, he waited in a shop doorway just across the road from the town hall and watched the hordes of young people leaving.\n\nYoung girls in pretty summer dresses, kiss curls lacquered to their cheeks, hair backcombed up to impossible heights, stiletto heels tapping out like castanets on the warm summer night. Boys in jeans, some in smart suits and winklepickers, others jumping astride their motor scooters, revving them up like some sort of odd mating dance.\n\nThey were in no hurry to go home. Some ran across the street to a coffee bar. Others bought chips and stood outside eating them from the newspaper, still more wandered past him hand in hand. He could see pleasure in their faces, a world he couldn't enter. He felt old, tired and bitter.\n\nMost of the crowd had gone now. About thirty or forty teenagers left standing outside the stage door, away from the main entrances now shut up.\n\nA cheer alerted him. First he saw a tall, dark man come out. The waiting girls flocked round him, pressing in on all sides.\n\nBehind them he saw Georgia. She was standing on the steps of the stage entrance wearing a white summer dress, her hair caught up with a ribbon, a few stray curls on her forehead. With the light behind her she was silhouetted in the doorway, slim and graceful.\n\nHe had hoped it wouldn't be her. A mistaken identity, just another dark girl with the same name. But as soon as her voice floated across the street he knew it was her.\n\n'Did you enjoy the show?' she asked the waiting kids. 'Thank you for coming.' Her voice sounded as if she were whispering in his ear. He could remember a time when she sat on his lap while Celia played the piano. He could feel her soft arms round his neck, breath sweet on his cheeks.\n\nHe moved closer, crossing the road, standing beside a parked van to conceal himself.\n\nAnother young man joined them. This one blond and frail looking. The two men stood either side of Georgia as if protecting her.\n\nBrian waited until the last piece of equipment was brought out, moving back over the road so he wouldn't be spotted. He heard their laughter, saw the familiar way they treated Georgia and as he watched so the pain in his heart seemed to get stronger.\n\nThe next day he read a review in the local paper.\n\n'Samson's strength is Georgia,' he read, with eyes welling up with tears of impotence. 'Catford Town Hall rang to her beautiful voice capturing the enthusiastic audience with her beauty and her power. Watch this little girl, she's going all the way.'\n\nThat was the start of the collection. One tiny review cut out and pinned to the wall. But it was soon to be added to.\n\nHe began to drink heavily, his lady friend disappeared. The flat grew dirty again. His shirts needed washing. The West Indians in the houses at the back of him seemed to play their drums even louder.\n\nWhen he read about the fire and the two boys who died he was glad. The band's name stopped appearing in music papers and he lost interest in buying them. The need for drink seemed to lessen and as his savings had almost gone he seriously considered moving somewhere smaller, nearer his job.\n\nHe got into a new routine at work. First the paper work, then a quick patrol round the warehouse checking doors, then back into the office for a cup of coffee to listen to Book at Bedtime on the radio.\n\nIt was cold and raining that night after the story, and he was loathe to leave the warm office for another check. He remembered twiddling the dial and finding music, settling down with his paper to do the crossword.\n\nViolins filled the office with a melancholy sound that fitted the rain and his mood perfectly. He turned up the radio, sat back and listened.\n\nAs guitars came in, he frowned, realizing it was a pop song after all.\n\n'There's no time, baby.'\n\nHe knew it was her the moment that voice filled the office. His hand reached out to switch off the set but somehow he couldn't do it.\n\nSlumped over the old desk, banging his forehead with rage, hating her voice, yet loving it too. She was taunting him with failure. Sneering at him over the radio. He would never be free of her.\n\nHe could see her before him as she had been that night of her birthday. The face upturned to receive Peter's kiss, pressing herself against him. She'd shut her father out then, she hadn't wanted him to join the party and later she'd called him pathetic.\n\nNightmare flashes. Prickling net in his hands, strangely mixed with satiny skin. Celia showing him a small back covered with weals from a cane. Georgia standing over him with that knife and blood spurting out. More blood, this time on Celia's face and all the time Georgia was telling the world that it was too late for her father.\n\nHe didn't remember the police picking him up by Lewisham Hospital where he was lying in a gutter. Neither did he remember not locking up the warehouse. All he could remember was that song and the melody that pounded into his head long after he lay vomiting in a police cell.\n\nAfter that everything became hazy. The police kept coming, questions and more questions. His landlord shouting abuse at him, some trouble with a woman downstairs. They said he'd conspired with the gang who broke in and stole thousands of pounds worth of goods. A man like him kept in Brixton prison with common criminals!\n\nIt made no difference that he was released after a few days and the case against him was quashed, the damage was done. Turned out of his flat, his mother's desk and Persian carpet taken for back rent.\n\nA hostel for the homeless, sleeping in a cubicle alongside filthy vagrants, a spell in hospital with pneumonia and finally someone pushed him into this room in Ladbroke Grove.\n\nNational Assistance kept him now, while the girl who was responsible for his sickness made millions.\n\nHe had just cheap sherry to keep him warm while she swanned about in limousines wrapped in furs.\n\nThere was no getting away from her success. She flew in and out of England like a film star. Every word she uttered, every place she visited was recorded. Pictures of her in every shop, newspaper and magazine. Game shows on television, interviews on the radio. He knew her escorts, her clothes, beauty hints and every appearance she made. He logged it down in a little book, and when she attended a film premi\u00e8re in the West End he waited in the shadows to see her.\n\nShe brushed past so close to him he smelt her perfume and almost touched her hair. It was in that crowd that his glasses got broken, crushed underfoot, just the way she had crushed him.\n\nHow clever she was at skirting round the truth about her earlier life. Never at any time did she speak about her childhood. Wasn't that proof enough for anyone that she had done something shameful?\n\nDid she know or care that her father watched her on television through the window of an electrical shop? Shivering and hungry, unable to hear her voice, only watch her dance and her lips move?\n\nBut of all the articles he read about her there was one that played on his mind above all the others.\n\nShe was posing on a white settee, bare arms wrapped round her knees, a short red dress revealing her brown thighs, hair tumbling down over her back, so shiny it looked like wet tar.\n\n'I love simple things,' she told her interviewer. 'Bright primary colours and lots of white paint. Huge vases of fresh flowers and the sun streaming in through the windows.'\n\nBrian thought about his old home. Walking in after a day at work to find the French windows open, the perfume of roses filling the room. Georgia on her swing, wrapped in a private world. Tea laid in the kitchen, flowers in the hall. Celia's voice calling out.\n\n'Come in now darling, Daddy's home.'\n\nThe sun never came in his windows. The paint in his room was brown and cracked and it had never seen a vase of flowers.\n\nHe'd found her flat in Chelsea. It took him all day to walk there. A three-sided block set around a central garden. The sort of plush place aristocrats kept for the Season. He looked at the polished doors, the gleaming brass and the uniformed porter, and knew he would never get inside. He was of less importance than the youngsters who gathered there, looking up at the gleaming windows, trying to guess which one of the flats was hers.\n\nHe had a pain in his stomach which wouldn't go away and sometimes when he vomited he brought up blood. Without his glasses he couldn't see clearly. His legs ached just on the walk to the post office. But if John Adams sold that story, all his troubles would be over.\n\nGeorgia would be finished.\n\n## Chapter 21\n\n'Have you heard this man play, Max?' Georgia stood in the office, a musical paper in her hands.\n\n'Who?' Max barely looked up. 'Do you think I've got time to listen to every musician in England?'\n\n'Don't be grumpy with me,' she grinned, perching on the edge of his desk and waving the paper under his nose. 'Sam Cameron, the saxophone player from the States. Ronnie Scott has extended his contract, he's played to packed houses nightly. Since when didn't you take an interest in something like that?'\n\n'Since you got back from the States and had stupid ideas about making a middle of the road album,' he snapped. 'Sometimes I don't think you've got a brain. Who in their right mind wants to buy a record full of old songs?'\n\n'Millions of people,' she retorted. 'A far greater number than the ones who want pop.'\n\nGeorgia was tired. The long tours, the one night stands, the travelling had become impossible. Her world was up there on the stage or in a recording studio. No time for friendships or even love affairs. She had been to so many countries, yet not one had she seen in any depth. She had so much money she could buy anything, yet the simple pleasure of a day's shopping, lunch in an ordinary pub or even a night in watching the television was denied her.\n\nAt twenty-one she'd seen everything, done everything. Yet nothing. Was she going to spend all her youth pursuing nothing more than money? When did she last meet a real human being to laugh and chat with?\n\nBodyguards to escort her everywhere, hairdressers fussing around her, fittings for fabulous dresses she wore once then put aside. Journalists hanging on her every word. People behaved as they thought she wanted them to. They fawned on her, smothered her. Every unguarded word was recorded. How much longer could she stand it? Wasn't that why so many famous people ended up with drug and drink problems?\n\nHer affair with Rod had changed things in so many ways. For a time it seemed impossible to stay in the same room as him. She knew he wanted her, she ached for him. But he'd been right, there was no future in it. Rod loved the high life, as soon as a gig was over he was out looking for more excitement. Her idea of a good night was a quiet dinner with people she could talk to. Compromise would have done no good, both of them would be unhappy half of the time.\n\nBut that closeness in Spain had left her hungry. How could she find that again when she was always on the move? Who could she trust enough, when half the men she met were only interested in her fortune, name and body?\n\nShe had a beautiful flat she rarely saw, a red Mercedes she hardly ever drove. Was she going to spend her life hoping for love, but never finding the time for that either?\n\nThere was a way out. To make beautiful albums for a different market. Her income would be assured, but by dropping out of the mainstream of pop she could gain time for herself. But of course Max would have none of that.\n\nRod and the rest of the band were restless. They wanted more now than being her backing group. She knew they were privately working on new material, an image for a band entirely different. Something that was theirs and theirs alone.\n\nThe Beatles and the Rolling Stones had paved the way. Samson wanted to do something equally innovative. She knew each one of them had experimented with drugs and it had altered their conception of sounds. They had moved in a different direction to her and they had a right to their freedom now.\n\nThe last thing Georgia wanted was to allow Max to push forward a new backing band, or even worse start booking her into cabaret work. She wanted to hold the reins herself, pick the musicians to go forward with. People she liked and trusted.\n\nListening to jazz in the States had started something. She had heard old classic songs revitalized and now she was burning to sing things she loved.\n\n'Come with me tonight to Ronnie Scott's?' She bent over Max and tickled his neck playfully. 'Don't be a grouch. Aren't I still your golden goose?'\n\nMax leaned back in his chair and smiled despite himself.\n\nHe'd met his match with this girl. However he tried he couldn't outmanoeuvre her. She took risks, she gambled on her fans' loyalty, she managed to get what she wanted every time.\n\nMax didn't need a crystal ball to see what she was up to. She was dredging up talent to join her on this harebrained scheme. She would do what she always did. Get something together and astound the men at Decca. But if she got her way in this, where would he stand?\n\nIt wasn't just the money now. He had enough. The one thing he couldn't face was Georgia going out of his life for good. She thought he didn't understand her, but he did. She wanted a quieter life. She needed a man to share it with. Before long someone would emerge from the woodwork and carry her off. That man might relegate him to the role of an old uncle. And that he couldn't bear.\n\nJust the way she sat on the desk gave him goosebumps of pleasure. The tiny mini skirt, the tight white boots and the skinny sweater might have been designed by Mary Quant but it would be Georgia who millions of young girls would copy slavishly. He had taken this girl, polished her up and promoted her. Now she was the hub of the wheel, music, fashion, all evolved round her. With her beside him Max was at the centre of the world and that's where he wanted to stay.\n\n'Okay,' he said wearily. 'I'll go with you. But because it's the only way I can have you to myself for a night.'\n\nIt was a wicked thing to think, but sometimes he almost wished something would bring her crashing down. He would trade his fortune, his huge house and his fleet of cars just to have her dependent on him again. Sometimes he had fantasies of being her only protector. Perhaps then she'd realize how much he loved her.\n\nMiriam and he were washed up now. The once unthinkable divorce had gone through. She'd found new happiness with a man she met in Athens. It didn't matter that there were countless young, pretty girls who adored him. He still wanted Georgia.\n\n'Don't block me about this album?' Georgia wheedled. 'I want your support. I can't go on with this frantic pace and you know it.'\n\n'Let's go and hear the man and then we'll see,' Max grinned. 'I suppose I should be grateful to think you care enough to want me with you.'\n\n'There'll always be a place in my life for you,' she said softly. 'I hate you sometimes, maybe even love you a bit. But we've been together too long to part now.'\n\nMax was important to Georgia. She took his blustering ways, his arrogance and even his bullying in her stride. He had been with her in the bad patches, and at the heights. Even that night in Kensington had its purpose, though she hadn't realized it until later. She had learned to stand up for herself, taken a little of his indestructible pride and cunning. If he'd been softer she wouldn't have all she had now. Learning to outwit him was the greatest lesson he'd given her. She knew too that it wasn't pure greed that motivated him any longer. Maybe both of them were unsure of what they were to one another, but it was something special.\n\nRonnie Scott's was packed, but still Max got them a table right down by the stage. Georgia wore just a plain black cr\u00eape trouser suit, hoping no one would recognize her.\n\n'I used to come here all the time,' Max said as he ordered them a drink. 'I don't know why I stopped coming, I've had more good times in here than I can count.'\n\nHe too was hoping Georgia could remain incognito. Jazz fans were in a world of their own and so far no one had even looked their way. Ronnie Scott maintained his distance from popular music and even though this place saw more stars in a month than anywhere else, he rarely publicised it. He hoped they would dance later, share supper or a drink. But he could never be sure of anything with this girl.\n\nGeorgia looked older tonight, her hair slicked back into a bun like a ballerina. She had lines under her eyes he hadn't noticed before and she was pale. She did need a holiday, that much was obvious, maybe if he was gentle with her, she could be persuaded to go somewhere warm with him.\n\nHow was it that not a breath of scandal had come out about her? What sort of iron will did she have to avoid dangerous relationships? The men who escorted her to parties, shows and theatre rarely got a second chance. She was never present when the lads got out of hand. Drink, drugs and sex didn't seem to figure in her life. Or was she just careful about it like him?\n\nThere had been something between her and Rod once in Spain, but that was over before Max heard about it. He'd seen a new maturity in Rod, a more caring attitude to women, yet a touch of pain in those slanty eyes. Thank God it had fizzled out, those two together would be a recipe for disaster.\n\nWhy did he always have this feeling there was some deep well in her he hadn't, managed to reach? Maybe that was the only thing needed to bring her to him?\n\n'They're coming on!' Georgia tapped his hand. 'I can't see the saxophone player though.'\n\nFive men in evening suits came on to the stage. Each one of them was over fifty. The man with the clarinet looked nearer seventy, he walked stiffly as if his leg hurt.\n\nThe drummer alone was black, he flashed a wide grin and played a roll on his drums, leading the quintet into a spirited version of a Glenn Miller number.\n\nThe trombone was good, the old clarinetist brilliant. The grey-haired man on the double bass was bent over his instrument as if unaware he was playing in public and the pianist one of the best she'd heard. Yet there was no real sparkle, or did she know too little of jazz to understand it?\n\nEveryone in the audience seemed delighted. Feet were tapping, fingers rapping on tables.\n\nAs the first number ended, Georgia sensed a charge of electricity. The pianist began an introduction to a slower number, melodic and almost haunting.\n\nFrom somewhere behind the stage Georgia heard the first saxophone notes, and a shudder of delight went down her spine. Rich, fruity tenor sounds, like nothing she'd heard before. She sat up, peering at the dimly lit stage. It was getting louder and she held her breath while first the saxophone appeared behind a curtain, a spotlight focusing on it.\n\nIt was a surprise to see the man playing wasn't a big man. Tall yes, but slender. His lips, hands and his entire body seemed to be playing the instrument. Never before had she seen anyone so at one with a saxophone.\n\nShe could feel herself melting inside, almost like an orgasm. His notes were so pure and delicious she barely heard or saw anyone else on the stage. Something primeval was swelling in her, her eyes filled with tears. She wanted to catch every last note and engrave it on her heart forever.\n\nMax felt her excitement. He too was enthralled as was everyone in the audience. But Georgia looked as if she had gone off to another planet.\n\nHer eyes that only minutes earlier had looked tired, now sparkled furiously. Her lips were parted, a glimpse of white teeth and pink tongue peeping out. She was flushed, tears rolling unnoticed down her cheeks.\n\nMax looked back at the man. He was golden brown, hair cropped closely to his head, eyes tightly shut, dark lashes curling on his cheeks. Expanded by air his chest seemed huge, yet his hips were slim and his hands long and fine like a surgeon's. His face was too contorted by playing for him to be handsome, yet Max knew he would be.\n\nWas her reaction purely the music, or was this something physical?\n\nThroughout the entire set, Max tried to sit back and watch objectively. This man was undeniably the greatest saxophone player he'd ever heard. His solo sent tremors down his spine, notes that almost shimmered under the spotlights, drifting off with the cigarette smoke, filling a hunger in the belly.\n\nIt was impossible to look away. Even when he rested while others played their solos, his face drew the eye.\n\nHigh cheek-bones, the straight proud nose and those full yet delicate lips. Yet there was humour in that handsome face, the way he listened, his head on one side, picking up each note and smiling when it pleased him.\n\nGeorgia's face fell as the set ended. Sam Cameron merely nodded at the audience and hurried off with the other men without even a backward glance.\n\nThe audience was going wild. Clapping, banging glasses and shouting. Middle-class people behaving like a crowd of teenagers at a pop concert.\n\n'I've got to make a recording with him,' was all she said, eyes as big as saucers. 'Can you arrange for me to speak to him?'\n\nMax ordered another drink to cover his confusion. On the one hand this man and her together could be dynamite. On the other he might just lose her. If he refused to help she would do it anyway. If he was enthusiastic he would be party to whatever came of it.\n\nOne thing was clear by her expression, she liked everything about this man. They even looked like a couple, something Max had never thought about anyone before.\n\n'I'll ring here tomorrow,' Max hedged his bets. 'From what I understand he's going home soon. So don't bank on anything.'\n\nMax left her for a moment to talk to someone he'd spotted across the club. Georgia took a gulp of her drink and looked at the curtain behind the stage. She had seen Max's shifty look and she had so much work piled up she had no idea when she'd find time to come here again. She had to act now.\n\nShe glanced round quickly. Everyone was engrossed in talking and drinking. She walked across the front of the stage, jumped up quickly in the corner and found her way through the curtain.\n\nStifling a giggle she found three steps down the back of the stage. She felt like a groupie. She had no idea of the layout in this club, but she'd find him.\n\n'You aren't allowed back here.' A little man in overalls came out of a door and blocked her way.\n\n'I am,' she smiled sweetly. 'I'm Georgia James and I've come to see Sam Cameron. Could you point out the dressing room?'\n\nHe faltered for a moment, but turned to one side, pointing further down the corridor. It was painted that green gloss paint they had backstage everywhere, chipped and stained, bare boards underfoot.\n\n'In there,' he said. 'I didn't see you if anyone asks.'\n\nShe could hear a low rumble of male voices. The sound of beer pouring into a glass. She tapped on the door and waited.\n\nThe elderly clarinetist opened it. He had a pint in one hand and he peered at her short sightedly.\n\n'Could I speak to Sam, please?' she said, feeling foolish now. Behind him she could see only legs stretched out, all the same in dark trousers and black shoes.\n\n'It's someone for you Sam.' He turned his head back to the room and she saw the double bass player's grey head come into view. His eyes widened. He ducked back and she heard a whisper. All at once one of the pairs of legs moved.\n\nHe was taller than he looked on stage. Well over six foot, his shoulders broader without his jacket. Muscles strained his badly-pressed shirt, a button was missing and his bow tie was slightly crooked.\n\n'I'm sorry to barge in like this,' she said nervously as she stood outside the door. 'But I had to speak to you.'\n\nHe was looking at her intently. Studying her as closely as she'd studied him earlier.\n\n'I'm Georgia,' she said, dropping her eyes from his dark brown ones.\n\n'I know,' he said. 'What brings you here?'\n\nHe was smiling, almost as if she was an old friend he expected. Yet she sensed shock too.\n\n'Did you know I was out there?' She was stumped for how to approach the subject, she felt odd. Like a girl on her first date.\n\n'No,' he said. 'But I'm glad you were.'\n\n'Why's that?'\n\n'Because I played good tonight. I'd have hated you to hear a bad set.'\n\n'You were brilliant,' the words came out even before she could think what she was going to say. 'I want you to do a recording with me. I was afraid if I didn't barge in you'd leave and I wouldn't get to speak to you.' She stopped short, aware how silly she must sound to a seasoned professional like him. She could feel sweat on her forehead, she was blushing. 'I mean, that is if you'd like to.'\n\nHe didn't reply for a moment. He leaned against the door post looking down at her.\n\n'Ma'am, I can't think of anything better,' he said. 'Just say the word and I'll be there.'\n\n'Are you laughing at me?' She could see his mouth twitching and from the silence behind the door she knew the other men were listening. Maybe her goosebumps were caused only by that delicious Southern accent, or was it those brown velvet eyes?\n\n'Not laughing,' he said, his mouth breaking into a wide grin. 'Just thinking maybe there is a God up there somewhere.'\n\n## Chapter 22\n\nSam stood in the doorway with Georgia's address and phone number in his hand. He wanted to laugh and cry all at once.\n\n'Well, you've got it made,' Dave the bass player came up behind him, playfully thumping him on the back. 'A year from now and you could be as big a star as her.'\n\nSam couldn't reply. What could any man say when he saw a dream about to come true?\n\nEverything had been topsy-turvy since his first day in England. First grief to find Katy was dead. Guilt that a child was out there somewhere, who could be theirs. Then the unexpected spotlight turning on him, critics raving about him, real success only inches away from his grasp.\n\nIn those first few days he had imagined it would be easy to find one war orphan, but that conviction was soon dashed.\n\nThe police offered no leads. No birth was recorded at Somerset House. Bureaucracy, despair and blind alleys. Social workers who shrugged their shoulders, the young priest in Whitechapel who suggested it would be better to put it behind him. The public library had revealed more news and photographs of the disaster, but still no report of a child.\n\nTorn between the acclaim he was getting nightly and days spent in searching. Afraid to tell anyone but Clive what was on his mind.\n\nHe tried the local council, who referred him to the children's department and at that point he discovered all war orphans were taken to a home in Billericay in Essex.\n\nNo one had bothered to tell him the home had been pulled down years earlier, or that all the old records were destroyed in a fire. Just another pointless journey, looking for someone no one seemed interested in.\n\nThe days were ticking by. He had to slot in his search between rehearsals, interviews and keeping himself prominent so he wouldn't be forgotten. His children's future depended on him becoming a lasting success. He couldn't risk failure by turning into a recluse.\n\nAlmost at the point of giving up, he went back to the library in Stepney. They had a local history department and one of the librarians, Miss Brice, was actually enthusiastic. 'Let's study the newspapers again,' she suggested. 'We might find something interesting, a name, an event that leads us on to something else.'\n\nMiss Brice reminded Sam of the women who helped out voluntarily during the war, manning tea stalls and organising dances for the troops. Silver grey hair, gold-rimmed spectacles and rosy, clear skin. He liked the way she used the words 'we' and 'us', it gave him the feeling she cared as much as he did.\n\nHe got really hung up on those old papers, making his way to the library almost every day, collecting scraps of information that were useless to him, but riveting all the same. He knew what was on at the movies. Men that got sent to prison, weddings, births and deaths, but nothing relating to an orphan. Time was slipping away. It was February already and he was due to leave on March 1st.\n\n##### *\n\n'Anything?' Miss Brice noticed immediately when something caught his eye.\n\n'Just reading this about a woman who was an Army nurse taking up a position in Stepney,' he said. 'Her life reads like a novel. She had travelled in China, Africa and India, nursing, then in field hospitals during the war.'\n\n'That'ud be Miss Hammick,' Miss Brice smiled. 'She was a regular Tartar. The kids used to shake in their shoes when she came along. She wouldn't stand for any nonsense. She ran the department like a battle campaign.'\n\n'What department was she in then?' Sam looked up. Just the mention of children made him take notice.\n\n'I can't remember what they called it then. She was like the forerunners of today's social workers.' Miss Brice came over to him at the table looking over his shoulder. 'She handled all sorts of things. Truants, child neglect. Unmarried mothers, family disputes. She wasn't noted for her diplomacy, but she got a lot of problems sorted while she was around here.'\n\n'She's not here any longer then?' Sam's spark of hope died as quickly as it was fired.\n\n'No, she retired a few years back. She must be at least seventy. But I think we may find out something more about her because they had a retirement \"do\" for her.'\n\n'I've got to go now,' Sam said reluctantly. 'I've got someone to see before going on to the club. But I'll be back tomorrow.'\n\n'I'll dig around Sam,' she assured him, making a note of it on a pad. 'This might be a real lead for us.'\n\nWhen Sam met Clive later he filled him in with this news.\n\n'Don't go building up your hopes Sam,' Clive said over a pint. 'Even if Miss Hammick knows something, this girl's grown up now. She may be married. She might not even be in England. Suppose after all that you find you aren't her dad?'\n\n'I'm sure I am,' Sam growled. He couldn't explain why exactly, but the more he dug into the events of the bombing, the more convinced he became.\n\n'Even if you are, she may have been brought up by white people. The shock of a black six-foot Yank may be too much.'\n\n'No daughter of mine would have those hang-ups,' Sam laughed. 'Her mother had too much guts to create a little dim-wit.'\n\nWhatever he said to Clive, he did have the same worry. How would Jasmine react to finding her mother years from now? And they were the same colour. The person who brought the child up was the one that counted surely? Could any girl find room for a father who left his pregnant girlfriend and never came back to look for her?\n\nHe knew Miss Brice had found something just as soon as he got in the library.\n\n'I've got it,' she said, almost jumping with excitement. Her hazel eyes glistened behind her glasses. She twitched at her strand of beads round her neck. 'I was so excited I telephoned her last night.'\n\n'Well ma'am, you are a surprise,' Sam chuckled. 'And they told me you Brits were cold-blooded.'\n\nShe was bursting with it, her lined face flushed salmon pink, contrasting vividly with her saxe green sweater.\n\n'She said to send you over there. She did remember a child of the right age. Last night she needed time to think it out.'\n\n'Where is she?' Sam said, ready to run out the door.\n\n'Buckhurst Hill,' she said, picking up the address and telephone number and pushing it into his hands. 'Please phone me and let me know.'\n\n'I'll do better than that,' he leaned forward and kissed her soft cheek. 'If I find her I'll bring her to see you.'\n\nSam straightened his jacket and rubbed his shoes on his trouser legs before pushing open the gate. It had seemed an interminable journey on the tube and he wished he had smarter clothes to see this Tartar.\n\nIt was a small bungalow, painted white with green shutters, set in a lawned garden. Everything was bare now, but it was the sort of garden he knew would be beautiful in summer. The door opened before he even reached it.\n\n'Mr Cameron I presume?' she said.\n\nSam wanted to laugh. She was the type of lady he had seen in British films. Snooty, tall, grey hair cut very short and a masculine set to her features. Her body was encased in men's cord trousers, with a thick navy sweater, through the thick material he knew it was large and muscular rather than fat. Even at seventy she looked formidable.\n\n'Pleased to meet you, ma'am,' Sam held out his hand. 'Thank you for seeing me.'\n\nHe couldn't make out what she was thinking. A woman of her age and background probably loathed black men, especially ones who impregnated young girls and left them alone to get killed.\n\n'Come in,' she said crisply. 'Wipe your feet.'\n\nSam followed her down the passage. She walked straight-backed, head held high. She turned into a room and he followed.\n\nIt was a beautiful room. Big French windows looked on to the garden. A green thick carpet which seemed to make it harder to see where the garden began or ended. A stone fireplace with a roaring log fire and fat comfortable-looking chairs. He could see no television or even radio. Just hundreds of books on shelves and an artist's easel standing by the window.\n\n'Do sit down,' she said, making a gesture to a chair. 'Last night when Miss Brice telephoned me I couldn't remember much. But I've thought it over and a little more has come back to me.'\n\n'Where is she?' Sam leaned forward in his chair.\n\n'I couldn't say now,' she said haughtily. 'I suggest you just listen to me and I'll tell you all I know.'\n\nThe way she barked out her orders reminded Sam of a woman he once worked for in New Orleans. She was German with an English husband, but she never let him get a word in edgewise either.\n\n'I did take a child away from foster parents. It was in 1946 and the child was around twenty months. She had been in Billericay war orphans home, and she was coloured.'\n\n'What was her name?'\n\n'I can't recall her Christian name, though I do remember it was the only thing which seemed to really belong to her. She took the surname Barlow from the people who fostered her. Whether she was abandoned or the same child as was found in Hughes Mansions I can't say for certain. I wasn't working in England during the war.'\n\nSam frowned. Miss Hammick read his mind.\n\n'It's no good you looking like that,' she said tersely. 'England and particularly the East End took a frightful hammering during the war. I know you Americans think you did it all, but I know better. People were buried hurriedly, records weren't kept that well. We had to conceal disasters to keep up morale.'\n\nSam nodded. 'I was here ma'am, I do understand, it's just painful to think a baby gets stuck in a home without anyone knowing how or why.'\n\n'If wouldn't have happened if I'd been there then,' she said with more than a hint of pride.\n\n'You say you took her from foster parents? To where and why?' Was she too old to remember clearly? Was this another wild goose chase?\n\n'She was too much of a handful for them,' Miss Hammick sniffed disdainfully. 'I can vouch for that in the one day I had her in my car. She never stopped moving.'\n\n'What did she look like?'\n\nAgain that strange haughty look.\n\n'Brown of course. With black curly hair. Half-caste I'd say,' she withered him with one glance. 'Anyway, I had an awful job placing her. No one knew anything about the child. Everywhere was full. I ended up at St Joseph's in Grove Park.'\n\n'Where's that?'\n\n'South London,' Miss Hammick sniffed again. 'Unfortunately that home has closed since. Not before time too as I believe, most of the nuns were totally unsuitable for child care.'\n\nSam blanched. If a woman like her thought someone was unsuitable they must have been fiends. This woman showed no emotion at all. Not even real curiosity.\n\n'How?' he had to ask.\n\n'Children were beaten. I dare say some of them merited it. Appalling diet, little medical attention. Had I known that then, of course I wouldn't have taken the child there. You have to understand there weren't the same standards in child care in those days. I made it my business to root out all those types of homes later on.'\n\n'Did you see her again after that day?'\n\n'No, I never did,' Miss Hammick spoke thoughtfully. 'When it was closed I believe most of the girls left were sent to Dr Barnardo's. But of course this child would have been thirteen or fourteen then. She might have been fostered out.'\n\n'Is there any way I could find out?'\n\n'Well,' she paused, as if she knew something but wasn't sure if she should reveal it. 'I rang Downham's children's department this morning. They told me all the names of girls taken over by Barnardo's, but they had no record of a girl called Barlow.'\n\n'Oh,' Sam's face fell. He wondered how a society could lose a child so easily.\n\n'But I did discover the whereabouts of one of the younger nuns from St Joseph's. In fact she was there the night I handed over the baby. She is with a convent in Hampstead.'\n\nAll these names of places were confusing Sam. Was Hampstead in London?\n\nHe looked blankly at Miss Hammick.\n\nNuns were something he feared, he wasn't sure why. Perhaps it had something to do with one who'd beaten him as a boy.\n\n'It's not a closed order or anything like that,' she said briskly. 'An unmarried mothers' home. From what I understand about Sister Mary you'll find her helpful and kind-hearted. It was she alone who kept any kind of standard in St Joseph's. She's still young enough to remember the details I may have forgotten.'\n\n'Can I go there now?' he said.\n\n'No, that wouldn't be prudent,' she said, pursing her lips. 'Write to Sister Mary giving her all the information I have given you. She'll arrange to meet you somewhere away from there.'\n\nShe paused again. 'Of course, it's very likely this child isn't yours. There is an even greater possibility that Sister Mary knows nothing of the girl's whereabouts. There were a great many war orphans. I don't know the percentage of coloured ones. But she wasn't the only one. Before you go claiming her, remember this. I'd hate to think I started something that ultimately brought disappointment.'\n\n'I'm very grateful to you ma'am,' Sam said. He knew the interview was terminated. He was even amused that she thought it unseemly for him to go to a home for unmarried mothers. Was she protecting him, or the girls?\n\n'We only call the Queen \"Ma'am\",' she said, rising to her feet. 'Miss Hammick is sufficient.'\n\nShe picked up a piece of paper from her writing desk.\n\n'The address in Hampstead,' she said. 'I'd be pleased to hear the outcome of this quest. For both your sakes I hope it will have a happy ending.'\n\nIt was torture waiting for Sister Mary to reply. Should he have suggested a meeting place, told her more about his background? Would he find his child was one who'd been ill-treated?\n\nOn Friday morning at nine, he stumbled down the stairs hopefully, as he had each morning for four days as soon as he heard the click of the letterbox.\n\nEndless bills for other tenants, an air letter for the woman below him. But amongst them was a blue one addressed to him. Thin, spidery writing that could only be Sister Mary, he tore it open and pulled out the single sheet.\n\n'Dear Mr Cameron,\n\nThank you for your letter. I do remember the child you spoke of and I would be happy to meet you to discuss this further.\n\nI usually go to Hampstead village on Saturday afternoons for some shopping. We could meet in the Half Moon tea shop up near the Heath. Shall we say at three thirty? I look forward to meeting you.\n\nYours sincerely,\n\nSister Mary\n\nIt was a temptation to go out and get roaring drunk, but reason got the better of him. Better to phone Miss Brice and Clive to share his news, than to risk a hangover when he met the nun.\n\nShe was already in the tea shop when he got there. A slight, dark little figure sitting at a table by the window, her eyes downcast in front of her.\n\n'Sister Mary?' he said, feeling too large and clumsy for such a small, quaint place.\n\nDelicate pink and white walls, dainty lace curtains on a brass rail. Snowy white cloths and a glass cabinet at the back of the shop piled high with home-made cakes.\n\n'Yes, Mr Cameron,' she looked up at him and took his outstretched hand. Hers was tiny, her eyes a bright blue set in a pink and white face which matched the decor.\n\n'I imagined you old,' he said awkwardly.\n\n'I am,' she laughed, like a trickle of water over pebbles. 'The Lord saw fit to give me a youthful face. But do sit down, I took the liberty of ordering tea.'\n\nHe could see at closer inspection she was well over fifty. Tiny lines round her eyes and mouth and her hands were veined and reddened from rough work. Her starched wimple gave her an ethereal look, enhanced by the lovely eyes.\n\n'I don't know where to begin,' he said. 'Tell me everything you know, quickly.'\n\nHer soft, small mouth curved into a smile. She said nothing, just looked at his face.\n\n'Well?' Sam felt a blush creeping up his neck, but still she studied him.\n\n'Just the way you are in such a hurry makes me sure we have the right child,' she said. 'But describe her mother please.'\n\nSam hadn't expected this. Once he'd had a photograph of her, but Ellie had destroyed it.\n\n'Small, slender,' he said. 'Dark shiny hair and eyes. Her face was kinda heart-shaped. You know, with one of those little pointy chins. She had long legs and a tiny waist. And a dimple here,' he pointed to his right cheek.\n\n'The chin and the dimple are enough,' she said. 'The rest I can see in your own face.'\n\nA sharp pain stabbed at Sam's heart. He hadn't felt anything like that since losing Katy.\n\n'You mean you believe she is my child?'\n\n'I do indeed, Mr Cameron,' she smiled gravely. 'I was reminded the moment you sprang in the door. What you have told me about Katy fills in the missing part.'\n\n'What was she called?'\n\n'I can't tell you that for a moment,' she said. 'First you must tell me exactly what happened between you, and your circumstances now.'\n\nIt was irritating to have to explain when and how they met. Even more distressing to explain their parting and his subsequent reasons for thinking she had changed her mind. He spoke of his marriage, his other two children and all the time Sister Mary listened carefully.\n\nWhen he'd finished, she poured the tea and passed a plate of hot buttered crumpets.\n\n'Do you know where she is now?'\n\nSam had this terrible feeling the woman was going to give him a name but nothing more. He hadn't much time left now. How many years did it take to find a missing person?\n\n'I know who she is,' Sister Mary said dropping her eyes from his. 'And it's because of this I feel I have to tread with caution.'\n\nA dozen different things ran through his mind. Was she in trouble? Had she been taken by someone who'd brought her up as their own?\n\n'So bad, huh?' he said, sipping his tea.\n\n'Oh no,' she smiled and shook her head. 'Your daughter is everything you could want. Beautiful, talented. But let me explain first.'\n\nSam listened to the story about how this woman called Celia Anderson took her away.\n\n'I have to admit I loved that child more than any other, even if I should feel ashamed for admitting favouritism.' She smiled as if the memory was very dear to her. 'I missed her so much. She was on my mind constantly. But I was happy because she was. Sometime after her fifteenth birthday I went to Blackheath on an errand, and taking a welcome chance to see her again I stopped off at her house.'\n\n'Did you see her?'\n\n'No, but what I did see worried me greatly. I'd never met her foster father until that day. He came to the door wrapped in a blanket. He looked ill, and he had been drinking. I asked for Mrs Anderson and he snapped at me. He said she had left him. I asked about the girl and he flew into a rage. He pulled up his shirt and showed me a fearful scar. He claimed she had stabbed him, then run away.'\n\nSam just stared. Nothing had prepared him for this.\n\n'I met Mrs Anderson on several occasions,' Sister Mary said gravely. 'She was a fine, honourable woman and I knew if she'd left that house, it was in disgust at something he had done to the girl. I believed at that time they were together somewhere. I tried to find Celia myself later on when I had given the matter more thought. Only then did I discover she'd left her job to search for the girl.'\n\n'Why do you keep calling her the girl?' Sam said gently.\n\n'Because her name is a household word now, and I have to be sure you can handle a reunion properly.'\n\n'I don't know what you mean.'\n\n'Your daughter changed her surname. She has never given anyone details of her background. Her foster father had a knife wound in here,' she touched her stomach. 'I'm not a worldly woman Mr Cameron. Yet I managed to reach an understanding of what happened in that house.'\n\nSam's eyes shot open wide.\n\n'You mean,' he covered his eyes with his hand.\n\n'Yes,' she said, softly lowering hers. 'I found out from the police that an incident had taken place.' She blushed furiously, her hand reaching for her rosary. 'No charges were ever made because she refused to speak about it. Mr Anderson claimed it was someone else and that she attacked him out of spite when he chastised her. Days later she ran away and was never found.'\n\n'But you have found her?'\n\n'I know who she is now, and follow her life with love, admiration and a sense of guilt that I was party to a child being placed with such a man. I believe she kept quiet in a misplaced sense of honour. Not for him so much, but her mother, and the boyfriend she had at the time.'\n\n'The b \u2013' Sam stopped short, his early upbringing reminding him he couldn't say words like that in front of a nun. 'I'll swing for him.'\n\n'You'll do no such thing,' she said firmly. 'I want you to be rational about this. Your daughter has kept that secret for six years. All her life she has thought she was an abandoned child. She doesn't even know which of her parents was black. I believe the shock of hearing you are her father could unbalance her. Fathers to her may still mean pain.'\n\n'What would you have me do?' He felt humbled by this little woman's understanding and strength of character.\n\n'You are well placed to get close to her,' she said. 'Try to get closer still and win her confidence. Be there for her, because she is surrounded by people who may not have her best interests at heart. God will guide you then.'\n\nShe picked up her rosary and held it between her worn red hands as if gaining strength from it.\n\n'Your daughter is Georgia James.'\n\nFor a moment the room spun round. The girl he had seen at the airport. That beautiful mulatto with the voice of an angel.\n\n'Are you absolutely sure of this?' he whispered. Tears were welling up in his eyes, he brushed them away almost angrily.\n\n'Look,' she said gently, pulling out of her habit a worn old photograph of a child with short curly hair.\n\n'This was my Georgia. I loved her as if I was her own mother. Can you tell me I'm mistaken now? She kept her real Christian name too. She dropped the Anderson because of its connections, but she couldn't change the name her mother gave her. She sang to me so often as a little girl. I recognised her voice on the radio, even without a picture of this new star. I knew her from the voice Sam, but her face confirms it.'\n\nSam took the picture in his hands. She was so thin, yet he could see Katy, and even Jasmine in the face before him.\n\n'Celia Anderson sent this one to me with a Christmas card.' Another picture appeared in her hands. 'She was almost fifteen then.'\n\nSam gasped. The face before him now was a dark-skinned version of Katy's. Her hair was past her shoulders, the dimple in her cheek, the heart-shaped face, all served to confirm her parentage. But her eyes were his, and the chiselled cheek-bones, like looking in a mirror and seeing his own reflection.\n\nSo many times he'd stopped to look at this same face in the record shop windows, idly wondering if she would get back to sit in at Ronnie Scott's before he left to go home. He'd listened to her records on the radio, even written to his kids and suggested they buy her latest album. Now this sweet little nun was telling him it was his daughter!\n\n'She's in the States now,' he whispered. 'I may have to go back. I've only got a contract for six weeks.'\n\n'I checked on you, too,' she smiled. 'You've been getting some startling reviews yourself. Surely you can engineer something to keep you here till she gets back?'\n\n'But how do I get to see her?' he whispered. 'No one gets near her from what I've read.'\n\n'I believe in good coming out of evil,' she said simply. 'Something made you come to England, then you went to check in the East End. Did you plan to do that before you came?'\n\n'No,' he said. 'But \u2013'\n\nShe stopped him with a touch on the hand. 'The hand of God,' she said. 'That same hand will bring her to you. Pray for his guidance when it happens.'\n\nSam had Georgia's address in his hands now. His daughter had come to him, just like Sister Mary had said. If he was a man who truly believed in the power of prayer he would get down on his knees and thank Him. But right now he was going to go back on the stage, get her to come up and do a number or two with the band.\n\n'Georgia,' he hummed the tune as he turned to go back into the dressing room. Sam remembered now, that was the number he'd been playing when he got his first glimpse of Katy back at Lakenheath.\n\n## Chapter 23\n\n'Fever, fever when you kiss me, fever when you hold me tight.' Georgia stood under the shower, singing at the top of her lungs, wet corkscrew curls like seaweed over her slender brown back.\n\nSix weeks ago, making the album of her dreams looked impossible, but at last all the problems had been overcome and today they were starting recording.\n\nMarch's weather had mirrored what was going on around her. Meeting Sam was like an early spring day. So much promise of good things to come, an unfolding of new leaves and flowers, unexpected warmth and sunshine. Max roared in like the March wind, laying waste all her plans. The press was Jack Frost, nipping at tender shoots, threatening to kill everything.\n\nRows, bad feeling, criticism. Roots put down years earlier, torn up. So much opposition to something she knew was right.\n\nSpeedy and Les were heavily into drugs and behaving like a pair of deprived, vacant louts. Norman sniping at his lost opportunities. Max turning into a demented, jealous old woman. Good pianists seemed extinct, even the press turned against her. There were times when she almost backed down.\n\n'If it hadn't been for Sam,' she said to herself as she stepped out of the shower, wrapping herself in a towel. 'You'd have cracked up.'\n\n##### *\n\nThe night in Ronnie Scott's was the beginning. Up until then the album had been a hazy dream. When Sam asked her up to sing with the band in the second half she felt a little presumptuous singing the old Billie Holliday number, 'That ole Devil called Love'. Yet she found her voice had the maturity and Sam's horn inspired her. Even though Max sat glowering at her from the audience, she didn't care.\n\nShe expected Sam to play hard to get when he came round to see her the next day. With the rave reviews he was getting, he could afford to take his time and be choosy before committing himself to any band or project. But instead she found him enthusiastic, open and straightforward.\n\n'I'm yours if you want me,' his soft dark eyes glimmered with an excitement she hadn't expected. 'My contract runs out at Scott's next week. As long as you can get my visa fixed up and pay me enough to send home for my kids, then I can stay for as long as it takes.'\n\nMax's attitude was quite the opposite when she called at his office later the same day and outlined her plans.\n\n'How dare you go behind my back and make arrangements? I haven't even agreed to this album,' he snapped. 'You know nothing about that guy and I suppose you rushed in there offering him the moon.'\n\n'Not the moon,' she said simply. 'I just told him I wanted to do a recording with him. The only credentials I care about is how well he plays his horn.'\n\n'Decca won't want to waste their time and money on an album like this,' he roared at her, purple in the face with anger. 'You're digging your own grave Georgia, it's vanity, nothing more. Stick to what you do best.'\n\n'There's nothing more boring than an entertainer who never moves on,' she shouted back at him. 'Making this album doesn't mean I won't make any more soul or rock records. It's just stretching myself, showing a new dimension. I could reach millions of new fans.'\n\n'The press will link your name with his,' Max snarled at her across his desk. 'Do you really think the public will be happy to see their golden girl with a big Yank nigger?'\n\n'You evil bastard,' she hissed back at him. 'Trust you to bring everything down to gutter level. Call anyone a nigger again, and this uppity one will walk out on you.'\n\nOf course he gave her all the rubbish about caring for her, trying to protect her. But she had hardly left his office before he was on the phone to Jack Levy, doing his best to block her.\n\nThen the press got a whiff of what was going on, and before she could talk to the band and outline her plans they had her stitched up.\n\n'Georgia goes it alone,' was the headline. 'No time for Samson now.'\n\n'Georgia is to split with Samson after six years'. 'We've grown apart,' she was quoted as saying. 'I'm in a position now when I don't need or want the responsibility of a full-time band. I want to experiment with other musicians and expand my career.'\n\n'I didn't ever say that,' she raged to Sam. 'They're making me out to be some sort of prima donna throwing off my old friends because I've outgrown them. Where did they get hold of such an idea?'\n\n'Max?' Sam raised one eyebrow. 'He's scared, honey. He wants his little girl right under his wing. But don't take too much notice of the press. The time to worry is when they don't bother to write about you, good or bad.'\n\nIn five years there'd been many squabbles, but this time it was serious. The boys closed ranks, refusing to speak to her on the telephone, ignoring even a letter she sent them explaining her plans.\n\nRod and John came round after Deirdre from the office intervened and admitted she'd overheard Max talking to someone from the press. But Les, Speedy and Norman chose to use the opportunity to make a final break from her.\n\nThere were moments she doubted her own judgement. Was it just inflated ego that made her think she could compete with singers like Aretha Franklin and Ella Fitzgerald? What if Jack Levy, Max and all the other men who had been in the business for years, were right? Suppose it was a flop, what would she do then?\n\nBut the doubts were gone now. She had swept away all the opposition to her plans. Fly or fall she was getting her chance, and she had no intention of falling.\n\nSam finally found a pianist in a pub in Barnes. He was a retired music teacher who played just twice a week in a jazz quartet. Harold Sweeting looked like everyone's favourite uncle. White-haired, rosy-cheeked, a jolly, roly-poly character with all the enthusiasm for music so many of the professional pianists they auditioned, lacked. He had wanted to be a concert pianist in his youth, but his wife and children had come before his own hopes and dreams. Now at sixty-five he had a lifetime of experience to fall back on, yet with a youthful exuberance that gave his playing a touch of magic.\n\nRod and John were joining the other session musicians too. Rod simply because he was the best drummer, and John begged to come in because he admired Sam's playing.\n\nPerhaps it was fortunate that Speedy, Les and Norman had refused to join her. It left her free to employ strings and a classical guitarist, without feeling guilty.\n\nAll they needed now was luck!\n\nGeorgia smiled ruefully as she pulled on her jeans and dragged a red sweater over her head.\n\nJack Levy was cunning. He had covered himself every which way. If she didn't finish recording within a week she would be in breach of contract and he could sue her. Like Max he was running scared, knowing her agreement with Decca ran out in a few months' time. It was thinly disguised blackmail, to make sure she wasn't tempted to sign up with another company offering her a better deal. If the recording was ready in time and then went on to be a smash hit, Jack would merely laugh all the way to the bank. If it failed however, he would blame her for going against his wishes. Before long he would having her back, recording another purely commercial record.\n\nLoyalty in this business was bought. No one really cared about talent. Half the singers who got into the top twenty were virtually manufactured, money changed hands to get them air time. Glossy promoters used hype to get their puppets noticed, and later when these one hit wonders no longer made them money, they were forgotten.\n\nThe music world was a giant ants' nest, everyone relying on the Queen to keep it fuelled. Just one slip from favour and she would be devoured by the drones and replaced.\n\nGeorgia sat down at her dressing table to dry her hair, pausing to look at a photograph of her and Samson.\n\nIt was her favourite one, taken just a few days before Ian and Alan died in the fire. The photographer James Ogilvy had been an unknown then, a pimply-faced weed who followed stars around looking for the picture that would make his fortune. He'd been after Adam Faith that day, not the support group, but he'd taken this one while he hung around waiting.\n\nThey were in a park close to the theatre, fooling around on a climbing frame. Rod sat right at the top, the other boys made a pyramid shape and she was in the middle hanging upside down over a rail, her hair hanging down to the floor. The boys looked funny now with their short hair cuts and wide shouldered jackets, and she looked positively ridiculous in that shapeless shift dress with a long pointed collar. It had taken so long to arrange herself so the dress stayed up over her knees.\n\n'What would you think of the boys now?' she said to Ian. He was laughing, his eyes crinkled up. Rod had just farted very noisily from his perch on the top and John was complaining it scorched his head. All eight of them had been so na\u00efve then, trusting children who would put up with anything for a few words of praise. That day they had no idea Ian and Alan's time with them was nearly at an end, or that fame would follow so soon after.\n\n'Would you stand by and watch Speedy and Les destroy themselves with drugs?' she asked.\n\nShe sighed and switched on the dryer. All of them, including herself, had experimented. It was as much part of the scene as groupies and drink. Purple hearts now and then to liven up the trip home or go on to a party. Smoking cannabis in the coach to relieve the boredom. But Speedy and Les had taken it a stage further since that holiday in Spain. Amphetamines to get them going, endless joints to calm them down, then out of their heads on cocaine half the night.\n\nIt was almost predictable that Les should be attracted to drugs, he was dim and he didn't have a great deal of personality. But why Speedy? He was the one with a real mind. A brilliant guitarist, handsome, charismatic and caring. What had made him prefer spending days spaced out, screwing every girl who passed his way, and every night in a West End club?\n\n'We've all changed,' she muttered. 'Perhaps the press are right. I am getting ruthless.'\n\nAll that fighting to keep them together and now fame and money were changing each one of them. Rod keeping up appearances as a rock star, strutting around town with hair down to his shoulders, in tight red trousers and embroidered jackets, spending more money in a day than he'd once earned in two years. John had lost his warm humour. Now he was full of astrology, meditation and every other half-chewed-over theory he happened to overhear. Norman had new friends now, society types who invited him for weekends in the country. Away from his old friends he could forget his East End origins, describe himself as a composer. He'd even been taking elocution lessons.\n\nEven Max was unbalanced. The man she had fought with, been in awe of, and perhaps even loved had been tough and unyielding. But he'd been consistent. Now he was riddled with jealousy about Sam, terrified she would dump him. One day depressed, another fanatical. Spiteful then solicitous. He took pills for an ulcer, handfuls of vitamins to allay his advancing years and surrounded himself with dolly birds who were only interested in his money.\n\nSam alone remained constant. A cool breeze on a hot day. A log fire when it was cold. He could talk about anything and everything. Laugh, make fun of her, but listen when she wanted to talk.\n\nHe'd filled her life all right. But not in the grubby way Max thought.\n\nThere had been moments that night in Ronnie Scott's that it seemed like the start of a love affair. She found herself gawping at him, and found him staring right back at her.\n\nShe loved the way he looked. From the close cropped hair, the golden brown skin and doleful eyes, to his broad shoulders and narrow hips. He was a dream of a man, but not in that way.\n\nAs the days ticked past, mutual admiration turned to close friendship. He was unmaterialistic, laughed at show business hype, demanded nothing of her. One morning he could turn up at her flat with a pile of secondhand jazz records for her to listen to. The next he was out in her kitchen cooking them a meal and insisting they went out later to explore some tourist place he hadn't seen. Unpredictable, serious, funny, affectionate and cool in turn. He told her about women he fancied, his ex-wife and his children. His past gradually unfurled in a series of hilarious stories that left her hungry for more.\n\nFrom G.I. to barman, truck driver to rat-catcher, he painted pictures so vivid she could see them.\n\n'You didn't kill rats? You're making it up,' she laughed as yet another talent came to light.\n\n'I did,' he insisted. 'Used to go round in a little van putting down poison, then round the next day to heave out the carcasses. Sometimes I even did gigs with a few bodies in the back. None of my buddies would get in it with me.'\n\nHe made light of everything. He presented his childhood as if it had all been running barefoot through meadows, fishing and climbing trees. But as she got closer to him, she guessed it had been tough.\n\nHe spoke of the racism in the States almost as if it was a joke. No trace of self-pity, or even bitterness, only sympathy for those who were trapped by it.\n\n'We're the lucky ones, honey,' he said. 'Up there on the stage people don't think about our colour, they only hear the music. Maybe by the time our kids are grown every black person will be valued for themselves.'\n\n'But how can you go back to it?' she asked. 'How can you bear Jasmine and Junior to grow up under that shadow?'\n\n'I hope I don't have to,' he said simply. 'I'd like to bring them here. Send them to good schools. England's a cool country.'\n\nShe wanted to give him the money to send for them right now. The thought of two children without either parent saddened her. She could see herself back in St Joseph's waiting while people came and looked her over, bypassing her, looking for the small, sweet blonde. It wasn't right for two children to have a father like Sam and not be with him.\n\nBut Sam was a proud man. He wanted to bring those children to a real home. But for him there would be no short cuts.\n\nSurrounded as she was by fawning sycophants his earthy opinions counted.\n\n'You don't have to worry about other people,' he said, when she told him her fears about the boys. 'They're grown men now. Be there for them, but don't try to hold them. You do what you know is right, and if for a while they fall off the path, then let them find their way back on to it.'\n\nTwo days of recording and everything was coming together. Perhaps the opposition to the album had made everyone stretch themselves just a little more. Sam had written all the brass arrangements and he and Steven the producer were at one in their ideas. Four tracks were already finished, another five well under way.\n\nSession men and the technicians had frightened Georgia once, but now she understood how it all worked there was no need for fear. Each one had their role and Steven put it all together.\n\nHarold was perfect. Every note he played sounded like a love affair. He could improvise like a jazz player, yet his classical background and knowledge of music was unsurpassed. His patience and humour made him a joy to work with.\n\n'Come and sing with me here,' he said, drawing up another stool for Georgia by the piano. 'Just relax and feel the music.'\n\nIt was like being ten again, joining Celia at her piano with the sun streaming in through the French windows. His slim long fingers danced over the keys, his white head nodding with the beat, the warmth of his rotund body, his encouraging smiles, dispelled any nervousness. She forgot it was a glassed-in studio way below the street, the pressure of getting it all tied up in a week, or the session men who just wanted to play their bits and go home. Harold with his hand-knitted yellow waistcoat, with red and white cravat and his huge stomach bulging over his thighs was an inspiration.\n\nOn the morning of the third day they began at six. The offices upstairs silent, typewriters covered, chairs tucked under desks. The night porter unlocked the doors for them, rubbing his eyes and yawning.\n\n'What a time to start,' he grumbled. 'I don't know how we'll get the studio cleaned. I suppose you'll still be here when I come back on tonight?'\n\n'Sure thing,' Georgia tickled him under the chin. 'Be a darling and put on the coffee?'\n\nBoth John and Rod were better for being away from the rest of the band. Rod had dropped his cocky stance, knowing the session men, Sam and Harold had not only years on him, but a far greater knowledge of music. John hadn't mentioned meditation once so far, he was engrossed in the sound, playing far better than she'd ever heard him before. Neither of them had complained about anything, not the early starts, or Steven's continual re-takes.\n\nShe had come to trust Stephen Albright implicitly. Ever since that first recording he had produced all her records. Like her he moved with the times, never falling into the trap of making each record sound the same. He was plumper now, so his height seemed less remarkable. A half-chewed pencil, eyes tightly closed behind his thick glasses were still his trademark. But his public school speech was peppered with cockney slang, scruffy clothes replaced by designer chic, he drove a Ferrari and lived in a penthouse in Mayfair, but he was still as uncompromising about music.\n\nGeorgia was halfway through 'Summer Time' when she saw Max's face pressed against the porthole in the studio door.\n\nIt was just after nine, and Max never normally surfaced before noon. Just one look at his bloated, angry face and the way he pummelled the glass with his fists, was enough to know something serious had happened.\n\n'Take five, Georgia,' Stephen's voice came through her headphones from the control room. 'Max is in a paddy about something.'\n\nGeorgia sighed, taking off her earphones. As she opened the studio door Max lunged forward.\n\n'Max wait. You can't go in there,' Stephen shouted behind him, clutching at Max's arm. He was too slight to create a real barrier between the bull-like man and Georgia, but he did his best.\n\n'What is it?' she asked, her face furrowed with irritation. 'Can't we even get on with this without interruptions?'\n\n'Interruption?' he roared. 'You're finished my girl, never mind interrupted.'\n\nGeorgia just stared. She had seen Max fighting mad many a time, but never quite like this. Black stubble on his chin, the shirt under his sweater looked suspiciously like pyjamas and his trousers could have been slept in. But it was his face that really unnerved her.\n\nIt was purple. Veins stood out on his forehead like ropes and he had dried spittle round his lips. Eyes blazing like a man about to kill someone.\n\n'Calm down,' she gingerly touched his arm. 'What's the matter?'\n\nShe was aware all the technicians were at the door of the control room, and behind her she could feel Rod, John and Sam. The silence from the session musicians proved they were all listening, still in their seats, their music open in front of them on stands.\n\nMax flicked her hand from his arm and pulled a newspaper from his back pocket.\n\n'This,' he almost slapped her with it. 'Why didn't you tell me?'\n\n'Calm down,' she snapped back at him. 'Why didn't I tell you what?'\n\n'That you were wanted for attempted murder!'\n\nEver since her first publicity she had half expected someone to confront her with her past. So many times she had intended to take Max aside to prepare him. But even in her worst nightmares she hadn't anticipated this.\n\nThe paper felt almost hot in her hands. The face of the man she had learned to forget, staring up at her.\n\nThe studio lights seemed to burn her, the walls buckled and moved.\n\n'Tell me it isn't true?' Max's plea seemed to come from a long way off. 'It's some sort of sick joke? It's made up?'\n\nThere was a roaring sound in her ears. She could feel the handle of the knife in her hand, see blood dripping from the blade and he was lying at her feet, fingers clutching at the wound in his white belly.\n\nSam instinctively knew what was in the paper, even before he got a glimpse of the headline.\n\nElbowing his way through the crowd in the doorway, he saw the colour drain from her face. She swayed, then crumpled to the floor.\n\nThe air was charged with emotion. Max's anger. Rod and John's shock. Stephen's eyes behind his thick glasses blinking with astonishment. The curiosity of the session men and technicians.\n\n'Out the way,' Sam reached Georgia in two strides. He knelt down beside her, stroking back her hair from her face. 'Get some water, damn it,' he yelled at Max.\n\nIt was Rod who ran for the water. Stephen found some smelling salts and rushed over with them. The rest of the men stood in a semi-circle around them, too shocked to speak.\n\n'All right honey,' Sam crooned softly, as he saw her eyes flicker open. 'I'm with you, baby.' He cradled her head in his arm, holding the glass to her lips.\n\n'I knew there was something about you,' Max's voice penetrated the hushed room. 'Why the fuck didn't you tell me?'\n\n'If you don't shut your mouth I'll shut it permanently,' Sam glowered up at Max. 'Are you so God damned dense you can't see the girl's in shock? Stop thinking about yourself and think of her.'\n\nHe rose, lifting Georgia in his arms as if she weighed nothing more than a few pounds.\n\n'Take me home Sam,' Georgia said weakly. 'I'll tell you everything then.'\n\n'It's me you should be telling,' Max barked at her. 'I've looked after you all these years, then you shut me out in favour of him.'\n\n'You bastard!' Sam's lips curled back showing his teeth. 'Call yourself a human being? I've known roaches with more sensitivity than you.'\n\nIt was then Sam knew Max was in love with Georgia. It wafted out of him, mixed with jealousy, suspicion and fear.\n\n'You can come with us,' Sam shot a warning glance at Max, holding Georgia tightly to his chest. 'But any more outbursts and I'll wring your neck.'\n\nIt was raining hard. Still holding Georgia in his arms, Sam made a dash for Max's white Rolls Royce parked outside. He lay her on the back seat, tucking a travel rug round her. A crowd of people making their way to work paused to watch under umbrellas, mouths gaping with surprise.\n\n'Get going,' Sam leapt into the front seat beside Max.\n\nThe traffic was jammed solid on Oxford Street. Max's face was grim, he held the steering wheel so tightly his knuckles shone white.\n\nFor ten minutes no one spoke.\n\nGeorgia broke the silence first.\n\n'Can I read what they've said about me?' Her hand came snaking between the front seats to reach for the paper.\n\nSam turned in his seat slightly to watch her. She knelt on the car floor, the paper spread out on the seat in front of her. She looked no older than sixteen in her tight little black mini-dress and long boots, hair falling over her face.\n\nThe headline read, 'Georgia wanted for attempted murder.'\n\nSam watched as she read the front page, shaking her head as though in disbelief, then turned the page, shuddering as she saw more pictures.\n\n'Bad, huh?' Sam said as she sat back on the seat and silently handed him the paper. Tears were welling in her eyes, threatening to spill over and trickle down those pale, drawn cheeks.\n\n'The worst,' she whispered, her full lips quivering. 'He's been watching me all this time.'\n\nSam didn't trust himself to speak. He had to read it. Pretend he knew less than Max. But inside he wanted to hold her, tell her this man was never her father. That he was the man who counted and he would take care of all this.\n\n'Mr Brian Anderson, a sick old man living in slum-like conditions reveals that the famous singer Georgia James is none other than the foster child who tried to stab him to death several years ago.\n\n'Yesterday in the offices of the _Mirror_ , Anderson related a story that makes a lie of everything the public believes of Georgia.\n\n'Brian Anderson and his wife Celia took Georgia from an orphanage when she was nine, to live in their beautiful home in Blackheath. They showered her with love, gave her singing and dancing lessons, bought her all the toys and clothes she'd never had.\n\n'The Andersons were liberal people. On her fifteenth birthday they allowed her to have a teenage party up in her old playroom and it was only when it got out of hand that Mr Anderson went upstairs to intervene.\n\n'He found his daughter in the throes of making love and indications that the party was nothing more than an excuse for wild, promiscuous behaviour. In his anger Mr Anderson, like any irate father, ordered her friends out the house.\n\n'Another girl would have felt shame. But not Georgia, instead she screamed abuse, told him she could do what she liked and then picked up a bread knife left on the party table and stabbed him in the stomach.\n\n'As he lay close to death in hospital and the police were waiting for him to recover consciousness, Georgia ran away rather than face prosecution. She left no apology for the couple who had given her everything, instead she took all the money left in the house, and disappeared without trace. Every effort of the police to find her was unsuccessful. The Andersons spent all their savings trying to trace her which culminated in Celia having a breakdown, and Brian losing his job as a bank manager.\n\n'While her fans have been listening to her songs of love and crediting her with sweetness, purity and sincerity, her father's life is in ruins. While she amasses great fortunes, he lives in squalor in Ladbroke Grove. Celia's heart was broken, her career and home lost and finally the long, happy marriage was over too.'\n\n'\"I still love her, even after everything she put us through,\" Brian Anderson wept as he told his story. \"I've collected every press clipping of her, every photograph. I always believed one day she would come back to me and say she was sorry.\"\n\n'Anderson's plight is a sad one. He is trapped in a vicious circle of sickness, poverty and appalling housing. He carries a childhood picture of her in his pocket. He watches her television appearances through a shop window as he cannot afford a tv of his own.'\n\nThere were four photographs. Georgia sitting between her foster parents at a Christmas dinner. One of her in a swimsuit taken on holiday. The house in Blackheath, and finally one of the man as he was now, old, broken and sick, standing in front of a dilapidated house.\n\nSam felt sick as he read it. It had a ring of truth about it. If he hadn't heard Sister Mary's story and got so close to Georgia in the past weeks, he might even have believed it himself. But he wasn't concerned now with what the public thought, only what it would do to Georgia. A snakepit she would have to go back into, just to clear herself.\n\nThe car stopped at traffic lights. Across the road Sam saw a newspaper stand.\n\n'Queen of Pop, wanted for attempted murder,' he read, and once again felt faint.\n\n'My God,' Max exploded once again as he too saw it. His thick, bull neck flushed purple with rage. He turned his head to Georgia, eyes blazing with renewed anger. 'I take you on, I feed you, train you and you think so little of me you don't let on about this.'\n\n'And you think so little of Georgia you prefer to believe this trash?' Sam snapped at Max with barely disguised disgust. 'I haven't known her as long as you have, but I recognize this for what it is.'\n\n'I don't want to believe it,' Max wiped his hand across his eyes. 'But when that plopped down on my mat this morning, I thought the world had ended.'\n\nThe Thames was black. Even the Albert Bridge looked forlorn. Across the river Battersea power station's four chimneys pumped out sulphurous fumes.\n\n'Oh shit,' Max exploded again as he drew close to Georgia's apartment block. 'Look at that lot!'\n\nThe pavements and the courtyard were blocked with an army of people. Reporters and photographers swarmed in groups despite the rain. Men in raincoats, women in high-heels and smart suits huddled under umbrellas. A television van was parked by the entrance, the doors open and men jumping out with film equipment. Older people paused to watch from the river side of the street. Youngsters took up vantage points on walls and railings.\n\n'We can't go in,' Max said. 'Let's go to a hotel?'\n\n'No,' Sam put a restraining hand on the steering wheel. 'We can't run from this. Besides, Georgia needs to be in her own home. We'll just barge on through them.'\n\nAs Max stopped in the middle of the road to make a right turn, so the crowd surged forward.\n\n'Keep down,' Sam urged Georgia. 'Put the rug over you.'\n\nBut the white Rolls was too obvious. People were running towards them, regardless of traffic.\n\n'Barge your way in,' Sam snapped. 'Go on!'\n\nSlowly the car inched forward, faces lunged at the windows, hands trying the doors. Max's breath was rasping as if he was having a seizure, but still he drove on, forcing the people to step aside.\n\nSam leapt out like a panther. He flung open the back door and reached for Georgia.\n\nA roar went up the moment they saw her. They ran across the gardens, leaping over fences. Like hounds after the fox.\n\nSam held his ground, tucked his arm firmly around Georgia, brushing away reporters as if they were flies. Striding across the courtyard his face set like a bronze sculpture, almost carrying her. Max came rushing after them, his wheezy breath audible even over the shouting.\n\n'Would you like to comment on the story about you in today's _Mirror_?' A young man with dark hair and an earnest face bounded up, running alongside them, quickly joined by the rest of the pack.\n\n'It's a fairy story told by a very bitter man,' Georgia tried to smile, but her mouth refused to co-operate.\n\n'Is that a denial?' another reporter shouted as cameras flashed all around her.\n\n'Miss James will tell her side of the story when she's ready,' Sam snarled. 'You jackals! Clear off and hassle the sick man who sold you all that shit!'\n\nThey had reached the doors of the foyer. The porter moved forward to unlock the door. One reporter tried to get himself in.\n\n'Out!' Sam put one hand on the man's collar and lifted him bodily out of the door.\n\nInside the block with the door locked behind them it was suddenly quiet. The wide staircase curving up round the old wrought-iron liftshaft, dark green carpet and polished mahogany doors were serenely comforting.\n\n'They'll be climbing the drain pipes by tonight,' Max said as they waited for the lift to come. His eyes darted about as if expecting someone to crawl out of the woodwork. 'They'll dig and dig till you can't fart without them reporting it.'\n\n'I won't let them inside.' Johnson the porter smiled at Georgia reassuringly, blocking her view of the doors so she couldn't see the crowds beyond.\n\nHe had read the story around the time the reporters began to arrive and he didn't believe a word of it. Hadn't she always had time for him and his missus? Only last Christmas she'd given him a hamper with fifty pounds tucked in a card. Some of the tenants in this place thought a porter was less than a speck of dirt. He might be close to sixty-five, with little education but he knew a good person when he met one. 'I told them it was all rubbish. I said you was a real kind girl. That man in the picture has been here before. I saw him two or three times outside. He's a loony.'\n\n'Thank you Johnson.' Georgia touched his arm.\n\nEven though Sam could see she was startled by the porter's revelation she was sensitive enough to understand he was offering his support. 'I'il tell you the whole story soon. But until then don't go talking to anyone will you?'\n\n'You can rely on me miss,' he said, patting his green serge uniform as proudly as if he was one of the Horse Guards. 'No one will get beyond that door while I'm here.'\n\nThe three of them sat in a semi-circle. The phone lying off the hook. Three empty brandy glasses and a large pottery ashtray full of Sam and Max's cigarette butts lay on the glass coffee-table between them. Georgia, putty-faced with traces of mascara staining her cheeks, her fingers picking at an imaginary thread on her dress.\n\nSam had to close his eyes as she haltingly whispered the true story. He could feel the agony suppressed for so many years, see the scene played out before him as if he'd been hidden in the playroom.\n\nThe rain outside cast an eerie, dirty light on the vivid yellow walls, the white carpet turned grey. Daffodils in tubs on the balcony bowed their heads just as Georgia was doing. An ugliness had crept uninvited into a room once full of bright, clear colour.\n\nSam's fingers clenched into fists, hatred burning in his gut. She knew all the words now, understood desire and passion and what they could do to men. But what did she know at fifteen?\n\nHe wanted to move closer. To take her in his arms and comfort her as she described her first night in Soho and later her abortion. How could any girl survive that without permanent scars?\n\nMax sat on the edge of an armchair, still red-faced, tense and angry. He reminded Sam of his sister's pressure cooker. Shaking, hissing, any moment now he would erupt unless someone could find the right words to cool him off.\n\n'You should have told me before,' he said. 'I could have done something.'\n\n'How could she?' Sam spoke out. 'How do you tell anyone something like that? \"Say Max! I was raped when I was fifteen. I stabbed him and ran away. Make it all right for me?'\" He half smiled at Max's discomfort. 'Of course you're mad, hurtin' because she didn't confide in you. But I bet there are plenty of secrets in your life you wouldn't tell?'\n\n'If you're so fucking clever tell me what to do now then?' Max's voice was rasping and wheezy, he looked older now, tired and frightened and Sam's slow Southern drawl was making him madder.\n\n'My gut feeling is to go out and find that creep. Beat the shit out of him until he tells the truth,' Sam said.\n\n'We daren't touch him,' Max raised his head in alarm. 'We need legal advice.'\n\n'A man who could rape a child in his care wants lynching,' Sam said quietly. 'You go and get legal advice Max. But if that don't work, don't bank on me sitting quietly.'\n\nSam wanted to be alone with Georgia. She was holding back the tears. Struggling to keep a grip on herself. He ached to tell her the truth about himself, to take over this situation as a father. But how could he blurt it out now? Another shock might just unhinge her. She was tough, but not indestructible.\n\nMax could hardly bear to look at Sam. Nothing had been the same since he turned up. He had wormed his way closer in six weeks than he had managed in so many years. He still didn't know if they were lovers for sure, but if not what was it between them? If it wasn't for that arrogant shit, Georgia might be clinging to him now.\n\nHe was scared too. A dark terror that this was the end of the road. He knew now why she had frozen on him that night and he also knew she really cared for that boy Peter. How long would it be before he turned up? He wasn't the type to worry about rejection, not when the girl he loved's future was at stake! Georgia might forgive a few shady deals, but she wouldn't overlook tampering with her personal life. A girl who could stick a knife in a rapist could do anything. The best thing he could do was get out of the way, lie low and just hope it would all wash over.\n\n'Even if Georgia tells the press the true story there's gonna be lots of people who'll believe him,' Max wanted Georgia to snap at him, start a row so he could be justified in running out on her. 'People love stories like these. Maybe it would be best to just shrug it off. Just say it's rubbish and refuse to comment further?'\n\n'Go on back to your office,' Sam snarled at him. 'You ain't doin' no good here. She needs time to think it out herself.'\n\nMax stood up. His ulcer was playing him up, his head hurt and he resented Sam's attitude.\n\n'I don't want you two running off half cocked and making things worse!' he said.\n\n'How much worse can it get?' Sam's lip curled back as he looked up at Max. 'Even now they're saying she's just another no-good nigger.'\n\n'Don't Sam,' Georgia put her hand on his arm. 'Max is only being realistic.'\n\n'There's only one way out of this,' Sam said quietly, a touch of venom in his tone. 'To tell the whole truth. There's people out there ready to help if asked.' He was thinking of Sister Mary, but he couldn't say her name now.\n\n'I'm going,' Max moved towards the door. 'I don't know how you expect me to sort this one!'\n\n'You mean you can't see how you can promote rape into a money spinner,' Sam stood up, shoulders back, eyes flashing dangerously. 'Piss off Max. Don't say anything to anyone until Georgia's decided how she's going to handle it.'\n\nSam expected Max to retaliate, but instead he could smell fear. He noticed the man had turned pale, he seemed to shrink as he buttoned up his overcoat and kept his eyes to the floor.\n\n'Keep the phone off the hook,' he said weakly. 'Put it back on at nine tonight. I'll phone you then.'\n\n'Breakfast,' Sam said once he heard the lift creaking back downstairs with Max in it. 'My mother always said the brain works best on a full stomach.' He patted Georgia's shoulder. 'Eggs and ham?'\n\nBy four in the afternoon Sam was getting worried. Georgia was too silent. She had picked at food, drank every cup of coffee he put in front of her. But still she wasn't talking.\n\nShe answered his questions. She went and made a bed up for him in one of her spare bedrooms. She took a shower and washed her hair, even filed her nails. But there was no real communication.\n\nHe knew she wasn't thinking or planning. She was locked in herself. Buried in the kind of black hole he'd been in himself when he lost Katy. But telling her now that he was her father wouldn't solve anything. She had to open up that black pit where she'd buried Anderson. She had to look at it for what it was and deal with it. Telling her something good would be like putting a lid on a pan of burning fat, it might halt a fire for a while, but soon it would blow up, blasting the lid away. However much he wanted to tell her, now wasn't the time.\n\n'Talk to me?' Sam knelt on the floor in front of her. She sat with her legs curled up under her, her head on a cushion, hair still damp in tight little ringlets. 'You look like Jasmine does when she's hurtin'. 'It's breaking me up.'\n\n'I'm finished now,' she said softly. 'Overnight I'm a bad smell. Decca won't want to be involved with a scandal. My fans will hate me.'\n\n'No honey,' Sam stroked back her hair. 'For one thing people love drama. Once the dust has settled and they know the truth, they'll admire you. You didn't allow yourself to wallow in self-pity. You didn't even allow the bastard to get away with it. You marked him for life. That's rough justice, but at the end of the day that's what everyone wants.'\n\n'But I made everything far worse,' she said in a small voice. 'I should have told the police and Celia what, really happened. Don't you think I was maybe responsible for it all?'\n\nShe had dark circles around her eyes as if she hadn't slept for days and her mouth was slack and lifeless.\n\nHe could smell guilt. She'd carried it on her shoulders the way she carried everyone's burdens. It was time she put down that burden.\n\n'No honey. You were never responsible. A little girl learns about men through her father. She climbs on his lap naked. She might get in the tub with him. She flirts and fights with him, but no normal man feels that way about his child, however desirable she may become as she grows up. You can feel pity for him because he's a sad little pervert. Anger because he took your youth and innocence. But never guilt. All that is his.'\n\nShe cried then. Huge sobs that racked her slim body, soaking the cushion under her head and distorting her face. Sam just sat there on the floor beside her, stroking her hair, waiting for the poison to drain out.\n\nSlowly the sobs subsided to mere hiccups. He passed her a box of tissues and waited again.\n\n'I feel so bad, Sam,' she sat up slightly and he moved next to her, sliding his arm round her. 'I've got everything haven't I? Money, this flat, I can go anywhere, do anything. But why do I feel so empty? What have I really achieved?'\n\nYou've given people yourself. Maybe that's why you feel so drained now. Maybe it's time you said, \"Hey, this is me, not a machine. I want to have fun. Be myself.\"' Sam lifted her face up to his, kissing her swollen eyes. 'You've worried about others too long baby.'\n\n'But everyone close to me gets hurt,' she whispered. 'Celia, Peter, Helen, Ian, Rod, even Max and Brian too. Now there's you. I thought I could help you and your family. Now you are right in the middle of a scandal. We'll never get that recording released, even if they let us finish it. Although my voice is a gift, it can be a curse too. If it wasn't for that I could just be an ordinary girl.'\n\n'We all feel we've hurt people,' Sam felt a lump growing in his throat. 'Being a musician has its own kind of guilt. We follow it because we love it and sometimes it feels downright selfish.\n\n'If I'd trained as a carpenter for as long as I have at playing my horn, I'd have a masterpiece to show the world. We put our heart and soul into each performance. Sometimes we even move people to tears, but it floats up in the sky along with the cigarette smoke and It's gone.\n\n'An artist has the finished canvas, the writer a manuscript, a carpenter the thing he has built. But your voice and my horn, they warm people for a moment and then It's gone.'\n\n'You mean It's worthless?' she turned a shocked face up to Sam.\n\n'Oh, no honey, not worthless. The musicians, the artists, the dancers, we're the ones that give food for the soul. Without us the world would be a dreary, dark place with no dreams or hope. We were given that talent for the same reason a rose was given its perfume, or a bird its song. Don't ever think it's worthless.'\n\nShe was silent for a time, sitting there with her head pressed into his chest and he knew the tide was turning.\n\n'You and I are so alike,' she whispered eventually. 'Do you think this was why we were brought together?'\n\nHe nearly told her then. He could see little Katy's face in Georgia's, just the way she'd looked that last leave when he said the troops were being mobilized. She was frightened he'd be killed or wounded, yet she had smiled to reassure him.\n\n'Don't be a hero.' She had held his face in her two hands and kissed his nose. 'I love you too much to lose you.'\n\nIt was raining hard, beating at the window. A grey miserable day that would turn into an even colder, darker night. He could remember nights like this as they went through France, fear making his skin prickle, wondering if each night might be his last. But he'd come through that unscathed. He'd even found his child after twenty years, surely he could wait another few weeks and make certain she was ready for it?\n\n'I'm sure the Almighty has a hand in it somewhere,' he smiled. 'Now what we have to do is put our heads together and come up with a plan.'\n\n## Chapter 24\n\n'What are we going to do?' Georgia asked sheepishly.\n\nTwo days had passed. Below in the courtyard reporters still hung around hoping for a glimpse of her, trampling on the flowers, throwing sandwich papers around, making the other tenants' lives a misery.\n\nIt was like being in a prison. Johnson the jailer, bringing up milk, bread and newspapers, passing on any information he'd heard.\n\nGeorgia spent much of the time sleeping, while Sam read, listened to music and cooked for them.\n\n'What do you want to do?' Sam asked. It had been torture for him to just sit and wait. Every bone in his body urged him to go out amongst those hyenas in the courtyard and tell the true story. Each time he looked in on Georgia and saw her curled up asleep, avoiding any confrontation, he felt more murderous towards the man who'd done this to her.\n\nBut even in his anger he knew he must bide his time. The slander would trickle to a halt eventually, and that was the time to reap revenge on all those who had a part in it.\n\nGeorgia looked better today. She was pale still but the circles had gone from under her eyes. She wore a long pink fluffy dressing-gown, her hair tied back, yet until he saw her actually get dressed he wouldn't be convinced she was fit to tackle anyone.\n\n'I should speak to Jack Levy,' she frowned. 'Someone's got to pay off the session men or at least give them some idea where we stand.'\n\n'Max should have done that.' Sam felt hate rising like bile from his stomach. 'My God, Georgia he's a rat, and we can't put him down with mere poison.'\n\nMax hadn't telephoned that first night. Instead he sent a note round the next day saying the studio was besieged with reporters and the board were having a meeting about what action they would take.\n\nThere had been no words of sympathy, not even a suggestion of concern. No telephone number for them to reach him, no promise to call round. He might just have easily said he didn't care about the outcome.\n\nHe was waiting for Georgia to find an answer. Distancing himself so if any blame came it wouldn't fall on his expensively-clothed shoulders.\n\n'I can see to paying the men,' Sam said. Once again she was worried about others. She knew none of them could afford to wait indefinitely and she couldn't bear to think she'd let them down. 'Jack Levy should come here and see you. You can't go there cap in hand.'\n\n'But I have to sort something out,' she sighed. 'What do you think is best?'\n\n'Seems to me,' Sam slid an arm round her small shoulders. 'You should write the whole story as it was. Then we get it to the papers. Maybe we can even ask their help to find Celia to back you up.' Sam had lain awake at night thinking up ways of getting help. He'd even slipped out to use Johnson's phone to ring Sister Mary for her advice.\n\nSister Mary was anxious to go and tell the press what she knew, but like Sam she knew Georgia had to tell the story herself. Bringing Celia into it had been an idea they'd cooked up between them, hoping that the excitement of attempting to find her might break Georgia out of her apathy.\n\n'Would they Sam?' her dark eyes gleamed with new hope.\n\n'Of course they will!' Sam grinned. 'Do you think any newspaper wouldn't jump at that kinda scoop? But first I reckon we sit tight. Let's watch and see which rats crawl out the holes first. Find out who's on your side out there!'\n\n'Okay,' she shrugged her shoulders, a faint smile playing on her lips. 'But first let me send a cheque to each of the men.'\n\nIt was painful to wake up each morning to find yet another slanderous story about her in the papers.\n\nIt seemed that anyone who had some minor grievance about her was prepared to slander her for financial gain.\n\nA landlady up in Scarborough spoke of a drunken orgy in her guest house. A barman in Lancashire claimed Georgia had pushed a broken glass in another girl's face. Stories about sex in changing rooms. Drugs taken openly. Shoplifting in Scotland. Young girls procured for the band, hotel rooms vandalized.\n\n'None of it's true!' she looked at Sam in horror. 'Why do they say these things?'\n\nThey had been drunk up in Scarborough. A girl had been slashed by a glass, but not by her. The boys were more than capable of procuring their own girls, without Georgia's help. Some of these stories had a grain of truth, distorted and embroidered, but most were pure fiction.\n\n'Laugh at it,' Sam suggested. 'These are people who would jump on any bandwagon that came along. If the newspapers said you had a religious experience in one of those places, they'd be nominating you for sainthood.'\n\nAt first it made her cry and go back to bed. But after a day or two she became so used to it she found herself laughing at the absurdity of it all, watching for the people who cared enough about her to contradict the accusations.\n\nRod was the first to go to the press. He spoke passionately in Georgia's defence, explaining her loyalty to the band, her loving nature and the lengths she'd gone to for their protection.\n\n'Me and the lads were a bit wild sometimes,' he said. 'But stick someone in front of me who says Georgia was involved, and I'll tear them to shreds.' When asked what he knew about her life before joining the band Rod got even more heated.\n\n'Hasn't it occurred to you there's another side to this? She'll tell you when she's good and ready, as she did me. All I can say to you is use your brains, work out what would make a fifteen-year-old stab a man, then run. Ask yourself what ordeal he put her through?'\n\nSpeedy was next, interviewed late at night on television. He blamed the press for causing a rift between them, laughed at the scandal-mongering and talked affectionately about her early days with the band.\n\n'He's straight,' Georgia said in surprise, leaning forward to the television set and peering at Speedy. His eyes were clear and unwavering, long hair trimmed, his face clean-shaven. 'Well, that's one good thing to come out of this.'\n\nThere were other people whose voices were heard. Bert and Babs from the caf\u00e9 spoke of the way she came into their lives, the comfort and love she showed Helen, her dignity and pride.\n\n'I don't care what rubbish people are saying,' Babs was quoted as saying. 'All of us in the market know the real Georgia. That girl's got more compassion in her than any of you will ever feel. You should be ashamed of yourselves!'\n\nBut even as that hit the papers so there was more slander from men who claimed to have slept with her in the days when the band was on the road.\n\n'I've never even seen any of these men, let alone slept with them,' Georgia gasped as she read about three-in-a-bed scenes from Edinburgh to Penzance. 'How could they say such a thing?'\n\n'Fantasy, honey,' Sam grinned. 'They've spent so long wishing they could do it, that now they think it's real. But just look at them. No one in their right mind would believe them.'\n\nIan's mother made claims that if Georgia hadn't treated her son so badly he wouldn't have got drunk and burned to death in that house. Rod's mother and stepfather backed up Mrs McShane by saying Georgia had turned their son against his family.\n\n'I don't believe anyone could make such things up,' Georgia gasped. 'Why doesn't Max stand up for me?'\n\nBut Max was caught by a photographer at London airport, leaving for America with a blonde dolly bird simpering on his arm.\n\n'The first I knew of this was when Anderson came forward.' Max looked furtive, embarrassed at being caught slipping away. 'I feel bitter she didn't take me into her confidence earlier, and although I know the whole story now, I'm not at liberty to say anything. I'm a business man with other clients to look out for. Georgia will handle it all in her own way.'\n\n'What can I say?' Sam spread his hands in a gesture of 'just as I thought'. 'The man's sitting on the fence. He hasn't got the guts to stay firmly on your side. Even now he's probably planning to sell his memoirs to the highest bidder.'\n\nIf it hadn't been for the scores of letters from the most unexpected people, she might have thought everyone was as uncaring as Max and Jack Levy. Harold and his wife wrote to invite her over to dinner. Norman sent a bouquet of flowers with a note offering his apologies for ever doubting her. Other musicians, girls and technicians from Decca, and even Deirdre, Max's receptionist sent letters of support and sympathy. Flowers arrived from Andreous, Steven her producer, from an American promoter, Ronnie Scott's and from Pop and the girls in Berwick Street.\n\n'Remember, I know the truth,' Janet wrote on a piece of bright blue paper, her spelling so bad Georgia could barely read it, her handwriting like a child's. 'If I'd had my way I would have gone to Fleet Street and told them exactly what that creep done to you. Pop says you've got your reasons for keeping quiet and if I go shooting my mouth off it might make things worse. But I want you to know we are all here for you when you want us. Who is this Sam? I hope he's good to you, if not he'll have me to reckon with.\n\nSal and all the others send their love, so do our kids. We play your records all the time.\n\nLove Janet.'\n\n'Real friends,' Georgia gulped back tears as she handed it to Sam. 'You find out the real ones when the going gets tough!'\n\nBut just when she thought they had run out of steam, Sam's face began to appear in the press. They sent someone to New Orleans to check out his home there and soon he was getting the treatment too.\n\n'Sam Cameron has two children, left with his sister while he holes up with Georgia,' they said. They found news of a bar-room brawl he'd been in. His ex-wife was interviewed and claimed Sam not only beat her, but kept her as little more than a slave. There was a picture of his two children, both in ragged clothes and bare feet, the house behind them looked like a shack.\n\n'I took that one myself,' Sam said, his dark eyes growing black with fury. 'They'd been out playing in some mud. I thought it was funny that they looked like a pair of waifs. Everyone has a snap of their children like that.'\n\nIt was cruel the picture they had painted of him. A heartless vagabond who cared nothing for his family. Somehow by hurting Sam they had succeeded in hurting Georgia still further.\n\n'How dare they?' she gasped. 'Oh Sam, what have I dragged you into?'\n\n'Don't worry about me,' Sam shrugged his shoulders. 'They're scraping the barrel now.'\n\nBut the next day there were more stories. One about Georgia spending four hundred pounds on shoes in one day and another claiming she didn't wash her clothes, but threw them away.\n\n'There can't be much else going on in England,' Georgia forced herself to laugh, even though she felt desperate. 'What are they going to dredge up next?'\n\n'Your headmistress,' Sam chuckled, opening another paper. 'Georgia was a real leader. A strong character we all remember well. She did leave school suddenly and although I heard rumours about her father, I remained convinced she wouldn't hurt anyone unless severely provoked. I hadn't actually realized the Georgia Anderson I knew and liked so much was in fact the famous singer. Had I known I would have written and expressed my pride in her. I wish her well and I urge her to tell the true story about these events in her past.'\n\nBut the few people who stood up for her were outweighed by the hate mail that arrived daily. After seeing a sample Georgia refused to even look further, and it was Sam who sorted through it, chuckling to himself.\n\n'You've gotta laugh,' he said in his defence. 'Just look at all this stuff. There's a woman who reckons you killed her dog, another who believes you had an affair with her husband. Even one who thinks you are the anti-Christ. The rest of them are from Rednecks who blame your colour for everything.' He shook his head in bewilderment. 'You sure stirred up a hornet's nest, honey!'\n\nWithout Sam she might have lost her mind. He somehow put it all into perspective and gave her the courage to write down the full story.\n\nIt was ten days after the news first broke that she saw a particularly cruel cartoon in one of the newspapers. A caricature of herself holding a knife over an old man, and the words 'There's no time Baby' coming in a balloon from her mouth.\n\nShe couldn't laugh now. Rage welled up in her, a desire to speak out and be heard.\n\n'It's time Sam,' she said, wiping away tears of frustration. 'I can't stay in here another day. I'm going to the press.'\n\nThe story was ready. A sharp, impassioned account, without exaggeration or embroidery. She spoke of her happiness with the Andersons until the rape. Her feelings for both her foster parents, then the shock and torment Brian put her through. Recalling the rape and stabbing was so painful it was tempting to gloss over it. She had to dig down deep within her, make herself remember each detail. The way he had laid sprawled on the landing, the knife in her hand as she came up the stairs, the blood as it spurted out of his belly. Once she'd faced that again it was easier to put down her explanation for running away. Her first few days in Soho, the abortion later, were softened by the people who helped her.\n\n'Want me to come too?' Sam said. He was making coffee in the kitchen, wearing just a pair of jeans, his feet and chest bare.\n\n'No,' she said shaking her head. 'I've got to do this on my own haven't I?'\n\n'I guess so,' he moved across the kitchen to her, putting his hands on her shoulders. 'They'll think more of you. Besides I couldn't trust myself to keep my trap shut.'\n\nShe took care over her appearance. For ten days she'd worn nothing but jeans and a sweatshirt. But now she had to look like a star.\n\nA white leather suit fitted the bill perfectly. The skirt was short and tight, the tiny jacket trimmed with silver stars went over the briefest skimpy silver top. She washed her hair and let it dry naturally in ringlets, adding star-shaped earrings studded with diamonds and a pair of long silver boots with cuban heels.\n\n'You look sensational,' Sam grinned up at her as she came back into the kitchen.\n\nShe faltered in the doorway for a moment.\n\n'Stage fright?' He poured her another cup of coffee and slid it across the table.\n\nSitting on the edge of a chair she lifted the coffee to her lips.\n\n'What if they still don't believe me?'\n\nSam's eyes crinkled up with laughter, he reached across the table and took her hand.\n\n'Everyone who really counts already believes you, honey.'\n\n'Do you think they will find Celia for me?' she whispered, her eyes wide with fear.\n\n'You make them find her,' Sam said fiercely. 'Don't forget for one moment, they owe you. Now off you go. Keep your head up, take deep breaths if you're nervous. I'll be here if you need me.'\n\nShe walked round the table and leaned over on to his shoulder, pressing her lips against his neck.\n\n'What would I have done without you, Sam?' she said softly. 'You've been mother, father and friend all in one. I can't tell you how important you are to me.'\n\nHis hand came up to caress her face and she saw his eyes were glistening with tears.\n\n'Off with you,' he said gruffly. 'Before I say something I might regret.'\n\nThrough the glass panel on the doors she could see a handful of reporters still patiently waiting, puffing on cigarettes, chatting in small groups.\n\nIt had been cold and wet most of the time she had been incarcerated in her flat, but now the sun was shining. Daffodils almost finished, tulips about to surpass them, and the almond tree was covered in delicate pinky-white blossom. Spring was finally here to stay.\n\nTaking a deep breath she opened the door and stepped out.\n\nShe wanted to laugh at the way they all jumped. Cigarettes stubbed underfoot, sandwiches shoved into pockets, fingers fumbling for pens, cameras lifted, every face wiped clean with surprise.\n\n'Good morning.' She waved the brown envelope containing her story, holding her car keys in readiness. 'Glad to see you haven't lost interest!'\n\nA small man darted forward. She had seen this one before on many occasions. He reminded her of a ferret, with his thin head, sharp nose and tiny eyes.\n\n'Have you any news for us?' he said, as the others quickly clustered round him.\n\n'I'm off to the _Mirror_ ,' she smiled more confidently than she felt and patted the envelope in her hand. 'The truth's in here. As they started the whole slanderous business, I expect them to lay it to rest too.'\n\n'Why have you taken so long to retaliate?' A woman's voice came through the crowd, Georgia could only see a pair of brown eyes and a mop of untidy red hair.\n\n'Timing,' Georgia grinned round at them. 'Giving you enough rope to hang yourselves. The story will be out tomorrow.' She paused to pose for the cameras. 'Go home now. There's nothing for you here.'\n\nThe sun was glinting on her red Mercedes. She opened the door and slipped in, winding down her window and turning her radio up loudly.\n\nAs she drove quickly out of the courtyard, she saw for once they were speechless, mouths open with shock.\n\nIt was sometime since she'd been to Holborn and she hadn't thought to check out where the building was. As she stopped at the lights, just past Gray's Inn Road, she noticed the huge modern building on her right.\n\nWithout even considering the heavy traffic going through to the West End she did a 'U' turn in the road, ignoring the other motorists who honked furiously at her and pulled up with one wheel on the kerb.\n\n'You can't stop here.' A young policeman came forward, a frown of irritation vanishing into a smile of delight as he recognized her.\n\n'You park it for me then,' she said dropping the keys into his hand. 'I've got important things to do.'\n\nThere were double ordinary doors up two steps or a revolving one to the side. She went through this so fast the door swung on round several times more behind her.\n\nShe got a last glimpse of the policeman staring after her, a bemused expression on his face, her keys still in his hand.\n\nThe foyer was all brown, highly-polished tiles and huge green plants in tubs. The corpulent uniformed porter looked up from his desk as she marched up to him. He knew her face was familiar, but he couldn't quite place it.\n\n'The Editor!' she snapped. 'Where is he?'\n\n'I'll just telephone his secretary,' he picked up the phone.\n\nGeorgia put out her hand and prevented him.\n\nShe could hear the lift coming down behind the porter, she wasn't anxious for anyone to recognize her just yet.\n\n'Just tell me which floor,' she barked. 'Now!'\n\n'Third, no fourth,' he stammered. 'But \u2013'\n\n'I'm going up there,' she said coolly. 'What's his name?'\n\n'Phillips,' he said weakly, his hand straying to the phone again, afraid he would lose his job.\n\n'You can warn him I'm on my way up,' she called back as she made for the stairs. 'Don't worry, I'll tell him it was all my doing!'\n\nShe was out of breath by the time she reached the fourth floor. Her face was flushed and her heart hammering nearly as loudly as her heels on the tiled corridor. She marched quickly down the corridor giving only a cursory glance into open doors where typewriters and teleprinters clattered, the girls who worked them looking up in astonishment as she passed.\n\nHis name was on the door. 'John Phillips, Editor.'\n\nShe didn't even knock, but opened the door wide and swept in.\n\nShe had expected a big man. Someone like Max in silk shirts and a fat cigar, full of bluster, a mouth like the Blackwall Tunnel. But the man behind the desk was short, thin, almost weedy, a boyish, open face. He wore corduroy trousers and a knitted tie, perhaps fifty, but he looked younger. His brown eyes blinked furiously, a small, gentle mouth opening in surprise.\n\n'Miss James,' he jumped up, holding out his hand.\n\n'I'm surprised you recognize me,' she snapped. 'After all the fiction you've been writing about me lately, I thought you had me mixed up with someone else.' She ignored the hand and plonked herself down in a chair, staring coolly at him.\n\nHe was disconcerted. One finger ran round the collar of his shirt, his face turning pink.\n\n'What can I do for you?' he said weakly. 'Would you like some coffee?'\n\n'You can print the truth,' she said, fixing her dark eyes on his pale brown ones. She noticed he had a small mole on his cheek, one of his front teeth was slightly broken off and he'd cut himself shaving. He didn't look as if he had a woman to look after him.\n\n'As far as we are concerned everything Mr Anderson said is the truth,' his eyes dropped from hers, tiny lines showing round his mouth.\n\n'How much did you pay him for that garbage?'\n\n'I, I \u2013,' he stammered.\n\nShe thought of how much she'd feared these people, thinking they were like gods who couldn't be beaten. Yet here was a little man in charge who couldn't even shave himself without making a hash of it.\n\n'All right, so you don't want to tell me that,' she was calm now. She might even enjoy tearing him and his paper to shreds. 'But I have the truth here,' she put the envelope on his desk. 'All you have heard is the outpouring of a bitter man who has misled you.'\n\nShe could smell Phillips' fear. His Adam's apple was leaping up and down, one eye was beginning to twitch. As his hand reached out for the envelope she remembered Celia.\n\n'Everything in there can be verified. I want you to find Mrs Anderson in return for that truth.'\n\nHe pulled out the six sheets of paper, flicked through them and put them down again.\n\n'You'll read it now,' she commanded. 'Not tomorrow, next week or when it suits you. Now, while I'm here.'\n\n'Of course,' he picked them up again, that Adam's apple threatening to get caught on his collar.\n\nOutside the window she could see only office windows and blue sky. A pair of pigeons were canoodling on a window-sill, the male spreading his tail and fluffing out his chest. The noise of traffic just a hum, subdued by the thick glass. It was strange to just sit silently while a total stranger read a story she had hardly been able to think about, much less tell.\n\nIt was a comfortable office, with a big grey desk and plants in tubs. His desk was covered in papers, two of the other chairs were piled high with cuttings and folded newspapers. There were large glossy photographs lying around, one of her, on a pile of papers, taken as she came back from America.\n\nShe could see Phillips was moved by the story. He bent closer to it, reading it slowly and carefully. His lips quivered, his fingers fiddled with his tie nervously, occasionally he glanced up at her, as if trying to fit the star in front of him into the story in his hands.\n\nShe heard a faint sigh as he finished it. An expression of profound sadness on his youthful face.\n\n'You write very well,' Phillips looked up at her, but he couldn't hold her steady gaze. 'Why did you wait so long to contact me?'\n\nShame poured out of him. He was even honest enough not to try and wriggle out of it by counter-attacking.\n\n'I watched and waited,' she said, but to her distress she could feel tears pricking her eyelids. 'I wanted to watch every last louse crawl out of the woodwork. Have you got any idea how painful it was for me to relive that night? Do you know what you've done to me?'\n\n'I do now,' Phillips' eyes caught hers. Sympathy and understanding, mixed with a stronger desire to set things straight. 'An apology seems futile.'\n\n'You will apologize, by printing the truth,' she wasn't going to let him off the hook just yet. 'I expect you to use all your connections to find my foster mother too.'\n\nHe cleared his throat. Fear of a law suit flickered across his watery eyes.\n\n'Do you understand our position? It's our obligation as a newspaper to print news as we are given it.' His voice was firm, yet there was an undeniable tone of shame in it. 'Mr Anderson's story was printed in good faith, he had photographs and evidence to support it.'\n\n'I'm sure he did,' she said. 'He may be many things but he was always plausible. I believed in him myself until he raped me. But didn't you even think of contacting me first?'\n\nPhillips shrugged his shoulders and waved his hands.\n\n'That's what a scoop is all about,' he said. 'We get a story, it sells our paper.'\n\n'Aren't you wondering why I've brought this to you?' she asked. 'I could have gone to one of your rivals, dug up dirt about how you got Anderson to sell his soul.'\n\n'Why didn't you?' he croaked. She knew he was expecting news of lawyers. His earlier pink flush was turning a little green. She wanted to play with him, make him suffer as she had.\n\n'Because I want your whole-hearted commitment.' She rapped one long nail on her story in front of him. 'If you succeed in clearing my name, bring that bastard to justice and find my mother, then maybe I'll just settle for a hefty donation.'\n\n'How much?' he looked up quickly.\n\nThe sharp expression in his eyes made her think of Max. Funny how money changed people!\n\n'You misunderstand,' she smirked. 'I have all the money I need, the donation can go to charity, one that deals with runaway kids.'\n\nRelief poured out of him. 'That's the least I can do.'\n\n'And you'll find my mother?'\n\n'Any idea where she might be?' There was a glow in his eyes, as if he relished the challenge.\n\n'No. I think she must have gone back to nursing. It must be somewhere remote or she would have read all this. I went back to Blackheath when I was sixteen, I tried to find her and Peter, but Peter's mother sent me away with a flea in my ear and Mum had left.'\n\nHe put the end of a pen in his mouth, sucking at it thoughtfully.\n\nThere was something troubling him, something in her story which had tripped a wire. She could almost hear and see his brain mulling it over. Had Brian said something more which hadn't been printed?\n\n'What is it?' she asked. 'I can see something's troubling you.'\n\nHe frowned. 'I don't know if it's important,' he looked down at her story again, then glanced back at her. 'We had a telephone call just after the story broke. A young man, he wanted your address.'\n\n'Not another of those fictitious lovers?' she laughed lightly. 'What did he have to offer?'\n\n'To be honest, we thought he was one of those,' Phillips looked uncomfortable. 'But the boyfriend, Peter, you mentioned \u2013'\n\n'It was Peter?'\n\nPhillips heard the catch in her voice, saw her eyes widen and sensed the emotion the name evoked.\n\n'What's his surname?'\n\n'Radcliffe.'\n\nPhillips' eyes closed for a moment. 'That's him. We had dozens of crank calls, so many we hardly listened to the filth they were saying, but \u2013'\n\n'What did he say?' Georgia was trembling now, every nerve-ending twitching. 'Tell me.'\n\nPhillips ran a finger round his collar, beads of perspiration were glistening on his upper lip.\n\n'Just that he was an old friend. He wanted your address or phone number.'\n\n'Why didn't you give it to him then?'\n\n'We never part with that kind of information.' Phillips looked shocked at the mere suggestion. 'The only reason I even remember his name was because he made no startling revelations. The girl who took the call said he was a teacher.'\n\n'Did you take down his number or address?'\n\n'Of course,' Phillips picked up a pen and fiddled with the point. 'We always log it down. We even tried to ring him back, but there was no reply. I think the girl suggested he could write to you care of this office.'\n\n'And has he?' Georgia's eyes were like glowing coals.\n\n'I don't think so, not yet.'\n\nGeorgia could feel her heart pounding, her palms sticky. The Peter she remembered wouldn't sit and read lies without doing something.\n\n'He's still special to you?' Phillips' voice softened.\n\n'Yes,' she dropped her eyes and blushed furiously. 'I never seem to be able to forget him. He might be married now, he certainly can't feel the same about me still. But even so.'\n\n'You'd still like to see him again?' Phillips raised one eyebrow.\n\n'Oh yes,' she sighed.\n\nPhillips could hardly believe what he was seeing and hearing. All through his interview with Anderson he had sensed something wasn't quite right, he'd had to force himself to forget he was a fan of Georgia's, give the public the story, putting aside his own qualms.\n\nHe knew he had the truth now, even without checking it out. But one thing was plain, he had to make amends for his paper's part in it, and he hadn't got to the position of Editor without knowing the value of emotional reunions.\n\n'Suppose I got him down to London?' Phillips smiled. 'Asked for his help. It would be easy for me to discover his circumstances without obligating him in anyway.'\n\n'Could you do that?' As much as she wanted to rush down into the office further down the corridor, force them to give her Peter's address and rush there immediately, she knew that wasn't practical. The memory of his mother's chilly face was still in her mind. 'Peter when I knew him was the sort that hated injustice. That's probably the only reason he rang here.'\n\n'I wouldn't say that,' Phillips laughed.\n\n'You don't know him,' she said quietly. 'Look, he's known who I was all along. If he'd still held a torch for me he would have got in touch. That means he has someone else doesn't it?'\n\n'I can hardly believe what I'm hearing,' Phillips sat back in his chair his lips twitching with amusement. 'You've been portrayed as a cross between Lizzie Borden and Lucrezia Borgia, yet you are apprehensive about causing a few ripples in an old love's life.'\n\n'I'll never forgive you if you turn it into a circus,' she threatened.\n\n'I promise you no one will know about this until you say the word,' he smiled. 'And that's one helluva promise for a newspaper man!'\n\nGeorgia stood up and held out her hand. 'Ring me at nine tonight. I'll put the phone back on specially.'\n\nHe took her hand and shook it.\n\n'Thank you for coming,' he said warmly. 'I'll get this story ready for tomorrow's paper and start the hunt for your mother. By the way,' he blushed bright pink. 'In all this I forgot to ask you. What are you going to do about Mr Anderson?'\n\n'I'd like to kill him,' she smiled sweetly. 'But please don't quote me on that!'\n\n## Chapter 25\n\nGeorgia walked up the steps of the President Hotel, through the double glass doors and paused in the foyer.\n\nShe had butterflies in her stomach, her pulse was racing and she was no longer certain she was doing the right thing.\n\n'He's at the President, in Bloomsbury,' Phillips said on the phone at nine. 'He wasn't keen to stay overnight. I had to twist his arm by pretending I needed more information tomorrow.'\n\nWhy hadn't she asked questions? Why hadn't she stopped to think? If Sam hadn't already left for a gig in the West End, he would have stopped her. The President wasn't even a cosy little boarding house as she'd imagined, but instead a huge, red brick Gothic hotel, the sort she hated.\n\nShe had grown accustomed to the grandeur of places like the Savoy and Claridges. The staff so well trained that they treated everyone with the same courtesy whether they were rich and famous or just stepping in to make an enquiry. But this was one that catered for international business men. The heavy red velvet drapes with ornate gold tassles, the rich red and gold carpet, the dark flocked wallpaper and the padded leather reception desk gave an impression of illicit encounters, deals and intrigue. The sort of place where the staff wouldn't balk at calling up a few journalists just to get their name in the paper.\n\n'I should have phoned and arranged to meet him somewhere,' she thought as she approached the reception desk.\n\nIt was almost ten at night. She was attracting speculative glances from a group of middle-aged Americans in the lounge to her right, and a swarthy porter lounging by the lift. Down some stairs to her right came the sound of male laughter and clink of glasses.\n\nWhy hadn't she asked Phillips how Peter reacted to his call? An hour ago it had been enough to know he'd got the first plane out of Manchester. But lack of hesitation on his part might only mean he had a day free.\n\n'I'd like to see Mr Radcliffe, please.'\n\nThe two women manning the reception desk were formidable fashion plates, in dark suits and candy-striped shirts. One peered at her suspiciously, glancing over the leather and wood counter at Georgia's jeans and white sweater and sniffed in disapproval.\n\nThe younger of the two opened the register, and slid one red talon down the page. Her hair was cut in geometric Mary Quant style, it swung forward over one eye, sleek and dark.\n\n'He's in 309,' she said in a bored voice. 'Would you like me to try his room?'\n\n'Yes please,' Georgia could see the second woman studying her closely. It was that same expression people often had when confronted with her. Her face looked familiar, but they couldn't quite place it.\n\n'No reply,' the dark haired woman put the phone down and flicked back her hair. 'Would you like to leave a message?'\n\nOn the fast drive across town Georgia hadn't considered for one moment Peter might go out. She had merely visualized knocking on a door and Peter opening it. Now what should she do?\n\n'Is his key there?' she asked. She could feel herself blushing and she knew the porter was now giving her bottom his undivided attention. Worse still, a man in a flashy checked suit had paused to consult a display of tourist information just to her left, and she sensed he was listening, about to offer her the kind of attention she didn't want.\n\nThe woman turned to examine the board behind the desk.\n\n'No,' she said curtly over her shoulder. 'But that doesn't mean anything, they always forget to hand them in.'\n\nGeorgia turned away in disappointment.\n\n'Excuse me!'\n\nGeorgia looked round. The second receptionist who had been studying her was leaning on the counter.\n\n'Is Mr Radcliffe young, tall, with blond hair?'\n\n'Yes,' Georgia's heart leapt, bringing a wide smile to her lips. In one bound she was back to reception, leaning on the desk. 'Do you know where he went?'\n\n'In the bar,' the woman smiled now, revealing a warmth that hadn't been there moments before. 'I'd forgotten until you looked so disappointed. He asked me earlier if he could borrow a street map, he took it in there with him to look at it. I could page him for you?'\n\n'I'll just go in there,' Georgia beamed at her. 'Thank you.'\n\nAs she made her way down the thickly-carpeted stairs to the bar, men's voices grew louder. A smell of cigar smoke wafted up to her and her knees were turning to jelly.\n\nThe stairs turned. In front of her she could see her reflection in yet another mirror framed by two tall imitation palms. Taller, more rounded than the night she ran with Peter across the heath, but her eyes were gleaming with excitement just as they had that night.\n\nTo her left, down just five more steps lay the bar. A rich, dark red carpet, a leather front to the bar, brass feet-rails and two business men deep in conversation, was all she could see. But judging by the noise it was crowded further in.\n\nShe stopped again in the doorway. The bar stretched along the wall in front of her, three deep with men. To her right was an archway leading to a smaller room with leather Chesterfields and low tables. To her right a larger area with small polished tables and straight-backed chairs.\n\nFew of the tables were occupied. Everyone was standing at the bar. In the main, business men in sober suits, faces flushed with drink, or was it merely the soft pinkish lighting?\n\nHeads turned in curiosity, smirks on their lips, eyes glinting as they sensed her embarrassment at breaking into a masculine world.\n\nGeorgia gulped. She could see ginger hair, blonds, bald heads and slickly-Brylcreemed heads. But not Peter.\n\nAs she walked down the bar conversations were halted. Whispered remarks, nudges, the kind of smiles that were an overture to conversation. She didn't dare let her eyes meet anyone's.\n\nThen she saw him.\n\nHe was leaning on the far end of the bar, deep in thought. One foot on the rail, his hand nursing a pint of beer. Golden hair caught under a light, wearing a denim jacket and a white open-necked shirt.\n\nShe had so many pictures of him trapped in her mind.\n\nAs a choir boy, in a white surplice with a ruffle round his neck, angelic and pure. The gawky schoolboy in scuffed shoes, grey slacks and a navy blazer, his cap pushed back on his head. In jeans and a sweater running with her, hand in hand across the frosty heath.\n\nThen there was the night of her party when he told her he loved her. Eyes as blue as a summer sky, golden skin and soft lips, his hair like ripe corn.\n\nThe boy who walked her home from choir had become a man. Wide muscular shoulders strained his jacket. His square jaw, tougher with a hint of stubble. Hair longer, badly cut, streaked from palest cream to deep gold. Yet his profile hadn't changed. Straight, proud nose, curving full lips, thick eyelashes fanning those peachy cheeks.\n\nShe forgot she was standing in a room full of curious strangers. A delicious flush of excitement crept over her.\n\nHe lifted his head as if aware he was being watched. His eyes flickered across to her, then shot open in shocked surprise.\n\n'Georgia!'\n\nFear of rejection forgotten, she found herself running towards him, hands outstretched, seeing only the blue eyes and a smile as wide as the Thames.\n\n'Peter!'\n\nFor a moment it was impossible to speak. His hands were holding hers. Blue and brown eyes searching one another. Two pairs of lips smiling, unable to find the right words.\n\n'What would you like to drink?' Peter's voice was gruff with emotion. 'Everyone's watching us,' he added in a whisper.\n\n'Just orange juice,' she smiled. 'And I don't give a toss about them.'\n\nHe ordered her a drink, keeping a tight hold on her hand, his thumb running across her fingers as if he was checking it was real.\n\n'Let's go over there,' he said softly, nodding to a table over in the corner. 'You might be used to audiences but it's new to me.'\n\nShe reached the table first, sitting down quickly so she could look again at him. He was much taller, perhaps six foot two. His face had filled out. He looked fit and athletic, skin with an unmistakable glow of the outdoors. Slim-hipped in his faded jeans, but his chest and thighs powerful.\n\n'It's all too much,' he said as he sat down. 'This morning I was having a lie-in, contemplating the marking, then the phone rang and before I knew it, I was on my way to London.'\n\n'Marking?' Georgia frowned.\n\n'I'm a teacher now,' he smiled, as if remembering how much time had passed. 'It was lucky it was school holidays or Phillips wouldn't have found me in.'\n\nA stab of guilt shot through her. Phillips had said he was a teacher, yet she hadn't considered what that meant. She was like a child herself, expecting people to come running when she needed them.\n\n'I wasn't sure you'd come,' she hung her head. 'I certainly didn't expect you here so quickly.'\n\n'You knew he was going to ask me then?' Peter gave her an odd look, disbelief mingled with pleasure.\n\nShe had to explain how Phillips had connected the boy in her story with an earlier caller.\n\n'You can't imagine how thrilled I was,' she said. 'I mean I thought I'd never see you again.'\n\n'Who dictated that, \"Push off, I'm not interested\" letter then?'\n\nGeorgia closed her eyes. There was no point in asking what he meant, or when it was. Whatever had happened Max had to be at the bottom of it.\n\n'I didn't know you'd written.' She reached out and touched Peter's hand, looking right into his blue eyes. 'Don't you know I would have been on the first train to see you if I had?'\n\n'How could I know that after what happened?' his eyes were guarded. 'I just assumed you'd written me out of your life.'\n\nIt was like a fencing match. Peter thrust accusations at her, she parried with explanations.\n\nHe told her about himself and Celia waiting for a phone call or letter up to the weekend after her sixteenth birthday.\n\n'But I went to your house, I tried to get Celia at her office. Your mother took my address and said she would give it to you. She said you weren't interested in me any longer.'\n\nHis eyes went dark with anger, his wide mouth trembled.\n\n'I had such misgivings about going out that morning,' he said softly. 'But you just don't credit a mother with being that cruel or underhand. She seemed kind of smug when I got home. Too nice, if you know what I mean.'\n\nHe told her how close Celia had come to breaking down. Brian's drinking, the violence and fights.\n\n'She had no choice but to leave her home,' he said, his lips trembling as he remembered. 'But we were so sure you'd write to me.'\n\nA terrible guilt crept over Georgia. For years she'd assumed all the pain was on her side, imagining Celia and Peter had carried on with their lives almost as if nothing had happened. Now she found her sudden departure had been like a hurricane, leaving untold devastation in its wake.\n\n'Didn't you realize what it would do to your mother?' Peter's blue eyes burned with anger. 'I saw her almost broken with grief. Not because she lost her home, or even her job, but because she couldn't bear to live without you.'\n\n'I thought she'd understand,' Georgia's eyes filled with tears. 'I went thinking it would be best for everyone.'\n\n'She even understood that,' Peter's voice softened, as if Celia's memory was still very dear to him. 'I was too young then to really comprehend how shattering rape is to women. I was so screwed up with bitterness, I guess I blamed you for not writing or phoning. But Celia just went on searching and loving you.'\n\n'Where is she now?'\n\n'I don't know,' he shook his head. 'She went to Africa to nurse when we couldn't find you. I persuaded her to, assuming you'd get in touch with me. We wrote for over a year, but then she stopped it.'\n\n'But why?'\n\nPeter sighed, as if he barely understood her reasons.\n\n'I think she was afraid for me. I was still looking for you in the holidays. My letters to her were always full of you. Several times she wrote telling me to forget you. To get on with my life, have fun and take out other girls. Not because she stopped loving you Georgia, but because she felt responsible for me too.'\n\n'She thought you were being obsessive?' She could imagine Celia in her mind's eye, putting on her glasses, tutting to herself as she read Peter's letters, weighing up the situation and deciding to be ruthless.\n\n'\"Enough's enough,\" was how she put it. \"Don't ever think I'm turning my back on you Peter. I just know it's unhealthy for a boy of your age to live in the past. Put Georgia and myself behind you. Look to the future.\"'\n\nThe quote from her mother brought her right into the crowded bar. Georgia smiled despite her sadness.\n\n'How did you feel?'\n\n'Relieved in one way,' he blushed, a tiny smile puckering the corners of his mouth. 'I mean it's hard to keep a broken heart when you're eighteen and surrounded by ravers.'\n\nShe could sense that he hadn't lived like a monk. A mischievous look in his eyes, the sensual lips, an assurance which showed in his straight back and wide shoulders. But rather than hurting her, it felt almost soothing.\n\n'So when was it you wrote? After I made the first record?'\n\n'No, before that, when I read about the fire. Some of my mates had seen your band in Hartlepool, so they were engrossed in the story. One of them described a girl singer who sounded like you. I did some checking and wrote to Celia first. When I got no reply, I wrote to you.'\n\n'I'm so sorry Peter. I can imagine how you felt.' She dropped her eyes from his. 'But I wonder why she didn't reply?'\n\n'I guess she'd moved on, I mean Africa isn't like Lewisham is it? Maybe my letters just lay in the post office. Phillips is checking out the organization she worked for. If anyone can find her, it's him.'\n\n'But she must have heard me sing?' Georgia whispered. 'My records have been played everywhere.'\n\n'Out in the bush?' Peter raised one blond eyebrow. 'The last place she was in was over a hundred miles from even a telephone. Syringes and medicine are more important there than music.'\n\n'I guess so,' Georgia smiled weakly.\n\n'We'll find her,' his voice had a confident ring. 'After all, I've actually got to speak to you at last. The way Max Menzies described you, it sounded as if you had a heart of stone.'\n\nGeorgia looked at him questioningly.\n\n'Max?'\n\nPeter shook his head in disbelief.\n\n'You didn't know that either? I went to his office two years ago.'\n\nGeorgia sat in stunned silence as Peter explained everything that had been said between them. Hatred for Max crept through her veins like a shot of whiskey on an empty stomach.\n\n'But why did you come after one rejection?' she said weakly.\n\n'Well, it's not everyone who can claim England's top star as their first love,' he laughed. 'I used to play your records, listen to the words and I got this feeling some of them were for me,' he blushed furiously. 'Daft isn't it? I expect every man who's met you thinks the same.'\n\n'They were for you,' she said softly. Her heart leapt crazily. He did still care. Even under the bravado, the anger and bitterness, a tiny flame was still flickering. '\"Crying\", especially.'\n\nSilence fell between them, so many questions as yet unasked, both waiting for the other to start again.\n\n'Are you going to marry Sam Cameron?'\n\nThis was the last thing she expected. It jolted her into realizing just how much ground they had to cover.\n\n'No. Of course not. He's just a friend.'\n\n'That's a relief,' he said, his mouth twisting slightly as if this of all questions was the most important. 'I mean, he sounds heavy duty.'\n\n'They've told almost as many lies about him as me,' she retorted quickly. 'Don't tell me you've become prejudiced too?'\n\n'I'm like everyone else in the world Georgia,' he said quietly. 'I believe what I'm told.'\n\n'Does that mean everything you read about me?'\n\nHe sighed. 'Some of it, I guess. I didn't want to. I kept telling myself it was all rumour and speculation, but I couldn't help thinking you must have changed.'\n\n'Yet you rang the _Mirror_ to offer help?'\n\n'I knew the truth about Anderson, remember. I never could stand liars. But when they blanked me out, wouldn't give me your address, I was as bad as everyone else. I didn't do anything.'\n\nThe bar showed no sign of closing, even though it was well after twelve. Some of the men at the bar were very drunk, their raucous laughter unnerving.\n\n'Would you like to come up to my room?' He blurted it out, his eyes firmly on the table. 'I don't mean. Well you know? I just want to get to the bottom of everything.'\n\n'So do I,' she whispered touching his hand tentatively. 'But here wouldn't be right. Come home with me?'\n\nHe didn't reply immediately, just closed his other hand over hers and squeezed it.\n\n'Just my luck,' he smiled ruefully. 'I get a free room in a posh hotel, then I get dragged out of it by the most dangerous woman in the world.'\n\n'You can lock up the knives if you like,' she joked. His hand over hers was making her melt inside and she wanted to kiss him so badly it hurt. 'I seem to remember my old Peter wishing he could be alone somewhere with me.'\n\n'Don't tease me Georgia,' his voice was suddenly gruff. 'Just seeing you again has brought back so many memories. But we were only kids then. We've both moved on.'\n\n'I don't feel as if I have.' He was overwhelmed by who she was, her money and fame. She had to convince him none of that mattered and she was still the same girl he knew. 'You've always been special to me. Come home and let's really talk.'\n\n'But is it safe?' he said, glancing round the bar. 'Won't there be reporters about?'\n\n'Not tonight,' she smiled. 'But they'll be round here in their droves tomorrow morning.' She stood up and held out her hand. 'Please, Peter?'\n\nPeter looked into those dark, magnetic eyes and felt as if he were drowning. When Phillips offered him an all-expenses trip to London he almost laughed at the man. He would have walked all the way, hitchhiked, even swum if there was a chance of seeing Georgia. Yet even in his wildest dreams he hadn't expected her to come here and take him off to her flat.\n\nShe was far more shapely, her little pointed chin more defiant, hair longer, and far more lustrous than he remembered. Away from him she'd grown into a woman, so desirable he could feel himself responding.\n\nThere had been many women in his life, some he'd even thought he loved for brief moments. Yet the moment Georgia stood in front of him, he understood why he had never been able to commit himself to anyone for more than a few weeks.\n\nBut he was scared now. What if her interest was only friendship? They were just kids six years ago. Was it possible to rekindle a small spark after all this time?\n\n'I don't know how I'm going to control my kids when I get back.' He had to make jokes, anything to avoid inadvertently saying what was in his mind. They were walking up the stairs now towards the foyer. Could he take her hand again, or would she see that as a pass?\n\n'Kids?' she stopped in her tracks. 'You got married then?'\n\nWas he imagining it, or was that panic in her voice?\n\n'The kids at school, silly. Do you think I would have shot down here so quickly if I had a wife and children?'\n\nShe giggled, that girlish sound he remembered so well.\n\n'I suppose not.'\n\n'Maybe they'll have new respect for boring old Radcliffe,' he said as he held the door open for her. 'But the Head won't be that thrilled to find his English teacher is a man with a past!'\n\nRussell Square was deserted now. Street lights cast yellow light over the railings, highlighting daffodils planted in the grass.\n\n'Did you ever tell anyone about me?' Georgia asked. Not once had Peter said anything that related to their relationship. How could she come out and ask if there was another girl?\n\n'No. I wanted to, especially when you made your first record. I guess I thought no one would believe me.'\n\nPeter stopped by her car, running his hand appreciatively over the sleek red bonnet.\n\n'That's some car,' he grinned round at her.\n\n'Would you like to drive it?' She took her keys out of her bag and waved them at him.\n\n'You mean it's yours?' His eyes grew huge with surprise, his lips trembling with schoolboyish wonder.\n\n'Well, I wouldn't suggest you drove someone else's car,' she laughed. 'You have got a licence?'\n\n'Yes. Celia taught me. Can I really?'\n\nAs he bent to unlock the door, Georgia had to touch him. She slid her hand under the short denim jacket and ran her fingers down his spine.\n\nHe straightened up and turned to her, so close she could feel his breath on her face.\n\nSlowly his hand came up, cupping her face, his eyes looking right into hers.\n\n'Oh Georgia,' he groaned.\n\nHis lips moved slowly towards hers, just like the dreams she had tormented herself with so often in the early days in Berwick Street. Her heart quickened, hands reaching out to hold him, body aching to touch his.\n\nSuch a slow, deep kiss, as if the years since her birthday were just a few hours. Every nerve tingled, yet at the same time the kiss held all the practised skill of a lifetime spent with each other.\n\n'People will think we're barmy,' he said at length, still holding her tightly, but covering her face with little kisses. He wanted to go on kissing her, tell her what was in his heart, but he was still wary of her. 'We walk out of a perfectly good hotel and stand kissing in the road.'\n\n'Let them watch,' she said, lifting her lips to his. 'I hope they're green with envy.'\n\n'I don't know which is best, kissing you or the thought of driving that car,' he whispered, holding her so close she could feel every line of his body. 'But I know we can't do both at once.'\n\n'And to think I was frightened of a rival in Manchester,' she playfully tapped his face. 'Come on then, drive!'\n\n'I haven't told you where I live,' she said as he took a road down towards Blackfriars.\n\n'I found that out today,' he turned to her and half smiled. 'I was torn between going straight home tomorrow or bearding you in your den.'\n\n'Which had you decided on then?'\n\n'I hadn't,' he grinned sheepishly. 'My friends back in Manchester would never let me forget it if I didn't see you. But then I reckoned it would be worse for my ego if you said \"thank you and goodnight.\"'\n\n'You didn't used to care what people thought,' she said. 'And I don't remember you having much ego either.'\n\n'I'm not the pure, idealistic boy you remember,' he said, glancing round at her. 'I've done things you wouldn't like, used people for my own ends. Maybe that's why I don't feel so angry with Max Menzies. I can identify a little with him.'\n\n'That's daft,' Georgia said briskly. 'You're nothing like him.'\n\n'Think about it,' he said quietly. 'We both wanted to escape our backgrounds. I used books, Max used his looks and wits.'\n\n'You'd never be as corrupt as him!' she said in horror. 'Besides how do you know so much about him?'\n\n'I made it my business to find out,' he smiled. 'He fascinated me. He was obviously in love with you. He's a millionaire, manipulative, handsome. A legend in his own time. I discovered that one of the reasons he is such a success, is because he does his homework. He checks people out, puts them under a microscope. I can appreciate that, I do it too.'\n\nGeorgia just sat there, watching the road ahead. Was Peter trying to tell her something? This wasn't what she had expected.\n\n'Don't look like that,' he reached across the seats and took her hand. 'I had to work like a dog to stay at university, building sites in the holidays, behind a bar at night. I've seen people who are a darn sight brighter than me end up in some boring job, and people with nothing but sharp wits get out there and make something of themselves. Making a pile of money isn't my goal. But doing something worthwhile, becoming a big person is. Do you understand?'\n\n'I think so,' she smiled weakly. 'But you teach. How does that fit in?'\n\n'Maybe it was Celia who put the germ of the idea in my head,' he said. 'Africa, India. All those people crying out for education. Right now it's Branscombe Road Secondary Modern, but the world doesn't end there.'\n\nPeter's face was a picture as they entered her flat. It was just the way it had been the first time she took him home to tea at Blackheath. Back then, he had stood wiping his feet on the doormat, staring around the hall with its thick carpets, the grandfather clock, the pictures on the wall, sniffing the smell of fresh-baked scones as if he'd entered a new world. Almost like a stray dog being taken into a real home.\n\n'This is the business!' His voice was husky with reverence. He looked up at the Italian lights, the vast expanse of apple-green carpet leading down the long hall. 'Can I be nosy?'\n\nShe took his hand in hers and led him first into the lounge, giggling as she put on all the lights.\n\nHe was so much bigger than he'd been that day over six years earlier in her parents' home. The child-like wonder was still there, but mixed now with maturity. He stood silently, feet apart, his eyes sweeping over the white settees, the stereo equipment, the antique writing desk and her collection of mementoes from every country she'd sung in.\n\n'It's best by day,' she said, clutching at his arm and taking him over to the window. She pulled on a cord and the long curtains drew back to reveal the small balcony and view over the Thames. To their left was Albert Bridge lit up like a glittering spider's web, to the right Chelsea, the river joining them like a slick of black tar, reflecting back silver lights on the far bank. 'The sun comes in here all day. At night in the summer I sit out there and watch boats going past.'\n\n'How long did it take to put me aside?' he said softly, switching the lights off and joining her at the window.\n\n'I didn't,' she said turning to him, putting one hand on his cheek and stroking it. 'Maybe the pain stopped after joining the band. But you were always there. I used to scan audiences when we did university gigs, hoping against hope you'd be amongst the crowd. Each record I made, I imagined you listening.'\n\n'It was the same for me,' he said softly, running one hand down her back. 'Every time I spotted a dark slender girl in a bar or club I'd rush over to check her out. But then when \"No time\" reached the charts your face was everywhere. I read everything, kept a box full of reviews, interviews and press pictures. Yet the more famous you became, somehow the more distant you seemed.'\n\n'And now?' she whispered.\n\n'I feel just like I did that night on the heath,' he said, running one finger down her cheek-bone. 'Wanting you so much, but afraid.'\n\nNothing compared with that moment. Not singing, applause or driving her Mercedes for the first time. His hands held her face, his lips came down on hers and it was as if Battersea fun fair with all its lights and music were turned on in her head.\n\nAll the pent up emotion of the day, all the years of missing one another vanished at his touch. They were back on that landing outside her playroom, the sounds of the Everly Brothers wafting out to them, two bodies and minds as one.\n\nBut this time there was no hesitation, no question of holding back. Peter's fingers were already under her sweater, reaching greedily for her small breasts.\n\nClothes torn off and tossed heedlessly away. Each touch electric. No time for careful foreplay. Just two bodies devouring each other, out of control, possession more important than subtlety.\n\nNot the settee or even the bedroom. Peter lowered her to the floor right there under the window, covering her naked body with his. Georgia wound her legs around him, clawing at his back, demanding his lips. It was animalistic, brutal, yet perfect. She arched her back to draw him into her, tears streaming down her face.\n\n'Georgia,' she heard him shout as he came, and although it had been over too soon, it was enough for the moment.\n\nAbove her head Georgia could see the moon shining in on them. Peter's body on hers was hot and sticky with sweat, his face buried in her shoulder and she knew he was crying too.\n\n'I love you Peter,' she said drawing his lips back to hers. 'Nothing's changed has it?'\n\n'I didn't dare even dream this,' his cheek against hers was damp. 'Just to see you was enough. Don't ever go out of my life again.'\n\nThey had a bath together later. Lying each end of the big tub. The room was once a bedroom, and the interior designer had pulled out all the stops to make it memorable. A mural of a jungle went right round the walls, and over a shower cubicle. Exotic birds, flowers, even a monkey grinned down at them. Real potted palms and a dark green carpet added to the illusion.\n\n'This is wonderful,' Peter rubbed soap over her breasts. 'The bath where I live you wouldn't take your dog, let alone your girl.'\n\n'I had a dose of places like that,' she said softly and bit by bit the past came out.\n\nHelen. The abortion. Her hopes with the band, the disappointments, Ian, Rod and Max, the men she'd dated, the places she'd seen.\n\nPeter told her about girls. Working on a building site, a summer on a Kibbutz in Israel, hitchhiking over Europe. Brief, vividly-painted sketches of six years packed with experiences so different to hers, yet enabling them to come together now as equals.\n\nThe water was getting cold. They climbed out and wrapped themselves in warm towels, moving on to the bedroom.\n\n'No one's ever been here,' she said as she sank down on to the big bed, covered in the Nottingham lace bedspread Max had bought her. 'This is all ours.'\n\nIt was a beautiful room, decorated in a very soft pink with cream. Totally feminine, from the soft pink lamps to the ruffled lace at the windows.\n\nPeter stood just looking at her. Dressed only in the white towel round his middle he looked like a Christian slave. His peachy skin, his legs and arms covered with fine blond hair, yet his broad chest as smooth and silky as a child's.\n\n'I think you are the most beautiful man I ever saw,' she said softly, getting up to move towards him. Her heart was racing again. She had that churning in her stomach the way she did when she first met him.\n\nHe came towards her slowly, reaching out and taking the band from her hair, then teasing it with his fingers till it fell over her naked shoulders.\n\n'You aren't just beautiful,' he murmured. 'You're exquisite. Can you possibly imagine how much I love you? For six years I've studied your pictures, listened to other men's fantasies about you and all the time I remembered the touch of your lips, the curve of your breasts and kept it to myself. I've told other women I loved them, but until now I didn't know what it meant.'\n\n'Oh Peter,' she stroked his chest lightly. 'I feel so happy.'\n\n'I used to try and imagine your breasts, when we were in the choir,' he said huskily, his fingers tugging at the towel covering them. 'I wondered what colour your nipples were, and whether I'd ever get to touch them.' The towel fell away, and he put one hand on each small pointed breast, running his fingers over her erect nipples. 'So they were dark, like chocolate buttons.' He knelt down on the floor in front of her, burying his head in them.\n\nGeorgia couldn't speak. She felt as if she could burst with tenderness and longing. The warmth of his body, gentle sensuous fingers, lips and tongue were driving her wild with desire.\n\n'Let me look at you,' she whispered, unwrapping his towel from round his waist. 'I want to watch it grow big.'\n\n'Men are so ugly next to women,' he said with a hint of embarrassment, but he stood up and let her take his towel away.\n\n'Not you,' she said softly, her hand going down to his penis. 'I used to wonder about this too in choir, it's a wonder we ever sang a note.'\n\nThere was no headlong rush now. His tongue and lips crept down her body, teasing and probing. He knelt in front of her just looking at her, touching, stroking and exploring. Time and time again she pulled him back to her, reaching for his lips, winding her legs around him. But still he made no attempt to enter her. He turned her on to her stomach licking her spine and massaging her back, then more kisses and stroking till she felt she would burst with longing.\n\nHe was heavier and stronger than Rod and Ian had been. He had Ian's sensitivity, Rod's passion and understanding of women, yet just enough of the brute in him to eclipse everything that had gone before. One moment so gentle she felt herself slipping away into a dream world. The next he crushed her into his arms as if he wanted to devour her.\n\nNever before had she taken the initiative in love-making, hers had always been the passive role. But now she wanted to please him and she moved down the bed to take him in her mouth.\n\nShe watched him as her tongue slid over his penis, his hands just touching her head, his mouth open, breathing heavily, a look of exquisite bliss in his eyes.\n\n'No more,' he groaned, 'It's too much.'\n\nGeorgia moved to climb on to him, but he grabbed her, rolling her onto her back and kneeling between her legs, pushing himself into her and enfolding her in his arms.\n\nA roller-coaster of pleasure, a feeling that any moment she would explode like a volcano. Moments of tenderness, more of savagery. Rolling together as if they were one person.\n\nShe remembered seeing his face at the moment her orgasm came, his eyes searching hers, a look of adoration more beautiful than anything she'd ever seen before, and his lips found hers at the moment her body erupted like a firework.\n\nHot and damp they lay entwined, Peter's head on her breasts. Tears welled up in her eyes and trickled down her cheeks.\n\n'Why the tears?' he whispered, wiping them away with one finger. 'Is it the pain of too much tenderness?'\n\n'That's a lovely way to put it,' she smiled up at him, tracing round his wide mouth with one finger.\n\n'It's not original,' he smiled sheepishly. 'I got it out of a book.'\n\n'Tell me about it?'\n\n'It's quite long and I don't know it all.' He rolled over and pulled her into his arms, nestling her head on his shoulder.\n\n'Go on,' she said softly. 'I want to hear it.'\n\nHis voice so deep and warm sent shivers down her spine.\n\n'\"Love has no other desire but to fulfil itself.\n\nBut if you love and must need have desires, let these be your desires.\n\nTo melt and be like a running brook that sings its melody to the night.\n\nTo know the pain of too much tenderness\".' He looked down at her, wiping away the last of her tears.\n\n'\"To be wounded by your own understanding of love, and to bleed willingly and joyfully.\n\nTo wake at dawn with a winged heart and give thanks for another day of loving.\n\nTo rest at the noon hour and meditate love's ecstasy.\n\nTo return home at eventide with gratitude.\n\nAnd then to sleep with a prayer for the beloved in your heart and a song of praise upon your lips.\"'\n\n'That's beautiful,' she sighed.\n\n'The whole chapter is even better,' he said softly. 'I used to read it and think of us. No one could ever touch that special place I kept for you.'\n\n'But Peter,' she whispered. 'You do know there are people out there who will try to come between us? My life isn't entirely my own any longer.'\n\n'Is anyone's?' He lifted one perfect eyebrow, turning over on his stomach to look down at her. 'I can't promise to be entirely yours either. I didn't spend all those years studying just to throw it away to be your consort.'\n\n'But will we be strong enough to stand it when people whisper things about our past? Old friends can be tactless.'\n\n'It's good to hear those incidents,' he said running his hand over her shoulders. 'Every person you meet and feel something for, leaves their mark. It's that which makes a well-rounded human being. So we couldn't share the last six years, but other people can put a new perspective to it.'\n\n'You are very wise,' she smiled. 'Is that what studying philosophy does for you?'\n\n'That was never more than a hobby and a second string,' he grinned. 'English is my main subject.'\n\n'There's so much about you I have to catch up on,' she said wistfully.\n\n'We've got the rest of our lives for that.'\n\n'Can we be that sure?'\n\n'Are you sure now?'\n\n'Yes,' she whispered. 'Absolutely!'\n\n'Well, that's all there is to it. Just take it one day at a time. Maybe in six weeks we'll find it was just a mirage. But for now it's real and beautiful. We don't need chains Georgia. The last few years have taught us that if nothing else.'\n\n'I love you,' she said, another tear dripping down her cheek. 'Nothing in my life has ever felt so right.'\n\nAll through the night they had to keep touching, as if to check that it was real. If they slept at all it was brief moments, only to wake to kiss again.\n\nAs the first rays of sunshine crept through the curtains, Georgia looked down at Peter. Her heart felt as if it could burst.\n\nShe was looking for a flaw, but she could see none. From his long knobbly toes, up the golden legs to his tight bottom, slim hips and smooth chest. Even the way he slept was perfection, legs splayed out, one curled up against her, one arm behind him, the other curled around his head. Golden lashes like brushes against peachy cheeks, even the stubble on his chin was blond. His lips were squashed against his arm, childlike and soft.\n\nShe knew then just how much power he had over her. If he asked her to give up singing, to live in a terraced house in Manchester as a teacher's wife, to give up her money, her car and never once again step out onto a stage, she would, willingly.\n\nAs if sensing her eyes on him, he woke, rubbed his eyes and smiled.\n\n'I love you,' he said, one sleepy hand coming up slowly to reach for her face.\n\n'You're so beautiful,' she said. 'I could watch you for ever.'\n\n'Just think what handsome devils our children will be,' he said pulling her back down to him.\n\n'Let's get up and run away?' she whispered. 'Even now those newspapers are plonking down on doormats everywhere. The phone will start ringing soon. It will be a circus and I've had enough of that.'\n\n'Where could we go?' His eyes lit up.\n\n'Somewhere warm,' she held him tightly against her.\n\n'But my passport's back in Manchester.'\n\n'So much the better,' she sighed. 'London airport is always swarming with press. We'll drive to Manchester now and catch the first flight.'\n\n'I haven't much money,' he retorted.\n\n'I've got lots,' she said. 'Now don't argue, just get in that shower. I'll ring Sam and tell him to keep the dogs off our scent.'\n\n'My, but you're bossy,' he grinned. 'What if I couldn't stomach being a rich woman's plaything?'\n\n'You aren't a plaything,' she said quickly, blushing as she realized she had been a little tactless.\n\n'So what am I?'\n\n'My love,' she answered with a toss of her head. 'So get going or I won't play with you anymore either.'\n\n## Chapter 26\n\nSam whistled cheerfully as he made his way up Berwick Street to Bert's caf\u00e9.\n\nSpring sunshine had finally found its way into the narrow streets of Soho. Shopkeepers were out sweeping their pavements. Stallholders were polishing apples, putting sale prices on winter woollies. The handbag stall was bright with pastel and white bags, flower stalls vivid with tulips. Even the pigeons had paused in their constant search for food to preen and coo in the sun.\n\n'She's my girl all right,' he thought to himself. 'Impulsive, hot-headed, but I wouldn't change her one bit.'\n\nLast night he had reluctantly left her to play at Ronnie Scott's. She insisted she didn't mind being alone; now she'd told her story she felt relaxed and secure. But he hadn't been sure. She looked tense and anxious to him, despite her delight that the editor had agreed to find Celia.\n\n'I can't keep you here wet-nursing me,' she laughed at his concern. 'Besides, it will be nice to be alone for a change.'\n\nWhen she rang him at his flat at seven in the morning, for a moment he thought some new disaster had erupted.\n\n'What is it? Has something happened?'\n\n'Oh yes,' she sort of sighed and he could hear happiness buzzing down the wire. 'I've found Peter, everything's wonderful.'\n\nHe didn't want to admit even to himself that he felt a pang of jealousy. But that was it, whether he liked it or not. His little girl had a man in her life now and maybe she'd never have room for him again.\n\n'That's wonderful, honey,' was all he could think of saying, as she rattled on about the pair of them running off to somewhere warm, Max's conspiracy to keep them apart and her conviction Peter was the love of her life.\n\n'Of course I'll sort things out here. Have a good time. You need time alone together.'\n\nHe couldn't go back to sleep again. Every time he shut his eyes, mean thoughts came to him. Suppose he was just another fortune hunter? What if she promised to marry him then regretted it later?\n\nHis other children plagued him too. He should get back to them. He had enough money now to bring them back to England and if he put himself about he could get enough work to keep them far more comfortably than they'd ever known. But somehow staying in England rested on admitting the truth to Georgia about himself.\n\nBy eight he was too wound up to stay in. He could see the newspaper man across the street waving papers and the caption on his box looked ominous.\n\n'RAPE! Georgia tells all.' It took him just a few minutes to jump into jeans and sneakers, then off across the road to pick up a paper.\n\nHe paused only briefly to check they had written her story correctly, then turning to the inside pages a picture of Peter Radcliffe leapt up at him.\n\nIt was a face of a real man, not a boy as he'd expected. Square jaw, bright eyes. An honest, open face.\n\n'I should have stayed with her that night,' Sam read. 'Anderson was drunk and in an odd mood, but you don't expect something like that to happen. Next morning when the police took me in for questioning was the worst day in my life. I may have only been seventeen, but I loved her. How could any man do something as animal as that to a child he'd brought up as his own?'\n\nHe spoke of the evening he discovered she'd run away.\n\n'Who could blame her? Her world was shattered, she knew the children's department would take her anyway. In her mind it was the only option. She believed that by removing herself from the picture, she could protect us all.'\n\nHe spoke of Celia, the beating she took from her husband when he returned home. The humiliation and the anguish of not knowing where her child was. The searching in clubs, bars and hostels.\n\nBut it was his final words that cut through all the doubts.\n\n'I never stopped caring. Maybe our worlds are too far apart for there to be anything but friendship between us. But I won't let that sick, tortured man hurt her further.'\n\nCharing Cross Road was heaving with people rushing to work. Traffic honked and snarled, exhaust fumes thick and choking. Yet for a moment Sam could have been standing in a garden.\n\nThis wasn't a boy wearing his heart on his sleeve, or some snivelling student with an eye to the main chance. It was a man who had held on to his love at all costs, who had searched for his girl. Someone who didn't give up, crumple under opposition as Sam had done himself. How could he have thought badly of the guy?\n\nHe took the paper into Leicester Square, sat on a bench and read and re-read the whole story.\n\nThere were a great deal of omissions, some facts bent. But it was written so well Sam could pass over that. It lingered over the horror, created an impression Max Menzies had been her Svengali, but in the main it was accurate.\n\nAs he sat there in spring sunshine he saw office workers flicking through the paper as they walked to work. By lunchtime the whole world would know the truth and with luck Georgia would be far enough away to escape further questions.\n\nSam knew where he wanted to be now. Up in the market where Georgia's real friends still lived. She had introduced him to so many of them while they were making the recording and he just knew the street would be buzzing with the news. It would be their opinion Georgia would want to know about when she telephoned again. So he'd just get himself up to the caf\u00e9 and join in the celebration.\n\nBabs was red-faced and flustered as she poured tea, fried eggs and brushed back her straggly hair all at once. Most of the tables were taken, filled with men in donkey-jackets and flat caps, huge greasy breakfasts in front of them and mugs of steaming tea. Her yellow apron was stained and greasy, a lank lacy collar, half in, half out of a matted blue sweater, yet there was new bounce to her normal shuffling gait.\n\nAs she saw Sam her round homely face burst into a wide grin.\n\n'Have you seen the news?' Her voice was squeaky with excitement. Heads turned to him, smiles of recognition on weatherbeaten faces. A sense of anticipation and a desire to know more.\n\n'Just read it,' Sam smiled, waved his paper at her and nodded to the men. 'I knew I could rely on you to be as happy as I am.'\n\n'I don't know why I feel so 'appy,' she wiped a tear away from her eye with the corner of her grubby apron. 'I started crying when I read what that man done to 'er. 'Er mate Janet always claimed that was what made 'er run away. I don't know why I didn't 'suss it out for meself. But it's that old boyfriend that's really got me going. Just look at him!'\n\nBabs held out the paper to Sam, pointing a wet finger at the picture.\n\n'A real movie star,' Sam grinned. 'Mind you, I've heard Georgia tell me how handsome he is for so long it's no surprise.'\n\nBabs gave him one of those long stares, like she was thinking something but didn't quite dare voice it.\n\n'Go on then, ask away!' Sam laughed cheerfully. 'Am I scared I'll lose her?'\n\nBabs blushed. 'Georgia said yous was just mates. Is that all?'\n\nBehind him he knew the men were waiting for his answer. His secret was bubbling inside him, like a child longing to tell a stranger it was his birthday. It would be so easy to tell Babs, she would burst into tears and give him one of those hugs she always gave Georgia. But he couldn't. It wouldn't be right.\n\n'Have you ever known her lie to you?'\n\n'No,' she giggled and blushed again, rushing over to rescue burning toast. 'She leaves things out, but doesn't lie,' she tossed over her shoulder.\n\n'Well she didn't leave anything out this time,' Sam retorted. 'It's true we are just mates. After all I'm \u2013' he stopped. He'd almost said it, old enough to be her father. 'An old man,' he added quickly.\n\n'I wouldn't say that,' she grinned wickedly, showing her broken front tooth. 'There's plenty round 'ere who wouldn't mind going a few rounds with you.'\n\n'I'm all tied up,' he joked. 'You've had my heart since I met you Babs.'\n\nShe giggled, showing more than a hint of the girl hidden beneath the slatternly apron.\n\n'Go on!' she reproved him. 'Me a married woman an' all. So do you think anything will come of it?' She leaned across the counter, her tired grey face alight with romance. 'I mean, he's clever, he's got letters after 'is name.'\n\n'Shall I tell you a secret?' Sam got up and leaned across the counter till his lips were right by her ear. She smelled of bacon and fried fat, a smell that took him right back to his own mother.\n\n'Go on,' she nodded, her eyes twinkling.\n\n'Something has come of it. She met him last night and they've run off together.'\n\nHer face mirrored his own pleasure. She slapped her hands over her mouth, tears sprang to her eyes.\n\n'Oh Sam,' she whispered. 'Really?'\n\nMaybe he should have been more discreet, yet in his heart he knew Georgia would have shouted it from the roof tops.\n\n'She rang me this morning,' Sam felt like a kid himself passing on an overheard secret. 'But don't you go saying anything to anyone, or I shan't tell you anything else.'\n\nHe had to go all over it again when Bert came in. What would happen at the studio? Would Anderson be charged with rape? Were Georgia and Peter getting married?\n\n'She ain't ever 'ad anyone to look after her.' Bert's customary gloomy face broke into a wide smile revealing blackened teeth. 'They've all made money out of her, worn her out, then wanted to kick her on the slag heap just because some old weirdo made up a load of lies. He deserves horse whipping.'\n\nGeorgia had told him so much about this pair. The frugal way they lived, the endless hours of work. A life that could be made easy by selling up and buying themselves a nice house. Yet he understood now why they stayed, they were the cornerstone of the community. They needed other people, the hustle and bustle. It wasn't money which kept them here, but long roots.\n\nThe caf\u00e9 had all the cosiness of his mother's kitchen when he was a boy. Gossip and speculation hanging in the air like damp washing. Each and every market trader was urged to look at the paper if they hadn't already seen it and as Sam sat eating eggs and bacon they plonked themselves down to question him.\n\n'What'll happen now? Will she be on the box tonight? When will we get to see her?'\n\n'No wonder Georgia loves it round here,' Sam said as he wiped up his egg with a slice of bread and butter. 'I never saw so many caring people in one place.'\n\n'Well she's our dream come true ain't she.' Babs shuffled round the counter to clear the tables. 'Not many people make it out of 'ere, not unless they're crooks. She's our pride ain't she?'\n\nIt was pride Sam's heart filled with too. That same feeling he felt back home in New Orleans when kids pointed to him in the street.\n\n'That's Sam Cameron, he plays a real mean horn.' These people might all be white but it was the same emotion that moved them as moved his folks. Screw the celebrities, the rich, and the tourists, making it was when your own people had that look on their face.\n\n'I bet the girls and Pop aren't doing a stroke of work this morning,' Bert said gleefully. 'They'll be all in 'ere soon, talkin' the hind legs off a donkey.'\n\n'Speaking of work,' Sam got up and felt in his pocket for some money. 'I guess I'd better go up to the studio and see how the land lies there. They say there is no such thing as bad publicity. I'll bet they'll be itching to get this new album out now, and they ain't gonna like it when they find their bird has flown.'\n\n'Stick that back in yer pocket,' Bert said shoving the money back across the counter. 'That's on me today, and if you've got time around six tonight come in and we'll have a few beers. Today we got some'at to celebrate!'\n\nSam was right. Everyone was in a turmoil at the studio. Phones were ringing, voices raised, teleprinters clattering. Max was in a meeting already with the chiefs and every desk had a copy of the paper spread out on it.\n\n'Do you know where Georgia is?' Ruth one of the secretaries rushed up to him. She had smears of mascara on her cheeks, eyes pink from recent tears, and this was a girl who was normally the blonde ice queen, efficient and unemotional. 'They are all going mad because they can't contact her.'\n\n'Sure,' he grinned. 'She rang me this morning. Shall I go in and put them out of their misery?'\n\n'Give her all our love,' she whispered, laying one cool hand on his arm. 'She deserves happiness after what that swine put her through.'\n\nOnce inside the boardroom Sam's elation vanished. His old jeans, grubby sweatshirt and sneakers, stubble on his chin, looked incongruous with their smart business suits. Apart from Jack Levy and Max he knew none of the other five men. All dark and Jewish, navy suits, white shirts and club ties. They cast suspicious glances at him over horn-rimmed glasses. Tight, humourless lips, faces that could have been born middle-aged. Plump, white hands rested on the polished table, water, fountain pens in readiness before them.\n\nDown in the market there had been joy of a wrong righted. Here there was nothing but the smell of money.\n\nHe remembered Georgia telling him about the day she signed her contract with them. No wonder she had found it so terrifying.\n\n'Good to see you Sam,' Max frowned as if it wasn't good at all. 'Where's Georgia?'\n\n'She's gone away,' Sam replied, pulling up a chair unasked.\n\n'What on earth does she think she's playing at?' Max exploded. 'She should be down here.'\n\nSeven indignant faces turning to him. Eyes narrowing at plans thwarted.\n\n'Now just one moment,' Sam felt a bubble of anger in his gut. 'Am I hearing right? You guys abandoned her, right at the time when she needed support.'\n\n'We did nothing of the sort.' Jack Levy drew himself up behind the table, dark eyes blinking furiously behind his glasses. 'We had no alternative but to suspend recording.'\n\n'Not one of you called on her, wrote her, sent her flowers,' Sam's eyes flashed from face to face. 'Did anyone of you go to the press and tell them what a nice girl she is? Did you hell!\n\n'You sat on your fat arses, panicking that the golden goose was finally about to be killed off. And probably had a meeting just like this one to decide who would get the carcass.'\n\n'That's not fair Sam,' Max flushed with anger. 'We had to act impartial.' He drew a cigar out of his breast pocket and sniffed it reflectively.\n\n'You acted like cowards,' Sam hissed at them. 'Even if she had done everything that creep said, after all she's given you, you should have stood up for her. I've only known her a few short weeks but it seems I was the only person who truly believed in her.'\n\n'Well, where is she now?' one of the directors said. His face showed no emotion, just irritation that she wasn't here, cap in hand.\n\n'She's doing what she should have done some time ago. Putting her own affairs in order,' Sam said, glaring round at each one of them in turn.\n\n'But the album,' Max said.\n\n'Screw the album,' Sam's voice was rising. 'I'd like just one of you to express concern for her. To show some emotion. She might find her mother. How ashamed you are that you doubted her? Or even ask about the boyfriend?'\n\n'So that's it,' Max said, a sly look in his eyes, taking out a gold cigar cutter. 'I suppose she's off screwing him.'\n\nSam leapt out of his chair and caught Max by the collar of his suit jacket, lifting him clean off the floor, scattering cigar and cutter to the floor.\n\n'You motherfucker,' he hissed. 'You stopped that boy from seeing her. You lied to them both. Why Max? Why?'\n\n'How was I to know?' Max's voice whined. 'Countless blokes claim to know her. I was protecting her.'\n\nSam let go of him, but his fists were clenched. He looked round at all the other men, searching their eyes to see if they had guessed the reason. There was interest, surprise, but no real understanding.\n\n'You wanted her yourself,' Sam snarled. 'I can understand that, but why then if you wanted her so badly didn't you stick up for her? You left the country with some blonde on your arm. What sort of a man are you?'\n\n'You've got it all wrong.' Max pulled at the revers of his jacket, his eyes flashed round the room trying to convey to the other men that Sam was mad. 'Georgia was like my own daughter.'\n\n'Then you,' Sam poked a finger at Max's belly, 'are just like Anderson. Because I've sure as hell heard you describe her as a cockraiser. Did you get to rape her too?'\n\nMax cowered back. Jack Levy leapt to his feet and moved swiftly over to Sam, putting a restraining hand on his arm.\n\n'That's enough, Sam,' he yelled. 'Enough!'\n\n'Enough?' Sam glowered round at the men. 'I ain't even started yet. Georgia is making what promises to be the finest album ever made,' he hissed. 'She's put her heart and soul into it, singing songs of love so beautiful they even move me to tears.' He pushed away Jack's hand, his mouth trembling with anger. 'But all you lot see is money,' he went on glaring at each of the men in turn. 'You don't see a little girl abused by a man she trusted. A girl with guts and fire that made things happen for herself without selling her soul in the process. Don't you know what you've got?'\n\nThe atmosphere was charged with electricity. Max shrank back against the wall, even Jack stopped short.\n\n'I'll tell you arseholes what you've got,' Sam yelled, waving his clenched fists at Jack. 'You've got a girl with more heart and guts than all us lot put together. Get off your chairs, get down in the studio and listen to those tapes. Forget how much money they'll make and listen to the message in her voice. When you've done that and found out what Georgia is all about, then maybe I'll tell you where she is!'\n\nHe paused. He saw the gaping mouths, sensed that his words had sunk in. Sam turned sharply, pulled open the door, and left, slamming it behind him.\n\nMax turned scarlet. 'I'm sorry about that.' His voice shook nearly as badly as his legs. He couldn't meet their eyes. For once he couldn't think of anything sharp to say. 'He's an artist, he can't help but be emotional.'\n\nJack Levy took off his glasses and polished them vigorously. He felt something he hadn't felt for years and he knew it was shame.\n\n'We'd better do what the man says.' He replaced his glasses on his nose. 'We'll talk again afterwards.'\n\nSam paused only long enough in the street to take a few deep breaths. He wasn't going to agonize over whether he'd gone too far, or whether he'd blocked all chances of his own career taking off. Right now he was going to ring that editor Phillips and make sure everyone was pulling out all the stops.\n\n'It's Sam Cameron,' he said, when they finally put him through. 'Georgia asked me to phone and find out the state of play.'\n\n'Where is she?'\n\nSam wanted to laugh, it was that same frantic question he'd heard in the boardroom, another would-be puppet master.\n\n'Why?'\n\n'I just found Peter had gone,' Phillips said. 'I just hoped they were together. As I see it that pair were made for each other.'\n\nSam felt all the anger inside him melt. At last someone had the right idea.\n\n'Between you and me, off to sunny places with Peter. The recording company is freaking out. They don't know where she is, so I'd be grateful if you didn't print that.'\n\nPhillips chuckled. 'Wonderful. That news makes it all worthwhile. Peter Radcliffe is a real human being. I'm keeping everything crossed for them.'\n\n'Yup, looks like you might get your big love story soon. But don't rush it, give them time. Now, about Mrs Anderson?'\n\n'We've managed to contact the health organization she is working for. The office is in Nairobi. Apparently she's way out in the bush running a small clinic and hospital.'\n\n'You don't say!' Sam's face broke into a broad smile.\n\n'We've got things in hand.' Phillips' voice had a ring of pure glee. 'It won't be instantaneous. Messages have to be sent by wire, the last lap will be by jeep over rough terrain. It could be a couple of weeks before they can get a replacement out for her.'\n\n'Did you tell them what it was about?'\n\n'Just the bare bones,' Phillips hesitated. 'I was apprehensive about the story being misinterpreted, so I promised to send full details by telex. I've just finished that.'\n\n'Georgia is going to flip,' Sam's voice was breaking with emotion. 'This is all too much. How soon before we know anything positive? I don't want to wind her up and then leave her dangling.'\n\n'I suggest you say nothing, yet,' Phillips' voice was more cautious. 'We don't want her rushing off to Nairobi and then missing her mother. As soon as we know she's on a flight home, that's the time to tell her. The poor kid's had enough grief to last most of us a lifetime.'\n\n'Sounds sensible,' Sam said. 'At least she's got Peter to take her mind off things. I'm very grateful to you.'\n\n'Has she said what action she's going to take about Anderson?' Once again he was just a reporter, wanting to be first with the news.\n\n'I couldn't say,' Sam said. 'If it were up to me I'd go round there and kick seven kinds of shit out of him. But Georgia isn't one for revenge. We'll just have to wait for that.'\n\n'Keep in touch,' Phillips' voice held warmth and sincerity. 'Let me know when your next gig is. I'd like to meet the man Georgia raved about.'\n\n'She spoke of me?'\n\n'Oh yes,' Phillips chuckled. 'You rate in importance along with Celia and Peter, but surely you knew that?'\n\nDown in the recording studio Jack Levy and his team were listening closely to the tapes Georgia had been working on. Heads bent forward, hands on knees, cigars, coffee and even note-making forgotten.\n\n'Sam was right.' Jack twisted his large gold ring around his finger during a pause, looking round at the other men with stunned eyes. 'This is some of the best stuff I've ever heard.'\n\nGeorgia's lush, rich voice filled the studio. The complex machinery, the plastic chairs, the glass partitions, the bright lights all softened in the music. With eyes closed, each one of them was transported to a place of beauty, memories and emotions long forgotten were stirred with her special magic.\n\nEvery note and instrument on the finished songs was impeccable. Sam's tenor sax sent shivers of delight down their spines, the strings, drums and piano all played their part in creating a masterpiece.\n\n'Sam's one hell of a player too,' one of the men said. 'We ought to get him under contract too, before he floats off back home.'\n\nAs the last note died away, Max got up. He felt drained, suddenly old and tired. He'd give anything to have Miriam back home to run to.\n\nWhy had he cheated Georgia, lied to her and held her back? Why couldn't he have been like Sam, listened, protected and encouraged? What made a man who had a bright and beautiful butterfly in his hand, crush it and still expect it to fly?\n\nGeorgia would merely laugh at the underhand things he'd done in the past. She accepted them all the way a zoo keeper expects the tiger to snarl at him. But by preventing that lad from seeing her, by turning his back on her when she was in trouble, that was when he dug his own grave.\n\n'What's up, Max?' Jack Levy squinted up at him through his glasses. 'Can't stand the heat any longer? She'll re-sign with us. They always do. We'll just have to offer her a better deal.'\n\n'I think I'm through with deals,' Max said. 'Sometimes they leave a nasty taste in your mouth.'\n\n## Chapter 27\n\nThe newspaper lay crumpled amongst empty sherry bottles, chip papers, cold cups of tea, and congealed greasy plates, hardly an inch of floor exposed from bed to window.\n\nThe stained china sink smelled like the lavatory he'd used it as, draining-board groaning with burned saucepans and jagged-edged empty cans.\n\nThe frayed brown curtains were no longer opened. The small table in front of the window strewn with pointers to a period of wealth. An empty whiskey bottle, a cigar box, the remains of an Indian takeaway meal thick with mould and a dead potted plant.\n\nDust, paper and food scraps were everywhere. Vomit lay on the floor just feet from where he lay huddled on his bed. The stench of himself, the vomit and rotting food combined to make the air unbreathable.\n\nOnly the wall covered in Georgia's pictures had any semblance of order and light.\n\nHe was sweating, so hot he felt he was on fire. But he knew soon the shivering would come back and nothing would warm him.\n\nDrink couldn't help now. Nothing could blot out the misery. No heat, money, drink or food. Trapped, sick and helpless.\n\nWhy was it that his mind had cleared now? Ever since that day when he was knocked down and robbed, the grey mist which stopped him hurting and thinking had vanished.\n\nHow long ago was it when those newspaper men came here? A week, two, maybe a month. He remembered putting a fiver in Mrs Dooley's hand and persuading her to clean up for him though.\n\n'Fancy her being your daughter,' she kept saying as she swept the floor and changed his bed. 'She should pay to put you in a home, you aren't capable of looking after yourself.'\n\nMrs Dooley wasn't the only one who treated him like a celebrity then. Someone left a bag of clothes on his doorstep, another neighbour brought him over a pot of stew. Every day the mail brought letters of sympathy, some with money inside them. Mrs Dooley was glad to clean for him. It gave her an opportunity to ask questions, she even referred to him as her 'poor old gentleman friend.'\n\nSo many visitors knocked on his door, offering invitations to their houses, sympathy, help and understanding.\n\nDown at the pub they all wanted to drink with him, they didn't refuse to serve him in the caf\u00e9, everyone said how badly Georgia had treated him.\n\nA taste of what it must be like to be her. A person people wanted to meet. He felt like shaving again, taking baths, eating proper meals, for a while drink hadn't been so necessary.\n\nChurch wardens came round and talked of re-housing him in a nice little flat. A lady down the road knitted him a blue pullover. Even the kids along the road began to smile at him.\n\nA hundred pounds seemed fair enough at first. He didn't know he'd been cheated until Adams made off with thousands. In that first week it had been enough to be important and know at last the public had turned against her.\n\nThe sickness had started the night he read just a few lines in the _Evening News_. A picture of the bitch standing by a flashy car, wearing a white mini skirt.\n\n'You can read the true story tomorrow,' she said.\n\nThat night even whiskey didn't help. A small voice kept whispering in his head, telling him things he didn't like. Later it turned to a gnawing pain in his stomach, just where she stabbed him. The nightmares came back too. Visions of him grabbing her, smooth skin under his hands, a rounded arse in front of him surrounded by harsh net petticoats. An act which until then he'd blotted from his mind.\n\nIt was Mrs Dooley that brought in the paper the next morning. The fat slut stood in his doorway, hands on hips, her hair in curlers, mouth like an angry red gash.\n\n'You bastard!' she screamed at him, flinging the paper in disgust. 'You filthy bastard! You'll rot in hell for what you did to her, and I'd gladly get you there a little quicker!'\n\nFunny that Georgia's retaliation didn't make him mad. He just lay there crying, remembering.\n\nWas it that mention of St Joseph's convent that made him think of her tiny, bony back, lacerated and weeping? What prompted the memory of guiding her down the pavement on her first bicycle, holding the saddle and urging her to steer and keep pedalling? Holding her on his shoulders to see penguins at the zoo. One hand under a smooth, soft tummy as he taught her to swim.\n\nPeter too. Sharp, clear pictures of him eating Christmas lunch with them. A red paper hat resting on his blond hair, talking about cricket, laughing at Brian's stories about people in the office. The good feeling at having male company.\n\nOther things wafted back. Georgia coming into the bedroom in her nightdress, with a tray of tea for him and Celia, her stocking under her arm.\n\n'I waited as long as I could.' She had that expression on her face that always made them smile. Wide-eyed, mouth trembling, a please-don't-be-cross face that worked everytime. 'Seven o'clock isn't that early?'\n\nOnce they'd put sugar mice in her stocking, tiny dolls, pens and pencils. That last year it had been makeup and stockings and a silly false nose and glasses she wore most of the morning.\n\nWhy was it now when he needed the grey mist, it didn't return? Sharp memories like Georgia sitting by his knee. The Christmas tree filling the room with the scent of pine, the fire banked up. Celia in a blue costume. Georgia in a tartan dress with a lace collar. He could see that book on photography she gave him. A shiny red and black jacket, the spine two inches thick, one he'd intended to buy for so long.\n\n'Mum didn't give me the money.' He could hear her soft voice shaking with excitement, feel her lips on his cheek, her arms round his neck. 'I saved it up myself.'\n\nHe knew when the reporters came back he would get no sympathy. He cowered in his bed listening to them scrabbling round the house, terrified they would burst in. He heard neighbours shout things outside the window.\n\n'Come on out you pervert! We'll show you how we deal with rapists round here.'\n\nJust enough strength to push a chair under the door, then stumble back to bed, hoping they wouldn't hurtle a brick or a fire bomb through his window. His chest, legs and stomach ached, but the worst hurt was inside his head.\n\nHow many times was it that he read that newspaper? Twenty, thirty? He lost count.\n\n'I don't know why he changed that night,' he read. 'One moment he was my dad, the sweetest, kindest man alive, the next like an evil stranger. Everything I knew about men came from him. I loved being in the car with him, holding his hand when we went for walks. The way he hugged me when he came home from work. He knew everything. He helped me with my homework, he taught me to swim. He clapped when I danced and sang. My mum and dad were the best parents anyone could have. I had nothing to rebel against. I felt loved. I didn't even mind when he got drunk and came up to the party. Everyone thought it was funny. I told him to go to bed after Peter had gone. But that's when he changed.'\n\nIt wasn't a nightmare after all. He really had done those things which haunted him. Soon the police would come for him. They'd lock him up, maybe even beat him. If only he had enough money to put in that meter, to turn the gas on and wait for oblivion.\n\nSomeone was insistently ringing the door bell. He heard Mrs Dooley shout to one of her children to answer it. Deep, male voices, too low to hear what they were saying.\n\n'He's in that room,' Mrs Dooley's Irish voice boomed out. 'He hasn't shown his face for nearly two weeks. But he's in there all right, more's the pity. Filthy bastard, you can smell him from the hall.'\n\n'Has anyone got a key?' The male voice was crisp and tough, the sort of voice belonging to someone with authority.\n\n'Don't think so.' Her voice was coming closer as if she was walking down the stairs. 'The landlord was supposed to be coming over to heave him out in a day or two. He hasn't even been out to use the toilet. God knows what you'll find in there.'\n\nHe buried his face when the rapping on the door started again. Was it night, or merely the dim light?\n\n'Mr Anderson!' That strong voice again. 'Mr Anderson, open the door or we'll have to break it down!'\n\nHe was sure it was the police. Reporters didn't threaten violence. He screwed up his eyes, huddled further under the blanket and waited, too sick and weak to make any protest.\n\nA thump and a splintering noise and they were in.\n\n'Bloody hell.' P.C. Blake clamped one hand over his nose and waved to his partner to open the window as he moved over to the bed. Cautiously he pulled back the thin blanket to find Anderson staring up at him blankly.\n\n'Are you all right mate?' he asked, his stomach churning.\n\nThere was no reply. Just those pale frightened eyes looking at him, a haggard, almost shrunken face glistening with sweat, flecks of white foam on his blue lips.\n\n'Get an ambulance,' Blake turned to the younger man standing gasping by the window. 'Warn them about the conditions. Book a fumigator afterwards.'\n\nAs the constable rushed back gagging to the door, Blake's professionalism got the better of revulsion. He lifted one scrawny wrist from the sopping bed and felt for a pulse. 'You've got yerself in a right state,' he said. 'It's hospital for you.'\n\n'I'm sorry for what I did,' Brian whimpered. He tried to sit up, but he was too weak. 'Will I go to prison?'\n\n'Don't look that way,' Blake moved away from the man's fetid breath. He glanced up at a picture cut from a glossy magazine. Georgia was sitting astride a cane chair, one arm leaning on the back, drinking a glass of milk wearing shorts and a T-shirt. 'It was her that asked us to check you out. Not a moment too soon I'd say.'\n\n'Georgia asked you?' Brian tried to focus his eyes. All he could see was silver buttons against blue serge as once again his bladder overflowed.\n\n'Welcome home,' Sam threw open the door as he heard the lift.\n\n'Sam!' Georgia launched herself towards him, arms wide to hug him. Peter was left in the lift with a suitcase.\n\n'I came over to make a meal for you,' Sam said. A lump came up in his throat, making it hard to speak. Her warm body pressed against him, the perfume of her hair, her lips against his neck. 'I felt I had to talk to you before everyone else grabbed you.'\n\nShe held him still, looking up at him, nose twitching, like a stray dog hoping for a meal, big eyes dancing.\n\n'It smells wonderful,' she said. 'But I'm being rude. This is Peter, I keep forgetting you haven't met before.'\n\nPeter in the flesh was far more striking than press photographs. Blue eyes alight with laughter, a rugged quality to his features. He seemed to fill the small hall; muscles straining under his thin jacket, blond hair streaked almost white by the sun, the golden tan, all gave the impression it had been achieved by a lifetime in rough country.\n\n'It's great to meet you at last,' Sam put out his hand and Peter gripped it firmly. 'Sorry I had to drag you away from the sun, but the people at Decca were getting frantic.'\n\n'We understood,' Peter grinned. 'I should be back at school anyway.'\n\nThey had been in the Canary Islands for nearly two weeks. Georgia looked black now, the whites of her eyes and her teeth flashing against her skin. She wore a red flouncy dress that made him think of gypsy dancers, bare feet in gold sandals.\n\nRest and love had done wonders for her. Skin glowing, eyes gleaming, she'd even put on a little weight. There was a calmer, softer look in her eyes.\n\n'I still don't understand what the panic is,' Georgia said as she bounced inside, gazing around her in delight. 'But whatever it is, it's nice to be home.'\n\nThe lounge was filled with late afternoon sunshine, lighting up the vivid primary colours of her Spanish rug and turning the white settees to pale gold.\n\nShe walked round the room, just reaching out and touching things as if telling them all she was back in charge.\n\nSam could see her eyes flitting out to the window-boxes on the balcony, her eyes lighting up at the clusters of giant pansies, blue and purple heads nodding at her as if in welcome.\n\n'I kept them watered,' Sam smiled. She was just like Katy, at heart a homemaker. Soon she would be running her fingers over ledges, making mental notes of jobs to be done. 'Now sit down and I'll make us a drink.'\n\nHe had to tell her tonight. Everything was moving so fast. He'd removed every possible obstacle to give him a clear field. He just had to hope no one came unexpectedly.\n\nPeter took the glass of beer and sank into a chair, but Georgia flitted in and out of the room looking at things as Peter described their hotel and the beach.\n\n'This is all a bit posh,' Georgia called out from the dining room across the hall. 'Come and see Peter, Sam's laid it all with flowers, and napkins. I didn't know you were so domesticated, Sam!'\n\n'There's lots you don't know about me yet Miss Smartypants,' Sam grinned, as he looked into see her straightening a knife here, a plate there. He too had been surprised to find a sideboard full of white bone china, polished silver cutlery in felt lined boxes and a wealth of starched tablecloths and napkins. Clearly Georgia hadn't rejected Celia Anderson's middle-class values. He wasn't going to admit that he had learned his skills while working as a waiter.\n\n'Now, I don't want you two to think I'm intruding on your last night together. I'll be off later.'\n\n'You don't have to go,' Peter touched Sam's elbow, his face full of concern that he might feel pushed out. 'We're both pleased to see you.'\n\nSam heard that deep voice, full of sincerity and knew this was a man he could respect. He wished he had time to get to know him the way he had Georgia, but there wasn't time for that now.\n\n'Thanks,' Sam grinned. 'We'll have lots of opportunities later to dig into each other. But first a drink and I'll dish up. I hope you like spicy food as it's about all I know how to cook.'\n\n'He's got something on his mind,' Peter said as Sam disappeared into the kitchen. He sat down on the settee while Georgia began sifting through records in the corner. 'Do you think he knows something about Celia?'\n\n'No, he would have told us immediately if he did,' she looked reflective. 'I hope there isn't something wrong with his kids. I couldn't bear him to leave England.'\n\nPeter shrugged.\n\n'He'll have to go sometime.'\n\n'Oh Peter,' she jumped up, dropping the record and bounded across the room to him. 'You aren't jealous are you?' She perched next to him, running one hand through his hair.\n\n'I guess so,' he grinned sheepishly. 'No, I'm not jealous of Sam exactly. Just a bit overwhelmed by your life. That welcoming reception at the airport, all those press hanging on to your every word. It makes me wonder about my role in your life.'\n\nHe'd read so often about Georgia being mobbed by fans, yet until he was in the thick of it himself it never seemed real. People grabbing his arm, microphones stuck right under his nose, the shouted questions, the flash of cameras, a feeling of terror that they could actually be crushed to death by this crowd.\n\n'You don't have a role in my life,' she smiled. 'You are my life.'\n\nThey had spent so much of their time away sounding out each other's ambitions. Georgia's went no further than finishing her album and finding a home out of London, but Peter's ideas were more altruistic. His dreams were filled with education for everyone, decent homes and proper health care and it was apparent to Georgia that the idealistic boy with missionary zeal had grown into a humanitarian.\n\n'Well, sweetness,' Peter put his hands on her neck, lifted her hair, then bent to kiss her ears. 'One thing's certain. Branscombe Road Secondary Modern isn't going to be thrilled at such an infamous teacher in their midst.'\n\n'If they're that small-minded it's the perfect excuse to walk out,' she grinned.\n\n'I don't walk out of anything until I'm ready,' Peter replied sternly. 'And I certainly won't leave them in the lurch just to be one of your acolytes!'\n\n'Well, what do you think of Creole cooking?' Sam said as finally their empty plates were pushed away.\n\n'Superb.' Georgia sat back in her chair, holding her stomach, grinning like a greedy child. Lazily she leaned forward, filling up the wine glasses again. 'If the music world lets you down you can always become my cook\/housekeeper.'\n\nSam had entertained them during the meal with gossip. The press's speculation about their future together. People from Berwick Street and the club scene too. He said how United Artists had offered the boys a contract, and Norman had written some brilliant music. Rod was taking singing lessons and claimed he was in love with a model called Patti. Speedy was straight still and the others were trying to influence Les to join him.\n\nSally and Janet had finally been offered new houses in Harlow, and Pop, faced with losing his two most reliable workers, was looking for small factory premises there too. Even Babs and Bert were seriously contemplating retiring.\n\n'You'd better let me meet them all soon,' Peter said. 'Otherwise they'll all be gone.'\n\n'We could throw a party,' Georgia's eyes lit up. 'How about Whitsun when you're on holiday, Peter?'\n\n'We'll talk about that some other time,' Peter groaned. 'I think Sam's got something on his mind.'\n\nIt had grown dark outside while they talked and Georgia stood up to draw the red curtains and turn on a small lamp on the sideboard. She'd lived in this flat for nearly two years, but this room was hardly ever used. It gave her a glow of pleasure to see how warm and inviting it could be.\n\n'Is it your kids?' Georgia touched Sam lightly on the shoulder before she sat down. 'Or has someone offered you a contract?'\n\nSam looked down at his empty plate. Chopin was playing softly from the lounge across the passage, the traffic down below had slowed to a mere purr. He couldn't stay here all night. He had to tell her now.\n\n'Neither of those,' he said. 'It's about you, honey. Hell, I don't exactly know how to put this,' he paused, biting his lip. 'I want you to think about your real background. I mean your natural mother and father.'\n\nGeorgia made a face, putting a finger in some sauce and licking it.\n\n'Is this a \"let's face the black side of yourself\" routine?' she said. 'A warning that mixed-race relationships are doomed from the start?'\n\nSam chuckled. 'No. There's sure as hell plenty of others will say that for me. I meant don't you ever wonder how you came to be abandoned, orphaned or whatever it was?'\n\n'Of course I do,' she smiled. 'But Celia tried to dig around, didn't she Peter? She didn't find much.'\n\n'All the records were destroyed,' Peter said. 'She was pretty certain Georgia was in the Billericay war orphans home, but there's not even any real evidence of that. What's made you bring this up?'\n\nHe could sense an undercurrent, something Sam had been brooding about for some time.\n\nSam cleared his throat nervously.\n\n'Before I met you Georgia, when I first arrived in England, I did some digging myself.' He paused looking at Georgia through thick curly lashes. His Southern drawl suddenly seemed more pronounced, or was it he was choosing every word carefully? 'I told you I was here during the war, and somehow it seemed important to just go down to my old haunts and look around.'\n\n'An old flame?' Peter smirked.\n\n'Yeah,' Sam was smiling, yet his eyes were sad. 'She just stopped writing, you see. It happened all the time. One day the girl's crazy about you, the next she's got cold feet.'\n\nPeter reached out and picked up the wine bottle, dividing up the remains between the three of them. He was sure this was leading to a serious warning about mixed marriages whatever Sam had said previously. Perhaps a pep talk to Georgia about accepting that Peter had a career too. He had noticed Sam studying him closely. Was he doubtful they could make it as a couple?\n\n'She was white?' Peter said, raising one eyebrow. 'Come on then, give us the whole story.'\n\nSam took a deep breath and began. His meeting with Katy at the base, falling in love and her parents' disgust that their only daughter should choose a black G.I. How Katy moved out and found a flat in the East End so they could be together and then on to his departure to France.\n\n'Everything was kinda frantic,' he said, his big lips trembling a little. 'We didn't know where we was goin', if or when we'd be back. I never told Katy just how hard it would be for her if she married me. We just kinda lived for the moment. All we had was letters and trust.'\n\n'Did she write?' Georgia asked.\n\nSam nodded. 'Every day. Sometimes I didn't get any for weeks, then I got a big bundle. She used to number them. A funny little figure on the back of an envelope with our names written round it. But then they stopped.'\n\nFor a moment Georgia thought he was going to cry. No tears, just a twitching in his cheeks as if he were fighting it.\n\n'It nearly broke me. I couldn't eat or sleep. I grew bitter. It kind of sapped all my energy.'\n\nPeter nodded. He knew exactly what Sam had gone through.\n\n'But why didn't you come back?'\n\n'I wanted to,' Sam bit his lip again. 'But everything was crazy in Germany. Then I got wounded, nothing real serious, just a bit of shrapnel in my arm. Just enough for the M.O. to decide I was to go back Stateside instead of staying like the others to clear up the mess.'\n\n'Did you write again?' Georgia asked. 'Did you tell her what was happening?'\n\n'I wrote over and over,' Sam shrugged his shoulders. 'What was I to think? Black guys had enough trouble getting willing girls over. What chance did I have with one who didn't even reply?'\n\n'So that's what you went to dig up?' Peter said.\n\nSam nodded. 'It had all changed. I found Hughes Mansions where she lived, all right. But something was different. Some people I met there told me about a V.2 dropping on it,' he said, watching Georgia's face. 'Katy was amongst the hundred or so killed.'\n\n'Oh Sam, you never knew?' Georgia's face fell. 'You thought she didn't love you, but all the time she was dead. I don't know which is worse.'\n\n'There is something worse,' Sam took Georgia's hand. 'To find the girl I loved didn't tell me she had my child.'\n\nGeorgia's hands flew up to her mouth, eyes filling with tears.\n\n'The child died too?'\n\n'No, it was rescued,' Sam said slowly. 'I went to the East End looking to just reminisce. Instead I find I've got another child.'\n\n'Where?' Georgia leaned forward. 'Have you found him?'\n\n'Her,' Sam corrected Georgia. 'That was somethin' else. Blind alleys, disappointment, hundreds of old papers to go through without even a name to help. Finally when I thought I couldn't go no further I found an old social worker who'd taken a child from a foster home, on to a convent.'\n\nSuddenly Peter saw the truth. Not just Sam's words, but the way he was looking at Georgia. The eyes were the same, round and large, two sets of identical dark chocolate, the same delicate eyelids, even the lashes like brushes.\n\nGeorgia couldn't see it. She was too immersed in Sam's tale, grieving over a woman she didn't know and the sadness of her child being orphaned.\n\n'What's her name, Sam?' Peter said softly. Someone had to help the man, he could see Sam wanted to blurt it out but couldn't find the right words.\n\nTwo weeks ago Peter had only the image the press had painted of this man in his head. A brute who beat his wife and abandoned his children. What would a man in his forties have in common with a girl like Georgia? Wasn't it more likely he was using her to further his own career, building up her trust so one day he could get his hands on her fortune?\n\nOf all the things he feared most about Georgia's life, this man was probably the thing which worried him the most.\n\nBut on holiday Georgia told him about her Sam. The talented musician, the caring father, the good friend. A man who made no advances to her, asked for nothing. The man who had stayed by her side when everyone else turned away.\n\nBoth images had stayed with him, like two pans on a scale. The pans had teetered up and down since meeting him. One moment he was sure Georgia's opinion was right, the next he had his doubts.\n\nBut now the pan was thumping down on the table, the image the press had put in his head, flipping out of the window with the force. He could see tears gathering in the man's eyes, feel the emotion in his heart.\n\n'Was it Georgia?' Peter asked.\n\nSam's eyes closed, a tear trapped by his lashes trickled down his dark cheek.\n\n'Yes,' he whispered. 'Oh yes.'\n\nFor a moment Georgia just sat there, stunned.\n\nShe rested her elbows on the table, holding her head in her hands. Her eyes moved from Peter to Sam, back to Peter's smiling face, then back to Sam.\n\n'Me?' she questioned. She looked like a frog. Huge bulging eyes and mouth gaping open. 'I don't understand. Are you sure?'\n\nSam opened his eyes again.\n\n'The nun was Sister Mary from St Joseph's,' he said.\n\nHer chair tipped over and crashed to the floor as she leapt up. She zoomed round the table, flinging her arms round Sam, burying her head in his shoulder, unable to say anything.\n\n'You're glad, ain't you,' Sam whispered against her hair lifting her onto his lap and holding her tightly. 'If I'd knowed I had a little girl I would have come back. There ain't been a day since I knew about the bomb I didn't regret not coming looking for Katy.'\n\n'Oh Sam,' Georgia's deep sigh was buried in his neck. 'It's the most wonderful thing. I knew you were special. I just knew it.' She cried then, small shoulders heaving, wrapping her arms around him tighter.\n\n'Those had better be happy tears.' Sam disentangled himself, cupping her face in his big hands and lifting it to his.\n\n'She only cries when she's happy,' Peter stood up and moved closer to them both, putting one hand on Sam's shoulder, the other stroking Georgia's hair. 'You should know she grows silent and grim-faced when she's sad. Hell, Sam that is one hell of a story!'\n\n'Why didn't Katy tell you about me?' Georgia's face was a study of glee, curiosity and sadness all at once.\n\n'I guess she didn't know till after I'd gone to France, honey,' Sam explained. 'After Christmas she wrote and said she had a surprise for me, but she never let on what it was. Maybe she was afraid I might go AWOL. I expect I would have too. I guess she knew me only too well.'\n\nThey moved into the lounge, leaving the dishes on the table and Sam went on to explain everything.\n\n'Sister Mary!' Georgia was quivering with excitement. 'How is she? Where is she? Did she know who I was?'\n\n'She was the one who led me to you,' Sam smiled, tucking her hand in his big brown one. 'She told me I must get close to you first and win your trust, that God would guide me when the time was right. You were in America then. I got all your records. I used to wallow in them all day. Wonderin' how I could get to meet you.'\n\n'And I came banging on your door,' Georgia laughed. 'No wonder you looked so shocked!'\n\n'Sister Mary's been waitin' for me to tell you, without her to talk to I might have lost my nerve. She's dying to see you, honey.'\n\n'Why did you wait so long to tell her Sam?' Peter frowned. 'I mean there must have been so many opportunities?'\n\n'I very nearly told her the day Anderson was plastered across the paper,' Sam half smiled. 'But I was afraid it might make things worse.'\n\n'Why now then, Sam?' Georgia looked round at him, a puzzled expression in her eyes. 'You seemed kind of in a hurry earlier. Are you going back to the States?'\n\n'Not till we've finished that album,' he grinned. 'I've sent Jasmine and Junior your records and said I have a special story about you. They already think you're just about the best thing in the world.'\n\n'My brother and sister,' she said pensively. 'I can hardly believe it.'\n\n'So what is the panic?' Peter asked.\n\n'No panic,' Sam smiled. 'But I knew in the next weeks I'd hardly get a chance to talk to you privately together. You two are so in love you've forgotten what's going on out there,' he paused to look at Peter. 'You've got to get back to Manchester. Georgia's got the album to finish. Your contract with Decca is running out, and there's Max and Anderson's fate hanging in the balance.'\n\n'Why do you lump Max and Anderson together?' Georgia asked. 'What's Max been doing while I was away?'\n\n'Keepin' his head down,' Sam smirked. 'Snakes both of them, and if I had my way I'd do the skinning.'\n\n'You had a fight with Max?' Georgia giggled, her hand over her mouth.\n\n'Me, a Southern gentleman?' Sam said dryly. 'No. I just marked his card.'\n\nGeorgia looked at Peter, who nodded as if it was time she too revealed all the things they had discussed.\n\n'I'm going to manage myself in future.'\n\n'Good idea, honey,' Sam grinned showing brilliant white teeth. 'Max'll hate that.'\n\nFor a moment she hesitated, looking at Peter for support.\n\n'Not to punish him,' she said softly. 'He's a shark. He can't help taking big bites out of everything. Maybe I don't want to be in the same pool as him any longer, but I can't forget how much he's taught me.'\n\n'He'll see it as the same thing,' Sam said, shrugging his shoulders.\n\nGeorgia shook her head.\n\n'No Sam, not once I've talked to him. He'll always have a place in my life.'\n\n'What about Decca?' Sam asked. 'Don't tell me you've gone soft on them too?'\n\n'I'll screw them so hard they may have to wave me goodbye,' she laughed. 'I don't feel any loyalty to them, it's a good deal or goodbye.'\n\n'And I only thought she'd inherited my musical talent,' Sam smirked. 'She's got the brains too!'\n\n'You'd better tell him what you've done about Anderson too,' Peter smiled. 'Maybe he won't be so cocky then.'\n\n'I spoke to the police,' she said in a small voice. 'I told them I wasn't going to press charges, it's all too long ago. But I asked them to check him out, put him on to social workers if he needs help.'\n\n'Shit!' Sam's eyes flashed with anger. 'Georgia, that guy deserves hell, nothing less.'\n\n'Perhaps he's already had that.' Her big dark eyes were full of pity. 'Don't try and bully me, Sam?'\n\n'She's crazy,' Peter put one restraining hand on Sam's arm. 'But it's a good kind of crazy. Phillips did say he reckoned the man had flipped.'\n\n'I've got so much,' Georgia said softly, looking at them both. 'I couldn't bear to see him again. But by the same token I can't pretend he doesn't exist. At least by footing the bill to have him dried out or whatever, I haven't just stood by.'\n\nSam shook his head and got up.\n\n'You are so very like Katy,' he smiled down at her where she sat curled up on the settee. 'She was a great one for lame dogs, doubt if she'd have even looked at me if I'd been like Peter. She'd have been so proud of you, honey.'\n\n'Don't go Sam!' she said. 'There's so much more to talk about.'\n\n'Not tonight,' he smiled, moving across the room to her and bending to kiss her forehead. 'We've got years ahead to catch up. You and Peter have only tonight, at least that's the way I remembered separations when I was your age.'\n\nShe jumped up then, flinging her arms round him.\n\n'I love you Sam. We're a family now. You get Jasmine and Junior and bring them back. There's plenty of room for them here.'\n\n'All in good time, honey.' He nuzzled his chin against her hair as he held her. 'You're one helluva daughter.'\n\nSam turned up his jacket collar as he walked along by the Thames. If he lived another twenty years in England he doubted he could get used to the climate. English people talked about this weather as warm, well perhaps they ought to try New Orleans. The smell was kinda the same as walking by the Mississippi, tangy and dirty, darkness turning it into a thing of beauty. But the Thames wasn't his river, just as London wasn't his town.\n\nIt was tempting to let Georgia take over his life. He could imagine her finding a house big enough for all of them, playing big sister to Jasmine and Junior, pushing him into the limelight. Perhaps forgetting Peter and herself as she tried to make everyone happy.\n\nHe'd learned a great deal about kids from watching Georgia. He had to go back and put all that into practice with Jasmine and Junior, win back their trust and love before he thrust them into a cauldron of new experience.\n\nGeorgia had grown up in a white world, learned the hard way how to deal with prejudice, rising above it without losing her deep understanding of human nature. His kids had been born into discrimination and segregation. By replanting them hastily, surrounded by people anxious to make them happy, they might grow up rootless, without that need to achieve anything for themselves.\n\nHe stopped for a moment by Albert Bridge. Each strut covered in lights, like a bridge to fairyland.\n\n'Sure is pretty,' he said aloud, looking out over the dark river. At home in New Orleans the place would be jumping now with music and people. Here it was silent and empty, just the odd man, out with his dog, and a pair of lovers further on the bridge, arms around each other. It was a good place to come to terms with his thoughts.\n\n'Finish the album. Then go home. There'll be holidays, time for us all to get acquainted. You've come this far without climbing on someone's back Sam, Jasmine and Junior have to learn that too.'\n\nMaybe he should have told Georgia about Celia. He had planned to. In long telephone conversations he'd come to care deeply for this brave little woman who'd given his daughter so much. But too much emotion in one day didn't make for restful sleep and tomorrow morning would be soon enough to break the good news.\n\n'Only another hour and we'll be landing.' Tania the stewardess stopped by Miss Tutthill's seat to reassure her.\n\nAll the crew knew who this lady was and where she was going. Tania just hoped she could get off the plane herself quick enough to see the reunion.\n\n'Maybe I should have waited another day or two?' Celia Tutthill looked up at the tall, willowy redhead, elegant in her cream blouse and skirt. 'I mean suppose she hasn't got back from her holiday yet?'\n\n'But you said her friend would make sure she did?' Tania perched on an empty seat across the aisle.\n\n'I know,' Celia smiled, her greeny-grey eyes wrinkling up with pleasure. In her telephone conversations to Sam Cameron she had heard so much that pleased her. All she could do was hope he did manage to engineer getting her home, and telling Georgia the truth about himself, without giving her daughter more anxiety. 'I'm just being an old worry-guts. I'm even worried Georgia won't recognize me!'\n\nShe was two stone lighter than she'd been for much of her life, something that started with African tummy and she never put back on. Her light brown hair was longer too, waving at her neck with streaks of gold from the sun. Gone were the days when she wore tailored suits, now her wardrobe consisted only of shorts and shirts. The green floral dress she wore today was new, bought hastily in Nairobi before flying out.\n\n'Of course she'll recognize you,' Tania chuckled.\n\nNairobi had been buzzing about this woman.\n\nAccording to the gossip she had arrived in Africa with only the sketchiest idea of what was ahead of her. Everyone had expected her to last a year at most, before the heat, flies, disease and lack of equipment sent her running home. But Nurse Tutthill took one look at Africa, rolled up her sleeves and adapted.\n\nTania was surprised to find Celia so small. The image the gossips had created was one of a big fierce woman who scorned bureaucracy, fought tooth and nail for supplies of medicine and almost singlehandedly had vaccinated thousands. When that dark tan faded she would be just another apple-cheeked middle-aged lady, just like her own mother, the sort of woman equally at home in church garden f\u00eates manning the cake stall.\n\n'I've got a lot more wrinkles,' Celia's eyes twinkled as she touched her face tentatively. Out in the bush there were no luxuries like mirrors. Faced with herself in a hotel room she had been shocked to see the changes. Crows' feet round her eyes, cheek-bones where once had been pads of flesh. Even her arms and legs seemed to belong to someone else, muscular, sinewy and an indecent dark brown. Yet despite the passing years she liked herself better. Even as a girl she was never pretty, but now at least people described her as striking.\n\n'If I look like you at your age I'll be delighted,' Tania smiled, teeth like an advertisement for dentistry. 'You look ten years younger than that old picture in the paper. But don't you think you should try and sleep for a while? It's going to be a long, emotional day for you.'\n\n'I think I've forgotten how to,' Celia said thoughtfully. 'Since the day that message came I haven't had more than a couple of hours' cat naps.'\n\n'You look good on it,' Tania got up and straightened her uniform. 'Would you like a drink?'\n\n'No thank you,' Celia grinned. 'You can't meet your daughter with alcohol on your breath!'\n\nThe message had come third hand. It had been relayed first by telephone, then by radio and the final few miles by a man on a pushbike.\n\nIt was early in the morning, already very warm, the sun rising up from behind the mountains when she saw the messenger.\n\nHe was young and lanky, wearing nothing but a pair of baggy khaki shorts. His skinny brown legs stood out like paddles as he came blundering down a track to the hospital.\n\nShe watched as he put his feet down to stop himself, threw his bike down on the bare earth and disappeared from her view.\n\nIt hadn't meant anything. Every day people flocked to the hospital. When she looked out over the bush for as far as the eye could see there was nothing but waving yellow grass, a few thorn trees, no sign of human habitation, yet day after day an endless procession of the old, the sick, the lame and the blind made their way here for treatment.\n\nShe hadn't even finished dressing when she heard feet running on the veranda towards her room.\n\n'What is it?' She hastily buttoned up her shirt and opened the door to see Carmel the Irish nurse coming to a halt, panting. She was fat, her striped dress bursting across her buxom chest, her apron dangling at her waist as if interrupted in her dressing too.\n\n'A message from Nairobi,' she huffed. 'You've got to go there immediately. Something to do with Georgia.'\n\nA wild ride in a beaten-up Landrover to meet her replacement at Buna, then into a mail plane as far as Archer's Post. Two days without sleep, surviving on just adrenalin. Fear and hope mingling like a lethal cocktail.\n\nUrgent messages to her only meant one thing. Death, sickness or disaster. By the time she got to a telephone at Archer's Post every bone in her body screamed for rest, but until someone could reassure her that Georgia wasn't dead she couldn't even pause for refreshment.\n\n'She's fine,' Hilary her old friend in Nairobi said. 'She's a famous singer now and a newspaper has been searching for you. It seems you and I are the only two people in the world who hadn't heard her.'\n\n'But \u2013'\n\nHilary cut her short. 'It's a long story, love. I can't tell you it all over the phone. Rest up tonight and get someone to drive you here tomorrow. She's safe and healthy. Don't worry anymore. I'll tell you everything soon.'\n\nAnother two days before she flopped into a chair in the Nairobi office. Dusty hair, red-rimmed eyes full of grit, bruised from the long hours in a truck, sweat stains covering her old shirt. Yet exhaustion faded as Hilary put a large gin in one hand and the telex in the other.\n\nThere, in the same steamy office where she'd started out in Africa, life turned a full circle.\n\nNow she hardly noticed the flies, the cane chairs which stuck in her legs, the bandages and syringes waiting for distribution, or the maddeningly slow fan that only served to churn up papers rather than air.\n\nIt was the first time she had cried since leaving England. All those years of uncertainty, all that grief held back came flooding out.\n\n'If you had told anyone but me about her,' Hilary said as she comforted her. 'They might have made the connection. But I'm as bad as you Celia, I don't read the papers or listen to music. Just fancy, your gel a star!'\n\nTears turned to laughter as Celia saw the absurdity of the situation. Two old nurses, who both fled from England to forget, suddenly aware how out of touch they were.\n\nHilary with her white cropped hair could have been a man wearing women's clothes. A print dress left over from the mid-forties, its demure lace trim revealing a scraggy, lined neck, and arms like a stevedore. While Celia in her khaki shorts and man's shirt looked more feminine than she ever had at home. No wonder the young girls in the office stared at them as if they were mad, drinking gin in the afternoon, crying and laughing alternately.\n\nYet those same girls were the ones who found old magazines, copies of Georgia's records and filled her in with everything that had happened in the last few years.\n\n'Why did you stop writing to Peter?' Hilary asked as once again Celia read the telex. 'I keep asking myself why I didn't open them instead of dumping them as you told me to do.'\n\nEven now she couldn't explain that fully, but at the time she got the letter from Mrs Radcliffe it seemed the honourable thing to do. She was just another mother worrying about her son.\n\nWhat was it she said? 'I am begging you as another mother to let my son forget. He isn't working as he should, each letter from you unsettles him. He has another girl now and they could be happy without reminders of the past. Please let him go, and I promise you if I ever hear of Georgia I will write immediately.'\n\nHow could she know then Mrs Radcliffe had no intention of helping? How foolish Celia was to think Peter's integrity had come from her!\n\n'Oh Hilary,' Celia sighed. 'If only you had! But then this whole business is built on \"if onlys\".'\n\nCelia leaned back in her seat and closed her eyes. She wasn't finished with Africa yet. A long overdue holiday. Time to hear Georgia sing and share in her life. A chance to recharge her batteries and she'd be back. Even Georgia re-entering her life couldn't make her stay in England.\n\nThe woman who'd left England five years ago had changed. Drought, famine, malnutrition and disease made the problems she'd encountered in London's East End seem trivial. How could any woman who'd seen people dying for want of clean water, possibly go back to treating verrucae, weighing healthy babies and filling in forms?\n\n'Still can't nod off?' Tania was back, offering her yet another drink.\n\n'Silly, isn't it?' Celia laughed. 'The most comfortable seat I've sat in for five years, all my old worries gone, and yet I'm still wide awake.'\n\n'Do you think she'll marry Peter?'\n\nCelia smiled. It was a bitter irony that people were now so fascinated by Georgia. Six years ago she hadn't even managed to convince the police this same girl was at risk. Was that what fame meant? In Africa people were dying of starvation, yet the rest of the world hung on news of a singer's wedding.\n\n'I hope so,' she said. 'If I could choose any man in the world for her, I'd still pick him.' She could see him so clearly. Those wide, honest blue eyes, his sensitive mouth, floppy blond hair and the proud conviction in his purpose.\n\nTania hovered, an unspoken question in her eyes. Celia knew she was trying to broach the subject of Brian Anderson, but diplomacy stopped her.\n\n'Speak up girl,' she said, imitating Hilary. 'You want to know about him?'\n\nTania blushed.\n\n'It's all right,' Celia said in a softer tone. 'Everyone's going to ask me that, why not you? But the answer is I don't know. I guess I'll have to see him at some stage, we are still married after all. Of course I have the advantage over everyone else. I always knew what he did.'\n\n'How do you feel about him now?' Tania said softly.\n\n'I ought to feel anger, I suppose,' Celia said slowly. 'Perhaps I've been exposed to too much horror over the years to feel that any longer. In a way I'm glad he surfaced, at least he brought Georgia back to me.'\n\n'You're a very strong lady,' Tania smiled, touching Celia's hand briefly. 'I can see now who moulded Georgia!'\n\nThe sun was warm on her shoulders as she made her way from the aircraft steps to the tarmac. Celia had been expecting, even welcomed, rain. On steamy nights out in the bush she often pictured the garden in Blackheath. Grass cool and damp under her feet, roses twining round the archway to the vegetable garden, the smell of wet soil, the dewy softness of it all. Her heartrate had been gradually quickening as the plane went over London. The vivid green grass, the silver Thames and all the majesty of Windsor Castle somehow embodied everything in England that was dear to her.\n\nThe tarmac spread on seemingly forever, surrounded by dark green grass. The air had freshness never felt in Africa. Luggage piled on trucks, men in blue uniforms standing in groups, an orderliness she wasn't used to. The plane behind her disgorged more passengers to add to the stream already making their way to the terminal. She wanted to run now, not wait while Customs men looked in her one small bag.\n\nLooking up at the glass and concrete building ahead of her she saw a figure behind glass. Masses of dark hair, a red dress, brown long legs, jumping up and down, waving with both hands.\n\nA lump came to her throat, tears blinded her. She raised her hand to her lips and blew a kiss.\n\nQuickening her pace, she dodged round an elderly couple and ran the rest of the way.\n\nDark-suited Customs men waved her through. She felt the smiles though she couldn't see them. All she saw was that finely polished corridor and a glimpse of people waiting beyond.\n\nHundreds of people pushing and shoving. Cameras flashing, voices shouting out greetings. But all she was aware of was one small voice.\n\n'Mum! Mum!'\n\nA mop of wild black hair, two dancing wet eyes, arms outstretched running towards her.\n\nThe longest minute, yet the quickest. She had grown from skinny-legged colt to an Arabian thoroughbred. So beautiful Celia could barely credit it was the same pitiful child she took away from St Joseph's that winter morning. A heart-shaped caramel face. Black curls tossed back, dazzling white teeth and a wide smiling mouth. Her red dress so short and tight it could have been sprayed on, matching tight boots that evoked an image of principal boys in pantomime.\n\nShe threw herself at Celia like a puppy, lifting Celia off her feet, crushing her into her arms.\n\n'Mummy,' was all she heard and all the missing years slipped away.\n\nCameras flashed like lightning. A barrage of shouted questions, an army of people advancing on them. Yet all they felt was two hearts pounding together. Two pairs of streaming eyes, two cheeks pressed against one another's.\n\nCelia felt Georgia take her waist, pushing her back to look at her.\n\n'You've shrunk, Mum! Or is it I've grown?' Georgia looked down at Celia, dark chocolate eyes melting with tears, then pulled her fiercely back to her shoulder, enveloping her in the kind of rocking hug she had once given Georgia.\n\n'My baby,' Celia whispered. 'I thought I'd lost you forever.'\n\n'I was never lost to you,' Georgia whispered back. 'I carried you inside me wherever I went, but let's get home now, away from all this.'\n\nWith her arm firmly around Celia's shoulder, Georgia turned to the fans and press.\n\n'I want you all to meet my mother,' she grinned. 'This is the happiest day of my life.'\n\nA mural of grinning faces. Another firing squad of photographs, Celia blinked, her lips trembling.\n\nGeorgia held up one hand, the crowd fell silent.\n\n'I know you all want to ask questions,' she said, glancing sideways at Celia. 'But my mother isn't used to this kind of publicity and just now we both want to go home and talk and rediscover one another.'\n\nAnother blast of flashing lights. Murmurs of disappointment, a predatory closing in.\n\n'I have just one announcement,' Georgia smiled round, head held high. 'Today you've seen the reunion with my mother, but last night I also learned who my real father is.'\n\nSilence fell. Two or three hundred people leaning forward to hear.\n\nGeorgia grinned, tossing back her hair, using all the timing she'd learned on stage.\n\n'It's a sad, wonderful story,' she said. 'One I'll tell you all in a day or two, but for now you'll all have to be content with just a name. That name is Sam Cameron.'\n\nShe didn't appear to hear the tumult that broke around her, or show any concern that she'd merely whetted their appetite for information. Georgia picked up Celia's bag and holding her mother closer to her, walked steadily ahead as if they were alone.\n\n'Well, Mum,' Georgia said as they swept out on to the forecourt where a liveried chauffeur held open the doors of a sleek black limousine. 'How does it feel to be home?'\n\nCelia didn't answer for a moment. She climbed into the back seat and sighed as she sank into the soft leather, waiting for Georgia to sit beside her.\n\n'Do you remember how you felt that first day when I opened the door at Blackheath and led you in?' she asked.\n\nGeorgia took her hand and lifted it to her cheek.\n\n'Like I was entering heaven,' she whispered.\n\n'Well, darling, that's just how I feel right now.' She reached out for Georgia and this time it was she who cradled her child against her breast. 'Then it was me taking you to my world, now it's you leading me to yours.'\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Table of Contents\n\nMastering OpenCV with Practical Computer Vision Projects\n\nCredits\n\nAbout the Authors\n\nAbout the Reviewers\n\nwww.PacktPub.com\n\nSupport files, eBooks, discount offers and more\n\nWhy Subscribe?\n\nFree Access for Packt account holders\n\nPreface\n\nWhat this book covers\n\nWhat you need for this book\n\nWho this book is for\n\nConventions\n\nReader feedback\n\nCustomer support\n\nDownloading the example code\n\nErrata\n\nPiracy\n\nQuestions\n\n1. Cartoonifier and Skin Changer for Android\n\nAccessing the webcam\n\nMain camera processing loop for a desktop app\n\nGenerating a black-and-white sketch\n\nGenerating a color painting and a cartoon\n\nGenerating an \"evil\" mode using edge filters\n\nGenerating an \"alien\" mode using skin detection\n\nSkin-detection algorithm\n\nShowing the user where to put their face\n\nImplementation of the skin-color changer\n\nPorting from desktop to Android\n\nSetting up an Android project that uses OpenCV\n\nColor formats used for image processing on Android\n\nInput color format from the camera\n\nOutput color format for display\n\nAdding the cartoonifier code to the Android NDK app\n\nReviewing the Android app\n\nCartoonifying the image when the user taps the screen\n\nSaving the image to a file and to the Android picture gallery\n\nShowing an Android notification message about a saved image\n\nChanging cartoon modes through the Android menu bar\n\nReducing the random pepper noise from the sketch image\n\nShowing the FPS of the app\n\nUsing a different camera resolution\n\nCustomizing the app\n\nSummary\n\n2. Marker-based Augmented Reality on iPhone or iPad\n\nCreating an iOS project that uses OpenCV\n\nAdding OpenCV framework\n\nIncluding OpenCV headers\n\nApplication architecture\n\nAccessing the camera\n\nMarker detection\n\nMarker identification\n\nGrayscale conversion\n\nImage binarization\n\nContours detection\n\nCandidates search\n\nMarker code recognition\n\nReading marker code\n\nMarker location refinement\n\nPlacing a marker in 3D\n\nCamera calibration\n\nMarker pose estimation\n\nRendering the 3D virtual object\n\nCreating the OpenGL rendering layer\n\nRendering an AR scene\n\nSummary\n\nReferences\n\n3. Marker-less Augmented Reality\n\nMarker-based versus marker-less AR\n\nUsing feature descriptors to find an arbitrary image on video\n\nFeature extraction\n\nDefinition of a pattern object\n\nMatching of feature points\n\nPatternDetector.cpp\n\nOutlier removal\n\nCross-match filter\n\nRatio test\n\nPatternDetector.cpp\n\nHomography estimation\n\nPatternDetector.cpp\n\nHomography refinement\n\nPatternDetector.cpp\n\nPutting it all together\n\nPattern pose estimation\n\nPatternDetector.cpp\n\nObtaining the camera-intrinsic matrix\n\nPattern.cpp\n\nApplication infrastructure\n\nARPipeline.hpp\n\nARPipeline.cpp\n\nEnabling support for 3D visualization in OpenCV\n\nCreating OpenGL windows using OpenCV\n\nVideo capture using OpenCV\n\nRendering augmented reality\n\nARDrawingContext.hpp\n\nARDrawingContext.cpp\n\nDemonstration\n\nmain.cpp\n\nSummary\n\nReferences\n\n4. Exploring Structure from Motion Using OpenCV\n\nStructure from Motion concepts\n\nEstimating the camera motion from a pair of images\n\nPoint matching using rich feature descriptors\n\nPoint matching using optical flow\n\nFinding camera matrices\n\nReconstructing the scene\n\nReconstruction from many views\n\nRefinement of the reconstruction\n\nVisualizing 3D point clouds with PCL\n\nUsing the example code\n\nSummary\n\nReferences\n\n5. Number Plate Recognition Using SVM and Neural Networks\n\nIntroduction to ANPR\n\nANPR algorithm\n\nPlate detection\n\nSegmentation\n\nClassification\n\nPlate recognition\n\nOCR segmentation\n\nFeature extraction\n\nOCR classification\n\nEvaluation\n\nSummary\n\n6. Non-rigid Face Tracking\n\nOverview\n\nUtilities\n\nObject-oriented design\n\nData collection: Image and video annotation\n\nTraining data types\n\nAnnotation tool\n\nPre-annotated data (The MUCT dataset)\n\nGeometrical constraints\n\nProcrustes analysis\n\nLinear shape models\n\nA combined local-global representation\n\nTraining and visualization\n\nFacial feature detectors\n\nCorrelation-based patch models\n\nLearning discriminative patch models\n\nGenerative versus discriminative patch models\n\nAccounting for global geometric transformations\n\nTraining and visualization\n\nFace detection and initialization\n\nFace tracking\n\nFace tracker implementation\n\nTraining and visualization\n\nGeneric versus person-specific models\n\nSummary\n\nReferences\n\n7. 3D Head Pose Estimation Using AAM and POSIT\n\nActive Appearance Models overview\n\nActive Shape Models\n\nGetting the feel of PCA\n\nTriangulation\n\nTriangle texture warping\n\nModel Instantiation \u2013 playing with the Active Appearance Model\n\nAAM search and fitting\n\nPOSIT\n\nDiving into POSIT\n\nPOSIT and head model\n\nTracking from webcam or video file\n\nSummary\n\nReferences\n\n8. Face Recognition using Eigenfaces or Fisherfaces\n\nIntroduction to face recognition and face detection\n\nStep 1: Face detection\n\nImplementing face detection using OpenCV\n\nLoading a Haar or LBP detector for object or face detection\n\nAccessing the webcam\n\nDetecting an object using the Haar or LBP Classifier\n\nGrayscale color conversion\n\nShrinking the camera image\n\nHistogram equalization\n\nDetecting the face\n\nStep 2: Face preprocessing\n\nEye detection\n\nEye search regions\n\nGeometrical transformation\n\nSeparate histogram equalization for left and right sides\n\nSmoothing\n\nElliptical mask\n\nStep 3: Collecting faces and learning from them\n\nCollecting preprocessed faces for training\n\nTraining the face recognition system from collected faces\n\nViewing the learned knowledge\n\nAverage face\n\nEigenvalues, Eigenfaces, and Fisherfaces\n\nStep 4: Face recognition\n\nFace identification: Recognizing people from their face\n\nFace verification: Validating that it is the claimed person\n\nFinishing touches: Saving and loading files\n\nFinishing touches: Making a nice and interactive GUI\n\nDrawing the GUI elements\n\nStartup mode\n\nDetection mode\n\nCollection mode\n\nTraining mode\n\nRecognition mode\n\nChecking and handling mouse clicks\n\nSummary\n\nReferences\n\nIndex\n\n#  **Mastering OpenCV with Practical Computer Vision Projects**\n\n* * *\n\n# Mastering OpenCV with Practical Computer Vision Projects\n\nCopyright (C) 2012 Packt Publishing\n\nAll rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior written permission of the publisher, except in the case of brief quotations embedded in critical articles or reviews.\n\nEvery effort has been made in the preparation of this book to ensure the accuracy of the information presented. However, the information contained in this book is sold without warranty, either express or implied. Neither the authors, nor Packt Publishing, and its dealers and distributors will be held liable for any damages caused or alleged to be caused directly or indirectly by this book.\n\nPackt Publishing has endeavored to provide trademark information about all of the companies and products mentioned in this book by the appropriate use of capitals. However, Packt Publishing cannot guarantee the accuracy of this information.\n\nFirst published: November 2012\n\nProduction Reference: 1161112\n\nPublished by Packt Publishing Ltd.\n\nLivery Place\n\n35 Livery Street\n\nBirmingham B3 2PB, UK.\n\nISBN 978-1-84951-782-9\n\nwww.packtpub.com\n\nCover Image by Neha Rajappan (`<neha.rajappan1@gmail.com>`)\n\n# Credits\n\n**Authors**\n\nDaniel Lelis Baggio\n\nShervin Emami\n\nDavid Millan Escriva\n\nKhvedchenia Ievgen\n\nNaureen Mahmood\n\nJason Saragih\n\nRoy Shilkrot\n\n**Reviewers**\n\nKirill Kornyakov\n\nLuis Diaz Mas\n\nSebastian Montabone\n\n**Acquisition Editor**\n\nUsha Iyer\n\n**Lead Technical Editor**\n\nAnkita Shashi\n\n**Technical Editors**\n\nSharvari Baet\n\nPrashant Salvi\n\n**Copy Editors**\n\nBrandt D'Mello\n\nAditya Nair\n\nAlfida Paiva\n\n**Project Coordinator**\n\nPriya Sharma\n\n**Proofreaders**\n\nChris Brown\n\nMartin Diver\n\n**Indexer**\n\nHemangini Bari\n\nTejal Soni\n\nRekha Nair\n\n**Graphics**\n\nValentina D'silva\n\nAditi Gajjar\n\n**Production Coordinator**\n\nArvindkumar Gupta\n\n**Cover Work**\n\nArvindkumar Gupta\n\n# About the Authors\n\n**Daniel L elis Baggio** started his work in computer vision through medical image processing at InCor (Instituto do Cora\u00e7ao - Heart Institute) in Sao Paulo, where he worked with intra-vascular ultrasound image segmentation. Since then, he has focused on GPGPU and ported the segmentation algorithm to work with NVIDIA's CUDA. He has also dived into six degrees of freedom head tracking with a natural user interface group through a project called ehci (<http:\/\/code.google.com\/p\/ehci\/>). He now works for the Brazilian Air Force.\n\n ****\n\nI'd like to thank God for the opportunity of working with computer vision. I try to understand the wonderful algorithms He has created for us to see. I also thank my family, and especially my wife, for all their support throughout the development of the book. I'd like to dedicate this book to my son Stefano.\n\n**Shervin Emami** (born in Iran) taught himself electronics and hobby robotics during his early teens in Australia. While building his first robot at the age of 15, he learned how RAM and CPUs work. He was so amazed by the concept that he soon designed and built a whole Z80 motherboard to control his robot, and wrote all the software purely in binary machine code using two push buttons for 0s and 1s. After learning that computers can be programmed in much easier ways such as assembly language and even high-level compilers, Shervin became hooked to computer programming and has been programming desktops, robots, and smartphones nearly every day since then. During his late teens he created Draw3D (<http:\/\/draw3d.shervinemami.info\/>), a 3D modeler with 30,000 lines of optimized C and assembly code that rendered 3D graphics faster than all the commercial alternatives of the time; but he lost interest in graphics programming when 3D hardware acceleration became available.\n\nIn University, Shervin took a subject on computer vision and became highly interested in it; so for his first thesis in 2003 he created a real-time face detection program based on Eigenfaces, using OpenCV (beta 3) for camera input. For his master's thesis in 2005 he created a visual navigation system for several mobile robots using OpenCV (v0.96). From 2008, he worked as a freelance Computer Vision Developer in Abu Dhabi and Philippines, using OpenCV for a large number of short-term commercial projects that included:\n\n  * Detecting faces using Haar or Eigenfaces\n  * Recognizing faces using Neural Networks, EHMM, or Eigenfaces\n  * Detecting the 3D position and orientation of a face from a single photo using AAM and POSIT\n  * Rotating a face in 3D using only a single photo\n  * Face preprocessing and artificial lighting using any 3D direction from a single photo\n  * Gender recognition\n  * Facial expression recognition\n  * Skin detection\n  * Iris detection\n  * Pupil detection\n  * Eye-gaze tracking\n  * Visual-saliency tracking\n  * Histogram matching\n  * Body-size detection\n  * Shirt and bikini detection\n  * Money recognition\n  * Video stabilization\n  * Face recognition on iPhone\n  * Food recognition on iPhone\n  * Marker-based augmented reality on iPhone (the second-fastest iPhone augmented reality app at the time).\n\nOpenCV was putting food on the table for Shervin's family, so he began giving back to OpenCV through regular advice on the forums and by posting free OpenCV tutorials on his website (<http:\/\/www.shervinemami.info\/openCV.html>). In 2011, he contacted the owners of other free OpenCV websites to write this book. He also began working on computer vision optimization for mobile devices at NVIDIA, working closely with the official OpenCV developers to produce an optimized version of OpenCV for Android. In 2012, he also joined the Khronos OpenVL committee for standardizing the hardware acceleration of computer vision for mobile devices, on which OpenCV will be based in the future.\n\n ****\n\nI thank my wife Gay and my baby Luna for enduring the stress while I juggled my time between this book, working fulltime, and raising a family. I also thank the developers of OpenCV, who worked hard for many years to provide a high-quality product for free.\n\n**David Mill an Escriva** was eight years old when he wrote his first program on an 8086 PC with Basic language, which enabled the 2D plotting of basic equations. In 2005, he finished his studies in IT through the Universitat Politecnica de Valencia with honors in human-computer interaction supported by computer vision with OpenCV (v0.96). He had a final project based on this subject and published it on HCI Spanish congress. He participated in Blender, an open source, 3D-software project, and worked in his first commercial movie _Plumiferos - Aventuras voladoras_ as a Computer Graphics Software Developer.\n\nDavid now has more than 10 years of experience in IT, with experience in computer vision, computer graphics, and pattern recognition, working on different projects and startups, applying his knowledge of computer vision, optical character recognition, and augmented reality. He is the author of the \"DamilesBlog\" (<http:\/\/blog.damiles.com>), where he publishes research articles and tutorials about OpenCV, computer vision in general, and Optical Character Recognition algorithms.\n\nDavid has reviewed the book _gnuPlot Cookbook_ by _Lee Phillips_ and published by _Packt Publishing_.\n\n ****\n\nThanks Izaskun and my daughter Eider for their patience and support. Os quiero peque\u00f1as.\n\nI also thank Shervin for giving me this opportunity, the OpenCV team for their work, the support of Artres, and the useful help provided by Augmate.\n\n**Khvedchenia Ievgen** is a computer vision expert from Ukraine. He started his career with research and development of a camera-based driver assistance system for Harman International. He then began working as a Computer Vision Consultant for ESG. Nowadays, he is a self-employed developer focusing on the development of augmented reality applications. Ievgen is the author of the _Computer Vision Talks_ blog (<http:\/\/computer-vision-talks.com> ), where he publishes research articles and tutorials pertaining to computer vision and augmented reality.\n\n ****\n\nI would like to say thanks to my father who inspired me to learn programming when I was 14. His help can't be overstated. And thanks to my mom, who always supported me in all my undertakings. You always gave me a freedom to choose my own way in this life. Thanks, parents!\n\nThanks to Kate, a woman who totally changed my life and made it extremely full. I'm happy we're together. Love you.\n\n**Naureen Mahmood** is a recent graduate from the Visualization department at Texas A&M University. She has experience working in various programming environments, animation software, and microcontroller electronics. Her work involves creating interactive applications using sensor-based electronics and software engineering. She has also worked on creating physics-based simulations and their use in special effects for animation.\n\n ****\n\nI wanted to especially mention the efforts of another student from Texas A &M, whose name you will undoubtedly come across in the code included for this book. Fluid Wall was developed as part of a student project by Austin Hines and myself. Major credit for the project goes to Austin, as he was the creative mind behind it. He was also responsible for the arduous job of implementing the fluid simulation code into our application. However, he wasn't able to participate in writing this book due to a number of work- and  study-related preoccupations.\n\n**Jason Saragih** received his B.Eng degree in mechatronics (with honors) and Ph.D. in computer science from the Australian National University, Canberra, Australia, in 2004 and 2008, respectively. From 2008 to 2010 he was a Postdoctoral fellow at the Robotics Institute of Carnegie Mellon University, Pittsburgh, PA. From 2010 to 2012 he worked at the Commonwealth Scientific and Industrial Research Organization (CSIRO) as a Research Scientist. He is currently a Senior Research Scientist at Visual Features, an Australian tech startup company.\n\nDr. Saragih has made a number of contributions to the field of computer vision, specifically on the topic of deformable model registration and modeling. He is the author of two non-profit open source libraries that are widely used in the scientific community; DeMoLib and FaceTracker, both of which make use of generic computer vision libraries including OpenCV.\n\n**Roy Shilkrot** is a researcher and professional in the area of computer vision and computer graphics. He obtained a B.Sc. in Computer Science from Tel-Aviv-Yaffo Academic College, and an M.Sc. from Tel-Aviv University. He is currently a PhD candidate in Media Laboratory of the Massachusetts Institute of Technology (MIT) in Cambridge.\n\nRoy has over seven years of experience as a Software Engineer in start-up companies and enterprises. Before joining the MIT Media Lab as a Research Assistant he worked as a Technology Strategist in the Innovation Laboratory of Comverse, a telecom solutions provider. He also dabbled in consultancy, and worked as an intern for Microsoft research at Redmond.\n\n ****\n\nThanks go to my wife for her limitless support and patience, my past and present advisors in both academia and industry for their wisdom, and my friends and colleagues for their challenging thoughts.\n\n# About the Reviewers\n\n**Kirill Kornyakov** is a Project Manager at Itseez, where he leads the development of OpenCV library for Android mobile devices. He manages activities for the mobile operating system's support and computer vision applications development, including performance optimization for NVIDIA's Tegra platform. Earlier he worked at Itseez on real-time computer vision systems for open source and commercial products, chief among them being stereo vision on GPU and face detection in complex environments. Kirill has a B.Sc. and an M.Sc. from Nizhniy Novgorod State University, Russia.\n\n ****\n\nI would like to thank my family for their support, my colleagues from Itseez, and Nizhniy Novgorod State University for productive discussions.\n\n**Luis D iaz Mas** considers himself a computer vision researcher and is passionate about open source and open-hardware communities. He has been working with image processing and computer vision algorithms since 2008 and is currently finishing his PhD on 3D reconstructions and action recognition. Currently he is working in CATEC (<http:\/\/www.catec.com.es\/en>), a research center for advanced aerospace technologies, where he mainly deals with the sensorial systems of UAVs. He has participated in several national and international projects where he has proven his skills in C\/C++ programming, application development for embedded systems with Qt libraries, and his experience with GNU\/Linux distribution configuration for embedded systems. Lately he is focusing his interest in ARM and CUDA development.\n\n**Sebastian Montabone** is a Computer Engineer with a Master of Science degree in computer vision. He is the author of scientific articles pertaining to image processing and has also authored a book, _Beginning Digital Image Processing: Using Free Tools for Photographers_.\n\nEmbedded systems have also been of interest to him, especially mobile phones. He created and taught a course about the development of applications for mobile phones, and has been recognized as a Nokia developer champion.\n\nCurrently he is a Software Consultant and Entrepreneur. You can visit his blog at www.samontab.com, where he shares his current projects with the world.\n\n# www.PacktPub.com\n\n# Support files, eBooks, discount offers and more\n\nYou might want to visit www.PacktPub.com for support files and downloads related to your book.\n\nDid you know that Packt offers eBook versions of every book published, with PDF and ePub files available? You can upgrade to the eBook version at www.PacktPub.com and as a print book customer, you are entitled to a discount on the eBook copy. 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Simply use your login credentials for immediate access.\n\n# Preface\n\n_Mastering OpenCV with Practical Computer Vision Projects_ contains nine chapters, where each chapter is a tutorial for an entire project from start to finish, based on OpenCV's C++ interface including full source code. The author of each chapter was chosen for their well-regarded online contributions to the OpenCV community on that topic, and the book was reviewed by one of the main OpenCV developers. Rather than explaining the basics of OpenCV functions, this is the first book that shows how to apply OpenCV to solve whole problems, including several 3D camera projects (augmented reality, 3D Structure from Motion, Kinect interaction) and several facial analysis projects (such as, skin detection, simple face and eye detection, complex facial feature tracking, 3D head orientation estimation, and face recognition), therefore it makes a great companion to existing OpenCV books.\n\n# What this book covers\n\nChapter 1, _Cartoonifier and Skin Changer for Android_ , contains a complete tutorial and source code for both a desktop application and an Android app that automatically generates a cartoon or painting from a real camera image, with several possible types of cartoons including a skin color changer.\n\nChapter 2, _Marker-based Augmented Reality on iPhone or iPad_ , contains a complete tutorial on how to build a marker-based augmented reality (AR) application for iPad and iPhone devices with an explanation of each step and source code.\n\nChapter 3, _Marker-less Augmented Reality_ , contains a complete tutorial on how to develop a marker-less augmented reality desktop application with an explanation of what marker-less AR is and source code.\n\nChapter 4, _Exploring Structure from Motion Using OpenCV_ , contains an introduction to Structure from Motion (SfM) via an implementation of SfM concepts in OpenCV. The reader will learn how to reconstruct 3D geometry from multiple 2D images and estimate camera positions.\n\nChapter 5, _Number Plate Recognition Using SVM and Neural Networks_ , contains a complete tutorial and source code to build an automatic number plate recognition application using pattern recognition algorithms using a support vector machine and Artificial Neural Networks. The reader will learn how to train and predict pattern-recognition algorithms to decide if an image is a number plate or not. It will also help classify a set of features into a character.\n\nChapter 6, _Non-rigid Face Tracking_ , contains a complete tutorial and source code to build a dynamic face tracking system that can model and track the many complex parts of a person's face.\n\nChapter 7, _3D Head Pose Estimation Using AAM and POSIT_ , contains all the background required to understand what **Active Appearance Models** ( **AAMs** ) are and how to create them with OpenCV using a set of face frames with different facial expressions. Besides, this chapter explains how to match a given frame through fitting capabilities offered by AAMs. Then, by applying the POSIT algorithm, one can find the 3D head pose.\n\nChapter 8, _Face Recognition using Eigenfaces or Fisherfaces_ , contains a complete tutorial and source code for a real-time face-recognition application that includes basic face and eye detection to handle the rotation of faces and varying lighting conditions in the images.\n\n_Chapter 9, Developing Fluid Wall Using the Microsoft Kinect_ , covers the complete development of an interactive fluid simulation called the Fluid Wall, which uses the Kinect sensor. The chapter will explain how to use Kinect data with OpenCV's optical flow methods and integrating it into a fluid solver.\n\nYou can download this chapter from: <http:\/\/www.packtpub.com\/sites\/default\/files\/downloads\/7829OS_Chapter9_Developing_Fluid_Wall_Using_the_Microsoft_Kinect.pdf>.\n\n# What you need for this book\n\nYou don't need to have special knowledge in computer vision to read this book, but you should have good C\/C++ programming skills and basic experience with OpenCV before reading this book. Readers without experience in OpenCV may wish to read the book _Learning OpenCV_ for an introduction to the OpenCV features, or read _OpenCV 2 Cookbook_ for examples on how to use OpenCV with recommended C\/C++ patterns, because _Mastering OpenCV with Practical Computer Vision Projects_ will show you how to solve real problems, assuming you are already familiar with the basics of OpenCV and C\/C++ development.\n\nIn addition to C\/C++ and OpenCV experience, you will also need a computer, and IDE of your choice (such as Visual Studio, XCode, Eclipse, or QtCreator, running on Windows, Mac or Linux). Some chapters have further requirements, in particular:\n\n  * To develop the Android app, you will need an Android device, Android development tools, and basic Android development experience.\n  * To develop the iOS app, you will need an iPhone, iPad, or iPod Touch device, iOS development tools (including an Apple computer, XCode IDE, and an Apple Developer Certificate), and basic iOS and Objective-C development experience.\n  * Several desktop projects require a webcam connected to your computer. Any common USB webcam should suffice, but a webcam of at least 1 megapixel may be desirable.\n  * CMake is used in some projects, including OpenCV itself, to build across operating systems and compilers. A basic understanding of build systems is required, and knowledge of cross-platform building is recommended.\n  * An understanding of linear algebra is expected, such as basic vector and matrix operations and eigen decomposition.\n\n# Who this book is for\n\n_Mastering OpenCV with Practical Computer Vision Projects_ is the perfect book for developers with basic OpenCV knowledge to create practical computer vision projects, as well as for seasoned OpenCV experts who want to add more computer vision topics to their skill set. It is aimed at senior computer science university students, graduates, researchers, and computer vision experts who wish to solve real problems using the OpenCV C++ interface, through practical step-by-step tutorials.\n\n# Conventions\n\nIn this book, you will find a number of styles of text that distinguish between different kinds of information. Here are some examples of these styles, and an explanation of their meaning.\n\nCode words in text are shown as follows: \"You should put most of the code of this chapter into the `cartoonifyImage()` function.\"\n\nA block of code is set as follows:\n\n    int cameraNumber = 0;\n    if (argc > 1)\n        cameraNumber = atoi(argv[1]);\n    \/\/ Get access to the camera.\n    cv::VideoCapture capture;\n\nWhen we wish to draw your attention to a particular part of a code block, the relevant lines or items are set in bold:\n\n    \/\/ Get access to the camera.\n    cv::VideoCapture capture;\n    **camera.open(cameraNumber);**\n    if (!camera.isOpened()) {\n        std::cerr << \"ERROR: Could not access the camera or video!\" <<\n\n**New terms** and **important words** are shown in bold. Words that you see on the screen, in menus or dialog boxes for example, appear in the text like this: \"clicking the **Next** button moves you to the next screen\".\n\n### Note\n\nWarnings or important notes appear in a box like this.\n\n### Tip\n\nTips and tricks appear like this.\n\n# Reader feedback\n\nFeedback from our readers is always welcome. Let us know what you think about this book--what you liked or may have disliked. Reader feedback is important for us to develop titles that you really get the most out of.\n\nTo send us general feedback, simply send an e-mail to `<feedback@packtpub.com>`, and mention the book title via the subject of your message.\n\nIf there is a topic that you have expertise in and you are interested in either writing or contributing to a book, see our author guide on www.packtpub.com\/authors.\n\n# Customer support\n\nNow that you are the proud owner of a Packt book, we have a number of things to help you to get the most from your purchase.\n\n## Downloading the example code\n\nYou can download the example code files for all Packt books you have purchased from your account at <http:\/\/www.PacktPub.com>. If you purchased this book elsewhere, you can visit <http:\/\/www.PacktPub.com\/support> and register to have the files e-mailed directly to you.\n\n## Errata\n\nAlthough we have taken every care to ensure the accuracy of our content, mistakes do happen. If you find a mistake in one of our books--maybe a mistake in the text or the code--we would be grateful if you would report this to us. By doing so, you can save other readers from frustration and help us improve subsequent versions of this book. If you find any errata, please report them by visiting <http:\/\/www.packtpub.com\/support>, selecting your book, clicking on the **errata** **submission** **form** link, and entering the details of your errata. Once your errata are verified, your submission will be accepted and the errata will be uploaded on our website, or added to any list of existing errata, under the Errata section of that title. Any existing errata can be viewed by selecting your title from <http:\/\/www.packtpub.com\/support>.\n\n## Piracy\n\nPiracy of copyright material on the Internet is an ongoing problem across all media. At Packt, we take the protection of our copyright and licenses very seriously. If you come across any illegal copies of our works, in any form, on the Internet, please provide us with the location address or website name immediately so that we can pursue a remedy.\n\nPlease contact us at `<copyright@packtpub.com>` with a link to the suspected pirated material.\n\nWe appreciate your help in protecting our authors, and our ability to bring you valuable content.\n\n## Questions\n\nYou can contact us at `<questions@packtpub.com>` if you are having a problem with any aspect of the book, and we will do our best to address it.\n\n# Chapter 1. Cartoonifier and Skin Changer for Android\n\nThis chapter will show you how to write some image-processing filters for Android smartphones and tablets, written first for desktop (in C\/C++) and then ported to Android (with the same C\/C++ code but with a Java GUI), since this is the recommended scenario when developing for mobile devices. This chapter will cover:\n\n  * How to convert a real-life image to a sketch drawing\n  * How to convert to a painting and overlay the sketch to produce a cartoon\n  * A scary \"evil\" mode to create bad characters instead of good characters\n  * A basic skin detector and skin color changer, to give someone green \"alien\" skin\n  * How to convert the project from a desktop app to a mobile app\n\nThe following screenshot shows the final Cartoonifier app running on an Android tablet:\n\nWe want to  make the real-world camera frames look like they are genuinely from a cartoon. The basic idea is to fill the flat parts with some color and then draw thick lines on the strong edges. In other words, the flat areas should become much more flat and the edges should become much more distinct. We will detect edges and smooth the flat areas, then draw enhanced edges back on top to produce a cartoon or comic book effect.\n\nWhen developing mobile computer vision apps, it is a good idea to build a fully working desktop version first before porting it to mobile, since it is much easier to develop and debug a desktop program than a mobile app! This chapter will therefore begin with a complete Cartoonifier desktop program that you can create using your favorite IDE (for example, Visual Studio, XCode, Eclipse, QtCreator, and so on). After it is working properly on the desktop, the last section shows how to port it to Android (or potentially iOS) with Eclipse. Since we will create two different projects that mostly share the same source code with different graphical user interfaces, you could create a library that is linked by both projects, but for simplicity we will put the desktop and Android projects next to each other, and set up the Android project to access some files (`cartoon.cpp` and `cartoon.h` , containing all the image processing code) from the `Desktop` folder. For example:\n\n  * `C:\\Cartoonifier_Desktop\\cartoon.cpp`\n  * `C:\\Cartoonifier_Desktop\\cartoon.h`\n  * `C:\\Cartoonifier_Desktop\\main_desktop.cpp`\n  * `C:\\Cartoonifier_Android\\...`\n\nThe desktop app uses an OpenCV GUI window, initializes the camera, and with each camera frame calls the `cartoonifyImage()` function containing most of the code in this chapter. It then displays the processed image on the GUI window. Similarly, the Android app uses an Android GUI window, initializes the camera using Java, and with each camera frame calls the exact same C++ `cartoonifyImage()` function as previously mentioned, but with Android menus and finger-touch input. This chapter will  explain how to create the desktop app from scratch, and the Android app from one of the OpenCV Android sample projects. So first you should create a desktop program in your favorite IDE, with a `main_desktop.cpp` file to hold the GUI code given in the following sections, such as the main loop, webcam functionality, and keyboard input, and you should create a `cartoon.cpp` file that will be shared between projects. You should put most of the code of this chapter into `cartoon.cpp` as a function called `cartoonifyImage()`.\n\n# Accessing the webcam\n\nTo access a computer's  webcam or camera device, you can simply call `open()` on a `cv::VideoCapture` object (OpenCV's method of accessing your camera device), and pass `0` as the default camera ID number. Some computers have multiple cameras attached or they do not work as default camera `0`; so it is common practice to allow the user to pass the desired camera number as a command-line argument, in case they want to try camera 1, 2, or -1, for example. We will also try to set the camera resolution to 640 x 480 using `cv::VideoCapture::set()` , in order to run faster on high-resolution cameras.\n\n### Note\n\nDepending on your camera model, driver, or system, OpenCV might not change the properties of your camera. It is not important for this project, so don't worry if it does not work with your camera.\n\nYou can put this code in the `main()` function of your `main_desktop.cpp`:\n\n    int cameraNumber = 0;\n    if (argc > 1)\n      cameraNumber = atoi(argv[1]);\n\n    \/\/ Get access to the camera.\n    cv::VideoCapture camera;\n    **camera.open(cameraNumber);**\n    if (!camera.isOpened()) {\n      std::cerr << \"ERROR: Could not access the camera or video!\" <<\n      std::endl;\n      exit(1);\n    }\n\n    \/\/ Try to set the camera resolution.\n    camera.set(cv::CV_CAP_PROP_FRAME_WIDTH, 640);\n    camera.set(cv::CV_CAP_PROP_FRAME_HEIGHT, 480);\n\nAfter the webcam has been initialized, you can grab the current camera image as a `cv::Mat` object (OpenCV's image container). You can grab each camera frame by using the C++ streaming operator from your `cv::VideoCapture` object into a `cv::Mat` object, just like if you were getting input from a console.\n\n### Note\n\nOpenCV makes it very easy to load a video file (such as an AVI or MPG file) and use it instead of a webcam. The only difference to your code would be that you should create the `cv::VideoCapture` object with the video filename, such as `camera.open(\"my_video.avi\")`, rather than the camera number, such as `camera.open(0)`. Both methods create a `cv::VideoCapture` object that can be used in the same way.\n\n# Main camera processing loop for a desktop app\n\nIf you  want to display a GUI window on the screen using OpenCV, you call `cv::imshow()` for each image, but you must also call `cv::waitKey()` once per frame, otherwise your windows will not update at all! Calling `cv::waitKey(0)` waits indefinitely until the user hits a key in the window, but a positive number such as `waitKey(20)` or higher will wait for at least that many milliseconds.\n\nPut this main loop in `main_desktop.cpp`, as the basis for your real-time camera app:\n\n    while (true) {\n      \/\/ Grab the next camera frame.\n      cv::Mat cameraFrame;\n    **camera >> cameraFrame;**\n      if (cameraFrame.empty()) {\n        std::cerr << \"ERROR: Couldn't grab a camera frame.\" <<\n        std::endl;\n        exit(1);\n      }\n      \/\/ Create a blank output image, that we will draw onto.\n      cv::Mat displayedFrame(cameraFrame.size(), cv::CV_8UC3);\n\n      \/\/ Run the cartoonifier filter on the camera frame.\n    **cartoonifyImage(cameraFrame, displayedFrame);**\n\n      \/\/ Display the processed image onto the screen.\n      imshow(\"Cartoonifier\", displayedFrame);\n\n      \/\/ IMPORTANT: Wait for at least 20 milliseconds,\n      \/\/ so that the image can be displayed on the screen!\n      \/\/ Also checks if a key was pressed in the GUI window.\n      \/\/ Note that it should be a \"char\" to support Linux.\n    **char keypress = cv::waitKey(20);  \/\/ Need this to see anything!**\n      if (keypress == 27) {   \/\/ Escape Key\n      \/\/ Quit the program!\n      break;\n      }\n    }\/\/end while\n\n# Generating a black-and-white sketch\n\nTo obtain a  sketch (black-and-white drawing) of the camera frame, we will use an edge-detection filter; whereas to obtain a color painting, we will use an edge-preserving filter (bilateral filter) to further smooth the flat regions while keeping the edges intact. By overlaying the sketch drawing on top of the color painting, we obtain a cartoon effect as shown earlier in the screenshot of the final app.\n\nThere are many different edge detection filters, such as Sobel, Scharr, Laplacian filters, or Canny-edge detector. We will use a Laplacian edge filter since it produces edges that look most similar to hand sketches compared to Sobel or Scharr, and that are quite consistent compared to a Canny-edge detector, which produces very clean line drawings but is affected more by random noise in the camera frames and the line drawings therefore often change drastically between frames.\n\nNevertheless, we still need to reduce the noise in the image before we use a Laplacian edge filter. We will use a Median filter because it is good at removing noise while keeping edges sharp; also, it is not as slow as a bilateral filter. Since Laplacian filters use grayscale images, we must convert from OpenCV's default BGR format to Grayscale. In your empty file `cartoon.cpp`, put this code at the top so you can access OpenCV and Standard C++  templates without typing `cv::` and `std::` everywhere:\n\n    \/\/ Include OpenCV's C++ Interface\n    #include \"opencv2\/opencv.hpp\"\n\n    using namespace cv;\n    using namespace std;\n\nPut this and all the remaining code in a `cartoonifyImage()` function in the `cartoon.cpp` file:\n\n    Mat gray;\n    cvtColor(srcColor, gray, CV_BGR2GRAY);\n    const int MEDIAN_BLUR_FILTER_SIZE = 7;\n    **medianBlur(gray, gray, MEDIAN_BLUR_FILTER_SIZE);**\n    Mat edges;\n    const int LAPLACIAN_FILTER_SIZE = 5;\n    **Laplacian(gray, edges, CV_8U, LAPLACIAN_FILTER_SIZE);**\n\nThe Laplacian filter produces edges with varying brightness, so to make the edges look more like a sketch we apply a binary threshold to make the edges either white or black:\n\n    Mat mask;\n    const int EDGES_THRESHOLD = 80;\n    **threshold(edges, mask, EDGES_THRESHOLD, 255, THRESH_BINARY_INV);**\n\nIn the following figure, you can see the original image (left side) and the generated edge mask (right side) that looks similar to a sketch drawing. After we generate a color painting (explained later), we can put this edge mask on top for black line drawings:\n\n# Generating a color painting and a cartoon\n\nA strong bilateral filter  smoothes flat regions while keeping edges sharp, and is therefore great as an automatic cartoonifier or painting filter, except that it is extremely slow (that is, measured in seconds or even minutes rather than milliseconds!). We will therefore use some tricks to obtain a nice cartoonifier that still runs at an acceptable speed. The most important trick we can use is to perform bilateral filtering at a lower resolution. It will have a similar effect as at full resolution, but will run much faster. Let's reduce the total number of pixels by a factor of four (for example, half width and half height):\n\n    Size size = srcColor.size();\n    Size smallSize;\n    smallSize.width = size.width\/2;\n    smallSize.height = size.height\/2;\n    Mat smallImg = Mat(smallSize, CV_8UC3);\n    **resize(srcColor, smallImg, smallSize, 0,0, INTER_LINEAR);**\n\nRather than applying a large bilateral filter, we will apply many small bilateral filters to produce a strong cartoon effect in less time. We will truncate the filter (see the following figure) so that instead of performing a whole filter (for example, a filter size of 21 x 21 when the bell curve is 21 pixels wide), it just uses the minimum filter size needed for a convincing result (for example, with a filter size of just 9 x 9 even if the bell curve is 21 pixels wide). This truncated filter will apply the major part of the filter (the gray area) without wasting time on the minor part of the filter (the white area under the curve), so it will run several times faster:\n\nWe have four parameters that control the bilateral filter: color strength, positional strength, size, and repetition count. We need a temp Mat since `bilateralFilter()` can't overwrite its input (referred to as \"in-place processing\"), but we can apply one filter storing a temp Mat and another filter storing back to the input:\n\n    Mat tmp = Mat(smallSize, CV_8UC3);\n    int repetitions = 7;  \/\/ Repetitions for strong cartoon effect.\n    for (int i=0; i<repetitions; i++) {\n      int ksize = 9;     \/\/ Filter size. Has a large effect on speed.\n      double sigmaColor = 9;    \/\/ Filter color strength.\n      double sigmaSpace = 7;    \/\/ Spatial strength. Affects speed.\n    **bilateralFilter(smallImg, tmp, ksize, sigmaColor, sigmaSpace);**\n    **bilateralFilter(tmp, smallImg, ksize, sigmaColor, sigmaSpace);**\n    }\n\nRemember  that this was applied to the shrunken image, so we need to expand the image back to the original size. Then we can overlay the edge mask that we found earlier. To overlay the edge mask \"sketch\" onto the bilateral filter \"painting\" (left-hand side of the following figure), we can start with a black background and copy the \"painting\" pixels that aren't edges in the \"sketch\" mask:\n\n    Mat bigImg;\n    resize(smallImg, bigImg, size, 0,0, INTER_LINEAR);\n    dst.setTo(0);\n    **bigImg.copyTo(dst, mask);**\n\nThe result is a cartoon version of the original photo, as shown on the right side of the figure, where the \"sketch\" mask is overlaid on the \"painting\":\n\n# Generating an \"evil\" mode using edge filters\n\nCartoons and  comics always have both good and bad characters. With the right combination of edge filters, a scary image can be generated from the most innocent-looking people! The trick is to use a small-edge filter that will find many edges all over the image, then merge the edges using a small Median filter.\n\nWe will perform this on a grayscale image with some noise reduction, so the previous code for converting the original image to grayscale and applying a 7 x 7 Median filter should be used again (the first image in the following figure shows the output of the grayscale Median blur). Instead of following it with a Laplacian filter and Binary threshold, we can get a scarier look if we apply a 3 x 3 Scharr gradient filter along x and y (the second image in the figure), and then apply a binary threshold with a very low cutoff (the third image in the figure) and a 3 x 3 Median blur, producing the final \"evil\" mask (the fourth image in the figure):\n\n    Mat gray;\n    cvtColor(srcColor, gray, CV_BGR2GRAY);\n    const int MEDIAN_BLUR_FILTER_SIZE = 7;\n    medianBlur(gray, gray, MEDIAN_BLUR_FILTER_SIZE);\n    Mat edges, edges2;\n    **Scharr(srcGray, edges, CV_8U, 1, 0);**\n    **Scharr(srcGray, edges2, CV_8U, 1, 0, -1);**\n    edges += edges2;     \/\/ Combine the x & y edges together.\n    const int EVIL_EDGE_THRESHOLD = 12;\n    **threshold(edges, mask, EVIL_EDGE_THRESHOLD, 255, THRESH_BINARY_INV);**\n    **medianBlur(mask, mask, 3);**\n\nNow that we have an \"evil\" mask, we can overlay this mask onto the cartoonified \"painting\" image like we did with the regular \"sketch\" edge mask. The final result is shown on the right side of the following figure:\n\n# Generating an \"alien\" mode using skin detection\n\nNow that we  have a sketch mode, a cartoon mode (painting + sketch mask), and an evil mode (painting + evil mask), for fun let's try something more complex: an \"alien\" mode, by detecting the skin regions of the face and then changing the skin color to be green.\n\n## Skin-detection algorithm\n\nThere are many different techniques used for detecting skin regions, from simple color thresholds using  **RGB** ( **Red-Green-Blue** ) or **HSV** ( **Hue-Saturation-Brightness** ) values or color histogram calculation and reprojection, to complex machine-learning algorithms of mixture models that need camera calibration in the CIELab color space and offline training with many sample faces, and so on. But even the complex methods don't necessarily work robustly across various camera and lighting conditions and skin types. Since we want our skin detection to run on a mobile device without any calibration or training, and we are just using skin detection  for a \"fun\" image filter, it is sufficient for us to use a simple skin-detection method. However, the color response from the tiny camera sensors in mobile devices tend to vary significantly, and we want to support skin detection for people of any skin color but without any calibration, so we need something more robust than simple color thresholds.\n\nFor example, a simple HSV skin detector can treat any pixel as skin if its hue is fairly red, saturation is fairly high but not extremely high, and its brightness is not too dark or too bright. But mobile cameras often have bad white balancing, and so a person's skin might look slightly blue instead of red, and so on, and this would be a major problem for simple HSV thresholding.\n\nA more robust solution is to perform face detection with a Haar or LBP cascade classifier (shown in Chapter 8, _Face Recognition using Eigenfaces_ ), and then look at the range of colors for the pixels in the middle of the detected face since you know that those pixels should be skin pixels of the actual person. You could then scan the whole image or the nearby region for pixels of a similar color as the center of the face. This has the advantage that it is very likely to find at least some of the true skin region of any detected person no matter what their skin color is or even if their skin appears somewhat blue or red in the camera image.\n\nUnfortunately, face detection using cascade classifiers is quite slow on current mobile devices, so this method might be less ideal for some real-time mobile applications. On the other hand, we can take advantage of the fact that for mobile apps it can be assumed that the user will be holding  the camera directly towards a person's face from close up, and since the user is holding the camera in their hand, which they can easily move, it is quite reasonable to ask the user to place their face at a specific location and distance, rather than try to detect the location and size of their face. This is the basis of many mobile phone apps where the app asks the user to place their face at a certain position or perhaps to manually drag points on the screen to show where the corners of their face are in a photo. So let's simply draw the outline of a face in the center of the screen and ask the user to move their face to the shown position and size.\n\n## Showing the user where to put their face\n\nWhen the alien mode is first started, we will draw the face outline on top of the camera frame so the user knows where to put their face. We will draw a big ellipse covering 70 percent of the image height, with a fixed aspect ratio of 0.72 so that the face will not become too skinny or fat depending on the aspect ratio of the camera:\n\n    \/\/ Draw the color face onto a black background.\n    Mat faceOutline = Mat::zeros(size, CV_8UC3);\n    Scalar color = CV_RGB(255,255,0);    \/\/ Yellow.\n    int thickness = 4;\n    \/\/ Use 70% of the screen height as the face height.\n    int sw = size.width;\n    int sh = size.height;\n    int faceH = sh\/2 * 70\/100;  \/\/ \"faceH\" is the radius of the ellipse.\n    \/\/ Scale the width to be the same shape for any screen width. int faceW = faceH * 72\/100;\n    \/\/ Draw the face outline.\n    **ellipse(faceOutline, Point(sw\/2, sh\/2), Size(faceW, faceH),**\n    **0, 0, 360, color, thickness, CV_AA);**\n\nTo make it more obvious that it is a face, let's also draw two eye outlines. Rather than drawing an eye as an ellipse, we can make it a bit more realistic (see the following figure) by drawing a truncated ellipse for the top of the eye and a truncated ellipse for the bottom of the eye, since we can specify the start and end angles when drawing with `ellipse()`:\n\n    \/\/ Draw the eye outlines, as 2 arcs per eye.\n    int eyeW = faceW * 23\/100;\n    int eyeH = faceH * 11\/100;\n    int eyeX = faceW * 48\/100;\n    int eyeY = faceH * 13\/100;\n    Size eyeSize = Size(eyeW, eyeH);\n    \/\/ Set the angle and shift for the eye half ellipses.\n    int eyeA = 15; \/\/ angle in degrees.\n    int eyeYshift = 11;\n    \/\/ Draw the top of the right eye.\n    **ellipse(faceOutline, Point(sw\/2 - eyeX, sh\/2 - eyeY),**\n    **eyeSize, 0, 180+eyeA, 360-eyeA, color, thickness, CV_AA);**\n    \/\/ Draw the bottom of the right eye.\n    **ellipse(faceOutline, Point(sw\/2 - eyeX, sh\/2 - eyeY - eyeYshift),**\n    **eyeSize, 0, 0+eyeA, 180-eyeA, color, thickness, CV_AA);**\n    \/\/ Draw the top of the left eye.\n    **ellipse(faceOutline, Point(sw\/2 + eyeX, sh\/2 - eyeY),**\n    **eyeSize, 0, 180+eyeA, 360-eyeA, color, thickness, CV_AA);**\n    \/\/ Draw the bottom of the left eye.\n    **ellipse(faceOutline, Point(sw\/2 + eyeX, sh\/2 - eyeY - eyeYshift),**\n    **eyeSize, 0, 0+eyeA, 180-eyeA, color, thickness, CV_AA);**\n\nWe can use the same method to draw the bottom lip of the mouth:\n\n    \/\/ Draw the bottom lip of the mouth.\n    int mouthY = faceH * 48\/100;\n    int mouthW = faceW * 45\/100;\n    int mouthH = faceH * 6\/100;\n    **ellipse(faceOutline, Point(sw\/2, sh\/2 + mouthY), Size(mouthW,**\n    **mouthH), 0, 0, 180, color, thickness, CV_AA);**\n\nTo make it even more obvious that the user should put their face where shown, let's write a message on the screen!\n\n    \/\/ Draw anti-aliased text.\n    int fontFace = FONT_HERSHEY_COMPLEX;\n    float fontScale = 1.0f;\n    int fontThickness = 2;\n    char *szMsg = \"Put your face here\";\n    **putText(faceOutline, szMsg, Point(sw * 23\/100, sh * 10\/100),**\n    **fontFace, fontScale, color, fontThickness, CV_AA);**\n\nNow that we have the face outline drawn, we can overlay it onto the displayed image by using alpha blending to combine the cartoonified image with this drawn outline:\n\n    addWeighted(dst, 1.0, faceOutline, 0.7, 0, dst, CV_8UC3);\n\nThis results in the outline on the following figure, showing the user where to put their face so we don't have to detect the face location:\n\n## Implementation of the skin-color changer\n\nRather than detecting the skin color and then the region with that skin color, we can use OpenCV's `floodFill()`, which is similar to the bucket fill tool in many image editing programs. We know that the regions in the middle of the screen should be skin pixels (since we asked the user to put their face in the middle), so to change the whole face to have green skin, we can just apply a green flood fill on the center pixel, which will always color at least some parts of the face as green. In reality, the color, saturation, and brightness is likely to be different in different parts of the face, so a flood fill will rarely cover all the skin pixels of a face unless the threshold is so low that it also covers unwanted pixels outside the face. So, instead of applying a single flood fill in the center of the image, let's apply a flood fill on six different points around the face that should be skin pixels.\n\nA nice feature of OpenCV's `floodFill()` function is that it can draw the flood fill into an external image rather than modifying the input image. So this feature can give us a mask image for adjusting the color of the skin pixels without necessarily changing the brightness or saturation, producing a more realistic image than if all skin pixels became an identical green pixel (losing significant face detail as a result).\n\nSkin-color changing does not work so well in the RGB color-space. This is because you want to allow brightness to vary in the face but not allow skin color to vary much, and RGB does not separate brightness from color. One solution is to use the Hue-Saturation-Brightness (HSV)  color-space, since it separates brightness from the color (hue) as well as the colorfulness (saturation). Unfortunately, HSV wraps the hue value around red, and since skin is mostly red it means that you need to work both with a hue of less than 10 percent and a hue greater than 90 percent, since these are both red. Accordingly, we will instead use the Y'CrCb color-space (the variant of YUV, which is in OpenCV), since it separates brightness from color, and only has a single range of values for typical skin color rather than two. Note that most cameras, images, and videos actually use some type of YUV as their color-space before conversion to RGB, so in many cases you can get a YUV image without having to convert it yourself.\n\nSince we want our alien mode to look like a cartoon, we will apply the alien filter after the image has already been cartoonified; in other words, we have access to the shrunken color image produced by the bilateral filter, and to the full-sized edge mask. Skin detection often works better at low resolutions, since it is the equivalent of analyzing the average value of each high-resolution pixel's neighbors (or the low-frequency signal instead of the high-frequency noisy signal). So let's work at the same shrunken scale as the bilateral filter (half width and half height). Let's convert the painting image to YUV:\n\n    Mat yuv = Mat(smallSize, CV_8UC3);\n    **cvtColor(smallImg, yuv, CV_BGR2YCrCb);**\n\nWe also need to shrink the edge mask so it is at the same scale as the painting image. There is a complication with OpenCV's `floodFill()` function when storing to a separate mask image, in that the mask should have a 1-pixel border around the whole image, so if the input image is _W_ x _H_ pixels in size, the separate mask image should be ( _W_ +2) x ( _H_ +2) pixels in size. But `floodFill()` also allows us to initialize the mask with edges that the flood-fill algorithm will ensure it does not cross. Let's use this feature in the hope that it helps prevent the flood fill from extending outside the face. So we need to provide two mask images: the edge mask that measures _W_ x _H_ in size, and the same edge mask but measuring ( _W_ +2) x ( _H_ +2) in size because it should include a border around the image. It is possible to have multiple `cv::Mat` objects (or headers) referencing the same data, or even to have a `cv::Mat `object that references a sub-region of another `cv::Mat` image. So instead of allocating two separate images and copying the edge mask pixels across, let's allocate a single mask image including the border, and create an extra `cv::Mat` header of _W_ x _H_ (that just references the region of interest in the flood-fill mask without the border). In other words, there is just one array of pixels of size ( _W_ +2) x ( _H_ +2) but two `cv::Mat` objects, where one is referencing the whole ( _W_ +2) x ( _H_ +2) image and the other is referencing the _W_ x _H_ region in the middle of that image:\n\n    int sw = smallSize.width;\n    int sh = smallSize.height;\n    Mat mask, maskPlusBorder;\n    **maskPlusBorder = Mat::zeros(sh+2, sw+2, CV_8UC1);**\n    **mask = maskPlusBorder(Rect(1,1,sw,sh)); \/\/ mask is in maskPlusBorder.**\n    resize(edge, mask, smallSize);         \/\/ Put edges in both of them.\n\nThe edge mask  (shown on the left-hand side of the following figure) is full of both strong and weak edges; but we only want strong edges, so we will apply a binary threshold (resulting in the middle image in the following figure). To join some gaps between edges we will then combine the morphological operators `dilate()` and `erode()` to remove some gaps (also referred to as the \"close\" operator), resulting in the right side of the figure:\n\n    const int EDGES_THRESHOLD = 80;\n    threshold(mask, mask, EDGES_THRESHOLD, 255, THRESH_BINARY);\n    dilate(mask, mask, Mat());\n    erode(mask, mask, Mat());\n\nAs mentioned earlier, we want to apply flood fills in numerous points around the face to make sure we include the various colors and shades of the whole face. Let's choose six points around the nose, cheeks, and forehead, as shown on the left side of the next figure. Note that these values are dependent on the face outline drawn earlier:\n\n    int const NUM_SKIN_POINTS = 6;\n    Point skinPts[NUM_SKIN_POINTS];\n    skinPts[0] = Point(sw\/2,          sh\/2 - sh\/6);\n    skinPts[1] = Point(sw\/2 - sw\/11,  sh\/2 - sh\/6);\n    skinPts[2] = Point(sw\/2 + sw\/11,  sh\/2 - sh\/6);\n    skinPts[3] = Point(sw\/2,          sh\/2 + sh\/16);\n    skinPts[4] = Point(sw\/2 - sw\/9,   sh\/2 + sh\/16);\n    skinPts[5] = Point(sw\/2 + sw\/9,   sh\/2 + sh\/16);\n\nNow we just  need to find some good lower and upper bounds for the flood fill. Remember that this is being performed in the Y'CrCb color space, so we basically decide how much the brightness, red component, and blue component can vary. We want to allow the brightness to vary a lot, to include shadows as well as highlights and reflections, but we don't want the colors to vary much at all:\n\n    const int LOWER_Y = 60;\n    const int UPPER_Y = 80;\n    const int LOWER_Cr = 25;\n    const int UPPER_Cr = 15;\n    const int LOWER_Cb = 20;\n    const int UPPER_Cb = 15;\n    Scalar lowerDiff = Scalar(LOWER_Y, LOWER_Cr, LOWER_Cb);\n    Scalar upperDiff = Scalar(UPPER_Y, UPPER_Cr, UPPER_Cb);\n\nWe will use `floodFill()` with its default flags, except that we want to store to an external mask, so we must specify `FLOODFILL_MASK_ONLY`:\n\n    const int CONNECTED_COMPONENTS = 4;  \/\/ To fill diagonally, use 8. const int flags = CONNECTED_COMPONENTS | FLOODFILL_FIXED_RANGE \\\n         | FLOODFILL_MASK_ONLY;\n    Mat edgeMask = mask.clone();    \/\/ Keep a copy of the edge mask.\n    \/\/ \"maskPlusBorder\" is initialized with edges to block floodFill().\n    for (int i=0; i< NUM_SKIN_POINTS; i++) {\n    **floodFill(yuv, maskPlusBorder, skinPts[i], Scalar(), NULL,**\n    **lowerDiff, upperDiff, flags);**\n    }\n\nIn the following figure, the left side shows the six flood-fill locations (shown as blue circles), and the right side of the figure shows the external mask that is generated, where skin is shown as gray and edges are shown as white. Note that the right-side image was modified for this book so that skin pixels (of value 1) are clearly visible:\n\nThe `mask` image (shown on the right side of the previous figure) now contains:\n\n  * pixels of value 255 for the edge pixels\n  * pixels of value 1 for the skin regions\n  * pixels of value 0 for the rest\n\nMeanwhile, `edgeMask` just  contains edge pixels (as value 255). So to get just the skin pixels, we can remove the edges from it:\n\n    **mask -= edgeMask;**\n\nThe `mask` image now just contains 1s for skin pixels and 0s for non-skin pixels. To change the skin color and brightness of the original image, we can use `cv::add()` with the skin mask to increase the green component in the original BGR image:\n\n    int Red = 0;\n    int Green = 70;\n    int Blue = 0;\n    **add(smallImgBGR, CV_RGB(Red, Green, Blue), smallImgBGR, mask);**\n\nThe following figure shows the original image on the left, and the final alien cartoon image on the right, where at least six parts of the face will now be green!\n\nNotice that we  have not only made the skin look green but also brighter (to look like an alien that glows in the dark). If you want to just change the skin color without making it brighter, you can use other color-changing methods, such as adding 70 to green while subtracting 70 from red and blue, or convert to HSV color space using `cvtColor(src, dst, \"CV_BGR2HSV_FULL\")`, and adjust the hue and saturation.\n\nThat's all! Run the app in the different modes until you are ready to port it to your mobile.\n\n# Porting from desktop to Android\n\nNow that the program  works on the desktop, we can make an Android or iOS app from it. The details given here are specific to Android, but also apply when porting to iOS for Apple iPhone and iPad or similar devices. When developing Android apps, OpenCV can be used directly from Java, but the result is unlikely to be as efficient as native C\/C++ code and doesn't allow the running of the same code on the desktop as it does for your mobile. So it is recommended to use C\/C++ for most OpenCV+Android app development (readers who want to write OpenCV apps purely in Java can use the JavaCV library by Samuel Audet, available at <http:\/\/code.google.com\/p\/javacv\/>, to run the same code on the desktop that we run on Android).\n\n### Note\n\nThis Android project uses a camera for live input, so it won't work on the Android Emulator. It needs a real Android 2.2 (Froyo) or later device with a camera.\n\nThe user  interface of an Android app should be written using Java, but for the image processing we will use the same `cartoon.cpp` C++ file that we used for the desktop. To use C\/C++ code in an Android app, we must use the **NDK** ( **Native Development Kit** ) that is based on  **JNI** ( **Java Native Interface** ). We will create a JNI wrapper for our `cartoonifyImage()` function so it can be used from Android with Java.\n\n## Setting up an Android project that uses OpenCV\n\nThe Android port of  OpenCV changes significantly each year, as does Android's method for camera access, so a book is not the best place to describe how it should be set up. Therefore the reader can follow the latest instructions at <http:\/\/opencv.org\/platforms\/android.html> to set up and build a native (NDK) Android app with OpenCV. OpenCV comes with an Android sample project called Sample3Native that accesses the camera using OpenCV and displays the modified image on the screen. This sample project is useful as a base for the Android app developed in this chapter, so readers should familiarize themselves with this sample app (currently available at <http:\/\/docs.opencv.org\/doc\/tutorials\/introduction\/android_binary_package\/android_binary_package_using_with_NDK.html>). We will then modify an Android OpenCV base project so that it can cartoonify the camera's video frames and display the resulting frames on the screen.\n\nIf you are stuck with OpenCV development for Android, for example if you are receiving a compile error or the camera always gives blank frames, try searching these websites for solutions:\n\n  1. The Android Binary Package NDK tutorial for OpenCV, mentioned previously.\n  2. The official Android-OpenCV Google group (<https:\/\/groups.google.com\/forum\/?fromgroups#!forum\/android-opencv>).\n  3. OpenCV's Q & A site (<http:\/\/answers.opencv.org>).\n  4. StackOverflow Q & A site (<http:\/\/stackoverflow.com\/questions\/tagged\/opencv+android>).\n  5. The Web (for example <http:\/\/www.google.com>).\n  6. If you still can't fix your problem after trying all of these, you should post a question on the Android-OpenCV Google group with details of the error message, and so on.\n\n### Color formats used for image processing on Android\n\nWhen developing for the  desktop, we only have to deal with BGR pixel format because the input (from camera, image, or video file) is in BGR format and so is the output (HighGUI window, image, or video file). But when developing for mobiles, you typically have to convert native color formats yourself.\n\n### Input color format from the camera\n\nLooking at the  sample code in `jni\\jni_part.cpp`, the `myuv` variable is the color image in Android's default camera format: `\"NV21\" YUV420sp`. The first part of the array is the grayscale pixel array, followed by a half-sized pixel array that alternates between the U and V color channels. So if we just want to access a grayscale image, we can get it directly from the first part of a `YUV420sp` semi-planar image without any conversions. But if we want a color image (for example, BGR or BGRA color format), we must convert the color format using `cvtColor()`.\n\n### Output color format for display\n\nLooking at the  Sample3Native code from OpenCV, the `mbgra` variable is the color image to be displayed on the Android device, in BGRA format. OpenCV's default format is BGR (the opposite byte order of RGB), and BGRA just adds an unused byte on the end of each pixel, so that each pixel is stored as Blue-Green-Red-Unused. You can either do all your processing in OpenCV's default BGR format and then convert your final output from BGR to BGRA before display on the screen, or you can ensure your image processing code can handle the BGRA format instead of or in addition to BGR format. This can often be simple to allow in OpenCV because many OpenCV functions accept the BGRA, but you must ensure that you create images with the same number of channels as the input, by seeing if the `Mat::channels()` value in your images are 3 or 4. Also, if you directly access pixels in your code, you would need separate code to handle 3-channel BGR and 4-channel BGRA images.\n\n### Note\n\nSome CV operations run faster with BGRA pixels (since it is aligned to 32-bit) while some run faster with BGR (since it requires less memory to read and write), so for maximum efficiency you should support both BGR and BGRA and then find which color format runs fastest overall in your app.\n\nLet's  begin with something simple: getting access to the camera frame in OpenCV but not processing it, and instead just displaying it on the screen. This can be done easily with Java code, but it is important to know how to do it using OpenCV too. As mentioned previously, the camera image arrives at our C++ code in `YUV420sp` format and should leave in BGRA format. So if we prepare our `cv::Mat` for input and output, we just need to convert from `YUV420sp` to BGRA using `cvtColor`. To write C\/C++ code for an Android Java app, we need to use special JNI function names that match the Java class and package name that will use that JNI function, in the format:\n\n    JNIEXPORT <Return> JNICALL Java_<Package>_<Class>_<Function>(JNIEnv* env, jobject, <Args>)\n\nSo let's create a `ShowPreview()` C\/C++ function that is used from a `CartoonifierView` Java class in a `Cartoonifier` Java package. Add this `ShowPreview()` C\/C++ function to `jni\\jni_part.cpp`:\n\n    \/\/ Just show the plain camera image without modifying it.\n    JNIEXPORT void JNICALL Java_com_Cartoonifier_CartoonifierView_ShowPreview(\n      JNIEnv* env, jobject,\n      jint width, jint height, jbyteArray yuv, jintArray bgra)\n    {\n      jbyte* _yuv  = env->GetByteArrayElements(yuv, 0);\n      jint*  _bgra = env->GetIntArrayElements(bgra, 0);\n\n      Mat myuv = Mat(height + height\/2, width, CV_8UC1, (uchar *)_yuv);\n      Mat mbgra = Mat(height, width, CV_8UC4, (uchar *)_bgra);\n\n      \/\/ Convert the color format from the camera's\n      \/\/ NV21 \"YUV420sp\" format to an Android BGRA color image.\n    **cvtColor(myuv, mbgra, CV_YUV420sp2BGRA);**\n\n      \/\/ OpenCV can now access\/modify the BGRA image \"mbgra\" ...\n\n      env->ReleaseIntArrayElements(bgra, _bgra, 0);\n      env->ReleaseByteArrayElements(yuv, _yuv, 0);\n    }\n\nWhile this code looks complex at first, the first two lines of the function just give us native access to the given Java arrays, the next two lines construct `cv::Mat` objects around the given pixel buffers (that is, they don't allocate new images, they make `myuv` access the pixels in the `_yuv` array, and so on), and the last two lines of the function release the native lock we placed on the Java arrays. The only real work we did in the function is to convert from YUV to BGRA format, so this function is the base that we can use for new functions. Now let's extend this to analyze and modify the BGRA `cv::Mat` before display.\n\n### Note\n\nThe `jni\\jni_part.cpp` sample code in OpenCV v2.4.2 uses this code:\n\n    cvtColor(myuv, mbgra, CV_YUV420sp2BGR, 4);\n\nThis looks like it converts to 3-channel BGR format (OpenCV's default format), but due to the \"`4\"` parameter it actually converts to 4-channel BGRA (Android's default output format) instead! So it's identical to this code, which is less confusing:\n\n    cvtColor(myuv, mbgra, CV_YUV420sp2BGRA);\n\nSince we  now have a BGRA image as input and output instead of OpenCV's default BGR, it leaves us with two options for how to process it:\n\n  * Convert from BGRA to BGR before we perform our image processing, do our processing in BGR, and then convert the output to BGRA so it can be displayed by Android\n  * Modify all our code to handle BGRA format in addition to (or instead of) BGR format, so we don't need to perform slow conversions between BGRA and BGR\n\nFor simplicity, we will just apply the color conversions from BGRA to BGR and back, rather than supporting both BGR and BGRA formats. If you are writing a real-time app, you should consider adding 4-channel BGRA support in your code to potentially improve performance. We will do one simple change to make things slightly faster: we are converting the input from `YUV420sp` to BGRA and then from BGRA to BGR, so we might as well just convert straight from `YUV420sp` to BGR!\n\nIt is a good idea to build and run with the `ShowPreview()` function (shown previously) on your device so you have something to go back to if you have problems with your C\/C++ code later. To call it from Java, we add the Java declaration just next to the Java declaration of `CartoonifyImage()` near the bottom of `CartoonifyView.java`:\n\n    public native void ShowPreview(int width, int height,\n    byte[] yuv, int[] rgba);\n\nWe can then call it just like the OpenCV sample code called `FindFeatures()`. Put this in the middle of the `processFrame()` function of `CartoonifierView.java`:\n\n    ShowPreview(getFrameWidth(), getFrameHeight(), data, rgba);\n\nYou should  build and run it now on your device, just to see the real-time camera preview.\n\n## Adding the cartoonifier code to the Android NDK app\n\nWe want to add the `cartoon.cpp` file that we used for the desktop app. The file `jni\\Android.mk` sets the C\/C++\/Assembly source files, header search paths, native libraries, and GCC compiler settings for your project:\n\n  1. Add `cartoon.cpp` (and `ImageUtils_0.7.cpp` if you want easier debugging) to `LOCAL_SRC_FILES`, but remember that they are in the desktop folder instead of the default `jni` folder. So add this after: `LOCAL_SRC_FILES := jni_part.cpp`:\n\n        LOCAL_SRC_FILES += ..\/..\/Cartoonifier_Desktop\/cartoon.cpp\n    LOCAL_SRC_FILES += ..\/..\/Cartoonifier_Desktop\/ImageUtils_0.7.cpp\n\n  2. Add the header file search path so it can find `cartoon.h` in the common parent folder:\n\n        LOCAL_C_INCLUDES += $(LOCAL_PATH)\/..\/..\/Cartoonifier_Desktop\n\n  3. In the file `jni\\jni_part.cpp`, insert this near the top instead of `#include <vector>`:\n\n        #include \"cartoon.h\"    \/\/ Cartoonifier.\n    #include \"ImageUtils.h\"    \/\/ (Optional) OpenCV debugging \/\/ functions.\n\n  4. Add a JNI function `CartoonifyImage()` to this file; this will cartoonify the image. We can start by duplicating the function `ShowPreview()` we created previously, which just shows the camera preview without modifying it. Notice that we convert directly from `YUV420sp` to BGR since we don't want to process BGRA images:\n\n        \/\/ Modify the camera image using the Cartoonifier filter.\n    JNIEXPORT void JNICALL Java_com_Cartoonifier_CartoonifierView_CartoonifyImage(\n        JNIEnv* env, jobject,\n       jint width, jint height, jbyteArray yuv, jintArray bgra)\n       {\n        \/\/ Get native access to the given Java arrays.\n        jbyte* _yuv  = env->GetByteArrayElements(yuv, 0);\n        jint* _bgra = env->GetIntArrayElements(bgra, 0);\n\n        \/\/ Create OpenCV wrappers around the input & output data.\n        Mat myuv(height + height\/2, width, CV_8UC1, (uchar *)_yuv);\n        Mat mbgra(height, width, CV_8UC4, (uchar *)_bgra);\n\n        \/\/ Convert the color format from the camera's YUV420sp \/\/ semi-planar\n        \/\/ format to OpenCV's default BGR color image.\n      **Mat mbgr(height, width, CV_8UC3);  \/\/ Allocate a new image buffer.**\n      **cvtColor(myuv, mbgr, CV_YUV420sp2BGR);**\n\n        \/\/ OpenCV can now access\/modify the BGR image \"mbgr\", and should\n        \/\/ store the output as the BGR image \"displayedFrame\".\n        Mat displayedFrame(mbgr.size(), CV_8UC3);\n\n        \/\/ TEMPORARY: Just show the camera image without modifying it.\n        displayedFrame = mbgr;\n\n          \/\/ Convert the output from OpenCV's BGR to Android's BGRA \/\/format.\n    **cvtColor(displayedFrame, mbgra, CV_BGR2BGRA);**\n\n    ****  \/\/ Release the native lock we placed on the Java arrays.\n    ****  env->ReleaseIntArrayElements(bgra, _bgra, 0);\n    ****  env->ReleaseByteArrayElements(yuv, _yuv, 0);\n    }\n\n  5. The previous code does not modify the image, but we want to process the image using the cartoonifier we developed earlier in this chapter. So now let's insert a call to our existing `cartoonifyImage()` function that we created in `cartoon.cpp` for the desktop app. Replace the temporary line of code `displayedFrame = mbgr` with this:\n\n        **cartoonifyImage(mbgr, displayedFrame);**\n\n  6. That's it! Build the code (Eclipse should compile the C\/C++ code for you using `ndk-build`) and run it on your device. You should have a working Cartoonifier Android app (right at the beginning of this chapter there is a sample screenshot showing what you should expect)! If it does not build or run, go back over the steps and fix the problems (look at the code provided with this book if you wish). Continue with the next steps once it is working.\n\n### Reviewing the Android app\n\nYou will quickly notice four issues with the app that is now running on your device:\n\n  * It is extremely slow; many seconds per frame! So we should just display the camera preview and only cartoonify a camera frame when the user has touched the screen to say it is a good photo.\n  * It needs to handle user input, such as to change modes between sketch, paint, evil, or alien modes. We will add these to the Android menu bar.\n  * It would be great if we could save the cartoonified result to image files, to share with others. Whenever the user touches the screen for a cartoonified image, we will save the result as an image file on the user's SD card and display it in the Android Gallery.\n  * There is a lot of random noise in the sketch edge detector. We will create a special \"pepper\" noise reduction filter to deal with this later.\n\n### Cartoonifying the image when the user taps the screen\n\nTo show the camera preview (until the user wants to cartoonify the selected camera frame), we can just call the `ShowPreview()` JNI function we wrote earlier. We will also wait for touch events from the user before cartoonifying the camera image. We only want to cartoonify one image when the user touches the screen; therefore we set a flag to say the next camera frame should be cartoonified and then that flag is reset, so it continues with the camera preview again. But this would mean the cartoonified image is only displayed for a fraction of a second and then the next camera preview will be displayed again. So we will use a second flag to say that the current image should be frozen on the screen for a few seconds before the camera frames overwrite it, to give the user some time to see it:\n\n  1. Add the following header imports near the top of the `CartoonifierApp.java` file in the `src\\com\\Cartoonifier` folder:\n\n        import android.view.View;\n    import android.view.View.OnTouchListener;\n    import android.view.MotionEvent;\n\n  2. Modify the class definition near the top of `CartoonifierApp.java`:\n\n        public class CartoonifierApp\n    extends Activity implements OnTouchListener {\n\n  3. Insert this code on the bottom of the `onCreate()` function:\n\n        \/\/ Call our \"onTouch()\" callback function whenever the user \/\/ touches the screen.\n    mView.setOnTouchListener(this);\n\n  4. Add the function `onTouch()` to process the touch event:\n\n        public boolean onTouch(View v, MotionEvent m) {\n        \/\/ Ignore finger movement event, we just care about when the \n        \/\/ finger first touches the screen.\n        if (m.getAction() != MotionEvent.ACTION_DOWN) {\n            return false; \/\/ We didn't use this touch movement event.\n        }\n        Log.i(TAG, \"onTouch down event\");\n        \/\/ Signal that we should cartoonify the next camera frame and save \n        \/\/ it, instead of just showing the preview.\n        mView.nextFrameShouldBeSaved(getBaseContext());\n        return true;\n    }\n\n  5. Now we need to add the `nextFrameShouldBeSaved()`function to `CartoonifierView.java`:\n\n        \/\/ Cartoonify the next camera frame & save it instead of preview.\n    protected void nextFrameShouldBeSaved(Context context) {\n        bSaveThisFrame = true;\n    }\n\n  6. Add these variables near the top of the `CartoonifierView` class:\n\n        private boolean bSaveThisFrame = false;\n    private boolean bFreezeOutput = false;\n    private static final int FREEZE_OUTPUT_MSECS = 3000;\n\n  7. The `processFrame()` function of `CartoonifierView` can now switch between cartoon and preview, but should also make sure to only display something if it is not trying to show a frozen cartoon image for a few seconds. So replace `processFrame()` with this:\n\n        @Override\n    protected Bitmap processFrame(byte[] data) {\n        \/\/ Store the output image to the RGBA member variable.\n        int[] rgba = mRGBA;\n        \/\/ Only process the camera or update the screen if we aren't \n        \/\/ supposed to just show the cartoon image.\n        if (bFreezeOutputbFreezeOutput) {\n            \/\/ Only needs to be triggered here once.\n            bFreezeOutput = false;\n            \/\/ Wait for several seconds, doing nothing!\n            try {\n                wait(FREEZE_OUTPUT_MSECS);\n            } catch (InterruptedException e) {\n                e.printStackTrace();\n            }\n            return null;\n        }\n        if (!bSaveThisFrame) {\n            ShowPreview(getFrameWidth(), getFrameHeight(), data, rgba);\n        }\n        else {\n            \/\/ Just do it once, then go back to preview mode.\n            bSaveThisFrame = false;\n            \/\/ Don't update the screen for a while, so the user can \/\/ see the cartoonifier output.\n    bFreezeOutput = true;\n\n    CartoonifyImage(getFrameWidth(), getFrameHeight(), data,\n            rgba, m_sketchMode, m_alienMode, m_evilMode,\n            m_debugMode);\n    }\n\n    \/\/ Put the processed image into the Bitmap object that will be \/\/ returned for display on the screen.\n    Bitmap bmp = mBitmap;\n    bmp.setPixels(rgba, 0, getFrameWidth(), 0, 0, getFrameWidth(),\n            getFrameHeight());\n\n    return bmp;\n    }\n\n  8. You should be able to build and run it to verify that the app works nicely now.\n\n### Saving the image to a file and to the Android picture gallery\n\nWe will save the output both as a PNG file and display in the Android picture gallery. The Android Gallery is designed for JPEG files, but JPEG is bad for cartoon images with solid colors and edges, so we'll use a tedious method to add PNG images to the gallery. We will create a Java function `savePNGImageToGallery()` to perform this for us. At the bottom of the `processFrame()` function just seen previously, we see that an Android `Bitmap` object is created with the output data; so we need a way to save the `Bitmap` object to a PNG file. OpenCV's `imwrite()` Java function can be used to save to a PNG file, but this would require linking to both OpenCV's Java API and OpenCV's C\/C++ API (just like the OpenCV4Android sample project \"tutorial-4-mixed\" does). Since we don't need the OpenCV Java API for anything else, the following code will just show how to save PNG files using the Android API instead of the OpenCV Java API:\n\n  1. Android's `Bitmap` class can save files to PNG format, so let's use it. Also, we need to choose a filename for the image. Let's use the current date and time, to allow saving many files and making it possible for the user to remember when it was taken. Insert this just before the `return bmp` statement of `processFrame()`:\n\n        if (bFreezeOutput) {\n    \/\/ Get the current date & time\n    SimpleDateFormat s = new SimpleDateFormat(\"yyyy-MM-dd,HH-mm-ss\");\n    String timestamp = s.format(new Date());\n    String baseFilename = \"Cartoon\" + timestamp + \".png\";\n\n    \/\/ Save the processed image as a PNG file on the SD card and show \/\/ it in the Android Gallery.\n    savePNGImageToGallery(bmp, mContext, baseFilename);\n    }\n\n  2. Add this to the top section of `CartoonifierView.java`:\n\n        \/\/ For saving Bitmaps to file and the Android picture gallery.\n    import android.graphics.Bitmap.CompressFormat;\n    import android.net.Uri;\n    import android.os.Environment;\n    import android.provider.MediaStore;\n    import android.provider.MediaStore.Images;\n    import android.text.format.DateFormat;\n    import android.util.Log;\n    import java.io.BufferedOutputStream;\n    import java.io.File;\n    import java.io.FileOutputStream;\n    import java.io.IOException;\n    import java.io.OutputStream;\n    import java.text.SimpleDateFormat;\n    import java.util.Date;\n\n  3. Insert this inside the `CartoonifierView` class, on the top:\n\n        private static final String TAG = \"CartoonifierView\";\n    private Context mContext;  \/\/ So we can access the Android \/\/ Gallery.\n\n  4. Add this to your `nextFrameShouldBeSaved()` function in `CartoonifierView`:\n\n        mContext = context;  \/\/ Save the Android context, for GUI \/\/ access.\n\n  5. Add the `savePNGImageToGallery()` function to `CartoonifierView`:\n\n        \/\/ Save the processed image as a PNG file on the SD card\n    \/\/ and shown in the Android Gallery.  \n    protected void savePNGImageToGallery(Bitmap bmp, Context context,\n            String baseFilename)\n    {\n        try {\n        \/\/ Get the file path to the SD card.\n        String baseFolder = \\\n        Environment.getExternalStoragePublicDirectory( \\\n        Environment.DIRECTORY_PICTURES).getAbsolutePath() \\\n        + \"\/\";\n        File file = new File(baseFolder + baseFilename);\n        Log.i(TAG, \"Saving the processed image to file [\" + \\\n        file.getAbsolutePath() + \"]\");\n\n        \/\/ Open the file.\n        OutputStream out = new BufferedOutputStream(\n        new FileOutputStream(file));\n        \/\/ Save the image file as PNG.\n        **bmp.compress(CompressFormat.PNG, 100, out);**\n        \/\/ Make sure it is saved to file soon, because we are about\n        \/\/ to add it to the Gallery.\n        out.flush();\n        out.close();\n\n        \/\/ Add the PNG file to the Android Gallery.\n        ContentValues image = new ContentValues();\n        image.put(Images.Media.TITLE, baseFilename);\n        image.put(Images.Media.DISPLAY_NAME, baseFilename);\n        image.put(Images.Media.DESCRIPTION,\n        \"Processed by the Cartoonifier App\");\n        image.put(Images.Media.DATE_TAKEN,\n        System.currentTimeMillis()); \/\/ msecs since 1970 UTC.\n        image.put(Images.Media.MIME_TYPE, \"image\/png\");\n        image.put(Images.Media.ORIENTATION, 0);\n        image.put(Images.Media.DATA, file.getAbsolutePath());\n        **Uri result = context.getContentResolver().insert(**\n        **MediaStore.Images.Media.EXTERNAL_CONTENT_URI,image);**\n        }\n    catch (Exception e) {\n        e.printStackTrace();\n        }\n    }\n\n  6. Android apps need permission from the user during installation if they need to store files on the device. So insert this line in `AndroidManifest.xml` just next to the similar line requesting permission for camera access:\n\n        <uses-permission android:name=\"android.permission.WRITE_EXTERNAL_STORAGE\"\/>\n\n  7. Build and run the app! When you touch the screen to save a photo, you should eventually see the cartoonified image shown on the screen (perhaps after 5 or 10 seconds of processing). Once it is shown on the screen, it means it should be saved to your SD card and to your photo gallery. Exit the Cartoonifier app, open the Android Gallery app, and view the Pictures album. You should see the cartoon image as a PNG image in your screen's full resolution.\n\n## Showing an Android notification message about a saved image\n\nIf you want to show a notification message whenever a new image is saved to the SD card and Android Gallery, follow these steps; otherwise feel free to skip this section:\n\n  1. Add the following to the top section of `CartoonifierView.java`:\n\n        \/\/ For showing a Notification message when saving a file.\n    import android.app.Notification;\n    import android.app.NotificationManager;\n    import android.app.PendingIntent;\n    import android.content.ContentValues;\n    import android.content.Intent;\n\n  2. Add this near the top section of `CartoonifierView`:\n\n        private int mNotificationID = 0;\n    \/\/ To show just 1 notification.\n\n  3. Insert this inside the `if` statement below the call to `savePNGImageToGallery()` in `processFrame()`:\n\n        showNotificationMessage(mContext, baseFilename);\n\n  4. Add the `showNotificationMessage()` function to `CartoonifierView`:\n\n        \/\/ Show a notification message, saying we've saved another image.\n    protected void showNotificationMessage(Context context,\n        String filename)\n    {\n    \/\/ Popup a notification message in the Android status   \n    \/\/ bar. To make sure a notification is shown for each \n    \/\/ image but only 1 is kept in the status bar at a time, \/\/ use a different ID each time\n    \/\/ but delete previous messages before creating it.\n    final NotificationManager mgr = (NotificationManager) \\\n    context.getSystemService(Context.NOTIFICATION_SERVICE);\n\n    \/\/ Close the previous popup message, so we only have 1 \/\/at a time, but it still shows a popup message for each \/\/one.\n    if (mNotificationID > 0)\n      mgr.cancel(mNotificationID);\n    mNotificationID++;\n\n    Notification notification = new Notification(R.drawable.icon,\n    \"Saving to gallery (image \" + mNotificationID + \") ...\",\n    System.currentTimeMillis());\n    Intent intent = new Intent(context, CartoonifierView.class);\n    \/\/ Close it if the user clicks on it.\n    notification.flags |= Notification.FLAG_AUTO_CANCEL;\n    PendingIntent pendingIntent = PendingIntent.getActivity(context,\n    0, intent, 0);\n    notification.setLatestEventInfo(context, \"Cartoonifier saved \" +\n    mNotificationID + \" images to Gallery\", \"Saved as '\" +\n    filename + \"'\", pendingIntent);\n    **mgr.notify(mNotificationID, notification);**\n    }\n\n  5. Once again, build and run the app! You should see a notification message pop up whenever you touch the screen for another saved image. If you want the notification message to pop up before the long delay of image processing rather than after, move the call to `showNotificationMessage()` before the call to `cartoonifyImage()`, and move the code for generating the date and time string so that the same string is given to the notification message and the actual file is saved.\n\n### Changing cartoon modes through the Android menu bar\n\nLet's allow the  user to change modes through the menu:\n\n  1. Add the following headers near the top of the file `src\\com\\Cartoonifier\\CartoonifierApp.java`:\n\n        import android.view.Menu;\n    import android.view.MenuItem;\n\n  2. Insert the following member variables inside the `CartoonifierApp` class:\n\n        \/\/ Items for the Android menu bar.\n    private MenuItem mMenuAlien;\n    private MenuItem mMenuEvil;\n    private MenuItem mMenuSketch;\n    private MenuItem mMenuDebug;\n\n  3. Add the following functions to `CartoonifierApp`:\n\n        \/** Called when the menu bar is being created by Android. *\/\n    public boolean onCreateOptionsMenu(Menu menu) {\n    Log.i(TAG, \"onCreateOptionsMenu\");\n    mMenuSketch = menu.add(\"Sketch or Painting\");\n    mMenuAlien = menu.add(\"Alien or Human\");\n    mMenuEvil = menu.add(\"Evil or Good\");\n    mMenuDebug = menu.add(\"[Debug mode]\");\n    return true;\n    }\n\n    \/** Called whenever the user pressed a menu item in the menu bar. *\/\n    public boolean onOptionsItemSelected(MenuItem item) {\n    Log.i(TAG, \"Menu Item selected: \" + item);\n    if (item == mMenuSketch)\n    mView.toggleSketchMode();\n    else if (item == mMenuAlien)\n    mView.toggleAlienMode();\n    else if (item == mMenuEvil)\n    mView.toggleEvilMode();\n    else if (item == mMenuDebug)\n    mView.toggleDebugMode();\n    return true;\n    }\n\n  4. Insert the following member variables inside the `CartoonifierView` class:\n\n        private boolean m_sketchMode = false;\n    private boolean m_alienMode = false;\n    private boolean m_evilMode = false;\n    private boolean m_debugMode = false;\n\n  5. Add the following functions to `CartoonifierView`:\n\n        protected void toggleSketchMode() {\n    m_sketchMode = !m_sketchMode;\n    }\n    protected void toggleAlienMode() {\n    m_alienMode = !m_alienMode;\n    }\n    protected void toggleEvilMode() {\n    m_evilMode = !m_evilMode;\n    }\n    protected void toggleDebugMode() {\n    m_debugMode = !m_debugMode;\n    }\n\n  6. We need to pass the mode values to the `cartoonifyImage()` JNI code, so let's send them as arguments. Modify the Java declaration of `CartoonifyImage()` in `CartoonifierView`:\n\n        public native void CartoonifyImage(int width, int height,byte[] yuv,\n    int[] rgba, boolean sketchMode, boolean alienMode,\n    boolean evilMode, boolean debugMode);\n\n  7. Now modify the Java code so we pass the current mode values in `processFrame()`:\n\n        CartoonifyImage(getFrameWidth(), getFrameHeight(), data,rgba,\n    m_sketchMode, m_alienMode, m_evilMode, m_debugMode);\n\n  8. The JNI declaration of `CartoonifyImage()` in `jni\\jni_part.cpp` should now be:\n\n        JNIEXPORT void JNICALL Java_com_Cartoonifier_CartoonifierView_CartoonifyImage(\n      JNIEnv* env, jobject, jint width, jint height,\n      jbyteArray yuv, jintArray bgra, jboolean sketchMode,\n      jboolean alienMode, jboolean evilMode, jboolean debugMode)\n\n  9. We then need to pass the modes to the C\/C++ code in `cartoon.cpp` from the JNI function in `jni\\jni_part.cpp`. When developing for Android we can only show one GUI window at a time, but on a desktop it is handy to show extra windows while debugging. So instead of taking a Boolean flag for `debugMode`, let's pass a number that would be `0` for non-debug, 1 for debug on mobile (where creating a GUI window in OpenCV would cause a crash!), and 2 for debug on desktop (where we can create as many extra windows as we want):\n\n        int debugType = 0;\n    if (debugMode)\n      debugType = 1;\n\n    cartoonifyImage(mbgr, displayedFrame, sketchMode, alienMode, evilMode, debugType);\n\n  10. Update the actual C\/C++ implementation in `cartoon.cpp`:\n\n        void cartoonifyImage(Mat srcColor, Mat dst, bool sketchMode,\n    bool alienMode, bool evilMode, int debugType)\n    {\n\n  11. And update the C\/C++ declaration in `cartoon.h`:\n\n        void cartoonifyImage(Mat srcColor, Mat dst, bool sketchMode,\n    bool alienMode, bool evilMode, int debugType);\n\n  12. Build and run it; then try pressing the small options-menu button on the bottom of the window. You should find that the sketch mode is real-time, whereas the paint mode has a large delay due to the bilateral filter.\n\n## Reducing the random pepper noise from the sketch image\n\nMost of the  cameras in current smartphones and tablets have significant image noise. This is normally acceptable, but it has a large effect on our 5 x 5 Laplacian-edge filter. The edge mask (shown as the sketch mode) will often have thousands of small blobs of black pixels called \"pepper\" noise, made of several black pixels next to each other in a white background. We are already using a Median filter, which is usually strong enough to remove pepper noise, but in our case it may not be strong enough. Our edge mask is mostly a pure white background (value of 255) with some black edges (value of 0) and the dots of noise (also values of 0). We could use a standard closing morphological operator, but it will remove a lot of edges. So, instead, we will apply a custom filter that removes small black regions that are surrounded completely by white pixels. This will remove a lot of noise while having little effect on actual edges.\n\nWe will scan the image for black pixels, and at each black pixel we'll check the border of the 5 x 5 square around it to see if all the 5 x 5 border pixels are white. If they are all white we know we have a small island of black noise, so we fill the whole block with white pixels to remove the black island. For simplicity in our 5 x 5 filter, we will ignore the two border pixels around the image and leave them as they are.\n\nThe following figure shows the original image from an Android tablet on the left side, with a sketch mode in the center (showing small black dots of pepper noise), and the result of our pepper-noise removal shown on the right side, where the skin looks cleaner:\n\nThe following code can be named as the function `removePepperNoise()`. This function will edit the image in place for simplicity:\n\n    void removePepperNoise(Mat &mask)\n    {\n    for (int y=2; y<mask.rows-2; y++) {\n      \/\/ Get access to each of the 5 rows near this pixel.\n      uchar *pUp2 = mask.ptr(y-2);\n      uchar *pUp1 = mask.ptr(y-1);\n      uchar *pThis = mask.ptr(y);\n      uchar *pDown1 = mask.ptr(y+1);\n      uchar *pDown2 = mask.ptr(y+2);\n\n      \/\/ Skip the first (and last) 2 pixels on each row.\n      pThis += 2;\n      pUp1 += 2;\n      pUp2 += 2;\n      pDown1 += 2;\n      pDown2 += 2;\n      for (int x=2; x<mask.cols-2; x++) {\n        uchar value = *pThis;  \/\/ Get this pixel value (0 or 255).\n        \/\/ Check if this is a black pixel that is surrounded by\n        \/\/ white pixels (ie: whether it is an \"island\" of black).\n    **if (value == 0) {**\n        bool above, left, below, right, surroundings;\n        above = *(pUp2 - 2) && *(pUp2 - 1) && *(pUp2) &&\n        *(pUp2 + 1) && *(pUp2 + 2);\n        left = *(pUp1 - 2) && *(pThis - 2) && *(pDown1 - 2);\n        below = *(pDown2 - 2) && *(pDown2 - 1) && *(pDown2) &&\n        *(pDown2 + 1) && *(pDown2 + 2);\n        right = *(pUp1 + 2) && *(pThis + 2) && *(pDown1 + 2);\n        surroundings = above && left && below && right;\n    **if (surroundings == true) {**\n          \/\/ Fill the whole 5x5 block as white. Since we know\n          \/\/ the 5x5 borders are already white, we just need to\n          \/\/ fill the 3x3 inner region.\n          *(pUp1 - 1) = 255;\n          *(pUp1 + 0) = 255;\n          *(pUp1 + 1) = 255;\n          *(pThis - 1) = 255;\n          *(pThis + 0) = 255;\n          *(pThis + 1) = 255;\n          *(pDown1 - 1) = 255;\n          *(pDown1 + 0) = 255;\n          *(pDown1 + 1) = 255;\n          \/\/ Since we just covered the whole 5x5 block with\n          \/\/ white, we know the next 2 pixels won't be black,\n         \/\/ so skip the next 2 pixels on the right.\n          pThis += 2;\n          pUp1 += 2;\n          pUp2 += 2;\n          pDown1 += 2;\n          pDown2 += 2;\n          }\n        }\n        \/\/ Move to the next pixel on the right.\n        pThis++;\n        pUp1++;\n        pUp2++;\n        pDown1++;\n        pDown2++;\n        }\n      }\n    }\n\n### Showing the FPS of the app\n\nIf you want to show the  frames per second (FPS **)** speed--which is less important for a slow app such as this, but still useful--on the screen, perform the following steps:\n\n  1. Copy the file `src\\org\\opencv\\samples\\imagemanipulations\\FpsMeter.java` from the ImageManipulations sample folder in OpenCV (for example, `C:\\OpenCV-2.4.1\\samples\\android\\image-manipulations`) to your `src\\com\\Cartoonifier` folder.\n  2. Replace the package name at the top of `FpsMeter.java` to be `com.Cartoonifier`.\n  3. In the file `CartoonifierViewBase.java`, declare your `FpsMeter` member variable after `private byte[] mBuffer;`:\n\n        private FpsMeter  mFps;\n\n  4. Initialize the `FpsMeter` object in the `CartoonifierViewBase()` constructor, after `mHolder.addCallback(this);`:\n\n        mFps = new FpsMeter();\n    mFps.init();\n\n  5. Measure the FPS of each frame in `run()` after the `try\/catch` block:\n\n        mFps.measure();\n\n  6. Draw the FPS onto the screen for each frame, in `run()` after the `canvas.drawBitmap()`function:\n\n        mFps.draw(canvas, (canvas.getWidth() - bmp.getWidth()) \/2, 0);\n\n### Using a different camera resolution\n\nIf you want your app to run faster, knowing that the quality will suffer, you should definitely consider either asking for a smaller camera image from the hardware or shrinking the image once you have it. The sample code that the Cartoonifier is based on uses the closest camera preview resolution to the screen height. So if your device has a 5 megapixel camera and the screen is just 640 x 480, it might use a camera resolution of 720 x 480, and so on. If you want to control which camera resolution is chosen, you can modify the parameters to `setupCamera()` in the `surfaceChanged()` function in `CartoonifierViewBase.java`. For example:\n\n    public void surfaceChanged(SurfaceHolder _holder, int format,\n      int width, int height) {\n      Log.i(TAG, \"Screen size: \" + width + \"x\" + height);\n      \/\/ Use a camera resolution of roughly half the screen height.\n    **setupCamera(width\/2, height\/2);**\n    }\n\nAn easy method to obtain the highest preview resolution from a camera is to pass a large size such as 10,000 x 10,000 and it will choose the maximum resolution available (note that it will only give the maximum preview resolution, which is the camera's video resolution and therefore is often much less than the camera's still-image resolution). Or if you want it to run really fast, pass 1 x 1 and it will find the lowest camera preview resolution (for example 160 x 120) for you.\n\n### Customizing the app\n\nNow that you have created a whole Android Cartoonifier app, you should know the basics of how it works and which parts do what; you should customize it! Change the GUI, the app behavior and workflow, the cartoonifier filter constants, the skin detector algorithm, or replace the cartoonifier code with your own ideas.\n\nYou can improve the skin-detection algorithm in many ways, such as by using a more complex skin-detection algorithm (for example, using trained Gaussian models from many recent CVPR or ICCV conference papers at <http:\/\/www.cvpapers.com>) or by adding face detection (see the _Face Detection_ section of Chapter 8, _Face Recognition using Eigenfaces_ ) to the skin detector, so that it detects where the user's face is rather than asking the user to put their face in the center of the screen. Beware that face detection may take many seconds on some devices or high-resolution cameras, so this approach may be limited by the comparatively slow processing speed, but smartphones and tablets are getting significantly faster every year, so this will become less of a problem.\n\nThe most significant way to speed up mobile computer vision apps is to reduce the camera resolution as much as possible (for example, 0.5 megapixel instead of 5 megapixel), allocate and free up images as rarely as possible, and do image conversions as rarely as possible (for instance, by supporting BGRA images throughout your code). You can also look for optimized image processing or math libraries from the CPU vendor of your device (for example, NVIDIA Tegra, Texas Instruments OMAP, Samsung Exynos, Apple A _x_ , or QualComm Snapdragon) or for your CPU family (for example, the ARM Cortex-A9). Remember, there may be an optimized version of OpenCV for your device.\n\nTo make customizing NDK and desktop image-processing code easier, this book comes with files `ImageUtils.cpp` and `ImageUtils.h` to help you experiment. It includes functions such as `printMatInfo()`, which prints a lot of information about a `cv::Mat` object, making debugging OpenCV much easier. There are also timing macros to easily add detailed timing statistics to your C\/C++ code. For example:\n\n    DECLARE_TIMING(myFilter);\n\n    void myImageFunction(Mat img) {\n      printMatInfo(img, \"input\");\n\n      START_TIMING(myFilter);\n      bilateralFilter(img,...);\n      STOP_TIMING(myFilter);\n      SHOW_TIMING(myFilter, \"My Filter\");\n    }\n\nYou would then see something like the following printed to your console:\n\n    **input: 800w600h 3ch 8bpp, range[19,255][17,243][47,251]**\n    **My Filter: time:  213ms   (ave=215ms min=197ms max=312ms, across 57 runs).**\n\nThis is useful when your OpenCV code is not working as expected; particularly for mobile development where it is often quite difficult to use an IDE debugger, and `printf()` statements generally won't work in Android NDK. However, the functions in `ImageUtils` work on both Android and desktop.\n\n# Summary\n\nThis chapter has shown several different types of image-processing filters that can be used to generate various cartoon effects: a plain sketch mode that looks like a pencil drawing, a paint mode that looks like a color painting, and a cartoon mode that overlays the sketch mode on top of the paint mode to make the image appear like a cartoon. It also shows that other fun effects can be obtained, such as the evil mode that greatly enhances noisy edges, and the alien mode that changes the skin of the face to appear bright green.\n\nThere are many commercial smartphone apps that perform similar fun effects on the user's face, such as cartoon filters and skin-color changers. There are also professional tools using similar concepts, such as skin-smoothing video post-processing tools that attempt to beautify women's faces by smoothing their skin while keeping the edges and non-skin regions sharp, in order to make their faces appear younger.\n\nThis chapter shows how to port the app from a desktop application to an Android mobile app, by following the recommended guidelines of developing a working desktop version first, porting it to a mobile app, and creating a user interface that is suitable for the mobile app. The image-processing code is shared between the two projects so that the reader can modify the cartoon filters for the desktop application, and by rebuilding the Android app it should automatically show their modifications in the Android app as well.\n\nThe steps required to use OpenCV4Android change regularly, and Android development itself is not static; so this chapter shows how to build the Android app by adding functionality to one of the OpenCV sample projects. It is expected that the reader can add the same functionality to an equivalent project in future versions of OpenCV4Android.\n\nThis book includes source code for both the desktop project and the Android project.\n\n# Chapter 2. Marker-based Augmented Reality on iPhone or iPad\n\n**Augmented reality** ( **AR** ) is a live view of a real-world environment whose elements are augmented by computer-generated graphics. As a result, the technology functions by enhancing one's current perception of reality. Augmentation is conventionally in real-time and in semantic context with environmental elements. With the help of advanced AR technology (for example, adding computer vision and object recognition) the information about the surrounding real world of the user becomes interactive and can be digitally manipulated. Artificial information about the environment and its objects can be overlaid on the real world.\n\nIn this chapter we will create an AR application for iPhone\/iPad devices. Starting from scratch, we will create an application that uses markers to draw some artificial objects on the images acquired from the camera. You will learn how to set up a project in XCode IDE and configure it to use OpenCV within your application. Also, aspects such as capturing a video from a built-in camera, 3D scene rendering using OpenGL ES, and building of a common AR application architecture are going to be explained.\n\nBefore we start, let me give you a brief list of knowledge and software you will need:\n\n  * You will need an Apple computer with XCode IDE installed. Development of applications for iPhone\/iPad is possible only with Apple's XCode IDE. This is the only way to build apps for this platform.\n  * You will need a model of iPhone, iPad, or iPod Touch devices. To run your applications on the device, you will have to purchase the Apple Developer Certificate for USD 99 per year. It's impossible to run developed applications on the device without this certificate.\n  * You will also need basic knowledge of XCode IDE. We will assume readers have some experience using this IDE.\n  * Basic knowledge of Objective-C and C++ programming languages is also necessary. However, all complex parts of application source code will be explained in detail.\n\nFrom this chapter you'll learn more about markers. The full detection routine is explained. After reading this chapter you will be able to write your own marker detection algorithm, estimate the marker pose in 3D world with regards to camera pose, and use this transformation between them to visualize arbitrary 3D objects.\n\nYou'll find the example project in this book's media for this chapter. It's a good starting point to create your first mobile Augmented Reality application.\n\nIn this chapter, we will cover the following topics:\n\n  * Creating an iOS project that uses OpenCV\n  * Application architecture\n  * Marker detection\n  * Marker identification\n  * Marker code recognition\n  * Placing a marker in 3D\n  * Rendering the 3D virtual object\n\n# Creating an iOS project that uses OpenCV\n\nIn this section we will create a demo application for iPhone\/iPad devices that will use the **OpenCV** ( **Open Source Computer Vision** ) library to detect markers in the camera frame and render 3D objects on it. This example will show  you how to get access to the raw video data stream from the device camera, perform image processing using the OpenCV library, find a marker in an image, and render an AR overlay.\n\nWe will start by first creating a new XCode project by choosing the iOS **Single View Application** template, as shown in the following screenshot:\n\nNow we have to add  OpenCV to our project. This step is necessary because in this application we will use a lot of functions from this library to detect markers and estimate position position.\n\nOpenCV is a library of programming functions for real-time computer vision. It was originally developed by Intel and is now supported by Willow Garage and Itseez. This library is written in C and C++ languages. It also has an official Python binding and unofficial bindings to Java and .NET languages.\n\n## Adding OpenCV framework\n\nFortunately  the library is cross-platform, so it can be used on iOS devices. Starting from version 2.4.2, OpenCV library is officially supported on the iOS platform and you can download the distribution package from the library website at  <http:\/\/opencv.org\/>. The **OpenCV for iOS** link points to the compressed OpenCV framework. Don't worry if you are new to iOS development; a framework is like a bundle of files. Usually each framework package contains a list of header files and list of statically linked libraries. Application frameworks provide an easy way to distribute precompiled libraries to developers.\n\nOf course, you  can build your own libraries from scratch. OpenCV documentation explains this process in detail. For simplicity, we follow the recommended way and use the framework for this chapter.\n\nAfter downloading the file we extract its content to the project folder, as shown in the following screenshot:\n\nTo inform the XCode IDE to use any framework during the build stage, click on **Project options** and locate the **Build phases** tab. From there we can add or remove the list of frameworks involved in the build process. Click on the plus sign to add a new framework, as shown in the following screenshot:\n\nFrom here we can choose from a list of standard frameworks. But to add a custom framework we should click on the **Add other** button. The open file dialog box will appear. Point it to **opencv2.framework** in the project folder as shown in the following screenshot:\n\n## Including OpenCV headers\n\nNow that we have added the OpenCV framework to the project, everything is almost done. One last thing--let's add  OpenCV headers to the project's precompiled headers. The precompiled headers are a great feature to speed up compilation time. By adding OpenCV headers to them, all your sources automatically include OpenCV headers as well. Find a `.pch` file in the project source tree and modify it in the following way.\n\nThe following code shows how to modify the `.pch` file in the project source tree:\n\n    \/\/\n    \/\/ Prefix header for all source files of the 'Example_MarkerBasedAR'\n    \/\/\n\n    #import <Availability.h>\n\n    #ifndef __IPHONE_5_0\n    #warning \"This project uses features only available in iOS SDK 5.0 and later.\"\n    #endif\n\n    #ifdef __cplusplus\n    #include <opencv2\/opencv.hpp>\n    #endif\n\n    #ifdef __OBJC__\n      #import <UIKit\/UIKit.h>\n      #import <Foundation\/Foundation.h>\n    #endif\n\nNow you can call any OpenCV function from any place in your project.\n\nThat's all. Our  project template is configured and we are ready to move further. Free advice: make a copy of this project; this will save you time when you are creating your next one!\n\n# Application architecture\n\nEach iOS application contains at least one instance of the `UIViewController` interface that handles all view events and manages the application's business logic. This class provides the fundamental view-management model for all iOS apps. A view controller manages a set of views that make up a portion of your app's user interface. As part of the controller layer of your app, a view controller coordinates its efforts with model objects and other controller objects--including other view controllers--so your app presents a single coherent user interface.\n\nThe application that we are going to write will have only one view; that's why we choose a **Single-View Application** template to create one. This view will be used to present the rendered picture. Our `ViewController` class will contain three major components that each AR application should have (see the next diagram):\n\n  * Video source\n  * Processing pipeline\n  * Visualization engine\n\nThe video source is responsible for providing new frames taken from the built-in camera to the user code. This means that the video source should be capable of choosing a camera device (front- or back-facing camera), adjusting its parameters (such as resolution of the captured video, white balance, and shutter speed), and grabbing frames without freezing the main UI.\n\nThe image processing routine will be encapsulated in the `MarkerDetector` class. This class provides a very thin interface to user code. Usually it's a set of functions like `processFrame` and `getResult`. Actually that's all that `ViewController` should know about. We must not expose low-level data structures and algorithms to the view layer without strong necessity. `VisualizationController` contains all logic concerned with visualization of the Augmented Reality on our view. `VisualizationController` is also a facade that hides a particular implementation of the rendering engine. Low code coherence gives us freedom to change these components without the need to rewrite the rest of your code.\n\nSuch an approach gives you the freedom to use independent modules on other platforms and compilers as well. For example, you can use the `MarkerDetector` class easily to develop desktop applications on Mac, Windows, and Linux systems without any changes to the code. Likewise, you can decide to port `VisualizationController` on the Windows platform and use Direct3D for rendering. In this case you should write only new `VisualizationController` implementation; other code parts will remain the same.\n\nThe main processing routine starts from receiving a new frame from the video source. This triggers video source to inform the user code about this event with a callback. `ViewController` handles this callback and performs the following operations:\n\n  1. Sends a new frame to the visualization controller.\n  2. Performs processing of the new frame using our pipeline.\n  3. Sends the detected markers to the visualization stage.\n  4. Renders a scene.\n\nLet's examine this routine in detail. The rendering of an AR scene includes the drawing of a background image that has a content of the last received frame; artificial 3D objects are drawn later on. When we send a new frame for visualization, we are copying image data to internal buffers of the rendering engine. This is not actual rendering yet; we are just updating the text with a new bitmap.\n\nThe second step is the processing of new frame and marker detection. We pass our image as input and as a result receive a list of the markers detected. on it. These markers are passed to the visualization controller, which knows how to deal with them. Let's take a look at the following sequence diagram where this routine is shown:\n\nWe start development by writing a video capture component. This class will be responsible for all frame grabbing and for sending notifications of captured frames via user callback. Later on we will write a marker detection algorithm. This detection routine is the core of your application. In this part of our program we will use a lot of OpenCV functions to process images, detect contours on them, find marker rectangles, and estimate their position. After that we will concentrate on visualization of our results using Augmented Reality. After bringing all these things together we will complete our first AR application. So let's move on!\n\n## Accessing the camera\n\nThe  Augmented Reality application is impossible to create without two major things: video capturing and AR visualization. The video capture stage consists of receiving frames from the device camera, performing necessary color conversion, and sending it to the processing pipeline. As the single frame processing time is so critical to AR applications, the capture process should be as efficient as possible. The best way to achieve maximum performance is to have direct access to the frames received from the camera. This became possible starting from iOS Version 4. Existing APIs from the AVFoundation framework provide the necessary functionality to read directly from image buffers in memory.\n\nYou can find a lot of examples that use the `AVCaptureVideoPreviewLayer` class and the `UIGetScreenImage` function to capture videos from the camera. This technique was used for iOS Version 3 and earlier. It has now become outdated and has two major disadvantages:\n\n  * Lack of direct access to frame data. To get a bitmap, you have to create an intermediate instance of `UIImage`, copy an image to it, and get it back. For AR applications this price is too high, because each millisecond matters. Losing a few frames per second (FPS) significantly decreases overall user experience.\n  * To draw an AR, you have to add a transparent overlay view that will present the AR. Referring to Apple guidelines, you should avoid non-opaque layers because their blending is hard for mobile processors.\n\nClasses `AVCaptureDevice` and `AVCaptureVideoDataOutput` allow you to configure, capture, and specify unprocessed video frames in 32 bpp BGRA format. Also you can set up the desired resolution of output frames. However, it does affect overall performance since the larger the frame the more processing time and memory is required.\n\nThere is a good alternative for high-performance video capture. The AVFoundation API offers a much faster and more elegant way to grab frames directly from the camera. But first, let's take a look at the following figure where the capturing process for iOS is shown:\n\n`AVCaptureSession` is a root capture object that we should create. Capture session requires two components--an input and an output. The input device can either be a physical device (camera) or a video file (not shown in diagram). In our case it's a built-in camera (front or back). The output device can be presented by one of the following interfaces:\n\n  * `AVCaptureMovieFileOutput`\n  * `AVCaptureStillImageOutput`\n  * `AVCaptureVideoPreviewLayer`\n  * `AVCaptureVideoDataOutput`\n\nThe `AVCaptureMovieFileOutput` interface is used to record video to the file, the `AVCaptureStillImageOutput` interface is used to to make still images, and the `AVCaptureVideoPreviewLayer` interface is used to play a video preview on the screen. We are interested in the `AVCaptureVideoDataOutput` interface because it gives you direct access to video data.\n\n### Note\n\nThe iOS platform is built on top of the Objective-C programming language. So to work with AVFoundation framework, our class also has to be written in Objective-C. In this section all code listings are in the Objective-C++ language.\n\nTo  encapsulate the video capturing process, we create the `VideoSource` interface as shown by the following code:\n\n    @protocol VideoSourceDelegate<NSObject>\n\n    -(void)frameReady:(BGRAVideoFrame) frame;\n\n    @end\n\n    @interface VideoSource : NSObject<AVCaptureVideoDataOutputSampleBufferDelegate>\n    {\n\n    }\n\n    @property (nonatomic, retain) AVCaptureSession *captureSession;\n    @property (nonatomic, retain) AVCaptureDeviceInput *deviceInput;\n    @property (nonatomic, retain) id<VideoSourceDelegate> delegate;\n\n    - (bool) startWithDevicePosition:(AVCaptureDevicePosition)devicePosition;\n    - (CameraCalibration) getCalibration;\n    - (CGSize) getFrameSize;\n\n    @end\n\nIn this callback we lock the image buffer to prevent modifications by any new frames, obtain a pointer to the image data and frame dimensions. Then we construct temporary BGRAVideoFrame object that is passed to outside via special delegate. This delegate has following prototype:\n\n    @protocol VideoSourceDelegate<NSObject>\n    -(void)frameReady:(BGRAVideoFrame) frame;\n\n    @end\n\nWithin `VideoSourceDelegate`, the `VideoSource` interface informs the user code that a new frame is available.\n\nThe step-by-step guide for the initialization of video capture is listed as follows:\n\n  1. Create an instance of `AVCaptureSession` and set the capture session quality preset.\n  2. Choose and create `AVCaptureDevice`. You can choose the front- or back-facing camera or use the default one.\n  3. Initialize `AVCaptureDeviceInput` using the created capture device and add it to the capture session.\n  4. Create an instance of `AVCaptureVideoDataOutput` and initialize it with format of video frame, callback delegate, and dispatch the queue.\n  5. Add the capture output to the capture session object.\n  6. Start the capture session.\n\nLet's explain some of these steps in more detail. After creating the capture session, we can specify the desired quality preset to ensure that we will obtain optimal performance. We don't need to process HD-quality video, so 640 x 480 or an even lesser frame resolution is a good choice:\n\n    - (id)init\n    {\n      if ((self = [super init]))\n      {\n        AVCaptureSession * capSession = [[AVCaptureSession alloc] init];\n\n        if ([capSession canSetSessionPreset:AVCaptureSessionPreset640x480])\n        {\n          [capSession setSessionPreset:AVCaptureSessionPreset640x480];\n          NSLog(@\"Set capture session preset AVCaptureSessionPreset640x480\");\n        }\n        else if ([capSession canSetSessionPreset:AVCaptureSessionPresetLow])\n        {\n          [capSession setSessionPreset:AVCaptureSessionPresetLow];\n          NSLog(@\"Set capture session preset AVCaptureSessionPresetLow\");\n        }\n\n        self.captureSession = capSession;\n      }\n      return self;\n    }\n\n### Note\n\nAlways check hardware capabilities using the appropriate API; there is no guarantee that every camera will be capable of setting a particular session preset.\n\nAfter  creating the capture session, we should add the capture input--the instance of `AVCaptureDeviceInput` will represent a physical camera device. The `cameraWithPosition` function is a helper function that returns the camera device for the requested position (front, back, or default):\n\n    - (bool) startWithDevicePosition:(AVCaptureDevicePosition)devicePosition\n    {\n      AVCaptureDevice *videoDevice = [self cameraWithPosition:devicePosition];\n\n      if (!videoDevice)\n        return FALSE;\n\n      {\n        NSError *error;\n\n        AVCaptureDeviceInput *videoIn = [AVCaptureDeviceInput \n        deviceInputWithDevice:videoDevice error:&error];\n        self.deviceInput = videoIn;\n\n        if (!error)\n        {\n          if ([[self captureSession] canAddInput:videoIn])\n          {\n            [[self captureSession] addInput:videoIn];\n          }\n          else\n          {\n            NSLog(@\"Couldn't add video input\");\n            return FALSE;\n          }\n        }\n        else\n        {\n          NSLog(@\"Couldn't create video input\");\n          return FALSE;\n        }\n      }\n\n      [self addRawViewOutput];\n      [captureSession startRunning];\n      return TRUE;\n    }\n\nPlease  notice the error handling code. Take care of return values for such an important thing as working with hardware setup is a good practice. Without this, your code can crash in unexpected cases without informing the user what has happened.\n\nWe created a capture session and added a source of the video frames. Now it's time to add a receiver--an object that will receive actual frame data. The `AVCaptureVideoDataOutput` class is used to process uncompressed frames from the video stream. The camera can provide frames in BGRA, CMYK, or simple grayscale color models. For our purposes the BGRA color model fits best of all, as we will use this frame for visualization and image processing. The following code shows the `addRawViewOutput` function:\n\n    - (void) addRawViewOutput\n    {\n      \/*We setupt the output*\/\n      AVCaptureVideoDataOutput *captureOutput = [[AVCaptureVideoDataOutput alloc] init];\n\n      \/*While a frame is processes in -captureOutput:didOutputSampleBuffer:fromConnection: delegate methods no other frames are added in the queue.\n        If you don't want this behaviour set the property to NO *\/\n      captureOutput.alwaysDiscardsLateVideoFrames = YES;\n\n      \/*We create a serial queue to handle the processing of our frames*\/\n      dispatch_queue_t queue;\n      queue = dispatch_queue_create(\"com.Example_MarkerBasedAR.cameraQueue\", \n      NULL);\n      [captureOutput setSampleBufferDelegate:self queue:queue];\n      dispatch_release(queue);\n\n      \/\/ Set the video output to store frame in BGRA (It is supposed to be faster)\n      NSString* key = (NSString*)kCVPixelBufferPixelFormatTypeKey;\n      NSNumber* value = [NSNumber \n      numberWithUnsignedInt:kCVPixelFormatType_32BGRA];\n\n      NSDictionary* videoSettings = [NSDictionary dictionaryWithObject:value \n      forKey:key];\n      [captureOutput setVideoSettings:videoSettings];\n\n      \/\/ Register an output\n      [self.captureSession addOutput:captureOutput];\n    }\n\nNow the capture session is finally configured. When started, it will capture frames from the camera and send it to user code. When the new frame is available, an `AVCaptureSession` object performs a` captureOutput: didOutputSampleBuffer:fromConnection` callback. In this function, we will perform a minor data conversion operation to get the image data in a more usable format and pass it to user code:\n\n    - (void)captureOutput:(AVCaptureOutput *)captureOutput \n    didOutputSampleBuffer:(CMSampleBufferRef)sampleBuffer \n           fromConnection:(AVCaptureConnection *)connection \n    { \n      \/\/ Get a image buffer holding video frame\n      CVImageBufferRef imageBuffer = CMSampleBufferGetImageBuffer(sampleBuffer);\n\n      \/\/ Lock the image buffer\n      CVPixelBufferLockBaseAddress(imageBuffer,0);\n\n      \/\/ Get information about the image\n      uint8_t *baseAddress = (uint8_t *)CVPixelBufferGetBaseAddress(imageBuffer); \n      size_t width = CVPixelBufferGetWidth(imageBuffer); \n      size_t height = CVPixelBufferGetHeight(imageBuffer);  \n      size_t stride = CVPixelBufferGetBytesPerRow(imageBuffer);\n\n      BGRAVideoFrame frame = {width, height, stride, baseAddress};\n      [delegate frameReady:frame];\n\n      \/*We unlock the  image buffer*\/\n      CVPixelBufferUnlockBaseAddress(imageBuffer,0);\n    }\n\nWe obtain a reference to the image buffer that stores our frame data. Then we lock it to prevent modifications by new frames. Now we have exclusive access to the frame data. With help of the CoreVideo API, we get the image dimensions, stride (number of pixels per row), and the pointer to the beginning of the image data.\n\n### Note\n\nI draw your attention to the `CVPixelBufferLockBaseAddress`\/ `CVPixelBufferUnlockBaseAddress` function call in the callback code. Until we hold a lock on the pixel buffer, it guarantees consistency and correctness of its data. Reading of pixels is available only after you have obtained a lock. When you're done, don't forget to unlock it to allow the OS to fill it with new data.\n\n# Marker detection\n\nA marker is usually designed as a rectangle image holding black and white areas inside it. Due to known limitations, the marker detection procedure is a simple one. First of all we need to find closed contours on the input image and unwarp the image inside it to a rectangle and then check this against our marker model.\n\nIn this sample the 5 x 5 marker will be used. Here is what it looks like:\n\nIn the sample project that you will find in this book, the marker detection routine is encapsulated in the `MarkerDetector` class:\n\n    \/**\n     * A top-level class that encapsulate marker detector algorithm\n     *\/\n    class MarkerDetector\n    {\n    public:\n\n      \/**\n       * Initialize a new instance of marker detector object\n       * @calibration[in] - Camera calibration necessary for pose estimation.\n       *\/\n      MarkerDetector(CameraCalibration calibration);\n\n      void processFrame(const BGRAVideoFrame& frame);\n\n      const std::vector<Transformation>& getTransformations() const;\n\n      protected:\n      bool findMarkers(const BGRAVideoFrame& frame, std::vector<Marker>& \n      detectedMarkers);\n\n      void prepareImage(const cv::Mat& bgraMat,\n                        cv::Mat& grayscale);\n\n      void performThreshold(const cv::Mat& grayscale,\n                            cv::Mat& thresholdImg);\n\n      void findContours(const cv::Mat& thresholdImg, \n                        std::vector<std::vector<cv::Point> >& contours,\n                        int minContourPointsAllowed);\n\n      void findMarkerCandidates(const std::vector<std::vector<cv::Point> >& \n      contours, std::vector<Marker>& detectedMarkers);\n\n      void detectMarkers(const cv::Mat& grayscale,\n                         std::vector<Marker>& detectedMarkers);\n\n      void estimatePosition(std::vector<Marker>& detectedMarkers);\n\n    private:\n    };\n\nTo help you better understand the marker detection routine, a step-by-step processing on one frame from a video will be shown. A source image taken from an iPad camera will be used as an example:\n\n## Marker identification\n\nHere is the workflow of the  marker detection routine:\n\n  1. Convert the input image to grayscale.\n  2. Perform binary threshold operation.\n  3. Detect contours.\n  4. Search for possible markers.\n  5. Detect and decode markers.\n  6. Estimate marker 3D pose.\n\n### Grayscale conversion\n\nThe conversion to  grayscale is necessary because markers usually contain only black and white blocks and it's much easier to operate with them on grayscale images. Fortunately, OpenCV color conversion is simple enough.\n\nPlease take a look at the following code listing in C++:\n\n    void MarkerDetector::prepareImage(const cv::Mat& bgraMat, cv::Mat& grayscale)\n    {\n      \/\/ Convert to grayscale\n      cv::cvtColor(bgraMat, grayscale, CV_BGRA2GRAY);\n    }\n\nThis function will convert the input BGRA image to grayscale (it will allocate image buffers if necessary) and place the result into the second argument. All further steps will be performed with the grayscale image.\n\n### Image binarization\n\nThe binarization operation  will transform each pixel of our image to black (zero intensity) or white (full intensity). This step is required to find contours. There are several threshold methods; each has strong and weak sides. The easiest and fastest method is absolute threshold. In this method the resulting value depends on current pixel intensity and some threshold value. If pixel intensity is greater than the threshold value, the result will be white (255); otherwise it will be black (0).\n\nThis method has a huge disadvantage--it depends on lighting conditions and soft intensity changes. The more preferable method is the adaptive threshold. The major difference of this method is the use of all pixels in given radius around the examined pixel. Using average intensity gives good results and secures more robust corner detection.\n\nThe following code snippet shows the `MarkerDetector` function:\n\n    void MarkerDetector::performThreshold(const cv::Mat& grayscale, cv::Mat& thresholdImg)\n    {\n      cv::adaptiveThreshold(grayscale,   \/\/ Input image\n                            thresholdImg,\/\/ Result binary image\n                            255,         \/\/\n                            cv::ADAPTIVE_THRESH_GAUSSIAN_C, \/\/\n                            cv::THRESH_BINARY_INV, \/\/\n                            7, \/\/\n                            7  \/\/\n                            );\n    }\n\nAfter applying adaptive threshold to the input image, the resulting image looks similar to the following one:\n\nEach marker  usually looks like a square figure with black and white areas inside it. So the best way to locate a marker is to find closed contours and approximate them with polygons of 4 vertices.\n\n### Contours detection\n\nThe `cv::findCountours` function will detect contours on the  input binary image:\n\n    void MarkerDetector::findContours(const cv::Mat& thresholdImg,\n                                      std::vector<std::vector<cv::Point> >& contours,\n    int minContourPointsAllowed)\n    {\n        std::vector< std::vector<cv::Point> > allContours;\n        cv::findContours(thresholdImg, allContours, CV_RETR_LIST, CV_CHAIN_APPROX_NONE);\n\n        contours.clear();\n        for (size_t i=0; i<allContours.size(); i++)\n        {\n            int contourSize = allContours[i].size();\n            if (contourSize > minContourPointsAllowed)\n            {\n                contours.push_back(allContours[i]);\n            }\n        }\n    }\n\nThe return value of this function is a list of polygons where each polygon represents a single contour. The function skips contours that have their perimeter in pixels value set to be less than the value of the `minContourPointsAllowed` variable. This is because we are not interested in small contours. (They will probably contain no marker, or the contour won't be able to be detected due to a small marker size.)\n\nThe following figure shows the visualization of detected contours:\n\n### Candidates search\n\nAfter finding contours, the polygon approximation stage is performed. This is done to decrease the number of points that describe the contour shape. It's a good quality check to filter out areas without markers because they can always be represented with a polygon that contains four vertices. If the approximated polygon has more than or fewer than 4 vertices, it's definitely not what we are looking for. The following code implements this idea:\n\n    void MarkerDetector::findCandidates\n    (\n        const ContoursVector& contours,\n        std::vector<Marker>& detectedMarkers\n    )\n    {\n        std::vector<cv::Point>  approxCurve;\n        std::vector<Marker>     possibleMarkers;\n\n        \/\/ For each contour, analyze if it is a parallelepiped likely to be the \n        marker\n        for (size_t i=0; i<contours.size(); i++)\n        {\n            \/\/ Approximate to a polygon\n            double eps = contours[i].size() * 0.05;\n            cv::approxPolyDP(contours[i], approxCurve, eps, true);\n\n            \/\/ We interested only in polygons that contains only four points\n            if (approxCurve.size() != 4)\n                continue;\n\n            \/\/ And they have to be convex\n            if (!cv::isContourConvex(approxCurve))\n                continue;\n\n            \/\/ Ensure that the distance between consecutive points is large enough\n            float minDist = std::numeric_limits<float>::max();\n\n            for (int i = 0; i < 4; i++)\n            {\n                cv::Point side = approxCurve[i] - approxCurve[(i+1)%4];\n                float squaredSideLength = side.dot(side);\n                minDist = std::min(minDist, squaredSideLength);\n            }\n\n            \/\/ Check that distance is not very small\n            if (minDist < m_minContourLengthAllowed)\n                continue;\n\n            \/\/ All tests are passed. Save marker candidate:\n            Marker m;\n\n            for (int i = 0; i<4; i++)\n                m.points.push_back( cv::Point2f(approxCurve[i].x,approxCurve[i].y) );\n\n            \/\/ Sort the points in anti-clockwise order\n            \/\/ Trace a line between the first and second point.\n            \/\/ If the third point is at the right side, then the points are anti-\n            clockwise\n            cv::Point v1 = m.points[1] - m.points[0];\n            cv::Point v2 = m.points[2] - m.points[0];\n\n            double o = (v1.x * v2.y) - (v1.y * v2.x);\n\n            if (o < 0.0)         \/\/if the third point is in the left side, then \n            sort in anti-clockwise order\n                std::swap(m.points[1], m.points[3]);\n\n            possibleMarkers.push_back(m);\n        }\n\n        \/\/ Remove these elements which corners are too close to each other.\n        \/\/ First detect candidates for removal:\n        std::vector< std::pair<int,int> > tooNearCandidates;\n        for (size_t i=0;i<possibleMarkers.size();i++)\n        { \n            const Marker& m1 = possibleMarkers[i];\n\n            \/\/calculate the average distance of each corner to the nearest corner \n            of the other marker candidate\n            for (size_t j=i+1;j<possibleMarkers.size();j++)\n            {\n                const Marker& m2 = possibleMarkers[j];\n\n                float distSquared = 0;\n\n                for (int c = 0; c < 4; c++)\n                {\n                    cv::Point v = m1.points[c] - m2.points[c];\n                    distSquared += v.dot(v);\n                }\n\n                distSquared \/= 4;\n\n                if (distSquared < 100)\n                {\n                    tooNearCandidates.push_back(std::pair<int,int>(i,j));\n                }\n            }                \n        }\n\n        \/\/ Mark for removal the element of the pair with smaller perimeter\n        std::vector<bool> removalMask (possibleMarkers.size(), false);\n\n        for (size_t i=0; i<tooNearCandidates.size(); i++)\n        {\n            float p1 = perimeter(possibleMarkers[tooNearCandidates[i].first \n            ].points);\n            float p2 = \n            perimeter(possibleMarkers[tooNearCandidates[i].second].points);\n\n            size_t removalIndex;\n            if (p1 > p2)\n                removalIndex = tooNearCandidates[i].second;\n            else\n                removalIndex = tooNearCandidates[i].first;\n\n            removalMask[removalIndex] = true;\n        }\n\n        \/\/ Return candidates\n        detectedMarkers.clear();\n        for (size_t i=0;i<possibleMarkers.size();i++)\n        {\n            if (!removalMask[i])\n                detectedMarkers.push_back(possibleMarkers[i]);\n        }\n    }\n\nNow we have  obtained a list of parallelepipeds that are likely to be the markers. To verify whether they are markers or not, we need to perform three steps:\n\n  1. First, we should remove the perspective projection so as to obtain a frontal view of the rectangle area.\n  2. Then we perform thresholding of the image using the Otsu algorithm. This algorithm assumes a bimodal distribution and finds the threshold value that maximizes the extra-class variance while keeping a low intra-class variance.\n  3. Finally we perform identification of the marker code. If it is a marker, it has an internal code. The marker is divided into a 7 x 7 grid, of which the internal 5 x 5 cells contain ID information. The rest correspond to the external black border. Here, we first check whether the external black border is present. Then we read the internal 5 x 5 cells and check if they provide a valid code. (It might be required to rotate the code to get the valid one.)\n\nTo get the rectangle marker image, we have to unwarp the input image using perspective transformation. This matrix can be calculated with the help of the `cv::getPerspectiveTransform` function. It finds the perspective transformation from four pairs of corresponding points. The first argument is the marker coordinates in image space and the second point corresponds to the coordinates of the square marker image. Estimated transformation will transform the marker to square form and let us analyze it:\n\n    cv::Mat canonicalMarker;\n    Marker& marker = detectedMarkers[i];\n\n    \/\/ Find the perspective transfomation that brings current marker to rectangular form\n    cv::Mat M = cv::getPerspectiveTransform(marker.points, m_markerCorners2d);\n\n    \/\/ Transform image to get a canonical marker image\n    cv::warpPerspective(grayscale, canonicalMarker,  M, markerSize);\n\nImage warping transforms our image to a rectangle form using perspective transformation:\n\nNow we can test  the image to verify if it is a valid marker image. Then we try to extract the bit mask with the marker code. As we expect our marker to contain only black and white colors, we can perform Otsu thresholding to remove gray pixels and leave only black and white pixels:\n\n    \/\/threshold image\n    cv::threshold(markerImage, markerImage, 125, 255, cv::THRESH_BINARY | cv::THRESH_OTSU);\n\n## Marker code recognition\n\nEach marker has an internal code given by 5 words of 5 bits each. The codification employed is a slight modification of the hamming code. In total, each word has only 2 bits of information out of the 5 bits employed. The other 3 are employed for error detection. As a consequence, we can have up to 1024 different IDs.\n\nThe main difference with the hamming code is that the first bit (parity of bits 3 and 5) is inverted. So, ID 0 (which in hamming code is 00000) becomes 10000 in our code. The idea is to prevent a completely black rectangle from being a valid marker ID, with the goal of reducing the likelihood of false positives with objects of the environment.\n\nCounting the number of black and white pixels for each cell gives us a 5 x 5-bit mask with marker code. To count the number of non-zero pixels on a certain image, the `cv::countNonZero` function is used. This function counts non-zero array elements from a given 1D or 2D array. The `cv::Mat` type can return a subimage view--a new instance of `cv::Mat` that contains a portion of the original image. For example, if you have a `cv::Mat` of size 400 x 400, the following piece of code will create a submatrix for the 50 x 50 image block starting from (10, 10):\n\n    cv::Mat src(400,400,CV_8UC1);\n    cv::Rect r(10,10,50,50);\n    cv::Mat subView = src(r);\n\n### Reading marker code\n\nUsing this  technique, we can easily find black and white cells on the marker board:\n\n     cv::Mat bitMatrix = cv::Mat::zeros(5,5,CV_8UC1);\n\n      \/\/get information(for each inner square, determine if it is  black or white)\n      for (int y=0;y<5;y++)\n      {\n        for (int x=0;x<5;x++)\n        {\n          int cellX = (x+1)*cellSize;\n          int cellY = (y+1)*cellSize;\n          cv::Mat cell = grey(cv::Rect(cellX,cellY,cellSize,cellSize));\n\n          int nZ = cv::countNonZero(cell);\n          if (nZ> (cellSize*cellSize) \/2)\n            bitMatrix.at<uchar>(y,x) = 1;\n        }\n      }\n\nTake a look at the following figure. The same marker can have four possible representations depending on the camera's point of view:\n\nAs there are four possible orientations of the marker picture, we have to find the correct marker position. Remember, we introduced three parity bits for each two bits of information. With their help we can find the hamming distance for each possible marker orientation. The correct marker position will have zero hamming distance error, while the other rotations won't.\n\nHere is a code snippet that rotates the bit matrix four times and finds the correct marker orientation:\n\n      \/\/check all possible rotations\n      cv::Mat rotations[4];\n      int distances[4];\n\n      rotations[0] = bitMatrix;  \n      distances[0] = hammDistMarker(rotations[0]);\n\n      std::pair<int,int> minDist(distances[0],0);\n\n      for (int i=1; i<4; i++)\n      {\n        \/\/get the hamming distance to the nearest possible word\n        rotations[i] = rotate(rotations[i-1]);\n        distances[i] = hammDistMarker(rotations[i]);\n\n        if (distances[i] < minDist.first)\n        {\n          minDist.first  = distances[i];\n          minDist.second = i;\n        }\n      }\n\nThis code finds the orientation of the bit matrix in such a way that it gives minimal error for the hamming distance metric. This error should be zero for correct marker ID; if it's not, it means that  we encountered a wrong marker pattern (corrupted image or false-positive marker detection).\n\n### Marker location refinement\n\nAfter finding the  right marker orientation, we rotate the marker's corners respectively to conform to their order:\n\n    \/\/sort the points so that they are always in the same order\n    \/\/ no matter the camera orientation\n    std::rotate(marker.points.begin(), marker.points.begin() + 4 - nRotations,\n    marker.points.end());\n\nAfter detecting a marker and decoding its ID, we will refine its corners. This operation will help us in the next step when we will estimate the marker position in 3D. To find the corner location with subpixel accuracy, the `cv::cornerSubPix` function is used:\n\n    std::vector<cv::Point2f> preciseCorners(4 * goodMarkers.size());\n\n    for (size_t i=0; i<goodMarkers.size(); i++)\n    {\n      Marker& marker = goodMarkers[i];\n\n      for (int c=0;c<4;c++)\n      {\n        preciseCorners[i*4+c] = marker.points[c];\n      }\n    }\n\n    cv::cornerSubPix(grayscale, preciseCorners, cvSize(5,5), cvSize(-1,-1), cvTermCriteria(CV_TERMCRIT_ITER,30,0.1));\n\n    \/\/copy back\n    for (size_t i=0;i<goodMarkers.size();i++)\n    {\n      Marker&marker = goodMarkers[i];\n\n      for (int c=0;c<4;c++)\n      {\n        marker.points[c] = preciseCorners[i*4+c];\n      }\n    }\n\nThe first step is to  prepare the input data for this function. We copy the list of vertices to the input array. Then we call `cv::cornerSubPix`, passing the actual image, list of points, and set of parameters that affect quality and performance of location refinement. When done, we copy the refined locations back to marker corners as shown in the following image.\n\nWe do not use `cornerSubPix` in the earlier stages of marker detection due to its complexity. It's very expensive to call this function for large numbers of points (in terms of computation time). Therefore we do this only for valid markers.\n\n# Placing a marker in 3D\n\nAugmented Reality tries  to fuse the real-world object with virtual content. To place a 3D model in a scene, we need to know its pose with regard to a camera that we use to obtain the video frames. We will use a Euclidian transformation in the Cartesian coordinate system to represent such a pose.\n\nThe position of the marker in 3D and its corresponding projection in 2D is restricted by the following equation:\n\nP = A * [R|T] * M;\n\nWhere:\n\n  *  _M_ denotes a point in a 3D space\n  *  _[R|T]_ denotes a [3|4] matrix representing a Euclidian transformation\n  *  _A_ denotes a camera matrix or a matrix of intrinsic parameters\n  *  _P_ denotes projection of _M_ in screen space\n\nAfter performing the marker detection step we now know the position of the four marker corners in 2D (projections in screen space). In the next section you will learn how to obtain the _A_ matrix and _M_ vector parameters and calculate the [R|T] transformation.\n\n## Camera calibration\n\nEach camera lens has unique parameters, such as focal length, principal point, and lens distortion model. The process of finding intrinsic camera parameters is called camera calibration. The camera calibration process is important for Augmented Reality applications because it describes the perspective transformation and lens distortion on an output image. To achieve the best user experience with Augmented Reality, visualization of an augmented object should be done using the same perspective projection.\n\nTo calibrate the camera, we need a special pattern image (chessboard plate or black circles on white background). The camera that is being calibrated takes 10-15 shots of this pattern from different points of view. A calibration algorithm then finds the optimal camera intrinsic parameters and the distortion vector:\n\nTo represent camera  calibration in our program, we use the `CameraCalibration` class:\n\n    \/**\n     * A camera calibration class that stores intrinsic matrix and distorsion coefficients.\n     *\/\n    class CameraCalibration\n    {\n      public:\n      CameraCalibration();\n      CameraCalibration(float fx, float fy, float cx, float cy);\n      CameraCalibration(float fx, float fy, float cx, float cy, float \n      distorsionCoeff[4]);\n\n      void getMatrix34(float cparam[3][4]) const;\n\n      const Matrix33& getIntrinsic() const;\n      const Vector4&  getDistorsion() const;\n\n      private:\n      Matrix33 m_intrinsic;\n      Vector4  m_distorsion;\n    };\n\nDetailed explanation of the calibration procedure is beyond the scope of this chapter. Please refer to _OpenCV camera_calibration sample_ or _OpenCV: Estimating_ _Projective Relations in Images_ at <http:\/\/www.packtpub.com\/article\/opencv-estimating-projective-relations-images> for additional information and source code.\n\nFor this sample  we provide internal parameters for all modern iOS devices (iPad 2, iPad 3, and iPhone 4).\n\n## Marker pose estimation\n\nWith the precise  location of marker corners, we can estimate a transformation between our camera and a marker in 3D space. This operation is known as pose estimation from 2D-3D correspondences. The pose estimation process finds a Euclidean transformation (that consists only of rotation and translation components) between the camera and the object.\n\nLet's take a look at the following figure:\n\nThe **C** is used to  denote the camera center. The **P1-P4** points are 3D points in the world coordinate system and the **p1-p4** points are their projections on the camera's image plane. Our goal is to find relative transformation between a known marker position in the 3D world ( **p1-p4** ) and the camera **C** using an intrinsic matrix and known point projections on image plane ( **P1-P4** ). But where do we get the coordinates of marker position in 3D space? We imagine them. As our marker always has a square form and all vertices lie in one plane, we can define their corners as follows:\n\nWe put our marker in the XY plane (Z component is zero) and the marker center corresponds to the (0.0, 0.0, 0.0) point. It's a great hint, because in this case the beginning of our coordinate system will be in the center of the marker (Z axis is perpendicular to the marker plane).\n\nTo find the camera location with the known 2D-3D correspondences, the `cv::solvePnP` function can be used:\n\n    void solvePnP(const Mat& objectPoints, const Mat& imagePoints, const Mat& \n    cameraMatrix, const Mat& distCoeffs, Mat& rvec, Mat& tvec, bool \n    useExtrinsicGuess=false);\n\nThe `objectPoints` array is an input array of object points in the object coordinate space. `std::vector<cv::Point3f>` can be passed here. The OpenCV matrix 3 x _N_ or _N_ x 3, where _N_ is the number of points, can also be passed as an input argument. Here we pass the list of marker coordinates in 3D space (a vector of four points).\n\nThe `imagePoints` array is an array of corresponding image points (or projections). This argument can also be `std::vector<cv::Point2f>` or `cv::Mat` of 2 x _N_ or _N_ x 2, where _N_ is the number of points. Here we pass the list of found marker corners.\n\n  * `cameraMatrix`: This is the 3 x 3 camera intrinsic matrix.\n  * `distCoeffs`: This is the input 4 x 1, 1 x 4, 5 x 1, or 1 x 5 vector of distortion coefficients (k1, k2, p1, p2, [k3]). If it is `NULL`, all of the distortion coefficients are set to 0.\n  * `rvec`: This is the output rotation vector that (together with `tvec`) brings points from the model coordinate system to the camera coordinate system.\n  * `tvec`: This is the output translation vector.\n  * `useExtrinsicGuess`: If true, the function will use the provided `rvec` and `tvec` vectors as the initial approximations of the rotation and translation vectors, respectively, and will further optimize them.\n\nThe function calculates the camera transformation in such a way that it minimizes reprojection error, that is, the sum of squared distances between the observed projection's `imagePoints` and the projected `objectPoints`.\n\nThe estimated transformation is defined by rotation (`rvec`) and translation components (`tvec`). This is also known as Euclidean transformation or rigid transformation.\n\nA rigid  transformation is formally defined as a transformation that, when acting on any vector _v_ , produces a transformed vector _T(v)_ of the form:\n\nT(v) = R _v_ \\+ _t_\n\nwhere RT = R-1 (that is, R is an orthogonal transformation), and _t_ is a vector giving the translation of the origin. A proper rigid transformation has, in addition,\n\ndet(R) = 1\n\nThis means that R does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation).\n\nTo obtain a 3 x 3 rotation matrix from the rotation vector, the function `cv::Rodrigues` is used. This function converts a rotation represented by a rotation vector and returns its equivalent rotation matrix.\n\n### Note\n\nBecause `cv::solvePnP` finds the camera position with regards to marker pose in 3D space, we have to invert the found transformation. The resulting transformation will describe a marker transformation in the camera coordinate system, which is much friendlier for the rendering process.\n\nHere is a listing  of the `estimatePosition` function, which finds the position of the detected markers:\n\n    void MarkerDetector::estimatePosition(std::vector<Marker>& detectedMarkers)\n    {\n      for (size_t i=0; i<detectedMarkers.size(); i++)\n      { \n        Marker& m = detectedMarkers[i];\n\n        cv::Mat Rvec;\n        cv::Mat_<float> Tvec;\n        cv::Mat raux,taux;\n        cv::solvePnP(m_markerCorners3d, m.points, camMatrix, distCoeff,raux,taux);\n        raux.convertTo(Rvec,CV_32F);\n        taux.convertTo(Tvec ,CV_32F);\n\n        cv::Mat_<float> rotMat(3,3); \n        cv::Rodrigues(Rvec, rotMat);\n\n        \/\/ Copy to transformation matrix\n        m.transformation = Transformation();\n\n        for (int col=0; col<3; col++)\n        {\n          for (int row=0; row<3; row++)\n          {\n            m.transformation.r().mat[row][col] = rotMat(row,col); \/\/ Copy rotation \n            component\n          }\n          m.transformation.t().data[col] = Tvec(col); \/\/ Copy translation \n          component\n        }\n\n        \/\/ Since solvePnP finds camera location, w.r.t to marker pose, to get \n        marker pose w.r.t to the camera we invert it.\n        m.transformation = m.transformation.getInverted();\n      }\n\n# Rendering the 3D virtual object\n\nSo, by now you already know how to find the markers on the image to calculate their exact position in space, relative to the camera. It's time to draw something. As already mentioned, to render the scene we will use OpenGL functions. 3D visualization is a core part of Augmented Reality. OpenGL provides all the basic features for creating high-quality rendering.\n\n### Note\n\nThere are a large number of commercial and open source 3D-engines (Unity, Unreal Engine, Ogre, and so on). But all these engines use either OpenGL or DirectX to pass commands to the video card. DirectX is a proprietary API and it's supported only on the Windows platform. For this reason, OpenGL is the first and last candidate for building cross-platform rendering systems.\n\nUnderstanding the principles of the rendering system will give you the necessary experience and knowledge to use these engines in the future or to write your own.\n\n## Creating the OpenGL rendering layer\n\nIn order to use  OpenGL functions in your application you should obtain an iOS graphics context surface, which will present the rendered scene to the user. This context is usually bound to **View** , which the user sees. The following screenshot shows the hierarchy of the application interface in XCode's **Interface Builder** :\n\nTo encapsulate the  OpenGL context initialization logic, we introduce the `EAGLView` class:\n\n    @class EAGLContext;\n\n    \/\/ This class wraps the CAEAGLLayer from CoreAnimation into a convenient UIView subclass.\n    \/\/ The view content is basically an EAGL surface you render your OpenGL scene into.\n    \/\/ Note that setting the view non-opaque will only work if the EAGL surface has an alpha channel.\n    @interface EAGLView : UIView\n    {\n    @private\n        \/\/ The OpenGL ES names for the framebuffer and renderbuffer used to render \n        to this view.\n        GLuint defaultFramebuffer, colorRenderbuffer;\n    }\n\n    @property (nonatomic, retain) EAGLContext *context;\n    \/\/ The pixel dimensions of the CAEAGLLayer.\n    @property (readonly) GLint framebufferWidth;\n    @property (readonly) GLint framebufferHeight;\n\n    - (void)setFramebuffer;\n    - (BOOL)presentFramebuffer;\n    - (void)initContext;\n    @end\n\nThis class is connected to our View in our interface definition file, so when the `NIB` file is loaded, the runtime will instantiate a new instance of our `EAGLView`. When created, it will receive events from iOS and initialize the OpenGL rendering context.\n\nThe following is a code listing showing the `initWithCoder` function:\n\n    \/\/The EAGL view is stored in the nib file. When it's unarchived it's sent -initWithCoder:.\n    - (id)initWithCoder:(NSCoder*)coder\n    {\n      self = [super initWithCoder:coder];\n      if (self) {\n        CAEAGLLayer *eaglLayer = (CAEAGLLayer *)self.layer;\n\n        eaglLayer.opaque = TRUE;\n        eaglLayer.drawableProperties = [NSDictionary dictionaryWithObjectsAndKeys:\n                                       [NSNumber numberWithBool:FALSE], \n                                       kEAGLDrawablePropertyRetainedBacking,\n                                       kEAGLColorFormatRGBA8, \n                                       kEAGLDrawablePropertyColorFormat,\n                                       nil];\n\n        [self initContext];\n      }\n\n      return self;\n    }\n\n    - (void)createFramebuffer\n    {\n      if (context && !defaultFramebuffer) {\n        [EAGLContext setCurrentContext:context];\n\n        \/\/ Create default framebuffer object.\n        glGenFramebuffers(1, &defaultFramebuffer);\n        glBindFramebuffer(GL_FRAMEBUFFER, defaultFramebuffer);\n\n        \/\/ Create color render buffer and allocate backing store.\n        glGenRenderbuffers(1, &colorRenderbuffer);\n        glBindRenderbuffer(GL_RENDERBUFFER, colorRenderbuffer);\n        [context renderbufferStorage:GL_RENDERBUFFER fromDrawable:(CAEAGLLayer *)self.layer];\n        glGetRenderbufferParameteriv(GL_RENDERBUFFER, GL_RENDERBUFFER_WIDTH, &framebufferWidth);\n        glGetRenderbufferParameteriv(GL_RENDERBUFFER, GL_RENDERBUFFER_HEIGHT, &framebufferHeight);\n\n        glFramebufferRenderbuffer(GL_FRAMEBUFFER, GL_COLOR_ATTACHMENT0, \n        GL_RENDERBUFFER, colorRenderbuffer);\n\n        if (glCheckFramebufferStatus(GL_FRAMEBUFFER) != GL_FRAMEBUFFER_COMPLETE)\n        NSLog(@\"Failed to make complete framebuffer object %x\", \n        glCheckFramebufferStatus(GL_FRAMEBUFFER));\n\n        \/\/glClearColor(0, 0, 0, 0);\n        NSLog(@\"Framebuffer created\");\n      }\n    }\n\n## Rendering an AR scene\n\nAs you can see, the `EAGLView` class does not contain methods for the visualization of 3D objects and video. This is done on purpose. The task of `EAGLView` is to provide rendering context. The separation of responsibilities allows us to change the logic of the visualization later.\n\nFor visualization of Augmented Reality, we will create a separate class called as `VisualizationController`:\n\n    @interface SimpleVisualizationController : NSObject<VisualizationController>\n    {\n      EAGLView * m_glview;\n      GLuint m_backgroundTextureId;\n      std::vector<Transformation> m_transformations;\n      CameraCalibration m_calibration;\n      CGSize m_frameSize;\n    }\n\n    -(id) initWithGLView:(EAGLView*)view calibration:(CameraCalibration) calibration frameSize:(CGSize) size;\n\n    -(void) drawFrame;\n    -(void) updateBackground:(BGRAVideoFrame) frame;\n    -(void) setTransformationList:(const std::vector<Transformation>&) transformations;\n\nThe `drawFrame` function performs rendering of the AR onto the given `EAGLView` target view. It performs the following steps:\n\n  1. Clears the scene.\n  2. Sets up orthographic projection for drawing the background.\n  3. Draws the latest received image from the camera on a viewport.\n  4. Sets up perspective projection with regards to a camera's intrinsic parameters.\n  5. For each detected marker, it moves the coordinate system to marker position in 3D. (It puts 4 x 4-transformation matrix to the OpenGl model-view matrix.)\n  6. Renders an arbitrary 3D object.\n  7. Shows the frame buffer.\n\nThe `drawFrame` function is called when the frame is ready to be drawn. It happens when a new camera frame has been uploaded to video memory and the marker detection stage has been completed.\n\nThe following code shows the `drawFrame` function:\n\n    - (void)drawFrame\n    {\n      \/\/ Set the active framebuffer\n      [m_glview setFramebuffer];\n\n      \/\/ Draw a video on the background\n      [self drawBackground];\n\n      \/\/ Draw 3D objects on the position of the detected markers\n      [self drawAR];\n\n      \/\/ Present framebuffer\n      bool ok = [m_glview presentFramebuffer];\n\n      int glErCode = glGetError();\n      if (!ok || glErCode != GL_NO_ERROR)\n      {\n        std::cerr << \"GL error detected. Error code:\" << glErCode << std::endl;\n      }\n    }\n\nDrawing a background is easy enough; we set the orthographic projection and draw a fullscreen texture with image from the current frame. Here is a code listing that uses the GLES 1 API to do this:\n\n    - (void) drawBackground\n    {\n      GLfloat w = m_glview.bounds.size.width;\n      GLfloat h = m_glview.bounds.size.height;\n      const GLfloat squareVertices[] =\n      {\n        0, 0,\n        w, 0,\n        0, h,\n        w, h\n      };\n\n      static const GLfloat textureVertices[] =\n      {\n        1, 0,\n        1, 1,\n        0, 0,\n        0, 1\n      };\n\n      static const GLfloat proj[] =\n      {\n        0, -2.f\/w, 0, 0,\n        -2.f\/h, 0, 0, 0,\n        0, 0, 1, 0,\n        1, 1, 0, 1\n      };\n\n      glMatrixMode(GL_PROJECTION);\n      glLoadMatrixf(proj);\n\n      glMatrixMode(GL_MODELVIEW);\n      glLoadIdentity();\n\n      glDisable(GL_COLOR_MATERIAL);\n\n      glEnable(GL_TEXTURE_2D);\n      glBindTexture(GL_TEXTURE_2D, m_backgroundTextureId);\n\n      \/\/ Update attribute values.\n      glVertexPointer(2, GL_FLOAT, 0, squareVertices);\n      glEnableClientState(GL_VERTEX_ARRAY);\n      glTexCoordPointer(2, GL_FLOAT, 0, textureVertices);\n      glEnableClientState(GL_TEXTURE_COORD_ARRAY);\n\n      glColor4f(1,1,1,1);\n      glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);\n\n      glDisableClientState(GL_VERTEX_ARRAY);\n      glDisableClientState(GL_TEXTURE_COORD_ARRAY);\n      glDisable(GL_TEXTURE_2D);\n    }\n\nRendering of artificial objects in a scene is somewhat tricky. First of all we have to adjust the OpenGL projection matrix with regards to the camera intrinsic (calibration) matrix. Without this step we will have the wrong perspective projection. Wrong perspective makes artificial objects look unnatural, as if they are \"flying in the air\" and not a part of the real world. Correct perspective is a must-have for any Augmented Reality application.\n\nHere is a code snippet that creates an OpenGL projection matrix from camera intrinsics:\n\n    - (void)buildProjectionMatrix:(Matrix33)cameraMatrix: (int)screen_width: (int)screen_height: (Matrix44&) projectionMatrix\n    {    \n      float near = 0.01;  \/\/ Near clipping distance\n      float far = 100;  \/\/ Far clipping distance\n\n      \/\/ Camera parameters\n      float f_x = cameraMatrix.data[0]; \/\/ Focal length in x axis\n      float f_y = cameraMatrix.data[4]; \/\/ Focal length in y axis (usually the \n      same?)\n      float c_x = cameraMatrix.data[2]; \/\/ Camera primary point x\n      float c_y = cameraMatrix.data[5]; \/\/ Camera primary point y\n\n      projectionMatrix.data[0] =  - 2.0 * f_x \/ screen_width;\n      projectionMatrix.data[1] = 0.0;\n      projectionMatrix.data[2] = 0.0;\n      projectionMatrix.data[3] = 0.0;\n\n      projectionMatrix.data[4] = 0.0;\n      projectionMatrix.data[5] = 2.0 * f_y \/ screen_height;\n      projectionMatrix.data[6] = 0.0;\n      projectionMatrix.data[7] = 0.0;\n\n      projectionMatrix.data[8] = 2.0 * c_x \/ screen_width - 1.0;\n      projectionMatrix.data[9] = 2.0 * c_y \/ screen_height - 1.0;   \n      projectionMatrix.data[10] = -( far+near ) \/ ( far - near );\n      projectionMatrix.data[11] = -1.0;\n\n      projectionMatrix.data[12] = 0.0;\n      projectionMatrix.data[13] = 0.0;\n      projectionMatrix.data[14] = -2.0 * far * near \/ ( far - near );\n      projectionMatrix.data[15] = 0.0;\n    }\n\nAfter we load this matrix to the OpenGL pipeline, it's time to draw some objects. Each transformation can be presented as a 4 x 4 matrix and loaded to the OpenGL model view matrix. This will move the coordinate system to the marker position in the world coordinate system.\n\nFor example, let's draw a coordinate axis on the top of each marker that will show its orientation in space, and a rectangle with gradient fill that overlays the whole marker. This visualization will give us visual feedback that our code is working as expected.\n\nThe following is a code snippet showing the `drawAR` function:\n\n    - (void) drawAR\n    {\n      Matrix44 projectionMatrix;\n      [self buildProjectionMatrix:m_calibration.getIntrinsic():m_frameSize.width\n      :m_frameSize.height :projectionMatrix];\n\n      glMatrixMode(GL_PROJECTION);\n      glLoadMatrixf(projectionMatrix.data);\n\n      glMatrixMode(GL_MODELVIEW);\n      glLoadIdentity();\n\n      glEnableClientState(GL_VERTEX_ARRAY);\n      glEnableClientState(GL_NORMAL_ARRAY);\n\n      glPushMatrix();\n      glLineWidth(3.0f);\n\n      float lineX[] = {0,0,0,1,0,0};\n      float lineY[] = {0,0,0,0,1,0};\n      float lineZ[] = {0,0,0,0,0,1};\n\n      const GLfloat squareVertices[] = {\n        -0.5f, -0.5f,\n        0.5f,  -0.5f,\n        -0.5f,  0.5f,\n        0.5f,   0.5f,\n      };\n      const GLubyte squareColors[] = {\n        255, 255,   0, 255,\n        0,   255, 255, 255,\n        0,     0,   0,   0,\n        255,   0, 255, 255,\n      };\n\n      for (size_t transformationIndex=0; \n      transformationIndex<m_transformations.size(); transformationIndex++)\n      {\n        const Transformation& transformation = \n        m_transformations[transformationIndex];\n\n        Matrix44 glMatrix = transformation.getInverted().getMat44();\n\n        glLoadMatrixf(reinterpret_cast<const GLfloat*>(&glMatrix.data[0]));\n\n        \/\/ draw data\n        glVertexPointer(2, GL_FLOAT, 0, squareVertices);\n        glEnableClientState(GL_VERTEX_ARRAY);\n        glColorPointer(4, GL_UNSIGNED_BYTE, 0, squareColors);\n        glEnableClientState(GL_COLOR_ARRAY);\n\n        glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);\n        glDisableClientState(GL_COLOR_ARRAY);\n\n        float scale = 0.5;\n        glScalef(scale, scale, scale);\n\n        glColor4f(1.0f, 0.0f, 0.0f, 1.0f);\n        glVertexPointer(3, GL_FLOAT, 0, lineX);\n        glDrawArrays(GL_LINES, 0, 2);\n\n        glColor4f(0.0f, 1.0f, 0.0f, 1.0f);\n        glVertexPointer(3, GL_FLOAT, 0, lineY);\n        glDrawArrays(GL_LINES, 0, 2);\n\n        glColor4f(0.0f, 0.0f, 1.0f, 1.0f);\n        glVertexPointer(3, GL_FLOAT, 0, lineZ);\n        glDrawArrays(GL_LINES, 0, 2);\n      }\n      glPopMatrix();\n      glDisableClientState(GL_VERTEX_ARRAY);\n    }\n\nIf you run the application, you will get the following figure:\n\nDespite the fact that we do not use a special 3D rendering engine for visualization of our scene, we have all the necessary data to do this by ourselves. Let's summarize the data we obtain:\n\n  * A frame from the camera device in BGRA format\n  * A correct projection matrix that gives us the right perspective projection for AR scene rendering\n  * A list of found marker poses\n\nYou can easily put this data to any 3D engine and create your own finished marker-based AR application\n\nAs you can see, the quads with gradient fill and pivots are placed exactly on the markers. This is the key feature of Augmented Reality--seamless fusion of real pictures and artificial objects.\n\n# Summary\n\nIn this chapter we learned how to create a mobile Augmented Reality application for iPhone\/iPad devices. You gained knowledge on how to use the OpenCV library within the XCode projects to create stunning state-of-the-art applications. Usage of OpenCV enables your application to perform complex image processing computations on mobile devices with real-time performance.\n\nFrom this chapter you also learned how to perform the initial image processing (translation in shades of gray and binarization), how to find closed contours in the image and approximate them with polygons, how to find markers in the image and decode them, how to compute the marker position in space, and the visualization of 3D objects in Augmented Reality.\n\n# References\n\n  *  _ArUco: a minimal library_ _for Augmented Reality applications based on OpenCV_ (<http:\/\/www.uco.es\/investiga\/grupos\/ava\/node\/26>)\n  *  _OpenCV Camera Calibration and 3D Reconstruc_ _tion_ (<http:\/\/opencv.itseez.com\/modules\/calib3d\/doc\/camera_calibration_and_3d_reconstruction.html>)\n  *  _OpenCV: Estimating Projective Relations in Images_ (<http:\/\/www.packtpub.com\/article\/opencv-estimatingprojective-relations-images>)\n  *  _Multiple View Geometry in Computer Vision_ ( _second edition_ ), _R.I. Hartley and A. Zisserman_ , _Cambridge University Press_ , _ISBN 0-521-54051-8_\n\n# Chapter 3. Marker-less Augmented Reality\n\nIn this chapter readers will learn how to create a standard real-time project using OpenCV (for desktop), and how to perform a new method of marker-less augmented reality, using the actual environment as the input instead of printed square markers. This chapter will cover some of the theory of marker-less AR and show how to apply it in useful projects.\n\nThe following is a list of topics that will be covered in this chapter:\n\n  * Marker-based versus marker-less AR\n  * Using feature descriptors to find an arbitrary image on video\n  * Pattern pose estimation\n  * Application infrastructure\n  * Enabling support for OpenGL visualization in OpenCV\n  * Rendering the augmented reality\n  * Demonstration\n\nBefore we start, let me give you a brief list of the knowledge required for this chapter and the software you will need:\n\n  * Basic knowledge of CMake. CMake is a cross-platform, open-source build system designed to build, test, and package software. Like the OpenCV library, the demonstration project for this chapter also uses the CMake build system. CMake can be downloaded from <http:\/\/www.cmake.org\/>.\n  * A basic knowledge of C++ programming language is also necessary. However, all complex parts of the application source code will be explained in detail.\n\n# Marker-based versus marker-less AR\n\nFrom the previous  chapter you've learned how to use special images called markers to augment a real scene. The strong aspects of the markers are as follows:\n\n  * Cheap detection algorithm\n  * Robust against lighting changes\n\nMarkers also have several weaknesses. They are as follows:\n\n  * Doesn't work if partially overlapped\n  * Marker image has to be black and white\n  * Has square form in most cases (because it's easy to detect)\n  * Non-esthetic visual look of the marker\n  * Has nothing in common with real-world objects\n\nSo, markers are a good point to start working with augmented reality; but if you want more, it's time to move on from marker-based to marker-less AR. Marker-less AR is a technique that is based on recognition of objects that exist in the real world. A few examples of a target for marker-less AR are: magazine covers, company logos, toys, and so on. In general, any object that has enough descriptive and discriminative information regarding the rest of the scene can be a target for marker-less AR.\n\nThe strong sides of the marker-less AR approach are:\n\n  * Can be used to detect real-world objects\n  * Works even if the target object is partially overlapped\n  * Can have arbitrary form and texture (except solid or smooth gradient textures)\n\nMarker-less AR systems can use real images and objects to position the camera in 3D space and present eye-catching effects on top of the real picture. The heart of the marker-less AR are image recognition and object detection algorithms. Unlike markers, whose shape and internal structure is fixed and known, real objects cannot be defined in such a way. Also, objects can have a complex shape and require modified pose estimation algorithms to find their correct 3D transformations.\n\n### Note\n\nTo give you an idea of marker-less AR, we will use a planar image as a target. Objects with complex shapes will not be considered here in detail. We will discuss the use of complex shapes for AR later in this chapter.\n\nMarker-less AR performs heavy CPU calculations, so a mobile device often is not capable to secure smooth  FPS. In this chapter, we will be targeting desktop platforms such as PC or Mac. For this purpose, we need a cross-platform build system. In this chapter we use the CMake build system.\n\n# Using feature descriptors to find an arbitrary image on video\n\nImage recognition is a computer vision technique that searches the input image for a particular bitmap pattern. Our image recognition algorithm should be able to detect the pattern even if it is scaled, rotated, or has different brightness than of the original image.\n\nHow do we  compare the pattern image against other images? As the pattern can be affected by perspective transformation, it's obvious that we can't directly compare pixels of the pattern and test image. The feature points and feature descriptors are helpful in this case. There is no universal or exact definition of what the feature is. The exact definition often depends on the problem or the type of application. Usually a feature is defined as an \"interesting\" part of an image, and features are used as a starting point for many computer vision algorithms. In this chapter we will use a **feature point** term, which is a part of the image defined by a center point, radius, and orientation. Each feature-detection algorithm tries to detect the same feature points regardless of the perspective transformation applied.\n\n## Feature extraction\n\nFeature detection is the method of finding areas of interest from the input image. There are a lot of feature-detection algorithms, which search for edges, corners, or blobs. In our case we are interested in corner detection. The corner detection is based on an analysis of the edges in the image. A corner-based edge detection algorithm searches for rapid changes in the image gradient. Usually it's done by looking for extremums of the first derivative of the image gradients in the X and Y directions.\n\nFeature-point orientation is usually computed as a direction of dominant image gradient in a particular area. When the image is rotated or scaled, the orientation of dominant gradient is recomputed by the feature-detection algorithm. This means that regardless of image rotation, the orientation of feature points will not change. Such features are called **rotation invariant**.\n\nAlso, I have to mention a few points about the size feature point. Some of the feature-detection algorithms use fixed-size features, while others calculate the optimal size for each keypoint separately. Knowing the feature size allows us to find the same feature points on scaled images. This makes features scale invariant.\n\nOpenCV has several feature-detection algorithms. All of them are derived from the base class `cv::FeatureDetector`. Creation of the feature-detection algorithm can be done in two ways:\n\n  * Via an explicit call of the concrete feature detector class constructor:\n\n        cv::Ptr<cv::FeatureDetector> detector = cv::Ptr<cv::FeatureDetector>(new cv::SurfFeatureDetector());\n\n  * Or by creating a feature detector by algorithm name:\n\n        cv::Ptr<cv::FeatureDetector> detector = cv::FeatureDetector::create(\"SURF\");\n\nBoth methods have their advantages, so choose the one you most prefer. The explicit class creation allows you to pass additional arguments to the feature detector constructor, while the creation by algorithm name makes it easier to switch the algorithm during runtime.\n\nTo detect feature points, you should call the `detect` method:\n\n    std::vector<cv::KeyPoint> keypoints;\n    detector->detect(image, keypoints);\n\nThe detected feature points are placed in the `keypoints` container. Each keypoint contains its center, radius, angle, and score, and has some correlation with the \"quality\" or \"strength\" of the feature point. Each feature-detection algorithm has its own score computation algorithm, so it's valid to compare scores of the keypoints detected by a particular detection algorithm.\n\n### Note\n\nCorner-based feature detectors use a grayscale image to find feature points. Descriptor-extraction algorithms also work with grayscale images. Of course, both of them can do color conversion implicitly. But in this case the color conversion will be done twice. We can improve performance by doing an explicit color conversion of the input image to grayscale and use that for feature detection and descriptor extraction.\n\nThe best results in pattern detection are achieved if the detector computes keypoint orientation and size. This makes keypoints invariant to rotation and scale. The most famous and robust keypoint detection algorithms are well known, they are used in SIFT and SURF feature detection\/description extraction. Unfortunately, they are patented; so they are not free for commercial use. However, their implementation is present in OpenCV, so you can evaluate them freely. But there are good and free replacements available. You can use the ORB or FREAK algorithm instead. The ORB detection is a modified FAST feature detector. The original FAST detector is amazingly fast but does not calculate the orientation or the size of the keypoint. Fortunately, the ORB algorithm does estimate keypoint orientation, but the feature size is still fixed. From the following paragraphs you will learn nice and cheap tricks of dealing with this. But first, let me explain why the feature point matters so much in image recognition.\n\nIf we deal with images, which usually have a color depth of 24 bits per pixel, for a resolution of 640 x 480, we have 912 KB of data. How do we find our pattern image in the real world? Pixel-to-pixel matching takes too long and we will have to deal with rotation and scaling too. It's definitely not an option. Using feature points can solve this problem. By detecting keypoints, we can be sure that returned features describe parts of the image that contains lot of information (that's because corner-based detectors return edges, corners, and other sharp figures). So to find correspondences between two frames, we only have to match keypoints.\n\nFrom the patch defined by the keypoint, we extract a vector called descriptor. It's a form of representation of the feature point. There are many methods of extraction of the descriptor from the feature point. All of them have their strengths and weaknesses. For example, SIFT and SURF descriptor-extraction algorithms are CPU-intensive but provide robust descriptors with good distinctiveness. In our sample project we use the ORB descriptor-extraction algorithm because we choose it as a feature detector too.\n\n### Note\n\nIt's always a good idea to use both feature detector and descriptor extractor from the same algorithm, as they will then fit each other perfectly.\n\nFeature descriptor is represented as a vector of fixed size (16 or more elements). Let's say our image has a resolution of 640 x 480 pixels and it has 1,500 feature points. Then, it will require `1500 * 16 * sizeof(float) = 96 KB` (for SURF). It's ten times smaller than the original image data. Also, it's much easier to operate with descriptors rather than with raster bitmaps. For two feature descriptors we can introduce a similarity score--a metric that defines the level of similarity between two vectors. Usually its L2 norm or hamming distance (based upon the kind of feature descriptor used).\n\nThe feature descriptor-extraction algorithms are derived from the `cv::DescriptorExtractor` base class. Likewise, as feature-detection algorithms they can be created by either specifying their name or with explicit constructor calls.\n\n## Definition of a pattern object\n\nTo describe a pattern object we introduce a class called `Pattern`, which holds a train image, list of features and extracted descriptors, and 2D and 3D correspondences for initial pattern position:\n\n    \/**\n     * Store the image data and computed descriptors of target pattern\n     *\/\n    struct Pattern\n    {\n      cv::Size                  size;\n      cv::Mat                   data;\n      std::vector<cv::KeyPoint> keypoints;\n      cv::Mat                   descriptors;\n\n      std::vector<cv::Point2f>  points2d;\n      std::vector<cv::Point3f>  points3d;\n    };\n\n## Matching of feature points\n\nThe process of finding frame-to-frame correspondences can be formulated as the search of the nearest neighbor from one set of descriptors for every element of another set. It's called the \"matching\" procedure. There are two main algorithms for descriptor matching in OpenCV:\n\n  * Brute-force matcher (`cv::BFMatcher`)\n  * Flann-based matcher (`cv::FlannBasedMatcher`)\n\nThe brute-force matcher looks for each descriptor in the first set and the closest descriptor in the second set by trying each one (exhaustive search). `cv::FlannBasedMatcher` uses the fast approximate nearest neighbor search algorithm to find correspondences (it uses fast third-party library for approximate nearest neighbors library for this).\n\nThe result of descriptor  matching is a list of correspondences between two sets of descriptors. The first set of descriptors is usually called the train set because it corresponds to our pattern image. The second set is called the query set as it belongs to the image where we will be looking for the pattern. The more correct matches found (more patterns to image correspondences exist) the more chances are that the pattern is present on the image.\n\nTo increase the matching speed, you can train a matcher before by calling the `match` function. The training stage can be used to optimize the performance of `cv::FlannBasedMatcher`. For this, the `train` class will build index trees for train descriptors. And this will increase the matching speed for large data sets (for example, if you want to find a match from hundreds of images). For `cv::BFMatcher` the `train` class does nothing as there is nothing to preprocess; it simply stores the train descriptors in the internal fields.\n\n### PatternDetector.cpp\n\nThe following code block trains the descriptor matcher using the pattern image:\n\n    void PatternDetector::train(const Pattern& pattern)\n    {\n        \/\/ Store the pattern object\n        m_pattern = pattern;\n\n        \/\/ API of cv::DescriptorMatcher is somewhat tricky\n        \/\/ First we clear old train data:\n        m_matcher->clear();\n\n        \/\/ That we add vector of descriptors \n        \/\/ (each descriptors matrix describe one image). \n        \/\/ This allows us to perform search across multiple images:\n        std::vector<cv::Mat> descriptors(1);\n        descriptors[0] = pattern.descriptors.clone(); \n        m_matcher->add(descriptors);\n\n        \/\/ After adding train data perform actual train:\n        m_matcher->train();\n    }\n\nTo match query descriptors, we can use one of the following methods of `cv::DescriptorMatcher`:\n\n  * To find the simple list of best matches:\n\n        void match(const Mat& queryDescriptors, vector<DMatch>& matches,\n               const vector<Mat>& masks=vector<Mat>() );\n\n  * To find _K_ nearest matches for each descriptor:\n\n        void knnMatch(const Mat& queryDescriptors, vector<vector<DMatch> >& matches, int k, const vector<Mat>& masks=vector<Mat>(),bool compactResult=false );\n\n  * To find correspondences whose distances are not farther than the specified distance:\n\n        void radiusMatch(const Mat& queryDescriptors, vector<vector<DMatch> >& matches, maxDistance, const vector<Mat>& masks=vector<Mat>(), bool compactResult=false );\n\n## Outlier removal\n\nMismatches during  the matching stage can happen. It's normal. There are two kinds of errors in matching:\n\n  *  **False-positive matches** : When the feature-point correspondence is wrong\n  *  **False-negative matches** : The absence of a match when the feature points are visible on both images\n\nFalse-negative matches are obviously bad. But we can't deal with them because the matching algorithm has rejected them. Our goal is therefore to minimize the number of false-positive matches. To reject wrong correspondences, we can use a cross-match technique. The idea is to match train descriptors with the query set and vice versa. Only common matches for these two matches are returned. Such techniques usually produce best results with minimal number of outliers when there are enough matches.\n\n### Cross-match filter\n\nCross-match  is available in the `cv::BFMatcher` class. To enable a cross-check test, create `cv::BFMatcher` with the second argument set to `true`:\n\n    cv::Ptr<cv::DescriptorMatcher> matcher(new cv::BFMatcher(cv::NORM_HAMMING, true));\n\nThe result of matching using cross-checks can be seen in the following screenshot:\n\n### Ratio test\n\nThe second  well-known outlier-removal technique is the ratio test. We perform KNN-matching first with K=2. Two nearest descriptors are returned for each match. The match is returned only if the distance ratio between the first and second matches is big enough (the ratio threshold is usually near two).\n\n#### PatternDetector.cpp\n\nThe following  code performs robust descriptor matching  using a ratio test:\n\n    void PatternDetector::getMatches(const cv::Mat& queryDescriptors, std::vector<cv::DMatch>& matches)\n    {\n        matches.clear();\n\n        if (enableRatioTest)\n        {\n            \/\/ To avoid NaNs when best match has \n            \/\/ zero distance we will use inverse ratio. \n            const float minRatio = 1.f \/ 1.5f;\n\n            \/\/ KNN match will return 2 nearest \n            \/\/ matches for each query descriptor\n            m_matcher->knnMatch(queryDescriptors, m_knnMatches, 2);\n\n            for (size_t i=0; i<m_knnMatches.size(); i++)\n            {\n                const cv::DMatch& bestMatch   = m_knnMatches[i][0];\n                const cv::DMatch& betterMatch = m_knnMatches[i][1];\n\n                float distanceRatio = bestMatch.distance \/ betterMatch.distance;\n\n                \/\/ Pass only matches where distance ratio between \n                \/\/ nearest matches is greater than 1.5 \n                \/\/ (distinct criteria)\n                if (distanceRatio < minRatio)\n                {\n                    matches.push_back(bestMatch);\n                }\n            }\n        }\n        else\n        {\n            \/\/ Perform regular match\n            m_matcher->match(queryDescriptors, matches);\n        }\n    }\n\nThe ratio test can remove almost all outliers. But in some cases, false-positive matches can pass through this test. In the next section, we will show you how to remove the rest of outliers and leave only correct matches.\n\n### Homography estimation\n\nTo improve our  matching even more, we can perform outlier filtration using the random sample consensus (RANSAC) method. As we're working with an image (a planar object) and we expect it to be rigid, it's ok to find the homography transformation between feature points on the pattern image and feature points on the query image. Homography transformations will bring points from a pattern to the query image coordinate system. To find this transformation, we use the `cv::findHomography` function. It uses RANSAC to find the best homography matrix by probing subsets of input points. As a side effect, this function marks each correspondence as either inlier or outlier, depending on the reprojection error for the  calculated homography matrix.\n\n#### PatternDetector.cpp\n\nThe following code  uses a homography matrix estimation using a RANSAC algorithm to filter out geometrically incorrect matches:\n\n    bool PatternDetector::refineMatchesWithHomography\n        (\n            const std::vector<cv::KeyPoint>& queryKeypoints,\n            const std::vector<cv::KeyPoint>& trainKeypoints, \n            float reprojectionThreshold,\n            std::vector<cv::DMatch>& matches,\n            cv::Mat& homography\n        )\n    {\n        const int minNumberMatchesAllowed = 8;\n\n        if (matches.size() < minNumberMatchesAllowed)\n            return false;\n\n        \/\/ Prepare data for cv::findHomography\n        std::vector<cv::Point2f> srcPoints(matches.size());\n        std::vector<cv::Point2f> dstPoints(matches.size());\n\n        for (size_t i = 0; i < matches.size(); i++)\n        {\n            srcPoints[i] = trainKeypoints[matches[i].trainIdx].pt;\n            dstPoints[i] = queryKeypoints[matches[i].queryIdx].pt;\n        }\n\n        \/\/ Find homography matrix and get inliers mask\n        std::vector<unsigned char> inliersMask(srcPoints.size());\n        homography = cv::findHomography(srcPoints, \n                                        dstPoints, \n                                        CV_FM_RANSAC, \n                                        reprojectionThreshold, \n                                        inliersMask);\n\n        std::vector<cv::DMatch> inliers;\n        for (size_t i=0; i<inliersMask.size(); i++)\n        {\n            if (inliersMask[i])\n                inliers.push_back(matches[i]);\n        }\n\n        matches.swap(inliers);\n        return matches.size() > minNumberMatchesAllowed;\n    }\n\nHere is a visualization of matches that were refined using this technique:\n\nThe homography search step is important because the obtained transformation is a key to find the pattern location in the query image.\n\n### Homography refinement\n\nWhen we look for  homography transformations, we already have all the necessary data to find their locations in 3D. However, we can improve its position even more by finding more accurate pattern corners. For this we warp the input image using estimated homography to obtain a pattern that has been found. The result should be very close to the source train image. Homography refinement can help to find more accurate homography transformations.\n\nThen we obtain another homography and another set of inlier features. The resultant precise homography will be the matrix product of the first (H1) and second (H2) homography.\n\n#### PatternDetector.cpp\n\nThe following code  block contains the final version of the pattern detection routine:\n\n    bool PatternDetector::findPattern(const cv::Mat& image, PatternTrackingInfo& info)\n    {\n        \/\/ Convert input image to gray\n        getGray(image, m_grayImg);\n\n        \/\/ Extract feature points from input gray image\n        extractFeatures(m_grayImg, m_queryKeypoints, m_queryDescriptors);\n\n        \/\/ Get matches with current pattern\n        getMatches(m_queryDescriptors, m_matches);\n\n        \/\/ Find homography transformation and detect good matches\n        bool homographyFound = refineMatchesWithHomography(\n            m_queryKeypoints, \n            m_pattern.keypoints, \n            homographyReprojectionThreshold, \n            m_matches, \n            m_roughHomography);\n\n        if (homographyFound)\n        {\n            \/\/ If homography refinement enabled \n            \/\/ improve found transformation\n            if (enableHomographyRefinement)\n            {\n                \/\/ Warp image using found homography\n                cv::warpPerspective(m_grayImg, m_warpedImg, m_roughHomography, m_pattern.size, cv::WARP_INVERSE_MAP | cv::INTER_CUBIC);\n\n                \/\/ Get refined matches:\n                std::vector<cv::KeyPoint> warpedKeypoints;\n                std::vector<cv::DMatch> refinedMatches;\n\n                \/\/ Detect features on warped image\n                extractFeatures(m_warpedImg, warpedKeypoints, m_queryDescriptors);\n\n                \/\/ Match with pattern\n                getMatches(m_queryDescriptors, refinedMatches);\n\n                \/\/ Estimate new refinement homography\n                homographyFound = refineMatchesWithHomography(\n                    warpedKeypoints, \n                    m_pattern.keypoints, \n                    homographyReprojectionThreshold, \n                    refinedMatches, \n                    m_refinedHomography);\n\n                \/\/ Get a result homography as result of matrix product \n                \/\/ of refined and rough homographies:\n                info.homography = m_roughHomography * m_refinedHomography;\n\n                \/\/ Transform contour with precise homography\n                cv::perspectiveTransform(m_pattern.points2d, info.points2d, info.homography);\n            }\n            else\n            {\n                info.homography = m_roughHomography;\n\n                \/\/ Transform contour with rough homography\n                cv::perspectiveTransform(m_pattern.points2d, info.points2d, m_roughHomography);\n            }\n        }\n\n        return homographyFound;\n    }\n\nIf, after all the outlier removal stages, the number of matches is still reasonably large (at least 25 percent of features from the pattern image have correspondences with the input one), you can be sure the pattern image is located correctly. If so, we proceed to the next stage--estimation of the 3D position of the pattern pose with regards to the camera.\n\n## Putting it all together\n\nTo hold instances of the feature detector, descriptor extractor, and matcher algorithms, we create a class `PatternMatcher`, which will encapsulate all this data. It takes ownership on the feature detection and descriptor-extraction algorithm, feature matching logic, and settings that control the detection process.\n\nThe class provides a method to compute all the necessary data to build a pattern structure from a given image:\n\n    void PatternDetector::computePatternFromImage(const cv::Mat& image, Pattern& pattern);\n\nThis method finds feature points on the input image and extracts descriptors using the specified detector and extractor algorithms, and fills the pattern structure with this data for later use.\n\nWhen `Pattern` is computed, we can train a detector with it by calling the `train` method:\n\n    void PatternDetector::train(const Pattern& pattern)\n\nThis function sets the argument as the current target pattern that we are going to find. Also, it trains a descriptor matcher with a pattern's descriptor set. After calling this method we are ready to find our train image. The pattern detection is done in the last public function `findPattern`. This method encapsulates the whole routine as described previously, including feature detection, descriptors extraction, and matching with outlier filtration.\n\nLet's conclude again with a brief list of the steps we performed:\n\n  1. Converted input image to grayscale.\n  2. Detected features on the query image using our feature-detection algorithm.\n  3. Extracted descriptors from the input image for the detected feature points.\n  4. Matched descriptors against pattern descriptors.\n  5. Used cross-checks or ratio tests to remove outliers.\n  6. Found the homography transformation using inlier matches.\n  7. Refined the homography by warping the query image with homography from the previous step.\n  8. Found the precise homography as a result of the multiplication of rough and refined homography.\n  9. Transformed the pattern corners to an image coordinate system to get pattern locations on the input image.\n\n# Pattern pose estimation\n\nThe pose estimation is done in a similar manner to marker pose estimation from the previous chapter. As usual we need 2D-3D correspondences to estimate the camera-extrinsic parameters. We assign four 3D points to coordinate with the corners of the unit rectangle that lies in the XY plane (the Z axis is up), and 2D points correspond to the corners of the image bitmap.\n\n## PatternDetector.cpp\n\nThe `buildPatternFromImage` class creates a `Pattern` object from the input image as follows:\n\n    void PatternDetector::buildPatternFromImage(const cv::Mat& image, Pattern& pattern) const\n    {\n        int numImages = 4;\n        float step = sqrtf(2.0f);\n\n        \/\/ Store original image in pattern structure\n        pattern.size = cv::Size(image.cols, image.rows);\n        pattern.frame = image.clone();\n        getGray(image, pattern.grayImg);\n\n        \/\/ Build 2d and 3d contours (3d contour lie in XY plane since \/\/ it's planar)\n        pattern.points2d.resize(4);\n        pattern.points3d.resize(4);\n\n        \/\/ Image dimensions\n        const float w = image.cols;\n        const float h = image.rows;\n\n        \/\/ Normalized dimensions:\n        const float maxSize = std::max(w,h);\n        const float unitW = w \/ maxSize;\n        const float unitH = h \/ maxSize;\n\n        pattern.points2d[0] = cv::Point2f(0,0);\n        pattern.points2d[1] = cv::Point2f(w,0);\n        pattern.points2d[2] = cv::Point2f(w,h);\n        pattern.points2d[3] = cv::Point2f(0,h);\n\n        pattern.points3d[0] = cv::Point3f(-unitW, -unitH, 0);\n        pattern.points3d[1] = cv::Point3f( unitW, -unitH, 0);\n        pattern.points3d[2] = cv::Point3f( unitW,  unitH, 0);\n        pattern.points3d[3] = cv::Point3f(-unitW,  unitH, 0);\n\n        extractFeatures(pattern.grayImg, pattern.keypoints, pattern.descriptors);\n    }\n\nThis configuration of corners is useful as this pattern coordinate system will be placed directly in the center of the pattern location lying in the XY plane, with the Z axis looking in the direction of the camera.\n\n## Obtaining the camera-intrinsic matrix\n\nThe camera-intrinsic parameters can be calculated using a sample program from the OpenCV distribution package called `camera_cailbration.exe`. This program will find the internal lens parameters such as focal length, principal point, and distortion coefficients using a series of pattern images. Let's say we have a set of eight calibration pattern images from various points of view, as follows:\n\nThen the command-line syntax to perform calibration will be as follows:\n\n    **imagelist_creator imagelist.yaml *.png**\n    **calibration -w 9 -h 6 -o camera_intrinsic.yaml imagelist.yaml**\n\nThe first command will create an image list of YAML format that the calibration tool expects as input from all PNG files in the current directory. You can use the exact file names, such as `img1.png`, `img2.png`, and `img3.png`. The generated file `imagelist.yaml` is then passed to the calibration application. Also, the calibration tool can take images from a regular web camera.\n\nWe specify the dimensions of the calibration pattern and input and output files where the calibration data will be written.\n\nAfter calibration is done, you'll get the following result in a YAML file:\n\n    **%YAML:1.0**\n    **calibration_time: \"06\/12\/12 11:17:56\"**\n    **image_width: 640**\n    **image_height: 480**\n    **board_width: 9**\n    **board_height: 6**\n    **square_size: 1.**\n    **flags: 0**\n    **camera_matrix: !!opencv-matrix**\n    **rows: 3**\n    **cols: 3**\n    **dt: d**\n    **data: [ 5.2658037684199849e+002, 0., 3.1841744018680112e+002, 0.,**\n    **5.2465577209994706e+002, 2.0296659047014398e+002, 0., 0., 1. ]**\n    **distortion_coefficients: !!opencv-matrix**\n    **rows: 5**\n    **cols: 1**\n    **dt: d**\n    **data: [ 7.3253671786835686e-002, -8.6143199924308911e-002,**\n    **-2.0800255026966759e-002, -6.8004894417795971e-004,**\n    **-1.7750733073535208e-001 ]**\n    **avg_reprojection_error: 3.6539552933501085e-001**\n\nWe are mainly interested in `camera_matrix`, which is the 3 x 3 camera-calibration matrix. It has the following notation:\n\nWe're mainly interested in four components: fx, fy, cx, and cy. With this data we can create an instance of the camera-calibration object using the following code for calibration:\n\n    CameraCalibration calibration(526.58037684199849e, 524.65577209994706e, 318.41744018680112, 202.96659047014398)\n\nWithout correct camera calibration it's impossible to create a natural-looking augmented reality. The estimated perspective transformation will differ from the transformation that the camera has. This will cause the augmented objects to look like they are too close or too far. The following is an example screenshot where the camera calibration was changed intentionally:\n\nAs you can see, the perspective look of the box differs from the overall scene.\n\nTo estimate the pattern position, we solve the PnP problem using the OpenCV function `cv::solvePnP`. You are probably familiar with this function because we used it in the marker-based AR too. We need the coordinates of the pattern corners on the current image, and its reference 3D coordinates we defined previously.\n\n### Note\n\nThe `cv::solvePnP` function can work with more than four points. Also, it's a key function if you want to create an AR with complex shape patterns. The idea remains the same--you just have to define a 3D structure of your pattern and the 2D find point correspondences. Of course, the homography estimation is not applicable here.\n\nWe take the reference 3D points from the trained pattern object and their corresponding projections in 2D from the `PatternTrackingInfo` structure; the camera calibration is stored in a `PatternDetector` private field.\n\n### Pattern.cpp\n\nThe pattern location in 3D space is estimated by the `computePose` function as follows:\n\n    void PatternTrackingInfo::computePose(const Pattern& pattern, const CameraCalibration& calibration)\n    {\n      cv::Mat camMatrix, distCoeff;\n      cv::Mat(3,3, CV_32F, const_cast<float*>(&calibration.getIntrinsic().data[0])).copyTo(camMatrix);\n      cv::Mat(4,1, CV_32F, const_cast<float*>(&calibration.getDistorsion().data[0])).copyTo(distCoeff);\n\n      cv::Mat Rvec;\n      cv::Mat_<float> Tvec;\n      cv::Mat raux,taux;\n      cv::solvePnP(pattern.points3d, points2d, camMatrix, distCoeff,raux,taux);\n      raux.convertTo(Rvec,CV_32F);\n      taux.convertTo(Tvec ,CV_32F);\n\n      cv::Mat_<float> rotMat(3,3); \n      cv::Rodrigues(Rvec, rotMat);\n\n      \/\/ Copy to transformation matrix\n      pose3d = Transformation();\n\n      for (int col=0; col<3; col++)\n      {\n        for (int row=0; row<3; row++)\n        {        \n         pose3d.r().mat[row][col] = rotMat(row,col); \n         \/\/ Copy rotation component\n        }\n        pose3d.t().data[col] = Tvec(col); \n        \/\/ Copy translation component\n      }\n\n      \/\/ Since solvePnP finds camera location, w.r.t to marker pose, \n      \/\/ to get marker pose w.r.t to the camera we invert it.\n      pose3d = pose3d.getInverted();\n    }\n\n# Application infrastructure\n\nSo far, we've learned how to detect a pattern and estimate its 3D position with regards to the camera. Now it's time to show how to put these algorithms into a real application. So our goal for this section is to show how to use OpenCV to capture a video from a web camera and create the visualization context for 3D rendering.\n\nAs our goal is to show how to use key features of marker-less AR, we will create a simple command-line application that will be capable of detecting arbitrary pattern images either in a video sequence or in still images.\n\nTo hold all image-processing logic and intermediate data, we introduce the `ARPipeline` class. It's a root object that holds all subcomponents necessary for augmented reality and performs all processing routines on the input frames. The following is a UML diagram of `ARPipeline` and its subcomponents:\n\nIt consists of:\n\n  * The camera-calibration object\n  * An Instance of the pattern-detector object\n  * A trained pattern object\n  * Intermediate data of pattern tracking\n\n## ARPipeline.hpp\n\nThe following code  contains a declaration of the `ARPipeline` class:\n\n    class ARPipeline\n    {\n    public:\n      ARPipeline(const cv::Mat& patternImage, const CameraCalibration& calibration);\n\n      bool processFrame(const cv::Mat& inputFrame);\n\n      const Transformation& getPatternLocation() const;\n\n    private:\n      CameraCalibration   m_calibration;\n      Pattern             m_pattern;\n      PatternTrackingInfo m_patternInfo;\n      PatternDetector     m_patternDetector;\n    };\n\nIn the `ARPipeline` constructor, a pattern object is initialized and the calibration data is saved to the `private` field. The `processFrame` function implements pattern detection and the person's pose-estimation routine. The return value indicates the success of pattern detection. You can get the calculated pattern pose by calling the `getPatternLocation` function.\n\n## ARPipeline.cpp\n\nThe following code contains the implementation of the `ARPipeline` class:\n\n    ARPipeline::ARPipeline(const cv::Mat& patternImage, const CameraCalibration& calibration)\n      : m_calibration(calibration)\n    {\n      m_patternDetector.buildPatternFromImage (patternImage, m_pattern);\n      m_patternDetector.train(m_pattern);\n    }\n\n    bool ARPipeline::processFrame(const cv::Mat& inputFrame)\n    {\n      bool patternFound = m_patternDetector.findPattern(inputFrame, m_patternInfo);\n\n      if (patternFound)\n      {\n        m_patternInfo.computePose(m_pattern, m_calibration);\n      }\n\n      return patternFound;\n    }\n\n    const Transformation& ARPipeline::getPatternLocation() const\n    {\n      return m_patternInfo.pose3d;\n    }\n\n## Enabling support for 3D visualization in OpenCV\n\nAs in the previous chapter, we will use OpenGL to render our 3D working. But unlike the iOS  environment, where we had to follow the iOS application architecture requirements, we now have much more freedom. On Windows and Mac you can choose from many 3D engines. In this chapter, we will learn how to create cross-platform 3D visualization using OpenCV. Starting from version 2.4.2, OpenCV has OpenGL's support in visualization windows. This means you can now easily render any 3D content in OpenCV.\n\nTo set up an OpenGL window in OpenCV, the first thing you need to do is to build OpenCV with OpenGL support. Otherwise, an exception will be thrown when you attempt to use the OpenGL-related functions of OpenCV. To enable OpenGL support, you should build the OpenCV library with the `ENABLE_OPENGL=YES` flag.\n\n### Note\n\nAs of the current version (OpenCV 2.4.2), OpenGL support is turned off by default. We cannot guarantee it, but OpenGL may be enabled by default in future releases. If so, there will be no need to build OpenCV manually.\n\nTo set up an OpenGL window in OpenCV, perform the following:\n\n  * Clone the OpenCV repository from GitHub (<https:\/\/github.com\/Itseez\/opencv>). You will need either command-line git tools or the GitHub Application installed on your computer to perform this step.\n  * Configure OpenCV and generate a workspace for your IDE. You will need a CMake application to complete this step. CMake can be freely downloaded from <http:\/\/www.cmake.org\/cmake\/resources\/software.html>.\n\nTo configure  OpenCV, you can either use the command-line CMake command as follows (run from the directory where you want the generated project to be placed):\n\n    **cmake -D ENABLE_OPENGL=YES <path to the OpenCV source directory>**\n\nOr, if you prefer GUI-style, use CMake-GUI for a more user-friendly project configuration:\n\nAfter the  generation of the OpenCV workspace for the selected IDE, open the project and execute the install target to build the library and install it. When this process is done, you can configure the sample project using the new OpenCV library you've just built.\n\n## Creating OpenGL windows using OpenCV\n\nNow that we have  OpenCV binaries with OpenGL support, it's time to create our first OpenGL window. The initialization of the OpenGL window starts with creating the named window with an OpenGL flag:\n\n    cv::namedWindow(ARWindowName, cv::WINDOW_OPENGL);\n\n`ARWindowName` is a string constant for the name of our window. We will use `Markerless AR` here. This call will create a window with the specified name. The `cv::WINDOW_OPENGL` flag indicates we're going to use OpenGL in this window. Then we set the desired window size:\n\n    cv::resizeWindow(ARWindowName, 640, 480);\n\nWe then set up the drawing context for this window:\n\n    cv::setOpenGlContext(ARWindowName);\n\nNow our window is ready for use. To draw something on it, we should register a callback function using the following method:\n\n    cv::setOpenGlDrawCallback(ARWindowName, drawAR, NULL);\n\nThis callback will be called on the repaint window. The first argument sets the window name, the second is a callback function, and the third optional argument will be passed to the callback function.\n\nThe `drawAR` function should have following signature:\n\n    void drawAR(void* param)\n    {\n      \/\/ Draw something using OpenGL here\n    }\n\nTo notify the system that you want to redraw your window, use the `cv::updateWindow` function:\n\n    cv::updateWindow(ARWindowName);\n\n## Video capture using OpenCV\n\nOpenCV allows  you to easily retrieve frames from almost every web camera and video file as well. To capture video from either a webcam or a video file, we can use the `cv::VideoCapture` class, as shown in the _Accessing the webcam_ section from Chapter 1, _Cartoonifier and Skin Changer for Android_.\n\n## Rendering augmented reality\n\nWe introduce the `ARDrawingContext` structure to hold all the necessary data that visualization may need:\n\n  * The most recent image taken from the camera\n  * The camera-calibration matrix\n  * The pattern pose in 3D (if present)\n  * The internal data related to OpenGL (texture ID and so on)\n\n### ARDrawingContext.hpp\n\nThe following code  contains a declaration of the `ARDrawingContext` class:\n\n    class ARDrawingContext\n    {\n    public:\n      ARDrawingContext(const CameraCalibration& c);\n\n      bool                patternPresent;\n      Transformation      patternPose;\n\n      \/\/! Request the redraw of the OpenGl window\n      void draw();\n\n      \/\/! Set the new frame for the background\n      void updateBackground(const cv::Mat& frame);\n\n    private:\n      \/\/! Draws the background with video\n      void drawCameraFrame ();\n\n      \/\/! Draws the AR\n      void drawAugmentedScene();\n\n      \/\/! Builds the right projection matrix \n      \/\/! from the camera calibration for AR\n      void buildProjectionMatrix(const Matrix33& calibration, int w, int h, Matrix44& result);\n\n      \/\/! Draws the coordinate axis \n      void drawCoordinateAxis();\n\n      \/\/! Draw the cube model\n      void drawCubeModel();\n\n    private:\n      bool               m_textureInitialized;\n      unsigned int       m_backgroundTextureId;\n      CameraCalibration  m_calibration;\n      cv::Mat            m_backgroundImage;\n    };\n\n### ARDrawingContext.cpp\n\nInitialization  of the OpenGL window is done in the constructor of the `ARDrawingContext` class as follows:\n\n    ARDrawingContext::ARDrawingContext(std::string windowName, cv::Size frameSize, const CameraCalibration& c)\n      : m_isTextureInitialized(false)\n      , m_calibration(c)\n      , m_windowName(windowName)\n    {\n      \/\/ Create window with OpenGL support\n      cv::namedWindow(windowName, cv::WINDOW_OPENGL);\n\n      \/\/ Resize it exactly to video size\n      cv::resizeWindow(windowName, frameSize.width, frameSize.height);\n\n      \/\/ Initialize OpenGL draw callback:\n      cv::setOpenGlContext(windowName);\n      cv::setOpenGlDrawCallback(windowName, ARDrawingContextDrawCallback, this);\n    }\n\nAs we now have a separate class for storing the visualization state, we modify the `cv::setOpenGlDrawCallback` call and pass an instance of `ARDrawingContext` as the parameter.\n\nThe modified callback function is as follows:\n\n    void ARDrawingContextDrawCallback(void* param)\n    {\n      ARDrawingContext * ctx = static_cast<ARDrawingContext*>(param);\n      if (ctx)\n      {\n        ctx->draw();\n      }\n    }\n\n`ARDrawingContext` takes all the responsibility of rendering the augmented reality. The frame rendering starts by drawing a background with an orthography projection. Then we render a 3D model with the correct perspective projection and model transformation. The following code contains the final version of the `draw` function:\n\n    void ARDrawingContext::draw()\n    {\n      \/\/ Clear entire screen\n      glClear(GL_DEPTH_BUFFER_BIT | GL_COLOR_BUFFER_BIT); \n      \/\/ Render background\n      drawCameraFrame();                                   \n      \/\/ Draw AR\n      drawAugmentedScene();                               \n    }\n\nAfter clearing the screen and depth buffer, we check if a texture for presenting a video is initialized. If so, we proceed to drawing a background, otherwise we create a new 2D texture by calling `glGenTextures`.\n\nTo draw a  background, we set up an orthographic projection and draw a solid rectangle that covers all the screen viewports. This rectangle is bound with a texture unit. This texture is filled with the content of an `m_backgroundImage` object. Its content is uploaded to the OpenGL memory beforehand. This function is identical to the function from the previous chapter, so we will omit its code here.\n\nAfter drawing the picture from a camera, we switch to drawing an AR. It's necessary to set the correct perspective projection that matches our camera calibration.\n\nThe following code shows how to build the correct OpenGL projection matrix from the camera calibration and render the scene:\n\n    void ARDrawingContext::drawAugmentedScene()\n    {\n      \/\/ Init augmentation projection\n      Matrix44 projectionMatrix;\n      int w = m_backgroundImage.cols;\n      int h = m_backgroundImage.rows;\n      buildProjectionMatrix(m_calibration, w, h, projectionMatrix);\n\n      glMatrixMode(GL_PROJECTION);\n      glLoadMatrixf(projectionMatrix.data);\n\n      glMatrixMode(GL_MODELVIEW);\n      glLoadIdentity();\n\n      if (isPatternPresent)\n      {\n        \/\/ Set the pattern transformation\n        Matrix44 glMatrix = patternPose.getMat44();\n        glLoadMatrixf(reinterpret_cast<const GLfloat*>(&glMatrix.data[0]));\n\n        \/\/ Render model\n        drawCoordinateAxis();\n        drawCubeModel();\n      }\n    }\n\nThe `buildProjectionMatrix` function was taken from the previous chapter, so it's the same. After applying perspective projection, we set the `GL_MODELVIEW` matrix to pattern transformation. To prove that our pose estimation works correctly, we draw a unit coordinate system in the pattern position.\n\nAlmost all  things are done. We create a pattern-detection algorithm and then we estimate the pose of the found pattern in 3D space, a visualization system to render the AR. Let's take a look at the following UML sequence diagram that demonstrates the frame-processing routine in our app:\n\n## Demonstration\n\nOur demonstration project supports the processing of still images, recorded videos, and live views from a web camera. We create two functions that help us with this.\n\n### main.cpp\n\nThe function `processVideo` handles the processing of the video and the function `processSingleImage` is used to process a single image, as follows:\n\n    void processVideo(const cv::Mat& patternImage, CameraCalibration& calibration, cv::VideoCapture& capture);\n\n    void processSingleImage(const cv::Mat& patternImage, CameraCalibration& calibration, const cv::Mat& image);\n\nFrom the function names it's clear that the first function processed the video source, and the second one works with a single image (this function is useful for debugging purposes). Both of them have a very common routine of image processing, pattern detection, scene rendering, and user interaction.\n\nThe `processFrame` function wraps these steps as follows:\n\n    \/**\n     * Performs full detection routine on camera frame \n    .* and draws the scene using drawing context. \n     * In addition, this function draw overlay with debug information \n    .* on top of the AR window. Returns true \n    .* if processing loop should be stopped; otherwise - false.\n     *\/\n    bool processFrame(const cv::Mat& cameraFrame, ARPipeline& pipeline, ARDrawingContext& drawingCtx)\n    {\n        \/\/ Clone image used for background (we will \n        \/\/ draw overlay on it)\n        cv::Mat img = cameraFrame.clone();\n\n        \/\/ Draw information:\n        if (pipeline.m_patternDetector.enableHomographyRefinement)\n            cv::putText(img, \"Pose refinement: On   ('h' to switch off)\", cv::Point(10,15), CV_FONT_HERSHEY_PLAIN, 1, CV_RGB(0,200,0));\n        else\n            cv::putText(img, \"Pose refinement: Off  ('h' to switch \n              on)\",  cv::Point(10,15), CV_FONT_HERSHEY_PLAIN, 1, CV_RGB(0,200,0));\n\n        cv::putText(img, \"RANSAC threshold: \" + ToString(pipeline.m_patternDetector.homographyReprojectionThreshold) + \"( Use'-'\/'+' to adjust)\", cv::Point(10, 30), CV_FONT_HERSHEY_PLAIN, 1, CV_RGB(0,200,0));\n        \/\/ Set a new camera frame:\n        drawingCtx.updateBackground(img);\n\n        \/\/ Find a pattern and update its detection status:\n        drawingCtx.isPatternPresent = pipeline.processFrame(cameraFrame);\n\n        \/\/ Update a pattern pose:\n        drawingCtx.patternPose = pipeline.getPatternLocation();\n\n        \/\/ Request redraw of the window:\n        drawingCtx.updateWindow();\n\n        \/\/ Read the keyboard input:\n        int keyCode = cv::waitKey(5);\n\n        bool shouldQuit = false;\n        if (keyCode == '+' || keyCode == '=')\n        {\n            pipeline.m_patternDetector.homographyReprojectionThreshold += 0.2f;\n            pipeline.m_patternDetector.homographyReprojectionThreshold = std::min(10.0f, pipeline.m_patternDetector.homographyReprojectionThreshold);\n        }\n        else if (keyCode == '-')\n        {\n            pipeline.m_patternDetector.homographyReprojectionThreshold -= 0.2f;\n            pipeline.m_patternDetector.homographyReprojectionThreshold = std::max(0.0f, pipeline.m_patternDetector.homographyReprojectionThreshold);\n        }\n        else if (keyCode == 'h')\n        {\n            pipeline.m_patternDetector.enableHomographyRefinement = !pipeline.m_patternDetector.enableHomographyRefinement;\n        }\n        else if (keyCode == 27 || keyCode == 'q')\n        {\n            shouldQuit = true;\n        }\n\n        return shouldQuit;\n    }\n\nThe initialization of `ARPipeline` and `ARDrawingContext` is done either in the `processSingleImage` or `processVideo` function as follows:\n\n    void processSingleImage(const cv::Mat& patternImage, CameraCalibration& calibration, const cv::Mat& image)\n    {\n        cv::Size frameSize(image.cols, image.rows);\n        ARPipeline pipeline(patternImage, calibration);\n        ARDrawingContext drawingCtx(\"Markerless AR\", frameSize, calibration);\n\n        bool shouldQuit = false;\n        do\n        {\n            shouldQuit = processFrame(image, pipeline, drawingCtx);\n        } while (!shouldQuit);\n    }\n\nWe create `ARPipeline` from the pattern image and calibration arguments. Then we initialize `ARDrawingContext` using calibration again. After these steps, the OpenGL window is created. Then we upload the query image into a drawing context and call `ARPipeline.processFrame` to find a pattern. If the pose pattern has been found, we copy its location to the drawing context for further frame rendering. If the pattern has not been detected, we render only the camera frame without any AR.\n\nYou can run the demo application in one of the following ways:\n\n  * To run on a single image call:\n\n        markerless_ar_demo pattern.png test_image.png\n\n  * To run on a recorded video call:\n\n        markerless_ar_demo pattern.png test_video.avi\n\n  * To run using live feed from a web camera, call:\n\n        **markerless_ar_demo pattern.png**\n\nThe result of augmenting a single image is shown in the following screenshot:\n\n# Summary\n\nIn this chapter you have learned about feature descriptors and how to use them to define a scale and a rotation invariant pattern description. This description can be used to find similar entries in other images. The strengths and weaknesses of most popular feature descriptors were also explained. In the second half of the chapter, we learned how to use OpenGL and OpenCV together for rendering augmented reality.\n\n# References\n\n  *  _Distinctive Image Features from Scale-Invariant Keypoints_ (<http:\/\/www.cs.ubc.ca\/~lowe\/papers\/ijcv04.pdf>)\n  *  _SURF: Speeded Up Robust Features_ (<http:\/\/www.vision.ee.ethz.ch\/~surf\/eccv06.pdf>)\n  *  _Model-Based Object Pose in 25 Lines of Code_ , _Dementhon and L.S Davis_ , _International Journal of Computer Vision_ , _edition 15_ , _pp. 123-141_ , _1995_\n  *  _Linear N-Point Camera Pose Determination, L.Quan_ , _IEEE Trans. on Pattern Analysis and Machine Intelligence_ , _vol. 21_ , _edition. 7_ , _July 1999_\n  *  _Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography_ , _M. Fischer and R. Bolles, Graphics and Image Processing_ , _vol. 24_ , _edition. 6_ , _pp. 381-395_ , _June 1981_\n  *  _Multiple View Geometry in Computer Vision_ , _R. Hartley and A.Zisserman_ , _Cambridge University Press_ (<http:\/\/www.umiacs.umd.edu\/~ramani\/cmsc828d\/lecture9.pdf>)\n  *  _Camera Pose Revisited - New Linear Algorithms_, _M_ _. Ameller, B.Triggs, L.Quan_ (<http:\/\/hal.inria.fr\/docs\/00\/54\/83\/06\/PDF\/Ameller-eccv00.pdf>)\n  *  _Closed-form solution of absolute orientation using unit quaternions_ , _Berthold K. P. Horn_ , _Journal of the Optical Society A_ , _vol._ _4_ , _629 -642_\n\n# Chapter 4. Exploring Structure from Motion Using OpenCV\n\nIn this chapter we will discuss the notion of **Structure from Motion** ( **SfM** ), or better put as extracting geometric structures from images taken through a camera's motion, using functions within OpenCV's API to help us. First, let us constrain the otherwise lengthy footpath of our approach to using a single camera, usually called a **monocular** approach, and a discrete and sparse set of frames rather than a continuous video stream. These two constrains will greatly simplify the system we will sketch in the coming pages, and help us understand the fundamentals of any SfM method. To implement our method we will follow in the footsteps of Hartley and Zisserman (hereafter referred to as H and Z), as documented in chapters 9 through 12 of their seminal book _Multiple View Geometry in Computer Vision_.\n\nIn this chapter we cover the following:\n\n  * Structure from Motion concepts\n  * Estimating the camera motion from a pair of images\n  * Reconstructing the scene\n  * Reconstruction from many views\n  * Refinement of the reconstruction\n  * Visualizing 3D point clouds\n\nThroughout the chapter we assume the use of a calibrated camera--one that was calibrated beforehand. **Calibration** is a ubiquitous operation in computer vision, fully supported in OpenCV using command-line tools and was discussed in previous chapters. We therefore assume the existence of the camera's intrinsic parameters embodied in the K matrix, one of the outputs from the calibration process.\n\nTo make things clear in terms of language, from this point on we will refer to a camera as a single view of the scene rather than to the optics and hardware taking the image. A camera has a position in space, and a direction of view. Between two cameras, there is a translation element (movement through space) and a rotation of the direction of view.\n\nWe will also unify the terms for the point in the scene, world, real, or 3D to be the same thing, a point that exists in our real world. The same goes for points in the image or 2D, which are points in the image coordinates, of some real 3D point that was projected on the camera sensor at that location and time.\n\nIn the chapter's code sections you will notice references to _Multiple View Geometry in Computer Vision_ , for example `\/\/ HZ 9.12`. This refers to equation number 12 of chapter 9 of the book. Also, the text will include excerpts of code only, while the complete runnable code is included in the material accompanied with the book.\n\n# Structure from Motion concepts\n\nThe first discrimination we should make is the difference between stereo (or indeed any multiview), 3D reconstruction using calibrated rigs, and SfM. While a rig of two or more cameras assume we already know what the motion between the cameras is, in SfM we don't actually know this motion and we wish to find it. Calibrated rigs, from a simplistic point of view, allow a much more accurate reconstruction of 3D geometry because there is no error in estimating the distance and rotation between the cameras--it is already known. The first step in implementing an SfM system is finding the motion between the cameras. OpenCV may help us in a number of ways to obtain this motion, specifically using the `findFundamentalMat` function.\n\nLet us think for one moment of the goal behind choosing an SfM algorithm. In most cases we wish to obtain the geometry of the scene, for example, where objects are in relation to the camera and what their form is. Assuming we already know the motion between the cameras picturing the same scene, from a reasonably similar point of view, we would now like to reconstruct the geometry. In computer vision jargon this is known as **triangulation** , and there are plenty of ways to go about it. It may be done by way of ray intersection, where we construct two rays: one from each camera's center of projection and a point on each of the image planes. The intersection of these rays in space will, ideally, intersect at one 3D point in the real world that was imaged in each camera, as shown in the following diagram:\n\nIn reality, ray intersection is highly unreliable; H and Z recommend against it. This is because the rays usually do not intersect, making us fall back to using the middle point on the shortest segment connecting the two rays. Instead, H and Z suggest a number of ways to triangulate 3D points, of which we will discuss a couple of them in the _Reconstructing the scene_ section. The current version of OpenCV does not contain a simple API for triangulation, so this part we will code on our own.\n\nAfter we have learned how to recover 3D geometry from two views, we will see how we can incorporate more views of the same scene to get an even richer reconstruction. At that point, most SfM methods try to optimize the bundle of estimated positions of our cameras and 3D points by means of Bundle Adjustment, in the _Refinement of the reconstruction_ section. OpenCV contains means for Bundle Adjustment in its new Image Stitching Toolbox. However, the beauty of working with OpenCV and C++ is the abundance of external tools that can be easily integrated into the pipeline. We will therefore see how to integrate an external bundle adjuster, the neat SSBA library.\n\nNow that we have sketched an outline of our approach to SfM using OpenCV, we will see how each element can be implemented.\n\n# Estimating the camera motion from a pair of images\n\nBefore we set out to actually find the motion between two cameras, let us examine the inputs and the tools we have at hand to perform this operation. First, we have two images of the same scene from (hopefully not extremely) different positions in space. This is a powerful asset, and we will make sure to use it. Now as far as tools go, we should take a look at mathematical objects that impose constraints over our images, cameras, and the scene.\n\nTwo very useful mathematical objects are the fundamental matrix (denoted by F) and the essential matrix (denoted by E). They are mostly similar, except that the essential matrix is assuming usage of calibrated cameras; this is the case for us, so we will choose it. OpenCV only allows us to find the fundamental matrix via the `findFundamentalMat` function; however, it is extremely simple to get the essential matrix from it using the calibration matrix `K` as follows:\n\n    Mat_<double> E = K.t() * F * K; \/\/according to HZ (9.12)\n\nThe essential matrix, a 3 x 3 sized matrix, imposes a constraint between a point in one image and a point in the other image with x'Ex=0, where x is a point in image one and x' is the corresponding point in image two. This is extremely useful, as we are about to see. Another important fact we use is that the essential matrix is all we need in order to recover both cameras for our images, although only up to scale; but we will get to that later. So, if we obtain the essential matrix, we know where each camera is positioned in space, and where it is looking. We can easily calculate the matrix if we have enough of those constraint equations, simply because each equation can be used to solve for a small part of the matrix. In fact, OpenCV allows us to calculate it using just seven point-pairs, but hopefully we will have many more pairs and get a more robust solution.\n\n## Point matching using rich feature descriptors\n\nNow we will make use of our constraint equations to calculate the essential matrix. To get our constraints, remember that for each point in image A we must find a corresponding point in image B. How can we achieve such a matching? Simply by using OpenCV's extensive feature-matching framework, which has greatly matured in the past few years.\n\nFeature extraction and descriptor matching is an essential process in computer vision, and is used in many methods to perform all sorts of operations. For example, detecting the position and orientation of an object in the image or searching a big database of images for similar images through a given query. In essence, **extraction** means selecting points in the image that would make the features good, and computing a descriptor for them. A **descriptor** is a vector of numbers that describes the surrounding environment around a feature point in an image. Different methods have different length and data type for their descriptor vectors. **Matching** is the process of finding a corresponding feature from one set in another using its descriptor. OpenCV provides very easy and powerful methods to support feature extraction and matching. More information about feature matching may be found in Chapter 3, _Marker-less Augmented Reality_.\n\nLet us examine a very simple feature extraction and matching scheme:\n\n    \/\/ detectingkeypoints\n    SurfFeatureDetectordetector();\n    vector<KeyPoint> keypoints1, keypoints2;\n    detector.detect(img1, keypoints1);\n    detector.detect(img2, keypoints2);\n\n    \/\/ computing descriptors\n    SurfDescriptorExtractor extractor;\n    Mat descriptors1, descriptors2;\n    extractor.compute(img1, keypoints1, descriptors1);\n    extractor.compute(img2, keypoints2, descriptors2);\n\n    \/\/ matching descriptors\n    BruteForceMatcher<L2<float>> matcher;\n    vector<DMatch> matches;\n    matcher.match(descriptors1, descriptors2, matches);\n\nYou may have already seen similar OpenCV code, but let us review it quickly. Our goal is to obtain three elements: Feature points for two images, descriptors for them, and a matching between the two sets of features. OpenCV provides a range of feature detectors, descriptor extractors, and matchers. In this simple example we use the `SurfFeatureDetector` function to get the 2D location of the **Speeded-Up Robust Features (SURF)** features, and the `SurfDescriptorExtractor` function to get the SURF descriptors. We use a brute-force matcher to get the matching, which is the most straightforward way to match two feature sets implemented by comparing each feature in the first set to each feature in the second set (hence the phrasing brute-force) and getting the best match.\n\nIn the next image we will see a matching of feature points on two images from the Fountain-P11 sequence found at <http:\/\/cvlab.epfl.ch\/~strecha\/multiview\/denseMVS.html>.\n\nPractically, raw matching like we just performed is good only up to a certain level, and many matches are probably erroneous. For that reason, most SfM methods perform some form of filtering on the matches to ensure correctness and reduce errors. One form of filtering, which is the built-in OpenCV's brute-force matcher, is cross-check filtering. That is, a match is considered true if a feature of the first image matched a feature of the second image, and the reverse check also matched the feature of the second image with the feature of the first image. Another common filtering mechanism, used in the provided code, is to filter based on the fact that the two images are of the same scene and have a certain stereo-view relationship between them. In practice, the filter tries to robustly calculate the fundamental matrix, of which we will learn in the _Finding camera matrices_ section, and retain those feature pairs that correspond with this calculation with small errors.\n\n## Point matching using optical flow\n\nAn alternative to using rich features, such as SURF, is using **optical flow** ( **OF** ). The following information box provides a short overview of optical flow. OpenCV recently extended its API for getting the flow field from two images and now it is faster and more powerful. We will try to use it as an alternative to matching features.\n\n### Note\n\n**Optical flow** is the process of matching selected points from one image to another, assuming both images are part of a sequence and relatively close to one another. Most optical flow methods compare a small region, known as the search window or patch, around each point from image A to the same area in image B. Following a very common rule in computer vision, called the brightness constancy constraint (and other names), the small patches of the image will not change drastically from one image to the other, and therefore the magnitude of their subtraction should be close to zero. In addition to matching patches, newer methods of optical flow use a number of additional methods to get better results. One is using image pyramids, which are smaller and smaller resized versions of the image, which allow for working \"from-coarse-to-fine\"--a very well-used trick in computer vision. Another method is to define global constraints on the flow field, assuming that the points close to each other \"move together\" in the same direction. A more in-depth review of optical flow methods in OpenCV can be found in _Chapter Developing Fluid Wall Using the Microsoft Kinect_ which is available on the Packt website..\n\nUsing optical flow in OpenCV is fairly easy by invoking the `calcOpticalFlowPyrLK` function. However, we would like to keep the result matching from OF similar to that using rich features, as in the future we would like the two approaches to be interchangeable. To that end, we must install a special matching method--one that is interchangeable with the previous feature-based method, which we are about to see in the code section that follows:\n\n    Vector<KeyPoint>left_keypoints,right_keypoints;\n\n    \/\/ Detect keypoints in the left and right images\n    FastFeatureDetectorffd;\n    ffd.detect(img1, left_keypoints);\n    ffd.detect(img2, right_keypoints);\n\n    vector<Point2f>left_points;\n    KeyPointsToPoints(left_keypoints,left_points);\n\n    vector<Point2f>right_points(left_points.size());\n\n    \/\/ making sure images are grayscale\n    Mat prevgray,gray;\n    if (img1.channels() == 3) {\n        cvtColor(img1,prevgray,CV_RGB2GRAY);\n        cvtColor(img2,gray,CV_RGB2GRAY);\n    } else {\n            prevgray = img1;\n            gray = img2;\n    }\n\n    \/\/ Calculate the optical flow field:\n    \/\/  how each left_point moved across the 2 images\n    vector<uchar>vstatus; vector<float>verror;\n    calcOpticalFlowPyrLK(prevgray, gray, left_points, right_points, vstatus, verror);\n\n    \/\/ First, filter out the points with high error\n    vector<Point2f>right_points_to_find;\n    vector<int>right_points_to_find_back_index;\n    for (unsigned inti=0; i<vstatus.size(); i++) {\n        if (vstatus[i] &&verror[i] < 12.0) {\n        \/\/ Keep the original index of the point in the\n        \/\/ optical flow array, for future use\n        right_points_to_find_back_index.push_back(i);\n        \/\/ Keep the feature point itself\n    right_points_to_find.push_back(j_pts[i]);\n    } else {\n            vstatus[i] = 0; \/\/ a bad flow\n    }\n    }\n\n    \/\/ for each right_point see which detected feature it belongs to\n    Mat right_points_to_find_flat = Mat(right_points_to_find).reshape(1,to_find.size()); \/\/flatten array\n\n    vector<Point2f>right_features; \/\/ detected features\n    KeyPointsToPoints(right_keypoints,right_features);\n\n    Mat right_features_flat = Mat(right_features).reshape(1,right_features.size());\n\n    \/\/ Look around each OF point in the right image\n    \/\/  for any features that were detected in its area\n    \/\/  and make a match.\n    BFMatchermatcher(CV_L2);\n    vector<vector<DMatch>>nearest_neighbors;\n    matcher.radiusMatch(\n    right_points_to_find_flat,\n    right_features_flat,\n    nearest_neighbors,\n    2.0f);\n\n    \/\/ Check that the found neighbors are unique (throw away neighbors\n    \/\/  that are too close together, as they may be confusing)\n    std::set<int>found_in_right_points; \/\/ for duplicate prevention\n    for(inti=0;i<nearest_neighbors.size();i++) {\n    DMatch _m;\n    if(nearest_neighbors[i].size()==1) {\n        _m = nearest_neighbors[i][0]; \/\/ only one neighbor\n    } else if(nearest_neighbors[i].size()>1) {\n            \/\/ 2 neighbors - check how close they are\n            double ratio = nearest_neighbors[i][0].distance \/ nearest_neighbors[i][1].distance;\n    if(ratio < 0.7) { \/\/ not too close\n        \/\/ take the closest (first) one\n        _m = nearest_neighbors[i][0];\n    } else { \/\/ too close - we cannot tell which is better\n            continue; \/\/ did not pass ratio test - throw away\n    }\n    } else {\n            continue; \/\/ no neighbors... :(\n    }\n\n    \/\/ prevent duplicates\n    if (found_in_right_points.find(_m.trainIdx) == found_in_right_points.end()) { \n        \/\/ The found neighbor was not yet used:\n        \/\/ We should match it with the original indexing \n        \/\/ ofthe left point\n        _m.queryIdx = right_points_to_find_back_index[_m.queryIdx]; \n        matches->push_back(_m); \/\/ add this match\n        found_in_right_points.insert(_m.trainIdx);\n        }\n    }\n    cout<<\"pruned \"<< matches->size() <<\" \/ \"<<nearest_neighbors.size() <<\" matches\"<<endl;\n\nThe functions `KeyPointsToPoints` and `PointsToKeyPoints` are simply convenience functions for conversion between the `cv::Point2f` and the `cv::KeyPoint` structs.\n\nIn the previous segment of code we can see a number of interesting things. The first thing to note is that when we use optical flow, our result shows a feature moved from a position in the image on the left-hand side to another position in the image on the right-hand side. But we have a new set of features detected in the image to the right-hand side, not necessarily aligning with the features that flowed from the image to the left-hand side in optical flow. We must align them. To find those lost features we use a **k-nearest neighbor** ( **kNN** ) radius search, which gives us up to two features that fall within a 2-pixel radius to the points of interest.\n\nOne more thing that we can see is an implementation of the ratio test for kNN, which is a common practice in SfM to reduce errors. In essence, it is a filter that removes confusing matches when we have a match between one feature in the left-hand side image and two features in the right-hand side image. If the two features in the right-hand side image are too close together, or the ratio between them is too big (close to 1.0), we consider them confusing and do not use them. We also install a duplicate prevention filter to further prune the matches.\n\nThe following image shows the flow field from one image to another. Pink arrows in the left-hand side image show the movement of patches from the left-hand side image to the right-hand side image. In the second image to the left, we see a small area of the flow field zoomed in. The pink arrows again show the motion of patches, and we can see it makes sense by looking at the two original image segments on the right-hand side. Visual features in the left-hand side image are moving leftwards across the image, in the directions of the pink arrows as shown in the following image:\n\nThe advantage of using optical flow in place of rich features is that the process is usually faster and can accommodate the matching many more points, making the reconstruction denser. In many optical flow methods there is also a monolithic model of the overall movement of patches, where matching rich features are usually not taken into account. The caveat in working with optical flow is that it works best for consecutive images taken by the same hardware, whereas rich features are mostly agnostic to that. The differences result from the fact that optical flow methods usually use very rudimentary features, like image patches around a keypoint, whereas higher-order richer features (for example, SURF) take into account higher-level information for each keypoint. Using optical flow or rich features is a decision the designer of the application should make depending on the input.\n\n## Finding camera matrices\n\nNow that we have obtained matches between keypoints, we can calculate the fundamental matrix and from that obtain the essential matrix. However, we must first align our matching points into two arrays, where an index in one array corresponds to the same index in the other. This is required by the `findFundamentalMat` function. We would also need to convert the `KeyPoint` structure to a `Point2f` structure. We must pay special attention to the `queryIdx` and `trainIdx` member variables of `DMatch`, the OpenCV struct that holds a match between two keypoints, as they must align with the way we used the `matcher.match()` function. The following code section shows how to align a matching into two corresponding sets of 2D points, and how these can be used to find the fundamental matrix:\n\n    vector<Point2f>imgpts1,imgpts2;\n    for( unsigned inti = 0; i<matches.size(); i++ )\n    {\n    \/\/ queryIdx is the \"left\" image\n    imgpts1.push_back(keypoints1[matches[i].queryIdx].pt);\n    \/\/ trainIdx is the \"right\" image\n    imgpts2.push_back(keypoints2[matches[i].trainIdx].pt);\n    }\n\n    Mat F = findFundamentalMat(imgpts1, imgpts2, FM_RANSAC, 0.1, 0.99, status);\n    Mat_<double> E = K.t() * F * K; \/\/according to HZ (9.12)\n\nWe may later use the `status` binary vector to prune those points that align with the recovered fundamental matrix. See the following image for an illustration of point matching after pruning with the fundamental matrix. The red arrows mark feature matches that were removed in the process of finding the `F` matrix, and the green arrows are feature matches that were kept.\n\nNow we are ready to find the camera matrices. This process is described at length in chapter 9 of H and Z's book; however, we are going to use a very straightforward and simplistic implementation of it, and OpenCV makes things very easy for us. But first, we will briefly examine the structure of the camera matrix we are going to use.\n\nThis is the model for our camera, it consists of two elements, rotation (denoted as **R** ) and translation (denoted as **t** ). The interesting thing about it is that it holds a very essential equation: x=PX, where x is a 2D point on the image and X is a 3D point in space. There is more to it, but this matrix gives us a very important relationship between the image points and the scene points. So, now that we have a motivation for finding the camera matrices, we will see how it can be done. The following code section shows how to decompose the essential matrix into the rotation and translation elements:\n\n    SVD svd(E);\n    Matx33d W(0,-1,0,\/\/HZ 9.13\n     1,0,0,\n     0,0,1);\n    Mat_<double> R = svd.u * Mat(W) * svd.vt; \/\/HZ 9.19\n    Mat_<double> t = svd.u.col(2); \/\/u3\n    Matx34d P1(  R(0,0),R(0,1), R(0,2), t(0),\n    R(1,0),R(1,1), R(1,2), t(1),\n    R(2,0),R(2,1), R(2,2), t(2));\n\nVery simple. All we had to do is take the **Singular Value Decomposition** ( **SVD** ) of the essential matrix we obtained from before, and multiply it by a special matrix `W`. Without going too deeply into the mathematical interpretation of what we did, we can say the `SVD` operation decomposed our matrix E into two parts, a rotation element and a translation element. In fact, the essential matrix was originally composed by the multiplication of these two elements. Strictly for satisfying our curiosity we can look at the following equation for the essential matrix, which appears in the literature: E=[t]xR. We see it is composed of (some form of) a translation element and a rotational element R.\n\nWe notice that what we just did only gives us one camera matrix, so where is the other camera matrix? Well, we perform this operation under the assumption that one camera matrix is fixed and canonical (no rotation and no translation). The next camera matrix is also canonical:\n\nThe other camera that we recovered from the essential matrix has moved and rotated in relation to the fixed one. This also means that any of the 3D points that we recover from these two camera matrices will have the first camera at the world origin point (0, 0, 0).\n\nThis, however, is not the complete solution. H and Z show in their book how and why this decomposition has in fact four possible camera matrices, but only one of them is the true one. The correct matrix is the one that will produce reconstructed points with a positive Z value (points that are in front of the camera). But we can only understand that after learning about triangulation and 3D reconstruction, which will be discussed in the next section.\n\nOne more thing we can think of adding to our method is error checking. Many a times the calculation of the fundamental matrix from the point matching is erroneous, and this affects the camera matrices. Continuing triangulation with faulty camera matrices is pointless. We can install a check to see if the rotation element is a valid rotation matrix. Keeping in mind that rotation matrices must have a determinant of 1 (or -1), we can simply do the following:\n\n    bool CheckCoherentRotation(cv::Mat_<double>& R) {\n    if(fabsf(determinant(R))-1.0 > 1e-07) {\n        cerr<<\"det(R) != +-1.0, this is not a rotation matrix\"<<endl;\n        return false; \n        }\n    return true;\n    }\n\nWe can now see how all these elements combine into a function that recovers the `P` matrices, as follows:\n\n    void FindCameraMatrices(const Mat& K,\n    const Mat& Kinv,\n    const vector<KeyPoint>& imgpts1,\n    const vector<KeyPoint>& imgpts2,\n    Matx34d& P,\n    Matx34d& P1,\n    vector<DMatch>& matches,\n    vector<CloudPoint>& outCloud\n    )\n    {\n    \/\/Find camera matrices\n\n    \/\/Get Fundamental Matrix\n    Mat F = GetFundamentalMat(imgpts1,imgpts2,matches);\n\n    \/\/Essential matrix: compute then extract cameras [R|t]\n    **Mat_ <double> E = K.t() * F * K; \/\/according to HZ (9.12)**\n\n    \/\/decompose E to P' , HZ (9.19)\n    SVD svd(E,SVD::MODIFY_A);\n    Mat svd_u = svd.u;\n    Mat svd_vt = svd.vt;\n    Mat svd_w = svd.w;\n\n    Matx33d W(0,-1,0,\/\/HZ 9.13\n    1,0,0,\n    0,0,1);\n    Mat_<double> R = svd_u * Mat(W) * svd_vt; \/\/HZ 9.19\n    Mat_<double> t = svd_u.col(2); \/\/u3\n\n    if (!CheckCoherentRotation(R)) {\n        cout<<\"resulting rotation is not coherent\\n\";\n        P1 = 0;\n        return;\n    }\n\n    P1 = Matx34d(R(0,0),R(0,1),R(0,2),t(0),\n    R(1,0),R(1,1),R(1,2),t(1),\n    R(2,0),R(2,1),R(2,2),t(2));\n    }\n\nAt this point we have the two cameras that we need in order to reconstruct the scene. The canonical first camera, in the `P` variable, and the second camera we calculated, form the fundamental matrix in the `P1` variable. The next section will reveal how we use these cameras to obtain a 3D structure of the scene.\n\n# Reconstructing the scene\n\nNext we look into the matter of recovering the 3D structure of the scene from the information we have acquired so far. As we had done before, we should look at the tools and information we have at hand to achieve this. In the preceding section we obtained two camera matrices from the essential and fundamental matrices; we already discussed how these tools will be useful for obtaining the 3D position of a point in space. Then, we can go back to our matched point pairs to fill in our equations with numerical data. The point pairs will also be useful in calculating the error we get from all our approximate calculations.\n\nThis is the time to see how we can perform triangulation using OpenCV. This time we will follow the steps Hartley and Sturm take in their article _Triangulation_ , where they implement and compare a few triangulation methods. We will implement one of their linear methods, as it is very simple to code with OpenCV.\n\nRemember we had two key equations arising from the 2D point matching and P matrices: x=PX and x'= P'X, where x and x' are matching 2D points and X is a real world 3D point imaged by the two cameras. If we rewrite the equations, we can formulate a system of linear equations that can be solved for the value of X, which is what we desire to find. Assuming X = (x, y, z, 1)t (a reasonable assumption for points that are not too close or too far from the camera center) creates an inhomogeneous linear equation system of the form AX = B. We can code and solve this equation system as follows:\n\n    Mat_<double> LinearLSTriangulation(\n    Point3d u,\/\/homogenous image point (u,v,1)\n    Matx34d P,\/\/camera 1 matrix\n    Point3d u1,\/\/homogenous image point in 2nd camera\n    Matx34d P1\/\/camera 2 matrix\n    )\n    {\n    \/\/build A matrix\n     Matx43d A(u.x*P(2,0)-P(0,0),u.x*P(2,1)-P(0,1),u.x*P(2,2)-P(0,2),\n    u.y*P(2,0)-P(1,0),u.y*P(2,1)-P(1,1),u.y*P(2,2)-P(1,2),\n    u1.x*P1(2,0)-P1(0,0), u1.x*P1(2,1)-P1(0,1),u1.x*P1(2,2)-P1(0,2),\n    u1.y*P1(2,0)-P1(1,0), u1.y*P1(2,1)-P1(1,1),u1.y*P1(2,2)-P1(1,2)\n      );\n    \/\/build B vector\n    Matx41d B(-(u.x*P(2,3)-P(0,3)),\n      -(u.y*P(2,3)-P(1,3)),\n      -(u1.x*P1(2,3)-P1(0,3)),\n      -(u1.y*P1(2,3)-P1(1,3)));\n\n    \/\/solve for X\n    Mat_<double> X;\n    solve(A,B,X,DECOMP_SVD);\n\n    return X;\n    }\n\nThis will give us an approximation for the 3D points arising from the two 2D points. One more thing to note is that the 2D points are represented in homogenous coordinates, meaning the x and y values are appended with a 1. We should make sure these points are in normalized coordinates, meaning that they were multiplied by the calibration matrix `K` beforehand. We may notice that instead of multiplying each point by the matrix `K` we can simply make use of the KP matrix (the `K` matrix multiplied by the `P` matrix), as H and Z do throughout chapter 9. We can now write a loop over the point matches to get a complete triangulation as follows:\n\n    double TriangulatePoints(\n    const vector<KeyPoint>& pt_set1,\n    const vector<KeyPoint>& pt_set2,\n    const Mat&Kinv,\n    const Matx34d& P,\n    const Matx34d& P1,\n    vector<Point3d>& pointcloud)\n    {\n    vector<double> reproj_error;\n    for (unsigned int i=0; i<pts_size; i++) {\n        \/\/convert to normalized homogeneous coordinates\n        Point2f kp = pt_set1[i].pt;\n        Point3d u(kp.x,kp.y,1.0);\n        Mat_<double> um = Kinv * Mat_<double>(u);\n        u = um.at<Point3d>(0);\n        Point2f kp1 = pt_set2[i].pt;\n        Point3d u1(kp1.x,kp1.y,1.0);\n        Mat_<double> um1 = Kinv * Mat_<double>(u1);\n        u1 = um1.at<Point3d>(0);\n\n        \/\/triangulate\n        Mat_<double> X = LinearLSTriangulation(u,P,u1,P1);\n\n        \/\/calculate reprojection error\n        Mat_<double> xPt_img = K * Mat(P1) * X;\n        Point2f xPt_img_(xPt_img(0)\/xPt_img(2),xPt_img(1)\/xPt_img(2));\n        reproj_error.push_back(norm(xPt_img_-kp1));\n\n        \/\/store 3D point\n        pointcloud.push_back(Point3d(X(0),X(1),X(2)));\n    }\n\n    \/\/return mean reprojection error\n    Scalar me = mean(reproj_error);\n    return me[0];\n    }\n\nIn the following image we will see a triangulation result of two images out of the Fountain P-11 sequence at <http:\/\/cvlab.epfl.ch\/~strecha\/multiview\/denseMVS.html>. The two images at the top are the original two views of the scene, and the bottom pair is the view of the reconstructed point cloud from the two views, including the estimated cameras looking at the fountain. We can see how the right-hand side section of the red brick wall was reconstructed, and also the fountain that protrudes from the wall.\n\nHowever, as we discussed earlier, we have an issue with the reconstruction being only up-to-scale. We should take a moment to understand what up-to-scale means. The motion we obtained between our two cameras is going to have an arbitrary unit of measurement, that is, it is not in centimeters or inches but simply a given unit of scale. Our reconstructed cameras we will be one unit of scale distance apart. This has big implications should we decide to recover more cameras later, as each pair of cameras will have their own units of scale, rather than a common one.\n\nWe will now discuss how the error measure that we set up may help us in finding a more robust reconstruction. First we should note that reprojection means we simply take the triangulated 3D point and reimage it on a camera to get a reprojected 2D point, we then compare the distance between the original 2D point and the reprojected 2D point. If this distance is large this means we may have an error in triangulation, so we may not want to include this point in the final result. Our global measure is the average reprojection distance and may give us a hint to how our triangulation performed overall. High average reprojection rates may point to a problem with the `P` matrices, and therefore a possible problem with the calculation of the essential matrix or the matched feature points.\n\nWe should briefly go back to our discussion of camera matrices in the previous section. We mentioned that composing the camera matrix `P1` can be performed in four different ways, but only one composition is correct. Now that we know how to triangulate a point, we can add a check to see which one of the four camera matrices is valid. We shall skip the implementation details at this point, as they are featured in the sample code attached to the book.\n\nNext we are going to take a look at recovering more cameras looking at the same scene, and combining the 3D reconstruction results.\n\n# Reconstruction from many views\n\nNow that we know how to recover the motion and scene geometry from two cameras, it would seem trivial to get the parameters of additional cameras and more scene points simply by applying the same process. This matter is in fact not so simple as we can only get a reconstruction that is up-to-scale, and each pair of pictures gives us a different scale.\n\nThere are a number of ways to correctly reconstruct the 3D scene data from multiple views. One way is of resection or camera pose estimation, also known as **Perspective N-Point** ( **PNP** ), where we try to solve for the position of a new camera using the scene points we have already found. Another way is to triangulate more points and see how they fit into our existing scene geometry; this will tell us the position of the new camera by means of the **Iterative Closest Point** ( **ICP** ) procedure. In this chapter we will discuss using OpenCV's `solvePnP` functions to achieve the first method.\n\nThe first step we choose in this kind of reconstruction--incremental 3D reconstruction with camera resection--is to get a baseline scene structure. As we are going to look for the position of any new camera based on a known structure of the scene, we need to find an initial structure and a baseline to work with. We can use the method we previously discussed--for example, between the first and second frames--to get a baseline by finding the camera matrices (using the `FindCameraMatrices` function) and triangulate the geometry (using the `TriangulatePoints` function).\n\nHaving found an initial structure, we may continue; however, our method requires quite a bit of bookkeeping. First we should note that the `solvePnP` function needs two aligned vectors of 3D and 2D points. Aligned vectors mean that the ith position in one vector aligns with the ith position in the other. To obtain these vectors we need to find those points among the 3D points that we recovered earlier, which align with the 2D points in our new frame. A simple way to do this is to attach, for each 3D point in the cloud, a vector denoting the 2D points it came from. We can then use feature matching to get a matching pair.\n\nLet us introduce a new structure for a 3D point as follows:\n\n    struct CloudPoint {\n    cv::Point3d pt;\n    std::vector<int>index_of_2d_origin;\n    };\n\nIt holds, on top of the 3D point, an index to the 2D point inside the vector of 2D points that each frame has, which had contributed to this 3D point. The information for `index_of_2d_origin` must be initialized when triangulating a new 3D point, recording which cameras were involved in the triangulation. We can then use it to trace back from our 3D point cloud to the 2D point in each frame, as follows:\n\n    std::vector<CloudPoint> pcloud; \/\/our global 3D point cloud\n\n    \/\/check for matches between i'th frame and 0'th frame (and thus the current cloud)\n    std::vector<cv::Point3f> ppcloud;\n    std::vector<cv::Point2f> imgPoints;\n    vector<int> pcloud_status(pcloud.size(),0);\n\n    \/\/scan the views we already used (good_views)\n    for (set<int>::iterator done_view = good_views.begin(); done_view != good_views.end(); ++done_view) \n    {\n        int old_view = *done_view; \/\/a view we already used for reconstrcution\n        \/\/check for matches_from_old_to_working between <working_view>'th frame and <old_view>'th frame (and thus the current cloud)\n        std::vector<cv::DMatch> matches_from_old_to_working = matches_matrix[std::make_pair(old_view,working_view)];\n        \/\/scan the 2D-2D matched-points\n        for (unsigned int match_from_old_view=0; match_from_old_view<matches_from_old_to_working.size(); match_from_old_view++) {\n        \/\/ the index of the matching 2D point in <old_view>\n        int idx_in_old_view = matches_from_old_to_working[match_from_old_view].queryIdx;\n\n        \/\/scan the existing cloud to see if this point from <old_view> exists for (unsigned int pcldp=0; pcldp<pcloud.size(); pcldp++) {\n        \/\/ see if this 2D point from <old_view> contributed to this 3D point in the cloud\n        if (idx_in_old_view == pcloud[pcldp].index_of_2d_origin[old_view] && pcloud_status[pcldp] == 0) \/\/prevent duplicates\n          {\n            \/\/3d point in cloud\n            ppcloud.push_back(pcloud[pcldp].pt);\n            \/\/2d point in image <working_view>\n            Point2d pt_ = imgpts[working_view][matches_from_old_to_working[match_from_old_view].trainIdx].pt;\n            imgPoints.push_back(pt_);\n\n            pcloud_status[pcldp] = 1;\n            break;\n            }\n        }\n      }\n    }\n    cout<<\"found \"<<ppcloud.size() <<\" 3d-2d point correspondences\"<<endl;\n\nNow we have an aligned pairing of 3D points in the scene to the 2D points in a new frame, and we can use them to recover the camera position as follows:\n\n    cv::Mat_<double> t,rvec,R;\n    cv::solvePnPRansac(ppcloud, imgPoints, K, distcoeff, rvec, t, false);\n\n    \/\/get rotation in 3x3 matrix form\n    Rodrigues(rvec, R);\n\n    P1 = cv::Matx34d(R(0,0),R(0,1),R(0,2),t(0),\n    R(1,0),R(1,1),R(1,2),t(1),\n    R(2,0),R(2,1),R(2,2),t(2));\n\nNote that we are using the `solvePnPRansac` function rather than the `solvePnP` function as it is more robust to outliers. Now that we have a new `P1` matrix, we can simply use the `TriangulatePoints` function we defined earlier again and populate our point cloud with more 3D points.\n\nIn the following image we see an incremental reconstruction of the Fountain-P11 scene at <http:\/\/cvlab.epfl.ch\/~strecha\/multiview\/denseMVS.html>, starting from the 4th image. The top-left image is the reconstruction after four images were used; the participating cameras are shown as red pyramids with a white line showing the direction. The other images show how more cameras add more points to the cloud.\n\n# Refinement of the reconstruction\n\nOne of the most important parts of an SfM method is refining and optimizing the reconstructed scene, also known as the process of **Bundle Adjustment** ( **BA** ). This is an optimizing step where all the data we gathered is fitted to a monolithic model. Both the position of the 3D points and the positions of cameras are optimized, so reprojection errors are minimized (that is, approximated 3D points are projected on the image close to the position of originating 2D points). This process usually entails the solving of very big linear equations of the order of tens of thousands of parameters. The process may be slightly laborious, but the steps we took earlier will allow for an easy integration with the bundle adjuster. Some things that seemed strange earlier may become clear; for example, the reason we retain the origin 2D points for each 3D point in the cloud.\n\nOne implementation of a bundle adjustment algorithm is the **Simple Sparse Bundle Adjustment** ( **SSBA** ) library; we will choose it as our BA optimizer as it has a simple API. It requires only a few input arguments that we can create rather easily from our data structures. The key object we will use from SSBA is the `CommonInternalsMetricBundleOptimizer` function, which performs the optimization. It needs the camera parameters, the 3D point cloud, the 2D image points that corresponds to each point in the point cloud, and cameras looking at the scene. By now it should be straightforward to come up with these parameters. We should note that this method of BA assumes all images were taken by the same hardware, hence the common internals, other modes of operation may not assume this. We can perform Bundle Adjustment as follows:\n\n    voidBundleAdjuster::adjustBundle(\n    vector<CloudPoint>&pointcloud,\n    const Mat&cam_intrinsics,\n    conststd::vector<std::vector<cv::KeyPoint>>&imgpts,\n    std::map<int ,cv::Matx34d>&Pmats\n    )\n    {\n    int N = Pmats.size(), M = pointcloud.size(), K = -1;\n\n    cout<<\"N (cams) = \"<< N <<\" M (points) = \"<< M <<\" K (measurements) = \"<< K <<endl;\n\n    StdDistortionFunction distortion;\n\n    \/\/ intrinsic parameters matrix\n    Matrix3x3d KMat;\n    makeIdentityMatrix(KMat);\n    KMat[0][0] = cam_intrinsics.at<double>(0,0);\n    KMat[0][1] = cam_intrinsics.at<double>(0,1);\n    KMat[0][2] = cam_intrinsics.at<double>(0,2);\n    KMat[1][1] = cam_intrinsics.at<double>(1,1);\n    KMat[1][2] = cam_intrinsics.at<double>(1,2);\n\n    ...\n\n    \/\/ 3D point cloud\n    vector<Vector3d >Xs(M);\n    for (int j = 0; j < M; ++j)\n    {\n    Xs[j][0] = pointcloud[j].pt.x;\n    Xs[j][1] = pointcloud[j].pt.y;\n    Xs[j][2] = pointcloud[j].pt.z;\n    }\n    cout<<\"Read the 3D points.\"<<endl;\n\n    \/\/ convert cameras to BA datastructs\n    vector<CameraMatrix> cams(N);\n    for (inti = 0; i< N; ++i)\n    {\n    intcamId = i;\n    Matrix3x3d R;\n    Vector3d T;\n\n    Matx34d& P = Pmats[i];\n\n    R[0][0] = P(0,0); R[0][1] = P(0,1); R[0][2] = P(0,2); T[0] = P(0,3);\n    R[1][0] = P(1,0); R[1][1] = P(1,1); R[1][2] = P(1,2); T[1] = P(1,3);\n    R[2][0] = P(2,0); R[2][1] = P(2,1); R[2][2] = P(2,2); T[2] = P(2,3);\n\n    cams[i].setIntrinsic(Knorm);\n    cams[i].setRotation(R);\n    cams[i].setTranslation(T);\n    }\n    cout<<\"Read the cameras.\"<<endl;\n\n    vector<Vector2d > measurements;\n    vector<int> correspondingView;\n    vector<int> correspondingPoint;\n\n    \/\/ 2D corresponding points\n    for (unsigned int k = 0; k <pointcloud.size(); ++k)\n    {\n    for (unsigned int i=0; i<pointcloud[k].imgpt_for_img.size(); i++) {\n    if (pointcloud[k].imgpt_for_img[i] >= 0) {\n    int view = i, point = k;\n    Vector3d p, np;\n\n    Point cvp = imgpts[i][pointcloud[k].imgpt_for_img[i]].pt;\n    p[0] = cvp.x;\n    p[1] = cvp.y;\n    p[2] = 1.0;\n\n    \/\/ Normalize the measurements to match the unit focal length.\n    scaleVectorIP(1.0\/f0, p);\n    measurements.push_back(Vector2d(p[0], p[1]));\n    correspondingView.push_back(view);\n    correspondingPoint.push_back(point);\n    }\n    }\n    } \/\/ end for (k)\n\n    K = measurements.size();\n\n    cout<<\"Read \"<< K <<\" valid 2D measurements.\"<<endl;\n\n    ...\n\n    \/\/ perform the bundle adjustment\n    {\n    CommonInternalsMetricBundleOptimizeropt(V3D::FULL_BUNDLE_FOCAL_LENGTH_PP, inlierThreshold, K0, distortion, cams, Xs, measurements, correspondingView, correspondingPoint);\n\n    opt.tau = 1e-3;\n    opt.maxIterations = 50;\n    opt.minimize();\n\n    cout<<\"optimizer status = \"<<opt.status<<endl;\n    }\n\n    ...\n\n    \/\/extract 3D points\n    for (unsigned int j = 0; j <Xs.size(); ++j)\n    {\n    pointcloud[j].pt.x = Xs[j][0];\n    pointcloud[j].pt.y = Xs[j][1];\n    pointcloud[j].pt.z = Xs[j][2];\n    }\n    \/\/extract adjusted cameras\n    for (int i = 0; i< N; ++i) \n    {\n    Matrix3x3d R = cams[i].getRotation();\n    Vector3d T = cams[i].getTranslation();\n    Matx34d P; \n    P(0,0) = R[0][0]; P(0,1) = R[0][1]; P(0,2) = R[0][2]; P(0,3) = T[0]; \n    P(1,0) = R[1][0]; P(1,1) = R[1][1]; P(1,2) = R[1][2]; P(1,3) = T[1]; \n    P(2,0) = R[2][0]; P(2,1) = R[2][1]; P(2,2) = R[2][2]; P(2,3) = T[2]; \n    Pmats[i] = P; \n    }\n    }\n\nThis code, albeit long, is primarily about converting our internal data structures to and from SSBA's data structures, and invoking the optimization process.\n\nThe following image shows the effects of BA. The two images on the left are the points of the point cloud before adjustment, from two perspectives, and the images on the right show the optimized cloud. The change is quite dramatic, and many misalignments between points triangulated from different views are now mostly consolidated. We can also notice how the adjustment created a far better reconstruction of flat surfaces.\n\n# Visualizing 3D point clouds with PCL\n\nWhile working with 3D data, it is hard to quickly understand if a result is correct simply by looking at reprojection error measures or raw point information. On the other hand, if we look at the point cloud itself we can immediately verify whether it makes sense or there was an error. For visualization we will use an up-and-coming sister project for OpenCV, called the **Point Cloud Library** ( **PCL** ). It comes with many tools for visualizing and also analyzing point clouds, such as finding flat surfaces, matching point clouds, segmenting objects, and eliminating outliers. These tools are highly useful if our goal is not a point cloud but rather some higher-order information such as a 3D model.\n\nFirst, we should represent our cloud (essentially a list of 3D points) in PCL's data structures. This can be done as follows:\n\n    pcl::PointCloud<pcl::PointXYZRGB>::Ptr cloud;\n\n    void PopulatePCLPointCloud(const vector<Point3d>& pointcloud,\n    const std::vector<cv::Vec3b>& pointcloud_RGB\n    )\n    \/\/Populate point cloud\n    {\n    cout<<\"Creating point cloud...\";\n    cloud.reset(new pcl::PointCloud<pcl::PointXYZRGB>);\n\n    for (unsigned int i=0; i<pointcloud.size(); i++) {\n    \/\/ get the RGB color value for the point\n    Vec3b rgbv(255,255,255);\n    if (pointcloud_RGB.size() >= i) {\n    rgbv = pointcloud_RGB[i];\n    }\n\n    \/\/ check for erroneous coordinates (NaN, Inf, etc.)\n    if (pointcloud[i].x != pointcloud[i].x || isnan(pointcloud[i].x) ||\n    pointcloud[i].y != pointcloud[i].y || isnan(pointcloud[i].y) ||\n    pointcloud[i].z != pointcloud[i].z || isnan(pointcloud[i].z) ||\n    fabsf(pointcloud[i].x) > 10.0 ||\n    fabsf(pointcloud[i].y) > 10.0 ||\n    fabsf(pointcloud[i].z) > 10.0) {\n    continue;\n    }\n\n    pcl::PointXYZRGB pclp;\n\n    \/\/ 3D coordinates\n    pclp.x = pointcloud[i].x;\n    pclp.y = pointcloud[i].y;\n    pclp.z = pointcloud[i].z;\n\n    \/\/ RGB color, needs to be represented as an integer\n    uint32_t rgb = ((uint32_t)rgbv[2] << 16 | (uint32_t)rgbv[1] << 8 | (uint32_t)rgbv[0]);\n    pclp.rgb = *reinterpret_cast<float*>(&rgb);\n\n    cloud->push_back(pclp);\n    }\n\n    cloud->width = (uint32_t) cloud->points.size(); \/\/ number of points\n    cloud->height = 1; \/\/ a list of points, one row of data\n    }\n\nTo have a nice effect for the purpose of visualization, we can also supply color data as RGB values taken from the images. We can also apply a filter to the raw cloud that will eliminate points that are likely to be outliers, using the **statistical outlier removal** ( **SOR** ) tool as follows:\n\n    Void SORFilter() {\n\n    pcl::PointCloud<pcl::PointXYZRGB>::Ptr cloud_filtered (new pcl::PointCloud<pcl::PointXYZRGB>);\n\n    std::cerr<<\"Cloud before SOR filtering: \"<< cloud->width * cloud->height <<\" data points\"<<std::endl;\n\n    \/\/ Create the filtering object\n    pcl::StatisticalOutlierRemoval<pcl::PointXYZRGB>sor;\n    sor.setInputCloud (cloud);\n    sor.setMeanK (50);\n    sor.setStddevMulThresh (1.0);\n    sor.filter (*cloud_filtered);\n\n    std::cerr<<\"Cloud after SOR filtering: \"<<cloud_filtered->width * cloud_filtered->height <<\" data points \"<<std::endl;\n\n    copyPointCloud(*cloud_filtered,*cloud);\n    }\n\nThen we can use PCL's API for running a simple point cloud visualizer as follows:\n\n    Void RunVisualization(const vector<cv::Point3d>& pointcloud,\n    const std::vector<cv::Vec3b>& pointcloud_RGB) {\n    PopulatePCLPointCloud(pointcloud,pointcloud_RGB);\n    SORFilter();\n    copyPointCloud(*cloud,*orig_cloud);\n\n    pcl::visualization::CloudViewer viewer(\"Cloud Viewer\");\n\n    \/\/ run the cloud viewer\n    viewer.showCloud(orig_cloud,\"orig\");\n\n    while (!viewer.wasStopped ())\n       {\n    \/\/ NOP\n    }\n    }\n\nThe following image shows the output after the statistical outlier removal tool has been used. The image on the left-hand side is the original resultant cloud of the SfM, with the cameras location and a zoomed-in view of a particular part of the cloud. The image on the right-hand side shows the filtered cloud after the SOR operation. We can notice some stray points were removed, leaving a cleaner point cloud:\n\n# Using the example code\n\nWe can find the example code for SfM with the supporting material of this book. We will now see how we can build, run, and make use of it. The code makes use of CMake, a cross-platform build environment similar to Maven or SCons. We should also make sure we have all the following prerequisites to build the application:\n\n  * OpenCV v2.3 or higher\n  * PCL v1.6 or higher\n  * SSBA v3.0 or higher\n\nFirst we must set up the build environment. To that end, we may create a folder named `build` in which all build-related files will go; we will now assume all command-line operations are within the `build\/`folder, although the process is similar (up to the locations of the files) even if not using the `build` folder.\n\nWe should make sure CMake can find SSBA and PCL. If PCL was installed properly, there should not be a problem; however, we must set the correct location to find SSBA's prebuilt binaries via the **-DSSBA_LIBRARY_DIR=...** build parameter. If we are using Windows as the operating system, we can use Microsoft Visual Studio to build; therefore, we should run the following command:\n\n    **cmake -G \"Visual Studio 10\" -DSSBA_LIBRARY_DIR=..\/3rdparty\/SSBA-3.0\/build\/ ..**\n\nIf we are using Linux, Mac OS, or another Unix-like operating system, we execute the following command:\n\n    **cmake -G \"Unix Makefiles\" -DSSBA_LIBRARY_DIR=..\/3rdparty\/SSBA-3.0\/build\/ ..**\n\nIf we prefer to use XCode on Mac OS, execute the following command:\n\n    **cmake -G Xcode -DSSBA_LIBRARY_DIR=..\/3rdparty\/SSBA-3.0\/build\/ ..**\n\nCMake also has the ability to build macros for Eclipse, Codeblocks, and more. After CMake is done creating the environment, we are ready to build. If we are using a Unix-like system we can simply execute the make utility, else we should use our development environment's building process.\n\nAfter the build has finished, we should be left with an executable named `ExploringSfMExec`, which runs the SfM process. Running it with no arguments will result in the following: **USAGE: .\/ExploringSfMExec <path_to_images>**\n\nTo execute the process over a set of images, we should supply a location on the drive to find image files. If a valid location is supplied, the process should start and we should see the progress and debug information on the screen. The process will end with a display of the point cloud that arises from the images. Pressing the _1_ and _2_ keys will switch between the adjusted and non-adjusted point cloud.\n\n# Summary\n\nIn this chapter we have seen how OpenCV can help us approach Structure from Motion in a manner that is both simple to code and to understand. OpenCV's API contains a number of useful functions and data structures that make our lives easier and also assist in a cleaner implementation.\n\nHowever, the state-of-the-art SfM methods are far more complex. There are many issues we choose to disregard in favor of simplicity, and plenty more error examinations that are usually in place. Our chosen methods for the different elements of SfM can also be revisited. For one, H and Z propose a highly accurate triangulation method that minimizes the reprojection error in the image domain. Some methods even use the N-view triangulation once they understand the relationship between the features in multiple images.\n\nIf we would like to extend and deepen our familiarity with SfM, we will certainly benefit from looking at other open-source SfM libraries. One particularly interesting project is libMV, which implements a vast array of SfM elements that may be interchanged to get the best results. There is a great body of work from University of Washington that provides tools for many flavors of SfM (Bundler and VisualSfM). This work inspired an online product from Microsoft called PhotoSynth. There are many more implementations of SfM readily available online, and one must only search to find quite a lot of them.\n\nAnother important relationship we have not discussed in depth is that of SfM and Visual Localization and Mapping, better known in as **Simultaneous Localization and Mapping (SLAM)** methods. In this chapter we have dealt with a given dataset of images and a video sequence, and using SfM is practical in those cases; however, some applications have no prerecorded dataset and must bootstrap the reconstruction on the fly. This process is better known as **Mapping** , and it is done while we are creating a 3D map of the world, using feature matching and tracking in 2D, and after triangulation.\n\nIn the next chapter we will see how OpenCV can be used for extracting license plate numbers from images, using various techniques in machine learning.\n\n# References\n\n  *  _Multiple View Geometry in Computer Vision_ , _Richard Hartley and Andrew Zisserman_ , _Cambridge University Press_\n  *  _Triangulation_ , _Richard I. Hartley and Peter Sturm_ , _Computer vision and image understanding_ , _Vol. 68, pp_ _. 146-157_\n  * <http:\/\/cvlab.epfl.ch\/~strecha\/multiview\/denseMVS.html>\n  *  _On Benchmarking Camera Calibration and Multi-View Stereo for High Resolution Imagery_ , _C. Strecha_ , _W. von Hansen_ , _L._ _Van Gool_ , _P. Fua_ , _and U. Thoennes_ _sen_ , _CVPR_\n  * <http:\/\/www.inf.ethz.ch\/personal\/chzach\/opensource.html>\n  * <http:\/\/www.ics.forth.gr\/~lourakis\/sba\/>\n  * <http:\/\/code.google.com\/p\/libmv\/>\n  * <http:\/\/www.cs.washington.edu\/homes\/ccwu\/vsfm\/>\n  * <http:\/\/phototour.cs.washington.edu\/bundler\/>\n  * <http:\/\/photosynth.net\/>\n  * <http:\/\/en.wikipedia.org\/wiki\/Simultaneous_localization_and_mapping>\n  * <http:\/\/pointclouds.org>\n  * <http:\/\/www.cmake.org>\n\n# Chapter 5. Number Plate Recognition Using SVM and Neural Networks\n\nThis chapter introduces us to the steps needed to create an application for **Automatic Number Plate Recognition** ( **ANPR** ). There are different approaches and techniques based on different situations, for example, IR cameras, fixed car positions, light conditions, and so on. We can proceed to construct an ANPR application to detect automobile license plates in a photograph taken between 2-3 meters from a car, in ambiguous light condition, and with non-parallel ground with minor perspective distortions of the automobile's plate.\n\nThe main purpose of this chapter is to introduce us to image segmentation and feature extraction, pattern recognition basics, and two important pattern recognition algorithms **Support Vector Machines** and  **Artificial Neural Networks**. In this chapter, we will cover:\n\n  * ANPR\n  * Plate detection\n  * Plate recognition\n\n# Introduction to ANPR\n\n**Automatic Number Plate Recognition** ( **ANPR** ), also known as  **Automatic License-Plate Recognition** ( **ALPR** ),  **Automatic Vehicle Identification** ( **AVI** ), or **Car Plate Recognition** ( **CPR** ), is a surveillance method that uses **Optical Character Recognition** ( **OCR** ) and other methods such as segmentations and detection to read vehicle registration plates.\n\nThe best results in an ANPR system can be obtained with an  **infrared (IR)** camera, because the segmentation steps for detection and OCR segmentation are easy, clean, and minimize errors. This is due to the laws of light, the basic one being that the angle of incidence equals the angle of reflection; we can see this basic reflection when we see a smooth surface such as a plane mirror. Reflection off of rough surfaces such as paper leads to a type of reflection known as diffuse or scatter reflection. The majority of number plates have a special characteristic named retro-reflection--the surface of the plate is made with a material that is covered with thousands of tiny hemispheres that cause light to be reflected back to the source as we can see in the following figure:\n\nIf we use a camera with a filter coupled with a structured infrared light projector, we can retrieve just the infrared light and then we have a very high-quality image to segment and subsequently detect, and recognize the plate number that is independent of any light environment as shown in the following figure:\n\nWe do not use IR photographs in this chapter; we use regular photographs. We do this so that we do not obtain the best results and get a higher level of detection errors and higher false recognition rate as opposed to the results we would expect if we used an IR camera; however, the steps for both are the same.\n\nEach country has different license plate sizes and specifications; it is useful to know these specifications in order to get the best results and reduce errors. The algorithms used in this chapter are intended to explain the basics of ANPR and the specifications for license plates from Spain, but we can extend them to any country or specification.\n\nIn this chapter, we will work with license plates from Spain. In Spain, there are three different sizes and shapes of license plates; we will only use the most common (large) license plate which is 520 x 110 mm. Two groups of characters are separated by a 41 mm space and then a 14 mm width separates each individual character. The first group of characters has four numeric digits, and the second group has three letters without the vowels A, E, I, O, U, nor the letters \u00d1 or Q; all characters have dimensions of 45 x 77 mm.\n\nThis data is important for character segmentation since we can check both the character and blank spaces to verify that we get a character and no other image segment. The following is a figure of one such license plate:\n\n# ANPR algorithm\n\nBefore explaining the ANPR code, we need to define the main steps and tasks in the ANPR algorithm. ANPR is divided in two main steps: plate detection and plate recognition. Plate detection has the purpose of detecting the location of the plate in the whole camera frame. When a plate is detected in an image, the plate segment is passed to the second step--plate recognition--which uses an OCR algorithm to determine the alphanumeric characters on the plate.\n\nIn the next figure we can see the two main algorithm steps, plate detection and plate recognition. After these steps the program draws over the camera frame the plate's characters that have been detected. The algorithms can return bad results or even no result:\n\nIn each step shown in the previous figure, we will define three additional steps that are commonly used in pattern recognition algorithms:\n\n  1.  **Segmentation** : This step detects and removes each patch\/region of interest in the image.\n  2.  **Feature extraction** : This step extracts from each patch a set of characteristics.\n  3.  **Classification** : This step extracts each character from the plate recognition step or classifies each image patch into \"plate\" or \"no plate\" in the plate-detection step.\n\nThe following figure shows us the pattern recognition steps in the whole algorithm application:\n\nAside from the main application, whose purpose is to detect and recognize a car's license plate number, we will briefly explain two more tasks that are usually not explained:\n\n  * How to train a pattern recognition system\n  * How to evaluate such a system\n\nThese tasks, however, can be more important than the main application itself, because if we do not train the pattern recognition system correctly, our system can fail and not work correctly; different patterns need different types of training and evaluation. We need to evaluate our system in different environments, conditions, and with different features to get the best results. These two tasks are sometimes used together since different features can produce different results that we can see in the evaluation section.\n\n# Plate detection\n\nIn this step we have to detect  all the plates in the current camera frame. To do this task, we divide it in two main steps: segmentation and segment classification. The feature step is not explained because we use the image patch as a vector feature.\n\nIn the first step (segmentation), we apply different filters, morphological operations, contour algorithms, and validations to retrieve those parts of the image that could have a plate.\n\nIn the second step (classification), we apply a **Support Vector Machine** ( **SVM** ) classifier to each image patch--our feature. Before creating our main application we train with two different classes--plate and non-plate. We work with parallel frontal-view color images that are 800 pixels wide and taken 2-4 meters from a car. These requirements are important to ensure correct segmentations. We can perform detection if we create a multi-scale image algorithm.\n\nIn the next image we have shown all the processes involved in plate detection:\n\n  * Sobel filter\n  * Threshold operation\n  * Close morphologic operation\n  * Mask of one filled area\n  * Possible detected plates marked in red (features images)\n  * Detected plates after the SVM classifier\n\n## Segmentation\n\nSegmentation is  the process of dividing an image into multiple segments. This process is to simplify the image for analysis and make feature extraction easier.\n\nOne important feature of plate segmentation is the high number of vertical edges in a license plate assuming that the image was taken frontally, and the plate is not rotated and is without perspective distortion. This feature can be exploited during the first segmentation step to eliminate regions that don't have any vertical edges.\n\nBefore finding vertical edges, we need to convert the color image to a grayscale image, (because color can't help us in this task), and remove possible noise generated by the camera or other ambient noise. We will apply a Gaussian blur of 5 x 5 and remove noise. If we don't apply a noise-removal method, we can get a lot of vertical edges that produce a falied detection.\n\n    \/\/convert image to gray\n    Mat img_gray;\n    cvtColor(input, img_gray, CV_BGR2GRAY);\n    blur(img_gray, img_gray, Size(5,5));\n\nTo find the vertical edges, we will use a Sobel filter and find the first horizontal derivative. The derivative is a mathematical function that allows us to find the vertical edges on an image. The definition of a Sobel function in OpenCV is:\n\n    void Sobel(InputArray src, OutputArray dst, int ddepth, int xorder, int yorder, int ksize=3, double scale=1, double delta=0, int borderType=BORDER_DEFAULT )\n\nHere, `ddepth` is the destination image depth, `xorder` is the order of the derivative by x, `yorder` is the order of derivative by y, `ksize` is the kernel size of either 1, 3, 5, or 7, `scale` is an optional factor for computed derivative values, `delta` is an optional value added to the result, and `borderType` is the pixel interpolation method.\n\nFor our case we can use a `xorder=1`, `yorder=0`, and a `ksize=3`:\n\n    \/\/Find vertical lines. Car plates have high density of vertical lines\n    Mat img_sobel;\n    Sobel(img_gray, img_sobel, CV_8U, 1, 0, 3, 1, 0);\n\nAfter a Sobel filter, we apply a threshold filter to obtain a binary image with a threshold value obtained through Otsu's method. Otsu's algorithm needs an 8-bit input image and Otsu's method automatically determines the optimal threshold value:\n\n    \/\/threshold image\n    Mat img_threshold;\n    threshold(img_sobel, img_threshold, 0, 255, CV_THRESH_OTSU+CV_THRESH_BINARY);\n\nTo define Otsu's method in the `threshold` function, if we combine the type parameter with the `CV_THRESH_OTSU` value, then the threshold value parameter is ignored.\n\n### Tip\n\nWhen the value of `CV_THRESH_OTSU` is defined, the threshold function returns the optimal threshold value obtained by the Otsu's algorithm.\n\nBy applying a close  morphological operation, we can remove blank spaces between each vertical edge line, and connect all regions that have a high number of edges. In this step we have the possible regions that can contain plates.\n\nFirst we define our structural element to use in our morphological operation. We will use the `getStructuringElement` function to define a structural rectangular element with a 17 x 3 dimension size in our case; this may be different in other image sizes:\n\n    Mat element = getStructuringElement(MORPH_RECT, Size(17, 3));\n\nAnd use this structural element in a close morphological operation using the `morphologyEx` function:\n\n    morphologyEx(img_threshold, img_threshold, CV_MOP_CLOSE, element);\n\nAfter applying these functions, we have regions in the image that could contain a plate; however, most of the regions will not contain license plates. These regions can be split with a connected-component analysis or by using the `findContours` function. This last function retrieves the contours of a binary image with different methods and results. We only need to get the external contours with any hierarchical relationship and any polygonal approximation results:\n\n    \/\/Find contours of possibles plates\n    vector< vector< Point> > contours;\n    findContours(img_threshold,\n                contours,           \/\/ a vector of contours\n                CV_RETR_EXTERNAL,   \/\/ retrieve the external contours\n                CV_CHAIN_APPROX_NONE); \/\/ all pixels of each contour\n\nFor each contour detected, extract the bounding rectangle of minimal area. OpenCV brings up the `minAreaRect` function for this task. This function returns a rotated rectangle class called `RotatedRect`. Then using a vector iterator over each contour, we can get the rotated rectangle and make some preliminary validations before we classify each region:\n\n    \/\/Start to iterate to each contour found\n    vector<vector<Point> >::iterator itc= contours.begin();\n    vector<RotatedRect> rects;\n\n    \/\/Remove patch that has  no inside limits of aspect ratio and area.\t\n    while (itc!=contours.end()) {\n    \/\/Create bounding rect of object\n      RotatedRect mr= minAreaRect(Mat(*itc));\n      if( !verifySizes(mr)){\n        itc= contours.erase(itc);\n      }else{\n      ++itc;\n      rects.push_back(mr);\n      }\n    }\n\nWe make  basic validations about the regions detected based on its area and aspect ratio. We only consider that a region can be a plate if the aspect ratio is approximately 520\/110 = 4.727272 (plate width divided by plate height) with an error margin of 40 percent and an area based on a minimum of 15 pixels and maximum of 125 pixels for the height of the plate. These values are calculated depending on the image sizes and camera position:\n\n    bool DetectRegions::verifySizes(RotatedRect candidate ){\n\n      float error=0.4;\n    \/\/Spain car plate size: 52x11 aspect 4,7272\n      const float aspect=4.7272;\n    \/\/Set a min and max area. All other patches are discarded\n      int min= 15*aspect*15; \/\/ minimum area\n      int max= 125*aspect*125; \/\/ maximum area\n    \/\/Get only patches that match to a respect ratio.\n      float rmin= aspect-aspect*error;\n      float rmax= aspect+aspect*error;\n\n      int area= candidate.size.height * candidate.size.width;\n      float r= (float)candidate.size.width \/ (float)candidate.size.height;\n      if(r<1)\n      r= 1\/r;\n\n      if(( area < min || area > max ) || ( r < rmin || r > rmax )){\n        return false;\n      }else{\n      return true;\n      }\n    }\n\nWe can make  more improvements using the license plate's white background property. All plates have the same background color and we can use a flood fill algorithm to retrieve the rotated rectangle for precise cropping.\n\nThe first step to crop the license plate is to get several seeds near the last rotated rectangle center. Then get the minimum size of plate between the width and height, and use it to generate random seeds near the patch center.\n\nWe want to select the white region and we need several seeds to touch at least one white pixel. Then for each seed, we use a `floodFill` function to draw a new mask image to store the new closest cropping region:\n\n    for(int i=0; i< rects.size(); i++){\n    \/\/For better rect cropping for each possible box\n    \/\/Make floodfill algorithm because the plate has white background\n    \/\/And then we can retrieve more clearly the contour box\n    circle(result, rects[i].center, 3, Scalar(0,255,0), -1);\n    \/\/get the min size between width and height\n    float minSize=(rects[i].size.width < rects[i].size.height)?rects[i].size.width:rects[i].size.height;\n    minSize=minSize-minSize*0.5;\n    \/\/initialize rand and get 5 points around center for floodfill algorithm\n    srand ( time(NULL) );\n    \/\/Initialize floodfill parameters and variables\n    Mat mask;\n    mask.create(input.rows + 2, input.cols + 2, CV_8UC1);\n    mask= Scalar::all(0);\n    int loDiff = 30;\n    int upDiff = 30;\n    int connectivity = 4;\n    int newMaskVal = 255;\n    int NumSeeds = 10;\n    Rect ccomp;\n    int flags = connectivity + (newMaskVal << 8 ) + CV_FLOODFILL_FIXED_RANGE + CV_FLOODFILL_MASK_ONLY;\n    for(int j=0; j<NumSeeds; j++){\n      Point seed;\n      seed.x=rects[i].center.x+rand()%(int)minSize-(minSize\/2);\n      seed.y=rects[i].center.y+rand()%(int)minSize-(minSize\/2);\n      circle(result, seed, 1, Scalar(0,255,255), -1);\n      int area = floodFill(input, mask, seed, Scalar(255,0,0), &ccomp,   Scalar(loDiff, loDiff, loDiff), Scalar(upDiff, upDiff, upDiff), flags);\n      }\n\nThe  `floodFill` function  fills a connected component with color into a mask image starting from a seed point, and sets maximal lower and upper brightness\/color difference between the pixel to fill and the pixel neighbors or seed pixel:\n\n    int floodFill(InputOutputArray image, InputOutputArray mask, Point seed, Scalar newVal, Rect* rect=0, Scalar loDiff=Scalar(), Scalar upDiff=Scalar(), int flags=4 )\n\nThe `newVal` parameter is the new color we want to put into the image when filling. Parameters `loDiff` and `upDiff` are the maximal lower and maximal upper brightness\/color difference between the pixel to fill and the pixel neighbors or seed pixel.\n\nThe `flag` parameter is a combination of:\n\n  * Lower bits: These bits contain connectivity value, 4 (by default), or 8, used within the function. Connectivity determines which neighbors of a pixel are considered.\n  * Upper bits: These can be 0 or a combination of the following values: `CV_FLOODFILL_FIXED_RANGE` and `CV_FLOODFILL_MASK_ONLY`.\n\n`CV_FLOODFILL_FIXED_RANGE` sets the difference between the current pixel and the seed pixel. `CV_FLOODFILL_MASK_ONLY` will only fill the image mask and not change the image itself.\n\nOnce we have a crop mask, we get a minimal area rectangle from the image-mask points and check the valid size again. For each mask, a white pixel gets the position and uses the `minAreaRect` function for retrieving the closest crop region:\n\n    \/\/Check new floodfill mask match for a correct patch.\n    \/\/Get all points detected for minimal rotated Rect\n    vector<Point> pointsInterest;\n    Mat_<uchar>::iterator itMask= mask.begin<uchar>();\n    Mat_<uchar>::iterator end= mask.end<uchar>();\n    for( ; itMask!=end; ++itMask)\n      if(*itMask==255)\n      pointsInterest.push_back(itMask.pos());\n      RotatedRect minRect = minAreaRect(pointsInterest);\n      if(verifySizes(minRect)){\n   ...\n\nNow that the segmentation  process is finished and we have valid regions, we can crop each detected region, remove any possible rotation, crop the image region, resize the image, and equalize the light of cropped image regions.\n\nFirst, we need to generate the transform matrix with `getRotationMatrix2D` to remove possible rotations in the detected region. We need to pay attention to height, because the `RotatedRect` class can be returned and rotated at 90 degrees, so we have to check the rectangle aspect, and if it is less than 1 then rotate it by 90 degrees:\n\n    \/\/Get rotation matrix\n    float r= (float)minRect.size.width \/ (float)minRect.size.height;\n    float angle=minRect.angle;\n    if(r<1)\n      angle=90+angle;\n      Mat rotmat= getRotationMatrix2D(minRect.center, angle,1);\n\nWith the transform matrix, we can now rotate the input image by an affine transformation (an affine transformation in geometry is a transformation that takes parallel lines to parallel lines) with the `warpAffine` function where we set the input and destination images, the transform matrix, the output size (same as the input in our case), and which interpolation method to use. We can define the border method and border value if needed:\n\n    \/\/Create and rotate image\n    Mat img_rotated;\n    warpAffine(input, img_rotated, rotmat, input.size(), CV_INTER_CUBIC);\n\nAfter we rotate the image, we crop the image with `getRectSubPix`, which crops and copies an image portion of given width and height centered in a point. If the image was rotated, we need to change the width and height sizes with the C++ `swap` function.\n\n    \/\/Crop image\n    Size rect_size=minRect.size;\n    if(r < 1)\n    swap(rect_size.width, rect_size.height);\n    Mat img_crop;\n    getRectSubPix(img_rotated, rect_size, minRect.center, img_crop);\n\nCropped images are not good for use in training and classification since they do not have the same size. Also, each image contains different light conditions, increasing their relative differences. To resolve this, we resize all images to the same width and height and apply light histogram equalization:\n\n    Mat resultResized;\n    resultResized.create(33,144, CV_8UC3);\n    **resize(img_crop, resultResized, resultResized.size(), 0, 0, INTER_CUBIC);**\n    \/\/Equalize cropped image\n    Mat grayResult;\n    cvtColor(resultResized, grayResult, CV_BGR2GRAY);\n    blur(grayResult, grayResult, Size(3,3));\n    **equalizeHist(grayResult, grayResult);**\n\nFor each detected region, we store the cropped image and its position in a vector:\n\n    output.push_back(Plate(grayResult,minRect.boundingRect()));\n\n## Classification\n\nAfter we preprocess and  segment all possible parts of an image, we now need to decide if each segment is (or is not) a license plate. To do this, we will use a  **Support Vector Machine** ( **SVM** ) algorithm.\n\nA Support Vector Machine is a pattern recognition algorithm included in a family of supervised-learning algorithms originally created for binary classification. Supervised learning is machine-learning algorithm that learns through the use of labeled data. We need to train the algorithm with an amount of data that is labeled; each data set needs to have a class.\n\nThe SVM creates one or more hyperplanes that are used to discriminate each class of the data.\n\nA classic example is a 2D point set that defines two classes; the SVM searches the optimal line that differentiates each class:\n\nThe first task before any classification is to train our classifier; this job is done prior to beginning the main application and it's named offline training. This is not an easy job because it requires a sufficient amount of data to train the system, but a bigger dataset does not always imply the best results. In our case, we do not have enough data due to the fact that there are no public license-plate databases. Because of this, we need to take hundreds of car photos and then preprocess and segment all the photos.\n\nWe trained our system with 75 license-plate images and 35 images without license plates of 144 x 33 pixels. We can see a sample of this data in the following image. This is not a large dataset, but it is sufficient enough to get decent results for our requirements. In a real application, we would need to train with more data:\n\nTo easily  understand how machine learning works, we proceed to use image pixel features of the classifier algorithm (keep in mind, there are better methods and features to train an SVM, such as Principal Components Analysis, Fourier transform, texture analysis, and so on).\n\nWe need to create the images to train our system using the `DetectRegions` class and set the `savingRegions` variable to true in order to save the images. We can use the `segmentAllFiles.sh` bash script to repeat the process on all image files under a folder. This can be taken from the source code of this book.\n\nTo make this easier, we store all image training data that is processed and prepared, into an XML file for use directly with the SVM function. The `trainSVM.cpp` application creates this file using the folders and number of image files.\n\n### Tip\n\nTraining data for a machine-learning OpenCV algorithm is stored in an _N_ x _M_ matrix with _N_ samples and _M_ features. Each data set is saved as a row in the training matrix.\n\nThe classes are stored in another matrix with _N_ x 1 size, where each class is identified by a float number.\n\nOpenCV has an easy way to manage a data file in XML or JSON format with the `FileStorage` class, this class lets us store and read OpenCV variables and structures or our custom variables. With this function, we can read the training-data matrix and training classes and save it in `SVM_TrainingData` and `SVM_Classes`:\n\n    FileStorage fs;\n    fs.open(\"SVM.xml\", FileStorage::READ);\n    Mat SVM_TrainingData;\n    Mat SVM_Classes;\n    fs[\"TrainingData\"] >> SVM_TrainingData;\n    fs[\"classes\"] >> SVM_Classes;\n\nNow we need to set the SVM parameters that define the basic parameters to use in an SVM algorithm; we will use the `CvSVMParams` structure to define it. It is a mapping done to the training data to improve its resemblance to a linearly separable set of data. This mapping consists of increasing the dimensionality of the data and is done efficiently using a kernel function. We choose here the `CvSVM::LINEAR` types which means that no mapping is done:\n\n    \/\/Set SVM params\n    CvSVMParams SVM_params;\n    SVM_params.kernel_type = CvSVM::LINEAR;\n\nWe then create and train our classifier. OpenCV defines the `CvSVM` class for the Support Vector Machine algorithm and we initialize it with the training data, classes, and parameter data:\n\n    CvSVM svmClassifier(SVM_TrainingData, SVM_Classes, Mat(), Mat(), SVM_params);\n\nOur  classifier is ready to predict a possible cropped image using the `predict` function of our SVM class; this function returns the class identifier `i`. In our case, we label a plate class with `1` and no plate class with `0`. Then for each detected region that can be a plate, we use SVM to classify it as a plate or no plate, and save only the correct responses. The following code is a part of main application, that is called online processing:\n\n    vector<Plate> plates;\n    for(int i=0; i< possible_regions.size(); i++)\n    {\n      Mat img=possible_regions[i].plateImg;\n      Mat p= img.reshape(1, 1);\/\/convert img to 1 row m features\n      p.convertTo(p, CV_32FC1);\n    **int response = (int)svmClassifier.predict( p );**\n      if(response==1)\n        plates.push_back(possible_regions[i]);\n    }\n\n# Plate recognition\n\nThe second step in license  plate recognition aims to retrieve the characters of the license plate with optical character recognition. For each detected plate, we proceed to segment the plate for each character, and use an **Artificial Neural Network** ( **ANN** ) machine-learning algorithm to recognize the character. Also in this section we will learn how to evaluate a classification algorithm.\n\n## OCR segmentation\n\nFirst, we obtain a plate image patch as the input to the  segmentation OCR function with an equalized histogram, we then need to apply a threshold filter and use this threshold image as the input of a  **Find contours** algorithm; we can see this process in the next figure:\n\nThis segmentation process is coded as:\n\n    Mat img_threshold;\n    threshold(input, img_threshold, 60, 255, CV_THRESH_BINARY_INV);\n    if(DEBUG)\n      imshow(\"Threshold plate\", img_threshold);\n    Mat img_contours;\n    img_threshold.copyTo(img_contours);\n    \/\/Find contours of possibles characters\n    vector< vector< Point> > contours;\n    findContours(img_contours,\n                contours,            \/\/ a vector of contours\n                CV_RETR_EXTERNAL,    \/\/ retrieve the external contours\n                CV_CHAIN_APPROX_NONE); \/\/ all pixels of each contour\n\nWe use the `CV_THRESH_BINARY_INV` parameter to invert the threshold output by turning the white input values black and black input values white. This is needed to get the contours of each character, because the contours algorithm looks for white pixels.\n\nFor each detected contour, we can make a size verification and remove all regions where the size is smaller or the aspect is not correct. In our case, the characters have a 45\/77 aspect, and we can accept a 35 percent error of aspect for rotated or distorted characters. If an area is higher than 80 percent, we consider that region to be a black block, and not a character. For counting the area, we can use the `countNonZero` function that counts the number of pixels with a value higher than 0:\n\n    bool OCR::verifySizes(Mat r)\n    {\n      \/\/Char sizes 45x77\n      float aspect=45.0f\/77.0f;\n      float charAspect= (float)r.cols\/(float)r.rows;\n      float error=0.35;\n      float minHeight=15;\n      float maxHeight=28;\n      \/\/We have a different aspect ratio for number 1, and it can be  \n      \/\/~0.2\n      float minAspect=0.2;\n      float maxAspect=aspect+aspect*error;\n      \/\/area of pixels\n      float area=countNonZero(r);\n      \/\/bb area\n      float bbArea=r.cols*r.rows;\n      \/\/% of pixel in area\n      float percPixels=area\/bbArea;\n      if(percPixels < 0.8 && charAspect > minAspect && charAspect <\n      maxAspect && r.rows >= minHeight && r.rows < maxHeight)\n      return true;\n      else\n      return false;\n    }\n\nIf a segmented  character is verified, we have to preprocess it to set the same size and position for all characters and save it in a vector with the auxiliary `CharSegment` class. This class saves the segmented character image and the position that we need to order the characters because the Find Contour algorithm does not return the contours in the required order.\n\n## Feature extraction\n\nThe next step for each  segmented character is to extract the features for training and classifying the Artificial Neural Network algorithm.\n\nUnlike the plate detection feature-extraction step that is used in SVM, we don't use all of the image pixels; we will apply more common features used in optical character recognition containing horizontal and vertical accumulation histograms and a low-resolution image sample. We can see this feature more graphically in the next image, where each image has a low-resolution 5 x 5 and the histogram accumulations:\n\nFor each character, we count the number of pixels in a row or column with a non-zero value using the `countNonZero` function and store it in a new data matrix called `mhist`. We normalize it by looking for the maximum value in the data matrix using the `minMaxLoc` function and divide all elements of `mhist` by the maximum value with the `convertTo` function. We create the `ProjectedHistogram` function to create the accumulation histograms that have as input a binary image and the type of histogram we need--horizontal or vertical:\n\n    Mat OCR::ProjectedHistogram(Mat img, int t)\n    {\n      int sz=(t)?img.rows:img.cols;\n      Mat mhist=Mat::zeros(1,sz,CV_32F);\n\n      for(int j=0; j<sz; j++){\n        Mat data=(t)?img.row(j):img.col(j);\n        mhist.at<float>(j)=countNonZero(data);\n      }\n\n    \/\/Normalize histogram\n    double min, max;\n    minMaxLoc(mhist, &min, &max);\n\n    if(max>0)\n      mhist.convertTo(mhist,-1 , 1.0f\/max, 0);\n\n      return mhist;\n    }\n\nOther  features use a low-resolution sample image. Instead of using the whole character image, we create a low-resolution character, for example 5 x 5. We train the system with 5 x 5, 10 x 10, 15 x 15, and 20 x 20 characters, and then evaluate which one returns the best result so that we can use it in our system. Once we have all the features, we create a matrix of _M_ columns by one row where the columns are the features:\n\n    Mat OCR::features(Mat in, int sizeData)\n    {\n    \/\/Histogram features\n    **Mat vhist=ProjectedHistogram(in,VERTICAL);**\n    **Mat hhist=ProjectedHistogram(in,HORIZONTAL);**\n    \/\/Low data feature\n    **Mat lowData;**\n    **resize(in, lowData, Size(sizeData, sizeData) );**\n      int numCols=vhist.cols + hhist.cols + lowData.cols * \n      lowData.cols;\n      Mat out=Mat::zeros(1,numCols,CV_32F);\n      \/\/Assign values to feature\n      int j=0;\n      for(int i=0; i<vhist.cols; i++)\n      {\n        out.at<float>(j)=vhist.at<float>(i); \n        j++;\n      }\n      for(int i=0; i<hhist.cols; i++)\n      {\n      out.at<float>(j)=hhist.at<float>(i);\n      j++;\n      }\n      for(int x=0; x<lowData.cols; x++)\n      {\n        for(int y=0; y<lowData.rows; y++)\n        {\n          out.at<float>(j)=(float)lowData.at<unsigned char>(x,y);\n          j++;\n        }\n      }\n      return out;\n    }\n\n## OCR classification\n\nIn the classification step, we  use an Artificial Neural Network machine-learning algorithm. More specifically, a **Multi-Layer Perceptron** ( **MLP** ), which is the most commonly used ANN algorithm.\n\nMLP consists of a network of neurons with an input layer, output layer, and one or more hidden layers. Each layer has one or more neurons connected with the previous and next layer.\n\nThe following example represents a 3-layer perceptron (it is a binary classifier that maps a real-valued vector input to a single binary value output) with three inputs, two outputs, and the hidden layer including five neurons:\n\nAll neurons in an  MLP are similar and each one has several inputs (the previous linked neurons) and several output links with the same value (the next linked neurons). Each neuron calculates the output value as a sum of the weighted inputs plus a bias term and is transformed by a selected activation function:\n\nThere are three widely used activation functions: Identity, Sigmoid, and Gaussian; the most common and default activation function is the Sigmoid function. It has an alpha and beta value set to 1:\n\nAn  ANN-trained network has a vector of input with features. It passes the values to the hidden layer and computes the results with the weights and activation function. It passes outputs further downstream until it gets the output layer that has the number of neuron classes.\n\nThe weight of each layer, synapses, and neuron is computed and learned by training the ANN algorithm. To train our classifier, we create two matrices of data as we did in the SVM training, but the training labels are a bit different. Instead of an _N_ x 1 matrix where _N_ stands for training data rows and 1 is the column, we use the label number identifier. We have to create an _N_ x _M_ matrix where _N_ is the training\/samples data and _M_ is the classes (10 digits + 20 letters in our case), and set 1 in a position ( _i_ , _j_ ) if the data row _i_ is classified with class _j_.\n\nWe create  an `OCR::train` function to create all the needed matrices and train our system, with the training data matrix, classes matrix, and the number of hidden neurons in the hidden layers. The training data is loaded from an XML file just as we did for the SVM training.\n\nWe have to define the number of neurons in each layer to initialize the ANN class. For our sample, we only use one hidden layer, then we define a matrix of 1 row and 3 columns. The first column position is the number of features, the second column position is the number of hidden neurons in the hidden layer, and the third column position is the number of classes.\n\nOpenCV defines a `CvANN_MLP` class for ANN. With the `create` function, we can initiate the class by defining the number of layers and neurons, the activation function, and the `alpha` and `beta` parameters:\n\n    void OCR::train(Mat TrainData, Mat classes, int nlayers)\n    {\n      Mat layerSizes(1,3,CV_32SC1);\n      layerSizes.at<int>(0)= TrainData.cols;\n      layerSizes.at<int>(1)= nlayers;\n      layerSizes.at<int>(2)= numCharacters;\n      ann.create(layerSizes, CvANN_MLP::SIGMOID_SYM, 1, 1); \/\/ann is  \n      global class variable\n\n      \/\/Prepare trainClasses\n      \/\/Create a mat with n trained data by m classes\n      Mat trainClasses;\n      trainClasses.create( TrainData.rows, numCharacters, CV_32FC1 );\n      for( int i = 0; i <  trainClasses.rows; i++ )\n      {\n        for( int k = 0; k < trainClasses.cols; k++ )\n        {\n        \/\/If class of data i is same than a k class\n        if( k == classes.at<int>(i) )\n        trainClasses.at<float>(i,k) = 1;\n        else\n        trainClasses.at<float>(i,k) = 0;\n        }\n      }\n      Mat weights( 1, TrainData.rows, CV_32FC1, Scalar::all(1) );\n\n      \/\/Learn classifier\n      ann.train( TrainData, trainClasses, weights );\n      trained=true;\n    }\n\nAfter training,  we can classify any segmented plate feature using the `OCR::classify` function:\n\n    int OCR::classify(Mat f)\n    {\n      int result=-1;\n      Mat output(1, numCharacters, CV_32FC1);\n    **ann.predict(f, output);**\n      Point maxLoc;\n      double maxVal;\n    **minMaxLoc(output, 0, &maxVal, 0, &maxLoc);**\n      \/\/We need to know where in output is the max val, the x (cols) is \n      \/\/the class.\n\n      return maxLoc.x;\n    }\n\nThe `CvANN_MLP` class uses the `predict` function for classifying a feature vector in a class. Unlike the SVM `classify` function, the ANN's `predict` function returns a row with the size equal to the number of classes with the probability of belonging to the input feature of each class.\n\nTo get the best result, we can use the `minMaxLoc` function to get the maximum and minimum response and the position in the matrix. The class of our character is specified by the x position of a higher value:\n\nTo finish  each plate detected, we order its characters and return a string using the `str()` function of the `Plate` class and we can draw it on the original image:\n\n    string licensePlate=plate.str();\n    rectangle(input_image, plate.position, Scalar(0,0,200));\n    putText(input_image, licensePlate, Point(plate.position.x, plate.position.y), CV_FONT_HERSHEY_SIMPLEX, 1, Scalar(0,0,200),2);\n\n## Evaluation\n\nOur project is finished, but when we train a machine-learning algorithm like OCR for example, we need to know the best features and parameters to use and how to correct the classification, recognition, and detection errors in our system.\n\nWe need to evaluate our system with different situations and parameters, and evaluate the errors produced, and get the best parameters that minimize those errors.\n\nIn this chapter, we evaluated the OCR task with the following variables: the size of low-level resolution image features and the number of hidden neurons in the hidden layer.\n\nWe have created the `evalOCR.cpp` application where we use the XML training data file generated by the `trainOCR.cpp` application. The `OCR.xml` file contains the training data matrix for 5 x 5, 10 x10, 15 x 15, and 20 x 20 downsampled image features.\n\n    Mat classes;\n    Mat trainingData;\n    \/\/Read file storage.\n    FileStorage fs;\n    fs.open(\"OCR.xml\", FileStorage::READ);\n    fs[data] >> trainingData;\n    fs[\"classes\"] >> classes;\n\nThe evaluation application gets each downsampled matrix feature and gets 100 random rows for training, as well as other rows for testing the ANN algorithm and checking the error.\n\nBefore training the system, we test each random sample and check if the response is correct. If the response is not correct, we increment the error-counter variable and then divide by the number of samples to evaluate. This indicates the error ratio between 0 and 1 for training with random data:\n\n    float test(Mat samples, Mat classes)\n    {\n      float errors=0;\n      for(int i=0; i<samples.rows; i++)\n      {\n        int result= ocr.classify(samples.row(i));\n        if(result!= classes.at<int>(i))\n        errors++;\n      }\n      return errors\/samples.rows;\n    }\n\nThe application returns the output command-line error ratio for each sample size. For a good evaluation, we need to train the application with different random training rows; this produces different test error values, then we can add up all errors and make an average. To do this task, we create the following `bash` Unix script to automate it:\n\n    #!\/bin\/bash\n    echo \"#ITS \\t 5 \\t 10 \\t 15 \\t 20\" > data.txt\n    folder=$(pwd)\n\n    for numNeurons in 10 20 30 40 50 60 70 80 90 100 120 150 200 500\n    do\n      s5=0;\n      s10=0;\n      s15=0;\n      s20=0;\n      for j  in {1..100}\n      do\n        echo $numNeurons $j\n        a=$($folder\/build\/evalOCR $numNeurons TrainingDataF5)\n        s5=$(echo \"scale=4; $s5+$a\" | bc -q 2>\/dev\/null)\n\n        a=$($folder\/build\/evalOCR $numNeurons TrainingDataF10)\n        s10=$(echo \"scale=4; $s10+$a\" | bc -q 2>\/dev\/null)\n\n        a=$($folder\/build\/evalOCR $numNeurons TrainingDataF15)\n        s15=$(echo \"scale=4; $s15+$a\" | bc -q 2>\/dev\/null)\n\n        a=$($folder\/build\/evalOCR $numNeurons TrainingDataF20)\n        s20=$(echo \"scale=4; $s20+$a\" | bc -q 2>\/dev\/null)\n      done\n\n      echo \"$i \\t $s5 \\t $s10 \\t $s15 \\t $s20\"\n      echo \"$i \\t $s5 \\t $s10 \\t $s15 \\t $s20\" >> data.txt\n    done\n\nThis script saves a `data.txt` file that contains all results for each size and neuron-hidden layer number. This file can be used for plotting with gnuplot. We can see the result in the following figure:\n\nWe can see that the lowest error is under 8 percent and is using 20 neurons in a hidden layer and characters' features extracted from a downscaled 10 x 10 image patch.\n\n# Summary\n\nIn this chapter, we learned how an Automatic License Plate Recognition program works, and its two important steps: plate localization and plate recognition.\n\nIn the first step we learned how to segment an image looking for patches where we can have a plate, and how to use a simple heuristics and Support Vector Machine algorithm to make a binary classification for patches with plates and no plates.\n\nIn the second step we learned how to segment with the Find Contours algorithm, extract feature vector from each character, and use an Artificial Neural Network to classify each feature in a character class.\n\nWe also learned how to evaluate a machine algorithm with training using random samples and evaluate it using different parameters and features.\n\n# Chapter 6. Non-rigid Face Tracking\n\nNon-rigid face tracking, which is the estimation of a quasi-dense set of facial features in each frame of a video stream, is a difficult problem for which modern approaches borrow ideas from a number of related fields, including computer vision, computational geometry, machine learning, and image processing. Non-rigidity here refers to the fact that relative distances between facial features vary between facial expression and across the population, and is distinct from face detection and tracking, which aims only to find the location of the face in each frame, rather than the configuration of facial features. Non-rigid face tracking is a popular research topic that has been pursued for over two decades, but it is only recently that various approaches have become robust enough, and processors fast enough, which makes the building of commercial applications possible.\n\nAlthough commercial-grade face tracking can be highly sophisticated and pose a challenge even for experienced computer vision scientists, in this chapter we will see that a face tracker that performs reasonably well under constrained settings can be devised using modest mathematical tools and OpenCV's substantial functionality in linear algebra, image processing, and visualization. This is particularly the case when the person to be tracked is known ahead of time, and training data in the form of images and landmark annotations are available. The techniques described henceforth will act as a useful starting point and a guide for further pursuits towards a more elaborate face-tracking system.\n\nAn outline of this chapter is as follows:\n\n  *  **Overview** : This section covers a brief history of face tracking.\n  *  **Utilities** : This section outlines the common structures and conventions used in this chapter. It includes the object-oriented design, data storage and representation, and a tool for data collection and annotation.\n  *  **Geometrical constraints** : This section describes how facial geometry and its variations are learned from the training data and utilized during tracking to constrain the solution. This includes modeling the face as a linear shape model and how global transformations can be integrated into its representation.\n  *  **Facial feature detectors** : This section describes how to learn the appearance of facial features in order to detect them in an image where the face is to be tracked.\n  *  **Face detection and initialization** : This section describes how to use face detection to initialize the tracking process.\n  *  **Face tracking** : This section combines all components described previously into a tracking system through the process of image alignment. A discussion on the settings in which the system can be expected to work best is also carried out.\n\nThe following block diagram illustrates the relationships between the various components of the system:\n\n### Note\n\nNote that all methods employed in this chapter follow a data-driven paradigm whereby all models used are learned from data rather than being designed by hand in a rule-based setting. As such, each component of the system will involve two components: training and testing. Training builds the models from data and testing employs these models on new unseen data.\n\n# Overview\n\nNon-rigid face tracking was first popularized in the early to mid 90s with the advent of **active shape models** ( **ASM** ) by Cootes and Taylor. Since then, a tremendous amount of research has been dedicated to solving the difficult problem of generic face tracking with many improvements over the original method that ASM proposed. The first milestone was the extension of ASM to **active appearance models** ( **AAM** ) in 2001, also by Cootes and Taylor. This approach was later formalized though the principled treatment of image warps by Baker and colleges in the the mid 2000s. Another strand of work along these lines was the **3D Morphable Model** ( **3DMM** ) by Blanz and Vetter, which like AAM, not only modeled image textures as opposed to profiles along object boundaries as in ASM, but took it one step further by representing the models with a highly dense 3D data learned from laser scans of faces. From the mid to the late 2000s, the focus of research on face tracking shifted away from how the face was parameterized to how the objective of the tracking algorithm was posed and optimized. Various techniques from the machine-learning community were applied with various degrees of success. Since the turn of the century, the focus has shifted once again, this time towards joint parameter and objective design strategies that guarantee global solutions.\n\nDespite the continued intense research into face tracking, there have been relatively few commercial applications that use it. There has also been a lag in uptake by hobbyists and enthusiasts, despite there being a number of freely available source-code packages for a number of common approaches. Nonetheless, in the past two years there has been a renewed interest in the public domain for the potential use of face tracking and commercial-grade products are beginning to emerge.\n\n# Utilities\n\nBefore diving into the intricacies of face tracking, a number of book-keeping tasks and conventions common to all face-tracking methods must first be introduced. The rest of this section will deal with these issues. An interested reader may want to skip this section at the first reading and go straight to the section on geometrical constraints.\n\n## Object-oriented design\n\nAs with face detection and recognition, programmatically, face tracking consists of two components: data and algorithms. The algorithms typically perform some kind of operation on the incoming (that is, online) data by referencing prestored (that is, offline) data as a guide. As such, an object-oriented design that couples algorithms with the data they rely on is a convenient design choice.\n\nIn OpenCV v2.x, a convenient XML\/YAML file storage class was introduced that greatly simplifies the task of organizing offline data for use in the algorithms. To leverage this feature, all classes described in this chapter will implement read-and write-serialization functions. An example of this is shown as follows for an imaginary class `foo`:\n\n    #include <opencv2\/opencv.hpp>\n    using namespace cv;\n    class foo{\n        public:\n            Mat a;\n            type_b b;\n            void write(FileStorage &fs) const{\n                  assert(fs.isOpened());\n                  fs << \"{\" << \"a\"  << a << \"b\"  << b << \"}\";\n            }\n            void read(const FileNode& node){\n                assert(node.type() == FileNode::MAP);\n                  node[\"a\"] >> a; node[\"b\"] >> b;\n            }\n    };\n\nHere, `Mat` is OpenCV's matrix class and `type_b` is an (imaginary) user-defined class that also has the serialization functionality defined. The I\/O functions `read` and `write` implement the serialization. The `FileStorage` class supports two types of data structures that can be serialized. For simplicity, in this chapter all classes will only utilize mappings, where each stored variable creates a `FileNode` object of type `FileNode::MAP`. This requires a unique key to be assigned to each element. Although the choice for this key is arbitrary, we will use the variable name as the label for consistency reasons. As illustrated in the preceding code snippet, the `read` and `write` functions take on a particularly simple form, whereby the streaming operators (`<<` and `>>`) are used to insert and extract data to the `FileStorage` object. Most OpenCV classes have implementations of the `read` and `write` functions, allowing the storage of the data that they contain to be done with ease.\n\nIn addition to defining the serialization functions, one must also define two additional functions for the serialization in the `FileStorage` class to work, as follows:\n\n    void write(FileStorage& fs, const string&, const foo& x){\n      x.write(fs);\n    }\n    void read(const FileNode& node, foo& x,const foo& default){\n      if(node.empty())x = d; else x.read(node);\n    }\n\nAs the functionality of these two functions remains the same for all classes we describe in this section, they are templated and defined in the `ft.hpp` header file found in the source code pertaining to this chapter. Finally, to easily save and load user-defined classes that utilize the serialization functionality, templated functions for these are also implemented in the header file as follows:\n\n    template <class T>\n    T load_ft(const char* fname){\n      T x; FileStorage f(fname,FileStorage::READ);\n      f[\"ft object\"] >> x; f.release(); return x;\n    }\n    template<class T>\n    void save_ft(const char* fname,const T& x){\n      FileStorage f(fname,FileStorage::WRITE);\n      f << \"ft object\" << x; f.release();\n    }\n\nNote that the label associated with the object is always the same (that is, `ft object`). With these functions defined, saving and loading object data is a painless process. This is shown with the help of the following example:\n\n    #include \"opencv_hotshots\/ft\/ft.hpp\"\n    #include \"foo.hpp\"\n    int main(){\n      ...\n      foo A; save_ft<foo>(\"foo.xml\",A);\n      ...\n      foo B = load_ft<foo>(\"foo.xml\");\n      ...\n    }\n\nNote that the `.xml` extension results in an XML-formatted data file. For any other extension it defaults to the (more human readable) YAML format.\n\n## Data collection: Image and video annotation\n\nModern face tracking techniques are almost entirely data driven, that is, the algorithms used to detect the locations of facial features in the image rely on models of the appearance of the facial features and the geometrical dependencies between their relative locations from a set of examples. The larger the set of examples, the more robust the algorithms behave, as they become more aware of the gamut of variability that faces can exhibit. Thus, the first step in building a face tracking algorithm is to create an image\/video annotation tool, where the user can specify the locations of the desired facial features in each example image.\n\n### Training data types\n\nThe data for training face tracking algorithms generally consists of four components:\n\n  *  **Images** : This component is a collection of images (still images or video frames) that contain an entire face. For best results, this collection should be specialized to the types of conditions (that is, identity, lighting, distance from camera, capturing device, among others) in which the tracker is later deployed. It is also crucial that the faces in the collection exhibit the range of head poses and facial expressions that the intended application expects.\n  *  **Annotations** : This component has ordered hand-labeled locations in each image that correspond to every facial feature to be tracked. More facial features often lead to a more robust tracker as the tracking algorithm can use their measurements to reinforce each other. The computational cost of common tracking algorithms typically scales linearly with the number of facial features.\n  *  **Symmetry indices** :  This component has an index for each facial feature point that defines its bilaterally symmetrical feature. This can be used to mirror the training images, effectively doubling the training set size and symmetrizing the data along the y axis.\n  *  **Connectivity indices** : This component has a set of index pairs of the annotations that define the semantic interpretation of the facial features. These connections are useful for visualizing the tracking results.\n\nA visualization of these four components is shown in the following image, where from left to right we have the raw image, facial feature annotations, color-coded bilateral symmetry points, mirrored image, and annotations and facial feature connectivity.\n\nTo conveniently manage such data, a class that implements storage and access functionality is a useful component. The `CvMLData` class in the `ml` module of OpenCV has the functionality for handling general data often used in machine-learning problems. However, it lacks the functionality required from the face-tracking data. As such, in this chapter we will use the `ft_data` class, declared in the `ft_data.hpp` header file, which is designed specifically with the peculiarity of face-tracking data in mind. All data elements are defined as public members of the class, as follows:\n\n    class ft_data{\n    public:\n      vector<int> symmetry;\n      vector<Vec2i> connections;\n      vector<string> imnames;\n      vector<vector<Point2f> > points;\n     ...\n    }\n\nThe `Vec2i` and `Point2f` types are OpenCV classes for vectors of two integers and 2D floating-point coordinates respectively. The `symmetry` vector has as many components as there are feature points on the face (as defined by the user). Each of the `connections` define a zero-based index pair of connected facial features. As the training set can potentially be very large, rather than storing the images directly, the class stores the filenames of each image in the `imnames` member variable (note that this requires the images to be located in the same relative path for the filenames to remain valid). Finally, for each training image, a collection of facial feature locations are stored as vectors of floating-point coordinates in the `points` member variable.\n\nThe `ft_data` class implements a number of convenience methods for accessing the data. To access an image in the dataset, the `get_image` function loads the image at the specified index, `idx`, and optionally mirrors it around the y axis as follows:\n\n    Mat\n    ft_data::get_image(\n    const int idx,   \/\/index of image to load from file\n    const int flag){ \/\/0=gray,1=gray+flip,2=rgb,3=rgb+flip\n      if((idx < 0) || (idx >= (int)imnames.size()))return Mat();\n      Mat img,im;\n      if(flag < 2)img = imread(imnames[idx],0);\n      else        img = imread(imnames[idx],1);\n      if(flag % 2 != 0)flip(img,im,1); else im = img;\n      return im;\n    }\n\nThe (`0`,`1`) flag passed to OpenCV's `imread` function specifies whether the image is loaded as a 3-channel color image or as a single-channel grayscale image. The flag passed to OpenCV's `flip` function specifies mirroring around the y axis.\n\nTo access a point set corresponding to an image at a particular index, the `get_points` function returns a vector of floating-point coordinates with the option of mirroring their indices as follows:\n\n    vector<Point2f>\n    ft_data::get_points(\n    const int idx,       \/\/index of image corresponding to pointsconst bool flipped){ \/\/is the image flipped around the y-axis?\n      if((idx < 0) || (idx >= (int)imnames.size()))\n        return vector<Point2f>();\n      vector<Point2f> p = points[idx];\n      if(flipped){\n        Mat im = this->get_image(idx,0); int n = p.size();\n        vector<Point2f> q(n);\n        for(int i = 0; i < n; i++){      \n          q[i].x = im.cols-1-p[symmetry[i]].x;\n          q[i].y = p[symmetry[i]].y;\n        }return q;\n      }else return p;\n    }\n\nNotice that when the mirroring flag is specified, this function calls the `get_image` function. This is required to determine the width of the image in order to correctly mirror the facial feature coordinates. A more efficient method could be devised by simply passing the image width as a variable. Finally, the utility of the `symmetry` member variable is illustrated in this function. The mirrored feature location of a particular index is simply the feature location at the index specified in the `symmetry` variable with its x coordinate flipped and biased.\n\nBoth the `get_image` and `get_points` functions return empty structures if the specified index is outside the one that exists for the dataset. It is also possible that not all images in the collection are annotated. Face tracking algorithms can be designed to handle missing data, however, these implementations are often quite involved and are outside the scope of this chapter. The `ft_data` class implements a function for removing samples from its collection that do not have corresponding annotations, as follows:\n\n    void\n    ft_data::rm_incomplete_samples(){\n      int n = points[0].size(),N = points.size();\n      for(int i = 1; i < N; i++)n = max(n,int(points[i].size()));\n      for(int i = 0; i < int(points.size()); i++){\n        if(int(points[i].size()) != n){\n          points.erase(points.begin()+i);\n           imnames.erase(imnames.begin()+i); i--;\n        }else{\n          int j = 0;\n          for(; j < n; j++){\n           if((points[i][j].x <= 0) ||\n            (points[i][j].y <= 0))break;\n          }\n          if(j < n){\n          points.erase(points.begin()+i);\n          imnames.erase(imnames.begin()+i); i--;\n          }\n        }\n      }\n    }\n\nThe sample instance that has the most number of annotations is assumed to be the canonical sample. All data instances that have a point set with less than that many number of points are removed from the collection using the vector's `erase` function. Also notice that points with (x, y) coordinates less than one are considered missing in their corresponding image (possibly due to occlusion, poor visibility, or ambiguity).\n\nThe `ft_data` class implements the serialization functions `read` and `write`, and can thus be stored and loaded easily. For example, saving a dataset can be done as simply as:\n\n    ft_data D;                        \/\/instantiate data structure\n   ...                                 \/\/populate data\n    save_ft<ft_data>(\"mydata.xml\",D); \/\/save data\n\nFor visualizing the dataset, `ft_data` implements a number of drawing functions. Their use is illustrated in the `visualize_annotations.cpp` file. This simple program loads annotation data stored in the file specified in the command line, removes the incomplete samples, and displays the training images with their corresponding annotations, symmetry, and connections superimposed. A few notable features of OpenCV's `highgui` module are demonstrated here. Although quite rudimentary and not well suited for complex user interfaces, the functionality in OpenCV's `highgui` module is extremely useful for loading and visualizing data and algorithmic outputs in computer vision applications. This is perhaps one of OpenCV's distinguishing qualities compared to other computer vision libraries.\n\n### Annotation tool\n\nTo aid in generating annotations for use with the code in this chapter, a rudimentary annotation tool can be found in the `annotate.cpp` file. The tool takes as input a video stream, either from a file or from the camera. The procedure for using the tool is listed in the following four steps:\n\n  1. **Capture images** : In this first step, the image stream is displayed on the screen and the user chooses the images to annotate by pressing the _S_ key. The best set of features to annotate are those that maximally span the range of facial behaviors that the face tracking system will be required to track.\n  2.  **Annotate first image** : In this second step, the user is presented with the first image selected in the previous stage. The user then proceeds to click on the image at the locations pertaining to the facial features that require tracking.\n  3.  **Annotate connectivity** : In this third step, to better visualize a shape, the connectivity structure of points needs to be defined. Here, the user is presented with the same image as in the previous stage, where the task now is to click a set of point pairs, one after the other, to build the connectivity structure for the face model.\n  4.  **Annotate symmetry** : In this step, still with the same image, the user selects pairs of points that exhibit bilateral symmetry.\n  5.  **Annotate remaining images** : In this final step, the procedure here is similar to that of step 2, except that the user can browse through the set of images and annotate them asynchronously.\n\nAn interested reader may want to improve on this tool by improving its usability or may even integrate an incremental learning procedure, whereby a tracking model is updated after each additional image is annotated and is subsequently used to initialize the points to reduce the burden of annotation.\n\nAlthough some publicly available datasets are available for use with the code developed in this chapter (see for example the description in the following section), the annotation tool can be used to build person-specific face tracking models, which often perform far better than their generic, person-independent, counterparts.\n\n### Pre-annotated data (The MUCT dataset)\n\nOne of the hindering factors of developing face tracking systems is the tedious and error-prone process of manually annotating a large collection of images, each with a large number of points. To ease this process for the purpose of following the work in this chapter, the publicly available MUCT dataset can be downloaded from: http:\/\/www\/milbo.org\/muct.\n\nThe dataset consists of 3,755 face images annotated with 76-point landmarks. The subjects in the dataset vary in age and ethnicity and are captured under a number of different lighting conditions and head poses.\n\nTo use the MUCT dataset with the code in this chapter, perform the following steps:\n\n  1.  **Download the image set** : In this step, all the images in the dataset can be obtained by downloading the files `muct-a-jpg-v1.tar.gz` to `muct-e-jpg-v1.tar.gz` and uncompressing them. This will generate a new folder in which all the images will be stored.\n  2.  **Download the annotations** : In this step, download the file containing the annotations `muct-landmarks-v1.tar.gz`. Save and uncompress this file in the same folder as the one in which the images were downloaded.\n  3.  **Define connections and symmetry using the annotation tool** : In this step, from the command line, issue the command `.\/annotate -m $mdir -d $odir`, where `$mdir` denotes the folder where the MUCT dataset was saved and `$odir` denotes the folder to which the `annotations.yaml` file, containing the data stored as a `ft_data` object will be written.\n\n### Tip\n\nUsage of the MUCT dataset is encouraged to get a quick introduction to the functionality of the face tracking code described in this chapter.\n\n# Geometrical constraints\n\nIn face tracking, geometry refers to the spatial configuration of a predefined set of points that correspond to physically consistent locations on the human face (such as eye corners, nose tip, and eyebrow edges). A particular choice of these points is application dependent, with some applications requiring a dense set of over 100 points and others requiring only a sparser selection. However, robustness of face tracking algorithms generally improves with an increased number of points, as their separate measurements can reinforce each other through their relative spatial dependencies. For example, knowing the location of an eye corner is a good indication of where to expect the nose to be located. However, there are limits to improvements in robustness gained by increasing the number of points, where performance typically plateaus after around 100 points. Furthermore, increasing the point set used to describe a face carries with it a linear increase in computational complexity. Thus, applications with strict constraints on computational load may fare better with fewer points.\n\nIt is also the case that faster tracking often leads to more accurate tracking in the online setting. This is because, when frames are dropped, the perceived motion between frames increases, and the optimization algorithm used to find the configuration of the face in each frame has to search a larger space of possible configurations of feature points; a process that often fails when displacement between frames becomes too large. In summary, although there are general guidelines on how to best design the selection of facial feature points, to get an optimal performance, this selection should be specialized to the application's domain.\n\nFacial geometry is often parameterized as a composition of two elements: a global (rigid) transformation and a local (non-rigid) deformation. The global transformation accounts for the overall placement of the face in the image, which is often allowed to vary without constraint (that is, the face can appear anywhere in the image). This includes the (x, y) location of the face in the image, the in-plane head rotation, and the size of the face in the image. Local deformations, on the other hand, account for differences between facial shapes across identities and between expressions. In contrast to the global transformation, these local deformations are often far more constrained largely due to the highly structured configuration of facial features. Global transformations are generic functions of 2D coordinates, applicable to any type of object, whereas local deformations are object specific and must be learned from a training dataset.\n\nIn this section we will describe the construction of a geometrical model of a facial structure, hereby referred to as the shape model. Depending on the application, it can capture expression variations of a single individual, differences between facial shapes across a population, or a combination of both. This model is implemented in the `shape_model` class that can be found in the `shape_model.hpp` and `shape_model.cpp` files. The following code snippet is a part of the header of the `shape_model` class that highlights its primary functionality:\n\n    class shape_model{ \/\/2d linear shape model\n    public:\n      Mat p; \/\/parameter vector (kx1) CV_32F\n      Mat V; \/\/linear subspace (2nxk) CV_32F\n      Mat e; \/\/parameter variance (kx1) CV_32F\n      Mat C; \/\/connectivity (cx2) CV_32S\n      ...\n      void calc_params(\n      const vector<Point2f> &pts,  \/\/points to compute parameters\n      const Mat &weight = Mat(),    \/\/weight\/point (nx1) CV_32F\n      const float c_factor = 3.0); \/\/clamping factor\n      ...\n      vector<Point2f>              \/\/shape described by parameters\n      calc_shape();\n      ...\n      void train(\n      const vector<vector<Point2f> > &p, \/\/N-example shapes\n      const vector<Vec2i> &con = vector<Vec2i>(),\/\/connectivity\n      const float frac = 0.95, \/\/fraction of variation to retain\n      const int kmax = 10);   \/\/maximum number of modes to retain\n      ...\n    }\n\nThe model that represents variations in face shapes is encoded in the subspace matrix `V` and variance vector `e`. The parameter vector `p` stores the encoding of a shape with respect to the model. The connectivity matrix `C` is also stored in this class as it pertains only to visualizing instances of the face's shape. The three functions of primary interest in this class are `calc_params`, `calc_shape`, and `train`. The `calc_params` function projects a set of points onto the space of plausible face shapes. It optionally provides separate confidence weights for each of the points to be projected. The `calc_shape` function generates a set of points by decoding the parameter vector `p` using the face model (encoded by `V` and `e`). The `train` function learns the encoding model from a dataset of face shapes, each of which consists of the same number of points. The parameters `frac` and `kmax` are parameters of the training procedure that can be specialized for the data at hand.\n\nThe functionality of this class will be elaborated in the sections that follow, where we begin by describing **Procrustes analysis** , a method for rigidly registering a point set, followed by the linear model used to represent local deformations. The programs in the `train_shape_model.cpp` and `visualize_shape_model.cpp` files train and visualize the shape model respectively. Their usage will be outlined at the end of this section.\n\n## Procrustes analysis\n\nIn order to build a deformation model of face shapes, we must first process the raw annotated data to remove components pertaining to global rigid motion. When modeling geometry in 2D, a rigid motion is often represented as a similarity transform; this includes the scale, in-plane rotation and translation. The following image illustrates the set of permissible motion types under a similarity transform. The process of removing global rigid motion from a collection of points is called Procrustes analysis.\n\nMathematically, the objective of Procrustes analysis is to simultaneously find a canonical shape and similarity transform each data instance that brings them into alignment with the canonical shape. Here, alignment is measured as the least-squares distance between each transformed shape with the canonical shape. An iterative procedure for fulfilling this objective is implemented in the `shape_model` class as follows:\n\n    #define fl at<float>\n    Mat shape_model::procrustes(\n    const Mat &X,       \/\/interleaved raw shape data as columns\n    const int itol,     \/\/maximum number of iterations to try\n    const float ftol)   \/\/convergence tolerance\n    {\n      int N = X.cols,n = X.rows\/2; Mat Co,P = X.clone();\/\/copy\n      for(int i = 0; i < N; i++){\n        Mat p = P.col(i);            \/\/i'th shape\n        float mx = 0,my = 0;         \/\/compute centre of mass...\n        for(int j = 0; j < n; j++){  \/\/for x and y separately\n          mx += p.fl(2*j); my += p.fl(2*j+1);\n        }\n        mx \/= n; my \/= n;\n        for(int j = 0; j < n; j++){ \/\/remove center of mass\n          p.fl(2*j) -= mx; p.fl(2*j+1) -= my;\n        }\n      }\n      for(int iter = 0; iter < itol; iter++){    \n        Mat C = P*Mat::ones(N,1,CV_32F)\/N; \/\/compute normalized...\n    **normalize(C,C);                    \/\/canonical shape**\n        if(iter > 0){if(norm(C,Co) < ftol)break;} \/\/converged?\n        Co = C.clone();               \/\/remember current estimate\n        for(int i = 0; i < N; i++){\n    **Mat R = this- >rot_scale_align(P.col(i),C);**\n          for(int j = 0; j < n; j++){ \/\/apply similarity transform\n           float x = P.fl(2*j,i),y = P.fl(2*j+1,i);\n           P.fl(2*j  ,i) = R.fl(0,0)*x + R.fl(0,1)*y;\n           P.fl(2*j+1,i) = R.fl(1,0)*x + R.fl(1,1)*y;\n          }\n        }\n      }return P; \/\/returned procrustes aligned shapes\n    }\n\nThe algorithm begins by subtracting the center of mass of each shape's instance followed by an iterative procedure that alternates between computing the canonical shape, as the normalized average of all shapes, and rotating and scaling each shape to best match the canonical shape. The normalization step of the estimated canonical shape is necessary to fix the scale of the problem and prevent it from shrinking all the shapes to zero. The choice of this anchor scale is arbitrary, here we have chosen to enforce the length of the canonical shape vector `C` to 1.0, as is the default behavior of OpenCV's `normalize` function. Computing the in-plane rotation and scaling that best aligns each shape's instance to the current estimate of the canonical shape is effected through the `rot_scale_align` function as follows:\n\n    Mat shape_model::rot_scale_align(\n    const Mat &src, \/\/[x1;y1;...;xn;yn] vector of source shape\n    const Mat &dst) \/\/destination shape\n    {\n      \/\/construct linear system\n      int n = src.rows\/2; float a=0,b=0,d=0;\n      for(int i = 0; i < n; i++){\n        d+= src.fl(2*i)*src.fl(2*i  )+src.fl(2*i+1)*src.fl(2*i+1);\n        a+= src.fl(2*i)*dst.fl(2*i  )+src.fl(2*i+1)*dst.fl(2*i+1);\n        b+= src.fl(2*i)*dst.fl(2*i+1)-src.fl(2*i+1)*dst.fl(2*i  );\n      }\n      a \/= d; b \/= d;\/\/solve linear system\n      return (Mat_<float>(2,2) << a,-b,b,a);\n    }\n\nThis function minimizes the following least-squares difference between the rotated and canonical shapes. Mathematically this can be written as:\n\nHere the solution to the least-squares problem takes on the closed-form solution shown in the following image on the right-hand side of the equation. Note that rather than solving for the scaling and in-plane rotation, which are nonlinearly related in the scaled 2D rotation matrix, we solve for the variables (a, b). These variables are related to the scale and rotation matrix as follows:\n\nA visualization of the effects of Procrustes analysis on raw annotated shape data is illustrated in the following image . Each facial feature is displayed with a unique color. After translation normalization, the structure of the face becomes apparent, where the locations of facial features cluster around their average locations. After the iterative scale and rotation normalization procedure, the feature clustering becomes more compact and their distribution becomes more representative of the variation induced by facial deformation. This last point is important as it is these deformations that we will attempt to model in the following section. Thus, the role of Procrustes analysis can be thought of as a preprocessing operation on the raw data that will allow better local deformation models of the face to be learned.\n\n## Linear shape models\n\nThe aim of facial-deformation modeling is to find a compact parametric representation of how the face's shape varies across identities and between expressions. There are many ways of achieving this goal with various levels of complexity. The simplest of these is to use a linear representation of facial geometry. Despite its simplicity, it has been shown to accurately capture the space of facial deformations, particularly when the faces in the dataset are largely in a frontal pose. It also has the advantage that inferring the parameters of its representation is an extremely simple and cheap operation, in contrast to its nonlinear counterparts. This plays an important role when deploying it to constrain the search procedure during tracking.\n\nThe main idea of linearly modeling facial shapes is illustrated in the following image. Here, a face shape, which consists of _N_ facial features, is modeled as a single point in a 2 _N_ -dimensional space. The aim of linear modeling is to find a low-dimensional hyperplane embedded within this 2 _N_ -dimensional space in which all the face shape points lie (that is, the green points in the image). As this hyperplane spans only a subset of the entire 2 _N_ -dimensional space it is often referred to as the subspace. The lower the dimensionality of the subspace the more compact the representation of the face is and the stronger the constraint that it places on the tracking procedure becomes. This often leads to more robust tracking. However, care should be taken in selecting the subspace's dimension so that it has enough capacity to span the space of all faces but not so much that non-face shapes lie within its span (that is, the red points in the image). It should be noted that when modeling data from a single person, the subspace that captures the face's variability is often far more compact than the one that models multiple identities. This is one of the reasons why person-specific trackers perform much better than generic ones.\n\nThe procedure for finding the best low-dimensional subspace that spans a dataset is called **Principal Component Analysis** ( **PCA** ). OpenCV implements a class for computing PCA, however, it requires the number of preserved subspace dimensions to be prespecified. As this is often difficult to determine a priori, a common heuristic is to choose it based on the fraction of the total amount of variation it accounts for. In the `shape_model::train` function, PCA is implemented as follows:\n\n    **SVD svd(dY*dY.t());**\n    int m = min(min(kmax,N-1),n-1);\n    float vsum = 0; for(int i = 0; i < m; i++)vsum += svd.w.fl(i);\n    float v = 0; int k = 0;\n    **for(k = 0; k < m; k++){**\n    **v += svd.w.fl(k); if(v\/vsum >= frac){k++; break;}**\n    **}**\n    if(k > m)k = m;\n    **Mat D = svd.u(Rect(0,0,k,2*n));**\n\nHere, each column of the `dY` variable denotes the mean-subtracted Procrustes-aligned shape. Thus, **singular value decomposition** ( **SVD** ) is effectively applied to the covariance matrix of the shape data (that is, `dY.t()*dY`). The `w` member of OpenCV's `SVD` class stores the variance in the major directions of variability of the data, ordered from largest to smallest. A common approach to choose the dimensionality of the subspace is to choose the smallest set of directions that preserve a fraction `frac` of the total energy of the data, which is represented by the entries of `svd.w`. As these entries are ordered from largest to smallest, it suffices to enumerate the subspace selection by greedily evaluating the energy in the top k directions of variability. The directions themselves are stored in the `u` member of the `SVD` class. The `svd.w` and `svd.u` components are generally referred to as the eigenspectrum and eigenvectors respectively. A visualization of these two components are shown in the following figure:\n\n### Note\n\nNotice that the eigenspectrum decreases rapidly, which suggests that most of the variation contained in the data can be modeled with a low-dimensional subspace.\n\n## A combined local-global representation\n\nA shape in the image frame is generated by the composition of a local deformation and a global transformation. Mathematically, this parameterization can be problematic, as the composition of these transformations results in a nonlinear function that does not admit a closed-form solution. A common way to circumvent this problem is to model the global transformation as a linear subspace and append it to the deformation subspace. For a fixed shape, a similarity transform can be modeled with a subspace as follows:\n\nIn the `shape_model` class, this subspace is generated using the `calc_rigid_basis` function. The shape from which the subspace is generated (that is, the x and y components in the preceding equation) is the mean shape ov++er the Procustes-aligned shape (that is, the canonical shape). In addition to constructing the subspace in the aforementioned form, each column of the matrix is normalized to unit length. In the `shape_model::train` function, the variable `dY` described in the previous section is computed by projecting out the components of the data that pertain to rigid motion, as follows:\n\n    Mat R = this->calc_rigid_basis(Y); \/\/compute rigid subspace\n    Mat P = R.t()*Y; Mat dY = Y - R*P; \/\/project-out rigidity\n\nNotice that this projection is implemented as a simple matrix multiplication. This is possible because the columns of the rigid subspace have been length normalized. This does not change the space spanned by the model, and means only that `R.t()*R` equals the identity matrix.\n\nAs the directions of variability stemming from rigid transformations have been removed from the data before learning the deformation model, the resulting deformation subspace will be orthogonal to the rigid transformation subspace. Thus, concatenating the two subspaces results in a combined local-global linear representation of facial shapes that is also orthonormal. Concatenation here can be performed by assigning the two subspace matrices to submatrices of the combined subspace matrix through the ROI extraction mechanism implemented in OpenCV's `Mat` class as follows:\n\n    V.create(2*n,4+k,CV_32F);                  \/\/combined subspace\n    Mat Vr = V(Rect(0,0,4,2*n)); R.copyTo(Vr); \/\/rigid subspace\n    Mat Vd = V(Rect(4,0,k,2*n)); D.copyTo(Vd); \/\/nonrigid subspace\n\nThe orthonormality of the resulting model means that the parameters describing a shape can be computed easily, as is done in the `shape_model::calc_params` function:\n\n    p = V.t()*s;\n\nHere `s` is a vectorized face shape and `p` stores the coordinates in the face subspace that represents it.\n\nA final point to note about linearly modeling facial shapes is how to constrain the subspace coordinates such that shapes generated using it remain valid. In the following image, instances of face shapes that lie within the subspace are shown for an increasing value of the coordinates in one of the directions of variability in increments of four standard deviations. Notice that for small values, the resulting shape remains face-like, but deteriorates as the values become too large.\n\nA simple way to prevent such deformation is to clamp the subspace coordinate values to lie within a permissible region as determined from the dataset. A common choice for this is a box constraint within \u00b1 3 standard deviations of the data, which accounts for 99.7 percent of variation in the data. These clamping values are computed in the `shape_model::train` function after the subspace is found, as follows:\n\n    Mat Q = V.t()*X;            \/\/project raw data onto subspace\n    for(int i = 0; i < N; i++){ \/\/normalize coordinates w.r.t scale\n    **float v = Q.fl(0,i); Mat q = Q.col(i); q \/= v;**\n    }\n    e.create(4+k,1,CV_32F); multiply(Q,Q,Q);\n    for(int i = 0; i < 4+k; i++){\n    **if(i < 4)e.fl(i) = -1; \/\/no clamping for rigid coefficients**\n      else e.fl(i) = Q.row(i).dot(Mat::ones(1,N,CV_32F))\/(N-1);\n    }\n\nNotice that the variance is computed over the subspace coordinate `Q` after normalizing with respect to the coordinate of the first dimension (that is, scale). This prevents data samples that have relatively large scale from dominating the estimate. Also, notice that a negative value is assigned to the variance of the coordinates of the rigid subspace (that is, the first four columns of `V`). The clamping function `shape_model::clamp` checks to see if the variance of a particular direction is negative and only applies clamping if it is not, as follows:\n\n    void shape_model::clamp(\n    const float c){ \/\/clamping as fraction of standard deviation\n      double scale = p.fl(0);        \/\/extract scale\n      for(int i = 0; i < e.rows; i++){\n    **if(e.fl(i) < 0)continue;     \/\/ignore rigid components**\n        float v = c*sqrt(e.fl(i));   \/\/c*standard deviations box\n        if(fabs(p.fl(i)\/scale) > v){ \/\/preserve sign of coordinate\n    **if(p.fl(i) > 0)p.fl(i) =  v*scale; \/\/positive threshold**\n    **else p.fl(i) = -v*scale; \/\/negative threshold**\n        }\n      }\n    }\n\nThe reason for this is that the training data is often captured under contrived settings where the face is upright and centered in the image at a particular scale. Clamping the rigid components of the shape model to adhere to the configurations in the training set would then be too restrictive. Finally, as the variance of each deformable coordinate is computed in the scale-normalized frame, the same scaling must be applied to the coordinates during clamping.\n\n## Training and visualization\n\nAn example program for training a shape model from the annotation data can be found in `train_shape_model.cpp`. With the command-line argument `argv[1]` containing the path to the annotation data, training begins by loading the data into memory and removing incomplete samples, as follows:\n\n    ft_data data = load_ft<ft_data>(argv[1]);\n    data.rm_incomplete_samples();\n\nThe annotations for each example, and optionally their mirrored counterparts, are then stored in a vector before passing them to the training function as follows:\n\n    vector<vector<Point2f> > points;\n    for(int i = 0; i < int(data.points.size()); i++){\n      points.push_back(data.get_points(i,false));\n      if(mirror)points.push_back(data.get_points(i,true));\n    }\n\nThe shape model is then trained by a single function call to `shape_model::train` as follows:\n\n    shape_model smodel; smodel.train(points,data.connections,frac,kmax);\n\nHere, `frac` (that is, the fraction of variation to retain) and `kmax` (that is, the maximum number of eigenvectors to retain) can be optionally set through command-line options, although the default settings of 0.95 and 20, respectively, tend to work well in most cases. Finally, with the command-line argument `argv[2]` containing the path to save the trained shape model to, saving can be performed by a single function call as follows:\n\n    save_ft(argv[2],smodel);\n\nThe simplicity of this step results from defining the `read` and `write` serialization functions for the `shape_model` class.\n\nTo visualize the trained shape model, the `visualize_shape_model.cpp` program animates the learned non-rigid deformations of each direction in turn. It begins by loading the shape model into memory as follows:\n\n    shape_model smodel = load_ft<shape_model>(argv[1]);\n\nThe rigid parameters that place the model at the center of the display window are computed as follows:\n\n    int n = smodel.V.rows\/2;\n    float scale = calc_scale(smodel.V.col(0),200);\n    float tranx =\n    n*150.0\/smodel.V.col(2).dot(Mat::ones(2*n,1,CV_32F));\n    float trany =\n    n*150.0\/smodel.V.col(3).dot(Mat::ones(2*n,1,CV_32F));\n\nHere, the `calc_scale` function finds the scaling coefficient that would generate face shapes with a width of 200 pixels. The translation components are computed by finding the coefficients that generate a translation of 150 pixels (that is, the model is mean-centered and the display window is 300 x 300 pixels in size).\n\n### Note\n\nNote that the first column of `shape_model::V` corresponds to scale and the third and fourth columns to x and y translations respectively.\n\nA trajectory of parameter values is then generated, which begins at zero, moves to the positive extreme, moves to the negative extreme, and then back to zero as follows:\n\n    vector<float> val;\n    for(int i = 0; i < 50; i++)val.push_back(float(i)\/50);\n    for(int i = 0; i < 50; i++)val.push_back(float(50-i)\/50);\n    for(int i = 0; i < 50; i++)val.push_back(-float(i)\/50);\n    for(int i = 0; i < 50; i++)val.push_back(-float(50-i)\/50);\n\nHere, each phase of the animation is composed of fifty increments. This trajectory is then used to animate the face model and render the results in a display window as follows:\n\n      Mat img(300,300,CV_8UC3); namedWindow(\"shape model\");\n      while(1){\n        for(int k = 4; k < smodel.V.cols; k++){\n          for(int j = 0; j < int(val.size()); j++){\n    **Mat p = Mat::zeros(smodel.V.cols,1,CV_32F);**\n    **p.at <float>(0) = scale;**\n    **p.at <float>(2) = tranx;**\n    **p.at <float>(3) = trany;**\n    **p.at <float>(k) = scale*val[j]*3.0***\n    **sqrt(smodel.e.at <float>(k));  **\n            p.copyTo(smodel.p); img = Scalar::all(255);\n            vector<Point2f> q = smodel.calc_shape();\n            draw_shape(img,q,smodel.C);\n            imshow(\"shape model\",img);\n            if(waitKey(10) == 'q')return 0;\n          }\n        }\n      }\n\n### Note\n\nNote that the rigid coefficients (that is, those corresponding to the first four columns of `shape_model::V`) are always set to the values computed previously, to place the face at the center of the display window. \n\n# Facial feature detectors\n\nDetecting facial features in images bares a strong resemblance to general object detection. OpenCV has a set of sophisticated functions for building general object detectors, the most well-known of which is the cascade of Haar-based feature detectors used in their implementation of the well-known Viola-Jones face detector. There are, however, a few distinguishing factors that make facial feature detection unique. These are as follows:\n\n  * **Precision versus robustness** : In generic object detection, the aim is to find the coarse position of the object in the image; facial feature detectors are required to give highly precise estimates of the location of the feature. An error of a few pixels is considered inconsequential in object detection but it can mean the difference between a smile and a frown in facial expression estimation through feature detections.\n  *  **Ambiguity from limited spatial support** : It is common to assume that the object of interest in generic object detection exhibits sufficient image structure such that it can be reliably discriminated from image regions that do not contain the object. This is often not the case for facial features, which typically have limited spatial support. This is because image regions that do not contain the object can often exhibit a very similar structure to facial features. For example, a feature on the periphery of the face, seen from a small bounding box centered at the feature, can be easily confused with any other image patch that contains a strong edge through its center.\n  *  **Computational complexity** : Generic object detection aims to find all instances of the object in an image. Face tracking, on the other hand, requires the locations of all facial features, which often ranges from around 20 to 100 features. Thus, the ability to evaluate each feature detector efficiently is paramount in building a face tracker that can run in real time.\n\nDue to these differences, the facial feature detectors used in face tracking are often specifically designed with that purpose in mind. There are, of course, many instances of generic object-detection techniques being applied to facial feature detectors in face tracking. However, there does not appear to be a consensus in the community about which representation is best suited for the problem.\n\nIn this section, we will build facial feature detectors using a representation that is perhaps the simplest model one would consider: a linear image patch. Despite its simplicity, with due care in designing its learning procedure, we will see that this representation can in fact give reasonable estimates of facial feature locations for use in a face tracking algorithm. Furthermore, their simplicity enables an extremely rapid evaluation that makes real-time face tracking possible. Due to their representation as an image patch, the facial feature detectors are hereby referred to as patch models. This model is implemented in the `patch_model` class that can be found in the `patch_model.hpp` and `patch_model.cpp` files. The following code snippet is of the header of the `patch_model` class that highlights its primary functionality:\n\n    class patch_model{\n    public:\n      Mat P; \/\/normalized patch\n      ...\n      Mat                          \/\/response map\n      calc_response(\n      const Mat &im,               \/\/image patch of search region\n      const bool sum2one = false); \/\/normalize to sum-to-one?\n      ...\n      void\n      train(const vector<Mat> &images, \/\/training image patches\n      const Size psize,                \/\/patch size\n      const float var = 1.0,           \/\/ideal response variance\n      const float lambda = 1e-6,       \/\/regularization weight\n      const float mu_init = 1e-3,      \/\/initial step size\n      const int nsamples = 1000,       \/\/number of samples\n      const bool visi = false);        \/\/visualize process?\n      ...\n    };\n\nThe patch model used to detect a facial feature is stored in the matrix `P`. The two functions of primary interest in this class are `calc_response` and `train`. The `calc_response` function evaluates the patch model's response at every integer displacements over the search region `im`. The `train` function learns the patch model `P` of size `psize` that, on an average, yields response maps over the training set that is as close as possible to the ideal response map. The parameters `var`, `lambda`, `mu_init`, and `nsamples` are parameters of the training procedure that can be tuned to optimize performance for the data at hand.\n\nThe functionality of this class will be elaborated in this section. We begin by discussing the correlation patch and its training procedure, which will be used to learn the patch model. Next, the `patch_models` class, which is a collection of the patch models for each facial feature and has functionality that accounts for global transformations, will be described. The programs in `train_patch_model.cpp` and `visualize_patch_model.cpp` train and visualize the patch models, respectively, and their usage will be outlined at the end of this section on facial feature detectors.\n\n## Correlation-based patch models\n\nIn learning detectors, there are two primary competing paradigms: generative and discriminative. Generative methods learn an underlying representation of image patches that can best generate the object appearance in all its manifestations. Discriminative methods, on the other hand, learn a representation that best discriminates instances of the object from other objects that the model will likely encounter when deployed. Generative methods have the advantage that the resulting model encodes properties specific to the object, allowing novel instances of the object to be visually inspected. A popular approach that falls within the paradigm of generative methods is the famous Eigenfaces method. Discriminative methods have the advantage that the full capacity of the model is geared directly towards the problem at hand; discriminating instances of the object from all others. Perhaps the most well-known of all discriminative methods is the support vector machine. Although both paradigms can work well in many situations, we will see that when modeling facial features as an image patch, the discriminative paradigm is far superior.\n\n### Note\n\nNote that the eigenfaces and support vector machine methods were originally developed for classification rather than detection or image alignment. However, their underlying mathematical concepts have been shown to be applicable to the face tracking domain.\n\n### Learning discriminative patch models\n\nGiven an annotated dataset, the feature detectors can be learned independently from each other. The learning objective of a discriminative patch model is to construct an image patch that, when cross-correlated with an image region containing the facial feature, yields a strong response at the feature location and weak responses everywhere else. Mathematically, this can be expressed as:\n\nHere, **P** denotes the patch model, **I** denotes the i'th training image I(a:b, c:d) denotes the rectangular region whose top-left and bottom-right corners are located at (a, c) and (b, d), respectively. The period symbol denotes the inner product operation and **R** denotes the ideal response map. The solution to this equation is a patch model that generates response maps that are, on average, closest to the ideal response map as measured using the least-squares criterion. An obvious choice for the ideal response map, R, is a matrix with zeros everywhere except at the center (assuming the training image patches are centered at the facial feature of interest). In practice, since the images are hand-labeled, there will always be an annotation error. To account for this, it is common to describe R as a decaying function of distance from the center. A good choice is the 2D-Gaussian distribution, which is equivalent to assuming the annotation error is Gaussian distributed. A visualization of this setup is shown in the following figure for the left outer eye corner:\n\nThe learning objective as written previously is in a form commonly referred to as linear least squares. As such, it affords a closed-form solution. However, the degrees of freedom of this problem, that is, the number of ways the variables can vary to solve the problem, is equal to the number of pixels in the patch. Thus, the computational cost and memory requirements of solving for the optimal patch model can be prohibitive, even for a moderately sized patch or example, a 40 x 40- patch model has 1,600 degrees of freedom).\n\nAn efficient alternative to solving the learning problem as a linear system of equations is a method called stochastic gradient descent. By visualizing the learning objective as an error terrain over the degrees of freedom of the patch model, stochastic gradient descent iteratively makes an approximate estimate of the gradient direction of the terrain and takes a small step in the opposite direction. For our problem, the approximation to gradient can be computed by considering only the gradient of the learning objective for a single, randomly chosen image from the training set:\n\nIn the `patch_model` class, this learning process is implemented in the `train` function:\n\n    void\n    patch_model::train(\n    const vector<Mat> &images, \/\/featured centered training images\n    const Size psize,          \/\/desired patch model size\n    const float var,           \/\/variance of annotation error\n    const float lambda,        \/\/regularization parameter\n    const float mu_init,       \/\/initial step size\n    const int nsamples,        \/\/number of stochastic samples\n    const bool visi){          \/\/visualise training process\n      int N = images.size(),n = psize.width*psize.height;\n      int dx = wsize.width-psize.width;   \/\/center of response map\n      int dy = wsize.height-psize.height; \/\/...\n    **Mat F(dy,dx,CV_32F); \/\/ideal response map**\n    **for(int y = 0; y < dy; y++){  float vy = (dy-1)\/2 - y;**\n    **for(int x = 0; x < dx; x++){float vx = (dx-1)\/2 - x;**\n    **F.fl(y,x) = exp(-0.5*(vx*vx+vy*vy)\/var); \/\/Gaussian**\n    **}**\n    **}**\n    **normalize(F,F,0,1,NORM_MINMAX); \/\/normalize to [0:1] range**\n\n      \/\/allocate memory\n      Mat I(wsize.height,wsize.width,CV_32F);\n      Mat dP(psize.height,psize.width,CV_32F);\n      Mat O = Mat::ones(psize.height,psize.width,CV_32F)\/n;\n      P = Mat::zeros(psize.height,psize.width,CV_32F);\n\n      \/\/optimise using stochastic gradient descent\n      RNG rn(getTickCount()); \/\/random number generator\n    **double mu=mu_init,step=pow(1e-8\/mu_init,1.0\/nsamples);**\n      for(int sample = 0; sample < nsamples; sample++){\n        int i = rn.uniform(0,N); \/\/randomly sample image index\n    **I = this- >convert_image(images[i]); dP = 0.0;**\n    **for(int y = 0; y < dy; y++){ \/\/compute stochastic gradient**\n    **for(int x = 0; x < dx; x++){**\n    **Mat Wi=I(Rect(x,y,psize.width,psize.height)).clone();**\n    **Wi -= Wi.dot(O); normalize(Wi,Wi); \/\/normalize**\n    **dP += (F.fl(y,x) - P.dot(Wi))*Wi;**\n    **}**\n    **}**\n    **P += mu*(dP - lambda*P); \/\/take a small step**\n        mu *= step;              \/\/reduce step size\n        ...\n      }return;\n    }\n\nThe first highlighted code snippet in the preceding code is where the ideal response map is computed. Since the images are centered on the facial feature of interest, the response map is the same for all samples. In the second highlighted code snippet, the decay rate, `step`, of the step sizes is determined such that after `nsamples` iterations, the step size would have decayed to a value close to zero. The third highlighted code snippet is where the stochastic gradient direction is computed and used to update the patch model. There are two things to note here. First, the images used in training are passed to the `patch_model::convert_image` function, which converts the image to a single-channel image (if it is a color image) and applies the natural logarithm to the image pixel intensities:\n\n    I += 1.0; log(I,I);\n\nA bias value of one is added to each pixel before applying the logarithm since the logarithm of zero is undefined. The reason for performing this pre-processing on the training images is because log-scale images are more robust against differences in contrast and changes in illumination conditions. The following figure shows images of two faces with different degrees of contrast in the facial region. The difference between the images is much less pronounced in the log-scale images than it is in the raw images.\n\nThe second point to note about the update equation is the subtraction of `lambda*P` from the update direction. This effectively regularizes the solution from growing too large; a procedure that is often applied in machine-learning algorithms to promote generalization to unseen data. The scaling factor `lambda` is user defined and is usually problem dependent. However, a small value typically works well for learning patch models for facial feature detection.\n\n### Generative versus discriminative patch models\n\nDespite the ease of which discriminative patch models can be learned as described previously, it is worth considering whether generative patch models and their corresponding training regimes are simpler enough to achieve similar results. The generative counterpart of the correlation patch model is the average patch. The learning objective for this model is to construct a single image patch that is as close as possible to all examples of the facial feature as measured via the least-squares criterion:\n\nThe solution to this problem is exactly the average of all the feature-centered training image patches. Thus, in a way, the solution afforded by this objective is far simpler.\n\nIn the following figure, a comparison is shown for the response maps obtained by cross-correlating the average and correlation patch models with an example image. The respective average and correlation patch models are also shown, where the range of pixel values are normalized for visualization purposes. Although the two patch model types exhibit some similarities, the response maps they generate differ substantially. While the correlation patch model generates response maps that are highly peaked around the feature location, the response map generated by the average patch model is overly smooth and does not strongly distinguish the feature location from those close by. Inspecting the patch models' appearance, the correlation patch model is mostly gray, which corresponds to zero in the un-normalized pixel range, with strong positive and negative values strategically placed around prominent areas of the facial feature. Thus, it preserves only components of the training patches useful for discriminating it from misaligned configuration, which leads to highly peaked responses. In contrast, the average patch model encodes no knowledge of misaligned data. As a result, it is not well suited to the task of facial feature localization, where the task is to discriminate an aligned image patch from locally shifted versions of itself.\n\n## Accounting for global geometric transformations\n\nSo far, we have assumed that the training images are centered at the facial feature and are normalized with respect to global scale and rotation. In practice, the face can appear at any scale and rotation within the image during tracking. Thus, a mechanism must be devised to account for this discrepancy between the training and testing conditions. One approach is to synthetically perturb the training images in scale and rotation within the ranges one expects to encounter during deployment. However, the simplistic form of the detector as a correlation patch model often lacks the capacity to generate useful response maps for that kind of data. On the other hand, the correlation patch model does exhibit a degree of robustness against small perturbations in scale and rotation. Since motion between consecutive frames in a video sequence is relatively small, one can leverage the estimated global transformation of the face in the previous frame to normalize the current image with respect to scale and rotation. All that is needed to enable this procedure is to select a reference frame in which the correlation patch models are learned.\n\nThe `patch_models` class stores the correlation patch models for each facial feature as well as the reference frame in which they are trained. It is the `patch_models` class, rather than the `patch_model` class, that the face tracker code interfaces with directly, to obtain the feature detections. The following code snippet of the declaration of this class highlights its primary functionality:\n\n    class patch_models{\n    public:\n      Mat reference;      \/\/reference shape [x1;y1;...;xn;yn]\n      vector<patch_model> patches; \/\/patch model\/facial feature\n      ...\n      void\n      train(ft_data &data,        \/\/annotated image and shape data\n      const vector<Point2f> &ref, \/\/reference shape\n      const Size psize,           \/\/desired patch size\n      const Size ssize,           \/\/training search window size\n      const bool mirror = false,  \/\/use mirrored training data\n      const float var = 1.0,      \/\/variance of annotation error\n      const float lambda = 1e-6,  \/\/regularisation weight\n      const float mu_init = 1e-3, \/\/initial step size\n      const int nsamples = 1000,  \/\/number of samples\n      const bool visi = false);   \/\/visualise training procedure?\n      ...\n      vector<Point2f>\/\/location of peak responses\/feature in image\n      calc_peaks(\n      const Mat &im,    \/\/image to detect features in\n      const vector<Point2f> &points, \/\/current estimate of shape\n      const Size ssize = Size(21,21)); \/\/search window size\n      ...\n    };\n\nThe `reference` shape is stored as an interleaved set of (x, y) coordinates that are used to normalize the scale and rotation of the training images, and later during deployment that of the test images. In the `patch_models::train` function, this is done by first computing the similarity transform between the `reference` shape and the annotated shape for a given image using the `patch_models::calc_simil` function, which solves a similar problem to that in the `shape_model::procrustes` function, albeit for a single pair of shapes. Since the rotation and scale is common across all facial features, the image normalization procedure only requires adjusting this similarity transform to account for the centers of each feature in the image and the center of the normalized image patch. In `patch_models::train`, this is implemented as follows:\n\n    Mat S = this->calc_simil(pt),A(2,3,CV_32F);\n    A.fl(0,0) = S.fl(0,0); A.fl(0,1) = S.fl(0,1);\n    A.fl(1,0) = S.fl(1,0); A.fl(1,1) = S.fl(1,1);\n    A.fl(0,2) = pt.fl(2*i  ) - (A.fl(0,0)*(wsize.width -1)\/2 +\n                                A.fl(0,1)*(wsize.height-1)\/2);\n    A.fl(1,2) = pt.fl(2*i+1) - (A.fl(1,0)*(wsize.width -1)\/2 +\n                                A.fl(1,1)*(wsize.height-1)\/2);\n    Mat I; warpAffine(im,I,A,wsize,INTER_LINEAR+WARP_INVERSE_MAP);\n\nHere, `wsize` is the total size of the normalized training image, which is the sum of the patch size and the search region size. As just mentioned, that the top-left (2 x 2) block of the similarity transform from the reference shape to the annotated shape `pt`, which corresponds to the scale and rotation component of the transformation, is preserved in the affine transform passed to OpenCV's `warpAffine` function. The last column of the affine transform `A` is an adjustment that will render the i'th facial feature location centered in the normalized image after warping (that is, the normalizing translation). Finally, the `cv::warpAffine` function has the default setting of warping from the image to the reference frame. Since the similarity transform was computed for transforming the `reference` shape to the image-space annotations `pt`, the `WARP_INVERSE_MAP` flag needs to be set to ensure the function applies the warp in the desired direction. Exactly the same procedure is performed in the `patch_models::calc_peaks` function, with the additional step that the computed similarity transform between the reference and the current shape in the image-frame is re-used to un-normalize the detected facial features, placing them appropriately in the image:\n\n    vector<Point2f>\n    patch_models::calc_peaks(const Mat &im,\n      const vector<Point2f> &points,const Size ssize){\n      int n = points.size(); assert(n == int(patches.size()));\n      Mat pt = Mat(points).reshape(1,2*n);\n    **Mat S = this- >calc_simil(pt);**\n    **Mat Si = this- >inv_simil(S);**\n    **vector <Point2f> pts = this->apply_simil(Si,points);**\n      for(int i = 0; i < n; i++){\n        Size wsize = ssize + patches[i].patch_size();\n        Mat A(2,3,CV_32F),I;     \n        A.fl(0,0) = S.fl(0,0); A.fl(0,1) = S.fl(0,1);\n        A.fl(1,0) = S.fl(1,0); A.fl(1,1) = S.fl(1,1);\n        A.fl(0,2) = pt.fl(2*i  ) - (A.fl(0,0)*(wsize.width -1)\/2 +\n                                    A.fl(0,1)*(wsize.height-1)\/2);\n        A.fl(1,2) = pt.fl(2*i+1) - (A.fl(1,0)*(wsize.width -1)\/2 +\n                                    A.fl(1,1)*(wsize.height-1)\/2);\n        warpAffine(im,I,A,wsize,INTER_LINEAR+WARP_INVERSE_MAP);\n        Mat R = patches[i].calc_response(I,false);\n        Point maxLoc; minMaxLoc(R,0,0,0,&maxLoc);\n    **pts[i] = Point2f(pts[i].x + maxLoc.x - 0.5*ssize.width,**\n    **pts[i].y + maxLoc.y - 0.5*ssize.height);**\n    **}return this- >apply_simil(S,pts);**\n\nIn the first highlighted code snippet in the preceding code, both the forward and inverse similarity transforms are computed. The reason why the inverse transform is required here is so that the peaks of the response map for each feature can be adjusted according to the normalized locations of the current shape estimate. This must be performed before reapplying the similarity transform to place the new estimates of the facial feature locations back into the image frame using the `patch_models::apply_simil` function.\n\n## Training and visualization\n\nAn example program for training the patch models from annotation data can be found in `train_patch_model.cpp`. With the command-line argument `argv[1]` containing the path to the annotation data, training begins by loading the data into memory and removing incomplete samples:\n\n    ft_data data = load_ft<ft_data>(argv[1]);\n    data.rm_incomplete_samples();\n\nThe simplest choice for the reference shape in the `patch_models` class is the average shape of the training set, scaled to a desired size. Assuming that a shape model has previously been trained for this dataset, the reference shape is computed by first loading the shape model stored in `argv[2]` as follows:\n\n    shape_model smodel = load_ft<shape_model>(argv[2]);\n\nThis is followed by the computation of the scaled centered average shape:\n\n    smodel.p = Scalar::all(0.0);\n    smodel.p.fl(0) = calc_scale(smodel.V.col(0),width);\n    vector<Point2f> r = smodel.calc_shape();\n\nThe `calc_scale` function computes the scaling factor to transform the average shape (that is, the first column of `shape_model::V`) to one with a width of `width`. Once the reference shape `r `is defined, training the set of patch models can be done with a single function call:\n\n    patch_models pmodel; pmodel.train(data,r,Size(psize,psize),Size(ssize,ssize));\n\nThe optimal choices for the parameters `width`, `psize`, and `ssize` are application dependent; however, the default values of 100, 11, and 11, respectively, give reasonable results in general.\n\nAlthough the training process is quite simple, it can still take some time to complete. Depending on the number of facial features, the size of the patches, and the number of stochastic samples in the optimization algorithm, the training process can take anywhere between a few minutes to over an hour. However, since the training of each patch can be performed independently of all others, this process can be sped up substantially by parallelizing the training process across multiple processor-cores or machines.\n\nOnce training has been completed, the program in `visualize_patch_model.cpp` can be used to visualize the resulting patch models. As with the `visualize_shape_model.cpp` program, the aim here is to visually inspect the results to verify if anything went wrong during the training process. The program generates a composite image of all the patch models, `patch_model::P`, each centered at their respective feature location in the reference shape, `patch_models::reference`, and displays a bounding rectangle around the patch whose index is currently active. The `cv::waitKey` function is used to get user input for selecting the active patch index and terminating the program. The following image shows three examples of composite patch images learned for patch model with varying spatial support. Despite using the same training data, modifying the spatial support of the patch model appears to change the structure of the patch models substantially. Visually inspecting the results in this way can lend intuition into how to modify the parameters of the training process, or even the training process itself, in order to optimize results for a particular application.\n\n# Face detection and initialization\n\nThe method for face tracking described thus far has assumed that the facial features in the image are located within a reasonable proximity to the current estimate. Although this assumption is reasonable during tracking, where face motion between frames is often quite small, we are still faced with the dilemma of how to initialize the model in the first frame of the sequence. An obvious choice for this is to use OpenCV's in-built cascade detector to find the face. However, the placement of the model within the detected bounding box will depend on the selection made for the facial features to track. In keeping with the data-driven paradigm we have followed so far in this chapter, a simple solution is to learn the geometrical relationship between the face detection's bounding box and the facial features.\n\nThe `face_detector` class implements exactly this solution. A snippet of its declaration that highlights its functionality is given as follows:\n\n    class face_detector{ \/\/face detector for initialisation\n    public:\n      string detector_fname; \/\/file containing cascade classifier\n      Vec3f detector_offset; \/\/offset from center of detection\n      Mat reference;         \/\/reference shape\n      CascadeClassifier detector; \/\/face detector\n\n      vector<Point2f>  \/\/points describing detected face in image\n      detect(const Mat &im,          \/\/image containing face\n       const float scaleFactor = 1.1,\/\/scale increment\n       const int minNeighbours = 2,  \/\/minimum neighborhood size\n       const Size minSize = Size(30,30));\/\/minimum window size\n\n      void\n      train(ft_data &data,               \/\/training data\n      const string fname,                \/\/cascade detector\n      const Mat &ref,                    \/\/reference shape\n      const bool mirror = false,         \/\/mirror data?\n      const bool visi = false,           \/\/visualize training?\n      const float frac = 0.8, \/\/fraction of points in detection\n      const float scaleFactor = 1.1, \/\/scale increment\n      const int minNeighbours = 2,   \/\/minimum neighbourhood size\n      const Size minSize = Size(30,30)); \/\/minimum window size\n      ...\n    };\n\nThe class has four public member variables: the path to an object of type `cv::CascadeClassifier` called `detector_fname`, a set of offsets from a detection bounding box to the location and scale of the face in the image `detector_offset`, a reference shape to place in the bounding box `reference`, and a face detector `detector`. The primary function of use to a face tracking system is `face_detector::detect`, which takes an image as the input, along with standard options for the `cv::CascadeClassifier` class, and returns a rough estimate of the facial feature locations in the image. Its implementation is as follows:\n\n    Mat gray; \/\/convert image to grayscale and histogram equalize\n    if(im.channels() == 1)gray = im;\n    else cvtColor(im,gray,CV_RGB2GRAY);\n    Mat eqIm; equalizeHist(gray,eqIm);\n    vector<Rect> faces; \/\/detect largest face in image\n     detector.detectMultiScale(eqIm,faces,scaleFactor,\n                               minNeighbours,0\n                             |CV_HAAR_FIND_BIGGEST_OBJECT\n                             |CV_HAAR_SCALE_IMAGE,minSize);\n    if(faces.size() < 1){return vector<Point2f>();}\n\n    **Rect R = faces[0]; Vec3f scale = detector_offset*R.width;**\n    **int n = reference.rows\/2; vector <Point2f> p(n);**\n    **for(int i = 0; i < n; i++){ \/\/predict face placement**\n    **p[i].x = scale[2]*reference.fl(2*i  ) +**\n    **R.x + 0.5 * R.width  + scale[0];**\n    **p[i].y = scale[2]*reference.fl(2*i+1) +**\n    **R.y + 0.5 * R.height + scale[1];**\n    **}return p;**\n\nThe face is detected in the image in the usual way, except that the `CV_HAAR_FIND_BIGGEST_OBJECT` flag is set so as to enable tracking the most prominent face in the image. The highlighted code is where the reference shape is placed in the image in accordance with the detected face's bounding box. The `detector_offset` member variable consists of three components: an (x, y) offset of the center of the face from the center of the detection's bounding box, and the scaling factor that resizes the reference shape to best fit the face in the image. All three components are a linear function of the bounding box's width.\n\nThe linear relationship between the bounding box's width and the `detector_offset` variable is learned from the annotated dataset in the `face_detector::train` function. The learning process is started by loading the training data into memory and assigning the reference shape:\n\n    detector.load(fname.c_str()); detector_fname = fname; reference = ref.clone();\n\nAs with the reference shape in the `patch_models` class, a convenient choice for the reference shape is the normalized average face shape in the dataset. The `cv::CascadeClassifier` is then applied to each image (and optionally its mirrored counterpart) in the dataset and the resulting detection is checked to ensure that enough annotated points lie within the detected bounding box (see the figure towards the end of this section) to prevent learning from misdetections:\n\n    if(this->enough_bounded_points(pt,faces[0],frac)){\n      Point2f center = this->center_of_mass(pt);\n      float w = faces[0].width;\n      xoffset.push_back((center.x -\n                        (faces[0].x+0.5*faces[0].width ))\/w);\n      yoffset.push_back((center.y -\n                         (faces[0].y+0.5*faces[0].height))\/w);\n      zoffset.push_back(this->calc_scale(pt)\/w);\n    }\n\nIf more than a fraction of `frac` of the annotated points lie within the bounding box, the linear relationship between its width and the offset parameters for that image are added as a new entry in an STL `vector` class object. Here, the `face_detector::center_of_mass` function computes the center of mass of the annotated point set for that image and the `face_detector::calc_scale` function computes the scaling factor for transforming the reference shape to the centered annotated shape. Once all images have been processed, the `detector_offset` variable is set to the median over all of the image-specific offsets:\n\n    Mat X = Mat(xoffset),Xsort,Y = Mat(yoffset),Ysort,Z = Mat(zoffset),Zsort;\n    cv::sort(X,Xsort,CV_SORT_EVERY_COLUMN|CV_SORT_ASCENDING);\n    int nx = Xsort.rows;\n    cv::sort(Y,Ysort,CV_SORT_EVERY_COLUMN|CV_SORT_ASCENDING);\n    int ny = Ysort.rows;\n    cv::sort(Z,Zsort,CV_SORT_EVERY_COLUMN|CV_SORT_ASCENDING);\n    int nz = Zsort.rows;\n    detector_offset =\n        Vec3f(Xsort.fl(nx\/2),Ysort.fl(ny\/2),Zsort.fl(nz\/2));\n\nAs with the shape and patch models, the simple program in `train_face_detector.cpp` is an example of how a `face_detector` object can be built and saved for later use in the tracker. It first loads the annotation data and the shape model, and sets the reference shape as the mean-centered average of the training data (that is, the identity shape of the `shape_model` class):\n\n    ft_data data = load_ft<ft_data>(argv[2]);\n    shape_model smodel = load_ft<shape_model>(argv[3]);\n    smodel.set_identity_params();\n    vector<Point2f> r = smodel.calc_shape();\n    Mat ref = Mat(r).reshape(1,2*r.size());\n\nTraining and saving the face detector, then, consists of two function calls:\n\n    face_detector detector;\n    detector.train(data,argv[1],ref,mirror,true,frac);\n    save_ft<face_detector>(argv[4],detector);\n\nTo test the performance of the resulting shape-placement procedure, the program in `visualize_face_detector.cpp` calls the `face_detector::detect` function for each image in the video or camera input stream and draws the results on screen. An example of the results using this approach is shown in the following figure Although the placed shape does not match the individual in the image, its placement is close enough so that face tracking can proceed using the approach described in the following section:\n\n# Face tracking\n\nThe problem of face tracking can be posed as that of finding an efficient and robust way to combine the independent detections of various facial features with the geometrical dependencies they exhibit in order to arrive at an accurate estimate of facial feature locations in each image of a sequence. With this in mind, it is perhaps worth considering whether geometrical dependencies are at all necessary. In the following figure, the results of detecting the facial features with and without geometrical constraints are shown. These results clearly highlight the benefit of capturing the spatial interdependencies between facial features. The relative performance of these two approaches is typical, whereby relying strictly on the detections leads to overly noisy solutions. The reason for this is that the response maps for each facial feature cannot be expected to always peak at the correct location. Whether due to image noise, lighting changes, or expression variation, the only way to overcome the limitations of facial feature detectors is by leveraging the geometrical relationship they share with each other.\n\nA particularly simple, but surprisingly effective, way to incorporate facial geometry into the tracking procedure is by projecting the output of the feature detections onto the linear shape model's subspace. This amounts to minimizing the distance between the original points and its closest plausible shape that lies on the subspace. Thus, when the spatial noise in the feature detections is close to being Gaussian distributed, the projection yields the maximum likely solution. In practice, the distribution of detection errors on occasion does not follow a Gaussian distribution and additional mechanisms need to be introduced to account for this.\n\n## Face tracker implementation\n\nAn implementation of the face tracking algorithm can be found in the `face_tracker` class (see `face_tracker.cpp` and `face_tracker.hpp`). The following code is a snippet of its header that highlights its primary functionality:\n\n    class face_tracker{\n    public:\n      bool tracking;          \/\/are we in tracking mode?\n      fps_timer timer;        \/\/frames\/second timer\n      vector<Point2f> points; \/\/current tracked points\n      face_detector detector; \/\/detector for initialisation\n      shape_model smodel;     \/\/shape model\n      patch_models pmodel;    \/\/feature detectors\n\n      face_tracker(){tracking = false;}\n\n      int                             \/\/0 = failure\n      track(const Mat &im,            \/\/image containing face\n      const face_tracker_params &p =  \/\/fitting parameters\n      face_tracker_params());     \/\/default tracking parameters\n\n      void\n      reset(){                            \/\/reset tracker\n        tracking = false; timer.reset();\n      }\n      ...\n    protected:\n      ...\n      vector<Point2f>   \/\/points for fitted face in image\n      fit(const Mat &image,\/\/image containing face\n          const vector<Point2f> &init,   \/\/initial point estimates\n          const Size ssize = Size(21,21),\/\/search region size\n          const bool robust = false,     \/\/use robust fitting?\n          const int itol = 10,    \/\/maximum number of iterations\n          const float ftol = 1e-3);      \/\/convergence tolerance\n    };\n\nThe class has public member instances of the `shape_model`, `patch_models`, and `face_detector` classes. It uses the functionality of these three classes to effect tracking. The `timer` variable is an instance of the `fps_timer` class that keeps track of the frame-rate at which the `face_tracker::track` function is called and is useful for analyzing the effects patch and shape model configurations on the computational complexity of the algorithm. The `tracking` member variable is a flag to indicate the current state of the tracking procedure. When this flag is set to `false`, as it is in the constructor and the `face_tracker::reset` function, the tracker enters a Detection mode whereby the `face_detector::detect` function is applied to the next incoming image to initialize the model. When in the tracking mode, the initial estimate used for inferring facial feature locations in the next incoming image is simply their location in the previous frame. The complete tracking algorithm is implemented simply as follows:\n\n    int face_tracker::\n    track(const Mat &im,const face_tracker_params &p){\n      Mat gray; \/\/convert image to grayscale\n      if(im.channels()==1)gray=im;\n      else cvtColor(im,gray,CV_RGB2GRAY);\n      if(!tracking) \/\/initialize\n        points = detector.detect(gray,p.scaleFactor,\n                                 p.minNeighbours,p.minSize);\n      if((int)points.size() != smodel.npts())return 0;\n    **for(int level = 0; level < int(p.ssize.size()); level++)**\n    **points = this- >fit(gray,points,p.ssize[level],**\n    **p.robust,p.itol,p.ftol);**\n      tracking = true; timer.increment();  return 1;\n    }\n\nOther than bookkeeping operations, such as setting the appropriate `tracking` state and incrementing the tracking time, the core of the tracking algorithm is the multi-level fitting procedure, which is highlighted in the preceding code snippet. The fitting algorithm, implemented in the `face_tracker::fit` function, is applied multiple times with the different search window sizes stored in `face_tracker_params::ssize`, where the output of the previous stage is used as input to the next. In its simplest setting, the `face_tracker_params::ssize` function performs the facial feature detection around the current estimate of the shape in the image:\n\n    smodel.calc_params(init);\n    vector<Point2f> pts = smodel.calc_shape();\n    vector<Point2f> peaks = pmodel.calc_peaks(image,pts,ssize);\n\nIt also projects the result onto the face shape's subspace:\n\n    smodel.calc_params(peaks);        \n    pts = smodel.calc_shape();\n\nTo account for gross outliers in the facial features' detected locations, a robust model's fitting procedure can be employed instead of a simple projection by setting the `robust` flag to `true`. However, in practice, when using a decaying search window size (that is, as set in `face_tracker_params::ssize`), this is often unnecessary as gross outliers typically remain far from its corresponding point in the projected shape, and will likely lie outside the search region of the next level of the fitting procedure. Thus, the rate at which the search region size is reduced acts as an incremental outlier rejection scheme.\n\n## Training and visualization\n\nUnlike the other classes detailed in this chapter, training a `face_tracker` object does not involve any learning process. It is implemented in `train_face_tracker.cpp` simply as:\n\n    face_tracker tracker;\n    tracker.smodel = load_ft<shape_model>(argv[1]);\n    tracker.pmodel = load_ft<patch_models>(argv[2]);\n    tracker.detector = load_ft<face_detector>(argv[3]);\n    save_ft<face_tracker>(argv[4],tracker);\n\nHere `arg[1]` to `argv[4]` contain the paths to the `shape_model`, `patch_model`, `face_detector`, and `face_tracker` objects, respectively. The visualization for the face tracker in `visualize_face_tracker.cpp` is equally simple. Obtaining its input image stream either from a camera or video file, through the `cv::VideoCapture` class, the program simply loops until the end of the stream or until the user presses the _Q_ key, tracking each frame as it comes in. The user also has the option of resetting the tracker by pressing the _D_ key at any time.\n\n## Generic versus person-specific models\n\nThere are a number of variables in the training and tracking process that can be tweaked to optimize the performance for a given application. However, one of the primary determinants of tracking quality is the range of shape and appearance variability the tracker has to model. As a case in point, consider the generic versus person-specific case. A generic model is trained using annotated data from multiple identities, expressions, lighting conditions, and other sources of variability. In contrast, person-specific models are trained specifically for a single individual. Thus, the amount of variability it needs to account for is far smaller. As a result, person-specific tracking is often more accurate than its generic counter part by a large magnitude.\n\nAn illustration of this is shown in the following image. Here the generic model was trained using the MUCT dataset. The person-specific model was learned from data generated using the annotation tool described earlier in this chapter. The results clearly show a substantially better tracking offered by the person-specific model, capable of capturing complex expressions and head-pose changes, whereas the generic model appears to struggle even for some of the simpler expressions:\n\nIt should be noted that the method for face tracking described in this chapter is a bare-bones approach that serves to highlight the various components utilized in most non-rigid face tracking algorithms. The numerous approaches to remedy some of the drawbacks of this method are beyond the scope of this book and require specialized mathematical tools that are not yet supported by OpenCV's functionality. The relatively few commercial-grade face-tracking software packages available is testament to the difficulty of this problem in the general setting. Nonetheless, the simple approach described in this chapter can work remarkably well in constrained settings.\n\n# Summary\n\nIn this chapter we have built a simple face tracker that can work reasonably in constrained settings using only modest mathematical tools and OpenCV's substantial functionality for basic image processing and linear algebraic operations. Improvements to this simple tracker can be achieved by employing more sophisticated techniques in each of the three components of the tracker: the shape model, the feature detectors, and the fitting algorithm. The modular design of the tracker described in this section should allow these three components to be modified without substantial disruptions to the functionality of the others.\n\n# References\n\n  *  _Procrustes Problems_ , _Gower, John C. and_ _Dijksterhuis, Garmt B_ , _Oxford University Press_ , _2004_.\n\n# Chapter 7. 3D Head Pose Estimation Using AAM and POSIT\n\nA good computer vision algorithm can't be complete without great, robust capabilities as well as wide generalization and a solid math foundation. All these features accompany the work mainly developed by Tim Cootes with Active Appearance Models. This chapter will teach you how to create an **Active Appearance Model** of your own using OpenCV as well as how to use it to search for the closest position your model is located at in a given frame. Besides, you will learn how to use the POSIT algorithm and how to fit your 3D model in the \"posed\" image. With all these tools, you will be able to track a 3D model in a video, in real time; ain't it great? Although the examples focus on head pose, virtually any deformable model could use the same approach.\n\nAs you read the sections, you will come across the following topics:\n\n  * Active Appearance Models overview\n  * Active Shape Models overview\n  * Model instantiation--playing with the Active Appearance Model\n  * AAM search and fitting\n  * POSIT\n\nThe following list has an explanation of the terms that you will come across in the chapter:\n\n  *  **Active Appearance Model** ( **AAM** ): An object model containing statistical information of its shape and texture. It is a powerful way of capturing shape and texture variation from objects.\n  *  **Active Shape Model** ( **ASM** ): A statistical model of the shape of an object. It is very useful for learning shape variation.\n  *  **Principal Component Analysis** ( **PCA** ): An orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. This procedure is often used in dimensionality reduction. When reducing the dimension of the original problem, one can use a faster-fitting algorithm.\n  *  **Delaunay Triangulation (DT)**: For a set of P points in a plane, it is a triangulation such that no point in P is inside the circumcircle of any triangle in the triangulation. It tends to avoid skinny triangles. The triangulation is required for texture mapping.\n  *  **Affine transformation** : Any transformation that can be expressed in the form of a matrix multiplication followed by a vector addition. This can be used for texture mapping.\n  *  **Pose from Orthography and Scaling with Iterations** ( **POSIT** ): A computer vision algorithm that performs 3D pose estimation.\n\n# Active Appearance Models overview\n\nIn few words, Active Appearance Models are a nice model parameterization of combined texture and shape coupled to an efficient search algorithm that can tell exactly where and how a model is located in a picture frame. In order to do that, we will start with the _Active Shape Models_ section and will see that they are more closely related to landmark positions. A principal component analysis and some hands-on experience will be better described in the following sections. Then, we will be able to get some help from OpenCV's Delaunay functions and learn some triangulation. From that we will evolve to applying piecewise affine warps in the triangle texture warping section, where we can get information from an object's texture.\n\nAs we get enough background to build a good model, we can play with the techniques in the model instantiation section. We will then be able to solve the inverse problem through AAM search and fitting. These by themselves are already very useful algorithms for 2D and maybe even 3D image matching. But when one is able to get it to work, why not bridge it to **POSIT** ( **Pose from Orthography and Scaling with Iterations** ), another rock solid algorithm for 3D model fitting? Diving into the POSIT section will give us enough background to work with it in OpenCV, and we will then learn how to couple a head model to it, in the following section. This way, we can use a 3D model to fit the already matched 2D frame. And if a sharp reader wants to know where this will take us, it is just a matter of combining AAM and Posit in a frame-by-frame fashion to get real-time 3D tracking by detection for deformable models! These details will be covered in the tracking from webcam or video file section.\n\nIt is said that a picture is worth a thousand words; imagine if we get _N_ pictures. This way, what we previously mentioned is easily tracked in the following screenshot:\n\n**Overview of the chapter algorithms** : Given an image (upper-left image in the preceding screenshot), we can use an Active Appearance search algorithm to find the 2D pose of the human head. The upper-right side figure in the screenshot shows a previously trained Active Appearance model used in the search algorithm. After a pose has been found, POSIT can be applied to extend the result to a 3D pose. If the procedure is applied to a video sequence, 3D tracking by detection will be obtained.\n\n# Active Shape Models\n\nAs mentioned previously, AAMs require a shape model, and this role is played by Active Shape Models (ASMs). In the coming sections, we will create an ASM that is a statistical model of shape variation. The shape model is generated through the combination of shape variations. A training set of labeled images is required, as described in the article _Active Shape Models - Their Training and Application_, by Timothy Cootes. In order to build a face-shape model, several images marked with points on key positions of a face are required to outline the main features. The following screenshot shows such an example:\n\nThere are 76 landmarks on a face, which are taken from the MUCT dataset. These landmarks are usually marked up by hand and they outline several face features, such as mouth contour, nose, eyes, eyebrows, and face shape, since they are easier to track.\n\n### Note\n\n**Procrustes Analysis** : A form of statistical shape analysis used to analyze the distribution of a set of shapes. Procrustes superimposition is performed by optimally translating, rotating, and uniformly scaling the objects.\n\nIf we have the previously mentioned set of images, we can generate a statistical model of shape variation. Since the labeled points on an object describe the shape of that object, we firstly align all the sets of points into a coordinate frame using Procrustes Analysis, if required, and represent each shape by a vector, _x_. Then, we apply Principal Component Analysis (PCA) to the data. We can then approximate any example using the following formula:\n\nx = x + Ps bs\n\nIn the preceding formula, _x_ is the mean shape, _Ps_ is a set of orthogonal modes of variation, and _bs_ is a set of shape parameters. Well, in order to understand that better, we will create a simple application in the rest of this section, which will show us how to deal with PCA and shape models.\n\nWhy use PCA at all? Because PCA is going to really help us when it comes to reducing the number of parameters of our model. We will also see how much that helps when searching for it in a given image later in this chapter. A web page URL should be given for the following quote (<http:\/\/en.wikipedia.org\/wiki\/Principal_component_analysis>):\n\n> PCA can supply the user with a lower-dimensional picture, a \"shadow\" of this object when viewed from its (in some sense) most informative viewpoint. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.\n\nThis becomes clear when we see a screenshot such as the following:\n\nImage source: <http:\/\/en.wikipedia.org\/wiki\/File:GaussianScatterPCA.png>\n\nThe preceding screenshot shows the PCA of a multivariate Gaussian distribution centered at (2,3). The vectors shown are the eigenvectors of the covariance matrix, shifted so their tails are at the mean.\n\nThis way, if we wanted to represent our model with a single parameter, taking the direction from the eigenvector that points to the upper-right part of the screenshot would be a good idea. Besides, by varying the parameter a bit, we can extrapolate data and get values similar to the ones we are looking for.\n\n## Getting the feel of PCA\n\nIn order to get a feeling of how PCA could help us with our face model, we will start with an active shape model and test some parameters.\n\nSince face detection and tracking has been studied for a while, several face databases are available online for research purposes. We are going to use a couple of samples from the IMM database.\n\nFirst, let's understand how the PCA class works in OpenCV. We can conclude from the documentation that the PCA class is used to compute a special basis for a set of vectors, which consists of eigenvectors of the covariance matrix computed from the input set of vectors. This class can also transform vectors to and from the new coordinate space, using `project` and `backproject` methods. This new coordinate system can be quite accurately approximated by taking just the first few of its components. This means we can represent the original vector from a high-dimensional space with a much shorter vector consisting of the projected vector's coordinates in the subspace.\n\nSince we want a parameterization in terms of a few scalar values, the main method we will use from the class is the `backproject` method. It takes principal component coordinates of projected vectors and reconstructs the original ones. We could retrieve the original vectors if we retained all the components, but the difference will be very small if we just use a couple of components; that's one of the reasons for using PCA. Since we want some variability around the original vectors, our parameterized scalars will be able to extrapolate the original data.\n\nBesides, the PCA class can transform vectors to and from the new coordinate space, defined by the basis. Mathematically, it means that we compute projection of the vector to a subspace formed by a few eigenvectors corresponding to the dominant eigenvalues of the covariance matrix, as one can see from the documentation.\n\nOur approach will be annotating our face images with landmarks yielding a training set for our **point distribution model** ( **PDM** ). If we have _k_ aligned landmarks in two dimensions, our shape description becomes:\n\nX = { x1, y1, x2, y2,..., xk, yk}\n\nIt's important to note that we need consistent labeling across all image samples. So, for instance, if the left part of the mouth is landmark number _3_ in the first image, it will need to be number _3_ in all other images.\n\nThese sequences of landmarks will now form the shape outlines, and a given training shape can be defined as a vector. We generally assume this scattering is Gaussian in this space, and we use PCA to compute normalized eigenvectors and eigenvalues of the covariance matrix across all training shapes. Using the top-center eigenvectors, we create a matrix of dimensions _2k * m_ , which we will call _P_. This way, each eigenvector describes a principal mode of variation along the set.\n\nNow we can define a new shape through the following equation:\n\nX' = X' + Pb\n\nHere, _X'_ is the mean shape across all training images--we just average each of the landmarks--and _b_ is a vector of scaling values for each principal component. This leads us to create a new shape modifying the value of _b_. It's common to set _b_ to vary within three standard deviations so that the generated shape can fall inside the training set.\n\nThe following screenshot shows point-annotated mouth landmarks for three different pictures:\n\nAs can be seen in the preceding screenshot, the shapes are described by their landmark sequences. One could use a program like GIMP or ImageJ as well as building a simple application in OpenCV, in order to annotate the training images. We will assume the user has completed this process and saved the points as sequences of _x_ and _y_ landmark positions for all training images in a text file, which will be used in our PCA analysis. We will then add two parameters to the first line of this file, which is the number of training images and the number of read columns. So, for _k_ 2D points, this number will be _2*k_.\n\nIn the following data, we have an instance of this file, which was obtained through the annotation of three images from IMM database, in which _k_ is equal to 5:\n\n    3 10\n    265 311 303 321 337 310 302 298 265 311\n    255 315 305 337 346 316 305 309 255 315\n    262 316 303 342 332 315 298 299 262 316\n\nNow that we have annotated images, let's turn this data into our shape model. Firstly, load this data into a matrix. This will be achieved through the function `loadPCA`. The following code snippet shows the use of the `loadPCA` function:\n\n    PCA loadPCA(char* fileName, int& rows, int& cols,Mat& pcaset){\n      FILE* in = fopen(fileName,\"r\");\n      int a;\n      fscanf(in,\"%d%d\",&rows,&cols);\n\n      pcaset = Mat::eye(rows,cols,CV_64F);\n      int i,j;\n\n      for(i=0;i<rows;i++){\n        for(j=0;j<cols;j++){\n          fscanf(in,\"%d\",&a);\n          pcaset.at<double>(i,j) = a;\n        }\n      }\n\n      PCA pca(pcaset, \/\/ pass the data\n        Mat(), \/\/ we do not have a pre-computed mean vector,\n        \/\/ so let the PCA engine compute it\n        CV_PCA_DATA_AS_ROW, \/\/ indicate that the vectors\n        \/\/ are stored as matrix rows\n        \/\/ (use CV_PCA_DATA_AS_COL if the vectors are\n        \/\/ the matrix columns)\n        pcaset.cols\/\/ specify, how many principal components to retain\n        );\n      return pca;\n    }\n\nNote that our matrix is created in the line `pcaset = Mat::eye(rows,cols,CV_64F)` and that enough space is allocated for _2*k_ values. After the two _for_ loops load the data into the matrix, the PCA constructor is called with the data--an empty matrix--that could be our precomputed mean vector, if we wish to make it only once. We also indicate that our vectors will be stored as matrix rows and that we wish to keep the same number of given rows as the number of components, though we could use just a few ones.\n\nNow that we have filled our PCA object with our training set, it has everything it needs to backproject our shape according to given parameters. We do so by invoking `PCA.backproject`, passing the parameters as a row vector, and receiving the backprojected vector into the second argument.\n\nThe two previous screenshots show two different shape configurations according to the selected parameters chosen from the slider. The yellow and green shapes show training data, while the red one reflects the shape generated from the chosen parameters.\n\nA sample program can be used to experiment with active shape models, as it allows the user to try different parameters for the model. One is able to note that varying only the first two scalar values through the slider (which correspond to the first and second modes of variation) we can achieve a shape that is very close to the trained ones. This variability will help us when searching for a model in AAM, since it provides interpolated shapes. We will discuss triangulation, texturing, AAM, and AAM-search in the following sections.\n\n## Triangulation\n\nAs the shape we are looking for might be distorted, such as an open mouth for instance, we are required to map our texture back to a mean shape and then apply PCA to this normalized texture. In order to do that, we will use triangulation. The concept is very simple: we will create triangles including our annotated points and then map from one triangle to another. OpenCV comes with a handy function called `cvCreateSubdivDelaunay2D`, which creates an empty Delaunay triangulation. You can just consider this a good triangulation that will avoid skinny triangles.\n\n### Note\n\nIn mathematics and computational geometry, a  **Delaunay triangulation** for a set _P_ of points in a plane is a triangulation DT(P) such that no point in _P_ is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid skinny triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934 onwards.\n\nAfter a Delaunay subdivision has been initialized, one will use `cvSubdivDelaunay2DInsert` to populate points into the subdivision. The following lines of code will elucidate what a direct use of triangulation would be like:\n\n    CvMemStorage* storage;\n    CvSubdiv2D* subdiv;\n    CvRect rect = { 0, 0, 640, 480 };\n\n    storage = cvCreateMemStorage(0);\n    subdiv = cvCreateSubdivDelaunay2D(rect,storage);\n\n    std::vector<CvPoint> points;\n\n    \/\/initialize points somehow\n    ...\n\n    \/\/iterate through points inserting them in the subdivision\n    for(int i=0;i<points.size();i++){    \n      float x = points.at(i).x;\n      float y = points.at(i).y;\n      CvPoint2D32f floatingPoint = cvPoint2D32f(x, y);\n      cvSubdivDelaunay2DInsert( subdiv, floatingPoint );\n    }\n\nNote that our points are going to be inside a rectangular frame that is passed as a parameter to `cvCreateSubdivDelaunay2D`. In order to create a subdivision, we also need to create and initialize a memory storage structure. This can be seen in the first five lines of the preceding code. Then, in order to create the triangulation, we need to insert points using the `cvSubdivDelaunay2DInsert` function. This happens inside the _for_ loop in the preceding code. Please note that the points should already have been initialized, since they are the ones we'll usually be using as inputs. The following screenshot shows what the triangulation could look like:\n\nThis screenshot is the output of the preceding code for a set of points that yield the triangulation using Delaunay algorithm.\n\nAlthough subdivision creation is a very handy function of OpenCV, it might not be very easy iterating through all the triangles. The following code shows how to iterate through the edges of a subdivision:\n\n    void iterate(CvSubdiv2D* subdiv, CvNextEdgeType triangleDirection){\n\n      CvSeqReader reader;\n      CvPoint buf[3];\n\n      int i, j, total = subdiv->edges->total;\n      int elem_size = subdiv->edges->elem_size;\n      cvStartReadSeq((CvSeq*)(subdiv->edges), &reader, 0);\n\n      for(i = 0; i < total; i++){\n        CvQuadEdge2D* edge = (CvQuadEdge2D*)(reader.ptr);\n\n        if(CV_IS_SET_ELEM(edge)){\n\n          CvSubdiv2DEdge t = (CvSubdiv2DEdge)edge;\n\n          for(j=0;j<3;j++){\n\n            CvSubdiv2DPoint* pt = cvSubdiv2DEdgeOrg(t);\n            if(!pt) break;\n            buf[j] = cvPoint(cvRound(pt->pt.x), cvRound(pt->pt.y));\n            t = cvSubdiv2DGetEdge(t, triangleDirection);\n          }\n        }\n        CV_NEXT_SEQ_ELEM(elem_size, reader);\n      }\n    }\n\nGiven a subdivision, we initialize its edge reader calling the `cvStartReadSeq` function. From OpenCV's documentation, we have the following quoted definition:\n\n> The function initializes the reader state. After that, all the sequence elements from the first one down to the last one can be read by subsequent calls of the macro CV_READ_SEQ_ELEM( read_elem, reader ) in the case of forward reading and by using CV_REV_READ_SEQ_ELEM( read_elem, reader ) in the case of reverse reading. Both macros put the sequence element to read_elem and move the reading pointer toward the next element.\n\nAn alternative way of getting the following element is by using the macro `CV_NEXT_SEQ_ELEM( elem_size, reader )`,which is preferred if sequence elements are large. In this case, we use `CvQuadEdge2D* edge = (CvQuadEdge2D*)(reader.ptr)` to access the edge, which is just a cast from a reader pointer to a `CvQuadEdge2D` pointer. The macro `CV_IS_SET_ELEM` only checks whether the specified edge is occupied or not. Given an edge, for us to get the source point we need to call the `cvSubdiv2DEdgeOrg` function. In order to run around a triangle, we repeatedly call `cvSubdiv2DGetEdge` and pass the triangle direction, which could be `CV_NEXT_AROUND_LEFT` or `CV_NEXT_AROUND_RIGHT`, for instance.\n\n## Triangle texture warping\n\nNow that we've been able to iterate through the triangles of a subdivision, we are able to warp one triangle from an original annotated image into a generated distorted one. This is useful for mapping the texture from the original shape to a distorted one. The following piece of code will guide the process:\n\n    void warpTextureFromTriangle(Point2f srcTri[3], Mat originalImage, Point2f dstTri[3], Mat warp_final){\n\n      Mat warp_mat(2, 3, CV_32FC1);\n      Mat warp_dst, warp_mask;\n      CvPoint trianglePoints[3];\n      trianglePoints[0] = dstTri[0];\n      trianglePoints[1] = dstTri[1];\n      trianglePoints[2] = dstTri[2];\n      warp_dst  = Mat::zeros(originalImage.rows, originalImage.cols, originalImage.type());\n      warp_mask = Mat::zeros(originalImage.rows, originalImage.cols, originalImage.type());\n\n      \/\/\/ Get the Affine Transform\n      warp_mat = getAffineTransform(srcTri, dstTri);\n\n      \/\/\/ Apply the Affine Transform to the src image\n      warpAffine(originalImage, warp_dst, warp_mat, warp_dst.size());\n      cvFillConvexPoly(new IplImage(warp_mask), trianglePoints, 3, CV_RGB(255,255,255), CV_AA, 0);  \n      warp_dst.copyTo(warp_final, warp_mask);\n\n    }\n\nThe preceding code assumes we have the triangle vertices packed in the `srcTri` array and the destination one packed in the `dstTri` array. The 2 x 3 `warp_mat` matrix is used to get the affine transformation from the source triangles to the destination ones. More information can be quoted from OpenCV's _cvGetAffineTransform_ documentation:\n\nThe function `cvGetAffineTransform` calculates the matrix of an affine transform such that:\n\nIn the preceding equation, destination _(i)_ is equal to (xi',yi'), source _(i)_ is equal to (xi, yi), and _i_ is equal to 0, 1, 2.\n\nAfter retrieving the affine matrix, we can apply the affine transformation to the source image. This is done through the  `warpAffine` function. Since we don't want to do it in the entire image--we want to focus on our triangle--a mask can be used for this task. This way, the last line copies only the triangle from our original image with our just-created mask, which was made through a `cvFillConvexPoly` call.\n\nThe following screenshot shows the result of applying this procedure to every triangle in an annotated image. Note that the triangles are mapped back to the alignment frame, which faces toward the viewer. This procedure is used to create the statistical texture of the AAM.\n\nThe preceding screenshot shows the result of warping all the mapped triangles in the left image to a mean reference frame.\n\n# Model Instantiation - playing with the Active Appearance Model\n\nAn interesting aspect of AAMs is their ability to easily interpolate the model that we trained our images on. We can get used to their amazing representational power through the adjustment of a couple of shape or model parameters. As we vary shape parameters, the destination of our warp changes according to the trained shape data. On the other hand, while appearance parameters are modified, the texture on the base shape is modified. Our warp transforms will take every triangle from the base shape to the modified destination shape so we can synthesize a closed mouth on top of an open mouth, as shown in the following screenshot:\n\nThis preceding screenshot shows a synthesized closed mouth obtained through active appearance model instantiation on top of another image. It shows how one could combine a smiling mouth with an admired face, extrapolating the trained images.\n\nThe preceding screenshot was obtained by changing only three parameters for shape and three for the texture, which is the goal of AAMs. A sample application has been developed and is available at <http:\/\/www.packtpub.com\/> for the reader to try out AAM. Instantiating a new model is just a question of sliding the equation parameters, as defined in the section _Getting the feel of PCA_. You should note that AAM search and fitting rely on this flexibility to find the best match for a given captured frame of our model in a different position from the trained ones. We will see this in the next section.\n\n# AAM search and fitting\n\nWith our fresh new combined shape and texture model, we have found a nice way to describe how a face could change not only in shape but also in appearance. Now we want to find which set of _p_ shape and _\u03bb_ appearance parameters will bring our model as close as possible to a given input image _I(x)_. We could naturally calculate the error between our instantiated model and the given input image in the coordinate frame of _I(x)_ , or map the points back to the base appearance and calculate the difference there. We are going to use the latter approach. This way, we want to minimize the following function:\n\nIn the preceding equation, _S0_ denotes the set of pixels _x_ is equal to _(x,y)T_ that lie inside the AAMs base mesh, _A_ _0_ _(x)_ is our base mesh texture, _A_ _i_ _(x)_ is appearance images from PCA, and _W(x;p)_ is the warp that takes pixels from the input image back to the base mesh frame.\n\nSeveral approaches have been proposed for this minimization through years of studying. The first idea was use an additive approach, in which _\u2206 p_ _i_ and _\u2206 \u03bb_ _i_ were calculated as linear functions of the error image and then shape parameter p and appearance \u03bb were updated as _p_ _i_ <- _p_ _i_ \\+ _\u2206 p_ _i_ and _\u03bb_ _i_ <- _\u03bb_ _i_ \\+ _\u2206 \u03bb_ _i_ , in the _i-th_ iteration. Although convergence can occur sometimes, the delta doesn't always depend on current parameters, and this might lead to divergence. Another approach--which was studied based on the gradient descent algorithms--was very slow, so another way of finding convergence was sought. Instead of updating the parameters, the whole warp could be updated. This way, a compositional approach was proposed by Ian Mathews and Simon Baker in a famous paper called _Active Appearance Models Revisited_. More details can be found in the paper, but the important contribution it gave to fitting was that it brought the most intensive computation to a pre-compute step, as seen in the following screenshot:\n\nNote that the update occurs in terms of a compositional step as seen in step **(9)** (see the previous screenshot). Equations **(40)** and **(41)** from the paper can be seen in the following screenshots:\n\nAlthough the algorithm just mentioned will mostly converge very well from a position near the final one, this might not be the case when there's a big difference in rotation, translation, or scale. We can bring more information to the convergence through the parameterization of a global 2D similarity transform. This is equation _42_ in the paper and is shown as follows:\n\nIn the preceding equation, the four parameters _q_ = ( _a_ , _b_ , _t_ _x_ , _t_ _y_ )T have the following interpretations. The first pair ( _a_ , _b_ ) are related to the scale _k_ and rotation \u03b8: _a_ is equal to k cos \u03b8 \\- 1, and _b_ = k sin \u03b8. The second pair ( _t_ _x_ , _t_ _y_ ) are the _x_ and _y_ translations, as proposed in the Active Appearance Models Revisited paper.\n\nWith a bit more of math transformations, one can finally use the preceding algorithm to find the best image fit with a global 2D transform.\n\nAs the warp compositional algorithm has several performance advantages, we will use the one described in the AAM Revisited paper, the inverse compositional project-out algorithm. Remember that in this method, the effects of appearance variation during fitting can be precomputed--or projected out-- improving AAM fitting performance.\n\nThe following screenshot shows convergence for different images from the MUCT dataset using the inverse compositional project-out AAM fitting algorithm.\n\nThe preceding screenshot shows successful convergences--over faces outside the AAM training set--using the inverse compositional project-out AAM fitting algorithm.\n\n# POSIT\n\nAfter we have found the 2D position of our landmark points, we can derive the 3D pose of our model using the POSIT. The pose _P_ of a 3D object is defined as the 3 x 3 rotation matrix _R_ and the 3D translation vector _T_ , hence _P_ is equal to _[ R | T ]_.\n\n### Note\n\nMost of this section is based on the _OpenCV POSIT_ tutorial by Javier Barandiaran.\n\nAs the name implies, POSIT uses the **Pose from Orthography and Scaling** ( **POS** ) algorithm in several iterations, so it is an acronym for POS with Iterations. The hypothesis for its working is that we can detect and match in the image four or more non-coplanar feature points of the object and that we know their relative geometry on the object.\n\nThe main idea of the algorithm is that we can find a good approximation to the object pose, supposing that all the model points are in the same plane, since their depths are not very different from one another if compared to the distance from the camera to a face. After the initial pose is obtained, the rotation matrix and translation vector of the object are found by solving a linear system. Then, the approximate pose is iteratively used to better compute scaled orthographic projections of the feature points, followed by POS application to these projections instead of the original ones. For more information, you can refer to the paper by DeMenton, _Model-Based Object Pose in 25 Lines of Code_.\n\n## Diving into POSIT\n\nIn order for POSIT to work, you need at least four non-coplanar 3D model points and their respective matchings in the 2D image. We add to that a termination criteria--since POSIT is an iterative algorithm--which generally is a number of iterations or a distance parameter. We then call the function `cvPOSIT`, which yields the rotation matrix and the translation vector.\n\nAs an example, we will follow the tutorial from Javier Barandiaran, which uses POSIT to obtain the pose of a cube. The model is created with four points. It is initialized with the following code:\n\n    float cubeSize = 10.0;\n    std::vector<CvPoint3D32f> modelPoints;\n    modelPoints.push_back(cvPoint3D32f(0.0f, 0.0f, 0.0f));\n    modelPoints.push_back(cvPoint3D32f(0.0f, 0.0f, cubeSize));\n    modelPoints.push_back(cvPoint3D32f(cubeSize, 0.0f, 0.0f));\n    modelPoints.push_back(cvPoint3D32f(0.0f, cubeSize, 0.0f));\n    CvPOSITObject *positObject = cvCreatePOSITObject( &modelPoints[0], static_cast<int>(modelPoints.size()) );\n\nNote that the model itself is created with the `cvCreatePOSITObject` method, which returns a `CvPOSITObject` method that will be used in the `cvPOSIT` function. Be aware that the pose will be calculated referring to the first model point, which makes it a good idea to put it at the origin.\n\nWe then need to put the 2D image points in another vector. Remember that they must be put in the array in the same order that the model points were inserted in; this way, the i'th 2D image point matches the i'th 3D model point. A catch here is that the origin for the 2D image points is located at the center of the image, which might require you to translate them. You can insert the following 2D image points (of course, they will vary according to the user's matching):\n\n      std::vector<CvPoint2D32f> srcImagePoints;\n      srcImagePoints.push_back( cvPoint2D32f( -48, -224 ) );\n      srcImagePoints.push_back( cvPoint2D32f( -287, -174 ) );\n      srcImagePoints.push_back( cvPoint2D32f( 132, -153 ) );\n      srcImagePoints.push_back( cvPoint2D32f( -52, 149 ) );\n\nNow, you only need to allocate memory for the matrixes and create termination criteria, followed by a call to `cvPOSIT`, as shown in the following code snippet:\n\n      \/\/Estimate the pose\n      CvMatr32f rotation_matrix = new float[9];\n      CvVect32f translation_vector = new float[3];  \n      CvTermCriteria criteria = cvTermCriteria(CV_TERMCRIT_EPS | CV_TERMCRIT_ITER, 100, 1.0e-4f);\n      cvPOSIT( positObject, &srcImagePoints[0], FOCAL_LENGTH, criteria, rotation_matrix, translation_vector );\n\nAfter the iterations, `cvPOSIT` will store the results in `rotation_matrix` and `translation_vector`. The following screenshot shows the inserted `srcImagePoints` with white circles as well as a coordinate axis showing the rotation and translation results:\n\nWith reference to the preceding screenshot, let's see the input points and results of running the POSIT algorithm:\n\n  * The white circles show input points, while the coordinate axes show the resulting model pose.\n  * Make sure you use the focal length of your camera as obtained through a calibration process. You might want to check one of the calibration procedures available in the _Camera calibration_ section in Chapter 2, _Marker-based Augmented Reality on iPhone or iPad_. The current implementation of POSIT will only allow square pixels, so there won't be room for focal length in the x and y axes.\n  * Expect the rotation matrix in the following format:\n\n[rot[0] rot[1] rot[2]]\n\n[rot[3] rot[4] rot[5]]\n\n[rot[6] rot[7] rot[8]]\n\n  * The translation vector will be in the following format:\n\n[trans[0]]\n\n[trans[1]]\n\n[trans[2]]\n\n## POSIT and head model\n\nIn order to use POSIT as a tool for head pose, you will need to use a 3D head model. There is one available from the Institute of Systems and Robotics of the University of Coimbra and can be found at <http:\/\/aifi.isr.uc.pt\/Downloads\/OpenGL\/glAnthropometric3DModel.cpp>. Note that the model can be obtained from where it says:\n\n    float Model3D[58][3]= {{-7.308957,0.913869,0.000000}, ...\n\nThe model can be seen in the following screenshot:\n\nThe preceding screenshot shows a 58-point 3D head model available for POSIT.\n\nIn order to get POSIT to work, the point corresponding to the 3D head model must be matched accordingly. Note that at least four non-coplanar 3D points and their corresponding 2D projections are required for POSIT to work, so these must be passed as parameters, pretty much as described in the _Diving into POSIT_ section. Note that this algorithm is linear in terms of the number of matched points. The following screenshot shows how matching should be done:\n\nThe preceding screenshot shows the correctly matched points of a 3D head model and an AAM mesh.\n\n## Tracking from webcam or video file\n\nNow that all the tools have been assembled to get 6 degrees of freedom head tracking, we can apply it to a camera stream or video file. OpenCV provides the `VideoCapture` class that can be used in the following manner (see the _Accessing the webcam_ section in Chapter 1, _Cartoonifier and Skin Changer for Android_ , for more details):\n\n    #include \"cv.h\"\n    #include \"highgui.h\"\n\n    using namespace cv;\n\n    int main(int, char**)\n    {\n       VideoCapture cap(0);\/\/ opens the default camera, could use a\n                            \/\/ video file path instead\n\n      if(!cap.isOpened()) \/\/ check if we succeeded\n            return -1;\n\n      AAM aam = loadPreviouslyTrainedAAM();\n      HeadModel headModel = load3DHeadModel();\n      Mapping mapping = mapAAMLandmarksToHeadModel();\n\n      Pose2D pose = detectFacePosition();\n\n        while(1)\n        {\n        Mat frame;\n        cap >> frame; \/\/ get a new frame from camera\n\n        Pose2D new2DPose = performAAMSearch(pose, aam);\n        Pose3D new3DPose = applyPOSIT(new2DPose, headModel, mapping);\n\n        if(waitKey(30) >= 0) break;\n        }\n\n        \/\/ the camera will be deinitialized automatically in VideoCapture \/\/ destructor\n        return 0;\n    }\n\nThe algorithm works like this. A video capture is initialized through `VideoCapture cap(0)`, so that the default webcam is used. Now that we have video capture working, we also need to load our trained active appearance model, which will occurs in the pseudocode `loadPreviouslyTrainedAAM` mapping. We also load the 3D head model for POSIT and the mapping of landmark points to 3D head points in our mapping variable.\n\nAfter everything we need has been loaded, we will need to initialize the algorithm from a known pose, which is a known 3D position, known rotation, and a known set of AAM parameters. This could be made automatically through OpenCV's highly documented Haar features classifier face detector (more details in the _Face Detection_ section of Chapter 6, _Non-rigid Face Tracking_ , or in OpenCV's cascade classifier documentation), or we could manually initialize the pose from a previously annotated frame. A brute-force approach, which would be to run an AAM fitting for every rectangle, could also be used, since it would be very slow only during the first frame. Note that by initialization we mean finding the 2D landmarks of the AAM through their parameters.\n\nWhen everything is loaded, we can iterate through the main loop delimited by the _while_ loop. In this loop, we first query the next grabbed frame, and we then run an active appearance model fit so that we can find landmarks on the next frame. Since the current position is very important at this step, we pass it as a parameter to the pseudocode function `performAAMSearch(pose,aam)`. If we find the current pose, which is signaled through error image convergence, we will get the next landmark positions so we can provide them to POSIT. This happens in the following line, `applyPOSIT(new2DPose, headModel, mapping)`, where the new 2D pose is passed as a parameter, as also our previously loaded `headModel` and the mapping. After that, we can render any 3D model in the obtained pose like a coordinate axis or an augmented reality model. As we have landmarks, more interesting effects can be obtained through model parameterization, such as opening a mouth or changing eyebrow position.\n\nAs this procedure relies on previous pose for next estimation, we could accumulate errors and diverge from head position. A workaround could be to reinitialize the procedure every time it happens, checking a given error image threshold. Another factor to pay attention to is the use of filters when tracking, since jittering can occur. A simple mean filter for each of the translation and rotation coordinates can give reasonable results.\n\n# Summary\n\nIn this chapter, we have discussed how active appearance models can be combined with the POSIT algorithm in order to obtain a 3D head pose. An overview on how to create, train, and manipulate AAMs has been given and the reader can use this background for any other field, such as medical, imaging, or industry. Besides dealing with AAMs, we got familiar to Delaunay subdivisions and learned how to use such an interesting structure as a triangulated mesh. We also showed how to perform texture mapping in the triangles using OpenCV functions. Another interesting topic was approached in AAM fitting. Although only the inverse compositional project-out algorithm was described, we could easily obtain the results of years of research by simply using its output.\n\nAfter enough theory and practice of AAMs, we dived into the details of POSIT in order to couple 2D measurements to 3D ones explaining how to fit a 3D model using matchings between model points. We concluded the chapter by showing how to use all the tools in an online face tracker by detection, which yields 6 degrees of freedom head pose--3 degrees for rotation-- and 3 for translation. The complete code for this chapter can be downloaded from <http:\/\/www.packtpub.com\/>.\n\n# References\n\n  *  _Active Appearance Models_ , _T.F. Cootes_ , _G. J. Edwards_ , _and_ _C. J. Taylor_ , _ECCV_ , _2:484 -498_, _1998_ (<http:\/\/www.cs.cmu.edu\/~efros\/courses\/AP06\/Papers\/cootes-eccv-98.pdf>)\n  *  _Active Shape Models - Their Training and Application_, _T.F. Cootes_ , _C.J. Taylor_ , _D.H. Cooper_ , _and_ _J. Graham_ , _Computer Vision and Image Understanding_ , _(61): 38 -59_, _1995_ (<http:\/\/www.wiau.man.ac.uk\/~bim\/Papers\/cviu95.pdf>)\n  *  _The MUCT Landmarked Face Database_ , _S. Milborrow_ , _J. Morkel_ , and _F. Nicolls_ , _Pattern Recognition Association of South Africa_ , _2010_ (<http:\/\/www.milbo.org\/muct\/>)\n  *  _The IMM Face Database - An Annotated Dataset of 240 Face Images_, _Michael M. Nordstrom_ , _Mads Larsen_ , _Janusz Sierakowski_ , _and_ _Mikkel B. Stegmann_ , _Informatics and Mathematical Modeling_ , _Technical University of Denmark_ , _2004_ , (<http:\/\/www2.imm.dtu.dk\/~aam\/datasets\/datasets.html>)\n  *  _Sur la sph ere vide_, _B. Delaunay_ , _Izvestia Akademii Nauk SSSR_ , _Otdelenie Matematicheskikh i Estestvennykh Nauk_ , _7:793 -800_, _1934_\n  *  _Active Appearance Models for Facial Expression Recognition and Monocular Head Pose Estimation Master Thesis_ , _P. Martins_ , _2008_\n  *  _Active Appearance Models Revisited_ , _International Journal of Computer Vision_ , _Vol. 60_ , _No. 2_ , _pp. 135 - 164_ , _I. Mathews_ _and_ _S. Baker_ , _November_ , _2004_ (<http:\/\/www.ri.cmu.edu\/pub_files\/pub4\/matthews_iain_2004_2\/matthews_iain_2004_2.pdf>)\n  *  _POSIT Tutorial_ , _Javier Barandiaran_ (<http:\/\/opencv.willowgarage.com\/wiki\/Posit>)\n  *  _Model-Based Object Pose in 25 Lines of Code, International Journal of Computer Vision_ , _15_ , _pp. 123-141_ , _Dementhon_ _and_ _L.S Davis_ , _1995_ (<http:\/\/www.cfar.umd.edu\/~daniel\/daniel_papersfordownload\/Pose25Lines.pdf>)\n\n# Chapter 8. Face Recognition using Eigenfaces or Fisherfaces\n\nThis chapter will introduce concepts in face detection and face recognition and provide a project to detect faces and recognize them when it sees them again. Face recognition is both a popular and a difficult topic, and many researchers devote years to the field of face recognition. So this chapter will explain simple methods of face recognition, giving the reader a good start if they want to explore more complex methods.\n\nIn this chapter, we cover the following:\n\n  * Face detection\n  * Face preprocessing\n  * Training a machine-learning algorithm from collected faces\n  * Face recognition\n  * Finishing touches\n\n# Introduction to face recognition and face detection\n\nFace recognition is the process of putting a label to a known face. Just like humans learn to recognize their family, friends and celebrities just by seeing their face, there are many techniques for a computer to learn to recognize a known face. These generally involve four main steps:\n\n  1.  **Face detection** : It is the process of locating a face region in an image (a large rectangle near the center of the following screenshot). This step does not care who the person is, just that it is a human face.\n  2.  **Face preprocessing** : It is the process of adjusting the face image to look more clear and similar to other faces (a small grayscale face in the top-center of the following screenshot).\n  3.  **Collect and learn faces** : It is the process of saving many preprocessed faces (for each person that should be recognized), and then learning how to recognize them.\n  4.  **Face recognition** : It is the process that checks which of the collected people are most similar to the face in the camera (a small rectangle on the top-right of the following screenshot).\n\n### Note\n\nNote that the phrase face recognition is often used by the general public for finding positions of faces (that is, face detection, as described in step 1), but this book will use the formal definition of face recognition referring to step 4 and face detection referring to step 1.\n\nThe following screenshot shows the final WebcamFaceRec project, including a small rectangle at the top-right corner highlighting the recognized person. Also notice the confidence bar that is next to the preprocessed face (a small face at the top-center of the rectangle marking the face), which in this case shows roughly 70 percent confidence that it has recognized the correct person.\n\nThe current face detection techniques are quite reliable in real-world conditions, whereas current face recognition techniques are much less reliable when used in real-world conditions. For example, it is easy to find research papers showing face recognition accuracy rates above 95 percent, but when testing those same algorithms yourself, you may often find that accuracy is lower than 50 percent. This comes from the fact that current face recognition techniques are very sensitive to exact conditions in the images, such as the type of lighting, direction of lighting and shadows, exact orientation of the face, expression of the face, and the current mood of the person. If they are all kept constant when training (collecting images) as well as when testing (from the camera image), then face recognition should work well, but if the person was standing to the left-hand side of the lights in a room when training, and then stood to the right-hand side while testing with the camera, it may give quite bad results. So the dataset used for training is very important.\n\nFace preprocessing (step 2) aims to reduce these problems, such as by making sure the face always appears to have similar brightness and contrast, and perhaps makes sure the features of the face will always be in the same position (such as aligning the eyes and\/or nose to certain positions). A good face preprocessing stage will help improve the reliability of the whole face recognition system, so this chapter will place some emphasis on face preprocessing methods.\n\nDespite the big claims about face recognition for security in the media, it is unlikely that the current face recognition methods alone are reliable enough for any true security system, but they can be used for purposes that don't need high reliability, such as playing personalized music for different people entering a room or a robot that says your name when it sees you. There are also various practical extensions to face recognition, such as gender recognition, age recognition, and emotion recogniion.\n\n## Step 1: Face detection\n\nUntil year 2000, there were many different techniques used for finding faces, but all of them were either very slow, very unreliable, or both. A major change came in 2001 when Viola and Jones invented the Haar-based cascade classifier for object detection, and in 2002 when it was improved by Lienhart and Maydt. The result is an object detector that is both fast (can detect faces in real time on a typical desktop with a VGA webcam) and reliable (detects approximately 95 percent of frontal faces correctly). This object detector revolutionized the field of face recognition (as well as that of robotics and computer vision in general), as it finally allowed real-time face detection and face recognition, especially as Lienhart himself wrote the object detector that comes free with OpenCV! It works not only for frontal faces but also side-view faces (referred to as profile faces), eyes, mouths, noses, company logos, and many other objects.\n\nThis object detector was extended in OpenCV v2.0 to also use LBP features for detection based on work by Ahonen, Hadid and Pietikainen in 2006, as LBP-based detectors are potentially several times faster than Haar-based detectors, and don't have the licensing issues that many Haar detectors have.\n\nThe basic idea of the Haar-based face detector is that if you look at most frontal faces, the region with the eyes should be darker than the forehead and cheeks, and the region with the mouth should be darker than cheeks, and so on. It typically performs about 20 stages of comparisons like this to decide if it is a face or not, but it must do this at each possible position in the image and for each possible size of the face, so in fact it often does thousands of checks per image. The basic idea of the LBP-based face detector is similar to the Haar-based one, but it uses histograms of pixel intensity comparisons, such as edges, corners, and flat regions.\n\nRather than have a person decide which comparisons would best define a face, both Haar- and LBP-based face detectors can be automatically trained to find faces from a large set of images, with the information stored as XML files to be used later. These cascade classifier detectors are typically trained using at least 1,000 unique face images and 10,000 non-face images (for example, photos of trees, cars, and text), and the training process can take a long time even on a multi-core desktop (typically a few hours for LBP but one week for Haar!). Luckily, OpenCV comes with some pretrained Haar and LBP detectors for you to use! In fact you can detect frontal faces, profile (side-view) faces, eyes, or noses just by loading different cascade classifier XML files to the object detector, and choose between the Haar or LBP detector, based on which XML file you choose.\n\n### Implementing face detection using OpenCV\n\nAs mentioned previously, OpenCV v2.4 comes with various pretrained XML detectors that you can use for different purposes. The following table lists some of the most popular XML files:\n\nType of cascade classifier | XML filename\n\n---|---\n\nFace detector (default) | `haarcascade_frontalface_default.xml`\n\nFace detector (fast Haar) | `haarcascade_frontalface_alt2.xml`\n\nFace detector (fast LBP) | `lbpcascade_frontalface.xml`\n\nProfile (side-looking) face detector | `haarcascade_profileface.xml`\n\nEye detector (separate for left and right) | `haarcascade_lefteye_2splits.xml`\n\nMouth detector | `haarcascade_mcs_mouth.xml`\n\nNose detector | `haarcascade_mcs_nose.xml`\n\nWhole person detector | `haarcascade_fullbody.xml`\n\nHaar-based detectors are stored in the folder `data\\haarcascades` and LBP-based detectors are stored in the folder `data\\lbpcascades` of the OpenCV root folder, such as `C:\\opencv\\data\\lbpcascades\\`.\n\nFor our face recognition project, we want to detect frontal faces, so let's use the LBP face detector because it is the fastest and doesn't have patent licensing issues. Note that this pretrained LBP face detector that comes with OpenCV v2.x is not tuned as well as the pretrained Haar face detectors, so if you want more reliable face detection then you may want to train your own LBP face detector or use a Haar face detector.\n\n### Loading a Haar or LBP detector for object or face detection\n\nTo perform object or face detection, first you must load the pretrained XML file using OpenCV's `CascadeClassifier` class as follows:\n\n    CascadeClassifier faceDetector;\n    faceDetector.load(faceCascadeFilename);\n\nThis can load Haar or LBP detectors just by giving a different filename. A very common mistake when using this is to provide the wrong folder or filename, but depending on your build environment, the `load()` method will either return `false` or generate a C++ exception (and exit your program with an assert error). So it is best to surround the `load()` method with a `try`\/`catch` block and display a nice error message to the user if something went wrong. Many beginners skip checking for errors, but it is crucial to show a help message to the user when something did not load correctly, otherwise you may spend a very long time debugging other parts of your code before eventually realizing something did not load. A simple error message can be displayed as follows:\n\n    CascadeClassifier faceDetector;\n    try {\n    **faceDetector.load(faceCascadeFilename);**\n    } catch (cv::Exception e) {}\n    if ( faceDetector.empty() ) {\n        cerr << \"ERROR: Couldn't load Face Detector (\";\n        cerr << faceCascadeFilename << \")!\" << endl;\n        exit(1);\n    }\n\n### Accessing the webcam\n\nTo grab frames from a computer's webcam or even from a video file, you can simply call the `VideoCapture::open()` function with the camera number or video filename, then grab the frames using the C++ stream operator, as mentioned in the section _Accessing the webcam_ in Chapter 1, _Cartoonifier and Skin Changer for Android_.\n\n### Detecting an object using the Haar or LBP Classifier\n\nNow that we have loaded the classifier (just once during initialization), we can use it to detect faces in each new camera frame. But first we should do some initial processing of the camera image just for face detection, by performing the following steps:\n\n  1.  **Grayscale color conversion** : Face detection only works on grayscale images. So we should convert the color camera frame to grayscale.\n  2.  **Shrinking the camera image** : The speed of face detection depends on the size of the input image (it is very slow for large images but fast for small images), and yet detection is still fairly reliable even at low resolutions. So we should shrink the camera image to a more reasonable size (or use a large value for `minFeatureSize` in the detector, as explained shortly).\n  3.  **Histogram equalization** : Face detection is not as reliable in low-light conditions. So we should perform histogram equalization to improve the contrast and brightness.\n\n#### Grayscale color conversion\n\nWe can easily convert an RGB color image to grayscale using the `cvtColor()` function. But we should only do this if we know we have a color image (that is, it is not a grayscale camera), and we must specify the format of our input image (usually 3-channel BGR on desktop or 4-channel BGRA on mobile). So we should allow three different input color formats, as shown in the following code:\n\n    Mat gray;\n    if (img.channels() == 3) {\n    **cvtColor(img, gray, CV_BGR2GRAY);**\n    }\n    else if (img.channels() == 4) {\n    **cvtColor(img, gray, CV_BGRA2GRAY);**\n    }\n    else {\n        \/\/ Access the grayscale input image directly.\n        gray = img;\n    }\n\n#### Shrinking the camera image\n\nWe can use the `resize()` function to shrink an image to a certain size or scale factor. Face detection usually works quite well for any image size greater than 240 x 240 pixels (unless you need to detect faces that are far away from the camera), because it will look for any faces larger than the `minFeatureSize` (typically 20 x 20 pixels). So let's shrink the camera image to be 320 pixels wide; it doesn't matter if the input is a VGA webcam or a 5 mega pixel HD camera. It is also important to remember and enlarge the detection results, because if you detect faces in a shrunk image then the results will also be shrunk. Note that instead of shrinking the input image, you could use a large `minFeatureSize` value in the detector instead. We must also ensure the image does not become fatter or thinner. For example, a widescreen 800 x 400 image when shrunk to 300 x 200 would make a person look thin. So we must keep the aspect ratio (the ratio of width to height) of the output same as the input. Let's calculate how much to shrink the image width by, then apply the same scale factor to the height as well, as follows:\n\n    const int DETECTION_WIDTH = 320;\n    \/\/ Possibly shrink the image, to run much faster.\n    Mat smallImg;\n    float scale = img.cols \/ (float) DETECTION_WIDTH;\n    if (img.cols > DETECTION_WIDTH) {\n        \/\/ Shrink the image while keeping the same aspect ratio.\n        int scaledHeight = cvRound(img.rows \/ scale);\n    **resize(img, smallImg, Size(DETECTION_WIDTH, scaledHeight));**\n    }\n    else {\n        \/\/ Access the input directly since it is already small.\n        smallImg = img;\n    }\n\n#### Histogram equalization\n\nWe can easily perform histogram equalization to improve the contrast and brightness of an image, using the `equalizeHist()` function (as explained in _Learning OpenCV: Computer Vision with the OpenCV Library_ ). Sometimes this will make the image look strange, but in general it should improve the brightness and contrast and help face detection. The `equalizeHist()` function is used as follows:\n\n    \/\/ Standardize the brightness & contrast, such as\n    \/\/ to improve dark images.\n    Mat equalizedImg;\n    **equalizeHist(inputImg, equalizedImg);**\n\n## Detecting the face\n\nNow that we have converted the image to grayscale, shrunk the image, and equalized the histogram, we are ready to detect the faces using the `CascadeClassifier::detectMultiScale()` function! There are many parameters that we pass to this function:\n\n  * `minFeatureSize`: This parameter determines the minimum face size that we care about, typically 20 x 20 or 30 x 30 pixels but this depends on your use case and image size. If you are performing face detection on a webcam or smartphone where the face will always be very close to the camera, you could enlarge this to 80 x 80 to have much faster detections, or if you want to detect far away faces, such as on a beach with friends, then leave this as 20 x 20.\n  * `searchScaleFactor`: The parameter determines how many different sizes of faces to look for; typically it would be `1.1` for good detection, or `1.2` for faster detection that does not find the face as often.\n  * `minNeighbors`: This parameter determines how sure the detector should be that it has detected a face, typically a value of `3` but you can set it higher if you want more reliable faces, even if many faces are not detected.\n  * `flags`: This parameter allows you to specify whether to look for all faces (default) or only look for the largest face (`CASCADE_FIND_BIGGEST_OBJECT`). If you only look for the largest face, it should run faster. There are several other parameters you can add to make the detection about one percent or two percent faster, such as `CASCADE_DO_ROUGH_SEARCH` or `CASCADE_SCALE_IMAGE`.\n\nThe output of the `detectMultiScale()` function will be a `std::vector` of the `cv::Rect` type object. For example, if it detects two faces then it will store an array of two rectangles in the output. The `detectMultiScale()` function is used as follows:\n\n    int flags = CASCADE_SCALE_IMAGE; \/\/ Search for many faces.\n    Size minFeatureSize(20, 20);     \/\/ Smallest face size.\n    float searchScaleFactor = 1.1f;  \/\/ How many sizes to search.\n    int minNeighbors = 4;            \/\/ Reliability vs many faces.\n\n    \/\/ Detect objects in the small grayscale image.\n    std::vector<Rect> faces;\n    **faceDetector.detectMultiScale(img, faces, searchScaleFactor,**\n    **minNeighbors, flags, minFeatureSize);**\n\nWe can see if any faces were detected by looking at the number of elements stored in our vector of rectangles, that is by using the `objects.size()` function.\n\nAs mentioned earlier, if we gave a shrunken image to the face detector, the results will also be shrunk, so we need to enlarge them if we want to know the face regions for the original image. We also need to make sure faces on the border of the image stay completely within the image, as OpenCV will now raise an exception if this happens, as shown by the following code:\n\n    \/\/ Enlarge the results if the image was temporarily shrunk.\n    if (img.cols > scaledWidth) {\n        for (int i = 0; i < (int)objects.size(); i++ ) {\n            objects[i].x = cvRound(objects[i].x * scale);\n            objects[i].y = cvRound(objects[i].y * scale);\n            objects[i].width = cvRound(objects[i].width * scale);\n            objects[i].height = cvRound(objects[i].height * scale);\n        }\n    }\n    \/\/ If the object is on a border, keep it in the image.\n    for (int i = 0; i < (int)objects.size(); i++ ) {\n        if (objects[i].x < 0)\n            objects[i].x = 0;\n        if (objects[i].y < 0)\n            objects[i].y = 0;\n        if (objects[i].x + objects[i].width > img.cols)\n            objects[i].x = img.cols - objects[i].width;\n        if (objects[i].y + objects[i].height > img.rows)\n            objects[i].y = img.rows - objects[i].height;\n    }\n\nNote that the preceding code will look for all faces in the image, but if you only care about one face, then you could change the flag variable as follows:\n\n    int flags = CASCADE_FIND_BIGGEST_OBJECT | \n                    CASCADE_DO_ROUGH_SEARCH;\n\nThe WebcamFaceRec project includes a wrapper around OpenCV's Haar or LBP detector, to make it easier to find a face or eye within an image. For example:\n\n    Rect faceRect;    \/\/ Stores the result of the detection, or -1.\n    int scaledWidth = 320;\t\/\/ Shrink the image before detection.\n    **detectLargestObject(cameraImg, faceDetector, faceRect,**\n    **scaledWidth);**\n    if (faceRect.width > 0)\n        cout << \"We detected a face!\" << endl;\n\nNow that we have a face rectangle, we can use it in many ways, such as to extract or crop the face image from the original image. The following code allows us to access the face:\n\n    \/\/ Access just the face within the camera image.\n    Mat faceImg = cameraImg(faceRect);\n\nThe following image shows the typical rectangular region given by the face detector:\n\n## Step 2: Face preprocessing\n\nAs mentioned earlier, Face recognition is extremely vulnerable to changes in lighting conditions, face orientation, face expression, and so on, so it is very important to reduce these differences as much as possible. Otherwise the face recognition algorithm will often think there is more similarity between faces of two different people in the same conditions than between two faces of the same person.\n\nThe easiest form of face preprocessing is just to apply histogram equalization using the `equalizeHist()` function, like we just did for face detection. This may be sufficient for some projects where the lighting and positional conditions won't change by much. But for reliability in real-world conditions, we need many sophisticated techniques, including facial feature detection (for example, detecting eyes, nose, mouth and eyebrows). For simplicity, this chapter will just use eye detection and ignore other facial features such as the mouth and nose, which are less useful. The following image shows an enlarged view of a typical preprocessed face, using the techniques that will be covered in this section:\n\n### Eye detection\n\nEye detection can be very useful for face preprocessing, because for frontal faces you can always assume a person's eyes should be horizontal and on opposite locations of the face and should have a fairly standard position and size within a face, despite changes in facial expressions, lighting conditions, camera properties, distance to camera, and so on. It is also useful to discard false positives when the face detector says it has detected a face and it is actually something else. It is rare that the face detector and two eye detectors will all be fooled at the same time, so if you only process images with a detected face and two detected eyes then it will not have many false positives (but will also give fewer faces for processing, as the eye detector will not work as often as the face detector).\n\nSome of the pretrained eye detectors that come with OpenCV v2.4 can detect an eye whether it is open or closed, whereas some of them can only detect open eyes.\n\nEye detectors that detect open or closed eyes are as follows:\n\n  * `haarcascade_mcs_lefteye.xml` (and `haarcascade_mcs_righteye.xml`)\n  * `haarcascade_lefteye_2splits.xml` (and `haarcascade_righteye_2splits.xml`)\n\nEye detectors that detect open eyes only are as follows:\n\n  * `haarcascade_eye.xml`\n  * `haarcascade_eye_tree_eyeglasses.xml`\n\n### Note\n\nAs the open or closed eye detectors specify which eye they are trained on, you need to use a different detector for the left and the right eye, whereas the detectors for just open eyes can use the same detector for left or right eyes.\n\nThe detector `haarcascade_eye_tree_eyeglasses.xml` can detect the eyes if the person is wearing glasses, but is not reliable if they don't wear glasses.\n\nIf the XML filename says \"left eye\", it means the actual left eye of the person, so in the camera image it would normally appear on the right-hand side of the face, not on the left-hand side!\n\nThe list of four eye detectors mentioned is ranked in approximate order from most reliable to least reliable, so if you know you don't need to find people with glasses then the first detector is probably the best choice.\n\n### Eye search regions\n\nFor eye detection, it is important to crop the input image to just show the approximate eye region, just like doing face detection and then cropping to just a small rectangle where the left eye should be (if you are using the left eye detector) and the same for the right rectangle for the right eye detector. If you just do eye detection on a whole face or whole photo then it will be much slower and less reliable. Different eye detectors are better suited to different regions of the face, for example, the `haarcascade_eye.xml` detector works best if it only searches in a very tight region around the actual eye, whereas the `haarcascade_mcs_lefteye.xml` and `haarcascade_lefteye_2splits.xml` detectors work best when there is a large region around the eye.\n\nThe following table lists some good search regions of the face for different eye detectors (when using the LBP face detector), using relative coordinates within the detected face rectangle:\n\nCascade Classifier | EYE_SX | EYE_SY | EYE_SW | EYE_SH\n\n---|---|---|---|---\n\n`haarcascade_eye.xml` | 0.16 | 0.26 | 0.30 | 0.28\n\n`haarcascade_mcs_lefteye.xml` | 0.10 | 0.19 | 0.40 | 0.36\n\n`haarcascade_lefteye_2splits.xml` | 0.12 | 0.17 | 0.37 | 0.36\n\nHere is the source code to extract the left-eye and right-eye regions from a detected face:\n\n    int leftX = cvRound(face.cols * EYE_SX);\n    int topY = cvRound(face.rows * EYE_SY);\n    int widthX = cvRound(face.cols * EYE_SW);\n    int heightY = cvRound(face.rows * EYE_SH);\n    int rightX = cvRound(face.cols * (1.0-EYE_SX-EYE_SW));\n\n    Mat topLeftOfFace = faceImg(Rect(leftX, topY, widthX,\n                    heightY));\n    Mat topRightOfFace = faceImg(Rect(rightX, topY, widthX, \n                    heightY));\n\nThe following image shows the ideal search regions for the different eye detectors, where `haarcascade_eye.xml` and `haarcascade_eye_tree_eyeglasses.xml` are best with the small search region, while `haarcascade_mcs_*eye.xml` and `haarcascade_*eye_2splits.xml` are best with larger search regions. Note that the detected face rectangle is also shown, to give an idea of how large the eye search regions are compared to the detected face rectangle:\n\nWhen using the eye search regions given in the preceding table, here are the approximate detection properties of the different eye detectors:\n\nCascade Classifier | Reliability* | Speed** | Eyes found | Glasses\n\n---|---|---|---|---\n\n`haarcascade_mcs_lefteye.xml` | 80% | 18 msec | Open or closed | no\n\n`haarcascade_lefteye_2splits.xml` | 60% | 7 msec | Open or closed | no\n\n`haarcascade_eye.xml` | 40% | 5 msec | Open only | no\n\n`haarcascade_eye_tree_eyeglasses.xml` | 15% | 10 msec | Open only | yes\n\n* Reliability values show how often both eyes will be detected after LBP frontal face detection when no eyeglasses are worn and both eyes are open. If eyes are closed then the reliability may drop, or if eyeglasses are worn then both reliability and speed will drop.\n\n** Speed values are in milliseconds for images scaled to the size of 320 x 240 pixels on an Intel Core i7 2.2 GHz (averaged across 1,000 photos). Speed is typically much faster when eyes are found than when eyes are not found, as it must scan the entire image, but the `haarcascade_mcs_lefteye.xml` is still much slower than the other eye detectors.\n\nFor example, if you shrink a photo to 320 x 240 pixels, perform a histogram equalization on it, use the LBP frontal face detector to get a face, then extract the left-eye-region and right-eye-region from the face using the `haarcascade_mcs_lefteye.xml` values, then perform a histogram equalization on each eye region. Then if you the `haarcascade_mcs_lefteye.xml` detector on the left eye (which is actually on the top-right side of your image) and use the `haarcascade_mcs_righteye.xml` detector on the right eye (the top-left part of your image), each eye detector should work in roughly 90 percent of photos with LBP-detected frontal faces. So if you want both eyes detected then it should work in roughly 80 percent of photos with LBP-detected frontal faces.\n\nNote that while it is recomended to shrink the camera image before detecting faces, you should detect eyes at the full camera resolution because eyes will obviously be much smaller than faces, so you need as much resolution as you can get.\n\n### Note\n\nBased on the table, it seems that when choosing an eye detector to use, you should decide whether you want to detect closed eyes or only open eyes. And remember that you can even use one eye detector, and if it does not detect an eye then you can try with another one.\n\nFor many tasks, it is useful to detect eyes whether they are opened or closed, so if speed is not crucial, it is best to search with the `mcs_*eye` detector first, and if it fails then search with the `eye_2splits` detector.\n\nBut for face recognition, a person will appear quite different if their eyes are closed, so it is best to search with the plain `haarcascade_eye` detector first, and if it fails then search with the `haarcascade_eye_tree_eyeglasses` detector.\n\nWe can use the same `detectLargestObject()`function we used for face detection to search for eyes, but instead of asking to shrink the images before eye detection, we specify the full eye region width to get a better eye detection. It is easy to search for the left eye using one detector, and if it fails then try another detector (same for right eye). The eye detection is done as follows:\n\n    CascadeClassifier eyeDetector1(\"haarcascade_eye.xml\");\n    CascadeClassifier \n             eyeDetector2(\"haarcascade_eye_tree_eyeglasses.xml\");\n    ...\n    Rect leftEyeRect;    \/\/ Stores the detected eye.\n    \/\/ Search the left region using the 1st eye detector.\n    **detectLargestObject(topLeftOfFace, eyeDetector1, leftEyeRect,**\n    **topLeftOfFace.cols);**\n    \/\/ If it failed, search the left region using the 2nd eye \n    \/\/ detector.\n    if (leftEyeRect.width <= 0)\n    **detectLargestObject(topLeftOfFace, eyeDetector2,**\n    **leftEyeRect, topLeftOfFace.cols);**\n    \/\/ Get the left eye center if one of the eye detectors worked.\n    Point leftEye = Point(-1,-1);\n    if (leftEyeRect.width <= 0) {\n        leftEye.x = leftEyeRect.x + leftEyeRect.width\/2 + leftX;\n        leftEye.y = leftEyeRect.y + leftEyeRect.height\/2 + topY;\n    }\n\n    \/\/ Do the same for the right-eye\n    ...\n\n    \/\/ Check if both eyes were detected.\n    if (leftEye.x >= 0 && rightEye.x >= 0) {\n        ...\n    }\n\nWith the face and both eyes detected, we'll perform face preprocessing by combining:\n\n  *  **Geometrical transformation and cropping** : This process would include scaling, rotating, and translating the images so that the eyes are aligned, followed by the removal of the forehead, chin, ears, and background from the face image.\n  *  **Separate histogram equalization for left and right sides** : This process standardizes the brightness and contrast on both the left- and right-hand sides of the face independently.\n  *  **Smoothing** : This process reduces the image noise using a bilateral filter.\n  *  **Elliptical mask** : The elliptical mask removes some remaining hair and background from the face image.\n\nThe following image shows the face preprocessing steps 1 to 4 applied to a detected face. Notice how the final image has good brightness and contrast on both sides of the face, whereas the original does not:\n\n#### Geometrical transformation\n\nIt is important that the faces are all aligned together, otherwise the face recognition algorithm might be comparing part of a nose with part of an eye, and so on. The output of face detection just seen will give aligned faces to some extent, but it is not very accurate (that is, the face rectangle will not always be starting from the same point on the forehead).\n\nTo have better alignment we will use eye detection to align the face so the positions of the two detected eyes line up perfectly in desired positions. We will do the geometrical transformation using the  `warpAffine()` function, which is a single operation that will do four things:\n\n  * Rotate the face so that the two eyes are horizontal.\n  * Scale the face so that the distance between the two eyes is always the same.\n  * Translate the face so that the eyes are always centered horizontally and at a desired height.\n  * Crop the outer parts of the face, since we want to crop away the image background, hair, forehead, ears, and chin.\n\nAffine Warping takes an affine matrix that transforms the two detected eye locations to the two desired eye locations, and then crops to a desired size and position. To generate this affine matrix, we will get the center between the eyes, calculate the angle at which the two detected eyes appear, and look at their distance apart as follows:\n\n    \/\/ Get the center between the 2 eyes.\n    Point2f eyesCenter;\n    eyesCenter.x = (leftEye.x + rightEye.x) * 0.5f;\n    eyesCenter.y = (leftEye.y + rightEye.y) * 0.5f;\n\n    \/\/ Get the angle between the 2 eyes.\n    double dy = (rightEye.y - leftEye.y);\n    double dx = (rightEye.x - leftEye.x);\n    double len = sqrt(dx*dx + dy*dy);\n    \/\/ Convert Radians to Degrees.\n    **double angle = atan2(dy, dx) * 180.0\/CV_PI;**\n\n    \/\/ Hand measurements shown that the left eye center should \n    \/\/ ideally be roughly at (0.16, 0.14) of a scaled face image.\n    const double DESIRED_LEFT_EYE_X = 0.16;\n    const double DESIRED_RIGHT_EYE_X = (1.0f - 0.16);\n    \/\/ Get the amount we need to scale the image to be the desired\n    \/\/ fixed size we want.\n    const int DESIRED_FACE_WIDTH = 70;\n    const int DESIRED_FACE_HEIGHT = 70;\n    double desiredLen = (DESIRED_RIGHT_EYE_X - 0.16);\n    **double scale = desiredLen * DESIRED_FACE_WIDTH \/ len;**\n\nNow we can transform the face (rotate, scale, and translate) to get the two detected eyes to be in the desired eye positions in an ideal face as follows:\n\n    \/\/ Get the transformation matrix for the desired angle & size.\n    Mat rot_mat = getRotationMatrix2D(eyesCenter, angle, scale);\n    \/\/ Shift the center of the eyes to be the desired center.\n    double ex = DESIRED_FACE_WIDTH * 0.5f - eyesCenter.x;\n    double ey = DESIRED_FACE_HEIGHT * DESIRED_LEFT_EYE_Y - \n                        eyesCenter.y;\n    rot_mat.at<double>(0, 2) += ex;\n    rot_mat.at<double>(1, 2) += ey;\n    \/\/ Transform the face image to the desired angle & size & \n    \/\/ position! Also clear the transformed image background to a \n    \/\/ default grey.\n    Mat warped = Mat(DESIRED_FACE_HEIGHT, DESIRED_FACE_WIDTH,\n                        CV_8U, Scalar(128));\n    **warpAffine(gray, warped, rot_mat, warped.size());**\n\n#### Separate histogram equalization for left and right sides\n\nIn real-world conditions, it is common to have strong lighting on one half of the face and weak lighting on the other. This has an enormous effect on the face recognition algorithm, as the left- and right-hand sides of the same face will seem like very different people. So we will perform histogram equalization separately on the left and right halves of the face, to have standardized brightness and contrast on each side of the face.\n\nIf we simply applied histogram equalization on the left half and then again on the right half, we would see a very distinct edge in the middle because the average brightness is likely to be different on the left and the right side, so to remove this edge, we will apply the two histogram equalizations gradually from the left-or right-hand side towards the center and mix it with a whole-face histogram equalization, so that the far left-hand side will use the left histogram equalization, the far right-hand side will use the right histogram equalization, and the center will use a smooth mix of left or right value and the whole-face equalized value.\n\nThe following image shows how the left-equalized, whole-equalized, and right-equalized images are blended together:\n\nTo perform this, we need copies of the whole face equalized as well as the left half equalized and the right half equalized, which is done as follows:\n\n    int w = faceImg.cols;\n    int h = faceImg.rows;\n    Mat wholeFace;\n    **equalizeHist(faceImg, wholeFace);**\n    int midX = w\/2;\n    Mat leftSide = faceImg(Rect(0,0, midX,h));\n    Mat rightSide = faceImg(Rect(midX,0, w-midX,h));\n    **equalizeHist(leftSide, leftSide);**\n    **equalizeHist(rightSide, rightSide);**\n\nNow we combine the three images together. As the images are small, we can easily access pixels directly using the `image.at<uchar>(y,x)` function even if it is slow; so let's merge the three images by directly accessing pixels in the three input images and output images, as follows:\n\n    for (int y=0; y<h; y++) {\n        for (int x=0; x<w; x++) {\n            int v;\n            if (x < w\/4) {\n                \/\/ Left 25%: just use the left face.\n    **v = leftSide.at <uchar>(y,x);**\n            }\n            else if (x < w*2\/4) {\n                \/\/ Mid-left 25%: blend the left face & whole face.\n                int lv = leftSide.at<uchar>(y,x);\n                int wv = wholeFace.at<uchar>(y,x);\n                \/\/ Blend more of the whole face as it moves\n                \/\/ further right along the face.\n                float f = (x - w*1\/4) \/ (float)(w\/4);\n    **v = cvRound((1.0f - f) * lv + (f) * wv);**\n            }\n            else if (x < w*3\/4) {\n                \/\/ Mid-right 25%: blend right face & whole face.\n                int rv = rightSide.at<uchar>(y,x-midX);\n                int wv = wholeFace.at<uchar>(y,x);\n                \/\/ Blend more of the right-side face as it moves\n                \/\/ further right along the face.\n                float f = (x - w*2\/4) \/ (float)(w\/4);\n    **v = cvRound((1.0f - f) * wv + (f) * rv);**\n            }\n            else {\n                \/\/ Right 25%: just use the right face.\n    **v = rightSide.at <uchar>(y,x-midX);**\n            }\n            faceImg.at<uchar>(y,x) = v;\n        }\/\/ end x loop\n    }\/\/end y loop\n\nThis separated histogram equalization should significantly help reduce the effect of different lighting on the left- and right-hand sides of the face, but we must understand that it won't completely remove the effect of one-sided lighting, since the face is a complex 3D shape with many shadows.\n\n#### Smoothing\n\nTo reduce the effect of pixel noise, we will use a bilateral filter on the face, as a bilateral filter is very good at smoothing most of an image while keeping edges sharp. Histogram equalization can significantly increase the pixel noise, so we will make the filter strength `20` to cover heavy pixel noise, but use a neighborhood of just two pixels as we want to heavily smooth the tiny pixel noise but not the large image regions, as follows:\n\n    Mat filtered = Mat(warped.size(), CV_8U);\n    **bilateralFilter(warped, filtered, 0, 20.0, 2.0);**\n\n#### Elliptical mask\n\nAlthough we have already removed most of the image background and forehead and hair when we did the geometrical transformation, we can apply an elliptical mask to remove some of the corner region such as the neck, which might be in shadow from the face, particularly if the face is not looking perfectly straight towards the camera. To create the mask, we will draw a black-filled ellipse onto a white image. One ellipse to perform this has a horizontal radius of 0.5 (that is, it covers the face width perfectly), a vertical radius of 0.8 (as faces are usually taller than they are wide), and centered at the coordinates 0.5, 0.4, as shown in the following image, where the elliptical mask has removed some unwanted corners from the face:\n\nWe can apply the mask when calling the `cv::setTo()` function, which would normally set a whole image to a certain pixel value, but as we will give a mask image, it will only set some parts to the given pixel value. We will fill the image in gray so that it should have less contrast to the rest of the face:\n\n    \/\/ Draw a black-filled ellipse in the middle of the image.\n    \/\/ First we initialize the mask image to white (255).\n    Mat mask = Mat(warped.size(), CV_8UC1, Scalar(255));\n    double dw = DESIRED_FACE_WIDTH;\n    double dh = DESIRED_FACE_HEIGHT;\n    Point faceCenter = Point( cvRound(dw * 0.5),\n                         cvRound(dh * 0.4) );\n    Size size = Size( cvRound(dw * 0.5), cvRound(dh * 0.8) );\n    **ellipse(mask, faceCenter, size, 0, 0, 360, Scalar(0),**\n    **CV_FILLED);**\n\n    \/\/ Apply the elliptical mask on the face, to remove corners.\n    \/\/ Sets corners to gray, without touching the inner face.\n    **filtered.setTo(Scalar(128), mask);**\n\nThe following enlarged image shows a sample result from all the face preprocessing stages. Notice it is much more consistent for face recognition in different brightness, face rotations, angle from camera, backgrounds, positions of lights, and so on. This preprocessed face will be used as input to the face recognition stages, both when collecting faces for training, and when trying to recognize input faces:\n\n## Step 3: Collecting faces and learning from them\n\nCollecting faces can be just as simple as putting each newly preprocessed face into an array of preprocessed faces from the camera, as well as putting a label into an array (to specify which person the face was taken from). For example, you could use 10 preprocessed faces of the first person and 10 preprocessed faces of a second person, so the input to the face recognition algorithm will be an array of 20 preprocessed faces and an array of 20 integers (where the first 10 numbers are 0 and the next 10 numbers are 1).\n\nThe face recognition algorithm will then learn how to distinguish between the faces of the different people. This is referred to as the training phase and the collected faces are referred to as the training set. After the face recognition algorithm has finished training, you could then save the generated knowledge to a file or memory and later use it to recognize which person is seen in front of the camera. This is referred to as the testing phase. If you used it directly from a camera input then the preprocessed face would be referred to as the test image, and if you tested with many images (such as from a folder of image files), it would be referred to as the testing set.\n\nIt is important that you provide a good training set that covers the types of variations you expect to occur in your testing set. For example, if you will only test with faces that are looking perfectly straight ahead (such as ID photos), then you only need to provide training images with faces that are looking perfectly straight ahead. But if the person might be looking to the left or up, then you should make sure the training set will also include faces of that person doing this, otherwise the face recognition algorithm will have trouble recognizing them, as their face will appear quite different. This also applies to other factors such as facial expression (for example, if the person is always smiling in the training set but not smiling in the testing set) or lighting direction (for example, a strong light is to the left-hand side in the training set but to the right-hand side in the testing set), then the face recognition algorithm will have difficulty recognizing them. The face preprocessing steps that we just saw will help reduce these issues, but it certainly won't remove these factors, particularly the direction in which the face is looking, as it has a large effect on the position of all elements in the face.\n\n### Note\n\nOne way to obtain a good training set that will cover many different real-world conditions is for each person to rotate their head from looking left, to up, to right, to down then looking directly straight. Then the person tilts their head sideways and then up and down, while also changing their facial expression, such as alternating between smiling, looking angry, and having a neutral face. If each person follows a routine such as this while collecting faces, then there is a much better chance of recognizing everyone in the real-world conditions.\n\nFor even better results, it should be performed again with one or two more locations or directions, such as by turning the camera around by 180 degrees and walking in the opposite direction of the camera then repeating the whole routine, so that the training set would include many different lighting conditions.\n\nSo in general, having 100 training faces for each person is likely to give better results than having just 10 training faces for each person, but if all 100 faces look almost identical then it will still perform badly because it is more important that the training set has enough variety to cover the testing set, rather than to just have a large number of faces. So to make sure the faces in the training set are not all too similar, we should add a noticeable delay between each collected face. For example, if the camera is running at 30 frames per second, then it might collect 100 faces in just several seconds when the person has not had time to move around, so it is better to collect just one face per second, while the person moves their face around. Another simple method to improve the variation in the training set is to only collect a face if it is noticeably different from the previously collected face.\n\n### Collecting preprocessed faces for training\n\nTo make sure there is at least a one-second gap between collecting new faces, we need to measure how much time has passed. This is done as follows:\n\n    \/\/ Check how long since the previous face was added.\n    double current_time = (double)getTickCount();\n    double timeDiff_seconds = (current_time -\n                    old_time) \/ getTickFrequency();\n\nTo compare the similarity of two images, pixel by pixel, you can find the relative L2 error, which just involves subtracting one image from the other, summing the squared value of it, and then getting the square root of it. So if the person had not moved at all, subtracting the current face with the previous face should give a very low number at each pixel, but if they had just moved slightly in any direction, subtracting the pixels would give a large number and so the L2 error will be high. As the result is summed over all pixels, the value will depend on the image resolution. So to get the mean error we should divide this value by the total number of pixels in the image. Let's put this in a handy function, `getSimilarity()`, as follows:\n\n    double getSimilarity(const Mat A, const Mat B) {\n        \/\/ Calculate the L2 relative error between the 2 images.\n    **double errorL2 = norm(A, B, CV_L2);**\n        \/\/ Scale the value since L2 is summed across all pixels.\n        double similarity = errorL2 \/ (double)(A.rows * A.cols);\n        return similarity;\n    }\n\n    ...\n\n    \/\/ Check if this face looks different from the previous face.\n    double imageDiff = MAX_DBL;\n    if (old_prepreprocessedFaceprepreprocessedFace.data) {\n        imageDiff = getSimilarity(preprocessedFace,\n                               old_prepreprocessedFace);\n    }\n\nThis similarity will often be less than 0.2 if the image did not move much, and higher than 0.4 if the image did move, so let's use 0.3 as our threshold for collecting a new face.\n\nThere are many tricks we can play to obtain more training data, such as using mirrored faces, adding random noise, shifting the face by a few pixels, scaling the face by a percentage, or rotating the face by a few degrees (even though we specifically tried to remove these effects when preprocessing the face!) Let's add mirrored faces to the training set, so that we have both, a larger training set as well as a reduction in the problems of asymmetrical faces or if a user is always oriented slightly to the left or right during training but not testing. This is done as follows:\n\n    \/\/ Only process the face if it's noticeably different from the\n    \/\/ previous frame and there has been a noticeable time gap.\n    **if ((imageDiff > 0.3) && (timeDiff_seconds > 1.0)) {**\n        \/\/ Also add the mirror image to the training set.\n        Mat mirroredFace;\n        flip(preprocessedFace, mirroredFace, 1);\n\n        \/\/ Add the face & mirrored face to the detected face lists.\n        preprocessedFaces.push_back(preprocessedFace);\n        preprocessedFaces.push_back(mirroredFace);\n        faceLabels.push_back(m_selectedPerson);\n        faceLabels.push_back(m_selectedPerson);\n\n        \/\/ Keep a copy of the processed face,\n        \/\/ to compare on next iteration.\n        old_prepreprocessedFace = preprocessedFace;\n        old_time = current_time;\n    }\n\nThis will collect the `std::vector` arrays `preprocessedFaces` and `faceLabels` for a preprocessed face as well as the label or ID number of that person (assuming it is in the integer `m_selectedPerson` variable).\n\nTo make it more obvious to the user that we have added their current face to the collection, you could provide a visual notification by either displaying a large white rectangle over the whole image or just displaying their face for just a fraction of a second so they realize a photo was taken. With OpenCV's C++ interface, you can use the `+` overloaded `cv::Mat` operator to add a value to every pixel in the image and have it clipped to 255 (using `saturate_cast`, so it doesn't overflow from white back to black!) Assuming `displayedFrame` will be a copy of the color camera frame that should be shown, insert this after the preceding code for face collection:\n\n    \/\/ Get access to the face region-of-interest.\n    Mat displayedFaceRegion = displayedFrame(faceRect);\n    \/\/ Add some brightness to each pixel of the face region.\n    displayedFaceRegion += CV_RGB(90,90,90);\n\n### Training the face recognition system from collected faces\n\nAfter you have collected enough faces for each person to recognize, you must train the system to learn the data using a machine-learning algorithm suited for face recognition. There are many different face recognition algorithms in literature, the simplest of which are Eigenfaces and Artificial Neural Networks. Eigenfaces tends to work better than ANNs, and despite its simplicity, it tends to work almost as well as many more complex face recognition algorithms, so it has become very popular as the basic face recognition algorithm for beginners as well as for new algorithms to be compared to.\n\nAny reader who wishes to work further on face recognition is recommended to read the theory behind:\n\n  * Eigenfaces (also referred to as **Principal Component Analysis** ( **PCA** )\n  * Fisherfaces (also referred to as **Linear Discriminant Analysis** ( **LDA** )\n  * Other classic face recognition algorithms (many are available at <http:\/\/www.face-rec.org\/algorithms\/>)\n  * Newer face recognition algorithms in recent Computer Vision research papers (such as CVPR and ICCV at <http:\/\/www.cvpapers.com\/>), as there are hundreds of face recognition papers published each year\n\nHowever, you don't need to understand the theory of these algorithms in order to use them as shown in this book. Thanks to the OpenCV team and Philipp Wagner's `libfacerec` contribution, OpenCV v2.4.1 provided `cv::Algo` `rithm` as a simple and generic method to perform face recognition using one of several different algorithms (even selectable at runtime) without necessarily understanding how they are implemented. You can find the available algorithms in your version of OpenCV by using the `Algorithm::getList()` function, such as with this code:\n\n    vector<string> algorithms;\n    **Algorithm::getList(algorithms);**\n    cout << \"Algorithms: \" << algorithms.size() << endl;\n    for (int i=0; i<algorithms.size(); i++) {\n        cout << algorithms[i] << endl;\n    }\n\nHere are the three face recognition algorithms available in OpenCV v2.4.1:\n\n  * `FaceRecognizer.Eigenfaces`: Eigenfaces, also referred to as PCA, first used by Turk and Pentland in 1991.\n  * `FaceRecognizer.Fisherfaces`: Fisherfaces, also referred to as LDA, invented by Belhumeur, Hespanha and Kriegman in 1997.\n  * `FaceRecognizer.LBPH`: Local Binary Pattern Histograms, invented by Ahonen, Hadid and Pietikainen in 2004.\n\n### Note\n\nMore information on these face recognition algorithm implementations can be found with documentation, samples, and Python equivalents for each of them on Philipp Wagner's website at <http:\/\/bytefish.de\/blog> and <http:\/\/bytefish.de\/dev\/libfacerec\/>.\n\nThese face recognition algorithms are available through the `FaceRecognizer` class in OpenCV's `contrib` module. Due to dynamic linking, it is possible that your program is linked to the `contrib` module but it is not actually loaded at runtime (if it was deemed as not required). So it is recommended to call the `cv::initModule_contrib()` function before trying to access the `FaceRecognizer` algorithms. This function is only available from OpenCV v2.4.1, so it also ensures that the face recognition algorithms are at least available to you at compile time:\n\n    \/\/ Load the \"contrib\" module is dynamically at runtime.\n    **bool haveContribModule = initModule_contrib();**\n    if (!haveContribModule) {\n        cerr << \"ERROR: The 'contrib' module is needed for \";\n        cerr << \"FaceRecognizer but hasn't been loaded to OpenCV!\";\n        cerr << endl;\n        exit(1);\n    }\n\nTo use one of the face recognition algorithms, we must create a `FaceRecognizer` object using the `cv::Algorithm::create<FaceRecognizer>()` function. We pass the name of the face recognition algorithm we want to use, as a string to this create function. This will give us access to that algorithm if it is available in the OpenCV version. So it may be used as a runtime error check to ensure the user has OpenCV v2.4.1 or newer. For example:\n\n    string facerecAlgorithm = \"FaceRecognizer.Fisherfaces\";\n    Ptr<FaceRecognizer> model;\n    \/\/ Use OpenCV's new FaceRecognizer in the \"contrib\" module:\n    **model = Algorithm::create <FaceRecognizer>(facerecAlgorithm);**\n    if (model.empty()) {\n        cerr << \"ERROR: The FaceRecognizer [\" << facerecAlgorithm;\n        cerr << \"] is not available in your version of OpenCV. \";\n        cerr << \"Please update to OpenCV v2.4.1 or newer.\" << endl;\n        exit(1);\n    }\n\nOnce we have loaded the `FaceRecognizer` algorithm, we simply call the `FaceRecognizer::train()` function with our collected face data as follows:\n\n    \/\/ Do the actual training from the collected faces.\n    **model- >train(preprocessedFaces, faceLabels);**\n\nThis one line of code will run the whole face recognition training algorithm that you selected (for example, Eigenfaces, Fisherfaces, or potentially other algorithms). If you have just a few people with less than 20 faces, then this training should return very quickly, but if you have many people with many faces, it is possible that `train()` function will take several seconds or even minutes to process all the data.\n\n### Viewing the learned knowledge\n\nWhile it is not necessary, it is quite useful to view the internal data structures that the face recognition algorithm generated when learning your training data, particularly if you understand the theory behind the algorithm you selected and want to verify if it worked or find why it is not working as you hoped. The internal data structures can be different for different algorithms, but luckily they are the same for Eigenfaces and Fisherfaces, so let's just look at those two. They are both based on 1D eigenvector matrices that appear somewhat like faces when viewed as 2D images, therefore it is common to refer eigenvectors as eigenfaces when using the Eigenface algorithm or as fisherfaces when using the Fisherface algorithm.\n\nIn simple terms, the basic principle of Eigenfaces is that it will calculate a set of special images (eigenfaces) and blending ratios (eigenvalues), which when combined in different ways can generate each of the images in the training set but can also be used to differentiate the many face images in the training set from each other. For example, if some of the faces in the training set had a moustache and some did not, then there would be at least one eigenface that shows a moustache, and so the training faces with a moustache would have a high blending ratio for that eigenface to show that it has a moustache, and the faces without a moustache would have a low blending ratio for that eigenvector. If the training set had 5 people with 20 faces for each person, then there would be 100 eigenfaces and eigenvalues to differentiate the 100 total faces in the training set, and in fact these would be sorted so the first few eigenfaces and eigenvalues would be the most critical differentiators, and the last few eigenfaces and eigenvalues would just be random pixel noises that don't actually help to differentiate the data. So it is common practice to discard some of the last eigenfaces and just keep the first 50 or so eigenfaces.\n\nIn comparison, the basic principle of Fisherfaces is that instead of calculating a special eigenvector and eigenvalue for each image in the training set, it only calculates one special eigenvector and eigenvalue for each person. So in the preceding example that has 5 people with 20 faces for each person, the Eigenfaces algorithm would use 100 eigenfaces and eigenvalues whereas the Fisherfaces algorithm would use just 5 fisherfaces and eigenvalues.\n\nTo access the internal data structures of the Eigenfaces and Fisherfaces algorithms, we must use the `cv::Algorithm::get()` function to obtain them at runtime, as there is no access to them at compile time. The data structures are used internally as part of mathematical calculations rather than for image processing, so they are usually stored as floating-point numbers typically ranging between 0.0 and 1.0, rather than 8-bit uchar pixels ranging from 0 to 255, similar to pixels in regular images. Also, they are often either a 1D row or column matrix or they make up one of the many 1D rows or columns of a larger matrix. So before you can display many of these internal data structures, you must reshape them to be the correct rectangular shape, and convert them to 8-bit uchar pixels between 0 and 255. As the matrix data might range from 0.0 to 1.0 or -1.0 to 1.0 or anything else, you can use the `cv::normalize()` function with the `cv::NORM_MINMAX` option to make sure it outputs data ranging between 0 and 255 no matter what the input range may be. Let's create a function to perform this reshaping to a rectangle and conversion to 8-bit pixels for us as follows:\n\n    \/\/ Convert the matrix row or column (float matrix) to a\n    \/\/ rectangular 8-bit image that can be displayed or saved.\n    \/\/ Scales the values to be between 0 to 255.\n    Mat getImageFrom1DFloatMat(const Mat matrixRow, int height)\n    {\n        \/\/ Make a rectangular shaped image instead of a single row.\n    **Mat rectangularMat = matrixRow.reshape(1, height);**\n        \/\/ Scale the values to be between 0 to 255 and store them \n        \/\/ as a regular 8-bit uchar image.\n        Mat dst;\n    **normalize(rectangularMat, dst, 0, 255, NORM_MINMAX,**\n    **CV_8UC1);**\n        return dst;\n    }\n\nTo make it easier to debug OpenCV code and even more so when internally debugging the `cv::Algorithm` data structure, we can use the `ImageUtils.cpp` and `ImageUtils.h` files to display information about a `cv::Mat` structure easily as follows:\n\n    Mat img = ...;\n    printMatInfo(img, \"My Image\");\n\nYou will see something similar to the following printed to your console:\n\n**My Image: 640w480h 3ch 8bpp, range[79,253][20,58][18,87]**\n\nThis tells you that it is 640 elements wide and 480 high (that is, a 640 x 480 image or a 480 x 640 matrix, depending on how you view it), with three channels per pixel that are 8-bits each (that is, a regular BGR image), and it shows the min and max value in the image for each of the color channels.\n\n### Note\n\nIt is also possible to print the actual contents of an image or matrix by using the `printMat()` function instead of the `printMatInfo()` function. This is quite handy for viewing matrices and multichannel-float matrices as these can be quite tricky to view for beginners.\n\nThe ImageUtils code is mostly for OpenCV's C interface, but is gradually including more of the C++ interface over time. The the most recent version can always be found at <http:\/\/shervinemami.info\/openCV.html>.\n\n### Average face\n\nBoth the Eigenfaces and Fisherfaces algorithms first calculate the average face that is the mathematical average of all the training images, so they can subtract the average image from each facial image to have better face recognition results. So let's view the average face from our training set. The average face is named `mean` in the Eigenfaces and Fisherfaces implementations, shown as follows:\n\n    **Mat averageFace = model- >get<Mat>(\"mean\");**\n    printMatInfo(averageFace, \"averageFace (row)\");\n    \/\/ Convert a 1D float row matrix to a regular 8-bit image.\n    **averageFace = getImageFrom1DFloatMat(averageFace, faceHeight);**\n    printMatInfo(averageFace, \"averageFace\");\n    imshow(\"averageFace\", averageFace);\n\nYou should now see an average face image on your screen similar to the following (enlarged) image that is a combination of a man, a woman, and a baby. You should also see similar text shown on your console:\n\n**averageFace (row): 4900w1h 1ch 64bpp, range[5.21,251.47]**\n\n**averageFace: 70w70h 1ch 8bpp, range[0,255]**\n\nThe image would appear as shown in the following screenshot:\n\nNotice that **averageFace (row)** was a single row matrix of 64-bit floats, whereas **averageFace** is a rectangular image with 8-bit pixels covering the full range from 0 to 255.\n\n### Eigenvalues, Eigenfaces, and Fisherfaces\n\nLet's view the actual component values in the eigenvalues (as text):\n\n    **Mat eigenvalues = model- >get<Mat>(\"eigenvalues\");**\n    printMat(eigenvalues, \"eigenvalues\");\n\nFor Eigenfaces, there is one eigenvalue for each face, so if we have three people with four faces each, we get a column vector with 12 eigenvalues sorted from best to worst as follows:\n\n    eigenvalues: 1w18h 1ch 64bpp, range[4.52e+04,2.02836e+06]\n    2.03e+06\n    1.09e+06\n    5.23e+05\n    4.04e+05\n    2.66e+05\n    2.31e+05\n    1.85e+05\n    1.23e+05\n    9.18e+04\n    7.61e+04\n    6.91e+04\n    4.52e+04\n\nFor Fisherfaces, there is just one eigenvalue for each extra person, so if there are three people with four faces each, we just get a row vector with two eigenvalues as follows:\n\n    eigenvalues: 2w1h 1ch 64bpp, range[152.4,316.6]\n    317, 152\n\nTo view the eigenvectors (as Eigenface or Fisherface images), we must extract them as columns from the big eigenvectors matrix. As data in OpenCV and C\/C++ is normally stored in matrices using row-major order, it means that to extract a column, we should use the `Mat::clone()` function to ensure the data will be continuous, otherwise we can't reshape the data to a rectangle. Once we have a continuous column Mat, we can display the eigenvectors using the `getImageFrom1DFloatMat()` function just like we did for the average face:\n\n    \/\/ Get the eigenvectors\n    **Mat eigenvectors = model- >get<Mat>(\"eigenvectors\");**\n    printMatInfo(eigenvectors, \"eigenvectors\");\n\n    \/\/ Show the best 20 eigenfaces\n    for (int i = 0; i < min(20, eigenvectors.cols); i++) {\n        \/\/ Create a continuous column vector from eigenvector #i.\n    **Mat eigenvector = eigenvectors.col(i).clone();**\n\n        Mat eigenface = getImageFrom1DFloatMat(eigenvector,\n                                    faceHeight);\n        imshow(format(\"Eigenface%d\", i), eigenface);\n    }\n\nThe following figure displays eigenvectors as images. You can see that for three people with four faces, there are 12 Eigenfaces (left-hand side of the figure) or two Fisherfaces (right-hand side).\n\nNotice that both Eigenfaces and Fisherfaces seem to have the resemblance of some facial features but they don't really look like faces. This is simply because the average face was subtracted from them, so they just show the differences for each Eigenface from the average face. The numbering shows which Eigenface it is, because they are always ordered from the most significant Eigenface to the least significant Eigenface, and if you have 50 or more Eigenfaces then the later Eigenfaces will often just show random image noise and therefore should be discarded.\n\n## Step 4: Face recognition\n\nNow that we have trained the Eigenfaces or Fisherfaces machine-learning algorithm with our set of training images and face labels, we are finally ready to figure out who a person is, just from a facial image! This last step is referred to as face recognition or face identification.\n\n### Face identification: Recognizing people from their face\n\nThanks to OpenCV's `FaceRecognizer` class, we can identify the person in a photo simply by calling the `FaceRecognizer::predict()` function on a facial image as follows:\n\n    int identity = model->predict(preprocessedFace);\n\nThis `identity` value will be the label number that we originally used when collecting faces for training. For example, 0 for the first person, 1 for the second person, and so on.\n\nThe problem with this identification is that it will always predict one of the given people, even if the input photo is of an unknown person or of a car. It would still tell you which person is the most likely person in that photo, so it can be difficult to trust the result! The solution is to obtain a confidence metric so we can judge how reliable the result is, and if it seems that the confidence is too low then we assume it is an unknown person.\n\n### Face verification: Validating that it is the claimed person\n\nTo confirm if the result of the prediction is reliable or whether it should be taken as an unknown person, we perform face verification (also referred to as face authentication), to obtain a confidence metric showing whether the single face image is similar to the claimed person (as opposed to face identification, which we just performed, comparing the single face image with many people).\n\nOpenCV's `FaceRecognizer` class can return a confidence metric when you call the `predict()` function but unfortunately the confidence metric is simply based on the distance in eigen-subspace, so it is not very reliable. The method we will use is to reconstruct the facial image using the eigenvectors and eigenvalues, and compare this reconstructed image with the input image. If the person had many of their faces included in the training set, then the reconstruction should work quite well from the learnt eigenvectors and eigenvalues, but if the person did not have any faces in the training set (or did not have any that have similar lighting and facial expressions as the test image), then the reconstructed face will look very different from the input face, signaling that it is probably an unknown face.\n\nRemember we said earlier that the Eigenfaces and Fisherfaces algorithms are based on the notion that an image can be roughly represented as a set of eigenvectors (special face images) and eigenvalues (blending ratios). So if we combine all the eigenvectors with the eigenvalues from one of the faces in the training set then we should obtain a fairly close replica of that original training image. The same applies with other images that are similar to the training set--if we combine the trained eigenvectors with the eigenvalues from a similar test image, we should be able to reconstruct an image that is somewhat a replica to the test image.\n\nOnce again, OpenCV's `FaceRecognizer` class makes it quite easy to generate a reconstructed face from any input image, by using the  `subspaceProject()` function to project onto the eigenspace and the `subspaceReconstruct()` function to go back from eigenspace to image space. The trick is that we need to convert it from a floating-point row matrix to a rectangular 8-bit image (like we did when displaying the average face and eigenfaces), but we don't want to normalize the data, as it is already in the ideal scale to compare with the original image. If we normalized the data, it would have a different brightness and contrast from the input image, and it would become difficult to compare the image similarity just by using the L2 relative error. This is done as follows:\n\n    \/\/ Get some required data from the FaceRecognizer model.\n    Mat eigenvectors = model->get<Mat>(\"eigenvectors\");\n    Mat averageFaceRow = model->get<Mat>(\"mean\");\n\n    \/\/ Project the input image onto the eigenspace.\n    **Mat projection = subspaceProject(eigenvectors, averageFaceRow,**\n    **preprocessedFace.reshape(1,1));**\n\n    \/\/ Generate the reconstructed face back from the eigenspace.\n    **Mat reconstructionRow = subspaceReconstruct(eigenvectors,**\n    **averageFaceRow, projection);**\n\n    \/\/ Make it a rectangular shaped image instead of a single row.\n    Mat reconstructionMat = reconstructionRow.reshape(1, \n                        faceHeight);\n    \/\/ Convert the floating-point pixels to regular 8-bit uchar.\n    Mat reconstructedFace = Mat(reconstructionMat.size(), CV_8U);\n    reconstructionMat.convertTo(reconstructedFace, CV_8U, 1, 0);\n\nThe following image shows two typical reconstructed faces. The face on the left-hand side was reconstructed well because it was from a known person, whereas the face on the right-hand side was reconstructed badly because it was from an unknown person or a known person but with unknown lighting conditions\/facial expression\/face direction.\n\nWe can now calculate how similar this reconstructed face is to the input face by using the same `getSimilarity()` function we created previously for comparing two images, where a value less than 0.3 implies that the two images are very similar. For Eigenfaces, there is one eigenvector for each face, so reconstruction tends to work well and therefore we can typically use a threshold of 0.5, but Fisherfaces has just one eigenvector for each person, so reconstruction will not work as well and therefore it needs a higher threshold, say 0.7. This is done as follows:\n\n    similarity = getSimilarity(preprocessedFace, \n                             reconstructedFace);\n    if (similarity > UNKNOWN_PERSON_THRESHOLD) {\n        identity = -1;    \/\/ Unknown person.\n    }\n\nNow you can just print the identity to the console, or use it for wherever your imagination takes you! Remember that this face recognition method and this face verification method are only reliable in the certain conditions that you train it for. So to obtain good recognition accuracy, you will need to ensure that the training set of each person covers the full range of lighting conditions, facial expressions, and angles that you expect to test with. The face preprocessing stage helped reduce some differences with lighting conditions and in-plane rotation (if the person tilts their head towards their left or right shoulder), but for other differences such as out-of-plane rotation (if the person turns their head towards the left-hand side or right-hand side), it will only work if it is covered well in your training set.\n\n## Finishing touches: Saving and loading files\n\nYou could potentially add a command-line based method that processes input files and saves them to the disk, or even perform face detection, face preprocessing and\/or face recognition as a web service, and so on. For these types of projects, it is quite easy to add the desired functionality by using the `save` and `load` functions of the `FaceRecognizer` class. You may also want to save the trained data and then load it on the program's start up.\n\nSaving the trained model to an XML or YML file is very easy:\n\n    **model- >save(\"trainedModel.yml\");**\n\nYou may also want to save the array of preprocessed faces and labels, if you will want to add more data to the training set later.\n\nFor example, here is some sample code for loading the trained model from a file. Note that you must specify the face recognition algorithm (for example `FaceRecognizer.Eigenfaces` or `FaceRecognizer.Fisherfaces`) that was originally used to create the trained model:\n\n    string facerecAlgorithm = \"FaceRecognizer.Fisherfaces\";\n    model = Algorithm::create<FaceRecognizer>(facerecAlgorithm);\n    Mat labels;\n    try {\n    **model- >load(\"trainedModel.yml\");**\n        labels = model->get<Mat>(\"labels\");\n    } catch (cv::Exception &e) {}\n    if (labels.rows <= 0) {\n        cerr << \"ERROR: Couldn't load trained data from \"\n                \"[trainedModel.yml]!\" << endl;\n        exit(1);\n    }\n\n## Finishing touches: Making a nice and interactive GUI\n\nWhile the code given so far in this chapter is sufficient for a whole face recognition system, there still needs to be a way to put the data into the system and a way to use it. Many face recognition systems for research will choose the ideal input to be text files listing where the static image files are stored on the computer, as well as other important data such as the true name or identity of the person and perhaps true pixel coordinates of regions of the face (such as ground truth of where the face and eye centers actually are). This would either be collected manually by another face recognition system.\n\nThe ideal output would then be a text file comparing the recognition results with the ground truth, so that statistics may be obtained for comparing the face recognition system with other face recognition systems.\n\nHowever, as the face recognition system in this chapter is designed for learning as well as practical fun purposes, rather than competing with the latest research methods it is useful to have an easy-to-use GUI that allows face collection, training, and testing, interactively from the webcam in real time. So this section will provide an interactive GUI providing these features. The reader is expected to either use this provided GUI that comes with this book, or to modify the GUI for their own purposes, or to ignore this GUI and design their own GUI to perform the face recognition techniques discussed so far.\n\nAs we need the GUI to perform multiple tasks, let's create a set of modes or states that the GUI will have, with buttons or mouse clicks for the user to change modes:\n\n  *  **Startup** : This state loads and initializes the data and webcam.\n  *  **Detection** : This state detects faces and shows them with preprocessing, until the user clicks on the **Add Person** button.\n  *  **Collection** : This state collects faces for the current person, until the user clicks anywhere in the window. This also shows the most recent face of each person. User clicks either one of the existing people or the **Add Person** button, to collect faces for different people.\n  *  **Training** : In this state, the system is trained with the help of all the collected faces of all the collected people.\n  *  **Recognition** : This consists of highlighting the recognized person and showing a confidence meter. The user clicks either one of the people or the **Add Person** button, to return to mode 2 (Collection).\n\nTo quit, the user can hit Escape in the window at any time. Let's also add a **Delete All** mode that restarts a new face recognition system, and a **Debug** button that toggles the display of extra debug info. We can create an enumerated `mode` variable to show the current mode.\n\n### Drawing the GUI elements\n\nTo display the current mode on the screen, let's create a function to draw text easily. OpenCV comes with a `cv::putText()` function with several fonts and anti-aliasing, but it can be tricky to place the text in the correct location that you want. Luckily, there is also a `cv::getTextSize()` function to calculate the bounding box around the text, so we can create a wrapper function to make it easier to place text. We want to be able to place text along any edge of the window and make sure it is completely visible and also to allow placing multiple lines or words of text next to each other without overwriting each other. So here is a wrapper function to allow you to specify either left-justified or right-justified, as well as to specify top-justified or bottom-justified, and return the bounding box, so we can easily draw multiple lines of text on any corner or edge of the window:\n\n    \/\/ Draw text into an image. Defaults to top-left-justified \n    \/\/ text, so give negative x coords for right-justified text,\n    \/\/ and\/or negative y coords for bottom-justified text.\n    \/\/ Returns the bounding rect around the drawn text.\n    Rect drawString(Mat img, string text, Point coord, Scalar \n                color, float fontScale = 0.6f, int thickness = 1,\n                int fontFace = FONT_HERSHEY_COMPLEX);\n\nNow to display the current mode on the GUI, as the background of the window will be the camera feed, it is quite possible that if we simply draw text over the camera feed, it might be the same color as the camera background! So let's just draw a black shadow of text that is just 1 pixel apart from the foreground text we want to draw. Let's also draw a line of helpful text below it, so the user knows the steps to follow. Here is an example of how to draw some text using the `drawString()` function:\n\n    string msg = \"Click [Add Person] when ready to collect faces.\";\n    \/\/ Draw it as black shadow & again as white text.\n    float txtSize = 0.4;\n    int BORDER = 10;\n    **drawString(displayedFrame, msg, Point(BORDER, -BORDER-2),**\n                CV_RGB(0,0,0), txtSize);\n    **Rect rcHelp = drawString(displayedFrame, msg, Point(BORDER+1,**\n                -BORDER-1), CV_RGB(255,255,255), txtSize);\n\nThe following partial screenshot shows the mode and info at the bottom of the GUI window, overlaid on top of the camera image:\n\nWe mentioned that we want a few GUI buttons, so let's create a function to draw a GUI button easily as follows:\n\n    \/\/ Draw a GUI button into the image, using drawString().\n    \/\/ Can give a minWidth to have several buttons of same width.\n    \/\/ Returns the bounding rect around the drawn button.\n    Rect drawButton(Mat img, string text, Point coord,\n                    int minWidth = 0)\n    {\n        const int B = 10;\n        Point textCoord = Point(coord.x + B, coord.y + B);\n        \/\/ Get the bounding box around the text.\n    **Rect rcText = drawString(img, text, textCoord,**\n    **CV_RGB(0,0,0));**\n        \/\/ Draw a filled rectangle around the text.\n        Rect rcButton = Rect(rcText.x - B, rcText.y - B,\n                        rcText.width + 2*B, rcText.height + 2*B);\n        \/\/ Set a minimum button width.\n        if (rcButton.width < minWidth)\n            rcButton.width = minWidth;\n        \/\/ Make a semi-transparent white rectangle.\n        Mat matButton = img(rcButton);\n        matButton += CV_RGB(90, 90, 90);\n        \/\/ Draw a non-transparent white border.\n    **rectangle(img, rcButton, CV_RGB(200,200,200), 1, CV_AA);**\n\n        \/\/ Draw the actual text that will be displayed.\n    **drawString(img, text, textCoord, CV_RGB(10,55,20));**\n\n        return rcButton;\n    }\n\nNow we create several clickable GUI buttons using the `drawButton()` function, which will always be shown at the top-left of the GUI, as shown in the following partial screenshot:\n\nAs we mentioned, the GUI program has some modes that it switches between (as a finite state machine), beginning with the Startup mode. We will store the current mode as the `m_mode` variable.\n\n#### Startup mode\n\nIn Startup mode, we just need to load the XML detector files to detect the face and eyes and initialize the webcam, which we've already covered. Let's also create a main GUI window with a mouse callback function that OpenCV will call whenever the user moves or clicks their mouse in our window. It may also be desirable to set the camera resolution to something reasonable, for example, 640 x 480, if the camera supports it. This is done as follows:\n\n    \/\/ Create a GUI window for display on the screen.\n    namedWindow(windowName);\n    \/\/ Call \"onMouse()\" when the user clicks in the window.\n    setMouseCallback(windowName, onMouse, 0);\n\n    \/\/ Set the camera resolution. Only works for some systems.\n    videoCapture.set(CV_CAP_PROP_FRAME_WIDTH, 640);\n    videoCapture.set(CV_CAP_PROP_FRAME_HEIGHT, 480);\n\n    \/\/ We're already initialized, so let's start in Detection mode.\n    m_mode = MODE_DETECTION;\n\n#### Detection mode\n\nIn Detection mode, we want to continuously detect faces and eyes, draw rectangles or circles around them to show the detection result, and show the current preprocessed face. In fact we will want these to be displayed no matter which mode we are in. The only thing special about Detection mode is that it will change to the next mode (Collection) when the user clicks the **Add Person** button.\n\nIf you remember from the detection step previously in this chapter, the output of our detection stage will be:\n\n  * `Mat preprocessedFace`: The preprocessed face (if face and eyes were detected)\n  * `Rect faceRect`: The detected face region coordinates\n  * `Point leftEye`, `rightEye`: The detected left and right eye center coordinates\n\nSo we should check if a preprocessed face was returned and draw a rectangle and circles around the face and eyes if they were detected as follows:\n\n    bool gotFaceAndEyes = false;\n    if (preprocessedFace.data)\n        gotFaceAndEyes = true;\n\n    if (faceRect.width > 0) {\n        \/\/ Draw an anti-aliased rectangle around the detected face.\n    **rectangle(displayedFrame, faceRect, CV_RGB(255, 255, 0), 2,**\n    **CV_AA);**\n\n        \/\/ Draw light-blue anti-aliased circles for the 2 eyes.\n        Scalar eyeColor = CV_RGB(0,255,255);\n        if (leftEye.x >= 0) {   \/\/ Check if the eye was detected\n    **circle(displayedFrame, Point(faceRect.x + leftEye.x,**\n    **faceRect.y + leftEye.y), 6, eyeColor, 1,**\n    **CV_AA);**\n        }\n        if (rightEye.x >= 0) {   \/\/ Check if the eye was detected\n    **circle(displayedFrame, Point(faceRect.x + rightEye.x,**\n    **faceRect.y + rightEye.y), 6, eyeColor, 1,**\n    **CV_AA);**\n        }\n    }\n\nWe will overlay the current preprocessed face at the top-center of the window as follows:\n\n    int cx = (displayedFrame.cols - faceWidth) \/ 2;\n    if (preprocessedFace.data) {\n        \/\/ Get a BGR version of the face, since the output is BGR.\n        Mat srcBGR = Mat(preprocessedFace.size(), CV_8UC3);\n        cvtColor(preprocessedFace, srcBGR, CV_GRAY2BGR);\n\n        \/\/ Get the destination ROI.\n        Rect dstRC = Rect(cx, BORDER, faceWidth, faceHeight);\n        Mat dstROI = displayedFrame(dstRC);\n\n        \/\/ Copy the pixels from src to dst.\n    **srcBGR.copyTo(dstROI);**\n    }\n    \/\/ Draw an anti-aliased border around the face.\n    **rectangle(displayedFrame, Rect(cx-1, BORDER-1, faceWidth+2,**\n                   faceHeight+2), CV_RGB(200,200,200), 1, CV_AA);\n\nThe following screenshot shows the displayed GUI when in Detection mode. The preprocessed face is shown at the top-center, and the detected face and eyes are marked:\n\n#### Collection mode\n\nWe enter Collection mode when the user clicks on the **Add Person** button to signal that they want to begin collecting faces for a new person. As mentioned previously, we have limited the face collection to one face per second and then only if it has changed noticeably from the previously collected face. And remember, we decided to collect not only the preprocessed face but also the mirror image of the preprocessed face.\n\nIn Collection mode, we want to show the most recent face of each known person and let the user click on one of those people to add more faces to them or click the **Add Person** button to add a new person to the collection. The user must click somewhere in the middle of the window to continue to the next (Training) mode.\n\nSo first we need to keep a reference to the latest face that was collected for each person. We'll do this by updating the `m_latestFaces` array of integers, which just stores the array index of each person, from the big `preprocessedFaces` array (that is, the collection of all faces of all the people). As we also store the mirrored face in that array, we want to reference the second last face, not the last face. This code should be appended to the code that adds a new face (and mirrored face) to the `preprocessedFaces` array as follows:\n\n    \/\/ Keep a reference to the latest face of each person.\n    m_latestFaces[m_selectedPerson] = preprocessedFaces.size() - 2;\n\nWe just have to remember to always grow or shrink the `m_latestFaces` array whenever a new person is added or deleted (for example, due to the user clicking on the **Add Person** button). Now let's display the most recent face for each of the collected people, on the right-hand side of the window (both in Collection mode and Recognition mode later) as follows:\n\n    m_gui_faces_left = displayedFrame.cols - BORDER - faceWidth;\n    m_gui_faces_top = BORDER;\n    for (int i=0; i<m_numPersons; i++) {\n        int index = m_latestFaces[i];\n        if (index >= 0 && index < (int)preprocessedFaces.size()) {\n            Mat srcGray = preprocessedFaces[index];\n            if (srcGray.data) {\n                \/\/ Get a BGR face, since the output is BGR.\n                Mat srcBGR = Mat(srcGray.size(), CV_8UC3);\n                cvtColor(srcGray, srcBGR, CV_GRAY2BGR);\n\n                \/\/ Get the destination ROI\n                int y = min(m_gui_faces_top + i * faceHeight,\n                            displayedFrame.rows - faceHeight);\n                Rect dstRC = Rect(m_gui_faces_left, y, faceWidth,\n                            faceHeight);\n                Mat dstROI = displayedFrame(dstRC);\n\n                \/\/ Copy the pixels from src to dst.\n    **srcBGR.copyTo(dstROI);**\n            }\n        }\n    }\n\nWe also want to highlight the current person being collected, using a thick red border around their face. This is done as follows:\n\n    if (m_mode == MODE_COLLECT_FACES) {\n        if (m_selectedPerson >= 0 &&\n                      m_selectedPerson < m_numPersons) {\n            int y = min(m_gui_faces_top + m_selectedPerson * \n                        faceHeight, displayedFrame.rows - \n                        faceHeight);\n            Rect rc = Rect(m_gui_faces_left, y, faceWidth, \n                        faceHeight);\n    **rectangle(displayedFrame, rc, CV_RGB(255,0,0), 3,**\n    **CV_AA);**\n        }\n    }\n\nThe following partial screenshot shows the typical display when faces for several people have been collected. The user can click any of the people at the top-right to collect more faces for that person.\n\n#### Training mode\n\nWhen the user finally clicks in the middle of the window, the face recognition algorithm will begin training on all the collected faces. But it is important to make sure there have been enough faces or people collected, otherwise the program may crash. In general, this just requires making sure there is at least one face in the training set (which implies there is at least one person). But the Fisherfaces algorithm looks for comparisons between people, so if there are less than two people in the training set, it will also crash. So we must check whether the selected face recognition algorithm is Fisherfaces. If it is, then we require at least two people with faces, otherwise we require at least one person with a face. If there isn't enough data, then the program goes back to Collection mode so the user can add more faces before training.\n\nTo check if there are at least two people with collected faces, we can make sure that when a user clicks on the **Add Person** button, a new person is only added if there isn't any empty person (that is, a person that was added but does not have any collected faces yet). We can then also make sure that if there are just two people and we are using the Fisherfaces algorithm, then we must make sure an `m_latestFaces` reference was set for the last person during the collection mode. `m_latestFaces[i]` is initialized to -1 when there still haven't been any faces added to that person, and then it becomes `0` or higher once faces for that person have been added. This is done as follows:\n\n    \/\/ Check if there is enough data to train from.\n    bool haveEnoughData = true;\n    if (!strcmp(facerecAlgorithm, \"FaceRecognizer.Fisherfaces\")) {\n    **if ((m_numPersons < 2) ||**\n    **(m_numPersons == 2 && m_latestFaces[1] < 0) ) {**\n            cout << \"Fisherfaces needs >= 2 people!\" << endl;\n            haveEnoughData = false;\n        }\n    }\n    **if (m_numPersons < 1 || preprocessedFaces.size() <= 0 ||**\n                preprocessedFaces.size() != faceLabels.size()) {\n        cout << \"Need data before it can be learnt!\" << endl;\n        haveEnoughData = false;\n    }\n\n    if (haveEnoughData) {\n        \/\/ Train collected faces using Eigenfaces or Fisherfaces.\n    **model = learnCollectedFaces(preprocessedFaces, faceLabels,**\n    **facerecAlgorithm);**\n\n        \/\/ Now that training is over, we can start recognizing!\n        m_mode = MODE_RECOGNITION;\n    }\n    else {\n        \/\/ Not enough training data, go back to Collection mode!\n        m_mode = MODE_COLLECT_FACES;\n    }\n\nThe training may take a fraction of a second or it may take several seconds or even minutes, depending on how much data is collected. Once the training of collected faces is complete, the face recognition system will automatically enter Recognition mode.\n\n#### Recognition mode\n\nIn Recognition mode, a confidence meter is shown next to the preprocessed face, so the user knows how reliable the recognition is. If the confidence level is higher than the unknown threshold, it will draw a green rectangle around the recognized person to show the result easily. The user can add more faces for further training if they click on the **Add Person** button or one of the existing people, which causes the program to return to the Collection mode.\n\nNow we have obtained the recognized identity and the similarity with the reconstructed face as mentioned earlier. To display the confidence meter, we know that the L2 similarity value is generally between 0 to 0.5 for high confidence and between 0.5 to 1.0 for low confidence, so we can just subtract it from 1.0 to get the confidence level between 0.0 to 1.0. Then we just draw a filled rectangle using the confidence level as the ratio shown as follows:\n\n    int cx = (displayedFrame.cols - faceWidth) \/ 2;\n    Point ptBottomRight = Point(cx - 5, BORDER + faceHeight);\n    Point ptTopLeft = Point(cx - 15, BORDER);\n\n    \/\/ Draw a gray line showing the threshold for \"unknown\" people.\n    Point ptThreshold = Point(ptTopLeft.x, ptBottomRight.y -\n                    (1.0 - UNKNOWN_PERSON_THRESHOLD) * faceHeight);\n    rectangle(displayedFrame, ptThreshold, Point(ptBottomRight.x,\n                    ptThreshold.y), CV_RGB(200,200,200), 1, CV_AA);\n\n    \/\/ Crop the confidence rating between 0 to 1 to fit in the bar.\n    double confidenceRatio = 1.0 - min(max(similarity, 0.0), 1.0);\n    Point ptConfidence = Point(ptTopLeft.x, ptBottomRight.y -\n                    confidenceRatio * faceHeight);\n\n    \/\/ Show the light-blue confidence bar.\n    **rectangle(displayedFrame, ptConfidence, ptBottomRight,**\n    **CV_RGB(0,255,255), CV_FILLED, CV_AA);**\n    \/\/ Show the gray border of the bar.\n    rectangle(displayedFrame, ptTopLeft, ptBottomRight,\n                    CV_RGB(200,200,200), 1, CV_AA);\n\nTo highlight the recognized person, we draw a green rectangle around their face as follows:\n\n    if (identity >= 0 && identity < 1000) {\n        int y = min(m_gui_faces_top + identity * faceHeight,\n                        displayedFrame.rows - faceHeight);\n        Rect rc = Rect(m_gui_faces_left, y, faceWidth, faceHeight);\n    **rectangle(displayedFrame, rc, CV_RGB(0,255,0), 3, CV_AA);**\n    }\n\nThe following partial screenshot shows a typical display when running in Recognition mode, showing the confidence meter next to the preprocessed face at the top-center, and highlighting the recognized person in the top-right corner.\n\n### Checking and handling mouse clicks\n\nNow that we have all our GUI elements drawn, we just need to process mouse events. When we initialized the display window, we told OpenCV that we want a mouse event callback to our  `onMouse` function. We don't care about mouse movement, only the mouse clicks, so first we skip the mouse events that aren't for the left-mouse-button click as follows:\n\n    void onMouse(int event, int x, int y, int, void*)\n    {\n        if (event != CV_EVENT_LBUTTONDOWN)\n            return;\n\n        Point pt = Point(x,y);\n\n        ... (handle mouse clicks) ...\n\n    }\n\nAs we obtained the drawn rectangle bounds of the buttons when drawing them, we just check if the mouse click location is in any of our button regions by calling OpenCV's `inside()` function. Now we can check for each button we have created.\n\nWhen the user clicks on the **Add Person** button, we just add 1 to the `m_numPersons` variable, allocate more space in the `m_latestFaces` variable, select the new person for collection, and begin Collection mode (no matter which mode we were previously in).\n\nBut there is one complication; to ensure that we have at least one face for each person when training, we will only allocate space for a new person if there isn't already a person with zero faces. This will ensure that we can always check the value of `m_latestFaces[m_numPersons-1]` to see if a face has been collected for every person. This is done as follows:\n\n    if (pt.inside(m_btnAddPerson)) {\n        \/\/ Ensure there isn't a person without collected faces.\n        if ((m_numPersons==0) ||\n                   (m_latestFaces[m_numPersons-1] >= 0)) {\n            \/\/ Add a new person.\n    **m_numPersons++;**\n            m_latestFaces.push_back(-1);\n        }\n        m_selectedPerson = m_numPersons - 1;\n    **m_mode = MODE_COLLECT_FACES;**\n    }\n\nThis method can be used to test for other button clicks, such as toggling the debug flag as follows:\n\n    else if (pt.inside(m_btnDebug)) {\n        m_debug = !m_debug;\n    }\n\nTo handle the **Delete All** button, we need to empty various data structures that are local to our main loop (that is, not accessible from the mouse event callback function), so we change to the **Delete All** mode and then we can delete everything from inside the main loop. We also must deal with the user clicking the main window (that is, not a button). If they clicked on one of the people on the right-hand side, then we want to select that person and change to Collection mode. Or if they clicked in the main window while in Collection mode, then we want to change to Training mode. This is done as follows:\n\n    else {\n        \/\/ Check if the user clicked on a face from the list.\n        int clickedPerson = -1;\n        for (int i=0; i<m_numPersons; i++) {\n            if (m_gui_faces_top >= 0) {\n                Rect rcFace = Rect(m_gui_faces_left, \n                             m_gui_faces_top + i * faceHeight, \n                             faceWidth, faceHeight);\n                if (pt.inside(rcFace)) {\n                    clickedPerson = i;\n                    break;\n                }\n            }\n        }\n        \/\/ Change the selected person, if the user clicked a face.\n        if (clickedPerson >= 0) {\n            \/\/ Change the current person & collect more photos.\n    **m_selectedPerson = clickedPerson;**\n    **m_mode = MODE_COLLECT_FACES;**\n        }\n        \/\/ Otherwise they clicked in the center.\n        else {\n            \/\/ Change to training mode if it was collecting faces.\n            if (m_mode == MODE_COLLECT_FACES) {\n    **m_mode = MODE_TRAINING;**\n            }\n        }\n    }\n\n# Summary\n\nThis chapter has shown you all the steps required to create a real time face recognition app, with enough preprocessing to allow some differences between the training set conditions and the testing set conditions, just using basic algorithms. We used face detection to find the location of a face within the camera image, followed by several forms of face preprocessing to reduce the effects of different lighting conditions, camera and face orientations, and facial expressions. We then trained an Eigenfaces or Fisherfaces machine-learning system with the preprocessed faces we collected, and finally we performed face recognition to see who the person is with face verification providing a confidence metric in case it is an unknown person.\n\nRather than providing a command-line tool that processes image files in an offline manner, we combined all the preceding steps into a self-contained real time GUI program to allow immediate use of the face recognition system. You should be able to modify the behavior of the system for your own purposes, such as to allow an automatic login of your computer, or if you are interested in improving the recognition reliability then you can read conference papers about recent advances in face recognition to potentially improve each step of the program until it is reliable enough for your specific needs. For example, you could improve the face preprocessing stages, or use a more advanced machine-learning algorithm, or an even better face verification algorithm, based on methods at <http:\/\/www.face-rec.org\/algorithms\/> and <http:\/\/www.cvpapers.com>\n\n# References\n\n  *  _Rapid Object Detection using a Boosted Cascade of Simple Features_ , _P. Viola and M.J. Jones_ , _Proceedings of the IEEE Transactions on CVPR 2001_ , _Vol. 1_ , _pp. 511-518_\n  *  _An Extended Set of Haar-like Features for Rapid Object Detection_ , _R. Lienhart and J. Maydt_ , _Proceedings of the IEEE Transactions on ICIP 2002_ , _Vol. 1_ , _pp. 900-903_\n  *  _Face Description with Local Binary Patterns: Application to Face Recognition_ , _T. Ahonen, A. Hadid and M. Pietik ainen_, _Proceedings of the IEEE Transactions on PAMI 2006_ , _Vol. 28_ , _Issue 12_ , _pp. 2037-2041_\n  *  _Learning OpenCV: Computer Vision with the OpenCV Library_ , _G. Bradski and A. Kaehler_ , _pp. 186-190_ , _O'Reilly Media_.\n  *  _Eigenfaces for recognition_ , _M. Turk and A. Pentland_ , _Journal of Cognitive Neuroscience 3_ , _pp. 71-86_\n  *  _Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection_ , _P.N. Belhumeur, J. Hespanha and D. Kriegman_ , _Proceedings of the IEEE Transactions on PAMI 1997_ , _Vol. 19_ , _Issue 7_ , _pp. 711 -720_\n  *  _Face Recognition with Local Binary Patterns_ , _T. Ahonen, A. Hadid and M. Pietik ainen_, _Computer Vision - ECCV 2004_ , _pp. 469 -48_\n\n# Index\n\n## A\n\n  * AAM \/ Overview\n    * overview \/ Active Appearance Models overview\n    * Model Instantiation \/ Model Instantiation \u2013 playing with the Active Appearance Model\n    * search \/ AAM search and fitting\n  * addRawViewOutput function \/ Accessing the camera\n  * Algorithm\n    * TopicngetList() function \/ Training the face recognition system from collected faces\n  * algorithms, for descriptor matching\n    * brute force matcher \/ Matching of feature points\n    * flann-based matcher \/ Matching of feature points\n  * alien mode\n    * generating, skin detection used \/ Generating an \"alien\" mode using skin detection\n  * ALPR\n    * about \/ Introduction to ANPR\n  * Android\n    * program, porting from desktop to \/ Porting from desktop to Android\n  * Android 2.2 (Froyo) \/ Porting from desktop to Android\n  * Android app\n    * reviewing \/ Reviewing the Android app\n    * Frames Per Second, displaying for \/ Showing the FPS of the app\n  * Android Cartoonifier app\n    * customizing \/ Customizing the app\n  * Android Emulator \/ Porting from desktop to Android\n  * Android gallery\n    * about \/ Saving the image to a file and to the Android picture gallery\n    * image, saving to \/ Saving the image to a file and to the Android picture gallery\n  * Android menu bar\n    * cartoon modes, modifying through \/ Changing cartoon modes through the Android menu bar\n  * Android NDK app\n    * cartoonifier code, adding to \/ Adding the cartoonifier code to the Android NDK app\n  * Android notification message\n    * displaying, for saved image \/ Showing an Android notification message about a saved image\n  * Android project\n    * setting up \/ Setting up an Android project that uses OpenCV\n    * color formats, used for image processing \/ Color formats used for image processing on Android\n    * color format, inputting from camera \/ Input color format from the camera\n    * output color format, for display \/ Output color format for display\n  * ANN algorithm\n    * about \/ OCR classification\n  * annotation tool\n    * using \/ Annotation tool\n  * ANPR\n    * about \/ Introduction to ANPR\n    * overview \/ Introduction to ANPR\n  * ANPR algorithm\n    * about \/ ANPR algorithm\n    * pattern recognition steps \/ ANPR algorithm\n    * plate detection \/ Plate detection\n    * plate recognition \/ Plate recognition\n  * application architecture\n    * about \/ Application architecture\n  * application infrastructure\n    * about \/ Application infrastructure\n    * ARPipeline.hpp \/ ARPipeline.hpp\n    * ARPipeline.cpp \/ ARPipeline.cpp\n    * support, enabling for 3D visualization in OpenCV \/ Enabling support for 3D visualization in OpenCV\n  * arbitrary image, on video\n    * searching, feature descriptors used \/ Using feature descriptors to find an arbitrary image on video\n  * ARDrawingContext.cpp file \/ ARDrawingContext.cpp\n  * ARDrawingContext.hpp file \/ ARDrawingContext.hpp\n  * ARDrawingContext class \/ ARDrawingContext.hpp, ARDrawingContext.cpp\n  * ARPipeline.cpp file\n    * about \/ ARPipeline.cpp\n  * ARPipeline.hpp file \/ ARPipeline.hpp\n  * ARPipeline class\n    * about \/ ARPipeline.hpp\n  * AR scene\n    * rendering \/ Rendering an AR scene\n  * Artificial Neural Network (ANN)\n    * about \/ Plate recognition\n  * ASM \/ Overview\n    * about \/ Active Shape Models\n    * PCA, working with \/ Getting the feel of PCA\n    * triangulation \/ Triangulation\n    * triangle texture warping \/ Triangle texture warping\n  * Augmented Reality (AR)\n    * rendering \/ Rendering augmented reality\n  * Augmented Reality (AR), rendering\n    * ARDrawingContext.hpp \/ ARDrawingContext.hpp\n    * ARDrawingContext.cpp \/ ARDrawingContext.cpp\n  * Augmented Reality (AR) application\n    * about \/ Application architecture\n    * components \/ Application architecture\n    * camera, accessing \/ Accessing the camera\n  * AVCaptureDevice class \/ Accessing the camera\n  * AVCaptureMovieFileOutput interface\n    * about \/ Accessing the camera\n  * AVCaptureSession object \/ Accessing the camera\n  * AVCaptureStillImageOutput interface\n    * about \/ Accessing the camera\n  * AVCaptureVideoDataOutput class \/ Accessing the camera\n  * AVCaptureVideoDataOutput interface\n    * about \/ Accessing the camera\n  * AVCaptureVideoPreviewLayer class \/ Accessing the camera\n  * AVCaptureVideoPreviewLayer interface\n    * about \/ Accessing the camera\n  * AVI\n    * about \/ Introduction to ANPR\n\n## B\n\n  * bilateralFilter() method \/ Generating a color painting and a cartoon\n  * black and white sketch\n    * generating \/ Generating a black-and-white sketch\n  * brute force matcher\n    * about \/ Matching of feature points\n  * bucket fill tool \/ Implementation of the skin-color changer\n  * buildPatternFromImage class\n    * about \/ PatternDetector.cpp\n  * buildProjectionMatrix function \/ ARDrawingContext.cpp\n  * Bundle Adjustment (BA) \/ Refinement of the reconstruction\n\n## C\n\n  * calc_scale function \/ Training and visualization, Training and visualization\n  * camera\n    * color format, inputting from \/ Input color format from the camera\n    * accessing \/ Accessing the camera\n  * camera-intrinsic matrix\n    * about \/ Obtaining the camera-intrinsic matrix\n    * obtaining \/ Obtaining the camera-intrinsic matrix\n  * CameraCalibration class\n    * about \/ Camera calibration\n  * camera calibration process\n    * about \/ Camera calibration\n  * camera matrices\n    * finding \/ Finding camera matrices\n  * camera motion\n    * estimating, from a pair of images \/ Estimating the camera motion from a pair of images\n  * camera motion estimation, from pair of images\n    * about \/ Estimating the camera motion from a pair of images\n    * point matching, rich feature descriptors used \/ Point matching using rich feature descriptors\n    * point matching, optical flow used \/ Point matching using optical flow\n    * camera matrices, finding \/ Finding camera matrices\n  * camera resolution\n    * using \/ Using a different camera resolution\n  * camera_cailbration.exe package \/ Obtaining the camera-intrinsic matrix\n  * candidates search \/ Candidates search\n  * Canny edge detector\n    * about \/ Generating a black-and-white sketch\n  * cartoon\n    * generating \/ Generating a color painting and a cartoon\n  * cartoon.cpp file \/ Generating a black-and-white sketch, Adding the cartoonifier code to the Android NDK app\n  * cartoonifier code\n    * adding, to Android NDK app \/ Adding the cartoonifier code to the Android NDK app\n  * CartoonifierView class \/ Saving the image to a file and to the Android picture gallery\n  * cartoonifyImage() function \/ Generating a black-and-white sketch, Porting from desktop to Android, Adding the cartoonifier code to the Android NDK app, Showing an Android notification message about a saved image\n  * CartoonifyImage() function \/ Adding the cartoonifier code to the Android NDK app\n  * cartoon modes\n    * modifying, through Android menu bar \/ Changing cartoon modes through the Android menu bar\n  * CharSegment class \/ OCR segmentation\n  * CIELab color space \/ Skin-detection algorithm\n  * classification\n    * about \/ Classification\n  * collection mode \/ Collection mode\n  * color format\n    * inputting, from camera \/ Input color format from the camera\n    * outputting, for display \/ Output color format for display\n  * color formats\n    * used, for image processing on Android \/ Color formats used for image processing on Android\n  * color painting\n    * generating \/ Generating a color painting and a cartoon\n  * components. Augmented Reality (AR) application\n    * video source \/ Application architecture\n    * image processing routine \/ Application architecture\n    * visualization engine \/ Application architecture\n  * computePose function \/ Pattern.cpp\n  * contours detection\n    * about \/ Contours detection\n  * convertTo function \/ Feature extraction\n  * corner-based feature detectors \/ Feature extraction\n  * correlation-based patch models\n    * about \/ Correlation-based patch models\n    * discriminative patch models \/ Learning discriminative patch models, Generative versus discriminative patch models\n    * generative patch models \/ Generative versus discriminative patch models\n  * countNonZero function \/ Feature extraction\n  * CPR\n    * about \/ Introduction to ANPR\n  * cross-match filter\n    * about \/ Cross-match filter\n  * cv\n    * TopicninitModule_contrib() function \/ Training the face recognition system from collected faces\n    * TopicnAlgorithmTopicn \/ Training the face recognition system from collected faces, Viewing the learned knowledge\n  * cv**BFMatcher class \/ Cross-match filter\n  * cv**cornerSubPix function \/ Marker location refinement\n  * cv**countNonZero function \/ Marker code recognition\n  * cv**FeatureDetector class \/ Feature extraction\n  * cv**findCountours function \/ Contours detection\n  * cv**findHomography function \/ Homography estimation\n  * cv**FlannBasedMatcher method \/ Matching of feature points\n  * cv**getPerspectiveTransform function \/ Candidates search\n  * cv**getTextSize() function \/ Drawing the GUI elements\n  * cv**imshow() method \/ Main camera processing loop for a desktop app\n  * cv**Mat object \/ Accessing the webcam\n  * cv**normalize() function \/ Viewing the learned knowledge\n  * cv**putText() function \/ Drawing the GUI elements\n  * cv**setOpenGlDrawCallback method \/ ARDrawingContext.cpp\n  * cv**solvePnP function \/ Obtaining the camera-intrinsic matrix\n  * cv**VideoCapture**set() method \/ Accessing the webcam\n  * cv**VideoCapture object \/ Accessing the webcam\n  * cv**waitKey() method \/ Main camera processing loop for a desktop app\n  * cv**WINDOW_OPENGL flag \/ Creating OpenGL windows using OpenCV\n  * cv\\\n    * \\TopicnVideoCapture class \/ Training and visualization\n\n## D\n\n  * 3D\n    * marker, placing in \/ Placing a marker in 3D\n  * 3DMM \/ Overview\n  * 3D point clouds\n    * visualizing, with PCL \/ Visualizing 3D point clouds with PCL\n  * 3D virtual object\n    * rendering \/ Rendering the 3D virtual object\n  * 3D visualization\n    * support, enabling in OpenCV \/ Enabling support for 3D visualization in OpenCV\n  * data.txt file \/ Evaluation\n  * Delaunay triangulation \/ Triangulation\n  * demonstration project\n    * about \/ Demonstration\n    * main.cpp \/ main.cpp\n  * descriptor\n    * about \/ Point matching using rich feature descriptors\n  * descriptor-extraction algorithms \/ Feature extraction\n  * desktop app\n    * main camera processing loop \/ Main camera processing loop for a desktop app\n  * detected feature points\n    * about \/ Feature extraction\n  * detection mode \/ Detection mode\n  * detection process\n    * controlling \/ Putting it all together\n  * detectLargestObject()function \/ Eye search regions\n  * detect method \/ Feature extraction\n  * DetectRegions class \/ Classification\n  * dilate() method \/ Implementation of the skin-color changer\n  * DirectX \/ Rendering the 3D virtual object\n  * display\n    * color format, outputting for \/ Output color format for display\n  * drawFrame function \/ Rendering an AR scene\n  * drawString() function \/ Drawing the GUI elements\n\n## E\n\n  * EAGLView class \/ Creating the OpenGL rendering layer, Rendering an AR scene\n  * edge filters\n    * used, for generating evil mode \/ Generating an \"evil\" mode using edge filters, Skin-detection algorithm, Showing the user where to put their face, Implementation of the skin-color changer\n  * ellipse() function \/ Showing the user where to put their face\n  * equalizeHist() function \/ Step 2: Face preprocessing\n  * erode() method \/ Implementation of the skin-color changer\n  * errors, matching\n    * false-positive matches \/ Outlier removal\n    * false-negative matches \/ Outlier removal\n  * estimatePosition function \/ Marker pose estimation\n  * evil mode\n    * generating, edge filters used \/ Generating an \"evil\" mode using edge filters, Skin-detection algorithm, Showing the user where to put their face, Implementation of the skin-color changer\n  * example code\n    * using \/ Using the example code\n  * extraction\n    * about \/ Point matching using rich feature descriptors\n  * eye search regions\n    * about \/ Eye search regions\n    * cascade classifier \/ Eye search regions\n    * example \/ Eye search regions\n    * face preprocessing, performing \/ Eye search regions\n    * geometrical transformation \/ Geometrical transformation, Separate histogram equalization for left and right sides\n    * histogram equalization \/ Separate histogram equalization for left and right sides\n    * pixel noise effect, reducing \/ Smoothing\n    * Elliptical mask \/ Elliptical mask\n\n## F\n\n  * face\n    * detecting \/ Detecting the face\n  * face detection\n    * about \/ Face detection and initialization, Introduction to face recognition and face detection, Step 1: Face detection\n    * implementing, OpenCV \/ Implementing face detection using OpenCV\n    * Haar for object, loading \/ Loading a Haar or LBP detector for object or face detection\n    * LBP detector for object, loading \/ Loading a Haar or LBP detector for object or face detection\n    * webcam, accessing \/ Accessing the webcam\n    * object detecting, LBP Classifier used \/ Detecting an object using the Haar or LBP Classifier\n    * object detecting, Haar used \/ Detecting an object using the Haar or LBP Classifier\n  * face preprocessing \/ Introduction to face recognition and face detection\n    * about \/ Step 2: Face preprocessing\n    * eye, detecting \/ Eye detection\n    * eye search regions \/ Eye search regions\n  * face recognition\n    * about \/ Introduction to face recognition and face detection, Step 4: Face recognition\n    * from face \/ Face identification: Recognizing people from their face\n    * claimed person, verifying \/ Face verification: Validating that it is the claimed person\n    * files, saving \/ Finishing touches: Saving and loading files\n    * files, loading \/ Finishing touches: Saving and loading files\n    * interactive GUI, creating \/ Finishing touches: Making a nice and interactive GUI\n  * FaceRecognizer**train() function \/ Training the face recognition system from collected faces\n  * faces\n    * collecting \/ Step 3: Collecting faces and learning from them\n    * researching \/ Step 3: Collecting faces and learning from them\n    * preprocessed faces, collecting \/ Collecting preprocessed faces for training\n    * face recognition system, training \/ Training the face recognition system from collected faces\n    * internal data structures, viewing \/ Viewing the learned knowledge\n    * average face \/ Average face\n    * eigenvalues \/ Eigenvalues, Eigenfaces, and Fisherfaces\n    * eigenfaces \/ Eigenvalues, Eigenfaces, and Fisherfaces\n    * Fisherfaces \/ Eigenvalues, Eigenfaces, and Fisherfaces\n  * face tracking\n    * about \/ Face tracking\n    * implementing \/ Face tracker implementation\n    * visualization \/ Training and visualization\n    * training \/ Training and visualization\n    * person-specific models \/ Generic versus person-specific models\n    * geometric-specific models \/ Generic versus person-specific models\n  * face tracking algorithms\n    * data training, image component \/ Training data types\n    * data training, annotations componentt \/ Training data types\n    * data training, symmetry indices component \/ Training data types\n    * data training, connectivity indices component \/ Training data types\n    * data training, components \/ Training data types\n  * face_detector class \/ Face detection and initialization\n  * facial feature detectors\n    * using \/ Facial feature detectors\n    * advantages \/ Facial feature detectors\n    * correlation-based patch models \/ Correlation-based patch models\n    * global geometric transformations \/ Accounting for global geometric transformations\n    * visualization \/ Training and visualization\n    * training \/ Training and visualization\n  * false-negative matches\n    * about \/ Outlier removal\n  * false-positive matches\n    * about \/ Outlier removal\n  * feature descriptors\n    * used, for searching arbitrary image on video \/ Using feature descriptors to find an arbitrary image on video\n    * about \/ Feature extraction\n  * feature detection\n    * about \/ Feature extraction\n  * feature extraction\n    * about \/ Feature extraction, Feature extraction\n  * feature point\n    * about \/ Using feature descriptors to find an arbitrary image on video\n  * feature point orientation\n    * about \/ Feature extraction\n  * feature points\n    * detecting \/ Feature extraction\n    * matching \/ Matching of feature points\n  * Find Contours algorithm\n    * about \/ OCR segmentation\n  * findContours function \/ Segmentation\n  * findFundamentalMat function \/ Estimating the camera motion from a pair of images\n  * flag parameter \/ Segmentation\n  * flags parameter \/ Detecting the face\n  * flann-based matcher\n    * about \/ Matching of feature points\n  * floodFill() function \/ Implementation of the skin-color changer\n  * floodFill function \/ Segmentation\n  * Frames Per Second speed\n    * displaying, of Android app \/ Showing the FPS of the app\n  * FREAK algorithm\n    * about \/ Feature extraction\n  * ft_data class \/ Training data types\n\n## G\n\n  * geometrical constraints\n    * about \/ Geometrical constraints\n    * procrustes analysis \/ Procrustes analysis\n    * linear shape models \/ Linear shape models\n    * combined local-global representation \/ A combined local-global representation\n    * training \/ Training and visualization\n    * visualization \/ Training and visualization\n  * getResult method \/ Application architecture\n  * getSimilarity() function \/ Face verification: Validating that it is the claimed person\n  * getStructuringElement function \/ Segmentation\n  * grayscale conversion \/ Grayscale conversion\n  * GUI\n    * creating \/ Finishing touches: Making a nice and interactive GUI\n    * modes, creating \/ Finishing touches: Making a nice and interactive GUI\n    * elements, drawing \/ Drawing the GUI elements\n    * mouse clicks, checking \/ Checking and handling mouse clicks\n    * mouse clicks, handling \/ Checking and handling mouse clicks\n  * GUI elements\n    * drawing \/ Drawing the GUI elements\n    * startup mode \/ Startup mode\n    * detection mode \/ Detection mode\n    * collection mode \/ Collection mode\n    * training mode \/ Training mode\n    * recognition mode \/ Recognition mode\n\n## H\n\n  * Hamming Code\n    * about \/ Marker code recognition\n  * homography estimation \/ Homography estimation\n    * PatternDetector.cpp \/ PatternDetector.cpp\n  * homography refinement\n    * about \/ Homography refinement\n    * PatternDetector.cpp \/ PatternDetector.cpp\n  * HSV (Hue-Saturation-Brightness) \/ Skin-detection algorithm, Implementation of the skin-color changer\n\n## I\n\n  * ICCV conference papers\n    * URL \/ Customizing the app\n  * image\n    * cartoonifying \/ Cartoonifying the image when the user taps the screen\n    * saving, to Android gallery \/ Saving the image to a file and to the Android picture gallery\n  * image binarization \/ Image binarization\n  * image recognition\n    * about \/ Using feature descriptors to find an arbitrary image on video\n  * imwrite() function \/ Saving the image to a file and to the Android picture gallery\n  * Infrared (IR) camera\n    * about \/ Introduction to ANPR\n  * initWithCoder function \/ Creating the OpenGL rendering layer\n  * inside() function \/ Checking and handling mouse clicks\n  * iOS project\n    * creating \/ Creating an iOS project that uses OpenCV\n    * OpenCV framework, adding \/ Adding OpenCV framework\n    * OpenCV headers, including \/ Including OpenCV headers\n  * Iterative Closest Point(ICP) procedure \/ Reconstruction from many views\n\n## J\n\n  * JavaCV library \/ Porting from desktop to Android\n  * JNI (Java Native Interface) \/ Porting from desktop to Android\n\n## K\n\n  * k-nearest neighbor (kNN) radius \/ Point matching using optical flow\n  * keypoints container\n    * about \/ Feature extraction\n\n## L\n\n  * Laplacian filters\n    * about \/ Generating a black-and-white sketch\n  * LDA \/ Training the face recognition system from collected faces\n  * loDiff parameter \/ Segmentation\n\n## M\n\n  * main camera processing loop\n    * for desktop app \/ Main camera processing loop for a desktop app\n  * marker\n    * about \/ Marker detection\n    * placing, in 3D \/ Placing a marker in 3D\n    * strengths \/ Marker-based versus marker-less AR\n    * limitations \/ Marker-based versus marker-less AR\n  * marker, in 3D\n    * camera calibration process \/ Camera calibration\n    * marker pose estimation \/ Marker pose estimation\n  * marker-based approach\n    * versus marker-less approach \/ Marker-based versus marker-less AR\n  * marker-less AR approach\n    * versus marked-based AR approach \/ Marker-based versus marker-less AR\n    * strengths \/ Marker-based versus marker-less AR\n  * marker code\n    * reading \/ Reading marker code\n  * marker code recognition\n    * about \/ Marker code recognition\n    * marker code, reading \/ Reading marker code\n    * location refinement \/ Marker location refinement\n  * marker detection procedure\n    * about \/ Marker detection\n    * marker identification \/ Marker identification\n    * contours detection \/ Contours detection\n    * candidates search \/ Candidates search\n  * marker detection routine\n    * about \/ Marker identification\n    * grayscale conversion \/ Grayscale conversion\n    * image binarization \/ Image binarization\n  * MarkerDetector class \/ Application architecture, Marker detection\n  * MarkerDetector function \/ Image binarization\n  * marker location refinement \/ Marker location refinement\n  * marker pose estimation\n    * about \/ Marker pose estimation\n  * Mat**channels() method \/ Output color format for display\n  * match function \/ Matching of feature points\n  * matching\n    * errors \/ Outlier removal\n    * about \/ Point matching using rich feature descriptors\n  * mbgra variable \/ Output color format for display\n  * minAreaRect function \/ Segmentation\n  * minContourPointsAllowed variable \/ Contours detection\n  * minFeatureSize parameter \/ Detecting the face\n  * minMaxLoc function \/ Feature extraction\n  * minNeighbors parameter \/ Detecting the face\n  * morphologyEx function \/ Segmentation\n  * MUCT dataset\n    * about \/ Pre-annotated data (The MUCT dataset)\")\n    * using \/ Pre-annotated data (The MUCT dataset)\")\n  * Multi-Layer Perceptron (MLP)\n    * about \/ OCR classification\n\n## N\n\n  * NDK (Native Development Kit) \/ Porting from desktop to Android\n  * newVal parameter \/ Segmentation\n  * nextFrameShouldBeSaved()function \/ Cartoonifying the image when the user taps the screen\n  * nextFrameShouldBeSaved() function \/ Saving the image to a file and to the Android picture gallery\n  * non-rigid face tracking\n    * overview \/ Overview\n\n## O\n\n  * object detection\n    * grayscale color conversion \/ Grayscale color conversion\n    * camera image, shrinking \/ Shrinking the camera image\n    * histogram equalization \/ Histogram equalization\n  * OCR\n    * about \/ Introduction to ANPR\n  * OCR**train function \/ OCR classification\n  * OCR.xml file \/ Evaluation\n  * OCR classification\n    * about \/ OCR classification\n  * OCR segmentation\n    * about \/ OCR segmentation\n  * Ogre \/ Rendering the 3D virtual object\n  * onMouse function \/ Checking and handling mouse clicks\n  * onTouch() function \/ Cartoonifying the image when the user taps the screen\n  * OpenCV\n    * webcam, accessing \/ Accessing the webcam\n    * about \/ Accessing the webcam, Creating an iOS project that uses OpenCV\n    * main camera processing loop, for desktop app \/ Main camera processing loop for a desktop app\n    * Android project, setting up \/ Setting up an Android project that uses OpenCV\n    * adding, to iOS project \/ Creating an iOS project that uses OpenCV\n    * URL \/ Adding OpenCV framework\n    * feature detection algorithms \/ Feature extraction\n    * support, enabling for 3D visualization \/ Enabling support for 3D visualization in OpenCV\n    * used, for creating OpenGL windows \/ Creating OpenGL windows using OpenCV\n    * used, for capturing videos \/ Video capture using OpenCV\n    * used, for face detection implementing \/ Implementing face detection using OpenCV\n  * OpenCV framework\n    * adding, to iOS project \/ Adding OpenCV framework\n  * OpenCV headers\n    * adding, to iOS project \/ Including OpenCV headers\n  * OpenCV v2.4.1\n    * recognition algorithms \/ Training the face recognition system from collected faces\n  * OpenGL \/ Rendering the 3D virtual object\n    * rendering layer, creating \/ Creating the OpenGL rendering layer\n  * OpenGL rendering layer\n    * creating \/ Creating the OpenGL rendering layer\n  * OpenGL windows\n    * creating, OpenCV used \/ Creating OpenGL windows using OpenCV\n  * optical flow (OF)\n    * used, for point matching \/ Point matching using optical flow\n  * ORB algorithm\n    * about \/ Feature extraction\n  * outlier removal\n    * about \/ Outlier removal\n    * cross-match filter \/ Cross-match filter\n    * ratio test \/ Ratio test\n    * homography estimation \/ Homography estimation\n    * homography refinement \/ Homography refinement\n\n## P\n\n  * .pch file \/ Including OpenCV headers\n  * Pattern.cpp \/ Pattern.cpp\n  * PatternDetector.cpp\n    * about \/ PatternDetector.cpp\n\/ PatternDetector.cpp, PatternDetector.cpp, PatternDetector.cpp\n  * PatternDetector.cpp file \/ PatternDetector.cpp\n  * PatternMatcher class \/ Putting it all together\n  * pattern object\n    * defining \/ Definition of a pattern object\n  * Pattern object\n    * about \/ PatternDetector.cpp\n  * pattern pose estimation\n    * about \/ Pattern pose estimation\n    * PatternDetector.cpp \/ PatternDetector.cpp\n  * pattern recognition algorithms, steps\n    * segmentation \/ ANPR algorithm\n    * feature extraction \/ ANPR algorithm\n    * classification \/ ANPR algorithm\n  * Perspective N-Point(PNP) \/ Reconstruction from many views\n  * plate detection procedure\n    * about \/ Plate detection\n    * segmentation \/ Segmentation\n    * classification \/ Classification\n  * plate recognition procedure\n    * about \/ Plate recognition\n    * OCR segmentation \/ OCR segmentation\n    * feature extraction \/ Feature extraction\n    * OCR classification \/ OCR classification\n    * evaluation \/ Evaluation\n  * Point Cloud Library (PCL)\n    * used, for visualizing 3D point clouds \/ Visualizing 3D point clouds with PCL\n    * about \/ Visualizing 3D point clouds with PCL\n  * POS \/ POSIT\n  * POSIT\n    * about \/ POSIT\n    * dividing into \/ Diving into POSIT\n    * head model \/ POSIT and head model\n    * video file, tracing from \/ Tracking from webcam or video file\n    * webcam, tracing from \/ Tracking from webcam or video file\n  * predict() function \/ Face verification: Validating that it is the claimed person\n  * printMat() function \/ Viewing the learned knowledge\n  * printMatInfo() function \/ Viewing the learned knowledge\n  * processFrame() function \/ Output color format for display, Cartoonifying the image when the user taps the screen, Saving the image to a file and to the Android picture gallery, Application architecture\n  * processFrame function \/ main.cpp\n  * processSingleImage function \/ main.cpp\n  * Procrustes Analysis \/ Active Shape Models\n  * program\n    * porting, from desktop to Android \/ Porting from desktop to Android\n  * ProjectedHistogram function \/ Feature extraction\n\n## Q\n\n  * query descriptors\n    * matching \/ PatternDetector.cpp\n\n## R\n\n  * random pepper noise\n    * reducing, from sketch image \/ Reducing the random pepper noise from the sketch image\n  * random sample consensus (RANSAC) method \/ Homography estimation\n  * ratio test\n    * about \/ Ratio test\n    * PatternDetector.cpp \/ PatternDetector.cpp\n    * used, for performing robust descriptor matching \/ PatternDetector.cpp\n  * real-life image\n    * converting, to sketch drawing \/ Generating a black-and-white sketch\n  * recognition mode \/ Recognition mode\n  * reconstruction, scene\n    * refining \/ Refinement of the reconstruction\n  * removePepperNoise() function \/ Reducing the random pepper noise from the sketch image\n  * RGB (Red-Green-Blue) \/ Skin-detection algorithm, Implementation of the skin-color changer\n  * robust descriptor matching\n    * performing, ratio test used \/ PatternDetector.cpp\n  * RotatedRect class \/ Segmentation\n  * rotation invariant\n    * about \/ Feature extraction\n\n## S\n\n  * saved image\n    * Android notification message, displaying for \/ Showing an Android notification message about a saved image\n  * savePNGImageToGallery() function \/ Saving the image to a file and to the Android picture gallery\n  * savingRegions variable \/ Classification\n  * scene\n    * reconstructing \/ Reconstructing the scene\n    * reconstructing, from multiple view \/ Reconstruction from many views\n  * Scharr\n    * about \/ Generating a black-and-white sketch\n  * searchScaleFactor parameter \/ Detecting the face\n  * segmentation\n    * about \/ Segmentation\n  * setupCamera() function \/ Using a different camera resolution\n  * showNotificationMessage() function \/ Showing an Android notification message about a saved image\n  * ShowPreview() function \/ Output color format for display, Adding the cartoonifier code to the Android NDK app, Cartoonifying the image when the user taps the screen\n  * SIFT\n    * about \/ Feature extraction\n  * Simple Sparse Bundle Adjustment (SSBA) library \/ Refinement of the reconstruction\n  * Singular Value Decomposition (SVD) \/ Finding camera matrices\n  * sketch drawing\n    * real-life image, converting to \/ Generating a black-and-white sketch\n  * sketch image\n    * random pepper noise, reducing from \/ Reducing the random pepper noise from the sketch image\n  * skin color changer\n    * implementing \/ Implementation of the skin-color changer\n  * skin detection\n    * used, for generating alien mode \/ Generating an \"alien\" mode using skin detection\n  * Skin detection algorithm\n    * about \/ Skin-detection algorithm\n  * Sobel\n    * about \/ Generating a black-and-white sketch\n  * Sobel filter\n    * about \/ Segmentation\n  * solvePnP functions \/ Reconstruction from many views\n  * solvePnPRansac function \/ Reconstruction from many views\n  * Speeded-Up Robust Features (SURF) features\n    * about \/ Point matching using rich feature descriptors\n  * Startup mode \/ Startup mode\n  * statistical outlier removal (SOR) tool \/ Visualizing 3D point clouds with PCL\n  * Structure from Motion (SfM)\n    * exploring \/ Structure from Motion concepts\n  * subspaceProject() function \/ Face verification: Validating that it is the claimed person\n  * subspaceReconstruct() function \/ Face verification: Validating that it is the claimed person\n  * Support Vector Machine (SVM)\n    * about \/ Plate detection\n  * Support Vector Machine (SVM) algorithm \/ Classification\n  * SURF\n    * about \/ Feature extraction\n  * surfaceChanged() function \/ Using a different camera resolution\n  * SVD \/ Linear shape models\n\n## T\n\n  * threshold function \/ Segmentation\n  * training mode \/ Training mode\n  * TriangulatePoints function \/ Reconstruction from many views\n\n## U\n\n  * UIGetScreenImage function \/ Accessing the camera\n  * UIViewController interface \/ Application architecture\n  * Unity \/ Rendering the 3D virtual object\n  * Unreal Engine \/ Rendering the 3D virtual object\n  * upDiff parameter \/ Segmentation\n  * user face\n    * placing, for alien mode \/ Showing the user where to put their face\n  * utilities\n    * about \/ Utilities\n    * Object-oriented design \/ Object-oriented design\n    * data collections \/ Data collection: Image and video annotation\n\n## V\n\n  * videos\n    * capturing, OpenCV used \/ Video capture using OpenCV\n  * VideoSource interface \/ Accessing the camera\n  * ViewController class \/ Application architecture\n  * VisualizationController \/ Application architecture\n  * VisualizationControllerclass \/ Rendering an AR scene\n\n## W\n\n  * warpAffine() function \/ Geometrical transformation\n  * warpAffine function \/ Segmentation, Triangle texture warping\n  * webcam\n    * accessing \/ Accessing the webcam\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Photo by Chip Harlan via Creative Commons\/Flickr. El Centro de la Raza is selling locally-grown Christmas trees and wreaths again this year, supplied by a local family-owned business. Cut your own christmas trees? Does anyone know some where in or around Lancashire where I will be able to go and cut down my own christmas tree?\nFollow. 8. Christmas Trees Manchester - Christmas trees-you choose and you cut, Christmas trees- you choose and we cut, Precut Christmas trees, Pot Grown (Living Christmas trees to plant later), Christmas wreaths, trees bagged (netting), trees tied, tree shaking provided, free tree trimmings, Beacon fell area pick and cut your own christmas tree, Preston, Lancashire. 578 likes. START THU 30\/11\/17. Open THUR& FRI 10-4, SAT& SUN 9-4. Don't. With four Signature artificial Christmas tree collections, including Aspen Christmas Signature, Vermont Signature, Napa Christmas Signature, and Smoky Mountain Christmas Signature, Balsam Hill aims to provide a beautiful unlit or pre-lit tree for any Christmas style.\nChristmas tree smell is a subjective topic, but the most aromatic tree is the classic Balsam fir. Douglas fir and Virginia pine are noted for their scents as well. White fir has a notably citrusy smell. The Beacon Fell Country Park consists of 110 hectares (271 acres) of woodland, moorland and farmland. The Beacon Fell summit is 266 metres (873 feet) above sea level and offers spectacular views of the Forest of Bowland and Morecambe Bay.\nPremium artificial Christmas trees made with the highest quality materials. Shop Balsam Hill for the perfect tree and inspire new holiday memories. Choose and Cut Christmas Trees at Richardson Christmas Tree Farm in Northern Illinois. While this snowy picture was taken years ago when snow actually fell in December, let the scene get you in the mood for where you can find a Christmas tree.\nThis Christmas, we invite you to find your perfect Christmas Tree and make lasting Christmas memories at one of New Jersey's local Christmas Tree Farms. Christmas Tree Shops. Sign up& save Be the first to receive special offers, new flyers, one day deals, and more from Christmas Tree Shops and That! First-time. Whether you want to cut your own tree, pick a live tree and have it cut for you, buy a tree already cut or buy a living tree you can plant, this page provides detailed listings of Northwest England's choose-and-cut Christmas tree farms, places to buy pre-cut (also called pre-harvested and fresh-cut) trees, stands, sleigh rides, Beacon fell christmas trees rides and related winter events and fun.\nThe award-winning Barton Grange Garden Centre is on the A6 at Brock between Preston and Garstang, set in beautiful Lancashire countryside and was named Destination Garden Centre of the Year in 2012. This website lists choose-and-cut your own Christmas tree farms, pre-cut tree farms, tree stands, tree lots sleigh rides, and other winter fun activities, resources.\nBeacon fell area pick and cut your own christmas tree shared Anna Scarisbrick's post. \u00b7 December 15, 2017 \u00b7 I think it looks lovely and it was the perfect size in the end. Beacon fell area pick and cut your own christmas tree \u00b7 December 17, 2017 \u00b7 We would like to say thank you to everyone who came for their tree and wish them a very Happy Christmas and New Year. Although Beacon Fell is only 873 feet above sea level its summit dominates the West Lancashire plain, with Blackpool Tower clearly visible on a good.\nUse these practical tips to be a more informed consumer and to ensure a great experience with your Real Tree. Click the plus sign (+) next to each item to view. B y the time Eureka earned its designation 80 years ago as \"Christmas Tree Capital of the World, \" the region's tree-cutting industry had already established roots dating back at least a.\nIs a fresh cut really necessary before putting a tree in a water stand?. Once the water level falls below the exposed surface on a tapered trunk, drying will begin. Hundreds gathered in downtown Oswego on Friday night for the annual tree-lighting ceremony followed by the Main Street Christmas Walk.\nThe demand for Montana Christmas trees led to Eureka being nicknamed the \"Christmas Tree Capitol of the World, \" but the demand began to decline in the 1970s because Douglas fir fell out of.\nSpecial offer on Christmas trees this December 2015 in Preston and all North West. After moving out of my tiny flat into my house, Ive finally got room for a Christmas tree. The big question is then do i go for a real one or fake? Beacon Fell Country Park.\nAs you go round the trail you will see the different species of trees that have been planted in the wood and the different habitats that. About The Beacon Fell Country Park consists of 110 hectares (271 acres) of woodland, moorland and farmland. The Beacon Fell summit is 266 metres (873 feet) above sea level and offers spectacular views of the Forest of Bowland and Morecambe Bay. Beacon Fell, whose moody sides you can see from the city, needs trees thinned to improve the quality and diversity of its habitat \u2013 according to Lancashire County Council.\nEaster Fall Halloween Christmas. LED Christmas Trees () Type. Pre-lit Trees (161). Northlight 7 ft. Mount Beacon Pine Slim Pre-lit Christmas Tree. The Christmas tree that adorns the Capitol lawn in Washington, D. C.this year will come from Idaho. as a beacon of holiday cheer. in your classroom this fall. Beacon Park Market Thursdays at Beacon Park Peruse the offerings or do a little open air shopping while you wait for your lunch order from the# DowntownStreetEats food trucks.\nEaster Fall Halloween Christmas. Pre-lit Christmas Trees () Type. Pre-lit Trees (486). Northlight 7 ft. Mount Beacon Pine Slim Pre-lit Christmas Tree. Fall D\u00e9cor Artificial Christmas Trees. Beacon Hill Cypress Topiary. 4' Clear LED - Battery. we offer a selection of affordable artificial Christmas trees under. Dec 04, 2007 \u00b7 Going to a tree farm means that ppl have planted trees, they have grown for 7 years, cleaning the air through their life style, and next spring the farm will plant another batch so that in 7 more years time the trees are ready.\nViews limited for tree growth. Beacon Fell (Longridge) Popular. presumably where you escape to from Preston when fed up with the relatives at Christmas. 1st. Buy\" Pre Lit Potted Christmas Trees\" products like National Tree 2-Foot Crestwood Spruce Pre-Lit Tree with White Lights, National Tree 2-Foot Crestwood Spruce Pre-Lit Tree with Clear Lights, Sterling 3.\n5-Foot Pre-Lit Potted Christmas Tree, National Tree 2-Foot Crestwood Spruce Pre-Lit Tree With Multicolor Lights, Northlight 6-Foot Pre-Lit Potted Pine Tree Shop our best selection of Pre-lit Christmas Trees to reflect your style and inspire your home this time of year.\nFind the perfect seasonal decor at Hayneedle, where you can buy online while you explore our room designs and curated looks for tips, ideas& inspiration to help you along the way. Posts to Beacon fell area pick and cut your own christmas tree the wind has been horrendous here. Over 20 huge trees down and although they weren't christmas trees they have destroyed some.\nWhere To Buy A Christmas Tree Near Beacon, NY. While the pictures taken below are from years ago when snow actually fell in December (not anymore it.\nBeacon fell area pick and cut your own christmas tree, Preston, Lancashire. 578 likes. START THU 30\/11\/17. Open THUR& FRI 10-4, SAT& SUN 9-4. Don't. Beacon Fell, whose moody sides you can see from the city, needs trees thinned to improve the quality and diversity of its habitat - according to Lancashire County Council. Christmas Tree Capital of The World. fell out of favor and customers grew drawn to the more symmetrical, plantation-raised varieties. The journey of the Capitol Christmas Tree included a.\nChristmas Tree Lighting Celebrations in Boston Find Out the Tree Lighting Schedule for the 2018 Holiday Season Christmas tree on Boston Common, with the State House in the background (the big tree is a special gift to Boston from Halifax) Beacon fell area pick and cut your own christmas tree, Preston, Lancashire. 582 likes \u00b7 1 talking about this. START THU 30\/11\/17. Open THUR& FRI 10-4.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Klaus Marius Hansen is currently with ISIS Katrinebjerg where he helps bring IT research to industry. He obtained his Ph.D. in Computer Science on the topic of experimental object-oriented modelling. Based on this research, he co-founded Ideogramic ApS, a start-up focussing on innovative diagramming solutions especially in combination with pen-based interfaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paccheri is a large, tubular pasta typically from the Campania region of southern Italy. This delicious example comes from the Gentile factory in Gragnano, the ancient pasta-making city of the Naples area. The use of exceptional ingredients and the traditional methods used make for excellent flavor and amazing presentation at the dinner table.\nPaccheri are a versatile shape of pasta, and can be used with any sauce. When in Umbria, the Paccheri are prepared with a tomato meat sauce, stuffed, covered with grated pecorino (sheep cheese) and baked. A drizzle of our Organic Crushed Basil or Wild Portobello Mushroom & Sage extra virgin olive oil provides the delicious finishing touch!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*It is important to thoroughly research essential oils before using them. Not all essential oils are beneficial to your health. Some may cause skin irritation, some may be dangerous if injested, or may cause an allergic reaction.\nThey are volatile liquids (meaning they turn from a liquid to a gas very quickly at room temperature or higher). They are derived from the leaves, stems, flowers, bark, roots, or other elements of a plant.\nEssential oils have been around for a long time. They were mankind's first medicine. Priests and physicians have been using oils for thousands of years. In Egypt recipes were recorded on the walls in hieroglypics. Romans used them for everything from scenting their hair and clothing to their military flags to using them in massage and baths. 188 references to essential oils can be found in The Bible.\nMost essential oils are high in antibacterial, anti-fungal, and anti-viral properties: This makes them an excellent addition to your homemade cleaning preparations. Oils that are best for cleaning are: Lemon, grapefruit, eucalyptus, peppermint, tea tree, lavender, and rosemary.\nEssential Oils have both physical and emotional level healing properties. Considered among the most therapeutic of all botanical extracts throughout the ages. Here are a few ways they can be used: to relax, sedate, balance, rejuvenate, invigorate, enhance memory, act as an anti-inflammatory, analgesic, anti-bacterial and anti-spasmodic.\nEssential oils are wholly natural and cannot be patented; which means that you'll never see an essential oil in a pharmaceutical drug.\n- More importantly, because essential oils cannot be patented, drug companies will not waste money studying them. This limits our scientific knowledge of essential oils GREATLY, and the majority of what we know about them are things that have been passed down through thousands of years of personal use and experimentation.\nEnormous amounts of plants are needed to produce essential oil. In fact, on the extreme end, it takes 4000 pounds of Bulgarian roses to produce 1 pound of essential oil. Other plants like lavender only take 100 pounds of plant material to produce a pound of essential oil.\nMost essential oils should never be used undiluted on the skin. They are Highly Concentrated -One drop of an essential oil is equivalent to thirty cups of tea in terms of concentration.\nInstead, they should be combined with \"real\" oils (called carrier oils), waxes, butters, alcohols, or other diluting measures. Because they're so concentrated, if you don't dilute, you may end up with an unfortunate reaction.\nNever use an undiluted essential oil on a baby or child. Children have much thinner, more delicate skin than adults have, and tend to be very sensitive to the potency of essential oils.\nAvoid the following essential oils while pregnant or nursing (and skip EOs completely in your first trimester): Aniseed, cedarwood, chamomile, cinnamon, clary sage, clove, ginger, jasmine, lemon, nutmeg, rosemary, sage.\nDo not take essential oils internally, especially oils like wintergreen and eucalyptus. While some essential oils may be used well-diluted in something like toothpaste with safety, it's generally recognized that there's no need to take essential oils internally. In fact, there are several toxic essential oils that should be avoided even through skin contact.\nQuality and quantity vary greatly depending on plant variety, time of harvest, soil condition, method of cultivation and method of extraction. The two methods of extraction used in \"true essential oils,\" are: steam distillation and expression. Beware of synthetic oils. Know the origin and reputation of the company before purchasing oils. Do not purchase oils labeled, \"perfume,\" \"fragrance,\" or \"potpourri.\" if you are looking to use the oil for aromatherapy purposes.\nEssential oils can penetrate the body through either the nose or skin. Inhalation of an oil can be a powerful way to affect memory, hormones and emotions. 1-3 drops of an oil is usually adequate when applying oils topically. The feet are the second fastest area to absorb oils because of the large pores. The wrists and behind the ears are also quick absorbing areas.\nUse Essential Oils To Help Your Mood. Lavender, peppermint, grapefruit, chamomile, lemon, ylang-ylang all help produce happy, joyous moods. Clary sage helps with PMS (although there have been reports that overuse of clary sage can lead to intoxication). Rosemary increases focus and concentration. Don't forget the mood benefits of essential oils.\nThe recommended usage of many essential oils is hotly contested throughout the aromatherapy profession. If you're ever unsure about an oil or its use, do the research you can, and if you still cannot make up your mind as to its safety \u2013 avoid it. But, by all means, do NOT be afraid of essential oils. Just use them with care and respect, and all will be fine.\nIf in doubt - ask the professionals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a specialized and insured moving company; that employ only licensed movers to offer you with moving and storage space services within East New York, in addition to anyplace in the country. Local Moving Company East New York's skilled relocation team is immensely satisfied of our proven star records; we regularly attend East New York customers' every necessity, want, or concern. Whether you are looking to move a few boxes, the contents of a whole house, or even a busy office \u2013 with our unique and specialized long distance relocating assistance, you will receive a prompt, courteous, and consistent service that is absolutely beyond belief. In addition to the first class of customer service you can obtain at Local Movers East New York, you will also discover that we have an impeccable record with the businesses governing agencies.\nMoving Company East New York's accredited movers can move and hoard a large selection of contents from any type of structure, anything from a small apartment to a full commercial procedure. We offer you all of the services that you would require from specialized long distance movers in New York. Whether you're only relocating within New York, or planning a long distance cross-country move, we'll always have just the exact solution for you. Our very skilled local movers or long distance movers will even handle each and every single detail for you \u2013 including planning, packing, relocating, storing, unpacking, and a good deal more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This morning, we woke up as we do every day. You drank your bottle, while I sipped my coffee. When you were done, I held you and told you it was Thanksgiving\u2026 your second. You stared at me, intently watching my lips. I told you I was grateful for you being here another year\u2026 I told you how amazing it is to have you here each day. You began to squirm as most kids would, and I let you get in your walker to play.\nThanksgiving is not my favorite of all the holidays. I'm not a huge fan of stuffing or gravy, and getting up super early to gut a large bird wouldn't be one of my most loved hobbies. But, much like everything else in my life, this holiday holds more sentiment than it did before you were born.\nI suppose technically it is your third Thanksgiving since you were in my belly for your first. You were safe and warm and perfect, as far as I knew. Your father and I were so completely unaware and unprepared for the next few years before us. Everything was ignorantly blissful and perfectly normal.\nYour next Thanksgiving you had grown so much. You were rolls upon rolls, and looked nothing like the tiny five pound baby I had held a few months ago. You lit up for your favorite movie as we spent our holiday in a new place. I was so absolutely proud of how far you had come.\nAnd now, this year, you are in a baby walker, cruising our living room, smacking your toys. You are becoming a twonager, bored quickly, and with very stern opinions. You throw things and laugh at yourself and smile when someone sings to you. You are everything to be thankful for.\nI can't help but think that you aren't even supposed to be here. Your heart shouldn't be beating. Your lungs shouldn't fill with air. According to the sheet of paper I was handed back when you were one month old, your massive amount of defects should have overwhelmed your supposedly frail body by now. You were supposed to die.\nBut you didn't. You gave us a few scares. There were nights you turned blue and I fought within myself if I had enough time to call 911 or if I needed to try and save you myself. One time we were in the car with your brothers when you started to choke on your own vomit. I pulled over while the boys kept yelling at me that something was wrong, and ripped you from your car seat. I remember screaming for a nasal bulb (as if your brothers knew what that was) and pounding your back to clear your throat and nose. Another night, I tried to suction you and you had a mucus plug block a nasal passage. You started to turn gray while I held you to my chest and tried to help you clear it. I wondered if I could run to my phone but I knew you'd be dead if I waited for an ambulance.\nBut we made it\u2026 and for that I am thankful. I am thankful for the early mornings started with your coos. I am thankful for the days full of therapies, specialists, and Mickey Mouse Clubhouse. I am thankful for the bathtimes and the playtimes and the crooked smiles. I am thankful for every second with you.\nHappy 2nd (or 3rd, but who's counting?) Thanksgiving, my fierce daughter. Every day is full of thanks with you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A significant contribution to the successful treatment of children with cancer and blood disorders comes from adopting a multi disciplinary team approach to patient care.\nOne of the pediatric oncologists\/hematologists is the primary physician responsible for directing the care of the child. This oncologist along with a nurse clinician and a social worker form the primary care team which liaises with the local community health care professionals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tom Hiddleston, everyone's lovable trickster Loki, may have just spoiled a thing or two about the Thor: Ragnarok story even before he knew where Thor 3 was headed. And he did it last year in an exclusive interview with MovieWeb a year ahead of Marvel Studios' confirmation that Thor 3 will be the Ragnarok storyline.\nGo the Next Page to read about Tom Hiddleston's possible reveals for Thor: Ragnarok!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Squeeze the meat out from half a length of Italian sausage and break it up with a fork. [How much is half a length? I don't know! Maybe the length of a Cumberland sausage.] Brown it in 2 tablesp olive oil. Chuck in a chopped red onion and brown this, too. Add about a litre of homemade tomato sauce [I did have some of that in the freezer!] and about 125 ml red wine. Season and add 300 gr sliced mushrooms. Let it all simmer, stirring now and then. Cook 400 gr rigatoni according to the instructions on the pack. Drain the pasta and place a layer of it in a large, oiled baking dish. Now add about 200 gr sliced mozzarella, half the rag\u00f9, then the rest of the pasta and a little more mozzarella. Finish off with the rest of the rag\u00f9 and sprinkle over about a teasp of breadcrumbs and some freshly grated parmesan. Cook at 180 C for 20 mins.\nYour troubles won't vanish but they will seem much smaller. Buon appetito!\nJust look at the plate of complimentary antipasti I got with my [non-alcoholic] aperitivo in Bar Cicara yesterday!\nI particularly liked the idea of spooning orange and onion salad onto bruschetta [left] - that's one I'm going to copy.\nSicily and Sardinia are popular tourist destinations these days but there is a smaller, quieter island which also has a lot to offer. Elba is a paradise in many ways and you can find some things to do there in this article.\nOur blog of the week was My Paesano, administered by Capri, an Italian American who is passionate about all things Italian. You'll find lots of travel advice and information on My Paesano too.\nNow, ladies, are you looking for a new bra? This one will set you back a mere $2 million. Or, if you'd rather spend your money on iPhone apps, this one is for you if you happen to have a church to run! Meanwhile, a regional council official in Abruzzo has come up with a unique solution to the area's prostitution problem and the Vatican has claimed those two renegades, Bart and Homer Simpson, as its own.\nFor my Patti Chiari column, I wrote about the elderly in Sicily. Are the elderly treated well and respected where you are?\nThis is Gino the artist's Christmas collage, in which he has used natural materials, such as carob pods, nuts and asparagus leaves [traditionally used to decorate crib scenes in Sicily] to suggest the shapes, colours and forms of Christmas. I know that Gino would love to receive some comments from you.\nA new bar, La Cicara - cicara being a Sicilian word for a little cup - has opened just across the road here and of course, I have been to explore it a couple of times. I must say that the service is good and it has a very pleasant d\u00e9cor and atmosphere. Today it was already crowded just before 1 pm as people stopped off for an aperitivo and to partake of the delicious complimentary antipasti on offer.\n\"Il bar non ha ricordi ma i ricordi portano inevitabilmente al bar.\"\n\"The bar has no memories but your memories will inevitably lead you to the bar.\"\nI wish Bar Cicara good luck. Their little gift made my day and traders who will go to so much trouble to please their customers deserve success.\nA fine lunch for those who had spent a hard morning thinking about olive picking!\nWith friends on Chiara's terrace for coffee, tea and cakes yesterday morning.\nI'm sure you'll all want to join me in congratulating Rosa, whose first grandchild, Adri, was born last week.\nWelcome to the world, little Adri. You will grow up with the love of a wonderful family and, as a bonus, you'll be able to enjoy your nonna Rosa's great cooking!\nI love to see mounds of chestnuts on sale and to catch the aroma of their roasting as I walk home on autumn evenings in Sicily. Broccoli, too, is abundant and the purple kind, especially, adds to the rich tapestry of colours here as you pass greengrocers' shops or fruitsellers' lorries.\nThe fruitseller who calls me bambina has returned from his extended summer break and who am I to argue when he tells me that his wares are in regalo [\"a gift\"] today at only \u20ac10.00 per crate? As usual, after asking the nearest passer-by to watch his lorry for a few minutes, he carries the crate home for me.\nFirst of all, there are only 65 shopping days left until Christmas and a good place to head in order to solve those present problems would be the Oh bej, Oh bej! Christmas market in Milan.\nStill with Christmas shopping, maybe the man in your life needs this and this CD is definitely going to be on my own wish list. The most amazing news of the week from Italy was this surgical breakthrough whilst a completely different kind of breakthrough occurred on Friday. I liked the honest logic of this Northern League supporter and, at the risk of losing my feminist credentials, I cannot help hoping that this film will soon be released outside France.\nFor my Patti Chiari column, I wrote in more detail about the Prickly Pear Festival at Militello and explained the linguistic confusion surrounding the term mostarda.\nToday bloggers in Italy are turning the blogosphere pink for the nastro rosa [pink ribbon] campaign against breast cancer.\nor visit the nastro rosa or LILT website.\nYou can find a full list of blogs participating in today's rivoluzione in rosa at the Mamma Felice site.\nMarinate 8 or so chicken breast escalopes - an Italian butcher will cut these very thinly for you but in the UK you may have to pound them thin - in the juice of 2 clementines, 2 tablesp olive oil and a half teasp saffron powder [or the contents of 2 envelopes of saffron powder in Italy].\nWith special love and thanks to you.\nWe'll start with the news stories tonight: The issue of the burqa made headlines again as Italy clarified its proposed new law which would ban the face veil. The children of Sakineh Mohammadi Ashtiani, the Iranian woman who has been condemned to death for adultery and the alleged murder of her husband, turned to Italy first to ask for asylum when they felt themselves to be under threat. I understand that they are now under arrest in Iran. This was also the week in which Italian surgeons gave a teenager a permanent artificial heart and what may be the world's smallest apartment went on sale in Rome. The organisers of the 2011 Sanremo Song Festival look set to abandon the tradition of the blonde female presenter or co-presenter and the Milanese argued over this statue.\nOur blog of the week was The Artichoke Blog and, as I always make a terrible mess when preparing these vegetables, I was relieved when Michael told me that this is OK! Staying with food, for my travel article I wrote about some of the best chestnut festivals in Italy at this time of year. And last but definitely not least, I very much enjoyed interviewing Katia Amore of the loveSicily Cookery School and blog for my Patti Chiari column. Do take a look to find out why Modicans began cooking with chocolate and what Katia thinks about the future of Sicilian cooking.\nLike people all over the world tonight, I am watching the news from Chile, holding my breath as each man reaches the surface and then crying as he embraces his family. I also find myself gasping at the feats of engineering and technology which are being achieved and, of course, admiring the bravery and resilience of miners and rescuers alike.\nComing from a country with a mining history, I am thinking ,on this night, of all my compatriots who lost their lives in the bowels of the earth and I cannot help wondering how many of them would have been saved had today's technology been available to them.\n\"There is patience in the Earth to allow us to go into her, and dig and hurt with tunnels and shafts, and if we put back the flesh we have torn from her and so make good what we have weakened, she is content to let us bleed her. But when we take, and leave her weak where we have taken, she has a soreness, and an anger that we should be so cruel to her and so thoughtless of her comfort. So she waits for us, and finding us, bears down, and bearing down, makes us part of her, flesh of her flesh, with our clay in place of the clay we have thoughtlessly shovelled away.\"\nSicily, of course, has a history of sulphur mining, so this island is no stranger to the hardships suffered by mine workers. Child labour was used in the sulphur mines until well into the twentieth century.\nAs the world watches and hopes, I feel sure that the Chilean miners have earned a special place in Welsh and Sicilian hearts.\nThis is another magazine recipe which I've adapted, mainly by roasting the veal in white wine and honey instead of Marsala. The grilled courgettes go well with the veal but the part I was most pleased with was the sauce of shallot, capers, balsamic vinegar, olive oil and white wine, all gently heated together. I do not have one of those meat-slicing machines so beloved of Italians so I couldn't cut the veal as finely as they would but this did not detract from the taste.\nBehold, reader, the lovely colour of this mandarin-flavoured honey that I found in Militello yesterday. Yes, it tastes as good as it looks!\nThe good folk of Militello in Val di Catania sure know how to have a festa and today my naval officer student Cristiano and I headed there for the Sagra del Fico d'India e della Mostarda or Prickly Pear and Mostarda Festival.\nThere were stalls as far as the eye could see, selling everything from clothing to ceramics to honey but the star attractions were the parade of traditional Sicilian carts and the band of the Bersaglieri, the crack troops of the Italian Army who run everywhere. Obviously, I could not photograph the latter but I can tell you they were a wonderful sight in their traditional black, plumed hats and, of all the bands on show, received the loudest and longest round of applause from the crowd.\nAll restaurants in Militello being solidly booked for lunch, we headed to Caltagirone and spent a pleasant few hours there, too. Near the famous ceramic steps, in the Church of San Giuseppe, there is a display of crib figurines from all over the world and this year, a group of volunteers has created a nocturnal crib scene with a model train running on the \"bridge\" which surrounds it. The train symbolises a messenger spreading the good news of Christ's coming around the world. All donations from visitors go to a fund for the vaccination of Tanzanian children against malaria.\nI very much enjoyed talking to Dana of the Stresa Sights blog and I learnt a lot from her about that lovely part of Italy.\nIf you like torrone, this festival is for you and, staying with food, I can't wait to try out Katia's recipe for chocolate fondue with chilli.\nThose of you lucky enough to be travelling in the Padua area during October will have a chance to see this interesting exhibition and, for cycling fans visiting Rome, there is the Fausto Coppi exhibition.\nThe saddest news story of the week was surely this and I was sorry to read on Monday that the search had been called off. This was the week in which Italy's Tourism Minister called for the removal of an iPhone application which plays on that old joke, the Mafia - except that it is not, of course, a joke - and I can't say I blame her. I had a great time summing up Milan Fashion Week and decided that, when I win the lottery, it will be Armani blue and Alviero Martini 1 Classe for me! It's always reassuring to know what Italy's Supreme Court is up to and that esteemed institution did not let us down last week.\nFinally, for my Patti Chiari column, I wrote about the \"unofficial saint of Modica\", Nino Baglieri, a man I wish I had known. Sadly, Nino is no longer with us but his spirit lives on here in his home town.\nWill someone please tell me why the Intimissimi stores have got hundreds of artificial cakes in their window displays? Everybody knows you can't get into their stuff if you eat cakes. It's not fair!\nBOOK REVIEW - \"LETTERA A MIA FIGLIA CHE VUOLE PORTARE IL VELO\"\nThis is the Italian edition of \"Lettre \u00e0 ma fille qui veut porter le voile\" [\"Letter to my Daughter who Wants to Wear the Veil\"] by the French-Algerian journalist Leila Djitli.\nAicha is a French-Algerrian woman who has fought hard for her freedoms and her place in French society. When her seventeen-year-old daughter, Nawel, suddenly decides that she wants to wear full Islamic dress, Aicha feels that these freedoms are threatened.\nShocked and upset, she resists the temptation to forbid her daughter to wear the veil and instead writes Nawel a long letter in which she explains her beliefs, history, hopes for her daughter's future and fears. She also sets down her thoughts on Islam and modern France.\nShe points out to Nawel that a Muslim man can wear a religious sign - such as the beard - without changing his whole life but that the moment a woman dons the full veil, the veil \"speaks about her and before her\". The veil, argues Aicha, demeans not only women but men, as it has implications for the way in which men perceive women.\n\"Religion can give you a lot, but not everything.\"\nWe do not know what Nawel's final decision will be but the book leaves the reader hoping that she will heed her mother and not be taken in by the peers who are pressurising her to don the veil for their own reasons. Through pen portraits of some of Aicha's friends, we also learn a lot about the lives of French-Algerian women today. But most of all this is Aicha's story: of immigration, of the battle for acceptance and of a woman who values freedom.\nAs the \"burqa debate\" continues to provoke strong feelings in both France and Italy, this is a timely book.\nSince the \"burqa debate\" is indeed, raging in Italy, I will add my thoughts on the whole sorry business here: I of course value women's freedom to wear what they like but in a country like France where the secular state is at the heart of hard-won liberty, it is not unreasonable to ask all to desist from wearing religious symbols in certain public institutions. In a country like Italy, whose own citizens are asked to formally identify themselves every day in order to carry out the simplest transactions, it is not unreasonable to ask a woman to leave her eyes uncovered. In a country like the UK, in which most schools have uniform policies, it is not unreasonable to ask all students to observe them. These matters apart, sisters, I care not whether you wish to wear the burqa or the bikini, provided the decision is truly your own. But remember that generations of women before you fought for your right to do so. Please respect that heritage because all our freedoms depend upon it.\nLove and the trouble it can cause have been subjects much on my mind lately so let's see what the wry Sicilians have to say about it all.\nMatch the proverbs 1 - 6 with their meanings a - f . For answers, highlight the space indicated at the end of the post and they will appear; they really will; it's magic, just like love!\n1. Amuri, biddizzi e denari su' tri cosi ch' un' si ponnu ammucciari.\n2. Amuri, tutti d\u00eccinu ch'\u00e8 amaru, e ognunu voli pruvari siddu \u00e8 veru.\n3. Lu veru amuri 'ntra lu cori appigghia megghiu di la linazza e di la pagghia.\n4. Pi fari beni l'amuri ci voli 'a forza r\u00ea picciuotti e 'u pitittu r\u00ea viecci.\n5. 'U veru amuri nun anvecchia mai.\n6. Nun c'\u00e8 rosa senza spini, nun c'\u00e8 amuri senza peni.\na. True love never grows old.\nb. There is no rose without a thorn and there is no love without pain.\nc. Love, beauty and money are three things which cannot be hidden.\nd. Everyone says love is bitter and everyone wants to find out if this is true.\ne. To make love well you need the strength of the young and the desire of the old.\nf. Real love ignites hearts better than flax or straw.\n1c; 2d; 3f; 4e; 5a; 6b.\nCook 300 gr risotto rice for the length of time stated on the pack in a large pan of water with a little oil, salt and a crushed vegetable stock cube. Drain it and put the rice into a lightly oiled mould [or 6 individual moulds if you prefer]. Put the mould in an oven heated to 180 C while you prepare the sauce [for no longer than 10 mins].\nChop a red onion and a clove of garlic and cook them in 2 tablesp olive oil in a pan. Remove 200 gr Italian sausage from its skin and mash it up with a fork. Add the sausage to the pan and brown it, stirring. Add a little white wine and a tablesp honey. [The magazine recipe says Marsala but I'm not keen on its flavour.] Also add a goodly dollop of 'strattu [tomato paste] diluted in a little homemade tomato sauce or water. Stir everything so that the 'strattu dissolves. [The magazine recipe uses chopped tomatoes but, as regular readers know, I'm mad about 'strattu!] Add seasalt and black pepper to taste and reduce the sauce so that it it quite thick.\nTurn the rice out onto a plate, pour the sauce around it, garnish as you like and serve.\nAntipasti: caponata; crocchettine di latte [vegetable croquettes]; verdure in pastella [vegetables in batter]; panelline [chickpea \"chips\"].\nPrimi: Involtini di melanzana [aubergine roulades]; risotto di frutti di mare [seafood risotto].\nSecondo: filetto in crosta [fillet in pastry]; vegetable sides.\nDolci: pistacchio and mandarin gelato; cassata siciliana.\nThe Pope will also be served a confection of almond paste and pistacchi in the shape of the Trinacria, prepared by the Fiasconaro Brothers of Castelbuono. This will be served with passito di Pantelleria, a liqueur made from pressed grapes.\nPope Benedict, by all accounts, is a man of modest eating habits so I'm sure he won't manage it all. I also hear that he drinks only orange juice at meals and at least he is arriving in the right season for that!\nI wish him buon appetito and a happy day in Sicily, where much enthusiasm and love await him.\nRemember, you read it on Sicily Scene first!\nGarden of Archbishop's Palace, Palermo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We had a big batch of trick-or-treaters show up at one time last night, about 9 kids age 12 and under.\n\"Who are you?\" I asked the first kid.\n\"The Hulk.\" I gave him some candy.\n\"Who are you?\" I asked the second kid.\n\"Thor.\" I gave him some candy.\n\"Who are you?\" I asked the third kid.\n\"Obama.\" He showed me a wadded-up Obama mask in his hand. I didn't give him any candy.\n\"Put the mask on,\" I said.\nMeanwhile, the other kids kept coming to the front and announcing their costumes . . .\n\"Superman.\" \"Batgirl.\" \"Pink lady from Grease.\" \"I'm John Cena.\" \"Witch.\" \"Minnie.\" They all got candy.\nFinally no one was left but me and Obama.\nWhy is Obama a Bad Politician?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Creativity is the spark that kindles innovation. That's the biggest reason that creativity in the workplace is so important. But how can you cultivate creativity, and more importantly, harness that energy into something productive in the workplace? Here's a quick step-by-step guide to help you do just that!\nIt takes the right kind of person with a very particular mindset to handle a creative job. That's why you should find creative ways to test prospective employees' mindsets. For example, you can ask that they choose and lead a warm-up game. The kind of game they choose to play can tell you a lot about their mindset.\nEncourage your team to grow.\nOnce you have the right team in place, give them the expectation that they should always be learning. Allow them the ability and time to stretch themselves and learn similar areas. A great way to do this is to have them work on collaborations.\nEmpower and embolden your team.\nCreativity needs a certain amount of freedom to function. When you lead a creative team, you must ensure that your team has the responsibility and freedom to use their creative solution skills. Be sure that you not only inspire them but remove unneeded obstructions from their path, as well.\nFor as wonderful as creativity is, it can be very fragile. Make sure that your team has access to all the right resources, including guidance from management and a reliable team.\nCelebrate both sides of effort.\nWith creativity, it's important to celebrate both failure and success. No matter how much you do for your team, something will fail at some point. Be sure to applaud the ideas that spring from failure just as much as a successful idea.\nNothing kills creativity faster than stagnation. Be sure that you continue to find new ways to up your team's creativity. Bring in new blood. Come up with new exercises. Be open to changes and embrace the positivity of creativity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover, read and share your new favorite magazines on issuu, all for free.\nWith issuu for iOS, you'll get access to more than 30 million publications in an incredibly immersive reading experience.\n\u2022 20,000 new magazines, catalogs and more in 30+ languages added every day. There's always so much to discover on issuu.\n\u2022 Read gorgeous, fascinating and unique content from around the globe on any topic you can think of \u2014 including photography, design, food & drink, business, fashion, gaming, technology, science, movies, vehicles, travel, style & beauty, real estate and more.\n\u2022 Use Clip to select any portion of a publication you'd like to save and share with your family and friends over email, Facebook, Pinterest, Twitter and more.\n\u2022 Stay up to date on your latest content by following and liking favorite publishers, as well as curating Stacks of publications you love.\n\u2022 Clips and likes show up on your profile page, so you can keep extra organized.\n\u2022 Read on issuu from your iPhone, iPad or via any web browser.\n\u2022 No WiFi, no problem. Save publications for later to Downloads.\n\u2022 Pick up where you left off within the issuu app when switching between iOS devices.\nWe are always working to make the app faster and more stable. If you are enjoying the app, please consider leaving a review or rating for us.\nThey have a feature where you can share apart of your editorial onto your instagram story. It's very easy to edit and translate into an easy mobile read.\nExcellent resource on astounding range of subjects.\nSome texts are teasers, and that's O.K., like a movie trailer. Other documents are comprehensive; hundreds of pages. If I can offer a suggestion; maybe I just having discovered it yet. I'd like to sort, categorize, or prioritize downloads. I'll go searching and find a lot of material. Not stopping at the first find, I would then like to organize it for future reference. Maybe use a personal rating to sort?\nHey thought I would write this review. I really enjoy using your app. I am always finding new information. One thing I would like to request is that you put the stack titles in alphabetical order. Every time that is use Issuu I'm saving at least a dozen or two new titles and now have amassed over 100 stacks and finding the stack I need is a bit of a struggle with the way you order the stacks now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cartouche Creations What in the world does that mean?\nApart from the fact that we are a video and television production company located in the heart of Greenwich Village, NYC, and that we produce ground-breaking videos for small and big firms, marketing, PR and, television broadcasters, Cartouche is the French word for cartridge, as in putting a brand new fresh cartouche of originality and imagination in the camera, ready to film, poised to explore new worlds, and thrilled to express new ways of seeing things.\nVideo concept, video production, video post production, video distribution, video strategy, social media, video optimization and SEO, video advertising, video Marketing, television commercials, television programs, story analysis and consultation.\nChat with one of our producers now. Oh, and OUI! We also work in French if that's your preferred language.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lent is the forty day period before Easter, excluding Sundays, which begins on Ash Wednesday (March 6) and ends on Holy Thursday (April 18).\nThe Stations of the Cross, which follow the path of Christ from Pontius Pilate's praetorium to Christ's tomb have been a popular devotion in parishes. In the 16th century, this pathway was officially entitled the \"Via Dolorosa\" (Sorrowful Way) or simply Way of the Cross or Stations of the Cross.\nAll season long, join your EWTN family as we celebrate Lent with some of our favorite programming. Don't miss out this Lenten Season! View our programming guide for a complete show listing.\nEvery day, EWTN will be adding new Lenten video reflections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since the presidential debate last week I've been trying to remember where I saw this movie before, as it were\u2014where a man considered a member of the elite, the cognoscenti, whose brilliance is considered by millions of admirers to be self evident and unimpeachable, got his clock thoroughly cleaned by someone speaking in plain, commonsense language; and the elitist reacted to being challenged by one of his presumed inferiors with utter petulance.\nWhere had I seen this? I just couldn't place it.\nThen reading Taranto's column today, I noticed his quip about Mary Matalin's schooling someone prominent, a member of the elite, the cognoscenti, etc., telling him, \"You're hardly credible on calling somebody else a liar.\" That someone was Paul Krugman, and Matalin's accurate riposte triggered an answer to where I'd seen the movie before. It was indeed a movie (of sorts): a Youtube of Paul Krugman at a conference in Spain earlier this year as one of a few invited economists.\nAfter Krugman delivers his remarks, Spanish professor Pedro Schwartz (of the Austrian school) surgically demolishes Krugman's arguments while Krugman, sitting behind him, can't hide his discomfort and displeasure. Then when Krugman does speak again to respond, it's with veiled outrage at having his arguments revealed as foolishness.\nProfessor Schwartz remains calm and respectful throughout, letting facts and numbers speak for themselves. Krugman, however, can't hide his petulance or thin skin. About the only difference in Krugman's reaction and Obama's is that Krugman refused to shake the victor's hand.\nThe whole thing is worth watching when you have an hour. If you want just a quick box of popcorn entertainment, fast forward to about 39 minutes.\nWe can only hope that Romney again plays the role of Schwartz next week. But it will be a complete surprise, and devastating letdown, if Paul Ryan doesn't do it Thursday night.\nI don't deliberately search out broadcasts the include Krugman. But I've still seen my share. And I cannot think of many if there is any challenge whatsoever, even if blandly mild, where he isn't petulant and condescending. Plus, he has unmistakable tells, I think, showing either his lack of confidence in what he is saying or his mania when challenged. The more his eyes dart and the more he fidgets with his hands, sometimes at near hummingbird speed\u2026, the more obvious he is. Only when he is allowed to deliver a soliloquy as if a sermon, does he seem calm.\nSo this event in Spain will likely be no different and will be beautiful to watch when I have time.\nI was thinking about the Schwartz video when I saw Mary Matalin take on Paul Krugman. Schwartz was masterful but Matalin came close enough to merit inclusion in the top Krugman take downs ever category. Krugman gets that same bug-eyed expression in reaction to both \u2013 priceless.\nPoor Prof. Krugman. Feeling so put down for not being intellectual enough or having the right credentials.\nHe must have put a silver foot in his mouth.\nI need a hanky, snif, snif \u2026. plus something to pick up the popcorn off the floor because I was laughing so hard at how he handled himself after Schwartz tore him to ribbons.\nWell, come to think of it, this video is a better way to create demand for a quicker picker upper versus by a government subsidy.\nThe Paul Krugman school of economics: shoot from the lip first, ask for a mop later.\nlegacyrepublican: Popcorn and comedy are a dangerous mix to which I can attest. Always make sure you are with someone who knows the Heimlich maneuver when mixing popcorn and very funny material.\nFrom Hollywood- \"Being There\" with the remarkable Peter Sellars!\nChauncey Gardiner foreshadowed the rise of Barack Obama.\nPaul Krugman is to Economics as a witch doctor is to the AMA.\nActually, given the AMA's shameful advocacy for Obamacare (like AARP's), the more correct analogy would be: Economics is to Krugman what the practice of medicine is to the AMA.\nKrugman was questioned towards the end of the video, about an article he wrote in 2002, actually advocating that Greenspan create a Housing Bubble to take the place of the Nasdaq Bubble. Krugman brushed the question off, saying that he was only joking in that article. I found it quite amusing, and so did the questioner, who chuckled to the person next to him. I searched out the article, which is written in typical hyper-partisan Krugman style, and did not pick up on his \"joke\". Maybe someone else can see it?!? http:\/\/www.nytimes.com\/2002\/08\/02\/opinion\/dubya-s-double-dip.html In any case, it is frightening to think that a handful of economic academics have the power to shape thinking, push demand-side and demonize supply-side, all with the help of the press.\nWatching Krugman squirm when it was pointed out that his speciality in international finance was completely irrelevant to economic analysis. It would be like a doctor that specialized in foot fungus treatment trying to weigh in heart surgery as AN AUTHORITY. Krugman goes on to say that the other side is illegitimately \"de-credentializing\" him. First off that wasn't the biggest blow he suffered, he was basically called a economic novelist with a speciality in science fiction romance. Schwartz correctly points out that saying that someone using a credential earned for something other than what one was weighing in for is an absurd use of \"appeal to authority\" debating tactic. So its not \"de-credentializing\", he is simply saying the credentials he does have aren't relevant to the discussion at hand. This is very important, because without that nobel credentials, Krugman sounds less like a very intelligent man trying to explain something deeply confusing and counter intuitive (eg demand driven government allocation of resources), and starts sounding like a raving lunatic trying to convince you the voices in his head are indeed real.\nSkip into the debate, during Krugman's presentation, he talks about that the root problem in Europe is that Europe is \"uncompetitive\". There is no real reason Europe is uncompetitive, its just that they are, and thus we have to spend tons of money on bailouts. Which makes this interesting is that Schwartz appears second and immediately answers and gives the reason why Europe is uncompetitive. Its simple, Europe is inefficient and wasteful with its capital investing heavily in demand driven policies (eg high taxes and government directed economy). Schwartz then points out how wasteful all the countries have been in their stimulus programs, diverting capital from productive capital to wasteful giveaways. When he talks about Supply driven economics, which means if you produce more your automatically wealthier as a society, because the more you produce, the more you have to share with one another. This means a smaller and more efficient government and more robust private sector. It was a logical, concise mystery free presentation. Schwartz schooled Krugman and like a miserly troll Krugman sniped back. delicious, it was like lord of the rings except at a spanish economic conference.\nI listened to the first set of exchanges. I am neither a supply sider nor an aggregate demand person \u2013 since both are grossly simplistic views. IMHO Prof Schwartz did not actually tear apart Krugman's position \u2013 though I wish someone would. He essentially missed the critical issue that should be a central tenet of the Austrian School \u2013 the possibilities and incentives for fundamental innovations and changes in the nature of entrepreneurial expectations. I am reminded of the misinterpretation of Henry Ford's policy of increasing his workers pay so that they could afford to buy his car. What is essential in this economic anecdote is that his workers had places to go and placed a significant value on their ability to save time in getting there.\nThe weakness of Krugman's solution in my opinion is that it dramatically increases the likelihood that entrepreneurs will stand on the side line or look for profit via rents and the modern equivalent of cartels and government sponsored monopolies. Romney's proposed policies of enabling energy independence by pursuing existing fossil fuels will do far more than Government investments hole digging and hole filling, not to mention the equivalent of perpetual motion machines, i.e., wind and solar.\nMy only problem with the comparison is that Krugman was a highly regarded economist at some point \u2013 before he went off the rails. Obama, by comparison, is a wholly manufactured entity. His elitist arrogance strikes me as being the result of confusing his innumerable participation trophies with actual achievement.\n\"VISUALIZE VICTORY\" for Ryan, for Romney, for the Republic.\nI am not counseling \"wishful thinking\", but rather an EXERCISE IN FAITH IN OUR CANDIDATES AND THEIR DIVINE CALLING TO SAVE THIS NATION!\nThus, our favorable comments, our public and private speech in support of Ryan [and many, many others in federal, state, and local levels], our CONTRIBUTIONS, and our PRAYERS to the God of Our Fathers serve to GENERATE POWER, AND INVOKE POWER FROM THE HEAVENS.\nThere is a stone rolling down the mountain whose direction and force is INCREASING SO QUICKLY that it WILL NOT FAIL TO CRUSH FREEDOM'S ENEMIES\u2013within and without our Republic.\nTHUS, VISUALIZE VICTORY, NEVER STOP BELIEVING, AND NEVER STOP FIGHTING, and we, conservatives, patriots, Tea Partiers, libertarians, AND SIMPLY FELLOW AMERICANS SHALL SHOUT A MIGHT SHOUT OF VICTORY AND DELIVERANCE\u2026in less than one (1) month!\nSay what you want about Krugman, he was funny as hell opposite Tony Randall in The Odd Couple.\nJust had time to watch a couple minutes. Now I really look forward to watching the whole thing, maybe this weekend.\nWith a name like Pedro Schwartz, it just has to be good.\nLukeHandCool (who, speaking of long videos and journeys and awkward segues, just a couple weeks or so after his disastrous bicycle journey to the southern tip of Honshu, decided to embark on a new bicycle journey headed north this time, to conquer his fears. So, one morning, he set off for the prefectural capital of Yamaguchi. After the long, strenuous ride, Luke remembered to steer clear of alcohol this time. He also watched the sky to make sure there was no hint of rain, and he watched the time carefully. But after sightseeing around the city, he parked himself on a park bench and proceeded to fall asleep. Luke awoke, perhaps an hour or so later, to find a disheveled, suspicious-looking young Japanese man sharing the park bench with him. Luke got up to stretch, and found that, once again, he had overdone it, and his knees felt like they were seizing up on him. He sat back down, sadly contemplating the painful ride home, berating himself with thoughts of, \"When will I ever learn? Why do I do this to myself?\" when the young man sharing the bench started speaking to Luke in almost incomprehensible English. Luke was not in the mood to be the victim of some bored young man's desire to practice his textbook English. Luke thought about faking a German or French accent and saying something like, \"Me \u2026 no English. English me no.\" But Luke thought that with the luck he had on his bicycle journey days, if he pretended to be from Portugal, the young man would no doubt turn out to be a former exchange student in Lisbon, fluent in Portuguese. So Luke forced a smile, and for 20 minutes or so, endured the young man's English-tainted gibberish. It got to be too excrutiating, so Luke politely told the young man he had to head back home. The disheveled little personal-space invader then wanted to exchange phone numbers. Luke told him he didn't have a phone. Then he wanted to exchange addresses. Luke assured him he hadn't remembered his address yet. He just knew by natural landmarks how to navigate his way to his apartment. The final straw was when the bench squatter pulled something from his pocket. It was a package of cough drops. He waved the package in Luke's face, insisting, \"Doozo, doozo!\" (\"Take some. Take some!\") Luke looked inside the package \u2026 there was one single cough drop left \u2026 completely covered in lint. On the physically agonizing ride home, Luke nonetheless enjoyed the sweet feeling of escape. And he tried to imagine the young man's dinner table telling later that evening about the rude foreigner he'd met in the park who refused his offering of a sweet treat).\nWhen I saw this video I was struck by how Krugman disowned the resulting debt from Keynesian borrowing as not part of Keynesian economic theory. I liken it to talking someone to jumping out of a window but then disowning the fact of the victim hitting the ground.\nThe former speaks to Krugman's hubris, and the latter speaks to the NYTimes bias.\nKudos to Prof. Schwartz for the elegant and effective slap down of Krugman and his ilk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Problems with your teeth or gums can sometimes be serious enough that they simply cannot wait. Incase of dental emergencies getting fast dental treatment can often mean the difference between losing and keeping a tooth, or between quickly clearing up an infection and facing more serious health risks. Our dental emergency team is available for calls 24 hours a day, 7 days a week. Call us today for more information regarding dental emergencies.\nKnowing that there's someone you can call if you need to, is a good feeling, because you never know what could happen. At Smilekraft Dentistry, we're always ready to help when a dental emergency arises. Situations can happen where a sudden flare-up of infection in a tooth can cause unbearable pain and injury can dislodge or loosen one of your teeth and only quick action can save it; a filling, veneer or crown can suddenly fall out.\nAll of these things are dental emergencies that often cannot wait until normal office hours. When the problem is extreme pain caused by an abscessed tooth or gum infection, we can usually prescribe you something to ease the pain until you can get to us during office hours. Broken teeth, broken dentures, or dislodged fillings and crowns have to be taken care of immediately in order to get the best possible results.\nThe most important thing to do when you have any type of tooth or gum emergency, is to call us so that we can assess the situation and let you know what you should do. Many times, there are things that you can do at home to help with the situation until you can see us, and we'll let you know what those things are.\nSwelling in or around the mouth is probably the most serious condition that requires attention from our very own emergency dentist at Smilekraft Dentistry, especially if it is accompanied by fever and\/or trouble swallowing or breathing. Swelling can be caused by an infection, a cyst, an injury, blocked salivary glands or even cancer. Cold compress can often help you feel more comfortable until you get to our office.\nSave any pieces of the broken tooth and rinse your mouth with warm water. If you have a swelling, apply a cold compress to the area to decrease the swelling and pain until you speak to us or visit us. If the fragment of the broken tooth is kept properly intact, then we can try fixing the same piece back. Or else we may do a tooth coloured restoration to cover for the lost part. If the fracture is deep, then we may first perform a root canal treatment to treat any possible infection followed by a tooth coloured cap.\nGently rinse off the knocked out tooth without removing any attached tissue. If possible, hold the tooth in place in the tooth socket. Otherwise, put the tooth in a glass of milk at room temperature and get to us right away. The first 20 to 40 minutes are the most critical in such a situation. We may have a chance to re-implant the same tooth if brought to us at the earliest. Otherwise, replacement of the tooth by an implant or a bridge would be the only remaining option.\nCarefully try to remove the object with dental floss. (Don't try using a sharp instrument instead!) If you are unable to dislodge the object with dental floss, contact us immediately. There might be some gap between the teeth or a cavity in the tooth. If it's not painful, then it may be treated with a simple filling.\nFor toothache relief rinse your mouth with warm water. Then use dental floss to make sure there isn't any food or other debris causing the pain. If the pain persists, call us as soon as possible. Until then for the tooth pain relief you may take an over-the-counter painkiller till the time you visit us.\nFor a broken or lost filling, as a temporary measure, stick a piece of sugarless gum into the cavity (sugar-filled gum will cause pain) and visit us as soon as possible. If a crown or veneer has fallen off, take an appointment with us immediately. If you can't visit us right away, then you may try to slip the crown back over the tooth. Before doing so, you may try coating the inner surface of the crown with a toothpaste or denture adhesive. Never ever use superglue. If the tooth is causing pain, use a cotton swab to apply a little clove oil that can be purchased at any local drug store on to the sensitive area. You can't do much on your own when any such restoration falls off, apart from seeing us at the earliest.\nIf a wire breaks or sticks out of a bracket or band and is poking your cheek, tongue, or gum, try using the eraser end of a pencil to push the wire into a more comfortable position. If you can't reposition the wire, cover the end with orthodontic wax, or a small cotton ball, or a piece of sugar free chewing gum until you can get to our office.\nIf you notice any pimple like swelling on your gums, rinse your mouth with salt water and immediately contact our office or Dr. Priyank directly. Dental abscesses should be taken seriously because they can lead to more severe infections if not treated promptly.\nIf any ulcer is bothering you while eating or is generally painful, you may apply any over-the-counter local anaesthetic gel over it before your meals to relieve you of the discomfort. You may also take a painkiller. Ulcers generally heal on their own, but if any ulcer is too painful or has stayed for more than a week, then contact Smilekraft Dentistry immediately to get it checked and treated.\nPain is your body's signal that something is wrong. It also means it's time to seek treatment for something you might have been putting off for a while.\nWhen you have sensitive teeth, activities such as brushing, flossing, eating and drinking can cause sharp, temporary pain in your teeth. Sensitive teeth are typically the result of worn tooth enamel or exposed tooth roots.\nRoot canals are most commonly a painless procedure (more than 90% to be precise at Smilekraft Dentistry) that will preserve your tooth and keep the structure from becoming damaged due to infection or trauma. In a nutshell a root canal is actually your tooth saviour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New allies joined the fight for supervised drug use sites in London this week, as religious groups, including the head of the Anglican Church in Southwestern Ontario, pledged support.\nBut public debate over the proposed locations continues. Many residents and neighbouring businesses continue to speak out against the sites. Meanwhile, an online petition in support of the facilities has registered almost 500 signatures.\nThe need for support, including harm reduction through injection sites, is a proven factor in assisting such healing.\nAnd the Diocese of Huron, which represents Anglican churches across a large swath of the province, has added its voice to the chorus advocating for supervised facilities.\n\"We would, of course, prefer to end the prevalence of drug addictions in our city,\" Bishop Linda Nicholls, head of the Anglican church in Southwestern Ontario, wrote in a letter to London city council.\nThe Diocese of Huron covers a wide region from Windsor to Owen Sound to Port Rowan.\nThe chair of the Middlesex Presbytery, which represents United Churches in the area, told The Free Press she's eager to see sites that would help improve safety and services for drug users.\n\"Whatever their struggles may be, we really should be looking after and looking out for vulnerable people in our community,\" Wendy Noble said, noting the United Church doesn't have an official position on the supervised injection sites.\nResidents at 241 Simcoe St., a public housing complex that's been pitched by the health unit as a potential location, continue to rail against the idea. Dozens of people who live at or near the site came out to last week's planning committee and public meeting to demand more consultation, and a small rally was held in front of city hall mid-afternoon on Tuesday.\nOther Londoners have expressed support for the sites, including leaders of agencies such as Big Brothers Big Sisters and United Way. They pointed to wider community benefits, like a decrease in discarded needles.\nThat's a key consideration for Noble. One of her churches is two blocks from 446 York St., the second proposed site.\n\"I would suggest that going to a safe injection site is a far better solution than the parking lot behind Dundas Street Centre (United Church),\" she said.\nCoun. Maureen Cassidy said she's not at all surprised to see religious institutions jumping in to support the cause.\nLondon still needs federal approval to open supervised sites. Future public meetings will also be required as part of the re-zoning process for the proposed sites. City council has to sign off on any change to the zoning bylaw to allow supervised drug use sites.\nThe health unit's application to open those sites also includes a bid to create a mobile service that would stop downtown and in the Old East Village to provide the same supports to drug users.\nThe province signed off last week on the two sites pitched by the health unit, and city council gave symbolic political support for the locations earlier this month.\nLondon is still waiting on the green light from the federal government, which must issue an exemption to allow illegal drugs at the supervised sites.\nZoning amendments to allow supervised drug use sites at 241 Simcoe St. and 446 York St. must be reviewed and approved by council. The process for a zoning change usually takes about four months.\nCommunity meetings must be held for everyone within 250 metres of the site before an application to change the zoning bylaw can be submitted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A high-quality Barlow lens with nearly no false colour. Doubles the focal length and therefore the magnification. 31.8mm connection and 26mm free aperture. Apochromatic design with 3 elements and fully multi-coated optics. Solid design, elegant finish with precise and gentle lock by brass compression ring. Filter thread for standard 31.8mm filters.\nTS-Optics APO TSShorty 2x Barlow lens 31,8mm combines good image quality with compact design. So this lens cannot only be used with Newtonians, but also with refractors and diagonal mirrors without touching the mirror surface. The three element design improves field correction and prevents from introducing chromatic aberration, retaining the image quality of your telescope. A brass compression ring locks accessories gently and with precision. The large clear aperture allows the use of eyepieces with longe focal length without vignetting. This Barlow lens fulfills highest expectations!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have had many success stories with our labels and label printing over the years. Feel free to look at our most recent label printing projects delivered to our satisfied customers, each with their own, unique requirements and budgets.\nAs one of the leading label printing companies in South Africa, we pride ourselves in creating eye-catching labels that will distinguish your product from your competition. Founded in 1997, we have created labels to suit any packaging need. We create any label from self-adhesive, digital, barcode to foil labels. Whether you are looking for a label to make your product shine or a security label to minimize theft, we are the label solution you are looking for. Whatever your industry, budget or requirements, we can help you with all your label needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Alternative Wall Decor Blanco American Wall Art by Uttermost at Howell Furniture in the Beaumont, Port Arthur, Lake Charles, Texas, Louisiana area. Product availability may vary. Contact us for the most current availability on this product.\nBrowse other items in the Alternative Wall Decor collection from Howell Furniture in the Beaumont, Port Arthur, Lake Charles, Texas, Louisiana area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvcef b/data_all_eng_slimpj/shuffled/split2/finalzzvcef
new file mode 100644
index 0000000000000000000000000000000000000000..58b6316935bd4b4ba07e419378036eabeebcd9d2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvcef
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+{"text":"NorFor has not, until now, given any recommendations concerning amino acids. There has been a demand for such recommendations and during 2013 NorFor has made a development work with data from the literature in order to set up recommendations for methionine, lysine and histidine.\nA dataset with data from 17 production trials where the effect of adding rumen protected amino acids was investigated and data from infusion trials, in total 27 trials with 128 treatments. The cows were 70 to 180 days in milk. Data about the cows and feedstuffs were put into NorFor in order to investigate relationships between amino acid supply and milk protein response.\nWhen only data from the production trials were used there was no relationship between the levels of methionine (% of total AAT) in control feed rations and the response in milk protein yield (g\/day). Note that the variation in methionine among the feed rations were small, between 2.1 to 2.35 % of total AAT. The conclusion from this analysis was that cows in mid- and late lactation with a content of methionine of 2.1 \u2013 2.3 % of total AAT don\u00b4t need any supplementation with rumen protected methionine to increase protein yield or milk yield.\nThe complete dataset were put into NorFor to see if milk protein yield was dependant of the methionine content of the ration. A logistic regression model was used and according to figure 1 the maximum milk protein yield will be achieved with a methionine content of 2,2 % of AAT for cows in mid or late lactation.\nThe dataset contained only a minor set of treatments with rumen protected lysine and most often in combination with rumen protected methionine. There was no response in milk protein yield by increasing the level of lysine in the feed ration, according to the studied data.\nThe cows in these trials were all in mid and late lactation. Data from literature indicate that milk production in cows in very early lactation can benefit from supplementing with rumen protected amino acids. However, a very limited number of studies are available.\nBased on these data analyses, a recommended minimum level of 2.2 % methionine of total AAT is now applied in the NorFor model. The recommendations for histidine and lysine are set relative to the content of methionine in milk. The content of histidine in milk protein is the same as the content of methionine, therefor NorFor recommends a minimum histidine level of 2.2 % total AAT. The lysine content in milk protein is 2.9 times higher than methionine. NorFor recommends a minimum lysine level of 6.4 % of total AAT.\nThis recommendation for amino acids was included in the latest update of the NorFor model, FRC 1.76.\nFigure 1. Relationship between milk protein yield (g\/d) and methionine supply (% of AAT) of the cow calculated in NorFor. The response function is modelled with logistic regression based on 128 treatments from 27 experiments, where methionine was added by infusion or as a rumen protected amino acids or in combination with other essential amino acids.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Real Lawn Fertilizer, we're ready to meet all your expectations for Lawn Fertilizer in Speer, IL. Our crew of highly skilled experts will supply the expert services that you require with the most innovative technologies in the industry. We make sure you get the very best solutions, the right selling price, and the finest quality products. We intend to help you to put together choices for the undertaking, address all your questions, and organize a meeting with our professionals when you call us by dialing 888-310-6631.\nYou have a price range to comply with, and you should cut costs. You'll still need professional quality services with Lawn Fertilizer in Speer, IL, and you're able to have confidence in our company to save you money while continually providing the highest quality services. Our efforts to cost less money are not going to sacrifice the excellence of our services. Our objective is to guarantee that you enjoy the best materials and a result which lasts throughout the years. As an example, we are alert to keep clear of costly errors, do the job promptly to help save hours, and ensure you get the most effective bargains on materials and work. Save your time and funds through getting in touch with Real Lawn Fertilizer right now. You will contact our team at 888-310-6631 to learn more.\nLawn Fertilizer are available in Speer, IL.\nTo make the most suitable decisions regarding Lawn Fertilizer in Speer, IL, you've got to be well informed. We will make sure that you understand what should be expected. This is exactly why we try to make every attempt to make sure that you are aware of the process and are not confronted by any sort of unexpected surprises. You can begin by going over your task with our customer support representatives when you dial 888-310-6631. During this call, you will get your concerns answered, and we'll establish a time to initiate work. We are going to work closely with you throughout the whole project, and our staff is going to arrive promptly and prepared.\nLots of reasons exist to choose Real Lawn Fertilizer regarding Lawn Fertilizer in Speer, IL. We have the highest customer satisfaction ratings, the best materials, and the most helpful and reliable money saving practices. Our company has the experience you will want to fulfill your objectives. Call 888-310-6631 whenever you need Lawn Fertilizer in Speer, and we are going to work with you to expertly accomplish your project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AriPro hydrating barrier mousse has been specially formulated for the management and relief of dry skin and irritated skin conditioners, including atopic dermatitis. Clinically proven, the unique formulation mimics the skins own composition, allowing for rapid absorption into the uppers layes of the epidermis. With revolutionary Proderm technology AriPro mousse gently and hygienically mositurises dry skin whilst providing a protective barrier, allowing the skin to breathe.\nPurifed Water, Propylene Glycol, Butane, Palmitic Acid, Stearic Acid, Isobutane, Propane, PVP, Glycerin, Polysorbate 20, Triethanolamine, Dimethicone.\nFor best results apply 2-3 times daily, AriPro mousse is suitable for adults, including the elderly and children from the age of 6 months.\nPressurised container. Keep out of the reach and sight of children. Avoid contact with eyes. Do not use in confined space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Accelerate your career within a growing private practice with a mixed caseload, great bonuses and opportunities for growth within the business.\nWe are currently seeking a passionate and motivated Physiotherapist to join a private practice in the Northern Suburbs of Sydney. You must be dedicated to providing a high standard of service and have a desire to continually improve yourself. You will be rewarded with a great base salary, ability to earn additional with bonuses, and ongoing investment into your education and personal development. We are looking for someone who wants to take the next step in their career to grow as our business grows.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We offer Sunday School from 9:45 \u2013 10:45AM for all Age groups every Sunday morning. Each of our Children's classes have core age appropriate teaching material, our high school class materials vary based on current topics and subjects, and our Adult classes topics vary and rotate on a quarterly schedule.\nEach Sunday morning, children get to participate in their own age appropriate classes. Bible lessons are taught in such a way that every child gets to participate and learn. Each of the class teachers is carefully screened, they love working with children and have a desire to help them in their walk with Christ.\nLife as an Christian adult can be difficult and sometimes we need encouragement from other believers to help us through the ups and downs we encounter. Our adult Sunday school classes help with various life situations and our class topics change every quarter. We offer a variety of class settings and topics including wen's only classes, women's only classes, classes for couples, parenting, and classes for the seasoned adult. Our classes including topics such as Marriage, Money, Getting Ready for Kids, Shepherding Children, struggles with living in an ungodly world, and digging deeper into Gods Word, just to name a few. Sunday school plays such a vital role in your walk with Christ.\nWe are always looking for full time and substitute teachers to teach. If you have a servants heart and a desire to teach, please talk with our Sunday School superintendent about teaching positions available. Please talk to our Sunday School Superintendent if you are interested in teaching a class.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"From left: Sheila Brown, Sophia Horwitz and Greg Woolner.\nSophia Horwitz is excited to ask the Dal community some big questions.\n\"What is learning that makes it so alive for you, and how do we integrate that into the larger idea of Dalhousie?\" she says.\nHorwitz is one of the facilitators who will guide the Dal community in conversation at the DALVision 2020 Senate Forum on Undergradate Education, which takes place on Tuesday, November 13.\nDALVision 2020 will be held in the McInnes Room of the Student Union Building during the day, with an evening session in Jenkins Hall at Truro's Agricultural Campus. It will also be webcast so faculty, staff and students who can't attend in person can be connected to the conversation.\nThe Senate Forum's goal is to start a discussion about academic innovation at Dalhousie, with the goal of preparing Dal undergraduate education for the students of 2020 and beyond.\nHorwitz, a Dal alum herself (IDS and community design), is the founder and director of the Co*Lab, a social innovation and creative engagement collective. She and her colleague Greg Woolner \u2014 another Dal grad, in sociology and music \u2014 will be guiding faculty, staff and students in World Caf\u00e9 and Open Space conversations throughout the day.\nThe other facilitator at DALVision 2020 is Sheila Brown, former president of Mount Saint Vincent University, who assists post-secondary institutions in discussions about educational issues, governance and best practices.\n\"We want a broad conversation, an inclusive conversation, where different perspectives are brought to the table,\" says Dr. Brown, who says that part of her role at the forum will be to contextualize the discussion within a large framework, one that takes into account similar conversations across the post-secondary sector.\nThe forum's morning keynote speaker will be Nick Mount, nationally acclaimed scholar and Dal alum, with a lecture titled \"The Talk About Teachers Today: Why scholarship thinks we're ignorant, Margaret Wente hates us, and the world still needs us.\" The afternoon keynote will be by Shelagh Crooks, who has worked extensively in education policy in her role with Saint Mary's University, speaking on \"Teaching as Inquiry.\" DSU President Jamie Arron will also speak on the student experience, and there will be a panel discussion towards the end of the day.\nBut it's between the keynotes that some of the most exciting talk will happen, as Hortwiz and Woolner lead attendees through World Caf\u00e9 and Open Space conversations about the issues raised, in small groups of no more than four or five people.\nIf you haven't had the chance to sign up for the DALVision 2020 Senate Forum, do so soon.\nFacebook: Join the FB event and spread the word!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"theoneyouwerewarnedabout is using Similar Worlds.\nSo the green goblin masquerades as a fishboy from Atlantis?\nWill humans evolve and grow an extra eyes in the top of their heads.. so the can walk and play on their phone at the same time?..\nYou can't ankle tap in rugby league anymore..\nWhy fish on Good Friday?\nThe initiation is a slice of toast buried in vegemite..\nAre most things on TV aimed at 15 year old girls?\nWhat is this trickery ?\nNew lovers are kinda inconsiderate huh?\nwhen Daenery was riding into town wearing fur.. ?.\nmy birthdays coming up.. you should mark that down in your calendars ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Sandoval County Human Resources Department's Mission is to foster a positive working environment that engenders professional growth and skills development among employees by providing information, training and assistance in all Human Resources and Safety related areas. In addition, our team works to mitigate loss and liability and to create a safe and healthy working environment by enhancing awareness and compliance through training, consultation and health, safety and wellness activities.\nThe Human Resources Department is available to serve residents of Sandoval County and employees with the County. We process employment applications, handle disciplinary actions and investigations, and provide risk management services. In addition, our department provides guidance and consultation to elected officials, directors, managers and supervisors in the areas of HR compliance, employee & labor relations and employment regulations & statutes.\nIn addition, the Human Resources Department also houses the Risk Management Division. Accidents do happen. If you suffer a loss due to County negligence or are involved in an accident with a County vehicle, report the incident to law enforcement officers and to the Risk Management Division at (505) 404-5866.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Voorkant Lampen, Parking Lights | Prins Truckshop B.V.\nReproduction original parklamp assembly for Chevrolet. Includes housing, lens, gasket, screws and wiring. Inquire for other years.\n(1969-70) PARKLAMP HOUSING ASSEMBLY-CLEAR-RIGHTReproduction original parklamp assembly for Chevrolet. Includes housing, lens, gasket, screws and wiring. Inquire for other years.\nAn excellent reproduction Chrome parking light bezel for the 1947-53 Chevrolet Pick-up truck front grill, fits either left or right side.\nAn excellent reproduction chrome parking light bezel for your 1958-59 Chevrolet trucks front grill, fits either left or right side.\nClear lense with 39 amber LED bulbs. Sold each.\nParklamp Lens for 1947-53 Chevy TruckThese are glass as original.\nAn excellent reproduction aluminum parking light bezel for your 1962-66 Chevrolet trucks front grill, fits either left or right side.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trimipramine is an oral tricyclic antidepressant. It is sold in the United States under the brand name Surmontil.\nTrimipramine is used primarily to treat depression and to treat the combination of symptoms of anxiety and depression. Like most antidepressants of this chemical and pharmacological class, trimipramine has also been used in limited numbers of patients to treat panic disorder , obsessive-compulsive disorder , attention-deficit\/hyperactivity disorder , enuresis (bed-wetting), eating disorders such as bulimia nervosa , cocaine dependency, and the depressive phase of bipolar (manic-depressive) disorder.\nTricyclic antidepressants act to change the balance of naturally occurring chemicals in the brain that regulate the transmission of nerve impulses between cells. Trimipramine acts primarily to increase the concentration of norepinephrine and serotonin (both chemicals that stimulate nerve cells) and, to a lesser extent, to block the action of another brain chemical, acetylcholine. Trimipramine shares most of the properties of other tricyclic antidepressants, such as amitriptyline , amoxapine , clomipramine , desipramine , imipramine , nortriptyline , and protriptyline . Studies comparing trimipramine with these other drugs have shown that trimipramine is no more or less effective than other antidepressants of its type. Its choice for treatment is as much a function of physician preference as any other factor.\nThe therapeutic effects of trimipramine, like other antidepressants, appear slowly. Maximum benefit is often not evident for at least two weeks after starting the drug. People taking trimipramine should be aware of this and continue taking the drug as directed even if they do not see immediate improvement.\nAs with any antidepressant, trimipramine must be carefully adjusted by a physician to produce the desired therapeutic effect. Trimipramine is available as 25-mg, 50-mg, and 100-mg oral capsules. Therapy is usually started at 75 to 100 mg per day and gradually increased up to 200 mg daily as needed. Hospitalized patients with more severe depression may require 300 mg per day. Amounts up to 200 mg may be given as a single dose. In people over age 60 and in adolescents, the therapeutic dose should start at 50 mg per day and is rarely increased beyond 100 mg per day.\nLike all tricyclic antidepressants, trimipramine should be used cautiously and with close physician supervision in people, especially the elderly, who have benign prostatic hypertrophy, urinary retention, and glaucoma, especially angle-closure glaucoma (the most severe form). Before starting treatment, people with these conditions should discuss the relative risks and benefits of treatment with their doctors to help determine if trimipramine is the right antidepressant for them.\nA common problem with tricyclic antidepressants is sedation (drowsiness, lack of physical and mental alertness). This side effect is especially noticeable early in therapy. In most patients, sedation decreases or disappears entirely with time, but until then patients taking trimipramine should not perform hazardous activities requiring mental alertness or coordination. The sedative effect is increased when trimipramine is taken with other central nervous system depressants, such as alcoholic beverages, sleeping medications, other sedatives, or antihistamines. It may be dangerous to take trimipramine in combination with these substances. Trimipramine may increase the possibility of having seizures . Patients should tell their physician if they have a history of seizures, including seizures brought on by the abuse of drugs or alcohol. These people should use trimipramine only with caution and be closely monitored by their physician.\nTrimipramine may increase heart rate and stress on the heart. It may be dangerous for people with cardiovascular disease, especially those who have recently had a heart attack, to take this drug or other antidepressants in the same pharmacological class. In rare cases where patients with cardiovascular disease must receive trimipramine, they should be monitored closely for cardiac rhythm disturbances and signs of cardiac stress or damage.\nTrimipramine shares side effects common to all tricyclic antidepressants. The most frequent of these are dry mouth, constipation, urinary retention, increased heart rate, sedation, irritability, dizziness, and decreased coordination. As with most side effects associated with tricyclic antidepressants, the intensity is highest at the beginning of therapy and tends to decrease with continued use.\nMen with prostate enlargement who take trimipramine may be especially likely to have problems with urinary retention. Symptoms include having difficulty starting a urine flow and more difficulty than usual passing urine. In most cases, urinary retention is managed with dose reduction or by switching to another type of antidepressant. In extreme cases, patients may require treatment with bethanechol, a drug that reverses this particular side effect. People who think they may be experiencing any side effects from this or any other medication should tell their physicians.\nDangerously high blood pressure has resulted from the combination of tricyclic antidepressants, such as trimipramine, and members of another class of antidepressants known as monoamine oxidase (MAO) inhibitors. Because of this, trimipramine should never be taken in combination with MAO inhibitors. Patient taking any MAO inhibitors, for example Nardil ( phenelzine sulfate) or Parmate ( tranylcypromine sulfate), should stop the MAO inhibitor then wait at least 14 days before starting trimipramine or any other tricyclic antidepressant. The same holds true when discontinuing trimipramine and starting an MAO inhibitor.\nTrimipramine may decrease the blood pressure\u2013lowering effects of clonidine . Patients who take both drugs should be monitored for loss of blood-pressure control and the dose of clonidine increased as needed.\nThe sedative effects of trimipramine are increased by other central nervous system depressants such as alcohol, sedatives, sleeping medications, or medications used for other mental disorders such as schizophrenia . The anticholinergic effects of trimipramine are additive with other anticholinergic drugs such as benztropine , biperiden , trihexyphenidyl , and antihistamines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Welcome to the Unofficial Thoroughbred Hall of Fame, which is dedicated to honoring the achievements of horse racing's finest athletes. This webpage will help you find information on over 500 noteworthy thoroughbred racehorses. Biographies are included for an increasing number of horses; this text is from my yet unpublished, but still copyrighted, work. If you are a new racing fan, you might want to read my Introduction to Thoroughbred Racing article.\nVideo property of Churchill Downs Inc.\nHelp me quit my job and finish this site!\nPlease feel free to e-mail any questions or comments. Also, contact me immediately if you have a problem with a company who advertises on my site, so that I may take appropriate action. Thanks!\nAnd for completely unrelated fun . . .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Top 30 ashley Furniture Burbank \u2013 If you want to know who published the photos, %krsauthor% is the only one who has a kind heart that published the images under the Ashley Furniture Burbank. Plenty other inspiration is also exist in the Ashley Furniture Burbank group. Keep in mind that our Ashley Furniture Burbank provides these images for free. You'll find them all in the category placed by author. Our goal to publish the images in the blog is to give you ideas and to boost your site performance.\nIn that case, we hope that each of our own posts can be your top guide and ideas for you.\nOur Ashley Furniture Burbank includes Ashley Furniture Burbank photograph series which might have got your Ashley Furniture Burbank search. So, we are truly happy to hear that you are amazed and motivated with this wallpaper, so you choose to make it as your top guide.\nIf you want to save it to your computer, simply use the 'right click' and then choose 'save images'. By this way, the Ashley Furniture Burbank will be your Ashley Furniture Burbank collection to improve your project. The final words for you, if you would like to explore our post to find your ideas, just scroll down below the post. Just click our Ashley Furniture Burbank to obtain additional update on the new ideas that you might like. Kindly regards, %krsauthor%.\nWe hope you can find what you need here. We always effort to show a picture with HD resolution or at least with perfect images. Ashley Furniture Burbank Street Fire In Waterville can be beneficial inspiration for those who seek an image according specific categories; you can find it in this site. Finally all pictures we have been displayed in this site will inspire you all..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Job vacancy as Hotel Management Trainee \/ Operation Assistant in Petaling Jaya Selangor.\nNearby to neighborhood, food and hostel can be get easily around this area. Salary is RM1600 and above, by the experience and quality of the candidate.\nWe are operating an affordable homestay-like accommodation in Petaling Jaya commercial area. We are looking for individual that is able to manage the property to welcome our guests.\nPlease contact \/ whatsapp 0163889228 if you are interested.\nhi.. wanita hamil boleh apply tak ya?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I wanted to take the time to give out some useful info for taking care of yourself in between massage sessions. Your body can only be as good as the things you put into it.\nWhen it comes to self care, there are a number of foundation principles of health that just cannot be ignored. There is no supplement, fancy gadget or magic pill that is better than the basics.\nIn today's blog, I will give a quick talk about the first three 'yin' principles. These are probably the most important, but I will give an outline of the other three 'yang' principles in the next installment.\nNutrition: with all the dietary confusion out there, it's not uncommon to hear conflicting information. The most important thing to note is that we are all unique in genetics and have unique capacities to handle each food that we ingest. To keep it very simple, keep it to wholefoods and organic if possible. Quality is number one and variety is quite important, eat with the seasons. If you couldn't farm it, catch or hunt it, then don't eat it. Have you ever seen doughnuts growing in the wild?\nHydration: Water is the only thing that we were really meant to consume. Anything else including tea, coffee, soft drinks, alcohol is not really necessary and will ultimately have an effect on your hormonal system and GI tract. Quality of water is important, consider getting a water filter to the best quality that you can afford. Aim to have 0.033 x body weight in kg. For example, a 100kg person would aim for 3.3L of water a day. This seems to be a good starting point, and the more you sweat during exercise, the more you would need. Many people often confuse hunger with thirst.\nSleep: Many people complain of poor quality sleep, and this can be quite a problem. Sleep is where we go into a deep repair and rest cycle, and where the immune system goes to work to help keep us functioning optimally. So we need to ensure approx 8hrs or so each night, Going to bed and waking at the same time each day is optimal. Aim to keep your room cool, dark, and free of EMF (phones, WIFI).\nHopefully you can incorporate some of these tips into your life. Remember aim to incorporate just one habit each week. For example, your focus for the first week may just be to drink your recommended amount of water each day. Then the next week, you can focus on a different habit. You might be suprised at the difficulty of some changes, but you may also be suprised how much your mental clarity, focus, movement, recovery and happiness improve. Your body will thank you when you start taking greater care of it more often.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As I am sure you know if you live in the city of Chicago, it's a foggy night out already. I just love the way the fog engulfs the city\u2026leaves the city with an erie feeling. This photo was taken of the skyline with the fog rolling over the skyline. A wonderful view from the Adler Planetarium.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I have found previous editions of this book to be the perfect companion for my first year module. The new edition is both interesting and informative, with a clear structure and useful case studies. I would recommend it for any first year module that focuses on the business environment within an international context.\nThis textbook is unique, in that it not only provides a comprehensive overview of the concepts students require to understand the international business environment, and to support further study, but also includes discussion on critical and ethical issues and case studies. This combination better prepares our students for the challenges of global business, and as a teacher, allows for a diverse range of topics and classroom activities to be included in the course.\nThe updated edition of Morrison's Global Business Environment is very useful and introduces timely and key themes of global business for Year 2 students. The opening cases are thought provoking and the review questions are well selected and could also be used to structure an interactive lecture with students. I highly recommend this text for use on any UG course of international business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \/ Life and other stories \/ Why i left my husband.\nWhy i left my husband.\nI am alone in Koh Phangan, a little island in Thailand. I have a little house close to the ocean and to a yoga-studio where I go everyday, it is quiet and nice and just what I needed. In the beginning I did not like it at all, I don\u00b4t know why, some how I always hate the place I go to the first day or so, maybe because it is not what I imagined, but then somehow I always manage to find the beauty in places.\nI have now been traveling with my husband, since we decided to sell everything we own, and just go for our dream. That is more than a year ago now.\nHas it been easy? Yes it was easy to sell everything we own, yes it was easy to say goodbye to our old life. But being together 24\/7 almost everyday, often living in small spaces around the world (tents, cars, small bungalows etc.), is not that easy. You become each others everything when you travel, that sounds really romantic, but believe me it is not.\nSomehow you melt together and become one ( again not romantic), you have to compromise so much that you kind of forget, who YOU are, and what you wanted out of this travel adventure. It all gets very blurry, and you cannot see clearly any more where you stop and he begins.\nYes, so we decided to take a month break from each other. Yep! My husband wanted to go and do Muay Thai training in Chaing Mai (place in Thailand in the jungle) to get really fit and sweaty with a bunch of other manly men, and I wanted the peace, beach and yoga, so of we went.\nThis is probably the healthiest thing we have ever done for our relationship, taking time apart, not breaking up, just giving each other space to breathe, think and find our way back to who we are as individuals.\nOf course I miss him, but I don\u00b4t let that over shine my wonderful experience, of being alone and finding back to my center.\nIn short she wanted have her own private space, you know like the men have there garage. But that was not the only thing she wanted, she also wanted 5 days a month, free from her family life, where she could recharge, find back to her center, and go out be with her girlfriends on her own time frame, selfish!? not at all, I think this a very healthy lifestyle she decided for her self.\nFunny enough I had a similar conversation, with a customer of mine a few years back, she told me if she could change one thing in her life, then she would have bought a small vacation home at the time her children where small, this would then have been her little escape place every now and then for 1 or 2 days, just to be my her self with a book, a cup tea, just breathe for moment and fill her cup up with self love again. She ended up saying, I think that could have saved my marriage. Selfish not at frocking all.\nWe women we do it all and we never give us time and space to just take care of our self, we just simply don\u00b4t allow our self to be selfish and recharge our energy.\nLike Elizbeth Gilbert so nicely said it in a interview, when we go into a relationship, instantly we become a servant and forget who we are and what we wanted out of life. Then years later you get your small midlife crises, where you get depressed or sick with something, because you for all these years have been hiding who you really are and what you wanted out of life.\nThen you end up at 'The Yoga Barn' in Ubud, in you mid 30s or 40s, depressed and desperate to find the meaning of life. Oh boy I've seen it, I listened to them at the women's circle, and it is not pretty. What do you think Eat Pray Love is all about??\nI am telling you this, so that you know If you ever had that thought of you own space, it is OKAY, you are not alone, many of us need a place to recharge.\nDemand your space, and start your journey today, find out who you really are and what dreams you have been hiding all these years. Start living and stop being a servant, become clear about who you are. NO it is not selfish.\nBig kiss from Thailand, I am on my journey, so join me \ud83d\ude42 Let\u00b4s live outside the box.\nI would love to here from you, let my know in the comments down below, what your first step is today, towards demanding your space and becoming clear about, if you are really living the life you truly want?\nUmiddelbart synes jeg, det er trist at h\u00f8re, at I har v\u00e6ret n\u00f8dt til at g\u00e5 hver til sit \u2013 men n\u00e5r det er sagt, s\u00e5 fungerer det jo ikke, hvis I ikke giver hinanden tilstr\u00e6kkelig n\u00e6ring til at \"spiralen g\u00e5r op-ad\". I er unge endnu og det er fint, at I har erkendt, Jeres veje m\u00e5 skilles \u2013 om ikke andet, s\u00e5 for en tid. Hver for sig har I mulighed for at finde ud af, om der er noget i savner ved hinanden. Efter nogen tid finder I jo nok ud af, om der er en essens af Jeres forhold, som berettiger til en eller anden relation, som bygger p\u00e5 gensidig tillid og respekt samt den personlige frihed, I har brug for. Who knows. Held og lykke med at finde \"den ny vej\" !\nJeg tror at du har misforst\u00e5et en smule hvad blog opslaget handlede om. Vi er ikke g\u00e5et fra hinanden, vi g\u00f8r bare ting hver for sig en m\u00e5ned.\nHele essensen af blog opslaget var s\u00e6tte focus p\u00e5 at det er okay tage tid til sig selv og give sig selv et frirum ogs\u00e5 n\u00e5r man er i et forhold.\nI think this doesn't apply to every woman. I don't think every woman gave up their dreams or lost themselves just because of a man. I think you can find your own space living with the partner you chose to be with. It's all about having balance in your daily life. Also, it's about styles, you might need more hours by yourself than maybe other woman and that's ok. Meditation helps to find your balance and you can do this by living or not living with your partner. I don't think \"separation\" is necessarily the answer to find yourself. We might just need to be more aware of who we are at all times, and learn how to have breaks without going to extremes. But again, if separation works for you that's fine, I just don't think this strategy can be generalize to every woman, it might not work and it might bring resentment to the people involved in the relationship. Every relationship needs to be observed independently and apply different strategies to not loose their individuality.\nOf course every woman and every relationship is different. What might work for me, might not work for you. But what the point is with the blog post, to say it is okay to give yourself freedom and space to find in to your center again. A relationship doesn't have to apply to certain rules, create your life and your relationship the way you want to, as long as you both a happy.\nI think you have to really travel with a partner for long stretches of time (months on end) to really understand what you are talking about. I had this happen when I traveled (backpacker\/flashpacker style) with another travel buddy, often times someone I met in my travels that we decided to travel together. I know EXACTLY what you are referring to where you loose yourself and feel as though you are acting and thinking as one. It happened with girls and guys I traveled with, it doesn't even need to be a romantic relationship thing, just a travel partner thing. My guess at why this happens is because when we are traveling in foreign countries and exploring and navigating foreign cultures we gain this weird inter-dependence with the other person since they become the only thing constant in your life as everything else feels upside down and backwards and you're struggling together to navigate this challenging life of travel. You end up doing EVERYTHING together. It's not the same as a normal couple or roommate\/ friendship relationship that you would have living normal lives in your home country together. You don't live separate lives, separate jobs and separate friends. You don't have other constants or comforts or familiarities to depend or turn to. I know this feeling you have described so well, it happened every time I parted ways with someone I had been traveling with for a while, you feel so open and vulnerable right after you separate like you are suddenly missing your other half, I was just trying to explain this to a friend the other day how suddenly you feel like you forgot how to travel and be alone all the sudden, almost like you forgot how to think for yourself. And unlike Marisol's understanding of your post, it's not a dependency on your man type of thing, it's about two travelers navigating foreign lands together too long, it's inevitable. The feeling always passed for me after the first few days alone so I can imagine why you weren't super happy in your first days in Koh Phangan. But I totally agree that you absolutely needed that space to find yourself again. It's therapeutic and healthy and parting ways every so often (especially in travel) is soooo important! Good post Nikki! I was even a little lost for a day or two after you left Bali and we were just staying at the same hostel, not even roommates lol. It just happens.\nThank you my sweet heart Heather, i really loved your comment ( our almost a whole new blog post \ud83d\ude42 ) Because you had some really good insides about how it is to travel with anybody, Partner or nor partner i liked that a lot, because it is not easy, because you have to communicate so well about every little detail.\nI missed you to actually after i left, still do sometimes.\nYou are not wrong for feeling the way you do!\nIt's Time to Stop Blaming Your Parents or Anybody Else!!\nStay a victim or go live your life.\nDream bigger, believe with all your heart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Twitter is nothing but a blog that limits your post to 140 characters\" and \"Instagram is just uploading photos and putting a filter over it, I can build that in a weekend\"... those are the kind of ignorant things I used to say. I used to think I was funny and was throwing appropriate jabs at companies I thought got too much money and attention for simple products. But I was wrong, yes you can build a micro blogging platform and a photo-sharing app in just a couple hours but they will not be Twitter or Instagram. No one or team can build a true clone of what twitter and instagram is today over a weekend. These products have taken thousands if not millions of hours of design and engineering refinement and they have the scars to prove it. I learned that the hard way when I, and a group of friends found out Houtsuite and Buffer were raking in tons of cash \"scheduling peoples' tweets\" then we tried to create our own app that failed never launched.\nI used to get upset when someone trivialized an app I built, the way I trivialized twitter and instagram but now I take it as a complement. The more I analyze software that has resonated with people the more I realize it is because of how simple they seem and also how hard it is to stay simple. If someone can look at an piece of software that took you weeks and months to build and they say, \"oh, that's just X mixed with Y, very simple\" I think you have executed correctly. I find that really good software rarely lose their way. It is easy for you to create a photo-sharing app that also becomes a music-sharing app and then also becomes a location-sharing app. It takes real discipline to know what the vision of the software is and every feature added does not expand the scope of the problems you are solving but make your app even better at solving that very first problem you set out to solve when you were creating your MVP (Minimum Viable Product).\nSo now when I catch myself about to criticize a popular app for being ridiculously simple, I stop and give respect instead. Because when you look closely the people behind it have a vision and they have executed it in possibly its simplest form, which is ridiculously hard to do. To me they are the REAL MVPs (Most Valuable Players).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are many things that are wrong in the world today, and a lot of reasons (if you choose) to despair.\nOr, we can work on ourselves to be the change that we want to see in the world. To eliminate the cruelty, selfishness, predjudice in ourselves. To be kinder, more understanding, and generous with one another.\nEven people that we don't know. Or don't look like us. Or don't live in our neighborhood\/city\/country.\nNo matter where we live, there is something to be grateful for.\nHaving a home, clothes to wear, food to eat, and water to drink. If you live in the West, there are all kinds of labor-saving devices that make your life easier. All kinds of gadgets to use to work, play, communicate, and other things. Just about all of them are within the financial reach of the average person. Physically, life doesn't get much better than this.\nWith all that we have physically, there is still the emotional and spiritual to work on. Can we become better people, so that our inner reality matches our physical reality? I think that we can. I think that I can; I'm willing to do my best to do so.\nCan we remember to use 'please', 'thank you', 'may I', and be kind, thoughtful and helpful to our immediate family members? Spread the love :-)!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This statement is made pursuant to Section 54 of the UK Modern Slavery Act 2015 and sets out the steps that Really Useful Products Ltd (the company) has taken to ensure that slavery and human trafficking is not taking place in our supply chains or in any part of our business.\nThe company is committed to preventing acts of modern slavery and human trafficking from occurring within its business and supply chain, and imposes the same high standard on its suppliers.\nReally Useful Products is a manufacturer and developer of plastic storage and organisational products. It employs 340 people and has business operations in the UK and the USA. We have assessed our risk to modern slavery and human trafficking as low. This assessment is made based on geography, the nature of the business and business transactions.\nWithin our business we have implemented various policies such as a Child Labour policy and in our staff handbook we set out our expectations in terms of issues such as working hours, child labour, discrimination and health and safety.\nIn addition, as a business our culture emphasises our focus on upholding our core values such as the commitment to our employees and standards for doing business. We adhere to current statutory employment legislation and we ensure that any supplier or third party follows the same fundamental principles. We expect all parts of the business to adhere to this Modern Slavery Statement.\nWe ensure there are provisions relating to modern slavery in the supply contracts we enter into.\nThese procedures are designed to establish, monitor and reduce potential areas of risk in our business and supply chains.\nWe train our staff on how to identify and report risks of modern slavery and human trafficking, however as we have assessed our risk to modern slavery and human trafficking as low the company does not feel we have to have a specific module dedicated to anti-slavery and human trafficking.\nOur CEO, Directors and our Human Resources department take responsibility for ensuring that this statement is up to date and reflects the actions Really Useful Products is taking to tackle slavery and human trafficking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Ithaca College School of Business holds several events each semester, ranging from guest speakers, student organization fundraisers and alumni visits. I joined the marketing team of the School of Business events coordinator during my senior year of college as the first student to hold this position. As the events coordinator, I was responsible for planning all aspects of each event, from organizing logistics to promoting the events to providing hospitality and management during the events. During my employment, I successfully planned several high-profile events with speakers including Ken Fisher (CEO of Fisher House), Hamdi Ulukaya (CEO of Chobani Yogurt), Mark French (Former Senior Vice President of NBC Everywhere; Currently CEO of Court Grip) and Stew Leonard Jr. (CEO of Stew Leonard's Grocery). In addition to planning events, I also assisted our school Marketing Communication Manager with marketing efforts for incoming students and alumni. I loved that my position at the School of Business is constantly evolving and changing, so there was never a dull day in the office!\nDuring the fall of my senior year, I started working as a Campus Representative for Kaplan Test Prep. In my position, I was responsible for promoting Kaplan Test Prep for students who were interested in graduate schools. As the only Ithaca College campus representative, I designed and implemented a marketing plan that included talking at student organization meetings and tabling in our campus center. Additionally, I also spoke at Cornell University events, like their junior premed meeting. During my time with Kaplan, I became the top-preforming college representative in my region.\nThe Campus Center and Events Services office is responsible for planning the logistics of all on campus events. During my time at CCES from my sophomore through junior year of college, I was responsible for planning events according to a client's demands, as well as managing my own events and those planned by my co-workers. To plan events, I created event summaries, which outlined all of the client's set-up, audio-visual, and catering needs. While managing events, I used the event summary to ensure that all of the client's requests are supplied. This position trained me to think on the spot and provide excellent customer service to our clients.\nStudents Consulting for Non-profit Organizations, or SCNO, is a national student organization that creates strategic plans for local nonprofits. Each semester, we enter semester-long engagements with local non-profits. Teams create a plan that addresses the organization's marketing specific needs. While working with SCNO I started out as a general member, but quickly moved up to a Public Relations Senior Consultant my sophomore year, where I consulted other groups on public relations questions. During my junior year in college, I served as a Project Manager, where I created an engagement agreement with a client and then led a team of 6-8 students to accomplish the quantifiable goals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Planning Place has successfully obtained development approval for Multiple Dwellings (4 Townhouses) at 40 Booligal Street, Carina. The approved development involves four two-storey townhouses with public and private spaces, and car parking. The townhouses will contribute to meeting Brisbane's CityShape guidelines by boosting housing choice and affordability in a well-serviced and accessible area. The townhouses will also exhibit a high level of architectural merit to add to the prevailing residential amenity.\nThe principal concern for the proposed development was regarding setback non-compliances. Specifically, boundary relaxations were sought for the front and rear boundaries and side boundary on units B and D. It was argued that the reduced setbacks were primarily a result of the angles of the lot and were considered minor as most the townhouse's walls were compliant.\nFor two-storey Multiple Dwellings less than 9.5m high located within the Low-Medium Density Residential, the application was determined to be Code Assessable. It was also determined via communications with Council that the proposed development application could be fast tracked through Brisbane City Council's RiskSMART service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Even the best of us get wrapped up in cooking that is defined by convenience. And there's nothing wrong with that. We're all getting more done in less time than ever before. But, just because you don't have time for a 43-ingredient, 5 preparation boeuf bourguignon...doesn't mean you don't appreciate the technique, precision and adherence to classical execution. Thumbing through the pages of a great cookbook...taking in the photos, the recipes, the tips on technique... is always a great escape, even if its just from a pile of dishes. So, here's one of our favorites...and we're sure you'll like it too.\nChef Thomas Keller is known for classical French flair expressed in modern Bistro dishes. His acclaimed restaurant, The French Laundry is located north of San Francisco in Napa Valley. Each and every day, he prepares a 9-course tasting menu in which no ingredient is ever repeated. The result is a series of small bites that intrigue, satisfy and inspire. He notes, \"What we want you to experience that sense of surprise when you taste something so new, so exciting, so comforting, so delicious, you think 'wow' and then its gone...We want you to say, 'I wish I had just one more bite of that.' And then the next plate arrives, the same thing happens, but in a different way.\"\nKeller's The French Laundry features 150 recipes from his famous restaurant, and some even refer to it as a \"culinary manifesto.\" His interpretation of modern dishes and ingredients such as Peas and Carrots (lobster paired with pea shoots and creamy ginger-carrot sauce) will delight any foodie. He's a modern-day classic himself. In addition to The French Laundry, Keller has also published Bouchon and Ad Hoc At Home. One of these books weighs 6.5 pounds! Discover which one for yourself.\nPS- Have you seen these books in a locally-owned bookstore? If so, please comment below!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"View, edit, download or restore quarantined files using your web browser.\nKernel based inotify real time file scanning of created\/modified\/moved files. Monitor your entire vhosts directory tree an instantly scan any changed files. All of its resources are inside kernel memory and has a very small cpu usage and userspace footprint in memory.\nSentinel allow you scan a domains web folder with just a few clicks. The scan can automatically quarantine detected threats (if enabled) or allow you to quarantine, clean malware or email a report the the customer.\nHTTP upload scanning through mod_security2 inspectFile hook.\nEnable or disable notifications for administrators, resellers, or clients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Global payments revenues increased by 11% in 2017, which was the highest point in the last 5 years. Unfortunately, security issues concerning the payments industry don't stop being a hot topic all over the world. If you're interested in chargeback insurance providers and want to discover the major threats to the payments space, just read this article.\nGlobal payments revenues accounted for $1.9 trillion in 2017. According to McKinsey & Company, an American worldwide management consulting firm, they're expected to reach $2 trillion by 2020.\nThe FBI reports that internal employees commit 90% of crimes. This is because of the excess access and privileges they have.\nBreach can never be small. Ensure your team takes security as a priority by all means.\nInternet-enabled devices can easily be hacked. Gartner reports that 50 trillion gigs of data will be transferred by IoT devices by 2020.\nOf course, the lack of an encryption technology means can make your card appear in a hacker's hands without difficulty, however, the use of such technology isn't everything.\nBe prepared for the cloud. Before hurrying to put your data into the cloud, know how to secure it there.\nSmarter phishing and \"spear phishing.\" The latter has to do with a language that's too specific to the recipient.\nWork with a reliable and secure payment processor. Today, it's critical to turn to only a true payment expert if you need to process payments securely. With a reputable credit card processor, merchants interested in chargeback insurance providers can enjoy the latest fraud protection and chargeback mitigation services at the lowest possible rates.\nNon-compliant and jail-broken devices aren't difficult to exploit.\nConsider information security and cyber risks too important. Don't fail to make education a priority.\nThe Ponemon Institute reports that 3rd party organizations were \"responsible\" for 42% of all data breaches.\nLack of application security programs. Such program can help you scan internal apps and do code reviews on a regular basis. Remember that security programs must be up to date.\nNot addressing patching demands on operating systems, as well as on applications and network infrastructure.\nDon't ignore the sophistication level of malware. Remember that bots can be installed in your system without your being aware of it.\nBe aware of the dangers of exposed credit card information. Such awareness can help you secure your business from fraud and theft effectively.\nDon't ignore the fact that the basic security protocols for your PC also apply to your business systems.\nIn our increasingly connected world, data is transferred through a variety of devices, such as personal computers and mobile phones, tablets and sensors, IoT, etc. So, no wonder that privacy and security breaches are a common thing these days. To avoid data breaches, which have been a threat to the payments sector for over 13 years, you should well of aware of these threats and take the right measures.\nAuthor Bio: Electronic payments expert Blair Thomas is the co-founder of high risk payment processing company eMerchantBroker that provides exceptional help to merchants interested in chargeback insurance providers. He's just as passionate about his business as he is with traveling and spending time with his dog Cooper.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A photo from Afghanistan, showing a range of vehicles operated by the Royal Marines. In the foreground is a WMIK Land Rover, followed by a Pinzgauer 4x4. In the background can be seen a Viking APC and a bulldozer belonging to the Royal Engineers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Within the pomp and glory of this week's Commonwealth Heads of Government Meeting, is an age-old institution looking to re-set its place in the world? Its future could be a bright one if the UK works with India \u2013 and the man of the moment, Prime Minister Narenda Modi \u2013 to capture the imagination of the next generation around what the Commonwealth can achieve, writes India Inc. Founder & CEO Manoj Ladwa.\nThe Commonwealth needs to reinvent itself if it wants to become a multilateral institution with global currency. This is against the backdrop where multilateral institutions across the world are struggling to stay relevant. President Trump's disdain for the US' contributions to NATO taskforces and the suspicion levelled at the EU by parts of the British political left and right show that these organisations are under fire to show how they can achieve more than solely the sum of their combined efforts. It is clear that multilateral organisations cannot continue as they have done.\nYet the Commonwealth is a huge force to be reckoned with. Commonwealth nations are globalising at an enormous pace, and during 2003-2013, trade in goods and services between Commonwealth countries expanded from $266bn to $592 billion. That figure is expected to pass $1trn by 2020.\nOn a trade and economic level, the Commonwealth can present a truly competitive alternative to the ASEAN and proposed RCEP. On every other level, it must lay the grounds for peace, collaboration and transformation that have allowed the EU and NATO countries to flourish. With a bold commitment from the UK and India, this is possible.\nLondon holds many crowns, but it should not be the only capital city of the Commonwealth. It needs to start sharing this responsibility and if any city should be given this honour, it should be New Delhi.\nOver half of the Commonwealth's 2.3bn citizens are Indians. A revived Commonwealth should reflect Indian interests and India's template for global transformation.\nIndia wants to engage with a rejuvenated Commonwealth, an emblem for the future that presents a major opportunity for its citizens \u2013 60 per cent of whom are under 30 years old.\nFortunately, political leaders in India and post-Brexit UK are in close agreement that the Commonwealth could possess an advantage for both nations as they redefine their position on the world stage in the coming years and decades. India seeks a proactive partner in the UK \u2013 one that can help it become a Permanent Member of the UN Security Council, for example.\nTogether with India, the UK can reimagine and reinvigorate the Commonwealth. In doing so, they will show that they are committed not only to strengthening their transactional partnership, but creating a transformational partnership, too.\nBrexit Britain and Global India both sense an opportunity to knock down barriers to free trade \u2013 widely heralded as the secret to Western Europe's greatest period of peace and prosperity since the Second World War.\nYet Commonwealth trade can only continue to grow if Commonwealth countries are ready to place new ideas at the heart of trade and investment: technological transformation in health, science, industry; high-yield impact investment and sustainable development driven by financial and technological innovations; and a recognition of the threats and challenges faced by the Commonwealth's youth.\nWhere India places great value on economic ties overseas, the Commonwealth presents a great opportunity. India has an ambitious target to increase trade with the African continent to $90bn and will not achieve this without agreements with the Commonwealth's 19 African nations.\nThe Modi Government also has a key target to attract greater inward investment, boost its export order books and to establish a healthy balance of payments by attracting more manufacturers to India.\nThe Make in India campaign has targeted an increase in manufacturing's share of India's GDP from 16 to 25 per cent by 2022 and lionises new technologies and digitally-integrated production lines that can cut costs and lead times for new parts.\nHowever, there is a greater ambition behind this move. India wants to become a leading manufacturer of new green technologies, from photovoltaic technologies to electric vehicles, which are fast becoming central to the way we live in cities and rural areas.\nThis is where PM Modi has shown international leadership and a yearning for action on a multilateral level.\nPM Modi launched the International Solar Alliance, his initiative to harness potential for solar energy in equatorial countries, which has enormous support from France and former French colonies in equatorial regions to create a belt of solar energy-rich nations. The UK's commitment to join, this week, is an acknowledgement of PM Modi's leadership in global renewable technology.\nIndia is not solely interested in trade transactions \u2013 it wants to play a transformative role placing green energy, digital technology, young leaders and balanced growth at the heart of development.\nThe Modi Government, which holds Indians overseas in great esteem, is fully aware of the overwhelmingly young demographic in the nation.\nOver half of Indian citizens are under 25 years old, battling with technological factors changing the nature of skilled work and even driving down the number of jobs in high-productivity industries, in some parts of the world.\nMany of India's youth have little access to higher education or vocational education to secure opportunities in emerging industries and help to build the prosperous India that PM Modi wishes to create.\nAgainst this backdrop, Indian students are becoming more mobile and many Indian students wish to study at the UK's world-leading universities, as well as institutions with which UK universities compete. Emmanuel Macron wants to double the number of Indian students studying in France, helping to unleash opportunities and skills for the next generation of Indians.\nIf the UK can match this commitment, more of the next generation of Indians will have the opportunity to become world leaders in their chosen field, not to mention lifelong ambassadors for the UK.\nThe Commonwealth is rich in heritage, but it desperately needs a new legacy \u2013 working with the India to lead the Commonwealth into a new phase where it champions peace, education and prosperity in some of the most aspirational countries in the world, could surely be the formula to sustain this wonderful institution for the future.\n*The above article was first published in International Business Times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Real estate is one of the highest costs associated with operating a business. Keel CRE provides its clients with comprehensive lease economic cost and cash flow analysis including pretax, post tax, GAAP and P & L.\nFrom our initial competitive proposal comparisons through final LOI and document negotiations, we provide decision makers and corporate financial analysts with sophisticated cash flow analysis and comparisons designed to support your decision-making process. Additionally, we compare and contrast many non-economic factors which can affect the decision-making process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use these instructions to care for a ZZ Plant. This guide will tell you how to water your ZZ Plant; its light, temperature, and humidity preferences; and any additional care your plant might need to help it grow.\nZZ plants do best in moderate to bright indirect light, but will do fine in extremely low light areas. If the light is too intense for your ZZ, you may see some scaling (brown spots) on the leaves. Curling leaves can also indicate too much light. This plant is ideal for a windowless office or bathroom where it will only receive a small amount of light from fluorescent lights.\nThe ZZ Plant is like a cactus in that it needs less, rather than more, water. Water your ZZ Plant only when the soil is dry\u2014perhaps every 1-2 weeks depending on your indoor climate. Let the water drain through the bottom, empty the saucer of any water, and then allow to dry out again. One of the only ways you can kill this low maintenance plant is to over water it. A ZZ Plant that turns yellow means that it is getting too much water and it may be rotting.\nFertilize your ZZ twice a year with a general purpose plant food.\nAs a survival technique, the ZZ Plant has evolved to start dropping its leaflets to conserve moisture during periods of severe drought. If you forget to water yours and see the leaves fall off, don't give up hope! Water your plant immediately and it should come back. This plant is shiny on its own, never spray with leaf shine. Ignore this plant and it will be happiest.\nZZ Plants are mildly toxic to pets and humans. Typically, ingestion will cause mouth and stomach irritation and possible vomiting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ways of the Teacher-Feminist: Text, schema, and stereotypes, oh my!\nI could write hundreds of posts about what it means to be a feminist and why feminism is important. For the purpose of this series, consider yourself a feminist if you believe that the economic, social, and political choices women make should not be unduly influenced by existing gender norms. (Not convinced gender matters? Check out this amazing visualization from the New York Times called Why Is Her Paycheck Smaller? for more information about gender-based wage gaps across industries or check out the research links below). My goal is to write one post about this subject per week.\nText, schema, and stereotypes, oh my!\nThe diversity of characters and social situations in children's literature has increased over time. Many books tackle issues of diversity in the plot. Other stories feature diverse characters in plots that are not about prejudice. I've realized reading these books to students and expecting them to soak up the social norms is not enough. Even in pre-school and Kindergarten, children have ideas about gender, race, class, language, ethnicity, and other things. In this article, I refer to these ideas as schema. Schema are the mental structures people use to organize information they have about the world and create a framework for processing future experiences. A child's gender schema is the mental framework they use understand how gender works in the world around them \u2013 they assign character traits, social position, and other values to people based on this schema.\nI see gender and race schemata in action everyday. When I co-taught a 2nd grade class, I had a calm, intellectual discussion with a diverse group of six students about race, language, and gender. A biography of Jackie Robinson sparked the conversation \u2013 one of the students asked \"Ok\u2026.so\u2026.these people are white and we are white\u2026..Are you black? What about ______? She's from Mexico \u2013 what is she?\". Without answering, I turned and asked the other students \"what do you think?\". From there, the students talked about how they think race changes over time and there are more races than white and black. I told them that I self-identify as a black person. One student asked \"why do we have the military if all they do is kill people?\". The other students gave him their ideas and one student started talking about how her mom is in the military. From there, students discussed how gender works in the military (girls do x, boys do x). The conversation could have continued forever. My role in the conversation was asking students to explain their thinking and asking other students to respond. After this experience, I realized no students are blank slates, and that all information we give them about social norms is compared with the schemata they already have.\nElementary school children were presented two Caldecott Award-winning picture books and asked to retell the stories and answer criterion-specific questions related to genderoconsistent\/inconsistent information contained in each story. Responses were qualitatively examined and revealed patterns of spurious recall. Children tended to misremember or distort gender-inconsistent story information. Story retellings included wrongful stereotyped interpretation of main character(s) behaviors and emotions. Follow-up interview questions revealed rationales for any misremembrance and generated responses consistent with gender bias.\nThe researchers noticed children's scheme has extraordinary power over their ability to recall stories. For example, when first and fourth graders retold Mirette on the Hire Wire, they were more likely to say that the character Bellini (male) did not want to take Mirette (female) on the high wire because of Mirette's fear. In the story, Bellini has the fear and Mirette persuades him to go on the high wire.\nWhat does this mean for teachers? Not only do teachers have to give students the opportunity to consume literature that gives students access to marginalized voices, and promotes positive norms, they have to teach children to be critical of themselves and the texts. This looks different at all levels but starts with us: we should preview the literature our children read both through our adult lens and their child lens. Then, we need to give students authentic opportunities to discuss these issues with their peers. Our goal is to provide them with experiences that challenge students' existing schemata.\nI ask for predictions. If I hear a prediction that seems to be based on gender\/race\/class\/etc schema, I ask the student to explain their thinking and invite other students to agree or disagree (verbally or with hand signals). Even if the conversation doesn't melt away all of the stereotypes, I know what schema my students might use, and can plan appropriate educational experiences.\nMissing narratives: I tell my students history is a story and try to make them aware of marginalized voices using authentic narratives, historical documents, and photographs. Howard Zinn has a history book series called A Young People's History of the United States that tells history through the perspective of women, indigenous peoples, enslaved peoples, and other marginalized voices (his A People's History of the United States is a must read). Have students write stories, journal entries, magazines, and newspaper articles from a traditionally marginalized perspective.\nHow are you a teacher-feminist? Have you noticed the impact of gender, race, and other social schemata in your classroom?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"When asked of what she thought her co-workers might say about her, Yuni offered, \"that I am helpful and demanding.\" We're not sure about the demanding part, but with so much riding on her work, it makes sense. Yuni handles Bronco's sales business in the Far East region, but we like to think of her work more like relationship management. She's friends with her clients on Facebook, and even answers when a problem comes through via text message late at night. She loves working with her Bronco family, and makes sure her customers feel that same family-style support at every touchpoint. And she's no stranger to stepping up and dreaming big. After attending Secretarial Academy in Jakarta, she broke into the oil and gas industry by way of Unocal (now Chevron). When she ventured to the US, she found a new home at Bronco, and we couldn't be happier.\n> When she's not pleasing customers halfway across the world, Yuni flexes her creative muscles with photography (her family is her muse).\n> At home, she wrangles two kids and a black lab.\n> When she decides to drop by 'home,' it's in Indonesia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wednesday August 22, 2018 - Amani National Congress (ANC) Party Leader, Musalia Mudavadi, has cleared the air over the main reason why he skipped the controversial mock swearing-in of Raila Odinga at Uhuru Park January 30th, 2018.\nSpeaking yesterday, Mudavadi said he was against the mock swearing-in of the Orange Democratic Movement (ODM) leader because it was against the Constitution.\nBesides, he said that he knew Raila would use his newly acquired status as the People's President to negotiate deals with President Uhuru Kenyatta which might see him join the Government and leave them in the cold, and as it turns out, he was right.\n\"I kept emphasizing 'tuwek break' (put it on hold),\" said Mudavadi.\n\"That is the reason I did not go to Uhuru Park,\" he added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During my training here in Seefeld, I experience the preparation for the home World Cup first hand.\nIt's really cool to know that this great event will be held here in 2 months.\nFor that I promised myself to give everything.\nThat meant also, instead of taking part at the 4 Hills Tournament, to continue training individually and intensively.\nWe work, above all, on a \"feeling-adaptation.\" This may sound like a minor thing and almost esoteric, but it makes a big difference for us in ski jumping: It's always about the right balance: from the start, while accelerating in the inrun, when taking off, how the skis move towards the body and while landing.\nThe body has to be perfectly balanced, so that I can achieve a top performance.\nWe work very methodically on the right feeling for this balance. On the technique, the set-up and the structure.\nThe set-up we are working on, can be compared to a golf swing.\nIf you don\u00b4t have the right feeling, each of the small movements within a swing becomes difficult, complicated and restrained.\nAnd above all, the ball does not fly as it could and should.\nIt was similar in my situation and my jumps.\nNow I am in the process of finding my right \"swing\" in ski jumping again. Every day. For that I invest a lot.\nThis started with training off the hill, visualizations, then training on small hills, now on larger hills.\nPhysically and mentally I feel really good and therefore I would like to compete again as soon as possible in the World Cup.\nBut before that, it is still necessary to create this basis.\nIt is challenging, but every day a great task of trying to make the most out of myself.\nThanks to the \u00d6SV. Above all, Mario Stecher, Andi Felder and Flo Liegl, who make this professional support possible.\nA very sincere thank you also to my partners, who have always believed in me.\nThis strong support continues to give me wings.\nThe plan for the upcoming days is still open.\nWe want to use the perfect conditions here in Seefeld for the next days.\nThen it will be time to decide on what's right for me.\nIn any case, the previous path makes me optimistic!\nI am looking forward to seeing you soon at the hill!\nIt\u00b4s time to change even more!\nThe last 4 years have been very intense for me. I've managed to learn more about my mental aspects.\nNow it\u00b4s important to intensify the developments in the sportive field.\nThis means, despite a lot of preparation, to change more than the usual.\nThe old system and the old technique that helped me succeed for a long time, needs change. Now I have the chance to continue working on my technique, and find out what works for me.\nWith more intuition and the development that is happening right now, I have full confidence that this is the right direction.\nChange is also challenge and I accept it. Fully!\nThe goal is that I develop a technique, once again, that will give me the chance to realize the full potential that I still feel inside of me.\nTogether with the coaching team, we decided to focus on training and optimizing the technique, rather than on the upcoming World Cups.\nFast, efficient, intensive and effective.\nWe analyzed the situation very intensely.\nFrom now on, I am completely focused on the solution and the associated change.\nI promise myself that I will try everything to achieve my goals.\nI am confident that I will succeed with the road ahead of me.\nTherefore starting today: full focus on my training.\nThis road is challenging, but it\u00b4s not going to stop me from being where I want to be.\nI am more ready than ever to take the necessary steps!\nI wish my skijumping colleagues and the whole team a lot of joy and success in the upcoming World Cups and I will follow it wholehearthedly.\nI wouldn't be here without all your support in the last years and now.\nKnowing you have my back means the world to me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello! I'm Rachel Allish-Zamora Cavanaugh. [And yes, if you include my middle name, this Costa Rican\/Canadian actually has five names.] I'm a 25-year-old wifey and mommy, residing in the good ole' state of Texas. First and foremost, I am a woman of faith. My journey has dramatically changed because of Jesus Christ, He's truly made all things new, and to Him alone I owe this wonderful life of mine. I believe we all have a responsibility to give\u2013 whether it's sharing our time, energy or resources.\nI consider myself a bit of an introvert and writing has always been my preferred pastime. I like to read several completely different books at a time. Kind of like flipping through channels\u2013 every day I'll read a chapter or two of each book until I finish them all. I love our two Pomeranian fur-children. Bear is two, and I've had PomPom since I was twelve. I enjoy singing in the shower, preferably anything Renee Olstead. With a background in modeling and fashion, I love playing with makeup, styling hair, redecorating our little home, attempting to keep up with fashion trends, and enjoy basically all things girly. I'll admit that I'm just now learning to cook. I am a bit of a neat freak\u2013one of those people who color-coordinate their clothes, make lists and revise them several times a day, and wish The Container Store gave out memberships. People who know me well know that quality time and coffee is the way to my heart.\nI believe joy is better than makeup, and a smile alone can light up any room. I believe fashion doesn't have to be provocative, nor must it beg for immodesty. I believe women should encourage and empower each other, instead of competing and comparing. I believe we should choose to find the beauty God's given us in every aspect of our lives\u2013 the good, the bad, and the seemingly ugly. I believe that eloquent furnishings may make a house, but only love makes it a home. I believe a cheerful heart trumps an exterior that looks like messy hair and sweatpants. I believe to survive in our society we must be genuine\u2013 I'll take real and raw over fake and perfect any day.\nI met my sweet Johnny Cavanaugh, in middle-school. Though we didn't date back then (the almost four-year age gap would have begged for awkwardness.) Our story (in short) is that he didn't remember me and hit on me online. Thankfully, I remembered him and after two years of friendship and dating, we married on the third of March, 2013. We spent our first few years as newlyweds traveling and exploring. We like to say we balance each other out as I'm a bit more cautious and timid, while Johnny is a full throttle adrenaline junkie through and through. We both come from huge families and have twelve siblings combined (not their including spouses.) We laugh way too much and often ask each other if everyone's really as weird as we are. He is my favourite person in the whole wide world, I'm absolutely crazy about him, and I am so honored to share this wonderful life with him. Together, we're the hubby + wife duo behind Cavie And Company\u2014 a custom and refinishing furniture company based out of Dallas, TX.\nI hope you enjoy this space, where I'll be posting the random tidbits of my life. Such as what God's doing in our hearts; my foodie finds, the cooking fails; what I'm learning; the fashion my little family is wearing; what I'm reading; and stuff our small company is building.\nThank you for taking the time to read a little more about me and this crazy-wonderful life I've been blessed with.\nHey rachel! I love this! It's awesome, I also really enjoyed your message, it's really true, it deffinitly spoke to me. I will see you Wednesday at the neo. P.S. I'm getting baptised Wednesday! :) love ya!\nyou're amazing! I am so inspired by your go-get-it personality. It helps me reach further!\nI haven't read them all but i will. You are an awesome orator. The way God reveals Himself to you will inspire millions. Get ready for this season. Stay in prayer, humility and live in His Grace. I love you, my daughter and I'm exicted to see how God is manifesting Himself in your life. Keep following hard after His righteousness. It will exalt a nation.\nKeep writing, Rachel. Words have more power than the greatest armies. It was with the word that God created the world and with His Word that he reconciled that world to himself. Whether you change one life or many, know the great responsibility that comes when you put that pen on the page and your fingers find the letters that paint your thoughts on paper. Write not too much or too little, but just enough. Blessings, Will.\nHey thanks for sharing!! Your words truly spoke to me\u2013and impacted me. Loved the blog where you were running and things all of a sudden changed. And you found out about where you were setting your trust on! Good stuff. Felt as if I was reading an excerpt from a well written inspiration book! Keep writing!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have missing or damaged teeth, you might need a dental crown to restore the aesthetic and function to your teeth. Call our dental office in Sugar Land for more information or to schedule an appointment with one of our dedicated dentists.\nThe price of a dental crown will depend on the scope and complexity of the work you need. Your dentist will provide you with a cost estimate during your appointment at our office in Sugar Land.\nA properly placed dental bridge shouldn't be painful. During the procedure, you'll be given anesthetics to ensure that you're comfortable.\nHow Much Will a Dental Bridge Cost Me?\nThe cost of your dental bridge will vary depending on the number of teeth involved, the complexity and location of the teeth, the material used, and your insurance plan coverage.\nWho Makes Dentures in Sugar Land?\nOur office in Sugar Land strives to provide quality, affordable dental work to our patients. We have several financing options available to ensure that you can receive the dental work that you need. If you need a set of dentures completed, contact us today.\nHow Much Will Full Dentures Cost Me?\nFull dentures are custom made appliances that will require at least two visits. We have financing plans available to help our patients and most insurance plans will cover at least part of the cost of dentures. Call our office in Sugar Land for more information or to schedule an appointment.\nA set of full dentures will have an appearance similar to that of your own teeth and gums. You may have a metal framework, depending on the type of denture you select or you may be able to use dental implants to anchor your dentures.\nWhat's Meant by Partial Dentures?\nA partial denture is similar to a bridge and is used to replace several teeth that are missing or severely damaged. A partial denture will appear similar to your own teeth.\nFlipper teeth are used as a temporary replacement for missing teeth. Flipper teeth aren't intended to be used on a long-term basis.\nAre Immediate Dentures Available in Sugar Land?\nAll of the dentists in our office in Sugar Land can provide you with immediate dentures. During your initial visit, your dentist will take measurements and make an impression of your teeth. Your permanent dentures will be ordered and after they arrive, you'll come in for your second visit. During the second visit, your natural teeth will be removed and your dentures will be fitted to your mouth. Most of our patients don't require any additional visits for immediate dentures.\nDental fillings shouldn't hurt after they're installed. During the installation procedure, you'll receive anesthetic as required to ensure that you're comfortable and not feeling pain.\nThe cost of your dental filling will depend on the size and location of the tooth being filled as well as the material used. Most insurance companies cover fillings because they're deemed medically necessary.\nA composite filling is the color of your natural teeth and is used for medium-sized and smaller cavities. Composite fillings will take longer to install than metal fillings.\nDental onlays are tooth-colored coverings that are used over fillings that need to be replaced or removed. They extend past the cusp of the teeth and provide more extensive coverage than a filling.\nWhat's the Cost of Onlays?\nThe cost for a dental onlay will depend on the location and size of the teeth to be covered as well as the number of teeth involved. Your dentist will provide you with a cost estimate during your appointment at our office in Sugar Land.\nA dental overlay works in a manner similar to an onlay but an overlay is used when more extensive coverage is necessary. An overlay will restore function and appearance to damaged teeth.\nHow Much Will A Dental Overlay Cost In Sugar Land?\nThe cost of your overlay will depend on the degree of restoration needed, the type of material used, and your insurance plan coverage. Call our office in Sugar Land for more information or to schedule an appointment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaeece b/data_all_eng_slimpj/shuffled/split2/finalzzzaeece
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+{"text":"Design, much like the film and technology industries, has widespread problems with gender discrimination. Designer Gabriel Ann Maher talks to NADJA about challenging the stereotypes and the impact of the #metoo campaign. In 2013 Gabriel undertook an analysis of Framed, a leading interior design magazine published in the Netherlands. Gabriel, (who uses the non-binary pronouns they\/them\/theirs) found that over the course of the year the representation of men to women was tremendously disproportionate, with a ratio of 80 to 20%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Another classic designed by Sackwear! This Beige Toyota Land Cruiser FJ40 will look great just about anywhere you stick it!! 4.5\" wide x 2.5\" tall this is a super clean and high quality embroidered patch manufactured by STEP 22 Gear. Top off your collection of Land Cruiser patches with this great addition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are there alternatives? As I don't know the brand and the design of it.\nThere are a ton of enclosures, we like this guy, but it's probably more than you need.\nAlso, while you can do what you want, you will be hampering the 850 some by typing it to the 840.\nThanks, maybe I can get a used 840.\nThe Netstor is a bit over the top for me as I could get a WD Mybook Pro as well including Drives (non-SSD) however for around 600\u20ac for 6GBs.\nThe would leave the option to move to RAID 5 at a later time.\nYes, put two in a four bay enclosure then. Use NewEgg reviews, there are hundreds of options I'd guess.\nFYI, Akitio Thunder2 Quad (which has a handle ) is not a RAID box, it's a simple JBOD. of course you can use the Disk Utility to create RAID0, but not raid5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After 6 years working in tourism in Panama Stuart and Will have made excellent contacts in San Blas and had noticed that many tourists found it hard to find reliable information about the San Blas islands with their amazing beaches and indigenous population. Their day tours of the San Blas Islands in Panama are a popular offering where you get to visit islands such as Isla Aguja, Isla Diablo and the popular Isla Perro, site of the famous San Blas ship wreck. On the day tour you will also visit a local Kuna community to get an insight into everyday life in Kuna Yala. They also offer an exclusive Panama Travel Unlimited three-day San Blas camping tour which explores the Lemon Keys islands as well as a unique day trip to the remote and not often visited Cayos Holandeses, a stretch of islands often difficult and expensive to visit.\nFor those looking for something with a bit more comfort overnight stays on the beach islands Isla Kuanidup, Isla Aguja, or a host of other islands are possible. Isla Kuanidup is one of the most popular destinations and the photos will show you why, its well organized and set in a stunning location with one of the most beautiful beaches. They offer day tours, good food and a quality of service above the norm in San Blas. For those more interested in the local Kuna culture then stays at Hostal Nadi, Cabanas Gabayai (owned by the local Kuna government) or a traditional homestay in Carti may be your thing. The Panama Travel Unlimited website has in depth descriptions and photo galleries on almost every Hotel\/Cabana operation in the San Blas islands and gives you honest advice and feedback on what to expect and what not to expect.\nTravel to the San Blas can be difficult with the main airport El Porvenir closed for renovations. Almost all clients to the region now travel with the Kuna controlled private transport company that has special licenses to enter the region. It's a 2-3 hr trip through the stunning mountain jungles, but can be taxing with up to 7 passengers in the vehicle. Unfortunately this is the only way in or out currently. Panama Travel do their best to provide clients with the most comfortable vehicles available. Some of the more luxurious Hotels are found further South in the Island chain such as Dad Ibe Lodge and these are accessed by flying to one of the airports in the region. They can help arrange this, book in advance as flight availability is limited.\nA word of caution when making a booking in the San Blas Islands, Panama. In reality you are entering a different country where few Panamanian laws are adhered to and service may be below what you expect. Only Kunas are allowed to conduct business in the islands and their customs and service ethic may differ greatly from what you are used to. Panama Travel has done its best to find the best operators in the islands and what they offer cannot be beaten on price or service. But we must also acknowledge that once you enter the Kuna Yala region it is out of thier hands. Go with an open mind and you should have a wonderful time in these idyllic islands.\nPanama Travel Unlimiteed offers a variety of Day Tours in and around Panama City. They have a number of vehicles and offer both shared and private tours dependent on your requirements and budget. They also offer luxury tours in our New Honda Pilot that can comfortably seat six clients. One of the most popular tours in Panama is \"The Embera Village Tour\" and they cannot be beaten on price with this fun and interesting day tour that will take you into the jungles on the shore of Lake Gatun and give you a wonderful insight into the Embera indigenous culture.\nThey are very flexible with their tours and can work with the client in arranging any tour they would like; as long as a guide and a vehicle are available they are happy to work on any itinerary to suit the client's needs. The Airport shuttle service is one of the cheapest in Panama and they aim to keep it that way. Available 24hrs a day it is a safe and reliable service and can cater to clients wishing to go to any of the Pacific Beach resorts, Colon Port, Portobelo, Isla Grande and many other destinations to and from Panama's Tocumen International airport. Also offer shuttle services within Panama on request.\nIn keeping with their commitment to provide a complete package of services for tourists travelling throughout Panama, Panama Travel Unlimited has also recently announced the opening of their first branch office in the popular Panamanian mountain town of Boquete. Operating from the same office as Boquete Mountain Safaris(on the Boquete main road, metres from the town square), the Panama Travel Unlimited office in Boquete aims to fill what has long been a gap in the market in Boquete. Travelers looking to find information on tours in San Blas as well as tours and hotel accommodation in Panama City itself have often been left to fend for themselves in Boquete. Now there is an office where they can sit and talk first hand to a travel consultant with immediate up to date information on what is available in terms of tours and hotels in San Blas, Panama City and other areas around Panama. If the consultant there happens to not know some information then they are connected directly to the Panama City office via the SKYPE chat service and able to find out whatever they need to know in real time. Often you will find that it's Will or Stuart themselves available to answer your harder questions. If you happen to find yourself in Boquete then feel free to drop by and say hi.\nComing soon, new office in Bocas Del Toro\u2026 To further enhance the Panama travel experience the website of Panama Travel Unlimited is also currently undergoing a revamp and a whole new look. The new layout makes it easier to find a myriad of information about Panama itself as well as Panama's most popular tourist destinations. Everything, from San Blas tours and hotels to Panama City day tours and hotels are clearly laid out and easily navigated. It's an ongoing process but more hotels, tours and packages in Panama City and the interior of Panama are being added all the time. Expect to find all that you'll need to plan an amazing trip to Panama. The bilingual and knowledgeable staff at Panama Travel Unlimited aim to make your stay in Panama as easy and enjoyable as possible.\nContact Information: For more information or to book a tour, please visit our website at www.panamatravelunlimited.com. You can contact us via email at: info@panamatravelunlimited.com. Our offices are located on the ground floor (Planta Baja), Edificio Stela, Calle Primera, Perejil, in Panama City. Our office telephone number is (507) 395 5014 or via cellular phone (507) 6673-0752.\nPhoto Slide Show: Please click on this link to see photos of some of the destinations we visit regularly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ISU's Insurance and Risk Management Program provides students with a broad understanding of the role of companies that bear the risk of insuring both public and private interests. Studies focus on all major aspects of insurance, risk management, and financial planning. ISU's courses also serve as the foundation for professional designations and licenses. Students who complete the program find careers as risk analysts and managers in large firms, government and regulatory agencies, and in consulting. Other career opportunities include positions as agents, brokers, underwriters, claims adjusters, financial planners, and employee benefits specialists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaggwg b/data_all_eng_slimpj/shuffled/split2/finalzzzaggwg
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index 0000000000000000000000000000000000000000..0ab9f491b88d34cece325519b5c75f7ce870ee2c
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+{"text":"As soon as I knew Ani DiFranco was coming to The Plaza Live in Orlando, I was clearly excited about seeing this gifted artist known for her folksy energetic music with a message. Ani has a huge strong and loyal underground following\u2013 and on Friday night it was apparent by the appearance of an older crowd of mostly females whom had likely followed her since her emergence in the scene in the 1990s. The venue was packed to the hilt, including the balconies.\nThe opening act for the evening was Ed Hamell AKA Hammel on Trial, Ani's longtime friend and Righteous Babe Records protege. Hamell On Trial played the acoustic guitar and had everyone in stitches with his rapid-fire quips and hilarious songs, often compared to Bill Hicks stand-up comedy routine.\nThen, Ani emerged to play her set accompanied by drums and a stand-up bass. It was exhilarating to finally see her live and well worth the wait! Her voice and energy were on point and the feel of the show was very intimate as she chatted with the audience and exuded much warmth. She spoke to the audience about the importance of voting but did not endorse any particular candidate. It is a well-known fact Ani has been a longtime activist for equal rights, justice, and the oppressed.\nThe focus of the evening was Ani's music. There was no glamour or light show. The show was all about music. Ani is a poet. A feminist. And her music spoke to everyone in attendance. She showed humility and appreciation for everyone who attended. It is really difficult to describe the aura of seeing her live\u2014but a performance I will soon not forget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2116143 Black genuine leather jacket for women \"Mizar-US\"\nStylish woman leather jacket with cords, 3D details \"Mizar-US\"\nproduct leather jacket made of genuine leather of excellent quality.\nJacket women with shortened sleeves. there mitts. The fitted jacket, medium length. The surface of the jacket relief.\nJacket women with shortened sleeves. detail detail. Completely handmade excellent.\nproduct leather jacket made of genuine leather of excellent quality. Jacket women with shortened sleeves. detail detail. Completely handmade excellent.\nThis instruction-how to dress the bracers!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the Virtuadopt Raffle! Every six hours, one Virtuadopt appears here. Here is the tip - you can win one! All you have to do is to buy a ticket... or two... or three... and keep your fingers crossed. Each ticket raises chances for winning - and the more overall tickets are bought, the slimmer the chance is. But, a Virtuadopt will find a new owner, and that's what really counts!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Anglesey is like Today.\nThe island of Anglesey is under 25 miles in length but there is a lot happening on Anglesey. We have lots of high resolution images, and have stories about the main towns like Holyhead, Amlwch, Beaumaris and Llangefni. We also try to keep you up to date with the things that are happening at the moment, particularly those things you wouldn't want to miss. Under 'Things to Do' there is information on many different things you might like to do on the island - e.g. Walking, Beaches, Cycling, Fishing, Bird watching, and much more besides.\nOn the ancient monuments pages we have photographed many of the island's ancient standing stones, burial chambers, and other neolithic items of interest. Anglesey probably has more ancient monuments per square acre than any other place on earth.\nIf you are interested in researching your family history then there is a page for you as well. If you wish to express yourself in Poetry, then let us have your entries for approval and posting to our site.\nAnd finally, should you need to find a business on the island then please turn to our Business pages above. It is these adverts that keep the site running, and we would appreciate your patronage.\nTo find out more about these and other places, click on a link above or in the right hand column.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Somebody said something very nasty to me on Saturday about Revd Kevin Annett, and as usual I didn't think of a reply until later. Here it is! (For background to Kevin Annett see: http:\/\/www.hiddenfromhistory.org\/ and watch the film Unrepentant free online.\nRemember me keeping on about a UFO photo I once took and went and lost? Well I'm happy to say that I've found it! HOORAY! I took it in September 1998 during a hiking holiday in Scotland. I used a Polaroid disposable camera and only made one print so I was annoyed at losing it. I didn't see anything at the time, but when I developed it this object could be seen on it. This photo was taken very close to the HMS Neptune Faslane Trident submarine base, and it's been observed that UFO's very often congregate near nuclear power and weapons facilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Last month we announced the latest addition to our portfolio of easy-to-manage IT products, Meraki Insight. This advanced new troubleshooting tool will be of interest to any organization working across a wide area network, whether they're connecting sites together or accessing resources in a remote data center. With more and more IT services migrating from on-site servers to virtual machines running in private or public clouds, it's never been more important to be able to quickly and easily troubleshoot the WAN and application server performance.\nMeraki Insight combines the deep packet inspection capabilities inherent to our MX platform with a robust, mature cloud-hosted architecture to analyze WAN traffic and server response times. By inspecting both the network and application layers, a picture soon emerges, enabling network administrators to quickly identify potential performance bottlenecks that may be leading to a diminished user experience. With IT support tickets being one of the most critical data points for measuring IT effectiveness, anything that helps to accelerate the closing of those tickets and even prevent them is welcome. A better network experience translates to happy users and customers who are more likely to return.\nWe've been delighted with the feedback we've received from early customers of Meraki Insight, and not just for the great user experience in the dashboard either. Real issues with real revenue implications are being resolved more quickly already, thanks to the capabilities unlocked by this new tool.\nIn one case, a well-known retailer with operations around the world was able to use Meraki Insight to rapidly identify and resolve a point-of-sale issue that was impeding sales during its busiest time of year: the holiday season. Another renowned storage company with over 1,500 locations was able to look at WAN performance data to help pinpoint a latency issue that was impacting business-related VoIP calls between branch sites.\nEven if alerting isn't being used to draw attention to an issue, one may be lurking. One Meraki client stumbled across a custom application with a low performance score. All clues pointed to a network layer issue, and sure enough the app was experiencing packet loss severe enough to affect user experience. After a call to their ISP, a support team was dispatched to fix interference on the line.\nReal world issues, impacting real business: these are the challenges Meraki Insight helps to address, thereby helping our customers increase revenue and reduce operational costs. For the first time the outstanding LAN troubleshooting tools that have helped build our success at Meraki are looking out to the WAN and the application servers we rely on every day. To gain better insight into issues that may be impacting your network and end-user experience, just click here to initiate a free trial.\nAre you are ready to get back to doing what you do best and leave your IT worries behind?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Info on affordable memorial jewelry for ashes.\nThere are a number of reasons for such a huge popularity of pandora jewelry. Very often the piece is likely to be scrapped or useful for yet another project. Knowing where to purchase cheap jewelry can truly provide you with the ability to spend less.\nKeyword phrases for this specific article are usually ring, affordable memorial jewelry for ashes, technique, imagination.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A barn style storage shed is often called a garden shed. A shed, which is typically simple, single-story structured and in the back garden or back yard is the garden shed. It is a slight structure built for shelter or storage. It is used as a workshop and can be attached as a 'lean-to' for a permanent building or separate. It is often with open front or sides. Sheds can be of various types depending on the complexity of their construction and their size. They vary from small open-sided tin-roofed structures to large wood-framed sheds with shingled roofs. Windows and electrical outlets can be attached to them. Check out our wide selection of shed styles at The Fireplace Showcase.\nPortable restrooms trailers have their own double-axle trailer and are delivered and set up by professionals in convenient and discreet areas. Portable restroom trailers are popular for outdoor or home weddings and are used typically when there is a need to offer a fancier, spacious, well equipped main toilet. These portable restrooms are designed luxuriously and are equipped with a lighted vanity and double sinks, heating and air conditioning, overhead lighting, a stereo system, and in some cases, baby changing stations, which make even the choosiest of guests feel comfortable. These units are an ideal solution for larger events and because they reduce lines and add a touch of elegance. This restroom facility is the perfect way to add convenience, comfort and style to any large gathering.\nKeep in mind that for small scale functions or events we offer a single mobile restroom. These units offer a flushable toilet, and wash basins with hot water. You can count on cleanliness, quality and service from Northeast Sanitation for your very special event. Call us today.\nA septic system is an underground wastewater treatment system that is used to dispose of household wastewater. If you have a septic system, maintenance is required to keep it running efficiently and this includes having your septic tank pumped regularly. How often you get your septic tank pumped depends on the size of your tank, how many rooms are in your house, and the number of people living there. It is recommended that you have it pumped every 1 to 3 years. The more a tank is used, the shorter the time should be between tank pumping. If you're unsure of when your tank needs to be pumped again, call Northeast Sanitation and we will help you determine what schedule is best for you based on the above factors and the septic system codes in your area.\nFollowing a few easy maintenance tasks and having your septic tank pumped regularly will preserve your septic system for 20-30 years. Having to replace your septic tank is extremely expensive, so avoid it by properly using and maintaining your septic system. Northeast Sanitation will pump your septic system and they will also help you maintain that system by sending you future reminders to ensure that you are on the proper septic tank cleaning schedule for your home.\nMost basements leak because there is or are either vertical-step or horizontal cracks on the outside of basement walls through which water enters and gets onto your floor. Water\/moisture will also enter the wall through and around where gas lines, electric lines, and plumbing enter the wall below ground. The only way you will completely keep all water from getting into the wall is to have these openings correctly sealed and backfilled on the outside of your foundation (basement wall). If you only have 1 crack then that is all you need to fix on the outside. You only need to waterproof about 5 linear feet on the outside. You most likely do not need the entire wall done or all the way around the house done as some companies will try hard to sell you. Basements leak because there are\"direct openings\" into the house on the outside of the house and you have to go to the outside to seal them, not inside unless you use the A-1 Foundation Crack Repair method. An inside water diverting method (jackhammer,drain tile) ONLY diverts water that has already entered the wall to under the floor. It doesn't stop water from entering walls. Therefore it cannot stop or prevent mold\/efflorescence from growing on your wall or behind your paneling\/drywall. The A-1 inside\/outside method done correctly will!\nYou need to consider some important factors when you are renting portable toilets for an event. When you are trying to determine the minimum amount of provisions, think about the length of the event, the estimated food and fluid consumed by those attending, and if you need some provisions for children, elderly, or disabled people who tend to take longer to use a facility.\nFor Municipal events, the maximum acceptable wait time for a portable toilet is a line of ten people. If alcohol is served, it is recommended that you increase the number of portable toilets by 15%. Also, one disabled unit per every two hundred people should be provided. A disabled portable toilet unit is a large restroom which can be used by people in wheelchairs and is ADA compliant.\nFor all of your municipal portable toilet needs, please contact Northeast Sanitation, your expert in sanitation facility rentals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A nightie finished in an intricate print you're going to love. This pretty pink number has been designed with a bird and floral print and a bow applique that gives it a romantic feel - one for the more girly among us. Its long length and soft to touch material is cosy though too, which is what it's all about when you're only in your nightwear.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scatterbrain.: A Sure-Fire Sign of the Apocalypse.\nI'm thinking \"Google: the Microsoft of the future\"\nHow long did it take to get back up?\nI tried to open Blogger an hour later and it was still down. It worked fine when I tried it the next day. i.e. I have no clue what the timing was.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I have always had a passion for writing - especially about the bad guys. Give me one that is barely fleshed out as far as motives and a back story and so much the better.\nThis is what I discovered in early 2015 when I fell obsessively, hopeless in love with a My Little Pony character named King Sombra. He is the perfect mixture of mystery, terrifying presence and potential for an origin that would make anyone's heart ache. And I am determined to do him justice by retelling his tale.\nJoin me on this journey as I weave our King's tales in Dark Crystal Fires: The King Sombra Chronicles.\nWhat warning tags do you need to see on our stories?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Text File Combiner makes it easy to combine multiple .txt and .csv files into a single file.\nA JavaFX based app that can be used on Win, Mac and Linux.\nany harm, use at your own particular RISK! We highly recommend to buy Text File Combiner from the The Developer ( gocoders ) website. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Got to love the engine for matching you with a hotel room in Virginia Beach. Choose location, amenities, etc. I ended up with the Sandcastle Hotel.\nOur latest poll is over and 62% are opposed to reelecting Rosemary Wilson to City Council next November. What I find intriguing is the 62%-38% spread is identical to regionwide for the 2002 Transportation referendum.\nOkay, wondering why I'd run an unscientific poll a year out on a Council race? I was trying to gauge the political persuasions of my readership. 62% want reform, while 38% are status quo. It was just a sneaky way of doing it.\nThe new poll is out of the Resort Area Master Plan Charete: where should the eastern terminus for light rail be? Both answers are covered in various options now on the table. Which one do you want?\n2. VBTA - realization that the political universe doesn't revolve around reducing the Property Tax rate. People are unhappy about it, but they care about many more issues.\n3. Rosemary Wilson - we have a mutual hatred society, but I care even less for the Deaniacs. With VBTA Vice Chair Bob Hedrick already announced as a challenger, I'll write the first attack ad for Rosemary.\nThe Deaniacs have been over-the-top on Virginia Beach's workforce housing program, which you'll try to take credit for. (Kind of like Al Gore inventing the Internet....) An ad with a picture of a concerned family huddled around. The caption, \"All They Want Is A Good Home. Bob Hedrick Thinks That's A Marxist Conspiracy.\" In the body of the ad, explain the program and quote the Deaniac rhetoric.\n5. Reid Greenmun - voted our biggest civic embarassment on this blog, I give him anger management classes. Maybe if he calms down, he'll come across sane politically.\n6. Sam Reid - persevernace for the incoming CCO President. The office has become a hot seat, with no recent President living up to expectations.\n7. Billy Almond - former RAC Chairman and Chair of the RAC's Planning\/Design Review Committee. He alone deserves credit for pushing Virginia Beach into the ongoing Resort Area Master Plan process.\nFor Billy: an apartment in one of those mixed-use developments in the Resort Area once built.\n8. Chris Stolle - a map of my home Aragona Precinct. Carolyn Weems won with 57% here; Joe Bouchard won with 56% here. Chris obviously doesn't know the place.\n9. Wawa - more transit-convienent locations in Hampton Roads. Drivers know them for their cheap gasoline. However, Wawa is also a great break for bus riders with clean restrooms, drinks, and good food.\n10. Transit riders - that Norfolk's Light Rail Starter Line continues to run under budget, and that Virginia Beach commission a Study for Newtown Road - Dome site in 2008.\nMerry Christmas. And don't forget to go to Christmas Mass.\nMonday and Tuesday saw a two-day opening to begin the crafting of the Resort Area Master Plan. A Charete was held at the convention center: 1-5 on Monday, 1-3 on Tuesday. On Monday there was an opening Presentation by the consulting team from Sasaki, followed by input. On Tuesday Sasaki brought back three draft plans for comment. Nobody seemed to like any of the plans in their entirety. A few of us tried to mix-and-match.\nWhat raised eyebrows is that Options 2 and 3 resurrected BRT as a Resort Area-only system. Don't like that? Option 1 had a streetcar running the length of Pacific Avenue. All three plans had light rail coming to the Resort Area from Norfolk, though with different routings and end points.\nThe development proposed was largely off-the-shelf: convention center HQ hotel, Rudee Loop, Dome site, mixed-use in RT-2 and RT-3, etc. It was the Transportation proposals that broke new ground.\nSasaki will be back in February at which time everyone will try to merge the three options into a single draft plan.\nBurton Station is a political quagmire waiting to happen for the Virginia Beach City Council, one they shouldn't want to enter with our first-ever November Council election less than a year away.\nIf the Editorial Board wanted to really do us a service, they would ask a simple question \"How can Norfolk redevelop Broad Creek without it being a political lightning rod, but Virginia Beach can't touch Burton Station?\" The answer, of course, is that Norfolk's ward system has led to the African-American community being seated at the table. In Virginia Beach, the African-American community is often left begging for crumbs from the table given the evil at-large voting system. Were they at the table, a rational plan could be negotiated. I myself have argued that the African-American community's leadership should bargain Burton Station away in exchange for legally protected status for Seatack. Seatack has greater historical significance, a larger population, and is threatened by Resort Area redevelopment.\nNow let's see if the Editorial Board can do a follow-up calling for political reform in Virginia Beach that would seat the African-American community at the table, through either a ward system or cumulative voting.\nI myself have received Social Security Disability Income (SSDI) for 15 years now. I won it on appeal, and the 1992 appeal took just over a month. Now read what's become of the system. What's so crucial about the disability payments is work clearance with such a condition. In my case, I had doctor's orders not to work for six months. For many, they never will be able to return to full-time work. It's the payments that cover the bills.\n1. Hire the additional appeals judges.\n2. Change the rules for the original cases. If two-thirds are winning on appeal (like I did), obviously the original case standards are too burdensome. That would lessen the number of appeals that need to be heard.\nI wish this story had appeared two days earlier. As my Congressperson, Thelma Drake, was at the Light Rail Groundbreaking, I would have given her an earful.\nEveryone prefers that a leftover grant from Bus Rapid Transit (BRT) be spent on a new hybrid bus for trolley replacement.\nThe Resort Advisory Commission's Transportation Committee had previously endorsed a position to use the grant to replace the Ticket Vending Machines (TVMs) for the VBWave trolley at the Oceanfront. Apparently the position doesn't have traction west of Birdneck Road.\n1. The only specific talk of expansion into Virginia Beach came from Norfolk Mayor Paul Fraim - who did it repeatedly.\n3. Messina's opening paragraph is spin. The loudest cheer of the day was the only standing ovation, given to Governor Tim Kaine at his introduction. The next two loudest were for Congresspersons Scott and Drake, when it was announced their roles in securing Federal funding.\n6. I spotted three Virginia Beach politicos there: John Welch, Bruce Williams, and Louisa Strayhorn.\n7. It was announced that Yvonne Miller has been named Senate Transportation Committee Chair by the Senate Democratic Caucus. Let's hope that means a greater emphasis on transit.\na. an increase in rail funding of over 60%.\nb. an increase in mass transit funding of over 40%.\nEven as a Republican, I have to admit Tim Kaine has been the most pro-mass transit Governor in Virginia history.\nThere's no formal citizen comment session at RAC meetings, but the Chair will normally allow anyone to speak who wishes. As soon as the Briefing was complete, Bob raised his hand. Wouldn't the hotel be in an AICUZ zone and thus draw fire from the Navy? City Council Liason John Uhrin pointed out the land is zoned RT-3, allowing by right development of a such a hotel up to 200 feet in height.\nWhat's so wrong with O'Connor's argument? Okay, he knee-jerkingly opposes any Resort spending. However, here is something disingenous: his use of the AICUZ argument. The 2005 R&HA Initiative by the Deaniacs (which O'Connor signed the petition for) sought to undermine any use of eminent domain to save Oceana under the BRAC order. In short, the Deaniacs wanted Oceana closed.\nNow in 2007 they want to pose as champions of Oceana to push their \"Demonize the Resort Area!\" agenda. They hope people have some very short memories.\nDecember 6 is Finnish Independence Day; I celebrated by going out for a nice dinner this evening. How many others locally celebrated?\nThis is the 90th Anniversary, as the Grand Duchy of Finland declared independence from the remnants of the Russian Empire on December 6, 1917. Russia was hit by Comrade Lenin's revolution in October, and the Finnish had no intention of sticking around. The Finnish assembly requested German troops, and Germany was happy to sever Finland from Russia. Russia was still at war with Germany (World War I).\nIn the old country, one would hit the sauna and get drunk on December 6. I don't even know of a sauna in Virginia Beach.\nThe linked story reports that an endorsement by Virginia Beach Vision (VBV) is looming. As much as I despise VBV, I'll take their clout and partnership in a coalition to bring light rail to Virginia Beach. Nearly every VBV pushed project has become a reality; if they endorse light rail in Virginia Beach, it's huge.\nHaving had a crucial role in getting Bob Dyer the AAPAC endorsement for City Council in 2004, I can testify how the group is non-partisan. (They also endorsed Glen Davis in 2006.) Given the huge hurdles the evil at-large voting system presents, it will be interesting to see exactly what the AAPAC does to promote African-American City Council candidates in 2008. However, with the elections now in November, minority voters will make up a larger share of the electorate than in May. Does that translate into wins?\nWell, I've been joking that every political junkee wants to see November, 2008 come along just to see how it changes the campaign model. However, too much shouldn't been read into the results, as November, 2010 will be quite different again. In 2010 there will be neither a Presidential nor Senatorial race on the ballot, changing the dynamics from 2008.\nJust a friendly reminder that the Groundbreaking Ceremony for Norfolk's Light Rail Starter Line is this Saturday, December 8. Activities will be from 1-3 P.M., with the rail spike driving and speeches at 1:30.\nDecember 8 is also the Feast of the Immaculate Conception. As a Catholic, I've spent the last few weeks trying to draw a paralell between the two; however, I can't come up with a line that isn't sacreligious.\nAggressive homeless persons in the Oceanfront parks has become an issue on the table at the Resort Advisory Commission (RAC).\nWednesday morning the Transportation Committee (which has oversight of the parks) was briefed on the problems there: panhandling, public urination, fornication, sleeping in the bushes, etc. The problem is worst at 17th Street and lessens as you move northward. The root of the problem at 17th Street is believed to be the number of programs for the homeless nearby: three churches serving food, the Lighthouse Center, and the Judeo-Christian Outreach Center. A proposal was made to close the parks, at least during some hours, but nobody on the Committee could support it.\nThe Oceanfront Enhancement Committee (which has oversight on Behavior issues) took up the issue on Friday morning. Virginia Beach Hotel\/Motel Association (VBHMA) Executive Director Nancy Perry had polled the VBHMA Board and nobody could support park closings. Council of Civic Organizations (CCO) RAC representative Henry Ryto informed the Committee that he had spoken to the CCO Executive Committee about the matter and that everyone was opposed to park closings. Virginia Beach Police Department Second Precinct commander Captain Tony Zuccaro called closing the parks \"scattering cats\" and reccomended that a security guard (not VBPD) be hired to patrol the Oceanfront parks. In the end, a subgroup was named to come up with a comprehensive solution to the problem.\nWatch this issue closely if you want to maintain access to our Oceanfront parks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take control of your education, choose courses that interest you, and prepare yourself for an interconnected world.\nBy earning a BIS degree, you can build your own path to graduation by selecting courses from across the disciplines that relate to your specific interests. Some students want to specialize in multiple areas without increasing their time to graduation. Others want to major in a field of study that SUU doesn't offer. The BIS major allows the most freedom to construct a degree path that is customized to your needs.\nThe Bachelor of Interdisciplinary Studies (BIS) is a rigorous degree that allows students to customize and develop a program of study to integrate interests from different disciplines. It emphasizes a broad learning experience and offers students the opportunity to participate in an interdisciplinary, academically sound, and intentional education directly relevant to the student's intellectual development and career goals.\nA student chooses two or three disciplines and constructs, under advisor supervision, a personalized program of study.\nPrograms of study need to be approved by the academic coordinator within the BIS program, department chair, and dean.\nOption 1: 2 disciplines: 21 credits in each, 15 must be upper-division credits in each discipline.\nOption 2: 3 disciplines: 15 credits in the first two tracks each, 9 must be upper-division credits in two disciplines, and 12 upper-division credits in the third discipline.\nStudents also take a common core of courses (9 credits) to facilitate integration.\nStudents must complete all General Education and other SUU requirements.\nStudents must complete a total of 120 credits, including 40 upper-division (3000 or higher) credits. Upper-division credits include the minimum 30 upper-division credits from disciplines and 6 upper-division BIS credits.\nA minimum of C- is required for any course counted toward the major.\nNo pass\/fail course may be used toward the BIS degree without prior approval.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking for a 3 bedroom brick home to occupy or for investment in a great location tucked away in a tranquil pocket? This must see home located in 5 North Pocket Avenell Heights maybe just the home you have been searching for.\nTo arrange your personal inspection of this great property call Aaron and Scott today before it is too late!\nRates: $1,600 per half year approx.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"You can enroll after you received your letter of acceptance. For your enrollment please bring all documents mentioned in your letter of acceptance.\nYou must have valid health insurance, to enroll at a German university.\nEU citizens do not need a German health insurance; European health insurance is valid and sufficient. You can order the European Health Insurance card from the insurance company in your home country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed to complement the legendary Redwood Room, the Velvet Room is a unique venue for corporate meetings and select nightlife events, while offering breakfast and dinner daily. Executive Chef Thomas Weibull has partnered with local farmers to produce seasonal menu items that are grown in a sustainable and organic manner. The elegant and dramatic room, complemented by lush velvet curtains, mahogany and leather banquettes and hand-blown Murano glass lamps also hosts an eclectic roster of events.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The controversies surrounding the remuneration of big business leaders have rebounded in the case of Carlos Ghosn. But would not these polemics respond to epidermal reactions in which History and human feelings mingle?\nWineLA, a wine club that embraces the wine culture of Los Angeles, recently held an afternoon Wine Tasting featuring the Dry Creek Valley wines at Culver City Smashbox studios benefiting the T.J. Martell Foundation.\nAs the holiday season quickly approaches, preparations for family gatherings including choosing the menu, that could include a variation of poultry, meats, seafood and pasta now include planning for leftovers which has become as popular as the main meal itself.\nPunxsutawney Phil, the Pennsylvania Groundhog who, since 1887, has become the annual predictor for winter's end, popped his head out of his hole-in-the-ground home and promptly saw his shadow, the indicator of another six more weeks of winter, snow, cold, ice and slush, sad news for those of us who are ready for winter's end.\n\"Cloudy with a Chance of Meatballs\", the animated version of the classic children's book previewed recently for select media and followed with a press junket at the world famous Dylan's Candy Store on Manhattan's Upper East Side.\nBeginning one week ago, President Donald Trump ordered the expulsion of dozens of Russian diplomats and the closure of the Russian Consulate in Seattle in response to the use of military grade chemical weapons on British soil.\nPirates of the Caribbean: On Stranger Tides, a fast action, high seas, comedic adventure from Walt Disney Pictures and Jerry Bruckheimer Films, recently hosted its world premiere at Disneyland in Anaheim, California.\nPreet Bharara, the United States Attorney for the Southern District of New York, announced that GENRIKH KOLONTYRSKIY, MOYSEY KUCHER, and DORA KUCHER pled guilty today in Manhattan federal court to conspiring to defraud programs administered by the Conference on Jewish Material Claims Against Germany, Inc. (the Claims Conference), established to aid the survivors of Nazi persecution, out of more than $57 million.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A warm welcome back to School to everyone. We had a great first day in Room 6 getting to know each other.\nWe were very lucky and got to pick new Duffy Books on our first day back.\nWe also did some counting and one to one matching. We made backgrounds for our posters that we are making about us and our Wh\u0101nau.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Chief Legal & Administrative Officer, Sheila Flaherty manages all global legal matters and talent management for Alert Logic. Flaherty brings extensive experience managing legal affairs at successful public and private corporations. She has taken numerous companies public on the NYSE and NASDAQ and co-founded a Special Purpose Acquisition Company. She has extensive experience with high growth companies, mergers and acquisitions, and expansive intellectual property portfolios. Most recently, Flaherty was Executive Vice President and Chief Legal Officer for e-commerce innovator Demandware, Inc. where, among other matters, she managed the successful initial public offering. Flaherty earned her bachelor's degree from the University of Massachusetts at Amherst, and Juris Doctorate from Georgetown University Law Center. She also studied law at Trinity College, in Dublin, Ireland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..a53a64d80a770f1f27684d09259b06f3f72b4322
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+{"text":"Does Your Team Need This Lifeline?\nA while back, after a much-needed vacation, I scheduled some time for writing and research. During this time, my interaction with my coaching and training clients was limited to text and phone conversations.\nAbout 10 days into this period, I noticed something unusual.\nI was starting to get a little down. Not an all-out depression, but I was noticing something declining in my overall mood. I felt like I was sinking. Nothing bad happened, really. In fact, I had just come off a very restful vacation and had plenty of work to accomplish.\nNonetheless, there it was; the feeling of not having enough connections that sustain love for my work.\nA lifeline is defined as \"a rope or line used for life-saving, typically one thrown to rescue someone in difficulties in water or one used by sailors to secure themselves to a boat.\" Things can happen to us in our lives that give us a similar feeling of sinking or being stuck. If we don't have some help to secure us, we can begin to feel alone and hopeless.\nFrom time to time, we all need a lifeline of care and compassion from others.\nIt is fairly common knowledge amongst psychologists that the feeling of isolation can be a key determinant for a wide range of human ailments, from depression all the way through to premature death.\nLoneliness a bigger health risk than smoking or being overweight?\nIf loneliness is a bigger health risk than cigarette smoking and obesity, then perhaps it is something we as leaders should pay closer attention to. Are there people in our sphere of influence that need a lifeline?\nA very insightful study was published last October in the Journal of the American Geriatrics Society. Researchers wanted to know the impacts and categories of social contact or lack thereof, that might predict clinical depression. In studying over 11 thousand people over the age of 50, the scientists found that only face-to-face interaction forestalled depression in older adults. Phone calls made a difference to people with a history of mood disorders but not to anyone else. Email and texts had no impact at all.\nIn the study, the frequency of gathering with friends and family\u2014or not\u2014was the key. What's more, the researchers discovered the more in-person contact the less likely that depression would occur in the future. Participants who had minimal social contacts had the highest depressive symptom rate, while those who connected with people in person at least three times a week had the lowest. It would seem that the more people gather in person, the better off they are.\nWhat could we as leaders do to become part of the solution?\nAs leaders, we can be intentional with those under our influence. Here is how I would adapt the above list for leader-follower interactions.\nSpend most of your time listening and asking questions, rather than being in \"solve mode\"\nHow often are you connecting with those you lead? How intentional are you in making connections? Who on your team seems a little down and needs to know you believe in them? Your lifeline of care and compassion might be what is needed to help your team reach peak performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AirWheel - New Dolly Shot Trend? - JSS Productions, Inc.JSS Productions, Inc.\nThe new dolly shot for your next film. AirWheel mixed with a Movi Gimbal = Smooth shots at 11-15 mph. Is this the new trend for film making?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MOSCOW (Sputnik) - Minnesota Democratic Senator Amy Klobuchar has announced she is running for president in the 2020 election.\n\"I stand before you as the granddaughter of an iron ore miner, as the daughter of a teacher and a newspaperman, as the first woman elected to the United States Senate from the State of Minnesota, to announce my candidacy for President of the United States,\" Klobuchar said in a speech.\nThe senator pledged that as the president she will \"focus on getting things done\" and \"lead from the heart.\"\n\"It is time to organize, time to galvanize, time to take back our democracy, it's time\u2026 to get the dark money out of our politics,\" Klobuchar underlined.\nKlobuchar, 58, served as a county attorney before before she was elected to the Senate to represent Minnesota in 2006.\nOver the past month, US Democratic Sen. Elizabeth Warren, member of Congress from the state of Hawaii Tulsi Gabbard and Democratic Sen. Kamala Harris of California have announced their presidential bid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Junius Spencer Morgan caused a sensation in 1876 when he paid the staggering sum of \u00a310,100 \u2014 more than the National Gallery of London's annual purchase grant \u2014 for Gainsborough's celebrated portrait of Georgiana, Duchess of Devonshire. Since then a seemingly interminable line of nouveaux-riches American 'Despoilers' has relieved the impecunious (and often only too willing) European aristocracy of the art treasures their ancestors had amassed over the centuries. The phenomenon prompted the foundation of the National Art Collections Fund in Britain in 1903 to help save such treasures for the nation, and was subtly dissected by Henry James eight years later in The Outcry. Revealingly, the essence of James's acquisitive American, Mr Breckenridge Bender, was his desire for 'an ideally expensive thing'. He had 'no use' for Lord Theign's \u00a310,000 Moretto; he was out to spend millions. He had come to England to bag the prized Sir Joshua.\nIt is a familiar enough story. Only now it is the Reynolds that costs the equivalent of \u00a310,000 and, almost inconceivably after more than a century of hoovering up masterpieces major and minor, it is the Breckenridge Benders that are selling up. Astonishingly, the Americans have turned from being net buyers of works of art into net sellers. America is, so to speak, the new Europe.\nAccording to The European Fine Art Foundation's (TEFAF) latest report, 'The International Art Market: A Survey of Europe in a Global Context', which includes dealer as well as auction sales, the tide turned in 2006. Sotheby's own figures put it earlier (its privately owned rival Christie's is not obliged to publish its accounts). The US auction house revealed that Americans have been selling more than they have been buying since 2005, with the original negative value of $106 million quadrupling the following year and rising to $702 million last year.\nThis marks a watershed in the history of the western art market, for the transatlantic flow of works of art has reversed. According to Sotheby's, the flow of art into the UK has risen tenfold since 2005 to $734 million last year. Art treasures are not, however, returning to the walls of Lord Theign's Dedborough Place. What has loosened them from their often illustrious old money American moorings is the seemingly inexhaustible supply of even newer money from Russia, China, the Middle East and India. A perfect illustration is Gauguin's 'Te Poipoi (Le Matin)', which sold from the collection of the late Joan Whitney Payson at Sotheby's New York last November to the Hong Kong property tycoon Joseph Lau for $39 million. Petrodollars, however, favour London.\nLong in danger of turning into a sleepy market backwater, London has been revived as a great international art entrepot. As the British government is only too aware, it has also become the preferred place of domicile of a significant number of the world's newest rich \u2014 or at least the place in which they keep their works of art. (Given the draconian export laws in their own countries, most Russian and Indian collectors choose to buy and keep their art overseas.) In fact, probably not since the likes of John Pierpont Morgan (son of Junius) and Henry Clay Frick assembled their picture collections in London to avoid the 20 per cent US import tax on works of art (books were excluded) has the capital offered refuge to so many distinguished foreign collections.\nThe main difference between now and the mid-20th century, however, when London was at its height as an art market, is the scale and breadth of the market itself. Never before have so many people from so many places around the globe competed to buy works of art. The art market is no longer simply international but global, and it is expanding at an extraordinary pace. Five years ago, for instance, Sotheby's top clients \u2014 those spending over $500,000 \u2014 originated from 36 countries; now they come from 58. The TEFAF report records the market's recent growth in terms of both transactions and value, the latter an astonishing 95 per cent between 2002\u20136 and rising from a global value of 26.7 billion euros to a record 43.3 billion euros. Striking, too, is the fact that China has now ousted Switzerland from its long-time ranking as the world's fourth largest national art market, and is just 1 per cent behind France in terms of market share. Its domestic auction market grew by a staggering 983 per cent between 2005 and 2006.\nBuyers of Russian origin accounted for 15 per cent of lots at Sotheby's London evening sale of Impressionist & Modern Art in February this year, including the top lot, Franz Marc's 'Grazing Horses III'. That sold for a record \u00a312.3 million against an estimate of \u00a36 million to \u00a38 million. Given the huge record price ($72.8 million) paid at Sotheby's New York last May for Rothko's not universally admired 'White Center (Yellow, Pink and Lavender on Rose)' \u2014 a picture consigned by another legendary American blue-blood, David Rockefeller \u2014 it is tempting to presume that it, too, went to a client from one of the emerging markets.\nUnlike his earlier European counterparts, Mr Rockefeller was not selling a major picture because he needed the money but because he realised that he could do some significant charitable work with the proceeds. It was the irresistible lure of extravagant sums that prised the painting off his wall nonetheless. One has a sense that, just like Breckenridge Bender, today's newest buyers are interested in the ideally expensive thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following a pogrom against the Jews in Kishinev in 1903, he founded the Jewish Self-Defense Organization in Odessa, an organization whose goal was to protect Jewish communities throughout the Russian Empire (\"Jewish youth, learn to shoot!\").\nHe soon left the world of journalism and traveled throughout Russia, spreading the ideas of Zionism and pleading with Jews to learn self-defense. In the same year he was elected Russia's delegate for the Sixth Zionist Congress.\nUpon the outbreak of WWI, Jabotinsky met Joseph Trumpledor and together proposed the idea to create a Jewish faction in the British Army. The Jewish Legion was officially established in 1917 and fought with General Allenby at the Jordan River crossing, where Jabotinsky was decorated for his bravery.\nIn 1920, he was elected to the first Assembly of Representatives in Mandatory Palestine and helped establish Keren Hayesod, the financial arm of the Zionist movement. Disagreeing with the moderate policies of the Zionist movement, Jabotinsky split from the mainstream and created the revisionist movement, a movement calling for the immediate establishment of the Jewish State on both sides of the Jordan River.\nIn 1923 he established Betar, the revisionist youth movement whose goal was to educate its members with ideals of nationalism, self-defense and education. In 1937 he became the Supreme Commander of the Irgun (Etzel), the military wing of the revisionist movement.\nDuring the 1930's, Jabotinsky grew uneasy about the anti-Semitic tendencies in Europe and proposed an evacuation plan of all Jewish residents. He met with leaders from Poland, Hungary, and Romania, who all accepted his proposal, but unfortunately, the British government vetoed the plan, rendering it obsolete.\nNonetheless, Jabotinsky continued to travel around Europe, begging Jews to immigrate to the Land of Israel. When the WWII broke out, he was active in England and America in trying to establish a Jewish army to fight alongside the Allies against the Nazis.\nOn August 4, 1940, while visiting a Betar camp in America, Jabotinsky had a heart attack and died. His legacy as one of the greatest Zionist leaders lives on to this day, so much so that there are more streets, parks and squares named after him in Israel than any other figure in Jewish or Israeli history.\nMichael Levin was a lone soldier in the IDF that was killed during the Second Lebanon War. Levin was born in Philadelphia, and at age 19, decided to leave all his friends and family behind and make Aliyah to enlist as a lone soldier in Battalion 890 of the Paratroopers Brigade.\nMichael got special permission during the summer of 2006 to fly back to America and visit his family, but while he was in America he heard of the outbreak of the war and flew straight back to Israel to join his friends in combat. On August 1, his battalion came under direct fire and Michael was shot along with two other soldiers. He was 21 years old.\nWhen Michael was in the army, he often expressed his dream that one day lone soldiers would have a place where they could go and receive help, support, meals and advice. In 2009, the Lone Soldier Center in Memory of Michael Levin was founded and has since helped thousands of lone soldiers throughout the country.\nPioneers from Ukraine founded Rishon LeZion on July 31, 1882, and hoped to earn a livelihood by working the land and growing crops. However, the inexperienced pioneers met hardships and were unable to overcome the rough conditions.\nThe settlers decided to reach out to Baron Edmond de Rothschild, who answered their pleas by sending money and experts who helped the people find water and cultivate the land.\nBacked by Baron Rothschild, Rishon LeZion flourished and attracted many notable Zionist pioneers: David Ben-Gurion worked at the Carmel Winery (one of the first modern wineries in the Land of Israel) and Eliezer Ben-Yehuda taught Hebrew in the first modern Hebrew school.\nHatikvah, Israel's national anthem, was written by Naftali Tzvi Imber while visiting Rishon LeZion, and the first Israeli flag was unfurled in a procession marking the settlement's third anniversary.\nWith a population of over 230,000, Rishon LeZion is currently Israel's fourth largest city and is one of the fastest-growing cities in the country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"25 best ideas about apartment living rooms on pinterest. Apartment decorating ideas: a brooklyn bedroom. Picture insights: small living room decorating ideas. Apartment artistic small studio decorating ideas with. 50 best small living room design ideas for 2017. Small nyc apartment living room ideas wwwlightneasynet. New small living room ideas temeculavalleyslowfood.\nSmall living room decorating ideas hometone. Free vector graphic: light, ceiling light, spotlight. Living room: living room movie theaters in portland oregon showtimes portland oregon movie times. Pinterest living room decorating ideas best on small rooms. 51 inspiring bohemian living room designs digsdigs. Living room decorating ideas features ergonomic seats. 40 cozy living room decorating ideas decoholic feedpuzzle. Value city living room sets value city furniture. Mix of white green victorian style house in london. Designers curtains for living room [peenmediacom]. Best 25 minimalist living rooms ideas on pinterest.\nLarge wall art for living rooms: ideas inspiration. Light brown couch living room ideas what colour curtains. Photo page hgtv. Living room paint colors for 2018. Transitional living room the house of silver lining. Couches for small living rooms image of review small. L shaped couch small living roomjpg (800600) interior.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Melbourne's first game against a side called St Kilda came in July 1859, but that side had no connection to the future VFL team.\nThe present day Saints were instead born from a merger between South Yarra (not related to the club who Melbourne played their first competitive match against) and the St Kilda Cricketers' Club in 1874.\nSometimes known as the Saints, but more often as the Seasiders, St Kilda were deplorable in their first few seasons in the league, failing to win a game in their first three seasons. They won their first ever VFL match against Melbourne on protest but won just one more in their first six seasons. Considering their early troubles it's no surprise that the Demons hold their best record against any foundation VFL club against the Saints.\nThey also protested against a Melbourne victory in Round 12, 1925 on the grounds that Gerry Donnelly wasn't eligible to be playing for the Redlegs. The VFL dismissed the match protest but forced Donnelly to cross to North Melbourne who he was residentially tied to.\nTwice in the modern era the Demons have caused St Kilda coaches to be sacked. Stan Alves after the 1998 Semi Final and Grant Thomas after the 2006 Elimination Final.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The McDowell Center for Children believes there is hope for all young people and that anyone can change, given the right guidance and structure. We understand that many adolescents referred to us have a long history of problematic behaviors at school, in the community, and in the family. McDowell is here to offer hope and encouragement to adolescents as well as their families and significant others.\nThe McDowell Center is a 28 bed residential treatment center for adolescent males ages 12-18 who are experiencing mental health, behavioral, and\/or substance abuse issues. The program is designed to ensure a supportive and structured environment that considers the individual's educational, emotional, and behavioral needs.\nWe are licensed by the Tennessee Department of Mental Health\/Developmental Disabilities and accredited by The Joint Commission. The McDowell Center is conveniently located to West Tennessee and Eastern Arkansas. For further information or directions to the center please call: 1.866.301.7076.\nPhysicians are on the medical staff of McDowell Center for Children, but, with limited exceptions, are independent practitioners who are not employees or agents of McDowell Center for Children. The facility shall not be liable for actions or treatments provided by physicians.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to our inaugural Opalesque Boston Roundtable.\nMore than New York or any other place in the U.S., Boston is probably the place where the two industries - traditional and alternative asset management - most closely co-exist. We asked ourselves, and the distinguished participants of this Roundtable: \"how do these sectors converge, confront, or cooperate? What is the relationship?\", and found surprising answers.\nWere you aware that the majority of hedge funds within the Bostonian alternatives community did actually NOT come from Fidelity, State Street or any of the myriads of large, established asset management organizations, but from somewhere completely different?\n... details on a joint venture between a hedge fund and traditional firm \u2013 a model for the future?\n... insiders' views about what really happened on May 6th, the day the Dow dropped 1000 points.\nThomas Callahan, CEO, NYSE Liffe U.S.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As many of you may know, I spent most of my life living in Cambridgeshire, although towards Huntingdon and only recently in Peterborough. In fact Creative Content Company was originally set up in Hartford, just outside Huntingdon, before moving to Peterborough when the company was just one month old!\nAnyhow, that has meant that I still have links (and heart strings), that pull me towards Huntingdon and the community and schools in that area. When I was invited to St Peters School I was unsure, I went to Hinchingbrooke School as a child, and if you are from that way you'll know the two schools have a friendly rivalry between them\u2026 so going to the school of the 'enemy' was a big step!\nHowever, I did it and have never looked back. It was a superb Careers Day and actually one of my most favourite of all career days that I have been along to. It was really well organised and the students hadn't been forced to go along \u2013 they wanted to.\nThe organiser, Laura, had chosen one business from each industry and then each student choose three industries they wanted to speak to and were then giving a form telling them the order in which they would see each industry.\nAs the students entered the room they quickly found the first table they were due to sit at and there were between 3 and 8 students per table, with the person from that industry. For that day I went as a promoter of Entrepreneurs and although what I did was discussed we went into details of how they could become entrepreneurs and thoughts they had for their futures.\nTo break the ice each table had a list of questions they could ask and then they could throw their own questions in too. I feel that because they were such small groups of children it was easier to ask questions and have a real chat that was beneficial to them and their ideas. I saw 7 groups that day and each conversation was completely different because they felt able to ask their own questions.\nthe qualifications and grades I had (a written down question on the list) and how much I earn (their own question).\nI am always keen to quash the rumour that if you don't get those amazing grades of As and A*s you will become nothing, so being able to be a 'real person' to be able to confirm this, instead of using the example of Richard Branson being dyslexic was a brilliant opportunity.\nAs much as I may be ashamed to admit it due to being an ex-Hinchingbrooke School student, I really enjoyed my time at St Peters School. The event was really well organised and the students were just superb. I may even be tempted to go back and work with them again \u2013 just don't let my old school know!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I've thought a lot, probably too much, about what it feels like begin a new project. When fueled by passion, the spark of inspiration can grow quickly into a roaring fire. Those moments are often unforgettable, but it's critical to remember what matters most is the work ahead. Inspiration is a pointless, momentary high unless you follow though.\nI was following though on the goals laid out in early 2016 when I recommitted to work as a full-time consultant. Or so I thought.\nWithin six months of moving, my personal life changed drastically as the genetic condition I've battled daily finally caught up with me, creating symptoms I could no longer ignore. Instead of pouring energy into finding customers and re-launching my consultancy in a new city, I was searching for doctors instead. It wasn't long before the hours required to operate as a one-man client-services agency became too much.\nI stopped talking about projects in public because I didn't know when, or if, I could finish them.\nNow in 2017, my recovery is well underway, but the lessons I learned while trying to handle client requests and manage a grueling medical schedule have taught me much. In fact, they're baked into our founding mission here at Uptime Ventures. These ideals are evident in our focus on internal projects, honing skills and products for small communities through continuous delivery before releasing our ideas into the wider world.\nVarious in-progress renditions off the Uptime Ventures arrow logomark in black, red, green, blue, and finally, the official yellow.\nWhile the expertise I've developed over years of consulting on diverse hardware systems running a plethora of languages will certainly be available to our clients, Uptime Ventures' main focus will be to find and exploit markets where applied technology can be maximally effective, choosing instead to operate as a trusted partner rather than a purely contract development team.\nWhether it's the mountain air, open skies, or something altogether unrelated; like others who came before me, I feel encouraged to dream bigger in the west.\nDuring the darkest period of recovery, I began reading the works of noted futurist Ray Kurzweil. I assumed that futurism, if anything, would offer hope and intellectual stimulation to my otherwise bored and depressed mind. I was right, though the ideas which sprang from my reading were unexpected to say the least.\nIn The Singularity is Near: When Humans Transcend Biology, Kurzweil observes that, \"Most short range forecasts of what is technically feasible in future time periods dramatically underestimate the power of future developments.\" In other words, because the greatest technological developments occurred in our relatively recent past, we tend to think the pace of innovation has always remained steady. In reality, history is full of many periods of intellectual stagnation. Kurzweil's work stands in stark relief, as he chooses to focus on what is referred to as the \"historical exponential,\" view of innovation, which plots major technological accomplishments within a comparatively recent, accelerated timeline.\nViewing modern history from Kurzweil's perspective, it becomes obvious we are living in a period of unprecidented innovation, beginning in the year 1900 and extending into the distant future.\nA core tenet of our mission at Uptime Ventures is that how and where technology is implemented is perhaps more important than the technology itself. With this purpose, I've created an agency designed to serve future needs. This belief also leads us to be a decisively multi-disciplinary firm; one that seems to seemlessly pair technical know-how with cultural awareness, thereby allowing us to safely explore the future of tech when combined with concepts from the arenas of medical, science, and artificial intellgence.\nIn the following weeks, I'll continue sharing more of the Uptime Ventures story, including projects that are being designed and built in-house, and also the open source work that has been contributed under the company banner. I hope you'll join me by subscribing via RSS or email. If you're so inclined, you can also follow us on Twitter, or like Uptime Ventures on Facebook.\nI'm excited about where this journey will lead, even though I don't know the way. The software industry has undergone many changes before now, and I sense that it will soon change again. Hopefully, that shift will carry us towards a brighter, more diverse future, of which Uptime Ventures intends to be a part.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It does not matter if or not you would possibly be searching for for brand-new sneakers, adidas sneakers on the net exercise routine session equipment, or tranquil sweatsuits, the Adidas sale has something you want. Now by way of November 27, it can prevent as a number of as on decide sneakers and garments. The sale abilities well-known variations akin to the Adidas Extremely Reinforce, AlphaBounce, NMD, Deerupt, on top of that a terrific deal additional.\nAdidas is seemingly heading for subtly employing this sort of range.adidas sneakers sale There are actually not any depictions of dragons or swords or Peter Dinklage visages adorning the trainers teased to this point. Pretty, Adidas seems for remaining dealing with coloration scheme to allude to point themes as well as in addition the fictional best relatives customers associates or homes - Belongings Stark, House Targaryen, etc. - while in the gathering. The beneath sneakers from Adidas reference Family Targaryen -- renowned and feared for its obtaining harnessed the toughness of dragons. Therefore, the fiery hues.\nJust soon after organizing colorways in August celebrating Goku and Frieza and two a lot more that functions Son Gohan and Cellular in September, Adidas carries on its range of Dragon Ball Z-themed sneakers with two new patterns surprised by Boo and Vegeta. Adidas Originals will launch its fifth and sixth Dragon Ball designs November 24 which has a tribute into the \"Majin Boo Saga\" building usage of the Kamanda and intensely Tech variants just like a canvas.\nFormulated in considerate collaboration with all the famous animated selection, Adidas \"Majin Boo Kamanda\" encompasses a pink Nubuck greater with perforated holes instead of the product is often a few stripes remaining an homage to all people today although while in the character is head and torso. A translucent rubber outsole evokes Boo is rippling trousers, and additional particulars around the tongue and gold foil emblem rejoice his abundant iconography.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TWO BRITISH WOMEN who went missing after a late-night swim in Spain have been found dead.\nSky News reports that Spain's Civil Protection Service said a search helicopter located the women off the coast of the Costa Brava, at the Lloret de Mar resort, 47 miles (75km) northeast of Barcelona.\nThe pair, aged 33 and 36, were with three other women on the beach. They were reported missing at around 4am after failing to return from the sea.\nTheir bodies were found at around noon today.\nA British Foreign Office spokesperson told Sky they are offering consular assistance.\nEmail \"British women found dead after going for late-night swim in Spain\".\nFeedback on \"British women found dead after going for late-night swim in Spain\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 6-day intensive program offers a deep, transformative experience of the Enneagram, focusing on the integration of psychology, spirituality and somatic. The program offers a full examination of the nine distinct type structures, with attention to the spiritual and somatic aspects of each type. Another key element of the training is how the types show up in relationships. The first three days include the basic proposition (psychology), the basics of self-reflection (spirituality), and body awareness (somatic).\nThe intensive program includes two rounds of type panels in the Narrative Tradition, an introduction to the Universal Growth Process, guided interactions related to current relationship issues, pre-recorded meditations by Helen Palmer, short didactics, self-observation and movement practices, and small-group exercises.\nFor more information, email Judy Tan at judytan@theenneagramsingapore.com or call +65-97815608.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you watched the Lion King (Rafiki the the monkey's tree) or Avatar (The tree of Souls) or spent time in rural Africa, the striking silhouette of a baobab tree at sunset should be familiar.\nIn fact, every part of the baobab tree is valuable - the bark can be turned into rope and clothing, the seeds can be used to make cosmetic oils, the leaves are edible and can store water and the fruit pulp is extraordinarily rich in nutrients. Women in Africa have turned to the baobab fruit as a natural source of health and beauty for centuries.\nBaobab is the only fruit in the world to dry naturally on the branch. This means the fruit simply needs to be harvested, deseeded and sieved to produce a 100% natural superfood powder that is exceptionally nutrient-dense - and super tasty!\nAs Baobab dries on the branch, its natural form is a 100% organic fruit powder.\nAduna Baobab Superfruit Powder has a delicious sweet and citrussy flavour. It is perfect for sprinkling onto cereal, granola, yoghurt or fruit or mixing into smoothies, juices or water.\nOur Baobab Raw Energy Bar is perfect for getting your daily dose of #baobablove on the go. Combing two teaspoons of baobab powder with cold-pressed pineapple and almonds, and free from sugar, gluten, wheat and dairy, its the perfect pick-me-up.\nAduna's Baobab Superfruit Powder and Baobab Raw Energy Bars were both winners at the 2015 Great Taste Awards: the first time a baobab powder has ever won this prestigious award.\nAduna's baobab products are best-sellers in leading health and beauty retailers around the world.\nOrder online now and we will deliver straight to your door. We offer free delivery for all orders over EUR 25 in Malta & Gozo. Shop now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"TORCZYNER-Harry. The World Federation of Diamond Bourses expresses its profound sorrow on the passing of its esteemed General Counsel, Harry Torczyner, and sends condolences to his family. Eli Izhakoff, President Gerson Goldschmidt, Secy. Gen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Catch The Entire Subtitled 13-Episode Series FOR FREE On Neon Alley Online Anime Streaming Platform!\nSan Francisco, CA, April 15, 2014 \u2013 VIZ Media, LLC (VIZ Media), the largest publisher, distributor and licensor of manga and anime in North America, brings the intergalactic action drama \u2013 GARGANTIA ON THE VERDUROUS PLANET \u2013 to its new Neon Alley FREE online streaming anime platform. The lushly animated 13-episode adventure was created by the famed animation studio Production I.G. and is available today in its entirety at NeonAlley.com. For $7.99 a month, U.S. fans can also access GARGANTIA ON THE VERDUROUS PLANET on mobile and digital devices through the subscription-based Hulu Plus Video-On-Demand service. The premiere of a dubbed version will follow later in the year.\nIn the distant future, Ledo, a young soldier from space wakes up and finds himself marooned on an ocean planet. Growing up in space, and knowing nothing but battle, he is a stranger in a strange land, full of odd customs and a foreign language, but he's not alone. Starting a new journey on this verdurous planet, Ledo must find a way to survive and reconcile the fact that he may never be able to return home!\nNeon Alley delivers the hottest in subbed and dubbed digital anime series, simulcast world premieres, exclusive on-demand access, and a massive legacy catalog of new as well as classic favorites (in HD when available). NeonAlley.com is a premier on-demand, online destination for streaming anime content in the U.S. The service is available for FREE online and also anytime, anywhere across a wide array of different devices, including PS4, PS3, Xbox One, Xbox 360, Roku, Chromecast, and a variety of other mobile devices through the subscription-based Hulu Plus Video-On-Demand service. Additional details on Hulu Plus-enabled devices can be found at http:\/\/www.hulu.com\/plus\/devices.\nAdditional information on Neon Alley is available at: www.NeonAlley.com.\nFor more information on anime titles from VIZ Media, please visit: www.VIZ.com.\nHeadquartered in San Francisco, California, VIZ Media distributes, markets and licenses the best anime and manga titles direct from Japan. Owned by three of Japan's largest manga and animation companies, Shueisha Inc., Shogakukan Inc., and Shogakukan-Shueisha Productions, Co., Ltd., VIZ Media has the most extensive library of anime and manga for English speaking audiences in North America, the United Kingdom, Ireland and South Africa. With its popular digital manga anthology WEEKLY SHONEN JUMP and blockbuster properties like NARUTO, BLEACH and INUYASHA, VIZ Media offers cutting-edge action, romance and family friendly properties for anime, manga, science fiction and fantasy fans of all ages. VIZ Media properties are available as graphic novels, DVDs, animated television series, feature films, downloadable and streaming video and a variety of consumer products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"our newest styles from bauxo, including the amy, zoe, and lisa! these lovely gals would make the sweetest (and most stylish) addition to this valentine's day!\nthis sweet, petite couple has been hangin' out in the store all day and we are just head-over-heels in love with them! stay tuned for more miniature moments . . .\nmake your own shadowbox necklace!\nthe fly london yepe is here!\njust in time for valentine's day . . . our best selling pillow!\na bit of snow is no problem for these northern fairies!\nour fairies won't let a little snow get in their way! they have been doing a great job keeping the walk clear for shoppers . . .\nwho you ask? cupid, of course! valentine's day is less than a month away, so start planning your romantic night and sweet little somethings for the special people in your life! and we believe this wonderful day isn't just for lovers . . . it's a great time to show friends, parents, children, grandparents and siblings that you care!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The interesting man with a plenty of positive qualities, but not deprived as well negative.\nWithout the family past and children.\nAge 26-31, growth from 1.80m, decent, reliable with sense of humour. The man, which wants and is capable to create strong happy family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shipping: Orders of all in stock items are normally shipped within 2-3 business days.\nReturns: If you have any problems with your order, please contact me as soon as possible. Your satisfaction is my number one priority.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Have you ever walked around a cemetery and noticed the dates with a tombstones? There include the dates the people were born, then long dashes and then your dates they perished. Those dashes are what James is talking about in this particular verse. He calls our life a vapor, something that appears and then vanishes in a moment. It's what's in-between the dates, the dashes, that tell about how you've spent living.\nIf you've received many sympathy flowers and cards from a group, much like your workplace or church or school, a person can send a short common give many thanks note into the organization and order them to present it to guarantee everyone look at it. Obtain send a new thank you note also known as a card every sender of sympathy flowers personally, if there undoubtedly are a few senders and you close all of them. As soon as you will the flowers you produces a note of selected and address, or ask someone have to be eliminated track with the name along with the address of the sender. A great deal more are comparatively free and in a better frame of mind, you can sit down and pen a message or easliy found . thank you note and send it to man or woman. People will appreciate it, though mention not really expect you to do it.\nI saw a lot of death as a minister. It sometimes was after the fact long enough to just bury somone in a great funeral service attire from a nice installing. Sometimes I found myself waiting on the side of a river while they searched for your lost much more taken along with morgue to roll the dead body of a young girl or friend out with regards to a drawer to have a private family look. I Even dug a grave once on the farm fat reduction waited household to arrive for incredible same day funeral and burial. I've picked in the cemains, ugh what a word, folks I had just spoken with a few days earlier, now reduced to about 10 lbs. of gray sand. I have transported the neatly wrapped body in the newborn option city on the backseat of my car, as the bride and groom could not afford for the funeral where you can find do the device.\nI would prefer not to be here, he explained to small. And for one shameful moment, he thought he said it out loud. But if he did, Aunt Laura, didn't hear the concept. Or she didn't care.\nThis place is as much smaller than I remember, he strategy. Scanning the stones, he counted six family plots, with anywhere between three and 7 stones per plot. A couple of the groupings each were tall obelisk jutting out of amongst them, and everyone of them had at least one white stone mix. An assorted collecting headstones many sizes filled out the remaining graves.\n\"That was beautiful,\" said a voice behind Tony adamowicz. He spun around, and there stood his Aunt Laura. She never looked more marvelous. Her eyes were simply radiant, and she or he looked to Tony become angelic.\nOFinally, be sure to read the obituary several times before sending it off to the newspaper to be published. This is how you proofread it and catch your grammar, spelling and punctuation errors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I used to hate when people told me grief would make me stronger. I didn't want strength, I wanted my dad. Yet, here I am. And here's how I transformed my life after tragically losing my dad.\nIt had been a fun Saturday with my boyfriend and his parents. We had gone to a basketball game at Madison Square Garden, indulged in a few cocktails at a fun rooftop bar, and were enjoying a delicious dinner at a swanky restaurant in the Meatpacking District of New York City. It was at dinner that I got a call from Victim Services. They told me that my dad had been skiing that afternoon and that he didn't make it down the last run of the day.\nMy entire body went into shock. I fell over the way a cardboard cut out would if you tipped it over- completely straight and with locked limbs. The restaurant staff supported me to exit through the back and my boyfriend's family helped carry me out.\nWe booked plane tickets and went to pack for my father's funeral, with seemingly every item I pulled from my closet soaked with tears by the time it arrived into my suitcase. Before I knew it, we were on a one-way flight to my hometown, where my biggest indicator of \"home\" was permanently gone.\nIt was honestly only a few days following my dad's death that an overwhelming urge to making something \"better\" of my life came over me. This surprised me. It was as if my dad departed briefly from the afterlife to visit me and planted a seed, an urge, inside of me, passing the baton from the risk-taker and visionary he was onto me.\nThe sentence itself is an oxymoron.\nBut that seed, that urge, that compulsion toward \"better\" persisted.\nAt that point in life, I was working as a coordinator at a non-profit. I lived with my boyfriend in New York City and we were happy. I had started a blog a few years before that was a creative outlet and an additional source of income around my 9-5 job. Things were good and I was happy.\nWhen my dad passed away, that urge grew into a complete obsession to transform my life. You'll often hear those that suffer from grief develop a new appreciation for life as they're shown quickly it can get taken from you.\nWhile I did experience this shift in perspective, I felt that my grief transcended a new appreciation. It manifested into a ruthless ferocity and utter refusal to accept anything for myself other than what made me feel alive. It shined a light on how frequently we as a society don't feel alive. How frequently we go through motions without any life in them at all.\nI started listening to fierceness within me closely, even though it didn't make sense. That ferocity to change my life was the most \"alive\" I had felt since my dad had died. It's no coincidence that following that feeling of life made the metaphorical seed within me grow. As it grew stronger, so did I.\nBuilt my website into a six-figure business.\nMore importantly, I got engaged to my boyfriend. We're planning our wedding in New York City, and I couldn't feel more confident about going into this chapter after having already endured such brutal trauma together.\nI know now that seed is, in fact, my dad. He is seeking expression to continue living through me.\nWe all have this from our parents, whether they're alive or not. That's what it means to be someone's legacy. To be his legacy is the honor and privilege that has kept me going through crippling grief.\nThe single greatest way you can honor yourself and your loved one who has left is by making the most of yourself. It doesn't mean that your sadness or grief will go away, but I have found that I don't want it to. It's your ability to keep going through the sadness that you become strong and better equipped to make something extraordinary out of your life.\nTrust me, I remember wanting to slap anyone across the face that told me that grief would make me strong. I didn't want to be strong. I wanted my dad. If you feel that way, that's okay. I still want my dad. That's what grief is.\nHonor that grief. It's your superpower. You can literally see through fear like the illusion and impostor it is, and you'll discover that you can truly live out whatever makes you feel alive on the other side. You've unlocked a strength within you that would make moving mountains seem simple.\nTake that strength to paint your life into the masterpiece you know it can be, and feel the presence of your departed loved one by doing so. There's a whole wonderful world out there waiting for you.\nIn loving memory of Paul Gervais. This is for you, dad!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Suits, hats and drinks were flowing at AC Bar for our showcase along side Goorin Bros and Santa Teresa Rum this past Wednesday. We had the opportunity to introduce our SS18 collection and raffle off a free shirt.\nOur friends from Santa Teresa Rum were pouring their special crafted drinks and Goorin Bros were showcasing their stylish hats, while we were presenting our new seasonal fabrics, design options and our new sartorial details. Within our sartorial details are our handmade made in Italy line, soft shoulder and our light weight canvas for a fresh summer look.\nWe had lady's and gents from all over town checking out our latest products and enjoying the night with friends. Thanks to all who made this night a success and for those who were able to come out. We are looking forward to our next event!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello there, I am Traveller \u2013Travel blogger, photographer, and Business Man by profession who's been exploring the world for over 2 years, with no plans to stop anytime soon.\nWelcome to my Adventurous blog! This is the place where I share entertaining stories, helpful travel tips, and amazing photography. It's a place for an individual like you who are searching for daily inspiration & motivation to live a life full of adventure and relaxing place where you can find your self in nature. Spending some quality time with nature.\nIn my early life, I was never a big traveler but was always excited to see places wherever my parents used to take me. When I started my early job life, I was exceptionally committed towards my work. Months passed on working and working and literally all my social life was to a halt. As time passed on, frustration started building. Thoughts like what the hell I was doing with my life. What was the use of all the earnings, if I can not use it for my own mental satisfaction?\nMost of the people are into a race of earning huge money and what they are using it for? Just to build a terrace to sleep!!! Work for years and years and keep paying bank loans just to build a house and buy some transport machines. I know many will not be willing from my perspective, but that's what my thinking has become now. I don't want to ruin my worthy life into all this. I am no more interested in saving just for collecting physical structures. I dreamed of more \u2014 a life filled with adventure. I want to explore this beautiful planet earth, spend time with different people and learn about their cultures, enjoy a variety of food and visit many places.\nI know there are various blogs to answer such queries but mine is a little different it shares experience, also help you find best of your locations. Many people avoid traveling as they think it would be very expensive and just one or two trips can ruin all their savings. Also if you check various packages provided by the travel agency, almost all are very expensive and not worthwhile at all. The main objective why I am writing all these blogs is to help people in finding out the cheapest way to manage their trips without pre-booking any tour packages and hotels at all. Also, I assure you that you don't have to compromise with the scenic beauty you are about to experience with such cheap trips.\nTo all my readers (especially those who comment regularly), this Adventurous blog wouldn't be what it is today without you. Thank you so much for reading!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"carries out customer centered service and regards quality first, service first and credibility first as its tenet.\nconsiders quality first, service first and credibility first as its tenet and is eager to cooperate and build strategic partnership with customers to keep pace with times together. we always offer you the best products and service wholeheartedly.\nMission: To create a win-win situation, is committed to building a stage for the mutual growth of as well as its staff, providing room for employees to realize their self value and dreams and earn fortune and social respect and letting all of us enjoy the success of our career and the happiness of work.\nEmployees are strategic partners of the company. advocates a loyal and cooperative spirit, emphasizes mutual efforts for common business and endeavors to make millionaires and billionaires and to achieve together in the future. It will try its best to create value for its share holders and clients.\nIntegrate culture with products, create a brand with quality products and win market through nice brand; only along with the sustainable growth of personnel can the company be developed sustainably.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Fighting for a major organization has made John Lineker ineligible for regional rankings.\nJohn Lineker is a Brazilian flyweight fighter currently signed to the UFC. He is a veteran of Shooto Brazil, Jungle Fight and the UFC. He has held bantamweight titles in Nitrix Fights Grand Prix and Jungle Fight. Although he has six losses on his record, five of them came when he took eleven fights in the first fifteen months of his career. Of his twenty three wins, fourteen have been finishes including eleven T\/KOs. Before his debut in the UFC, he was riding a thirteen fight win streak. John has missed weight for three of his UFC fights. He came in at 127 pounds for his debut against Louis Gaudinot, 129 pounds against Jose Maria, and 128 pounds against Phil Harris.\nLineker began his career fighting on the regional circuit in his native Brazil. After a rocky 6-5 start, he would go on a three year, thirteen fight win streak which earned him a spot in the UFC. He was tapped as one of the initial members of the UFC's flyweight division. He would make his promotional debut against TUF 14 competitor, Louis Gaudinot, in what would go on to earn 'Fight of the Night' honors. Lineker and Gaudinot spent many various moments throughout the fight in wild punching exchanges, until Gaudinot choked the Brazilian to sleep in the second round with a guillotine. Lineker would get back to his winning ways by winning three straight over Yasuhiro Urushitani, Azamat Gashimov and Jose Maria Tome proving that he could be a force in the flyweight division.\nJohn took on Phil Harris at UFC Fight Night 30 in Manchester, England and was able to dispatch the Brit in the first round to earn a TKO victory.. This was his fourth UFC fight in less than a year. Lineker again missed weight for this fight marking the second consecutive and third overall UFC fight that he has not made weight. It is unclear what, if any, the consequences will be for his repeated weight misses. He would take on Ali Bagautinov at UFC 169 in what was likely to be a number one contender fight. He would lose to the Russian by unanimous decision.\nLineker bounce back with a TKO over Alp Ozkilic in the third round. He would follow that up with an upset over Ian McCall picking up the Unanimous Decision at UFC 183. Lineker would miss weight for the fourth time against McCall prompting Dana White to say that Lineker will never fight at flyweight again he will have to move up to bantamweight.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Latest episode of Ballers written by - wait for it - Rashard \"Socrates\" Mendenhall!!!!!!!!!!!!!!!!!!!!!!!\nI remember they approached him a few years back for some work.\nGood for him. He was a decent player here, glad to see a football player actually having a career when they are done with playing.\nHe was good until he quit on the team.\nYeah, I didn't realize that.\nBut he was actually credited for this show (the script), so not just a contributor. And it was a pretty good episode.\nThe guy realized football was not the life for him, so he moved on with his life's work. It's not like he pulled a Vontae Davis or anything. Good for him.\nHe had like 3-4 good years, was a starter on a SB team. Nothing to be ashamed of.\nDon't understand the hate for him. Really had one bad play where I think Kemo or one of those replacement level OLinemen got blown up in the Super Bowl. Yeah his passion wasn't football, but he wasn't a distraction either. Good for him anyway.\nWell that's all relative. Compared to the shitstorm goin on now w\/ the Steelers, Socrates looks like a choir boy.\nWriting credits are very, very negotiable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Blumine Island Battery Historic Area.\nCopyright: NZ Historic Places Trust. Date: 11\/12\/2003.\nThe historic area encompasses the northern half of Blumine Island where the northern battery and associated features were concentrated.\n4) Second accommodation camp comprising approximately 217 concrete steps, 20-30 concrete foundations of accommodation and ablution buildings, and two large water tanks).\nBuilt as one of a complex of installations to protect the Marlborough Sounds, Blumine Island Battery sits in the middle of Queen Charlotte Sound and its guns cover the northern entrance. The Battery was built by the Public Works Department, with a scow delivering the cement directly from the Tarakohe Cement Works to the site. After World War II the island was farmed until the 1960s.\nDuring World War II New Zealand made provision for the defence of a great range of places and Blumine Island is representative of the kind of close defence batteries requested for shipping channels anchorages, as well as ports . \"This relatively remote site was erected to defend the outer portion of Queen Charlotte Sound and was the key defensive installation for the Sound. The two emplacements were augmented with magazines, engine room and observation posts, and there were two accommodation camps, water reticulation, roads and a wharf. Although a number of the buildings have gone, the island illustrates the kind of the complex needed for a remote location defence installation\" .The batteries on Blumine Island are the most extensive of the Marlborough Sounds Coastal Batteries and they are regarded as the best in terms of their integrity .\nThis battery was to be the key defence for Queen Charlotte Sound. The technology used at Blumine Island is not exceptional but it remains very significant for its integrity and condition. Although some associated buildings have gone, all the concrete buildings, including the emplacements, are intact and in reasonable condition. There also remains a remarkable set of concrete steps, approximately 217 steps high, which climbs to the site of the second accommodation camp. The walls of some of the remaining buildings contain graffiti, some contemporary and including, interestingly, sketches of the above-mentioned steps. The amount of graffiti has been restricted due to the remote location of the batteries remote location. Of added significance is the manner in which the battery was vacated by the army. It carefully capped the gun mountings, in contrast to a number of other sites around the country where it simply removed the guns and abandoned the site .\n\"Although remote, Blumine Island - a scenic reserve - is visited occasionally by the public and is relatively well known in the region. In recognition of its significance the Department of Conservation maintains this site and intends to protect the structures against further degradation\" .DoC has an information sheet available to the public. Yacht charter companies have information for visitors on how to moor and visit the island and its historic features. Punga Lodge, Endeavour Inlet has information on the island on its notice board.\nThe British military recommended establishing major coastal defences in the Marlborough Sounds in late 1941. Plans to provide a secure anchorage for the US navy in Queen Charlotte Sound began after Japan entered the war in December 1941 . It was also contemplated for training allied forces in marine operations for further campaigns . These batteries are part of New Zealand's military history and exemplify how provisions were made for the defence of a great range of places. Blumine Island is representative of the kind of close defence batteries requested for shipping channels, anchorages and ports.\nThe batteries were to play purely an anti-invasion role: to deny to the enemy the use of the sheltered waters as anchorages. The War Cabinet approved a provisional estimate of \u00a3220,000 in February 1942 for the work. This was for two 6-inch guns each covering the Pelorus and Queen Charlotte Sounds and Tory Channel. Each site chosen was to have a camp for 31 men, a BOP, a command post and a war shelter . The final figure varies between accounts. Cooke in his book on defence sites in the Sounds puts the cost to build all the sites in 1942 to be \u00a3450,150 . However, the military history of New Zealand defence force records the amount for the total cost for these coastal batteries to be \u00a3230,150 .\nA wide range of underwater defences - minefields, booms and submarine detection devices - were added to the scheme from April 1942. No searchlights were intended and originally some anti-aircraft guns were allotted, but soon withdrawn. These were to be twelve 40mm Bofors guns and were allocated in July. A reconnaissance of the Sounds was carried out in December 1942 to site them but the number available was thought to be inadequate. They were also later deemed unnecessary, as camouflage was good at 1.219km (4000ft) and LMG AA guns were issued instead .\nBlumine Island in 1942 was farmed and had little tree cover. Consequently, particular attention was paid to camouflage for air defence. \"Past experience has shown that little or no attempt has been made to disguise the fact that work of a special nature is proceeding,\" the QMG quibbled in February 1942 . After the war, it was decided that Queen Charlotte Sound was to be regarded as an extension of Wellington Harbour in wartime. The Chiefs of Staff concurred with the Defence of Bases Committee and gave the Sounds a Category 'C' rating, meaning only mobile defences would cover the anchorage .\nThe defence service describes the construction of the gun emplacements and accommodation buildings in the Marlborough Sounds as one of the \"most difficult defence works undertaken in the northern portion of the South Island\" . This was due to their remoteness and inaccessibility. For example, all materials for Blumine Island had to be delivered by sea. The Defence Department hired two scows and a number of punts for this purpose . However, the island's very steep slopes and soft rock caused delay in the construction of the batteries.\nThe buildings of the batteries in the Sounds was authorised in February 1942 and work on the gun emplacements began immediately. The building work was carried out by the Defence Department's own forces and was not finished until December 1942 . The guns had been mounted on 17th and 19th June 1942 and were test fired on 8th March 1943. The guns and concrete foundations were recorded to stand up to the proofing range 14.63km (16,000yds) satisfactorily. The Blumine emplacements each had a 6-inch BL MkVII gun on a naval P.III mount, 156-160 ft ASL, range c13.5km .\nThe nature of the work commissioned for two sites on Blumine Island was the construction of gun emplacements (and associated buildings) and the provision of accommodation for the battery personnel. The two on Blumine Island were built as part of series of coastal batteries in the Marlborough Sounds. The others were Maud Island and Post Office Point, Pelorus Sound, Tory Channel and Maraetai .\nThe jetties and roads constructed during the work improved access to the island. Other services for the forces positioned at the batteries were also built - water supplies, sewerage, drainage and electricity from generating sets. By October 1942 the erection of accommodation buildings was able to proceed. Most of this was undertaken by contract, under the master schedule system and the work was completed by February 1943 .\nThe photo (p7a) shows the difficulties of building the battery at Blumine Island East. When this photograph was taken, 8 November 1942, the gunners of 84 Hy Bty had to be ready to use the gun at any time, even though the protective overhead cover was still in the process of being built. Also visible is rudimentary camouflage and the steep terrain . The site now looks very different from its original context as it is surrounded by native bush. On a site visit by the Trust in April 2003 it was noted that three of the soldiers who had assisted building the second gun emplacement had inscribed their names in the cement on the gun mount. A diary of the installation of equipment describes the difficulties of delivering gun 1438 to Blumine Island. The tides did not allow the scow to go along side the breastwork so it had to be returned to Picton. The following day the gun was able to be unloaded but unable to be mounted as it had no pedestal, which was brought the following day. The gun placed at the eastern emplacement was number 1770 .\nThe 84 Heavy Battery had its headquarters in Broadway Street, Picton. Captain H.G. Thomas was in charge of overseeing the military input into the public works construction of the permanent emplacements . Two officers and up to 20 men of the 84 battery usually were stationed on Blumine Island. From late 1943 the number dropped to one non-commissioned officer and seven other ranks as most of the day to day work was the monitoring of ship movements .\nAfter the Battle of Midway in June 1942 the government decided that the threat of enemy invasion had declined and from September 1943, the battery was designated as for \"care and maintenance'. On October 21st 1943 the both guns at Blumine Island were dismounted and sent to Picton where they were later shipped to Auckland Naval Reserve . The War Assets Realisation Board removed most of the structures during 1944-45 and Blumine and the other batteries were formally disestablished in December 1945 .\nBlumine Island had been declared a Pilot and Signal Station Reserve in 1865 but thereafter used for sheep farming . In 1912 a 113-acre scenic reserve was established at the southern end of the island (not the area in which the defence batteries were built). After World War II and the batteries had been disestablished the army passed the land to the Department of Lands and Survey. It is now administered as a Scenic Reserve by the Department of Conservation . As part of its active management DoC organised specifications for the repair and maintenance work to be carried out at the two observation posts, magazine and gun emplacement .\nMarlborough Sounds Resource Management Plan Vol. 2 Rules, February 2003.\nKerry Neal, Nola Leov, 1999, The Price of Vigilance: the building of Gun Emplacements in the Marlborough Sounds, 1942, Privately Printed: Nelson.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The great value 2-Park SeaWorld and Aquatica Ticket offers unlimited admission to both SeaWorld\u00ae Orlando and Aquatica\u00ae for up to 14 consecutive days. This great value ticket costs little more than a one day ticket with the added bonus that you can visit both parks as many times as you like for the duration of the ticket.\nThe NBA has it all - the most attitude, the brightest stars, the biggest dunks and even the most highly charged fans!\nIf you are visiting Orlando during the NBA season, which runs from October to April, a trip to a basketball game is highly recommended and a memorable way to spend an evening or afternoon.\nYou don't have to be a basketball fan or even a sports enthusiast to enjoy taking in an Orlando Magic game at one of North America's best arenas! And with your transport taken care of, all you have to concentrate on is the action taking place on and off the court.\nWonderWorks is an amusement park for the mind and features over 100 interactive exhibits for visitors of all ages to experience.\nHere you can participate in some of the most imaginative displays and exhibits found in any facility in the United States starting with the inversion tunnel to be turned upside-down!\nUpgrade your WonderWorks ticket to include the amusing Outta Control Magic Comedy Dinner Show. Orlando's top-rated dinner show is located inside the WonderWorks building and combines high-energy comedy improvisation, spectacular magic and lots of audience participation to thoroughly entertain audiences of all ages.\nGet ready to embark on a journey into the world of Gator Gardens, a fictional rival park that takes over a swamp buggy business from a new Gatorland character named Swampy Sal.\nFor over half a century, Gatorland has been creating fond holiday memories for millions of visitors who have entered through its gaping gator mouth entrance.\nWith your Gatorland Admission Ticket you can enjoy a full day's admission to Gatorland Orlando. The 'Alligator Capital of the World' is set in 110 acres and showcases an amazing display of alligators and crocodiles and also features a number of fantastic family shows including the unique alligator wrestling show and the world famous Gator Jumparoo.\nIn addition to all day admission to Gatorland, the Gatorland Zip Line ticket includes the 2.5 hour 'Screamin' Gator Zip Line Experience'. It's not just a ride; it's an adventure that takes you on a thrilling journey along 1,200 feet of high wires from seven platforms designed to merge with Gatorland's natural and scenic habitats.\nEnhance your Gatorland experience by upgrading your ticket to include the all new 'Screamin' Gator Zip Line'. This one-of-a-kind ride takes you on a breathtaking journey through the park along 1,200 feet of high flying thrills at speeds close to 30 miles per hour. It's one wild adventure!\nHave you ever held a baby alligator? Well here's your chance! How about a backstage tour that gets you up close and personal with some of the largest alligators and crocodiles? You'll get to do that too!\nYour day begins meeting some of the smallest members of the Gatorland family. Next, is a hands-on meeting with some of the reptiles and creepy crawlers from the Upclose Encounters Show.\nJoin the Trainer for a Day program and get a glimpse into what it takes to be able to work with, and around dangerous animals.\nIn addition to all day admission to Gatorland, you can now enjoy a Single Zip Line experience over the Alligator Breeding Marsh. That's one wild ride!\nSoar 350 feet through the air and defy gravity while gliding over the Alligator Breeding Marsh, complete with one hundred and thirty giant alligators watching from below. Dubbed Gator Gauntlet, this is the only zip line available to guests with mobility challenges and other disabilities.\nThe ICON Orlando\u2122 Fly & Dine is a play-at-your-own-pace kind of package, from the views to conversations around the table with friends.\nIt starts with the wheel, but the fun keeps going around and around long after the ride.\nRelax and enjoy the best views of Orlando from 400 feet in the air, and enjoy a meal at your choice of fast casual or full service dining all in one package. The observation wheel takes guests on an aerial journey of Central Florida, offering 360-degree breathtaking views of nearby theme parks and attractions, downtown Orlando and Kennedy Space Center on Florida's east coast.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A much-needed resource for the long-distance dads and dads-to-be who are seeking information and advice on how to be involved with their family before, during, and after their deployment.\nUnfortunately, there are very few resources for military (and deployed civilian) fathers, who are looking for guidance on how to be in close touch with their families back home. The Military Father, written by the country's leading authority on fathers and families, fills that gap, providing deployed dads with everything they need to know to stay (or become) involved with and connected to their family regardless of the distance that separates them.\nPart I, Pre-Deployment of this essential sourcebook explores the profound effect a dad's absence will have on his spouse, his children, and himself. It also provides extensive pre-deployment checklists and detailed tips aimed at preparing the dad and his family for separation and long-distance communication.\nPart II, During Deployment, explains how to stay involved and connected when you're far away, and includes specific strategies and activities designed to help dads and their family remain close across time and distance.\nPart III, Coming Home, offers advice on post-deployment times from preparing to come home and surmounting the challenges of returning to military or civilian life. In addition, the book includes a chapter on how dads can support a spouse when she's the one being deployed, and a comprehensive listing of resources available to soldiers and their families.\nA former Marine, Armin A. Brott has devoted the last 15 years to providing men with the tools, support, and knowledge to help them become the fathers they want to be\u2014and their families need them to be. His seven critically acclaimed books for fathers have sold well over a million copies. He has written on fatherhood for hundreds of newspapers and magazines and is a frequent guest on such television programs as the Today Show. He lives with his family in Oakland, California.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Can I Print my NewPanda Newsletter?\nHome \u2192 NewPanda \u2192 eNewsletter How-To's and Help Topics \u2192 Can I Print my NewPanda Newsletter?\n6.17. Can I Print my NewPanda Newsletter?\nPlease visit our Knowledge Books page on printing email messages.\nNewPanda > eMessages How-To's and Help Topics > Can I Print My Email Messages?\nNewPanda > eNewsletter How-To's and Help Topics > Can I edit the NewPanda- or broker-provided content for my eNewsletter?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A previous paper presented long term bleaching data on various glasses exposed to 10.6 krad of ionizing radiation. All the glasses mported except FK 51 have readily available \"G\" glass equivalents that are stabilized to the natural space environment. Yet, FK 51, because of its location on the Abbe diagram is extremely useful in certain lens design applications. To more fully explore the bleaching ofFK 51, after the initial dose of 10.6 had at 1 1.8 krad\/hour, we inadiated three more samples at a similar dose rate but to different total doses. Since the dose rate for this study was significantly higher than the dose rate anticipated for glasses in a shielded space-based lens system (-3 md\/day), additional data wei obtained at a lower rate of 7 md\/hour. While this dose rate is still higher than the anticipated operational rate, it is more than 1000 times lower than the dose rate used for our initial studies. The bleaching rate for the samples exposed at the lower dose rate is considerably less than for the samples exposed at the higher rate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AirBNB vs Agoda: Where Should You Book Your Next Room Rental?\nWhenever I travel to a new city, I find myself browsing through various online booking apps to find an affordable, clean room in a safe part of the city.\nThanks to modern technology, you can find rooms at the tip of your fingertips using your smartphone. My two favorite online booking apps are: AirBNB and Agoda.\nI used to use Booking.com in the past but I use Paypal for the majority of my rental bookings. I still think Booking.com is a good choice but Agoda accepts Paypal and offers a great affiliate program where you receive free award bonuses based on number of bookings.\nIn this article, I'm going to compare AirBNB and Agoda based on price & fees, ease of use, payment options, and overall value.\nBoth Agoda and AirBNB charge service fees to use their booking platforms. This is one of the most annoying aspects of using a 3rd party platform (it's often much cheaper to book a room directly from a real estate agent or in person).\nSome hotels, condos, and apartments offer cheaper walk-in rates if you pay in cash. Sometimes, the walk-in rate is more expensive than the online quote. Shop around a bit and call up the hotel\/condo to ask about the walk-in rate to see if you can save some money.\nIf not, then you'll want to use a 3rd party app.\nOn average, you'll spend around 10% extra using 3rd party booking sites. The 10% service charge is the standard way booking platforms generate revenue for their shareholders.\nHowever, Agoda is cheaper because many AirBNB hosts charge a cleaning fee in additional to AirBNB's service charge. The cleaning fee is usually between $5 to $20 depending on how big the room is. Cleaning fees are annoying, but it's standard practice nowadays since most hosts either clean the unit themselves or hire a maid to do the work. There is no hotel\/apartment staff to handle this job for them.\nOverall, those extra AirBNB fees add up and Agoda is often much cheaper per night than using AirBNB. The only time AirBNB is cheaper is when you book for a long period of time (1 week or 1 month) or the host offers a first time host discount to attract guests.\nAgoda not only offers a larger amount of cheaper rooms, but has a partner program where you receive cash back on your bookings. You can join the Agoda Partner program to start saving additional money on your bookings.\nAirBNB and Agoda both have excellent search and matchings features that help you find a good room. AirBNB has very good filtering options (kitchen, WIFI, washer) if you are looking for specific amenities in your room. The user interface is pretty simple and you can view lots of photos and details about a room from the main search results.\nAgoda doesn't have the same filtering options, but they allow you to easily filter rooms according to \"Lowest Price\". It's super easy to view all the prices starting from lowest to highest according to your budget. One nice benefit is that Agoda offers cheaper prices on their App. You can save an extra 1 to 2% off your booking if you download their Android\/Iphone app. However, Agoda often makes mistakes with room amenities and the pictures sometimes look nothing like the actual room when you arrive. There is plenty of false advertising on the Agoda App.\nOverall, AirBNB offers more precise filtering options and what you see is what you get. It's easier to book a room on AirBNB and end up satisfied with your results. Agoda is good, but you will sometimes arrive and not receive the amenities that were listed. Just recently, I booked a room through Agoda that offered Cable TV but when I arrived, there was no cable installed.\nBoth sites accept major credit (Visa, Mastercard and American Express) as well as Paypal. However, you must go through a lot of verification measures using AirBNB if you use Paypal. I've had to email their support staff statements of my Paypal account before they accepted my bookings. I've never experienced this problem with Agoda.\nPerhaps AirBNB experiences a lot of fraud or they enjoy making it more difficult to take my money.\nBoth sites offers excellent room options but I think your preference will depend on the length and nature of your stay.\nLong term rentals: AirBNB is best suited for rentals longer than 1 week or so. Hosts offer discounts if you book a long term stay.\nIt's hard to decide between both sites because they are both excellent in their own ways.\nFor tourists, Agoda is a very good option and you'll find rooms in new city with ease.\nFor long term expats, AirBNB is a wonderful resource for finding a condo\/apartment and staying in a modern-type room without dealing with long term contracts\/leases.\nWhich do you prefer: AirBNB or Agoda? Leave your thoughts in the comments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was one of those rare situations that hardly anyone tackled.\nNot Blackhawks fans, but believing fans that the Hawks would not have won the 2015 Stanley Cup without the emergence of their backup goalie in the Nashville series. They wanted their child from his hometown to be paid and they wanted to see him reach the peak of his career. Not supporters of the Hurricanes, but believing fans that if their team could be content to keep the average goalkeepers of the league, she would attend the playoffs for the first time in nine years. Certainly not the organization that has an investment of $ 16.6 million over four years in the modified commercial clause in this area.\nNo, no one was rooting against the well-documented Comeback Story. Then everyone is back on board for Comeback Story II.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apr 18: Who will succeed Carney to run Britain's central bank?\nApr 18: Meanwhile: Forget the Easter Bunny. Let's Celebrate the Easter Bug.\nApr 18: He Built, Then Nearly Broke, a Successful Beauty Start-Up. Can It Go on Without Him?\nApr 18: Cannabis Watch: Cannabis stocks rally as Canopy secures future foothold in the U.S.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Norwich Business Group (NBG), which this year celebrates its 21st year of running monthly business networking events in the city, is putting on a Charity Dinner Dance in aid of two worthy causes \u2013 Nelson's Journey and Action for Blind People.\nThe black tie event will be held on Friday 19 October from 7.30pm at The Mercure Hotel, Boundary Road and will feature live music, sponsored by Steeles Law, and an auction, overseen by NBG committee member Nigel Morgan of Spalding & Co.\nTickets sales for the event are going well and NBG are looking for individuals and businesses to take the final few tables. Priced at \u00a325 each, tickets can be purchased from Nelson's Journey (01603 431788), Action for Blind People (01603 455767) or Steeles Law (01603 598000).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find a dentist that can see you immediately!\nChoose The Right DENTIST is a professional network of dentists. We can help you find the PERFECT dentist close to you! We are a free service that has been helping thousands of patients find a gentle dentist since 1991.\nIf you are searching for a KIND & GENTLE dentist for yourself and your family, we are happy to help Choose The Right DENTIST for you and your family that will be the perfect fit! We can also help those with emergency dental needs.\nThere is NO CHARGE for this referral service, and NO internet forms to fill out!\nMost dental insurances accepted and CareCredit financing is available!\nThank you so much CTR DENTIST. I had a toothache and I'm new to the area and didn't know which dentist to choose. You connected me with a great office and I look forward to a long relationship with them. Thanks Again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gathering the photographs is an emphasize of my week, and the series is well-received by blog visitors. However, some commenters made an important point: the vast majority of the projects spotlighted are construction-based.\nDo you know what? They're right. That's an inherent flaw in my own Extreme Makeovers show. It only includes real tasks \u2014 the ones that provide a permanent transformation to places like baseball diamonds, community areas and historical cemeteries.\nI'm a fan of construction-based tasks; my very own included making a wooden walkway and benches. But i am aware that numerous jobs can't be depicted with before-and-after pictures.\nExamples of these generally include preparation and performing a blood drive, organizing a communitywide event, producing and circulating a security pamphlet, and a whole lot more.\nFor evidence that non-construction projects could make a large difference, aim to Shane Uribe. The California Eagle Scout won this year's Glenn A. and Melinda W. Adams National Eagle Scout provider Project of the season Award.\nThe thing is that, Shane invested his early many years in several foster houses. He transported most of his belongings from place to invest a black trash case.\nNot wanting other individuals to endure that same fate, he planned a duffel-bag drive. He loaded 100 top-quality duffel bags with covers, flashlights and stuffed pets. No trash bags of these children.\nTheir project was such successful that north park County decided to allow it to be a permanent system. Today additional than 100 kids will benefit from Shane's project.\nNo place in Boy Scouts of America's assistance with Eagle Scout projects \u2014 maybe not in Guide to Advancement, perhaps not into the Eagle Scout provider Project Workbook \u2014 does it mandate that Eagle jobs be construction-based or permanent.\nThat renders an intentionally wide range of prospective tasks, and it includes jobs that are construction-based and projects that aren't.\nWhile there are principles \u2014 no projects that advantage the BSA, no fundraisers, no routine labor \u2014 there are many more projects a Scout can do than people he can't.\nIt is necessary to not categorically decline tasks that, on top, may not seem to need enough planning and development. Consider, like, a blood drive. Often refused beyond control, this task, if done properly, could possibly be appropriate. Couple of would matter the beneficiary. Blood banking institutions save lives\u2014thousands of them: maybe yours, maybe that a family member. In the event that candidate proposes to use a collection of \"canned\" instructions from lender, implemented with no additional planning, the look effort wouldn't normally meet up with the test.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Need some extra help adjusting your training plan for your hectic schedule or outlining your season? Want to learn the rationale behind the training plan design? Do you have extra time to train and want to discuss custom revisions to make your plan better?\nWe offer 30-minute phone consultations for $75. Just share your TrainingPeaks info with Coach Frank, [Enter his email frank@fascatcoaching.com under account > Add Coaches], in TrainingPeaks and we can work with you to get you on track for a stellar season!\nWe offer 30 minute phone consultations for $75. Just share your TrainingPeaks info with Coach Frank, [Enter his email frank@fascatcoaching.com under account > Add Coaches], in TrainingPeaks and we can work with you to get you on track for a stellar season!\nCall us Monday-Friday, 9-5 MST. Please leave a message if we can't answer right away. Please be in front of your TrainingPeaks for the call.\nI have just recently purchased FasCat's Mtn bike Interval training plan and had questions about incorporating some planed races scheduled during the same time period. I wanted to race but also follow the plan so I could be at my best for my semi local event, I also had questions about my training history as I had just been \"freelancing\" and wanted to make sure I had \"done the work\" so I could handle the interval program. Frank was able to look at my racing schedule and past training through my training peaks account and told me how to \"tweak\" the plan to incorporate the races. To me, the training plan along w\/ the phone Consultation is probably going to be the best money I've spent for performance gains as now I feel I have a good solid plan to follow and not worry if I'm doing the right training..\nI have used several FasCat training plans and loved them for their simplicity and the results I got. I recently was nearing the end of one plan and ready to start the next. I had questions about how to incorporate some races into the plan, figure out how to rearrange some workouts and just needed to chat about my overall plans for the season.\nI used the phone consultation and Frank was able to take care of the changes plus answer all my questions and give me some tips for my upcoming race!\nSuper helpful! I have a weird work schedule and Frank really helped me customize my training plan to accommodate that. He was also helpful in answering questions about workouts and power meters.\nVery helpful and attentive. Was able to direct me to make the best use of the investments in plans already made and make a mid-season pivot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Grand Theft Auto 2 by Rockstar Games - Should I Remove It?\nShould I remove Grand Theft Auto 2?\nGrand Theft Auto 2 is a software program developed by Rockstar Games. The primary executable is named gta2.exe. The setup package generally installs about 11 files and is usually about 384.51 MB (403,189,999 bytes). Relative to the overall usage of users who have this installed on their PCs, most are running Windows 7 (SP1) and Windows 10. While about 48% of users of Grand Theft Auto 2 come from the United States, it is also popular in United Kingdom and Canada.\nDmavideo.dll - The DMA Direct Draw screen and mode handler.\nHow do I remove Grand Theft Auto 2?\nQuickly and completely remove Grand Theft Auto 2 from your computer by downloading \"Should I Remove It?\", its 100% FREE and installs in seconds (click the button below).\nOr, you can uninstall Grand Theft Auto 2 from your computer by using the Add\/Remove Program feature in the Window's Control Panel.\nFollow the prompts. A progress bar shows you how long it will take to remove Grand Theft Auto 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauhot b/data_all_eng_slimpj/shuffled/split2/finalzzzauhot
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index 0000000000000000000000000000000000000000..22c2d312f68438d0f785a9ed5b8da7cabc1bbfa8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Huawei's sub-brand Honor has now announced a four-day 'Honor Days Sale' where it is listing a host of its smartphones with attractive offers on Flipkart. The four-day sale will begin on Tuesday, September 18, and go on till Friday, September 21, and will be held exclusively on Flipkart. Smartphones that are a part of the sale include the Honor 9 Lite, Honor 10, Honor 9N, and Honor 9i. The company has also announced two new colour options for the Honor 9N \u2013 Robin Egg and Lavender Purple.\nUnder the Honor Days Sale on Flipkart,, the Honor 9 Lite 4GB RAM\/ 64GB storage variant will be available for Rs. 14,999, at its original price, but with an exchange offer of Rs. 3,000. The Honor 10 6GB RAM\/ 128GB storage variant will get a price cut of Rs. 5,000 and be available for a price of Rs. 27,999 during the four-day period. The original price of the handset is Rs. 32,999. The Honor 9i, the company's first smartphone with a four-camera setup, will be available at a discounted price of Rs. 16,999 (originally priced at Rs. 17,999).\nThe Honor 9N 3GB RAM\/ 32GB storage and 4GB RAM\/ 64GB storage variants, priced at Rs. 11,999 and Rs. 13,999 respectively, will be available at an exchange discount of Rs. 1,000. Furthermore, the company has also now made the Honor 9N 4GB RAM\/ 64GB storage variant available in two more colour options \u2013 Lavender Purple and Robin Egg Blue colours. The two colour options will be available starting tomorrow on Flipkart for the same price tag Rs. 13,999. The other two colour options already available are Sapphire Blue and Midnight Black.\nFurthermore, the recently launched Honor 7S will be up for sale on September 19 and September 21. The smartphone's price in India has been set at Rs. 6,999 for the 2GB RAM and 16GB storage configuration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home > What Does Page RPM Mean in Google Adsense?\nWhat Does Page RPM Mean in Google Adsense?\nYou may be wondering why your page RPM is so low. In this article, I will reveal the myth around what does the page RPM means in Google Adsense. And how to increase page RPM for better revenues.\nFirst, Adsense is a great way to monetize a site. Unlike affiliate marketing, displaying advertisements on your site makes you money in a totally passive way. That's why it's one of the best monetization methods available.\nIf you are planning to build a website, you may possibly be interested in using Google ads for revenue generation. Google Adsense is the central hub for worldwide publishers and brands.\nHere, I'm going to give you more details on how Adsense works and how the page RPM being calculated. Let's get started and see how it all works on your website.\nWhat Does Page RPM Mean?\nPage RPM is the newest addition to the Google Adsense system. RPM is the literal abbreviation of revenue-per-mile. This means the revenue per thousand views on your website.\nLet's say you've displayed Google Advertisements on your website. The RPM is the term calculating the revenue per each 1,000 advertisement impressions on your website.\nThis helps to set up predictions on how much to earn. And accordingly, you can set up plans for your website content and ad placements. Page RPM varies from a page to another on your website. And of course, it varies from a website to another.\nThe page RPM calculations are pretty simple to understand. Page revenue per thousand impressions is calculated by dividing the estimated earnings of your site by the number of page views of the page. And then multiply it by a 1 thousand number.\nThis is how it's being calculated. Google Adsense has integrated a feature that will calculate it for you in your account dashboard. you will get these numbers right there for you to build decisions upon.\nThe same calculations go for Youtube since it relies on the same Adsense platform in calculating the publisher's revenue. Overall, it should be clear now on how it's being calculated for you.\nHere is a reputable reference for calculating page RPM: Google Adsense RPM Calculator. You can use this tool to calculate and predict your RPM based on Daily page impressions, CTR %, Cost per Click inputs.\nThe page RPM metric is a very crucial factor in determining a website's success. It gives the owner an idea about the performance of different pages on the website individually.\nIt shows the performance of pages separately. This will definitely help the website owner to know which pages are bringing in better revenue than the others. Accordingly, this will give him more data to enhance other pages.\nGoogle Adsense has become way easier than before. And the overall overview has changed drastically to accommodate the user's metrics. A user-friendly interface is of the utmost importance in Adsense's Dashboard.\nHow To Increase Page RPM?\nYou may be wondering why is my Adsense RPM is so low. Actually, there are a number of reasons for that. And it all depends on your case whether it was high and dropped or it has been so low since ever.\nWhat you should then care about is increasing your page RPM. There are a number of ways to do so. It's indeed necessary if you want to make more profits and revenues out of your website.\nTo increase the page RPM, the first and foremost thing to do is to focus on generating quality and targeted content. Generally, people are visiting these websites that provide content of the highest quality.\nWriting high-quality content will definitely increase your RPM. Furthermore, it will help to attract more visitors from the search engines. And accordingly, you will get more traffic and more people to see your ads.\nThe content developer should focus on using keywords that pay usually higher. It should be kept in mind that this should not reduce the quality of the content.\nLower paying CPC keywords and advertisements should be avoided at all costs. Targetting high CPC keywords will definitely help to attract relative advertisements with higher revenues.\nThe number of ads posted on a single page is a big factor. You should avoid posting many ads on one page. the more the ads mean the dirtier the website.\nNevertheless, the traffic quality will be low. And as a result, the overall website quality score will drop significantly. The optimum number of ads per page should be between 2-3.\nFor me, I tend to put only one ad on a single page. If you're using WordPress, you may use Advanced Ads plugin to display a number of ads on the same page using the same exact spot.\nWhat is a Good Average Adsense RPM?\nThe average page RPM for Google Adsense varies greatly from a page to another or a website to another. It's all based on a number of factors such as your niche, content quality, website performance, traffic source, and indeed the number of advertisers in your niche.\nGenerally speaking, your page RPM in the Google Adsense platform should be around $2 to $5. And in some niches, it may go all the way up to $10 per thousand impressions (RPM).\nThe page RPM is certainly an important tool that guides you about your forecasted earnings per page, and also gives an idea about the direction the website is going in.\nWhy Most Bloggers Fail & How To Make Money?\nHow to Earn Money from AdFly Url Shortner?\nYou should always be monetizing and analyzing the data analytics of your website. This way, you will get a better chance to get more earnings out of your website.\nThat was it all about the Google Adsense RPM meaning. Please feel free to leave your comment below with your own personal experience. I would love to hear back from you.\nI do not know what to say except that I have enjoyed reading your article. I will keep visiting this blog very often. I'll use this information for my work.\nReally I didn't know about Page RPM, but now I knew thanks for sharing such useful information.\nReally I wasnt know about Page RPM, but now I knew thanks for sharing such useful information.\nI have found here lots of interesting information for my knowledge I need. all the details you provide to us, it was very helpful and useful, thanks for sharing this amazing post.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here you may find the instructions to access the machine(s) avaialable for the project.\nThe final SPM project is structured in 6 different alternatives. Students should agree which one they want to choose with the Professor before actually starting to implement the project.\nThis is video filtering application. Video frames should be read from file\/device via OpenCV calls. Some kind of filtering (up to student or suggested by professor) has to be applied to each one of the frames. Eventually a new video has to be rebuilt from streams using again OpenCV calls.\nThe application finds the solution of a system of linear equations using the Jabcobi iterative method.\nAny application of interest for the student. TO be discussed and agreed with the professor before actually start working on it.\nImplement a generic D&C support, with a clean and seamless C++ interface.\nThe pattern process a stream of input tasks and a global state value. For each one of the input tasks x one or more output tasks f(x,s) may be produced onto the input stream and the state value s is updated to s'(x,s). The function updating the state is monotonic. The computation of an f(x,s) with a non updated state s does not produce any damage\/side effect (but using additional computational resources).\nStudents are encourage to propose and discuss with the professor different alternatives, possibly individuated because of personal interests, subjects already tackled in other courses etc.\nApplications should be implemented using FastFlow. Alternative programming frameworks may be discussed with the professor. A comparison implementation using plain C++ threads will be helpful and appreciated.\nRun time supports should be implemented using C++ threads. Alternative proramming frameworks may be discussed with the professor.\nThe students using OpenCV for the Video processing application will get a different account on a machine with openCV pre-installed, in case.\nat the very beginning, please state clearly in a single sentence the problem solved, the programming framework(s) used, their version (if significant) and the architecture(s) used to run the experiments.\nThis project is valid for all the exam terms in A.Y. 2016\/2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have a fun marquee light box that I like to use for various occasions, and Ava set it up by the Christmas tree saying \"Merry Xmas\". Elfie obviously decided that it needed to be changed as we woke up on the morning of the 23rd to find that it reading \"Elves Rule OK!\". The Merry Xmas letters were strewn on the television cabinet in front of the light. Elfie... we can't trust you with anything!\nOn Christmas Eve the voucher in the advent calendar was for reading a Christmas story. Elfie must've sneaked a peek as we found him sitting on the dining table with a wrapped up parcel that looked suspiciously like a book. He had a pair of reading glasses with him and a coffee cup from Starbucks. He was even leaning against a Starbucks Gingerbread Cafe kit. Looks like he was getting ready for a day with coffee and a good book! The girls unwrapped the gift and sure enough, it was a Christmas book from Elfie. The girls and I read the book, and later in the day we assembled the gingerbread Starbucks - which came out not quite as nice as the one on the box, but was still fun!\nThe big day! Like every year on Christmas morning, Elfie was found sitting in the advent calendar holding letters for the girls from Santa. The girls always stop to collect these letters before they make their way to the Christmas tree. It's sort of become a tradition. At the end of the day, the girls wrote thank you letters to Santa and gave them to Elfie to take back to the North Pole.\nNow that Christmas is over for another year Elfie Pokie Nokie has gone back to the North Pole to stay with Santa. Last year Mathilde was upset about him leaving so he stayed with us until new year and she carried him around everywhere during that time. This year she wasn't so upset about him leaving so he was able to fly back straight away.\nGood bye Elfie Pokie Nokie. See you next December!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paintball is an interesting sport that comes in all shapes and styles. For instance, there is Woodsball, Mil-Sim\/Scenario and Speedball. Selection of a paintball gun that fits is dependent on the feel and the looks of the gun itself. Picking a good paintball gun from the hundreds of brands currently available in the market is a daunting task. Different people that play paintball have varying tastes in paintball guns but it is safe to say that the brand: Azodin has everybody covered with regard to the paintball guns they release into the market.\nThe Azodin Kaos Semi-Auto Paintball Marker Gun is a quality product from the brand: Azodin and is a paintball gun that is designed for optimum performance.\nColor \u2013 This paintball gun is available in aesthetically appealing colors such as Black, Blue, Black\/Gold, Gun Metal Blue and Black 2.\nWeight \u2013 This product is very light in weight facilitating portability and easy storage as it weighs only 2 pounds.\nBarrel \u2013 It is a 12-inch one piece barrel paintball gun making it easy to handle while on the battle field.\nBolt \u2013 It is equipped with a self lubricate delrin bolt.\nPanels \u2013 It is designed with rubber grip panels that are non-slip. These eliminates the possibility of the paintball unexpectedly slipping out of the hands in the battle field.\nFeedneck \u2013 This product is equipped with a low rise twist lock feed neck that is quite convenient while playing.\nDouble ball dents \u2013 This eliminates the risk of double feeding in the breech.\nThe Azodin Kaos 2 Paintball Gun can be described as a modern rendition of the Kaos Series which is the signature of the Azodin brand.\nWeight - This product is very light in weight facilitating portability and easy storage as it weighs only 6 pounds at shipping.\nDesign \u2013 This paintball gun has an all-aluminum frame and body. It has a single trigger frame.\nDual ball dents \u2013 This eliminates the risk of double feeding. In the breech.\nCocking rods \u2013 It has dual cocking rods for improved reliability and for a smooth pump stroke.\nFeedneck \u2013 It is aluminum clamped to achieve a firm attachment.\nQuick release bolt \u2013 It has a quick pull-pin release bolt that is easier to handle and ensures optimum performance.\nColor \u2013 It is available in Black, White\/Gold, White\/Purple and Blue\/Green colors that are aesthetically appealing.\nAir supply \u2013 Air in this paintball gun is supplied to the Rock Steady Regulator through a streamlined on\/ off ASA which as a swivel fitted macroline.\nThe Azodin KP3 Kaos Pump 3 Paintball Marker is a product that promises to take over the pump paintball scene. Sleek operation and accuracy is the definition of this paintball gun.\nDesign \u2013 This product is more ergonomic and sleek when compared to its predecessor. Its ergonomic features promote safety during play and its sleek design contributes to its aesthetics and performance.\nReceiver- It has been streamlined to make the gun lighter in weight. This has also enhanced acuity when potential targets are being sighted.\nFeedneck \u2013 It has a Feedneck that is screw-locked to ensure firm attachment.\nBarrel \u2013 This paintball gun has a 12-inch multiphase honed barrel. This type of barrel is strictly for optimum performance as can be deduced from the nature of its build.\nWeight \u2013 It has a lighter body to facilitate portability and ease in storage as it weighs only 8.2 pounds on shipping.\nFrame \u2013 Its double trigger frame is a feature that can come in quite handy in the battle field. It is a clear indication that this gun is built for nothing else but battle.\nGrip \u2013 It has a 45 degree gas-thru foregrip that allows for easy handling of the gun.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..a1445f827670650406dd7d3c849b67374594b7c1
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Gowdy # If you have any new entries please submit them via # # send. Canon dr 2080c driver download. # # List of USB ID' s # # Maintained by Stephen J.\nView and Download Canon DR- C startup manual online. Canon Scanner DR- C Manual.\nDR- C Scanner pdf manual download. imageFORMULA is not the name of the ' s the name of the Canon product line.\nIn other words, your scanner is the imageFORMULA DR- C.\nCan' t find a driver for your Canon scanner? VueScan is here to help Don' t you hate how Canon stop releasing drivers whenever a new operating system comes out? View and Download Canon DR 2580C - Document Scanner operation manual online.\nDR 2580C - Document Scanner Scanner pdf manual download. Also for: 1941patible Scanner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"jasminetck fuss free deal. nice seller to deal with. prompt reply and she arrange a location for meet up which is conveinent for me.\nbreadandcheese Great seller to deal with! Very flexible and accommodating. Thank you so much for your swift replies! Item was delivered as described. Highly recommended seller! Thanks again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dry body brushing is one of my favourite morning rituals.\nI get asked a lot about the best technique to use. Dry body brushing is one of those wonderfully easy practices to invigorate your body, stimulate the lymphatic system and gives you smooth, brighter, glowing skin.\nFind more information about the benefits of body brushing here.\nAfter testing out many different body brushes over the years, I have found using a brush with natural plant based bristles such as sisal from the agave plant is the most effective for invigorating the skin. These bristles provide the best exfoliation and stimulation for your skin. They are the highest quality, long lasting and don't scratch the skin.\nYou can dry body brush anytime of the day before bathing. First thing in the morning is especially invigorating, giving you a boost for the day ahead. If you are brushing in the evening you can use a more gentle stroke; which can result into beautiful meditative self-care ritual.\nUse light pressure on areas where the skin is softer and more pressure on those harder areas. Take care of the more sensitive areas of the body such as your chest, breasts and stomach. A slight pinkish colour may occur at first on your skin due to increased blood circulation. This is normal and shows that your skin brushing is working. Over time your body adjusts and it starts the crave the sensation.\nAfter you have dried off use an organic sweet almond, sesame or coconut oil on you skin with an invigorating essential oil blend to keep you uplifted and your body glowing.\nYou will feel light, energised and your skin will be glowing. Your body will love you for it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nicole Kidman stunned everyone with her gorgeous red gown on last night's Emmys red carpet. After so many years of attending awards shows, she's got the whole getting-ready process down and her routine is actually very low maintenance. Prior to snagging her wins, Nicole shared how she preps for big nights in Hollywood. If there's one thing she's known for aside from her acting chops, it's her gorgeous skin. She swears by sunscreen every day for protection, but her antiaging product of choice is rather surprising.\nShe's a fan of retinol for its wrinkle-diminishing abilities, but Nicole doesn't opt for a luxury option. Instead she stocks up on a drugstore staple\u2014Neutrogena's Rapid Wrinkle Repair Regenerating Cream. She insists it's the formula that works best for her sensitive skin. According to customer reviews, skincare junkies all over are huge fans too. If you're looking to kick-start or switch up your antiaging routine, try Nicole's affordable favorite for yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Homewood Suites by Hilton Boston \/ Andover, an extended-stay Boston hotel. Our all-suite hotel is located near downtown Boston and Logan and Manchester airports. Enjoy easy access to I-93, Routes 95\/128, and the Massachusetts Turnpike (I-90). Nestled in the quiet, wooded Merrimack Valley, our Andover, MA hotel is a gateway to many New England activities. If you want to attend a Red Sox game, we're just 30 minutes from downtown Boston. Take a short drive to the historic cities of Lexington, Concord, and Salem. Business travelers will appreciate our proximity to many Boston-area corporations and office parks.\nRelax or catch up on work in a suite that features a separate living room with a sofa bed, work and dining table, TVs in each room, and premium cable channels. Prepare dinner or snacks in the fully equipped kitchen. Plunge into the seasonal outdoor pool or work out in our fitness center. Buy a cold beverage in our hotel convenience store.\nCheck emails with free internet access in each suite or our business center. Host a business meeting or event for up to 50 people. Wake up to a hot full breakfast served daily, and unwind at the end of the day with an evening social served Monday through Thursday, both included in your extended stay in Boston. For business or leisure, Homewood Suites by Hilton Boston \/ Andover hotel is your home away from home.\nRelax in one of our studio, one- or two-bedroom suites knowing we've worried about the details so you don't have to. Spread out in the separate living area with a full-sized sleeper sofa. Prepare dinner or a snack on the two-burner stovetop or in the microwave in your fully equipped kitchen. Finish a report or stay connected with your family at your work desk using complimentary internet access. Watch a favorite program on one of the TVs in each room. At the end of a long day, relax on your king-sized bed and set the alarm for your complimentary hot full breakfast the next morning.\nAt our Homewood Suites by Hilton hotel, you'll enjoy a range of inclusive perks to help you feel right at home. Indulge in a hot full breakfast each morning and evening social Monday through Thursday. We offer a complimentary grocery shopping service - and if you forget something, there's a 24-hour on-site convenience store. Other benefits include newspaper delivery every weekday, and complimentary WiFi. It's all you need to make yourself at home.\nBook today to earn double points on your next stay.\n100% non smoking, A fee of up to 250 USD will be assessed for smoking in a non-smoking room. Please ask the Front Desk for locations of designated outdoor smoking areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Nothing feels better than the perfect tee with the most effortless fit. True to form, White + Warren has given us the must-have tee for any season, complete with luxe detailing, simple yet exquisite design, and the highest quality cashmere. Whether you're hoping to refresh your seasonal wardrobe or are on the lookout for your next year-round fave, this pearl white cashmere tee shirt is the perfect place to start.\nNot only is this pearl white cashmere tee shirt beautifully designed, it also captures this brand's commitment to creating clothing that feels truly fantastic to wear. Soft to the touch and made from 100% high-quality cashmere, this piece is an immediate wardrobe staple. Reflecting the brand's ethos to creatively combine upscale fabrics and timeless designs, this essential tee is all about infusing a bit of luxury into the everyday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's more than 20 years now from the first collective industry recognition (Energy Star ratings) and the desire to do something about reducing power consumption by IT equipment. The cycles of heightened awareness and hype coincide with economic cycles or at times a paradigm change brought about by new technology or use of existing technology like virtualization or cloud computing. So when one of the organizers asked me to talk about Green IT, I was perplexed on the unexpected resurgence; or was it a red herring or just a slot filler ?\nResearching the market beyond personal experience to understand the current traction and prominence of Green IT, I started talking to CIOs to understand their journey or milestones on the subject; I was hoping to use some of the insights in my presentation to an audience of CIOs. Thus over the next few weeks the data points that I gathered did not focus on public or private clouds, but to understand if Green was still a discussion in different industries and industry leaders or has Green fatigue set in ?\nLess than a decade ago carbon credits were a boardroom discussion and some people made a lot of money trading them; then they just vanished. Data Centers started touting Power Usage Effectiveness (PUE) or Data Center Infrastructure Efficiency to demonstrate how Green they were. Overhead non-compute power usage for cooling and other uses ranged from typical 80% to highly efficient 10%. Power used by servers and computers for the same task has been reducing almost linearly over time challenging PUE.\nCIOs finished fixing the data center quite early and also created efficiency for end compute devices with power management policies and improved awareness. From there on they focused energies on office infrastructure, ambient lighting and temperatures, electronic documentation, communication and as a result saved trees. A few also took control of energy usage monitoring with embedded and targeted sensors linked to BMS (Building Management Systems) and reduced power consumption in good measure.\nThe question is, with so much getting done and almost every enterprise reaping the benefits over the last decade, is Green still a relevant discussion ? Who if anyone is tasked with overall enterprise Green initiatives ? Is it only about controlling power used or creating a movement to conserve natural resources ? Is Green an integral part of decision making explicitly mentioned in evaluation parameters or implicitly used in every decision making ? Have people stopped flying around for meetings and embraced Video Conferencing ? The answer is Yes and No.\nI don't believe that CIOs are measuring and reporting metrics around Green. None of the CIOs I met did, but there is a discussion around power efficiency and overall cost of running BAU or keeping the lights on. CXOs are not enamored by earlier conventional solutions which are now basic hygiene as are new LEED certified buildings. Reality is that for technology and compute power, the bar keeps shifting every 4-5 years; what was green in 2000 or for that matter even 5 years back appears archaic when compared to currently available hardware.\nAt the conference the audience engaged in a discussion and debate on the relevance, challenges and opportunities; one of the participants sought suggestions to break the cultural vice around personal printers which gave his global centurion company a person to printer ratio of close to one. Someone shared extreme automation to switch off lights and air-conditioners when there was no movement resulting in hilarious and occasionally dark and sweaty situations while people were in the room working !\nGreen is here to stay and awareness of the new generation brings new opportunities. For CIOs it is important to seek avenues beyond the conventional interventions of the past. Cloud computing is shrinking the data center and mobility is driving the workforce out of offices. Power generation is focusing on renewable energy sources which reduce the global carbon footprint. Some of the power hungry enterprises are going to embrace these as the world works towards creating a better tomorrow.\nThe role you play in this evolution is up to you !\n\"You can't just ask customers what they want and then try to give that to them. By the time you get it built, they'll want something new.\" So said the most iconic leader in the IT industry and stuck to this philosophy as his company built some of the most vied for products; the success that followed remains unparalleled though the bull run has slowed down a bit. Everyone wondered exasperatedly on how they can replicate this model as it is contrary to conventional wisdom and what they were taught in B-school.\nUntil not too long ago the software industry churned out products with features and functionality based on internal discussions on what the customer may need or in some cases based on what their initial customers asked for. With generic solutions not fulfilling the expanding needs, over time they started hiring industry domain experts to create vertically aligned solutions. This did address the gap partly for a while and then customers started demanding better aligned solutions for their specific problems and opportunities.\nSome companies recognized the need early and started creating Customer Advisory Boards (CAB) with CIOs of their large customers to participate in the product roadmap. This was extended to include some of the innovative adopters of their solutions though they may not have been high revenue customers but brought value to the discussion. The ensuing engagement, discussion and debate influenced the prioritization of new features and in some cases the positioning of their solutions resulting in a win-win situation.\nSome of the services vendors took the cue and hired from the industry to strengthen their industry practices; consulting companies followed suite thereby changing the discussion with their customers. They determined that the need was to embed the resources internally and not limit to an advisory role. Now the software industry is going through a transition with even mid-sized companies thinking of CAB to gain the benefit of customer connect and better alignment of their product features and evolution to what the industry wants.\nInterestingly hardware manufacturers have remained disconnected; they continue to launch products with the philosophy of the icon attributed with the famous quote. Past practice of customer focus groups has largely been discarded by marketing teams. Faster processors, bigger, brighter and higher resolution screens; consumers love it and they do more of the same. Then they have attempted to push the same products to enterprise customers and wondered why it is not gaining traction the way consumers are lapping them up.\nCIOs are not excited; what else do you want has been the lament ? Over the years the clear message from many CIOs to the IT industry enamored by all things mobile (phones, tablets, and applications) has been that the faster, better, cheaper does not connect with enterprise use cases. Enterprises need manageability, serviceability backed by service levels, and reasonable (measured in years not months) longevity. Consumer devices require additional investments to make them work in our environments.\nConsumer applications and games are great; couple of apps on the app store for some customers or pilots on industry specific use cases does not make you an enterprise ready development partner. We don't want to explain everything from the basics to your team; how are you going to fill in the gap between what we say and what your team understands. Do some homework and more than anything else listen before you start crafting solutions; you have an advantage over your big competitors, use it well.\nThe question is then, is CAB the way to go for companies who want better traction of their solutions or services in the enterprise space ? It is a model that may work for the larger IT companies; how does a smaller outfit get the benefit of the experience ? In \"Scaling Startups\" I had referred to a mentoring model and role that CIOs can play; maybe it is time for IT companies to embrace CIOs to help them forge ahead. What is important is the change in mindset and philosophy with internal agreement on the new way of working. I hope IT companies understand this sooner than later.\nAs a CIO, are you up to the game ?\nThe country is going through the most complex exercise of voting for and electing new representatives to the government. Selecting amongst the candidates is difficult; some of them are easy to disregard as they have no visible credible experience to stake a claim to the seat. Handfuls have relevant experience and on paper they look like good options. References to past work demonstrate their ability to deliver and execute; a couple have the backing of their respective political parties who lend the promise of a collective manifesto.\nEvery 5 years this process repeats itself, sometimes a little earlier if the incumbent government is unable to serve out a full term. Soon we will have a new government, a new head, a set of ministers who will vie for the most visible and high profile ministries. The correlation between portfolios and core competencies is always a good thing to do; however many a times that does not happen. Bureaucrats and the staff within the ministries does not change much, they follow the new directions set by the ministers irrespective of expertise.\nRunning an enterprise is very much like running the country, especially a large diversified group with interests in varied businesses. Each company and function has a head that is selected from outside more often than inside. The selection is most of the time purportedly on core competency and merit. The difference between a government and an enterprise is normally the available options from which a candidate is chosen. In a government, the candidate is from the elected party, in the other case almost everyone can apply.\nGovernment portfolios are presumably distributed by the head of the winning party and head of the country collaboratively. Most often decisions are based on seniority, past experience, credibility and interest. Cross functional movements are the norm and it is expected that the person would do equally well in the new function too. The rationale here is that a leader need not be a functional expert; the team has adequate skills to advise the leader on the best options when taking a decision. We know how well this process works.\nIn the corporate world the skew is more towards functional expertise while selecting a person. Cross movements do happen at beginning and mid-careers; moving up the ladder, these are rare. Leaders at the top take on additional responsibilities at times; lateral moves occur but are not frequent. Success rates are higher in comparison as everything is expected to be time bound. It is not a sure shot recipe for success; we do observe failures across the board, many attributable to the leader not being effective or fitting in.\nIt is not necessarily the interviewer or selectors inability to assess that result in a bad selection. Drawing parallels, it is evident that the best person does not always get selected for the job; favoritism and at times other factors like past workplace association, belonging to same religious sect or geographical area, been to the same school\/college, having the same ideologies, result in sub-optimal choices. While not always avoidable, enterprises do watch out for such hires critically lest they end up with unwanted baggage.\nIt is relatively easier to replace a person within an enterprise as compared to the government; performance appraisals even though many a times skewed do elevate non-performance. On the other hand, recently observed citizen activism has its place in creating change. Cabinet reshuffles however move the problem from one area to another; this is rare in the enterprise though not unheard of. Are there learning that can be applied to enterprises to not follow the same path that ails many parts of the government ?\nI believe that when we choose an elected representative or a new hire, in both cases rigorous due diligence is essential. Our choices can come back and haunt us not just in the short-term but also in the long-run. Both impact our lives and future; we tend to spend more effort in our workplace due to impact proximity and blame bad decisions to the ill choice of others. If we want control of our future and destiny, we have to exercise our rights and influence outcomes. Can we afford not to ?\nYou decide, you have a choice !\nNot too long ago one of the consulting companies' classified people by their digital proficiency and created three groups; digital dinosaurs, digital migrants, and digital natives. This was picked up by many people who used this classification to target their solutions or services to their respective benefit. The resultant divide did not matter to the bourgeois or the elite, they anyway continued to demonstrate behaviors that they did; the world continued to evolve, and gave way to a new species, which is gaining ground.\nCharacteristics of the first three categories were quite easy to comprehend and evident with their names. The Dinosaurs are the people born before the advent of technology (today's 60+ generation) who use basic technology to stay connected. Migrants (born before 1990 or thereabouts) embraced the new technology wave sometimes a bit uncomfortable, but more or less adept. The Natives are born into the new world and do not know of a life without social media, mobility, ubiquitous connectivity and instant gratification.\nAs is with all kinds of evolution, there are genetic mutations, exceptions and the differently abled who do not partake in the normal. They have all the means, the environment, the facilities, the stimulus, the desire; they show a lot of promise and demonstrate a stray spark of brilliance. These individuals are mostly found in the cusp of the natives or wannabe migrants (migrants who believe they are better natives than the natives themselves). It does not take too much of an effort to spot them, they are visibly obvious.\nIn a digital conference with average participant age less than 24, we tried spotting the pseudo natives; one of the sparsely haired speakers in mid-40s was desperately trying to impress the audience with his knowledge and investments into digital startups. A mobile cloud, take it anywhere you go; make 2G work at 3G speeds; family only social media, were some of the references to his invested concepts which he believed were ahead of the evolution curve. No one had heard of his ventures but the crowd humored him to get his wallet share.\nA corporate 20-something CDO (Chief Digital Officer) reveled in the fact that he was ahead of his peers having already achieved the pinnacle position in an industry which was always in the forefront of digital technology. With an MBA from a prestigious institute, he strutted around to the envy of others who wanted to get there. He was on stage being interviewed on his vision and predictions of where the digital world was headed. Not that he wanted to entertain the audience; his responses had most people snickering.\nHis stated digital strategy for the company revolved around leveraging an existing portal which had lost traction with customers. Is mobility part of your strategy ? Off course, we are deploying apps ! How do you plan to differentiate ? We are looking at global trends ! Do more of the same, do it a little differently; will it change the world or bring around a revolution ? As the Q&A progressed, the audience became restless wanting to be rid of the listless conversation. One anonymous listener shouted across the room, \"Get off the stage, you \u2026\" !\nTitles aside the general enthusiasm around digital everything ranged from wearable technology to esoteric business cases; the general feeling was that if you are a migrant, there's a generation gap, if you are older, why are you here ? You don't understand our language; you don't belong to this new wave. Don't slow us down; we know where we are going. While some oldies tried to moderate the irrational exuberance reminiscent of the dotcom era, the young believers retaliated with cries of frustration with the dinosaurs.\nI think that giving some latitude to sprouting innovation connect to real life use cases is the need of the hour. It would serve the well-wishers to espouse the cause while staying out of the way of the emerging digital tsunami. The velocity and variety of change challenges legacy thinking; to understand and appreciate the thinking process of the new generation entrepreneurs requires unlearning and new mindset failing which the older generation and some from the current too run the risk of being alienated or being labeled Digital Morons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Risk factors for cervical precancer detection among previously unscreened HIV-infected women in Western Kenya.\nHIV and cervical cancer are intersecting epidemics in many low-resource settings, yet there are few accurate estimates of the scope of this public health challenge. To understand disease prevalence and risk factors for cervical intraepithelial neoplasia 2 or greater (CIN2+), we conducted a cross-sectional study of women undergoing cervical cancer screening as part of routine HIV care in Kisumu, Kenya. Women were offered screening with visual inspection with acetic acid, followed by confirmation with colposcopy and biopsy as needed. Univariable and multivariable analyses were carried out to determine clinical and demographic predictors of prevalent CIN2+. Among 3,241 women screened, 287 (9%) had an initial diagnosis of biopsy-confirmed CIN2+. On multivariable analysis, combined oral contraceptives remained significantly associated with detection of CIN2+ among women on HAART (AOR 1.84, CI 1.20-2.82), and not on HAART (AOR 1.72, 95% CI 1.08-2.73), while use of a progesterone implant was associated with increased detection of CIN2+ (AOR 9.43, 95% CI 2.85-31.20) only among women not on HAART. CD4+ nadir over 500 cells\/mm(3) was associated with reduced detection of CIN2+ (AOR 0.61, CI 0.38, 0.97) in the overall group, but current CD4+ was only associated with reduced detection of CIN2+ among women not on HAART (AOR 0.42, CI 0.22, 0.80). In conclusion, a history of less severe immunosuppression appeared to reduce the risk of CIN2+ detection, but current CD4+ count was significant only in non-HAART users. The association of CIN2+ with hormonal contraception should be explored more in prospective studies designed to better control for confounding factors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Palisades football player and Easton field hockey player are voted as Coordinated Health Athletes of the Week.\nPirates quarterback Ethan Brader earned the vote after collecting 13 carries for 246 yards and scoring three touchdowns in a 56-7 win over Pen Argyl.\nRed Rover Ellie McIntyre scored the overtime game-winner in an East Penn Conference field hockey semifinal and scored the first goal in the EPC title game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For those suffering from chronic sleep disorders, it may seem as if there's little to do for relief. It is estimated that millions of Americans of all ages are affected by sleep disorders each year and the number is continually growing.\nWhile most people try medication, exercise, and certain mindfulness routines to help themselves wind down, it may not always work. Hypnosis is an alternative treatment not widely known.\nThough it has not been comprehensively studied for the more severe sleep problems, such as, insomnia and similar conditions, the research that has been conducted seems promising for helping induce deeper sleep in individuals with chronic disorders.\nHypnosis, or hypnotherapy, is a natural treatment for different chronic disorders and conditions. Sleep disorders are often a result of a restless mind, which is why hypnosis, which focuses mainly in the mind, is believed to help with this problem.\nSleep disorders can be incredibly disruptive to one's life, causing them emotional stress such as irritability, paranoia, depression, and anxiety as well as physical stressors, such as, headaches, stomach aches, soreness, chronic fatigue, poor concentration and much more.\nHypnosis may be able to help these symptoms and more in individuals with chronic sleep disorders in need of relief.\nWhen experiencing hypnosis, a person will experience relaxation, focused attention, and become more impressionable and open to suggestion due to being placed in a trance. This state of mind is essential for performing successful hypnosis because it helps the individual increase their responsiveness as well as alter their behavior and reaction.\nWhen used for the treatment of chronic sleep disorders and other problems, this openness to suggestion allows the hypnotherapist to try different procedures or practices that may illuminate the cause of the patient's chronic health problem.\nThere are ways to hypnotize oneself, and this is called self-hypnosis. The techniques practiced in this procedure are most commonly used in order to treat chronic insomnia. The idea behind this particular treatment is that while the individual practices self-hypnosis, they are actually pulling attention away from their anxieties and producing thoughts that encourage relaxation which can then lead to restful sleep.\nAfter first consulting a doctor to make sure that one's sleep disorder is not caused by an underlying problem, one can begin exploring how to practice this treatment. Most doctors may suggest a hypnotherapist to get things rolling and even evaluate the severity of the patient's condition.\nOnce the chronic sleep problem is clearly defined, practitioners are better able to decide how to proceed. Usually in addition to hypnosis, patients also try natural approaches to sleep such as supplements, exercise, and certain foods as well as mind-body therapies that promote relaxation and focused mindfulness.\nWhen used for sleep, hypnosis can have many wonderful effects on an individual.\nNightmares: Dreams that are distressing and frightening in nature. Many people experience recurring nightmares which may cause anxiety and fear and later develop into insomnia.\nSleep Terrors: Most commonly defined by extremely distressing and disturbing nightmares that are accompanied by screaming and other physical responses. Often the result of traumatic and unresolved experiences.\nBedwetting: This often occurs at the height of a distressing dream and happens when the individual is experiencing fitful sleep and deep dreams.\nSleepwalking: When the brain fails to release the mild paralysis needed to keep the body still while asleep, sleepwalking can occur. Sleepwalkers may physically move about (i.e. walking and talking) while completely asleep due to being deeply engaged in their dream.\nWhile research suggests that hypnosis may help these problems, the findings and studies are very small. Still, hypnosis has proven effective for sounder sleep in most individuals. Due to this, the treatment of sleep disorders has begun to include this alternative treatment for patients looking to try everything they can to alleviate their chronic disorder.\nHypnosis is also effective as an exploratory treatment, meaning that it can help identify other underlying problems causing chronic sleep and health problems.\nWith the help of hypnosis, other symptoms and causes have been found and treated by other medications and treatments, helping to improve or completely eradicate some sleep disorders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Born in Auckland in 1930, the young Russell Kerr displayed an equal talent for piano and dance. He settled on dance, where his career began as a student of Auckland ballet teacher Kathleen Whitford.\nHe was awarded a New Zealand Government Bursary in 1950 to travel to Europe. After making his European debut with the Jose Greco Spanish Company, he danced with the Sadler's Wells (now) Royal Ballet, with Ballet Rambert, and as a soloist with London's Festival Ballet. He made a name for himself as a dancer with deeply informed musicality and outstanding dramatic expression.\nAfter returning to New Zealand in 1957, Russell began to work for the New Zealand Ballet in 1959, of which he later became Artistic Director. An exemplary work from that period is Prismatic Variations. Jointly choreographed with Poul Gnatt, this work carried distinctly neo-classical tones, while its novelty also gave indication of the company's assured originality.\nRussell is regarded as a consummate professional who has held many key positions in New Zealand dance: he was Director of the Nettleton-Edwards-Kerr School of Ballet (1959-61), Artistic Director of the New Zealand Ballet (1962-69), Director of the New Zealand Dance Centre (1969-77), and Director of the Southern Ballet Theatre (1978-90). His teaching and choreographic skills provide a continuing source of inspiration.\nHe was Vice Patron of the Auckland Dance Company, Patron of the International Ballet Academy and a Trustee of the Christchurch Dance Education Bursary Trust.\nHe was awarded a QEII Arts Council Fellowship in 1977, followed in 1986 by a Queen's Service Medal for Services to Ballet in the New Zealand dance community and, in 2000, was appointed an Officer of the New Zealand Order of Merit.\nIn 2005 Russell Kerr created a new full-length, two-act ballet based onthe secret life of Hans Christian Andersen for The International Ballet Academy, Christchurch. In June 2007, the third return season of Russell's version of Swan Lake toured the country. Raewyn White, NZ Herald, describes the work as \"showing us once again the hallmarks that make him one of the great ballet choreographers. Kerr's rich musicality ensures we miss none of the emotional nuances of the score in the dancing\".\nRussell lives in Christchurch and continues to teach and choreograph.\nArts Foundation Icon Award;Created a new full-length ballet based upon the secret life of Hans Christian Anderson.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Oral Health Foundation is delighted to announce that BioMin F has become the first toothpaste to gain accreditation from its expert panel for both sensitivity reduction and tooth remineralisation.\nResearch suggests more than 40% of adults experience tooth sensitivity at some stage in their life.\nTo combat this, BioMin F toothpaste slowly releases calcium, phosphate and fluoride ions to form a protective layer on the tooth surface.\nThis layer effectively seals any exposed and open dentine tubules, which are a key cause of tooth sensitivity due to their direct link with the central nervous system.\nBioMin Chief Executive, Richard Whatley, expressed his satisfaction at the conclusion of the accreditation process, saying: \"Oral Health Foundation accreditation is a hugely important endorsement of our toothpaste. It assures consumers that BioMin F has been evaluated by an independent panel of internationally-recognised dental experts and academics.\nCommenting on the news, Chief Executive of the Oral Health Foundation, Dr Nigel Carter OBE said: \"It is a great pleasure to be adding our seal of approval to the BioMin F.\nThe Oral Health Foundation has the largest oral healthcare product accreditation scheme in the world.\nThe charity's independent and internationally-recognised panel of experts examine product claims to make sure they are reliable and can be supported by scientific evidence.\nEach product that passes through the programme is awarded with a kite mark to show that its claims have been rigorously checked and effectively verified.\nTo see the latest list of products approved by the Oral Health Foundation, follow the link to our accreditation page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"College students pick their majors for any number of reasons - interest in a particular field of study, ability to get a job post-graduation, amount of education or training needed, and more.\nAnother factor students often consider is how much they might earn once they enter the workforce. While it's hard to imagine that any student would actively seek a low-paying major, some students may weigh salary heavier than others when making a decision on a degree.According to the most recent salary survey by the National Association of Colleges and Employers, the average starting salary for a Class of 2012 graduate is $44,442. To break it down further, here's a list of the highest-paying bachelor's degrees within common academic disciplines. Also included is the percent change in starting salaries from the Class of 2011 to the Class of 2012, along with examples of occupations graduates within each major might pursue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download OraLoader 4.9 - OraLoader, Save Oracle import\/export wrok.\nAdd support to field contains \"\". Add support to field contains line break. Improve reading Excel file. Improve support for more datetime format. Improve support for more large integer field. Improve support for value \"(null)\". Improve writing Excel file.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This post was originally written for DigitalOcean.\nAnother common type of loop you will encounter is the for statement, which executes a set number of times. while and do...while loops are conditionally based, and therefore it is not necessary to know beforehand how many times the loop will run.\nFor an example, we will pretend we have an aquarium that has a population limit. For each iteration of the loop, we will add one fish. Once the aquarium has 10 fish, the population limit will be reached, and the program will cease to add more fish.\nThe aquarium has room for 9 more fish.\nThe aquarium has room for 8 more fish.\nThe aquarium has room for 7 more fish.\nThe aquarium has room for 6 more fish.\nThe aquarium has room for 5 more fish.\nThe aquarium has room for 4 more fish.\nThe aquarium has room for 3 more fish.\nThe aquarium has room for 2 more fish.\nThe aquarium has room for 1 more fish.\nThe aquarium has room for 0 more fish.\nIn our example, we set out while loop to run as long as the number of fish was less than the population limit of the aquarium. For each iteration, one fish would be added to the aquarium until all 10 spots were filled. At that point, the loop would stop running.\nAn infinite loop, as the name suggests, is a loop that will keep running forever. If you accidentally make an infinite loop at some point, it will likely crash your browser or computer. It is important to be aware of inifite loops so you can make sure to avoid them.\nThe simplest example of an infinite loop is simply setting the condition of the while statement to true. Below is an example of code that will run forever. It is not necessary to test any infinite loops.\nAn infinite loop will run forever, but the program can be terminated with the break keyword. In the below example, we will add an if statement to the while loop, and when that condition is met, we will terminate the loop with break.\nThere are 5 polar bears.\nThere are 4 polar bears.\nThere are 3 polar bears.\nThere are 2 polar bears.\nThere are 1 polar bears.\nThere are no polar bears left.\nNote that this is not necessarily a practical method of creating and terminating a loop, but break is a useful keyword to be aware of.\nWe already learned about the while loop, which executes a block of code for as long as a specified condition is true. Builing on that is the do...while statement, which is very similar to while with the major difference being that a do...while loop will always execute once, even if the condition is never true.\nBelow we will demonstrate the syntax of the do...while loop.\nAs you can see, the do portion of the loop comes first, and is follwed by while (condition). The code block will run, then the condition will be tested as a normal while loop.\nTo test this very quickly, we can set a variable to 0, increment it inside the do, and set our condition to false.\nOur output came out to 1, meaning that the code block ran once before it was stopped by an unsuccessful (false) while condition. Otherwise, the do...while loop can be used for the same purposes as a while loop.\nIn this article, we learned about the while loop, the do...while loop, and infinite loops. Automation of repetitive tasks is an extremely important part of programming, and these loops can help make your programs more efficient and concise. To learn more, read about the while and do...while loops on the Mozilla Developer Network.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazkln b/data_all_eng_slimpj/shuffled/split2/finalzzzazkln
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"JALALABAD (Pajhwok): During his trip to eastern Nangarhar province, President Ashraf Ghani on Sunday met local Islamic scholars, university trachers and students.\nMaulvi Abdul Basir Sirat, an scholar who attended the meeting at the Shahi Palace in Jalalabad, backed the president's policy on the peace process and appreciated his efforts for stability.\nHe sought land for a graveyard, a greater Haj quota for ulema , opening of a library and building of a conference hall in Nangarhar. Management of the Eidgah Mosque should shift from the municipality to the Ministry of Hajj , he proposed.\nGhani thanked ulema's unprecedented fatwa on peace and said he had met Islamic scholars and Haj directors from all 34 provinces of the country. He listened to their problems in the holy month of Ramadan .\nHe said the government had a countrywide programme for seminaries and Quran memorisation centres. Distribution of one million copies of the holy book across the country has been ordered.\nThe president said the government paid serious attention to the reconstruction of mosques and he had directed local officials to build a grand mosque in Badakhshan province.\nGhani directed the Nangarhar governor to investigate the grabbing of Qasimabad Township land and present a detailed report to the Presidential Palace. Legal ownership documents could not be issued for grabbed land, he added.\nGhani promised major changes in the Ministry of Haj and Religious Affairs and the Afghanistan Ulema Council head, Maulvi Kashaf was currently working on opening a Darul Iftaa (house of fatwa).\nThe president also met Nangarhar University lecturers and a number of schoolteachers. Chancellor Babrak Miakhel shared with the president information on university affairs and problems in acquisition of land. He also floated suggestions for expanding university faculties and improving security.\nGhani said: \"I am ready to allocate land for the university and provide necessary facilities for it.\" He directed the officials concerned to appoint a delegation for the acquisition of land for the university.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In New York, we are about to get a record breaking blizzard. They are estimating a snow fall between 24 and 36 inches. That is 3 feet people! So, what does one do in preparation for a snow storm like this? Bread, milk and eggs of course. I am not sure why those are the essential items that one buys in a storm like this.. I mean really, french toast? There was no bread left yesterday, and I accidentally dropped the eggs in the parking lot, leaving them splattered all over the pavement.\nWe stocked up on food last night, really because we wanted some snacks to munch on while snowed in. The only issue is, now my husband was mandated to sleep at his job (hospital) and I will be all alone with all these snacks! :) During our grocery shopping last night, we got ingredients for risotto, pork tenderloin, chop meat, frozen pizzas, water, etc. We made sure to have batteries at home for flash lights, and a new lighter for candles and our gas stove\/oven - just in case we lose power. I even stopped at Petco to stock up on my feathered children's food!\nBeing we had the ingredients to make a delicious dinner, I decided to go with basil risotto (one of our favorites) and a perfect pork tenderloin recipe. You will not be sorry when you taste the butter-like texture of this pork and flavorful honey mustard\/honey glaze. It melts in your mouth! You can find the basil risotto recipe here. Below is the recipe for the pork only.\n*optional: carrots, fingerling potatoes, etc.\n1. Season pork generously with salt & pepper.\n2. Heat a medium\/large non-stick pan. On medium to high heat, heat olive oil and butter in pan. Place the tenderloin in the pan and brown each side for 3 minutes (total of 6 minutes).\n3. Put the garlic cloves and shallots in a casserole dish or roasting pan, leaving some room in the center for the pork. Once your pork is browned, place into roasting dish. Brush the honey mustard and honey on top of the pork.\n4. Bake for 40 minutes on 350 degrees. Add chicken stock into dish and continue baking for another 10-15 minutes (total of 55 minutes). Broil at the end for a minute or two to crisp the top.\n5. Remove from oven and let sit for 3-5 minutes. Slice pork about 1 inch thickness and serve.\nWe always have sundaes on Sunday so here is a quick pic of our fudge ripple sundaes! Kevin doesn't like sprinkles so you can assume which one is mine!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The bacteria that live in and on us are an integral part of our immune system and influence much of our health and well being \u2013 they even talk directly to our brain and influence our mood and cravings. They protect us from infections, regulate and activate our immune system and even regulate our weight and metabolism.\nHowever as a result of our modern diets, sedentary lifestyles and over exposure to stress, toxins and medications our gut health is deteriorating globally. This deterioration has been linked with the obesity crisis, rising levels of diabetes and metabolic syndrome \u2013 and a whole range of auto immune-related diseases from asthma and allergies, inflammatory bowel disease and IBS, to Rheumatoid Arthritis, Parkinsons and MS.\nSugar in our diet feeds the types of bacteria and pathogens we want to keep in check \u2013 so a diet rich in sugar can lead to an over growth of these bacteria and a resulting in balance and dysbiosis in our guts. Further more a study at Oregon State University found that altered gut bacteria as a result of a high-sugar diet, appeared to impact \"cognitive flexibility,\" in ones ability to adjust to changing situations \u2013 and also showed an impairment of early learning for both long-term and short-term memory. Its also worth noting that sugar is addictive and refined sugar contains no nutrients \u2013only calories which are quickly absorbed causing blood sugar levels to jump and then collapse \u2013 causing hunger and further cravings. Artificial Sweeteners Concerns over the health effects of artificial sweeteners have long been raised \u2013 and that consuming them prompts hunger and further cravings when the expected sugar hit doesn't arrive \u2013 however recent studies have shown that artificial sweeteners also harm the balance of our gut bacteria \u2013 promoting growth in the undesirable strains of bacteria associated with obesity and diabetes \u2013 as well as inflaming the gut lining.\nPreservatives are added to most packaged foods inorder to slow or prevent the processes of oxidation and bacterial growth \u2013 or put another way they are designed to kill bacteria. There are many natural preservatives such as salt and natural acids, oils and vinegars which are easily digested and do no harm however most preservatives in our food are synthetic manmade \u2013 and studies have shown they could be doing serious harm to our gut Emulsifiers are very common in packaged foods which also extend the shelf life of foods and keep ingredients\u2014often oils and fats\u2014from separating. Emulsifiers are found in many common household foods from mayonnaise to ice cream, biscuits to peanut butter with common emulsifiers including ingredients such as polysorbate 80, carboxymethylcellulose, lecithin, carrageenan, polyglycerols, and xanthan. Some major studies have shown how emulsifiers negatively affect the make up of our gut bacteria and disrupt the protective mucous layer that shields our intestinal tract, resulting in inflammation and can lead to bacterial infection. Another side effect is an interference with the signal of 'satiety' \u2013 or feeling full \u2013 leading to overeating and get fatter.\nOlive oil is a source of one of the healthiest fats known to man, rich in a range of polyphenols and antioxidants, which help reduce inflammation in the gut and elsewhere in the body. One major problem with olive oil is that the quality varies and some lower quality versions can be extracted using chemicals, or diluted with cheaper oils. Therefore, buying the right type of extra virgin olive oil from a trusted brand or source is incredibly important Fish Oily fish like olive oil, is full of good fats which are anti-inflammatory. The key ingredient in oily fish is omega 3 which is well renowned for its health benefits. However there's a big difference in the omega-3 content of different varieties of fish. Fish caught in the wild rather than farmed, will have higher levels of the fatty acid.\nThe approximate percentages of omega-3 oil content by type of fish are as follows: \u2022 Group I: mackerel (1.8%), trout (1.6%), herring (1.5%), sardines (1.4%), albacore tuna (1.3%), salmon (1.1%) \u2022 Group II: halibut (0.6%), river trout (0.5%), catfish (0.4%) \u2022 Group III: cod (0.3%), snapper (0.2%), tuna packed in water (0.2%) Meat: As discussed in 2, the quantity of meat in your diet should be reduced to make way for more plant based foods \u2013 but that is not to say meat needs to be avoided altogether as \u2013 it is the quality and source of the meat is important. Exclusively Grass fed beef for example has twice as much omega3 fatty acids than intensively farmed beef, although this still pales in comparison to mackerel or trout. Better reasons to eat organic meat is the absence of antibiotics- and when pastures are well managed, the production is generally less energy-intensive, less polluting, and more humane.\nPlants are an essential part of our diet and should comprise the significant majority of what we eat if we are to restore and maintain our gut health. We recommend eating a wide variety of different fruits and vegetables nourishes to promote diversity of our gut flora and provide the fibre our bodies need to feed our microbes and clean our gut. There are however some fruits and vegetables that are particularly beneficial Phytonutrients: The skins of most vegetables contain phytonutrients \u2013 which naturally occurring bioactive chemicals that give fruits and vegetables their colours. These bioactive compounds all contribute to a healthy gut by promoting growth of the different strains and species of microbes such as Bifidobacteria and Lactobacilli. In turn, the increased diversity of our gut microbes helps us extract more phytonutrients from our diet promoting further growth and increasing the absorption of these beneficial compounds by our bodies.\nGood sources of Phytonutrients include: Cruciferous Vegetables Cruciferous vegetables, which include cabbage, cauliflower and broccoli. contain sulfur-containing metabolites, known as glucosinolates, essential for the production of glutathione, an important antioxidant, as well as amino acids that are essential for building tissue and muscle. However brassicas may not be suitable for those with gut inflammation or leaky gut as while sulphur is beneficial for a normal healthy it can also aggravate inflamed guts. Dark Fruits, Vegetables & Spices Dark Fruits, Vegetables & Spices are rich in flavonoids, thought to help with allergies, inflammation and infection. These can be found in a variety of fruit, vegetables and spices including black tea, red wine, apples, chocolate, peanuts, blackberries, blueberries, purple carrots, red cabbage and eggplant. Blueberries are a widely touted superfoods for their general health benefits as well as encouraging the growth of 'good' bacteria such as Bifidobacterium and Lactobacillus. Leafy Greens Leafy Greens such as spinach, Kale and lettuce are also an excellent source of essential minerals, including magnesium, manganese and potassium. Spinach is also a good source of follate Green and Black Teas Studies have shown consuming black and green teas results in less of the bacteria associated with the obesity and more of the 'good' bacteria linked with lean body mass. tea is a good source of polyphenols, and while the polyphenols in green tea are more effectively absorbed Allium Vegetables Garlic, white onions, shallots and leeks are all rich in Phytonutrients, alliums and allyl sulphur compounds. Garlic in particular is well known as a remedy for killing 'bad' microbes. (The next time you have a sore throat, try eating a food containing raw garlic and you should start to feel better in minutes). These vegetables are also a valuable source of inulin \u2013 one of the most popular prebiotics. Jerusalem Artichokes High in Inulin, an insoluble fiber that bypasses the small intestine, this fibre will find its way to the colon largely undigested where it is fermented by and feeds our good bacteria. As with other foods high in fibre \u2013it may be neccessary to introduce this food carefully and gradually if your gut is sensitive or inflamed. Asparagus: Asparagus is another vegetable rich in Inulin and also in protein. While seasonality limits availability to a brief period it gives you a good excuse to indulge! Apples As the old saying suggests \u2013 an apple a day will indeed feed your gut health and keep the doctor away. High in fibre and low in sugar apples also help generate butyrate upon digestion feeding our good bacteria. Seaweed Seaweed is an excellent source of prebiotics, fibre and omega 3 fatty acids. Interestingly seaweed water is often used in farming and gardening to feed the microbes in the soil and support the plants. Chocolate Good news for all chocoholics \u2013 dark chocolate is good for our gut bacteria health \u2013 as it feeds some of the 'good' bacteria living in our large intestine and colon. Cocoa is broken down to produce nitric oxide, which expands your arteries and is good for your cardiovascular system. Cocoa is also an excellent source of flavonoids and polyphenols which in turn feed our gut bacteria. Other health benefits includeincreasing insulin sensitivity and when consumed daily, lowering blood pressure an average of two to three points. Whole Grains & Fibre Barley & Oats Both Barley and Oats have been an old staple in our diets for centuries or millennia and are both a great source of beta-glucan \u2013 another well regarded prebiotic to feed the 'good' bacteria in our gut. This prebiotic also helps lower LDL cholesterol levels, by stopping it from being absorbed into the blood. Flaxseeds Flaxseeds are both high in insoluble fibre \u2013 helping to regulate bowel movements and also Omega-3 \u2013 which is important for our gut microbiota. Resistant Starch While we advise cutting down or removing sugary foods and refined starch altogether \u2013 resistant starch on the other hand is a good way to feed our good bacteria and strengthen the gut. While refined starch is broken down and absorbed rapidly upsetting the balance of our microbiota and piling on the calories, resistant starch is hard to digest so will pass through the stomach and small intestine largely undigested to reach the large intestine. Here it feeds the good bacteria releasing butyrate which in turn helps reduce inflammation and strengthen the intestinal lining. Resistant starch can be found in grains, seeds, legumes, pulses and in unripe bananas and green peas. Interestingly however the starch present in both pasta and rice can be restructured into resistant starch by cooling and reheating after being initially cooked.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u203a Christians in Malaysia \u203a MyJihad: Can Malaysia do as much?\nMyJihad: Can Malaysia do as much?\nMalaysia needs someone like Imam Mazhar Shaheen to model civilized speech and behavior of this kind in place of the trash talk, bluster, bullying and mindless squabbles about who has a lock on using \"Allah\" and who is a real Malaysian.\nThis kind of action is what would give \"One Malaysia\" substance and give Malaysians \u2013of all faiths and all ethnicities\u2013a shot at building a strong nation.\nThe question is: do politicians and religious leaders in Malaysia have it in them to provide the public space for the kind of interfaith understanding and mutual, respectful acceptance shown in this video?\nWe had it once. We could have it again.\nSimple. Beautiful. So hard to do. Can be done. Yes. It can be done.\nBeautiful. Absolutely beautiful. Gives us all reason to hope.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Online stores are becoming more popular among customers and customers are switching their lifestyle and started to prefer online stores as compared to the other retail store. The most common benefit of an online store is that they are time saving and convenient. Amazon Prime Day Discounts is an online store popular for its products and extraordinary services. it offers a wide range of products at a reasonable and affordable price. On their online website, we can easily find the required product with huge Amazon Prime Day Discounts discounts. Also, their website is continuously being updated on daily basis with Amazon Prime Day Discounts promo codes for the promotion and advertisement of their new and quality products. Amazon Prime Day Discounts management primarily focuses on achieving customer satisfaction through their quality products and services. therefore, to benefit customer Amazon Prime Day Discounts provide them multiple bonuses on completing the different tasks which include sign up bonus, subscription bonus, referral bonus, and also they provide a bonus for their loyal customers on completing specific purchase targets. They also have different Amazon Prime Day Discounts sales throughout the year on different occasion and events. These sales offers are specifically designed and highly advertised to attract potential customer to their online store. With their extraordinary services Amazon Prime Day Discounts hold the recording for successfully transforming their new customers into long-lasting and loyal customers.\nThey are yours only if you want it! Get this Amazon Prime Day Discounts coupon to access excellent savings.\nEnjoy big savings today with Amazon Prime Day Discounts coupon. Grab it now!\nApply Amazon Prime Day Discounts coupon to have the groundbreaking shopping experience. This sale will end soon!\nLimited time only! You will receive an extra discount by using Amazon Prime Day Discounts coupon.\nAwesome offers available at Amazon Prime Day Discounts. Take action and make an excellent deal now.\nGet discount on your order at Amazon Prime Day Discounts. What are you waiting for?\nEnjoy huge price discounts at Amazon Prime Day Discounts! Act quickly before this Amazon Prime Day Discounts coupon is gone!\nFind incredible products with good prices at this store today with Amazon Prime Day Discounts. Click your mouse to shop.\nGet a great sale up to 60% off. Pomp up the fun with this Amazon Prime Day Discounts coupon.\nShop and save money with this awesome deal from Amazon Prime Day Discounts. Act immediately before the sale ends.\nEnjoy huge price discounts at Amazon Prime Day Discounts! This is where families shop.\nChoose from a wide range of selected goods at Amazon Prime Day Discounts and receive amazing discount. A higher form of shopping.\nSpend less and save more with Amazon Prime Day Discounts coupon reach up to 70% off. You heard that!\nThis Amazon Prime Day Discounts coupon will help you save your order. Grasp. Grasp. Grasp it!\nYou can't miss such a great deal at Amazon Prime Day Discounts. Must have it? We've got it.\nShop and save money with this awesome deal from Amazon Prime Day Discounts. Do not miss this amazing opportunity!\nCheck out 50% discount before this amazing Amazon Prime Day Discounts coupon ends!\nClick through to enjoy amazing savings with Amazon Prime Day Discounts coupon. Don't hesitate any longer.\nBest Amazon Prime Day Discounts coupon to save your order up to 60% off. Don't sleep on this offer.\nCheck it out! When is the best time? Now. Want to save more? Use our Amazon Prime Day Discounts coupon. Don't eye it any longer.\nCheck out the steep discounts with Amazon Prime Day Discounts coupon code! More stores. More value.\nGet it immediately, instead of regret later! Save more by using this Amazon Prime Day Discounts coupon.\nA big deal from Amazon Prime Day Discounts that you can't miss! Stack coupons for maximum savings.\nDon't miss it!! Save 20% off on All orders from Amazon Prime Day Discounts !!!\nAwesome offers available at Amazon Prime Day Discounts. You will only find the best deals here!\nTake action and make an excellent deal now! Shop with this Amazon Prime Day Discounts coupon and enjoy attractive discounts!\nFor today time only, Amazon Prime Day Discounts is offering bargains at never-before-seen prices. Groundbreaking bargain for only a limited period.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazrmj b/data_all_eng_slimpj/shuffled/split2/finalzzzazrmj
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index 0000000000000000000000000000000000000000..c0430fe814fbc53e4121fbadc42fbeb80b878562
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@@ -0,0 +1,5 @@
+{"text":"The weather outside is frightful or so the forecast predictions for this winter say.\nWith the added pressure on an already stretched NHS and local authorities, the focus is all about planning for this busy winter period. High costs from illnesses and injuries from trips and falls, means hospitals tend to see a rise in the number of frail and vulnerable patients.\nHandicare has planned and prepared key equipment to manage the overall rise in patients, facilitate easier discharge and assist patients to stay in comfort at home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PICA Australia is pleased to provide you access to pica.com.au. Access to this site is subject to the terms and conditions as stated below.\nPlease read them carefully and let us know if you have any questions.\nPICA Australia and its associates provide their services to you subject to the following conditions. If you visit or shop within this website, you accept these conditions. Please read them carefully and let us know if you have any questions.\nWhen you visit pica.com.au or send e-mails to us, you are communicating with us electronically. You consent to receive communications from us electronically. We will communicate with you by e-mail or by posting notices on this site. 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Arbitration under this agreement shall be conducted under the rules then prevailing of the American Arbitration Association. The arbitrators award shall be binding and may be entered as a judgment in any court of competent jurisdiction. To the fullest extent permitted by applicable law, no arbitration under this Agreement shall be joined to an arbitration involving any other party subject to this Agreement, whether through class arbitration proceedings or otherwise.\nPlease review our other policies, such as our Shipping and Returns policy, posted on this site. These policies also govern your visit to PICA Australia. We reserve the right to make changes to our site, policies, and these Conditions of Use at any time. If any of these conditions shall be deemed invalid, void, or for any reason unenforceable, that condition shall be deemed severable and shall not affect the validity and enforceability of any remaining condition.\nThis policy was updated on July 1, 2015.\nBecause we value your privacy we have taken the necessary precautions to offer an Online Privacy Guarantee. We therefore will not distribute your personal information to outside parties without your consent.\nYour product satisfaction is important to PICA Australia, so we've developed a Consumer Warranty Policy that covers how we handle warranty for end users. Please take a moment to familiarise yourself with our warranty practices and let us know if you have any questions.\nWhere we give you a limited warranty, this is our promise to repair or replace products if they are faulty and faults or defects have not arisen from misuse or normal wear and tear. Any modifications to products will void the warranty. Items not covered by warranty are components expected to be replaced regularly, without limitation, including: batteries, screen guards, cleaning cloths, etc. The warrantor has the right to request the return of the product at your expense. The warranty does not cover any consequential loss or damage.\nOur limited warranty and its benefits are in addition to any legal rights you may have (such as a consumer guarantee under the Australian Consumer Law).\nIf you are a consumer under the Australian Consumer Law, our goods come with guarantees that cannot be excluded under that law. You are entitled to a replacement or refund for a major failure and compensation for any other reasonably foreseeable loss or damage. You are also entitled to have the goods repaired or replaced if the goods fail to be of acceptable quality and the failure does not amount to a major failure. You may not be able to claim some or all of your costs and in these cases you are responsible for these. Unless otherwise specified warranty is limited to twelve (12) months from date of purchase.\nIf you have a faulty product and it is covered by a limited warranty, please email Returns@pica.com.au and you will receive a reply to your email with 2 business days. In order to make a claim under limited warranty, you will need to present the product and original Tax Invoice as supplied by PICA Australia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Having a large family, you can only imagine the amount of Band-Aids we go through. The tiniest of scratch my girls need a Band-Aid.\nThis Christmas, Santa will be stocking their stockings with some fun new prints from Band-Aid including Inside Out, Peanuts, Star Wars and Oh Joy.\nFor my son he will get the new COMPEED\u00ae Blister Cushion which is perfect for him and sports.\nNeosporin is a staple in our house and in my diaper bag. With my accident prone kids it always comes in handy and makes the perfect stocking stuffer for them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Situated 1,500 metres above sea level in B\u00fcrchen, on the south-facing slope of the Rhone valley, the Hotel Restaurant B\u00fcrchnerhof offers views of the alpine landscape. Most of the nicely furnished rooms feature a balcony. A hot tub and a sauna are also provided and massages are available on request.\nMoosalpe price per day from CHF 107 .\nJasmin price per day from CHF 110 .\nEgga C \/ Hilles price per day from CHF 70 .\nZimmer Mamma Mia price per day from CHF 85 Offering a convenient location only 500 metres from the Moosalp Bergbahnen Ski Lift in B\u00fcrchen in the Swiss canton of Valais, Zimmer Mamma Mia provides free WiFi, free shuttle service from the bus station and an \u00e0 la carte restaurant, which serves Italian and local Swiss specialities. Combining traditional Alpine style with elegant modern details, this small family-run hotel offers relaxed atmosphere..\nBietschhorn price per day from \u20ac 69 Set in the Valais Alps, hotel Bietschhorn offers spacious rooms with free Wi-Fi and stunning views of the surrounding Alps. A large children's playground can also be found on site. Guests of Bietschhorn hotel can rent bicycles in summer and explore the area's numerous hiking trails..\nTown of Unterbach (~2 km). 2+ accommodation units !\nTown of Eischoll (~3.1 km). 5+ apartments !\nTown of Raron (~3.4 km). 3+ villas\/holiday houses !\nTown of Zeneggen (~3.9 km).\nTown of Baltschieder (~4.8 km).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sundays 10:30am (9:30am july-Labor Day weekend, see calender). Wednesday services during Lent at 7:30pm preceded by Soup and Bread Suppers at 6:30pm.\nCommunion served twice per month and holy days.\nAdult Sunday School, Sunday mornings at 9:15am.\nConfirmation is a 2 year program for 7th and 8th graders weekly on Mondays from 6:00pm to 7:30pm utilizing the Faith Incubators curriculum. Confirmation events are held as Learning, Service, or Fellowship events. The rite of confirmation is held for 9th graders in October after 3 additional sessions typically held over the summer.\nSunday School and Confirmation programs run from September through May and go on break over the summer.\nAdult Confirmation classes are offered as necessary, with two offerings in the last four years. It is offered as a 10 session program using a curriculum from Sola publishing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazskc b/data_all_eng_slimpj/shuffled/split2/finalzzzazskc
new file mode 100644
index 0000000000000000000000000000000000000000..851ea0717316fb110f315fa0c115c862a3a11440
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@@ -0,0 +1,5 @@
+{"text":"When People magazine's Most Beautiful cover girl dishes on her coveted skincare secrets, the world is bound to listen -- but will they make you look just as flawless? Learn what Oscar-winning actress Lupita Nyong'o uses on her skin.\nWhen People Magazine's Most Beautiful cover girl dishes on her coveted skincare secrets, the world is bound to listen -- but will they make you look just as flawless?\nOscar-winning actress Lupita Nyong'o, who's also famous for her colorful ensembles and silky smooth skin, revealed to Glamour Magazine that she loves to slather Hawaiian kukui oil on her face.\n\"I like to keep things natural, simple, and straightforward,\" says the 31-year-old Mexico-born, Kenya-raised star.\nWhile the product may sound like a new beauty invention, it's actually been used for centuries. According to a 1993 report from the University of Hawaii, kukui oil is produced in the island by cold pressing oil from the nut of the kukui, specifically to protect the delicate skin of newborns.\nSound too good to be true? A 2005 study also reveals that kukui nut oil is loaded with fatty acids, which helps skin remain smooth, supple, and blemish-free. It's no wonder the cosmetics industry is now catching on.\nOne expert agrees with the latest fascination over this golden must-have.\nHowever, another expert is wary of kukui nut oil's sudden popularity and recommends that shoppers think twice before splurging.\n\"There's really not much scientific studies on the use of kukui oil in the dermatologic literature, and the one study I know of failed to demonstrate any differences between kukui oil and the placebo in treating mild psoriasis,\" says Julia Tzu, MD, clinical assistant professor of dermatology at New York University School of Medicine.\nBut the oil does hold some promise. Tzu also says there is some \"anecdotal evidence from its usage in traditional Hawaiian medicine\" that it can help alleviate some common skin problems.\nEven Tanzi admits that while it can quench parched, dry features, kukui oil could possibly clog pores, potentially leading to unwanted breakouts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to outbound sales 101!\nOutbound sales is the process of having sales representatives make phone calls to solicit business. Even with the rise in popularity of other marketing methods, outbound sales continues to be an effective approach. Modern technology has provided a variety of methods to automate and enhance the process.\nPredictive dialers improve efficiency by only delivering live calls to telemarketing staff. The software identifies unanswered calls, calls sent to voice mail and busy signals and continues to make calls until a line is answered. Predictive dialers keep outbound sales representatives focused on making sales rather than dialing numbers, thereby helping to improve selling skills and keep staff focused.\nOutbound sales efforts are enhanced by the use of software that keeps records of each prospect's contact information. This includes address, phone number and any relevant details pertaining to activity such as product purchases, past conversations and scheduled calls. The contact management software works in conjunction with predictive dialing: as the predictive dialer makes calls, contact information is displayed. Automated call logging software such as Tenfold can be used as well to add further efficiency to the process. Tenfold records call activity without the need for manual logging of each call by a sales representative.\nSelecting a database of prospects to be called plays a big part in outbound sales success. When deciding on a database, factors such as past results, demographic details and other such data should be taken into account. Performing sample runs with a portion of a list before purchasing the whole database is one way to improve the cost efficacy of the process.\nScripts outlining their pitch to prospective customers should be provided to sales representatives. A variety of scripts should be tested before deciding on which one is most effective. One way to improve scripts over time is by getting feedback from sales reps to identify improvements that can be made to particular scripts.\nThe performance of outbound sales staff can be gauged using metrics calculating the amount of sales closed in relation to calls made. Other metrics such as callbacks scheduled and time spent per call can be evaluated as well. Training should be conducted to identify areas of improvement for outbound sales staff as a whole. Underperforming representatives can be given extra training to help improve results.\nRead on about how you can take a more calculated approach to sales and exceed your goals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you wondering who should attend our banksman training in Hull?\nAnybody who is responsible for the manoeuvring and loading\/unloading of vehicles should attend this course.\nThe course duration and delivery of our banksman training in Hull is 1 day delivered here at our state-of-the-art training facilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The music is irritating. Specific information in its place would be more useful\/meaningful.\nI liked it. Only thing I would suggest is slowing it down as I could barely keep up with what you were showing. I realize the song kind of dicated the pace, maybe a tad slower song or following every other beat on this one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These leather gloves are very similar to our cow hide gloves but they are made with a coarser slightly thicker Buffalo leather. They are very supple and will help protect your hands from the bees. They have cotton twill gauntlets and are elasticated at the hem for extra protection. The gloves are washable and can be tumble dried on a low heat, however we recommend you leave them to drip dry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazxvc b/data_all_eng_slimpj/shuffled/split2/finalzzzazxvc
new file mode 100644
index 0000000000000000000000000000000000000000..560ae8c771454c051434bfea1ccb8e01278b7068
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@@ -0,0 +1,5 @@
+{"text":"2. And a certain centurion's servant, who was dear unto him, was sick, and ready to die.\n17. And this rumour of him went forth throughout all Judaea, and throughout all the region round about.\n21. And in that same hour he cured many of their infirmities and plagues, and of evil spirits; and unto many that were blind he gave sight.\n25. But what went ye out for to see? A man clothed in soft raiment? Behold, they which are gorgeously apparelled, and live delicately, are in kings' courts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a rule, buying flight tickets from Abu Dhabi to Cairo expends a lot. Using our service of searching for cheap flight tickets, you will be able to save a significant sum of money! We will help tip to buy flight tickets from Abu Dhabi to Cairo at the most favorable prices.\nOut service helps you to find tickets from Abu Dhabi to Cairo in the most reliable travel agencies and booking systems, including all the airline companies. We don't sell tickets and don't do any surcharges, we just find the best offers - you choose yourself where you want to buy flight tickets Abu Dhabi - Cairo at the best price!\nWe also recommend you to look through the prices for return tickets from Cairo to Abu Dhabi.\nThe distance between the airports in Abu Dhabi and Cairo is about 2364km.\nThe cheapest flight from Abu Dhabi to Cairo is $245 on Etihad Airways, departing on 08.06.2019 and returning on 15.07.2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With our range of dairy cakes we offer different protein levels from 16 to 24 %. Our cakes are designed to match your farming system. We have compounds for high performance cows, mid and low input systems. You can select different cakes to match grass based systems and different silage qualities. We can supply compounds suitable for robotic systems and have ones specific for transition cows.\nWe offer blends for dairy, beef and sheep systems. We can make blends to meet your individual needs; you can select combinations of ingredients available in our blend shed. We can formulate blends based on your advice or we can provide nutritional advice if needed.\nWe can adjust your blend so that you can move production forward and also ensure that it is cost-effective. Blends are made from only the finest feed assured ingredients, giving full traceability. Our blends are available either mineralized or unmineralized.\nDiets for high yielding dairy cows need to be packed full of high quality ingredients. For cows to milk well, remain healthy and get pregnant, quality feed needs to be used.\nFarmers on low input systems will want great value for money, whilst feeding cakes that suit their cows. A reduced input dairy farm can include increased grazing, outwintering, cross breeding and many other possibilities.\nHigh quality home-grown forages can be the most cost-effective feeds for dairy cows, making a large contribution to farm profit. Our grass match cakes complement good quality grazing, making sure you get the most from grass.\nOur cakes are very palatable, making them ideal for use in robotic systems. Cows milked on robots will need high quality, palatable cake, attracting them into the robot.\nAs a vital link from one lactation to the next and preparation for calving, the dry period must be managed positively to ensure cows start their new lactation in the best possible nutritional status.\nOur experts can help advise you on all your dairy farming needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very popular playground. Plenty of parking. Several slides, playhouses, lots of climbing and hiding spots, sandbox. Tire swing, regular swings. Climbing wall, climbing ropes. Separate area for younger children. Woodchip and sand base. Plenty of seating; shady and sunny picnic areas. Right near Truro Public Library.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Teen Health Week, an annual week-long designation to raise the profile of adolescent health, will be celebrated globally this year (for the first time) from March 18-24, 2018. We spoke with Dr. Laura Offutt, a physician and teen health advocate, about how she started the event and what is planned for this year. For more information on Teen Health Week, be sure to visit www.TeenHealthWeek.org .\nQ: How did Teen Health Week get started?\nDr. Offutt: The idea for a Teen Health Week occurred to me the summer of 2015 when I was working with a group of youth advisors who were helping me improve Real Talk with Dr. Offutt (my digital teen health resource). I realized that although there are specific observances for specific teen-health topics (Teen Dating Violence Prevention Month), there was no observance that focused on the holistic aspects of teen health. While so much health communication to this age group is fear mongering and preachy (and in turn, ineffective), I thought health should be fun \u2013 after all, what is more enjoyable than feeling and staying well? Putting together a week that felt more like a celebration (lime green spirit wear, youth planning and engagement) captured what I love about health and encouraging others to be healthy as a physician. It also had the potential to inspire young people to be more engaged in their own health and to learn to advocate for themselves.\nQ: What themes and activities are part of the 2018 Teen Health Week?\nDr. Offutt: One of the best parts about this event is how different participants are making Teen Health Week their own. Around the world there are events such as: a yoga\/mindfulness kickoff party, a fun-filled gathering for high school students about Teen Mental Health, a substance abuse panel, many assemblies and guest speakers, a Teen Summit for local schools through a medical system in Nigeria, a theme-based educational event at a school in western Hungary, and the list goes on! Participating regions are highlighted on our global map .\nThe following are suggested themes for each day of Teen Health Week, although individual organizations are invited to focus on one that is in line with their mission. The themes are: Violence Prevention (Sunday, 3\/18); Preventive Care and Vaccines (Monday, 3\/19); Healthy Diet and Exercise (Tuesday, 3\/20); Mental Health (Wednesday, 3\/21); Sexual Development and Health (Thursday, 3\/22); Substance Use and Abuse (Friday, 3\/23); and Oral Health (Saturday, 3\/24).\nQ: How does health care transition fit into Teen Health Week?\nDr. Offutt: Health care transition is an important aspect of Teen Health Week. One of the main goals of the week is to engage adolescents in their own health, and providing them the tools they need to decide when and how to transition from pediatric to adult care is a critical aspect of their health self-advocacy learning. Preventive Care is one of the week's 7 themes, and we have incorporated Got Transition resources and information into the Preventive Care and Vaccines toolkit. For example, a suggested activity for older teens in health class is for them to think about when they might be ready to transition from pediatric to adult care by completing the Health Care Transition Quiz in class.\nQ: After Teen Health Week, what can individuals\/organizations do going forward to advocate for these teen health issues?\nDr. Offutt: We hope that Teen Health Week will be a good opportunity for organizations to pilot initiatives that they will then roll out to a larger audience. At the most basic level, Teen Health Week ideally inspires young people to ask for help, ask questions, and advocate for their own health, which would not end based on a calendar. Health experts can reach out to and inform government officials about teen health issues, shaping policy that affects our youth. It would be wonderful if schools are inspired to start Teen Health Week clubs where young people can plan for future Teen Health Weeks and be peer health leaders in their school communities. In the end, we hope that Teen Health Week draws greater attention to the need for a broader focus on adolescent health and wellness around the world.\nThe transition from pediatric to adult health care is a critical time in the lives of millions of young adults. To engage young adults in their health care transition, Got Transition and its National Young Adult Transition Advisory Group have developed a social media toolkit to promote its new Health Care Transition Quiz, \" Are You Ready to Transition to Adult Care? \" The quiz asks key questions to help young adults prepare for changes that happen usually between 18 and 22. The quiz includes tips and linked resources to support young adults to be self-advocates. These resources include information on how to add medical information to a smartphone, what legal changes occur at age 18, how to sign up for health insurance, and more.\nTo promote the Health Care Transition Quiz and Got Transition's website ( www.GotTransition.org ), the Young Adult Social Media Toolkit includes sample newsletter text, Facebook posts, Tweets, and youth-friendly memes (to be incorporated into Facebook or Twitter posts). We encourage you to incorporate these samples into your social media and other media platforms.\nRegistration is now open for Got Transition's five-part webinar series on Health Care Transition & Title V Care Coordination Initiatives . The series features state Title V, clinical experts, and young adult leaders who will provide examples of best practices among state Title V agencies, tools and resources, and problem-solving strategies.\nThe first session, \"Starting a Transition Improvement Process Using the Six Core Elements of Health Care Transition,\" laid the groundwork for creating an evidence-driven approach to health care transition, building on Title V's rich history in care coordination. Additional session topics will include: Transition Preparation (March 28th, 3-4 pm ET); Transfer to Adult Care (April 26th, 3-4 pm ET); Integration into Adult Care (May 31st, 3-4 pm ET); and Youth, Young Adult, & Parent Engagement (June 28th, 3-4 pm ET).\nBe sure to visit our Webinar page for registration information on upcoming sessions, as well as the recording and handouts from the first webinar session.\nThe American Academy of Pediatrics (AAP) and Got Transition collaborated on a series of 7 ECHO virtual community sessions for the Bureau of Maternal and Child Health's (MCHB) Epilepsy grantees and other health care professionals on incorporating health care transition from pediatric to adult-oriented care into their practice. Each of the 7 sessions includes a case, a brief didactic slide deck, and a discussion. Topics include how to begin a transition process, how to find and partner with adult providers, how to engage families, and more. Click HERE for the 7 sessions and additional transition resources.\nThe National Alliance to Advance Adolescent Health, which runs the Got Transition program, is proud to unveil its newly-updated website . The site includes research, policy communications, and other resources on a host of adolescent health issues that include adolescent-centered care, mental health services, insurance coverage, and health care transition. Please visit www.TheNationalAlliance.org for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"This is a placeholder page for Melinda Bond, which means this person is not currently on this site. We do suggest using the tools below to find Melinda Bond.\nYou are visiting the placeholder page for Melinda Bond. This page is here because someone used our placeholder utility to look for Melinda Bond. We created this page automatically in hopes Melinda Bond would find it. If you are not Melinda Bond, but are an alumni of Cass High School, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The coach went on fire while travelling on the A71 in West Lothian on Monday morning.\nA school bus had to be evacuated during a morning run in West Lothian after a fire broke out on board.\nThe coach, operated by E&M Horsburgh, went up in flames on the A71 at the Wilderness Roundabout at Bellsquarry near Livingston shortly after 8.30am on Monday.\nIt is understood a number of West Calder High School pupils were on board, however no one was injured in the incident.\nConcerned father Daniel Fuller feared his daughter was on the bus and immediately drove from his home in Bathgate to the scene.\nHe told STV News: \"Luckily my daughter wasn't on it.\n\"Fortunately no one was hurt, but the flames spread right through the bus.\n\"It could have been worse - that's the scary thing about it.\"\nWest Lothian Council advised drivers to avoid the area, posting on social media: \"Everyone is safe and emergency services are on the scene.\"\nThe Scottish Fire and Rescue Service confirmed three engines responded.\nA spokeswoman said: \"The Scottish Fire & Rescue Service was alerted at 8.31am to reports of a bus on fire in Livingston.\n\"All people were accounted for.\"\nThe fire crews issued the stop call at 9.42am and thereafter left the scene.\nA spokesperson from E&M Horsburgh said: \"We can confirm that one of our vehicles was unfortunately involved in a fire incident this morning.\n\"There were no passenger injuries as a result of this fire. The cause of the fire is now under investigation.\n\"E&M Horsburgh was established in 1986 and holds an exemplary safety record.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"None of my TV Templates were installed. How do I get them?\nYou can get the TV Templates by downloading them from the download section of this tech support website, or reinstall the software.\nI installed a foreign dictionary, but when I try to select that language under Spell > Select Language, it says that the dictionary is not installed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Career: 20+ years as a Notary Public and Signing Agent. Extensive knowledge of all types of loan documents, Title\/Escrow documents along with a variety of Deeds, and Affidavits.\nThis Notary has revered experience and reliable service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marta Ramos is graduated in Painting from the School of Fine Arts (FBAUL; University of Libon), a post-graduate degree in Illustration by Superior Artistic School of Guimar\u00e3es (ESAG) and a Master's Degree in Visual Arts teaching for the 3rd cycle of Basic Education and Secondary Education in Lus\u00f3fona University of Lisbon (ULHT). Having training and professional experience in photography, illustration, animation, communication design and webdesign. Currently integrates the Gamilearning Project research team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbazc b/data_all_eng_slimpj/shuffled/split2/finalzzzbbazc
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index 0000000000000000000000000000000000000000..84fff4a9d048dcbf0132d922efef5a44f284d753
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@@ -0,0 +1,5 @@
+{"text":"Fans of the original CHIPS tv series may very well find this movie remake absolutely blasphemous. Personally, I don't remember it at all, even though I'm sure I've watched more than a few episodes growing up. However, I'm pretty sure it wasn't crass, vulgar, a tad bit stupid and more than a little suggestive, aka everything this movie is. But gosh darn I thought the movie version was pretty freakin hilarious.\nYes, it's slapstick, and not the smartest (or stupidest) thing you'll watch on any sized screen, but it was fun, had me cracking up and was a good way to spend a couple of flight hours. I thought Dax Shepard and Michael Pe\u00f1a pulled off their characters really well, and there were some moments in the film that were absolutely worthy of laughing out loud, if only I were in the comforts of my own home instead of sharing a plane with hundreds of other strangers! (Ok I don't have the exact number!) And hey, a few minutes of Josh Duhamel can't be a bad thing either, right?\nI mean, I enjoyed it. You'll have to watch the movie for yourself to figure out how you feel about it. Forget the CHIPS of the past. Just take the movie as is and maybe you'll find yourself giggling too. I mean, even Erik Estrada was cool to roll with it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A leading Property firm based in the city is recruiting for an HR Business Partner to support a key project for a period of 12 months (which could go perm thereafter).\nYou will be the primary HR interface with the business to manage the HR relationship with the business unit and to challenge, support and add value to the business. Additionally as HRBP you will create HR plans that drive optimal business performance in terms of Change Management, Talent Management, Organisational Design, Senior Resourcing, Performance Management and Leadership Capability and ensure these are in line with HR polices and standards.\nThis role requires someone with in depth experience working in complex matrixed structures from a retail environment. You will have previously worked in large blue chip organisations and shown you have worked project based roles.\nExperience of organisational and complex business change together with an understanding of organisation development.\nEvidence of working with leadership\/senior teams, ability to establish formal and informal networks and influence change.\nExperience of building capability, managing cultural and behavioural change.\nExposure to international organisational\/people issues.\nIf this role matches your skill set apply immediately to Marcus at Frazer Jones.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The team, which included researchers from universities in Spain, Israel, and the United States, including the University of Washington, focused on the thorax \u2014 the area of the body containing the rib cage and upper spine, which forms a cavity to house the heart and lungs. Using CT scans of fossils from an approximately 60,000-year-old male skeleton known as Kebara 2, researchers were able to create a 3D model of the chest \u2014 one that is different from the longstanding image of the barrel-chested, hunched-over \"caveman.\" The conclusions point to what may have been an upright individual with greater lung capacity and a straighter spine than today's modern human.\nThe study is published Oct. 30 in Nature Communications.\n\"The shape of the thorax is key to understanding how Neandertals moved in their environment because it informs us about their breathing and balance,\" said Asier Gomez-Olivencia, an Ikerbasque Fellow at the University of the Basque Country and the study's lead author.\nAnd how Neandertals moved would have had a direct impact on their ability to survive on the resources available to them, said Patricia Kramer, professor in the UW Department of Anthropology and corresponding author on the paper.\nNeandertals are a type of human that emerged about 400,000 years ago, living mostly from what is today Western Europe to Central Asia. They were hunter-gatherers who, in some areas, lived in caves and who weathered several glacial periods before going extinct about 40,000 years ago. Studies in recent years have suggested that Neandertals and early Homo sapiens interbred, because evidence of Neandertal DNA has turned up in many populations.\nOver the past 150 years, Neandertal remains have been found at many sites in Europe, Asia and the Middle East. This team worked with a skeleton labeled Kebara 2, also known as \"Moshe,\" which was found in Kebara Cave in Northern Israel's Carmel mountain range in 1983. Though the cranium is missing, the remains of the young adult male are considered one of the most complete Neandertal skeleton ever found. Two different forms of dating of the surrounding soil, thermoluminescence and electron spin resonance, put the age at somewhere between 59,000 and 64,000 years.\nDiscoveries and studies of other Neandertal remains in the 19th and early 20th centuries gave rise to theories and images of a stereotypical, hunched-over caveman. Over time, further research clarified scientific understanding of many Neandertal traits, but some debate has lingered over the structure of the thorax, the capacity of the lungs and what conditions Neandertals might have been able to adapt to, or not.\nOver the past decade, virtual reconstruction has become more commonplace in biological anthropology, Kramer explained. The approach is useful with fossils such as the thorax, where fragile bones make physical reconstruction difficult and risky.\nNearly two years ago, the same research team created a virtual reconstruction of the Kebara 2 spine, the first step in updating theories of Neandertal biomechanics. The team's paper, published in the book \"Human Paleontology and Prehistory,\" reaffirmed the likelihood of upright posture but pointed to a straighter spine than that of modern humans.\nThey then used a technique called morphometric analysis to compare the images of Neandertal bones with medical scans of bones from present-day adult men. \"In the reconstruction process, it was necessary to virtually 'cut' and realign some of the parts that showed deformation, and mirror-image some of those that were not so well-preserved in order to get a complete thorax,\" said Gomez-Olivencia.\nThe reconstruction of the thorax, coupled with the team's earlier finding, shows ribs that connect to the spine in an inward direction, forcing the chest cavity outward and allowing the spine to tilt slightly back, with little of the lumbar curve that is part of the modern human skeletal structure. \"The differences between a Neandertal and modern human thorax are striking,\" said Markus Bastir, senior research scientist at the Laboratory of Virtual Anthropology at the National Museum of Natural History in Spain.\n\"The Neandertal spine is located more inside the thorax, which provides more stability,\" said Gomez-Olivencia. \"Also, the thorax is wider in its lower part.\" This shape of the rib cage suggests a larger diaphragm and thus, greater lung capacity.\nWhat that means for how Kebara 2 lived is ripe for further research, Kramer said. How did Neandertals breathe, and for what physical demands might they have needed powerful lungs? What does that tell us about how they moved, and the environment in which they lived? Did any of these physical traits make them more or less adaptive to climate change?\nReconstructing the thorax was an exercise in starting from scratch, deliberately trying to avoid being influenced by past theories of how Neandertals looked or lived, Kramer said.\nOther authors of the study were Daniel Garcia-Martinez of the National Museum of Natural History and Mikel Arlegi, of the University of the Basque Country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amzallag, A. ; Vaillant, C. ; Jacob, M. ; Unser, M. ; Bednar, J. ; Kahn, J.D. ; Dubochet, J. ; Stasiak, A. ; Maddocks, J.H.\nWe use cryo-electron microscopy to compare 3D shapes of 158 bp long DNA minicircles that differ only in the sequence within an 18 bp block containing either a TATA box or a catabolite activator protein binding site. We present a sorting algorithm that correlates the reconstructed shapes and groups them into distinct categories. We conclude that the presence of the TATA box sequence, which is believed to be easily bent, does not significantly affect the observed shapes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This week, Trig is attending the Southern Farm Show at Raleigh's North Carolina State Fairgrounds from February 1-3. Over 300 exhibiting companies from the world of farm equipment will make the Southern Farm Show the largest exhibition of its kind in the Carolinas\/Virginia area of the country.\nIn addition to building our depth of insight to this industry, we'll be attending on behalf of a longtime trusted client who is exhibiting at the show. This client has been one of the foundational collaborators who helped us grow into the branding and marketing services space, giving us the opportunity to help them with their messaging, logos, market research, and help define a visual brand language for their design elements. They have helped us realize our ability to translate product knowledge and customer research into strategic marketing deliverables that have been a driver for growth for their own business.\nIn our first engagement with this client, we developed a comprehensive family of product logos, visual brand language manual for each of their 13 product lines, and supporting printed sales collateral. Next, we worked with them to develop push-marketing techniques through direct mail and digital campaigns to build new channels for communicating with their customers.\nOur current work revolves around the launch of an innovative product line of cutting-edge technology that is truly the next step in precision agriculture. To support the launch, they asked us to create a brand identity and messaging platform that will quickly communicate the enhanced capabilities, flexible product configurations, and clean user experience \u2013 for a total package that will shake up the industry.\nIt is one thing to develop a brand identity that resonates internally, but there is no substitute for market testing to see if you are on the right track. Over the course of the show, we will be iterating on the messaging platform as we interact with customers. This gives us the opportunity to build more effective sales tools that will help to communicate the improvements this product can make to the customer's financial performance.\nTo that end, the Trig\u00ae Brand team is issuing this challenge to the new product development community: you should partner with a branding team that understands how essential customer alignment is to effective innovation programs, especially at the product launch. We invite you to contact us to partner on your next successful launch.\nWe're an innovation consulting, product design, and marketing firm that offers comprehensive insight-to-commercialization services for start-ups, mid-sized, and Fortune 500 companies across the United States. Based in Chapel Hill, the Trig\u00ae team can be found in Charlotte, Raleigh, and Richmond, Virginia.\nThe latest thinking in Innovation, design, marketing, and creative performance optimization for your business\u2014 straight to your inbox.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Eastern NC Stop Human Trafficking Now supports Fair Trade as a Human Trafficking prevention strategy.\nWNCT-TV Channel 9 Greenville, N.C.\nWNCT-TV Channel 9 covered the first Human Trafficking 101 ENC Stop Human Trafficking Now held in Beaufort County as an initiative to expand the community education program. READ MORE ABOUT OUR HUMAN TRAFFICKING 101.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"938 total views, 5 today.\nAbout: in the mood,feel like it,just wanna get nasty.\n860 total views, 7 today.\nAbout: Hi.Im from a small town I enjoy walking and camping in the out doors I live nature and getting out in it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everything you need to know before purchasing your first Tractive GPS Tracker.\nTractive GPS features, finding your pet with Tractive, troubleshooting guide.\nAccount Settings, Subscription Plans, Replacements, Orders & Shipping, MOTION, & Dog Walk.\nYes! All Tractive GPS devices have no sim-card opening or USB port and are therefore not only protected against splash water, but are also absolutely waterproof (up to 1 meter depth, 30 minutes) according to IPX7-standard. As a result, Tractive GPS devices are perfectly suitable for water-loving pets and can, of course, be used in the rain.\nCan I use the Tractive GPS Tracker for my cat?\nYes! The original Tractive GPS Pet Tracker can be used for cats, dogs, and other animals. It was built by pet lovers for pet lovers; to help you ensure the safety and wellbeing of your canine and feline friends.\nNote: Since this article was published, the Tractive GPS CAT Tracker has been made available in select countries. Click here to purchase the Tractive GPS CAT or otherwise subscribe to the Tractive newsletter to receive updates on CAT tracker availability.\nNot only can you assure that your cat will be located with the help of a Tractive GPS Tracker, you can also have fun viewing where your cat has been when she's out and about exploring the local neighborhood or countryside.\nAt 35g and with dimensions of only 51 x 41 x 15mm (2.0 x 1.6 x 0.6in), the tracker is recommended for pets weighing 4.5 kg or above and can be attached via most normal collars or harnesses.\nWhat do our existing customers say about the Tractive GPS Tracker for cats?\n\"I have this on my large (22lbs) cat and it works excellent! I can always see where he is. I get notified when the battery is low and when he goes out of the area I have set on the app. I have had absolutely no problems with it! This is a well built product! Also it has never came off the collar clip used to secure it. Well done!!\"\n\"We bought the Tractive to use for our indoor cat. She was meowing FOR MONTHS....wanting to be let outside. Buying the Tractive has given us our sanity back. She wears a small harness and it is attached to her harness. I know where her favorite spot is and can find her within seconds.\"\n\"I am using this product on a 3.5kg cat. It is an excellent product for any dog\/cat owner for complete peace of mind. The product does exactly what it says and more...\"\nSo what are you waiting for? Invest in the Tractive GPS Tracker for Cats and Dogs today or continue reading all about the Tractive GPS Tracker.\nThe size of the Tractive GPS Tracker varies by model. Continue reading below to find the size, weight, and recommended pet size for each available Tractive GPS Tracker.\nThe original and best-selling Tractive GPS (pictured below) has dimensions of 51 x 41 x 15mm (2.0 x 1.6 x 0.6in) and weighs 35g (1.23 oz); it is therefore recommended for pets weighing 4.5kg (9.9 pounds) and above. These dimensions also apply to the special Hunters Edition and Tractive GPS 3G Tracker.\nCurious if this tracker will be purrfectly suitable for your cat? Click here to read real customer reviews of the original Tractive GPS for cats.\nThe Tractive GPS XL Tracker is the largest Tractive GPS tracking device and therefore offers the longest battery lifetime. With dimensions of 80 x 49 x 29mm (3.1 x 1.9 x 1.1in) and a weight of 142g (5 oz), the tracker is recommended for all pets weighing 20kg (44 pounds) and above. The battery lasts up to 6 weeks on a single charge.\nThe new Tractive GPS CAT Tracker weighs 50g (1.76 oz), fits necks with a circumference ranging from 19 to 30 cm (7.5 to 11.8 in), and is recommended for all cats and small dogs weighing 3.5 kg (7.7 pounds) and above.\nThe new Tractive GPS DOG Tracker has dimensions of 72 x 29 x 16 mm and weighs 35g. It is suitable for all pets 4kg and above.\nMore information about the different Tractive GPS devices can be found here.\nCan't find the answer to your question in our Help Center? Please use the contact button below.\nOur Customer Happiness team is always ready and available to assist you.\nYour request has been submitted successfully and someone will be following up with you shortly.\nIf you want to give us additional information, please reply directly in the ticket confirmation email you have received.\nGathering feedback from our customers helps to improve Tractive's products and services.\nYour idea will be noted for future products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LITTLE ROCK \u2013 Arkansas' public schools have continued to make healthy changes to their nutrition policies and practices, according to the seventh annual evaluation of Act 1220 of 2003, a comprehensive and coordinated approach to addressing childhood obesity in schools and communities.\nThe survey and report, conducted by the University of Arkansas for Medical Sciences (UAMS) Fay W. Boozman College of Public Health, identifies several key trends, including a sustained effort from school districts to prohibit junk foods, most significantly a continuation of the steady decline of school districts that have contracts with soft drink companies. The 80 percent of school districts in the state that had such contracts in 2004 declined to 64 percent in 2010.\nThe percentage of schools reporting compliance with state regulations prohibiting student access to vending machines during lunch periods increased. The report shows 77 percent of schools with vending machines curbed student access in 2010, a 6 percent improvement from the previous year.\nFull reports for each year can be found online at www.uams.edu\/coph\/reports\/#Obesity.\nThe UAMS College of Public Health used interviews and surveys of teachers, administrators, students and their families and a variety of other key groups to evaluate the impact of Act 1220. The evaluation's lead investigators are College of Public Health Dean Jim Raczynski, Ph.D., and Martha Phillips, Ph.D., assistant professor in the UAMS Departments of Psychiatry and Epidemiology. The research and the production of the report were supported by a grant from the Robert Wood Johnson Foundation.\nThe current evaluation also found that school superintendents are working to find solutions to time constraints during the school day to allow more time for students to participate in physical activity and to meet physical education requirements. Only 58 percent of elementary schools in 2004 had regularly scheduled recess compared to 70 percent in 2010. That number is up from 64 percent the previous year.\nParents also show improved commitment to Act 1220's goals as percentages rose nearly 5 percent from 2009 to 2010 in those that prefer three or more days a week of physical education at elementary (75 to 79 percent), middle (79 to 83 percent) and high schools (70 to 75 percent).\nRaczynski said that overall since its creation in 2004, Act 1220 reports have shown greatest improvements and progress in the first three to four years after the passage of the legislation. But continued long-term studies are necessary to continue identifying important trends that lead to better implementation of programs and initiatives to make Arkansas schools healthier places.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The global Powder Coatings Market is likely to gain a significant CAGR in the forthcoming period. The powder coating process is much identical to the painting process with the only difference that, the paint material used as powder coating is dry than liquid. The powder attaches to the parts due to electrostatic charge and grounding. However, several advantages such as reuse and durability are achieved with these whereas the downsides demand less levelling, i.e. needs more peel and curing needs more energy.\nThe powder coatings market is driven by market expansion of construction sector and real estate. The need for aftermarket service in the construction domain is highly prevalent. Jobs like wall painting, house renovation, and furniture maintenance are likely to result in high use of powder coatings to fulfill these jobs. Apart from the residential space, powder coatings find industrial use in the manufacture and packaging sector. Powder coatings are used in industries to enhance the product features and give a finer finish.\nGeographical segmentation for powder coating market includes North America, South America, Europe, Asia-Pacific, Middle East and Africa. Asia-Pacific's powder coatings industry is likely to gain a significant CAGR in the forthcoming period due to the presence of manufacturing sector and broader use of powder coating for varied applications. Rise in demand, increasing use in industries and rise in rate of manufacture are likely to propel the APAC market growth.\nNorth American and European regions are likely to gain a higher CAGR in the forecast period due to increasing use of powder coatings for furniture and construction industry. Apart, government regulations pertaining to manufacture of powder coatings by using eco-friendly products are likely to pump the powder coatings market growth in the near future. The key players in the powder coating industry include Asian Paints Limited, Axalta Coating Systems, Jotun A\/S, Berger Paints India Limited, Kansai Paint Co, PPG Industries Inc, The Valspar Corporation, The Sherwin-Williams Company and the Somar Corporation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Cartoon Sandwich . Clip-art, Illustration.\nSandwich set isolated on a white background.\nColorful cartoon sandwich icon poster. Comfort fast food vector illustration for gift card certificate banner sticker, badge sign, stamp, logo, icon label.\nColorful cartoon sandwich juise fast food set. Comfort fast food vector illustration for gift card certificate banner sticker, badge sign, stamp, logo, icon, label poster.\nA cartoon sandwich before and after a bite.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Returning for a second summer run, Impossible, the glossy West End magic showcase, has a lot in common with an old-fashioned variety night. There are seven performers on the bill, with the 2016 show containing some new faces and some familiar, some new routines and some old.\nAttention has evidently been paid to some of the criticisms leveled at last year's production because the line-up now includes Women. Two of them. Sabine van Diemen gets to saw a man in half. She does it while wearing hot-pants, but I suppose this is progress. Of sorts.\nThe big draw here is the presence of Lance Corporal Richard Jones, the first magician to win Britain's Got Talent, but while he performs a serviceable card trick to a (very) cute kid from the audience, he's lacking in panache and stage presence.\nBen Hart, who featured on the BBC's Killer Magic, is clearly a bit of a magic geek; he has an evident interest in its history \u2013 he dons Victorian costume for one of the stand-out illusions of the night \u2013 but some of his close-up tricks, while elegant, get a little lost, despite the presence of video screens around the theatre.\nThe most solid offerings come from the returnees. Chris Cox's mind reading shtick is slick and escapologist Jonathan Goodwin knows how to wow a crowd \u2013 you have to hand it to a man who will willingly sets his own crotch on fire. Hip-hop street magician Magical Bones, another newcomer, pulls off a couple of impressive tricks \u2013 this is a man can backflip while doing card magic \u2013 but he could really do with dialling down the smarm.\nThe whole thing feels like a kind of budget British magic Avengers. Only with too many Hawkeyes (one of them even has a crossbow), one Cap, two interchangeable Black Widows and the kind of dialogue that would make Joss Whedon weep into his beard. Seriously, some of the writing whiffs like Epoisses that's been left to sit for a bit.\nDespite the glitz, the video, and the Vegas lasers, the show highlights just how difficult it is to make magic work in a pros arch theatre, (and what a spectacularly good job Derren Brown does of it). Van Diemen and escapologist Josephine Lee feel fairly secondary; Lee's water torture cell sequence in particular is weirdly lacking in peril considering it involves her being locked in a glass tank full of water.\nDespite some astonishing moments, the show as whole has a throw-back quality which is fine \u2013 it's clearly pitched at a family audience \u2013 but there's only so much soft rock and portentous voiceover one can take, only so many times you can watch silent women having pointy things thrown at them and being locked in boxes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"October  brought me two very different gigs\u2014one on the world stage and one in a Brooklyn community center.\nPeople were really listening\u2014to the ambassador from Argentina (Argentina sponsored the event) and to my colleagues on the panel that followed: Burhan Gafoor, the ambassador of Singapore; Alana Officer of the World Health Organization in Geneva; and Necodemus Chipfupa, the South Africa regional director for HelpAge International\u2014what fantastic company!\nThe second gig was in a linoleum-floored meeting room on staff-development day at the Fort Greene Council (FGC). The audience consisted of 80 workers from the 13 senior centers FGC sponsors across Brooklyn, not just managers but food service, maintenance and transportation staff. I was one of three white people in the room. The invitation came from the FGC's executive director, Claudette Macey, who had attended the New York City Department for the Aging's launch of its \"End Ageism Now!\" initiative on September 21. That event kicked off with a conversation between Commissioner Donna Corrado and me, and the commissioner gave out copies of my book, This Chair Rocks: A Manifesto Against Ageism, to staff city-wide.\nMacey read it and decided to make history: hers is the first agency to implement a concerted effort to confront ageism in all its senior centers. She bought copies of the book for her entire staff, asked me to sign them and talk about its message, and is assigning a chapter for discussion at a staff meeting each month. The Fort Greene Council will be measuring outcomes, holding a \"graduation\" and establishing a model that other organizations can follow. That's thrilling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The CAAHEP accredited* Diagnostic Medical Sonography program at South Hills will launch you in a career working closely with radiologists, cardiologists, and other medical professionals. You will learn to use high-tech equipment utilizing ultrasound to assess human body. South Hills offers 3 pathways to enroll in sonography based on an individual's background. Within each program, a student may choose between three specializations: General Sonography (OB\/GYN, Abdominal), Cardiac Sonography (echocardiography\/heart), and Vascular Sonography.\nOur DMS graduates hold titles including Cardiac Sonographer, Vascular Sonographer, General\/Abdominal Sonographer, OB\/Gyn Sonographer, Ultrasound Department Manager, Imaging Services Manager, and Applications Specialist.\nThe Diagnostic Medical Sonography program (DMS) is a three-year program in which graduates earn an associate's degree in specialized technology. This program is for those individuals that either do not have any post-secondary education or those that have minimal college credits earned.\nSuccessful students in this program are dedicated and enthusiastic about learning, practice good time management skills, and have good communication skills.\nThis CAAHEP accredited* two-year (6 terms) diploma program is designed for applicants who already have a previous Bachelor's degree in a field other than medical imaging. It includes coursework in anatomy, physiology, medical terminology, patient care, and medical imaging sciences.\nThis CAAHEP accredited* 1.5-year (5 terms) diploma program is designed for applicants who already have a previous 2-year or 4-year degree in a medical imaging specialty such as radiologic technology or nuclear medicine technology. It requires a strong background in anatomy, physiology, medical terminology, patient care, and medical imaging sciences.\n* The DMS, DPP, & DMP programs for General, Cardiac, and Vascular Sonography are accredited by the Commission on Accreditation for Allied Health Education Programs (CAAHEP) upon recommendation of the Joint Review Commission for Diagnostic Medical Sonography.\nThis program information appears in the 2019-20 school catalog.\nTricia Turner received her BS Degree in Exercise and Sports Science from Penn State in 1996 and is a 1999 graduate of the South Hills Diagnostic Medical Sonography program. After working in a clinical environment, Trish joined the South Hills faculty in 2002. She teaches obstetrics, gynecology, abdomen, and vascular sonography. Trish became the DMS Program Director in 2015.\nStephanie Wilson has been at South Hills School of Business & Technology since 2005. She received her Diagnostic Medical Sonography specialized associate degree from South Hills in the late 1990s and went on to earn her bachelor of science degree in Vascular Technology from the Oregon Institute of Technology. Stephanie served as a Senior Vascular Sonographer within the Penn State Heart and Vascular Institute prior to joining the South Hills team. She has been on the board of Directors for the Society for Vascular Ultrasound since 2012 and on the executive committee since 2016. Stephanie has also been on the Joint Review Committee on Education for Cardiovascular Technology (JRCCVT) Board of Directors since 2014.\nA 1983 graduate of Penn State University, Beth Lampe received her Sonography training at the Maryland Institute of Ultrasound Technology in Baltimore, Maryland. Beth has been active in all aspects of Sonography for more than 30 years, focusing on adult and pediatric echo for the last 15 years.\nLiz Ladrido joined the South Hills team in 2009, after being employed as a clinical sonographer at The Johns Hopkins Hospital in Baltimore, Maryland. Liz holds a Bachelor's degree from Penn State University, and she received her sonography training at South Hills. Liz now teaches abdominal and vascular sonography and ultrasound physics.\nGregory Tressler has been in the medical imaging field since 1978. In addition to general sonography, Greg has experience in general radiography, Computed Tomography, and Magnetic Resonance imaging. Greg brings a wealth of clinical knowledge and experience to the Diagnostic Medical Sonography program.\nFor more information about our graduation rates, the median debt of students who completed the program, and other important information, please view the Diagnostic Medical Sonography Disclosures, Diagnostic Medical Sonography Professional Plus Disclosures, or Diagnostic Medical Sonography Professional Disclosures pages on our website.\n* As of 2019, reporting has changed from attrition to retention.\n** Sample size less than 10 causing results to be statistically unreliable.\nThe CAAHEP Annual Survey outcome statistics are based on the students who graduated June 2017. The job placement rate includes graduates hired for jobs within their field and related fields, and were hired within 9 months of completion. The graduates are tracked via graduate and employer surveys.\n* Sample size less than 10 causing results to be statistically unreliable.\nThe CAAHEP Annual Survey outcome statistics are based on the students who graduated June 2016. The job placement rate includes graduates hired for jobs within their field and related fields, and were hired within 9 months of completion. The graduates are tracked via graduate and employer surveys.\nThe CAAHEP Annual Survey outcome statistics are based on the students who graduated June 2015. The job placement rate includes graduates hired for jobs within their field and related fields, and were hired within 9 months of completion. The graduates are tracked via graduate and employer surveys.\nThe CAAHEP Annual Survey outcome statistics are based on the students who graduated June 2014. The job placement rate includes graduates hired for jobs within their field and related fields, and were hired within 9 months of completion. The graduates are tracked via graduate and employer surveys.\nThe CAAHEP Annual Survey outcome statistics are based on the students who graduated June 2013. The job placement rate includes graduates hired for jobs within their field and related fields, and were hired within 9 months of completion. The graduates are tracked via graduate and employer surveys.\nDMS Graduates Can Earn a Bachelor's Degree Online!\nSouth Hills students graduating with the Associate in Specialized Technology Degree in Diagnostic Medical Sonography are eligible to articulate into the Bachelor of Science program in Applied Health Sciences at Pennsylvania College of Technology (PCT). PCT is an accredited college offering Bachelor's and Master's degree programs in a variety of disciplines. All courses required for the degree completion are available online. Go to our Articulation Agreements page or visit www.pct.edu for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take one small step to build your platform and enter to win!\nI've been busy going through all the exciting info gleaned from my three days at Harvard Medical School's publishing course. I'm planning several blog posts on my private conversations with literary agents and publishers on what they're looking for, the state of publishing, what authors can do, etc.\n1. She signed her e-mails with an aol address. Use your website e-mail, such as [email protected] on all e-mails. This is free advertising and many people will look up your website based on your e-mail address. If you haven't set up an e-mail address for your website, just ask your webmaster to do it. You should be able to view and respond to those e-mails in whatever manner you usually view other e-mails.\n2. Her website received 150 visitors a month, but she had no way of continuing to communicate with those visitors. You need some way to convert visitors to followers, whether it's having them subscribe to your blog, having them provide their e-mail address and first name in exchange for a free special report or tip list from you. You want to build a relationship with your visitors and have a way to communicate to them when you have something to offer them.\n3. She wasn't branding herself. In fact, she specialized in several diverse areas with both parents and children. If you want to be seen as an expert, it's best to pick one area to focus on, at least on your home page, in order to send a clear message.\nCan you find one idea from the above to implement in your marketing and platform-building strategy? Let me know what it is and when you've implemented it. Point us to your page. The best implementation will receive a free CD of Writing in the Zone: Tap into Your Creative Genius: a Visualization for Writers, or a free e-kit: The Insider's Guide to Writing Your Best Book in 60 Days.\nPat, when you market yourself as those things, you are telling people the different ways they can access you, but they all have the same marketing message or brand\u2013you put them in touch with their faith; you teach them how to pray and get answered prayer. So your message is consistent, and your market is consistent, even though there are different ways you can do that for people\u2013through your book, workshops, counseling, etc. Great question.\nI've focused on #2 for the last few years. My debut novel releases in September. My goal is to have 1,000 subscribers by then. Considering the amount of time it's taken to capture the newsletter subscribers I have so far, I'm very glad to have started early!\nI think my mother\/co-writer (of other books) and I have found a niche in the \"writing-related newsletter market\" by focusing on our humor. In fact, if you google \"infrequent\" and \"humorous,\" the number one link is to our newsletter.\nHey Lisa, great contest. These are simple ideas that go a long way from JUST being online to being seen by the world as a professional and focused business person.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Ajari Yukei, a wandering monk from Nachi, travels with a mountain guide on a religious pilgrimage. As the sunsets and the day grows late, they find themselves in the Shimotsuke Province in a small village. The two ask one of the local villagers if they might stay the night, but the villager refuses, and Yukei and his guide are left no choice but to spend the night outside.\nWhile they are asleep, they two are paralyzed by a strange power. When they question the villagers about it, they come to find that it was the work of a fox demon which has been terrorizing the village.\nYukei and his guide are then attacked by the demon but, while Yukei manages to flee, the mountain guide is not so lucky and falls prey to the demon. It is not until two skilled archers, Miura-no-suke and Kazusa-no-suke, appear on the scene that the fierce fox demon is finally defeated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All Entries Tagged With: \"antibiotic stewardship\"\nThe fight against antimicrobial-resistant (AMR) infections has become a hot topic in 2016, and the United Nations (UN) has now joined the fray. This year, The Joint Commission and CMS making antimicrobial stewardship programs (ASP) mandatory for all healthcare facilities. At the World Economic Forum in Davos, 74 drug makers, 11 diagnostic test manufacturers, and nine industry groups signed \"The \"Declaration on Combating Antimicrobial Resistance.\" Two different disease strains were found to be resistant to the \"last-resort\" antibiotic colistin.\nDuring the 71st session of the UN General Assembly in New York City last week, the world's governments discussed the increasing dangers posed by AMR infections and doubled down on the need for national and international AMR action plans. This makes AMR infections the fourth health issue in history to be taken up by the U.N. General Assembly after HIV, noncommunicable diseases, and Ebola.\nThe Joint Commission released the standard a month after attending the White House Forum on Antibiotic Stewardship, which focused on implementing changes over the next five years to slow the emergence of antibiotic-resistant bacteria, detect resistant strains, promote stewardship of existing antibiotics, and prevent the spread of resistant infections. Representatives from 150 retailers, food organizations, healthcare organizations, and animal health organizations were in attendance.\nOnly 40% of U.S. hospitals have an antibiotic stewardship program and an estimated 30% to 50% of prescribed antibiotics are unnecessary or inappropriate. In the U.S., drug-resistant diseases cause 23,000 deaths and 2 million illnesses each year. That number is expected to increase exponentially in the upcoming decades.\nLast week, CMS published a list of proposed rule changes for hospitals and laboratories. One of the more notable proposals would make antibiotic stewardship programs [ASP] mandatory. ASPs are viewed by many as a way to prevent antibiotic misuse and the spread of drug-resistant disease.\n\"We propose to change the introductory paragraph [in Infection Control Condition of Participation \u00a7482.42] to require that a hospital's infection prevention and control and antibiotic stewardship programs be active and hospitalwide for the surveillance, prevention, and control of HAIs and other infectious diseases, and for the optimization of antibiotic use through stewardship,\" according to the proposed rule.\nOnly 40% of U.S. hospitals have an antibiotic stewardship program and an estimated 30% to 50% of prescribed antibiotics are unnecessary or inappropriate. In the U.S., drug-resistant diseases cause 23,000 deaths and 2 million illnesses each year.\nThe document also changed included changes to patient rights, restraint usage, physician assistants, medical records, diet orders, periodic evaluations, and several more. To see the full list of changes, click here.\nHealthcare professionals can comment on the proposed rules until 5 p.m. on August 15, 2016. Comments can be sent by traditional mail to CMS or done online.\nIn the wake of a CMS ruling that will make antibiotic stewardship programs [ASP] mandatory, The Joint Commission recently announced that it will roll out a similar standard. Effective January 1, 2017, the new Medication Management standard requires facilities to create an effective ASP. The standard applies to hospitals, critical access hospitals, and nursing care centers.\nThe new standard can be viewed in the July issue of Joint Commission Perspectives.\nHealthcare professionals have 60 days to comment on a proposed CMS rule to make antibiotic stewardship programs (ASP) mandatory for all hospitals and critical access hospitals.\n\"We propose to change the introductory paragraph [in Infection Control Condition of Participation \u00a7482.42] to require that a hospital's infection prevention and control and antibiotic stewardship programs be active and hospital-wide for the surveillance, prevention, and control of HAIs and other infectious diseases, and for the optimization of antibiotic use through stewardship,\" according to the proposed rule.\nReleased late last month, the National Quality Forum's (NQF) Antibiotic Stewardship in Acute Care: A Practical Playbook is broken into five core elements focused on proper antibiotic usage: Leadership commitment, accountability, drug expertise, action, tracking.\nCurrently, only 40% of U.S. hospitals have an antibiotic stewardship program. The book was created by experts from the NQF, Centers for Disease Control and Prevention (CDC), and the Hospital Corporation of America and is based on the CDC's Core Elements of Hospital Antibiotic Stewardship Programs. The playbook also aligns with upcoming Joint Commission standards.\nRead the full article on the NQF playbook at HealthLeaders Media.\nFirst, the good news. Between 2008 and 2014 there was a 50% and 9% drop in central line-associated bloodstream infections at short-term care (STC) facilities and long-term acute care (LTAC) facilities, respectively. Surgical site infection rates are also down by 17% in STC facilities, while LTAC facilities saw a 11% decline in catheter-associated urinary tract infections.\nNow the bad news. Even as hospitals reduce hospital-acquired infections (HAI), there have been more cases of antibiotic-resistant (AR) bacteria. A new Vital Signs report released by the Centers for Disease Control and Prevention (CDC) found that one in seven HAIs at a STC facility is caused by an AR infection. At LTAC facilities, there's a one in four chance that an HAI is caused by a AR infection.\n8% of Enterobacteriaceae are extended-spectrum \u03b2-lactamase-producing.\nThe CDC encourages the healthcare community to continue to focus on preventing HAIs by stronger adherence to existing best practices. The agency has also come up with a new web app that allows users can make customizable, interactive maps and tables with regional, state and national on HAIs caused by AR bacteria.\nOn March 7, the Centers for Disease Control and Prevention (CDC) the release of a new web app; Antibiotic Resistance Patient Safety Atlas (AR Atlas).\nThe AR Atlas includes information from 2011-2014 and data collated from 3,676 acute care hospitals, 506 long-term acute care hospitals and 221 inpatient rehabilitation facilities. Click here to learn more about the app and its uses.\nDespite strong efforts to reduce hospital acquired infections, some of the best teaching hospitals in the U.S. have seen a rapid rise in Clostridium difficile (C. diff) infections. In 2014 alone, 101,074 patients contracted C. diff at a hospital, a 4% increase from the previous year. C. diff infects 450,000 people annually and plays a role in 29,000 deaths.\nAll 24 teaching hospitals have at least 500 beds and over 200,000 patient days a year. Representatives from several of the larger hospitals in question have said that the higher infection rates might be because they treat a higher ratio of sicker patients than non-teaching hospitals or that they may be better at detecting and testing for C. diff.\nThe report suggests better hygiene and antimicrobial stewardship is needed to fight C. diff infections. Antibiotics can kill off the \"good\" bacteria in a patient's stomach that normally keep C. diff at bay, and antibiotic misuse is thought to be one of the main causes in growing C. diff rates. The bacteria is also able to live outside the body for weeks, making hand hygiene and sanitation a high priority.\nOnly 28% of hospitals received top scores in C. diff prevention, four of which were large teaching hospitals; Mount Sinai St. Luke's, Maimonides Medical Center, Maine Medical Center, Harris Health System.\nClick here to see the Consumer Reports interactive map to see how your local hospital did on the report.\nDeveloping new antibiotics is a notoriously difficult task and the rampant misuse of existing antibiotics have made them ineffective against many types of diseases, known as superbugs. Superbugs have become a global health concern, with governments and drug makers joining forces to tackle the problem.\nSuggested solutions have included behavioral changes to how doctors prescribe antibiotics, increased collaboration between medical researchers and no longer giving antibiotics to livestock.\nThe United Kingdom's Review on Antimicrobial Resistance (RAR), which hosts the industry's \"Declaration on Combating Antimicrobial Resistance\" on its website, issued a new report on February 11 saying increasing the use and spread of vaccines can drop the number of superbug cases.\nThe report urges governments and drug makers worldwide to increase the distribution of existing vaccines, as well as develop new vaccines for common ailments. O'Neill singled out the pneumococcal conjugate vaccine, which treats Streptococcus pneumonia, as an example. The vaccine is widely used in parts of the world. O'Neill wrote that global coverage could prevent 800,000 annual deaths of children under five and reduce antibiotic usage.\n\"Universal coverage by a pneumococcal conjugate vaccine could potentially avert 11.4 million days of antibiotic use per year in children younger than five, roughly a 47% reduction in the amount of antibiotics used for pneumonia cases caused by S. pneumonia,\"the report states.\nThe RAR's report estimates that antimicrobial resistance could kill 10 million people per year and cost up to $100 trillion by 2050 if not brought under control.\nClick here to read the full RAR vaccine report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The above statements highlight the need for more close integration between telecom companies and social media. Here's how they can utilize the power of social media.\nHow can telecom companies harness social media tools?\nSocial media tools help telecom companies to design campaigns that attract more subscribers and become the market leader. This is done with quick communication through an enterprise social network that allows different telecom marketing members like creative team, media buyer, field sales reps and senior executives to collaborate on vital decisions. As it is done in real time, telecom providers save money and time by reducing the time needed for reviews and approvals.\nTelecom companies can harness social media to get vital consumer information like service suggestions, questions etc. This will help them design new products and schemes that customers would find attractive. They are already aware of the customer's needs by monitoring their page traffic, digital campaign's responses etc.\nToday, a robust social media presence is not an option, but a must have. Gone are the days when you could depend on traditional mediums of television and print media for public relation (PR) management. With mobile internet browsing increasing at 61% annually, it is imperative for telecom companies to have strong interactive online social media presence. With social media tools, PR teams can filter out news feeds mentioning their telecom company and respond rapidly to any critical situation.\nThe capabilities provided by social media tools negate the need for setting up dedicated resources for complaint and feedback mechanism. They allow you to capture complaints real-time and respond quickly and effectively. Feedback on new services rolled out are maintained and acted upon in future releases. This in turn increases customer engagement. With new features like peer to peer customer support, telecom can reduce complaint calls, thus freeing vital resources. One of our leading telecom clients was able to enjoy a call deflection rate of 44%.\nTelecom companies can enable users to make and receive real-time updates on orders, tickets, alerts, contacts and billing information. With the help of analytic reports, they can determine spending ratios to support new digital services. Executives can access detailed alerts through social notifications to stay on top of network performance. Subscribers can thus avoid unwanted performance and billing surprises.\nTelecom companies can reap the benefits of social CRM to boost their subscriber base. A well designed CRM for telecom will have powerful social tools inbuilt that provides all the above listed features.\nA telecom CRM blog that is wikipedia for business growth!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Men, Woman and Chainsaws,\" a cinema media studies textbook on horror film genres.\nValley. And what better place than West Hills.I have also taken the role of serving as the WHNC Film Liaison to the City to promote the growth of and also address concerns about filming in our communty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cameron, Katharine: The flowers I love. A series of twenty-four drawings in colour by Katharine Cameron. With an anthology of flower poems selected by Edward Thomas.\nThe flowers I love. A series of twenty-four drawings in colour by Katharine Cameron. With an anthology of flower poems selected by Edward Thomas.\nLondon & Edinburgh, T.C. & E.C. Jack, , pp. xi, 82, 24 fine mounted colour-plates after paintings with title-leaf, 4to (180 x 263 mm), uncut, top-edge gilt, finely silk-backed gilt-decorated boards with flower motifs in art nouveau style (signed KC). Spine-ends slightly rubbed, but a fine copy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Designed by S.L. MacGregor Mathers, the head of The Order of the Golden Dawn and combining the elements, the planets and the 22 paths this extraordinary cross can be the key to initiation on the Path to High Magick. To be empowered to any High Purpose.\nApproximate Size: 2?\" x 2\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Omega looks back on a successful past and looks forward to a promising future!\nOmega Technology Plastic \u2013 In the plastics industry this name is synonymous with first class engineered components made from various types of plastic.\nOMEGA TECHNOLOGY PLASTIC is a company name that in the plastics industry is synonymous with first class engineered components made from various types of plastic. These components are predominantly used in the automotive industry, in mechanical engineering applications and in agricultural vehicle technology. It is doubtful that the average driver ever stops to consider how much work goes into such a small component as he adjusts the fresh air vent in his car. There is actually a considerable amount of precision involved to ensure that the surface matches the entire dashboard in terms of colour and structure and that the dimensions fit absolutely perfectly otherwise the part cannot be installed and many other criteria also need to be fulfilled. OMEGA specialises in meeting precisely these types of requirements irrespective of whether the part needs to fit an operating panel in a crane or whether a door handle is required to fit a garage door.\nOMEGA currently holds nearly one thousand different types of tools in stock. These are used to manufacture engineered plastic components which are sent to customers all over the world. How did it all begin? The company was originally founded as P + S Plast- und Schaumtechnik and then the entire injection moulding division was taken over by the company's current owner and managing director Walter D. Olbert in 1995. From this point, the company positioned itself in the marketplace under the name of OMEGA TECHNOLOGY PLASTIC. This name represents the company's clear corporate philosophy which cannot be fully appreciated by merely looking at a glossy brochure but is lived and breathed each and every day by the company's current workforce of over 140 employees. The company stands out from the crowd due to the fact that it always fulfils its promises to its customers to deliver outstanding quality, that it is committed to environmentally-friendly business practices and the management concept is firmly rooted in ethical principles.\nCustomers from the automotive industry in particular require suppliers to focus on adhering to the most recent environmental standards. Attempts are being made to place all articles in reusable containers and to deliver them in reusable packaging and transport equipment. Another concrete example of an environmental measure that has been implemented by the company is the use of waste heat from the machinery to heat the large warehouse.\nThe project team led by our very experienced project manager Friedrich Kenneweg develops tailor-made product ideas for customers. When it comes to developing new cars, issues such as quality and durability are of utmost importance, however, it is now also important to ensure that each component fulfils the requirements stated in the technical specifications that relate to weight reduction. The team at OMEGA is always keen to embrace new challenges and it is therefore important that all employees are well trained in order to meet these requirements. OMEGA has earned an excellent reputation in the Diepholz area in recent years as an accredited training centre for technical and commercial apprenticeships. The willingness to delegate tasks and responsibilities to apprentices at an early stage in their training is a good example of the company's excellent training practices. OMEGA also gets involved in educational programmes with other local companies and provides training in the company's meeting rooms in order to supplement the curricula offered at the vocational colleges.\nWhat goals is the company setting for the future? As Managing Director Walter D. Olbert explains, 'We want to achieve strong and above-average growth and develop our manufacturing facilities in Diepholz in a gradual and secure way.' This has been demonstrated by the various measures that have been undertaken at OMEGA in the past few months which include new items of machinery, a new office area, a 3D measuring device and the newly constructed production and tooling building. These are all impressive examples that demonstrate the company is achieving strong growth and is looking forward to a promising future!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"POSTPONED (USA) (Bay 1997-Stud 2002). 4 wins-1 at 2, Belmont Peter Pan S., Gr.2. Sire of 440 rnrs, 246 wnrs, 14 SW, inc. Dorabella (New Zealand One Thousand Guineas, Gr.1), Consensus, Pasta Post, Wait a Sec, Postmans Daughter, Naturo, Duckworth Lewis, Mr O'Ceirin, Deferment - Farson (Mac.), Post Thyme, Big Hold Up, Moveovr, Roxon, Delayedreaction, SP Trojan Warrior, Mountain Jack, Dorian Grey, Poste Restante, Vittoria, Blue Grass, Referred, etc.\nSIGNORINA, by Conatus. Winner at 1200m in NZ. This is her first foal.\nBoy's Talk. 10 wins to 1200m, A$173,335, SAJC Toro Irrigation 3YO H., Geelong RC Dual Choice P., SAJC Leading Laboratory H., Lakewood Stud H., 2d SAJC SA Volunteers Day H., BTC Sunny Queen Eggs H., 3d MRC Biscay H., BTC Carlton Draught H.\nDonna Rossa. 5 wins. See above.\nEsprit Rossa. 5 wins from 1100m to 1600m, A$173,670, MVRC Drain Man Country Cup, 2d VRC Flemington Vase H., MRC Browns Sawdust & Shavings H., MVRC Nelson Alexander Real Estate H., 3d MRC City of Greater Dandenong H., MVRC Dr Sheahan H.\nAxis Point. 3 wins from 1100m to 1600m to 2016-17 in Aust.\nEndeavour Bay. Winner at 1300m in 2016-17 in Aust.\nGoing Strong. 2 wins at 1350m in Aust.\nIllogical. Winner at 1150m in Aust.\nWritten in Stone. Winner at 1100m in Aust.\nBionic Lad. Placed at 3 in Aust.\nKeep it Secret. 2 wins at 1700m, 1850m in Aust. and China.\nMr Tanaka. Winner at 1250m, STC Bankstown Club P., 3d AJC Emirates H.\nShigeru Mayasan. Winner at 1400m in Japan.\nAunt Ruby. 3 wins at 1100m, 1200m in Aust.\nTrimace. Winner at 1300m in Aust.\nTons of Lute. 5 wins to 1450m in Aust.\nTime Goes By. 5 wins to 1400m in Aust. and Macau, 3d AJC Staff H.\nPrince of Rubiton. Winner at 1200m in Aust.\nMascot (Manihi). 6 wins from 1200m to 1650m, BATC Graduation S., Improvers H., 2d SAJC Queen Adelaide S., L, VRC Byron Moore S., L, 3d SAJC Breeders' S., Gr.2.\nDon Quixote. 4 wins to 1200m in Aust.\nGraceful Dancer. Winner. See above.\nBallet Slippers. Placed at 2, 2d STC 2yo Fillies H.\nAction Time. 3 wins to 1200m in Aust.\nJewel of the Ranch. 2 wins at 800m, 850m in Aust.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home > Gyan > Perks_of_PM > Thoughts > VIP Security > Protecting Indian PM \u2013 Special Protection Group!!\nProtecting Indian PM \u2013 Special Protection Group!!\nThat is what we spend on securing Indian Prime Minister. Read on to know how and where it is spent.\nSecuring Prime Minister of India has always been a critical issue for the security agencies. But, no one thought of handling this in a professional manner before 1981. Prior to 1981, PM's security was handled by a special group within the Delhi Police which provided both proximate security and also protected the residence and offices of the PM. In 1981, a special task force was constituted within the Intelligence Bureau to take over the proximate security in and out of the capital. However, this was still not commensurate with the criticality of the security for the leader of world's largest democracy. The assassination of Indira Gandhi in 1984 brought this loophole to limelight and security for PM was considered from scratch. And thus was sown the seeds for a specialized organization dedicated to needs for the security of the Prime Minister and his family members.\nAs a first short term measure in this direction, high-level committee of secretaries reviewed the security needs and constituted a special task force under dedicated leadership for the purpose. Based on the recommendation of the Birbal Nath Committee to create a Special Protection Unit, the President of India created around 800 posts for the unit under the Cabinet Secretariat. Shortly thereafter the SPU was then re-christened as Special Protection Group (SPG) led by a director of a rank equivalent to the Inspector General of Police. Thus, 8th April 1985 became the foundation day for this elite agency SPG with Dr. S. Subramaniam, then Joint Director (VIP Security) in the IB, as its first Director.\nContrary to what many may think, it was not just a matter of constitution of a new agency under a director. A special team responsible for the proximate security of PM having vast powers to empower it for the tasks was a very new concept for the security agencies. Thus, creation of SPG required elaborate exercise to decide protocols for their activities, developing operating procedures and delineating responsibilities of multiple agencies involved in various aspects of the security. . The provisions contained in the Blue Book, which lays down security guidelines for the protection of the Prime Minister, had to be harmoniously blended with this new concept of proximate security. IB and the State\/UT Police concerned were responsible for coordination, collection, and dissemination of intelligence affecting VIP security. State\/UT Police and the SPG were responsible for providing physical security arrangements for the Prime Minister while the IB was to provide the required intelligence inputs to these operational agencies.\nAs it is today, SPG is headed administratively by a Secretary (Security) who is responsible for monitoring the security arrangements for the Prime Minister. SPG today is comprised of about 3000 personnel and is entrusted with the sole responsibility of protection of the Prime Minister, former PMs, and their close family members. Recruits for SPG are sourced from local Police and the NSG Commandos and they are trained online of the elite US Secret Service. SPG is responsible especially for the proximate security for its protectees including protection provided from close quarters, journey on foot, road, rail, aircraft, watercraft or any other means of transport, and includes places of functions, engagements, residence, etc. They comprise ring round teams, isolation cordons, the sterile zone, and the rostrum & access control to the person or members of his immediate family. This is purely a defensive force for which protection of the VIP is more important than taking down the attacker.\nOperationally, this security group is divided into the four departments or teams named Operations (looks after the actual protection duties and includes the Communications Wing, Technical Wing and Transport Wing), Training (deals with the training of personnel on a continuous basis including training in physical efficiency, firing, anti-sabotage checks, communication and other operative aspects connected with close protection drills), Intelligence & Tours (looks after the threat assessment, internal intelligence pertaining to personnel, verification of character and antecedents, tours and other allied jobs) and Administration (deals with personnel, procurement and other related matters).\nAs one would expect, this agency with such critical responsibility is provided with the best in class equipment to help them fulfill their duties. Belgian seems to be a favorite source for this agency as far as weaponry is considered. A 5.56x45mm NATO caliber, Belgium-made rifle is the weapon of choice for this agency. This 3.5kg rifle fires 850 rounds per minute and has an effective range of 500m. Commandoes are also equipped with an FN Five-Seven semi-automatic pistol with an effective range to fire 5.7mm caliber bullets up to 50m which is also of Belgian make. However, this is not their only equipment. Personnel with such a huge responsibility cannot work merely with attack weapons. A lot more goes into their gears. SPG commandoes use tactical eyewear which provides them protection against splinters and a ballistic shield to prevent any distraction from foreign objects. Agency procures light weight and high-grade bulletproof vests categorized as level-3 Kevlar. These specialized vests weighing over 2.2 kgs can withstand a 7.62- caliber bullet fired from 10m from an AK-47 which can be considered an impressive capability. Further special knee and elbow pads along with non-skid combat boots are part of the gear.\nSecurity of elite is not a one-time plan and needs to evolve with time. Keeping with this requirement, SPG has grown over time in terms of size and specialization. Their protocols and training are reviewed and updated frequently. Their equipment is maintained at par with the best in the world. The scope of agency also undergoes changes with time. The assassination of Rajiv Gandhi brought forth the need for securing a PM after he relinquishes the office and hence a provision was allowed to provide SPG protection to prime ministers and their family for up to 5 years after office. A special case was made for the family of Rajiv Gandhi based on threat perception of central agencies. Similarly, with changing demographics and trends in the country where women have started taking up key positions and availability of female workforce, the agency has recently empaneled a team of women commandoes ready to take up any challenge which comes in their way of protecting a VIP. Prime Minister Manmohan Singh's wife has become the first Special Protection Group (SPG) protectee to have women commandos in her immediate security cordon.\nThere are many who may raise eyebrows at the expenditure that goes in maintaining this agency which is over 400 crore rupees, but I believe that there are certain expenses which are inherent in running a government. Security of the executive head of a country is one of them. Any threat to the head of a country of such vast expanse as India needs to be responded to and his security ensured. Any incident threatening PM's life brings a bad name to the nation in entirely. I would agree that there are politicians who use their security as a status symbol and this needs to be curbed. However, securing PM is not an avoidable overhead but a cost of conducting business. Being the head of a nation of over 1.2 billion is itself a symbol of status and PM needs no security to prove his power. This is just a necessity in today's environment and I hope we maintain this agency at best standards available for such security forces.\nThanks a lot for such a great information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This free application for your smart phone or computer helps you plan and practice conversations with your child about drinking. It provides real life scenarios and offers tips and advice on how to have these important conversations with your child through an interactive simulation.\nRead the article \"Why Small Conversations Make a Big Impression\" to learn more about the value of having conversations with your child about drinking.\nWatch the video first to learn how to application works then download the app at the Apple Store or on GooglePlay.\nYou will first be asked to select parent and child 'avatars' to play out different scenarios.\n\"The Talk\" takes your 'character' through the experience of having a conversation with your child about drinking. You will be able to learn more about the scenario when you click on \"Get the Story\". By using the \"Start Talking\" function, you have ability to play out different parts of the conversation including how to bring up specific points with your child.\nThe \"Goals\" section provides insight into the five goals that you should strive for when having a conversation with your child. It offers tips on what to and what not to say to achieve those goals.\n\"Achievements\" tracks your progress throughout the conversation process.\nThere is a settings and help sections as well, and includes a closed caption option and technical support information.\nParents of young adolescents (12-16) and of older teens\/young adults (17-25) should use this resource to learn more about the importance of having conversations about alcohol at an early age and throughout your child's teen and young adult years. Research shows that parents are the leading influence on their child's decisions about alcohol. The conversations you have with your child do matter.\nThis resource presents information made available through The Substance Abuse and Mental Health Services Administration (SAMHSA), the agency within the U.S. Department of Health and Human Services that leads public health efforts to advance the behavioral health of the nation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Between getting groceries, working on P.T. at Wellness Center, helping DH after dental surgery today, pick up prescription, I threw these additional blocks together. 27 done, need 63. I'm really liking it!\nI already have ideas for the leftovers! Oh dear - when will I have enuf?\nI am really enjoying your progress on these pineapple blocks. It is on my bucket list.\nWow are these cute! Do they work up as fast as they seem to? It seems like everyone else's projects work up much faster than my own. Probably because I am working on 10 things at once.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our GMAT Recording Library is comprised of hours of recordings. It includes teaching sessions from our expert instructors. Each session has a mix of lectures and practice problems from official guides and our own teaching material. When viewing them, you can either press Pause or simply practice along with the class in the video timed by the instructor.\nMake sure to search for the keyword GMAT in the Topic. Each topic category has a different password and you are only given the password(s) to access recordings of a specific topic.\nView more GMAT Teaching Videos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Was there life before tourism?\nThis project was funded, in part, through U.S. Embassy grant. The opinions, findings and conclusions or recommendations expressed herein are those of the Author(s) and do not necessarily reflect those of the Department of State.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The other day, my partner and I were driving through my neighbourhood, Mount Hope. This little suburb is the link that connects downtown Kitchener to uptown Waterloo; while the streets are lined with well-kept century homes, Mount Hope's most well known protagonist is the Mount Hope Cemetery on Moore Avenue. A space that welcomes dog-walkers and acts as a home to several squirrels.\n\"I love my neighbourhood,\" I cooed, watching the houses pass from the car window.\n\"It does have a lot of mature trees and nice architecture,\" he replied, a smug smile spreading across his face. This comment was a cute inside joke, a private secret that he and I share.\nSeveral months earlier, when we had our first conversation on Tinder, he asked me where in Kitchener I lived. I told him I was close to the Google building \u2014 specific enough to determine our proximity, yet vague enough to protect me from potential axe murderers on a dating app.\nOur conversation ran into the next day \u2014 once again, it was vague, the type of initial conversation you have on those apps. I asked him what he was doing that afternoon and he said he was going to take his dog for a walk.\n\"I was thinking of walking her by the Google building,\" he wrote.\nMy \"joke\" \u2014 if that's what it was \u2014 went over well. About an hour later, I met him in the Google parking lot to join him as he walked his dog. Mount Hope became a sort of character in our first date (as did his dog) and it actually challenged my perception of a concrete parking lot. It was in that parking lot that we met, my hands nervously shaking as he watched me walk towards him, unaware of the impact we'd both have on each other.\nSo if a neighbourhood and a parking lot can contribute to this anecdote that we both hold so close to our hearts, what other stories linger in unsuspecting places in Waterloo Region?\nI've been following The New York Times' Modern Love column for some time, and due to the vast volume of submissions, they recently launched Tiny Love Stories \u2014 a space for readers to submit their love stories in 100 words or less. Shamelessly, I copied the idea, asking you, residents and visitors of Waterloo Region, to share with the Community Edition your love stories, using Waterloo Region as a backdrop.\nThe rules were simple: your tiny love story must take place somewhere in Waterloo Region, it cannot be more than 100 words, you can sign off using your real name, or choose to be anonymous, stories told in first person are best and we weren't going to publish all of them, only the ones that we thought best fit our vision.\nThank you for trusting us with the intimacy and intricacies of your private love stories.\nI met you online. It didn't matter to you that we lived 108.2 miles (you) or 174.1 kilometres (me) away from each other. I wasn't so convinced, but I agreed to keep chatting until your \"Good morning, sunshine\" and \"Good night, sweet pea\" texts became the best part of my every day.\nOne day you suggested making the two hour drive from Buffalo, New York to Waterloo, Ontario to meet, lunch, no pressure. But I already knew when you showed up at my door all shy with coffee stained jeans that I'd never want you to leave again.\nWe were visiting town for the second time. Outside the Princess Cinema in 2004, I cried in his car. The movie had felt too real. It was silent, and my nerves grew. I focused on the little streams of water trickling down his windshield as the rain grew heavier. He said my name, nervously, stumbled over it like he hadn't meant to fall and needed my name to steady himself. I love you. I love you too. 14 years have gone by, and we have planted roots here with our daughter. It feels like home the most when it rains.\nWe first met at the KW Humane Society. As I leaned against the glass I whispered, \"You are magnificent.\" I knew this to be true. I didn't know was that I was staring directly at my new family member who would change our lives. We were beyond lucky to have met you, to have stumbled across you, and even more lucky to be the ones to love you \u2014 and for you to love us back. Adopt don't shop. Adopt senior pets.\nI fell in love with a bold 'n' brilliant woman while away in university but couldn't be open about it while here in my hometown.\nAfter a lengthy and complicated experience coming out, I will never forget the first time we were publicly out as a couple in KW. It was at Oktoberfest a few years ago, surrounded by German oompah-pahs and schnitzel, that we danced and held hands. Even though we were leered at when we kissed, I felt brave. We fought for that night \u2014 a celebratory night filled with our profound, simple, queer love.\nI met one of my best friends in the alley beside the Jane Bond. A mutual friend had put us on each other's radar, but then one night when I was walking from the Original Princess to the Princess Twin this guy called out to me from the dark. I went over, said hi, and we've been friends ever since. We'd go on to publish numerous issues of a zine together, go on trips, and be in each other's weddings, but it all goes back to that dark little alley.\nA good part of my relationships have ended in green spaces. With dog and heart wrapped on leash beside the downtown splash pad. Under Huron trees, sitting on burned summer grass, spilling hurt and fears.\nBut a better part have started in them as well. Waterloo Park walk, rain race, first kiss. Seized by impulse, a private moonlight countdown that new year's eve, when we ran into an open field muffled with snow, our teeth chattering over swigs from a bottle of Asti.\nYou can't get away from beginnings and ends in Waterloo. Not if you know where to look.\nWe met at Smile Tiger for our first date, but realized quickly that caffeine wasn't ideal given the time, and instead, suggested we grab a beer at Abe Erb down the street. That walk over was terribly awkward; conversation was minimal. But as soon as we were seated, we couldn't stop talking. At the end of the night, I offered to walk her home and she gratefully agreed, suggesting we walk through Victoria Park where we then shared our first kiss. Almost two years now, we have an apartment together in downtown Kitchener, and still frequent those spots.\nIn a tiny hospital room in Grand River our lives would be forever changed by you, and while we welcomed you and lost you all in the same breath, our love for you is the only thing that keeps us here.\nMost nights we take to the streets, driving for hours, wondering what it would be like to have raised you in this city, the city we feel you in everyday.\nLove isn't always butterflies and dusty rose cheeks, sometimes its pure agony, but to have known you for a minute makes it worth it.\nMissed opportunities, or was it in my head? That's the problem with cute strangers: if you don't speak up, you might never see them again. A few years ago, you worked at Games on Tap. I attended university outside town, yet was there with friends. Cards against Humanity was what we mostly played, two of us not into games. But, it was my birthday. What I said went; had I asked for your number, would you have said yes? Cashing out, you asked if I was from town. I had to say no. But, if you remember \u2014 I'm here now.\nWe both worked at the Gateway Chapters (RIP), and she asked me to see a movie. I wanted to hold her hand, but since I was moving away in a couple weeks, I weirdly just pinched her arm instead. While I was away, we texted and Skyped non-stop, and as soon as I came back we started dating. Less than a year later we were married, and now six years and lots of adventures later, we have a beautiful one-year-old daughter, a growing collection of books, and just the start of a lifetime of wonderful memories.\nI met my girlfriend of almost four years across at a flip-cup table during a Laurier MLSB kegger. She smoked me on the opening draw and gave me a wry smile \u2014 I was hooked immediately. Directly after the game, the entire table departed as if it were mysteriously orchestrated. We were suddenly alone and she complimented my Ghostbusters sweater; my pupils began to dilate. A year later at a co-op job in Toronto over coffee with a manager he mentioned he met his wife at an MLSB kegger at Laurier 13 years prior. Just a bunch of lucky Golden Hawks.\nTo keep reading, check out Chapter Two and Chapter Three.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Reviews \u2013 The HL-5170DN from Brother is a system prepared monochrome laser printer with an astounding print picture and various expandability alternatives. Resolutions keep running as high as 2,400 x 600 dpi, on account of Brother's HQ1200 innovation, bringing about precise and fresh illustrations and content. Print paces of up to 21 ppm convey occupations fast so you can concentrate on different assignments.\nWith the 133 MHz Fujitsu SPARClite processor and 32 MB of inner memory (upgradeable to 160 MB), the HL-5170DN can deal with medium to substantial print employments. Programmed duplexing permits you to effortlessly print twofold sided records. A 250-sheet standard information plate is joined by a 50-sheet multi-reason plate and a 150-sheet yield plate. A discretionary 250-sheet plate is likewise bolstered.\nIn the event that you claim a Brother HL-5170DN laser printer you are most likely completely fulfilled that cash was well spent on this amazing machine. Be that as it may, in the event that you need to appreciate extraordinary quality for cash over the working existence of your printer then the most ideal approach to make awesome reserve funds is to purchase Brother HL-5170DN toners online at Cartridge Save. We work with low running expenses and our proceeded with development has implied that the funds we are making can be gone on to our clients as shabby Brother HL5170DN toners. Disregard shopping on the high road on the grounds that these stores will typically be not able offer you a markdown on Brother HL 5170DN toner cartridges. All things considered, who might point the finger at them considering they need to meet their higher running expenses that frequently will incorporate leasing costly premises. Get yourself a superior arrangement by requesting our less expensive Brother cartridges from the solace of your own home or business and make the most of our fast 24 hour conveyance administration.\nWe offer free conveyance as standard so on the off chance that you are hoping to appreciate a rebate when purchasing Brother HL-5170 toners online without being charged any additional items you have go to the correct spot. We pride ourselves on the nature of our items so you can simply hope to get awesome results when utilizing our unique or good Brother HL5170DN cartridges. There is never any danger included on the grounds that our 30 day cash back gauarantee can be utilized to assert a full discount in the far-fetched occasion of accepting defective or harmed products. Unwind and palce your request for Brother HL 5170DN toners in the sheltered hands of our grant winning client consideration group who will guarantee that you get complete consumer loyalty each time you arrange from Cartridge Save.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"La giralda, one of the largest cathedrals of the world, right up there with St. Peters in Rome and St. Pauls in London. Such an amazing structure, incorporates parts of an old mosque, it is Gothic and Moorish, all mixed up like every thing else in this city. I have been circling the outside everyday and being amazed. Today I went in to see what there was to see. It is grand and has the usual bloody history of cardinals and kings. Rich, rich in every corner, large, imposing and though an amazing structure it did not have the peace of a church, the Walls and stones were too steeped in gore and stories. Who ruled? Kings or Cardinals, it was a thin line of power, there where wheels within wheels, whispers, betrayals and Conspiracies. I wish the walls could tell. Guides have such a rote story that I never listen, one has to be very lucky to find one who actually knows the history and not one who has just learnt up the regular babble to spout forth.\nThere is romance and history and so many stories, some I know and more to be read at leisure and put in perspective now that I have seen the sites.\nI was then on my way to the river and the bus stand to buy my ticket for tomorrow, but everything is so on top of each other and I just get so lucky that I found myself at the plaza de Los Torros. La Maestranza in Sevilla, Is the oldest bullring which was first used for bullfighting in 1765.\nThough I don't think I ever could watch a bull fight, it is such a huge tradition here that I had to go see where and how it happens. So, to see again\u2026 The guide was a young girl whose English sounded distinctly Spanish, I actually understood her Spanish better than the English!\nA tradition that emerged to keep soldiers busy when they were not at war. Right up alongside jousts and hunts along came the bull fight. Did you know that if a bull ( that poor beset animal, that is hounded and poked half to death before ever the toreador gets to him) actually kills the man, the bull is then killed and his head is mounted, not only his, but also his mother's because she bred a killer!\nOn entering that ring I can see how it would truly be a spectacle to observe, with the colour, pomp and show. However I will give it a miss, would probably sit there weeping my eyes out like Malu's Kanika. That it is an art form, I do not doubt, that the toreador has to be an artist trained and brave, too is a reality, however who talks about the bravery off that poor animal, his bewilderment and agony? Somehow it is not man that emerges larger here, the torero has help, that bull is alone in the field with a million men yelling all around him and stupid men on prancing horses poking and prodding at him. By the time that prouncing, dancing man gets there the poor animal is hugely beset and quite lost, what chances? What sport? Cannot see!\nI did eventually get to the bus station and was very glad that I had not left it for tomorrow, such long queues, would have miscalculated for sure and missed the bus!\nHonestly necessary to beware of pick pockets and what here they call the Romanies, and we the gypsies. They look like us \u2013 unfortunately \u2013 foreign us, if that makes any sense. Like every pinch faced person you would avoid at any of our big city stations.\nI walk miles in this city, in the heat, and then I do the good Spanish thing and go up to my air-conditioned, little room and have a siesta. I should actually have it like myriad others under an orange tree 'en alguna plaza', but somewhere I have to draw the line between a teenage backpacking fantasy and my real, today self. The air-conditioning is most welcome.\nThis evening I got nice and lost for about half an hour trying to find another of the recommended tapas bars in la plaza de Los terceros. Have become so familiar with the little pedestrian streets that when I hit equally little, trafficked ones, I got lost. Just being amazed that cars could actually drive through.\nHowever all these lovely people here soon had me walking straight and right into exactly where I wanted to be, sitting under orange trees in a tiny square, drinking great Sangria and eating delicious Tapas at a bar called La Huerta, they specialise in veg tapas, a delightful oxymoron. Recommended: Pavia de bacalao, a fried cod and Quiche de Puerros, leak quiche, also Queso con marmalade de pimientos rojos.\nGot home early. The square in front of the hotel is one that I had not sat in and wanted to from the first day here. However there is so much to see and discover in this city that it is necessary to wander. I am here now, there are lovely tiled benches all around, the lit up tower of La Giralda peeps above one corner, more orange trees and the typical little fountain. How utterly charming does it get? I guess this is what people travel the world for, the charms that we discover.\nThere are a number of restaurants that share space on the square and it all seems to work effortlessly. A tiny bodega in the corner has only two tables but the kids who work in there are charming and have become friends \u2013 telling me where to go and what to see. The waiters in my hotel are friends too and sneak me beers while I sit here writing to tell you all my news.\nI am going to miss this place. I am either too easy to please or else I am finding the most amazing people and places, either way, what a blessing.\nThe expert Sevillano! I am it, have discovered all the magical bits of this city. The Plaza Santa Cruz has a very good Flamenco show at Los Gallos. They have two shows one at 20.00 hrs and the other at 22.00 hrs. A good troupe of dancers and singers, not cheap but you get a drink included so I sat there loving every minute of it and drinking an extremely tall glass of Sangria.\nAfter, I found all the best bodegas in Santa Cruz, it was like they just appeared around the corner. Bar la Estrella, Bar las Teresas and Bar la Columna. Jerez Amontillado in the first and a solomillo whisky (pork loin cooked in brandy or whisky). Another Sangria in the next with some amazing camembert frito \u2013 (fried camembert with raspberry jam).\nI thought I was heading home after, but first I saw a lovely store that sold only white cotton clothes, I have acquired a dress and a shirt that I do not need, but wanted so much. Then I was finally headed home when I walked past La Columna, rocking, but I had no money and you can't buy one tiny drink and a tapa on the card. So I thought \u2026This is going to be one of those few things that Maya said to leave behind!\nTurned the corner and saw La Giralda lighting up the sky and knew that the ATM was somewhere close by\u2026. Another walk around the magically lit monuments into the busy street \u2013 directly to the ATM ( across from the tram stop) and ' hombre que si' I was ready to hit the next bar.\nWondering when I turned into a foody? When I walk and walk and get hungry. Remember these are only tapas so I am doing very well. That was a three course dinner, as far as I was concerned, nicely spaced out.\nThe se\u00f1or was locking up the door to the hotel when I slid in.\nA morning wandering the halls, and gardens of the Alcazar palace. It's much like the Mughal architecture at home, with mixtures of Gothic, the gardens and water ways are pure Mughal, Moorish, whatever we term it!\nStreets have orange trees growing along them, a guitarist sat playing beautifully in the plaza de Los Reyes. Artists had set up around the square. All in all just another great day in Spain.\nToday I took a bus from San Feliu to Barcelona, and am now on the Ave, or the super fast train to Sevilla. It is a wonder to me how anyone who does not speak Spanish manages to travel here. I have met about 3 persons who speak a smattering of English and no others.\nI would have been lost indeed if my Castellano was not returning in leaps and bounds.\nThis train ride is taking me virtually all across Spain and is wonderful. We passed Tarragona and Zaragossa, then we rode through a wilderness of mountains with tiny villages and are now in the plateau beyond. An old hacienda and then a modern installation, fields and hills and all of it sort of reminiscent of India \u2013 in terms of topography, but distinctly Spanish too. Am so glad I chose not to fly.\n! I am in Sevilla! I found a lady taxista, who gave me a constant run down of all the places we were passing and also told me that the hotel I have chosen is a long walk from anywhere that a car might take me. Thank the stars for the Ryan Air, 10 kg restriction!\nHowever she ended up dropping me closer than I thought. I had to walk down cobbled streets narrower than any in Chandni Chowk with music pouring out of every second door because it is a bar, bodega, restaurant or hotel. Past plazas with quaint fountains, only two actually, full of people sitting at candlelit tables. Then I walk down a tiny, tiny alley, seems like a dead end..it opens into a lovely plaza, lots of shady trees ringed with Casa de do\u00f1a Elvira, bodega de no se qual and there at one end is another minuscule alley called calle Gloria, and down that is La Casa de Dona Lina, which is where I am staying. There is a courtyard with tables, stairs going up one side with two rooms off each floor all looking down into the courtyard. I have a tiny room with a very sufficient bathroom and air-conditioning that is trying to work. The temperature is about 40 degrees but it is all totally enchanting. Can't believe how lucky I got finding this tiny corner.\nHave had a lovely late dinner sitting in the square. Walked down another little alley to discover myself at the Alcazar. Victorias clip clipping around the square, magically lit up fairy tale buildings. Indescribable!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I like to say we are in the generation of \"instant coffee\" - we want it all and we want it now! Look around and the material world reflects and supports exactly that - off course as all things have a dual purpose, there are some things about the instant world that I at least quite like - the connectivity it has facilitated as an example, I am sure there are many more endearing factors.\nShe looked at herself appraisingly in the mirror. Her feet painted with an intricate henna design and framed with her delicate anklets. Her long skirt a dark green with tiny crystals twinkling in the light. Her blouse of a fabric woven with varying shades of green and blue and all of this standing out in sharp contrast to the bright yellow of her dupatta, veil. She smeared some kajal into her eyes and placed a red dot of vermillion on her forehead.\nI believe freedom is within each one of us - as much our birthright as the skin on our bodies. And what does freedom mean to me? - the ability to make a choice in every moment about what I think, whether or not I think, how I feel and therefore how I interact with my environment, other beings and myself. If one could know oneself so well and become so aware of oneself that one truly had these choices in every aspect of life - then that is freedom for me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"31 Mar Thirty is a lot of states, all with varying levels of protections for their Cities in the likes of Wyoming and Kentucky and Arkansas are the best .. tiny Lake Michigan towns have the Midwest's largest gay and lesbian resort.\nIt's never been a better time to be gay in the USA. GLBT travelers will find lots of places where they can be themselves without thinking twice. Beaches and big.\n9 Jan The River Market District is the main gay area, and many businesses that And looking at the criteria \u2014 elected LGBT city officials, transgender protections, UM, salt lake city, utah is the top ranked most gayest city in america? really? I guess it's nice we have gay-friendly churches BUT a lot of the LGBT.\nDescription:Walmart is part of a wave of companies that use their considerable economic clout to force progressive change. When her organization launched in , its priority was encouraging employees to come out at work and broach domestic-partner benefits with HR. Saunders started promoting Eureka Springs to friends and family members across the state and created a boomtown. Walmart representatives did not respond to several requests to be interviewed for this story. In only two years, thousands of homes and commercial enterprises were constructed. The organization continues to present an annual summer opera festival in Eureka Springs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This martini has been the sleek steakhouse's signature cocktail for nearly 30 years. We understand why. Vodka infused with fresh pineapple is chilled and served straight up to create a timeless classic.\nMaker's Mark and a honey liqueur are the foundation for this inventive cocktail which gets some color thanks to blackberry compote and is finished with ginger ale, lemon juice and fresh mint, garnished with a lemon twist.\nCimmaron Tequila that has been infused with thyme is combined with a blood orange liqueur, lime juice and some club soda for a little acidity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eighteenth Bank Limited (The) operates in the Commercial banks sector. In addition to historical fundamental analyses, the complete report available to purchase compares The Eighteenth Bank Limited with three other companies in this sector in Japan: Akita Bank Ltd (2018 sales of 47.02 billion Japanese Yen [US$419.92 million] of which 87% was Banking), Towa Bank Ltd (47.70 billion Japanese Yen [US$426.00 million] of which 100% was Banking & Related Activities), and Hokuetsu Bank Limited (48.15 billion Japanese Yen [US$430.02 million] of which 88% was Banking).\nSales Analysis. The Eighteenth Bank Limited reported sales of \u00a547.35 billion (US$422.83 million) for the fiscal year ending March of 2018. This represents a decrease of 6.6% versus 2017, when the company's sales were \u00a550.71 billion. The sales level in 2018 was fairly close to the level five years ago: in 2013, The Eighteenth Bank Limited had sales of \u00a547.89 billion. Contributing to the drop in overall sales was the 10.1% decline in Banking, from \u00a542.88 billion to \u00a538.54 billion. . However, these declines were partially offset by the increase in sales of Leasing Business (up 1.9% to \u00a57.63 billion) and Other (up 7.8% to \u00a51.12 billion) and Assurance Industry (up 7.5% to \u00a5446.00 million) .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oh Ebay. You were there for me when I was 12 and able to convince my mother that I absolutely needed a crappy alarm clock with James Marsters' face on it, and you're here for me now as I struggle to buy college books and make rent. Unfortunately, it looks like you won't have my back when I graduate next year and need a spell that will ensure me a well-paying job right out of college: in their 2012 Fall Seller Update, Ebay announced that it will ban all sales of magic potions, charms, and spells effective August 30th.\nSo I suppose that this is an effort on Ebay's part to protect their users, as it's probably safe to say that if a woman is trying to auction off her soul for a minimum of $2000, she's probably trying to swindle you. On the other hand, I'm sure a lot of folks make a living off hawking their psychic prowess, which, in my opinion, usually boils down to being empathetic and giving good advice (which is a newly prohibited commodity as well). So what will become of these witches, wizards, and all-around magical folk? According to Ebay, they'll have until September 1st to sell the rest of their items, after which their listings will be shut down and removed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Harlequin Football Club announced that it has reached an agreement in principle to become a minority shareholder and strategic partner in USA Rugby's commercial subsidiary, Rugby International Marketing (RIM), and in the US governing body's digital video platform, The Rugby Channel.\nRIM was established in 2015 as the commercial arm of USA Rugby, representing its commercial rights with the responsibility to identify and secure new revenue streams that will support the development of the game in the United States.\nTo build interest in the game, RIM launched The Rugby Channel (TRC) earlier this year, an 'over the top' digital video platform, which features live and on-demand rugby content from the US and around the world. Other strategic investors in RIM include the Rugby Football Union and CSM Sport and Entertainment.\nUnder the terms of its investment, Harlequins will commit the Club's content and programming expertise to enhance The Rugby Channel's value proposition and engage more American sports fans in rugby.\nAmong the first programs to be delivered under the partnership will be Harlequins' 150th Anniversary celebration match against the Maori All Blacks, which The Rugby Channel will stream exclusively live from The Twickenham Stoop on 16th November.\nHarlequins sees the United States as an exciting and promising rugby market, as the game is now the fastest growing team sport in the country with participation numbers in excess of 1.3 million. As the country's men's and women's teams become increasingly competitive around the world, that interest is sure to grow.\nThe American teams' impact in the rugby sevens at the Rio Olympics in August and USA Rugby's selection to host the 2018 Rugby World Cup Sevens in San Francisco represent significant milestones in that process.\nHaving entered into an initial partnership with USA Rugby in 2015, Harlequins' investment into RIM and TRC will expedite the fulfilment of the club's strategic growth plans in America.\nThose plans include pre-season tours and related matches; the identification of US rugby talent and the creation of effective pathways for talented players to realise their potential within the domestic game; the development of a new grassroots rugby competition in California; and the expansion of all aspects of the youth and elite game, from facilities through coaching and playing.\nRIM will represent Harlequins' commercial interests in the USA and will be appointed as the Club's sales agency of record for the country.\nIn addition, as part of the partnership, USA Rugby will use The Twickenham Stoop as a \"home away from home\" for its national teams when touring in the UK and Europe.\nThe Club and the governing body will also collaborate on the delivery of a new \"home and away\" event to excite the sport and engage new and existing supporters.\nCommenting on today's announcement, David Ellis, Chief Executive of Harlequins, said: \"We are delighted to confirm our investment into RIM and TRC as our sport continues to grow at unprecedented levels in America.\n\"Following the signing of our MOU with USA Rugby ahead of last year's World Cup we have been working closely together to support our respective on and off field development. Our pledge to help grow the game in America has been the driver for our work, which to date has been in the form of pre-season tours and supporting the development of grass roots programmes.\nThis sounds like a great investment\/partnership with the Quins! I'm intrigued by the announcement of a new grassroots competition in California. Any one know if this is the new California Cup? I'm also excited to hear about the training partnership as well. Hopefully, that will also turn into an annual match against the Quins in their preseason against our domestic Eagles.\nI do agree. I have asked what the end game is if RIM ends up competing with PRO as they broadcast games (whether by itself or via an obscure partner), NBC Sports, and ESPN as they broadcast Super Rugby and Mitre 10. Shouldn't the point be to get casual fans by getting games on cable channels? Unless they can't get Eagles matches on those channels.\nCan anyone help by telling me how they watch games in the US. I have NBC Sports and Watch ESPN. Any other sources Inam missing? Many thanks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've added an additional parameter to allow passing a PSCredential object to this function\/cmdlet. This is helpful when the user running the script\/cmdlet does not have permissions on the target computer. Another enhancement in this update to to trap errors resulting from failure to WMI connect to the target computer like in case of 'Access Denied', or 'RPC Server not available' errors.\nGet-SBWMI PowerShell function is part of the SB-Tools PowerShell module. It provides similar functionality to the native Get-WMIObject cmdlet with added option: Timeout.\nIn seconds. The default is 20.\nThis is a particularly helpful option especially if you're in the situation of trying to query many hundreds or thousands of computers, where the vast majority respond quickly but few can drag out for several minutes or not respond at all causing the script to hang if using the native Get-WMIObject cmdlet.\nSB-Tools contains functions that depend on POSH-SSH module, and they're typically installed together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjquq b/data_all_eng_slimpj/shuffled/split2/finalzzzbjquq
new file mode 100644
index 0000000000000000000000000000000000000000..90afb425d69183eaf8cc86b4927369444cfa33c5
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@@ -0,0 +1,5 @@
+{"text":"This is a placeholder page for Jon Lisk, which means this person is not currently on this site. We do suggest using the tools below to find Jon Lisk.\nYou are visiting the placeholder page for Jon Lisk. This page is here because someone used our placeholder utility to look for Jon Lisk. We created this page automatically in hopes Jon Lisk would find it. If you are not Jon Lisk, but are an alumni of Palma High School, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We provide effective marketing services that connect with your best prospects and nurture them to become your loyal patients, growing your practice the way that you want it to grow.\nHow do I launch a wellness practice?\nHow do I grow my wellness practice?\nHow can I make a living delivering wellness services?\nHow do I become a recognized expert?\nHow do I relocate my wellness practice?\nBija PracticeBuilder\u2122 delivers business development strategies and tactics.\nBija PracticeStar\u2122 helps you become a recognized expert and celebrity. If you want to speak, publish, professionally consult, and gain celebrity in your area of expertise, we'll coach you through the whole process.\nPracticeAccelerator\u2122 provides additional services to help you achieve your advanced goals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some of the jumpers headed to the final Jumpernite of the winter season on March 22nd and came home with great show miles and ribbons.\nThere were many top ribbons from the EVF hunters. Great job team!\nThe second week at Monroe was also full of great rides and top ribbons. Way to go jumpers!\nExcellent riding by two of our juniors!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"mizonokuchi.me 9 out of 10 based on 232 ratings. 1,819 user reviews.\nHello, I have a 2004 Wrangler that I just purchased a hardtop for. I also purchased a Mopar wire kit. I followed the directions to the letter but the defroster switch light is on and will not turn off. The top is not on the jeep right now so the rear connection is unpulugged.\nYou have a difficult situation there. The hard top wiring harnesses for '97 TJs are no longer sold and even if you could find one, they don't just plug into a newer 2004 2006 hard top... the connectors and circuits are different.\nI have a 2004 jeep wrangler and a hard top that has a rear windshield wiper but on the jeep i do not have the switch to oporate the wipers. Also there is no wires gong to the back to connect to the hardtop. What do i need to install and how would i do so in order to get the rear windshield and defroster working on my hard top.\nThis brand new OE MOPAR\u00ae Hard Top wiring kit includes everything you need to install a Brand New OE Hard top on your Wrangler TJ. Kit includes wiring harness, washer bottle, rear washer pump, defroster switch and wiper washer switch.\nFind great deals on eBay for jeep wiring harness 2004. Shop with confidence.\n2004 Jeep Wrangler Wiring Harness Diagram Third Level In Tj is one from many image from this website. you can find the latest images of 2004 Jeep Wrangler Wiring Harness Diagram Third Level In Tj on this website and other images. we hope this picture can be useful for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's long been thought that when they bound towards you and start licking your face, it's your dog's special way of showing how much they love you.\nBut hold on a second: aren't we getting a little too soppy here? Could it simply be that they recognise that we're their meal ticket - and that showing us affection is nothing more than a survival strategy?\nActually, recent research seems to back up what dog owners kind of knew already: that dogs really do love us. Neuroscientist, Dr. Gregory Berns of Emory University, Atlanta has been studying the \"reward centre\" within canine brains. One of his experiments involves giving dogs sausages in some situations and praise in others. He found that the response to praise was just as strong as the response to food - and that in 20% of cases, it was actually stronger.\nAccording to Berns, dogs on the whole love us \"at least as much as food\" - we'll take that!\nLicking is something that your pooch associates with happiness. Studies have shown that when a dog licks something (or someone), it releases endorphins - i.e. pleasure hormones. They see something they like and that they associate with \"good times\" - so licking it is the most natural thing in the world; it reinforces that feeling of happiness.\nIt's their way of saying hello - and asking about your day. On top of this, it's worth bearing in mind that dogs in packs didn't have access to brushes. So that act of licking performed two very useful functions at the same time: it helped reinforce those all-important bonds between pack members - and provided a chance to clear each other's faces and bodies of ticks, fleas and grime. So your dog may think they're doing you a big favour on the hygiene front.\nAll that salt, sweat, make-up, soap, aftershave, dead skin, traffic fume residue - maybe even food debris: it makes for an interesting and irresistible cocktail. When there's no actual food about, you're the next best thing. Why wouldn't they lick you?\nA friendly lick is one thing - and it's perfectly natural. But there are some situations where licking can be excessive and can be a sign that something's wrong.\nA quick welcome lick when you arrive home is great - but when the greeting seems to last forever - and when you finally shake them off, they retreat to a corner and spend hours licking themselves, is something wrong?\nAs we've mentioned, licking releases endorphins. So it could be that licking is your dog's way of releasing anxiety. And in some situations, compulsive licking can be the sign of an obsessive disorder. Has there been a recent change in your buddy's environment that could be making them feel anxious? Are they left alone for long periods?\nObsessive licking can lead to physical problems (e.g. infections and build ups of painful tissue known as granulomas) and failure to address the anxiety issue is likely to only make it worse. So be sure to consult with your vet if there seems to be something out of the ordinary here. Some specialist behavioural training might be in order.\nLicking could also be a sign of illness. Constant licking of the skin accompanied by head jerks could indicate a flea infestation or other parasitic invasion. If your buddy seems to be licking everything (e.g. you, the walls, the floor!), there might be a gastrointestinal upset. Especially if it's accompanied by diarrhoea, vomiting or listlessness and it doesn't clear up after 24 hours, speak to your vet.\nSo should you let your dog lick you?\nThere's no getting away from it: dog's mouths aren't exactly the cleanest of places.\nIf you're a dog, your own saliva is useful stuff: it contains proteins that help with self-cleansing and even with the healing of wounds. But it's all the other \"nasties\" that sometimes reside in there that ought to cause some concern - many of which are zoonotic - i.e. they can potentially pass to humans and cause illness.\nSalmonella, E-coli and campylobacter can all be present in saliva. And because dogs are prone to eating their own and other dogs' faeces, there's also the risk of passing on certain parasitic infections, such as hookworms and roundworms.\nBut is there a danger of getting overly sensitive about the odd lick? For one thing, pathogens don't absorb through normal, unbroken skin - so the chances of problems caused by being licked on the hand or cheek are actually extremely low.\nThat said, harmful organisms can be absorbed more easily through mucous membranes - so this includes the mouth, nose and eyes. So when your dog comes up to you for a kiss, you might want to offer up your cheek and steer your buddy away from the rest of your face.\nIn general, a healthy, happy doggy can mean fewer health risk for all the family. So just make sure you keep on top of grooming, vaccines and deworming!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkfxf b/data_all_eng_slimpj/shuffled/split2/finalzzzbkfxf
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index 0000000000000000000000000000000000000000..69b0af087553055637855dca682db39b755b4291
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@@ -0,0 +1,5 @@
+{"text":"Here, we report a new approach to the systematization of health screening for employees at a hospital that involves developing a new system requiring little capital investment. A new system was developed with FileMaker Pro using existing order entry and radiological health screening systems. The effectiveness of this new system was examined by comparing the implementation efficiency of health screening for employees before and after its introduction, and by comparison with another such system currently in place at other medical institutions with regard to efficiency. Although the new system has been in place for more than three years, no problems have yet been encountered. This new system is very useful in this hospital and may also be useful for institutions where it is difficult to introduce expensive new systems for systematic health screening because of problems regarding capital management.\nHealth screening, employee, order entry, radiography, FileMaker Pro, workflows, existing systems, systematization, hospital, capital management.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dining room black white ideas chair is one images from 20 fresh white and black rooms of Lentine Marine photos gallery. This image has dimension 800x600 Pixel and File Size 66 KB, you can click the image above to see the large or full size photo. Previous photo in the gallery is decorating ideas black white bedroom room. For next photo in the gallery is modern black white bedroom ideas. You are viewing image #5 of 20, you can see the complete gallery at the bottom below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":".One piece casting of vise bed and flexed jaw.\n.Down thrust \"Semis-sphere Segment\" mechanism-eliminates jaw lift and work tilt.\n.Rigid material-vise body of ductile iron FCD60JIS (equal to GGG60) with 60 kgs\/mm2 or 80,000 psi tensile strength.\n.Vise bed flame with harden treatment up to HRC 45o and up takes up wear and maintain accuracy for long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let us help you or your business with all its self storage needs.\nUnlike nationwide chains, Blacks Road Self Storage is a family owned and operated business. We are committed to providing a secure, top quality facility at a very reasonable price.\nWith monthly rates starting as low as $59.99, you can easily reserve your storage unit, or pay your bill online through our secure website. Remember to look for us on the right-hand side of Blacks Road heading East off of Route 10 in Cheshire, Connecticut.\nAt Blacks Road we value your overall investment. We offer one month free when you pre-pay for 12 months.\nNot sure how much storage you need? Don't panic, we have multiple sizes to choose from.\nNothing can be easier than reserving your self storage unit online. Don't delay, act now to get started.\nNew to self storage services? Have a question about our facility, check out our list of frequently asked questions.\nWhat size storage unit do you need for your stuff?\nBlacks Road Self Storage knows that determining the right size unit is one of the most important considerations when selecting your self storage space. We have an experienced team who will guide you in person every step of the way.\nSelect a storage unit size below to see the highlights, or for more detailed information including unit photos view our storage sizes.\n5 x 10 size unit yields 50 square feet of space and are small, about the size of a large walk-in closet.\nBut still you can store many items as they are an excellent in storing all of your items like yard and garden tools, seasonal items, office supplies, couch, chair, or miscellaneous boxes, etc.\n5 x 15 size unit provides 75 square feet of space, about the size of a very small room.\nPerfect for storing bedroom or office furniture, boxes, ladders, small lawn tractor, motorcycle, etc.\nAre you looking to remodel a room or office? You can have the space to keep all of the contents during construction instead of clogging up other rooms!\n10 x 10 size unit yields 100 square feet of space, about the size of a small bedroom, or 1\/2 the size of a one car garage.\nPerfect for furniture and boxes from a one bedroom apartment or home with refrigerator or washer and dryer are perfect items for this size unit.\n10 x 15 size unit provides 150 square feet of space, about the size of medium bedroom, or 2\/3 of a one car garage.\nGreat for large furniture and boxes from a one bedroom home, or excellent for storage of business records and excess office furniture\/equipment.\n10 x 20 size unit yields 200 square feet of space, the size of a one car garage.\nGreat for furniture and boxes from a two bedroom apartment or home with appliances, or a car or truck, plus additional small items.\n10 x 30 size unit provides 300 square feet of space, the size of a 1.5 car garage.\nPerfect for furniture and boxes from a 3-4 bedroom apartment or home with appliances, patio furniture, bikes, seasonal items, or a large car, truck or boat.\n* This size unit must be requested in advance of occupancy.\nAffordable price ranges according to your requirements.\nKeep your important items safe in our fully fenced in facility.\nStore all your small and big carton boxes free from the elements.\nBy special request, we are available round the clock for you.\nEasily store all your home items, products, and appliances.\nStore your important assets as your business expands.\nWe have car storage with safety measures for all the seasons.\nStore your boat and recreational items during off-season periods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After the success of the recently launched Mi Band 3, Xiaomi has now started crowdfunding the Black Plus Smartband, which features a 0.95-inch AMOLED touchscreen with a resolution of 240\u00d7120.\nThe display is capable of 24-bit true color, and has support for custom dial, caller ID and it can even display messages from popular services such as WeChat, QQ, Alipay, Weibo and Taobao directly onto the device.\nOne of the most unique features of the Black Plus Smartband is that it features multi-function NFC which can be used for a number of handy things, such as simulating your work card and access control card which will allow you to pass your office security checks without pulling out your employee pass. The smartband also supports NFC smart door locks, so in case you have a smart door at your home or workplace, you'll easily be able to lock\/unlock it without pulling out your smartphone.\nFor those in China, the Black Plus Smartband also supports the Beijing-Tianjin-Hebei Intercommunication bus card which allows users to effortlessly travel in 148 cities across the country, with the company planning to add more cities in the near future. Other than that, the Black Plus Smartband is powered by a 120mAh rechargeable battery which has been rated for 18 days of continuous use thanks to its ultra-low power technology and efficient power optimizations.\nFurthermore, the Black Plus Smartband also features a PPG heart rate senor which will accurately track your vitals and a high-precision three-axis sensor coupled with an anti-motion interference monitoring algorithm to accurately track motion data. The Black Plus Smartband is said to be waterproof for up to 50 meters, which means that you'll be able to take it swimming with you.\nUsers will also be able to connect their Black Plus Smartbands to the Mijia app, allowing them to control any Mijia smart home product using the smartband. The Black Plus Smartband is currently being offered at a crowdfunding exclusive price of 229 Yuan (~$33) and it has already reached 165 percent of its crowdfunding goal. The Black Plus Smartband is expected to hit the Chinese market on September 20, with no further information regarding a global release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkvsm b/data_all_eng_slimpj/shuffled/split2/finalzzzbkvsm
new file mode 100644
index 0000000000000000000000000000000000000000..902cfe439778da43d727ea118053b257eb4e8dec
--- /dev/null
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+{"text":"Barbara Thorson, a bunny-ear-wearing and D&D obsessed fifth-grader, is, according to her teachers and classmates, pretty weird. She's fiercely independent and sarcastic, has occasional violent bursts of anger, and proudly declares to a career counsellor who's talking to her class that she already has a career, thank you very much\u2014she's a giant-killer. As the story progresses, readers begin to see the loneliness that hides behind Barbara's independence, to glimpse the fear behind her sarcasm, and to understand that there are very good reasons for her anger. Barbara is trying to kill the biggest giant of them all. But some giants, alas, cannot be killed.\nAnd that, I'm afraid, is all I can tell you about the story, because I Kill Giants is highly spoilable. The revelation it builds up to is a gradual rather than a sudden one, but there was still a panel that caused me to completely break down in tears. (I was reading this book at work during my lunch break\u2014it's times like these that make me glad I'm alone in my office all day). This panel - the whole story, really - was so very moving, so painful, and so very real.\nBarbara's giant has a name, but even before we learn it, the glimpses the book gives us of Barbara's world and its mix of fantasy and reality hint at the serious things that remain unsaid. My favourite thing about I Kill Giants was probably how well it captured the psychological processes of a child facing something so horrifying she can't quite grasp it; how well it portrays her sense of helplessness and her attempts not to be helpless. But as Barbara learns, sometimes being brave means realising that there are things you cannot fight.\nThere is no way around how sad this story is, but there's a sense of acceptance mixed with the sadness, a determination to make the most of her circumstances on Barbara's part, that turns what could have been a devastating ending into a bittersweet one. There's no denying the pain, but as Barbara tells herself in the final panel, we are stronger than we think.\nI honestly don't have enough superlatives to describe I Kill Giants. The story's sheer power, the expressive artwork, the excellent characterization, and the sensitive way in which the characters' emotions are handled made it an instant addition to my list of favourite graphic novels. You know you want to read it too.\nThe artwork is stunning, I must say. You have been on a real roll here lately...not too many duds! I'm intrigued by this book. I might try to find it for the GN Challenge. The character REALLY reminds me of a girl in my daughter's class. She is basically in her own little world, highly brilliant, and ostracized by her peers. I have a soft spot for her.\nYour description, a bunny-ear-wearing fifth-grader, is enough to make me want to read this! I was sort of like that myself (although my bunny ears were not actually visible). Thanks (or no thanks! :--) )for bringing this book that now I want to read to my attention!\nOh Ana this sounds AMAZING. So amazing, in fact, that I said nevermind to my January plans and went to my library's website to request it. Unfortunately, they don't own it (Arg! Horrible GN selection!), and no stores anywhere near me carry it either. :( So I'm on the waiting list at bookmooch.\nWow...how come I've never heard of this book before?!! (Oh yeah, probably that living in a cave thing, huh?) It sounds like such an incredible book, Ana. I still have gift card...this will be mine soon. Thanks, Ana.\nYou know, I'm tempted to scold you for the amount of damage you've done to my wish list already this year, but how can I scold you when you introduce me to a book as wonderful as this, huh?\nWithout knowing what Barbara's giant is, I love her character and feel for her. Your review is fantastic!\nYou have convinced me! I'm going to see if my library has this one. I love the bunny ears and think that Barbara sounds so special although I'm afraid of it too because it sounds like it will make me cry!\nWow! This sounds like a very powerful read and you have totally sold me on reading this graphic novel. I don't read many of them, but I have a feeling that this will be one that moves me very much. I like the artwork too. Very solid and clean looking. Great review, Nymeth. Once again, you recommend a book that I can't deny wanting to get my hands on!!\nI like the artwork! I would like to pay more attention to graphic novels one day. They're quite costly over here, though. Thanks for the review, Ana!\nI've seen my future. I will add this book to the list and covet it mightily. Neither my bookshop nor my library will have it, and I am on a book-buying ban anyway. Many months will go by, and I will forget all about it, and then around October, it will appear at my library.\nWithout having read the book, and with your practical vagueness in the review, this reminds me of what you were saying about Mary Poppins the other day - only older, and more serious.\nYou're so evil (and I mean that in a good, Bad Blogger way of course). I.Want.This.Book.Now!\nOk, so I'm still really happy that I got Blankets the other night when I went graphic novel shopping, but now I REALLY wish that I would've splurged and just got this too :p I just wish graphic novels weren't so expensive!! I would've ended up spending like $50 if I would've gotten both :( But damn this sounds so good. Like I said, I flipped through it and I can see everything you said. It does look like a very emotionally packed book and the art really is so beautiful!\nThis is an incredible book, I loved it and I cried.\nMy library just bought this, so I am looking forward to reading it when it comes in!\nOK this sounds like a must have for me. Will try and convince my hubby to guy it for me, I will play up the D&D angle! Thanks for the review, I hadn't heard of it before.\nSandy: It's mostly that I wanted to tell you about the books I really loved first :P I have a few meh reads saved for next week. And awww, I have a soft spot for that girl too just from hearing you talk about her!\nRebecca: I hope you enjoy it if you do, though enjoy might not be the best word. The perspective worked really really well.\nJill: They explain in the afterword how the ears were not in the script, but the artist wanted something that would make her visibly different, and took the idea from Japanese fashion trends. I think it was a great idea!\nAmanda: I'm so sorry they don't have it :( Hopefully a copy will appear on BookMooch before too long.\nKathy, your heart will break when you do find out :( It's such a sad book, but SO worth it.\nBelle: Who's the short story by? I want to read it!\nZibilee: I think this would be a great intro to the GN medium, so absolutely go for it!\nAlice: Yeah, same here :( And to make matters worse libraries don't carry them at all.\nLightheaded: lol! Evil in a good way, eh? :P Well, so are you!\nHagelrat: It absolutely is. Thank you for your comment urging me to read it asap the other day!\nKailana: yay! Can't wait to hear what you think.\nI was so impressed with the artwork that I picked this up as it came out in individual issues. You are right, it is powerful and very well done. An excellent review, Nymeth. You didn't give anything away as a spoiler and yet able to convey what the story is about. No one picking this up should be surprised by the subject matter nor, I hope, will they disagree that this is well worth the read.\nAwesome review. Another one to be read! I'm sure this is one I will love.\nI don't know if you've already read it, but I think The Complete Persepolis by Marjane Satrapi would by worth checking out for your graphic novel challenge, or just to read. It's a coming of age memoir set in Tehran during the Islamic Revolution in comic form.\nSo I finally got around to reading this. It went out of stock in Amazon when The Boyfriend tried to get it for me earlier this year, but he found it again last week, and can I just say Thank You for your review, because I loved this. And I never would have picked it up without your review.\nSuch a great blend of words & art. Really great.\nI just found out about this book yesterday, and I couldn't be happier. This is one of the best stand-alone graphic novels I've read recently; a pristine example of visual sequential storytelling, that everyone should read.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In case you want to market your own products within the EU, you need to transfer the respective Marketing Authorization to the EU in first place. You may do this by establishing your own branch office. In this case you can also get support from us to set-up a EU-GMP compliant GMP system for the branch. Alternatively, you may choose our comfort option \"Company-As-A-Service\" and we take care of the whole MAH transfer for you.\nThe second step is the GMP-inspection of your manufacturing site. Within our \"Inspection-As-A-Service\" package we perform an intense pre-audit to assess your GMP-status and propose necessary actions for improvement. Furthermore, we offer to manage the inspection process from application to follow-up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"kitchen cabinet doors for sale full size of modern frosted glass kitchen cabinet doors images price for only and agreeable g easy kitchen cabinet doors raised.\nkitchen cabinet doors for sale glass kitchen cabinet doors glass panels for kitchen cabinets within seeded cabinet doors door used glass replacement kitchen cabinet doors glass front as your reference upper cabinets for sale kitchen without upper cabinets new kitchen cabinet doors for sale tags glass.\nkitchen cabinet doors for sale used kitchen cabinet doors full size of kitchen wood kitchen cabinets drawers painting used colors owner used kitchen cabinet doors cabinet doors handmade cabinet doors kitchen cabinet doors cabinet doors handmade cabinet doors kitchen cabinet doors wholesale cabinet doors.\nkitchen cabinet doors for sale flat panel kitchen kitchen cabinet doors spectacular in wow home decoration kitchen painted cabinet doors pros and cons medium size of kitchen cabinets best wood mdf.\nkitchen cabinet doors for sale enlarge planed finish kitchen cabinets barnwood kitchen cabinets barnwood barnwood kitchen cabinets for sale beautiful white lacquer paint kitchen cabinets barnwood kitchen cabinets barnwood kitchen.\nkitchen cabinet doors for sale cupboard contemporary glazed kitchen cupboard doors awesome new kitchen cupboard doors for sale mdeca group kitchen cabinet door handles and knobs bobitaovodainfo kitchen cabinet door handles and knobs kitchen metal kitchen cabinets for sale red kitchen cabinet door.\nkitchen cabinet doors for sale i bought tile on sale at lowes clearance for 300 each and i am going to try this on one kitchen door of the most popular kitchen cabinet door styles here are the most popular kitchen cabinet door styles.\nkitchen cabinet doors for sale cabinet door service cabinet door service cabinet door service cabinet doors kitchen cabinet doors for sale kitchen cabinet doors custom made modern aluminum frame cabinet kitchen cabinet doors custom made modern 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doors door styles sizes island ideas flat panel kitchen cabinets white cabinet doors.\nkitchen cabinet doors for sale glass kitchen cabinet doors for sale only home depot how to add confessions of a serial do it licious k glass kitchen cabinet doors for sale only home depot how to add full size of modern frosted glass kitchen cabinet doors images price for only and agreeable g.\nkitchen cabinet doors for sale leaded glass door inserts glass kitchen cabinet doors for sale only home depot how to add full size of glass kitchen cabinet doors pictures pinterest home depot innovative and wood only appealing.\nkitchen cabinet doors for sale glass kitchen cabinet doors choose glass kitchen cabinet doors modern stunning intended for used glass kitchen glass kitchen cabinet doors kitchen cabinets should you replace or reface hgtv shop this look.\nkitchen cabinet doors for sale used cabinet doors oak kitchen cabinet doors only for sale used replacement cupboard kitchen cabinet doors kitchen 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doors kitchen cabinets inspirational kitchen cabinet doors for sale kitchen cabinets contemporary kitchen cabinet doors for sale lovely kitchen cabinet doors nj ideas buy.\nkitchen cabinet doors for sale raised paint grade cabinet doors as low as acmecabinetdoorscom paint grade cabinet doors as low as.\nkitchen cabinet doors for sale click this picture below to price and order from our most popular cheap cabinet doors also shop cheap cabinet doors on our sister website cabinetdoor kitchen cubboard doors cream gloss kitchen cream gloss kitchen kitchen cubboard doors cream gloss kitchen cream gloss kitchen cupboard doors kitchen cupboard doors for sale.\nkitchen cabinet doors for sale kitchen cabinet glass cupboard doors for sale modern display cabinets cases small trophy case fronted the hickory cabinet doors rustic hickory kitchen cabinets for sale in a hickory cabinet doors rustic hickory kitchen cabinets for sale in a natural stain show tons of.\nkitchen cabinet doors for sale cabinet doors for sale near me kitchen cabinet doors buy new cabinet doors natural wood kitchen cabinet doors for sale near me kitchen buying guide kitchen cabinets at the home depot buy kitchen cabinets.\nkitchen cabinet doors for sale open frame and mullion door cabinets will arrive prepared for glass or other insert material you purchase locally flat panel kitchen cabinets white cabinet doors n for sale white flat panel kitchen cabinets google search for sale cabinet doors door styles sizes island ideas flat panel kitchen cabinets white cabinet doors.\nkitchen cabinet doors for sale kitchen cabinets glass doors beach residence a more info used glass kitchen cabinet doors for sale diy kitchen cabinets ideas plans that are easy cheap to build the chicken wire cabinet doors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 6kg vented tumble dryer allows you to keep track of the cycle with an LED progress indicator. It can also prevent the over-drying of your clothes, thanks to sensor programmes that stop when the optimal dryness level is achieved.\nIdeal for smaller families, this 6kg capacity tumble dryer can dry up to 17 hand towels in one cycle, helping you save time and money.\nYes we do offer a delivery service for an additional charge. The charge for this service depends on the location you require it to be delivered to.\nTo check the price for delivery to your area, you can enter your postcode into a search box under the 'See all delivery & collection options' tab to the right of the product image where all options will be listed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wednesday Dec 11th Just the usual dull day, with Reveille at 7:30. Practice all morning & lazy afternoon.\nI wonder how they organised Reveille with the soldiers billeted all through the village? Did the bugler go to the centre of the village and play?\nMonday Dec 11th another warm and dry looking day. I pity the firefighters down south and have great sympathy for the townspeople awaiting the arrival of the fires.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbllpi b/data_all_eng_slimpj/shuffled/split2/finalzzzbllpi
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+{"text":"SGS Chlorinators offers three innovative products for swimming pool chlorination: the Breeze 320, Breeze 540 and Breeze 760 salt chlorine generators.The Breeze Professional Series is proudly made in the USA, and it is designed with the consumer in mind with its simplicity of operation, dependability and increased longevity.\nYou will no longer have to experience dry, irritated skin, red burning eyes and a heavy chlorine smell due to by-products of typical chlorine sanitizing methods. With the Breeze Series salt chlorine generator, your pool will be continually \"shocked\" by a much milder and safer type of free chlorine that will result in water that feels soft and luxurious.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's some more little-known Bernstein. His 1953 musical \"Wonderful Town,\" with lyrics by Betty Comden and Adolph Green, was based on the popular play called \"My Sister Eileen,\" which tells the story of two sisters from Ohio who come to New York to pursue careers as a writer and an actress. They move into a dumpy Greenwich Village basement and shortly after arriving there sing this song, longing for home.\nAssistant Concertmaster Chair Now Endowed!\nPacific Symphony Presents: \"The Music Man\" In Concert!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kinfonet Quote of the Day: If you really think about it, you will see that your whole life is based on the pursuit of individual security, safety and comfort.\nIf you really think about it, you will see that your whole life is based on the pursuit of individual security, safety and comfort. In this search for security, naturally there is born fear. When you are seeking comfort, when the mind is trying to evade struggle, conflict, sorrow, it must create various avenues of escape, and these avenues of escape become our illusions. So fear, which is the outcome of individual search for security, is the breeder of illusions. This drives you from one religious sect to another, from one philosophy to another, from one teacher to another, to seek that security, that comfort. This you call the search for truth, for happiness.\nNow, there is no security, no comfort, but only clarity of thought which brings about the understanding of the fundamental cause of suffering, which alone will liberate man. In this liberation lies the blessedness of the present. I say that there is an eternal reality which can be discovered only when the mind is free from all illusion. So beware of the person who offers you comfort, for in this there must be exploitation; he creates a snare in which you are caught like a fish in a net.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Place Used To Be A Home To Nightmares. Now? It's All Fun And Adventure.\nThe most recent one I noticed was robbiehurz. I'll miss his encouraging memes with Bible verses, and other memes. It's sad.\nWhy I procrastinate cutting my fingernails.\nHe Approached A Stray Dog On The Street. What He Did Next Will Warm Your Heart!\nThis Guy's Roommate Left Him With A Complete Mess. What He Did In Return Is Totally Awesome.\nThis Bug Protects Itself By Looking Like Something Much Scarier Than It Is.\nOh who am I kidding. No one else has had this happen to them\u2026 right?\nBecause I own two mansions and a condo in the friend zone.\nMy Battery Is 4% Ahh\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since its beginnings in 1876, the Crawford AuSable School District has been serving the needs of students in our community. Located in Grayling, Michigan -- the county seat of Crawford County in the middle of northern Michigan -- we operate three schools that serve over 1,600 students in pre-school through grade 12.\nOur administrative offices are located within Grayling High School at 1135 Old U.S. 27 North in Grayling.\nOur school district and individual schools have received many awards of recognition throughout the past several years that are described in the Points of Pride section of our site's home page.\nThe Crawford AuSable School district, as a partner with our families and community, educates students in a safe, challenging and supportive environment, graduating citizens prepared for life in a global society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Mitsubishi is talking bullishly about the Outlander Plug-in Hybrid Electric Vehicle being a \"game changer\" in the segment, and it's easy to see why.\nCombining a petrol engine with an electric motor, it's capable of around 32.5 miles on electric power alone before switching to a petrol-electric hybrid. That technology offers figures of 44g\/km of CO2, putting the car in the 5% BIK band for this year.\nAs Mitsubishi proclaims, it's a full-size four-wheel drive SUV with five seats and a big boot, and for drivers doing less than 30 miles between charges, it won't use any fuel. And best of all is the price, which, post-Government \u00a35000 grant, is comparable with the equivalent diesel Outlander, rather than being saddled with the kind of price premium that is putting some people off plug-in cars. Employers' National Insurance expense will also be reduced as a result of the Outlander's ultra low-emission vehicle status.\nThis category was one of the most impressive in terms of depth of entries, so we've chosen two Highly Commendeds.\nHyundai has moved into leadership of the hydrogen fuel cell arena by developing a version of its ix35 crossover model powered by the zero tailpipe-emission fuel. The first fuel cell to enter series production, Hyundai is already running hydrogen-powered cars with fleets in Europe, with seven cars due to be operational in London imminently.\nWe also highly commended Tusker for its efforts to help fleets cut their environmental impact in salary sacrifice through both its impressive low-emission vehicle microsite offering a wealth of information and help to drivers considering ULEVs, and by developing a carbon offsetting facility during the order process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Christy Essien Igbokwe, an actress, a singer and the former President of the Performing Musicians Association of Nigeria is a delight any day. In this interview with ADAEZE AMOS, she talks about her experience as the PMAN president and her humble beginning.\nYou have been in the cooler for some time now, and why haven't you released new albums?\nThe problem is this: sometimes things happen without you knowing why, or without you having a tangible explanation why they happen. I remember a woman of God once told me when I asked her why things should happen like this to some people. She just laughed and asked me what happened. I explained to her what happened.\nWhat did you tell her happened to you?\nNo, I won't tell you. I don't want to talk about it.\nWhy not just say a part, not all?\nCan you continue with your story please?\nOkay, so the woman now said sometimes God could delay things, especially if he is not through with you. That if he stills wants to use you to touch people's lives and glorify his name, that he could cause some little upsets in your business and in your life. But the only thing is that he will not take your life nor allow you to be sorrowful, but he could cause some upsets. And that if things keep happening to you smoothly, you ought to get worried. I later believed her.\nYes, because anything can happen; you may just die and go. When she told me this, I didn't believe immediately because nothing was happening to me that was that bad. I came back home and I thought over what she said, still doubting. Then when I dreamt, I was told in my dream that what the woman said was right. That life is all about ups and downs, that life is about experiences. When you get an experience that you do not expect, there is something there for you to learn and if you fail to, then you are not being realistic. God is perfect; it is only him that knows us in and out.\nYou were the president of the Performing Musicians Association of Nigeria. What was the experience like?\nYes, let me start by saying how I became the president of PMAN. Some people came and begged me to come and lead the union. I never said I wanted to be the president. By that time I was supposed to be on tour to the U.S. It was only by God's grace that I was not sued by the company because I got the visa stamped to stay in the U.S. for six months. It was some old men that came to me and begged me to be the president of PMAN. They said that since I formed PMAN, I should come and be the mother of PMAN. I told them that I was supposed to be on top. I told them I had gone for the visa (to the U.S.) and how I had wished I hadn't gone for the visa. That didn't deter them. They continued begging for two weeks and I didn't know what to do. I had to recall my husband where he was then and told him, 'please come, I'm in trouble.' Do you know why I later got worried? My father had told me that when elders come beckoning to you, begging you to do something, if you don't do it, you are finished. So, I said to my husband, 'please come.' I also called a family friend and they all met the seven old men. They kept pleading and my husband and my family friend said I should accept it. I then accepted. It's not as if I went for it.\nYour experience as the PMAN president?\nYes, when I got there, God knew I was selfless. Only God can vindicate me, nobody else can. I don't care what anybody says. But you see, I keep saying that they should go and check my records. I'm the first PMAN president that handed over in writing what I saw, what I'm owing before I left. Nobody has ever done that. I'm the first president that didn't use the union's money for anything \u2013 accommodation, feeding or even allowance. I even used the commissions due to me in helping the union, which I also accounted for.\nThis is because I kept an open account that was operated by me and the national treasurer, which of course shouldn't have been so. By right, I was due to my commissions, but I opened an account that the treasurer was a co-signatory to with me and I used that money to take care of the musicians who were in trouble.\nBut I hope the musicians appreciated what you did?\nFor where? Musicians are very ungrateful people.\nBut you are a musician, aren't you?\nYes I am. I'm part of them, but I'm not ungrateful. I wanted to make a difference in the union. I didn't go there to take their money. I didn't need their money. But very soon God will vindicate me. When I was there, they said all sorts of things, but I just laughed at them without talking. People said why? I told them that my grandmother told me that if you are sure of yourself, what you have done for the good of the people and with the fear of God, don't talk or be hasty to talk. She said you should let them finish talking; that God would vindicate you and that was what I did. It is thieves that call others thieves. If you are not a thief, you can't say somebody is a thief. Although I have left the throne as the president, I still wish them well. I have always been there for them.\nWhy is it that you have not been participating actively in home videos as an actress?\nWell, don't worry about that, but I have acted some home videos. I was in Flesh and Blood. Then I also acted in Tears of Womanhood, which was launched by Mrs. Jerry Rawlings in Ghana. You see, one thing with me is that people bring scripts to me. I can show you many scripts here with me, but you see, if I don't believe in the story line and the part that is given to me, I won't act it. Take it or leave it, I'm a role model. There are some parts I shouldn't do.\nWhy haven't you been able to produce your own movie?\nThere is time for everything. I sang it now. It's one of my tracks. Let me sing it for you... 'There is time, time, time for everything (sings).\nSo that then means you are waiting for that time?\nYes. When that time comes, I will do everything that God wants me to do.\nWhat are you doing to maintain your voice?\nI don't do anything. God gave me the voice. God told me that I'm a member of the choir in heaven. He didn't only say that to me, he also told other people that I'm a member of the choir and that is why my voice has remained even without me taking care of it. I don't need to; God said I shouldn't bother; that I should enjoy myself and do what I want to do; that my voice will remain protected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From 2007 to 2011 Katherine was undertaking design research, observing a textile development project in Burkina Faso in West Africa. The charity has been working with the indigenous strip weavers who practise an ancient form of weaving that is found all over West Africa. More information on this technique can be found on the post here. Using a variety of local textiles Katherine created samples that the charity could use as part of their contemporary collection.\nKatherine's own collection, BAMAKO, was exhibited at the ICFF New York in May 2011 as part of her PhD practice. These products were a hybrid of contemporary digital manufacturing and traditional African imagery that explored notions of 'authenticity' and 'African-ness' in consumers' perceptions.\nThe animal cushions and toys for children are also part of the BAMAKO collection. The designs are based on traditional appliqu\u00e9d textiles of the Dahomey peoples of northern Benin in West Africa.\nTheir meaning, as a cultural product, has been lost over the years. They have journeyed from being symbols of various royal kings \u2013 or 'fons' \u2013 of the Dahomey people of Benin, with only tightly controlled family guilds allowed to produce them, to tourist craft sold in every market in the country. The original technique was taught to the Dahomey royal artisans by Brazilian Portuguese settlers, and this has been recorded as far back as the 1840s by European travellers. Thus the 'authenticity' of these animals is a moveable and fluid definition, obscured by cross-cultural relationships that are woven through their history.\nIn 1997 Katherine Ladd co-founded lighting design consultancy, GECKO. Specialising in architectural public space lighting, the company also designed and manufactured a range of domestic lighting. They were commissioned to provide the lighting at the 2000 Vanity Fair post-Oscars party in Los Angeles, and won further restaurant lighting contracts all over the US.\nWorking with British architects such as Wilkinson Eyre, GECKO designed installations for The Empress State Building in Earl's Court, London, the bar at Diageo's headquarters (below) in north London, and many other commissions.\nDear Helen, please do send me photos of this loom. And any fabrics you might have, collected at the time? These artefacts are becoming increasingly rare, and I would be delighted to try and get them into a museum for analysis and storage. Where are you based?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Homes are built on a 'cushion' that can be filled with a water or gas to raise them in the event of an earthquake, causing them to hover while the buffer material takes the brunt of the quake. a new version will use powerful magnets to do the same job, the firm said.\nResearchers have revealed plans for a 'hover home' than can simply rise up off the ground in the event of an earthquake.\nIt would use giant magnets to lift homes for the duration of a quake, avoiding damage.\nThe firm behind the project is also developing a hoverboard - and said it hopes to apply the same technology to houses.\nGreg Henderson, co-founder and CEO at Arx Pax told Business Insider that although the firm has patents for a system using water or gas to raise homes, a new version would use magnets instead of liquid.\n'Our goal is to eliminate structural movement by pinpointing the exact time an object or building's 'landing gear' should retract and activate the hover engines.' he said.\nTo hover a three-story house for the average earthquake length of 90 seconds, it would use the energy of five car batteries, or cost around $13 using PG&E's rates, Henderson calculated.\nInstead of building large water-filled bases and foundations, Arx Pax's new vision would require installation of the hover engines and an earthquake-proof base.\n'When we receive warning of an earthquake, the computer turns on the hover engines,' Henderson explained.\nIt consists of a containment vessel, a buffer medium and a construction platform.\nThe construction platform rests on the buffer medium which rests on the containment vessel.\nA building can then be built on the construction platform, according to the patent.\nThe buffer medium can be a fluid, a gas or a liquefiable solid.\n'In the case of a liquid buffer medium, the construction platform can be designed to displace enough of the buffer medium such that the platform and any building constructed on the platform 'float' on the buffer medium.\n'The buffer medium can be selected such that seismic forces, and in particular lateral forces which are known to be most damaging to building an earthquake, are not greatly transmitted through the medium.\nArx Pax, creator of the Hendo Hoverboard also said that it is integrating the ShakeAlert earthquake early-warning software into its plan.\n'Weaving ShakeAlert into our MFA seismic isolation solution provides a valuable new tool to architects, engineers, and developers who are looking for a better way to build in areas affected by earthquakes.\nShakeAlert uses the geophysical networks operated by University of California Berkeley, California Institute of Technology, University of Washington, and the USGS, which has successfully detected California earthquakes and predicted the shaking before it could be felt.\n'Arx Pax's new MFA base isolation technology combined with the ShakeAlert early-warning system will allow state-of-the-art seismic protection and vibration control for buildings, operating rooms, highly calibrated instruments and much more,' said Dr. Jennifer Strauss, external relations officer at Berkeley Seismological Laboratory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last year, when I was in Australia visiting my brother, I had an interesting experience; yet another non-coincidence.\nI kinda thought I was hallucinating. No one else saw it, but I could have sworn I had just seen my kids names painted, with a heart, on the side of some building.\nA few days later we took the same train and I made everyone else watch for it too. Indeed, there it was. I wasn't hallucinating. Unfortunately, we moved by so quickly there was no possible way to take a picture.\nI'm not saying that finding this graffiti is any sort of special sign or anything. I just thought it was pretty cool. And, honestly, I think the chances of me coincidentally looking in that direction at that exact moment were pretty slim. I'm sure whoever painted it was referring to a different \"Peter and Katie\", but it still made me smile.\nYou've probably never heard me refer to Kate as \"Katie\" because I never called her Katie. She didn't really like that nickname, although there were two people she allowed to call her that: her paternal grandfather, Dziadziu, and her Kindergarten teacher, Mrs. S. She loved when they called her Katie.\nI have been wanting a picture of this graffiti since I was in Sydney, but I had no way of getting one. Last night I was randomly googling my kids - I mean, when you're looking for someone, isn't the internet the first place to start?! - when I decided to try \"peter katie sydney graffiti pic\". It worked! Thankfully, this person took a picture and uploaded it to Flickr. I don't you, but whoever you are, thank you!\nSo, while I haven't exactly found my kids, I have found a little something. Maybe that means I'm on the right track. Thank you all for your encouragement and advice about my journey to find them. I appreciate it.\nI am so glad for you that you found this.\nThat gave me chills. A definite sign. Keep going Amy..\nI had a huge smile reading this! Then all the other comments were smiling too!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbobkw b/data_all_eng_slimpj/shuffled/split2/finalzzzbobkw
new file mode 100644
index 0000000000000000000000000000000000000000..a098cf8a83d4651e4db24733debc23e7cea4e12d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I already have some posts up mentioning hotel condoms, or condoms given out in hotels due to some special occasions like the Olympic Games condoms that were given out in Beijing in 2008. Back then 300'000 condoms were distributed in the city hotels.\nRecently I have come across the images of condom packagings distributed by the Tenface Hotel in Bangkok. I thought the illustrations and the details were well worth to be shared with my readers.\nDid you notice the girls' naughty look on the left-hand studded condoms packaging version versus the right-hand giant condom excitement? Fun.\nImage source Tenface Hotel Bangkok.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to our database1, Gunnar W. Klau authored at least 85 papers between 1998 and 2019.\nThe Copy-Number Tree Mixture Deconvolution Problem and Applications to Multi-sample Bulk Sequencing Tumor Data.\nA Guided Tour to Computational Haplotyping.\nOn the Minimum Error Correction Problem for Haplotype Assembly in Diploid and Polyploid Genomes.\nHapCol: accurate and memory-efficient haplotype assembly from long reads.\nmetaModules identifies key functional subnetworks in microbiome-related disease.\nWhatsHap: Weighted Haplotype Assembly for Future-Generation Sequencing Reads.\nGenome sequence analysis with MonetDB - A case study on Ebola virus diversity.\nxHeinz: an algorithm for mining cross-species network modules under a flexible conservation model.\nNatalie 2.0: Sparse Global Network Alignment as a Special Case of Quadratic Assignment.\nAutomatic Classification of Protein Structure Using the Maximum Contact Map Overlap Metric.\nCIDANE: Comprehensive Isoform Discovery and Abundance Estimation.\nOn the Fixed Parameter Tractability and Approximability of the Minimum Error Correction Problem.\nGenome sequence analysis with monetdb: a case study on ebola virus diversity.\nTowards the ground truth: Exact algorithms for bioinformatics research (NII Shonan Meeting 2014-2).\nA web server for protein-protein interaction network querying.\neXamine: Exploring annotated modules in networks.\nWhatsHap: Haplotype Assembly for Future-Generation Sequencing Reads.\nExact Protein Structure Classification Using the Maximum Contact Map Overlap Metric.\nDALIX: Optimal DALI Protein Structure Alignment.\nMapping proteins in the presence of paralogs using units of coevolution.\nReconstructing Consensus Bayesian Network Structures with Application to Learning Molecular Interaction Networks.\nAntilope - A Lagrangian Relaxation Approach to the de novo Peptide Sequencing Problem.\nCSA: comprehensive comparison of pairwise protein structure alignments.\nAlgorithm engineering for optimal alignment of protein structure distance matrices.\nLagrangian Relaxation Applied to Sparse Global Network Alignment.\nOn Tree-Constrained Matchings and Generalizations.\nSparsifying Distance Matrices for Protein-Protein Structure Alignments.\nIntroduction to the Special Theme: Computational Biology.\nTowards optimal alignment of protein structure distance matrices.\nBioNet: an R-Package for the functional analysis of biological networks.\nFighting Lymph Cancer with Mathematics.\nPAUL: protein structural alignment using integer linear programming and Lagrangian relaxation.\nA new graph-based method for pairwise global network alignment.\nOn optimal comparability editing with applications to molecular diagnostics.\nAligning Protein Structures Using Distance Matrices and Combinatorial Optimization.\nInteger Linear Programming in Computational Biology.\nA branch-and-cut approach to the crossing number problem.\nAn Exact Mathematical Programming Approach to Multiple RNA Sequence-Structure Alignment.\nInteger linear programming approaches for non-unique probe selection.\nAccurate multiple sequence-structure alignment of RNA sequences using combinatorial optimization.\nA General Paradigm for Fast, Adaptive Clustering of Biological Sequences.\nAn Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem.\nInteger Linear Programs for Discovering Approximate Gene Clusters.\nAccelerated microRNA-Precursor Detection Using the Smith-Waterman Algorithm on FPGAs.\nMultiple Structural RNA Alignment with Lagrangian Relaxation.\nNon-planar Orthogonal Drawings with Fixed Topology.\nFast and Accurate Structural RNA Alignment by Progressive Lagrangian Optimization.\nSolving the Prize-Collecting Steiner Tree Problem to Optimality.\nStructural Alignment of Two RNA Sequences with Lagrangian Relaxation.\nCombining a Memetic Algorithm with Integer Programming to Solve the Prize-Collecting Steiner Tree Problem.\nLabel Number Maximization in the Slider Model.\nAGD - A Library of Algorithms for Graph Drawing.\nOptimal labeling of point features in rectangular labeling models.\nThe Fractional Prize-Collecting Steiner Tree Problem on Trees: Extended Abstract.\nA Combinatorial Approach to Orthogonal Placement Problems.\nThe HuGS platform: a toolkit for interactive optimization.\nAGD: A Library of Algorithms for Graph Drawing.\nGraph Drawing Algorithm Engineering with AGD.\nAn Experimental Comparison of Orthogonal Compaction Algorithms (Extended Abstract).\nOptimal Labelling of Point Features in the Slider Model.\nOptimal Compaction of Orthogonal Grid Drawings.\nCombining Graph Labeling and Compaction.\nA Library of Algorithms for Graph Drawing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brazil has confirmed recent cases of Chikungunya. Risk of Chikungunya is present throughout the country and is highest in the states of Cear\u00e1 and in Roraima. Peak transmission occurs from January through June.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My 3 Words: Stretch, Focus and Learn.\nSwitch, talked about needing three elements to bring about change: a rider (your plans and intents), the elephant (what your mood will do no matter what your plans say), and the path (the environment within which you intend to implement those changes). The concept of the three words is like the path.\nI am resolution averse. However, I have had a measure of success lately with resolutions: in 2010 by not answering kids when they yell at me from some other room, in 2011 by starting this blog and in 2012 by baking cookies. This year my goal feels fuzzier: I want to write more or better or be read more widely. Hard to pin down; hard to define and therefore hard to accomplish. But, let's see what I can do with Brogan's technique.\nSo, I'd like to write. The elephants are crowding the room. I am too busy. I shoot down my efforts. I lack skills in grammar, style and writing technique. I get distracted. What is my path?\nStretch means to make myself a bit uncomfortable. I am short on time. I hate getting out of bed in the dark but, perhaps finding time to write means doing so. Writing regularly means writing less than perfect pieces at times. Or, asking for feedback. Or, writing about difficult topics. Stretching means risking.\nLearn is easy to define. I went to a liberal arts college which at the time had no general education requirements. I took political science, philosophy and science\u2026 no English. Therefore I have a load to learn. Starting with grammar; for example, how to use a semicolon.\nAt work I have two exam rooms. This card hangs in the one reserved for teen patients so that I can see it daily. Hopefully some of them read it and I'll bet many of those that do wonder why on earth number 6 is on the list but, I know.\nThanks go to Bryan Vartabedian whose blog post at 33 Charts got me started with my 3.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Played college basketball at Texas A&M (2003-07).\nSigned for the 2007-08 season by Atlanta Hawks.\nSigned for the 2009-10 season by Golden State Warriors.\nOn November'09 traded to Charlotte Bobcats.\nOn Feburary'10, signed by Chicago Bulls.\nSigned for the 2010-11 season by Memphis Grizzlies.\nOn December'10 signed by Golden State Warriors.\nOn January'12 moved to Greece, signed by Olympiacos Piraeus.\nNamed to the 2006-07 All-Big 12 1st Team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbourv b/data_all_eng_slimpj/shuffled/split2/finalzzzbourv
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+{"text":"and this is a very good thing.\nLet not the future of our children be left to the disdain of politics.\nA clear, timeless perspective \u2013 possibly overlooked in the wave of cool and epic forward thinking \u2013 simple, basic foundational values.\nI use \"our\" loosely. American youth belong to us\u2026all of us. Inherently, these little ones will someday be ruling our nation and making decisions as we fade into the generations of long since past, leaving only a legacy of what we once stood for. It's our responsibility to look out for the whole\u2026not just those we consider our own. In doing good for others we are ultimately doing good for ourselves.\nThey'll hold strong through generations. Evidence will be apparent for years to come as the love from one is passed down to another and another. Can we say we're leaving this a better place with what we give of ourselves today? Time, money, passions\u2026are they aligned with core values? What we believe today will be evident in tomorrow.\nI was rather disappointed in a recent article I read on Douglas County Board of Education's $200 million bond and $20 million mill levy override for the November ballot. The tone of the article was rather negatively leading and incorporated the voucher debate rather than focusing on the issue at hand. If I had simply read the article and not looked further into the facts, I may have considered it a lost cause and possible jumped on the \"no thanks\" band wagon.\nThe issue is simple, our public schools need funds to make improvements and pay teachers. Where will we find these dollars and make the burden less upon the individual. Through taxes. Small contributions by many for the bennefit of all.\nI don't know about you, but I would gladly make small sacrifices for the betterment of our country and for the good of the whole. I know I can't do it all on my own, but together we can make an amazing difference in the lives of many.\nI understand that the mention of a proposed tax increase can be daunting when viewed in the short-term, but with a futuristic approach and full perspective of realistic needs and benefits, a little sacrifice now doesn't sound like such a burden. In opposition to the specific article I stumbled across, I'd advise for each individual to seek education on the issues and don't make assumptions, be mislead by haughty promises or distracted with side issues\u2026do have faith in our educational system and the caring, loving individuals whom have given their lives to improve the future for the sake of all of us\u2026and absolutely assess your core values when considering what you can do to help improve our community.\nWe all work together as one body\u2026and with each of us doing our own part even when we don't agree or like what others are doing, our contribution still results in an astounding big picture. I'm not a fan of negative media influences or politically charged debates\u2026be aware of the source when collecting information and don't let the small disparities divide us in our hearts \u2013 in the values that profess who we truly are.\nI am in favor of 3A & 3B to improve our schools, pay our teachers and give hope to our future leaders. It is one thing I can do, as an individual, to help improve our community. Yes, arguments can be had \u2013 and some within reason \u2013 but based on my beliefs, my values, my faith\u2026the answer is simple for me.\nOh, and on a completely selfish level\u2026I live in Douglas County and benefit from the increased property value in neighborhoods with fantastic schools \u2013 it is one of the very reasons we bought our home in Highlands Ranch\u2026but that's not the core value I'm basing my decision on \u2013 just being real honest here, wink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"London Heathrow Airport is a British airport. This is the largest airport in Europe and it is also the airport with the most connections to other airports. At world level, this is the fourth airport in 2011 in terms of persons handled, after Atlanta \/ Hartsfield, Beijing \/ Capital and Chicago \/ O'Hare. But due to the large number of international connections, Heathrow has the world's largest number of international passenger services.\nThe floors of this enormous complex are maintained and cleaned by means of UNICLEAN LOW FOAMING by AGS One. AGS One offers specialist cleaning, surface renovation and safety critical work in collaboration with the public sector and organizations using public transport.\nThe product MAVROSEAL SC is used to protect the joints and tiles on the floors. This ensures that the contamination on the floors is manageable and clean.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Delhi: After a day long strike by workers' unions, normalcy returned to the Gurgaon-Manesar- Bawal industrial belt in Haryana on Wednesday with employees of various companies reporting back for duty.\nOnly Rico Auto Industries, whose Gurgaon plant was the epicentre of the strike after the death a worker in a group clash on Sunday, continued to be under lock-out.\n\"Workers are now back at their duty in all places, barring Rico Auto where there is a lock-out,\" All India Trade Union Congress Gurgaon District Secretary Suresh Gaud said.\nHe said a delegation of workers' unions is going to meet the Additional District Collector during the day to resolve the labour dispute at Rico Auto, which led to the initial flare up.\nThousands of workers from various industries in the belt went for a day long strike yesterday after a 26-year-old employee of Rico Auto, Ajit Yadav, died during a group clash outside the company's Gurgaon facility on Sunday.\n\"We are going to raise our issues, including \u20b9 50 lakh compensation for the kin of the deceased, action against Rico's management and withdrawal of false FI \u20b9 against workers. If our demands are not met, we will continue with the agitation,\" Gaud said.\nOfficials at Rico Auto could not be contacted despite repeated calls.\nMajor auto makers, including Hero Honda Motors and Honda Motorcycle and Scooter India (HMSI), which were affected by the strike reported normal activity this morning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This website is designed by us. Super Smart Card is designed and developed by Super Smart Card (India) a market leader in sales promotion and loyalty programs. We offer exclusive and unique deals of discounts from most sought after brand in food & dinning, beauty, fashion, super markets, healthcare, retail, institutes, entertainment, travel, electronic & home appliances and much more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ahead from the Adidas NMD_R1 \"Energy Ink\" launch, adidas nmd shoes nz the 3-Stripes will continue on to include loud patterns to its relatively refined silhouette, readying another colorway dropping later this thirty day period. Another tackle the runner contains a new \"Green Marble\" print throughout its Primeknit upper. Additional specifics consist of a black eyestay and heel tab, with white medial panels. Adding distinction, the shoe sits atop a white full-length Raise midsole.\nadidas nmd runner nz Among the more common iterations is coming to your Adidas NMD_R1, with leaked products photographs revealing the lifestyle runner inside a black and pink (Bred) colorway established to reach in September. Though it looks like 2018 just started, Xmas is about the corner, and Adidas Originals is serving up that yuletide emotion that has a festive new make-up of the NMD_R1 STLT Primeknit.\nWhile no formal connections for the vacation are noticeable, adidas nmd r1 nz the lifestyle runner's unorthodox combination of crimson, white and environmentally friendly have acquired it the fan-chosen nickname of \"Christmas.\" Down below sits full-length Increase cushioning in white, bold green stabilizers and a black rubber outsole.A lot more colorways with the Adidas NMD_R1 Primeknit are releasing, as well as latest drop will definitely flip some heads.\nadidas nmd womens nz The life-style runner will aspect an eye-catching \"Energy Ink\" colorway sporting a purple Primeknit higher with black hits to the eyestay, heel tab, sitting down atop full-length Increase cushioning along with a black outsole. Based within the hashtags of various silhouettes utilized within the social networking system, the corporate has rated the very best thirty most Instagrammed sneaker types. Topping the chart may be the adidas Originals NMD, that is highlighted in about 5,709,871 posts, accompanied by the really sought-after YEEZY Raise 350, which sees an amassed volume of 4,198,238 posts. The ever traditional Vans Outdated Skool arrives 3rd, remaining tagged in additional than three,594,318 posts to the system. In opposite to well known belief, the Nike Air Pressure 1 is just identified to the 17th spot, straight away followed by the Air Jordan 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboypp b/data_all_eng_slimpj/shuffled/split2/finalzzzboypp
new file mode 100644
index 0000000000000000000000000000000000000000..9f834be30aa00595f7646ebd6ff61f4c989b7337
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzboypp
@@ -0,0 +1,5 @@
+{"text":"Breast augmentation surgery has one of the highest patient satisfaction rates of any plastic surgery. It can lead to a boost in self-confidence, self-esteem, and a better self-image. The addition of breast implants can help a woman achieve the breasts and the physical shape she has always wanted.\nAfter the surgery, your breasts will be swollen, sore, and wrapped in bandages. That tight little tank top likely won't fit you at this point, nor would it be very comfortable if you did try to wear it home.\nDress in comfortable, loose-fitting clothes to your breast augmentation surgery so you can have the right thing to wear on your trip home.\nBefore your surgery, make sure to drink plenty of water to keep your body hydrated throughout the process. Any major surgical procedure that uses anesthetics can leave you feeling nausea or disoriented.\nDrinking water ahead of time can sometimes help prevent some of these side effects. Make sure you follow your plastic surgeon's advice on eating and drinking before your surgery.\nOn the day of your surgery, have a friend or family member drive you to and from your surgical appointment. Following your procedure, you will still be heavily sedated and unable to drive on your own.\nTaxis and public transportation are also not considered a safe way for you to get home, so your doctor won't allow it. Remember that this is for your safety, not your inconvenience, so make arrangements ahead of time for a slow, successful transition home to recover.\nSome pain medications can lead to nausea or vomiting, especially if they are taken on an empty stomach. Unless your doctor or pharmacist specify that your medication must be taken without food, you should try to pair your medications with a meal or snack. After your breast augmentation surgery, you won't want the extra strain of nausea or vomiting on your body as it heals.\nSome medications, vitamins, and herbal supplements can actually interfere with anesthesia, pain medication, proper healing, and more. To avoid unnecessary complications throughout your surgery and healing process, make sure you skip any medications that your plastic advises against taking.\nYou should stay away from anything on your list for at least two weeks prior and two weeks following your breast augmentation surgery.\nAfter your surgery, you will be given a supportive garment or special medical bra to wear throughout your recovery process. You won't be able to wear a regular bra for a little while and even then, you may not know what size to buy.\nSkip the pre-surgery shopping trip and instead wait until your breasts have healed and the swelling has gone down, so you can be properly measured. After your surgery, it is essential that you wear a well-fitted bra with enough support to maintain the look and feel of your new breasts for years to come.\nYou might be thinking you'll just jump into your usual routine a day or so after you have your breast augmentation surgery, but it's impossible to predict how long your body will need to properly heal. It's also unlikely that anyone could heal that fast.\nInstead of leaving so many things to do after your surgery, plan ahead and check off your regular errands and housework before your surgery date. Besides, minimizing your stress after surgery is essential to a successful recovery.\nA breast augmentation offers many benefits. Dr. Vitenas and his staff will provide more complete information during the initial consultation. He will also go over your complete medical history, answer any other questions, and design the treatment to suit your unique needs.\nTo answer any questions or concerns that you may have, book a consultation with Dr. Vitenas by contacting our office 8 a.m. to 5 p.m., Monday through Friday, by dialing 281-484-0088. Or email us anytime, and one of our patient coordinators will get right back to you.\nWe are conveniently located in the heart of Houston. If you are not in the area, we can consult with you online and assist with travel arrangements. Financing options are always available.\nDr. Paul Vitenas, Jr., M.D., F.A.C.S. is a Board Certified Plastic Surgeon and founder of Vitenas Cosmetic Surgery and the Fine Art of Natural Cosmetic Surgery. He has practiced in the Houston area since 1990. He was named one of Houston's Top Doctors and America's Top Plastic Surgeons and awarded a Top Doctor for Women and Houston's Most Compassionate Doctor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Directors, Producers, Playwrights, and Actors, each time we step onto a stage or into a classroom we affirm the value and purpose of American theater as part of a well-rounded school curriculum and contributor to an area's quality of life.\nOur rigorous program focuses on expanding the actors' physical, vocal and imaginative capabilities as an ensemble.\nPhase Eight is committed to creating a financially supportive environment for our actors, writers, directors, dancers, technicians, and designers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Zionsville, Indiana, Freedom Foods will deliver your produce basket right to your door each week. To make getting natural produce to your table that much easier, we take time to learn what we can do to make our delivery process most convenient for you.\nProduce delivery to homes in Zionsville, Indiana is on Tuesday each week.\nGet natural fruits, vegetables, and eggs from the farm to your front door with Freedom Foods produce delivery.\nNutrition comes first for your family, but we know that fresh, natural produce can be difficult to find at your local grocery store. We're happy to help you avoid the hassle by delivering our fresh baskets straight to your home.\nWould your fitness center in Zionsville, Indiana make a great delivery location for Freedom Foods baskets? Contact us to talk about delivery options!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Psychology and therapy practice in downtown Los Angeles, California (DTLA). Individual therapy and counseling for stress, anxiety, depression, child anxiety, autism spectrum, and mindfulness-based coping. CBT (Cognitive-behavioral therapy), ACT, DBT, and evidence-based treatments.\nDowntown L.A.\/Los Angeles CBT and evidence-based therapy practice of psychologists for adults, children, and teens\/adolescents.\nSubmit a general inquiry or specify which clinician you'd like to hear from.\nThank you for contacting me! I will be in touch with you very soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hendra, Chairman IMERC urged the Indonesian people to support the cause of Kashmiris self-determination right.\nQuoting various UN Resolutions, Hendra, Chairman, IMERC urged the Indonesian people and international community to support the cause of self-determination of Kashmiris people.\nProjects in the pipeline also include collaboration with the Beaufort Laboratory UCC located at the IMERC Campus in Ringaskiddy, the National Sensor Centre in DCU, Tyndall Research Institute in UCC, the Eindhoven Technical University and the University of Central Florida.","meta":{"redpajama_set_name":"RedPajamaC4"}}